LMFDB - Embedded newform 5625.2.a.w.1.8

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:	$8$
Coefficient field:	8.8.33620000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{6} + 30x^{4} - 25x^{2} + 5 $ x^8 - 10x^6 + 30x^4 - 25*x^2 + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.16942$ 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)

$f(q)$	$=$	$q+2.16942 q^{2} +2.70636 q^{4} -3.32440 q^{7} +1.53239 q^{8} +O(q^{10})$	$q+2.16942 q^{2} +2.70636 q^{4} -3.32440 q^{7} +1.53239 q^{8} +3.82024 q^{11} +2.65177 q^{13} -7.21200 q^{14} -2.08833 q^{16} -4.73253 q^{17} -3.65177 q^{19} +8.28768 q^{22} -3.61815 q^{23} +5.75280 q^{26} -8.99702 q^{28} -0.0347749 q^{29} -10.1062 q^{31} -7.59524 q^{32} -10.2668 q^{34} +10.4643 q^{37} -7.92221 q^{38} -1.90917 q^{41} -2.27279 q^{43} +10.3389 q^{44} -7.84928 q^{46} -1.47613 q^{47} +4.05161 q^{49} +7.17666 q^{52} +8.92498 q^{53} -5.09428 q^{56} -0.0754413 q^{58} -12.4004 q^{59} -5.17666 q^{61} -21.9245 q^{62} -12.3006 q^{64} -5.43173 q^{67} -12.8079 q^{68} -12.1211 q^{71} +12.2588 q^{73} +22.7015 q^{74} -9.88302 q^{76} -12.7000 q^{77} -7.24895 q^{79} -4.14178 q^{82} +5.62790 q^{83} -4.93062 q^{86} +5.85410 q^{88} -17.8158 q^{89} -8.81554 q^{91} -9.79204 q^{92} -3.20233 q^{94} -8.27777 q^{97} +8.78962 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4}+O(q^{10}) $ 8 * q + 4 * q^4	$ 8 q + 4 q^{4} + 10 q^{13} - 8 q^{16} - 18 q^{19} - 30 q^{28} - 26 q^{31} - 20 q^{34} - 40 q^{43} - 30 q^{46} - 16 q^{49} + 40 q^{52} + 10 q^{58} - 24 q^{61} - 34 q^{64} - 40 q^{67} + 40 q^{73} - 44 q^{76} - 42 q^{79} + 60 q^{82} + 20 q^{88} - 40 q^{91} - 10 q^{94} - 40 q^{97}+O(q^{100}) $ 8 * q + 4 * q^4 + 10 * q^13 - 8 * q^16 - 18 * q^19 - 30 * q^28 - 26 * q^31 - 20 * q^34 - 40 * q^43 - 30 * q^46 - 16 * q^49 + 40 * q^52 + 10 * q^58 - 24 * q^61 - 34 * q^64 - 40 * q^67 + 40 * q^73 - 44 * q^76 - 42 * q^79 + 60 * q^82 + 20 * q^88 - 40 * q^91 - 10 * q^94 - 40 * q^97

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 $

$2$	2.16942	1.53401	0.767004	−	0.641642i	$-0.221747\pi$
$2$	2.16942	1.53401	0.767004	+	0.641642i	$0.221747\pi$
$3$	0	0
$3$	0	0
$4$	2.70636	1.35318
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.32440	−1.25650	−0.628252	−	0.778010i	$-0.716229\pi$
$7$	−3.32440	−1.25650	−0.628252	+	0.778010i	$0.716229\pi$
$8$	1.53239	0.541783
$9$	0	0
$10$	0	0
$11$	3.82024	1.15184	0.575922	−	0.817504i	$-0.304643\pi$
$11$	3.82024	1.15184	0.575922	+	0.817504i	$0.304643\pi$
$12$	0	0
$13$	2.65177	0.735469	0.367735	−	0.929931i	$-0.380133\pi$
$13$	2.65177	0.735469	0.367735	+	0.929931i	$0.380133\pi$
$14$	−7.21200	−1.92749
$15$	0	0
$16$	−2.08833	−0.522082
$17$	−4.73253	−1.14781	−0.573904	−	0.818923i	$-0.694572\pi$
$17$	−4.73253	−1.14781	−0.573904	+	0.818923i	$0.694572\pi$
$18$	0	0
$19$	−3.65177	−0.837774	−0.418887	−	0.908038i	$-0.637580\pi$
$19$	−3.65177	−0.837774	−0.418887	+	0.908038i	$0.637580\pi$
$20$	0	0
$21$	0	0
$22$	8.28768	1.76694
$23$	−3.61815	−0.754437	−0.377219	−	0.926124i	$-0.623120\pi$
$23$	−3.61815	−0.754437	−0.377219	+	0.926124i	$0.623120\pi$
$24$	0	0
$25$	0	0
$26$	5.75280	1.12822
$27$	0	0
$28$	−8.99702	−1.70028
$29$	−0.0347749	−0.00645754	−0.00322877	−	0.999995i	$-0.501028\pi$
$29$	−0.0347749	−0.00645754	−0.00322877	+	0.999995i	$0.501028\pi$
$30$	0	0
$31$	−10.1062	−1.81513	−0.907564	−	0.419915i	$-0.862060\pi$
$31$	−10.1062	−1.81513	−0.907564	+	0.419915i	$0.862060\pi$
$32$	−7.59524	−1.34266
$33$	0	0
$34$	−10.2668	−1.76075
$35$	0	0
$36$	0	0
$37$	10.4643	1.72033	0.860163	−	0.510019i	$-0.170362\pi$
$37$	10.4643	1.72033	0.860163	+	0.510019i	$0.170362\pi$
$38$	−7.92221	−1.28515
$39$	0	0
$40$	0	0
$41$	−1.90917	−0.298162	−0.149081	−	0.988825i	$-0.547632\pi$
$41$	−1.90917	−0.298162	−0.149081	+	0.988825i	$0.547632\pi$
$42$	0	0
$43$	−2.27279	−0.346597	−0.173298	−	0.984869i	$-0.555442\pi$
$43$	−2.27279	−0.346597	−0.173298	+	0.984869i	$0.555442\pi$
$44$	10.3389	1.55865
$45$	0	0
$46$	−7.84928	−1.15731
$47$	−1.47613	−0.215315	−0.107658	−	0.994188i	$-0.534335\pi$
$47$	−1.47613	−0.215315	−0.107658	+	0.994188i	$0.534335\pi$
$48$	0	0
$49$	4.05161	0.578801
$50$	0	0
$51$	0	0
$52$	7.17666	0.995223
$53$	8.92498	1.22594	0.612970	−	0.790106i	$-0.289975\pi$
$53$	8.92498	1.22594	0.612970	+	0.790106i	$0.289975\pi$
$54$	0	0
$55$	0	0
$56$	−5.09428	−0.680752
$57$	0	0
$58$	−0.0754413	−0.00990592
$59$	−12.4004	−1.61439	−0.807197	−	0.590283i	$-0.799016\pi$
$59$	−12.4004	−1.61439	−0.807197	+	0.590283i	$0.799016\pi$
$60$	0	0
$61$	−5.17666	−0.662803	−0.331401	−	0.943490i	$-0.607521\pi$
$61$	−5.17666	−0.662803	−0.331401	+	0.943490i	$0.607521\pi$
$62$	−21.9245	−2.78442
$63$	0	0
$64$	−12.3006	−1.53757
$65$	0	0
$66$	0	0
$67$	−5.43173	−0.663592	−0.331796	−	0.943351i	$-0.607655\pi$
$67$	−5.43173	−0.663592	−0.331796	+	0.943351i	$0.607655\pi$
$68$	−12.8079	−1.55319
$69$	0	0
$70$	0	0
$71$	−12.1211	−1.43851	−0.719256	−	0.694745i	$-0.755517\pi$
$71$	−12.1211	−1.43851	−0.719256	+	0.694745i	$0.755517\pi$
$72$	0	0
$73$	12.2588	1.43478	0.717390	−	0.696672i	$-0.245337\pi$
$73$	12.2588	1.43478	0.717390	+	0.696672i	$0.245337\pi$
$74$	22.7015	2.63899
$75$	0	0
$76$	−9.88302	−1.13366
$77$	−12.7000	−1.44730
$78$	0	0
$79$	−7.24895	−0.815571	−0.407786	−	0.913078i	$-0.633699\pi$
$79$	−7.24895	−0.815571	−0.407786	+	0.913078i	$0.633699\pi$
$80$	0	0
$81$	0	0
$82$	−4.14178	−0.457383
$83$	5.62790	0.617743	0.308871	−	0.951104i	$-0.400049\pi$
$83$	5.62790	0.617743	0.308871	+	0.951104i	$0.400049\pi$
$84$	0	0
$85$	0	0
$86$	−4.93062	−0.531682
$87$	0	0
$88$	5.85410	0.624049
$89$	−17.8158	−1.88847	−0.944233	−	0.329278i	$-0.893195\pi$
$89$	−17.8158	−1.88847	−0.944233	+	0.329278i	$0.893195\pi$
$90$	0	0
$91$	−8.81554	−0.924120
$92$	−9.79204	−1.02089
$93$	0	0
$94$	−3.20233	−0.330295
$95$	0	0
$96$	0	0
$97$	−8.27777	−0.840480	−0.420240	−	0.907413i	$-0.638054\pi$
$97$	−8.27777	−0.840480	−0.420240	+	0.907413i	$0.638054\pi$
$98$	8.78962	0.887886
$99$	0	0
$100$	0	0
$101$	−3.57692	−0.355916	−0.177958	−	0.984038i	$-0.556949\pi$
$101$	−3.57692	−0.355916	−0.177958	+	0.984038i	$0.556949\pi$
$102$	0	0
$103$	12.4595	1.22767	0.613836	−	0.789433i	$-0.289626\pi$
$103$	12.4595	1.22767	0.613836	+	0.789433i	$0.289626\pi$
$104$	4.06356	0.398465
$105$	0	0
$106$	19.3620	1.88060
$107$	0.574861	0.0555740	0.0277870	−	0.999614i	$-0.491154\pi$
$107$	0.574861	0.0555740	0.0277870	+	0.999614i	$0.491154\pi$
$108$	0	0
$109$	11.3522	1.08734	0.543671	−	0.839299i	$-0.317034\pi$
$109$	11.3522	1.08734	0.543671	+	0.839299i	$0.317034\pi$
$110$	0	0
$111$	0	0
$112$	6.94243	0.655998
$113$	17.2802	1.62559	0.812794	−	0.582552i	$-0.197946\pi$
$113$	17.2802	1.62559	0.812794	+	0.582552i	$0.197946\pi$
$114$	0	0
$115$	0	0
$116$	−0.0941136	−0.00873822
$117$	0	0
$118$	−26.9016	−2.47649
$119$	15.7328	1.44222
$120$	0	0
$121$	3.59420	0.326746
$122$	−11.2303	−1.01675
$123$	0	0
$124$	−27.3510	−2.45620
$125$	0	0
$126$	0	0
$127$	−9.48204	−0.841395	−0.420698	−	0.907201i	$-0.638215\pi$
$127$	−9.48204	−0.841395	−0.420698	+	0.907201i	$0.638215\pi$
$128$	−11.4946	−1.01599
$129$	0	0
$130$	0	0
$131$	−9.62930	−0.841316	−0.420658	−	0.907219i	$-0.638201\pi$
$131$	−9.62930	−0.841316	−0.420658	+	0.907219i	$0.638201\pi$
$132$	0	0
$133$	12.1399	1.05267
$134$	−11.7837	−1.01796
$135$	0	0
$136$	−7.25210	−0.621862
$137$	−1.42443	−0.121697	−0.0608484	−	0.998147i	$-0.519381\pi$
$137$	−1.42443	−0.121697	−0.0608484	+	0.998147i	$0.519381\pi$
$138$	0	0
$139$	−1.64669	−0.139670	−0.0698351	−	0.997559i	$-0.522247\pi$
$139$	−1.64669	−0.139670	−0.0698351	+	0.997559i	$0.522247\pi$
$140$	0	0
$141$	0	0
$142$	−26.2957	−2.20669
$143$	10.1304	0.847146
$144$	0	0
$145$	0	0
$146$	26.5943	2.20096
$147$	0	0
$148$	28.3203	2.32791
$149$	9.87569	0.809048	0.404524	−	0.914527i	$-0.367437\pi$
$149$	9.87569	0.809048	0.404524	+	0.914527i	$0.367437\pi$
$150$	0	0
$151$	−11.6003	−0.944021	−0.472011	−	0.881593i	$-0.656472\pi$
$151$	−11.6003	−0.944021	−0.472011	+	0.881593i	$0.656472\pi$
$152$	−5.59595	−0.453891
$153$	0	0
$154$	−27.5515	−2.22017
$155$	0	0
$156$	0	0
$157$	5.64993	0.450914	0.225457	−	0.974253i	$-0.427613\pi$
$157$	5.64993	0.450914	0.225457	+	0.974253i	$0.427613\pi$
$158$	−15.7260	−1.25109
$159$	0	0
$160$	0	0
$161$	12.0282	0.947953
$162$	0	0
$163$	−12.1498	−0.951649	−0.475825	−	0.879540i	$-0.657850\pi$
$163$	−12.1498	−0.951649	−0.475825	+	0.879540i	$0.657850\pi$
$164$	−5.16690	−0.403467
$165$	0	0
$166$	12.2093	0.947622
$167$	−9.87944	−0.764494	−0.382247	−	0.924060i	$-0.624850\pi$
$167$	−9.87944	−0.764494	−0.382247	+	0.924060i	$0.624850\pi$
$168$	0	0
$169$	−5.96810	−0.459085
$170$	0	0
$171$	0	0
$172$	−6.15098	−0.469008
$173$	−0.187054	−0.0142214	−0.00711072	−	0.999975i	$-0.502263\pi$
$173$	−0.187054	−0.0142214	−0.00711072	+	0.999975i	$0.502263\pi$
$174$	0	0
$175$	0	0
$176$	−7.97791	−0.601357
$177$	0	0
$178$	−38.6498	−2.89692
$179$	15.9180	1.18976	0.594882	−	0.803813i	$-0.297199\pi$
$179$	15.9180	1.18976	0.594882	+	0.803813i	$0.297199\pi$
$180$	0	0
$181$	−13.2331	−0.983608	−0.491804	−	0.870706i	$-0.663662\pi$
$181$	−13.2331	−0.983608	−0.491804	+	0.870706i	$0.663662\pi$
$182$	−19.1246	−1.41761
$183$	0	0
$184$	−5.54443	−0.408741
$185$	0	0
$186$	0	0
$187$	−18.0794	−1.32210
$188$	−3.99493	−0.291360
$189$	0	0
$190$	0	0
$191$	2.64734	0.191555	0.0957774	−	0.995403i	$-0.469466\pi$
$191$	2.64734	0.191555	0.0957774	+	0.995403i	$0.469466\pi$
$192$	0	0
$193$	−8.29846	−0.597336	−0.298668	−	0.954357i	$-0.596542\pi$
$193$	−8.29846	−0.597336	−0.298668	+	0.954357i	$0.596542\pi$
$194$	−17.9579	−1.28930
$195$	0	0
$196$	10.9651	0.783223
$197$	−20.8357	−1.48448	−0.742239	−	0.670135i	$-0.766236\pi$
$197$	−20.8357	−1.48448	−0.742239	+	0.670135i	$0.766236\pi$
$198$	0	0
$199$	−3.31747	−0.235169	−0.117585	−	0.993063i	$-0.537515\pi$
$199$	−3.31747	−0.235169	−0.117585	+	0.993063i	$0.537515\pi$
$200$	0	0
$201$	0	0
$202$	−7.75981	−0.545979
$203$	0.115606	0.00811393
$204$	0	0
$205$	0	0
$206$	27.0299	1.88326
$207$	0	0
$208$	−5.53777	−0.383975
$209$	−13.9506	−0.964985
$210$	0	0
$211$	10.6766	0.735005	0.367503	−	0.930022i	$-0.380213\pi$
$211$	10.6766	0.735005	0.367503	+	0.930022i	$0.380213\pi$
$212$	24.1542	1.65892
$213$	0	0
$214$	1.24711	0.0852509
$215$	0	0
$216$	0	0
$217$	33.5970	2.28071
$218$	24.6276	1.66799
$219$	0	0
$220$	0	0
$221$	−12.5496	−0.844177
$222$	0	0
$223$	0.548553	0.0367338	0.0183669	−	0.999831i	$-0.494153\pi$
$223$	0.548553	0.0367338	0.0183669	+	0.999831i	$0.494153\pi$
$224$	25.2496	1.68706
$225$	0	0
$226$	37.4880	2.49366
$227$	−3.68396	−0.244513	−0.122256	−	0.992499i	$-0.539013\pi$
$227$	−3.68396	−0.244513	−0.122256	+	0.992499i	$0.539013\pi$
$228$	0	0
$229$	−1.72213	−0.113801	−0.0569007	−	0.998380i	$-0.518122\pi$
$229$	−1.72213	−0.113801	−0.0569007	+	0.998380i	$0.518122\pi$
$230$	0	0
$231$	0	0
$232$	−0.0532889	−0.00349858
$233$	17.3095	1.13398	0.566992	−	0.823723i	$-0.308107\pi$
$233$	17.3095	1.13398	0.566992	+	0.823723i	$0.308107\pi$
$234$	0	0
$235$	0	0
$236$	−33.5599	−2.18457
$237$	0	0
$238$	34.1310	2.21238
$239$	10.1897	0.659119	0.329560	−	0.944135i	$-0.393100\pi$
$239$	10.1897	0.659119	0.329560	+	0.944135i	$0.393100\pi$
$240$	0	0
$241$	12.0119	0.773755	0.386878	−	0.922131i	$-0.373554\pi$
$241$	12.0119	0.773755	0.386878	+	0.922131i	$0.373554\pi$
$242$	7.79732	0.501231
$243$	0	0
$244$	−14.0099	−0.896892
$245$	0	0
$246$	0	0
$247$	−9.68367	−0.616157
$248$	−15.4867	−0.983404
$249$	0	0
$250$	0	0
$251$	8.68448	0.548159	0.274080	−	0.961707i	$-0.411627\pi$
$251$	8.68448	0.548159	0.274080	+	0.961707i	$0.411627\pi$
$252$	0	0
$253$	−13.8222	−0.868995
$254$	−20.5705	−1.29071
$255$	0	0
$256$	−0.335341	−0.0209588
$257$	−8.45069	−0.527140	−0.263570	−	0.964640i	$-0.584900\pi$
$257$	−8.45069	−0.527140	−0.263570	+	0.964640i	$0.584900\pi$
$258$	0	0
$259$	−34.7876	−2.16160
$260$	0	0
$261$	0	0
$262$	−20.8899	−1.29059
$263$	−8.16646	−0.503565	−0.251783	−	0.967784i	$-0.581017\pi$
$263$	−8.16646	−0.503565	−0.251783	+	0.967784i	$0.581017\pi$
$264$	0	0
$265$	0	0
$266$	26.3366	1.61480
$267$	0	0
$268$	−14.7002	−0.897960
$269$	−13.6366	−0.831437	−0.415719	−	0.909493i	$-0.636470\pi$
$269$	−13.6366	−0.831437	−0.415719	+	0.909493i	$0.636470\pi$
$270$	0	0
$271$	−1.07826	−0.0654995	−0.0327498	−	0.999464i	$-0.510426\pi$
$271$	−1.07826	−0.0654995	−0.0327498	+	0.999464i	$0.510426\pi$
$272$	9.88308	0.599250
$273$	0	0
$274$	−3.09017	−0.186684
$275$	0	0
$276$	0	0
$277$	17.6518	1.06059	0.530296	−	0.847813i	$-0.322081\pi$
$277$	17.6518	1.06059	0.530296	+	0.847813i	$0.322081\pi$
$278$	−3.57235	−0.214255
$279$	0	0
$280$	0	0
$281$	25.3621	1.51298	0.756488	−	0.654007i	$-0.226913\pi$
$281$	25.3621	1.51298	0.756488	+	0.654007i	$0.226913\pi$
$282$	0	0
$283$	−10.8899	−0.647340	−0.323670	−	0.946170i	$-0.604917\pi$
$283$	−10.8899	−0.647340	−0.323670	+	0.946170i	$0.604917\pi$
$284$	−32.8041	−1.94657
$285$	0	0
$286$	21.9770	1.29953
$287$	6.34683	0.374642
$288$	0	0
$289$	5.39686	0.317462
$290$	0	0
$291$	0	0
$292$	33.1766	1.94152
$293$	−8.22487	−0.480502	−0.240251	−	0.970711i	$-0.577230\pi$
$293$	−8.22487	−0.480502	−0.240251	+	0.970711i	$0.577230\pi$
$294$	0	0
$295$	0	0
$296$	16.0355	0.932043
$297$	0	0
$298$	21.4245	1.24109
$299$	−9.59452	−0.554866
$300$	0	0
$301$	7.55564	0.435500
$302$	−25.1659	−1.44814
$303$	0	0
$304$	7.62610	0.437387
$305$	0	0
$306$	0	0
$307$	6.38809	0.364587	0.182294	−	0.983244i	$-0.441648\pi$
$307$	6.38809	0.364587	0.182294	+	0.983244i	$0.441648\pi$
$308$	−34.3707	−1.95845
$309$	0	0
$310$	0	0
$311$	18.7871	1.06532	0.532660	−	0.846329i	$-0.321192\pi$
$311$	18.7871	1.06532	0.532660	+	0.846329i	$0.321192\pi$
$312$	0	0
$313$	21.8101	1.23278	0.616391	−	0.787441i	$-0.288594\pi$
$313$	21.8101	1.23278	0.616391	+	0.787441i	$0.288594\pi$
$314$	12.2570	0.691705
$315$	0	0
$316$	−19.6183	−1.10362
$317$	11.0107	0.618425	0.309212	−	0.950993i	$-0.399935\pi$
$317$	11.0107	0.618425	0.309212	+	0.950993i	$0.399935\pi$
$318$	0	0
$319$	−0.132848	−0.00743809
$320$	0	0
$321$	0	0
$322$	26.0941	1.45417
$323$	17.2821	0.961603
$324$	0	0
$325$	0	0
$326$	−26.3581	−1.45984
$327$	0	0
$328$	−2.92560	−0.161539
$329$	4.90723	0.270544
$330$	0	0
$331$	8.75683	0.481319	0.240660	−	0.970610i	$-0.422636\pi$
$331$	8.75683	0.481319	0.240660	+	0.970610i	$0.422636\pi$
$332$	15.2311	0.835917
$333$	0	0
$334$	−21.4326	−1.17274
$335$	0	0
$336$	0	0
$337$	−30.5031	−1.66161	−0.830804	−	0.556566i	$-0.812119\pi$
$337$	−30.5031	−1.66161	−0.830804	+	0.556566i	$0.812119\pi$
$338$	−12.9473	−0.704240
$339$	0	0
$340$	0	0
$341$	−38.6081	−2.09074
$342$	0	0
$343$	9.80162	0.529238
$344$	−3.48280	−0.187780
$345$	0	0
$346$	−0.405798	−0.0218158
$347$	−12.7260	−0.683169	−0.341585	−	0.939851i	$-0.610964\pi$
$347$	−12.7260	−0.683169	−0.341585	+	0.939851i	$0.610964\pi$
$348$	0	0
$349$	−22.2817	−1.19271	−0.596357	−	0.802720i	$-0.703386\pi$
$349$	−22.2817	−1.19271	−0.596357	+	0.802720i	$0.703386\pi$
$350$	0	0
$351$	0	0
$352$	−29.0156	−1.54654
$353$	−17.0086	−0.905276	−0.452638	−	0.891694i	$-0.649517\pi$
$353$	−17.0086	−0.905276	−0.452638	+	0.891694i	$0.649517\pi$
$354$	0	0
$355$	0	0
$356$	−48.2159	−2.55544
$357$	0	0
$358$	34.5327	1.82511
$359$	20.2116	1.06673	0.533363	−	0.845887i	$-0.320928\pi$
$359$	20.2116	1.06673	0.533363	+	0.845887i	$0.320928\pi$
$360$	0	0
$361$	−5.66456	−0.298135
$362$	−28.7081	−1.50886
$363$	0	0
$364$	−23.8580	−1.25050
$365$	0	0
$366$	0	0
$367$	14.5674	0.760412	0.380206	−	0.924902i	$-0.375853\pi$
$367$	14.5674	0.760412	0.380206	+	0.924902i	$0.375853\pi$
$368$	7.55589	0.393878
$369$	0	0
$370$	0	0
$371$	−29.6702	−1.54040
$372$	0	0
$373$	−5.66796	−0.293476	−0.146738	−	0.989175i	$-0.546877\pi$
$373$	−5.66796	−0.293476	−0.146738	+	0.989175i	$0.546877\pi$
$374$	−39.2217	−2.02811
$375$	0	0
$376$	−2.26200	−0.116654
$377$	−0.0922152	−0.00474932
$378$	0	0
$379$	8.98922	0.461745	0.230873	−	0.972984i	$-0.425842\pi$
$379$	8.98922	0.461745	0.230873	+	0.972984i	$0.425842\pi$
$380$	0	0
$381$	0	0
$382$	5.74318	0.293847
$383$	23.2164	1.18630	0.593152	−	0.805090i	$-0.297883\pi$
$383$	23.2164	1.18630	0.593152	+	0.805090i	$0.297883\pi$
$384$	0	0
$385$	0	0
$386$	−18.0028	−0.916319
$387$	0	0
$388$	−22.4026	−1.13732
$389$	10.5936	0.537119	0.268560	−	0.963263i	$-0.413452\pi$
$389$	10.5936	0.537119	0.268560	+	0.963263i	$0.413452\pi$
$390$	0	0
$391$	17.1230	0.865949
$392$	6.20866	0.313585
$393$	0	0
$394$	−45.2012	−2.27720
$395$	0	0
$396$	0	0
$397$	−2.71118	−0.136070	−0.0680352	−	0.997683i	$-0.521673\pi$
$397$	−2.71118	−0.136070	−0.0680352	+	0.997683i	$0.521673\pi$
$398$	−7.19697	−0.360751
$399$	0	0
$400$	0	0
$401$	−8.49932	−0.424436	−0.212218	−	0.977222i	$-0.568069\pi$
$401$	−8.49932	−0.424436	−0.212218	+	0.977222i	$0.568069\pi$
$402$	0	0
$403$	−26.7993	−1.33497
$404$	−9.68043	−0.481619
$405$	0	0
$406$	0.250797	0.0124468
$407$	39.9762	1.98155
$408$	0	0
$409$	16.0537	0.793805	0.396903	−	0.917861i	$-0.370085\pi$
$409$	16.0537	0.793805	0.396903	+	0.917861i	$0.370085\pi$
$410$	0	0
$411$	0	0
$412$	33.7200	1.66126
$413$	41.2238	2.02849
$414$	0	0
$415$	0	0
$416$	−20.1408	−0.987486
$417$	0	0
$418$	−30.2647	−1.48030
$419$	27.2568	1.33158	0.665791	−	0.746138i	$-0.268094\pi$
$419$	27.2568	1.33158	0.665791	+	0.746138i	$0.268094\pi$
$420$	0	0
$421$	2.99388	0.145913	0.0729563	−	0.997335i	$-0.476757\pi$
$421$	2.99388	0.145913	0.0729563	+	0.997335i	$0.476757\pi$
$422$	23.1619	1.12750
$423$	0	0
$424$	13.6766	0.664193
$425$	0	0
$426$	0	0
$427$	17.2093	0.832814
$428$	1.55578	0.0752016
$429$	0	0
$430$	0	0
$431$	34.2582	1.65016	0.825080	−	0.565016i	$-0.191130\pi$
$431$	34.2582	1.65016	0.825080	+	0.565016i	$0.191130\pi$
$432$	0	0
$433$	−10.2956	−0.494777	−0.247388	−	0.968916i	$-0.579572\pi$
$433$	−10.2956	−0.494777	−0.247388	+	0.968916i	$0.579572\pi$
$434$	72.8859	3.49863
$435$	0	0
$436$	30.7231	1.47137
$437$	13.2127	0.632048
$438$	0	0
$439$	−39.3047	−1.87591	−0.937955	−	0.346757i	$-0.887283\pi$
$439$	−39.3047	−1.87591	−0.937955	+	0.346757i	$0.887283\pi$
$440$	0	0
$441$	0	0
$442$	−27.2253	−1.29497
$443$	−0.749909	−0.0356293	−0.0178146	−	0.999841i	$-0.505671\pi$
$443$	−0.749909	−0.0356293	−0.0178146	+	0.999841i	$0.505671\pi$
$444$	0	0
$445$	0	0
$446$	1.19004	0.0563500
$447$	0	0
$448$	40.8919	1.93196
$449$	−6.79987	−0.320906	−0.160453	−	0.987043i	$-0.551295\pi$
$449$	−6.79987	−0.320906	−0.160453	+	0.987043i	$0.551295\pi$
$450$	0	0
$451$	−7.29348	−0.343436
$452$	46.7666	2.19971
$453$	0	0
$454$	−7.99204	−0.375085
$455$	0	0
$456$	0	0
$457$	17.4455	0.816065	0.408033	−	0.912967i	$-0.366215\pi$
$457$	17.4455	0.816065	0.408033	+	0.912967i	$0.366215\pi$
$458$	−3.73601	−0.174572
$459$	0	0
$460$	0	0
$461$	38.3433	1.78582	0.892912	−	0.450232i	$-0.148659\pi$
$461$	38.3433	1.78582	0.892912	+	0.450232i	$0.148659\pi$
$462$	0	0
$463$	15.0973	0.701629	0.350815	−	0.936445i	$-0.385905\pi$
$463$	15.0973	0.701629	0.350815	+	0.936445i	$0.385905\pi$
$464$	0.0726215	0.00337137
$465$	0	0
$466$	37.5515	1.73954
$467$	5.58149	0.258281	0.129140	−	0.991626i	$-0.458778\pi$
$467$	5.58149	0.258281	0.129140	+	0.991626i	$0.458778\pi$
$468$	0	0
$469$	18.0572	0.833806
$470$	0	0
$471$	0	0
$472$	−19.0023	−0.874650
$473$	−8.68258	−0.399225
$474$	0	0
$475$	0	0
$476$	42.5787	1.95159
$477$	0	0
$478$	22.1058	1.01109
$479$	2.94296	0.134467	0.0672337	−	0.997737i	$-0.478583\pi$
$479$	2.94296	0.134467	0.0672337	+	0.997737i	$0.478583\pi$
$480$	0	0
$481$	27.7490	1.26525
$482$	26.0588	1.18695
$483$	0	0
$484$	9.72721	0.442146
$485$	0	0
$486$	0	0
$487$	−5.70988	−0.258739	−0.129370	−	0.991596i	$-0.541295\pi$
$487$	−5.70988	−0.258739	−0.129370	+	0.991596i	$0.541295\pi$
$488$	−7.93267	−0.359095
$489$	0	0
$490$	0	0
$491$	41.2647	1.86225	0.931126	−	0.364699i	$-0.118828\pi$
$491$	41.2647	1.86225	0.931126	+	0.364699i	$0.118828\pi$
$492$	0	0
$493$	0.164573	0.00741202
$494$	−21.0079	−0.945190
$495$	0	0
$496$	21.1051	0.947645
$497$	40.2954	1.80750
$498$	0	0
$499$	34.8039	1.55804	0.779018	−	0.627002i	$-0.215718\pi$
$499$	34.8039	1.55804	0.779018	+	0.627002i	$0.215718\pi$
$500$	0	0
$501$	0	0
$502$	18.8402	0.840881
$503$	28.1187	1.25375	0.626876	−	0.779119i	$-0.284333\pi$
$503$	28.1187	1.25375	0.626876	+	0.779119i	$0.284333\pi$
$504$	0	0
$505$	0	0
$506$	−29.9861	−1.33304
$507$	0	0
$508$	−25.6618	−1.13856
$509$	31.7323	1.40651	0.703255	−	0.710937i	$-0.251729\pi$
$509$	31.7323	1.40651	0.703255	+	0.710937i	$0.251729\pi$
$510$	0	0
$511$	−40.7530	−1.80281
$512$	22.2616	0.983834
$513$	0	0
$514$	−18.3331	−0.808637
$515$	0	0
$516$	0	0
$517$	−5.63915	−0.248009
$518$	−75.4687	−3.31591
$519$	0	0
$520$	0	0
$521$	−1.31306	−0.0575264	−0.0287632	−	0.999586i	$-0.509157\pi$
$521$	−1.31306	−0.0575264	−0.0287632	+	0.999586i	$0.509157\pi$
$522$	0	0
$523$	−0.723367	−0.0316306	−0.0158153	−	0.999875i	$-0.505034\pi$
$523$	−0.723367	−0.0316306	−0.0158153	+	0.999875i	$0.505034\pi$
$524$	−26.0604	−1.13845
$525$	0	0
$526$	−17.7164	−0.772473
$527$	47.8279	2.08342
$528$	0	0
$529$	−9.90896	−0.430824
$530$	0	0
$531$	0	0
$532$	32.8551	1.42445
$533$	−5.06268	−0.219289
$534$	0	0
$535$	0	0
$536$	−8.32355	−0.359523
$537$	0	0
$538$	−29.5834	−1.27543
$539$	15.4781	0.666689
$540$	0	0
$541$	−43.7724	−1.88192	−0.940962	−	0.338513i	$-0.890076\pi$
$541$	−43.7724	−1.88192	−0.940962	+	0.338513i	$0.890076\pi$
$542$	−2.33919	−0.100477
$543$	0	0
$544$	35.9447	1.54112
$545$	0	0
$546$	0	0
$547$	−22.2588	−0.951716	−0.475858	−	0.879522i	$-0.657862\pi$
$547$	−22.2588	−0.951716	−0.475858	+	0.879522i	$0.657862\pi$
$548$	−3.85501	−0.164678
$549$	0	0
$550$	0	0
$551$	0.126990	0.00540996
$552$	0	0
$553$	24.0984	1.02477
$554$	38.2940	1.62696
$555$	0	0
$556$	−4.45653	−0.188999
$557$	−40.8497	−1.73086	−0.865429	−	0.501032i	$-0.832954\pi$
$557$	−40.8497	−1.73086	−0.865429	+	0.501032i	$0.832954\pi$
$558$	0	0
$559$	−6.02691	−0.254911
$560$	0	0
$561$	0	0
$562$	55.0209	2.32092
$563$	−37.7591	−1.59136	−0.795678	−	0.605719i	$-0.792885\pi$
$563$	−37.7591	−1.59136	−0.795678	+	0.605719i	$0.792885\pi$
$564$	0	0
$565$	0	0
$566$	−23.6248	−0.993025
$567$	0	0
$568$	−18.5743	−0.779361
$569$	−27.2580	−1.14271	−0.571357	−	0.820701i	$-0.693583\pi$
$569$	−27.2580	−1.14271	−0.571357	+	0.820701i	$0.693583\pi$
$570$	0	0
$571$	−40.0883	−1.67764	−0.838822	−	0.544406i	$-0.816755\pi$
$571$	−40.0883	−1.67764	−0.838822	+	0.544406i	$0.816755\pi$
$572$	27.4165	1.14634
$573$	0	0
$574$	13.7689	0.574703
$575$	0	0
$576$	0	0
$577$	16.6281	0.692237	0.346119	−	0.938191i	$-0.387500\pi$
$577$	16.6281	0.692237	0.346119	+	0.938191i	$0.387500\pi$
$578$	11.7080	0.486990
$579$	0	0
$580$	0	0
$581$	−18.7094	−0.776196
$582$	0	0
$583$	34.0955	1.41209
$584$	18.7852	0.777339
$585$	0	0
$586$	−17.8432	−0.737094
$587$	10.7184	0.442397	0.221199	−	0.975229i	$-0.429003\pi$
$587$	10.7184	0.442397	0.221199	+	0.975229i	$0.429003\pi$
$588$	0	0
$589$	36.9055	1.52067
$590$	0	0
$591$	0	0
$592$	−21.8530	−0.898151
$593$	30.3399	1.24591	0.622954	−	0.782258i	$-0.285932\pi$
$593$	30.3399	1.24591	0.622954	+	0.782258i	$0.285932\pi$
$594$	0	0
$595$	0	0
$596$	26.7272	1.09479
$597$	0	0
$598$	−20.8145	−0.851168
$599$	−38.5303	−1.57431	−0.787153	−	0.616758i	$-0.788446\pi$
$599$	−38.5303	−1.57431	−0.787153	+	0.616758i	$0.788446\pi$
$600$	0	0
$601$	−12.5110	−0.510333	−0.255166	−	0.966897i	$-0.582130\pi$
$601$	−12.5110	−0.510333	−0.255166	+	0.966897i	$0.582130\pi$
$602$	16.3913	0.668061
$603$	0	0
$604$	−31.3947	−1.27743
$605$	0	0
$606$	0	0
$607$	−8.69260	−0.352822	−0.176411	−	0.984317i	$-0.556449\pi$
$607$	−8.69260	−0.352822	−0.176411	+	0.984317i	$0.556449\pi$
$608$	27.7361	1.12485
$609$	0	0
$610$	0	0
$611$	−3.91435	−0.158358
$612$	0	0
$613$	−2.85919	−0.115481	−0.0577407	−	0.998332i	$-0.518390\pi$
$613$	−2.85919	−0.115481	−0.0577407	+	0.998332i	$0.518390\pi$
$614$	13.8584	0.559280
$615$	0	0
$616$	−19.4614	−0.784120
$617$	39.6954	1.59808	0.799039	−	0.601280i	$-0.205342\pi$
$617$	39.6954	1.59808	0.799039	+	0.601280i	$0.205342\pi$
$618$	0	0
$619$	21.7072	0.872487	0.436244	−	0.899829i	$-0.356309\pi$
$619$	21.7072	0.872487	0.436244	+	0.899829i	$0.356309\pi$
$620$	0	0
$621$	0	0
$622$	40.7571	1.63421
$623$	59.2266	2.37286
$624$	0	0
$625$	0	0
$626$	47.3152	1.89110
$627$	0	0
$628$	15.2908	0.610168
$629$	−49.5228	−1.97460
$630$	0	0
$631$	−4.23775	−0.168702	−0.0843510	−	0.996436i	$-0.526882\pi$
$631$	−4.23775	−0.168702	−0.0843510	+	0.996436i	$0.526882\pi$
$632$	−11.1082	−0.441862
$633$	0	0
$634$	23.8869	0.948669
$635$	0	0
$636$	0	0
$637$	10.7439	0.425691
$638$	−0.288203	−0.0114101
$639$	0	0
$640$	0	0
$641$	25.8746	1.02198	0.510992	−	0.859585i	$-0.329278\pi$
$641$	25.8746	1.02198	0.510992	+	0.859585i	$0.329278\pi$
$642$	0	0
$643$	3.38669	0.133558	0.0667790	−	0.997768i	$-0.478728\pi$
$643$	3.38669	0.133558	0.0667790	+	0.997768i	$0.478728\pi$
$644$	32.5526	1.28275
$645$	0	0
$646$	37.4921	1.47511
$647$	6.65156	0.261500	0.130750	−	0.991415i	$-0.458262\pi$
$647$	6.65156	0.261500	0.130750	+	0.991415i	$0.458262\pi$
$648$	0	0
$649$	−47.3724	−1.85953
$650$	0	0
$651$	0	0
$652$	−32.8819	−1.28775
$653$	19.6601	0.769359	0.384679	−	0.923050i	$-0.374312\pi$
$653$	19.6601	0.769359	0.384679	+	0.923050i	$0.374312\pi$
$654$	0	0
$655$	0	0
$656$	3.98697	0.155665
$657$	0	0
$658$	10.6458	0.415017
$659$	0.251530	0.00979822	0.00489911	−	0.999988i	$-0.498441\pi$
$659$	0.251530	0.00979822	0.00489911	+	0.999988i	$0.498441\pi$
$660$	0	0
$661$	−47.0832	−1.83133	−0.915663	−	0.401947i	$-0.868333\pi$
$661$	−47.0832	−1.83133	−0.915663	+	0.401947i	$0.868333\pi$
$662$	18.9972	0.738347
$663$	0	0
$664$	8.62416	0.334682
$665$	0	0
$666$	0	0
$667$	0.125821	0.00487181
$668$	−26.7373	−1.03450
$669$	0	0
$670$	0	0
$671$	−19.7760	−0.763446
$672$	0	0
$673$	15.0647	0.580700	0.290350	−	0.956921i	$-0.406228\pi$
$673$	15.0647	0.580700	0.290350	+	0.956921i	$0.406228\pi$
$674$	−66.1738	−2.54892
$675$	0	0
$676$	−16.1518	−0.621225
$677$	6.18681	0.237779	0.118889	−	0.992908i	$-0.462067\pi$
$677$	6.18681	0.237779	0.118889	+	0.992908i	$0.462067\pi$
$678$	0	0
$679$	27.5186	1.05607
$680$	0	0
$681$	0	0
$682$	−83.7569	−3.20722
$683$	12.4499	0.476384	0.238192	−	0.971218i	$-0.423445\pi$
$683$	12.4499	0.476384	0.238192	+	0.971218i	$0.423445\pi$
$684$	0	0
$685$	0	0
$686$	21.2638	0.811855
$687$	0	0
$688$	4.74632	0.180952
$689$	23.6670	0.901641
$690$	0	0
$691$	−27.3248	−1.03949	−0.519743	−	0.854323i	$-0.673972\pi$
$691$	−27.3248	−1.03949	−0.519743	+	0.854323i	$0.673972\pi$
$692$	−0.506236	−0.0192442
$693$	0	0
$694$	−27.6081	−1.04799
$695$	0	0
$696$	0	0
$697$	9.03520	0.342233
$698$	−48.3383	−1.82963
$699$	0	0
$700$	0	0
$701$	−27.7130	−1.04671	−0.523353	−	0.852116i	$-0.675319\pi$
$701$	−27.7130	−1.04671	−0.523353	+	0.852116i	$0.675319\pi$
$702$	0	0
$703$	−38.2134	−1.44124
$704$	−46.9911	−1.77104
$705$	0	0
$706$	−36.8987	−1.38870
$707$	11.8911	0.447210
$708$	0	0
$709$	8.02059	0.301220	0.150610	−	0.988593i	$-0.451876\pi$
$709$	8.02059	0.301220	0.150610	+	0.988593i	$0.451876\pi$
$710$	0	0
$711$	0	0
$712$	−27.3007	−1.02314
$713$	36.5658	1.36940
$714$	0	0
$715$	0	0
$716$	43.0798	1.60997
$717$	0	0
$718$	43.8473	1.63637
$719$	−16.0324	−0.597908	−0.298954	−	0.954268i	$-0.596638\pi$
$719$	−16.0324	−0.597908	−0.298954	+	0.954268i	$0.596638\pi$
$720$	0	0
$721$	−41.4204	−1.54257
$722$	−12.2888	−0.457341
$723$	0	0
$724$	−35.8135	−1.33100
$725$	0	0
$726$	0	0
$727$	−3.78819	−0.140496	−0.0702481	−	0.997530i	$-0.522379\pi$
$727$	−3.78819	−0.140496	−0.0702481	+	0.997530i	$0.522379\pi$
$728$	−13.5089	−0.500672
$729$	0	0
$730$	0	0
$731$	10.7560	0.397826
$732$	0	0
$733$	−42.4612	−1.56834	−0.784170	−	0.620546i	$-0.786911\pi$
$733$	−42.4612	−1.56834	−0.784170	+	0.620546i	$0.786911\pi$
$734$	31.6027	1.16648
$735$	0	0
$736$	27.4807	1.01295
$737$	−20.7505	−0.764355
$738$	0	0
$739$	−1.15212	−0.0423815	−0.0211907	−	0.999775i	$-0.506746\pi$
$739$	−1.15212	−0.0423815	−0.0211907	+	0.999775i	$0.506746\pi$
$740$	0	0
$741$	0	0
$742$	−64.3669	−2.36298
$743$	39.7658	1.45887	0.729433	−	0.684053i	$-0.239784\pi$
$743$	39.7658	1.45887	0.729433	+	0.684053i	$0.239784\pi$
$744$	0	0
$745$	0	0
$746$	−12.2962	−0.450195
$747$	0	0
$748$	−48.9294	−1.78904
$749$	−1.91107	−0.0698289
$750$	0	0
$751$	−26.2947	−0.959506	−0.479753	−	0.877404i	$-0.659274\pi$
$751$	−26.2947	−0.959506	−0.479753	+	0.877404i	$0.659274\pi$
$752$	3.08264	0.112412
$753$	0	0
$754$	−0.200053	−0.00728550
$755$	0	0
$756$	0	0
$757$	−24.5334	−0.891681	−0.445840	−	0.895113i	$-0.647095\pi$
$757$	−24.5334	−0.891681	−0.445840	+	0.895113i	$0.647095\pi$
$758$	19.5013	0.708321
$759$	0	0
$760$	0	0
$761$	18.8035	0.681626	0.340813	−	0.940131i	$-0.389298\pi$
$761$	18.8035	0.681626	0.340813	+	0.940131i	$0.389298\pi$
$762$	0	0
$763$	−37.7391	−1.36625
$764$	7.16466	0.259208
$765$	0	0
$766$	50.3661	1.81980
$767$	−32.8830	−1.18734
$768$	0	0
$769$	−31.9012	−1.15039	−0.575193	−	0.818018i	$-0.695073\pi$
$769$	−31.9012	−1.15039	−0.575193	+	0.818018i	$0.695073\pi$
$770$	0	0
$771$	0	0
$772$	−22.4586	−0.808304
$773$	−5.08382	−0.182852	−0.0914262	−	0.995812i	$-0.529143\pi$
$773$	−5.08382	−0.182852	−0.0914262	+	0.995812i	$0.529143\pi$
$774$	0	0
$775$	0	0
$776$	−12.6848	−0.455358
$777$	0	0
$778$	22.9820	0.823945
$779$	6.97185	0.249792
$780$	0	0
$781$	−46.3055	−1.65694
$782$	37.1470	1.32837
$783$	0	0
$784$	−8.46109	−0.302182
$785$	0	0
$786$	0	0
$787$	−31.5066	−1.12309	−0.561544	−	0.827447i	$-0.689793\pi$
$787$	−31.5066	−1.12309	−0.561544	+	0.827447i	$0.689793\pi$
$788$	−56.3888	−2.00877
$789$	0	0
$790$	0	0
$791$	−57.4463	−2.04256
$792$	0	0
$793$	−13.7273	−0.487471
$794$	−5.88168	−0.208733
$795$	0	0
$796$	−8.97827	−0.318226
$797$	17.5860	0.622927	0.311463	−	0.950258i	$-0.399181\pi$
$797$	17.5860	0.622927	0.311463	+	0.950258i	$0.399181\pi$
$798$	0	0
$799$	6.98581	0.247140
$800$	0	0
$801$	0	0
$802$	−18.4386	−0.651088
$803$	46.8314	1.65264
$804$	0	0
$805$	0	0
$806$	−58.1389	−2.04786
$807$	0	0
$808$	−5.48124	−0.192829
$809$	14.4567	0.508270	0.254135	−	0.967169i	$-0.418209\pi$
$809$	14.4567	0.508270	0.254135	+	0.967169i	$0.418209\pi$
$810$	0	0
$811$	25.5964	0.898810	0.449405	−	0.893328i	$-0.351636\pi$
$811$	25.5964	0.898810	0.449405	+	0.893328i	$0.351636\pi$
$812$	0.312871	0.0109796
$813$	0	0
$814$	86.7250	3.03971
$815$	0	0
$816$	0	0
$817$	8.29970	0.290370
$818$	34.8272	1.21770
$819$	0	0
$820$	0	0
$821$	11.4860	0.400864	0.200432	−	0.979708i	$-0.435765\pi$
$821$	11.4860	0.400864	0.200432	+	0.979708i	$0.435765\pi$
$822$	0	0
$823$	−55.7742	−1.94417	−0.972083	−	0.234636i	$-0.924610\pi$
$823$	−55.7742	−1.94417	−0.972083	+	0.234636i	$0.924610\pi$
$824$	19.0929	0.665132
$825$	0	0
$826$	89.4316	3.11172
$827$	5.19086	0.180504	0.0902520	−	0.995919i	$-0.471233\pi$
$827$	5.19086	0.180504	0.0902520	+	0.995919i	$0.471233\pi$
$828$	0	0
$829$	29.8892	1.03810	0.519048	−	0.854745i	$-0.326286\pi$
$829$	29.8892	1.03810	0.519048	+	0.854745i	$0.326286\pi$
$830$	0	0
$831$	0	0
$832$	−32.6183	−1.13084
$833$	−19.1744	−0.664353
$834$	0	0
$835$	0	0
$836$	−37.7555	−1.30580
$837$	0	0
$838$	59.1314	2.04266
$839$	−42.2678	−1.45925	−0.729624	−	0.683849i	$-0.760305\pi$
$839$	−42.2678	−1.45925	−0.729624	+	0.683849i	$0.760305\pi$
$840$	0	0
$841$	−28.9988	−0.999958
$842$	6.49496	0.223831
$843$	0	0
$844$	28.8947	0.994595
$845$	0	0
$846$	0	0
$847$	−11.9486	−0.410557
$848$	−18.6383	−0.640041
$849$	0	0
$850$	0	0
$851$	−37.8616	−1.29788
$852$	0	0
$853$	−5.58228	−0.191134	−0.0955669	−	0.995423i	$-0.530466\pi$
$853$	−5.58228	−0.191134	−0.0955669	+	0.995423i	$0.530466\pi$
$854$	37.3340	1.27754
$855$	0	0
$856$	0.880914	0.0301090
$857$	40.9218	1.39786	0.698930	−	0.715190i	$-0.253660\pi$
$857$	40.9218	1.39786	0.698930	+	0.715190i	$0.253660\pi$
$858$	0	0
$859$	6.29742	0.214865	0.107433	−	0.994212i	$-0.465737\pi$
$859$	6.29742	0.214865	0.107433	+	0.994212i	$0.465737\pi$
$860$	0	0
$861$	0	0
$862$	74.3203	2.53136
$863$	19.1377	0.651457	0.325728	−	0.945463i	$-0.394391\pi$
$863$	19.1377	0.651457	0.325728	+	0.945463i	$0.394391\pi$
$864$	0	0
$865$	0	0
$866$	−22.3355	−0.758992
$867$	0	0
$868$	90.9257	3.08622
$869$	−27.6927	−0.939411
$870$	0	0
$871$	−14.4037	−0.488052
$872$	17.3960	0.589103
$873$	0	0
$874$	28.6638	0.969567
$875$	0	0
$876$	0	0
$877$	13.3276	0.450042	0.225021	−	0.974354i	$-0.427755\pi$
$877$	13.3276	0.450042	0.225021	+	0.974354i	$0.427755\pi$
$878$	−85.2682	−2.87766
$879$	0	0
$880$	0	0
$881$	32.8256	1.10592	0.552962	−	0.833207i	$-0.313498\pi$
$881$	32.8256	1.10592	0.552962	+	0.833207i	$0.313498\pi$
$882$	0	0
$883$	−7.28848	−0.245277	−0.122638	−	0.992451i	$-0.539136\pi$
$883$	−7.28848	−0.245277	−0.122638	+	0.992451i	$0.539136\pi$
$884$	−33.9638	−1.14232
$885$	0	0
$886$	−1.62686	−0.0546556
$887$	−31.8681	−1.07003	−0.535013	−	0.844844i	$-0.679693\pi$
$887$	−31.8681	−1.07003	−0.535013	+	0.844844i	$0.679693\pi$
$888$	0	0
$889$	31.5221	1.05722
$890$	0	0
$891$	0	0
$892$	1.48458	0.0497075
$893$	5.39048	0.180385
$894$	0	0
$895$	0	0
$896$	38.2125	1.27659
$897$	0	0
$898$	−14.7517	−0.492272
$899$	0.351442	0.0117213
$900$	0	0
$901$	−42.2377	−1.40714
$902$	−15.8226	−0.526834
$903$	0	0
$904$	26.4801	0.880715
$905$	0	0
$906$	0	0
$907$	−40.7506	−1.35310	−0.676551	−	0.736396i	$-0.736526\pi$
$907$	−40.7506	−1.35310	−0.676551	+	0.736396i	$0.736526\pi$
$908$	−9.97013	−0.330870
$909$	0	0
$910$	0	0
$911$	−25.1883	−0.834527	−0.417263	−	0.908786i	$-0.637011\pi$
$911$	−25.1883	−0.834527	−0.417263	+	0.908786i	$0.637011\pi$
$912$	0	0
$913$	21.4999	0.711543
$914$	37.8465	1.25185
$915$	0	0
$916$	−4.66070	−0.153994
$917$	32.0116	1.05712
$918$	0	0
$919$	−2.40754	−0.0794174	−0.0397087	−	0.999211i	$-0.512643\pi$
$919$	−2.40754	−0.0794174	−0.0397087	+	0.999211i	$0.512643\pi$
$920$	0	0
$921$	0	0
$922$	83.1824	2.73947
$923$	−32.1425	−1.05798
$924$	0	0
$925$	0	0
$926$	32.7522	1.07630
$927$	0	0
$928$	0.264124	0.00867029
$929$	−11.0372	−0.362119	−0.181059	−	0.983472i	$-0.557953\pi$
$929$	−11.0372	−0.362119	−0.181059	+	0.983472i	$0.557953\pi$
$930$	0	0
$931$	−14.7956	−0.484905
$932$	46.8458	1.53449
$933$	0	0
$934$	12.1086	0.396205
$935$	0	0
$936$	0	0
$937$	1.59262	0.0520288	0.0260144	−	0.999662i	$-0.491718\pi$
$937$	1.59262	0.0520288	0.0260144	+	0.999662i	$0.491718\pi$
$938$	39.1736	1.27906
$939$	0	0
$940$	0	0
$941$	−8.02279	−0.261535	−0.130768	−	0.991413i	$-0.541744\pi$
$941$	−8.02279	−0.261535	−0.130768	+	0.991413i	$0.541744\pi$
$942$	0	0
$943$	6.90767	0.224945
$944$	25.8961	0.842846
$945$	0	0
$946$	−18.8361	−0.612415
$947$	−11.6107	−0.377298	−0.188649	−	0.982045i	$-0.560411\pi$
$947$	−11.6107	−0.377298	−0.188649	+	0.982045i	$0.560411\pi$
$948$	0	0
$949$	32.5074	1.05524
$950$	0	0
$951$	0	0
$952$	24.1088	0.781372
$953$	−0.918512	−0.0297535	−0.0148768	−	0.999889i	$-0.504736\pi$
$953$	−0.918512	−0.0297535	−0.0148768	+	0.999889i	$0.504736\pi$
$954$	0	0
$955$	0	0
$956$	27.5771	0.891908
$957$	0	0
$958$	6.38451	0.206274
$959$	4.73535	0.152913
$960$	0	0
$961$	71.1353	2.29469
$962$	60.1992	1.94090
$963$	0	0
$964$	32.5086	1.04703
$965$	0	0
$966$	0	0
$967$	22.5457	0.725022	0.362511	−	0.931979i	$-0.381920\pi$
$967$	22.5457	0.725022	0.362511	+	0.931979i	$0.381920\pi$
$968$	5.50773	0.177025
$969$	0	0
$970$	0	0
$971$	−56.1611	−1.80229	−0.901147	−	0.433513i	$-0.857274\pi$
$971$	−56.1611	−1.80229	−0.901147	+	0.433513i	$0.857274\pi$
$972$	0	0
$973$	5.47424	0.175496
$974$	−12.3871	−0.396909
$975$	0	0
$976$	10.8106	0.346038
$977$	−49.6098	−1.58716	−0.793580	−	0.608466i	$-0.791785\pi$
$977$	−49.6098	−1.58716	−0.793580	+	0.608466i	$0.791785\pi$
$978$	0	0
$979$	−68.0604	−2.17522
$980$	0	0
$981$	0	0
$982$	89.5203	2.85671
$983$	−56.2029	−1.79259	−0.896297	−	0.443454i	$-0.853753\pi$
$983$	−56.2029	−1.79259	−0.896297	+	0.443454i	$0.853753\pi$
$984$	0	0
$985$	0	0
$986$	0.357028	0.0113701
$987$	0	0
$988$	−26.2075	−0.833772
$989$	8.22329	0.261486
$990$	0	0
$991$	−17.2327	−0.547414	−0.273707	−	0.961813i	$-0.588250\pi$
$991$	−17.2327	−0.547414	−0.273707	+	0.961813i	$0.588250\pi$
$992$	76.7590	2.43710
$993$	0	0
$994$	87.4175	2.77271
$995$	0	0
$996$	0	0
$997$	1.59296	0.0504495	0.0252247	−	0.999682i	$-0.491970\pi$
$997$	1.59296	0.0504495	0.0252247	+	0.999682i	$0.491970\pi$
$998$	75.5041	2.39004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.w.1.8		8
3.2	odd	2	inner	5625.2.a.w.1.1		8
5.4	even	2		5625.2.a.v.1.1		8
15.14	odd	2		5625.2.a.v.1.8		8
25.11	even	5		225.2.h.e.46.4	yes	16
25.16	even	5		225.2.h.e.181.4	yes	16
75.11	odd	10		225.2.h.e.46.1	✓	16
75.41	odd	10		225.2.h.e.181.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.h.e.46.1	✓	16	75.11	odd	10
225.2.h.e.46.4	yes	16	25.11	even	5
225.2.h.e.181.1	yes	16	75.41	odd	10
225.2.h.e.181.4	yes	16	25.16	even	5
5625.2.a.v.1.1		8	5.4	even	2
5625.2.a.v.1.8		8	15.14	odd	2
5625.2.a.w.1.1		8	3.2	odd	2	inner
5625.2.a.w.1.8		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.16942 q^{2} +2.70636 q^{4} -3.32440 q^{7} +1.53239 q^{8} +O(q^{10})\)	\(q+2.16942 q^{2} +2.70636 q^{4} -3.32440 q^{7} +1.53239 q^{8} +3.82024 q^{11} +2.65177 q^{13} -7.21200 q^{14} -2.08833 q^{16} -4.73253 q^{17} -3.65177 q^{19} +8.28768 q^{22} -3.61815 q^{23} +5.75280 q^{26} -8.99702 q^{28} -0.0347749 q^{29} -10.1062 q^{31} -7.59524 q^{32} -10.2668 q^{34} +10.4643 q^{37} -7.92221 q^{38} -1.90917 q^{41} -2.27279 q^{43} +10.3389 q^{44} -7.84928 q^{46} -1.47613 q^{47} +4.05161 q^{49} +7.17666 q^{52} +8.92498 q^{53} -5.09428 q^{56} -0.0754413 q^{58} -12.4004 q^{59} -5.17666 q^{61} -21.9245 q^{62} -12.3006 q^{64} -5.43173 q^{67} -12.8079 q^{68} -12.1211 q^{71} +12.2588 q^{73} +22.7015 q^{74} -9.88302 q^{76} -12.7000 q^{77} -7.24895 q^{79} -4.14178 q^{82} +5.62790 q^{83} -4.93062 q^{86} +5.85410 q^{88} -17.8158 q^{89} -8.81554 q^{91} -9.79204 q^{92} -3.20233 q^{94} -8.27777 q^{97} +8.78962 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \) 8 * q + 4 * q^4	\( 8 q + 4 q^{4} + 10 q^{13} - 8 q^{16} - 18 q^{19} - 30 q^{28} - 26 q^{31} - 20 q^{34} - 40 q^{43} - 30 q^{46} - 16 q^{49} + 40 q^{52} + 10 q^{58} - 24 q^{61} - 34 q^{64} - 40 q^{67} + 40 q^{73} - 44 q^{76} - 42 q^{79} + 60 q^{82} + 20 q^{88} - 40 q^{91} - 10 q^{94} - 40 q^{97}+O(q^{100}) \) 8 * q + 4 * q^4 + 10 * q^13 - 8 * q^16 - 18 * q^19 - 30 * q^28 - 26 * q^31 - 20 * q^34 - 40 * q^43 - 30 * q^46 - 16 * q^49 + 40 * q^52 + 10 * q^58 - 24 * q^61 - 34 * q^64 - 40 * q^67 + 40 * q^73 - 44 * q^76 - 42 * q^79 + 60 * q^82 + 20 * q^88 - 40 * q^91 - 10 * q^94 - 40 * q^97

Introduction

L-functions

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