LMFDB - Embedded newform 7569.2.a.bm.1.1

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			1.1
Root			$3.19178$ 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)

$f(q)$	$=$	$q-2.19178 q^{2} +2.80388 q^{4} +2.39677 q^{5} +1.35191 q^{7} -1.76193 q^{8} -5.25319 q^{10} +3.50954 q^{11} +6.03467 q^{13} -2.96308 q^{14} -1.74601 q^{16} +3.05437 q^{17} -5.63574 q^{19} +6.72027 q^{20} -7.69212 q^{22} -1.01478 q^{23} +0.744514 q^{25} -13.2266 q^{26} +3.79059 q^{28} -8.11166 q^{31} +7.35072 q^{32} -6.69450 q^{34} +3.24021 q^{35} -5.49645 q^{37} +12.3523 q^{38} -4.22294 q^{40} -6.85782 q^{41} +7.80197 q^{43} +9.84033 q^{44} +2.22416 q^{46} -0.418254 q^{47} -5.17235 q^{49} -1.63181 q^{50} +16.9205 q^{52} +13.3425 q^{53} +8.41156 q^{55} -2.38197 q^{56} -3.56102 q^{59} -7.20954 q^{61} +17.7790 q^{62} -12.6191 q^{64} +14.4637 q^{65} +8.31317 q^{67} +8.56410 q^{68} -7.10182 q^{70} +9.78678 q^{71} +0.303730 q^{73} +12.0470 q^{74} -15.8019 q^{76} +4.74457 q^{77} -4.00638 q^{79} -4.18478 q^{80} +15.0308 q^{82} +4.31434 q^{83} +7.32063 q^{85} -17.1002 q^{86} -6.18356 q^{88} +13.9924 q^{89} +8.15831 q^{91} -2.84532 q^{92} +0.916719 q^{94} -13.5076 q^{95} +3.80357 q^{97} +11.3366 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.19178	−1.54982	−0.774910	−	0.632072i	$-0.782205\pi$
$2$	−2.19178	−1.54982	−0.774910	+	0.632072i	$0.782205\pi$
$3$	0	0
$3$	0	0
$4$	2.80388	1.40194
$5$	2.39677	1.07187	0.535934	−	0.844260i	$-0.319959\pi$
$5$	2.39677	1.07187	0.535934	+	0.844260i	$0.319959\pi$
$6$	0	0
$7$	1.35191	0.510973	0.255486	−	0.966813i	$-0.417764\pi$
$7$	1.35191	0.510973	0.255486	+	0.966813i	$0.417764\pi$
$8$	−1.76193	−0.622936
$9$	0	0
$10$	−5.25319	−1.66120
$11$	3.50954	1.05817	0.529083	−	0.848570i	$-0.322536\pi$
$11$	3.50954	1.05817	0.529083	+	0.848570i	$0.322536\pi$
$12$	0	0
$13$	6.03467	1.67372	0.836858	−	0.547420i	$-0.184390\pi$
$13$	6.03467	1.67372	0.836858	+	0.547420i	$0.184390\pi$
$14$	−2.96308	−0.791916
$15$	0	0
$16$	−1.74601	−0.436502
$17$	3.05437	0.740794	0.370397	−	0.928874i	$-0.379222\pi$
$17$	3.05437	0.740794	0.370397	+	0.928874i	$0.379222\pi$
$18$	0	0
$19$	−5.63574	−1.29293	−0.646463	−	0.762945i	$-0.723753\pi$
$19$	−5.63574	−1.29293	−0.646463	+	0.762945i	$0.723753\pi$
$20$	6.72027	1.50270
$21$	0	0
$22$	−7.69212	−1.63997
$23$	−1.01478	−0.211596	−0.105798	−	0.994388i	$-0.533740\pi$
$23$	−1.01478	−0.211596	−0.105798	+	0.994388i	$0.533740\pi$
$24$	0	0
$25$	0.744514	0.148903
$26$	−13.2266	−2.59396
$27$	0	0
$28$	3.79059	0.716354
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−8.11166	−1.45690	−0.728449	−	0.685100i	$-0.759758\pi$
$31$	−8.11166	−1.45690	−0.728449	+	0.685100i	$0.759758\pi$
$32$	7.35072	1.29944
$33$	0	0
$34$	−6.69450	−1.14810
$35$	3.24021	0.547696
$36$	0	0
$37$	−5.49645	−0.903610	−0.451805	−	0.892117i	$-0.649220\pi$
$37$	−5.49645	−0.903610	−0.451805	+	0.892117i	$0.649220\pi$
$38$	12.3523	2.00380
$39$	0	0
$40$	−4.22294	−0.667706
$41$	−6.85782	−1.07101	−0.535506	−	0.844531i	$-0.679879\pi$
$41$	−6.85782	−1.07101	−0.535506	+	0.844531i	$0.679879\pi$
$42$	0	0
$43$	7.80197	1.18979	0.594895	−	0.803804i	$-0.297194\pi$
$43$	7.80197	1.18979	0.594895	+	0.803804i	$0.297194\pi$
$44$	9.84033	1.48349
$45$	0	0
$46$	2.22416	0.327935
$47$	−0.418254	−0.0610086	−0.0305043	−	0.999535i	$-0.509711\pi$
$47$	−0.418254	−0.0610086	−0.0305043	+	0.999535i	$0.509711\pi$
$48$	0	0
$49$	−5.17235	−0.738907
$50$	−1.63181	−0.230773
$51$	0	0
$52$	16.9205	2.34645
$53$	13.3425	1.83273	0.916367	−	0.400340i	$-0.131108\pi$
$53$	13.3425	1.83273	0.916367	+	0.400340i	$0.131108\pi$
$54$	0	0
$55$	8.41156	1.13421
$56$	−2.38197	−0.318304
$57$	0	0
$58$	0	0
$59$	−3.56102	−0.463606	−0.231803	−	0.972763i	$-0.574462\pi$
$59$	−3.56102	−0.463606	−0.231803	+	0.972763i	$0.574462\pi$
$60$	0	0
$61$	−7.20954	−0.923086	−0.461543	−	0.887118i	$-0.652704\pi$
$61$	−7.20954	−0.923086	−0.461543	+	0.887118i	$0.652704\pi$
$62$	17.7790	2.25793
$63$	0	0
$64$	−12.6191	−1.57739
$65$	14.4637	1.79400
$66$	0	0
$67$	8.31317	1.01562	0.507808	−	0.861471i	$-0.330456\pi$
$67$	8.31317	1.01562	0.507808	+	0.861471i	$0.330456\pi$
$68$	8.56410	1.03855
$69$	0	0
$70$	−7.10182	−0.848830
$71$	9.78678	1.16148	0.580739	−	0.814090i	$-0.302764\pi$
$71$	9.78678	1.16148	0.580739	+	0.814090i	$0.302764\pi$
$72$	0	0
$73$	0.303730	0.0355489	0.0177745	−	0.999842i	$-0.494342\pi$
$73$	0.303730	0.0355489	0.0177745	+	0.999842i	$0.494342\pi$
$74$	12.0470	1.40043
$75$	0	0
$76$	−15.8019	−1.81261
$77$	4.74457	0.540694
$78$	0	0
$79$	−4.00638	−0.450753	−0.225377	−	0.974272i	$-0.572361\pi$
$79$	−4.00638	−0.450753	−0.225377	+	0.974272i	$0.572361\pi$
$80$	−4.18478	−0.467873
$81$	0	0
$82$	15.0308	1.65988
$83$	4.31434	0.473560	0.236780	−	0.971563i	$-0.423908\pi$
$83$	4.31434	0.473560	0.236780	+	0.971563i	$0.423908\pi$
$84$	0	0
$85$	7.32063	0.794034
$86$	−17.1002	−1.84396
$87$	0	0
$88$	−6.18356	−0.659170
$89$	13.9924	1.48320	0.741598	−	0.670844i	$-0.234068\pi$
$89$	13.9924	1.48320	0.741598	+	0.670844i	$0.234068\pi$
$90$	0	0
$91$	8.15831	0.855224
$92$	−2.84532	−0.296645
$93$	0	0
$94$	0.916719	0.0945523
$95$	−13.5076	−1.38585
$96$	0	0
$97$	3.80357	0.386194	0.193097	−	0.981180i	$-0.438147\pi$
$97$	3.80357	0.386194	0.193097	+	0.981180i	$0.438147\pi$
$98$	11.3366	1.14517
$99$	0	0
$100$	2.08753	0.208753
$101$	−0.0422219	−0.00420124	−0.00210062	−	0.999998i	$-0.500669\pi$
$101$	−0.0422219	−0.00420124	−0.00210062	+	0.999998i	$0.500669\pi$
$102$	0	0
$103$	11.7804	1.16075	0.580377	−	0.814348i	$-0.302905\pi$
$103$	11.7804	1.16075	0.580377	+	0.814348i	$0.302905\pi$
$104$	−10.6327	−1.04262
$105$	0	0
$106$	−29.2438	−2.84041
$107$	6.26717	0.605870	0.302935	−	0.953011i	$-0.402033\pi$
$107$	6.26717	0.605870	0.302935	+	0.953011i	$0.402033\pi$
$108$	0	0
$109$	−14.8073	−1.41828	−0.709141	−	0.705067i	$-0.750917\pi$
$109$	−14.8073	−1.41828	−0.709141	+	0.705067i	$0.750917\pi$
$110$	−18.4363	−1.75783
$111$	0	0
$112$	−2.36044	−0.223041
$113$	11.8933	1.11882	0.559411	−	0.828890i	$-0.311027\pi$
$113$	11.8933	1.11882	0.559411	+	0.828890i	$0.311027\pi$
$114$	0	0
$115$	−2.43219	−0.226803
$116$	0	0
$117$	0	0
$118$	7.80497	0.718506
$119$	4.12923	0.378526
$120$	0	0
$121$	1.31686	0.119714
$122$	15.8017	1.43062
$123$	0	0
$124$	−22.7442	−2.04249
$125$	−10.1994	−0.912265
$126$	0	0
$127$	15.0938	1.33935	0.669677	−	0.742652i	$-0.266433\pi$
$127$	15.0938	1.33935	0.669677	+	0.742652i	$0.266433\pi$
$128$	12.9568	1.14523
$129$	0	0
$130$	−31.7012	−2.78038
$131$	10.4468	0.912739	0.456369	−	0.889790i	$-0.349150\pi$
$131$	10.4468	0.912739	0.456369	+	0.889790i	$0.349150\pi$
$132$	0	0
$133$	−7.61899	−0.660650
$134$	−18.2206	−1.57402
$135$	0	0
$136$	−5.38159	−0.461468
$137$	−3.67212	−0.313731	−0.156865	−	0.987620i	$-0.550139\pi$
$137$	−3.67212	−0.313731	−0.156865	+	0.987620i	$0.550139\pi$
$138$	0	0
$139$	5.78134	0.490367	0.245183	−	0.969477i	$-0.421152\pi$
$139$	5.78134	0.490367	0.245183	+	0.969477i	$0.421152\pi$
$140$	9.08518	0.767838
$141$	0	0
$142$	−21.4504	−1.80008
$143$	21.1789	1.77107
$144$	0	0
$145$	0	0
$146$	−0.665709	−0.0550944
$147$	0	0
$148$	−15.4114	−1.26681
$149$	8.58416	0.703242	0.351621	−	0.936142i	$-0.385631\pi$
$149$	8.58416	0.703242	0.351621	+	0.936142i	$0.385631\pi$
$150$	0	0
$151$	11.9406	0.971714	0.485857	−	0.874038i	$-0.338508\pi$
$151$	11.9406	0.971714	0.485857	+	0.874038i	$0.338508\pi$
$152$	9.92977	0.805411
$153$	0	0
$154$	−10.3990	−0.837978
$155$	−19.4418	−1.56160
$156$	0	0
$157$	12.2535	0.977936	0.488968	−	0.872302i	$-0.337374\pi$
$157$	12.2535	0.977936	0.488968	+	0.872302i	$0.337374\pi$
$158$	8.78109	0.698586
$159$	0	0
$160$	17.6180	1.39282
$161$	−1.37189	−0.108120
$162$	0	0
$163$	7.93140	0.621235	0.310618	−	0.950535i	$-0.399464\pi$
$163$	7.93140	0.621235	0.310618	+	0.950535i	$0.399464\pi$
$164$	−19.2285	−1.50150
$165$	0	0
$166$	−9.45607	−0.733933
$167$	−12.9244	−1.00012	−0.500061	−	0.865990i	$-0.666689\pi$
$167$	−12.9244	−1.00012	−0.500061	+	0.865990i	$0.666689\pi$
$168$	0	0
$169$	23.4172	1.80133
$170$	−16.0452	−1.23061
$171$	0	0
$172$	21.8758	1.66801
$173$	−2.65357	−0.201747	−0.100874	−	0.994899i	$-0.532164\pi$
$173$	−2.65357	−0.201747	−0.100874	+	0.994899i	$0.532164\pi$
$174$	0	0
$175$	1.00651	0.0760853
$176$	−6.12768	−0.461891
$177$	0	0
$178$	−30.6683	−2.29869
$179$	13.3988	1.00147	0.500735	−	0.865601i	$-0.333063\pi$
$179$	13.3988	1.00147	0.500735	+	0.865601i	$0.333063\pi$
$180$	0	0
$181$	1.51389	0.112527	0.0562634	−	0.998416i	$-0.482081\pi$
$181$	1.51389	0.112527	0.0562634	+	0.998416i	$0.482081\pi$
$182$	−17.8812	−1.32544
$183$	0	0
$184$	1.78797	0.131811
$185$	−13.1737	−0.968552
$186$	0	0
$187$	10.7194	0.783883
$188$	−1.17273	−0.0855305
$189$	0	0
$190$	29.6056	2.14781
$191$	−11.6187	−0.840696	−0.420348	−	0.907363i	$-0.638092\pi$
$191$	−11.6187	−0.840696	−0.420348	+	0.907363i	$0.638092\pi$
$192$	0	0
$193$	6.16179	0.443536	0.221768	−	0.975100i	$-0.428817\pi$
$193$	6.16179	0.443536	0.221768	+	0.975100i	$0.428817\pi$
$194$	−8.33658	−0.598532
$195$	0	0
$196$	−14.5027	−1.03590
$197$	2.52672	0.180021	0.0900107	−	0.995941i	$-0.471310\pi$
$197$	2.52672	0.180021	0.0900107	+	0.995941i	$0.471310\pi$
$198$	0	0
$199$	13.2818	0.941524	0.470762	−	0.882260i	$-0.343979\pi$
$199$	13.2818	0.941524	0.470762	+	0.882260i	$0.343979\pi$
$200$	−1.31178	−0.0927570
$201$	0	0
$202$	0.0925410	0.00651116
$203$	0	0
$204$	0	0
$205$	−16.4366	−1.14798
$206$	−25.8199	−1.79896
$207$	0	0
$208$	−10.5366	−0.730580
$209$	−19.7788	−1.36813
$210$	0	0
$211$	8.30478	0.571724	0.285862	−	0.958271i	$-0.407720\pi$
$211$	8.30478	0.571724	0.285862	+	0.958271i	$0.407720\pi$
$212$	37.4108	2.56938
$213$	0	0
$214$	−13.7362	−0.938990
$215$	18.6995	1.27530
$216$	0	0
$217$	−10.9662	−0.744435
$218$	32.4543	2.19808
$219$	0	0
$220$	23.5850	1.59010
$221$	18.4321	1.23988
$222$	0	0
$223$	25.0119	1.67492	0.837461	−	0.546496i	$-0.184039\pi$
$223$	25.0119	1.67492	0.837461	+	0.546496i	$0.184039\pi$
$224$	9.93749	0.663977
$225$	0	0
$226$	−26.0673	−1.73397
$227$	−3.04387	−0.202029	−0.101014	−	0.994885i	$-0.532209\pi$
$227$	−3.04387	−0.202029	−0.101014	+	0.994885i	$0.532209\pi$
$228$	0	0
$229$	21.9086	1.44776	0.723879	−	0.689927i	$-0.242357\pi$
$229$	21.9086	1.44776	0.723879	+	0.689927i	$0.242357\pi$
$230$	5.33082	0.351504
$231$	0	0
$232$	0	0
$233$	−9.96464	−0.652805	−0.326403	−	0.945231i	$-0.605837\pi$
$233$	−9.96464	−0.652805	−0.326403	+	0.945231i	$0.605837\pi$
$234$	0	0
$235$	−1.00246	−0.0653932
$236$	−9.98469	−0.649948
$237$	0	0
$238$	−9.05034	−0.586647
$239$	7.78097	0.503309	0.251655	−	0.967817i	$-0.419025\pi$
$239$	7.78097	0.503309	0.251655	+	0.967817i	$0.419025\pi$
$240$	0	0
$241$	−23.1891	−1.49374	−0.746872	−	0.664968i	$-0.768445\pi$
$241$	−23.1891	−1.49374	−0.746872	+	0.664968i	$0.768445\pi$
$242$	−2.88626	−0.185536
$243$	0	0
$244$	−20.2147	−1.29411
$245$	−12.3969	−0.792011
$246$	0	0
$247$	−34.0098	−2.16399
$248$	14.2922	0.907555
$249$	0	0
$250$	22.3549	1.41385
$251$	1.18054	0.0745150	0.0372575	−	0.999306i	$-0.488138\pi$
$251$	1.18054	0.0745150	0.0372575	+	0.999306i	$0.488138\pi$
$252$	0	0
$253$	−3.56140	−0.223903
$254$	−33.0821	−2.07576
$255$	0	0
$256$	−3.16025	−0.197516
$257$	−15.9745	−0.996461	−0.498231	−	0.867044i	$-0.666017\pi$
$257$	−15.9745	−0.996461	−0.498231	+	0.867044i	$0.666017\pi$
$258$	0	0
$259$	−7.43069	−0.461720
$260$	40.5546	2.51509
$261$	0	0
$262$	−22.8970	−1.41458
$263$	14.6106	0.900927	0.450463	−	0.892795i	$-0.351259\pi$
$263$	14.6106	0.900927	0.450463	+	0.892795i	$0.351259\pi$
$264$	0	0
$265$	31.9789	1.96445
$266$	16.6991	1.02389
$267$	0	0
$268$	23.3091	1.42383
$269$	22.1146	1.34835	0.674174	−	0.738573i	$-0.264500\pi$
$269$	22.1146	1.34835	0.674174	+	0.738573i	$0.264500\pi$
$270$	0	0
$271$	23.1151	1.40414	0.702072	−	0.712106i	$-0.252258\pi$
$271$	23.1151	1.40414	0.702072	+	0.712106i	$0.252258\pi$
$272$	−5.33296	−0.323358
$273$	0	0
$274$	8.04847	0.486226
$275$	2.61290	0.157564
$276$	0	0
$277$	18.7130	1.12435	0.562177	−	0.827017i	$-0.309964\pi$
$277$	18.7130	1.12435	0.562177	+	0.827017i	$0.309964\pi$
$278$	−12.6714	−0.759980
$279$	0	0
$280$	−5.70903	−0.341180
$281$	8.66587	0.516962	0.258481	−	0.966016i	$-0.416778\pi$
$281$	8.66587	0.516962	0.258481	+	0.966016i	$0.416778\pi$
$282$	0	0
$283$	5.58496	0.331991	0.165996	−	0.986126i	$-0.446916\pi$
$283$	5.58496	0.331991	0.165996	+	0.986126i	$0.446916\pi$
$284$	27.4410	1.62832
$285$	0	0
$286$	−46.4194	−2.74484
$287$	−9.27114	−0.547258
$288$	0	0
$289$	−7.67081	−0.451224
$290$	0	0
$291$	0	0
$292$	0.851624	0.0498375
$293$	−4.71899	−0.275686	−0.137843	−	0.990454i	$-0.544017\pi$
$293$	−4.71899	−0.275686	−0.137843	+	0.990454i	$0.544017\pi$
$294$	0	0
$295$	−8.53496	−0.496925
$296$	9.68436	0.562892
$297$	0	0
$298$	−18.8146	−1.08990
$299$	−6.12385	−0.354151
$300$	0	0
$301$	10.5475	0.607950
$302$	−26.1712	−1.50598
$303$	0	0
$304$	9.84004	0.564365
$305$	−17.2796	−0.989428
$306$	0	0
$307$	−11.7362	−0.669821	−0.334910	−	0.942250i	$-0.608706\pi$
$307$	−11.7362	−0.669821	−0.334910	+	0.942250i	$0.608706\pi$
$308$	13.3032	0.758021
$309$	0	0
$310$	42.6121	2.42020
$311$	16.8038	0.952858	0.476429	−	0.879213i	$-0.341931\pi$
$311$	16.8038	0.952858	0.476429	+	0.879213i	$0.341931\pi$
$312$	0	0
$313$	−11.9599	−0.676012	−0.338006	−	0.941144i	$-0.609752\pi$
$313$	−11.9599	−0.676012	−0.338006	+	0.941144i	$0.609752\pi$
$314$	−26.8569	−1.51562
$315$	0	0
$316$	−11.2334	−0.631929
$317$	−14.1360	−0.793954	−0.396977	−	0.917829i	$-0.629941\pi$
$317$	−14.1360	−0.793954	−0.396977	+	0.917829i	$0.629941\pi$
$318$	0	0
$319$	0	0
$320$	−30.2451	−1.69075
$321$	0	0
$322$	3.00686	0.167566
$323$	−17.2136	−0.957792
$324$	0	0
$325$	4.49290	0.249221
$326$	−17.3839	−0.962803
$327$	0	0
$328$	12.0830	0.667172
$329$	−0.565441	−0.0311737
$330$	0	0
$331$	−23.0236	−1.26549	−0.632747	−	0.774359i	$-0.718073\pi$
$331$	−23.0236	−1.26549	−0.632747	+	0.774359i	$0.718073\pi$
$332$	12.0969	0.663904
$333$	0	0
$334$	28.3275	1.55001
$335$	19.9248	1.08861
$336$	0	0
$337$	2.55630	0.139250	0.0696252	−	0.997573i	$-0.477820\pi$
$337$	2.55630	0.139250	0.0696252	+	0.997573i	$0.477820\pi$
$338$	−51.3253	−2.79173
$339$	0	0
$340$	20.5262	1.11319
$341$	−28.4682	−1.54164
$342$	0	0
$343$	−16.4559	−0.888534
$344$	−13.7465	−0.741163
$345$	0	0
$346$	5.81604	0.312672
$347$	0.891885	0.0478789	0.0239394	−	0.999713i	$-0.492379\pi$
$347$	0.891885	0.0478789	0.0239394	+	0.999713i	$0.492379\pi$
$348$	0	0
$349$	−0.755575	−0.0404450	−0.0202225	−	0.999796i	$-0.506437\pi$
$349$	−0.755575	−0.0404450	−0.0202225	+	0.999796i	$0.506437\pi$
$350$	−2.20605	−0.117919
$351$	0	0
$352$	25.7976	1.37502
$353$	−19.2406	−1.02407	−0.512036	−	0.858964i	$-0.671109\pi$
$353$	−19.2406	−1.02407	−0.512036	+	0.858964i	$0.671109\pi$
$354$	0	0
$355$	23.4567	1.24495
$356$	39.2332	2.07935
$357$	0	0
$358$	−29.3671	−1.55210
$359$	17.6146	0.929661	0.464830	−	0.885400i	$-0.346115\pi$
$359$	17.6146	0.929661	0.464830	+	0.885400i	$0.346115\pi$
$360$	0	0
$361$	12.7615	0.671658
$362$	−3.31812	−0.174396
$363$	0	0
$364$	22.8750	1.19897
$365$	0.727972	0.0381038
$366$	0	0
$367$	−3.79787	−0.198247	−0.0991235	−	0.995075i	$-0.531604\pi$
$367$	−3.79787	−0.198247	−0.0991235	+	0.995075i	$0.531604\pi$
$368$	1.77181	0.0923620
$369$	0	0
$370$	28.8739	1.50108
$371$	18.0378	0.936477
$372$	0	0
$373$	−8.29264	−0.429377	−0.214688	−	0.976683i	$-0.568874\pi$
$373$	−8.29264	−0.429377	−0.214688	+	0.976683i	$0.568874\pi$
$374$	−23.4946	−1.21488
$375$	0	0
$376$	0.736934	0.0380045
$377$	0	0
$378$	0	0
$379$	8.93238	0.458826	0.229413	−	0.973329i	$-0.426319\pi$
$379$	8.93238	0.458826	0.229413	+	0.973329i	$0.426319\pi$
$380$	−37.8736	−1.94288
$381$	0	0
$382$	25.4655	1.30293
$383$	−35.8530	−1.83200	−0.916001	−	0.401176i	$-0.868602\pi$
$383$	−35.8530	−1.83200	−0.916001	+	0.401176i	$0.868602\pi$
$384$	0	0
$385$	11.3717	0.579553
$386$	−13.5053	−0.687400
$387$	0	0
$388$	10.6648	0.541422
$389$	11.3446	0.575194	0.287597	−	0.957752i	$-0.407144\pi$
$389$	11.3446	0.575194	0.287597	+	0.957752i	$0.407144\pi$
$390$	0	0
$391$	−3.09951	−0.156749
$392$	9.11331	0.460292
$393$	0	0
$394$	−5.53800	−0.279001
$395$	−9.60238	−0.483148
$396$	0	0
$397$	−24.1353	−1.21131	−0.605657	−	0.795726i	$-0.707090\pi$
$397$	−24.1353	−1.21131	−0.605657	+	0.795726i	$0.707090\pi$
$398$	−29.1108	−1.45919
$399$	0	0
$400$	−1.29993	−0.0649964
$401$	−19.3564	−0.966611	−0.483305	−	0.875452i	$-0.660564\pi$
$401$	−19.3564	−0.966611	−0.483305	+	0.875452i	$0.660564\pi$
$402$	0	0
$403$	−48.9512	−2.43843
$404$	−0.118385	−0.00588989
$405$	0	0
$406$	0	0
$407$	−19.2900	−0.956169
$408$	0	0
$409$	−11.4459	−0.565963	−0.282981	−	0.959125i	$-0.591323\pi$
$409$	−11.4459	−0.565963	−0.282981	+	0.959125i	$0.591323\pi$
$410$	36.0254	1.77917
$411$	0	0
$412$	33.0308	1.62731
$413$	−4.81418	−0.236890
$414$	0	0
$415$	10.3405	0.507595
$416$	44.3592	2.17489
$417$	0	0
$418$	43.3508	2.12036
$419$	21.1185	1.03171	0.515854	−	0.856677i	$-0.327475\pi$
$419$	21.1185	1.03171	0.515854	+	0.856677i	$0.327475\pi$
$420$	0	0
$421$	−27.5131	−1.34090	−0.670452	−	0.741953i	$-0.733900\pi$
$421$	−27.5131	−1.34090	−0.670452	+	0.741953i	$0.733900\pi$
$422$	−18.2022	−0.886070
$423$	0	0
$424$	−23.5086	−1.14168
$425$	2.27402	0.110306
$426$	0	0
$427$	−9.74662	−0.471672
$428$	17.5724	0.849394
$429$	0	0
$430$	−40.9852	−1.97648
$431$	1.94954	0.0939060	0.0469530	−	0.998897i	$-0.485049\pi$
$431$	1.94954	0.0939060	0.0469530	+	0.998897i	$0.485049\pi$
$432$	0	0
$433$	−7.83478	−0.376516	−0.188258	−	0.982120i	$-0.560284\pi$
$433$	−7.83478	−0.376516	−0.188258	+	0.982120i	$0.560284\pi$
$434$	24.0355	1.15374
$435$	0	0
$436$	−41.5179	−1.98835
$437$	5.71902	0.273578
$438$	0	0
$439$	2.38989	0.114063	0.0570317	−	0.998372i	$-0.481836\pi$
$439$	2.38989	0.114063	0.0570317	+	0.998372i	$0.481836\pi$
$440$	−14.8206	−0.706544
$441$	0	0
$442$	−40.3991	−1.92159
$443$	38.3497	1.82205	0.911024	−	0.412354i	$-0.135293\pi$
$443$	38.3497	1.82205	0.911024	+	0.412354i	$0.135293\pi$
$444$	0	0
$445$	33.5367	1.58979
$446$	−54.8206	−2.59583
$447$	0	0
$448$	−17.0599	−0.806003
$449$	−31.3441	−1.47922	−0.739609	−	0.673037i	$-0.764990\pi$
$449$	−31.3441	−1.47922	−0.739609	+	0.673037i	$0.764990\pi$
$450$	0	0
$451$	−24.0678	−1.13331
$452$	33.3473	1.56852
$453$	0	0
$454$	6.67148	0.313108
$455$	19.5536	0.916688
$456$	0	0
$457$	−32.2643	−1.50926	−0.754630	−	0.656151i	$-0.772183\pi$
$457$	−32.2643	−1.50926	−0.754630	+	0.656151i	$0.772183\pi$
$458$	−48.0187	−2.24376
$459$	0	0
$460$	−6.81957	−0.317964
$461$	−5.67806	−0.264454	−0.132227	−	0.991219i	$-0.542213\pi$
$461$	−5.67806	−0.264454	−0.132227	+	0.991219i	$0.542213\pi$
$462$	0	0
$463$	23.2339	1.07977	0.539886	−	0.841738i	$-0.318467\pi$
$463$	23.2339	1.07977	0.539886	+	0.841738i	$0.318467\pi$
$464$	0	0
$465$	0	0
$466$	21.8403	1.01173
$467$	12.0771	0.558861	0.279430	−	0.960166i	$-0.409854\pi$
$467$	12.0771	0.558861	0.279430	+	0.960166i	$0.409854\pi$
$468$	0	0
$469$	11.2386	0.518952
$470$	2.19717	0.101348
$471$	0	0
$472$	6.27428	0.288797
$473$	27.3813	1.25899
$474$	0	0
$475$	−4.19588	−0.192520
$476$	11.5779	0.530671
$477$	0	0
$478$	−17.0541	−0.780038
$479$	−30.5109	−1.39408	−0.697039	−	0.717033i	$-0.745500\pi$
$479$	−30.5109	−1.39408	−0.697039	+	0.717033i	$0.745500\pi$
$480$	0	0
$481$	−33.1692	−1.51239
$482$	50.8254	2.31503
$483$	0	0
$484$	3.69231	0.167832
$485$	9.11630	0.413950
$486$	0	0
$487$	−23.5248	−1.06601	−0.533005	−	0.846112i	$-0.678937\pi$
$487$	−23.5248	−1.06601	−0.533005	+	0.846112i	$0.678937\pi$
$488$	12.7027	0.575024
$489$	0	0
$490$	27.1713	1.22747
$491$	27.5887	1.24506	0.622531	−	0.782595i	$-0.286104\pi$
$491$	27.5887	1.24506	0.622531	+	0.782595i	$0.286104\pi$
$492$	0	0
$493$	0	0
$494$	74.5419	3.35380
$495$	0	0
$496$	14.1630	0.635939
$497$	13.2308	0.593483
$498$	0	0
$499$	−38.3679	−1.71758	−0.858791	−	0.512327i	$-0.828784\pi$
$499$	−38.3679	−1.71758	−0.858791	+	0.512327i	$0.828784\pi$
$500$	−28.5980	−1.27894
$501$	0	0
$502$	−2.58748	−0.115485
$503$	−4.95969	−0.221142	−0.110571	−	0.993868i	$-0.535268\pi$
$503$	−4.95969	−0.221142	−0.110571	+	0.993868i	$0.535268\pi$
$504$	0	0
$505$	−0.101196	−0.00450318
$506$	7.80579	0.347010
$507$	0	0
$508$	42.3211	1.87770
$509$	16.4294	0.728220	0.364110	−	0.931356i	$-0.381373\pi$
$509$	16.4294	0.728220	0.364110	+	0.931356i	$0.381373\pi$
$510$	0	0
$511$	0.410615	0.0181645
$512$	−18.9871	−0.839119
$513$	0	0
$514$	35.0125	1.54434
$515$	28.2348	1.24418
$516$	0	0
$517$	−1.46788	−0.0645572
$518$	16.2864	0.715583
$519$	0	0
$520$	−25.4841	−1.11755
$521$	30.7040	1.34517	0.672583	−	0.740021i	$-0.265185\pi$
$521$	30.7040	1.34517	0.672583	+	0.740021i	$0.265185\pi$
$522$	0	0
$523$	37.0601	1.62052	0.810262	−	0.586068i	$-0.199325\pi$
$523$	37.0601	1.62052	0.810262	+	0.586068i	$0.199325\pi$
$524$	29.2915	1.27961
$525$	0	0
$526$	−32.0231	−1.39627
$527$	−24.7760	−1.07926
$528$	0	0
$529$	−21.9702	−0.955227
$530$	−70.0906	−3.04454
$531$	0	0
$532$	−21.3628	−0.926193
$533$	−41.3847	−1.79257
$534$	0	0
$535$	15.0210	0.649413
$536$	−14.6472	−0.632664
$537$	0	0
$538$	−48.4701	−2.08970
$539$	−18.1525	−0.781886
$540$	0	0
$541$	−34.0254	−1.46287	−0.731434	−	0.681913i	$-0.761148\pi$
$541$	−34.0254	−1.46287	−0.731434	+	0.681913i	$0.761148\pi$
$542$	−50.6632	−2.17617
$543$	0	0
$544$	22.4518	0.962614
$545$	−35.4897	−1.52021
$546$	0	0
$547$	17.1964	0.735265	0.367633	−	0.929971i	$-0.380168\pi$
$547$	17.1964	0.735265	0.367633	+	0.929971i	$0.380168\pi$
$548$	−10.2962	−0.439832
$549$	0	0
$550$	−5.72689	−0.244196
$551$	0	0
$552$	0	0
$553$	−5.41626	−0.230323
$554$	−41.0147	−1.74255
$555$	0	0
$556$	16.2102	0.687466
$557$	29.6847	1.25778	0.628890	−	0.777494i	$-0.283510\pi$
$557$	29.6847	1.25778	0.628890	+	0.777494i	$0.283510\pi$
$558$	0	0
$559$	47.0823	1.99137
$560$	−5.65744	−0.239070
$561$	0	0
$562$	−18.9936	−0.801198
$563$	−27.8621	−1.17425	−0.587125	−	0.809497i	$-0.699740\pi$
$563$	−27.8621	−1.17425	−0.587125	+	0.809497i	$0.699740\pi$
$564$	0	0
$565$	28.5054	1.19923
$566$	−12.2410	−0.514527
$567$	0	0
$568$	−17.2436	−0.723526
$569$	−9.78846	−0.410354	−0.205177	−	0.978725i	$-0.565777\pi$
$569$	−9.78846	−0.410354	−0.205177	+	0.978725i	$0.565777\pi$
$570$	0	0
$571$	−32.3247	−1.35275	−0.676374	−	0.736559i	$-0.736450\pi$
$571$	−32.3247	−1.35275	−0.676374	+	0.736559i	$0.736450\pi$
$572$	59.3831	2.48293
$573$	0	0
$574$	20.3203	0.848151
$575$	−0.755516	−0.0315072
$576$	0	0
$577$	43.6057	1.81533	0.907664	−	0.419698i	$-0.137864\pi$
$577$	43.6057	1.81533	0.907664	+	0.419698i	$0.137864\pi$
$578$	16.8127	0.699316
$579$	0	0
$580$	0	0
$581$	5.83259	0.241977
$582$	0	0
$583$	46.8260	1.93934
$584$	−0.535152	−0.0221447
$585$	0	0
$586$	10.3430	0.427264
$587$	43.6633	1.80218	0.901088	−	0.433636i	$-0.142769\pi$
$587$	43.6633	1.80218	0.901088	+	0.433636i	$0.142769\pi$
$588$	0	0
$589$	45.7152	1.88366
$590$	18.7067	0.770144
$591$	0	0
$592$	9.59684	0.394428
$593$	−24.0076	−0.985873	−0.492937	−	0.870065i	$-0.664077\pi$
$593$	−24.0076	−0.985873	−0.492937	+	0.870065i	$0.664077\pi$
$594$	0	0
$595$	9.89682	0.405730
$596$	24.0690	0.985904
$597$	0	0
$598$	13.4221	0.548870
$599$	3.63501	0.148522	0.0742612	−	0.997239i	$-0.476340\pi$
$599$	3.63501	0.148522	0.0742612	+	0.997239i	$0.476340\pi$
$600$	0	0
$601$	15.2601	0.622474	0.311237	−	0.950332i	$-0.399257\pi$
$601$	15.2601	0.622474	0.311237	+	0.950332i	$0.399257\pi$
$602$	−23.1178	−0.942213
$603$	0	0
$604$	33.4801	1.36229
$605$	3.15621	0.128318
$606$	0	0
$607$	2.89009	0.117305	0.0586526	−	0.998278i	$-0.481320\pi$
$607$	2.89009	0.117305	0.0586526	+	0.998278i	$0.481320\pi$
$608$	−41.4267	−1.68007
$609$	0	0
$610$	37.8730	1.53343
$611$	−2.52402	−0.102111
$612$	0	0
$613$	−4.40402	−0.177877	−0.0889384	−	0.996037i	$-0.528347\pi$
$613$	−4.40402	−0.177877	−0.0889384	+	0.996037i	$0.528347\pi$
$614$	25.7231	1.03810
$615$	0	0
$616$	−8.35960	−0.336818
$617$	6.95938	0.280174	0.140087	−	0.990139i	$-0.455262\pi$
$617$	6.95938	0.280174	0.140087	+	0.990139i	$0.455262\pi$
$618$	0	0
$619$	−13.1480	−0.528461	−0.264230	−	0.964460i	$-0.585118\pi$
$619$	−13.1480	−0.528461	−0.264230	+	0.964460i	$0.585118\pi$
$620$	−54.5125	−2.18928
$621$	0	0
$622$	−36.8302	−1.47676
$623$	18.9165	0.757873
$624$	0	0
$625$	−28.1683	−1.12673
$626$	26.2134	1.04770
$627$	0	0
$628$	34.3574	1.37101
$629$	−16.7882	−0.669389
$630$	0	0
$631$	−13.2938	−0.529217	−0.264609	−	0.964356i	$-0.585243\pi$
$631$	−13.2938	−0.529217	−0.264609	+	0.964356i	$0.585243\pi$
$632$	7.05896	0.280790
$633$	0	0
$634$	30.9828	1.23049
$635$	36.1763	1.43561
$636$	0	0
$637$	−31.2134	−1.23672
$638$	0	0
$639$	0	0
$640$	31.0546	1.22754
$641$	20.1373	0.795376	0.397688	−	0.917521i	$-0.369813\pi$
$641$	20.1373	0.795376	0.397688	+	0.917521i	$0.369813\pi$
$642$	0	0
$643$	44.1506	1.74113	0.870565	−	0.492053i	$-0.163753\pi$
$643$	44.1506	1.74113	0.870565	+	0.492053i	$0.163753\pi$
$644$	−3.84660	−0.151577
$645$	0	0
$646$	37.7284	1.48440
$647$	−18.2323	−0.716784	−0.358392	−	0.933571i	$-0.616675\pi$
$647$	−18.2323	−0.716784	−0.358392	+	0.933571i	$0.616675\pi$
$648$	0	0
$649$	−12.4976	−0.490572
$650$	−9.84742	−0.386248
$651$	0	0
$652$	22.2387	0.870935
$653$	−15.1966	−0.594688	−0.297344	−	0.954770i	$-0.596101\pi$
$653$	−15.1966	−0.594688	−0.297344	+	0.954770i	$0.596101\pi$
$654$	0	0
$655$	25.0385	0.978336
$656$	11.9738	0.467499
$657$	0	0
$658$	1.23932	0.0483137
$659$	−16.7811	−0.653699	−0.326849	−	0.945076i	$-0.605987\pi$
$659$	−16.7811	−0.653699	−0.326849	+	0.945076i	$0.605987\pi$
$660$	0	0
$661$	−1.34387	−0.0522705	−0.0261352	−	0.999658i	$-0.508320\pi$
$661$	−1.34387	−0.0522705	−0.0261352	+	0.999658i	$0.508320\pi$
$662$	50.4627	1.96129
$663$	0	0
$664$	−7.60157	−0.294998
$665$	−18.2610	−0.708131
$666$	0	0
$667$	0	0
$668$	−36.2386	−1.40211
$669$	0	0
$670$	−43.6706	−1.68714
$671$	−25.3021	−0.976778
$672$	0	0
$673$	11.9193	0.459457	0.229728	−	0.973255i	$-0.426216\pi$
$673$	11.9193	0.459457	0.229728	+	0.973255i	$0.426216\pi$
$674$	−5.60283	−0.215813
$675$	0	0
$676$	65.6592	2.52535
$677$	16.2061	0.622850	0.311425	−	0.950271i	$-0.399194\pi$
$677$	16.2061	0.622850	0.311425	+	0.950271i	$0.399194\pi$
$678$	0	0
$679$	5.14208	0.197335
$680$	−12.8984	−0.494633
$681$	0	0
$682$	62.3959	2.38926
$683$	28.8129	1.10249	0.551247	−	0.834342i	$-0.314152\pi$
$683$	28.8129	1.10249	0.551247	+	0.834342i	$0.314152\pi$
$684$	0	0
$685$	−8.80124	−0.336278
$686$	36.0676	1.37707
$687$	0	0
$688$	−13.6223	−0.519345
$689$	80.5176	3.06747
$690$	0	0
$691$	−25.3535	−0.964492	−0.482246	−	0.876036i	$-0.660179\pi$
$691$	−25.3535	−0.964492	−0.482246	+	0.876036i	$0.660179\pi$
$692$	−7.44031	−0.282838
$693$	0	0
$694$	−1.95481	−0.0742037
$695$	13.8566	0.525609
$696$	0	0
$697$	−20.9463	−0.793399
$698$	1.65605	0.0626825
$699$	0	0
$700$	2.82215	0.106667
$701$	−13.2108	−0.498965	−0.249482	−	0.968379i	$-0.580260\pi$
$701$	−13.2108	−0.498965	−0.249482	+	0.968379i	$0.580260\pi$
$702$	0	0
$703$	30.9765	1.16830
$704$	−44.2873	−1.66914
$705$	0	0
$706$	42.1710	1.58713
$707$	−0.0570801	−0.00214672
$708$	0	0
$709$	−35.6702	−1.33962	−0.669811	−	0.742532i	$-0.733625\pi$
$709$	−35.6702	−1.33962	−0.669811	+	0.742532i	$0.733625\pi$
$710$	−51.4118	−1.92945
$711$	0	0
$712$	−24.6537	−0.923937
$713$	8.23153	0.308273
$714$	0	0
$715$	50.7610	1.89835
$716$	37.5685	1.40400
$717$	0	0
$718$	−38.6072	−1.44081
$719$	−2.02313	−0.0754500	−0.0377250	−	0.999288i	$-0.512011\pi$
$719$	−2.02313	−0.0754500	−0.0377250	+	0.999288i	$0.512011\pi$
$720$	0	0
$721$	15.9260	0.593114
$722$	−27.9704	−1.04095
$723$	0	0
$724$	4.24478	0.157756
$725$	0	0
$726$	0	0
$727$	10.5449	0.391091	0.195545	−	0.980695i	$-0.437352\pi$
$727$	10.5449	0.391091	0.195545	+	0.980695i	$0.437352\pi$
$728$	−14.3744	−0.532750
$729$	0	0
$730$	−1.59555	−0.0590540
$731$	23.8301	0.881389
$732$	0	0
$733$	19.0185	0.702464	0.351232	−	0.936289i	$-0.385763\pi$
$733$	19.0185	0.702464	0.351232	+	0.936289i	$0.385763\pi$
$734$	8.32408	0.307247
$735$	0	0
$736$	−7.45934	−0.274955
$737$	29.1754	1.07469
$738$	0	0
$739$	43.0359	1.58310	0.791550	−	0.611104i	$-0.209274\pi$
$739$	43.0359	1.58310	0.791550	+	0.611104i	$0.209274\pi$
$740$	−36.9376	−1.35785
$741$	0	0
$742$	−39.5349	−1.45137
$743$	−3.32534	−0.121995	−0.0609974	−	0.998138i	$-0.519428\pi$
$743$	−3.32534	−0.121995	−0.0609974	+	0.998138i	$0.519428\pi$
$744$	0	0
$745$	20.5743	0.753783
$746$	18.1756	0.665456
$747$	0	0
$748$	30.0560	1.09896
$749$	8.47263	0.309583
$750$	0	0
$751$	−20.4786	−0.747274	−0.373637	−	0.927575i	$-0.621889\pi$
$751$	−20.4786	−0.747274	−0.373637	+	0.927575i	$0.621889\pi$
$752$	0.730275	0.0266304
$753$	0	0
$754$	0	0
$755$	28.6190	1.04155
$756$	0	0
$757$	8.42636	0.306261	0.153131	−	0.988206i	$-0.451064\pi$
$757$	8.42636	0.306261	0.153131	+	0.988206i	$0.451064\pi$
$758$	−19.5778	−0.711097
$759$	0	0
$760$	23.7994	0.863295
$761$	26.1414	0.947625	0.473812	−	0.880626i	$-0.342877\pi$
$761$	26.1414	0.947625	0.473812	+	0.880626i	$0.342877\pi$
$762$	0	0
$763$	−20.0181	−0.724703
$764$	−32.5773	−1.17861
$765$	0	0
$766$	78.5817	2.83927
$767$	−21.4896	−0.775945
$768$	0	0
$769$	−41.1420	−1.48362	−0.741808	−	0.670612i	$-0.766032\pi$
$769$	−41.1420	−1.48362	−0.741808	+	0.670612i	$0.766032\pi$
$770$	−24.9241	−0.898203
$771$	0	0
$772$	17.2769	0.621811
$773$	34.8669	1.25407	0.627037	−	0.778990i	$-0.284268\pi$
$773$	34.8669	1.25407	0.627037	+	0.778990i	$0.284268\pi$
$774$	0	0
$775$	−6.03925	−0.216936
$776$	−6.70163	−0.240575
$777$	0	0
$778$	−24.8648	−0.891446
$779$	38.6489	1.38474
$780$	0	0
$781$	34.3471	1.22904
$782$	6.79343	0.242932
$783$	0	0
$784$	9.03096	0.322534
$785$	29.3688	1.04822
$786$	0	0
$787$	15.9557	0.568759	0.284380	−	0.958712i	$-0.408212\pi$
$787$	15.9557	0.568759	0.284380	+	0.958712i	$0.408212\pi$
$788$	7.08463	0.252379
$789$	0	0
$790$	21.0463	0.748792
$791$	16.0786	0.571688
$792$	0	0
$793$	−43.5072	−1.54498
$794$	52.8991	1.87732
$795$	0	0
$796$	37.2407	1.31996
$797$	38.0735	1.34863	0.674316	−	0.738443i	$-0.264439\pi$
$797$	38.0735	1.34863	0.674316	+	0.738443i	$0.264439\pi$
$798$	0	0
$799$	−1.27750	−0.0451948
$800$	5.47271	0.193490
$801$	0	0
$802$	42.4248	1.49807
$803$	1.06595	0.0376167
$804$	0	0
$805$	−3.28810	−0.115890
$806$	107.290	3.77913
$807$	0	0
$808$	0.0743921	0.00261710
$809$	−52.1314	−1.83284	−0.916421	−	0.400216i	$-0.868935\pi$
$809$	−52.1314	−1.83284	−0.916421	+	0.400216i	$0.868935\pi$
$810$	0	0
$811$	6.35528	0.223164	0.111582	−	0.993755i	$-0.464408\pi$
$811$	6.35528	0.223164	0.111582	+	0.993755i	$0.464408\pi$
$812$	0	0
$813$	0	0
$814$	42.2793	1.48189
$815$	19.0098	0.665883
$816$	0	0
$817$	−43.9698	−1.53831
$818$	25.0868	0.877140
$819$	0	0
$820$	−46.0864	−1.60941
$821$	−0.213766	−0.00746047	−0.00373024	−	0.999993i	$-0.501187\pi$
$821$	−0.213766	−0.00746047	−0.00373024	+	0.999993i	$0.501187\pi$
$822$	0	0
$823$	−25.4184	−0.886029	−0.443015	−	0.896514i	$-0.646091\pi$
$823$	−25.4184	−0.886029	−0.443015	+	0.896514i	$0.646091\pi$
$824$	−20.7562	−0.723076
$825$	0	0
$826$	10.5516	0.367137
$827$	39.3267	1.36752	0.683761	−	0.729706i	$-0.260343\pi$
$827$	39.3267	1.36752	0.683761	+	0.729706i	$0.260343\pi$
$828$	0	0
$829$	−2.14332	−0.0744405	−0.0372203	−	0.999307i	$-0.511850\pi$
$829$	−2.14332	−0.0744405	−0.0372203	+	0.999307i	$0.511850\pi$
$830$	−22.6640	−0.786680
$831$	0	0
$832$	−76.1522	−2.64010
$833$	−15.7983	−0.547378
$834$	0	0
$835$	−30.9769	−1.07200
$836$	−55.4575	−1.91804
$837$	0	0
$838$	−46.2871	−1.59896
$839$	−15.6125	−0.539002	−0.269501	−	0.963000i	$-0.586859\pi$
$839$	−15.6125	−0.539002	−0.269501	+	0.963000i	$0.586859\pi$
$840$	0	0
$841$	0	0
$842$	60.3025	2.07816
$843$	0	0
$844$	23.2856	0.801524
$845$	56.1257	1.93078
$846$	0	0
$847$	1.78027	0.0611708
$848$	−23.2961	−0.799992
$849$	0	0
$850$	−4.98415	−0.170955
$851$	5.57767	0.191200
$852$	0	0
$853$	−36.4571	−1.24827	−0.624133	−	0.781318i	$-0.714548\pi$
$853$	−36.4571	−1.24827	−0.624133	+	0.781318i	$0.714548\pi$
$854$	21.3624	0.731007
$855$	0	0
$856$	−11.0423	−0.377419
$857$	0.490297	0.0167482	0.00837412	−	0.999965i	$-0.497334\pi$
$857$	0.490297	0.0167482	0.00837412	+	0.999965i	$0.497334\pi$
$858$	0	0
$859$	36.7217	1.25293	0.626465	−	0.779450i	$-0.284501\pi$
$859$	36.7217	1.25293	0.626465	+	0.779450i	$0.284501\pi$
$860$	52.4313	1.78789
$861$	0	0
$862$	−4.27295	−0.145537
$863$	−40.0757	−1.36419	−0.682096	−	0.731263i	$-0.738931\pi$
$863$	−40.0757	−1.36419	−0.682096	+	0.731263i	$0.738931\pi$
$864$	0	0
$865$	−6.36001	−0.216247
$866$	17.1721	0.583531
$867$	0	0
$868$	−30.7480	−1.04365
$869$	−14.0605	−0.476971
$870$	0	0
$871$	50.1672	1.69985
$872$	26.0894	0.883499
$873$	0	0
$874$	−12.5348	−0.423996
$875$	−13.7887	−0.466143
$876$	0	0
$877$	29.2270	0.986925	0.493463	−	0.869767i	$-0.335731\pi$
$877$	29.2270	0.986925	0.493463	+	0.869767i	$0.335731\pi$
$878$	−5.23811	−0.176778
$879$	0	0
$880$	−14.6867	−0.495087
$881$	23.3679	0.787285	0.393643	−	0.919264i	$-0.371215\pi$
$881$	23.3679	0.787285	0.393643	+	0.919264i	$0.371215\pi$
$882$	0	0
$883$	−22.0878	−0.743314	−0.371657	−	0.928370i	$-0.621210\pi$
$883$	−22.0878	−0.743314	−0.371657	+	0.928370i	$0.621210\pi$
$884$	51.6815	1.73824
$885$	0	0
$886$	−84.0539	−2.82384
$887$	16.2336	0.545072	0.272536	−	0.962146i	$-0.412138\pi$
$887$	16.2336	0.545072	0.272536	+	0.962146i	$0.412138\pi$
$888$	0	0
$889$	20.4054	0.684374
$890$	−73.5049	−2.46389
$891$	0	0
$892$	70.1305	2.34814
$893$	2.35717	0.0788796
$894$	0	0
$895$	32.1138	1.07344
$896$	17.5164	0.585183
$897$	0	0
$898$	68.6992	2.29252
$899$	0	0
$900$	0	0
$901$	40.7530	1.35768
$902$	52.7512	1.75642
$903$	0	0
$904$	−20.9551	−0.696956
$905$	3.62846	0.120614
$906$	0	0
$907$	3.44533	0.114400	0.0572002	−	0.998363i	$-0.481783\pi$
$907$	3.44533	0.114400	0.0572002	+	0.998363i	$0.481783\pi$
$908$	−8.53466	−0.283233
$909$	0	0
$910$	−42.8571	−1.42070
$911$	−18.9945	−0.629315	−0.314658	−	0.949205i	$-0.601890\pi$
$911$	−18.9945	−0.629315	−0.314658	+	0.949205i	$0.601890\pi$
$912$	0	0
$913$	15.1413	0.501105
$914$	70.7161	2.33908
$915$	0	0
$916$	61.4290	2.02967
$917$	14.1231	0.466385
$918$	0	0
$919$	−16.7391	−0.552171	−0.276086	−	0.961133i	$-0.589037\pi$
$919$	−16.7391	−0.552171	−0.276086	+	0.961133i	$0.589037\pi$
$920$	4.28535	0.141284
$921$	0	0
$922$	12.4450	0.409856
$923$	59.0600	1.94398
$924$	0	0
$925$	−4.09218	−0.134550
$926$	−50.9236	−1.67345
$927$	0	0
$928$	0	0
$929$	57.7025	1.89316	0.946578	−	0.322474i	$-0.104515\pi$
$929$	57.7025	1.89316	0.946578	+	0.322474i	$0.104515\pi$
$930$	0	0
$931$	29.1500	0.955352
$932$	−27.9397	−0.915194
$933$	0	0
$934$	−26.4703	−0.866133
$935$	25.6920	0.840219
$936$	0	0
$937$	−40.1165	−1.31055	−0.655274	−	0.755391i	$-0.727447\pi$
$937$	−40.1165	−1.31055	−0.655274	+	0.755391i	$0.727447\pi$
$938$	−24.6326	−0.804282
$939$	0	0
$940$	−2.81078	−0.0916774
$941$	−37.2722	−1.21504	−0.607520	−	0.794305i	$-0.707835\pi$
$941$	−37.2722	−1.21504	−0.607520	+	0.794305i	$0.707835\pi$
$942$	0	0
$943$	6.95916	0.226622
$944$	6.21758	0.202365
$945$	0	0
$946$	−60.0137	−1.95121
$947$	−27.4624	−0.892408	−0.446204	−	0.894931i	$-0.647224\pi$
$947$	−27.4624	−0.892408	−0.446204	+	0.894931i	$0.647224\pi$
$948$	0	0
$949$	1.83291	0.0594988
$950$	9.19644	0.298372
$951$	0	0
$952$	−7.27541	−0.235797
$953$	−2.42289	−0.0784851	−0.0392425	−	0.999230i	$-0.512494\pi$
$953$	−2.42289	−0.0784851	−0.0392425	+	0.999230i	$0.512494\pi$
$954$	0	0
$955$	−27.8473	−0.901116
$956$	21.8169	0.705610
$957$	0	0
$958$	66.8731	2.16057
$959$	−4.96437	−0.160308
$960$	0	0
$961$	34.7991	1.12255
$962$	72.6995	2.34393
$963$	0	0
$964$	−65.0196	−2.09414
$965$	14.7684	0.475412
$966$	0	0
$967$	−37.8518	−1.21723	−0.608616	−	0.793465i	$-0.708275\pi$
$967$	−37.8518	−1.21723	−0.608616	+	0.793465i	$0.708275\pi$
$968$	−2.32021	−0.0745744
$969$	0	0
$970$	−19.9809	−0.641548
$971$	−12.8802	−0.413346	−0.206673	−	0.978410i	$-0.566264\pi$
$971$	−12.8802	−0.413346	−0.206673	+	0.978410i	$0.566264\pi$
$972$	0	0
$973$	7.81584	0.250564
$974$	51.5611	1.65212
$975$	0	0
$976$	12.5879	0.402929
$977$	−1.41211	−0.0451775	−0.0225887	−	0.999745i	$-0.507191\pi$
$977$	−1.41211	−0.0451775	−0.0225887	+	0.999745i	$0.507191\pi$
$978$	0	0
$979$	49.1070	1.56947
$980$	−34.7595	−1.11035
$981$	0	0
$982$	−60.4683	−1.92962
$983$	12.0413	0.384059	0.192030	−	0.981389i	$-0.438493\pi$
$983$	12.0413	0.384059	0.192030	+	0.981389i	$0.438493\pi$
$984$	0	0
$985$	6.05597	0.192959
$986$	0	0
$987$	0	0
$988$	−95.3595	−3.03379
$989$	−7.91726	−0.251754
$990$	0	0
$991$	−44.9003	−1.42630	−0.713152	−	0.701009i	$-0.752733\pi$
$991$	−44.9003	−1.42630	−0.713152	+	0.701009i	$0.752733\pi$
$992$	−59.6266	−1.89315
$993$	0	0
$994$	−28.9990	−0.919792
$995$	31.8335	1.00919
$996$	0	0
$997$	−18.2114	−0.576762	−0.288381	−	0.957516i	$-0.593117\pi$
$997$	−18.2114	−0.576762	−0.288381	+	0.957516i	$0.593117\pi$
$998$	84.0938	2.66194
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bm.1.1		9
3.2	odd	2		2523.2.a.o.1.9		9
29.5	even	14		261.2.k.c.199.1		18
29.6	even	14		261.2.k.c.181.1		18
29.28	even	2		7569.2.a.bj.1.9		9
87.5	odd	14		87.2.g.a.25.3	yes	18
87.35	odd	14		87.2.g.a.7.3	✓	18
87.86	odd	2		2523.2.a.r.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.7.3	✓	18	87.35	odd	14
87.2.g.a.25.3	yes	18	87.5	odd	14
261.2.k.c.181.1		18	29.6	even	14
261.2.k.c.199.1		18	29.5	even	14
2523.2.a.o.1.9		9	3.2	odd	2
2523.2.a.r.1.1		9	87.86	odd	2
7569.2.a.bj.1.9		9	29.28	even	2
7569.2.a.bm.1.1		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)

\(f(q)\)	\(=\)	\(q-2.19178 q^{2} +2.80388 q^{4} +2.39677 q^{5} +1.35191 q^{7} -1.76193 q^{8} -5.25319 q^{10} +3.50954 q^{11} +6.03467 q^{13} -2.96308 q^{14} -1.74601 q^{16} +3.05437 q^{17} -5.63574 q^{19} +6.72027 q^{20} -7.69212 q^{22} -1.01478 q^{23} +0.744514 q^{25} -13.2266 q^{26} +3.79059 q^{28} -8.11166 q^{31} +7.35072 q^{32} -6.69450 q^{34} +3.24021 q^{35} -5.49645 q^{37} +12.3523 q^{38} -4.22294 q^{40} -6.85782 q^{41} +7.80197 q^{43} +9.84033 q^{44} +2.22416 q^{46} -0.418254 q^{47} -5.17235 q^{49} -1.63181 q^{50} +16.9205 q^{52} +13.3425 q^{53} +8.41156 q^{55} -2.38197 q^{56} -3.56102 q^{59} -7.20954 q^{61} +17.7790 q^{62} -12.6191 q^{64} +14.4637 q^{65} +8.31317 q^{67} +8.56410 q^{68} -7.10182 q^{70} +9.78678 q^{71} +0.303730 q^{73} +12.0470 q^{74} -15.8019 q^{76} +4.74457 q^{77} -4.00638 q^{79} -4.18478 q^{80} +15.0308 q^{82} +4.31434 q^{83} +7.32063 q^{85} -17.1002 q^{86} -6.18356 q^{88} +13.9924 q^{89} +8.15831 q^{91} -2.84532 q^{92} +0.916719 q^{94} -13.5076 q^{95} +3.80357 q^{97} +11.3366 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more