LMFDB - Embedded newform 5625.2.a.c.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:	$1$
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 625)
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)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} +3.85410 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-1.61803 q^{2} +0.618034 q^{4} +3.85410 q^{7} +2.23607 q^{8} -0.618034 q^{11} -5.47214 q^{13} -6.23607 q^{14} -4.85410 q^{16} +1.47214 q^{17} -0.854102 q^{19} +1.00000 q^{22} +1.85410 q^{23} +8.85410 q^{26} +2.38197 q^{28} +2.76393 q^{29} +2.00000 q^{31} +3.38197 q^{32} -2.38197 q^{34} +3.00000 q^{37} +1.38197 q^{38} -6.47214 q^{41} -0.472136 q^{43} -0.381966 q^{44} -3.00000 q^{46} -11.6180 q^{47} +7.85410 q^{49} -3.38197 q^{52} -8.47214 q^{53} +8.61803 q^{56} -4.47214 q^{58} +13.9443 q^{59} -8.85410 q^{61} -3.23607 q^{62} +4.23607 q^{64} -11.4721 q^{67} +0.909830 q^{68} +3.32624 q^{71} -0.145898 q^{73} -4.85410 q^{74} -0.527864 q^{76} -2.38197 q^{77} -6.70820 q^{79} +10.4721 q^{82} -12.9443 q^{83} +0.763932 q^{86} -1.38197 q^{88} -3.61803 q^{89} -21.0902 q^{91} +1.14590 q^{92} +18.7984 q^{94} -10.6180 q^{97} -12.7082 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + q^{11} - 2 q^{13} - 8 q^{14} - 3 q^{16} - 6 q^{17} + 5 q^{19} + 2 q^{22} - 3 q^{23} + 11 q^{26} + 7 q^{28} + 10 q^{29} + 4 q^{31} + 9 q^{32} - 7 q^{34} + 6 q^{37} + 5 q^{38} - 4 q^{41} + 8 q^{43} - 3 q^{44} - 6 q^{46} - 21 q^{47} + 9 q^{49} - 9 q^{52} - 8 q^{53} + 15 q^{56} + 10 q^{59} - 11 q^{61} - 2 q^{62} + 4 q^{64} - 14 q^{67} + 13 q^{68} - 9 q^{71} - 7 q^{73} - 3 q^{74} - 10 q^{76} - 7 q^{77} + 12 q^{82} - 8 q^{83} + 6 q^{86} - 5 q^{88} - 5 q^{89} - 31 q^{91} + 9 q^{92} + 13 q^{94} - 19 q^{97} - 12 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + q^7 + q^11 - 2 * q^13 - 8 * q^14 - 3 * q^16 - 6 * q^17 + 5 * q^19 + 2 * q^22 - 3 * q^23 + 11 * q^26 + 7 * q^28 + 10 * q^29 + 4 * q^31 + 9 * q^32 - 7 * q^34 + 6 * q^37 + 5 * q^38 - 4 * q^41 + 8 * q^43 - 3 * q^44 - 6 * q^46 - 21 * q^47 + 9 * q^49 - 9 * q^52 - 8 * q^53 + 15 * q^56 + 10 * q^59 - 11 * q^61 - 2 * q^62 + 4 * q^64 - 14 * q^67 + 13 * q^68 - 9 * q^71 - 7 * q^73 - 3 * q^74 - 10 * q^76 - 7 * q^77 + 12 * q^82 - 8 * q^83 + 6 * q^86 - 5 * q^88 - 5 * q^89 - 31 * q^91 + 9 * q^92 + 13 * q^94 - 19 * q^97 - 12 * 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
$7$	3.85410	1.45671	0.728357	−	0.685198i	$-0.240284\pi$
$7$	3.85410	1.45671	0.728357	+	0.685198i	$0.240284\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	0	0
$11$	−0.618034	−0.186344	−0.0931721	−	0.995650i	$-0.529701\pi$
$11$	−0.618034	−0.186344	−0.0931721	+	0.995650i	$0.529701\pi$
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	−6.23607	−1.66666
$15$	0	0
$16$	−4.85410	−1.21353
$17$	1.47214	0.357045	0.178523	−	0.983936i	$-0.442868\pi$
$17$	1.47214	0.357045	0.178523	+	0.983936i	$0.442868\pi$
$18$	0	0
$19$	−0.854102	−0.195944	−0.0979722	−	0.995189i	$-0.531236\pi$
$19$	−0.854102	−0.195944	−0.0979722	+	0.995189i	$0.531236\pi$
$20$	0	0
$21$	0	0
$22$	1.00000	0.213201
$23$	1.85410	0.386607	0.193303	−	0.981139i	$-0.438080\pi$
$23$	1.85410	0.386607	0.193303	+	0.981139i	$0.438080\pi$
$24$	0	0
$25$	0	0
$26$	8.85410	1.73643
$27$	0	0
$28$	2.38197	0.450149
$29$	2.76393	0.513249	0.256625	−	0.966511i	$-0.417390\pi$
$29$	2.76393	0.513249	0.256625	+	0.966511i	$0.417390\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	−2.38197	−0.408504
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	1.38197	0.224184
$39$	0	0
$40$	0	0
$41$	−6.47214	−1.01078	−0.505389	−	0.862892i	$-0.668651\pi$
$41$	−6.47214	−1.01078	−0.505389	+	0.862892i	$0.668651\pi$
$42$	0	0
$43$	−0.472136	−0.0720001	−0.0360000	−	0.999352i	$-0.511462\pi$
$43$	−0.472136	−0.0720001	−0.0360000	+	0.999352i	$0.511462\pi$
$44$	−0.381966	−0.0575835
$45$	0	0
$46$	−3.00000	−0.442326
$47$	−11.6180	−1.69466	−0.847332	−	0.531063i	$-0.821793\pi$
$47$	−11.6180	−1.69466	−0.847332	+	0.531063i	$0.821793\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	0	0
$51$	0	0
$52$	−3.38197	−0.468994
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	0	0
$56$	8.61803	1.15163
$57$	0	0
$58$	−4.47214	−0.587220
$59$	13.9443	1.81539	0.907695	−	0.419631i	$-0.137841\pi$
$59$	13.9443	1.81539	0.907695	+	0.419631i	$0.137841\pi$
$60$	0	0
$61$	−8.85410	−1.13365	−0.566826	−	0.823838i	$-0.691829\pi$
$61$	−8.85410	−1.13365	−0.566826	+	0.823838i	$0.691829\pi$
$62$	−3.23607	−0.410981
$63$	0	0
$64$	4.23607	0.529508
$65$	0	0
$66$	0	0
$67$	−11.4721	−1.40154	−0.700772	−	0.713385i	$-0.747161\pi$
$67$	−11.4721	−1.40154	−0.700772	+	0.713385i	$0.747161\pi$
$68$	0.909830	0.110333
$69$	0	0
$70$	0	0
$71$	3.32624	0.394752	0.197376	−	0.980328i	$-0.436758\pi$
$71$	3.32624	0.394752	0.197376	+	0.980328i	$0.436758\pi$
$72$	0	0
$73$	−0.145898	−0.0170761	−0.00853804	−	0.999964i	$-0.502718\pi$
$73$	−0.145898	−0.0170761	−0.00853804	+	0.999964i	$0.502718\pi$
$74$	−4.85410	−0.564278
$75$	0	0
$76$	−0.527864	−0.0605502
$77$	−2.38197	−0.271450
$78$	0	0
$79$	−6.70820	−0.754732	−0.377366	−	0.926064i	$-0.623170\pi$
$79$	−6.70820	−0.754732	−0.377366	+	0.926064i	$0.623170\pi$
$80$	0	0
$81$	0	0
$82$	10.4721	1.15645
$83$	−12.9443	−1.42082	−0.710409	−	0.703789i	$-0.751490\pi$
$83$	−12.9443	−1.42082	−0.710409	+	0.703789i	$0.751490\pi$
$84$	0	0
$85$	0	0
$86$	0.763932	0.0823769
$87$	0	0
$88$	−1.38197	−0.147318
$89$	−3.61803	−0.383511	−0.191755	−	0.981443i	$-0.561418\pi$
$89$	−3.61803	−0.383511	−0.191755	+	0.981443i	$0.561418\pi$
$90$	0	0
$91$	−21.0902	−2.21085
$92$	1.14590	0.119468
$93$	0	0
$94$	18.7984	1.93890
$95$	0	0
$96$	0	0
$97$	−10.6180	−1.07810	−0.539049	−	0.842274i	$-0.681216\pi$
$97$	−10.6180	−1.07810	−0.539049	+	0.842274i	$0.681216\pi$
$98$	−12.7082	−1.28372
$99$	0	0
$100$	0	0
$101$	−9.76393	−0.971548	−0.485774	−	0.874085i	$-0.661462\pi$
$101$	−9.76393	−0.971548	−0.485774	+	0.874085i	$0.661462\pi$
$102$	0	0
$103$	−5.14590	−0.507040	−0.253520	−	0.967330i	$-0.581588\pi$
$103$	−5.14590	−0.507040	−0.253520	+	0.967330i	$0.581588\pi$
$104$	−12.2361	−1.19985
$105$	0	0
$106$	13.7082	1.33146
$107$	11.7984	1.14059	0.570296	−	0.821439i	$-0.306829\pi$
$107$	11.7984	1.14059	0.570296	+	0.821439i	$0.306829\pi$
$108$	0	0
$109$	15.8541	1.51855	0.759274	−	0.650771i	$-0.225554\pi$
$109$	15.8541	1.51855	0.759274	+	0.650771i	$0.225554\pi$
$110$	0	0
$111$	0	0
$112$	−18.7082	−1.76776
$113$	−0.708204	−0.0666222	−0.0333111	−	0.999445i	$-0.510605\pi$
$113$	−0.708204	−0.0666222	−0.0333111	+	0.999445i	$0.510605\pi$
$114$	0	0
$115$	0	0
$116$	1.70820	0.158603
$117$	0	0
$118$	−22.5623	−2.07703
$119$	5.67376	0.520113
$120$	0	0
$121$	−10.6180	−0.965276
$122$	14.3262	1.29704
$123$	0	0
$124$	1.23607	0.111002
$125$	0	0
$126$	0	0
$127$	7.14590	0.634096	0.317048	−	0.948410i	$-0.397308\pi$
$127$	7.14590	0.634096	0.317048	+	0.948410i	$0.397308\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	0	0
$131$	0.763932	0.0667451	0.0333725	−	0.999443i	$-0.489375\pi$
$131$	0.763932	0.0667451	0.0333725	+	0.999443i	$0.489375\pi$
$132$	0	0
$133$	−3.29180	−0.285435
$134$	18.5623	1.60354
$135$	0	0
$136$	3.29180	0.282269
$137$	−20.2361	−1.72888	−0.864442	−	0.502733i	$-0.832328\pi$
$137$	−20.2361	−1.72888	−0.864442	+	0.502733i	$0.832328\pi$
$138$	0	0
$139$	8.41641	0.713870	0.356935	−	0.934129i	$-0.383822\pi$
$139$	8.41641	0.713870	0.356935	+	0.934129i	$0.383822\pi$
$140$	0	0
$141$	0	0
$142$	−5.38197	−0.451645
$143$	3.38197	0.282814
$144$	0	0
$145$	0	0
$146$	0.236068	0.0195371
$147$	0	0
$148$	1.85410	0.152406
$149$	13.6180	1.11563	0.557816	−	0.829964i	$-0.311639\pi$
$149$	13.6180	1.11563	0.557816	+	0.829964i	$0.311639\pi$
$150$	0	0
$151$	15.9443	1.29753	0.648763	−	0.760990i	$-0.275287\pi$
$151$	15.9443	1.29753	0.648763	+	0.760990i	$0.275287\pi$
$152$	−1.90983	−0.154908
$153$	0	0
$154$	3.85410	0.310572
$155$	0	0
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	10.8541	0.863506
$159$	0	0
$160$	0	0
$161$	7.14590	0.563176
$162$	0	0
$163$	−1.85410	−0.145224	−0.0726122	−	0.997360i	$-0.523134\pi$
$163$	−1.85410	−0.145224	−0.0726122	+	0.997360i	$0.523134\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	20.9443	1.62559
$167$	−17.4721	−1.35203	−0.676017	−	0.736886i	$-0.736296\pi$
$167$	−17.4721	−1.35203	−0.676017	+	0.736886i	$0.736296\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	0	0
$171$	0	0
$172$	−0.291796	−0.0222492
$173$	−15.7082	−1.19427	−0.597136	−	0.802140i	$-0.703695\pi$
$173$	−15.7082	−1.19427	−0.597136	+	0.802140i	$0.703695\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	5.85410	0.438783
$179$	20.1246	1.50418	0.752092	−	0.659058i	$-0.229045\pi$
$179$	20.1246	1.50418	0.752092	+	0.659058i	$0.229045\pi$
$180$	0	0
$181$	−13.3262	−0.990531	−0.495266	−	0.868742i	$-0.664929\pi$
$181$	−13.3262	−0.990531	−0.495266	+	0.868742i	$0.664929\pi$
$182$	34.1246	2.52948
$183$	0	0
$184$	4.14590	0.305640
$185$	0	0
$186$	0	0
$187$	−0.909830	−0.0665334
$188$	−7.18034	−0.523680
$189$	0	0
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	22.9443	1.65156	0.825782	−	0.563989i	$-0.190734\pi$
$193$	22.9443	1.65156	0.825782	+	0.563989i	$0.190734\pi$
$194$	17.1803	1.23348
$195$	0	0
$196$	4.85410	0.346722
$197$	16.4721	1.17359	0.586796	−	0.809735i	$-0.300389\pi$
$197$	16.4721	1.17359	0.586796	+	0.809735i	$0.300389\pi$
$198$	0	0
$199$	−10.8541	−0.769427	−0.384713	−	0.923036i	$-0.625700\pi$
$199$	−10.8541	−0.769427	−0.384713	+	0.923036i	$0.625700\pi$
$200$	0	0
$201$	0	0
$202$	15.7984	1.11157
$203$	10.6525	0.747657
$204$	0	0
$205$	0	0
$206$	8.32624	0.580116
$207$	0	0
$208$	26.5623	1.84176
$209$	0.527864	0.0365131
$210$	0	0
$211$	13.1803	0.907372	0.453686	−	0.891162i	$-0.350109\pi$
$211$	13.1803	0.907372	0.453686	+	0.891162i	$0.350109\pi$
$212$	−5.23607	−0.359615
$213$	0	0
$214$	−19.0902	−1.30498
$215$	0	0
$216$	0	0
$217$	7.70820	0.523267
$218$	−25.6525	−1.73740
$219$	0	0
$220$	0	0
$221$	−8.05573	−0.541887
$222$	0	0
$223$	−0.145898	−0.00977005	−0.00488503	−	0.999988i	$-0.501555\pi$
$223$	−0.145898	−0.00977005	−0.00488503	+	0.999988i	$0.501555\pi$
$224$	13.0344	0.870900
$225$	0	0
$226$	1.14590	0.0762240
$227$	−0.763932	−0.0507039	−0.0253520	−	0.999679i	$-0.508071\pi$
$227$	−0.763932	−0.0507039	−0.0253520	+	0.999679i	$0.508071\pi$
$228$	0	0
$229$	−3.29180	−0.217528	−0.108764	−	0.994068i	$-0.534689\pi$
$229$	−3.29180	−0.217528	−0.108764	+	0.994068i	$0.534689\pi$
$230$	0	0
$231$	0	0
$232$	6.18034	0.405759
$233$	−22.0902	−1.44718	−0.723588	−	0.690233i	$-0.757508\pi$
$233$	−22.0902	−1.44718	−0.723588	+	0.690233i	$0.757508\pi$
$234$	0	0
$235$	0	0
$236$	8.61803	0.560986
$237$	0	0
$238$	−9.18034	−0.595073
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−13.8541	−0.892421	−0.446211	−	0.894928i	$-0.647227\pi$
$241$	−13.8541	−0.892421	−0.446211	+	0.894928i	$0.647227\pi$
$242$	17.1803	1.10439
$243$	0	0
$244$	−5.47214	−0.350318
$245$	0	0
$246$	0	0
$247$	4.67376	0.297384
$248$	4.47214	0.283981
$249$	0	0
$250$	0	0
$251$	9.18034	0.579458	0.289729	−	0.957109i	$-0.406435\pi$
$251$	9.18034	0.579458	0.289729	+	0.957109i	$0.406435\pi$
$252$	0	0
$253$	−1.14590	−0.0720420
$254$	−11.5623	−0.725484
$255$	0	0
$256$	13.5623	0.847644
$257$	3.70820	0.231311	0.115656	−	0.993289i	$-0.463103\pi$
$257$	3.70820	0.231311	0.115656	+	0.993289i	$0.463103\pi$
$258$	0	0
$259$	11.5623	0.718447
$260$	0	0
$261$	0	0
$262$	−1.23607	−0.0763645
$263$	−19.3262	−1.19171	−0.595853	−	0.803093i	$-0.703186\pi$
$263$	−19.3262	−1.19171	−0.595853	+	0.803093i	$0.703186\pi$
$264$	0	0
$265$	0	0
$266$	5.32624	0.326573
$267$	0	0
$268$	−7.09017	−0.433101
$269$	−3.81966	−0.232889	−0.116444	−	0.993197i	$-0.537150\pi$
$269$	−3.81966	−0.232889	−0.116444	+	0.993197i	$0.537150\pi$
$270$	0	0
$271$	−13.0000	−0.789694	−0.394847	−	0.918747i	$-0.629202\pi$
$271$	−13.0000	−0.789694	−0.394847	+	0.918747i	$0.629202\pi$
$272$	−7.14590	−0.433284
$273$	0	0
$274$	32.7426	1.97806
$275$	0	0
$276$	0	0
$277$	−15.2918	−0.918795	−0.459397	−	0.888231i	$-0.651935\pi$
$277$	−15.2918	−0.918795	−0.459397	+	0.888231i	$0.651935\pi$
$278$	−13.6180	−0.816755
$279$	0	0
$280$	0	0
$281$	−19.3607	−1.15496	−0.577481	−	0.816404i	$-0.695964\pi$
$281$	−19.3607	−1.15496	−0.577481	+	0.816404i	$0.695964\pi$
$282$	0	0
$283$	−23.3607	−1.38865	−0.694324	−	0.719662i	$-0.744297\pi$
$283$	−23.3607	−1.38865	−0.694324	+	0.719662i	$0.744297\pi$
$284$	2.05573	0.121985
$285$	0	0
$286$	−5.47214	−0.323574
$287$	−24.9443	−1.47241
$288$	0	0
$289$	−14.8328	−0.872519
$290$	0	0
$291$	0	0
$292$	−0.0901699	−0.00527680
$293$	−14.3262	−0.836948	−0.418474	−	0.908229i	$-0.637435\pi$
$293$	−14.3262	−0.836948	−0.418474	+	0.908229i	$0.637435\pi$
$294$	0	0
$295$	0	0
$296$	6.70820	0.389906
$297$	0	0
$298$	−22.0344	−1.27642
$299$	−10.1459	−0.586752
$300$	0	0
$301$	−1.81966	−0.104883
$302$	−25.7984	−1.48453
$303$	0	0
$304$	4.14590	0.237784
$305$	0	0
$306$	0	0
$307$	13.9787	0.797807	0.398904	−	0.916993i	$-0.369391\pi$
$307$	13.9787	0.797807	0.398904	+	0.916993i	$0.369391\pi$
$308$	−1.47214	−0.0838827
$309$	0	0
$310$	0	0
$311$	−14.5623	−0.825753	−0.412876	−	0.910787i	$-0.635476\pi$
$311$	−14.5623	−0.825753	−0.412876	+	0.910787i	$0.635476\pi$
$312$	0	0
$313$	21.0344	1.18894	0.594468	−	0.804119i	$-0.297362\pi$
$313$	21.0344	1.18894	0.594468	+	0.804119i	$0.297362\pi$
$314$	−21.0344	−1.18704
$315$	0	0
$316$	−4.14590	−0.233225
$317$	6.79837	0.381835	0.190917	−	0.981606i	$-0.438854\pi$
$317$	6.79837	0.381835	0.190917	+	0.981606i	$0.438854\pi$
$318$	0	0
$319$	−1.70820	−0.0956411
$320$	0	0
$321$	0	0
$322$	−11.5623	−0.644342
$323$	−1.25735	−0.0699611
$324$	0	0
$325$	0	0
$326$	3.00000	0.166155
$327$	0	0
$328$	−14.4721	−0.799090
$329$	−44.7771	−2.46864
$330$	0	0
$331$	15.2918	0.840513	0.420257	−	0.907405i	$-0.361940\pi$
$331$	15.2918	0.840513	0.420257	+	0.907405i	$0.361940\pi$
$332$	−8.00000	−0.439057
$333$	0	0
$334$	28.2705	1.54689
$335$	0	0
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	−27.4164	−1.49126
$339$	0	0
$340$	0	0
$341$	−1.23607	−0.0669368
$342$	0	0
$343$	3.29180	0.177740
$344$	−1.05573	−0.0569210
$345$	0	0
$346$	25.4164	1.36639
$347$	−5.76393	−0.309424	−0.154712	−	0.987960i	$-0.549445\pi$
$347$	−5.76393	−0.309424	−0.154712	+	0.987960i	$0.549445\pi$
$348$	0	0
$349$	−20.1246	−1.07725	−0.538623	−	0.842547i	$-0.681055\pi$
$349$	−20.1246	−1.07725	−0.538623	+	0.842547i	$0.681055\pi$
$350$	0	0
$351$	0	0
$352$	−2.09017	−0.111406
$353$	6.65248	0.354076	0.177038	−	0.984204i	$-0.443349\pi$
$353$	6.65248	0.354076	0.177038	+	0.984204i	$0.443349\pi$
$354$	0	0
$355$	0	0
$356$	−2.23607	−0.118511
$357$	0	0
$358$	−32.5623	−1.72097
$359$	25.6525	1.35389	0.676943	−	0.736035i	$-0.263304\pi$
$359$	25.6525	1.35389	0.676943	+	0.736035i	$0.263304\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	21.5623	1.13329
$363$	0	0
$364$	−13.0344	−0.683190
$365$	0	0
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	−9.00000	−0.469157
$369$	0	0
$370$	0	0
$371$	−32.6525	−1.69523
$372$	0	0
$373$	−5.14590	−0.266445	−0.133222	−	0.991086i	$-0.542532\pi$
$373$	−5.14590	−0.266445	−0.133222	+	0.991086i	$0.542532\pi$
$374$	1.47214	0.0761223
$375$	0	0
$376$	−25.9787	−1.33975
$377$	−15.1246	−0.778957
$378$	0	0
$379$	2.76393	0.141974	0.0709868	−	0.997477i	$-0.477385\pi$
$379$	2.76393	0.141974	0.0709868	+	0.997477i	$0.477385\pi$
$380$	0	0
$381$	0	0
$382$	−4.85410	−0.248357
$383$	−25.3820	−1.29696	−0.648479	−	0.761233i	$-0.724595\pi$
$383$	−25.3820	−1.29696	−0.648479	+	0.761233i	$0.724595\pi$
$384$	0	0
$385$	0	0
$386$	−37.1246	−1.88959
$387$	0	0
$388$	−6.56231	−0.333151
$389$	−10.3262	−0.523561	−0.261781	−	0.965127i	$-0.584310\pi$
$389$	−10.3262	−0.523561	−0.261781	+	0.965127i	$0.584310\pi$
$390$	0	0
$391$	2.72949	0.138036
$392$	17.5623	0.887030
$393$	0	0
$394$	−26.6525	−1.34273
$395$	0	0
$396$	0	0
$397$	11.4164	0.572973	0.286487	−	0.958084i	$-0.407513\pi$
$397$	11.4164	0.572973	0.286487	+	0.958084i	$0.407513\pi$
$398$	17.5623	0.880319
$399$	0	0
$400$	0	0
$401$	−14.5623	−0.727207	−0.363603	−	0.931554i	$-0.618454\pi$
$401$	−14.5623	−0.727207	−0.363603	+	0.931554i	$0.618454\pi$
$402$	0	0
$403$	−10.9443	−0.545173
$404$	−6.03444	−0.300225
$405$	0	0
$406$	−17.2361	−0.855412
$407$	−1.85410	−0.0919044
$408$	0	0
$409$	−31.7082	−1.56787	−0.783935	−	0.620843i	$-0.786790\pi$
$409$	−31.7082	−1.56787	−0.783935	+	0.620843i	$0.786790\pi$
$410$	0	0
$411$	0	0
$412$	−3.18034	−0.156684
$413$	53.7426	2.64450
$414$	0	0
$415$	0	0
$416$	−18.5066	−0.907360
$417$	0	0
$418$	−0.854102	−0.0417755
$419$	32.2361	1.57483	0.787417	−	0.616420i	$-0.211418\pi$
$419$	32.2361	1.57483	0.787417	+	0.616420i	$0.211418\pi$
$420$	0	0
$421$	13.1803	0.642370	0.321185	−	0.947016i	$-0.395919\pi$
$421$	13.1803	0.642370	0.321185	+	0.947016i	$0.395919\pi$
$422$	−21.3262	−1.03815
$423$	0	0
$424$	−18.9443	−0.920015
$425$	0	0
$426$	0	0
$427$	−34.1246	−1.65141
$428$	7.29180	0.352462
$429$	0	0
$430$	0	0
$431$	4.58359	0.220784	0.110392	−	0.993888i	$-0.464789\pi$
$431$	4.58359	0.220784	0.110392	+	0.993888i	$0.464789\pi$
$432$	0	0
$433$	−0.145898	−0.00701141	−0.00350571	−	0.999994i	$-0.501116\pi$
$433$	−0.145898	−0.00701141	−0.00350571	+	0.999994i	$0.501116\pi$
$434$	−12.4721	−0.598682
$435$	0	0
$436$	9.79837	0.469257
$437$	−1.58359	−0.0757535
$438$	0	0
$439$	9.27051	0.442457	0.221229	−	0.975222i	$-0.428993\pi$
$439$	9.27051	0.442457	0.221229	+	0.975222i	$0.428993\pi$
$440$	0	0
$441$	0	0
$442$	13.0344	0.619985
$443$	27.7082	1.31646	0.658228	−	0.752818i	$-0.271306\pi$
$443$	27.7082	1.31646	0.658228	+	0.752818i	$0.271306\pi$
$444$	0	0
$445$	0	0
$446$	0.236068	0.0111781
$447$	0	0
$448$	16.3262	0.771342
$449$	−29.0689	−1.37185	−0.685923	−	0.727674i	$-0.740601\pi$
$449$	−29.0689	−1.37185	−0.685923	+	0.727674i	$0.740601\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	−0.437694	−0.0205874
$453$	0	0
$454$	1.23607	0.0580115
$455$	0	0
$456$	0	0
$457$	−34.8885	−1.63202	−0.816009	−	0.578040i	$-0.803818\pi$
$457$	−34.8885	−1.63202	−0.816009	+	0.578040i	$0.803818\pi$
$458$	5.32624	0.248879
$459$	0	0
$460$	0	0
$461$	14.9098	0.694420	0.347210	−	0.937787i	$-0.387129\pi$
$461$	14.9098	0.694420	0.347210	+	0.937787i	$0.387129\pi$
$462$	0	0
$463$	3.67376	0.170734	0.0853671	−	0.996350i	$-0.472794\pi$
$463$	3.67376	0.170734	0.0853671	+	0.996350i	$0.472794\pi$
$464$	−13.4164	−0.622841
$465$	0	0
$466$	35.7426	1.65575
$467$	−3.32624	−0.153920	−0.0769600	−	0.997034i	$-0.524521\pi$
$467$	−3.32624	−0.153920	−0.0769600	+	0.997034i	$0.524521\pi$
$468$	0	0
$469$	−44.2148	−2.04165
$470$	0	0
$471$	0	0
$472$	31.1803	1.43519
$473$	0.291796	0.0134168
$474$	0	0
$475$	0	0
$476$	3.50658	0.160724
$477$	0	0
$478$	0	0
$479$	−22.7639	−1.04011	−0.520055	−	0.854133i	$-0.674089\pi$
$479$	−22.7639	−1.04011	−0.520055	+	0.854133i	$0.674089\pi$
$480$	0	0
$481$	−16.4164	−0.748524
$482$	22.4164	1.02104
$483$	0	0
$484$	−6.56231	−0.298287
$485$	0	0
$486$	0	0
$487$	5.36068	0.242916	0.121458	−	0.992597i	$-0.461243\pi$
$487$	5.36068	0.242916	0.121458	+	0.992597i	$0.461243\pi$
$488$	−19.7984	−0.896230
$489$	0	0
$490$	0	0
$491$	−30.9443	−1.39650	−0.698248	−	0.715856i	$-0.746037\pi$
$491$	−30.9443	−1.39650	−0.698248	+	0.715856i	$0.746037\pi$
$492$	0	0
$493$	4.06888	0.183253
$494$	−7.56231	−0.340244
$495$	0	0
$496$	−9.70820	−0.435911
$497$	12.8197	0.575040
$498$	0	0
$499$	−28.4164	−1.27209	−0.636047	−	0.771651i	$-0.719431\pi$
$499$	−28.4164	−1.27209	−0.636047	+	0.771651i	$0.719431\pi$
$500$	0	0
$501$	0	0
$502$	−14.8541	−0.662971
$503$	−9.00000	−0.401290	−0.200645	−	0.979664i	$-0.564304\pi$
$503$	−9.00000	−0.401290	−0.200645	+	0.979664i	$0.564304\pi$
$504$	0	0
$505$	0	0
$506$	1.85410	0.0824249
$507$	0	0
$508$	4.41641	0.195946
$509$	−31.3820	−1.39098	−0.695491	−	0.718535i	$-0.744813\pi$
$509$	−31.3820	−1.39098	−0.695491	+	0.718535i	$0.744813\pi$
$510$	0	0
$511$	−0.562306	−0.0248749
$512$	5.29180	0.233867
$513$	0	0
$514$	−6.00000	−0.264649
$515$	0	0
$516$	0	0
$517$	7.18034	0.315791
$518$	−18.7082	−0.821991
$519$	0	0
$520$	0	0
$521$	33.0000	1.44576	0.722878	−	0.690976i	$-0.242819\pi$
$521$	33.0000	1.44576	0.722878	+	0.690976i	$0.242819\pi$
$522$	0	0
$523$	−21.1246	−0.923715	−0.461857	−	0.886954i	$-0.652817\pi$
$523$	−21.1246	−0.923715	−0.461857	+	0.886954i	$0.652817\pi$
$524$	0.472136	0.0206254
$525$	0	0
$526$	31.2705	1.36346
$527$	2.94427	0.128254
$528$	0	0
$529$	−19.5623	−0.850535
$530$	0	0
$531$	0	0
$532$	−2.03444	−0.0882042
$533$	35.4164	1.53405
$534$	0	0
$535$	0	0
$536$	−25.6525	−1.10802
$537$	0	0
$538$	6.18034	0.266453
$539$	−4.85410	−0.209081
$540$	0	0
$541$	23.7082	1.01930	0.509648	−	0.860383i	$-0.329776\pi$
$541$	23.7082	1.01930	0.509648	+	0.860383i	$0.329776\pi$
$542$	21.0344	0.903507
$543$	0	0
$544$	4.97871	0.213461
$545$	0	0
$546$	0	0
$547$	−5.29180	−0.226261	−0.113130	−	0.993580i	$-0.536088\pi$
$547$	−5.29180	−0.226261	−0.113130	+	0.993580i	$0.536088\pi$
$548$	−12.5066	−0.534255
$549$	0	0
$550$	0	0
$551$	−2.36068	−0.100568
$552$	0	0
$553$	−25.8541	−1.09943
$554$	24.7426	1.05121
$555$	0	0
$556$	5.20163	0.220598
$557$	−14.3820	−0.609383	−0.304692	−	0.952451i	$-0.598553\pi$
$557$	−14.3820	−0.609383	−0.304692	+	0.952451i	$0.598553\pi$
$558$	0	0
$559$	2.58359	0.109274
$560$	0	0
$561$	0	0
$562$	31.3262	1.32142
$563$	−35.8328	−1.51017	−0.755087	−	0.655625i	$-0.772405\pi$
$563$	−35.8328	−1.51017	−0.755087	+	0.655625i	$0.772405\pi$
$564$	0	0
$565$	0	0
$566$	37.7984	1.58878
$567$	0	0
$568$	7.43769	0.312079
$569$	12.2361	0.512963	0.256481	−	0.966549i	$-0.417437\pi$
$569$	12.2361	0.512963	0.256481	+	0.966549i	$0.417437\pi$
$570$	0	0
$571$	15.2918	0.639942	0.319971	−	0.947427i	$-0.396327\pi$
$571$	15.2918	0.639942	0.319971	+	0.947427i	$0.396327\pi$
$572$	2.09017	0.0873944
$573$	0	0
$574$	40.3607	1.68462
$575$	0	0
$576$	0	0
$577$	−16.5967	−0.690932	−0.345466	−	0.938431i	$-0.612279\pi$
$577$	−16.5967	−0.690932	−0.345466	+	0.938431i	$0.612279\pi$
$578$	24.0000	0.998268
$579$	0	0
$580$	0	0
$581$	−49.8885	−2.06973
$582$	0	0
$583$	5.23607	0.216856
$584$	−0.326238	−0.0134998
$585$	0	0
$586$	23.1803	0.957571
$587$	1.14590	0.0472963	0.0236481	−	0.999720i	$-0.492472\pi$
$587$	1.14590	0.0472963	0.0236481	+	0.999720i	$0.492472\pi$
$588$	0	0
$589$	−1.70820	−0.0703853
$590$	0	0
$591$	0	0
$592$	−14.5623	−0.598507
$593$	21.0000	0.862367	0.431183	−	0.902264i	$-0.358096\pi$
$593$	21.0000	0.862367	0.431183	+	0.902264i	$0.358096\pi$
$594$	0	0
$595$	0	0
$596$	8.41641	0.344749
$597$	0	0
$598$	16.4164	0.671317
$599$	−7.23607	−0.295658	−0.147829	−	0.989013i	$-0.547228\pi$
$599$	−7.23607	−0.295658	−0.147829	+	0.989013i	$0.547228\pi$
$600$	0	0
$601$	−24.1803	−0.986337	−0.493168	−	0.869934i	$-0.664161\pi$
$601$	−24.1803	−0.986337	−0.493168	+	0.869934i	$0.664161\pi$
$602$	2.94427	0.120000
$603$	0	0
$604$	9.85410	0.400958
$605$	0	0
$606$	0	0
$607$	4.58359	0.186042	0.0930211	−	0.995664i	$-0.470348\pi$
$607$	4.58359	0.186042	0.0930211	+	0.995664i	$0.470348\pi$
$608$	−2.88854	−0.117146
$609$	0	0
$610$	0	0
$611$	63.5755	2.57199
$612$	0	0
$613$	−27.7082	−1.11912	−0.559562	−	0.828789i	$-0.689031\pi$
$613$	−27.7082	−1.11912	−0.559562	+	0.828789i	$0.689031\pi$
$614$	−22.6180	−0.912790
$615$	0	0
$616$	−5.32624	−0.214600
$617$	42.3262	1.70399	0.851995	−	0.523550i	$-0.175393\pi$
$617$	42.3262	1.70399	0.851995	+	0.523550i	$0.175393\pi$
$618$	0	0
$619$	26.3050	1.05729	0.528643	−	0.848844i	$-0.322701\pi$
$619$	26.3050	1.05729	0.528643	+	0.848844i	$0.322701\pi$
$620$	0	0
$621$	0	0
$622$	23.5623	0.944762
$623$	−13.9443	−0.558665
$624$	0	0
$625$	0	0
$626$	−34.0344	−1.36029
$627$	0	0
$628$	8.03444	0.320609
$629$	4.41641	0.176094
$630$	0	0
$631$	−47.4721	−1.88984	−0.944918	−	0.327307i	$-0.893859\pi$
$631$	−47.4721	−1.88984	−0.944918	+	0.327307i	$0.893859\pi$
$632$	−15.0000	−0.596668
$633$	0	0
$634$	−11.0000	−0.436866
$635$	0	0
$636$	0	0
$637$	−42.9787	−1.70288
$638$	2.76393	0.109425
$639$	0	0
$640$	0	0
$641$	36.5410	1.44328	0.721642	−	0.692267i	$-0.243388\pi$
$641$	36.5410	1.44328	0.721642	+	0.692267i	$0.243388\pi$
$642$	0	0
$643$	−18.2361	−0.719160	−0.359580	−	0.933114i	$-0.617080\pi$
$643$	−18.2361	−0.719160	−0.359580	+	0.933114i	$0.617080\pi$
$644$	4.41641	0.174031
$645$	0	0
$646$	2.03444	0.0800440
$647$	16.5967	0.652485	0.326243	−	0.945286i	$-0.394217\pi$
$647$	16.5967	0.652485	0.326243	+	0.945286i	$0.394217\pi$
$648$	0	0
$649$	−8.61803	−0.338287
$650$	0	0
$651$	0	0
$652$	−1.14590	−0.0448768
$653$	34.7426	1.35958	0.679792	−	0.733405i	$-0.262070\pi$
$653$	34.7426	1.35958	0.679792	+	0.733405i	$0.262070\pi$
$654$	0	0
$655$	0	0
$656$	31.4164	1.22660
$657$	0	0
$658$	72.4508	2.82443
$659$	20.4508	0.796652	0.398326	−	0.917244i	$-0.369591\pi$
$659$	20.4508	0.796652	0.398326	+	0.917244i	$0.369591\pi$
$660$	0	0
$661$	20.2918	0.789259	0.394630	−	0.918840i	$-0.370873\pi$
$661$	20.2918	0.789259	0.394630	+	0.918840i	$0.370873\pi$
$662$	−24.7426	−0.961650
$663$	0	0
$664$	−28.9443	−1.12326
$665$	0	0
$666$	0	0
$667$	5.12461	0.198426
$668$	−10.7984	−0.417802
$669$	0	0
$670$	0	0
$671$	5.47214	0.211249
$672$	0	0
$673$	−14.4164	−0.555712	−0.277856	−	0.960623i	$-0.589624\pi$
$673$	−14.4164	−0.555712	−0.277856	+	0.960623i	$0.589624\pi$
$674$	3.23607	0.124649
$675$	0	0
$676$	10.4721	0.402774
$677$	−14.5066	−0.557533	−0.278767	−	0.960359i	$-0.589926\pi$
$677$	−14.5066	−0.557533	−0.278767	+	0.960359i	$0.589926\pi$
$678$	0	0
$679$	−40.9230	−1.57048
$680$	0	0
$681$	0	0
$682$	2.00000	0.0765840
$683$	10.6738	0.408420	0.204210	−	0.978927i	$-0.434537\pi$
$683$	10.6738	0.408420	0.204210	+	0.978927i	$0.434537\pi$
$684$	0	0
$685$	0	0
$686$	−5.32624	−0.203357
$687$	0	0
$688$	2.29180	0.0873739
$689$	46.3607	1.76620
$690$	0	0
$691$	15.5410	0.591208	0.295604	−	0.955311i	$-0.404479\pi$
$691$	15.5410	0.591208	0.295604	+	0.955311i	$0.404479\pi$
$692$	−9.70820	−0.369051
$693$	0	0
$694$	9.32624	0.354019
$695$	0	0
$696$	0	0
$697$	−9.52786	−0.360894
$698$	32.5623	1.23250
$699$	0	0
$700$	0	0
$701$	11.9443	0.451129	0.225564	−	0.974228i	$-0.427577\pi$
$701$	11.9443	0.451129	0.225564	+	0.974228i	$0.427577\pi$
$702$	0	0
$703$	−2.56231	−0.0966392
$704$	−2.61803	−0.0986709
$705$	0	0
$706$	−10.7639	−0.405106
$707$	−37.6312	−1.41527
$708$	0	0
$709$	0.124612	0.00467989	0.00233995	−	0.999997i	$-0.499255\pi$
$709$	0.124612	0.00467989	0.00233995	+	0.999997i	$0.499255\pi$
$710$	0	0
$711$	0	0
$712$	−8.09017	−0.303192
$713$	3.70820	0.138873
$714$	0	0
$715$	0	0
$716$	12.4377	0.464818
$717$	0	0
$718$	−41.5066	−1.54901
$719$	−43.2148	−1.61164	−0.805820	−	0.592161i	$-0.798275\pi$
$719$	−43.2148	−1.61164	−0.805820	+	0.592161i	$0.798275\pi$
$720$	0	0
$721$	−19.8328	−0.738613
$722$	29.5623	1.10020
$723$	0	0
$724$	−8.23607	−0.306091
$725$	0	0
$726$	0	0
$727$	28.9787	1.07476	0.537381	−	0.843340i	$-0.319414\pi$
$727$	28.9787	1.07476	0.537381	+	0.843340i	$0.319414\pi$
$728$	−47.1591	−1.74783
$729$	0	0
$730$	0	0
$731$	−0.695048	−0.0257073
$732$	0	0
$733$	−27.8328	−1.02803	−0.514014	−	0.857782i	$-0.671842\pi$
$733$	−27.8328	−1.02803	−0.514014	+	0.857782i	$0.671842\pi$
$734$	−29.1246	−1.07501
$735$	0	0
$736$	6.27051	0.231134
$737$	7.09017	0.261170
$738$	0	0
$739$	29.4721	1.08415	0.542075	−	0.840330i	$-0.317639\pi$
$739$	29.4721	1.08415	0.542075	+	0.840330i	$0.317639\pi$
$740$	0	0
$741$	0	0
$742$	52.8328	1.93955
$743$	−25.9098	−0.950539	−0.475270	−	0.879840i	$-0.657650\pi$
$743$	−25.9098	−0.950539	−0.475270	+	0.879840i	$0.657650\pi$
$744$	0	0
$745$	0	0
$746$	8.32624	0.304845
$747$	0	0
$748$	−0.562306	−0.0205599
$749$	45.4721	1.66152
$750$	0	0
$751$	−20.0344	−0.731067	−0.365534	−	0.930798i	$-0.619113\pi$
$751$	−20.0344	−0.731067	−0.365534	+	0.930798i	$0.619113\pi$
$752$	56.3951	2.05652
$753$	0	0
$754$	24.4721	0.891223
$755$	0	0
$756$	0	0
$757$	−53.8328	−1.95659	−0.978293	−	0.207224i	$-0.933557\pi$
$757$	−53.8328	−1.95659	−0.978293	+	0.207224i	$0.933557\pi$
$758$	−4.47214	−0.162435
$759$	0	0
$760$	0	0
$761$	27.2705	0.988555	0.494278	−	0.869304i	$-0.335433\pi$
$761$	27.2705	0.988555	0.494278	+	0.869304i	$0.335433\pi$
$762$	0	0
$763$	61.1033	2.21209
$764$	1.85410	0.0670791
$765$	0	0
$766$	41.0689	1.48388
$767$	−76.3050	−2.75521
$768$	0	0
$769$	−33.5410	−1.20952	−0.604760	−	0.796408i	$-0.706731\pi$
$769$	−33.5410	−1.20952	−0.604760	+	0.796408i	$0.706731\pi$
$770$	0	0
$771$	0	0
$772$	14.1803	0.510362
$773$	−43.7214	−1.57255	−0.786274	−	0.617878i	$-0.787993\pi$
$773$	−43.7214	−1.57255	−0.786274	+	0.617878i	$0.787993\pi$
$774$	0	0
$775$	0	0
$776$	−23.7426	−0.852311
$777$	0	0
$778$	16.7082	0.599018
$779$	5.52786	0.198056
$780$	0	0
$781$	−2.05573	−0.0735597
$782$	−4.41641	−0.157930
$783$	0	0
$784$	−38.1246	−1.36159
$785$	0	0
$786$	0	0
$787$	30.8885	1.10106	0.550529	−	0.834816i	$-0.314426\pi$
$787$	30.8885	1.10106	0.550529	+	0.834816i	$0.314426\pi$
$788$	10.1803	0.362660
$789$	0	0
$790$	0	0
$791$	−2.72949	−0.0970495
$792$	0	0
$793$	48.4508	1.72054
$794$	−18.4721	−0.655552
$795$	0	0
$796$	−6.70820	−0.237766
$797$	−40.6869	−1.44120	−0.720602	−	0.693349i	$-0.756135\pi$
$797$	−40.6869	−1.44120	−0.720602	+	0.693349i	$0.756135\pi$
$798$	0	0
$799$	−17.1033	−0.605072
$800$	0	0
$801$	0	0
$802$	23.5623	0.832014
$803$	0.0901699	0.00318203
$804$	0	0
$805$	0	0
$806$	17.7082	0.623745
$807$	0	0
$808$	−21.8328	−0.768076
$809$	35.4508	1.24639	0.623193	−	0.782068i	$-0.285835\pi$
$809$	35.4508	1.24639	0.623193	+	0.782068i	$0.285835\pi$
$810$	0	0
$811$	2.12461	0.0746052	0.0373026	−	0.999304i	$-0.488123\pi$
$811$	2.12461	0.0746052	0.0373026	+	0.999304i	$0.488123\pi$
$812$	6.58359	0.231039
$813$	0	0
$814$	3.00000	0.105150
$815$	0	0
$816$	0	0
$817$	0.403252	0.0141080
$818$	51.3050	1.79384
$819$	0	0
$820$	0	0
$821$	22.4721	0.784283	0.392141	−	0.919905i	$-0.371734\pi$
$821$	22.4721	0.784283	0.392141	+	0.919905i	$0.371734\pi$
$822$	0	0
$823$	0.708204	0.0246864	0.0123432	−	0.999924i	$-0.496071\pi$
$823$	0.708204	0.0246864	0.0123432	+	0.999924i	$0.496071\pi$
$824$	−11.5066	−0.400851
$825$	0	0
$826$	−86.9574	−3.02564
$827$	6.87539	0.239081	0.119540	−	0.992829i	$-0.461858\pi$
$827$	6.87539	0.239081	0.119540	+	0.992829i	$0.461858\pi$
$828$	0	0
$829$	−14.8754	−0.516644	−0.258322	−	0.966059i	$-0.583169\pi$
$829$	−14.8754	−0.516644	−0.258322	+	0.966059i	$0.583169\pi$
$830$	0	0
$831$	0	0
$832$	−23.1803	−0.803634
$833$	11.5623	0.400610
$834$	0	0
$835$	0	0
$836$	0.326238	0.0112832
$837$	0	0
$838$	−52.1591	−1.80180
$839$	26.5066	0.915109	0.457554	−	0.889182i	$-0.348726\pi$
$839$	26.5066	0.915109	0.457554	+	0.889182i	$0.348726\pi$
$840$	0	0
$841$	−21.3607	−0.736575
$842$	−21.3262	−0.734951
$843$	0	0
$844$	8.14590	0.280393
$845$	0	0
$846$	0	0
$847$	−40.9230	−1.40613
$848$	41.1246	1.41222
$849$	0	0
$850$	0	0
$851$	5.56231	0.190673
$852$	0	0
$853$	51.6869	1.76973	0.884863	−	0.465851i	$-0.154252\pi$
$853$	51.6869	1.76973	0.884863	+	0.465851i	$0.154252\pi$
$854$	55.2148	1.88941
$855$	0	0
$856$	26.3820	0.901717
$857$	−22.0689	−0.753859	−0.376929	−	0.926242i	$-0.623020\pi$
$857$	−22.0689	−0.753859	−0.376929	+	0.926242i	$0.623020\pi$
$858$	0	0
$859$	−7.76393	−0.264902	−0.132451	−	0.991190i	$-0.542285\pi$
$859$	−7.76393	−0.264902	−0.132451	+	0.991190i	$0.542285\pi$
$860$	0	0
$861$	0	0
$862$	−7.41641	−0.252604
$863$	31.2492	1.06374	0.531868	−	0.846827i	$-0.321490\pi$
$863$	31.2492	1.06374	0.531868	+	0.846827i	$0.321490\pi$
$864$	0	0
$865$	0	0
$866$	0.236068	0.00802192
$867$	0	0
$868$	4.76393	0.161698
$869$	4.14590	0.140640
$870$	0	0
$871$	62.7771	2.12712
$872$	35.4508	1.20052
$873$	0	0
$874$	2.56231	0.0866713
$875$	0	0
$876$	0	0
$877$	−17.9787	−0.607098	−0.303549	−	0.952816i	$-0.598172\pi$
$877$	−17.9787	−0.607098	−0.303549	+	0.952816i	$0.598172\pi$
$878$	−15.0000	−0.506225
$879$	0	0
$880$	0	0
$881$	−32.4508	−1.09330	−0.546648	−	0.837362i	$-0.684097\pi$
$881$	−32.4508	−1.09330	−0.546648	+	0.837362i	$0.684097\pi$
$882$	0	0
$883$	−11.0000	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$884$	−4.97871	−0.167452
$885$	0	0
$886$	−44.8328	−1.50619
$887$	16.0689	0.539540	0.269770	−	0.962925i	$-0.413052\pi$
$887$	16.0689	0.539540	0.269770	+	0.962925i	$0.413052\pi$
$888$	0	0
$889$	27.5410	0.923696
$890$	0	0
$891$	0	0
$892$	−0.0901699	−0.00301911
$893$	9.92299	0.332060
$894$	0	0
$895$	0	0
$896$	−52.4853	−1.75341
$897$	0	0
$898$	47.0344	1.56956
$899$	5.52786	0.184365
$900$	0	0
$901$	−12.4721	−0.415507
$902$	−6.47214	−0.215499
$903$	0	0
$904$	−1.58359	−0.0526695
$905$	0	0
$906$	0	0
$907$	−52.8541	−1.75499	−0.877496	−	0.479584i	$-0.840787\pi$
$907$	−52.8541	−1.75499	−0.877496	+	0.479584i	$0.840787\pi$
$908$	−0.472136	−0.0156684
$909$	0	0
$910$	0	0
$911$	−32.5279	−1.07770	−0.538848	−	0.842403i	$-0.681140\pi$
$911$	−32.5279	−1.07770	−0.538848	+	0.842403i	$0.681140\pi$
$912$	0	0
$913$	8.00000	0.264761
$914$	56.4508	1.86723
$915$	0	0
$916$	−2.03444	−0.0672199
$917$	2.94427	0.0972284
$918$	0	0
$919$	57.3607	1.89215	0.946077	−	0.323941i	$-0.105008\pi$
$919$	57.3607	1.89215	0.946077	+	0.323941i	$0.105008\pi$
$920$	0	0
$921$	0	0
$922$	−24.1246	−0.794502
$923$	−18.2016	−0.599114
$924$	0	0
$925$	0	0
$926$	−5.94427	−0.195341
$927$	0	0
$928$	9.34752	0.306848
$929$	−15.6525	−0.513541	−0.256771	−	0.966472i	$-0.582658\pi$
$929$	−15.6525	−0.513541	−0.256771	+	0.966472i	$0.582658\pi$
$930$	0	0
$931$	−6.70820	−0.219853
$932$	−13.6525	−0.447202
$933$	0	0
$934$	5.38197	0.176103
$935$	0	0
$936$	0	0
$937$	4.70820	0.153810	0.0769052	−	0.997038i	$-0.475496\pi$
$937$	4.70820	0.153810	0.0769052	+	0.997038i	$0.475496\pi$
$938$	71.5410	2.33590
$939$	0	0
$940$	0	0
$941$	−1.06888	−0.0348446	−0.0174223	−	0.999848i	$-0.505546\pi$
$941$	−1.06888	−0.0348446	−0.0174223	+	0.999848i	$0.505546\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	−67.6869	−2.20302
$945$	0	0
$946$	−0.472136	−0.0153505
$947$	−3.00000	−0.0974869	−0.0487435	−	0.998811i	$-0.515522\pi$
$947$	−3.00000	−0.0974869	−0.0487435	+	0.998811i	$0.515522\pi$
$948$	0	0
$949$	0.798374	0.0259163
$950$	0	0
$951$	0	0
$952$	12.6869	0.411185
$953$	−43.9230	−1.42281	−0.711403	−	0.702785i	$-0.751940\pi$
$953$	−43.9230	−1.42281	−0.711403	+	0.702785i	$0.751940\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	36.8328	1.19001
$959$	−77.9919	−2.51849
$960$	0	0
$961$	−27.0000	−0.870968
$962$	26.5623	0.856403
$963$	0	0
$964$	−8.56231	−0.275773
$965$	0	0
$966$	0	0
$967$	−31.0689	−0.999108	−0.499554	−	0.866283i	$-0.666503\pi$
$967$	−31.0689	−0.999108	−0.499554	+	0.866283i	$0.666503\pi$
$968$	−23.7426	−0.763118
$969$	0	0
$970$	0	0
$971$	−4.68692	−0.150410	−0.0752052	−	0.997168i	$-0.523961\pi$
$971$	−4.68692	−0.150410	−0.0752052	+	0.997168i	$0.523961\pi$
$972$	0	0
$973$	32.4377	1.03990
$974$	−8.67376	−0.277925
$975$	0	0
$976$	42.9787	1.37572
$977$	−27.0689	−0.866010	−0.433005	−	0.901391i	$-0.642547\pi$
$977$	−27.0689	−0.866010	−0.433005	+	0.901391i	$0.642547\pi$
$978$	0	0
$979$	2.23607	0.0714650
$980$	0	0
$981$	0	0
$982$	50.0689	1.59776
$983$	−35.0557	−1.11810	−0.559052	−	0.829133i	$-0.688835\pi$
$983$	−35.0557	−1.11810	−0.559052	+	0.829133i	$0.688835\pi$
$984$	0	0
$985$	0	0
$986$	−6.58359	−0.209664
$987$	0	0
$988$	2.88854	0.0918968
$989$	−0.875388	−0.0278357
$990$	0	0
$991$	−36.5410	−1.16076	−0.580382	−	0.814344i	$-0.697097\pi$
$991$	−36.5410	−1.16076	−0.580382	+	0.814344i	$0.697097\pi$
$992$	6.76393	0.214755
$993$	0	0
$994$	−20.7426	−0.657917
$995$	0	0
$996$	0	0
$997$	5.43769	0.172214	0.0861068	−	0.996286i	$-0.472557\pi$
$997$	5.43769	0.172214	0.0861068	+	0.996286i	$0.472557\pi$
$998$	45.9787	1.45543
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.c.1.1		2
3.2	odd	2		625.2.a.d.1.2	yes	2
5.4	even	2		5625.2.a.e.1.2		2
12.11	even	2		10000.2.a.b.1.1		2
15.2	even	4		625.2.b.b.624.4		4
15.8	even	4		625.2.b.b.624.1		4
15.14	odd	2		625.2.a.a.1.1	✓	2
60.59	even	2		10000.2.a.m.1.2		2
75.2	even	20		625.2.e.f.124.2		8
75.8	even	20		625.2.e.e.374.2		8
75.11	odd	10		625.2.d.f.501.1		4
75.14	odd	10		625.2.d.e.501.1		4
75.17	even	20		625.2.e.e.374.1		8
75.23	even	20		625.2.e.f.124.1		8
75.29	odd	10		625.2.d.i.376.1		4
75.38	even	20		625.2.e.f.499.2		8
75.41	odd	10		625.2.d.f.126.1		4
75.44	odd	10		625.2.d.i.251.1		4
75.47	even	20		625.2.e.e.249.2		8
75.53	even	20		625.2.e.e.249.1		8
75.56	odd	10		625.2.d.c.251.1		4
75.59	odd	10		625.2.d.e.126.1		4
75.62	even	20		625.2.e.f.499.1		8
75.71	odd	10		625.2.d.c.376.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.a.1.1	✓	2	15.14	odd	2
625.2.a.d.1.2	yes	2	3.2	odd	2
625.2.b.b.624.1		4	15.8	even	4
625.2.b.b.624.4		4	15.2	even	4
625.2.d.c.251.1		4	75.56	odd	10
625.2.d.c.376.1		4	75.71	odd	10
625.2.d.e.126.1		4	75.59	odd	10
625.2.d.e.501.1		4	75.14	odd	10
625.2.d.f.126.1		4	75.41	odd	10
625.2.d.f.501.1		4	75.11	odd	10
625.2.d.i.251.1		4	75.44	odd	10
625.2.d.i.376.1		4	75.29	odd	10
625.2.e.e.249.1		8	75.53	even	20
625.2.e.e.249.2		8	75.47	even	20
625.2.e.e.374.1		8	75.17	even	20
625.2.e.e.374.2		8	75.8	even	20
625.2.e.f.124.1		8	75.23	even	20
625.2.e.f.124.2		8	75.2	even	20
625.2.e.f.499.1		8	75.62	even	20
625.2.e.f.499.2		8	75.38	even	20
5625.2.a.c.1.1		2	1.1	even	1	trivial
5625.2.a.e.1.2		2	5.4	even	2
10000.2.a.b.1.1		2	12.11	even	2
10000.2.a.m.1.2		2	60.59	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)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} +3.85410 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} +3.85410 q^{7} +2.23607 q^{8} -0.618034 q^{11} -5.47214 q^{13} -6.23607 q^{14} -4.85410 q^{16} +1.47214 q^{17} -0.854102 q^{19} +1.00000 q^{22} +1.85410 q^{23} +8.85410 q^{26} +2.38197 q^{28} +2.76393 q^{29} +2.00000 q^{31} +3.38197 q^{32} -2.38197 q^{34} +3.00000 q^{37} +1.38197 q^{38} -6.47214 q^{41} -0.472136 q^{43} -0.381966 q^{44} -3.00000 q^{46} -11.6180 q^{47} +7.85410 q^{49} -3.38197 q^{52} -8.47214 q^{53} +8.61803 q^{56} -4.47214 q^{58} +13.9443 q^{59} -8.85410 q^{61} -3.23607 q^{62} +4.23607 q^{64} -11.4721 q^{67} +0.909830 q^{68} +3.32624 q^{71} -0.145898 q^{73} -4.85410 q^{74} -0.527864 q^{76} -2.38197 q^{77} -6.70820 q^{79} +10.4721 q^{82} -12.9443 q^{83} +0.763932 q^{86} -1.38197 q^{88} -3.61803 q^{89} -21.0902 q^{91} +1.14590 q^{92} +18.7984 q^{94} -10.6180 q^{97} -12.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + q^7	\( 2 q - q^{2} - q^{4} + q^{7} + q^{11} - 2 q^{13} - 8 q^{14} - 3 q^{16} - 6 q^{17} + 5 q^{19} + 2 q^{22} - 3 q^{23} + 11 q^{26} + 7 q^{28} + 10 q^{29} + 4 q^{31} + 9 q^{32} - 7 q^{34} + 6 q^{37} + 5 q^{38} - 4 q^{41} + 8 q^{43} - 3 q^{44} - 6 q^{46} - 21 q^{47} + 9 q^{49} - 9 q^{52} - 8 q^{53} + 15 q^{56} + 10 q^{59} - 11 q^{61} - 2 q^{62} + 4 q^{64} - 14 q^{67} + 13 q^{68} - 9 q^{71} - 7 q^{73} - 3 q^{74} - 10 q^{76} - 7 q^{77} + 12 q^{82} - 8 q^{83} + 6 q^{86} - 5 q^{88} - 5 q^{89} - 31 q^{91} + 9 q^{92} + 13 q^{94} - 19 q^{97} - 12 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + q^7 + q^11 - 2 * q^13 - 8 * q^14 - 3 * q^16 - 6 * q^17 + 5 * q^19 + 2 * q^22 - 3 * q^23 + 11 * q^26 + 7 * q^28 + 10 * q^29 + 4 * q^31 + 9 * q^32 - 7 * q^34 + 6 * q^37 + 5 * q^38 - 4 * q^41 + 8 * q^43 - 3 * q^44 - 6 * q^46 - 21 * q^47 + 9 * q^49 - 9 * q^52 - 8 * q^53 + 15 * q^56 + 10 * q^59 - 11 * q^61 - 2 * q^62 + 4 * q^64 - 14 * q^67 + 13 * q^68 - 9 * q^71 - 7 * q^73 - 3 * q^74 - 10 * q^76 - 7 * q^77 + 12 * q^82 - 8 * q^83 + 6 * q^86 - 5 * q^88 - 5 * q^89 - 31 * q^91 + 9 * q^92 + 13 * q^94 - 19 * q^97 - 12 * q^98

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