LMFDB - Embedded newform 5625.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.46980000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 80x^{4} - 180x^{2} + 145 $ x^8 - 15x^6 + 80x^4 - 180*x^2 + 145
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44055$ 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)

$f(q)$	$=$	$q-2.44055 q^{2} +3.95630 q^{4} +0.591023 q^{7} -4.77444 q^{8} +O(q^{10})$	$q-2.44055 q^{2} +3.95630 q^{4} +0.591023 q^{7} -4.77444 q^{8} -4.26423 q^{11} +5.06316 q^{13} -1.44242 q^{14} +3.73968 q^{16} -5.39132 q^{17} -5.73968 q^{19} +10.4071 q^{22} +2.12522 q^{23} -12.3569 q^{26} +2.33826 q^{28} -9.48296 q^{29} -1.38761 q^{31} +0.421994 q^{32} +13.1578 q^{34} -11.3502 q^{37} +14.0080 q^{38} +0.403551 q^{41} +5.30211 q^{43} -16.8705 q^{44} -5.18672 q^{46} +8.60289 q^{47} -6.65069 q^{49} +20.0314 q^{52} +0.337629 q^{53} -2.82180 q^{56} +23.1437 q^{58} +2.73744 q^{59} +9.01478 q^{61} +3.38654 q^{62} -8.50926 q^{64} -5.86133 q^{67} -21.3296 q^{68} -10.7281 q^{71} +5.02136 q^{73} +27.7007 q^{74} -22.7079 q^{76} -2.52026 q^{77} +8.44321 q^{79} -0.984886 q^{82} -11.0136 q^{83} -12.9401 q^{86} +20.3593 q^{88} +13.1464 q^{89} +2.99244 q^{91} +8.40801 q^{92} -20.9958 q^{94} +7.24927 q^{97} +16.2314 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) $ 8 * q + 14 * q^4 + 10 * q^7	$ 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} - 18 q^{16} + 2 q^{19} + 20 q^{22} + 10 q^{28} + 4 q^{31} + 50 q^{34} + 50 q^{43} - 30 q^{46} - 6 q^{49} + 30 q^{52} + 60 q^{58} + 46 q^{61} - 14 q^{64} + 40 q^{67} + 50 q^{73} - 34 q^{76} - 12 q^{79} + 60 q^{82} + 70 q^{88} - 10 q^{91} - 20 q^{94} + 50 q^{97}+O(q^{100}) $ 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 - 18 * q^16 + 2 * q^19 + 20 * q^22 + 10 * q^28 + 4 * q^31 + 50 * q^34 + 50 * q^43 - 30 * q^46 - 6 * q^49 + 30 * q^52 + 60 * q^58 + 46 * q^61 - 14 * q^64 + 40 * q^67 + 50 * q^73 - 34 * q^76 - 12 * q^79 + 60 * q^82 + 70 * q^88 - 10 * q^91 - 20 * q^94 + 50 * 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 $

$2$	−2.44055	−1.72573	−0.862866	−	0.505434i	$-0.831333\pi$
$2$	−2.44055	−1.72573	−0.862866	+	0.505434i	$0.831333\pi$
$3$	0	0
$3$	0	0
$4$	3.95630	1.97815
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.591023	0.223386	0.111693	−	0.993743i	$-0.464373\pi$
$7$	0.591023	0.223386	0.111693	+	0.993743i	$0.464373\pi$
$8$	−4.77444	−1.68802
$9$	0	0
$10$	0	0
$11$	−4.26423	−1.28571	−0.642856	−	0.765987i	$-0.722251\pi$
$11$	−4.26423	−1.28571	−0.642856	+	0.765987i	$0.722251\pi$
$12$	0	0
$13$	5.06316	1.40427	0.702134	−	0.712045i	$-0.252231\pi$
$13$	5.06316	1.40427	0.702134	+	0.712045i	$0.252231\pi$
$14$	−1.44242	−0.385504
$15$	0	0
$16$	3.73968	0.934920
$17$	−5.39132	−1.30759	−0.653793	−	0.756673i	$-0.726823\pi$
$17$	−5.39132	−1.30759	−0.653793	+	0.756673i	$0.726823\pi$
$18$	0	0
$19$	−5.73968	−1.31677	−0.658387	−	0.752680i	$-0.728761\pi$
$19$	−5.73968	−1.31677	−0.658387	+	0.752680i	$0.728761\pi$
$20$	0	0
$21$	0	0
$22$	10.4071	2.21879
$23$	2.12522	0.443140	0.221570	−	0.975145i	$-0.428882\pi$
$23$	2.12522	0.443140	0.221570	+	0.975145i	$0.428882\pi$
$24$	0	0
$25$	0	0
$26$	−12.3569	−2.42339
$27$	0	0
$28$	2.33826	0.441890
$29$	−9.48296	−1.76094	−0.880471	−	0.474101i	$-0.842773\pi$
$29$	−9.48296	−1.76094	−0.880471	+	0.474101i	$0.842773\pi$
$30$	0	0
$31$	−1.38761	−0.249223	−0.124611	−	0.992206i	$-0.539768\pi$
$31$	−1.38761	−0.249223	−0.124611	+	0.992206i	$0.539768\pi$
$32$	0.421994	0.0745986
$33$	0	0
$34$	13.1578	2.25654
$35$	0	0
$36$	0	0
$37$	−11.3502	−1.86595	−0.932977	−	0.359935i	$-0.882799\pi$
$37$	−11.3502	−1.86595	−0.932977	+	0.359935i	$0.882799\pi$
$38$	14.0080	2.27240
$39$	0	0
$40$	0	0
$41$	0.403551	0.0630240	0.0315120	−	0.999503i	$-0.489968\pi$
$41$	0.403551	0.0630240	0.0315120	+	0.999503i	$0.489968\pi$
$42$	0	0
$43$	5.30211	0.808565	0.404282	−	0.914634i	$-0.367521\pi$
$43$	5.30211	0.808565	0.404282	+	0.914634i	$0.367521\pi$
$44$	−16.8705	−2.54333
$45$	0	0
$46$	−5.18672	−0.764740
$47$	8.60289	1.25486	0.627430	−	0.778673i	$-0.284107\pi$
$47$	8.60289	1.25486	0.627430	+	0.778673i	$0.284107\pi$
$48$	0	0
$49$	−6.65069	−0.950099
$50$	0	0
$51$	0	0
$52$	20.0314	2.77785
$53$	0.337629	0.0463769	0.0231884	−	0.999731i	$-0.492618\pi$
$53$	0.337629	0.0463769	0.0231884	+	0.999731i	$0.492618\pi$
$54$	0	0
$55$	0	0
$56$	−2.82180	−0.377079
$57$	0	0
$58$	23.1437	3.03891
$59$	2.73744	0.356384	0.178192	−	0.983996i	$-0.442975\pi$
$59$	2.73744	0.356384	0.178192	+	0.983996i	$0.442975\pi$
$60$	0	0
$61$	9.01478	1.15422	0.577112	−	0.816665i	$-0.304179\pi$
$61$	9.01478	1.15422	0.577112	+	0.816665i	$0.304179\pi$
$62$	3.38654	0.430091
$63$	0	0
$64$	−8.50926	−1.06366
$65$	0	0
$66$	0	0
$67$	−5.86133	−0.716075	−0.358038	−	0.933707i	$-0.616554\pi$
$67$	−5.86133	−0.716075	−0.358038	+	0.933707i	$0.616554\pi$
$68$	−21.3296	−2.58660
$69$	0	0
$70$	0	0
$71$	−10.7281	−1.27319	−0.636596	−	0.771197i	$-0.719658\pi$
$71$	−10.7281	−1.27319	−0.636596	+	0.771197i	$0.719658\pi$
$72$	0	0
$73$	5.02136	0.587706	0.293853	−	0.955851i	$-0.405062\pi$
$73$	5.02136	0.587706	0.293853	+	0.955851i	$0.405062\pi$
$74$	27.7007	3.22014
$75$	0	0
$76$	−22.7079	−2.60477
$77$	−2.52026	−0.287210
$78$	0	0
$79$	8.44321	0.949936	0.474968	−	0.880003i	$-0.342460\pi$
$79$	8.44321	0.949936	0.474968	+	0.880003i	$0.342460\pi$
$80$	0	0
$81$	0	0
$82$	−0.984886	−0.108762
$83$	−11.0136	−1.20890	−0.604450	−	0.796643i	$-0.706607\pi$
$83$	−11.0136	−1.20890	−0.604450	+	0.796643i	$0.706607\pi$
$84$	0	0
$85$	0	0
$86$	−12.9401	−1.39537
$87$	0	0
$88$	20.3593	2.17031
$89$	13.1464	1.39351	0.696756	−	0.717308i	$-0.254626\pi$
$89$	13.1464	1.39351	0.696756	+	0.717308i	$0.254626\pi$
$90$	0	0
$91$	2.99244	0.313693
$92$	8.40801	0.876595
$93$	0	0
$94$	−20.9958	−2.16555
$95$	0	0
$96$	0	0
$97$	7.24927	0.736052	0.368026	−	0.929815i	$-0.380034\pi$
$97$	7.24927	0.736052	0.368026	+	0.929815i	$0.380034\pi$
$98$	16.2314	1.63962
$99$	0	0
$100$	0	0
$101$	10.2203	1.01696	0.508478	−	0.861075i	$-0.330208\pi$
$101$	10.2203	1.01696	0.508478	+	0.861075i	$0.330208\pi$
$102$	0	0
$103$	11.6781	1.15068	0.575339	−	0.817915i	$-0.304870\pi$
$103$	11.6781	1.15068	0.575339	+	0.817915i	$0.304870\pi$
$104$	−24.1738	−2.37043
$105$	0	0
$106$	−0.824000	−0.0800340
$107$	1.07495	0.103920	0.0519598	−	0.998649i	$-0.483453\pi$
$107$	1.07495	0.103920	0.0519598	+	0.998649i	$0.483453\pi$
$108$	0	0
$109$	−0.279773	−0.0267974	−0.0133987	−	0.999910i	$-0.504265\pi$
$109$	−0.279773	−0.0267974	−0.0133987	+	0.999910i	$0.504265\pi$
$110$	0	0
$111$	0	0
$112$	2.21024	0.208848
$113$	13.6813	1.28703	0.643513	−	0.765436i	$-0.277476\pi$
$113$	13.6813	1.28703	0.643513	+	0.765436i	$0.277476\pi$
$114$	0	0
$115$	0	0
$116$	−37.5174	−3.48340
$117$	0	0
$118$	−6.68086	−0.615023
$119$	−3.18639	−0.292096
$120$	0	0
$121$	7.18363	0.653057
$122$	−22.0011	−1.99188
$123$	0	0
$124$	−5.48981	−0.492999
$125$	0	0
$126$	0	0
$127$	14.9264	1.32450	0.662252	−	0.749281i	$-0.269601\pi$
$127$	14.9264	1.32450	0.662252	+	0.749281i	$0.269601\pi$
$128$	19.9233	1.76099
$129$	0	0
$130$	0	0
$131$	−20.2324	−1.76771	−0.883856	−	0.467758i	$-0.845062\pi$
$131$	−20.2324	−1.76771	−0.883856	+	0.467758i	$0.845062\pi$
$132$	0	0
$133$	−3.39228	−0.294148
$134$	14.3049	1.23575
$135$	0	0
$136$	25.7405	2.20723
$137$	−8.00831	−0.684196	−0.342098	−	0.939664i	$-0.611138\pi$
$137$	−8.00831	−0.684196	−0.342098	+	0.939664i	$0.611138\pi$
$138$	0	0
$139$	10.0009	0.848261	0.424131	−	0.905601i	$-0.360580\pi$
$139$	10.0009	0.848261	0.424131	+	0.905601i	$0.360580\pi$
$140$	0	0
$141$	0	0
$142$	26.1825	2.19719
$143$	−21.5905	−1.80548
$144$	0	0
$145$	0	0
$146$	−12.2549	−1.01422
$147$	0	0
$148$	−44.9046	−3.69113
$149$	20.2500	1.65895	0.829474	−	0.558546i	$-0.188640\pi$
$149$	20.2500	1.65895	0.829474	+	0.558546i	$0.188640\pi$
$150$	0	0
$151$	−0.0600683	−0.00488829	−0.00244414	−	0.999997i	$-0.500778\pi$
$151$	−0.0600683	−0.00488829	−0.00244414	+	0.999997i	$0.500778\pi$
$152$	27.4038	2.22274
$153$	0	0
$154$	6.15082	0.495647
$155$	0	0
$156$	0	0
$157$	3.69886	0.295201	0.147601	−	0.989047i	$-0.452845\pi$
$157$	3.69886	0.295201	0.147601	+	0.989047i	$0.452845\pi$
$158$	−20.6061	−1.63933
$159$	0	0
$160$	0	0
$161$	1.25606	0.0989910
$162$	0	0
$163$	4.88965	0.382987	0.191493	−	0.981494i	$-0.438667\pi$
$163$	4.88965	0.382987	0.191493	+	0.981494i	$0.438667\pi$
$164$	1.59656	0.124671
$165$	0	0
$166$	26.8793	2.08624
$167$	16.3419	1.26457	0.632286	−	0.774735i	$-0.282117\pi$
$167$	16.3419	1.26457	0.632286	+	0.774735i	$0.282117\pi$
$168$	0	0
$169$	12.6356	0.971968
$170$	0	0
$171$	0	0
$172$	20.9767	1.59946
$173$	−11.1117	−0.844811	−0.422405	−	0.906407i	$-0.638814\pi$
$173$	−11.1117	−0.844811	−0.422405	+	0.906407i	$0.638814\pi$
$174$	0	0
$175$	0	0
$176$	−15.9468	−1.20204
$177$	0	0
$178$	−32.0844	−2.40483
$179$	18.0527	1.34932	0.674659	−	0.738129i	$-0.264291\pi$
$179$	18.0527	1.34932	0.674659	+	0.738129i	$0.264291\pi$
$180$	0	0
$181$	−3.11291	−0.231381	−0.115690	−	0.993285i	$-0.536908\pi$
$181$	−3.11291	−0.231381	−0.115690	+	0.993285i	$0.536908\pi$
$182$	−7.30321	−0.541350
$183$	0	0
$184$	−10.1467	−0.748028
$185$	0	0
$186$	0	0
$187$	22.9898	1.68118
$188$	34.0356	2.48230
$189$	0	0
$190$	0	0
$191$	15.0422	1.08842	0.544208	−	0.838950i	$-0.316830\pi$
$191$	15.0422	1.08842	0.544208	+	0.838950i	$0.316830\pi$
$192$	0	0
$193$	−5.39017	−0.387993	−0.193996	−	0.981002i	$-0.562145\pi$
$193$	−5.39017	−0.387993	−0.193996	+	0.981002i	$0.562145\pi$
$194$	−17.6922	−1.27023
$195$	0	0
$196$	−26.3121	−1.87944
$197$	12.5879	0.896848	0.448424	−	0.893821i	$-0.351985\pi$
$197$	12.5879	0.896848	0.448424	+	0.893821i	$0.351985\pi$
$198$	0	0
$199$	16.6365	1.17933	0.589666	−	0.807648i	$-0.299260\pi$
$199$	16.6365	1.17933	0.589666	+	0.807648i	$0.299260\pi$
$200$	0	0
$201$	0	0
$202$	−24.9431	−1.75499
$203$	−5.60465	−0.393369
$204$	0	0
$205$	0	0
$206$	−28.5010	−1.98576
$207$	0	0
$208$	18.9346	1.31288
$209$	24.4753	1.69299
$210$	0	0
$211$	−23.3539	−1.60775	−0.803874	−	0.594799i	$-0.797232\pi$
$211$	−23.3539	−1.60775	−0.803874	+	0.594799i	$0.797232\pi$
$212$	1.33576	0.0917403
$213$	0	0
$214$	−2.62348	−0.179337
$215$	0	0
$216$	0	0
$217$	−0.820111	−0.0556728
$218$	0.682800	0.0462451
$219$	0	0
$220$	0	0
$221$	−27.2971	−1.83620
$222$	0	0
$223$	0.829977	0.0555794	0.0277897	−	0.999614i	$-0.491153\pi$
$223$	0.829977	0.0555794	0.0277897	+	0.999614i	$0.491153\pi$
$224$	0.249408	0.0166643
$225$	0	0
$226$	−33.3898	−2.22106
$227$	−22.7061	−1.50706	−0.753530	−	0.657413i	$-0.771651\pi$
$227$	−22.7061	−1.50706	−0.753530	+	0.657413i	$0.771651\pi$
$228$	0	0
$229$	−27.7726	−1.83527	−0.917633	−	0.397429i	$-0.869903\pi$
$229$	−27.7726	−1.83527	−0.917633	+	0.397429i	$0.869903\pi$
$230$	0	0
$231$	0	0
$232$	45.2758	2.97250
$233$	−1.14087	−0.0747411	−0.0373706	−	0.999301i	$-0.511898\pi$
$233$	−1.14087	−0.0747411	−0.0373706	+	0.999301i	$0.511898\pi$
$234$	0	0
$235$	0	0
$236$	10.8301	0.704981
$237$	0	0
$238$	7.77656	0.504079
$239$	7.82335	0.506051	0.253025	−	0.967460i	$-0.418574\pi$
$239$	7.82335	0.506051	0.253025	+	0.967460i	$0.418574\pi$
$240$	0	0
$241$	−0.865353	−0.0557423	−0.0278711	−	0.999612i	$-0.508873\pi$
$241$	−0.865353	−0.0557423	−0.0278711	+	0.999612i	$0.508873\pi$
$242$	−17.5320	−1.12700
$243$	0	0
$244$	35.6651	2.28323
$245$	0	0
$246$	0	0
$247$	−29.0609	−1.84910
$248$	6.62508	0.420693
$249$	0	0
$250$	0	0
$251$	−6.09930	−0.384984	−0.192492	−	0.981299i	$-0.561657\pi$
$251$	−6.09930	−0.384984	−0.192492	+	0.981299i	$0.561657\pi$
$252$	0	0
$253$	−9.06243	−0.569750
$254$	−36.4287	−2.28574
$255$	0	0
$256$	−31.6054	−1.97533
$257$	18.2063	1.13568	0.567839	−	0.823140i	$-0.307780\pi$
$257$	18.2063	1.13568	0.567839	+	0.823140i	$0.307780\pi$
$258$	0	0
$259$	−6.70820	−0.416828
$260$	0	0
$261$	0	0
$262$	49.3782	3.05060
$263$	−11.7262	−0.723071	−0.361536	−	0.932358i	$-0.617747\pi$
$263$	−11.7262	−0.723071	−0.361536	+	0.932358i	$0.617747\pi$
$264$	0	0
$265$	0	0
$266$	8.27904	0.507621
$267$	0	0
$268$	−23.1891	−1.41650
$269$	12.0486	0.734617	0.367309	−	0.930099i	$-0.380279\pi$
$269$	12.0486	0.734617	0.367309	+	0.930099i	$0.380279\pi$
$270$	0	0
$271$	−19.0195	−1.15535	−0.577675	−	0.816267i	$-0.696040\pi$
$271$	−19.0195	−1.15535	−0.577675	+	0.816267i	$0.696040\pi$
$272$	−20.1618	−1.22249
$273$	0	0
$274$	19.5447	1.18074
$275$	0	0
$276$	0	0
$277$	−7.22640	−0.434192	−0.217096	−	0.976150i	$-0.569658\pi$
$277$	−7.22640	−0.434192	−0.217096	+	0.976150i	$0.569658\pi$
$278$	−24.4076	−1.46387
$279$	0	0
$280$	0	0
$281$	19.2675	1.14940	0.574700	−	0.818364i	$-0.305119\pi$
$281$	19.2675	1.14940	0.574700	+	0.818364i	$0.305119\pi$
$282$	0	0
$283$	23.2436	1.38169	0.690843	−	0.723004i	$-0.257239\pi$
$283$	23.2436	1.38169	0.690843	+	0.723004i	$0.257239\pi$
$284$	−42.4436	−2.51856
$285$	0	0
$286$	52.6926	3.11578
$287$	0.238508	0.0140787
$288$	0	0
$289$	12.0663	0.709783
$290$	0	0
$291$	0	0
$292$	19.8660	1.16257
$293$	16.9450	0.989936	0.494968	−	0.868911i	$-0.335180\pi$
$293$	16.9450	0.989936	0.494968	+	0.868911i	$0.335180\pi$
$294$	0	0
$295$	0	0
$296$	54.1907	3.14977
$297$	0	0
$298$	−49.4213	−2.86290
$299$	10.7603	0.622286
$300$	0	0
$301$	3.13367	0.180622
$302$	0.146600	0.00843587
$303$	0	0
$304$	−21.4646	−1.23108
$305$	0	0
$306$	0	0
$307$	7.48318	0.427088	0.213544	−	0.976933i	$-0.431499\pi$
$307$	7.48318	0.427088	0.213544	+	0.976933i	$0.431499\pi$
$308$	−9.97087	−0.568143
$309$	0	0
$310$	0	0
$311$	−29.3081	−1.66191	−0.830955	−	0.556339i	$-0.812206\pi$
$311$	−29.3081	−1.66191	−0.830955	+	0.556339i	$0.812206\pi$
$312$	0	0
$313$	−20.6261	−1.16585	−0.582927	−	0.812524i	$-0.698093\pi$
$313$	−20.6261	−1.16585	−0.582927	+	0.812524i	$0.698093\pi$
$314$	−9.02727	−0.509438
$315$	0	0
$316$	33.4039	1.87911
$317$	−29.7154	−1.66898	−0.834490	−	0.551023i	$-0.814238\pi$
$317$	−29.7154	−1.66898	−0.834490	+	0.551023i	$0.814238\pi$
$318$	0	0
$319$	40.4375	2.26406
$320$	0	0
$321$	0	0
$322$	−3.06547	−0.170832
$323$	30.9445	1.72180
$324$	0	0
$325$	0	0
$326$	−11.9334	−0.660932
$327$	0	0
$328$	−1.92673	−0.106386
$329$	5.08451	0.280318
$330$	0	0
$331$	33.8462	1.86036	0.930178	−	0.367109i	$-0.119652\pi$
$331$	33.8462	1.86036	0.930178	+	0.367109i	$0.119652\pi$
$332$	−43.5731	−2.39138
$333$	0	0
$334$	−39.8832	−2.18231
$335$	0	0
$336$	0	0
$337$	−5.39168	−0.293704	−0.146852	−	0.989159i	$-0.546914\pi$
$337$	−5.39168	−0.293704	−0.146852	+	0.989159i	$0.546914\pi$
$338$	−30.8378	−1.67735
$339$	0	0
$340$	0	0
$341$	5.91710	0.320429
$342$	0	0
$343$	−8.06787	−0.435624
$344$	−25.3146	−1.36487
$345$	0	0
$346$	27.1188	1.45792
$347$	20.1190	1.08004	0.540022	−	0.841651i	$-0.318416\pi$
$347$	20.1190	1.08004	0.540022	+	0.841651i	$0.318416\pi$
$348$	0	0
$349$	−15.9106	−0.851672	−0.425836	−	0.904800i	$-0.640020\pi$
$349$	−15.9106	−0.851672	−0.425836	+	0.904800i	$0.640020\pi$
$350$	0	0
$351$	0	0
$352$	−1.79948	−0.0959124
$353$	7.21261	0.383888	0.191944	−	0.981406i	$-0.438521\pi$
$353$	7.21261	0.383888	0.191944	+	0.981406i	$0.438521\pi$
$354$	0	0
$355$	0	0
$356$	52.0109	2.75657
$357$	0	0
$358$	−44.0584	−2.32856
$359$	−3.37048	−0.177887	−0.0889436	−	0.996037i	$-0.528349\pi$
$359$	−3.37048	−0.177887	−0.0889436	+	0.996037i	$0.528349\pi$
$360$	0	0
$361$	13.9439	0.733892
$362$	7.59722	0.399301
$363$	0	0
$364$	11.8390	0.620532
$365$	0	0
$366$	0	0
$367$	−5.37420	−0.280531	−0.140266	−	0.990114i	$-0.544796\pi$
$367$	−5.37420	−0.280531	−0.140266	+	0.990114i	$0.544796\pi$
$368$	7.94765	0.414300
$369$	0	0
$370$	0	0
$371$	0.199546	0.0103599
$372$	0	0
$373$	−16.0783	−0.832505	−0.416252	−	0.909249i	$-0.636657\pi$
$373$	−16.0783	−0.832505	−0.416252	+	0.909249i	$0.636657\pi$
$374$	−56.1078	−2.90127
$375$	0	0
$376$	−41.0740	−2.11823
$377$	−48.0137	−2.47283
$378$	0	0
$379$	−27.6162	−1.41855	−0.709274	−	0.704933i	$-0.750977\pi$
$379$	−27.6162	−1.41855	−0.709274	+	0.704933i	$0.750977\pi$
$380$	0	0
$381$	0	0
$382$	−36.7113	−1.87831
$383$	23.2577	1.18841	0.594207	−	0.804312i	$-0.297466\pi$
$383$	23.2577	1.18841	0.594207	+	0.804312i	$0.297466\pi$
$384$	0	0
$385$	0	0
$386$	13.1550	0.669571
$387$	0	0
$388$	28.6803	1.45602
$389$	−12.8126	−0.649624	−0.324812	−	0.945779i	$-0.605301\pi$
$389$	−12.8126	−0.649624	−0.324812	+	0.945779i	$0.605301\pi$
$390$	0	0
$391$	−11.4578	−0.579443
$392$	31.7533	1.60379
$393$	0	0
$394$	−30.7214	−1.54772
$395$	0	0
$396$	0	0
$397$	−10.8608	−0.545085	−0.272543	−	0.962144i	$-0.587865\pi$
$397$	−10.8608	−0.545085	−0.272543	+	0.962144i	$0.587865\pi$
$398$	−40.6023	−2.03521
$399$	0	0
$400$	0	0
$401$	23.0220	1.14966	0.574831	−	0.818272i	$-0.305068\pi$
$401$	23.0220	1.14966	0.574831	+	0.818272i	$0.305068\pi$
$402$	0	0
$403$	−7.02570	−0.349975
$404$	40.4345	2.01169
$405$	0	0
$406$	13.6784	0.678849
$407$	48.3996	2.39908
$408$	0	0
$409$	−20.3469	−1.00609	−0.503044	−	0.864261i	$-0.667787\pi$
$409$	−20.3469	−1.00609	−0.503044	+	0.864261i	$0.667787\pi$
$410$	0	0
$411$	0	0
$412$	46.2020	2.27621
$413$	1.61789	0.0796111
$414$	0	0
$415$	0	0
$416$	2.13662	0.104756
$417$	0	0
$418$	−59.7332	−2.92165
$419$	21.6227	1.05634	0.528169	−	0.849140i	$-0.322879\pi$
$419$	21.6227	1.05634	0.528169	+	0.849140i	$0.322879\pi$
$420$	0	0
$421$	−5.15561	−0.251269	−0.125635	−	0.992077i	$-0.540097\pi$
$421$	−5.15561	−0.251269	−0.125635	+	0.992077i	$0.540097\pi$
$422$	56.9964	2.77454
$423$	0	0
$424$	−1.61199	−0.0782850
$425$	0	0
$426$	0	0
$427$	5.32794	0.257837
$428$	4.25283	0.205568
$429$	0	0
$430$	0	0
$431$	−1.21065	−0.0583150	−0.0291575	−	0.999575i	$-0.509282\pi$
$431$	−1.21065	−0.0583150	−0.0291575	+	0.999575i	$0.509282\pi$
$432$	0	0
$433$	−0.0648649	−0.00311721	−0.00155860	−	0.999999i	$-0.500496\pi$
$433$	−0.0648649	−0.00311721	−0.00155860	+	0.999999i	$0.500496\pi$
$434$	2.00152	0.0960762
$435$	0	0
$436$	−1.10686	−0.0530092
$437$	−12.1981	−0.583514
$438$	0	0
$439$	−11.7704	−0.561770	−0.280885	−	0.959741i	$-0.590628\pi$
$439$	−11.7704	−0.561770	−0.280885	+	0.959741i	$0.590628\pi$
$440$	0	0
$441$	0	0
$442$	66.6200	3.16879
$443$	19.1683	0.910716	0.455358	−	0.890309i	$-0.349511\pi$
$443$	19.1683	0.910716	0.455358	+	0.890309i	$0.349511\pi$
$444$	0	0
$445$	0	0
$446$	−2.02560	−0.0959151
$447$	0	0
$448$	−5.02917	−0.237606
$449$	−0.536369	−0.0253128	−0.0126564	−	0.999920i	$-0.504029\pi$
$449$	−0.536369	−0.0253128	−0.0126564	+	0.999920i	$0.504029\pi$
$450$	0	0
$451$	−1.72083	−0.0810308
$452$	54.1271	2.54593
$453$	0	0
$454$	55.4155	2.60078
$455$	0	0
$456$	0	0
$457$	35.1952	1.64636	0.823181	−	0.567779i	$-0.192197\pi$
$457$	35.1952	1.64636	0.823181	+	0.567779i	$0.192197\pi$
$458$	67.7805	3.16718
$459$	0	0
$460$	0	0
$461$	1.79752	0.0837188	0.0418594	−	0.999124i	$-0.486672\pi$
$461$	1.79752	0.0837188	0.0418594	+	0.999124i	$0.486672\pi$
$462$	0	0
$463$	15.8682	0.737460	0.368730	−	0.929537i	$-0.379793\pi$
$463$	15.8682	0.737460	0.368730	+	0.929537i	$0.379793\pi$
$464$	−35.4632	−1.64634
$465$	0	0
$466$	2.78436	0.128983
$467$	−41.1577	−1.90455	−0.952275	−	0.305240i	$-0.901263\pi$
$467$	−41.1577	−1.90455	−0.952275	+	0.305240i	$0.901263\pi$
$468$	0	0
$469$	−3.46418	−0.159961
$470$	0	0
$471$	0	0
$472$	−13.0697	−0.601584
$473$	−22.6094	−1.03958
$474$	0	0
$475$	0	0
$476$	−12.6063	−0.577809
$477$	0	0
$478$	−19.0933	−0.873307
$479$	15.1052	0.690176	0.345088	−	0.938570i	$-0.387849\pi$
$479$	15.1052	0.690176	0.345088	+	0.938570i	$0.387849\pi$
$480$	0	0
$481$	−57.4677	−2.62030
$482$	2.11194	0.0961962
$483$	0	0
$484$	28.4205	1.29184
$485$	0	0
$486$	0	0
$487$	−4.90873	−0.222436	−0.111218	−	0.993796i	$-0.535475\pi$
$487$	−4.90873	−0.222436	−0.111218	+	0.993796i	$0.535475\pi$
$488$	−43.0405	−1.94835
$489$	0	0
$490$	0	0
$491$	43.7750	1.97554	0.987769	−	0.155925i	$-0.0498359\pi$
$491$	43.7750	1.97554	0.987769	+	0.155925i	$0.0498359\pi$
$492$	0	0
$493$	51.1257	2.30258
$494$	70.9247	3.19105
$495$	0	0
$496$	−5.18923	−0.233003
$497$	−6.34056	−0.284413
$498$	0	0
$499$	6.00199	0.268686	0.134343	−	0.990935i	$-0.457108\pi$
$499$	6.00199	0.268686	0.134343	+	0.990935i	$0.457108\pi$
$500$	0	0
$501$	0	0
$502$	14.8857	0.664379
$503$	28.8130	1.28471	0.642354	−	0.766408i	$-0.277958\pi$
$503$	28.8130	1.28471	0.642354	+	0.766408i	$0.277958\pi$
$504$	0	0
$505$	0	0
$506$	22.1173	0.983235
$507$	0	0
$508$	59.0532	2.62006
$509$	−12.3745	−0.548492	−0.274246	−	0.961660i	$-0.588428\pi$
$509$	−12.3745	−0.548492	−0.274246	+	0.961660i	$0.588428\pi$
$510$	0	0
$511$	2.96774	0.131285
$512$	37.2879	1.64791
$513$	0	0
$514$	−44.4334	−1.95987
$515$	0	0
$516$	0	0
$517$	−36.6847	−1.61339
$518$	16.3717	0.719332
$519$	0	0
$520$	0	0
$521$	−16.0357	−0.702535	−0.351268	−	0.936275i	$-0.614249\pi$
$521$	−16.0357	−0.702535	−0.351268	+	0.936275i	$0.614249\pi$
$522$	0	0
$523$	34.7701	1.52039	0.760195	−	0.649695i	$-0.225103\pi$
$523$	34.7701	1.52039	0.760195	+	0.649695i	$0.225103\pi$
$524$	−80.0453	−3.49680
$525$	0	0
$526$	28.6185	1.24783
$527$	7.48106	0.325880
$528$	0	0
$529$	−18.4834	−0.803627
$530$	0	0
$531$	0	0
$532$	−13.4209	−0.581869
$533$	2.04324	0.0885026
$534$	0	0
$535$	0	0
$536$	27.9846	1.20875
$537$	0	0
$538$	−29.4053	−1.26775
$539$	28.3601	1.22155
$540$	0	0
$541$	−10.0421	−0.431745	−0.215872	−	0.976422i	$-0.569260\pi$
$541$	−10.0421	−0.431745	−0.215872	+	0.976422i	$0.569260\pi$
$542$	46.4180	1.99382
$543$	0	0
$544$	−2.27510	−0.0975442
$545$	0	0
$546$	0	0
$547$	34.1914	1.46192	0.730959	−	0.682421i	$-0.239073\pi$
$547$	34.1914	1.46192	0.730959	+	0.682421i	$0.239073\pi$
$548$	−31.6832	−1.35344
$549$	0	0
$550$	0	0
$551$	54.4292	2.31876
$552$	0	0
$553$	4.99013	0.212202
$554$	17.6364	0.749299
$555$	0	0
$556$	39.5663	1.67799
$557$	−1.35501	−0.0574135	−0.0287068	−	0.999588i	$-0.509139\pi$
$557$	−1.35501	−0.0574135	−0.0287068	+	0.999588i	$0.509139\pi$
$558$	0	0
$559$	26.8454	1.13544
$560$	0	0
$561$	0	0
$562$	−47.0233	−1.98356
$563$	−14.5306	−0.612391	−0.306195	−	0.951969i	$-0.599056\pi$
$563$	−14.5306	−0.612391	−0.306195	+	0.951969i	$0.599056\pi$
$564$	0	0
$565$	0	0
$566$	−56.7271	−2.38442
$567$	0	0
$568$	51.2207	2.14917
$569$	19.6012	0.821727	0.410863	−	0.911697i	$-0.365227\pi$
$569$	19.6012	0.821727	0.410863	+	0.911697i	$0.365227\pi$
$570$	0	0
$571$	32.8810	1.37603	0.688013	−	0.725698i	$-0.258483\pi$
$571$	32.8810	1.37603	0.688013	+	0.725698i	$0.258483\pi$
$572$	−85.4182	−3.57151
$573$	0	0
$574$	−0.582090	−0.0242960
$575$	0	0
$576$	0	0
$577$	7.26819	0.302579	0.151289	−	0.988490i	$-0.451657\pi$
$577$	7.26819	0.302579	0.151289	+	0.988490i	$0.451657\pi$
$578$	−29.4485	−1.22490
$579$	0	0
$580$	0	0
$581$	−6.50929	−0.270051
$582$	0	0
$583$	−1.43972	−0.0596273
$584$	−23.9742	−0.992060
$585$	0	0
$586$	−41.3551	−1.70836
$587$	9.51042	0.392537	0.196268	−	0.980550i	$-0.437118\pi$
$587$	9.51042	0.392537	0.196268	+	0.980550i	$0.437118\pi$
$588$	0	0
$589$	7.96446	0.328170
$590$	0	0
$591$	0	0
$592$	−42.4460	−1.74452
$593$	9.75081	0.400418	0.200209	−	0.979753i	$-0.435838\pi$
$593$	9.75081	0.400418	0.200209	+	0.979753i	$0.435838\pi$
$594$	0	0
$595$	0	0
$596$	80.1151	3.28164
$597$	0	0
$598$	−26.2612	−1.07390
$599$	−6.63034	−0.270908	−0.135454	−	0.990784i	$-0.543249\pi$
$599$	−6.63034	−0.270908	−0.135454	+	0.990784i	$0.543249\pi$
$600$	0	0
$601$	−25.8619	−1.05493	−0.527465	−	0.849577i	$-0.676857\pi$
$601$	−25.8619	−1.05493	−0.527465	+	0.849577i	$0.676857\pi$
$602$	−7.64789	−0.311705
$603$	0	0
$604$	−0.237648	−0.00966975
$605$	0	0
$606$	0	0
$607$	11.3597	0.461075	0.230537	−	0.973063i	$-0.425952\pi$
$607$	11.3597	0.461075	0.230537	+	0.973063i	$0.425952\pi$
$608$	−2.42211	−0.0982295
$609$	0	0
$610$	0	0
$611$	43.5578	1.76216
$612$	0	0
$613$	30.2599	1.22218	0.611092	−	0.791559i	$-0.290730\pi$
$613$	30.2599	1.22218	0.611092	+	0.791559i	$0.290730\pi$
$614$	−18.2631	−0.737039
$615$	0	0
$616$	12.0328	0.484816
$617$	33.1940	1.33634	0.668170	−	0.744009i	$-0.267078\pi$
$617$	33.1940	1.33634	0.668170	+	0.744009i	$0.267078\pi$
$618$	0	0
$619$	−11.1705	−0.448979	−0.224489	−	0.974477i	$-0.572071\pi$
$619$	−11.1705	−0.448979	−0.224489	+	0.974477i	$0.572071\pi$
$620$	0	0
$621$	0	0
$622$	71.5280	2.86801
$623$	7.76980	0.311291
$624$	0	0
$625$	0	0
$626$	50.3390	2.01195
$627$	0	0
$628$	14.6338	0.583952
$629$	61.1923	2.43990
$630$	0	0
$631$	−13.7542	−0.547546	−0.273773	−	0.961794i	$-0.588272\pi$
$631$	−13.7542	−0.547546	−0.273773	+	0.961794i	$0.588272\pi$
$632$	−40.3116	−1.60351
$633$	0	0
$634$	72.5219	2.88021
$635$	0	0
$636$	0	0
$637$	−33.6735	−1.33419
$638$	−98.6898	−3.90717
$639$	0	0
$640$	0	0
$641$	26.6242	1.05159	0.525797	−	0.850610i	$-0.323767\pi$
$641$	26.6242	1.05159	0.525797	+	0.850610i	$0.323767\pi$
$642$	0	0
$643$	−21.6827	−0.855082	−0.427541	−	0.903996i	$-0.640620\pi$
$643$	−21.6827	−0.855082	−0.427541	+	0.903996i	$0.640620\pi$
$644$	4.96933	0.195819
$645$	0	0
$646$	−75.5216	−2.97136
$647$	13.6216	0.535520	0.267760	−	0.963486i	$-0.413717\pi$
$647$	13.6216	0.535520	0.267760	+	0.963486i	$0.413717\pi$
$648$	0	0
$649$	−11.6731	−0.458208
$650$	0	0
$651$	0	0
$652$	19.3449	0.757604
$653$	−30.4783	−1.19271	−0.596355	−	0.802721i	$-0.703385\pi$
$653$	−30.4783	−1.19271	−0.596355	+	0.802721i	$0.703385\pi$
$654$	0	0
$655$	0	0
$656$	1.50915	0.0589224
$657$	0	0
$658$	−12.4090	−0.483753
$659$	17.7054	0.689705	0.344852	−	0.938657i	$-0.387929\pi$
$659$	17.7054	0.689705	0.344852	+	0.938657i	$0.387929\pi$
$660$	0	0
$661$	7.21561	0.280655	0.140327	−	0.990105i	$-0.455184\pi$
$661$	7.21561	0.280655	0.140327	+	0.990105i	$0.455184\pi$
$662$	−82.6035	−3.21047
$663$	0	0
$664$	52.5838	2.04065
$665$	0	0
$666$	0	0
$667$	−20.1534	−0.780343
$668$	64.6533	2.50151
$669$	0	0
$670$	0	0
$671$	−38.4411	−1.48400
$672$	0	0
$673$	41.3984	1.59579	0.797896	−	0.602796i	$-0.205947\pi$
$673$	41.3984	1.59579	0.797896	+	0.602796i	$0.205947\pi$
$674$	13.1587	0.506853
$675$	0	0
$676$	49.9901	1.92270
$677$	−4.67244	−0.179576	−0.0897882	−	0.995961i	$-0.528619\pi$
$677$	−4.67244	−0.179576	−0.0897882	+	0.995961i	$0.528619\pi$
$678$	0	0
$679$	4.28449	0.164423
$680$	0	0
$681$	0	0
$682$	−14.4410	−0.552974
$683$	0.695173	0.0266001	0.0133000	−	0.999912i	$-0.495766\pi$
$683$	0.695173	0.0266001	0.0133000	+	0.999912i	$0.495766\pi$
$684$	0	0
$685$	0	0
$686$	19.6901	0.751770
$687$	0	0
$688$	19.8282	0.755944
$689$	1.70947	0.0651255
$690$	0	0
$691$	16.0208	0.609461	0.304731	−	0.952439i	$-0.401434\pi$
$691$	16.0208	0.609461	0.304731	+	0.952439i	$0.401434\pi$
$692$	−43.9614	−1.67116
$693$	0	0
$694$	−49.1015	−1.86387
$695$	0	0
$696$	0	0
$697$	−2.17567	−0.0824094
$698$	38.8305	1.46976
$699$	0	0
$700$	0	0
$701$	33.8563	1.27873	0.639367	−	0.768902i	$-0.279197\pi$
$701$	33.8563	1.27873	0.639367	+	0.768902i	$0.279197\pi$
$702$	0	0
$703$	65.1463	2.45704
$704$	36.2854	1.36756
$705$	0	0
$706$	−17.6028	−0.662488
$707$	6.04042	0.227173
$708$	0	0
$709$	−14.4041	−0.540959	−0.270479	−	0.962726i	$-0.587182\pi$
$709$	−14.4041	−0.540959	−0.270479	+	0.962726i	$0.587182\pi$
$710$	0	0
$711$	0	0
$712$	−62.7666	−2.35228
$713$	−2.94899	−0.110440
$714$	0	0
$715$	0	0
$716$	71.4216	2.66915
$717$	0	0
$718$	8.22584	0.306985
$719$	28.5156	1.06345	0.531726	−	0.846916i	$-0.321543\pi$
$719$	28.5156	1.06345	0.531726	+	0.846916i	$0.321543\pi$
$720$	0	0
$721$	6.90203	0.257045
$722$	−34.0309	−1.26650
$723$	0	0
$724$	−12.3156	−0.457705
$725$	0	0
$726$	0	0
$727$	47.5108	1.76208	0.881040	−	0.473042i	$-0.156844\pi$
$727$	47.5108	1.76208	0.881040	+	0.473042i	$0.156844\pi$
$728$	−14.2872	−0.529520
$729$	0	0
$730$	0	0
$731$	−28.5854	−1.05727
$732$	0	0
$733$	24.3105	0.897928	0.448964	−	0.893550i	$-0.351793\pi$
$733$	24.3105	0.897928	0.448964	+	0.893550i	$0.351793\pi$
$734$	13.1160	0.484121
$735$	0	0
$736$	0.896830	0.0330576
$737$	24.9940	0.920667
$738$	0	0
$739$	41.7625	1.53626	0.768130	−	0.640294i	$-0.221187\pi$
$739$	41.7625	1.53626	0.768130	+	0.640294i	$0.221187\pi$
$740$	0	0
$741$	0	0
$742$	−0.487003	−0.0178784
$743$	19.7341	0.723972	0.361986	−	0.932183i	$-0.382099\pi$
$743$	19.7341	0.723972	0.361986	+	0.932183i	$0.382099\pi$
$744$	0	0
$745$	0	0
$746$	39.2400	1.43668
$747$	0	0
$748$	90.9544	3.32562
$749$	0.635321	0.0232141
$750$	0	0
$751$	18.2348	0.665397	0.332699	−	0.943033i	$-0.392041\pi$
$751$	18.2348	0.665397	0.332699	+	0.943033i	$0.392041\pi$
$752$	32.1721	1.17319
$753$	0	0
$754$	117.180	4.26744
$755$	0	0
$756$	0	0
$757$	−15.5166	−0.563962	−0.281981	−	0.959420i	$-0.590992\pi$
$757$	−15.5166	−0.563962	−0.281981	+	0.959420i	$0.590992\pi$
$758$	67.3988	2.44803
$759$	0	0
$760$	0	0
$761$	−11.7633	−0.426419	−0.213210	−	0.977006i	$-0.568392\pi$
$761$	−11.7633	−0.426419	−0.213210	+	0.977006i	$0.568392\pi$
$762$	0	0
$763$	−0.165352	−0.00598615
$764$	59.5114	2.15305
$765$	0	0
$766$	−56.7617	−2.05088
$767$	13.8601	0.500459
$768$	0	0
$769$	−9.10438	−0.328312	−0.164156	−	0.986434i	$-0.552490\pi$
$769$	−9.10438	−0.328312	−0.164156	+	0.986434i	$0.552490\pi$
$770$	0	0
$771$	0	0
$772$	−21.3251	−0.767507
$773$	−25.6603	−0.922937	−0.461468	−	0.887157i	$-0.652677\pi$
$773$	−25.6603	−0.922937	−0.461468	+	0.887157i	$0.652677\pi$
$774$	0	0
$775$	0	0
$776$	−34.6112	−1.24247
$777$	0	0
$778$	31.2698	1.12108
$779$	−2.31625	−0.0829883
$780$	0	0
$781$	45.7471	1.63696
$782$	27.9632	0.999963
$783$	0	0
$784$	−24.8715	−0.888267
$785$	0	0
$786$	0	0
$787$	−32.6096	−1.16241	−0.581204	−	0.813758i	$-0.697418\pi$
$787$	−32.6096	−1.16241	−0.581204	+	0.813758i	$0.697418\pi$
$788$	49.8013	1.77410
$789$	0	0
$790$	0	0
$791$	8.08594	0.287503
$792$	0	0
$793$	45.6433	1.62084
$794$	26.5062	0.940671
$795$	0	0
$796$	65.8190	2.33289
$797$	29.7470	1.05369	0.526846	−	0.849961i	$-0.323374\pi$
$797$	29.7470	1.05369	0.526846	+	0.849961i	$0.323374\pi$
$798$	0	0
$799$	−46.3809	−1.64084
$800$	0	0
$801$	0	0
$802$	−56.1863	−1.98401
$803$	−21.4122	−0.755621
$804$	0	0
$805$	0	0
$806$	17.1466	0.603963
$807$	0	0
$808$	−48.7961	−1.71664
$809$	28.9282	1.01706	0.508530	−	0.861044i	$-0.330189\pi$
$809$	28.9282	1.01706	0.508530	+	0.861044i	$0.330189\pi$
$810$	0	0
$811$	9.02896	0.317050	0.158525	−	0.987355i	$-0.449326\pi$
$811$	9.02896	0.317050	0.158525	+	0.987355i	$0.449326\pi$
$812$	−22.1736	−0.778142
$813$	0	0
$814$	−118.122	−4.14017
$815$	0	0
$816$	0	0
$817$	−30.4324	−1.06470
$818$	49.6576	1.73624
$819$	0	0
$820$	0	0
$821$	−22.0570	−0.769796	−0.384898	−	0.922959i	$-0.625763\pi$
$821$	−22.0570	−0.769796	−0.384898	+	0.922959i	$0.625763\pi$
$822$	0	0
$823$	−17.9828	−0.626840	−0.313420	−	0.949615i	$-0.601475\pi$
$823$	−17.9828	−0.626840	−0.313420	+	0.949615i	$0.601475\pi$
$824$	−55.7564	−1.94237
$825$	0	0
$826$	−3.94854	−0.137387
$827$	−13.4856	−0.468939	−0.234470	−	0.972123i	$-0.575335\pi$
$827$	−13.4856	−0.468939	−0.234470	+	0.972123i	$0.575335\pi$
$828$	0	0
$829$	34.6056	1.20190	0.600952	−	0.799285i	$-0.294788\pi$
$829$	34.6056	1.20190	0.600952	+	0.799285i	$0.294788\pi$
$830$	0	0
$831$	0	0
$832$	−43.0837	−1.49366
$833$	35.8560	1.24234
$834$	0	0
$835$	0	0
$836$	96.8315	3.34899
$837$	0	0
$838$	−52.7713	−1.82295
$839$	−25.5209	−0.881080	−0.440540	−	0.897733i	$-0.645213\pi$
$839$	−25.5209	−0.881080	−0.440540	+	0.897733i	$0.645213\pi$
$840$	0	0
$841$	60.9265	2.10091
$842$	12.5825	0.433623
$843$	0	0
$844$	−92.3949	−3.18036
$845$	0	0
$846$	0	0
$847$	4.24569	0.145884
$848$	1.26262	0.0433587
$849$	0	0
$850$	0	0
$851$	−24.1216	−0.826878
$852$	0	0
$853$	−3.05949	−0.104755	−0.0523775	−	0.998627i	$-0.516680\pi$
$853$	−3.05949	−0.104755	−0.0523775	+	0.998627i	$0.516680\pi$
$854$	−13.0031	−0.444958
$855$	0	0
$856$	−5.13229	−0.175418
$857$	−29.0297	−0.991635	−0.495817	−	0.868427i	$-0.665131\pi$
$857$	−29.0297	−0.991635	−0.495817	+	0.868427i	$0.665131\pi$
$858$	0	0
$859$	−11.5059	−0.392576	−0.196288	−	0.980546i	$-0.562889\pi$
$859$	−11.5059	−0.392576	−0.196288	+	0.980546i	$0.562889\pi$
$860$	0	0
$861$	0	0
$862$	2.95466	0.100636
$863$	−10.0121	−0.340816	−0.170408	−	0.985374i	$-0.554509\pi$
$863$	−10.0121	−0.340816	−0.170408	+	0.985374i	$0.554509\pi$
$864$	0	0
$865$	0	0
$866$	0.158306	0.00537946
$867$	0	0
$868$	−3.24460	−0.110129
$869$	−36.0038	−1.22134
$870$	0	0
$871$	−29.6768	−1.00556
$872$	1.33576	0.0452345
$873$	0	0
$874$	29.7701	1.00699
$875$	0	0
$876$	0	0
$877$	16.6844	0.563392	0.281696	−	0.959504i	$-0.409103\pi$
$877$	16.6844	0.563392	0.281696	+	0.959504i	$0.409103\pi$
$878$	28.7262	0.969464
$879$	0	0
$880$	0	0
$881$	−9.74584	−0.328346	−0.164173	−	0.986432i	$-0.552495\pi$
$881$	−9.74584	−0.328346	−0.164173	+	0.986432i	$0.552495\pi$
$882$	0	0
$883$	19.7999	0.666319	0.333160	−	0.942870i	$-0.391885\pi$
$883$	19.7999	0.666319	0.333160	+	0.942870i	$0.391885\pi$
$884$	−107.995	−3.63228
$885$	0	0
$886$	−46.7813	−1.57165
$887$	−16.0655	−0.539427	−0.269714	−	0.962941i	$-0.586929\pi$
$887$	−16.0655	−0.539427	−0.269714	+	0.962941i	$0.586929\pi$
$888$	0	0
$889$	8.82184	0.295875
$890$	0	0
$891$	0	0
$892$	3.28364	0.109944
$893$	−49.3779	−1.65237
$894$	0	0
$895$	0	0
$896$	11.7751	0.393380
$897$	0	0
$898$	1.30904	0.0436831
$899$	13.1587	0.438866
$900$	0	0
$901$	−1.82026	−0.0606418
$902$	4.19978	0.139837
$903$	0	0
$904$	−65.3204	−2.17252
$905$	0	0
$906$	0	0
$907$	17.0483	0.566080	0.283040	−	0.959108i	$-0.408657\pi$
$907$	17.0483	0.566080	0.283040	+	0.959108i	$0.408657\pi$
$908$	−89.8322	−2.98119
$909$	0	0
$910$	0	0
$911$	−30.9582	−1.02569	−0.512846	−	0.858481i	$-0.671409\pi$
$911$	−30.9582	−1.02569	−0.512846	+	0.858481i	$0.671409\pi$
$912$	0	0
$913$	46.9645	1.55430
$914$	−85.8957	−2.84118
$915$	0	0
$916$	−109.877	−3.63043
$917$	−11.9578	−0.394882
$918$	0	0
$919$	17.0914	0.563794	0.281897	−	0.959445i	$-0.409036\pi$
$919$	17.0914	0.563794	0.281897	+	0.959445i	$0.409036\pi$
$920$	0	0
$921$	0	0
$922$	−4.38694	−0.144476
$923$	−54.3181	−1.78790
$924$	0	0
$925$	0	0
$926$	−38.7273	−1.27266
$927$	0	0
$928$	−4.00175	−0.131364
$929$	−3.86068	−0.126665	−0.0633323	−	0.997992i	$-0.520173\pi$
$929$	−3.86068	−0.126665	−0.0633323	+	0.997992i	$0.520173\pi$
$930$	0	0
$931$	38.1729	1.25106
$932$	−4.51363	−0.147849
$933$	0	0
$934$	100.447	3.28674
$935$	0	0
$936$	0	0
$937$	7.05231	0.230389	0.115194	−	0.993343i	$-0.463251\pi$
$937$	7.05231	0.230389	0.115194	+	0.993343i	$0.463251\pi$
$938$	8.45451	0.276050
$939$	0	0
$940$	0	0
$941$	11.6549	0.379937	0.189969	−	0.981790i	$-0.439161\pi$
$941$	11.6549	0.379937	0.189969	+	0.981790i	$0.439161\pi$
$942$	0	0
$943$	0.857635	0.0279284
$944$	10.2371	0.333191
$945$	0	0
$946$	55.1794	1.79404
$947$	31.9057	1.03680	0.518398	−	0.855140i	$-0.326529\pi$
$947$	31.9057	1.03680	0.518398	+	0.855140i	$0.326529\pi$
$948$	0	0
$949$	25.4240	0.825297
$950$	0	0
$951$	0	0
$952$	15.2132	0.493064
$953$	−26.3431	−0.853337	−0.426668	−	0.904408i	$-0.640313\pi$
$953$	−26.3431	−0.853337	−0.426668	+	0.904408i	$0.640313\pi$
$954$	0	0
$955$	0	0
$956$	30.9515	1.00104
$957$	0	0
$958$	−36.8651	−1.19106
$959$	−4.73310	−0.152840
$960$	0	0
$961$	−29.0745	−0.937888
$962$	140.253	4.52193
$963$	0	0
$964$	−3.42359	−0.110266
$965$	0	0
$966$	0	0
$967$	−2.32875	−0.0748876	−0.0374438	−	0.999299i	$-0.511922\pi$
$967$	−2.32875	−0.0748876	−0.0374438	+	0.999299i	$0.511922\pi$
$968$	−34.2978	−1.10237
$969$	0	0
$970$	0	0
$971$	43.6099	1.39951	0.699755	−	0.714383i	$-0.253293\pi$
$971$	43.6099	1.39951	0.699755	+	0.714383i	$0.253293\pi$
$972$	0	0
$973$	5.91073	0.189489
$974$	11.9800	0.383864
$975$	0	0
$976$	33.7124	1.07911
$977$	−36.5836	−1.17041	−0.585207	−	0.810884i	$-0.698987\pi$
$977$	−36.5836	−1.17041	−0.585207	+	0.810884i	$0.698987\pi$
$978$	0	0
$979$	−56.0591	−1.79166
$980$	0	0
$981$	0	0
$982$	−106.835	−3.40925
$983$	−26.5522	−0.846885	−0.423442	−	0.905923i	$-0.639178\pi$
$983$	−26.5522	−0.846885	−0.423442	+	0.905923i	$0.639178\pi$
$984$	0	0
$985$	0	0
$986$	−124.775	−3.97364
$987$	0	0
$988$	−114.974	−3.65780
$989$	11.2682	0.358307
$990$	0	0
$991$	−52.9441	−1.68182	−0.840912	−	0.541172i	$-0.817981\pi$
$991$	−52.9441	−1.68182	−0.840912	+	0.541172i	$0.817981\pi$
$992$	−0.585564	−0.0185917
$993$	0	0
$994$	15.4745	0.490820
$995$	0	0
$996$	0	0
$997$	−49.7519	−1.57566	−0.787828	−	0.615895i	$-0.788795\pi$
$997$	−49.7519	−1.57566	−0.787828	+	0.615895i	$0.788795\pi$
$998$	−14.6482	−0.463680
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.ba.1.1	yes	8
3.2	odd	2	inner	5625.2.a.ba.1.8	yes	8
5.4	even	2		5625.2.a.y.1.8	yes	8
15.14	odd	2		5625.2.a.y.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.y.1.1	✓	8	15.14	odd	2
5625.2.a.y.1.8	yes	8	5.4	even	2
5625.2.a.ba.1.1	yes	8	1.1	even	1	trivial
5625.2.a.ba.1.8	yes	8	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.44055 q^{2} +3.95630 q^{4} +0.591023 q^{7} -4.77444 q^{8} +O(q^{10})\)	\(q-2.44055 q^{2} +3.95630 q^{4} +0.591023 q^{7} -4.77444 q^{8} -4.26423 q^{11} +5.06316 q^{13} -1.44242 q^{14} +3.73968 q^{16} -5.39132 q^{17} -5.73968 q^{19} +10.4071 q^{22} +2.12522 q^{23} -12.3569 q^{26} +2.33826 q^{28} -9.48296 q^{29} -1.38761 q^{31} +0.421994 q^{32} +13.1578 q^{34} -11.3502 q^{37} +14.0080 q^{38} +0.403551 q^{41} +5.30211 q^{43} -16.8705 q^{44} -5.18672 q^{46} +8.60289 q^{47} -6.65069 q^{49} +20.0314 q^{52} +0.337629 q^{53} -2.82180 q^{56} +23.1437 q^{58} +2.73744 q^{59} +9.01478 q^{61} +3.38654 q^{62} -8.50926 q^{64} -5.86133 q^{67} -21.3296 q^{68} -10.7281 q^{71} +5.02136 q^{73} +27.7007 q^{74} -22.7079 q^{76} -2.52026 q^{77} +8.44321 q^{79} -0.984886 q^{82} -11.0136 q^{83} -12.9401 q^{86} +20.3593 q^{88} +13.1464 q^{89} +2.99244 q^{91} +8.40801 q^{92} -20.9958 q^{94} +7.24927 q^{97} +16.2314 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) \) 8 * q + 14 * q^4 + 10 * q^7	\( 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} - 18 q^{16} + 2 q^{19} + 20 q^{22} + 10 q^{28} + 4 q^{31} + 50 q^{34} + 50 q^{43} - 30 q^{46} - 6 q^{49} + 30 q^{52} + 60 q^{58} + 46 q^{61} - 14 q^{64} + 40 q^{67} + 50 q^{73} - 34 q^{76} - 12 q^{79} + 60 q^{82} + 70 q^{88} - 10 q^{91} - 20 q^{94} + 50 q^{97}+O(q^{100}) \) 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 - 18 * q^16 + 2 * q^19 + 20 * q^22 + 10 * q^28 + 4 * q^31 + 50 * q^34 + 50 * q^43 - 30 * q^46 - 6 * q^49 + 30 * q^52 + 60 * q^58 + 46 * q^61 - 14 * q^64 + 40 * q^67 + 50 * q^73 - 34 * q^76 - 12 * q^79 + 60 * q^82 + 70 * q^88 - 10 * q^91 - 20 * q^94 + 50 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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