LMFDB - Embedded newform 5625.2.a.o.1.3

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:	$6$
Coefficient field:	6.6.46840000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 $ x^6 - x^5 - 11x^4 + 8x^3 + 31x^2 - 15x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.858825$ 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)

$f(q)$	$=$	$q-0.858825 q^{2} -1.26242 q^{4} +3.88045 q^{7} +2.80185 q^{8} +O(q^{10})$	$q-0.858825 q^{2} -1.26242 q^{4} +3.88045 q^{7} +2.80185 q^{8} +1.39825 q^{11} +3.36204 q^{13} -3.33263 q^{14} +0.118543 q^{16} -3.11590 q^{17} -2.70595 q^{19} -1.20085 q^{22} +6.43989 q^{23} -2.88740 q^{26} -4.89876 q^{28} -8.26242 q^{29} -6.34027 q^{31} -5.70550 q^{32} +2.67601 q^{34} -7.49949 q^{37} +2.32394 q^{38} -11.3783 q^{41} -3.39725 q^{43} -1.76518 q^{44} -5.53074 q^{46} -8.38291 q^{47} +8.05792 q^{49} -4.24430 q^{52} -11.4431 q^{53} +10.8724 q^{56} +7.09597 q^{58} -7.64074 q^{59} +10.8219 q^{61} +5.44518 q^{62} +4.66294 q^{64} +3.54377 q^{67} +3.93358 q^{68} -1.18356 q^{71} +2.39461 q^{73} +6.44075 q^{74} +3.41605 q^{76} +5.42585 q^{77} +10.6245 q^{79} +9.77199 q^{82} -1.40894 q^{83} +2.91764 q^{86} +3.91769 q^{88} -4.62972 q^{89} +13.0462 q^{91} -8.12984 q^{92} +7.19945 q^{94} -1.51516 q^{97} -6.92034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8	$ 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38} - 33 q^{41} - 3 q^{43} + 11 q^{44} - 12 q^{46} + 6 q^{47} + 4 q^{49} - 33 q^{52} - 14 q^{53} + 30 q^{56} - q^{58} + 8 q^{59} + 34 q^{61} - 31 q^{62} + 12 q^{64} + 2 q^{67} - 27 q^{68} + 3 q^{71} - 36 q^{73} - 36 q^{74} + 27 q^{76} - 16 q^{77} + 25 q^{79} + 36 q^{82} + 12 q^{83} + 30 q^{86} + 56 q^{88} - 18 q^{89} + 28 q^{91} - 3 q^{92} - 50 q^{94} + 7 q^{97} - 15 q^{98}+O(q^{100}) $ 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8 + 4 * q^14 + 17 * q^16 - 2 * q^17 - 2 * q^19 + 9 * q^22 + q^23 - 37 * q^26 - 44 * q^28 - 31 * q^29 - 2 * q^31 - 33 * q^32 + 37 * q^34 - 22 * q^37 - 27 * q^38 - 33 * q^41 - 3 * q^43 + 11 * q^44 - 12 * q^46 + 6 * q^47 + 4 * q^49 - 33 * q^52 - 14 * q^53 + 30 * q^56 - q^58 + 8 * q^59 + 34 * q^61 - 31 * q^62 + 12 * q^64 + 2 * q^67 - 27 * q^68 + 3 * q^71 - 36 * q^73 - 36 * q^74 + 27 * q^76 - 16 * q^77 + 25 * q^79 + 36 * q^82 + 12 * q^83 + 30 * q^86 + 56 * q^88 - 18 * q^89 + 28 * q^91 - 3 * q^92 - 50 * q^94 + 7 * q^97 - 15 * 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 $

$2$	−0.858825	−0.607281	−0.303640	−	0.952787i	$-0.598202\pi$
$2$	−0.858825	−0.607281	−0.303640	+	0.952787i	$0.598202\pi$
$3$	0	0
$3$	0	0
$4$	−1.26242	−0.631210
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.88045	1.46667	0.733337	−	0.679865i	$-0.237962\pi$
$7$	3.88045	1.46667	0.733337	+	0.679865i	$0.237962\pi$
$8$	2.80185	0.990603
$9$	0	0
$10$	0	0
$11$	1.39825	0.421589	0.210794	−	0.977530i	$-0.432395\pi$
$11$	1.39825	0.421589	0.210794	+	0.977530i	$0.432395\pi$
$12$	0	0
$13$	3.36204	0.932461	0.466230	−	0.884663i	$-0.345612\pi$
$13$	3.36204	0.932461	0.466230	+	0.884663i	$0.345612\pi$
$14$	−3.33263	−0.890683
$15$	0	0
$16$	0.118543	0.0296359
$17$	−3.11590	−0.755717	−0.377859	−	0.925863i	$-0.623339\pi$
$17$	−3.11590	−0.755717	−0.377859	+	0.925863i	$0.623339\pi$
$18$	0	0
$19$	−2.70595	−0.620788	−0.310394	−	0.950608i	$-0.600461\pi$
$19$	−2.70595	−0.620788	−0.310394	+	0.950608i	$0.600461\pi$
$20$	0	0
$21$	0	0
$22$	−1.20085	−0.256023
$23$	6.43989	1.34281	0.671405	−	0.741091i	$-0.265691\pi$
$23$	6.43989	1.34281	0.671405	+	0.741091i	$0.265691\pi$
$24$	0	0
$25$	0	0
$26$	−2.88740	−0.566266
$27$	0	0
$28$	−4.89876	−0.925779
$29$	−8.26242	−1.53429	−0.767146	−	0.641472i	$-0.778324\pi$
$29$	−8.26242	−1.53429	−0.767146	+	0.641472i	$0.778324\pi$
$30$	0	0
$31$	−6.34027	−1.13875	−0.569373	−	0.822079i	$-0.692814\pi$
$31$	−6.34027	−1.13875	−0.569373	+	0.822079i	$0.692814\pi$
$32$	−5.70550	−1.00860
$33$	0	0
$34$	2.67601	0.458933
$35$	0	0
$36$	0	0
$37$	−7.49949	−1.23291	−0.616454	−	0.787391i	$-0.711432\pi$
$37$	−7.49949	−1.23291	−0.616454	+	0.787391i	$0.711432\pi$
$38$	2.32394	0.376993
$39$	0	0
$40$	0	0
$41$	−11.3783	−1.77700	−0.888498	−	0.458881i	$-0.848250\pi$
$41$	−11.3783	−1.77700	−0.888498	+	0.458881i	$0.848250\pi$
$42$	0	0
$43$	−3.39725	−0.518076	−0.259038	−	0.965867i	$-0.583405\pi$
$43$	−3.39725	−0.518076	−0.259038	+	0.965867i	$0.583405\pi$
$44$	−1.76518	−0.266111
$45$	0	0
$46$	−5.53074	−0.815463
$47$	−8.38291	−1.22277	−0.611387	−	0.791332i	$-0.709388\pi$
$47$	−8.38291	−1.22277	−0.611387	+	0.791332i	$0.709388\pi$
$48$	0	0
$49$	8.05792	1.15113
$50$	0	0
$51$	0	0
$52$	−4.24430	−0.588579
$53$	−11.4431	−1.57184	−0.785918	−	0.618330i	$-0.787809\pi$
$53$	−11.4431	−1.57184	−0.785918	+	0.618330i	$0.787809\pi$
$54$	0	0
$55$	0	0
$56$	10.8724	1.45289
$57$	0	0
$58$	7.09597	0.931747
$59$	−7.64074	−0.994740	−0.497370	−	0.867539i	$-0.665701\pi$
$59$	−7.64074	−0.994740	−0.497370	+	0.867539i	$0.665701\pi$
$60$	0	0
$61$	10.8219	1.38560	0.692798	−	0.721132i	$-0.256378\pi$
$61$	10.8219	1.38560	0.692798	+	0.721132i	$0.256378\pi$
$62$	5.44518	0.691539
$63$	0	0
$64$	4.66294	0.582868
$65$	0	0
$66$	0	0
$67$	3.54377	0.432940	0.216470	−	0.976289i	$-0.430546\pi$
$67$	3.54377	0.432940	0.216470	+	0.976289i	$0.430546\pi$
$68$	3.93358	0.477016
$69$	0	0
$70$	0	0
$71$	−1.18356	−0.140463	−0.0702317	−	0.997531i	$-0.522374\pi$
$71$	−1.18356	−0.140463	−0.0702317	+	0.997531i	$0.522374\pi$
$72$	0	0
$73$	2.39461	0.280268	0.140134	−	0.990133i	$-0.455247\pi$
$73$	2.39461	0.280268	0.140134	+	0.990133i	$0.455247\pi$
$74$	6.44075	0.748722
$75$	0	0
$76$	3.41605	0.391847
$77$	5.42585	0.618333
$78$	0	0
$79$	10.6245	1.19534	0.597672	−	0.801740i	$-0.296092\pi$
$79$	10.6245	1.19534	0.597672	+	0.801740i	$0.296092\pi$
$80$	0	0
$81$	0	0
$82$	9.77199	1.07914
$83$	−1.40894	−0.154651	−0.0773255	−	0.997006i	$-0.524638\pi$
$83$	−1.40894	−0.154651	−0.0773255	+	0.997006i	$0.524638\pi$
$84$	0	0
$85$	0	0
$86$	2.91764	0.314617
$87$	0	0
$88$	3.91769	0.417627
$89$	−4.62972	−0.490750	−0.245375	−	0.969428i	$-0.578911\pi$
$89$	−4.62972	−0.490750	−0.245375	+	0.969428i	$0.578911\pi$
$90$	0	0
$91$	13.0462	1.36762
$92$	−8.12984	−0.847595
$93$	0	0
$94$	7.19945	0.742567
$95$	0	0
$96$	0	0
$97$	−1.51516	−0.153841	−0.0769204	−	0.997037i	$-0.524509\pi$
$97$	−1.51516	−0.153841	−0.0769204	+	0.997037i	$0.524509\pi$
$98$	−6.92034	−0.699060
$99$	0	0
$100$	0	0
$101$	−17.2376	−1.71521	−0.857604	−	0.514310i	$-0.828048\pi$
$101$	−17.2376	−1.71521	−0.857604	+	0.514310i	$0.828048\pi$
$102$	0	0
$103$	−10.9905	−1.08292	−0.541461	−	0.840726i	$-0.682129\pi$
$103$	−10.9905	−1.08292	−0.541461	+	0.840726i	$0.682129\pi$
$104$	9.41991	0.923698
$105$	0	0
$106$	9.82766	0.954546
$107$	−6.80699	−0.658056	−0.329028	−	0.944320i	$-0.606721\pi$
$107$	−6.80699	−0.658056	−0.329028	+	0.944320i	$0.606721\pi$
$108$	0	0
$109$	−11.4451	−1.09624	−0.548121	−	0.836399i	$-0.684657\pi$
$109$	−11.4451	−1.09624	−0.548121	+	0.836399i	$0.684657\pi$
$110$	0	0
$111$	0	0
$112$	0.460003	0.0434662
$113$	16.2686	1.53042	0.765210	−	0.643781i	$-0.222635\pi$
$113$	16.2686	1.53042	0.765210	+	0.643781i	$0.222635\pi$
$114$	0	0
$115$	0	0
$116$	10.4306	0.968461
$117$	0	0
$118$	6.56206	0.604087
$119$	−12.0911	−1.10839
$120$	0	0
$121$	−9.04489	−0.822263
$122$	−9.29408	−0.841446
$123$	0	0
$124$	8.00409	0.718788
$125$	0	0
$126$	0	0
$127$	−16.2477	−1.44175	−0.720875	−	0.693065i	$-0.756260\pi$
$127$	−16.2477	−1.44175	−0.720875	+	0.693065i	$0.756260\pi$
$128$	7.40636	0.654636
$129$	0	0
$130$	0	0
$131$	15.3119	1.33781	0.668903	−	0.743349i	$-0.266764\pi$
$131$	15.3119	1.33781	0.668903	+	0.743349i	$0.266764\pi$
$132$	0	0
$133$	−10.5003	−0.910493
$134$	−3.04348	−0.262916
$135$	0	0
$136$	−8.73028	−0.748615
$137$	−6.86417	−0.586445	−0.293223	−	0.956044i	$-0.594728\pi$
$137$	−6.86417	−0.586445	−0.293223	+	0.956044i	$0.594728\pi$
$138$	0	0
$139$	13.8225	1.17241	0.586203	−	0.810164i	$-0.300622\pi$
$139$	13.8225	1.17241	0.586203	+	0.810164i	$0.300622\pi$
$140$	0	0
$141$	0	0
$142$	1.01647	0.0853007
$143$	4.70097	0.393115
$144$	0	0
$145$	0	0
$146$	−2.05655	−0.170201
$147$	0	0
$148$	9.46751	0.778224
$149$	−7.09597	−0.581325	−0.290662	−	0.956826i	$-0.593876\pi$
$149$	−7.09597	−0.581325	−0.290662	+	0.956826i	$0.593876\pi$
$150$	0	0
$151$	19.1474	1.55819	0.779097	−	0.626903i	$-0.215678\pi$
$151$	19.1474	1.55819	0.779097	+	0.626903i	$0.215678\pi$
$152$	−7.58166	−0.614954
$153$	0	0
$154$	−4.65986	−0.375502
$155$	0	0
$156$	0	0
$157$	−11.8114	−0.942653	−0.471326	−	0.881959i	$-0.656225\pi$
$157$	−11.8114	−0.942653	−0.471326	+	0.881959i	$0.656225\pi$
$158$	−9.12455	−0.725910
$159$	0	0
$160$	0	0
$161$	24.9897	1.96946
$162$	0	0
$163$	8.92635	0.699165	0.349583	−	0.936906i	$-0.386323\pi$
$163$	8.92635	0.699165	0.349583	+	0.936906i	$0.386323\pi$
$164$	14.3642	1.12166
$165$	0	0
$166$	1.21003	0.0939166
$167$	−15.8114	−1.22352	−0.611762	−	0.791042i	$-0.709539\pi$
$167$	−15.8114	−1.22352	−0.611762	+	0.791042i	$0.709539\pi$
$168$	0	0
$169$	−1.69672	−0.130517
$170$	0	0
$171$	0	0
$172$	4.28876	0.327015
$173$	−16.3259	−1.24124	−0.620619	−	0.784112i	$-0.713119\pi$
$173$	−16.3259	−1.24124	−0.620619	+	0.784112i	$0.713119\pi$
$174$	0	0
$175$	0	0
$176$	0.165754	0.0124942
$177$	0	0
$178$	3.97612	0.298023
$179$	0.852080	0.0636875	0.0318437	−	0.999493i	$-0.489862\pi$
$179$	0.852080	0.0636875	0.0318437	+	0.999493i	$0.489862\pi$
$180$	0	0
$181$	21.9643	1.63260	0.816298	−	0.577631i	$-0.196023\pi$
$181$	21.9643	1.63260	0.816298	+	0.577631i	$0.196023\pi$
$182$	−11.2044	−0.830527
$183$	0	0
$184$	18.0436	1.33019
$185$	0	0
$186$	0	0
$187$	−4.35682	−0.318602
$188$	10.5828	0.771827
$189$	0	0
$190$	0	0
$191$	1.15408	0.0835065	0.0417532	−	0.999128i	$-0.486706\pi$
$191$	1.15408	0.0835065	0.0417532	+	0.999128i	$0.486706\pi$
$192$	0	0
$193$	−0.983419	−0.0707880	−0.0353940	−	0.999373i	$-0.511269\pi$
$193$	−0.983419	−0.0707880	−0.0353940	+	0.999373i	$0.511269\pi$
$194$	1.30125	0.0934246
$195$	0	0
$196$	−10.1725	−0.726606
$197$	7.40726	0.527745	0.263873	−	0.964558i	$-0.415000\pi$
$197$	7.40726	0.527745	0.263873	+	0.964558i	$0.415000\pi$
$198$	0	0
$199$	−13.5887	−0.963274	−0.481637	−	0.876371i	$-0.659958\pi$
$199$	−13.5887	−0.963274	−0.481637	+	0.876371i	$0.659958\pi$
$200$	0	0
$201$	0	0
$202$	14.8041	1.04161
$203$	−32.0619	−2.25031
$204$	0	0
$205$	0	0
$206$	9.43888	0.657638
$207$	0	0
$208$	0.398547	0.0276343
$209$	−3.78360	−0.261717
$210$	0	0
$211$	26.2982	1.81044	0.905221	−	0.424941i	$-0.139705\pi$
$211$	26.2982	1.81044	0.905221	+	0.424941i	$0.139705\pi$
$212$	14.4461	0.992159
$213$	0	0
$214$	5.84601	0.399625
$215$	0	0
$216$	0	0
$217$	−24.6031	−1.67017
$218$	9.82934	0.665727
$219$	0	0
$220$	0	0
$221$	−10.4758	−0.704677
$222$	0	0
$223$	−13.5470	−0.907176	−0.453588	−	0.891212i	$-0.649856\pi$
$223$	−13.5470	−0.907176	−0.453588	+	0.891212i	$0.649856\pi$
$224$	−22.1399	−1.47929
$225$	0	0
$226$	−13.9719	−0.929395
$227$	−0.265434	−0.0176175	−0.00880874	−	0.999961i	$-0.502804\pi$
$227$	−0.265434	−0.0176175	−0.00880874	+	0.999961i	$0.502804\pi$
$228$	0	0
$229$	4.28115	0.282906	0.141453	−	0.989945i	$-0.454823\pi$
$229$	4.28115	0.282906	0.141453	+	0.989945i	$0.454823\pi$
$230$	0	0
$231$	0	0
$232$	−23.1500	−1.51987
$233$	6.74720	0.442024	0.221012	−	0.975271i	$-0.429064\pi$
$233$	6.74720	0.442024	0.221012	+	0.975271i	$0.429064\pi$
$234$	0	0
$235$	0	0
$236$	9.64582	0.627890
$237$	0	0
$238$	10.3841	0.673104
$239$	0.307310	0.0198782	0.00993912	−	0.999951i	$-0.496836\pi$
$239$	0.307310	0.0198782	0.00993912	+	0.999951i	$0.496836\pi$
$240$	0	0
$241$	−11.1233	−0.716513	−0.358256	−	0.933623i	$-0.616629\pi$
$241$	−11.1233	−0.716513	−0.358256	+	0.933623i	$0.616629\pi$
$242$	7.76798	0.499344
$243$	0	0
$244$	−13.6617	−0.874602
$245$	0	0
$246$	0	0
$247$	−9.09751	−0.578860
$248$	−17.7645	−1.12805
$249$	0	0
$250$	0	0
$251$	20.5499	1.29710	0.648549	−	0.761173i	$-0.275376\pi$
$251$	20.5499	1.29710	0.648549	+	0.761173i	$0.275376\pi$
$252$	0	0
$253$	9.00459	0.566114
$254$	13.9539	0.875548
$255$	0	0
$256$	−15.6866	−0.980415
$257$	2.43352	0.151799	0.0758996	−	0.997115i	$-0.475817\pi$
$257$	2.43352	0.151799	0.0758996	+	0.997115i	$0.475817\pi$
$258$	0	0
$259$	−29.1014	−1.80827
$260$	0	0
$261$	0	0
$262$	−13.1502	−0.812425
$263$	20.1748	1.24403	0.622017	−	0.783004i	$-0.286314\pi$
$263$	20.1748	1.24403	0.622017	+	0.783004i	$0.286314\pi$
$264$	0	0
$265$	0	0
$266$	9.01794	0.552925
$267$	0	0
$268$	−4.47372	−0.273276
$269$	18.6834	1.13915	0.569574	−	0.821940i	$-0.307108\pi$
$269$	18.6834	1.13915	0.569574	+	0.821940i	$0.307108\pi$
$270$	0	0
$271$	7.61184	0.462387	0.231193	−	0.972908i	$-0.425737\pi$
$271$	7.61184	0.462387	0.231193	+	0.972908i	$0.425737\pi$
$272$	−0.369370	−0.0223963
$273$	0	0
$274$	5.89512	0.356137
$275$	0	0
$276$	0	0
$277$	−8.17364	−0.491107	−0.245553	−	0.969383i	$-0.578970\pi$
$277$	−8.17364	−0.491107	−0.245553	+	0.969383i	$0.578970\pi$
$278$	−11.8711	−0.711980
$279$	0	0
$280$	0	0
$281$	−14.8891	−0.888211	−0.444105	−	0.895975i	$-0.646478\pi$
$281$	−14.8891	−0.888211	−0.444105	+	0.895975i	$0.646478\pi$
$282$	0	0
$283$	−1.35203	−0.0803698	−0.0401849	−	0.999192i	$-0.512795\pi$
$283$	−1.35203	−0.0803698	−0.0401849	+	0.999192i	$0.512795\pi$
$284$	1.49416	0.0886618
$285$	0	0
$286$	−4.03731	−0.238731
$287$	−44.1531	−2.60627
$288$	0	0
$289$	−7.29115	−0.428891
$290$	0	0
$291$	0	0
$292$	−3.02300	−0.176908
$293$	10.4773	0.612091	0.306045	−	0.952017i	$-0.400994\pi$
$293$	10.4773	0.612091	0.306045	+	0.952017i	$0.400994\pi$
$294$	0	0
$295$	0	0
$296$	−21.0124	−1.22132
$297$	0	0
$298$	6.09420	0.353027
$299$	21.6511	1.25212
$300$	0	0
$301$	−13.1829	−0.759848
$302$	−16.4443	−0.946262
$303$	0	0
$304$	−0.320773	−0.0183976
$305$	0	0
$306$	0	0
$307$	10.4991	0.599216	0.299608	−	0.954062i	$-0.403144\pi$
$307$	10.4991	0.599216	0.299608	+	0.954062i	$0.403144\pi$
$308$	−6.84971	−0.390298
$309$	0	0
$310$	0	0
$311$	4.68539	0.265684	0.132842	−	0.991137i	$-0.457590\pi$
$311$	4.68539	0.265684	0.132842	+	0.991137i	$0.457590\pi$
$312$	0	0
$313$	−17.2431	−0.974635	−0.487317	−	0.873225i	$-0.662025\pi$
$313$	−17.2431	−0.974635	−0.487317	+	0.873225i	$0.662025\pi$
$314$	10.1439	0.572455
$315$	0	0
$316$	−13.4125	−0.754513
$317$	8.94648	0.502484	0.251242	−	0.967924i	$-0.419161\pi$
$317$	8.94648	0.502484	0.251242	+	0.967924i	$0.419161\pi$
$318$	0	0
$319$	−11.5529	−0.646841
$320$	0	0
$321$	0	0
$322$	−21.4618	−1.19602
$323$	8.43148	0.469140
$324$	0	0
$325$	0	0
$326$	−7.66617	−0.424590
$327$	0	0
$328$	−31.8803	−1.76030
$329$	−32.5295	−1.79341
$330$	0	0
$331$	12.0810	0.664034	0.332017	−	0.943273i	$-0.392271\pi$
$331$	12.0810	0.664034	0.332017	+	0.943273i	$0.392271\pi$
$332$	1.77867	0.0976172
$333$	0	0
$334$	13.5792	0.743022
$335$	0	0
$336$	0	0
$337$	15.2014	0.828074	0.414037	−	0.910260i	$-0.364118\pi$
$337$	15.2014	0.828074	0.414037	+	0.910260i	$0.364118\pi$
$338$	1.45718	0.0792603
$339$	0	0
$340$	0	0
$341$	−8.86530	−0.480083
$342$	0	0
$343$	4.10522	0.221661
$344$	−9.51857	−0.513207
$345$	0	0
$346$	14.0211	0.753780
$347$	8.51395	0.457053	0.228526	−	0.973538i	$-0.426609\pi$
$347$	8.51395	0.457053	0.228526	+	0.973538i	$0.426609\pi$
$348$	0	0
$349$	−6.72703	−0.360090	−0.180045	−	0.983658i	$-0.557624\pi$
$349$	−6.72703	−0.360090	−0.180045	+	0.983658i	$0.557624\pi$
$350$	0	0
$351$	0	0
$352$	−7.97773	−0.425215
$353$	−13.3066	−0.708239	−0.354119	−	0.935200i	$-0.615219\pi$
$353$	−13.3066	−0.708239	−0.354119	+	0.935200i	$0.615219\pi$
$354$	0	0
$355$	0	0
$356$	5.84465	0.309766
$357$	0	0
$358$	−0.731788	−0.0386762
$359$	27.1766	1.43433	0.717163	−	0.696905i	$-0.245440\pi$
$359$	27.1766	1.43433	0.717163	+	0.696905i	$0.245440\pi$
$360$	0	0
$361$	−11.6778	−0.614622
$362$	−18.8635	−0.991445
$363$	0	0
$364$	−16.4698	−0.863253
$365$	0	0
$366$	0	0
$367$	15.9939	0.834877	0.417439	−	0.908705i	$-0.362928\pi$
$367$	15.9939	0.834877	0.417439	+	0.908705i	$0.362928\pi$
$368$	0.763407	0.0397953
$369$	0	0
$370$	0	0
$371$	−44.4046	−2.30537
$372$	0	0
$373$	−31.2294	−1.61700	−0.808499	−	0.588497i	$-0.799720\pi$
$373$	−31.2294	−1.61700	−0.808499	+	0.588497i	$0.799720\pi$
$374$	3.74174	0.193481
$375$	0	0
$376$	−23.4876	−1.21128
$377$	−27.7785	−1.43067
$378$	0	0
$379$	−32.2022	−1.65412	−0.827059	−	0.562115i	$-0.809988\pi$
$379$	−32.2022	−1.65412	−0.827059	+	0.562115i	$0.809988\pi$
$380$	0	0
$381$	0	0
$382$	−0.991154	−0.0507119
$383$	19.4271	0.992678	0.496339	−	0.868129i	$-0.334677\pi$
$383$	19.4271	0.992678	0.496339	+	0.868129i	$0.334677\pi$
$384$	0	0
$385$	0	0
$386$	0.844584	0.0429882
$387$	0	0
$388$	1.91276	0.0971059
$389$	−7.87404	−0.399230	−0.199615	−	0.979874i	$-0.563969\pi$
$389$	−7.87404	−0.399230	−0.199615	+	0.979874i	$0.563969\pi$
$390$	0	0
$391$	−20.0661	−1.01478
$392$	22.5771	1.14031
$393$	0	0
$394$	−6.36154	−0.320490
$395$	0	0
$396$	0	0
$397$	28.4801	1.42938	0.714689	−	0.699443i	$-0.246568\pi$
$397$	28.4801	1.42938	0.714689	+	0.699443i	$0.246568\pi$
$398$	11.6703	0.584978
$399$	0	0
$400$	0	0
$401$	−23.2147	−1.15929	−0.579643	−	0.814870i	$-0.696808\pi$
$401$	−23.2147	−1.15929	−0.579643	+	0.814870i	$0.696808\pi$
$402$	0	0
$403$	−21.3162	−1.06184
$404$	21.7611	1.08266
$405$	0	0
$406$	27.5356	1.36657
$407$	−10.4862	−0.519781
$408$	0	0
$409$	4.32217	0.213718	0.106859	−	0.994274i	$-0.465921\pi$
$409$	4.32217	0.213718	0.106859	+	0.994274i	$0.465921\pi$
$410$	0	0
$411$	0	0
$412$	13.8746	0.683551
$413$	−29.6495	−1.45896
$414$	0	0
$415$	0	0
$416$	−19.1821	−0.940480
$417$	0	0
$418$	3.24945	0.158936
$419$	1.96426	0.0959604	0.0479802	−	0.998848i	$-0.484722\pi$
$419$	1.96426	0.0959604	0.0479802	+	0.998848i	$0.484722\pi$
$420$	0	0
$421$	2.13023	0.103821	0.0519106	−	0.998652i	$-0.483469\pi$
$421$	2.13023	0.103821	0.0519106	+	0.998652i	$0.483469\pi$
$422$	−22.5855	−1.09945
$423$	0	0
$424$	−32.0619	−1.55707
$425$	0	0
$426$	0	0
$427$	41.9937	2.03222
$428$	8.59328	0.415372
$429$	0	0
$430$	0	0
$431$	−0.0662228	−0.00318984	−0.00159492	−	0.999999i	$-0.500508\pi$
$431$	−0.0662228	−0.00318984	−0.00159492	+	0.999999i	$0.500508\pi$
$432$	0	0
$433$	−1.22844	−0.0590352	−0.0295176	−	0.999564i	$-0.509397\pi$
$433$	−1.22844	−0.0590352	−0.0295176	+	0.999564i	$0.509397\pi$
$434$	21.1298	1.01426
$435$	0	0
$436$	14.4485	0.691959
$437$	−17.4260	−0.833600
$438$	0	0
$439$	−37.8322	−1.80563	−0.902816	−	0.430028i	$-0.858504\pi$
$439$	−37.8322	−1.80563	−0.902816	+	0.430028i	$0.858504\pi$
$440$	0	0
$441$	0	0
$442$	8.99686	0.427937
$443$	−26.9171	−1.27887	−0.639435	−	0.768845i	$-0.720832\pi$
$443$	−26.9171	−1.27887	−0.639435	+	0.768845i	$0.720832\pi$
$444$	0	0
$445$	0	0
$446$	11.6345	0.550910
$447$	0	0
$448$	18.0943	0.854877
$449$	25.3090	1.19440	0.597202	−	0.802091i	$-0.296279\pi$
$449$	25.3090	1.19440	0.597202	+	0.802091i	$0.296279\pi$
$450$	0	0
$451$	−15.9098	−0.749162
$452$	−20.5378	−0.966016
$453$	0	0
$454$	0.227961	0.0106988
$455$	0	0
$456$	0	0
$457$	−38.0944	−1.78198	−0.890990	−	0.454023i	$-0.849988\pi$
$457$	−38.0944	−1.78198	−0.890990	+	0.454023i	$0.849988\pi$
$458$	−3.67676	−0.171804
$459$	0	0
$460$	0	0
$461$	−3.53623	−0.164699	−0.0823494	−	0.996604i	$-0.526242\pi$
$461$	−3.53623	−0.164699	−0.0823494	+	0.996604i	$0.526242\pi$
$462$	0	0
$463$	23.7206	1.10239	0.551195	−	0.834377i	$-0.314172\pi$
$463$	23.7206	1.10239	0.551195	+	0.834377i	$0.314172\pi$
$464$	−0.979456	−0.0454701
$465$	0	0
$466$	−5.79466	−0.268433
$467$	−19.0985	−0.883774	−0.441887	−	0.897071i	$-0.645691\pi$
$467$	−19.0985	−0.883774	−0.441887	+	0.897071i	$0.645691\pi$
$468$	0	0
$469$	13.7514	0.634982
$470$	0	0
$471$	0	0
$472$	−21.4082	−0.985392
$473$	−4.75021	−0.218415
$474$	0	0
$475$	0	0
$476$	15.2641	0.699627
$477$	0	0
$478$	−0.263926	−0.0120717
$479$	−18.3034	−0.836302	−0.418151	−	0.908378i	$-0.637322\pi$
$479$	−18.3034	−0.836302	−0.418151	+	0.908378i	$0.637322\pi$
$480$	0	0
$481$	−25.2136	−1.14964
$482$	9.55294	0.435124
$483$	0	0
$484$	11.4184	0.519020
$485$	0	0
$486$	0	0
$487$	−7.01638	−0.317943	−0.158971	−	0.987283i	$-0.550818\pi$
$487$	−7.01638	−0.317943	−0.158971	+	0.987283i	$0.550818\pi$
$488$	30.3212	1.37258
$489$	0	0
$490$	0	0
$491$	4.63076	0.208983	0.104492	−	0.994526i	$-0.466678\pi$
$491$	4.63076	0.208983	0.104492	+	0.994526i	$0.466678\pi$
$492$	0	0
$493$	25.7449	1.15949
$494$	7.81316	0.351531
$495$	0	0
$496$	−0.751598	−0.0337477
$497$	−4.59277	−0.206014
$498$	0	0
$499$	−5.44561	−0.243779	−0.121890	−	0.992544i	$-0.538895\pi$
$499$	−5.44561	−0.243779	−0.121890	+	0.992544i	$0.538895\pi$
$500$	0	0
$501$	0	0
$502$	−17.6488	−0.787703
$503$	−17.2645	−0.769788	−0.384894	−	0.922961i	$-0.625762\pi$
$503$	−17.2645	−0.769788	−0.384894	+	0.922961i	$0.625762\pi$
$504$	0	0
$505$	0	0
$506$	−7.73336	−0.343790
$507$	0	0
$508$	20.5114	0.910047
$509$	10.1973	0.451989	0.225994	−	0.974129i	$-0.427437\pi$
$509$	10.1973	0.451989	0.225994	+	0.974129i	$0.427437\pi$
$510$	0	0
$511$	9.29217	0.411061
$512$	−1.34063	−0.0592481
$513$	0	0
$514$	−2.08997	−0.0921847
$515$	0	0
$516$	0	0
$517$	−11.7214	−0.515508
$518$	24.9930	1.09813
$519$	0	0
$520$	0	0
$521$	−38.5575	−1.68923	−0.844617	−	0.535371i	$-0.820172\pi$
$521$	−38.5575	−1.68923	−0.844617	+	0.535371i	$0.820172\pi$
$522$	0	0
$523$	−6.40172	−0.279928	−0.139964	−	0.990157i	$-0.544699\pi$
$523$	−6.40172	−0.279928	−0.139964	+	0.990157i	$0.544699\pi$
$524$	−19.3300	−0.844437
$525$	0	0
$526$	−17.3266	−0.755478
$527$	19.7557	0.860570
$528$	0	0
$529$	18.4722	0.803137
$530$	0	0
$531$	0	0
$532$	13.2558	0.574712
$533$	−38.2543	−1.65698
$534$	0	0
$535$	0	0
$536$	9.92910	0.428872
$537$	0	0
$538$	−16.0458	−0.691783
$539$	11.2670	0.485304
$540$	0	0
$541$	28.7378	1.23553	0.617766	−	0.786362i	$-0.288038\pi$
$541$	28.7378	1.23553	0.617766	+	0.786362i	$0.288038\pi$
$542$	−6.53724	−0.280799
$543$	0	0
$544$	17.7778	0.762216
$545$	0	0
$546$	0	0
$547$	5.20272	0.222452	0.111226	−	0.993795i	$-0.464522\pi$
$547$	5.20272	0.222452	0.111226	+	0.993795i	$0.464522\pi$
$548$	8.66546	0.370170
$549$	0	0
$550$	0	0
$551$	22.3577	0.952470
$552$	0	0
$553$	41.2277	1.75318
$554$	7.01973	0.298240
$555$	0	0
$556$	−17.4498	−0.740035
$557$	33.0079	1.39859	0.699296	−	0.714833i	$-0.253497\pi$
$557$	33.0079	1.39859	0.699296	+	0.714833i	$0.253497\pi$
$558$	0	0
$559$	−11.4217	−0.483085
$560$	0	0
$561$	0	0
$562$	12.7872	0.539393
$563$	−15.1684	−0.639270	−0.319635	−	0.947541i	$-0.603560\pi$
$563$	−15.1684	−0.639270	−0.319635	+	0.947541i	$0.603560\pi$
$564$	0	0
$565$	0	0
$566$	1.16116	0.0488070
$567$	0	0
$568$	−3.31617	−0.139143
$569$	29.5024	1.23681	0.618403	−	0.785861i	$-0.287780\pi$
$569$	29.5024	1.23681	0.618403	+	0.785861i	$0.287780\pi$
$570$	0	0
$571$	3.12393	0.130733	0.0653663	−	0.997861i	$-0.479178\pi$
$571$	3.12393	0.130733	0.0653663	+	0.997861i	$0.479178\pi$
$572$	−5.93460	−0.248138
$573$	0	0
$574$	37.9197	1.58274
$575$	0	0
$576$	0	0
$577$	28.0748	1.16877	0.584385	−	0.811476i	$-0.301336\pi$
$577$	28.0748	1.16877	0.584385	+	0.811476i	$0.301336\pi$
$578$	6.26182	0.260458
$579$	0	0
$580$	0	0
$581$	−5.46732	−0.226823
$582$	0	0
$583$	−16.0004	−0.662669
$584$	6.70933	0.277634
$585$	0	0
$586$	−8.99817	−0.371711
$587$	35.9054	1.48197	0.740987	−	0.671520i	$-0.234358\pi$
$587$	35.9054	1.48197	0.740987	+	0.671520i	$0.234358\pi$
$588$	0	0
$589$	17.1565	0.706920
$590$	0	0
$591$	0	0
$592$	−0.889016	−0.0365383
$593$	15.3084	0.628641	0.314321	−	0.949317i	$-0.398223\pi$
$593$	15.3084	0.628641	0.314321	+	0.949317i	$0.398223\pi$
$594$	0	0
$595$	0	0
$596$	8.95810	0.366938
$597$	0	0
$598$	−18.5945	−0.760387
$599$	16.2538	0.664114	0.332057	−	0.943259i	$-0.392257\pi$
$599$	16.2538	0.664114	0.332057	+	0.943259i	$0.392257\pi$
$600$	0	0
$601$	−0.233686	−0.00953223	−0.00476612	−	0.999989i	$-0.501517\pi$
$601$	−0.233686	−0.00953223	−0.00476612	+	0.999989i	$0.501517\pi$
$602$	11.3218	0.461441
$603$	0	0
$604$	−24.1721	−0.983548
$605$	0	0
$606$	0	0
$607$	−10.6590	−0.432637	−0.216318	−	0.976323i	$-0.569405\pi$
$607$	−10.6590	−0.432637	−0.216318	+	0.976323i	$0.569405\pi$
$608$	15.4388	0.626127
$609$	0	0
$610$	0	0
$611$	−28.1836	−1.14019
$612$	0	0
$613$	17.9094	0.723353	0.361677	−	0.932304i	$-0.382204\pi$
$613$	17.9094	0.723353	0.361677	+	0.932304i	$0.382204\pi$
$614$	−9.01689	−0.363892
$615$	0	0
$616$	15.2024	0.612523
$617$	−31.6418	−1.27385	−0.636926	−	0.770925i	$-0.719794\pi$
$617$	−31.6418	−1.27385	−0.636926	+	0.770925i	$0.719794\pi$
$618$	0	0
$619$	21.8317	0.877488	0.438744	−	0.898612i	$-0.355423\pi$
$619$	21.8317	0.877488	0.438744	+	0.898612i	$0.355423\pi$
$620$	0	0
$621$	0	0
$622$	−4.02393	−0.161345
$623$	−17.9654	−0.719769
$624$	0	0
$625$	0	0
$626$	14.8088	0.591877
$627$	0	0
$628$	14.9110	0.595012
$629$	23.3677	0.931730
$630$	0	0
$631$	1.62102	0.0645318	0.0322659	−	0.999479i	$-0.489728\pi$
$631$	1.62102	0.0645318	0.0322659	+	0.999479i	$0.489728\pi$
$632$	29.7681	1.18411
$633$	0	0
$634$	−7.68346	−0.305149
$635$	0	0
$636$	0	0
$637$	27.0910	1.07339
$638$	9.92196	0.392814
$639$	0	0
$640$	0	0
$641$	−4.53285	−0.179037	−0.0895183	−	0.995985i	$-0.528533\pi$
$641$	−4.53285	−0.179037	−0.0895183	+	0.995985i	$0.528533\pi$
$642$	0	0
$643$	12.6562	0.499111	0.249556	−	0.968360i	$-0.419715\pi$
$643$	12.6562	0.499111	0.249556	+	0.968360i	$0.419715\pi$
$644$	−31.5475	−1.24314
$645$	0	0
$646$	−7.24117	−0.284900
$647$	22.3268	0.877757	0.438879	−	0.898546i	$-0.355376\pi$
$647$	22.3268	0.877757	0.438879	+	0.898546i	$0.355376\pi$
$648$	0	0
$649$	−10.6837	−0.419371
$650$	0	0
$651$	0	0
$652$	−11.2688	−0.441320
$653$	29.7988	1.16612	0.583058	−	0.812431i	$-0.301856\pi$
$653$	29.7988	1.16612	0.583058	+	0.812431i	$0.301856\pi$
$654$	0	0
$655$	0	0
$656$	−1.34883	−0.0526628
$657$	0	0
$658$	27.9371	1.08910
$659$	−50.2976	−1.95932	−0.979659	−	0.200671i	$-0.935688\pi$
$659$	−50.2976	−1.95932	−0.979659	+	0.200671i	$0.935688\pi$
$660$	0	0
$661$	−20.8701	−0.811753	−0.405877	−	0.913928i	$-0.633034\pi$
$661$	−20.8701	−0.811753	−0.405877	+	0.913928i	$0.633034\pi$
$662$	−10.3755	−0.403255
$663$	0	0
$664$	−3.94763	−0.153198
$665$	0	0
$666$	0	0
$667$	−53.2091	−2.06026
$668$	19.9606	0.772300
$669$	0	0
$670$	0	0
$671$	15.1317	0.584152
$672$	0	0
$673$	−4.50092	−0.173498	−0.0867488	−	0.996230i	$-0.527648\pi$
$673$	−4.50092	−0.173498	−0.0867488	+	0.996230i	$0.527648\pi$
$674$	−13.0554	−0.502874
$675$	0	0
$676$	2.14197	0.0823834
$677$	13.5082	0.519161	0.259580	−	0.965721i	$-0.416416\pi$
$677$	13.5082	0.519161	0.259580	+	0.965721i	$0.416416\pi$
$678$	0	0
$679$	−5.87950	−0.225634
$680$	0	0
$681$	0	0
$682$	7.61374	0.291545
$683$	6.98466	0.267261	0.133630	−	0.991031i	$-0.457337\pi$
$683$	6.98466	0.267261	0.133630	+	0.991031i	$0.457337\pi$
$684$	0	0
$685$	0	0
$686$	−3.52566	−0.134610
$687$	0	0
$688$	−0.402722	−0.0153536
$689$	−38.4723	−1.46568
$690$	0	0
$691$	−25.6891	−0.977261	−0.488630	−	0.872491i	$-0.662503\pi$
$691$	−25.6891	−0.977261	−0.488630	+	0.872491i	$0.662503\pi$
$692$	20.6102	0.783482
$693$	0	0
$694$	−7.31200	−0.277560
$695$	0	0
$696$	0	0
$697$	35.4537	1.34291
$698$	5.77734	0.218676
$699$	0	0
$700$	0	0
$701$	−2.35830	−0.0890717	−0.0445358	−	0.999008i	$-0.514181\pi$
$701$	−2.35830	−0.0890717	−0.0445358	+	0.999008i	$0.514181\pi$
$702$	0	0
$703$	20.2933	0.765375
$704$	6.51997	0.245731
$705$	0	0
$706$	11.4280	0.430100
$707$	−66.8898	−2.51565
$708$	0	0
$709$	13.1277	0.493022	0.246511	−	0.969140i	$-0.420716\pi$
$709$	13.1277	0.493022	0.246511	+	0.969140i	$0.420716\pi$
$710$	0	0
$711$	0	0
$712$	−12.9718	−0.486138
$713$	−40.8306	−1.52912
$714$	0	0
$715$	0	0
$716$	−1.07568	−0.0402002
$717$	0	0
$718$	−23.3399	−0.871039
$719$	29.5306	1.10131	0.550653	−	0.834735i	$-0.314379\pi$
$719$	29.5306	1.10131	0.550653	+	0.834735i	$0.314379\pi$
$720$	0	0
$721$	−42.6480	−1.58829
$722$	10.0292	0.373248
$723$	0	0
$724$	−27.7282	−1.03051
$725$	0	0
$726$	0	0
$727$	−45.2103	−1.67676	−0.838378	−	0.545089i	$-0.816496\pi$
$727$	−45.2103	−1.67676	−0.838378	+	0.545089i	$0.816496\pi$
$728$	36.5535	1.35476
$729$	0	0
$730$	0	0
$731$	10.5855	0.391519
$732$	0	0
$733$	−20.6475	−0.762633	−0.381317	−	0.924444i	$-0.624529\pi$
$733$	−20.6475	−0.762633	−0.381317	+	0.924444i	$0.624529\pi$
$734$	−13.7360	−0.507005
$735$	0	0
$736$	−36.7428	−1.35436
$737$	4.95508	0.182523
$738$	0	0
$739$	26.6229	0.979339	0.489670	−	0.871908i	$-0.337117\pi$
$739$	26.6229	0.979339	0.489670	+	0.871908i	$0.337117\pi$
$740$	0	0
$741$	0	0
$742$	38.1358	1.40001
$743$	−34.2594	−1.25686	−0.628428	−	0.777868i	$-0.716301\pi$
$743$	−34.2594	−1.25686	−0.628428	+	0.777868i	$0.716301\pi$
$744$	0	0
$745$	0	0
$746$	26.8206	0.981972
$747$	0	0
$748$	5.50013	0.201105
$749$	−26.4142	−0.965154
$750$	0	0
$751$	34.9423	1.27506	0.637531	−	0.770425i	$-0.279956\pi$
$751$	34.9423	1.27506	0.637531	+	0.770425i	$0.279956\pi$
$752$	−0.993739	−0.0362380
$753$	0	0
$754$	23.8569	0.868817
$755$	0	0
$756$	0	0
$757$	25.4445	0.924796	0.462398	−	0.886673i	$-0.346989\pi$
$757$	25.4445	0.924796	0.462398	+	0.886673i	$0.346989\pi$
$758$	27.6561	1.00451
$759$	0	0
$760$	0	0
$761$	14.7352	0.534149	0.267075	−	0.963676i	$-0.413943\pi$
$761$	14.7352	0.534149	0.267075	+	0.963676i	$0.413943\pi$
$762$	0	0
$763$	−44.4122	−1.60783
$764$	−1.45694	−0.0527101
$765$	0	0
$766$	−16.6845	−0.602834
$767$	−25.6884	−0.927556
$768$	0	0
$769$	28.9603	1.04434	0.522168	−	0.852843i	$-0.325123\pi$
$769$	28.9603	1.04434	0.522168	+	0.852843i	$0.325123\pi$
$770$	0	0
$771$	0	0
$772$	1.24149	0.0446821
$773$	−19.7129	−0.709022	−0.354511	−	0.935052i	$-0.615353\pi$
$773$	−19.7129	−0.709022	−0.354511	+	0.935052i	$0.615353\pi$
$774$	0	0
$775$	0	0
$776$	−4.24524	−0.152395
$777$	0	0
$778$	6.76242	0.242445
$779$	30.7892	1.10314
$780$	0	0
$781$	−1.65492	−0.0592178
$782$	17.2332	0.616259
$783$	0	0
$784$	0.955214	0.0341148
$785$	0	0
$786$	0	0
$787$	−18.0212	−0.642385	−0.321193	−	0.947014i	$-0.604084\pi$
$787$	−18.0212	−0.642385	−0.321193	+	0.947014i	$0.604084\pi$
$788$	−9.35107	−0.333118
$789$	0	0
$790$	0	0
$791$	63.1295	2.24463
$792$	0	0
$793$	36.3835	1.29201
$794$	−24.4595	−0.868033
$795$	0	0
$796$	17.1546	0.608028
$797$	−17.7479	−0.628663	−0.314332	−	0.949313i	$-0.601780\pi$
$797$	−17.7479	−0.628663	−0.314332	+	0.949313i	$0.601780\pi$
$798$	0	0
$799$	26.1203	0.924071
$800$	0	0
$801$	0	0
$802$	19.9374	0.704012
$803$	3.34827	0.118158
$804$	0	0
$805$	0	0
$806$	18.3069	0.644833
$807$	0	0
$808$	−48.2972	−1.69909
$809$	−24.7020	−0.868477	−0.434238	−	0.900798i	$-0.642982\pi$
$809$	−24.7020	−0.868477	−0.434238	+	0.900798i	$0.642982\pi$
$810$	0	0
$811$	−13.6855	−0.480561	−0.240281	−	0.970703i	$-0.577239\pi$
$811$	−13.6855	−0.480561	−0.240281	+	0.970703i	$0.577239\pi$
$812$	40.4756	1.42042
$813$	0	0
$814$	9.00579	0.315653
$815$	0	0
$816$	0	0
$817$	9.19279	0.321615
$818$	−3.71199	−0.129787
$819$	0	0
$820$	0	0
$821$	56.1647	1.96016	0.980081	−	0.198600i	$-0.0636396\pi$
$821$	56.1647	1.96016	0.980081	+	0.198600i	$0.0636396\pi$
$822$	0	0
$823$	−15.1705	−0.528810	−0.264405	−	0.964412i	$-0.585175\pi$
$823$	−15.1705	−0.528810	−0.264405	+	0.964412i	$0.585175\pi$
$824$	−30.7936	−1.07275
$825$	0	0
$826$	25.4638	0.885998
$827$	12.2655	0.426515	0.213257	−	0.976996i	$-0.431593\pi$
$827$	12.2655	0.426515	0.213257	+	0.976996i	$0.431593\pi$
$828$	0	0
$829$	29.4191	1.02177	0.510884	−	0.859650i	$-0.329318\pi$
$829$	29.4191	1.02177	0.510884	+	0.859650i	$0.329318\pi$
$830$	0	0
$831$	0	0
$832$	15.6770	0.543501
$833$	−25.1077	−0.869930
$834$	0	0
$835$	0	0
$836$	4.77650	0.165199
$837$	0	0
$838$	−1.68696	−0.0582749
$839$	7.01431	0.242161	0.121080	−	0.992643i	$-0.461364\pi$
$839$	7.01431	0.242161	0.121080	+	0.992643i	$0.461364\pi$
$840$	0	0
$841$	39.2676	1.35405
$842$	−1.82950	−0.0630486
$843$	0	0
$844$	−33.1994	−1.14277
$845$	0	0
$846$	0	0
$847$	−35.0983	−1.20599
$848$	−1.35651	−0.0465827
$849$	0	0
$850$	0	0
$851$	−48.2959	−1.65556
$852$	0	0
$853$	−27.1094	−0.928208	−0.464104	−	0.885781i	$-0.653624\pi$
$853$	−27.1094	−0.928208	−0.464104	+	0.885781i	$0.653624\pi$
$854$	−36.0652	−1.23413
$855$	0	0
$856$	−19.0721	−0.651872
$857$	5.19405	0.177425	0.0887127	−	0.996057i	$-0.471725\pi$
$857$	5.19405	0.177425	0.0887127	+	0.996057i	$0.471725\pi$
$858$	0	0
$859$	−15.2545	−0.520478	−0.260239	−	0.965544i	$-0.583801\pi$
$859$	−15.2545	−0.520478	−0.260239	+	0.965544i	$0.583801\pi$
$860$	0	0
$861$	0	0
$862$	0.0568738	0.00193713
$863$	46.8290	1.59408	0.797038	−	0.603929i	$-0.206399\pi$
$863$	46.8290	1.59408	0.797038	+	0.603929i	$0.206399\pi$
$864$	0	0
$865$	0	0
$866$	1.05502	0.0358510
$867$	0	0
$868$	31.0595	1.05423
$869$	14.8557	0.503944
$870$	0	0
$871$	11.9143	0.403700
$872$	−32.0674	−1.08594
$873$	0	0
$874$	14.9659	0.506229
$875$	0	0
$876$	0	0
$877$	−0.979616	−0.0330793	−0.0165396	−	0.999863i	$-0.505265\pi$
$877$	−0.979616	−0.0330793	−0.0165396	+	0.999863i	$0.505265\pi$
$878$	32.4912	1.09653
$879$	0	0
$880$	0	0
$881$	−39.0186	−1.31457	−0.657286	−	0.753641i	$-0.728296\pi$
$881$	−39.0186	−1.31457	−0.657286	+	0.753641i	$0.728296\pi$
$882$	0	0
$883$	8.90989	0.299842	0.149921	−	0.988698i	$-0.452098\pi$
$883$	8.90989	0.299842	0.149921	+	0.988698i	$0.452098\pi$
$884$	13.2248	0.444799
$885$	0	0
$886$	23.1171	0.776633
$887$	51.4215	1.72656	0.863282	−	0.504721i	$-0.168405\pi$
$887$	51.4215	1.72656	0.863282	+	0.504721i	$0.168405\pi$
$888$	0	0
$889$	−63.0485	−2.11458
$890$	0	0
$891$	0	0
$892$	17.1020	0.572618
$893$	22.6838	0.759083
$894$	0	0
$895$	0	0
$896$	28.7400	0.960137
$897$	0	0
$898$	−21.7360	−0.725339
$899$	52.3860	1.74717
$900$	0	0
$901$	35.6557	1.18786
$902$	13.6637	0.454952
$903$	0	0
$904$	45.5821	1.51604
$905$	0	0
$906$	0	0
$907$	18.0057	0.597871	0.298935	−	0.954273i	$-0.403369\pi$
$907$	18.0057	0.597871	0.298935	+	0.954273i	$0.403369\pi$
$908$	0.335089	0.0111203
$909$	0	0
$910$	0	0
$911$	−42.3831	−1.40421	−0.702107	−	0.712071i	$-0.747757\pi$
$911$	−42.3831	−1.40421	−0.702107	+	0.712071i	$0.747757\pi$
$912$	0	0
$913$	−1.97005	−0.0651991
$914$	32.7164	1.08216
$915$	0	0
$916$	−5.40460	−0.178573
$917$	59.4171	1.96213
$918$	0	0
$919$	15.9252	0.525325	0.262662	−	0.964888i	$-0.415399\pi$
$919$	15.9252	0.525325	0.262662	+	0.964888i	$0.415399\pi$
$920$	0	0
$921$	0	0
$922$	3.03700	0.100018
$923$	−3.97919	−0.130977
$924$	0	0
$925$	0	0
$926$	−20.3718	−0.669460
$927$	0	0
$928$	47.1413	1.54749
$929$	−7.47386	−0.245210	−0.122605	−	0.992456i	$-0.539125\pi$
$929$	−7.47386	−0.245210	−0.122605	+	0.992456i	$0.539125\pi$
$930$	0	0
$931$	−21.8043	−0.714609
$932$	−8.51780	−0.279010
$933$	0	0
$934$	16.4023	0.536699
$935$	0	0
$936$	0	0
$937$	−26.3843	−0.861937	−0.430969	−	0.902367i	$-0.641828\pi$
$937$	−26.3843	−0.861937	−0.430969	+	0.902367i	$0.641828\pi$
$938$	−11.8101	−0.385612
$939$	0	0
$940$	0	0
$941$	−40.8626	−1.33208	−0.666041	−	0.745915i	$-0.732012\pi$
$941$	−40.8626	−1.33208	−0.666041	+	0.745915i	$0.732012\pi$
$942$	0	0
$943$	−73.2751	−2.38617
$944$	−0.905760	−0.0294800
$945$	0	0
$946$	4.07960	0.132639
$947$	44.3060	1.43975	0.719876	−	0.694103i	$-0.244199\pi$
$947$	44.3060	1.43975	0.719876	+	0.694103i	$0.244199\pi$
$948$	0	0
$949$	8.05076	0.261339
$950$	0	0
$951$	0	0
$952$	−33.8775	−1.09797
$953$	21.0768	0.682744	0.341372	−	0.939928i	$-0.389108\pi$
$953$	21.0768	0.682744	0.341372	+	0.939928i	$0.389108\pi$
$954$	0	0
$955$	0	0
$956$	−0.387954	−0.0125473
$957$	0	0
$958$	15.7194	0.507870
$959$	−26.6361	−0.860124
$960$	0	0
$961$	9.19905	0.296744
$962$	21.6540	0.698154
$963$	0	0
$964$	14.0422	0.452270
$965$	0	0
$966$	0	0
$967$	9.34590	0.300544	0.150272	−	0.988645i	$-0.451985\pi$
$967$	9.34590	0.300544	0.150272	+	0.988645i	$0.451985\pi$
$968$	−25.3424	−0.814536
$969$	0	0
$970$	0	0
$971$	49.2708	1.58118	0.790588	−	0.612348i	$-0.209775\pi$
$971$	49.2708	1.58118	0.790588	+	0.612348i	$0.209775\pi$
$972$	0	0
$973$	53.6374	1.71954
$974$	6.02584	0.193080
$975$	0	0
$976$	1.28286	0.0410634
$977$	54.8767	1.75566	0.877830	−	0.478971i	$-0.158990\pi$
$977$	54.8767	1.75566	0.877830	+	0.478971i	$0.158990\pi$
$978$	0	0
$979$	−6.47352	−0.206895
$980$	0	0
$981$	0	0
$982$	−3.97701	−0.126912
$983$	15.9788	0.509646	0.254823	−	0.966988i	$-0.417983\pi$
$983$	15.9788	0.509646	0.254823	+	0.966988i	$0.417983\pi$
$984$	0	0
$985$	0	0
$986$	−22.1104	−0.704137
$987$	0	0
$988$	11.4849	0.365382
$989$	−21.8779	−0.695677
$990$	0	0
$991$	−29.2757	−0.929973	−0.464986	−	0.885318i	$-0.653941\pi$
$991$	−29.2757	−0.929973	−0.464986	+	0.885318i	$0.653941\pi$
$992$	36.1744	1.14854
$993$	0	0
$994$	3.94438	0.125108
$995$	0	0
$996$	0	0
$997$	−19.6838	−0.623392	−0.311696	−	0.950182i	$-0.600897\pi$
$997$	−19.6838	−0.623392	−0.311696	+	0.950182i	$0.600897\pi$
$998$	4.67683	0.148042
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.o.1.3		6
3.2	odd	2		1875.2.a.l.1.4	yes	6
5.4	even	2		5625.2.a.r.1.4		6
15.2	even	4		1875.2.b.e.1249.8		12
15.8	even	4		1875.2.b.e.1249.5		12
15.14	odd	2		1875.2.a.i.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.3	✓	6	15.14	odd	2
1875.2.a.l.1.4	yes	6	3.2	odd	2
1875.2.b.e.1249.5		12	15.8	even	4
1875.2.b.e.1249.8		12	15.2	even	4
5625.2.a.o.1.3		6	1.1	even	1	trivial
5625.2.a.r.1.4		6	5.4	even	2

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)

\(f(q)\)	\(=\)	\(q-0.858825 q^{2} -1.26242 q^{4} +3.88045 q^{7} +2.80185 q^{8} +O(q^{10})\)	\(q-0.858825 q^{2} -1.26242 q^{4} +3.88045 q^{7} +2.80185 q^{8} +1.39825 q^{11} +3.36204 q^{13} -3.33263 q^{14} +0.118543 q^{16} -3.11590 q^{17} -2.70595 q^{19} -1.20085 q^{22} +6.43989 q^{23} -2.88740 q^{26} -4.89876 q^{28} -8.26242 q^{29} -6.34027 q^{31} -5.70550 q^{32} +2.67601 q^{34} -7.49949 q^{37} +2.32394 q^{38} -11.3783 q^{41} -3.39725 q^{43} -1.76518 q^{44} -5.53074 q^{46} -8.38291 q^{47} +8.05792 q^{49} -4.24430 q^{52} -11.4431 q^{53} +10.8724 q^{56} +7.09597 q^{58} -7.64074 q^{59} +10.8219 q^{61} +5.44518 q^{62} +4.66294 q^{64} +3.54377 q^{67} +3.93358 q^{68} -1.18356 q^{71} +2.39461 q^{73} +6.44075 q^{74} +3.41605 q^{76} +5.42585 q^{77} +10.6245 q^{79} +9.77199 q^{82} -1.40894 q^{83} +2.91764 q^{86} +3.91769 q^{88} -4.62972 q^{89} +13.0462 q^{91} -8.12984 q^{92} +7.19945 q^{94} -1.51516 q^{97} -6.92034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8	\( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38} - 33 q^{41} - 3 q^{43} + 11 q^{44} - 12 q^{46} + 6 q^{47} + 4 q^{49} - 33 q^{52} - 14 q^{53} + 30 q^{56} - q^{58} + 8 q^{59} + 34 q^{61} - 31 q^{62} + 12 q^{64} + 2 q^{67} - 27 q^{68} + 3 q^{71} - 36 q^{73} - 36 q^{74} + 27 q^{76} - 16 q^{77} + 25 q^{79} + 36 q^{82} + 12 q^{83} + 30 q^{86} + 56 q^{88} - 18 q^{89} + 28 q^{91} - 3 q^{92} - 50 q^{94} + 7 q^{97} - 15 q^{98}+O(q^{100}) \) 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8 + 4 * q^14 + 17 * q^16 - 2 * q^17 - 2 * q^19 + 9 * q^22 + q^23 - 37 * q^26 - 44 * q^28 - 31 * q^29 - 2 * q^31 - 33 * q^32 + 37 * q^34 - 22 * q^37 - 27 * q^38 - 33 * q^41 - 3 * q^43 + 11 * q^44 - 12 * q^46 + 6 * q^47 + 4 * q^49 - 33 * q^52 - 14 * q^53 + 30 * q^56 - q^58 + 8 * q^59 + 34 * q^61 - 31 * q^62 + 12 * q^64 + 2 * q^67 - 27 * q^68 + 3 * q^71 - 36 * q^73 - 36 * q^74 + 27 * q^76 - 16 * q^77 + 25 * q^79 + 36 * q^82 + 12 * q^83 + 30 * q^86 + 56 * q^88 - 18 * q^89 + 28 * q^91 - 3 * q^92 - 50 * q^94 + 7 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

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