LMFDB - Embedded newform 6534.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6534,2,Mod(1,6534)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6534, 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("6534.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6534 = 2 \cdot 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6534.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:	$52.1742526807$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 594)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6534.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.236068 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.236068 q^{7} +1.00000 q^{8} -1.00000 q^{10} -4.61803 q^{13} +0.236068 q^{14} +1.00000 q^{16} +2.38197 q^{17} +1.38197 q^{19} -1.00000 q^{20} -3.47214 q^{23} -4.00000 q^{25} -4.61803 q^{26} +0.236068 q^{28} -4.09017 q^{29} +7.47214 q^{31} +1.00000 q^{32} +2.38197 q^{34} -0.236068 q^{35} +0.381966 q^{37} +1.38197 q^{38} -1.00000 q^{40} +6.47214 q^{41} +2.09017 q^{43} -3.47214 q^{46} -1.76393 q^{47} -6.94427 q^{49} -4.00000 q^{50} -4.61803 q^{52} +3.85410 q^{53} +0.236068 q^{56} -4.09017 q^{58} -8.14590 q^{59} -13.7984 q^{61} +7.47214 q^{62} +1.00000 q^{64} +4.61803 q^{65} -1.29180 q^{67} +2.38197 q^{68} -0.236068 q^{70} -7.61803 q^{71} -0.236068 q^{73} +0.381966 q^{74} +1.38197 q^{76} +4.47214 q^{79} -1.00000 q^{80} +6.47214 q^{82} -16.6525 q^{83} -2.38197 q^{85} +2.09017 q^{86} -13.7082 q^{89} -1.09017 q^{91} -3.47214 q^{92} -1.76393 q^{94} -1.38197 q^{95} -10.3820 q^{97} -6.94427 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} - 2 q^{10} - 7 q^{13} - 4 q^{14} + 2 q^{16} + 7 q^{17} + 5 q^{19} - 2 q^{20} + 2 q^{23} - 8 q^{25} - 7 q^{26} - 4 q^{28} + 3 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{34} + 4 q^{35} + 3 q^{37} + 5 q^{38} - 2 q^{40} + 4 q^{41} - 7 q^{43} + 2 q^{46} - 8 q^{47} + 4 q^{49} - 8 q^{50} - 7 q^{52} + q^{53} - 4 q^{56} + 3 q^{58} - 23 q^{59} - 3 q^{61} + 6 q^{62} + 2 q^{64} + 7 q^{65} - 16 q^{67} + 7 q^{68} + 4 q^{70} - 13 q^{71} + 4 q^{73} + 3 q^{74} + 5 q^{76} - 2 q^{80} + 4 q^{82} - 2 q^{83} - 7 q^{85} - 7 q^{86} - 14 q^{89} + 9 q^{91} + 2 q^{92} - 8 q^{94} - 5 q^{95} - 23 q^{97} + 4 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8 - 2 * q^10 - 7 * q^13 - 4 * q^14 + 2 * q^16 + 7 * q^17 + 5 * q^19 - 2 * q^20 + 2 * q^23 - 8 * q^25 - 7 * q^26 - 4 * q^28 + 3 * q^29 + 6 * q^31 + 2 * q^32 + 7 * q^34 + 4 * q^35 + 3 * q^37 + 5 * q^38 - 2 * q^40 + 4 * q^41 - 7 * q^43 + 2 * q^46 - 8 * q^47 + 4 * q^49 - 8 * q^50 - 7 * q^52 + q^53 - 4 * q^56 + 3 * q^58 - 23 * q^59 - 3 * q^61 + 6 * q^62 + 2 * q^64 + 7 * q^65 - 16 * q^67 + 7 * q^68 + 4 * q^70 - 13 * q^71 + 4 * q^73 + 3 * q^74 + 5 * q^76 - 2 * q^80 + 4 * q^82 - 2 * q^83 - 7 * q^85 - 7 * q^86 - 14 * q^89 + 9 * q^91 + 2 * q^92 - 8 * q^94 - 5 * q^95 - 23 * q^97 + 4 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0.236068	0.0892253	0.0446127	−	0.999004i	$-0.485795\pi$
$7$	0.236068	0.0892253	0.0446127	+	0.999004i	$0.485795\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.61803	−1.28081	−0.640406	−	0.768036i	$-0.721234\pi$
$13$	−4.61803	−1.28081	−0.640406	+	0.768036i	$0.721234\pi$
$14$	0.236068	0.0630918
$15$	0	0
$16$	1.00000	0.250000
$17$	2.38197	0.577712	0.288856	−	0.957373i	$-0.406725\pi$
$17$	2.38197	0.577712	0.288856	+	0.957373i	$0.406725\pi$
$18$	0	0
$19$	1.38197	0.317045	0.158522	−	0.987355i	$-0.449327\pi$
$19$	1.38197	0.317045	0.158522	+	0.987355i	$0.449327\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	0	0
$23$	−3.47214	−0.723990	−0.361995	−	0.932180i	$-0.617904\pi$
$23$	−3.47214	−0.723990	−0.361995	+	0.932180i	$0.617904\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−4.61803	−0.905671
$27$	0	0
$28$	0.236068	0.0446127
$29$	−4.09017	−0.759525	−0.379763	−	0.925084i	$-0.623994\pi$
$29$	−4.09017	−0.759525	−0.379763	+	0.925084i	$0.623994\pi$
$30$	0	0
$31$	7.47214	1.34204	0.671018	−	0.741441i	$-0.265857\pi$
$31$	7.47214	1.34204	0.671018	+	0.741441i	$0.265857\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.38197	0.408504
$35$	−0.236068	−0.0399028
$36$	0	0
$37$	0.381966	0.0627948	0.0313974	−	0.999507i	$-0.490004\pi$
$37$	0.381966	0.0627948	0.0313974	+	0.999507i	$0.490004\pi$
$38$	1.38197	0.224184
$39$	0	0
$40$	−1.00000	−0.158114
$41$	6.47214	1.01078	0.505389	−	0.862892i	$-0.331349\pi$
$41$	6.47214	1.01078	0.505389	+	0.862892i	$0.331349\pi$
$42$	0	0
$43$	2.09017	0.318748	0.159374	−	0.987218i	$-0.449052\pi$
$43$	2.09017	0.318748	0.159374	+	0.987218i	$0.449052\pi$
$44$	0	0
$45$	0	0
$46$	−3.47214	−0.511939
$47$	−1.76393	−0.257296	−0.128648	−	0.991690i	$-0.541064\pi$
$47$	−1.76393	−0.257296	−0.128648	+	0.991690i	$0.541064\pi$
$48$	0	0
$49$	−6.94427	−0.992039
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−4.61803	−0.640406
$53$	3.85410	0.529402	0.264701	−	0.964331i	$-0.414727\pi$
$53$	3.85410	0.529402	0.264701	+	0.964331i	$0.414727\pi$
$54$	0	0
$55$	0	0
$56$	0.236068	0.0315459
$57$	0	0
$58$	−4.09017	−0.537066
$59$	−8.14590	−1.06051	−0.530253	−	0.847840i	$-0.677903\pi$
$59$	−8.14590	−1.06051	−0.530253	+	0.847840i	$0.677903\pi$
$60$	0	0
$61$	−13.7984	−1.76670	−0.883350	−	0.468713i	$-0.844718\pi$
$61$	−13.7984	−1.76670	−0.883350	+	0.468713i	$0.844718\pi$
$62$	7.47214	0.948962
$63$	0	0
$64$	1.00000	0.125000
$65$	4.61803	0.572797
$66$	0	0
$67$	−1.29180	−0.157818	−0.0789090	−	0.996882i	$-0.525144\pi$
$67$	−1.29180	−0.157818	−0.0789090	+	0.996882i	$0.525144\pi$
$68$	2.38197	0.288856
$69$	0	0
$70$	−0.236068	−0.0282155
$71$	−7.61803	−0.904094	−0.452047	−	0.891994i	$-0.649306\pi$
$71$	−7.61803	−0.904094	−0.452047	+	0.891994i	$0.649306\pi$
$72$	0	0
$73$	−0.236068	−0.0276297	−0.0138148	−	0.999905i	$-0.504398\pi$
$73$	−0.236068	−0.0276297	−0.0138148	+	0.999905i	$0.504398\pi$
$74$	0.381966	0.0444026
$75$	0	0
$76$	1.38197	0.158522
$77$	0	0
$78$	0	0
$79$	4.47214	0.503155	0.251577	−	0.967837i	$-0.419051\pi$
$79$	4.47214	0.503155	0.251577	+	0.967837i	$0.419051\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.47214	0.714728
$83$	−16.6525	−1.82785	−0.913923	−	0.405887i	$-0.866963\pi$
$83$	−16.6525	−1.82785	−0.913923	+	0.405887i	$0.866963\pi$
$84$	0	0
$85$	−2.38197	−0.258360
$86$	2.09017	0.225389
$87$	0	0
$88$	0	0
$89$	−13.7082	−1.45307	−0.726533	−	0.687131i	$-0.758870\pi$
$89$	−13.7082	−1.45307	−0.726533	+	0.687131i	$0.758870\pi$
$90$	0	0
$91$	−1.09017	−0.114281
$92$	−3.47214	−0.361995
$93$	0	0
$94$	−1.76393	−0.181936
$95$	−1.38197	−0.141787
$96$	0	0
$97$	−10.3820	−1.05413	−0.527064	−	0.849825i	$-0.676707\pi$
$97$	−10.3820	−1.05413	−0.527064	+	0.849825i	$0.676707\pi$
$98$	−6.94427	−0.701477
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−10.5279	−1.04756	−0.523781	−	0.851853i	$-0.675479\pi$
$101$	−10.5279	−1.04756	−0.523781	+	0.851853i	$0.675479\pi$
$102$	0	0
$103$	13.1803	1.29870	0.649349	−	0.760491i	$-0.275042\pi$
$103$	13.1803	1.29870	0.649349	+	0.760491i	$0.275042\pi$
$104$	−4.61803	−0.452835
$105$	0	0
$106$	3.85410	0.374343
$107$	1.23607	0.119495	0.0597476	−	0.998214i	$-0.480970\pi$
$107$	1.23607	0.119495	0.0597476	+	0.998214i	$0.480970\pi$
$108$	0	0
$109$	13.5623	1.29903	0.649517	−	0.760347i	$-0.274971\pi$
$109$	13.5623	1.29903	0.649517	+	0.760347i	$0.274971\pi$
$110$	0	0
$111$	0	0
$112$	0.236068	0.0223063
$113$	−12.2361	−1.15107	−0.575536	−	0.817776i	$-0.695207\pi$
$113$	−12.2361	−1.15107	−0.575536	+	0.817776i	$0.695207\pi$
$114$	0	0
$115$	3.47214	0.323778
$116$	−4.09017	−0.379763
$117$	0	0
$118$	−8.14590	−0.749891
$119$	0.562306	0.0515465
$120$	0	0
$121$	0	0
$122$	−13.7984	−1.24925
$123$	0	0
$124$	7.47214	0.671018
$125$	9.00000	0.804984
$126$	0	0
$127$	−8.56231	−0.759782	−0.379891	−	0.925031i	$-0.624038\pi$
$127$	−8.56231	−0.759782	−0.379891	+	0.925031i	$0.624038\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.61803	0.405028
$131$	3.70820	0.323987	0.161994	−	0.986792i	$-0.448208\pi$
$131$	3.70820	0.323987	0.161994	+	0.986792i	$0.448208\pi$
$132$	0	0
$133$	0.326238	0.0282884
$134$	−1.29180	−0.111594
$135$	0	0
$136$	2.38197	0.204252
$137$	−13.8541	−1.18364	−0.591818	−	0.806072i	$-0.701590\pi$
$137$	−13.8541	−1.18364	−0.591818	+	0.806072i	$0.701590\pi$
$138$	0	0
$139$	5.47214	0.464141	0.232070	−	0.972699i	$-0.425450\pi$
$139$	5.47214	0.464141	0.232070	+	0.972699i	$0.425450\pi$
$140$	−0.236068	−0.0199514
$141$	0	0
$142$	−7.61803	−0.639291
$143$	0	0
$144$	0	0
$145$	4.09017	0.339670
$146$	−0.236068	−0.0195371
$147$	0	0
$148$	0.381966	0.0313974
$149$	−14.7639	−1.20951	−0.604754	−	0.796412i	$-0.706729\pi$
$149$	−14.7639	−1.20951	−0.604754	+	0.796412i	$0.706729\pi$
$150$	0	0
$151$	−16.4721	−1.34048	−0.670242	−	0.742143i	$-0.733810\pi$
$151$	−16.4721	−1.34048	−0.670242	+	0.742143i	$0.733810\pi$
$152$	1.38197	0.112092
$153$	0	0
$154$	0	0
$155$	−7.47214	−0.600176
$156$	0	0
$157$	3.76393	0.300394	0.150197	−	0.988656i	$-0.452009\pi$
$157$	3.76393	0.300394	0.150197	+	0.988656i	$0.452009\pi$
$158$	4.47214	0.355784
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−0.819660	−0.0645983
$162$	0	0
$163$	6.00000	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$163$	6.00000	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$164$	6.47214	0.505389
$165$	0	0
$166$	−16.6525	−1.29248
$167$	4.94427	0.382599	0.191300	−	0.981532i	$-0.438730\pi$
$167$	4.94427	0.382599	0.191300	+	0.981532i	$0.438730\pi$
$168$	0	0
$169$	8.32624	0.640480
$170$	−2.38197	−0.182688
$171$	0	0
$172$	2.09017	0.159374
$173$	17.5623	1.33524	0.667619	−	0.744503i	$-0.267314\pi$
$173$	17.5623	1.33524	0.667619	+	0.744503i	$0.267314\pi$
$174$	0	0
$175$	−0.944272	−0.0713802
$176$	0	0
$177$	0	0
$178$	−13.7082	−1.02747
$179$	19.3607	1.44709	0.723543	−	0.690280i	$-0.242513\pi$
$179$	19.3607	1.44709	0.723543	+	0.690280i	$0.242513\pi$
$180$	0	0
$181$	6.38197	0.474368	0.237184	−	0.971465i	$-0.423776\pi$
$181$	6.38197	0.474368	0.237184	+	0.971465i	$0.423776\pi$
$182$	−1.09017	−0.0808088
$183$	0	0
$184$	−3.47214	−0.255969
$185$	−0.381966	−0.0280827
$186$	0	0
$187$	0	0
$188$	−1.76393	−0.128648
$189$	0	0
$190$	−1.38197	−0.100258
$191$	−22.7426	−1.64560	−0.822800	−	0.568331i	$-0.807589\pi$
$191$	−22.7426	−1.64560	−0.822800	+	0.568331i	$0.807589\pi$
$192$	0	0
$193$	8.23607	0.592845	0.296423	−	0.955057i	$-0.404206\pi$
$193$	8.23607	0.592845	0.296423	+	0.955057i	$0.404206\pi$
$194$	−10.3820	−0.745382
$195$	0	0
$196$	−6.94427	−0.496019
$197$	−3.18034	−0.226590	−0.113295	−	0.993561i	$-0.536140\pi$
$197$	−3.18034	−0.226590	−0.113295	+	0.993561i	$0.536140\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−10.5279	−0.740738
$203$	−0.965558	−0.0677689
$204$	0	0
$205$	−6.47214	−0.452034
$206$	13.1803	0.918318
$207$	0	0
$208$	−4.61803	−0.320203
$209$	0	0
$210$	0	0
$211$	18.2361	1.25542	0.627711	−	0.778446i	$-0.283992\pi$
$211$	18.2361	1.25542	0.627711	+	0.778446i	$0.283992\pi$
$212$	3.85410	0.264701
$213$	0	0
$214$	1.23607	0.0844959
$215$	−2.09017	−0.142548
$216$	0	0
$217$	1.76393	0.119744
$218$	13.5623	0.918555
$219$	0	0
$220$	0	0
$221$	−11.0000	−0.739940
$222$	0	0
$223$	−28.7426	−1.92475	−0.962375	−	0.271725i	$-0.912406\pi$
$223$	−28.7426	−1.92475	−0.962375	+	0.271725i	$0.912406\pi$
$224$	0.236068	0.0157730
$225$	0	0
$226$	−12.2361	−0.813931
$227$	20.5623	1.36477	0.682384	−	0.730994i	$-0.260943\pi$
$227$	20.5623	1.36477	0.682384	+	0.730994i	$0.260943\pi$
$228$	0	0
$229$	1.94427	0.128481	0.0642406	−	0.997934i	$-0.479538\pi$
$229$	1.94427	0.128481	0.0642406	+	0.997934i	$0.479538\pi$
$230$	3.47214	0.228946
$231$	0	0
$232$	−4.09017	−0.268533
$233$	−26.2361	−1.71878	−0.859391	−	0.511319i	$-0.829157\pi$
$233$	−26.2361	−1.71878	−0.859391	+	0.511319i	$0.829157\pi$
$234$	0	0
$235$	1.76393	0.115066
$236$	−8.14590	−0.530253
$237$	0	0
$238$	0.562306	0.0364489
$239$	17.2361	1.11491	0.557454	−	0.830208i	$-0.311778\pi$
$239$	17.2361	1.11491	0.557454	+	0.830208i	$0.311778\pi$
$240$	0	0
$241$	6.41641	0.413317	0.206659	−	0.978413i	$-0.433741\pi$
$241$	6.41641	0.413317	0.206659	+	0.978413i	$0.433741\pi$
$242$	0	0
$243$	0	0
$244$	−13.7984	−0.883350
$245$	6.94427	0.443653
$246$	0	0
$247$	−6.38197	−0.406075
$248$	7.47214	0.474481
$249$	0	0
$250$	9.00000	0.569210
$251$	−26.6525	−1.68229	−0.841145	−	0.540810i	$-0.818118\pi$
$251$	−26.6525	−1.68229	−0.841145	+	0.540810i	$0.818118\pi$
$252$	0	0
$253$	0	0
$254$	−8.56231	−0.537247
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0344	−1.87350	−0.936749	−	0.350003i	$-0.886180\pi$
$257$	−30.0344	−1.87350	−0.936749	+	0.350003i	$0.886180\pi$
$258$	0	0
$259$	0.0901699	0.00560289
$260$	4.61803	0.286398
$261$	0	0
$262$	3.70820	0.229094
$263$	−9.41641	−0.580641	−0.290320	−	0.956930i	$-0.593762\pi$
$263$	−9.41641	−0.580641	−0.290320	+	0.956930i	$0.593762\pi$
$264$	0	0
$265$	−3.85410	−0.236756
$266$	0.326238	0.0200029
$267$	0	0
$268$	−1.29180	−0.0789090
$269$	−17.1803	−1.04750	−0.523752	−	0.851871i	$-0.675468\pi$
$269$	−17.1803	−1.04750	−0.523752	+	0.851871i	$0.675468\pi$
$270$	0	0
$271$	−15.2918	−0.928910	−0.464455	−	0.885597i	$-0.653750\pi$
$271$	−15.2918	−0.928910	−0.464455	+	0.885597i	$0.653750\pi$
$272$	2.38197	0.144428
$273$	0	0
$274$	−13.8541	−0.836957
$275$	0	0
$276$	0	0
$277$	−15.7082	−0.943815	−0.471907	−	0.881648i	$-0.656434\pi$
$277$	−15.7082	−0.943815	−0.471907	+	0.881648i	$0.656434\pi$
$278$	5.47214	0.328197
$279$	0	0
$280$	−0.236068	−0.0141078
$281$	−11.5623	−0.689749	−0.344875	−	0.938649i	$-0.612079\pi$
$281$	−11.5623	−0.689749	−0.344875	+	0.938649i	$0.612079\pi$
$282$	0	0
$283$	8.61803	0.512289	0.256144	−	0.966639i	$-0.417548\pi$
$283$	8.61803	0.512289	0.256144	+	0.966639i	$0.417548\pi$
$284$	−7.61803	−0.452047
$285$	0	0
$286$	0	0
$287$	1.52786	0.0901870
$288$	0	0
$289$	−11.3262	−0.666249
$290$	4.09017	0.240183
$291$	0	0
$292$	−0.236068	−0.0138148
$293$	11.8541	0.692524	0.346262	−	0.938138i	$-0.387451\pi$
$293$	11.8541	0.692524	0.346262	+	0.938138i	$0.387451\pi$
$294$	0	0
$295$	8.14590	0.474273
$296$	0.381966	0.0222013
$297$	0	0
$298$	−14.7639	−0.855252
$299$	16.0344	0.927296
$300$	0	0
$301$	0.493422	0.0284404
$302$	−16.4721	−0.947865
$303$	0	0
$304$	1.38197	0.0792612
$305$	13.7984	0.790093
$306$	0	0
$307$	−10.8885	−0.621442	−0.310721	−	0.950501i	$-0.600570\pi$
$307$	−10.8885	−0.621442	−0.310721	+	0.950501i	$0.600570\pi$
$308$	0	0
$309$	0	0
$310$	−7.47214	−0.424389
$311$	−15.3820	−0.872231	−0.436116	−	0.899891i	$-0.643646\pi$
$311$	−15.3820	−0.872231	−0.436116	+	0.899891i	$0.643646\pi$
$312$	0	0
$313$	−27.8328	−1.57320	−0.786602	−	0.617461i	$-0.788162\pi$
$313$	−27.8328	−1.57320	−0.786602	+	0.617461i	$0.788162\pi$
$314$	3.76393	0.212411
$315$	0	0
$316$	4.47214	0.251577
$317$	25.8885	1.45405	0.727023	−	0.686613i	$-0.240903\pi$
$317$	25.8885	1.45405	0.727023	+	0.686613i	$0.240903\pi$
$318$	0	0
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−0.819660	−0.0456779
$323$	3.29180	0.183160
$324$	0	0
$325$	18.4721	1.02465
$326$	6.00000	0.332309
$327$	0	0
$328$	6.47214	0.357364
$329$	−0.416408	−0.0229573
$330$	0	0
$331$	29.0689	1.59777	0.798885	−	0.601484i	$-0.205423\pi$
$331$	29.0689	1.59777	0.798885	+	0.601484i	$0.205423\pi$
$332$	−16.6525	−0.913923
$333$	0	0
$334$	4.94427	0.270539
$335$	1.29180	0.0705784
$336$	0	0
$337$	29.3262	1.59750	0.798751	−	0.601662i	$-0.205494\pi$
$337$	29.3262	1.59750	0.798751	+	0.601662i	$0.205494\pi$
$338$	8.32624	0.452888
$339$	0	0
$340$	−2.38197	−0.129180
$341$	0	0
$342$	0	0
$343$	−3.29180	−0.177740
$344$	2.09017	0.112694
$345$	0	0
$346$	17.5623	0.944155
$347$	−16.1246	−0.865615	−0.432807	−	0.901486i	$-0.642477\pi$
$347$	−16.1246	−0.865615	−0.432807	+	0.901486i	$0.642477\pi$
$348$	0	0
$349$	23.6525	1.26609	0.633044	−	0.774116i	$-0.281805\pi$
$349$	23.6525	1.26609	0.633044	+	0.774116i	$0.281805\pi$
$350$	−0.944272	−0.0504735
$351$	0	0
$352$	0	0
$353$	−14.9443	−0.795403	−0.397702	−	0.917515i	$-0.630192\pi$
$353$	−14.9443	−0.795403	−0.397702	+	0.917515i	$0.630192\pi$
$354$	0	0
$355$	7.61803	0.404323
$356$	−13.7082	−0.726533
$357$	0	0
$358$	19.3607	1.02324
$359$	28.3050	1.49388	0.746939	−	0.664892i	$-0.231523\pi$
$359$	28.3050	1.49388	0.746939	+	0.664892i	$0.231523\pi$
$360$	0	0
$361$	−17.0902	−0.899483
$362$	6.38197	0.335429
$363$	0	0
$364$	−1.09017	−0.0571404
$365$	0.236068	0.0123564
$366$	0	0
$367$	19.0557	0.994701	0.497350	−	0.867550i	$-0.334306\pi$
$367$	19.0557	0.994701	0.497350	+	0.867550i	$0.334306\pi$
$368$	−3.47214	−0.180998
$369$	0	0
$370$	−0.381966	−0.0198575
$371$	0.909830	0.0472360
$372$	0	0
$373$	12.5279	0.648668	0.324334	−	0.945943i	$-0.394860\pi$
$373$	12.5279	0.648668	0.324334	+	0.945943i	$0.394860\pi$
$374$	0	0
$375$	0	0
$376$	−1.76393	−0.0909678
$377$	18.8885	0.972809
$378$	0	0
$379$	−20.2918	−1.04232	−0.521160	−	0.853459i	$-0.674500\pi$
$379$	−20.2918	−1.04232	−0.521160	+	0.853459i	$0.674500\pi$
$380$	−1.38197	−0.0708934
$381$	0	0
$382$	−22.7426	−1.16361
$383$	16.5967	0.848054	0.424027	−	0.905650i	$-0.360616\pi$
$383$	16.5967	0.848054	0.424027	+	0.905650i	$0.360616\pi$
$384$	0	0
$385$	0	0
$386$	8.23607	0.419205
$387$	0	0
$388$	−10.3820	−0.527064
$389$	13.7426	0.696780	0.348390	−	0.937350i	$-0.386729\pi$
$389$	13.7426	0.696780	0.348390	+	0.937350i	$0.386729\pi$
$390$	0	0
$391$	−8.27051	−0.418258
$392$	−6.94427	−0.350739
$393$	0	0
$394$	−3.18034	−0.160223
$395$	−4.47214	−0.225018
$396$	0	0
$397$	18.6180	0.934412	0.467206	−	0.884148i	$-0.345261\pi$
$397$	18.6180	0.934412	0.467206	+	0.884148i	$0.345261\pi$
$398$	3.00000	0.150376
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	0	0
$403$	−34.5066	−1.71890
$404$	−10.5279	−0.523781
$405$	0	0
$406$	−0.965558	−0.0479198
$407$	0	0
$408$	0	0
$409$	37.2361	1.84121	0.920603	−	0.390501i	$-0.127698\pi$
$409$	37.2361	1.84121	0.920603	+	0.390501i	$0.127698\pi$
$410$	−6.47214	−0.319636
$411$	0	0
$412$	13.1803	0.649349
$413$	−1.92299	−0.0946239
$414$	0	0
$415$	16.6525	0.817438
$416$	−4.61803	−0.226418
$417$	0	0
$418$	0	0
$419$	−36.8328	−1.79940	−0.899700	−	0.436508i	$-0.856215\pi$
$419$	−36.8328	−1.79940	−0.899700	+	0.436508i	$0.856215\pi$
$420$	0	0
$421$	−10.1459	−0.494481	−0.247240	−	0.968954i	$-0.579524\pi$
$421$	−10.1459	−0.494481	−0.247240	+	0.968954i	$0.579524\pi$
$422$	18.2361	0.887718
$423$	0	0
$424$	3.85410	0.187172
$425$	−9.52786	−0.462169
$426$	0	0
$427$	−3.25735	−0.157634
$428$	1.23607	0.0597476
$429$	0	0
$430$	−2.09017	−0.100797
$431$	−38.2148	−1.84074	−0.920371	−	0.391047i	$-0.872113\pi$
$431$	−38.2148	−1.84074	−0.920371	+	0.391047i	$0.872113\pi$
$432$	0	0
$433$	8.61803	0.414156	0.207078	−	0.978324i	$-0.433605\pi$
$433$	8.61803	0.414156	0.207078	+	0.978324i	$0.433605\pi$
$434$	1.76393	0.0846714
$435$	0	0
$436$	13.5623	0.649517
$437$	−4.79837	−0.229537
$438$	0	0
$439$	5.67376	0.270794	0.135397	−	0.990791i	$-0.456769\pi$
$439$	5.67376	0.270794	0.135397	+	0.990791i	$0.456769\pi$
$440$	0	0
$441$	0	0
$442$	−11.0000	−0.523217
$443$	21.7984	1.03567	0.517836	−	0.855480i	$-0.326738\pi$
$443$	21.7984	1.03567	0.517836	+	0.855480i	$0.326738\pi$
$444$	0	0
$445$	13.7082	0.649831
$446$	−28.7426	−1.36100
$447$	0	0
$448$	0.236068	0.0111532
$449$	10.9656	0.517497	0.258748	−	0.965945i	$-0.416690\pi$
$449$	10.9656	0.517497	0.258748	+	0.965945i	$0.416690\pi$
$450$	0	0
$451$	0	0
$452$	−12.2361	−0.575536
$453$	0	0
$454$	20.5623	0.965037
$455$	1.09017	0.0511080
$456$	0	0
$457$	−21.9787	−1.02812	−0.514060	−	0.857754i	$-0.671859\pi$
$457$	−21.9787	−1.02812	−0.514060	+	0.857754i	$0.671859\pi$
$458$	1.94427	0.0908499
$459$	0	0
$460$	3.47214	0.161889
$461$	17.9787	0.837352	0.418676	−	0.908136i	$-0.362494\pi$
$461$	17.9787	0.837352	0.418676	+	0.908136i	$0.362494\pi$
$462$	0	0
$463$	−1.90983	−0.0887573	−0.0443787	−	0.999015i	$-0.514131\pi$
$463$	−1.90983	−0.0887573	−0.0443787	+	0.999015i	$0.514131\pi$
$464$	−4.09017	−0.189881
$465$	0	0
$466$	−26.2361	−1.21536
$467$	−7.56231	−0.349942	−0.174971	−	0.984574i	$-0.555983\pi$
$467$	−7.56231	−0.349942	−0.174971	+	0.984574i	$0.555983\pi$
$468$	0	0
$469$	−0.304952	−0.0140814
$470$	1.76393	0.0813641
$471$	0	0
$472$	−8.14590	−0.374945
$473$	0	0
$474$	0	0
$475$	−5.52786	−0.253636
$476$	0.562306	0.0257732
$477$	0	0
$478$	17.2361	0.788359
$479$	−0.965558	−0.0441175	−0.0220587	−	0.999757i	$-0.507022\pi$
$479$	−0.965558	−0.0441175	−0.0220587	+	0.999757i	$0.507022\pi$
$480$	0	0
$481$	−1.76393	−0.0804284
$482$	6.41641	0.292259
$483$	0	0
$484$	0	0
$485$	10.3820	0.471421
$486$	0	0
$487$	32.8328	1.48780	0.743898	−	0.668293i	$-0.232975\pi$
$487$	32.8328	1.48780	0.743898	+	0.668293i	$0.232975\pi$
$488$	−13.7984	−0.624623
$489$	0	0
$490$	6.94427	0.313710
$491$	−41.2492	−1.86155	−0.930776	−	0.365591i	$-0.880867\pi$
$491$	−41.2492	−1.86155	−0.930776	+	0.365591i	$0.880867\pi$
$492$	0	0
$493$	−9.74265	−0.438787
$494$	−6.38197	−0.287138
$495$	0	0
$496$	7.47214	0.335509
$497$	−1.79837	−0.0806681
$498$	0	0
$499$	−36.7082	−1.64328	−0.821642	−	0.570003i	$-0.806942\pi$
$499$	−36.7082	−1.64328	−0.821642	+	0.570003i	$0.806942\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	−26.6525	−1.18956
$503$	23.0689	1.02859	0.514295	−	0.857613i	$-0.328054\pi$
$503$	23.0689	1.02859	0.514295	+	0.857613i	$0.328054\pi$
$504$	0	0
$505$	10.5279	0.468484
$506$	0	0
$507$	0	0
$508$	−8.56231	−0.379891
$509$	−26.3262	−1.16689	−0.583445	−	0.812153i	$-0.698296\pi$
$509$	−26.3262	−1.16689	−0.583445	+	0.812153i	$0.698296\pi$
$510$	0	0
$511$	−0.0557281	−0.00246527
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.0344	−1.32476
$515$	−13.1803	−0.580795
$516$	0	0
$517$	0	0
$518$	0.0901699	0.00396184
$519$	0	0
$520$	4.61803	0.202514
$521$	29.8328	1.30700	0.653500	−	0.756927i	$-0.273300\pi$
$521$	29.8328	1.30700	0.653500	+	0.756927i	$0.273300\pi$
$522$	0	0
$523$	5.65248	0.247166	0.123583	−	0.992334i	$-0.460562\pi$
$523$	5.65248	0.247166	0.123583	+	0.992334i	$0.460562\pi$
$524$	3.70820	0.161994
$525$	0	0
$526$	−9.41641	−0.410575
$527$	17.7984	0.775309
$528$	0	0
$529$	−10.9443	−0.475838
$530$	−3.85410	−0.167411
$531$	0	0
$532$	0.326238	0.0141442
$533$	−29.8885	−1.29462
$534$	0	0
$535$	−1.23607	−0.0534399
$536$	−1.29180	−0.0557971
$537$	0	0
$538$	−17.1803	−0.740697
$539$	0	0
$540$	0	0
$541$	32.1803	1.38354	0.691770	−	0.722117i	$-0.256831\pi$
$541$	32.1803	1.38354	0.691770	+	0.722117i	$0.256831\pi$
$542$	−15.2918	−0.656839
$543$	0	0
$544$	2.38197	0.102126
$545$	−13.5623	−0.580945
$546$	0	0
$547$	−8.20163	−0.350676	−0.175338	−	0.984508i	$-0.556102\pi$
$547$	−8.20163	−0.350676	−0.175338	+	0.984508i	$0.556102\pi$
$548$	−13.8541	−0.591818
$549$	0	0
$550$	0	0
$551$	−5.65248	−0.240804
$552$	0	0
$553$	1.05573	0.0448941
$554$	−15.7082	−0.667378
$555$	0	0
$556$	5.47214	0.232070
$557$	39.5066	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$557$	39.5066	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$558$	0	0
$559$	−9.65248	−0.408256
$560$	−0.236068	−0.00997569
$561$	0	0
$562$	−11.5623	−0.487726
$563$	14.4164	0.607579	0.303790	−	0.952739i	$-0.401748\pi$
$563$	14.4164	0.607579	0.303790	+	0.952739i	$0.401748\pi$
$564$	0	0
$565$	12.2361	0.514775
$566$	8.61803	0.362243
$567$	0	0
$568$	−7.61803	−0.319646
$569$	−19.2918	−0.808754	−0.404377	−	0.914592i	$-0.632512\pi$
$569$	−19.2918	−0.808754	−0.404377	+	0.914592i	$0.632512\pi$
$570$	0	0
$571$	24.6525	1.03167	0.515837	−	0.856687i	$-0.327481\pi$
$571$	24.6525	1.03167	0.515837	+	0.856687i	$0.327481\pi$
$572$	0	0
$573$	0	0
$574$	1.52786	0.0637718
$575$	13.8885	0.579192
$576$	0	0
$577$	−11.1803	−0.465444	−0.232722	−	0.972543i	$-0.574763\pi$
$577$	−11.1803	−0.465444	−0.232722	+	0.972543i	$0.574763\pi$
$578$	−11.3262	−0.471109
$579$	0	0
$580$	4.09017	0.169835
$581$	−3.93112	−0.163090
$582$	0	0
$583$	0	0
$584$	−0.236068	−0.00976856
$585$	0	0
$586$	11.8541	0.489688
$587$	−14.9098	−0.615395	−0.307697	−	0.951484i	$-0.599558\pi$
$587$	−14.9098	−0.615395	−0.307697	+	0.951484i	$0.599558\pi$
$588$	0	0
$589$	10.3262	0.425485
$590$	8.14590	0.335361
$591$	0	0
$592$	0.381966	0.0156987
$593$	−25.2148	−1.03545	−0.517723	−	0.855548i	$-0.673220\pi$
$593$	−25.2148	−1.03545	−0.517723	+	0.855548i	$0.673220\pi$
$594$	0	0
$595$	−0.562306	−0.0230523
$596$	−14.7639	−0.604754
$597$	0	0
$598$	16.0344	0.655697
$599$	−19.3820	−0.791926	−0.395963	−	0.918267i	$-0.629589\pi$
$599$	−19.3820	−0.791926	−0.395963	+	0.918267i	$0.629589\pi$
$600$	0	0
$601$	24.6180	1.00419	0.502095	−	0.864812i	$-0.332563\pi$
$601$	24.6180	1.00419	0.502095	+	0.864812i	$0.332563\pi$
$602$	0.493422	0.0201104
$603$	0	0
$604$	−16.4721	−0.670242
$605$	0	0
$606$	0	0
$607$	34.3050	1.39240	0.696198	−	0.717850i	$-0.254874\pi$
$607$	34.3050	1.39240	0.696198	+	0.717850i	$0.254874\pi$
$608$	1.38197	0.0560461
$609$	0	0
$610$	13.7984	0.558680
$611$	8.14590	0.329548
$612$	0	0
$613$	42.7082	1.72497	0.862484	−	0.506084i	$-0.168908\pi$
$613$	42.7082	1.72497	0.862484	+	0.506084i	$0.168908\pi$
$614$	−10.8885	−0.439426
$615$	0	0
$616$	0	0
$617$	−21.1459	−0.851302	−0.425651	−	0.904887i	$-0.639955\pi$
$617$	−21.1459	−0.851302	−0.425651	+	0.904887i	$0.639955\pi$
$618$	0	0
$619$	16.5279	0.664311	0.332155	−	0.943225i	$-0.392224\pi$
$619$	16.5279	0.664311	0.332155	+	0.943225i	$0.392224\pi$
$620$	−7.47214	−0.300088
$621$	0	0
$622$	−15.3820	−0.616761
$623$	−3.23607	−0.129650
$624$	0	0
$625$	11.0000	0.440000
$626$	−27.8328	−1.11242
$627$	0	0
$628$	3.76393	0.150197
$629$	0.909830	0.0362773
$630$	0	0
$631$	−47.6869	−1.89839	−0.949193	−	0.314694i	$-0.898098\pi$
$631$	−47.6869	−1.89839	−0.949193	+	0.314694i	$0.898098\pi$
$632$	4.47214	0.177892
$633$	0	0
$634$	25.8885	1.02817
$635$	8.56231	0.339785
$636$	0	0
$637$	32.0689	1.27062
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−30.2705	−1.19561	−0.597807	−	0.801640i	$-0.703961\pi$
$641$	−30.2705	−1.19561	−0.597807	+	0.801640i	$0.703961\pi$
$642$	0	0
$643$	−41.0344	−1.61824	−0.809120	−	0.587643i	$-0.800056\pi$
$643$	−41.0344	−1.61824	−0.809120	+	0.587643i	$0.800056\pi$
$644$	−0.819660	−0.0322991
$645$	0	0
$646$	3.29180	0.129514
$647$	26.0000	1.02217	0.511083	−	0.859532i	$-0.329245\pi$
$647$	26.0000	1.02217	0.511083	+	0.859532i	$0.329245\pi$
$648$	0	0
$649$	0	0
$650$	18.4721	0.724537
$651$	0	0
$652$	6.00000	0.234978
$653$	−33.8328	−1.32398	−0.661990	−	0.749512i	$-0.730288\pi$
$653$	−33.8328	−1.32398	−0.661990	+	0.749512i	$0.730288\pi$
$654$	0	0
$655$	−3.70820	−0.144892
$656$	6.47214	0.252694
$657$	0	0
$658$	−0.416408	−0.0162333
$659$	−4.41641	−0.172039	−0.0860194	−	0.996293i	$-0.527415\pi$
$659$	−4.41641	−0.172039	−0.0860194	+	0.996293i	$0.527415\pi$
$660$	0	0
$661$	−6.88854	−0.267933	−0.133967	−	0.990986i	$-0.542772\pi$
$661$	−6.88854	−0.267933	−0.133967	+	0.990986i	$0.542772\pi$
$662$	29.0689	1.12979
$663$	0	0
$664$	−16.6525	−0.646241
$665$	−0.326238	−0.0126510
$666$	0	0
$667$	14.2016	0.549889
$668$	4.94427	0.191300
$669$	0	0
$670$	1.29180	0.0499064
$671$	0	0
$672$	0	0
$673$	−9.34752	−0.360321	−0.180160	−	0.983637i	$-0.557662\pi$
$673$	−9.34752	−0.360321	−0.180160	+	0.983637i	$0.557662\pi$
$674$	29.3262	1.12960
$675$	0	0
$676$	8.32624	0.320240
$677$	4.32624	0.166271	0.0831354	−	0.996538i	$-0.473507\pi$
$677$	4.32624	0.166271	0.0831354	+	0.996538i	$0.473507\pi$
$678$	0	0
$679$	−2.45085	−0.0940550
$680$	−2.38197	−0.0913442
$681$	0	0
$682$	0	0
$683$	16.6525	0.637189	0.318595	−	0.947891i	$-0.396789\pi$
$683$	16.6525	0.637189	0.318595	+	0.947891i	$0.396789\pi$
$684$	0	0
$685$	13.8541	0.529338
$686$	−3.29180	−0.125681
$687$	0	0
$688$	2.09017	0.0796870
$689$	−17.7984	−0.678064
$690$	0	0
$691$	−3.02129	−0.114935	−0.0574676	−	0.998347i	$-0.518303\pi$
$691$	−3.02129	−0.114935	−0.0574676	+	0.998347i	$0.518303\pi$
$692$	17.5623	0.667619
$693$	0	0
$694$	−16.1246	−0.612082
$695$	−5.47214	−0.207570
$696$	0	0
$697$	15.4164	0.583938
$698$	23.6525	0.895259
$699$	0	0
$700$	−0.944272	−0.0356901
$701$	−16.9443	−0.639976	−0.319988	−	0.947422i	$-0.603679\pi$
$701$	−16.9443	−0.639976	−0.319988	+	0.947422i	$0.603679\pi$
$702$	0	0
$703$	0.527864	0.0199088
$704$	0	0
$705$	0	0
$706$	−14.9443	−0.562435
$707$	−2.48529	−0.0934690
$708$	0	0
$709$	35.1803	1.32123	0.660613	−	0.750727i	$-0.270297\pi$
$709$	35.1803	1.32123	0.660613	+	0.750727i	$0.270297\pi$
$710$	7.61803	0.285900
$711$	0	0
$712$	−13.7082	−0.513737
$713$	−25.9443	−0.971621
$714$	0	0
$715$	0	0
$716$	19.3607	0.723543
$717$	0	0
$718$	28.3050	1.05633
$719$	8.90983	0.332281	0.166140	−	0.986102i	$-0.446870\pi$
$719$	8.90983	0.332281	0.166140	+	0.986102i	$0.446870\pi$
$720$	0	0
$721$	3.11146	0.115877
$722$	−17.0902	−0.636030
$723$	0	0
$724$	6.38197	0.237184
$725$	16.3607	0.607620
$726$	0	0
$727$	38.2705	1.41937	0.709687	−	0.704517i	$-0.248836\pi$
$727$	38.2705	1.41937	0.709687	+	0.704517i	$0.248836\pi$
$728$	−1.09017	−0.0404044
$729$	0	0
$730$	0.236068	0.00873727
$731$	4.97871	0.184144
$732$	0	0
$733$	−20.2705	−0.748708	−0.374354	−	0.927286i	$-0.622136\pi$
$733$	−20.2705	−0.748708	−0.374354	+	0.927286i	$0.622136\pi$
$734$	19.0557	0.703360
$735$	0	0
$736$	−3.47214	−0.127985
$737$	0	0
$738$	0	0
$739$	21.6525	0.796499	0.398250	−	0.917277i	$-0.369618\pi$
$739$	21.6525	0.796499	0.398250	+	0.917277i	$0.369618\pi$
$740$	−0.381966	−0.0140413
$741$	0	0
$742$	0.909830	0.0334009
$743$	40.5967	1.48935	0.744675	−	0.667427i	$-0.232604\pi$
$743$	40.5967	1.48935	0.744675	+	0.667427i	$0.232604\pi$
$744$	0	0
$745$	14.7639	0.540909
$746$	12.5279	0.458678
$747$	0	0
$748$	0	0
$749$	0.291796	0.0106620
$750$	0	0
$751$	−11.0902	−0.404686	−0.202343	−	0.979315i	$-0.564856\pi$
$751$	−11.0902	−0.404686	−0.202343	+	0.979315i	$0.564856\pi$
$752$	−1.76393	−0.0643240
$753$	0	0
$754$	18.8885	0.687880
$755$	16.4721	0.599482
$756$	0	0
$757$	−9.00000	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$757$	−9.00000	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$758$	−20.2918	−0.737031
$759$	0	0
$760$	−1.38197	−0.0501292
$761$	46.1246	1.67202	0.836008	−	0.548717i	$-0.184884\pi$
$761$	46.1246	1.67202	0.836008	+	0.548717i	$0.184884\pi$
$762$	0	0
$763$	3.20163	0.115907
$764$	−22.7426	−0.822800
$765$	0	0
$766$	16.5967	0.599665
$767$	37.6180	1.35831
$768$	0	0
$769$	−22.3951	−0.807589	−0.403794	−	0.914850i	$-0.632309\pi$
$769$	−22.3951	−0.807589	−0.403794	+	0.914850i	$0.632309\pi$
$770$	0	0
$771$	0	0
$772$	8.23607	0.296423
$773$	48.0344	1.72768	0.863839	−	0.503767i	$-0.168053\pi$
$773$	48.0344	1.72768	0.863839	+	0.503767i	$0.168053\pi$
$774$	0	0
$775$	−29.8885	−1.07363
$776$	−10.3820	−0.372691
$777$	0	0
$778$	13.7426	0.492698
$779$	8.94427	0.320462
$780$	0	0
$781$	0	0
$782$	−8.27051	−0.295753
$783$	0	0
$784$	−6.94427	−0.248010
$785$	−3.76393	−0.134340
$786$	0	0
$787$	47.3607	1.68823	0.844113	−	0.536165i	$-0.180128\pi$
$787$	47.3607	1.68823	0.844113	+	0.536165i	$0.180128\pi$
$788$	−3.18034	−0.113295
$789$	0	0
$790$	−4.47214	−0.159111
$791$	−2.88854	−0.102705
$792$	0	0
$793$	63.7214	2.26281
$794$	18.6180	0.660729
$795$	0	0
$796$	3.00000	0.106332
$797$	35.5967	1.26090	0.630451	−	0.776229i	$-0.282870\pi$
$797$	35.5967	1.26090	0.630451	+	0.776229i	$0.282870\pi$
$798$	0	0
$799$	−4.20163	−0.148643
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−5.00000	−0.176556
$803$	0	0
$804$	0	0
$805$	0.819660	0.0288892
$806$	−34.5066	−1.21544
$807$	0	0
$808$	−10.5279	−0.370369
$809$	22.8328	0.802759	0.401380	−	0.915912i	$-0.368531\pi$
$809$	22.8328	0.802759	0.401380	+	0.915912i	$0.368531\pi$
$810$	0	0
$811$	−15.3607	−0.539386	−0.269693	−	0.962946i	$-0.586922\pi$
$811$	−15.3607	−0.539386	−0.269693	+	0.962946i	$0.586922\pi$
$812$	−0.965558	−0.0338844
$813$	0	0
$814$	0	0
$815$	−6.00000	−0.210171
$816$	0	0
$817$	2.88854	0.101057
$818$	37.2361	1.30193
$819$	0	0
$820$	−6.47214	−0.226017
$821$	−50.3050	−1.75565	−0.877827	−	0.478977i	$-0.841008\pi$
$821$	−50.3050	−1.75565	−0.877827	+	0.478977i	$0.841008\pi$
$822$	0	0
$823$	42.3262	1.47540	0.737700	−	0.675129i	$-0.235912\pi$
$823$	42.3262	1.47540	0.737700	+	0.675129i	$0.235912\pi$
$824$	13.1803	0.459159
$825$	0	0
$826$	−1.92299	−0.0669092
$827$	5.83282	0.202827	0.101413	−	0.994844i	$-0.467664\pi$
$827$	5.83282	0.202827	0.101413	+	0.994844i	$0.467664\pi$
$828$	0	0
$829$	18.3262	0.636497	0.318248	−	0.948007i	$-0.396905\pi$
$829$	18.3262	0.636497	0.318248	+	0.948007i	$0.396905\pi$
$830$	16.6525	0.578016
$831$	0	0
$832$	−4.61803	−0.160102
$833$	−16.5410	−0.573112
$834$	0	0
$835$	−4.94427	−0.171104
$836$	0	0
$837$	0	0
$838$	−36.8328	−1.27237
$839$	−27.8541	−0.961630	−0.480815	−	0.876822i	$-0.659659\pi$
$839$	−27.8541	−0.961630	−0.480815	+	0.876822i	$0.659659\pi$
$840$	0	0
$841$	−12.2705	−0.423121
$842$	−10.1459	−0.349651
$843$	0	0
$844$	18.2361	0.627711
$845$	−8.32624	−0.286431
$846$	0	0
$847$	0	0
$848$	3.85410	0.132350
$849$	0	0
$850$	−9.52786	−0.326803
$851$	−1.32624	−0.0454629
$852$	0	0
$853$	−7.74265	−0.265103	−0.132552	−	0.991176i	$-0.542317\pi$
$853$	−7.74265	−0.265103	−0.132552	+	0.991176i	$0.542317\pi$
$854$	−3.25735	−0.111464
$855$	0	0
$856$	1.23607	0.0422479
$857$	−8.25735	−0.282066	−0.141033	−	0.990005i	$-0.545042\pi$
$857$	−8.25735	−0.282066	−0.141033	+	0.990005i	$0.545042\pi$
$858$	0	0
$859$	19.0689	0.650622	0.325311	−	0.945607i	$-0.394531\pi$
$859$	19.0689	0.650622	0.325311	+	0.945607i	$0.394531\pi$
$860$	−2.09017	−0.0712742
$861$	0	0
$862$	−38.2148	−1.30160
$863$	−4.88854	−0.166408	−0.0832040	−	0.996533i	$-0.526515\pi$
$863$	−4.88854	−0.166408	−0.0832040	+	0.996533i	$0.526515\pi$
$864$	0	0
$865$	−17.5623	−0.597136
$866$	8.61803	0.292853
$867$	0	0
$868$	1.76393	0.0598718
$869$	0	0
$870$	0	0
$871$	5.96556	0.202135
$872$	13.5623	0.459278
$873$	0	0
$874$	−4.79837	−0.162307
$875$	2.12461	0.0718250
$876$	0	0
$877$	54.3607	1.83563	0.917815	−	0.397009i	$-0.129952\pi$
$877$	54.3607	1.83563	0.917815	+	0.397009i	$0.129952\pi$
$878$	5.67376	0.191480
$879$	0	0
$880$	0	0
$881$	16.7082	0.562914	0.281457	−	0.959574i	$-0.409182\pi$
$881$	16.7082	0.562914	0.281457	+	0.959574i	$0.409182\pi$
$882$	0	0
$883$	−14.3475	−0.482833	−0.241416	−	0.970422i	$-0.577612\pi$
$883$	−14.3475	−0.482833	−0.241416	+	0.970422i	$0.577612\pi$
$884$	−11.0000	−0.369970
$885$	0	0
$886$	21.7984	0.732331
$887$	−28.9787	−0.973010	−0.486505	−	0.873678i	$-0.661728\pi$
$887$	−28.9787	−0.973010	−0.486505	+	0.873678i	$0.661728\pi$
$888$	0	0
$889$	−2.02129	−0.0677918
$890$	13.7082	0.459500
$891$	0	0
$892$	−28.7426	−0.962375
$893$	−2.43769	−0.0815743
$894$	0	0
$895$	−19.3607	−0.647156
$896$	0.236068	0.00788648
$897$	0	0
$898$	10.9656	0.365925
$899$	−30.5623	−1.01931
$900$	0	0
$901$	9.18034	0.305841
$902$	0	0
$903$	0	0
$904$	−12.2361	−0.406966
$905$	−6.38197	−0.212144
$906$	0	0
$907$	−12.0557	−0.400304	−0.200152	−	0.979765i	$-0.564144\pi$
$907$	−12.0557	−0.400304	−0.200152	+	0.979765i	$0.564144\pi$
$908$	20.5623	0.682384
$909$	0	0
$910$	1.09017	0.0361388
$911$	15.4164	0.510768	0.255384	−	0.966840i	$-0.417798\pi$
$911$	15.4164	0.510768	0.255384	+	0.966840i	$0.417798\pi$
$912$	0	0
$913$	0	0
$914$	−21.9787	−0.726991
$915$	0	0
$916$	1.94427	0.0642406
$917$	0.875388	0.0289079
$918$	0	0
$919$	40.9787	1.35176	0.675882	−	0.737010i	$-0.263763\pi$
$919$	40.9787	1.35176	0.675882	+	0.737010i	$0.263763\pi$
$920$	3.47214	0.114473
$921$	0	0
$922$	17.9787	0.592097
$923$	35.1803	1.15797
$924$	0	0
$925$	−1.52786	−0.0502359
$926$	−1.90983	−0.0627609
$927$	0	0
$928$	−4.09017	−0.134266
$929$	53.6656	1.76071	0.880356	−	0.474313i	$-0.157304\pi$
$929$	53.6656	1.76071	0.880356	+	0.474313i	$0.157304\pi$
$930$	0	0
$931$	−9.59675	−0.314521
$932$	−26.2361	−0.859391
$933$	0	0
$934$	−7.56231	−0.247446
$935$	0	0
$936$	0	0
$937$	36.0902	1.17901	0.589507	−	0.807763i	$-0.299322\pi$
$937$	36.0902	1.17901	0.589507	+	0.807763i	$0.299322\pi$
$938$	−0.304952	−0.00995703
$939$	0	0
$940$	1.76393	0.0575331
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	−22.4721	−0.731793
$944$	−8.14590	−0.265126
$945$	0	0
$946$	0	0
$947$	10.9230	0.354949	0.177475	−	0.984125i	$-0.443207\pi$
$947$	10.9230	0.354949	0.177475	+	0.984125i	$0.443207\pi$
$948$	0	0
$949$	1.09017	0.0353884
$950$	−5.52786	−0.179348
$951$	0	0
$952$	0.562306	0.0182244
$953$	52.8673	1.71254	0.856269	−	0.516530i	$-0.172776\pi$
$953$	52.8673	1.71254	0.856269	+	0.516530i	$0.172776\pi$
$954$	0	0
$955$	22.7426	0.735935
$956$	17.2361	0.557454
$957$	0	0
$958$	−0.965558	−0.0311958
$959$	−3.27051	−0.105610
$960$	0	0
$961$	24.8328	0.801059
$962$	−1.76393	−0.0568715
$963$	0	0
$964$	6.41641	0.206659
$965$	−8.23607	−0.265128
$966$	0	0
$967$	−7.85410	−0.252571	−0.126285	−	0.991994i	$-0.540306\pi$
$967$	−7.85410	−0.252571	−0.126285	+	0.991994i	$0.540306\pi$
$968$	0	0
$969$	0	0
$970$	10.3820	0.333345
$971$	17.7639	0.570072	0.285036	−	0.958517i	$-0.407994\pi$
$971$	17.7639	0.570072	0.285036	+	0.958517i	$0.407994\pi$
$972$	0	0
$973$	1.29180	0.0414131
$974$	32.8328	1.05203
$975$	0	0
$976$	−13.7984	−0.441675
$977$	−9.11146	−0.291501	−0.145751	−	0.989321i	$-0.546560\pi$
$977$	−9.11146	−0.291501	−0.145751	+	0.989321i	$0.546560\pi$
$978$	0	0
$979$	0	0
$980$	6.94427	0.221827
$981$	0	0
$982$	−41.2492	−1.31632
$983$	22.0000	0.701691	0.350846	−	0.936433i	$-0.385894\pi$
$983$	22.0000	0.701691	0.350846	+	0.936433i	$0.385894\pi$
$984$	0	0
$985$	3.18034	0.101334
$986$	−9.74265	−0.310269
$987$	0	0
$988$	−6.38197	−0.203037
$989$	−7.25735	−0.230770
$990$	0	0
$991$	−3.58359	−0.113837	−0.0569183	−	0.998379i	$-0.518127\pi$
$991$	−3.58359	−0.113837	−0.0569183	+	0.998379i	$0.518127\pi$
$992$	7.47214	0.237241
$993$	0	0
$994$	−1.79837	−0.0570410
$995$	−3.00000	−0.0951064
$996$	0	0
$997$	−2.87539	−0.0910645	−0.0455322	−	0.998963i	$-0.514498\pi$
$997$	−2.87539	−0.0910645	−0.0455322	+	0.998963i	$0.514498\pi$
$998$	−36.7082	−1.16198
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6534.2.a.bz.1.2		2
3.2	odd	2		6534.2.a.bt.1.2		2
11.7	odd	10		594.2.f.h.379.1	yes	4
11.8	odd	10		594.2.f.h.163.1	yes	4
11.10	odd	2		6534.2.a.bg.1.1		2
33.8	even	10		594.2.f.c.163.1	✓	4
33.29	even	10		594.2.f.c.379.1	yes	4
33.32	even	2		6534.2.a.cn.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
594.2.f.c.163.1	✓	4	33.8	even	10
594.2.f.c.379.1	yes	4	33.29	even	10
594.2.f.h.163.1	yes	4	11.8	odd	10
594.2.f.h.379.1	yes	4	11.7	odd	10
6534.2.a.bg.1.1		2	11.10	odd	2
6534.2.a.bt.1.2		2	3.2	odd	2
6534.2.a.bz.1.2		2	1.1	even	1	trivial
6534.2.a.cn.1.1		2	33.32	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:	\( N \)	\(=\)	\( 6534 = 2 \cdot 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6534.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.236068 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.236068 q^{7} +1.00000 q^{8} -1.00000 q^{10} -4.61803 q^{13} +0.236068 q^{14} +1.00000 q^{16} +2.38197 q^{17} +1.38197 q^{19} -1.00000 q^{20} -3.47214 q^{23} -4.00000 q^{25} -4.61803 q^{26} +0.236068 q^{28} -4.09017 q^{29} +7.47214 q^{31} +1.00000 q^{32} +2.38197 q^{34} -0.236068 q^{35} +0.381966 q^{37} +1.38197 q^{38} -1.00000 q^{40} +6.47214 q^{41} +2.09017 q^{43} -3.47214 q^{46} -1.76393 q^{47} -6.94427 q^{49} -4.00000 q^{50} -4.61803 q^{52} +3.85410 q^{53} +0.236068 q^{56} -4.09017 q^{58} -8.14590 q^{59} -13.7984 q^{61} +7.47214 q^{62} +1.00000 q^{64} +4.61803 q^{65} -1.29180 q^{67} +2.38197 q^{68} -0.236068 q^{70} -7.61803 q^{71} -0.236068 q^{73} +0.381966 q^{74} +1.38197 q^{76} +4.47214 q^{79} -1.00000 q^{80} +6.47214 q^{82} -16.6525 q^{83} -2.38197 q^{85} +2.09017 q^{86} -13.7082 q^{89} -1.09017 q^{91} -3.47214 q^{92} -1.76393 q^{94} -1.38197 q^{95} -10.3820 q^{97} -6.94427 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 2 q^{8} - 2 q^{10} - 7 q^{13} - 4 q^{14} + 2 q^{16} + 7 q^{17} + 5 q^{19} - 2 q^{20} + 2 q^{23} - 8 q^{25} - 7 q^{26} - 4 q^{28} + 3 q^{29} + 6 q^{31} + 2 q^{32} + 7 q^{34} + 4 q^{35} + 3 q^{37} + 5 q^{38} - 2 q^{40} + 4 q^{41} - 7 q^{43} + 2 q^{46} - 8 q^{47} + 4 q^{49} - 8 q^{50} - 7 q^{52} + q^{53} - 4 q^{56} + 3 q^{58} - 23 q^{59} - 3 q^{61} + 6 q^{62} + 2 q^{64} + 7 q^{65} - 16 q^{67} + 7 q^{68} + 4 q^{70} - 13 q^{71} + 4 q^{73} + 3 q^{74} + 5 q^{76} - 2 q^{80} + 4 q^{82} - 2 q^{83} - 7 q^{85} - 7 q^{86} - 14 q^{89} + 9 q^{91} + 2 q^{92} - 8 q^{94} - 5 q^{95} - 23 q^{97} + 4 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 2 * q^8 - 2 * q^10 - 7 * q^13 - 4 * q^14 + 2 * q^16 + 7 * q^17 + 5 * q^19 - 2 * q^20 + 2 * q^23 - 8 * q^25 - 7 * q^26 - 4 * q^28 + 3 * q^29 + 6 * q^31 + 2 * q^32 + 7 * q^34 + 4 * q^35 + 3 * q^37 + 5 * q^38 - 2 * q^40 + 4 * q^41 - 7 * q^43 + 2 * q^46 - 8 * q^47 + 4 * q^49 - 8 * q^50 - 7 * q^52 + q^53 - 4 * q^56 + 3 * q^58 - 23 * q^59 - 3 * q^61 + 6 * q^62 + 2 * q^64 + 7 * q^65 - 16 * q^67 + 7 * q^68 + 4 * q^70 - 13 * q^71 + 4 * q^73 + 3 * q^74 + 5 * q^76 - 2 * q^80 + 4 * q^82 - 2 * q^83 - 7 * q^85 - 7 * q^86 - 14 * q^89 + 9 * q^91 + 2 * q^92 - 8 * q^94 - 5 * q^95 - 23 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

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