LMFDB - Embedded newform 3465.2.a.bb.1.3

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
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			1.3
Root			$-1.81361$ 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)

$f(q)$	$=$	$q+1.81361 q^{2} +1.28917 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.28917 q^{8} +O(q^{10})$	\(q+1.81361 q^{2} +1.28917 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.28917 q^{8} -1.81361 q^{10} -1.00000 q^{11} +5.10278 q^{13} -1.81361 q^{14} -4.91638 q^{16} +1.91638 q^{17} +3.33804 q^{19} -1.28917 q^{20} -1.81361 q^{22} +2.28917 q^{23} +1.00000 q^{25} +9.25443 q^{26} -1.28917 q^{28} -1.18639 q^{29} -2.52444 q^{31} -6.33804 q^{32} +3.47556 q^{34} +1.00000 q^{35} +3.94610 q^{37} +6.05390 q^{38} +1.28917 q^{40} +1.68111 q^{41} +7.01916 q^{43} -1.28917 q^{44} +4.15165 q^{46} -2.52444 q^{47} +1.00000 q^{49} +1.81361 q^{50} +6.57834 q^{52} -0.289169 q^{53} +1.00000 q^{55} +1.28917 q^{56} -2.15165 q^{58} +10.4408 q^{59} +6.49472 q^{61} -4.57834 q^{62} -1.66196 q^{64} -5.10278 q^{65} -0.578337 q^{67} +2.47054 q^{68} +1.81361 q^{70} +13.3083 q^{71} +0.372787 q^{73} +7.15667 q^{74} +4.30330 q^{76} +1.00000 q^{77} +15.4061 q^{79} +4.91638 q^{80} +3.04888 q^{82} -15.7491 q^{83} -1.91638 q^{85} +12.7300 q^{86} +1.28917 q^{88} +0.0297193 q^{89} -5.10278 q^{91} +2.95112 q^{92} -4.57834 q^{94} -3.33804 q^{95} +15.8030 q^{97} +1.81361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8} + q^{10} - 3 q^{11} + 8 q^{13} + q^{14} - q^{16} - 8 q^{17} - 2 q^{19} - 3 q^{20} + q^{22} + 6 q^{23} + 3 q^{25} + 2 q^{26} - 3 q^{28} - 10 q^{29} - 2 q^{31} - 7 q^{32} + 16 q^{34} + 3 q^{35} + 8 q^{37} + 22 q^{38} + 3 q^{40} - 4 q^{41} - 3 q^{44} - 6 q^{46} - 2 q^{47} + 3 q^{49} - q^{50} + 18 q^{52} + 3 q^{55} + 3 q^{56} + 12 q^{58} + 12 q^{59} + 4 q^{61} - 12 q^{62} - 17 q^{64} - 8 q^{65} - 2 q^{68} - q^{70} + 18 q^{71} + 14 q^{73} + 18 q^{74} - 24 q^{76} + 3 q^{77} + 2 q^{79} + q^{80} - 2 q^{82} - 6 q^{83} + 8 q^{85} + 18 q^{86} + 3 q^{88} + 10 q^{89} - 8 q^{91} + 20 q^{92} - 12 q^{94} + 2 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 3 * q^8 + q^10 - 3 * q^11 + 8 * q^13 + q^14 - q^16 - 8 * q^17 - 2 * q^19 - 3 * q^20 + q^22 + 6 * q^23 + 3 * q^25 + 2 * q^26 - 3 * q^28 - 10 * q^29 - 2 * q^31 - 7 * q^32 + 16 * q^34 + 3 * q^35 + 8 * q^37 + 22 * q^38 + 3 * q^40 - 4 * q^41 - 3 * q^44 - 6 * q^46 - 2 * q^47 + 3 * q^49 - q^50 + 18 * q^52 + 3 * q^55 + 3 * q^56 + 12 * q^58 + 12 * q^59 + 4 * q^61 - 12 * q^62 - 17 * q^64 - 8 * q^65 - 2 * q^68 - q^70 + 18 * q^71 + 14 * q^73 + 18 * q^74 - 24 * q^76 + 3 * q^77 + 2 * q^79 + q^80 - 2 * q^82 - 6 * q^83 + 8 * q^85 + 18 * q^86 + 3 * q^88 + 10 * q^89 - 8 * q^91 + 20 * q^92 - 12 * q^94 + 2 * q^95 + 10 * q^97 - 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 $

$2$	1.81361	1.28241	0.641207	−	0.767368i	$-0.278434\pi$
$2$	1.81361	1.28241	0.641207	+	0.767368i	$0.278434\pi$
$3$	0	0
$3$	0	0
$4$	1.28917	0.644584
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.28917	−0.455790
$9$	0	0
$10$	−1.81361	−0.573513
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.10278	1.41526	0.707628	−	0.706586i	$-0.249765\pi$
$13$	5.10278	1.41526	0.707628	+	0.706586i	$0.249765\pi$
$14$	−1.81361	−0.484707
$15$	0	0
$16$	−4.91638	−1.22910
$17$	1.91638	0.464791	0.232395	−	0.972621i	$-0.425344\pi$
$17$	1.91638	0.464791	0.232395	+	0.972621i	$0.425344\pi$
$18$	0	0
$19$	3.33804	0.765800	0.382900	−	0.923790i	$-0.374925\pi$
$19$	3.33804	0.765800	0.382900	+	0.923790i	$0.374925\pi$
$20$	−1.28917	−0.288267
$21$	0	0
$22$	−1.81361	−0.386662
$23$	2.28917	0.477325	0.238662	−	0.971103i	$-0.423291\pi$
$23$	2.28917	0.477325	0.238662	+	0.971103i	$0.423291\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	9.25443	1.81494
$27$	0	0
$28$	−1.28917	−0.243630
$29$	−1.18639	−0.220308	−0.110154	−	0.993915i	$-0.535134\pi$
$29$	−1.18639	−0.220308	−0.110154	+	0.993915i	$0.535134\pi$
$30$	0	0
$31$	−2.52444	−0.453402	−0.226701	−	0.973964i	$-0.572794\pi$
$31$	−2.52444	−0.453402	−0.226701	+	0.973964i	$0.572794\pi$
$32$	−6.33804	−1.12042
$33$	0	0
$34$	3.47556	0.596054
$35$	1.00000	0.169031
$36$	0	0
$37$	3.94610	0.648735	0.324367	−	0.945931i	$-0.394849\pi$
$37$	3.94610	0.648735	0.324367	+	0.945931i	$0.394849\pi$
$38$	6.05390	0.982072
$39$	0	0
$40$	1.28917	0.203835
$41$	1.68111	0.262546	0.131273	−	0.991346i	$-0.458094\pi$
$41$	1.68111	0.262546	0.131273	+	0.991346i	$0.458094\pi$
$42$	0	0
$43$	7.01916	1.07041	0.535206	−	0.844722i	$-0.320234\pi$
$43$	7.01916	1.07041	0.535206	+	0.844722i	$0.320234\pi$
$44$	−1.28917	−0.194349
$45$	0	0
$46$	4.15165	0.612128
$47$	−2.52444	−0.368227	−0.184114	−	0.982905i	$-0.558941\pi$
$47$	−2.52444	−0.368227	−0.184114	+	0.982905i	$0.558941\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.81361	0.256483
$51$	0	0
$52$	6.57834	0.912251
$53$	−0.289169	−0.0397204	−0.0198602	−	0.999803i	$-0.506322\pi$
$53$	−0.289169	−0.0397204	−0.0198602	+	0.999803i	$0.506322\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.28917	0.172272
$57$	0	0
$58$	−2.15165	−0.282526
$59$	10.4408	1.35928	0.679639	−	0.733546i	$-0.262136\pi$
$59$	10.4408	1.35928	0.679639	+	0.733546i	$0.262136\pi$
$60$	0	0
$61$	6.49472	0.831564	0.415782	−	0.909464i	$-0.363508\pi$
$61$	6.49472	0.831564	0.415782	+	0.909464i	$0.363508\pi$
$62$	−4.57834	−0.581449
$63$	0	0
$64$	−1.66196	−0.207744
$65$	−5.10278	−0.632921
$66$	0	0
$67$	−0.578337	−0.0706551	−0.0353276	−	0.999376i	$-0.511247\pi$
$67$	−0.578337	−0.0706551	−0.0353276	+	0.999376i	$0.511247\pi$
$68$	2.47054	0.299597
$69$	0	0
$70$	1.81361	0.216767
$71$	13.3083	1.57941	0.789704	−	0.613488i	$-0.210234\pi$
$71$	13.3083	1.57941	0.789704	+	0.613488i	$0.210234\pi$
$72$	0	0
$73$	0.372787	0.0436314	0.0218157	−	0.999762i	$-0.493055\pi$
$73$	0.372787	0.0436314	0.0218157	+	0.999762i	$0.493055\pi$
$74$	7.15667	0.831946
$75$	0	0
$76$	4.30330	0.493623
$77$	1.00000	0.113961
$78$	0	0
$79$	15.4061	1.73332	0.866660	−	0.498900i	$-0.166263\pi$
$79$	15.4061	1.73332	0.866660	+	0.498900i	$0.166263\pi$
$80$	4.91638	0.549668
$81$	0	0
$82$	3.04888	0.336692
$83$	−15.7491	−1.72869	−0.864347	−	0.502897i	$-0.832268\pi$
$83$	−15.7491	−1.72869	−0.864347	+	0.502897i	$0.832268\pi$
$84$	0	0
$85$	−1.91638	−0.207861
$86$	12.7300	1.37271
$87$	0	0
$88$	1.28917	0.137426
$89$	0.0297193	0.00315024	0.00157512	−	0.999999i	$-0.499499\pi$
$89$	0.0297193	0.00315024	0.00157512	+	0.999999i	$0.499499\pi$
$90$	0	0
$91$	−5.10278	−0.534916
$92$	2.95112	0.307676
$93$	0	0
$94$	−4.57834	−0.472219
$95$	−3.33804	−0.342476
$96$	0	0
$97$	15.8030	1.60456	0.802278	−	0.596951i	$-0.203621\pi$
$97$	15.8030	1.60456	0.802278	+	0.596951i	$0.203621\pi$
$98$	1.81361	0.183202
$99$	0	0
$100$	1.28917	0.128917
$101$	6.41110	0.637928	0.318964	−	0.947767i	$-0.396665\pi$
$101$	6.41110	0.637928	0.318964	+	0.947767i	$0.396665\pi$
$102$	0	0
$103$	4.81361	0.474299	0.237149	−	0.971473i	$-0.423787\pi$
$103$	4.81361	0.474299	0.237149	+	0.971473i	$0.423787\pi$
$104$	−6.57834	−0.645059
$105$	0	0
$106$	−0.524438	−0.0509379
$107$	13.8867	1.34247	0.671237	−	0.741243i	$-0.265763\pi$
$107$	13.8867	1.34247	0.671237	+	0.741243i	$0.265763\pi$
$108$	0	0
$109$	−2.25945	−0.216416	−0.108208	−	0.994128i	$-0.534511\pi$
$109$	−2.25945	−0.216416	−0.108208	+	0.994128i	$0.534511\pi$
$110$	1.81361	0.172921
$111$	0	0
$112$	4.91638	0.464554
$113$	−9.17081	−0.862717	−0.431359	−	0.902181i	$-0.641966\pi$
$113$	−9.17081	−0.862717	−0.431359	+	0.902181i	$0.641966\pi$
$114$	0	0
$115$	−2.28917	−0.213466
$116$	−1.52946	−0.142007
$117$	0	0
$118$	18.9355	1.74316
$119$	−1.91638	−0.175674
$120$	0	0
$121$	1.00000	0.0909091
$122$	11.7789	1.06641
$123$	0	0
$124$	−3.25443	−0.292256
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.5975	1.38405	0.692027	−	0.721872i	$-0.256718\pi$
$127$	15.5975	1.38405	0.692027	+	0.721872i	$0.256718\pi$
$128$	9.66196	0.854004
$129$	0	0
$130$	−9.25443	−0.811667
$131$	5.36776	0.468984	0.234492	−	0.972118i	$-0.424657\pi$
$131$	5.36776	0.468984	0.234492	+	0.972118i	$0.424657\pi$
$132$	0	0
$133$	−3.33804	−0.289445
$134$	−1.04888	−0.0906091
$135$	0	0
$136$	−2.47054	−0.211847
$137$	−19.2927	−1.64829	−0.824145	−	0.566379i	$-0.808344\pi$
$137$	−19.2927	−1.64829	−0.824145	+	0.566379i	$0.808344\pi$
$138$	0	0
$139$	−3.52946	−0.299365	−0.149682	−	0.988734i	$-0.547825\pi$
$139$	−3.52946	−0.299365	−0.149682	+	0.988734i	$0.547825\pi$
$140$	1.28917	0.108955
$141$	0	0
$142$	24.1361	2.02545
$143$	−5.10278	−0.426715
$144$	0	0
$145$	1.18639	0.0985246
$146$	0.676089	0.0559535
$147$	0	0
$148$	5.08719	0.418164
$149$	−13.3622	−1.09468	−0.547338	−	0.836912i	$-0.684359\pi$
$149$	−13.3622	−1.09468	−0.547338	+	0.836912i	$0.684359\pi$
$150$	0	0
$151$	12.5783	1.02361	0.511805	−	0.859101i	$-0.328977\pi$
$151$	12.5783	1.02361	0.511805	+	0.859101i	$0.328977\pi$
$152$	−4.30330	−0.349044
$153$	0	0
$154$	1.81361	0.146145
$155$	2.52444	0.202768
$156$	0	0
$157$	−1.59749	−0.127494	−0.0637469	−	0.997966i	$-0.520305\pi$
$157$	−1.59749	−0.127494	−0.0637469	+	0.997966i	$0.520305\pi$
$158$	27.9406	2.22283
$159$	0	0
$160$	6.33804	0.501066
$161$	−2.28917	−0.180412
$162$	0	0
$163$	−8.62219	−0.675342	−0.337671	−	0.941264i	$-0.609639\pi$
$163$	−8.62219	−0.675342	−0.337671	+	0.941264i	$0.609639\pi$
$164$	2.16724	0.169233
$165$	0	0
$166$	−28.5628	−2.21690
$167$	−17.6272	−1.36403	−0.682017	−	0.731336i	$-0.738897\pi$
$167$	−17.6272	−1.36403	−0.682017	+	0.731336i	$0.738897\pi$
$168$	0	0
$169$	13.0383	1.00295
$170$	−3.47556	−0.266563
$171$	0	0
$172$	9.04888	0.689970
$173$	−20.0383	−1.52348	−0.761742	−	0.647880i	$-0.775656\pi$
$173$	−20.0383	−1.52348	−0.761742	+	0.647880i	$0.775656\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	4.91638	0.370586
$177$	0	0
$178$	0.0538991	0.00403991
$179$	4.89722	0.366036	0.183018	−	0.983110i	$-0.441413\pi$
$179$	4.89722	0.366036	0.183018	+	0.983110i	$0.441413\pi$
$180$	0	0
$181$	−6.41110	−0.476533	−0.238267	−	0.971200i	$-0.576579\pi$
$181$	−6.41110	−0.476533	−0.238267	+	0.971200i	$0.576579\pi$
$182$	−9.25443	−0.685984
$183$	0	0
$184$	−2.95112	−0.217560
$185$	−3.94610	−0.290123
$186$	0	0
$187$	−1.91638	−0.140140
$188$	−3.25443	−0.237353
$189$	0	0
$190$	−6.05390	−0.439196
$191$	10.0978	0.730648	0.365324	−	0.930880i	$-0.380958\pi$
$191$	10.0978	0.730648	0.365324	+	0.930880i	$0.380958\pi$
$192$	0	0
$193$	7.04888	0.507389	0.253695	−	0.967284i	$-0.418354\pi$
$193$	7.04888	0.507389	0.253695	+	0.967284i	$0.418354\pi$
$194$	28.6605	2.05770
$195$	0	0
$196$	1.28917	0.0920835
$197$	0.372787	0.0265600	0.0132800	−	0.999912i	$-0.495773\pi$
$197$	0.372787	0.0265600	0.0132800	+	0.999912i	$0.495773\pi$
$198$	0	0
$199$	−23.6655	−1.67760	−0.838802	−	0.544436i	$-0.816744\pi$
$199$	−23.6655	−1.67760	−0.838802	+	0.544436i	$0.816744\pi$
$200$	−1.28917	−0.0911580
$201$	0	0
$202$	11.6272	0.818088
$203$	1.18639	0.0832685
$204$	0	0
$205$	−1.68111	−0.117414
$206$	8.72999	0.608247
$207$	0	0
$208$	−25.0872	−1.73948
$209$	−3.33804	−0.230897
$210$	0	0
$211$	3.52946	0.242978	0.121489	−	0.992593i	$-0.461233\pi$
$211$	3.52946	0.242978	0.121489	+	0.992593i	$0.461233\pi$
$212$	−0.372787	−0.0256031
$213$	0	0
$214$	25.1849	1.72161
$215$	−7.01916	−0.478703
$216$	0	0
$217$	2.52444	0.171370
$218$	−4.09775	−0.277535
$219$	0	0
$220$	1.28917	0.0869157
$221$	9.77886	0.657798
$222$	0	0
$223$	−26.6847	−1.78694	−0.893469	−	0.449124i	$-0.851736\pi$
$223$	−26.6847	−1.78694	−0.893469	+	0.449124i	$0.851736\pi$
$224$	6.33804	0.423478
$225$	0	0
$226$	−16.6322	−1.10636
$227$	24.6902	1.63875	0.819374	−	0.573260i	$-0.194321\pi$
$227$	24.6902	1.63875	0.819374	+	0.573260i	$0.194321\pi$
$228$	0	0
$229$	−18.8972	−1.24876	−0.624382	−	0.781119i	$-0.714649\pi$
$229$	−18.8972	−1.24876	−0.624382	+	0.781119i	$0.714649\pi$
$230$	−4.15165	−0.273752
$231$	0	0
$232$	1.52946	0.100414
$233$	15.0333	0.984864	0.492432	−	0.870351i	$-0.336108\pi$
$233$	15.0333	0.984864	0.492432	+	0.870351i	$0.336108\pi$
$234$	0	0
$235$	2.52444	0.164676
$236$	13.4600	0.876170
$237$	0	0
$238$	−3.47556	−0.225287
$239$	−0.0680325	−0.00440066	−0.00220033	−	0.999998i	$-0.500700\pi$
$239$	−0.0680325	−0.00440066	−0.00220033	+	0.999998i	$0.500700\pi$
$240$	0	0
$241$	−15.9305	−1.02617	−0.513087	−	0.858336i	$-0.671498\pi$
$241$	−15.9305	−1.02617	−0.513087	+	0.858336i	$0.671498\pi$
$242$	1.81361	0.116583
$243$	0	0
$244$	8.37279	0.536013
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	17.0333	1.08380
$248$	3.25443	0.206656
$249$	0	0
$250$	−1.81361	−0.114703
$251$	−8.41110	−0.530904	−0.265452	−	0.964124i	$-0.585521\pi$
$251$	−8.41110	−0.530904	−0.265452	+	0.964124i	$0.585521\pi$
$252$	0	0
$253$	−2.28917	−0.143919
$254$	28.2877	1.77493
$255$	0	0
$256$	20.8469	1.30293
$257$	−2.72999	−0.170292	−0.0851460	−	0.996368i	$-0.527136\pi$
$257$	−2.72999	−0.170292	−0.0851460	+	0.996368i	$0.527136\pi$
$258$	0	0
$259$	−3.94610	−0.245199
$260$	−6.57834	−0.407971
$261$	0	0
$262$	9.73501	0.601431
$263$	−12.7144	−0.784004	−0.392002	−	0.919964i	$-0.628217\pi$
$263$	−12.7144	−0.784004	−0.392002	+	0.919964i	$0.628217\pi$
$264$	0	0
$265$	0.289169	0.0177635
$266$	−6.05390	−0.371188
$267$	0	0
$268$	−0.745574	−0.0455432
$269$	−25.1552	−1.53374	−0.766870	−	0.641802i	$-0.778187\pi$
$269$	−25.1552	−1.53374	−0.766870	+	0.641802i	$0.778187\pi$
$270$	0	0
$271$	−15.8086	−0.960303	−0.480152	−	0.877186i	$-0.659418\pi$
$271$	−15.8086	−0.960303	−0.480152	+	0.877186i	$0.659418\pi$
$272$	−9.42166	−0.571272
$273$	0	0
$274$	−34.9894	−2.11379
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−14.2439	−0.855831	−0.427915	−	0.903819i	$-0.640752\pi$
$277$	−14.2439	−0.855831	−0.427915	+	0.903819i	$0.640752\pi$
$278$	−6.40105	−0.383910
$279$	0	0
$280$	−1.28917	−0.0770426
$281$	−11.9305	−0.711715	−0.355857	−	0.934540i	$-0.615811\pi$
$281$	−11.9305	−0.711715	−0.355857	+	0.934540i	$0.615811\pi$
$282$	0	0
$283$	−7.98441	−0.474624	−0.237312	−	0.971433i	$-0.576266\pi$
$283$	−7.98441	−0.474624	−0.237312	+	0.971433i	$0.576266\pi$
$284$	17.1567	1.01806
$285$	0	0
$286$	−9.25443	−0.547226
$287$	−1.68111	−0.0992329
$288$	0	0
$289$	−13.3275	−0.783970
$290$	2.15165	0.126349
$291$	0	0
$292$	0.480585	0.0281241
$293$	33.3069	1.94581	0.972904	−	0.231209i	$-0.0742681\pi$
$293$	33.3069	1.94581	0.972904	+	0.231209i	$0.0742681\pi$
$294$	0	0
$295$	−10.4408	−0.607888
$296$	−5.08719	−0.295687
$297$	0	0
$298$	−24.2338	−1.40383
$299$	11.6811	0.675536
$300$	0	0
$301$	−7.01916	−0.404577
$302$	22.8122	1.31269
$303$	0	0
$304$	−16.4111	−0.941241
$305$	−6.49472	−0.371887
$306$	0	0
$307$	20.4705	1.16832	0.584158	−	0.811640i	$-0.301425\pi$
$307$	20.4705	1.16832	0.584158	+	0.811640i	$0.301425\pi$
$308$	1.28917	0.0734572
$309$	0	0
$310$	4.57834	0.260032
$311$	−3.89220	−0.220707	−0.110353	−	0.993892i	$-0.535198\pi$
$311$	−3.89220	−0.220707	−0.110353	+	0.993892i	$0.535198\pi$
$312$	0	0
$313$	−13.4303	−0.759123	−0.379561	−	0.925167i	$-0.623925\pi$
$313$	−13.4303	−0.759123	−0.379561	+	0.925167i	$0.623925\pi$
$314$	−2.89722	−0.163500
$315$	0	0
$316$	19.8610	1.11727
$317$	29.0872	1.63370	0.816850	−	0.576851i	$-0.195719\pi$
$317$	29.0872	1.63370	0.816850	+	0.576851i	$0.195719\pi$
$318$	0	0
$319$	1.18639	0.0664253
$320$	1.66196	0.0929061
$321$	0	0
$322$	−4.15165	−0.231362
$323$	6.39697	0.355937
$324$	0	0
$325$	5.10278	0.283051
$326$	−15.6373	−0.866068
$327$	0	0
$328$	−2.16724	−0.119666
$329$	2.52444	0.139177
$330$	0	0
$331$	−19.8086	−1.08878	−0.544389	−	0.838833i	$-0.683238\pi$
$331$	−19.8086	−1.08878	−0.544389	+	0.838833i	$0.683238\pi$
$332$	−20.3033	−1.11429
$333$	0	0
$334$	−31.9688	−1.74926
$335$	0.578337	0.0315979
$336$	0	0
$337$	−21.4303	−1.16738	−0.583690	−	0.811976i	$-0.698392\pi$
$337$	−21.4303	−1.16738	−0.583690	+	0.811976i	$0.698392\pi$
$338$	23.6464	1.28619
$339$	0	0
$340$	−2.47054	−0.133984
$341$	2.52444	0.136706
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−9.04888	−0.487883
$345$	0	0
$346$	−36.3416	−1.95374
$347$	30.6167	1.64359	0.821794	−	0.569785i	$-0.192973\pi$
$347$	30.6167	1.64359	0.821794	+	0.569785i	$0.192973\pi$
$348$	0	0
$349$	−6.38692	−0.341884	−0.170942	−	0.985281i	$-0.554681\pi$
$349$	−6.38692	−0.341884	−0.170942	+	0.985281i	$0.554681\pi$
$350$	−1.81361	−0.0969413
$351$	0	0
$352$	6.33804	0.337819
$353$	−10.6378	−0.566192	−0.283096	−	0.959092i	$-0.591361\pi$
$353$	−10.6378	−0.566192	−0.283096	+	0.959092i	$0.591361\pi$
$354$	0	0
$355$	−13.3083	−0.706333
$356$	0.0383132	0.00203060
$357$	0	0
$358$	8.88164	0.469409
$359$	32.3713	1.70849	0.854247	−	0.519868i	$-0.174019\pi$
$359$	32.3713	1.70849	0.854247	+	0.519868i	$0.174019\pi$
$360$	0	0
$361$	−7.85746	−0.413550
$362$	−11.6272	−0.611113
$363$	0	0
$364$	−6.57834	−0.344799
$365$	−0.372787	−0.0195126
$366$	0	0
$367$	19.7053	1.02861	0.514304	−	0.857608i	$-0.328050\pi$
$367$	19.7053	1.02861	0.514304	+	0.857608i	$0.328050\pi$
$368$	−11.2544	−0.586678
$369$	0	0
$370$	−7.15667	−0.372058
$371$	0.289169	0.0150129
$372$	0	0
$373$	5.12695	0.265464	0.132732	−	0.991152i	$-0.457625\pi$
$373$	5.12695	0.265464	0.132732	+	0.991152i	$0.457625\pi$
$374$	−3.47556	−0.179717
$375$	0	0
$376$	3.25443	0.167834
$377$	−6.05390	−0.311792
$378$	0	0
$379$	9.07306	0.466052	0.233026	−	0.972471i	$-0.425137\pi$
$379$	9.07306	0.466052	0.233026	+	0.972471i	$0.425137\pi$
$380$	−4.30330	−0.220755
$381$	0	0
$382$	18.3133	0.936992
$383$	6.16170	0.314848	0.157424	−	0.987531i	$-0.449681\pi$
$383$	6.16170	0.314848	0.157424	+	0.987531i	$0.449681\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	12.7839	0.650683
$387$	0	0
$388$	20.3728	1.03427
$389$	−10.9739	−0.556396	−0.278198	−	0.960524i	$-0.589737\pi$
$389$	−10.9739	−0.556396	−0.278198	+	0.960524i	$0.589737\pi$
$390$	0	0
$391$	4.38692	0.221856
$392$	−1.28917	−0.0651128
$393$	0	0
$394$	0.676089	0.0340609
$395$	−15.4061	−0.775164
$396$	0	0
$397$	31.9305	1.60255	0.801273	−	0.598298i	$-0.204156\pi$
$397$	31.9305	1.60255	0.801273	+	0.598298i	$0.204156\pi$
$398$	−42.9200	−2.15138
$399$	0	0
$400$	−4.91638	−0.245819
$401$	14.7300	0.735581	0.367790	−	0.929909i	$-0.380114\pi$
$401$	14.7300	0.735581	0.367790	+	0.929909i	$0.380114\pi$
$402$	0	0
$403$	−12.8816	−0.641680
$404$	8.26499	0.411199
$405$	0	0
$406$	2.15165	0.106785
$407$	−3.94610	−0.195601
$408$	0	0
$409$	−2.36274	−0.116830	−0.0584150	−	0.998292i	$-0.518605\pi$
$409$	−2.36274	−0.116830	−0.0584150	+	0.998292i	$0.518605\pi$
$410$	−3.04888	−0.150573
$411$	0	0
$412$	6.20555	0.305726
$413$	−10.4408	−0.513759
$414$	0	0
$415$	15.7491	0.773095
$416$	−32.3416	−1.58568
$417$	0	0
$418$	−6.05390	−0.296106
$419$	11.5975	0.566575	0.283287	−	0.959035i	$-0.408575\pi$
$419$	11.5975	0.566575	0.283287	+	0.959035i	$0.408575\pi$
$420$	0	0
$421$	15.4842	0.754652	0.377326	−	0.926081i	$-0.376844\pi$
$421$	15.4842	0.754652	0.377326	+	0.926081i	$0.376844\pi$
$422$	6.40105	0.311598
$423$	0	0
$424$	0.372787	0.0181041
$425$	1.91638	0.0929582
$426$	0	0
$427$	−6.49472	−0.314301
$428$	17.9022	0.865338
$429$	0	0
$430$	−12.7300	−0.613895
$431$	23.9688	1.15454	0.577269	−	0.816554i	$-0.304118\pi$
$431$	23.9688	1.15454	0.577269	+	0.816554i	$0.304118\pi$
$432$	0	0
$433$	4.73553	0.227575	0.113787	−	0.993505i	$-0.463702\pi$
$433$	4.73553	0.227575	0.113787	+	0.993505i	$0.463702\pi$
$434$	4.57834	0.219767
$435$	0	0
$436$	−2.91281	−0.139498
$437$	7.64135	0.365535
$438$	0	0
$439$	−13.3764	−0.638419	−0.319209	−	0.947684i	$-0.603417\pi$
$439$	−13.3764	−0.638419	−0.319209	+	0.947684i	$0.603417\pi$
$440$	−1.28917	−0.0614587
$441$	0	0
$442$	17.7350	0.843568
$443$	−11.1950	−0.531890	−0.265945	−	0.963988i	$-0.585684\pi$
$443$	−11.1950	−0.531890	−0.265945	+	0.963988i	$0.585684\pi$
$444$	0	0
$445$	−0.0297193	−0.00140883
$446$	−48.3955	−2.29159
$447$	0	0
$448$	1.66196	0.0785200
$449$	−9.23884	−0.436008	−0.218004	−	0.975948i	$-0.569955\pi$
$449$	−9.23884	−0.436008	−0.218004	+	0.975948i	$0.569955\pi$
$450$	0	0
$451$	−1.68111	−0.0791605
$452$	−11.8227	−0.556094
$453$	0	0
$454$	44.7783	2.10155
$455$	5.10278	0.239222
$456$	0	0
$457$	13.7053	0.641107	0.320553	−	0.947230i	$-0.396131\pi$
$457$	13.7053	0.641107	0.320553	+	0.947230i	$0.396131\pi$
$458$	−34.2721	−1.60143
$459$	0	0
$460$	−2.95112	−0.137597
$461$	12.6167	0.587616	0.293808	−	0.955864i	$-0.405077\pi$
$461$	12.6167	0.587616	0.293808	+	0.955864i	$0.405077\pi$
$462$	0	0
$463$	21.0972	0.980472	0.490236	−	0.871590i	$-0.336911\pi$
$463$	21.0972	0.980472	0.490236	+	0.871590i	$0.336911\pi$
$464$	5.83276	0.270779
$465$	0	0
$466$	27.2645	1.26300
$467$	29.7944	1.37872	0.689361	−	0.724418i	$-0.257891\pi$
$467$	29.7944	1.37872	0.689361	+	0.724418i	$0.257891\pi$
$468$	0	0
$469$	0.578337	0.0267051
$470$	4.57834	0.211183
$471$	0	0
$472$	−13.4600	−0.619546
$473$	−7.01916	−0.322741
$474$	0	0
$475$	3.33804	0.153160
$476$	−2.47054	−0.113237
$477$	0	0
$478$	−0.123384	−0.00564346
$479$	−30.0822	−1.37449	−0.687245	−	0.726426i	$-0.741180\pi$
$479$	−30.0822	−1.37449	−0.687245	+	0.726426i	$0.741180\pi$
$480$	0	0
$481$	20.1361	0.918126
$482$	−28.8917	−1.31598
$483$	0	0
$484$	1.28917	0.0585986
$485$	−15.8030	−0.717579
$486$	0	0
$487$	−23.8016	−1.07855	−0.539277	−	0.842129i	$-0.681302\pi$
$487$	−23.8016	−1.07855	−0.539277	+	0.842129i	$0.681302\pi$
$488$	−8.37279	−0.379018
$489$	0	0
$490$	−1.81361	−0.0819304
$491$	32.9980	1.48918	0.744590	−	0.667522i	$-0.232645\pi$
$491$	32.9980	1.48918	0.744590	+	0.667522i	$0.232645\pi$
$492$	0	0
$493$	−2.27358	−0.102397
$494$	30.8917	1.38988
$495$	0	0
$496$	12.4111	0.557275
$497$	−13.3083	−0.596960
$498$	0	0
$499$	7.44584	0.333322	0.166661	−	0.986014i	$-0.446701\pi$
$499$	7.44584	0.333322	0.166661	+	0.986014i	$0.446701\pi$
$500$	−1.28917	−0.0576534
$501$	0	0
$502$	−15.2544	−0.680838
$503$	−13.7592	−0.613492	−0.306746	−	0.951791i	$-0.599240\pi$
$503$	−13.7592	−0.613492	−0.306746	+	0.951791i	$0.599240\pi$
$504$	0	0
$505$	−6.41110	−0.285290
$506$	−4.15165	−0.184563
$507$	0	0
$508$	20.1078	0.892139
$509$	9.43026	0.417989	0.208994	−	0.977917i	$-0.432981\pi$
$509$	9.43026	0.417989	0.208994	+	0.977917i	$0.432981\pi$
$510$	0	0
$511$	−0.372787	−0.0164911
$512$	18.4842	0.816892
$513$	0	0
$514$	−4.95112	−0.218385
$515$	−4.81361	−0.212113
$516$	0	0
$517$	2.52444	0.111025
$518$	−7.15667	−0.314446
$519$	0	0
$520$	6.57834	0.288479
$521$	−15.9703	−0.699671	−0.349835	−	0.936811i	$-0.613762\pi$
$521$	−15.9703	−0.699671	−0.349835	+	0.936811i	$0.613762\pi$
$522$	0	0
$523$	−13.6756	−0.597991	−0.298996	−	0.954255i	$-0.596652\pi$
$523$	−13.6756	−0.597991	−0.298996	+	0.954255i	$0.596652\pi$
$524$	6.91995	0.302300
$525$	0	0
$526$	−23.0589	−1.00542
$527$	−4.83779	−0.210737
$528$	0	0
$529$	−17.7597	−0.772161
$530$	0.524438	0.0227801
$531$	0	0
$532$	−4.30330	−0.186572
$533$	8.57834	0.371569
$534$	0	0
$535$	−13.8867	−0.600373
$536$	0.745574	0.0322039
$537$	0	0
$538$	−45.6217	−1.96689
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	13.7406	0.590752	0.295376	−	0.955381i	$-0.404555\pi$
$541$	13.7406	0.590752	0.295376	+	0.955381i	$0.404555\pi$
$542$	−28.6705	−1.23151
$543$	0	0
$544$	−12.1461	−0.520760
$545$	2.25945	0.0967842
$546$	0	0
$547$	−37.8797	−1.61962	−0.809809	−	0.586694i	$-0.800429\pi$
$547$	−37.8797	−1.61962	−0.809809	+	0.586694i	$0.800429\pi$
$548$	−24.8716	−1.06246
$549$	0	0
$550$	−1.81361	−0.0773324
$551$	−3.96023	−0.168712
$552$	0	0
$553$	−15.4061	−0.655133
$554$	−25.8328	−1.09753
$555$	0	0
$556$	−4.55007	−0.192966
$557$	22.3799	0.948268	0.474134	−	0.880453i	$-0.342761\pi$
$557$	22.3799	0.948268	0.474134	+	0.880453i	$0.342761\pi$
$558$	0	0
$559$	35.8172	1.51491
$560$	−4.91638	−0.207755
$561$	0	0
$562$	−21.6373	−0.912713
$563$	−24.5783	−1.03585	−0.517927	−	0.855425i	$-0.673296\pi$
$563$	−24.5783	−1.03585	−0.517927	+	0.855425i	$0.673296\pi$
$564$	0	0
$565$	9.17081	0.385819
$566$	−14.4806	−0.608664
$567$	0	0
$568$	−17.1567	−0.719878
$569$	−14.5980	−0.611980	−0.305990	−	0.952035i	$-0.598987\pi$
$569$	−14.5980	−0.611980	−0.305990	+	0.952035i	$0.598987\pi$
$570$	0	0
$571$	−18.7995	−0.786733	−0.393367	−	0.919382i	$-0.628690\pi$
$571$	−18.7995	−0.786733	−0.393367	+	0.919382i	$0.628690\pi$
$572$	−6.57834	−0.275054
$573$	0	0
$574$	−3.04888	−0.127258
$575$	2.28917	0.0954649
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−24.1708	−1.00537
$579$	0	0
$580$	1.52946	0.0635074
$581$	15.7491	0.653385
$582$	0	0
$583$	0.289169	0.0119761
$584$	−0.480585	−0.0198868
$585$	0	0
$586$	60.4056	2.49533
$587$	2.79947	0.115547	0.0577733	−	0.998330i	$-0.481600\pi$
$587$	2.79947	0.115547	0.0577733	+	0.998330i	$0.481600\pi$
$588$	0	0
$589$	−8.42669	−0.347216
$590$	−18.9355	−0.779564
$591$	0	0
$592$	−19.4005	−0.797357
$593$	−25.4983	−1.04709	−0.523545	−	0.851998i	$-0.675391\pi$
$593$	−25.4983	−1.04709	−0.523545	+	0.851998i	$0.675391\pi$
$594$	0	0
$595$	1.91638	0.0785640
$596$	−17.2262	−0.705611
$597$	0	0
$598$	21.1849	0.866317
$599$	−23.1950	−0.947721	−0.473861	−	0.880600i	$-0.657140\pi$
$599$	−23.1950	−0.947721	−0.473861	+	0.880600i	$0.657140\pi$
$600$	0	0
$601$	2.63078	0.107312	0.0536560	−	0.998559i	$-0.482913\pi$
$601$	2.63078	0.107312	0.0536560	+	0.998559i	$0.482913\pi$
$602$	−12.7300	−0.518836
$603$	0	0
$604$	16.2156	0.659803
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	45.4882	1.84631	0.923155	−	0.384427i	$-0.125601\pi$
$607$	45.4882	1.84631	0.923155	+	0.384427i	$0.125601\pi$
$608$	−21.1567	−0.858016
$609$	0	0
$610$	−11.7789	−0.476912
$611$	−12.8816	−0.521135
$612$	0	0
$613$	34.4111	1.38985	0.694926	−	0.719082i	$-0.255437\pi$
$613$	34.4111	1.38985	0.694926	+	0.719082i	$0.255437\pi$
$614$	37.1255	1.49826
$615$	0	0
$616$	−1.28917	−0.0519421
$617$	42.8505	1.72509	0.862547	−	0.505976i	$-0.168868\pi$
$617$	42.8505	1.72509	0.862547	+	0.505976i	$0.168868\pi$
$618$	0	0
$619$	26.6761	1.07220	0.536101	−	0.844154i	$-0.319897\pi$
$619$	26.6761	1.07220	0.536101	+	0.844154i	$0.319897\pi$
$620$	3.25443	0.130701
$621$	0	0
$622$	−7.05892	−0.283037
$623$	−0.0297193	−0.00119068
$624$	0	0
$625$	1.00000	0.0400000
$626$	−24.3572	−0.973510
$627$	0	0
$628$	−2.05944	−0.0821805
$629$	7.56223	0.301526
$630$	0	0
$631$	15.1708	0.603940	0.301970	−	0.953317i	$-0.402356\pi$
$631$	15.1708	0.603940	0.301970	+	0.953317i	$0.402356\pi$
$632$	−19.8610	−0.790029
$633$	0	0
$634$	52.7527	2.09508
$635$	−15.5975	−0.618968
$636$	0	0
$637$	5.10278	0.202179
$638$	2.15165	0.0851847
$639$	0	0
$640$	−9.66196	−0.381922
$641$	−17.6655	−0.697746	−0.348873	−	0.937170i	$-0.613436\pi$
$641$	−17.6655	−0.697746	−0.348873	+	0.937170i	$0.613436\pi$
$642$	0	0
$643$	21.9108	0.864079	0.432040	−	0.901855i	$-0.357794\pi$
$643$	21.9108	0.864079	0.432040	+	0.901855i	$0.357794\pi$
$644$	−2.95112	−0.116291
$645$	0	0
$646$	11.6016	0.456458
$647$	44.0455	1.73161	0.865803	−	0.500385i	$-0.166808\pi$
$647$	44.0455	1.73161	0.865803	+	0.500385i	$0.166808\pi$
$648$	0	0
$649$	−10.4408	−0.409838
$650$	9.25443	0.362988
$651$	0	0
$652$	−11.1155	−0.435315
$653$	24.7003	0.966596	0.483298	−	0.875456i	$-0.339439\pi$
$653$	24.7003	0.966596	0.483298	+	0.875456i	$0.339439\pi$
$654$	0	0
$655$	−5.36776	−0.209736
$656$	−8.26499	−0.322694
$657$	0	0
$658$	4.57834	0.178482
$659$	−31.6741	−1.23385	−0.616924	−	0.787023i	$-0.711622\pi$
$659$	−31.6741	−1.23385	−0.616924	+	0.787023i	$0.711622\pi$
$660$	0	0
$661$	1.63275	0.0635067	0.0317534	−	0.999496i	$-0.489891\pi$
$661$	1.63275	0.0635067	0.0317534	+	0.999496i	$0.489891\pi$
$662$	−35.9250	−1.39626
$663$	0	0
$664$	20.3033	0.787921
$665$	3.33804	0.129444
$666$	0	0
$667$	−2.71585	−0.105158
$668$	−22.7244	−0.879235
$669$	0	0
$670$	1.04888	0.0405216
$671$	−6.49472	−0.250726
$672$	0	0
$673$	−15.5280	−0.598561	−0.299280	−	0.954165i	$-0.596747\pi$
$673$	−15.5280	−0.598561	−0.299280	+	0.954165i	$0.596747\pi$
$674$	−38.8661	−1.49706
$675$	0	0
$676$	16.8086	0.646484
$677$	−5.33804	−0.205158	−0.102579	−	0.994725i	$-0.532709\pi$
$677$	−5.33804	−0.205158	−0.102579	+	0.994725i	$0.532709\pi$
$678$	0	0
$679$	−15.8030	−0.606465
$680$	2.47054	0.0947408
$681$	0	0
$682$	4.57834	0.175314
$683$	32.4111	1.24018	0.620088	−	0.784532i	$-0.287097\pi$
$683$	32.4111	1.24018	0.620088	+	0.784532i	$0.287097\pi$
$684$	0	0
$685$	19.2927	0.737138
$686$	−1.81361	−0.0692438
$687$	0	0
$688$	−34.5089	−1.31564
$689$	−1.47556	−0.0562144
$690$	0	0
$691$	1.73501	0.0660029	0.0330015	−	0.999455i	$-0.489493\pi$
$691$	1.73501	0.0660029	0.0330015	+	0.999455i	$0.489493\pi$
$692$	−25.8328	−0.982014
$693$	0	0
$694$	55.5266	2.10776
$695$	3.52946	0.133880
$696$	0	0
$697$	3.22165	0.122029
$698$	−11.5834	−0.438437
$699$	0	0
$700$	−1.28917	−0.0487260
$701$	1.59749	0.0603365	0.0301683	−	0.999545i	$-0.490396\pi$
$701$	1.59749	0.0603365	0.0301683	+	0.999545i	$0.490396\pi$
$702$	0	0
$703$	13.1723	0.496801
$704$	1.66196	0.0626373
$705$	0	0
$706$	−19.2927	−0.726092
$707$	−6.41110	−0.241114
$708$	0	0
$709$	27.3280	1.02632	0.513162	−	0.858292i	$-0.328474\pi$
$709$	27.3280	1.02632	0.513162	+	0.858292i	$0.328474\pi$
$710$	−24.1361	−0.905810
$711$	0	0
$712$	−0.0383132	−0.00143585
$713$	−5.77886	−0.216420
$714$	0	0
$715$	5.10278	0.190833
$716$	6.31335	0.235941
$717$	0	0
$718$	58.7089	2.19100
$719$	−3.32246	−0.123907	−0.0619534	−	0.998079i	$-0.519733\pi$
$719$	−3.32246	−0.123907	−0.0619534	+	0.998079i	$0.519733\pi$
$720$	0	0
$721$	−4.81361	−0.179268
$722$	−14.2503	−0.530343
$723$	0	0
$724$	−8.26499	−0.307166
$725$	−1.18639	−0.0440615
$726$	0	0
$727$	−35.4897	−1.31624	−0.658120	−	0.752913i	$-0.728648\pi$
$727$	−35.4897	−1.31624	−0.658120	+	0.752913i	$0.728648\pi$
$728$	6.57834	0.243809
$729$	0	0
$730$	−0.676089	−0.0250232
$731$	13.4514	0.497517
$732$	0	0
$733$	−41.6344	−1.53780	−0.768900	−	0.639369i	$-0.779196\pi$
$733$	−41.6344	−1.53780	−0.768900	+	0.639369i	$0.779196\pi$
$734$	35.7376	1.31910
$735$	0	0
$736$	−14.5089	−0.534803
$737$	0.578337	0.0213033
$738$	0	0
$739$	−33.7038	−1.23982	−0.619908	−	0.784675i	$-0.712830\pi$
$739$	−33.7038	−1.23982	−0.619908	+	0.784675i	$0.712830\pi$
$740$	−5.08719	−0.187009
$741$	0	0
$742$	0.524438	0.0192527
$743$	6.15719	0.225885	0.112943	−	0.993602i	$-0.463972\pi$
$743$	6.15719	0.225885	0.112943	+	0.993602i	$0.463972\pi$
$744$	0	0
$745$	13.3622	0.489554
$746$	9.29828	0.340434
$747$	0	0
$748$	−2.47054	−0.0903318
$749$	−13.8867	−0.507408
$750$	0	0
$751$	9.95469	0.363252	0.181626	−	0.983368i	$-0.441864\pi$
$751$	9.95469	0.363252	0.181626	+	0.983368i	$0.441864\pi$
$752$	12.4111	0.452586
$753$	0	0
$754$	−10.9794	−0.399846
$755$	−12.5783	−0.457773
$756$	0	0
$757$	−25.1466	−0.913970	−0.456985	−	0.889474i	$-0.651071\pi$
$757$	−25.1466	−0.913970	−0.456985	+	0.889474i	$0.651071\pi$
$758$	16.4550	0.597671
$759$	0	0
$760$	4.30330	0.156097
$761$	9.25443	0.335473	0.167736	−	0.985832i	$-0.446354\pi$
$761$	9.25443	0.335473	0.167736	+	0.985832i	$0.446354\pi$
$762$	0	0
$763$	2.25945	0.0817976
$764$	13.0177	0.470964
$765$	0	0
$766$	11.1749	0.403765
$767$	53.2772	1.92373
$768$	0	0
$769$	−18.0524	−0.650988	−0.325494	−	0.945544i	$-0.605531\pi$
$769$	−18.0524	−0.650988	−0.325494	+	0.945544i	$0.605531\pi$
$770$	−1.81361	−0.0653578
$771$	0	0
$772$	9.08719	0.327055
$773$	−33.8922	−1.21902	−0.609509	−	0.792779i	$-0.708633\pi$
$773$	−33.8922	−1.21902	−0.609509	+	0.792779i	$0.708633\pi$
$774$	0	0
$775$	−2.52444	−0.0906805
$776$	−20.3728	−0.731340
$777$	0	0
$778$	−19.9022	−0.713530
$779$	5.61163	0.201057
$780$	0	0
$781$	−13.3083	−0.476209
$782$	7.95615	0.284511
$783$	0	0
$784$	−4.91638	−0.175585
$785$	1.59749	0.0570170
$786$	0	0
$787$	−37.1567	−1.32449	−0.662246	−	0.749286i	$-0.730397\pi$
$787$	−37.1567	−1.32449	−0.662246	+	0.749286i	$0.730397\pi$
$788$	0.480585	0.0171201
$789$	0	0
$790$	−27.9406	−0.994081
$791$	9.17081	0.326076
$792$	0	0
$793$	33.1411	1.17687
$794$	57.9094	2.05513
$795$	0	0
$796$	−30.5089	−1.08136
$797$	−18.7144	−0.662898	−0.331449	−	0.943473i	$-0.607538\pi$
$797$	−18.7144	−0.662898	−0.331449	+	0.943473i	$0.607538\pi$
$798$	0	0
$799$	−4.83779	−0.171149
$800$	−6.33804	−0.224084
$801$	0	0
$802$	26.7144	0.943318
$803$	−0.372787	−0.0131554
$804$	0	0
$805$	2.28917	0.0806826
$806$	−23.3622	−0.822899
$807$	0	0
$808$	−8.26499	−0.290761
$809$	34.5955	1.21631	0.608157	−	0.793817i	$-0.291909\pi$
$809$	34.5955	1.21631	0.608157	+	0.793817i	$0.291909\pi$
$810$	0	0
$811$	25.0177	0.878490	0.439245	−	0.898367i	$-0.355246\pi$
$811$	25.0177	0.878490	0.439245	+	0.898367i	$0.355246\pi$
$812$	1.52946	0.0536736
$813$	0	0
$814$	−7.15667	−0.250841
$815$	8.62219	0.302022
$816$	0	0
$817$	23.4303	0.819721
$818$	−4.28508	−0.149824
$819$	0	0
$820$	−2.16724	−0.0756832
$821$	−32.7330	−1.14239	−0.571196	−	0.820814i	$-0.693520\pi$
$821$	−32.7330	−1.14239	−0.571196	+	0.820814i	$0.693520\pi$
$822$	0	0
$823$	−41.3366	−1.44090	−0.720452	−	0.693505i	$-0.756065\pi$
$823$	−41.3366	−1.44090	−0.720452	+	0.693505i	$0.756065\pi$
$824$	−6.20555	−0.216181
$825$	0	0
$826$	−18.9355	−0.658852
$827$	−4.30330	−0.149640	−0.0748202	−	0.997197i	$-0.523838\pi$
$827$	−4.30330	−0.149640	−0.0748202	+	0.997197i	$0.523838\pi$
$828$	0	0
$829$	−10.6378	−0.369465	−0.184733	−	0.982789i	$-0.559142\pi$
$829$	−10.6378	−0.369465	−0.184733	+	0.982789i	$0.559142\pi$
$830$	28.5628	0.991428
$831$	0	0
$832$	−8.48059	−0.294011
$833$	1.91638	0.0663987
$834$	0	0
$835$	17.6272	0.610015
$836$	−4.30330	−0.148833
$837$	0	0
$838$	21.0333	0.726583
$839$	13.9703	0.482308	0.241154	−	0.970487i	$-0.422474\pi$
$839$	13.9703	0.482308	0.241154	+	0.970487i	$0.422474\pi$
$840$	0	0
$841$	−27.5925	−0.951464
$842$	28.0822	0.967775
$843$	0	0
$844$	4.55007	0.156620
$845$	−13.0383	−0.448532
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	1.42166	0.0488201
$849$	0	0
$850$	3.47556	0.119211
$851$	9.03329	0.309657
$852$	0	0
$853$	2.62219	0.0897821	0.0448910	−	0.998992i	$-0.485706\pi$
$853$	2.62219	0.0897821	0.0448910	+	0.998992i	$0.485706\pi$
$854$	−11.7789	−0.403064
$855$	0	0
$856$	−17.9022	−0.611886
$857$	−45.8610	−1.56658	−0.783291	−	0.621655i	$-0.786461\pi$
$857$	−45.8610	−1.56658	−0.783291	+	0.621655i	$0.786461\pi$
$858$	0	0
$859$	12.3078	0.419937	0.209969	−	0.977708i	$-0.432664\pi$
$859$	12.3078	0.419937	0.209969	+	0.977708i	$0.432664\pi$
$860$	−9.04888	−0.308564
$861$	0	0
$862$	43.4700	1.48059
$863$	15.8086	0.538130	0.269065	−	0.963122i	$-0.413285\pi$
$863$	15.8086	0.538130	0.269065	+	0.963122i	$0.413285\pi$
$864$	0	0
$865$	20.0383	0.681323
$866$	8.58838	0.291845
$867$	0	0
$868$	3.25443	0.110462
$869$	−15.4061	−0.522615
$870$	0	0
$871$	−2.95112	−0.0999950
$872$	2.91281	0.0986402
$873$	0	0
$874$	13.8584	0.468767
$875$	1.00000	0.0338062
$876$	0	0
$877$	20.4892	0.691870	0.345935	−	0.938258i	$-0.387562\pi$
$877$	20.4892	0.691870	0.345935	+	0.938258i	$0.387562\pi$
$878$	−24.2594	−0.818717
$879$	0	0
$880$	−4.91638	−0.165731
$881$	2.03977	0.0687215	0.0343607	−	0.999409i	$-0.489060\pi$
$881$	2.03977	0.0687215	0.0343607	+	0.999409i	$0.489060\pi$
$882$	0	0
$883$	−49.7522	−1.67429	−0.837147	−	0.546977i	$-0.815778\pi$
$883$	−49.7522	−1.67429	−0.837147	+	0.546977i	$0.815778\pi$
$884$	12.6066	0.424006
$885$	0	0
$886$	−20.3033	−0.682103
$887$	55.9446	1.87844	0.939219	−	0.343319i	$-0.111551\pi$
$887$	55.9446	1.87844	0.939219	+	0.343319i	$0.111551\pi$
$888$	0	0
$889$	−15.5975	−0.523123
$890$	−0.0538991	−0.00180670
$891$	0	0
$892$	−34.4011	−1.15183
$893$	−8.42669	−0.281988
$894$	0	0
$895$	−4.89722	−0.163696
$896$	−9.66196	−0.322783
$897$	0	0
$898$	−16.7556	−0.559142
$899$	2.99498	0.0998881
$900$	0	0
$901$	−0.554157	−0.0184617
$902$	−3.04888	−0.101516
$903$	0	0
$904$	11.8227	0.393218
$905$	6.41110	0.213112
$906$	0	0
$907$	−41.7789	−1.38724	−0.693622	−	0.720339i	$-0.743986\pi$
$907$	−41.7789	−1.38724	−0.693622	+	0.720339i	$0.743986\pi$
$908$	31.8299	1.05631
$909$	0	0
$910$	9.25443	0.306781
$911$	−26.6167	−0.881849	−0.440924	−	0.897544i	$-0.645349\pi$
$911$	−26.6167	−0.881849	−0.440924	+	0.897544i	$0.645349\pi$
$912$	0	0
$913$	15.7491	0.521221
$914$	24.8560	0.822164
$915$	0	0
$916$	−24.3617	−0.804933
$917$	−5.36776	−0.177259
$918$	0	0
$919$	32.7738	1.08111	0.540555	−	0.841309i	$-0.318214\pi$
$919$	32.7738	1.08111	0.540555	+	0.841309i	$0.318214\pi$
$920$	2.95112	0.0972957
$921$	0	0
$922$	22.8816	0.753567
$923$	67.9094	2.23526
$924$	0	0
$925$	3.94610	0.129747
$926$	38.2621	1.25737
$927$	0	0
$928$	7.51941	0.246837
$929$	0.264989	0.00869400	0.00434700	−	0.999991i	$-0.498616\pi$
$929$	0.264989	0.00869400	0.00434700	+	0.999991i	$0.498616\pi$
$930$	0	0
$931$	3.33804	0.109400
$932$	19.3804	0.634828
$933$	0	0
$934$	54.0354	1.76809
$935$	1.91638	0.0626724
$936$	0	0
$937$	26.0610	0.851377	0.425689	−	0.904870i	$-0.360032\pi$
$937$	26.0610	0.851377	0.425689	+	0.904870i	$0.360032\pi$
$938$	1.04888	0.0342470
$939$	0	0
$940$	3.25443	0.106148
$941$	−30.3472	−0.989289	−0.494644	−	0.869095i	$-0.664702\pi$
$941$	−30.3472	−0.989289	−0.494644	+	0.869095i	$0.664702\pi$
$942$	0	0
$943$	3.84835	0.125319
$944$	−51.3311	−1.67068
$945$	0	0
$946$	−12.7300	−0.413888
$947$	43.3380	1.40830	0.704149	−	0.710053i	$-0.251329\pi$
$947$	43.3380	1.40830	0.704149	+	0.710053i	$0.251329\pi$
$948$	0	0
$949$	1.90225	0.0617496
$950$	6.05390	0.196414
$951$	0	0
$952$	2.47054	0.0800706
$953$	−15.9022	−0.515124	−0.257562	−	0.966262i	$-0.582919\pi$
$953$	−15.9022	−0.515124	−0.257562	+	0.966262i	$0.582919\pi$
$954$	0	0
$955$	−10.0978	−0.326756
$956$	−0.0877054	−0.00283659
$957$	0	0
$958$	−54.5572	−1.76266
$959$	19.2927	0.622995
$960$	0	0
$961$	−24.6272	−0.794426
$962$	36.5189	1.17742
$963$	0	0
$964$	−20.5371	−0.661456
$965$	−7.04888	−0.226911
$966$	0	0
$967$	−23.6852	−0.761665	−0.380832	−	0.924644i	$-0.624362\pi$
$967$	−23.6852	−0.761665	−0.380832	+	0.924644i	$0.624362\pi$
$968$	−1.28917	−0.0414354
$969$	0	0
$970$	−28.6605	−0.920233
$971$	−18.9597	−0.608446	−0.304223	−	0.952601i	$-0.598397\pi$
$971$	−18.9597	−0.608446	−0.304223	+	0.952601i	$0.598397\pi$
$972$	0	0
$973$	3.52946	0.113149
$974$	−43.1667	−1.38315
$975$	0	0
$976$	−31.9305	−1.02207
$977$	23.9063	0.764831	0.382416	−	0.923990i	$-0.375092\pi$
$977$	23.9063	0.764831	0.382416	+	0.923990i	$0.375092\pi$
$978$	0	0
$979$	−0.0297193	−0.000949833 0
$980$	−1.28917	−0.0411810
$981$	0	0
$982$	59.8454	1.90974
$983$	1.62721	0.0519000	0.0259500	−	0.999663i	$-0.491739\pi$
$983$	1.62721	0.0519000	0.0259500	+	0.999663i	$0.491739\pi$
$984$	0	0
$985$	−0.372787	−0.0118780
$986$	−4.12338	−0.131315
$987$	0	0
$988$	21.9588	0.698602
$989$	16.0680	0.510934
$990$	0	0
$991$	−20.9058	−0.664095	−0.332048	−	0.943263i	$-0.607739\pi$
$991$	−20.9058	−0.664095	−0.332048	+	0.943263i	$0.607739\pi$
$992$	16.0000	0.508001
$993$	0	0
$994$	−24.1361	−0.765549
$995$	23.6655	0.750248
$996$	0	0
$997$	0.0977518	0.00309583	0.00154792	−	0.999999i	$-0.499507\pi$
$997$	0.0977518	0.00309583	0.00154792	+	0.999999i	$0.499507\pi$
$998$	13.5038	0.427456
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bb.1.3		3
3.2	odd	2		1155.2.a.t.1.1	✓	3
15.14	odd	2		5775.2.a.bq.1.3		3
21.20	even	2		8085.2.a.bl.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.t.1.1	✓	3	3.2	odd	2
3465.2.a.bb.1.3		3	1.1	even	1	trivial
5775.2.a.bq.1.3		3	15.14	odd	2
8085.2.a.bl.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.81361 q^{2} +1.28917 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.28917 q^{8} +O(q^{10})\)	\(q+1.81361 q^{2} +1.28917 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.28917 q^{8} -1.81361 q^{10} -1.00000 q^{11} +5.10278 q^{13} -1.81361 q^{14} -4.91638 q^{16} +1.91638 q^{17} +3.33804 q^{19} -1.28917 q^{20} -1.81361 q^{22} +2.28917 q^{23} +1.00000 q^{25} +9.25443 q^{26} -1.28917 q^{28} -1.18639 q^{29} -2.52444 q^{31} -6.33804 q^{32} +3.47556 q^{34} +1.00000 q^{35} +3.94610 q^{37} +6.05390 q^{38} +1.28917 q^{40} +1.68111 q^{41} +7.01916 q^{43} -1.28917 q^{44} +4.15165 q^{46} -2.52444 q^{47} +1.00000 q^{49} +1.81361 q^{50} +6.57834 q^{52} -0.289169 q^{53} +1.00000 q^{55} +1.28917 q^{56} -2.15165 q^{58} +10.4408 q^{59} +6.49472 q^{61} -4.57834 q^{62} -1.66196 q^{64} -5.10278 q^{65} -0.578337 q^{67} +2.47054 q^{68} +1.81361 q^{70} +13.3083 q^{71} +0.372787 q^{73} +7.15667 q^{74} +4.30330 q^{76} +1.00000 q^{77} +15.4061 q^{79} +4.91638 q^{80} +3.04888 q^{82} -15.7491 q^{83} -1.91638 q^{85} +12.7300 q^{86} +1.28917 q^{88} +0.0297193 q^{89} -5.10278 q^{91} +2.95112 q^{92} -4.57834 q^{94} -3.33804 q^{95} +15.8030 q^{97} +1.81361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8} + q^{10} - 3 q^{11} + 8 q^{13} + q^{14} - q^{16} - 8 q^{17} - 2 q^{19} - 3 q^{20} + q^{22} + 6 q^{23} + 3 q^{25} + 2 q^{26} - 3 q^{28} - 10 q^{29} - 2 q^{31} - 7 q^{32} + 16 q^{34} + 3 q^{35} + 8 q^{37} + 22 q^{38} + 3 q^{40} - 4 q^{41} - 3 q^{44} - 6 q^{46} - 2 q^{47} + 3 q^{49} - q^{50} + 18 q^{52} + 3 q^{55} + 3 q^{56} + 12 q^{58} + 12 q^{59} + 4 q^{61} - 12 q^{62} - 17 q^{64} - 8 q^{65} - 2 q^{68} - q^{70} + 18 q^{71} + 14 q^{73} + 18 q^{74} - 24 q^{76} + 3 q^{77} + 2 q^{79} + q^{80} - 2 q^{82} - 6 q^{83} + 8 q^{85} + 18 q^{86} + 3 q^{88} + 10 q^{89} - 8 q^{91} + 20 q^{92} - 12 q^{94} + 2 q^{95} + 10 q^{97} - q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^4 - 3 * q^5 - 3 * q^7 - 3 * q^8 + q^10 - 3 * q^11 + 8 * q^13 + q^14 - q^16 - 8 * q^17 - 2 * q^19 - 3 * q^20 + q^22 + 6 * q^23 + 3 * q^25 + 2 * q^26 - 3 * q^28 - 10 * q^29 - 2 * q^31 - 7 * q^32 + 16 * q^34 + 3 * q^35 + 8 * q^37 + 22 * q^38 + 3 * q^40 - 4 * q^41 - 3 * q^44 - 6 * q^46 - 2 * q^47 + 3 * q^49 - q^50 + 18 * q^52 + 3 * q^55 + 3 * q^56 + 12 * q^58 + 12 * q^59 + 4 * q^61 - 12 * q^62 - 17 * q^64 - 8 * q^65 - 2 * q^68 - q^70 + 18 * q^71 + 14 * q^73 + 18 * q^74 - 24 * q^76 + 3 * q^77 + 2 * q^79 + q^80 - 2 * q^82 - 6 * q^83 + 8 * q^85 + 18 * q^86 + 3 * q^88 + 10 * q^89 - 8 * q^91 + 20 * q^92 - 12 * q^94 + 2 * q^95 + 10 * q^97 - q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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