LMFDB - Embedded newform 522.4.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [522,4,Mod(1,522)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(522, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("522.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 522 = 2 \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	522.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:	$30.7989970230$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.19816.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 42x - 54 $ x^3 - x^2 - 42*x - 54
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.39712$ of defining polynomial
Character	$\chi$	$=$	522.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} +3.28077 q^{5} +33.9461 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} +3.28077 q^{5} +33.9461 q^{7} -8.00000 q^{8} -6.56153 q^{10} -14.5759 q^{11} -86.5287 q^{13} -67.8922 q^{14} +16.0000 q^{16} -102.550 q^{17} -105.688 q^{19} +13.1231 q^{20} +29.1519 q^{22} +135.456 q^{23} -114.237 q^{25} +173.057 q^{26} +135.784 q^{28} -29.0000 q^{29} +223.883 q^{31} -32.0000 q^{32} +205.100 q^{34} +111.369 q^{35} -239.723 q^{37} +211.377 q^{38} -26.2461 q^{40} -219.331 q^{41} +18.9838 q^{43} -58.3037 q^{44} -270.911 q^{46} -147.922 q^{47} +809.338 q^{49} +228.473 q^{50} -346.115 q^{52} -613.202 q^{53} -47.8202 q^{55} -271.569 q^{56} +58.0000 q^{58} -184.206 q^{59} -13.6914 q^{61} -447.766 q^{62} +64.0000 q^{64} -283.880 q^{65} +328.733 q^{67} -410.199 q^{68} -222.738 q^{70} -5.15186 q^{71} -428.481 q^{73} +479.445 q^{74} -422.753 q^{76} -494.796 q^{77} -392.819 q^{79} +52.4922 q^{80} +438.661 q^{82} +454.172 q^{83} -336.442 q^{85} -37.9675 q^{86} +116.607 q^{88} +811.929 q^{89} -2937.31 q^{91} +541.822 q^{92} +295.844 q^{94} -346.738 q^{95} -11.3513 q^{97} -1618.68 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8}+O(q^{10}) $ 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8	$ 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26} + 96 q^{28} - 87 q^{29} - 86 q^{31} - 96 q^{32} - 132 q^{34} - 24 q^{35} - 42 q^{37} + 328 q^{38} + 160 q^{40} - 562 q^{41} + 18 q^{43} - 40 q^{44} - 408 q^{46} - 654 q^{47} + 539 q^{49} - 158 q^{50} - 16 q^{52} - 712 q^{53} + 142 q^{55} - 192 q^{56} + 174 q^{58} - 184 q^{59} + 322 q^{61} + 172 q^{62} + 192 q^{64} - 1494 q^{65} - 228 q^{67} + 264 q^{68} + 48 q^{70} + 52 q^{71} - 494 q^{73} + 84 q^{74} - 656 q^{76} - 872 q^{77} - 2110 q^{79} - 320 q^{80} + 1124 q^{82} + 288 q^{83} - 2704 q^{85} - 36 q^{86} + 80 q^{88} - 914 q^{89} - 2984 q^{91} + 816 q^{92} + 1308 q^{94} + 1900 q^{95} + 218 q^{97} - 1078 q^{98}+O(q^{100}) $ 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 + 40 * q^10 - 10 * q^11 - 4 * q^13 - 48 * q^14 + 48 * q^16 + 66 * q^17 - 164 * q^19 - 80 * q^20 + 20 * q^22 + 204 * q^23 + 79 * q^25 + 8 * q^26 + 96 * q^28 - 87 * q^29 - 86 * q^31 - 96 * q^32 - 132 * q^34 - 24 * q^35 - 42 * q^37 + 328 * q^38 + 160 * q^40 - 562 * q^41 + 18 * q^43 - 40 * q^44 - 408 * q^46 - 654 * q^47 + 539 * q^49 - 158 * q^50 - 16 * q^52 - 712 * q^53 + 142 * q^55 - 192 * q^56 + 174 * q^58 - 184 * q^59 + 322 * q^61 + 172 * q^62 + 192 * q^64 - 1494 * q^65 - 228 * q^67 + 264 * q^68 + 48 * q^70 + 52 * q^71 - 494 * q^73 + 84 * q^74 - 656 * q^76 - 872 * q^77 - 2110 * q^79 - 320 * q^80 + 1124 * q^82 + 288 * q^83 - 2704 * q^85 - 36 * q^86 + 80 * q^88 - 914 * q^89 - 2984 * q^91 + 816 * q^92 + 1308 * q^94 + 1900 * q^95 + 218 * q^97 - 1078 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	3.28077	0.293441	0.146720	−	0.989178i	$-0.453128\pi$
$5$	3.28077	0.293441	0.146720	+	0.989178i	$0.453128\pi$
$6$	0	0
$7$	33.9461	1.83292	0.916459	−	0.400130i	$-0.131035\pi$
$7$	33.9461	1.83292	0.916459	+	0.400130i	$0.131035\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−6.56153	−0.207494
$11$	−14.5759	−0.399528	−0.199764	−	0.979844i	$-0.564018\pi$
$11$	−14.5759	−0.399528	−0.199764	+	0.979844i	$0.564018\pi$
$12$	0	0
$13$	−86.5287	−1.84606	−0.923029	−	0.384730i	$-0.874294\pi$
$13$	−86.5287	−1.84606	−0.923029	+	0.384730i	$0.874294\pi$
$14$	−67.8922	−1.29607
$15$	0	0
$16$	16.0000	0.250000
$17$	−102.550	−1.46306	−0.731529	−	0.681810i	$-0.761193\pi$
$17$	−102.550	−1.46306	−0.731529	+	0.681810i	$0.761193\pi$
$18$	0	0
$19$	−105.688	−1.27613	−0.638067	−	0.769981i	$-0.720266\pi$
$19$	−105.688	−1.27613	−0.638067	+	0.769981i	$0.720266\pi$
$20$	13.1231	0.146720
$21$	0	0
$22$	29.1519	0.282509
$23$	135.456	1.22802	0.614010	−	0.789299i	$-0.289556\pi$
$23$	135.456	1.22802	0.614010	+	0.789299i	$0.289556\pi$
$24$	0	0
$25$	−114.237	−0.913893
$26$	173.057	1.30536
$27$	0	0
$28$	135.784	0.916459
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	223.883	1.29712	0.648558	−	0.761165i	$-0.275372\pi$
$31$	223.883	1.29712	0.648558	+	0.761165i	$0.275372\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	205.100	1.03454
$35$	111.369	0.537852
$36$	0	0
$37$	−239.723	−1.06514	−0.532569	−	0.846386i	$-0.678773\pi$
$37$	−239.723	−1.06514	−0.532569	+	0.846386i	$0.678773\pi$
$38$	211.377	0.902363
$39$	0	0
$40$	−26.2461	−0.103747
$41$	−219.331	−0.835456	−0.417728	−	0.908572i	$-0.637173\pi$
$41$	−219.331	−0.835456	−0.417728	+	0.908572i	$0.637173\pi$
$42$	0	0
$43$	18.9838	0.0673255	0.0336627	−	0.999433i	$-0.489283\pi$
$43$	18.9838	0.0673255	0.0336627	+	0.999433i	$0.489283\pi$
$44$	−58.3037	−0.199764
$45$	0	0
$46$	−270.911	−0.868341
$47$	−147.922	−0.459077	−0.229539	−	0.973300i	$-0.573722\pi$
$47$	−147.922	−0.459077	−0.229539	+	0.973300i	$0.573722\pi$
$48$	0	0
$49$	809.338	2.35959
$50$	228.473	0.646220
$51$	0	0
$52$	−346.115	−0.923029
$53$	−613.202	−1.58924	−0.794621	−	0.607106i	$-0.792330\pi$
$53$	−613.202	−1.58924	−0.794621	+	0.607106i	$0.792330\pi$
$54$	0	0
$55$	−47.8202	−0.117238
$56$	−271.569	−0.648034
$57$	0	0
$58$	58.0000	0.131306
$59$	−184.206	−0.406468	−0.203234	−	0.979130i	$-0.565145\pi$
$59$	−184.206	−0.406468	−0.203234	+	0.979130i	$0.565145\pi$
$60$	0	0
$61$	−13.6914	−0.0287377	−0.0143689	−	0.999897i	$-0.504574\pi$
$61$	−13.6914	−0.0287377	−0.0143689	+	0.999897i	$0.504574\pi$
$62$	−447.766	−0.917200
$63$	0	0
$64$	64.0000	0.125000
$65$	−283.880	−0.541708
$66$	0	0
$67$	328.733	0.599421	0.299710	−	0.954030i	$-0.403110\pi$
$67$	328.733	0.599421	0.299710	+	0.954030i	$0.403110\pi$
$68$	−410.199	−0.731529
$69$	0	0
$70$	−222.738	−0.380319
$71$	−5.15186	−0.00861145	−0.00430573	−	0.999991i	$-0.501371\pi$
$71$	−5.15186	−0.00861145	−0.00430573	+	0.999991i	$0.501371\pi$
$72$	0	0
$73$	−428.481	−0.686984	−0.343492	−	0.939156i	$-0.611610\pi$
$73$	−428.481	−0.686984	−0.343492	+	0.939156i	$0.611610\pi$
$74$	479.445	0.753167
$75$	0	0
$76$	−422.753	−0.638067
$77$	−494.796	−0.732302
$78$	0	0
$79$	−392.819	−0.559437	−0.279719	−	0.960082i	$-0.590241\pi$
$79$	−392.819	−0.559437	−0.279719	+	0.960082i	$0.590241\pi$
$80$	52.4922	0.0733601
$81$	0	0
$82$	438.661	0.590757
$83$	454.172	0.600624	0.300312	−	0.953841i	$-0.402909\pi$
$83$	454.172	0.600624	0.300312	+	0.953841i	$0.402909\pi$
$84$	0	0
$85$	−336.442	−0.429321
$86$	−37.9675	−0.0476063
$87$	0	0
$88$	116.607	0.141254
$89$	811.929	0.967015	0.483507	−	0.875340i	$-0.339363\pi$
$89$	811.929	0.967015	0.483507	+	0.875340i	$0.339363\pi$
$90$	0	0
$91$	−2937.31	−3.38367
$92$	541.822	0.614010
$93$	0	0
$94$	295.844	0.324617
$95$	−346.738	−0.374470
$96$	0	0
$97$	−11.3513	−0.0118820	−0.00594098	−	0.999982i	$-0.501891\pi$
$97$	−11.3513	−0.0118820	−0.00594098	+	0.999982i	$0.501891\pi$
$98$	−1618.68	−1.66848
$99$	0	0
$100$	−456.946	−0.456946
$101$	−1548.46	−1.52552	−0.762762	−	0.646679i	$-0.776157\pi$
$101$	−1548.46	−1.52552	−0.762762	+	0.646679i	$0.776157\pi$
$102$	0	0
$103$	861.888	0.824508	0.412254	−	0.911069i	$-0.364742\pi$
$103$	861.888	0.824508	0.412254	+	0.911069i	$0.364742\pi$
$104$	692.230	0.652680
$105$	0	0
$106$	1226.40	1.12376
$107$	1374.40	1.24176	0.620880	−	0.783906i	$-0.286775\pi$
$107$	1374.40	1.24176	0.620880	+	0.783906i	$0.286775\pi$
$108$	0	0
$109$	−467.521	−0.410829	−0.205415	−	0.978675i	$-0.565854\pi$
$109$	−467.521	−0.410829	−0.205415	+	0.978675i	$0.565854\pi$
$110$	95.6404	0.0828996
$111$	0	0
$112$	543.138	0.458229
$113$	−655.223	−0.545471	−0.272736	−	0.962089i	$-0.587928\pi$
$113$	−655.223	−0.545471	−0.272736	+	0.962089i	$0.587928\pi$
$114$	0	0
$115$	444.398	0.360351
$116$	−116.000	−0.0928477
$117$	0	0
$118$	368.412	0.287416
$119$	−3481.17	−2.68166
$120$	0	0
$121$	−1118.54	−0.840377
$122$	27.3828	0.0203207
$123$	0	0
$124$	895.533	0.648558
$125$	−784.879	−0.561614
$126$	0	0
$127$	830.607	0.580350	0.290175	−	0.956974i	$-0.406286\pi$
$127$	830.607	0.580350	0.290175	+	0.956974i	$0.406286\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	567.761	0.383046
$131$	−783.195	−0.522351	−0.261176	−	0.965291i	$-0.584110\pi$
$131$	−783.195	−0.522351	−0.261176	+	0.965291i	$0.584110\pi$
$132$	0	0
$133$	−3587.71	−2.33905
$134$	−657.467	−0.423854
$135$	0	0
$136$	820.399	0.517269
$137$	1362.28	0.849545	0.424773	−	0.905300i	$-0.360354\pi$
$137$	1362.28	0.849545	0.424773	+	0.905300i	$0.360354\pi$
$138$	0	0
$139$	−874.798	−0.533809	−0.266904	−	0.963723i	$-0.586001\pi$
$139$	−874.798	−0.533809	−0.266904	+	0.963723i	$0.586001\pi$
$140$	445.477	0.268926
$141$	0	0
$142$	10.3037	0.00608922
$143$	1261.24	0.737552
$144$	0	0
$145$	−95.1422	−0.0544905
$146$	856.961	0.485771
$147$	0	0
$148$	−958.890	−0.532569
$149$	−1938.20	−1.06566	−0.532830	−	0.846222i	$-0.678871\pi$
$149$	−1938.20	−1.06566	−0.532830	+	0.846222i	$0.678871\pi$
$150$	0	0
$151$	−1460.33	−0.787018	−0.393509	−	0.919321i	$-0.628739\pi$
$151$	−1460.33	−0.787018	−0.393509	+	0.919321i	$0.628739\pi$
$152$	845.506	0.451182
$153$	0	0
$154$	989.592	0.517816
$155$	734.508	0.380627
$156$	0	0
$157$	421.035	0.214027	0.107014	−	0.994258i	$-0.465871\pi$
$157$	421.035	0.214027	0.107014	+	0.994258i	$0.465871\pi$
$158$	785.637	0.395582
$159$	0	0
$160$	−104.984	−0.0518735
$161$	4598.19	2.25086
$162$	0	0
$163$	−796.182	−0.382588	−0.191294	−	0.981533i	$-0.561268\pi$
$163$	−796.182	−0.382588	−0.191294	+	0.981533i	$0.561268\pi$
$164$	−877.323	−0.417728
$165$	0	0
$166$	−908.344	−0.424706
$167$	−3625.72	−1.68004	−0.840020	−	0.542556i	$-0.817457\pi$
$167$	−3625.72	−1.68004	−0.840020	+	0.542556i	$0.817457\pi$
$168$	0	0
$169$	5290.22	2.40793
$170$	672.884	0.303576
$171$	0	0
$172$	75.9350	0.0336627
$173$	−587.037	−0.257986	−0.128993	−	0.991645i	$-0.541175\pi$
$173$	−587.037	−0.257986	−0.128993	+	0.991645i	$0.541175\pi$
$174$	0	0
$175$	−3877.89	−1.67509
$176$	−233.215	−0.0998820
$177$	0	0
$178$	−1623.86	−0.683783
$179$	908.839	0.379496	0.189748	−	0.981833i	$-0.439233\pi$
$179$	908.839	0.379496	0.189748	+	0.981833i	$0.439233\pi$
$180$	0	0
$181$	2139.11	0.878447	0.439223	−	0.898378i	$-0.355254\pi$
$181$	2139.11	0.878447	0.439223	+	0.898378i	$0.355254\pi$
$182$	5874.63	2.39262
$183$	0	0
$184$	−1083.64	−0.434170
$185$	−786.473	−0.312555
$186$	0	0
$187$	1494.76	0.584533
$188$	−591.688	−0.229539
$189$	0	0
$190$	693.477	0.264790
$191$	1715.31	0.649819	0.324910	−	0.945745i	$-0.394666\pi$
$191$	1715.31	0.649819	0.324910	+	0.945745i	$0.394666\pi$
$192$	0	0
$193$	−2509.88	−0.936090	−0.468045	−	0.883705i	$-0.655042\pi$
$193$	−2509.88	−0.936090	−0.468045	+	0.883705i	$0.655042\pi$
$194$	22.7026	0.00840181
$195$	0	0
$196$	3237.35	1.17979
$197$	688.689	0.249071	0.124536	−	0.992215i	$-0.460256\pi$
$197$	688.689	0.249071	0.124536	+	0.992215i	$0.460256\pi$
$198$	0	0
$199$	1106.28	0.394079	0.197040	−	0.980396i	$-0.436867\pi$
$199$	1106.28	0.394079	0.197040	+	0.980396i	$0.436867\pi$
$200$	913.893	0.323110
$201$	0	0
$202$	3096.93	1.07871
$203$	−984.437	−0.340364
$204$	0	0
$205$	−719.572	−0.245157
$206$	−1723.78	−0.583015
$207$	0	0
$208$	−1384.46	−0.461514
$209$	1540.51	0.509851
$210$	0	0
$211$	−2094.08	−0.683234	−0.341617	−	0.939839i	$-0.610975\pi$
$211$	−2094.08	−0.683234	−0.341617	+	0.939839i	$0.610975\pi$
$212$	−2452.81	−0.794621
$213$	0	0
$214$	−2748.80	−0.878057
$215$	62.2812	0.0197560
$216$	0	0
$217$	7599.96	2.37751
$218$	935.042	0.290500
$219$	0	0
$220$	−191.281	−0.0586189
$221$	8873.51	2.70089
$222$	0	0
$223$	6033.11	1.81169	0.905845	−	0.423610i	$-0.139237\pi$
$223$	6033.11	1.81169	0.905845	+	0.423610i	$0.139237\pi$
$224$	−1086.28	−0.324017
$225$	0	0
$226$	1310.45	0.385706
$227$	−189.492	−0.0554053	−0.0277027	−	0.999616i	$-0.508819\pi$
$227$	−189.492	−0.0554053	−0.0277027	+	0.999616i	$0.508819\pi$
$228$	0	0
$229$	−2347.80	−0.677498	−0.338749	−	0.940877i	$-0.610004\pi$
$229$	−2347.80	−0.677498	−0.338749	+	0.940877i	$0.610004\pi$
$230$	−888.796	−0.254806
$231$	0	0
$232$	232.000	0.0656532
$233$	−551.777	−0.155142	−0.0775710	−	0.996987i	$-0.524716\pi$
$233$	−551.777	−0.155142	−0.0775710	+	0.996987i	$0.524716\pi$
$234$	0	0
$235$	−485.297	−0.134712
$236$	−736.825	−0.203234
$237$	0	0
$238$	6962.33	1.89622
$239$	4533.45	1.22696	0.613482	−	0.789708i	$-0.289768\pi$
$239$	4533.45	1.22696	0.613482	+	0.789708i	$0.289768\pi$
$240$	0	0
$241$	1148.28	0.306917	0.153458	−	0.988155i	$-0.450959\pi$
$241$	1148.28	0.306917	0.153458	+	0.988155i	$0.450959\pi$
$242$	2237.08	0.594237
$243$	0	0
$244$	−54.7655	−0.0143689
$245$	2655.25	0.692398
$246$	0	0
$247$	9145.07	2.35582
$248$	−1791.07	−0.458600
$249$	0	0
$250$	1569.76	0.397121
$251$	7236.42	1.81976	0.909878	−	0.414876i	$-0.136175\pi$
$251$	7236.42	1.81976	0.909878	+	0.414876i	$0.136175\pi$
$252$	0	0
$253$	−1974.39	−0.490628
$254$	−1661.21	−0.410369
$255$	0	0
$256$	256.000	0.0625000
$257$	−5051.89	−1.22618	−0.613089	−	0.790013i	$-0.710073\pi$
$257$	−5051.89	−1.22618	−0.613089	+	0.790013i	$0.710073\pi$
$258$	0	0
$259$	−8137.64	−1.95231
$260$	−1135.52	−0.270854
$261$	0	0
$262$	1566.39	0.369358
$263$	−6368.81	−1.49322	−0.746611	−	0.665260i	$-0.768320\pi$
$263$	−6368.81	−1.49322	−0.746611	+	0.665260i	$0.768320\pi$
$264$	0	0
$265$	−2011.77	−0.466348
$266$	7175.41	1.65396
$267$	0	0
$268$	1314.93	0.299710
$269$	−3299.89	−0.747948	−0.373974	−	0.927439i	$-0.622005\pi$
$269$	−3299.89	−0.747948	−0.373974	+	0.927439i	$0.622005\pi$
$270$	0	0
$271$	−2235.21	−0.501031	−0.250515	−	0.968113i	$-0.580600\pi$
$271$	−2235.21	−0.501031	−0.250515	+	0.968113i	$0.580600\pi$
$272$	−1640.80	−0.365765
$273$	0	0
$274$	−2724.57	−0.600719
$275$	1665.10	0.365126
$276$	0	0
$277$	−2354.61	−0.510739	−0.255369	−	0.966844i	$-0.582197\pi$
$277$	−2354.61	−0.510739	−0.255369	+	0.966844i	$0.582197\pi$
$278$	1749.60	0.377460
$279$	0	0
$280$	−890.953	−0.190159
$281$	6858.62	1.45605	0.728027	−	0.685549i	$-0.240438\pi$
$281$	6858.62	1.45605	0.728027	+	0.685549i	$0.240438\pi$
$282$	0	0
$283$	−3031.29	−0.636719	−0.318360	−	0.947970i	$-0.603132\pi$
$283$	−3031.29	−0.636719	−0.318360	+	0.947970i	$0.603132\pi$
$284$	−20.6074	−0.00430573
$285$	0	0
$286$	−2522.47	−0.521528
$287$	−7445.42	−1.53132
$288$	0	0
$289$	5603.47	1.14054
$290$	190.284	0.0385306
$291$	0	0
$292$	−1713.92	−0.343492
$293$	−1921.35	−0.383094	−0.191547	−	0.981483i	$-0.561350\pi$
$293$	−1921.35	−0.383094	−0.191547	+	0.981483i	$0.561350\pi$
$294$	0	0
$295$	−604.337	−0.119274
$296$	1917.78	0.376583
$297$	0	0
$298$	3876.39	0.753535
$299$	−11720.8	−2.26699
$300$	0	0
$301$	644.424	0.123402
$302$	2920.65	0.556506
$303$	0	0
$304$	−1691.01	−0.319034
$305$	−44.9182	−0.00843282
$306$	0	0
$307$	−8734.92	−1.62387	−0.811935	−	0.583747i	$-0.801586\pi$
$307$	−8734.92	−1.62387	−0.811935	+	0.583747i	$0.801586\pi$
$308$	−1979.18	−0.366151
$309$	0	0
$310$	−1469.02	−0.269144
$311$	198.321	0.0361599	0.0180800	−	0.999837i	$-0.494245\pi$
$311$	198.321	0.0361599	0.0180800	+	0.999837i	$0.494245\pi$
$312$	0	0
$313$	−3759.23	−0.678864	−0.339432	−	0.940631i	$-0.610235\pi$
$313$	−3759.23	−0.678864	−0.339432	+	0.940631i	$0.610235\pi$
$314$	−842.070	−0.151340
$315$	0	0
$316$	−1571.27	−0.279719
$317$	5543.84	0.982249	0.491124	−	0.871090i	$-0.336586\pi$
$317$	5543.84	0.982249	0.491124	+	0.871090i	$0.336586\pi$
$318$	0	0
$319$	422.702	0.0741905
$320$	209.969	0.0366801
$321$	0	0
$322$	−9196.38	−1.59160
$323$	10838.3	1.86706
$324$	0	0
$325$	9884.75	1.68710
$326$	1592.36	0.270530
$327$	0	0
$328$	1754.65	0.295378
$329$	−5021.37	−0.841451
$330$	0	0
$331$	−2875.96	−0.477574	−0.238787	−	0.971072i	$-0.576750\pi$
$331$	−2875.96	−0.477574	−0.238787	+	0.971072i	$0.576750\pi$
$332$	1816.69	0.300312
$333$	0	0
$334$	7251.44	1.18797
$335$	1078.50	0.175894
$336$	0	0
$337$	8141.60	1.31603	0.658014	−	0.753006i	$-0.271397\pi$
$337$	8141.60	1.31603	0.658014	+	0.753006i	$0.271397\pi$
$338$	−10580.4	−1.70266
$339$	0	0
$340$	−1345.77	−0.214660
$341$	−3263.31	−0.518234
$342$	0	0
$343$	15830.3	2.49201
$344$	−151.870	−0.0238031
$345$	0	0
$346$	1174.07	0.182424
$347$	8941.88	1.38336	0.691679	−	0.722205i	$-0.256871\pi$
$347$	8941.88	1.38336	0.691679	+	0.722205i	$0.256871\pi$
$348$	0	0
$349$	−3061.39	−0.469548	−0.234774	−	0.972050i	$-0.575435\pi$
$349$	−3061.39	−0.469548	−0.234774	+	0.972050i	$0.575435\pi$
$350$	7755.77	1.18447
$351$	0	0
$352$	466.430	0.0706272
$353$	−2871.64	−0.432979	−0.216490	−	0.976285i	$-0.569461\pi$
$353$	−2871.64	−0.432979	−0.216490	+	0.976285i	$0.569461\pi$
$354$	0	0
$355$	−16.9020	−0.00252695
$356$	3247.72	0.483507
$357$	0	0
$358$	−1817.68	−0.268344
$359$	9773.24	1.43680	0.718401	−	0.695629i	$-0.244874\pi$
$359$	9773.24	1.43680	0.718401	+	0.695629i	$0.244874\pi$
$360$	0	0
$361$	4311.01	0.628519
$362$	−4278.22	−0.621156
$363$	0	0
$364$	−11749.3	−1.69184
$365$	−1405.74	−0.201589
$366$	0	0
$367$	−6064.61	−0.862588	−0.431294	−	0.902211i	$-0.641943\pi$
$367$	−6064.61	−0.862588	−0.431294	+	0.902211i	$0.641943\pi$
$368$	2167.29	0.307005
$369$	0	0
$370$	1572.95	0.221010
$371$	−20815.8	−2.91295
$372$	0	0
$373$	2684.28	0.372618	0.186309	−	0.982491i	$-0.440347\pi$
$373$	2684.28	0.372618	0.186309	+	0.982491i	$0.440347\pi$
$374$	−2989.52	−0.413327
$375$	0	0
$376$	1183.38	0.162308
$377$	2509.33	0.342804
$378$	0	0
$379$	2069.54	0.280489	0.140244	−	0.990117i	$-0.455211\pi$
$379$	2069.54	0.280489	0.140244	+	0.990117i	$0.455211\pi$
$380$	−1386.95	−0.187235
$381$	0	0
$382$	−3430.62	−0.459492
$383$	−8550.46	−1.14075	−0.570376	−	0.821384i	$-0.693203\pi$
$383$	−8550.46	−1.14075	−0.570376	+	0.821384i	$0.693203\pi$
$384$	0	0
$385$	−1623.31	−0.214887
$386$	5019.77	0.661916
$387$	0	0
$388$	−45.4052	−0.00594098
$389$	10974.4	1.43040	0.715199	−	0.698921i	$-0.246336\pi$
$389$	10974.4	1.43040	0.715199	+	0.698921i	$0.246336\pi$
$390$	0	0
$391$	−13890.9	−1.79666
$392$	−6474.70	−0.834239
$393$	0	0
$394$	−1377.38	−0.176120
$395$	−1288.75	−0.164162
$396$	0	0
$397$	973.767	0.123103	0.0615516	−	0.998104i	$-0.480395\pi$
$397$	973.767	0.123103	0.0615516	+	0.998104i	$0.480395\pi$
$398$	−2212.55	−0.278656
$399$	0	0
$400$	−1827.79	−0.228473
$401$	−2196.32	−0.273514	−0.136757	−	0.990605i	$-0.543668\pi$
$401$	−2196.32	−0.273514	−0.136757	+	0.990605i	$0.543668\pi$
$402$	0	0
$403$	−19372.3	−2.39455
$404$	−6193.86	−0.762762
$405$	0	0
$406$	1968.87	0.240674
$407$	3494.18	0.425553
$408$	0	0
$409$	7847.40	0.948726	0.474363	−	0.880329i	$-0.342678\pi$
$409$	7847.40	0.948726	0.474363	+	0.880329i	$0.342678\pi$
$410$	1439.14	0.173352
$411$	0	0
$412$	3447.55	0.412254
$413$	−6253.08	−0.745022
$414$	0	0
$415$	1490.03	0.176248
$416$	2768.92	0.326340
$417$	0	0
$418$	−3081.01	−0.360519
$419$	6943.21	0.809541	0.404771	−	0.914418i	$-0.367351\pi$
$419$	6943.21	0.809541	0.404771	+	0.914418i	$0.367351\pi$
$420$	0	0
$421$	−10527.3	−1.21869	−0.609343	−	0.792907i	$-0.708567\pi$
$421$	−10527.3	−1.21869	−0.609343	+	0.792907i	$0.708567\pi$
$422$	4188.16	0.483120
$423$	0	0
$424$	4905.62	0.561882
$425$	11714.9	1.33708
$426$	0	0
$427$	−464.769	−0.0526739
$428$	5497.60	0.620880
$429$	0	0
$430$	−124.562	−0.0139696
$431$	1735.58	0.193967	0.0969836	−	0.995286i	$-0.469081\pi$
$431$	1735.58	0.193967	0.0969836	+	0.995286i	$0.469081\pi$
$432$	0	0
$433$	−9380.15	−1.04107	−0.520533	−	0.853842i	$-0.674267\pi$
$433$	−9380.15	−1.04107	−0.520533	+	0.853842i	$0.674267\pi$
$434$	−15199.9	−1.68115
$435$	0	0
$436$	−1870.08	−0.205415
$437$	−14316.1	−1.56712
$438$	0	0
$439$	−4099.58	−0.445700	−0.222850	−	0.974853i	$-0.571536\pi$
$439$	−4099.58	−0.445700	−0.222850	+	0.974853i	$0.571536\pi$
$440$	382.562	0.0414498
$441$	0	0
$442$	−17747.0	−1.90982
$443$	−5718.20	−0.613273	−0.306636	−	0.951827i	$-0.599204\pi$
$443$	−5718.20	−0.613273	−0.306636	+	0.951827i	$0.599204\pi$
$444$	0	0
$445$	2663.75	0.283761
$446$	−12066.2	−1.28106
$447$	0	0
$448$	2172.55	0.229115
$449$	−2474.77	−0.260115	−0.130057	−	0.991506i	$-0.541516\pi$
$449$	−2474.77	−0.260115	−0.130057	+	0.991506i	$0.541516\pi$
$450$	0	0
$451$	3196.95	0.333788
$452$	−2620.89	−0.272736
$453$	0	0
$454$	378.983	0.0391775
$455$	−9636.63	−0.992906
$456$	0	0
$457$	10932.4	1.11903	0.559513	−	0.828822i	$-0.310988\pi$
$457$	10932.4	1.11903	0.559513	+	0.828822i	$0.310988\pi$
$458$	4695.60	0.479063
$459$	0	0
$460$	1777.59	0.180175
$461$	−16589.3	−1.67602	−0.838008	−	0.545659i	$-0.816280\pi$
$461$	−16589.3	−1.67602	−0.838008	+	0.545659i	$0.816280\pi$
$462$	0	0
$463$	894.244	0.0897604	0.0448802	−	0.998992i	$-0.485709\pi$
$463$	894.244	0.0897604	0.0448802	+	0.998992i	$0.485709\pi$
$464$	−464.000	−0.0464238
$465$	0	0
$466$	1103.55	0.109702
$467$	8011.09	0.793809	0.396905	−	0.917860i	$-0.370084\pi$
$467$	8011.09	0.793809	0.396905	+	0.917860i	$0.370084\pi$
$468$	0	0
$469$	11159.2	1.09869
$470$	970.595	0.0952557
$471$	0	0
$472$	1473.65	0.143708
$473$	−276.706	−0.0268984
$474$	0	0
$475$	12073.5	1.16625
$476$	−13924.7	−1.34083
$477$	0	0
$478$	−9066.90	−0.867595
$479$	14558.7	1.38873	0.694366	−	0.719622i	$-0.255685\pi$
$479$	14558.7	1.38873	0.694366	+	0.719622i	$0.255685\pi$
$480$	0	0
$481$	20742.9	1.96631
$482$	−2296.55	−0.217023
$483$	0	0
$484$	−4474.17	−0.420189
$485$	−37.2409	−0.00348665
$486$	0	0
$487$	−17666.8	−1.64386	−0.821928	−	0.569591i	$-0.807102\pi$
$487$	−17666.8	−1.64386	−0.821928	+	0.569591i	$0.807102\pi$
$488$	109.531	0.0101603
$489$	0	0
$490$	−5310.49	−0.489599
$491$	4904.36	0.450775	0.225388	−	0.974269i	$-0.427635\pi$
$491$	4904.36	0.450775	0.225388	+	0.974269i	$0.427635\pi$
$492$	0	0
$493$	2973.95	0.271683
$494$	−18290.1	−1.66582
$495$	0	0
$496$	3582.13	0.324279
$497$	−174.886	−0.0157841
$498$	0	0
$499$	2496.99	0.224009	0.112005	−	0.993708i	$-0.464273\pi$
$499$	2496.99	0.224009	0.112005	+	0.993708i	$0.464273\pi$
$500$	−3139.52	−0.280807
$501$	0	0
$502$	−14472.8	−1.28676
$503$	968.191	0.0858241	0.0429120	−	0.999079i	$-0.486336\pi$
$503$	968.191	0.0858241	0.0429120	+	0.999079i	$0.486336\pi$
$504$	0	0
$505$	−5080.15	−0.447651
$506$	3948.78	0.346926
$507$	0	0
$508$	3322.43	0.290175
$509$	−3379.07	−0.294253	−0.147126	−	0.989118i	$-0.547002\pi$
$509$	−3379.07	−0.294253	−0.147126	+	0.989118i	$0.547002\pi$
$510$	0	0
$511$	−14545.2	−1.25919
$512$	−512.000	−0.0441942
$513$	0	0
$514$	10103.8	0.867039
$515$	2827.65	0.241944
$516$	0	0
$517$	2156.10	0.183414
$518$	16275.3	1.38049
$519$	0	0
$520$	2271.04	0.191523
$521$	−10941.9	−0.920098	−0.460049	−	0.887893i	$-0.652168\pi$
$521$	−10941.9	−0.920098	−0.460049	+	0.887893i	$0.652168\pi$
$522$	0	0
$523$	8947.20	0.748057	0.374029	−	0.927417i	$-0.377976\pi$
$523$	8947.20	0.748057	0.374029	+	0.927417i	$0.377976\pi$
$524$	−3132.78	−0.261176
$525$	0	0
$526$	12737.6	1.05587
$527$	−22959.2	−1.89776
$528$	0	0
$529$	6181.21	0.508031
$530$	4023.54	0.329758
$531$	0	0
$532$	−14350.8	−1.16952
$533$	18978.4	1.54230
$534$	0	0
$535$	4509.09	0.364383
$536$	−2629.87	−0.211927
$537$	0	0
$538$	6599.79	0.528879
$539$	−11796.8	−0.942720
$540$	0	0
$541$	9008.10	0.715875	0.357938	−	0.933746i	$-0.383480\pi$
$541$	9008.10	0.715875	0.357938	+	0.933746i	$0.383480\pi$
$542$	4470.42	0.354282
$543$	0	0
$544$	3281.59	0.258635
$545$	−1533.83	−0.120554
$546$	0	0
$547$	12980.3	1.01462	0.507312	−	0.861763i	$-0.330639\pi$
$547$	12980.3	1.01462	0.507312	+	0.861763i	$0.330639\pi$
$548$	5449.13	0.424773
$549$	0	0
$550$	−3330.21	−0.258183
$551$	3064.96	0.236972
$552$	0	0
$553$	−13334.7	−1.02540
$554$	4709.21	0.361147
$555$	0	0
$556$	−3499.19	−0.266904
$557$	8777.82	0.667734	0.333867	−	0.942620i	$-0.391646\pi$
$557$	8777.82	0.667734	0.333867	+	0.942620i	$0.391646\pi$
$558$	0	0
$559$	−1642.64	−0.124287
$560$	1781.91	0.134463
$561$	0	0
$562$	−13717.2	−1.02959
$563$	−3734.91	−0.279588	−0.139794	−	0.990181i	$-0.544644\pi$
$563$	−3734.91	−0.279588	−0.139794	+	0.990181i	$0.544644\pi$
$564$	0	0
$565$	−2149.63	−0.160063
$566$	6062.58	0.450229
$567$	0	0
$568$	41.2149	0.00304461
$569$	14373.7	1.05901	0.529504	−	0.848307i	$-0.322378\pi$
$569$	14373.7	1.05901	0.529504	+	0.848307i	$0.322378\pi$
$570$	0	0
$571$	8147.54	0.597135	0.298567	−	0.954389i	$-0.403491\pi$
$571$	8147.54	0.597135	0.298567	+	0.954389i	$0.403491\pi$
$572$	5044.95	0.368776
$573$	0	0
$574$	14890.8	1.08281
$575$	−15474.0	−1.12228
$576$	0	0
$577$	9438.43	0.680982	0.340491	−	0.940248i	$-0.389407\pi$
$577$	9438.43	0.680982	0.340491	+	0.940248i	$0.389407\pi$
$578$	−11206.9	−0.806483
$579$	0	0
$580$	−380.569	−0.0272453
$581$	15417.4	1.10089
$582$	0	0
$583$	8937.99	0.634946
$584$	3427.84	0.242886
$585$	0	0
$586$	3842.71	0.270889
$587$	3498.97	0.246027	0.123014	−	0.992405i	$-0.460744\pi$
$587$	3498.97	0.246027	0.123014	+	0.992405i	$0.460744\pi$
$588$	0	0
$589$	−23661.8	−1.65530
$590$	1208.67	0.0843396
$591$	0	0
$592$	−3835.56	−0.266285
$593$	−7383.92	−0.511335	−0.255667	−	0.966765i	$-0.582295\pi$
$593$	−7383.92	−0.511335	−0.255667	+	0.966765i	$0.582295\pi$
$594$	0	0
$595$	−11420.9	−0.786909
$596$	−7752.79	−0.532830
$597$	0	0
$598$	23441.6	1.60301
$599$	−14673.8	−1.00093	−0.500463	−	0.865758i	$-0.666837\pi$
$599$	−14673.8	−1.00093	−0.500463	+	0.865758i	$0.666837\pi$
$600$	0	0
$601$	−10950.2	−0.743209	−0.371605	−	0.928391i	$-0.621192\pi$
$601$	−10950.2	−0.743209	−0.371605	+	0.928391i	$0.621192\pi$
$602$	−1288.85	−0.0872584
$603$	0	0
$604$	−5841.31	−0.393509
$605$	−3669.67	−0.246601
$606$	0	0
$607$	17469.1	1.16812	0.584059	−	0.811712i	$-0.301464\pi$
$607$	17469.1	1.16812	0.584059	+	0.811712i	$0.301464\pi$
$608$	3382.03	0.225591
$609$	0	0
$610$	89.8364	0.00596290
$611$	12799.5	0.847484
$612$	0	0
$613$	5866.70	0.386548	0.193274	−	0.981145i	$-0.438089\pi$
$613$	5866.70	0.386548	0.193274	+	0.981145i	$0.438089\pi$
$614$	17469.8	1.14825
$615$	0	0
$616$	3958.37	0.258908
$617$	14383.8	0.938528	0.469264	−	0.883058i	$-0.344519\pi$
$617$	14383.8	0.938528	0.469264	+	0.883058i	$0.344519\pi$
$618$	0	0
$619$	28631.0	1.85909	0.929545	−	0.368710i	$-0.120200\pi$
$619$	28631.0	1.85909	0.929545	+	0.368710i	$0.120200\pi$
$620$	2938.03	0.190313
$621$	0	0
$622$	−396.642	−0.0255689
$623$	27561.8	1.77246
$624$	0	0
$625$	11704.6	0.749092
$626$	7518.47	0.480029
$627$	0	0
$628$	1684.14	0.107014
$629$	24583.5	1.55836
$630$	0	0
$631$	−2860.48	−0.180466	−0.0902329	−	0.995921i	$-0.528761\pi$
$631$	−2860.48	−0.180466	−0.0902329	+	0.995921i	$0.528761\pi$
$632$	3142.55	0.197791
$633$	0	0
$634$	−11087.7	−0.694555
$635$	2725.03	0.170298
$636$	0	0
$637$	−70031.0	−4.35593
$638$	−845.404	−0.0524606
$639$	0	0
$640$	−419.938	−0.0259367
$641$	−6003.37	−0.369920	−0.184960	−	0.982746i	$-0.559216\pi$
$641$	−6003.37	−0.369920	−0.184960	+	0.982746i	$0.559216\pi$
$642$	0	0
$643$	6690.61	0.410345	0.205173	−	0.978726i	$-0.434224\pi$
$643$	6690.61	0.410345	0.205173	+	0.978726i	$0.434224\pi$
$644$	18392.8	1.12543
$645$	0	0
$646$	−21676.6	−1.32021
$647$	8697.55	0.528495	0.264247	−	0.964455i	$-0.414876\pi$
$647$	8697.55	0.528495	0.264247	+	0.964455i	$0.414876\pi$
$648$	0	0
$649$	2684.98	0.162395
$650$	−19769.5	−1.19296
$651$	0	0
$652$	−3184.73	−0.191294
$653$	−21260.9	−1.27413	−0.637063	−	0.770812i	$-0.719851\pi$
$653$	−21260.9	−1.27413	−0.637063	+	0.770812i	$0.719851\pi$
$654$	0	0
$655$	−2569.48	−0.153279
$656$	−3509.29	−0.208864
$657$	0	0
$658$	10042.7	0.594996
$659$	−11717.3	−0.692625	−0.346313	−	0.938119i	$-0.612566\pi$
$659$	−11717.3	−0.692625	−0.346313	+	0.938119i	$0.612566\pi$
$660$	0	0
$661$	27523.1	1.61955	0.809775	−	0.586740i	$-0.199589\pi$
$661$	27523.1	1.61955	0.809775	+	0.586740i	$0.199589\pi$
$662$	5751.92	0.337696
$663$	0	0
$664$	−3633.37	−0.212353
$665$	−11770.4	−0.686372
$666$	0	0
$667$	−3928.21	−0.228037
$668$	−14502.9	−0.840020
$669$	0	0
$670$	−2156.99	−0.124376
$671$	199.565	0.0114815
$672$	0	0
$673$	−30200.5	−1.72978	−0.864891	−	0.501960i	$-0.832612\pi$
$673$	−30200.5	−1.72978	−0.864891	+	0.501960i	$0.832612\pi$
$674$	−16283.2	−0.930572
$675$	0	0
$676$	21160.9	1.20396
$677$	−19986.2	−1.13461	−0.567306	−	0.823507i	$-0.692014\pi$
$677$	−19986.2	−1.13461	−0.567306	+	0.823507i	$0.692014\pi$
$678$	0	0
$679$	−385.332	−0.0217786
$680$	2691.54	0.151788
$681$	0	0
$682$	6526.61	0.366447
$683$	6037.11	0.338219	0.169110	−	0.985597i	$-0.445911\pi$
$683$	6037.11	0.338219	0.169110	+	0.985597i	$0.445911\pi$
$684$	0	0
$685$	4469.33	0.249291
$686$	−31660.7	−1.76211
$687$	0	0
$688$	303.740	0.0168314
$689$	53059.6	2.93383
$690$	0	0
$691$	−28078.2	−1.54579	−0.772897	−	0.634531i	$-0.781193\pi$
$691$	−28078.2	−1.54579	−0.772897	+	0.634531i	$0.781193\pi$
$692$	−2348.15	−0.128993
$693$	0	0
$694$	−17883.8	−0.978182
$695$	−2870.01	−0.156641
$696$	0	0
$697$	22492.3	1.22232
$698$	6122.78	0.332021
$699$	0	0
$700$	−15511.5	−0.837545
$701$	−32839.5	−1.76938	−0.884688	−	0.466184i	$-0.845628\pi$
$701$	−32839.5	−1.76938	−0.884688	+	0.466184i	$0.845628\pi$
$702$	0	0
$703$	25335.9	1.35926
$704$	−932.860	−0.0499410
$705$	0	0
$706$	5743.27	0.306163
$707$	−52564.3	−2.79616
$708$	0	0
$709$	30391.9	1.60986	0.804932	−	0.593367i	$-0.202202\pi$
$709$	30391.9	1.60986	0.804932	+	0.593367i	$0.202202\pi$
$710$	33.8041	0.00178682
$711$	0	0
$712$	−6495.43	−0.341891
$713$	30326.2	1.59288
$714$	0	0
$715$	4137.82	0.216428
$716$	3635.36	0.189748
$717$	0	0
$718$	−19546.5	−1.01597
$719$	28162.9	1.46078	0.730390	−	0.683031i	$-0.239338\pi$
$719$	28162.9	1.46078	0.730390	+	0.683031i	$0.239338\pi$
$720$	0	0
$721$	29257.7	1.51125
$722$	−8622.03	−0.444430
$723$	0	0
$724$	8556.44	0.439223
$725$	3312.86	0.169706
$726$	0	0
$727$	−36419.3	−1.85794	−0.928968	−	0.370161i	$-0.879302\pi$
$727$	−36419.3	−1.85794	−0.928968	+	0.370161i	$0.879302\pi$
$728$	23498.5	1.19631
$729$	0	0
$730$	2811.49	0.142545
$731$	−1946.78	−0.0985011
$732$	0	0
$733$	8525.28	0.429589	0.214794	−	0.976659i	$-0.431092\pi$
$733$	8525.28	0.429589	0.214794	+	0.976659i	$0.431092\pi$
$734$	12129.2	0.609942
$735$	0	0
$736$	−4334.58	−0.217085
$737$	−4791.60	−0.239485
$738$	0	0
$739$	18309.2	0.911389	0.455695	−	0.890136i	$-0.349391\pi$
$739$	18309.2	0.911389	0.455695	+	0.890136i	$0.349391\pi$
$740$	−3145.89	−0.156277
$741$	0	0
$742$	41631.6	2.05976
$743$	−12992.8	−0.641534	−0.320767	−	0.947158i	$-0.603941\pi$
$743$	−12992.8	−0.641534	−0.320767	+	0.947158i	$0.603941\pi$
$744$	0	0
$745$	−6358.77	−0.312708
$746$	−5368.56	−0.263481
$747$	0	0
$748$	5979.04	0.292266
$749$	46655.5	2.27604
$750$	0	0
$751$	−3443.18	−0.167302	−0.0836508	−	0.996495i	$-0.526658\pi$
$751$	−3443.18	−0.167302	−0.0836508	+	0.996495i	$0.526658\pi$
$752$	−2366.75	−0.114769
$753$	0	0
$754$	−5018.67	−0.242399
$755$	−4790.99	−0.230943
$756$	0	0
$757$	3498.73	0.167983	0.0839917	−	0.996466i	$-0.473233\pi$
$757$	3498.73	0.167983	0.0839917	+	0.996466i	$0.473233\pi$
$758$	−4139.09	−0.198336
$759$	0	0
$760$	2773.91	0.132395
$761$	21460.2	1.02225	0.511124	−	0.859507i	$-0.329229\pi$
$761$	21460.2	1.02225	0.511124	+	0.859507i	$0.329229\pi$
$762$	0	0
$763$	−15870.5	−0.753016
$764$	6861.24	0.324910
$765$	0	0
$766$	17100.9	0.806634
$767$	15939.1	0.750363
$768$	0	0
$769$	−8766.41	−0.411086	−0.205543	−	0.978648i	$-0.565896\pi$
$769$	−8766.41	−0.411086	−0.205543	+	0.978648i	$0.565896\pi$
$770$	3246.62	0.151948
$771$	0	0
$772$	−10039.5	−0.468045
$773$	17969.5	0.836116	0.418058	−	0.908420i	$-0.362711\pi$
$773$	17969.5	0.836116	0.418058	+	0.908420i	$0.362711\pi$
$774$	0	0
$775$	−25575.7	−1.18543
$776$	90.8104	0.00420091
$777$	0	0
$778$	−21948.8	−1.01144
$779$	23180.7	1.06615
$780$	0	0
$781$	75.0931	0.00344052
$782$	27781.9	1.27043
$783$	0	0
$784$	12949.4	0.589896
$785$	1381.32	0.0628042
$786$	0	0
$787$	−2799.53	−0.126801	−0.0634005	−	0.997988i	$-0.520195\pi$
$787$	−2799.53	−0.126801	−0.0634005	+	0.997988i	$0.520195\pi$
$788$	2754.76	0.124536
$789$	0	0
$790$	2577.49	0.116080
$791$	−22242.3	−0.999803
$792$	0	0
$793$	1184.70	0.0530515
$794$	−1947.53	−0.0870471
$795$	0	0
$796$	4425.10	0.197040
$797$	21771.2	0.967597	0.483799	−	0.875179i	$-0.339257\pi$
$797$	21771.2	0.967597	0.483799	+	0.875179i	$0.339257\pi$
$798$	0	0
$799$	15169.4	0.671657
$800$	3655.57	0.161555
$801$	0	0
$802$	4392.65	0.193404
$803$	6245.50	0.274469
$804$	0	0
$805$	15085.6	0.660493
$806$	38744.7	1.69320
$807$	0	0
$808$	12387.7	0.539354
$809$	28676.9	1.24626	0.623130	−	0.782118i	$-0.285861\pi$
$809$	28676.9	1.24626	0.623130	+	0.782118i	$0.285861\pi$
$810$	0	0
$811$	12021.0	0.520486	0.260243	−	0.965543i	$-0.416197\pi$
$811$	12021.0	0.520486	0.260243	+	0.965543i	$0.416197\pi$
$812$	−3937.75	−0.170182
$813$	0	0
$814$	−6988.36	−0.300911
$815$	−2612.09	−0.112267
$816$	0	0
$817$	−2006.36	−0.0859164
$818$	−15694.8	−0.670850
$819$	0	0
$820$	−2878.29	−0.122578
$821$	14455.7	0.614504	0.307252	−	0.951628i	$-0.400591\pi$
$821$	14455.7	0.614504	0.307252	+	0.951628i	$0.400591\pi$
$822$	0	0
$823$	−43192.9	−1.82941	−0.914707	−	0.404117i	$-0.867579\pi$
$823$	−43192.9	−1.82941	−0.914707	+	0.404117i	$0.867579\pi$
$824$	−6895.10	−0.291508
$825$	0	0
$826$	12506.2	0.526810
$827$	−10474.2	−0.440415	−0.220207	−	0.975453i	$-0.570673\pi$
$827$	−10474.2	−0.440415	−0.220207	+	0.975453i	$0.570673\pi$
$828$	0	0
$829$	−13114.2	−0.549425	−0.274713	−	0.961526i	$-0.588583\pi$
$829$	−13114.2	−0.549425	−0.274713	+	0.961526i	$0.588583\pi$
$830$	−2980.06	−0.124626
$831$	0	0
$832$	−5537.84	−0.230757
$833$	−82997.5	−3.45221
$834$	0	0
$835$	−11895.1	−0.492992
$836$	6162.02	0.254926
$837$	0	0
$838$	−13886.4	−0.572432
$839$	−4806.78	−0.197793	−0.0988965	−	0.995098i	$-0.531531\pi$
$839$	−4806.78	−0.197793	−0.0988965	+	0.995098i	$0.531531\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	21054.5	0.861741
$843$	0	0
$844$	−8376.32	−0.341617
$845$	17356.0	0.706584
$846$	0	0
$847$	−37970.1	−1.54034
$848$	−9811.23	−0.397310
$849$	0	0
$850$	−23429.9	−0.945457
$851$	−32471.8	−1.30801
$852$	0	0
$853$	49150.9	1.97291	0.986456	−	0.164026i	$-0.0524481\pi$
$853$	49150.9	1.97291	0.986456	+	0.164026i	$0.0524481\pi$
$854$	929.538	0.0372461
$855$	0	0
$856$	−10995.2	−0.439028
$857$	−44886.9	−1.78916	−0.894578	−	0.446911i	$-0.852524\pi$
$857$	−44886.9	−1.78916	−0.894578	+	0.446911i	$0.852524\pi$
$858$	0	0
$859$	−28327.3	−1.12516	−0.562582	−	0.826742i	$-0.690192\pi$
$859$	−28327.3	−1.12516	−0.562582	+	0.826742i	$0.690192\pi$
$860$	249.125	0.00987801
$861$	0	0
$862$	−3471.16	−0.137155
$863$	−28693.4	−1.13179	−0.565895	−	0.824477i	$-0.691469\pi$
$863$	−28693.4	−1.13179	−0.565895	+	0.824477i	$0.691469\pi$
$864$	0	0
$865$	−1925.93	−0.0757036
$866$	18760.3	0.736144
$867$	0	0
$868$	30399.8	1.18875
$869$	5725.70	0.223511
$870$	0	0
$871$	−28444.9	−1.10657
$872$	3740.17	0.145250
$873$	0	0
$874$	28632.1	1.10812
$875$	−26643.6	−1.02939
$876$	0	0
$877$	−29417.1	−1.13266	−0.566332	−	0.824177i	$-0.691638\pi$
$877$	−29417.1	−1.13266	−0.566332	+	0.824177i	$0.691638\pi$
$878$	8199.15	0.315157
$879$	0	0
$880$	−765.123	−0.0293094
$881$	6154.67	0.235365	0.117682	−	0.993051i	$-0.462454\pi$
$881$	6154.67	0.235365	0.117682	+	0.993051i	$0.462454\pi$
$882$	0	0
$883$	28371.7	1.08130	0.540648	−	0.841249i	$-0.318179\pi$
$883$	28371.7	1.08130	0.540648	+	0.841249i	$0.318179\pi$
$884$	35494.0	1.35045
$885$	0	0
$886$	11436.4	0.433649
$887$	−36685.3	−1.38869	−0.694346	−	0.719641i	$-0.744306\pi$
$887$	−36685.3	−1.38869	−0.694346	+	0.719641i	$0.744306\pi$
$888$	0	0
$889$	28195.9	1.06373
$890$	−5327.50	−0.200650
$891$	0	0
$892$	24132.4	0.905845
$893$	15633.6	0.585845
$894$	0	0
$895$	2981.69	0.111360
$896$	−4345.10	−0.162009
$897$	0	0
$898$	4949.54	0.183929
$899$	−6492.61	−0.240868
$900$	0	0
$901$	62883.8	2.32515
$902$	−6393.90	−0.236024
$903$	0	0
$904$	5241.79	0.192853
$905$	7017.92	0.257772
$906$	0	0
$907$	−38189.3	−1.39808	−0.699039	−	0.715084i	$-0.746388\pi$
$907$	−38189.3	−1.39808	−0.699039	+	0.715084i	$0.746388\pi$
$908$	−757.967	−0.0277027
$909$	0	0
$910$	19273.3	0.702091
$911$	−6696.40	−0.243537	−0.121768	−	0.992559i	$-0.538856\pi$
$911$	−6696.40	−0.243537	−0.121768	+	0.992559i	$0.538856\pi$
$912$	0	0
$913$	−6619.98	−0.239966
$914$	−21864.7	−0.791271
$915$	0	0
$916$	−9391.20	−0.338749
$917$	−26586.4	−0.957427
$918$	0	0
$919$	48279.1	1.73295	0.866474	−	0.499222i	$-0.166381\pi$
$919$	48279.1	1.73295	0.866474	+	0.499222i	$0.166381\pi$
$920$	−3555.18	−0.127403
$921$	0	0
$922$	33178.7	1.18512
$923$	445.784	0.0158972
$924$	0	0
$925$	27385.1	0.973423
$926$	−1788.49	−0.0634702
$927$	0	0
$928$	928.000	0.0328266
$929$	−22375.0	−0.790206	−0.395103	−	0.918637i	$-0.629291\pi$
$929$	−22375.0	−0.790206	−0.395103	+	0.918637i	$0.629291\pi$
$930$	0	0
$931$	−85537.5	−3.01115
$932$	−2207.11	−0.0775710
$933$	0	0
$934$	−16022.2	−0.561308
$935$	4903.95	0.171526
$936$	0	0
$937$	−27929.2	−0.973754	−0.486877	−	0.873470i	$-0.661864\pi$
$937$	−27929.2	−0.973754	−0.486877	+	0.873470i	$0.661864\pi$
$938$	−22318.4	−0.776890
$939$	0	0
$940$	−1941.19	−0.0673560
$941$	11456.3	0.396881	0.198440	−	0.980113i	$-0.436412\pi$
$941$	11456.3	0.396881	0.198440	+	0.980113i	$0.436412\pi$
$942$	0	0
$943$	−29709.6	−1.02596
$944$	−2947.30	−0.101617
$945$	0	0
$946$	553.412	0.0190200
$947$	38730.6	1.32901	0.664506	−	0.747283i	$-0.268642\pi$
$947$	38730.6	1.32901	0.664506	+	0.747283i	$0.268642\pi$
$948$	0	0
$949$	37075.9	1.26821
$950$	−24146.9	−0.824663
$951$	0	0
$952$	27849.3	0.948112
$953$	−56157.0	−1.90882	−0.954408	−	0.298504i	$-0.903512\pi$
$953$	−56157.0	−1.90882	−0.954408	+	0.298504i	$0.903512\pi$
$954$	0	0
$955$	5627.53	0.190683
$956$	18133.8	0.613482
$957$	0	0
$958$	−29117.4	−0.981982
$959$	46244.2	1.55715
$960$	0	0
$961$	20332.7	0.682511
$962$	−41485.8	−1.39039
$963$	0	0
$964$	4593.11	0.153458
$965$	−8234.34	−0.274687
$966$	0	0
$967$	7078.24	0.235389	0.117694	−	0.993050i	$-0.462450\pi$
$967$	7078.24	0.235389	0.117694	+	0.993050i	$0.462450\pi$
$968$	8948.34	0.297118
$969$	0	0
$970$	74.4819	0.00246543
$971$	49719.3	1.64322	0.821610	−	0.570049i	$-0.193076\pi$
$971$	49719.3	1.64322	0.821610	+	0.570049i	$0.193076\pi$
$972$	0	0
$973$	−29696.0	−0.978427
$974$	35333.6	1.16238
$975$	0	0
$976$	−219.062	−0.00718444
$977$	−41123.3	−1.34662	−0.673312	−	0.739358i	$-0.735129\pi$
$977$	−41123.3	−1.34662	−0.673312	+	0.739358i	$0.735129\pi$
$978$	0	0
$979$	−11834.6	−0.386349
$980$	10621.0	0.346199
$981$	0	0
$982$	−9808.72	−0.318746
$983$	13824.2	0.448549	0.224275	−	0.974526i	$-0.427999\pi$
$983$	13824.2	0.448549	0.224275	+	0.974526i	$0.427999\pi$
$984$	0	0
$985$	2259.43	0.0730876
$986$	−5947.89	−0.192109
$987$	0	0
$988$	36580.3	1.17791
$989$	2571.46	0.0826770
$990$	0	0
$991$	−13606.6	−0.436154	−0.218077	−	0.975932i	$-0.569978\pi$
$991$	−13606.6	−0.436154	−0.218077	+	0.975932i	$0.569978\pi$
$992$	−7164.26	−0.229300
$993$	0	0
$994$	349.771	0.0111610
$995$	3629.43	0.115639
$996$	0	0
$997$	−41867.4	−1.32995	−0.664973	−	0.746868i	$-0.731557\pi$
$997$	−41867.4	−1.32995	−0.664973	+	0.746868i	$0.731557\pi$
$998$	−4993.98	−0.158398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	522.4.a.k.1.3		3
3.2	odd	2		58.4.a.d.1.2	✓	3
12.11	even	2		464.4.a.i.1.2		3
15.14	odd	2		1450.4.a.h.1.2		3
24.5	odd	2		1856.4.a.r.1.2		3
24.11	even	2		1856.4.a.s.1.2		3
87.86	odd	2		1682.4.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.d.1.2	✓	3	3.2	odd	2
464.4.a.i.1.2		3	12.11	even	2
522.4.a.k.1.3		3	1.1	even	1	trivial
1450.4.a.h.1.2		3	15.14	odd	2
1682.4.a.d.1.2		3	87.86	odd	2
1856.4.a.r.1.2		3	24.5	odd	2
1856.4.a.s.1.2		3	24.11	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:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	522.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +3.28077 q^{5} +33.9461 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} +3.28077 q^{5} +33.9461 q^{7} -8.00000 q^{8} -6.56153 q^{10} -14.5759 q^{11} -86.5287 q^{13} -67.8922 q^{14} +16.0000 q^{16} -102.550 q^{17} -105.688 q^{19} +13.1231 q^{20} +29.1519 q^{22} +135.456 q^{23} -114.237 q^{25} +173.057 q^{26} +135.784 q^{28} -29.0000 q^{29} +223.883 q^{31} -32.0000 q^{32} +205.100 q^{34} +111.369 q^{35} -239.723 q^{37} +211.377 q^{38} -26.2461 q^{40} -219.331 q^{41} +18.9838 q^{43} -58.3037 q^{44} -270.911 q^{46} -147.922 q^{47} +809.338 q^{49} +228.473 q^{50} -346.115 q^{52} -613.202 q^{53} -47.8202 q^{55} -271.569 q^{56} +58.0000 q^{58} -184.206 q^{59} -13.6914 q^{61} -447.766 q^{62} +64.0000 q^{64} -283.880 q^{65} +328.733 q^{67} -410.199 q^{68} -222.738 q^{70} -5.15186 q^{71} -428.481 q^{73} +479.445 q^{74} -422.753 q^{76} -494.796 q^{77} -392.819 q^{79} +52.4922 q^{80} +438.661 q^{82} +454.172 q^{83} -336.442 q^{85} -37.9675 q^{86} +116.607 q^{88} +811.929 q^{89} -2937.31 q^{91} +541.822 q^{92} +295.844 q^{94} -346.738 q^{95} -11.3513 q^{97} -1618.68 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8}+O(q^{10}) \) 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8	\( 3 q - 6 q^{2} + 12 q^{4} - 20 q^{5} + 24 q^{7} - 24 q^{8} + 40 q^{10} - 10 q^{11} - 4 q^{13} - 48 q^{14} + 48 q^{16} + 66 q^{17} - 164 q^{19} - 80 q^{20} + 20 q^{22} + 204 q^{23} + 79 q^{25} + 8 q^{26} + 96 q^{28} - 87 q^{29} - 86 q^{31} - 96 q^{32} - 132 q^{34} - 24 q^{35} - 42 q^{37} + 328 q^{38} + 160 q^{40} - 562 q^{41} + 18 q^{43} - 40 q^{44} - 408 q^{46} - 654 q^{47} + 539 q^{49} - 158 q^{50} - 16 q^{52} - 712 q^{53} + 142 q^{55} - 192 q^{56} + 174 q^{58} - 184 q^{59} + 322 q^{61} + 172 q^{62} + 192 q^{64} - 1494 q^{65} - 228 q^{67} + 264 q^{68} + 48 q^{70} + 52 q^{71} - 494 q^{73} + 84 q^{74} - 656 q^{76} - 872 q^{77} - 2110 q^{79} - 320 q^{80} + 1124 q^{82} + 288 q^{83} - 2704 q^{85} - 36 q^{86} + 80 q^{88} - 914 q^{89} - 2984 q^{91} + 816 q^{92} + 1308 q^{94} + 1900 q^{95} + 218 q^{97} - 1078 q^{98}+O(q^{100}) \) 3 * q - 6 * q^2 + 12 * q^4 - 20 * q^5 + 24 * q^7 - 24 * q^8 + 40 * q^10 - 10 * q^11 - 4 * q^13 - 48 * q^14 + 48 * q^16 + 66 * q^17 - 164 * q^19 - 80 * q^20 + 20 * q^22 + 204 * q^23 + 79 * q^25 + 8 * q^26 + 96 * q^28 - 87 * q^29 - 86 * q^31 - 96 * q^32 - 132 * q^34 - 24 * q^35 - 42 * q^37 + 328 * q^38 + 160 * q^40 - 562 * q^41 + 18 * q^43 - 40 * q^44 - 408 * q^46 - 654 * q^47 + 539 * q^49 - 158 * q^50 - 16 * q^52 - 712 * q^53 + 142 * q^55 - 192 * q^56 + 174 * q^58 - 184 * q^59 + 322 * q^61 + 172 * q^62 + 192 * q^64 - 1494 * q^65 - 228 * q^67 + 264 * q^68 + 48 * q^70 + 52 * q^71 - 494 * q^73 + 84 * q^74 - 656 * q^76 - 872 * q^77 - 2110 * q^79 - 320 * q^80 + 1124 * q^82 + 288 * q^83 - 2704 * q^85 - 36 * q^86 + 80 * q^88 - 914 * q^89 - 2984 * q^91 + 816 * q^92 + 1308 * q^94 + 1900 * q^95 + 218 * q^97 - 1078 * q^98

Introduction

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