LMFDB - Embedded newform 5625.2.a.x.1.4

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:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.183172$ 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)

$f(q)$	$=$	$q-0.183172 q^{2} -1.96645 q^{4} +3.26086 q^{7} +0.726543 q^{8} +O(q^{10})$	$q-0.183172 q^{2} -1.96645 q^{4} +3.26086 q^{7} +0.726543 q^{8} -2.00000 q^{11} +0.296379 q^{13} -0.597298 q^{14} +3.79981 q^{16} -5.16297 q^{17} +1.73038 q^{19} +0.366344 q^{22} +0.879192 q^{23} -0.0542883 q^{26} -6.41230 q^{28} -5.91216 q^{29} +6.09953 q^{31} -2.14910 q^{32} +0.945712 q^{34} -8.08800 q^{37} -0.316957 q^{38} +1.01515 q^{41} +3.24199 q^{43} +3.93290 q^{44} -0.161043 q^{46} +4.21996 q^{47} +3.63318 q^{49} -0.582813 q^{52} -8.10072 q^{53} +2.36915 q^{56} +1.08294 q^{58} -5.93635 q^{59} +0.915615 q^{61} -1.11726 q^{62} -7.20597 q^{64} -6.88806 q^{67} +10.1527 q^{68} +5.96878 q^{71} -8.83341 q^{73} +1.48150 q^{74} -3.40270 q^{76} -6.52171 q^{77} -7.76067 q^{79} -0.185946 q^{82} +14.5154 q^{83} -0.593842 q^{86} -1.45309 q^{88} -7.52642 q^{89} +0.966448 q^{91} -1.72889 q^{92} -0.772978 q^{94} +6.72649 q^{97} -0.665497 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4}+O(q^{10}) $ 8 * q + 6 * q^4	$ 8 q + 6 q^{4} - 16 q^{11} - 12 q^{14} - 2 q^{16} + 10 q^{19} - 6 q^{26} - 20 q^{29} + 16 q^{31} + 2 q^{34} - 26 q^{41} - 12 q^{44} + 6 q^{46} - 14 q^{49} - 10 q^{56} - 30 q^{59} + 6 q^{61} - 44 q^{64} - 46 q^{71} - 12 q^{74} - 20 q^{76} + 10 q^{79} + 14 q^{86} - 30 q^{89} - 14 q^{91} - 68 q^{94}+O(q^{100}) $ 8 * q + 6 * q^4 - 16 * q^11 - 12 * q^14 - 2 * q^16 + 10 * q^19 - 6 * q^26 - 20 * q^29 + 16 * q^31 + 2 * q^34 - 26 * q^41 - 12 * q^44 + 6 * q^46 - 14 * q^49 - 10 * q^56 - 30 * q^59 + 6 * q^61 - 44 * q^64 - 46 * q^71 - 12 * q^74 - 20 * q^76 + 10 * q^79 + 14 * q^86 - 30 * q^89 - 14 * q^91 - 68 * q^94

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 $

$2$	−0.183172	−0.129522	−0.0647611	−	0.997901i	$-0.520629\pi$
$2$	−0.183172	−0.129522	−0.0647611	+	0.997901i	$0.520629\pi$
$3$	0	0
$3$	0	0
$4$	−1.96645	−0.983224
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.26086	1.23249	0.616244	−	0.787555i	$-0.288654\pi$
$7$	3.26086	1.23249	0.616244	+	0.787555i	$0.288654\pi$
$8$	0.726543	0.256872
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0.296379	0.0822006	0.0411003	−	0.999155i	$-0.486914\pi$
$13$	0.296379	0.0822006	0.0411003	+	0.999155i	$0.486914\pi$
$14$	−0.597298	−0.159635
$15$	0	0
$16$	3.79981	0.949953
$17$	−5.16297	−1.25220	−0.626102	−	0.779741i	$-0.715351\pi$
$17$	−5.16297	−1.25220	−0.626102	+	0.779741i	$0.715351\pi$
$18$	0	0
$19$	1.73038	0.396976	0.198488	−	0.980103i	$-0.436397\pi$
$19$	1.73038	0.396976	0.198488	+	0.980103i	$0.436397\pi$
$20$	0	0
$21$	0	0
$22$	0.366344	0.0781048
$23$	0.879192	0.183324	0.0916621	−	0.995790i	$-0.470782\pi$
$23$	0.879192	0.183324	0.0916621	+	0.995790i	$0.470782\pi$
$24$	0	0
$25$	0	0
$26$	−0.0542883	−0.0106468
$27$	0	0
$28$	−6.41230	−1.21181
$29$	−5.91216	−1.09786	−0.548930	−	0.835868i	$-0.684965\pi$
$29$	−5.91216	−1.09786	−0.548930	+	0.835868i	$0.684965\pi$
$30$	0	0
$31$	6.09953	1.09551	0.547754	−	0.836639i	$-0.315483\pi$
$31$	6.09953	1.09551	0.547754	+	0.836639i	$0.315483\pi$
$32$	−2.14910	−0.379912
$33$	0	0
$34$	0.945712	0.162188
$35$	0	0
$36$	0	0
$37$	−8.08800	−1.32966	−0.664830	−	0.746995i	$-0.731496\pi$
$37$	−8.08800	−1.32966	−0.664830	+	0.746995i	$0.731496\pi$
$38$	−0.316957	−0.0514173
$39$	0	0
$40$	0	0
$41$	1.01515	0.158539	0.0792695	−	0.996853i	$-0.474741\pi$
$41$	1.01515	0.158539	0.0792695	+	0.996853i	$0.474741\pi$
$42$	0	0
$43$	3.24199	0.494399	0.247200	−	0.968965i	$-0.420490\pi$
$43$	3.24199	0.494399	0.247200	+	0.968965i	$0.420490\pi$
$44$	3.93290	0.592906
$45$	0	0
$46$	−0.161043	−0.0237446
$47$	4.21996	0.615544	0.307772	−	0.951460i	$-0.400417\pi$
$47$	4.21996	0.615544	0.307772	+	0.951460i	$0.400417\pi$
$48$	0	0
$49$	3.63318	0.519026
$50$	0	0
$51$	0	0
$52$	−0.582813	−0.0808216
$53$	−8.10072	−1.11272	−0.556360	−	0.830941i	$-0.687802\pi$
$53$	−8.10072	−1.11272	−0.556360	+	0.830941i	$0.687802\pi$
$54$	0	0
$55$	0	0
$56$	2.36915	0.316591
$57$	0	0
$58$	1.08294	0.142197
$59$	−5.93635	−0.772847	−0.386424	−	0.922321i	$-0.626290\pi$
$59$	−5.93635	−0.772847	−0.386424	+	0.922321i	$0.626290\pi$
$60$	0	0
$61$	0.915615	0.117232	0.0586162	−	0.998281i	$-0.481331\pi$
$61$	0.915615	0.117232	0.0586162	+	0.998281i	$0.481331\pi$
$62$	−1.11726	−0.141893
$63$	0	0
$64$	−7.20597	−0.900746
$65$	0	0
$66$	0	0
$67$	−6.88806	−0.841510	−0.420755	−	0.907174i	$-0.638235\pi$
$67$	−6.88806	−0.841510	−0.420755	+	0.907174i	$0.638235\pi$
$68$	10.1527	1.23120
$69$	0	0
$70$	0	0
$71$	5.96878	0.708364	0.354182	−	0.935177i	$-0.384759\pi$
$71$	5.96878	0.708364	0.354182	+	0.935177i	$0.384759\pi$
$72$	0	0
$73$	−8.83341	−1.03387	−0.516936	−	0.856024i	$-0.672928\pi$
$73$	−8.83341	−1.03387	−0.516936	+	0.856024i	$0.672928\pi$
$74$	1.48150	0.172220
$75$	0	0
$76$	−3.40270	−0.390317
$77$	−6.52171	−0.743218
$78$	0	0
$79$	−7.76067	−0.873144	−0.436572	−	0.899669i	$-0.643808\pi$
$79$	−7.76067	−0.873144	−0.436572	+	0.899669i	$0.643808\pi$
$80$	0	0
$81$	0	0
$82$	−0.185946	−0.0205343
$83$	14.5154	1.59327	0.796635	−	0.604461i	$-0.206612\pi$
$83$	14.5154	1.59327	0.796635	+	0.604461i	$0.206612\pi$
$84$	0	0
$85$	0	0
$86$	−0.593842	−0.0640357
$87$	0	0
$88$	−1.45309	−0.154899
$89$	−7.52642	−0.797799	−0.398900	−	0.916995i	$-0.630608\pi$
$89$	−7.52642	−0.797799	−0.398900	+	0.916995i	$0.630608\pi$
$90$	0	0
$91$	0.966448	0.101311
$92$	−1.72889	−0.180249
$93$	0	0
$94$	−0.772978	−0.0797266
$95$	0	0
$96$	0	0
$97$	6.72649	0.682971	0.341486	−	0.939887i	$-0.389070\pi$
$97$	6.72649	0.682971	0.341486	+	0.939887i	$0.389070\pi$
$98$	−0.665497	−0.0672254
$99$	0	0
$100$	0	0
$101$	12.1955	1.21350	0.606748	−	0.794894i	$-0.292474\pi$
$101$	12.1955	1.21350	0.606748	+	0.794894i	$0.292474\pi$
$102$	0	0
$103$	−1.38140	−0.136114	−0.0680569	−	0.997681i	$-0.521680\pi$
$103$	−1.38140	−0.136114	−0.0680569	+	0.997681i	$0.521680\pi$
$104$	0.215332	0.0211150
$105$	0	0
$106$	1.48383	0.144122
$107$	15.8285	1.53020	0.765101	−	0.643911i	$-0.222689\pi$
$107$	15.8285	1.53020	0.765101	+	0.643911i	$0.222689\pi$
$108$	0	0
$109$	−2.00377	−0.191926	−0.0959632	−	0.995385i	$-0.530593\pi$
$109$	−2.00377	−0.191926	−0.0959632	+	0.995385i	$0.530593\pi$
$110$	0	0
$111$	0	0
$112$	12.3906	1.17081
$113$	−10.4275	−0.980935	−0.490468	−	0.871459i	$-0.663174\pi$
$113$	−10.4275	−0.980935	−0.490468	+	0.871459i	$0.663174\pi$
$114$	0	0
$115$	0	0
$116$	11.6260	1.07944
$117$	0	0
$118$	1.08737	0.100101
$119$	−16.8357	−1.54333
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−0.167715	−0.0151842
$123$	0	0
$124$	−11.9944	−1.07713
$125$	0	0
$126$	0	0
$127$	5.85007	0.519110	0.259555	−	0.965728i	$-0.416424\pi$
$127$	5.85007	0.519110	0.259555	+	0.965728i	$0.416424\pi$
$128$	5.61814	0.496578
$129$	0	0
$130$	0	0
$131$	1.49664	0.130762	0.0653811	−	0.997860i	$-0.479174\pi$
$131$	1.49664	0.130762	0.0653811	+	0.997860i	$0.479174\pi$
$132$	0	0
$133$	5.64252	0.489268
$134$	1.26170	0.108994
$135$	0	0
$136$	−3.75112	−0.321656
$137$	−7.84887	−0.670574	−0.335287	−	0.942116i	$-0.608833\pi$
$137$	−7.84887	−0.670574	−0.335287	+	0.942116i	$0.608833\pi$
$138$	0	0
$139$	5.39152	0.457303	0.228651	−	0.973508i	$-0.426568\pi$
$139$	5.39152	0.457303	0.228651	+	0.973508i	$0.426568\pi$
$140$	0	0
$141$	0	0
$142$	−1.09331	−0.0917488
$143$	−0.592757	−0.0495689
$144$	0	0
$145$	0	0
$146$	1.61803	0.133909
$147$	0	0
$148$	15.9046	1.30735
$149$	−18.8229	−1.54203	−0.771015	−	0.636817i	$-0.780251\pi$
$149$	−18.8229	−1.54203	−0.771015	+	0.636817i	$0.780251\pi$
$150$	0	0
$151$	−3.88797	−0.316398	−0.158199	−	0.987407i	$-0.550569\pi$
$151$	−3.88797	−0.316398	−0.158199	+	0.987407i	$0.550569\pi$
$152$	1.25719	0.101972
$153$	0	0
$154$	1.19460	0.0962632
$155$	0	0
$156$	0	0
$157$	−4.28378	−0.341883	−0.170941	−	0.985281i	$-0.554681\pi$
$157$	−4.28378	−0.341883	−0.170941	+	0.985281i	$0.554681\pi$
$158$	1.42154	0.113092
$159$	0	0
$160$	0	0
$161$	2.86692	0.225945
$162$	0	0
$163$	−15.7263	−1.23178	−0.615890	−	0.787832i	$-0.711204\pi$
$163$	−15.7263	−1.23178	−0.615890	+	0.787832i	$0.711204\pi$
$164$	−1.99623	−0.155879
$165$	0	0
$166$	−2.65881	−0.206364
$167$	21.0330	1.62758	0.813792	−	0.581156i	$-0.197399\pi$
$167$	21.0330	1.62758	0.813792	+	0.581156i	$0.197399\pi$
$168$	0	0
$169$	−12.9122	−0.993243
$170$	0	0
$171$	0	0
$172$	−6.37521	−0.486105
$173$	7.16663	0.544869	0.272434	−	0.962174i	$-0.412171\pi$
$173$	7.16663	0.544869	0.272434	+	0.962174i	$0.412171\pi$
$174$	0	0
$175$	0	0
$176$	−7.59963	−0.572843
$177$	0	0
$178$	1.37863	0.103333
$179$	−8.03934	−0.600888	−0.300444	−	0.953799i	$-0.597135\pi$
$179$	−8.03934	−0.600888	−0.300444	+	0.953799i	$0.597135\pi$
$180$	0	0
$181$	−20.6171	−1.53246	−0.766229	−	0.642568i	$-0.777869\pi$
$181$	−20.6171	−1.53246	−0.766229	+	0.642568i	$0.777869\pi$
$182$	−0.177026	−0.0131221
$183$	0	0
$184$	0.638770	0.0470908
$185$	0	0
$186$	0	0
$187$	10.3259	0.755107
$188$	−8.29833	−0.605218
$189$	0	0
$190$	0	0
$191$	−18.0303	−1.30463	−0.652313	−	0.757950i	$-0.726201\pi$
$191$	−18.0303	−1.30463	−0.652313	+	0.757950i	$0.726201\pi$
$192$	0	0
$193$	−6.78859	−0.488653	−0.244327	−	0.969693i	$-0.578567\pi$
$193$	−6.78859	−0.488653	−0.244327	+	0.969693i	$0.578567\pi$
$194$	−1.23210	−0.0884599
$195$	0	0
$196$	−7.14446	−0.510318
$197$	7.99537	0.569647	0.284823	−	0.958580i	$-0.408065\pi$
$197$	7.99537	0.569647	0.284823	+	0.958580i	$0.408065\pi$
$198$	0	0
$199$	−5.20485	−0.368962	−0.184481	−	0.982836i	$-0.559060\pi$
$199$	−5.20485	−0.368962	−0.184481	+	0.982836i	$0.559060\pi$
$200$	0	0
$201$	0	0
$202$	−2.23387	−0.157175
$203$	−19.2787	−1.35310
$204$	0	0
$205$	0	0
$206$	0.253035	0.0176298
$207$	0	0
$208$	1.12618	0.0780868
$209$	−3.46076	−0.239386
$210$	0	0
$211$	16.6020	1.14293	0.571463	−	0.820628i	$-0.306376\pi$
$211$	16.6020	1.14293	0.571463	+	0.820628i	$0.306376\pi$
$212$	15.9296	1.09405
$213$	0	0
$214$	−2.89934	−0.198195
$215$	0	0
$216$	0	0
$217$	19.8897	1.35020
$218$	0.367035	0.0248587
$219$	0	0
$220$	0	0
$221$	−1.53019	−0.102932
$222$	0	0
$223$	−6.62808	−0.443849	−0.221925	−	0.975064i	$-0.571234\pi$
$223$	−6.62808	−0.443849	−0.221925	+	0.975064i	$0.571234\pi$
$224$	−7.00792	−0.468236
$225$	0	0
$226$	1.91002	0.127053
$227$	13.3919	0.888853	0.444427	−	0.895815i	$-0.353407\pi$
$227$	13.3919	0.888853	0.444427	+	0.895815i	$0.353407\pi$
$228$	0	0
$229$	10.0867	0.666549	0.333274	−	0.942830i	$-0.391846\pi$
$229$	10.0867	0.666549	0.333274	+	0.942830i	$0.391846\pi$
$230$	0	0
$231$	0	0
$232$	−4.29544	−0.282009
$233$	22.0055	1.44163	0.720814	−	0.693129i	$-0.243768\pi$
$233$	22.0055	1.44163	0.720814	+	0.693129i	$0.243768\pi$
$234$	0	0
$235$	0	0
$236$	11.6735	0.759882
$237$	0	0
$238$	3.08383	0.199895
$239$	−7.56029	−0.489035	−0.244517	−	0.969645i	$-0.578629\pi$
$239$	−7.56029	−0.489035	−0.244517	+	0.969645i	$0.578629\pi$
$240$	0	0
$241$	−20.3945	−1.31373	−0.656864	−	0.754009i	$-0.728117\pi$
$241$	−20.3945	−1.31373	−0.656864	+	0.754009i	$0.728117\pi$
$242$	1.28220	0.0824232
$243$	0	0
$244$	−1.80051	−0.115266
$245$	0	0
$246$	0	0
$247$	0.512848	0.0326317
$248$	4.43157	0.281405
$249$	0	0
$250$	0	0
$251$	−10.5717	−0.667278	−0.333639	−	0.942701i	$-0.608277\pi$
$251$	−10.5717	−0.667278	−0.333639	+	0.942701i	$0.608277\pi$
$252$	0	0
$253$	−1.75838	−0.110549
$254$	−1.07157	−0.0672362
$255$	0	0
$256$	13.3829	0.836428
$257$	−20.2700	−1.26441	−0.632205	−	0.774801i	$-0.717850\pi$
$257$	−20.2700	−1.26441	−0.632205	+	0.774801i	$0.717850\pi$
$258$	0	0
$259$	−26.3738	−1.63879
$260$	0	0
$261$	0	0
$262$	−0.274143	−0.0169366
$263$	−28.0909	−1.73216	−0.866079	−	0.499906i	$-0.833368\pi$
$263$	−28.0909	−1.73216	−0.866079	+	0.499906i	$0.833368\pi$
$264$	0	0
$265$	0	0
$266$	−1.03355	−0.0633711
$267$	0	0
$268$	13.5450	0.827393
$269$	−20.3230	−1.23911	−0.619557	−	0.784952i	$-0.712688\pi$
$269$	−20.3230	−1.23911	−0.619557	+	0.784952i	$0.712688\pi$
$270$	0	0
$271$	31.4467	1.91025	0.955125	−	0.296202i	$-0.0957200\pi$
$271$	31.4467	1.91025	0.955125	+	0.296202i	$0.0957200\pi$
$272$	−19.6183	−1.18954
$273$	0	0
$274$	1.43769	0.0868543
$275$	0	0
$276$	0	0
$277$	−13.9305	−0.837001	−0.418501	−	0.908217i	$-0.637444\pi$
$277$	−13.9305	−0.837001	−0.418501	+	0.908217i	$0.637444\pi$
$278$	−0.987576	−0.0592309
$279$	0	0
$280$	0	0
$281$	−25.8777	−1.54373	−0.771866	−	0.635785i	$-0.780677\pi$
$281$	−25.8777	−1.54373	−0.771866	+	0.635785i	$0.780677\pi$
$282$	0	0
$283$	23.7316	1.41070	0.705350	−	0.708859i	$-0.250790\pi$
$283$	23.7316	1.41070	0.705350	+	0.708859i	$0.250790\pi$
$284$	−11.7373	−0.696480
$285$	0	0
$286$	0.108577	0.00642027
$287$	3.31024	0.195397
$288$	0	0
$289$	9.65625	0.568014
$290$	0	0
$291$	0	0
$292$	17.3704	1.01653
$293$	12.3029	0.718742	0.359371	−	0.933195i	$-0.382991\pi$
$293$	12.3029	0.718742	0.359371	+	0.933195i	$0.382991\pi$
$294$	0	0
$295$	0	0
$296$	−5.87628	−0.341552
$297$	0	0
$298$	3.44783	0.199727
$299$	0.260574	0.0150694
$300$	0	0
$301$	10.5717	0.609341
$302$	0.712167	0.0409806
$303$	0	0
$304$	6.57512	0.377109
$305$	0	0
$306$	0	0
$307$	4.28249	0.244415	0.122207	−	0.992505i	$-0.461003\pi$
$307$	4.28249	0.244415	0.122207	+	0.992505i	$0.461003\pi$
$308$	12.8246	0.730750
$309$	0	0
$310$	0	0
$311$	−25.6467	−1.45429	−0.727145	−	0.686484i	$-0.759153\pi$
$311$	−25.6467	−1.45429	−0.727145	+	0.686484i	$0.759153\pi$
$312$	0	0
$313$	−22.3513	−1.26337	−0.631684	−	0.775226i	$-0.717636\pi$
$313$	−22.3513	−1.26337	−0.631684	+	0.775226i	$0.717636\pi$
$314$	0.784668	0.0442814
$315$	0	0
$316$	15.2610	0.858496
$317$	−21.9228	−1.23131	−0.615655	−	0.788016i	$-0.711108\pi$
$317$	−21.9228	−1.23131	−0.615655	+	0.788016i	$0.711108\pi$
$318$	0	0
$319$	11.8243	0.662035
$320$	0	0
$321$	0	0
$322$	−0.525139	−0.0292649
$323$	−8.93390	−0.497095
$324$	0	0
$325$	0	0
$326$	2.88062	0.159543
$327$	0	0
$328$	0.737546	0.0407242
$329$	13.7607	0.758650
$330$	0	0
$331$	−8.96299	−0.492651	−0.246325	−	0.969187i	$-0.579223\pi$
$331$	−8.96299	−0.492651	−0.246325	+	0.969187i	$0.579223\pi$
$332$	−28.5437	−1.56654
$333$	0	0
$334$	−3.85266	−0.210808
$335$	0	0
$336$	0	0
$337$	−29.0836	−1.58428	−0.792141	−	0.610338i	$-0.791034\pi$
$337$	−29.0836	−1.58428	−0.792141	+	0.610338i	$0.791034\pi$
$338$	2.36515	0.128647
$339$	0	0
$340$	0	0
$341$	−12.1991	−0.660616
$342$	0	0
$343$	−10.9787	−0.592795
$344$	2.35544	0.126997
$345$	0	0
$346$	−1.31273	−0.0705726
$347$	−14.2972	−0.767513	−0.383757	−	0.923434i	$-0.625370\pi$
$347$	−14.2972	−0.767513	−0.383757	+	0.923434i	$0.625370\pi$
$348$	0	0
$349$	5.62382	0.301036	0.150518	−	0.988607i	$-0.451906\pi$
$349$	5.62382	0.301036	0.150518	+	0.988607i	$0.451906\pi$
$350$	0	0
$351$	0	0
$352$	4.29821	0.229095
$353$	−1.90997	−0.101658	−0.0508288	−	0.998707i	$-0.516186\pi$
$353$	−1.90997	−0.101658	−0.0508288	+	0.998707i	$0.516186\pi$
$354$	0	0
$355$	0	0
$356$	14.8003	0.784415
$357$	0	0
$358$	1.47258	0.0778284
$359$	−22.2047	−1.17192	−0.585958	−	0.810341i	$-0.699282\pi$
$359$	−22.2047	−1.17192	−0.585958	+	0.810341i	$0.699282\pi$
$360$	0	0
$361$	−16.0058	−0.842410
$362$	3.77648	0.198487
$363$	0	0
$364$	−1.90047	−0.0996117
$365$	0	0
$366$	0	0
$367$	−21.4450	−1.11942	−0.559709	−	0.828689i	$-0.689087\pi$
$367$	−21.4450	−1.11942	−0.559709	+	0.828689i	$0.689087\pi$
$368$	3.34077	0.174149
$369$	0	0
$370$	0	0
$371$	−26.4153	−1.37141
$372$	0	0
$373$	23.1399	1.19814	0.599070	−	0.800697i	$-0.295537\pi$
$373$	23.1399	1.19814	0.599070	+	0.800697i	$0.295537\pi$
$374$	−1.89142	−0.0978032
$375$	0	0
$376$	3.06598	0.158116
$377$	−1.75224	−0.0902448
$378$	0	0
$379$	−21.6501	−1.11209	−0.556047	−	0.831151i	$-0.687682\pi$
$379$	−21.6501	−1.11209	−0.556047	+	0.831151i	$0.687682\pi$
$380$	0	0
$381$	0	0
$382$	3.30265	0.168978
$383$	−5.65481	−0.288947	−0.144474	−	0.989509i	$-0.546149\pi$
$383$	−5.65481	−0.288947	−0.144474	+	0.989509i	$0.546149\pi$
$384$	0	0
$385$	0	0
$386$	1.24348	0.0632915
$387$	0	0
$388$	−13.2273	−0.671514
$389$	8.12605	0.412007	0.206004	−	0.978551i	$-0.433954\pi$
$389$	8.12605	0.412007	0.206004	+	0.978551i	$0.433954\pi$
$390$	0	0
$391$	−4.53924	−0.229559
$392$	2.63966	0.133323
$393$	0	0
$394$	−1.46453	−0.0737819
$395$	0	0
$396$	0	0
$397$	−20.7689	−1.04236	−0.521180	−	0.853447i	$-0.674508\pi$
$397$	−20.7689	−1.04236	−0.521180	+	0.853447i	$0.674508\pi$
$398$	0.953382	0.0477887
$399$	0	0
$400$	0	0
$401$	−30.1195	−1.50410	−0.752049	−	0.659107i	$-0.770934\pi$
$401$	−30.1195	−1.50410	−0.752049	+	0.659107i	$0.770934\pi$
$402$	0	0
$403$	1.80777	0.0900515
$404$	−23.9818	−1.19314
$405$	0	0
$406$	3.53132	0.175256
$407$	16.1760	0.801815
$408$	0	0
$409$	11.0452	0.546152	0.273076	−	0.961992i	$-0.411959\pi$
$409$	11.0452	0.546152	0.273076	+	0.961992i	$0.411959\pi$
$410$	0	0
$411$	0	0
$412$	2.71646	0.133830
$413$	−19.3576	−0.952524
$414$	0	0
$415$	0	0
$416$	−0.636949	−0.0312290
$417$	0	0
$418$	0.633915	0.0310058
$419$	32.9212	1.60830	0.804152	−	0.594424i	$-0.202620\pi$
$419$	32.9212	1.60830	0.804152	+	0.594424i	$0.202620\pi$
$420$	0	0
$421$	−20.2647	−0.987642	−0.493821	−	0.869564i	$-0.664400\pi$
$421$	−20.2647	−0.987642	−0.493821	+	0.869564i	$0.664400\pi$
$422$	−3.04101	−0.148034
$423$	0	0
$424$	−5.88552	−0.285826
$425$	0	0
$426$	0	0
$427$	2.98569	0.144488
$428$	−31.1260	−1.50453
$429$	0	0
$430$	0	0
$431$	−7.15526	−0.344657	−0.172328	−	0.985040i	$-0.555129\pi$
$431$	−7.15526	−0.344657	−0.172328	+	0.985040i	$0.555129\pi$
$432$	0	0
$433$	−5.48264	−0.263479	−0.131739	−	0.991284i	$-0.542056\pi$
$433$	−5.48264	−0.263479	−0.131739	+	0.991284i	$0.542056\pi$
$434$	−3.64324	−0.174881
$435$	0	0
$436$	3.94031	0.188707
$437$	1.52134	0.0727754
$438$	0	0
$439$	−30.7561	−1.46791	−0.733954	−	0.679199i	$-0.762327\pi$
$439$	−30.7561	−1.46791	−0.733954	+	0.679199i	$0.762327\pi$
$440$	0	0
$441$	0	0
$442$	0.280289	0.0133320
$443$	−11.3527	−0.539381	−0.269691	−	0.962947i	$-0.586921\pi$
$443$	−11.3527	−0.539381	−0.269691	+	0.962947i	$0.586921\pi$
$444$	0	0
$445$	0	0
$446$	1.21408	0.0574883
$447$	0	0
$448$	−23.4976	−1.11016
$449$	15.7661	0.744050	0.372025	−	0.928223i	$-0.378664\pi$
$449$	15.7661	0.744050	0.372025	+	0.928223i	$0.378664\pi$
$450$	0	0
$451$	−2.03029	−0.0956027
$452$	20.5051	0.964479
$453$	0	0
$454$	−2.45303	−0.115126
$455$	0	0
$456$	0	0
$457$	−4.16714	−0.194931	−0.0974653	−	0.995239i	$-0.531074\pi$
$457$	−4.16714	−0.194931	−0.0974653	+	0.995239i	$0.531074\pi$
$458$	−1.84760	−0.0863329
$459$	0	0
$460$	0	0
$461$	23.9714	1.11646	0.558230	−	0.829686i	$-0.311481\pi$
$461$	23.9714	1.11646	0.558230	+	0.829686i	$0.311481\pi$
$462$	0	0
$463$	41.3247	1.92052	0.960260	−	0.279107i	$-0.0900385\pi$
$463$	41.3247	1.92052	0.960260	+	0.279107i	$0.0900385\pi$
$464$	−22.4651	−1.04292
$465$	0	0
$466$	−4.03079	−0.186723
$467$	10.3966	0.481097	0.240548	−	0.970637i	$-0.422673\pi$
$467$	10.3966	0.481097	0.240548	+	0.970637i	$0.422673\pi$
$468$	0	0
$469$	−22.4610	−1.03715
$470$	0	0
$471$	0	0
$472$	−4.31301	−0.198522
$473$	−6.48398	−0.298134
$474$	0	0
$475$	0	0
$476$	33.1065	1.51743
$477$	0	0
$478$	1.38483	0.0633408
$479$	−36.0081	−1.64525	−0.822626	−	0.568582i	$-0.807492\pi$
$479$	−36.0081	−1.64525	−0.822626	+	0.568582i	$0.807492\pi$
$480$	0	0
$481$	−2.39711	−0.109299
$482$	3.73571	0.170157
$483$	0	0
$484$	13.7651	0.625688
$485$	0	0
$486$	0	0
$487$	10.6378	0.482045	0.241023	−	0.970520i	$-0.422517\pi$
$487$	10.6378	0.482045	0.241023	+	0.970520i	$0.422517\pi$
$488$	0.665233	0.0301137
$489$	0	0
$490$	0	0
$491$	17.6693	0.797402	0.398701	−	0.917081i	$-0.369461\pi$
$491$	17.6693	0.797402	0.398701	+	0.917081i	$0.369461\pi$
$492$	0	0
$493$	30.5243	1.37475
$494$	−0.0939394	−0.00422653
$495$	0	0
$496$	23.1771	1.04068
$497$	19.4633	0.873049
$498$	0	0
$499$	−9.41734	−0.421578	−0.210789	−	0.977532i	$-0.567603\pi$
$499$	−9.41734	−0.421578	−0.210789	+	0.977532i	$0.567603\pi$
$500$	0	0
$501$	0	0
$502$	1.93643	0.0864273
$503$	18.0133	0.803172	0.401586	−	0.915821i	$-0.368459\pi$
$503$	18.0133	0.803172	0.401586	+	0.915821i	$0.368459\pi$
$504$	0	0
$505$	0	0
$506$	0.322087	0.0143185
$507$	0	0
$508$	−11.5039	−0.510401
$509$	16.0485	0.711337	0.355669	−	0.934612i	$-0.384253\pi$
$509$	16.0485	0.711337	0.355669	+	0.934612i	$0.384253\pi$
$510$	0	0
$511$	−28.8045	−1.27423
$512$	−13.6877	−0.604914
$513$	0	0
$514$	3.71291	0.163769
$515$	0	0
$516$	0	0
$517$	−8.43991	−0.371187
$518$	4.83095	0.212260
$519$	0	0
$520$	0	0
$521$	2.20069	0.0964142	0.0482071	−	0.998837i	$-0.484649\pi$
$521$	2.20069	0.0964142	0.0482071	+	0.998837i	$0.484649\pi$
$522$	0	0
$523$	−7.43588	−0.325148	−0.162574	−	0.986696i	$-0.551980\pi$
$523$	−7.43588	−0.325148	−0.162574	+	0.986696i	$0.551980\pi$
$524$	−2.94307	−0.128569
$525$	0	0
$526$	5.14547	0.224353
$527$	−31.4917	−1.37180
$528$	0	0
$529$	−22.2270	−0.966392
$530$	0	0
$531$	0	0
$532$	−11.0957	−0.481061
$533$	0.300867	0.0130320
$534$	0	0
$535$	0	0
$536$	−5.00447	−0.216160
$537$	0	0
$538$	3.72260	0.160493
$539$	−7.26636	−0.312984
$540$	0	0
$541$	20.6474	0.887701	0.443850	−	0.896101i	$-0.353612\pi$
$541$	20.6474	0.887701	0.443850	+	0.896101i	$0.353612\pi$
$542$	−5.76016	−0.247420
$543$	0	0
$544$	11.0958	0.475727
$545$	0	0
$546$	0	0
$547$	31.3762	1.34155	0.670774	−	0.741662i	$-0.265962\pi$
$547$	31.3762	1.34155	0.670774	+	0.741662i	$0.265962\pi$
$548$	15.4344	0.659325
$549$	0	0
$550$	0	0
$551$	−10.2303	−0.435825
$552$	0	0
$553$	−25.3064	−1.07614
$554$	2.55167	0.108410
$555$	0	0
$556$	−10.6021	−0.449631
$557$	22.3515	0.947064	0.473532	−	0.880776i	$-0.342979\pi$
$557$	22.3515	0.947064	0.473532	+	0.880776i	$0.342979\pi$
$558$	0	0
$559$	0.960857	0.0406399
$560$	0	0
$561$	0	0
$562$	4.74007	0.199948
$563$	34.3663	1.44837	0.724184	−	0.689607i	$-0.242217\pi$
$563$	34.3663	1.44837	0.724184	+	0.689607i	$0.242217\pi$
$564$	0	0
$565$	0	0
$566$	−4.34697	−0.182717
$567$	0	0
$568$	4.33657	0.181958
$569$	−32.1662	−1.34848	−0.674239	−	0.738513i	$-0.735528\pi$
$569$	−32.1662	−1.34848	−0.674239	+	0.738513i	$0.735528\pi$
$570$	0	0
$571$	−27.1668	−1.13690	−0.568448	−	0.822719i	$-0.692456\pi$
$571$	−27.1668	−1.13690	−0.568448	+	0.822719i	$0.692456\pi$
$572$	1.16563	0.0487373
$573$	0	0
$574$	−0.606344	−0.0253083
$575$	0	0
$576$	0	0
$577$	13.8503	0.576596	0.288298	−	0.957541i	$-0.406911\pi$
$577$	13.8503	0.576596	0.288298	+	0.957541i	$0.406911\pi$
$578$	−1.76875	−0.0735705
$579$	0	0
$580$	0	0
$581$	47.3325	1.96368
$582$	0	0
$583$	16.2014	0.670995
$584$	−6.41785	−0.265572
$585$	0	0
$586$	−2.25354	−0.0930930
$587$	−44.1499	−1.82226	−0.911131	−	0.412118i	$-0.864789\pi$
$587$	−44.1499	−1.82226	−0.911131	+	0.412118i	$0.864789\pi$
$588$	0	0
$589$	10.5545	0.434891
$590$	0	0
$591$	0	0
$592$	−30.7329	−1.26311
$593$	16.2531	0.667437	0.333718	−	0.942673i	$-0.391697\pi$
$593$	16.2531	0.667437	0.333718	+	0.942673i	$0.391697\pi$
$594$	0	0
$595$	0	0
$596$	37.0142	1.51616
$597$	0	0
$598$	−0.0477298	−0.00195182
$599$	−30.4822	−1.24547	−0.622734	−	0.782433i	$-0.713978\pi$
$599$	−30.4822	−1.24547	−0.622734	+	0.782433i	$0.713978\pi$
$600$	0	0
$601$	−28.9162	−1.17952	−0.589758	−	0.807580i	$-0.700777\pi$
$601$	−28.9162	−1.17952	−0.589758	+	0.807580i	$0.700777\pi$
$602$	−1.93643	−0.0789232
$603$	0	0
$604$	7.64549	0.311090
$605$	0	0
$606$	0	0
$607$	8.23276	0.334157	0.167079	−	0.985944i	$-0.446567\pi$
$607$	8.23276	0.334157	0.167079	+	0.985944i	$0.446567\pi$
$608$	−3.71877	−0.150816
$609$	0	0
$610$	0	0
$611$	1.25071	0.0505981
$612$	0	0
$613$	4.79811	0.193794	0.0968969	−	0.995294i	$-0.469108\pi$
$613$	4.79811	0.193794	0.0968969	+	0.995294i	$0.469108\pi$
$614$	−0.784432	−0.0316571
$615$	0	0
$616$	−4.73830	−0.190912
$617$	2.03184	0.0817986	0.0408993	−	0.999163i	$-0.486978\pi$
$617$	2.03184	0.0817986	0.0408993	+	0.999163i	$0.486978\pi$
$618$	0	0
$619$	8.14100	0.327215	0.163607	−	0.986526i	$-0.447687\pi$
$619$	8.14100	0.327215	0.163607	+	0.986526i	$0.447687\pi$
$620$	0	0
$621$	0	0
$622$	4.69776	0.188363
$623$	−24.5426	−0.983278
$624$	0	0
$625$	0	0
$626$	4.09413	0.163634
$627$	0	0
$628$	8.42382	0.336147
$629$	41.7581	1.66500
$630$	0	0
$631$	33.2847	1.32504	0.662522	−	0.749042i	$-0.269486\pi$
$631$	33.2847	1.32504	0.662522	+	0.749042i	$0.269486\pi$
$632$	−5.63846	−0.224286
$633$	0	0
$634$	4.01565	0.159482
$635$	0	0
$636$	0	0
$637$	1.07680	0.0426642
$638$	−2.16589	−0.0857482
$639$	0	0
$640$	0	0
$641$	40.0481	1.58180	0.790902	−	0.611943i	$-0.209612\pi$
$641$	40.0481	1.58180	0.790902	+	0.611943i	$0.209612\pi$
$642$	0	0
$643$	−11.6870	−0.460890	−0.230445	−	0.973085i	$-0.574018\pi$
$643$	−11.6870	−0.460890	−0.230445	+	0.973085i	$0.574018\pi$
$644$	−5.63764	−0.222154
$645$	0	0
$646$	1.63644	0.0643849
$647$	7.94936	0.312522	0.156261	−	0.987716i	$-0.450056\pi$
$647$	7.94936	0.312522	0.156261	+	0.987716i	$0.450056\pi$
$648$	0	0
$649$	11.8727	0.466044
$650$	0	0
$651$	0	0
$652$	30.9250	1.21112
$653$	−3.33736	−0.130601	−0.0653005	−	0.997866i	$-0.520801\pi$
$653$	−3.33736	−0.130601	−0.0653005	+	0.997866i	$0.520801\pi$
$654$	0	0
$655$	0	0
$656$	3.85736	0.150605
$657$	0	0
$658$	−2.52057	−0.0982621
$659$	30.0508	1.17061	0.585306	−	0.810812i	$-0.300974\pi$
$659$	30.0508	1.17061	0.585306	+	0.810812i	$0.300974\pi$
$660$	0	0
$661$	−6.58417	−0.256094	−0.128047	−	0.991768i	$-0.540871\pi$
$661$	−6.58417	−0.256094	−0.128047	+	0.991768i	$0.540871\pi$
$662$	1.64177	0.0638092
$663$	0	0
$664$	10.5460	0.409266
$665$	0	0
$666$	0	0
$667$	−5.19792	−0.201264
$668$	−41.3603	−1.60028
$669$	0	0
$670$	0	0
$671$	−1.83123	−0.0706939
$672$	0	0
$673$	6.73140	0.259476	0.129738	−	0.991548i	$-0.458586\pi$
$673$	6.73140	0.259476	0.129738	+	0.991548i	$0.458586\pi$
$674$	5.32730	0.205200
$675$	0	0
$676$	25.3911	0.976580
$677$	−13.6478	−0.524529	−0.262265	−	0.964996i	$-0.584469\pi$
$677$	−13.6478	−0.524529	−0.262265	+	0.964996i	$0.584469\pi$
$678$	0	0
$679$	21.9341	0.841753
$680$	0	0
$681$	0	0
$682$	2.23453	0.0855645
$683$	−1.17220	−0.0448529	−0.0224265	−	0.999748i	$-0.507139\pi$
$683$	−1.17220	−0.0448529	−0.0224265	+	0.999748i	$0.507139\pi$
$684$	0	0
$685$	0	0
$686$	2.01099	0.0767801
$687$	0	0
$688$	12.3190	0.469656
$689$	−2.40088	−0.0914663
$690$	0	0
$691$	12.2674	0.466673	0.233336	−	0.972396i	$-0.425036\pi$
$691$	12.2674	0.466673	0.233336	+	0.972396i	$0.425036\pi$
$692$	−14.0928	−0.535728
$693$	0	0
$694$	2.61885	0.0994100
$695$	0	0
$696$	0	0
$697$	−5.24116	−0.198523
$698$	−1.03013	−0.0389909
$699$	0	0
$700$	0	0
$701$	20.0271	0.756415	0.378207	−	0.925721i	$-0.376541\pi$
$701$	20.0271	0.756415	0.378207	+	0.925721i	$0.376541\pi$
$702$	0	0
$703$	−13.9953	−0.527843
$704$	14.4119	0.543171
$705$	0	0
$706$	0.349854	0.0131669
$707$	39.7677	1.49562
$708$	0	0
$709$	−4.39510	−0.165061	−0.0825306	−	0.996589i	$-0.526300\pi$
$709$	−4.39510	−0.165061	−0.0825306	+	0.996589i	$0.526300\pi$
$710$	0	0
$711$	0	0
$712$	−5.46827	−0.204932
$713$	5.36266	0.200833
$714$	0	0
$715$	0	0
$716$	15.8089	0.590808
$717$	0	0
$718$	4.06727	0.151789
$719$	10.9283	0.407557	0.203779	−	0.979017i	$-0.434678\pi$
$719$	10.9283	0.407557	0.203779	+	0.979017i	$0.434678\pi$
$720$	0	0
$721$	−4.50456	−0.167759
$722$	2.93181	0.109111
$723$	0	0
$724$	40.5425	1.50675
$725$	0	0
$726$	0	0
$727$	−32.9032	−1.22031	−0.610156	−	0.792281i	$-0.708893\pi$
$727$	−32.9032	−1.22031	−0.610156	+	0.792281i	$0.708893\pi$
$728$	0.702166	0.0260240
$729$	0	0
$730$	0	0
$731$	−16.7383	−0.619088
$732$	0	0
$733$	−13.7498	−0.507859	−0.253929	−	0.967223i	$-0.581723\pi$
$733$	−13.7498	−0.507859	−0.253929	+	0.967223i	$0.581723\pi$
$734$	3.92812	0.144990
$735$	0	0
$736$	−1.88948	−0.0696470
$737$	13.7761	0.507450
$738$	0	0
$739$	−43.1893	−1.58874	−0.794372	−	0.607432i	$-0.792200\pi$
$739$	−43.1893	−1.58874	−0.794372	+	0.607432i	$0.792200\pi$
$740$	0	0
$741$	0	0
$742$	4.83854	0.177628
$743$	31.8479	1.16838	0.584192	−	0.811615i	$-0.301411\pi$
$743$	31.8479	1.16838	0.584192	+	0.811615i	$0.301411\pi$
$744$	0	0
$745$	0	0
$746$	−4.23859	−0.155186
$747$	0	0
$748$	−20.3054	−0.742440
$749$	51.6145	1.88595
$750$	0	0
$751$	−29.5952	−1.07995	−0.539973	−	0.841682i	$-0.681565\pi$
$751$	−29.5952	−1.07995	−0.539973	+	0.841682i	$0.681565\pi$
$752$	16.0351	0.584738
$753$	0	0
$754$	0.320961	0.0116887
$755$	0	0
$756$	0	0
$757$	0.0984401	0.00357786	0.00178893	−	0.999998i	$-0.499431\pi$
$757$	0.0984401	0.00357786	0.00178893	+	0.999998i	$0.499431\pi$
$758$	3.96570	0.144041
$759$	0	0
$760$	0	0
$761$	3.54402	0.128471	0.0642353	−	0.997935i	$-0.479539\pi$
$761$	3.54402	0.128471	0.0642353	+	0.997935i	$0.479539\pi$
$762$	0	0
$763$	−6.53400	−0.236547
$764$	35.4556	1.28274
$765$	0	0
$766$	1.03580	0.0374251
$767$	−1.75941	−0.0635285
$768$	0	0
$769$	1.42759	0.0514802	0.0257401	−	0.999669i	$-0.491806\pi$
$769$	1.42759	0.0514802	0.0257401	+	0.999669i	$0.491806\pi$
$770$	0	0
$771$	0	0
$772$	13.3494	0.480456
$773$	33.7807	1.21501	0.607504	−	0.794317i	$-0.292171\pi$
$773$	33.7807	1.21501	0.607504	+	0.794317i	$0.292171\pi$
$774$	0	0
$775$	0	0
$776$	4.88708	0.175436
$777$	0	0
$778$	−1.48847	−0.0533641
$779$	1.75659	0.0629363
$780$	0	0
$781$	−11.9376	−0.427159
$782$	0.831462	0.0297330
$783$	0	0
$784$	13.8054	0.493050
$785$	0	0
$786$	0	0
$787$	−2.18030	−0.0777192	−0.0388596	−	0.999245i	$-0.512373\pi$
$787$	−2.18030	−0.0777192	−0.0388596	+	0.999245i	$0.512373\pi$
$788$	−15.7225	−0.560090
$789$	0	0
$790$	0	0
$791$	−34.0025	−1.20899
$792$	0	0
$793$	0.271369	0.00963659
$794$	3.80428	0.135009
$795$	0	0
$796$	10.2351	0.362772
$797$	23.4919	0.832124	0.416062	−	0.909336i	$-0.363410\pi$
$797$	23.4919	0.832124	0.416062	+	0.909336i	$0.363410\pi$
$798$	0	0
$799$	−21.7875	−0.770787
$800$	0	0
$801$	0	0
$802$	5.51706	0.194814
$803$	17.6668	0.623449
$804$	0	0
$805$	0	0
$806$	−0.331133	−0.0116637
$807$	0	0
$808$	8.86054	0.311713
$809$	38.1509	1.34132	0.670658	−	0.741767i	$-0.266012\pi$
$809$	38.1509	1.34132	0.670658	+	0.741767i	$0.266012\pi$
$810$	0	0
$811$	47.9069	1.68224	0.841120	−	0.540849i	$-0.181897\pi$
$811$	47.9069	1.68224	0.841120	+	0.540849i	$0.181897\pi$
$812$	37.9106	1.33040
$813$	0	0
$814$	−2.96299	−0.103853
$815$	0	0
$816$	0	0
$817$	5.60988	0.196265
$818$	−2.02318	−0.0707388
$819$	0	0
$820$	0	0
$821$	−18.9808	−0.662434	−0.331217	−	0.943555i	$-0.607459\pi$
$821$	−18.9808	−0.662434	−0.331217	+	0.943555i	$0.607459\pi$
$822$	0	0
$823$	22.1681	0.772732	0.386366	−	0.922345i	$-0.373730\pi$
$823$	22.1681	0.772732	0.386366	+	0.922345i	$0.373730\pi$
$824$	−1.00365	−0.0349638
$825$	0	0
$826$	3.54577	0.123373
$827$	−4.72938	−0.164457	−0.0822283	−	0.996614i	$-0.526204\pi$
$827$	−4.72938	−0.164457	−0.0822283	+	0.996614i	$0.526204\pi$
$828$	0	0
$829$	16.4400	0.570986	0.285493	−	0.958381i	$-0.407843\pi$
$829$	16.4400	0.570986	0.285493	+	0.958381i	$0.407843\pi$
$830$	0	0
$831$	0	0
$832$	−2.13570	−0.0740419
$833$	−18.7580	−0.649926
$834$	0	0
$835$	0	0
$836$	6.80540	0.235370
$837$	0	0
$838$	−6.03024	−0.208311
$839$	5.60083	0.193362	0.0966811	−	0.995315i	$-0.469177\pi$
$839$	5.60083	0.193362	0.0966811	+	0.995315i	$0.469177\pi$
$840$	0	0
$841$	5.95363	0.205298
$842$	3.71193	0.127922
$843$	0	0
$844$	−32.6469	−1.12375
$845$	0	0
$846$	0	0
$847$	−22.8260	−0.784310
$848$	−30.7812	−1.05703
$849$	0	0
$850$	0	0
$851$	−7.11091	−0.243759
$852$	0	0
$853$	17.8905	0.612559	0.306279	−	0.951942i	$-0.400916\pi$
$853$	17.8905	0.612559	0.306279	+	0.951942i	$0.400916\pi$
$854$	−0.546895	−0.0187144
$855$	0	0
$856$	11.5001	0.393065
$857$	3.19536	0.109151	0.0545757	−	0.998510i	$-0.482619\pi$
$857$	3.19536	0.109151	0.0545757	+	0.998510i	$0.482619\pi$
$858$	0	0
$859$	43.4196	1.48146	0.740728	−	0.671805i	$-0.234481\pi$
$859$	43.4196	1.48146	0.740728	+	0.671805i	$0.234481\pi$
$860$	0	0
$861$	0	0
$862$	1.31064	0.0446407
$863$	43.2724	1.47301	0.736504	−	0.676433i	$-0.236475\pi$
$863$	43.2724	1.47301	0.736504	+	0.676433i	$0.236475\pi$
$864$	0	0
$865$	0	0
$866$	1.00427	0.0341264
$867$	0	0
$868$	−39.1120	−1.32755
$869$	15.5213	0.526525
$870$	0	0
$871$	−2.04147	−0.0691727
$872$	−1.45582	−0.0493004
$873$	0	0
$874$	−0.278666	−0.00942603
$875$	0	0
$876$	0	0
$877$	−34.2339	−1.15600	−0.577999	−	0.816038i	$-0.696166\pi$
$877$	−34.2339	−1.15600	−0.577999	+	0.816038i	$0.696166\pi$
$878$	5.63366	0.190127
$879$	0	0
$880$	0	0
$881$	−27.6270	−0.930779	−0.465389	−	0.885106i	$-0.654086\pi$
$881$	−27.6270	−0.930779	−0.465389	+	0.885106i	$0.654086\pi$
$882$	0	0
$883$	−25.8990	−0.871571	−0.435786	−	0.900051i	$-0.643529\pi$
$883$	−25.8990	−0.871571	−0.435786	+	0.900051i	$0.643529\pi$
$884$	3.00905	0.101205
$885$	0	0
$886$	2.07949	0.0698618
$887$	17.2449	0.579027	0.289514	−	0.957174i	$-0.406506\pi$
$887$	17.2449	0.579027	0.289514	+	0.957174i	$0.406506\pi$
$888$	0	0
$889$	19.0762	0.639796
$890$	0	0
$891$	0	0
$892$	13.0338	0.436403
$893$	7.30213	0.244356
$894$	0	0
$895$	0	0
$896$	18.3200	0.612027
$897$	0	0
$898$	−2.88792	−0.0963710
$899$	−36.0614	−1.20271
$900$	0	0
$901$	41.8238	1.39335
$902$	0.371893	0.0123827
$903$	0	0
$904$	−7.57601	−0.251974
$905$	0	0
$906$	0	0
$907$	57.0465	1.89420	0.947099	−	0.320940i	$-0.103999\pi$
$907$	57.0465	1.89420	0.947099	+	0.320940i	$0.103999\pi$
$908$	−26.3345	−0.873942
$909$	0	0
$910$	0	0
$911$	−19.8683	−0.658267	−0.329133	−	0.944283i	$-0.606757\pi$
$911$	−19.8683	−0.658267	−0.329133	+	0.944283i	$0.606757\pi$
$912$	0	0
$913$	−29.0307	−0.960777
$914$	0.763304	0.0252478
$915$	0	0
$916$	−19.8350	−0.655367
$917$	4.88033	0.161163
$918$	0	0
$919$	27.9785	0.922924	0.461462	−	0.887160i	$-0.347325\pi$
$919$	27.9785	0.922924	0.461462	+	0.887160i	$0.347325\pi$
$920$	0	0
$921$	0	0
$922$	−4.39090	−0.144606
$923$	1.76902	0.0582279
$924$	0	0
$925$	0	0
$926$	−7.56952	−0.248750
$927$	0	0
$928$	12.7059	0.417090
$929$	14.1249	0.463424	0.231712	−	0.972784i	$-0.425567\pi$
$929$	14.1249	0.463424	0.231712	+	0.972784i	$0.425567\pi$
$930$	0	0
$931$	6.28678	0.206041
$932$	−43.2727	−1.41744
$933$	0	0
$934$	−1.90437	−0.0623127
$935$	0	0
$936$	0	0
$937$	−35.9546	−1.17459	−0.587293	−	0.809374i	$-0.699806\pi$
$937$	−35.9546	−1.17459	−0.587293	+	0.809374i	$0.699806\pi$
$938$	4.11422	0.134334
$939$	0	0
$940$	0	0
$941$	−26.6979	−0.870327	−0.435164	−	0.900351i	$-0.643309\pi$
$941$	−26.6979	−0.870327	−0.435164	+	0.900351i	$0.643309\pi$
$942$	0	0
$943$	0.892508	0.0290640
$944$	−22.5570	−0.734169
$945$	0	0
$946$	1.18768	0.0386150
$947$	17.3102	0.562506	0.281253	−	0.959634i	$-0.409250\pi$
$947$	17.3102	0.562506	0.281253	+	0.959634i	$0.409250\pi$
$948$	0	0
$949$	−2.61803	−0.0849850
$950$	0	0
$951$	0	0
$952$	−12.2318	−0.396436
$953$	24.2622	0.785931	0.392965	−	0.919553i	$-0.371449\pi$
$953$	24.2622	0.785931	0.392965	+	0.919553i	$0.371449\pi$
$954$	0	0
$955$	0	0
$956$	14.8669	0.480830
$957$	0	0
$958$	6.59568	0.213097
$959$	−25.5940	−0.826475
$960$	0	0
$961$	6.20427	0.200138
$962$	0.439084	0.0141566
$963$	0	0
$964$	40.1048	1.29169
$965$	0	0
$966$	0	0
$967$	31.1143	1.00057	0.500284	−	0.865861i	$-0.333229\pi$
$967$	31.1143	1.00057	0.500284	+	0.865861i	$0.333229\pi$
$968$	−5.08580	−0.163464
$969$	0	0
$970$	0	0
$971$	17.3721	0.557497	0.278749	−	0.960364i	$-0.410080\pi$
$971$	17.3721	0.557497	0.278749	+	0.960364i	$0.410080\pi$
$972$	0	0
$973$	17.5810	0.563620
$974$	−1.94855	−0.0624356
$975$	0	0
$976$	3.47917	0.111365
$977$	−42.8929	−1.37227	−0.686133	−	0.727476i	$-0.740693\pi$
$977$	−42.8929	−1.37227	−0.686133	+	0.727476i	$0.740693\pi$
$978$	0	0
$979$	15.0528	0.481091
$980$	0	0
$981$	0	0
$982$	−3.23651	−0.103281
$983$	−37.4533	−1.19457	−0.597287	−	0.802028i	$-0.703755\pi$
$983$	−37.4533	−1.19457	−0.597287	+	0.802028i	$0.703755\pi$
$984$	0	0
$985$	0	0
$986$	−5.59120	−0.178060
$987$	0	0
$988$	−1.00849	−0.0320843
$989$	2.85033	0.0906353
$990$	0	0
$991$	16.7624	0.532476	0.266238	−	0.963907i	$-0.414219\pi$
$991$	16.7624	0.532476	0.266238	+	0.963907i	$0.414219\pi$
$992$	−13.1085	−0.416196
$993$	0	0
$994$	−3.56514	−0.113079
$995$	0	0
$996$	0	0
$997$	59.8595	1.89577	0.947884	−	0.318615i	$-0.103218\pi$
$997$	59.8595	1.89577	0.947884	+	0.318615i	$0.103218\pi$
$998$	1.72499	0.0546037
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.x.1.4		8
3.2	odd	2		625.2.a.f.1.5		8
5.4	even	2	inner	5625.2.a.x.1.5		8
12.11	even	2		10000.2.a.bj.1.3		8
15.2	even	4		625.2.b.c.624.5		8
15.8	even	4		625.2.b.c.624.4		8
15.14	odd	2		625.2.a.f.1.4		8
25.2	odd	20		225.2.m.a.154.1		8
25.13	odd	20		225.2.m.a.19.1		8
60.59	even	2		10000.2.a.bj.1.6		8
75.2	even	20		25.2.e.a.4.2	✓	8
75.8	even	20		625.2.e.a.374.2		8
75.11	odd	10		125.2.d.b.101.3		16
75.14	odd	10		125.2.d.b.101.2		16
75.17	even	20		625.2.e.i.374.1		8
75.23	even	20		125.2.e.b.24.1		8
75.29	odd	10		625.2.d.o.376.3		16
75.38	even	20		25.2.e.a.19.2	yes	8
75.41	odd	10		125.2.d.b.26.3		16
75.44	odd	10		625.2.d.o.251.3		16
75.47	even	20		625.2.e.a.249.2		8
75.53	even	20		625.2.e.i.249.1		8
75.56	odd	10		625.2.d.o.251.2		16
75.59	odd	10		125.2.d.b.26.2		16
75.62	even	20		125.2.e.b.99.1		8
75.71	odd	10		625.2.d.o.376.2		16
300.227	odd	20		400.2.y.c.129.2		8
300.263	odd	20		400.2.y.c.369.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.4.2	✓	8	75.2	even	20
25.2.e.a.19.2	yes	8	75.38	even	20
125.2.d.b.26.2		16	75.59	odd	10
125.2.d.b.26.3		16	75.41	odd	10
125.2.d.b.101.2		16	75.14	odd	10
125.2.d.b.101.3		16	75.11	odd	10
125.2.e.b.24.1		8	75.23	even	20
125.2.e.b.99.1		8	75.62	even	20
225.2.m.a.19.1		8	25.13	odd	20
225.2.m.a.154.1		8	25.2	odd	20
400.2.y.c.129.2		8	300.227	odd	20
400.2.y.c.369.2		8	300.263	odd	20
625.2.a.f.1.4		8	15.14	odd	2
625.2.a.f.1.5		8	3.2	odd	2
625.2.b.c.624.4		8	15.8	even	4
625.2.b.c.624.5		8	15.2	even	4
625.2.d.o.251.2		16	75.56	odd	10
625.2.d.o.251.3		16	75.44	odd	10
625.2.d.o.376.2		16	75.71	odd	10
625.2.d.o.376.3		16	75.29	odd	10
625.2.e.a.249.2		8	75.47	even	20
625.2.e.a.374.2		8	75.8	even	20
625.2.e.i.249.1		8	75.53	even	20
625.2.e.i.374.1		8	75.17	even	20
5625.2.a.x.1.4		8	1.1	even	1	trivial
5625.2.a.x.1.5		8	5.4	even	2	inner
10000.2.a.bj.1.3		8	12.11	even	2
10000.2.a.bj.1.6		8	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)

\(f(q)\)	\(=\)	\(q-0.183172 q^{2} -1.96645 q^{4} +3.26086 q^{7} +0.726543 q^{8} +O(q^{10})\)	\(q-0.183172 q^{2} -1.96645 q^{4} +3.26086 q^{7} +0.726543 q^{8} -2.00000 q^{11} +0.296379 q^{13} -0.597298 q^{14} +3.79981 q^{16} -5.16297 q^{17} +1.73038 q^{19} +0.366344 q^{22} +0.879192 q^{23} -0.0542883 q^{26} -6.41230 q^{28} -5.91216 q^{29} +6.09953 q^{31} -2.14910 q^{32} +0.945712 q^{34} -8.08800 q^{37} -0.316957 q^{38} +1.01515 q^{41} +3.24199 q^{43} +3.93290 q^{44} -0.161043 q^{46} +4.21996 q^{47} +3.63318 q^{49} -0.582813 q^{52} -8.10072 q^{53} +2.36915 q^{56} +1.08294 q^{58} -5.93635 q^{59} +0.915615 q^{61} -1.11726 q^{62} -7.20597 q^{64} -6.88806 q^{67} +10.1527 q^{68} +5.96878 q^{71} -8.83341 q^{73} +1.48150 q^{74} -3.40270 q^{76} -6.52171 q^{77} -7.76067 q^{79} -0.185946 q^{82} +14.5154 q^{83} -0.593842 q^{86} -1.45309 q^{88} -7.52642 q^{89} +0.966448 q^{91} -1.72889 q^{92} -0.772978 q^{94} +6.72649 q^{97} -0.665497 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4}+O(q^{10}) \) 8 * q + 6 * q^4	\( 8 q + 6 q^{4} - 16 q^{11} - 12 q^{14} - 2 q^{16} + 10 q^{19} - 6 q^{26} - 20 q^{29} + 16 q^{31} + 2 q^{34} - 26 q^{41} - 12 q^{44} + 6 q^{46} - 14 q^{49} - 10 q^{56} - 30 q^{59} + 6 q^{61} - 44 q^{64} - 46 q^{71} - 12 q^{74} - 20 q^{76} + 10 q^{79} + 14 q^{86} - 30 q^{89} - 14 q^{91} - 68 q^{94}+O(q^{100}) \) 8 * q + 6 * q^4 - 16 * q^11 - 12 * q^14 - 2 * q^16 + 10 * q^19 - 6 * q^26 - 20 * q^29 + 16 * q^31 + 2 * q^34 - 26 * q^41 - 12 * q^44 + 6 * q^46 - 14 * q^49 - 10 * q^56 - 30 * q^59 + 6 * q^61 - 44 * q^64 - 46 * q^71 - 12 * q^74 - 20 * q^76 + 10 * q^79 + 14 * q^86 - 30 * q^89 - 14 * q^91 - 68 * q^94

Introduction

L-functions

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