LMFDB - Embedded newform 7569.2.a.bl.1.7

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} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 $ x^9 - x^8 - 14x^7 + 9x^6 + 70x^5 - 23x^4 - 141x^3 + 14x^2 + 84*x - 7
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			1.7
Root			$1.98078$ 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)

$f(q)$	$=$	$q+1.98078 q^{2} +1.92348 q^{4} -1.21902 q^{5} +0.393238 q^{7} -0.151564 q^{8} +O(q^{10})$	\(q+1.98078 q^{2} +1.92348 q^{4} -1.21902 q^{5} +0.393238 q^{7} -0.151564 q^{8} -2.41460 q^{10} +2.49378 q^{11} +3.30428 q^{13} +0.778916 q^{14} -4.14718 q^{16} +4.30028 q^{17} +5.37118 q^{19} -2.34476 q^{20} +4.93963 q^{22} -3.00189 q^{23} -3.51400 q^{25} +6.54504 q^{26} +0.756386 q^{28} -10.0037 q^{31} -7.91152 q^{32} +8.51790 q^{34} -0.479363 q^{35} +3.64927 q^{37} +10.6391 q^{38} +0.184759 q^{40} +7.82011 q^{41} +2.34873 q^{43} +4.79675 q^{44} -5.94608 q^{46} +8.54106 q^{47} -6.84536 q^{49} -6.96045 q^{50} +6.35572 q^{52} +10.9212 q^{53} -3.03996 q^{55} -0.0596006 q^{56} -10.2851 q^{59} +3.72270 q^{61} -19.8151 q^{62} -7.37660 q^{64} -4.02797 q^{65} +5.14680 q^{67} +8.27152 q^{68} -0.949512 q^{70} +8.09127 q^{71} +8.75660 q^{73} +7.22839 q^{74} +10.3314 q^{76} +0.980649 q^{77} +7.29925 q^{79} +5.05548 q^{80} +15.4899 q^{82} +3.74978 q^{83} -5.24211 q^{85} +4.65232 q^{86} -0.377968 q^{88} -10.4120 q^{89} +1.29936 q^{91} -5.77409 q^{92} +16.9179 q^{94} -6.54755 q^{95} -16.2019 q^{97} -13.5591 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * 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 $

$2$	1.98078	1.40062	0.700311	−	0.713838i	$-0.253045\pi$
$2$	1.98078	1.40062	0.700311	+	0.713838i	$0.253045\pi$
$3$	0	0
$3$	0	0
$4$	1.92348	0.961741
$5$	−1.21902	−0.545161	−0.272580	−	0.962133i	$-0.587877\pi$
$5$	−1.21902	−0.545161	−0.272580	+	0.962133i	$0.587877\pi$
$6$	0	0
$7$	0.393238	0.148630	0.0743149	−	0.997235i	$-0.476323\pi$
$7$	0.393238	0.148630	0.0743149	+	0.997235i	$0.476323\pi$
$8$	−0.151564	−0.0535860
$9$	0	0
$10$	−2.41460	−0.763564
$11$	2.49378	0.751904	0.375952	−	0.926639i	$-0.377316\pi$
$11$	2.49378	0.751904	0.375952	+	0.926639i	$0.377316\pi$
$12$	0	0
$13$	3.30428	0.916441	0.458221	−	0.888839i	$-0.348487\pi$
$13$	3.30428	0.916441	0.458221	+	0.888839i	$0.348487\pi$
$14$	0.778916	0.208174
$15$	0	0
$16$	−4.14718	−1.03679
$17$	4.30028	1.04297	0.521486	−	0.853260i	$-0.325378\pi$
$17$	4.30028	1.04297	0.521486	+	0.853260i	$0.325378\pi$
$18$	0	0
$19$	5.37118	1.23223	0.616116	−	0.787655i	$-0.288705\pi$
$19$	5.37118	1.23223	0.616116	+	0.787655i	$0.288705\pi$
$20$	−2.34476	−0.524304
$21$	0	0
$22$	4.93963	1.05313
$23$	−3.00189	−0.625938	−0.312969	−	0.949763i	$-0.601324\pi$
$23$	−3.00189	−0.625938	−0.312969	+	0.949763i	$0.601324\pi$
$24$	0	0
$25$	−3.51400	−0.702800
$26$	6.54504	1.28359
$27$	0	0
$28$	0.756386	0.142943
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−10.0037	−1.79672	−0.898359	−	0.439262i	$-0.855240\pi$
$31$	−10.0037	−1.79672	−0.898359	+	0.439262i	$0.855240\pi$
$32$	−7.91152	−1.39857
$33$	0	0
$34$	8.51790	1.46081
$35$	−0.479363	−0.0810271
$36$	0	0
$37$	3.64927	0.599936	0.299968	−	0.953949i	$-0.403024\pi$
$37$	3.64927	0.599936	0.299968	+	0.953949i	$0.403024\pi$
$38$	10.6391	1.72589
$39$	0	0
$40$	0.184759	0.0292130
$41$	7.82011	1.22130	0.610648	−	0.791902i	$-0.290909\pi$
$41$	7.82011	1.22130	0.610648	+	0.791902i	$0.290909\pi$
$42$	0	0
$43$	2.34873	0.358178	0.179089	−	0.983833i	$-0.442685\pi$
$43$	2.34873	0.358178	0.179089	+	0.983833i	$0.442685\pi$
$44$	4.79675	0.723137
$45$	0	0
$46$	−5.94608	−0.876702
$47$	8.54106	1.24584	0.622921	−	0.782285i	$-0.285946\pi$
$47$	8.54106	1.24584	0.622921	+	0.782285i	$0.285946\pi$
$48$	0	0
$49$	−6.84536	−0.977909
$50$	−6.96045	−0.984357
$51$	0	0
$52$	6.35572	0.881379
$53$	10.9212	1.50014	0.750069	−	0.661360i	$-0.230020\pi$
$53$	10.9212	1.50014	0.750069	+	0.661360i	$0.230020\pi$
$54$	0	0
$55$	−3.03996	−0.409908
$56$	−0.0596006	−0.00796447
$57$	0	0
$58$	0	0
$59$	−10.2851	−1.33901	−0.669503	−	0.742809i	$-0.733493\pi$
$59$	−10.2851	−1.33901	−0.669503	+	0.742809i	$0.733493\pi$
$60$	0	0
$61$	3.72270	0.476642	0.238321	−	0.971186i	$-0.423403\pi$
$61$	3.72270	0.476642	0.238321	+	0.971186i	$0.423403\pi$
$62$	−19.8151	−2.51652
$63$	0	0
$64$	−7.37660	−0.922075
$65$	−4.02797	−0.499608
$66$	0	0
$67$	5.14680	0.628781	0.314391	−	0.949294i	$-0.398200\pi$
$67$	5.14680	0.628781	0.314391	+	0.949294i	$0.398200\pi$
$68$	8.27152	1.00307
$69$	0	0
$70$	−0.949512	−0.113488
$71$	8.09127	0.960257	0.480128	−	0.877198i	$-0.340590\pi$
$71$	8.09127	0.960257	0.480128	+	0.877198i	$0.340590\pi$
$72$	0	0
$73$	8.75660	1.02488	0.512441	−	0.858722i	$-0.328741\pi$
$73$	8.75660	1.02488	0.512441	+	0.858722i	$0.328741\pi$
$74$	7.22839	0.840283
$75$	0	0
$76$	10.3314	1.18509
$77$	0.980649	0.111755
$78$	0	0
$79$	7.29925	0.821230	0.410615	−	0.911809i	$-0.365314\pi$
$79$	7.29925	0.821230	0.410615	+	0.911809i	$0.365314\pi$
$80$	5.05548	0.565220
$81$	0	0
$82$	15.4899	1.71057
$83$	3.74978	0.411592	0.205796	−	0.978595i	$-0.434022\pi$
$83$	3.74978	0.411592	0.205796	+	0.978595i	$0.434022\pi$
$84$	0	0
$85$	−5.24211	−0.568587
$86$	4.65232	0.501672
$87$	0	0
$88$	−0.377968	−0.0402915
$89$	−10.4120	−1.10367	−0.551837	−	0.833952i	$-0.686073\pi$
$89$	−10.4120	−1.10367	−0.551837	+	0.833952i	$0.686073\pi$
$90$	0	0
$91$	1.29936	0.136210
$92$	−5.77409	−0.601990
$93$	0	0
$94$	16.9179	1.74495
$95$	−6.54755	−0.671765
$96$	0	0
$97$	−16.2019	−1.64506	−0.822529	−	0.568723i	$-0.807438\pi$
$97$	−16.2019	−1.64506	−0.822529	+	0.568723i	$0.807438\pi$
$98$	−13.5591	−1.36968
$99$	0	0
$100$	−6.75912	−0.675912
$101$	2.76424	0.275052	0.137526	−	0.990498i	$-0.456085\pi$
$101$	2.76424	0.275052	0.137526	+	0.990498i	$0.456085\pi$
$102$	0	0
$103$	3.78612	0.373057	0.186529	−	0.982450i	$-0.440276\pi$
$103$	3.78612	0.373057	0.186529	+	0.982450i	$0.440276\pi$
$104$	−0.500809	−0.0491084
$105$	0	0
$106$	21.6324	2.10113
$107$	11.2962	1.09204	0.546020	−	0.837772i	$-0.316142\pi$
$107$	11.2962	1.09204	0.546020	+	0.837772i	$0.316142\pi$
$108$	0	0
$109$	14.8783	1.42508	0.712542	−	0.701629i	$-0.247544\pi$
$109$	14.8783	1.42508	0.712542	+	0.701629i	$0.247544\pi$
$110$	−6.02149	−0.574127
$111$	0	0
$112$	−1.63083	−0.154099
$113$	0.0175163	0.00164780	0.000823898	−	1.00000i	$-0.499738\pi$
$113$	0.0175163	0.00164780	0.000823898		1.00000i	$0.499738\pi$
$114$	0	0
$115$	3.65936	0.341237
$116$	0	0
$117$	0	0
$118$	−20.3725	−1.87544
$119$	1.69103	0.155017
$120$	0	0
$121$	−4.78105	−0.434641
$122$	7.37384	0.667596
$123$	0	0
$124$	−19.2420	−1.72798
$125$	10.3787	0.928299
$126$	0	0
$127$	10.2560	0.910070	0.455035	−	0.890473i	$-0.349627\pi$
$127$	10.2560	0.910070	0.455035	+	0.890473i	$0.349627\pi$
$128$	1.21162	0.107093
$129$	0	0
$130$	−7.97851	−0.699761
$131$	−4.95269	−0.432719	−0.216359	−	0.976314i	$-0.569418\pi$
$131$	−4.95269	−0.432719	−0.216359	+	0.976314i	$0.569418\pi$
$132$	0	0
$133$	2.11215	0.183146
$134$	10.1947	0.880685
$135$	0	0
$136$	−0.651768	−0.0558886
$137$	13.5688	1.15926	0.579631	−	0.814879i	$-0.303197\pi$
$137$	13.5688	1.15926	0.579631	+	0.814879i	$0.303197\pi$
$138$	0	0
$139$	16.4724	1.39717	0.698584	−	0.715528i	$-0.253814\pi$
$139$	16.4724	1.39717	0.698584	+	0.715528i	$0.253814\pi$
$140$	−0.922046	−0.0779271
$141$	0	0
$142$	16.0270	1.34496
$143$	8.24015	0.689076
$144$	0	0
$145$	0	0
$146$	17.3449	1.43547
$147$	0	0
$148$	7.01930	0.576983
$149$	−7.98689	−0.654312	−0.327156	−	0.944970i	$-0.606090\pi$
$149$	−7.98689	−0.654312	−0.327156	+	0.944970i	$0.606090\pi$
$150$	0	0
$151$	−20.3034	−1.65227	−0.826133	−	0.563475i	$-0.809464\pi$
$151$	−20.3034	−1.65227	−0.826133	+	0.563475i	$0.809464\pi$
$152$	−0.814077	−0.0660304
$153$	0	0
$154$	1.94245	0.156527
$155$	12.1947	0.979500
$156$	0	0
$157$	−4.08988	−0.326408	−0.163204	−	0.986592i	$-0.552183\pi$
$157$	−4.08988	−0.326408	−0.163204	+	0.986592i	$0.552183\pi$
$158$	14.4582	1.15023
$159$	0	0
$160$	9.64427	0.762446
$161$	−1.18046	−0.0930330
$162$	0	0
$163$	19.2776	1.50993	0.754967	−	0.655762i	$-0.227653\pi$
$163$	19.2776	1.50993	0.754967	+	0.655762i	$0.227653\pi$
$164$	15.0418	1.17457
$165$	0	0
$166$	7.42749	0.576485
$167$	3.24714	0.251271	0.125635	−	0.992076i	$-0.459903\pi$
$167$	3.24714	0.251271	0.125635	+	0.992076i	$0.459903\pi$
$168$	0	0
$169$	−2.08176	−0.160136
$170$	−10.3835	−0.796375
$171$	0	0
$172$	4.51775	0.344475
$173$	8.02956	0.610476	0.305238	−	0.952276i	$-0.401264\pi$
$173$	8.02956	0.610476	0.305238	+	0.952276i	$0.401264\pi$
$174$	0	0
$175$	−1.38184	−0.104457
$176$	−10.3422	−0.779570
$177$	0	0
$178$	−20.6239	−1.54583
$179$	10.5652	0.789681	0.394840	−	0.918750i	$-0.370800\pi$
$179$	10.5652	0.789681	0.394840	+	0.918750i	$0.370800\pi$
$180$	0	0
$181$	−12.2440	−0.910088	−0.455044	−	0.890469i	$-0.650376\pi$
$181$	−12.2440	−0.910088	−0.455044	+	0.890469i	$0.650376\pi$
$182$	2.57375	0.190779
$183$	0	0
$184$	0.454979	0.0335415
$185$	−4.44852	−0.327061
$186$	0	0
$187$	10.7240	0.784214
$188$	16.4286	1.19818
$189$	0	0
$190$	−12.9692	−0.940888
$191$	−6.79213	−0.491461	−0.245731	−	0.969338i	$-0.579028\pi$
$191$	−6.79213	−0.491461	−0.245731	+	0.969338i	$0.579028\pi$
$192$	0	0
$193$	−24.4529	−1.76016	−0.880080	−	0.474826i	$-0.842511\pi$
$193$	−24.4529	−1.76016	−0.880080	+	0.474826i	$0.842511\pi$
$194$	−32.0925	−2.30410
$195$	0	0
$196$	−13.1669	−0.940496
$197$	7.52376	0.536045	0.268023	−	0.963413i	$-0.413630\pi$
$197$	7.52376	0.536045	0.268023	+	0.963413i	$0.413630\pi$
$198$	0	0
$199$	−23.0582	−1.63455	−0.817275	−	0.576248i	$-0.804516\pi$
$199$	−23.0582	−1.63455	−0.817275	+	0.576248i	$0.804516\pi$
$200$	0.532596	0.0376602
$201$	0	0
$202$	5.47534	0.385244
$203$	0	0
$204$	0	0
$205$	−9.53284	−0.665802
$206$	7.49946	0.522512
$207$	0	0
$208$	−13.7034	−0.950161
$209$	13.3945	0.926520
$210$	0	0
$211$	6.53288	0.449742	0.224871	−	0.974389i	$-0.427804\pi$
$211$	6.53288	0.449742	0.224871	+	0.974389i	$0.427804\pi$
$212$	21.0067	1.44274
$213$	0	0
$214$	22.3752	1.52954
$215$	−2.86314	−0.195265
$216$	0	0
$217$	−3.93383	−0.267046
$218$	29.4707	1.99600
$219$	0	0
$220$	−5.84731	−0.394226
$221$	14.2093	0.955822
$222$	0	0
$223$	−12.2164	−0.818070	−0.409035	−	0.912519i	$-0.634135\pi$
$223$	−12.2164	−0.818070	−0.409035	+	0.912519i	$0.634135\pi$
$224$	−3.11110	−0.207869
$225$	0	0
$226$	0.0346959	0.00230794
$227$	14.4701	0.960417	0.480209	−	0.877154i	$-0.340561\pi$
$227$	14.4701	0.960417	0.480209	+	0.877154i	$0.340561\pi$
$228$	0	0
$229$	10.4194	0.688530	0.344265	−	0.938872i	$-0.388128\pi$
$229$	10.4194	0.688530	0.344265	+	0.938872i	$0.388128\pi$
$230$	7.24837	0.477944
$231$	0	0
$232$	0	0
$233$	18.5595	1.21587	0.607935	−	0.793987i	$-0.291998\pi$
$233$	18.5595	1.21587	0.607935	+	0.793987i	$0.291998\pi$
$234$	0	0
$235$	−10.4117	−0.679184
$236$	−19.7832	−1.28778
$237$	0	0
$238$	3.34956	0.217120
$239$	23.8345	1.54173	0.770864	−	0.637000i	$-0.219825\pi$
$239$	23.8345	1.54173	0.770864	+	0.637000i	$0.219825\pi$
$240$	0	0
$241$	18.2561	1.17598	0.587990	−	0.808868i	$-0.299919\pi$
$241$	18.2561	1.17598	0.587990	+	0.808868i	$0.299919\pi$
$242$	−9.47019	−0.608767
$243$	0	0
$244$	7.16054	0.458407
$245$	8.34461	0.533118
$246$	0	0
$247$	17.7478	1.12927
$248$	1.51620	0.0962789
$249$	0	0
$250$	20.5579	1.30020
$251$	17.1613	1.08321	0.541606	−	0.840632i	$-0.317816\pi$
$251$	17.1613	1.08321	0.541606	+	0.840632i	$0.317816\pi$
$252$	0	0
$253$	−7.48607	−0.470645
$254$	20.3148	1.27466
$255$	0	0
$256$	17.1532	1.07207
$257$	1.64299	0.102487	0.0512435	−	0.998686i	$-0.483682\pi$
$257$	1.64299	0.102487	0.0512435	+	0.998686i	$0.483682\pi$
$258$	0	0
$259$	1.43503	0.0891683
$260$	−7.74772	−0.480493
$261$	0	0
$262$	−9.81019	−0.606075
$263$	−6.87423	−0.423883	−0.211942	−	0.977282i	$-0.567979\pi$
$263$	−6.87423	−0.423883	−0.211942	+	0.977282i	$0.567979\pi$
$264$	0	0
$265$	−13.3131	−0.817816
$266$	4.18370	0.256519
$267$	0	0
$268$	9.89978	0.604725
$269$	4.03745	0.246167	0.123084	−	0.992396i	$-0.460722\pi$
$269$	4.03745	0.246167	0.123084	+	0.992396i	$0.460722\pi$
$270$	0	0
$271$	12.8985	0.783530	0.391765	−	0.920065i	$-0.371865\pi$
$271$	12.8985	0.783530	0.391765	+	0.920065i	$0.371865\pi$
$272$	−17.8340	−1.08135
$273$	0	0
$274$	26.8768	1.62369
$275$	−8.76315	−0.528438
$276$	0	0
$277$	−3.20273	−0.192433	−0.0962166	−	0.995360i	$-0.530674\pi$
$277$	−3.20273	−0.192433	−0.0962166	+	0.995360i	$0.530674\pi$
$278$	32.6281	1.95690
$279$	0	0
$280$	0.0726542	0.00434192
$281$	−19.2355	−1.14749	−0.573746	−	0.819033i	$-0.694510\pi$
$281$	−19.2355	−1.14749	−0.573746	+	0.819033i	$0.694510\pi$
$282$	0	0
$283$	28.2362	1.67847	0.839235	−	0.543770i	$-0.183003\pi$
$283$	28.2362	1.67847	0.839235	+	0.543770i	$0.183003\pi$
$284$	15.5634	0.923518
$285$	0	0
$286$	16.3219	0.965134
$287$	3.07516	0.181521
$288$	0	0
$289$	1.49242	0.0877892
$290$	0	0
$291$	0	0
$292$	16.8432	0.985672
$293$	33.3244	1.94683	0.973417	−	0.229039i	$-0.0735583\pi$
$293$	33.3244	1.94683	0.973417	+	0.229039i	$0.0735583\pi$
$294$	0	0
$295$	12.5377	0.729974
$296$	−0.553097	−0.0321481
$297$	0	0
$298$	−15.8203	−0.916443
$299$	−9.91908	−0.573635
$300$	0	0
$301$	0.923610	0.0532360
$302$	−40.2165	−2.31420
$303$	0	0
$304$	−22.2752	−1.27757
$305$	−4.53803	−0.259847
$306$	0	0
$307$	−0.940065	−0.0536523	−0.0268262	−	0.999640i	$-0.508540\pi$
$307$	−0.940065	−0.0536523	−0.0268262	+	0.999640i	$0.508540\pi$
$308$	1.88626	0.107480
$309$	0	0
$310$	24.1550	1.37191
$311$	−8.75219	−0.496291	−0.248146	−	0.968723i	$-0.579821\pi$
$311$	−8.75219	−0.496291	−0.248146	+	0.968723i	$0.579821\pi$
$312$	0	0
$313$	−32.9372	−1.86172	−0.930861	−	0.365372i	$-0.880942\pi$
$313$	−32.9372	−1.86172	−0.930861	+	0.365372i	$0.880942\pi$
$314$	−8.10115	−0.457174
$315$	0	0
$316$	14.0400	0.789811
$317$	23.4587	1.31757	0.658785	−	0.752331i	$-0.271071\pi$
$317$	23.4587	1.31757	0.658785	+	0.752331i	$0.271071\pi$
$318$	0	0
$319$	0	0
$320$	8.99219	0.502679
$321$	0	0
$322$	−2.33822	−0.130304
$323$	23.0976	1.28518
$324$	0	0
$325$	−11.6112	−0.644075
$326$	38.1846	2.11485
$327$	0	0
$328$	−1.18525	−0.0654443
$329$	3.35866	0.185169
$330$	0	0
$331$	−26.4233	−1.45236	−0.726178	−	0.687506i	$-0.758705\pi$
$331$	−26.4233	−1.45236	−0.726178	+	0.687506i	$0.758705\pi$
$332$	7.21264	0.395845
$333$	0	0
$334$	6.43186	0.351936
$335$	−6.27403	−0.342787
$336$	0	0
$337$	−6.42229	−0.349844	−0.174922	−	0.984582i	$-0.555967\pi$
$337$	−6.42229	−0.349844	−0.174922	+	0.984582i	$0.555967\pi$
$338$	−4.12351	−0.224290
$339$	0	0
$340$	−10.0831	−0.546834
$341$	−24.9471	−1.35096
$342$	0	0
$343$	−5.44452	−0.293976
$344$	−0.355983	−0.0191933
$345$	0	0
$346$	15.9048	0.855046
$347$	5.30429	0.284749	0.142375	−	0.989813i	$-0.454526\pi$
$347$	5.30429	0.284749	0.142375	+	0.989813i	$0.454526\pi$
$348$	0	0
$349$	13.3972	0.717134	0.358567	−	0.933504i	$-0.383265\pi$
$349$	13.3972	0.717134	0.358567	+	0.933504i	$0.383265\pi$
$350$	−2.73711	−0.146305
$351$	0	0
$352$	−19.7296	−1.05159
$353$	16.0910	0.856437	0.428219	−	0.903675i	$-0.359141\pi$
$353$	16.0910	0.856437	0.428219	+	0.903675i	$0.359141\pi$
$354$	0	0
$355$	−9.86339	−0.523494
$356$	−20.0274	−1.06145
$357$	0	0
$358$	20.9273	1.10604
$359$	12.0867	0.637913	0.318957	−	0.947769i	$-0.396668\pi$
$359$	12.0867	0.637913	0.318957	+	0.947769i	$0.396668\pi$
$360$	0	0
$361$	9.84954	0.518397
$362$	−24.2526	−1.27469
$363$	0	0
$364$	2.49931	0.130999
$365$	−10.6744	−0.558726
$366$	0	0
$367$	−7.32140	−0.382174	−0.191087	−	0.981573i	$-0.561201\pi$
$367$	−7.32140	−0.382174	−0.191087	+	0.981573i	$0.561201\pi$
$368$	12.4494	0.648969
$369$	0	0
$370$	−8.81152	−0.458089
$371$	4.29461	0.222965
$372$	0	0
$373$	−8.27558	−0.428493	−0.214247	−	0.976780i	$-0.568730\pi$
$373$	−8.27558	−0.428493	−0.214247	+	0.976780i	$0.568730\pi$
$374$	21.2418	1.09839
$375$	0	0
$376$	−1.29452	−0.0667596
$377$	0	0
$378$	0	0
$379$	−3.85809	−0.198177	−0.0990885	−	0.995079i	$-0.531593\pi$
$379$	−3.85809	−0.198177	−0.0990885	+	0.995079i	$0.531593\pi$
$380$	−12.5941	−0.646064
$381$	0	0
$382$	−13.4537	−0.688351
$383$	5.53107	0.282625	0.141312	−	0.989965i	$-0.454868\pi$
$383$	5.53107	0.282625	0.141312	+	0.989965i	$0.454868\pi$
$384$	0	0
$385$	−1.19543	−0.0609246
$386$	−48.4358	−2.46532
$387$	0	0
$388$	−31.1642	−1.58212
$389$	−8.73816	−0.443042	−0.221521	−	0.975156i	$-0.571102\pi$
$389$	−8.73816	−0.443042	−0.221521	+	0.975156i	$0.571102\pi$
$390$	0	0
$391$	−12.9090	−0.652835
$392$	1.03751	0.0524022
$393$	0	0
$394$	14.9029	0.750797
$395$	−8.89791	−0.447702
$396$	0	0
$397$	−6.75150	−0.338848	−0.169424	−	0.985543i	$-0.554191\pi$
$397$	−6.75150	−0.338848	−0.169424	+	0.985543i	$0.554191\pi$
$398$	−45.6731	−2.28939
$399$	0	0
$400$	14.5732	0.728659
$401$	23.4914	1.17311	0.586553	−	0.809911i	$-0.300485\pi$
$401$	23.4914	1.17311	0.586553	+	0.809911i	$0.300485\pi$
$402$	0	0
$403$	−33.0550	−1.64659
$404$	5.31696	0.264529
$405$	0	0
$406$	0	0
$407$	9.10048	0.451094
$408$	0	0
$409$	−24.4037	−1.20668	−0.603341	−	0.797483i	$-0.706164\pi$
$409$	−24.4037	−1.20668	−0.603341	+	0.797483i	$0.706164\pi$
$410$	−18.8824	−0.932537
$411$	0	0
$412$	7.28253	0.358784
$413$	−4.04449	−0.199016
$414$	0	0
$415$	−4.57105	−0.224384
$416$	−26.1418	−1.28171
$417$	0	0
$418$	26.5316	1.29770
$419$	−16.7208	−0.816864	−0.408432	−	0.912789i	$-0.633924\pi$
$419$	−16.7208	−0.816864	−0.408432	+	0.912789i	$0.633924\pi$
$420$	0	0
$421$	−23.1473	−1.12813	−0.564066	−	0.825730i	$-0.690764\pi$
$421$	−23.1473	−1.12813	−0.564066	+	0.825730i	$0.690764\pi$
$422$	12.9402	0.629919
$423$	0	0
$424$	−1.65526	−0.0803863
$425$	−15.1112	−0.733000
$426$	0	0
$427$	1.46390	0.0708433
$428$	21.7280	1.05026
$429$	0	0
$430$	−5.67125	−0.273492
$431$	−37.5513	−1.80878	−0.904391	−	0.426705i	$-0.859674\pi$
$431$	−37.5513	−1.80878	−0.904391	+	0.426705i	$0.859674\pi$
$432$	0	0
$433$	17.1657	0.824929	0.412464	−	0.910974i	$-0.364668\pi$
$433$	17.1657	0.824929	0.412464	+	0.910974i	$0.364668\pi$
$434$	−7.79205	−0.374030
$435$	0	0
$436$	28.6182	1.37056
$437$	−16.1237	−0.771301
$438$	0	0
$439$	5.29840	0.252879	0.126439	−	0.991974i	$-0.459645\pi$
$439$	5.29840	0.252879	0.126439	+	0.991974i	$0.459645\pi$
$440$	0.460749	0.0219653
$441$	0	0
$442$	28.1455	1.33874
$443$	−13.8119	−0.656225	−0.328113	−	0.944639i	$-0.606413\pi$
$443$	−13.8119	−0.656225	−0.328113	+	0.944639i	$0.606413\pi$
$444$	0	0
$445$	12.6924	0.601680
$446$	−24.1980	−1.14581
$447$	0	0
$448$	−2.90076	−0.137048
$449$	−17.3095	−0.816886	−0.408443	−	0.912784i	$-0.633928\pi$
$449$	−17.3095	−0.816886	−0.408443	+	0.912784i	$0.633928\pi$
$450$	0	0
$451$	19.5017	0.918297
$452$	0.0336923	0.00158475
$453$	0	0
$454$	28.6621	1.34518
$455$	−1.58395	−0.0742566
$456$	0	0
$457$	−20.4082	−0.954657	−0.477328	−	0.878725i	$-0.658395\pi$
$457$	−20.4082	−0.954657	−0.477328	+	0.878725i	$0.658395\pi$
$458$	20.6384	0.964370
$459$	0	0
$460$	7.03871	0.328181
$461$	−16.9405	−0.788995	−0.394498	−	0.918897i	$-0.629081\pi$
$461$	−16.9405	−0.788995	−0.394498	+	0.918897i	$0.629081\pi$
$462$	0	0
$463$	26.1995	1.21759	0.608796	−	0.793327i	$-0.291653\pi$
$463$	26.1995	1.21759	0.608796	+	0.793327i	$0.291653\pi$
$464$	0	0
$465$	0	0
$466$	36.7622	1.70297
$467$	−28.3546	−1.31209	−0.656047	−	0.754720i	$-0.727773\pi$
$467$	−28.3546	−1.31209	−0.656047	+	0.754720i	$0.727773\pi$
$468$	0	0
$469$	2.02391	0.0934557
$470$	−20.6232	−0.951279
$471$	0	0
$472$	1.55885	0.0717519
$473$	5.85723	0.269316
$474$	0	0
$475$	−18.8743	−0.866013
$476$	3.25267	0.149086
$477$	0	0
$478$	47.2109	2.15938
$479$	−33.6686	−1.53836	−0.769179	−	0.639033i	$-0.779335\pi$
$479$	−33.6686	−1.53836	−0.769179	+	0.639033i	$0.779335\pi$
$480$	0	0
$481$	12.0582	0.549806
$482$	36.1613	1.64710
$483$	0	0
$484$	−9.19626	−0.418012
$485$	19.7504	0.896821
$486$	0	0
$487$	−23.2890	−1.05532	−0.527662	−	0.849454i	$-0.676931\pi$
$487$	−23.2890	−1.05532	−0.527662	+	0.849454i	$0.676931\pi$
$488$	−0.564227	−0.0255413
$489$	0	0
$490$	16.5288	0.746696
$491$	14.8333	0.669418	0.334709	−	0.942322i	$-0.391362\pi$
$491$	14.8333	0.669418	0.334709	+	0.942322i	$0.391362\pi$
$492$	0	0
$493$	0	0
$494$	35.1545	1.58168
$495$	0	0
$496$	41.4872	1.86283
$497$	3.18179	0.142723
$498$	0	0
$499$	23.1693	1.03720	0.518599	−	0.855017i	$-0.326454\pi$
$499$	23.1693	1.03720	0.518599	+	0.855017i	$0.326454\pi$
$500$	19.9633	0.892784
$501$	0	0
$502$	33.9927	1.51717
$503$	15.5235	0.692158	0.346079	−	0.938205i	$-0.387513\pi$
$503$	15.5235	0.692158	0.346079	+	0.938205i	$0.387513\pi$
$504$	0	0
$505$	−3.36965	−0.149948
$506$	−14.8282	−0.659196
$507$	0	0
$508$	19.7272	0.875252
$509$	−32.3172	−1.43244	−0.716218	−	0.697876i	$-0.754129\pi$
$509$	−32.3172	−1.43244	−0.716218	+	0.697876i	$0.754129\pi$
$510$	0	0
$511$	3.44342	0.152328
$512$	31.5534	1.39447
$513$	0	0
$514$	3.25440	0.143545
$515$	−4.61534	−0.203376
$516$	0	0
$517$	21.2995	0.936753
$518$	2.84247	0.124891
$519$	0	0
$520$	0.610494	0.0267720
$521$	0.674556	0.0295529	0.0147764	−	0.999891i	$-0.495296\pi$
$521$	0.674556	0.0295529	0.0147764	+	0.999891i	$0.495296\pi$
$522$	0	0
$523$	−6.73109	−0.294330	−0.147165	−	0.989112i	$-0.547015\pi$
$523$	−6.73109	−0.294330	−0.147165	+	0.989112i	$0.547015\pi$
$524$	−9.52642	−0.416164
$525$	0	0
$526$	−13.6163	−0.593700
$527$	−43.0187	−1.87393
$528$	0	0
$529$	−13.9886	−0.608202
$530$	−26.3703	−1.14545
$531$	0	0
$532$	4.06268	0.176140
$533$	25.8398	1.11925
$534$	0	0
$535$	−13.7702	−0.595338
$536$	−0.780069	−0.0336939
$537$	0	0
$538$	7.99728	0.344787
$539$	−17.0709	−0.735294
$540$	0	0
$541$	−15.6659	−0.673531	−0.336766	−	0.941588i	$-0.609333\pi$
$541$	−15.6659	−0.673531	−0.336766	+	0.941588i	$0.609333\pi$
$542$	25.5491	1.09743
$543$	0	0
$544$	−34.0217	−1.45867
$545$	−18.1369	−0.776900
$546$	0	0
$547$	−12.9377	−0.553177	−0.276589	−	0.960988i	$-0.589204\pi$
$547$	−12.9377	−0.553177	−0.276589	+	0.960988i	$0.589204\pi$
$548$	26.0994	1.11491
$549$	0	0
$550$	−17.3579	−0.740142
$551$	0	0
$552$	0	0
$553$	2.87034	0.122059
$554$	−6.34389	−0.269526
$555$	0	0
$556$	31.6843	1.34371
$557$	−9.74915	−0.413085	−0.206542	−	0.978438i	$-0.566221\pi$
$557$	−9.74915	−0.413085	−0.206542	+	0.978438i	$0.566221\pi$
$558$	0	0
$559$	7.76086	0.328249
$560$	1.98800	0.0840085
$561$	0	0
$562$	−38.1012	−1.60720
$563$	−39.5811	−1.66814	−0.834072	−	0.551656i	$-0.813996\pi$
$563$	−39.5811	−1.66814	−0.834072	+	0.551656i	$0.813996\pi$
$564$	0	0
$565$	−0.0213527	−0.000898314 0
$566$	55.9297	2.35090
$567$	0	0
$568$	−1.22634	−0.0514563
$569$	28.9833	1.21504	0.607521	−	0.794304i	$-0.292164\pi$
$569$	28.9833	1.21504	0.607521	+	0.794304i	$0.292164\pi$
$570$	0	0
$571$	−30.2218	−1.26474	−0.632372	−	0.774665i	$-0.717919\pi$
$571$	−30.2218	−1.26474	−0.632372	+	0.774665i	$0.717919\pi$
$572$	15.8498	0.662712
$573$	0	0
$574$	6.09121	0.254242
$575$	10.5487	0.439909
$576$	0	0
$577$	−25.9505	−1.08033	−0.540167	−	0.841558i	$-0.681639\pi$
$577$	−25.9505	−1.08033	−0.540167	+	0.841558i	$0.681639\pi$
$578$	2.95615	0.122960
$579$	0	0
$580$	0	0
$581$	1.47456	0.0611749
$582$	0	0
$583$	27.2350	1.12796
$584$	−1.32719	−0.0549193
$585$	0	0
$586$	66.0083	2.72678
$587$	−0.775462	−0.0320067	−0.0160034	−	0.999872i	$-0.505094\pi$
$587$	−0.775462	−0.0320067	−0.0160034	+	0.999872i	$0.505094\pi$
$588$	0	0
$589$	−53.7317	−2.21397
$590$	24.8344	1.02242
$591$	0	0
$592$	−15.1342	−0.622010
$593$	27.4401	1.12683	0.563415	−	0.826174i	$-0.309487\pi$
$593$	27.4401	1.12683	0.563415	+	0.826174i	$0.309487\pi$
$594$	0	0
$595$	−2.06140	−0.0845090
$596$	−15.3627	−0.629279
$597$	0	0
$598$	−19.6475	−0.803446
$599$	−43.2574	−1.76745	−0.883724	−	0.468008i	$-0.844972\pi$
$599$	−43.2574	−1.76745	−0.883724	+	0.468008i	$0.844972\pi$
$600$	0	0
$601$	−33.0371	−1.34761	−0.673806	−	0.738908i	$-0.735342\pi$
$601$	−33.0371	−1.34761	−0.673806	+	0.738908i	$0.735342\pi$
$602$	1.82947	0.0745635
$603$	0	0
$604$	−39.0532	−1.58905
$605$	5.82817	0.236949
$606$	0	0
$607$	−17.6236	−0.715319	−0.357659	−	0.933852i	$-0.616425\pi$
$607$	−17.6236	−0.715319	−0.357659	+	0.933852i	$0.616425\pi$
$608$	−42.4942	−1.72337
$609$	0	0
$610$	−8.98883	−0.363947
$611$	28.2220	1.14174
$612$	0	0
$613$	38.0570	1.53711	0.768554	−	0.639785i	$-0.220977\pi$
$613$	38.0570	1.53711	0.768554	+	0.639785i	$0.220977\pi$
$614$	−1.86206	−0.0751466
$615$	0	0
$616$	−0.148631	−0.00598852
$617$	5.15982	0.207727	0.103863	−	0.994592i	$-0.466880\pi$
$617$	5.15982	0.207727	0.103863	+	0.994592i	$0.466880\pi$
$618$	0	0
$619$	−29.6683	−1.19247	−0.596236	−	0.802809i	$-0.703338\pi$
$619$	−29.6683	−1.19247	−0.596236	+	0.802809i	$0.703338\pi$
$620$	23.4563	0.942026
$621$	0	0
$622$	−17.3362	−0.695116
$623$	−4.09441	−0.164039
$624$	0	0
$625$	4.91819	0.196727
$626$	−65.2414	−2.60757
$627$	0	0
$628$	−7.86682	−0.313920
$629$	15.6929	0.625716
$630$	0	0
$631$	9.97005	0.396901	0.198451	−	0.980111i	$-0.436409\pi$
$631$	9.97005	0.396901	0.198451	+	0.980111i	$0.436409\pi$
$632$	−1.10630	−0.0440064
$633$	0	0
$634$	46.4664	1.84542
$635$	−12.5022	−0.496135
$636$	0	0
$637$	−22.6190	−0.896196
$638$	0	0
$639$	0	0
$640$	−1.47699	−0.0583832
$641$	−22.3786	−0.883903	−0.441951	−	0.897039i	$-0.645714\pi$
$641$	−22.3786	−0.883903	−0.441951	+	0.897039i	$0.645714\pi$
$642$	0	0
$643$	−5.92551	−0.233679	−0.116840	−	0.993151i	$-0.537276\pi$
$643$	−5.92551	−0.233679	−0.116840	+	0.993151i	$0.537276\pi$
$644$	−2.27059	−0.0894737
$645$	0	0
$646$	45.7512	1.80006
$647$	−15.3902	−0.605050	−0.302525	−	0.953141i	$-0.597830\pi$
$647$	−15.3902	−0.605050	−0.302525	+	0.953141i	$0.597830\pi$
$648$	0	0
$649$	−25.6488	−1.00680
$650$	−22.9993	−0.902105
$651$	0	0
$652$	37.0800	1.45217
$653$	−33.5424	−1.31262	−0.656308	−	0.754493i	$-0.727883\pi$
$653$	−33.5424	−1.31262	−0.656308	+	0.754493i	$0.727883\pi$
$654$	0	0
$655$	6.03741	0.235901
$656$	−32.4314	−1.26623
$657$	0	0
$658$	6.65277	0.259352
$659$	−10.2318	−0.398575	−0.199288	−	0.979941i	$-0.563863\pi$
$659$	−10.2318	−0.398575	−0.199288	+	0.979941i	$0.563863\pi$
$660$	0	0
$661$	13.3227	0.518193	0.259097	−	0.965851i	$-0.416575\pi$
$661$	13.3227	0.518193	0.259097	+	0.965851i	$0.416575\pi$
$662$	−52.3387	−2.03420
$663$	0	0
$664$	−0.568332	−0.0220556
$665$	−2.57474	−0.0998443
$666$	0	0
$667$	0	0
$668$	6.24581	0.241658
$669$	0	0
$670$	−12.4275	−0.480115
$671$	9.28360	0.358389
$672$	0	0
$673$	38.0338	1.46610	0.733048	−	0.680177i	$-0.238097\pi$
$673$	38.0338	1.46610	0.733048	+	0.680177i	$0.238097\pi$
$674$	−12.7211	−0.489999
$675$	0	0
$676$	−4.00424	−0.154009
$677$	−8.81473	−0.338778	−0.169389	−	0.985549i	$-0.554179\pi$
$677$	−8.81473	−0.338778	−0.169389	+	0.985549i	$0.554179\pi$
$678$	0	0
$679$	−6.37121	−0.244505
$680$	0.794515	0.0304683
$681$	0	0
$682$	−49.4146	−1.89218
$683$	10.1501	0.388382	0.194191	−	0.980964i	$-0.437792\pi$
$683$	10.1501	0.388382	0.194191	+	0.980964i	$0.437792\pi$
$684$	0	0
$685$	−16.5406	−0.631985
$686$	−10.7844	−0.411750
$687$	0	0
$688$	−9.74062	−0.371358
$689$	36.0865	1.37479
$690$	0	0
$691$	−0.144207	−0.00548590	−0.00274295	−	0.999996i	$-0.500873\pi$
$691$	−0.144207	−0.00548590	−0.00274295	+	0.999996i	$0.500873\pi$
$692$	15.4447	0.587120
$693$	0	0
$694$	10.5066	0.398826
$695$	−20.0801	−0.761681
$696$	0	0
$697$	33.6287	1.27378
$698$	26.5368	1.00443
$699$	0	0
$700$	−2.65794	−0.100461
$701$	−48.2313	−1.82167	−0.910835	−	0.412771i	$-0.864561\pi$
$701$	−48.2313	−1.82167	−0.910835	+	0.412771i	$0.864561\pi$
$702$	0	0
$703$	19.6009	0.739260
$704$	−18.3956	−0.693312
$705$	0	0
$706$	31.8727	1.19954
$707$	1.08700	0.0408809
$708$	0	0
$709$	10.5595	0.396571	0.198286	−	0.980144i	$-0.436463\pi$
$709$	10.5595	0.396571	0.198286	+	0.980144i	$0.436463\pi$
$710$	−19.5372	−0.733217
$711$	0	0
$712$	1.57809	0.0591414
$713$	30.0301	1.12463
$714$	0	0
$715$	−10.0449	−0.375657
$716$	20.3220	0.759468
$717$	0	0
$718$	23.9411	0.893475
$719$	25.2961	0.943386	0.471693	−	0.881763i	$-0.343643\pi$
$719$	25.2961	0.943386	0.471693	+	0.881763i	$0.343643\pi$
$720$	0	0
$721$	1.48884	0.0554474
$722$	19.5098	0.726078
$723$	0	0
$724$	−23.5511	−0.875269
$725$	0	0
$726$	0	0
$727$	−49.0795	−1.82026	−0.910129	−	0.414326i	$-0.864017\pi$
$727$	−49.0795	−1.82026	−0.910129	+	0.414326i	$0.864017\pi$
$728$	−0.196937	−0.00729897
$729$	0	0
$730$	−21.1437	−0.782563
$731$	10.1002	0.373570
$732$	0	0
$733$	−3.40131	−0.125630	−0.0628151	−	0.998025i	$-0.520008\pi$
$733$	−3.40131	−0.125630	−0.0628151	+	0.998025i	$0.520008\pi$
$734$	−14.5021	−0.535281
$735$	0	0
$736$	23.7495	0.875419
$737$	12.8350	0.472783
$738$	0	0
$739$	7.22771	0.265876	0.132938	−	0.991124i	$-0.457559\pi$
$739$	7.22771	0.265876	0.132938	+	0.991124i	$0.457559\pi$
$740$	−8.55664	−0.314548
$741$	0	0
$742$	8.50668	0.312290
$743$	−22.2471	−0.816166	−0.408083	−	0.912945i	$-0.633803\pi$
$743$	−22.2471	−0.816166	−0.408083	+	0.912945i	$0.633803\pi$
$744$	0	0
$745$	9.73615	0.356705
$746$	−16.3921	−0.600157
$747$	0	0
$748$	20.6274	0.754211
$749$	4.44207	0.162310
$750$	0	0
$751$	25.7262	0.938761	0.469381	−	0.882996i	$-0.344477\pi$
$751$	25.7262	0.938761	0.469381	+	0.882996i	$0.344477\pi$
$752$	−35.4213	−1.29168
$753$	0	0
$754$	0	0
$755$	24.7502	0.900750
$756$	0	0
$757$	−26.5887	−0.966382	−0.483191	−	0.875515i	$-0.660522\pi$
$757$	−26.5887	−0.966382	−0.483191	+	0.875515i	$0.660522\pi$
$758$	−7.64203	−0.277571
$759$	0	0
$760$	0.992373	0.0359972
$761$	8.21201	0.297685	0.148843	−	0.988861i	$-0.452445\pi$
$761$	8.21201	0.297685	0.148843	+	0.988861i	$0.452445\pi$
$762$	0	0
$763$	5.85071	0.211810
$764$	−13.0645	−0.472659
$765$	0	0
$766$	10.9558	0.395850
$767$	−33.9848	−1.22712
$768$	0	0
$769$	−41.9873	−1.51410	−0.757050	−	0.653357i	$-0.773360\pi$
$769$	−41.9873	−1.51410	−0.757050	+	0.653357i	$0.773360\pi$
$770$	−2.36788	−0.0853323
$771$	0	0
$772$	−47.0348	−1.69282
$773$	48.1727	1.73265	0.866327	−	0.499478i	$-0.166475\pi$
$773$	48.1727	1.73265	0.866327	+	0.499478i	$0.166475\pi$
$774$	0	0
$775$	35.1530	1.26273
$776$	2.45563	0.0881520
$777$	0	0
$778$	−17.3084	−0.620535
$779$	42.0032	1.50492
$780$	0	0
$781$	20.1779	0.722021
$782$	−25.5698	−0.914375
$783$	0	0
$784$	28.3890	1.01389
$785$	4.98563	0.177945
$786$	0	0
$787$	17.3814	0.619581	0.309791	−	0.950805i	$-0.399741\pi$
$787$	17.3814	0.619581	0.309791	+	0.950805i	$0.399741\pi$
$788$	14.4718	0.515537
$789$	0	0
$790$	−17.6248	−0.627061
$791$	0.00688807	0.000244912 0
$792$	0	0
$793$	12.3008	0.436815
$794$	−13.3732	−0.474598
$795$	0	0
$796$	−44.3520	−1.57201
$797$	33.2762	1.17870	0.589352	−	0.807876i	$-0.299383\pi$
$797$	33.2762	1.17870	0.589352	+	0.807876i	$0.299383\pi$
$798$	0	0
$799$	36.7289	1.29938
$800$	27.8011	0.982916
$801$	0	0
$802$	46.5313	1.64308
$803$	21.8371	0.770613
$804$	0	0
$805$	1.43900	0.0507180
$806$	−65.4746	−2.30625
$807$	0	0
$808$	−0.418959	−0.0147389
$809$	−5.66329	−0.199111	−0.0995553	−	0.995032i	$-0.531742\pi$
$809$	−5.66329	−0.199111	−0.0995553	+	0.995032i	$0.531742\pi$
$810$	0	0
$811$	15.4044	0.540922	0.270461	−	0.962731i	$-0.412824\pi$
$811$	15.4044	0.540922	0.270461	+	0.962731i	$0.412824\pi$
$812$	0	0
$813$	0	0
$814$	18.0260	0.631812
$815$	−23.4997	−0.823157
$816$	0	0
$817$	12.6155	0.441359
$818$	−48.3382	−1.69011
$819$	0	0
$820$	−18.3363	−0.640330
$821$	−0.807130	−0.0281690	−0.0140845	−	0.999901i	$-0.504483\pi$
$821$	−0.807130	−0.0281690	−0.0140845	+	0.999901i	$0.504483\pi$
$822$	0	0
$823$	−38.9003	−1.35598	−0.677989	−	0.735072i	$-0.737149\pi$
$823$	−38.9003	−1.35598	−0.677989	+	0.735072i	$0.737149\pi$
$824$	−0.573839	−0.0199906
$825$	0	0
$826$	−8.01123	−0.278746
$827$	−1.80340	−0.0627103	−0.0313552	−	0.999508i	$-0.509982\pi$
$827$	−1.80340	−0.0627103	−0.0313552	+	0.999508i	$0.509982\pi$
$828$	0	0
$829$	7.75729	0.269422	0.134711	−	0.990885i	$-0.456989\pi$
$829$	7.75729	0.269422	0.134711	+	0.990885i	$0.456989\pi$
$830$	−9.05423	−0.314277
$831$	0	0
$832$	−24.3743	−0.845027
$833$	−29.4370	−1.01993
$834$	0	0
$835$	−3.95831	−0.136983
$836$	25.7642	0.891073
$837$	0	0
$838$	−33.1202	−1.14412
$839$	−33.0135	−1.13975	−0.569876	−	0.821731i	$-0.693009\pi$
$839$	−33.0135	−1.13975	−0.569876	+	0.821731i	$0.693009\pi$
$840$	0	0
$841$	0	0
$842$	−45.8497	−1.58009
$843$	0	0
$844$	12.5659	0.432536
$845$	2.53771	0.0872997
$846$	0	0
$847$	−1.88009	−0.0646005
$848$	−45.2920	−1.55534
$849$	0	0
$850$	−29.9319	−1.02666
$851$	−10.9547	−0.375523
$852$	0	0
$853$	−5.02035	−0.171893	−0.0859467	−	0.996300i	$-0.527391\pi$
$853$	−5.02035	−0.171893	−0.0859467	+	0.996300i	$0.527391\pi$
$854$	2.89967	0.0992246
$855$	0	0
$856$	−1.71209	−0.0585180
$857$	−8.87355	−0.303115	−0.151557	−	0.988448i	$-0.548429\pi$
$857$	−8.87355	−0.303115	−0.151557	+	0.988448i	$0.548429\pi$
$858$	0	0
$859$	−36.4701	−1.24434	−0.622172	−	0.782880i	$-0.713750\pi$
$859$	−36.4701	−1.24434	−0.622172	+	0.782880i	$0.713750\pi$
$860$	−5.50721	−0.187794
$861$	0	0
$862$	−74.3808	−2.53342
$863$	8.86240	0.301680	0.150840	−	0.988558i	$-0.451802\pi$
$863$	8.86240	0.301680	0.150840	+	0.988558i	$0.451802\pi$
$864$	0	0
$865$	−9.78816	−0.332807
$866$	34.0014	1.15541
$867$	0	0
$868$	−7.56666	−0.256829
$869$	18.2027	0.617486
$870$	0	0
$871$	17.0064	0.576241
$872$	−2.25502	−0.0763645
$873$	0	0
$874$	−31.9375	−1.08030
$875$	4.08130	0.137973
$876$	0	0
$877$	−41.5401	−1.40271	−0.701355	−	0.712812i	$-0.747421\pi$
$877$	−41.5401	−1.40271	−0.701355	+	0.712812i	$0.747421\pi$
$878$	10.4950	0.354187
$879$	0	0
$880$	12.6073	0.424991
$881$	−14.7088	−0.495552	−0.247776	−	0.968817i	$-0.579700\pi$
$881$	−14.7088	−0.495552	−0.247776	+	0.968817i	$0.579700\pi$
$882$	0	0
$883$	−31.6654	−1.06563	−0.532813	−	0.846233i	$-0.678865\pi$
$883$	−31.6654	−1.06563	−0.532813	+	0.846233i	$0.678865\pi$
$884$	27.3314	0.919253
$885$	0	0
$886$	−27.3584	−0.919123
$887$	37.7959	1.26906	0.634531	−	0.772897i	$-0.281193\pi$
$887$	37.7959	1.26906	0.634531	+	0.772897i	$0.281193\pi$
$888$	0	0
$889$	4.03303	0.135264
$890$	25.1409	0.842726
$891$	0	0
$892$	−23.4980	−0.786772
$893$	45.8755	1.53517
$894$	0	0
$895$	−12.8792	−0.430503
$896$	0.476456	0.0159173
$897$	0	0
$898$	−34.2863	−1.14415
$899$	0	0
$900$	0	0
$901$	46.9641	1.56460
$902$	38.6285	1.28619
$903$	0	0
$904$	−0.00265484	−8.82987e−5 0
$905$	14.9256	0.496144
$906$	0	0
$907$	4.99667	0.165912	0.0829559	−	0.996553i	$-0.473564\pi$
$907$	4.99667	0.165912	0.0829559	+	0.996553i	$0.473564\pi$
$908$	27.8331	0.923673
$909$	0	0
$910$	−3.13745	−0.104005
$911$	−12.9724	−0.429795	−0.214898	−	0.976637i	$-0.568942\pi$
$911$	−12.9724	−0.429795	−0.214898	+	0.976637i	$0.568942\pi$
$912$	0	0
$913$	9.35115	0.309478
$914$	−40.4242	−1.33711
$915$	0	0
$916$	20.0414	0.662188
$917$	−1.94758	−0.0643149
$918$	0	0
$919$	53.4139	1.76196	0.880981	−	0.473151i	$-0.156883\pi$
$919$	53.4139	1.76196	0.880981	+	0.473151i	$0.156883\pi$
$920$	−0.554627	−0.0182855
$921$	0	0
$922$	−33.5553	−1.10508
$923$	26.7358	0.880019
$924$	0	0
$925$	−12.8235	−0.421635
$926$	51.8953	1.70539
$927$	0	0
$928$	0	0
$929$	−46.1682	−1.51473	−0.757365	−	0.652992i	$-0.773513\pi$
$929$	−46.1682	−1.51473	−0.757365	+	0.652992i	$0.773513\pi$
$930$	0	0
$931$	−36.7677	−1.20501
$932$	35.6988	1.16935
$933$	0	0
$934$	−56.1641	−1.83775
$935$	−13.0727	−0.427523
$936$	0	0
$937$	−10.1380	−0.331193	−0.165596	−	0.986194i	$-0.552955\pi$
$937$	−10.1380	−0.331193	−0.165596	+	0.986194i	$0.552955\pi$
$938$	4.00892	0.130896
$939$	0	0
$940$	−20.0267	−0.653199
$941$	6.79516	0.221516	0.110758	−	0.993847i	$-0.464672\pi$
$941$	6.79516	0.221516	0.110758	+	0.993847i	$0.464672\pi$
$942$	0	0
$943$	−23.4751	−0.764455
$944$	42.6542	1.38828
$945$	0	0
$946$	11.6019	0.377209
$947$	−3.79087	−0.123187	−0.0615933	−	0.998101i	$-0.519618\pi$
$947$	−3.79087	−0.123187	−0.0615933	+	0.998101i	$0.519618\pi$
$948$	0	0
$949$	28.9342	0.939245
$950$	−37.3858	−1.21296
$951$	0	0
$952$	−0.256300	−0.00830671
$953$	−51.2275	−1.65942	−0.829711	−	0.558193i	$-0.811495\pi$
$953$	−51.2275	−1.65942	−0.829711	+	0.558193i	$0.811495\pi$
$954$	0	0
$955$	8.27972	0.267925
$956$	45.8453	1.48274
$957$	0	0
$958$	−66.6901	−2.15466
$959$	5.33577	0.172301
$960$	0	0
$961$	69.0741	2.22820
$962$	23.8846	0.770070
$963$	0	0
$964$	35.1153	1.13099
$965$	29.8085	0.959570
$966$	0	0
$967$	21.1563	0.680342	0.340171	−	0.940364i	$-0.389515\pi$
$967$	21.1563	0.680342	0.340171	+	0.940364i	$0.389515\pi$
$968$	0.724634	0.0232906
$969$	0	0
$970$	39.1212	1.25611
$971$	51.3515	1.64795	0.823974	−	0.566628i	$-0.191752\pi$
$971$	51.3515	1.64795	0.823974	+	0.566628i	$0.191752\pi$
$972$	0	0
$973$	6.47755	0.207661
$974$	−46.1303	−1.47811
$975$	0	0
$976$	−15.4387	−0.494180
$977$	−55.4930	−1.77538	−0.887689	−	0.460443i	$-0.847690\pi$
$977$	−55.4930	−1.77538	−0.887689	+	0.460443i	$0.847690\pi$
$978$	0	0
$979$	−25.9654	−0.829857
$980$	16.0507	0.512721
$981$	0	0
$982$	29.3815	0.937601
$983$	4.53823	0.144747	0.0723735	−	0.997378i	$-0.476943\pi$
$983$	4.53823	0.144747	0.0723735	+	0.997378i	$0.476943\pi$
$984$	0	0
$985$	−9.17158	−0.292231
$986$	0	0
$987$	0	0
$988$	34.1377	1.08606
$989$	−7.05064	−0.224197
$990$	0	0
$991$	7.37723	0.234345	0.117173	−	0.993112i	$-0.462617\pi$
$991$	7.37723	0.234345	0.117173	+	0.993112i	$0.462617\pi$
$992$	79.1445	2.51284
$993$	0	0
$994$	6.30242	0.199901
$995$	28.1083	0.891092
$996$	0	0
$997$	30.8978	0.978542	0.489271	−	0.872132i	$-0.337263\pi$
$997$	30.8978	0.978542	0.489271	+	0.872132i	$0.337263\pi$
$998$	45.8932	1.45272
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.7		9
3.2	odd	2		2523.2.a.p.1.3		9
29.7	even	7		261.2.k.b.136.1		18
29.25	even	7		261.2.k.b.190.1		18
29.28	even	2		7569.2.a.bk.1.3		9
87.65	odd	14		87.2.g.b.49.3	yes	18
87.83	odd	14		87.2.g.b.16.3	✓	18
87.86	odd	2		2523.2.a.q.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.16.3	✓	18	87.83	odd	14
87.2.g.b.49.3	yes	18	87.65	odd	14
261.2.k.b.136.1		18	29.7	even	7
261.2.k.b.190.1		18	29.25	even	7
2523.2.a.p.1.3		9	3.2	odd	2
2523.2.a.q.1.7		9	87.86	odd	2
7569.2.a.bk.1.3		9	29.28	even	2
7569.2.a.bl.1.7		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)

\(f(q)\)	\(=\)	\(q+1.98078 q^{2} +1.92348 q^{4} -1.21902 q^{5} +0.393238 q^{7} -0.151564 q^{8} +O(q^{10})\)	\(q+1.98078 q^{2} +1.92348 q^{4} -1.21902 q^{5} +0.393238 q^{7} -0.151564 q^{8} -2.41460 q^{10} +2.49378 q^{11} +3.30428 q^{13} +0.778916 q^{14} -4.14718 q^{16} +4.30028 q^{17} +5.37118 q^{19} -2.34476 q^{20} +4.93963 q^{22} -3.00189 q^{23} -3.51400 q^{25} +6.54504 q^{26} +0.756386 q^{28} -10.0037 q^{31} -7.91152 q^{32} +8.51790 q^{34} -0.479363 q^{35} +3.64927 q^{37} +10.6391 q^{38} +0.184759 q^{40} +7.82011 q^{41} +2.34873 q^{43} +4.79675 q^{44} -5.94608 q^{46} +8.54106 q^{47} -6.84536 q^{49} -6.96045 q^{50} +6.35572 q^{52} +10.9212 q^{53} -3.03996 q^{55} -0.0596006 q^{56} -10.2851 q^{59} +3.72270 q^{61} -19.8151 q^{62} -7.37660 q^{64} -4.02797 q^{65} +5.14680 q^{67} +8.27152 q^{68} -0.949512 q^{70} +8.09127 q^{71} +8.75660 q^{73} +7.22839 q^{74} +10.3314 q^{76} +0.980649 q^{77} +7.29925 q^{79} +5.05548 q^{80} +15.4899 q^{82} +3.74978 q^{83} -5.24211 q^{85} +4.65232 q^{86} -0.377968 q^{88} -10.4120 q^{89} +1.29936 q^{91} -5.77409 q^{92} +16.9179 q^{94} -6.54755 q^{95} -16.2019 q^{97} -13.5591 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

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Database

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