LMFDB - Embedded newform 8379.2.a.bq.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [3,-1,0,9,0,0,0,9,0,-10,-4,0,-2,0,0,13,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.65544$ 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)

$f(q)$	$=$	$q-1.39593 q^{2} -0.0513742 q^{4} +3.05137 q^{5} +2.86358 q^{8} -4.25951 q^{10} -5.31088 q^{11} +3.31088 q^{13} -3.89461 q^{16} +0.948626 q^{17} +1.00000 q^{19} -0.156762 q^{20} +7.41363 q^{22} -1.31088 q^{23} +4.31088 q^{25} -4.62177 q^{26} +7.84324 q^{29} +4.79186 q^{31} -0.290544 q^{32} -1.32422 q^{34} -8.62177 q^{37} -1.39593 q^{38} +8.73785 q^{40} +11.3109 q^{41} +3.20814 q^{43} +0.272843 q^{44} +1.82991 q^{46} -5.84324 q^{47} -6.01770 q^{50} -0.170094 q^{52} +2.77853 q^{53} -16.2055 q^{55} -10.9486 q^{58} -8.20550 q^{59} -12.6218 q^{61} -6.68912 q^{62} +8.19480 q^{64} +10.1027 q^{65} -4.51902 q^{67} -0.0487349 q^{68} +13.5704 q^{71} +4.10275 q^{73} +12.0354 q^{74} -0.0513742 q^{76} -8.20550 q^{79} -11.8839 q^{80} -15.7892 q^{82} -6.36226 q^{83} +2.89461 q^{85} -4.47834 q^{86} -15.2081 q^{88} +15.5164 q^{89} +0.0673457 q^{92} +8.15676 q^{94} +3.05137 q^{95} -11.5837 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 9 q^{4} + 9 q^{8} - 10 q^{10} - 4 q^{11} - 2 q^{13} + 13 q^{16} + 12 q^{17} + 3 q^{19} - 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 10 q^{26} + 8 q^{29} + 8 q^{31} + 27 q^{32} + 6 q^{34}+ \cdots - 22 q^{97}+O(q^{100}) $ 3 * q - q^2 + 9 * q^4 + 9 * q^8 - 10 * q^10 - 4 * q^11 - 2 * q^13 + 13 * q^16 + 12 * q^17 + 3 * q^19 - 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 + 10 * q^26 + 8 * q^29 + 8 * q^31 + 27 * q^32 + 6 * q^34 - 2 * q^37 - q^38 - 14 * q^40 + 22 * q^41 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 9 * q^50 - 18 * q^52 - 12 * q^55 - 42 * q^58 + 12 * q^59 - 14 * q^61 - 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 + 52 * q^68 + 26 * q^71 - 6 * q^73 - 18 * q^74 + 9 * q^76 + 12 * q^79 - 52 * q^80 + 2 * q^82 + 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 - 2 * q^89 + 36 * q^92 + 40 * q^94 - 22 * q^97

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 $

$2$	−1.39593	−0.987073	−0.493536	−	0.869725i	$-0.664296\pi$
$2$	−1.39593	−0.987073	−0.493536	+	0.869725i	$0.664296\pi$
$3$	0	0
$3$	0	0
$4$	−0.0513742	−0.0256871
$5$	3.05137	1.36462	0.682308	−	0.731065i	$-0.260976\pi$
$5$	3.05137	1.36462	0.682308	+	0.731065i	$0.260976\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.86358	1.01243
$9$	0	0
$10$	−4.25951	−1.34698
$11$	−5.31088	−1.60129	−0.800646	−	0.599138i	$-0.795510\pi$
$11$	−5.31088	−1.60129	−0.800646	+	0.599138i	$0.795510\pi$
$12$	0	0
$13$	3.31088	0.918274	0.459137	−	0.888365i	$-0.348159\pi$
$13$	3.31088	0.918274	0.459137	+	0.888365i	$0.348159\pi$
$14$	0	0
$15$	0	0
$16$	−3.89461	−0.973653
$17$	0.948626	0.230076	0.115038	−	0.993361i	$-0.463301\pi$
$17$	0.948626	0.230076	0.115038	+	0.993361i	$0.463301\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.156762	−0.0350531
$21$	0	0
$22$	7.41363	1.58059
$23$	−1.31088	−0.273338	−0.136669	−	0.990617i	$-0.543640\pi$
$23$	−1.31088	−0.273338	−0.136669	+	0.990617i	$0.543640\pi$
$24$	0	0
$25$	4.31088	0.862177
$26$	−4.62177	−0.906404
$27$	0	0
$28$	0	0
$29$	7.84324	1.45645	0.728226	−	0.685337i	$-0.240345\pi$
$29$	7.84324	1.45645	0.728226	+	0.685337i	$0.240345\pi$
$30$	0	0
$31$	4.79186	0.860644	0.430322	−	0.902675i	$-0.358400\pi$
$31$	4.79186	0.860644	0.430322	+	0.902675i	$0.358400\pi$
$32$	−0.290544	−0.0513614
$33$	0	0
$34$	−1.32422	−0.227101
$35$	0	0
$36$	0	0
$37$	−8.62177	−1.41741	−0.708705	−	0.705505i	$-0.750720\pi$
$37$	−8.62177	−1.41741	−0.708705	+	0.705505i	$0.750720\pi$
$38$	−1.39593	−0.226450
$39$	0	0
$40$	8.73785	1.38158
$41$	11.3109	1.76646	0.883232	−	0.468937i	$-0.155363\pi$
$41$	11.3109	1.76646	0.883232	+	0.468937i	$0.155363\pi$
$42$	0	0
$43$	3.20814	0.489236	0.244618	−	0.969620i	$-0.421337\pi$
$43$	3.20814	0.489236	0.244618	+	0.969620i	$0.421337\pi$
$44$	0.272843	0.0411326
$45$	0	0
$46$	1.82991	0.269805
$47$	−5.84324	−0.852324	−0.426162	−	0.904647i	$-0.640135\pi$
$47$	−5.84324	−0.852324	−0.426162	+	0.904647i	$0.640135\pi$
$48$	0	0
$49$	0	0
$50$	−6.01770	−0.851031
$51$	0	0
$52$	−0.170094	−0.0235878
$53$	2.77853	0.381661	0.190830	−	0.981623i	$-0.438882\pi$
$53$	2.77853	0.381661	0.190830	+	0.981623i	$0.438882\pi$
$54$	0	0
$55$	−16.2055	−2.18515
$56$	0	0
$57$	0	0
$58$	−10.9486	−1.43762
$59$	−8.20550	−1.06826	−0.534132	−	0.845401i	$-0.679362\pi$
$59$	−8.20550	−1.06826	−0.534132	+	0.845401i	$0.679362\pi$
$60$	0	0
$61$	−12.6218	−1.61605	−0.808026	−	0.589147i	$-0.799464\pi$
$61$	−12.6218	−1.61605	−0.808026	+	0.589147i	$0.799464\pi$
$62$	−6.68912	−0.849518
$63$	0	0
$64$	8.19480	1.02435
$65$	10.1027	1.25309
$66$	0	0
$67$	−4.51902	−0.552086	−0.276043	−	0.961145i	$-0.589023\pi$
$67$	−4.51902	−0.552086	−0.276043	+	0.961145i	$0.589023\pi$
$68$	−0.0487349	−0.00590998
$69$	0	0
$70$	0	0
$71$	13.5704	1.61051	0.805255	−	0.592929i	$-0.202028\pi$
$71$	13.5704	1.61051	0.805255	+	0.592929i	$0.202028\pi$
$72$	0	0
$73$	4.10275	0.480190	0.240095	−	0.970749i	$-0.422821\pi$
$73$	4.10275	0.480190	0.240095	+	0.970749i	$0.422821\pi$
$74$	12.0354	1.39909
$75$	0	0
$76$	−0.0513742	−0.00589303
$77$	0	0
$78$	0	0
$79$	−8.20550	−0.923191	−0.461595	−	0.887091i	$-0.652723\pi$
$79$	−8.20550	−0.923191	−0.461595	+	0.887091i	$0.652723\pi$
$80$	−11.8839	−1.32866
$81$	0	0
$82$	−15.7892	−1.74363
$83$	−6.36226	−0.698349	−0.349174	−	0.937058i	$-0.613538\pi$
$83$	−6.36226	−0.698349	−0.349174	+	0.937058i	$0.613538\pi$
$84$	0	0
$85$	2.89461	0.313965
$86$	−4.47834	−0.482912
$87$	0	0
$88$	−15.2081	−1.62119
$89$	15.5164	1.64473	0.822367	−	0.568958i	$-0.192653\pi$
$89$	15.5164	1.64473	0.822367	+	0.568958i	$0.192653\pi$
$90$	0	0
$91$	0	0
$92$	0.0673457	0.00702128
$93$	0	0
$94$	8.15676	0.841306
$95$	3.05137	0.313064
$96$	0	0
$97$	−11.5837	−1.17615	−0.588075	−	0.808807i	$-0.700114\pi$
$97$	−11.5837	−1.17615	−0.588075	+	0.808807i	$0.700114\pi$
$98$	0	0
$99$	0	0
$100$	−0.221468	−0.0221468
$101$	9.67314	0.962514	0.481257	−	0.876580i	$-0.340180\pi$
$101$	9.67314	0.962514	0.481257	+	0.876580i	$0.340180\pi$
$102$	0	0
$103$	2.62177	0.258331	0.129165	−	0.991623i	$-0.458770\pi$
$103$	2.62177	0.258331	0.129165	+	0.991623i	$0.458770\pi$
$104$	9.48098	0.929686
$105$	0	0
$106$	−3.87864	−0.376727
$107$	10.9486	1.05844	0.529222	−	0.848484i	$-0.322484\pi$
$107$	10.9486	1.05844	0.529222	+	0.848484i	$0.322484\pi$
$108$	0	0
$109$	18.7245	1.79348	0.896742	−	0.442554i	$-0.145928\pi$
$109$	18.7245	1.79348	0.896742	+	0.442554i	$0.145928\pi$
$110$	22.6218	2.15690
$111$	0	0
$112$	0	0
$113$	3.84324	0.361541	0.180771	−	0.983525i	$-0.442141\pi$
$113$	3.84324	0.361541	0.180771	+	0.983525i	$0.442141\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	−0.402940	−0.0374121
$117$	0	0
$118$	11.4543	1.05446
$119$	0	0
$120$	0	0
$121$	17.2055	1.56414
$122$	17.6191	1.59516
$123$	0	0
$124$	−0.246178	−0.0221075
$125$	−2.10275	−0.188076
$126$	0	0
$127$	7.48098	0.663830	0.331915	−	0.943309i	$-0.392305\pi$
$127$	7.48098	0.663830	0.331915	+	0.943309i	$0.392305\pi$
$128$	−10.8583	−0.959747
$129$	0	0
$130$	−14.1027	−1.23689
$131$	5.32422	0.465179	0.232589	−	0.972575i	$-0.425280\pi$
$131$	5.32422	0.465179	0.232589	+	0.972575i	$0.425280\pi$
$132$	0	0
$133$	0	0
$134$	6.30825	0.544949
$135$	0	0
$136$	2.71646	0.232935
$137$	−0.416273	−0.0355646	−0.0177823	−	0.999842i	$-0.505661\pi$
$137$	−0.416273	−0.0355646	−0.0177823	+	0.999842i	$0.505661\pi$
$138$	0	0
$139$	−10.1027	−0.856904	−0.428452	−	0.903565i	$-0.640941\pi$
$139$	−10.1027	−0.856904	−0.428452	+	0.903565i	$0.640941\pi$
$140$	0	0
$141$	0	0
$142$	−18.9433	−1.58969
$143$	−17.5837	−1.47043
$144$	0	0
$145$	23.9327	1.98750
$146$	−5.72716	−0.473983
$147$	0	0
$148$	0.442937	0.0364092
$149$	13.6865	1.12124	0.560620	−	0.828073i	$-0.310563\pi$
$149$	13.6865	1.12124	0.560620	+	0.828073i	$0.310563\pi$
$150$	0	0
$151$	−15.1408	−1.23214	−0.616070	−	0.787691i	$-0.711276\pi$
$151$	−15.1408	−1.23214	−0.616070	+	0.787691i	$0.711276\pi$
$152$	2.86358	0.232267
$153$	0	0
$154$	0	0
$155$	14.6218	1.17445
$156$	0	0
$157$	−6.20550	−0.495253	−0.247626	−	0.968856i	$-0.579651\pi$
$157$	−6.20550	−0.495253	−0.247626	+	0.968856i	$0.579651\pi$
$158$	11.4543	0.911256
$159$	0	0
$160$	−0.886559	−0.0700886
$161$	0	0
$162$	0	0
$163$	7.41363	0.580681	0.290340	−	0.956923i	$-0.406232\pi$
$163$	7.41363	0.580681	0.290340	+	0.956923i	$0.406232\pi$
$164$	−0.581088	−0.0453754
$165$	0	0
$166$	8.88128	0.689321
$167$	24.9300	1.92914	0.964571	−	0.263823i	$-0.0849833\pi$
$167$	24.9300	1.92914	0.964571	+	0.263823i	$0.0849833\pi$
$168$	0	0
$169$	−2.03804	−0.156772
$170$	−4.04068	−0.309906
$171$	0	0
$172$	−0.164816	−0.0125671
$173$	19.5164	1.48380	0.741902	−	0.670509i	$-0.233924\pi$
$173$	19.5164	1.48380	0.741902	+	0.670509i	$0.233924\pi$
$174$	0	0
$175$	0	0
$176$	20.6838	1.55910
$177$	0	0
$178$	−21.6598	−1.62347
$179$	−7.15412	−0.534724	−0.267362	−	0.963596i	$-0.586152\pi$
$179$	−7.15412	−0.534724	−0.267362	+	0.963596i	$0.586152\pi$
$180$	0	0
$181$	−15.5837	−1.15833	−0.579165	−	0.815211i	$-0.696621\pi$
$181$	−15.5837	−1.15833	−0.579165	+	0.815211i	$0.696621\pi$
$182$	0	0
$183$	0	0
$184$	−3.75382	−0.276735
$185$	−26.3082	−1.93422
$186$	0	0
$187$	−5.03804	−0.368418
$188$	0.300192	0.0218937
$189$	0	0
$190$	−4.25951	−0.309017
$191$	−13.5164	−0.978011	−0.489006	−	0.872281i	$-0.662640\pi$
$191$	−13.5164	−0.978011	−0.489006	+	0.872281i	$0.662640\pi$
$192$	0	0
$193$	7.03804	0.506609	0.253305	−	0.967387i	$-0.418482\pi$
$193$	7.03804	0.506609	0.253305	+	0.967387i	$0.418482\pi$
$194$	16.1701	1.16095
$195$	0	0
$196$	0	0
$197$	−19.7892	−1.40992	−0.704962	−	0.709245i	$-0.749036\pi$
$197$	−19.7892	−1.40992	−0.704962	+	0.709245i	$0.749036\pi$
$198$	0	0
$199$	11.6865	0.828432	0.414216	−	0.910179i	$-0.364056\pi$
$199$	11.6865	0.828432	0.414216	+	0.910179i	$0.364056\pi$
$200$	12.3446	0.872892
$201$	0	0
$202$	−13.5030	−0.950071
$203$	0	0
$204$	0	0
$205$	34.5137	2.41054
$206$	−3.65981	−0.254991
$207$	0	0
$208$	−12.8946	−0.894080
$209$	−5.31088	−0.367362
$210$	0	0
$211$	18.8273	1.29612	0.648061	−	0.761588i	$-0.275580\pi$
$211$	18.8273	1.29612	0.648061	+	0.761588i	$0.275580\pi$
$212$	−0.142745	−0.00980376
$213$	0	0
$214$	−15.2835	−1.04476
$215$	9.78922	0.667620
$216$	0	0
$217$	0	0
$218$	−26.1382	−1.77030
$219$	0	0
$220$	0.832545	0.0561302
$221$	3.14079	0.211272
$222$	0	0
$223$	5.82991	0.390399	0.195200	−	0.980764i	$-0.437465\pi$
$223$	5.82991	0.390399	0.195200	+	0.980764i	$0.437465\pi$
$224$	0	0
$225$	0	0
$226$	−5.36490	−0.356868
$227$	−5.06471	−0.336156	−0.168078	−	0.985774i	$-0.553756\pi$
$227$	−5.06471	−0.336156	−0.168078	+	0.985774i	$0.553756\pi$
$228$	0	0
$229$	21.6865	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$229$	21.6865	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$230$	5.58373	0.368180
$231$	0	0
$232$	22.4597	1.47455
$233$	12.3082	0.806340	0.403170	−	0.915125i	$-0.367908\pi$
$233$	12.3082	0.806340	0.403170	+	0.915125i	$0.367908\pi$
$234$	0	0
$235$	−17.8299	−1.16309
$236$	0.421551	0.0274406
$237$	0	0
$238$	0	0
$239$	−14.0354	−0.907875	−0.453937	−	0.891034i	$-0.649981\pi$
$239$	−14.0354	−0.907875	−0.453937	+	0.891034i	$0.649981\pi$
$240$	0	0
$241$	−3.58373	−0.230848	−0.115424	−	0.993316i	$-0.536823\pi$
$241$	−3.58373	−0.230848	−0.115424	+	0.993316i	$0.536823\pi$
$242$	−24.0177	−1.54392
$243$	0	0
$244$	0.648434	0.0415117
$245$	0	0
$246$	0	0
$247$	3.31088	0.210667
$248$	13.7219	0.871340
$249$	0	0
$250$	2.93529	0.185644
$251$	−15.0868	−0.952269	−0.476134	−	0.879372i	$-0.657962\pi$
$251$	−15.0868	−0.952269	−0.476134	+	0.879372i	$0.657962\pi$
$252$	0	0
$253$	6.96196	0.437695
$254$	−10.4429	−0.655248
$255$	0	0
$256$	−1.23216	−0.0770101
$257$	−18.4517	−1.15098	−0.575492	−	0.817807i	$-0.695189\pi$
$257$	−18.4517	−1.15098	−0.575492	+	0.817807i	$0.695189\pi$
$258$	0	0
$259$	0	0
$260$	−0.519021	−0.0321883
$261$	0	0
$262$	−7.43224	−0.459166
$263$	16.7919	1.03543	0.517715	−	0.855553i	$-0.326783\pi$
$263$	16.7919	1.03543	0.517715	+	0.855553i	$0.326783\pi$
$264$	0	0
$265$	8.47834	0.520820
$266$	0	0
$267$	0	0
$268$	0.232161	0.0141815
$269$	18.4517	1.12502	0.562509	−	0.826791i	$-0.309836\pi$
$269$	18.4517	1.12502	0.562509	+	0.826791i	$0.309836\pi$
$270$	0	0
$271$	−6.62177	−0.402244	−0.201122	−	0.979566i	$-0.564459\pi$
$271$	−6.62177	−0.402244	−0.201122	+	0.979566i	$0.564459\pi$
$272$	−3.69453	−0.224014
$273$	0	0
$274$	0.581088	0.0351048
$275$	−22.8946	−1.38060
$276$	0	0
$277$	−8.89461	−0.534425	−0.267213	−	0.963638i	$-0.586103\pi$
$277$	−8.89461	−0.534425	−0.267213	+	0.963638i	$0.586103\pi$
$278$	14.1027	0.845827
$279$	0	0
$280$	0	0
$281$	−17.7405	−1.05831	−0.529154	−	0.848526i	$-0.677491\pi$
$281$	−17.7405	−1.05831	−0.529154	+	0.848526i	$0.677491\pi$
$282$	0	0
$283$	−0.859209	−0.0510747	−0.0255373	−	0.999674i	$-0.508130\pi$
$283$	−0.859209	−0.0510747	−0.0255373	+	0.999674i	$0.508130\pi$
$284$	−0.697169	−0.0413694
$285$	0	0
$286$	24.5457	1.45142
$287$	0	0
$288$	0	0
$289$	−16.1001	−0.947065
$290$	−33.4084	−1.96181
$291$	0	0
$292$	−0.210776	−0.0123347
$293$	23.3109	1.36184	0.680918	−	0.732359i	$-0.261581\pi$
$293$	23.3109	1.36184	0.680918	+	0.732359i	$0.261581\pi$
$294$	0	0
$295$	−25.0380	−1.45777
$296$	−24.6891	−1.43503
$297$	0	0
$298$	−19.1054	−1.10675
$299$	−4.34019	−0.251000
$300$	0	0
$301$	0	0
$302$	21.1355	1.21621
$303$	0	0
$304$	−3.89461	−0.223371
$305$	−38.5137	−2.20529
$306$	0	0
$307$	−11.4136	−0.651410	−0.325705	−	0.945471i	$-0.605602\pi$
$307$	−11.4136	−0.651410	−0.325705	+	0.945471i	$0.605602\pi$
$308$	0	0
$309$	0	0
$310$	−20.4110	−1.15927
$311$	19.9460	1.13103	0.565517	−	0.824737i	$-0.308677\pi$
$311$	19.9460	1.13103	0.565517	+	0.824737i	$0.308677\pi$
$312$	0	0
$313$	13.1675	0.744269	0.372134	−	0.928179i	$-0.378626\pi$
$313$	13.1675	0.744269	0.372134	+	0.928179i	$0.378626\pi$
$314$	8.66245	0.488850
$315$	0	0
$316$	0.421551	0.0237141
$317$	26.4650	1.48642	0.743211	−	0.669057i	$-0.233302\pi$
$317$	26.4650	1.48642	0.743211	+	0.669057i	$0.233302\pi$
$318$	0	0
$319$	−41.6545	−2.33221
$320$	25.0054	1.39785
$321$	0	0
$322$	0	0
$323$	0.948626	0.0527829
$324$	0	0
$325$	14.2728	0.791715
$326$	−10.3489	−0.573174
$327$	0	0
$328$	32.3896	1.78842
$329$	0	0
$330$	0	0
$331$	−0.519021	−0.0285280	−0.0142640	−	0.999898i	$-0.504541\pi$
$331$	−0.519021	−0.0285280	−0.0142640	+	0.999898i	$0.504541\pi$
$332$	0.326856	0.0179386
$333$	0	0
$334$	−34.8006	−1.90420
$335$	−13.7892	−0.753386
$336$	0	0
$337$	9.48098	0.516462	0.258231	−	0.966083i	$-0.416860\pi$
$337$	9.48098	0.516462	0.258231	+	0.966083i	$0.416860\pi$
$338$	2.84497	0.154746
$339$	0	0
$340$	−0.148709	−0.00806485
$341$	−25.4490	−1.37814
$342$	0	0
$343$	0	0
$344$	9.18675	0.495316
$345$	0	0
$346$	−27.2435	−1.46462
$347$	31.4136	1.68637	0.843186	−	0.537622i	$-0.180677\pi$
$347$	31.4136	1.68637	0.843186	+	0.537622i	$0.180677\pi$
$348$	0	0
$349$	32.8273	1.75720	0.878602	−	0.477555i	$-0.158477\pi$
$349$	32.8273	1.75720	0.878602	+	0.477555i	$0.158477\pi$
$350$	0	0
$351$	0	0
$352$	1.54305	0.0822446
$353$	−7.25687	−0.386244	−0.193122	−	0.981175i	$-0.561861\pi$
$353$	−7.25687	−0.386244	−0.193122	+	0.981175i	$0.561861\pi$
$354$	0	0
$355$	41.4084	2.19773
$356$	−0.797142	−0.0422485
$357$	0	0
$358$	9.98667	0.527812
$359$	−22.5544	−1.19038	−0.595188	−	0.803586i	$-0.702923\pi$
$359$	−22.5544	−1.19038	−0.595188	+	0.803586i	$0.702923\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	21.7538	1.14336
$363$	0	0
$364$	0	0
$365$	12.5190	0.655276
$366$	0	0
$367$	−6.93529	−0.362019	−0.181010	−	0.983481i	$-0.557937\pi$
$367$	−6.93529	−0.362019	−0.181010	+	0.983481i	$0.557937\pi$
$368$	5.10539	0.266137
$369$	0	0
$370$	36.7245	1.90922
$371$	0	0
$372$	0	0
$373$	−21.1675	−1.09601	−0.548005	−	0.836475i	$-0.684612\pi$
$373$	−21.1675	−1.09601	−0.548005	+	0.836475i	$0.684612\pi$
$374$	7.03276	0.363656
$375$	0	0
$376$	−16.7326	−0.862916
$377$	25.9681	1.33742
$378$	0	0
$379$	7.34629	0.377353	0.188677	−	0.982039i	$-0.439580\pi$
$379$	7.34629	0.377353	0.188677	+	0.982039i	$0.439580\pi$
$380$	−0.156762	−0.00804172
$381$	0	0
$382$	18.8679	0.965368
$383$	−23.3463	−1.19294	−0.596470	−	0.802636i	$-0.703430\pi$
$383$	−23.3463	−1.19294	−0.596470	+	0.802636i	$0.703430\pi$
$384$	0	0
$385$	0	0
$386$	−9.82463	−0.500060
$387$	0	0
$388$	0.595105	0.0302119
$389$	−0.102748	−0.00520955	−0.00260478	−	0.999997i	$-0.500829\pi$
$389$	−0.102748	−0.00520955	−0.00260478	+	0.999997i	$0.500829\pi$
$390$	0	0
$391$	−1.24354	−0.0628885
$392$	0	0
$393$	0	0
$394$	27.6244	1.39170
$395$	−25.0380	−1.25980
$396$	0	0
$397$	−23.4490	−1.17687	−0.588437	−	0.808543i	$-0.700256\pi$
$397$	−23.4490	−1.17687	−0.588437	+	0.808543i	$0.700256\pi$
$398$	−16.3135	−0.817723
$399$	0	0
$400$	−16.7892	−0.839461
$401$	10.9840	0.548516	0.274258	−	0.961656i	$-0.411568\pi$
$401$	10.9840	0.548516	0.274258	+	0.961656i	$0.411568\pi$
$402$	0	0
$403$	15.8653	0.790307
$404$	−0.496950	−0.0247242
$405$	0	0
$406$	0	0
$407$	45.7892	2.26969
$408$	0	0
$409$	6.27284	0.310172	0.155086	−	0.987901i	$-0.450435\pi$
$409$	6.27284	0.310172	0.155086	+	0.987901i	$0.450435\pi$
$410$	−48.1788	−2.37938
$411$	0	0
$412$	−0.134691	−0.00663577
$413$	0	0
$414$	0	0
$415$	−19.4136	−0.952978
$416$	−0.961958	−0.0471639
$417$	0	0
$418$	7.41363	0.362613
$419$	−7.19480	−0.351489	−0.175745	−	0.984436i	$-0.556233\pi$
$419$	−7.19480	−0.351489	−0.175745	+	0.984436i	$0.556233\pi$
$420$	0	0
$421$	−20.8539	−1.01636	−0.508179	−	0.861251i	$-0.669681\pi$
$421$	−20.8539	−1.01636	−0.508179	+	0.861251i	$0.669681\pi$
$422$	−26.2816	−1.27937
$423$	0	0
$424$	7.95654	0.386404
$425$	4.08942	0.198366
$426$	0	0
$427$	0	0
$428$	−0.562477	−0.0271884
$429$	0	0
$430$	−13.6651	−0.658989
$431$	24.3977	1.17519	0.587597	−	0.809154i	$-0.300074\pi$
$431$	24.3977	1.17519	0.587597	+	0.809154i	$0.300074\pi$
$432$	0	0
$433$	−18.0000	−0.865025	−0.432512	−	0.901628i	$-0.642373\pi$
$433$	−18.0000	−0.865025	−0.432512	+	0.901628i	$0.642373\pi$
$434$	0	0
$435$	0	0
$436$	−0.961958	−0.0460694
$437$	−1.31088	−0.0627081
$438$	0	0
$439$	23.9593	1.14352	0.571758	−	0.820422i	$-0.306262\pi$
$439$	23.9593	1.14352	0.571758	+	0.820422i	$0.306262\pi$
$440$	−46.4057	−2.21231
$441$	0	0
$442$	−4.38433	−0.208541
$443$	28.6572	1.36154	0.680772	−	0.732496i	$-0.261645\pi$
$443$	28.6572	1.36154	0.680772	+	0.732496i	$0.261645\pi$
$444$	0	0
$445$	47.3463	2.24443
$446$	−8.13815	−0.385353
$447$	0	0
$448$	0	0
$449$	−16.6758	−0.786979	−0.393489	−	0.919329i	$-0.628732\pi$
$449$	−16.6758	−0.786979	−0.393489	+	0.919329i	$0.628732\pi$
$450$	0	0
$451$	−60.0708	−2.82862
$452$	−0.197443	−0.00928696
$453$	0	0
$454$	7.06999	0.331811
$455$	0	0
$456$	0	0
$457$	−27.3109	−1.27755	−0.638775	−	0.769394i	$-0.720558\pi$
$457$	−27.3109	−1.27755	−0.638775	+	0.769394i	$0.720558\pi$
$458$	−30.2728	−1.41456
$459$	0	0
$460$	0.205497	0.00958135
$461$	10.7112	0.498870	0.249435	−	0.968392i	$-0.419755\pi$
$461$	10.7112	0.498870	0.249435	+	0.968392i	$0.419755\pi$
$462$	0	0
$463$	−28.2055	−1.31082	−0.655410	−	0.755273i	$-0.727504\pi$
$463$	−28.2055	−1.31082	−0.655410	+	0.755273i	$0.727504\pi$
$464$	−30.5464	−1.41808
$465$	0	0
$466$	−17.1815	−0.795916
$467$	−12.2595	−0.567302	−0.283651	−	0.958928i	$-0.591546\pi$
$467$	−12.2595	−0.567302	−0.283651	+	0.958928i	$0.591546\pi$
$468$	0	0
$469$	0	0
$470$	24.8893	1.14806
$471$	0	0
$472$	−23.4971	−1.08154
$473$	−17.0380	−0.783410
$474$	0	0
$475$	4.31088	0.197797
$476$	0	0
$477$	0	0
$478$	19.5925	0.896139
$479$	−31.4003	−1.43472	−0.717358	−	0.696705i	$-0.754649\pi$
$479$	−31.4003	−1.43472	−0.717358	+	0.696705i	$0.754649\pi$
$480$	0	0
$481$	−28.5457	−1.30157
$482$	5.00264	0.227864
$483$	0	0
$484$	−0.883919	−0.0401782
$485$	−35.3463	−1.60499
$486$	0	0
$487$	22.2816	1.00967	0.504837	−	0.863214i	$-0.331552\pi$
$487$	22.2816	1.00967	0.504837	+	0.863214i	$0.331552\pi$
$488$	−36.1434	−1.63614
$489$	0	0
$490$	0	0
$491$	31.4136	1.41768	0.708839	−	0.705371i	$-0.249219\pi$
$491$	31.4136	1.41768	0.708839	+	0.705371i	$0.249219\pi$
$492$	0	0
$493$	7.44030	0.335094
$494$	−4.62177	−0.207943
$495$	0	0
$496$	−18.6625	−0.837969
$497$	0	0
$498$	0	0
$499$	16.7512	0.749886	0.374943	−	0.927048i	$-0.377662\pi$
$499$	16.7512	0.749886	0.374943	+	0.927048i	$0.377662\pi$
$500$	0.108027	0.00483112
$501$	0	0
$502$	21.0601	0.939959
$503$	25.7352	1.14748	0.573738	−	0.819039i	$-0.305493\pi$
$503$	25.7352	1.14748	0.573738	+	0.819039i	$0.305493\pi$
$504$	0	0
$505$	29.5164	1.31346
$506$	−9.71842	−0.432036
$507$	0	0
$508$	−0.384330	−0.0170519
$509$	−5.54832	−0.245925	−0.122963	−	0.992411i	$-0.539240\pi$
$509$	−5.54832	−0.245925	−0.122963	+	0.992411i	$0.539240\pi$
$510$	0	0
$511$	0	0
$512$	23.4366	1.03576
$513$	0	0
$514$	25.7573	1.13610
$515$	8.00000	0.352522
$516$	0	0
$517$	31.0328	1.36482
$518$	0	0
$519$	0	0
$520$	28.9300	1.26867
$521$	11.3375	0.496707	0.248354	−	0.968669i	$-0.420111\pi$
$521$	11.3375	0.496707	0.248354	+	0.968669i	$0.420111\pi$
$522$	0	0
$523$	28.8627	1.26208	0.631038	−	0.775752i	$-0.282629\pi$
$523$	28.8627	1.26208	0.631038	+	0.775752i	$0.282629\pi$
$524$	−0.273528	−0.0119491
$525$	0	0
$526$	−23.4403	−1.02205
$527$	4.54569	0.198013
$528$	0	0
$529$	−21.2816	−0.925286
$530$	−11.8352	−0.514088
$531$	0	0
$532$	0	0
$533$	37.4490	1.62210
$534$	0	0
$535$	33.4084	1.44437
$536$	−12.9406	−0.558948
$537$	0	0
$538$	−25.7573	−1.11048
$539$	0	0
$540$	0	0
$541$	41.5784	1.78760	0.893799	−	0.448469i	$-0.148030\pi$
$541$	41.5784	1.78760	0.893799	+	0.448469i	$0.148030\pi$
$542$	9.24354	0.397044
$543$	0	0
$544$	−0.275618	−0.0118170
$545$	57.1355	2.44742
$546$	0	0
$547$	3.16745	0.135431	0.0677153	−	0.997705i	$-0.478429\pi$
$547$	3.16745	0.135431	0.0677153	+	0.997705i	$0.478429\pi$
$548$	0.0213857	0.000913551 0
$549$	0	0
$550$	31.9593	1.36275
$551$	7.84324	0.334133
$552$	0	0
$553$	0	0
$554$	12.4163	0.527517
$555$	0	0
$556$	0.519021	0.0220114
$557$	26.3843	1.11794	0.558970	−	0.829188i	$-0.311197\pi$
$557$	26.3843	1.11794	0.558970	+	0.829188i	$0.311197\pi$
$558$	0	0
$559$	10.6218	0.449253
$560$	0	0
$561$	0	0
$562$	24.7645	1.04463
$563$	−12.5190	−0.527614	−0.263807	−	0.964576i	$-0.584978\pi$
$563$	−12.5190	−0.527614	−0.263807	+	0.964576i	$0.584978\pi$
$564$	0	0
$565$	11.7272	0.493365
$566$	1.19940	0.0504145
$567$	0	0
$568$	38.8599	1.63052
$569$	39.7085	1.66467	0.832334	−	0.554274i	$-0.187004\pi$
$569$	39.7085	1.66467	0.832334	+	0.554274i	$0.187004\pi$
$570$	0	0
$571$	20.5457	0.859810	0.429905	−	0.902874i	$-0.358547\pi$
$571$	20.5457	0.859810	0.429905	+	0.902874i	$0.358547\pi$
$572$	0.903351	0.0377710
$573$	0	0
$574$	0	0
$575$	−5.65107	−0.235666
$576$	0	0
$577$	−30.4110	−1.26603	−0.633013	−	0.774141i	$-0.718182\pi$
$577$	−30.4110	−1.26603	−0.633013	+	0.774141i	$0.718182\pi$
$578$	22.4747	0.934822
$579$	0	0
$580$	−1.22952	−0.0510531
$581$	0	0
$582$	0	0
$583$	−14.7565	−0.611150
$584$	11.7485	0.486158
$585$	0	0
$586$	−32.5404	−1.34423
$587$	−18.2542	−0.753433	−0.376716	−	0.926329i	$-0.622947\pi$
$587$	−18.2542	−0.753433	−0.376716	+	0.926329i	$0.622947\pi$
$588$	0	0
$589$	4.79186	0.197445
$590$	34.9514	1.43893
$591$	0	0
$592$	33.5784	1.38007
$593$	26.7112	1.09690	0.548448	−	0.836184i	$-0.315219\pi$
$593$	26.7112	1.09690	0.548448	+	0.836184i	$0.315219\pi$
$594$	0	0
$595$	0	0
$596$	−0.703132	−0.0288014
$597$	0	0
$598$	6.05861	0.247755
$599$	−14.8139	−0.605281	−0.302640	−	0.953105i	$-0.597868\pi$
$599$	−14.8139	−0.605281	−0.302640	+	0.953105i	$0.597868\pi$
$600$	0	0
$601$	−5.79450	−0.236363	−0.118181	−	0.992992i	$-0.537706\pi$
$601$	−5.79450	−0.236363	−0.118181	+	0.992992i	$0.537706\pi$
$602$	0	0
$603$	0	0
$604$	0.777847	0.0316501
$605$	52.5004	2.13445
$606$	0	0
$607$	−34.6218	−1.40525	−0.702627	−	0.711558i	$-0.747990\pi$
$607$	−34.6218	−1.40525	−0.702627	+	0.711558i	$0.747990\pi$
$608$	−0.290544	−0.0117831
$609$	0	0
$610$	53.7626	2.17678
$611$	−19.3463	−0.782667
$612$	0	0
$613$	47.5164	1.91917	0.959584	−	0.281421i	$-0.0908059\pi$
$613$	47.5164	1.91917	0.959584	+	0.281421i	$0.0908059\pi$
$614$	15.9327	0.642990
$615$	0	0
$616$	0	0
$617$	18.2055	0.732926	0.366463	−	0.930433i	$-0.380569\pi$
$617$	18.2055	0.732926	0.366463	+	0.930433i	$0.380569\pi$
$618$	0	0
$619$	−2.10275	−0.0845166	−0.0422583	−	0.999107i	$-0.513455\pi$
$619$	−2.10275	−0.0845166	−0.0422583	+	0.999107i	$0.513455\pi$
$620$	−0.751182	−0.0301682
$621$	0	0
$622$	−27.8432	−1.11641
$623$	0	0
$624$	0	0
$625$	−27.9707	−1.11883
$626$	−18.3809	−0.734647
$627$	0	0
$628$	0.318803	0.0127216
$629$	−8.17883	−0.326111
$630$	0	0
$631$	−11.0026	−0.438008	−0.219004	−	0.975724i	$-0.570281\pi$
$631$	−11.0026	−0.438008	−0.219004	+	0.975724i	$0.570281\pi$
$632$	−23.4971	−0.934664
$633$	0	0
$634$	−36.9433	−1.46721
$635$	22.8273	0.905872
$636$	0	0
$637$	0	0
$638$	58.1469	2.30206
$639$	0	0
$640$	−33.1327	−1.30969
$641$	21.9460	0.866814	0.433407	−	0.901198i	$-0.357311\pi$
$641$	21.9460	0.866814	0.433407	+	0.901198i	$0.357311\pi$
$642$	0	0
$643$	−14.1027	−0.556158	−0.278079	−	0.960558i	$-0.589698\pi$
$643$	−14.1027	−0.556158	−0.278079	+	0.960558i	$0.589698\pi$
$644$	0	0
$645$	0	0
$646$	−1.32422	−0.0521006
$647$	13.3242	0.523829	0.261915	−	0.965091i	$-0.415646\pi$
$647$	13.3242	0.523829	0.261915	+	0.965091i	$0.415646\pi$
$648$	0	0
$649$	43.5784	1.71060
$650$	−19.9239	−0.781480
$651$	0	0
$652$	−0.380870	−0.0149160
$653$	8.96196	0.350709	0.175354	−	0.984505i	$-0.443893\pi$
$653$	8.96196	0.350709	0.175354	+	0.984505i	$0.443893\pi$
$654$	0	0
$655$	16.2462	0.634791
$656$	−44.0515	−1.71992
$657$	0	0
$658$	0	0
$659$	−27.1541	−1.05777	−0.528887	−	0.848692i	$-0.677391\pi$
$659$	−27.1541	−1.05777	−0.528887	+	0.848692i	$0.677391\pi$
$660$	0	0
$661$	1.23480	0.0480282	0.0240141	−	0.999712i	$-0.492355\pi$
$661$	1.23480	0.0480282	0.0240141	+	0.999712i	$0.492355\pi$
$662$	0.724518	0.0281592
$663$	0	0
$664$	−18.2188	−0.707028
$665$	0	0
$666$	0	0
$667$	−10.2816	−0.398104
$668$	−1.28076	−0.0495541
$669$	0	0
$670$	19.2488	0.743647
$671$	67.0328	2.58777
$672$	0	0
$673$	47.4490	1.82903	0.914513	−	0.404557i	$-0.132574\pi$
$673$	47.4490	1.82903	0.914513	+	0.404557i	$0.132574\pi$
$674$	−13.2348	−0.509785
$675$	0	0
$676$	0.104703	0.00402703
$677$	1.41363	0.0543303	0.0271652	−	0.999631i	$-0.491352\pi$
$677$	1.41363	0.0543303	0.0271652	+	0.999631i	$0.491352\pi$
$678$	0	0
$679$	0	0
$680$	8.28895	0.317867
$681$	0	0
$682$	35.5251	1.36033
$683$	7.80784	0.298759	0.149379	−	0.988780i	$-0.452272\pi$
$683$	7.80784	0.298759	0.149379	+	0.988780i	$0.452272\pi$
$684$	0	0
$685$	−1.27020	−0.0485320
$686$	0	0
$687$	0	0
$688$	−12.4944	−0.476346
$689$	9.19940	0.350469
$690$	0	0
$691$	37.7892	1.43757	0.718785	−	0.695233i	$-0.244699\pi$
$691$	37.7892	1.43757	0.718785	+	0.695233i	$0.244699\pi$
$692$	−1.00264	−0.0381146
$693$	0	0
$694$	−43.8513	−1.66457
$695$	−30.8273	−1.16934
$696$	0	0
$697$	10.7298	0.406420
$698$	−45.8246	−1.73449
$699$	0	0
$700$	0	0
$701$	−44.8273	−1.69310	−0.846551	−	0.532307i	$-0.821325\pi$
$701$	−44.8273	−1.69310	−0.846551	+	0.532307i	$0.821325\pi$
$702$	0	0
$703$	−8.62177	−0.325176
$704$	−43.5217	−1.64028
$705$	0	0
$706$	10.1301	0.381251
$707$	0	0
$708$	0	0
$709$	50.5491	1.89841	0.949206	−	0.314654i	$-0.101888\pi$
$709$	50.5491	1.89841	0.949206	+	0.314654i	$0.101888\pi$
$710$	−57.8032	−2.16932
$711$	0	0
$712$	44.4324	1.66517
$713$	−6.28158	−0.235247
$714$	0	0
$715$	−53.6545	−2.00657
$716$	0.367538	0.0137355
$717$	0	0
$718$	31.4844	1.17499
$719$	1.43224	0.0534137	0.0267068	−	0.999643i	$-0.491498\pi$
$719$	1.43224	0.0534137	0.0267068	+	0.999643i	$0.491498\pi$
$720$	0	0
$721$	0	0
$722$	−1.39593	−0.0519512
$723$	0	0
$724$	0.800602	0.0297541
$725$	33.8113	1.25572
$726$	0	0
$727$	−47.5784	−1.76459	−0.882293	−	0.470700i	$-0.844002\pi$
$727$	−47.5784	−1.76459	−0.882293	+	0.470700i	$0.844002\pi$
$728$	0	0
$729$	0	0
$730$	−17.4757	−0.646805
$731$	3.04332	0.112561
$732$	0	0
$733$	19.2702	0.711761	0.355881	−	0.934531i	$-0.384181\pi$
$733$	19.2702	0.711761	0.355881	+	0.934531i	$0.384181\pi$
$734$	9.68120	0.357340
$735$	0	0
$736$	0.380870	0.0140390
$737$	24.0000	0.884051
$738$	0	0
$739$	30.3756	1.11738	0.558692	−	0.829375i	$-0.311303\pi$
$739$	30.3756	1.11738	0.558692	+	0.829375i	$0.311303\pi$
$740$	1.35157	0.0496846
$741$	0	0
$742$	0	0
$743$	50.5271	1.85366	0.926829	−	0.375483i	$-0.122523\pi$
$743$	50.5271	1.85366	0.926829	+	0.375483i	$0.122523\pi$
$744$	0	0
$745$	41.7626	1.53006
$746$	29.5483	1.08184
$747$	0	0
$748$	0.258826	0.00946360
$749$	0	0
$750$	0	0
$751$	28.4377	1.03770	0.518852	−	0.854864i	$-0.326359\pi$
$751$	28.4377	1.03770	0.518852	+	0.854864i	$0.326359\pi$
$752$	22.7571	0.829868
$753$	0	0
$754$	−36.2496	−1.32013
$755$	−46.2002	−1.68140
$756$	0	0
$757$	−37.1675	−1.35087	−0.675437	−	0.737418i	$-0.736045\pi$
$757$	−37.1675	−1.35087	−0.675437	+	0.737418i	$0.736045\pi$
$758$	−10.2549	−0.372475
$759$	0	0
$760$	8.73785	0.316955
$761$	35.6679	1.29296	0.646480	−	0.762931i	$-0.276241\pi$
$761$	35.6679	1.29296	0.646480	+	0.762931i	$0.276241\pi$
$762$	0	0
$763$	0	0
$764$	0.694394	0.0251223
$765$	0	0
$766$	32.5898	1.17752
$767$	−27.1675	−0.980960
$768$	0	0
$769$	−1.79450	−0.0647114	−0.0323557	−	0.999476i	$-0.510301\pi$
$769$	−1.79450	−0.0647114	−0.0323557	+	0.999476i	$0.510301\pi$
$770$	0	0
$771$	0	0
$772$	−0.361574	−0.0130133
$773$	10.4783	0.376880	0.188440	−	0.982085i	$-0.439657\pi$
$773$	10.4783	0.376880	0.188440	+	0.982085i	$0.439657\pi$
$774$	0	0
$775$	20.6572	0.742028
$776$	−33.1709	−1.19077
$777$	0	0
$778$	0.143430	0.00514221
$779$	11.3109	0.405255
$780$	0	0
$781$	−72.0708	−2.57890
$782$	1.73590	0.0620755
$783$	0	0
$784$	0	0
$785$	−18.9353	−0.675830
$786$	0	0
$787$	−28.8627	−1.02884	−0.514422	−	0.857537i	$-0.671993\pi$
$787$	−28.8627	−1.02884	−0.514422	+	0.857537i	$0.671993\pi$
$788$	1.01666	0.0362169
$789$	0	0
$790$	34.9514	1.24352
$791$	0	0
$792$	0	0
$793$	−41.7892	−1.48398
$794$	32.7333	1.16166
$795$	0	0
$796$	−0.600384	−0.0212800
$797$	−28.7599	−1.01873	−0.509364	−	0.860551i	$-0.670119\pi$
$797$	−28.7599	−1.01873	−0.509364	+	0.860551i	$0.670119\pi$
$798$	0	0
$799$	−5.54305	−0.196099
$800$	−1.25250	−0.0442826
$801$	0	0
$802$	−15.3330	−0.541425
$803$	−21.7892	−0.768925
$804$	0	0
$805$	0	0
$806$	−22.1469	−0.780091
$807$	0	0
$808$	27.6998	0.974476
$809$	−42.9567	−1.51028	−0.755138	−	0.655566i	$-0.772430\pi$
$809$	−42.9567	−1.51028	−0.755138	+	0.655566i	$0.772430\pi$
$810$	0	0
$811$	0.340188	0.0119456	0.00597281	−	0.999982i	$-0.498099\pi$
$811$	0.340188	0.0119456	0.00597281	+	0.999982i	$0.498099\pi$
$812$	0	0
$813$	0	0
$814$	−63.9186	−2.24035
$815$	22.6218	0.792406
$816$	0	0
$817$	3.20814	0.112238
$818$	−8.75646	−0.306162
$819$	0	0
$820$	−1.77312	−0.0619199
$821$	23.7626	0.829319	0.414660	−	0.909977i	$-0.363901\pi$
$821$	23.7626	0.829319	0.414660	+	0.909977i	$0.363901\pi$
$822$	0	0
$823$	34.2409	1.19356	0.596781	−	0.802404i	$-0.296446\pi$
$823$	34.2409	1.19356	0.596781	+	0.802404i	$0.296446\pi$
$824$	7.50764	0.261541
$825$	0	0
$826$	0	0
$827$	−35.0461	−1.21867	−0.609336	−	0.792912i	$-0.708564\pi$
$827$	−35.0461	−1.21867	−0.609336	+	0.792912i	$0.708564\pi$
$828$	0	0
$829$	32.4163	1.12586	0.562932	−	0.826503i	$-0.309673\pi$
$829$	32.4163	1.12586	0.562932	+	0.826503i	$0.309673\pi$
$830$	27.1001	0.940659
$831$	0	0
$832$	27.1321	0.940635
$833$	0	0
$834$	0	0
$835$	76.0708	2.63254
$836$	0.272843	0.00943646
$837$	0	0
$838$	10.0435	0.346945
$839$	−8.41099	−0.290380	−0.145190	−	0.989404i	$-0.546379\pi$
$839$	−8.41099	−0.290380	−0.145190	+	0.989404i	$0.546379\pi$
$840$	0	0
$841$	32.5164	1.12125
$842$	29.1107	1.00322
$843$	0	0
$844$	−0.967237	−0.0332937
$845$	−6.21883	−0.213934
$846$	0	0
$847$	0	0
$848$	−10.8213	−0.371605
$849$	0	0
$850$	−5.70855	−0.195802
$851$	11.3021	0.387433
$852$	0	0
$853$	29.6598	1.01553	0.507766	−	0.861495i	$-0.330471\pi$
$853$	29.6598	1.01553	0.507766	+	0.861495i	$0.330471\pi$
$854$	0	0
$855$	0	0
$856$	31.3523	1.07160
$857$	26.1382	0.892862	0.446431	−	0.894818i	$-0.352695\pi$
$857$	26.1382	0.892862	0.446431	+	0.894818i	$0.352695\pi$
$858$	0	0
$859$	−1.78922	−0.0610475	−0.0305238	−	0.999534i	$-0.509718\pi$
$859$	−1.78922	−0.0610475	−0.0305238	+	0.999534i	$0.509718\pi$
$860$	−0.502914	−0.0171492
$861$	0	0
$862$	−34.0575	−1.16000
$863$	46.2949	1.57590	0.787949	−	0.615741i	$-0.211143\pi$
$863$	46.2949	1.57590	0.787949	+	0.615741i	$0.211143\pi$
$864$	0	0
$865$	59.5518	2.02482
$866$	25.1268	0.853843
$867$	0	0
$868$	0	0
$869$	43.5784	1.47830
$870$	0	0
$871$	−14.9620	−0.506967
$872$	53.6191	1.81577
$873$	0	0
$874$	1.82991	0.0618975
$875$	0	0
$876$	0	0
$877$	−19.4490	−0.656747	−0.328374	−	0.944548i	$-0.606501\pi$
$877$	−19.4490	−0.656747	−0.328374	+	0.944548i	$0.606501\pi$
$878$	−33.4456	−1.12873
$879$	0	0
$880$	63.1141	2.12758
$881$	4.94863	0.166723	0.0833617	−	0.996519i	$-0.473434\pi$
$881$	4.94863	0.166723	0.0833617	+	0.996519i	$0.473434\pi$
$882$	0	0
$883$	33.7892	1.13710	0.568549	−	0.822649i	$-0.307505\pi$
$883$	33.7892	1.13710	0.568549	+	0.822649i	$0.307505\pi$
$884$	−0.161356	−0.00542698
$885$	0	0
$886$	−40.0035	−1.34394
$887$	−23.3463	−0.783892	−0.391946	−	0.919988i	$-0.628198\pi$
$887$	−23.3463	−0.783892	−0.391946	+	0.919988i	$0.628198\pi$
$888$	0	0
$889$	0	0
$890$	−66.0922	−2.21542
$891$	0	0
$892$	−0.299507	−0.0100282
$893$	−5.84324	−0.195537
$894$	0	0
$895$	−21.8299	−0.729693
$896$	0	0
$897$	0	0
$898$	23.2783	0.776805
$899$	37.5837	1.25349
$900$	0	0
$901$	2.63579	0.0878108
$902$	83.8548	2.79206
$903$	0	0
$904$	11.0054	0.366035
$905$	−47.5518	−1.58067
$906$	0	0
$907$	36.4110	1.20901	0.604504	−	0.796602i	$-0.293371\pi$
$907$	36.4110	1.20901	0.604504	+	0.796602i	$0.293371\pi$
$908$	0.260195	0.00863489
$909$	0	0
$910$	0	0
$911$	−0.192165	−0.00636670	−0.00318335	−	0.999995i	$-0.501013\pi$
$911$	−0.192165	−0.00636670	−0.00318335	+	0.999995i	$0.501013\pi$
$912$	0	0
$913$	33.7892	1.11826
$914$	38.1241	1.26103
$915$	0	0
$916$	−1.11413	−0.0368118
$917$	0	0
$918$	0	0
$919$	−18.4871	−0.609832	−0.304916	−	0.952379i	$-0.598628\pi$
$919$	−18.4871	−0.609832	−0.304916	+	0.952379i	$0.598628\pi$
$920$	−11.4543	−0.377638
$921$	0	0
$922$	−14.9521	−0.492421
$923$	44.9300	1.47889
$924$	0	0
$925$	−37.1675	−1.22206
$926$	39.3730	1.29388
$927$	0	0
$928$	−2.27881	−0.0748055
$929$	−19.7492	−0.647951	−0.323976	−	0.946065i	$-0.605020\pi$
$929$	−19.7492	−0.647951	−0.323976	+	0.946065i	$0.605020\pi$
$930$	0	0
$931$	0	0
$932$	−0.632327	−0.0207126
$933$	0	0
$934$	17.1134	0.559969
$935$	−15.3730	−0.502749
$936$	0	0
$937$	51.6545	1.68748	0.843740	−	0.536752i	$-0.180349\pi$
$937$	51.6545	1.68748	0.843740	+	0.536752i	$0.180349\pi$
$938$	0	0
$939$	0	0
$940$	0.915998	0.0298766
$941$	−52.1062	−1.69861	−0.849307	−	0.527899i	$-0.822980\pi$
$941$	−52.1062	−1.69861	−0.849307	+	0.527899i	$0.822980\pi$
$942$	0	0
$943$	−14.8273	−0.482842
$944$	31.9572	1.04012
$945$	0	0
$946$	23.7839	0.773283
$947$	31.8513	1.03503	0.517514	−	0.855675i	$-0.326858\pi$
$947$	31.8513	1.03503	0.517514	+	0.855675i	$0.326858\pi$
$948$	0	0
$949$	13.5837	0.440946
$950$	−6.01770	−0.195240
$951$	0	0
$952$	0	0
$953$	−28.8813	−0.935556	−0.467778	−	0.883846i	$-0.654945\pi$
$953$	−28.8813	−0.935556	−0.467778	+	0.883846i	$0.654945\pi$
$954$	0	0
$955$	−41.2435	−1.33461
$956$	0.721058	0.0233207
$957$	0	0
$958$	43.8327	1.41617
$959$	0	0
$960$	0	0
$961$	−8.03804	−0.259292
$962$	39.8478	1.28475
$963$	0	0
$964$	0.184111	0.00592983
$965$	21.4757	0.691327
$966$	0	0
$967$	−8.16482	−0.262563	−0.131281	−	0.991345i	$-0.541909\pi$
$967$	−8.16482	−0.262563	−0.131281	+	0.991345i	$0.541909\pi$
$968$	49.2693	1.58358
$969$	0	0
$970$	49.3410	1.58424
$971$	−32.6979	−1.04932	−0.524662	−	0.851311i	$-0.675808\pi$
$971$	−32.6979	−1.04932	−0.524662	+	0.851311i	$0.675808\pi$
$972$	0	0
$973$	0	0
$974$	−31.1036	−0.996623
$975$	0	0
$976$	49.1569	1.57347
$977$	5.11344	0.163593	0.0817967	−	0.996649i	$-0.473934\pi$
$977$	5.11344	0.163593	0.0817967	+	0.996649i	$0.473934\pi$
$978$	0	0
$979$	−82.4057	−2.63370
$980$	0	0
$981$	0	0
$982$	−43.8513	−1.39935
$983$	22.9620	0.732373	0.366186	−	0.930542i	$-0.380663\pi$
$983$	22.9620	0.732373	0.366186	+	0.930542i	$0.380663\pi$
$984$	0	0
$985$	−60.3843	−1.92400
$986$	−10.3861	−0.330762
$987$	0	0
$988$	−0.170094	−0.00541142
$989$	−4.20550	−0.133727
$990$	0	0
$991$	−35.1675	−1.11713	−0.558566	−	0.829460i	$-0.688648\pi$
$991$	−35.1675	−1.11713	−0.558566	+	0.829460i	$0.688648\pi$
$992$	−1.39225	−0.0442039
$993$	0	0
$994$	0	0
$995$	35.6598	1.13049
$996$	0	0
$997$	24.0761	0.762497	0.381249	−	0.924473i	$-0.375494\pi$
$997$	24.0761	0.762497	0.381249	+	0.924473i	$0.375494\pi$
$998$	−23.3835	−0.740192
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bq.1.2		3
3.2	odd	2		2793.2.a.w.1.2		3
7.6	odd	2		1197.2.a.m.1.2		3
21.20	even	2		399.2.a.e.1.2	✓	3
84.83	odd	2		6384.2.a.bu.1.3		3
105.104	even	2		9975.2.a.x.1.2		3
399.398	odd	2		7581.2.a.l.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.e.1.2	✓	3	21.20	even	2
1197.2.a.m.1.2		3	7.6	odd	2
2793.2.a.w.1.2		3	3.2	odd	2
6384.2.a.bu.1.3		3	84.83	odd	2
7581.2.a.l.1.2		3	399.398	odd	2
8379.2.a.bq.1.2		3	1.1	even	1	trivial
9975.2.a.x.1.2		3	105.104	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q-1.39593 q^{2} -0.0513742 q^{4} +3.05137 q^{5} +2.86358 q^{8} -4.25951 q^{10} -5.31088 q^{11} +3.31088 q^{13} -3.89461 q^{16} +0.948626 q^{17} +1.00000 q^{19} -0.156762 q^{20} +7.41363 q^{22} -1.31088 q^{23} +4.31088 q^{25} -4.62177 q^{26} +7.84324 q^{29} +4.79186 q^{31} -0.290544 q^{32} -1.32422 q^{34} -8.62177 q^{37} -1.39593 q^{38} +8.73785 q^{40} +11.3109 q^{41} +3.20814 q^{43} +0.272843 q^{44} +1.82991 q^{46} -5.84324 q^{47} -6.01770 q^{50} -0.170094 q^{52} +2.77853 q^{53} -16.2055 q^{55} -10.9486 q^{58} -8.20550 q^{59} -12.6218 q^{61} -6.68912 q^{62} +8.19480 q^{64} +10.1027 q^{65} -4.51902 q^{67} -0.0487349 q^{68} +13.5704 q^{71} +4.10275 q^{73} +12.0354 q^{74} -0.0513742 q^{76} -8.20550 q^{79} -11.8839 q^{80} -15.7892 q^{82} -6.36226 q^{83} +2.89461 q^{85} -4.47834 q^{86} -15.2081 q^{88} +15.5164 q^{89} +0.0673457 q^{92} +8.15676 q^{94} +3.05137 q^{95} -11.5837 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 9 q^{4} + 9 q^{8} - 10 q^{10} - 4 q^{11} - 2 q^{13} + 13 q^{16} + 12 q^{17} + 3 q^{19} - 16 q^{20} - 8 q^{22} + 8 q^{23} + q^{25} + 10 q^{26} + 8 q^{29} + 8 q^{31} + 27 q^{32} + 6 q^{34}+ \cdots - 22 q^{97}+O(q^{100}) \) 3 * q - q^2 + 9 * q^4 + 9 * q^8 - 10 * q^10 - 4 * q^11 - 2 * q^13 + 13 * q^16 + 12 * q^17 + 3 * q^19 - 16 * q^20 - 8 * q^22 + 8 * q^23 + q^25 + 10 * q^26 + 8 * q^29 + 8 * q^31 + 27 * q^32 + 6 * q^34 - 2 * q^37 - q^38 - 14 * q^40 + 22 * q^41 + 16 * q^43 - 12 * q^46 - 2 * q^47 + 9 * q^50 - 18 * q^52 - 12 * q^55 - 42 * q^58 + 12 * q^59 - 14 * q^61 - 32 * q^62 + 29 * q^64 + 12 * q^65 - 8 * q^67 + 52 * q^68 + 26 * q^71 - 6 * q^73 - 18 * q^74 + 9 * q^76 + 12 * q^79 - 52 * q^80 + 2 * q^82 + 2 * q^83 - 16 * q^85 + 24 * q^86 - 52 * q^88 - 2 * q^89 + 36 * q^92 + 40 * q^94 - 22 * q^97

Introduction

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