LMFDB - Embedded newform 225.4.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.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:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{10}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 10 $ x^2 - 10
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			1.2
Root			$3.16228$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16228 q^{2} +2.00000 q^{4} +15.0000 q^{7} -18.9737 q^{8} +O(q^{10})$	$q+3.16228 q^{2} +2.00000 q^{4} +15.0000 q^{7} -18.9737 q^{8} +63.2456 q^{11} +35.0000 q^{13} +47.4342 q^{14} -76.0000 q^{16} +88.5438 q^{17} +91.0000 q^{19} +200.000 q^{22} -113.842 q^{23} +110.680 q^{26} +30.0000 q^{28} -63.2456 q^{29} -147.000 q^{31} -88.5438 q^{32} +280.000 q^{34} +370.000 q^{37} +287.767 q^{38} -442.719 q^{41} +335.000 q^{43} +126.491 q^{44} -360.000 q^{46} -177.088 q^{47} -118.000 q^{49} +70.0000 q^{52} +88.5438 q^{53} -284.605 q^{56} -200.000 q^{58} -885.438 q^{59} +427.000 q^{61} -464.855 q^{62} +328.000 q^{64} +15.0000 q^{67} +177.088 q^{68} -63.2456 q^{71} -70.0000 q^{73} +1170.04 q^{74} +182.000 q^{76} +948.683 q^{77} -876.000 q^{79} -1400.00 q^{82} -531.263 q^{83} +1059.36 q^{86} -1200.00 q^{88} +525.000 q^{91} -227.684 q^{92} -560.000 q^{94} -1085.00 q^{97} -373.149 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} + 30 q^{7}+O(q^{10}) $ 2 * q + 4 * q^4 + 30 * q^7	$ 2 q + 4 q^{4} + 30 q^{7} + 70 q^{13} - 152 q^{16} + 182 q^{19} + 400 q^{22} + 60 q^{28} - 294 q^{31} + 560 q^{34} + 740 q^{37} + 670 q^{43} - 720 q^{46} - 236 q^{49} + 140 q^{52} - 400 q^{58} + 854 q^{61} + 656 q^{64} + 30 q^{67} - 140 q^{73} + 364 q^{76} - 1752 q^{79} - 2800 q^{82} - 2400 q^{88} + 1050 q^{91} - 1120 q^{94} - 2170 q^{97}+O(q^{100}) $ 2 * q + 4 * q^4 + 30 * q^7 + 70 * q^13 - 152 * q^16 + 182 * q^19 + 400 * q^22 + 60 * q^28 - 294 * q^31 + 560 * q^34 + 740 * q^37 + 670 * q^43 - 720 * q^46 - 236 * q^49 + 140 * q^52 - 400 * q^58 + 854 * q^61 + 656 * q^64 + 30 * q^67 - 140 * q^73 + 364 * q^76 - 1752 * q^79 - 2800 * q^82 - 2400 * q^88 + 1050 * q^91 - 1120 * q^94 - 2170 * 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 $

$2$	3.16228	1.11803	0.559017	−	0.829156i	$-0.311179\pi$
$2$	3.16228	1.11803	0.559017	+	0.829156i	$0.311179\pi$
$3$	0	0
$3$	0	0
$4$	2.00000	0.250000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	15.0000	0.809924	0.404962	−	0.914334i	$-0.367285\pi$
$7$	15.0000	0.809924	0.404962	+	0.914334i	$0.367285\pi$
$8$	−18.9737	−0.838525
$9$	0	0
$10$	0	0
$11$	63.2456	1.73357	0.866784	−	0.498683i	$-0.166183\pi$
$11$	63.2456	1.73357	0.866784	+	0.498683i	$0.166183\pi$
$12$	0	0
$13$	35.0000	0.746712	0.373356	−	0.927688i	$-0.378207\pi$
$13$	35.0000	0.746712	0.373356	+	0.927688i	$0.378207\pi$
$14$	47.4342	0.905522
$15$	0	0
$16$	−76.0000	−1.18750
$17$	88.5438	1.26324	0.631618	−	0.775280i	$-0.282391\pi$
$17$	88.5438	1.26324	0.631618	+	0.775280i	$0.282391\pi$
$18$	0	0
$19$	91.0000	1.09878	0.549390	−	0.835566i	$-0.314860\pi$
$19$	91.0000	1.09878	0.549390	+	0.835566i	$0.314860\pi$
$20$	0	0
$21$	0	0
$22$	200.000	1.93819
$23$	−113.842	−1.03207	−0.516037	−	0.856566i	$-0.672593\pi$
$23$	−113.842	−1.03207	−0.516037	+	0.856566i	$0.672593\pi$
$24$	0	0
$25$	0	0
$26$	110.680	0.834849
$27$	0	0
$28$	30.0000	0.202481
$29$	−63.2456	−0.404979	−0.202490	−	0.979284i	$-0.564903\pi$
$29$	−63.2456	−0.404979	−0.202490	+	0.979284i	$0.564903\pi$
$30$	0	0
$31$	−147.000	−0.851677	−0.425838	−	0.904799i	$-0.640021\pi$
$31$	−147.000	−0.851677	−0.425838	+	0.904799i	$0.640021\pi$
$32$	−88.5438	−0.489140
$33$	0	0
$34$	280.000	1.41234
$35$	0	0
$36$	0	0
$37$	370.000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	370.000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	287.767	1.22847
$39$	0	0
$40$	0	0
$41$	−442.719	−1.68637	−0.843184	−	0.537625i	$-0.819321\pi$
$41$	−442.719	−1.68637	−0.843184	+	0.537625i	$0.819321\pi$
$42$	0	0
$43$	335.000	1.18807	0.594035	−	0.804439i	$-0.297534\pi$
$43$	335.000	1.18807	0.594035	+	0.804439i	$0.297534\pi$
$44$	126.491	0.433392
$45$	0	0
$46$	−360.000	−1.15389
$47$	−177.088	−0.549593	−0.274797	−	0.961502i	$-0.588610\pi$
$47$	−177.088	−0.549593	−0.274797	+	0.961502i	$0.588610\pi$
$48$	0	0
$49$	−118.000	−0.344023
$50$	0	0
$51$	0	0
$52$	70.0000	0.186678
$53$	88.5438	0.229480	0.114740	−	0.993396i	$-0.463397\pi$
$53$	88.5438	0.229480	0.114740	+	0.993396i	$0.463397\pi$
$54$	0	0
$55$	0	0
$56$	−284.605	−0.679142
$57$	0	0
$58$	−200.000	−0.452781
$59$	−885.438	−1.95380	−0.976900	−	0.213698i	$-0.931449\pi$
$59$	−885.438	−1.95380	−0.976900	+	0.213698i	$0.931449\pi$
$60$	0	0
$61$	427.000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	427.000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−464.855	−0.952204
$63$	0	0
$64$	328.000	0.640625
$65$	0	0
$66$	0	0
$67$	15.0000	0.0273514	0.0136757	−	0.999906i	$-0.495647\pi$
$67$	15.0000	0.0273514	0.0136757	+	0.999906i	$0.495647\pi$
$68$	177.088	0.315809
$69$	0	0
$70$	0	0
$71$	−63.2456	−0.105716	−0.0528582	−	0.998602i	$-0.516833\pi$
$71$	−63.2456	−0.105716	−0.0528582	+	0.998602i	$0.516833\pi$
$72$	0	0
$73$	−70.0000	−0.112231	−0.0561156	−	0.998424i	$-0.517872\pi$
$73$	−70.0000	−0.112231	−0.0561156	+	0.998424i	$0.517872\pi$
$74$	1170.04	1.83804
$75$	0	0
$76$	182.000	0.274695
$77$	948.683	1.40406
$78$	0	0
$79$	−876.000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$79$	−876.000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$80$	0	0
$81$	0	0
$82$	−1400.00	−1.88542
$83$	−531.263	−0.702574	−0.351287	−	0.936268i	$-0.614256\pi$
$83$	−531.263	−0.702574	−0.351287	+	0.936268i	$0.614256\pi$
$84$	0	0
$85$	0	0
$86$	1059.36	1.32830
$87$	0	0
$88$	−1200.00	−1.45364
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	525.000	0.604780
$92$	−227.684	−0.258018
$93$	0	0
$94$	−560.000	−0.614464
$95$	0	0
$96$	0	0
$97$	−1085.00	−1.13572	−0.567861	−	0.823124i	$-0.692229\pi$
$97$	−1085.00	−1.13572	−0.567861	+	0.823124i	$0.692229\pi$
$98$	−373.149	−0.384630
$99$	0	0
$100$	0	0
$101$	1328.16	1.30848	0.654240	−	0.756287i	$-0.272989\pi$
$101$	1328.16	1.30848	0.654240	+	0.756287i	$0.272989\pi$
$102$	0	0
$103$	1540.00	1.47321	0.736605	−	0.676323i	$-0.236428\pi$
$103$	1540.00	1.47321	0.736605	+	0.676323i	$0.236428\pi$
$104$	−664.078	−0.626137
$105$	0	0
$106$	280.000	0.256566
$107$	796.894	0.719987	0.359994	−	0.932955i	$-0.382779\pi$
$107$	796.894	0.719987	0.359994	+	0.932955i	$0.382779\pi$
$108$	0	0
$109$	−811.000	−0.712658	−0.356329	−	0.934361i	$-0.615972\pi$
$109$	−811.000	−0.712658	−0.356329	+	0.934361i	$0.615972\pi$
$110$	0	0
$111$	0	0
$112$	−1140.00	−0.961785
$113$	−1619.09	−1.34788	−0.673942	−	0.738785i	$-0.735400\pi$
$113$	−1619.09	−1.34788	−0.673942	+	0.738785i	$0.735400\pi$
$114$	0	0
$115$	0	0
$116$	−126.491	−0.101245
$117$	0	0
$118$	−2800.00	−2.18441
$119$	1328.16	1.02313
$120$	0	0
$121$	2669.00	2.00526
$122$	1350.29	1.00205
$123$	0	0
$124$	−294.000	−0.212919
$125$	0	0
$126$	0	0
$127$	260.000	0.181664	0.0908318	−	0.995866i	$-0.471047\pi$
$127$	260.000	0.181664	0.0908318	+	0.995866i	$0.471047\pi$
$128$	1745.58	1.20538
$129$	0	0
$130$	0	0
$131$	−1328.16	−0.885814	−0.442907	−	0.896568i	$-0.646053\pi$
$131$	−1328.16	−0.885814	−0.442907	+	0.896568i	$0.646053\pi$
$132$	0	0
$133$	1365.00	0.889929
$134$	47.4342	0.0305798
$135$	0	0
$136$	−1680.00	−1.05926
$137$	−2656.31	−1.65653	−0.828263	−	0.560339i	$-0.810671\pi$
$137$	−2656.31	−1.65653	−0.828263	+	0.560339i	$0.810671\pi$
$138$	0	0
$139$	−784.000	−0.478403	−0.239201	−	0.970970i	$-0.576886\pi$
$139$	−784.000	−0.478403	−0.239201	+	0.970970i	$0.576886\pi$
$140$	0	0
$141$	0	0
$142$	−200.000	−0.118195
$143$	2213.59	1.29448
$144$	0	0
$145$	0	0
$146$	−221.359	−0.125478
$147$	0	0
$148$	740.000	0.410997
$149$	−2150.35	−1.18230	−0.591152	−	0.806560i	$-0.701327\pi$
$149$	−2150.35	−1.18230	−0.591152	+	0.806560i	$0.701327\pi$
$150$	0	0
$151$	−797.000	−0.429529	−0.214765	−	0.976666i	$-0.568898\pi$
$151$	−797.000	−0.429529	−0.214765	+	0.976666i	$0.568898\pi$
$152$	−1726.60	−0.921356
$153$	0	0
$154$	3000.00	1.56978
$155$	0	0
$156$	0	0
$157$	1225.00	0.622711	0.311356	−	0.950293i	$-0.399217\pi$
$157$	1225.00	0.622711	0.311356	+	0.950293i	$0.399217\pi$
$158$	−2770.16	−1.39482
$159$	0	0
$160$	0	0
$161$	−1707.63	−0.835901
$162$	0	0
$163$	−275.000	−0.132145	−0.0660726	−	0.997815i	$-0.521047\pi$
$163$	−275.000	−0.132145	−0.0660726	+	0.997815i	$0.521047\pi$
$164$	−885.438	−0.421592
$165$	0	0
$166$	−1680.00	−0.785502
$167$	265.631	0.123085	0.0615424	−	0.998104i	$-0.480398\pi$
$167$	265.631	0.123085	0.0615424	+	0.998104i	$0.480398\pi$
$168$	0	0
$169$	−972.000	−0.442421
$170$	0	0
$171$	0	0
$172$	670.000	0.297018
$173$	3984.47	1.75106	0.875531	−	0.483162i	$-0.160512\pi$
$173$	3984.47	1.75106	0.875531	+	0.483162i	$0.160512\pi$
$174$	0	0
$175$	0	0
$176$	−4806.66	−2.05861
$177$	0	0
$178$	0	0
$179$	1770.88	0.739449	0.369725	−	0.929141i	$-0.379452\pi$
$179$	1770.88	0.739449	0.369725	+	0.929141i	$0.379452\pi$
$180$	0	0
$181$	−1687.00	−0.692783	−0.346391	−	0.938090i	$-0.612593\pi$
$181$	−1687.00	−0.692783	−0.346391	+	0.938090i	$0.612593\pi$
$182$	1660.20	0.676164
$183$	0	0
$184$	2160.00	0.865420
$185$	0	0
$186$	0	0
$187$	5600.00	2.18991
$188$	−354.175	−0.137398
$189$	0	0
$190$	0	0
$191$	2213.59	0.838587	0.419293	−	0.907851i	$-0.362278\pi$
$191$	2213.59	0.838587	0.419293	+	0.907851i	$0.362278\pi$
$192$	0	0
$193$	3625.00	1.35199	0.675993	−	0.736908i	$-0.263715\pi$
$193$	3625.00	1.35199	0.675993	+	0.736908i	$0.263715\pi$
$194$	−3431.07	−1.26978
$195$	0	0
$196$	−236.000	−0.0860058
$197$	−1745.58	−0.631306	−0.315653	−	0.948875i	$-0.602224\pi$
$197$	−1745.58	−0.631306	−0.315653	+	0.948875i	$0.602224\pi$
$198$	0	0
$199$	3199.00	1.13955	0.569777	−	0.821800i	$-0.307030\pi$
$199$	3199.00	1.13955	0.569777	+	0.821800i	$0.307030\pi$
$200$	0	0
$201$	0	0
$202$	4200.00	1.46293
$203$	−948.683	−0.328003
$204$	0	0
$205$	0	0
$206$	4869.91	1.64710
$207$	0	0
$208$	−2660.00	−0.886720
$209$	5755.35	1.90481
$210$	0	0
$211$	887.000	0.289401	0.144700	−	0.989476i	$-0.453778\pi$
$211$	887.000	0.289401	0.144700	+	0.989476i	$0.453778\pi$
$212$	177.088	0.0573699
$213$	0	0
$214$	2520.00	0.804970
$215$	0	0
$216$	0	0
$217$	−2205.00	−0.689793
$218$	−2564.61	−0.796776
$219$	0	0
$220$	0	0
$221$	3099.03	0.943274
$222$	0	0
$223$	3535.00	1.06153	0.530765	−	0.847519i	$-0.321905\pi$
$223$	3535.00	1.06153	0.530765	+	0.847519i	$0.321905\pi$
$224$	−1328.16	−0.396166
$225$	0	0
$226$	−5120.00	−1.50698
$227$	−88.5438	−0.0258892	−0.0129446	−	0.999916i	$-0.504121\pi$
$227$	−88.5438	−0.0258892	−0.0129446	+	0.999916i	$0.504121\pi$
$228$	0	0
$229$	−5019.00	−1.44832	−0.724159	−	0.689633i	$-0.757772\pi$
$229$	−5019.00	−1.44832	−0.724159	+	0.689633i	$0.757772\pi$
$230$	0	0
$231$	0	0
$232$	1200.00	0.339586
$233$	−3605.00	−1.01361	−0.506805	−	0.862061i	$-0.669174\pi$
$233$	−3605.00	−1.01361	−0.506805	+	0.862061i	$0.669174\pi$
$234$	0	0
$235$	0	0
$236$	−1770.88	−0.488450
$237$	0	0
$238$	4200.00	1.14389
$239$	442.719	0.119821	0.0599103	−	0.998204i	$-0.480919\pi$
$239$	442.719	0.119821	0.0599103	+	0.998204i	$0.480919\pi$
$240$	0	0
$241$	−623.000	−0.166518	−0.0832592	−	0.996528i	$-0.526533\pi$
$241$	−623.000	−0.166518	−0.0832592	+	0.996528i	$0.526533\pi$
$242$	8440.12	2.24195
$243$	0	0
$244$	854.000	0.224065
$245$	0	0
$246$	0	0
$247$	3185.00	0.820472
$248$	2789.13	0.714153
$249$	0	0
$250$	0	0
$251$	2656.31	0.667988	0.333994	−	0.942575i	$-0.391603\pi$
$251$	2656.31	0.667988	0.333994	+	0.942575i	$0.391603\pi$
$252$	0	0
$253$	−7200.00	−1.78917
$254$	822.192	0.203106
$255$	0	0
$256$	2896.00	0.707031
$257$	3718.84	0.902626	0.451313	−	0.892366i	$-0.350956\pi$
$257$	3718.84	0.902626	0.451313	+	0.892366i	$0.350956\pi$
$258$	0	0
$259$	5550.00	1.33151
$260$	0	0
$261$	0	0
$262$	−4200.00	−0.990370
$263$	−2896.65	−0.679144	−0.339572	−	0.940580i	$-0.610282\pi$
$263$	−2896.65	−0.679144	−0.339572	+	0.940580i	$0.610282\pi$
$264$	0	0
$265$	0	0
$266$	4316.51	0.994970
$267$	0	0
$268$	30.0000	0.00683784
$269$	442.719	0.100346	0.0501729	−	0.998741i	$-0.484023\pi$
$269$	442.719	0.100346	0.0501729	+	0.998741i	$0.484023\pi$
$270$	0	0
$271$	−308.000	−0.0690394	−0.0345197	−	0.999404i	$-0.510990\pi$
$271$	−308.000	−0.0690394	−0.0345197	+	0.999404i	$0.510990\pi$
$272$	−6729.33	−1.50009
$273$	0	0
$274$	−8400.00	−1.85205
$275$	0	0
$276$	0	0
$277$	−7485.00	−1.62357	−0.811787	−	0.583953i	$-0.801505\pi$
$277$	−7485.00	−1.62357	−0.811787	+	0.583953i	$0.801505\pi$
$278$	−2479.23	−0.534871
$279$	0	0
$280$	0	0
$281$	−948.683	−0.201401	−0.100701	−	0.994917i	$-0.532108\pi$
$281$	−948.683	−0.201401	−0.100701	+	0.994917i	$0.532108\pi$
$282$	0	0
$283$	525.000	0.110276	0.0551378	−	0.998479i	$-0.482440\pi$
$283$	525.000	0.110276	0.0551378	+	0.998479i	$0.482440\pi$
$284$	−126.491	−0.0264291
$285$	0	0
$286$	7000.00	1.44727
$287$	−6640.78	−1.36583
$288$	0	0
$289$	2927.00	0.595766
$290$	0	0
$291$	0	0
$292$	−140.000	−0.0280578
$293$	6375.15	1.27113	0.635564	−	0.772048i	$-0.280768\pi$
$293$	6375.15	1.27113	0.635564	+	0.772048i	$0.280768\pi$
$294$	0	0
$295$	0	0
$296$	−7020.26	−1.37853
$297$	0	0
$298$	−6800.00	−1.32186
$299$	−3984.47	−0.770662
$300$	0	0
$301$	5025.00	0.962246
$302$	−2520.34	−0.480228
$303$	0	0
$304$	−6916.00	−1.30480
$305$	0	0
$306$	0	0
$307$	−2695.00	−0.501016	−0.250508	−	0.968115i	$-0.580598\pi$
$307$	−2695.00	−0.501016	−0.250508	+	0.968115i	$0.580598\pi$
$308$	1897.37	0.351015
$309$	0	0
$310$	0	0
$311$	−5312.63	−0.968654	−0.484327	−	0.874887i	$-0.660936\pi$
$311$	−5312.63	−0.968654	−0.484327	+	0.874887i	$0.660936\pi$
$312$	0	0
$313$	−2555.00	−0.461397	−0.230698	−	0.973025i	$-0.574101\pi$
$313$	−2555.00	−0.461397	−0.230698	+	0.973025i	$0.574101\pi$
$314$	3873.79	0.696212
$315$	0	0
$316$	−1752.00	−0.311891
$317$	7462.98	1.32228	0.661140	−	0.750263i	$-0.270073\pi$
$317$	7462.98	1.32228	0.661140	+	0.750263i	$0.270073\pi$
$318$	0	0
$319$	−4000.00	−0.702060
$320$	0	0
$321$	0	0
$322$	−5400.00	−0.934566
$323$	8057.48	1.38802
$324$	0	0
$325$	0	0
$326$	−869.626	−0.147743
$327$	0	0
$328$	8400.00	1.41406
$329$	−2656.31	−0.445129
$330$	0	0
$331$	−5088.00	−0.844900	−0.422450	−	0.906386i	$-0.638830\pi$
$331$	−5088.00	−0.844900	−0.422450	+	0.906386i	$0.638830\pi$
$332$	−1062.53	−0.175644
$333$	0	0
$334$	840.000	0.137613
$335$	0	0
$336$	0	0
$337$	−11485.0	−1.85646	−0.928231	−	0.372004i	$-0.878671\pi$
$337$	−11485.0	−1.85646	−0.928231	+	0.372004i	$0.878671\pi$
$338$	−3073.73	−0.494642
$339$	0	0
$340$	0	0
$341$	−9297.10	−1.47644
$342$	0	0
$343$	−6915.00	−1.08856
$344$	−6356.18	−0.996227
$345$	0	0
$346$	12600.0	1.95775
$347$	−6400.45	−0.990185	−0.495092	−	0.868840i	$-0.664866\pi$
$347$	−6400.45	−0.990185	−0.495092	+	0.868840i	$0.664866\pi$
$348$	0	0
$349$	−9674.00	−1.48377	−0.741887	−	0.670525i	$-0.766069\pi$
$349$	−9674.00	−1.48377	−0.741887	+	0.670525i	$0.766069\pi$
$350$	0	0
$351$	0	0
$352$	−5600.00	−0.847957
$353$	−5578.26	−0.841078	−0.420539	−	0.907274i	$-0.638159\pi$
$353$	−5578.26	−0.841078	−0.420539	+	0.907274i	$0.638159\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	5600.00	0.826730
$359$	−8917.62	−1.31101	−0.655507	−	0.755189i	$-0.727545\pi$
$359$	−8917.62	−1.31101	−0.655507	+	0.755189i	$0.727545\pi$
$360$	0	0
$361$	1422.00	0.207319
$362$	−5334.76	−0.774555
$363$	0	0
$364$	1050.00	0.151195
$365$	0	0
$366$	0	0
$367$	8505.00	1.20969	0.604847	−	0.796342i	$-0.293234\pi$
$367$	8505.00	1.20969	0.604847	+	0.796342i	$0.293234\pi$
$368$	8651.99	1.22559
$369$	0	0
$370$	0	0
$371$	1328.16	0.185861
$372$	0	0
$373$	−6775.00	−0.940472	−0.470236	−	0.882541i	$-0.655831\pi$
$373$	−6775.00	−0.940472	−0.470236	+	0.882541i	$0.655831\pi$
$374$	17708.8	2.44839
$375$	0	0
$376$	3360.00	0.460848
$377$	−2213.59	−0.302403
$378$	0	0
$379$	9241.00	1.25245	0.626225	−	0.779643i	$-0.284599\pi$
$379$	9241.00	1.25245	0.626225	+	0.779643i	$0.284599\pi$
$380$	0	0
$381$	0	0
$382$	7000.00	0.937568
$383$	−9031.46	−1.20493	−0.602463	−	0.798147i	$-0.705814\pi$
$383$	−9031.46	−1.20493	−0.602463	+	0.798147i	$0.705814\pi$
$384$	0	0
$385$	0	0
$386$	11463.3	1.51157
$387$	0	0
$388$	−2170.00	−0.283931
$389$	−4363.94	−0.568794	−0.284397	−	0.958707i	$-0.591793\pi$
$389$	−4363.94	−0.568794	−0.284397	+	0.958707i	$0.591793\pi$
$390$	0	0
$391$	−10080.0	−1.30375
$392$	2238.89	0.288472
$393$	0	0
$394$	−5520.00	−0.705821
$395$	0	0
$396$	0	0
$397$	−4795.00	−0.606182	−0.303091	−	0.952962i	$-0.598019\pi$
$397$	−4795.00	−0.606182	−0.303091	+	0.952962i	$0.598019\pi$
$398$	10116.1	1.27406
$399$	0	0
$400$	0	0
$401$	7020.26	0.874252	0.437126	−	0.899400i	$-0.355996\pi$
$401$	7020.26	0.874252	0.437126	+	0.899400i	$0.355996\pi$
$402$	0	0
$403$	−5145.00	−0.635957
$404$	2656.31	0.327120
$405$	0	0
$406$	−3000.00	−0.366718
$407$	23400.9	2.84997
$408$	0	0
$409$	6881.00	0.831891	0.415946	−	0.909389i	$-0.363451\pi$
$409$	6881.00	0.831891	0.415946	+	0.909389i	$0.363451\pi$
$410$	0	0
$411$	0	0
$412$	3080.00	0.368303
$413$	−13281.6	−1.58243
$414$	0	0
$415$	0	0
$416$	−3099.03	−0.365247
$417$	0	0
$418$	18200.0	2.12964
$419$	12838.8	1.49694	0.748471	−	0.663167i	$-0.230788\pi$
$419$	12838.8	1.49694	0.748471	+	0.663167i	$0.230788\pi$
$420$	0	0
$421$	6418.00	0.742979	0.371490	−	0.928437i	$-0.378847\pi$
$421$	6418.00	0.742979	0.371490	+	0.928437i	$0.378847\pi$
$422$	2804.94	0.323560
$423$	0	0
$424$	−1680.00	−0.192425
$425$	0	0
$426$	0	0
$427$	6405.00	0.725901
$428$	1593.79	0.179997
$429$	0	0
$430$	0	0
$431$	1328.16	0.148434	0.0742170	−	0.997242i	$-0.476354\pi$
$431$	1328.16	0.148434	0.0742170	+	0.997242i	$0.476354\pi$
$432$	0	0
$433$	−8155.00	−0.905091	−0.452545	−	0.891741i	$-0.649484\pi$
$433$	−8155.00	−0.905091	−0.452545	+	0.891741i	$0.649484\pi$
$434$	−6972.82	−0.771212
$435$	0	0
$436$	−1622.00	−0.178164
$437$	−10359.6	−1.13402
$438$	0	0
$439$	14749.0	1.60349	0.801744	−	0.597667i	$-0.203906\pi$
$439$	14749.0	1.60349	0.801744	+	0.597667i	$0.203906\pi$
$440$	0	0
$441$	0	0
$442$	9800.00	1.05461
$443$	−11156.5	−1.19653	−0.598264	−	0.801299i	$-0.704143\pi$
$443$	−11156.5	−1.19653	−0.598264	+	0.801299i	$0.704143\pi$
$444$	0	0
$445$	0	0
$446$	11178.7	1.18683
$447$	0	0
$448$	4920.00	0.518857
$449$	−9233.85	−0.970540	−0.485270	−	0.874364i	$-0.661279\pi$
$449$	−9233.85	−0.970540	−0.485270	+	0.874364i	$0.661279\pi$
$450$	0	0
$451$	−28000.0	−2.92343
$452$	−3238.17	−0.336971
$453$	0	0
$454$	−280.000	−0.0289450
$455$	0	0
$456$	0	0
$457$	−1030.00	−0.105430	−0.0527148	−	0.998610i	$-0.516787\pi$
$457$	−1030.00	−0.105430	−0.0527148	+	0.998610i	$0.516787\pi$
$458$	−15871.5	−1.61927
$459$	0	0
$460$	0	0
$461$	9297.10	0.939282	0.469641	−	0.882858i	$-0.344383\pi$
$461$	9297.10	0.939282	0.469641	+	0.882858i	$0.344383\pi$
$462$	0	0
$463$	11940.0	1.19849	0.599243	−	0.800567i	$-0.295468\pi$
$463$	11940.0	1.19849	0.599243	+	0.800567i	$0.295468\pi$
$464$	4806.66	0.480913
$465$	0	0
$466$	−11400.0	−1.13325
$467$	12130.5	1.20200	0.600998	−	0.799250i	$-0.294770\pi$
$467$	12130.5	1.20200	0.600998	+	0.799250i	$0.294770\pi$
$468$	0	0
$469$	225.000	0.0221525
$470$	0	0
$471$	0	0
$472$	16800.0	1.63831
$473$	21187.3	2.05960
$474$	0	0
$475$	0	0
$476$	2656.31	0.255781
$477$	0	0
$478$	1400.00	0.133963
$479$	−2656.31	−0.253382	−0.126691	−	0.991942i	$-0.540436\pi$
$479$	−2656.31	−0.253382	−0.126691	+	0.991942i	$0.540436\pi$
$480$	0	0
$481$	12950.0	1.22759
$482$	−1970.10	−0.186173
$483$	0	0
$484$	5338.00	0.501315
$485$	0	0
$486$	0	0
$487$	14125.0	1.31430	0.657151	−	0.753759i	$-0.271761\pi$
$487$	14125.0	1.31430	0.657151	+	0.753759i	$0.271761\pi$
$488$	−8101.76	−0.751535
$489$	0	0
$490$	0	0
$491$	−4047.72	−0.372038	−0.186019	−	0.982546i	$-0.559559\pi$
$491$	−4047.72	−0.372038	−0.186019	+	0.982546i	$0.559559\pi$
$492$	0	0
$493$	−5600.00	−0.511585
$494$	10071.9	0.917316
$495$	0	0
$496$	11172.0	1.01137
$497$	−948.683	−0.0856223
$498$	0	0
$499$	1279.00	0.114741	0.0573706	−	0.998353i	$-0.481728\pi$
$499$	1279.00	0.114741	0.0573706	+	0.998353i	$0.481728\pi$
$500$	0	0
$501$	0	0
$502$	8400.00	0.746833
$503$	16380.6	1.45204	0.726019	−	0.687675i	$-0.241369\pi$
$503$	16380.6	1.45204	0.726019	+	0.687675i	$0.241369\pi$
$504$	0	0
$505$	0	0
$506$	−22768.4	−2.00035
$507$	0	0
$508$	520.000	0.0454159
$509$	4427.19	0.385524	0.192762	−	0.981246i	$-0.438255\pi$
$509$	4427.19	0.385524	0.192762	+	0.981246i	$0.438255\pi$
$510$	0	0
$511$	−1050.00	−0.0908988
$512$	−4806.66	−0.414895
$513$	0	0
$514$	11760.0	1.00917
$515$	0	0
$516$	0	0
$517$	−11200.0	−0.952757
$518$	17550.6	1.48867
$519$	0	0
$520$	0	0
$521$	−11068.0	−0.930704	−0.465352	−	0.885126i	$-0.654072\pi$
$521$	−11068.0	−0.930704	−0.465352	+	0.885126i	$0.654072\pi$
$522$	0	0
$523$	10745.0	0.898367	0.449184	−	0.893439i	$-0.351715\pi$
$523$	10745.0	0.898367	0.449184	+	0.893439i	$0.351715\pi$
$524$	−2656.31	−0.221453
$525$	0	0
$526$	−9160.00	−0.759306
$527$	−13015.9	−1.07587
$528$	0	0
$529$	793.000	0.0651763
$530$	0	0
$531$	0	0
$532$	2730.00	0.222482
$533$	−15495.2	−1.25923
$534$	0	0
$535$	0	0
$536$	−284.605	−0.0229348
$537$	0	0
$538$	1400.00	0.112190
$539$	−7462.98	−0.596388
$540$	0	0
$541$	−9423.00	−0.748847	−0.374424	−	0.927258i	$-0.622159\pi$
$541$	−9423.00	−0.748847	−0.374424	+	0.927258i	$0.622159\pi$
$542$	−973.982	−0.0771884
$543$	0	0
$544$	−7840.00	−0.617899
$545$	0	0
$546$	0	0
$547$	9080.00	0.709749	0.354875	−	0.934914i	$-0.384524\pi$
$547$	9080.00	0.709749	0.354875	+	0.934914i	$0.384524\pi$
$548$	−5312.63	−0.414132
$549$	0	0
$550$	0	0
$551$	−5755.35	−0.444984
$552$	0	0
$553$	−13140.0	−1.01043
$554$	−23669.6	−1.81521
$555$	0	0
$556$	−1568.00	−0.119601
$557$	−8348.41	−0.635069	−0.317535	−	0.948247i	$-0.602855\pi$
$557$	−8348.41	−0.635069	−0.317535	+	0.948247i	$0.602855\pi$
$558$	0	0
$559$	11725.0	0.887146
$560$	0	0
$561$	0	0
$562$	−3000.00	−0.225173
$563$	7703.31	0.576653	0.288327	−	0.957532i	$-0.406901\pi$
$563$	7703.31	0.576653	0.288327	+	0.957532i	$0.406901\pi$
$564$	0	0
$565$	0	0
$566$	1660.20	0.123292
$567$	0	0
$568$	1200.00	0.0886459
$569$	−6261.31	−0.461314	−0.230657	−	0.973035i	$-0.574088\pi$
$569$	−6261.31	−0.461314	−0.230657	+	0.973035i	$0.574088\pi$
$570$	0	0
$571$	587.000	0.0430213	0.0215107	−	0.999769i	$-0.493152\pi$
$571$	587.000	0.0430213	0.0215107	+	0.999769i	$0.493152\pi$
$572$	4427.19	0.323619
$573$	0	0
$574$	−21000.0	−1.52704
$575$	0	0
$576$	0	0
$577$	−15995.0	−1.15404	−0.577020	−	0.816730i	$-0.695784\pi$
$577$	−15995.0	−1.15404	−0.577020	+	0.816730i	$0.695784\pi$
$578$	9255.99	0.666087
$579$	0	0
$580$	0	0
$581$	−7968.94	−0.569032
$582$	0	0
$583$	5600.00	0.397819
$584$	1328.16	0.0941088
$585$	0	0
$586$	20160.0	1.42116
$587$	−20453.6	−1.43818	−0.719089	−	0.694918i	$-0.755441\pi$
$587$	−20453.6	−1.43818	−0.719089	+	0.694918i	$0.755441\pi$
$588$	0	0
$589$	−13377.0	−0.935806
$590$	0	0
$591$	0	0
$592$	−28120.0	−1.95224
$593$	11599.2	0.803244	0.401622	−	0.915806i	$-0.368447\pi$
$593$	11599.2	0.803244	0.401622	+	0.915806i	$0.368447\pi$
$594$	0	0
$595$	0	0
$596$	−4300.70	−0.295576
$597$	0	0
$598$	−12600.0	−0.861626
$599$	19100.2	1.30286	0.651428	−	0.758710i	$-0.274170\pi$
$599$	19100.2	1.30286	0.651428	+	0.758710i	$0.274170\pi$
$600$	0	0
$601$	3143.00	0.213321	0.106660	−	0.994296i	$-0.465984\pi$
$601$	3143.00	0.213321	0.106660	+	0.994296i	$0.465984\pi$
$602$	15890.4	1.07582
$603$	0	0
$604$	−1594.00	−0.107382
$605$	0	0
$606$	0	0
$607$	2660.00	0.177868	0.0889342	−	0.996038i	$-0.471654\pi$
$607$	2660.00	0.177868	0.0889342	+	0.996038i	$0.471654\pi$
$608$	−8057.48	−0.537457
$609$	0	0
$610$	0	0
$611$	−6198.06	−0.410388
$612$	0	0
$613$	−6670.00	−0.439476	−0.219738	−	0.975559i	$-0.570520\pi$
$613$	−6670.00	−0.439476	−0.219738	+	0.975559i	$0.570520\pi$
$614$	−8522.34	−0.560152
$615$	0	0
$616$	−18000.0	−1.17734
$617$	11042.7	0.720521	0.360260	−	0.932852i	$-0.382688\pi$
$617$	11042.7	0.720521	0.360260	+	0.932852i	$0.382688\pi$
$618$	0	0
$619$	5579.00	0.362260	0.181130	−	0.983459i	$-0.442025\pi$
$619$	5579.00	0.362260	0.181130	+	0.983459i	$0.442025\pi$
$620$	0	0
$621$	0	0
$622$	−16800.0	−1.08299
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−8079.62	−0.515857
$627$	0	0
$628$	2450.00	0.155678
$629$	32761.2	2.07675
$630$	0	0
$631$	23617.0	1.48998	0.744990	−	0.667075i	$-0.232454\pi$
$631$	23617.0	1.48998	0.744990	+	0.667075i	$0.232454\pi$
$632$	16620.9	1.04612
$633$	0	0
$634$	23600.0	1.47835
$635$	0	0
$636$	0	0
$637$	−4130.00	−0.256886
$638$	−12649.1	−0.784926
$639$	0	0
$640$	0	0
$641$	31496.3	1.94076	0.970381	−	0.241579i	$-0.0776655\pi$
$641$	31496.3	1.94076	0.970381	+	0.241579i	$0.0776655\pi$
$642$	0	0
$643$	−14280.0	−0.875814	−0.437907	−	0.899020i	$-0.644280\pi$
$643$	−14280.0	−0.875814	−0.437907	+	0.899020i	$0.644280\pi$
$644$	−3415.26	−0.208975
$645$	0	0
$646$	25480.0	1.55185
$647$	−12396.1	−0.753234	−0.376617	−	0.926369i	$-0.622913\pi$
$647$	−12396.1	−0.753234	−0.376617	+	0.926369i	$0.622913\pi$
$648$	0	0
$649$	−56000.0	−3.38705
$650$	0	0
$651$	0	0
$652$	−550.000	−0.0330363
$653$	−12042.0	−0.721651	−0.360825	−	0.932633i	$-0.617505\pi$
$653$	−12042.0	−0.721651	−0.360825	+	0.932633i	$0.617505\pi$
$654$	0	0
$655$	0	0
$656$	33646.6	2.00256
$657$	0	0
$658$	−8400.00	−0.497669
$659$	5755.35	0.340207	0.170104	−	0.985426i	$-0.445590\pi$
$659$	5755.35	0.340207	0.170104	+	0.985426i	$0.445590\pi$
$660$	0	0
$661$	25438.0	1.49686	0.748429	−	0.663215i	$-0.230808\pi$
$661$	25438.0	1.49686	0.748429	+	0.663215i	$0.230808\pi$
$662$	−16089.7	−0.944626
$663$	0	0
$664$	10080.0	0.589126
$665$	0	0
$666$	0	0
$667$	7200.00	0.417969
$668$	531.263	0.0307712
$669$	0	0
$670$	0	0
$671$	27005.9	1.55372
$672$	0	0
$673$	15150.0	0.867741	0.433870	−	0.900975i	$-0.357148\pi$
$673$	15150.0	0.867741	0.433870	+	0.900975i	$0.357148\pi$
$674$	−36318.8	−2.07559
$675$	0	0
$676$	−1944.00	−0.110605
$677$	20984.9	1.19131	0.595653	−	0.803242i	$-0.296893\pi$
$677$	20984.9	1.19131	0.595653	+	0.803242i	$0.296893\pi$
$678$	0	0
$679$	−16275.0	−0.919849
$680$	0	0
$681$	0	0
$682$	−29400.0	−1.65071
$683$	−1973.26	−0.110549	−0.0552743	−	0.998471i	$-0.517603\pi$
$683$	−1973.26	−0.110549	−0.0552743	+	0.998471i	$0.517603\pi$
$684$	0	0
$685$	0	0
$686$	−21867.2	−1.21704
$687$	0	0
$688$	−25460.0	−1.41083
$689$	3099.03	0.171355
$690$	0	0
$691$	−20552.0	−1.13145	−0.565727	−	0.824592i	$-0.691404\pi$
$691$	−20552.0	−1.13145	−0.565727	+	0.824592i	$0.691404\pi$
$692$	7968.94	0.437765
$693$	0	0
$694$	−20240.0	−1.10706
$695$	0	0
$696$	0	0
$697$	−39200.0	−2.13028
$698$	−30591.9	−1.65891
$699$	0	0
$700$	0	0
$701$	−8411.66	−0.453215	−0.226608	−	0.973986i	$-0.572764\pi$
$701$	−8411.66	−0.453215	−0.226608	+	0.973986i	$0.572764\pi$
$702$	0	0
$703$	33670.0	1.80638
$704$	20744.5	1.11057
$705$	0	0
$706$	−17640.0	−0.940354
$707$	19922.3	1.05977
$708$	0	0
$709$	17281.0	0.915376	0.457688	−	0.889113i	$-0.348678\pi$
$709$	17281.0	0.915376	0.457688	+	0.889113i	$0.348678\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16734.8	0.878993
$714$	0	0
$715$	0	0
$716$	3541.75	0.184862
$717$	0	0
$718$	−28200.0	−1.46576
$719$	−9297.10	−0.482230	−0.241115	−	0.970497i	$-0.577513\pi$
$719$	−9297.10	−0.482230	−0.241115	+	0.970497i	$0.577513\pi$
$720$	0	0
$721$	23100.0	1.19319
$722$	4496.76	0.231790
$723$	0	0
$724$	−3374.00	−0.173196
$725$	0	0
$726$	0	0
$727$	30415.0	1.55162	0.775811	−	0.630965i	$-0.217341\pi$
$727$	30415.0	1.55162	0.775811	+	0.630965i	$0.217341\pi$
$728$	−9961.17	−0.507123
$729$	0	0
$730$	0	0
$731$	29662.2	1.50081
$732$	0	0
$733$	18550.0	0.934734	0.467367	−	0.884063i	$-0.345203\pi$
$733$	18550.0	0.934734	0.467367	+	0.884063i	$0.345203\pi$
$734$	26895.2	1.35248
$735$	0	0
$736$	10080.0	0.504828
$737$	948.683	0.0474155
$738$	0	0
$739$	24016.0	1.19546	0.597729	−	0.801699i	$-0.296070\pi$
$739$	24016.0	1.19546	0.597729	+	0.801699i	$0.296070\pi$
$740$	0	0
$741$	0	0
$742$	4200.00	0.207799
$743$	−12396.1	−0.612072	−0.306036	−	0.952020i	$-0.599003\pi$
$743$	−12396.1	−0.612072	−0.306036	+	0.952020i	$0.599003\pi$
$744$	0	0
$745$	0	0
$746$	−21424.4	−1.05148
$747$	0	0
$748$	11200.0	0.547477
$749$	11953.4	0.583135
$750$	0	0
$751$	−7772.00	−0.377636	−0.188818	−	0.982012i	$-0.560466\pi$
$751$	−7772.00	−0.377636	−0.188818	+	0.982012i	$0.560466\pi$
$752$	13458.7	0.652642
$753$	0	0
$754$	−7000.00	−0.338097
$755$	0	0
$756$	0	0
$757$	−3865.00	−0.185569	−0.0927846	−	0.995686i	$-0.529577\pi$
$757$	−3865.00	−0.185569	−0.0927846	+	0.995686i	$0.529577\pi$
$758$	29222.6	1.40028
$759$	0	0
$760$	0	0
$761$	−15937.9	−0.759195	−0.379598	−	0.925152i	$-0.623938\pi$
$761$	−15937.9	−0.759195	−0.379598	+	0.925152i	$0.623938\pi$
$762$	0	0
$763$	−12165.0	−0.577199
$764$	4427.19	0.209647
$765$	0	0
$766$	−28560.0	−1.34715
$767$	−30990.3	−1.45893
$768$	0	0
$769$	−7441.00	−0.348933	−0.174466	−	0.984663i	$-0.555820\pi$
$769$	−7441.00	−0.348933	−0.174466	+	0.984663i	$0.555820\pi$
$770$	0	0
$771$	0	0
$772$	7250.00	0.337996
$773$	25766.2	1.19890	0.599448	−	0.800413i	$-0.295387\pi$
$773$	25766.2	1.19890	0.599448	+	0.800413i	$0.295387\pi$
$774$	0	0
$775$	0	0
$776$	20586.4	0.952332
$777$	0	0
$778$	−13800.0	−0.635931
$779$	−40287.4	−1.85295
$780$	0	0
$781$	−4000.00	−0.183267
$782$	−31875.8	−1.45764
$783$	0	0
$784$	8968.00	0.408528
$785$	0	0
$786$	0	0
$787$	6125.00	0.277424	0.138712	−	0.990333i	$-0.455704\pi$
$787$	6125.00	0.277424	0.138712	+	0.990333i	$0.455704\pi$
$788$	−3491.15	−0.157826
$789$	0	0
$790$	0	0
$791$	−24286.3	−1.09168
$792$	0	0
$793$	14945.0	0.669247
$794$	−15163.1	−0.677732
$795$	0	0
$796$	6398.00	0.284888
$797$	−20010.9	−0.889363	−0.444681	−	0.895689i	$-0.646683\pi$
$797$	−20010.9	−0.889363	−0.444681	+	0.895689i	$0.646683\pi$
$798$	0	0
$799$	−15680.0	−0.694266
$800$	0	0
$801$	0	0
$802$	22200.0	0.977443
$803$	−4427.19	−0.194561
$804$	0	0
$805$	0	0
$806$	−16269.9	−0.711022
$807$	0	0
$808$	−25200.0	−1.09719
$809$	−2719.56	−0.118189	−0.0590943	−	0.998252i	$-0.518821\pi$
$809$	−2719.56	−0.118189	−0.0590943	+	0.998252i	$0.518821\pi$
$810$	0	0
$811$	28623.0	1.23932	0.619661	−	0.784870i	$-0.287270\pi$
$811$	28623.0	1.23932	0.619661	+	0.784870i	$0.287270\pi$
$812$	−1897.37	−0.0820006
$813$	0	0
$814$	74000.0	3.18636
$815$	0	0
$816$	0	0
$817$	30485.0	1.30543
$818$	21759.6	0.930083
$819$	0	0
$820$	0	0
$821$	−26563.1	−1.12918	−0.564592	−	0.825370i	$-0.690966\pi$
$821$	−26563.1	−1.12918	−0.564592	+	0.825370i	$0.690966\pi$
$822$	0	0
$823$	1135.00	0.0480724	0.0240362	−	0.999711i	$-0.492348\pi$
$823$	1135.00	0.0480724	0.0240362	+	0.999711i	$0.492348\pi$
$824$	−29219.4	−1.23532
$825$	0	0
$826$	−42000.0	−1.76921
$827$	999.280	0.0420174	0.0210087	−	0.999779i	$-0.493312\pi$
$827$	999.280	0.0420174	0.0210087	+	0.999779i	$0.493312\pi$
$828$	0	0
$829$	−1974.00	−0.0827019	−0.0413509	−	0.999145i	$-0.513166\pi$
$829$	−1974.00	−0.0827019	−0.0413509	+	0.999145i	$0.513166\pi$
$830$	0	0
$831$	0	0
$832$	11480.0	0.478362
$833$	−10448.2	−0.434583
$834$	0	0
$835$	0	0
$836$	11510.7	0.476203
$837$	0	0
$838$	40600.0	1.67363
$839$	42058.3	1.73065	0.865324	−	0.501213i	$-0.167113\pi$
$839$	42058.3	1.73065	0.865324	+	0.501213i	$0.167113\pi$
$840$	0	0
$841$	−20389.0	−0.835992
$842$	20295.5	0.830676
$843$	0	0
$844$	1774.00	0.0723502
$845$	0	0
$846$	0	0
$847$	40035.0	1.62411
$848$	−6729.33	−0.272507
$849$	0	0
$850$	0	0
$851$	−42121.5	−1.69672
$852$	0	0
$853$	36645.0	1.47093	0.735464	−	0.677564i	$-0.236964\pi$
$853$	36645.0	1.47093	0.735464	+	0.677564i	$0.236964\pi$
$854$	20254.4	0.811582
$855$	0	0
$856$	−15120.0	−0.603728
$857$	−10802.3	−0.430573	−0.215286	−	0.976551i	$-0.569069\pi$
$857$	−10802.3	−0.430573	−0.215286	+	0.976551i	$0.569069\pi$
$858$	0	0
$859$	20104.0	0.798533	0.399266	−	0.916835i	$-0.369265\pi$
$859$	20104.0	0.798533	0.399266	+	0.916835i	$0.369265\pi$
$860$	0	0
$861$	0	0
$862$	4200.00	0.165954
$863$	2150.35	0.0848189	0.0424095	−	0.999100i	$-0.486497\pi$
$863$	2150.35	0.0848189	0.0424095	+	0.999100i	$0.486497\pi$
$864$	0	0
$865$	0	0
$866$	−25788.4	−1.01192
$867$	0	0
$868$	−4410.00	−0.172448
$869$	−55403.1	−2.16274
$870$	0	0
$871$	525.000	0.0204236
$872$	15387.6	0.597582
$873$	0	0
$874$	−32760.0	−1.26788
$875$	0	0
$876$	0	0
$877$	−21975.0	−0.846115	−0.423058	−	0.906103i	$-0.639043\pi$
$877$	−21975.0	−0.846115	−0.423058	+	0.906103i	$0.639043\pi$
$878$	46640.4	1.79275
$879$	0	0
$880$	0	0
$881$	−885.438	−0.0338606	−0.0169303	−	0.999857i	$-0.505389\pi$
$881$	−885.438	−0.0338606	−0.0169303	+	0.999857i	$0.505389\pi$
$882$	0	0
$883$	34915.0	1.33067	0.665336	−	0.746544i	$-0.268288\pi$
$883$	34915.0	1.33067	0.665336	+	0.746544i	$0.268288\pi$
$884$	6198.06	0.235818
$885$	0	0
$886$	−35280.0	−1.33776
$887$	49407.4	1.87028	0.935140	−	0.354277i	$-0.115273\pi$
$887$	49407.4	1.87028	0.935140	+	0.354277i	$0.115273\pi$
$888$	0	0
$889$	3900.00	0.147134
$890$	0	0
$891$	0	0
$892$	7070.00	0.265382
$893$	−16115.0	−0.603882
$894$	0	0
$895$	0	0
$896$	26183.7	0.976266
$897$	0	0
$898$	−29200.0	−1.08510
$899$	9297.10	0.344912
$900$	0	0
$901$	7840.00	0.289887
$902$	−88543.8	−3.26850
$903$	0	0
$904$	30720.0	1.13023
$905$	0	0
$906$	0	0
$907$	17880.0	0.654571	0.327285	−	0.944926i	$-0.393866\pi$
$907$	17880.0	0.654571	0.327285	+	0.944926i	$0.393866\pi$
$908$	−177.088	−0.00647231
$909$	0	0
$910$	0	0
$911$	17329.3	0.630236	0.315118	−	0.949053i	$-0.397956\pi$
$911$	17329.3	0.630236	0.315118	+	0.949053i	$0.397956\pi$
$912$	0	0
$913$	−33600.0	−1.21796
$914$	−3257.15	−0.117874
$915$	0	0
$916$	−10038.0	−0.362080
$917$	−19922.3	−0.717442
$918$	0	0
$919$	−25829.0	−0.927117	−0.463558	−	0.886066i	$-0.653428\pi$
$919$	−25829.0	−0.927117	−0.463558	+	0.886066i	$0.653428\pi$
$920$	0	0
$921$	0	0
$922$	29400.0	1.05015
$923$	−2213.59	−0.0789397
$924$	0	0
$925$	0	0
$926$	37757.6	1.33995
$927$	0	0
$928$	5600.00	0.198092
$929$	−3099.03	−0.109447	−0.0547233	−	0.998502i	$-0.517428\pi$
$929$	−3099.03	−0.109447	−0.0547233	+	0.998502i	$0.517428\pi$
$930$	0	0
$931$	−10738.0	−0.378006
$932$	−7209.99	−0.253403
$933$	0	0
$934$	38360.0	1.34387
$935$	0	0
$936$	0	0
$937$	40005.0	1.39478	0.697389	−	0.716693i	$-0.254345\pi$
$937$	40005.0	1.39478	0.697389	+	0.716693i	$0.254345\pi$
$938$	711.512	0.0247673
$939$	0	0
$940$	0	0
$941$	30104.9	1.04292	0.521462	−	0.853275i	$-0.325387\pi$
$941$	30104.9	1.04292	0.521462	+	0.853275i	$0.325387\pi$
$942$	0	0
$943$	50400.0	1.74046
$944$	67293.3	2.32014
$945$	0	0
$946$	67000.0	2.30270
$947$	43297.9	1.48574	0.742868	−	0.669437i	$-0.233465\pi$
$947$	43297.9	1.48574	0.742868	+	0.669437i	$0.233465\pi$
$948$	0	0
$949$	−2450.00	−0.0838044
$950$	0	0
$951$	0	0
$952$	−25200.0	−0.857917
$953$	−53682.8	−1.82472	−0.912360	−	0.409390i	$-0.865742\pi$
$953$	−53682.8	−1.82472	−0.912360	+	0.409390i	$0.865742\pi$
$954$	0	0
$955$	0	0
$956$	885.438	0.0299551
$957$	0	0
$958$	−8400.00	−0.283290
$959$	−39844.7	−1.34166
$960$	0	0
$961$	−8182.00	−0.274647
$962$	40951.5	1.37248
$963$	0	0
$964$	−1246.00	−0.0416296
$965$	0	0
$966$	0	0
$967$	−20540.0	−0.683063	−0.341531	−	0.939870i	$-0.610946\pi$
$967$	−20540.0	−0.683063	−0.341531	+	0.939870i	$0.610946\pi$
$968$	−50640.7	−1.68146
$969$	0	0
$970$	0	0
$971$	10625.3	0.351164	0.175582	−	0.984465i	$-0.443819\pi$
$971$	10625.3	0.351164	0.175582	+	0.984465i	$0.443819\pi$
$972$	0	0
$973$	−11760.0	−0.387470
$974$	44667.2	1.46943
$975$	0	0
$976$	−32452.0	−1.06431
$977$	28359.3	0.928654	0.464327	−	0.885664i	$-0.346296\pi$
$977$	28359.3	0.928654	0.464327	+	0.885664i	$0.346296\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	−12800.0	−0.415952
$983$	35063.3	1.13769	0.568844	−	0.822446i	$-0.307391\pi$
$983$	35063.3	1.13769	0.568844	+	0.822446i	$0.307391\pi$
$984$	0	0
$985$	0	0
$986$	−17708.8	−0.571969
$987$	0	0
$988$	6370.00	0.205118
$989$	−38137.1	−1.22618
$990$	0	0
$991$	−18283.0	−0.586053	−0.293027	−	0.956104i	$-0.594662\pi$
$991$	−18283.0	−0.586053	−0.293027	+	0.956104i	$0.594662\pi$
$992$	13015.9	0.416589
$993$	0	0
$994$	−3000.00	−0.0957286
$995$	0	0
$996$	0	0
$997$	−13230.0	−0.420259	−0.210130	−	0.977674i	$-0.567389\pi$
$997$	−13230.0	−0.420259	−0.210130	+	0.977674i	$0.567389\pi$
$998$	4044.55	0.128285
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.m.1.2	yes	2
3.2	odd	2	inner	225.4.a.m.1.1	yes	2
5.2	odd	4		225.4.b.i.199.4		4
5.3	odd	4		225.4.b.i.199.1		4
5.4	even	2		225.4.a.l.1.1	✓	2
15.2	even	4		225.4.b.i.199.2		4
15.8	even	4		225.4.b.i.199.3		4
15.14	odd	2		225.4.a.l.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.4.a.l.1.1	✓	2	5.4	even	2
225.4.a.l.1.2	yes	2	15.14	odd	2
225.4.a.m.1.1	yes	2	3.2	odd	2	inner
225.4.a.m.1.2	yes	2	1.1	even	1	trivial
225.4.b.i.199.1		4	5.3	odd	4
225.4.b.i.199.2		4	15.2	even	4
225.4.b.i.199.3		4	15.8	even	4
225.4.b.i.199.4		4	5.2	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q+3.16228 q^{2} +2.00000 q^{4} +15.0000 q^{7} -18.9737 q^{8} +O(q^{10})\)	\(q+3.16228 q^{2} +2.00000 q^{4} +15.0000 q^{7} -18.9737 q^{8} +63.2456 q^{11} +35.0000 q^{13} +47.4342 q^{14} -76.0000 q^{16} +88.5438 q^{17} +91.0000 q^{19} +200.000 q^{22} -113.842 q^{23} +110.680 q^{26} +30.0000 q^{28} -63.2456 q^{29} -147.000 q^{31} -88.5438 q^{32} +280.000 q^{34} +370.000 q^{37} +287.767 q^{38} -442.719 q^{41} +335.000 q^{43} +126.491 q^{44} -360.000 q^{46} -177.088 q^{47} -118.000 q^{49} +70.0000 q^{52} +88.5438 q^{53} -284.605 q^{56} -200.000 q^{58} -885.438 q^{59} +427.000 q^{61} -464.855 q^{62} +328.000 q^{64} +15.0000 q^{67} +177.088 q^{68} -63.2456 q^{71} -70.0000 q^{73} +1170.04 q^{74} +182.000 q^{76} +948.683 q^{77} -876.000 q^{79} -1400.00 q^{82} -531.263 q^{83} +1059.36 q^{86} -1200.00 q^{88} +525.000 q^{91} -227.684 q^{92} -560.000 q^{94} -1085.00 q^{97} -373.149 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} + 30 q^{7}+O(q^{10}) \) 2 * q + 4 * q^4 + 30 * q^7	\( 2 q + 4 q^{4} + 30 q^{7} + 70 q^{13} - 152 q^{16} + 182 q^{19} + 400 q^{22} + 60 q^{28} - 294 q^{31} + 560 q^{34} + 740 q^{37} + 670 q^{43} - 720 q^{46} - 236 q^{49} + 140 q^{52} - 400 q^{58} + 854 q^{61} + 656 q^{64} + 30 q^{67} - 140 q^{73} + 364 q^{76} - 1752 q^{79} - 2800 q^{82} - 2400 q^{88} + 1050 q^{91} - 1120 q^{94} - 2170 q^{97}+O(q^{100}) \) 2 * q + 4 * q^4 + 30 * q^7 + 70 * q^13 - 152 * q^16 + 182 * q^19 + 400 * q^22 + 60 * q^28 - 294 * q^31 + 560 * q^34 + 740 * q^37 + 670 * q^43 - 720 * q^46 - 236 * q^49 + 140 * q^52 - 400 * q^58 + 854 * q^61 + 656 * q^64 + 30 * q^67 - 140 * q^73 + 364 * q^76 - 1752 * q^79 - 2800 * q^82 - 2400 * q^88 + 1050 * q^91 - 1120 * q^94 - 2170 * q^97

Introduction

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