LMFDB - Embedded newform 5625.2.a.bf.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:	$24$
Twist minimal:	no (minimal twist has level 225)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.64061 q^{2} +4.97280 q^{4} -4.34551 q^{7} -7.84999 q^{8} +O(q^{10})$	$q-2.64061 q^{2} +4.97280 q^{4} -4.34551 q^{7} -7.84999 q^{8} -2.53375 q^{11} +2.22691 q^{13} +11.4748 q^{14} +10.7831 q^{16} -4.51356 q^{17} +0.864346 q^{19} +6.69063 q^{22} -0.357197 q^{23} -5.88039 q^{26} -21.6093 q^{28} +4.40749 q^{29} -0.455324 q^{31} -12.7740 q^{32} +11.9185 q^{34} +2.37618 q^{37} -2.28240 q^{38} -2.77736 q^{41} +10.2326 q^{43} -12.5998 q^{44} +0.943217 q^{46} -7.92283 q^{47} +11.8834 q^{49} +11.0740 q^{52} -6.45146 q^{53} +34.1122 q^{56} -11.6385 q^{58} -6.91672 q^{59} -6.10294 q^{61} +1.20233 q^{62} +12.1649 q^{64} -11.0361 q^{67} -22.4450 q^{68} -7.37508 q^{71} -13.6036 q^{73} -6.27455 q^{74} +4.29822 q^{76} +11.0104 q^{77} +10.7830 q^{79} +7.33391 q^{82} -0.336984 q^{83} -27.0204 q^{86} +19.8899 q^{88} -13.0652 q^{89} -9.67705 q^{91} -1.77627 q^{92} +20.9211 q^{94} -7.78850 q^{97} -31.3795 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 32 q^{4}+O(q^{10}) $ 24 * q + 32 * q^4	$ 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) $ 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * 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 $

$2$	−2.64061	−1.86719	−0.933595	−	0.358330i	$-0.883346\pi$
$2$	−2.64061	−1.86719	−0.933595	+	0.358330i	$0.883346\pi$
$3$	0	0
$3$	0	0
$4$	4.97280	2.48640
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.34551	−1.64245	−0.821224	−	0.570606i	$-0.806708\pi$
$7$	−4.34551	−1.64245	−0.821224	+	0.570606i	$0.806708\pi$
$8$	−7.84999	−2.77539
$9$	0	0
$10$	0	0
$11$	−2.53375	−0.763954	−0.381977	−	0.924172i	$-0.624757\pi$
$11$	−2.53375	−0.763954	−0.381977	+	0.924172i	$0.624757\pi$
$12$	0	0
$13$	2.22691	0.617633	0.308817	−	0.951122i	$-0.400067\pi$
$13$	2.22691	0.617633	0.308817	+	0.951122i	$0.400067\pi$
$14$	11.4748	3.06676
$15$	0	0
$16$	10.7831	2.69578
$17$	−4.51356	−1.09470	−0.547350	−	0.836904i	$-0.684363\pi$
$17$	−4.51356	−1.09470	−0.547350	+	0.836904i	$0.684363\pi$
$18$	0	0
$19$	0.864346	0.198295	0.0991473	−	0.995073i	$-0.468389\pi$
$19$	0.864346	0.198295	0.0991473	+	0.995073i	$0.468389\pi$
$20$	0	0
$21$	0	0
$22$	6.69063	1.42645
$23$	−0.357197	−0.0744808	−0.0372404	−	0.999306i	$-0.511857\pi$
$23$	−0.357197	−0.0744808	−0.0372404	+	0.999306i	$0.511857\pi$
$24$	0	0
$25$	0	0
$26$	−5.88039	−1.15324
$27$	0	0
$28$	−21.6093	−4.08378
$29$	4.40749	0.818451	0.409225	−	0.912433i	$-0.365799\pi$
$29$	4.40749	0.818451	0.409225	+	0.912433i	$0.365799\pi$
$30$	0	0
$31$	−0.455324	−0.0817786	−0.0408893	−	0.999164i	$-0.513019\pi$
$31$	−0.455324	−0.0817786	−0.0408893	+	0.999164i	$0.513019\pi$
$32$	−12.7740	−2.25815
$33$	0	0
$34$	11.9185	2.04401
$35$	0	0
$36$	0	0
$37$	2.37618	0.390641	0.195321	−	0.980739i	$-0.437425\pi$
$37$	2.37618	0.390641	0.195321	+	0.980739i	$0.437425\pi$
$38$	−2.28240	−0.370254
$39$	0	0
$40$	0	0
$41$	−2.77736	−0.433750	−0.216875	−	0.976199i	$-0.569586\pi$
$41$	−2.77736	−0.433750	−0.216875	+	0.976199i	$0.569586\pi$
$42$	0	0
$43$	10.2326	1.56046	0.780232	−	0.625490i	$-0.215101\pi$
$43$	10.2326	1.56046	0.780232	+	0.625490i	$0.215101\pi$
$44$	−12.5998	−1.89949
$45$	0	0
$46$	0.943217	0.139070
$47$	−7.92283	−1.15566	−0.577832	−	0.816156i	$-0.696101\pi$
$47$	−7.92283	−1.15566	−0.577832	+	0.816156i	$0.696101\pi$
$48$	0	0
$49$	11.8834	1.69764
$50$	0	0
$51$	0	0
$52$	11.0740	1.53568
$53$	−6.45146	−0.886176	−0.443088	−	0.896478i	$-0.646117\pi$
$53$	−6.45146	−0.886176	−0.443088	+	0.896478i	$0.646117\pi$
$54$	0	0
$55$	0	0
$56$	34.1122	4.55843
$57$	0	0
$58$	−11.6385	−1.52820
$59$	−6.91672	−0.900481	−0.450240	−	0.892907i	$-0.648662\pi$
$59$	−6.91672	−0.900481	−0.450240	+	0.892907i	$0.648662\pi$
$60$	0	0
$61$	−6.10294	−0.781401	−0.390701	−	0.920518i	$-0.627767\pi$
$61$	−6.10294	−0.781401	−0.390701	+	0.920518i	$0.627767\pi$
$62$	1.20233	0.152696
$63$	0	0
$64$	12.1649	1.52061
$65$	0	0
$66$	0	0
$67$	−11.0361	−1.34828	−0.674140	−	0.738604i	$-0.735485\pi$
$67$	−11.0361	−1.34828	−0.674140	+	0.738604i	$0.735485\pi$
$68$	−22.4450	−2.72186
$69$	0	0
$70$	0	0
$71$	−7.37508	−0.875261	−0.437631	−	0.899155i	$-0.644182\pi$
$71$	−7.37508	−0.875261	−0.437631	+	0.899155i	$0.644182\pi$
$72$	0	0
$73$	−13.6036	−1.59218	−0.796090	−	0.605178i	$-0.793102\pi$
$73$	−13.6036	−1.59218	−0.796090	+	0.605178i	$0.793102\pi$
$74$	−6.27455	−0.729402
$75$	0	0
$76$	4.29822	0.493039
$77$	11.0104	1.25475
$78$	0	0
$79$	10.7830	1.21319	0.606593	−	0.795013i	$-0.292536\pi$
$79$	10.7830	1.21319	0.606593	+	0.795013i	$0.292536\pi$
$80$	0	0
$81$	0	0
$82$	7.33391	0.809894
$83$	−0.336984	−0.0369888	−0.0184944	−	0.999829i	$-0.505887\pi$
$83$	−0.336984	−0.0369888	−0.0184944	+	0.999829i	$0.505887\pi$
$84$	0	0
$85$	0	0
$86$	−27.0204	−2.91368
$87$	0	0
$88$	19.8899	2.12027
$89$	−13.0652	−1.38491	−0.692456	−	0.721460i	$-0.743471\pi$
$89$	−13.0652	−1.38491	−0.692456	+	0.721460i	$0.743471\pi$
$90$	0	0
$91$	−9.67705	−1.01443
$92$	−1.77627	−0.185189
$93$	0	0
$94$	20.9211	2.15784
$95$	0	0
$96$	0	0
$97$	−7.78850	−0.790802	−0.395401	−	0.918509i	$-0.629394\pi$
$97$	−7.78850	−0.790802	−0.395401	+	0.918509i	$0.629394\pi$
$98$	−31.3795	−3.16981
$99$	0	0
$100$	0	0
$101$	−16.6461	−1.65635	−0.828176	−	0.560468i	$-0.810621\pi$
$101$	−16.6461	−1.65635	−0.828176	+	0.560468i	$0.810621\pi$
$102$	0	0
$103$	1.80337	0.177691	0.0888456	−	0.996045i	$-0.471682\pi$
$103$	1.80337	0.177691	0.0888456	+	0.996045i	$0.471682\pi$
$104$	−17.4812	−1.71417
$105$	0	0
$106$	17.0358	1.65466
$107$	14.5670	1.40824	0.704122	−	0.710079i	$-0.251341\pi$
$107$	14.5670	1.40824	0.704122	+	0.710079i	$0.251341\pi$
$108$	0	0
$109$	−9.12779	−0.874284	−0.437142	−	0.899393i	$-0.644009\pi$
$109$	−9.12779	−0.874284	−0.437142	+	0.899393i	$0.644009\pi$
$110$	0	0
$111$	0	0
$112$	−46.8582	−4.42768
$113$	−2.17825	−0.204912	−0.102456	−	0.994738i	$-0.532670\pi$
$113$	−2.17825	−0.204912	−0.102456	+	0.994738i	$0.532670\pi$
$114$	0	0
$115$	0	0
$116$	21.9176	2.03500
$117$	0	0
$118$	18.2643	1.68137
$119$	19.6137	1.79799
$120$	0	0
$121$	−4.58011	−0.416374
$122$	16.1155	1.45902
$123$	0	0
$124$	−2.26423	−0.203334
$125$	0	0
$126$	0	0
$127$	−6.18568	−0.548890	−0.274445	−	0.961603i	$-0.588494\pi$
$127$	−6.18568	−0.548890	−0.274445	+	0.961603i	$0.588494\pi$
$128$	−6.57457	−0.581116
$129$	0	0
$130$	0	0
$131$	6.33233	0.553258	0.276629	−	0.960977i	$-0.410783\pi$
$131$	6.33233	0.553258	0.276629	+	0.960977i	$0.410783\pi$
$132$	0	0
$133$	−3.75602	−0.325688
$134$	29.1421	2.51749
$135$	0	0
$136$	35.4314	3.03822
$137$	17.8381	1.52401	0.762005	−	0.647571i	$-0.224215\pi$
$137$	17.8381	1.52401	0.762005	+	0.647571i	$0.224215\pi$
$138$	0	0
$139$	10.9106	0.925429	0.462715	−	0.886507i	$-0.346875\pi$
$139$	10.9106	0.925429	0.462715	+	0.886507i	$0.346875\pi$
$140$	0	0
$141$	0	0
$142$	19.4747	1.63428
$143$	−5.64243	−0.471843
$144$	0	0
$145$	0	0
$146$	35.9217	2.97290
$147$	0	0
$148$	11.8163	0.971291
$149$	−16.2808	−1.33378	−0.666889	−	0.745157i	$-0.732374\pi$
$149$	−16.2808	−1.33378	−0.666889	+	0.745157i	$0.732374\pi$
$150$	0	0
$151$	19.1931	1.56191	0.780957	−	0.624585i	$-0.214732\pi$
$151$	19.1931	1.56191	0.780957	+	0.624585i	$0.214732\pi$
$152$	−6.78510	−0.550345
$153$	0	0
$154$	−29.0742	−2.34287
$155$	0	0
$156$	0	0
$157$	−8.59302	−0.685798	−0.342899	−	0.939372i	$-0.611409\pi$
$157$	−8.59302	−0.685798	−0.342899	+	0.939372i	$0.611409\pi$
$158$	−28.4737	−2.26525
$159$	0	0
$160$	0	0
$161$	1.55220	0.122331
$162$	0	0
$163$	6.85021	0.536550	0.268275	−	0.963342i	$-0.413546\pi$
$163$	6.85021	0.536550	0.268275	+	0.963342i	$0.413546\pi$
$164$	−13.8112	−1.07848
$165$	0	0
$166$	0.889841	0.0690651
$167$	13.9035	1.07588	0.537942	−	0.842982i	$-0.319202\pi$
$167$	13.9035	1.07588	0.537942	+	0.842982i	$0.319202\pi$
$168$	0	0
$169$	−8.04088	−0.618529
$170$	0	0
$171$	0	0
$172$	50.8849	3.87994
$173$	15.2451	1.15907	0.579533	−	0.814949i	$-0.303235\pi$
$173$	15.2451	1.15907	0.579533	+	0.814949i	$0.303235\pi$
$174$	0	0
$175$	0	0
$176$	−27.3217	−2.05945
$177$	0	0
$178$	34.5001	2.58589
$179$	6.83738	0.511050	0.255525	−	0.966803i	$-0.417752\pi$
$179$	6.83738	0.511050	0.255525	+	0.966803i	$0.417752\pi$
$180$	0	0
$181$	12.8487	0.955039	0.477519	−	0.878621i	$-0.341536\pi$
$181$	12.8487	0.955039	0.477519	+	0.878621i	$0.341536\pi$
$182$	25.5533	1.89413
$183$	0	0
$184$	2.80399	0.206713
$185$	0	0
$186$	0	0
$187$	11.4362	0.836301
$188$	−39.3986	−2.87344
$189$	0	0
$190$	0	0
$191$	−0.175077	−0.0126682	−0.00633408	−	0.999980i	$-0.502016\pi$
$191$	−0.175077	−0.0126682	−0.00633408	+	0.999980i	$0.502016\pi$
$192$	0	0
$193$	−10.0004	−0.719844	−0.359922	−	0.932982i	$-0.617197\pi$
$193$	−10.0004	−0.719844	−0.359922	+	0.932982i	$0.617197\pi$
$194$	20.5664	1.47658
$195$	0	0
$196$	59.0940	4.22100
$197$	18.3527	1.30757	0.653786	−	0.756679i	$-0.273180\pi$
$197$	18.3527	1.30757	0.653786	+	0.756679i	$0.273180\pi$
$198$	0	0
$199$	2.11923	0.150228	0.0751139	−	0.997175i	$-0.476068\pi$
$199$	2.11923	0.150228	0.0751139	+	0.997175i	$0.476068\pi$
$200$	0	0
$201$	0	0
$202$	43.9559	3.09273
$203$	−19.1528	−1.34426
$204$	0	0
$205$	0	0
$206$	−4.76199	−0.331783
$207$	0	0
$208$	24.0130	1.66500
$209$	−2.19004	−0.151488
$210$	0	0
$211$	2.91678	0.200799	0.100400	−	0.994947i	$-0.467988\pi$
$211$	2.91678	0.200799	0.100400	+	0.994947i	$0.467988\pi$
$212$	−32.0818	−2.20339
$213$	0	0
$214$	−38.4657	−2.62946
$215$	0	0
$216$	0	0
$217$	1.97861	0.134317
$218$	24.1029	1.63245
$219$	0	0
$220$	0	0
$221$	−10.0513	−0.676123
$222$	0	0
$223$	0.617321	0.0413388	0.0206694	−	0.999786i	$-0.493420\pi$
$223$	0.617321	0.0413388	0.0206694	+	0.999786i	$0.493420\pi$
$224$	55.5095	3.70889
$225$	0	0
$226$	5.75189	0.382610
$227$	17.2458	1.14464	0.572322	−	0.820029i	$-0.306043\pi$
$227$	17.2458	1.14464	0.572322	+	0.820029i	$0.306043\pi$
$228$	0	0
$229$	12.0681	0.797481	0.398741	−	0.917064i	$-0.369447\pi$
$229$	12.0681	0.797481	0.398741	+	0.917064i	$0.369447\pi$
$230$	0	0
$231$	0	0
$232$	−34.5988	−2.27152
$233$	−2.80161	−0.183540	−0.0917698	−	0.995780i	$-0.529252\pi$
$233$	−2.80161	−0.183540	−0.0917698	+	0.995780i	$0.529252\pi$
$234$	0	0
$235$	0	0
$236$	−34.3955	−2.23895
$237$	0	0
$238$	−51.7921	−3.35718
$239$	−12.7863	−0.827077	−0.413538	−	0.910487i	$-0.635707\pi$
$239$	−12.7863	−0.827077	−0.413538	+	0.910487i	$0.635707\pi$
$240$	0	0
$241$	3.21958	0.207392	0.103696	−	0.994609i	$-0.466933\pi$
$241$	3.21958	0.207392	0.103696	+	0.994609i	$0.466933\pi$
$242$	12.0943	0.777449
$243$	0	0
$244$	−30.3487	−1.94288
$245$	0	0
$246$	0	0
$247$	1.92482	0.122473
$248$	3.57429	0.226968
$249$	0	0
$250$	0	0
$251$	18.8989	1.19289	0.596444	−	0.802655i	$-0.296580\pi$
$251$	18.8989	1.19289	0.596444	+	0.802655i	$0.296580\pi$
$252$	0	0
$253$	0.905048	0.0568999
$254$	16.3339	1.02488
$255$	0	0
$256$	−6.96885	−0.435553
$257$	−24.5521	−1.53152	−0.765758	−	0.643128i	$-0.777636\pi$
$257$	−24.5521	−1.53152	−0.765758	+	0.643128i	$0.777636\pi$
$258$	0	0
$259$	−10.3257	−0.641608
$260$	0	0
$261$	0	0
$262$	−16.7212	−1.03304
$263$	−8.23440	−0.507755	−0.253877	−	0.967236i	$-0.581706\pi$
$263$	−8.23440	−0.507755	−0.253877	+	0.967236i	$0.581706\pi$
$264$	0	0
$265$	0	0
$266$	9.91817	0.608122
$267$	0	0
$268$	−54.8805	−3.35236
$269$	1.30450	0.0795368	0.0397684	−	0.999209i	$-0.487338\pi$
$269$	1.30450	0.0795368	0.0397684	+	0.999209i	$0.487338\pi$
$270$	0	0
$271$	−1.98566	−0.120620	−0.0603100	−	0.998180i	$-0.519209\pi$
$271$	−1.98566	−0.120620	−0.0603100	+	0.998180i	$0.519209\pi$
$272$	−48.6703	−2.95107
$273$	0	0
$274$	−47.1033	−2.84562
$275$	0	0
$276$	0	0
$277$	−13.7586	−0.826675	−0.413338	−	0.910578i	$-0.635637\pi$
$277$	−13.7586	−0.826675	−0.413338	+	0.910578i	$0.635637\pi$
$278$	−28.8107	−1.72795
$279$	0	0
$280$	0	0
$281$	28.8763	1.72261	0.861307	−	0.508086i	$-0.169647\pi$
$281$	28.8763	1.72261	0.861307	+	0.508086i	$0.169647\pi$
$282$	0	0
$283$	31.6927	1.88394	0.941968	−	0.335704i	$-0.108974\pi$
$283$	31.6927	1.88394	0.941968	+	0.335704i	$0.108974\pi$
$284$	−36.6748	−2.17625
$285$	0	0
$286$	14.8994	0.881021
$287$	12.0690	0.712412
$288$	0	0
$289$	3.37225	0.198368
$290$	0	0
$291$	0	0
$292$	−67.6479	−3.95880
$293$	−5.57990	−0.325981	−0.162991	−	0.986628i	$-0.552114\pi$
$293$	−5.57990	−0.325981	−0.162991	+	0.986628i	$0.552114\pi$
$294$	0	0
$295$	0	0
$296$	−18.6530	−1.08418
$297$	0	0
$298$	42.9913	2.49042
$299$	−0.795445	−0.0460018
$300$	0	0
$301$	−44.4661	−2.56298
$302$	−50.6814	−2.91639
$303$	0	0
$304$	9.32035	0.534559
$305$	0	0
$306$	0	0
$307$	18.1679	1.03690	0.518448	−	0.855109i	$-0.326510\pi$
$307$	18.1679	1.03690	0.518448	+	0.855109i	$0.326510\pi$
$308$	54.7526	3.11982
$309$	0	0
$310$	0	0
$311$	14.3412	0.813216	0.406608	−	0.913603i	$-0.366712\pi$
$311$	14.3412	0.813216	0.406608	+	0.913603i	$0.366712\pi$
$312$	0	0
$313$	−9.06837	−0.512574	−0.256287	−	0.966601i	$-0.582499\pi$
$313$	−9.06837	−0.512574	−0.256287	+	0.966601i	$0.582499\pi$
$314$	22.6908	1.28052
$315$	0	0
$316$	53.6218	3.01646
$317$	20.1498	1.13172	0.565862	−	0.824500i	$-0.308544\pi$
$317$	20.1498	1.13172	0.565862	+	0.824500i	$0.308544\pi$
$318$	0	0
$319$	−11.1675	−0.625259
$320$	0	0
$321$	0	0
$322$	−4.09876	−0.228415
$323$	−3.90128	−0.217073
$324$	0	0
$325$	0	0
$326$	−18.0887	−1.00184
$327$	0	0
$328$	21.8022	1.20383
$329$	34.4287	1.89812
$330$	0	0
$331$	−30.3788	−1.66977	−0.834884	−	0.550426i	$-0.814465\pi$
$331$	−30.3788	−1.66977	−0.834884	+	0.550426i	$0.814465\pi$
$332$	−1.67575	−0.0919689
$333$	0	0
$334$	−36.7136	−2.00888
$335$	0	0
$336$	0	0
$337$	−13.9608	−0.760495	−0.380248	−	0.924885i	$-0.624161\pi$
$337$	−13.9608	−0.760495	−0.380248	+	0.924885i	$0.624161\pi$
$338$	21.2328	1.15491
$339$	0	0
$340$	0	0
$341$	1.15368	0.0624751
$342$	0	0
$343$	−21.2211	−1.14583
$344$	−80.3261	−4.33090
$345$	0	0
$346$	−40.2564	−2.16420
$347$	12.4485	0.668270	0.334135	−	0.942525i	$-0.391556\pi$
$347$	12.4485	0.668270	0.334135	+	0.942525i	$0.391556\pi$
$348$	0	0
$349$	−10.1110	−0.541228	−0.270614	−	0.962688i	$-0.587227\pi$
$349$	−10.1110	−0.541228	−0.270614	+	0.962688i	$0.587227\pi$
$350$	0	0
$351$	0	0
$352$	32.3661	1.72512
$353$	35.4392	1.88624	0.943118	−	0.332457i	$-0.107878\pi$
$353$	35.4392	1.88624	0.943118	+	0.332457i	$0.107878\pi$
$354$	0	0
$355$	0	0
$356$	−64.9708	−3.44344
$357$	0	0
$358$	−18.0548	−0.954227
$359$	−31.5562	−1.66547	−0.832736	−	0.553670i	$-0.813227\pi$
$359$	−31.5562	−1.66547	−0.832736	+	0.553670i	$0.813227\pi$
$360$	0	0
$361$	−18.2529	−0.960679
$362$	−33.9284	−1.78324
$363$	0	0
$364$	−48.1220	−2.52228
$365$	0	0
$366$	0	0
$367$	12.9433	0.675632	0.337816	−	0.941212i	$-0.390312\pi$
$367$	12.9433	0.675632	0.337816	+	0.941212i	$0.390312\pi$
$368$	−3.85170	−0.200784
$369$	0	0
$370$	0	0
$371$	28.0349	1.45550
$372$	0	0
$373$	−8.19546	−0.424345	−0.212172	−	0.977232i	$-0.568054\pi$
$373$	−8.19546	−0.424345	−0.212172	+	0.977232i	$0.568054\pi$
$374$	−30.1986	−1.56153
$375$	0	0
$376$	62.1941	3.20742
$377$	9.81508	0.505502
$378$	0	0
$379$	−17.9537	−0.922218	−0.461109	−	0.887343i	$-0.652548\pi$
$379$	−17.9537	−0.922218	−0.461109	+	0.887343i	$0.652548\pi$
$380$	0	0
$381$	0	0
$382$	0.462311	0.0236539
$383$	20.0827	1.02618	0.513088	−	0.858336i	$-0.328501\pi$
$383$	20.0827	1.02618	0.513088	+	0.858336i	$0.328501\pi$
$384$	0	0
$385$	0	0
$386$	26.4071	1.34409
$387$	0	0
$388$	−38.7306	−1.96625
$389$	19.1250	0.969678	0.484839	−	0.874603i	$-0.338878\pi$
$389$	19.1250	0.969678	0.484839	+	0.874603i	$0.338878\pi$
$390$	0	0
$391$	1.61223	0.0815341
$392$	−93.2849	−4.71160
$393$	0	0
$394$	−48.4621	−2.44149
$395$	0	0
$396$	0	0
$397$	−2.88366	−0.144727	−0.0723634	−	0.997378i	$-0.523054\pi$
$397$	−2.88366	−0.144727	−0.0723634	+	0.997378i	$0.523054\pi$
$398$	−5.59604	−0.280504
$399$	0	0
$400$	0	0
$401$	5.50842	0.275077	0.137539	−	0.990496i	$-0.456081\pi$
$401$	5.50842	0.275077	0.137539	+	0.990496i	$0.456081\pi$
$402$	0	0
$403$	−1.01396	−0.0505092
$404$	−82.7779	−4.11835
$405$	0	0
$406$	50.5750	2.50999
$407$	−6.02064	−0.298432
$408$	0	0
$409$	37.2351	1.84116	0.920579	−	0.390556i	$-0.127717\pi$
$409$	37.2351	1.84116	0.920579	+	0.390556i	$0.127717\pi$
$410$	0	0
$411$	0	0
$412$	8.96779	0.441811
$413$	30.0567	1.47899
$414$	0	0
$415$	0	0
$416$	−28.4465	−1.39471
$417$	0	0
$418$	5.78302	0.282857
$419$	23.8229	1.16382	0.581912	−	0.813252i	$-0.302305\pi$
$419$	23.8229	1.16382	0.581912	+	0.813252i	$0.302305\pi$
$420$	0	0
$421$	8.92684	0.435068	0.217534	−	0.976053i	$-0.430199\pi$
$421$	8.92684	0.435068	0.217534	+	0.976053i	$0.430199\pi$
$422$	−7.70207	−0.374931
$423$	0	0
$424$	50.6439	2.45948
$425$	0	0
$426$	0	0
$427$	26.5204	1.28341
$428$	72.4387	3.50146
$429$	0	0
$430$	0	0
$431$	−26.0347	−1.25405	−0.627025	−	0.778999i	$-0.715728\pi$
$431$	−26.0347	−1.25405	−0.627025	+	0.778999i	$0.715728\pi$
$432$	0	0
$433$	−5.28263	−0.253867	−0.126933	−	0.991911i	$-0.540513\pi$
$433$	−5.28263	−0.253867	−0.126933	+	0.991911i	$0.540513\pi$
$434$	−5.22474	−0.250796
$435$	0	0
$436$	−45.3906	−2.17382
$437$	−0.308742	−0.0147691
$438$	0	0
$439$	17.0961	0.815951	0.407976	−	0.912993i	$-0.366235\pi$
$439$	17.0961	0.815951	0.407976	+	0.912993i	$0.366235\pi$
$440$	0	0
$441$	0	0
$442$	26.5415	1.26245
$443$	−7.99868	−0.380029	−0.190014	−	0.981781i	$-0.560853\pi$
$443$	−7.99868	−0.380029	−0.190014	+	0.981781i	$0.560853\pi$
$444$	0	0
$445$	0	0
$446$	−1.63010	−0.0771875
$447$	0	0
$448$	−52.8625	−2.49752
$449$	−22.6881	−1.07072	−0.535358	−	0.844625i	$-0.679823\pi$
$449$	−22.6881	−1.07072	−0.535358	+	0.844625i	$0.679823\pi$
$450$	0	0
$451$	7.03713	0.331365
$452$	−10.8320	−0.509494
$453$	0	0
$454$	−45.5393	−2.13727
$455$	0	0
$456$	0	0
$457$	−7.79373	−0.364575	−0.182288	−	0.983245i	$-0.558350\pi$
$457$	−7.79373	−0.364575	−0.182288	+	0.983245i	$0.558350\pi$
$458$	−31.8670	−1.48905
$459$	0	0
$460$	0	0
$461$	20.1897	0.940328	0.470164	−	0.882579i	$-0.344195\pi$
$461$	20.1897	0.940328	0.470164	+	0.882579i	$0.344195\pi$
$462$	0	0
$463$	29.6830	1.37949	0.689744	−	0.724054i	$-0.257723\pi$
$463$	29.6830	1.37949	0.689744	+	0.724054i	$0.257723\pi$
$464$	47.5265	2.20636
$465$	0	0
$466$	7.39795	0.342703
$467$	33.0268	1.52830	0.764150	−	0.645039i	$-0.223159\pi$
$467$	33.0268	1.52830	0.764150	+	0.645039i	$0.223159\pi$
$468$	0	0
$469$	47.9576	2.21448
$470$	0	0
$471$	0	0
$472$	54.2962	2.49918
$473$	−25.9270	−1.19212
$474$	0	0
$475$	0	0
$476$	97.5351	4.47051
$477$	0	0
$478$	33.7636	1.54431
$479$	29.3680	1.34186	0.670930	−	0.741521i	$-0.265895\pi$
$479$	29.3680	1.34186	0.670930	+	0.741521i	$0.265895\pi$
$480$	0	0
$481$	5.29153	0.241273
$482$	−8.50165	−0.387240
$483$	0	0
$484$	−22.7760	−1.03527
$485$	0	0
$486$	0	0
$487$	−21.6984	−0.983250	−0.491625	−	0.870807i	$-0.663597\pi$
$487$	−21.6984	−0.983250	−0.491625	+	0.870807i	$0.663597\pi$
$488$	47.9080	2.16869
$489$	0	0
$490$	0	0
$491$	27.8598	1.25730	0.628648	−	0.777690i	$-0.283609\pi$
$491$	27.8598	1.25730	0.628648	+	0.777690i	$0.283609\pi$
$492$	0	0
$493$	−19.8935	−0.895958
$494$	−5.08269	−0.228681
$495$	0	0
$496$	−4.90982	−0.220457
$497$	32.0485	1.43757
$498$	0	0
$499$	−9.83737	−0.440381	−0.220191	−	0.975457i	$-0.570668\pi$
$499$	−9.83737	−0.440381	−0.220191	+	0.975457i	$0.570668\pi$
$500$	0	0
$501$	0	0
$502$	−49.9045	−2.22735
$503$	−26.5708	−1.18473	−0.592366	−	0.805669i	$-0.701806\pi$
$503$	−26.5708	−1.18473	−0.592366	+	0.805669i	$0.701806\pi$
$504$	0	0
$505$	0	0
$506$	−2.38988	−0.106243
$507$	0	0
$508$	−30.7601	−1.36476
$509$	12.6392	0.560224	0.280112	−	0.959967i	$-0.409628\pi$
$509$	12.6392	0.560224	0.280112	+	0.959967i	$0.409628\pi$
$510$	0	0
$511$	59.1146	2.61507
$512$	31.5511	1.39438
$513$	0	0
$514$	64.8324	2.85963
$515$	0	0
$516$	0	0
$517$	20.0745	0.882874
$518$	27.2661	1.19800
$519$	0	0
$520$	0	0
$521$	−24.0542	−1.05383	−0.526917	−	0.849917i	$-0.676652\pi$
$521$	−24.0542	−1.05383	−0.526917	+	0.849917i	$0.676652\pi$
$522$	0	0
$523$	−4.32020	−0.188909	−0.0944546	−	0.995529i	$-0.530111\pi$
$523$	−4.32020	−0.188909	−0.0944546	+	0.995529i	$0.530111\pi$
$524$	31.4894	1.37562
$525$	0	0
$526$	21.7438	0.948075
$527$	2.05513	0.0895230
$528$	0	0
$529$	−22.8724	−0.994453
$530$	0	0
$531$	0	0
$532$	−18.6779	−0.809791
$533$	−6.18492	−0.267899
$534$	0	0
$535$	0	0
$536$	86.6336	3.74200
$537$	0	0
$538$	−3.44467	−0.148510
$539$	−30.1097	−1.29692
$540$	0	0
$541$	7.85543	0.337731	0.168866	−	0.985639i	$-0.445990\pi$
$541$	7.85543	0.337731	0.168866	+	0.985639i	$0.445990\pi$
$542$	5.24334	0.225221
$543$	0	0
$544$	57.6563	2.47199
$545$	0	0
$546$	0	0
$547$	−16.7797	−0.717450	−0.358725	−	0.933443i	$-0.616788\pi$
$547$	−16.7797	−0.717450	−0.358725	+	0.933443i	$0.616788\pi$
$548$	88.7052	3.78930
$549$	0	0
$550$	0	0
$551$	3.80960	0.162294
$552$	0	0
$553$	−46.8578	−1.99259
$554$	36.3311	1.54356
$555$	0	0
$556$	54.2564	2.30099
$557$	32.4327	1.37422	0.687109	−	0.726554i	$-0.258879\pi$
$557$	32.4327	1.37422	0.687109	+	0.726554i	$0.258879\pi$
$558$	0	0
$559$	22.7872	0.963794
$560$	0	0
$561$	0	0
$562$	−76.2508	−3.21645
$563$	−27.2108	−1.14680	−0.573400	−	0.819276i	$-0.694376\pi$
$563$	−27.2108	−1.14680	−0.573400	+	0.819276i	$0.694376\pi$
$564$	0	0
$565$	0	0
$566$	−83.6879	−3.51767
$567$	0	0
$568$	57.8943	2.42919
$569$	46.1814	1.93602	0.968012	−	0.250905i	$-0.0807283\pi$
$569$	46.1814	1.93602	0.968012	+	0.250905i	$0.0807283\pi$
$570$	0	0
$571$	6.50973	0.272424	0.136212	−	0.990680i	$-0.456507\pi$
$571$	6.50973	0.272424	0.136212	+	0.990680i	$0.456507\pi$
$572$	−28.0587	−1.17319
$573$	0	0
$574$	−31.8696	−1.33021
$575$	0	0
$576$	0	0
$577$	−29.3322	−1.22111	−0.610557	−	0.791972i	$-0.709054\pi$
$577$	−29.3322	−1.22111	−0.610557	+	0.791972i	$0.709054\pi$
$578$	−8.90479	−0.370390
$579$	0	0
$580$	0	0
$581$	1.46437	0.0607521
$582$	0	0
$583$	16.3464	0.676998
$584$	106.788	4.41892
$585$	0	0
$586$	14.7343	0.608669
$587$	3.30184	0.136282	0.0681408	−	0.997676i	$-0.478293\pi$
$587$	3.30184	0.136282	0.0681408	+	0.997676i	$0.478293\pi$
$588$	0	0
$589$	−0.393558	−0.0162163
$590$	0	0
$591$	0	0
$592$	25.6226	1.05308
$593$	21.5576	0.885263	0.442631	−	0.896704i	$-0.354045\pi$
$593$	21.5576	0.885263	0.442631	+	0.896704i	$0.354045\pi$
$594$	0	0
$595$	0	0
$596$	−80.9613	−3.31630
$597$	0	0
$598$	2.10046	0.0858941
$599$	−32.2761	−1.31877	−0.659383	−	0.751807i	$-0.729182\pi$
$599$	−32.2761	−1.31877	−0.659383	+	0.751807i	$0.729182\pi$
$600$	0	0
$601$	−7.60616	−0.310262	−0.155131	−	0.987894i	$-0.549580\pi$
$601$	−7.60616	−0.310262	−0.155131	+	0.987894i	$0.549580\pi$
$602$	117.417	4.78557
$603$	0	0
$604$	95.4434	3.88354
$605$	0	0
$606$	0	0
$607$	31.1356	1.26375	0.631877	−	0.775069i	$-0.282285\pi$
$607$	31.1356	1.26375	0.631877	+	0.775069i	$0.282285\pi$
$608$	−11.0412	−0.447778
$609$	0	0
$610$	0	0
$611$	−17.6434	−0.713776
$612$	0	0
$613$	22.1646	0.895218	0.447609	−	0.894229i	$-0.352276\pi$
$613$	22.1646	0.895218	0.447609	+	0.894229i	$0.352276\pi$
$614$	−47.9743	−1.93608
$615$	0	0
$616$	−86.4317	−3.48243
$617$	−11.9749	−0.482093	−0.241047	−	0.970514i	$-0.577491\pi$
$617$	−11.9749	−0.482093	−0.241047	+	0.970514i	$0.577491\pi$
$618$	0	0
$619$	31.8848	1.28156	0.640779	−	0.767725i	$-0.278611\pi$
$619$	31.8848	1.28156	0.640779	+	0.767725i	$0.278611\pi$
$620$	0	0
$621$	0	0
$622$	−37.8695	−1.51843
$623$	56.7751	2.27465
$624$	0	0
$625$	0	0
$626$	23.9460	0.957074
$627$	0	0
$628$	−42.7314	−1.70517
$629$	−10.7250	−0.427635
$630$	0	0
$631$	−8.00382	−0.318627	−0.159313	−	0.987228i	$-0.550928\pi$
$631$	−8.00382	−0.318627	−0.159313	+	0.987228i	$0.550928\pi$
$632$	−84.6467	−3.36706
$633$	0	0
$634$	−53.2076	−2.11314
$635$	0	0
$636$	0	0
$637$	26.4633	1.04852
$638$	29.4889	1.16748
$639$	0	0
$640$	0	0
$641$	−48.3672	−1.91039	−0.955196	−	0.295975i	$-0.904355\pi$
$641$	−48.3672	−1.91039	−0.955196	+	0.295975i	$0.904355\pi$
$642$	0	0
$643$	34.0606	1.34322	0.671609	−	0.740906i	$-0.265604\pi$
$643$	34.0606	1.34322	0.671609	+	0.740906i	$0.265604\pi$
$644$	7.71880	0.304163
$645$	0	0
$646$	10.3017	0.405317
$647$	−8.73128	−0.343262	−0.171631	−	0.985161i	$-0.554904\pi$
$647$	−8.73128	−0.343262	−0.171631	+	0.985161i	$0.554904\pi$
$648$	0	0
$649$	17.5252	0.687926
$650$	0	0
$651$	0	0
$652$	34.0647	1.33408
$653$	8.48473	0.332033	0.166017	−	0.986123i	$-0.446909\pi$
$653$	8.48473	0.332033	0.166017	+	0.986123i	$0.446909\pi$
$654$	0	0
$655$	0	0
$656$	−29.9486	−1.16930
$657$	0	0
$658$	−90.9127	−3.54414
$659$	26.8779	1.04701	0.523507	−	0.852021i	$-0.324623\pi$
$659$	26.8779	1.04701	0.523507	+	0.852021i	$0.324623\pi$
$660$	0	0
$661$	−29.3620	−1.14205	−0.571025	−	0.820933i	$-0.693454\pi$
$661$	−29.3620	−1.14205	−0.571025	+	0.820933i	$0.693454\pi$
$662$	80.2183	3.11777
$663$	0	0
$664$	2.64532	0.102658
$665$	0	0
$666$	0	0
$667$	−1.57434	−0.0609589
$668$	69.1392	2.67508
$669$	0	0
$670$	0	0
$671$	15.4633	0.596955
$672$	0	0
$673$	−25.2406	−0.972952	−0.486476	−	0.873694i	$-0.661718\pi$
$673$	−25.2406	−0.972952	−0.486476	+	0.873694i	$0.661718\pi$
$674$	36.8651	1.41999
$675$	0	0
$676$	−39.9857	−1.53791
$677$	−0.0600999	−0.00230983	−0.00115491	−	0.999999i	$-0.500368\pi$
$677$	−0.0600999	−0.00230983	−0.00115491	+	0.999999i	$0.500368\pi$
$678$	0	0
$679$	33.8450	1.29885
$680$	0	0
$681$	0	0
$682$	−3.04641	−0.116653
$683$	32.0602	1.22675	0.613375	−	0.789792i	$-0.289811\pi$
$683$	32.0602	1.22675	0.613375	+	0.789792i	$0.289811\pi$
$684$	0	0
$685$	0	0
$686$	56.0364	2.13948
$687$	0	0
$688$	110.340	4.20667
$689$	−14.3668	−0.547332
$690$	0	0
$691$	37.9022	1.44187	0.720933	−	0.693005i	$-0.243714\pi$
$691$	37.9022	1.44187	0.720933	+	0.693005i	$0.243714\pi$
$692$	75.8109	2.88190
$693$	0	0
$694$	−32.8716	−1.24779
$695$	0	0
$696$	0	0
$697$	12.5358	0.474826
$698$	26.6991	1.01058
$699$	0	0
$700$	0	0
$701$	−12.6083	−0.476209	−0.238104	−	0.971240i	$-0.576526\pi$
$701$	−12.6083	−0.476209	−0.238104	+	0.971240i	$0.576526\pi$
$702$	0	0
$703$	2.05384	0.0774621
$704$	−30.8227	−1.16167
$705$	0	0
$706$	−93.5809	−3.52196
$707$	72.3359	2.72047
$708$	0	0
$709$	42.8213	1.60819	0.804094	−	0.594502i	$-0.202651\pi$
$709$	42.8213	1.60819	0.804094	+	0.594502i	$0.202651\pi$
$710$	0	0
$711$	0	0
$712$	102.562	3.84367
$713$	0.162641	0.00609093
$714$	0	0
$715$	0	0
$716$	34.0009	1.27067
$717$	0	0
$718$	83.3275	3.10975
$719$	28.1122	1.04841	0.524204	−	0.851593i	$-0.324363\pi$
$719$	28.1122	1.04841	0.524204	+	0.851593i	$0.324363\pi$
$720$	0	0
$721$	−7.83656	−0.291849
$722$	48.1987	1.79377
$723$	0	0
$724$	63.8942	2.37461
$725$	0	0
$726$	0	0
$727$	10.9496	0.406098	0.203049	−	0.979169i	$-0.434915\pi$
$727$	10.9496	0.406098	0.203049	+	0.979169i	$0.434915\pi$
$728$	75.9647	2.81544
$729$	0	0
$730$	0	0
$731$	−46.1857	−1.70824
$732$	0	0
$733$	29.2496	1.08036	0.540179	−	0.841550i	$-0.318357\pi$
$733$	29.2496	1.08036	0.540179	+	0.841550i	$0.318357\pi$
$734$	−34.1780	−1.26153
$735$	0	0
$736$	4.56284	0.168188
$737$	27.9628	1.03002
$738$	0	0
$739$	−11.2931	−0.415425	−0.207713	−	0.978190i	$-0.566602\pi$
$739$	−11.2931	−0.415425	−0.207713	+	0.978190i	$0.566602\pi$
$740$	0	0
$741$	0	0
$742$	−74.0291	−2.71769
$743$	4.14745	0.152155	0.0760777	−	0.997102i	$-0.475760\pi$
$743$	4.14745	0.152155	0.0760777	+	0.997102i	$0.475760\pi$
$744$	0	0
$745$	0	0
$746$	21.6410	0.792332
$747$	0	0
$748$	56.8701	2.07938
$749$	−63.3010	−2.31297
$750$	0	0
$751$	−3.25169	−0.118656	−0.0593279	−	0.998239i	$-0.518896\pi$
$751$	−3.25169	−0.118656	−0.0593279	+	0.998239i	$0.518896\pi$
$752$	−85.4328	−3.11541
$753$	0	0
$754$	−25.9178	−0.943869
$755$	0	0
$756$	0	0
$757$	28.7514	1.04499	0.522494	−	0.852643i	$-0.325002\pi$
$757$	28.7514	1.04499	0.522494	+	0.852643i	$0.325002\pi$
$758$	47.4086	1.72196
$759$	0	0
$760$	0	0
$761$	−14.6353	−0.530530	−0.265265	−	0.964176i	$-0.585459\pi$
$761$	−14.6353	−0.530530	−0.265265	+	0.964176i	$0.585459\pi$
$762$	0	0
$763$	39.6649	1.43597
$764$	−0.870625	−0.0314981
$765$	0	0
$766$	−53.0304	−1.91607
$767$	−15.4029	−0.556167
$768$	0	0
$769$	−11.1730	−0.402909	−0.201454	−	0.979498i	$-0.564567\pi$
$769$	−11.1730	−0.402909	−0.201454	+	0.979498i	$0.564567\pi$
$770$	0	0
$771$	0	0
$772$	−49.7300	−1.78982
$773$	−46.4955	−1.67233	−0.836163	−	0.548481i	$-0.815206\pi$
$773$	−46.4955	−1.67233	−0.836163	+	0.548481i	$0.815206\pi$
$774$	0	0
$775$	0	0
$776$	61.1396	2.19478
$777$	0	0
$778$	−50.5017	−1.81057
$779$	−2.40060	−0.0860103
$780$	0	0
$781$	18.6866	0.668659
$782$	−4.25727	−0.152240
$783$	0	0
$784$	128.141	4.57645
$785$	0	0
$786$	0	0
$787$	2.83022	0.100886	0.0504432	−	0.998727i	$-0.483937\pi$
$787$	2.83022	0.100886	0.0504432	+	0.998727i	$0.483937\pi$
$788$	91.2640	3.25115
$789$	0	0
$790$	0	0
$791$	9.46559	0.336558
$792$	0	0
$793$	−13.5907	−0.482619
$794$	7.61462	0.270233
$795$	0	0
$796$	10.5385	0.373526
$797$	−1.73900	−0.0615987	−0.0307993	−	0.999526i	$-0.509805\pi$
$797$	−1.73900	−0.0615987	−0.0307993	+	0.999526i	$0.509805\pi$
$798$	0	0
$799$	35.7602	1.26510
$800$	0	0
$801$	0	0
$802$	−14.5456	−0.513621
$803$	34.4681	1.21635
$804$	0	0
$805$	0	0
$806$	2.67748	0.0943103
$807$	0	0
$808$	130.672	4.59702
$809$	21.5190	0.756569	0.378285	−	0.925689i	$-0.376514\pi$
$809$	21.5190	0.756569	0.378285	+	0.925689i	$0.376514\pi$
$810$	0	0
$811$	−7.90459	−0.277568	−0.138784	−	0.990323i	$-0.544319\pi$
$811$	−7.90459	−0.277568	−0.138784	+	0.990323i	$0.544319\pi$
$812$	−95.2430	−3.34237
$813$	0	0
$814$	15.8981	0.557230
$815$	0	0
$816$	0	0
$817$	8.84455	0.309432
$818$	−98.3233	−3.43779
$819$	0	0
$820$	0	0
$821$	−25.9417	−0.905370	−0.452685	−	0.891670i	$-0.649534\pi$
$821$	−25.9417	−0.905370	−0.452685	+	0.891670i	$0.649534\pi$
$822$	0	0
$823$	−2.65720	−0.0926242	−0.0463121	−	0.998927i	$-0.514747\pi$
$823$	−2.65720	−0.0926242	−0.0463121	+	0.998927i	$0.514747\pi$
$824$	−14.1564	−0.493162
$825$	0	0
$826$	−79.3678	−2.76156
$827$	3.04566	0.105908	0.0529541	−	0.998597i	$-0.483136\pi$
$827$	3.04566	0.105908	0.0529541	+	0.998597i	$0.483136\pi$
$828$	0	0
$829$	23.7338	0.824309	0.412154	−	0.911114i	$-0.364776\pi$
$829$	23.7338	0.824309	0.412154	+	0.911114i	$0.364776\pi$
$830$	0	0
$831$	0	0
$832$	27.0900	0.939177
$833$	−53.6367	−1.85840
$834$	0	0
$835$	0	0
$836$	−10.8906	−0.376659
$837$	0	0
$838$	−62.9069	−2.17308
$839$	5.90055	0.203709	0.101855	−	0.994799i	$-0.467522\pi$
$839$	5.90055	0.203709	0.101855	+	0.994799i	$0.467522\pi$
$840$	0	0
$841$	−9.57400	−0.330138
$842$	−23.5723	−0.812354
$843$	0	0
$844$	14.5046	0.499268
$845$	0	0
$846$	0	0
$847$	19.9029	0.683873
$848$	−69.5669	−2.38894
$849$	0	0
$850$	0	0
$851$	−0.848765	−0.0290953
$852$	0	0
$853$	−44.4569	−1.52218	−0.761088	−	0.648648i	$-0.775335\pi$
$853$	−44.4569	−1.52218	−0.761088	+	0.648648i	$0.775335\pi$
$854$	−70.0298	−2.39637
$855$	0	0
$856$	−114.351	−3.90843
$857$	17.1872	0.587103	0.293551	−	0.955943i	$-0.405163\pi$
$857$	17.1872	0.587103	0.293551	+	0.955943i	$0.405163\pi$
$858$	0	0
$859$	30.1365	1.02825	0.514123	−	0.857717i	$-0.328118\pi$
$859$	30.1365	1.02825	0.514123	+	0.857717i	$0.328118\pi$
$860$	0	0
$861$	0	0
$862$	68.7475	2.34155
$863$	14.0786	0.479241	0.239621	−	0.970867i	$-0.422977\pi$
$863$	14.0786	0.479241	0.239621	+	0.970867i	$0.422977\pi$
$864$	0	0
$865$	0	0
$866$	13.9493	0.474018
$867$	0	0
$868$	9.83925	0.333966
$869$	−27.3215	−0.926818
$870$	0	0
$871$	−24.5765	−0.832742
$872$	71.6530	2.42648
$873$	0	0
$874$	0.815266	0.0275768
$875$	0	0
$876$	0	0
$877$	38.0628	1.28529	0.642645	−	0.766164i	$-0.277837\pi$
$877$	38.0628	1.28529	0.642645	+	0.766164i	$0.277837\pi$
$878$	−45.1440	−1.52354
$879$	0	0
$880$	0	0
$881$	−28.9338	−0.974803	−0.487402	−	0.873178i	$-0.662055\pi$
$881$	−28.9338	−0.974803	−0.487402	+	0.873178i	$0.662055\pi$
$882$	0	0
$883$	−13.7208	−0.461743	−0.230871	−	0.972984i	$-0.574158\pi$
$883$	−13.7208	−0.461743	−0.230871	+	0.972984i	$0.574158\pi$
$884$	−49.9830	−1.68111
$885$	0	0
$886$	21.1214	0.709586
$887$	−14.9778	−0.502907	−0.251453	−	0.967869i	$-0.580908\pi$
$887$	−14.9778	−0.502907	−0.251453	+	0.967869i	$0.580908\pi$
$888$	0	0
$889$	26.8799	0.901524
$890$	0	0
$891$	0	0
$892$	3.06981	0.102785
$893$	−6.84806	−0.229162
$894$	0	0
$895$	0	0
$896$	28.5699	0.954452
$897$	0	0
$898$	59.9102	1.99923
$899$	−2.00684	−0.0669318
$900$	0	0
$901$	29.1191	0.970097
$902$	−18.5823	−0.618722
$903$	0	0
$904$	17.0992	0.568711
$905$	0	0
$906$	0	0
$907$	−19.4807	−0.646848	−0.323424	−	0.946254i	$-0.604834\pi$
$907$	−19.4807	−0.646848	−0.323424	+	0.946254i	$0.604834\pi$
$908$	85.7598	2.84604
$909$	0	0
$910$	0	0
$911$	−38.8443	−1.28697	−0.643484	−	0.765459i	$-0.722512\pi$
$911$	−38.8443	−1.28697	−0.643484	+	0.765459i	$0.722512\pi$
$912$	0	0
$913$	0.853832	0.0282577
$914$	20.5802	0.680731
$915$	0	0
$916$	60.0121	1.98286
$917$	−27.5172	−0.908697
$918$	0	0
$919$	31.5224	1.03983	0.519914	−	0.854219i	$-0.325964\pi$
$919$	31.5224	1.03983	0.519914	+	0.854219i	$0.325964\pi$
$920$	0	0
$921$	0	0
$922$	−53.3130	−1.75577
$923$	−16.4236	−0.540590
$924$	0	0
$925$	0	0
$926$	−78.3812	−2.57577
$927$	0	0
$928$	−56.3013	−1.84818
$929$	22.6619	0.743514	0.371757	−	0.928330i	$-0.378755\pi$
$929$	22.6619	0.743514	0.371757	+	0.928330i	$0.378755\pi$
$930$	0	0
$931$	10.2714	0.336632
$932$	−13.9318	−0.456352
$933$	0	0
$934$	−87.2109	−2.85363
$935$	0	0
$936$	0	0
$937$	7.99594	0.261216	0.130608	−	0.991434i	$-0.458307\pi$
$937$	7.99594	0.261216	0.130608	+	0.991434i	$0.458307\pi$
$938$	−126.637	−4.13485
$939$	0	0
$940$	0	0
$941$	13.5377	0.441317	0.220658	−	0.975351i	$-0.429179\pi$
$941$	13.5377	0.441317	0.220658	+	0.975351i	$0.429179\pi$
$942$	0	0
$943$	0.992064	0.0323061
$944$	−74.5839	−2.42750
$945$	0	0
$946$	68.4629	2.22592
$947$	−35.6664	−1.15900	−0.579502	−	0.814971i	$-0.696753\pi$
$947$	−35.6664	−1.15900	−0.579502	+	0.814971i	$0.696753\pi$
$948$	0	0
$949$	−30.2940	−0.983384
$950$	0	0
$951$	0	0
$952$	−153.967	−4.99012
$953$	28.8285	0.933847	0.466923	−	0.884298i	$-0.345362\pi$
$953$	28.8285	0.933847	0.466923	+	0.884298i	$0.345362\pi$
$954$	0	0
$955$	0	0
$956$	−63.5837	−2.05644
$957$	0	0
$958$	−77.5494	−2.50551
$959$	−77.5155	−2.50311
$960$	0	0
$961$	−30.7927	−0.993312
$962$	−13.9729	−0.450503
$963$	0	0
$964$	16.0103	0.515658
$965$	0	0
$966$	0	0
$967$	−32.4097	−1.04223	−0.521113	−	0.853488i	$-0.674483\pi$
$967$	−32.4097	−1.04223	−0.521113	+	0.853488i	$0.674483\pi$
$968$	35.9538	1.15560
$969$	0	0
$970$	0	0
$971$	43.0622	1.38193	0.690966	−	0.722888i	$-0.257186\pi$
$971$	43.0622	1.38193	0.690966	+	0.722888i	$0.257186\pi$
$972$	0	0
$973$	−47.4123	−1.51997
$974$	57.2970	1.83591
$975$	0	0
$976$	−65.8087	−2.10649
$977$	11.3350	0.362640	0.181320	−	0.983424i	$-0.441963\pi$
$977$	11.3350	0.362640	0.181320	+	0.983424i	$0.441963\pi$
$978$	0	0
$979$	33.1040	1.05801
$980$	0	0
$981$	0	0
$982$	−73.5668	−2.34761
$983$	−22.7836	−0.726684	−0.363342	−	0.931656i	$-0.618364\pi$
$983$	−22.7836	−0.726684	−0.363342	+	0.931656i	$0.618364\pi$
$984$	0	0
$985$	0	0
$986$	52.5309	1.67292
$987$	0	0
$988$	9.57174	0.304517
$989$	−3.65507	−0.116225
$990$	0	0
$991$	−50.5633	−1.60620	−0.803099	−	0.595846i	$-0.796817\pi$
$991$	−50.5633	−1.60620	−0.803099	+	0.595846i	$0.796817\pi$
$992$	5.81631	0.184668
$993$	0	0
$994$	−84.6274	−2.68422
$995$	0	0
$996$	0	0
$997$	60.3040	1.90985	0.954924	−	0.296850i	$-0.0959362\pi$
$997$	60.3040	1.90985	0.954924	+	0.296850i	$0.0959362\pi$
$998$	25.9766	0.822276
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bf.1.1		24
3.2	odd	2	inner	5625.2.a.bf.1.23		24
5.4	even	2	inner	5625.2.a.bf.1.24		24
15.14	odd	2	inner	5625.2.a.bf.1.2		24
25.3	odd	20		225.2.m.c.109.6	yes	24
25.17	odd	20		225.2.m.c.64.6	yes	24
75.17	even	20		225.2.m.c.64.1	✓	24
75.53	even	20		225.2.m.c.109.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.m.c.64.1	✓	24	75.17	even	20
225.2.m.c.64.6	yes	24	25.17	odd	20
225.2.m.c.109.1	yes	24	75.53	even	20
225.2.m.c.109.6	yes	24	25.3	odd	20
5625.2.a.bf.1.1		24	1.1	even	1	trivial
5625.2.a.bf.1.2		24	15.14	odd	2	inner
5625.2.a.bf.1.23		24	3.2	odd	2	inner
5625.2.a.bf.1.24		24	5.4	even	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)

\(f(q)\)	\(=\)	\(q-2.64061 q^{2} +4.97280 q^{4} -4.34551 q^{7} -7.84999 q^{8} +O(q^{10})\)	\(q-2.64061 q^{2} +4.97280 q^{4} -4.34551 q^{7} -7.84999 q^{8} -2.53375 q^{11} +2.22691 q^{13} +11.4748 q^{14} +10.7831 q^{16} -4.51356 q^{17} +0.864346 q^{19} +6.69063 q^{22} -0.357197 q^{23} -5.88039 q^{26} -21.6093 q^{28} +4.40749 q^{29} -0.455324 q^{31} -12.7740 q^{32} +11.9185 q^{34} +2.37618 q^{37} -2.28240 q^{38} -2.77736 q^{41} +10.2326 q^{43} -12.5998 q^{44} +0.943217 q^{46} -7.92283 q^{47} +11.8834 q^{49} +11.0740 q^{52} -6.45146 q^{53} +34.1122 q^{56} -11.6385 q^{58} -6.91672 q^{59} -6.10294 q^{61} +1.20233 q^{62} +12.1649 q^{64} -11.0361 q^{67} -22.4450 q^{68} -7.37508 q^{71} -13.6036 q^{73} -6.27455 q^{74} +4.29822 q^{76} +11.0104 q^{77} +10.7830 q^{79} +7.33391 q^{82} -0.336984 q^{83} -27.0204 q^{86} +19.8899 q^{88} -13.0652 q^{89} -9.67705 q^{91} -1.77627 q^{92} +20.9211 q^{94} -7.78850 q^{97} -31.3795 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 32 q^{4}+O(q^{10}) \) 24 * q + 32 * q^4	\( 24 q + 32 q^{4} + 56 q^{16} + 36 q^{19} + 52 q^{31} + 60 q^{34} + 60 q^{46} + 72 q^{49} + 68 q^{61} + 108 q^{64} + 88 q^{76} + 84 q^{79} + 80 q^{91} + 100 q^{94}+O(q^{100}) \) 24 * q + 32 * q^4 + 56 * q^16 + 36 * q^19 + 52 * q^31 + 60 * q^34 + 60 * q^46 + 72 * q^49 + 68 * q^61 + 108 * q^64 + 88 * q^76 + 84 * q^79 + 80 * q^91 + 100 * q^94

Introduction

L-functions

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