LMFDB - Embedded newform 625.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.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:	$4.99065012633$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -0.618034 q^{7} -2.23607 q^{8} -2.00000 q^{9} -5.23607 q^{11} -0.618034 q^{12} -1.85410 q^{13} -1.00000 q^{14} -4.85410 q^{16} +5.23607 q^{17} -3.23607 q^{18} +0.854102 q^{19} +0.618034 q^{21} -8.47214 q^{22} -3.76393 q^{23} +2.23607 q^{24} -3.00000 q^{26} +5.00000 q^{27} -0.381966 q^{28} -3.61803 q^{29} -3.00000 q^{31} -3.38197 q^{32} +5.23607 q^{33} +8.47214 q^{34} -1.23607 q^{36} +0.236068 q^{37} +1.38197 q^{38} +1.85410 q^{39} -0.763932 q^{41} +1.00000 q^{42} +4.85410 q^{43} -3.23607 q^{44} -6.09017 q^{46} -0.618034 q^{47} +4.85410 q^{48} -6.61803 q^{49} -5.23607 q^{51} -1.14590 q^{52} +3.47214 q^{53} +8.09017 q^{54} +1.38197 q^{56} -0.854102 q^{57} -5.85410 q^{58} -10.8541 q^{59} +8.70820 q^{61} -4.85410 q^{62} +1.23607 q^{63} +4.23607 q^{64} +8.47214 q^{66} -4.76393 q^{67} +3.23607 q^{68} +3.76393 q^{69} -6.61803 q^{71} +4.47214 q^{72} +9.00000 q^{73} +0.381966 q^{74} +0.527864 q^{76} +3.23607 q^{77} +3.00000 q^{78} -8.09017 q^{79} +1.00000 q^{81} -1.23607 q^{82} +6.23607 q^{83} +0.381966 q^{84} +7.85410 q^{86} +3.61803 q^{87} +11.7082 q^{88} -8.94427 q^{89} +1.14590 q^{91} -2.32624 q^{92} +3.00000 q^{93} -1.00000 q^{94} +3.38197 q^{96} +3.85410 q^{97} -10.7082 q^{98} +10.4721 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} - 6 q^{11} + q^{12} + 3 q^{13} - 2 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{18} - 5 q^{19} - q^{21} - 8 q^{22} - 12 q^{23} - 6 q^{26} + 10 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + q^7 - 4 * q^9 - 6 * q^11 + q^12 + 3 * q^13 - 2 * q^14 - 3 * q^16 + 6 * q^17 - 2 * q^18 - 5 * q^19 - q^21 - 8 * q^22 - 12 * q^23 - 6 * q^26 + 10 * q^27 - 3 * q^28 - 5 * q^29 - 6 * q^31 - 9 * q^32 + 6 * q^33 + 8 * q^34 + 2 * q^36 - 4 * q^37 + 5 * q^38 - 3 * q^39 - 6 * q^41 + 2 * q^42 + 3 * q^43 - 2 * q^44 - q^46 + q^47 + 3 * q^48 - 11 * q^49 - 6 * q^51 - 9 * q^52 - 2 * q^53 + 5 * q^54 + 5 * q^56 + 5 * q^57 - 5 * q^58 - 15 * q^59 + 4 * q^61 - 3 * q^62 - 2 * q^63 + 4 * q^64 + 8 * q^66 - 14 * q^67 + 2 * q^68 + 12 * q^69 - 11 * q^71 + 18 * q^73 + 3 * q^74 + 10 * q^76 + 2 * q^77 + 6 * q^78 - 5 * q^79 + 2 * q^81 + 2 * q^82 + 8 * q^83 + 3 * q^84 + 9 * q^86 + 5 * q^87 + 10 * q^88 + 9 * q^91 + 11 * q^92 + 6 * q^93 - 2 * q^94 + 9 * q^96 + q^97 - 8 * q^98 + 12 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	−1.61803	−0.660560
$7$	−0.618034	−0.233595	−0.116797	−	0.993156i	$-0.537263\pi$
$7$	−0.618034	−0.233595	−0.116797	+	0.993156i	$0.537263\pi$
$8$	−2.23607	−0.790569
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	−0.618034	−0.178411
$13$	−1.85410	−0.514235	−0.257118	−	0.966380i	$-0.582773\pi$
$13$	−1.85410	−0.514235	−0.257118	+	0.966380i	$0.582773\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	−4.85410	−1.21353
$17$	5.23607	1.26993	0.634967	−	0.772540i	$-0.281014\pi$
$17$	5.23607	1.26993	0.634967	+	0.772540i	$0.281014\pi$
$18$	−3.23607	−0.762749
$19$	0.854102	0.195944	0.0979722	−	0.995189i	$-0.468764\pi$
$19$	0.854102	0.195944	0.0979722	+	0.995189i	$0.468764\pi$
$20$	0	0
$21$	0.618034	0.134866
$22$	−8.47214	−1.80627
$23$	−3.76393	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$23$	−3.76393	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−3.00000	−0.588348
$27$	5.00000	0.962250
$28$	−0.381966	−0.0721848
$29$	−3.61803	−0.671852	−0.335926	−	0.941888i	$-0.609049\pi$
$29$	−3.61803	−0.671852	−0.335926	+	0.941888i	$0.609049\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−3.38197	−0.597853
$33$	5.23607	0.911482
$34$	8.47214	1.45296
$35$	0	0
$36$	−1.23607	−0.206011
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	1.38197	0.224184
$39$	1.85410	0.296894
$40$	0	0
$41$	−0.763932	−0.119306	−0.0596531	−	0.998219i	$-0.518999\pi$
$41$	−0.763932	−0.119306	−0.0596531	+	0.998219i	$0.518999\pi$
$42$	1.00000	0.154303
$43$	4.85410	0.740244	0.370122	−	0.928983i	$-0.379316\pi$
$43$	4.85410	0.740244	0.370122	+	0.928983i	$0.379316\pi$
$44$	−3.23607	−0.487856
$45$	0	0
$46$	−6.09017	−0.897947
$47$	−0.618034	−0.0901495	−0.0450748	−	0.998984i	$-0.514353\pi$
$47$	−0.618034	−0.0901495	−0.0450748	+	0.998984i	$0.514353\pi$
$48$	4.85410	0.700629
$49$	−6.61803	−0.945433
$50$	0	0
$51$	−5.23607	−0.733196
$52$	−1.14590	−0.158907
$53$	3.47214	0.476935	0.238467	−	0.971151i	$-0.423355\pi$
$53$	3.47214	0.476935	0.238467	+	0.971151i	$0.423355\pi$
$54$	8.09017	1.10093
$55$	0	0
$56$	1.38197	0.184673
$57$	−0.854102	−0.113129
$58$	−5.85410	−0.768681
$59$	−10.8541	−1.41308	−0.706542	−	0.707671i	$-0.749746\pi$
$59$	−10.8541	−1.41308	−0.706542	+	0.707671i	$0.749746\pi$
$60$	0	0
$61$	8.70820	1.11497	0.557486	−	0.830187i	$-0.311766\pi$
$61$	8.70820	1.11497	0.557486	+	0.830187i	$0.311766\pi$
$62$	−4.85410	−0.616472
$63$	1.23607	0.155730
$64$	4.23607	0.529508
$65$	0	0
$66$	8.47214	1.04285
$67$	−4.76393	−0.582007	−0.291003	−	0.956722i	$-0.593989\pi$
$67$	−4.76393	−0.582007	−0.291003	+	0.956722i	$0.593989\pi$
$68$	3.23607	0.392431
$69$	3.76393	0.453124
$70$	0	0
$71$	−6.61803	−0.785416	−0.392708	−	0.919663i	$-0.628462\pi$
$71$	−6.61803	−0.785416	−0.392708	+	0.919663i	$0.628462\pi$
$72$	4.47214	0.527046
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0.381966	0.0444026
$75$	0	0
$76$	0.527864	0.0605502
$77$	3.23607	0.368784
$78$	3.00000	0.339683
$79$	−8.09017	−0.910215	−0.455108	−	0.890436i	$-0.650399\pi$
$79$	−8.09017	−0.910215	−0.455108	+	0.890436i	$0.650399\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.23607	−0.136501
$83$	6.23607	0.684497	0.342249	−	0.939609i	$-0.388811\pi$
$83$	6.23607	0.684497	0.342249	+	0.939609i	$0.388811\pi$
$84$	0.381966	0.0416759
$85$	0	0
$86$	7.85410	0.846930
$87$	3.61803	0.387894
$88$	11.7082	1.24810
$89$	−8.94427	−0.948091	−0.474045	−	0.880500i	$-0.657207\pi$
$89$	−8.94427	−0.948091	−0.474045	+	0.880500i	$0.657207\pi$
$90$	0	0
$91$	1.14590	0.120123
$92$	−2.32624	−0.242527
$93$	3.00000	0.311086
$94$	−1.00000	−0.103142
$95$	0	0
$96$	3.38197	0.345170
$97$	3.85410	0.391325	0.195662	−	0.980671i	$-0.437314\pi$
$97$	3.85410	0.391325	0.195662	+	0.980671i	$0.437314\pi$
$98$	−10.7082	−1.08169
$99$	10.4721	1.05249
$100$	0	0
$101$	1.47214	0.146483	0.0732415	−	0.997314i	$-0.476666\pi$
$101$	1.47214	0.146483	0.0732415	+	0.997314i	$0.476666\pi$
$102$	−8.47214	−0.838866
$103$	−8.56231	−0.843669	−0.421835	−	0.906673i	$-0.638614\pi$
$103$	−8.56231	−0.843669	−0.421835	+	0.906673i	$0.638614\pi$
$104$	4.14590	0.406539
$105$	0	0
$106$	5.61803	0.545672
$107$	16.4164	1.58703	0.793517	−	0.608548i	$-0.208248\pi$
$107$	16.4164	1.58703	0.793517	+	0.608548i	$0.208248\pi$
$108$	3.09017	0.297352
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−0.236068	−0.0224066
$112$	3.00000	0.283473
$113$	−16.8541	−1.58550	−0.792750	−	0.609547i	$-0.791352\pi$
$113$	−16.8541	−1.58550	−0.792750	+	0.609547i	$0.791352\pi$
$114$	−1.38197	−0.129433
$115$	0	0
$116$	−2.23607	−0.207614
$117$	3.70820	0.342824
$118$	−17.5623	−1.61674
$119$	−3.23607	−0.296650
$120$	0	0
$121$	16.4164	1.49240
$122$	14.0902	1.27566
$123$	0.763932	0.0688814
$124$	−1.85410	−0.166503
$125$	0	0
$126$	2.00000	0.178174
$127$	−19.8885	−1.76482	−0.882411	−	0.470479i	$-0.844081\pi$
$127$	−19.8885	−1.76482	−0.882411	+	0.470479i	$0.844081\pi$
$128$	13.6180	1.20368
$129$	−4.85410	−0.427380
$130$	0	0
$131$	6.79837	0.593977	0.296988	−	0.954881i	$-0.404018\pi$
$131$	6.79837	0.593977	0.296988	+	0.954881i	$0.404018\pi$
$132$	3.23607	0.281664
$133$	−0.527864	−0.0457716
$134$	−7.70820	−0.665887
$135$	0	0
$136$	−11.7082	−1.00397
$137$	11.9443	1.02047	0.510234	−	0.860036i	$-0.329559\pi$
$137$	11.9443	1.02047	0.510234	+	0.860036i	$0.329559\pi$
$138$	6.09017	0.518430
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	0.618034	0.0520479
$142$	−10.7082	−0.898613
$143$	9.70820	0.811841
$144$	9.70820	0.809017
$145$	0	0
$146$	14.5623	1.20519
$147$	6.61803	0.545846
$148$	0.145898	0.0119927
$149$	−3.94427	−0.323127	−0.161564	−	0.986862i	$-0.551654\pi$
$149$	−3.94427	−0.323127	−0.161564	+	0.986862i	$0.551654\pi$
$150$	0	0
$151$	14.5623	1.18506	0.592532	−	0.805547i	$-0.298128\pi$
$151$	14.5623	1.18506	0.592532	+	0.805547i	$0.298128\pi$
$152$	−1.90983	−0.154908
$153$	−10.4721	−0.846622
$154$	5.23607	0.421934
$155$	0	0
$156$	1.14590	0.0917453
$157$	−13.1803	−1.05191	−0.525953	−	0.850514i	$-0.676291\pi$
$157$	−13.1803	−1.05191	−0.525953	+	0.850514i	$0.676291\pi$
$158$	−13.0902	−1.04140
$159$	−3.47214	−0.275358
$160$	0	0
$161$	2.32624	0.183333
$162$	1.61803	0.127125
$163$	−11.0000	−0.861586	−0.430793	−	0.902451i	$-0.641766\pi$
$163$	−11.0000	−0.861586	−0.430793	+	0.902451i	$0.641766\pi$
$164$	−0.472136	−0.0368676
$165$	0	0
$166$	10.0902	0.783149
$167$	−14.5623	−1.12687	−0.563433	−	0.826162i	$-0.690520\pi$
$167$	−14.5623	−1.12687	−0.563433	+	0.826162i	$0.690520\pi$
$168$	−1.38197	−0.106621
$169$	−9.56231	−0.735562
$170$	0	0
$171$	−1.70820	−0.130630
$172$	3.00000	0.228748
$173$	−18.8885	−1.43607	−0.718035	−	0.696007i	$-0.754958\pi$
$173$	−18.8885	−1.43607	−0.718035	+	0.696007i	$0.754958\pi$
$174$	5.85410	0.443798
$175$	0	0
$176$	25.4164	1.91583
$177$	10.8541	0.815844
$178$	−14.4721	−1.08473
$179$	−0.527864	−0.0394544	−0.0197272	−	0.999805i	$-0.506280\pi$
$179$	−0.527864	−0.0394544	−0.0197272	+	0.999805i	$0.506280\pi$
$180$	0	0
$181$	0.291796	0.0216890	0.0108445	−	0.999941i	$-0.496548\pi$
$181$	0.291796	0.0216890	0.0108445	+	0.999941i	$0.496548\pi$
$182$	1.85410	0.137435
$183$	−8.70820	−0.643729
$184$	8.41641	0.620466
$185$	0	0
$186$	4.85410	0.355920
$187$	−27.4164	−2.00489
$188$	−0.381966	−0.0278577
$189$	−3.09017	−0.224777
$190$	0	0
$191$	−1.81966	−0.131666	−0.0658330	−	0.997831i	$-0.520970\pi$
$191$	−1.81966	−0.131666	−0.0658330	+	0.997831i	$0.520970\pi$
$192$	−4.23607	−0.305712
$193$	−7.70820	−0.554849	−0.277424	−	0.960747i	$-0.589481\pi$
$193$	−7.70820	−0.554849	−0.277424	+	0.960747i	$0.589481\pi$
$194$	6.23607	0.447724
$195$	0	0
$196$	−4.09017	−0.292155
$197$	−3.70820	−0.264199	−0.132099	−	0.991236i	$-0.542172\pi$
$197$	−3.70820	−0.264199	−0.132099	+	0.991236i	$0.542172\pi$
$198$	16.9443	1.20418
$199$	−17.5623	−1.24496	−0.622479	−	0.782636i	$-0.713875\pi$
$199$	−17.5623	−1.24496	−0.622479	+	0.782636i	$0.713875\pi$
$200$	0	0
$201$	4.76393	0.336022
$202$	2.38197	0.167595
$203$	2.23607	0.156941
$204$	−3.23607	−0.226570
$205$	0	0
$206$	−13.8541	−0.965261
$207$	7.52786	0.523223
$208$	9.00000	0.624038
$209$	−4.47214	−0.309344
$210$	0	0
$211$	−9.18034	−0.632001	−0.316000	−	0.948759i	$-0.602340\pi$
$211$	−9.18034	−0.632001	−0.316000	+	0.948759i	$0.602340\pi$
$212$	2.14590	0.147381
$213$	6.61803	0.453460
$214$	26.5623	1.81576
$215$	0	0
$216$	−11.1803	−0.760726
$217$	1.85410	0.125865
$218$	16.1803	1.09587
$219$	−9.00000	−0.608164
$220$	0	0
$221$	−9.70820	−0.653044
$222$	−0.381966	−0.0256359
$223$	0.180340	0.0120765	0.00603823	−	0.999982i	$-0.498078\pi$
$223$	0.180340	0.0120765	0.00603823	+	0.999982i	$0.498078\pi$
$224$	2.09017	0.139655
$225$	0	0
$226$	−27.2705	−1.81401
$227$	−14.7639	−0.979917	−0.489958	−	0.871746i	$-0.662988\pi$
$227$	−14.7639	−0.979917	−0.489958	+	0.871746i	$0.662988\pi$
$228$	−0.527864	−0.0349587
$229$	21.7082	1.43452	0.717259	−	0.696806i	$-0.245396\pi$
$229$	21.7082	1.43452	0.717259	+	0.696806i	$0.245396\pi$
$230$	0	0
$231$	−3.23607	−0.212918
$232$	8.09017	0.531146
$233$	2.94427	0.192886	0.0964428	−	0.995339i	$-0.469254\pi$
$233$	2.94427	0.192886	0.0964428	+	0.995339i	$0.469254\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	−6.70820	−0.436667
$237$	8.09017	0.525513
$238$	−5.23607	−0.339404
$239$	−20.5279	−1.32784	−0.663919	−	0.747805i	$-0.731108\pi$
$239$	−20.5279	−1.32784	−0.663919	+	0.747805i	$0.731108\pi$
$240$	0	0
$241$	2.52786	0.162834	0.0814170	−	0.996680i	$-0.474055\pi$
$241$	2.52786	0.162834	0.0814170	+	0.996680i	$0.474055\pi$
$242$	26.5623	1.70749
$243$	−16.0000	−1.02640
$244$	5.38197	0.344545
$245$	0	0
$246$	1.23607	0.0788088
$247$	−1.58359	−0.100762
$248$	6.70820	0.425971
$249$	−6.23607	−0.395195
$250$	0	0
$251$	−29.1803	−1.84185	−0.920923	−	0.389744i	$-0.872564\pi$
$251$	−29.1803	−1.84185	−0.920923	+	0.389744i	$0.872564\pi$
$252$	0.763932	0.0481232
$253$	19.7082	1.23904
$254$	−32.1803	−2.01917
$255$	0	0
$256$	13.5623	0.847644
$257$	−22.8541	−1.42560	−0.712800	−	0.701367i	$-0.752573\pi$
$257$	−22.8541	−1.42560	−0.712800	+	0.701367i	$0.752573\pi$
$258$	−7.85410	−0.488975
$259$	−0.145898	−0.00906566
$260$	0	0
$261$	7.23607	0.447901
$262$	11.0000	0.679582
$263$	10.9098	0.672729	0.336364	−	0.941732i	$-0.390803\pi$
$263$	10.9098	0.672729	0.336364	+	0.941732i	$0.390803\pi$
$264$	−11.7082	−0.720590
$265$	0	0
$266$	−0.854102	−0.0523684
$267$	8.94427	0.547381
$268$	−2.94427	−0.179850
$269$	−12.7639	−0.778231	−0.389115	−	0.921189i	$-0.627219\pi$
$269$	−12.7639	−0.778231	−0.389115	+	0.921189i	$0.627219\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−25.4164	−1.54110
$273$	−1.14590	−0.0693529
$274$	19.3262	1.16754
$275$	0	0
$276$	2.32624	0.140023
$277$	24.7082	1.48457	0.742286	−	0.670083i	$-0.233742\pi$
$277$	24.7082	1.48457	0.742286	+	0.670083i	$0.233742\pi$
$278$	8.09017	0.485216
$279$	6.00000	0.359211
$280$	0	0
$281$	10.0902	0.601929	0.300965	−	0.953635i	$-0.402691\pi$
$281$	10.0902	0.601929	0.300965	+	0.953635i	$0.402691\pi$
$282$	1.00000	0.0595491
$283$	29.8541	1.77464	0.887321	−	0.461152i	$-0.152564\pi$
$283$	29.8541	1.77464	0.887321	+	0.461152i	$0.152564\pi$
$284$	−4.09017	−0.242707
$285$	0	0
$286$	15.7082	0.928846
$287$	0.472136	0.0278693
$288$	6.76393	0.398569
$289$	10.4164	0.612730
$290$	0	0
$291$	−3.85410	−0.225931
$292$	5.56231	0.325509
$293$	19.5279	1.14083	0.570415	−	0.821357i	$-0.306782\pi$
$293$	19.5279	1.14083	0.570415	+	0.821357i	$0.306782\pi$
$294$	10.7082	0.624515
$295$	0	0
$296$	−0.527864	−0.0306815
$297$	−26.1803	−1.51914
$298$	−6.38197	−0.369697
$299$	6.97871	0.403589
$300$	0	0
$301$	−3.00000	−0.172917
$302$	23.5623	1.35586
$303$	−1.47214	−0.0845720
$304$	−4.14590	−0.237784
$305$	0	0
$306$	−16.9443	−0.968640
$307$	−9.23607	−0.527130	−0.263565	−	0.964642i	$-0.584898\pi$
$307$	−9.23607	−0.527130	−0.263565	+	0.964642i	$0.584898\pi$
$308$	2.00000	0.113961
$309$	8.56231	0.487093
$310$	0	0
$311$	8.50658	0.482364	0.241182	−	0.970480i	$-0.422465\pi$
$311$	8.50658	0.482364	0.241182	+	0.970480i	$0.422465\pi$
$312$	−4.14590	−0.234715
$313$	16.7639	0.947553	0.473777	−	0.880645i	$-0.342890\pi$
$313$	16.7639	0.947553	0.473777	+	0.880645i	$0.342890\pi$
$314$	−21.3262	−1.20351
$315$	0	0
$316$	−5.00000	−0.281272
$317$	−7.65248	−0.429806	−0.214903	−	0.976635i	$-0.568944\pi$
$317$	−7.65248	−0.429806	−0.214903	+	0.976635i	$0.568944\pi$
$318$	−5.61803	−0.315044
$319$	18.9443	1.06068
$320$	0	0
$321$	−16.4164	−0.916275
$322$	3.76393	0.209756
$323$	4.47214	0.248836
$324$	0.618034	0.0343352
$325$	0	0
$326$	−17.7984	−0.985761
$327$	−10.0000	−0.553001
$328$	1.70820	0.0943198
$329$	0.381966	0.0210585
$330$	0	0
$331$	−23.1246	−1.27104	−0.635522	−	0.772083i	$-0.719215\pi$
$331$	−23.1246	−1.27104	−0.635522	+	0.772083i	$0.719215\pi$
$332$	3.85410	0.211521
$333$	−0.472136	−0.0258729
$334$	−23.5623	−1.28927
$335$	0	0
$336$	−3.00000	−0.163663
$337$	−7.85410	−0.427840	−0.213920	−	0.976851i	$-0.568623\pi$
$337$	−7.85410	−0.427840	−0.213920	+	0.976851i	$0.568623\pi$
$338$	−15.4721	−0.841573
$339$	16.8541	0.915389
$340$	0	0
$341$	15.7082	0.850647
$342$	−2.76393	−0.149456
$343$	8.41641	0.454443
$344$	−10.8541	−0.585214
$345$	0	0
$346$	−30.5623	−1.64304
$347$	19.9098	1.06882	0.534408	−	0.845227i	$-0.320535\pi$
$347$	19.9098	1.06882	0.534408	+	0.845227i	$0.320535\pi$
$348$	2.23607	0.119866
$349$	21.7082	1.16201	0.581007	−	0.813899i	$-0.302659\pi$
$349$	21.7082	1.16201	0.581007	+	0.813899i	$0.302659\pi$
$350$	0	0
$351$	−9.27051	−0.494823
$352$	17.7082	0.943850
$353$	−12.9098	−0.687121	−0.343560	−	0.939131i	$-0.611633\pi$
$353$	−12.9098	−0.687121	−0.343560	+	0.939131i	$0.611633\pi$
$354$	17.5623	0.933426
$355$	0	0
$356$	−5.52786	−0.292976
$357$	3.23607	0.171271
$358$	−0.854102	−0.0451407
$359$	13.7426	0.725309	0.362655	−	0.931924i	$-0.381870\pi$
$359$	13.7426	0.725309	0.362655	+	0.931924i	$0.381870\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	0.472136	0.0248149
$363$	−16.4164	−0.861638
$364$	0.708204	0.0371200
$365$	0	0
$366$	−14.0902	−0.736505
$367$	25.5623	1.33434	0.667171	−	0.744905i	$-0.267505\pi$
$367$	25.5623	1.33434	0.667171	+	0.744905i	$0.267505\pi$
$368$	18.2705	0.952416
$369$	1.52786	0.0795374
$370$	0	0
$371$	−2.14590	−0.111409
$372$	1.85410	0.0961307
$373$	28.2705	1.46379	0.731896	−	0.681417i	$-0.238636\pi$
$373$	28.2705	1.46379	0.731896	+	0.681417i	$0.238636\pi$
$374$	−44.3607	−2.29384
$375$	0	0
$376$	1.38197	0.0712695
$377$	6.70820	0.345490
$378$	−5.00000	−0.257172
$379$	14.5967	0.749785	0.374892	−	0.927068i	$-0.377680\pi$
$379$	14.5967	0.749785	0.374892	+	0.927068i	$0.377680\pi$
$380$	0	0
$381$	19.8885	1.01892
$382$	−2.94427	−0.150642
$383$	−33.3607	−1.70465	−0.852326	−	0.523012i	$-0.824808\pi$
$383$	−33.3607	−1.70465	−0.852326	+	0.523012i	$0.824808\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	−12.4721	−0.634815
$387$	−9.70820	−0.493496
$388$	2.38197	0.120926
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	−19.7082	−0.996687
$392$	14.7984	0.747431
$393$	−6.79837	−0.342933
$394$	−6.00000	−0.302276
$395$	0	0
$396$	6.47214	0.325237
$397$	−29.0344	−1.45720	−0.728598	−	0.684941i	$-0.759828\pi$
$397$	−29.0344	−1.45720	−0.728598	+	0.684941i	$0.759828\pi$
$398$	−28.4164	−1.42439
$399$	0.527864	0.0264263
$400$	0	0
$401$	26.5967	1.32818	0.664089	−	0.747653i	$-0.268820\pi$
$401$	26.5967	1.32818	0.664089	+	0.747653i	$0.268820\pi$
$402$	7.70820	0.384450
$403$	5.56231	0.277078
$404$	0.909830	0.0452657
$405$	0	0
$406$	3.61803	0.179560
$407$	−1.23607	−0.0612696
$408$	11.7082	0.579642
$409$	1.58359	0.0783036	0.0391518	−	0.999233i	$-0.487534\pi$
$409$	1.58359	0.0783036	0.0391518	+	0.999233i	$0.487534\pi$
$410$	0	0
$411$	−11.9443	−0.589167
$412$	−5.29180	−0.260708
$413$	6.70820	0.330089
$414$	12.1803	0.598631
$415$	0	0
$416$	6.27051	0.307437
$417$	−5.00000	−0.244851
$418$	−7.23607	−0.353928
$419$	−9.47214	−0.462744	−0.231372	−	0.972865i	$-0.574321\pi$
$419$	−9.47214	−0.462744	−0.231372	+	0.972865i	$0.574321\pi$
$420$	0	0
$421$	32.0000	1.55958	0.779792	−	0.626038i	$-0.215325\pi$
$421$	32.0000	1.55958	0.779792	+	0.626038i	$0.215325\pi$
$422$	−14.8541	−0.723086
$423$	1.23607	0.0600997
$424$	−7.76393	−0.377050
$425$	0	0
$426$	10.7082	0.518814
$427$	−5.38197	−0.260452
$428$	10.1459	0.490420
$429$	−9.70820	−0.468717
$430$	0	0
$431$	−29.8328	−1.43700	−0.718498	−	0.695529i	$-0.755170\pi$
$431$	−29.8328	−1.43700	−0.718498	+	0.695529i	$0.755170\pi$
$432$	−24.2705	−1.16772
$433$	−26.8541	−1.29053	−0.645263	−	0.763961i	$-0.723252\pi$
$433$	−26.8541	−1.29053	−0.645263	+	0.763961i	$0.723252\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	6.18034	0.295985
$437$	−3.21478	−0.153784
$438$	−14.5623	−0.695814
$439$	−40.9787	−1.95581	−0.977904	−	0.209056i	$-0.932961\pi$
$439$	−40.9787	−1.95581	−0.977904	+	0.209056i	$0.932961\pi$
$440$	0	0
$441$	13.2361	0.630289
$442$	−15.7082	−0.747163
$443$	−29.9443	−1.42270	−0.711348	−	0.702840i	$-0.751915\pi$
$443$	−29.9443	−1.42270	−0.711348	+	0.702840i	$0.751915\pi$
$444$	−0.145898	−0.00692401
$445$	0	0
$446$	0.291796	0.0138169
$447$	3.94427	0.186558
$448$	−2.61803	−0.123690
$449$	4.67376	0.220568	0.110284	−	0.993900i	$-0.464824\pi$
$449$	4.67376	0.220568	0.110284	+	0.993900i	$0.464824\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	−10.4164	−0.489947
$453$	−14.5623	−0.684197
$454$	−23.8885	−1.12114
$455$	0	0
$456$	1.90983	0.0894360
$457$	21.4164	1.00182	0.500909	−	0.865500i	$-0.332999\pi$
$457$	21.4164	1.00182	0.500909	+	0.865500i	$0.332999\pi$
$458$	35.1246	1.64127
$459$	26.1803	1.22199
$460$	0	0
$461$	0.819660	0.0381754	0.0190877	−	0.999818i	$-0.493924\pi$
$461$	0.819660	0.0381754	0.0190877	+	0.999818i	$0.493924\pi$
$462$	−5.23607	−0.243604
$463$	24.1246	1.12117	0.560583	−	0.828098i	$-0.310577\pi$
$463$	24.1246	1.12117	0.560583	+	0.828098i	$0.310577\pi$
$464$	17.5623	0.815310
$465$	0	0
$466$	4.76393	0.220685
$467$	−27.4508	−1.27027	−0.635137	−	0.772400i	$-0.719056\pi$
$467$	−27.4508	−1.27027	−0.635137	+	0.772400i	$0.719056\pi$
$468$	2.29180	0.105938
$469$	2.94427	0.135954
$470$	0	0
$471$	13.1803	0.607318
$472$	24.2705	1.11714
$473$	−25.4164	−1.16865
$474$	13.0902	0.601251
$475$	0	0
$476$	−2.00000	−0.0916698
$477$	−6.94427	−0.317956
$478$	−33.2148	−1.51921
$479$	−10.8541	−0.495937	−0.247968	−	0.968768i	$-0.579763\pi$
$479$	−10.8541	−0.495937	−0.247968	+	0.968768i	$0.579763\pi$
$480$	0	0
$481$	−0.437694	−0.0199571
$482$	4.09017	0.186302
$483$	−2.32624	−0.105847
$484$	10.1459	0.461177
$485$	0	0
$486$	−25.8885	−1.17433
$487$	36.4164	1.65018	0.825092	−	0.564998i	$-0.191123\pi$
$487$	36.4164	1.65018	0.825092	+	0.564998i	$0.191123\pi$
$488$	−19.4721	−0.881462
$489$	11.0000	0.497437
$490$	0	0
$491$	−43.2492	−1.95181	−0.975905	−	0.218196i	$-0.929983\pi$
$491$	−43.2492	−1.95181	−0.975905	+	0.218196i	$0.929983\pi$
$492$	0.472136	0.0212855
$493$	−18.9443	−0.853207
$494$	−2.56231	−0.115284
$495$	0	0
$496$	14.5623	0.653867
$497$	4.09017	0.183469
$498$	−10.0902	−0.452151
$499$	7.56231	0.338535	0.169268	−	0.985570i	$-0.445860\pi$
$499$	7.56231	0.338535	0.169268	+	0.985570i	$0.445860\pi$
$500$	0	0
$501$	14.5623	0.650596
$502$	−47.2148	−2.10730
$503$	37.4164	1.66832	0.834158	−	0.551526i	$-0.185954\pi$
$503$	37.4164	1.66832	0.834158	+	0.551526i	$0.185954\pi$
$504$	−2.76393	−0.123115
$505$	0	0
$506$	31.8885	1.41762
$507$	9.56231	0.424677
$508$	−12.2918	−0.545360
$509$	−20.3262	−0.900945	−0.450472	−	0.892790i	$-0.648744\pi$
$509$	−20.3262	−0.900945	−0.450472	+	0.892790i	$0.648744\pi$
$510$	0	0
$511$	−5.56231	−0.246062
$512$	−5.29180	−0.233867
$513$	4.27051	0.188548
$514$	−36.9787	−1.63106
$515$	0	0
$516$	−3.00000	−0.132068
$517$	3.23607	0.142322
$518$	−0.236068	−0.0103722
$519$	18.8885	0.829115
$520$	0	0
$521$	29.3607	1.28631	0.643157	−	0.765734i	$-0.277624\pi$
$521$	29.3607	1.28631	0.643157	+	0.765734i	$0.277624\pi$
$522$	11.7082	0.512454
$523$	13.1459	0.574830	0.287415	−	0.957806i	$-0.407204\pi$
$523$	13.1459	0.574830	0.287415	+	0.957806i	$0.407204\pi$
$524$	4.20163	0.183549
$525$	0	0
$526$	17.6525	0.769685
$527$	−15.7082	−0.684260
$528$	−25.4164	−1.10611
$529$	−8.83282	−0.384035
$530$	0	0
$531$	21.7082	0.942056
$532$	−0.326238	−0.0141442
$533$	1.41641	0.0613514
$534$	14.4721	0.626271
$535$	0	0
$536$	10.6525	0.460117
$537$	0.527864	0.0227790
$538$	−20.6525	−0.890391
$539$	34.6525	1.49259
$540$	0	0
$541$	27.1246	1.16618	0.583089	−	0.812408i	$-0.301844\pi$
$541$	27.1246	1.16618	0.583089	+	0.812408i	$0.301844\pi$
$542$	−12.9443	−0.556004
$543$	−0.291796	−0.0125222
$544$	−17.7082	−0.759233
$545$	0	0
$546$	−1.85410	−0.0793482
$547$	21.2918	0.910371	0.455186	−	0.890397i	$-0.349573\pi$
$547$	21.2918	0.910371	0.455186	+	0.890397i	$0.349573\pi$
$548$	7.38197	0.315342
$549$	−17.4164	−0.743314
$550$	0	0
$551$	−3.09017	−0.131646
$552$	−8.41641	−0.358226
$553$	5.00000	0.212622
$554$	39.9787	1.69853
$555$	0	0
$556$	3.09017	0.131052
$557$	−4.76393	−0.201854	−0.100927	−	0.994894i	$-0.532181\pi$
$557$	−4.76393	−0.201854	−0.100927	+	0.994894i	$0.532181\pi$
$558$	9.70820	0.410981
$559$	−9.00000	−0.380659
$560$	0	0
$561$	27.4164	1.15752
$562$	16.3262	0.688681
$563$	−7.38197	−0.311113	−0.155556	−	0.987827i	$-0.549717\pi$
$563$	−7.38197	−0.311113	−0.155556	+	0.987827i	$0.549717\pi$
$564$	0.381966	0.0160837
$565$	0	0
$566$	48.3050	2.03041
$567$	−0.618034	−0.0259550
$568$	14.7984	0.620926
$569$	20.5279	0.860573	0.430286	−	0.902692i	$-0.358413\pi$
$569$	20.5279	0.860573	0.430286	+	0.902692i	$0.358413\pi$
$570$	0	0
$571$	−8.12461	−0.340004	−0.170002	−	0.985444i	$-0.554377\pi$
$571$	−8.12461	−0.340004	−0.170002	+	0.985444i	$0.554377\pi$
$572$	6.00000	0.250873
$573$	1.81966	0.0760174
$574$	0.763932	0.0318859
$575$	0	0
$576$	−8.47214	−0.353006
$577$	33.7771	1.40616	0.703079	−	0.711111i	$-0.251808\pi$
$577$	33.7771	1.40616	0.703079	+	0.711111i	$0.251808\pi$
$578$	16.8541	0.701038
$579$	7.70820	0.320342
$580$	0	0
$581$	−3.85410	−0.159895
$582$	−6.23607	−0.258493
$583$	−18.1803	−0.752953
$584$	−20.1246	−0.832762
$585$	0	0
$586$	31.5967	1.30525
$587$	−5.29180	−0.218416	−0.109208	−	0.994019i	$-0.534831\pi$
$587$	−5.29180	−0.218416	−0.109208	+	0.994019i	$0.534831\pi$
$588$	4.09017	0.168676
$589$	−2.56231	−0.105578
$590$	0	0
$591$	3.70820	0.152535
$592$	−1.14590	−0.0470961
$593$	10.9098	0.448013	0.224007	−	0.974588i	$-0.428086\pi$
$593$	10.9098	0.448013	0.224007	+	0.974588i	$0.428086\pi$
$594$	−42.3607	−1.73808
$595$	0	0
$596$	−2.43769	−0.0998518
$597$	17.5623	0.718777
$598$	11.2918	0.461756
$599$	−9.47214	−0.387021	−0.193510	−	0.981098i	$-0.561987\pi$
$599$	−9.47214	−0.387021	−0.193510	+	0.981098i	$0.561987\pi$
$600$	0	0
$601$	2.72949	0.111338	0.0556691	−	0.998449i	$-0.482271\pi$
$601$	2.72949	0.111338	0.0556691	+	0.998449i	$0.482271\pi$
$602$	−4.85410	−0.197838
$603$	9.52786	0.388005
$604$	9.00000	0.366205
$605$	0	0
$606$	−2.38197	−0.0967608
$607$	35.5623	1.44343	0.721715	−	0.692191i	$-0.243354\pi$
$607$	35.5623	1.44343	0.721715	+	0.692191i	$0.243354\pi$
$608$	−2.88854	−0.117146
$609$	−2.23607	−0.0906100
$610$	0	0
$611$	1.14590	0.0463581
$612$	−6.47214	−0.261621
$613$	14.9787	0.604985	0.302492	−	0.953152i	$-0.402181\pi$
$613$	14.9787	0.604985	0.302492	+	0.953152i	$0.402181\pi$
$614$	−14.9443	−0.603102
$615$	0	0
$616$	−7.23607	−0.291549
$617$	−14.2361	−0.573123	−0.286561	−	0.958062i	$-0.592512\pi$
$617$	−14.2361	−0.573123	−0.286561	+	0.958062i	$0.592512\pi$
$618$	13.8541	0.557294
$619$	30.5279	1.22702	0.613509	−	0.789688i	$-0.289757\pi$
$619$	30.5279	1.22702	0.613509	+	0.789688i	$0.289757\pi$
$620$	0	0
$621$	−18.8197	−0.755207
$622$	13.7639	0.551883
$623$	5.52786	0.221469
$624$	−9.00000	−0.360288
$625$	0	0
$626$	27.1246	1.08412
$627$	4.47214	0.178600
$628$	−8.14590	−0.325057
$629$	1.23607	0.0492853
$630$	0	0
$631$	−10.2361	−0.407491	−0.203746	−	0.979024i	$-0.565312\pi$
$631$	−10.2361	−0.407491	−0.203746	+	0.979024i	$0.565312\pi$
$632$	18.0902	0.719588
$633$	9.18034	0.364886
$634$	−12.3820	−0.491751
$635$	0	0
$636$	−2.14590	−0.0850904
$637$	12.2705	0.486175
$638$	30.6525	1.21354
$639$	13.2361	0.523611
$640$	0	0
$641$	−1.09017	−0.0430591	−0.0215296	−	0.999768i	$-0.506854\pi$
$641$	−1.09017	−0.0430591	−0.0215296	+	0.999768i	$0.506854\pi$
$642$	−26.5623	−1.04833
$643$	30.8328	1.21593	0.607964	−	0.793965i	$-0.291987\pi$
$643$	30.8328	1.21593	0.607964	+	0.793965i	$0.291987\pi$
$644$	1.43769	0.0566531
$645$	0	0
$646$	7.23607	0.284699
$647$	36.5410	1.43658	0.718288	−	0.695746i	$-0.244926\pi$
$647$	36.5410	1.43658	0.718288	+	0.695746i	$0.244926\pi$
$648$	−2.23607	−0.0878410
$649$	56.8328	2.23088
$650$	0	0
$651$	−1.85410	−0.0726680
$652$	−6.79837	−0.266245
$653$	−19.0902	−0.747056	−0.373528	−	0.927619i	$-0.621852\pi$
$653$	−19.0902	−0.747056	−0.373528	+	0.927619i	$0.621852\pi$
$654$	−16.1803	−0.632701
$655$	0	0
$656$	3.70820	0.144781
$657$	−18.0000	−0.702247
$658$	0.618034	0.0240935
$659$	15.5279	0.604880	0.302440	−	0.953168i	$-0.402199\pi$
$659$	15.5279	0.604880	0.302440	+	0.953168i	$0.402199\pi$
$660$	0	0
$661$	19.6869	0.765732	0.382866	−	0.923804i	$-0.374937\pi$
$661$	19.6869	0.765732	0.382866	+	0.923804i	$0.374937\pi$
$662$	−37.4164	−1.45423
$663$	9.70820	0.377035
$664$	−13.9443	−0.541143
$665$	0	0
$666$	−0.763932	−0.0296018
$667$	13.6180	0.527292
$668$	−9.00000	−0.348220
$669$	−0.180340	−0.00697234
$670$	0	0
$671$	−45.5967	−1.76024
$672$	−2.09017	−0.0806301
$673$	−12.1803	−0.469518	−0.234759	−	0.972054i	$-0.575430\pi$
$673$	−12.1803	−0.469518	−0.234759	+	0.972054i	$0.575430\pi$
$674$	−12.7082	−0.489502
$675$	0	0
$676$	−5.90983	−0.227301
$677$	−10.6180	−0.408084	−0.204042	−	0.978962i	$-0.565408\pi$
$677$	−10.6180	−0.408084	−0.204042	+	0.978962i	$0.565408\pi$
$678$	27.2705	1.04732
$679$	−2.38197	−0.0914115
$680$	0	0
$681$	14.7639	0.565755
$682$	25.4164	0.973245
$683$	13.4721	0.515497	0.257748	−	0.966212i	$-0.417019\pi$
$683$	13.4721	0.515497	0.257748	+	0.966212i	$0.417019\pi$
$684$	−1.05573	−0.0403668
$685$	0	0
$686$	13.6180	0.519939
$687$	−21.7082	−0.828220
$688$	−23.5623	−0.898304
$689$	−6.43769	−0.245257
$690$	0	0
$691$	36.2705	1.37980	0.689898	−	0.723907i	$-0.257656\pi$
$691$	36.2705	1.37980	0.689898	+	0.723907i	$0.257656\pi$
$692$	−11.6738	−0.443770
$693$	−6.47214	−0.245856
$694$	32.2148	1.22286
$695$	0	0
$696$	−8.09017	−0.306657
$697$	−4.00000	−0.151511
$698$	35.1246	1.32949
$699$	−2.94427	−0.111363
$700$	0	0
$701$	−41.0132	−1.54905	−0.774523	−	0.632546i	$-0.782010\pi$
$701$	−41.0132	−1.54905	−0.774523	+	0.632546i	$0.782010\pi$
$702$	−15.0000	−0.566139
$703$	0.201626	0.00760447
$704$	−22.1803	−0.835953
$705$	0	0
$706$	−20.8885	−0.786151
$707$	−0.909830	−0.0342177
$708$	6.70820	0.252110
$709$	−33.5410	−1.25966	−0.629830	−	0.776733i	$-0.716875\pi$
$709$	−33.5410	−1.25966	−0.629830	+	0.776733i	$0.716875\pi$
$710$	0	0
$711$	16.1803	0.606810
$712$	20.0000	0.749532
$713$	11.2918	0.422881
$714$	5.23607	0.195955
$715$	0	0
$716$	−0.326238	−0.0121921
$717$	20.5279	0.766627
$718$	22.2361	0.829843
$719$	−23.2918	−0.868637	−0.434319	−	0.900759i	$-0.643011\pi$
$719$	−23.2918	−0.868637	−0.434319	+	0.900759i	$0.643011\pi$
$720$	0	0
$721$	5.29180	0.197077
$722$	−29.5623	−1.10020
$723$	−2.52786	−0.0940123
$724$	0.180340	0.00670228
$725$	0	0
$726$	−26.5623	−0.985820
$727$	−24.5623	−0.910966	−0.455483	−	0.890245i	$-0.650533\pi$
$727$	−24.5623	−0.910966	−0.455483	+	0.890245i	$0.650533\pi$
$728$	−2.56231	−0.0949654
$729$	13.0000	0.481481
$730$	0	0
$731$	25.4164	0.940060
$732$	−5.38197	−0.198923
$733$	19.9787	0.737931	0.368965	−	0.929443i	$-0.379712\pi$
$733$	19.9787	0.737931	0.368965	+	0.929443i	$0.379712\pi$
$734$	41.3607	1.52665
$735$	0	0
$736$	12.7295	0.469215
$737$	24.9443	0.918834
$738$	2.47214	0.0910006
$739$	15.9787	0.587786	0.293893	−	0.955838i	$-0.405049\pi$
$739$	15.9787	0.587786	0.293893	+	0.955838i	$0.405049\pi$
$740$	0	0
$741$	1.58359	0.0581747
$742$	−3.47214	−0.127466
$743$	−28.3607	−1.04045	−0.520226	−	0.854029i	$-0.674152\pi$
$743$	−28.3607	−1.04045	−0.520226	+	0.854029i	$0.674152\pi$
$744$	−6.70820	−0.245935
$745$	0	0
$746$	45.7426	1.67476
$747$	−12.4721	−0.456332
$748$	−16.9443	−0.619544
$749$	−10.1459	−0.370723
$750$	0	0
$751$	−5.11146	−0.186520	−0.0932598	−	0.995642i	$-0.529729\pi$
$751$	−5.11146	−0.186520	−0.0932598	+	0.995642i	$0.529729\pi$
$752$	3.00000	0.109399
$753$	29.1803	1.06339
$754$	10.8541	0.395283
$755$	0	0
$756$	−1.90983	−0.0694598
$757$	−30.4164	−1.10550	−0.552752	−	0.833346i	$-0.686422\pi$
$757$	−30.4164	−1.10550	−0.552752	+	0.833346i	$0.686422\pi$
$758$	23.6180	0.857846
$759$	−19.7082	−0.715362
$760$	0	0
$761$	−18.4508	−0.668843	−0.334421	−	0.942424i	$-0.608541\pi$
$761$	−18.4508	−0.668843	−0.334421	+	0.942424i	$0.608541\pi$
$762$	32.1803	1.16577
$763$	−6.18034	−0.223743
$764$	−1.12461	−0.0406870
$765$	0	0
$766$	−53.9787	−1.95033
$767$	20.1246	0.726658
$768$	−13.5623	−0.489388
$769$	13.4164	0.483808	0.241904	−	0.970300i	$-0.422228\pi$
$769$	13.4164	0.483808	0.241904	+	0.970300i	$0.422228\pi$
$770$	0	0
$771$	22.8541	0.823070
$772$	−4.76393	−0.171458
$773$	36.1591	1.30055	0.650275	−	0.759699i	$-0.274654\pi$
$773$	36.1591	1.30055	0.650275	+	0.759699i	$0.274654\pi$
$774$	−15.7082	−0.564620
$775$	0	0
$776$	−8.61803	−0.309369
$777$	0.145898	0.00523406
$778$	24.2705	0.870140
$779$	−0.652476	−0.0233774
$780$	0	0
$781$	34.6525	1.23996
$782$	−31.8885	−1.14033
$783$	−18.0902	−0.646490
$784$	32.1246	1.14731
$785$	0	0
$786$	−11.0000	−0.392357
$787$	11.8197	0.421325	0.210663	−	0.977559i	$-0.432438\pi$
$787$	11.8197	0.421325	0.210663	+	0.977559i	$0.432438\pi$
$788$	−2.29180	−0.0816419
$789$	−10.9098	−0.388400
$790$	0	0
$791$	10.4164	0.370365
$792$	−23.4164	−0.832066
$793$	−16.1459	−0.573358
$794$	−46.9787	−1.66721
$795$	0	0
$796$	−10.8541	−0.384713
$797$	−9.76393	−0.345856	−0.172928	−	0.984934i	$-0.555323\pi$
$797$	−9.76393	−0.345856	−0.172928	+	0.984934i	$0.555323\pi$
$798$	0.854102	0.0302349
$799$	−3.23607	−0.114484
$800$	0	0
$801$	17.8885	0.632061
$802$	43.0344	1.51960
$803$	−47.1246	−1.66299
$804$	2.94427	0.103836
$805$	0	0
$806$	9.00000	0.317011
$807$	12.7639	0.449312
$808$	−3.29180	−0.115805
$809$	30.9787	1.08915	0.544577	−	0.838711i	$-0.316690\pi$
$809$	30.9787	1.08915	0.544577	+	0.838711i	$0.316690\pi$
$810$	0	0
$811$	−14.7082	−0.516475	−0.258237	−	0.966081i	$-0.583142\pi$
$811$	−14.7082	−0.516475	−0.258237	+	0.966081i	$0.583142\pi$
$812$	1.38197	0.0484975
$813$	8.00000	0.280572
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	25.4164	0.889752
$817$	4.14590	0.145047
$818$	2.56231	0.0895889
$819$	−2.29180	−0.0800818
$820$	0	0
$821$	−40.6869	−1.41998	−0.709992	−	0.704210i	$-0.751301\pi$
$821$	−40.6869	−1.41998	−0.709992	+	0.704210i	$0.751301\pi$
$822$	−19.3262	−0.674080
$823$	−47.7082	−1.66300	−0.831502	−	0.555522i	$-0.812518\pi$
$823$	−47.7082	−1.66300	−0.831502	+	0.555522i	$0.812518\pi$
$824$	19.1459	0.666979
$825$	0	0
$826$	10.8541	0.377663
$827$	0.965558	0.0335757	0.0167879	−	0.999859i	$-0.494656\pi$
$827$	0.965558	0.0335757	0.0167879	+	0.999859i	$0.494656\pi$
$828$	4.65248	0.161685
$829$	−35.8541	−1.24526	−0.622632	−	0.782515i	$-0.713937\pi$
$829$	−35.8541	−1.24526	−0.622632	+	0.782515i	$0.713937\pi$
$830$	0	0
$831$	−24.7082	−0.857118
$832$	−7.85410	−0.272292
$833$	−34.6525	−1.20064
$834$	−8.09017	−0.280140
$835$	0	0
$836$	−2.76393	−0.0955926
$837$	−15.0000	−0.518476
$838$	−15.3262	−0.529436
$839$	10.8541	0.374725	0.187363	−	0.982291i	$-0.440006\pi$
$839$	10.8541	0.374725	0.187363	+	0.982291i	$0.440006\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	51.7771	1.78436
$843$	−10.0902	−0.347524
$844$	−5.67376	−0.195299
$845$	0	0
$846$	2.00000	0.0687614
$847$	−10.1459	−0.348617
$848$	−16.8541	−0.578772
$849$	−29.8541	−1.02459
$850$	0	0
$851$	−0.888544	−0.0304589
$852$	4.09017	0.140127
$853$	15.3050	0.524032	0.262016	−	0.965064i	$-0.415613\pi$
$853$	15.3050	0.524032	0.262016	+	0.965064i	$0.415613\pi$
$854$	−8.70820	−0.297989
$855$	0	0
$856$	−36.7082	−1.25466
$857$	−19.6869	−0.672492	−0.336246	−	0.941774i	$-0.609157\pi$
$857$	−19.6869	−0.672492	−0.336246	+	0.941774i	$0.609157\pi$
$858$	−15.7082	−0.536269
$859$	−1.58359	−0.0540315	−0.0270157	−	0.999635i	$-0.508600\pi$
$859$	−1.58359	−0.0540315	−0.0270157	+	0.999635i	$0.508600\pi$
$860$	0	0
$861$	−0.472136	−0.0160904
$862$	−48.2705	−1.64410
$863$	21.4377	0.729748	0.364874	−	0.931057i	$-0.381112\pi$
$863$	21.4377	0.729748	0.364874	+	0.931057i	$0.381112\pi$
$864$	−16.9098	−0.575284
$865$	0	0
$866$	−43.4508	−1.47652
$867$	−10.4164	−0.353760
$868$	1.14590	0.0388943
$869$	42.3607	1.43699
$870$	0	0
$871$	8.83282	0.299289
$872$	−22.3607	−0.757228
$873$	−7.70820	−0.260883
$874$	−5.20163	−0.175948
$875$	0	0
$876$	−5.56231	−0.187933
$877$	36.5410	1.23390	0.616951	−	0.787001i	$-0.288368\pi$
$877$	36.5410	1.23390	0.616951	+	0.787001i	$0.288368\pi$
$878$	−66.3050	−2.23768
$879$	−19.5279	−0.658659
$880$	0	0
$881$	−40.3607	−1.35979	−0.679893	−	0.733311i	$-0.737974\pi$
$881$	−40.3607	−1.35979	−0.679893	+	0.733311i	$0.737974\pi$
$882$	21.4164	0.721128
$883$	20.5836	0.692693	0.346347	−	0.938107i	$-0.387422\pi$
$883$	20.5836	0.692693	0.346347	+	0.938107i	$0.387422\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	−48.4508	−1.62774
$887$	−29.8885	−1.00356	−0.501780	−	0.864996i	$-0.667321\pi$
$887$	−29.8885	−1.00356	−0.501780	+	0.864996i	$0.667321\pi$
$888$	0.527864	0.0177140
$889$	12.2918	0.412254
$890$	0	0
$891$	−5.23607	−0.175415
$892$	0.111456	0.00373183
$893$	−0.527864	−0.0176643
$894$	6.38197	0.213445
$895$	0	0
$896$	−8.41641	−0.281172
$897$	−6.97871	−0.233012
$898$	7.56231	0.252357
$899$	10.8541	0.362005
$900$	0	0
$901$	18.1803	0.605675
$902$	6.47214	0.215499
$903$	3.00000	0.0998337
$904$	37.6869	1.25345
$905$	0	0
$906$	−23.5623	−0.782805
$907$	33.2492	1.10402	0.552011	−	0.833837i	$-0.313861\pi$
$907$	33.2492	1.10402	0.552011	+	0.833837i	$0.313861\pi$
$908$	−9.12461	−0.302811
$909$	−2.94427	−0.0976553
$910$	0	0
$911$	−40.2361	−1.33308	−0.666540	−	0.745469i	$-0.732226\pi$
$911$	−40.2361	−1.33308	−0.666540	+	0.745469i	$0.732226\pi$
$912$	4.14590	0.137284
$913$	−32.6525	−1.08064
$914$	34.6525	1.14620
$915$	0	0
$916$	13.4164	0.443291
$917$	−4.20163	−0.138750
$918$	42.3607	1.39811
$919$	−53.2148	−1.75539	−0.877697	−	0.479216i	$-0.840921\pi$
$919$	−53.2148	−1.75539	−0.877697	+	0.479216i	$0.840921\pi$
$920$	0	0
$921$	9.23607	0.304339
$922$	1.32624	0.0436773
$923$	12.2705	0.403889
$924$	−2.00000	−0.0657952
$925$	0	0
$926$	39.0344	1.28275
$927$	17.1246	0.562446
$928$	12.2361	0.401669
$929$	41.6312	1.36588	0.682938	−	0.730477i	$-0.260702\pi$
$929$	41.6312	1.36588	0.682938	+	0.730477i	$0.260702\pi$
$930$	0	0
$931$	−5.65248	−0.185252
$932$	1.81966	0.0596049
$933$	−8.50658	−0.278493
$934$	−44.4164	−1.45335
$935$	0	0
$936$	−8.29180	−0.271026
$937$	−17.7295	−0.579197	−0.289599	−	0.957148i	$-0.593522\pi$
$937$	−17.7295	−0.579197	−0.289599	+	0.957148i	$0.593522\pi$
$938$	4.76393	0.155548
$939$	−16.7639	−0.547070
$940$	0	0
$941$	−46.4164	−1.51313	−0.756566	−	0.653918i	$-0.773124\pi$
$941$	−46.4164	−1.51313	−0.756566	+	0.653918i	$0.773124\pi$
$942$	21.3262	0.694846
$943$	2.87539	0.0936355
$944$	52.6869	1.71481
$945$	0	0
$946$	−41.1246	−1.33708
$947$	−2.65248	−0.0861939	−0.0430969	−	0.999071i	$-0.513722\pi$
$947$	−2.65248	−0.0861939	−0.0430969	+	0.999071i	$0.513722\pi$
$948$	5.00000	0.162392
$949$	−16.6869	−0.541680
$950$	0	0
$951$	7.65248	0.248149
$952$	7.23607	0.234522
$953$	7.74265	0.250809	0.125404	−	0.992106i	$-0.459977\pi$
$953$	7.74265	0.250809	0.125404	+	0.992106i	$0.459977\pi$
$954$	−11.2361	−0.363781
$955$	0	0
$956$	−12.6869	−0.410324
$957$	−18.9443	−0.612381
$958$	−17.5623	−0.567412
$959$	−7.38197	−0.238376
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−0.708204	−0.0228334
$963$	−32.8328	−1.05802
$964$	1.56231	0.0503185
$965$	0	0
$966$	−3.76393	−0.121103
$967$	−4.11146	−0.132216	−0.0661078	−	0.997812i	$-0.521058\pi$
$967$	−4.11146	−0.132216	−0.0661078	+	0.997812i	$0.521058\pi$
$968$	−36.7082	−1.17985
$969$	−4.47214	−0.143666
$970$	0	0
$971$	5.61803	0.180291	0.0901456	−	0.995929i	$-0.471267\pi$
$971$	5.61803	0.180291	0.0901456	+	0.995929i	$0.471267\pi$
$972$	−9.88854	−0.317175
$973$	−3.09017	−0.0990663
$974$	58.9230	1.88801
$975$	0	0
$976$	−42.2705	−1.35305
$977$	2.34752	0.0751040	0.0375520	−	0.999295i	$-0.488044\pi$
$977$	2.34752	0.0751040	0.0375520	+	0.999295i	$0.488044\pi$
$978$	17.7984	0.569129
$979$	46.8328	1.49678
$980$	0	0
$981$	−20.0000	−0.638551
$982$	−69.9787	−2.23311
$983$	−9.61803	−0.306768	−0.153384	−	0.988167i	$-0.549017\pi$
$983$	−9.61803	−0.306768	−0.153384	+	0.988167i	$0.549017\pi$
$984$	−1.70820	−0.0544556
$985$	0	0
$986$	−30.6525	−0.976174
$987$	−0.381966	−0.0121581
$988$	−0.978714	−0.0311370
$989$	−18.2705	−0.580968
$990$	0	0
$991$	−15.3607	−0.487948	−0.243974	−	0.969782i	$-0.578451\pi$
$991$	−15.3607	−0.487948	−0.243974	+	0.969782i	$0.578451\pi$
$992$	10.1459	0.322133
$993$	23.1246	0.733837
$994$	6.61803	0.209911
$995$	0	0
$996$	−3.85410	−0.122122
$997$	−24.8885	−0.788228	−0.394114	−	0.919062i	$-0.628949\pi$
$997$	−24.8885	−0.788228	−0.394114	+	0.919062i	$0.628949\pi$
$998$	12.2361	0.387326
$999$	1.18034	0.0373443

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.c.1.2		2
3.2	odd	2		5625.2.a.d.1.1		2
4.3	odd	2		10000.2.a.l.1.2		2
5.2	odd	4		625.2.b.a.624.4		4
5.3	odd	4		625.2.b.a.624.1		4
5.4	even	2		625.2.a.b.1.1		2
15.14	odd	2		5625.2.a.f.1.2		2
20.19	odd	2		10000.2.a.c.1.1		2
25.2	odd	20		125.2.e.a.24.2		8
25.3	odd	20		625.2.e.c.249.1		8
25.4	even	10		625.2.d.h.376.1		4
25.6	even	5		625.2.d.b.251.1		4
25.8	odd	20		625.2.e.c.374.2		8
25.9	even	10		25.2.d.a.6.1	✓	4
25.11	even	5		125.2.d.a.101.1		4
25.12	odd	20		125.2.e.a.99.1		8
25.13	odd	20		125.2.e.a.99.2		8
25.14	even	10		25.2.d.a.21.1	yes	4
25.16	even	5		125.2.d.a.26.1		4
25.17	odd	20		625.2.e.c.374.1		8
25.19	even	10		625.2.d.h.251.1		4
25.21	even	5		625.2.d.b.376.1		4
25.22	odd	20		625.2.e.c.249.2		8
25.23	odd	20		125.2.e.a.24.1		8
75.14	odd	10		225.2.h.b.46.1		4
75.59	odd	10		225.2.h.b.181.1		4
100.39	odd	10		400.2.u.b.321.1		4
100.59	odd	10		400.2.u.b.81.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.d.a.6.1	✓	4	25.9	even	10
25.2.d.a.21.1	yes	4	25.14	even	10
125.2.d.a.26.1		4	25.16	even	5
125.2.d.a.101.1		4	25.11	even	5
125.2.e.a.24.1		8	25.23	odd	20
125.2.e.a.24.2		8	25.2	odd	20
125.2.e.a.99.1		8	25.12	odd	20
125.2.e.a.99.2		8	25.13	odd	20
225.2.h.b.46.1		4	75.14	odd	10
225.2.h.b.181.1		4	75.59	odd	10
400.2.u.b.81.1		4	100.59	odd	10
400.2.u.b.321.1		4	100.39	odd	10
625.2.a.b.1.1		2	5.4	even	2
625.2.a.c.1.2		2	1.1	even	1	trivial
625.2.b.a.624.1		4	5.3	odd	4
625.2.b.a.624.4		4	5.2	odd	4
625.2.d.b.251.1		4	25.6	even	5
625.2.d.b.376.1		4	25.21	even	5
625.2.d.h.251.1		4	25.19	even	10
625.2.d.h.376.1		4	25.4	even	10
625.2.e.c.249.1		8	25.3	odd	20
625.2.e.c.249.2		8	25.22	odd	20
625.2.e.c.374.1		8	25.17	odd	20
625.2.e.c.374.2		8	25.8	odd	20
5625.2.a.d.1.1		2	3.2	odd	2
5625.2.a.f.1.2		2	15.14	odd	2
10000.2.a.c.1.1		2	20.19	odd	2
10000.2.a.l.1.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -1.61803 q^{6} -0.618034 q^{7} -2.23607 q^{8} -2.00000 q^{9} -5.23607 q^{11} -0.618034 q^{12} -1.85410 q^{13} -1.00000 q^{14} -4.85410 q^{16} +5.23607 q^{17} -3.23607 q^{18} +0.854102 q^{19} +0.618034 q^{21} -8.47214 q^{22} -3.76393 q^{23} +2.23607 q^{24} -3.00000 q^{26} +5.00000 q^{27} -0.381966 q^{28} -3.61803 q^{29} -3.00000 q^{31} -3.38197 q^{32} +5.23607 q^{33} +8.47214 q^{34} -1.23607 q^{36} +0.236068 q^{37} +1.38197 q^{38} +1.85410 q^{39} -0.763932 q^{41} +1.00000 q^{42} +4.85410 q^{43} -3.23607 q^{44} -6.09017 q^{46} -0.618034 q^{47} +4.85410 q^{48} -6.61803 q^{49} -5.23607 q^{51} -1.14590 q^{52} +3.47214 q^{53} +8.09017 q^{54} +1.38197 q^{56} -0.854102 q^{57} -5.85410 q^{58} -10.8541 q^{59} +8.70820 q^{61} -4.85410 q^{62} +1.23607 q^{63} +4.23607 q^{64} +8.47214 q^{66} -4.76393 q^{67} +3.23607 q^{68} +3.76393 q^{69} -6.61803 q^{71} +4.47214 q^{72} +9.00000 q^{73} +0.381966 q^{74} +0.527864 q^{76} +3.23607 q^{77} +3.00000 q^{78} -8.09017 q^{79} +1.00000 q^{81} -1.23607 q^{82} +6.23607 q^{83} +0.381966 q^{84} +7.85410 q^{86} +3.61803 q^{87} +11.7082 q^{88} -8.94427 q^{89} +1.14590 q^{91} -2.32624 q^{92} +3.00000 q^{93} -1.00000 q^{94} +3.38197 q^{96} +3.85410 q^{97} -10.7082 q^{98} +10.4721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} - q^{4} - q^{6} + q^{7} - 4 q^{9} - 6 q^{11} + q^{12} + 3 q^{13} - 2 q^{14} - 3 q^{16} + 6 q^{17} - 2 q^{18} - 5 q^{19} - q^{21} - 8 q^{22} - 12 q^{23} - 6 q^{26} + 10 q^{27}+ \cdots + 12 q^{99}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 - q^4 - q^6 + q^7 - 4 * q^9 - 6 * q^11 + q^12 + 3 * q^13 - 2 * q^14 - 3 * q^16 + 6 * q^17 - 2 * q^18 - 5 * q^19 - q^21 - 8 * q^22 - 12 * q^23 - 6 * q^26 + 10 * q^27 - 3 * q^28 - 5 * q^29 - 6 * q^31 - 9 * q^32 + 6 * q^33 + 8 * q^34 + 2 * q^36 - 4 * q^37 + 5 * q^38 - 3 * q^39 - 6 * q^41 + 2 * q^42 + 3 * q^43 - 2 * q^44 - q^46 + q^47 + 3 * q^48 - 11 * q^49 - 6 * q^51 - 9 * q^52 - 2 * q^53 + 5 * q^54 + 5 * q^56 + 5 * q^57 - 5 * q^58 - 15 * q^59 + 4 * q^61 - 3 * q^62 - 2 * q^63 + 4 * q^64 + 8 * q^66 - 14 * q^67 + 2 * q^68 + 12 * q^69 - 11 * q^71 + 18 * q^73 + 3 * q^74 + 10 * q^76 + 2 * q^77 + 6 * q^78 - 5 * q^79 + 2 * q^81 + 2 * q^82 + 8 * q^83 + 3 * q^84 + 9 * q^86 + 5 * q^87 + 10 * q^88 + 9 * q^91 + 11 * q^92 + 6 * q^93 - 2 * q^94 + 9 * q^96 + q^97 - 8 * q^98 + 12 * q^99

Introduction

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Database

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