LMFDB - Embedded newform 2646.2.d.f.2645.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2646,2,Mod(2645,2646)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2646, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2646.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2646 = 2 \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2646.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$21.1284163748$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2645.14
Root			$-0.608761 + 0.793353i$ of defining polynomial
Character	$\chi$	$=$	2646.2645
Dual form			2646.2.d.f.2645.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.966684 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +0.966684 q^{5} -1.00000i q^{8} +0.966684i q^{10} -2.87476i q^{11} -2.13246i q^{13} +1.00000 q^{16} -2.44949 q^{17} -0.528589i q^{19} -0.966684 q^{20} +2.87476 q^{22} +1.61463i q^{23} -4.06552 q^{25} +2.13246 q^{26} -2.43751i q^{29} -0.279344i q^{31} +1.00000i q^{32} -2.44949i q^{34} -10.9691 q^{37} +0.528589 q^{38} -0.966684i q^{40} -8.32540 q^{41} -3.69764 q^{43} +2.87476i q^{44} -1.61463 q^{46} +12.0586 q^{47} -4.06552i q^{50} +2.13246i q^{52} -6.27558i q^{53} -2.77898i q^{55} +2.43751 q^{58} -11.4390 q^{59} +10.1252i q^{61} +0.279344 q^{62} -1.00000 q^{64} -2.06142i q^{65} -9.72648 q^{67} +2.44949 q^{68} -14.2062i q^{71} +1.31381i q^{73} -10.9691i q^{74} +0.528589i q^{76} -5.27004 q^{79} +0.966684 q^{80} -8.32540i q^{82} -8.58425 q^{83} -2.36788 q^{85} -3.69764i q^{86} -2.87476 q^{88} +3.59764 q^{89} -1.61463i q^{92} +12.0586i q^{94} -0.510979i q^{95} +4.27942i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{4}+O(q^{10}) $ 16 * q - 16 * q^4	$ 16 q - 16 q^{4} + 16 q^{16} + 16 q^{22} + 16 q^{37} - 32 q^{43} + 48 q^{46} - 32 q^{58} - 16 q^{64} - 32 q^{67} - 16 q^{88}+O(q^{100}) $ 16 * q - 16 * q^4 + 16 * q^16 + 16 * q^22 + 16 * q^37 - 32 * q^43 + 48 * q^46 - 32 * q^58 - 16 * q^64 - 32 * q^67 - 16 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$
$\chi(n)$	$-1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0.966684	0.432314	0.216157	−	0.976359i	$-0.430648\pi$
$5$	0.966684	0.432314	0.216157	+	0.976359i	$0.430648\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0.966684i	0.305692i
$11$	− 2.87476i	− 0.866772i	−0.901208	−	0.433386i	$-0.857319\pi$
$11$	− 2.87476i	− 0.866772i	0.901208	−	0.433386i	$-0.142681\pi$
$12$	0	0
$13$	− 2.13246i	− 0.591439i	−0.955275	−	0.295720i	$-0.904441\pi$
$13$	− 2.13246i	− 0.591439i	0.955275	−	0.295720i	$-0.0955594\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.44949	−0.594089	−0.297044	−	0.954864i	$-0.596001\pi$
$17$	−2.44949	−0.594089	−0.297044	+	0.954864i	$0.596001\pi$
$18$	0	0
$19$	− 0.528589i	− 0.121267i	−0.998160	−	0.0606334i	$-0.980688\pi$
$19$	− 0.528589i	− 0.121267i	0.998160	−	0.0606334i	$-0.0193120\pi$
$20$	−0.966684	−0.216157
$21$	0	0
$22$	2.87476	0.612901
$23$	1.61463i	0.336673i	0.985730	+	0.168336i	$0.0538395\pi$
$23$	1.61463i	0.336673i	−0.985730	+	0.168336i	$0.946161\pi$
$24$	0	0
$25$	−4.06552	−0.813104
$26$	2.13246	0.418211
$27$	0	0
$28$	0	0
$29$	− 2.43751i	− 0.452634i	−0.974054	−	0.226317i	$-0.927332\pi$
$29$	− 2.43751i	− 0.452634i	0.974054	−	0.226317i	$-0.0726685\pi$
$30$	0	0
$31$	− 0.279344i	− 0.0501717i	−0.999685	−	0.0250859i	$-0.992014\pi$
$31$	− 0.279344i	− 0.0501717i	0.999685	−	0.0250859i	$-0.00798592\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	− 2.44949i	− 0.420084i
$35$	0	0
$36$	0	0
$37$	−10.9691	−1.80331	−0.901656	−	0.432454i	$-0.857648\pi$
$37$	−10.9691	−1.80331	−0.901656	+	0.432454i	$0.857648\pi$
$38$	0.528589	0.0857485
$39$	0	0
$40$	− 0.966684i	− 0.152846i
$41$	−8.32540	−1.30021	−0.650105	−	0.759845i	$-0.725275\pi$
$41$	−8.32540	−1.30021	−0.650105	+	0.759845i	$0.725275\pi$
$42$	0	0
$43$	−3.69764	−0.563885	−0.281942	−	0.959431i	$-0.590979\pi$
$43$	−3.69764	−0.563885	−0.281942	+	0.959431i	$0.590979\pi$
$44$	2.87476i	0.433386i
$45$	0	0
$46$	−1.61463	−0.238064
$47$	12.0586	1.75893	0.879464	−	0.475966i	$-0.157901\pi$
$47$	12.0586	1.75893	0.879464	+	0.475966i	$0.157901\pi$
$48$	0	0
$49$	0	0
$50$	− 4.06552i	− 0.574952i
$51$	0	0
$52$	2.13246i	0.295720i
$53$	− 6.27558i	− 0.862018i	−0.902348	−	0.431009i	$-0.858158\pi$
$53$	− 6.27558i	− 0.862018i	0.902348	−	0.431009i	$-0.141842\pi$
$54$	0	0
$55$	− 2.77898i	− 0.374718i
$56$	0	0
$57$	0	0
$58$	2.43751	0.320060
$59$	−11.4390	−1.48924	−0.744618	−	0.667491i	$-0.767368\pi$
$59$	−11.4390	−1.48924	−0.744618	+	0.667491i	$0.767368\pi$
$60$	0	0
$61$	10.1252i	1.29640i	0.761469	+	0.648202i	$0.224479\pi$
$61$	10.1252i	1.29640i	−0.761469	+	0.648202i	$0.775521\pi$
$62$	0.279344	0.0354768
$63$	0	0
$64$	−1.00000	−0.125000
$65$	− 2.06142i	− 0.255688i
$66$	0	0
$67$	−9.72648	−1.18828	−0.594139	−	0.804362i	$-0.702507\pi$
$67$	−9.72648	−1.18828	−0.594139	+	0.804362i	$0.702507\pi$
$68$	2.44949	0.297044
$69$	0	0
$70$	0	0
$71$	− 14.2062i	− 1.68597i	−0.537939	−	0.842984i	$-0.680797\pi$
$71$	− 14.2062i	− 1.68597i	0.537939	−	0.842984i	$-0.319203\pi$
$72$	0	0
$73$	1.31381i	0.153770i	0.997040	+	0.0768848i	$0.0244974\pi$
$73$	1.31381i	0.153770i	−0.997040	+	0.0768848i	$0.975503\pi$
$74$	− 10.9691i	− 1.27513i
$75$	0	0
$76$	0.528589i	0.0606334i
$77$	0	0
$78$	0	0
$79$	−5.27004	−0.592926	−0.296463	−	0.955044i	$-0.595807\pi$
$79$	−5.27004	−0.592926	−0.296463	+	0.955044i	$0.595807\pi$
$80$	0.966684	0.108079
$81$	0	0
$82$	− 8.32540i	− 0.919387i
$83$	−8.58425	−0.942244	−0.471122	−	0.882068i	$-0.656151\pi$
$83$	−8.58425	−0.942244	−0.471122	+	0.882068i	$0.656151\pi$
$84$	0	0
$85$	−2.36788	−0.256833
$86$	− 3.69764i	− 0.398727i
$87$	0	0
$88$	−2.87476	−0.306450
$89$	3.59764	0.381349	0.190674	−	0.981653i	$-0.438933\pi$
$89$	3.59764	0.381349	0.190674	+	0.981653i	$0.438933\pi$
$90$	0	0
$91$	0	0
$92$	− 1.61463i	− 0.168336i
$93$	0	0
$94$	12.0586i	1.24375i
$95$	− 0.510979i	− 0.0524253i
$96$	0	0
$97$	4.27942i	0.434509i	0.976115	+	0.217255i	$0.0697102\pi$
$97$	4.27942i	0.434509i	−0.976115	+	0.217255i	$0.930290\pi$
$98$	0	0
$99$	0	0
$100$	4.06552	0.406552
$101$	15.7746	1.56963	0.784816	−	0.619729i	$-0.212757\pi$
$101$	15.7746	1.56963	0.784816	+	0.619729i	$0.212757\pi$
$102$	0	0
$103$	5.97601i	0.588834i	0.955677	+	0.294417i	$0.0951254\pi$
$103$	5.97601i	0.588834i	−0.955677	+	0.294417i	$0.904875\pi$
$104$	−2.13246	−0.209105
$105$	0	0
$106$	6.27558	0.609539
$107$	5.24819i	0.507361i	0.967288	+	0.253681i	$0.0816412\pi$
$107$	5.24819i	0.507361i	−0.967288	+	0.253681i	$0.918359\pi$
$108$	0	0
$109$	8.55465	0.819387	0.409693	−	0.912223i	$-0.365636\pi$
$109$	8.55465	0.819387	0.409693	+	0.912223i	$0.365636\pi$
$110$	2.77898	0.264966
$111$	0	0
$112$	0	0
$113$	− 18.9965i	− 1.78704i	−0.449021	−	0.893521i	$-0.648227\pi$
$113$	− 18.9965i	− 1.78704i	0.449021	−	0.893521i	$-0.351773\pi$
$114$	0	0
$115$	1.56083i	0.145548i
$116$	2.43751i	0.226317i
$117$	0	0
$118$	− 11.4390i	− 1.05305i
$119$	0	0
$120$	0	0
$121$	2.73576	0.248706
$122$	−10.1252	−0.916695
$123$	0	0
$124$	0.279344i	0.0250859i
$125$	−8.76349	−0.783831
$126$	0	0
$127$	−12.6723	−1.12449	−0.562243	−	0.826972i	$-0.690061\pi$
$127$	−12.6723	−1.12449	−0.562243	+	0.826972i	$0.690061\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	2.06142	0.180798
$131$	−0.744398	−0.0650384	−0.0325192	−	0.999471i	$-0.510353\pi$
$131$	−0.744398	−0.0650384	−0.0325192	+	0.999471i	$0.510353\pi$
$132$	0	0
$133$	0	0
$134$	− 9.72648i	− 0.840240i
$135$	0	0
$136$	2.44949i	0.210042i
$137$	− 16.8165i	− 1.43673i	−0.695667	−	0.718364i	$-0.744891\pi$
$137$	− 16.8165i	− 1.43673i	0.695667	−	0.718364i	$-0.255109\pi$
$138$	0	0
$139$	− 0.707834i	− 0.0600377i	−0.999549	−	0.0300188i	$-0.990443\pi$
$139$	− 0.707834i	− 0.0600377i	0.999549	−	0.0300188i	$-0.00955673\pi$
$140$	0	0
$141$	0	0
$142$	14.2062	1.19216
$143$	−6.13032	−0.512643
$144$	0	0
$145$	− 2.35630i	− 0.195680i
$146$	−1.31381	−0.108732
$147$	0	0
$148$	10.9691	0.901656
$149$	− 3.33930i	− 0.273566i	−0.990601	−	0.136783i	$-0.956324\pi$
$149$	− 3.33930i	− 0.273566i	0.990601	−	0.136783i	$-0.0436763\pi$
$150$	0	0
$151$	−1.26013	−0.102548	−0.0512740	−	0.998685i	$-0.516328\pi$
$151$	−1.26013	−0.102548	−0.0512740	+	0.998685i	$0.516328\pi$
$152$	−0.528589	−0.0428743
$153$	0	0
$154$	0	0
$155$	− 0.270038i	− 0.0216900i
$156$	0	0
$157$	− 9.15052i	− 0.730291i	−0.930950	−	0.365146i	$-0.881019\pi$
$157$	− 9.15052i	− 0.730291i	0.930950	−	0.365146i	$-0.118981\pi$
$158$	− 5.27004i	− 0.419262i
$159$	0	0
$160$	0.966684i	0.0764231i
$161$	0	0
$162$	0	0
$163$	−15.6338	−1.22454	−0.612268	−	0.790651i	$-0.709742\pi$
$163$	−15.6338	−1.22454	−0.612268	+	0.790651i	$0.709742\pi$
$164$	8.32540	0.650105
$165$	0	0
$166$	− 8.58425i	− 0.666267i
$167$	−10.9439	−0.846863	−0.423432	−	0.905928i	$-0.639175\pi$
$167$	−10.9439	−0.846863	−0.423432	+	0.905928i	$0.639175\pi$
$168$	0	0
$169$	8.45260	0.650200
$170$	− 2.36788i	− 0.181608i
$171$	0	0
$172$	3.69764	0.281942
$173$	19.5121	1.48347	0.741737	−	0.670691i	$-0.234002\pi$
$173$	19.5121	1.48347	0.741737	+	0.670691i	$0.234002\pi$
$174$	0	0
$175$	0	0
$176$	− 2.87476i	− 0.216693i
$177$	0	0
$178$	3.59764i	0.269654i
$179$	− 2.28923i	− 0.171105i	−0.996334	−	0.0855525i	$-0.972734\pi$
$179$	− 2.28923i	− 0.171105i	0.996334	−	0.0855525i	$-0.0272655\pi$
$180$	0	0
$181$	− 6.01369i	− 0.446994i	−0.974705	−	0.223497i	$-0.928253\pi$
$181$	− 6.01369i	− 0.446994i	0.974705	−	0.223497i	$-0.0717473\pi$
$182$	0	0
$183$	0	0
$184$	1.61463	0.119032
$185$	−10.6037	−0.779597
$186$	0	0
$187$	7.04169i	0.514939i
$188$	−12.0586	−0.879464
$189$	0	0
$190$	0.510979	0.0370703
$191$	− 4.09991i	− 0.296659i	−0.988938	−	0.148329i	$-0.952610\pi$
$191$	− 4.09991i	− 0.296659i	0.988938	−	0.148329i	$-0.0473896\pi$
$192$	0	0
$193$	11.3030	0.813606	0.406803	−	0.913516i	$-0.366643\pi$
$193$	11.3030	0.813606	0.406803	+	0.913516i	$0.366643\pi$
$194$	−4.27942	−0.307244
$195$	0	0
$196$	0	0
$197$	− 11.8765i	− 0.846163i	−0.906092	−	0.423081i	$-0.860948\pi$
$197$	− 11.8765i	− 0.846163i	0.906092	−	0.423081i	$-0.139052\pi$
$198$	0	0
$199$	− 10.6070i	− 0.751911i	−0.926638	−	0.375955i	$-0.877315\pi$
$199$	− 10.6070i	− 0.751911i	0.926638	−	0.375955i	$-0.122685\pi$
$200$	4.06552i	0.287476i
$201$	0	0
$202$	15.7746i	1.10990i
$203$	0	0
$204$	0	0
$205$	−8.04803	−0.562099
$206$	−5.97601	−0.416368
$207$	0	0
$208$	− 2.13246i	− 0.147860i
$209$	−1.51957	−0.105111
$210$	0	0
$211$	−18.5223	−1.27513	−0.637565	−	0.770397i	$-0.720058\pi$
$211$	−18.5223	−1.27513	−0.637565	+	0.770397i	$0.720058\pi$
$212$	6.27558i	0.431009i
$213$	0	0
$214$	−5.24819	−0.358759
$215$	−3.57445	−0.243775
$216$	0	0
$217$	0	0
$218$	8.55465i	0.579394i
$219$	0	0
$220$	2.77898i	0.187359i
$221$	5.22345i	0.351367i
$222$	0	0
$223$	26.3299i	1.76318i	0.472015	+	0.881591i	$0.343527\pi$
$223$	26.3299i	1.76318i	−0.472015	+	0.881591i	$0.656473\pi$
$224$	0	0
$225$	0	0
$226$	18.9965	1.26363
$227$	20.3566	1.35111	0.675556	−	0.737308i	$-0.263904\pi$
$227$	20.3566	1.35111	0.675556	+	0.737308i	$0.263904\pi$
$228$	0	0
$229$	22.3049i	1.47395i	0.675920	+	0.736975i	$0.263746\pi$
$229$	22.3049i	1.47395i	−0.675920	+	0.736975i	$0.736254\pi$
$230$	−1.56083	−0.102918
$231$	0	0
$232$	−2.43751	−0.160030
$233$	− 4.49723i	− 0.294623i	−0.989090	−	0.147312i	$-0.952938\pi$
$233$	− 4.49723i	− 0.294623i	0.989090	−	0.147312i	$-0.0470620\pi$
$234$	0	0
$235$	11.6569	0.760409
$236$	11.4390	0.744618
$237$	0	0
$238$	0	0
$239$	− 18.7605i	− 1.21352i	−0.794887	−	0.606758i	$-0.792470\pi$
$239$	− 18.7605i	− 1.21352i	0.794887	−	0.606758i	$-0.207530\pi$
$240$	0	0
$241$	− 24.1774i	− 1.55740i	−0.627395	−	0.778702i	$-0.715879\pi$
$241$	− 24.1774i	− 1.55740i	0.627395	−	0.778702i	$-0.284121\pi$
$242$	2.73576i	0.175862i
$243$	0	0
$244$	− 10.1252i	− 0.648202i
$245$	0	0
$246$	0	0
$247$	−1.12720	−0.0717219
$248$	−0.279344	−0.0177384
$249$	0	0
$250$	− 8.76349i	− 0.554252i
$251$	16.0357	1.01216	0.506082	−	0.862486i	$-0.331093\pi$
$251$	16.0357	1.01216	0.506082	+	0.862486i	$0.331093\pi$
$252$	0	0
$253$	4.64166	0.291819
$254$	− 12.6723i	− 0.795131i
$255$	0	0
$256$	1.00000	0.0625000
$257$	−7.75599	−0.483805	−0.241903	−	0.970301i	$-0.577771\pi$
$257$	−7.75599	−0.483805	−0.241903	+	0.970301i	$0.577771\pi$
$258$	0	0
$259$	0	0
$260$	2.06142i	0.127844i
$261$	0	0
$262$	− 0.744398i	− 0.0459891i
$263$	− 24.2173i	− 1.49330i	−0.665215	−	0.746652i	$-0.731660\pi$
$263$	− 24.2173i	− 1.49330i	0.665215	−	0.746652i	$-0.268340\pi$
$264$	0	0
$265$	− 6.06651i	− 0.372662i
$266$	0	0
$267$	0	0
$268$	9.72648	0.594139
$269$	−18.5186	−1.12910	−0.564550	−	0.825399i	$-0.690950\pi$
$269$	−18.5186	−1.12910	−0.564550	+	0.825399i	$0.690950\pi$
$270$	0	0
$271$	− 24.9149i	− 1.51347i	−0.653721	−	0.756736i	$-0.726793\pi$
$271$	− 24.9149i	− 1.51347i	0.653721	−	0.756736i	$-0.273207\pi$
$272$	−2.44949	−0.148522
$273$	0	0
$274$	16.8165	1.01592
$275$	11.6874i	0.704776i
$276$	0	0
$277$	3.00206	0.180376	0.0901882	−	0.995925i	$-0.471253\pi$
$277$	3.00206	0.180376	0.0901882	+	0.995925i	$0.471253\pi$
$278$	0.707834	0.0424530
$279$	0	0
$280$	0	0
$281$	31.5374i	1.88136i	0.339289	+	0.940682i	$0.389814\pi$
$281$	31.5374i	1.88136i	−0.339289	+	0.940682i	$0.610186\pi$
$282$	0	0
$283$	− 24.6583i	− 1.46578i	−0.680347	−	0.732891i	$-0.738171\pi$
$283$	− 24.6583i	− 1.46578i	0.680347	−	0.732891i	$-0.261829\pi$
$284$	14.2062i	0.842984i
$285$	0	0
$286$	− 6.13032i	− 0.362493i
$287$	0	0
$288$	0	0
$289$	−11.0000	−0.647059
$290$	2.35630	0.138367
$291$	0	0
$292$	− 1.31381i	− 0.0768848i
$293$	20.6424	1.20594	0.602970	−	0.797764i	$-0.293984\pi$
$293$	20.6424	1.20594	0.602970	+	0.797764i	$0.293984\pi$
$294$	0	0
$295$	−11.0579	−0.643818
$296$	10.9691i	0.637567i
$297$	0	0
$298$	3.33930	0.193440
$299$	3.44313	0.199122
$300$	0	0
$301$	0	0
$302$	− 1.26013i	− 0.0725125i
$303$	0	0
$304$	− 0.528589i	− 0.0303167i
$305$	9.78790i	0.560453i
$306$	0	0
$307$	32.3969i	1.84899i	0.381196	+	0.924494i	$0.375512\pi$
$307$	32.3969i	1.84899i	−0.381196	+	0.924494i	$0.624488\pi$
$308$	0	0
$309$	0	0
$310$	0.270038	0.0153371
$311$	−16.5933	−0.940918	−0.470459	−	0.882422i	$-0.655912\pi$
$311$	−16.5933	−0.940918	−0.470459	+	0.882422i	$0.655912\pi$
$312$	0	0
$313$	3.01152i	0.170221i	0.996372	+	0.0851105i	$0.0271243\pi$
$313$	3.01152i	0.170221i	−0.996372	+	0.0851105i	$0.972876\pi$
$314$	9.15052	0.516394
$315$	0	0
$316$	5.27004	0.296463
$317$	8.48240i	0.476419i	0.971214	+	0.238209i	$0.0765605\pi$
$317$	8.48240i	0.476419i	−0.971214	+	0.238209i	$0.923440\pi$
$318$	0	0
$319$	−7.00725	−0.392330
$320$	−0.966684	−0.0540393
$321$	0	0
$322$	0	0
$323$	1.29477i	0.0720432i
$324$	0	0
$325$	8.66958i	0.480902i
$326$	− 15.6338i	− 0.865877i
$327$	0	0
$328$	8.32540i	0.459693i
$329$	0	0
$330$	0	0
$331$	15.5567	0.855074	0.427537	−	0.903998i	$-0.359381\pi$
$331$	15.5567	0.855074	0.427537	+	0.903998i	$0.359381\pi$
$332$	8.58425	0.471122
$333$	0	0
$334$	− 10.9439i	− 0.598823i
$335$	−9.40243	−0.513710
$336$	0	0
$337$	−10.9188	−0.594785	−0.297392	−	0.954755i	$-0.596117\pi$
$337$	−10.9188	−0.594785	−0.297392	+	0.954755i	$0.596117\pi$
$338$	8.45260i	0.459761i
$339$	0	0
$340$	2.36788	0.128416
$341$	−0.803048	−0.0434875
$342$	0	0
$343$	0	0
$344$	3.69764i	0.199363i
$345$	0	0
$346$	19.5121i	1.04897i
$347$	32.3932i	1.73896i	0.493968	+	0.869480i	$0.335546\pi$
$347$	32.3932i	1.73896i	−0.493968	+	0.869480i	$0.664454\pi$
$348$	0	0
$349$	− 8.54769i	− 0.457547i	−0.973480	−	0.228774i	$-0.926528\pi$
$349$	− 8.54769i	− 0.457547i	0.973480	−	0.228774i	$-0.0734715\pi$
$350$	0	0
$351$	0	0
$352$	2.87476	0.153225
$353$	26.2115	1.39510	0.697549	−	0.716537i	$-0.254274\pi$
$353$	26.2115	1.39510	0.697549	+	0.716537i	$0.254274\pi$
$354$	0	0
$355$	− 13.7329i	− 0.728868i
$356$	−3.59764	−0.190674
$357$	0	0
$358$	2.28923	0.120990
$359$	2.25048i	0.118776i	0.998235	+	0.0593880i	$0.0189149\pi$
$359$	2.25048i	0.118776i	−0.998235	+	0.0593880i	$0.981085\pi$
$360$	0	0
$361$	18.7206	0.985294
$362$	6.01369	0.316073
$363$	0	0
$364$	0	0
$365$	1.27004i	0.0664768i
$366$	0	0
$367$	10.3011i	0.537715i	0.963180	+	0.268858i	$0.0866461\pi$
$367$	10.3011i	0.537715i	−0.963180	+	0.268858i	$0.913354\pi$
$368$	1.61463i	0.0841682i
$369$	0	0
$370$	− 10.6037i	− 0.551259i
$371$	0	0
$372$	0	0
$373$	12.8896	0.667398	0.333699	−	0.942680i	$-0.391703\pi$
$373$	12.8896	0.667398	0.333699	+	0.942680i	$0.391703\pi$
$374$	−7.04169	−0.364117
$375$	0	0
$376$	− 12.0586i	− 0.621875i
$377$	−5.19790	−0.267705
$378$	0	0
$379$	3.95341	0.203073	0.101537	−	0.994832i	$-0.467624\pi$
$379$	3.95341	0.203073	0.101537	+	0.994832i	$0.467624\pi$
$380$	0.510979i	0.0262127i
$381$	0	0
$382$	4.09991	0.209770
$383$	−17.1961	−0.878680	−0.439340	−	0.898321i	$-0.644788\pi$
$383$	−17.1961	−0.878680	−0.439340	+	0.898321i	$0.644788\pi$
$384$	0	0
$385$	0	0
$386$	11.3030i	0.575307i
$387$	0	0
$388$	− 4.27942i	− 0.217255i
$389$	− 7.94812i	− 0.402986i	−0.979490	−	0.201493i	$-0.935421\pi$
$389$	− 7.94812i	− 0.402986i	0.979490	−	0.201493i	$-0.0645793\pi$
$390$	0	0
$391$	− 3.95501i	− 0.200013i
$392$	0	0
$393$	0	0
$394$	11.8765	0.598327
$395$	−5.09446	−0.256330
$396$	0	0
$397$	19.5529i	0.981331i	0.871348	+	0.490665i	$0.163246\pi$
$397$	19.5529i	0.981331i	−0.871348	+	0.490665i	$0.836754\pi$
$398$	10.6070	0.531681
$399$	0	0
$400$	−4.06552	−0.203276
$401$	14.7972i	0.738936i	0.929243	+	0.369468i	$0.120460\pi$
$401$	14.7972i	0.738936i	−0.929243	+	0.369468i	$0.879540\pi$
$402$	0	0
$403$	−0.595692	−0.0296735
$404$	−15.7746	−0.784816
$405$	0	0
$406$	0	0
$407$	31.5336i	1.56306i
$408$	0	0
$409$	− 31.0851i	− 1.53706i	−0.639814	−	0.768530i	$-0.720989\pi$
$409$	− 31.0851i	− 1.53706i	0.639814	−	0.768530i	$-0.279011\pi$
$410$	− 8.04803i	− 0.397464i
$411$	0	0
$412$	− 5.97601i	− 0.294417i
$413$	0	0
$414$	0	0
$415$	−8.29826	−0.407345
$416$	2.13246	0.104553
$417$	0	0
$418$	− 1.51957i	− 0.0743244i
$419$	21.2123	1.03629	0.518143	−	0.855294i	$-0.326623\pi$
$419$	21.2123	1.03629	0.518143	+	0.855294i	$0.326623\pi$
$420$	0	0
$421$	20.1139	0.980292	0.490146	−	0.871640i	$-0.336943\pi$
$421$	20.1139	0.980292	0.490146	+	0.871640i	$0.336943\pi$
$422$	− 18.5223i	− 0.901653i
$423$	0	0
$424$	−6.27558	−0.304769
$425$	9.95845	0.483056
$426$	0	0
$427$	0	0
$428$	− 5.24819i	− 0.253681i
$429$	0	0
$430$	− 3.57445i	− 0.172375i
$431$	17.5145i	0.843643i	0.906679	+	0.421822i	$0.138609\pi$
$431$	17.5145i	0.843643i	−0.906679	+	0.421822i	$0.861391\pi$
$432$	0	0
$433$	7.13421i	0.342848i	0.985197	+	0.171424i	$0.0548368\pi$
$433$	7.13421i	0.342848i	−0.985197	+	0.171424i	$0.945163\pi$
$434$	0	0
$435$	0	0
$436$	−8.55465	−0.409693
$437$	0.853474	0.0408272
$438$	0	0
$439$	16.3774i	0.781653i	0.920464	+	0.390826i	$0.127811\pi$
$439$	16.3774i	0.781653i	−0.920464	+	0.390826i	$0.872189\pi$
$440$	−2.77898	−0.132483
$441$	0	0
$442$	−5.22345	−0.248454
$443$	− 29.3172i	− 1.39290i	−0.717604	−	0.696451i	$-0.754761\pi$
$443$	− 29.3172i	− 1.39290i	0.717604	−	0.696451i	$-0.245239\pi$
$444$	0	0
$445$	3.47778	0.164863
$446$	−26.3299	−1.24676
$447$	0	0
$448$	0	0
$449$	32.7436i	1.54527i	0.634852	+	0.772634i	$0.281061\pi$
$449$	32.7436i	1.54527i	−0.634852	+	0.772634i	$0.718939\pi$
$450$	0	0
$451$	23.9335i	1.12699i
$452$	18.9965i	0.893521i
$453$	0	0
$454$	20.3566i	0.955381i
$455$	0	0
$456$	0	0
$457$	−16.4912	−0.771425	−0.385713	−	0.922619i	$-0.626044\pi$
$457$	−16.4912	−0.771425	−0.385713	+	0.922619i	$0.626044\pi$
$458$	−22.3049	−1.04224
$459$	0	0
$460$	− 1.56083i	− 0.0727742i
$461$	12.7292	0.592859	0.296430	−	0.955055i	$-0.404204\pi$
$461$	12.7292	0.592859	0.296430	+	0.955055i	$0.404204\pi$
$462$	0	0
$463$	−23.2392	−1.08002	−0.540008	−	0.841660i	$-0.681579\pi$
$463$	−23.2392	−1.08002	−0.540008	+	0.841660i	$0.681579\pi$
$464$	− 2.43751i	− 0.113158i
$465$	0	0
$466$	4.49723	0.208330
$467$	−28.5783	−1.32245	−0.661223	−	0.750190i	$-0.729962\pi$
$467$	−28.5783	−1.32245	−0.661223	+	0.750190i	$0.729962\pi$
$468$	0	0
$469$	0	0
$470$	11.6569i	0.537691i
$471$	0	0
$472$	11.4390i	0.526524i
$473$	10.6298i	0.488760i
$474$	0	0
$475$	2.14899i	0.0986025i
$476$	0	0
$477$	0	0
$478$	18.7605	0.858085
$479$	21.3931	0.977474	0.488737	−	0.872431i	$-0.337458\pi$
$479$	21.3931	0.977474	0.488737	+	0.872431i	$0.337458\pi$
$480$	0	0
$481$	23.3913i	1.06655i
$482$	24.1774	1.10125
$483$	0	0
$484$	−2.73576	−0.124353
$485$	4.13685i	0.187845i
$486$	0	0
$487$	−18.6245	−0.843958	−0.421979	−	0.906606i	$-0.638664\pi$
$487$	−18.6245	−0.843958	−0.421979	+	0.906606i	$0.638664\pi$
$488$	10.1252	0.458348
$489$	0	0
$490$	0	0
$491$	16.5243i	0.745730i	0.927886	+	0.372865i	$0.121625\pi$
$491$	16.5243i	0.745730i	−0.927886	+	0.372865i	$0.878375\pi$
$492$	0	0
$493$	5.97065i	0.268905i
$494$	− 1.12720i	− 0.0507150i
$495$	0	0
$496$	− 0.279344i	− 0.0125429i
$497$	0	0
$498$	0	0
$499$	−7.54562	−0.337788	−0.168894	−	0.985634i	$-0.554020\pi$
$499$	−7.54562	−0.337788	−0.168894	+	0.985634i	$0.554020\pi$
$500$	8.76349	0.391915
$501$	0	0
$502$	16.0357i	0.715708i
$503$	−10.9142	−0.486642	−0.243321	−	0.969946i	$-0.578237\pi$
$503$	−10.9142	−0.486642	−0.243321	+	0.969946i	$0.578237\pi$
$504$	0	0
$505$	15.2491	0.678574
$506$	4.64166i	0.206347i
$507$	0	0
$508$	12.6723	0.562243
$509$	−19.9071	−0.882368	−0.441184	−	0.897417i	$-0.645441\pi$
$509$	−19.9071	−0.882368	−0.441184	+	0.897417i	$0.645441\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	− 7.75599i	− 0.342102i
$515$	5.77691i	0.254561i
$516$	0	0
$517$	− 34.6656i	− 1.52459i
$518$	0	0
$519$	0	0
$520$	−2.06142	−0.0903992
$521$	0.362091	0.0158635	0.00793175	−	0.999969i	$-0.497475\pi$
$521$	0.362091	0.0158635	0.00793175	+	0.999969i	$0.497475\pi$
$522$	0	0
$523$	0.737933i	0.0322676i	0.999870	+	0.0161338i	$0.00513576\pi$
$523$	0.737933i	0.0322676i	−0.999870	+	0.0161338i	$0.994864\pi$
$524$	0.744398	0.0325192
$525$	0	0
$526$	24.2173	1.05592
$527$	0.684251i	0.0298065i
$528$	0	0
$529$	20.3930	0.886651
$530$	6.06651	0.263512
$531$	0	0
$532$	0	0
$533$	17.7536i	0.768995i
$534$	0	0
$535$	5.07334i	0.219340i
$536$	9.72648i	0.420120i
$537$	0	0
$538$	− 18.5186i	− 0.798394i
$539$	0	0
$540$	0	0
$541$	−3.56840	−0.153418	−0.0767088	−	0.997054i	$-0.524441\pi$
$541$	−3.56840	−0.153418	−0.0767088	+	0.997054i	$0.524441\pi$
$542$	24.9149	1.07019
$543$	0	0
$544$	− 2.44949i	− 0.105021i
$545$	8.26964	0.354233
$546$	0	0
$547$	−36.5512	−1.56282	−0.781408	−	0.624021i	$-0.785498\pi$
$547$	−36.5512	−1.56282	−0.781408	+	0.624021i	$0.785498\pi$
$548$	16.8165i	0.718364i
$549$	0	0
$550$	−11.6874	−0.498352
$551$	−1.28844	−0.0548894
$552$	0	0
$553$	0	0
$554$	3.00206i	0.127545i
$555$	0	0
$556$	0.707834i	0.0300188i
$557$	43.4448i	1.84081i	0.390961	+	0.920407i	$0.372143\pi$
$557$	43.4448i	1.84081i	−0.390961	+	0.920407i	$0.627857\pi$
$558$	0	0
$559$	7.88509i	0.333504i
$560$	0	0
$561$	0	0
$562$	−31.5374	−1.33033
$563$	−1.29951	−0.0547680	−0.0273840	−	0.999625i	$-0.508718\pi$
$563$	−1.29951	−0.0547680	−0.0273840	+	0.999625i	$0.508718\pi$
$564$	0	0
$565$	− 18.3636i	− 0.772564i
$566$	24.6583	1.03646
$567$	0	0
$568$	−14.2062	−0.596080
$569$	− 7.71320i	− 0.323354i	−0.986844	−	0.161677i	$-0.948310\pi$
$569$	− 7.71320i	− 0.323354i	0.986844	−	0.161677i	$-0.0516903\pi$
$570$	0	0
$571$	−29.9398	−1.25294	−0.626471	−	0.779445i	$-0.715501\pi$
$571$	−29.9398	−1.25294	−0.626471	+	0.779445i	$0.715501\pi$
$572$	6.13032	0.256322
$573$	0	0
$574$	0	0
$575$	− 6.56430i	− 0.273750i
$576$	0	0
$577$	− 16.7661i	− 0.697980i	−0.937126	−	0.348990i	$-0.886525\pi$
$577$	− 16.7661i	− 0.697980i	0.937126	−	0.348990i	$-0.113475\pi$
$578$	− 11.0000i	− 0.457540i
$579$	0	0
$580$	2.35630i	0.0978400i
$581$	0	0
$582$	0	0
$583$	−18.0408	−0.747173
$584$	1.31381	0.0543658
$585$	0	0
$586$	20.6424i	0.852729i
$587$	10.9865	0.453459	0.226730	−	0.973958i	$-0.427197\pi$
$587$	10.9865	0.453459	0.226730	+	0.973958i	$0.427197\pi$
$588$	0	0
$589$	−0.147659	−0.00608416
$590$	− 11.0579i	− 0.455248i
$591$	0	0
$592$	−10.9691	−0.450828
$593$	−22.6090	−0.928440	−0.464220	−	0.885720i	$-0.653665\pi$
$593$	−22.6090	−0.928440	−0.464220	+	0.885720i	$0.653665\pi$
$594$	0	0
$595$	0	0
$596$	3.33930i	0.136783i
$597$	0	0
$598$	3.44313i	0.140800i
$599$	− 1.29537i	− 0.0529275i	−0.999650	−	0.0264637i	$-0.991575\pi$
$599$	− 1.29537i	− 0.0529275i	0.999650	−	0.0264637i	$-0.00842465\pi$
$600$	0	0
$601$	0.327469i	0.0133577i	0.999978	+	0.00667887i	$0.00212597\pi$
$601$	0.327469i	0.0133577i	−0.999978	+	0.00667887i	$0.997874\pi$
$602$	0	0
$603$	0	0
$604$	1.26013	0.0512740
$605$	2.64462	0.107519
$606$	0	0
$607$	− 24.4016i	− 0.990431i	−0.868770	−	0.495216i	$-0.835089\pi$
$607$	− 24.4016i	− 0.990431i	0.868770	−	0.495216i	$-0.164911\pi$
$608$	0.528589	0.0214371
$609$	0	0
$610$	−9.78790	−0.396300
$611$	− 25.7145i	− 1.04030i
$612$	0	0
$613$	−7.62298	−0.307889	−0.153945	−	0.988079i	$-0.549198\pi$
$613$	−7.62298	−0.307889	−0.153945	+	0.988079i	$0.549198\pi$
$614$	−32.3969	−1.30743
$615$	0	0
$616$	0	0
$617$	− 6.32695i	− 0.254713i	−0.991857	−	0.127357i	$-0.959351\pi$
$617$	− 6.32695i	− 0.254713i	0.991857	−	0.127357i	$-0.0406493\pi$
$618$	0	0
$619$	34.5191i	1.38744i	0.720245	+	0.693720i	$0.244030\pi$
$619$	34.5191i	1.38744i	−0.720245	+	0.693720i	$0.755970\pi$
$620$	0.270038i	0.0108450i
$621$	0	0
$622$	− 16.5933i	− 0.665329i
$623$	0	0
$624$	0	0
$625$	11.8561	0.474243
$626$	−3.01152	−0.120364
$627$	0	0
$628$	9.15052i	0.365146i
$629$	26.8687	1.07133
$630$	0	0
$631$	10.4498	0.415999	0.208000	−	0.978129i	$-0.433305\pi$
$631$	10.4498	0.415999	0.208000	+	0.978129i	$0.433305\pi$
$632$	5.27004i	0.209631i
$633$	0	0
$634$	−8.48240	−0.336879
$635$	−12.2501	−0.486131
$636$	0	0
$637$	0	0
$638$	− 7.00725i	− 0.277419i
$639$	0	0
$640$	− 0.966684i	− 0.0382115i
$641$	0.464648i	0.0183525i	0.999958	+	0.00917624i	$0.00292093\pi$
$641$	0.464648i	0.0183525i	−0.999958	+	0.00917624i	$0.997079\pi$
$642$	0	0
$643$	− 14.6722i	− 0.578614i	−0.957236	−	0.289307i	$-0.906575\pi$
$643$	− 14.6722i	− 0.578614i	0.957236	−	0.289307i	$-0.0934248\pi$
$644$	0	0
$645$	0	0
$646$	−1.29477	−0.0509422
$647$	32.1782	1.26506	0.632528	−	0.774537i	$-0.282017\pi$
$647$	32.1782	1.26506	0.632528	+	0.774537i	$0.282017\pi$
$648$	0	0
$649$	32.8845i	1.29083i
$650$	−8.66958	−0.340049
$651$	0	0
$652$	15.6338	0.612268
$653$	− 18.8075i	− 0.735993i	−0.929827	−	0.367996i	$-0.880044\pi$
$653$	− 18.8075i	− 0.735993i	0.929827	−	0.367996i	$-0.119956\pi$
$654$	0	0
$655$	−0.719598	−0.0281170
$656$	−8.32540	−0.325052
$657$	0	0
$658$	0	0
$659$	33.2677i	1.29593i	0.761671	+	0.647964i	$0.224379\pi$
$659$	33.2677i	1.29593i	−0.761671	+	0.647964i	$0.775621\pi$
$660$	0	0
$661$	36.9343i	1.43658i	0.695746	+	0.718288i	$0.255074\pi$
$661$	36.9343i	1.43658i	−0.695746	+	0.718288i	$0.744926\pi$
$662$	15.5567i	0.604629i
$663$	0	0
$664$	8.58425i	0.333134i
$665$	0	0
$666$	0	0
$667$	3.93566	0.152389
$668$	10.9439	0.423432
$669$	0	0
$670$	− 9.40243i	− 0.363248i
$671$	29.1076	1.12369
$672$	0	0
$673$	45.4719	1.75281	0.876406	−	0.481572i	$-0.159934\pi$
$673$	45.4719	1.75281	0.876406	+	0.481572i	$0.159934\pi$
$674$	− 10.9188i	− 0.420576i
$675$	0	0
$676$	−8.45260	−0.325100
$677$	−16.3819	−0.629608	−0.314804	−	0.949157i	$-0.601939\pi$
$677$	−16.3819	−0.629608	−0.314804	+	0.949157i	$0.601939\pi$
$678$	0	0
$679$	0	0
$680$	2.36788i	0.0908041i
$681$	0	0
$682$	− 0.803048i	− 0.0307503i
$683$	2.38948i	0.0914308i	0.998955	+	0.0457154i	$0.0145567\pi$
$683$	2.38948i	0.0914308i	−0.998955	+	0.0457154i	$0.985443\pi$
$684$	0	0
$685$	− 16.2562i	− 0.621118i
$686$	0	0
$687$	0	0
$688$	−3.69764	−0.140971
$689$	−13.3825	−0.509831
$690$	0	0
$691$	− 40.7640i	− 1.55073i	−0.631511	−	0.775367i	$-0.717565\pi$
$691$	− 40.7640i	− 1.55073i	0.631511	−	0.775367i	$-0.282435\pi$
$692$	−19.5121	−0.741737
$693$	0	0
$694$	−32.3932	−1.22963
$695$	− 0.684251i	− 0.0259551i
$696$	0	0
$697$	20.3930	0.772439
$698$	8.54769	0.323535
$699$	0	0
$700$	0	0
$701$	13.5881i	0.513216i	0.966516	+	0.256608i	$0.0826049\pi$
$701$	13.5881i	0.513216i	−0.966516	+	0.256608i	$0.917395\pi$
$702$	0	0
$703$	5.79816i	0.218682i
$704$	2.87476i	0.108347i
$705$	0	0
$706$	26.2115i	0.986483i
$707$	0	0
$708$	0	0
$709$	−23.5911	−0.885982	−0.442991	−	0.896526i	$-0.646083\pi$
$709$	−23.5911	−0.885982	−0.442991	+	0.896526i	$0.646083\pi$
$710$	13.7329	0.515387
$711$	0	0
$712$	− 3.59764i	− 0.134827i
$713$	0.451037	0.0168915
$714$	0	0
$715$	−5.92608	−0.221623
$716$	2.28923i	0.0855525i
$717$	0	0
$718$	−2.25048	−0.0839873
$719$	7.41597	0.276569	0.138285	−	0.990393i	$-0.455841\pi$
$719$	7.41597	0.276569	0.138285	+	0.990393i	$0.455841\pi$
$720$	0	0
$721$	0	0
$722$	18.7206i	0.696708i
$723$	0	0
$724$	6.01369i	0.223497i
$725$	9.90974i	0.368039i
$726$	0	0
$727$	1.57044i	0.0582443i	0.999576	+	0.0291222i	$0.00927119\pi$
$727$	1.57044i	0.0582443i	−0.999576	+	0.0291222i	$0.990729\pi$
$728$	0	0
$729$	0	0
$730$	−1.27004	−0.0470062
$731$	9.05733	0.334997
$732$	0	0
$733$	− 3.63688i	− 0.134331i	−0.997742	−	0.0671656i	$-0.978604\pi$
$733$	− 3.63688i	− 0.134331i	0.997742	−	0.0671656i	$-0.0213956\pi$
$734$	−10.3011	−0.380222
$735$	0	0
$736$	−1.61463	−0.0595159
$737$	27.9613i	1.02997i
$738$	0	0
$739$	−47.4352	−1.74493	−0.872466	−	0.488674i	$-0.837481\pi$
$739$	−47.4352	−1.74493	−0.872466	+	0.488674i	$0.837481\pi$
$740$	10.6037	0.389799
$741$	0	0
$742$	0	0
$743$	14.9167i	0.547242i	0.961838	+	0.273621i	$0.0882214\pi$
$743$	14.9167i	0.547242i	−0.961838	+	0.273621i	$0.911779\pi$
$744$	0	0
$745$	− 3.22805i	− 0.118267i
$746$	12.8896i	0.471921i
$747$	0	0
$748$	− 7.04169i	− 0.257470i
$749$	0	0
$750$	0	0
$751$	46.2662	1.68828	0.844139	−	0.536124i	$-0.180112\pi$
$751$	46.2662	1.68828	0.844139	+	0.536124i	$0.180112\pi$
$752$	12.0586	0.439732
$753$	0	0
$754$	− 5.19790i	− 0.189296i
$755$	−1.21815	−0.0443330
$756$	0	0
$757$	32.6851	1.18796	0.593980	−	0.804480i	$-0.297556\pi$
$757$	32.6851	1.18796	0.593980	+	0.804480i	$0.297556\pi$
$758$	3.95341i	0.143594i
$759$	0	0
$760$	−0.510979	−0.0185352
$761$	−49.8168	−1.80586	−0.902930	−	0.429789i	$-0.858588\pi$
$761$	−49.8168	−1.80586	−0.902930	+	0.429789i	$0.858588\pi$
$762$	0	0
$763$	0	0
$764$	4.09991i	0.148329i
$765$	0	0
$766$	− 17.1961i	− 0.621320i
$767$	24.3933i	0.880793i
$768$	0	0
$769$	− 40.2091i	− 1.44998i	−0.688761	−	0.724989i	$-0.741845\pi$
$769$	− 40.2091i	− 1.44998i	0.688761	−	0.724989i	$-0.258155\pi$
$770$	0	0
$771$	0	0
$772$	−11.3030	−0.406803
$773$	−23.8448	−0.857637	−0.428818	−	0.903391i	$-0.641070\pi$
$773$	−23.8448	−0.857637	−0.428818	+	0.903391i	$0.641070\pi$
$774$	0	0
$775$	1.13568i	0.0407949i
$776$	4.27942	0.153622
$777$	0	0
$778$	7.94812	0.284954
$779$	4.40072i	0.157672i
$780$	0	0
$781$	−40.8394	−1.46135
$782$	3.95501	0.141431
$783$	0	0
$784$	0	0
$785$	− 8.84566i	− 0.315715i
$786$	0	0
$787$	− 33.5961i	− 1.19757i	−0.800909	−	0.598785i	$-0.795650\pi$
$787$	− 33.5961i	− 1.19757i	0.800909	−	0.598785i	$-0.204350\pi$
$788$	11.8765i	0.423081i
$789$	0	0
$790$	− 5.09446i	− 0.181253i
$791$	0	0
$792$	0	0
$793$	21.5917	0.766744
$794$	−19.5529	−0.693906
$795$	0	0
$796$	10.6070i	0.375955i
$797$	−39.5333	−1.40034	−0.700171	−	0.713975i	$-0.746893\pi$
$797$	−39.5333	−1.40034	−0.700171	+	0.713975i	$0.746893\pi$
$798$	0	0
$799$	−29.5374	−1.04496
$800$	− 4.06552i	− 0.143738i
$801$	0	0
$802$	−14.7972	−0.522507
$803$	3.77688	0.133283
$804$	0	0
$805$	0	0
$806$	− 0.595692i	− 0.0209824i
$807$	0	0
$808$	− 15.7746i	− 0.554949i
$809$	− 35.5988i	− 1.25159i	−0.779988	−	0.625794i	$-0.784775\pi$
$809$	− 35.5988i	− 1.25159i	0.779988	−	0.625794i	$-0.215225\pi$
$810$	0	0
$811$	36.7362i	1.28998i	0.764189	+	0.644992i	$0.223139\pi$
$811$	36.7362i	1.28998i	−0.764189	+	0.644992i	$0.776861\pi$
$812$	0	0
$813$	0	0
$814$	−31.5336	−1.10525
$815$	−15.1130	−0.529384
$816$	0	0
$817$	1.95453i	0.0683805i
$818$	31.0851	1.08687
$819$	0	0
$820$	8.04803	0.281049
$821$	− 30.2290i	− 1.05500i	−0.849555	−	0.527500i	$-0.823130\pi$
$821$	− 30.2290i	− 1.05500i	0.849555	−	0.527500i	$-0.176870\pi$
$822$	0	0
$823$	−33.0574	−1.15231	−0.576155	−	0.817340i	$-0.695448\pi$
$823$	−33.0574	−1.15231	−0.576155	+	0.817340i	$0.695448\pi$
$824$	5.97601	0.208184
$825$	0	0
$826$	0	0
$827$	16.7550i	0.582627i	0.956628	+	0.291313i	$0.0940923\pi$
$827$	16.7550i	0.582627i	−0.956628	+	0.291313i	$0.905908\pi$
$828$	0	0
$829$	34.2232i	1.18862i	0.804235	+	0.594311i	$0.202575\pi$
$829$	34.2232i	1.18862i	−0.804235	+	0.594311i	$0.797425\pi$
$830$	− 8.29826i	− 0.288037i
$831$	0	0
$832$	2.13246i	0.0739299i
$833$	0	0
$834$	0	0
$835$	−10.5793	−0.366111
$836$	1.51957	0.0525553
$837$	0	0
$838$	21.2123i	0.732765i
$839$	41.9748	1.44913	0.724565	−	0.689206i	$-0.242040\pi$
$839$	41.9748	1.44913	0.724565	+	0.689206i	$0.242040\pi$
$840$	0	0
$841$	23.0586	0.795123
$842$	20.1139i	0.693171i
$843$	0	0
$844$	18.5223	0.637565
$845$	8.17099	0.281091
$846$	0	0
$847$	0	0
$848$	− 6.27558i	− 0.215504i
$849$	0	0
$850$	9.95845i	0.341572i
$851$	− 17.7110i	− 0.607126i
$852$	0	0
$853$	− 14.3240i	− 0.490443i	−0.969467	−	0.245222i	$-0.921139\pi$
$853$	− 14.3240i	− 0.490443i	0.969467	−	0.245222i	$-0.0788607\pi$
$854$	0	0
$855$	0	0
$856$	5.24819	0.179379
$857$	0.818643	0.0279643	0.0139822	−	0.999902i	$-0.495549\pi$
$857$	0.818643	0.0279643	0.0139822	+	0.999902i	$0.495549\pi$
$858$	0	0
$859$	− 21.2281i	− 0.724292i	−0.932121	−	0.362146i	$-0.882044\pi$
$859$	− 21.2281i	− 0.724292i	0.932121	−	0.362146i	$-0.117956\pi$
$860$	3.57445	0.121888
$861$	0	0
$862$	−17.5145	−0.596546
$863$	13.1579i	0.447901i	0.974601	+	0.223951i	$0.0718954\pi$
$863$	13.1579i	0.447901i	−0.974601	+	0.223951i	$0.928105\pi$
$864$	0	0
$865$	18.8620	0.641327
$866$	−7.13421	−0.242430
$867$	0	0
$868$	0	0
$869$	15.1501i	0.513931i
$870$	0	0
$871$	20.7414i	0.702795i
$872$	− 8.55465i	− 0.289697i
$873$	0	0
$874$	0.853474i	0.0288692i
$875$	0	0
$876$	0	0
$877$	−18.9404	−0.639571	−0.319786	−	0.947490i	$-0.603611\pi$
$877$	−18.9404	−0.639571	−0.319786	+	0.947490i	$0.603611\pi$
$878$	−16.3774	−0.552712
$879$	0	0
$880$	− 2.77898i	− 0.0936795i
$881$	−1.57128	−0.0529377	−0.0264688	−	0.999650i	$-0.508426\pi$
$881$	−1.57128	−0.0529377	−0.0264688	+	0.999650i	$0.508426\pi$
$882$	0	0
$883$	−20.4882	−0.689483	−0.344741	−	0.938698i	$-0.612033\pi$
$883$	−20.4882	−0.689483	−0.344741	+	0.938698i	$0.612033\pi$
$884$	− 5.22345i	− 0.175684i
$885$	0	0
$886$	29.3172	0.984931
$887$	39.7781	1.33562	0.667809	−	0.744332i	$-0.267232\pi$
$887$	39.7781	1.33562	0.667809	+	0.744332i	$0.267232\pi$
$888$	0	0
$889$	0	0
$890$	3.47778i	0.116575i
$891$	0	0
$892$	− 26.3299i	− 0.881591i
$893$	− 6.37405i	− 0.213299i
$894$	0	0
$895$	− 2.21296i	− 0.0739711i
$896$	0	0
$897$	0	0
$898$	−32.7436	−1.09267
$899$	−0.680904	−0.0227094
$900$	0	0
$901$	15.3720i	0.512115i
$902$	−23.9335	−0.796899
$903$	0	0
$904$	−18.9965	−0.631815
$905$	− 5.81334i	− 0.193242i
$906$	0	0
$907$	26.6336	0.884353	0.442176	−	0.896928i	$-0.354207\pi$
$907$	26.6336	0.884353	0.442176	+	0.896928i	$0.354207\pi$
$908$	−20.3566	−0.675556
$909$	0	0
$910$	0	0
$911$	− 17.6486i	− 0.584726i	−0.956307	−	0.292363i	$-0.905558\pi$
$911$	− 17.6486i	− 0.584726i	0.956307	−	0.292363i	$-0.0944415\pi$
$912$	0	0
$913$	24.6776i	0.816711i
$914$	− 16.4912i	− 0.545480i
$915$	0	0
$916$	− 22.3049i	− 0.736975i
$917$	0	0
$918$	0	0
$919$	0.863410	0.0284813	0.0142406	−	0.999899i	$-0.495467\pi$
$919$	0.863410	0.0284813	0.0142406	+	0.999899i	$0.495467\pi$
$920$	1.56083	0.0514591
$921$	0	0
$922$	12.7292i	0.419215i
$923$	−30.2942	−0.997147
$924$	0	0
$925$	44.5952	1.46628
$926$	− 23.2392i	− 0.763686i
$927$	0	0
$928$	2.43751	0.0800151
$929$	14.7462	0.483806	0.241903	−	0.970300i	$-0.422228\pi$
$929$	14.7462	0.483806	0.241903	+	0.970300i	$0.422228\pi$
$930$	0	0
$931$	0	0
$932$	4.49723i	0.147312i
$933$	0	0
$934$	− 28.5783i	− 0.935110i
$935$	6.80709i	0.222616i
$936$	0	0
$937$	22.8194i	0.745479i	0.927936	+	0.372739i	$0.121581\pi$
$937$	22.8194i	0.745479i	−0.927936	+	0.372739i	$0.878419\pi$
$938$	0	0
$939$	0	0
$940$	−11.6569	−0.380205
$941$	41.9661	1.36806	0.684028	−	0.729456i	$-0.260227\pi$
$941$	41.9661	1.36806	0.684028	+	0.729456i	$0.260227\pi$
$942$	0	0
$943$	− 13.4424i	− 0.437745i
$944$	−11.4390	−0.372309
$945$	0	0
$946$	−10.6298	−0.345605
$947$	52.2387i	1.69753i	0.528771	+	0.848764i	$0.322653\pi$
$947$	52.2387i	1.69753i	−0.528771	+	0.848764i	$0.677347\pi$
$948$	0	0
$949$	2.80165	0.0909454
$950$	−2.14899	−0.0697225
$951$	0	0
$952$	0	0
$953$	− 47.1712i	− 1.52803i	−0.645201	−	0.764013i	$-0.723226\pi$
$953$	− 47.1712i	− 1.52803i	0.645201	−	0.764013i	$-0.276774\pi$
$954$	0	0
$955$	− 3.96331i	− 0.128250i
$956$	18.7605i	0.606758i
$957$	0	0
$958$	21.3931i	0.691178i
$959$	0	0
$960$	0	0
$961$	30.9220	0.997483
$962$	−23.3913	−0.754164
$963$	0	0
$964$	24.1774i	0.778702i
$965$	10.9264	0.351734
$966$	0	0
$967$	−18.7457	−0.602820	−0.301410	−	0.953495i	$-0.597457\pi$
$967$	−18.7457	−0.602820	−0.301410	+	0.953495i	$0.597457\pi$
$968$	− 2.73576i	− 0.0879308i
$969$	0	0
$970$	−4.13685	−0.132826
$971$	5.11543	0.164162	0.0820810	−	0.996626i	$-0.473843\pi$
$971$	5.11543	0.164162	0.0820810	+	0.996626i	$0.473843\pi$
$972$	0	0
$973$	0	0
$974$	− 18.6245i	− 0.596768i
$975$	0	0
$976$	10.1252i	0.324101i
$977$	31.3200i	1.00202i	0.865443	+	0.501008i	$0.167037\pi$
$977$	31.3200i	1.00202i	−0.865443	+	0.501008i	$0.832963\pi$
$978$	0	0
$979$	− 10.3423i	− 0.330543i
$980$	0	0
$981$	0	0
$982$	−16.5243	−0.527311
$983$	−23.6521	−0.754386	−0.377193	−	0.926135i	$-0.623111\pi$
$983$	−23.6521	−0.754386	−0.377193	+	0.926135i	$0.623111\pi$
$984$	0	0
$985$	− 11.4808i	− 0.365808i
$986$	−5.97065	−0.190144
$987$	0	0
$988$	1.12720	0.0358610
$989$	− 5.97031i	− 0.189845i
$990$	0	0
$991$	4.68726	0.148896	0.0744479	−	0.997225i	$-0.476281\pi$
$991$	4.68726	0.148896	0.0744479	+	0.997225i	$0.476281\pi$
$992$	0.279344	0.00886919
$993$	0	0
$994$	0	0
$995$	− 10.2536i	− 0.325062i
$996$	0	0
$997$	− 47.3814i	− 1.50058i	−0.661107	−	0.750291i	$-0.729913\pi$
$997$	− 47.3814i	− 1.50058i	0.661107	−	0.750291i	$-0.270087\pi$
$998$	− 7.54562i	− 0.238852i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2646.2.d.f.2645.14	yes	16
3.2	odd	2	inner	2646.2.d.f.2645.3	✓	16
7.6	odd	2	inner	2646.2.d.f.2645.11	yes	16
21.20	even	2	inner	2646.2.d.f.2645.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2646.2.d.f.2645.3	✓	16	3.2	odd	2	inner
2646.2.d.f.2645.6	yes	16	21.20	even	2	inner
2646.2.d.f.2645.11	yes	16	7.6	odd	2	inner
2646.2.d.f.2645.14	yes	16	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2646.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.966684 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.966684 q^{5} -1.00000i q^{8} +0.966684i q^{10} -2.87476i q^{11} -2.13246i q^{13} +1.00000 q^{16} -2.44949 q^{17} -0.528589i q^{19} -0.966684 q^{20} +2.87476 q^{22} +1.61463i q^{23} -4.06552 q^{25} +2.13246 q^{26} -2.43751i q^{29} -0.279344i q^{31} +1.00000i q^{32} -2.44949i q^{34} -10.9691 q^{37} +0.528589 q^{38} -0.966684i q^{40} -8.32540 q^{41} -3.69764 q^{43} +2.87476i q^{44} -1.61463 q^{46} +12.0586 q^{47} -4.06552i q^{50} +2.13246i q^{52} -6.27558i q^{53} -2.77898i q^{55} +2.43751 q^{58} -11.4390 q^{59} +10.1252i q^{61} +0.279344 q^{62} -1.00000 q^{64} -2.06142i q^{65} -9.72648 q^{67} +2.44949 q^{68} -14.2062i q^{71} +1.31381i q^{73} -10.9691i q^{74} +0.528589i q^{76} -5.27004 q^{79} +0.966684 q^{80} -8.32540i q^{82} -8.58425 q^{83} -2.36788 q^{85} -3.69764i q^{86} -2.87476 q^{88} +3.59764 q^{89} -1.61463i q^{92} +12.0586i q^{94} -0.510979i q^{95} +4.27942i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{4}+O(q^{10}) \) 16 * q - 16 * q^4	\( 16 q - 16 q^{4} + 16 q^{16} + 16 q^{22} + 16 q^{37} - 32 q^{43} + 48 q^{46} - 32 q^{58} - 16 q^{64} - 32 q^{67} - 16 q^{88}+O(q^{100}) \) 16 * q - 16 * q^4 + 16 * q^16 + 16 * q^22 + 16 * q^37 - 32 * q^43 + 48 * q^46 - 32 * q^58 - 16 * q^64 - 32 * q^67 - 16 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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