LMFDB - Embedded newform 3750.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3750.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.a (trivial)

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:	yes
Analytic conductor:	$29.9439007580$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3750.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.61803 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.61803 q^{11} -1.00000 q^{12} -6.47214 q^{13} +2.61803 q^{14} +1.00000 q^{16} +1.23607 q^{17} +1.00000 q^{18} -5.70820 q^{19} -2.61803 q^{21} +3.61803 q^{22} +4.47214 q^{23} -1.00000 q^{24} -6.47214 q^{26} -1.00000 q^{27} +2.61803 q^{28} +8.47214 q^{29} +6.61803 q^{31} +1.00000 q^{32} -3.61803 q^{33} +1.23607 q^{34} +1.00000 q^{36} +8.00000 q^{37} -5.70820 q^{38} +6.47214 q^{39} +5.70820 q^{41} -2.61803 q^{42} -7.70820 q^{43} +3.61803 q^{44} +4.47214 q^{46} +1.70820 q^{47} -1.00000 q^{48} -0.145898 q^{49} -1.23607 q^{51} -6.47214 q^{52} -2.09017 q^{53} -1.00000 q^{54} +2.61803 q^{56} +5.70820 q^{57} +8.47214 q^{58} -3.61803 q^{59} +2.76393 q^{61} +6.61803 q^{62} +2.61803 q^{63} +1.00000 q^{64} -3.61803 q^{66} -1.52786 q^{67} +1.23607 q^{68} -4.47214 q^{69} +5.52786 q^{71} +1.00000 q^{72} +3.52786 q^{73} +8.00000 q^{74} -5.70820 q^{76} +9.47214 q^{77} +6.47214 q^{78} -5.61803 q^{79} +1.00000 q^{81} +5.70820 q^{82} +2.14590 q^{83} -2.61803 q^{84} -7.70820 q^{86} -8.47214 q^{87} +3.61803 q^{88} +3.52786 q^{89} -16.9443 q^{91} +4.47214 q^{92} -6.61803 q^{93} +1.70820 q^{94} -1.00000 q^{96} -3.38197 q^{97} -0.145898 q^{98} +3.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 4 q^{13} + 3 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 2 q^{19} - 3 q^{21} + 5 q^{22} - 2 q^{24} - 4 q^{26}+ \cdots + 5 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 3 * q^7 + 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 - 4 * q^13 + 3 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^18 + 2 * q^19 - 3 * q^21 + 5 * q^22 - 2 * q^24 - 4 * q^26 - 2 * q^27 + 3 * q^28 + 8 * q^29 + 11 * q^31 + 2 * q^32 - 5 * q^33 - 2 * q^34 + 2 * q^36 + 16 * q^37 + 2 * q^38 + 4 * q^39 - 2 * q^41 - 3 * q^42 - 2 * q^43 + 5 * q^44 - 10 * q^47 - 2 * q^48 - 7 * q^49 + 2 * q^51 - 4 * q^52 + 7 * q^53 - 2 * q^54 + 3 * q^56 - 2 * q^57 + 8 * q^58 - 5 * q^59 + 10 * q^61 + 11 * q^62 + 3 * q^63 + 2 * q^64 - 5 * q^66 - 12 * q^67 - 2 * q^68 + 20 * q^71 + 2 * q^72 + 16 * q^73 + 16 * q^74 + 2 * q^76 + 10 * q^77 + 4 * q^78 - 9 * q^79 + 2 * q^81 - 2 * q^82 + 11 * q^83 - 3 * q^84 - 2 * q^86 - 8 * q^87 + 5 * q^88 + 16 * q^89 - 16 * q^91 - 11 * q^93 - 10 * q^94 - 2 * q^96 - 9 * q^97 - 7 * q^98 + 5 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	2.61803	0.989524	0.494762	−	0.869029i	$-0.335255\pi$
$7$	2.61803	0.989524	0.494762	+	0.869029i	$0.335255\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.61803	1.09088	0.545439	−	0.838150i	$-0.316363\pi$
$11$	3.61803	1.09088	0.545439	+	0.838150i	$0.316363\pi$
$12$	−1.00000	−0.288675
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	2.61803	0.699699
$15$	0	0
$16$	1.00000	0.250000
$17$	1.23607	0.299791	0.149895	−	0.988702i	$-0.452106\pi$
$17$	1.23607	0.299791	0.149895	+	0.988702i	$0.452106\pi$
$18$	1.00000	0.235702
$19$	−5.70820	−1.30955	−0.654776	−	0.755823i	$-0.727237\pi$
$19$	−5.70820	−1.30955	−0.654776	+	0.755823i	$0.727237\pi$
$20$	0	0
$21$	−2.61803	−0.571302
$22$	3.61803	0.771367
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.47214	−1.26929
$27$	−1.00000	−0.192450
$28$	2.61803	0.494762
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	6.61803	1.18863	0.594317	−	0.804231i	$-0.297422\pi$
$31$	6.61803	1.18863	0.594317	+	0.804231i	$0.297422\pi$
$32$	1.00000	0.176777
$33$	−3.61803	−0.629819
$34$	1.23607	0.211984
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−5.70820	−0.925993
$39$	6.47214	1.03637
$40$	0	0
$41$	5.70820	0.891472	0.445736	−	0.895165i	$-0.352942\pi$
$41$	5.70820	0.891472	0.445736	+	0.895165i	$0.352942\pi$
$42$	−2.61803	−0.403971
$43$	−7.70820	−1.17549	−0.587745	−	0.809046i	$-0.699984\pi$
$43$	−7.70820	−1.17549	−0.587745	+	0.809046i	$0.699984\pi$
$44$	3.61803	0.545439
$45$	0	0
$46$	4.47214	0.659380
$47$	1.70820	0.249167	0.124584	−	0.992209i	$-0.460241\pi$
$47$	1.70820	0.249167	0.124584	+	0.992209i	$0.460241\pi$
$48$	−1.00000	−0.144338
$49$	−0.145898	−0.0208426
$50$	0	0
$51$	−1.23607	−0.173084
$52$	−6.47214	−0.897524
$53$	−2.09017	−0.287107	−0.143553	−	0.989643i	$-0.545853\pi$
$53$	−2.09017	−0.287107	−0.143553	+	0.989643i	$0.545853\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.61803	0.349850
$57$	5.70820	0.756070
$58$	8.47214	1.11245
$59$	−3.61803	−0.471028	−0.235514	−	0.971871i	$-0.575677\pi$
$59$	−3.61803	−0.471028	−0.235514	+	0.971871i	$0.575677\pi$
$60$	0	0
$61$	2.76393	0.353885	0.176943	−	0.984221i	$-0.443379\pi$
$61$	2.76393	0.353885	0.176943	+	0.984221i	$0.443379\pi$
$62$	6.61803	0.840491
$63$	2.61803	0.329841
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.61803	−0.445349
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	1.23607	0.149895
$69$	−4.47214	−0.538382
$70$	0	0
$71$	5.52786	0.656037	0.328018	−	0.944671i	$-0.393619\pi$
$71$	5.52786	0.656037	0.328018	+	0.944671i	$0.393619\pi$
$72$	1.00000	0.117851
$73$	3.52786	0.412905	0.206453	−	0.978457i	$-0.433808\pi$
$73$	3.52786	0.412905	0.206453	+	0.978457i	$0.433808\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−5.70820	−0.654776
$77$	9.47214	1.07945
$78$	6.47214	0.732825
$79$	−5.61803	−0.632078	−0.316039	−	0.948746i	$-0.602353\pi$
$79$	−5.61803	−0.632078	−0.316039	+	0.948746i	$0.602353\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	5.70820	0.630366
$83$	2.14590	0.235543	0.117771	−	0.993041i	$-0.462425\pi$
$83$	2.14590	0.235543	0.117771	+	0.993041i	$0.462425\pi$
$84$	−2.61803	−0.285651
$85$	0	0
$86$	−7.70820	−0.831197
$87$	−8.47214	−0.908308
$88$	3.61803	0.385684
$89$	3.52786	0.373953	0.186976	−	0.982364i	$-0.440131\pi$
$89$	3.52786	0.373953	0.186976	+	0.982364i	$0.440131\pi$
$90$	0	0
$91$	−16.9443	−1.77624
$92$	4.47214	0.466252
$93$	−6.61803	−0.686258
$94$	1.70820	0.176188
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−3.38197	−0.343387	−0.171693	−	0.985150i	$-0.554924\pi$
$97$	−3.38197	−0.343387	−0.171693	+	0.985150i	$0.554924\pi$
$98$	−0.145898	−0.0147379
$99$	3.61803	0.363626
$100$	0	0
$101$	4.38197	0.436022	0.218011	−	0.975946i	$-0.430043\pi$
$101$	4.38197	0.436022	0.218011	+	0.975946i	$0.430043\pi$
$102$	−1.23607	−0.122389
$103$	14.3262	1.41161	0.705803	−	0.708408i	$-0.250586\pi$
$103$	14.3262	1.41161	0.705803	+	0.708408i	$0.250586\pi$
$104$	−6.47214	−0.634645
$105$	0	0
$106$	−2.09017	−0.203015
$107$	−11.6180	−1.12316	−0.561579	−	0.827423i	$-0.689806\pi$
$107$	−11.6180	−1.12316	−0.561579	+	0.827423i	$0.689806\pi$
$108$	−1.00000	−0.0962250
$109$	17.2361	1.65092	0.825458	−	0.564464i	$-0.190917\pi$
$109$	17.2361	1.65092	0.825458	+	0.564464i	$0.190917\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	2.61803	0.247381
$113$	−19.2361	−1.80958	−0.904789	−	0.425861i	$-0.859971\pi$
$113$	−19.2361	−1.80958	−0.904789	+	0.425861i	$0.859971\pi$
$114$	5.70820	0.534622
$115$	0	0
$116$	8.47214	0.786618
$117$	−6.47214	−0.598349
$118$	−3.61803	−0.333067
$119$	3.23607	0.296650
$120$	0	0
$121$	2.09017	0.190015
$122$	2.76393	0.250235
$123$	−5.70820	−0.514691
$124$	6.61803	0.594317
$125$	0	0
$126$	2.61803	0.233233
$127$	13.6180	1.20841	0.604203	−	0.796831i	$-0.293492\pi$
$127$	13.6180	1.20841	0.604203	+	0.796831i	$0.293492\pi$
$128$	1.00000	0.0883883
$129$	7.70820	0.678670
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	−3.61803	−0.314909
$133$	−14.9443	−1.29583
$134$	−1.52786	−0.131987
$135$	0	0
$136$	1.23607	0.105992
$137$	−12.1803	−1.04064	−0.520318	−	0.853972i	$-0.674187\pi$
$137$	−12.1803	−1.04064	−0.520318	+	0.853972i	$0.674187\pi$
$138$	−4.47214	−0.380693
$139$	10.4721	0.888235	0.444117	−	0.895969i	$-0.353517\pi$
$139$	10.4721	0.888235	0.444117	+	0.895969i	$0.353517\pi$
$140$	0	0
$141$	−1.70820	−0.143857
$142$	5.52786	0.463888
$143$	−23.4164	−1.95818
$144$	1.00000	0.0833333
$145$	0	0
$146$	3.52786	0.291968
$147$	0.145898	0.0120335
$148$	8.00000	0.657596
$149$	22.0902	1.80970	0.904849	−	0.425733i	$-0.139984\pi$
$149$	22.0902	1.80970	0.904849	+	0.425733i	$0.139984\pi$
$150$	0	0
$151$	18.3262	1.49137	0.745684	−	0.666300i	$-0.232123\pi$
$151$	18.3262	1.49137	0.745684	+	0.666300i	$0.232123\pi$
$152$	−5.70820	−0.462996
$153$	1.23607	0.0999302
$154$	9.47214	0.763286
$155$	0	0
$156$	6.47214	0.518186
$157$	8.65248	0.690543	0.345271	−	0.938503i	$-0.387787\pi$
$157$	8.65248	0.690543	0.345271	+	0.938503i	$0.387787\pi$
$158$	−5.61803	−0.446947
$159$	2.09017	0.165761
$160$	0	0
$161$	11.7082	0.922736
$162$	1.00000	0.0785674
$163$	−0.472136	−0.0369805	−0.0184903	−	0.999829i	$-0.505886\pi$
$163$	−0.472136	−0.0369805	−0.0184903	+	0.999829i	$0.505886\pi$
$164$	5.70820	0.445736
$165$	0	0
$166$	2.14590	0.166554
$167$	11.7082	0.906008	0.453004	−	0.891508i	$-0.350352\pi$
$167$	11.7082	0.906008	0.453004	+	0.891508i	$0.350352\pi$
$168$	−2.61803	−0.201986
$169$	28.8885	2.22220
$170$	0	0
$171$	−5.70820	−0.436517
$172$	−7.70820	−0.587745
$173$	5.09017	0.386998	0.193499	−	0.981100i	$-0.438016\pi$
$173$	5.09017	0.386998	0.193499	+	0.981100i	$0.438016\pi$
$174$	−8.47214	−0.642271
$175$	0	0
$176$	3.61803	0.272720
$177$	3.61803	0.271948
$178$	3.52786	0.264425
$179$	9.85410	0.736530	0.368265	−	0.929721i	$-0.379952\pi$
$179$	9.85410	0.736530	0.368265	+	0.929721i	$0.379952\pi$
$180$	0	0
$181$	−6.65248	−0.494475	−0.247237	−	0.968955i	$-0.579523\pi$
$181$	−6.65248	−0.494475	−0.247237	+	0.968955i	$0.579523\pi$
$182$	−16.9443	−1.25599
$183$	−2.76393	−0.204316
$184$	4.47214	0.329690
$185$	0	0
$186$	−6.61803	−0.485258
$187$	4.47214	0.327035
$188$	1.70820	0.124584
$189$	−2.61803	−0.190434
$190$	0	0
$191$	4.29180	0.310543	0.155272	−	0.987872i	$-0.450375\pi$
$191$	4.29180	0.310543	0.155272	+	0.987872i	$0.450375\pi$
$192$	−1.00000	−0.0721688
$193$	17.8541	1.28517	0.642583	−	0.766216i	$-0.277863\pi$
$193$	17.8541	1.28517	0.642583	+	0.766216i	$0.277863\pi$
$194$	−3.38197	−0.242811
$195$	0	0
$196$	−0.145898	−0.0104213
$197$	−17.0902	−1.21762	−0.608812	−	0.793314i	$-0.708354\pi$
$197$	−17.0902	−1.21762	−0.608812	+	0.793314i	$0.708354\pi$
$198$	3.61803	0.257122
$199$	19.5066	1.38278	0.691392	−	0.722480i	$-0.256998\pi$
$199$	19.5066	1.38278	0.691392	+	0.722480i	$0.256998\pi$
$200$	0	0
$201$	1.52786	0.107767
$202$	4.38197	0.308314
$203$	22.1803	1.55675
$204$	−1.23607	−0.0865421
$205$	0	0
$206$	14.3262	0.998156
$207$	4.47214	0.310835
$208$	−6.47214	−0.448762
$209$	−20.6525	−1.42856
$210$	0	0
$211$	−3.41641	−0.235195	−0.117598	−	0.993061i	$-0.537519\pi$
$211$	−3.41641	−0.235195	−0.117598	+	0.993061i	$0.537519\pi$
$212$	−2.09017	−0.143553
$213$	−5.52786	−0.378763
$214$	−11.6180	−0.794192
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	17.3262	1.17618
$218$	17.2361	1.16737
$219$	−3.52786	−0.238391
$220$	0	0
$221$	−8.00000	−0.538138
$222$	−8.00000	−0.536925
$223$	−8.09017	−0.541758	−0.270879	−	0.962613i	$-0.587314\pi$
$223$	−8.09017	−0.541758	−0.270879	+	0.962613i	$0.587314\pi$
$224$	2.61803	0.174925
$225$	0	0
$226$	−19.2361	−1.27956
$227$	−22.2705	−1.47815	−0.739073	−	0.673626i	$-0.764736\pi$
$227$	−22.2705	−1.47815	−0.739073	+	0.673626i	$0.764736\pi$
$228$	5.70820	0.378035
$229$	−5.52786	−0.365292	−0.182646	−	0.983179i	$-0.558466\pi$
$229$	−5.52786	−0.365292	−0.182646	+	0.983179i	$0.558466\pi$
$230$	0	0
$231$	−9.47214	−0.623221
$232$	8.47214	0.556223
$233$	18.6525	1.22196	0.610982	−	0.791644i	$-0.290775\pi$
$233$	18.6525	1.22196	0.610982	+	0.791644i	$0.290775\pi$
$234$	−6.47214	−0.423097
$235$	0	0
$236$	−3.61803	−0.235514
$237$	5.61803	0.364931
$238$	3.23607	0.209763
$239$	12.2918	0.795090	0.397545	−	0.917583i	$-0.369862\pi$
$239$	12.2918	0.795090	0.397545	+	0.917583i	$0.369862\pi$
$240$	0	0
$241$	9.56231	0.615962	0.307981	−	0.951392i	$-0.400347\pi$
$241$	9.56231	0.615962	0.307981	+	0.951392i	$0.400347\pi$
$242$	2.09017	0.134361
$243$	−1.00000	−0.0641500
$244$	2.76393	0.176943
$245$	0	0
$246$	−5.70820	−0.363942
$247$	36.9443	2.35071
$248$	6.61803	0.420246
$249$	−2.14590	−0.135991
$250$	0	0
$251$	13.5623	0.856045	0.428023	−	0.903768i	$-0.359210\pi$
$251$	13.5623	0.856045	0.428023	+	0.903768i	$0.359210\pi$
$252$	2.61803	0.164921
$253$	16.1803	1.01725
$254$	13.6180	0.854471
$255$	0	0
$256$	1.00000	0.0625000
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	7.70820	0.479892
$259$	20.9443	1.30141
$260$	0	0
$261$	8.47214	0.524412
$262$	−17.8885	−1.10516
$263$	−11.7082	−0.721959	−0.360979	−	0.932574i	$-0.617558\pi$
$263$	−11.7082	−0.721959	−0.360979	+	0.932574i	$0.617558\pi$
$264$	−3.61803	−0.222675
$265$	0	0
$266$	−14.9443	−0.916292
$267$	−3.52786	−0.215902
$268$	−1.52786	−0.0933292
$269$	−9.09017	−0.554237	−0.277119	−	0.960836i	$-0.589380\pi$
$269$	−9.09017	−0.554237	−0.277119	+	0.960836i	$0.589380\pi$
$270$	0	0
$271$	−18.0344	−1.09551	−0.547757	−	0.836637i	$-0.684518\pi$
$271$	−18.0344	−1.09551	−0.547757	+	0.836637i	$0.684518\pi$
$272$	1.23607	0.0749476
$273$	16.9443	1.02551
$274$	−12.1803	−0.735841
$275$	0	0
$276$	−4.47214	−0.269191
$277$	16.9443	1.01808	0.509041	−	0.860742i	$-0.330000\pi$
$277$	16.9443	1.01808	0.509041	+	0.860742i	$0.330000\pi$
$278$	10.4721	0.628077
$279$	6.61803	0.396211
$280$	0	0
$281$	5.88854	0.351281	0.175641	−	0.984454i	$-0.443800\pi$
$281$	5.88854	0.351281	0.175641	+	0.984454i	$0.443800\pi$
$282$	−1.70820	−0.101722
$283$	4.58359	0.272466	0.136233	−	0.990677i	$-0.456500\pi$
$283$	4.58359	0.272466	0.136233	+	0.990677i	$0.456500\pi$
$284$	5.52786	0.328018
$285$	0	0
$286$	−23.4164	−1.38464
$287$	14.9443	0.882132
$288$	1.00000	0.0589256
$289$	−15.4721	−0.910126
$290$	0	0
$291$	3.38197	0.198254
$292$	3.52786	0.206453
$293$	−22.0902	−1.29052	−0.645261	−	0.763962i	$-0.723251\pi$
$293$	−22.0902	−1.29052	−0.645261	+	0.763962i	$0.723251\pi$
$294$	0.145898	0.00850895
$295$	0	0
$296$	8.00000	0.464991
$297$	−3.61803	−0.209940
$298$	22.0902	1.27965
$299$	−28.9443	−1.67389
$300$	0	0
$301$	−20.1803	−1.16318
$302$	18.3262	1.05456
$303$	−4.38197	−0.251737
$304$	−5.70820	−0.327388
$305$	0	0
$306$	1.23607	0.0706613
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	9.47214	0.539725
$309$	−14.3262	−0.814991
$310$	0	0
$311$	−1.05573	−0.0598648	−0.0299324	−	0.999552i	$-0.509529\pi$
$311$	−1.05573	−0.0598648	−0.0299324	+	0.999552i	$0.509529\pi$
$312$	6.47214	0.366413
$313$	−16.2705	−0.919664	−0.459832	−	0.888006i	$-0.652090\pi$
$313$	−16.2705	−0.919664	−0.459832	+	0.888006i	$0.652090\pi$
$314$	8.65248	0.488287
$315$	0	0
$316$	−5.61803	−0.316039
$317$	−20.5623	−1.15489	−0.577447	−	0.816428i	$-0.695951\pi$
$317$	−20.5623	−1.15489	−0.577447	+	0.816428i	$0.695951\pi$
$318$	2.09017	0.117211
$319$	30.6525	1.71621
$320$	0	0
$321$	11.6180	0.648455
$322$	11.7082	0.652473
$323$	−7.05573	−0.392591
$324$	1.00000	0.0555556
$325$	0	0
$326$	−0.472136	−0.0261492
$327$	−17.2361	−0.953157
$328$	5.70820	0.315183
$329$	4.47214	0.246557
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	2.14590	0.117771
$333$	8.00000	0.438397
$334$	11.7082	0.640644
$335$	0	0
$336$	−2.61803	−0.142825
$337$	0.909830	0.0495616	0.0247808	−	0.999693i	$-0.492111\pi$
$337$	0.909830	0.0495616	0.0247808	+	0.999693i	$0.492111\pi$
$338$	28.8885	1.57133
$339$	19.2361	1.04476
$340$	0	0
$341$	23.9443	1.29666
$342$	−5.70820	−0.308664
$343$	−18.7082	−1.01015
$344$	−7.70820	−0.415599
$345$	0	0
$346$	5.09017	0.273649
$347$	20.5623	1.10384	0.551921	−	0.833896i	$-0.313895\pi$
$347$	20.5623	1.10384	0.551921	+	0.833896i	$0.313895\pi$
$348$	−8.47214	−0.454154
$349$	−27.8885	−1.49284	−0.746420	−	0.665475i	$-0.768229\pi$
$349$	−27.8885	−1.49284	−0.746420	+	0.665475i	$0.768229\pi$
$350$	0	0
$351$	6.47214	0.345457
$352$	3.61803	0.192842
$353$	−13.2361	−0.704485	−0.352242	−	0.935909i	$-0.614581\pi$
$353$	−13.2361	−0.704485	−0.352242	+	0.935909i	$0.614581\pi$
$354$	3.61803	0.192296
$355$	0	0
$356$	3.52786	0.186976
$357$	−3.23607	−0.171271
$358$	9.85410	0.520805
$359$	22.1803	1.17063	0.585317	−	0.810805i	$-0.300970\pi$
$359$	22.1803	1.17063	0.585317	+	0.810805i	$0.300970\pi$
$360$	0	0
$361$	13.5836	0.714926
$362$	−6.65248	−0.349646
$363$	−2.09017	−0.109705
$364$	−16.9443	−0.888121
$365$	0	0
$366$	−2.76393	−0.144473
$367$	−30.6180	−1.59825	−0.799124	−	0.601166i	$-0.794703\pi$
$367$	−30.6180	−1.59825	−0.799124	+	0.601166i	$0.794703\pi$
$368$	4.47214	0.233126
$369$	5.70820	0.297157
$370$	0	0
$371$	−5.47214	−0.284099
$372$	−6.61803	−0.343129
$373$	−11.5279	−0.596890	−0.298445	−	0.954427i	$-0.596468\pi$
$373$	−11.5279	−0.596890	−0.298445	+	0.954427i	$0.596468\pi$
$374$	4.47214	0.231249
$375$	0	0
$376$	1.70820	0.0880939
$377$	−54.8328	−2.82403
$378$	−2.61803	−0.134657
$379$	−20.1803	−1.03659	−0.518297	−	0.855201i	$-0.673434\pi$
$379$	−20.1803	−1.03659	−0.518297	+	0.855201i	$0.673434\pi$
$380$	0	0
$381$	−13.6180	−0.697673
$382$	4.29180	0.219587
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	17.8541	0.908750
$387$	−7.70820	−0.391830
$388$	−3.38197	−0.171693
$389$	−12.3820	−0.627791	−0.313895	−	0.949458i	$-0.601634\pi$
$389$	−12.3820	−0.627791	−0.313895	+	0.949458i	$0.601634\pi$
$390$	0	0
$391$	5.52786	0.279556
$392$	−0.145898	−0.00736896
$393$	17.8885	0.902358
$394$	−17.0902	−0.860990
$395$	0	0
$396$	3.61803	0.181813
$397$	−30.7639	−1.54400	−0.771999	−	0.635624i	$-0.780743\pi$
$397$	−30.7639	−1.54400	−0.771999	+	0.635624i	$0.780743\pi$
$398$	19.5066	0.977776
$399$	14.9443	0.748149
$400$	0	0
$401$	25.7082	1.28381	0.641903	−	0.766786i	$-0.278145\pi$
$401$	25.7082	1.28381	0.641903	+	0.766786i	$0.278145\pi$
$402$	1.52786	0.0762029
$403$	−42.8328	−2.13365
$404$	4.38197	0.218011
$405$	0	0
$406$	22.1803	1.10079
$407$	28.9443	1.43471
$408$	−1.23607	−0.0611945
$409$	5.79837	0.286711	0.143356	−	0.989671i	$-0.454211\pi$
$409$	5.79837	0.286711	0.143356	+	0.989671i	$0.454211\pi$
$410$	0	0
$411$	12.1803	0.600812
$412$	14.3262	0.705803
$413$	−9.47214	−0.466093
$414$	4.47214	0.219793
$415$	0	0
$416$	−6.47214	−0.317323
$417$	−10.4721	−0.512823
$418$	−20.6525	−1.01015
$419$	−3.09017	−0.150965	−0.0754823	−	0.997147i	$-0.524050\pi$
$419$	−3.09017	−0.150965	−0.0754823	+	0.997147i	$0.524050\pi$
$420$	0	0
$421$	−22.7639	−1.10945	−0.554723	−	0.832035i	$-0.687176\pi$
$421$	−22.7639	−1.10945	−0.554723	+	0.832035i	$0.687176\pi$
$422$	−3.41641	−0.166308
$423$	1.70820	0.0830557
$424$	−2.09017	−0.101508
$425$	0	0
$426$	−5.52786	−0.267826
$427$	7.23607	0.350178
$428$	−11.6180	−0.561579
$429$	23.4164	1.13055
$430$	0	0
$431$	16.8328	0.810808	0.405404	−	0.914138i	$-0.367131\pi$
$431$	16.8328	0.810808	0.405404	+	0.914138i	$0.367131\pi$
$432$	−1.00000	−0.0481125
$433$	15.5066	0.745199	0.372599	−	0.927992i	$-0.378467\pi$
$433$	15.5066	0.745199	0.372599	+	0.927992i	$0.378467\pi$
$434$	17.3262	0.831686
$435$	0	0
$436$	17.2361	0.825458
$437$	−25.5279	−1.22116
$438$	−3.52786	−0.168568
$439$	−23.5066	−1.12191	−0.560954	−	0.827847i	$-0.689566\pi$
$439$	−23.5066	−1.12191	−0.560954	+	0.827847i	$0.689566\pi$
$440$	0	0
$441$	−0.145898	−0.00694753
$442$	−8.00000	−0.380521
$443$	16.6180	0.789547	0.394773	−	0.918779i	$-0.370823\pi$
$443$	16.6180	0.789547	0.394773	+	0.918779i	$0.370823\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	−8.09017	−0.383081
$447$	−22.0902	−1.04483
$448$	2.61803	0.123690
$449$	−27.7082	−1.30763	−0.653815	−	0.756654i	$-0.726833\pi$
$449$	−27.7082	−1.30763	−0.653815	+	0.756654i	$0.726833\pi$
$450$	0	0
$451$	20.6525	0.972487
$452$	−19.2361	−0.904789
$453$	−18.3262	−0.861042
$454$	−22.2705	−1.04521
$455$	0	0
$456$	5.70820	0.267311
$457$	−4.09017	−0.191330	−0.0956650	−	0.995414i	$-0.530498\pi$
$457$	−4.09017	−0.191330	−0.0956650	+	0.995414i	$0.530498\pi$
$458$	−5.52786	−0.258300
$459$	−1.23607	−0.0576947
$460$	0	0
$461$	7.56231	0.352212	0.176106	−	0.984371i	$-0.443650\pi$
$461$	7.56231	0.352212	0.176106	+	0.984371i	$0.443650\pi$
$462$	−9.47214	−0.440684
$463$	−22.8328	−1.06113	−0.530565	−	0.847644i	$-0.678020\pi$
$463$	−22.8328	−1.06113	−0.530565	+	0.847644i	$0.678020\pi$
$464$	8.47214	0.393309
$465$	0	0
$466$	18.6525	0.864059
$467$	10.6738	0.493923	0.246961	−	0.969025i	$-0.420568\pi$
$467$	10.6738	0.493923	0.246961	+	0.969025i	$0.420568\pi$
$468$	−6.47214	−0.299175
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−8.65248	−0.398685
$472$	−3.61803	−0.166534
$473$	−27.8885	−1.28232
$474$	5.61803	0.258045
$475$	0	0
$476$	3.23607	0.148325
$477$	−2.09017	−0.0957023
$478$	12.2918	0.562214
$479$	−29.1246	−1.33074	−0.665369	−	0.746515i	$-0.731726\pi$
$479$	−29.1246	−1.33074	−0.665369	+	0.746515i	$0.731726\pi$
$480$	0	0
$481$	−51.7771	−2.36083
$482$	9.56231	0.435551
$483$	−11.7082	−0.532742
$484$	2.09017	0.0950077
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	17.7984	0.806521	0.403261	−	0.915085i	$-0.367877\pi$
$487$	17.7984	0.806521	0.403261	+	0.915085i	$0.367877\pi$
$488$	2.76393	0.125117
$489$	0.472136	0.0213507
$490$	0	0
$491$	−25.2705	−1.14044	−0.570221	−	0.821491i	$-0.693142\pi$
$491$	−25.2705	−1.14044	−0.570221	+	0.821491i	$0.693142\pi$
$492$	−5.70820	−0.257346
$493$	10.4721	0.471641
$494$	36.9443	1.66220
$495$	0	0
$496$	6.61803	0.297158
$497$	14.4721	0.649164
$498$	−2.14590	−0.0961600
$499$	35.5967	1.59353	0.796765	−	0.604290i	$-0.206543\pi$
$499$	35.5967	1.59353	0.796765	+	0.604290i	$0.206543\pi$
$500$	0	0
$501$	−11.7082	−0.523084
$502$	13.5623	0.605315
$503$	23.8885	1.06514	0.532569	−	0.846387i	$-0.321227\pi$
$503$	23.8885	1.06514	0.532569	+	0.846387i	$0.321227\pi$
$504$	2.61803	0.116617
$505$	0	0
$506$	16.1803	0.719304
$507$	−28.8885	−1.28299
$508$	13.6180	0.604203
$509$	−17.5066	−0.775965	−0.387983	−	0.921667i	$-0.626828\pi$
$509$	−17.5066	−0.775965	−0.387983	+	0.921667i	$0.626828\pi$
$510$	0	0
$511$	9.23607	0.408580
$512$	1.00000	0.0441942
$513$	5.70820	0.252023
$514$	−22.0000	−0.970378
$515$	0	0
$516$	7.70820	0.339335
$517$	6.18034	0.271811
$518$	20.9443	0.920238
$519$	−5.09017	−0.223434
$520$	0	0
$521$	14.1803	0.621252	0.310626	−	0.950532i	$-0.399461\pi$
$521$	14.1803	0.621252	0.310626	+	0.950532i	$0.399461\pi$
$522$	8.47214	0.370815
$523$	11.0557	0.483433	0.241717	−	0.970347i	$-0.422290\pi$
$523$	11.0557	0.483433	0.241717	+	0.970347i	$0.422290\pi$
$524$	−17.8885	−0.781465
$525$	0	0
$526$	−11.7082	−0.510502
$527$	8.18034	0.356341
$528$	−3.61803	−0.157455
$529$	−3.00000	−0.130435
$530$	0	0
$531$	−3.61803	−0.157009
$532$	−14.9443	−0.647916
$533$	−36.9443	−1.60023
$534$	−3.52786	−0.152666
$535$	0	0
$536$	−1.52786	−0.0659937
$537$	−9.85410	−0.425236
$538$	−9.09017	−0.391905
$539$	−0.527864	−0.0227367
$540$	0	0
$541$	−26.1803	−1.12558	−0.562790	−	0.826600i	$-0.690272\pi$
$541$	−26.1803	−1.12558	−0.562790	+	0.826600i	$0.690272\pi$
$542$	−18.0344	−0.774646
$543$	6.65248	0.285485
$544$	1.23607	0.0529960
$545$	0	0
$546$	16.9443	0.725148
$547$	9.70820	0.415093	0.207546	−	0.978225i	$-0.433452\pi$
$547$	9.70820	0.415093	0.207546	+	0.978225i	$0.433452\pi$
$548$	−12.1803	−0.520318
$549$	2.76393	0.117962
$550$	0	0
$551$	−48.3607	−2.06023
$552$	−4.47214	−0.190347
$553$	−14.7082	−0.625456
$554$	16.9443	0.719893
$555$	0	0
$556$	10.4721	0.444117
$557$	−18.3262	−0.776508	−0.388254	−	0.921552i	$-0.626922\pi$
$557$	−18.3262	−0.776508	−0.388254	+	0.921552i	$0.626922\pi$
$558$	6.61803	0.280164
$559$	49.8885	2.11006
$560$	0	0
$561$	−4.47214	−0.188814
$562$	5.88854	0.248393
$563$	21.2705	0.896445	0.448223	−	0.893922i	$-0.352057\pi$
$563$	21.2705	0.896445	0.448223	+	0.893922i	$0.352057\pi$
$564$	−1.70820	−0.0719284
$565$	0	0
$566$	4.58359	0.192663
$567$	2.61803	0.109947
$568$	5.52786	0.231944
$569$	−27.7771	−1.16448	−0.582238	−	0.813018i	$-0.697823\pi$
$569$	−27.7771	−1.16448	−0.582238	+	0.813018i	$0.697823\pi$
$570$	0	0
$571$	7.88854	0.330125	0.165063	−	0.986283i	$-0.447217\pi$
$571$	7.88854	0.330125	0.165063	+	0.986283i	$0.447217\pi$
$572$	−23.4164	−0.979089
$573$	−4.29180	−0.179292
$574$	14.9443	0.623762
$575$	0	0
$576$	1.00000	0.0416667
$577$	8.97871	0.373789	0.186894	−	0.982380i	$-0.440158\pi$
$577$	8.97871	0.373789	0.186894	+	0.982380i	$0.440158\pi$
$578$	−15.4721	−0.643556
$579$	−17.8541	−0.741991
$580$	0	0
$581$	5.61803	0.233075
$582$	3.38197	0.140187
$583$	−7.56231	−0.313199
$584$	3.52786	0.145984
$585$	0	0
$586$	−22.0902	−0.912537
$587$	−0.965558	−0.0398528	−0.0199264	−	0.999801i	$-0.506343\pi$
$587$	−0.965558	−0.0398528	−0.0199264	+	0.999801i	$0.506343\pi$
$588$	0.145898	0.00601673
$589$	−37.7771	−1.55658
$590$	0	0
$591$	17.0902	0.702996
$592$	8.00000	0.328798
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	−3.61803	−0.148450
$595$	0	0
$596$	22.0902	0.904849
$597$	−19.5066	−0.798351
$598$	−28.9443	−1.18362
$599$	0.472136	0.0192910	0.00964548	−	0.999953i	$-0.496930\pi$
$599$	0.472136	0.0192910	0.00964548	+	0.999953i	$0.496930\pi$
$600$	0	0
$601$	−7.72949	−0.315292	−0.157646	−	0.987496i	$-0.550391\pi$
$601$	−7.72949	−0.315292	−0.157646	+	0.987496i	$0.550391\pi$
$602$	−20.1803	−0.822489
$603$	−1.52786	−0.0622194
$604$	18.3262	0.745684
$605$	0	0
$606$	−4.38197	−0.178005
$607$	6.56231	0.266356	0.133178	−	0.991092i	$-0.457482\pi$
$607$	6.56231	0.266356	0.133178	+	0.991092i	$0.457482\pi$
$608$	−5.70820	−0.231498
$609$	−22.1803	−0.898793
$610$	0	0
$611$	−11.0557	−0.447267
$612$	1.23607	0.0499651
$613$	48.2492	1.94877	0.974384	−	0.224891i	$-0.0722027\pi$
$613$	48.2492	1.94877	0.974384	+	0.224891i	$0.0722027\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	9.47214	0.381643
$617$	2.11146	0.0850040	0.0425020	−	0.999096i	$-0.486467\pi$
$617$	2.11146	0.0850040	0.0425020	+	0.999096i	$0.486467\pi$
$618$	−14.3262	−0.576286
$619$	−9.41641	−0.378477	−0.189239	−	0.981931i	$-0.560602\pi$
$619$	−9.41641	−0.378477	−0.189239	+	0.981931i	$0.560602\pi$
$620$	0	0
$621$	−4.47214	−0.179461
$622$	−1.05573	−0.0423308
$623$	9.23607	0.370035
$624$	6.47214	0.259093
$625$	0	0
$626$	−16.2705	−0.650300
$627$	20.6525	0.824780
$628$	8.65248	0.345271
$629$	9.88854	0.394282
$630$	0	0
$631$	46.4721	1.85003	0.925013	−	0.379935i	$-0.124054\pi$
$631$	46.4721	1.85003	0.925013	+	0.379935i	$0.124054\pi$
$632$	−5.61803	−0.223473
$633$	3.41641	0.135790
$634$	−20.5623	−0.816633
$635$	0	0
$636$	2.09017	0.0828806
$637$	0.944272	0.0374134
$638$	30.6525	1.21354
$639$	5.52786	0.218679
$640$	0	0
$641$	−13.8197	−0.545844	−0.272922	−	0.962036i	$-0.587990\pi$
$641$	−13.8197	−0.545844	−0.272922	+	0.962036i	$0.587990\pi$
$642$	11.6180	0.458527
$643$	21.8885	0.863200	0.431600	−	0.902065i	$-0.357949\pi$
$643$	21.8885	0.863200	0.431600	+	0.902065i	$0.357949\pi$
$644$	11.7082	0.461368
$645$	0	0
$646$	−7.05573	−0.277604
$647$	−22.9443	−0.902032	−0.451016	−	0.892516i	$-0.648938\pi$
$647$	−22.9443	−0.902032	−0.451016	+	0.892516i	$0.648938\pi$
$648$	1.00000	0.0392837
$649$	−13.0902	−0.513834
$650$	0	0
$651$	−17.3262	−0.679069
$652$	−0.472136	−0.0184903
$653$	−3.85410	−0.150823	−0.0754113	−	0.997153i	$-0.524027\pi$
$653$	−3.85410	−0.150823	−0.0754113	+	0.997153i	$0.524027\pi$
$654$	−17.2361	−0.673984
$655$	0	0
$656$	5.70820	0.222868
$657$	3.52786	0.137635
$658$	4.47214	0.174342
$659$	−20.6180	−0.803165	−0.401582	−	0.915823i	$-0.631540\pi$
$659$	−20.6180	−0.803165	−0.401582	+	0.915823i	$0.631540\pi$
$660$	0	0
$661$	−30.4721	−1.18523	−0.592614	−	0.805486i	$-0.701904\pi$
$661$	−30.4721	−1.18523	−0.592614	+	0.805486i	$0.701904\pi$
$662$	−18.0000	−0.699590
$663$	8.00000	0.310694
$664$	2.14590	0.0832770
$665$	0	0
$666$	8.00000	0.309994
$667$	37.8885	1.46705
$668$	11.7082	0.453004
$669$	8.09017	0.312784
$670$	0	0
$671$	10.0000	0.386046
$672$	−2.61803	−0.100993
$673$	−31.4508	−1.21234	−0.606171	−	0.795335i	$-0.707295\pi$
$673$	−31.4508	−1.21234	−0.606171	+	0.795335i	$0.707295\pi$
$674$	0.909830	0.0350453
$675$	0	0
$676$	28.8885	1.11110
$677$	29.0902	1.11803	0.559013	−	0.829159i	$-0.311180\pi$
$677$	29.0902	1.11803	0.559013	+	0.829159i	$0.311180\pi$
$678$	19.2361	0.738757
$679$	−8.85410	−0.339789
$680$	0	0
$681$	22.2705	0.853408
$682$	23.9443	0.916874
$683$	33.5066	1.28209	0.641047	−	0.767502i	$-0.278500\pi$
$683$	33.5066	1.28209	0.641047	+	0.767502i	$0.278500\pi$
$684$	−5.70820	−0.218259
$685$	0	0
$686$	−18.7082	−0.714283
$687$	5.52786	0.210901
$688$	−7.70820	−0.293873
$689$	13.5279	0.515371
$690$	0	0
$691$	11.2361	0.427440	0.213720	−	0.976895i	$-0.431442\pi$
$691$	11.2361	0.427440	0.213720	+	0.976895i	$0.431442\pi$
$692$	5.09017	0.193499
$693$	9.47214	0.359817
$694$	20.5623	0.780534
$695$	0	0
$696$	−8.47214	−0.321135
$697$	7.05573	0.267255
$698$	−27.8885	−1.05560
$699$	−18.6525	−0.705501
$700$	0	0
$701$	40.8328	1.54223	0.771117	−	0.636693i	$-0.219698\pi$
$701$	40.8328	1.54223	0.771117	+	0.636693i	$0.219698\pi$
$702$	6.47214	0.244275
$703$	−45.6656	−1.72231
$704$	3.61803	0.136360
$705$	0	0
$706$	−13.2361	−0.498146
$707$	11.4721	0.431454
$708$	3.61803	0.135974
$709$	−43.7082	−1.64150	−0.820748	−	0.571290i	$-0.806443\pi$
$709$	−43.7082	−1.64150	−0.820748	+	0.571290i	$0.806443\pi$
$710$	0	0
$711$	−5.61803	−0.210693
$712$	3.52786	0.132212
$713$	29.5967	1.10841
$714$	−3.23607	−0.121107
$715$	0	0
$716$	9.85410	0.368265
$717$	−12.2918	−0.459046
$718$	22.1803	0.827763
$719$	−16.5836	−0.618464	−0.309232	−	0.950987i	$-0.600072\pi$
$719$	−16.5836	−0.618464	−0.309232	+	0.950987i	$0.600072\pi$
$720$	0	0
$721$	37.5066	1.39682
$722$	13.5836	0.505529
$723$	−9.56231	−0.355626
$724$	−6.65248	−0.247237
$725$	0	0
$726$	−2.09017	−0.0775735
$727$	−8.58359	−0.318348	−0.159174	−	0.987251i	$-0.550883\pi$
$727$	−8.58359	−0.318348	−0.159174	+	0.987251i	$0.550883\pi$
$728$	−16.9443	−0.627996
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.52786	−0.352401
$732$	−2.76393	−0.102158
$733$	−35.8885	−1.32557	−0.662787	−	0.748808i	$-0.730626\pi$
$733$	−35.8885	−1.32557	−0.662787	+	0.748808i	$0.730626\pi$
$734$	−30.6180	−1.13013
$735$	0	0
$736$	4.47214	0.164845
$737$	−5.52786	−0.203621
$738$	5.70820	0.210122
$739$	−48.1803	−1.77234	−0.886171	−	0.463358i	$-0.846644\pi$
$739$	−48.1803	−1.77234	−0.886171	+	0.463358i	$0.846644\pi$
$740$	0	0
$741$	−36.9443	−1.35718
$742$	−5.47214	−0.200888
$743$	−39.0132	−1.43125	−0.715627	−	0.698483i	$-0.753859\pi$
$743$	−39.0132	−1.43125	−0.715627	+	0.698483i	$0.753859\pi$
$744$	−6.61803	−0.242629
$745$	0	0
$746$	−11.5279	−0.422065
$747$	2.14590	0.0785143
$748$	4.47214	0.163517
$749$	−30.4164	−1.11139
$750$	0	0
$751$	24.7984	0.904906	0.452453	−	0.891788i	$-0.350549\pi$
$751$	24.7984	0.904906	0.452453	+	0.891788i	$0.350549\pi$
$752$	1.70820	0.0622918
$753$	−13.5623	−0.494238
$754$	−54.8328	−1.99689
$755$	0	0
$756$	−2.61803	−0.0952170
$757$	−17.1246	−0.622405	−0.311202	−	0.950344i	$-0.600732\pi$
$757$	−17.1246	−0.622405	−0.311202	+	0.950344i	$0.600732\pi$
$758$	−20.1803	−0.732983
$759$	−16.1803	−0.587309
$760$	0	0
$761$	1.41641	0.0513447	0.0256724	−	0.999670i	$-0.491827\pi$
$761$	1.41641	0.0513447	0.0256724	+	0.999670i	$0.491827\pi$
$762$	−13.6180	−0.493329
$763$	45.1246	1.63362
$764$	4.29180	0.155272
$765$	0	0
$766$	−20.0000	−0.722629
$767$	23.4164	0.845517
$768$	−1.00000	−0.0360844
$769$	24.6869	0.890233	0.445117	−	0.895473i	$-0.353162\pi$
$769$	24.6869	0.890233	0.445117	+	0.895473i	$0.353162\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	17.8541	0.642583
$773$	−1.14590	−0.0412151	−0.0206075	−	0.999788i	$-0.506560\pi$
$773$	−1.14590	−0.0412151	−0.0206075	+	0.999788i	$0.506560\pi$
$774$	−7.70820	−0.277066
$775$	0	0
$776$	−3.38197	−0.121406
$777$	−20.9443	−0.751372
$778$	−12.3820	−0.443915
$779$	−32.5836	−1.16743
$780$	0	0
$781$	20.0000	0.715656
$782$	5.52786	0.197676
$783$	−8.47214	−0.302769
$784$	−0.145898	−0.00521064
$785$	0	0
$786$	17.8885	0.638063
$787$	28.0689	1.00055	0.500274	−	0.865867i	$-0.333233\pi$
$787$	28.0689	1.00055	0.500274	+	0.865867i	$0.333233\pi$
$788$	−17.0902	−0.608812
$789$	11.7082	0.416823
$790$	0	0
$791$	−50.3607	−1.79062
$792$	3.61803	0.128561
$793$	−17.8885	−0.635241
$794$	−30.7639	−1.09177
$795$	0	0
$796$	19.5066	0.691392
$797$	1.50658	0.0533657	0.0266829	−	0.999644i	$-0.491506\pi$
$797$	1.50658	0.0533657	0.0266829	+	0.999644i	$0.491506\pi$
$798$	14.9443	0.529021
$799$	2.11146	0.0746979
$800$	0	0
$801$	3.52786	0.124651
$802$	25.7082	0.907788
$803$	12.7639	0.450429
$804$	1.52786	0.0538836
$805$	0	0
$806$	−42.8328	−1.50872
$807$	9.09017	0.319989
$808$	4.38197	0.154157
$809$	−30.8328	−1.08402	−0.542012	−	0.840371i	$-0.682337\pi$
$809$	−30.8328	−1.08402	−0.542012	+	0.840371i	$0.682337\pi$
$810$	0	0
$811$	−5.70820	−0.200442	−0.100221	−	0.994965i	$-0.531955\pi$
$811$	−5.70820	−0.200442	−0.100221	+	0.994965i	$0.531955\pi$
$812$	22.1803	0.778377
$813$	18.0344	0.632495
$814$	28.9443	1.01450
$815$	0	0
$816$	−1.23607	−0.0432710
$817$	44.0000	1.53937
$818$	5.79837	0.202735
$819$	−16.9443	−0.592081
$820$	0	0
$821$	−45.0902	−1.57366	−0.786829	−	0.617171i	$-0.788279\pi$
$821$	−45.0902	−1.57366	−0.786829	+	0.617171i	$0.788279\pi$
$822$	12.1803	0.424838
$823$	2.96556	0.103373	0.0516864	−	0.998663i	$-0.483540\pi$
$823$	2.96556	0.103373	0.0516864	+	0.998663i	$0.483540\pi$
$824$	14.3262	0.499078
$825$	0	0
$826$	−9.47214	−0.329578
$827$	44.1033	1.53362	0.766811	−	0.641872i	$-0.221842\pi$
$827$	44.1033	1.53362	0.766811	+	0.641872i	$0.221842\pi$
$828$	4.47214	0.155417
$829$	25.8885	0.899146	0.449573	−	0.893244i	$-0.351576\pi$
$829$	25.8885	0.899146	0.449573	+	0.893244i	$0.351576\pi$
$830$	0	0
$831$	−16.9443	−0.587790
$832$	−6.47214	−0.224381
$833$	−0.180340	−0.00624841
$834$	−10.4721	−0.362620
$835$	0	0
$836$	−20.6525	−0.714281
$837$	−6.61803	−0.228753
$838$	−3.09017	−0.106748
$839$	6.36068	0.219595	0.109798	−	0.993954i	$-0.464980\pi$
$839$	6.36068	0.219595	0.109798	+	0.993954i	$0.464980\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	−22.7639	−0.784497
$843$	−5.88854	−0.202812
$844$	−3.41641	−0.117598
$845$	0	0
$846$	1.70820	0.0587293
$847$	5.47214	0.188025
$848$	−2.09017	−0.0717767
$849$	−4.58359	−0.157308
$850$	0	0
$851$	35.7771	1.22642
$852$	−5.52786	−0.189382
$853$	−43.5967	−1.49272	−0.746362	−	0.665540i	$-0.768201\pi$
$853$	−43.5967	−1.49272	−0.746362	+	0.665540i	$0.768201\pi$
$854$	7.23607	0.247613
$855$	0	0
$856$	−11.6180	−0.397096
$857$	−16.0689	−0.548903	−0.274451	−	0.961601i	$-0.588496\pi$
$857$	−16.0689	−0.548903	−0.274451	+	0.961601i	$0.588496\pi$
$858$	23.4164	0.799423
$859$	−2.29180	−0.0781951	−0.0390975	−	0.999235i	$-0.512448\pi$
$859$	−2.29180	−0.0781951	−0.0390975	+	0.999235i	$0.512448\pi$
$860$	0	0
$861$	−14.9443	−0.509299
$862$	16.8328	0.573328
$863$	−10.1803	−0.346543	−0.173271	−	0.984874i	$-0.555434\pi$
$863$	−10.1803	−0.346543	−0.173271	+	0.984874i	$0.555434\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	15.5066	0.526935
$867$	15.4721	0.525461
$868$	17.3262	0.588091
$869$	−20.3262	−0.689520
$870$	0	0
$871$	9.88854	0.335061
$872$	17.2361	0.583687
$873$	−3.38197	−0.114462
$874$	−25.5279	−0.863493
$875$	0	0
$876$	−3.52786	−0.119195
$877$	−37.1246	−1.25361	−0.626805	−	0.779177i	$-0.715638\pi$
$877$	−37.1246	−1.25361	−0.626805	+	0.779177i	$0.715638\pi$
$878$	−23.5066	−0.793309
$879$	22.0902	0.745083
$880$	0	0
$881$	−23.5967	−0.794995	−0.397497	−	0.917603i	$-0.630121\pi$
$881$	−23.5967	−0.794995	−0.397497	+	0.917603i	$0.630121\pi$
$882$	−0.145898	−0.00491264
$883$	−30.0689	−1.01190	−0.505949	−	0.862563i	$-0.668858\pi$
$883$	−30.0689	−1.01190	−0.505949	+	0.862563i	$0.668858\pi$
$884$	−8.00000	−0.269069
$885$	0	0
$886$	16.6180	0.558294
$887$	23.4164	0.786246	0.393123	−	0.919486i	$-0.371395\pi$
$887$	23.4164	0.786246	0.393123	+	0.919486i	$0.371395\pi$
$888$	−8.00000	−0.268462
$889$	35.6525	1.19575
$890$	0	0
$891$	3.61803	0.121209
$892$	−8.09017	−0.270879
$893$	−9.75078	−0.326297
$894$	−22.0902	−0.738806
$895$	0	0
$896$	2.61803	0.0874624
$897$	28.9443	0.966421
$898$	−27.7082	−0.924635
$899$	56.0689	1.87000
$900$	0	0
$901$	−2.58359	−0.0860719
$902$	20.6525	0.687652
$903$	20.1803	0.671560
$904$	−19.2361	−0.639782
$905$	0	0
$906$	−18.3262	−0.608848
$907$	30.4721	1.01181	0.505905	−	0.862589i	$-0.331159\pi$
$907$	30.4721	1.01181	0.505905	+	0.862589i	$0.331159\pi$
$908$	−22.2705	−0.739073
$909$	4.38197	0.145341
$910$	0	0
$911$	18.1803	0.602342	0.301171	−	0.953570i	$-0.402623\pi$
$911$	18.1803	0.602342	0.301171	+	0.953570i	$0.402623\pi$
$912$	5.70820	0.189018
$913$	7.76393	0.256949
$914$	−4.09017	−0.135291
$915$	0	0
$916$	−5.52786	−0.182646
$917$	−46.8328	−1.54656
$918$	−1.23607	−0.0407963
$919$	−40.7214	−1.34327	−0.671637	−	0.740881i	$-0.734408\pi$
$919$	−40.7214	−1.34327	−0.671637	+	0.740881i	$0.734408\pi$
$920$	0	0
$921$	−10.0000	−0.329511
$922$	7.56231	0.249051
$923$	−35.7771	−1.17762
$924$	−9.47214	−0.311610
$925$	0	0
$926$	−22.8328	−0.750333
$927$	14.3262	0.470535
$928$	8.47214	0.278111
$929$	−19.0132	−0.623801	−0.311901	−	0.950115i	$-0.600966\pi$
$929$	−19.0132	−0.623801	−0.311901	+	0.950115i	$0.600966\pi$
$930$	0	0
$931$	0.832816	0.0272944
$932$	18.6525	0.610982
$933$	1.05573	0.0345630
$934$	10.6738	0.349256
$935$	0	0
$936$	−6.47214	−0.211548
$937$	−8.68692	−0.283789	−0.141895	−	0.989882i	$-0.545319\pi$
$937$	−8.68692	−0.283789	−0.141895	+	0.989882i	$0.545319\pi$
$938$	−4.00000	−0.130605
$939$	16.2705	0.530968
$940$	0	0
$941$	−4.43769	−0.144665	−0.0723323	−	0.997381i	$-0.523044\pi$
$941$	−4.43769	−0.144665	−0.0723323	+	0.997381i	$0.523044\pi$
$942$	−8.65248	−0.281913
$943$	25.5279	0.831302
$944$	−3.61803	−0.117757
$945$	0	0
$946$	−27.8885	−0.906735
$947$	−33.5623	−1.09063	−0.545314	−	0.838232i	$-0.683590\pi$
$947$	−33.5623	−1.09063	−0.545314	+	0.838232i	$0.683590\pi$
$948$	5.61803	0.182465
$949$	−22.8328	−0.741185
$950$	0	0
$951$	20.5623	0.666778
$952$	3.23607	0.104882
$953$	5.30495	0.171844	0.0859221	−	0.996302i	$-0.472616\pi$
$953$	5.30495	0.171844	0.0859221	+	0.996302i	$0.472616\pi$
$954$	−2.09017	−0.0676718
$955$	0	0
$956$	12.2918	0.397545
$957$	−30.6525	−0.990854
$958$	−29.1246	−0.940973
$959$	−31.8885	−1.02973
$960$	0	0
$961$	12.7984	0.412851
$962$	−51.7771	−1.66936
$963$	−11.6180	−0.374386
$964$	9.56231	0.307981
$965$	0	0
$966$	−11.7082	−0.376705
$967$	10.9098	0.350836	0.175418	−	0.984494i	$-0.443872\pi$
$967$	10.9098	0.350836	0.175418	+	0.984494i	$0.443872\pi$
$968$	2.09017	0.0671806
$969$	7.05573	0.226663
$970$	0	0
$971$	−14.2148	−0.456174	−0.228087	−	0.973641i	$-0.573247\pi$
$971$	−14.2148	−0.456174	−0.228087	+	0.973641i	$0.573247\pi$
$972$	−1.00000	−0.0320750
$973$	27.4164	0.878930
$974$	17.7984	0.570297
$975$	0	0
$976$	2.76393	0.0884713
$977$	−0.763932	−0.0244404	−0.0122202	−	0.999925i	$-0.503890\pi$
$977$	−0.763932	−0.0244404	−0.0122202	+	0.999925i	$0.503890\pi$
$978$	0.472136	0.0150972
$979$	12.7639	0.407937
$980$	0	0
$981$	17.2361	0.550305
$982$	−25.2705	−0.806414
$983$	40.1803	1.28155	0.640777	−	0.767727i	$-0.278612\pi$
$983$	40.1803	1.28155	0.640777	+	0.767727i	$0.278612\pi$
$984$	−5.70820	−0.181971
$985$	0	0
$986$	10.4721	0.333501
$987$	−4.47214	−0.142350
$988$	36.9443	1.17535
$989$	−34.4721	−1.09615
$990$	0	0
$991$	29.4508	0.935537	0.467769	−	0.883851i	$-0.345058\pi$
$991$	29.4508	0.935537	0.467769	+	0.883851i	$0.345058\pi$
$992$	6.61803	0.210123
$993$	18.0000	0.571213
$994$	14.4721	0.459028
$995$	0	0
$996$	−2.14590	−0.0679954
$997$	−5.59675	−0.177251	−0.0886254	−	0.996065i	$-0.528247\pi$
$997$	−5.59675	−0.177251	−0.0886254	+	0.996065i	$0.528247\pi$
$998$	35.5967	1.12680
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.a.f.1.2		2
5.2	odd	4		3750.2.c.b.1249.4		4
5.3	odd	4		3750.2.c.b.1249.1		4
5.4	even	2		3750.2.a.d.1.1		2
25.3	odd	20		750.2.h.b.49.1		8
25.4	even	10		750.2.g.b.451.1		4
25.6	even	5		150.2.g.a.61.1	✓	4
25.8	odd	20		750.2.h.b.199.2		8
25.17	odd	20		750.2.h.b.199.1		8
25.19	even	10		750.2.g.b.301.1		4
25.21	even	5		150.2.g.a.91.1	yes	4
25.22	odd	20		750.2.h.b.49.2		8
75.56	odd	10		450.2.h.c.361.1		4
75.71	odd	10		450.2.h.c.91.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.g.a.61.1	✓	4	25.6	even	5
150.2.g.a.91.1	yes	4	25.21	even	5
450.2.h.c.91.1		4	75.71	odd	10
450.2.h.c.361.1		4	75.56	odd	10
750.2.g.b.301.1		4	25.19	even	10
750.2.g.b.451.1		4	25.4	even	10
750.2.h.b.49.1		8	25.3	odd	20
750.2.h.b.49.2		8	25.22	odd	20
750.2.h.b.199.1		8	25.17	odd	20
750.2.h.b.199.2		8	25.8	odd	20
3750.2.a.d.1.1		2	5.4	even	2
3750.2.a.f.1.2		2	1.1	even	1	trivial
3750.2.c.b.1249.1		4	5.3	odd	4
3750.2.c.b.1249.4		4	5.2	odd	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:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.61803 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.61803 q^{11} -1.00000 q^{12} -6.47214 q^{13} +2.61803 q^{14} +1.00000 q^{16} +1.23607 q^{17} +1.00000 q^{18} -5.70820 q^{19} -2.61803 q^{21} +3.61803 q^{22} +4.47214 q^{23} -1.00000 q^{24} -6.47214 q^{26} -1.00000 q^{27} +2.61803 q^{28} +8.47214 q^{29} +6.61803 q^{31} +1.00000 q^{32} -3.61803 q^{33} +1.23607 q^{34} +1.00000 q^{36} +8.00000 q^{37} -5.70820 q^{38} +6.47214 q^{39} +5.70820 q^{41} -2.61803 q^{42} -7.70820 q^{43} +3.61803 q^{44} +4.47214 q^{46} +1.70820 q^{47} -1.00000 q^{48} -0.145898 q^{49} -1.23607 q^{51} -6.47214 q^{52} -2.09017 q^{53} -1.00000 q^{54} +2.61803 q^{56} +5.70820 q^{57} +8.47214 q^{58} -3.61803 q^{59} +2.76393 q^{61} +6.61803 q^{62} +2.61803 q^{63} +1.00000 q^{64} -3.61803 q^{66} -1.52786 q^{67} +1.23607 q^{68} -4.47214 q^{69} +5.52786 q^{71} +1.00000 q^{72} +3.52786 q^{73} +8.00000 q^{74} -5.70820 q^{76} +9.47214 q^{77} +6.47214 q^{78} -5.61803 q^{79} +1.00000 q^{81} +5.70820 q^{82} +2.14590 q^{83} -2.61803 q^{84} -7.70820 q^{86} -8.47214 q^{87} +3.61803 q^{88} +3.52786 q^{89} -16.9443 q^{91} +4.47214 q^{92} -6.61803 q^{93} +1.70820 q^{94} -1.00000 q^{96} -3.38197 q^{97} -0.145898 q^{98} +3.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 3 q^{7} + 2 q^{8} + 2 q^{9} + 5 q^{11} - 2 q^{12} - 4 q^{13} + 3 q^{14} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 2 q^{19} - 3 q^{21} + 5 q^{22} - 2 q^{24} - 4 q^{26}+ \cdots + 5 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 3 * q^7 + 2 * q^8 + 2 * q^9 + 5 * q^11 - 2 * q^12 - 4 * q^13 + 3 * q^14 + 2 * q^16 - 2 * q^17 + 2 * q^18 + 2 * q^19 - 3 * q^21 + 5 * q^22 - 2 * q^24 - 4 * q^26 - 2 * q^27 + 3 * q^28 + 8 * q^29 + 11 * q^31 + 2 * q^32 - 5 * q^33 - 2 * q^34 + 2 * q^36 + 16 * q^37 + 2 * q^38 + 4 * q^39 - 2 * q^41 - 3 * q^42 - 2 * q^43 + 5 * q^44 - 10 * q^47 - 2 * q^48 - 7 * q^49 + 2 * q^51 - 4 * q^52 + 7 * q^53 - 2 * q^54 + 3 * q^56 - 2 * q^57 + 8 * q^58 - 5 * q^59 + 10 * q^61 + 11 * q^62 + 3 * q^63 + 2 * q^64 - 5 * q^66 - 12 * q^67 - 2 * q^68 + 20 * q^71 + 2 * q^72 + 16 * q^73 + 16 * q^74 + 2 * q^76 + 10 * q^77 + 4 * q^78 - 9 * q^79 + 2 * q^81 - 2 * q^82 + 11 * q^83 - 3 * q^84 - 2 * q^86 - 8 * q^87 + 5 * q^88 + 16 * q^89 - 16 * q^91 - 11 * q^93 - 10 * q^94 - 2 * q^96 - 9 * q^97 - 7 * q^98 + 5 * q^99

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