LMFDB - Embedded newform 1521.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 169)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.801938 q^{2} -1.35690 q^{4} -0.246980 q^{5} -2.35690 q^{7} +2.69202 q^{8} +0.198062 q^{10} +4.24698 q^{11} +1.89008 q^{14} +0.554958 q^{16} -2.15883 q^{17} -0.0881460 q^{19} +0.335126 q^{20} -3.40581 q^{22} -1.49396 q^{23} -4.93900 q^{25} +3.19806 q^{28} -4.63102 q^{29} -6.63102 q^{31} -5.82908 q^{32} +1.73125 q^{34} +0.582105 q^{35} +5.69202 q^{37} +0.0706876 q^{38} -0.664874 q^{40} +11.5918 q^{41} -0.295897 q^{43} -5.76271 q^{44} +1.19806 q^{46} +7.35690 q^{47} -1.44504 q^{49} +3.96077 q^{50} +10.3937 q^{53} -1.04892 q^{55} -6.34481 q^{56} +3.71379 q^{58} +6.78017 q^{59} +3.47219 q^{61} +5.31767 q^{62} +3.56465 q^{64} +7.67994 q^{67} +2.92931 q^{68} -0.466812 q^{70} +8.66487 q^{71} +6.73556 q^{73} -4.56465 q^{74} +0.119605 q^{76} -10.0097 q^{77} +9.97046 q^{79} -0.137063 q^{80} -9.29590 q^{82} -1.60925 q^{83} +0.533188 q^{85} +0.237291 q^{86} +11.4330 q^{88} +2.88471 q^{89} +2.02715 q^{92} -5.89977 q^{94} +0.0217703 q^{95} -8.05861 q^{97} +1.15883 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{10} + 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} + 14 q^{28} + q^{29} - 5 q^{31} - 7 q^{32} + 13 q^{34} - 4 q^{35}+ \cdots - 5 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8 + 5 * q^10 + 8 * q^11 + 5 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 5 * q^23 - 5 * q^25 + 14 * q^28 + q^29 - 5 * q^31 - 7 * q^32 + 13 * q^34 - 4 * q^35 + 12 * q^37 - 12 * q^38 - 3 * q^40 + 7 * q^41 + 13 * q^43 + 8 * q^46 + 18 * q^47 - 4 * q^49 - q^50 - q^53 + 6 * q^55 + 4 * q^56 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 11 * q^64 - q^67 + 21 * q^68 + 2 * q^70 + 27 * q^71 + 9 * q^73 + 8 * q^74 - 21 * q^76 - 8 * q^77 - 5 * q^79 + 5 * q^80 - 14 * q^82 + 7 * q^83 + 5 * q^85 + 18 * q^86 + 15 * q^88 + 11 * q^89 + 5 * q^94 - 3 * q^95 + 7 * q^97 - 5 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.801938	−0.567056	−0.283528	−	0.958964i	$-0.591505\pi$
$2$	−0.801938	−0.567056	−0.283528	+	0.958964i	$0.591505\pi$
$3$	0	0
$3$	0	0
$4$	−1.35690	−0.678448
$5$	−0.246980	−0.110453	−0.0552263	−	0.998474i	$-0.517588\pi$
$5$	−0.246980	−0.110453	−0.0552263	+	0.998474i	$0.517588\pi$
$6$	0	0
$7$	−2.35690	−0.890823	−0.445411	−	0.895326i	$-0.646943\pi$
$7$	−2.35690	−0.890823	−0.445411	+	0.895326i	$0.646943\pi$
$8$	2.69202	0.951773
$9$	0	0
$10$	0.198062	0.0626328
$11$	4.24698	1.28051	0.640256	−	0.768161i	$-0.278828\pi$
$11$	4.24698	1.28051	0.640256	+	0.768161i	$0.278828\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	1.89008	0.505146
$15$	0	0
$16$	0.554958	0.138740
$17$	−2.15883	−0.523594	−0.261797	−	0.965123i	$-0.584315\pi$
$17$	−2.15883	−0.523594	−0.261797	+	0.965123i	$0.584315\pi$
$18$	0	0
$19$	−0.0881460	−0.0202221	−0.0101110	−	0.999949i	$-0.503218\pi$
$19$	−0.0881460	−0.0202221	−0.0101110	+	0.999949i	$0.503218\pi$
$20$	0.335126	0.0749364
$21$	0	0
$22$	−3.40581	−0.726122
$23$	−1.49396	−0.311512	−0.155756	−	0.987796i	$-0.549781\pi$
$23$	−1.49396	−0.311512	−0.155756	+	0.987796i	$0.549781\pi$
$24$	0	0
$25$	−4.93900	−0.987800
$26$	0	0
$27$	0	0
$28$	3.19806	0.604377
$29$	−4.63102	−0.859959	−0.429980	−	0.902839i	$-0.641479\pi$
$29$	−4.63102	−0.859959	−0.429980	+	0.902839i	$0.641479\pi$
$30$	0	0
$31$	−6.63102	−1.19097	−0.595483	−	0.803368i	$-0.703039\pi$
$31$	−6.63102	−1.19097	−0.595483	+	0.803368i	$0.703039\pi$
$32$	−5.82908	−1.03045
$33$	0	0
$34$	1.73125	0.296907
$35$	0.582105	0.0983937
$36$	0	0
$37$	5.69202	0.935763	0.467881	−	0.883791i	$-0.345017\pi$
$37$	5.69202	0.935763	0.467881	+	0.883791i	$0.345017\pi$
$38$	0.0706876	0.0114670
$39$	0	0
$40$	−0.664874	−0.105126
$41$	11.5918	1.81033	0.905167	−	0.425056i	$-0.139746\pi$
$41$	11.5918	1.81033	0.905167	+	0.425056i	$0.139746\pi$
$42$	0	0
$43$	−0.295897	−0.0451239	−0.0225619	−	0.999745i	$-0.507182\pi$
$43$	−0.295897	−0.0451239	−0.0225619	+	0.999745i	$0.507182\pi$
$44$	−5.76271	−0.868761
$45$	0	0
$46$	1.19806	0.176645
$47$	7.35690	1.07311	0.536557	−	0.843864i	$-0.319725\pi$
$47$	7.35690	1.07311	0.536557	+	0.843864i	$0.319725\pi$
$48$	0	0
$49$	−1.44504	−0.206435
$50$	3.96077	0.560138
$51$	0	0
$52$	0	0
$53$	10.3937	1.42769	0.713844	−	0.700304i	$-0.246952\pi$
$53$	10.3937	1.42769	0.713844	+	0.700304i	$0.246952\pi$
$54$	0	0
$55$	−1.04892	−0.141436
$56$	−6.34481	−0.847861
$57$	0	0
$58$	3.71379	0.487645
$59$	6.78017	0.882703	0.441351	−	0.897334i	$-0.354499\pi$
$59$	6.78017	0.882703	0.441351	+	0.897334i	$0.354499\pi$
$60$	0	0
$61$	3.47219	0.444568	0.222284	−	0.974982i	$-0.428649\pi$
$61$	3.47219	0.444568	0.222284	+	0.974982i	$0.428649\pi$
$62$	5.31767	0.675344
$63$	0	0
$64$	3.56465	0.445581
$65$	0	0
$66$	0	0
$67$	7.67994	0.938254	0.469127	−	0.883131i	$-0.344569\pi$
$67$	7.67994	0.938254	0.469127	+	0.883131i	$0.344569\pi$
$68$	2.92931	0.355231
$69$	0	0
$70$	−0.466812	−0.0557947
$71$	8.66487	1.02833	0.514166	−	0.857691i	$-0.328102\pi$
$71$	8.66487	1.02833	0.514166	+	0.857691i	$0.328102\pi$
$72$	0	0
$73$	6.73556	0.788338	0.394169	−	0.919038i	$-0.371032\pi$
$73$	6.73556	0.788338	0.394169	+	0.919038i	$0.371032\pi$
$74$	−4.56465	−0.530629
$75$	0	0
$76$	0.119605	0.0137196
$77$	−10.0097	−1.14071
$78$	0	0
$79$	9.97046	1.12176	0.560882	−	0.827896i	$-0.310462\pi$
$79$	9.97046	1.12176	0.560882	+	0.827896i	$0.310462\pi$
$80$	−0.137063	−0.0153241
$81$	0	0
$82$	−9.29590	−1.02656
$83$	−1.60925	−0.176638	−0.0883192	−	0.996092i	$-0.528150\pi$
$83$	−1.60925	−0.176638	−0.0883192	+	0.996092i	$0.528150\pi$
$84$	0	0
$85$	0.533188	0.0578323
$86$	0.237291	0.0255877
$87$	0	0
$88$	11.4330	1.21876
$89$	2.88471	0.305778	0.152889	−	0.988243i	$-0.451142\pi$
$89$	2.88471	0.305778	0.152889	+	0.988243i	$0.451142\pi$
$90$	0	0
$91$	0	0
$92$	2.02715	0.211345
$93$	0	0
$94$	−5.89977	−0.608515
$95$	0.0217703	0.00223358
$96$	0	0
$97$	−8.05861	−0.818227	−0.409114	−	0.912483i	$-0.634162\pi$
$97$	−8.05861	−0.818227	−0.409114	+	0.912483i	$0.634162\pi$
$98$	1.15883	0.117060
$99$	0	0
$100$	6.70171	0.670171
$101$	13.3545	1.32882	0.664411	−	0.747367i	$-0.268682\pi$
$101$	13.3545	1.32882	0.664411	+	0.747367i	$0.268682\pi$
$102$	0	0
$103$	1.36227	0.134229	0.0671144	−	0.997745i	$-0.478621\pi$
$103$	1.36227	0.134229	0.0671144	+	0.997745i	$0.478621\pi$
$104$	0	0
$105$	0	0
$106$	−8.33513	−0.809579
$107$	−3.26875	−0.316002	−0.158001	−	0.987439i	$-0.550505\pi$
$107$	−3.26875	−0.316002	−0.158001	+	0.987439i	$0.550505\pi$
$108$	0	0
$109$	15.7017	1.50395	0.751976	−	0.659191i	$-0.229101\pi$
$109$	15.7017	1.50395	0.751976	+	0.659191i	$0.229101\pi$
$110$	0.841166	0.0802021
$111$	0	0
$112$	−1.30798	−0.123592
$113$	−12.0489	−1.13347	−0.566733	−	0.823901i	$-0.691793\pi$
$113$	−12.0489	−1.13347	−0.566733	+	0.823901i	$0.691793\pi$
$114$	0	0
$115$	0.368977	0.0344073
$116$	6.28382	0.583438
$117$	0	0
$118$	−5.43727	−0.500541
$119$	5.08815	0.466430
$120$	0	0
$121$	7.03684	0.639712
$122$	−2.78448	−0.252095
$123$	0	0
$124$	8.99761	0.808009
$125$	2.45473	0.219558
$126$	0	0
$127$	−9.80731	−0.870258	−0.435129	−	0.900368i	$-0.643297\pi$
$127$	−9.80731	−0.870258	−0.435129	+	0.900368i	$0.643297\pi$
$128$	8.79954	0.777777
$129$	0	0
$130$	0	0
$131$	6.57673	0.574611	0.287306	−	0.957839i	$-0.407240\pi$
$131$	6.57673	0.574611	0.287306	+	0.957839i	$0.407240\pi$
$132$	0	0
$133$	0.207751	0.0180143
$134$	−6.15883	−0.532042
$135$	0	0
$136$	−5.81163	−0.498343
$137$	−6.21983	−0.531396	−0.265698	−	0.964056i	$-0.585602\pi$
$137$	−6.21983	−0.531396	−0.265698	+	0.964056i	$0.585602\pi$
$138$	0	0
$139$	−14.7071	−1.24744	−0.623719	−	0.781648i	$-0.714379\pi$
$139$	−14.7071	−1.24744	−0.623719	+	0.781648i	$0.714379\pi$
$140$	−0.789856	−0.0667550
$141$	0	0
$142$	−6.94869	−0.583121
$143$	0	0
$144$	0	0
$145$	1.14377	0.0949848
$146$	−5.40150	−0.447031
$147$	0	0
$148$	−7.72348	−0.634866
$149$	−4.33513	−0.355147	−0.177574	−	0.984108i	$-0.556825\pi$
$149$	−4.33513	−0.355147	−0.177574	+	0.984108i	$0.556825\pi$
$150$	0	0
$151$	3.94438	0.320989	0.160494	−	0.987037i	$-0.448691\pi$
$151$	3.94438	0.320989	0.160494	+	0.987037i	$0.448691\pi$
$152$	−0.237291	−0.0192468
$153$	0	0
$154$	8.02715	0.646846
$155$	1.63773	0.131545
$156$	0	0
$157$	4.45473	0.355526	0.177763	−	0.984073i	$-0.443114\pi$
$157$	4.45473	0.355526	0.177763	+	0.984073i	$0.443114\pi$
$158$	−7.99569	−0.636103
$159$	0	0
$160$	1.43967	0.113816
$161$	3.52111	0.277502
$162$	0	0
$163$	16.1588	1.26566	0.632829	−	0.774292i	$-0.281894\pi$
$163$	16.1588	1.26566	0.632829	+	0.774292i	$0.281894\pi$
$164$	−15.7289	−1.22822
$165$	0	0
$166$	1.29052	0.100164
$167$	−16.1172	−1.24719	−0.623594	−	0.781749i	$-0.714328\pi$
$167$	−16.1172	−1.24719	−0.623594	+	0.781749i	$0.714328\pi$
$168$	0	0
$169$	0	0
$170$	−0.427583	−0.0327942
$171$	0	0
$172$	0.401501	0.0306142
$173$	21.5362	1.63736	0.818682	−	0.574247i	$-0.194705\pi$
$173$	21.5362	1.63736	0.818682	+	0.574247i	$0.194705\pi$
$174$	0	0
$175$	11.6407	0.879955
$176$	2.35690	0.177658
$177$	0	0
$178$	−2.31336	−0.173393
$179$	−11.4330	−0.854540	−0.427270	−	0.904124i	$-0.640525\pi$
$179$	−11.4330	−0.854540	−0.427270	+	0.904124i	$0.640525\pi$
$180$	0	0
$181$	20.9705	1.55872	0.779361	−	0.626575i	$-0.215544\pi$
$181$	20.9705	1.55872	0.779361	+	0.626575i	$0.215544\pi$
$182$	0	0
$183$	0	0
$184$	−4.02177	−0.296489
$185$	−1.40581	−0.103357
$186$	0	0
$187$	−9.16852	−0.670469
$188$	−9.98254	−0.728052
$189$	0	0
$190$	−0.0174584	−0.00126657
$191$	14.4373	1.04464	0.522322	−	0.852748i	$-0.325066\pi$
$191$	14.4373	1.04464	0.522322	+	0.852748i	$0.325066\pi$
$192$	0	0
$193$	−13.5797	−0.977489	−0.488745	−	0.872427i	$-0.662545\pi$
$193$	−13.5797	−0.977489	−0.488745	+	0.872427i	$0.662545\pi$
$194$	6.46250	0.463980
$195$	0	0
$196$	1.96077	0.140055
$197$	0.560335	0.0399222	0.0199611	−	0.999801i	$-0.493646\pi$
$197$	0.560335	0.0399222	0.0199611	+	0.999801i	$0.493646\pi$
$198$	0	0
$199$	11.4916	0.814616	0.407308	−	0.913291i	$-0.366468\pi$
$199$	11.4916	0.814616	0.407308	+	0.913291i	$0.366468\pi$
$200$	−13.2959	−0.940162
$201$	0	0
$202$	−10.7095	−0.753516
$203$	10.9148	0.766071
$204$	0	0
$205$	−2.86294	−0.199956
$206$	−1.09246	−0.0761151
$207$	0	0
$208$	0	0
$209$	−0.374354	−0.0258946
$210$	0	0
$211$	8.78448	0.604748	0.302374	−	0.953189i	$-0.402221\pi$
$211$	8.78448	0.604748	0.302374	+	0.953189i	$0.402221\pi$
$212$	−14.1032	−0.968613
$213$	0	0
$214$	2.62133	0.179191
$215$	0.0730805	0.00498405
$216$	0	0
$217$	15.6286	1.06094
$218$	−12.5918	−0.852824
$219$	0	0
$220$	1.42327	0.0959570
$221$	0	0
$222$	0	0
$223$	−2.25906	−0.151278	−0.0756390	−	0.997135i	$-0.524100\pi$
$223$	−2.25906	−0.151278	−0.0756390	+	0.997135i	$0.524100\pi$
$224$	13.7385	0.917945
$225$	0	0
$226$	9.66248	0.642739
$227$	6.96615	0.462359	0.231180	−	0.972911i	$-0.425741\pi$
$227$	6.96615	0.462359	0.231180	+	0.972911i	$0.425741\pi$
$228$	0	0
$229$	−24.1739	−1.59746	−0.798728	−	0.601692i	$-0.794493\pi$
$229$	−24.1739	−1.59746	−0.798728	+	0.601692i	$0.794493\pi$
$230$	−0.295897	−0.0195109
$231$	0	0
$232$	−12.4668	−0.818486
$233$	3.06100	0.200533	0.100266	−	0.994961i	$-0.468031\pi$
$233$	3.06100	0.200533	0.100266	+	0.994961i	$0.468031\pi$
$234$	0	0
$235$	−1.81700	−0.118528
$236$	−9.19998	−0.598868
$237$	0	0
$238$	−4.08038	−0.264492
$239$	25.1468	1.62661	0.813304	−	0.581839i	$-0.197667\pi$
$239$	25.1468	1.62661	0.813304	+	0.581839i	$0.197667\pi$
$240$	0	0
$241$	−20.2664	−1.30547	−0.652735	−	0.757586i	$-0.726379\pi$
$241$	−20.2664	−1.30547	−0.652735	+	0.757586i	$0.726379\pi$
$242$	−5.64310	−0.362752
$243$	0	0
$244$	−4.71140	−0.301616
$245$	0.356896	0.0228012
$246$	0	0
$247$	0	0
$248$	−17.8509	−1.13353
$249$	0	0
$250$	−1.96854	−0.124501
$251$	23.7211	1.49726	0.748631	−	0.662987i	$-0.230712\pi$
$251$	23.7211	1.49726	0.748631	+	0.662987i	$0.230712\pi$
$252$	0	0
$253$	−6.34481	−0.398895
$254$	7.86486	0.493485
$255$	0	0
$256$	−14.1860	−0.886624
$257$	−14.2241	−0.887278	−0.443639	−	0.896206i	$-0.646313\pi$
$257$	−14.2241	−0.887278	−0.443639	+	0.896206i	$0.646313\pi$
$258$	0	0
$259$	−13.4155	−0.833599
$260$	0	0
$261$	0	0
$262$	−5.27413	−0.325837
$263$	17.0954	1.05415	0.527075	−	0.849819i	$-0.323289\pi$
$263$	17.0954	1.05415	0.527075	+	0.849819i	$0.323289\pi$
$264$	0	0
$265$	−2.56704	−0.157692
$266$	−0.166603	−0.0102151
$267$	0	0
$268$	−10.4209	−0.636556
$269$	6.46681	0.394288	0.197144	−	0.980374i	$-0.436833\pi$
$269$	6.46681	0.394288	0.197144	+	0.980374i	$0.436833\pi$
$270$	0	0
$271$	6.44803	0.391690	0.195845	−	0.980635i	$-0.437255\pi$
$271$	6.44803	0.391690	0.195845	+	0.980635i	$0.437255\pi$
$272$	−1.19806	−0.0726432
$273$	0	0
$274$	4.98792	0.301331
$275$	−20.9758	−1.26489
$276$	0	0
$277$	13.4601	0.808739	0.404370	−	0.914596i	$-0.367491\pi$
$277$	13.4601	0.808739	0.404370	+	0.914596i	$0.367491\pi$
$278$	11.7942	0.707367
$279$	0	0
$280$	1.56704	0.0936485
$281$	−5.03684	−0.300472	−0.150236	−	0.988650i	$-0.548003\pi$
$281$	−5.03684	−0.300472	−0.150236	+	0.988650i	$0.548003\pi$
$282$	0	0
$283$	22.1280	1.31537	0.657686	−	0.753293i	$-0.271536\pi$
$283$	22.1280	1.31537	0.657686	+	0.753293i	$0.271536\pi$
$284$	−11.7573	−0.697669
$285$	0	0
$286$	0	0
$287$	−27.3207	−1.61269
$288$	0	0
$289$	−12.3394	−0.725849
$290$	−0.917231	−0.0538616
$291$	0	0
$292$	−9.13946	−0.534846
$293$	−14.9463	−0.873172	−0.436586	−	0.899663i	$-0.643813\pi$
$293$	−14.9463	−0.873172	−0.436586	+	0.899663i	$0.643813\pi$
$294$	0	0
$295$	−1.67456	−0.0974968
$296$	15.3230	0.890634
$297$	0	0
$298$	3.47650	0.201388
$299$	0	0
$300$	0	0
$301$	0.697398	0.0401974
$302$	−3.16315	−0.182019
$303$	0	0
$304$	−0.0489173	−0.00280560
$305$	−0.857560	−0.0491037
$306$	0	0
$307$	19.1293	1.09177	0.545883	−	0.837861i	$-0.316194\pi$
$307$	19.1293	1.09177	0.545883	+	0.837861i	$0.316194\pi$
$308$	13.5821	0.773912
$309$	0	0
$310$	−1.31336	−0.0745936
$311$	0.269815	0.0152998	0.00764990	−	0.999971i	$-0.497565\pi$
$311$	0.269815	0.0152998	0.00764990	+	0.999971i	$0.497565\pi$
$312$	0	0
$313$	−23.3937	−1.32229	−0.661146	−	0.750257i	$-0.729930\pi$
$313$	−23.3937	−1.32229	−0.661146	+	0.750257i	$0.729930\pi$
$314$	−3.57242	−0.201603
$315$	0	0
$316$	−13.5289	−0.761059
$317$	13.9952	0.786050	0.393025	−	0.919528i	$-0.371429\pi$
$317$	13.9952	0.786050	0.393025	+	0.919528i	$0.371429\pi$
$318$	0	0
$319$	−19.6679	−1.10119
$320$	−0.880395	−0.0492156
$321$	0	0
$322$	−2.82371	−0.157359
$323$	0.190293	0.0105882
$324$	0	0
$325$	0	0
$326$	−12.9584	−0.717698
$327$	0	0
$328$	31.2054	1.72303
$329$	−17.3394	−0.955954
$330$	0	0
$331$	17.8213	0.979548	0.489774	−	0.871849i	$-0.337079\pi$
$331$	17.8213	0.979548	0.489774	+	0.871849i	$0.337079\pi$
$332$	2.18359	0.119840
$333$	0	0
$334$	12.9250	0.707225
$335$	−1.89679	−0.103633
$336$	0	0
$337$	−27.8485	−1.51700	−0.758501	−	0.651672i	$-0.774068\pi$
$337$	−27.8485	−1.51700	−0.758501	+	0.651672i	$0.774068\pi$
$338$	0	0
$339$	0	0
$340$	−0.723480	−0.0392362
$341$	−28.1618	−1.52505
$342$	0	0
$343$	19.9041	1.07472
$344$	−0.796561	−0.0429477
$345$	0	0
$346$	−17.2707	−0.928477
$347$	−1.50365	−0.0807200	−0.0403600	−	0.999185i	$-0.512850\pi$
$347$	−1.50365	−0.0807200	−0.0403600	+	0.999185i	$0.512850\pi$
$348$	0	0
$349$	−14.1860	−0.759358	−0.379679	−	0.925118i	$-0.623966\pi$
$349$	−14.1860	−0.759358	−0.379679	+	0.925118i	$0.623966\pi$
$350$	−9.33513	−0.498983
$351$	0	0
$352$	−24.7560	−1.31950
$353$	7.16852	0.381542	0.190771	−	0.981635i	$-0.438901\pi$
$353$	7.16852	0.381542	0.190771	+	0.981635i	$0.438901\pi$
$354$	0	0
$355$	−2.14005	−0.113582
$356$	−3.91425	−0.207455
$357$	0	0
$358$	9.16852	0.484571
$359$	−19.8853	−1.04951	−0.524753	−	0.851255i	$-0.675842\pi$
$359$	−19.8853	−1.04951	−0.524753	+	0.851255i	$0.675842\pi$
$360$	0	0
$361$	−18.9922	−0.999591
$362$	−16.8170	−0.883882
$363$	0	0
$364$	0	0
$365$	−1.66355	−0.0870740
$366$	0	0
$367$	1.08383	0.0565757	0.0282878	−	0.999600i	$-0.490994\pi$
$367$	1.08383	0.0565757	0.0282878	+	0.999600i	$0.490994\pi$
$368$	−0.829085	−0.0432190
$369$	0	0
$370$	1.12737	0.0586094
$371$	−24.4969	−1.27182
$372$	0	0
$373$	−6.13036	−0.317418	−0.158709	−	0.987325i	$-0.550733\pi$
$373$	−6.13036	−0.317418	−0.158709	+	0.987325i	$0.550733\pi$
$374$	7.35258	0.380193
$375$	0	0
$376$	19.8049	1.02136
$377$	0	0
$378$	0	0
$379$	−2.40880	−0.123732	−0.0618658	−	0.998084i	$-0.519705\pi$
$379$	−2.40880	−0.123732	−0.0618658	+	0.998084i	$0.519705\pi$
$380$	−0.0295400	−0.00151537
$381$	0	0
$382$	−11.5778	−0.592371
$383$	30.3913	1.55292	0.776462	−	0.630164i	$-0.217012\pi$
$383$	30.3913	1.55292	0.776462	+	0.630164i	$0.217012\pi$
$384$	0	0
$385$	2.47219	0.125994
$386$	10.8901	0.554291
$387$	0	0
$388$	10.9347	0.555125
$389$	15.9409	0.808237	0.404118	−	0.914707i	$-0.367578\pi$
$389$	15.9409	0.808237	0.404118	+	0.914707i	$0.367578\pi$
$390$	0	0
$391$	3.22521	0.163106
$392$	−3.89008	−0.196479
$393$	0	0
$394$	−0.449354	−0.0226381
$395$	−2.46250	−0.123902
$396$	0	0
$397$	16.9148	0.848931	0.424466	−	0.905444i	$-0.360462\pi$
$397$	16.9148	0.848931	0.424466	+	0.905444i	$0.360462\pi$
$398$	−9.21552	−0.461932
$399$	0	0
$400$	−2.74094	−0.137047
$401$	−26.6625	−1.33146	−0.665730	−	0.746192i	$-0.731880\pi$
$401$	−26.6625	−1.33146	−0.665730	+	0.746192i	$0.731880\pi$
$402$	0	0
$403$	0	0
$404$	−18.1207	−0.901537
$405$	0	0
$406$	−8.75302	−0.434405
$407$	24.1739	1.19826
$408$	0	0
$409$	28.5163	1.41004	0.705021	−	0.709187i	$-0.250938\pi$
$409$	28.5163	1.41004	0.705021	+	0.709187i	$0.250938\pi$
$410$	2.29590	0.113386
$411$	0	0
$412$	−1.84846	−0.0910672
$413$	−15.9801	−0.786332
$414$	0	0
$415$	0.397452	0.0195102
$416$	0	0
$417$	0	0
$418$	0.300209	0.0146837
$419$	29.6093	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$419$	29.6093	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$420$	0	0
$421$	−11.6606	−0.568301	−0.284151	−	0.958780i	$-0.591712\pi$
$421$	−11.6606	−0.568301	−0.284151	+	0.958780i	$0.591712\pi$
$422$	−7.04461	−0.342926
$423$	0	0
$424$	27.9801	1.35884
$425$	10.6625	0.517206
$426$	0	0
$427$	−8.18359	−0.396032
$428$	4.43535	0.214391
$429$	0	0
$430$	−0.0586060	−0.00282623
$431$	−4.34913	−0.209490	−0.104745	−	0.994499i	$-0.533403\pi$
$431$	−4.34913	−0.209490	−0.104745	+	0.994499i	$0.533403\pi$
$432$	0	0
$433$	−14.3884	−0.691460	−0.345730	−	0.938334i	$-0.612369\pi$
$433$	−14.3884	−0.691460	−0.345730	+	0.938334i	$0.612369\pi$
$434$	−12.5332	−0.601612
$435$	0	0
$436$	−21.3056	−1.02035
$437$	0.131687	0.00629942
$438$	0	0
$439$	−20.2325	−0.965645	−0.482822	−	0.875718i	$-0.660388\pi$
$439$	−20.2325	−0.965645	−0.482822	+	0.875718i	$0.660388\pi$
$440$	−2.82371	−0.134615
$441$	0	0
$442$	0	0
$443$	−8.12200	−0.385888	−0.192944	−	0.981210i	$-0.561804\pi$
$443$	−8.12200	−0.385888	−0.192944	+	0.981210i	$0.561804\pi$
$444$	0	0
$445$	−0.712464	−0.0337740
$446$	1.81163	0.0857830
$447$	0	0
$448$	−8.40150	−0.396934
$449$	−12.4916	−0.589513	−0.294757	−	0.955572i	$-0.595239\pi$
$449$	−12.4916	−0.589513	−0.294757	+	0.955572i	$0.595239\pi$
$450$	0	0
$451$	49.2301	2.31816
$452$	16.3491	0.768998
$453$	0	0
$454$	−5.58642	−0.262184
$455$	0	0
$456$	0	0
$457$	5.98121	0.279789	0.139895	−	0.990166i	$-0.455324\pi$
$457$	5.98121	0.279789	0.139895	+	0.990166i	$0.455324\pi$
$458$	19.3860	0.905847
$459$	0	0
$460$	−0.500664	−0.0233436
$461$	2.05669	0.0957895	0.0478947	−	0.998852i	$-0.484749\pi$
$461$	2.05669	0.0957895	0.0478947	+	0.998852i	$0.484749\pi$
$462$	0	0
$463$	−8.44935	−0.392675	−0.196337	−	0.980536i	$-0.562905\pi$
$463$	−8.44935	−0.392675	−0.196337	+	0.980536i	$0.562905\pi$
$464$	−2.57002	−0.119310
$465$	0	0
$466$	−2.45473	−0.113713
$467$	−33.5139	−1.55084	−0.775420	−	0.631446i	$-0.782462\pi$
$467$	−33.5139	−1.55084	−0.775420	+	0.631446i	$0.782462\pi$
$468$	0	0
$469$	−18.1008	−0.835818
$470$	1.45712	0.0672121
$471$	0	0
$472$	18.2524	0.840133
$473$	−1.25667	−0.0577817
$474$	0	0
$475$	0.435353	0.0199754
$476$	−6.90408	−0.316448
$477$	0	0
$478$	−20.1661	−0.922377
$479$	24.7313	1.13000	0.565000	−	0.825091i	$-0.308876\pi$
$479$	24.7313	1.13000	0.565000	+	0.825091i	$0.308876\pi$
$480$	0	0
$481$	0	0
$482$	16.2524	0.740275
$483$	0	0
$484$	−9.54825	−0.434012
$485$	1.99031	0.0903754
$486$	0	0
$487$	−37.7555	−1.71087	−0.855433	−	0.517913i	$-0.826709\pi$
$487$	−37.7555	−1.71087	−0.855433	+	0.517913i	$0.826709\pi$
$488$	9.34721	0.423128
$489$	0	0
$490$	−0.286208	−0.0129296
$491$	−31.3110	−1.41304	−0.706522	−	0.707691i	$-0.749737\pi$
$491$	−31.3110	−1.41304	−0.706522	+	0.707691i	$0.749737\pi$
$492$	0	0
$493$	9.99761	0.450270
$494$	0	0
$495$	0	0
$496$	−3.67994	−0.165234
$497$	−20.4222	−0.916061
$498$	0	0
$499$	21.4873	0.961902	0.480951	−	0.876748i	$-0.340292\pi$
$499$	21.4873	0.961902	0.480951	+	0.876748i	$0.340292\pi$
$500$	−3.33081	−0.148959
$501$	0	0
$502$	−19.0228	−0.849031
$503$	−37.5924	−1.67616	−0.838081	−	0.545546i	$-0.816322\pi$
$503$	−37.5924	−1.67616	−0.838081	+	0.545546i	$0.816322\pi$
$504$	0	0
$505$	−3.29829	−0.146772
$506$	5.08815	0.226196
$507$	0	0
$508$	13.3075	0.590425
$509$	17.1075	0.758278	0.379139	−	0.925340i	$-0.376220\pi$
$509$	17.1075	0.758278	0.379139	+	0.925340i	$0.376220\pi$
$510$	0	0
$511$	−15.8750	−0.702269
$512$	−6.22282	−0.275012
$513$	0	0
$514$	11.4069	0.503136
$515$	−0.336454	−0.0148259
$516$	0	0
$517$	31.2446	1.37414
$518$	10.7584	0.472697
$519$	0	0
$520$	0	0
$521$	19.8465	0.869493	0.434746	−	0.900553i	$-0.356838\pi$
$521$	19.8465	0.869493	0.434746	+	0.900553i	$0.356838\pi$
$522$	0	0
$523$	−11.4300	−0.499798	−0.249899	−	0.968272i	$-0.580397\pi$
$523$	−11.4300	−0.499798	−0.249899	+	0.968272i	$0.580397\pi$
$524$	−8.92394	−0.389844
$525$	0	0
$526$	−13.7095	−0.597762
$527$	14.3153	0.623583
$528$	0	0
$529$	−20.7681	−0.902960
$530$	2.05861	0.0894201
$531$	0	0
$532$	−0.281896	−0.0122218
$533$	0	0
$534$	0	0
$535$	0.807315	0.0349033
$536$	20.6746	0.893005
$537$	0	0
$538$	−5.18598	−0.223584
$539$	−6.13706	−0.264342
$540$	0	0
$541$	16.1884	0.695993	0.347996	−	0.937496i	$-0.386862\pi$
$541$	16.1884	0.695993	0.347996	+	0.937496i	$0.386862\pi$
$542$	−5.17092	−0.222110
$543$	0	0
$544$	12.5840	0.539536
$545$	−3.87800	−0.166115
$546$	0	0
$547$	5.33081	0.227929	0.113965	−	0.993485i	$-0.463645\pi$
$547$	5.33081	0.227929	0.113965	+	0.993485i	$0.463645\pi$
$548$	8.43967	0.360525
$549$	0	0
$550$	16.8213	0.717263
$551$	0.408206	0.0173902
$552$	0	0
$553$	−23.4993	−0.999293
$554$	−10.7942	−0.458600
$555$	0	0
$556$	19.9560	0.846322
$557$	−7.39075	−0.313156	−0.156578	−	0.987666i	$-0.550046\pi$
$557$	−7.39075	−0.313156	−0.156578	+	0.987666i	$0.550046\pi$
$558$	0	0
$559$	0	0
$560$	0.323044	0.0136511
$561$	0	0
$562$	4.03923	0.170385
$563$	9.47889	0.399488	0.199744	−	0.979848i	$-0.435989\pi$
$563$	9.47889	0.399488	0.199744	+	0.979848i	$0.435989\pi$
$564$	0	0
$565$	2.97584	0.125194
$566$	−17.7453	−0.745889
$567$	0	0
$568$	23.3260	0.978738
$569$	10.1438	0.425249	0.212624	−	0.977134i	$-0.431799\pi$
$569$	10.1438	0.425249	0.212624	+	0.977134i	$0.431799\pi$
$570$	0	0
$571$	−14.0925	−0.589751	−0.294876	−	0.955536i	$-0.595278\pi$
$571$	−14.0925	−0.589751	−0.294876	+	0.955536i	$0.595278\pi$
$572$	0	0
$573$	0	0
$574$	21.9095	0.914483
$575$	7.37867	0.307712
$576$	0	0
$577$	25.1545	1.04720	0.523598	−	0.851965i	$-0.324589\pi$
$577$	25.1545	1.04720	0.523598	+	0.851965i	$0.324589\pi$
$578$	9.89546	0.411597
$579$	0	0
$580$	−1.55197	−0.0644422
$581$	3.79284	0.157354
$582$	0	0
$583$	44.1420	1.82817
$584$	18.1323	0.750319
$585$	0	0
$586$	11.9860	0.495137
$587$	−43.8353	−1.80928	−0.904639	−	0.426180i	$-0.859859\pi$
$587$	−43.8353	−1.80928	−0.904639	+	0.426180i	$0.859859\pi$
$588$	0	0
$589$	0.584498	0.0240838
$590$	1.34290	0.0552861
$591$	0	0
$592$	3.15883	0.129827
$593$	24.9965	1.02648	0.513242	−	0.858244i	$-0.328444\pi$
$593$	24.9965	1.02648	0.513242	+	0.858244i	$0.328444\pi$
$594$	0	0
$595$	−1.25667	−0.0515184
$596$	5.88231	0.240949
$597$	0	0
$598$	0	0
$599$	6.24027	0.254971	0.127485	−	0.991840i	$-0.459309\pi$
$599$	6.24027	0.254971	0.127485	+	0.991840i	$0.459309\pi$
$600$	0	0
$601$	6.32975	0.258196	0.129098	−	0.991632i	$-0.458792\pi$
$601$	6.32975	0.258196	0.129098	+	0.991632i	$0.458792\pi$
$602$	−0.559270	−0.0227941
$603$	0	0
$604$	−5.35211	−0.217774
$605$	−1.73795	−0.0706579
$606$	0	0
$607$	−43.6480	−1.77162	−0.885809	−	0.464050i	$-0.846396\pi$
$607$	−43.6480	−1.77162	−0.885809	+	0.464050i	$0.846396\pi$
$608$	0.513811	0.0208378
$609$	0	0
$610$	0.687710	0.0278445
$611$	0	0
$612$	0	0
$613$	−25.9541	−1.04827	−0.524137	−	0.851634i	$-0.675612\pi$
$613$	−25.9541	−1.04827	−0.524137	+	0.851634i	$0.675612\pi$
$614$	−15.3405	−0.619092
$615$	0	0
$616$	−26.9463	−1.08570
$617$	−45.9396	−1.84946	−0.924729	−	0.380626i	$-0.875709\pi$
$617$	−45.9396	−1.84946	−0.924729	+	0.380626i	$0.875709\pi$
$618$	0	0
$619$	6.73556	0.270725	0.135363	−	0.990796i	$-0.456780\pi$
$619$	6.73556	0.270725	0.135363	+	0.990796i	$0.456780\pi$
$620$	−2.22223	−0.0892467
$621$	0	0
$622$	−0.216375	−0.00867583
$623$	−6.79895	−0.272394
$624$	0	0
$625$	24.0887	0.963549
$626$	18.7603	0.749813
$627$	0	0
$628$	−6.04461	−0.241206
$629$	−12.2881	−0.489960
$630$	0	0
$631$	45.0998	1.79539	0.897696	−	0.440614i	$-0.145239\pi$
$631$	45.0998	1.79539	0.897696	+	0.440614i	$0.145239\pi$
$632$	26.8407	1.06767
$633$	0	0
$634$	−11.2233	−0.445734
$635$	2.42221	0.0961223
$636$	0	0
$637$	0	0
$638$	15.7724	0.624435
$639$	0	0
$640$	−2.17331	−0.0859075
$641$	−32.5821	−1.28692	−0.643458	−	0.765482i	$-0.722501\pi$
$641$	−32.5821	−1.28692	−0.643458	+	0.765482i	$0.722501\pi$
$642$	0	0
$643$	25.5754	1.00860	0.504298	−	0.863530i	$-0.331751\pi$
$643$	25.5754	1.00860	0.504298	+	0.863530i	$0.331751\pi$
$644$	−4.77777	−0.188271
$645$	0	0
$646$	−0.152603	−0.00600408
$647$	30.1715	1.18616	0.593082	−	0.805142i	$-0.297911\pi$
$647$	30.1715	1.18616	0.593082	+	0.805142i	$0.297911\pi$
$648$	0	0
$649$	28.7952	1.13031
$650$	0	0
$651$	0	0
$652$	−21.9259	−0.858683
$653$	−36.9028	−1.44412	−0.722058	−	0.691832i	$-0.756804\pi$
$653$	−36.9028	−1.44412	−0.722058	+	0.691832i	$0.756804\pi$
$654$	0	0
$655$	−1.62432	−0.0634673
$656$	6.43296	0.251165
$657$	0	0
$658$	13.9051	0.542079
$659$	−23.6866	−0.922701	−0.461350	−	0.887218i	$-0.652635\pi$
$659$	−23.6866	−0.922701	−0.461350	+	0.887218i	$0.652635\pi$
$660$	0	0
$661$	−31.7590	−1.23528	−0.617641	−	0.786460i	$-0.711911\pi$
$661$	−31.7590	−1.23528	−0.617641	+	0.786460i	$0.711911\pi$
$662$	−14.2916	−0.555458
$663$	0	0
$664$	−4.33214	−0.168120
$665$	−0.0513102	−0.00198973
$666$	0	0
$667$	6.91856	0.267888
$668$	21.8694	0.846152
$669$	0	0
$670$	1.52111	0.0587655
$671$	14.7463	0.569275
$672$	0	0
$673$	−7.50232	−0.289193	−0.144597	−	0.989491i	$-0.546188\pi$
$673$	−7.50232	−0.289193	−0.144597	+	0.989491i	$0.546188\pi$
$674$	22.3327	0.860225
$675$	0	0
$676$	0	0
$677$	35.0315	1.34637	0.673184	−	0.739475i	$-0.264926\pi$
$677$	35.0315	1.34637	0.673184	+	0.739475i	$0.264926\pi$
$678$	0	0
$679$	18.9933	0.728896
$680$	1.43535	0.0550433
$681$	0	0
$682$	22.5840	0.864787
$683$	24.0834	0.921524	0.460762	−	0.887524i	$-0.347576\pi$
$683$	24.0834	0.921524	0.460762	+	0.887524i	$0.347576\pi$
$684$	0	0
$685$	1.53617	0.0586941
$686$	−15.9618	−0.609426
$687$	0	0
$688$	−0.164210	−0.00626046
$689$	0	0
$690$	0	0
$691$	2.01447	0.0766342	0.0383171	−	0.999266i	$-0.487800\pi$
$691$	2.01447	0.0766342	0.0383171	+	0.999266i	$0.487800\pi$
$692$	−29.2223	−1.11087
$693$	0	0
$694$	1.20583	0.0457728
$695$	3.63235	0.137783
$696$	0	0
$697$	−25.0248	−0.947880
$698$	11.3763	0.430598
$699$	0	0
$700$	−15.7952	−0.597004
$701$	48.8189	1.84387	0.921933	−	0.387350i	$-0.126610\pi$
$701$	48.8189	1.84387	0.921933	+	0.387350i	$0.126610\pi$
$702$	0	0
$703$	−0.501729	−0.0189231
$704$	15.1390	0.570572
$705$	0	0
$706$	−5.74871	−0.216355
$707$	−31.4752	−1.18375
$708$	0	0
$709$	−20.8060	−0.781385	−0.390693	−	0.920521i	$-0.627764\pi$
$709$	−20.8060	−0.781385	−0.390693	+	0.920521i	$0.627764\pi$
$710$	1.71618	0.0644073
$711$	0	0
$712$	7.76569	0.291032
$713$	9.90648	0.371000
$714$	0	0
$715$	0	0
$716$	15.5133	0.579761
$717$	0	0
$718$	15.9468	0.595128
$719$	−21.4306	−0.799225	−0.399613	−	0.916684i	$-0.630855\pi$
$719$	−21.4306	−0.799225	−0.399613	+	0.916684i	$0.630855\pi$
$720$	0	0
$721$	−3.21073	−0.119574
$722$	15.2306	0.566824
$723$	0	0
$724$	−28.4547	−1.05751
$725$	22.8726	0.849468
$726$	0	0
$727$	13.4862	0.500175	0.250088	−	0.968223i	$-0.419541\pi$
$727$	13.4862	0.500175	0.250088	+	0.968223i	$0.419541\pi$
$728$	0	0
$729$	0	0
$730$	1.33406	0.0493758
$731$	0.638792	0.0236266
$732$	0	0
$733$	−43.5424	−1.60828	−0.804138	−	0.594443i	$-0.797373\pi$
$733$	−43.5424	−1.60828	−0.804138	+	0.594443i	$0.797373\pi$
$734$	−0.869167	−0.0320816
$735$	0	0
$736$	8.70841	0.320996
$737$	32.6165	1.20145
$738$	0	0
$739$	20.0543	0.737709	0.368855	−	0.929487i	$-0.379750\pi$
$739$	20.0543	0.737709	0.368855	+	0.929487i	$0.379750\pi$
$740$	1.90754	0.0701226
$741$	0	0
$742$	19.6450	0.721191
$743$	33.1685	1.21684	0.608418	−	0.793617i	$-0.291805\pi$
$743$	33.1685	1.21684	0.608418	+	0.793617i	$0.291805\pi$
$744$	0	0
$745$	1.07069	0.0392270
$746$	4.91617	0.179994
$747$	0	0
$748$	12.4407	0.454878
$749$	7.70410	0.281502
$750$	0	0
$751$	39.2814	1.43340	0.716700	−	0.697382i	$-0.245652\pi$
$751$	39.2814	1.43340	0.716700	+	0.697382i	$0.245652\pi$
$752$	4.08277	0.148883
$753$	0	0
$754$	0	0
$755$	−0.974181	−0.0354541
$756$	0	0
$757$	−46.6426	−1.69526	−0.847628	−	0.530592i	$-0.821970\pi$
$757$	−46.6426	−1.69526	−0.847628	+	0.530592i	$0.821970\pi$
$758$	1.93171	0.0701627
$759$	0	0
$760$	0.0586060	0.00212586
$761$	21.8984	0.793818	0.396909	−	0.917858i	$-0.370083\pi$
$761$	21.8984	0.793818	0.396909	+	0.917858i	$0.370083\pi$
$762$	0	0
$763$	−37.0073	−1.33975
$764$	−19.5899	−0.708737
$765$	0	0
$766$	−24.3720	−0.880595
$767$	0	0
$768$	0	0
$769$	46.7096	1.68439	0.842196	−	0.539172i	$-0.181263\pi$
$769$	46.7096	1.68439	0.842196	+	0.539172i	$0.181263\pi$
$770$	−1.98254	−0.0714458
$771$	0	0
$772$	18.4263	0.663175
$773$	30.2416	1.08771	0.543857	−	0.839178i	$-0.316963\pi$
$773$	30.2416	1.08771	0.543857	+	0.839178i	$0.316963\pi$
$774$	0	0
$775$	32.7506	1.17644
$776$	−21.6939	−0.778767
$777$	0	0
$778$	−12.7836	−0.458315
$779$	−1.02177	−0.0366087
$780$	0	0
$781$	36.7995	1.31679
$782$	−2.58642	−0.0924901
$783$	0	0
$784$	−0.801938	−0.0286406
$785$	−1.10023	−0.0392688
$786$	0	0
$787$	28.7023	1.02313	0.511563	−	0.859246i	$-0.329067\pi$
$787$	28.7023	1.02313	0.511563	+	0.859246i	$0.329067\pi$
$788$	−0.760316	−0.0270851
$789$	0	0
$790$	1.97477	0.0702592
$791$	28.3980	1.00972
$792$	0	0
$793$	0	0
$794$	−13.5646	−0.481391
$795$	0	0
$796$	−15.5929	−0.552674
$797$	18.5418	0.656785	0.328392	−	0.944541i	$-0.393493\pi$
$797$	18.5418	0.656785	0.328392	+	0.944541i	$0.393493\pi$
$798$	0	0
$799$	−15.8823	−0.561876
$800$	28.7899	1.01788
$801$	0	0
$802$	21.3817	0.755012
$803$	28.6058	1.00948
$804$	0	0
$805$	−0.869641	−0.0306508
$806$	0	0
$807$	0	0
$808$	35.9506	1.26474
$809$	10.0677	0.353962	0.176981	−	0.984214i	$-0.443367\pi$
$809$	10.0677	0.353962	0.176981	+	0.984214i	$0.443367\pi$
$810$	0	0
$811$	−10.0285	−0.352147	−0.176074	−	0.984377i	$-0.556340\pi$
$811$	−10.0285	−0.352147	−0.176074	+	0.984377i	$0.556340\pi$
$812$	−14.8103	−0.519740
$813$	0	0
$814$	−19.3860	−0.679478
$815$	−3.99090	−0.139795
$816$	0	0
$817$	0.0260821	0.000912498 0
$818$	−22.8683	−0.799572
$819$	0	0
$820$	3.88471	0.135660
$821$	26.1704	0.913355	0.456677	−	0.889632i	$-0.349039\pi$
$821$	26.1704	0.913355	0.456677	+	0.889632i	$0.349039\pi$
$822$	0	0
$823$	1.82238	0.0635242	0.0317621	−	0.999495i	$-0.489888\pi$
$823$	1.82238	0.0635242	0.0317621	+	0.999495i	$0.489888\pi$
$824$	3.66727	0.127755
$825$	0	0
$826$	12.8151	0.445894
$827$	32.2941	1.12298	0.561488	−	0.827485i	$-0.310229\pi$
$827$	32.2941	1.12298	0.561488	+	0.827485i	$0.310229\pi$
$828$	0	0
$829$	15.1002	0.524453	0.262226	−	0.965006i	$-0.415543\pi$
$829$	15.1002	0.524453	0.262226	+	0.965006i	$0.415543\pi$
$830$	−0.318732	−0.0110634
$831$	0	0
$832$	0	0
$833$	3.11960	0.108088
$834$	0	0
$835$	3.98062	0.137755
$836$	0.507960	0.0175682
$837$	0	0
$838$	−23.7448	−0.820250
$839$	32.9965	1.13917	0.569584	−	0.821933i	$-0.307104\pi$
$839$	32.9965	1.13917	0.569584	+	0.821933i	$0.307104\pi$
$840$	0	0
$841$	−7.55363	−0.260470
$842$	9.35105	0.322258
$843$	0	0
$844$	−11.9196	−0.410290
$845$	0	0
$846$	0	0
$847$	−16.5851	−0.569870
$848$	5.76809	0.198077
$849$	0	0
$850$	−8.55065	−0.293285
$851$	−8.50365	−0.291501
$852$	0	0
$853$	−37.7802	−1.29357	−0.646784	−	0.762673i	$-0.723887\pi$
$853$	−37.7802	−1.29357	−0.646784	+	0.762673i	$0.723887\pi$
$854$	6.56273	0.224572
$855$	0	0
$856$	−8.79954	−0.300762
$857$	−27.3623	−0.934677	−0.467339	−	0.884078i	$-0.654787\pi$
$857$	−27.3623	−0.934677	−0.467339	+	0.884078i	$0.654787\pi$
$858$	0	0
$859$	−20.0629	−0.684538	−0.342269	−	0.939602i	$-0.611195\pi$
$859$	−20.0629	−0.684538	−0.342269	+	0.939602i	$0.611195\pi$
$860$	−0.0991626	−0.00338142
$861$	0	0
$862$	3.48773	0.118792
$863$	6.14483	0.209173	0.104586	−	0.994516i	$-0.466648\pi$
$863$	6.14483	0.209173	0.104586	+	0.994516i	$0.466648\pi$
$864$	0	0
$865$	−5.31900	−0.180851
$866$	11.5386	0.392096
$867$	0	0
$868$	−21.2064	−0.719793
$869$	42.3443	1.43643
$870$	0	0
$871$	0	0
$872$	42.2693	1.43142
$873$	0	0
$874$	−0.105604	−0.00357212
$875$	−5.78554	−0.195587
$876$	0	0
$877$	−13.5077	−0.456123	−0.228061	−	0.973647i	$-0.573239\pi$
$877$	−13.5077	−0.456123	−0.228061	+	0.973647i	$0.573239\pi$
$878$	16.2252	0.547574
$879$	0	0
$880$	−0.582105	−0.0196228
$881$	5.23431	0.176348	0.0881741	−	0.996105i	$-0.471897\pi$
$881$	5.23431	0.176348	0.0881741	+	0.996105i	$0.471897\pi$
$882$	0	0
$883$	−4.57301	−0.153894	−0.0769470	−	0.997035i	$-0.524517\pi$
$883$	−4.57301	−0.153894	−0.0769470	+	0.997035i	$0.524517\pi$
$884$	0	0
$885$	0	0
$886$	6.51334	0.218820
$887$	1.64071	0.0550897	0.0275448	−	0.999621i	$-0.491231\pi$
$887$	1.64071	0.0550897	0.0275448	+	0.999621i	$0.491231\pi$
$888$	0	0
$889$	23.1148	0.775246
$890$	0.571352	0.0191517
$891$	0	0
$892$	3.06531	0.102634
$893$	−0.648481	−0.0217006
$894$	0	0
$895$	2.82371	0.0943861
$896$	−20.7396	−0.692862
$897$	0	0
$898$	10.0175	0.334287
$899$	30.7084	1.02418
$900$	0	0
$901$	−22.4383	−0.747529
$902$	−39.4795	−1.31452
$903$	0	0
$904$	−32.4359	−1.07880
$905$	−5.17928	−0.172165
$906$	0	0
$907$	8.10215	0.269027	0.134514	−	0.990912i	$-0.457053\pi$
$907$	8.10215	0.269027	0.134514	+	0.990912i	$0.457053\pi$
$908$	−9.45234	−0.313687
$909$	0	0
$910$	0	0
$911$	9.18119	0.304187	0.152093	−	0.988366i	$-0.451399\pi$
$911$	9.18119	0.304187	0.152093	+	0.988366i	$0.451399\pi$
$912$	0	0
$913$	−6.83446	−0.226188
$914$	−4.79656	−0.158656
$915$	0	0
$916$	32.8015	1.08379
$917$	−15.5007	−0.511877
$918$	0	0
$919$	27.5036	0.907262	0.453631	−	0.891190i	$-0.350128\pi$
$919$	27.5036	0.907262	0.453631	+	0.891190i	$0.350128\pi$
$920$	0.993295	0.0327480
$921$	0	0
$922$	−1.64933	−0.0543180
$923$	0	0
$924$	0	0
$925$	−28.1129	−0.924346
$926$	6.77586	0.222668
$927$	0	0
$928$	26.9946	0.886142
$929$	−24.2131	−0.794407	−0.397203	−	0.917731i	$-0.630019\pi$
$929$	−24.2131	−0.794407	−0.397203	+	0.917731i	$0.630019\pi$
$930$	0	0
$931$	0.127375	0.00417454
$932$	−4.15346	−0.136051
$933$	0	0
$934$	26.8761	0.879412
$935$	2.26444	0.0740550
$936$	0	0
$937$	11.1830	0.365333	0.182666	−	0.983175i	$-0.441527\pi$
$937$	11.1830	0.365333	0.182666	+	0.983175i	$0.441527\pi$
$938$	14.5157	0.473955
$939$	0	0
$940$	2.46548	0.0804152
$941$	15.9638	0.520404	0.260202	−	0.965554i	$-0.416211\pi$
$941$	15.9638	0.520404	0.260202	+	0.965554i	$0.416211\pi$
$942$	0	0
$943$	−17.3177	−0.563941
$944$	3.76271	0.122466
$945$	0	0
$946$	1.00777	0.0327654
$947$	−6.51466	−0.211698	−0.105849	−	0.994382i	$-0.533756\pi$
$947$	−6.51466	−0.211698	−0.105849	+	0.994382i	$0.533756\pi$
$948$	0	0
$949$	0	0
$950$	−0.349126	−0.0113271
$951$	0	0
$952$	13.6974	0.443935
$953$	−47.6469	−1.54344	−0.771718	−	0.635965i	$-0.780602\pi$
$953$	−47.6469	−1.54344	−0.771718	+	0.635965i	$0.780602\pi$
$954$	0	0
$955$	−3.56571	−0.115384
$956$	−34.1215	−1.10357
$957$	0	0
$958$	−19.8329	−0.640773
$959$	14.6595	0.473380
$960$	0	0
$961$	12.9705	0.418402
$962$	0	0
$963$	0	0
$964$	27.4993	0.885694
$965$	3.35391	0.107966
$966$	0	0
$967$	−43.8122	−1.40891	−0.704453	−	0.709751i	$-0.748808\pi$
$967$	−43.8122	−1.40891	−0.704453	+	0.709751i	$0.748808\pi$
$968$	18.9433	0.608861
$969$	0	0
$970$	−1.59611	−0.0512479
$971$	−4.29483	−0.137828	−0.0689139	−	0.997623i	$-0.521953\pi$
$971$	−4.29483	−0.137828	−0.0689139	+	0.997623i	$0.521953\pi$
$972$	0	0
$973$	34.6631	1.11125
$974$	30.2776	0.970156
$975$	0	0
$976$	1.92692	0.0616792
$977$	−26.8019	−0.857470	−0.428735	−	0.903430i	$-0.641041\pi$
$977$	−26.8019	−0.857470	−0.428735	+	0.903430i	$0.641041\pi$
$978$	0	0
$979$	12.2513	0.391553
$980$	−0.484271	−0.0154695
$981$	0	0
$982$	25.1094	0.801275
$983$	−27.2495	−0.869124	−0.434562	−	0.900642i	$-0.643097\pi$
$983$	−27.2495	−0.869124	−0.434562	+	0.900642i	$0.643097\pi$
$984$	0	0
$985$	−0.138391	−0.00440951
$986$	−8.01746	−0.255328
$987$	0	0
$988$	0	0
$989$	0.442058	0.0140566
$990$	0	0
$991$	24.3889	0.774740	0.387370	−	0.921924i	$-0.373384\pi$
$991$	24.3889	0.774740	0.387370	+	0.921924i	$0.373384\pi$
$992$	38.6528	1.22723
$993$	0	0
$994$	16.3773	0.519458
$995$	−2.83818	−0.0899764
$996$	0	0
$997$	31.3207	0.991935	0.495967	−	0.868341i	$-0.334814\pi$
$997$	31.3207	0.991935	0.495967	+	0.868341i	$0.334814\pi$
$998$	−17.2314	−0.545452
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.r.1.1		3
3.2	odd	2		169.2.a.b.1.3	✓	3
12.11	even	2		2704.2.a.z.1.3		3
13.5	odd	4		1521.2.b.l.1351.5		6
13.8	odd	4		1521.2.b.l.1351.2		6
13.12	even	2		1521.2.a.o.1.3		3
15.14	odd	2		4225.2.a.bg.1.1		3
21.20	even	2		8281.2.a.bf.1.3		3
39.2	even	12		169.2.e.b.147.5		12
39.5	even	4		169.2.b.b.168.2		6
39.8	even	4		169.2.b.b.168.5		6
39.11	even	12		169.2.e.b.147.2		12
39.17	odd	6		169.2.c.b.146.3		6
39.20	even	12		169.2.e.b.23.5		12
39.23	odd	6		169.2.c.b.22.3		6
39.29	odd	6		169.2.c.c.22.1		6
39.32	even	12		169.2.e.b.23.2		12
39.35	odd	6		169.2.c.c.146.1		6
39.38	odd	2		169.2.a.c.1.1	yes	3
156.47	odd	4		2704.2.f.o.337.6		6
156.83	odd	4		2704.2.f.o.337.5		6
156.155	even	2		2704.2.a.ba.1.3		3
195.194	odd	2		4225.2.a.bb.1.3		3
273.272	even	2		8281.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.3	✓	3	3.2	odd	2
169.2.a.c.1.1	yes	3	39.38	odd	2
169.2.b.b.168.2		6	39.5	even	4
169.2.b.b.168.5		6	39.8	even	4
169.2.c.b.22.3		6	39.23	odd	6
169.2.c.b.146.3		6	39.17	odd	6
169.2.c.c.22.1		6	39.29	odd	6
169.2.c.c.146.1		6	39.35	odd	6
169.2.e.b.23.2		12	39.32	even	12
169.2.e.b.23.5		12	39.20	even	12
169.2.e.b.147.2		12	39.11	even	12
169.2.e.b.147.5		12	39.2	even	12
1521.2.a.o.1.3		3	13.12	even	2
1521.2.a.r.1.1		3	1.1	even	1	trivial
1521.2.b.l.1351.2		6	13.8	odd	4
1521.2.b.l.1351.5		6	13.5	odd	4
2704.2.a.z.1.3		3	12.11	even	2
2704.2.a.ba.1.3		3	156.155	even	2
2704.2.f.o.337.5		6	156.83	odd	4
2704.2.f.o.337.6		6	156.47	odd	4
4225.2.a.bb.1.3		3	195.194	odd	2
4225.2.a.bg.1.1		3	15.14	odd	2
8281.2.a.bf.1.3		3	21.20	even	2
8281.2.a.bj.1.1		3	273.272	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-0.801938 q^{2} -1.35690 q^{4} -0.246980 q^{5} -2.35690 q^{7} +2.69202 q^{8} +0.198062 q^{10} +4.24698 q^{11} +1.89008 q^{14} +0.554958 q^{16} -2.15883 q^{17} -0.0881460 q^{19} +0.335126 q^{20} -3.40581 q^{22} -1.49396 q^{23} -4.93900 q^{25} +3.19806 q^{28} -4.63102 q^{29} -6.63102 q^{31} -5.82908 q^{32} +1.73125 q^{34} +0.582105 q^{35} +5.69202 q^{37} +0.0706876 q^{38} -0.664874 q^{40} +11.5918 q^{41} -0.295897 q^{43} -5.76271 q^{44} +1.19806 q^{46} +7.35690 q^{47} -1.44504 q^{49} +3.96077 q^{50} +10.3937 q^{53} -1.04892 q^{55} -6.34481 q^{56} +3.71379 q^{58} +6.78017 q^{59} +3.47219 q^{61} +5.31767 q^{62} +3.56465 q^{64} +7.67994 q^{67} +2.92931 q^{68} -0.466812 q^{70} +8.66487 q^{71} +6.73556 q^{73} -4.56465 q^{74} +0.119605 q^{76} -10.0097 q^{77} +9.97046 q^{79} -0.137063 q^{80} -9.29590 q^{82} -1.60925 q^{83} +0.533188 q^{85} +0.237291 q^{86} +11.4330 q^{88} +2.88471 q^{89} +2.02715 q^{92} -5.89977 q^{94} +0.0217703 q^{95} -8.05861 q^{97} +1.15883 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 4 q^{5} - 3 q^{7} + 3 q^{8} + 5 q^{10} + 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} + 14 q^{28} + q^{29} - 5 q^{31} - 7 q^{32} + 13 q^{34} - 4 q^{35}+ \cdots - 5 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 4 * q^5 - 3 * q^7 + 3 * q^8 + 5 * q^10 + 8 * q^11 + 5 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 + 3 * q^22 + 5 * q^23 - 5 * q^25 + 14 * q^28 + q^29 - 5 * q^31 - 7 * q^32 + 13 * q^34 - 4 * q^35 + 12 * q^37 - 12 * q^38 - 3 * q^40 + 7 * q^41 + 13 * q^43 + 8 * q^46 + 18 * q^47 - 4 * q^49 - q^50 - q^53 + 6 * q^55 + 4 * q^56 + 3 * q^58 + 19 * q^59 + 4 * q^61 - q^62 - 11 * q^64 - q^67 + 21 * q^68 + 2 * q^70 + 27 * q^71 + 9 * q^73 + 8 * q^74 - 21 * q^76 - 8 * q^77 - 5 * q^79 + 5 * q^80 - 14 * q^82 + 7 * q^83 + 5 * q^85 + 18 * q^86 + 15 * q^88 + 11 * q^89 + 5 * q^94 - 3 * q^95 + 7 * q^97 - 5 * q^98

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