LMFDB - Embedded newform 175.4.b.d.99.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [175,4,Mod(99,175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("175.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 175 = 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	175.b (of order $2$, degree $1$, not 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:	$10.3253342510$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$-2.70156i$ of defining polynomial
Character	$\chi$	$=$	175.99
Dual form			175.4.b.d.99.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.70156i q^{2} -0.701562i q^{3} +0.701562 q^{4} -1.89531 q^{6} +7.00000i q^{7} -23.5078i q^{8} +26.5078 q^{9} +4.01562 q^{11} -0.492189i q^{12} -51.6125i q^{13} +18.9109 q^{14} -57.8953 q^{16} -67.5078i q^{17} -71.6125i q^{18} +50.9109 q^{19} +4.91093 q^{21} -10.8485i q^{22} -0.507811i q^{23} -16.4922 q^{24} -139.434 q^{26} -37.5391i q^{27} +4.91093i q^{28} +120.058 q^{29} -292.303 q^{31} -31.6547i q^{32} -2.81721i q^{33} -182.377 q^{34} +18.5969 q^{36} -144.989i q^{37} -137.539i q^{38} -36.2094 q^{39} -57.2047 q^{41} -13.2672i q^{42} +283.020i q^{43} +2.81721 q^{44} -1.37188 q^{46} -233.769i q^{47} +40.6172i q^{48} -49.0000 q^{49} -47.3609 q^{51} -36.2094i q^{52} +406.334i q^{53} -101.414 q^{54} +164.555 q^{56} -35.7172i q^{57} -324.344i q^{58} +577.328 q^{59} +322.116 q^{61} +789.675i q^{62} +185.555i q^{63} -548.680 q^{64} -7.61086 q^{66} +985.459i q^{67} -47.3609i q^{68} -0.356261 q^{69} +1033.57 q^{71} -623.141i q^{72} -692.720i q^{73} -391.697 q^{74} +35.7172 q^{76} +28.1093i q^{77} +97.8219i q^{78} +428.236 q^{79} +689.375 q^{81} +154.542i q^{82} +537.592i q^{83} +3.44533 q^{84} +764.597 q^{86} -84.2280i q^{87} -94.3985i q^{88} +802.073 q^{89} +361.287 q^{91} -0.356261i q^{92} +205.069i q^{93} -631.541 q^{94} -22.2077 q^{96} +1752.82i q^{97} +132.377i q^{98} +106.445 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4} - 46 q^{6} + 42 q^{9} - 112 q^{11} - 14 q^{14} - 270 q^{16} + 114 q^{19} - 70 q^{21} - 130 q^{24} - 276 q^{26} + 826 q^{29} - 324 q^{31} - 102 q^{34} + 100 q^{36} - 68 q^{39} - 1010 q^{41}+ \cdots + 874 q^{99}+O(q^{100}) $ 4 * q - 10 * q^4 - 46 * q^6 + 42 * q^9 - 112 * q^11 - 14 * q^14 - 270 * q^16 + 114 * q^19 - 70 * q^21 - 130 * q^24 - 276 * q^26 + 826 * q^29 - 324 * q^31 - 102 * q^34 + 100 * q^36 - 68 * q^39 - 1010 * q^41 + 690 * q^44 - 236 * q^46 - 196 * q^49 + 310 * q^51 - 1110 * q^54 + 210 * q^56 - 380 * q^59 + 1980 * q^61 - 722 * q^64 + 2518 * q^66 - 360 * q^69 + 254 * q^71 - 2412 * q^74 + 2 * q^76 + 2238 * q^79 - 316 * q^81 + 462 * q^84 + 3084 * q^86 + 3426 * q^89 + 728 * q^91 - 4652 * q^94 + 2434 * q^96 + 874 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/175\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 2.70156i	− 0.955146i	−0.878592	−	0.477573i	$-0.841517\pi$
$2$	− 2.70156i	− 0.955146i	0.878592	−	0.477573i	$-0.158483\pi$
$3$	− 0.701562i	− 0.135016i	−0.997719	−	0.0675078i	$-0.978495\pi$
$3$	− 0.701562i	− 0.135016i	0.997719	−	0.0675078i	$-0.0215048\pi$
$4$	0.701562	0.0876953
$5$	0	0
$5$	0	0
$6$	−1.89531	−0.128960
$7$	7.00000i	0.377964i
$7$	7.00000i	0.377964i
$8$	− 23.5078i	− 1.03891i
$9$	26.5078	0.981771
$10$	0	0
$11$	4.01562	0.110069	0.0550343	−	0.998484i	$-0.482473\pi$
$11$	4.01562	0.110069	0.0550343	+	0.998484i	$0.482473\pi$
$12$	− 0.492189i	− 0.0118402i
$13$	− 51.6125i	− 1.10113i	−0.834791	−	0.550567i	$-0.814412\pi$
$13$	− 51.6125i	− 1.10113i	0.834791	−	0.550567i	$-0.185588\pi$
$14$	18.9109	0.361011
$15$	0	0
$16$	−57.8953	−0.904614
$17$	− 67.5078i	− 0.963121i	−0.876413	−	0.481560i	$-0.840070\pi$
$17$	− 67.5078i	− 0.963121i	0.876413	−	0.481560i	$-0.159930\pi$
$18$	− 71.6125i	− 0.937735i
$19$	50.9109	0.614725	0.307362	−	0.951593i	$-0.400554\pi$
$19$	50.9109	0.614725	0.307362	+	0.951593i	$0.400554\pi$
$20$	0	0
$21$	4.91093	0.0510311
$22$	− 10.8485i	− 0.105132i
$23$	− 0.507811i	− 0.00460373i	−0.999997	−	0.00230187i	$-0.999267\pi$
$23$	− 0.507811i	− 0.00460373i	0.999997	−	0.00230187i	$-0.000732707\pi$
$24$	−16.4922	−0.140269
$25$	0	0
$26$	−139.434	−1.05174
$27$	− 37.5391i	− 0.267570i
$28$	4.91093i	0.0331457i
$29$	120.058	0.768765	0.384382	−	0.923174i	$-0.374414\pi$
$29$	120.058	0.768765	0.384382	+	0.923174i	$0.374414\pi$
$30$	0	0
$31$	−292.303	−1.69352	−0.846761	−	0.531973i	$-0.821451\pi$
$31$	−292.303	−1.69352	−0.846761	+	0.531973i	$0.821451\pi$
$32$	− 31.6547i	− 0.174869i
$33$	− 2.81721i	− 0.0148610i
$34$	−182.377	−0.919921
$35$	0	0
$36$	18.5969	0.0860966
$37$	− 144.989i	− 0.644218i	−0.946703	−	0.322109i	$-0.895608\pi$
$37$	− 144.989i	− 0.644218i	0.946703	−	0.322109i	$-0.104392\pi$
$38$	− 137.539i	− 0.587152i
$39$	−36.2094	−0.148670
$40$	0	0
$41$	−57.2047	−0.217899	−0.108950	−	0.994047i	$-0.534749\pi$
$41$	−57.2047	−0.217899	−0.108950	+	0.994047i	$0.534749\pi$
$42$	− 13.2672i	− 0.0487422i
$43$	283.020i	1.00373i	0.864947	+	0.501863i	$0.167352\pi$
$43$	283.020i	1.00373i	−0.864947	+	0.501863i	$0.832648\pi$
$44$	2.81721	0.00965250
$45$	0	0
$46$	−1.37188	−0.00439724
$47$	− 233.769i	− 0.725504i	−0.931886	−	0.362752i	$-0.881837\pi$
$47$	− 233.769i	− 0.725504i	0.931886	−	0.362752i	$-0.118163\pi$
$48$	40.6172i	0.122137i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−47.3609	−0.130036
$52$	− 36.2094i	− 0.0965642i
$53$	406.334i	1.05310i	0.850144	+	0.526550i	$0.176515\pi$
$53$	406.334i	1.05310i	−0.850144	+	0.526550i	$0.823485\pi$
$54$	−101.414	−0.255569
$55$	0	0
$56$	164.555	0.392670
$57$	− 35.7172i	− 0.0829975i
$58$	− 324.344i	− 0.734283i
$59$	577.328	1.27393	0.636964	−	0.770894i	$-0.280190\pi$
$59$	577.328	1.27393	0.636964	+	0.770894i	$0.280190\pi$
$60$	0	0
$61$	322.116	0.676110	0.338055	−	0.941126i	$-0.390231\pi$
$61$	322.116	0.676110	0.338055	+	0.941126i	$0.390231\pi$
$62$	789.675i	1.61756i
$63$	185.555i	0.371074i
$64$	−548.680	−1.07164
$65$	0	0
$66$	−7.61086	−0.0141944
$67$	985.459i	1.79691i	0.439065	+	0.898455i	$0.355310\pi$
$67$	985.459i	1.79691i	−0.439065	+	0.898455i	$0.644690\pi$
$68$	− 47.3609i	− 0.0844611i
$69$	−0.356261	−0.000621576 0
$70$	0	0
$71$	1033.57	1.72764	0.863821	−	0.503799i	$-0.168065\pi$
$71$	1033.57	1.72764	0.863821	+	0.503799i	$0.168065\pi$
$72$	− 623.141i	− 1.01997i
$73$	− 692.720i	− 1.11064i	−0.831637	−	0.555320i	$-0.812596\pi$
$73$	− 692.720i	− 1.11064i	0.831637	−	0.555320i	$-0.187404\pi$
$74$	−391.697	−0.615322
$75$	0	0
$76$	35.7172	0.0539084
$77$	28.1093i	0.0416020i
$78$	97.8219i	0.142002i
$79$	428.236	0.609877	0.304939	−	0.952372i	$-0.401364\pi$
$79$	428.236	0.609877	0.304939	+	0.952372i	$0.401364\pi$
$80$	0	0
$81$	689.375	0.945645
$82$	154.542i	0.208126i
$83$	537.592i	0.710945i	0.934687	+	0.355472i	$0.115680\pi$
$83$	537.592i	0.710945i	−0.934687	+	0.355472i	$0.884320\pi$
$84$	3.44533	0.00447519
$85$	0	0
$86$	764.597	0.958705
$87$	− 84.2280i	− 0.103795i
$88$	− 94.3985i	− 0.114351i
$89$	802.073	0.955277	0.477638	−	0.878557i	$-0.341493\pi$
$89$	802.073	0.955277	0.477638	+	0.878557i	$0.341493\pi$
$90$	0	0
$91$	361.287	0.416189
$92$	− 0.356261i	0 0.000403725i
$93$	205.069i	0.228652i
$94$	−631.541	−0.692962
$95$	0	0
$96$	−22.2077	−0.0236101
$97$	1752.82i	1.83477i	0.398004	+	0.917384i	$0.369703\pi$
$97$	1752.82i	1.83477i	−0.398004	+	0.917384i	$0.630297\pi$
$98$	132.377i	0.136449i
$99$	106.445	0.108062
$100$	0	0
$101$	−1987.85	−1.95840	−0.979200	−	0.202896i	$-0.934965\pi$
$101$	−1987.85	−1.95840	−0.979200	+	0.202896i	$0.934965\pi$
$102$	127.948i	0.124204i
$103$	389.122i	0.372246i	0.982526	+	0.186123i	$0.0595923\pi$
$103$	389.122i	0.372246i	−0.982526	+	0.186123i	$0.940408\pi$
$104$	−1213.30	−1.14398
$105$	0	0
$106$	1097.74	1.00586
$107$	− 508.595i	− 0.459512i	−0.973248	−	0.229756i	$-0.926207\pi$
$107$	− 508.595i	− 0.459512i	0.973248	−	0.229756i	$-0.0737928\pi$
$108$	− 26.3360i	− 0.0234646i
$109$	−344.130	−0.302400	−0.151200	−	0.988503i	$-0.548314\pi$
$109$	−344.130	−0.302400	−0.151200	+	0.988503i	$0.548314\pi$
$110$	0	0
$111$	−101.719	−0.0869795
$112$	− 405.267i	− 0.341912i
$113$	1218.12i	1.01408i	0.861923	+	0.507040i	$0.169260\pi$
$113$	1218.12i	1.01408i	−0.861923	+	0.507040i	$0.830740\pi$
$114$	−96.4922	−0.0792748
$115$	0	0
$116$	84.2280	0.0674170
$117$	− 1368.13i	− 1.08106i
$118$	− 1559.69i	− 1.21679i
$119$	472.555	0.364025
$120$	0	0
$121$	−1314.87	−0.987885
$122$	− 870.215i	− 0.645784i
$123$	40.1327i	0.0294198i
$124$	−205.069	−0.148514
$125$	0	0
$126$	501.287	0.354430
$127$	− 281.652i	− 0.196792i	−0.995147	−	0.0983958i	$-0.968629\pi$
$127$	− 281.652i	− 0.196792i	0.995147	−	0.0983958i	$-0.0313711\pi$
$128$	1229.05i	0.848704i
$129$	198.556	0.135519
$130$	0	0
$131$	−699.706	−0.466669	−0.233334	−	0.972397i	$-0.574964\pi$
$131$	−699.706	−0.466669	−0.233334	+	0.972397i	$0.574964\pi$
$132$	− 1.97645i	− 0.00130324i
$133$	356.377i	0.232344i
$134$	2662.28	1.71631
$135$	0	0
$136$	−1586.96	−1.00059
$137$	1588.77i	0.990789i	0.868668	+	0.495394i	$0.164976\pi$
$137$	1588.77i	0.990789i	−0.868668	+	0.495394i	$0.835024\pi$
$138$	0.962460i	0 0.000593696i
$139$	−2564.09	−1.56463	−0.782315	−	0.622884i	$-0.785961\pi$
$139$	−2564.09	−1.56463	−0.782315	+	0.622884i	$0.785961\pi$
$140$	0	0
$141$	−164.003	−0.0979544
$142$	− 2792.26i	− 1.65015i
$143$	− 207.256i	− 0.121200i
$144$	−1534.68	−0.888124
$145$	0	0
$146$	−1871.43	−1.06082
$147$	34.3765i	0.0192880i
$148$	− 101.719i	− 0.0564948i
$149$	3386.03	1.86171	0.930854	−	0.365390i	$-0.119065\pi$
$149$	3386.03	1.86171	0.930854	+	0.365390i	$0.119065\pi$
$150$	0	0
$151$	−2301.98	−1.24061	−0.620306	−	0.784360i	$-0.712992\pi$
$151$	−2301.98	−1.24061	−0.620306	+	0.784360i	$0.712992\pi$
$152$	− 1196.80i	− 0.638643i
$153$	− 1789.48i	− 0.945564i
$154$	75.9392	0.0397360
$155$	0	0
$156$	−25.4031	−0.0130377
$157$	587.325i	0.298558i	0.988795	+	0.149279i	$0.0476953\pi$
$157$	587.325i	0.298558i	−0.988795	+	0.149279i	$0.952305\pi$
$158$	− 1156.91i	− 0.582522i
$159$	285.069	0.142185
$160$	0	0
$161$	3.55467	0.00174005
$162$	− 1862.39i	− 0.903229i
$163$	− 914.189i	− 0.439293i	−0.975579	−	0.219647i	$-0.929510\pi$
$163$	− 914.189i	− 0.439293i	0.975579	−	0.219647i	$-0.0704905\pi$
$164$	−40.1327	−0.0191087
$165$	0	0
$166$	1452.34	0.679056
$167$	3316.66i	1.53683i	0.639951	+	0.768416i	$0.278955\pi$
$167$	3316.66i	1.53683i	−0.639951	+	0.768416i	$0.721045\pi$
$168$	− 115.445i	− 0.0530167i
$169$	−466.850	−0.212494
$170$	0	0
$171$	1349.54	0.603519
$172$	198.556i	0.0880220i
$173$	− 1542.16i	− 0.677737i	−0.940834	−	0.338868i	$-0.889956\pi$
$173$	− 1542.16i	− 0.677737i	0.940834	−	0.338868i	$-0.110044\pi$
$174$	−227.547	−0.0991397
$175$	0	0
$176$	−232.486	−0.0995697
$177$	− 405.031i	− 0.172000i
$178$	− 2166.85i	− 0.912429i
$179$	550.755	0.229974	0.114987	−	0.993367i	$-0.463317\pi$
$179$	550.755	0.229974	0.114987	+	0.993367i	$0.463317\pi$
$180$	0	0
$181$	−3562.15	−1.46283	−0.731416	−	0.681931i	$-0.761140\pi$
$181$	−3562.15	−1.46283	−0.731416	+	0.681931i	$0.761140\pi$
$182$	− 976.041i	− 0.397522i
$183$	− 225.984i	− 0.0912854i
$184$	−11.9375	−0.00478285
$185$	0	0
$186$	554.006	0.218396
$187$	− 271.086i	− 0.106009i
$188$	− 164.003i	− 0.0636232i
$189$	262.773	0.101132
$190$	0	0
$191$	3436.00	1.30168	0.650838	−	0.759217i	$-0.274418\pi$
$191$	3436.00	1.30168	0.650838	+	0.759217i	$0.274418\pi$
$192$	384.933i	0.144688i
$193$	− 1047.12i	− 0.390534i	−0.980750	−	0.195267i	$-0.937443\pi$
$193$	− 1047.12i	− 0.390534i	0.980750	−	0.195267i	$-0.0625573\pi$
$194$	4735.37	1.75247
$195$	0	0
$196$	−34.3765	−0.0125279
$197$	− 3205.61i	− 1.15934i	−0.814851	−	0.579670i	$-0.803181\pi$
$197$	− 3205.61i	− 1.15934i	0.814851	−	0.579670i	$-0.196819\pi$
$198$	− 287.569i	− 0.103215i
$199$	−22.0532	−0.00785581	−0.00392791	−	0.999992i	$-0.501250\pi$
$199$	−22.0532	−0.00785581	−0.00392791	+	0.999992i	$0.501250\pi$
$200$	0	0
$201$	691.361	0.242611
$202$	5370.30i	1.87056i
$203$	840.405i	0.290566i
$204$	−33.2266	−0.0114036
$205$	0	0
$206$	1051.24	0.355549
$207$	− 13.4609i	− 0.00451981i
$208$	2988.12i	0.996101i
$209$	204.439	0.0676619
$210$	0	0
$211$	2362.52	0.770819	0.385410	−	0.922746i	$-0.374060\pi$
$211$	2362.52	0.770819	0.385410	+	0.922746i	$0.374060\pi$
$212$	285.069i	0.0923519i
$213$	− 725.116i	− 0.233259i
$214$	−1374.00	−0.438901
$215$	0	0
$216$	−882.461	−0.277981
$217$	− 2046.12i	− 0.640091i
$218$	929.688i	0.288837i
$219$	−485.986	−0.149954
$220$	0	0
$221$	−3484.25	−1.06052
$222$	274.800i	0.0830781i
$223$	1428.67i	0.429016i	0.976722	+	0.214508i	$0.0688148\pi$
$223$	1428.67i	0.429016i	−0.976722	+	0.214508i	$0.931185\pi$
$224$	221.583	0.0660943
$225$	0	0
$226$	3290.82	0.968594
$227$	5979.38i	1.74831i	0.485651	+	0.874153i	$0.338582\pi$
$227$	5979.38i	1.74831i	−0.485651	+	0.874153i	$0.661418\pi$
$228$	− 25.0578i	− 0.00727849i
$229$	−6377.77	−1.84042	−0.920208	−	0.391430i	$-0.871980\pi$
$229$	−6377.77	−1.84042	−0.920208	+	0.391430i	$0.871980\pi$
$230$	0	0
$231$	19.7205	0.00561693
$232$	− 2822.30i	− 0.798676i
$233$	4947.18i	1.39099i	0.718531	+	0.695495i	$0.244815\pi$
$233$	4947.18i	1.39099i	−0.718531	+	0.695495i	$0.755185\pi$
$234$	−3696.10	−1.03257
$235$	0	0
$236$	405.031	0.111717
$237$	− 300.434i	− 0.0823430i
$238$	− 1276.64i	− 0.347698i
$239$	348.213	0.0942427	0.0471213	−	0.998889i	$-0.484995\pi$
$239$	348.213	0.0942427	0.0471213	+	0.998889i	$0.484995\pi$
$240$	0	0
$241$	4702.23	1.25683	0.628417	−	0.777876i	$-0.283703\pi$
$241$	4702.23	1.25683	0.628417	+	0.777876i	$0.283703\pi$
$242$	3552.22i	0.943575i
$243$	− 1497.19i	− 0.395247i
$244$	225.984	0.0592916
$245$	0	0
$246$	108.421	0.0281003
$247$	− 2627.64i	− 0.676894i
$248$	6871.41i	1.75941i
$249$	377.154	0.0959887
$250$	0	0
$251$	−2431.64	−0.611489	−0.305745	−	0.952114i	$-0.598905\pi$
$251$	−2431.64	−0.611489	−0.305745	+	0.952114i	$0.598905\pi$
$252$	130.178i	0.0325415i
$253$	− 2.03917i	0 0.000506727i
$254$	−760.899	−0.187965
$255$	0	0
$256$	−1069.07	−0.261003
$257$	− 1631.76i	− 0.396055i	−0.980196	−	0.198028i	$-0.936546\pi$
$257$	− 1631.76i	− 0.396055i	0.980196	−	0.198028i	$-0.0634535\pi$
$258$	− 536.412i	− 0.129440i
$259$	1014.92	0.243491
$260$	0	0
$261$	3182.47	0.754751
$262$	1890.30i	0.445737i
$263$	3563.67i	0.835534i	0.908554	+	0.417767i	$0.137187\pi$
$263$	3563.67i	0.835534i	−0.908554	+	0.417767i	$0.862813\pi$
$264$	−66.2264	−0.0154392
$265$	0	0
$266$	962.773	0.221923
$267$	− 562.704i	− 0.128977i
$268$	691.361i	0.157581i
$269$	−791.631	−0.179430	−0.0897149	−	0.995967i	$-0.528596\pi$
$269$	−791.631	−0.179430	−0.0897149	+	0.995967i	$0.528596\pi$
$270$	0	0
$271$	7823.17	1.75359	0.876796	−	0.480862i	$-0.159676\pi$
$271$	7823.17	1.75359	0.876796	+	0.480862i	$0.159676\pi$
$272$	3908.39i	0.871253i
$273$	− 253.466i	− 0.0561921i
$274$	4292.17	0.946348
$275$	0	0
$276$	−0.249939	−5.45093e−5 0
$277$	− 1554.16i	− 0.337114i	−0.985692	−	0.168557i	$-0.946089\pi$
$277$	− 1554.16i	− 0.337114i	0.985692	−	0.168557i	$-0.0539107\pi$
$278$	6927.05i	1.49445i
$279$	−7748.32	−1.66265
$280$	0	0
$281$	−5043.72	−1.07076	−0.535379	−	0.844612i	$-0.679831\pi$
$281$	−5043.72	−1.07076	−0.535379	+	0.844612i	$0.679831\pi$
$282$	443.065i	0.0935608i
$283$	5897.15i	1.23869i	0.785119	+	0.619345i	$0.212602\pi$
$283$	5897.15i	1.23869i	−0.785119	+	0.619345i	$0.787398\pi$
$284$	725.116	0.151506
$285$	0	0
$286$	−559.916	−0.115764
$287$	− 400.433i	− 0.0823582i
$288$	− 839.097i	− 0.171681i
$289$	355.696	0.0723988
$290$	0	0
$291$	1229.72	0.247722
$292$	− 485.986i	− 0.0973979i
$293$	− 8592.25i	− 1.71319i	−0.515990	−	0.856595i	$-0.672576\pi$
$293$	− 8592.25i	− 1.71319i	0.515990	−	0.856595i	$-0.327424\pi$
$294$	92.8704	0.0184228
$295$	0	0
$296$	−3408.37	−0.669283
$297$	− 150.743i	− 0.0294511i
$298$	− 9147.58i	− 1.77820i
$299$	−26.2094	−0.00506932
$300$	0	0
$301$	−1981.14	−0.379373
$302$	6218.94i	1.18497i
$303$	1394.60i	0.264415i
$304$	−2947.50	−0.556089
$305$	0	0
$306$	−4834.40	−0.903152
$307$	− 3682.20i	− 0.684542i	−0.939601	−	0.342271i	$-0.888804\pi$
$307$	− 3682.20i	− 0.684542i	0.939601	−	0.342271i	$-0.111196\pi$
$308$	19.7205i	0.00364830i
$309$	272.993	0.0502590
$310$	0	0
$311$	6866.18	1.25191	0.625957	−	0.779858i	$-0.284709\pi$
$311$	6866.18	1.25191	0.625957	+	0.779858i	$0.284709\pi$
$312$	851.203i	0.154455i
$313$	− 2958.60i	− 0.534281i	−0.963658	−	0.267140i	$-0.913921\pi$
$313$	− 2958.60i	− 0.534281i	0.963658	−	0.267140i	$-0.0860788\pi$
$314$	1586.69	0.285167
$315$	0	0
$316$	300.434	0.0534834
$317$	1585.21i	0.280866i	0.990090	+	0.140433i	$0.0448494\pi$
$317$	1585.21i	0.280866i	−0.990090	+	0.140433i	$0.955151\pi$
$318$	− 770.131i	− 0.135808i
$319$	482.107	0.0846169
$320$	0	0
$321$	−356.811	−0.0620413
$322$	− 9.60317i	− 0.00166200i
$323$	− 3436.89i	− 0.592054i
$324$	483.639	0.0829286
$325$	0	0
$326$	−2469.74	−0.419589
$327$	241.428i	0.0408288i
$328$	1344.76i	0.226377i
$329$	1636.38	0.274215
$330$	0	0
$331$	6045.84	1.00396	0.501978	−	0.864880i	$-0.332606\pi$
$331$	6045.84	1.00396	0.501978	+	0.864880i	$0.332606\pi$
$332$	377.154i	0.0623465i
$333$	− 3843.34i	− 0.632474i
$334$	8960.16	1.46790
$335$	0	0
$336$	−284.320	−0.0461635
$337$	− 9205.64i	− 1.48802i	−0.668168	−	0.744010i	$-0.732921\pi$
$337$	− 9205.64i	− 1.48802i	0.668168	−	0.744010i	$-0.267079\pi$
$338$	1261.22i	0.202963i
$339$	854.586	0.136917
$340$	0	0
$341$	−1173.78	−0.186404
$342$	− 3645.86i	− 0.576449i
$343$	− 343.000i	− 0.0539949i
$344$	6653.19	1.04278
$345$	0	0
$346$	−4166.25	−0.647338
$347$	− 3092.28i	− 0.478393i	−0.970971	−	0.239196i	$-0.923116\pi$
$347$	− 3092.28i	− 0.478393i	0.970971	−	0.239196i	$-0.0768839\pi$
$348$	− 59.0912i	− 0.00910236i
$349$	5231.61	0.802412	0.401206	−	0.915988i	$-0.368591\pi$
$349$	5231.61	0.802412	0.401206	+	0.915988i	$0.368591\pi$
$350$	0	0
$351$	−1937.48	−0.294630
$352$	− 127.113i	− 0.0192476i
$353$	− 9013.71i	− 1.35907i	−0.733644	−	0.679534i	$-0.762182\pi$
$353$	− 9013.71i	− 1.35907i	0.733644	−	0.679534i	$-0.237818\pi$
$354$	−1094.22	−0.164285
$355$	0	0
$356$	562.704	0.0837732
$357$	− 331.526i	− 0.0491491i
$358$	− 1487.90i	− 0.219659i
$359$	4779.38	0.702635	0.351317	−	0.936256i	$-0.385734\pi$
$359$	4779.38	0.702635	0.351317	+	0.936256i	$0.385734\pi$
$360$	0	0
$361$	−4267.08	−0.622114
$362$	9623.38i	1.39722i
$363$	922.466i	0.133380i
$364$	253.466	0.0364978
$365$	0	0
$366$	−610.510	−0.0871909
$367$	175.769i	0.0250001i	0.999922	+	0.0125001i	$0.00397900\pi$
$367$	175.769i	0.0250001i	−0.999922	+	0.0125001i	$0.996021\pi$
$368$	29.3999i	0.00416460i
$369$	−1516.37	−0.213927
$370$	0	0
$371$	−2844.34	−0.398034
$372$	143.868i	0.0200517i
$373$	− 7982.98i	− 1.10816i	−0.832464	−	0.554079i	$-0.813070\pi$
$373$	− 7982.98i	− 1.10816i	0.832464	−	0.554079i	$-0.186930\pi$
$374$	−732.355	−0.101254
$375$	0	0
$376$	−5495.39	−0.753732
$377$	− 6196.48i	− 0.846512i
$378$	− 709.899i	− 0.0965959i
$379$	−12663.1	−1.71626	−0.858129	−	0.513434i	$-0.828373\pi$
$379$	−12663.1	−1.71626	−0.858129	+	0.513434i	$0.828373\pi$
$380$	0	0
$381$	−197.596	−0.0265700
$382$	− 9282.56i	− 1.24329i
$383$	− 4678.68i	− 0.624202i	−0.950049	−	0.312101i	$-0.898967\pi$
$383$	− 4678.68i	− 0.624202i	0.950049	−	0.312101i	$-0.101033\pi$
$384$	862.258	0.114588
$385$	0	0
$386$	−2828.85	−0.373017
$387$	7502.25i	0.985428i
$388$	1229.72i	0.160900i
$389$	50.2546	0.00655015	0.00327508	−	0.999995i	$-0.498958\pi$
$389$	50.2546	0.00655015	0.00327508	+	0.999995i	$0.498958\pi$
$390$	0	0
$391$	−34.2812	−0.00443395
$392$	1151.88i	0.148415i
$393$	490.887i	0.0630076i
$394$	−8660.15	−1.10734
$395$	0	0
$396$	74.6780	0.00947654
$397$	− 11059.9i	− 1.39819i	−0.715029	−	0.699095i	$-0.753587\pi$
$397$	− 11059.9i	− 1.39819i	0.715029	−	0.699095i	$-0.246413\pi$
$398$	59.5780i	0.00750345i
$399$	250.020	0.0313701
$400$	0	0
$401$	−13667.8	−1.70209	−0.851046	−	0.525092i	$-0.824031\pi$
$401$	−13667.8	−1.70209	−0.851046	+	0.525092i	$0.824031\pi$
$402$	− 1867.75i	− 0.231729i
$403$	15086.5i	1.86479i
$404$	−1394.60	−0.171742
$405$	0	0
$406$	2270.41	0.277533
$407$	− 582.221i	− 0.0709082i
$408$	1113.35i	0.135096i
$409$	10990.5	1.32872	0.664359	−	0.747414i	$-0.268705\pi$
$409$	10990.5	1.32872	0.664359	+	0.747414i	$0.268705\pi$
$410$	0	0
$411$	1114.62	0.133772
$412$	272.993i	0.0326442i
$413$	4041.30i	0.481499i
$414$	−36.3656	−0.00431708
$415$	0	0
$416$	−1633.78	−0.192554
$417$	1798.87i	0.211249i
$418$	− 552.305i	− 0.0646271i
$419$	−1115.39	−0.130049	−0.0650246	−	0.997884i	$-0.520713\pi$
$419$	−1115.39	−0.130049	−0.0650246	+	0.997884i	$0.520713\pi$
$420$	0	0
$421$	−2395.26	−0.277287	−0.138643	−	0.990342i	$-0.544274\pi$
$421$	−2395.26	−0.277287	−0.138643	+	0.990342i	$0.544274\pi$
$422$	− 6382.50i	− 0.736245i
$423$	− 6196.70i	− 0.712278i
$424$	9552.03	1.09407
$425$	0	0
$426$	−1958.95	−0.222796
$427$	2254.81i	0.255545i
$428$	− 356.811i	− 0.0402970i
$429$	−145.403	−0.0163639
$430$	0	0
$431$	1684.62	0.188272	0.0941360	−	0.995559i	$-0.469991\pi$
$431$	1684.62	0.188272	0.0941360	+	0.995559i	$0.469991\pi$
$432$	2173.34i	0.242048i
$433$	13355.7i	1.48230i	0.671340	+	0.741150i	$0.265719\pi$
$433$	13355.7i	1.48230i	−0.671340	+	0.741150i	$0.734281\pi$
$434$	−5527.72	−0.611381
$435$	0	0
$436$	−241.428	−0.0265191
$437$	− 25.8531i	− 0.00283003i
$438$	1312.92i	0.143228i
$439$	−13817.4	−1.50220	−0.751101	−	0.660188i	$-0.770477\pi$
$439$	−13817.4	−1.50220	−0.751101	+	0.660188i	$0.770477\pi$
$440$	0	0
$441$	−1298.88	−0.140253
$442$	9412.91i	1.01296i
$443$	− 6305.19i	− 0.676227i	−0.941105	−	0.338114i	$-0.890211\pi$
$443$	− 6305.19i	− 0.676227i	0.941105	−	0.338114i	$-0.109789\pi$
$444$	−71.3621	−0.00762769
$445$	0	0
$446$	3859.63	0.409773
$447$	− 2375.51i	− 0.251360i
$448$	− 3840.76i	− 0.405042i
$449$	−5446.85	−0.572501	−0.286250	−	0.958155i	$-0.592409\pi$
$449$	−5446.85	−0.572501	−0.286250	+	0.958155i	$0.592409\pi$
$450$	0	0
$451$	−229.712	−0.0239839
$452$	854.586i	0.0889300i
$453$	1614.98i	0.167502i
$454$	16153.7	1.66989
$455$	0	0
$456$	−839.633	−0.0862268
$457$	1221.02i	0.124982i	0.998046	+	0.0624910i	$0.0199045\pi$
$457$	1221.02i	0.124982i	−0.998046	+	0.0624910i	$0.980096\pi$
$458$	17230.0i	1.75787i
$459$	−2534.18	−0.257702
$460$	0	0
$461$	9967.46	1.00701	0.503504	−	0.863993i	$-0.332044\pi$
$461$	9967.46	1.00701	0.503504	+	0.863993i	$0.332044\pi$
$462$	− 53.2760i	− 0.00536499i
$463$	6309.54i	0.633324i	0.948538	+	0.316662i	$0.102562\pi$
$463$	6309.54i	0.633324i	−0.948538	+	0.316662i	$0.897438\pi$
$464$	−6950.79	−0.695436
$465$	0	0
$466$	13365.1	1.32860
$467$	7784.79i	0.771386i	0.922627	+	0.385693i	$0.126038\pi$
$467$	7784.79i	0.771386i	−0.922627	+	0.385693i	$0.873962\pi$
$468$	− 959.831i	− 0.0948039i
$469$	−6898.22	−0.679168
$470$	0	0
$471$	412.045	0.0403100
$472$	− 13571.7i	− 1.32349i
$473$	1136.50i	0.110479i
$474$	−811.641	−0.0786496
$475$	0	0
$476$	331.526	0.0319233
$477$	10771.0i	1.03390i
$478$	− 940.718i	− 0.0900156i
$479$	4425.28	0.422122	0.211061	−	0.977473i	$-0.432308\pi$
$479$	4425.28	0.422122	0.211061	+	0.977473i	$0.432308\pi$
$480$	0	0
$481$	−7483.25	−0.709369
$482$	− 12703.4i	− 1.20046i
$483$	− 2.49382i	0 0.000234934i
$484$	−922.466	−0.0866328
$485$	0	0
$486$	−4044.76	−0.377519
$487$	13075.3i	1.21663i	0.793695	+	0.608315i	$0.208154\pi$
$487$	13075.3i	1.21663i	−0.793695	+	0.608315i	$0.791846\pi$
$488$	− 7572.23i	− 0.702416i
$489$	−641.360	−0.0593115
$490$	0	0
$491$	11455.7	1.05293	0.526463	−	0.850198i	$-0.323518\pi$
$491$	11455.7	1.05293	0.526463	+	0.850198i	$0.323518\pi$
$492$	28.1556i	0.00257998i
$493$	− 8104.84i	− 0.740413i
$494$	−7098.73	−0.646533
$495$	0	0
$496$	16923.0	1.53198
$497$	7235.01i	0.652987i
$498$	− 1018.91i	− 0.0916833i
$499$	−5521.10	−0.495307	−0.247654	−	0.968849i	$-0.579660\pi$
$499$	−5521.10	−0.495307	−0.247654	+	0.968849i	$0.579660\pi$
$500$	0	0
$501$	2326.84	0.207496
$502$	6569.23i	0.584062i
$503$	11491.6i	1.01866i	0.860573	+	0.509328i	$0.170106\pi$
$503$	11491.6i	1.01866i	−0.860573	+	0.509328i	$0.829894\pi$
$504$	4361.98	0.385512
$505$	0	0
$506$	−5.50896	−0.000483998 0
$507$	327.524i	0.0286901i
$508$	− 197.596i	− 0.0172577i
$509$	9207.84	0.801828	0.400914	−	0.916116i	$-0.368693\pi$
$509$	9207.84	0.801828	0.400914	+	0.916116i	$0.368693\pi$
$510$	0	0
$511$	4849.04	0.419783
$512$	12720.6i	1.09800i
$513$	− 1911.15i	− 0.164482i
$514$	−4408.29	−0.378291
$515$	0	0
$516$	139.300	0.0118843
$517$	− 938.727i	− 0.0798552i
$518$	− 2741.88i	− 0.232570i
$519$	−1081.92	−0.0915051
$520$	0	0
$521$	−11598.0	−0.975271	−0.487636	−	0.873047i	$-0.662141\pi$
$521$	−11598.0	−0.975271	−0.487636	+	0.873047i	$0.662141\pi$
$522$	− 8597.64i	− 0.720898i
$523$	− 4596.93i	− 0.384340i	−0.981362	−	0.192170i	$-0.938448\pi$
$523$	− 4596.93i	− 0.384340i	0.981362	−	0.192170i	$-0.0615525\pi$
$524$	−490.887	−0.0409246
$525$	0	0
$526$	9627.49	0.798058
$527$	19732.7i	1.63107i
$528$	163.103i	0.0134435i
$529$	12166.7	0.999979
$530$	0	0
$531$	15303.7	1.25070
$532$	250.020i	0.0203755i
$533$	2952.48i	0.239936i
$534$	−1520.18	−0.123192
$535$	0	0
$536$	23166.0	1.86683
$537$	− 386.389i	− 0.0310501i
$538$	2138.64i	0.171382i
$539$	−196.765	−0.0157241
$540$	0	0
$541$	−8427.65	−0.669747	−0.334874	−	0.942263i	$-0.608694\pi$
$541$	−8427.65	−0.669747	−0.334874	+	0.942263i	$0.608694\pi$
$542$	− 21134.8i	− 1.67494i
$543$	2499.07i	0.197505i
$544$	−2136.94	−0.168420
$545$	0	0
$546$	−684.753	−0.0536717
$547$	12864.6i	1.00557i	0.864410	+	0.502787i	$0.167692\pi$
$547$	12864.6i	1.00557i	−0.864410	+	0.502787i	$0.832308\pi$
$548$	1114.62i	0.0868875i
$549$	8538.58	0.663785
$550$	0	0
$551$	6112.26	0.472579
$552$	8.37491i	0 0.000645760i
$553$	2997.65i	0.230512i
$554$	−4198.67	−0.321993
$555$	0	0
$556$	−1798.87	−0.137211
$557$	− 19219.4i	− 1.46203i	−0.682360	−	0.731016i	$-0.739046\pi$
$557$	− 19219.4i	− 1.46203i	0.682360	−	0.731016i	$-0.260954\pi$
$558$	20932.6i	1.58807i
$559$	14607.4	1.10524
$560$	0	0
$561$	−190.184	−0.0143129
$562$	13625.9i	1.02273i
$563$	− 17253.3i	− 1.29155i	−0.763529	−	0.645774i	$-0.776535\pi$
$563$	− 17253.3i	− 1.29155i	0.763529	−	0.645774i	$-0.223465\pi$
$564$	−115.058	−0.00859013
$565$	0	0
$566$	15931.5	1.18313
$567$	4825.62i	0.357420i
$568$	− 24297.0i	− 1.79486i
$569$	6242.80	0.459950	0.229975	−	0.973197i	$-0.426136\pi$
$569$	6242.80	0.459950	0.229975	+	0.973197i	$0.426136\pi$
$570$	0	0
$571$	5904.17	0.432718	0.216359	−	0.976314i	$-0.430582\pi$
$571$	5904.17	0.432718	0.216359	+	0.976314i	$0.430582\pi$
$572$	− 145.403i	− 0.0106287i
$573$	− 2410.57i	− 0.175747i
$574$	−1081.79	−0.0786642
$575$	0	0
$576$	−14544.3	−1.05210
$577$	11390.2i	0.821806i	0.911679	+	0.410903i	$0.134786\pi$
$577$	11390.2i	0.821806i	−0.911679	+	0.410903i	$0.865214\pi$
$578$	− 960.934i	− 0.0691515i
$579$	−734.617	−0.0527282
$580$	0	0
$581$	−3763.15	−0.268712
$582$	− 3322.15i	− 0.236611i
$583$	1631.68i	0.115913i
$584$	−16284.3	−1.15385
$585$	0	0
$586$	−23212.5	−1.63635
$587$	− 20459.5i	− 1.43859i	−0.694704	−	0.719295i	$-0.744465\pi$
$587$	− 20459.5i	− 1.43859i	0.694704	−	0.719295i	$-0.255535\pi$
$588$	24.1173i	0.00169146i
$589$	−14881.4	−1.04105
$590$	0	0
$591$	−2248.93	−0.156529
$592$	8394.19i	0.582768i
$593$	− 20051.1i	− 1.38853i	−0.719719	−	0.694266i	$-0.755729\pi$
$593$	− 20051.1i	− 1.38853i	0.719719	−	0.694266i	$-0.244271\pi$
$594$	−407.241	−0.0281301
$595$	0	0
$596$	2375.51	0.163263
$597$	15.4717i	0.00106066i
$598$	70.8062i	0.00484194i
$599$	−10578.4	−0.721572	−0.360786	−	0.932649i	$-0.617492\pi$
$599$	−10578.4	−0.721572	−0.360786	+	0.932649i	$0.617492\pi$
$600$	0	0
$601$	−4619.46	−0.313530	−0.156765	−	0.987636i	$-0.550107\pi$
$601$	−4619.46	−0.313530	−0.156765	+	0.987636i	$0.550107\pi$
$602$	5352.18i	0.362356i
$603$	26122.4i	1.76415i
$604$	−1614.98	−0.108796
$605$	0	0
$606$	3767.60	0.252555
$607$	5724.50i	0.382785i	0.981514	+	0.191392i	$0.0613003\pi$
$607$	5724.50i	0.382785i	−0.981514	+	0.191392i	$0.938700\pi$
$608$	− 1611.57i	− 0.107496i
$609$	589.596	0.0392309
$610$	0	0
$611$	−12065.4	−0.798876
$612$	− 1255.43i	− 0.0829214i
$613$	19286.4i	1.27075i	0.772204	+	0.635375i	$0.219154\pi$
$613$	19286.4i	1.27075i	−0.772204	+	0.635375i	$0.780846\pi$
$614$	−9947.70	−0.653838
$615$	0	0
$616$	660.789	0.0432207
$617$	6864.84i	0.447922i	0.974598	+	0.223961i	$0.0718989\pi$
$617$	6864.84i	0.447922i	−0.974598	+	0.223961i	$0.928101\pi$
$618$	− 737.508i	− 0.0480047i
$619$	17559.4	1.14018	0.570090	−	0.821582i	$-0.306908\pi$
$619$	17559.4	1.14018	0.570090	+	0.821582i	$0.306908\pi$
$620$	0	0
$621$	−19.0627	−0.00123182
$622$	− 18549.4i	− 1.19576i
$623$	5614.51i	0.361061i
$624$	2096.35	0.134489
$625$	0	0
$626$	−7992.84	−0.510317
$627$	− 143.427i	− 0.00913542i
$628$	412.045i	0.0261821i
$629$	−9787.89	−0.620459
$630$	0	0
$631$	−24780.6	−1.56339	−0.781694	−	0.623662i	$-0.785644\pi$
$631$	−24780.6	−1.56339	−0.781694	+	0.623662i	$0.785644\pi$
$632$	− 10066.9i	− 0.633607i
$633$	− 1657.46i	− 0.104073i
$634$	4282.55	0.268268
$635$	0	0
$636$	199.993	0.0124690
$637$	2529.01i	0.157305i
$638$	− 1302.44i	− 0.0808215i
$639$	27397.8	1.69615
$640$	0	0
$641$	−21724.3	−1.33863	−0.669313	−	0.742980i	$-0.733412\pi$
$641$	−21724.3	−1.33863	−0.669313	+	0.742980i	$0.733412\pi$
$642$	963.947i	0.0592585i
$643$	18736.1i	1.14911i	0.818466	+	0.574555i	$0.194825\pi$
$643$	18736.1i	1.14911i	−0.818466	+	0.574555i	$0.805175\pi$
$644$	2.49382	0.000152594 0
$645$	0	0
$646$	−9284.96	−0.565498
$647$	− 25687.8i	− 1.56088i	−0.625228	−	0.780442i	$-0.714994\pi$
$647$	− 25687.8i	− 1.56088i	0.625228	−	0.780442i	$-0.285006\pi$
$648$	− 16205.7i	− 0.982438i
$649$	2318.33	0.140219
$650$	0	0
$651$	−1435.48	−0.0864224
$652$	− 641.360i	− 0.0385239i
$653$	− 15450.7i	− 0.925929i	−0.886377	−	0.462964i	$-0.846786\pi$
$653$	− 15450.7i	− 0.925929i	0.886377	−	0.462964i	$-0.153214\pi$
$654$	652.234	0.0389975
$655$	0	0
$656$	3311.88	0.197115
$657$	− 18362.5i	− 1.09039i
$658$	− 4420.78i	− 0.261915i
$659$	1402.85	0.0829246	0.0414623	−	0.999140i	$-0.486798\pi$
$659$	1402.85	0.0829246	0.0414623	+	0.999140i	$0.486798\pi$
$660$	0	0
$661$	6896.75	0.405828	0.202914	−	0.979197i	$-0.434959\pi$
$661$	6896.75	0.405828	0.202914	+	0.979197i	$0.434959\pi$
$662$	− 16333.2i	− 0.958926i
$663$	2444.42i	0.143187i
$664$	12637.6	0.738606
$665$	0	0
$666$	−10383.0	−0.604105
$667$	− 60.9666i	− 0.00353919i
$668$	2326.84i	0.134773i
$669$	1002.30	0.0579239
$670$	0	0
$671$	1293.49	0.0744185
$672$	− 155.454i	− 0.00892377i
$673$	3869.29i	0.221620i	0.993842	+	0.110810i	$0.0353445\pi$
$673$	3869.29i	0.221620i	−0.993842	+	0.110810i	$0.964656\pi$
$674$	−24869.6	−1.42128
$675$	0	0
$676$	−327.524	−0.0186347
$677$	711.604i	0.0403976i	0.999796	+	0.0201988i	$0.00642991\pi$
$677$	711.604i	0.0403976i	−0.999796	+	0.0201988i	$0.993570\pi$
$678$	− 2308.72i	− 0.130775i
$679$	−12269.8	−0.693477
$680$	0	0
$681$	4194.91	0.236049
$682$	3171.04i	0.178043i
$683$	13112.0i	0.734577i	0.930107	+	0.367288i	$0.119714\pi$
$683$	13112.0i	0.734577i	−0.930107	+	0.367288i	$0.880286\pi$
$684$	946.784	0.0529257
$685$	0	0
$686$	−926.636	−0.0515731
$687$	4474.40i	0.248485i
$688$	− 16385.5i	− 0.907984i
$689$	20971.9	1.15960
$690$	0	0
$691$	−12573.5	−0.692211	−0.346105	−	0.938196i	$-0.612496\pi$
$691$	−12573.5	−0.692211	−0.346105	+	0.938196i	$0.612496\pi$
$692$	− 1081.92i	− 0.0594343i
$693$	745.117i	0.0408437i
$694$	−8353.98	−0.456935
$695$	0	0
$696$	−1980.02	−0.107834
$697$	3861.76i	0.209863i
$698$	− 14133.5i	− 0.766421i
$699$	3470.75	0.187805
$700$	0	0
$701$	−7489.24	−0.403516	−0.201758	−	0.979435i	$-0.564665\pi$
$701$	−7489.24	−0.403516	−0.201758	+	0.979435i	$0.564665\pi$
$702$	5234.23i	0.281415i
$703$	− 7381.53i	− 0.396016i
$704$	−2203.29	−0.117954
$705$	0	0
$706$	−24351.1	−1.29811
$707$	− 13914.9i	− 0.740206i
$708$	− 284.155i	− 0.0150836i
$709$	−20869.6	−1.10547	−0.552733	−	0.833358i	$-0.686415\pi$
$709$	−20869.6	−1.10547	−0.552733	+	0.833358i	$0.686415\pi$
$710$	0	0
$711$	11351.6	0.598760
$712$	− 18855.0i	− 0.992445i
$713$	148.435i	0.00779652i
$714$	−895.639	−0.0469446
$715$	0	0
$716$	386.389	0.0201676
$717$	− 244.293i	− 0.0127242i
$718$	− 12911.8i	− 0.671119i
$719$	5889.90	0.305503	0.152751	−	0.988265i	$-0.451187\pi$
$719$	5889.90	0.305503	0.152751	+	0.988265i	$0.451187\pi$
$720$	0	0
$721$	−2723.85	−0.140696
$722$	11527.8i	0.594210i
$723$	− 3298.91i	− 0.169692i
$724$	−2499.07	−0.128284
$725$	0	0
$726$	2492.10	0.127397
$727$	24760.2i	1.26314i	0.775318	+	0.631571i	$0.217589\pi$
$727$	24760.2i	1.26314i	−0.775318	+	0.631571i	$0.782411\pi$
$728$	− 8493.08i	− 0.432382i
$729$	17562.7	0.892280
$730$	0	0
$731$	19106.1	0.966708
$732$	− 158.542i	− 0.00800530i
$733$	4537.53i	0.228646i	0.993444	+	0.114323i	$0.0364699\pi$
$733$	4537.53i	0.228646i	−0.993444	+	0.114323i	$0.963530\pi$
$734$	474.850	0.0238788
$735$	0	0
$736$	−16.0746	−0.000805051 0
$737$	3957.23i	0.197784i
$738$	4096.57i	0.204332i
$739$	24010.0	1.19516	0.597579	−	0.801810i	$-0.296130\pi$
$739$	24010.0	1.19516	0.597579	+	0.801810i	$0.296130\pi$
$740$	0	0
$741$	−1843.45	−0.0913913
$742$	7684.16i	0.380181i
$743$	− 25175.4i	− 1.24306i	−0.783389	−	0.621532i	$-0.786511\pi$
$743$	− 25175.4i	− 1.24306i	0.783389	−	0.621532i	$-0.213489\pi$
$744$	4820.72	0.237549
$745$	0	0
$746$	−21566.5	−1.05845
$747$	14250.4i	0.697985i
$748$	− 190.184i	− 0.00929652i
$749$	3560.17	0.173679
$750$	0	0
$751$	24920.0	1.21085	0.605423	−	0.795904i	$-0.293004\pi$
$751$	24920.0	1.21085	0.605423	+	0.795904i	$0.293004\pi$
$752$	13534.1i	0.656301i
$753$	1705.95i	0.0825607i
$754$	−16740.2	−0.808543
$755$	0	0
$756$	184.352	0.00886880
$757$	− 28274.4i	− 1.35753i	−0.734356	−	0.678765i	$-0.762516\pi$
$757$	− 28274.4i	− 1.35753i	0.734356	−	0.678765i	$-0.237484\pi$
$758$	34210.3i	1.63928i
$759$	−1.43061	−6.84160e−5 0
$760$	0	0
$761$	12377.9	0.589616	0.294808	−	0.955557i	$-0.404744\pi$
$761$	12377.9	0.589616	0.294808	+	0.955557i	$0.404744\pi$
$762$	533.818i	0.0253782i
$763$	− 2408.91i	− 0.114297i
$764$	2410.57	0.114151
$765$	0	0
$766$	−12639.7	−0.596204
$767$	− 29797.3i	− 1.40276i
$768$	750.019i	0.0352396i
$769$	31845.7	1.49335	0.746674	−	0.665190i	$-0.231650\pi$
$769$	31845.7	1.49335	0.746674	+	0.665190i	$0.231650\pi$
$770$	0	0
$771$	−1144.78	−0.0534736
$772$	− 734.617i	− 0.0342480i
$773$	− 7342.21i	− 0.341631i	−0.985303	−	0.170816i	$-0.945360\pi$
$773$	− 7342.21i	− 0.341631i	0.985303	−	0.170816i	$-0.0546403\pi$
$774$	20267.8	0.941228
$775$	0	0
$776$	41205.1	1.90615
$777$	− 712.032i	− 0.0328752i
$778$	− 135.766i	− 0.00625636i
$779$	−2912.35	−0.133948
$780$	0	0
$781$	4150.44	0.190159
$782$	92.6127i	0.00423507i
$783$	− 4506.86i	− 0.205699i
$784$	2836.87	0.129231
$785$	0	0
$786$	1326.16	0.0601815
$787$	36457.4i	1.65129i	0.564188	+	0.825646i	$0.309189\pi$
$787$	36457.4i	1.65129i	−0.564188	+	0.825646i	$0.690811\pi$
$788$	− 2248.93i	− 0.101669i
$789$	2500.14	0.112810
$790$	0	0
$791$	−8526.83	−0.383286
$792$	− 2502.30i	− 0.112267i
$793$	− 16625.2i	− 0.744487i
$794$	−29879.0	−1.33548
$795$	0	0
$796$	−15.4717	−0.000688918 0
$797$	− 9358.08i	− 0.415910i	−0.978138	−	0.207955i	$-0.933319\pi$
$797$	− 9358.08i	− 0.415910i	0.978138	−	0.207955i	$-0.0666808\pi$
$798$	− 675.445i	− 0.0299630i
$799$	−15781.2	−0.698747
$800$	0	0
$801$	21261.2	0.937863
$802$	36924.5i	1.62575i
$803$	− 2781.70i	− 0.122247i
$804$	485.033	0.0212758
$805$	0	0
$806$	40757.1	1.78115
$807$	555.378i	0.0242258i
$808$	46730.0i	2.03460i
$809$	−28189.6	−1.22509	−0.612543	−	0.790437i	$-0.709853\pi$
$809$	−28189.6	−1.22509	−0.612543	+	0.790437i	$0.709853\pi$
$810$	0	0
$811$	−22909.7	−0.991946	−0.495973	−	0.868338i	$-0.665189\pi$
$811$	−22909.7	−0.991946	−0.495973	+	0.868338i	$0.665189\pi$
$812$	589.596i	0.0254812i
$813$	− 5488.44i	− 0.236763i
$814$	−1572.91	−0.0677277
$815$	0	0
$816$	2741.98	0.117633
$817$	14408.8i	0.617015i
$818$	− 29691.5i	− 1.26912i
$819$	9576.94	0.408602
$820$	0	0
$821$	20512.6	0.871980	0.435990	−	0.899952i	$-0.356398\pi$
$821$	20512.6	0.871980	0.435990	+	0.899952i	$0.356398\pi$
$822$	− 3011.22i	− 0.127772i
$823$	− 7882.78i	− 0.333872i	−0.985968	−	0.166936i	$-0.946613\pi$
$823$	− 7882.78i	− 0.333872i	0.985968	−	0.166936i	$-0.0533873\pi$
$824$	9147.40	0.386729
$825$	0	0
$826$	10917.8	0.459902
$827$	− 19276.5i	− 0.810531i	−0.914199	−	0.405265i	$-0.867179\pi$
$827$	− 19276.5i	− 0.810531i	0.914199	−	0.405265i	$-0.132821\pi$
$828$	− 9.44369i	0 0.000396366i
$829$	−13771.7	−0.576972	−0.288486	−	0.957484i	$-0.593152\pi$
$829$	−13771.7	−0.576972	−0.288486	+	0.957484i	$0.593152\pi$
$830$	0	0
$831$	−1090.34	−0.0455157
$832$	28318.7i	1.18002i
$833$	3307.88i	0.137589i
$834$	4859.76	0.201774
$835$	0	0
$836$	143.427	0.00593363
$837$	10972.8i	0.453136i
$838$	3013.31i	0.124216i
$839$	−23175.4	−0.953640	−0.476820	−	0.879001i	$-0.658211\pi$
$839$	−23175.4	−0.953640	−0.476820	+	0.879001i	$0.658211\pi$
$840$	0	0
$841$	−9975.12	−0.409001
$842$	6470.94i	0.264849i
$843$	3538.48i	0.144569i
$844$	1657.46	0.0675972
$845$	0	0
$846$	−16740.8	−0.680330
$847$	− 9204.12i	− 0.373385i
$848$	− 23524.9i	− 0.952650i
$849$	4137.22	0.167243
$850$	0	0
$851$	−73.6270	−0.00296581
$852$	− 508.714i	− 0.0204557i
$853$	31431.7i	1.26166i	0.775920	+	0.630832i	$0.217286\pi$
$853$	31431.7i	1.26166i	−0.775920	+	0.630832i	$0.782714\pi$
$854$	6091.51	0.244083
$855$	0	0
$856$	−11956.0	−0.477391
$857$	3992.52i	0.159139i	0.996829	+	0.0795693i	$0.0253545\pi$
$857$	3992.52i	0.159139i	−0.996829	+	0.0795693i	$0.974646\pi$
$858$	392.816i	0.0156300i
$859$	−12909.8	−0.512778	−0.256389	−	0.966574i	$-0.582533\pi$
$859$	−12909.8	−0.512778	−0.256389	+	0.966574i	$0.582533\pi$
$860$	0	0
$861$	−280.929	−0.0111197
$862$	− 4551.10i	− 0.179827i
$863$	40128.5i	1.58284i	0.611274	+	0.791419i	$0.290657\pi$
$863$	40128.5i	1.58284i	−0.611274	+	0.791419i	$0.709343\pi$
$864$	−1188.29	−0.0467898
$865$	0	0
$866$	36081.3	1.41581
$867$	− 249.542i	− 0.00977498i
$868$	− 1435.48i	− 0.0561330i
$869$	1719.63	0.0671284
$870$	0	0
$871$	50862.0	1.97864
$872$	8089.73i	0.314166i
$873$	46463.6i	1.80132i
$874$	−69.8438	−0.00270309
$875$	0	0
$876$	−340.950	−0.0131502
$877$	3222.42i	0.124074i	0.998074	+	0.0620372i	$0.0197598\pi$
$877$	3222.42i	0.124074i	−0.998074	+	0.0620372i	$0.980240\pi$
$878$	37328.4i	1.43482i
$879$	−6028.00	−0.231307
$880$	0	0
$881$	−18712.4	−0.715591	−0.357795	−	0.933800i	$-0.616471\pi$
$881$	−18712.4	−0.715591	−0.357795	+	0.933800i	$0.616471\pi$
$882$	3509.01i	0.133962i
$883$	17924.0i	0.683114i	0.939861	+	0.341557i	$0.110954\pi$
$883$	17924.0i	0.683114i	−0.939861	+	0.341557i	$0.889046\pi$
$884$	−2444.42	−0.0930029
$885$	0	0
$886$	−17033.9	−0.645896
$887$	9873.21i	0.373743i	0.982384	+	0.186871i	$0.0598348\pi$
$887$	9873.21i	0.373743i	−0.982384	+	0.186871i	$0.940165\pi$
$888$	2391.19i	0.0903637i
$889$	1971.56	0.0743803
$890$	0	0
$891$	2768.27	0.104086
$892$	1002.30i	0.0376226i
$893$	− 11901.4i	− 0.445985i
$894$	−6417.59	−0.240086
$895$	0	0
$896$	−8603.38	−0.320780
$897$	18.3875i	0 0.000684438i
$898$	14715.0i	0.546822i
$899$	−35093.3	−1.30192
$900$	0	0
$901$	27430.7	1.01426
$902$	620.582i	0.0229081i
$903$	1389.89i	0.0512212i
$904$	28635.3	1.05354
$905$	0	0
$906$	4362.97	0.159989
$907$	− 30129.2i	− 1.10300i	−0.834174	−	0.551502i	$-0.814055\pi$
$907$	− 30129.2i	− 1.10300i	0.834174	−	0.551502i	$-0.185945\pi$
$908$	4194.91i	0.153318i
$909$	−52693.5	−1.92270
$910$	0	0
$911$	−37831.8	−1.37588	−0.687938	−	0.725769i	$-0.741484\pi$
$911$	−37831.8	−1.37588	−0.687938	+	0.725769i	$0.741484\pi$
$912$	2067.86i	0.0750807i
$913$	2158.77i	0.0782527i
$914$	3298.65	0.119376
$915$	0	0
$916$	−4474.40	−0.161396
$917$	− 4897.94i	− 0.176384i
$918$	6846.24i	0.246143i
$919$	−4771.81	−0.171281	−0.0856407	−	0.996326i	$-0.527294\pi$
$919$	−4771.81	−0.171281	−0.0856407	+	0.996326i	$0.527294\pi$
$920$	0	0
$921$	−2583.30	−0.0924240
$922$	− 26927.7i	− 0.961841i
$923$	− 53345.3i	− 1.90236i
$924$	13.8351	0.000492578 0
$925$	0	0
$926$	17045.6	0.604917
$927$	10314.8i	0.365460i
$928$	− 3800.39i	− 0.134433i
$929$	45138.3	1.59412	0.797061	−	0.603899i	$-0.206387\pi$
$929$	45138.3	1.59412	0.797061	+	0.603899i	$0.206387\pi$
$930$	0	0
$931$	−2494.64	−0.0878178
$932$	3470.75i	0.121983i
$933$	− 4817.05i	− 0.169028i
$934$	21031.1	0.736786
$935$	0	0
$936$	−32161.8	−1.12312
$937$	22634.7i	0.789161i	0.918861	+	0.394580i	$0.129110\pi$
$937$	22634.7i	0.789161i	−0.918861	+	0.394580i	$0.870890\pi$
$938$	18636.0i	0.648705i
$939$	−2075.64	−0.0721363
$940$	0	0
$941$	11695.3	0.405160	0.202580	−	0.979266i	$-0.435067\pi$
$941$	11695.3	0.405160	0.202580	+	0.979266i	$0.435067\pi$
$942$	− 1113.16i	− 0.0385020i
$943$	29.0492i	0.00100315i
$944$	−33424.6	−1.15241
$945$	0	0
$946$	3070.33	0.105523
$947$	− 45429.1i	− 1.55887i	−0.626485	−	0.779434i	$-0.715507\pi$
$947$	− 45429.1i	− 1.55887i	0.626485	−	0.779434i	$-0.284493\pi$
$948$	− 210.773i	− 0.00722109i
$949$	−35753.0	−1.22296
$950$	0	0
$951$	1112.13	0.0379213
$952$	− 11108.7i	− 0.378189i
$953$	39025.9i	1.32652i	0.748390	+	0.663259i	$0.230827\pi$
$953$	39025.9i	1.32652i	−0.748390	+	0.663259i	$0.769173\pi$
$954$	29098.6	0.987529
$955$	0	0
$956$	244.293	0.00826464
$957$	− 338.228i	− 0.0114246i
$958$	− 11955.2i	− 0.403188i
$959$	−11121.4	−0.374483
$960$	0	0
$961$	55650.1	1.86802
$962$	20216.5i	0.677552i
$963$	− 13481.7i	− 0.451135i
$964$	3298.91	0.110218
$965$	0	0
$966$	−6.73722	−0.000224396 0
$967$	− 12986.1i	− 0.431857i	−0.976409	−	0.215929i	$-0.930722\pi$
$967$	− 12986.1i	− 0.431857i	0.976409	−	0.215929i	$-0.0692779\pi$
$968$	30909.8i	1.02632i
$969$	−2411.19	−0.0799366
$970$	0	0
$971$	−15044.2	−0.497210	−0.248605	−	0.968605i	$-0.579972\pi$
$971$	−15044.2	−0.497210	−0.248605	+	0.968605i	$0.579972\pi$
$972$	− 1050.37i	− 0.0346613i
$973$	− 17948.6i	− 0.591374i
$974$	35323.8	1.16206
$975$	0	0
$976$	−18649.0	−0.611618
$977$	26593.4i	0.870828i	0.900230	+	0.435414i	$0.143398\pi$
$977$	26593.4i	0.870828i	−0.900230	+	0.435414i	$0.856602\pi$
$978$	1732.68i	0.0566512i
$979$	3220.82	0.105146
$980$	0	0
$981$	−9122.12	−0.296888
$982$	− 30948.2i	− 1.00570i
$983$	− 36050.3i	− 1.16971i	−0.811137	−	0.584856i	$-0.801151\pi$
$983$	− 36050.3i	− 1.16971i	0.811137	−	0.584856i	$-0.198849\pi$
$984$	943.431	0.0305645
$985$	0	0
$986$	−21895.7	−0.707203
$987$	− 1148.02i	− 0.0370233i
$988$	− 1843.45i	− 0.0593604i
$989$	143.721	0.00462088
$990$	0	0
$991$	−54789.8	−1.75626	−0.878131	−	0.478420i	$-0.841210\pi$
$991$	−54789.8	−1.75626	−0.878131	+	0.478420i	$0.841210\pi$
$992$	9252.77i	0.296145i
$993$	− 4241.53i	− 0.135550i
$994$	19545.8	0.623699
$995$	0	0
$996$	264.597	0.00841775
$997$	23309.0i	0.740426i	0.928947	+	0.370213i	$0.120715\pi$
$997$	23309.0i	0.740426i	−0.928947	+	0.370213i	$0.879285\pi$
$998$	14915.6i	0.473091i
$999$	−5442.75	−0.172373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	175.4.b.d.99.2		4
5.2	odd	4		175.4.a.d.1.2	✓	2
5.3	odd	4		175.4.a.e.1.1	yes	2
5.4	even	2	inner	175.4.b.d.99.3		4
15.2	even	4		1575.4.a.v.1.1		2
15.8	even	4		1575.4.a.s.1.2		2
35.13	even	4		1225.4.a.t.1.1		2
35.27	even	4		1225.4.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.4.a.d.1.2	✓	2	5.2	odd	4
175.4.a.e.1.1	yes	2	5.3	odd	4
175.4.b.d.99.2		4	1.1	even	1	trivial
175.4.b.d.99.3		4	5.4	even	2	inner
1225.4.a.r.1.2		2	35.27	even	4
1225.4.a.t.1.1		2	35.13	even	4
1575.4.a.s.1.2		2	15.8	even	4
1575.4.a.v.1.1		2	15.2	even	4

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:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.70156i q^{2} -0.701562i q^{3} +0.701562 q^{4} -1.89531 q^{6} +7.00000i q^{7} -23.5078i q^{8} +26.5078 q^{9} +4.01562 q^{11} -0.492189i q^{12} -51.6125i q^{13} +18.9109 q^{14} -57.8953 q^{16} -67.5078i q^{17} -71.6125i q^{18} +50.9109 q^{19} +4.91093 q^{21} -10.8485i q^{22} -0.507811i q^{23} -16.4922 q^{24} -139.434 q^{26} -37.5391i q^{27} +4.91093i q^{28} +120.058 q^{29} -292.303 q^{31} -31.6547i q^{32} -2.81721i q^{33} -182.377 q^{34} +18.5969 q^{36} -144.989i q^{37} -137.539i q^{38} -36.2094 q^{39} -57.2047 q^{41} -13.2672i q^{42} +283.020i q^{43} +2.81721 q^{44} -1.37188 q^{46} -233.769i q^{47} +40.6172i q^{48} -49.0000 q^{49} -47.3609 q^{51} -36.2094i q^{52} +406.334i q^{53} -101.414 q^{54} +164.555 q^{56} -35.7172i q^{57} -324.344i q^{58} +577.328 q^{59} +322.116 q^{61} +789.675i q^{62} +185.555i q^{63} -548.680 q^{64} -7.61086 q^{66} +985.459i q^{67} -47.3609i q^{68} -0.356261 q^{69} +1033.57 q^{71} -623.141i q^{72} -692.720i q^{73} -391.697 q^{74} +35.7172 q^{76} +28.1093i q^{77} +97.8219i q^{78} +428.236 q^{79} +689.375 q^{81} +154.542i q^{82} +537.592i q^{83} +3.44533 q^{84} +764.597 q^{86} -84.2280i q^{87} -94.3985i q^{88} +802.073 q^{89} +361.287 q^{91} -0.356261i q^{92} +205.069i q^{93} -631.541 q^{94} -22.2077 q^{96} +1752.82i q^{97} +132.377i q^{98} +106.445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} - 46 q^{6} + 42 q^{9} - 112 q^{11} - 14 q^{14} - 270 q^{16} + 114 q^{19} - 70 q^{21} - 130 q^{24} - 276 q^{26} + 826 q^{29} - 324 q^{31} - 102 q^{34} + 100 q^{36} - 68 q^{39} - 1010 q^{41}+ \cdots + 874 q^{99}+O(q^{100}) \) 4 * q - 10 * q^4 - 46 * q^6 + 42 * q^9 - 112 * q^11 - 14 * q^14 - 270 * q^16 + 114 * q^19 - 70 * q^21 - 130 * q^24 - 276 * q^26 + 826 * q^29 - 324 * q^31 - 102 * q^34 + 100 * q^36 - 68 * q^39 - 1010 * q^41 + 690 * q^44 - 236 * q^46 - 196 * q^49 + 310 * q^51 - 1110 * q^54 + 210 * q^56 - 380 * q^59 + 1980 * q^61 - 722 * q^64 + 2518 * q^66 - 360 * q^69 + 254 * q^71 - 2412 * q^74 + 2 * q^76 + 2238 * q^79 - 316 * q^81 + 462 * q^84 + 3084 * q^86 + 3426 * q^89 + 728 * q^91 - 4652 * q^94 + 2434 * q^96 + 874 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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