LMFDB - Embedded newform 3025.2.a.bk.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.1480160000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 27x^{4} - 31x^{2} + 11 $ x^8 - 9x^6 + 27x^4 - 31*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.23399$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23399 q^{2} +0.363982 q^{3} -0.477260 q^{4} +0.449152 q^{6} +2.58558 q^{7} -3.05692 q^{8} -2.86752 q^{9} +O(q^{10})$	\(q+1.23399 q^{2} +0.363982 q^{3} -0.477260 q^{4} +0.449152 q^{6} +2.58558 q^{7} -3.05692 q^{8} -2.86752 q^{9} -0.173714 q^{12} +2.75929 q^{13} +3.19059 q^{14} -2.81770 q^{16} -3.85124 q^{17} -3.53850 q^{18} +0.277591 q^{19} +0.941105 q^{21} -8.40180 q^{23} -1.11267 q^{24} +3.40495 q^{26} -2.13567 q^{27} -1.23399 q^{28} -3.32307 q^{29} +0.564263 q^{31} +2.63682 q^{32} -4.75241 q^{34} +1.36855 q^{36} -0.522583 q^{37} +0.342546 q^{38} +1.00433 q^{39} +5.11188 q^{41} +1.16132 q^{42} -2.54457 q^{43} -10.3678 q^{46} -4.92477 q^{47} -1.02559 q^{48} -0.314780 q^{49} -1.40178 q^{51} -1.31690 q^{52} -8.72086 q^{53} -2.63541 q^{54} -7.90392 q^{56} +0.101038 q^{57} -4.10065 q^{58} +7.50726 q^{59} -14.1791 q^{61} +0.696297 q^{62} -7.41419 q^{63} +8.88922 q^{64} +3.20618 q^{67} +1.83804 q^{68} -3.05810 q^{69} -8.40099 q^{71} +8.76578 q^{72} +13.0353 q^{73} -0.644864 q^{74} -0.132483 q^{76} +1.23934 q^{78} -9.70425 q^{79} +7.82520 q^{81} +6.30802 q^{82} -3.29699 q^{83} -0.449152 q^{84} -3.13998 q^{86} -1.20954 q^{87} +2.48823 q^{89} +7.13437 q^{91} +4.00984 q^{92} +0.205382 q^{93} -6.07713 q^{94} +0.959755 q^{96} -10.9014 q^{97} -0.388437 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10}) $ 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9	$ 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} - 12 q^{19} - 4 q^{21} - 2 q^{24} + 10 q^{26} - 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} - 34 q^{41} - 24 q^{46} - 30 q^{49} - 54 q^{51} - 20 q^{54} - 10 q^{56} - 6 q^{59} - 20 q^{61} + 14 q^{64} + 32 q^{69} - 42 q^{71} + 4 q^{74} - 28 q^{76} - 16 q^{79} - 36 q^{81} + 6 q^{84} + 46 q^{86} - 12 q^{89} + 20 q^{91} + 42 q^{94} + 8 q^{96}+O(q^{100}) $ 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9 - 4 * q^14 - 22 * q^16 - 12 * q^19 - 4 * q^21 - 2 * q^24 + 10 * q^26 - 24 * q^29 + 14 * q^31 + 8 * q^34 + 20 * q^36 - 30 * q^39 - 34 * q^41 - 24 * q^46 - 30 * q^49 - 54 * q^51 - 20 * q^54 - 10 * q^56 - 6 * q^59 - 20 * q^61 + 14 * q^64 + 32 * q^69 - 42 * q^71 + 4 * q^74 - 28 * q^76 - 16 * q^79 - 36 * q^81 + 6 * q^84 + 46 * q^86 - 12 * q^89 + 20 * q^91 + 42 * q^94 + 8 * q^96

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 $

$2$	1.23399	0.872565	0.436283	−	0.899810i	$-0.356295\pi$
$2$	1.23399	0.872565	0.436283	+	0.899810i	$0.356295\pi$
$3$	0.363982	0.210145	0.105073	−	0.994465i	$-0.466492\pi$
$3$	0.363982	0.210145	0.105073	+	0.994465i	$0.466492\pi$
$4$	−0.477260	−0.238630
$5$	0	0
$5$	0	0
$6$	0.449152	0.183365
$7$	2.58558	0.977257	0.488629	−	0.872492i	$-0.337497\pi$
$7$	2.58558	0.977257	0.488629	+	0.872492i	$0.337497\pi$
$8$	−3.05692	−1.08079
$9$	−2.86752	−0.955839
$10$	0	0
$11$	0	0
$11$	0	0
$12$	−0.173714	−0.0501470
$13$	2.75929	0.765290	0.382645	−	0.923895i	$-0.375013\pi$
$13$	2.75929	0.765290	0.382645	+	0.923895i	$0.375013\pi$
$14$	3.19059	0.852721
$15$	0	0
$16$	−2.81770	−0.704426
$17$	−3.85124	−0.934063	−0.467031	−	0.884241i	$-0.654677\pi$
$17$	−3.85124	−0.934063	−0.467031	+	0.884241i	$0.654677\pi$
$18$	−3.53850	−0.834032
$19$	0.277591	0.0636838	0.0318419	−	0.999493i	$-0.489863\pi$
$19$	0.277591	0.0636838	0.0318419	+	0.999493i	$0.489863\pi$
$20$	0	0
$21$	0.941105	0.205366
$22$	0	0
$23$	−8.40180	−1.75190	−0.875948	−	0.482406i	$-0.839763\pi$
$23$	−8.40180	−1.75190	−0.875948	+	0.482406i	$0.839763\pi$
$24$	−1.11267	−0.227122
$25$	0	0
$26$	3.40495	0.667766
$27$	−2.13567	−0.411010
$28$	−1.23399	−0.233203
$29$	−3.32307	−0.617079	−0.308539	−	0.951212i	$-0.599840\pi$
$29$	−3.32307	−0.617079	−0.308539	+	0.951212i	$0.599840\pi$
$30$	0	0
$31$	0.564263	0.101345	0.0506723	−	0.998715i	$-0.483864\pi$
$31$	0.564263	0.101345	0.0506723	+	0.998715i	$0.483864\pi$
$32$	2.63682	0.466128
$33$	0	0
$34$	−4.75241	−0.815031
$35$	0	0
$36$	1.36855	0.228092
$37$	−0.522583	−0.0859121	−0.0429560	−	0.999077i	$-0.513678\pi$
$37$	−0.522583	−0.0859121	−0.0429560	+	0.999077i	$0.513678\pi$
$38$	0.342546	0.0555682
$39$	1.00433	0.160822
$40$	0	0
$41$	5.11188	0.798341	0.399170	−	0.916877i	$-0.369298\pi$
$41$	5.11188	0.798341	0.399170	+	0.916877i	$0.369298\pi$
$42$	1.16132	0.179195
$43$	−2.54457	−0.388043	−0.194022	−	0.980997i	$-0.562153\pi$
$43$	−2.54457	−0.388043	−0.194022	+	0.980997i	$0.562153\pi$
$44$	0	0
$45$	0	0
$46$	−10.3678	−1.52864
$47$	−4.92477	−0.718351	−0.359176	−	0.933270i	$-0.616942\pi$
$47$	−4.92477	−0.718351	−0.359176	+	0.933270i	$0.616942\pi$
$48$	−1.02559	−0.148032
$49$	−0.314780	−0.0449686
$50$	0	0
$51$	−1.40178	−0.196289
$52$	−1.31690	−0.182621
$53$	−8.72086	−1.19790	−0.598951	−	0.800786i	$-0.704416\pi$
$53$	−8.72086	−1.19790	−0.598951	+	0.800786i	$0.704416\pi$
$54$	−2.63541	−0.358633
$55$	0	0
$56$	−7.90392	−1.05621
$57$	0.101038	0.0133828
$58$	−4.10065	−0.538441
$59$	7.50726	0.977362	0.488681	−	0.872463i	$-0.337478\pi$
$59$	7.50726	0.977362	0.488681	+	0.872463i	$0.337478\pi$
$60$	0	0
$61$	−14.1791	−1.81544	−0.907721	−	0.419573i	$-0.862180\pi$
$61$	−14.1791	−1.81544	−0.907721	+	0.419573i	$0.862180\pi$
$62$	0.696297	0.0884298
$63$	−7.41419	−0.934100
$64$	8.88922	1.11115
$65$	0	0
$66$	0	0
$67$	3.20618	0.391698	0.195849	−	0.980634i	$-0.437254\pi$
$67$	3.20618	0.391698	0.195849	+	0.980634i	$0.437254\pi$
$68$	1.83804	0.222895
$69$	−3.05810	−0.368153
$70$	0	0
$71$	−8.40099	−0.997014	−0.498507	−	0.866886i	$-0.666118\pi$
$71$	−8.40099	−0.997014	−0.498507	+	0.866886i	$0.666118\pi$
$72$	8.76578	1.03306
$73$	13.0353	1.52566	0.762831	−	0.646598i	$-0.223809\pi$
$73$	13.0353	1.52566	0.762831	+	0.646598i	$0.223809\pi$
$74$	−0.644864	−0.0749639
$75$	0	0
$76$	−0.132483	−0.0151969
$77$	0	0
$78$	1.23934	0.140328
$79$	−9.70425	−1.09181	−0.545907	−	0.837846i	$-0.683815\pi$
$79$	−9.70425	−1.09181	−0.545907	+	0.837846i	$0.683815\pi$
$80$	0	0
$81$	7.82520	0.869467
$82$	6.30802	0.696604
$83$	−3.29699	−0.361892	−0.180946	−	0.983493i	$-0.557916\pi$
$83$	−3.29699	−0.361892	−0.180946	+	0.983493i	$0.557916\pi$
$84$	−0.449152	−0.0490065
$85$	0	0
$86$	−3.13998	−0.338593
$87$	−1.20954	−0.129676
$88$	0	0
$89$	2.48823	0.263752	0.131876	−	0.991266i	$-0.457900\pi$
$89$	2.48823	0.263752	0.131876	+	0.991266i	$0.457900\pi$
$90$	0	0
$91$	7.13437	0.747885
$92$	4.00984	0.418055
$93$	0.205382	0.0212971
$94$	−6.07713	−0.626808
$95$	0	0
$96$	0.959755	0.0979546
$97$	−10.9014	−1.10687	−0.553437	−	0.832891i	$-0.686684\pi$
$97$	−10.9014	−1.10687	−0.553437	+	0.832891i	$0.686684\pi$
$98$	−0.388437	−0.0392380
$99$	0	0
$100$	0	0
$101$	−13.8382	−1.37696	−0.688478	−	0.725257i	$-0.741721\pi$
$101$	−13.8382	−1.37696	−0.688478	+	0.725257i	$0.741721\pi$
$102$	−1.72979	−0.171275
$103$	13.3899	1.31935	0.659673	−	0.751553i	$-0.270695\pi$
$103$	13.3899	1.31935	0.659673	+	0.751553i	$0.270695\pi$
$104$	−8.43495	−0.827115
$105$	0	0
$106$	−10.7615	−1.04525
$107$	−10.2594	−0.991815	−0.495907	−	0.868375i	$-0.665164\pi$
$107$	−10.2594	−0.991815	−0.495907	+	0.868375i	$0.665164\pi$
$108$	1.01927	0.0980794
$109$	4.94262	0.473417	0.236708	−	0.971581i	$-0.423931\pi$
$109$	4.94262	0.473417	0.236708	+	0.971581i	$0.423931\pi$
$110$	0	0
$111$	−0.190211	−0.0180540
$112$	−7.28539	−0.688405
$113$	−9.92408	−0.933579	−0.466789	−	0.884369i	$-0.654589\pi$
$113$	−9.92408	−0.933579	−0.466789	+	0.884369i	$0.654589\pi$
$114$	0.124681	0.0116774
$115$	0	0
$116$	1.58597	0.147254
$117$	−7.91232	−0.731494
$118$	9.26391	0.852812
$119$	−9.95769	−0.912820
$120$	0	0
$121$	0	0
$122$	−17.4969	−1.58409
$123$	1.86063	0.167768
$124$	−0.269300	−0.0241839
$125$	0	0
$126$	−9.14906	−0.815063
$127$	−11.1357	−0.988138	−0.494069	−	0.869423i	$-0.664491\pi$
$127$	−11.1357	−0.988138	−0.494069	+	0.869423i	$0.664491\pi$
$128$	5.69561	0.503425
$129$	−0.926179	−0.0815455
$130$	0	0
$131$	−10.1649	−0.888114	−0.444057	−	0.895999i	$-0.646461\pi$
$131$	−10.1649	−0.888114	−0.444057	+	0.895999i	$0.646461\pi$
$132$	0	0
$133$	0.717734	0.0622354
$134$	3.95641	0.341782
$135$	0	0
$136$	11.7729	1.00952
$137$	−4.34606	−0.371309	−0.185655	−	0.982615i	$-0.559441\pi$
$137$	−4.34606	−0.371309	−0.185655	+	0.982615i	$0.559441\pi$
$138$	−3.77368	−0.321237
$139$	8.30246	0.704206	0.352103	−	0.935961i	$-0.385467\pi$
$139$	8.30246	0.704206	0.352103	+	0.935961i	$0.385467\pi$
$140$	0	0
$141$	−1.79253	−0.150958
$142$	−10.3668	−0.869960
$143$	0	0
$144$	8.07981	0.673318
$145$	0	0
$146$	16.0854	1.33124
$147$	−0.114574	−0.00944994
$148$	0.249408	0.0205012
$149$	−8.46690	−0.693636	−0.346818	−	0.937933i	$-0.612738\pi$
$149$	−8.46690	−0.693636	−0.346818	+	0.937933i	$0.612738\pi$
$150$	0	0
$151$	−11.1023	−0.903493	−0.451746	−	0.892146i	$-0.649199\pi$
$151$	−11.1023	−0.903493	−0.451746	+	0.892146i	$0.649199\pi$
$152$	−0.848575	−0.0688285
$153$	11.0435	0.892814
$154$	0	0
$155$	0	0
$156$	−0.479328	−0.0383770
$157$	13.7662	1.09867	0.549333	−	0.835604i	$-0.314882\pi$
$157$	13.7662	1.09867	0.549333	+	0.835604i	$0.314882\pi$
$158$	−11.9750	−0.952678
$159$	−3.17424	−0.251734
$160$	0	0
$161$	−21.7235	−1.71205
$162$	9.65625	0.758667
$163$	8.94093	0.700308	0.350154	−	0.936692i	$-0.386129\pi$
$163$	8.94093	0.700308	0.350154	+	0.936692i	$0.386129\pi$
$164$	−2.43969	−0.190508
$165$	0	0
$166$	−4.06846	−0.315774
$167$	16.9182	1.30917	0.654583	−	0.755990i	$-0.272844\pi$
$167$	16.9182	1.30917	0.654583	+	0.755990i	$0.272844\pi$
$168$	−2.87689	−0.221957
$169$	−5.38630	−0.414331
$170$	0	0
$171$	−0.795997	−0.0608714
$172$	1.21442	0.0925988
$173$	−3.37990	−0.256969	−0.128484	−	0.991712i	$-0.541011\pi$
$173$	−3.37990	−0.256969	−0.128484	+	0.991712i	$0.541011\pi$
$174$	−1.49256	−0.113151
$175$	0	0
$176$	0	0
$177$	2.73251	0.205388
$178$	3.07046	0.230141
$179$	13.8239	1.03325	0.516624	−	0.856212i	$-0.327188\pi$
$179$	13.8239	1.03325	0.516624	+	0.856212i	$0.327188\pi$
$180$	0	0
$181$	−2.59901	−0.193183	−0.0965913	−	0.995324i	$-0.530794\pi$
$181$	−2.59901	−0.193183	−0.0965913	+	0.995324i	$0.530794\pi$
$182$	8.80377	0.652579
$183$	−5.16093	−0.381507
$184$	25.6836	1.89342
$185$	0	0
$186$	0.253440	0.0185831
$187$	0	0
$188$	2.35039	0.171420
$189$	−5.52195	−0.401663
$190$	0	0
$191$	2.21832	0.160512	0.0802560	−	0.996774i	$-0.474426\pi$
$191$	2.21832	0.160512	0.0802560	+	0.996774i	$0.474426\pi$
$192$	3.23552	0.233504
$193$	10.4765	0.754112	0.377056	−	0.926190i	$-0.376936\pi$
$193$	10.4765	0.754112	0.377056	+	0.926190i	$0.376936\pi$
$194$	−13.4523	−0.965820
$195$	0	0
$196$	0.150232	0.0107309
$197$	1.32667	0.0945210	0.0472605	−	0.998883i	$-0.484951\pi$
$197$	1.32667	0.0945210	0.0472605	+	0.998883i	$0.484951\pi$
$198$	0	0
$199$	−5.20321	−0.368846	−0.184423	−	0.982847i	$-0.559042\pi$
$199$	−5.20321	−0.368846	−0.184423	+	0.982847i	$0.559042\pi$
$200$	0	0
$201$	1.16699	0.0823134
$202$	−17.0763	−1.20148
$203$	−8.59206	−0.603045
$204$	0.669015	0.0468404
$205$	0	0
$206$	16.5230	1.15122
$207$	24.0923	1.67453
$208$	−7.77487	−0.539090
$209$	0	0
$210$	0	0
$211$	−18.9165	−1.30227	−0.651134	−	0.758963i	$-0.725706\pi$
$211$	−18.9165	−1.30227	−0.651134	+	0.758963i	$0.725706\pi$
$212$	4.16212	0.285855
$213$	−3.05781	−0.209518
$214$	−12.6600	−0.865423
$215$	0	0
$216$	6.52859	0.444214
$217$	1.45895	0.0990398
$218$	6.09916	0.413087
$219$	4.74460	0.320611
$220$	0	0
$221$	−10.6267	−0.714829
$222$	−0.234719	−0.0157533
$223$	21.8723	1.46468	0.732338	−	0.680942i	$-0.238429\pi$
$223$	21.8723	1.46468	0.732338	+	0.680942i	$0.238429\pi$
$224$	6.81770	0.455527
$225$	0	0
$226$	−12.2462	−0.814608
$227$	2.43865	0.161859	0.0809294	−	0.996720i	$-0.474211\pi$
$227$	2.43865	0.161859	0.0809294	+	0.996720i	$0.474211\pi$
$228$	−0.0482215	−0.00319355
$229$	7.30119	0.482476	0.241238	−	0.970466i	$-0.422447\pi$
$229$	7.30119	0.482476	0.241238	+	0.970466i	$0.422447\pi$
$230$	0	0
$231$	0	0
$232$	10.1584	0.666930
$233$	−4.45831	−0.292073	−0.146037	−	0.989279i	$-0.546652\pi$
$233$	−4.45831	−0.292073	−0.146037	+	0.989279i	$0.546652\pi$
$234$	−9.76375	−0.638276
$235$	0	0
$236$	−3.58291	−0.233228
$237$	−3.53217	−0.229439
$238$	−12.2877	−0.796495
$239$	−24.1747	−1.56373	−0.781867	−	0.623446i	$-0.785732\pi$
$239$	−24.1747	−1.56373	−0.781867	+	0.623446i	$0.785732\pi$
$240$	0	0
$241$	12.0393	0.775522	0.387761	−	0.921760i	$-0.373249\pi$
$241$	12.0393	0.775522	0.387761	+	0.921760i	$0.373249\pi$
$242$	0	0
$243$	9.25525	0.593725
$244$	6.76710	0.433219
$245$	0	0
$246$	2.29601	0.146388
$247$	0.765955	0.0487366
$248$	−1.72491	−0.109532
$249$	−1.20005	−0.0760498
$250$	0	0
$251$	−0.536388	−0.0338565	−0.0169283	−	0.999857i	$-0.505389\pi$
$251$	−0.536388	−0.0338565	−0.0169283	+	0.999857i	$0.505389\pi$
$252$	3.53850	0.222904
$253$	0	0
$254$	−13.7414	−0.862214
$255$	0	0
$256$	−10.7501	−0.671882
$257$	23.7199	1.47961	0.739805	−	0.672822i	$-0.234918\pi$
$257$	23.7199	1.47961	0.739805	+	0.672822i	$0.234918\pi$
$258$	−1.14290	−0.0711538
$259$	−1.35118	−0.0839582
$260$	0	0
$261$	9.52896	0.589828
$262$	−12.5435	−0.774937
$263$	4.97643	0.306860	0.153430	−	0.988160i	$-0.450968\pi$
$263$	4.97643	0.306860	0.153430	+	0.988160i	$0.450968\pi$
$264$	0	0
$265$	0	0
$266$	0.885679	0.0543044
$267$	0.905671	0.0554262
$268$	−1.53018	−0.0934708
$269$	28.8046	1.75625	0.878124	−	0.478434i	$-0.158795\pi$
$269$	28.8046	1.75625	0.878124	+	0.478434i	$0.158795\pi$
$270$	0	0
$271$	16.0762	0.976559	0.488280	−	0.872687i	$-0.337625\pi$
$271$	16.0762	0.976559	0.488280	+	0.872687i	$0.337625\pi$
$272$	10.8517	0.657978
$273$	2.59679	0.157165
$274$	−5.36301	−0.323992
$275$	0	0
$276$	1.45951	0.0878522
$277$	−18.0038	−1.08174	−0.540871	−	0.841105i	$-0.681905\pi$
$277$	−18.0038	−1.08174	−0.540871	+	0.841105i	$0.681905\pi$
$278$	10.2452	0.614465
$279$	−1.61803	−0.0968692
$280$	0	0
$281$	−13.7150	−0.818167	−0.409084	−	0.912497i	$-0.634152\pi$
$281$	−13.7150	−0.818167	−0.409084	+	0.912497i	$0.634152\pi$
$282$	−2.21197	−0.131721
$283$	20.5044	1.21886	0.609431	−	0.792839i	$-0.291398\pi$
$283$	20.5044	1.21886	0.609431	+	0.792839i	$0.291398\pi$
$284$	4.00946	0.237918
$285$	0	0
$286$	0	0
$287$	13.2172	0.780184
$288$	−7.56112	−0.445543
$289$	−2.16795	−0.127526
$290$	0	0
$291$	−3.96793	−0.232604
$292$	−6.22121	−0.364069
$293$	−22.4133	−1.30940	−0.654699	−	0.755890i	$-0.727204\pi$
$293$	−22.4133	−1.30940	−0.654699	+	0.755890i	$0.727204\pi$
$294$	−0.141384	−0.00824569
$295$	0	0
$296$	1.59750	0.0928525
$297$	0	0
$298$	−10.4481	−0.605242
$299$	−23.1830	−1.34071
$300$	0	0
$301$	−6.57919	−0.379218
$302$	−13.7002	−0.788356
$303$	−5.03688	−0.289361
$304$	−0.782169	−0.0448605
$305$	0	0
$306$	13.6276	0.779038
$307$	20.3044	1.15883	0.579416	−	0.815032i	$-0.303281\pi$
$307$	20.3044	1.15883	0.579416	+	0.815032i	$0.303281\pi$
$308$	0	0
$309$	4.87369	0.277254
$310$	0	0
$311$	−8.75881	−0.496666	−0.248333	−	0.968675i	$-0.579883\pi$
$311$	−8.75881	−0.496666	−0.248333	+	0.968675i	$0.579883\pi$
$312$	−3.07017	−0.173814
$313$	1.73123	0.0978550	0.0489275	−	0.998802i	$-0.484420\pi$
$313$	1.73123	0.0978550	0.0489275	+	0.998802i	$0.484420\pi$
$314$	16.9875	0.958657
$315$	0	0
$316$	4.63145	0.260539
$317$	17.4764	0.981573	0.490786	−	0.871280i	$-0.336710\pi$
$317$	17.4764	0.981573	0.490786	+	0.871280i	$0.336710\pi$
$318$	−3.91699	−0.219654
$319$	0	0
$320$	0	0
$321$	−3.73424	−0.208425
$322$	−26.8067	−1.49388
$323$	−1.06907	−0.0594846
$324$	−3.73466	−0.207481
$325$	0	0
$326$	11.0331	0.611064
$327$	1.79902	0.0994863
$328$	−15.6266	−0.862835
$329$	−12.7334	−0.702014
$330$	0	0
$331$	12.6193	0.693620	0.346810	−	0.937935i	$-0.387265\pi$
$331$	12.6193	0.693620	0.346810	+	0.937935i	$0.387265\pi$
$332$	1.57352	0.0863582
$333$	1.49852	0.0821181
$334$	20.8769	1.14233
$335$	0	0
$336$	−2.65175	−0.144665
$337$	11.8599	0.646047	0.323024	−	0.946391i	$-0.395301\pi$
$337$	11.8599	0.646047	0.323024	+	0.946391i	$0.395301\pi$
$338$	−6.64666	−0.361531
$339$	−3.61219	−0.196187
$340$	0	0
$341$	0	0
$342$	−0.982255	−0.0531143
$343$	−18.9129	−1.02120
$344$	7.77856	0.419392
$345$	0	0
$346$	−4.17077	−0.224222
$347$	22.3040	1.19734	0.598672	−	0.800994i	$-0.295695\pi$
$347$	22.3040	1.19734	0.598672	+	0.800994i	$0.295695\pi$
$348$	0.577265	0.0309446
$349$	−15.9679	−0.854740	−0.427370	−	0.904077i	$-0.640560\pi$
$349$	−15.9679	−0.854740	−0.427370	+	0.904077i	$0.640560\pi$
$350$	0	0
$351$	−5.89295	−0.314542
$352$	0	0
$353$	24.1406	1.28488	0.642439	−	0.766337i	$-0.277923\pi$
$353$	24.1406	1.28488	0.642439	+	0.766337i	$0.277923\pi$
$354$	3.37190	0.179214
$355$	0	0
$356$	−1.18753	−0.0629391
$357$	−3.62442	−0.191825
$358$	17.0586	0.901577
$359$	−20.3784	−1.07553	−0.537764	−	0.843095i	$-0.680731\pi$
$359$	−20.3784	−1.07553	−0.537764	+	0.843095i	$0.680731\pi$
$360$	0	0
$361$	−18.9229	−0.995944
$362$	−3.20716	−0.168564
$363$	0	0
$364$	−3.40495	−0.178468
$365$	0	0
$366$	−6.36855	−0.332889
$367$	−5.40430	−0.282102	−0.141051	−	0.990002i	$-0.545048\pi$
$367$	−5.40430	−0.282102	−0.141051	+	0.990002i	$0.545048\pi$
$368$	23.6738	1.23408
$369$	−14.6584	−0.763085
$370$	0	0
$371$	−22.5485	−1.17066
$372$	−0.0980205	−0.00508213
$373$	−17.0982	−0.885311	−0.442656	−	0.896692i	$-0.645964\pi$
$373$	−17.0982	−0.885311	−0.442656	+	0.896692i	$0.645964\pi$
$374$	0	0
$375$	0	0
$376$	15.0546	0.776384
$377$	−9.16933	−0.472244
$378$	−6.81405	−0.350477
$379$	−8.59196	−0.441339	−0.220670	−	0.975349i	$-0.570824\pi$
$379$	−8.59196	−0.441339	−0.220670	+	0.975349i	$0.570824\pi$
$380$	0	0
$381$	−4.05321	−0.207652
$382$	2.73739	0.140057
$383$	9.66958	0.494093	0.247046	−	0.969004i	$-0.420540\pi$
$383$	9.66958	0.494093	0.247046	+	0.969004i	$0.420540\pi$
$384$	2.07310	0.105792
$385$	0	0
$386$	12.9279	0.658012
$387$	7.29660	0.370907
$388$	5.20283	0.264133
$389$	−27.3693	−1.38768	−0.693841	−	0.720129i	$-0.744083\pi$
$389$	−27.3693	−1.38768	−0.693841	+	0.720129i	$0.744083\pi$
$390$	0	0
$391$	32.3573	1.63638
$392$	0.962259	0.0486014
$393$	−3.69985	−0.186633
$394$	1.63710	0.0824758
$395$	0	0
$396$	0	0
$397$	10.6518	0.534596	0.267298	−	0.963614i	$-0.413869\pi$
$397$	10.6518	0.534596	0.267298	+	0.963614i	$0.413869\pi$
$398$	−6.42073	−0.321842
$399$	0.261242	0.0130785
$400$	0	0
$401$	13.7146	0.684875	0.342437	−	0.939541i	$-0.388748\pi$
$401$	13.7146	0.684875	0.342437	+	0.939541i	$0.388748\pi$
$402$	1.44006	0.0718238
$403$	1.55697	0.0775581
$404$	6.60444	0.328583
$405$	0	0
$406$	−10.6026	−0.526196
$407$	0	0
$408$	4.28514	0.212146
$409$	22.3083	1.10308	0.551538	−	0.834150i	$-0.314041\pi$
$409$	22.3083	1.10308	0.551538	+	0.834150i	$0.314041\pi$
$410$	0	0
$411$	−1.58189	−0.0780289
$412$	−6.39046	−0.314836
$413$	19.4106	0.955134
$414$	29.7297	1.46114
$415$	0	0
$416$	7.27576	0.356723
$417$	3.02195	0.147986
$418$	0	0
$419$	−0.510725	−0.0249506	−0.0124753	−	0.999922i	$-0.503971\pi$
$419$	−0.510725	−0.0249506	−0.0124753	+	0.999922i	$0.503971\pi$
$420$	0	0
$421$	13.2150	0.644058	0.322029	−	0.946730i	$-0.395635\pi$
$421$	13.2150	0.644058	0.322029	+	0.946730i	$0.395635\pi$
$422$	−23.3429	−1.13631
$423$	14.1219	0.686628
$424$	26.6590	1.29468
$425$	0	0
$426$	−3.77332	−0.182818
$427$	−36.6611	−1.77415
$428$	4.89641	0.236677
$429$	0	0
$430$	0	0
$431$	29.4064	1.41645	0.708227	−	0.705985i	$-0.249495\pi$
$431$	29.4064	1.41645	0.708227	+	0.705985i	$0.249495\pi$
$432$	6.01769	0.289526
$433$	−10.2165	−0.490975	−0.245488	−	0.969400i	$-0.578948\pi$
$433$	−10.2165	−0.490975	−0.245488	+	0.969400i	$0.578948\pi$
$434$	1.80033	0.0864187
$435$	0	0
$436$	−2.35891	−0.112971
$437$	−2.33226	−0.111567
$438$	5.85481	0.279754
$439$	1.53306	0.0731691	0.0365846	−	0.999331i	$-0.488352\pi$
$439$	1.53306	0.0731691	0.0365846	+	0.999331i	$0.488352\pi$
$440$	0	0
$441$	0.902638	0.0429827
$442$	−13.1133	−0.623735
$443$	3.39254	0.161185	0.0805923	−	0.996747i	$-0.474319\pi$
$443$	3.39254	0.161185	0.0805923	+	0.996747i	$0.474319\pi$
$444$	0.0907801	0.00430823
$445$	0	0
$446$	26.9902	1.27802
$447$	−3.08180	−0.145764
$448$	22.9838	1.08588
$449$	−32.3501	−1.52669	−0.763347	−	0.645989i	$-0.776445\pi$
$449$	−32.3501	−1.52669	−0.763347	+	0.645989i	$0.776445\pi$
$450$	0	0
$451$	0	0
$452$	4.73637	0.222780
$453$	−4.04104	−0.189865
$454$	3.00928	0.141232
$455$	0	0
$456$	−0.308866	−0.0144640
$457$	−12.1730	−0.569427	−0.284714	−	0.958613i	$-0.591899\pi$
$457$	−12.1730	−0.569427	−0.284714	+	0.958613i	$0.591899\pi$
$458$	9.00962	0.420992
$459$	8.22499	0.383910
$460$	0	0
$461$	16.5699	0.771739	0.385869	−	0.922553i	$-0.373902\pi$
$461$	16.5699	0.771739	0.385869	+	0.922553i	$0.373902\pi$
$462$	0	0
$463$	14.6302	0.679924	0.339962	−	0.940439i	$-0.389586\pi$
$463$	14.6302	0.679924	0.339962	+	0.940439i	$0.389586\pi$
$464$	9.36343	0.434686
$465$	0	0
$466$	−5.50152	−0.254853
$467$	30.4853	1.41069	0.705347	−	0.708862i	$-0.250791\pi$
$467$	30.4853	1.41069	0.705347	+	0.708862i	$0.250791\pi$
$468$	3.77623	0.174556
$469$	8.28984	0.382789
$470$	0	0
$471$	5.01067	0.230879
$472$	−22.9491	−1.05632
$473$	0	0
$474$	−4.35868	−0.200201
$475$	0	0
$476$	4.75241	0.217826
$477$	25.0072	1.14500
$478$	−29.8315	−1.36446
$479$	−13.9993	−0.639643	−0.319822	−	0.947478i	$-0.603623\pi$
$479$	−13.9993	−0.639643	−0.319822	+	0.947478i	$0.603623\pi$
$480$	0	0
$481$	−1.44196	−0.0657477
$482$	14.8565	0.676693
$483$	−7.90697	−0.359780
$484$	0	0
$485$	0	0
$486$	11.4209	0.518064
$487$	34.1311	1.54663	0.773313	−	0.634024i	$-0.218598\pi$
$487$	34.1311	1.54663	0.773313	+	0.634024i	$0.218598\pi$
$488$	43.3443	1.96210
$489$	3.25434	0.147166
$490$	0	0
$491$	3.19431	0.144157	0.0720786	−	0.997399i	$-0.477037\pi$
$491$	3.19431	0.144157	0.0720786	+	0.997399i	$0.477037\pi$
$492$	−0.888005	−0.0400344
$493$	12.7979	0.576390
$494$	0.945184	0.0425258
$495$	0	0
$496$	−1.58993	−0.0713898
$497$	−21.7214	−0.974339
$498$	−1.48085	−0.0663584
$499$	6.52850	0.292256	0.146128	−	0.989266i	$-0.453319\pi$
$499$	6.52850	0.292256	0.146128	+	0.989266i	$0.453319\pi$
$500$	0	0
$501$	6.15791	0.275115
$502$	−0.661899	−0.0295420
$503$	−1.00387	−0.0447603	−0.0223802	−	0.999750i	$-0.507124\pi$
$503$	−1.00387	−0.0447603	−0.0223802	+	0.999750i	$0.507124\pi$
$504$	22.6646	1.00956
$505$	0	0
$506$	0	0
$507$	−1.96052	−0.0870697
$508$	5.31465	0.235799
$509$	−4.27401	−0.189442	−0.0947211	−	0.995504i	$-0.530196\pi$
$509$	−4.27401	−0.189442	−0.0947211	+	0.995504i	$0.530196\pi$
$510$	0	0
$511$	33.7037	1.49096
$512$	−24.6568	−1.08969
$513$	−0.592844	−0.0261747
$514$	29.2703	1.29106
$515$	0	0
$516$	0.442028	0.0194592
$517$	0	0
$518$	−1.66735	−0.0732590
$519$	−1.23022	−0.0540008
$520$	0	0
$521$	−9.88817	−0.433208	−0.216604	−	0.976259i	$-0.569498\pi$
$521$	−9.88817	−0.433208	−0.216604	+	0.976259i	$0.569498\pi$
$522$	11.7587	0.514663
$523$	−44.1670	−1.93129	−0.965644	−	0.259870i	$-0.916320\pi$
$523$	−44.1670	−1.93129	−0.965644	+	0.259870i	$0.916320\pi$
$524$	4.85131	0.211931
$525$	0	0
$526$	6.14089	0.267755
$527$	−2.17311	−0.0946623
$528$	0	0
$529$	47.5902	2.06914
$530$	0	0
$531$	−21.5272	−0.934200
$532$	−0.342546	−0.0148512
$533$	14.1052	0.610963
$534$	1.11759	0.0483630
$535$	0	0
$536$	−9.80105	−0.423341
$537$	5.03166	0.217132
$538$	35.5447	1.53244
$539$	0	0
$540$	0	0
$541$	21.2695	0.914449	0.457224	−	0.889351i	$-0.348844\pi$
$541$	21.2695	0.914449	0.457224	+	0.889351i	$0.348844\pi$
$542$	19.8379	0.852112
$543$	−0.945992	−0.0405964
$544$	−10.1550	−0.435393
$545$	0	0
$546$	3.20442	0.137136
$547$	−4.13393	−0.176754	−0.0883771	−	0.996087i	$-0.528168\pi$
$547$	−4.13393	−0.176754	−0.0883771	+	0.996087i	$0.528168\pi$
$548$	2.07420	0.0886055
$549$	40.6587	1.73527
$550$	0	0
$551$	−0.922455	−0.0392979
$552$	9.34839	0.397894
$553$	−25.0911	−1.06698
$554$	−22.2166	−0.943891
$555$	0	0
$556$	−3.96243	−0.168045
$557$	−31.6671	−1.34178	−0.670890	−	0.741557i	$-0.734088\pi$
$557$	−31.6671	−1.34178	−0.670890	+	0.741557i	$0.734088\pi$
$558$	−1.99664	−0.0845247
$559$	−7.02122	−0.296966
$560$	0	0
$561$	0	0
$562$	−16.9242	−0.713904
$563$	12.7467	0.537210	0.268605	−	0.963250i	$-0.413437\pi$
$563$	12.7467	0.537210	0.268605	+	0.963250i	$0.413437\pi$
$564$	0.855502	0.0360231
$565$	0	0
$566$	25.3023	1.06354
$567$	20.2327	0.849693
$568$	25.6812	1.07756
$569$	−15.6352	−0.655461	−0.327730	−	0.944771i	$-0.606284\pi$
$569$	−15.6352	−0.655461	−0.327730	+	0.944771i	$0.606284\pi$
$570$	0	0
$571$	3.61999	0.151492	0.0757460	−	0.997127i	$-0.475866\pi$
$571$	3.61999	0.151492	0.0757460	+	0.997127i	$0.475866\pi$
$572$	0	0
$573$	0.807429	0.0337308
$574$	16.3099	0.680762
$575$	0	0
$576$	−25.4900	−1.06208
$577$	−23.8352	−0.992273	−0.496136	−	0.868245i	$-0.665248\pi$
$577$	−23.8352	−0.992273	−0.496136	+	0.868245i	$0.665248\pi$
$578$	−2.67523	−0.111275
$579$	3.81325	0.158473
$580$	0	0
$581$	−8.52463	−0.353661
$582$	−4.89641	−0.202963
$583$	0	0
$584$	−39.8478	−1.64891
$585$	0	0
$586$	−27.6578	−1.14253
$587$	−1.24698	−0.0514684	−0.0257342	−	0.999669i	$-0.508192\pi$
$587$	−1.24698	−0.0514684	−0.0257342	+	0.999669i	$0.508192\pi$
$588$	0.0546818	0.00225504
$589$	0.156634	0.00645401
$590$	0	0
$591$	0.482883	0.0198631
$592$	1.47248	0.0605187
$593$	18.5288	0.760886	0.380443	−	0.924804i	$-0.375772\pi$
$593$	18.5288	0.760886	0.380443	+	0.924804i	$0.375772\pi$
$594$	0	0
$595$	0	0
$596$	4.04091	0.165522
$597$	−1.89388	−0.0775112
$598$	−28.6077	−1.16986
$599$	33.8359	1.38250	0.691248	−	0.722617i	$-0.257061\pi$
$599$	33.8359	1.38250	0.691248	+	0.722617i	$0.257061\pi$
$600$	0	0
$601$	21.3026	0.868951	0.434475	−	0.900684i	$-0.356934\pi$
$601$	21.3026	0.868951	0.434475	+	0.900684i	$0.356934\pi$
$602$	−8.11868	−0.330893
$603$	−9.19378	−0.374400
$604$	5.29869	0.215601
$605$	0	0
$606$	−6.21547	−0.252486
$607$	37.4601	1.52046	0.760230	−	0.649654i	$-0.225086\pi$
$607$	37.4601	1.52046	0.760230	+	0.649654i	$0.225086\pi$
$608$	0.731957	0.0296848
$609$	−3.12736	−0.126727
$610$	0	0
$611$	−13.5889	−0.549747
$612$	−5.27062	−0.213052
$613$	−20.0047	−0.807983	−0.403991	−	0.914763i	$-0.632377\pi$
$613$	−20.0047	−0.807983	−0.403991	+	0.914763i	$0.632377\pi$
$614$	25.0555	1.01116
$615$	0	0
$616$	0	0
$617$	−34.7932	−1.40072	−0.700360	−	0.713790i	$-0.746977\pi$
$617$	−34.7932	−1.40072	−0.700360	+	0.713790i	$0.746977\pi$
$618$	6.01410	0.241922
$619$	−46.0840	−1.85227	−0.926136	−	0.377189i	$-0.876891\pi$
$619$	−46.0840	−1.85227	−0.926136	+	0.377189i	$0.876891\pi$
$620$	0	0
$621$	17.9435	0.720047
$622$	−10.8083	−0.433374
$623$	6.43351	0.257753
$624$	−2.82991	−0.113287
$625$	0	0
$626$	2.13633	0.0853849
$627$	0	0
$628$	−6.57008	−0.262175
$629$	2.01259	0.0802473
$630$	0	0
$631$	−20.5380	−0.817604	−0.408802	−	0.912623i	$-0.634053\pi$
$631$	−20.5380	−0.817604	−0.408802	+	0.912623i	$0.634053\pi$
$632$	29.6651	1.18002
$633$	−6.88529	−0.273666
$634$	21.5658	0.856486
$635$	0	0
$636$	1.51494	0.0600712
$637$	−0.868571	−0.0344140
$638$	0	0
$639$	24.0900	0.952985
$640$	0	0
$641$	15.9095	0.628389	0.314194	−	0.949359i	$-0.398266\pi$
$641$	15.9095	0.628389	0.314194	+	0.949359i	$0.398266\pi$
$642$	−4.60803	−0.181865
$643$	−42.3039	−1.66830	−0.834151	−	0.551536i	$-0.814042\pi$
$643$	−42.3039	−1.66830	−0.834151	+	0.551536i	$0.814042\pi$
$644$	10.3678	0.408547
$645$	0	0
$646$	−1.31923	−0.0519042
$647$	7.48891	0.294419	0.147210	−	0.989105i	$-0.452971\pi$
$647$	7.48891	0.294419	0.147210	+	0.989105i	$0.452971\pi$
$648$	−23.9210	−0.939707
$649$	0	0
$650$	0	0
$651$	0.531031	0.0208127
$652$	−4.26715	−0.167114
$653$	−16.4702	−0.644531	−0.322265	−	0.946649i	$-0.604444\pi$
$653$	−16.4702	−0.644531	−0.322265	+	0.946649i	$0.604444\pi$
$654$	2.21999	0.0868083
$655$	0	0
$656$	−14.4037	−0.562372
$657$	−37.3788	−1.45829
$658$	−15.7129	−0.612553
$659$	−23.7359	−0.924619	−0.462310	−	0.886719i	$-0.652979\pi$
$659$	−23.7359	−0.924619	−0.462310	+	0.886719i	$0.652979\pi$
$660$	0	0
$661$	13.4183	0.521911	0.260956	−	0.965351i	$-0.415962\pi$
$661$	13.4183	0.521911	0.260956	+	0.965351i	$0.415962\pi$
$662$	15.5721	0.605229
$663$	−3.86793	−0.150218
$664$	10.0786	0.391127
$665$	0	0
$666$	1.84916	0.0716534
$667$	27.9198	1.08106
$668$	−8.07436	−0.312407
$669$	7.96112	0.307795
$670$	0	0
$671$	0	0
$672$	2.48152	0.0957268
$673$	48.2870	1.86133	0.930664	−	0.365875i	$-0.119230\pi$
$673$	48.2870	1.86133	0.930664	+	0.365875i	$0.119230\pi$
$674$	14.6350	0.563718
$675$	0	0
$676$	2.57067	0.0988718
$677$	24.3637	0.936374	0.468187	−	0.883629i	$-0.344907\pi$
$677$	24.3637	0.936374	0.468187	+	0.883629i	$0.344907\pi$
$678$	−4.45742	−0.171186
$679$	−28.1866	−1.08170
$680$	0	0
$681$	0.887625	0.0340139
$682$	0	0
$683$	−38.9856	−1.49174	−0.745871	−	0.666090i	$-0.767966\pi$
$683$	−38.9856	−1.49174	−0.745871	+	0.666090i	$0.767966\pi$
$684$	0.379898	0.0145257
$685$	0	0
$686$	−23.3384	−0.891066
$687$	2.65750	0.101390
$688$	7.16984	0.273348
$689$	−24.0634	−0.916743
$690$	0	0
$691$	5.91854	0.225152	0.112576	−	0.993643i	$-0.464090\pi$
$691$	5.91854	0.225152	0.112576	+	0.993643i	$0.464090\pi$
$692$	1.61309	0.0613205
$693$	0	0
$694$	27.5230	1.04476
$695$	0	0
$696$	3.69747	0.140152
$697$	−19.6871	−0.745701
$698$	−19.7042	−0.745817
$699$	−1.62275	−0.0613779
$700$	0	0
$701$	−11.9672	−0.451996	−0.225998	−	0.974128i	$-0.572564\pi$
$701$	−11.9672	−0.451996	−0.225998	+	0.974128i	$0.572564\pi$
$702$	−7.27186	−0.274459
$703$	−0.145064	−0.00547120
$704$	0	0
$705$	0	0
$706$	29.7894	1.12114
$707$	−35.7799	−1.34564
$708$	−1.30412	−0.0490117
$709$	6.80850	0.255699	0.127849	−	0.991794i	$-0.459193\pi$
$709$	6.80850	0.255699	0.127849	+	0.991794i	$0.459193\pi$
$710$	0	0
$711$	27.8271	1.04360
$712$	−7.60633	−0.285059
$713$	−4.74082	−0.177545
$714$	−4.47251	−0.167380
$715$	0	0
$716$	−6.59761	−0.246564
$717$	−8.79917	−0.328611
$718$	−25.1468	−0.938469
$719$	−9.26411	−0.345493	−0.172746	−	0.984966i	$-0.555264\pi$
$719$	−9.26411	−0.345493	−0.172746	+	0.984966i	$0.555264\pi$
$720$	0	0
$721$	34.6206	1.28934
$722$	−23.3508	−0.869026
$723$	4.38210	0.162972
$724$	1.24040	0.0460992
$725$	0	0
$726$	0	0
$727$	−19.4121	−0.719956	−0.359978	−	0.932961i	$-0.617216\pi$
$727$	−19.4121	−0.719956	−0.359978	+	0.932961i	$0.617216\pi$
$728$	−21.8092	−0.808304
$729$	−20.1069	−0.744699
$730$	0	0
$731$	9.79975	0.362457
$732$	2.46310	0.0910390
$733$	11.5620	0.427054	0.213527	−	0.976937i	$-0.431505\pi$
$733$	11.5620	0.427054	0.213527	+	0.976937i	$0.431505\pi$
$734$	−6.66887	−0.246153
$735$	0	0
$736$	−22.1540	−0.816608
$737$	0	0
$738$	−18.0884	−0.665842
$739$	−9.10736	−0.335020	−0.167510	−	0.985870i	$-0.553573\pi$
$739$	−9.10736	−0.335020	−0.167510	+	0.985870i	$0.553573\pi$
$740$	0	0
$741$	0.278794	0.0102418
$742$	−27.8247	−1.02148
$743$	0.0731008	0.00268181	0.00134090	−	0.999999i	$-0.499573\pi$
$743$	0.0731008	0.00268181	0.00134090	+	0.999999i	$0.499573\pi$
$744$	−0.627836	−0.0230176
$745$	0	0
$746$	−21.0991	−0.772492
$747$	9.45417	0.345910
$748$	0	0
$749$	−26.5265	−0.969258
$750$	0	0
$751$	−38.5239	−1.40576	−0.702878	−	0.711310i	$-0.748102\pi$
$751$	−38.5239	−1.40576	−0.702878	+	0.711310i	$0.748102\pi$
$752$	13.8765	0.506025
$753$	−0.195236	−0.00711479
$754$	−11.3149	−0.412064
$755$	0	0
$756$	2.63541	0.0958488
$757$	−28.1617	−1.02355	−0.511777	−	0.859118i	$-0.671013\pi$
$757$	−28.1617	−1.02355	−0.511777	+	0.859118i	$0.671013\pi$
$758$	−10.6024	−0.385097
$759$	0	0
$760$	0	0
$761$	29.4894	1.06899	0.534496	−	0.845171i	$-0.320502\pi$
$761$	29.4894	1.06899	0.534496	+	0.845171i	$0.320502\pi$
$762$	−5.00164	−0.181190
$763$	12.7795	0.462650
$764$	−1.05871	−0.0383030
$765$	0	0
$766$	11.9322	0.431128
$767$	20.7147	0.747965
$768$	−3.91285	−0.141193
$769$	−8.42410	−0.303781	−0.151890	−	0.988397i	$-0.548536\pi$
$769$	−8.42410	−0.303781	−0.151890	+	0.988397i	$0.548536\pi$
$770$	0	0
$771$	8.63364	0.310933
$772$	−5.00000	−0.179954
$773$	−0.897483	−0.0322802	−0.0161401	−	0.999870i	$-0.505138\pi$
$773$	−0.897483	−0.0322802	−0.0161401	+	0.999870i	$0.505138\pi$
$774$	9.00396	0.323641
$775$	0	0
$776$	33.3249	1.19629
$777$	−0.491805	−0.0176434
$778$	−33.7736	−1.21084
$779$	1.41901	0.0508413
$780$	0	0
$781$	0	0
$782$	39.9287	1.42785
$783$	7.09699	0.253626
$784$	0.886957	0.0316770
$785$	0	0
$786$	−4.56559	−0.162849
$787$	−37.8007	−1.34745	−0.673724	−	0.738983i	$-0.735306\pi$
$787$	−37.8007	−1.34745	−0.673724	+	0.738983i	$0.735306\pi$
$788$	−0.633165	−0.0225556
$789$	1.81133	0.0644852
$790$	0	0
$791$	−25.6595	−0.912346
$792$	0	0
$793$	−39.1242	−1.38934
$794$	13.1442	0.466470
$795$	0	0
$796$	2.48329	0.0880177
$797$	−15.1906	−0.538078	−0.269039	−	0.963129i	$-0.586706\pi$
$797$	−15.1906	−0.538078	−0.269039	+	0.963129i	$0.586706\pi$
$798$	0.322371	0.0114118
$799$	18.9665	0.670985
$800$	0	0
$801$	−7.13504	−0.252104
$802$	16.9237	0.597598
$803$	0	0
$804$	−0.556959	−0.0196424
$805$	0	0
$806$	1.92129	0.0676745
$807$	10.4844	0.369067
$808$	42.3024	1.48819
$809$	39.7109	1.39616	0.698080	−	0.716020i	$-0.254038\pi$
$809$	39.7109	1.39616	0.698080	+	0.716020i	$0.254038\pi$
$810$	0	0
$811$	−8.81679	−0.309599	−0.154800	−	0.987946i	$-0.549473\pi$
$811$	−8.81679	−0.309599	−0.154800	+	0.987946i	$0.549473\pi$
$812$	4.10065	0.143905
$813$	5.85145	0.205219
$814$	0	0
$815$	0	0
$816$	3.94981	0.138271
$817$	−0.706350	−0.0247121
$818$	27.5283	0.962506
$819$	−20.4579	−0.714858
$820$	0	0
$821$	8.42070	0.293884	0.146942	−	0.989145i	$-0.453057\pi$
$821$	8.42070	0.293884	0.146942	+	0.989145i	$0.453057\pi$
$822$	−1.95204	−0.0680853
$823$	−23.9040	−0.833241	−0.416621	−	0.909080i	$-0.636786\pi$
$823$	−23.9040	−0.833241	−0.416621	+	0.909080i	$0.636786\pi$
$824$	−40.9319	−1.42593
$825$	0	0
$826$	23.9526	0.833416
$827$	−47.5118	−1.65215	−0.826074	−	0.563561i	$-0.809431\pi$
$827$	−47.5118	−1.65215	−0.826074	+	0.563561i	$0.809431\pi$
$828$	−11.4983	−0.399593
$829$	−28.2003	−0.979437	−0.489718	−	0.871881i	$-0.662900\pi$
$829$	−28.2003	−0.979437	−0.489718	+	0.871881i	$0.662900\pi$
$830$	0	0
$831$	−6.55306	−0.227323
$832$	24.5280	0.850354
$833$	1.21229	0.0420035
$834$	3.72907	0.129127
$835$	0	0
$836$	0	0
$837$	−1.20508	−0.0416537
$838$	−0.630232	−0.0217710
$839$	22.3406	0.771284	0.385642	−	0.922648i	$-0.373980\pi$
$839$	22.3406	0.771284	0.385642	+	0.922648i	$0.373980\pi$
$840$	0	0
$841$	−17.9572	−0.619214
$842$	16.3072	0.561983
$843$	−4.99201	−0.171934
$844$	9.02811	0.310760
$845$	0	0
$846$	17.4263	0.599128
$847$	0	0
$848$	24.5728	0.843833
$849$	7.46325	0.256138
$850$	0	0
$851$	4.39063	0.150509
$852$	1.45937	0.0499973
$853$	−8.15265	−0.279141	−0.139571	−	0.990212i	$-0.544572\pi$
$853$	−8.15265	−0.279141	−0.139571	+	0.990212i	$0.544572\pi$
$854$	−45.2395	−1.54807
$855$	0	0
$856$	31.3622	1.07194
$857$	4.42433	0.151132	0.0755662	−	0.997141i	$-0.475924\pi$
$857$	4.42433	0.151132	0.0755662	+	0.997141i	$0.475924\pi$
$858$	0	0
$859$	2.90501	0.0991176	0.0495588	−	0.998771i	$-0.484218\pi$
$859$	2.90501	0.0991176	0.0495588	+	0.998771i	$0.484218\pi$
$860$	0	0
$861$	4.81081	0.163952
$862$	36.2873	1.23595
$863$	17.7101	0.602858	0.301429	−	0.953489i	$-0.402536\pi$
$863$	17.7101	0.602858	0.301429	+	0.953489i	$0.402536\pi$
$864$	−5.63138	−0.191583
$865$	0	0
$866$	−12.6071	−0.428408
$867$	−0.789095	−0.0267991
$868$	−0.696297	−0.0236339
$869$	0	0
$870$	0	0
$871$	8.84680	0.299762
$872$	−15.1092	−0.511662
$873$	31.2601	1.05799
$874$	−2.87800	−0.0973497
$875$	0	0
$876$	−2.26441	−0.0765073
$877$	51.3798	1.73497	0.867487	−	0.497461i	$-0.165734\pi$
$877$	51.3798	1.73497	0.867487	+	0.497461i	$0.165734\pi$
$878$	1.89179	0.0638448
$879$	−8.15803	−0.275164
$880$	0	0
$881$	33.6727	1.13446	0.567231	−	0.823559i	$-0.308015\pi$
$881$	33.6727	1.13446	0.567231	+	0.823559i	$0.308015\pi$
$882$	1.11385	0.0375052
$883$	13.9106	0.468128	0.234064	−	0.972221i	$-0.424797\pi$
$883$	13.9106	0.468128	0.234064	+	0.972221i	$0.424797\pi$
$884$	5.07170	0.170580
$885$	0	0
$886$	4.18638	0.140644
$887$	44.7006	1.50090	0.750450	−	0.660927i	$-0.229837\pi$
$887$	44.7006	1.50090	0.750450	+	0.660927i	$0.229837\pi$
$888$	0.581460	0.0195125
$889$	−28.7923	−0.965664
$890$	0	0
$891$	0	0
$892$	−10.4388	−0.349516
$893$	−1.36707	−0.0457473
$894$	−3.80292	−0.127189
$895$	0	0
$896$	14.7264	0.491976
$897$	−8.43821	−0.281744
$898$	−39.9198	−1.33214
$899$	−1.87509	−0.0625376
$900$	0	0
$901$	33.5861	1.11892
$902$	0	0
$903$	−2.39471	−0.0796909
$904$	30.3371	1.00900
$905$	0	0
$906$	−4.98662	−0.165669
$907$	−22.7980	−0.756994	−0.378497	−	0.925602i	$-0.623559\pi$
$907$	−22.7980	−0.756994	−0.378497	+	0.925602i	$0.623559\pi$
$908$	−1.16387	−0.0386244
$909$	39.6814	1.31615
$910$	0	0
$911$	37.5625	1.24450	0.622250	−	0.782819i	$-0.286219\pi$
$911$	37.5625	1.24450	0.622250	+	0.782819i	$0.286219\pi$
$912$	−0.284696	−0.00942722
$913$	0	0
$914$	−15.0214	−0.496863
$915$	0	0
$916$	−3.48456	−0.115133
$917$	−26.2822	−0.867915
$918$	10.1496	0.334986
$919$	−39.7124	−1.30999	−0.654996	−	0.755632i	$-0.727330\pi$
$919$	−39.7124	−1.30999	−0.654996	+	0.755632i	$0.727330\pi$
$920$	0	0
$921$	7.39044	0.243523
$922$	20.4472	0.673392
$923$	−23.1808	−0.763005
$924$	0	0
$925$	0	0
$926$	18.0536	0.593278
$927$	−38.3958	−1.26108
$928$	−8.76234	−0.287638
$929$	20.7298	0.680123	0.340061	−	0.940403i	$-0.389552\pi$
$929$	20.7298	0.680123	0.340061	+	0.940403i	$0.389552\pi$
$930$	0	0
$931$	−0.0873802	−0.00286377
$932$	2.12777	0.0696975
$933$	−3.18805	−0.104372
$934$	37.6187	1.23092
$935$	0	0
$936$	24.1874	0.790588
$937$	55.8506	1.82456	0.912280	−	0.409568i	$-0.134320\pi$
$937$	55.8506	1.82456	0.912280	+	0.409568i	$0.134320\pi$
$938$	10.2296	0.334009
$939$	0.630138	0.0205638
$940$	0	0
$941$	13.3644	0.435667	0.217834	−	0.975986i	$-0.430101\pi$
$941$	13.3644	0.435667	0.217834	+	0.975986i	$0.430101\pi$
$942$	6.18313	0.201457
$943$	−42.9489	−1.39861
$944$	−21.1532	−0.688479
$945$	0	0
$946$	0	0
$947$	42.2245	1.37211	0.686055	−	0.727550i	$-0.259341\pi$
$947$	42.2245	1.37211	0.686055	+	0.727550i	$0.259341\pi$
$948$	1.68577	0.0547511
$949$	35.9681	1.16757
$950$	0	0
$951$	6.36110	0.206273
$952$	30.4399	0.986562
$953$	14.7188	0.476790	0.238395	−	0.971168i	$-0.423379\pi$
$953$	14.7188	0.476790	0.238395	+	0.971168i	$0.423379\pi$
$954$	30.8587	0.999089
$955$	0	0
$956$	11.5376	0.373154
$957$	0	0
$958$	−17.2750	−0.558131
$959$	−11.2371	−0.362865
$960$	0	0
$961$	−30.6816	−0.989729
$962$	−1.77937	−0.0573691
$963$	29.4190	0.948015
$964$	−5.74589	−0.185063
$965$	0	0
$966$	−9.75715	−0.313931
$967$	−32.2786	−1.03801	−0.519005	−	0.854771i	$-0.673697\pi$
$967$	−32.2786	−1.03801	−0.519005	+	0.854771i	$0.673697\pi$
$968$	0	0
$969$	−0.389123	−0.0125004
$970$	0	0
$971$	−42.8008	−1.37354	−0.686771	−	0.726874i	$-0.740973\pi$
$971$	−42.8008	−1.37354	−0.686771	+	0.726874i	$0.740973\pi$
$972$	−4.41716	−0.141681
$973$	21.4667	0.688190
$974$	42.1175	1.34953
$975$	0	0
$976$	39.9524	1.27884
$977$	−12.1972	−0.390222	−0.195111	−	0.980781i	$-0.562507\pi$
$977$	−12.1972	−0.390222	−0.195111	+	0.980781i	$0.562507\pi$
$978$	4.01584	0.128412
$979$	0	0
$980$	0	0
$981$	−14.1730	−0.452510
$982$	3.94176	0.125787
$983$	−14.0671	−0.448670	−0.224335	−	0.974512i	$-0.572021\pi$
$983$	−14.0671	−0.448670	−0.224335	+	0.974512i	$0.572021\pi$
$984$	−5.68781	−0.181321
$985$	0	0
$986$	15.7926	0.502938
$987$	−4.63472	−0.147525
$988$	−0.365560	−0.0116300
$989$	21.3790	0.679811
$990$	0	0
$991$	−22.9455	−0.728887	−0.364444	−	0.931225i	$-0.618741\pi$
$991$	−22.9455	−0.728887	−0.364444	+	0.931225i	$0.618741\pi$
$992$	1.48786	0.0472396
$993$	4.59321	0.145761
$994$	−26.8041	−0.850175
$995$	0	0
$996$	0.572734	0.0181478
$997$	3.63447	0.115105	0.0575525	−	0.998342i	$-0.481670\pi$
$997$	3.63447	0.115105	0.0575525	+	0.998342i	$0.481670\pi$
$998$	8.05612	0.255012
$999$	1.11607	0.0353108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bk.1.6		8
5.2	odd	4		605.2.b.f.364.6		8
5.3	odd	4		605.2.b.f.364.3		8
5.4	even	2	inner	3025.2.a.bk.1.3		8
11.2	odd	10		275.2.h.d.26.3		16
11.6	odd	10		275.2.h.d.201.3		16
11.10	odd	2		3025.2.a.bl.1.3		8
55.2	even	20		55.2.j.a.4.2	✓	16
55.3	odd	20		605.2.j.g.9.2		16
55.7	even	20		605.2.j.h.269.3		16
55.8	even	20		605.2.j.h.9.3		16
55.13	even	20		55.2.j.a.4.3	yes	16
55.17	even	20		55.2.j.a.14.3	yes	16
55.18	even	20		605.2.j.h.269.2		16
55.24	odd	10		275.2.h.d.26.2		16
55.27	odd	20		605.2.j.d.124.2		16
55.28	even	20		55.2.j.a.14.2	yes	16
55.32	even	4		605.2.b.g.364.3		8
55.37	odd	20		605.2.j.g.269.2		16
55.38	odd	20		605.2.j.d.124.3		16
55.39	odd	10		275.2.h.d.201.2		16
55.42	odd	20		605.2.j.d.444.3		16
55.43	even	4		605.2.b.g.364.6		8
55.47	odd	20		605.2.j.g.9.3		16
55.48	odd	20		605.2.j.g.269.3		16
55.52	even	20		605.2.j.h.9.2		16
55.53	odd	20		605.2.j.d.444.2		16
55.54	odd	2		3025.2.a.bl.1.6		8
165.2	odd	20		495.2.ba.a.334.3		16
165.17	odd	20		495.2.ba.a.289.2		16
165.68	odd	20		495.2.ba.a.334.2		16
165.83	odd	20		495.2.ba.a.289.3		16
220.83	odd	20		880.2.cd.c.289.3		16
220.123	odd	20		880.2.cd.c.609.2		16
220.127	odd	20		880.2.cd.c.289.2		16
220.167	odd	20		880.2.cd.c.609.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.j.a.4.2	✓	16	55.2	even	20
55.2.j.a.4.3	yes	16	55.13	even	20
55.2.j.a.14.2	yes	16	55.28	even	20
55.2.j.a.14.3	yes	16	55.17	even	20
275.2.h.d.26.2		16	55.24	odd	10
275.2.h.d.26.3		16	11.2	odd	10
275.2.h.d.201.2		16	55.39	odd	10
275.2.h.d.201.3		16	11.6	odd	10
495.2.ba.a.289.2		16	165.17	odd	20
495.2.ba.a.289.3		16	165.83	odd	20
495.2.ba.a.334.2		16	165.68	odd	20
495.2.ba.a.334.3		16	165.2	odd	20
605.2.b.f.364.3		8	5.3	odd	4
605.2.b.f.364.6		8	5.2	odd	4
605.2.b.g.364.3		8	55.32	even	4
605.2.b.g.364.6		8	55.43	even	4
605.2.j.d.124.2		16	55.27	odd	20
605.2.j.d.124.3		16	55.38	odd	20
605.2.j.d.444.2		16	55.53	odd	20
605.2.j.d.444.3		16	55.42	odd	20
605.2.j.g.9.2		16	55.3	odd	20
605.2.j.g.9.3		16	55.47	odd	20
605.2.j.g.269.2		16	55.37	odd	20
605.2.j.g.269.3		16	55.48	odd	20
605.2.j.h.9.2		16	55.52	even	20
605.2.j.h.9.3		16	55.8	even	20
605.2.j.h.269.2		16	55.18	even	20
605.2.j.h.269.3		16	55.7	even	20
880.2.cd.c.289.2		16	220.127	odd	20
880.2.cd.c.289.3		16	220.83	odd	20
880.2.cd.c.609.2		16	220.123	odd	20
880.2.cd.c.609.3		16	220.167	odd	20
3025.2.a.bk.1.3		8	5.4	even	2	inner
3025.2.a.bk.1.6		8	1.1	even	1	trivial
3025.2.a.bl.1.3		8	11.10	odd	2
3025.2.a.bl.1.6		8	55.54	odd	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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23399 q^{2} +0.363982 q^{3} -0.477260 q^{4} +0.449152 q^{6} +2.58558 q^{7} -3.05692 q^{8} -2.86752 q^{9} +O(q^{10})\)	\(q+1.23399 q^{2} +0.363982 q^{3} -0.477260 q^{4} +0.449152 q^{6} +2.58558 q^{7} -3.05692 q^{8} -2.86752 q^{9} -0.173714 q^{12} +2.75929 q^{13} +3.19059 q^{14} -2.81770 q^{16} -3.85124 q^{17} -3.53850 q^{18} +0.277591 q^{19} +0.941105 q^{21} -8.40180 q^{23} -1.11267 q^{24} +3.40495 q^{26} -2.13567 q^{27} -1.23399 q^{28} -3.32307 q^{29} +0.564263 q^{31} +2.63682 q^{32} -4.75241 q^{34} +1.36855 q^{36} -0.522583 q^{37} +0.342546 q^{38} +1.00433 q^{39} +5.11188 q^{41} +1.16132 q^{42} -2.54457 q^{43} -10.3678 q^{46} -4.92477 q^{47} -1.02559 q^{48} -0.314780 q^{49} -1.40178 q^{51} -1.31690 q^{52} -8.72086 q^{53} -2.63541 q^{54} -7.90392 q^{56} +0.101038 q^{57} -4.10065 q^{58} +7.50726 q^{59} -14.1791 q^{61} +0.696297 q^{62} -7.41419 q^{63} +8.88922 q^{64} +3.20618 q^{67} +1.83804 q^{68} -3.05810 q^{69} -8.40099 q^{71} +8.76578 q^{72} +13.0353 q^{73} -0.644864 q^{74} -0.132483 q^{76} +1.23934 q^{78} -9.70425 q^{79} +7.82520 q^{81} +6.30802 q^{82} -3.29699 q^{83} -0.449152 q^{84} -3.13998 q^{86} -1.20954 q^{87} +2.48823 q^{89} +7.13437 q^{91} +4.00984 q^{92} +0.205382 q^{93} -6.07713 q^{94} +0.959755 q^{96} -10.9014 q^{97} -0.388437 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9}+O(q^{10}) \) 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9	\( 8 q + 2 q^{4} - 6 q^{6} + 4 q^{9} - 4 q^{14} - 22 q^{16} - 12 q^{19} - 4 q^{21} - 2 q^{24} + 10 q^{26} - 24 q^{29} + 14 q^{31} + 8 q^{34} + 20 q^{36} - 30 q^{39} - 34 q^{41} - 24 q^{46} - 30 q^{49} - 54 q^{51} - 20 q^{54} - 10 q^{56} - 6 q^{59} - 20 q^{61} + 14 q^{64} + 32 q^{69} - 42 q^{71} + 4 q^{74} - 28 q^{76} - 16 q^{79} - 36 q^{81} + 6 q^{84} + 46 q^{86} - 12 q^{89} + 20 q^{91} + 42 q^{94} + 8 q^{96}+O(q^{100}) \) 8 * q + 2 * q^4 - 6 * q^6 + 4 * q^9 - 4 * q^14 - 22 * q^16 - 12 * q^19 - 4 * q^21 - 2 * q^24 + 10 * q^26 - 24 * q^29 + 14 * q^31 + 8 * q^34 + 20 * q^36 - 30 * q^39 - 34 * q^41 - 24 * q^46 - 30 * q^49 - 54 * q^51 - 20 * q^54 - 10 * q^56 - 6 * q^59 - 20 * q^61 + 14 * q^64 + 32 * q^69 - 42 * q^71 + 4 * q^74 - 28 * q^76 - 16 * q^79 - 36 * q^81 + 6 * q^84 + 46 * q^86 - 12 * q^89 + 20 * q^91 + 42 * q^94 + 8 * q^96

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