LMFDB - Embedded newform 538.4.b.a.537.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,4,Mod(537,538)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("538.537");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 538 = 2 \cdot 269 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	538.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$31.7430275831$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			537.2
Character	$\chi$	$=$	538.537
Dual form			538.4.b.a.537.67

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -9.56359i q^{3} -4.00000 q^{4} -14.2576 q^{5} -19.1272 q^{6} -26.9416i q^{7} +8.00000i q^{8} -64.4623 q^{9} +28.5153i q^{10} -0.346824 q^{11} +38.2544i q^{12} -23.3585 q^{13} -53.8833 q^{14} +136.354i q^{15} +16.0000 q^{16} -24.9509i q^{17} +128.925i q^{18} +47.1782i q^{19} +57.0305 q^{20} -257.659 q^{21} +0.693648i q^{22} +35.8457 q^{23} +76.5087 q^{24} +78.2799 q^{25} +46.7170i q^{26} +358.274i q^{27} +107.767i q^{28} -302.462i q^{29} +272.708 q^{30} -273.996i q^{31} -32.0000i q^{32} +3.31688i q^{33} -49.9019 q^{34} +384.124i q^{35} +257.849 q^{36} -72.6312 q^{37} +94.3564 q^{38} +223.391i q^{39} -114.061i q^{40} -65.4647 q^{41} +515.318i q^{42} +331.031 q^{43} +1.38730 q^{44} +919.079 q^{45} -71.6914i q^{46} -303.561 q^{47} -153.017i q^{48} -382.852 q^{49} -156.560i q^{50} -238.620 q^{51} +93.4341 q^{52} -276.628 q^{53} +716.548 q^{54} +4.94489 q^{55} +215.533 q^{56} +451.193 q^{57} -604.924 q^{58} +245.103i q^{59} -545.416i q^{60} +653.364 q^{61} -547.991 q^{62} +1736.72i q^{63} -64.0000 q^{64} +333.037 q^{65} +6.63377 q^{66} -201.060 q^{67} +99.8037i q^{68} -342.814i q^{69} +768.248 q^{70} -443.585i q^{71} -515.698i q^{72} -216.738 q^{73} +145.262i q^{74} -748.637i q^{75} -188.713i q^{76} +9.34401i q^{77} +446.783 q^{78} +857.697 q^{79} -228.122 q^{80} +1685.90 q^{81} +130.929i q^{82} -804.747i q^{83} +1030.64 q^{84} +355.741i q^{85} -662.061i q^{86} -2892.62 q^{87} -2.77459i q^{88} +1065.44 q^{89} -1838.16i q^{90} +629.317i q^{91} -143.383 q^{92} -2620.38 q^{93} +607.123i q^{94} -672.649i q^{95} -306.035 q^{96} +933.748 q^{97} +765.705i q^{98} +22.3571 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 68 q - 272 q^{4} + 38 q^{5} - 4 q^{6} - 594 q^{9} + 18 q^{11} - 114 q^{13} + 8 q^{14} + 1088 q^{16} - 152 q^{20} - 20 q^{21} - 224 q^{23} + 16 q^{24} + 1098 q^{25} + 384 q^{30} - 600 q^{34} + 2376 q^{36}+ \cdots + 3370 q^{99}+O(q^{100}) $ 68 * q - 272 * q^4 + 38 * q^5 - 4 * q^6 - 594 * q^9 + 18 * q^11 - 114 * q^13 + 8 * q^14 + 1088 * q^16 - 152 * q^20 - 20 * q^21 - 224 * q^23 + 16 * q^24 + 1098 * q^25 + 384 * q^30 - 600 * q^34 + 2376 * q^36 - 302 * q^37 + 436 * q^38 - 624 * q^41 - 654 * q^43 - 72 * q^44 - 1074 * q^45 + 692 * q^47 - 3052 * q^49 + 596 * q^51 + 456 * q^52 + 326 * q^53 + 1504 * q^54 + 828 * q^55 - 32 * q^56 - 1024 * q^57 - 100 * q^58 + 226 * q^61 + 312 * q^62 - 4352 * q^64 + 316 * q^65 + 2480 * q^66 - 1318 * q^67 + 608 * q^70 - 2028 * q^73 - 1152 * q^78 - 1444 * q^79 + 608 * q^80 + 3204 * q^81 + 80 * q^84 - 5600 * q^87 - 1284 * q^89 + 896 * q^92 - 2528 * q^93 - 64 * q^96 - 288 * q^97 + 3370 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/538\mathbb{Z}\right)^\times$.

$n$	$271$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.00000i		−	0.707107i
$2$		−	2.00000i		−	0.707107i
$3$		−	9.56359i		−	1.84051i	−0.391315	−	0.920257i	$-0.627980\pi$
$3$		−	9.56359i		−	1.84051i	0.391315	−	0.920257i	$-0.372020\pi$
$4$	−4.00000			−0.500000
$5$	−14.2576			−1.27524			−0.637620	−	0.770351i	$-0.720081\pi$
$5$	−14.2576			−1.27524			−0.637620	+	0.770351i	$0.720081\pi$
$6$	−19.1272			−1.30144
$7$		−	26.9416i		−	1.45471i	−0.686260	−	0.727356i	$-0.740749\pi$
$7$		−	26.9416i		−	1.45471i	0.686260	−	0.727356i	$-0.259251\pi$
$8$			8.00000i			0.353553i
$9$	−64.4623			−2.38749
$10$			28.5153i			0.901731i
$11$	−0.346824			−0.00950649			−0.00475325	−	0.999989i	$-0.501513\pi$
$11$	−0.346824			−0.00950649			−0.00475325	+	0.999989i	$0.501513\pi$
$12$			38.2544i			0.920257i
$13$	−23.3585			−0.498345			−0.249173	−	0.968459i	$-0.580159\pi$
$13$	−23.3585			−0.498345			−0.249173	+	0.968459i	$0.580159\pi$
$14$	−53.8833			−1.02864
$15$			136.354i			2.34710i
$16$	16.0000			0.250000
$17$		−	24.9509i		−	0.355970i	−0.984033	−	0.177985i	$-0.943042\pi$
$17$		−	24.9509i		−	0.355970i	0.984033	−	0.177985i	$-0.0569579\pi$
$18$			128.925i			1.68821i
$19$			47.1782i			0.569654i	0.958579	+	0.284827i	$0.0919362\pi$
$19$			47.1782i			0.569654i	−0.958579	+	0.284827i	$0.908064\pi$
$20$	57.0305			0.637620
$21$	−257.659			−2.67742
$22$			0.693648i			0.00672211i
$23$	35.8457			0.324972			0.162486	−	0.986711i	$-0.448049\pi$
$23$	35.8457			0.324972			0.162486	+	0.986711i	$0.448049\pi$
$24$	76.5087			0.650720
$25$	78.2799			0.626239
$26$			46.7170i			0.352383i
$27$			358.274i			2.55370i
$28$			107.767i			0.727356i
$29$		−	302.462i		−	1.93675i	−0.249500	−	0.968375i	$-0.580266\pi$
$29$		−	302.462i		−	1.93675i	0.249500	−	0.968375i	$-0.419734\pi$
$30$	272.708			1.65965
$31$		−	273.996i		−	1.58745i	−0.608274	−	0.793727i	$-0.708138\pi$
$31$		−	273.996i		−	1.58745i	0.608274	−	0.793727i	$-0.291862\pi$
$32$		−	32.0000i		−	0.176777i
$33$			3.31688i			0.0174968i
$34$	−49.9019			−0.251709
$35$			384.124i			1.85511i
$36$	257.849			1.19375
$37$	−72.6312			−0.322716			−0.161358	−	0.986896i	$-0.551587\pi$
$37$	−72.6312			−0.322716			−0.161358	+	0.986896i	$0.551587\pi$
$38$	94.3564			0.402806
$39$			223.391i			0.917211i
$40$		−	114.061i		−	0.450866i
$41$	−65.4647			−0.249363			−0.124681	−	0.992197i	$-0.539791\pi$
$41$	−65.4647			−0.249363			−0.124681	+	0.992197i	$0.539791\pi$
$42$			515.318i			1.89322i
$43$	331.031			1.17399			0.586996	−	0.809590i	$-0.300310\pi$
$43$	331.031			1.17399			0.586996	+	0.809590i	$0.300310\pi$
$44$	1.38730			0.00475325
$45$	919.079			3.04463
$46$		−	71.6914i		−	0.229790i
$47$	−303.561			−0.942106			−0.471053	−	0.882105i	$-0.656126\pi$
$47$	−303.561			−0.942106			−0.471053	+	0.882105i	$0.656126\pi$
$48$		−	153.017i		−	0.460128i
$49$	−382.852			−1.11619
$50$		−	156.560i		−	0.442818i
$51$	−238.620			−0.655168
$52$	93.4341			0.249173
$53$	−276.628			−0.716939			−0.358470	−	0.933541i	$-0.616701\pi$
$53$	−276.628			−0.716939			−0.358470	+	0.933541i	$0.616701\pi$
$54$	716.548			1.80574
$55$	4.94489			0.0121231
$56$	215.533			0.514318
$57$	451.193			1.04846
$58$	−604.924			−1.36949
$59$			245.103i			0.540842i	0.962742	+	0.270421i	$0.0871629\pi$
$59$			245.103i			0.540842i	−0.962742	+	0.270421i	$0.912837\pi$
$60$		−	545.416i		−	1.17355i
$61$	653.364			1.37139			0.685694	−	0.727890i	$-0.259499\pi$
$61$	653.364			1.37139			0.685694	+	0.727890i	$0.259499\pi$
$62$	−547.991			−1.12250
$63$			1736.72i			3.47311i
$64$	−64.0000			−0.125000
$65$	333.037			0.635510
$66$	6.63377			0.0123721
$67$	−201.060			−0.366617			−0.183309	−	0.983055i	$-0.558681\pi$
$67$	−201.060			−0.366617			−0.183309	+	0.983055i	$0.558681\pi$
$68$			99.8037i			0.177985i
$69$		−	342.814i		−	0.598115i
$70$	768.248			1.31176
$71$		−	443.585i		−	0.741463i	−0.928740	−	0.370732i	$-0.879107\pi$
$71$		−	443.585i		−	0.741463i	0.928740	−	0.370732i	$-0.120893\pi$
$72$		−	515.698i		−	0.844106i
$73$	−216.738			−0.347496			−0.173748	−	0.984790i	$-0.555588\pi$
$73$	−216.738			−0.347496			−0.173748	+	0.984790i	$0.555588\pi$
$74$			145.262i			0.228195i
$75$		−	748.637i		−	1.15260i
$76$		−	188.713i		−	0.284827i
$77$			9.34401i			0.0138292i
$78$	446.783			0.648566
$79$	857.697			1.22150			0.610749	−	0.791824i	$-0.290868\pi$
$79$	857.697			1.22150			0.610749	+	0.791824i	$0.290868\pi$
$80$	−228.122			−0.318810
$81$	1685.90			2.31262
$82$			130.929i			0.176326i
$83$		−	804.747i		−	1.06425i	−0.846667	−	0.532123i	$-0.821394\pi$
$83$		−	804.747i		−	1.06425i	0.846667	−	0.532123i	$-0.178606\pi$
$84$	1030.64			1.33871
$85$			355.741i			0.453947i
$86$		−	662.061i		−	0.830138i
$87$	−2892.62			−3.56461
$88$		−	2.77459i		−	0.00336105i
$89$	1065.44			1.26895			0.634476	−	0.772942i	$-0.281216\pi$
$89$	1065.44			1.26895			0.634476	+	0.772942i	$0.281216\pi$
$90$		−	1838.16i		−	2.15288i
$91$			629.317i			0.724949i
$92$	−143.383			−0.162486
$93$	−2620.38			−2.92173
$94$			607.123i			0.666169i
$95$		−	672.649i		−	0.726446i
$96$	−306.035			−0.325360
$97$	933.748			0.977399			0.488700	−	0.872452i	$-0.337471\pi$
$97$	933.748			0.977399			0.488700	+	0.872452i	$0.337471\pi$
$98$			765.705i			0.789264i
$99$	22.3571			0.0226967
$100$	−313.120			−0.313120
$101$		−	185.401i		−	0.182655i	−0.995821	−	0.0913274i	$-0.970889\pi$
$101$		−	185.401i		−	0.182655i	0.995821	−	0.0913274i	$-0.0291110\pi$
$102$			477.241i			0.463274i
$103$	−514.660			−0.492339			−0.246170	−	0.969227i	$-0.579172\pi$
$103$	−514.660			−0.492339			−0.246170	+	0.969227i	$0.579172\pi$
$104$		−	186.868i		−	0.176192i
$105$	3673.60			3.41435
$106$			553.256i			0.506953i
$107$		−	392.691i		−	0.354794i	−0.984139	−	0.177397i	$-0.943232\pi$
$107$		−	392.691i		−	0.354794i	0.984139	−	0.177397i	$-0.0567676\pi$
$108$		−	1433.10i		−	1.27685i
$109$		−	1235.67i		−	1.08583i	−0.839787	−	0.542917i	$-0.817320\pi$
$109$		−	1235.67i		−	1.08583i	0.839787	−	0.542917i	$-0.182680\pi$
$110$		−	9.88978i		−	0.00857230i
$111$			694.615i			0.593963i
$112$		−	431.066i		−	0.363678i
$113$		−	928.456i		−	0.772937i	−0.922303	−	0.386468i	$-0.873695\pi$
$113$		−	928.456i		−	0.772937i	0.922303	−	0.386468i	$-0.126305\pi$
$114$		−	902.386i		−	0.741370i
$115$	−511.075			−0.414417
$116$			1209.85i			0.968375i
$117$	1505.74			1.18979
$118$	490.206			0.382433
$119$	−672.219			−0.517834
$120$	−1090.83			−0.829825
$121$	−1330.88			−0.999910
$122$		−	1306.73i		−	0.969718i
$123$			626.078i			0.458956i
$124$			1095.98i			0.793727i
$125$	666.118			0.476635
$126$	3473.44			2.45586
$127$	504.854			0.352744			0.176372	−	0.984324i	$-0.443564\pi$
$127$	504.854			0.352744			0.176372	+	0.984324i	$0.443564\pi$
$128$			128.000i			0.0883883i
$129$		−	3165.84i		−	2.16075i
$130$		−	666.074i		−	0.449373i
$131$	−2146.56			−1.43165			−0.715825	−	0.698280i	$-0.753949\pi$
$131$	−2146.56			−1.43165			−0.715825	+	0.698280i	$0.753949\pi$
$132$		−	13.2675i		−	0.00874842i
$133$	1271.06			0.828683
$134$			402.120i			0.259238i
$135$		−	5108.13i		−	3.25658i
$136$	199.607			0.125854
$137$			2998.64i			1.87001i	0.354633	+	0.935005i	$0.384606\pi$
$137$			2998.64i			1.87001i	−0.354633	+	0.935005i	$0.615394\pi$
$138$	−685.627			−0.422931
$139$			2373.35i			1.44823i	0.689677	+	0.724117i	$0.257753\pi$
$139$			2373.35i			1.44823i	−0.689677	+	0.724117i	$0.742247\pi$
$140$		−	1536.50i		−	0.927554i
$141$			2903.14i			1.73396i
$142$	−887.171			−0.524294
$143$	8.10130			0.00473751
$144$	−1031.40			−0.596873
$145$			4312.39i			2.46982i
$146$			433.475i			0.245717i
$147$			3661.44i			2.05436i
$148$	290.525			0.161358
$149$	−81.2644			−0.0446808			−0.0223404	−	0.999750i	$-0.507112\pi$
$149$	−81.2644			−0.0446808			−0.0223404	+	0.999750i	$0.507112\pi$
$150$	−1497.27			−0.815012
$151$	2683.65			1.44631			0.723154	−	0.690686i	$-0.242691\pi$
$151$	2683.65			1.44631			0.723154	+	0.690686i	$0.242691\pi$
$152$	−377.426			−0.201403
$153$			1608.39i			0.849875i
$154$	18.6880			0.00977873
$155$			3906.53i			2.02439i
$156$		−	893.565i		−	0.458606i
$157$		−	268.464i		−	0.136470i	−0.997669	−	0.0682348i	$-0.978263\pi$
$157$		−	268.464i		−	0.136470i	0.997669	−	0.0682348i	$-0.0217367\pi$
$158$		−	1715.39i		−	0.863730i
$159$			2645.56i			1.31954i
$160$			456.244i			0.225433i
$161$		−	965.742i		−	0.472740i
$162$		−	3371.81i		−	1.63527i
$163$			2770.72i			1.33141i	0.746215	+	0.665705i	$0.231869\pi$
$163$			2770.72i			1.33141i	−0.746215	+	0.665705i	$0.768131\pi$
$164$	261.859			0.124681
$165$		−	47.2909i		−	0.0223127i
$166$	−1609.49			−0.752535
$167$			3441.58i			1.59472i	0.603507	+	0.797358i	$0.293770\pi$
$167$			3441.58i			1.59472i	−0.603507	+	0.797358i	$0.706230\pi$
$168$		−	2061.27i		−	0.946610i
$169$	−1651.38			−0.751652
$170$	711.482			0.320989
$171$		−	3041.22i		−	1.36004i
$172$	−1324.12			−0.586996
$173$	1187.14			0.521716			0.260858	−	0.965377i	$-0.415995\pi$
$173$	1187.14			0.521716			0.260858	+	0.965377i	$0.415995\pi$
$174$			5785.24i			2.52056i
$175$		−	2108.99i		−	0.910998i
$176$	−5.54919			−0.00237662
$177$	2344.06			0.995427
$178$		−	2130.89i		−	0.897285i
$179$			848.139i			0.354150i	0.984197	+	0.177075i	$0.0566636\pi$
$179$			848.139i			0.354150i	−0.984197	+	0.177075i	$0.943336\pi$
$180$	−3676.32			−1.52231
$181$			2805.89i			1.15227i	0.817356	+	0.576133i	$0.195439\pi$
$181$			2805.89i			1.15227i	−0.817356	+	0.576133i	$0.804561\pi$
$182$	1258.63			0.512616
$183$		−	6248.51i		−	2.52406i
$184$			286.766i			0.114895i
$185$	1035.55			0.411541
$186$			5240.76i			2.06598i
$187$			8.65359i			0.00338403i
$188$	1214.25			0.471053
$189$	9652.49			3.71489
$190$	−1345.30			−0.513675
$191$	−4713.20			−1.78552			−0.892762	−	0.450529i	$-0.851235\pi$
$191$	−4713.20			−1.78552			−0.892762	+	0.450529i	$0.851235\pi$
$192$			612.070i			0.230064i
$193$		−	2310.77i		−	0.861829i	−0.902393	−	0.430914i	$-0.858191\pi$
$193$		−	2310.77i		−	0.861829i	0.902393	−	0.430914i	$-0.141809\pi$
$194$		−	1867.50i		−	0.691126i
$195$		−	3185.03i		−	1.16967i
$196$	1531.41			0.558094
$197$			5266.35i			1.90463i	0.305120	+	0.952314i	$0.401304\pi$
$197$			5266.35i			1.90463i	−0.305120	+	0.952314i	$0.598696\pi$
$198$		−	44.7141i		−	0.0160490i
$199$	4285.74			1.52667			0.763337	−	0.646000i	$-0.223560\pi$
$199$	4285.74			1.52667			0.763337	+	0.646000i	$0.223560\pi$
$200$			626.239i			0.221409i
$201$			1922.85i			0.674765i
$202$	−370.803			−0.129156
$203$	−8148.82			−2.81741
$204$	954.482			0.327584
$205$	933.372			0.317998
$206$			1029.32i			0.348136i
$207$	−2310.70			−0.775867
$208$	−373.736			−0.124586
$209$		−	16.3625i		−	0.00541541i
$210$		−	7347.21i		−	2.41431i
$211$	−1808.55			−0.590074			−0.295037	−	0.955486i	$-0.595332\pi$
$211$	−1808.55			−0.590074			−0.295037	+	0.955486i	$0.595332\pi$
$212$	1106.51			0.358470
$213$	−4242.27			−1.36467
$214$	−785.383			−0.250877
$215$	−4719.71			−1.49712
$216$	−2866.19			−0.902868
$217$	−7381.89			−2.30929
$218$	−2471.34			−0.767800
$219$			2072.79i			0.639571i
$220$	−19.7796			−0.00606153
$221$			582.817i			0.177396i
$222$	1389.23			0.419996
$223$		−	1282.99i		−	0.385271i	−0.981270	−	0.192635i	$-0.938297\pi$
$223$		−	1282.99i		−	0.385271i	0.981270	−	0.192635i	$-0.0617034\pi$
$224$	−862.133			−0.257159
$225$	−5046.10			−1.49514
$226$	−1856.91			−0.546549
$227$		−	729.687i		−	0.213353i	−0.994294	−	0.106676i	$-0.965979\pi$
$227$		−	729.687i		−	0.213353i	0.994294	−	0.106676i	$-0.0340208\pi$
$228$	−1804.77			−0.524228
$229$		−	4127.32i		−	1.19101i	−0.803352	−	0.595504i	$-0.796952\pi$
$229$		−	4127.32i		−	1.19101i	0.803352	−	0.595504i	$-0.203048\pi$
$230$			1022.15i			0.293037i
$231$	89.3623			0.0254529
$232$	2419.69			0.684744
$233$	−6507.96			−1.82983			−0.914916	−	0.403644i	$-0.867743\pi$
$233$	−6507.96			−1.82983			−0.914916	+	0.403644i	$0.867743\pi$
$234$		−	3011.49i		−	0.841312i
$235$	4328.06			1.20141
$236$		−	980.412i		−	0.270421i
$237$		−	8202.66i		−	2.24819i
$238$			1344.44i			0.366164i
$239$	3981.03			1.07745			0.538727	−	0.842480i	$-0.318905\pi$
$239$	3981.03			1.07745			0.538727	+	0.842480i	$0.318905\pi$
$240$			2181.67i			0.586775i
$241$		−	4314.62i		−	1.15323i	−0.817015	−	0.576616i	$-0.804373\pi$
$241$		−	4314.62i		−	1.15323i	0.817015	−	0.576616i	$-0.195627\pi$
$242$			2661.76i			0.707043i
$243$		−	6449.89i		−	1.70272i
$244$	−2613.46			−0.685694
$245$	5458.57			1.42341
$246$	1252.16			0.324531
$247$		−	1102.01i		−	0.283884i
$248$	2191.97			0.561250
$249$	−7696.27			−1.95876
$250$		−	1332.24i		−	0.337032i
$251$		−	6343.86i		−	1.59530i	−0.603118	−	0.797652i	$-0.706075\pi$
$251$		−	6343.86i		−	1.59530i	0.603118	−	0.797652i	$-0.293925\pi$
$252$		−	6946.88i		−	1.73656i
$253$	−12.4322			−0.00308934
$254$		−	1009.71i		−	0.249428i
$255$	3402.16			0.835497
$256$	256.000			0.0625000
$257$		−	4333.01i		−	1.05170i	−0.850578	−	0.525848i	$-0.823748\pi$
$257$		−	4333.01i		−	1.05170i	0.850578	−	0.525848i	$-0.176252\pi$
$258$	−6331.68			−1.52788
$259$			1956.80i			0.469459i
$260$	−1332.15			−0.317755
$261$			19497.4i			4.62397i
$262$			4293.13i			1.01233i
$263$	−1683.63			−0.394741			−0.197370	−	0.980329i	$-0.563240\pi$
$263$	−1683.63			−0.394741			−0.197370	+	0.980329i	$0.563240\pi$
$264$	−26.5351			−0.00618606
$265$	3944.06			0.914270
$266$		−	2542.12i		−	0.585967i
$267$		−	10189.5i		−	2.33553i
$268$	804.239			0.183309
$269$	900.216	+	4319.11i	0.204041	+	0.978962i
$269$	900.216	+	4319.11i	0.204041	+	0.978962i
$270$	−10216.3			−2.30275
$271$			1844.59i			0.413473i	0.978397	+	0.206736i	$0.0662842\pi$
$271$			1844.59i			0.413473i	−0.978397	+	0.206736i	$0.933716\pi$
$272$		−	399.215i		−	0.0889925i
$273$	6018.53			1.33428
$274$	5997.29			1.32230
$275$	−27.1494			−0.00595334
$276$			1371.25i			0.299057i
$277$			1833.01i			0.397599i	0.980040	+	0.198800i	$0.0637043\pi$
$277$			1833.01i			0.397599i	−0.980040	+	0.198800i	$0.936296\pi$
$278$	4746.69			1.02406
$279$			17662.4i			3.79003i
$280$	−3072.99			−0.655880
$281$			7945.63i			1.68682i	0.537269	+	0.843411i	$0.319456\pi$
$281$			7945.63i			1.68682i	−0.537269	+	0.843411i	$0.680544\pi$
$282$	5806.27			1.22609
$283$	3932.96			0.826114			0.413057	−	0.910705i	$-0.364461\pi$
$283$	3932.96			0.826114			0.413057	+	0.910705i	$0.364461\pi$
$284$			1774.34i			0.370732i
$285$	−6432.94			−1.33703
$286$		−	16.2026i		−	0.00334993i
$287$			1763.73i			0.362751i
$288$			2062.79i			0.422053i
$289$	4290.45			0.873285
$290$	8624.77			1.74643
$291$		−	8929.98i		−	1.79892i
$292$	866.950			0.173748
$293$	−4206.12			−0.838649			−0.419325	−	0.907836i	$-0.637733\pi$
$293$	−4206.12			−0.838649			−0.419325	+	0.907836i	$0.637733\pi$
$294$	7322.89			1.45265
$295$		−	3494.59i		−	0.689704i
$296$		−	581.050i		−	0.114097i
$297$		−	124.258i		−	0.0242767i
$298$			162.529i			0.0315941i
$299$	−837.303			−0.161948
$300$			2994.55i			0.576301i
$301$		−	8918.51i		−	1.70782i
$302$		−	5367.31i		−	1.02269i
$303$	−1773.10			−0.336179
$304$			754.852i			0.142414i
$305$	−9315.42			−1.74885
$306$	3216.79			0.600953
$307$	4616.12			0.858162			0.429081	−	0.903266i	$-0.358838\pi$
$307$	4616.12			0.858162			0.429081	+	0.903266i	$0.358838\pi$
$308$		−	37.3761i		−	0.00691461i
$309$			4922.00i			0.906157i
$310$	7813.05			1.43146
$311$			6447.94i			1.17566i	0.808986	+	0.587828i	$0.200017\pi$
$311$			6447.94i			1.17566i	−0.808986	+	0.587828i	$0.799983\pi$
$312$	−1787.13			−0.324283
$313$	−9115.22			−1.64608			−0.823040	−	0.567984i	$-0.807723\pi$
$313$	−9115.22			−1.64608			−0.823040	+	0.567984i	$0.807723\pi$
$314$	−536.927			−0.0964986
$315$		−	24761.5i		−	4.42906i
$316$	−3430.79			−0.610749
$317$		−	4851.22i		−	0.859531i	−0.902940	−	0.429766i	$-0.858596\pi$
$317$		−	4851.22i		−	0.859531i	0.902940	−	0.429766i	$-0.141404\pi$
$318$	5291.12			0.933054
$319$			104.901i			0.0184117i
$320$	912.488			0.159405
$321$	−3755.54			−0.653003
$322$	−1931.48			−0.334278
$323$	1177.14			0.202780
$324$	−6743.61			−1.15631
$325$	−1828.50			−0.312083
$326$	5541.45			0.941449
$327$	−11817.5			−1.99849
$328$		−	523.718i		−	0.0881631i
$329$			8178.44i			1.37049i
$330$	−94.5818			−0.0157774
$331$	−10073.2			−1.67273			−0.836363	−	0.548176i	$-0.815322\pi$
$331$	−10073.2			−1.67273			−0.836363	+	0.548176i	$0.815322\pi$
$332$			3218.99i			0.532123i
$333$	4681.97			0.770482
$334$	6883.16			1.12763
$335$	2866.64			0.467526
$336$	−4122.54			−0.669355
$337$			716.368i			0.115795i	0.998323	+	0.0578977i	$0.0184397\pi$
$337$			716.368i			0.115795i	−0.998323	+	0.0578977i	$0.981560\pi$
$338$			3302.76i			0.531498i
$339$	−8879.38			−1.42260
$340$		−	1422.96i		−	0.226974i
$341$			95.0283i			0.0150911i
$342$	−6082.43			−0.961696
$343$			1073.69i			0.169020i
$344$			2648.24i			0.415069i
$345$			4887.71i			0.762740i
$346$		−	2374.29i		−	0.368909i
$347$	4793.79			0.741626			0.370813	−	0.928707i	$-0.379079\pi$
$347$	4793.79			0.741626			0.370813	+	0.928707i	$0.379079\pi$
$348$	11570.5			1.78231
$349$	8842.28			1.35621			0.678104	−	0.734966i	$-0.262802\pi$
$349$	8842.28			1.35621			0.678104	+	0.734966i	$0.262802\pi$
$350$	−4217.98			−0.644173
$351$		−	8368.74i		−	1.27262i
$352$			11.0984i			0.00168053i
$353$	−8455.54			−1.27491			−0.637454	−	0.770488i	$-0.720013\pi$
$353$	−8455.54			−1.27491			−0.637454	+	0.770488i	$0.720013\pi$
$354$		−	4688.13i		−	0.703873i
$355$			6324.47i			0.945544i
$356$	−4261.77			−0.634476
$357$			6428.83i			0.953080i
$358$	1696.28			0.250422
$359$			9150.79i			1.34529i	0.739964	+	0.672646i	$0.234842\pi$
$359$			9150.79i			1.34529i	−0.739964	+	0.672646i	$0.765158\pi$
$360$			7352.63i			1.07644i
$361$	4633.22			0.675494
$362$	5611.78			0.814775
$363$			12728.0i			1.84035i
$364$		−	2517.27i		−	0.362474i
$365$	3090.16			0.443141
$366$	−12497.0			−1.78478
$367$		−	7829.22i		−	1.11357i	−0.830655	−	0.556787i	$-0.812034\pi$
$367$		−	7829.22i		−	1.11357i	0.830655	−	0.556787i	$-0.187966\pi$
$368$	573.531			0.0812429
$369$	4220.01			0.595352
$370$		−	2071.10i		−	0.291003i
$371$			7452.82i			1.04294i
$372$	10481.5			1.46087
$373$			5.42489i			0.000753056i	1.00000		0.000376528i	$0.000119853\pi$
$373$			5.42489i			0.000753056i	−1.00000		0.000376528i	$0.999880\pi$
$374$	17.3072			0.00239287
$375$		−	6370.48i		−	0.877254i
$376$		−	2428.49i		−	0.333085i
$377$			7065.06i			0.965170i
$378$		−	19305.0i		−	2.62683i
$379$		−	7786.46i		−	1.05531i	−0.849458	−	0.527656i	$-0.823071\pi$
$379$		−	7786.46i		−	1.05531i	0.849458	−	0.527656i	$-0.176929\pi$
$380$			2690.60i			0.363223i
$381$		−	4828.22i		−	0.649231i
$382$			9426.40i			1.26256i
$383$			3165.06i			0.422264i	0.977458	+	0.211132i	$0.0677150\pi$
$383$			3165.06i			0.422264i	−0.977458	+	0.211132i	$0.932285\pi$
$384$	1224.14			0.162680
$385$		−	133.223i		−	0.0176356i
$386$	−4621.54			−0.609405
$387$	−21339.0			−2.80290
$388$	−3734.99			−0.488700
$389$	−2591.46			−0.337769			−0.168885	−	0.985636i	$-0.554017\pi$
$389$	−2591.46			−0.337769			−0.168885	+	0.985636i	$0.554017\pi$
$390$	−6370.06			−0.827078
$391$		−	894.384i		−	0.115680i
$392$		−	3062.82i		−	0.394632i
$393$			20528.9i			2.63497i
$394$	10532.7			1.34678
$395$	−12228.7			−1.55771
$396$	−89.4283			−0.0113483
$397$		−	14460.4i		−	1.82808i	−0.405628	−	0.914038i	$-0.632947\pi$
$397$		−	14460.4i		−	1.82808i	0.405628	−	0.914038i	$-0.367053\pi$
$398$		−	8571.48i		−	1.07952i
$399$		−	12155.9i		−	1.52520i
$400$	1252.48			0.156560
$401$		−	7839.19i		−	0.976236i	−0.872778	−	0.488118i	$-0.837684\pi$
$401$		−	7839.19i		−	0.976236i	0.872778	−	0.488118i	$-0.162316\pi$
$402$	3845.71			0.477131
$403$			6400.13i			0.791100i
$404$			741.606i			0.0913274i
$405$	−24037.0			−2.94915
$406$			16297.6i			1.99221i
$407$	25.1903			0.00306790
$408$		−	1908.96i		−	0.231637i
$409$		−	14087.3i		−	1.70311i	−0.524264	−	0.851556i	$-0.675659\pi$
$409$		−	14087.3i		−	1.70311i	0.524264	−	0.851556i	$-0.324341\pi$
$410$		−	1866.74i		−	0.224858i
$411$	28677.8			3.44178
$412$	2058.64			0.246170
$413$	6603.48			0.786770
$414$			4621.39i			0.548621i
$415$			11473.8i			1.35717i
$416$			747.473i			0.0880958i
$417$	22697.7			2.66550
$418$	−32.7251			−0.00382927
$419$	−517.862			−0.0603799			−0.0301900	−	0.999544i	$-0.509611\pi$
$419$	−517.862			−0.0603799			−0.0301900	+	0.999544i	$0.509611\pi$
$420$	−14694.4			−1.70718
$421$	14856.4			1.71986			0.859928	−	0.510416i	$-0.170509\pi$
$421$	14856.4			1.71986			0.859928	+	0.510416i	$0.170509\pi$
$422$			3617.10i			0.417245i
$423$	19568.2			2.24927
$424$		−	2213.02i		−	0.253476i
$425$		−	1953.16i		−	0.222922i
$426$			8484.54i			0.964970i
$427$		−	17602.7i		−	1.99498i
$428$			1570.77i			0.177397i
$429$		−	77.4775i		−	0.00871946i
$430$			9439.42i			1.05863i
$431$			4099.39i			0.458145i	0.973409	+	0.229073i	$0.0735693\pi$
$431$			4099.39i			0.458145i	−0.973409	+	0.229073i	$0.926431\pi$
$432$			5732.38i			0.638424i
$433$	42.9027			0.00476160			0.00238080	−	0.999997i	$-0.499242\pi$
$433$	42.9027			0.00476160			0.00238080	+	0.999997i	$0.499242\pi$
$434$			14763.8i			1.63291i
$435$	41241.9			4.54574
$436$			4942.69i			0.542917i
$437$			1691.14i			0.185121i
$438$	4145.58			0.452245
$439$	−10516.7			−1.14336			−0.571678	−	0.820478i	$-0.693707\pi$
$439$	−10516.7			−1.14336			−0.571678	+	0.820478i	$0.693707\pi$
$440$			39.5591i			0.00428615i
$441$	24679.5			2.66489
$442$	1165.63			0.125438
$443$			7146.01i			0.766405i	0.923664	+	0.383202i	$0.125179\pi$
$443$			7146.01i			0.766405i	−0.923664	+	0.383202i	$0.874821\pi$
$444$		−	2778.46i		−	0.296982i
$445$	−15190.7			−1.61822
$446$	−2565.98			−0.272427
$447$			777.180i			0.0822357i
$448$			1724.27i			0.181839i
$449$	4319.81			0.454041			0.227020	−	0.973890i	$-0.427102\pi$
$449$	4319.81			0.454041			0.227020	+	0.973890i	$0.427102\pi$
$450$			10092.2i			1.05722i
$451$	22.7048			0.00237057
$452$			3713.83i			0.386468i
$453$		−	25665.4i		−	2.66195i
$454$	−1459.37			−0.150863
$455$		−	8972.57i		−	0.924484i
$456$			3609.55i			0.370685i
$457$	−8335.47			−0.853209			−0.426605	−	0.904438i	$-0.640290\pi$
$457$	−8335.47			−0.853209			−0.426605	+	0.904438i	$0.640290\pi$
$458$	−8254.64			−0.842170
$459$	8939.26			0.909039
$460$	2044.30			0.207209
$461$			9688.52i			0.978827i	0.872052	+	0.489413i	$0.162789\pi$
$461$			9688.52i			0.978827i	−0.872052	+	0.489413i	$0.837211\pi$
$462$		−	178.725i		−	0.0179979i
$463$		−	12914.8i		−	1.29633i	−0.761500	−	0.648165i	$-0.775537\pi$
$463$		−	12914.8i		−	1.29633i	0.761500	−	0.648165i	$-0.224463\pi$
$464$		−	4839.39i		−	0.484187i
$465$	37360.4			3.72591
$466$			13015.9i			1.29389i
$467$		−	15570.3i		−	1.54284i	−0.636324	−	0.771422i	$-0.719546\pi$
$467$		−	15570.3i		−	1.54284i	0.636324	−	0.771422i	$-0.280454\pi$
$468$	−6022.97			−0.594897
$469$			5416.88i			0.533323i
$470$		−	8656.13i		−	0.849526i
$471$	−2567.48			−0.251174
$472$	−1960.82			−0.191217
$473$	−114.809			−0.0111606
$474$	−16405.3			−1.58971
$475$			3693.11i			0.356740i
$476$	2688.88			0.258917
$477$	17832.1			1.71169
$478$		−	7962.07i		−	0.761875i
$479$		−	2367.56i		−	0.225839i	−0.993604	−	0.112919i	$-0.963980\pi$
$479$		−	2367.56i		−	0.225839i	0.993604	−	0.112919i	$-0.0360201\pi$
$480$	4363.33			0.414912
$481$	1696.56			0.160824
$482$	−8629.24			−0.815458
$483$	−9235.97			−0.870085
$484$	5323.52			0.499955
$485$	−13313.0			−1.24642
$486$	−12899.8			−1.20400
$487$	−9266.89			−0.862265			−0.431132	−	0.902289i	$-0.641886\pi$
$487$	−9266.89			−0.862265			−0.431132	+	0.902289i	$0.641886\pi$
$488$			5226.91i			0.484859i
$489$	26498.1			2.45048
$490$		−	10917.1i		−	1.00650i
$491$	−6708.71			−0.616619			−0.308309	−	0.951286i	$-0.599763\pi$
$491$	−6708.71			−0.616619			−0.308309	+	0.951286i	$0.599763\pi$
$492$		−	2504.31i		−	0.229478i
$493$	−7546.70			−0.689425
$494$	−2204.03			−0.200737
$495$	−318.759			−0.0289437
$496$		−	4383.93i		−	0.396863i
$497$	−11950.9			−1.07862
$498$			15392.5i			1.38505i
$499$			4033.98i			0.361895i	0.983493	+	0.180948i	$0.0579165\pi$
$499$			4033.98i			0.361895i	−0.983493	+	0.180948i	$0.942084\pi$
$500$	−2664.47			−0.238318
$501$	32913.9			2.93510
$502$	−12687.7			−1.12805
$503$		−	8071.90i		−	0.715523i	−0.933813	−	0.357762i	$-0.883540\pi$
$503$		−	8071.90i		−	0.715523i	0.933813	−	0.357762i	$-0.116460\pi$
$504$	−13893.8			−1.22793
$505$			2643.38i			0.232929i
$506$			24.8643i			0.00218449i
$507$			15793.1i			1.38343i
$508$	−2019.42			−0.176372
$509$			8417.36i			0.732992i	0.930420	+	0.366496i	$0.119443\pi$
$509$			8417.36i			0.732992i	−0.930420	+	0.366496i	$0.880557\pi$
$510$		−	6804.32i		−	0.590785i
$511$			5839.27i			0.505507i
$512$		−	512.000i		−	0.0441942i
$513$	−16902.7			−1.45472
$514$	−8666.03			−0.743662
$515$	7337.83			0.627851
$516$			12663.4i			1.08038i
$517$	105.282			0.00895612
$518$	3913.61			0.331958
$519$		−	11353.4i		−	0.960225i
$520$			2664.30i			0.224687i
$521$		−	20975.0i		−	1.76378i	−0.471453	−	0.881891i	$-0.656270\pi$
$521$		−	20975.0i		−	1.76378i	0.471453	−	0.881891i	$-0.343730\pi$
$522$	38994.7			3.26964
$523$		−	562.357i		−	0.0470175i	−0.999724	−	0.0235088i	$-0.992516\pi$
$523$		−	562.357i		−	0.0470175i	0.999724	−	0.0235088i	$-0.00748376\pi$
$524$	8586.25			0.715825
$525$	−20169.5			−1.67670
$526$			3367.25i			0.279124i
$527$	−6836.45			−0.565086
$528$			53.0701i			0.00437421i
$529$	−10882.1			−0.894393
$530$		−	7888.12i		−	0.646487i
$531$		−	15799.9i		−	1.29126i
$532$	−5084.24			−0.414341
$533$	1529.16			0.124269
$534$	−20378.9			−1.65147
$535$			5598.85i			0.452447i
$536$		−	1608.48i		−	0.129619i
$537$	8111.26			0.651818
$538$	8638.22	−	1800.43i	0.692231	−	0.144279i
$539$	132.782			0.0106110
$540$			20432.5i			1.62829i
$541$			8812.88i			0.700361i	0.936682	+	0.350180i	$0.113880\pi$
$541$			8812.88i			0.700361i	−0.936682	+	0.350180i	$0.886120\pi$
$542$	3689.19			0.292369
$543$	26834.4			2.12076
$544$	−798.430			−0.0629272
$545$			17617.7i			1.38470i
$546$		−	12037.1i		−	0.943477i
$547$	13171.3			1.02955			0.514775	−	0.857325i	$-0.327876\pi$
$547$	13171.3			1.02955			0.514775	+	0.857325i	$0.327876\pi$
$548$		−	11994.6i		−	0.935005i
$549$	−42117.3			−3.27418
$550$			54.2987i			0.00420964i
$551$	14269.6			1.10328
$552$	2742.51			0.211465
$553$		−	23107.8i		−	1.77693i
$554$	3666.02			0.281145
$555$		−	9903.56i		−	0.757446i
$556$		−	9493.39i		−	0.724117i
$557$		−	7094.31i		−	0.539669i	−0.962907	−	0.269834i	$-0.913031\pi$
$557$		−	7094.31i		−	0.539669i	0.962907	−	0.269834i	$-0.0869689\pi$
$558$	35324.8			2.67996
$559$	−7732.38			−0.585054
$560$			6145.98i			0.463777i
$561$	82.7594			0.00622835
$562$	15891.3			1.19276
$563$	5851.12			0.438002			0.219001	−	0.975725i	$-0.429720\pi$
$563$	5851.12			0.438002			0.219001	+	0.975725i	$0.429720\pi$
$564$		−	11612.5i		−	0.866979i
$565$			13237.6i			0.985680i
$566$		−	7865.92i		−	0.584151i
$567$		−	45421.0i		−	3.36420i
$568$	3548.68			0.262147
$569$		−	11893.5i		−	0.876274i	−0.898908	−	0.438137i	$-0.855639\pi$
$569$		−	11893.5i		−	0.876274i	0.898908	−	0.438137i	$-0.144361\pi$
$570$			12865.9i			0.945426i
$571$			15906.0i			1.16575i	0.812562	+	0.582875i	$0.198072\pi$
$571$			15906.0i			1.16575i	−0.812562	+	0.582875i	$0.801928\pi$
$572$	−32.4052			−0.00236876
$573$			45075.1i			3.28628i
$574$	3527.46			0.256504
$575$	2806.00			0.203510
$576$	4125.59			0.298436
$577$		−	320.673i		−	0.0231366i	−0.999933	−	0.0115683i	$-0.996318\pi$
$577$		−	320.673i		−	0.0231366i	0.999933	−	0.0115683i	$-0.00368238\pi$
$578$		−	8580.90i		−	0.617506i
$579$	−22099.3			−1.58621
$580$		−	17249.5i		−	1.23491i
$581$	−21681.2			−1.54817
$582$	−17860.0			−1.27203
$583$	95.9413			0.00681558
$584$		−	1733.90i		−	0.122858i
$585$	−21468.3			−1.51727
$586$			8412.24i			0.593014i
$587$	−9129.42			−0.641928			−0.320964	−	0.947091i	$-0.604007\pi$
$587$	−9129.42			−0.641928			−0.320964	+	0.947091i	$0.604007\pi$
$588$		−	14645.8i		−	1.02718i
$589$	12926.6			0.904299
$590$	−6989.17			−0.487694
$591$	50365.2			3.50549
$592$	−1162.10			−0.0806790
$593$	−1090.60			−0.0755237			−0.0377619	−	0.999287i	$-0.512023\pi$
$593$	−1090.60			−0.0755237			−0.0377619	+	0.999287i	$0.512023\pi$
$594$	−248.516			−0.0171662
$595$	9584.25			0.660363
$596$	325.058			0.0223404
$597$		−	40987.1i		−	2.80987i
$598$			1674.61i			0.114515i
$599$	−4370.22			−0.298101			−0.149050	−	0.988830i	$-0.547622\pi$
$599$	−4370.22			−0.298101			−0.149050	+	0.988830i	$0.547622\pi$
$600$	5989.09			0.407506
$601$		−	10814.8i		−	0.734020i	−0.930217	−	0.367010i	$-0.880382\pi$
$601$		−	10814.8i		−	0.734020i	0.930217	−	0.367010i	$-0.119618\pi$
$602$	−17837.0			−1.20761
$603$	12960.8			0.875296
$604$	−10734.6			−0.723154
$605$	18975.2			1.27513
$606$			3546.21i			0.237714i
$607$		−	28235.1i		−	1.88802i	−0.329923	−	0.944008i	$-0.607023\pi$
$607$		−	28235.1i		−	1.88802i	0.329923	−	0.944008i	$-0.392977\pi$
$608$	1509.70			0.100702
$609$			77932.0i			5.18549i
$610$			18630.8i			1.23662i
$611$	7090.74			0.469494
$612$		−	6433.57i		−	0.424938i
$613$			25067.1i			1.65163i	0.563939	+	0.825816i	$0.309285\pi$
$613$			25067.1i			1.65163i	−0.563939	+	0.825816i	$0.690715\pi$
$614$		−	9232.23i		−	0.606812i
$615$		−	8926.39i		−	0.585279i
$616$	−74.7521			−0.00488936
$617$	−21859.9			−1.42633			−0.713165	−	0.700997i	$-0.752739\pi$
$617$	−21859.9			−1.42633			−0.713165	+	0.700997i	$0.752739\pi$
$618$	9843.99			0.640750
$619$	21636.1			1.40489			0.702446	−	0.711737i	$-0.252091\pi$
$619$	21636.1			1.40489			0.702446	+	0.711737i	$0.252091\pi$
$620$		−	15626.1i		−	1.01219i
$621$			12842.6i			0.829879i
$622$	12895.9			0.831314
$623$		−	28704.8i		−	1.84596i
$624$			3574.26i			0.229303i
$625$	−19282.2			−1.23406
$626$			18230.4i			1.16395i
$627$	−156.485			−0.00996714
$628$			1073.85i			0.0682348i
$629$			1812.22i			0.114877i
$630$	−49523.0			−3.13182
$631$	−14629.4			−0.922961			−0.461481	−	0.887150i	$-0.652682\pi$
$631$	−14629.4			−0.922961			−0.461481	+	0.887150i	$0.652682\pi$
$632$			6861.57i			0.431865i
$633$			17296.2i			1.08604i
$634$	−9702.43			−0.607781
$635$	−7198.02			−0.449834
$636$		−	10582.2i		−	0.659769i
$637$	8942.87			0.556247
$638$	209.802			0.0130190
$639$			28594.5i			1.77024i
$640$		−	1824.98i		−	0.112716i
$641$	−30108.7			−1.85526			−0.927629	−	0.373502i	$-0.878157\pi$
$641$	−30108.7			−1.85526			−0.927629	+	0.373502i	$0.878157\pi$
$642$			7511.08i			0.461743i
$643$	11114.6			0.681675			0.340837	−	0.940122i	$-0.389289\pi$
$643$	11114.6			0.681675			0.340837	+	0.940122i	$0.389289\pi$
$644$			3862.97i			0.236370i
$645$			45137.4i			2.75548i
$646$		−	2354.28i		−	0.143387i
$647$		−	6161.95i		−	0.374422i	−0.982320	−	0.187211i	$-0.940055\pi$
$647$		−	6161.95i		−	0.374422i	0.982320	−	0.187211i	$-0.0599448\pi$
$648$			13487.2i			0.817636i
$649$		−	85.0076i		−	0.00514151i
$650$			3657.00i			0.220676i
$651$			70597.4i			4.25028i
$652$		−	11082.9i		−	0.665705i
$653$	−18594.3			−1.11432			−0.557160	−	0.830405i	$-0.688109\pi$
$653$	−18594.3			−1.11432			−0.557160	+	0.830405i	$0.688109\pi$
$654$			23634.9i			1.41315i
$655$	30604.9			1.82570
$656$	−1047.44			−0.0623407
$657$	13971.4			0.829644
$658$	16356.9			0.969085
$659$	19648.2			1.16143			0.580716	−	0.814106i	$-0.302772\pi$
$659$	19648.2			1.16143			0.580716	+	0.814106i	$0.302772\pi$
$660$			189.164i			0.0111563i
$661$		−	4200.79i		−	0.247189i	−0.992333	−	0.123594i	$-0.960558\pi$
$661$		−	4200.79i		−	0.247189i	0.992333	−	0.123594i	$-0.0394421\pi$
$662$			20146.4i			1.18280i
$663$	5573.82			0.326500
$664$	6437.97			0.376268
$665$	−18122.3			−1.05677
$666$		−	9363.94i		−	0.544813i
$667$		−	10842.0i		−	0.629389i
$668$		−	13766.3i		−	0.797358i
$669$	−12270.0			−0.709096
$670$		−	5733.27i		−	0.330590i
$671$	−226.602			−0.0130371
$672$			8245.08i			0.473305i
$673$			1662.17i			0.0952035i	0.998866	+	0.0476018i	$0.0151578\pi$
$673$			1662.17i			0.0952035i	−0.998866	+	0.0476018i	$0.984842\pi$
$674$	1432.74			0.0818797
$675$			28045.6i			1.59922i
$676$	6605.52			0.375826
$677$			19318.9i			1.09673i	0.836239	+	0.548365i	$0.184749\pi$
$677$			19318.9i			1.09673i	−0.836239	+	0.548365i	$0.815251\pi$
$678$			17758.8i			1.00593i
$679$		−	25156.7i		−	1.42184i
$680$	−2845.93			−0.160495
$681$	−6978.43			−0.392678
$682$	190.057			0.0106710
$683$		−	13463.9i		−	0.754293i	−0.926154	−	0.377146i	$-0.876905\pi$
$683$		−	13463.9i		−	0.754293i	0.926154	−	0.377146i	$-0.123095\pi$
$684$			12164.9i			0.680022i
$685$		−	42753.6i		−	2.38471i
$686$	2147.38			0.119515
$687$	−39472.0			−2.19207
$688$	5296.49			0.293498
$689$	6461.62			0.357283
$690$	9775.42			0.539339
$691$			14909.1i			0.820792i	0.911907	+	0.410396i	$0.134610\pi$
$691$			14909.1i			0.820792i	−0.911907	+	0.410396i	$0.865390\pi$
$692$	−4748.57			−0.260858
$693$		−	602.336i		−	0.0330171i
$694$		−	9587.59i		−	0.524409i
$695$		−	33838.3i		−	1.84685i
$696$		−	23141.0i		−	1.26028i
$697$			1633.41i			0.0887657i
$698$		−	17684.6i		−	0.958984i
$699$			62239.5i			3.36783i
$700$			8435.96i			0.455499i
$701$		−	25048.6i		−	1.34960i	−0.737999	−	0.674802i	$-0.764229\pi$
$701$		−	25048.6i		−	1.34960i	0.737999	−	0.674802i	$-0.235771\pi$
$702$	−16737.5			−0.899880
$703$		−	3426.61i		−	0.183836i
$704$	22.1967			0.00118831
$705$		−	41391.8i		−	2.21121i
$706$			16911.1i			0.901497i
$707$	−4995.02			−0.265710
$708$	−9376.26			−0.497714
$709$			34140.1i			1.80840i	0.427107	+	0.904201i	$0.359533\pi$
$709$			34140.1i			1.80840i	−0.427107	+	0.904201i	$0.640467\pi$
$710$	12648.9			0.668601
$711$	−55289.1			−2.91632
$712$			8523.55i			0.448643i
$713$		−	9821.57i		−	0.515877i
$714$	12857.7			0.673930
$715$	−115.505			−0.00604147
$716$		−	3392.56i		−	0.177075i
$717$		−	38073.0i		−	1.98307i
$718$	18301.6			0.951265
$719$			8695.64i			0.451033i	0.974239	+	0.225517i	$0.0724070\pi$
$719$			8695.64i			0.451033i	−0.974239	+	0.225517i	$0.927593\pi$
$720$	14705.3			0.761157
$721$			13865.8i			0.716212i
$722$		−	9266.43i		−	0.477647i
$723$	−41263.2			−2.12254
$724$		−	11223.6i		−	0.576133i
$725$		−	23676.7i		−	1.21287i
$726$	25456.0			1.30132
$727$	17895.4			0.912934			0.456467	−	0.889740i	$-0.349115\pi$
$727$	17895.4			0.912934			0.456467	+	0.889740i	$0.349115\pi$
$728$	−5034.54			−0.256308
$729$	−16164.7			−0.821254
$730$		−	6180.33i		−	0.313348i
$731$		−	8259.52i		−	0.417906i
$732$			24994.0i			1.26203i
$733$			23167.4i			1.16741i	0.811967	+	0.583703i	$0.198397\pi$
$733$			23167.4i			1.16741i	−0.811967	+	0.583703i	$0.801603\pi$
$734$	−15658.4			−0.787416
$735$		−	52203.5i		−	2.61980i
$736$		−	1147.06i		−	0.0574474i
$737$	69.7324			0.00348525
$738$		−	8440.01i		−	0.420977i
$739$		−	6585.41i		−	0.327805i	−0.986477	−	0.163903i	$-0.947592\pi$
$739$		−	6585.41i		−	0.327805i	0.986477	−	0.163903i	$-0.0524083\pi$
$740$	−4142.19			−0.205770
$741$	−10539.2			−0.522493
$742$	14905.6			0.737470
$743$	−10429.5			−0.514968			−0.257484	−	0.966283i	$-0.582893\pi$
$743$	−10429.5			−0.514968			−0.257484	+	0.966283i	$0.582893\pi$
$744$		−	20963.1i		−	1.03299i
$745$	1158.64			0.0569788
$746$	10.8498			0.000532491
$747$			51875.8i			2.54088i
$748$		−	34.6143i		−	0.00169201i
$749$	−10579.8			−0.516123
$750$	−12741.0			−0.620312
$751$	11440.4			0.555881			0.277941	−	0.960598i	$-0.410348\pi$
$751$	11440.4			0.555881			0.277941	+	0.960598i	$0.410348\pi$
$752$	−4856.98			−0.235526
$753$	−60670.1			−2.93618
$754$	14130.1			0.682478
$755$	−38262.5			−1.84439
$756$	−38609.9			−1.85745
$757$		−	13434.2i		−	0.645011i	−0.946568	−	0.322506i	$-0.895475\pi$
$757$		−	13434.2i		−	0.645011i	0.946568	−	0.322506i	$-0.104525\pi$
$758$	−15572.9			−0.746219
$759$			118.896i			0.00568597i
$760$	5381.20			0.256837
$761$		−	11860.5i		−	0.564972i	−0.959271	−	0.282486i	$-0.908841\pi$
$761$		−	11860.5i		−	0.564972i	0.959271	−	0.282486i	$-0.0911590\pi$
$762$	−9656.43			−0.459076
$763$	−33291.0			−1.57958
$764$	18852.8			0.892762
$765$		−	22931.9i		−	1.08380i
$766$	6330.12			0.298586
$767$		−	5725.24i		−	0.269526i
$768$		−	2448.28i		−	0.115032i
$769$	35760.1			1.67691			0.838453	−	0.544974i	$-0.183460\pi$
$769$	35760.1			1.67691			0.838453	+	0.544974i	$0.183460\pi$
$770$	−266.447			−0.0124702
$771$	−41439.2			−1.93566
$772$			9243.09i			0.430914i
$773$	21358.1			0.993785			0.496893	−	0.867812i	$-0.334474\pi$
$773$	21358.1			0.993785			0.496893	+	0.867812i	$0.334474\pi$
$774$			42678.0i			1.98195i
$775$		−	21448.3i		−	0.994126i
$776$			7469.98i			0.345563i
$777$	18714.1			0.864046
$778$			5182.92i			0.238839i
$779$		−	3088.51i		−	0.142051i
$780$			12740.1i			0.584833i
$781$			153.846i			0.00704872i
$782$	−1788.77			−0.0817982
$783$	108364.			4.94587
$784$	−6125.64			−0.279047
$785$			3827.65i			0.174032i
$786$	41057.7			1.86321
$787$	4530.70			0.205212			0.102606	−	0.994722i	$-0.467282\pi$
$787$	4530.70			0.205212			0.102606	+	0.994722i	$0.467282\pi$
$788$		−	21065.4i		−	0.952314i
$789$			16101.5i			0.726526i
$790$			24457.4i			1.10146i
$791$	−25014.1			−1.12440
$792$			178.857i			0.00802448i
$793$	−15261.6			−0.683425
$794$	−28920.8			−1.29265
$795$		−	37719.4i		−	1.68273i
$796$	−17143.0			−0.763337
$797$			26456.9i			1.17585i	0.808916	+	0.587924i	$0.200055\pi$
$797$			26456.9i			1.17585i	−0.808916	+	0.587924i	$0.799945\pi$
$798$	−24311.8			−1.07848
$799$			7574.14i			0.335361i
$800$		−	2504.96i		−	0.110704i
$801$	−68680.9			−3.02961
$802$	−15678.4			−0.690303
$803$	75.1698			0.00330347
$804$		−	7691.42i		−	0.337382i
$805$			13769.2i			0.602858i
$806$	12800.3			0.559392
$807$	41306.2	−	8609.30i	1.80179	−	0.375541i
$808$	1483.21			0.0645782
$809$			2255.60i			0.0980254i	0.998798	+	0.0490127i	$0.0156075\pi$
$809$			2255.60i			0.0980254i	−0.998798	+	0.0490127i	$0.984393\pi$
$810$			48073.9i			2.08537i
$811$	−27661.7			−1.19770			−0.598848	−	0.800862i	$-0.704375\pi$
$811$	−27661.7			−1.19770			−0.598848	+	0.800862i	$0.704375\pi$
$812$	32595.3			1.40871
$813$	17640.9			0.761002
$814$		−	50.3805i		−	0.00216933i
$815$		−	39503.9i		−	1.69787i
$816$	−3817.93			−0.163792
$817$			15617.4i			0.668770i
$818$	−28174.6			−1.20428
$819$		−	40567.2i		−	1.73081i
$820$	−3733.49			−0.158999
$821$	11811.7			0.502107			0.251053	−	0.967973i	$-0.419223\pi$
$821$	11811.7			0.502107			0.251053	+	0.967973i	$0.419223\pi$
$822$		−	57355.6i		−	2.43371i
$823$	−19909.1			−0.843241			−0.421620	−	0.906772i	$-0.638539\pi$
$823$	−19909.1			−0.843241			−0.421620	+	0.906772i	$0.638539\pi$
$824$		−	4117.28i		−	0.174068i
$825$			259.645i			0.0109572i
$826$		−	13207.0i		−	0.556330i
$827$	−13270.1			−0.557977			−0.278988	−	0.960294i	$-0.589999\pi$
$827$	−13270.1			−0.557977			−0.278988	+	0.960294i	$0.589999\pi$
$828$	9242.78			0.387933
$829$		−	32425.2i		−	1.35847i	−0.733920	−	0.679236i	$-0.762311\pi$
$829$		−	32425.2i		−	1.35847i	0.733920	−	0.679236i	$-0.237689\pi$
$830$	22947.5			0.959664
$831$	17530.2			0.731787
$832$	1494.95			0.0622931
$833$			9552.52i			0.397329i
$834$		−	45395.4i		−	1.88479i
$835$		−	49068.8i		−	2.03365i
$836$			65.4502i			0.00270771i
$837$	98165.5			4.05388
$838$			1035.72i			0.0426951i
$839$			17115.1i			0.704265i	0.935950	+	0.352132i	$0.114543\pi$
$839$			17115.1i			0.704265i	−0.935950	+	0.352132i	$0.885457\pi$
$840$			29388.8i			1.20716i
$841$	−67094.1			−2.75100
$842$		−	29712.9i		−	1.21612i
$843$	75988.8			3.10462
$844$	7234.20			0.295037
$845$	23544.8			0.958537
$846$		−	39136.5i		−	1.59047i
$847$			35856.1i			1.45458i
$848$	−4426.05			−0.179235
$849$		−	37613.2i		−	1.52047i
$850$	−3906.31			−0.157630
$851$	−2603.52			−0.104874
$852$	16969.1			0.682337
$853$		−	8631.92i		−	0.346485i	−0.984879	−	0.173242i	$-0.944576\pi$
$853$		−	8631.92i		−	0.346485i	0.984879	−	0.173242i	$-0.0554244\pi$
$854$	−35205.4			−1.41066
$855$			43360.5i			1.73438i
$856$	3141.53			0.125438
$857$		−	1518.59i		−	0.0605299i	−0.999542	−	0.0302649i	$-0.990365\pi$
$857$		−	1518.59i		−	0.0605299i	0.999542	−	0.0302649i	$-0.00963510\pi$
$858$	−154.955			−0.00616559
$859$	36266.8			1.44052			0.720260	−	0.693704i	$-0.244023\pi$
$859$	36266.8			1.44052			0.720260	+	0.693704i	$0.244023\pi$
$860$	18878.8			0.748562
$861$	16867.6			0.667649
$862$	8198.77			0.323957
$863$	−20517.9			−0.809314			−0.404657	−	0.914468i	$-0.632609\pi$
$863$	−20517.9			−0.809314			−0.404657	+	0.914468i	$0.632609\pi$
$864$	11464.8			0.451434
$865$	−16925.8			−0.665313
$866$		−	85.8054i		−	0.00336696i
$867$		−	41032.1i		−	1.60729i
$868$	29527.6			1.15464
$869$	−297.470			−0.0116122
$870$		−	82483.8i		−	3.21433i
$871$	4696.46			0.182702
$872$	9885.37			0.383900
$873$	−60191.5			−2.33353
$874$	3382.27			0.130901
$875$		−	17946.3i		−	0.693367i
$876$		−	8291.16i		−	0.319786i
$877$	46547.6			1.79225			0.896123	−	0.443806i	$-0.146372\pi$
$877$	46547.6			1.79225			0.896123	+	0.443806i	$0.146372\pi$
$878$			21033.4i			0.808475i
$879$			40225.6i			1.54355i
$880$	79.1182			0.00303077
$881$		−	17266.9i		−	0.660313i	−0.943926	−	0.330156i	$-0.892899\pi$
$881$		−	17266.9i		−	0.660313i	0.943926	−	0.330156i	$-0.107101\pi$
$882$		−	49359.1i		−	1.88436i
$883$			43073.3i			1.64160i	0.571217	+	0.820799i	$0.306471\pi$
$883$			43073.3i			1.64160i	−0.571217	+	0.820799i	$0.693529\pi$
$884$		−	2331.27i		−	0.0886980i
$885$	−33420.8			−1.26941
$886$	14292.0			0.541930
$887$	2416.16			0.0914620			0.0457310	−	0.998954i	$-0.485438\pi$
$887$	2416.16			0.0914620			0.0457310	+	0.998954i	$0.485438\pi$
$888$	−5556.92			−0.209998
$889$		−	13601.6i		−	0.513142i
$890$			30381.4i			1.14425i
$891$	−584.712			−0.0219849
$892$			5131.96i			0.192635i
$893$		−	14321.5i		−	0.536674i
$894$	1554.36			0.0581494
$895$		−	12092.4i		−	0.451627i
$896$	3448.53			0.128580
$897$			8007.62i			0.298068i
$898$		−	8639.62i		−	0.321055i
$899$	−82873.2			−3.07450
$900$	20184.4			0.747570
$901$			6902.13i			0.255209i
$902$		−	45.4095i		−	0.00167624i
$903$	−85293.0			−3.14327
$904$	7427.65			0.273274
$905$		−	40005.3i		−	1.46942i
$906$	−51330.7			−1.88228
$907$	−13571.6			−0.496845			−0.248423	−	0.968652i	$-0.579912\pi$
$907$	−13571.6			−0.496845			−0.248423	+	0.968652i	$0.579912\pi$
$908$			2918.75i			0.106676i
$909$			11951.4i			0.436087i
$910$	−17945.1			−0.653709
$911$		−	40223.9i		−	1.46287i	−0.681909	−	0.731437i	$-0.738850\pi$
$911$		−	40223.9i		−	1.46287i	0.681909	−	0.731437i	$-0.261150\pi$
$912$	7219.09			0.262114
$913$			279.106i			0.0101172i
$914$			16670.9i			0.603310i
$915$			89088.8i			3.21878i
$916$			16509.3i			0.595504i
$917$			57832.0i			2.08264i
$918$		−	17878.5i		−	0.642788i
$919$		−	37047.5i		−	1.32980i	−0.746933	−	0.664899i	$-0.768475\pi$
$919$		−	37047.5i		−	1.32980i	0.746933	−	0.664899i	$-0.231525\pi$
$920$		−	4088.60i		−	0.146519i
$921$		−	44146.6i		−	1.57946i
$922$	19377.0			0.692135
$923$			10361.5i			0.369505i
$924$	−357.449			−0.0127264
$925$	−5685.56			−0.202097
$926$	−25829.6			−0.916644
$927$	33176.1			1.17546
$928$	−9678.78			−0.342372
$929$			48867.0i			1.72581i	0.505369	+	0.862903i	$0.331356\pi$
$929$			48867.0i			1.72581i	−0.505369	+	0.862903i	$0.668644\pi$
$930$		−	74720.9i		−	2.63462i
$931$		−	18062.3i		−	0.635841i
$932$	26031.9			0.914916
$933$	61665.5			2.16381
$934$	−31140.6			−1.09095
$935$		−	123.380i		−	0.00431545i
$936$			12045.9i			0.420656i
$937$		−	2338.49i		−	0.0815318i	−0.999169	−	0.0407659i	$-0.987020\pi$
$937$		−	2338.49i		−	0.0815318i	0.999169	−	0.0407659i	$-0.0129798\pi$
$938$	10833.8			0.377116
$939$			87174.2i			3.02963i
$940$	−17312.3			−0.600706
$941$		−	42784.1i		−	1.48217i	−0.671411	−	0.741085i	$-0.734311\pi$
$941$		−	42784.1i		−	1.48217i	0.671411	−	0.741085i	$-0.265689\pi$
$942$			5134.95i			0.177607i
$943$	−2346.63			−0.0810358
$944$			3921.65i			0.135211i
$945$	−137622.			−4.73738
$946$			229.619i			0.00789170i
$947$			1892.81i			0.0649503i	0.999473	+	0.0324752i	$0.0103390\pi$
$947$			1892.81i			0.0649503i	−0.999473	+	0.0324752i	$0.989661\pi$
$948$			32810.6i			1.12409i
$949$	5062.67			0.173173
$950$	7386.21			0.252253
$951$	−46395.1			−1.58198
$952$		−	5377.75i		−	0.183082i
$953$			44072.0i			1.49804i	0.662548	+	0.749019i	$0.269475\pi$
$953$			44072.0i			1.49804i	−0.662548	+	0.749019i	$0.730525\pi$
$954$		−	35664.1i		−	1.21035i
$955$	67199.0			2.27697
$956$	−15924.1			−0.538727
$957$	1003.23			0.0338870
$958$	−4735.12			−0.159692
$959$	80788.4			2.72033
$960$		−	8726.66i		−	0.293387i
$961$	−45282.6			−1.52001
$962$		−	3393.11i		−	0.113720i
$963$			25313.8i			0.847067i
$964$			17258.5i			0.576616i
$965$			32946.1i			1.09904i
$966$			18471.9i			0.615243i
$967$			21965.8i			0.730477i	0.930914	+	0.365239i	$0.119013\pi$
$967$			21965.8i			0.730477i	−0.930914	+	0.365239i	$0.880987\pi$
$968$		−	10647.0i		−	0.353521i
$969$		−	11257.7i		−	0.373219i
$970$			26626.1i			0.881352i
$971$	6115.06			0.202103			0.101051	−	0.994881i	$-0.467779\pi$
$971$	6115.06			0.202103			0.101051	+	0.994881i	$0.467779\pi$
$972$			25799.6i			0.851359i
$973$	63941.9			2.10677
$974$			18533.8i			0.609713i
$975$			17487.0i			0.574393i
$976$	10453.8			0.342847
$977$	−27007.3			−0.884382			−0.442191	−	0.896921i	$-0.645799\pi$
$977$	−27007.3			−0.884382			−0.442191	+	0.896921i	$0.645799\pi$
$978$		−	52996.1i		−	1.73275i
$979$	−369.522			−0.0120633
$980$	−21834.3			−0.711704
$981$			79654.2i			2.59242i
$982$			13417.4i			0.436015i
$983$	−15755.3			−0.511207			−0.255603	−	0.966782i	$-0.582274\pi$
$983$	−15755.3			−0.511207			−0.255603	+	0.966782i	$0.582274\pi$
$984$	−5008.62			−0.162265
$985$		−	75085.6i		−	2.42886i
$986$			15093.4i			0.487497i
$987$	78215.3			2.52241
$988$			4408.05i			0.141942i
$989$	11866.0			0.381514
$990$			637.518i			0.0204663i
$991$			9810.59i			0.314474i	0.987561	+	0.157237i	$0.0502587\pi$
$991$			9810.59i			0.314474i	−0.987561	+	0.157237i	$0.949741\pi$
$992$	−8767.86			−0.280625
$993$			96335.8i			3.07867i
$994$			23901.8i			0.762697i
$995$	−61104.5			−1.94688
$996$	30785.1			0.979380
$997$	32678.8			1.03806			0.519031	−	0.854756i	$-0.326293\pi$
$997$	32678.8			1.03806			0.519031	+	0.854756i	$0.326293\pi$
$998$	8067.96			0.255899
$999$		−	26021.9i		−	0.824119i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.4.b.a.537.2	✓	68
269.268	even	2	inner	538.4.b.a.537.67	yes	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.4.b.a.537.2	✓	68	1.1	even	1	trivial
538.4.b.a.537.67	yes	68	269.268	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	538.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -9.56359i q^{3} -4.00000 q^{4} -14.2576 q^{5} -19.1272 q^{6} -26.9416i q^{7} +8.00000i q^{8} -64.4623 q^{9} +28.5153i q^{10} -0.346824 q^{11} +38.2544i q^{12} -23.3585 q^{13} -53.8833 q^{14} +136.354i q^{15} +16.0000 q^{16} -24.9509i q^{17} +128.925i q^{18} +47.1782i q^{19} +57.0305 q^{20} -257.659 q^{21} +0.693648i q^{22} +35.8457 q^{23} +76.5087 q^{24} +78.2799 q^{25} +46.7170i q^{26} +358.274i q^{27} +107.767i q^{28} -302.462i q^{29} +272.708 q^{30} -273.996i q^{31} -32.0000i q^{32} +3.31688i q^{33} -49.9019 q^{34} +384.124i q^{35} +257.849 q^{36} -72.6312 q^{37} +94.3564 q^{38} +223.391i q^{39} -114.061i q^{40} -65.4647 q^{41} +515.318i q^{42} +331.031 q^{43} +1.38730 q^{44} +919.079 q^{45} -71.6914i q^{46} -303.561 q^{47} -153.017i q^{48} -382.852 q^{49} -156.560i q^{50} -238.620 q^{51} +93.4341 q^{52} -276.628 q^{53} +716.548 q^{54} +4.94489 q^{55} +215.533 q^{56} +451.193 q^{57} -604.924 q^{58} +245.103i q^{59} -545.416i q^{60} +653.364 q^{61} -547.991 q^{62} +1736.72i q^{63} -64.0000 q^{64} +333.037 q^{65} +6.63377 q^{66} -201.060 q^{67} +99.8037i q^{68} -342.814i q^{69} +768.248 q^{70} -443.585i q^{71} -515.698i q^{72} -216.738 q^{73} +145.262i q^{74} -748.637i q^{75} -188.713i q^{76} +9.34401i q^{77} +446.783 q^{78} +857.697 q^{79} -228.122 q^{80} +1685.90 q^{81} +130.929i q^{82} -804.747i q^{83} +1030.64 q^{84} +355.741i q^{85} -662.061i q^{86} -2892.62 q^{87} -2.77459i q^{88} +1065.44 q^{89} -1838.16i q^{90} +629.317i q^{91} -143.383 q^{92} -2620.38 q^{93} +607.123i q^{94} -672.649i q^{95} -306.035 q^{96} +933.748 q^{97} +765.705i q^{98} +22.3571 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 272 q^{4} + 38 q^{5} - 4 q^{6} - 594 q^{9} + 18 q^{11} - 114 q^{13} + 8 q^{14} + 1088 q^{16} - 152 q^{20} - 20 q^{21} - 224 q^{23} + 16 q^{24} + 1098 q^{25} + 384 q^{30} - 600 q^{34} + 2376 q^{36}+ \cdots + 3370 q^{99}+O(q^{100}) \) 68 * q - 272 * q^4 + 38 * q^5 - 4 * q^6 - 594 * q^9 + 18 * q^11 - 114 * q^13 + 8 * q^14 + 1088 * q^16 - 152 * q^20 - 20 * q^21 - 224 * q^23 + 16 * q^24 + 1098 * q^25 + 384 * q^30 - 600 * q^34 + 2376 * q^36 - 302 * q^37 + 436 * q^38 - 624 * q^41 - 654 * q^43 - 72 * q^44 - 1074 * q^45 + 692 * q^47 - 3052 * q^49 + 596 * q^51 + 456 * q^52 + 326 * q^53 + 1504 * q^54 + 828 * q^55 - 32 * q^56 - 1024 * q^57 - 100 * q^58 + 226 * q^61 + 312 * q^62 - 4352 * q^64 + 316 * q^65 + 2480 * q^66 - 1318 * q^67 + 608 * q^70 - 2028 * q^73 - 1152 * q^78 - 1444 * q^79 + 608 * q^80 + 3204 * q^81 + 80 * q^84 - 5600 * q^87 - 1284 * q^89 + 896 * q^92 - 2528 * q^93 - 64 * q^96 - 288 * q^97 + 3370 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more