LMFDB - Embedded newform 5625.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
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:	$\Z[a_1, a_2]$
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			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−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
$9$	0	0
$10$	0	0
$11$	5.23607	1.57873	0.789367	−	0.613922i	$-0.210409\pi$
$11$	5.23607	1.57873	0.789367	+	0.613922i	$0.210409\pi$
$12$	0	0
$13$	−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	$-0.718986\pi$
$17$	−5.23607	−1.26993	−0.634967	+	0.772540i	$0.718986\pi$
$18$	0	0
$19$	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	0
$22$	−8.47214	−1.80627
$23$	3.76393	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$23$	3.76393	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$24$	0	0
$25$	0	0
$26$	3.00000	0.588348
$27$	0	0
$28$	−0.381966	−0.0721848
$29$	3.61803	0.671852	0.335926	−	0.941888i	$-0.390951\pi$
$29$	3.61803	0.671852	0.335926	+	0.941888i	$0.390951\pi$
$30$	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
$33$	0	0
$34$	8.47214	1.45296
$35$	0	0
$36$	0	0
$37$	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
$39$	0	0
$40$	0	0
$41$	0.763932	0.119306	0.0596531	−	0.998219i	$-0.481001\pi$
$41$	0.763932	0.119306	0.0596531	+	0.998219i	$0.481001\pi$
$42$	0	0
$43$	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	$-0.485647\pi$
$47$	0.618034	0.0901495	0.0450748	+	0.998984i	$0.485647\pi$
$48$	0	0
$49$	−6.61803	−0.945433
$50$	0	0
$51$	0	0
$52$	−1.14590	−0.158907
$53$	−3.47214	−0.476935	−0.238467	−	0.971151i	$-0.576645\pi$
$53$	−3.47214	−0.476935	−0.238467	+	0.971151i	$0.576645\pi$
$54$	0	0
$55$	0	0
$56$	−1.38197	−0.184673
$57$	0	0
$58$	−5.85410	−0.768681
$59$	10.8541	1.41308	0.706542	−	0.707671i	$-0.250254\pi$
$59$	10.8541	1.41308	0.706542	+	0.707671i	$0.250254\pi$
$60$	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
$63$	0	0
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	−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
$69$	0	0
$70$	0	0
$71$	6.61803	0.785416	0.392708	−	0.919663i	$-0.371538\pi$
$71$	6.61803	0.785416	0.392708	+	0.919663i	$0.371538\pi$
$72$	0	0
$73$	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
$78$	0	0
$79$	−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
$81$	0	0
$82$	−1.23607	−0.136501
$83$	−6.23607	−0.684497	−0.342249	−	0.939609i	$-0.611189\pi$
$83$	−6.23607	−0.684497	−0.342249	+	0.939609i	$0.611189\pi$
$84$	0	0
$85$	0	0
$86$	−7.85410	−0.846930
$87$	0	0
$88$	11.7082	1.24810
$89$	8.94427	0.948091	0.474045	−	0.880500i	$-0.342793\pi$
$89$	8.94427	0.948091	0.474045	+	0.880500i	$0.342793\pi$
$90$	0	0
$91$	1.14590	0.120123
$92$	2.32624	0.242527
$93$	0	0
$94$	−1.00000	−0.103142
$95$	0	0
$96$	0	0
$97$	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
$99$	0	0
$100$	0	0
$101$	−1.47214	−0.146483	−0.0732415	−	0.997314i	$-0.523334\pi$
$101$	−1.47214	−0.146483	−0.0732415	+	0.997314i	$0.523334\pi$
$102$	0	0
$103$	−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	$-0.791752\pi$
$107$	−16.4164	−1.58703	−0.793517	+	0.608548i	$0.791752\pi$
$108$	0	0
$109$	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	0
$112$	3.00000	0.283473
$113$	16.8541	1.58550	0.792750	−	0.609547i	$-0.208648\pi$
$113$	16.8541	1.58550	0.792750	+	0.609547i	$0.208648\pi$
$114$	0	0
$115$	0	0
$116$	2.23607	0.207614
$117$	0	0
$118$	−17.5623	−1.61674
$119$	3.23607	0.296650
$120$	0	0
$121$	16.4164	1.49240
$122$	−14.0902	−1.27566
$123$	0	0
$124$	−1.85410	−0.166503
$125$	0	0
$126$	0	0
$127$	−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
$129$	0	0
$130$	0	0
$131$	−6.79837	−0.593977	−0.296988	−	0.954881i	$-0.595982\pi$
$131$	−6.79837	−0.593977	−0.296988	+	0.954881i	$0.595982\pi$
$132$	0	0
$133$	−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	$-0.670441\pi$
$137$	−11.9443	−1.02047	−0.510234	+	0.860036i	$0.670441\pi$
$138$	0	0
$139$	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	0
$142$	−10.7082	−0.898613
$143$	−9.70820	−0.811841
$144$	0	0
$145$	0	0
$146$	−14.5623	−1.20519
$147$	0	0
$148$	0.145898	0.0119927
$149$	3.94427	0.323127	0.161564	−	0.986862i	$-0.448346\pi$
$149$	3.94427	0.323127	0.161564	+	0.986862i	$0.448346\pi$
$150$	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
$153$	0	0
$154$	5.23607	0.421934
$155$	0	0
$156$	0	0
$157$	−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
$159$	0	0
$160$	0	0
$161$	−2.32624	−0.183333
$162$	0	0
$163$	−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	$-0.309480\pi$
$167$	14.5623	1.12687	0.563433	+	0.826162i	$0.309480\pi$
$168$	0	0
$169$	−9.56231	−0.735562
$170$	0	0
$171$	0	0
$172$	3.00000	0.228748
$173$	18.8885	1.43607	0.718035	−	0.696007i	$-0.245042\pi$
$173$	18.8885	1.43607	0.718035	+	0.696007i	$0.245042\pi$
$174$	0	0
$175$	0	0
$176$	−25.4164	−1.91583
$177$	0	0
$178$	−14.4721	−1.08473
$179$	0.527864	0.0394544	0.0197272	−	0.999805i	$-0.493720\pi$
$179$	0.527864	0.0394544	0.0197272	+	0.999805i	$0.493720\pi$
$180$	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
$183$	0	0
$184$	8.41641	0.620466
$185$	0	0
$186$	0	0
$187$	−27.4164	−2.00489
$188$	0.381966	0.0278577
$189$	0	0
$190$	0	0
$191$	1.81966	0.131666	0.0658330	−	0.997831i	$-0.479030\pi$
$191$	1.81966	0.131666	0.0658330	+	0.997831i	$0.479030\pi$
$192$	0	0
$193$	−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	$-0.457828\pi$
$197$	3.70820	0.264199	0.132099	+	0.991236i	$0.457828\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	2.38197	0.167595
$203$	−2.23607	−0.156941
$204$	0	0
$205$	0	0
$206$	13.8541	0.965261
$207$	0	0
$208$	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
$213$	0	0
$214$	26.5623	1.81576
$215$	0	0
$216$	0	0
$217$	1.85410	0.125865
$218$	−16.1803	−1.09587
$219$	0	0
$220$	0	0
$221$	9.70820	0.653044
$222$	0	0
$223$	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	$-0.337012\pi$
$227$	14.7639	0.979917	0.489958	+	0.871746i	$0.337012\pi$
$228$	0	0
$229$	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
$231$	0	0
$232$	8.09017	0.531146
$233$	−2.94427	−0.192886	−0.0964428	−	0.995339i	$-0.530746\pi$
$233$	−2.94427	−0.192886	−0.0964428	+	0.995339i	$0.530746\pi$
$234$	0	0
$235$	0	0
$236$	6.70820	0.436667
$237$	0	0
$238$	−5.23607	−0.339404
$239$	20.5279	1.32784	0.663919	−	0.747805i	$-0.268892\pi$
$239$	20.5279	1.32784	0.663919	+	0.747805i	$0.268892\pi$
$240$	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
$243$	0	0
$244$	5.38197	0.344545
$245$	0	0
$246$	0	0
$247$	−1.58359	−0.100762
$248$	−6.70820	−0.425971
$249$	0	0
$250$	0	0
$251$	29.1803	1.84185	0.920923	−	0.389744i	$-0.127436\pi$
$251$	29.1803	1.84185	0.920923	+	0.389744i	$0.127436\pi$
$252$	0	0
$253$	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	$-0.247427\pi$
$257$	22.8541	1.42560	0.712800	+	0.701367i	$0.247427\pi$
$258$	0	0
$259$	−0.145898	−0.00906566
$260$	0	0
$261$	0	0
$262$	11.0000	0.679582
$263$	−10.9098	−0.672729	−0.336364	−	0.941732i	$-0.609197\pi$
$263$	−10.9098	−0.672729	−0.336364	+	0.941732i	$0.609197\pi$
$264$	0	0
$265$	0	0
$266$	0.854102	0.0523684
$267$	0	0
$268$	−2.94427	−0.179850
$269$	12.7639	0.778231	0.389115	−	0.921189i	$-0.372781\pi$
$269$	12.7639	0.778231	0.389115	+	0.921189i	$0.372781\pi$
$270$	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
$273$	0	0
$274$	19.3262	1.16754
$275$	0	0
$276$	0	0
$277$	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
$279$	0	0
$280$	0	0
$281$	−10.0902	−0.601929	−0.300965	−	0.953635i	$-0.597309\pi$
$281$	−10.0902	−0.601929	−0.300965	+	0.953635i	$0.597309\pi$
$282$	0	0
$283$	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
$288$	0	0
$289$	10.4164	0.612730
$290$	0	0
$291$	0	0
$292$	5.56231	0.325509
$293$	−19.5279	−1.14083	−0.570415	−	0.821357i	$-0.693218\pi$
$293$	−19.5279	−1.14083	−0.570415	+	0.821357i	$0.693218\pi$
$294$	0	0
$295$	0	0
$296$	0.527864	0.0306815
$297$	0	0
$298$	−6.38197	−0.369697
$299$	−6.97871	−0.403589
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−23.5623	−1.35586
$303$	0	0
$304$	−4.14590	−0.237784
$305$	0	0
$306$	0	0
$307$	−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
$309$	0	0
$310$	0	0
$311$	−8.50658	−0.482364	−0.241182	−	0.970480i	$-0.577535\pi$
$311$	−8.50658	−0.482364	−0.241182	+	0.970480i	$0.577535\pi$
$312$	0	0
$313$	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	$-0.431056\pi$
$317$	7.65248	0.429806	0.214903	+	0.976635i	$0.431056\pi$
$318$	0	0
$319$	18.9443	1.06068
$320$	0	0
$321$	0	0
$322$	3.76393	0.209756
$323$	−4.47214	−0.248836
$324$	0	0
$325$	0	0
$326$	17.7984	0.985761
$327$	0	0
$328$	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	0
$334$	−23.5623	−1.28927
$335$	0	0
$336$	0	0
$337$	−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
$339$	0	0
$340$	0	0
$341$	−15.7082	−0.850647
$342$	0	0
$343$	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	$-0.679465\pi$
$347$	−19.9098	−1.06882	−0.534408	+	0.845227i	$0.679465\pi$
$348$	0	0
$349$	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
$351$	0	0
$352$	17.7082	0.943850
$353$	12.9098	0.687121	0.343560	−	0.939131i	$-0.388367\pi$
$353$	12.9098	0.687121	0.343560	+	0.939131i	$0.388367\pi$
$354$	0	0
$355$	0	0
$356$	5.52786	0.292976
$357$	0	0
$358$	−0.854102	−0.0451407
$359$	−13.7426	−0.725309	−0.362655	−	0.931924i	$-0.618130\pi$
$359$	−13.7426	−0.725309	−0.362655	+	0.931924i	$0.618130\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	−0.472136	−0.0248149
$363$	0	0
$364$	0.708204	0.0371200
$365$	0	0
$366$	0	0
$367$	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
$369$	0	0
$370$	0	0
$371$	2.14590	0.111409
$372$	0	0
$373$	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
$378$	0	0
$379$	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
$381$	0	0
$382$	−2.94427	−0.150642
$383$	33.3607	1.70465	0.852326	−	0.523012i	$-0.175192\pi$
$383$	33.3607	1.70465	0.852326	+	0.523012i	$0.175192\pi$
$384$	0	0
$385$	0	0
$386$	12.4721	0.634815
$387$	0	0
$388$	2.38197	0.120926
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	−19.7082	−0.996687
$392$	−14.7984	−0.747431
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	−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	0
$400$	0	0
$401$	−26.5967	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$401$	−26.5967	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$402$	0	0
$403$	5.56231	0.277078
$404$	−0.909830	−0.0452657
$405$	0	0
$406$	3.61803	0.179560
$407$	1.23607	0.0612696
$408$	0	0
$409$	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
$411$	0	0
$412$	−5.29180	−0.260708
$413$	−6.70820	−0.330089
$414$	0	0
$415$	0	0
$416$	−6.27051	−0.307437
$417$	0	0
$418$	−7.23607	−0.353928
$419$	9.47214	0.462744	0.231372	−	0.972865i	$-0.425679\pi$
$419$	9.47214	0.462744	0.231372	+	0.972865i	$0.425679\pi$
$420$	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
$423$	0	0
$424$	−7.76393	−0.377050
$425$	0	0
$426$	0	0
$427$	−5.38197	−0.260452
$428$	−10.1459	−0.490420
$429$	0	0
$430$	0	0
$431$	29.8328	1.43700	0.718498	−	0.695529i	$-0.244830\pi$
$431$	29.8328	1.43700	0.718498	+	0.695529i	$0.244830\pi$
$432$	0	0
$433$	−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
$438$	0	0
$439$	−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
$441$	0	0
$442$	−15.7082	−0.747163
$443$	29.9443	1.42270	0.711348	−	0.702840i	$-0.248085\pi$
$443$	29.9443	1.42270	0.711348	+	0.702840i	$0.248085\pi$
$444$	0	0
$445$	0	0
$446$	−0.291796	−0.0138169
$447$	0	0
$448$	−2.61803	−0.123690
$449$	−4.67376	−0.220568	−0.110284	−	0.993900i	$-0.535176\pi$
$449$	−4.67376	−0.220568	−0.110284	+	0.993900i	$0.535176\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	10.4164	0.489947
$453$	0	0
$454$	−23.8885	−1.12114
$455$	0	0
$456$	0	0
$457$	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
$459$	0	0
$460$	0	0
$461$	−0.819660	−0.0381754	−0.0190877	−	0.999818i	$-0.506076\pi$
$461$	−0.819660	−0.0381754	−0.0190877	+	0.999818i	$0.506076\pi$
$462$	0	0
$463$	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	$-0.280944\pi$
$467$	27.4508	1.27027	0.635137	+	0.772400i	$0.280944\pi$
$468$	0	0
$469$	2.94427	0.135954
$470$	0	0
$471$	0	0
$472$	24.2705	1.11714
$473$	25.4164	1.16865
$474$	0	0
$475$	0	0
$476$	2.00000	0.0916698
$477$	0	0
$478$	−33.2148	−1.51921
$479$	10.8541	0.495937	0.247968	−	0.968768i	$-0.420237\pi$
$479$	10.8541	0.495937	0.247968	+	0.968768i	$0.420237\pi$
$480$	0	0
$481$	−0.437694	−0.0199571
$482$	−4.09017	−0.186302
$483$	0	0
$484$	10.1459	0.461177
$485$	0	0
$486$	0	0
$487$	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
$489$	0	0
$490$	0	0
$491$	43.2492	1.95181	0.975905	−	0.218196i	$-0.0700171\pi$
$491$	43.2492	1.95181	0.975905	+	0.218196i	$0.0700171\pi$
$492$	0	0
$493$	−18.9443	−0.853207
$494$	2.56231	0.115284
$495$	0	0
$496$	14.5623	0.653867
$497$	−4.09017	−0.183469
$498$	0	0
$499$	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
$501$	0	0
$502$	−47.2148	−2.10730
$503$	−37.4164	−1.66832	−0.834158	−	0.551526i	$-0.814046\pi$
$503$	−37.4164	−1.66832	−0.834158	+	0.551526i	$0.814046\pi$
$504$	0	0
$505$	0	0
$506$	−31.8885	−1.41762
$507$	0	0
$508$	−12.2918	−0.545360
$509$	20.3262	0.900945	0.450472	−	0.892790i	$-0.351256\pi$
$509$	20.3262	0.900945	0.450472	+	0.892790i	$0.351256\pi$
$510$	0	0
$511$	−5.56231	−0.246062
$512$	5.29180	0.233867
$513$	0	0
$514$	−36.9787	−1.63106
$515$	0	0
$516$	0	0
$517$	3.23607	0.142322
$518$	0.236068	0.0103722
$519$	0	0
$520$	0	0
$521$	−29.3607	−1.28631	−0.643157	−	0.765734i	$-0.722376\pi$
$521$	−29.3607	−1.28631	−0.643157	+	0.765734i	$0.722376\pi$
$522$	0	0
$523$	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
$528$	0	0
$529$	−8.83282	−0.384035
$530$	0	0
$531$	0	0
$532$	−0.326238	−0.0141442
$533$	−1.41641	−0.0613514
$534$	0	0
$535$	0	0
$536$	−10.6525	−0.460117
$537$	0	0
$538$	−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	0
$544$	−17.7082	−0.759233
$545$	0	0
$546$	0	0
$547$	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
$549$	0	0
$550$	0	0
$551$	3.09017	0.131646
$552$	0	0
$553$	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	$-0.467819\pi$
$557$	4.76393	0.201854	0.100927	+	0.994894i	$0.467819\pi$
$558$	0	0
$559$	−9.00000	−0.380659
$560$	0	0
$561$	0	0
$562$	16.3262	0.688681
$563$	7.38197	0.311113	0.155556	−	0.987827i	$-0.450283\pi$
$563$	7.38197	0.311113	0.155556	+	0.987827i	$0.450283\pi$
$564$	0	0
$565$	0	0
$566$	−48.3050	−2.03041
$567$	0	0
$568$	14.7984	0.620926
$569$	−20.5279	−0.860573	−0.430286	−	0.902692i	$-0.641587\pi$
$569$	−20.5279	−0.860573	−0.430286	+	0.902692i	$0.641587\pi$
$570$	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
$573$	0	0
$574$	0.763932	0.0318859
$575$	0	0
$576$	0	0
$577$	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
$579$	0	0
$580$	0	0
$581$	3.85410	0.159895
$582$	0	0
$583$	−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	$-0.465169\pi$
$587$	5.29180	0.218416	0.109208	+	0.994019i	$0.465169\pi$
$588$	0	0
$589$	−2.56231	−0.105578
$590$	0	0
$591$	0	0
$592$	−1.14590	−0.0470961
$593$	−10.9098	−0.448013	−0.224007	−	0.974588i	$-0.571914\pi$
$593$	−10.9098	−0.448013	−0.224007	+	0.974588i	$0.571914\pi$
$594$	0	0
$595$	0	0
$596$	2.43769	0.0998518
$597$	0	0
$598$	11.2918	0.461756
$599$	9.47214	0.387021	0.193510	−	0.981098i	$-0.438013\pi$
$599$	9.47214	0.387021	0.193510	+	0.981098i	$0.438013\pi$
$600$	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
$603$	0	0
$604$	9.00000	0.366205
$605$	0	0
$606$	0	0
$607$	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
$609$	0	0
$610$	0	0
$611$	−1.14590	−0.0463581
$612$	0	0
$613$	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	$-0.407488\pi$
$617$	14.2361	0.573123	0.286561	+	0.958062i	$0.407488\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	13.7639	0.551883
$623$	−5.52786	−0.221469
$624$	0	0
$625$	0	0
$626$	−27.1246	−1.08412
$627$	0	0
$628$	−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
$633$	0	0
$634$	−12.3820	−0.491751
$635$	0	0
$636$	0	0
$637$	12.2705	0.486175
$638$	−30.6525	−1.21354
$639$	0	0
$640$	0	0
$641$	1.09017	0.0430591	0.0215296	−	0.999768i	$-0.493146\pi$
$641$	1.09017	0.0430591	0.0215296	+	0.999768i	$0.493146\pi$
$642$	0	0
$643$	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	$-0.755074\pi$
$647$	−36.5410	−1.43658	−0.718288	+	0.695746i	$0.755074\pi$
$648$	0	0
$649$	56.8328	2.23088
$650$	0	0
$651$	0	0
$652$	−6.79837	−0.266245
$653$	19.0902	0.747056	0.373528	−	0.927619i	$-0.378148\pi$
$653$	19.0902	0.747056	0.373528	+	0.927619i	$0.378148\pi$
$654$	0	0
$655$	0	0
$656$	−3.70820	−0.144781
$657$	0	0
$658$	0.618034	0.0240935
$659$	−15.5279	−0.604880	−0.302440	−	0.953168i	$-0.597801\pi$
$659$	−15.5279	−0.604880	−0.302440	+	0.953168i	$0.597801\pi$
$660$	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
$663$	0	0
$664$	−13.9443	−0.541143
$665$	0	0
$666$	0	0
$667$	13.6180	0.527292
$668$	9.00000	0.348220
$669$	0	0
$670$	0	0
$671$	45.5967	1.76024
$672$	0	0
$673$	−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	$-0.434592\pi$
$677$	10.6180	0.408084	0.204042	+	0.978962i	$0.434592\pi$
$678$	0	0
$679$	−2.38197	−0.0914115
$680$	0	0
$681$	0	0
$682$	25.4164	0.973245
$683$	−13.4721	−0.515497	−0.257748	−	0.966212i	$-0.582981\pi$
$683$	−13.4721	−0.515497	−0.257748	+	0.966212i	$0.582981\pi$
$684$	0	0
$685$	0	0
$686$	−13.6180	−0.519939
$687$	0	0
$688$	−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
$693$	0	0
$694$	32.2148	1.22286
$695$	0	0
$696$	0	0
$697$	−4.00000	−0.151511
$698$	−35.1246	−1.32949
$699$	0	0
$700$	0	0
$701$	41.0132	1.54905	0.774523	−	0.632546i	$-0.217990\pi$
$701$	41.0132	1.54905	0.774523	+	0.632546i	$0.217990\pi$
$702$	0	0
$703$	0.201626	0.00760447
$704$	22.1803	0.835953
$705$	0	0
$706$	−20.8885	−0.786151
$707$	0.909830	0.0342177
$708$	0	0
$709$	−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
$711$	0	0
$712$	20.0000	0.749532
$713$	−11.2918	−0.422881
$714$	0	0
$715$	0	0
$716$	0.326238	0.0121921
$717$	0	0
$718$	22.2361	0.829843
$719$	23.2918	0.868637	0.434319	−	0.900759i	$-0.356989\pi$
$719$	23.2918	0.868637	0.434319	+	0.900759i	$0.356989\pi$
$720$	0	0
$721$	5.29180	0.197077
$722$	29.5623	1.10020
$723$	0	0
$724$	0.180340	0.00670228
$725$	0	0
$726$	0	0
$727$	−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
$729$	0	0
$730$	0	0
$731$	−25.4164	−0.940060
$732$	0	0
$733$	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
$738$	0	0
$739$	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
$741$	0	0
$742$	−3.47214	−0.127466
$743$	28.3607	1.04045	0.520226	−	0.854029i	$-0.325848\pi$
$743$	28.3607	1.04045	0.520226	+	0.854029i	$0.325848\pi$
$744$	0	0
$745$	0	0
$746$	−45.7426	−1.67476
$747$	0	0
$748$	−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
$753$	0	0
$754$	10.8541	0.395283
$755$	0	0
$756$	0	0
$757$	−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
$759$	0	0
$760$	0	0
$761$	18.4508	0.668843	0.334421	−	0.942424i	$-0.391459\pi$
$761$	18.4508	0.668843	0.334421	+	0.942424i	$0.391459\pi$
$762$	0	0
$763$	−6.18034	−0.223743
$764$	1.12461	0.0406870
$765$	0	0
$766$	−53.9787	−1.95033
$767$	−20.1246	−0.726658
$768$	0	0
$769$	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
$771$	0	0
$772$	−4.76393	−0.171458
$773$	−36.1591	−1.30055	−0.650275	−	0.759699i	$-0.725346\pi$
$773$	−36.1591	−1.30055	−0.650275	+	0.759699i	$0.725346\pi$
$774$	0	0
$775$	0	0
$776$	8.61803	0.309369
$777$	0	0
$778$	24.2705	0.870140
$779$	0.652476	0.0233774
$780$	0	0
$781$	34.6525	1.23996
$782$	31.8885	1.14033
$783$	0	0
$784$	32.1246	1.14731
$785$	0	0
$786$	0	0
$787$	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
$789$	0	0
$790$	0	0
$791$	−10.4164	−0.370365
$792$	0	0
$793$	−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	$-0.444677\pi$
$797$	9.76393	0.345856	0.172928	+	0.984934i	$0.444677\pi$
$798$	0	0
$799$	−3.23607	−0.114484
$800$	0	0
$801$	0	0
$802$	43.0344	1.51960
$803$	47.1246	1.66299
$804$	0	0
$805$	0	0
$806$	−9.00000	−0.317011
$807$	0	0
$808$	−3.29180	−0.115805
$809$	−30.9787	−1.08915	−0.544577	−	0.838711i	$-0.683310\pi$
$809$	−30.9787	−1.08915	−0.544577	+	0.838711i	$0.683310\pi$
$810$	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
$813$	0	0
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	0	0
$817$	4.14590	0.145047
$818$	−2.56231	−0.0895889
$819$	0	0
$820$	0	0
$821$	40.6869	1.41998	0.709992	−	0.704210i	$-0.248699\pi$
$821$	40.6869	1.41998	0.709992	+	0.704210i	$0.248699\pi$
$822$	0	0
$823$	−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	$-0.505344\pi$
$827$	−0.965558	−0.0335757	−0.0167879	+	0.999859i	$0.505344\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	−7.85410	−0.272292
$833$	34.6525	1.20064
$834$	0	0
$835$	0	0
$836$	2.76393	0.0955926
$837$	0	0
$838$	−15.3262	−0.529436
$839$	−10.8541	−0.374725	−0.187363	−	0.982291i	$-0.559994\pi$
$839$	−10.8541	−0.374725	−0.187363	+	0.982291i	$0.559994\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	−51.7771	−1.78436
$843$	0	0
$844$	−5.67376	−0.195299
$845$	0	0
$846$	0	0
$847$	−10.1459	−0.348617
$848$	16.8541	0.578772
$849$	0	0
$850$	0	0
$851$	0.888544	0.0304589
$852$	0	0
$853$	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	$-0.390843\pi$
$857$	19.6869	0.672492	0.336246	+	0.941774i	$0.390843\pi$
$858$	0	0
$859$	−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	0
$862$	−48.2705	−1.64410
$863$	−21.4377	−0.729748	−0.364874	−	0.931057i	$-0.618888\pi$
$863$	−21.4377	−0.729748	−0.364874	+	0.931057i	$0.618888\pi$
$864$	0	0
$865$	0	0
$866$	43.4508	1.47652
$867$	0	0
$868$	1.14590	0.0388943
$869$	−42.3607	−1.43699
$870$	0	0
$871$	8.83282	0.299289
$872$	22.3607	0.757228
$873$	0	0
$874$	−5.20163	−0.175948
$875$	0	0
$876$	0	0
$877$	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
$879$	0	0
$880$	0	0
$881$	40.3607	1.35979	0.679893	−	0.733311i	$-0.262026\pi$
$881$	40.3607	1.35979	0.679893	+	0.733311i	$0.262026\pi$
$882$	0	0
$883$	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	$-0.332679\pi$
$887$	29.8885	1.00356	0.501780	+	0.864996i	$0.332679\pi$
$888$	0	0
$889$	12.2918	0.412254
$890$	0	0
$891$	0	0
$892$	0.111456	0.00373183
$893$	0.527864	0.0176643
$894$	0	0
$895$	0	0
$896$	8.41641	0.281172
$897$	0	0
$898$	7.56231	0.252357
$899$	−10.8541	−0.362005
$900$	0	0
$901$	18.1803	0.605675
$902$	−6.47214	−0.215499
$903$	0	0
$904$	37.6869	1.25345
$905$	0	0
$906$	0	0
$907$	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
$909$	0	0
$910$	0	0
$911$	40.2361	1.33308	0.666540	−	0.745469i	$-0.267774\pi$
$911$	40.2361	1.33308	0.666540	+	0.745469i	$0.267774\pi$
$912$	0	0
$913$	−32.6525	−1.08064
$914$	−34.6525	−1.14620
$915$	0	0
$916$	13.4164	0.443291
$917$	4.20163	0.138750
$918$	0	0
$919$	−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
$921$	0	0
$922$	1.32624	0.0436773
$923$	−12.2705	−0.403889
$924$	0	0
$925$	0	0
$926$	−39.0344	−1.28275
$927$	0	0
$928$	12.2361	0.401669
$929$	−41.6312	−1.36588	−0.682938	−	0.730477i	$-0.739298\pi$
$929$	−41.6312	−1.36588	−0.682938	+	0.730477i	$0.739298\pi$
$930$	0	0
$931$	−5.65248	−0.185252
$932$	−1.81966	−0.0596049
$933$	0	0
$934$	−44.4164	−1.45335
$935$	0	0
$936$	0	0
$937$	−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
$939$	0	0
$940$	0	0
$941$	46.4164	1.51313	0.756566	−	0.653918i	$-0.226876\pi$
$941$	46.4164	1.51313	0.756566	+	0.653918i	$0.226876\pi$
$942$	0	0
$943$	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	$-0.486278\pi$
$947$	2.65248	0.0861939	0.0430969	+	0.999071i	$0.486278\pi$
$948$	0	0
$949$	−16.6869	−0.541680
$950$	0	0
$951$	0	0
$952$	7.23607	0.234522
$953$	−7.74265	−0.250809	−0.125404	−	0.992106i	$-0.540023\pi$
$953$	−7.74265	−0.250809	−0.125404	+	0.992106i	$0.540023\pi$
$954$	0	0
$955$	0	0
$956$	12.6869	0.410324
$957$	0	0
$958$	−17.5623	−0.567412
$959$	7.38197	0.238376
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0.708204	0.0228334
$963$	0	0
$964$	1.56231	0.0503185
$965$	0	0
$966$	0	0
$967$	−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
$969$	0	0
$970$	0	0
$971$	−5.61803	−0.180291	−0.0901456	−	0.995929i	$-0.528733\pi$
$971$	−5.61803	−0.180291	−0.0901456	+	0.995929i	$0.528733\pi$
$972$	0	0
$973$	−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	$-0.511956\pi$
$977$	−2.34752	−0.0751040	−0.0375520	+	0.999295i	$0.511956\pi$
$978$	0	0
$979$	46.8328	1.49678
$980$	0	0
$981$	0	0
$982$	−69.9787	−2.23311
$983$	9.61803	0.306768	0.153384	−	0.988167i	$-0.450983\pi$
$983$	9.61803	0.306768	0.153384	+	0.988167i	$0.450983\pi$
$984$	0	0
$985$	0	0
$986$	30.6525	0.976174
$987$	0	0
$988$	−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
$993$	0	0
$994$	6.61803	0.209911
$995$	0	0
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.d.1.1		2
3.2	odd	2		625.2.a.c.1.2		2
5.4	even	2		5625.2.a.f.1.2		2
12.11	even	2		10000.2.a.l.1.2		2
15.2	even	4		625.2.b.a.624.4		4
15.8	even	4		625.2.b.a.624.1		4
15.14	odd	2		625.2.a.b.1.1		2
25.9	even	10		225.2.h.b.181.1		4
25.14	even	10		225.2.h.b.46.1		4
60.59	even	2		10000.2.a.c.1.1		2
75.2	even	20		125.2.e.a.24.2		8
75.8	even	20		625.2.e.c.374.2		8
75.11	odd	10		125.2.d.a.101.1		4
75.14	odd	10		25.2.d.a.21.1	yes	4
75.17	even	20		625.2.e.c.374.1		8
75.23	even	20		125.2.e.a.24.1		8
75.29	odd	10		625.2.d.h.376.1		4
75.38	even	20		125.2.e.a.99.2		8
75.41	odd	10		125.2.d.a.26.1		4
75.44	odd	10		625.2.d.h.251.1		4
75.47	even	20		625.2.e.c.249.2		8
75.53	even	20		625.2.e.c.249.1		8
75.56	odd	10		625.2.d.b.251.1		4
75.59	odd	10		25.2.d.a.6.1	✓	4
75.62	even	20		125.2.e.a.99.1		8
75.71	odd	10		625.2.d.b.376.1		4
300.59	even	10		400.2.u.b.81.1		4
300.239	even	10		400.2.u.b.321.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.d.a.6.1	✓	4	75.59	odd	10
25.2.d.a.21.1	yes	4	75.14	odd	10
125.2.d.a.26.1		4	75.41	odd	10
125.2.d.a.101.1		4	75.11	odd	10
125.2.e.a.24.1		8	75.23	even	20
125.2.e.a.24.2		8	75.2	even	20
125.2.e.a.99.1		8	75.62	even	20
125.2.e.a.99.2		8	75.38	even	20
225.2.h.b.46.1		4	25.14	even	10
225.2.h.b.181.1		4	25.9	even	10
400.2.u.b.81.1		4	300.59	even	10
400.2.u.b.321.1		4	300.239	even	10
625.2.a.b.1.1		2	15.14	odd	2
625.2.a.c.1.2		2	3.2	odd	2
625.2.b.a.624.1		4	15.8	even	4
625.2.b.a.624.4		4	15.2	even	4
625.2.d.b.251.1		4	75.56	odd	10
625.2.d.b.376.1		4	75.71	odd	10
625.2.d.h.251.1		4	75.44	odd	10
625.2.d.h.376.1		4	75.29	odd	10
625.2.e.c.249.1		8	75.53	even	20
625.2.e.c.249.2		8	75.47	even	20
625.2.e.c.374.1		8	75.17	even	20
625.2.e.c.374.2		8	75.8	even	20
5625.2.a.d.1.1		2	1.1	even	1	trivial
5625.2.a.f.1.2		2	5.4	even	2
10000.2.a.c.1.1		2	60.59	even	2
10000.2.a.l.1.2		2	12.11	even	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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

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

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