LMFDB - Embedded newform 5625.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5625,2,Mod(1,5625)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,0,-2,0,0,0,6,0,0,-2,0,9,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{4} -4.47214 q^{7} +3.00000 q^{8} +1.23607 q^{11} +5.61803 q^{13} +4.47214 q^{14} -1.00000 q^{16} -3.85410 q^{17} +1.23607 q^{19} -1.23607 q^{22} -4.47214 q^{23} -5.61803 q^{26} +4.47214 q^{28} -6.61803 q^{29} +2.76393 q^{31} -5.00000 q^{32} +3.85410 q^{34} -3.09017 q^{37} -1.23607 q^{38} +3.61803 q^{41} +7.70820 q^{43} -1.23607 q^{44} +4.47214 q^{46} +0.763932 q^{47} +13.0000 q^{49} -5.61803 q^{52} -3.61803 q^{53} -13.4164 q^{56} +6.61803 q^{58} +4.00000 q^{59} +1.61803 q^{61} -2.76393 q^{62} +7.00000 q^{64} +0.763932 q^{67} +3.85410 q^{68} +5.23607 q^{71} -8.09017 q^{73} +3.09017 q^{74} -1.23607 q^{76} -5.52786 q^{77} -3.61803 q^{82} -12.4721 q^{83} -7.70820 q^{86} +3.70820 q^{88} -5.38197 q^{89} -25.1246 q^{91} +4.47214 q^{92} -0.763932 q^{94} +2.14590 q^{97} -13.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8} - 2 q^{11} + 9 q^{13} - 2 q^{16} - q^{17} - 2 q^{19} + 2 q^{22} - 9 q^{26} - 11 q^{29} + 10 q^{31} - 10 q^{32} + q^{34} + 5 q^{37} + 2 q^{38} + 5 q^{41} + 2 q^{43} + 2 q^{44}+ \cdots - 26 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8 - 2 * q^11 + 9 * q^13 - 2 * q^16 - q^17 - 2 * q^19 + 2 * q^22 - 9 * q^26 - 11 * q^29 + 10 * q^31 - 10 * q^32 + q^34 + 5 * q^37 + 2 * q^38 + 5 * q^41 + 2 * q^43 + 2 * q^44 + 6 * q^47 + 26 * q^49 - 9 * q^52 - 5 * q^53 + 11 * q^58 + 8 * q^59 + q^61 - 10 * q^62 + 14 * q^64 + 6 * q^67 + q^68 + 6 * q^71 - 5 * q^73 - 5 * q^74 + 2 * q^76 - 20 * q^77 - 5 * q^82 - 16 * q^83 - 2 * q^86 - 6 * q^88 - 13 * q^89 - 10 * q^91 - 6 * q^94 + 11 * q^97 - 26 * q^98

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	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.47214	−1.69031	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	−4.47214	−1.69031	−0.845154	+	0.534522i	$0.820491\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	5.61803	1.55816	0.779081	−	0.626923i	$-0.215686\pi$
$13$	5.61803	1.55816	0.779081	+	0.626923i	$0.215686\pi$
$14$	4.47214	1.19523
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−3.85410	−0.934757	−0.467379	−	0.884057i	$-0.654801\pi$
$17$	−3.85410	−0.934757	−0.467379	+	0.884057i	$0.654801\pi$
$18$	0	0
$19$	1.23607	0.283573	0.141787	−	0.989897i	$-0.454715\pi$
$19$	1.23607	0.283573	0.141787	+	0.989897i	$0.454715\pi$
$20$	0	0
$21$	0	0
$22$	−1.23607	−0.263531
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	0	0
$25$	0	0
$26$	−5.61803	−1.10179
$27$	0	0
$28$	4.47214	0.845154
$29$	−6.61803	−1.22894	−0.614469	−	0.788941i	$-0.710630\pi$
$29$	−6.61803	−1.22894	−0.614469	+	0.788941i	$0.710630\pi$
$30$	0	0
$31$	2.76393	0.496417	0.248208	−	0.968707i	$-0.420158\pi$
$31$	2.76393	0.496417	0.248208	+	0.968707i	$0.420158\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	3.85410	0.660973
$35$	0	0
$36$	0	0
$37$	−3.09017	−0.508021	−0.254010	−	0.967201i	$-0.581750\pi$
$37$	−3.09017	−0.508021	−0.254010	+	0.967201i	$0.581750\pi$
$38$	−1.23607	−0.200517
$39$	0	0
$40$	0	0
$41$	3.61803	0.565042	0.282521	−	0.959261i	$-0.408829\pi$
$41$	3.61803	0.565042	0.282521	+	0.959261i	$0.408829\pi$
$42$	0	0
$43$	7.70820	1.17549	0.587745	−	0.809046i	$-0.300016\pi$
$43$	7.70820	1.17549	0.587745	+	0.809046i	$0.300016\pi$
$44$	−1.23607	−0.186344
$45$	0	0
$46$	4.47214	0.659380
$47$	0.763932	0.111431	0.0557155	−	0.998447i	$-0.482256\pi$
$47$	0.763932	0.111431	0.0557155	+	0.998447i	$0.482256\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	0	0
$51$	0	0
$52$	−5.61803	−0.779081
$53$	−3.61803	−0.496975	−0.248488	−	0.968635i	$-0.579934\pi$
$53$	−3.61803	−0.496975	−0.248488	+	0.968635i	$0.579934\pi$
$54$	0	0
$55$	0	0
$56$	−13.4164	−1.79284
$57$	0	0
$58$	6.61803	0.868990
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	1.61803	0.207168	0.103584	−	0.994621i	$-0.466969\pi$
$61$	1.61803	0.207168	0.103584	+	0.994621i	$0.466969\pi$
$62$	−2.76393	−0.351020
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	0.763932	0.0933292	0.0466646	−	0.998911i	$-0.485141\pi$
$67$	0.763932	0.0933292	0.0466646	+	0.998911i	$0.485141\pi$
$68$	3.85410	0.467379
$69$	0	0
$70$	0	0
$71$	5.23607	0.621407	0.310703	−	0.950507i	$-0.399435\pi$
$71$	5.23607	0.621407	0.310703	+	0.950507i	$0.399435\pi$
$72$	0	0
$73$	−8.09017	−0.946883	−0.473441	−	0.880825i	$-0.656988\pi$
$73$	−8.09017	−0.946883	−0.473441	+	0.880825i	$0.656988\pi$
$74$	3.09017	0.359225
$75$	0	0
$76$	−1.23607	−0.141787
$77$	−5.52786	−0.629959
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	−3.61803	−0.399545
$83$	−12.4721	−1.36899	−0.684497	−	0.729015i	$-0.739978\pi$
$83$	−12.4721	−1.36899	−0.684497	+	0.729015i	$0.739978\pi$
$84$	0	0
$85$	0	0
$86$	−7.70820	−0.831197
$87$	0	0
$88$	3.70820	0.395296
$89$	−5.38197	−0.570487	−0.285244	−	0.958455i	$-0.592075\pi$
$89$	−5.38197	−0.570487	−0.285244	+	0.958455i	$0.592075\pi$
$90$	0	0
$91$	−25.1246	−2.63377
$92$	4.47214	0.466252
$93$	0	0
$94$	−0.763932	−0.0787936
$95$	0	0
$96$	0	0
$97$	2.14590	0.217883	0.108941	−	0.994048i	$-0.465254\pi$
$97$	2.14590	0.217883	0.108941	+	0.994048i	$0.465254\pi$
$98$	−13.0000	−1.31320
$99$	0	0
$100$	0	0
$101$	−3.56231	−0.354463	−0.177231	−	0.984169i	$-0.556714\pi$
$101$	−3.56231	−0.354463	−0.177231	+	0.984169i	$0.556714\pi$
$102$	0	0
$103$	3.23607	0.318859	0.159430	−	0.987209i	$-0.449034\pi$
$103$	3.23607	0.318859	0.159430	+	0.987209i	$0.449034\pi$
$104$	16.8541	1.65268
$105$	0	0
$106$	3.61803	0.351415
$107$	−7.52786	−0.727746	−0.363873	−	0.931449i	$-0.618546\pi$
$107$	−7.52786	−0.727746	−0.363873	+	0.931449i	$0.618546\pi$
$108$	0	0
$109$	−7.90983	−0.757624	−0.378812	−	0.925474i	$-0.623667\pi$
$109$	−7.90983	−0.757624	−0.378812	+	0.925474i	$0.623667\pi$
$110$	0	0
$111$	0	0
$112$	4.47214	0.422577
$113$	−18.3262	−1.72399	−0.861994	−	0.506919i	$-0.830784\pi$
$113$	−18.3262	−1.72399	−0.861994	+	0.506919i	$0.830784\pi$
$114$	0	0
$115$	0	0
$116$	6.61803	0.614469
$117$	0	0
$118$	−4.00000	−0.368230
$119$	17.2361	1.58003
$120$	0	0
$121$	−9.47214	−0.861103
$122$	−1.61803	−0.146490
$123$	0	0
$124$	−2.76393	−0.248208
$125$	0	0
$126$	0	0
$127$	−3.70820	−0.329050	−0.164525	−	0.986373i	$-0.552609\pi$
$127$	−3.70820	−0.329050	−0.164525	+	0.986373i	$0.552609\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	0	0
$131$	8.18034	0.714720	0.357360	−	0.933967i	$-0.383677\pi$
$131$	8.18034	0.714720	0.357360	+	0.933967i	$0.383677\pi$
$132$	0	0
$133$	−5.52786	−0.479327
$134$	−0.763932	−0.0659937
$135$	0	0
$136$	−11.5623	−0.991460
$137$	7.61803	0.650853	0.325426	−	0.945567i	$-0.394492\pi$
$137$	7.61803	0.650853	0.325426	+	0.945567i	$0.394492\pi$
$138$	0	0
$139$	−22.9443	−1.94611	−0.973054	−	0.230578i	$-0.925938\pi$
$139$	−22.9443	−1.94611	−0.973054	+	0.230578i	$0.925938\pi$
$140$	0	0
$141$	0	0
$142$	−5.23607	−0.439401
$143$	6.94427	0.580709
$144$	0	0
$145$	0	0
$146$	8.09017	0.669547
$147$	0	0
$148$	3.09017	0.254010
$149$	18.8541	1.54459	0.772294	−	0.635265i	$-0.219109\pi$
$149$	18.8541	1.54459	0.772294	+	0.635265i	$0.219109\pi$
$150$	0	0
$151$	7.52786	0.612609	0.306304	−	0.951934i	$-0.400907\pi$
$151$	7.52786	0.612609	0.306304	+	0.951934i	$0.400907\pi$
$152$	3.70820	0.300775
$153$	0	0
$154$	5.52786	0.445448
$155$	0	0
$156$	0	0
$157$	10.7984	0.861804	0.430902	−	0.902399i	$-0.358195\pi$
$157$	10.7984	0.861804	0.430902	+	0.902399i	$0.358195\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.0000	1.57622
$162$	0	0
$163$	−14.9443	−1.17053	−0.585263	−	0.810844i	$-0.699009\pi$
$163$	−14.9443	−1.17053	−0.585263	+	0.810844i	$0.699009\pi$
$164$	−3.61803	−0.282521
$165$	0	0
$166$	12.4721	0.968025
$167$	−3.41641	−0.264370	−0.132185	−	0.991225i	$-0.542199\pi$
$167$	−3.41641	−0.264370	−0.132185	+	0.991225i	$0.542199\pi$
$168$	0	0
$169$	18.5623	1.42787
$170$	0	0
$171$	0	0
$172$	−7.70820	−0.587745
$173$	−21.0902	−1.60346	−0.801728	−	0.597689i	$-0.796086\pi$
$173$	−21.0902	−1.60346	−0.801728	+	0.597689i	$0.796086\pi$
$174$	0	0
$175$	0	0
$176$	−1.23607	−0.0931721
$177$	0	0
$178$	5.38197	0.403395
$179$	16.1803	1.20938	0.604688	−	0.796463i	$-0.293298\pi$
$179$	16.1803	1.20938	0.604688	+	0.796463i	$0.293298\pi$
$180$	0	0
$181$	14.7984	1.09995	0.549977	−	0.835180i	$-0.314636\pi$
$181$	14.7984	1.09995	0.549977	+	0.835180i	$0.314636\pi$
$182$	25.1246	1.86236
$183$	0	0
$184$	−13.4164	−0.989071
$185$	0	0
$186$	0	0
$187$	−4.76393	−0.348373
$188$	−0.763932	−0.0557155
$189$	0	0
$190$	0	0
$191$	−6.65248	−0.481356	−0.240678	−	0.970605i	$-0.577370\pi$
$191$	−6.65248	−0.481356	−0.240678	+	0.970605i	$0.577370\pi$
$192$	0	0
$193$	23.3262	1.67906	0.839530	−	0.543314i	$-0.182831\pi$
$193$	23.3262	1.67906	0.839530	+	0.543314i	$0.182831\pi$
$194$	−2.14590	−0.154067
$195$	0	0
$196$	−13.0000	−0.928571
$197$	−7.61803	−0.542762	−0.271381	−	0.962472i	$-0.587480\pi$
$197$	−7.61803	−0.542762	−0.271381	+	0.962472i	$0.587480\pi$
$198$	0	0
$199$	12.6525	0.896910	0.448455	−	0.893805i	$-0.351974\pi$
$199$	12.6525	0.896910	0.448455	+	0.893805i	$0.351974\pi$
$200$	0	0
$201$	0	0
$202$	3.56231	0.250643
$203$	29.5967	2.07728
$204$	0	0
$205$	0	0
$206$	−3.23607	−0.225468
$207$	0	0
$208$	−5.61803	−0.389541
$209$	1.52786	0.105685
$210$	0	0
$211$	17.8885	1.23150	0.615749	−	0.787942i	$-0.288854\pi$
$211$	17.8885	1.23150	0.615749	+	0.787942i	$0.288854\pi$
$212$	3.61803	0.248488
$213$	0	0
$214$	7.52786	0.514594
$215$	0	0
$216$	0	0
$217$	−12.3607	−0.839098
$218$	7.90983	0.535721
$219$	0	0
$220$	0	0
$221$	−21.6525	−1.45650
$222$	0	0
$223$	8.18034	0.547796	0.273898	−	0.961759i	$-0.411687\pi$
$223$	8.18034	0.547796	0.273898	+	0.961759i	$0.411687\pi$
$224$	22.3607	1.49404
$225$	0	0
$226$	18.3262	1.21904
$227$	20.0000	1.32745	0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000	1.32745	0.663723	+	0.747978i	$0.268975\pi$
$228$	0	0
$229$	21.9787	1.45239	0.726197	−	0.687487i	$-0.241286\pi$
$229$	21.9787	1.45239	0.726197	+	0.687487i	$0.241286\pi$
$230$	0	0
$231$	0	0
$232$	−19.8541	−1.30349
$233$	−12.3820	−0.811170	−0.405585	−	0.914057i	$-0.632932\pi$
$233$	−12.3820	−0.811170	−0.405585	+	0.914057i	$0.632932\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−17.2361	−1.11725
$239$	24.9443	1.61351	0.806755	−	0.590886i	$-0.201222\pi$
$239$	24.9443	1.61351	0.806755	+	0.590886i	$0.201222\pi$
$240$	0	0
$241$	28.0344	1.80586	0.902929	−	0.429791i	$-0.141413\pi$
$241$	28.0344	1.80586	0.902929	+	0.429791i	$0.141413\pi$
$242$	9.47214	0.608892
$243$	0	0
$244$	−1.61803	−0.103584
$245$	0	0
$246$	0	0
$247$	6.94427	0.441853
$248$	8.29180	0.526530
$249$	0	0
$250$	0	0
$251$	9.05573	0.571592	0.285796	−	0.958290i	$-0.407742\pi$
$251$	9.05573	0.571592	0.285796	+	0.958290i	$0.407742\pi$
$252$	0	0
$253$	−5.52786	−0.347534
$254$	3.70820	0.232673
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−11.7984	−0.735962	−0.367981	−	0.929833i	$-0.619951\pi$
$257$	−11.7984	−0.735962	−0.367981	+	0.929833i	$0.619951\pi$
$258$	0	0
$259$	13.8197	0.858712
$260$	0	0
$261$	0	0
$262$	−8.18034	−0.505383
$263$	23.8885	1.47303	0.736515	−	0.676421i	$-0.236470\pi$
$263$	23.8885	1.47303	0.736515	+	0.676421i	$0.236470\pi$
$264$	0	0
$265$	0	0
$266$	5.52786	0.338935
$267$	0	0
$268$	−0.763932	−0.0466646
$269$	6.85410	0.417902	0.208951	−	0.977926i	$-0.432995\pi$
$269$	6.85410	0.417902	0.208951	+	0.977926i	$0.432995\pi$
$270$	0	0
$271$	−0.180340	−0.0109549	−0.00547743	−	0.999985i	$-0.501744\pi$
$271$	−0.180340	−0.0109549	−0.00547743	+	0.999985i	$0.501744\pi$
$272$	3.85410	0.233689
$273$	0	0
$274$	−7.61803	−0.460222
$275$	0	0
$276$	0	0
$277$	18.7984	1.12948	0.564742	−	0.825267i	$-0.308976\pi$
$277$	18.7984	1.12948	0.564742	+	0.825267i	$0.308976\pi$
$278$	22.9443	1.37611
$279$	0	0
$280$	0	0
$281$	19.6180	1.17031	0.585157	−	0.810920i	$-0.301033\pi$
$281$	19.6180	1.17031	0.585157	+	0.810920i	$0.301033\pi$
$282$	0	0
$283$	13.7082	0.814868	0.407434	−	0.913235i	$-0.366424\pi$
$283$	13.7082	0.814868	0.407434	+	0.913235i	$0.366424\pi$
$284$	−5.23607	−0.310703
$285$	0	0
$286$	−6.94427	−0.410623
$287$	−16.1803	−0.955095
$288$	0	0
$289$	−2.14590	−0.126229
$290$	0	0
$291$	0	0
$292$	8.09017	0.473441
$293$	20.7984	1.21505	0.607527	−	0.794299i	$-0.292162\pi$
$293$	20.7984	1.21505	0.607527	+	0.794299i	$0.292162\pi$
$294$	0	0
$295$	0	0
$296$	−9.27051	−0.538837
$297$	0	0
$298$	−18.8541	−1.09219
$299$	−25.1246	−1.45299
$300$	0	0
$301$	−34.4721	−1.98694
$302$	−7.52786	−0.433180
$303$	0	0
$304$	−1.23607	−0.0708934
$305$	0	0
$306$	0	0
$307$	32.6525	1.86358	0.931788	−	0.363004i	$-0.118249\pi$
$307$	32.6525	1.86358	0.931788	+	0.363004i	$0.118249\pi$
$308$	5.52786	0.314979
$309$	0	0
$310$	0	0
$311$	−17.7082	−1.00414	−0.502070	−	0.864827i	$-0.667428\pi$
$311$	−17.7082	−1.00414	−0.502070	+	0.864827i	$0.667428\pi$
$312$	0	0
$313$	−0.472136	−0.0266867	−0.0133434	−	0.999911i	$-0.504247\pi$
$313$	−0.472136	−0.0266867	−0.0133434	+	0.999911i	$0.504247\pi$
$314$	−10.7984	−0.609387
$315$	0	0
$316$	0	0
$317$	−28.8328	−1.61941	−0.809706	−	0.586836i	$-0.800373\pi$
$317$	−28.8328	−1.61941	−0.809706	+	0.586836i	$0.800373\pi$
$318$	0	0
$319$	−8.18034	−0.458011
$320$	0	0
$321$	0	0
$322$	−20.0000	−1.11456
$323$	−4.76393	−0.265072
$324$	0	0
$325$	0	0
$326$	14.9443	0.827687
$327$	0	0
$328$	10.8541	0.599318
$329$	−3.41641	−0.188353
$330$	0	0
$331$	−6.94427	−0.381692	−0.190846	−	0.981620i	$-0.561123\pi$
$331$	−6.94427	−0.381692	−0.190846	+	0.981620i	$0.561123\pi$
$332$	12.4721	0.684497
$333$	0	0
$334$	3.41641	0.186938
$335$	0	0
$336$	0	0
$337$	2.94427	0.160385	0.0801924	−	0.996779i	$-0.474447\pi$
$337$	2.94427	0.160385	0.0801924	+	0.996779i	$0.474447\pi$
$338$	−18.5623	−1.00966
$339$	0	0
$340$	0	0
$341$	3.41641	0.185009
$342$	0	0
$343$	−26.8328	−1.44884
$344$	23.1246	1.24680
$345$	0	0
$346$	21.0902	1.13381
$347$	20.2918	1.08932	0.544660	−	0.838657i	$-0.316659\pi$
$347$	20.2918	1.08932	0.544660	+	0.838657i	$0.316659\pi$
$348$	0	0
$349$	−4.03444	−0.215959	−0.107979	−	0.994153i	$-0.534438\pi$
$349$	−4.03444	−0.215959	−0.107979	+	0.994153i	$0.534438\pi$
$350$	0	0
$351$	0	0
$352$	−6.18034	−0.329413
$353$	−14.3607	−0.764342	−0.382171	−	0.924092i	$-0.624823\pi$
$353$	−14.3607	−0.764342	−0.382171	+	0.924092i	$0.624823\pi$
$354$	0	0
$355$	0	0
$356$	5.38197	0.285244
$357$	0	0
$358$	−16.1803	−0.855158
$359$	37.4164	1.97476	0.987381	−	0.158361i	$-0.0506211\pi$
$359$	37.4164	1.97476	0.987381	+	0.158361i	$0.0506211\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	−14.7984	−0.777785
$363$	0	0
$364$	25.1246	1.31689
$365$	0	0
$366$	0	0
$367$	−6.00000	−0.313197	−0.156599	−	0.987662i	$-0.550053\pi$
$367$	−6.00000	−0.313197	−0.156599	+	0.987662i	$0.550053\pi$
$368$	4.47214	0.233126
$369$	0	0
$370$	0	0
$371$	16.1803	0.840041
$372$	0	0
$373$	2.58359	0.133773	0.0668867	−	0.997761i	$-0.478693\pi$
$373$	2.58359	0.133773	0.0668867	+	0.997761i	$0.478693\pi$
$374$	4.76393	0.246337
$375$	0	0
$376$	2.29180	0.118190
$377$	−37.1803	−1.91488
$378$	0	0
$379$	−3.41641	−0.175489	−0.0877445	−	0.996143i	$-0.527966\pi$
$379$	−3.41641	−0.175489	−0.0877445	+	0.996143i	$0.527966\pi$
$380$	0	0
$381$	0	0
$382$	6.65248	0.340370
$383$	32.8328	1.67768	0.838839	−	0.544379i	$-0.183235\pi$
$383$	32.8328	1.67768	0.838839	+	0.544379i	$0.183235\pi$
$384$	0	0
$385$	0	0
$386$	−23.3262	−1.18727
$387$	0	0
$388$	−2.14590	−0.108941
$389$	25.9098	1.31368	0.656840	−	0.754030i	$-0.271893\pi$
$389$	25.9098	1.31368	0.656840	+	0.754030i	$0.271893\pi$
$390$	0	0
$391$	17.2361	0.871665
$392$	39.0000	1.96980
$393$	0	0
$394$	7.61803	0.383791
$395$	0	0
$396$	0	0
$397$	−6.94427	−0.348523	−0.174262	−	0.984699i	$-0.555754\pi$
$397$	−6.94427	−0.348523	−0.174262	+	0.984699i	$0.555754\pi$
$398$	−12.6525	−0.634211
$399$	0	0
$400$	0	0
$401$	22.4508	1.12114	0.560571	−	0.828106i	$-0.310582\pi$
$401$	22.4508	1.12114	0.560571	+	0.828106i	$0.310582\pi$
$402$	0	0
$403$	15.5279	0.773498
$404$	3.56231	0.177231
$405$	0	0
$406$	−29.5967	−1.46886
$407$	−3.81966	−0.189334
$408$	0	0
$409$	1.20163	0.0594166	0.0297083	−	0.999559i	$-0.490542\pi$
$409$	1.20163	0.0594166	0.0297083	+	0.999559i	$0.490542\pi$
$410$	0	0
$411$	0	0
$412$	−3.23607	−0.159430
$413$	−17.8885	−0.880238
$414$	0	0
$415$	0	0
$416$	−28.0902	−1.37723
$417$	0	0
$418$	−1.52786	−0.0747303
$419$	−15.0557	−0.735520	−0.367760	−	0.929921i	$-0.619875\pi$
$419$	−15.0557	−0.735520	−0.367760	+	0.929921i	$0.619875\pi$
$420$	0	0
$421$	29.8541	1.45500	0.727500	−	0.686108i	$-0.240682\pi$
$421$	29.8541	1.45500	0.727500	+	0.686108i	$0.240682\pi$
$422$	−17.8885	−0.870801
$423$	0	0
$424$	−10.8541	−0.527122
$425$	0	0
$426$	0	0
$427$	−7.23607	−0.350178
$428$	7.52786	0.363873
$429$	0	0
$430$	0	0
$431$	−20.6525	−0.994795	−0.497397	−	0.867523i	$-0.665711\pi$
$431$	−20.6525	−0.994795	−0.497397	+	0.867523i	$0.665711\pi$
$432$	0	0
$433$	−21.9787	−1.05623	−0.528115	−	0.849173i	$-0.677101\pi$
$433$	−21.9787	−1.05623	−0.528115	+	0.849173i	$0.677101\pi$
$434$	12.3607	0.593332
$435$	0	0
$436$	7.90983	0.378812
$437$	−5.52786	−0.264434
$438$	0	0
$439$	16.1803	0.772245	0.386123	−	0.922447i	$-0.373814\pi$
$439$	16.1803	0.772245	0.386123	+	0.922447i	$0.373814\pi$
$440$	0	0
$441$	0	0
$442$	21.6525	1.02990
$443$	35.2361	1.67412	0.837058	−	0.547114i	$-0.184274\pi$
$443$	35.2361	1.67412	0.837058	+	0.547114i	$0.184274\pi$
$444$	0	0
$445$	0	0
$446$	−8.18034	−0.387350
$447$	0	0
$448$	−31.3050	−1.47902
$449$	−16.7984	−0.792764	−0.396382	−	0.918086i	$-0.629734\pi$
$449$	−16.7984	−0.792764	−0.396382	+	0.918086i	$0.629734\pi$
$450$	0	0
$451$	4.47214	0.210585
$452$	18.3262	0.861994
$453$	0	0
$454$	−20.0000	−0.938647
$455$	0	0
$456$	0	0
$457$	−22.3607	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$457$	−22.3607	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$458$	−21.9787	−1.02700
$459$	0	0
$460$	0	0
$461$	32.7984	1.52757	0.763786	−	0.645469i	$-0.223338\pi$
$461$	32.7984	1.52757	0.763786	+	0.645469i	$0.223338\pi$
$462$	0	0
$463$	18.7639	0.872034	0.436017	−	0.899938i	$-0.356389\pi$
$463$	18.7639	0.872034	0.436017	+	0.899938i	$0.356389\pi$
$464$	6.61803	0.307235
$465$	0	0
$466$	12.3820	0.573583
$467$	−28.1803	−1.30403	−0.652015	−	0.758206i	$-0.726076\pi$
$467$	−28.1803	−1.30403	−0.652015	+	0.758206i	$0.726076\pi$
$468$	0	0
$469$	−3.41641	−0.157755
$470$	0	0
$471$	0	0
$472$	12.0000	0.552345
$473$	9.52786	0.438092
$474$	0	0
$475$	0	0
$476$	−17.2361	−0.790014
$477$	0	0
$478$	−24.9443	−1.14092
$479$	−18.9443	−0.865586	−0.432793	−	0.901493i	$-0.642472\pi$
$479$	−18.9443	−0.865586	−0.432793	+	0.901493i	$0.642472\pi$
$480$	0	0
$481$	−17.3607	−0.791579
$482$	−28.0344	−1.27693
$483$	0	0
$484$	9.47214	0.430552
$485$	0	0
$486$	0	0
$487$	9.70820	0.439921	0.219960	−	0.975509i	$-0.429407\pi$
$487$	9.70820	0.439921	0.219960	+	0.975509i	$0.429407\pi$
$488$	4.85410	0.219735
$489$	0	0
$490$	0	0
$491$	−5.88854	−0.265746	−0.132873	−	0.991133i	$-0.542420\pi$
$491$	−5.88854	−0.265746	−0.132873	+	0.991133i	$0.542420\pi$
$492$	0	0
$493$	25.5066	1.14876
$494$	−6.94427	−0.312438
$495$	0	0
$496$	−2.76393	−0.124104
$497$	−23.4164	−1.05037
$498$	0	0
$499$	−6.00000	−0.268597	−0.134298	−	0.990941i	$-0.542878\pi$
$499$	−6.00000	−0.268597	−0.134298	+	0.990941i	$0.542878\pi$
$500$	0	0
$501$	0	0
$502$	−9.05573	−0.404177
$503$	−8.58359	−0.382723	−0.191362	−	0.981520i	$-0.561290\pi$
$503$	−8.58359	−0.382723	−0.191362	+	0.981520i	$0.561290\pi$
$504$	0	0
$505$	0	0
$506$	5.52786	0.245744
$507$	0	0
$508$	3.70820	0.164525
$509$	−20.1459	−0.892951	−0.446476	−	0.894796i	$-0.647321\pi$
$509$	−20.1459	−0.892951	−0.446476	+	0.894796i	$0.647321\pi$
$510$	0	0
$511$	36.1803	1.60052
$512$	11.0000	0.486136
$513$	0	0
$514$	11.7984	0.520404
$515$	0	0
$516$	0	0
$517$	0.944272	0.0415290
$518$	−13.8197	−0.607201
$519$	0	0
$520$	0	0
$521$	27.0344	1.18440	0.592200	−	0.805791i	$-0.298259\pi$
$521$	27.0344	1.18440	0.592200	+	0.805791i	$0.298259\pi$
$522$	0	0
$523$	−25.7082	−1.12414	−0.562071	−	0.827089i	$-0.689995\pi$
$523$	−25.7082	−1.12414	−0.562071	+	0.827089i	$0.689995\pi$
$524$	−8.18034	−0.357360
$525$	0	0
$526$	−23.8885	−1.04159
$527$	−10.6525	−0.464029
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	0	0
$532$	5.52786	0.239663
$533$	20.3262	0.880427
$534$	0	0
$535$	0	0
$536$	2.29180	0.0989905
$537$	0	0
$538$	−6.85410	−0.295501
$539$	16.0689	0.692136
$540$	0	0
$541$	−19.6738	−0.845841	−0.422921	−	0.906167i	$-0.638995\pi$
$541$	−19.6738	−0.845841	−0.422921	+	0.906167i	$0.638995\pi$
$542$	0.180340	0.00774626
$543$	0	0
$544$	19.2705	0.826216
$545$	0	0
$546$	0	0
$547$	42.3607	1.81121	0.905606	−	0.424120i	$-0.139417\pi$
$547$	42.3607	1.81121	0.905606	+	0.424120i	$0.139417\pi$
$548$	−7.61803	−0.325426
$549$	0	0
$550$	0	0
$551$	−8.18034	−0.348494
$552$	0	0
$553$	0	0
$554$	−18.7984	−0.798666
$555$	0	0
$556$	22.9443	0.973054
$557$	11.2705	0.477547	0.238773	−	0.971075i	$-0.423255\pi$
$557$	11.2705	0.477547	0.238773	+	0.971075i	$0.423255\pi$
$558$	0	0
$559$	43.3050	1.83160
$560$	0	0
$561$	0	0
$562$	−19.6180	−0.827537
$563$	26.4721	1.11567	0.557834	−	0.829953i	$-0.311633\pi$
$563$	26.4721	1.11567	0.557834	+	0.829953i	$0.311633\pi$
$564$	0	0
$565$	0	0
$566$	−13.7082	−0.576199
$567$	0	0
$568$	15.7082	0.659102
$569$	−3.27051	−0.137107	−0.0685535	−	0.997647i	$-0.521838\pi$
$569$	−3.27051	−0.137107	−0.0685535	+	0.997647i	$0.521838\pi$
$570$	0	0
$571$	−16.7639	−0.701549	−0.350774	−	0.936460i	$-0.614082\pi$
$571$	−16.7639	−0.701549	−0.350774	+	0.936460i	$0.614082\pi$
$572$	−6.94427	−0.290355
$573$	0	0
$574$	16.1803	0.675354
$575$	0	0
$576$	0	0
$577$	−13.0557	−0.543517	−0.271759	−	0.962365i	$-0.587605\pi$
$577$	−13.0557	−0.543517	−0.271759	+	0.962365i	$0.587605\pi$
$578$	2.14590	0.0892576
$579$	0	0
$580$	0	0
$581$	55.7771	2.31402
$582$	0	0
$583$	−4.47214	−0.185217
$584$	−24.2705	−1.00432
$585$	0	0
$586$	−20.7984	−0.859173
$587$	41.4164	1.70944	0.854719	−	0.519091i	$-0.173729\pi$
$587$	41.4164	1.70944	0.854719	+	0.519091i	$0.173729\pi$
$588$	0	0
$589$	3.41641	0.140771
$590$	0	0
$591$	0	0
$592$	3.09017	0.127005
$593$	−9.74265	−0.400083	−0.200041	−	0.979787i	$-0.564108\pi$
$593$	−9.74265	−0.400083	−0.200041	+	0.979787i	$0.564108\pi$
$594$	0	0
$595$	0	0
$596$	−18.8541	−0.772294
$597$	0	0
$598$	25.1246	1.02742
$599$	3.52786	0.144145	0.0720723	−	0.997399i	$-0.477039\pi$
$599$	3.52786	0.144145	0.0720723	+	0.997399i	$0.477039\pi$
$600$	0	0
$601$	−8.67376	−0.353810	−0.176905	−	0.984228i	$-0.556609\pi$
$601$	−8.67376	−0.353810	−0.176905	+	0.984228i	$0.556609\pi$
$602$	34.4721	1.40498
$603$	0	0
$604$	−7.52786	−0.306304
$605$	0	0
$606$	0	0
$607$	−34.7639	−1.41102	−0.705512	−	0.708698i	$-0.749283\pi$
$607$	−34.7639	−1.41102	−0.705512	+	0.708698i	$0.749283\pi$
$608$	−6.18034	−0.250646
$609$	0	0
$610$	0	0
$611$	4.29180	0.173627
$612$	0	0
$613$	1.49342	0.0603188	0.0301594	−	0.999545i	$-0.490399\pi$
$613$	1.49342	0.0603188	0.0301594	+	0.999545i	$0.490399\pi$
$614$	−32.6525	−1.31775
$615$	0	0
$616$	−16.5836	−0.668172
$617$	19.9787	0.804313	0.402156	−	0.915571i	$-0.368261\pi$
$617$	19.9787	0.804313	0.402156	+	0.915571i	$0.368261\pi$
$618$	0	0
$619$	−16.3607	−0.657591	−0.328796	−	0.944401i	$-0.606643\pi$
$619$	−16.3607	−0.657591	−0.328796	+	0.944401i	$0.606643\pi$
$620$	0	0
$621$	0	0
$622$	17.7082	0.710034
$623$	24.0689	0.964299
$624$	0	0
$625$	0	0
$626$	0.472136	0.0188703
$627$	0	0
$628$	−10.7984	−0.430902
$629$	11.9098	0.474876
$630$	0	0
$631$	23.7082	0.943809	0.471904	−	0.881650i	$-0.343567\pi$
$631$	23.7082	0.943809	0.471904	+	0.881650i	$0.343567\pi$
$632$	0	0
$633$	0	0
$634$	28.8328	1.14510
$635$	0	0
$636$	0	0
$637$	73.0344	2.89373
$638$	8.18034	0.323863
$639$	0	0
$640$	0	0
$641$	−21.0557	−0.831651	−0.415826	−	0.909444i	$-0.636507\pi$
$641$	−21.0557	−0.831651	−0.415826	+	0.909444i	$0.636507\pi$
$642$	0	0
$643$	−21.8885	−0.863200	−0.431600	−	0.902065i	$-0.642051\pi$
$643$	−21.8885	−0.863200	−0.431600	+	0.902065i	$0.642051\pi$
$644$	−20.0000	−0.788110
$645$	0	0
$646$	4.76393	0.187434
$647$	34.3607	1.35086	0.675429	−	0.737425i	$-0.263959\pi$
$647$	34.3607	1.35086	0.675429	+	0.737425i	$0.263959\pi$
$648$	0	0
$649$	4.94427	0.194080
$650$	0	0
$651$	0	0
$652$	14.9443	0.585263
$653$	42.1033	1.64763	0.823815	−	0.566858i	$-0.191841\pi$
$653$	42.1033	1.64763	0.823815	+	0.566858i	$0.191841\pi$
$654$	0	0
$655$	0	0
$656$	−3.61803	−0.141260
$657$	0	0
$658$	3.41641	0.133185
$659$	33.4164	1.30172	0.650859	−	0.759198i	$-0.274409\pi$
$659$	33.4164	1.30172	0.650859	+	0.759198i	$0.274409\pi$
$660$	0	0
$661$	−1.41641	−0.0550919	−0.0275459	−	0.999621i	$-0.508769\pi$
$661$	−1.41641	−0.0550919	−0.0275459	+	0.999621i	$0.508769\pi$
$662$	6.94427	0.269897
$663$	0	0
$664$	−37.4164	−1.45204
$665$	0	0
$666$	0	0
$667$	29.5967	1.14599
$668$	3.41641	0.132185
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	5.20163	0.200508	0.100254	−	0.994962i	$-0.468034\pi$
$673$	5.20163	0.200508	0.100254	+	0.994962i	$0.468034\pi$
$674$	−2.94427	−0.113409
$675$	0	0
$676$	−18.5623	−0.713935
$677$	−10.5836	−0.406760	−0.203380	−	0.979100i	$-0.565193\pi$
$677$	−10.5836	−0.406760	−0.203380	+	0.979100i	$0.565193\pi$
$678$	0	0
$679$	−9.59675	−0.368289
$680$	0	0
$681$	0	0
$682$	−3.41641	−0.130821
$683$	19.4164	0.742948	0.371474	−	0.928443i	$-0.378852\pi$
$683$	19.4164	0.742948	0.371474	+	0.928443i	$0.378852\pi$
$684$	0	0
$685$	0	0
$686$	26.8328	1.02448
$687$	0	0
$688$	−7.70820	−0.293873
$689$	−20.3262	−0.774368
$690$	0	0
$691$	−16.7639	−0.637730	−0.318865	−	0.947800i	$-0.603302\pi$
$691$	−16.7639	−0.637730	−0.318865	+	0.947800i	$0.603302\pi$
$692$	21.0902	0.801728
$693$	0	0
$694$	−20.2918	−0.770266
$695$	0	0
$696$	0	0
$697$	−13.9443	−0.528177
$698$	4.03444	0.152706
$699$	0	0
$700$	0	0
$701$	−20.5623	−0.776628	−0.388314	−	0.921527i	$-0.626942\pi$
$701$	−20.5623	−0.776628	−0.388314	+	0.921527i	$0.626942\pi$
$702$	0	0
$703$	−3.81966	−0.144061
$704$	8.65248	0.326102
$705$	0	0
$706$	14.3607	0.540471
$707$	15.9311	0.599151
$708$	0	0
$709$	−40.2705	−1.51239	−0.756195	−	0.654346i	$-0.772944\pi$
$709$	−40.2705	−1.51239	−0.756195	+	0.654346i	$0.772944\pi$
$710$	0	0
$711$	0	0
$712$	−16.1459	−0.605093
$713$	−12.3607	−0.462911
$714$	0	0
$715$	0	0
$716$	−16.1803	−0.604688
$717$	0	0
$718$	−37.4164	−1.39637
$719$	0.111456	0.00415661	0.00207831	−	0.999998i	$-0.499338\pi$
$719$	0.111456	0.00415661	0.00207831	+	0.999998i	$0.499338\pi$
$720$	0	0
$721$	−14.4721	−0.538971
$722$	17.4721	0.650246
$723$	0	0
$724$	−14.7984	−0.549977
$725$	0	0
$726$	0	0
$727$	46.6525	1.73024	0.865122	−	0.501561i	$-0.167241\pi$
$727$	46.6525	1.73024	0.865122	+	0.501561i	$0.167241\pi$
$728$	−75.3738	−2.79354
$729$	0	0
$730$	0	0
$731$	−29.7082	−1.09880
$732$	0	0
$733$	29.4164	1.08652	0.543260	−	0.839565i	$-0.317190\pi$
$733$	29.4164	1.08652	0.543260	+	0.839565i	$0.317190\pi$
$734$	6.00000	0.221464
$735$	0	0
$736$	22.3607	0.824226
$737$	0.944272	0.0347827
$738$	0	0
$739$	4.29180	0.157876	0.0789381	−	0.996880i	$-0.474847\pi$
$739$	4.29180	0.157876	0.0789381	+	0.996880i	$0.474847\pi$
$740$	0	0
$741$	0	0
$742$	−16.1803	−0.593999
$743$	−41.1246	−1.50872	−0.754358	−	0.656463i	$-0.772052\pi$
$743$	−41.1246	−1.50872	−0.754358	+	0.656463i	$0.772052\pi$
$744$	0	0
$745$	0	0
$746$	−2.58359	−0.0945920
$747$	0	0
$748$	4.76393	0.174187
$749$	33.6656	1.23012
$750$	0	0
$751$	36.6525	1.33747	0.668734	−	0.743502i	$-0.266837\pi$
$751$	36.6525	1.33747	0.668734	+	0.743502i	$0.266837\pi$
$752$	−0.763932	−0.0278577
$753$	0	0
$754$	37.1803	1.35403
$755$	0	0
$756$	0	0
$757$	29.6180	1.07649	0.538243	−	0.842790i	$-0.319088\pi$
$757$	29.6180	1.07649	0.538243	+	0.842790i	$0.319088\pi$
$758$	3.41641	0.124090
$759$	0	0
$760$	0	0
$761$	−32.7426	−1.18692	−0.593460	−	0.804863i	$-0.702238\pi$
$761$	−32.7426	−1.18692	−0.593460	+	0.804863i	$0.702238\pi$
$762$	0	0
$763$	35.3738	1.28062
$764$	6.65248	0.240678
$765$	0	0
$766$	−32.8328	−1.18630
$767$	22.4721	0.811422
$768$	0	0
$769$	−50.3607	−1.81605	−0.908026	−	0.418913i	$-0.862411\pi$
$769$	−50.3607	−1.81605	−0.908026	+	0.418913i	$0.862411\pi$
$770$	0	0
$771$	0	0
$772$	−23.3262	−0.839530
$773$	−28.9230	−1.04029	−0.520144	−	0.854079i	$-0.674122\pi$
$773$	−28.9230	−1.04029	−0.520144	+	0.854079i	$0.674122\pi$
$774$	0	0
$775$	0	0
$776$	6.43769	0.231100
$777$	0	0
$778$	−25.9098	−0.928912
$779$	4.47214	0.160231
$780$	0	0
$781$	6.47214	0.231591
$782$	−17.2361	−0.616361
$783$	0	0
$784$	−13.0000	−0.464286
$785$	0	0
$786$	0	0
$787$	−14.1803	−0.505475	−0.252737	−	0.967535i	$-0.581331\pi$
$787$	−14.1803	−0.505475	−0.252737	+	0.967535i	$0.581331\pi$
$788$	7.61803	0.271381
$789$	0	0
$790$	0	0
$791$	81.9574	2.91407
$792$	0	0
$793$	9.09017	0.322801
$794$	6.94427	0.246443
$795$	0	0
$796$	−12.6525	−0.448455
$797$	49.1033	1.73933	0.869665	−	0.493643i	$-0.164335\pi$
$797$	49.1033	1.73933	0.869665	+	0.493643i	$0.164335\pi$
$798$	0	0
$799$	−2.94427	−0.104161
$800$	0	0
$801$	0	0
$802$	−22.4508	−0.792767
$803$	−10.0000	−0.352892
$804$	0	0
$805$	0	0
$806$	−15.5279	−0.546946
$807$	0	0
$808$	−10.6869	−0.375964
$809$	−28.2148	−0.991979	−0.495989	−	0.868329i	$-0.665195\pi$
$809$	−28.2148	−0.991979	−0.495989	+	0.868329i	$0.665195\pi$
$810$	0	0
$811$	4.76393	0.167284	0.0836421	−	0.996496i	$-0.473345\pi$
$811$	4.76393	0.167284	0.0836421	+	0.996496i	$0.473345\pi$
$812$	−29.5967	−1.03864
$813$	0	0
$814$	3.81966	0.133879
$815$	0	0
$816$	0	0
$817$	9.52786	0.333338
$818$	−1.20163	−0.0420139
$819$	0	0
$820$	0	0
$821$	−22.9443	−0.800761	−0.400380	−	0.916349i	$-0.631122\pi$
$821$	−22.9443	−0.800761	−0.400380	+	0.916349i	$0.631122\pi$
$822$	0	0
$823$	−19.2361	−0.670527	−0.335264	−	0.942124i	$-0.608825\pi$
$823$	−19.2361	−0.670527	−0.335264	+	0.942124i	$0.608825\pi$
$824$	9.70820	0.338201
$825$	0	0
$826$	17.8885	0.622422
$827$	36.0689	1.25424	0.627119	−	0.778923i	$-0.284234\pi$
$827$	36.0689	1.25424	0.627119	+	0.778923i	$0.284234\pi$
$828$	0	0
$829$	11.5066	0.399640	0.199820	−	0.979833i	$-0.435964\pi$
$829$	11.5066	0.399640	0.199820	+	0.979833i	$0.435964\pi$
$830$	0	0
$831$	0	0
$832$	39.3262	1.36339
$833$	−50.1033	−1.73598
$834$	0	0
$835$	0	0
$836$	−1.52786	−0.0528423
$837$	0	0
$838$	15.0557	0.520091
$839$	−14.0689	−0.485712	−0.242856	−	0.970062i	$-0.578084\pi$
$839$	−14.0689	−0.485712	−0.242856	+	0.970062i	$0.578084\pi$
$840$	0	0
$841$	14.7984	0.510289
$842$	−29.8541	−1.02884
$843$	0	0
$844$	−17.8885	−0.615749
$845$	0	0
$846$	0	0
$847$	42.3607	1.45553
$848$	3.61803	0.124244
$849$	0	0
$850$	0	0
$851$	13.8197	0.473732
$852$	0	0
$853$	23.1459	0.792500	0.396250	−	0.918143i	$-0.370311\pi$
$853$	23.1459	0.792500	0.396250	+	0.918143i	$0.370311\pi$
$854$	7.23607	0.247613
$855$	0	0
$856$	−22.5836	−0.771891
$857$	12.4721	0.426040	0.213020	−	0.977048i	$-0.431670\pi$
$857$	12.4721	0.426040	0.213020	+	0.977048i	$0.431670\pi$
$858$	0	0
$859$	−23.4164	−0.798958	−0.399479	−	0.916742i	$-0.630809\pi$
$859$	−23.4164	−0.798958	−0.399479	+	0.916742i	$0.630809\pi$
$860$	0	0
$861$	0	0
$862$	20.6525	0.703426
$863$	−10.8754	−0.370203	−0.185101	−	0.982719i	$-0.559261\pi$
$863$	−10.8754	−0.370203	−0.185101	+	0.982719i	$0.559261\pi$
$864$	0	0
$865$	0	0
$866$	21.9787	0.746867
$867$	0	0
$868$	12.3607	0.419549
$869$	0	0
$870$	0	0
$871$	4.29180	0.145422
$872$	−23.7295	−0.803582
$873$	0	0
$874$	5.52786	0.186983
$875$	0	0
$876$	0	0
$877$	−33.2148	−1.12158	−0.560792	−	0.827957i	$-0.689503\pi$
$877$	−33.2148	−1.12158	−0.560792	+	0.827957i	$0.689503\pi$
$878$	−16.1803	−0.546060
$879$	0	0
$880$	0	0
$881$	−42.9443	−1.44683	−0.723415	−	0.690414i	$-0.757428\pi$
$881$	−42.9443	−1.44683	−0.723415	+	0.690414i	$0.757428\pi$
$882$	0	0
$883$	22.1803	0.746428	0.373214	−	0.927745i	$-0.378256\pi$
$883$	22.1803	0.746428	0.373214	+	0.927745i	$0.378256\pi$
$884$	21.6525	0.728252
$885$	0	0
$886$	−35.2361	−1.18378
$887$	23.1246	0.776448	0.388224	−	0.921565i	$-0.373089\pi$
$887$	23.1246	0.776448	0.388224	+	0.921565i	$0.373089\pi$
$888$	0	0
$889$	16.5836	0.556196
$890$	0	0
$891$	0	0
$892$	−8.18034	−0.273898
$893$	0.944272	0.0315989
$894$	0	0
$895$	0	0
$896$	−13.4164	−0.448211
$897$	0	0
$898$	16.7984	0.560569
$899$	−18.2918	−0.610066
$900$	0	0
$901$	13.9443	0.464551
$902$	−4.47214	−0.148906
$903$	0	0
$904$	−54.9787	−1.82856
$905$	0	0
$906$	0	0
$907$	7.12461	0.236569	0.118284	−	0.992980i	$-0.462261\pi$
$907$	7.12461	0.236569	0.118284	+	0.992980i	$0.462261\pi$
$908$	−20.0000	−0.663723
$909$	0	0
$910$	0	0
$911$	−18.1803	−0.602342	−0.301171	−	0.953570i	$-0.597377\pi$
$911$	−18.1803	−0.602342	−0.301171	+	0.953570i	$0.597377\pi$
$912$	0	0
$913$	−15.4164	−0.510209
$914$	22.3607	0.739626
$915$	0	0
$916$	−21.9787	−0.726197
$917$	−36.5836	−1.20810
$918$	0	0
$919$	49.1935	1.62274	0.811372	−	0.584530i	$-0.198721\pi$
$919$	49.1935	1.62274	0.811372	+	0.584530i	$0.198721\pi$
$920$	0	0
$921$	0	0
$922$	−32.7984	−1.08016
$923$	29.4164	0.968253
$924$	0	0
$925$	0	0
$926$	−18.7639	−0.616621
$927$	0	0
$928$	33.0902	1.08624
$929$	9.09017	0.298239	0.149119	−	0.988819i	$-0.452356\pi$
$929$	9.09017	0.298239	0.149119	+	0.988819i	$0.452356\pi$
$930$	0	0
$931$	16.0689	0.526636
$932$	12.3820	0.405585
$933$	0	0
$934$	28.1803	0.922089
$935$	0	0
$936$	0	0
$937$	−48.6869	−1.59053	−0.795266	−	0.606260i	$-0.792669\pi$
$937$	−48.6869	−1.59053	−0.795266	+	0.606260i	$0.792669\pi$
$938$	3.41641	0.111550
$939$	0	0
$940$	0	0
$941$	10.3262	0.336626	0.168313	−	0.985734i	$-0.446168\pi$
$941$	10.3262	0.336626	0.168313	+	0.985734i	$0.446168\pi$
$942$	0	0
$943$	−16.1803	−0.526904
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−9.52786	−0.309778
$947$	−2.58359	−0.0839555	−0.0419777	−	0.999119i	$-0.513366\pi$
$947$	−2.58359	−0.0839555	−0.0419777	+	0.999119i	$0.513366\pi$
$948$	0	0
$949$	−45.4508	−1.47540
$950$	0	0
$951$	0	0
$952$	51.7082	1.67587
$953$	1.09017	0.0353141	0.0176570	−	0.999844i	$-0.494379\pi$
$953$	1.09017	0.0353141	0.0176570	+	0.999844i	$0.494379\pi$
$954$	0	0
$955$	0	0
$956$	−24.9443	−0.806755
$957$	0	0
$958$	18.9443	0.612062
$959$	−34.0689	−1.10014
$960$	0	0
$961$	−23.3607	−0.753570
$962$	17.3607	0.559731
$963$	0	0
$964$	−28.0344	−0.902929
$965$	0	0
$966$	0	0
$967$	−18.8328	−0.605623	−0.302811	−	0.953051i	$-0.597925\pi$
$967$	−18.8328	−0.605623	−0.302811	+	0.953051i	$0.597925\pi$
$968$	−28.4164	−0.913338
$969$	0	0
$970$	0	0
$971$	33.4164	1.07238	0.536192	−	0.844096i	$-0.319862\pi$
$971$	33.4164	1.07238	0.536192	+	0.844096i	$0.319862\pi$
$972$	0	0
$973$	102.610	3.28952
$974$	−9.70820	−0.311071
$975$	0	0
$976$	−1.61803	−0.0517920
$977$	3.25735	0.104212	0.0521060	−	0.998642i	$-0.483407\pi$
$977$	3.25735	0.104212	0.0521060	+	0.998642i	$0.483407\pi$
$978$	0	0
$979$	−6.65248	−0.212614
$980$	0	0
$981$	0	0
$982$	5.88854	0.187911
$983$	−8.29180	−0.264467	−0.132234	−	0.991219i	$-0.542215\pi$
$983$	−8.29180	−0.264467	−0.132234	+	0.991219i	$0.542215\pi$
$984$	0	0
$985$	0	0
$986$	−25.5066	−0.812295
$987$	0	0
$988$	−6.94427	−0.220927
$989$	−34.4721	−1.09615
$990$	0	0
$991$	37.1246	1.17930	0.589651	−	0.807658i	$-0.299265\pi$
$991$	37.1246	1.17930	0.589651	+	0.807658i	$0.299265\pi$
$992$	−13.8197	−0.438775
$993$	0	0
$994$	23.4164	0.742723
$995$	0	0
$996$	0	0
$997$	−17.4164	−0.551583	−0.275792	−	0.961217i	$-0.588940\pi$
$997$	−17.4164	−0.551583	−0.275792	+	0.961217i	$0.588940\pi$
$998$	6.00000	0.189927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.a.1.1		2
3.2	odd	2		1875.2.a.d.1.1		2
5.4	even	2		5625.2.a.h.1.2		2
15.2	even	4		1875.2.b.b.1249.3		4
15.8	even	4		1875.2.b.b.1249.2		4
15.14	odd	2		1875.2.a.a.1.2		2
25.11	even	5		225.2.h.a.46.1		4
25.16	even	5		225.2.h.a.181.1		4
75.2	even	20		375.2.i.a.274.2		8
75.11	odd	10		75.2.g.a.46.1	yes	4
75.14	odd	10		375.2.g.a.226.1		4
75.23	even	20		375.2.i.a.274.1		8
75.38	even	20		375.2.i.a.349.2		8
75.41	odd	10		75.2.g.a.31.1	✓	4
75.59	odd	10		375.2.g.a.151.1		4
75.62	even	20		375.2.i.a.349.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.g.a.31.1	✓	4	75.41	odd	10
75.2.g.a.46.1	yes	4	75.11	odd	10
225.2.h.a.46.1		4	25.11	even	5
225.2.h.a.181.1		4	25.16	even	5
375.2.g.a.151.1		4	75.59	odd	10
375.2.g.a.226.1		4	75.14	odd	10
375.2.i.a.274.1		8	75.23	even	20
375.2.i.a.274.2		8	75.2	even	20
375.2.i.a.349.1		8	75.62	even	20
375.2.i.a.349.2		8	75.38	even	20
1875.2.a.a.1.2		2	15.14	odd	2
1875.2.a.d.1.1		2	3.2	odd	2
1875.2.b.b.1249.2		4	15.8	even	4
1875.2.b.b.1249.3		4	15.2	even	4
5625.2.a.a.1.1		2	1.1	even	1	trivial
5625.2.a.h.1.2		2	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{4} -4.47214 q^{7} +3.00000 q^{8} +1.23607 q^{11} +5.61803 q^{13} +4.47214 q^{14} -1.00000 q^{16} -3.85410 q^{17} +1.23607 q^{19} -1.23607 q^{22} -4.47214 q^{23} -5.61803 q^{26} +4.47214 q^{28} -6.61803 q^{29} +2.76393 q^{31} -5.00000 q^{32} +3.85410 q^{34} -3.09017 q^{37} -1.23607 q^{38} +3.61803 q^{41} +7.70820 q^{43} -1.23607 q^{44} +4.47214 q^{46} +0.763932 q^{47} +13.0000 q^{49} -5.61803 q^{52} -3.61803 q^{53} -13.4164 q^{56} +6.61803 q^{58} +4.00000 q^{59} +1.61803 q^{61} -2.76393 q^{62} +7.00000 q^{64} +0.763932 q^{67} +3.85410 q^{68} +5.23607 q^{71} -8.09017 q^{73} +3.09017 q^{74} -1.23607 q^{76} -5.52786 q^{77} -3.61803 q^{82} -12.4721 q^{83} -7.70820 q^{86} +3.70820 q^{88} -5.38197 q^{89} -25.1246 q^{91} +4.47214 q^{92} -0.763932 q^{94} +2.14590 q^{97} -13.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8} - 2 q^{11} + 9 q^{13} - 2 q^{16} - q^{17} - 2 q^{19} + 2 q^{22} - 9 q^{26} - 11 q^{29} + 10 q^{31} - 10 q^{32} + q^{34} + 5 q^{37} + 2 q^{38} + 5 q^{41} + 2 q^{43} + 2 q^{44}+ \cdots - 26 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8 - 2 * q^11 + 9 * q^13 - 2 * q^16 - q^17 - 2 * q^19 + 2 * q^22 - 9 * q^26 - 11 * q^29 + 10 * q^31 - 10 * q^32 + q^34 + 5 * q^37 + 2 * q^38 + 5 * q^41 + 2 * q^43 + 2 * q^44 + 6 * q^47 + 26 * q^49 - 9 * q^52 - 5 * q^53 + 11 * q^58 + 8 * q^59 + q^61 - 10 * q^62 + 14 * q^64 + 6 * q^67 + q^68 + 6 * q^71 - 5 * q^73 - 5 * q^74 + 2 * q^76 - 20 * q^77 - 5 * q^82 - 16 * q^83 - 2 * q^86 - 6 * q^88 - 13 * q^89 - 10 * q^91 - 6 * q^94 + 11 * q^97 - 26 * q^98

Introduction

L-functions

Modular forms

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Database

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