LMFDB - Embedded newform 3465.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3465,2,Mod(1,3465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3465, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3465.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:	$27.6681643004$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	3465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -9.46410 q^{8} +O(q^{10})$	\(q-2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -9.46410 q^{8} -2.73205 q^{10} +1.00000 q^{11} +6.19615 q^{13} -2.73205 q^{14} +14.9282 q^{16} -0.464102 q^{17} -8.46410 q^{19} +5.46410 q^{20} -2.73205 q^{22} -1.53590 q^{23} +1.00000 q^{25} -16.9282 q^{26} +5.46410 q^{28} -2.26795 q^{29} -8.73205 q^{31} -21.8564 q^{32} +1.26795 q^{34} +1.00000 q^{35} +6.19615 q^{37} +23.1244 q^{38} -9.46410 q^{40} -9.66025 q^{41} -1.73205 q^{43} +5.46410 q^{44} +4.19615 q^{46} -4.73205 q^{47} +1.00000 q^{49} -2.73205 q^{50} +33.8564 q^{52} +2.46410 q^{53} +1.00000 q^{55} -9.46410 q^{56} +6.19615 q^{58} -10.2679 q^{59} -9.39230 q^{61} +23.8564 q^{62} +29.8564 q^{64} +6.19615 q^{65} -8.92820 q^{67} -2.53590 q^{68} -2.73205 q^{70} +9.66025 q^{71} -12.3923 q^{73} -16.9282 q^{74} -46.2487 q^{76} +1.00000 q^{77} -13.6603 q^{79} +14.9282 q^{80} +26.3923 q^{82} +4.46410 q^{83} -0.464102 q^{85} +4.73205 q^{86} -9.46410 q^{88} +2.66025 q^{89} +6.19615 q^{91} -8.39230 q^{92} +12.9282 q^{94} -8.46410 q^{95} -5.73205 q^{97} -2.73205 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8	$ 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 16 q^{16} + 6 q^{17} - 10 q^{19} + 4 q^{20} - 2 q^{22} - 10 q^{23} + 2 q^{25} - 20 q^{26} + 4 q^{28} - 8 q^{29} - 14 q^{31} - 16 q^{32} + 6 q^{34} + 2 q^{35} + 2 q^{37} + 22 q^{38} - 12 q^{40} - 2 q^{41} + 4 q^{44} - 2 q^{46} - 6 q^{47} + 2 q^{49} - 2 q^{50} + 40 q^{52} - 2 q^{53} + 2 q^{55} - 12 q^{56} + 2 q^{58} - 24 q^{59} + 2 q^{61} + 20 q^{62} + 32 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 2 q^{70} + 2 q^{71} - 4 q^{73} - 20 q^{74} - 44 q^{76} + 2 q^{77} - 10 q^{79} + 16 q^{80} + 32 q^{82} + 2 q^{83} + 6 q^{85} + 6 q^{86} - 12 q^{88} - 12 q^{89} + 2 q^{91} + 4 q^{92} + 12 q^{94} - 10 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 - 2 * q^10 + 2 * q^11 + 2 * q^13 - 2 * q^14 + 16 * q^16 + 6 * q^17 - 10 * q^19 + 4 * q^20 - 2 * q^22 - 10 * q^23 + 2 * q^25 - 20 * q^26 + 4 * q^28 - 8 * q^29 - 14 * q^31 - 16 * q^32 + 6 * q^34 + 2 * q^35 + 2 * q^37 + 22 * q^38 - 12 * q^40 - 2 * q^41 + 4 * q^44 - 2 * q^46 - 6 * q^47 + 2 * q^49 - 2 * q^50 + 40 * q^52 - 2 * q^53 + 2 * q^55 - 12 * q^56 + 2 * q^58 - 24 * q^59 + 2 * q^61 + 20 * q^62 + 32 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 2 * q^70 + 2 * q^71 - 4 * q^73 - 20 * q^74 - 44 * q^76 + 2 * q^77 - 10 * q^79 + 16 * q^80 + 32 * q^82 + 2 * q^83 + 6 * q^85 + 6 * q^86 - 12 * q^88 - 12 * q^89 + 2 * q^91 + 4 * q^92 + 12 * q^94 - 10 * q^95 - 8 * q^97 - 2 * 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 $

$2$	−2.73205	−1.93185	−0.965926	−	0.258819i	$-0.916667\pi$
$2$	−2.73205	−1.93185	−0.965926	+	0.258819i	$0.916667\pi$
$3$	0	0
$3$	0	0
$4$	5.46410	2.73205
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−9.46410	−3.34607
$9$	0	0
$10$	−2.73205	−0.863950
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.19615	1.71850	0.859252	−	0.511553i	$-0.170930\pi$
$13$	6.19615	1.71850	0.859252	+	0.511553i	$0.170930\pi$
$14$	−2.73205	−0.730171
$15$	0	0
$16$	14.9282	3.73205
$17$	−0.464102	−0.112561	−0.0562806	−	0.998415i	$-0.517924\pi$
$17$	−0.464102	−0.112561	−0.0562806	+	0.998415i	$0.517924\pi$
$18$	0	0
$19$	−8.46410	−1.94180	−0.970899	−	0.239489i	$-0.923020\pi$
$19$	−8.46410	−1.94180	−0.970899	+	0.239489i	$0.923020\pi$
$20$	5.46410	1.22181
$21$	0	0
$22$	−2.73205	−0.582475
$23$	−1.53590	−0.320257	−0.160128	−	0.987096i	$-0.551191\pi$
$23$	−1.53590	−0.320257	−0.160128	+	0.987096i	$0.551191\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−16.9282	−3.31989
$27$	0	0
$28$	5.46410	1.03262
$29$	−2.26795	−0.421148	−0.210574	−	0.977578i	$-0.567533\pi$
$29$	−2.26795	−0.421148	−0.210574	+	0.977578i	$0.567533\pi$
$30$	0	0
$31$	−8.73205	−1.56832	−0.784161	−	0.620557i	$-0.786907\pi$
$31$	−8.73205	−1.56832	−0.784161	+	0.620557i	$0.786907\pi$
$32$	−21.8564	−3.86370
$33$	0	0
$34$	1.26795	0.217451
$35$	1.00000	0.169031
$36$	0	0
$37$	6.19615	1.01864	0.509321	−	0.860577i	$-0.329897\pi$
$37$	6.19615	1.01864	0.509321	+	0.860577i	$0.329897\pi$
$38$	23.1244	3.75127
$39$	0	0
$40$	−9.46410	−1.49641
$41$	−9.66025	−1.50868	−0.754339	−	0.656485i	$-0.772043\pi$
$41$	−9.66025	−1.50868	−0.754339	+	0.656485i	$0.772043\pi$
$42$	0	0
$43$	−1.73205	−0.264135	−0.132068	−	0.991241i	$-0.542162\pi$
$43$	−1.73205	−0.264135	−0.132068	+	0.991241i	$0.542162\pi$
$44$	5.46410	0.823744
$45$	0	0
$46$	4.19615	0.618689
$47$	−4.73205	−0.690241	−0.345120	−	0.938558i	$-0.612162\pi$
$47$	−4.73205	−0.690241	−0.345120	+	0.938558i	$0.612162\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.73205	−0.386370
$51$	0	0
$52$	33.8564	4.69504
$53$	2.46410	0.338470	0.169235	−	0.985576i	$-0.445870\pi$
$53$	2.46410	0.338470	0.169235	+	0.985576i	$0.445870\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−9.46410	−1.26469
$57$	0	0
$58$	6.19615	0.813595
$59$	−10.2679	−1.33677	−0.668387	−	0.743814i	$-0.733015\pi$
$59$	−10.2679	−1.33677	−0.668387	+	0.743814i	$0.733015\pi$
$60$	0	0
$61$	−9.39230	−1.20256	−0.601281	−	0.799038i	$-0.705343\pi$
$61$	−9.39230	−1.20256	−0.601281	+	0.799038i	$0.705343\pi$
$62$	23.8564	3.02977
$63$	0	0
$64$	29.8564	3.73205
$65$	6.19615	0.768538
$66$	0	0
$67$	−8.92820	−1.09075	−0.545377	−	0.838191i	$-0.683613\pi$
$67$	−8.92820	−1.09075	−0.545377	+	0.838191i	$0.683613\pi$
$68$	−2.53590	−0.307523
$69$	0	0
$70$	−2.73205	−0.326543
$71$	9.66025	1.14646	0.573231	−	0.819394i	$-0.305690\pi$
$71$	9.66025	1.14646	0.573231	+	0.819394i	$0.305690\pi$
$72$	0	0
$73$	−12.3923	−1.45041	−0.725205	−	0.688533i	$-0.758255\pi$
$73$	−12.3923	−1.45041	−0.725205	+	0.688533i	$0.758255\pi$
$74$	−16.9282	−1.96786
$75$	0	0
$76$	−46.2487	−5.30509
$77$	1.00000	0.113961
$78$	0	0
$79$	−13.6603	−1.53690	−0.768449	−	0.639911i	$-0.778971\pi$
$79$	−13.6603	−1.53690	−0.768449	+	0.639911i	$0.778971\pi$
$80$	14.9282	1.66902
$81$	0	0
$82$	26.3923	2.91454
$83$	4.46410	0.489999	0.244999	−	0.969523i	$-0.421212\pi$
$83$	4.46410	0.489999	0.244999	+	0.969523i	$0.421212\pi$
$84$	0	0
$85$	−0.464102	−0.0503389
$86$	4.73205	0.510270
$87$	0	0
$88$	−9.46410	−1.00888
$89$	2.66025	0.281986	0.140993	−	0.990011i	$-0.454970\pi$
$89$	2.66025	0.281986	0.140993	+	0.990011i	$0.454970\pi$
$90$	0	0
$91$	6.19615	0.649533
$92$	−8.39230	−0.874958
$93$	0	0
$94$	12.9282	1.33344
$95$	−8.46410	−0.868399
$96$	0	0
$97$	−5.73205	−0.582002	−0.291001	−	0.956723i	$-0.593988\pi$
$97$	−5.73205	−0.582002	−0.291001	+	0.956723i	$0.593988\pi$
$98$	−2.73205	−0.275979
$99$	0	0
$100$	5.46410	0.546410
$101$	−3.07180	−0.305655	−0.152828	−	0.988253i	$-0.548838\pi$
$101$	−3.07180	−0.305655	−0.152828	+	0.988253i	$0.548838\pi$
$102$	0	0
$103$	−17.7321	−1.74719	−0.873595	−	0.486653i	$-0.838218\pi$
$103$	−17.7321	−1.74719	−0.873595	+	0.486653i	$0.838218\pi$
$104$	−58.6410	−5.75022
$105$	0	0
$106$	−6.73205	−0.653875
$107$	−8.73205	−0.844159	−0.422080	−	0.906559i	$-0.638700\pi$
$107$	−8.73205	−0.844159	−0.422080	+	0.906559i	$0.638700\pi$
$108$	0	0
$109$	−8.73205	−0.836379	−0.418189	−	0.908360i	$-0.637335\pi$
$109$	−8.73205	−0.836379	−0.418189	+	0.908360i	$0.637335\pi$
$110$	−2.73205	−0.260491
$111$	0	0
$112$	14.9282	1.41058
$113$	16.8564	1.58572	0.792859	−	0.609406i	$-0.208592\pi$
$113$	16.8564	1.58572	0.792859	+	0.609406i	$0.208592\pi$
$114$	0	0
$115$	−1.53590	−0.143223
$116$	−12.3923	−1.15060
$117$	0	0
$118$	28.0526	2.58245
$119$	−0.464102	−0.0425441
$120$	0	0
$121$	1.00000	0.0909091
$122$	25.6603	2.32317
$123$	0	0
$124$	−47.7128	−4.28474
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.12436	0.188506	0.0942530	−	0.995548i	$-0.469954\pi$
$127$	2.12436	0.188506	0.0942530	+	0.995548i	$0.469954\pi$
$128$	−37.8564	−3.34607
$129$	0	0
$130$	−16.9282	−1.48470
$131$	−5.80385	−0.507085	−0.253542	−	0.967324i	$-0.581596\pi$
$131$	−5.80385	−0.507085	−0.253542	+	0.967324i	$0.581596\pi$
$132$	0	0
$133$	−8.46410	−0.733931
$134$	24.3923	2.10717
$135$	0	0
$136$	4.39230	0.376637
$137$	13.4641	1.15032	0.575158	−	0.818042i	$-0.304941\pi$
$137$	13.4641	1.15032	0.575158	+	0.818042i	$0.304941\pi$
$138$	0	0
$139$	4.53590	0.384730	0.192365	−	0.981323i	$-0.438384\pi$
$139$	4.53590	0.384730	0.192365	+	0.981323i	$0.438384\pi$
$140$	5.46410	0.461801
$141$	0	0
$142$	−26.3923	−2.21479
$143$	6.19615	0.518148
$144$	0	0
$145$	−2.26795	−0.188343
$146$	33.8564	2.80198
$147$	0	0
$148$	33.8564	2.78298
$149$	−8.53590	−0.699288	−0.349644	−	0.936883i	$-0.613697\pi$
$149$	−8.53590	−0.699288	−0.349644	+	0.936883i	$0.613697\pi$
$150$	0	0
$151$	15.8564	1.29038	0.645188	−	0.764024i	$-0.276779\pi$
$151$	15.8564	1.29038	0.645188	+	0.764024i	$0.276779\pi$
$152$	80.1051	6.49738
$153$	0	0
$154$	−2.73205	−0.220155
$155$	−8.73205	−0.701375
$156$	0	0
$157$	10.1244	0.808012	0.404006	−	0.914756i	$-0.367618\pi$
$157$	10.1244	0.808012	0.404006	+	0.914756i	$0.367618\pi$
$158$	37.3205	2.96906
$159$	0	0
$160$	−21.8564	−1.72790
$161$	−1.53590	−0.121046
$162$	0	0
$163$	−5.12436	−0.401371	−0.200685	−	0.979656i	$-0.564317\pi$
$163$	−5.12436	−0.401371	−0.200685	+	0.979656i	$0.564317\pi$
$164$	−52.7846	−4.12179
$165$	0	0
$166$	−12.1962	−0.946605
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	0	0
$169$	25.3923	1.95325
$170$	1.26795	0.0972473
$171$	0	0
$172$	−9.46410	−0.721631
$173$	18.5359	1.40926	0.704629	−	0.709576i	$-0.251113\pi$
$173$	18.5359	1.40926	0.704629	+	0.709576i	$0.251113\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	14.9282	1.12526
$177$	0	0
$178$	−7.26795	−0.544756
$179$	−18.0526	−1.34931	−0.674656	−	0.738132i	$-0.735708\pi$
$179$	−18.0526	−1.34931	−0.674656	+	0.738132i	$0.735708\pi$
$180$	0	0
$181$	20.9282	1.55558	0.777791	−	0.628524i	$-0.216340\pi$
$181$	20.9282	1.55558	0.777791	+	0.628524i	$0.216340\pi$
$182$	−16.9282	−1.25480
$183$	0	0
$184$	14.5359	1.07160
$185$	6.19615	0.455550
$186$	0	0
$187$	−0.464102	−0.0339385
$188$	−25.8564	−1.88577
$189$	0	0
$190$	23.1244	1.67762
$191$	14.9282	1.08017	0.540083	−	0.841611i	$-0.318393\pi$
$191$	14.9282	1.08017	0.540083	+	0.841611i	$0.318393\pi$
$192$	0	0
$193$	−0.535898	−0.0385748	−0.0192874	−	0.999814i	$-0.506140\pi$
$193$	−0.535898	−0.0385748	−0.0192874	+	0.999814i	$0.506140\pi$
$194$	15.6603	1.12434
$195$	0	0
$196$	5.46410	0.390293
$197$	1.46410	0.104313	0.0521565	−	0.998639i	$-0.483391\pi$
$197$	1.46410	0.104313	0.0521565	+	0.998639i	$0.483391\pi$
$198$	0	0
$199$	−1.07180	−0.0759777	−0.0379888	−	0.999278i	$-0.512095\pi$
$199$	−1.07180	−0.0759777	−0.0379888	+	0.999278i	$0.512095\pi$
$200$	−9.46410	−0.669213
$201$	0	0
$202$	8.39230	0.590481
$203$	−2.26795	−0.159179
$204$	0	0
$205$	−9.66025	−0.674701
$206$	48.4449	3.37531
$207$	0	0
$208$	92.4974	6.41354
$209$	−8.46410	−0.585474
$210$	0	0
$211$	11.4641	0.789221	0.394611	−	0.918848i	$-0.370879\pi$
$211$	11.4641	0.789221	0.394611	+	0.918848i	$0.370879\pi$
$212$	13.4641	0.924718
$213$	0	0
$214$	23.8564	1.63079
$215$	−1.73205	−0.118125
$216$	0	0
$217$	−8.73205	−0.592770
$218$	23.8564	1.61576
$219$	0	0
$220$	5.46410	0.368390
$221$	−2.87564	−0.193437
$222$	0	0
$223$	11.1962	0.749750	0.374875	−	0.927075i	$-0.377686\pi$
$223$	11.1962	0.749750	0.374875	+	0.927075i	$0.377686\pi$
$224$	−21.8564	−1.46034
$225$	0	0
$226$	−46.0526	−3.06337
$227$	−15.3923	−1.02162	−0.510812	−	0.859693i	$-0.670655\pi$
$227$	−15.3923	−1.02162	−0.510812	+	0.859693i	$0.670655\pi$
$228$	0	0
$229$	17.8038	1.17651	0.588256	−	0.808675i	$-0.299815\pi$
$229$	17.8038	1.17651	0.588256	+	0.808675i	$0.299815\pi$
$230$	4.19615	0.276686
$231$	0	0
$232$	21.4641	1.40919
$233$	−0.196152	−0.0128504	−0.00642519	−	0.999979i	$-0.502045\pi$
$233$	−0.196152	−0.0128504	−0.00642519	+	0.999979i	$0.502045\pi$
$234$	0	0
$235$	−4.73205	−0.308685
$236$	−56.1051	−3.65213
$237$	0	0
$238$	1.26795	0.0821889
$239$	6.12436	0.396152	0.198076	−	0.980187i	$-0.436531\pi$
$239$	6.12436	0.396152	0.198076	+	0.980187i	$0.436531\pi$
$240$	0	0
$241$	13.0718	0.842028	0.421014	−	0.907054i	$-0.361674\pi$
$241$	13.0718	0.842028	0.421014	+	0.907054i	$0.361674\pi$
$242$	−2.73205	−0.175623
$243$	0	0
$244$	−51.3205	−3.28546
$245$	1.00000	0.0638877
$246$	0	0
$247$	−52.4449	−3.33699
$248$	82.6410	5.24771
$249$	0	0
$250$	−2.73205	−0.172790
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	−1.53590	−0.0965611
$254$	−5.80385	−0.364166
$255$	0	0
$256$	43.7128	2.73205
$257$	−24.1962	−1.50931	−0.754657	−	0.656119i	$-0.772197\pi$
$257$	−24.1962	−1.50931	−0.754657	+	0.656119i	$0.772197\pi$
$258$	0	0
$259$	6.19615	0.385010
$260$	33.8564	2.09969
$261$	0	0
$262$	15.8564	0.979612
$263$	23.3205	1.43800	0.719002	−	0.695008i	$-0.244599\pi$
$263$	23.3205	1.43800	0.719002	+	0.695008i	$0.244599\pi$
$264$	0	0
$265$	2.46410	0.151369
$266$	23.1244	1.41785
$267$	0	0
$268$	−48.7846	−2.97999
$269$	23.5885	1.43821	0.719107	−	0.694900i	$-0.244551\pi$
$269$	23.5885	1.43821	0.719107	+	0.694900i	$0.244551\pi$
$270$	0	0
$271$	−4.07180	−0.247344	−0.123672	−	0.992323i	$-0.539467\pi$
$271$	−4.07180	−0.247344	−0.123672	+	0.992323i	$0.539467\pi$
$272$	−6.92820	−0.420084
$273$	0	0
$274$	−36.7846	−2.22224
$275$	1.00000	0.0603023
$276$	0	0
$277$	−22.9282	−1.37762	−0.688811	−	0.724941i	$-0.741867\pi$
$277$	−22.9282	−1.37762	−0.688811	+	0.724941i	$0.741867\pi$
$278$	−12.3923	−0.743241
$279$	0	0
$280$	−9.46410	−0.565588
$281$	−30.9282	−1.84502	−0.922511	−	0.385971i	$-0.873867\pi$
$281$	−30.9282	−1.84502	−0.922511	+	0.385971i	$0.873867\pi$
$282$	0	0
$283$	10.1962	0.606098	0.303049	−	0.952975i	$-0.401995\pi$
$283$	10.1962	0.606098	0.303049	+	0.952975i	$0.401995\pi$
$284$	52.7846	3.13219
$285$	0	0
$286$	−16.9282	−1.00099
$287$	−9.66025	−0.570227
$288$	0	0
$289$	−16.7846	−0.987330
$290$	6.19615	0.363851
$291$	0	0
$292$	−67.7128	−3.96259
$293$	−9.00000	−0.525786	−0.262893	−	0.964825i	$-0.584677\pi$
$293$	−9.00000	−0.525786	−0.262893	+	0.964825i	$0.584677\pi$
$294$	0	0
$295$	−10.2679	−0.597823
$296$	−58.6410	−3.40844
$297$	0	0
$298$	23.3205	1.35092
$299$	−9.51666	−0.550363
$300$	0	0
$301$	−1.73205	−0.0998337
$302$	−43.3205	−2.49282
$303$	0	0
$304$	−126.354	−7.24689
$305$	−9.39230	−0.537802
$306$	0	0
$307$	10.3923	0.593120	0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923	0.593120	0.296560	+	0.955014i	$0.404160\pi$
$308$	5.46410	0.311346
$309$	0	0
$310$	23.8564	1.35495
$311$	27.3205	1.54920	0.774602	−	0.632449i	$-0.217950\pi$
$311$	27.3205	1.54920	0.774602	+	0.632449i	$0.217950\pi$
$312$	0	0
$313$	−1.73205	−0.0979013	−0.0489506	−	0.998801i	$-0.515588\pi$
$313$	−1.73205	−0.0979013	−0.0489506	+	0.998801i	$0.515588\pi$
$314$	−27.6603	−1.56096
$315$	0	0
$316$	−74.6410	−4.19889
$317$	−5.07180	−0.284860	−0.142430	−	0.989805i	$-0.545492\pi$
$317$	−5.07180	−0.284860	−0.142430	+	0.989805i	$0.545492\pi$
$318$	0	0
$319$	−2.26795	−0.126981
$320$	29.8564	1.66902
$321$	0	0
$322$	4.19615	0.233842
$323$	3.92820	0.218571
$324$	0	0
$325$	6.19615	0.343701
$326$	14.0000	0.775388
$327$	0	0
$328$	91.4256	5.04814
$329$	−4.73205	−0.260886
$330$	0	0
$331$	−26.3205	−1.44671	−0.723353	−	0.690478i	$-0.757400\pi$
$331$	−26.3205	−1.44671	−0.723353	+	0.690478i	$0.757400\pi$
$332$	24.3923	1.33870
$333$	0	0
$334$	28.3923	1.55356
$335$	−8.92820	−0.487800
$336$	0	0
$337$	6.66025	0.362807	0.181404	−	0.983409i	$-0.441936\pi$
$337$	6.66025	0.362807	0.181404	+	0.983409i	$0.441936\pi$
$338$	−69.3731	−3.77340
$339$	0	0
$340$	−2.53590	−0.137528
$341$	−8.73205	−0.472867
$342$	0	0
$343$	1.00000	0.0539949
$344$	16.3923	0.883814
$345$	0	0
$346$	−50.6410	−2.72248
$347$	−5.46410	−0.293328	−0.146664	−	0.989186i	$-0.546854\pi$
$347$	−5.46410	−0.293328	−0.146664	+	0.989186i	$0.546854\pi$
$348$	0	0
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	−2.73205	−0.146034
$351$	0	0
$352$	−21.8564	−1.16495
$353$	−8.53590	−0.454320	−0.227160	−	0.973857i	$-0.572944\pi$
$353$	−8.53590	−0.454320	−0.227160	+	0.973857i	$0.572944\pi$
$354$	0	0
$355$	9.66025	0.512713
$356$	14.5359	0.770401
$357$	0	0
$358$	49.3205	2.60667
$359$	−10.5167	−0.555048	−0.277524	−	0.960719i	$-0.589514\pi$
$359$	−10.5167	−0.555048	−0.277524	+	0.960719i	$0.589514\pi$
$360$	0	0
$361$	52.6410	2.77058
$362$	−57.1769	−3.00515
$363$	0	0
$364$	33.8564	1.77456
$365$	−12.3923	−0.648643
$366$	0	0
$367$	27.1962	1.41963	0.709814	−	0.704389i	$-0.248779\pi$
$367$	27.1962	1.41963	0.709814	+	0.704389i	$0.248779\pi$
$368$	−22.9282	−1.19522
$369$	0	0
$370$	−16.9282	−0.880055
$371$	2.46410	0.127930
$372$	0	0
$373$	18.8038	0.973626	0.486813	−	0.873506i	$-0.338159\pi$
$373$	18.8038	0.973626	0.486813	+	0.873506i	$0.338159\pi$
$374$	1.26795	0.0655641
$375$	0	0
$376$	44.7846	2.30959
$377$	−14.0526	−0.723744
$378$	0	0
$379$	−6.85641	−0.352190	−0.176095	−	0.984373i	$-0.556347\pi$
$379$	−6.85641	−0.352190	−0.176095	+	0.984373i	$0.556347\pi$
$380$	−46.2487	−2.37251
$381$	0	0
$382$	−40.7846	−2.08672
$383$	−29.2679	−1.49552	−0.747761	−	0.663968i	$-0.768871\pi$
$383$	−29.2679	−1.49552	−0.747761	+	0.663968i	$0.768871\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	1.46410	0.0745208
$387$	0	0
$388$	−31.3205	−1.59006
$389$	−29.5167	−1.49655	−0.748277	−	0.663386i	$-0.769119\pi$
$389$	−29.5167	−1.49655	−0.748277	+	0.663386i	$0.769119\pi$
$390$	0	0
$391$	0.712813	0.0360485
$392$	−9.46410	−0.478009
$393$	0	0
$394$	−4.00000	−0.201517
$395$	−13.6603	−0.687322
$396$	0	0
$397$	10.1436	0.509092	0.254546	−	0.967061i	$-0.418074\pi$
$397$	10.1436	0.509092	0.254546	+	0.967061i	$0.418074\pi$
$398$	2.92820	0.146778
$399$	0	0
$400$	14.9282	0.746410
$401$	8.87564	0.443229	0.221614	−	0.975134i	$-0.428867\pi$
$401$	8.87564	0.443229	0.221614	+	0.975134i	$0.428867\pi$
$402$	0	0
$403$	−54.1051	−2.69517
$404$	−16.7846	−0.835066
$405$	0	0
$406$	6.19615	0.307510
$407$	6.19615	0.307132
$408$	0	0
$409$	−22.9282	−1.13373	−0.566863	−	0.823812i	$-0.691843\pi$
$409$	−22.9282	−1.13373	−0.566863	+	0.823812i	$0.691843\pi$
$410$	26.3923	1.30342
$411$	0	0
$412$	−96.8897	−4.77341
$413$	−10.2679	−0.505253
$414$	0	0
$415$	4.46410	0.219134
$416$	−135.426	−6.63979
$417$	0	0
$418$	23.1244	1.13105
$419$	−26.2679	−1.28327	−0.641637	−	0.767009i	$-0.721744\pi$
$419$	−26.2679	−1.28327	−0.641637	+	0.767009i	$0.721744\pi$
$420$	0	0
$421$	−30.7128	−1.49685	−0.748425	−	0.663219i	$-0.769190\pi$
$421$	−30.7128	−1.49685	−0.748425	+	0.663219i	$0.769190\pi$
$422$	−31.3205	−1.52466
$423$	0	0
$424$	−23.3205	−1.13254
$425$	−0.464102	−0.0225122
$426$	0	0
$427$	−9.39230	−0.454525
$428$	−47.7128	−2.30629
$429$	0	0
$430$	4.73205	0.228200
$431$	3.60770	0.173777	0.0868883	−	0.996218i	$-0.472308\pi$
$431$	3.60770	0.173777	0.0868883	+	0.996218i	$0.472308\pi$
$432$	0	0
$433$	−25.3205	−1.21683	−0.608413	−	0.793621i	$-0.708194\pi$
$433$	−25.3205	−1.21683	−0.608413	+	0.793621i	$0.708194\pi$
$434$	23.8564	1.14514
$435$	0	0
$436$	−47.7128	−2.28503
$437$	13.0000	0.621874
$438$	0	0
$439$	28.7128	1.37039	0.685194	−	0.728361i	$-0.259717\pi$
$439$	28.7128	1.37039	0.685194	+	0.728361i	$0.259717\pi$
$440$	−9.46410	−0.451183
$441$	0	0
$442$	7.85641	0.373691
$443$	16.5359	0.785644	0.392822	−	0.919614i	$-0.371499\pi$
$443$	16.5359	0.785644	0.392822	+	0.919614i	$0.371499\pi$
$444$	0	0
$445$	2.66025	0.126108
$446$	−30.5885	−1.44841
$447$	0	0
$448$	29.8564	1.41058
$449$	22.0526	1.04072	0.520362	−	0.853946i	$-0.325797\pi$
$449$	22.0526	1.04072	0.520362	+	0.853946i	$0.325797\pi$
$450$	0	0
$451$	−9.66025	−0.454884
$452$	92.1051	4.33226
$453$	0	0
$454$	42.0526	1.97362
$455$	6.19615	0.290480
$456$	0	0
$457$	14.6603	0.685778	0.342889	−	0.939376i	$-0.388595\pi$
$457$	14.6603	0.685778	0.342889	+	0.939376i	$0.388595\pi$
$458$	−48.6410	−2.27285
$459$	0	0
$460$	−8.39230	−0.391293
$461$	5.32051	0.247801	0.123900	−	0.992295i	$-0.460460\pi$
$461$	5.32051	0.247801	0.123900	+	0.992295i	$0.460460\pi$
$462$	0	0
$463$	19.4641	0.904574	0.452287	−	0.891873i	$-0.350608\pi$
$463$	19.4641	0.904574	0.452287	+	0.891873i	$0.350608\pi$
$464$	−33.8564	−1.57174
$465$	0	0
$466$	0.535898	0.0248250
$467$	−7.32051	−0.338753	−0.169376	−	0.985551i	$-0.554175\pi$
$467$	−7.32051	−0.338753	−0.169376	+	0.985551i	$0.554175\pi$
$468$	0	0
$469$	−8.92820	−0.412266
$470$	12.9282	0.596334
$471$	0	0
$472$	97.1769	4.47293
$473$	−1.73205	−0.0796398
$474$	0	0
$475$	−8.46410	−0.388360
$476$	−2.53590	−0.116233
$477$	0	0
$478$	−16.7321	−0.765306
$479$	3.26795	0.149316	0.0746582	−	0.997209i	$-0.476213\pi$
$479$	3.26795	0.149316	0.0746582	+	0.997209i	$0.476213\pi$
$480$	0	0
$481$	38.3923	1.75054
$482$	−35.7128	−1.62667
$483$	0	0
$484$	5.46410	0.248368
$485$	−5.73205	−0.260279
$486$	0	0
$487$	7.46410	0.338231	0.169115	−	0.985596i	$-0.445909\pi$
$487$	7.46410	0.338231	0.169115	+	0.985596i	$0.445909\pi$
$488$	88.8897	4.02385
$489$	0	0
$490$	−2.73205	−0.123421
$491$	−20.6603	−0.932384	−0.466192	−	0.884684i	$-0.654374\pi$
$491$	−20.6603	−0.932384	−0.466192	+	0.884684i	$0.654374\pi$
$492$	0	0
$493$	1.05256	0.0474049
$494$	143.282	6.44656
$495$	0	0
$496$	−130.354	−5.85306
$497$	9.66025	0.433322
$498$	0	0
$499$	8.60770	0.385333	0.192667	−	0.981264i	$-0.438286\pi$
$499$	8.60770	0.385333	0.192667	+	0.981264i	$0.438286\pi$
$500$	5.46410	0.244362
$501$	0	0
$502$	18.9282	0.844807
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	0	0
$505$	−3.07180	−0.136693
$506$	4.19615	0.186542
$507$	0	0
$508$	11.6077	0.515008
$509$	6.80385	0.301575	0.150788	−	0.988566i	$-0.451819\pi$
$509$	6.80385	0.301575	0.150788	+	0.988566i	$0.451819\pi$
$510$	0	0
$511$	−12.3923	−0.548203
$512$	−43.7128	−1.93185
$513$	0	0
$514$	66.1051	2.91577
$515$	−17.7321	−0.781368
$516$	0	0
$517$	−4.73205	−0.208115
$518$	−16.9282	−0.743783
$519$	0	0
$520$	−58.6410	−2.57158
$521$	−29.0526	−1.27282	−0.636408	−	0.771353i	$-0.719580\pi$
$521$	−29.0526	−1.27282	−0.636408	+	0.771353i	$0.719580\pi$
$522$	0	0
$523$	7.32051	0.320103	0.160052	−	0.987109i	$-0.448834\pi$
$523$	7.32051	0.320103	0.160052	+	0.987109i	$0.448834\pi$
$524$	−31.7128	−1.38538
$525$	0	0
$526$	−63.7128	−2.77801
$527$	4.05256	0.176532
$528$	0	0
$529$	−20.6410	−0.897435
$530$	−6.73205	−0.292422
$531$	0	0
$532$	−46.2487	−2.00514
$533$	−59.8564	−2.59267
$534$	0	0
$535$	−8.73205	−0.377519
$536$	84.4974	3.64973
$537$	0	0
$538$	−64.4449	−2.77842
$539$	1.00000	0.0430730
$540$	0	0
$541$	4.05256	0.174233	0.0871166	−	0.996198i	$-0.472235\pi$
$541$	4.05256	0.174233	0.0871166	+	0.996198i	$0.472235\pi$
$542$	11.1244	0.477832
$543$	0	0
$544$	10.1436	0.434903
$545$	−8.73205	−0.374040
$546$	0	0
$547$	−7.58846	−0.324459	−0.162230	−	0.986753i	$-0.551868\pi$
$547$	−7.58846	−0.324459	−0.162230	+	0.986753i	$0.551868\pi$
$548$	73.5692	3.14272
$549$	0	0
$550$	−2.73205	−0.116495
$551$	19.1962	0.817784
$552$	0	0
$553$	−13.6603	−0.580893
$554$	62.6410	2.66136
$555$	0	0
$556$	24.7846	1.05110
$557$	−15.4641	−0.655235	−0.327618	−	0.944810i	$-0.606246\pi$
$557$	−15.4641	−0.655235	−0.327618	+	0.944810i	$0.606246\pi$
$558$	0	0
$559$	−10.7321	−0.453917
$560$	14.9282	0.630832
$561$	0	0
$562$	84.4974	3.56431
$563$	11.0718	0.466621	0.233310	−	0.972402i	$-0.425044\pi$
$563$	11.0718	0.466621	0.233310	+	0.972402i	$0.425044\pi$
$564$	0	0
$565$	16.8564	0.709154
$566$	−27.8564	−1.17089
$567$	0	0
$568$	−91.4256	−3.83613
$569$	−12.6603	−0.530745	−0.265373	−	0.964146i	$-0.585495\pi$
$569$	−12.6603	−0.530745	−0.265373	+	0.964146i	$0.585495\pi$
$570$	0	0
$571$	−7.12436	−0.298145	−0.149073	−	0.988826i	$-0.547629\pi$
$571$	−7.12436	−0.298145	−0.149073	+	0.988826i	$0.547629\pi$
$572$	33.8564	1.41561
$573$	0	0
$574$	26.3923	1.10159
$575$	−1.53590	−0.0640514
$576$	0	0
$577$	−20.9282	−0.871253	−0.435626	−	0.900128i	$-0.643473\pi$
$577$	−20.9282	−0.871253	−0.435626	+	0.900128i	$0.643473\pi$
$578$	45.8564	1.90738
$579$	0	0
$580$	−12.3923	−0.514562
$581$	4.46410	0.185202
$582$	0	0
$583$	2.46410	0.102053
$584$	117.282	4.85317
$585$	0	0
$586$	24.5885	1.01574
$587$	−12.5885	−0.519581	−0.259791	−	0.965665i	$-0.583654\pi$
$587$	−12.5885	−0.519581	−0.259791	+	0.965665i	$0.583654\pi$
$588$	0	0
$589$	73.9090	3.04537
$590$	28.0526	1.15491
$591$	0	0
$592$	92.4974	3.80162
$593$	−16.0000	−0.657041	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	−16.0000	−0.657041	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	−0.464102	−0.0190263
$596$	−46.6410	−1.91049
$597$	0	0
$598$	26.0000	1.06322
$599$	20.2487	0.827340	0.413670	−	0.910427i	$-0.364247\pi$
$599$	20.2487	0.827340	0.413670	+	0.910427i	$0.364247\pi$
$600$	0	0
$601$	−23.7846	−0.970194	−0.485097	−	0.874460i	$-0.661216\pi$
$601$	−23.7846	−0.970194	−0.485097	+	0.874460i	$0.661216\pi$
$602$	4.73205	0.192864
$603$	0	0
$604$	86.6410	3.52537
$605$	1.00000	0.0406558
$606$	0	0
$607$	−39.1769	−1.59014	−0.795071	−	0.606516i	$-0.792566\pi$
$607$	−39.1769	−1.59014	−0.795071	+	0.606516i	$0.792566\pi$
$608$	184.995	7.50253
$609$	0	0
$610$	25.6603	1.03895
$611$	−29.3205	−1.18618
$612$	0	0
$613$	−6.78461	−0.274028	−0.137014	−	0.990569i	$-0.543751\pi$
$613$	−6.78461	−0.274028	−0.137014	+	0.990569i	$0.543751\pi$
$614$	−28.3923	−1.14582
$615$	0	0
$616$	−9.46410	−0.381320
$617$	−22.6410	−0.911493	−0.455746	−	0.890110i	$-0.650628\pi$
$617$	−22.6410	−0.911493	−0.455746	+	0.890110i	$0.650628\pi$
$618$	0	0
$619$	−11.0718	−0.445013	−0.222507	−	0.974931i	$-0.571424\pi$
$619$	−11.0718	−0.445013	−0.222507	+	0.974931i	$0.571424\pi$
$620$	−47.7128	−1.91619
$621$	0	0
$622$	−74.6410	−2.99283
$623$	2.66025	0.106581
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.73205	0.189131
$627$	0	0
$628$	55.3205	2.20753
$629$	−2.87564	−0.114659
$630$	0	0
$631$	38.7128	1.54113	0.770566	−	0.637360i	$-0.219973\pi$
$631$	38.7128	1.54113	0.770566	+	0.637360i	$0.219973\pi$
$632$	129.282	5.14256
$633$	0	0
$634$	13.8564	0.550308
$635$	2.12436	0.0843025
$636$	0	0
$637$	6.19615	0.245500
$638$	6.19615	0.245308
$639$	0	0
$640$	−37.8564	−1.49641
$641$	−40.6410	−1.60522	−0.802612	−	0.596502i	$-0.796557\pi$
$641$	−40.6410	−1.60522	−0.802612	+	0.596502i	$0.796557\pi$
$642$	0	0
$643$	−17.1962	−0.678150	−0.339075	−	0.940759i	$-0.610114\pi$
$643$	−17.1962	−0.678150	−0.339075	+	0.940759i	$0.610114\pi$
$644$	−8.39230	−0.330703
$645$	0	0
$646$	−10.7321	−0.422247
$647$	32.1051	1.26218	0.631091	−	0.775709i	$-0.282607\pi$
$647$	32.1051	1.26218	0.631091	+	0.775709i	$0.282607\pi$
$648$	0	0
$649$	−10.2679	−0.403052
$650$	−16.9282	−0.663979
$651$	0	0
$652$	−28.0000	−1.09656
$653$	36.4641	1.42695	0.713475	−	0.700680i	$-0.247120\pi$
$653$	36.4641	1.42695	0.713475	+	0.700680i	$0.247120\pi$
$654$	0	0
$655$	−5.80385	−0.226775
$656$	−144.210	−5.63046
$657$	0	0
$658$	12.9282	0.503994
$659$	5.87564	0.228883	0.114441	−	0.993430i	$-0.463492\pi$
$659$	5.87564	0.228883	0.114441	+	0.993430i	$0.463492\pi$
$660$	0	0
$661$	0.732051	0.0284735	0.0142367	−	0.999899i	$-0.495468\pi$
$661$	0.732051	0.0284735	0.0142367	+	0.999899i	$0.495468\pi$
$662$	71.9090	2.79482
$663$	0	0
$664$	−42.2487	−1.63957
$665$	−8.46410	−0.328224
$666$	0	0
$667$	3.48334	0.134875
$668$	−56.7846	−2.19706
$669$	0	0
$670$	24.3923	0.942357
$671$	−9.39230	−0.362586
$672$	0	0
$673$	13.5885	0.523797	0.261898	−	0.965095i	$-0.415652\pi$
$673$	13.5885	0.523797	0.261898	+	0.965095i	$0.415652\pi$
$674$	−18.1962	−0.700890
$675$	0	0
$676$	138.746	5.33639
$677$	−9.24871	−0.355457	−0.177728	−	0.984080i	$-0.556875\pi$
$677$	−9.24871	−0.355457	−0.177728	+	0.984080i	$0.556875\pi$
$678$	0	0
$679$	−5.73205	−0.219976
$680$	4.39230	0.168437
$681$	0	0
$682$	23.8564	0.913509
$683$	33.0718	1.26546	0.632729	−	0.774374i	$-0.281935\pi$
$683$	33.0718	1.26546	0.632729	+	0.774374i	$0.281935\pi$
$684$	0	0
$685$	13.4641	0.514437
$686$	−2.73205	−0.104310
$687$	0	0
$688$	−25.8564	−0.985766
$689$	15.2679	0.581663
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	101.282	3.85017
$693$	0	0
$694$	14.9282	0.566667
$695$	4.53590	0.172056
$696$	0	0
$697$	4.48334	0.169819
$698$	40.9808	1.55114
$699$	0	0
$700$	5.46410	0.206524
$701$	−22.8038	−0.861289	−0.430645	−	0.902522i	$-0.641714\pi$
$701$	−22.8038	−0.861289	−0.430645	+	0.902522i	$0.641714\pi$
$702$	0	0
$703$	−52.4449	−1.97800
$704$	29.8564	1.12526
$705$	0	0
$706$	23.3205	0.877679
$707$	−3.07180	−0.115527
$708$	0	0
$709$	29.5359	1.10924	0.554622	−	0.832102i	$-0.312863\pi$
$709$	29.5359	1.10924	0.554622	+	0.832102i	$0.312863\pi$
$710$	−26.3923	−0.990486
$711$	0	0
$712$	−25.1769	−0.943545
$713$	13.4115	0.502266
$714$	0	0
$715$	6.19615	0.231723
$716$	−98.6410	−3.68639
$717$	0	0
$718$	28.7321	1.07227
$719$	−15.3397	−0.572076	−0.286038	−	0.958218i	$-0.592338\pi$
$719$	−15.3397	−0.572076	−0.286038	+	0.958218i	$0.592338\pi$
$720$	0	0
$721$	−17.7321	−0.660376
$722$	−143.818	−5.35235
$723$	0	0
$724$	114.354	4.24993
$725$	−2.26795	−0.0842295
$726$	0	0
$727$	−22.9090	−0.849646	−0.424823	−	0.905276i	$-0.639664\pi$
$727$	−22.9090	−0.849646	−0.424823	+	0.905276i	$0.639664\pi$
$728$	−58.6410	−2.17338
$729$	0	0
$730$	33.8564	1.25308
$731$	0.803848	0.0297314
$732$	0	0
$733$	−11.1769	−0.412829	−0.206414	−	0.978465i	$-0.566179\pi$
$733$	−11.1769	−0.412829	−0.206414	+	0.978465i	$0.566179\pi$
$734$	−74.3013	−2.74251
$735$	0	0
$736$	33.5692	1.23738
$737$	−8.92820	−0.328875
$738$	0	0
$739$	15.4641	0.568856	0.284428	−	0.958697i	$-0.408196\pi$
$739$	15.4641	0.568856	0.284428	+	0.958697i	$0.408196\pi$
$740$	33.8564	1.24459
$741$	0	0
$742$	−6.73205	−0.247141
$743$	43.1769	1.58401	0.792004	−	0.610516i	$-0.209038\pi$
$743$	43.1769	1.58401	0.792004	+	0.610516i	$0.209038\pi$
$744$	0	0
$745$	−8.53590	−0.312731
$746$	−51.3731	−1.88090
$747$	0	0
$748$	−2.53590	−0.0927216
$749$	−8.73205	−0.319062
$750$	0	0
$751$	−18.3205	−0.668525	−0.334262	−	0.942480i	$-0.608487\pi$
$751$	−18.3205	−0.668525	−0.334262	+	0.942480i	$0.608487\pi$
$752$	−70.6410	−2.57601
$753$	0	0
$754$	38.3923	1.39817
$755$	15.8564	0.577074
$756$	0	0
$757$	15.4641	0.562052	0.281026	−	0.959700i	$-0.409325\pi$
$757$	15.4641	0.562052	0.281026	+	0.959700i	$0.409325\pi$
$758$	18.7321	0.680379
$759$	0	0
$760$	80.1051	2.90572
$761$	24.9282	0.903647	0.451823	−	0.892107i	$-0.350774\pi$
$761$	24.9282	0.903647	0.451823	+	0.892107i	$0.350774\pi$
$762$	0	0
$763$	−8.73205	−0.316121
$764$	81.5692	2.95107
$765$	0	0
$766$	79.9615	2.88913
$767$	−63.6218	−2.29725
$768$	0	0
$769$	17.7846	0.641329	0.320665	−	0.947193i	$-0.396094\pi$
$769$	17.7846	0.641329	0.320665	+	0.947193i	$0.396094\pi$
$770$	−2.73205	−0.0984563
$771$	0	0
$772$	−2.92820	−0.105388
$773$	12.2487	0.440556	0.220278	−	0.975437i	$-0.429304\pi$
$773$	12.2487	0.440556	0.220278	+	0.975437i	$0.429304\pi$
$774$	0	0
$775$	−8.73205	−0.313665
$776$	54.2487	1.94742
$777$	0	0
$778$	80.6410	2.89112
$779$	81.7654	2.92955
$780$	0	0
$781$	9.66025	0.345671
$782$	−1.94744	−0.0696404
$783$	0	0
$784$	14.9282	0.533150
$785$	10.1244	0.361354
$786$	0	0
$787$	26.6410	0.949650	0.474825	−	0.880080i	$-0.342511\pi$
$787$	26.6410	0.949650	0.474825	+	0.880080i	$0.342511\pi$
$788$	8.00000	0.284988
$789$	0	0
$790$	37.3205	1.32780
$791$	16.8564	0.599345
$792$	0	0
$793$	−58.1962	−2.06661
$794$	−27.7128	−0.983491
$795$	0	0
$796$	−5.85641	−0.207575
$797$	−51.1769	−1.81278	−0.906390	−	0.422443i	$-0.861173\pi$
$797$	−51.1769	−1.81278	−0.906390	+	0.422443i	$0.861173\pi$
$798$	0	0
$799$	2.19615	0.0776943
$800$	−21.8564	−0.772741
$801$	0	0
$802$	−24.2487	−0.856252
$803$	−12.3923	−0.437315
$804$	0	0
$805$	−1.53590	−0.0541333
$806$	147.818	5.20666
$807$	0	0
$808$	29.0718	1.02274
$809$	−15.4641	−0.543689	−0.271844	−	0.962341i	$-0.587634\pi$
$809$	−15.4641	−0.543689	−0.271844	+	0.962341i	$0.587634\pi$
$810$	0	0
$811$	−36.7846	−1.29168	−0.645841	−	0.763472i	$-0.723493\pi$
$811$	−36.7846	−1.29168	−0.645841	+	0.763472i	$0.723493\pi$
$812$	−12.3923	−0.434885
$813$	0	0
$814$	−16.9282	−0.593333
$815$	−5.12436	−0.179498
$816$	0	0
$817$	14.6603	0.512897
$818$	62.6410	2.19019
$819$	0	0
$820$	−52.7846	−1.84332
$821$	−41.9808	−1.46514	−0.732569	−	0.680692i	$-0.761679\pi$
$821$	−41.9808	−1.46514	−0.732569	+	0.680692i	$0.761679\pi$
$822$	0	0
$823$	−40.4449	−1.40982	−0.704910	−	0.709297i	$-0.749012\pi$
$823$	−40.4449	−1.40982	−0.704910	+	0.709297i	$0.749012\pi$
$824$	167.818	5.84621
$825$	0	0
$826$	28.0526	0.976073
$827$	−6.53590	−0.227275	−0.113638	−	0.993522i	$-0.536250\pi$
$827$	−6.53590	−0.227275	−0.113638	+	0.993522i	$0.536250\pi$
$828$	0	0
$829$	27.1769	0.943893	0.471947	−	0.881627i	$-0.343551\pi$
$829$	27.1769	0.943893	0.471947	+	0.881627i	$0.343551\pi$
$830$	−12.1962	−0.423335
$831$	0	0
$832$	184.995	6.41354
$833$	−0.464102	−0.0160802
$834$	0	0
$835$	−10.3923	−0.359641
$836$	−46.2487	−1.59955
$837$	0	0
$838$	71.7654	2.47909
$839$	−32.6603	−1.12756	−0.563779	−	0.825926i	$-0.690653\pi$
$839$	−32.6603	−1.12756	−0.563779	+	0.825926i	$0.690653\pi$
$840$	0	0
$841$	−23.8564	−0.822635
$842$	83.9090	2.89169
$843$	0	0
$844$	62.6410	2.15619
$845$	25.3923	0.873522
$846$	0	0
$847$	1.00000	0.0343604
$848$	36.7846	1.26319
$849$	0	0
$850$	1.26795	0.0434903
$851$	−9.51666	−0.326227
$852$	0	0
$853$	35.1244	1.20264	0.601318	−	0.799010i	$-0.294643\pi$
$853$	35.1244	1.20264	0.601318	+	0.799010i	$0.294643\pi$
$854$	25.6603	0.878076
$855$	0	0
$856$	82.6410	2.82461
$857$	5.71281	0.195146	0.0975730	−	0.995228i	$-0.468892\pi$
$857$	5.71281	0.195146	0.0975730	+	0.995228i	$0.468892\pi$
$858$	0	0
$859$	−17.4115	−0.594074	−0.297037	−	0.954866i	$-0.595999\pi$
$859$	−17.4115	−0.594074	−0.297037	+	0.954866i	$0.595999\pi$
$860$	−9.46410	−0.322723
$861$	0	0
$862$	−9.85641	−0.335711
$863$	32.4641	1.10509	0.552545	−	0.833483i	$-0.313657\pi$
$863$	32.4641	1.10509	0.552545	+	0.833483i	$0.313657\pi$
$864$	0	0
$865$	18.5359	0.630239
$866$	69.1769	2.35073
$867$	0	0
$868$	−47.7128	−1.61948
$869$	−13.6603	−0.463392
$870$	0	0
$871$	−55.3205	−1.87446
$872$	82.6410	2.79858
$873$	0	0
$874$	−35.5167	−1.20137
$875$	1.00000	0.0338062
$876$	0	0
$877$	−54.3731	−1.83605	−0.918024	−	0.396525i	$-0.870216\pi$
$877$	−54.3731	−1.83605	−0.918024	+	0.396525i	$0.870216\pi$
$878$	−78.4449	−2.64739
$879$	0	0
$880$	14.9282	0.503230
$881$	48.9090	1.64778	0.823892	−	0.566746i	$-0.191798\pi$
$881$	48.9090	1.64778	0.823892	+	0.566746i	$0.191798\pi$
$882$	0	0
$883$	25.1769	0.847271	0.423635	−	0.905833i	$-0.360754\pi$
$883$	25.1769	0.847271	0.423635	+	0.905833i	$0.360754\pi$
$884$	−15.7128	−0.528479
$885$	0	0
$886$	−45.1769	−1.51775
$887$	0.607695	0.0204044	0.0102022	−	0.999948i	$-0.496752\pi$
$887$	0.607695	0.0204044	0.0102022	+	0.999948i	$0.496752\pi$
$888$	0	0
$889$	2.12436	0.0712486
$890$	−7.26795	−0.243622
$891$	0	0
$892$	61.1769	2.04835
$893$	40.0526	1.34031
$894$	0	0
$895$	−18.0526	−0.603430
$896$	−37.8564	−1.26469
$897$	0	0
$898$	−60.2487	−2.01053
$899$	19.8038	0.660495
$900$	0	0
$901$	−1.14359	−0.0380986
$902$	26.3923	0.878768
$903$	0	0
$904$	−159.531	−5.30591
$905$	20.9282	0.695677
$906$	0	0
$907$	35.6603	1.18408	0.592040	−	0.805909i	$-0.298323\pi$
$907$	35.6603	1.18408	0.592040	+	0.805909i	$0.298323\pi$
$908$	−84.1051	−2.79113
$909$	0	0
$910$	−16.9282	−0.561164
$911$	−52.3923	−1.73583	−0.867917	−	0.496709i	$-0.834542\pi$
$911$	−52.3923	−1.73583	−0.867917	+	0.496709i	$0.834542\pi$
$912$	0	0
$913$	4.46410	0.147740
$914$	−40.0526	−1.32482
$915$	0	0
$916$	97.2820	3.21429
$917$	−5.80385	−0.191660
$918$	0	0
$919$	50.4974	1.66576	0.832878	−	0.553456i	$-0.186691\pi$
$919$	50.4974	1.66576	0.832878	+	0.553456i	$0.186691\pi$
$920$	14.5359	0.479234
$921$	0	0
$922$	−14.5359	−0.478714
$923$	59.8564	1.97020
$924$	0	0
$925$	6.19615	0.203728
$926$	−53.1769	−1.74750
$927$	0	0
$928$	49.5692	1.62719
$929$	−39.7128	−1.30294	−0.651468	−	0.758676i	$-0.725846\pi$
$929$	−39.7128	−1.30294	−0.651468	+	0.758676i	$0.725846\pi$
$930$	0	0
$931$	−8.46410	−0.277400
$932$	−1.07180	−0.0351079
$933$	0	0
$934$	20.0000	0.654420
$935$	−0.464102	−0.0151777
$936$	0	0
$937$	−24.9808	−0.816086	−0.408043	−	0.912963i	$-0.633789\pi$
$937$	−24.9808	−0.816086	−0.408043	+	0.912963i	$0.633789\pi$
$938$	24.3923	0.796437
$939$	0	0
$940$	−25.8564	−0.843343
$941$	52.8372	1.72244	0.861221	−	0.508230i	$-0.169700\pi$
$941$	52.8372	1.72244	0.861221	+	0.508230i	$0.169700\pi$
$942$	0	0
$943$	14.8372	0.483165
$944$	−153.282	−4.98891
$945$	0	0
$946$	4.73205	0.153852
$947$	−42.7128	−1.38798	−0.693990	−	0.719985i	$-0.744149\pi$
$947$	−42.7128	−1.38798	−0.693990	+	0.719985i	$0.744149\pi$
$948$	0	0
$949$	−76.7846	−2.49253
$950$	23.1244	0.750253
$951$	0	0
$952$	4.39230	0.142355
$953$	4.92820	0.159640	0.0798201	−	0.996809i	$-0.474565\pi$
$953$	4.92820	0.159640	0.0798201	+	0.996809i	$0.474565\pi$
$954$	0	0
$955$	14.9282	0.483065
$956$	33.4641	1.08231
$957$	0	0
$958$	−8.92820	−0.288457
$959$	13.4641	0.434779
$960$	0	0
$961$	45.2487	1.45964
$962$	−104.890	−3.38178
$963$	0	0
$964$	71.4256	2.30046
$965$	−0.535898	−0.0172512
$966$	0	0
$967$	−14.5167	−0.466824	−0.233412	−	0.972378i	$-0.574989\pi$
$967$	−14.5167	−0.466824	−0.233412	+	0.972378i	$0.574989\pi$
$968$	−9.46410	−0.304188
$969$	0	0
$970$	15.6603	0.502820
$971$	−25.3397	−0.813191	−0.406596	−	0.913608i	$-0.633284\pi$
$971$	−25.3397	−0.813191	−0.406596	+	0.913608i	$0.633284\pi$
$972$	0	0
$973$	4.53590	0.145414
$974$	−20.3923	−0.653412
$975$	0	0
$976$	−140.210	−4.48802
$977$	9.39230	0.300486	0.150243	−	0.988649i	$-0.451994\pi$
$977$	9.39230	0.300486	0.150243	+	0.988649i	$0.451994\pi$
$978$	0	0
$979$	2.66025	0.0850221
$980$	5.46410	0.174544
$981$	0	0
$982$	56.4449	1.80123
$983$	−45.3205	−1.44550	−0.722750	−	0.691110i	$-0.757122\pi$
$983$	−45.3205	−1.44550	−0.722750	+	0.691110i	$0.757122\pi$
$984$	0	0
$985$	1.46410	0.0466502
$986$	−2.87564	−0.0915792
$987$	0	0
$988$	−286.564	−9.11682
$989$	2.66025	0.0845912
$990$	0	0
$991$	52.5692	1.66992	0.834958	−	0.550313i	$-0.185492\pi$
$991$	52.5692	1.66992	0.834958	+	0.550313i	$0.185492\pi$
$992$	190.851	6.05953
$993$	0	0
$994$	−26.3923	−0.837113
$995$	−1.07180	−0.0339782
$996$	0	0
$997$	25.7128	0.814333	0.407166	−	0.913354i	$-0.366517\pi$
$997$	25.7128	0.814333	0.407166	+	0.913354i	$0.366517\pi$
$998$	−23.5167	−0.744407
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.v.1.1		2
3.2	odd	2		1155.2.a.r.1.2	✓	2
15.14	odd	2		5775.2.a.bc.1.1		2
21.20	even	2		8085.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.r.1.2	✓	2	3.2	odd	2
3465.2.a.v.1.1		2	1.1	even	1	trivial
5775.2.a.bc.1.1		2	15.14	odd	2
8085.2.a.bh.1.2		2	21.20	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:	\( N \)	\(=\)	\( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3465.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -9.46410 q^{8} +O(q^{10})\)	\(q-2.73205 q^{2} +5.46410 q^{4} +1.00000 q^{5} +1.00000 q^{7} -9.46410 q^{8} -2.73205 q^{10} +1.00000 q^{11} +6.19615 q^{13} -2.73205 q^{14} +14.9282 q^{16} -0.464102 q^{17} -8.46410 q^{19} +5.46410 q^{20} -2.73205 q^{22} -1.53590 q^{23} +1.00000 q^{25} -16.9282 q^{26} +5.46410 q^{28} -2.26795 q^{29} -8.73205 q^{31} -21.8564 q^{32} +1.26795 q^{34} +1.00000 q^{35} +6.19615 q^{37} +23.1244 q^{38} -9.46410 q^{40} -9.66025 q^{41} -1.73205 q^{43} +5.46410 q^{44} +4.19615 q^{46} -4.73205 q^{47} +1.00000 q^{49} -2.73205 q^{50} +33.8564 q^{52} +2.46410 q^{53} +1.00000 q^{55} -9.46410 q^{56} +6.19615 q^{58} -10.2679 q^{59} -9.39230 q^{61} +23.8564 q^{62} +29.8564 q^{64} +6.19615 q^{65} -8.92820 q^{67} -2.53590 q^{68} -2.73205 q^{70} +9.66025 q^{71} -12.3923 q^{73} -16.9282 q^{74} -46.2487 q^{76} +1.00000 q^{77} -13.6603 q^{79} +14.9282 q^{80} +26.3923 q^{82} +4.46410 q^{83} -0.464102 q^{85} +4.73205 q^{86} -9.46410 q^{88} +2.66025 q^{89} +6.19615 q^{91} -8.39230 q^{92} +12.9282 q^{94} -8.46410 q^{95} -5.73205 q^{97} -2.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8	\( 2 q - 2 q^{2} + 4 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{13} - 2 q^{14} + 16 q^{16} + 6 q^{17} - 10 q^{19} + 4 q^{20} - 2 q^{22} - 10 q^{23} + 2 q^{25} - 20 q^{26} + 4 q^{28} - 8 q^{29} - 14 q^{31} - 16 q^{32} + 6 q^{34} + 2 q^{35} + 2 q^{37} + 22 q^{38} - 12 q^{40} - 2 q^{41} + 4 q^{44} - 2 q^{46} - 6 q^{47} + 2 q^{49} - 2 q^{50} + 40 q^{52} - 2 q^{53} + 2 q^{55} - 12 q^{56} + 2 q^{58} - 24 q^{59} + 2 q^{61} + 20 q^{62} + 32 q^{64} + 2 q^{65} - 4 q^{67} - 12 q^{68} - 2 q^{70} + 2 q^{71} - 4 q^{73} - 20 q^{74} - 44 q^{76} + 2 q^{77} - 10 q^{79} + 16 q^{80} + 32 q^{82} + 2 q^{83} + 6 q^{85} + 6 q^{86} - 12 q^{88} - 12 q^{89} + 2 q^{91} + 4 q^{92} + 12 q^{94} - 10 q^{95} - 8 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^4 + 2 * q^5 + 2 * q^7 - 12 * q^8 - 2 * q^10 + 2 * q^11 + 2 * q^13 - 2 * q^14 + 16 * q^16 + 6 * q^17 - 10 * q^19 + 4 * q^20 - 2 * q^22 - 10 * q^23 + 2 * q^25 - 20 * q^26 + 4 * q^28 - 8 * q^29 - 14 * q^31 - 16 * q^32 + 6 * q^34 + 2 * q^35 + 2 * q^37 + 22 * q^38 - 12 * q^40 - 2 * q^41 + 4 * q^44 - 2 * q^46 - 6 * q^47 + 2 * q^49 - 2 * q^50 + 40 * q^52 - 2 * q^53 + 2 * q^55 - 12 * q^56 + 2 * q^58 - 24 * q^59 + 2 * q^61 + 20 * q^62 + 32 * q^64 + 2 * q^65 - 4 * q^67 - 12 * q^68 - 2 * q^70 + 2 * q^71 - 4 * q^73 - 20 * q^74 - 44 * q^76 + 2 * q^77 - 10 * q^79 + 16 * q^80 + 32 * q^82 + 2 * q^83 + 6 * q^85 + 6 * q^86 - 12 * q^88 - 12 * q^89 + 2 * q^91 + 4 * q^92 + 12 * q^94 - 10 * q^95 - 8 * q^97 - 2 * q^98

Introduction

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