LMFDB - Embedded newform 5625.2.a.m.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:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
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			$-0.209057$ 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)

$f(q)$	$=$	$q+1.33826 q^{2} -0.209057 q^{4} -1.27977 q^{7} -2.95630 q^{8} +O(q^{10})$	$q+1.33826 q^{2} -0.209057 q^{4} -1.27977 q^{7} -2.95630 q^{8} -1.16535 q^{11} +3.61048 q^{13} -1.71267 q^{14} -3.53818 q^{16} +5.35772 q^{17} -7.61048 q^{19} -1.55955 q^{22} +3.41811 q^{23} +4.83176 q^{26} +0.267545 q^{28} +3.21661 q^{29} -1.09306 q^{31} +1.17758 q^{32} +7.17002 q^{34} +7.80126 q^{37} -10.1848 q^{38} +3.00565 q^{41} +3.42278 q^{43} +0.243625 q^{44} +4.57433 q^{46} -9.41462 q^{47} -5.36218 q^{49} -0.754795 q^{52} -7.64760 q^{53} +3.78339 q^{56} +4.30467 q^{58} -12.7500 q^{59} -10.1506 q^{61} -1.46280 q^{62} +8.65227 q^{64} -8.78903 q^{67} -1.12007 q^{68} +15.0647 q^{71} -13.1533 q^{73} +10.4401 q^{74} +1.59102 q^{76} +1.49139 q^{77} -11.5155 q^{79} +4.02234 q^{82} -10.7189 q^{83} +4.58058 q^{86} +3.44512 q^{88} +4.51584 q^{89} -4.62059 q^{91} -0.714580 q^{92} -12.5992 q^{94} -18.0135 q^{97} -7.17600 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8	$ 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8} + 6 q^{11} - 7 q^{13} + 10 q^{14} - 9 q^{16} - 7 q^{17} - 9 q^{19} - 6 q^{22} + 10 q^{23} + 2 q^{26} - 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} + 10 q^{37} - 6 q^{38} - q^{43} + 9 q^{44} + 5 q^{46} - 23 q^{47} - 3 q^{49} - 13 q^{52} + 2 q^{58} - 4 q^{59} - 43 q^{61} - 10 q^{62} - 7 q^{64} - 8 q^{67} - 3 q^{68} + 27 q^{71} - 15 q^{73} - 5 q^{74} + 9 q^{76} - 15 q^{77} + 10 q^{79} + 20 q^{82} + 3 q^{83} - 24 q^{86} + 3 q^{88} + 9 q^{89} + 5 q^{91} - 15 q^{92} - 22 q^{94} - 13 q^{97} - 42 q^{98}+O(q^{100}) $ 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8 + 6 * q^11 - 7 * q^13 + 10 * q^14 - 9 * q^16 - 7 * q^17 - 9 * q^19 - 6 * q^22 + 10 * q^23 + 2 * q^26 - 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 + 10 * q^37 - 6 * q^38 - q^43 + 9 * q^44 + 5 * q^46 - 23 * q^47 - 3 * q^49 - 13 * q^52 + 2 * q^58 - 4 * q^59 - 43 * q^61 - 10 * q^62 - 7 * q^64 - 8 * q^67 - 3 * q^68 + 27 * q^71 - 15 * q^73 - 5 * q^74 + 9 * q^76 - 15 * q^77 + 10 * q^79 + 20 * q^82 + 3 * q^83 - 24 * q^86 + 3 * q^88 + 9 * q^89 + 5 * q^91 - 15 * q^92 - 22 * q^94 - 13 * q^97 - 42 * 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 $

$2$	1.33826	0.946294	0.473147	−	0.880984i	$-0.343118\pi$
$2$	1.33826	0.946294	0.473147	+	0.880984i	$0.343118\pi$
$3$	0	0
$3$	0	0
$4$	−0.209057	−0.104528
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.27977	−0.483709	−0.241854	−	0.970313i	$-0.577756\pi$
$7$	−1.27977	−0.483709	−0.241854	+	0.970313i	$0.577756\pi$
$8$	−2.95630	−1.04521
$9$	0	0
$10$	0	0
$11$	−1.16535	−0.351367	−0.175683	−	0.984447i	$-0.556214\pi$
$11$	−1.16535	−0.351367	−0.175683	+	0.984447i	$0.556214\pi$
$12$	0	0
$13$	3.61048	1.00137	0.500683	−	0.865631i	$-0.333082\pi$
$13$	3.61048	1.00137	0.500683	+	0.865631i	$0.333082\pi$
$14$	−1.71267	−0.457730
$15$	0	0
$16$	−3.53818	−0.884545
$17$	5.35772	1.29944	0.649718	−	0.760175i	$-0.274887\pi$
$17$	5.35772	1.29944	0.649718	+	0.760175i	$0.274887\pi$
$18$	0	0
$19$	−7.61048	−1.74596	−0.872982	−	0.487753i	$-0.837817\pi$
$19$	−7.61048	−1.74596	−0.872982	+	0.487753i	$0.837817\pi$
$20$	0	0
$21$	0	0
$22$	−1.55955	−0.332496
$23$	3.41811	0.712726	0.356363	−	0.934348i	$-0.384017\pi$
$23$	3.41811	0.712726	0.356363	+	0.934348i	$0.384017\pi$
$24$	0	0
$25$	0	0
$26$	4.83176	0.947586
$27$	0	0
$28$	0.267545	0.0505613
$29$	3.21661	0.597310	0.298655	−	0.954361i	$-0.403462\pi$
$29$	3.21661	0.597310	0.298655	+	0.954361i	$0.403462\pi$
$30$	0	0
$31$	−1.09306	−0.196319	−0.0981594	−	0.995171i	$-0.531295\pi$
$31$	−1.09306	−0.196319	−0.0981594	+	0.995171i	$0.531295\pi$
$32$	1.17758	0.208169
$33$	0	0
$34$	7.17002	1.22965
$35$	0	0
$36$	0	0
$37$	7.80126	1.28252	0.641260	−	0.767324i	$-0.278412\pi$
$37$	7.80126	1.28252	0.641260	+	0.767324i	$0.278412\pi$
$38$	−10.1848	−1.65219
$39$	0	0
$40$	0	0
$41$	3.00565	0.469403	0.234702	−	0.972067i	$-0.424589\pi$
$41$	3.00565	0.469403	0.234702	+	0.972067i	$0.424589\pi$
$42$	0	0
$43$	3.42278	0.521970	0.260985	−	0.965343i	$-0.415953\pi$
$43$	3.42278	0.521970	0.260985	+	0.965343i	$0.415953\pi$
$44$	0.243625	0.0367278
$45$	0	0
$46$	4.57433	0.674448
$47$	−9.41462	−1.37326	−0.686632	−	0.727005i	$-0.740912\pi$
$47$	−9.41462	−1.37326	−0.686632	+	0.727005i	$0.740912\pi$
$48$	0	0
$49$	−5.36218	−0.766026
$50$	0	0
$51$	0	0
$52$	−0.754795	−0.104671
$53$	−7.64760	−1.05048	−0.525239	−	0.850954i	$-0.676024\pi$
$53$	−7.64760	−1.05048	−0.525239	+	0.850954i	$0.676024\pi$
$54$	0	0
$55$	0	0
$56$	3.78339	0.505576
$57$	0	0
$58$	4.30467	0.565231
$59$	−12.7500	−1.65991	−0.829954	−	0.557832i	$-0.811634\pi$
$59$	−12.7500	−1.65991	−0.829954	+	0.557832i	$0.811634\pi$
$60$	0	0
$61$	−10.1506	−1.29965	−0.649824	−	0.760085i	$-0.725157\pi$
$61$	−10.1506	−1.29965	−0.649824	+	0.760085i	$0.725157\pi$
$62$	−1.46280	−0.185775
$63$	0	0
$64$	8.65227	1.08153
$65$	0	0
$66$	0	0
$67$	−8.78903	−1.07375	−0.536876	−	0.843661i	$-0.680396\pi$
$67$	−8.78903	−1.07375	−0.536876	+	0.843661i	$0.680396\pi$
$68$	−1.12007	−0.135828
$69$	0	0
$70$	0	0
$71$	15.0647	1.78786	0.893928	−	0.448211i	$-0.147939\pi$
$71$	15.0647	1.78786	0.893928	+	0.448211i	$0.147939\pi$
$72$	0	0
$73$	−13.1533	−1.53948	−0.769740	−	0.638357i	$-0.779614\pi$
$73$	−13.1533	−1.53948	−0.769740	+	0.638357i	$0.779614\pi$
$74$	10.4401	1.21364
$75$	0	0
$76$	1.59102	0.182503
$77$	1.49139	0.169959
$78$	0	0
$79$	−11.5155	−1.29560	−0.647798	−	0.761812i	$-0.724310\pi$
$79$	−11.5155	−1.29560	−0.647798	+	0.761812i	$0.724310\pi$
$80$	0	0
$81$	0	0
$82$	4.02234	0.444193
$83$	−10.7189	−1.17655	−0.588277	−	0.808659i	$-0.700194\pi$
$83$	−10.7189	−1.17655	−0.588277	+	0.808659i	$0.700194\pi$
$84$	0	0
$85$	0	0
$86$	4.58058	0.493937
$87$	0	0
$88$	3.44512	0.367252
$89$	4.51584	0.478678	0.239339	−	0.970936i	$-0.423069\pi$
$89$	4.51584	0.478678	0.239339	+	0.970936i	$0.423069\pi$
$90$	0	0
$91$	−4.62059	−0.484369
$92$	−0.714580	−0.0745002
$93$	0	0
$94$	−12.5992	−1.29951
$95$	0	0
$96$	0	0
$97$	−18.0135	−1.82899	−0.914496	−	0.404596i	$-0.867412\pi$
$97$	−18.0135	−1.82899	−0.914496	+	0.404596i	$0.867412\pi$
$98$	−7.17600	−0.724885
$99$	0	0
$100$	0	0
$101$	6.54634	0.651385	0.325693	−	0.945476i	$-0.394403\pi$
$101$	6.54634	0.651385	0.325693	+	0.945476i	$0.394403\pi$
$102$	0	0
$103$	−11.1588	−1.09951	−0.549753	−	0.835327i	$-0.685278\pi$
$103$	−11.1588	−1.09951	−0.549753	+	0.835327i	$0.685278\pi$
$104$	−10.6736	−1.04664
$105$	0	0
$106$	−10.2345	−0.994061
$107$	−3.72681	−0.360284	−0.180142	−	0.983641i	$-0.557656\pi$
$107$	−3.72681	−0.360284	−0.180142	+	0.983641i	$0.557656\pi$
$108$	0	0
$109$	−6.99533	−0.670031	−0.335016	−	0.942213i	$-0.608742\pi$
$109$	−6.99533	−0.670031	−0.335016	+	0.942213i	$0.608742\pi$
$110$	0	0
$111$	0	0
$112$	4.52807	0.427862
$113$	−5.59858	−0.526670	−0.263335	−	0.964704i	$-0.584822\pi$
$113$	−5.59858	−0.526670	−0.263335	+	0.964704i	$0.584822\pi$
$114$	0	0
$115$	0	0
$116$	−0.672455	−0.0624359
$117$	0	0
$118$	−17.0628	−1.57076
$119$	−6.85666	−0.628549
$120$	0	0
$121$	−9.64195	−0.876541
$122$	−13.5841	−1.22985
$123$	0	0
$124$	0.228511	0.0205209
$125$	0	0
$126$	0	0
$127$	4.38919	0.389478	0.194739	−	0.980855i	$-0.437614\pi$
$127$	4.38919	0.389478	0.194739	+	0.980855i	$0.437614\pi$
$128$	9.22384	0.815280
$129$	0	0
$130$	0	0
$131$	4.99244	0.436192	0.218096	−	0.975927i	$-0.430015\pi$
$131$	4.99244	0.436192	0.218096	+	0.975927i	$0.430015\pi$
$132$	0	0
$133$	9.73968	0.844537
$134$	−11.7620	−1.01608
$135$	0	0
$136$	−15.8390	−1.35818
$137$	12.2056	1.04279	0.521396	−	0.853315i	$-0.325411\pi$
$137$	12.2056	1.04279	0.521396	+	0.853315i	$0.325411\pi$
$138$	0	0
$139$	−5.20617	−0.441582	−0.220791	−	0.975321i	$-0.570864\pi$
$139$	−5.20617	−0.441582	−0.220791	+	0.975321i	$0.570864\pi$
$140$	0	0
$141$	0	0
$142$	20.1606	1.69184
$143$	−4.20748	−0.351847
$144$	0	0
$145$	0	0
$146$	−17.6026	−1.45680
$147$	0	0
$148$	−1.63091	−0.134060
$149$	21.6208	1.77124	0.885622	−	0.464406i	$-0.153732\pi$
$149$	21.6208	1.77124	0.885622	+	0.464406i	$0.153732\pi$
$150$	0	0
$151$	11.8918	0.967743	0.483872	−	0.875139i	$-0.339230\pi$
$151$	11.8918	0.967743	0.483872	+	0.875139i	$0.339230\pi$
$152$	22.4988	1.82490
$153$	0	0
$154$	1.99586	0.160831
$155$	0	0
$156$	0	0
$157$	12.1112	0.966579	0.483290	−	0.875461i	$-0.339442\pi$
$157$	12.1112	0.966579	0.483290	+	0.875461i	$0.339442\pi$
$158$	−15.4108	−1.22601
$159$	0	0
$160$	0	0
$161$	−4.37441	−0.344752
$162$	0	0
$163$	−8.26913	−0.647688	−0.323844	−	0.946111i	$-0.604975\pi$
$163$	−8.26913	−0.647688	−0.323844	+	0.946111i	$0.604975\pi$
$164$	−0.628351	−0.0490660
$165$	0	0
$166$	−14.3447	−1.11337
$167$	6.26564	0.484849	0.242425	−	0.970170i	$-0.422057\pi$
$167$	6.26564	0.484849	0.242425	+	0.970170i	$0.422057\pi$
$168$	0	0
$169$	0.0355444	0.00273418
$170$	0	0
$171$	0	0
$172$	−0.715557	−0.0545607
$173$	6.65609	0.506053	0.253027	−	0.967459i	$-0.418574\pi$
$173$	6.65609	0.506053	0.253027	+	0.967459i	$0.418574\pi$
$174$	0	0
$175$	0	0
$176$	4.12323	0.310800
$177$	0	0
$178$	6.04337	0.452970
$179$	−21.7154	−1.62309	−0.811544	−	0.584292i	$-0.801372\pi$
$179$	−21.7154	−1.62309	−0.811544	+	0.584292i	$0.801372\pi$
$180$	0	0
$181$	12.9474	0.962375	0.481188	−	0.876618i	$-0.340206\pi$
$181$	12.9474	0.962375	0.481188	+	0.876618i	$0.340206\pi$
$182$	−6.18356	−0.458356
$183$	0	0
$184$	−10.1050	−0.744947
$185$	0	0
$186$	0	0
$187$	−6.24362	−0.456579
$188$	1.96819	0.143545
$189$	0	0
$190$	0	0
$191$	−15.7683	−1.14095	−0.570476	−	0.821314i	$-0.693241\pi$
$191$	−15.7683	−1.14095	−0.570476	+	0.821314i	$0.693241\pi$
$192$	0	0
$193$	−6.93514	−0.499202	−0.249601	−	0.968349i	$-0.580299\pi$
$193$	−6.93514	−0.499202	−0.249601	+	0.968349i	$0.580299\pi$
$194$	−24.1067	−1.73076
$195$	0	0
$196$	1.12100	0.0800715
$197$	−7.43948	−0.530041	−0.265020	−	0.964243i	$-0.585379\pi$
$197$	−7.43948	−0.530041	−0.265020	+	0.964243i	$0.585379\pi$
$198$	0	0
$199$	−14.2398	−1.00943	−0.504716	−	0.863285i	$-0.668403\pi$
$199$	−14.2398	−1.00943	−0.504716	+	0.863285i	$0.668403\pi$
$200$	0	0
$201$	0	0
$202$	8.76072	0.616402
$203$	−4.11653	−0.288924
$204$	0	0
$205$	0	0
$206$	−14.9334	−1.04046
$207$	0	0
$208$	−12.7745	−0.885754
$209$	8.86889	0.613474
$210$	0	0
$211$	6.89569	0.474719	0.237360	−	0.971422i	$-0.423718\pi$
$211$	6.89569	0.474719	0.237360	+	0.971422i	$0.423718\pi$
$212$	1.59878	0.109805
$213$	0	0
$214$	−4.98744	−0.340935
$215$	0	0
$216$	0	0
$217$	1.39886	0.0949611
$218$	−9.36158	−0.634046
$219$	0	0
$220$	0	0
$221$	19.3439	1.30121
$222$	0	0
$223$	−13.3983	−0.897219	−0.448609	−	0.893728i	$-0.648080\pi$
$223$	−13.3983	−0.897219	−0.448609	+	0.893728i	$0.648080\pi$
$224$	−1.50703	−0.100693
$225$	0	0
$226$	−7.49236	−0.498385
$227$	−0.849831	−0.0564053	−0.0282026	−	0.999602i	$-0.508978\pi$
$227$	−0.849831	−0.0564053	−0.0282026	+	0.999602i	$0.508978\pi$
$228$	0	0
$229$	−8.79561	−0.581231	−0.290615	−	0.956840i	$-0.593860\pi$
$229$	−8.79561	−0.581231	−0.290615	+	0.956840i	$0.593860\pi$
$230$	0	0
$231$	0	0
$232$	−9.50926	−0.624314
$233$	21.2093	1.38947	0.694734	−	0.719267i	$-0.255522\pi$
$233$	21.2093	1.38947	0.694734	+	0.719267i	$0.255522\pi$
$234$	0	0
$235$	0	0
$236$	2.66548	0.173508
$237$	0	0
$238$	−9.17600	−0.594792
$239$	16.7972	1.08652	0.543261	−	0.839564i	$-0.317189\pi$
$239$	16.7972	1.08652	0.543261	+	0.839564i	$0.317189\pi$
$240$	0	0
$241$	9.51860	0.613147	0.306573	−	0.951847i	$-0.400817\pi$
$241$	9.51860	0.613147	0.306573	+	0.951847i	$0.400817\pi$
$242$	−12.9035	−0.829465
$243$	0	0
$244$	2.12205	0.135850
$245$	0	0
$246$	0	0
$247$	−27.4775	−1.74835
$248$	3.23140	0.205194
$249$	0	0
$250$	0	0
$251$	−12.9486	−0.817309	−0.408655	−	0.912689i	$-0.634002\pi$
$251$	−12.9486	−0.817309	−0.408655	+	0.912689i	$0.634002\pi$
$252$	0	0
$253$	−3.98331	−0.250428
$254$	5.87389	0.368560
$255$	0	0
$256$	−4.96064	−0.310040
$257$	−16.6837	−1.04070	−0.520352	−	0.853952i	$-0.674199\pi$
$257$	−16.6837	−1.04070	−0.520352	+	0.853952i	$0.674199\pi$
$258$	0	0
$259$	−9.98384	−0.620366
$260$	0	0
$261$	0	0
$262$	6.68119	0.412765
$263$	−9.45426	−0.582975	−0.291487	−	0.956575i	$-0.594150\pi$
$263$	−9.45426	−0.582975	−0.291487	+	0.956575i	$0.594150\pi$
$264$	0	0
$265$	0	0
$266$	13.0342	0.799180
$267$	0	0
$268$	1.83741	0.112238
$269$	−31.7384	−1.93513	−0.967563	−	0.252631i	$-0.918704\pi$
$269$	−31.7384	−1.93513	−0.967563	+	0.252631i	$0.918704\pi$
$270$	0	0
$271$	19.0556	1.15755	0.578773	−	0.815489i	$-0.303532\pi$
$271$	19.0556	1.15755	0.578773	+	0.815489i	$0.303532\pi$
$272$	−18.9566	−1.14941
$273$	0	0
$274$	16.3342	0.986787
$275$	0	0
$276$	0	0
$277$	2.71929	0.163387	0.0816933	−	0.996658i	$-0.473967\pi$
$277$	2.71929	0.163387	0.0816933	+	0.996658i	$0.473967\pi$
$278$	−6.96722	−0.417866
$279$	0	0
$280$	0	0
$281$	14.6076	0.871416	0.435708	−	0.900088i	$-0.356498\pi$
$281$	14.6076	0.871416	0.435708	+	0.900088i	$0.356498\pi$
$282$	0	0
$283$	−18.0890	−1.07528	−0.537639	−	0.843175i	$-0.680684\pi$
$283$	−18.0890	−1.07528	−0.537639	+	0.843175i	$0.680684\pi$
$284$	−3.14939	−0.186882
$285$	0	0
$286$	−5.63070	−0.332950
$287$	−3.84655	−0.227054
$288$	0	0
$289$	11.7051	0.688536
$290$	0	0
$291$	0	0
$292$	2.74979	0.160920
$293$	−0.293910	−0.0171704	−0.00858521	−	0.999963i	$-0.502733\pi$
$293$	−0.293910	−0.0171704	−0.00858521	+	0.999963i	$0.502733\pi$
$294$	0	0
$295$	0	0
$296$	−23.0628	−1.34050
$297$	0	0
$298$	28.9343	1.67612
$299$	12.3410	0.713700
$300$	0	0
$301$	−4.38039	−0.252481
$302$	15.9144	0.915769
$303$	0	0
$304$	26.9272	1.54438
$305$	0	0
$306$	0	0
$307$	1.34411	0.0767125	0.0383563	−	0.999264i	$-0.487788\pi$
$307$	1.34411	0.0767125	0.0383563	+	0.999264i	$0.487788\pi$
$308$	−0.311785	−0.0177656
$309$	0	0
$310$	0	0
$311$	−19.9569	−1.13165	−0.565826	−	0.824525i	$-0.691443\pi$
$311$	−19.9569	−1.13165	−0.565826	+	0.824525i	$0.691443\pi$
$312$	0	0
$313$	−14.1446	−0.799500	−0.399750	−	0.916624i	$-0.630903\pi$
$313$	−14.1446	−0.799500	−0.399750	+	0.916624i	$0.630903\pi$
$314$	16.2080	0.914668
$315$	0	0
$316$	2.40740	0.135427
$317$	17.5897	0.987937	0.493968	−	0.869480i	$-0.335546\pi$
$317$	17.5897	0.987937	0.493968	+	0.869480i	$0.335546\pi$
$318$	0	0
$319$	−3.74849	−0.209875
$320$	0	0
$321$	0	0
$322$	−5.85410	−0.326236
$323$	−40.7748	−2.26877
$324$	0	0
$325$	0	0
$326$	−11.0662	−0.612903
$327$	0	0
$328$	−8.88558	−0.490624
$329$	12.0486	0.664260
$330$	0	0
$331$	8.00683	0.440095	0.220048	−	0.975489i	$-0.429379\pi$
$331$	8.00683	0.440095	0.220048	+	0.975489i	$0.429379\pi$
$332$	2.24086	0.122983
$333$	0	0
$334$	8.38506	0.458810
$335$	0	0
$336$	0	0
$337$	−10.8474	−0.590895	−0.295448	−	0.955359i	$-0.595469\pi$
$337$	−10.8474	−0.590895	−0.295448	+	0.955359i	$0.595469\pi$
$338$	0.0475677	0.00258734
$339$	0	0
$340$	0	0
$341$	1.27380	0.0689799
$342$	0	0
$343$	15.8208	0.854242
$344$	−10.1188	−0.545567
$345$	0	0
$346$	8.90759	0.478875
$347$	1.86153	0.0999323	0.0499662	−	0.998751i	$-0.484089\pi$
$347$	1.86153	0.0999323	0.0499662	+	0.998751i	$0.484089\pi$
$348$	0	0
$349$	24.4623	1.30944	0.654719	−	0.755872i	$-0.272787\pi$
$349$	24.4623	1.30944	0.654719	+	0.755872i	$0.272787\pi$
$350$	0	0
$351$	0	0
$352$	−1.37229	−0.0731436
$353$	−37.3621	−1.98858	−0.994291	−	0.106704i	$-0.965970\pi$
$353$	−37.3621	−1.98858	−0.994291	+	0.106704i	$0.965970\pi$
$354$	0	0
$355$	0	0
$356$	−0.944068	−0.0500355
$357$	0	0
$358$	−29.0609	−1.53592
$359$	25.3360	1.33718	0.668592	−	0.743629i	$-0.266897\pi$
$359$	25.3360	1.33718	0.668592	+	0.743629i	$0.266897\pi$
$360$	0	0
$361$	38.9194	2.04839
$362$	17.3270	0.910689
$363$	0	0
$364$	0.965966	0.0506304
$365$	0	0
$366$	0	0
$367$	−23.7057	−1.23743	−0.618714	−	0.785616i	$-0.712346\pi$
$367$	−23.7057	−1.23743	−0.618714	+	0.785616i	$0.712346\pi$
$368$	−12.0939	−0.630438
$369$	0	0
$370$	0	0
$371$	9.78719	0.508126
$372$	0	0
$373$	2.93244	0.151836	0.0759181	−	0.997114i	$-0.475811\pi$
$373$	2.93244	0.151836	0.0759181	+	0.997114i	$0.475811\pi$
$374$	−8.35560	−0.432058
$375$	0	0
$376$	27.8324	1.43535
$377$	11.6135	0.598126
$378$	0	0
$379$	−17.7349	−0.910980	−0.455490	−	0.890241i	$-0.650536\pi$
$379$	−17.7349	−0.910980	−0.455490	+	0.890241i	$0.650536\pi$
$380$	0	0
$381$	0	0
$382$	−21.1021	−1.07968
$383$	−4.40524	−0.225097	−0.112549	−	0.993646i	$-0.535901\pi$
$383$	−4.40524	−0.225097	−0.112549	+	0.993646i	$0.535901\pi$
$384$	0	0
$385$	0	0
$386$	−9.28102	−0.472392
$387$	0	0
$388$	3.76584	0.191182
$389$	31.3215	1.58806	0.794031	−	0.607877i	$-0.207979\pi$
$389$	31.3215	1.58806	0.794031	+	0.607877i	$0.207979\pi$
$390$	0	0
$391$	18.3133	0.926142
$392$	15.8522	0.800657
$393$	0	0
$394$	−9.95596	−0.501574
$395$	0	0
$396$	0	0
$397$	16.4281	0.824501	0.412250	−	0.911071i	$-0.364743\pi$
$397$	16.4281	0.824501	0.412250	+	0.911071i	$0.364743\pi$
$398$	−19.0566	−0.955220
$399$	0	0
$400$	0	0
$401$	15.1813	0.758120	0.379060	−	0.925372i	$-0.376247\pi$
$401$	15.1813	0.758120	0.379060	+	0.925372i	$0.376247\pi$
$402$	0	0
$403$	−3.94646	−0.196587
$404$	−1.36856	−0.0680883
$405$	0	0
$406$	−5.50900	−0.273407
$407$	−9.09122	−0.450635
$408$	0	0
$409$	−14.8421	−0.733897	−0.366948	−	0.930241i	$-0.619597\pi$
$409$	−14.8421	−0.733897	−0.366948	+	0.930241i	$0.619597\pi$
$410$	0	0
$411$	0	0
$412$	2.33282	0.114930
$413$	16.3171	0.802912
$414$	0	0
$415$	0	0
$416$	4.25162	0.208453
$417$	0	0
$418$	11.8689	0.580526
$419$	35.2636	1.72274	0.861370	−	0.507978i	$-0.169607\pi$
$419$	35.2636	1.72274	0.861370	+	0.507978i	$0.169607\pi$
$420$	0	0
$421$	−11.2827	−0.549887	−0.274943	−	0.961460i	$-0.588659\pi$
$421$	−11.2827	−0.549887	−0.274943	+	0.961460i	$0.588659\pi$
$422$	9.22824	0.449224
$423$	0	0
$424$	22.6086	1.09797
$425$	0	0
$426$	0	0
$427$	12.9904	0.628651
$428$	0.779115	0.0376599
$429$	0	0
$430$	0	0
$431$	−25.8249	−1.24394	−0.621971	−	0.783041i	$-0.713668\pi$
$431$	−25.8249	−1.24394	−0.621971	+	0.783041i	$0.713668\pi$
$432$	0	0
$433$	9.99902	0.480522	0.240261	−	0.970708i	$-0.422767\pi$
$433$	9.99902	0.480522	0.240261	+	0.970708i	$0.422767\pi$
$434$	1.87205	0.0898610
$435$	0	0
$436$	1.46242	0.0700373
$437$	−26.0135	−1.24439
$438$	0	0
$439$	9.20385	0.439276	0.219638	−	0.975581i	$-0.429512\pi$
$439$	9.20385	0.439276	0.219638	+	0.975581i	$0.429512\pi$
$440$	0	0
$441$	0	0
$442$	25.8872	1.23133
$443$	−9.12790	−0.433680	−0.216840	−	0.976207i	$-0.569575\pi$
$443$	−9.12790	−0.433680	−0.216840	+	0.976207i	$0.569575\pi$
$444$	0	0
$445$	0	0
$446$	−17.9305	−0.849032
$447$	0	0
$448$	−11.0729	−0.523147
$449$	−35.4831	−1.67455	−0.837275	−	0.546781i	$-0.815853\pi$
$449$	−35.4831	−1.67455	−0.837275	+	0.546781i	$0.815853\pi$
$450$	0	0
$451$	−3.50264	−0.164933
$452$	1.17042	0.0550520
$453$	0	0
$454$	−1.13730	−0.0533759
$455$	0	0
$456$	0	0
$457$	−18.7516	−0.877162	−0.438581	−	0.898692i	$-0.644519\pi$
$457$	−18.7516	−0.877162	−0.438581	+	0.898692i	$0.644519\pi$
$458$	−11.7708	−0.550015
$459$	0	0
$460$	0	0
$461$	16.0877	0.749280	0.374640	−	0.927170i	$-0.377766\pi$
$461$	16.0877	0.749280	0.374640	+	0.927170i	$0.377766\pi$
$462$	0	0
$463$	21.1040	0.980787	0.490393	−	0.871501i	$-0.336853\pi$
$463$	21.1040	0.980787	0.490393	+	0.871501i	$0.336853\pi$
$464$	−11.3810	−0.528348
$465$	0	0
$466$	28.3836	1.31484
$467$	18.0122	0.833504	0.416752	−	0.909020i	$-0.363168\pi$
$467$	18.0122	0.833504	0.416752	+	0.909020i	$0.363168\pi$
$468$	0	0
$469$	11.2480	0.519383
$470$	0	0
$471$	0	0
$472$	37.6928	1.73495
$473$	−3.98875	−0.183403
$474$	0	0
$475$	0	0
$476$	1.43343	0.0657012
$477$	0	0
$478$	22.4791	1.02817
$479$	−7.31414	−0.334191	−0.167096	−	0.985941i	$-0.553439\pi$
$479$	−7.31414	−0.334191	−0.167096	+	0.985941i	$0.553439\pi$
$480$	0	0
$481$	28.1663	1.28427
$482$	12.7384	0.580217
$483$	0	0
$484$	2.01572	0.0916235
$485$	0	0
$486$	0	0
$487$	−11.9433	−0.541202	−0.270601	−	0.962692i	$-0.587222\pi$
$487$	−11.9433	−0.541202	−0.270601	+	0.962692i	$0.587222\pi$
$488$	30.0081	1.35840
$489$	0	0
$490$	0	0
$491$	−11.1658	−0.503904	−0.251952	−	0.967740i	$-0.581072\pi$
$491$	−11.1658	−0.503904	−0.251952	+	0.967740i	$0.581072\pi$
$492$	0	0
$493$	17.2337	0.776167
$494$	−36.7720	−1.65445
$495$	0	0
$496$	3.86743	0.173653
$497$	−19.2794	−0.864801
$498$	0	0
$499$	30.6045	1.37005	0.685023	−	0.728522i	$-0.259792\pi$
$499$	30.6045	1.37005	0.685023	+	0.728522i	$0.259792\pi$
$500$	0	0
$501$	0	0
$502$	−17.3286	−0.773414
$503$	−1.90950	−0.0851404	−0.0425702	−	0.999093i	$-0.513555\pi$
$503$	−1.90950	−0.0851404	−0.0425702	+	0.999093i	$0.513555\pi$
$504$	0	0
$505$	0	0
$506$	−5.33070	−0.236979
$507$	0	0
$508$	−0.917591	−0.0407115
$509$	−2.52518	−0.111927	−0.0559634	−	0.998433i	$-0.517823\pi$
$509$	−2.52518	−0.111927	−0.0559634	+	0.998433i	$0.517823\pi$
$510$	0	0
$511$	16.8333	0.744660
$512$	−25.0863	−1.10867
$513$	0	0
$514$	−22.3272	−0.984811
$515$	0	0
$516$	0	0
$517$	10.9714	0.482520
$518$	−13.3610	−0.587048
$519$	0	0
$520$	0	0
$521$	−3.65479	−0.160119	−0.0800595	−	0.996790i	$-0.525511\pi$
$521$	−3.65479	−0.160119	−0.0800595	+	0.996790i	$0.525511\pi$
$522$	0	0
$523$	36.3215	1.58823	0.794114	−	0.607768i	$-0.207935\pi$
$523$	36.3215	1.58823	0.794114	+	0.607768i	$0.207935\pi$
$524$	−1.04370	−0.0455945
$525$	0	0
$526$	−12.6523	−0.551665
$527$	−5.85629	−0.255104
$528$	0	0
$529$	−11.3165	−0.492022
$530$	0	0
$531$	0	0
$532$	−2.03615	−0.0882782
$533$	10.8518	0.470044
$534$	0	0
$535$	0	0
$536$	25.9830	1.12229
$537$	0	0
$538$	−42.4743	−1.83120
$539$	6.24883	0.269156
$540$	0	0
$541$	−13.3159	−0.572496	−0.286248	−	0.958156i	$-0.592408\pi$
$541$	−13.3159	−0.572496	−0.286248	+	0.958156i	$0.592408\pi$
$542$	25.5014	1.09538
$543$	0	0
$544$	6.30914	0.270502
$545$	0	0
$546$	0	0
$547$	29.6657	1.26841	0.634207	−	0.773163i	$-0.281327\pi$
$547$	29.6657	1.26841	0.634207	+	0.773163i	$0.281327\pi$
$548$	−2.55166	−0.109001
$549$	0	0
$550$	0	0
$551$	−24.4800	−1.04288
$552$	0	0
$553$	14.7372	0.626691
$554$	3.63913	0.154612
$555$	0	0
$556$	1.08839	0.0461578
$557$	−42.4585	−1.79902	−0.899512	−	0.436897i	$-0.856078\pi$
$557$	−42.4585	−1.79902	−0.899512	+	0.436897i	$0.856078\pi$
$558$	0	0
$559$	12.3579	0.522683
$560$	0	0
$561$	0	0
$562$	19.5488	0.824615
$563$	−41.0132	−1.72850	−0.864250	−	0.503063i	$-0.832206\pi$
$563$	−41.0132	−1.72850	−0.864250	+	0.503063i	$0.832206\pi$
$564$	0	0
$565$	0	0
$566$	−24.2078	−1.01753
$567$	0	0
$568$	−44.5358	−1.86868
$569$	29.0619	1.21834	0.609169	−	0.793040i	$-0.291503\pi$
$569$	29.0619	1.21834	0.609169	+	0.793040i	$0.291503\pi$
$570$	0	0
$571$	−0.230751	−0.00965665	−0.00482832	−	0.999988i	$-0.501537\pi$
$571$	−0.230751	−0.00965665	−0.00482832	+	0.999988i	$0.501537\pi$
$572$	0.879602	0.0367780
$573$	0	0
$574$	−5.14768	−0.214860
$575$	0	0
$576$	0	0
$577$	37.4944	1.56091	0.780456	−	0.625211i	$-0.214987\pi$
$577$	37.4944	1.56091	0.780456	+	0.625211i	$0.214987\pi$
$578$	15.6645	0.651557
$579$	0	0
$580$	0	0
$581$	13.7178	0.569110
$582$	0	0
$583$	8.91215	0.369103
$584$	38.8851	1.60908
$585$	0	0
$586$	−0.393329	−0.0162483
$587$	−15.3719	−0.634464	−0.317232	−	0.948348i	$-0.602753\pi$
$587$	−15.3719	−0.634464	−0.317232	+	0.948348i	$0.602753\pi$
$588$	0	0
$589$	8.31868	0.342765
$590$	0	0
$591$	0	0
$592$	−27.6023	−1.13445
$593$	8.30710	0.341132	0.170566	−	0.985346i	$-0.445440\pi$
$593$	8.30710	0.341132	0.170566	+	0.985346i	$0.445440\pi$
$594$	0	0
$595$	0	0
$596$	−4.51998	−0.185145
$597$	0	0
$598$	16.5155	0.675369
$599$	29.0293	1.18610	0.593052	−	0.805164i	$-0.297923\pi$
$599$	29.0293	1.18610	0.593052	+	0.805164i	$0.297923\pi$
$600$	0	0
$601$	−43.2988	−1.76620	−0.883098	−	0.469188i	$-0.844547\pi$
$601$	−43.2988	−1.76620	−0.883098	+	0.469188i	$0.844547\pi$
$602$	−5.86210	−0.238921
$603$	0	0
$604$	−2.48607	−0.101157
$605$	0	0
$606$	0	0
$607$	−14.4484	−0.586443	−0.293222	−	0.956044i	$-0.594727\pi$
$607$	−14.4484	−0.586443	−0.293222	+	0.956044i	$0.594727\pi$
$608$	−8.96194	−0.363455
$609$	0	0
$610$	0	0
$611$	−33.9913	−1.37514
$612$	0	0
$613$	8.63973	0.348955	0.174478	−	0.984661i	$-0.444176\pi$
$613$	8.63973	0.348955	0.174478	+	0.984661i	$0.444176\pi$
$614$	1.79877	0.0725926
$615$	0	0
$616$	−4.40898	−0.177643
$617$	−32.9268	−1.32559	−0.662793	−	0.748803i	$-0.730629\pi$
$617$	−32.9268	−1.32559	−0.662793	+	0.748803i	$0.730629\pi$
$618$	0	0
$619$	9.82425	0.394870	0.197435	−	0.980316i	$-0.436739\pi$
$619$	9.82425	0.394870	0.197435	+	0.980316i	$0.436739\pi$
$620$	0	0
$621$	0	0
$622$	−26.7075	−1.07087
$623$	−5.77925	−0.231541
$624$	0	0
$625$	0	0
$626$	−18.9292	−0.756561
$627$	0	0
$628$	−2.53193	−0.101035
$629$	41.7969	1.66655
$630$	0	0
$631$	2.88051	0.114671	0.0573356	−	0.998355i	$-0.481739\pi$
$631$	2.88051	0.114671	0.0573356	+	0.998355i	$0.481739\pi$
$632$	34.0432	1.35417
$633$	0	0
$634$	23.5396	0.934878
$635$	0	0
$636$	0	0
$637$	−19.3600	−0.767072
$638$	−5.01646	−0.198603
$639$	0	0
$640$	0	0
$641$	18.6464	0.736490	0.368245	−	0.929729i	$-0.379959\pi$
$641$	18.6464	0.736490	0.368245	+	0.929729i	$0.379959\pi$
$642$	0	0
$643$	−1.96215	−0.0773795	−0.0386897	−	0.999251i	$-0.512318\pi$
$643$	−1.96215	−0.0773795	−0.0386897	+	0.999251i	$0.512318\pi$
$644$	0.914501	0.0360364
$645$	0	0
$646$	−54.5673	−2.14692
$647$	−1.72726	−0.0679055	−0.0339528	−	0.999423i	$-0.510810\pi$
$647$	−1.72726	−0.0679055	−0.0339528	+	0.999423i	$0.510810\pi$
$648$	0	0
$649$	14.8582	0.583237
$650$	0	0
$651$	0	0
$652$	1.72872	0.0677018
$653$	8.37928	0.327907	0.163953	−	0.986468i	$-0.447575\pi$
$653$	8.37928	0.327907	0.163953	+	0.986468i	$0.447575\pi$
$654$	0	0
$655$	0	0
$656$	−10.6345	−0.415208
$657$	0	0
$658$	16.1241	0.628585
$659$	−31.2624	−1.21781	−0.608905	−	0.793243i	$-0.708391\pi$
$659$	−31.2624	−1.21781	−0.608905	+	0.793243i	$0.708391\pi$
$660$	0	0
$661$	−10.8798	−0.423175	−0.211588	−	0.977359i	$-0.567863\pi$
$661$	−10.8798	−0.423175	−0.211588	+	0.977359i	$0.567863\pi$
$662$	10.7152	0.416459
$663$	0	0
$664$	31.6883	1.22974
$665$	0	0
$666$	0	0
$667$	10.9948	0.425719
$668$	−1.30987	−0.0506806
$669$	0	0
$670$	0	0
$671$	11.8290	0.456653
$672$	0	0
$673$	−28.9803	−1.11711	−0.558554	−	0.829468i	$-0.688644\pi$
$673$	−28.9803	−1.11711	−0.558554	+	0.829468i	$0.688644\pi$
$674$	−14.5166	−0.559160
$675$	0	0
$676$	−0.00743080	−0.000285800 0
$677$	−5.89919	−0.226724	−0.113362	−	0.993554i	$-0.536162\pi$
$677$	−5.89919	−0.226724	−0.113362	+	0.993554i	$0.536162\pi$
$678$	0	0
$679$	23.0532	0.884699
$680$	0	0
$681$	0	0
$682$	1.70467	0.0652752
$683$	45.9083	1.75663	0.878316	−	0.478081i	$-0.158668\pi$
$683$	45.9083	1.75663	0.878316	+	0.478081i	$0.158668\pi$
$684$	0	0
$685$	0	0
$686$	21.1723	0.808364
$687$	0	0
$688$	−12.1104	−0.461706
$689$	−27.6115	−1.05191
$690$	0	0
$691$	20.9475	0.796880	0.398440	−	0.917194i	$-0.369552\pi$
$691$	20.9475	0.796880	0.398440	+	0.917194i	$0.369552\pi$
$692$	−1.39150	−0.0528970
$693$	0	0
$694$	2.49122	0.0945653
$695$	0	0
$696$	0	0
$697$	16.1034	0.609960
$698$	32.7370	1.23911
$699$	0	0
$700$	0	0
$701$	−52.2223	−1.97241	−0.986204	−	0.165533i	$-0.947065\pi$
$701$	−52.2223	−1.97241	−0.986204	+	0.165533i	$0.947065\pi$
$702$	0	0
$703$	−59.3713	−2.23923
$704$	−10.0829	−0.380015
$705$	0	0
$706$	−50.0002	−1.88178
$707$	−8.37783	−0.315081
$708$	0	0
$709$	−46.9974	−1.76503	−0.882513	−	0.470288i	$-0.844150\pi$
$709$	−46.9974	−1.76503	−0.882513	+	0.470288i	$0.844150\pi$
$710$	0	0
$711$	0	0
$712$	−13.3502	−0.500318
$713$	−3.73619	−0.139921
$714$	0	0
$715$	0	0
$716$	4.53976	0.169659
$717$	0	0
$718$	33.9062	1.26537
$719$	4.69037	0.174921	0.0874607	−	0.996168i	$-0.472125\pi$
$719$	4.69037	0.174921	0.0874607	+	0.996168i	$0.472125\pi$
$720$	0	0
$721$	14.2807	0.531841
$722$	52.0843	1.93838
$723$	0	0
$724$	−2.70675	−0.100596
$725$	0	0
$726$	0	0
$727$	−27.7594	−1.02954	−0.514769	−	0.857329i	$-0.672122\pi$
$727$	−27.7594	−1.02954	−0.514769	+	0.857329i	$0.672122\pi$
$728$	13.6598	0.506267
$729$	0	0
$730$	0	0
$731$	18.3383	0.678267
$732$	0	0
$733$	30.6040	1.13038	0.565192	−	0.824960i	$-0.308802\pi$
$733$	30.6040	1.13038	0.565192	+	0.824960i	$0.308802\pi$
$734$	−31.7244	−1.17097
$735$	0	0
$736$	4.02510	0.148367
$737$	10.2423	0.377281
$738$	0	0
$739$	4.93383	0.181494	0.0907469	−	0.995874i	$-0.471075\pi$
$739$	4.93383	0.181494	0.0907469	+	0.995874i	$0.471075\pi$
$740$	0	0
$741$	0	0
$742$	13.0978	0.480836
$743$	21.7109	0.796496	0.398248	−	0.917278i	$-0.369618\pi$
$743$	21.7109	0.796496	0.398248	+	0.917278i	$0.369618\pi$
$744$	0	0
$745$	0	0
$746$	3.92438	0.143682
$747$	0	0
$748$	1.30527	0.0477255
$749$	4.76947	0.174273
$750$	0	0
$751$	−33.6080	−1.22637	−0.613186	−	0.789939i	$-0.710112\pi$
$751$	−33.6080	−1.22637	−0.613186	+	0.789939i	$0.710112\pi$
$752$	33.3106	1.21471
$753$	0	0
$754$	15.5419	0.566003
$755$	0	0
$756$	0	0
$757$	4.88834	0.177670	0.0888349	−	0.996046i	$-0.471686\pi$
$757$	4.88834	0.177670	0.0888349	+	0.996046i	$0.471686\pi$
$758$	−23.7339	−0.862054
$759$	0	0
$760$	0	0
$761$	30.2366	1.09608	0.548038	−	0.836453i	$-0.315375\pi$
$761$	30.2366	1.09608	0.548038	+	0.836453i	$0.315375\pi$
$762$	0	0
$763$	8.95243	0.324100
$764$	3.29647	0.119262
$765$	0	0
$766$	−5.89536	−0.213008
$767$	−46.0336	−1.66218
$768$	0	0
$769$	51.0523	1.84099	0.920497	−	0.390751i	$-0.127785\pi$
$769$	51.0523	1.84099	0.920497	+	0.390751i	$0.127785\pi$
$770$	0	0
$771$	0	0
$772$	1.44984	0.0521808
$773$	−6.54233	−0.235311	−0.117656	−	0.993054i	$-0.537538\pi$
$773$	−6.54233	−0.235311	−0.117656	+	0.993054i	$0.537538\pi$
$774$	0	0
$775$	0	0
$776$	53.2532	1.91168
$777$	0	0
$778$	41.9163	1.50277
$779$	−22.8744	−0.819561
$780$	0	0
$781$	−17.5557	−0.628193
$782$	24.5080	0.876403
$783$	0	0
$784$	18.9724	0.677585
$785$	0	0
$786$	0	0
$787$	−10.9765	−0.391269	−0.195635	−	0.980677i	$-0.562677\pi$
$787$	−10.9765	−0.391269	−0.195635	+	0.980677i	$0.562677\pi$
$788$	1.55527	0.0554044
$789$	0	0
$790$	0	0
$791$	7.16491	0.254755
$792$	0	0
$793$	−36.6484	−1.30142
$794$	21.9850	0.780220
$795$	0	0
$796$	2.97693	0.105514
$797$	12.6891	0.449473	0.224736	−	0.974420i	$-0.427848\pi$
$797$	12.6891	0.449473	0.224736	+	0.974420i	$0.427848\pi$
$798$	0	0
$799$	−50.4409	−1.78447
$800$	0	0
$801$	0	0
$802$	20.3166	0.717404
$803$	15.3283	0.540923
$804$	0	0
$805$	0	0
$806$	−5.28139	−0.186029
$807$	0	0
$808$	−19.3529	−0.680833
$809$	17.7818	0.625174	0.312587	−	0.949889i	$-0.398805\pi$
$809$	17.7818	0.625174	0.312587	+	0.949889i	$0.398805\pi$
$810$	0	0
$811$	37.0811	1.30210	0.651048	−	0.759037i	$-0.274330\pi$
$811$	37.0811	1.30210	0.651048	+	0.759037i	$0.274330\pi$
$812$	0.860590	0.0302008
$813$	0	0
$814$	−12.1664	−0.426433
$815$	0	0
$816$	0	0
$817$	−26.0490	−0.911340
$818$	−19.8627	−0.694482
$819$	0	0
$820$	0	0
$821$	0.337853	0.0117911	0.00589557	−	0.999983i	$-0.498123\pi$
$821$	0.337853	0.0117911	0.00589557	+	0.999983i	$0.498123\pi$
$822$	0	0
$823$	−26.6629	−0.929409	−0.464704	−	0.885466i	$-0.653839\pi$
$823$	−26.6629	−0.929409	−0.464704	+	0.885466i	$0.653839\pi$
$824$	32.9886	1.14921
$825$	0	0
$826$	21.8365	0.759791
$827$	−43.4655	−1.51144	−0.755722	−	0.654893i	$-0.772714\pi$
$827$	−43.4655	−1.51144	−0.755722	+	0.654893i	$0.772714\pi$
$828$	0	0
$829$	18.9493	0.658137	0.329068	−	0.944306i	$-0.393265\pi$
$829$	18.9493	0.658137	0.329068	+	0.944306i	$0.393265\pi$
$830$	0	0
$831$	0	0
$832$	31.2388	1.08301
$833$	−28.7290	−0.995402
$834$	0	0
$835$	0	0
$836$	−1.85410	−0.0641255
$837$	0	0
$838$	47.1919	1.63022
$839$	−22.7357	−0.784922	−0.392461	−	0.919769i	$-0.628376\pi$
$839$	−22.7357	−0.784922	−0.392461	+	0.919769i	$0.628376\pi$
$840$	0	0
$841$	−18.6534	−0.643221
$842$	−15.0992	−0.520355
$843$	0	0
$844$	−1.44159	−0.0496217
$845$	0	0
$846$	0	0
$847$	12.3395	0.423991
$848$	27.0586	0.929196
$849$	0	0
$850$	0	0
$851$	26.6656	0.914085
$852$	0	0
$853$	35.9922	1.23235	0.616175	−	0.787610i	$-0.288682\pi$
$853$	35.9922	1.23235	0.616175	+	0.787610i	$0.288682\pi$
$854$	17.3846	0.594888
$855$	0	0
$856$	11.0175	0.376572
$857$	−20.1724	−0.689076	−0.344538	−	0.938772i	$-0.611964\pi$
$857$	−20.1724	−0.689076	−0.344538	+	0.938772i	$0.611964\pi$
$858$	0	0
$859$	39.0077	1.33093	0.665464	−	0.746430i	$-0.268234\pi$
$859$	39.0077	1.33093	0.665464	+	0.746430i	$0.268234\pi$
$860$	0	0
$861$	0	0
$862$	−34.5605	−1.17713
$863$	24.4962	0.833861	0.416930	−	0.908938i	$-0.363106\pi$
$863$	24.4962	0.833861	0.416930	+	0.908938i	$0.363106\pi$
$864$	0	0
$865$	0	0
$866$	13.3813	0.454715
$867$	0	0
$868$	−0.292442	−0.00992613
$869$	13.4196	0.455230
$870$	0	0
$871$	−31.7326	−1.07522
$872$	20.6803	0.700322
$873$	0	0
$874$	−34.8128	−1.17756
$875$	0	0
$876$	0	0
$877$	−29.2278	−0.986952	−0.493476	−	0.869759i	$-0.664274\pi$
$877$	−29.2278	−0.986952	−0.493476	+	0.869759i	$0.664274\pi$
$878$	12.3172	0.415684
$879$	0	0
$880$	0	0
$881$	3.72194	0.125395	0.0626977	−	0.998033i	$-0.480030\pi$
$881$	3.72194	0.125395	0.0626977	+	0.998033i	$0.480030\pi$
$882$	0	0
$883$	7.01518	0.236080	0.118040	−	0.993009i	$-0.462339\pi$
$883$	7.01518	0.236080	0.118040	+	0.993009i	$0.462339\pi$
$884$	−4.04398	−0.136014
$885$	0	0
$886$	−12.2155	−0.410388
$887$	27.4537	0.921806	0.460903	−	0.887451i	$-0.347526\pi$
$887$	27.4537	0.921806	0.460903	+	0.887451i	$0.347526\pi$
$888$	0	0
$889$	−5.61717	−0.188394
$890$	0	0
$891$	0	0
$892$	2.80101	0.0937849
$893$	71.6498	2.39767
$894$	0	0
$895$	0	0
$896$	−11.8044	−0.394358
$897$	0	0
$898$	−47.4857	−1.58462
$899$	−3.51594	−0.117263
$900$	0	0
$901$	−40.9737	−1.36503
$902$	−4.68744	−0.156075
$903$	0	0
$904$	16.5511	0.550480
$905$	0	0
$906$	0	0
$907$	6.09167	0.202271	0.101135	−	0.994873i	$-0.467752\pi$
$907$	6.09167	0.202271	0.101135	+	0.994873i	$0.467752\pi$
$908$	0.177663	0.00589595
$909$	0	0
$910$	0	0
$911$	2.77586	0.0919682	0.0459841	−	0.998942i	$-0.485358\pi$
$911$	2.77586	0.0919682	0.0459841	+	0.998942i	$0.485358\pi$
$912$	0	0
$913$	12.4913	0.413402
$914$	−25.0945	−0.830053
$915$	0	0
$916$	1.83878	0.0607551
$917$	−6.38919	−0.210990
$918$	0	0
$919$	−23.7566	−0.783659	−0.391829	−	0.920038i	$-0.628158\pi$
$919$	−23.7566	−0.783659	−0.391829	+	0.920038i	$0.628158\pi$
$920$	0	0
$921$	0	0
$922$	21.5296	0.709039
$923$	54.3909	1.79030
$924$	0	0
$925$	0	0
$926$	28.2427	0.928112
$927$	0	0
$928$	3.78782	0.124341
$929$	−28.7158	−0.942136	−0.471068	−	0.882097i	$-0.656131\pi$
$929$	−28.7158	−0.942136	−0.471068	+	0.882097i	$0.656131\pi$
$930$	0	0
$931$	40.8088	1.33745
$932$	−4.43395	−0.145239
$933$	0	0
$934$	24.1050	0.788739
$935$	0	0
$936$	0	0
$937$	13.1368	0.429161	0.214580	−	0.976706i	$-0.431162\pi$
$937$	13.1368	0.429161	0.214580	+	0.976706i	$0.431162\pi$
$938$	15.0527	0.491489
$939$	0	0
$940$	0	0
$941$	39.4446	1.28586	0.642929	−	0.765926i	$-0.277719\pi$
$941$	39.4446	1.28586	0.642929	+	0.765926i	$0.277719\pi$
$942$	0	0
$943$	10.2736	0.334556
$944$	45.1118	1.46826
$945$	0	0
$946$	−5.33799	−0.173553
$947$	25.1266	0.816504	0.408252	−	0.912869i	$-0.366138\pi$
$947$	25.1266	0.816504	0.408252	+	0.912869i	$0.366138\pi$
$948$	0	0
$949$	−47.4898	−1.54158
$950$	0	0
$951$	0	0
$952$	20.2703	0.656964
$953$	−6.14850	−0.199169	−0.0995847	−	0.995029i	$-0.531751\pi$
$953$	−6.14850	−0.199169	−0.0995847	+	0.995029i	$0.531751\pi$
$954$	0	0
$955$	0	0
$956$	−3.51158	−0.113573
$957$	0	0
$958$	−9.78823	−0.316243
$959$	−15.6204	−0.504407
$960$	0	0
$961$	−29.8052	−0.961459
$962$	37.6938	1.21530
$963$	0	0
$964$	−1.98993	−0.0640913
$965$	0	0
$966$	0	0
$967$	49.4461	1.59008	0.795040	−	0.606557i	$-0.207450\pi$
$967$	49.4461	1.59008	0.795040	+	0.606557i	$0.207450\pi$
$968$	28.5045	0.916168
$969$	0	0
$970$	0	0
$971$	−51.7332	−1.66020	−0.830099	−	0.557616i	$-0.811716\pi$
$971$	−51.7332	−1.66020	−0.830099	+	0.557616i	$0.811716\pi$
$972$	0	0
$973$	6.66272	0.213597
$974$	−15.9832	−0.512136
$975$	0	0
$976$	35.9146	1.14960
$977$	−36.3167	−1.16187	−0.580937	−	0.813948i	$-0.697314\pi$
$977$	−36.3167	−1.16187	−0.580937	+	0.813948i	$0.697314\pi$
$978$	0	0
$979$	−5.26254	−0.168192
$980$	0	0
$981$	0	0
$982$	−14.9427	−0.476841
$983$	52.0481	1.66008	0.830039	−	0.557706i	$-0.188318\pi$
$983$	52.0481	1.66008	0.830039	+	0.557706i	$0.188318\pi$
$984$	0	0
$985$	0	0
$986$	23.0632	0.734482
$987$	0	0
$988$	5.74435	0.182752
$989$	11.6995	0.372021
$990$	0	0
$991$	−29.4843	−0.936599	−0.468300	−	0.883570i	$-0.655133\pi$
$991$	−29.4843	−0.936599	−0.468300	+	0.883570i	$0.655133\pi$
$992$	−1.28716	−0.0408674
$993$	0	0
$994$	−25.8009	−0.818356
$995$	0	0
$996$	0	0
$997$	−42.4390	−1.34406	−0.672029	−	0.740525i	$-0.734577\pi$
$997$	−42.4390	−1.34406	−0.672029	+	0.740525i	$0.734577\pi$
$998$	40.9568	1.29647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.m.1.3		4
3.2	odd	2		1875.2.a.f.1.2	✓	4
5.4	even	2		5625.2.a.j.1.2		4
15.2	even	4		1875.2.b.d.1249.3		8
15.8	even	4		1875.2.b.d.1249.6		8
15.14	odd	2		1875.2.a.g.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.2	✓	4	3.2	odd	2
1875.2.a.g.1.3	yes	4	15.14	odd	2
1875.2.b.d.1249.3		8	15.2	even	4
1875.2.b.d.1249.6		8	15.8	even	4
5625.2.a.j.1.2		4	5.4	even	2
5625.2.a.m.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.33826 q^{2} -0.209057 q^{4} -1.27977 q^{7} -2.95630 q^{8} +O(q^{10})\)	\(q+1.33826 q^{2} -0.209057 q^{4} -1.27977 q^{7} -2.95630 q^{8} -1.16535 q^{11} +3.61048 q^{13} -1.71267 q^{14} -3.53818 q^{16} +5.35772 q^{17} -7.61048 q^{19} -1.55955 q^{22} +3.41811 q^{23} +4.83176 q^{26} +0.267545 q^{28} +3.21661 q^{29} -1.09306 q^{31} +1.17758 q^{32} +7.17002 q^{34} +7.80126 q^{37} -10.1848 q^{38} +3.00565 q^{41} +3.42278 q^{43} +0.243625 q^{44} +4.57433 q^{46} -9.41462 q^{47} -5.36218 q^{49} -0.754795 q^{52} -7.64760 q^{53} +3.78339 q^{56} +4.30467 q^{58} -12.7500 q^{59} -10.1506 q^{61} -1.46280 q^{62} +8.65227 q^{64} -8.78903 q^{67} -1.12007 q^{68} +15.0647 q^{71} -13.1533 q^{73} +10.4401 q^{74} +1.59102 q^{76} +1.49139 q^{77} -11.5155 q^{79} +4.02234 q^{82} -10.7189 q^{83} +4.58058 q^{86} +3.44512 q^{88} +4.51584 q^{89} -4.62059 q^{91} -0.714580 q^{92} -12.5992 q^{94} -18.0135 q^{97} -7.17600 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8	\( 4 q + q^{2} + q^{4} - 5 q^{7} - 3 q^{8} + 6 q^{11} - 7 q^{13} + 10 q^{14} - 9 q^{16} - 7 q^{17} - 9 q^{19} - 6 q^{22} + 10 q^{23} + 2 q^{26} - 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} + 10 q^{37} - 6 q^{38} - q^{43} + 9 q^{44} + 5 q^{46} - 23 q^{47} - 3 q^{49} - 13 q^{52} + 2 q^{58} - 4 q^{59} - 43 q^{61} - 10 q^{62} - 7 q^{64} - 8 q^{67} - 3 q^{68} + 27 q^{71} - 15 q^{73} - 5 q^{74} + 9 q^{76} - 15 q^{77} + 10 q^{79} + 20 q^{82} + 3 q^{83} - 24 q^{86} + 3 q^{88} + 9 q^{89} + 5 q^{91} - 15 q^{92} - 22 q^{94} - 13 q^{97} - 42 q^{98}+O(q^{100}) \) 4 * q + q^2 + q^4 - 5 * q^7 - 3 * q^8 + 6 * q^11 - 7 * q^13 + 10 * q^14 - 9 * q^16 - 7 * q^17 - 9 * q^19 - 6 * q^22 + 10 * q^23 + 2 * q^26 - 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 + 10 * q^37 - 6 * q^38 - q^43 + 9 * q^44 + 5 * q^46 - 23 * q^47 - 3 * q^49 - 13 * q^52 + 2 * q^58 - 4 * q^59 - 43 * q^61 - 10 * q^62 - 7 * q^64 - 8 * q^67 - 3 * q^68 + 27 * q^71 - 15 * q^73 - 5 * q^74 + 9 * q^76 - 15 * q^77 + 10 * q^79 + 20 * q^82 + 3 * q^83 - 24 * q^86 + 3 * q^88 + 9 * q^89 + 5 * q^91 - 15 * q^92 - 22 * q^94 - 13 * q^97 - 42 * q^98

Introduction

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