LMFDB - Embedded newform 7569.2.a.bm.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, 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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 $ x^9 - 4x^8 - 6x^7 + 33x^6 + 6x^5 - 90x^4 + 21x^3 + 84x^2 - 36x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.58860$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58860 q^{2} +4.70087 q^{4} +1.14894 q^{5} +2.76526 q^{7} +6.99148 q^{8} +2.97416 q^{10} -4.35418 q^{11} +5.62914 q^{13} +7.15816 q^{14} +8.69643 q^{16} +2.41516 q^{17} -1.87296 q^{19} +5.40103 q^{20} -11.2712 q^{22} +8.82048 q^{23} -3.67993 q^{25} +14.5716 q^{26} +12.9991 q^{28} -1.56053 q^{31} +8.52865 q^{32} +6.25190 q^{34} +3.17713 q^{35} -7.76330 q^{37} -4.84834 q^{38} +8.03281 q^{40} +3.16072 q^{41} +1.46397 q^{43} -20.4684 q^{44} +22.8327 q^{46} +0.357351 q^{47} +0.646666 q^{49} -9.52588 q^{50} +26.4618 q^{52} +1.63498 q^{53} -5.00270 q^{55} +19.3333 q^{56} +4.85026 q^{59} -1.55510 q^{61} -4.03961 q^{62} +4.68444 q^{64} +6.46755 q^{65} -9.66950 q^{67} +11.3534 q^{68} +8.22432 q^{70} +8.18108 q^{71} -11.3786 q^{73} -20.0961 q^{74} -8.80452 q^{76} -12.0404 q^{77} +8.13075 q^{79} +9.99170 q^{80} +8.18185 q^{82} +0.481835 q^{83} +2.77488 q^{85} +3.78965 q^{86} -30.4421 q^{88} -1.99621 q^{89} +15.5660 q^{91} +41.4639 q^{92} +0.925040 q^{94} -2.15192 q^{95} -4.63516 q^{97} +1.67396 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * 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 $

$2$	2.58860	1.83042	0.915210	−	0.402978i	$-0.132025\pi$
$2$	2.58860	1.83042	0.915210	+	0.402978i	$0.132025\pi$
$3$	0	0
$3$	0	0
$4$	4.70087	2.35043
$5$	1.14894	0.513823	0.256911	−	0.966435i	$-0.417295\pi$
$5$	1.14894	0.513823	0.256911	+	0.966435i	$0.417295\pi$
$6$	0	0
$7$	2.76526	1.04517	0.522585	−	0.852587i	$-0.324968\pi$
$7$	2.76526	1.04517	0.522585	+	0.852587i	$0.324968\pi$
$8$	6.99148	2.47186
$9$	0	0
$10$	2.97416	0.940511
$11$	−4.35418	−1.31283	−0.656417	−	0.754399i	$-0.727929\pi$
$11$	−4.35418	−1.31283	−0.656417	+	0.754399i	$0.727929\pi$
$12$	0	0
$13$	5.62914	1.56124	0.780621	−	0.625005i	$-0.214903\pi$
$13$	5.62914	1.56124	0.780621	+	0.625005i	$0.214903\pi$
$14$	7.15816	1.91310
$15$	0	0
$16$	8.69643	2.17411
$17$	2.41516	0.585763	0.292881	−	0.956149i	$-0.405386\pi$
$17$	2.41516	0.585763	0.292881	+	0.956149i	$0.405386\pi$
$18$	0	0
$19$	−1.87296	−0.429686	−0.214843	−	0.976649i	$-0.568924\pi$
$19$	−1.87296	−0.429686	−0.214843	+	0.976649i	$0.568924\pi$
$20$	5.40103	1.20771
$21$	0	0
$22$	−11.2712	−2.40304
$23$	8.82048	1.83920	0.919598	−	0.392860i	$-0.128514\pi$
$23$	8.82048	1.83920	0.919598	+	0.392860i	$0.128514\pi$
$24$	0	0
$25$	−3.67993	−0.735986
$26$	14.5716	2.85773
$27$	0	0
$28$	12.9991	2.45660
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−1.56053	−0.280280	−0.140140	−	0.990132i	$-0.544755\pi$
$31$	−1.56053	−0.280280	−0.140140	+	0.990132i	$0.544755\pi$
$32$	8.52865	1.50767
$33$	0	0
$34$	6.25190	1.07219
$35$	3.17713	0.537032
$36$	0	0
$37$	−7.76330	−1.27628	−0.638139	−	0.769921i	$-0.720295\pi$
$37$	−7.76330	−1.27628	−0.638139	+	0.769921i	$0.720295\pi$
$38$	−4.84834	−0.786505
$39$	0	0
$40$	8.03281	1.27010
$41$	3.16072	0.493622	0.246811	−	0.969064i	$-0.420617\pi$
$41$	3.16072	0.493622	0.246811	+	0.969064i	$0.420617\pi$
$42$	0	0
$43$	1.46397	0.223254	0.111627	−	0.993750i	$-0.464394\pi$
$43$	1.46397	0.223254	0.111627	+	0.993750i	$0.464394\pi$
$44$	−20.4684	−3.08573
$45$	0	0
$46$	22.8327	3.36650
$47$	0.357351	0.0521250	0.0260625	−	0.999660i	$-0.491703\pi$
$47$	0.357351	0.0521250	0.0260625	+	0.999660i	$0.491703\pi$
$48$	0	0
$49$	0.646666	0.0923809
$50$	−9.52588	−1.34716
$51$	0	0
$52$	26.4618	3.66960
$53$	1.63498	0.224582	0.112291	−	0.993675i	$-0.464181\pi$
$53$	1.63498	0.224582	0.112291	+	0.993675i	$0.464181\pi$
$54$	0	0
$55$	−5.00270	−0.674563
$56$	19.3333	2.58352
$57$	0	0
$58$	0	0
$59$	4.85026	0.631450	0.315725	−	0.948851i	$-0.397752\pi$
$59$	4.85026	0.631450	0.315725	+	0.948851i	$0.397752\pi$
$60$	0	0
$61$	−1.55510	−0.199110	−0.0995551	−	0.995032i	$-0.531742\pi$
$61$	−1.55510	−0.199110	−0.0995551	+	0.995032i	$0.531742\pi$
$62$	−4.03961	−0.513030
$63$	0	0
$64$	4.68444	0.585555
$65$	6.46755	0.802201
$66$	0	0
$67$	−9.66950	−1.18132	−0.590659	−	0.806922i	$-0.701132\pi$
$67$	−9.66950	−1.18132	−0.590659	+	0.806922i	$0.701132\pi$
$68$	11.3534	1.37680
$69$	0	0
$70$	8.22432	0.982994
$71$	8.18108	0.970915	0.485458	−	0.874260i	$-0.338653\pi$
$71$	8.18108	0.970915	0.485458	+	0.874260i	$0.338653\pi$
$72$	0	0
$73$	−11.3786	−1.33177	−0.665884	−	0.746055i	$-0.731946\pi$
$73$	−11.3786	−1.33177	−0.665884	+	0.746055i	$0.731946\pi$
$74$	−20.0961	−2.33612
$75$	0	0
$76$	−8.80452	−1.00995
$77$	−12.0404	−1.37213
$78$	0	0
$79$	8.13075	0.914781	0.457391	−	0.889266i	$-0.348784\pi$
$79$	8.13075	0.914781	0.457391	+	0.889266i	$0.348784\pi$
$80$	9.99170	1.11711
$81$	0	0
$82$	8.18185	0.903535
$83$	0.481835	0.0528882	0.0264441	−	0.999650i	$-0.491582\pi$
$83$	0.481835	0.0528882	0.0264441	+	0.999650i	$0.491582\pi$
$84$	0	0
$85$	2.77488	0.300978
$86$	3.78965	0.408648
$87$	0	0
$88$	−30.4421	−3.24514
$89$	−1.99621	−0.211598	−0.105799	−	0.994388i	$-0.533740\pi$
$89$	−1.99621	−0.211598	−0.105799	+	0.994388i	$0.533740\pi$
$90$	0	0
$91$	15.5660	1.63176
$92$	41.4639	4.32291
$93$	0	0
$94$	0.925040	0.0954106
$95$	−2.15192	−0.220782
$96$	0	0
$97$	−4.63516	−0.470629	−0.235315	−	0.971919i	$-0.575612\pi$
$97$	−4.63516	−0.470629	−0.235315	+	0.971919i	$0.575612\pi$
$98$	1.67396	0.169096
$99$	0	0
$100$	−17.2989	−1.72989
$101$	−1.89907	−0.188964	−0.0944822	−	0.995527i	$-0.530120\pi$
$101$	−1.89907	−0.188964	−0.0944822	+	0.995527i	$0.530120\pi$
$102$	0	0
$103$	−8.22122	−0.810061	−0.405030	−	0.914303i	$-0.632739\pi$
$103$	−8.22122	−0.810061	−0.405030	+	0.914303i	$0.632739\pi$
$104$	39.3560	3.85917
$105$	0	0
$106$	4.23233	0.411080
$107$	−4.45778	−0.430950	−0.215475	−	0.976509i	$-0.569130\pi$
$107$	−4.45778	−0.430950	−0.215475	+	0.976509i	$0.569130\pi$
$108$	0	0
$109$	17.5256	1.67865	0.839324	−	0.543631i	$-0.182951\pi$
$109$	17.5256	1.67865	0.839324	+	0.543631i	$0.182951\pi$
$110$	−12.9500	−1.23473
$111$	0	0
$112$	24.0479	2.27231
$113$	7.93117	0.746102	0.373051	−	0.927811i	$-0.378312\pi$
$113$	7.93117	0.746102	0.373051	+	0.927811i	$0.378312\pi$
$114$	0	0
$115$	10.1342	0.945021
$116$	0	0
$117$	0	0
$118$	12.5554	1.15582
$119$	6.67855	0.612222
$120$	0	0
$121$	7.95884	0.723531
$122$	−4.02554	−0.364455
$123$	0	0
$124$	−7.33587	−0.658780
$125$	−9.97274	−0.891989
$126$	0	0
$127$	3.35542	0.297745	0.148873	−	0.988856i	$-0.452436\pi$
$127$	3.35542	0.297745	0.148873	+	0.988856i	$0.452436\pi$
$128$	−4.93114	−0.435855
$129$	0	0
$130$	16.7419	1.46836
$131$	−8.69320	−0.759528	−0.379764	−	0.925083i	$-0.623995\pi$
$131$	−8.69320	−0.759528	−0.379764	+	0.925083i	$0.623995\pi$
$132$	0	0
$133$	−5.17921	−0.449095
$134$	−25.0305	−2.16231
$135$	0	0
$136$	16.8856	1.44792
$137$	3.22382	0.275429	0.137715	−	0.990472i	$-0.456024\pi$
$137$	3.22382	0.275429	0.137715	+	0.990472i	$0.456024\pi$
$138$	0	0
$139$	−22.4884	−1.90744	−0.953721	−	0.300693i	$-0.902782\pi$
$139$	−22.4884	−1.90744	−0.953721	+	0.300693i	$0.902782\pi$
$140$	14.9352	1.26226
$141$	0	0
$142$	21.1776	1.77718
$143$	−24.5102	−2.04965
$144$	0	0
$145$	0	0
$146$	−29.4548	−2.43769
$147$	0	0
$148$	−36.4942	−2.99981
$149$	−1.28934	−0.105627	−0.0528133	−	0.998604i	$-0.516819\pi$
$149$	−1.28934	−0.105627	−0.0528133	+	0.998604i	$0.516819\pi$
$150$	0	0
$151$	6.55195	0.533190	0.266595	−	0.963809i	$-0.414101\pi$
$151$	6.55195	0.533190	0.266595	+	0.963809i	$0.414101\pi$
$152$	−13.0947	−1.06212
$153$	0	0
$154$	−31.1679	−2.51158
$155$	−1.79296	−0.144014
$156$	0	0
$157$	24.1178	1.92481	0.962407	−	0.271612i	$-0.0875566\pi$
$157$	24.1178	1.92481	0.962407	+	0.271612i	$0.0875566\pi$
$158$	21.0473	1.67443
$159$	0	0
$160$	9.79893	0.774674
$161$	24.3909	1.92227
$162$	0	0
$163$	−16.1434	−1.26445	−0.632223	−	0.774786i	$-0.717857\pi$
$163$	−16.1434	−1.26445	−0.632223	+	0.774786i	$0.717857\pi$
$164$	14.8581	1.16023
$165$	0	0
$166$	1.24728	0.0968076
$167$	5.42045	0.419447	0.209723	−	0.977761i	$-0.432744\pi$
$167$	5.42045	0.419447	0.209723	+	0.977761i	$0.432744\pi$
$168$	0	0
$169$	18.6872	1.43747
$170$	7.18307	0.550916
$171$	0	0
$172$	6.88195	0.524744
$173$	11.2923	0.858537	0.429268	−	0.903177i	$-0.358771\pi$
$173$	11.2923	0.858537	0.429268	+	0.903177i	$0.358771\pi$
$174$	0	0
$175$	−10.1760	−0.769231
$176$	−37.8658	−2.85424
$177$	0	0
$178$	−5.16739	−0.387312
$179$	10.6541	0.796323	0.398161	−	0.917315i	$-0.369648\pi$
$179$	10.6541	0.796323	0.398161	+	0.917315i	$0.369648\pi$
$180$	0	0
$181$	−12.2934	−0.913763	−0.456882	−	0.889527i	$-0.651034\pi$
$181$	−12.2934	−0.913763	−0.456882	+	0.889527i	$0.651034\pi$
$182$	40.2943	2.98681
$183$	0	0
$184$	61.6682	4.54624
$185$	−8.91958	−0.655781
$186$	0	0
$187$	−10.5160	−0.769009
$188$	1.67986	0.122516
$189$	0	0
$190$	−5.57047	−0.404124
$191$	−1.68311	−0.121786	−0.0608929	−	0.998144i	$-0.519395\pi$
$191$	−1.68311	−0.121786	−0.0608929	+	0.998144i	$0.519395\pi$
$192$	0	0
$193$	−10.5761	−0.761284	−0.380642	−	0.924722i	$-0.624297\pi$
$193$	−10.5761	−0.761284	−0.380642	+	0.924722i	$0.624297\pi$
$194$	−11.9986	−0.861449
$195$	0	0
$196$	3.03989	0.217135
$197$	12.1150	0.863161	0.431581	−	0.902074i	$-0.357956\pi$
$197$	12.1150	0.863161	0.431581	+	0.902074i	$0.357956\pi$
$198$	0	0
$199$	4.99592	0.354151	0.177076	−	0.984197i	$-0.443336\pi$
$199$	4.99592	0.354151	0.177076	+	0.984197i	$0.443336\pi$
$200$	−25.7282	−1.81926
$201$	0	0
$202$	−4.91594	−0.345884
$203$	0	0
$204$	0	0
$205$	3.63149	0.253634
$206$	−21.2815	−1.48275
$207$	0	0
$208$	48.9534	3.39431
$209$	8.15518	0.564106
$210$	0	0
$211$	19.2108	1.32253	0.661263	−	0.750154i	$-0.270021\pi$
$211$	19.2108	1.32253	0.661263	+	0.750154i	$0.270021\pi$
$212$	7.68585	0.527866
$213$	0	0
$214$	−11.5394	−0.788819
$215$	1.68202	0.114713
$216$	0	0
$217$	−4.31528	−0.292941
$218$	45.3668	3.07263
$219$	0	0
$220$	−23.5170	−1.58552
$221$	13.5953	0.914517
$222$	0	0
$223$	−21.2442	−1.42262	−0.711308	−	0.702881i	$-0.751897\pi$
$223$	−21.2442	−1.42262	−0.711308	+	0.702881i	$0.751897\pi$
$224$	23.5840	1.57577
$225$	0	0
$226$	20.5307	1.36568
$227$	−19.7559	−1.31125	−0.655624	−	0.755088i	$-0.727594\pi$
$227$	−19.7559	−1.31125	−0.655624	+	0.755088i	$0.727594\pi$
$228$	0	0
$229$	−20.7139	−1.36881	−0.684405	−	0.729102i	$-0.739938\pi$
$229$	−20.7139	−1.36881	−0.684405	+	0.729102i	$0.739938\pi$
$230$	26.2335	1.72978
$231$	0	0
$232$	0	0
$233$	−20.7745	−1.36098	−0.680492	−	0.732756i	$-0.738234\pi$
$233$	−20.7745	−1.36098	−0.680492	+	0.732756i	$0.738234\pi$
$234$	0	0
$235$	0.410576	0.0267830
$236$	22.8004	1.48418
$237$	0	0
$238$	17.2881	1.12062
$239$	2.27481	0.147145	0.0735727	−	0.997290i	$-0.476560\pi$
$239$	2.27481	0.147145	0.0735727	+	0.997290i	$0.476560\pi$
$240$	0	0
$241$	11.7783	0.758706	0.379353	−	0.925252i	$-0.376147\pi$
$241$	11.7783	0.758706	0.379353	+	0.925252i	$0.376147\pi$
$242$	20.6023	1.32437
$243$	0	0
$244$	−7.31032	−0.467995
$245$	0.742982	0.0474674
$246$	0	0
$247$	−10.5431	−0.670843
$248$	−10.9104	−0.692814
$249$	0	0
$250$	−25.8155	−1.63271
$251$	−1.42714	−0.0900805	−0.0450402	−	0.998985i	$-0.514342\pi$
$251$	−1.42714	−0.0900805	−0.0450402	+	0.998985i	$0.514342\pi$
$252$	0	0
$253$	−38.4059	−2.41456
$254$	8.68585	0.544999
$255$	0	0
$256$	−22.1337	−1.38335
$257$	28.6450	1.78683	0.893414	−	0.449234i	$-0.148303\pi$
$257$	28.6450	1.78683	0.893414	+	0.449234i	$0.148303\pi$
$258$	0	0
$259$	−21.4675	−1.33393
$260$	30.4031	1.88552
$261$	0	0
$262$	−22.5032	−1.39025
$263$	18.5023	1.14090	0.570451	−	0.821332i	$-0.306769\pi$
$263$	18.5023	1.14090	0.570451	+	0.821332i	$0.306769\pi$
$264$	0	0
$265$	1.87850	0.115396
$266$	−13.4069	−0.822032
$267$	0	0
$268$	−45.4550	−2.77661
$269$	−28.7074	−1.75032	−0.875160	−	0.483834i	$-0.839244\pi$
$269$	−28.7074	−1.75032	−0.875160	+	0.483834i	$0.839244\pi$
$270$	0	0
$271$	−1.80441	−0.109610	−0.0548050	−	0.998497i	$-0.517454\pi$
$271$	−1.80441	−0.109610	−0.0548050	+	0.998497i	$0.517454\pi$
$272$	21.0033	1.27351
$273$	0	0
$274$	8.34518	0.504151
$275$	16.0231	0.966227
$276$	0	0
$277$	−14.1949	−0.852890	−0.426445	−	0.904513i	$-0.640234\pi$
$277$	−14.1949	−0.852890	−0.426445	+	0.904513i	$0.640234\pi$
$278$	−58.2136	−3.49142
$279$	0	0
$280$	22.2128	1.32747
$281$	−18.8216	−1.12280	−0.561400	−	0.827545i	$-0.689737\pi$
$281$	−18.8216	−1.12280	−0.561400	+	0.827545i	$0.689737\pi$
$282$	0	0
$283$	21.8832	1.30082	0.650410	−	0.759584i	$-0.274597\pi$
$283$	21.8832	1.30082	0.650410	+	0.759584i	$0.274597\pi$
$284$	38.4582	2.28207
$285$	0	0
$286$	−63.4473	−3.75172
$287$	8.74022	0.515919
$288$	0	0
$289$	−11.1670	−0.656882
$290$	0	0
$291$	0	0
$292$	−53.4895	−3.13023
$293$	0.00776592	0.000453690 0	0.000226845	−	1.00000i	$-0.499928\pi$
$293$	0.00776592	0.000453690 0	0.000226845		1.00000i	$0.499928\pi$
$294$	0	0
$295$	5.57267	0.324453
$296$	−54.2769	−3.15478
$297$	0	0
$298$	−3.33758	−0.193341
$299$	49.6517	2.87143
$300$	0	0
$301$	4.04827	0.233338
$302$	16.9604	0.975961
$303$	0	0
$304$	−16.2880	−0.934183
$305$	−1.78672	−0.102307
$306$	0	0
$307$	−0.214941	−0.0122673	−0.00613367	−	0.999981i	$-0.501952\pi$
$307$	−0.214941	−0.0122673	−0.00613367	+	0.999981i	$0.501952\pi$
$308$	−56.6005	−3.22511
$309$	0	0
$310$	−4.64127	−0.263607
$311$	28.2266	1.60058	0.800291	−	0.599612i	$-0.204678\pi$
$311$	28.2266	1.60058	0.800291	+	0.599612i	$0.204678\pi$
$312$	0	0
$313$	−10.2302	−0.578244	−0.289122	−	0.957292i	$-0.593363\pi$
$313$	−10.2302	−0.578244	−0.289122	+	0.957292i	$0.593363\pi$
$314$	62.4316	3.52322
$315$	0	0
$316$	38.2216	2.15013
$317$	23.7789	1.33555	0.667777	−	0.744361i	$-0.267246\pi$
$317$	23.7789	1.33555	0.667777	+	0.744361i	$0.267246\pi$
$318$	0	0
$319$	0	0
$320$	5.38216	0.300872
$321$	0	0
$322$	63.1384	3.51857
$323$	−4.52349	−0.251694
$324$	0	0
$325$	−20.7148	−1.14905
$326$	−41.7888	−2.31447
$327$	0	0
$328$	22.0981	1.22016
$329$	0.988168	0.0544795
$330$	0	0
$331$	7.07856	0.389073	0.194536	−	0.980895i	$-0.437680\pi$
$331$	7.07856	0.389073	0.194536	+	0.980895i	$0.437680\pi$
$332$	2.26504	0.124310
$333$	0	0
$334$	14.0314	0.767763
$335$	−11.1097	−0.606987
$336$	0	0
$337$	7.02402	0.382623	0.191311	−	0.981529i	$-0.438726\pi$
$337$	7.02402	0.382623	0.191311	+	0.981529i	$0.438726\pi$
$338$	48.3737	2.63118
$339$	0	0
$340$	13.0444	0.707429
$341$	6.79484	0.367961
$342$	0	0
$343$	−17.5686	−0.948617
$344$	10.2353	0.551852
$345$	0	0
$346$	29.2313	1.57148
$347$	−10.0795	−0.541094	−0.270547	−	0.962707i	$-0.587205\pi$
$347$	−10.0795	−0.541094	−0.270547	+	0.962707i	$0.587205\pi$
$348$	0	0
$349$	−1.64246	−0.0879187	−0.0439593	−	0.999033i	$-0.513997\pi$
$349$	−1.64246	−0.0879187	−0.0439593	+	0.999033i	$0.513997\pi$
$350$	−26.3416	−1.40802
$351$	0	0
$352$	−37.1353	−1.97932
$353$	−24.1232	−1.28395	−0.641975	−	0.766726i	$-0.721885\pi$
$353$	−24.1232	−1.28395	−0.641975	+	0.766726i	$0.721885\pi$
$354$	0	0
$355$	9.39959	0.498878
$356$	−9.38391	−0.497346
$357$	0	0
$358$	27.5792	1.45760
$359$	−12.8958	−0.680612	−0.340306	−	0.940315i	$-0.610531\pi$
$359$	−12.8958	−0.680612	−0.340306	+	0.940315i	$0.610531\pi$
$360$	0	0
$361$	−15.4920	−0.815370
$362$	−31.8228	−1.67257
$363$	0	0
$364$	73.1739	3.83535
$365$	−13.0734	−0.684293
$366$	0	0
$367$	31.7420	1.65692	0.828459	−	0.560050i	$-0.189218\pi$
$367$	31.7420	1.65692	0.828459	+	0.560050i	$0.189218\pi$
$368$	76.7067	3.99861
$369$	0	0
$370$	−23.0893	−1.20035
$371$	4.52116	0.234727
$372$	0	0
$373$	4.75511	0.246210	0.123105	−	0.992394i	$-0.460715\pi$
$373$	4.75511	0.246210	0.123105	+	0.992394i	$0.460715\pi$
$374$	−27.2218	−1.40761
$375$	0	0
$376$	2.49841	0.128846
$377$	0	0
$378$	0	0
$379$	−2.11960	−0.108877	−0.0544383	−	0.998517i	$-0.517337\pi$
$379$	−2.11960	−0.108877	−0.0544383	+	0.998517i	$0.517337\pi$
$380$	−10.1159	−0.518934
$381$	0	0
$382$	−4.35691	−0.222919
$383$	−14.6407	−0.748103	−0.374051	−	0.927408i	$-0.622032\pi$
$383$	−14.6407	−0.748103	−0.374051	+	0.927408i	$0.622032\pi$
$384$	0	0
$385$	−13.8338	−0.705034
$386$	−27.3773	−1.39347
$387$	0	0
$388$	−21.7893	−1.10618
$389$	1.93401	0.0980583	0.0490292	−	0.998797i	$-0.484387\pi$
$389$	1.93401	0.0980583	0.0490292	+	0.998797i	$0.484387\pi$
$390$	0	0
$391$	21.3029	1.07733
$392$	4.52115	0.228353
$393$	0	0
$394$	31.3611	1.57995
$395$	9.34177	0.470035
$396$	0	0
$397$	−30.0191	−1.50662	−0.753309	−	0.657667i	$-0.771543\pi$
$397$	−30.0191	−1.50662	−0.753309	+	0.657667i	$0.771543\pi$
$398$	12.9324	0.648245
$399$	0	0
$400$	−32.0023	−1.60011
$401$	−15.7085	−0.784444	−0.392222	−	0.919871i	$-0.628293\pi$
$401$	−15.7085	−0.784444	−0.392222	+	0.919871i	$0.628293\pi$
$402$	0	0
$403$	−8.78446	−0.437585
$404$	−8.92728	−0.444149
$405$	0	0
$406$	0	0
$407$	33.8027	1.67554
$408$	0	0
$409$	4.87216	0.240913	0.120456	−	0.992719i	$-0.461564\pi$
$409$	4.87216	0.240913	0.120456	+	0.992719i	$0.461564\pi$
$410$	9.40048	0.464257
$411$	0	0
$412$	−38.6469	−1.90399
$413$	13.4122	0.659973
$414$	0	0
$415$	0.553600	0.0271752
$416$	48.0090	2.35383
$417$	0	0
$418$	21.1105	1.03255
$419$	−19.0131	−0.928853	−0.464426	−	0.885612i	$-0.653739\pi$
$419$	−19.0131	−0.928853	−0.464426	+	0.885612i	$0.653739\pi$
$420$	0	0
$421$	−12.5171	−0.610047	−0.305023	−	0.952345i	$-0.598664\pi$
$421$	−12.5171	−0.610047	−0.305023	+	0.952345i	$0.598664\pi$
$422$	49.7292	2.42078
$423$	0	0
$424$	11.4310	0.555136
$425$	−8.88763	−0.431113
$426$	0	0
$427$	−4.30026	−0.208104
$428$	−20.9554	−1.01292
$429$	0	0
$430$	4.35409	0.209973
$431$	−35.3970	−1.70501	−0.852507	−	0.522716i	$-0.824919\pi$
$431$	−35.3970	−1.70501	−0.852507	+	0.522716i	$0.824919\pi$
$432$	0	0
$433$	−5.85494	−0.281371	−0.140685	−	0.990054i	$-0.544931\pi$
$433$	−5.85494	−0.281371	−0.140685	+	0.990054i	$0.544931\pi$
$434$	−11.1706	−0.536204
$435$	0	0
$436$	82.3856	3.94555
$437$	−16.5204	−0.790277
$438$	0	0
$439$	24.8297	1.18506	0.592530	−	0.805549i	$-0.298129\pi$
$439$	24.8297	1.18506	0.592530	+	0.805549i	$0.298129\pi$
$440$	−34.9762	−1.66743
$441$	0	0
$442$	35.1928	1.67395
$443$	23.7488	1.12834	0.564170	−	0.825659i	$-0.309196\pi$
$443$	23.7488	1.12834	0.564170	+	0.825659i	$0.309196\pi$
$444$	0	0
$445$	−2.29353	−0.108724
$446$	−54.9928	−2.60398
$447$	0	0
$448$	12.9537	0.612005
$449$	6.44854	0.304325	0.152163	−	0.988355i	$-0.451376\pi$
$449$	6.44854	0.304325	0.152163	+	0.988355i	$0.451376\pi$
$450$	0	0
$451$	−13.7623	−0.648043
$452$	37.2834	1.75366
$453$	0	0
$454$	−51.1403	−2.40013
$455$	17.8845	0.838437
$456$	0	0
$457$	−2.62569	−0.122824	−0.0614122	−	0.998112i	$-0.519560\pi$
$457$	−2.62569	−0.122824	−0.0614122	+	0.998112i	$0.519560\pi$
$458$	−53.6200	−2.50550
$459$	0	0
$460$	47.6396	2.22121
$461$	−7.15794	−0.333378	−0.166689	−	0.986009i	$-0.553308\pi$
$461$	−7.15794	−0.333378	−0.166689	+	0.986009i	$0.553308\pi$
$462$	0	0
$463$	−18.2411	−0.847737	−0.423868	−	0.905724i	$-0.639328\pi$
$463$	−18.2411	−0.847737	−0.423868	+	0.905724i	$0.639328\pi$
$464$	0	0
$465$	0	0
$466$	−53.7770	−2.49117
$467$	35.4677	1.64125	0.820624	−	0.571468i	$-0.193626\pi$
$467$	35.4677	1.64125	0.820624	+	0.571468i	$0.193626\pi$
$468$	0	0
$469$	−26.7387	−1.23468
$470$	1.06282	0.0490241
$471$	0	0
$472$	33.9105	1.56086
$473$	−6.37440	−0.293095
$474$	0	0
$475$	6.89235	0.316243
$476$	31.3950	1.43899
$477$	0	0
$478$	5.88859	0.269338
$479$	26.1004	1.19256	0.596280	−	0.802777i	$-0.296645\pi$
$479$	26.1004	1.19256	0.596280	+	0.802777i	$0.296645\pi$
$480$	0	0
$481$	−43.7006	−1.99258
$482$	30.4893	1.38875
$483$	0	0
$484$	37.4135	1.70061
$485$	−5.32553	−0.241820
$486$	0	0
$487$	22.1493	1.00368	0.501839	−	0.864961i	$-0.332657\pi$
$487$	22.1493	1.00368	0.501839	+	0.864961i	$0.332657\pi$
$488$	−10.8724	−0.492173
$489$	0	0
$490$	1.92329	0.0868852
$491$	−11.7483	−0.530193	−0.265096	−	0.964222i	$-0.585404\pi$
$491$	−11.7483	−0.530193	−0.265096	+	0.964222i	$0.585404\pi$
$492$	0	0
$493$	0	0
$494$	−27.2920	−1.22792
$495$	0	0
$496$	−13.5711	−0.609359
$497$	22.6228	1.01477
$498$	0	0
$499$	−16.3753	−0.733060	−0.366530	−	0.930406i	$-0.619454\pi$
$499$	−16.3753	−0.733060	−0.366530	+	0.930406i	$0.619454\pi$
$500$	−46.8805	−2.09656
$501$	0	0
$502$	−3.69431	−0.164885
$503$	14.8354	0.661476	0.330738	−	0.943723i	$-0.392702\pi$
$503$	14.8354	0.661476	0.330738	+	0.943723i	$0.392702\pi$
$504$	0	0
$505$	−2.18192	−0.0970942
$506$	−99.4177	−4.41965
$507$	0	0
$508$	15.7734	0.699831
$509$	−31.5733	−1.39946	−0.699732	−	0.714406i	$-0.746697\pi$
$509$	−31.5733	−1.39946	−0.699732	+	0.714406i	$0.746697\pi$
$510$	0	0
$511$	−31.4649	−1.39192
$512$	−47.4330	−2.09626
$513$	0	0
$514$	74.1507	3.27065
$515$	−9.44571	−0.416228
$516$	0	0
$517$	−1.55597	−0.0684314
$518$	−55.5709	−2.44165
$519$	0	0
$520$	45.2178	1.98293
$521$	8.03875	0.352184	0.176092	−	0.984374i	$-0.443654\pi$
$521$	8.03875	0.352184	0.176092	+	0.984374i	$0.443654\pi$
$522$	0	0
$523$	−1.48633	−0.0649927	−0.0324963	−	0.999472i	$-0.510346\pi$
$523$	−1.48633	−0.0649927	−0.0324963	+	0.999472i	$0.510346\pi$
$524$	−40.8656	−1.78522
$525$	0	0
$526$	47.8951	2.08833
$527$	−3.76894	−0.164178
$528$	0	0
$529$	54.8008	2.38264
$530$	4.86270	0.211222
$531$	0	0
$532$	−24.3468	−1.05557
$533$	17.7921	0.770663
$534$	0	0
$535$	−5.12173	−0.221432
$536$	−67.6041	−2.92005
$537$	0	0
$538$	−74.3120	−3.20382
$539$	−2.81570	−0.121281
$540$	0	0
$541$	−38.8306	−1.66946	−0.834728	−	0.550662i	$-0.814375\pi$
$541$	−38.8306	−1.66946	−0.834728	+	0.550662i	$0.814375\pi$
$542$	−4.67090	−0.200632
$543$	0	0
$544$	20.5981	0.883135
$545$	20.1359	0.862528
$546$	0	0
$547$	−17.3495	−0.741809	−0.370905	−	0.928671i	$-0.620952\pi$
$547$	−17.3495	−0.741809	−0.370905	+	0.928671i	$0.620952\pi$
$548$	15.1547	0.647378
$549$	0	0
$550$	41.4774	1.76860
$551$	0	0
$552$	0	0
$553$	22.4837	0.956102
$554$	−36.7450	−1.56115
$555$	0	0
$556$	−105.715	−4.48332
$557$	−30.4259	−1.28919	−0.644593	−	0.764526i	$-0.722973\pi$
$557$	−30.4259	−1.28919	−0.644593	+	0.764526i	$0.722973\pi$
$558$	0	0
$559$	8.24091	0.348553
$560$	27.6296	1.16757
$561$	0	0
$562$	−48.7215	−2.05519
$563$	−12.1911	−0.513793	−0.256897	−	0.966439i	$-0.582700\pi$
$563$	−12.1911	−0.513793	−0.256897	+	0.966439i	$0.582700\pi$
$564$	0	0
$565$	9.11246	0.383364
$566$	56.6469	2.38104
$567$	0	0
$568$	57.1978	2.39997
$569$	−3.15785	−0.132384	−0.0661920	−	0.997807i	$-0.521085\pi$
$569$	−3.15785	−0.132384	−0.0661920	+	0.997807i	$0.521085\pi$
$570$	0	0
$571$	24.3414	1.01865	0.509327	−	0.860573i	$-0.329894\pi$
$571$	24.3414	1.01865	0.509327	+	0.860573i	$0.329894\pi$
$572$	−115.219	−4.81757
$573$	0	0
$574$	22.6250	0.944347
$575$	−32.4588	−1.35362
$576$	0	0
$577$	−7.99079	−0.332661	−0.166330	−	0.986070i	$-0.553192\pi$
$577$	−7.99079	−0.332661	−0.166330	+	0.986070i	$0.553192\pi$
$578$	−28.9069	−1.20237
$579$	0	0
$580$	0	0
$581$	1.33240	0.0552772
$582$	0	0
$583$	−7.11901	−0.294839
$584$	−79.5535	−3.29195
$585$	0	0
$586$	0.0201029	0.000830443 0
$587$	42.3655	1.74861	0.874305	−	0.485377i	$-0.161318\pi$
$587$	42.3655	1.74861	0.874305	+	0.485377i	$0.161318\pi$
$588$	0	0
$589$	2.92281	0.120432
$590$	14.4254	0.593886
$591$	0	0
$592$	−67.5130	−2.77477
$593$	26.3761	1.08314	0.541569	−	0.840656i	$-0.317831\pi$
$593$	26.3761	1.08314	0.541569	+	0.840656i	$0.317831\pi$
$594$	0	0
$595$	7.67327	0.314573
$596$	−6.06100	−0.248268
$597$	0	0
$598$	128.528	5.25592
$599$	6.35768	0.259768	0.129884	−	0.991529i	$-0.458540\pi$
$599$	6.35768	0.259768	0.129884	+	0.991529i	$0.458540\pi$
$600$	0	0
$601$	16.9016	0.689430	0.344715	−	0.938707i	$-0.387976\pi$
$601$	16.9016	0.689430	0.344715	+	0.938707i	$0.387976\pi$
$602$	10.4794	0.427107
$603$	0	0
$604$	30.7999	1.25323
$605$	9.14425	0.371767
$606$	0	0
$607$	7.41550	0.300986	0.150493	−	0.988611i	$-0.451914\pi$
$607$	7.41550	0.300986	0.150493	+	0.988611i	$0.451914\pi$
$608$	−15.9738	−0.647823
$609$	0	0
$610$	−4.62511	−0.187265
$611$	2.01158	0.0813797
$612$	0	0
$613$	−28.9255	−1.16829	−0.584145	−	0.811649i	$-0.698570\pi$
$613$	−28.9255	−1.16829	−0.584145	+	0.811649i	$0.698570\pi$
$614$	−0.556398	−0.0224544
$615$	0	0
$616$	−84.1804	−3.39173
$617$	−34.9128	−1.40554	−0.702768	−	0.711419i	$-0.748053\pi$
$617$	−34.9128	−1.40554	−0.702768	+	0.711419i	$0.748053\pi$
$618$	0	0
$619$	−45.1342	−1.81410	−0.907048	−	0.421028i	$-0.861669\pi$
$619$	−45.1342	−1.81410	−0.907048	+	0.421028i	$0.861669\pi$
$620$	−8.42849	−0.338496
$621$	0	0
$622$	73.0674	2.92974
$623$	−5.52003	−0.221155
$624$	0	0
$625$	6.94155	0.277662
$626$	−26.4819	−1.05843
$627$	0	0
$628$	113.375	4.52415
$629$	−18.7496	−0.747596
$630$	0	0
$631$	29.2602	1.16483	0.582415	−	0.812892i	$-0.302108\pi$
$631$	29.2602	1.16483	0.582415	+	0.812892i	$0.302108\pi$
$632$	56.8460	2.26121
$633$	0	0
$634$	61.5540	2.44462
$635$	3.85518	0.152988
$636$	0	0
$637$	3.64017	0.144229
$638$	0	0
$639$	0	0
$640$	−5.66560	−0.223952
$641$	17.7234	0.700031	0.350016	−	0.936744i	$-0.386176\pi$
$641$	17.7234	0.700031	0.350016	+	0.936744i	$0.386176\pi$
$642$	0	0
$643$	38.4443	1.51609	0.758047	−	0.652200i	$-0.226154\pi$
$643$	38.4443	1.51609	0.758047	+	0.652200i	$0.226154\pi$
$644$	114.659	4.51818
$645$	0	0
$646$	−11.7095	−0.460705
$647$	−40.5910	−1.59580	−0.797899	−	0.602792i	$-0.794055\pi$
$647$	−40.5910	−1.59580	−0.797899	+	0.602792i	$0.794055\pi$
$648$	0	0
$649$	−21.1189	−0.828989
$650$	−53.6225	−2.10325
$651$	0	0
$652$	−75.8879	−2.97200
$653$	−26.6169	−1.04160	−0.520800	−	0.853679i	$-0.674366\pi$
$653$	−26.6169	−1.04160	−0.520800	+	0.853679i	$0.674366\pi$
$654$	0	0
$655$	−9.98798	−0.390263
$656$	27.4870	1.07319
$657$	0	0
$658$	2.55798	0.0997203
$659$	−23.1105	−0.900259	−0.450130	−	0.892963i	$-0.648622\pi$
$659$	−23.1105	−0.900259	−0.450130	+	0.892963i	$0.648622\pi$
$660$	0	0
$661$	23.4613	0.912537	0.456269	−	0.889842i	$-0.349186\pi$
$661$	23.4613	0.912537	0.456269	+	0.889842i	$0.349186\pi$
$662$	18.3236	0.712166
$663$	0	0
$664$	3.36874	0.130732
$665$	−5.95062	−0.230755
$666$	0	0
$667$	0	0
$668$	25.4808	0.985882
$669$	0	0
$670$	−28.7586	−1.11104
$671$	6.77118	0.261398
$672$	0	0
$673$	−26.9621	−1.03931	−0.519656	−	0.854376i	$-0.673940\pi$
$673$	−26.9621	−1.03931	−0.519656	+	0.854376i	$0.673940\pi$
$674$	18.1824	0.700360
$675$	0	0
$676$	87.8460	3.37869
$677$	33.5314	1.28872	0.644358	−	0.764724i	$-0.277125\pi$
$677$	33.5314	1.28872	0.644358	+	0.764724i	$0.277125\pi$
$678$	0	0
$679$	−12.8174	−0.491888
$680$	19.4005	0.743976
$681$	0	0
$682$	17.5891	0.673523
$683$	12.3343	0.471957	0.235979	−	0.971758i	$-0.424170\pi$
$683$	12.3343	0.471957	0.235979	+	0.971758i	$0.424170\pi$
$684$	0	0
$685$	3.70398	0.141522
$686$	−45.4782	−1.73637
$687$	0	0
$688$	12.7313	0.485378
$689$	9.20355	0.350627
$690$	0	0
$691$	2.02123	0.0768913	0.0384457	−	0.999261i	$-0.487759\pi$
$691$	2.02123	0.0768913	0.0384457	+	0.999261i	$0.487759\pi$
$692$	53.0836	2.01793
$693$	0	0
$694$	−26.0918	−0.990429
$695$	−25.8379	−0.980087
$696$	0	0
$697$	7.63365	0.289145
$698$	−4.25167	−0.160928
$699$	0	0
$700$	−47.8359	−1.80803
$701$	33.6980	1.27275	0.636377	−	0.771378i	$-0.280432\pi$
$701$	33.6980	1.27275	0.636377	+	0.771378i	$0.280432\pi$
$702$	0	0
$703$	14.5403	0.548398
$704$	−20.3969	−0.768737
$705$	0	0
$706$	−62.4455	−2.35017
$707$	−5.25142	−0.197500
$708$	0	0
$709$	12.6531	0.475198	0.237599	−	0.971363i	$-0.423640\pi$
$709$	12.6531	0.475198	0.237599	+	0.971363i	$0.423640\pi$
$710$	24.3318	0.913157
$711$	0	0
$712$	−13.9564	−0.523040
$713$	−13.7647	−0.515490
$714$	0	0
$715$	−28.1609	−1.05316
$716$	50.0834	1.87170
$717$	0	0
$718$	−33.3820	−1.24581
$719$	−13.3377	−0.497411	−0.248706	−	0.968579i	$-0.580005\pi$
$719$	−13.3377	−0.497411	−0.248706	+	0.968579i	$0.580005\pi$
$720$	0	0
$721$	−22.7338	−0.846651
$722$	−40.1027	−1.49247
$723$	0	0
$724$	−57.7898	−2.14774
$725$	0	0
$726$	0	0
$727$	41.7683	1.54910	0.774551	−	0.632512i	$-0.217976\pi$
$727$	41.7683	1.54910	0.774551	+	0.632512i	$0.217976\pi$
$728$	108.830	4.03349
$729$	0	0
$730$	−33.8418	−1.25254
$731$	3.53573	0.130774
$732$	0	0
$733$	42.5635	1.57212	0.786059	−	0.618152i	$-0.212118\pi$
$733$	42.5635	1.57212	0.786059	+	0.618152i	$0.212118\pi$
$734$	82.1674	3.03285
$735$	0	0
$736$	75.2268	2.77290
$737$	42.1027	1.55087
$738$	0	0
$739$	−22.2405	−0.818129	−0.409064	−	0.912506i	$-0.634145\pi$
$739$	−22.2405	−0.818129	−0.409064	+	0.912506i	$0.634145\pi$
$740$	−41.9298	−1.54137
$741$	0	0
$742$	11.7035	0.429648
$743$	−28.0714	−1.02984	−0.514920	−	0.857238i	$-0.672178\pi$
$743$	−28.0714	−1.02984	−0.514920	+	0.857238i	$0.672178\pi$
$744$	0	0
$745$	−1.48137	−0.0542733
$746$	12.3091	0.450668
$747$	0	0
$748$	−49.4345	−1.80750
$749$	−12.3269	−0.450416
$750$	0	0
$751$	−28.5813	−1.04295	−0.521473	−	0.853268i	$-0.674617\pi$
$751$	−28.5813	−1.04295	−0.521473	+	0.853268i	$0.674617\pi$
$752$	3.10768	0.113325
$753$	0	0
$754$	0	0
$755$	7.52781	0.273965
$756$	0	0
$757$	21.4521	0.779689	0.389844	−	0.920881i	$-0.372529\pi$
$757$	21.4521	0.779689	0.389844	+	0.920881i	$0.372529\pi$
$758$	−5.48680	−0.199290
$759$	0	0
$760$	−15.0451	−0.545743
$761$	21.4222	0.776553	0.388277	−	0.921543i	$-0.373071\pi$
$761$	21.4222	0.776553	0.388277	+	0.921543i	$0.373071\pi$
$762$	0	0
$763$	48.4629	1.75447
$764$	−7.91209	−0.286249
$765$	0	0
$766$	−37.8989	−1.36934
$767$	27.3028	0.985846
$768$	0	0
$769$	22.7152	0.819130	0.409565	−	0.912281i	$-0.365680\pi$
$769$	22.7152	0.819130	0.409565	+	0.912281i	$0.365680\pi$
$770$	−35.8101	−1.29051
$771$	0	0
$772$	−49.7168	−1.78935
$773$	−0.180228	−0.00648235	−0.00324118	−	0.999995i	$-0.501032\pi$
$773$	−0.180228	−0.00648235	−0.00324118	+	0.999995i	$0.501032\pi$
$774$	0	0
$775$	5.74266	0.206282
$776$	−32.4066	−1.16333
$777$	0	0
$778$	5.00639	0.179488
$779$	−5.91989	−0.212102
$780$	0	0
$781$	−35.6219	−1.27465
$782$	55.1447	1.97197
$783$	0	0
$784$	5.62369	0.200846
$785$	27.7100	0.989013
$786$	0	0
$787$	44.4923	1.58598	0.792989	−	0.609236i	$-0.208524\pi$
$787$	44.4923	1.58598	0.792989	+	0.609236i	$0.208524\pi$
$788$	56.9513	2.02880
$789$	0	0
$790$	24.1821	0.860362
$791$	21.9318	0.779803
$792$	0	0
$793$	−8.75387	−0.310859
$794$	−77.7076	−2.75774
$795$	0	0
$796$	23.4851	0.832409
$797$	−2.28072	−0.0807872	−0.0403936	−	0.999184i	$-0.512861\pi$
$797$	−2.28072	−0.0807872	−0.0403936	+	0.999184i	$0.512861\pi$
$798$	0	0
$799$	0.863060	0.0305329
$800$	−31.3849	−1.10962
$801$	0	0
$802$	−40.6630	−1.43586
$803$	49.5446	1.74839
$804$	0	0
$805$	28.0238	0.987708
$806$	−22.7395	−0.800964
$807$	0	0
$808$	−13.2773	−0.467094
$809$	11.4774	0.403523	0.201761	−	0.979435i	$-0.435333\pi$
$809$	11.4774	0.403523	0.201761	+	0.979435i	$0.435333\pi$
$810$	0	0
$811$	27.4524	0.963985	0.481993	−	0.876175i	$-0.339913\pi$
$811$	27.4524	0.963985	0.481993	+	0.876175i	$0.339913\pi$
$812$	0	0
$813$	0	0
$814$	87.5019	3.06694
$815$	−18.5478	−0.649701
$816$	0	0
$817$	−2.74196	−0.0959290
$818$	12.6121	0.440971
$819$	0	0
$820$	17.0711	0.596150
$821$	25.1785	0.878734	0.439367	−	0.898308i	$-0.355203\pi$
$821$	25.1785	0.878734	0.439367	+	0.898308i	$0.355203\pi$
$822$	0	0
$823$	−2.12913	−0.0742169	−0.0371085	−	0.999311i	$-0.511815\pi$
$823$	−2.12913	−0.0742169	−0.0371085	+	0.999311i	$0.511815\pi$
$824$	−57.4785	−2.00236
$825$	0	0
$826$	34.7190	1.20803
$827$	−18.3427	−0.637837	−0.318918	−	0.947782i	$-0.603320\pi$
$827$	−18.3427	−0.637837	−0.318918	+	0.947782i	$0.603320\pi$
$828$	0	0
$829$	−52.9373	−1.83859	−0.919295	−	0.393570i	$-0.871240\pi$
$829$	−52.9373	−1.83859	−0.919295	+	0.393570i	$0.871240\pi$
$830$	1.43305	0.0497419
$831$	0	0
$832$	26.3694	0.914194
$833$	1.56180	0.0541133
$834$	0	0
$835$	6.22778	0.215521
$836$	38.3364	1.32589
$837$	0	0
$838$	−49.2175	−1.70019
$839$	−26.8505	−0.926984	−0.463492	−	0.886101i	$-0.653404\pi$
$839$	−26.8505	−0.926984	−0.463492	+	0.886101i	$0.653404\pi$
$840$	0	0
$841$	0	0
$842$	−32.4018	−1.11664
$843$	0	0
$844$	90.3075	3.10851
$845$	21.4705	0.738607
$846$	0	0
$847$	22.0083	0.756213
$848$	14.2185	0.488266
$849$	0	0
$850$	−23.0065	−0.789118
$851$	−68.4760	−2.34733
$852$	0	0
$853$	−39.8554	−1.36462	−0.682311	−	0.731062i	$-0.739025\pi$
$853$	−39.8554	−1.36462	−0.682311	+	0.731062i	$0.739025\pi$
$854$	−11.1317	−0.380918
$855$	0	0
$856$	−31.1665	−1.06525
$857$	−50.9827	−1.74154	−0.870768	−	0.491694i	$-0.836378\pi$
$857$	−50.9827	−1.74154	−0.870768	+	0.491694i	$0.836378\pi$
$858$	0	0
$859$	−14.8714	−0.507404	−0.253702	−	0.967282i	$-0.581648\pi$
$859$	−14.8714	−0.507404	−0.253702	+	0.967282i	$0.581648\pi$
$860$	7.90696	0.269625
$861$	0	0
$862$	−91.6288	−3.12089
$863$	20.4603	0.696478	0.348239	−	0.937406i	$-0.386780\pi$
$863$	20.4603	0.696478	0.348239	+	0.937406i	$0.386780\pi$
$864$	0	0
$865$	12.9742	0.441136
$866$	−15.1561	−0.515026
$867$	0	0
$868$	−20.2856	−0.688538
$869$	−35.4027	−1.20096
$870$	0	0
$871$	−54.4309	−1.84432
$872$	122.530	4.14939
$873$	0	0
$874$	−42.7647	−1.44654
$875$	−27.5772	−0.932280
$876$	0	0
$877$	−15.4211	−0.520732	−0.260366	−	0.965510i	$-0.583843\pi$
$877$	−15.4211	−0.520732	−0.260366	+	0.965510i	$0.583843\pi$
$878$	64.2744	2.16916
$879$	0	0
$880$	−43.5056	−1.46657
$881$	55.0390	1.85431	0.927155	−	0.374678i	$-0.122247\pi$
$881$	55.0390	1.85431	0.927155	+	0.374678i	$0.122247\pi$
$882$	0	0
$883$	37.9954	1.27865	0.639324	−	0.768938i	$-0.279214\pi$
$883$	37.9954	1.27865	0.639324	+	0.768938i	$0.279214\pi$
$884$	63.9096	2.14951
$885$	0	0
$886$	61.4762	2.06533
$887$	18.2699	0.613444	0.306722	−	0.951799i	$-0.400768\pi$
$887$	18.2699	0.613444	0.306722	+	0.951799i	$0.400768\pi$
$888$	0	0
$889$	9.27861	0.311195
$890$	−5.93703	−0.199010
$891$	0	0
$892$	−99.8661	−3.34376
$893$	−0.669303	−0.0223974
$894$	0	0
$895$	12.2409	0.409169
$896$	−13.6359	−0.455543
$897$	0	0
$898$	16.6927	0.557043
$899$	0	0
$900$	0	0
$901$	3.94875	0.131552
$902$	−35.6252	−1.18619
$903$	0	0
$904$	55.4506	1.84426
$905$	−14.1244	−0.469512
$906$	0	0
$907$	−25.2735	−0.839192	−0.419596	−	0.907711i	$-0.637828\pi$
$907$	−25.2735	−0.839192	−0.419596	+	0.907711i	$0.637828\pi$
$908$	−92.8701	−3.08200
$909$	0	0
$910$	46.2958	1.53469
$911$	34.3336	1.13752	0.568762	−	0.822502i	$-0.307422\pi$
$911$	34.3336	1.13752	0.568762	+	0.822502i	$0.307422\pi$
$912$	0	0
$913$	−2.09799	−0.0694334
$914$	−6.79686	−0.224820
$915$	0	0
$916$	−97.3731	−3.21730
$917$	−24.0390	−0.793836
$918$	0	0
$919$	−26.8764	−0.886572	−0.443286	−	0.896380i	$-0.646187\pi$
$919$	−26.8764	−0.886572	−0.443286	+	0.896380i	$0.646187\pi$
$920$	70.8532	2.33596
$921$	0	0
$922$	−18.5291	−0.610222
$923$	46.0524	1.51583
$924$	0	0
$925$	28.5684	0.939323
$926$	−47.2190	−1.55171
$927$	0	0
$928$	0	0
$929$	27.4218	0.899679	0.449839	−	0.893109i	$-0.351481\pi$
$929$	27.4218	0.899679	0.449839	+	0.893109i	$0.351481\pi$
$930$	0	0
$931$	−1.21118	−0.0396947
$932$	−97.6582	−3.19890
$933$	0	0
$934$	91.8118	3.00417
$935$	−12.0823	−0.395134
$936$	0	0
$937$	−10.3107	−0.336836	−0.168418	−	0.985716i	$-0.553866\pi$
$937$	−10.3107	−0.336836	−0.168418	+	0.985716i	$0.553866\pi$
$938$	−69.2158	−2.25998
$939$	0	0
$940$	1.93006	0.0629517
$941$	−16.9225	−0.551657	−0.275828	−	0.961207i	$-0.588952\pi$
$941$	−16.9225	−0.551657	−0.275828	+	0.961207i	$0.588952\pi$
$942$	0	0
$943$	27.8791	0.907867
$944$	42.1800	1.37284
$945$	0	0
$946$	−16.5008	−0.536487
$947$	−21.2231	−0.689659	−0.344829	−	0.938665i	$-0.612063\pi$
$947$	−21.2231	−0.689659	−0.344829	+	0.938665i	$0.612063\pi$
$948$	0	0
$949$	−64.0519	−2.07921
$950$	17.8416	0.578857
$951$	0	0
$952$	46.6929	1.51333
$953$	8.78444	0.284556	0.142278	−	0.989827i	$-0.454557\pi$
$953$	8.78444	0.284556	0.142278	+	0.989827i	$0.454557\pi$
$954$	0	0
$955$	−1.93380	−0.0625763
$956$	10.6936	0.345856
$957$	0	0
$958$	67.5637	2.18288
$959$	8.91469	0.287870
$960$	0	0
$961$	−28.5647	−0.921443
$962$	−113.124	−3.64725
$963$	0	0
$964$	55.3682	1.78329
$965$	−12.1513	−0.391165
$966$	0	0
$967$	25.9334	0.833961	0.416981	−	0.908915i	$-0.363088\pi$
$967$	25.9334	0.833961	0.416981	+	0.908915i	$0.363088\pi$
$968$	55.6441	1.78847
$969$	0	0
$970$	−13.7857	−0.442632
$971$	−58.1391	−1.86577	−0.932886	−	0.360172i	$-0.882718\pi$
$971$	−58.1391	−1.86577	−0.932886	+	0.360172i	$0.882718\pi$
$972$	0	0
$973$	−62.1863	−1.99360
$974$	57.3357	1.83715
$975$	0	0
$976$	−13.5238	−0.432887
$977$	−29.9259	−0.957414	−0.478707	−	0.877975i	$-0.658894\pi$
$977$	−29.9259	−0.957414	−0.478707	+	0.877975i	$0.658894\pi$
$978$	0	0
$979$	8.69184	0.277792
$980$	3.49266	0.111569
$981$	0	0
$982$	−30.4117	−0.970475
$983$	−40.2091	−1.28247	−0.641236	−	0.767344i	$-0.721578\pi$
$983$	−40.2091	−1.28247	−0.641236	+	0.767344i	$0.721578\pi$
$984$	0	0
$985$	13.9195	0.443512
$986$	0	0
$987$	0	0
$988$	−49.5619	−1.57677
$989$	12.9129	0.410608
$990$	0	0
$991$	39.1707	1.24430	0.622149	−	0.782898i	$-0.286260\pi$
$991$	39.1707	1.24430	0.622149	+	0.782898i	$0.286260\pi$
$992$	−13.3093	−0.422569
$993$	0	0
$994$	58.5615	1.85746
$995$	5.74002	0.181971
$996$	0	0
$997$	−34.4756	−1.09185	−0.545927	−	0.837833i	$-0.683823\pi$
$997$	−34.4756	−1.09185	−0.545927	+	0.837833i	$0.683823\pi$
$998$	−42.3892	−1.34181
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bm.1.8		9
3.2	odd	2		2523.2.a.o.1.2		9
29.9	even	14		261.2.k.c.226.1		18
29.13	even	14		261.2.k.c.82.1		18
29.28	even	2		7569.2.a.bj.1.2		9
87.38	odd	14		87.2.g.a.52.3	✓	18
87.71	odd	14		87.2.g.a.82.3	yes	18
87.86	odd	2		2523.2.a.r.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.52.3	✓	18	87.38	odd	14
87.2.g.a.82.3	yes	18	87.71	odd	14
261.2.k.c.82.1		18	29.13	even	14
261.2.k.c.226.1		18	29.9	even	14
2523.2.a.o.1.2		9	3.2	odd	2
2523.2.a.r.1.8		9	87.86	odd	2
7569.2.a.bj.1.2		9	29.28	even	2
7569.2.a.bm.1.8		9	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58860 q^{2} +4.70087 q^{4} +1.14894 q^{5} +2.76526 q^{7} +6.99148 q^{8} +2.97416 q^{10} -4.35418 q^{11} +5.62914 q^{13} +7.15816 q^{14} +8.69643 q^{16} +2.41516 q^{17} -1.87296 q^{19} +5.40103 q^{20} -11.2712 q^{22} +8.82048 q^{23} -3.67993 q^{25} +14.5716 q^{26} +12.9991 q^{28} -1.56053 q^{31} +8.52865 q^{32} +6.25190 q^{34} +3.17713 q^{35} -7.76330 q^{37} -4.84834 q^{38} +8.03281 q^{40} +3.16072 q^{41} +1.46397 q^{43} -20.4684 q^{44} +22.8327 q^{46} +0.357351 q^{47} +0.646666 q^{49} -9.52588 q^{50} +26.4618 q^{52} +1.63498 q^{53} -5.00270 q^{55} +19.3333 q^{56} +4.85026 q^{59} -1.55510 q^{61} -4.03961 q^{62} +4.68444 q^{64} +6.46755 q^{65} -9.66950 q^{67} +11.3534 q^{68} +8.22432 q^{70} +8.18108 q^{71} -11.3786 q^{73} -20.0961 q^{74} -8.80452 q^{76} -12.0404 q^{77} +8.13075 q^{79} +9.99170 q^{80} +8.18185 q^{82} +0.481835 q^{83} +2.77488 q^{85} +3.78965 q^{86} -30.4421 q^{88} -1.99621 q^{89} +15.5660 q^{91} +41.4639 q^{92} +0.925040 q^{94} -2.15192 q^{95} -4.63516 q^{97} +1.67396 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * q^98

Introduction

L-functions

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