LMFDB - Embedded newform 5625.2.a.bf.1.2

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			1.2
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	$-0.193292\pi$
$7$	4.34551	1.64245	0.821224	+	0.570606i	$0.193292\pi$
$8$	−7.84999	−2.77539
$9$	0	0
$10$	0	0
$11$	2.53375	0.763954	0.381977	−	0.924172i	$-0.375243\pi$
$11$	2.53375	0.763954	0.381977	+	0.924172i	$0.375243\pi$
$12$	0	0
$13$	−2.22691	−0.617633	−0.308817	−	0.951122i	$-0.599933\pi$
$13$	−2.22691	−0.617633	−0.308817	+	0.951122i	$0.599933\pi$
$14$	−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	$-0.634201\pi$
$29$	−4.40749	−0.818451	−0.409225	+	0.912433i	$0.634201\pi$
$30$	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	$-0.562575\pi$
$37$	−2.37618	−0.390641	−0.195321	+	0.980739i	$0.562575\pi$
$38$	−2.28240	−0.370254
$39$	0	0
$40$	0	0
$41$	2.77736	0.433750	0.216875	−	0.976199i	$-0.430414\pi$
$41$	2.77736	0.433750	0.216875	+	0.976199i	$0.430414\pi$
$42$	0	0
$43$	−10.2326	−1.56046	−0.780232	−	0.625490i	$-0.784899\pi$
$43$	−10.2326	−1.56046	−0.780232	+	0.625490i	$0.784899\pi$
$44$	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	$-0.351338\pi$
$59$	6.91672	0.900481	0.450240	+	0.892907i	$0.351338\pi$
$60$	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	$-0.264515\pi$
$67$	11.0361	1.34828	0.674140	+	0.738604i	$0.264515\pi$
$68$	−22.4450	−2.72186
$69$	0	0
$70$	0	0
$71$	7.37508	0.875261	0.437631	−	0.899155i	$-0.355818\pi$
$71$	7.37508	0.875261	0.437631	+	0.899155i	$0.355818\pi$
$72$	0	0
$73$	13.6036	1.59218	0.796090	−	0.605178i	$-0.206898\pi$
$73$	13.6036	1.59218	0.796090	+	0.605178i	$0.206898\pi$
$74$	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	$-0.256529\pi$
$89$	13.0652	1.38491	0.692456	+	0.721460i	$0.256529\pi$
$90$	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	$-0.370606\pi$
$97$	7.78850	0.790802	0.395401	+	0.918509i	$0.370606\pi$
$98$	−31.3795	−3.16981
$99$	0	0
$100$	0	0
$101$	16.6461	1.65635	0.828176	−	0.560468i	$-0.189379\pi$
$101$	16.6461	1.65635	0.828176	+	0.560468i	$0.189379\pi$
$102$	0	0
$103$	−1.80337	−0.177691	−0.0888456	−	0.996045i	$-0.528318\pi$
$103$	−1.80337	−0.177691	−0.0888456	+	0.996045i	$0.528318\pi$
$104$	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	$-0.411506\pi$
$127$	6.18568	0.548890	0.274445	+	0.961603i	$0.411506\pi$
$128$	−6.57457	−0.581116
$129$	0	0
$130$	0	0
$131$	−6.33233	−0.553258	−0.276629	−	0.960977i	$-0.589217\pi$
$131$	−6.33233	−0.553258	−0.276629	+	0.960977i	$0.589217\pi$
$132$	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	$-0.267626\pi$
$149$	16.2808	1.33378	0.666889	+	0.745157i	$0.267626\pi$
$150$	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	$-0.388591\pi$
$157$	8.59302	0.685798	0.342899	+	0.939372i	$0.388591\pi$
$158$	−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	$-0.586454\pi$
$163$	−6.85021	−0.536550	−0.268275	+	0.963342i	$0.586454\pi$
$164$	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	$-0.582248\pi$
$179$	−6.83738	−0.511050	−0.255525	+	0.966803i	$0.582248\pi$
$180$	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	$-0.497984\pi$
$191$	0.175077	0.0126682	0.00633408	+	0.999980i	$0.497984\pi$
$192$	0	0
$193$	10.0004	0.719844	0.359922	−	0.932982i	$-0.382803\pi$
$193$	10.0004	0.719844	0.359922	+	0.932982i	$0.382803\pi$
$194$	−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	$-0.506580\pi$
$223$	−0.617321	−0.0413388	−0.0206694	+	0.999786i	$0.506580\pi$
$224$	−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	$-0.364293\pi$
$239$	12.7863	0.827077	0.413538	+	0.910487i	$0.364293\pi$
$240$	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	$-0.703420\pi$
$251$	−18.8989	−1.19289	−0.596444	+	0.802655i	$0.703420\pi$
$252$	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	$-0.512662\pi$
$269$	−1.30450	−0.0795368	−0.0397684	+	0.999209i	$0.512662\pi$
$270$	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	$-0.364363\pi$
$277$	13.7586	0.826675	0.413338	+	0.910578i	$0.364363\pi$
$278$	−28.8107	−1.72795
$279$	0	0
$280$	0	0
$281$	−28.8763	−1.72261	−0.861307	−	0.508086i	$-0.830353\pi$
$281$	−28.8763	−1.72261	−0.861307	+	0.508086i	$0.830353\pi$
$282$	0	0
$283$	−31.6927	−1.88394	−0.941968	−	0.335704i	$-0.891026\pi$
$283$	−31.6927	−1.88394	−0.941968	+	0.335704i	$0.891026\pi$
$284$	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	$-0.673490\pi$
$307$	−18.1679	−1.03690	−0.518448	+	0.855109i	$0.673490\pi$
$308$	54.7526	3.11982
$309$	0	0
$310$	0	0
$311$	−14.3412	−0.813216	−0.406608	−	0.913603i	$-0.633288\pi$
$311$	−14.3412	−0.813216	−0.406608	+	0.913603i	$0.633288\pi$
$312$	0	0
$313$	9.06837	0.512574	0.256287	−	0.966601i	$-0.417501\pi$
$313$	9.06837	0.512574	0.256287	+	0.966601i	$0.417501\pi$
$314$	−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	$-0.375839\pi$
$337$	13.9608	0.760495	0.380248	+	0.924885i	$0.375839\pi$
$338$	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	$-0.186773\pi$
$359$	31.5562	1.66547	0.832736	+	0.553670i	$0.186773\pi$
$360$	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	$-0.609688\pi$
$367$	−12.9433	−0.675632	−0.337816	+	0.941212i	$0.609688\pi$
$368$	−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	$-0.431946\pi$
$373$	8.19546	0.424345	0.212172	+	0.977232i	$0.431946\pi$
$374$	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	$-0.661122\pi$
$389$	−19.1250	−0.969678	−0.484839	+	0.874603i	$0.661122\pi$
$390$	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	$-0.476946\pi$
$397$	2.88366	0.144727	0.0723634	+	0.997378i	$0.476946\pi$
$398$	−5.59604	−0.280504
$399$	0	0
$400$	0	0
$401$	−5.50842	−0.275077	−0.137539	−	0.990496i	$-0.543919\pi$
$401$	−5.50842	−0.275077	−0.137539	+	0.990496i	$0.543919\pi$
$402$	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	$-0.697695\pi$
$419$	−23.8229	−1.16382	−0.581912	+	0.813252i	$0.697695\pi$
$420$	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	$-0.284272\pi$
$431$	26.0347	1.25405	0.627025	+	0.778999i	$0.284272\pi$
$432$	0	0
$433$	5.28263	0.253867	0.126933	−	0.991911i	$-0.459487\pi$
$433$	5.28263	0.253867	0.126933	+	0.991911i	$0.459487\pi$
$434$	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	$-0.320177\pi$
$449$	22.6881	1.07072	0.535358	+	0.844625i	$0.320177\pi$
$450$	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	$-0.441650\pi$
$457$	7.79373	0.364575	0.182288	+	0.983245i	$0.441650\pi$
$458$	−31.8670	−1.48905
$459$	0	0
$460$	0	0
$461$	−20.1897	−0.940328	−0.470164	−	0.882579i	$-0.655805\pi$
$461$	−20.1897	−0.940328	−0.470164	+	0.882579i	$0.655805\pi$
$462$	0	0
$463$	−29.6830	−1.37949	−0.689744	−	0.724054i	$-0.742277\pi$
$463$	−29.6830	−1.37949	−0.689744	+	0.724054i	$0.742277\pi$
$464$	−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	$-0.734105\pi$
$479$	−29.3680	−1.34186	−0.670930	+	0.741521i	$0.734105\pi$
$480$	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	$-0.336403\pi$
$487$	21.6984	0.983250	0.491625	+	0.870807i	$0.336403\pi$
$488$	47.9080	2.16869
$489$	0	0
$490$	0	0
$491$	−27.8598	−1.25730	−0.628648	−	0.777690i	$-0.716391\pi$
$491$	−27.8598	−1.25730	−0.628648	+	0.777690i	$0.716391\pi$
$492$	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	$-0.590372\pi$
$509$	−12.6392	−0.560224	−0.280112	+	0.959967i	$0.590372\pi$
$510$	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	$-0.323348\pi$
$521$	24.0542	1.05383	0.526917	+	0.849917i	$0.323348\pi$
$522$	0	0
$523$	4.32020	0.188909	0.0944546	−	0.995529i	$-0.469889\pi$
$523$	4.32020	0.188909	0.0944546	+	0.995529i	$0.469889\pi$
$524$	−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	$-0.383212\pi$
$547$	16.7797	0.717450	0.358725	+	0.933443i	$0.383212\pi$
$548$	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	$-0.919272\pi$
$569$	−46.1814	−1.93602	−0.968012	+	0.250905i	$0.919272\pi$
$570$	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	$-0.290946\pi$
$577$	29.3322	1.22111	0.610557	+	0.791972i	$0.290946\pi$
$578$	−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	$-0.270818\pi$
$599$	32.2761	1.31877	0.659383	+	0.751807i	$0.270818\pi$
$600$	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	$-0.717715\pi$
$607$	−31.1356	−1.26375	−0.631877	+	0.775069i	$0.717715\pi$
$608$	−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	$-0.647724\pi$
$613$	−22.1646	−0.895218	−0.447609	+	0.894229i	$0.647724\pi$
$614$	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	$-0.0956446\pi$
$641$	48.3672	1.91039	0.955196	+	0.295975i	$0.0956446\pi$
$642$	0	0
$643$	−34.0606	−1.34322	−0.671609	−	0.740906i	$-0.734396\pi$
$643$	−34.0606	−1.34322	−0.671609	+	0.740906i	$0.734396\pi$
$644$	−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	$-0.675377\pi$
$659$	−26.8779	−1.04701	−0.523507	+	0.852021i	$0.675377\pi$
$660$	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	$-0.338282\pi$
$673$	25.2406	0.972952	0.486476	+	0.873694i	$0.338282\pi$
$674$	−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	$-0.423474\pi$
$701$	12.6083	0.476209	0.238104	+	0.971240i	$0.423474\pi$
$702$	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	$-0.675637\pi$
$719$	−28.1122	−1.04841	−0.524204	+	0.851593i	$0.675637\pi$
$720$	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	$-0.565085\pi$
$727$	−10.9496	−0.406098	−0.203049	+	0.979169i	$0.565085\pi$
$728$	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	$-0.681643\pi$
$733$	−29.2496	−1.08036	−0.540179	+	0.841550i	$0.681643\pi$
$734$	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	$-0.674998\pi$
$757$	−28.7514	−1.04499	−0.522494	+	0.852643i	$0.674998\pi$
$758$	47.4086	1.72196
$759$	0	0
$760$	0	0
$761$	14.6353	0.530530	0.265265	−	0.964176i	$-0.414541\pi$
$761$	14.6353	0.530530	0.265265	+	0.964176i	$0.414541\pi$
$762$	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	$-0.516063\pi$
$787$	−2.83022	−0.100886	−0.0504432	+	0.998727i	$0.516063\pi$
$788$	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	$-0.623486\pi$
$809$	−21.5190	−0.756569	−0.378285	+	0.925689i	$0.623486\pi$
$810$	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	$-0.350466\pi$
$821$	25.9417	0.905370	0.452685	+	0.891670i	$0.350466\pi$
$822$	0	0
$823$	2.65720	0.0926242	0.0463121	−	0.998927i	$-0.485253\pi$
$823$	2.65720	0.0926242	0.0463121	+	0.998927i	$0.485253\pi$
$824$	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	$-0.532478\pi$
$839$	−5.90055	−0.203709	−0.101855	+	0.994799i	$0.532478\pi$
$840$	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	$-0.224665\pi$
$853$	44.4569	1.52218	0.761088	+	0.648648i	$0.224665\pi$
$854$	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	$-0.722163\pi$
$877$	−38.0628	−1.28529	−0.642645	+	0.766164i	$0.722163\pi$
$878$	−45.1440	−1.52354
$879$	0	0
$880$	0	0
$881$	28.9338	0.974803	0.487402	−	0.873178i	$-0.337945\pi$
$881$	28.9338	0.974803	0.487402	+	0.873178i	$0.337945\pi$
$882$	0	0
$883$	13.7208	0.461743	0.230871	−	0.972984i	$-0.425842\pi$
$883$	13.7208	0.461743	0.230871	+	0.972984i	$0.425842\pi$
$884$	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	$-0.395166\pi$
$907$	19.4807	0.646848	0.323424	+	0.946254i	$0.395166\pi$
$908$	85.7598	2.84604
$909$	0	0
$910$	0	0
$911$	38.8443	1.28697	0.643484	−	0.765459i	$-0.277488\pi$
$911$	38.8443	1.28697	0.643484	+	0.765459i	$0.277488\pi$
$912$	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	$-0.621245\pi$
$929$	−22.6619	−0.743514	−0.371757	+	0.928330i	$0.621245\pi$
$930$	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	$-0.541693\pi$
$937$	−7.99594	−0.261216	−0.130608	+	0.991434i	$0.541693\pi$
$938$	−126.637	−4.13485
$939$	0	0
$940$	0	0
$941$	−13.5377	−0.441317	−0.220658	−	0.975351i	$-0.570821\pi$
$941$	−13.5377	−0.441317	−0.220658	+	0.975351i	$0.570821\pi$
$942$	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	$-0.325517\pi$
$967$	32.4097	1.04223	0.521113	+	0.853488i	$0.325517\pi$
$968$	35.9538	1.15560
$969$	0	0
$970$	0	0
$971$	−43.0622	−1.38193	−0.690966	−	0.722888i	$-0.742814\pi$
$971$	−43.0622	−1.38193	−0.690966	+	0.722888i	$0.742814\pi$
$972$	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	$-0.904064\pi$
$997$	−60.3040	−1.90985	−0.954924	+	0.296850i	$0.904064\pi$
$998$	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.1.2		24
3.2	odd	2	inner	5625.2.a.bf.1.24		24
5.4	even	2	inner	5625.2.a.bf.1.23		24
15.14	odd	2	inner	5625.2.a.bf.1.1		24
25.8	odd	20		225.2.m.c.64.1	✓	24
25.22	odd	20		225.2.m.c.109.1	yes	24
75.8	even	20		225.2.m.c.64.6	yes	24
75.47	even	20		225.2.m.c.109.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
225.2.m.c.64.1	✓	24	25.8	odd	20
225.2.m.c.64.6	yes	24	75.8	even	20
225.2.m.c.109.1	yes	24	25.22	odd	20
225.2.m.c.109.6	yes	24	75.47	even	20
5625.2.a.bf.1.1		24	15.14	odd	2	inner
5625.2.a.bf.1.2		24	1.1	even	1	trivial
5625.2.a.bf.1.23		24	5.4	even	2	inner
5625.2.a.bf.1.24		24	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)

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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more