LMFDB - Embedded newform 5625.2.a.bd.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:	$1$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.35083$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.35083 q^{2} -0.175259 q^{4} -1.59580 q^{7} +2.93840 q^{8} +O(q^{10})$	$q-1.35083 q^{2} -0.175259 q^{4} -1.59580 q^{7} +2.93840 q^{8} -3.33277 q^{11} -7.05132 q^{13} +2.15565 q^{14} -3.61877 q^{16} +4.09625 q^{17} +0.567535 q^{19} +4.50200 q^{22} +6.30400 q^{23} +9.52513 q^{26} +0.279678 q^{28} +2.78357 q^{29} -0.995824 q^{31} -0.988473 q^{32} -5.53333 q^{34} +3.55334 q^{37} -0.766643 q^{38} -1.16293 q^{41} -0.117022 q^{43} +0.584098 q^{44} -8.51563 q^{46} +7.64173 q^{47} -4.45343 q^{49} +1.23581 q^{52} +0.523635 q^{53} -4.68910 q^{56} -3.76013 q^{58} -0.983998 q^{59} +10.6137 q^{61} +1.34519 q^{62} +8.57279 q^{64} +15.2159 q^{67} -0.717905 q^{68} +10.6639 q^{71} -5.55832 q^{73} -4.79996 q^{74} -0.0994657 q^{76} +5.31842 q^{77} -14.5969 q^{79} +1.57091 q^{82} +5.02398 q^{83} +0.158076 q^{86} -9.79302 q^{88} -2.82350 q^{89} +11.2525 q^{91} -1.10483 q^{92} -10.3227 q^{94} +1.70592 q^{97} +6.01583 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) $ 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8	$ 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8} - 2 q^{11} - 16 q^{13} - 6 q^{14} + 16 q^{17} - 14 q^{19} - 12 q^{22} + 14 q^{23} - 6 q^{26} - 16 q^{28} - 2 q^{29} - 22 q^{31} - 2 q^{32} - 12 q^{34} - 28 q^{37} - 16 q^{38} - 8 q^{41} - 20 q^{43} - 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{52} + 44 q^{53} - 30 q^{56} - 8 q^{58} - 14 q^{59} - 20 q^{61} + 16 q^{62} + 6 q^{64} - 16 q^{67} - 2 q^{68} - 16 q^{71} - 24 q^{73} - 26 q^{74} - 16 q^{76} + 46 q^{77} - 30 q^{79} - 16 q^{82} + 12 q^{83} - 32 q^{86} - 32 q^{88} - 16 q^{89} - 12 q^{91} - 2 q^{92} + 14 q^{94} - 16 q^{97} + 4 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8 - 2 * q^11 - 16 * q^13 - 6 * q^14 + 16 * q^17 - 14 * q^19 - 12 * q^22 + 14 * q^23 - 6 * q^26 - 16 * q^28 - 2 * q^29 - 22 * q^31 - 2 * q^32 - 12 * q^34 - 28 * q^37 - 16 * q^38 - 8 * q^41 - 20 * q^43 - 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^52 + 44 * q^53 - 30 * q^56 - 8 * q^58 - 14 * q^59 - 20 * q^61 + 16 * q^62 + 6 * q^64 - 16 * q^67 - 2 * q^68 - 16 * q^71 - 24 * q^73 - 26 * q^74 - 16 * q^76 + 46 * q^77 - 30 * q^79 - 16 * q^82 + 12 * q^83 - 32 * q^86 - 32 * q^88 - 16 * q^89 - 12 * q^91 - 2 * q^92 + 14 * q^94 - 16 * q^97 + 4 * q^98

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 $

$2$	−1.35083	−0.955181	−0.477590	−	0.878583i	$-0.658490\pi$
$2$	−1.35083	−0.955181	−0.477590	+	0.878583i	$0.658490\pi$
$3$	0	0
$3$	0	0
$4$	−0.175259	−0.0876296
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.59580	−0.603155	−0.301577	−	0.953442i	$-0.597513\pi$
$7$	−1.59580	−0.603155	−0.301577	+	0.953442i	$0.597513\pi$
$8$	2.93840	1.03888
$9$	0	0
$10$	0	0
$11$	−3.33277	−1.00487	−0.502434	−	0.864616i	$-0.667562\pi$
$11$	−3.33277	−1.00487	−0.502434	+	0.864616i	$0.667562\pi$
$12$	0	0
$13$	−7.05132	−1.95568	−0.977842	−	0.209345i	$-0.932867\pi$
$13$	−7.05132	−1.95568	−0.977842	+	0.209345i	$0.932867\pi$
$14$	2.15565	0.576122
$15$	0	0
$16$	−3.61877	−0.904691
$17$	4.09625	0.993486	0.496743	−	0.867898i	$-0.334529\pi$
$17$	4.09625	0.993486	0.496743	+	0.867898i	$0.334529\pi$
$18$	0	0
$19$	0.567535	0.130201	0.0651007	−	0.997879i	$-0.479263\pi$
$19$	0.567535	0.130201	0.0651007	+	0.997879i	$0.479263\pi$
$20$	0	0
$21$	0	0
$22$	4.50200	0.959830
$23$	6.30400	1.31448	0.657238	−	0.753683i	$-0.271725\pi$
$23$	6.30400	1.31448	0.657238	+	0.753683i	$0.271725\pi$
$24$	0	0
$25$	0	0
$26$	9.52513	1.86803
$27$	0	0
$28$	0.279678	0.0528542
$29$	2.78357	0.516897	0.258448	−	0.966025i	$-0.416789\pi$
$29$	2.78357	0.516897	0.258448	+	0.966025i	$0.416789\pi$
$30$	0	0
$31$	−0.995824	−0.178855	−0.0894276	−	0.995993i	$-0.528504\pi$
$31$	−0.995824	−0.178855	−0.0894276	+	0.995993i	$0.528504\pi$
$32$	−0.988473	−0.174739
$33$	0	0
$34$	−5.53333	−0.948959
$35$	0	0
$36$	0	0
$37$	3.55334	0.584165	0.292083	−	0.956393i	$-0.405652\pi$
$37$	3.55334	0.584165	0.292083	+	0.956393i	$0.405652\pi$
$38$	−0.766643	−0.124366
$39$	0	0
$40$	0	0
$41$	−1.16293	−0.181618	−0.0908092	−	0.995868i	$-0.528945\pi$
$41$	−1.16293	−0.181618	−0.0908092	+	0.995868i	$0.528945\pi$
$42$	0	0
$43$	−0.117022	−0.0178456	−0.00892281	−	0.999960i	$-0.502840\pi$
$43$	−0.117022	−0.0178456	−0.00892281	+	0.999960i	$0.502840\pi$
$44$	0.584098	0.0880561
$45$	0	0
$46$	−8.51563	−1.25556
$47$	7.64173	1.11466	0.557331	−	0.830291i	$-0.311826\pi$
$47$	7.64173	1.11466	0.557331	+	0.830291i	$0.311826\pi$
$48$	0	0
$49$	−4.45343	−0.636205
$50$	0	0
$51$	0	0
$52$	1.23581	0.171376
$53$	0.523635	0.0719268	0.0359634	−	0.999353i	$-0.488550\pi$
$53$	0.523635	0.0719268	0.0359634	+	0.999353i	$0.488550\pi$
$54$	0	0
$55$	0	0
$56$	−4.68910	−0.626607
$57$	0	0
$58$	−3.76013	−0.493730
$59$	−0.983998	−0.128106	−0.0640528	−	0.997947i	$-0.520403\pi$
$59$	−0.983998	−0.128106	−0.0640528	+	0.997947i	$0.520403\pi$
$60$	0	0
$61$	10.6137	1.35895	0.679473	−	0.733701i	$-0.262208\pi$
$61$	10.6137	1.35895	0.679473	+	0.733701i	$0.262208\pi$
$62$	1.34519	0.170839
$63$	0	0
$64$	8.57279	1.07160
$65$	0	0
$66$	0	0
$67$	15.2159	1.85892	0.929461	−	0.368920i	$-0.120272\pi$
$67$	15.2159	1.85892	0.929461	+	0.368920i	$0.120272\pi$
$68$	−0.717905	−0.0870588
$69$	0	0
$70$	0	0
$71$	10.6639	1.26558	0.632788	−	0.774325i	$-0.281910\pi$
$71$	10.6639	1.26558	0.632788	+	0.774325i	$0.281910\pi$
$72$	0	0
$73$	−5.55832	−0.650552	−0.325276	−	0.945619i	$-0.605457\pi$
$73$	−5.55832	−0.650552	−0.325276	+	0.945619i	$0.605457\pi$
$74$	−4.79996	−0.557984
$75$	0	0
$76$	−0.0994657	−0.0114095
$77$	5.31842	0.606090
$78$	0	0
$79$	−14.5969	−1.64227	−0.821137	−	0.570731i	$-0.806660\pi$
$79$	−14.5969	−1.64227	−0.821137	+	0.570731i	$0.806660\pi$
$80$	0	0
$81$	0	0
$82$	1.57091	0.173478
$83$	5.02398	0.551453	0.275727	−	0.961236i	$-0.411082\pi$
$83$	5.02398	0.551453	0.275727	+	0.961236i	$0.411082\pi$
$84$	0	0
$85$	0	0
$86$	0.158076	0.0170458
$87$	0	0
$88$	−9.79302	−1.04394
$89$	−2.82350	−0.299291	−0.149645	−	0.988740i	$-0.547813\pi$
$89$	−2.82350	−0.299291	−0.149645	+	0.988740i	$0.547813\pi$
$90$	0	0
$91$	11.2525	1.17958
$92$	−1.10483	−0.115187
$93$	0	0
$94$	−10.3227	−1.06470
$95$	0	0
$96$	0	0
$97$	1.70592	0.173210	0.0866049	−	0.996243i	$-0.472398\pi$
$97$	1.70592	0.173210	0.0866049	+	0.996243i	$0.472398\pi$
$98$	6.01583	0.607690
$99$	0	0
$100$	0	0
$101$	−13.1747	−1.31093	−0.655464	−	0.755226i	$-0.727527\pi$
$101$	−13.1747	−1.31093	−0.655464	+	0.755226i	$0.727527\pi$
$102$	0	0
$103$	−10.8720	−1.07125	−0.535624	−	0.844456i	$-0.679924\pi$
$103$	−10.8720	−1.07125	−0.535624	+	0.844456i	$0.679924\pi$
$104$	−20.7196	−2.03173
$105$	0	0
$106$	−0.707341	−0.0687031
$107$	9.37236	0.906060	0.453030	−	0.891495i	$-0.350343\pi$
$107$	9.37236	0.906060	0.453030	+	0.891495i	$0.350343\pi$
$108$	0	0
$109$	−15.5899	−1.49324	−0.746622	−	0.665249i	$-0.768326\pi$
$109$	−15.5899	−1.49324	−0.746622	+	0.665249i	$0.768326\pi$
$110$	0	0
$111$	0	0
$112$	5.77482	0.545669
$113$	−10.0481	−0.945243	−0.472622	−	0.881265i	$-0.656692\pi$
$113$	−10.0481	−0.945243	−0.472622	+	0.881265i	$0.656692\pi$
$114$	0	0
$115$	0	0
$116$	−0.487847	−0.0452954
$117$	0	0
$118$	1.32921	0.122364
$119$	−6.53678	−0.599226
$120$	0	0
$121$	0.107347	0.00975880
$122$	−14.3373	−1.29804
$123$	0	0
$124$	0.174527	0.0156730
$125$	0	0
$126$	0	0
$127$	−0.976784	−0.0866756	−0.0433378	−	0.999060i	$-0.513799\pi$
$127$	−0.976784	−0.0866756	−0.0433378	+	0.999060i	$0.513799\pi$
$128$	−9.60343	−0.848832
$129$	0	0
$130$	0	0
$131$	−10.0616	−0.879086	−0.439543	−	0.898221i	$-0.644860\pi$
$131$	−10.0616	−0.879086	−0.439543	+	0.898221i	$0.644860\pi$
$132$	0	0
$133$	−0.905670	−0.0785316
$134$	−20.5541	−1.77561
$135$	0	0
$136$	12.0364	1.03212
$137$	4.87244	0.416281	0.208140	−	0.978099i	$-0.433259\pi$
$137$	4.87244	0.416281	0.208140	+	0.978099i	$0.433259\pi$
$138$	0	0
$139$	0.185784	0.0157580	0.00787898	−	0.999969i	$-0.497492\pi$
$139$	0.185784	0.0157580	0.00787898	+	0.999969i	$0.497492\pi$
$140$	0	0
$141$	0	0
$142$	−14.4052	−1.20885
$143$	23.5004	1.96520
$144$	0	0
$145$	0	0
$146$	7.50834	0.621395
$147$	0	0
$148$	−0.622755	−0.0511902
$149$	−3.88889	−0.318590	−0.159295	−	0.987231i	$-0.550922\pi$
$149$	−3.88889	−0.318590	−0.159295	+	0.987231i	$0.550922\pi$
$150$	0	0
$151$	−22.1146	−1.79966	−0.899829	−	0.436242i	$-0.856309\pi$
$151$	−22.1146	−1.79966	−0.899829	+	0.436242i	$0.856309\pi$
$152$	1.66765	0.135264
$153$	0	0
$154$	−7.18428	−0.578926
$155$	0	0
$156$	0	0
$157$	13.6058	1.08586	0.542931	−	0.839777i	$-0.317314\pi$
$157$	13.6058	1.08586	0.542931	+	0.839777i	$0.317314\pi$
$158$	19.7179	1.56867
$159$	0	0
$160$	0	0
$161$	−10.0599	−0.792832
$162$	0	0
$163$	8.62895	0.675871	0.337936	−	0.941169i	$-0.390271\pi$
$163$	8.62895	0.675871	0.337936	+	0.941169i	$0.390271\pi$
$164$	0.203813	0.0159151
$165$	0	0
$166$	−6.78654	−0.526737
$167$	−6.59891	−0.510639	−0.255319	−	0.966857i	$-0.582181\pi$
$167$	−6.59891	−0.510639	−0.255319	+	0.966857i	$0.582181\pi$
$168$	0	0
$169$	36.7211	2.82470
$170$	0	0
$171$	0	0
$172$	0.0205091	0.00156381
$173$	−13.6595	−1.03851	−0.519257	−	0.854618i	$-0.673791\pi$
$173$	−13.6595	−1.03851	−0.519257	+	0.854618i	$0.673791\pi$
$174$	0	0
$175$	0	0
$176$	12.0605	0.909095
$177$	0	0
$178$	3.81407	0.285877
$179$	−9.82880	−0.734639	−0.367320	−	0.930095i	$-0.619724\pi$
$179$	−9.82880	−0.734639	−0.367320	+	0.930095i	$0.619724\pi$
$180$	0	0
$181$	17.8687	1.32817	0.664085	−	0.747657i	$-0.268821\pi$
$181$	17.8687	1.32817	0.664085	+	0.747657i	$0.268821\pi$
$182$	−15.2002	−1.12671
$183$	0	0
$184$	18.5237	1.36559
$185$	0	0
$186$	0	0
$187$	−13.6518	−0.998322
$188$	−1.33928	−0.0976773
$189$	0	0
$190$	0	0
$191$	−0.325813	−0.0235750	−0.0117875	−	0.999931i	$-0.503752\pi$
$191$	−0.325813	−0.0235750	−0.0117875	+	0.999931i	$0.503752\pi$
$192$	0	0
$193$	−2.90187	−0.208881	−0.104441	−	0.994531i	$-0.533305\pi$
$193$	−2.90187	−0.208881	−0.104441	+	0.994531i	$0.533305\pi$
$194$	−2.30441	−0.165447
$195$	0	0
$196$	0.780505	0.0557503
$197$	18.1220	1.29114	0.645568	−	0.763702i	$-0.276620\pi$
$197$	18.1220	1.29114	0.645568	+	0.763702i	$0.276620\pi$
$198$	0	0
$199$	1.53256	0.108640	0.0543201	−	0.998524i	$-0.482701\pi$
$199$	1.53256	0.108640	0.0543201	+	0.998524i	$0.482701\pi$
$200$	0	0
$201$	0	0
$202$	17.7967	1.25217
$203$	−4.44202	−0.311769
$204$	0	0
$205$	0	0
$206$	14.6862	1.02324
$207$	0	0
$208$	25.5171	1.76929
$209$	−1.89146	−0.130835
$210$	0	0
$211$	−11.3698	−0.782727	−0.391363	−	0.920236i	$-0.627996\pi$
$211$	−11.3698	−0.782727	−0.391363	+	0.920236i	$0.627996\pi$
$212$	−0.0917718	−0.00630291
$213$	0	0
$214$	−12.6605	−0.865451
$215$	0	0
$216$	0	0
$217$	1.58913	0.107877
$218$	21.0593	1.42632
$219$	0	0
$220$	0	0
$221$	−28.8839	−1.94294
$222$	0	0
$223$	16.9507	1.13510	0.567550	−	0.823339i	$-0.307891\pi$
$223$	16.9507	1.13510	0.567550	+	0.823339i	$0.307891\pi$
$224$	1.57740	0.105395
$225$	0	0
$226$	13.5732	0.902878
$227$	−14.1117	−0.936627	−0.468314	−	0.883562i	$-0.655138\pi$
$227$	−14.1117	−0.936627	−0.468314	+	0.883562i	$0.655138\pi$
$228$	0	0
$229$	0.0619945	0.00409671	0.00204835	−	0.999998i	$-0.499348\pi$
$229$	0.0619945	0.00409671	0.00204835	+	0.999998i	$0.499348\pi$
$230$	0	0
$231$	0	0
$232$	8.17927	0.536995
$233$	26.0191	1.70457	0.852283	−	0.523081i	$-0.175218\pi$
$233$	26.0191	1.70457	0.852283	+	0.523081i	$0.175218\pi$
$234$	0	0
$235$	0	0
$236$	0.172455	0.0112258
$237$	0	0
$238$	8.83008	0.572369
$239$	−19.4970	−1.26116	−0.630579	−	0.776125i	$-0.717183\pi$
$239$	−19.4970	−1.26116	−0.630579	+	0.776125i	$0.717183\pi$
$240$	0	0
$241$	−4.09860	−0.264014	−0.132007	−	0.991249i	$-0.542142\pi$
$241$	−4.09860	−0.264014	−0.132007	+	0.991249i	$0.542142\pi$
$242$	−0.145007	−0.00932141
$243$	0	0
$244$	−1.86015	−0.119084
$245$	0	0
$246$	0	0
$247$	−4.00187	−0.254633
$248$	−2.92613	−0.185810
$249$	0	0
$250$	0	0
$251$	1.02933	0.0649704	0.0324852	−	0.999472i	$-0.489658\pi$
$251$	1.02933	0.0649704	0.0324852	+	0.999472i	$0.489658\pi$
$252$	0	0
$253$	−21.0098	−1.32087
$254$	1.31947	0.0827909
$255$	0	0
$256$	−4.17298	−0.260811
$257$	18.5597	1.15772	0.578862	−	0.815426i	$-0.303497\pi$
$257$	18.5597	1.15772	0.578862	+	0.815426i	$0.303497\pi$
$258$	0	0
$259$	−5.67041	−0.352342
$260$	0	0
$261$	0	0
$262$	13.5915	0.839686
$263$	−12.3938	−0.764231	−0.382116	−	0.924114i	$-0.624804\pi$
$263$	−12.3938	−0.764231	−0.382116	+	0.924114i	$0.624804\pi$
$264$	0	0
$265$	0	0
$266$	1.22341	0.0750119
$267$	0	0
$268$	−2.66673	−0.162897
$269$	5.30032	0.323166	0.161583	−	0.986859i	$-0.448340\pi$
$269$	5.30032	0.323166	0.161583	+	0.986859i	$0.448340\pi$
$270$	0	0
$271$	0.797428	0.0484403	0.0242201	−	0.999707i	$-0.492290\pi$
$271$	0.797428	0.0484403	0.0242201	+	0.999707i	$0.492290\pi$
$272$	−14.8234	−0.898798
$273$	0	0
$274$	−6.58184	−0.397624
$275$	0	0
$276$	0	0
$277$	−4.74425	−0.285054	−0.142527	−	0.989791i	$-0.545523\pi$
$277$	−4.74425	−0.285054	−0.142527	+	0.989791i	$0.545523\pi$
$278$	−0.250962	−0.0150517
$279$	0	0
$280$	0	0
$281$	18.7398	1.11792	0.558961	−	0.829194i	$-0.311200\pi$
$281$	18.7398	1.11792	0.558961	+	0.829194i	$0.311200\pi$
$282$	0	0
$283$	11.7612	0.699130	0.349565	−	0.936912i	$-0.386329\pi$
$283$	11.7612	0.699130	0.349565	+	0.936912i	$0.386329\pi$
$284$	−1.86895	−0.110902
$285$	0	0
$286$	−31.7451	−1.87712
$287$	1.85579	0.109544
$288$	0	0
$289$	−0.220750	−0.0129853
$290$	0	0
$291$	0	0
$292$	0.974146	0.0570076
$293$	−22.2819	−1.30172	−0.650860	−	0.759198i	$-0.725592\pi$
$293$	−22.2819	−1.30172	−0.650860	+	0.759198i	$0.725592\pi$
$294$	0	0
$295$	0	0
$296$	10.4411	0.606879
$297$	0	0
$298$	5.25322	0.304311
$299$	−44.4515	−2.57070
$300$	0	0
$301$	0.186743	0.0107637
$302$	29.8730	1.71900
$303$	0	0
$304$	−2.05378	−0.117792
$305$	0	0
$306$	0	0
$307$	−15.3063	−0.873574	−0.436787	−	0.899565i	$-0.643884\pi$
$307$	−15.3063	−0.873574	−0.436787	+	0.899565i	$0.643884\pi$
$308$	−0.932102	−0.0531115
$309$	0	0
$310$	0	0
$311$	12.8545	0.728913	0.364456	−	0.931220i	$-0.381255\pi$
$311$	12.8545	0.728913	0.364456	+	0.931220i	$0.381255\pi$
$312$	0	0
$313$	−10.5072	−0.593901	−0.296951	−	0.954893i	$-0.595970\pi$
$313$	−10.5072	−0.593901	−0.296951	+	0.954893i	$0.595970\pi$
$314$	−18.3791	−1.03720
$315$	0	0
$316$	2.55823	0.143912
$317$	19.4806	1.09414	0.547071	−	0.837086i	$-0.315743\pi$
$317$	19.4806	1.09414	0.547071	+	0.837086i	$0.315743\pi$
$318$	0	0
$319$	−9.27701	−0.519413
$320$	0	0
$321$	0	0
$322$	13.5892	0.757298
$323$	2.32476	0.129353
$324$	0	0
$325$	0	0
$326$	−11.6562	−0.645579
$327$	0	0
$328$	−3.41714	−0.188680
$329$	−12.1947	−0.672313
$330$	0	0
$331$	−14.4925	−0.796579	−0.398289	−	0.917260i	$-0.630396\pi$
$331$	−14.4925	−0.796579	−0.398289	+	0.917260i	$0.630396\pi$
$332$	−0.880498	−0.0483236
$333$	0	0
$334$	8.91400	0.487753
$335$	0	0
$336$	0	0
$337$	9.33225	0.508360	0.254180	−	0.967157i	$-0.418194\pi$
$337$	9.33225	0.508360	0.254180	+	0.967157i	$0.418194\pi$
$338$	−49.6039	−2.69810
$339$	0	0
$340$	0	0
$341$	3.31885	0.179726
$342$	0	0
$343$	18.2774	0.986884
$344$	−0.343857	−0.0185395
$345$	0	0
$346$	18.4517	0.991968
$347$	1.05341	0.0565499	0.0282750	−	0.999600i	$-0.490999\pi$
$347$	1.05341	0.0565499	0.0282750	+	0.999600i	$0.490999\pi$
$348$	0	0
$349$	−13.0715	−0.699700	−0.349850	−	0.936806i	$-0.613767\pi$
$349$	−13.0715	−0.699700	−0.349850	+	0.936806i	$0.613767\pi$
$350$	0	0
$351$	0	0
$352$	3.29435	0.175590
$353$	−33.9473	−1.80683	−0.903415	−	0.428767i	$-0.858948\pi$
$353$	−33.9473	−1.80683	−0.903415	+	0.428767i	$0.858948\pi$
$354$	0	0
$355$	0	0
$356$	0.494845	0.0262267
$357$	0	0
$358$	13.2770	0.701713
$359$	6.09450	0.321656	0.160828	−	0.986982i	$-0.448584\pi$
$359$	6.09450	0.321656	0.160828	+	0.986982i	$0.448584\pi$
$360$	0	0
$361$	−18.6779	−0.983048
$362$	−24.1376	−1.26864
$363$	0	0
$364$	−1.97210	−0.103366
$365$	0	0
$366$	0	0
$367$	−21.8636	−1.14127	−0.570636	−	0.821203i	$-0.693303\pi$
$367$	−21.8636	−1.14127	−0.570636	+	0.821203i	$0.693303\pi$
$368$	−22.8127	−1.18919
$369$	0	0
$370$	0	0
$371$	−0.835615	−0.0433830
$372$	0	0
$373$	−24.4559	−1.26628	−0.633140	−	0.774037i	$-0.718234\pi$
$373$	−24.4559	−1.26628	−0.633140	+	0.774037i	$0.718234\pi$
$374$	18.4413	0.953578
$375$	0	0
$376$	22.4545	1.15800
$377$	−19.6279	−1.01089
$378$	0	0
$379$	−6.27821	−0.322490	−0.161245	−	0.986914i	$-0.551551\pi$
$379$	−6.27821	−0.322490	−0.161245	+	0.986914i	$0.551551\pi$
$380$	0	0
$381$	0	0
$382$	0.440118	0.0225184
$383$	−24.6876	−1.26148	−0.630738	−	0.775996i	$-0.717248\pi$
$383$	−24.6876	−1.26148	−0.630738	+	0.775996i	$0.717248\pi$
$384$	0	0
$385$	0	0
$386$	3.91993	0.199519
$387$	0	0
$388$	−0.298978	−0.0151783
$389$	12.9236	0.655251	0.327626	−	0.944808i	$-0.393752\pi$
$389$	12.9236	0.655251	0.327626	+	0.944808i	$0.393752\pi$
$390$	0	0
$391$	25.8228	1.30591
$392$	−13.0860	−0.660942
$393$	0	0
$394$	−24.4797	−1.23327
$395$	0	0
$396$	0	0
$397$	−29.0214	−1.45654	−0.728272	−	0.685288i	$-0.759676\pi$
$397$	−29.0214	−1.45654	−0.728272	+	0.685288i	$0.759676\pi$
$398$	−2.07022	−0.103771
$399$	0	0
$400$	0	0
$401$	−23.3926	−1.16817	−0.584084	−	0.811693i	$-0.698546\pi$
$401$	−23.3926	−1.16817	−0.584084	+	0.811693i	$0.698546\pi$
$402$	0	0
$403$	7.02187	0.349784
$404$	2.30898	0.114876
$405$	0	0
$406$	6.00041	0.297795
$407$	−11.8425	−0.587009
$408$	0	0
$409$	−15.9918	−0.790742	−0.395371	−	0.918521i	$-0.629384\pi$
$409$	−15.9918	−0.790742	−0.395371	+	0.918521i	$0.629384\pi$
$410$	0	0
$411$	0	0
$412$	1.90542	0.0938731
$413$	1.57026	0.0772675
$414$	0	0
$415$	0	0
$416$	6.97004	0.341734
$417$	0	0
$418$	2.55504	0.124971
$419$	−32.3769	−1.58172	−0.790858	−	0.611999i	$-0.790366\pi$
$419$	−32.3769	−1.58172	−0.790858	+	0.611999i	$0.790366\pi$
$420$	0	0
$421$	18.0520	0.879801	0.439900	−	0.898047i	$-0.355014\pi$
$421$	18.0520	0.879801	0.439900	+	0.898047i	$0.355014\pi$
$422$	15.3586	0.747646
$423$	0	0
$424$	1.53865	0.0747235
$425$	0	0
$426$	0	0
$427$	−16.9373	−0.819654
$428$	−1.64259	−0.0793977
$429$	0	0
$430$	0	0
$431$	−33.1353	−1.59607	−0.798035	−	0.602612i	$-0.794127\pi$
$431$	−33.1353	−1.59607	−0.798035	+	0.602612i	$0.794127\pi$
$432$	0	0
$433$	−22.6653	−1.08922	−0.544612	−	0.838688i	$-0.683323\pi$
$433$	−22.6653	−1.08922	−0.544612	+	0.838688i	$0.683323\pi$
$434$	−2.14665	−0.103042
$435$	0	0
$436$	2.73228	0.130852
$437$	3.57774	0.171147
$438$	0	0
$439$	−8.50436	−0.405891	−0.202945	−	0.979190i	$-0.565051\pi$
$439$	−8.50436	−0.405891	−0.202945	+	0.979190i	$0.565051\pi$
$440$	0	0
$441$	0	0
$442$	39.0173	1.85586
$443$	−6.35768	−0.302063	−0.151031	−	0.988529i	$-0.548259\pi$
$443$	−6.35768	−0.302063	−0.151031	+	0.988529i	$0.548259\pi$
$444$	0	0
$445$	0	0
$446$	−22.8975	−1.08423
$447$	0	0
$448$	−13.6804	−0.646340
$449$	6.25726	0.295298	0.147649	−	0.989040i	$-0.452829\pi$
$449$	6.25726	0.295298	0.147649	+	0.989040i	$0.452829\pi$
$450$	0	0
$451$	3.87576	0.182502
$452$	1.76102	0.0828313
$453$	0	0
$454$	19.0625	0.894648
$455$	0	0
$456$	0	0
$457$	−11.0441	−0.516620	−0.258310	−	0.966062i	$-0.583166\pi$
$457$	−11.0441	−0.516620	−0.258310	+	0.966062i	$0.583166\pi$
$458$	−0.0837440	−0.00391310
$459$	0	0
$460$	0	0
$461$	23.6622	1.10206	0.551029	−	0.834486i	$-0.314235\pi$
$461$	23.6622	1.10206	0.551029	+	0.834486i	$0.314235\pi$
$462$	0	0
$463$	−6.22442	−0.289273	−0.144637	−	0.989485i	$-0.546201\pi$
$463$	−6.22442	−0.289273	−0.144637	+	0.989485i	$0.546201\pi$
$464$	−10.0731	−0.467632
$465$	0	0
$466$	−35.1473	−1.62817
$467$	−4.94679	−0.228910	−0.114455	−	0.993428i	$-0.536512\pi$
$467$	−4.94679	−0.228910	−0.114455	+	0.993428i	$0.536512\pi$
$468$	0	0
$469$	−24.2815	−1.12122
$470$	0	0
$471$	0	0
$472$	−2.89138	−0.133087
$473$	0.390006	0.0179325
$474$	0	0
$475$	0	0
$476$	1.14563	0.0525099
$477$	0	0
$478$	26.3372	1.20463
$479$	30.0898	1.37484	0.687419	−	0.726261i	$-0.258744\pi$
$479$	30.0898	1.37484	0.687419	+	0.726261i	$0.258744\pi$
$480$	0	0
$481$	−25.0557	−1.14244
$482$	5.53651	0.252181
$483$	0	0
$484$	−0.0188135	−0.000855159 0
$485$	0	0
$486$	0	0
$487$	−34.2499	−1.55201	−0.776006	−	0.630726i	$-0.782757\pi$
$487$	−34.2499	−1.55201	−0.776006	+	0.630726i	$0.782757\pi$
$488$	31.1874	1.41179
$489$	0	0
$490$	0	0
$491$	10.4193	0.470218	0.235109	−	0.971969i	$-0.424455\pi$
$491$	10.4193	0.470218	0.235109	+	0.971969i	$0.424455\pi$
$492$	0	0
$493$	11.4022	0.513530
$494$	5.40584	0.243220
$495$	0	0
$496$	3.60365	0.161809
$497$	−17.0175	−0.763338
$498$	0	0
$499$	8.83514	0.395515	0.197757	−	0.980251i	$-0.436634\pi$
$499$	8.83514	0.395515	0.197757	+	0.980251i	$0.436634\pi$
$500$	0	0
$501$	0	0
$502$	−1.39044	−0.0620585
$503$	−21.3734	−0.952994	−0.476497	−	0.879176i	$-0.658094\pi$
$503$	−21.3734	−0.952994	−0.476497	+	0.879176i	$0.658094\pi$
$504$	0	0
$505$	0	0
$506$	28.3806	1.26167
$507$	0	0
$508$	0.171190	0.00759535
$509$	16.7800	0.743759	0.371879	−	0.928281i	$-0.378714\pi$
$509$	16.7800	0.743759	0.371879	+	0.928281i	$0.378714\pi$
$510$	0	0
$511$	8.86994	0.392383
$512$	24.8438	1.09795
$513$	0	0
$514$	−25.0710	−1.10584
$515$	0	0
$516$	0	0
$517$	−25.4681	−1.12009
$518$	7.65976	0.336550
$519$	0	0
$520$	0	0
$521$	4.60508	0.201752	0.100876	−	0.994899i	$-0.467835\pi$
$521$	4.60508	0.201752	0.100876	+	0.994899i	$0.467835\pi$
$522$	0	0
$523$	12.9634	0.566852	0.283426	−	0.958994i	$-0.408529\pi$
$523$	12.9634	0.566852	0.283426	+	0.958994i	$0.408529\pi$
$524$	1.76339	0.0770340
$525$	0	0
$526$	16.7418	0.729979
$527$	−4.07914	−0.177690
$528$	0	0
$529$	16.7405	0.727846
$530$	0	0
$531$	0	0
$532$	0.158727	0.00688169
$533$	8.20015	0.355188
$534$	0	0
$535$	0	0
$536$	44.7106	1.93120
$537$	0	0
$538$	−7.15983	−0.308682
$539$	14.8423	0.639301
$540$	0	0
$541$	−27.3641	−1.17647	−0.588237	−	0.808688i	$-0.700178\pi$
$541$	−27.3641	−1.17647	−0.588237	+	0.808688i	$0.700178\pi$
$542$	−1.07719	−0.0462692
$543$	0	0
$544$	−4.04903	−0.173601
$545$	0	0
$546$	0	0
$547$	27.6453	1.18203	0.591015	−	0.806661i	$-0.298728\pi$
$547$	27.6453	1.18203	0.591015	+	0.806661i	$0.298728\pi$
$548$	−0.853941	−0.0364785
$549$	0	0
$550$	0	0
$551$	1.57978	0.0673007
$552$	0	0
$553$	23.2936	0.990545
$554$	6.40868	0.272279
$555$	0	0
$556$	−0.0325603	−0.00138086
$557$	6.17333	0.261572	0.130786	−	0.991411i	$-0.458250\pi$
$557$	6.17333	0.261572	0.130786	+	0.991411i	$0.458250\pi$
$558$	0	0
$559$	0.825156	0.0349004
$560$	0	0
$561$	0	0
$562$	−25.3143	−1.06782
$563$	−5.69934	−0.240198	−0.120099	−	0.992762i	$-0.538321\pi$
$563$	−5.69934	−0.240198	−0.120099	+	0.992762i	$0.538321\pi$
$564$	0	0
$565$	0	0
$566$	−15.8874	−0.667796
$567$	0	0
$568$	31.3350	1.31479
$569$	−20.1708	−0.845605	−0.422802	−	0.906222i	$-0.638954\pi$
$569$	−20.1708	−0.845605	−0.422802	+	0.906222i	$0.638954\pi$
$570$	0	0
$571$	−15.7554	−0.659341	−0.329671	−	0.944096i	$-0.606938\pi$
$571$	−15.7554	−0.659341	−0.329671	+	0.944096i	$0.606938\pi$
$572$	−4.11866	−0.172210
$573$	0	0
$574$	−2.50686	−0.104634
$575$	0	0
$576$	0	0
$577$	−16.1062	−0.670509	−0.335255	−	0.942128i	$-0.608822\pi$
$577$	−16.1062	−0.670509	−0.335255	+	0.942128i	$0.608822\pi$
$578$	0.298195	0.0124033
$579$	0	0
$580$	0	0
$581$	−8.01725	−0.332611
$582$	0	0
$583$	−1.74515	−0.0722769
$584$	−16.3326	−0.675847
$585$	0	0
$586$	30.0990	1.24338
$587$	−2.00072	−0.0825786	−0.0412893	−	0.999147i	$-0.513147\pi$
$587$	−2.00072	−0.0825786	−0.0412893	+	0.999147i	$0.513147\pi$
$588$	0	0
$589$	−0.565165	−0.0232872
$590$	0	0
$591$	0	0
$592$	−12.8587	−0.528489
$593$	26.8231	1.10149	0.550747	−	0.834672i	$-0.314343\pi$
$593$	26.8231	1.10149	0.550747	+	0.834672i	$0.314343\pi$
$594$	0	0
$595$	0	0
$596$	0.681563	0.0279179
$597$	0	0
$598$	60.0464	2.45548
$599$	44.8025	1.83058	0.915290	−	0.402796i	$-0.131962\pi$
$599$	44.8025	1.83058	0.915290	+	0.402796i	$0.131962\pi$
$600$	0	0
$601$	−14.2298	−0.580446	−0.290223	−	0.956959i	$-0.593729\pi$
$601$	−14.2298	−0.580446	−0.290223	+	0.956959i	$0.593729\pi$
$602$	−0.252258	−0.0102813
$603$	0	0
$604$	3.87578	0.157703
$605$	0	0
$606$	0	0
$607$	−20.3346	−0.825356	−0.412678	−	0.910877i	$-0.635406\pi$
$607$	−20.3346	−0.825356	−0.412678	+	0.910877i	$0.635406\pi$
$608$	−0.560993	−0.0227513
$609$	0	0
$610$	0	0
$611$	−53.8843	−2.17993
$612$	0	0
$613$	−20.7921	−0.839783	−0.419892	−	0.907574i	$-0.637932\pi$
$613$	−20.7921	−0.839783	−0.419892	+	0.907574i	$0.637932\pi$
$614$	20.6761	0.834421
$615$	0	0
$616$	15.6277	0.629657
$617$	−4.94634	−0.199132	−0.0995660	−	0.995031i	$-0.531745\pi$
$617$	−4.94634	−0.199132	−0.0995660	+	0.995031i	$0.531745\pi$
$618$	0	0
$619$	−45.9527	−1.84700	−0.923498	−	0.383603i	$-0.874683\pi$
$619$	−45.9527	−1.84700	−0.923498	+	0.383603i	$0.874683\pi$
$620$	0	0
$621$	0	0
$622$	−17.3643	−0.696243
$623$	4.50574	0.180519
$624$	0	0
$625$	0	0
$626$	14.1934	0.567283
$627$	0	0
$628$	−2.38454	−0.0951537
$629$	14.5554	0.580360
$630$	0	0
$631$	−21.6636	−0.862414	−0.431207	−	0.902253i	$-0.641912\pi$
$631$	−21.6636	−0.862414	−0.431207	+	0.902253i	$0.641912\pi$
$632$	−42.8915	−1.70613
$633$	0	0
$634$	−26.3150	−1.04510
$635$	0	0
$636$	0	0
$637$	31.4026	1.24421
$638$	12.5317	0.496133
$639$	0	0
$640$	0	0
$641$	−36.3870	−1.43720	−0.718600	−	0.695424i	$-0.755217\pi$
$641$	−36.3870	−1.43720	−0.718600	+	0.695424i	$0.755217\pi$
$642$	0	0
$643$	−1.01349	−0.0399682	−0.0199841	−	0.999800i	$-0.506362\pi$
$643$	−1.01349	−0.0399682	−0.0199841	+	0.999800i	$0.506362\pi$
$644$	1.76309	0.0694755
$645$	0	0
$646$	−3.14036	−0.123556
$647$	−13.7007	−0.538629	−0.269315	−	0.963052i	$-0.586797\pi$
$647$	−13.7007	−0.538629	−0.269315	+	0.963052i	$0.586797\pi$
$648$	0	0
$649$	3.27944	0.128729
$650$	0	0
$651$	0	0
$652$	−1.51230	−0.0592263
$653$	23.8372	0.932822	0.466411	−	0.884568i	$-0.345547\pi$
$653$	23.8372	0.932822	0.466411	+	0.884568i	$0.345547\pi$
$654$	0	0
$655$	0	0
$656$	4.20835	0.164309
$657$	0	0
$658$	16.4729	0.642181
$659$	−14.7758	−0.575583	−0.287791	−	0.957693i	$-0.592921\pi$
$659$	−14.7758	−0.575583	−0.287791	+	0.957693i	$0.592921\pi$
$660$	0	0
$661$	7.55471	0.293844	0.146922	−	0.989148i	$-0.453063\pi$
$661$	7.55471	0.293844	0.146922	+	0.989148i	$0.453063\pi$
$662$	19.5769	0.760877
$663$	0	0
$664$	14.7625	0.572895
$665$	0	0
$666$	0	0
$667$	17.5477	0.679448
$668$	1.15652	0.0447471
$669$	0	0
$670$	0	0
$671$	−35.3730	−1.36556
$672$	0	0
$673$	−12.1947	−0.470069	−0.235035	−	0.971987i	$-0.575520\pi$
$673$	−12.1947	−0.470069	−0.235035	+	0.971987i	$0.575520\pi$
$674$	−12.6063	−0.485576
$675$	0	0
$676$	−6.43571	−0.247527
$677$	−17.9507	−0.689900	−0.344950	−	0.938621i	$-0.612104\pi$
$677$	−17.9507	−0.689900	−0.344950	+	0.938621i	$0.612104\pi$
$678$	0	0
$679$	−2.72230	−0.104472
$680$	0	0
$681$	0	0
$682$	−4.48320	−0.171671
$683$	−10.0273	−0.383685	−0.191842	−	0.981426i	$-0.561446\pi$
$683$	−10.0273	−0.383685	−0.191842	+	0.981426i	$0.561446\pi$
$684$	0	0
$685$	0	0
$686$	−24.6896	−0.942653
$687$	0	0
$688$	0.423474	0.0161448
$689$	−3.69231	−0.140666
$690$	0	0
$691$	19.1296	0.727722	0.363861	−	0.931453i	$-0.381458\pi$
$691$	19.1296	0.727722	0.363861	+	0.931453i	$0.381458\pi$
$692$	2.39396	0.0910045
$693$	0	0
$694$	−1.42297	−0.0540154
$695$	0	0
$696$	0	0
$697$	−4.76363	−0.180435
$698$	17.6573	0.668340
$699$	0	0
$700$	0	0
$701$	3.81920	0.144249	0.0721246	−	0.997396i	$-0.477022\pi$
$701$	3.81920	0.144249	0.0721246	+	0.997396i	$0.477022\pi$
$702$	0	0
$703$	2.01664	0.0760592
$704$	−28.5711	−1.07681
$705$	0	0
$706$	45.8570	1.72585
$707$	21.0241	0.790692
$708$	0	0
$709$	38.0535	1.42913	0.714565	−	0.699569i	$-0.246625\pi$
$709$	38.0535	1.42913	0.714565	+	0.699569i	$0.246625\pi$
$710$	0	0
$711$	0	0
$712$	−8.29660	−0.310928
$713$	−6.27768	−0.235101
$714$	0	0
$715$	0	0
$716$	1.72259	0.0643761
$717$	0	0
$718$	−8.23264	−0.307239
$719$	−19.4340	−0.724766	−0.362383	−	0.932029i	$-0.618037\pi$
$719$	−19.4340	−0.724766	−0.362383	+	0.932029i	$0.618037\pi$
$720$	0	0
$721$	17.3495	0.646129
$722$	25.2307	0.938988
$723$	0	0
$724$	−3.13165	−0.116387
$725$	0	0
$726$	0	0
$727$	−38.0739	−1.41208	−0.706042	−	0.708170i	$-0.749521\pi$
$727$	−38.0739	−1.41208	−0.706042	+	0.708170i	$0.749521\pi$
$728$	33.0643	1.22544
$729$	0	0
$730$	0	0
$731$	−0.479350	−0.0177294
$732$	0	0
$733$	16.9591	0.626400	0.313200	−	0.949687i	$-0.398599\pi$
$733$	16.9591	0.626400	0.313200	+	0.949687i	$0.398599\pi$
$734$	29.5341	1.09012
$735$	0	0
$736$	−6.23134	−0.229690
$737$	−50.7112	−1.86797
$738$	0	0
$739$	13.2040	0.485716	0.242858	−	0.970062i	$-0.421915\pi$
$739$	13.2040	0.485716	0.242858	+	0.970062i	$0.421915\pi$
$740$	0	0
$741$	0	0
$742$	1.12877	0.0414386
$743$	−42.2364	−1.54950	−0.774752	−	0.632265i	$-0.782125\pi$
$743$	−42.2364	−1.54950	−0.774752	+	0.632265i	$0.782125\pi$
$744$	0	0
$745$	0	0
$746$	33.0358	1.20953
$747$	0	0
$748$	2.39261	0.0874825
$749$	−14.9564	−0.546494
$750$	0	0
$751$	−1.04801	−0.0382426	−0.0191213	−	0.999817i	$-0.506087\pi$
$751$	−1.04801	−0.0382426	−0.0191213	+	0.999817i	$0.506087\pi$
$752$	−27.6536	−1.00842
$753$	0	0
$754$	26.5139	0.965579
$755$	0	0
$756$	0	0
$757$	17.0074	0.618146	0.309073	−	0.951038i	$-0.399981\pi$
$757$	17.0074	0.618146	0.309073	+	0.951038i	$0.399981\pi$
$758$	8.48079	0.308036
$759$	0	0
$760$	0	0
$761$	24.1244	0.874508	0.437254	−	0.899338i	$-0.355951\pi$
$761$	24.1244	0.874508	0.437254	+	0.899338i	$0.355951\pi$
$762$	0	0
$763$	24.8783	0.900657
$764$	0.0571018	0.00206587
$765$	0	0
$766$	33.3487	1.20494
$767$	6.93848	0.250534
$768$	0	0
$769$	49.7819	1.79518	0.897591	−	0.440830i	$-0.145316\pi$
$769$	49.7819	1.79518	0.897591	+	0.440830i	$0.145316\pi$
$770$	0	0
$771$	0	0
$772$	0.508580	0.0183042
$773$	31.4743	1.13205	0.566025	−	0.824388i	$-0.308480\pi$
$773$	31.4743	1.13205	0.566025	+	0.824388i	$0.308480\pi$
$774$	0	0
$775$	0	0
$776$	5.01268	0.179945
$777$	0	0
$778$	−17.4575	−0.625883
$779$	−0.660001	−0.0236470
$780$	0	0
$781$	−35.5404	−1.27174
$782$	−34.8822	−1.24738
$783$	0	0
$784$	16.1159	0.575569
$785$	0	0
$786$	0	0
$787$	−19.1415	−0.682321	−0.341161	−	0.940005i	$-0.610820\pi$
$787$	−19.1415	−0.682321	−0.341161	+	0.940005i	$0.610820\pi$
$788$	−3.17604	−0.113142
$789$	0	0
$790$	0	0
$791$	16.0347	0.570128
$792$	0	0
$793$	−74.8406	−2.65767
$794$	39.2030	1.39126
$795$	0	0
$796$	−0.268595	−0.00952009
$797$	7.21437	0.255546	0.127773	−	0.991803i	$-0.459217\pi$
$797$	7.21437	0.255546	0.127773	+	0.991803i	$0.459217\pi$
$798$	0	0
$799$	31.3024	1.10740
$800$	0	0
$801$	0	0
$802$	31.5994	1.11581
$803$	18.5246	0.653718
$804$	0	0
$805$	0	0
$806$	−9.48535	−0.334107
$807$	0	0
$808$	−38.7125	−1.36190
$809$	−1.05941	−0.0372469	−0.0186234	−	0.999827i	$-0.505928\pi$
$809$	−1.05941	−0.0372469	−0.0186234	+	0.999827i	$0.505928\pi$
$810$	0	0
$811$	21.6400	0.759883	0.379941	−	0.925011i	$-0.375944\pi$
$811$	21.6400	0.759883	0.379941	+	0.925011i	$0.375944\pi$
$812$	0.778504	0.0273202
$813$	0	0
$814$	15.9971	0.560700
$815$	0	0
$816$	0	0
$817$	−0.0664138	−0.00232353
$818$	21.6022	0.755302
$819$	0	0
$820$	0	0
$821$	23.1828	0.809085	0.404543	−	0.914519i	$-0.367431\pi$
$821$	23.1828	0.809085	0.404543	+	0.914519i	$0.367431\pi$
$822$	0	0
$823$	−27.9769	−0.975214	−0.487607	−	0.873063i	$-0.662130\pi$
$823$	−27.9769	−0.975214	−0.487607	+	0.873063i	$0.662130\pi$
$824$	−31.9463	−1.11290
$825$	0	0
$826$	−2.12116	−0.0738044
$827$	41.3045	1.43630	0.718150	−	0.695889i	$-0.244989\pi$
$827$	41.3045	1.43630	0.718150	+	0.695889i	$0.244989\pi$
$828$	0	0
$829$	47.5687	1.65213	0.826064	−	0.563577i	$-0.190575\pi$
$829$	47.5687	1.65213	0.826064	+	0.563577i	$0.190575\pi$
$830$	0	0
$831$	0	0
$832$	−60.4495	−2.09571
$833$	−18.2424	−0.632060
$834$	0	0
$835$	0	0
$836$	0.331496	0.0114650
$837$	0	0
$838$	43.7357	1.51083
$839$	−49.0322	−1.69278	−0.846389	−	0.532565i	$-0.821228\pi$
$839$	−49.0322	−1.69278	−0.846389	+	0.532565i	$0.821228\pi$
$840$	0	0
$841$	−21.2517	−0.732818
$842$	−24.3852	−0.840369
$843$	0	0
$844$	1.99266	0.0685900
$845$	0	0
$846$	0	0
$847$	−0.171304	−0.00588606
$848$	−1.89491	−0.0650715
$849$	0	0
$850$	0	0
$851$	22.4003	0.767871
$852$	0	0
$853$	−32.5376	−1.11407	−0.557033	−	0.830490i	$-0.688061\pi$
$853$	−32.5376	−1.11407	−0.557033	+	0.830490i	$0.688061\pi$
$854$	22.8794	0.782918
$855$	0	0
$856$	27.5398	0.941290
$857$	30.4813	1.04122	0.520610	−	0.853794i	$-0.325704\pi$
$857$	30.4813	1.04122	0.520610	+	0.853794i	$0.325704\pi$
$858$	0	0
$859$	4.12215	0.140646	0.0703231	−	0.997524i	$-0.477597\pi$
$859$	4.12215	0.140646	0.0703231	+	0.997524i	$0.477597\pi$
$860$	0	0
$861$	0	0
$862$	44.7601	1.52453
$863$	18.4184	0.626969	0.313485	−	0.949593i	$-0.398504\pi$
$863$	18.4184	0.626969	0.313485	+	0.949593i	$0.398504\pi$
$864$	0	0
$865$	0	0
$866$	30.6169	1.04041
$867$	0	0
$868$	−0.278510	−0.00945325
$869$	48.6479	1.65027
$870$	0	0
$871$	−107.292	−3.63546
$872$	−45.8095	−1.55131
$873$	0	0
$874$	−4.83292	−0.163476
$875$	0	0
$876$	0	0
$877$	−18.9126	−0.638634	−0.319317	−	0.947648i	$-0.603453\pi$
$877$	−18.9126	−0.638634	−0.319317	+	0.947648i	$0.603453\pi$
$878$	11.4879	0.387699
$879$	0	0
$880$	0	0
$881$	−34.6336	−1.16684	−0.583418	−	0.812172i	$-0.698285\pi$
$881$	−34.6336	−1.16684	−0.583418	+	0.812172i	$0.698285\pi$
$882$	0	0
$883$	−32.6874	−1.10002	−0.550009	−	0.835158i	$-0.685376\pi$
$883$	−32.6874	−1.10002	−0.550009	+	0.835158i	$0.685376\pi$
$884$	5.06218	0.170259
$885$	0	0
$886$	8.58814	0.288524
$887$	−48.7918	−1.63827	−0.819134	−	0.573602i	$-0.805545\pi$
$887$	−48.7918	−1.63827	−0.819134	+	0.573602i	$0.805545\pi$
$888$	0	0
$889$	1.55875	0.0522788
$890$	0	0
$891$	0	0
$892$	−2.97076	−0.0994684
$893$	4.33695	0.145131
$894$	0	0
$895$	0	0
$896$	15.3251	0.511977
$897$	0	0
$898$	−8.45249	−0.282063
$899$	−2.77195	−0.0924497
$900$	0	0
$901$	2.14494	0.0714582
$902$	−5.23549	−0.174323
$903$	0	0
$904$	−29.5253	−0.981997
$905$	0	0
$906$	0	0
$907$	45.0367	1.49542	0.747710	−	0.664025i	$-0.231153\pi$
$907$	45.0367	1.49542	0.747710	+	0.664025i	$0.231153\pi$
$908$	2.47321	0.0820762
$909$	0	0
$910$	0	0
$911$	−12.9449	−0.428882	−0.214441	−	0.976737i	$-0.568793\pi$
$911$	−12.9449	−0.428882	−0.214441	+	0.976737i	$0.568793\pi$
$912$	0	0
$913$	−16.7438	−0.554137
$914$	14.9187	0.493466
$915$	0	0
$916$	−0.0108651	−0.000358993 0
$917$	16.0563	0.530225
$918$	0	0
$919$	14.5719	0.480682	0.240341	−	0.970689i	$-0.422741\pi$
$919$	14.5719	0.480682	0.240341	+	0.970689i	$0.422741\pi$
$920$	0	0
$921$	0	0
$922$	−31.9636	−1.05266
$923$	−75.1948	−2.47507
$924$	0	0
$925$	0	0
$926$	8.40813	0.276308
$927$	0	0
$928$	−2.75149	−0.0903220
$929$	31.3591	1.02886	0.514430	−	0.857532i	$-0.328004\pi$
$929$	31.3591	1.02886	0.514430	+	0.857532i	$0.328004\pi$
$930$	0	0
$931$	−2.52748	−0.0828347
$932$	−4.56008	−0.149370
$933$	0	0
$934$	6.68227	0.218651
$935$	0	0
$936$	0	0
$937$	−48.7726	−1.59333	−0.796666	−	0.604420i	$-0.793405\pi$
$937$	−48.7726	−1.59333	−0.796666	+	0.604420i	$0.793405\pi$
$938$	32.8002	1.07097
$939$	0	0
$940$	0	0
$941$	32.2359	1.05086	0.525431	−	0.850836i	$-0.323904\pi$
$941$	32.2359	1.05086	0.525431	+	0.850836i	$0.323904\pi$
$942$	0	0
$943$	−7.33108	−0.238733
$944$	3.56086	0.115896
$945$	0	0
$946$	−0.526832	−0.0171288
$947$	51.9235	1.68729	0.843644	−	0.536903i	$-0.180406\pi$
$947$	51.9235	1.68729	0.843644	+	0.536903i	$0.180406\pi$
$948$	0	0
$949$	39.1935	1.27227
$950$	0	0
$951$	0	0
$952$	−19.2077	−0.622525
$953$	39.7788	1.28856	0.644281	−	0.764789i	$-0.277157\pi$
$953$	39.7788	1.28856	0.644281	+	0.764789i	$0.277157\pi$
$954$	0	0
$955$	0	0
$956$	3.41703	0.110515
$957$	0	0
$958$	−40.6462	−1.31322
$959$	−7.77543	−0.251082
$960$	0	0
$961$	−30.0083	−0.968011
$962$	33.8460	1.09124
$963$	0	0
$964$	0.718317	0.0231354
$965$	0	0
$966$	0	0
$967$	−19.8817	−0.639353	−0.319677	−	0.947527i	$-0.603574\pi$
$967$	−19.8817	−0.639353	−0.319677	+	0.947527i	$0.603574\pi$
$968$	0.315428	0.0101382
$969$	0	0
$970$	0	0
$971$	41.3841	1.32808	0.664039	−	0.747698i	$-0.268841\pi$
$971$	41.3841	1.32808	0.664039	+	0.747698i	$0.268841\pi$
$972$	0	0
$973$	−0.296473	−0.00950449
$974$	46.2658	1.48245
$975$	0	0
$976$	−38.4085	−1.22943
$977$	3.61991	0.115811	0.0579056	−	0.998322i	$-0.481558\pi$
$977$	3.61991	0.115811	0.0579056	+	0.998322i	$0.481558\pi$
$978$	0	0
$979$	9.41009	0.300748
$980$	0	0
$981$	0	0
$982$	−14.0747	−0.449143
$983$	−20.1099	−0.641405	−0.320702	−	0.947180i	$-0.603919\pi$
$983$	−20.1099	−0.641405	−0.320702	+	0.947180i	$0.603919\pi$
$984$	0	0
$985$	0	0
$986$	−15.4024	−0.490514
$987$	0	0
$988$	0.701364	0.0223134
$989$	−0.737705	−0.0234576
$990$	0	0
$991$	7.43990	0.236336	0.118168	−	0.992994i	$-0.462298\pi$
$991$	7.43990	0.236336	0.118168	+	0.992994i	$0.462298\pi$
$992$	0.984345	0.0312530
$993$	0	0
$994$	22.9877	0.729126
$995$	0	0
$996$	0	0
$997$	46.1472	1.46150	0.730748	−	0.682648i	$-0.239172\pi$
$997$	46.1472	1.46150	0.730748	+	0.682648i	$0.239172\pi$
$998$	−11.9348	−0.377788
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bd.1.2		8
3.2	odd	2		1875.2.a.m.1.7		8
5.4	even	2		5625.2.a.t.1.7		8
15.2	even	4		1875.2.b.h.1249.12		16
15.8	even	4		1875.2.b.h.1249.5		16
15.14	odd	2		1875.2.a.p.1.2		8
25.2	odd	20		225.2.m.b.154.2		16
25.13	odd	20		225.2.m.b.19.2		16
75.2	even	20		75.2.i.a.4.3	✓	16
75.11	odd	10		375.2.g.e.226.4		16
75.14	odd	10		375.2.g.d.226.1		16
75.23	even	20		375.2.i.c.274.2		16
75.38	even	20		75.2.i.a.19.3	yes	16
75.41	odd	10		375.2.g.e.151.4		16
75.59	odd	10		375.2.g.d.151.1		16
75.62	even	20		375.2.i.c.349.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.3	✓	16	75.2	even	20
75.2.i.a.19.3	yes	16	75.38	even	20
225.2.m.b.19.2		16	25.13	odd	20
225.2.m.b.154.2		16	25.2	odd	20
375.2.g.d.151.1		16	75.59	odd	10
375.2.g.d.226.1		16	75.14	odd	10
375.2.g.e.151.4		16	75.41	odd	10
375.2.g.e.226.4		16	75.11	odd	10
375.2.i.c.274.2		16	75.23	even	20
375.2.i.c.349.2		16	75.62	even	20
1875.2.a.m.1.7		8	3.2	odd	2
1875.2.a.p.1.2		8	15.14	odd	2
1875.2.b.h.1249.5		16	15.8	even	4
1875.2.b.h.1249.12		16	15.2	even	4
5625.2.a.t.1.7		8	5.4	even	2
5625.2.a.bd.1.2		8	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.35083 q^{2} -0.175259 q^{4} -1.59580 q^{7} +2.93840 q^{8} +O(q^{10})\)	\(q-1.35083 q^{2} -0.175259 q^{4} -1.59580 q^{7} +2.93840 q^{8} -3.33277 q^{11} -7.05132 q^{13} +2.15565 q^{14} -3.61877 q^{16} +4.09625 q^{17} +0.567535 q^{19} +4.50200 q^{22} +6.30400 q^{23} +9.52513 q^{26} +0.279678 q^{28} +2.78357 q^{29} -0.995824 q^{31} -0.988473 q^{32} -5.53333 q^{34} +3.55334 q^{37} -0.766643 q^{38} -1.16293 q^{41} -0.117022 q^{43} +0.584098 q^{44} -8.51563 q^{46} +7.64173 q^{47} -4.45343 q^{49} +1.23581 q^{52} +0.523635 q^{53} -4.68910 q^{56} -3.76013 q^{58} -0.983998 q^{59} +10.6137 q^{61} +1.34519 q^{62} +8.57279 q^{64} +15.2159 q^{67} -0.717905 q^{68} +10.6639 q^{71} -5.55832 q^{73} -4.79996 q^{74} -0.0994657 q^{76} +5.31842 q^{77} -14.5969 q^{79} +1.57091 q^{82} +5.02398 q^{83} +0.158076 q^{86} -9.79302 q^{88} -2.82350 q^{89} +11.2525 q^{91} -1.10483 q^{92} -10.3227 q^{94} +1.70592 q^{97} +6.01583 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8}+O(q^{10}) \) 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8	\( 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8} - 2 q^{11} - 16 q^{13} - 6 q^{14} + 16 q^{17} - 14 q^{19} - 12 q^{22} + 14 q^{23} - 6 q^{26} - 16 q^{28} - 2 q^{29} - 22 q^{31} - 2 q^{32} - 12 q^{34} - 28 q^{37} - 16 q^{38} - 8 q^{41} - 20 q^{43} - 22 q^{44} - 2 q^{46} + 10 q^{47} - 16 q^{52} + 44 q^{53} - 30 q^{56} - 8 q^{58} - 14 q^{59} - 20 q^{61} + 16 q^{62} + 6 q^{64} - 16 q^{67} - 2 q^{68} - 16 q^{71} - 24 q^{73} - 26 q^{74} - 16 q^{76} + 46 q^{77} - 30 q^{79} - 16 q^{82} + 12 q^{83} - 32 q^{86} - 32 q^{88} - 16 q^{89} - 12 q^{91} - 2 q^{92} + 14 q^{94} - 16 q^{97} + 4 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8 - 2 * q^11 - 16 * q^13 - 6 * q^14 + 16 * q^17 - 14 * q^19 - 12 * q^22 + 14 * q^23 - 6 * q^26 - 16 * q^28 - 2 * q^29 - 22 * q^31 - 2 * q^32 - 12 * q^34 - 28 * q^37 - 16 * q^38 - 8 * q^41 - 20 * q^43 - 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^52 + 44 * q^53 - 30 * q^56 - 8 * q^58 - 14 * q^59 - 20 * q^61 + 16 * q^62 + 6 * q^64 - 16 * q^67 - 2 * q^68 - 16 * q^71 - 24 * q^73 - 26 * q^74 - 16 * q^76 + 46 * q^77 - 30 * q^79 - 16 * q^82 + 12 * q^83 - 32 * q^86 - 32 * q^88 - 16 * q^89 - 12 * q^91 - 2 * q^92 + 14 * q^94 - 16 * q^97 + 4 * q^98

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