LMFDB - Embedded newform 7569.2.a.f.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
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			$1.61803$ 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)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} +3.23607 q^{5} -4.23607 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-1.61803 q^{2} +0.618034 q^{4} +3.23607 q^{5} -4.23607 q^{7} +2.23607 q^{8} -5.23607 q^{10} +2.23607 q^{11} +1.00000 q^{13} +6.85410 q^{14} -4.85410 q^{16} -3.00000 q^{17} -7.70820 q^{19} +2.00000 q^{20} -3.61803 q^{22} -1.23607 q^{23} +5.47214 q^{25} -1.61803 q^{26} -2.61803 q^{28} +7.23607 q^{31} +3.38197 q^{32} +4.85410 q^{34} -13.7082 q^{35} +4.76393 q^{37} +12.4721 q^{38} +7.23607 q^{40} +4.47214 q^{41} +0.472136 q^{43} +1.38197 q^{44} +2.00000 q^{46} +10.2361 q^{47} +10.9443 q^{49} -8.85410 q^{50} +0.618034 q^{52} -6.76393 q^{53} +7.23607 q^{55} -9.47214 q^{56} -8.94427 q^{59} -2.76393 q^{61} -11.7082 q^{62} +4.23607 q^{64} +3.23607 q^{65} -4.70820 q^{67} -1.85410 q^{68} +22.1803 q^{70} +4.76393 q^{71} -7.70820 q^{73} -7.70820 q^{74} -4.76393 q^{76} -9.47214 q^{77} +2.29180 q^{79} -15.7082 q^{80} -7.23607 q^{82} +6.00000 q^{83} -9.70820 q^{85} -0.763932 q^{86} +5.00000 q^{88} -0.0557281 q^{89} -4.23607 q^{91} -0.763932 q^{92} -16.5623 q^{94} -24.9443 q^{95} +18.1803 q^{97} -17.7082 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 4 * q^7	$ 2 q - q^{2} - q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{10} + 2 q^{13} + 7 q^{14} - 3 q^{16} - 6 q^{17} - 2 q^{19} + 4 q^{20} - 5 q^{22} + 2 q^{23} + 2 q^{25} - q^{26} - 3 q^{28} + 10 q^{31} + 9 q^{32} + 3 q^{34} - 14 q^{35} + 14 q^{37} + 16 q^{38} + 10 q^{40} - 8 q^{43} + 5 q^{44} + 4 q^{46} + 16 q^{47} + 4 q^{49} - 11 q^{50} - q^{52} - 18 q^{53} + 10 q^{55} - 10 q^{56} - 10 q^{61} - 10 q^{62} + 4 q^{64} + 2 q^{65} + 4 q^{67} + 3 q^{68} + 22 q^{70} + 14 q^{71} - 2 q^{73} - 2 q^{74} - 14 q^{76} - 10 q^{77} + 18 q^{79} - 18 q^{80} - 10 q^{82} + 12 q^{83} - 6 q^{85} - 6 q^{86} + 10 q^{88} - 18 q^{89} - 4 q^{91} - 6 q^{92} - 13 q^{94} - 32 q^{95} + 14 q^{97} - 22 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 2 * q^5 - 4 * q^7 - 6 * q^10 + 2 * q^13 + 7 * q^14 - 3 * q^16 - 6 * q^17 - 2 * q^19 + 4 * q^20 - 5 * q^22 + 2 * q^23 + 2 * q^25 - q^26 - 3 * q^28 + 10 * q^31 + 9 * q^32 + 3 * q^34 - 14 * q^35 + 14 * q^37 + 16 * q^38 + 10 * q^40 - 8 * q^43 + 5 * q^44 + 4 * q^46 + 16 * q^47 + 4 * q^49 - 11 * q^50 - q^52 - 18 * q^53 + 10 * q^55 - 10 * q^56 - 10 * q^61 - 10 * q^62 + 4 * q^64 + 2 * q^65 + 4 * q^67 + 3 * q^68 + 22 * q^70 + 14 * q^71 - 2 * q^73 - 2 * q^74 - 14 * q^76 - 10 * q^77 + 18 * q^79 - 18 * q^80 - 10 * q^82 + 12 * q^83 - 6 * q^85 - 6 * q^86 + 10 * q^88 - 18 * q^89 - 4 * q^91 - 6 * q^92 - 13 * q^94 - 32 * q^95 + 14 * q^97 - 22 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−5.23607	−1.65579
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	6.85410	1.83184
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	−7.70820	−1.76838	−0.884192	−	0.467124i	$-0.845290\pi$
$19$	−7.70820	−1.76838	−0.884192	+	0.467124i	$0.845290\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−3.61803	−0.771367
$23$	−1.23607	−0.257738	−0.128869	−	0.991662i	$-0.541135\pi$
$23$	−1.23607	−0.257738	−0.128869	+	0.991662i	$0.541135\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	−1.61803	−0.317323
$27$	0	0
$28$	−2.61803	−0.494762
$29$	0	0
$29$	0	0
$30$	0	0
$31$	7.23607	1.29964	0.649818	−	0.760090i	$-0.274845\pi$
$31$	7.23607	1.29964	0.649818	+	0.760090i	$0.274845\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	4.85410	0.832472
$35$	−13.7082	−2.31711
$36$	0	0
$37$	4.76393	0.783186	0.391593	−	0.920139i	$-0.371924\pi$
$37$	4.76393	0.783186	0.391593	+	0.920139i	$0.371924\pi$
$38$	12.4721	2.02325
$39$	0	0
$40$	7.23607	1.14412
$41$	4.47214	0.698430	0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214	0.698430	0.349215	+	0.937043i	$0.386448\pi$
$42$	0	0
$43$	0.472136	0.0720001	0.0360000	−	0.999352i	$-0.488538\pi$
$43$	0.472136	0.0720001	0.0360000	+	0.999352i	$0.488538\pi$
$44$	1.38197	0.208339
$45$	0	0
$46$	2.00000	0.294884
$47$	10.2361	1.49308	0.746542	−	0.665338i	$-0.231713\pi$
$47$	10.2361	1.49308	0.746542	+	0.665338i	$0.231713\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	−8.85410	−1.25216
$51$	0	0
$52$	0.618034	0.0857059
$53$	−6.76393	−0.929098	−0.464549	−	0.885548i	$-0.653783\pi$
$53$	−6.76393	−0.929098	−0.464549	+	0.885548i	$0.653783\pi$
$54$	0	0
$55$	7.23607	0.975711
$56$	−9.47214	−1.26577
$57$	0	0
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	−11.7082	−1.48694
$63$	0	0
$64$	4.23607	0.529508
$65$	3.23607	0.401385
$66$	0	0
$67$	−4.70820	−0.575199	−0.287599	−	0.957751i	$-0.592857\pi$
$67$	−4.70820	−0.575199	−0.287599	+	0.957751i	$0.592857\pi$
$68$	−1.85410	−0.224843
$69$	0	0
$70$	22.1803	2.65106
$71$	4.76393	0.565375	0.282687	−	0.959212i	$-0.408774\pi$
$71$	4.76393	0.565375	0.282687	+	0.959212i	$0.408774\pi$
$72$	0	0
$73$	−7.70820	−0.902177	−0.451089	−	0.892479i	$-0.648964\pi$
$73$	−7.70820	−0.902177	−0.451089	+	0.892479i	$0.648964\pi$
$74$	−7.70820	−0.896061
$75$	0	0
$76$	−4.76393	−0.546460
$77$	−9.47214	−1.07945
$78$	0	0
$79$	2.29180	0.257847	0.128924	−	0.991655i	$-0.458848\pi$
$79$	2.29180	0.257847	0.128924	+	0.991655i	$0.458848\pi$
$80$	−15.7082	−1.75623
$81$	0	0
$82$	−7.23607	−0.799090
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−9.70820	−1.05300
$86$	−0.763932	−0.0823769
$87$	0	0
$88$	5.00000	0.533002
$89$	−0.0557281	−0.00590717	−0.00295358	−	0.999996i	$-0.500940\pi$
$89$	−0.0557281	−0.00590717	−0.00295358	+	0.999996i	$0.500940\pi$
$90$	0	0
$91$	−4.23607	−0.444061
$92$	−0.763932	−0.0796454
$93$	0	0
$94$	−16.5623	−1.70827
$95$	−24.9443	−2.55923
$96$	0	0
$97$	18.1803	1.84593	0.922967	−	0.384879i	$-0.125757\pi$
$97$	18.1803	1.84593	0.922967	+	0.384879i	$0.125757\pi$
$98$	−17.7082	−1.78880
$99$	0	0
$100$	3.38197	0.338197
$101$	−3.94427	−0.392470	−0.196235	−	0.980557i	$-0.562871\pi$
$101$	−3.94427	−0.392470	−0.196235	+	0.980557i	$0.562871\pi$
$102$	0	0
$103$	−19.4164	−1.91316	−0.956578	−	0.291477i	$-0.905853\pi$
$103$	−19.4164	−1.91316	−0.956578	+	0.291477i	$0.905853\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	10.9443	1.06300
$107$	−0.763932	−0.0738521	−0.0369260	−	0.999318i	$-0.511757\pi$
$107$	−0.763932	−0.0738521	−0.0369260	+	0.999318i	$0.511757\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	−11.7082	−1.11633
$111$	0	0
$112$	20.5623	1.94296
$113$	−4.52786	−0.425946	−0.212973	−	0.977058i	$-0.568315\pi$
$113$	−4.52786	−0.425946	−0.212973	+	0.977058i	$0.568315\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	14.4721	1.33227
$119$	12.7082	1.16496
$120$	0	0
$121$	−6.00000	−0.545455
$122$	4.47214	0.404888
$123$	0	0
$124$	4.47214	0.401610
$125$	1.52786	0.136656
$126$	0	0
$127$	12.4721	1.10672	0.553362	−	0.832941i	$-0.313345\pi$
$127$	12.4721	1.10672	0.553362	+	0.832941i	$0.313345\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	−5.23607	−0.459234
$131$	−12.2361	−1.06907	−0.534535	−	0.845146i	$-0.679513\pi$
$131$	−12.2361	−1.06907	−0.534535	+	0.845146i	$0.679513\pi$
$132$	0	0
$133$	32.6525	2.83133
$134$	7.61803	0.658098
$135$	0	0
$136$	−6.70820	−0.575224
$137$	−3.52786	−0.301406	−0.150703	−	0.988579i	$-0.548154\pi$
$137$	−3.52786	−0.301406	−0.150703	+	0.988579i	$0.548154\pi$
$138$	0	0
$139$	−6.70820	−0.568982	−0.284491	−	0.958679i	$-0.591825\pi$
$139$	−6.70820	−0.568982	−0.284491	+	0.958679i	$0.591825\pi$
$140$	−8.47214	−0.716026
$141$	0	0
$142$	−7.70820	−0.646858
$143$	2.23607	0.186989
$144$	0	0
$145$	0	0
$146$	12.4721	1.03220
$147$	0	0
$148$	2.94427	0.242018
$149$	−5.52786	−0.452860	−0.226430	−	0.974027i	$-0.572705\pi$
$149$	−5.52786	−0.452860	−0.226430	+	0.974027i	$0.572705\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−17.2361	−1.39803
$153$	0	0
$154$	15.3262	1.23502
$155$	23.4164	1.88085
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−3.70820	−0.295009
$159$	0	0
$160$	10.9443	0.865221
$161$	5.23607	0.412660
$162$	0	0
$163$	7.41641	0.580898	0.290449	−	0.956890i	$-0.406195\pi$
$163$	7.41641	0.580898	0.290449	+	0.956890i	$0.406195\pi$
$164$	2.76393	0.215827
$165$	0	0
$166$	−9.70820	−0.753503
$167$	15.8885	1.22949	0.614746	−	0.788725i	$-0.289258\pi$
$167$	15.8885	1.22949	0.614746	+	0.788725i	$0.289258\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	15.7082	1.20476
$171$	0	0
$172$	0.291796	0.0222492
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−23.1803	−1.75227
$176$	−10.8541	−0.818159
$177$	0	0
$178$	0.0901699	0.00675852
$179$	−17.2361	−1.28828	−0.644142	−	0.764906i	$-0.722785\pi$
$179$	−17.2361	−1.28828	−0.644142	+	0.764906i	$0.722785\pi$
$180$	0	0
$181$	20.4164	1.51754	0.758770	−	0.651359i	$-0.225801\pi$
$181$	20.4164	1.51754	0.758770	+	0.651359i	$0.225801\pi$
$182$	6.85410	0.508060
$183$	0	0
$184$	−2.76393	−0.203760
$185$	15.4164	1.13344
$186$	0	0
$187$	−6.70820	−0.490552
$188$	6.32624	0.461388
$189$	0	0
$190$	40.3607	2.92807
$191$	8.94427	0.647185	0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	0	0
$193$	−17.4164	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$193$	−17.4164	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$194$	−29.4164	−2.11198
$195$	0	0
$196$	6.76393	0.483138
$197$	3.70820	0.264199	0.132099	−	0.991236i	$-0.457828\pi$
$197$	3.70820	0.264199	0.132099	+	0.991236i	$0.457828\pi$
$198$	0	0
$199$	−20.1246	−1.42660	−0.713298	−	0.700861i	$-0.752799\pi$
$199$	−20.1246	−1.42660	−0.713298	+	0.700861i	$0.752799\pi$
$200$	12.2361	0.865221
$201$	0	0
$202$	6.38197	0.449034
$203$	0	0
$204$	0	0
$205$	14.4721	1.01078
$206$	31.4164	2.18888
$207$	0	0
$208$	−4.85410	−0.336571
$209$	−17.2361	−1.19224
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	−4.18034	−0.287107
$213$	0	0
$214$	1.23607	0.0844959
$215$	1.52786	0.104199
$216$	0	0
$217$	−30.6525	−2.08083
$218$	8.09017	0.547935
$219$	0	0
$220$	4.47214	0.301511
$221$	−3.00000	−0.201802
$222$	0	0
$223$	−12.7082	−0.851004	−0.425502	−	0.904957i	$-0.639903\pi$
$223$	−12.7082	−0.851004	−0.425502	+	0.904957i	$0.639903\pi$
$224$	−14.3262	−0.957212
$225$	0	0
$226$	7.32624	0.487334
$227$	−9.05573	−0.601050	−0.300525	−	0.953774i	$-0.597162\pi$
$227$	−9.05573	−0.601050	−0.300525	+	0.953774i	$0.597162\pi$
$228$	0	0
$229$	22.1803	1.46572	0.732859	−	0.680380i	$-0.238185\pi$
$229$	22.1803	1.46572	0.732859	+	0.680380i	$0.238185\pi$
$230$	6.47214	0.426760
$231$	0	0
$232$	0	0
$233$	10.4721	0.686052	0.343026	−	0.939326i	$-0.388548\pi$
$233$	10.4721	0.686052	0.343026	+	0.939326i	$0.388548\pi$
$234$	0	0
$235$	33.1246	2.16081
$236$	−5.52786	−0.359833
$237$	0	0
$238$	−20.5623	−1.33286
$239$	−26.1803	−1.69347	−0.846733	−	0.532019i	$-0.821434\pi$
$239$	−26.1803	−1.69347	−0.846733	+	0.532019i	$0.821434\pi$
$240$	0	0
$241$	3.00000	0.193247	0.0966235	−	0.995321i	$-0.469196\pi$
$241$	3.00000	0.193247	0.0966235	+	0.995321i	$0.469196\pi$
$242$	9.70820	0.624067
$243$	0	0
$244$	−1.70820	−0.109357
$245$	35.4164	2.26267
$246$	0	0
$247$	−7.70820	−0.490461
$248$	16.1803	1.02745
$249$	0	0
$250$	−2.47214	−0.156352
$251$	−12.2361	−0.772334	−0.386167	−	0.922429i	$-0.626201\pi$
$251$	−12.2361	−0.772334	−0.386167	+	0.922429i	$0.626201\pi$
$252$	0	0
$253$	−2.76393	−0.173767
$254$	−20.1803	−1.26623
$255$	0	0
$256$	13.5623	0.847644
$257$	26.4721	1.65129	0.825643	−	0.564193i	$-0.190812\pi$
$257$	26.4721	1.65129	0.825643	+	0.564193i	$0.190812\pi$
$258$	0	0
$259$	−20.1803	−1.25395
$260$	2.00000	0.124035
$261$	0	0
$262$	19.7984	1.22315
$263$	−4.94427	−0.304877	−0.152438	−	0.988313i	$-0.548713\pi$
$263$	−4.94427	−0.304877	−0.152438	+	0.988313i	$0.548713\pi$
$264$	0	0
$265$	−21.8885	−1.34460
$266$	−52.8328	−3.23939
$267$	0	0
$268$	−2.90983	−0.177746
$269$	21.3607	1.30238	0.651192	−	0.758913i	$-0.274269\pi$
$269$	21.3607	1.30238	0.651192	+	0.758913i	$0.274269\pi$
$270$	0	0
$271$	−23.4164	−1.42245	−0.711223	−	0.702967i	$-0.751858\pi$
$271$	−23.4164	−1.42245	−0.711223	+	0.702967i	$0.751858\pi$
$272$	14.5623	0.882969
$273$	0	0
$274$	5.70820	0.344845
$275$	12.2361	0.737863
$276$	0	0
$277$	6.41641	0.385525	0.192762	−	0.981245i	$-0.438255\pi$
$277$	6.41641	0.385525	0.192762	+	0.981245i	$0.438255\pi$
$278$	10.8541	0.650986
$279$	0	0
$280$	−30.6525	−1.83184
$281$	1.41641	0.0844958	0.0422479	−	0.999107i	$-0.486548\pi$
$281$	1.41641	0.0844958	0.0422479	+	0.999107i	$0.486548\pi$
$282$	0	0
$283$	13.8885	0.825588	0.412794	−	0.910824i	$-0.364553\pi$
$283$	13.8885	0.825588	0.412794	+	0.910824i	$0.364553\pi$
$284$	2.94427	0.174710
$285$	0	0
$286$	−3.61803	−0.213939
$287$	−18.9443	−1.11825
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	−4.76393	−0.278788
$293$	−29.9443	−1.74936	−0.874682	−	0.484698i	$-0.838929\pi$
$293$	−29.9443	−1.74936	−0.874682	+	0.484698i	$0.838929\pi$
$294$	0	0
$295$	−28.9443	−1.68520
$296$	10.6525	0.619163
$297$	0	0
$298$	8.94427	0.518128
$299$	−1.23607	−0.0714837
$300$	0	0
$301$	−2.00000	−0.115278
$302$	19.4164	1.11729
$303$	0	0
$304$	37.4164	2.14598
$305$	−8.94427	−0.512148
$306$	0	0
$307$	−28.1803	−1.60834	−0.804168	−	0.594401i	$-0.797389\pi$
$307$	−28.1803	−1.60834	−0.804168	+	0.594401i	$0.797389\pi$
$308$	−5.85410	−0.333568
$309$	0	0
$310$	−37.8885	−2.15192
$311$	−15.6525	−0.887570	−0.443785	−	0.896133i	$-0.646365\pi$
$311$	−15.6525	−0.887570	−0.443785	+	0.896133i	$0.646365\pi$
$312$	0	0
$313$	−5.47214	−0.309303	−0.154652	−	0.987969i	$-0.549426\pi$
$313$	−5.47214	−0.309303	−0.154652	+	0.987969i	$0.549426\pi$
$314$	−3.23607	−0.182622
$315$	0	0
$316$	1.41641	0.0796792
$317$	−23.8328	−1.33858	−0.669292	−	0.742999i	$-0.733403\pi$
$317$	−23.8328	−1.33858	−0.669292	+	0.742999i	$0.733403\pi$
$318$	0	0
$319$	0	0
$320$	13.7082	0.766312
$321$	0	0
$322$	−8.47214	−0.472134
$323$	23.1246	1.28669
$324$	0	0
$325$	5.47214	0.303539
$326$	−12.0000	−0.664619
$327$	0	0
$328$	10.0000	0.552158
$329$	−43.3607	−2.39055
$330$	0	0
$331$	−8.29180	−0.455758	−0.227879	−	0.973689i	$-0.573179\pi$
$331$	−8.29180	−0.455758	−0.227879	+	0.973689i	$0.573179\pi$
$332$	3.70820	0.203514
$333$	0	0
$334$	−25.7082	−1.40669
$335$	−15.2361	−0.832435
$336$	0	0
$337$	19.2361	1.04786	0.523928	−	0.851763i	$-0.324466\pi$
$337$	19.2361	1.04786	0.523928	+	0.851763i	$0.324466\pi$
$338$	19.4164	1.05611
$339$	0	0
$340$	−6.00000	−0.325396
$341$	16.1803	0.876215
$342$	0	0
$343$	−16.7082	−0.902158
$344$	1.05573	0.0569210
$345$	0	0
$346$	9.70820	0.521916
$347$	−22.6525	−1.21605	−0.608024	−	0.793918i	$-0.708038\pi$
$347$	−22.6525	−1.21605	−0.608024	+	0.793918i	$0.708038\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	37.5066	2.00481
$351$	0	0
$352$	7.56231	0.403072
$353$	4.65248	0.247626	0.123813	−	0.992306i	$-0.460488\pi$
$353$	4.65248	0.247626	0.123813	+	0.992306i	$0.460488\pi$
$354$	0	0
$355$	15.4164	0.818218
$356$	−0.0344419	−0.00182541
$357$	0	0
$358$	27.8885	1.47396
$359$	−25.3050	−1.33554	−0.667772	−	0.744366i	$-0.732752\pi$
$359$	−25.3050	−1.33554	−0.667772	+	0.744366i	$0.732752\pi$
$360$	0	0
$361$	40.4164	2.12718
$362$	−33.0344	−1.73625
$363$	0	0
$364$	−2.61803	−0.137222
$365$	−24.9443	−1.30564
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−24.9443	−1.29679
$371$	28.6525	1.48756
$372$	0	0
$373$	−3.88854	−0.201341	−0.100671	−	0.994920i	$-0.532099\pi$
$373$	−3.88854	−0.201341	−0.100671	+	0.994920i	$0.532099\pi$
$374$	10.8541	0.561252
$375$	0	0
$376$	22.8885	1.18039
$377$	0	0
$378$	0	0
$379$	−14.2918	−0.734120	−0.367060	−	0.930197i	$-0.619636\pi$
$379$	−14.2918	−0.734120	−0.367060	+	0.930197i	$0.619636\pi$
$380$	−15.4164	−0.790845
$381$	0	0
$382$	−14.4721	−0.740459
$383$	−5.34752	−0.273246	−0.136623	−	0.990623i	$-0.543625\pi$
$383$	−5.34752	−0.273246	−0.136623	+	0.990623i	$0.543625\pi$
$384$	0	0
$385$	−30.6525	−1.56219
$386$	28.1803	1.43434
$387$	0	0
$388$	11.2361	0.570425
$389$	13.4721	0.683064	0.341532	−	0.939870i	$-0.389054\pi$
$389$	13.4721	0.683064	0.341532	+	0.939870i	$0.389054\pi$
$390$	0	0
$391$	3.70820	0.187532
$392$	24.4721	1.23603
$393$	0	0
$394$	−6.00000	−0.302276
$395$	7.41641	0.373160
$396$	0	0
$397$	6.94427	0.348523	0.174262	−	0.984699i	$-0.444246\pi$
$397$	6.94427	0.348523	0.174262	+	0.984699i	$0.444246\pi$
$398$	32.5623	1.63220
$399$	0	0
$400$	−26.5623	−1.32812
$401$	−33.7082	−1.68331	−0.841654	−	0.540018i	$-0.818418\pi$
$401$	−33.7082	−1.68331	−0.841654	+	0.540018i	$0.818418\pi$
$402$	0	0
$403$	7.23607	0.360454
$404$	−2.43769	−0.121280
$405$	0	0
$406$	0	0
$407$	10.6525	0.528024
$408$	0	0
$409$	10.5836	0.523325	0.261662	−	0.965159i	$-0.415729\pi$
$409$	10.5836	0.523325	0.261662	+	0.965159i	$0.415729\pi$
$410$	−23.4164	−1.15645
$411$	0	0
$412$	−12.0000	−0.591198
$413$	37.8885	1.86437
$414$	0	0
$415$	19.4164	0.953114
$416$	3.38197	0.165815
$417$	0	0
$418$	27.8885	1.36407
$419$	3.81966	0.186603	0.0933013	−	0.995638i	$-0.470258\pi$
$419$	3.81966	0.186603	0.0933013	+	0.995638i	$0.470258\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	−16.1803	−0.787647
$423$	0	0
$424$	−15.1246	−0.734516
$425$	−16.4164	−0.796313
$426$	0	0
$427$	11.7082	0.566600
$428$	−0.472136	−0.0228216
$429$	0	0
$430$	−2.47214	−0.119217
$431$	−14.3607	−0.691730	−0.345865	−	0.938284i	$-0.612414\pi$
$431$	−14.3607	−0.691730	−0.345865	+	0.938284i	$0.612414\pi$
$432$	0	0
$433$	−14.2918	−0.686820	−0.343410	−	0.939186i	$-0.611582\pi$
$433$	−14.2918	−0.686820	−0.343410	+	0.939186i	$0.611582\pi$
$434$	49.5967	2.38072
$435$	0	0
$436$	−3.09017	−0.147992
$437$	9.52786	0.455780
$438$	0	0
$439$	−13.2918	−0.634383	−0.317191	−	0.948362i	$-0.602740\pi$
$439$	−13.2918	−0.634383	−0.317191	+	0.948362i	$0.602740\pi$
$440$	16.1803	0.771367
$441$	0	0
$442$	4.85410	0.230886
$443$	−9.65248	−0.458603	−0.229301	−	0.973355i	$-0.573644\pi$
$443$	−9.65248	−0.458603	−0.229301	+	0.973355i	$0.573644\pi$
$444$	0	0
$445$	−0.180340	−0.00854893
$446$	20.5623	0.973653
$447$	0	0
$448$	−17.9443	−0.847787
$449$	18.8885	0.891405	0.445703	−	0.895181i	$-0.352954\pi$
$449$	18.8885	0.891405	0.445703	+	0.895181i	$0.352954\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	−2.79837	−0.131624
$453$	0	0
$454$	14.6525	0.687675
$455$	−13.7082	−0.642651
$456$	0	0
$457$	−6.41641	−0.300147	−0.150073	−	0.988675i	$-0.547951\pi$
$457$	−6.41641	−0.300147	−0.150073	+	0.988675i	$0.547951\pi$
$458$	−35.8885	−1.67696
$459$	0	0
$460$	−2.47214	−0.115264
$461$	18.9443	0.882323	0.441161	−	0.897428i	$-0.354567\pi$
$461$	18.9443	0.882323	0.441161	+	0.897428i	$0.354567\pi$
$462$	0	0
$463$	−0.708204	−0.0329130	−0.0164565	−	0.999865i	$-0.505239\pi$
$463$	−0.708204	−0.0329130	−0.0164565	+	0.999865i	$0.505239\pi$
$464$	0	0
$465$	0	0
$466$	−16.9443	−0.784928
$467$	−13.5279	−0.625995	−0.312997	−	0.949754i	$-0.601333\pi$
$467$	−13.5279	−0.625995	−0.312997	+	0.949754i	$0.601333\pi$
$468$	0	0
$469$	19.9443	0.920941
$470$	−53.5967	−2.47223
$471$	0	0
$472$	−20.0000	−0.920575
$473$	1.05573	0.0485424
$474$	0	0
$475$	−42.1803	−1.93537
$476$	7.85410	0.359992
$477$	0	0
$478$	42.3607	1.93753
$479$	−10.4721	−0.478484	−0.239242	−	0.970960i	$-0.576899\pi$
$479$	−10.4721	−0.478484	−0.239242	+	0.970960i	$0.576899\pi$
$480$	0	0
$481$	4.76393	0.217217
$482$	−4.85410	−0.221098
$483$	0	0
$484$	−3.70820	−0.168555
$485$	58.8328	2.67146
$486$	0	0
$487$	20.3607	0.922630	0.461315	−	0.887236i	$-0.347378\pi$
$487$	20.3607	0.922630	0.461315	+	0.887236i	$0.347378\pi$
$488$	−6.18034	−0.279771
$489$	0	0
$490$	−57.3050	−2.58877
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	12.4721	0.561148
$495$	0	0
$496$	−35.1246	−1.57714
$497$	−20.1803	−0.905212
$498$	0	0
$499$	−21.1803	−0.948162	−0.474081	−	0.880481i	$-0.657220\pi$
$499$	−21.1803	−0.948162	−0.474081	+	0.880481i	$0.657220\pi$
$500$	0.944272	0.0422291
$501$	0	0
$502$	19.7984	0.883645
$503$	−25.1803	−1.12274	−0.561368	−	0.827566i	$-0.689725\pi$
$503$	−25.1803	−1.12274	−0.561368	+	0.827566i	$0.689725\pi$
$504$	0	0
$505$	−12.7639	−0.567988
$506$	4.47214	0.198811
$507$	0	0
$508$	7.70820	0.341996
$509$	26.8328	1.18934	0.594672	−	0.803969i	$-0.297282\pi$
$509$	26.8328	1.18934	0.594672	+	0.803969i	$0.297282\pi$
$510$	0	0
$511$	32.6525	1.44446
$512$	5.29180	0.233867
$513$	0	0
$514$	−42.8328	−1.88927
$515$	−62.8328	−2.76874
$516$	0	0
$517$	22.8885	1.00664
$518$	32.6525	1.43467
$519$	0	0
$520$	7.23607	0.317323
$521$	−3.52786	−0.154559	−0.0772793	−	0.997009i	$-0.524623\pi$
$521$	−3.52786	−0.154559	−0.0772793	+	0.997009i	$0.524623\pi$
$522$	0	0
$523$	33.0689	1.44600	0.723001	−	0.690847i	$-0.242762\pi$
$523$	33.0689	1.44600	0.723001	+	0.690847i	$0.242762\pi$
$524$	−7.56231	−0.330361
$525$	0	0
$526$	8.00000	0.348817
$527$	−21.7082	−0.945624
$528$	0	0
$529$	−21.4721	−0.933571
$530$	35.4164	1.53839
$531$	0	0
$532$	20.1803	0.874929
$533$	4.47214	0.193710
$534$	0	0
$535$	−2.47214	−0.106880
$536$	−10.5279	−0.454734
$537$	0	0
$538$	−34.5623	−1.49009
$539$	24.4721	1.05409
$540$	0	0
$541$	10.6525	0.457986	0.228993	−	0.973428i	$-0.426457\pi$
$541$	10.6525	0.457986	0.228993	+	0.973428i	$0.426457\pi$
$542$	37.8885	1.62745
$543$	0	0
$544$	−10.1459	−0.435002
$545$	−16.1803	−0.693090
$546$	0	0
$547$	−6.34752	−0.271401	−0.135700	−	0.990750i	$-0.543328\pi$
$547$	−6.34752	−0.271401	−0.135700	+	0.990750i	$0.543328\pi$
$548$	−2.18034	−0.0931395
$549$	0	0
$550$	−19.7984	−0.844205
$551$	0	0
$552$	0	0
$553$	−9.70820	−0.412835
$554$	−10.3820	−0.441087
$555$	0	0
$556$	−4.14590	−0.175825
$557$	−16.4721	−0.697947	−0.348973	−	0.937133i	$-0.613470\pi$
$557$	−16.4721	−0.697947	−0.348973	+	0.937133i	$0.613470\pi$
$558$	0	0
$559$	0.472136	0.0199692
$560$	66.5410	2.81187
$561$	0	0
$562$	−2.29180	−0.0966736
$563$	−13.0689	−0.550788	−0.275394	−	0.961331i	$-0.588808\pi$
$563$	−13.0689	−0.550788	−0.275394	+	0.961331i	$0.588808\pi$
$564$	0	0
$565$	−14.6525	−0.616434
$566$	−22.4721	−0.944574
$567$	0	0
$568$	10.6525	0.446968
$569$	−21.0000	−0.880366	−0.440183	−	0.897908i	$-0.645086\pi$
$569$	−21.0000	−0.880366	−0.440183	+	0.897908i	$0.645086\pi$
$570$	0	0
$571$	18.8328	0.788129	0.394064	−	0.919083i	$-0.371069\pi$
$571$	18.8328	0.788129	0.394064	+	0.919083i	$0.371069\pi$
$572$	1.38197	0.0577829
$573$	0	0
$574$	30.6525	1.27941
$575$	−6.76393	−0.282075
$576$	0	0
$577$	−24.8328	−1.03380	−0.516902	−	0.856045i	$-0.672915\pi$
$577$	−24.8328	−1.03380	−0.516902	+	0.856045i	$0.672915\pi$
$578$	12.9443	0.538411
$579$	0	0
$580$	0	0
$581$	−25.4164	−1.05445
$582$	0	0
$583$	−15.1246	−0.626397
$584$	−17.2361	−0.713234
$585$	0	0
$586$	48.4508	2.00149
$587$	0.944272	0.0389743	0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272	0.0389743	0.0194871	+	0.999810i	$0.493797\pi$
$588$	0	0
$589$	−55.7771	−2.29825
$590$	46.8328	1.92808
$591$	0	0
$592$	−23.1246	−0.950416
$593$	40.1803	1.65001	0.825004	−	0.565126i	$-0.191173\pi$
$593$	40.1803	1.65001	0.825004	+	0.565126i	$0.191173\pi$
$594$	0	0
$595$	41.1246	1.68594
$596$	−3.41641	−0.139942
$597$	0	0
$598$	2.00000	0.0817861
$599$	−1.76393	−0.0720723	−0.0360362	−	0.999350i	$-0.511473\pi$
$599$	−1.76393	−0.0720723	−0.0360362	+	0.999350i	$0.511473\pi$
$600$	0	0
$601$	18.5410	0.756304	0.378152	−	0.925744i	$-0.376560\pi$
$601$	18.5410	0.756304	0.378152	+	0.925744i	$0.376560\pi$
$602$	3.23607	0.131892
$603$	0	0
$604$	−7.41641	−0.301769
$605$	−19.4164	−0.789389
$606$	0	0
$607$	25.4164	1.03162	0.515810	−	0.856703i	$-0.327491\pi$
$607$	25.4164	1.03162	0.515810	+	0.856703i	$0.327491\pi$
$608$	−26.0689	−1.05723
$609$	0	0
$610$	14.4721	0.585960
$611$	10.2361	0.414107
$612$	0	0
$613$	24.4164	0.986169	0.493085	−	0.869981i	$-0.335869\pi$
$613$	24.4164	0.986169	0.493085	+	0.869981i	$0.335869\pi$
$614$	45.5967	1.84013
$615$	0	0
$616$	−21.1803	−0.853380
$617$	−19.8885	−0.800683	−0.400341	−	0.916366i	$-0.631108\pi$
$617$	−19.8885	−0.800683	−0.400341	+	0.916366i	$0.631108\pi$
$618$	0	0
$619$	−32.5410	−1.30793	−0.653967	−	0.756523i	$-0.726896\pi$
$619$	−32.5410	−1.30793	−0.653967	+	0.756523i	$0.726896\pi$
$620$	14.4721	0.581215
$621$	0	0
$622$	25.3262	1.01549
$623$	0.236068	0.00945786
$624$	0	0
$625$	−22.4164	−0.896656
$626$	8.85410	0.353881
$627$	0	0
$628$	1.23607	0.0493245
$629$	−14.2918	−0.569851
$630$	0	0
$631$	−28.7082	−1.14286	−0.571428	−	0.820652i	$-0.693610\pi$
$631$	−28.7082	−1.14286	−0.571428	+	0.820652i	$0.693610\pi$
$632$	5.12461	0.203846
$633$	0	0
$634$	38.5623	1.53150
$635$	40.3607	1.60166
$636$	0	0
$637$	10.9443	0.433628
$638$	0	0
$639$	0	0
$640$	−44.0689	−1.74198
$641$	−19.4721	−0.769103	−0.384552	−	0.923104i	$-0.625644\pi$
$641$	−19.4721	−0.769103	−0.384552	+	0.923104i	$0.625644\pi$
$642$	0	0
$643$	−0.708204	−0.0279288	−0.0139644	−	0.999902i	$-0.504445\pi$
$643$	−0.708204	−0.0279288	−0.0139644	+	0.999902i	$0.504445\pi$
$644$	3.23607	0.127519
$645$	0	0
$646$	−37.4164	−1.47213
$647$	17.5967	0.691800	0.345900	−	0.938271i	$-0.387574\pi$
$647$	17.5967	0.691800	0.345900	+	0.938271i	$0.387574\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	−8.85410	−0.347286
$651$	0	0
$652$	4.58359	0.179507
$653$	32.3050	1.26419	0.632095	−	0.774891i	$-0.282195\pi$
$653$	32.3050	1.26419	0.632095	+	0.774891i	$0.282195\pi$
$654$	0	0
$655$	−39.5967	−1.54717
$656$	−21.7082	−0.847563
$657$	0	0
$658$	70.1591	2.73508
$659$	−26.2361	−1.02201	−0.511006	−	0.859577i	$-0.670727\pi$
$659$	−26.2361	−1.02201	−0.511006	+	0.859577i	$0.670727\pi$
$660$	0	0
$661$	−29.8328	−1.16036	−0.580181	−	0.814488i	$-0.697018\pi$
$661$	−29.8328	−1.16036	−0.580181	+	0.814488i	$0.697018\pi$
$662$	13.4164	0.521443
$663$	0	0
$664$	13.4164	0.520658
$665$	105.666	4.09754
$666$	0	0
$667$	0	0
$668$	9.81966	0.379934
$669$	0	0
$670$	24.6525	0.952408
$671$	−6.18034	−0.238589
$672$	0	0
$673$	4.41641	0.170240	0.0851200	−	0.996371i	$-0.472873\pi$
$673$	4.41641	0.170240	0.0851200	+	0.996371i	$0.472873\pi$
$674$	−31.1246	−1.19888
$675$	0	0
$676$	−7.41641	−0.285246
$677$	−18.0557	−0.693938	−0.346969	−	0.937877i	$-0.612789\pi$
$677$	−18.0557	−0.693938	−0.346969	+	0.937877i	$0.612789\pi$
$678$	0	0
$679$	−77.0132	−2.95549
$680$	−21.7082	−0.832472
$681$	0	0
$682$	−26.1803	−1.00250
$683$	−32.9443	−1.26058	−0.630289	−	0.776361i	$-0.717064\pi$
$683$	−32.9443	−1.26058	−0.630289	+	0.776361i	$0.717064\pi$
$684$	0	0
$685$	−11.4164	−0.436199
$686$	27.0344	1.03218
$687$	0	0
$688$	−2.29180	−0.0873739
$689$	−6.76393	−0.257685
$690$	0	0
$691$	−30.2361	−1.15023	−0.575117	−	0.818071i	$-0.695044\pi$
$691$	−30.2361	−1.15023	−0.575117	+	0.818071i	$0.695044\pi$
$692$	−3.70820	−0.140965
$693$	0	0
$694$	36.6525	1.39131
$695$	−21.7082	−0.823439
$696$	0	0
$697$	−13.4164	−0.508183
$698$	−16.1803	−0.612435
$699$	0	0
$700$	−14.3262	−0.541481
$701$	8.18034	0.308967	0.154484	−	0.987995i	$-0.450629\pi$
$701$	8.18034	0.308967	0.154484	+	0.987995i	$0.450629\pi$
$702$	0	0
$703$	−36.7214	−1.38497
$704$	9.47214	0.356995
$705$	0	0
$706$	−7.52786	−0.283315
$707$	16.7082	0.628377
$708$	0	0
$709$	13.4164	0.503864	0.251932	−	0.967745i	$-0.418934\pi$
$709$	13.4164	0.503864	0.251932	+	0.967745i	$0.418934\pi$
$710$	−24.9443	−0.936142
$711$	0	0
$712$	−0.124612	−0.00467002
$713$	−8.94427	−0.334966
$714$	0	0
$715$	7.23607	0.270614
$716$	−10.6525	−0.398102
$717$	0	0
$718$	40.9443	1.52803
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	82.2492	3.06312
$722$	−65.3951	−2.43375
$723$	0	0
$724$	12.6180	0.468946
$725$	0	0
$726$	0	0
$727$	−5.81966	−0.215839	−0.107920	−	0.994160i	$-0.534419\pi$
$727$	−5.81966	−0.215839	−0.107920	+	0.994160i	$0.534419\pi$
$728$	−9.47214	−0.351061
$729$	0	0
$730$	40.3607	1.49382
$731$	−1.41641	−0.0523877
$732$	0	0
$733$	8.47214	0.312925	0.156463	−	0.987684i	$-0.449991\pi$
$733$	8.47214	0.312925	0.156463	+	0.987684i	$0.449991\pi$
$734$	−29.1246	−1.07501
$735$	0	0
$736$	−4.18034	−0.154089
$737$	−10.5279	−0.387799
$738$	0	0
$739$	−12.1803	−0.448061	−0.224031	−	0.974582i	$-0.571922\pi$
$739$	−12.1803	−0.448061	−0.224031	+	0.974582i	$0.571922\pi$
$740$	9.52786	0.350251
$741$	0	0
$742$	−46.3607	−1.70195
$743$	−43.0689	−1.58004	−0.790022	−	0.613078i	$-0.789931\pi$
$743$	−43.0689	−1.58004	−0.790022	+	0.613078i	$0.789931\pi$
$744$	0	0
$745$	−17.8885	−0.655386
$746$	6.29180	0.230359
$747$	0	0
$748$	−4.14590	−0.151589
$749$	3.23607	0.118243
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	−49.6869	−1.81190
$753$	0	0
$754$	0	0
$755$	−38.8328	−1.41327
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	23.1246	0.839924
$759$	0	0
$760$	−55.7771	−2.02325
$761$	−2.65248	−0.0961522	−0.0480761	−	0.998844i	$-0.515309\pi$
$761$	−2.65248	−0.0961522	−0.0480761	+	0.998844i	$0.515309\pi$
$762$	0	0
$763$	21.1803	0.766780
$764$	5.52786	0.199991
$765$	0	0
$766$	8.65248	0.312627
$767$	−8.94427	−0.322959
$768$	0	0
$769$	−40.1803	−1.44894	−0.724470	−	0.689306i	$-0.757915\pi$
$769$	−40.1803	−1.44894	−0.724470	+	0.689306i	$0.757915\pi$
$770$	49.5967	1.78734
$771$	0	0
$772$	−10.7639	−0.387402
$773$	31.8885	1.14695	0.573476	−	0.819223i	$-0.305595\pi$
$773$	31.8885	1.14695	0.573476	+	0.819223i	$0.305595\pi$
$774$	0	0
$775$	39.5967	1.42236
$776$	40.6525	1.45934
$777$	0	0
$778$	−21.7984	−0.781510
$779$	−34.4721	−1.23509
$780$	0	0
$781$	10.6525	0.381176
$782$	−6.00000	−0.214560
$783$	0	0
$784$	−53.1246	−1.89731
$785$	6.47214	0.231000
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	2.29180	0.0816419
$789$	0	0
$790$	−12.0000	−0.426941
$791$	19.1803	0.681974
$792$	0	0
$793$	−2.76393	−0.0981501
$794$	−11.2361	−0.398753
$795$	0	0
$796$	−12.4377	−0.440842
$797$	−44.8328	−1.58806	−0.794030	−	0.607879i	$-0.792021\pi$
$797$	−44.8328	−1.58806	−0.794030	+	0.607879i	$0.792021\pi$
$798$	0	0
$799$	−30.7082	−1.08638
$800$	18.5066	0.654306
$801$	0	0
$802$	54.5410	1.92591
$803$	−17.2361	−0.608248
$804$	0	0
$805$	16.9443	0.597207
$806$	−11.7082	−0.412404
$807$	0	0
$808$	−8.81966	−0.310275
$809$	−43.3607	−1.52448	−0.762240	−	0.647294i	$-0.775900\pi$
$809$	−43.3607	−1.52448	−0.762240	+	0.647294i	$0.775900\pi$
$810$	0	0
$811$	−46.5967	−1.63623	−0.818117	−	0.575052i	$-0.804982\pi$
$811$	−46.5967	−1.63623	−0.818117	+	0.575052i	$0.804982\pi$
$812$	0	0
$813$	0	0
$814$	−17.2361	−0.604124
$815$	24.0000	0.840683
$816$	0	0
$817$	−3.63932	−0.127324
$818$	−17.1246	−0.598748
$819$	0	0
$820$	8.94427	0.312348
$821$	−12.4721	−0.435281	−0.217640	−	0.976029i	$-0.569836\pi$
$821$	−12.4721	−0.435281	−0.217640	+	0.976029i	$0.569836\pi$
$822$	0	0
$823$	−26.3607	−0.918876	−0.459438	−	0.888210i	$-0.651949\pi$
$823$	−26.3607	−0.918876	−0.459438	+	0.888210i	$0.651949\pi$
$824$	−43.4164	−1.51248
$825$	0	0
$826$	−61.3050	−2.13307
$827$	−29.8885	−1.03933	−0.519663	−	0.854371i	$-0.673943\pi$
$827$	−29.8885	−1.03933	−0.519663	+	0.854371i	$0.673943\pi$
$828$	0	0
$829$	−27.3050	−0.948340	−0.474170	−	0.880433i	$-0.657252\pi$
$829$	−27.3050	−0.948340	−0.474170	+	0.880433i	$0.657252\pi$
$830$	−31.4164	−1.09048
$831$	0	0
$832$	4.23607	0.146859
$833$	−32.8328	−1.13759
$834$	0	0
$835$	51.4164	1.77934
$836$	−10.6525	−0.368424
$837$	0	0
$838$	−6.18034	−0.213496
$839$	0.708204	0.0244499	0.0122250	−	0.999925i	$-0.496109\pi$
$839$	0.708204	0.0244499	0.0122250	+	0.999925i	$0.496109\pi$
$840$	0	0
$841$	0	0
$842$	−32.3607	−1.11522
$843$	0	0
$844$	6.18034	0.212736
$845$	−38.8328	−1.33589
$846$	0	0
$847$	25.4164	0.873318
$848$	32.8328	1.12748
$849$	0	0
$850$	26.5623	0.911080
$851$	−5.88854	−0.201857
$852$	0	0
$853$	−45.1935	−1.54740	−0.773698	−	0.633555i	$-0.781595\pi$
$853$	−45.1935	−1.54740	−0.773698	+	0.633555i	$0.781595\pi$
$854$	−18.9443	−0.648260
$855$	0	0
$856$	−1.70820	−0.0583852
$857$	−37.5967	−1.28428	−0.642140	−	0.766587i	$-0.721953\pi$
$857$	−37.5967	−1.28428	−0.642140	+	0.766587i	$0.721953\pi$
$858$	0	0
$859$	−36.3607	−1.24061	−0.620305	−	0.784361i	$-0.712991\pi$
$859$	−36.3607	−1.24061	−0.620305	+	0.784361i	$0.712991\pi$
$860$	0.944272	0.0321994
$861$	0	0
$862$	23.2361	0.791424
$863$	−46.6525	−1.58807	−0.794034	−	0.607873i	$-0.792023\pi$
$863$	−46.6525	−1.58807	−0.794034	+	0.607873i	$0.792023\pi$
$864$	0	0
$865$	−19.4164	−0.660178
$866$	23.1246	0.785806
$867$	0	0
$868$	−18.9443	−0.643010
$869$	5.12461	0.173841
$870$	0	0
$871$	−4.70820	−0.159531
$872$	−11.1803	−0.378614
$873$	0	0
$874$	−15.4164	−0.521468
$875$	−6.47214	−0.218798
$876$	0	0
$877$	32.4721	1.09651	0.548253	−	0.836312i	$-0.315293\pi$
$877$	32.4721	1.09651	0.548253	+	0.836312i	$0.315293\pi$
$878$	21.5066	0.725812
$879$	0	0
$880$	−35.1246	−1.18405
$881$	−55.2492	−1.86139	−0.930697	−	0.365792i	$-0.880798\pi$
$881$	−55.2492	−1.86139	−0.930697	+	0.365792i	$0.880798\pi$
$882$	0	0
$883$	−34.2492	−1.15258	−0.576289	−	0.817246i	$-0.695500\pi$
$883$	−34.2492	−1.15258	−0.576289	+	0.817246i	$0.695500\pi$
$884$	−1.85410	−0.0623602
$885$	0	0
$886$	15.6180	0.524698
$887$	−17.0689	−0.573117	−0.286559	−	0.958063i	$-0.592511\pi$
$887$	−17.0689	−0.573117	−0.286559	+	0.958063i	$0.592511\pi$
$888$	0	0
$889$	−52.8328	−1.77196
$890$	0.291796	0.00978103
$891$	0	0
$892$	−7.85410	−0.262975
$893$	−78.9017	−2.64034
$894$	0	0
$895$	−55.7771	−1.86442
$896$	57.6869	1.92718
$897$	0	0
$898$	−30.5623	−1.01988
$899$	0	0
$900$	0	0
$901$	20.2918	0.676018
$902$	−16.1803	−0.538746
$903$	0	0
$904$	−10.1246	−0.336740
$905$	66.0689	2.19620
$906$	0	0
$907$	26.4721	0.878993	0.439496	−	0.898244i	$-0.355157\pi$
$907$	26.4721	0.878993	0.439496	+	0.898244i	$0.355157\pi$
$908$	−5.59675	−0.185735
$909$	0	0
$910$	22.1803	0.735271
$911$	51.1803	1.69568	0.847840	−	0.530252i	$-0.177903\pi$
$911$	51.1803	1.69568	0.847840	+	0.530252i	$0.177903\pi$
$912$	0	0
$913$	13.4164	0.444018
$914$	10.3820	0.343405
$915$	0	0
$916$	13.7082	0.452932
$917$	51.8328	1.71167
$918$	0	0
$919$	−25.6525	−0.846197	−0.423099	−	0.906084i	$-0.639058\pi$
$919$	−25.6525	−0.846197	−0.423099	+	0.906084i	$0.639058\pi$
$920$	−8.94427	−0.294884
$921$	0	0
$922$	−30.6525	−1.00949
$923$	4.76393	0.156807
$924$	0	0
$925$	26.0689	0.857140
$926$	1.14590	0.0376565
$927$	0	0
$928$	0	0
$929$	32.3607	1.06172	0.530860	−	0.847460i	$-0.321869\pi$
$929$	32.3607	1.06172	0.530860	+	0.847460i	$0.321869\pi$
$930$	0	0
$931$	−84.3607	−2.76481
$932$	6.47214	0.212002
$933$	0	0
$934$	21.8885	0.716215
$935$	−21.7082	−0.709934
$936$	0	0
$937$	−33.0000	−1.07806	−0.539032	−	0.842286i	$-0.681210\pi$
$937$	−33.0000	−1.07806	−0.539032	+	0.842286i	$0.681210\pi$
$938$	−32.2705	−1.05367
$939$	0	0
$940$	20.4721	0.667727
$941$	−44.8328	−1.46151	−0.730754	−	0.682641i	$-0.760831\pi$
$941$	−44.8328	−1.46151	−0.730754	+	0.682641i	$0.760831\pi$
$942$	0	0
$943$	−5.52786	−0.180012
$944$	43.4164	1.41308
$945$	0	0
$946$	−1.70820	−0.0555385
$947$	−16.8197	−0.546566	−0.273283	−	0.961934i	$-0.588109\pi$
$947$	−16.8197	−0.546566	−0.273283	+	0.961934i	$0.588109\pi$
$948$	0	0
$949$	−7.70820	−0.250219
$950$	68.2492	2.21430
$951$	0	0
$952$	28.4164	0.920981
$953$	59.6656	1.93276	0.966380	−	0.257119i	$-0.0827733\pi$
$953$	59.6656	1.93276	0.966380	+	0.257119i	$0.0827733\pi$
$954$	0	0
$955$	28.9443	0.936615
$956$	−16.1803	−0.523310
$957$	0	0
$958$	16.9443	0.547445
$959$	14.9443	0.482576
$960$	0	0
$961$	21.3607	0.689054
$962$	−7.70820	−0.248522
$963$	0	0
$964$	1.85410	0.0597166
$965$	−56.3607	−1.81431
$966$	0	0
$967$	27.1246	0.872269	0.436134	−	0.899882i	$-0.356347\pi$
$967$	27.1246	0.872269	0.436134	+	0.899882i	$0.356347\pi$
$968$	−13.4164	−0.431220
$969$	0	0
$970$	−95.1935	−3.05648
$971$	16.5836	0.532193	0.266096	−	0.963946i	$-0.414266\pi$
$971$	16.5836	0.532193	0.266096	+	0.963946i	$0.414266\pi$
$972$	0	0
$973$	28.4164	0.910988
$974$	−32.9443	−1.05560
$975$	0	0
$976$	13.4164	0.429449
$977$	−51.0132	−1.63206	−0.816028	−	0.578013i	$-0.803828\pi$
$977$	−51.0132	−1.63206	−0.816028	+	0.578013i	$0.803828\pi$
$978$	0	0
$979$	−0.124612	−0.00398261
$980$	21.8885	0.699204
$981$	0	0
$982$	0	0
$983$	13.8885	0.442976	0.221488	−	0.975163i	$-0.428909\pi$
$983$	13.8885	0.442976	0.221488	+	0.975163i	$0.428909\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	−4.76393	−0.151561
$989$	−0.583592	−0.0185572
$990$	0	0
$991$	21.2918	0.676356	0.338178	−	0.941082i	$-0.390189\pi$
$991$	21.2918	0.676356	0.338178	+	0.941082i	$0.390189\pi$
$992$	24.4721	0.776991
$993$	0	0
$994$	32.6525	1.03567
$995$	−65.1246	−2.06459
$996$	0	0
$997$	24.5836	0.778570	0.389285	−	0.921117i	$-0.372722\pi$
$997$	24.5836	0.778570	0.389285	+	0.921117i	$0.372722\pi$
$998$	34.2705	1.08481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.f.1.1		2
3.2	odd	2		2523.2.a.e.1.2		2
29.12	odd	4		261.2.c.b.28.1		4
29.17	odd	4		261.2.c.b.28.4		4
29.28	even	2		7569.2.a.n.1.2		2
87.17	even	4		87.2.c.a.28.1	✓	4
87.41	even	4		87.2.c.a.28.4	yes	4
87.86	odd	2		2523.2.a.d.1.1		2
116.75	even	4		4176.2.o.l.289.2		4
116.99	even	4		4176.2.o.l.289.1		4
348.191	odd	4		1392.2.o.i.289.2		4
348.215	odd	4		1392.2.o.i.289.4		4
435.17	odd	4		2175.2.f.b.724.3		4
435.104	even	4		2175.2.d.e.376.4		4
435.128	odd	4		2175.2.f.b.724.4		4
435.278	odd	4		2175.2.f.a.724.2		4
435.302	odd	4		2175.2.f.a.724.1		4
435.389	even	4		2175.2.d.e.376.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.c.a.28.1	✓	4	87.17	even	4
87.2.c.a.28.4	yes	4	87.41	even	4
261.2.c.b.28.1		4	29.12	odd	4
261.2.c.b.28.4		4	29.17	odd	4
1392.2.o.i.289.2		4	348.191	odd	4
1392.2.o.i.289.4		4	348.215	odd	4
2175.2.d.e.376.1		4	435.389	even	4
2175.2.d.e.376.4		4	435.104	even	4
2175.2.f.a.724.1		4	435.302	odd	4
2175.2.f.a.724.2		4	435.278	odd	4
2175.2.f.b.724.3		4	435.17	odd	4
2175.2.f.b.724.4		4	435.128	odd	4
2523.2.a.d.1.1		2	87.86	odd	2
2523.2.a.e.1.2		2	3.2	odd	2
4176.2.o.l.289.1		4	116.99	even	4
4176.2.o.l.289.2		4	116.75	even	4
7569.2.a.f.1.1		2	1.1	even	1	trivial
7569.2.a.n.1.2		2	29.28	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} +3.23607 q^{5} -4.23607 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} +3.23607 q^{5} -4.23607 q^{7} +2.23607 q^{8} -5.23607 q^{10} +2.23607 q^{11} +1.00000 q^{13} +6.85410 q^{14} -4.85410 q^{16} -3.00000 q^{17} -7.70820 q^{19} +2.00000 q^{20} -3.61803 q^{22} -1.23607 q^{23} +5.47214 q^{25} -1.61803 q^{26} -2.61803 q^{28} +7.23607 q^{31} +3.38197 q^{32} +4.85410 q^{34} -13.7082 q^{35} +4.76393 q^{37} +12.4721 q^{38} +7.23607 q^{40} +4.47214 q^{41} +0.472136 q^{43} +1.38197 q^{44} +2.00000 q^{46} +10.2361 q^{47} +10.9443 q^{49} -8.85410 q^{50} +0.618034 q^{52} -6.76393 q^{53} +7.23607 q^{55} -9.47214 q^{56} -8.94427 q^{59} -2.76393 q^{61} -11.7082 q^{62} +4.23607 q^{64} +3.23607 q^{65} -4.70820 q^{67} -1.85410 q^{68} +22.1803 q^{70} +4.76393 q^{71} -7.70820 q^{73} -7.70820 q^{74} -4.76393 q^{76} -9.47214 q^{77} +2.29180 q^{79} -15.7082 q^{80} -7.23607 q^{82} +6.00000 q^{83} -9.70820 q^{85} -0.763932 q^{86} +5.00000 q^{88} -0.0557281 q^{89} -4.23607 q^{91} -0.763932 q^{92} -16.5623 q^{94} -24.9443 q^{95} +18.1803 q^{97} -17.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 4 * q^7	\( 2 q - q^{2} - q^{4} + 2 q^{5} - 4 q^{7} - 6 q^{10} + 2 q^{13} + 7 q^{14} - 3 q^{16} - 6 q^{17} - 2 q^{19} + 4 q^{20} - 5 q^{22} + 2 q^{23} + 2 q^{25} - q^{26} - 3 q^{28} + 10 q^{31} + 9 q^{32} + 3 q^{34} - 14 q^{35} + 14 q^{37} + 16 q^{38} + 10 q^{40} - 8 q^{43} + 5 q^{44} + 4 q^{46} + 16 q^{47} + 4 q^{49} - 11 q^{50} - q^{52} - 18 q^{53} + 10 q^{55} - 10 q^{56} - 10 q^{61} - 10 q^{62} + 4 q^{64} + 2 q^{65} + 4 q^{67} + 3 q^{68} + 22 q^{70} + 14 q^{71} - 2 q^{73} - 2 q^{74} - 14 q^{76} - 10 q^{77} + 18 q^{79} - 18 q^{80} - 10 q^{82} + 12 q^{83} - 6 q^{85} - 6 q^{86} + 10 q^{88} - 18 q^{89} - 4 q^{91} - 6 q^{92} - 13 q^{94} - 32 q^{95} + 14 q^{97} - 22 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 2 * q^5 - 4 * q^7 - 6 * q^10 + 2 * q^13 + 7 * q^14 - 3 * q^16 - 6 * q^17 - 2 * q^19 + 4 * q^20 - 5 * q^22 + 2 * q^23 + 2 * q^25 - q^26 - 3 * q^28 + 10 * q^31 + 9 * q^32 + 3 * q^34 - 14 * q^35 + 14 * q^37 + 16 * q^38 + 10 * q^40 - 8 * q^43 + 5 * q^44 + 4 * q^46 + 16 * q^47 + 4 * q^49 - 11 * q^50 - q^52 - 18 * q^53 + 10 * q^55 - 10 * q^56 - 10 * q^61 - 10 * q^62 + 4 * q^64 + 2 * q^65 + 4 * q^67 + 3 * q^68 + 22 * q^70 + 14 * q^71 - 2 * q^73 - 2 * q^74 - 14 * q^76 - 10 * q^77 + 18 * q^79 - 18 * q^80 - 10 * q^82 + 12 * q^83 - 6 * q^85 - 6 * q^86 + 10 * q^88 - 18 * q^89 - 4 * q^91 - 6 * q^92 - 13 * q^94 - 32 * q^95 + 14 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more