LMFDB - Embedded newform 5625.2.a.be.1.6

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:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.32610$ 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)

$f(q)$	$=$	$q+2.32610 q^{2} +3.41075 q^{4} -3.59425 q^{7} +3.28154 q^{8} +O(q^{10})$	$q+2.32610 q^{2} +3.41075 q^{4} -3.59425 q^{7} +3.28154 q^{8} +0.497788 q^{11} -2.64789 q^{13} -8.36058 q^{14} +0.811708 q^{16} +5.10719 q^{17} -0.987277 q^{19} +1.15790 q^{22} +6.41382 q^{23} -6.15926 q^{26} -12.2591 q^{28} +5.57001 q^{29} +6.05507 q^{31} -4.67497 q^{32} +11.8798 q^{34} +4.59612 q^{37} -2.29651 q^{38} +2.87475 q^{41} +9.48858 q^{43} +1.69783 q^{44} +14.9192 q^{46} +5.36834 q^{47} +5.91861 q^{49} -9.03129 q^{52} -0.307600 q^{53} -11.7947 q^{56} +12.9564 q^{58} +1.26645 q^{59} -6.22625 q^{61} +14.0847 q^{62} -12.4979 q^{64} +5.28626 q^{67} +17.4193 q^{68} +0.151963 q^{71} +14.8741 q^{73} +10.6910 q^{74} -3.36735 q^{76} -1.78917 q^{77} -16.5886 q^{79} +6.68696 q^{82} +14.5960 q^{83} +22.0714 q^{86} +1.63351 q^{88} -11.3822 q^{89} +9.51717 q^{91} +21.8759 q^{92} +12.4873 q^{94} -0.849192 q^{97} +13.7673 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8	$ 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8} - q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} + 15 q^{17} - 10 q^{19} + 5 q^{22} + 30 q^{23} - 11 q^{26} + 5 q^{28} - 10 q^{29} - 9 q^{31} + 30 q^{32} + 7 q^{34} + 10 q^{37} + 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} + 30 q^{47} - 4 q^{49} - 5 q^{52} + 10 q^{53} + 30 q^{58} + 5 q^{59} + 6 q^{61} + 10 q^{62} - 9 q^{64} - 10 q^{67} + 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} + 5 q^{77} - 20 q^{79} + 45 q^{82} + 40 q^{83} + 24 q^{86} + 40 q^{88} + 5 q^{89} + 6 q^{91} + 15 q^{92} + 47 q^{94} - 30 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8 - q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 + 15 * q^17 - 10 * q^19 + 5 * q^22 + 30 * q^23 - 11 * q^26 + 5 * q^28 - 10 * q^29 - 9 * q^31 + 30 * q^32 + 7 * q^34 + 10 * q^37 + 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 + 30 * q^47 - 4 * q^49 - 5 * q^52 + 10 * q^53 + 30 * q^58 + 5 * q^59 + 6 * q^61 + 10 * q^62 - 9 * q^64 - 10 * q^67 + 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 + 5 * q^77 - 20 * q^79 + 45 * q^82 + 40 * q^83 + 24 * q^86 + 40 * q^88 + 5 * q^89 + 6 * q^91 + 15 * q^92 + 47 * q^94 - 30 * 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 $

$2$	2.32610	1.64480	0.822401	−	0.568908i	$-0.192634\pi$
$2$	2.32610	1.64480	0.822401	+	0.568908i	$0.192634\pi$
$3$	0	0
$3$	0	0
$4$	3.41075	1.70537
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.59425	−1.35850	−0.679249	−	0.733908i	$-0.737694\pi$
$7$	−3.59425	−1.35850	−0.679249	+	0.733908i	$0.737694\pi$
$8$	3.28154	1.16020
$9$	0	0
$10$	0	0
$11$	0.497788	0.150089	0.0750443	−	0.997180i	$-0.476090\pi$
$11$	0.497788	0.150089	0.0750443	+	0.997180i	$0.476090\pi$
$12$	0	0
$13$	−2.64789	−0.734392	−0.367196	−	0.930143i	$-0.619682\pi$
$13$	−2.64789	−0.734392	−0.367196	+	0.930143i	$0.619682\pi$
$14$	−8.36058	−2.23446
$15$	0	0
$16$	0.811708	0.202927
$17$	5.10719	1.23867	0.619337	−	0.785125i	$-0.287401\pi$
$17$	5.10719	1.23867	0.619337	+	0.785125i	$0.287401\pi$
$18$	0	0
$19$	−0.987277	−0.226497	−0.113248	−	0.993567i	$-0.536126\pi$
$19$	−0.987277	−0.226497	−0.113248	+	0.993567i	$0.536126\pi$
$20$	0	0
$21$	0	0
$22$	1.15790	0.246866
$23$	6.41382	1.33737	0.668687	−	0.743544i	$-0.266857\pi$
$23$	6.41382	1.33737	0.668687	+	0.743544i	$0.266857\pi$
$24$	0	0
$25$	0	0
$26$	−6.15926	−1.20793
$27$	0	0
$28$	−12.2591	−2.31675
$29$	5.57001	1.03432	0.517162	−	0.855887i	$-0.326988\pi$
$29$	5.57001	1.03432	0.517162	+	0.855887i	$0.326988\pi$
$30$	0	0
$31$	6.05507	1.08752	0.543762	−	0.839240i	$-0.317000\pi$
$31$	6.05507	1.08752	0.543762	+	0.839240i	$0.317000\pi$
$32$	−4.67497	−0.826426
$33$	0	0
$34$	11.8798	2.03738
$35$	0	0
$36$	0	0
$37$	4.59612	0.755598	0.377799	−	0.925888i	$-0.376681\pi$
$37$	4.59612	0.755598	0.377799	+	0.925888i	$0.376681\pi$
$38$	−2.29651	−0.372543
$39$	0	0
$40$	0	0
$41$	2.87475	0.448960	0.224480	−	0.974479i	$-0.427932\pi$
$41$	2.87475	0.448960	0.224480	+	0.974479i	$0.427932\pi$
$42$	0	0
$43$	9.48858	1.44700	0.723498	−	0.690327i	$-0.242533\pi$
$43$	9.48858	1.44700	0.723498	+	0.690327i	$0.242533\pi$
$44$	1.69783	0.255957
$45$	0	0
$46$	14.9192	2.19972
$47$	5.36834	0.783053	0.391527	−	0.920167i	$-0.371947\pi$
$47$	5.36834	0.783053	0.391527	+	0.920167i	$0.371947\pi$
$48$	0	0
$49$	5.91861	0.845515
$50$	0	0
$51$	0	0
$52$	−9.03129	−1.25241
$53$	−0.307600	−0.0422521	−0.0211261	−	0.999777i	$-0.506725\pi$
$53$	−0.307600	−0.0422521	−0.0211261	+	0.999777i	$0.506725\pi$
$54$	0	0
$55$	0	0
$56$	−11.7947	−1.57613
$57$	0	0
$58$	12.9564	1.70126
$59$	1.26645	0.164878	0.0824389	−	0.996596i	$-0.473729\pi$
$59$	1.26645	0.164878	0.0824389	+	0.996596i	$0.473729\pi$
$60$	0	0
$61$	−6.22625	−0.797190	−0.398595	−	0.917127i	$-0.630502\pi$
$61$	−6.22625	−0.797190	−0.398595	+	0.917127i	$0.630502\pi$
$62$	14.0847	1.78876
$63$	0	0
$64$	−12.4979	−1.56223
$65$	0	0
$66$	0	0
$67$	5.28626	0.645819	0.322910	−	0.946430i	$-0.395339\pi$
$67$	5.28626	0.645819	0.322910	+	0.946430i	$0.395339\pi$
$68$	17.4193	2.11240
$69$	0	0
$70$	0	0
$71$	0.151963	0.0180347	0.00901734	−	0.999959i	$-0.497130\pi$
$71$	0.151963	0.0180347	0.00901734	+	0.999959i	$0.497130\pi$
$72$	0	0
$73$	14.8741	1.74088	0.870439	−	0.492276i	$-0.163835\pi$
$73$	14.8741	1.74088	0.870439	+	0.492276i	$0.163835\pi$
$74$	10.6910	1.24281
$75$	0	0
$76$	−3.36735	−0.386262
$77$	−1.78917	−0.203895
$78$	0	0
$79$	−16.5886	−1.86636	−0.933181	−	0.359406i	$-0.882979\pi$
$79$	−16.5886	−1.86636	−0.933181	+	0.359406i	$0.882979\pi$
$80$	0	0
$81$	0	0
$82$	6.68696	0.738451
$83$	14.5960	1.60212	0.801058	−	0.598587i	$-0.204271\pi$
$83$	14.5960	1.60212	0.801058	+	0.598587i	$0.204271\pi$
$84$	0	0
$85$	0	0
$86$	22.0714	2.38002
$87$	0	0
$88$	1.63351	0.174133
$89$	−11.3822	−1.20652	−0.603258	−	0.797546i	$-0.706131\pi$
$89$	−11.3822	−1.20652	−0.603258	+	0.797546i	$0.706131\pi$
$90$	0	0
$91$	9.51717	0.997670
$92$	21.8759	2.28072
$93$	0	0
$94$	12.4873	1.28797
$95$	0	0
$96$	0	0
$97$	−0.849192	−0.0862223	−0.0431112	−	0.999070i	$-0.513727\pi$
$97$	−0.849192	−0.0862223	−0.0431112	+	0.999070i	$0.513727\pi$
$98$	13.7673	1.39070
$99$	0	0
$100$	0	0
$101$	13.2498	1.31841	0.659203	−	0.751965i	$-0.270894\pi$
$101$	13.2498	1.31841	0.659203	+	0.751965i	$0.270894\pi$
$102$	0	0
$103$	−0.830909	−0.0818719	−0.0409360	−	0.999162i	$-0.513034\pi$
$103$	−0.830909	−0.0818719	−0.0409360	+	0.999162i	$0.513034\pi$
$104$	−8.68917	−0.852043
$105$	0	0
$106$	−0.715509	−0.0694964
$107$	−0.0722844	−0.00698800	−0.00349400	−	0.999994i	$-0.501112\pi$
$107$	−0.0722844	−0.00698800	−0.00349400	+	0.999994i	$0.501112\pi$
$108$	0	0
$109$	5.59621	0.536020	0.268010	−	0.963416i	$-0.413634\pi$
$109$	5.59621	0.536020	0.268010	+	0.963416i	$0.413634\pi$
$110$	0	0
$111$	0	0
$112$	−2.91748	−0.275676
$113$	−2.60239	−0.244812	−0.122406	−	0.992480i	$-0.539061\pi$
$113$	−2.60239	−0.244812	−0.122406	+	0.992480i	$0.539061\pi$
$114$	0	0
$115$	0	0
$116$	18.9979	1.76391
$117$	0	0
$118$	2.94589	0.271191
$119$	−18.3565	−1.68274
$120$	0	0
$121$	−10.7522	−0.977473
$122$	−14.4829	−1.31122
$123$	0	0
$124$	20.6523	1.85463
$125$	0	0
$126$	0	0
$127$	−4.94483	−0.438783	−0.219391	−	0.975637i	$-0.570407\pi$
$127$	−4.94483	−0.438783	−0.219391	+	0.975637i	$0.570407\pi$
$128$	−19.7214	−1.74314
$129$	0	0
$130$	0	0
$131$	−2.70342	−0.236199	−0.118099	−	0.993002i	$-0.537680\pi$
$131$	−2.70342	−0.236199	−0.118099	+	0.993002i	$0.537680\pi$
$132$	0	0
$133$	3.54852	0.307695
$134$	12.2964	1.06224
$135$	0	0
$136$	16.7595	1.43711
$137$	−2.24838	−0.192092	−0.0960459	−	0.995377i	$-0.530620\pi$
$137$	−2.24838	−0.192092	−0.0960459	+	0.995377i	$0.530620\pi$
$138$	0	0
$139$	−10.8032	−0.916317	−0.458159	−	0.888870i	$-0.651491\pi$
$139$	−10.8032	−0.916317	−0.458159	+	0.888870i	$0.651491\pi$
$140$	0	0
$141$	0	0
$142$	0.353481	0.0296635
$143$	−1.31809	−0.110224
$144$	0	0
$145$	0	0
$146$	34.5986	2.86340
$147$	0	0
$148$	15.6762	1.28858
$149$	−12.1878	−0.998460	−0.499230	−	0.866469i	$-0.666384\pi$
$149$	−12.1878	−0.998460	−0.499230	+	0.866469i	$0.666384\pi$
$150$	0	0
$151$	17.0860	1.39044	0.695220	−	0.718797i	$-0.255307\pi$
$151$	17.0860	1.39044	0.695220	+	0.718797i	$0.255307\pi$
$152$	−3.23979	−0.262782
$153$	0	0
$154$	−4.16179	−0.335367
$155$	0	0
$156$	0	0
$157$	7.49835	0.598433	0.299217	−	0.954185i	$-0.403275\pi$
$157$	7.49835	0.598433	0.299217	+	0.954185i	$0.403275\pi$
$158$	−38.5868	−3.06980
$159$	0	0
$160$	0	0
$161$	−23.0529	−1.81682
$162$	0	0
$163$	−1.95259	−0.152939	−0.0764693	−	0.997072i	$-0.524365\pi$
$163$	−1.95259	−0.152939	−0.0764693	+	0.997072i	$0.524365\pi$
$164$	9.80504	0.765645
$165$	0	0
$166$	33.9517	2.63516
$167$	−0.356578	−0.0275928	−0.0137964	−	0.999905i	$-0.504392\pi$
$167$	−0.356578	−0.0275928	−0.0137964	+	0.999905i	$0.504392\pi$
$168$	0	0
$169$	−5.98868	−0.460668
$170$	0	0
$171$	0	0
$172$	32.3632	2.46767
$173$	−9.95032	−0.756509	−0.378254	−	0.925702i	$-0.623476\pi$
$173$	−9.95032	−0.756509	−0.378254	+	0.925702i	$0.623476\pi$
$174$	0	0
$175$	0	0
$176$	0.404058	0.0304570
$177$	0	0
$178$	−26.4763	−1.98448
$179$	−15.3824	−1.14973	−0.574867	−	0.818247i	$-0.694946\pi$
$179$	−15.3824	−1.14973	−0.574867	+	0.818247i	$0.694946\pi$
$180$	0	0
$181$	8.91917	0.662957	0.331478	−	0.943463i	$-0.392453\pi$
$181$	8.91917	0.662957	0.331478	+	0.943463i	$0.392453\pi$
$182$	22.1379	1.64097
$183$	0	0
$184$	21.0472	1.55162
$185$	0	0
$186$	0	0
$187$	2.54230	0.185911
$188$	18.3101	1.33540
$189$	0	0
$190$	0	0
$191$	11.3887	0.824056	0.412028	−	0.911171i	$-0.364821\pi$
$191$	11.3887	0.824056	0.412028	+	0.911171i	$0.364821\pi$
$192$	0	0
$193$	−17.3321	−1.24759	−0.623795	−	0.781588i	$-0.714410\pi$
$193$	−17.3321	−1.24759	−0.623795	+	0.781588i	$0.714410\pi$
$194$	−1.97531	−0.141819
$195$	0	0
$196$	20.1869	1.44192
$197$	25.9609	1.84964	0.924820	−	0.380405i	$-0.124215\pi$
$197$	25.9609	1.84964	0.924820	+	0.380405i	$0.124215\pi$
$198$	0	0
$199$	−8.38571	−0.594447	−0.297223	−	0.954808i	$-0.596061\pi$
$199$	−8.38571	−0.594447	−0.297223	+	0.954808i	$0.596061\pi$
$200$	0	0
$201$	0	0
$202$	30.8204	2.16852
$203$	−20.0200	−1.40513
$204$	0	0
$205$	0	0
$206$	−1.93278	−0.134663
$207$	0	0
$208$	−2.14931	−0.149028
$209$	−0.491454	−0.0339946
$210$	0	0
$211$	15.9135	1.09553	0.547765	−	0.836632i	$-0.315479\pi$
$211$	15.9135	1.09553	0.547765	+	0.836632i	$0.315479\pi$
$212$	−1.04915	−0.0720557
$213$	0	0
$214$	−0.168141	−0.0114939
$215$	0	0
$216$	0	0
$217$	−21.7634	−1.47740
$218$	13.0174	0.881647
$219$	0	0
$220$	0	0
$221$	−13.5233	−0.909674
$222$	0	0
$223$	0.379706	0.0254270	0.0127135	−	0.999919i	$-0.495953\pi$
$223$	0.379706	0.0254270	0.0127135	+	0.999919i	$0.495953\pi$
$224$	16.8030	1.12270
$225$	0	0
$226$	−6.05343	−0.402668
$227$	20.3009	1.34742	0.673710	−	0.738996i	$-0.264700\pi$
$227$	20.3009	1.34742	0.673710	+	0.738996i	$0.264700\pi$
$228$	0	0
$229$	−11.9107	−0.787078	−0.393539	−	0.919308i	$-0.628749\pi$
$229$	−11.9107	−0.787078	−0.393539	+	0.919308i	$0.628749\pi$
$230$	0	0
$231$	0	0
$232$	18.2782	1.20002
$233$	−19.0610	−1.24873	−0.624364	−	0.781133i	$-0.714642\pi$
$233$	−19.0610	−1.24873	−0.624364	+	0.781133i	$0.714642\pi$
$234$	0	0
$235$	0	0
$236$	4.31954	0.281178
$237$	0	0
$238$	−42.6991	−2.76777
$239$	−4.12493	−0.266819	−0.133410	−	0.991061i	$-0.542593\pi$
$239$	−4.12493	−0.266819	−0.133410	+	0.991061i	$0.542593\pi$
$240$	0	0
$241$	16.1858	1.04262	0.521310	−	0.853367i	$-0.325444\pi$
$241$	16.1858	1.04262	0.521310	+	0.853367i	$0.325444\pi$
$242$	−25.0107	−1.60775
$243$	0	0
$244$	−21.2362	−1.35951
$245$	0	0
$246$	0	0
$247$	2.61420	0.166338
$248$	19.8700	1.26175
$249$	0	0
$250$	0	0
$251$	−11.8718	−0.749344	−0.374672	−	0.927157i	$-0.622245\pi$
$251$	−11.8718	−0.749344	−0.374672	+	0.927157i	$0.622245\pi$
$252$	0	0
$253$	3.19272	0.200725
$254$	−11.5022	−0.721711
$255$	0	0
$256$	−20.8782	−1.30489
$257$	−18.9164	−1.17997	−0.589986	−	0.807414i	$-0.700867\pi$
$257$	−18.9164	−1.17997	−0.589986	+	0.807414i	$0.700867\pi$
$258$	0	0
$259$	−16.5196	−1.02648
$260$	0	0
$261$	0	0
$262$	−6.28843	−0.388501
$263$	−6.74703	−0.416040	−0.208020	−	0.978125i	$-0.566702\pi$
$263$	−6.74703	−0.416040	−0.208020	+	0.978125i	$0.566702\pi$
$264$	0	0
$265$	0	0
$266$	8.25421	0.506098
$267$	0	0
$268$	18.0301	1.10136
$269$	25.3329	1.54457	0.772286	−	0.635275i	$-0.219113\pi$
$269$	25.3329	1.54457	0.772286	+	0.635275i	$0.219113\pi$
$270$	0	0
$271$	9.40735	0.571456	0.285728	−	0.958311i	$-0.407765\pi$
$271$	9.40735	0.571456	0.285728	+	0.958311i	$0.407765\pi$
$272$	4.14555	0.251361
$273$	0	0
$274$	−5.22995	−0.315953
$275$	0	0
$276$	0	0
$277$	5.96736	0.358544	0.179272	−	0.983800i	$-0.442626\pi$
$277$	5.96736	0.358544	0.179272	+	0.983800i	$0.442626\pi$
$278$	−25.1294	−1.50716
$279$	0	0
$280$	0	0
$281$	−11.0723	−0.660516	−0.330258	−	0.943891i	$-0.607136\pi$
$281$	−11.0723	−0.660516	−0.330258	+	0.943891i	$0.607136\pi$
$282$	0	0
$283$	3.02406	0.179762	0.0898810	−	0.995953i	$-0.471351\pi$
$283$	3.02406	0.179762	0.0898810	+	0.995953i	$0.471351\pi$
$284$	0.518307	0.0307559
$285$	0	0
$286$	−3.06600	−0.181297
$287$	−10.3326	−0.609911
$288$	0	0
$289$	9.08337	0.534316
$290$	0	0
$291$	0	0
$292$	50.7317	2.96885
$293$	6.26426	0.365962	0.182981	−	0.983116i	$-0.441425\pi$
$293$	6.26426	0.365962	0.182981	+	0.983116i	$0.441425\pi$
$294$	0	0
$295$	0	0
$296$	15.0824	0.876646
$297$	0	0
$298$	−28.3500	−1.64227
$299$	−16.9831	−0.982158
$300$	0	0
$301$	−34.1043	−1.96574
$302$	39.7438	2.28700
$303$	0	0
$304$	−0.801381	−0.0459623
$305$	0	0
$306$	0	0
$307$	−25.8734	−1.47667	−0.738337	−	0.674432i	$-0.764388\pi$
$307$	−25.8734	−1.47667	−0.738337	+	0.674432i	$0.764388\pi$
$308$	−6.10241	−0.347717
$309$	0	0
$310$	0	0
$311$	9.00277	0.510500	0.255250	−	0.966875i	$-0.417842\pi$
$311$	9.00277	0.510500	0.255250	+	0.966875i	$0.417842\pi$
$312$	0	0
$313$	33.2274	1.87812	0.939062	−	0.343748i	$-0.111696\pi$
$313$	33.2274	1.87812	0.939062	+	0.343748i	$0.111696\pi$
$314$	17.4419	0.984304
$315$	0	0
$316$	−56.5795	−3.18285
$317$	−25.9693	−1.45858	−0.729290	−	0.684204i	$-0.760150\pi$
$317$	−25.9693	−1.45858	−0.729290	+	0.684204i	$0.760150\pi$
$318$	0	0
$319$	2.77268	0.155240
$320$	0	0
$321$	0	0
$322$	−53.6233	−2.98831
$323$	−5.04221	−0.280556
$324$	0	0
$325$	0	0
$326$	−4.54192	−0.251554
$327$	0	0
$328$	9.43361	0.520884
$329$	−19.2951	−1.06378
$330$	0	0
$331$	12.1284	0.666636	0.333318	−	0.942814i	$-0.391832\pi$
$331$	12.1284	0.666636	0.333318	+	0.942814i	$0.391832\pi$
$332$	49.7832	2.73221
$333$	0	0
$334$	−0.829436	−0.0453847
$335$	0	0
$336$	0	0
$337$	30.2519	1.64793	0.823963	−	0.566644i	$-0.191759\pi$
$337$	30.2519	1.64793	0.823963	+	0.566644i	$0.191759\pi$
$338$	−13.9303	−0.757707
$339$	0	0
$340$	0	0
$341$	3.01414	0.163225
$342$	0	0
$343$	3.88680	0.209867
$344$	31.1372	1.67881
$345$	0	0
$346$	−23.1455	−1.24431
$347$	−13.3283	−0.715502	−0.357751	−	0.933817i	$-0.616456\pi$
$347$	−13.3283	−0.715502	−0.357751	+	0.933817i	$0.616456\pi$
$348$	0	0
$349$	−27.4444	−1.46906	−0.734532	−	0.678574i	$-0.762598\pi$
$349$	−27.4444	−1.46906	−0.734532	+	0.678574i	$0.762598\pi$
$350$	0	0
$351$	0	0
$352$	−2.32714	−0.124037
$353$	15.5564	0.827983	0.413992	−	0.910281i	$-0.364134\pi$
$353$	15.5564	0.827983	0.413992	+	0.910281i	$0.364134\pi$
$354$	0	0
$355$	0	0
$356$	−38.8220	−2.05756
$357$	0	0
$358$	−35.7810	−1.89109
$359$	21.9829	1.16021	0.580107	−	0.814540i	$-0.303011\pi$
$359$	21.9829	1.16021	0.580107	+	0.814540i	$0.303011\pi$
$360$	0	0
$361$	−18.0253	−0.948699
$362$	20.7469	1.09043
$363$	0	0
$364$	32.4607	1.70140
$365$	0	0
$366$	0	0
$367$	−12.1103	−0.632155	−0.316078	−	0.948733i	$-0.602366\pi$
$367$	−12.1103	−0.632155	−0.316078	+	0.948733i	$0.602366\pi$
$368$	5.20615	0.271389
$369$	0	0
$370$	0	0
$371$	1.10559	0.0573994
$372$	0	0
$373$	−7.08604	−0.366901	−0.183451	−	0.983029i	$-0.558727\pi$
$373$	−7.08604	−0.366901	−0.183451	+	0.983029i	$0.558727\pi$
$374$	5.91364	0.305787
$375$	0	0
$376$	17.6164	0.908499
$377$	−14.7488	−0.759600
$378$	0	0
$379$	−21.5030	−1.10453	−0.552266	−	0.833668i	$-0.686237\pi$
$379$	−21.5030	−1.10453	−0.552266	+	0.833668i	$0.686237\pi$
$380$	0	0
$381$	0	0
$382$	26.4912	1.35541
$383$	24.8816	1.27139	0.635696	−	0.771940i	$-0.280713\pi$
$383$	24.8816	1.27139	0.635696	+	0.771940i	$0.280713\pi$
$384$	0	0
$385$	0	0
$386$	−40.3161	−2.05204
$387$	0	0
$388$	−2.89638	−0.147041
$389$	35.4142	1.79557	0.897785	−	0.440434i	$-0.145175\pi$
$389$	35.4142	1.79557	0.897785	+	0.440434i	$0.145175\pi$
$390$	0	0
$391$	32.7566	1.65657
$392$	19.4222	0.980967
$393$	0	0
$394$	60.3878	3.04229
$395$	0	0
$396$	0	0
$397$	−5.49705	−0.275889	−0.137944	−	0.990440i	$-0.544050\pi$
$397$	−5.49705	−0.275889	−0.137944	+	0.990440i	$0.544050\pi$
$398$	−19.5060	−0.977748
$399$	0	0
$400$	0	0
$401$	−24.8463	−1.24077	−0.620383	−	0.784299i	$-0.713023\pi$
$401$	−24.8463	−1.24077	−0.620383	+	0.784299i	$0.713023\pi$
$402$	0	0
$403$	−16.0332	−0.798669
$404$	45.1918	2.24838
$405$	0	0
$406$	−46.5685	−2.31116
$407$	2.28789	0.113407
$408$	0	0
$409$	3.93157	0.194404	0.0972019	−	0.995265i	$-0.469011\pi$
$409$	3.93157	0.194404	0.0972019	+	0.995265i	$0.469011\pi$
$410$	0	0
$411$	0	0
$412$	−2.83402	−0.139622
$413$	−4.55194	−0.223986
$414$	0	0
$415$	0	0
$416$	12.3788	0.606921
$417$	0	0
$418$	−1.14317	−0.0559144
$419$	4.76604	0.232836	0.116418	−	0.993200i	$-0.462859\pi$
$419$	4.76604	0.232836	0.116418	+	0.993200i	$0.462859\pi$
$420$	0	0
$421$	16.0581	0.782625	0.391312	−	0.920258i	$-0.372021\pi$
$421$	16.0581	0.782625	0.391312	+	0.920258i	$0.372021\pi$
$422$	37.0164	1.80193
$423$	0	0
$424$	−1.00940	−0.0490209
$425$	0	0
$426$	0	0
$427$	22.3787	1.08298
$428$	−0.246544	−0.0119172
$429$	0	0
$430$	0	0
$431$	41.3693	1.99269	0.996344	−	0.0854360i	$-0.0272283\pi$
$431$	41.3693	1.99269	0.996344	+	0.0854360i	$0.0272283\pi$
$432$	0	0
$433$	−15.1854	−0.729763	−0.364881	−	0.931054i	$-0.618890\pi$
$433$	−15.1854	−0.729763	−0.364881	+	0.931054i	$0.618890\pi$
$434$	−50.6239	−2.43003
$435$	0	0
$436$	19.0873	0.914115
$437$	−6.33222	−0.302911
$438$	0	0
$439$	17.1194	0.817063	0.408532	−	0.912744i	$-0.366041\pi$
$439$	17.1194	0.817063	0.408532	+	0.912744i	$0.366041\pi$
$440$	0	0
$441$	0	0
$442$	−31.4565	−1.49623
$443$	35.3909	1.68147	0.840736	−	0.541445i	$-0.182123\pi$
$443$	35.3909	1.68147	0.840736	+	0.541445i	$0.182123\pi$
$444$	0	0
$445$	0	0
$446$	0.883234	0.0418223
$447$	0	0
$448$	44.9204	2.12229
$449$	−20.6830	−0.976090	−0.488045	−	0.872818i	$-0.662290\pi$
$449$	−20.6830	−0.976090	−0.488045	+	0.872818i	$0.662290\pi$
$450$	0	0
$451$	1.43101	0.0673838
$452$	−8.87610	−0.417497
$453$	0	0
$454$	47.2220	2.21624
$455$	0	0
$456$	0	0
$457$	−41.7664	−1.95375	−0.976876	−	0.213807i	$-0.931414\pi$
$457$	−41.7664	−1.95375	−0.976876	+	0.213807i	$0.931414\pi$
$458$	−27.7054	−1.29459
$459$	0	0
$460$	0	0
$461$	10.6783	0.497340	0.248670	−	0.968588i	$-0.420007\pi$
$461$	10.6783	0.497340	0.248670	+	0.968588i	$0.420007\pi$
$462$	0	0
$463$	7.83033	0.363906	0.181953	−	0.983307i	$-0.441758\pi$
$463$	7.83033	0.363906	0.181953	+	0.983307i	$0.441758\pi$
$464$	4.52122	0.209892
$465$	0	0
$466$	−44.3379	−2.05391
$467$	4.98100	0.230493	0.115246	−	0.993337i	$-0.463234\pi$
$467$	4.98100	0.230493	0.115246	+	0.993337i	$0.463234\pi$
$468$	0	0
$469$	−19.0001	−0.877344
$470$	0	0
$471$	0	0
$472$	4.15591	0.191291
$473$	4.72330	0.217178
$474$	0	0
$475$	0	0
$476$	−62.6094	−2.86970
$477$	0	0
$478$	−9.59500	−0.438865
$479$	16.3981	0.749247	0.374623	−	0.927177i	$-0.377772\pi$
$479$	16.3981	0.749247	0.374623	+	0.927177i	$0.377772\pi$
$480$	0	0
$481$	−12.1700	−0.554906
$482$	37.6498	1.71490
$483$	0	0
$484$	−36.6731	−1.66696
$485$	0	0
$486$	0	0
$487$	−0.876508	−0.0397184	−0.0198592	−	0.999803i	$-0.506322\pi$
$487$	−0.876508	−0.0397184	−0.0198592	+	0.999803i	$0.506322\pi$
$488$	−20.4317	−0.924901
$489$	0	0
$490$	0	0
$491$	−12.2118	−0.551112	−0.275556	−	0.961285i	$-0.588862\pi$
$491$	−12.2118	−0.551112	−0.275556	+	0.961285i	$0.588862\pi$
$492$	0	0
$493$	28.4471	1.28119
$494$	6.08090	0.273592
$495$	0	0
$496$	4.91495	0.220688
$497$	−0.546192	−0.0245001
$498$	0	0
$499$	−3.34603	−0.149789	−0.0748945	−	0.997191i	$-0.523862\pi$
$499$	−3.34603	−0.149789	−0.0748945	+	0.997191i	$0.523862\pi$
$500$	0	0
$501$	0	0
$502$	−27.6151	−1.23252
$503$	11.8820	0.529792	0.264896	−	0.964277i	$-0.414662\pi$
$503$	11.8820	0.529792	0.264896	+	0.964277i	$0.414662\pi$
$504$	0	0
$505$	0	0
$506$	7.42660	0.330152
$507$	0	0
$508$	−16.8656	−0.748289
$509$	−36.3183	−1.60978	−0.804890	−	0.593424i	$-0.797776\pi$
$509$	−36.3183	−1.60978	−0.804890	+	0.593424i	$0.797776\pi$
$510$	0	0
$511$	−53.4611	−2.36498
$512$	−9.12203	−0.403140
$513$	0	0
$514$	−44.0014	−1.94082
$515$	0	0
$516$	0	0
$517$	2.67229	0.117527
$518$	−38.4263	−1.68835
$519$	0	0
$520$	0	0
$521$	−0.204224	−0.00894722	−0.00447361	−	0.999990i	$-0.501424\pi$
$521$	−0.204224	−0.00894722	−0.00447361	+	0.999990i	$0.501424\pi$
$522$	0	0
$523$	−28.0312	−1.22572	−0.612859	−	0.790192i	$-0.709981\pi$
$523$	−28.0312	−1.22572	−0.612859	+	0.790192i	$0.709981\pi$
$524$	−9.22069	−0.402808
$525$	0	0
$526$	−15.6943	−0.684303
$527$	30.9244	1.34709
$528$	0	0
$529$	18.1371	0.788570
$530$	0	0
$531$	0	0
$532$	12.1031	0.524736
$533$	−7.61202	−0.329713
$534$	0	0
$535$	0	0
$536$	17.3471	0.749280
$537$	0	0
$538$	58.9268	2.54052
$539$	2.94621	0.126902
$540$	0	0
$541$	−9.32216	−0.400791	−0.200396	−	0.979715i	$-0.564223\pi$
$541$	−9.32216	−0.400791	−0.200396	+	0.979715i	$0.564223\pi$
$542$	21.8824	0.939931
$543$	0	0
$544$	−23.8760	−1.02367
$545$	0	0
$546$	0	0
$547$	20.0378	0.856755	0.428378	−	0.903600i	$-0.359085\pi$
$547$	20.0378	0.856755	0.428378	+	0.903600i	$0.359085\pi$
$548$	−7.66865	−0.327588
$549$	0	0
$550$	0	0
$551$	−5.49914	−0.234271
$552$	0	0
$553$	59.6235	2.53545
$554$	13.8807	0.589734
$555$	0	0
$556$	−36.8471	−1.56266
$557$	−28.9839	−1.22809	−0.614043	−	0.789272i	$-0.710458\pi$
$557$	−28.9839	−1.22809	−0.614043	+	0.789272i	$0.710458\pi$
$558$	0	0
$559$	−25.1247	−1.06266
$560$	0	0
$561$	0	0
$562$	−25.7552	−1.08642
$563$	13.7025	0.577492	0.288746	−	0.957406i	$-0.406762\pi$
$563$	13.7025	0.577492	0.288746	+	0.957406i	$0.406762\pi$
$564$	0	0
$565$	0	0
$566$	7.03428	0.295673
$567$	0	0
$568$	0.498673	0.0209239
$569$	−34.6152	−1.45115	−0.725573	−	0.688146i	$-0.758425\pi$
$569$	−34.6152	−1.45115	−0.725573	+	0.688146i	$0.758425\pi$
$570$	0	0
$571$	−42.4197	−1.77521	−0.887604	−	0.460607i	$-0.847632\pi$
$571$	−42.4197	−1.77521	−0.887604	+	0.460607i	$0.847632\pi$
$572$	−4.49566	−0.187973
$573$	0	0
$574$	−24.0346	−1.00318
$575$	0	0
$576$	0	0
$577$	34.3049	1.42813	0.714067	−	0.700078i	$-0.246851\pi$
$577$	34.3049	1.42813	0.714067	+	0.700078i	$0.246851\pi$
$578$	21.1288	0.878844
$579$	0	0
$580$	0	0
$581$	−52.4615	−2.17647
$582$	0	0
$583$	−0.153120	−0.00634156
$584$	48.8099	2.01977
$585$	0	0
$586$	14.5713	0.601935
$587$	29.7521	1.22800	0.614001	−	0.789305i	$-0.289559\pi$
$587$	29.7521	1.22800	0.614001	+	0.789305i	$0.289559\pi$
$588$	0	0
$589$	−5.97803	−0.246321
$590$	0	0
$591$	0	0
$592$	3.73071	0.153331
$593$	−14.8105	−0.608195	−0.304098	−	0.952641i	$-0.598355\pi$
$593$	−14.8105	−0.608195	−0.304098	+	0.952641i	$0.598355\pi$
$594$	0	0
$595$	0	0
$596$	−41.5694	−1.70275
$597$	0	0
$598$	−39.5044	−1.61545
$599$	−27.2394	−1.11297	−0.556486	−	0.830857i	$-0.687851\pi$
$599$	−27.2394	−1.11297	−0.556486	+	0.830857i	$0.687851\pi$
$600$	0	0
$601$	33.1682	1.35296	0.676480	−	0.736461i	$-0.263505\pi$
$601$	33.1682	1.35296	0.676480	+	0.736461i	$0.263505\pi$
$602$	−79.3301	−3.23325
$603$	0	0
$604$	58.2761	2.37122
$605$	0	0
$606$	0	0
$607$	−5.79849	−0.235353	−0.117677	−	0.993052i	$-0.537545\pi$
$607$	−5.79849	−0.235353	−0.117677	+	0.993052i	$0.537545\pi$
$608$	4.61549	0.187183
$609$	0	0
$610$	0	0
$611$	−14.2148	−0.575068
$612$	0	0
$613$	−4.04653	−0.163438	−0.0817189	−	0.996655i	$-0.526041\pi$
$613$	−4.04653	−0.163438	−0.0817189	+	0.996655i	$0.526041\pi$
$614$	−60.1842	−2.42884
$615$	0	0
$616$	−5.87125	−0.236559
$617$	37.1749	1.49661	0.748303	−	0.663357i	$-0.230869\pi$
$617$	37.1749	1.49661	0.748303	+	0.663357i	$0.230869\pi$
$618$	0	0
$619$	−37.7118	−1.51576	−0.757882	−	0.652391i	$-0.773766\pi$
$619$	−37.7118	−1.51576	−0.757882	+	0.652391i	$0.773766\pi$
$620$	0	0
$621$	0	0
$622$	20.9413	0.839672
$623$	40.9106	1.63905
$624$	0	0
$625$	0	0
$626$	77.2903	3.08914
$627$	0	0
$628$	25.5750	1.02055
$629$	23.4733	0.935940
$630$	0	0
$631$	18.5225	0.737368	0.368684	−	0.929555i	$-0.379808\pi$
$631$	18.5225	0.737368	0.368684	+	0.929555i	$0.379808\pi$
$632$	−54.4362	−2.16536
$633$	0	0
$634$	−60.4072	−2.39908
$635$	0	0
$636$	0	0
$637$	−15.6718	−0.620940
$638$	6.44954	0.255340
$639$	0	0
$640$	0	0
$641$	19.7116	0.778560	0.389280	−	0.921120i	$-0.372724\pi$
$641$	19.7116	0.778560	0.389280	+	0.921120i	$0.372724\pi$
$642$	0	0
$643$	−42.5897	−1.67957	−0.839787	−	0.542916i	$-0.817320\pi$
$643$	−42.5897	−1.67957	−0.839787	+	0.542916i	$0.817320\pi$
$644$	−78.6275	−3.09836
$645$	0	0
$646$	−11.7287	−0.461459
$647$	27.4018	1.07728	0.538638	−	0.842537i	$-0.318939\pi$
$647$	27.4018	1.07728	0.538638	+	0.842537i	$0.318939\pi$
$648$	0	0
$649$	0.630424	0.0247463
$650$	0	0
$651$	0	0
$652$	−6.65979	−0.260818
$653$	21.4414	0.839068	0.419534	−	0.907740i	$-0.362193\pi$
$653$	21.4414	0.839068	0.419534	+	0.907740i	$0.362193\pi$
$654$	0	0
$655$	0	0
$656$	2.33346	0.0911062
$657$	0	0
$658$	−44.8825	−1.74970
$659$	36.7342	1.43096	0.715480	−	0.698633i	$-0.246208\pi$
$659$	36.7342	1.43096	0.715480	+	0.698633i	$0.246208\pi$
$660$	0	0
$661$	−15.7515	−0.612664	−0.306332	−	0.951925i	$-0.599102\pi$
$661$	−15.7515	−0.612664	−0.306332	+	0.951925i	$0.599102\pi$
$662$	28.2118	1.09648
$663$	0	0
$664$	47.8973	1.85878
$665$	0	0
$666$	0	0
$667$	35.7250	1.38328
$668$	−1.21620	−0.0470561
$669$	0	0
$670$	0	0
$671$	−3.09935	−0.119649
$672$	0	0
$673$	−31.0107	−1.19537	−0.597687	−	0.801730i	$-0.703913\pi$
$673$	−31.0107	−1.19537	−0.597687	+	0.801730i	$0.703913\pi$
$674$	70.3690	2.71051
$675$	0	0
$676$	−20.4259	−0.785611
$677$	−9.59882	−0.368912	−0.184456	−	0.982841i	$-0.559052\pi$
$677$	−9.59882	−0.368912	−0.184456	+	0.982841i	$0.559052\pi$
$678$	0	0
$679$	3.05220	0.117133
$680$	0	0
$681$	0	0
$682$	7.01120	0.268473
$683$	−0.868350	−0.0332265	−0.0166132	−	0.999862i	$-0.505288\pi$
$683$	−0.868350	−0.0332265	−0.0166132	+	0.999862i	$0.505288\pi$
$684$	0	0
$685$	0	0
$686$	9.04109	0.345190
$687$	0	0
$688$	7.70196	0.293635
$689$	0.814491	0.0310296
$690$	0	0
$691$	−13.1375	−0.499772	−0.249886	−	0.968275i	$-0.580393\pi$
$691$	−13.1375	−0.499772	−0.249886	+	0.968275i	$0.580393\pi$
$692$	−33.9380	−1.29013
$693$	0	0
$694$	−31.0030	−1.17686
$695$	0	0
$696$	0	0
$697$	14.6819	0.556116
$698$	−63.8384	−2.41632
$699$	0	0
$700$	0	0
$701$	−7.13602	−0.269524	−0.134762	−	0.990878i	$-0.543027\pi$
$701$	−7.13602	−0.269524	−0.134762	+	0.990878i	$0.543027\pi$
$702$	0	0
$703$	−4.53765	−0.171141
$704$	−6.22129	−0.234474
$705$	0	0
$706$	36.1858	1.36187
$707$	−47.6231	−1.79105
$708$	0	0
$709$	−8.00513	−0.300639	−0.150319	−	0.988637i	$-0.548030\pi$
$709$	−8.00513	−0.300639	−0.150319	+	0.988637i	$0.548030\pi$
$710$	0	0
$711$	0	0
$712$	−37.3513	−1.39980
$713$	38.8362	1.45443
$714$	0	0
$715$	0	0
$716$	−52.4655	−1.96073
$717$	0	0
$718$	51.1345	1.90832
$719$	41.2567	1.53861	0.769307	−	0.638879i	$-0.220602\pi$
$719$	41.2567	1.53861	0.769307	+	0.638879i	$0.220602\pi$
$720$	0	0
$721$	2.98649	0.111223
$722$	−41.9286	−1.56042
$723$	0	0
$724$	30.4210	1.13059
$725$	0	0
$726$	0	0
$727$	−39.2602	−1.45608	−0.728040	−	0.685534i	$-0.759569\pi$
$727$	−39.2602	−1.45608	−0.728040	+	0.685534i	$0.759569\pi$
$728$	31.2310	1.15750
$729$	0	0
$730$	0	0
$731$	48.4600	1.79236
$732$	0	0
$733$	−0.912055	−0.0336875	−0.0168438	−	0.999858i	$-0.505362\pi$
$733$	−0.912055	−0.0336875	−0.0168438	+	0.999858i	$0.505362\pi$
$734$	−28.1699	−1.03977
$735$	0	0
$736$	−29.9844	−1.10524
$737$	2.63143	0.0969301
$738$	0	0
$739$	−11.6941	−0.430174	−0.215087	−	0.976595i	$-0.569003\pi$
$739$	−11.6941	−0.430174	−0.215087	+	0.976595i	$0.569003\pi$
$740$	0	0
$741$	0	0
$742$	2.57171	0.0944106
$743$	11.5631	0.424211	0.212105	−	0.977247i	$-0.431968\pi$
$743$	11.5631	0.424211	0.212105	+	0.977247i	$0.431968\pi$
$744$	0	0
$745$	0	0
$746$	−16.4828	−0.603480
$747$	0	0
$748$	8.67113	0.317048
$749$	0.259808	0.00949318
$750$	0	0
$751$	−27.2430	−0.994112	−0.497056	−	0.867718i	$-0.665586\pi$
$751$	−27.2430	−0.994112	−0.497056	+	0.867718i	$0.665586\pi$
$752$	4.35753	0.158903
$753$	0	0
$754$	−34.3071	−1.24939
$755$	0	0
$756$	0	0
$757$	−17.1080	−0.621800	−0.310900	−	0.950443i	$-0.600630\pi$
$757$	−17.1080	−0.621800	−0.310900	+	0.950443i	$0.600630\pi$
$758$	−50.0181	−1.81674
$759$	0	0
$760$	0	0
$761$	−37.7173	−1.36725	−0.683625	−	0.729833i	$-0.739598\pi$
$761$	−37.7173	−1.36725	−0.683625	+	0.729833i	$0.739598\pi$
$762$	0	0
$763$	−20.1142	−0.728182
$764$	38.8439	1.40532
$765$	0	0
$766$	57.8772	2.09119
$767$	−3.35342	−0.121085
$768$	0	0
$769$	−5.93347	−0.213966	−0.106983	−	0.994261i	$-0.534119\pi$
$769$	−5.93347	−0.213966	−0.106983	+	0.994261i	$0.534119\pi$
$770$	0	0
$771$	0	0
$772$	−59.1153	−2.12761
$773$	23.5908	0.848503	0.424252	−	0.905544i	$-0.360537\pi$
$773$	23.5908	0.848503	0.424252	+	0.905544i	$0.360537\pi$
$774$	0	0
$775$	0	0
$776$	−2.78666	−0.100035
$777$	0	0
$778$	82.3770	2.95336
$779$	−2.83817	−0.101688
$780$	0	0
$781$	0.0756453	0.00270680
$782$	76.1952	2.72473
$783$	0	0
$784$	4.80418	0.171578
$785$	0	0
$786$	0	0
$787$	18.2307	0.649854	0.324927	−	0.945739i	$-0.394660\pi$
$787$	18.2307	0.649854	0.324927	+	0.945739i	$0.394660\pi$
$788$	88.5462	3.15433
$789$	0	0
$790$	0	0
$791$	9.35364	0.332577
$792$	0	0
$793$	16.4864	0.585451
$794$	−12.7867	−0.453783
$795$	0	0
$796$	−28.6015	−1.01375
$797$	−10.8296	−0.383603	−0.191801	−	0.981434i	$-0.561433\pi$
$797$	−10.8296	−0.383603	−0.191801	+	0.981434i	$0.561433\pi$
$798$	0	0
$799$	27.4171	0.969948
$800$	0	0
$801$	0	0
$802$	−57.7950	−2.04081
$803$	7.40413	0.261286
$804$	0	0
$805$	0	0
$806$	−37.2948	−1.31365
$807$	0	0
$808$	43.4799	1.52962
$809$	−44.8177	−1.57571	−0.787854	−	0.615863i	$-0.788808\pi$
$809$	−44.8177	−1.57571	−0.787854	+	0.615863i	$0.788808\pi$
$810$	0	0
$811$	−3.06296	−0.107555	−0.0537775	−	0.998553i	$-0.517126\pi$
$811$	−3.06296	−0.107555	−0.0537775	+	0.998553i	$0.517126\pi$
$812$	−68.2831	−2.39627
$813$	0	0
$814$	5.32187	0.186532
$815$	0	0
$816$	0	0
$817$	−9.36786	−0.327740
$818$	9.14524	0.319756
$819$	0	0
$820$	0	0
$821$	−20.3210	−0.709208	−0.354604	−	0.935017i	$-0.615384\pi$
$821$	−20.3210	−0.709208	−0.354604	+	0.935017i	$0.615384\pi$
$822$	0	0
$823$	50.0369	1.74418	0.872089	−	0.489348i	$-0.162765\pi$
$823$	50.0369	1.74418	0.872089	+	0.489348i	$0.162765\pi$
$824$	−2.72666	−0.0949879
$825$	0	0
$826$	−10.5883	−0.368413
$827$	−31.6293	−1.09986	−0.549929	−	0.835211i	$-0.685345\pi$
$827$	−31.6293	−1.09986	−0.549929	+	0.835211i	$0.685345\pi$
$828$	0	0
$829$	33.1369	1.15089	0.575446	−	0.817840i	$-0.304829\pi$
$829$	33.1369	1.15089	0.575446	+	0.817840i	$0.304829\pi$
$830$	0	0
$831$	0	0
$832$	33.0930	1.14729
$833$	30.2274	1.04732
$834$	0	0
$835$	0	0
$836$	−1.67623	−0.0579735
$837$	0	0
$838$	11.0863	0.382970
$839$	−10.3854	−0.358543	−0.179271	−	0.983800i	$-0.557374\pi$
$839$	−10.3854	−0.358543	−0.179271	+	0.983800i	$0.557374\pi$
$840$	0	0
$841$	2.02500	0.0698275
$842$	37.3528	1.28726
$843$	0	0
$844$	54.2769	1.86829
$845$	0	0
$846$	0	0
$847$	38.6461	1.32790
$848$	−0.249681	−0.00857410
$849$	0	0
$850$	0	0
$851$	29.4787	1.01052
$852$	0	0
$853$	9.31456	0.318924	0.159462	−	0.987204i	$-0.449024\pi$
$853$	9.31456	0.318924	0.159462	+	0.987204i	$0.449024\pi$
$854$	52.0551	1.78129
$855$	0	0
$856$	−0.237204	−0.00810748
$857$	−34.0314	−1.16249	−0.581245	−	0.813729i	$-0.697434\pi$
$857$	−34.0314	−1.16249	−0.581245	+	0.813729i	$0.697434\pi$
$858$	0	0
$859$	33.4243	1.14042	0.570211	−	0.821498i	$-0.306861\pi$
$859$	33.4243	1.14042	0.570211	+	0.821498i	$0.306861\pi$
$860$	0	0
$861$	0	0
$862$	96.2291	3.27758
$863$	16.5900	0.564730	0.282365	−	0.959307i	$-0.408881\pi$
$863$	16.5900	0.564730	0.282365	+	0.959307i	$0.408881\pi$
$864$	0	0
$865$	0	0
$866$	−35.3227	−1.20032
$867$	0	0
$868$	−74.2296	−2.51952
$869$	−8.25760	−0.280120
$870$	0	0
$871$	−13.9974	−0.474285
$872$	18.3642	0.621891
$873$	0	0
$874$	−14.7294	−0.498229
$875$	0	0
$876$	0	0
$877$	32.9538	1.11277	0.556385	−	0.830925i	$-0.312188\pi$
$877$	32.9538	1.11277	0.556385	+	0.830925i	$0.312188\pi$
$878$	39.8214	1.34391
$879$	0	0
$880$	0	0
$881$	50.0755	1.68709	0.843544	−	0.537061i	$-0.180465\pi$
$881$	50.0755	1.68709	0.843544	+	0.537061i	$0.180465\pi$
$882$	0	0
$883$	−12.5335	−0.421786	−0.210893	−	0.977509i	$-0.567637\pi$
$883$	−12.5335	−0.421786	−0.210893	+	0.977509i	$0.567637\pi$
$884$	−46.1245	−1.55133
$885$	0	0
$886$	82.3228	2.76569
$887$	26.9358	0.904415	0.452207	−	0.891913i	$-0.350637\pi$
$887$	26.9358	0.904415	0.452207	+	0.891913i	$0.350637\pi$
$888$	0	0
$889$	17.7729	0.596085
$890$	0	0
$891$	0	0
$892$	1.29508	0.0433625
$893$	−5.30004	−0.177359
$894$	0	0
$895$	0	0
$896$	70.8835	2.36805
$897$	0	0
$898$	−48.1107	−1.60548
$899$	33.7268	1.12485
$900$	0	0
$901$	−1.57097	−0.0523366
$902$	3.32869	0.110833
$903$	0	0
$904$	−8.53986	−0.284032
$905$	0	0
$906$	0	0
$907$	28.6510	0.951342	0.475671	−	0.879623i	$-0.342205\pi$
$907$	28.6510	0.951342	0.475671	+	0.879623i	$0.342205\pi$
$908$	69.2414	2.29786
$909$	0	0
$910$	0	0
$911$	8.90071	0.294894	0.147447	−	0.989070i	$-0.452894\pi$
$911$	8.90071	0.294894	0.147447	+	0.989070i	$0.452894\pi$
$912$	0	0
$913$	7.26569	0.240459
$914$	−97.1530	−3.21354
$915$	0	0
$916$	−40.6242	−1.34226
$917$	9.71676	0.320876
$918$	0	0
$919$	17.1901	0.567049	0.283525	−	0.958965i	$-0.408496\pi$
$919$	17.1901	0.567049	0.283525	+	0.958965i	$0.408496\pi$
$920$	0	0
$921$	0	0
$922$	24.8389	0.818025
$923$	−0.402381	−0.0132445
$924$	0	0
$925$	0	0
$926$	18.2141	0.598554
$927$	0	0
$928$	−26.0396	−0.854793
$929$	−34.7789	−1.14106	−0.570530	−	0.821277i	$-0.693262\pi$
$929$	−34.7789	−1.14106	−0.570530	+	0.821277i	$0.693262\pi$
$930$	0	0
$931$	−5.84330	−0.191507
$932$	−65.0123	−2.12955
$933$	0	0
$934$	11.5863	0.379115
$935$	0	0
$936$	0	0
$937$	16.5386	0.540292	0.270146	−	0.962819i	$-0.412928\pi$
$937$	16.5386	0.540292	0.270146	+	0.962819i	$0.412928\pi$
$938$	−44.1962	−1.44306
$939$	0	0
$940$	0	0
$941$	−41.1982	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$941$	−41.1982	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$942$	0	0
$943$	18.4381	0.600428
$944$	1.02799	0.0334582
$945$	0	0
$946$	10.9869	0.357214
$947$	−37.1110	−1.20594	−0.602972	−	0.797762i	$-0.706017\pi$
$947$	−37.1110	−1.20594	−0.602972	+	0.797762i	$0.706017\pi$
$948$	0	0
$949$	−39.3849	−1.27849
$950$	0	0
$951$	0	0
$952$	−60.2376	−1.95231
$953$	20.8962	0.676895	0.338447	−	0.940985i	$-0.390098\pi$
$953$	20.8962	0.676895	0.338447	+	0.940985i	$0.390098\pi$
$954$	0	0
$955$	0	0
$956$	−14.0691	−0.455027
$957$	0	0
$958$	38.1436	1.23236
$959$	8.08122	0.260956
$960$	0	0
$961$	5.66391	0.182707
$962$	−28.3087	−0.912710
$963$	0	0
$964$	55.2057	1.77806
$965$	0	0
$966$	0	0
$967$	−8.28057	−0.266285	−0.133143	−	0.991097i	$-0.542507\pi$
$967$	−8.28057	−0.266285	−0.133143	+	0.991097i	$0.542507\pi$
$968$	−35.2838	−1.13407
$969$	0	0
$970$	0	0
$971$	26.4077	0.847463	0.423731	−	0.905788i	$-0.360720\pi$
$971$	26.4077	0.847463	0.423731	+	0.905788i	$0.360720\pi$
$972$	0	0
$973$	38.8294	1.24481
$974$	−2.03885	−0.0653289
$975$	0	0
$976$	−5.05390	−0.161771
$977$	25.3280	0.810314	0.405157	−	0.914247i	$-0.367217\pi$
$977$	25.3280	0.810314	0.405157	+	0.914247i	$0.367217\pi$
$978$	0	0
$979$	−5.66594	−0.181084
$980$	0	0
$981$	0	0
$982$	−28.4059	−0.906470
$983$	−5.12811	−0.163561	−0.0817807	−	0.996650i	$-0.526061\pi$
$983$	−5.12811	−0.163561	−0.0817807	+	0.996650i	$0.526061\pi$
$984$	0	0
$985$	0	0
$986$	66.1708	2.10731
$987$	0	0
$988$	8.91638	0.283668
$989$	60.8581	1.93517
$990$	0	0
$991$	−26.5396	−0.843059	−0.421530	−	0.906815i	$-0.638507\pi$
$991$	−26.5396	−0.843059	−0.421530	+	0.906815i	$0.638507\pi$
$992$	−28.3073	−0.898758
$993$	0	0
$994$	−1.27050	−0.0402978
$995$	0	0
$996$	0	0
$997$	−44.4973	−1.40924	−0.704622	−	0.709583i	$-0.748883\pi$
$997$	−44.4973	−1.40924	−0.704622	+	0.709583i	$0.748883\pi$
$998$	−7.78321	−0.246373
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.be.1.6		8
3.2	odd	2		625.2.a.e.1.3	✓	8
5.4	even	2		5625.2.a.s.1.3		8
12.11	even	2		10000.2.a.bn.1.1		8
15.2	even	4		625.2.b.d.624.3		16
15.8	even	4		625.2.b.d.624.14		16
15.14	odd	2		625.2.a.g.1.6	yes	8
60.59	even	2		10000.2.a.be.1.8		8
75.2	even	20		625.2.e.k.124.2		32
75.8	even	20		625.2.e.j.374.2		32
75.11	odd	10		625.2.d.p.501.2		16
75.14	odd	10		625.2.d.n.501.3		16
75.17	even	20		625.2.e.j.374.7		32
75.23	even	20		625.2.e.k.124.7		32
75.29	odd	10		625.2.d.m.376.2		16
75.38	even	20		625.2.e.k.499.2		32
75.41	odd	10		625.2.d.p.126.2		16
75.44	odd	10		625.2.d.m.251.2		16
75.47	even	20		625.2.e.j.249.2		32
75.53	even	20		625.2.e.j.249.7		32
75.56	odd	10		625.2.d.q.251.3		16
75.59	odd	10		625.2.d.n.126.3		16
75.62	even	20		625.2.e.k.499.7		32
75.71	odd	10		625.2.d.q.376.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.3	✓	8	3.2	odd	2
625.2.a.g.1.6	yes	8	15.14	odd	2
625.2.b.d.624.3		16	15.2	even	4
625.2.b.d.624.14		16	15.8	even	4
625.2.d.m.251.2		16	75.44	odd	10
625.2.d.m.376.2		16	75.29	odd	10
625.2.d.n.126.3		16	75.59	odd	10
625.2.d.n.501.3		16	75.14	odd	10
625.2.d.p.126.2		16	75.41	odd	10
625.2.d.p.501.2		16	75.11	odd	10
625.2.d.q.251.3		16	75.56	odd	10
625.2.d.q.376.3		16	75.71	odd	10
625.2.e.j.249.2		32	75.47	even	20
625.2.e.j.249.7		32	75.53	even	20
625.2.e.j.374.2		32	75.8	even	20
625.2.e.j.374.7		32	75.17	even	20
625.2.e.k.124.2		32	75.2	even	20
625.2.e.k.124.7		32	75.23	even	20
625.2.e.k.499.2		32	75.38	even	20
625.2.e.k.499.7		32	75.62	even	20
5625.2.a.s.1.3		8	5.4	even	2
5625.2.a.be.1.6		8	1.1	even	1	trivial
10000.2.a.be.1.8		8	60.59	even	2
10000.2.a.bn.1.1		8	12.11	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)

\(f(q)\)	\(=\)	\(q+2.32610 q^{2} +3.41075 q^{4} -3.59425 q^{7} +3.28154 q^{8} +O(q^{10})\)	\(q+2.32610 q^{2} +3.41075 q^{4} -3.59425 q^{7} +3.28154 q^{8} +0.497788 q^{11} -2.64789 q^{13} -8.36058 q^{14} +0.811708 q^{16} +5.10719 q^{17} -0.987277 q^{19} +1.15790 q^{22} +6.41382 q^{23} -6.15926 q^{26} -12.2591 q^{28} +5.57001 q^{29} +6.05507 q^{31} -4.67497 q^{32} +11.8798 q^{34} +4.59612 q^{37} -2.29651 q^{38} +2.87475 q^{41} +9.48858 q^{43} +1.69783 q^{44} +14.9192 q^{46} +5.36834 q^{47} +5.91861 q^{49} -9.03129 q^{52} -0.307600 q^{53} -11.7947 q^{56} +12.9564 q^{58} +1.26645 q^{59} -6.22625 q^{61} +14.0847 q^{62} -12.4979 q^{64} +5.28626 q^{67} +17.4193 q^{68} +0.151963 q^{71} +14.8741 q^{73} +10.6910 q^{74} -3.36735 q^{76} -1.78917 q^{77} -16.5886 q^{79} +6.68696 q^{82} +14.5960 q^{83} +22.0714 q^{86} +1.63351 q^{88} -11.3822 q^{89} +9.51717 q^{91} +21.8759 q^{92} +12.4873 q^{94} -0.849192 q^{97} +13.7673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8	\( 8 q + 5 q^{2} + 11 q^{4} - 10 q^{7} + 15 q^{8} - q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} + 15 q^{17} - 10 q^{19} + 5 q^{22} + 30 q^{23} - 11 q^{26} + 5 q^{28} - 10 q^{29} - 9 q^{31} + 30 q^{32} + 7 q^{34} + 10 q^{37} + 20 q^{38} + 4 q^{41} + 18 q^{44} - 9 q^{46} + 30 q^{47} - 4 q^{49} - 5 q^{52} + 10 q^{53} + 30 q^{58} + 5 q^{59} + 6 q^{61} + 10 q^{62} - 9 q^{64} - 10 q^{67} + 40 q^{68} + 9 q^{71} + 18 q^{74} - 10 q^{76} + 5 q^{77} - 20 q^{79} + 45 q^{82} + 40 q^{83} + 24 q^{86} + 40 q^{88} + 5 q^{89} + 6 q^{91} + 15 q^{92} + 47 q^{94} - 30 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 11 * q^4 - 10 * q^7 + 15 * q^8 - q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 + 15 * q^17 - 10 * q^19 + 5 * q^22 + 30 * q^23 - 11 * q^26 + 5 * q^28 - 10 * q^29 - 9 * q^31 + 30 * q^32 + 7 * q^34 + 10 * q^37 + 20 * q^38 + 4 * q^41 + 18 * q^44 - 9 * q^46 + 30 * q^47 - 4 * q^49 - 5 * q^52 + 10 * q^53 + 30 * q^58 + 5 * q^59 + 6 * q^61 + 10 * q^62 - 9 * q^64 - 10 * q^67 + 40 * q^68 + 9 * q^71 + 18 * q^74 - 10 * q^76 + 5 * q^77 - 20 * q^79 + 45 * q^82 + 40 * q^83 + 24 * q^86 + 40 * q^88 + 5 * q^89 + 6 * q^91 + 15 * q^92 + 47 * q^94 - 30 * q^98

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