LMFDB - Embedded newform 8379.2.a.ck.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8379,2,Mod(1,8379)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8379, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8379.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.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:	$66.9066518536$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 4x^{3} - 12x^{2} - x + 1 $ x^7 - 3x^6 - 6x^5 + 18x^4 + 4x^3 - 12*x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.862998$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40650 q^{2} -0.0217491 q^{4} +0.295752 q^{5} -2.84360 q^{8} +O(q^{10})$	$q+1.40650 q^{2} -0.0217491 q^{4} +0.295752 q^{5} -2.84360 q^{8} +0.415977 q^{10} +0.372561 q^{11} -3.46904 q^{13} -3.95603 q^{16} +3.29575 q^{17} -1.00000 q^{19} -0.00643235 q^{20} +0.524008 q^{22} +7.41280 q^{23} -4.91253 q^{25} -4.87922 q^{26} -3.21894 q^{29} +3.96455 q^{31} +0.123026 q^{32} +4.63549 q^{34} +9.83661 q^{37} -1.40650 q^{38} -0.841001 q^{40} +2.81301 q^{41} -6.08278 q^{43} -0.00810286 q^{44} +10.4261 q^{46} +6.82129 q^{47} -6.90949 q^{50} +0.0754486 q^{52} -10.4357 q^{53} +0.110186 q^{55} -4.52745 q^{58} -4.68799 q^{59} -7.09816 q^{61} +5.57615 q^{62} +8.08509 q^{64} -1.02598 q^{65} -7.66601 q^{67} -0.0716797 q^{68} -1.88170 q^{71} +1.01791 q^{73} +13.8352 q^{74} +0.0217491 q^{76} -10.6056 q^{79} -1.17001 q^{80} +3.95650 q^{82} -12.3958 q^{83} +0.974727 q^{85} -8.55545 q^{86} -1.05941 q^{88} -6.18557 q^{89} -0.161222 q^{92} +9.59417 q^{94} -0.295752 q^{95} +0.235136 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 10 q^{4} - 2 q^{5} - 12 q^{8}+O(q^{10}) $ 7 * q - 2 * q^2 + 10 * q^4 - 2 * q^5 - 12 * q^8	$ 7 q - 2 q^{2} + 10 q^{4} - 2 q^{5} - 12 q^{8} - 7 q^{11} - 6 q^{13} + 24 q^{16} + 19 q^{17} - 7 q^{19} - 8 q^{20} - 6 q^{22} + q^{23} - 3 q^{25} + 12 q^{26} - 24 q^{29} - 26 q^{32} - 6 q^{34} + 8 q^{37} + 2 q^{38} + 10 q^{40} - 4 q^{41} + 4 q^{43} - 26 q^{44} - 16 q^{46} + 5 q^{47} - 16 q^{50} - 42 q^{52} - 20 q^{53} + 30 q^{55} - 16 q^{59} - 5 q^{61} + 24 q^{62} + 32 q^{64} - 26 q^{65} - 4 q^{67} + 22 q^{68} - 12 q^{71} - 3 q^{73} + 4 q^{74} - 10 q^{76} - 20 q^{79} - 4 q^{80} + 48 q^{82} - 11 q^{83} + 26 q^{85} - 36 q^{86} - 32 q^{88} - 10 q^{89} + 30 q^{92} - 16 q^{94} + 2 q^{95} + 4 q^{97}+O(q^{100}) $ 7 * q - 2 * q^2 + 10 * q^4 - 2 * q^5 - 12 * q^8 - 7 * q^11 - 6 * q^13 + 24 * q^16 + 19 * q^17 - 7 * q^19 - 8 * q^20 - 6 * q^22 + q^23 - 3 * q^25 + 12 * q^26 - 24 * q^29 - 26 * q^32 - 6 * q^34 + 8 * q^37 + 2 * q^38 + 10 * q^40 - 4 * q^41 + 4 * q^43 - 26 * q^44 - 16 * q^46 + 5 * q^47 - 16 * q^50 - 42 * q^52 - 20 * q^53 + 30 * q^55 - 16 * q^59 - 5 * q^61 + 24 * q^62 + 32 * q^64 - 26 * q^65 - 4 * q^67 + 22 * q^68 - 12 * q^71 - 3 * q^73 + 4 * q^74 - 10 * q^76 - 20 * q^79 - 4 * q^80 + 48 * q^82 - 11 * q^83 + 26 * q^85 - 36 * q^86 - 32 * q^88 - 10 * q^89 + 30 * q^92 - 16 * q^94 + 2 * q^95 + 4 * q^97

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 $

$2$	1.40650	0.994548	0.497274	−	0.867594i	$-0.334334\pi$
$2$	1.40650	0.994548	0.497274	+	0.867594i	$0.334334\pi$
$3$	0	0
$3$	0	0
$4$	−0.0217491	−0.0108746
$5$	0.295752	0.132265	0.0661323	−	0.997811i	$-0.478934\pi$
$5$	0.295752	0.132265	0.0661323	+	0.997811i	$0.478934\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.84360	−1.00536
$9$	0	0
$10$	0.415977	0.131543
$11$	0.372561	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$11$	0.372561	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$12$	0	0
$13$	−3.46904	−0.962140	−0.481070	−	0.876682i	$-0.659752\pi$
$13$	−3.46904	−0.962140	−0.481070	+	0.876682i	$0.659752\pi$
$14$	0	0
$15$	0	0
$16$	−3.95603	−0.989007
$17$	3.29575	0.799337	0.399669	−	0.916660i	$-0.369125\pi$
$17$	3.29575	0.799337	0.399669	+	0.916660i	$0.369125\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.00643235	−0.00143832
$21$	0	0
$22$	0.524008	0.111719
$23$	7.41280	1.54568	0.772838	−	0.634604i	$-0.218837\pi$
$23$	7.41280	1.54568	0.772838	+	0.634604i	$0.218837\pi$
$24$	0	0
$25$	−4.91253	−0.982506
$26$	−4.87922	−0.956894
$27$	0	0
$28$	0	0
$29$	−3.21894	−0.597743	−0.298871	−	0.954293i	$-0.596610\pi$
$29$	−3.21894	−0.597743	−0.298871	+	0.954293i	$0.596610\pi$
$30$	0	0
$31$	3.96455	0.712054	0.356027	−	0.934476i	$-0.384131\pi$
$31$	3.96455	0.712054	0.356027	+	0.934476i	$0.384131\pi$
$32$	0.123026	0.0217482
$33$	0	0
$34$	4.63549	0.794979
$35$	0	0
$36$	0	0
$37$	9.83661	1.61713	0.808564	−	0.588408i	$-0.200245\pi$
$37$	9.83661	1.61713	0.808564	+	0.588408i	$0.200245\pi$
$38$	−1.40650	−0.228165
$39$	0	0
$40$	−0.841001	−0.132974
$41$	2.81301	0.439318	0.219659	−	0.975577i	$-0.429506\pi$
$41$	2.81301	0.439318	0.219659	+	0.975577i	$0.429506\pi$
$42$	0	0
$43$	−6.08278	−0.927615	−0.463808	−	0.885936i	$-0.653517\pi$
$43$	−6.08278	−0.927615	−0.463808	+	0.885936i	$0.653517\pi$
$44$	−0.00810286	−0.00122155
$45$	0	0
$46$	10.4261	1.53725
$47$	6.82129	0.994988	0.497494	−	0.867468i	$-0.334254\pi$
$47$	6.82129	0.994988	0.497494	+	0.867468i	$0.334254\pi$
$48$	0	0
$49$	0	0
$50$	−6.90949	−0.977149
$51$	0	0
$52$	0.0754486	0.0104628
$53$	−10.4357	−1.43345	−0.716724	−	0.697356i	$-0.754359\pi$
$53$	−10.4357	−1.43345	−0.716724	+	0.697356i	$0.754359\pi$
$54$	0	0
$55$	0.110186	0.0148574
$56$	0	0
$57$	0	0
$58$	−4.52745	−0.594484
$59$	−4.68799	−0.610324	−0.305162	−	0.952300i	$-0.598711\pi$
$59$	−4.68799	−0.610324	−0.305162	+	0.952300i	$0.598711\pi$
$60$	0	0
$61$	−7.09816	−0.908827	−0.454413	−	0.890791i	$-0.650151\pi$
$61$	−7.09816	−0.908827	−0.454413	+	0.890791i	$0.650151\pi$
$62$	5.57615	0.708172
$63$	0	0
$64$	8.08509	1.01064
$65$	−1.02598	−0.127257
$66$	0	0
$67$	−7.66601	−0.936552	−0.468276	−	0.883582i	$-0.655125\pi$
$67$	−7.66601	−0.936552	−0.468276	+	0.883582i	$0.655125\pi$
$68$	−0.0716797	−0.00869244
$69$	0	0
$70$	0	0
$71$	−1.88170	−0.223317	−0.111658	−	0.993747i	$-0.535616\pi$
$71$	−1.88170	−0.223317	−0.111658	+	0.993747i	$0.535616\pi$
$72$	0	0
$73$	1.01791	0.119138	0.0595688	−	0.998224i	$-0.481027\pi$
$73$	1.01791	0.119138	0.0595688	+	0.998224i	$0.481027\pi$
$74$	13.8352	1.60831
$75$	0	0
$76$	0.0217491	0.00249480
$77$	0	0
$78$	0	0
$79$	−10.6056	−1.19323	−0.596614	−	0.802529i	$-0.703487\pi$
$79$	−10.6056	−1.19323	−0.596614	+	0.802529i	$0.703487\pi$
$80$	−1.17001	−0.130811
$81$	0	0
$82$	3.95650	0.436923
$83$	−12.3958	−1.36061	−0.680306	−	0.732928i	$-0.738153\pi$
$83$	−12.3958	−1.36061	−0.680306	+	0.732928i	$0.738153\pi$
$84$	0	0
$85$	0.974727	0.105724
$86$	−8.55545	−0.922558
$87$	0	0
$88$	−1.05941	−0.112934
$89$	−6.18557	−0.655669	−0.327834	−	0.944735i	$-0.606319\pi$
$89$	−6.18557	−0.655669	−0.327834	+	0.944735i	$0.606319\pi$
$90$	0	0
$91$	0	0
$92$	−0.161222	−0.0168085
$93$	0	0
$94$	9.59417	0.989563
$95$	−0.295752	−0.0303436
$96$	0	0
$97$	0.235136	0.0238745	0.0119372	−	0.999929i	$-0.496200\pi$
$97$	0.235136	0.0238745	0.0119372	+	0.999929i	$0.496200\pi$
$98$	0	0
$99$	0	0
$100$	0.106843	0.0106843
$101$	0.771429	0.0767601	0.0383800	−	0.999263i	$-0.487780\pi$
$101$	0.771429	0.0767601	0.0383800	+	0.999263i	$0.487780\pi$
$102$	0	0
$103$	−10.6191	−1.04633	−0.523167	−	0.852230i	$-0.675250\pi$
$103$	−10.6191	−1.04633	−0.523167	+	0.852230i	$0.675250\pi$
$104$	9.86456	0.967300
$105$	0	0
$106$	−14.6778	−1.42563
$107$	−8.14803	−0.787700	−0.393850	−	0.919175i	$-0.628857\pi$
$107$	−8.14803	−0.787700	−0.393850	+	0.919175i	$0.628857\pi$
$108$	0	0
$109$	6.70289	0.642020	0.321010	−	0.947076i	$-0.395978\pi$
$109$	6.70289	0.642020	0.321010	+	0.947076i	$0.395978\pi$
$110$	0.154977	0.0147764
$111$	0	0
$112$	0	0
$113$	−15.6066	−1.46814	−0.734072	−	0.679072i	$-0.762383\pi$
$113$	−15.6066	−1.46814	−0.734072	+	0.679072i	$0.762383\pi$
$114$	0	0
$115$	2.19235	0.204438
$116$	0.0700092	0.00650019
$117$	0	0
$118$	−6.59367	−0.606996
$119$	0	0
$120$	0	0
$121$	−10.8612	−0.987382
$122$	−9.98359	−0.903872
$123$	0	0
$124$	−0.0862254	−0.00774327
$125$	−2.93165	−0.262215
$126$	0	0
$127$	−3.34348	−0.296686	−0.148343	−	0.988936i	$-0.547394\pi$
$127$	−3.34348	−0.296686	−0.148343	+	0.988936i	$0.547394\pi$
$128$	11.1257	0.983378
$129$	0	0
$130$	−1.44304	−0.126563
$131$	−15.7131	−1.37286	−0.686430	−	0.727196i	$-0.740823\pi$
$131$	−15.7131	−1.37286	−0.686430	+	0.727196i	$0.740823\pi$
$132$	0	0
$133$	0	0
$134$	−10.7823	−0.931445
$135$	0	0
$136$	−9.37179	−0.803624
$137$	−13.5984	−1.16179	−0.580893	−	0.813980i	$-0.697297\pi$
$137$	−13.5984	−1.16179	−0.580893	+	0.813980i	$0.697297\pi$
$138$	0	0
$139$	21.3319	1.80935	0.904675	−	0.426102i	$-0.140114\pi$
$139$	21.3319	1.80935	0.904675	+	0.426102i	$0.140114\pi$
$140$	0	0
$141$	0	0
$142$	−2.64662	−0.222099
$143$	−1.29243	−0.108078
$144$	0	0
$145$	−0.952011	−0.0790602
$146$	1.43170	0.118488
$147$	0	0
$148$	−0.213938	−0.0175856
$149$	−10.2208	−0.837320	−0.418660	−	0.908143i	$-0.637500\pi$
$149$	−10.2208	−0.837320	−0.418660	+	0.908143i	$0.637500\pi$
$150$	0	0
$151$	−2.58011	−0.209966	−0.104983	−	0.994474i	$-0.533479\pi$
$151$	−2.58011	−0.209966	−0.104983	+	0.994474i	$0.533479\pi$
$152$	2.84360	0.230646
$153$	0	0
$154$	0	0
$155$	1.17252	0.0941795
$156$	0	0
$157$	21.6002	1.72388	0.861941	−	0.507009i	$-0.169249\pi$
$157$	21.6002	1.72388	0.861941	+	0.507009i	$0.169249\pi$
$158$	−14.9169	−1.18672
$159$	0	0
$160$	0.0363853	0.00287651
$161$	0	0
$162$	0	0
$163$	−15.4704	−1.21174	−0.605868	−	0.795565i	$-0.707174\pi$
$163$	−15.4704	−1.21174	−0.605868	+	0.795565i	$0.707174\pi$
$164$	−0.0611804	−0.00477739
$165$	0	0
$166$	−17.4347	−1.35319
$167$	7.96646	0.616463	0.308231	−	0.951311i	$-0.400263\pi$
$167$	7.96646	0.616463	0.308231	+	0.951311i	$0.400263\pi$
$168$	0	0
$169$	−0.965736	−0.0742874
$170$	1.37096	0.105148
$171$	0	0
$172$	0.132295	0.0100874
$173$	−5.61526	−0.426920	−0.213460	−	0.976952i	$-0.568473\pi$
$173$	−5.61526	−0.426920	−0.213460	+	0.976952i	$0.568473\pi$
$174$	0	0
$175$	0	0
$176$	−1.47386	−0.111096
$177$	0	0
$178$	−8.70002	−0.652094
$179$	−7.43709	−0.555874	−0.277937	−	0.960599i	$-0.589651\pi$
$179$	−7.43709	−0.555874	−0.277937	+	0.960599i	$0.589651\pi$
$180$	0	0
$181$	8.99507	0.668598	0.334299	−	0.942467i	$-0.391500\pi$
$181$	8.99507	0.668598	0.334299	+	0.942467i	$0.391500\pi$
$182$	0	0
$183$	0	0
$184$	−21.0790	−1.55396
$185$	2.90920	0.213889
$186$	0	0
$187$	1.22787	0.0897906
$188$	−0.148357	−0.0108201
$189$	0	0
$190$	−0.415977	−0.0301781
$191$	−22.1575	−1.60326	−0.801629	−	0.597822i	$-0.796033\pi$
$191$	−22.1575	−1.60326	−0.801629	+	0.597822i	$0.796033\pi$
$192$	0	0
$193$	0.407408	0.0293259	0.0146629	−	0.999892i	$-0.495332\pi$
$193$	0.407408	0.0293259	0.0146629	+	0.999892i	$0.495332\pi$
$194$	0.330720	0.0237443
$195$	0	0
$196$	0	0
$197$	20.5203	1.46201	0.731006	−	0.682371i	$-0.239051\pi$
$197$	20.5203	1.46201	0.731006	+	0.682371i	$0.239051\pi$
$198$	0	0
$199$	7.01563	0.497325	0.248662	−	0.968590i	$-0.420009\pi$
$199$	7.01563	0.497325	0.248662	+	0.968590i	$0.420009\pi$
$200$	13.9693	0.987775
$201$	0	0
$202$	1.08502	0.0763416
$203$	0	0
$204$	0	0
$205$	0.831953	0.0581061
$206$	−14.9358	−1.04063
$207$	0	0
$208$	13.7236	0.951563
$209$	−0.372561	−0.0257706
$210$	0	0
$211$	−15.2256	−1.04817	−0.524087	−	0.851665i	$-0.675593\pi$
$211$	−15.2256	−1.04817	−0.524087	+	0.851665i	$0.675593\pi$
$212$	0.226967	0.0155881
$213$	0	0
$214$	−11.4602	−0.783405
$215$	−1.79900	−0.122691
$216$	0	0
$217$	0	0
$218$	9.42763	0.638520
$219$	0	0
$220$	−0.00239644	−0.000161568 0
$221$	−11.4331	−0.769074
$222$	0	0
$223$	0.0430042	0.00287977	0.00143989	−	0.999999i	$-0.499542\pi$
$223$	0.0430042	0.00287977	0.00143989	+	0.999999i	$0.499542\pi$
$224$	0	0
$225$	0	0
$226$	−21.9507	−1.46014
$227$	15.7366	1.04448	0.522238	−	0.852800i	$-0.325097\pi$
$227$	15.7366	1.04448	0.522238	+	0.852800i	$0.325097\pi$
$228$	0	0
$229$	10.5762	0.698898	0.349449	−	0.936955i	$-0.386369\pi$
$229$	10.5762	0.698898	0.349449	+	0.936955i	$0.386369\pi$
$230$	3.08355	0.203323
$231$	0	0
$232$	9.15338	0.600949
$233$	0.526501	0.0344922	0.0172461	−	0.999851i	$-0.494510\pi$
$233$	0.526501	0.0344922	0.0172461	+	0.999851i	$0.494510\pi$
$234$	0	0
$235$	2.01741	0.131602
$236$	0.101960	0.00663701
$237$	0	0
$238$	0	0
$239$	4.56967	0.295588	0.147794	−	0.989018i	$-0.452783\pi$
$239$	4.56967	0.295588	0.147794	+	0.989018i	$0.452783\pi$
$240$	0	0
$241$	22.8596	1.47252	0.736258	−	0.676701i	$-0.236591\pi$
$241$	22.8596	1.47252	0.736258	+	0.676701i	$0.236591\pi$
$242$	−15.2763	−0.981998
$243$	0	0
$244$	0.154379	0.00988309
$245$	0	0
$246$	0	0
$247$	3.46904	0.220730
$248$	−11.2736	−0.715873
$249$	0	0
$250$	−4.12338	−0.260786
$251$	−8.49789	−0.536382	−0.268191	−	0.963366i	$-0.586426\pi$
$251$	−8.49789	−0.536382	−0.268191	+	0.963366i	$0.586426\pi$
$252$	0	0
$253$	2.76172	0.173628
$254$	−4.70261	−0.295068
$255$	0	0
$256$	−0.521917	−0.0326198
$257$	−11.3461	−0.707749	−0.353875	−	0.935293i	$-0.615136\pi$
$257$	−11.3461	−0.707749	−0.353875	+	0.935293i	$0.615136\pi$
$258$	0	0
$259$	0	0
$260$	0.0223141	0.00138386
$261$	0	0
$262$	−22.1005	−1.36537
$263$	4.91687	0.303187	0.151594	−	0.988443i	$-0.451559\pi$
$263$	4.91687	0.303187	0.151594	+	0.988443i	$0.451559\pi$
$264$	0	0
$265$	−3.08637	−0.189594
$266$	0	0
$267$	0	0
$268$	0.166729	0.0101846
$269$	−28.8234	−1.75739	−0.878696	−	0.477381i	$-0.841586\pi$
$269$	−28.8234	−1.75739	−0.878696	+	0.477381i	$0.841586\pi$
$270$	0	0
$271$	2.65446	0.161247	0.0806234	−	0.996745i	$-0.474309\pi$
$271$	2.65446	0.161247	0.0806234	+	0.996745i	$0.474309\pi$
$272$	−13.0381	−0.790550
$273$	0	0
$274$	−19.1261	−1.15545
$275$	−1.83022	−0.110366
$276$	0	0
$277$	−3.59970	−0.216285	−0.108143	−	0.994135i	$-0.534490\pi$
$277$	−3.59970	−0.216285	−0.108143	+	0.994135i	$0.534490\pi$
$278$	30.0034	1.79949
$279$	0	0
$280$	0	0
$281$	−1.35683	−0.0809415	−0.0404708	−	0.999181i	$-0.512886\pi$
$281$	−1.35683	−0.0809415	−0.0404708	+	0.999181i	$0.512886\pi$
$282$	0	0
$283$	6.05006	0.359639	0.179819	−	0.983700i	$-0.442449\pi$
$283$	6.05006	0.359639	0.179819	+	0.983700i	$0.442449\pi$
$284$	0.0409253	0.00242847
$285$	0	0
$286$	−1.81781	−0.107489
$287$	0	0
$288$	0	0
$289$	−6.13802	−0.361060
$290$	−1.33901	−0.0786291
$291$	0	0
$292$	−0.0221387	−0.00129557
$293$	−3.98610	−0.232871	−0.116435	−	0.993198i	$-0.537147\pi$
$293$	−3.98610	−0.232871	−0.116435	+	0.993198i	$0.537147\pi$
$294$	0	0
$295$	−1.38648	−0.0807242
$296$	−27.9713	−1.62580
$297$	0	0
$298$	−14.3756	−0.832755
$299$	−25.7153	−1.48716
$300$	0	0
$301$	0	0
$302$	−3.62893	−0.208822
$303$	0	0
$304$	3.95603	0.226894
$305$	−2.09930	−0.120206
$306$	0	0
$307$	−17.9160	−1.02252	−0.511259	−	0.859427i	$-0.670821\pi$
$307$	−17.9160	−1.02252	−0.511259	+	0.859427i	$0.670821\pi$
$308$	0	0
$309$	0	0
$310$	1.64916	0.0936660
$311$	18.8468	1.06870	0.534352	−	0.845262i	$-0.320556\pi$
$311$	18.8468	1.06870	0.534352	+	0.845262i	$0.320556\pi$
$312$	0	0
$313$	18.8133	1.06339	0.531694	−	0.846936i	$-0.321556\pi$
$313$	18.8133	1.06339	0.531694	+	0.846936i	$0.321556\pi$
$314$	30.3807	1.71448
$315$	0	0
$316$	0.230663	0.0129758
$317$	−14.0847	−0.791078	−0.395539	−	0.918449i	$-0.629442\pi$
$317$	−14.0847	−0.791078	−0.395539	+	0.918449i	$0.629442\pi$
$318$	0	0
$319$	−1.19925	−0.0671452
$320$	2.39119	0.133671
$321$	0	0
$322$	0	0
$323$	−3.29575	−0.183381
$324$	0	0
$325$	17.0418	0.945308
$326$	−21.7592	−1.20513
$327$	0	0
$328$	−7.99905	−0.441674
$329$	0	0
$330$	0	0
$331$	7.13182	0.392000	0.196000	−	0.980604i	$-0.437205\pi$
$331$	7.13182	0.392000	0.196000	+	0.980604i	$0.437205\pi$
$332$	0.269597	0.0147961
$333$	0	0
$334$	11.2048	0.613102
$335$	−2.26724	−0.123873
$336$	0	0
$337$	−6.45355	−0.351547	−0.175774	−	0.984431i	$-0.556243\pi$
$337$	−6.45355	−0.351547	−0.175774	+	0.984431i	$0.556243\pi$
$338$	−1.35831	−0.0738824
$339$	0	0
$340$	−0.0211994	−0.00114970
$341$	1.47703	0.0799859
$342$	0	0
$343$	0	0
$344$	17.2970	0.932590
$345$	0	0
$346$	−7.89788	−0.424593
$347$	−4.87230	−0.261559	−0.130779	−	0.991411i	$-0.541748\pi$
$347$	−4.87230	−0.261559	−0.130779	+	0.991411i	$0.541748\pi$
$348$	0	0
$349$	−11.2362	−0.601459	−0.300729	−	0.953709i	$-0.597230\pi$
$349$	−11.2362	−0.601459	−0.300729	+	0.953709i	$0.597230\pi$
$350$	0	0
$351$	0	0
$352$	0.0458347	0.00244300
$353$	6.20110	0.330051	0.165026	−	0.986289i	$-0.447229\pi$
$353$	6.20110	0.330051	0.165026	+	0.986289i	$0.447229\pi$
$354$	0	0
$355$	−0.556517	−0.0295369
$356$	0.134531	0.00713011
$357$	0	0
$358$	−10.4603	−0.552844
$359$	−19.4926	−1.02878	−0.514390	−	0.857557i	$-0.671982\pi$
$359$	−19.4926	−1.02878	−0.514390	+	0.857557i	$0.671982\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	12.6516	0.664953
$363$	0	0
$364$	0	0
$365$	0.301050	0.0157577
$366$	0	0
$367$	−23.3201	−1.21730	−0.608649	−	0.793440i	$-0.708288\pi$
$367$	−23.3201	−1.21730	−0.608649	+	0.793440i	$0.708288\pi$
$368$	−29.3252	−1.52868
$369$	0	0
$370$	4.09180	0.212723
$371$	0	0
$372$	0	0
$373$	−0.251690	−0.0130320	−0.00651601	−	0.999979i	$-0.502074\pi$
$373$	−0.251690	−0.0130320	−0.00651601	+	0.999979i	$0.502074\pi$
$374$	1.72700	0.0893010
$375$	0	0
$376$	−19.3970	−1.00032
$377$	11.1667	0.575112
$378$	0	0
$379$	−6.23458	−0.320249	−0.160125	−	0.987097i	$-0.551190\pi$
$379$	−6.23458	−0.320249	−0.160125	+	0.987097i	$0.551190\pi$
$380$	0.00643235	0.000329973 0
$381$	0	0
$382$	−31.1645	−1.59452
$383$	−11.5020	−0.587723	−0.293862	−	0.955848i	$-0.594940\pi$
$383$	−11.5020	−0.587723	−0.293862	+	0.955848i	$0.594940\pi$
$384$	0	0
$385$	0	0
$386$	0.573021	0.0291660
$387$	0	0
$388$	−0.00511401	−0.000259624 0
$389$	34.8083	1.76485	0.882426	−	0.470452i	$-0.155909\pi$
$389$	34.8083	1.76485	0.882426	+	0.470452i	$0.155909\pi$
$390$	0	0
$391$	24.4307	1.23552
$392$	0	0
$393$	0	0
$394$	28.8619	1.45404
$395$	−3.13664	−0.157822
$396$	0	0
$397$	−29.1524	−1.46311	−0.731557	−	0.681780i	$-0.761206\pi$
$397$	−29.1524	−1.46311	−0.731557	+	0.681780i	$0.761206\pi$
$398$	9.86750	0.494613
$399$	0	0
$400$	19.4341	0.971706
$401$	14.5746	0.727820	0.363910	−	0.931434i	$-0.381442\pi$
$401$	14.5746	0.727820	0.363910	+	0.931434i	$0.381442\pi$
$402$	0	0
$403$	−13.7532	−0.685095
$404$	−0.0167779	−0.000834732 0
$405$	0	0
$406$	0	0
$407$	3.66473	0.181654
$408$	0	0
$409$	9.60370	0.474872	0.237436	−	0.971403i	$-0.423693\pi$
$409$	9.60370	0.474872	0.237436	+	0.971403i	$0.423693\pi$
$410$	1.17015	0.0577893
$411$	0	0
$412$	0.230957	0.0113784
$413$	0	0
$414$	0	0
$415$	−3.66608	−0.179961
$416$	−0.426783	−0.0209248
$417$	0	0
$418$	−0.524008	−0.0256301
$419$	−17.2401	−0.842234	−0.421117	−	0.907006i	$-0.638362\pi$
$419$	−17.2401	−0.842234	−0.421117	+	0.907006i	$0.638362\pi$
$420$	0	0
$421$	15.3528	0.748251	0.374126	−	0.927378i	$-0.377943\pi$
$421$	15.3528	0.748251	0.374126	+	0.927378i	$0.377943\pi$
$422$	−21.4149	−1.04246
$423$	0	0
$424$	29.6748	1.44114
$425$	−16.1905	−0.785354
$426$	0	0
$427$	0	0
$428$	0.177212	0.00856589
$429$	0	0
$430$	−2.53029	−0.122022
$431$	5.98348	0.288214	0.144107	−	0.989562i	$-0.453969\pi$
$431$	5.98348	0.288214	0.144107	+	0.989562i	$0.453969\pi$
$432$	0	0
$433$	15.2443	0.732596	0.366298	−	0.930498i	$-0.380625\pi$
$433$	15.2443	0.732596	0.366298	+	0.930498i	$0.380625\pi$
$434$	0	0
$435$	0	0
$436$	−0.145782	−0.00698169
$437$	−7.41280	−0.354602
$438$	0	0
$439$	31.9500	1.52489	0.762445	−	0.647053i	$-0.223999\pi$
$439$	31.9500	1.52489	0.762445	+	0.647053i	$0.223999\pi$
$440$	−0.313324	−0.0149371
$441$	0	0
$442$	−16.0807	−0.764881
$443$	−33.7439	−1.60322	−0.801610	−	0.597848i	$-0.796023\pi$
$443$	−33.7439	−1.60322	−0.801610	+	0.597848i	$0.796023\pi$
$444$	0	0
$445$	−1.82940	−0.0867217
$446$	0.0604855	0.00286407
$447$	0	0
$448$	0	0
$449$	−21.1734	−0.999234	−0.499617	−	0.866246i	$-0.666526\pi$
$449$	−21.1734	−0.999234	−0.499617	+	0.866246i	$0.666526\pi$
$450$	0	0
$451$	1.04802	0.0493491
$452$	0.339430	0.0159654
$453$	0	0
$454$	22.1336	1.03878
$455$	0	0
$456$	0	0
$457$	−8.06009	−0.377035	−0.188518	−	0.982070i	$-0.560368\pi$
$457$	−8.06009	−0.377035	−0.188518	+	0.982070i	$0.560368\pi$
$458$	14.8755	0.695088
$459$	0	0
$460$	−0.0476817	−0.00222317
$461$	12.4647	0.580537	0.290268	−	0.956945i	$-0.406255\pi$
$461$	12.4647	0.580537	0.290268	+	0.956945i	$0.406255\pi$
$462$	0	0
$463$	14.8402	0.689685	0.344842	−	0.938661i	$-0.387932\pi$
$463$	14.8402	0.689685	0.344842	+	0.938661i	$0.387932\pi$
$464$	12.7342	0.591172
$465$	0	0
$466$	0.740525	0.0343042
$467$	−2.49691	−0.115543	−0.0577715	−	0.998330i	$-0.518399\pi$
$467$	−2.49691	−0.115543	−0.0577715	+	0.998330i	$0.518399\pi$
$468$	0	0
$469$	0	0
$470$	2.83750	0.130884
$471$	0	0
$472$	13.3307	0.613597
$473$	−2.26620	−0.104200
$474$	0	0
$475$	4.91253	0.225402
$476$	0	0
$477$	0	0
$478$	6.42726	0.293976
$479$	−15.7163	−0.718097	−0.359049	−	0.933319i	$-0.616899\pi$
$479$	−15.7163	−0.718097	−0.359049	+	0.933319i	$0.616899\pi$
$480$	0	0
$481$	−34.1236	−1.55590
$482$	32.1521	1.46449
$483$	0	0
$484$	0.236222	0.0107373
$485$	0.0695421	0.00315775
$486$	0	0
$487$	14.8609	0.673412	0.336706	−	0.941610i	$-0.390687\pi$
$487$	14.8609	0.673412	0.336706	+	0.941610i	$0.390687\pi$
$488$	20.1843	0.913701
$489$	0	0
$490$	0	0
$491$	2.79345	0.126067	0.0630334	−	0.998011i	$-0.479923\pi$
$491$	2.79345	0.126067	0.0630334	+	0.998011i	$0.479923\pi$
$492$	0	0
$493$	−10.6088	−0.477798
$494$	4.87922	0.219527
$495$	0	0
$496$	−15.6839	−0.704226
$497$	0	0
$498$	0	0
$499$	31.0687	1.39082	0.695412	−	0.718611i	$-0.255222\pi$
$499$	31.0687	1.39082	0.695412	+	0.718611i	$0.255222\pi$
$500$	0.0637609	0.00285147
$501$	0	0
$502$	−11.9523	−0.533458
$503$	−9.93140	−0.442819	−0.221410	−	0.975181i	$-0.571066\pi$
$503$	−9.93140	−0.442819	−0.221410	+	0.975181i	$0.571066\pi$
$504$	0	0
$505$	0.228152	0.0101526
$506$	3.88436	0.172681
$507$	0	0
$508$	0.0727177	0.00322633
$509$	14.3464	0.635894	0.317947	−	0.948108i	$-0.397007\pi$
$509$	14.3464	0.635894	0.317947	+	0.948108i	$0.397007\pi$
$510$	0	0
$511$	0	0
$512$	−22.9854	−1.01582
$513$	0	0
$514$	−15.9583	−0.703891
$515$	−3.14063	−0.138393
$516$	0	0
$517$	2.54134	0.111768
$518$	0	0
$519$	0	0
$520$	2.91747	0.127939
$521$	35.4447	1.55286	0.776431	−	0.630203i	$-0.217028\pi$
$521$	35.4447	1.55286	0.776431	+	0.630203i	$0.217028\pi$
$522$	0	0
$523$	33.8791	1.48143	0.740714	−	0.671820i	$-0.234487\pi$
$523$	33.8791	1.48143	0.740714	+	0.671820i	$0.234487\pi$
$524$	0.341746	0.0149292
$525$	0	0
$526$	6.91559	0.301534
$527$	13.0662	0.569171
$528$	0	0
$529$	31.9496	1.38911
$530$	−4.34099	−0.188561
$531$	0	0
$532$	0	0
$533$	−9.75844	−0.422685
$534$	0	0
$535$	−2.40980	−0.104185
$536$	21.7990	0.941574
$537$	0	0
$538$	−40.5402	−1.74781
$539$	0	0
$540$	0	0
$541$	−31.4110	−1.35046	−0.675232	−	0.737606i	$-0.735956\pi$
$541$	−31.4110	−1.35046	−0.675232	+	0.737606i	$0.735956\pi$
$542$	3.73350	0.160368
$543$	0	0
$544$	0.405464	0.0173841
$545$	1.98240	0.0849165
$546$	0	0
$547$	−39.3436	−1.68221	−0.841106	−	0.540870i	$-0.818095\pi$
$547$	−39.3436	−1.68221	−0.841106	+	0.540870i	$0.818095\pi$
$548$	0.295752	0.0126339
$549$	0	0
$550$	−2.57420	−0.109764
$551$	3.21894	0.137132
$552$	0	0
$553$	0	0
$554$	−5.06300	−0.215106
$555$	0	0
$556$	−0.463951	−0.0196759
$557$	−22.2814	−0.944091	−0.472046	−	0.881574i	$-0.656484\pi$
$557$	−22.2814	−0.944091	−0.472046	+	0.881574i	$0.656484\pi$
$558$	0	0
$559$	21.1014	0.892495
$560$	0	0
$561$	0	0
$562$	−1.90838	−0.0805002
$563$	−22.9600	−0.967649	−0.483825	−	0.875165i	$-0.660753\pi$
$563$	−22.9600	−0.967649	−0.483825	+	0.875165i	$0.660753\pi$
$564$	0	0
$565$	−4.61569	−0.194183
$566$	8.50943	0.357678
$567$	0	0
$568$	5.35080	0.224514
$569$	5.71545	0.239604	0.119802	−	0.992798i	$-0.461774\pi$
$569$	5.71545	0.239604	0.119802	+	0.992798i	$0.461774\pi$
$570$	0	0
$571$	20.7502	0.868370	0.434185	−	0.900824i	$-0.357036\pi$
$571$	20.7502	0.868370	0.434185	+	0.900824i	$0.357036\pi$
$572$	0.0281092	0.00117530
$573$	0	0
$574$	0	0
$575$	−36.4156	−1.51864
$576$	0	0
$577$	−3.83652	−0.159717	−0.0798583	−	0.996806i	$-0.525447\pi$
$577$	−3.83652	−0.159717	−0.0798583	+	0.996806i	$0.525447\pi$
$578$	−8.63314	−0.359091
$579$	0	0
$580$	0.0207054	0.000859745 0
$581$	0	0
$582$	0	0
$583$	−3.88792	−0.161021
$584$	−2.89453	−0.119777
$585$	0	0
$586$	−5.60647	−0.231601
$587$	36.8951	1.52282	0.761412	−	0.648269i	$-0.224507\pi$
$587$	36.8951	1.52282	0.761412	+	0.648269i	$0.224507\pi$
$588$	0	0
$589$	−3.96455	−0.163356
$590$	−1.95009	−0.0802841
$591$	0	0
$592$	−38.9139	−1.59935
$593$	−37.4036	−1.53598	−0.767990	−	0.640462i	$-0.778743\pi$
$593$	−37.4036	−1.53598	−0.767990	+	0.640462i	$0.778743\pi$
$594$	0	0
$595$	0	0
$596$	0.222293	0.00910549
$597$	0	0
$598$	−36.1687	−1.47905
$599$	14.1382	0.577672	0.288836	−	0.957379i	$-0.406732\pi$
$599$	14.1382	0.577672	0.288836	+	0.957379i	$0.406732\pi$
$600$	0	0
$601$	−42.8102	−1.74627	−0.873133	−	0.487482i	$-0.837916\pi$
$601$	−42.8102	−1.74627	−0.873133	+	0.487482i	$0.837916\pi$
$602$	0	0
$603$	0	0
$604$	0.0561151	0.00228329
$605$	−3.21223	−0.130596
$606$	0	0
$607$	−13.3315	−0.541107	−0.270554	−	0.962705i	$-0.587207\pi$
$607$	−13.3315	−0.541107	−0.270554	+	0.962705i	$0.587207\pi$
$608$	−0.123026	−0.00498937
$609$	0	0
$610$	−2.95267	−0.119550
$611$	−23.6634	−0.957317
$612$	0	0
$613$	3.39651	0.137184	0.0685919	−	0.997645i	$-0.478149\pi$
$613$	3.39651	0.137184	0.0685919	+	0.997645i	$0.478149\pi$
$614$	−25.1989	−1.01694
$615$	0	0
$616$	0	0
$617$	26.1522	1.05285	0.526424	−	0.850222i	$-0.323533\pi$
$617$	26.1522	1.05285	0.526424	+	0.850222i	$0.323533\pi$
$618$	0	0
$619$	−46.8265	−1.88211	−0.941057	−	0.338247i	$-0.890166\pi$
$619$	−46.8265	−1.88211	−0.941057	+	0.338247i	$0.890166\pi$
$620$	−0.0255014	−0.00102416
$621$	0	0
$622$	26.5081	1.06288
$623$	0	0
$624$	0	0
$625$	23.6956	0.947824
$626$	26.4609	1.05759
$627$	0	0
$628$	−0.469785	−0.0187465
$629$	32.4190	1.29263
$630$	0	0
$631$	12.1061	0.481935	0.240968	−	0.970533i	$-0.422535\pi$
$631$	12.1061	0.481935	0.240968	+	0.970533i	$0.422535\pi$
$632$	30.1581	1.19963
$633$	0	0
$634$	−19.8102	−0.786765
$635$	−0.988842	−0.0392410
$636$	0	0
$637$	0	0
$638$	−1.68675	−0.0667791
$639$	0	0
$640$	3.29044	0.130066
$641$	−17.4856	−0.690638	−0.345319	−	0.938485i	$-0.612229\pi$
$641$	−17.4856	−0.690638	−0.345319	+	0.938485i	$0.612229\pi$
$642$	0	0
$643$	−0.553783	−0.0218391	−0.0109195	−	0.999940i	$-0.503476\pi$
$643$	−0.553783	−0.0218391	−0.0109195	+	0.999940i	$0.503476\pi$
$644$	0	0
$645$	0	0
$646$	−4.63549	−0.182381
$647$	13.1248	0.515991	0.257995	−	0.966146i	$-0.416938\pi$
$647$	13.1248	0.515991	0.257995	+	0.966146i	$0.416938\pi$
$648$	0	0
$649$	−1.74656	−0.0685585
$650$	23.9693	0.940154
$651$	0	0
$652$	0.336468	0.0131771
$653$	−3.42592	−0.134067	−0.0670333	−	0.997751i	$-0.521353\pi$
$653$	−3.42592	−0.134067	−0.0670333	+	0.997751i	$0.521353\pi$
$654$	0	0
$655$	−4.64719	−0.181581
$656$	−11.1283	−0.434488
$657$	0	0
$658$	0	0
$659$	−3.34771	−0.130408	−0.0652041	−	0.997872i	$-0.520770\pi$
$659$	−3.34771	−0.130408	−0.0652041	+	0.997872i	$0.520770\pi$
$660$	0	0
$661$	−25.2423	−0.981810	−0.490905	−	0.871213i	$-0.663334\pi$
$661$	−25.2423	−0.981810	−0.490905	+	0.871213i	$0.663334\pi$
$662$	10.0309	0.389863
$663$	0	0
$664$	35.2486	1.36791
$665$	0	0
$666$	0	0
$667$	−23.8614	−0.923916
$668$	−0.173263	−0.00670376
$669$	0	0
$670$	−3.18888	−0.123197
$671$	−2.64450	−0.102090
$672$	0	0
$673$	−5.61847	−0.216576	−0.108288	−	0.994120i	$-0.534537\pi$
$673$	−5.61847	−0.216576	−0.108288	+	0.994120i	$0.534537\pi$
$674$	−9.07693	−0.349630
$675$	0	0
$676$	0.0210039	0.000807843 0
$677$	−51.6090	−1.98349	−0.991747	−	0.128209i	$-0.959077\pi$
$677$	−51.6090	−1.98349	−0.991747	+	0.128209i	$0.959077\pi$
$678$	0	0
$679$	0	0
$680$	−2.77173	−0.106291
$681$	0	0
$682$	2.07745	0.0795498
$683$	−16.4961	−0.631207	−0.315603	−	0.948891i	$-0.602207\pi$
$683$	−16.4961	−0.631207	−0.315603	+	0.948891i	$0.602207\pi$
$684$	0	0
$685$	−4.02175	−0.153663
$686$	0	0
$687$	0	0
$688$	24.0637	0.917418
$689$	36.2018	1.37918
$690$	0	0
$691$	−11.7493	−0.446965	−0.223482	−	0.974708i	$-0.571743\pi$
$691$	−11.7493	−0.446965	−0.223482	+	0.974708i	$0.571743\pi$
$692$	0.122127	0.00464257
$693$	0	0
$694$	−6.85290	−0.260133
$695$	6.30897	0.239313
$696$	0	0
$697$	9.27097	0.351163
$698$	−15.8037	−0.598180
$699$	0	0
$700$	0	0
$701$	28.7953	1.08758	0.543792	−	0.839220i	$-0.316988\pi$
$701$	28.7953	1.08758	0.543792	+	0.839220i	$0.316988\pi$
$702$	0	0
$703$	−9.83661	−0.370995
$704$	3.01219	0.113526
$705$	0	0
$706$	8.72187	0.328252
$707$	0	0
$708$	0	0
$709$	18.1688	0.682345	0.341172	−	0.940001i	$-0.389176\pi$
$709$	18.1688	0.682345	0.341172	+	0.940001i	$0.389176\pi$
$710$	−0.782743	−0.0293758
$711$	0	0
$712$	17.5893	0.659185
$713$	29.3884	1.10060
$714$	0	0
$715$	−0.382239	−0.0142949
$716$	0.161750	0.00604489
$717$	0	0
$718$	−27.4164	−1.02317
$719$	34.2823	1.27851	0.639256	−	0.768994i	$-0.279242\pi$
$719$	34.2823	1.27851	0.639256	+	0.768994i	$0.279242\pi$
$720$	0	0
$721$	0	0
$722$	1.40650	0.0523446
$723$	0	0
$724$	−0.195635	−0.00727071
$725$	15.8132	0.587286
$726$	0	0
$727$	13.1214	0.486646	0.243323	−	0.969945i	$-0.421763\pi$
$727$	13.1214	0.486646	0.243323	+	0.969945i	$0.421763\pi$
$728$	0	0
$729$	0	0
$730$	0.423428	0.0156718
$731$	−20.0473	−0.741478
$732$	0	0
$733$	6.06773	0.224117	0.112058	−	0.993702i	$-0.464256\pi$
$733$	6.06773	0.224117	0.112058	+	0.993702i	$0.464256\pi$
$734$	−32.7997	−1.21066
$735$	0	0
$736$	0.911968	0.0336156
$737$	−2.85605	−0.105204
$738$	0	0
$739$	23.7399	0.873288	0.436644	−	0.899634i	$-0.356167\pi$
$739$	23.7399	0.873288	0.436644	+	0.899634i	$0.356167\pi$
$740$	−0.0632726	−0.00232595
$741$	0	0
$742$	0	0
$743$	33.1525	1.21625	0.608124	−	0.793842i	$-0.291922\pi$
$743$	33.1525	1.21625	0.608124	+	0.793842i	$0.291922\pi$
$744$	0	0
$745$	−3.02283	−0.110748
$746$	−0.354003	−0.0129610
$747$	0	0
$748$	−0.0267050	−0.000976433 0
$749$	0	0
$750$	0	0
$751$	37.5315	1.36954	0.684772	−	0.728757i	$-0.259902\pi$
$751$	37.5315	1.36954	0.684772	+	0.728757i	$0.259902\pi$
$752$	−26.9852	−0.984050
$753$	0	0
$754$	15.7059	0.571977
$755$	−0.763074	−0.0277711
$756$	0	0
$757$	14.2165	0.516708	0.258354	−	0.966050i	$-0.416820\pi$
$757$	14.2165	0.516708	0.258354	+	0.966050i	$0.416820\pi$
$758$	−8.76896	−0.318503
$759$	0	0
$760$	0.841001	0.0305063
$761$	51.4365	1.86457	0.932285	−	0.361725i	$-0.117812\pi$
$761$	51.4365	1.86457	0.932285	+	0.361725i	$0.117812\pi$
$762$	0	0
$763$	0	0
$764$	0.481905	0.0174347
$765$	0	0
$766$	−16.1776	−0.584519
$767$	16.2628	0.587217
$768$	0	0
$769$	−29.5161	−1.06438	−0.532188	−	0.846626i	$-0.678630\pi$
$769$	−29.5161	−1.06438	−0.532188	+	0.846626i	$0.678630\pi$
$770$	0	0
$771$	0	0
$772$	−0.00886077	−0.000318906 0
$773$	−47.3158	−1.70183	−0.850916	−	0.525301i	$-0.823953\pi$
$773$	−47.3158	−1.70183	−0.850916	+	0.525301i	$0.823953\pi$
$774$	0	0
$775$	−19.4760	−0.699597
$776$	−0.668633	−0.0240025
$777$	0	0
$778$	48.9580	1.75523
$779$	−2.81301	−0.100786
$780$	0	0
$781$	−0.701047	−0.0250854
$782$	34.3619	1.22878
$783$	0	0
$784$	0	0
$785$	6.38830	0.228008
$786$	0	0
$787$	20.6740	0.736948	0.368474	−	0.929638i	$-0.379880\pi$
$787$	20.6740	0.736948	0.368474	+	0.929638i	$0.379880\pi$
$788$	−0.446299	−0.0158987
$789$	0	0
$790$	−4.41170	−0.156961
$791$	0	0
$792$	0	0
$793$	24.6238	0.874418
$794$	−41.0029	−1.45514
$795$	0	0
$796$	−0.152584	−0.00540819
$797$	−19.0955	−0.676398	−0.338199	−	0.941075i	$-0.609818\pi$
$797$	−19.0955	−0.676398	−0.338199	+	0.941075i	$0.609818\pi$
$798$	0	0
$799$	22.4813	0.795331
$800$	−0.604370	−0.0213677
$801$	0	0
$802$	20.4992	0.723851
$803$	0.379234	0.0133829
$804$	0	0
$805$	0	0
$806$	−19.3439	−0.681360
$807$	0	0
$808$	−2.19363	−0.0771717
$809$	−7.56887	−0.266107	−0.133054	−	0.991109i	$-0.542478\pi$
$809$	−7.56887	−0.266107	−0.133054	+	0.991109i	$0.542478\pi$
$810$	0	0
$811$	−10.5995	−0.372200	−0.186100	−	0.982531i	$-0.559585\pi$
$811$	−10.5995	−0.372200	−0.186100	+	0.982531i	$0.559585\pi$
$812$	0	0
$813$	0	0
$814$	5.15446	0.180664
$815$	−4.57541	−0.160270
$816$	0	0
$817$	6.08278	0.212810
$818$	13.5076	0.472283
$819$	0	0
$820$	−0.0180943	−0.000631879 0
$821$	−5.17484	−0.180603	−0.0903016	−	0.995914i	$-0.528783\pi$
$821$	−5.17484	−0.180603	−0.0903016	+	0.995914i	$0.528783\pi$
$822$	0	0
$823$	−11.4852	−0.400348	−0.200174	−	0.979760i	$-0.564151\pi$
$823$	−11.4852	−0.400348	−0.200174	+	0.979760i	$0.564151\pi$
$824$	30.1965	1.05195
$825$	0	0
$826$	0	0
$827$	4.29977	0.149518	0.0747588	−	0.997202i	$-0.476181\pi$
$827$	4.29977	0.149518	0.0747588	+	0.997202i	$0.476181\pi$
$828$	0	0
$829$	−56.5964	−1.96567	−0.982837	−	0.184474i	$-0.940942\pi$
$829$	−56.5964	−1.96567	−0.982837	+	0.184474i	$0.940942\pi$
$830$	−5.15635	−0.178980
$831$	0	0
$832$	−28.0475	−0.972374
$833$	0	0
$834$	0	0
$835$	2.35610	0.0815362
$836$	0.00810286	0.000280243 0
$837$	0	0
$838$	−24.2483	−0.837642
$839$	−8.94673	−0.308875	−0.154438	−	0.988003i	$-0.549357\pi$
$839$	−8.94673	−0.308875	−0.154438	+	0.988003i	$0.549357\pi$
$840$	0	0
$841$	−18.6384	−0.642703
$842$	21.5938	0.744172
$843$	0	0
$844$	0.331143	0.0113984
$845$	−0.285619	−0.00982558
$846$	0	0
$847$	0	0
$848$	41.2838	1.41769
$849$	0	0
$850$	−22.7720	−0.781072
$851$	72.9168	2.49956
$852$	0	0
$853$	21.1366	0.723702	0.361851	−	0.932236i	$-0.382145\pi$
$853$	21.1366	0.723702	0.361851	+	0.932236i	$0.382145\pi$
$854$	0	0
$855$	0	0
$856$	23.1697	0.791924
$857$	57.6877	1.97058	0.985288	−	0.170903i	$-0.0546683\pi$
$857$	57.6877	1.97058	0.985288	+	0.170903i	$0.0546683\pi$
$858$	0	0
$859$	−28.3309	−0.966638	−0.483319	−	0.875444i	$-0.660569\pi$
$859$	−28.3309	−0.966638	−0.483319	+	0.875444i	$0.660569\pi$
$860$	0.0391266	0.00133421
$861$	0	0
$862$	8.41578	0.286643
$863$	−8.74556	−0.297702	−0.148851	−	0.988860i	$-0.547558\pi$
$863$	−8.74556	−0.297702	−0.148851	+	0.988860i	$0.547558\pi$
$864$	0	0
$865$	−1.66073	−0.0564664
$866$	21.4412	0.728602
$867$	0	0
$868$	0	0
$869$	−3.95124	−0.134037
$870$	0	0
$871$	26.5937	0.901093
$872$	−19.0603	−0.645463
$873$	0	0
$874$	−10.4261	−0.352669
$875$	0	0
$876$	0	0
$877$	33.7988	1.14131	0.570653	−	0.821191i	$-0.306690\pi$
$877$	33.7988	1.14131	0.570653	+	0.821191i	$0.306690\pi$
$878$	44.9378	1.51658
$879$	0	0
$880$	−0.435898	−0.0146941
$881$	−7.64434	−0.257544	−0.128772	−	0.991674i	$-0.541104\pi$
$881$	−7.64434	−0.257544	−0.128772	+	0.991674i	$0.541104\pi$
$882$	0	0
$883$	35.1172	1.18179	0.590894	−	0.806749i	$-0.298775\pi$
$883$	35.1172	1.18179	0.590894	+	0.806749i	$0.298775\pi$
$884$	0.248660	0.00836334
$885$	0	0
$886$	−47.4608	−1.59448
$887$	38.9233	1.30692	0.653459	−	0.756962i	$-0.273317\pi$
$887$	38.9233	1.30692	0.653459	+	0.756962i	$0.273317\pi$
$888$	0	0
$889$	0	0
$890$	−2.57305	−0.0862489
$891$	0	0
$892$	−0.000935303 0	−3.13163e−5 0
$893$	−6.82129	−0.228266
$894$	0	0
$895$	−2.19954	−0.0735225
$896$	0	0
$897$	0	0
$898$	−29.7804	−0.993786
$899$	−12.7617	−0.425625
$900$	0	0
$901$	−34.3934	−1.14581
$902$	1.47404	0.0490801
$903$	0	0
$904$	44.3788	1.47602
$905$	2.66031	0.0884318
$906$	0	0
$907$	−34.8962	−1.15871	−0.579355	−	0.815075i	$-0.696696\pi$
$907$	−34.8962	−1.15871	−0.579355	+	0.815075i	$0.696696\pi$
$908$	−0.342258	−0.0113582
$909$	0	0
$910$	0	0
$911$	−15.0789	−0.499588	−0.249794	−	0.968299i	$-0.580363\pi$
$911$	−15.0789	−0.499588	−0.249794	+	0.968299i	$0.580363\pi$
$912$	0	0
$913$	−4.61818	−0.152839
$914$	−11.3365	−0.374980
$915$	0	0
$916$	−0.230024	−0.00760021
$917$	0	0
$918$	0	0
$919$	−39.1404	−1.29112	−0.645562	−	0.763708i	$-0.723377\pi$
$919$	−39.1404	−1.29112	−0.645562	+	0.763708i	$0.723377\pi$
$920$	−6.23417	−0.205534
$921$	0	0
$922$	17.5316	0.577372
$923$	6.52770	0.214862
$924$	0	0
$925$	−48.3226	−1.58884
$926$	20.8729	0.685925
$927$	0	0
$928$	−0.396014	−0.0129998
$929$	45.5626	1.49486	0.747430	−	0.664341i	$-0.231288\pi$
$929$	45.5626	1.49486	0.747430	+	0.664341i	$0.231288\pi$
$930$	0	0
$931$	0	0
$932$	−0.0114509	−0.000375088 0
$933$	0	0
$934$	−3.51191	−0.114913
$935$	0.363145	0.0118761
$936$	0	0
$937$	34.5829	1.12977	0.564887	−	0.825168i	$-0.308920\pi$
$937$	34.5829	1.12977	0.564887	+	0.825168i	$0.308920\pi$
$938$	0	0
$939$	0	0
$940$	−0.0438770	−0.00143111
$941$	37.4816	1.22186	0.610932	−	0.791683i	$-0.290795\pi$
$941$	37.4816	1.22186	0.610932	+	0.791683i	$0.290795\pi$
$942$	0	0
$943$	20.8522	0.679042
$944$	18.5458	0.603615
$945$	0	0
$946$	−3.18742	−0.103632
$947$	31.6815	1.02951	0.514755	−	0.857337i	$-0.327883\pi$
$947$	31.6815	1.02951	0.514755	+	0.857337i	$0.327883\pi$
$948$	0	0
$949$	−3.53118	−0.114627
$950$	6.90949	0.224173
$951$	0	0
$952$	0	0
$953$	−48.2237	−1.56212	−0.781060	−	0.624456i	$-0.785321\pi$
$953$	−48.2237	−1.56212	−0.781060	+	0.624456i	$0.785321\pi$
$954$	0	0
$955$	−6.55312	−0.212054
$956$	−0.0993864	−0.00321439
$957$	0	0
$958$	−22.1051	−0.714182
$959$	0	0
$960$	0	0
$961$	−15.2824	−0.492979
$962$	−47.9950	−1.54742
$963$	0	0
$964$	−0.497176	−0.0160130
$965$	0.120492	0.00387877
$966$	0	0
$967$	33.3518	1.07252	0.536260	−	0.844053i	$-0.319836\pi$
$967$	33.3518	1.07252	0.536260	+	0.844053i	$0.319836\pi$
$968$	30.8849	0.992677
$969$	0	0
$970$	0.0978112	0.00314053
$971$	−52.4761	−1.68404	−0.842019	−	0.539448i	$-0.818633\pi$
$971$	−52.4761	−1.68404	−0.842019	+	0.539448i	$0.818633\pi$
$972$	0	0
$973$	0	0
$974$	20.9019	0.669740
$975$	0	0
$976$	28.0805	0.898836
$977$	−21.5232	−0.688588	−0.344294	−	0.938862i	$-0.611882\pi$
$977$	−21.5232	−0.688588	−0.344294	+	0.938862i	$0.611882\pi$
$978$	0	0
$979$	−2.30450	−0.0736521
$980$	0	0
$981$	0	0
$982$	3.92900	0.125380
$983$	−9.88684	−0.315341	−0.157671	−	0.987492i	$-0.550398\pi$
$983$	−9.88684	−0.315341	−0.157671	+	0.987492i	$0.550398\pi$
$984$	0	0
$985$	6.06893	0.193372
$986$	−14.9214	−0.475193
$987$	0	0
$988$	−0.0754486	−0.00240034
$989$	−45.0904	−1.43379
$990$	0	0
$991$	25.5753	0.812426	0.406213	−	0.913778i	$-0.366849\pi$
$991$	25.5753	0.812426	0.406213	+	0.913778i	$0.366849\pi$
$992$	0.487743	0.0154859
$993$	0	0
$994$	0	0
$995$	2.07489	0.0657784
$996$	0	0
$997$	5.06941	0.160550	0.0802750	−	0.996773i	$-0.474420\pi$
$997$	5.06941	0.160550	0.0802750	+	0.996773i	$0.474420\pi$
$998$	43.6982	1.38324
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.ck.1.6		7
3.2	odd	2		931.2.a.o.1.2		7
7.3	odd	6		1197.2.j.l.856.2		14
7.5	odd	6		1197.2.j.l.172.2		14
7.6	odd	2		8379.2.a.cl.1.6		7
21.2	odd	6		931.2.f.p.704.6		14
21.5	even	6		133.2.f.d.39.6	✓	14
21.11	odd	6		931.2.f.p.324.6		14
21.17	even	6		133.2.f.d.58.6	yes	14
21.20	even	2		931.2.a.n.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.f.d.39.6	✓	14	21.5	even	6
133.2.f.d.58.6	yes	14	21.17	even	6
931.2.a.n.1.2		7	21.20	even	2
931.2.a.o.1.2		7	3.2	odd	2
931.2.f.p.324.6		14	21.11	odd	6
931.2.f.p.704.6		14	21.2	odd	6
1197.2.j.l.172.2		14	7.5	odd	6
1197.2.j.l.856.2		14	7.3	odd	6
8379.2.a.ck.1.6		7	1.1	even	1	trivial
8379.2.a.cl.1.6		7	7.6	odd	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:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+1.40650 q^{2} -0.0217491 q^{4} +0.295752 q^{5} -2.84360 q^{8} +O(q^{10})\)	\(q+1.40650 q^{2} -0.0217491 q^{4} +0.295752 q^{5} -2.84360 q^{8} +0.415977 q^{10} +0.372561 q^{11} -3.46904 q^{13} -3.95603 q^{16} +3.29575 q^{17} -1.00000 q^{19} -0.00643235 q^{20} +0.524008 q^{22} +7.41280 q^{23} -4.91253 q^{25} -4.87922 q^{26} -3.21894 q^{29} +3.96455 q^{31} +0.123026 q^{32} +4.63549 q^{34} +9.83661 q^{37} -1.40650 q^{38} -0.841001 q^{40} +2.81301 q^{41} -6.08278 q^{43} -0.00810286 q^{44} +10.4261 q^{46} +6.82129 q^{47} -6.90949 q^{50} +0.0754486 q^{52} -10.4357 q^{53} +0.110186 q^{55} -4.52745 q^{58} -4.68799 q^{59} -7.09816 q^{61} +5.57615 q^{62} +8.08509 q^{64} -1.02598 q^{65} -7.66601 q^{67} -0.0716797 q^{68} -1.88170 q^{71} +1.01791 q^{73} +13.8352 q^{74} +0.0217491 q^{76} -10.6056 q^{79} -1.17001 q^{80} +3.95650 q^{82} -12.3958 q^{83} +0.974727 q^{85} -8.55545 q^{86} -1.05941 q^{88} -6.18557 q^{89} -0.161222 q^{92} +9.59417 q^{94} -0.295752 q^{95} +0.235136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 10 q^{4} - 2 q^{5} - 12 q^{8}+O(q^{10}) \) 7 * q - 2 * q^2 + 10 * q^4 - 2 * q^5 - 12 * q^8	\( 7 q - 2 q^{2} + 10 q^{4} - 2 q^{5} - 12 q^{8} - 7 q^{11} - 6 q^{13} + 24 q^{16} + 19 q^{17} - 7 q^{19} - 8 q^{20} - 6 q^{22} + q^{23} - 3 q^{25} + 12 q^{26} - 24 q^{29} - 26 q^{32} - 6 q^{34} + 8 q^{37} + 2 q^{38} + 10 q^{40} - 4 q^{41} + 4 q^{43} - 26 q^{44} - 16 q^{46} + 5 q^{47} - 16 q^{50} - 42 q^{52} - 20 q^{53} + 30 q^{55} - 16 q^{59} - 5 q^{61} + 24 q^{62} + 32 q^{64} - 26 q^{65} - 4 q^{67} + 22 q^{68} - 12 q^{71} - 3 q^{73} + 4 q^{74} - 10 q^{76} - 20 q^{79} - 4 q^{80} + 48 q^{82} - 11 q^{83} + 26 q^{85} - 36 q^{86} - 32 q^{88} - 10 q^{89} + 30 q^{92} - 16 q^{94} + 2 q^{95} + 4 q^{97}+O(q^{100}) \) 7 * q - 2 * q^2 + 10 * q^4 - 2 * q^5 - 12 * q^8 - 7 * q^11 - 6 * q^13 + 24 * q^16 + 19 * q^17 - 7 * q^19 - 8 * q^20 - 6 * q^22 + q^23 - 3 * q^25 + 12 * q^26 - 24 * q^29 - 26 * q^32 - 6 * q^34 + 8 * q^37 + 2 * q^38 + 10 * q^40 - 4 * q^41 + 4 * q^43 - 26 * q^44 - 16 * q^46 + 5 * q^47 - 16 * q^50 - 42 * q^52 - 20 * q^53 + 30 * q^55 - 16 * q^59 - 5 * q^61 + 24 * q^62 + 32 * q^64 - 26 * q^65 - 4 * q^67 + 22 * q^68 - 12 * q^71 - 3 * q^73 + 4 * q^74 - 10 * q^76 - 20 * q^79 - 4 * q^80 + 48 * q^82 - 11 * q^83 + 26 * q^85 - 36 * q^86 - 32 * q^88 - 10 * q^89 + 30 * q^92 - 16 * q^94 + 2 * q^95 + 4 * q^97

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