LMFDB - Embedded newform 8379.2.a.bv.1.3

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 = [4,0,0,2,8,0,0,-6,0,-10,6,0,-2,0,0,2,8] 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:	$4$
Coefficient field:	4.4.5744.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 931)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.751024$ 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)

$f(q)$	$=$	$q+0.751024 q^{2} -1.43596 q^{4} +4.26543 q^{5} -2.58049 q^{8} +3.20344 q^{10} -4.09899 q^{11} +2.18699 q^{13} +0.933914 q^{16} +0.590042 q^{17} +1.00000 q^{19} -6.12500 q^{20} -3.07844 q^{22} +4.36988 q^{23} +13.1939 q^{25} +1.64248 q^{26} +7.36442 q^{29} -4.51850 q^{31} +5.86237 q^{32} +0.443136 q^{34} +8.95446 q^{37} +0.751024 q^{38} -11.0069 q^{40} -2.17744 q^{41} -8.69594 q^{43} +5.88600 q^{44} +3.28188 q^{46} -11.8333 q^{47} +9.90893 q^{50} -3.14043 q^{52} -4.40996 q^{53} -17.4840 q^{55} +5.53086 q^{58} +10.7178 q^{59} +2.43051 q^{61} -3.39350 q^{62} +2.53496 q^{64} +9.32844 q^{65} -5.65348 q^{67} -0.847279 q^{68} -8.32434 q^{71} +10.6795 q^{73} +6.72502 q^{74} -1.43596 q^{76} +8.56377 q^{79} +3.98355 q^{80} -1.63531 q^{82} -4.25826 q^{83} +2.51678 q^{85} -6.53086 q^{86} +10.5774 q^{88} +3.79110 q^{89} -6.27498 q^{92} -8.88709 q^{94} +4.26543 q^{95} -2.11026 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 8 q^{5} - 6 q^{8} - 10 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{16} + 8 q^{17} + 4 q^{19} - 14 q^{22} + 8 q^{23} + 20 q^{25} + 16 q^{26} - 2 q^{29} - 2 q^{32} + 22 q^{34} + 10 q^{37} - 22 q^{40}+ \cdots - 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 + 8 * q^5 - 6 * q^8 - 10 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^16 + 8 * q^17 + 4 * q^19 - 14 * q^22 + 8 * q^23 + 20 * q^25 + 16 * q^26 - 2 * q^29 - 2 * q^32 + 22 * q^34 + 10 * q^37 - 22 * q^40 + 12 * q^41 + 4 * q^43 + 14 * q^44 - 8 * q^46 + 16 * q^47 - 12 * q^50 - 28 * q^52 - 12 * q^53 + 8 * q^55 + 4 * q^58 + 14 * q^59 - 20 * q^61 - 20 * q^62 - 20 * q^64 - 10 * q^65 + 2 * q^67 - 24 * q^68 + 2 * q^71 + 16 * q^73 + 26 * q^74 + 2 * q^76 - 8 * q^79 + 28 * q^80 + 12 * q^82 + 20 * q^83 - 14 * q^85 - 8 * q^86 - 2 * q^88 + 16 * q^89 - 26 * q^92 - 10 * q^94 + 8 * q^95 - 2 * 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 $

$2$	0.751024	0.531054	0.265527	−	0.964103i	$-0.414454\pi$
$2$	0.751024	0.531054	0.265527	+	0.964103i	$0.414454\pi$
$3$	0	0
$3$	0	0
$4$	−1.43596	−0.717981
$5$	4.26543	1.90756	0.953779	−	0.300509i	$-0.0971565\pi$
$5$	4.26543	1.90756	0.953779	+	0.300509i	$0.0971565\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.58049	−0.912341
$9$	0	0
$10$	3.20344	1.01302
$11$	−4.09899	−1.23589	−0.617946	−	0.786220i	$-0.712035\pi$
$11$	−4.09899	−1.23589	−0.617946	+	0.786220i	$0.712035\pi$
$12$	0	0
$13$	2.18699	0.606561	0.303281	−	0.952901i	$-0.401918\pi$
$13$	2.18699	0.606561	0.303281	+	0.952901i	$0.401918\pi$
$14$	0	0
$15$	0	0
$16$	0.933914	0.233479
$17$	0.590042	0.143106	0.0715531	−	0.997437i	$-0.477204\pi$
$17$	0.590042	0.143106	0.0715531	+	0.997437i	$0.477204\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−6.12500	−1.36959
$21$	0	0
$22$	−3.07844	−0.656326
$23$	4.36988	0.911182	0.455591	−	0.890189i	$-0.349428\pi$
$23$	4.36988	0.911182	0.455591	+	0.890189i	$0.349428\pi$
$24$	0	0
$25$	13.1939	2.63878
$26$	1.64248	0.322117
$27$	0	0
$28$	0	0
$29$	7.36442	1.36754	0.683769	−	0.729698i	$-0.260339\pi$
$29$	7.36442	1.36754	0.683769	+	0.729698i	$0.260339\pi$
$30$	0	0
$31$	−4.51850	−0.811547	−0.405773	−	0.913974i	$-0.632998\pi$
$31$	−4.51850	−0.811547	−0.405773	+	0.913974i	$0.632998\pi$
$32$	5.86237	1.03633
$33$	0	0
$34$	0.443136	0.0759972
$35$	0	0
$36$	0	0
$37$	8.95446	1.47210	0.736052	−	0.676924i	$-0.236688\pi$
$37$	8.95446	1.47210	0.736052	+	0.676924i	$0.236688\pi$
$38$	0.751024	0.121832
$39$	0	0
$40$	−11.0069	−1.74034
$41$	−2.17744	−0.340058	−0.170029	−	0.985439i	$-0.554386\pi$
$41$	−2.17744	−0.340058	−0.170029	+	0.985439i	$0.554386\pi$
$42$	0	0
$43$	−8.69594	−1.32612	−0.663059	−	0.748567i	$-0.730742\pi$
$43$	−8.69594	−1.32612	−0.663059	+	0.748567i	$0.730742\pi$
$44$	5.88600	0.887348
$45$	0	0
$46$	3.28188	0.483887
$47$	−11.8333	−1.72606	−0.863032	−	0.505150i	$-0.831437\pi$
$47$	−11.8333	−1.72606	−0.863032	+	0.505150i	$0.831437\pi$
$48$	0	0
$49$	0	0
$50$	9.90893	1.40133
$51$	0	0
$52$	−3.14043	−0.435500
$53$	−4.40996	−0.605754	−0.302877	−	0.953030i	$-0.597947\pi$
$53$	−4.40996	−0.605754	−0.302877	+	0.953030i	$0.597947\pi$
$54$	0	0
$55$	−17.4840	−2.35754
$56$	0	0
$57$	0	0
$58$	5.53086	0.726237
$59$	10.7178	1.39534	0.697672	−	0.716417i	$-0.254219\pi$
$59$	10.7178	1.39534	0.697672	+	0.716417i	$0.254219\pi$
$60$	0	0
$61$	2.43051	0.311195	0.155597	−	0.987821i	$-0.450270\pi$
$61$	2.43051	0.311195	0.155597	+	0.987821i	$0.450270\pi$
$62$	−3.39350	−0.430975
$63$	0	0
$64$	2.53496	0.316869
$65$	9.32844	1.15705
$66$	0	0
$67$	−5.65348	−0.690682	−0.345341	−	0.938477i	$-0.612237\pi$
$67$	−5.65348	−0.690682	−0.345341	+	0.938477i	$0.612237\pi$
$68$	−0.847279	−0.102748
$69$	0	0
$70$	0	0
$71$	−8.32434	−0.987918	−0.493959	−	0.869485i	$-0.664451\pi$
$71$	−8.32434	−0.987918	−0.493959	+	0.869485i	$0.664451\pi$
$72$	0	0
$73$	10.6795	1.24994	0.624970	−	0.780649i	$-0.285111\pi$
$73$	10.6795	1.24994	0.624970	+	0.780649i	$0.285111\pi$
$74$	6.72502	0.781768
$75$	0	0
$76$	−1.43596	−0.164716
$77$	0	0
$78$	0	0
$79$	8.56377	0.963499	0.481750	−	0.876309i	$-0.340002\pi$
$79$	8.56377	0.963499	0.481750	+	0.876309i	$0.340002\pi$
$80$	3.98355	0.445374
$81$	0	0
$82$	−1.63531	−0.180589
$83$	−4.25826	−0.467404	−0.233702	−	0.972308i	$-0.575084\pi$
$83$	−4.25826	−0.467404	−0.233702	+	0.972308i	$0.575084\pi$
$84$	0	0
$85$	2.51678	0.272983
$86$	−6.53086	−0.704241
$87$	0	0
$88$	10.5774	1.12756
$89$	3.79110	0.401856	0.200928	−	0.979606i	$-0.435604\pi$
$89$	3.79110	0.401856	0.200928	+	0.979606i	$0.435604\pi$
$90$	0	0
$91$	0	0
$92$	−6.27498	−0.654212
$93$	0	0
$94$	−8.88709	−0.916633
$95$	4.26543	0.437624
$96$	0	0
$97$	−2.11026	−0.214265	−0.107132	−	0.994245i	$-0.534167\pi$
$97$	−2.11026	−0.214265	−0.107132	+	0.994245i	$0.534167\pi$
$98$	0	0
$99$	0	0
$100$	−18.9459	−1.89459
$101$	15.0606	1.49859	0.749294	−	0.662237i	$-0.230393\pi$
$101$	15.0606	1.49859	0.749294	+	0.662237i	$0.230393\pi$
$102$	0	0
$103$	−4.89383	−0.482204	−0.241102	−	0.970500i	$-0.577509\pi$
$103$	−4.89383	−0.482204	−0.241102	+	0.970500i	$0.577509\pi$
$104$	−5.64350	−0.553391
$105$	0	0
$106$	−3.31198	−0.321688
$107$	−0.888650	−0.0859090	−0.0429545	−	0.999077i	$-0.513677\pi$
$107$	−0.888650	−0.0859090	−0.0429545	+	0.999077i	$0.513677\pi$
$108$	0	0
$109$	−1.11135	−0.106448	−0.0532240	−	0.998583i	$-0.516950\pi$
$109$	−1.11135	−0.106448	−0.0532240	+	0.998583i	$0.516950\pi$
$110$	−13.1309	−1.25198
$111$	0	0
$112$	0	0
$113$	12.9161	1.21504	0.607522	−	0.794303i	$-0.292164\pi$
$113$	12.9161	1.21504	0.607522	+	0.794303i	$0.292164\pi$
$114$	0	0
$115$	18.6394	1.73813
$116$	−10.5750	−0.981867
$117$	0	0
$118$	8.04936	0.741004
$119$	0	0
$120$	0	0
$121$	5.80174	0.527431
$122$	1.82537	0.165261
$123$	0	0
$124$	6.48840	0.582676
$125$	34.9505	3.12606
$126$	0	0
$127$	−3.11373	−0.276299	−0.138149	−	0.990411i	$-0.544115\pi$
$127$	−3.11373	−0.276299	−0.138149	+	0.990411i	$0.544115\pi$
$128$	−9.82094	−0.868056
$129$	0	0
$130$	7.00588	0.614457
$131$	3.70721	0.323900	0.161950	−	0.986799i	$-0.448222\pi$
$131$	3.70721	0.323900	0.161950	+	0.986799i	$0.448222\pi$
$132$	0	0
$133$	0	0
$134$	−4.24590	−0.366790
$135$	0	0
$136$	−1.52260	−0.130562
$137$	−1.16098	−0.0991894	−0.0495947	−	0.998769i	$-0.515793\pi$
$137$	−1.16098	−0.0991894	−0.0495947	+	0.998769i	$0.515793\pi$
$138$	0	0
$139$	4.24633	0.360169	0.180084	−	0.983651i	$-0.442363\pi$
$139$	4.24633	0.360169	0.180084	+	0.983651i	$0.442363\pi$
$140$	0	0
$141$	0	0
$142$	−6.25178	−0.524638
$143$	−8.96444	−0.749644
$144$	0	0
$145$	31.4124	2.60866
$146$	8.02055	0.663785
$147$	0	0
$148$	−12.8583	−1.05694
$149$	23.9932	1.96560	0.982799	−	0.184677i	$-0.0591239\pi$
$149$	23.9932	1.96560	0.982799	+	0.184677i	$0.0591239\pi$
$150$	0	0
$151$	8.50069	0.691776	0.345888	−	0.938276i	$-0.387578\pi$
$151$	8.50069	0.691776	0.345888	+	0.938276i	$0.387578\pi$
$152$	−2.58049	−0.209305
$153$	0	0
$154$	0	0
$155$	−19.2734	−1.54807
$156$	0	0
$157$	−4.43877	−0.354252	−0.177126	−	0.984188i	$-0.556680\pi$
$157$	−4.43877	−0.354252	−0.177126	+	0.984188i	$0.556680\pi$
$158$	6.43160	0.511670
$159$	0	0
$160$	25.0055	1.97686
$161$	0	0
$162$	0	0
$163$	−0.146906	−0.0115066	−0.00575329	−	0.999983i	$-0.501831\pi$
$163$	−0.146906	−0.0115066	−0.00575329	+	0.999983i	$0.501831\pi$
$164$	3.12672	0.244156
$165$	0	0
$166$	−3.19805	−0.248217
$167$	0.885641	0.0685329	0.0342665	−	0.999413i	$-0.489091\pi$
$167$	0.885641	0.0685329	0.0342665	+	0.999413i	$0.489091\pi$
$168$	0	0
$169$	−8.21709	−0.632084
$170$	1.89017	0.144969
$171$	0	0
$172$	12.4870	0.952128
$173$	24.2638	1.84474	0.922371	−	0.386305i	$-0.126249\pi$
$173$	24.2638	1.84474	0.922371	+	0.386305i	$0.126249\pi$
$174$	0	0
$175$	0	0
$176$	−3.82811	−0.288555
$177$	0	0
$178$	2.84721	0.213408
$179$	12.9466	0.967677	0.483838	−	0.875157i	$-0.339242\pi$
$179$	12.9466	0.967677	0.483838	+	0.875157i	$0.339242\pi$
$180$	0	0
$181$	−18.6034	−1.38278	−0.691391	−	0.722481i	$-0.743002\pi$
$181$	−18.6034	−1.38278	−0.691391	+	0.722481i	$0.743002\pi$
$182$	0	0
$183$	0	0
$184$	−11.2764	−0.831309
$185$	38.1946	2.80813
$186$	0	0
$187$	−2.41858	−0.176864
$188$	16.9922	1.23928
$189$	0	0
$190$	3.20344	0.232402
$191$	23.2888	1.68512	0.842559	−	0.538605i	$-0.181048\pi$
$191$	23.2888	1.68512	0.842559	+	0.538605i	$0.181048\pi$
$192$	0	0
$193$	3.18843	0.229509	0.114754	−	0.993394i	$-0.463392\pi$
$193$	3.18843	0.229509	0.114754	+	0.993394i	$0.463392\pi$
$194$	−1.58486	−0.113786
$195$	0	0
$196$	0	0
$197$	8.25178	0.587915	0.293958	−	0.955818i	$-0.405028\pi$
$197$	8.25178	0.587915	0.293958	+	0.955818i	$0.405028\pi$
$198$	0	0
$199$	19.6024	1.38958	0.694789	−	0.719214i	$-0.255498\pi$
$199$	19.6024	1.38958	0.694789	+	0.719214i	$0.255498\pi$
$200$	−34.0467	−2.40747
$201$	0	0
$202$	11.3109	0.795832
$203$	0	0
$204$	0	0
$205$	−9.28770	−0.648681
$206$	−3.67539	−0.256076
$207$	0	0
$208$	2.04246	0.141619
$209$	−4.09899	−0.283533
$210$	0	0
$211$	9.49078	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$211$	9.49078	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$212$	6.33254	0.434920
$213$	0	0
$214$	−0.667398	−0.0456224
$215$	−37.0919	−2.52965
$216$	0	0
$217$	0	0
$218$	−0.834651	−0.0565297
$219$	0	0
$220$	25.1063	1.69267
$221$	1.29041	0.0868027
$222$	0	0
$223$	18.8829	1.26449	0.632247	−	0.774767i	$-0.282133\pi$
$223$	18.8829	1.26449	0.632247	+	0.774767i	$0.282133\pi$
$224$	0	0
$225$	0	0
$226$	9.70030	0.645255
$227$	−20.5405	−1.36332	−0.681660	−	0.731669i	$-0.738742\pi$
$227$	−20.5405	−1.36332	−0.681660	+	0.731669i	$0.738742\pi$
$228$	0	0
$229$	6.98499	0.461581	0.230791	−	0.973003i	$-0.425869\pi$
$229$	6.98499	0.461581	0.230791	+	0.973003i	$0.425869\pi$
$230$	13.9986	0.923043
$231$	0	0
$232$	−19.0038	−1.24766
$233$	−19.5265	−1.27922	−0.639612	−	0.768698i	$-0.720905\pi$
$233$	−19.5265	−1.27922	−0.639612	+	0.768698i	$0.720905\pi$
$234$	0	0
$235$	−50.4741	−3.29257
$236$	−15.3904	−1.00183
$237$	0	0
$238$	0	0
$239$	17.8056	1.15175	0.575873	−	0.817539i	$-0.304662\pi$
$239$	17.8056	1.15175	0.575873	+	0.817539i	$0.304662\pi$
$240$	0	0
$241$	15.2268	0.980844	0.490422	−	0.871485i	$-0.336843\pi$
$241$	15.2268	0.980844	0.490422	+	0.871485i	$0.336843\pi$
$242$	4.35725	0.280095
$243$	0	0
$244$	−3.49012	−0.223432
$245$	0	0
$246$	0	0
$247$	2.18699	0.139155
$248$	11.6600	0.740408
$249$	0	0
$250$	26.2486	1.66011
$251$	−6.70311	−0.423097	−0.211548	−	0.977368i	$-0.567851\pi$
$251$	−6.70311	−0.423097	−0.211548	+	0.977368i	$0.567851\pi$
$252$	0	0
$253$	−17.9121	−1.12612
$254$	−2.33848	−0.146730
$255$	0	0
$256$	−12.4457	−0.777854
$257$	−8.25135	−0.514705	−0.257353	−	0.966318i	$-0.582850\pi$
$257$	−8.25135	−0.514705	−0.257353	+	0.966318i	$0.582850\pi$
$258$	0	0
$259$	0	0
$260$	−13.3953	−0.830741
$261$	0	0
$262$	2.78420	0.172009
$263$	0.700305	0.0431827	0.0215913	−	0.999767i	$-0.493127\pi$
$263$	0.700305	0.0431827	0.0215913	+	0.999767i	$0.493127\pi$
$264$	0	0
$265$	−18.8104	−1.15551
$266$	0	0
$267$	0	0
$268$	8.11819	0.495897
$269$	−13.9652	−0.851473	−0.425736	−	0.904847i	$-0.639985\pi$
$269$	−13.9652	−0.851473	−0.425736	+	0.904847i	$0.639985\pi$
$270$	0	0
$271$	−8.32570	−0.505750	−0.252875	−	0.967499i	$-0.581376\pi$
$271$	−8.32570	−0.505750	−0.252875	+	0.967499i	$0.581376\pi$
$272$	0.551049	0.0334122
$273$	0	0
$274$	−0.871925	−0.0526749
$275$	−54.0817	−3.26125
$276$	0	0
$277$	−15.1501	−0.910280	−0.455140	−	0.890420i	$-0.650411\pi$
$277$	−15.1501	−0.910280	−0.455140	+	0.890420i	$0.650411\pi$
$278$	3.18909	0.191269
$279$	0	0
$280$	0	0
$281$	10.2038	0.608708	0.304354	−	0.952559i	$-0.401559\pi$
$281$	10.2038	0.608708	0.304354	+	0.952559i	$0.401559\pi$
$282$	0	0
$283$	−6.29116	−0.373971	−0.186985	−	0.982363i	$-0.559872\pi$
$283$	−6.29116	−0.373971	−0.186985	+	0.982363i	$0.559872\pi$
$284$	11.9534	0.709306
$285$	0	0
$286$	−6.73251	−0.398102
$287$	0	0
$288$	0	0
$289$	−16.6519	−0.979521
$290$	23.5915	1.38534
$291$	0	0
$292$	−15.3353	−0.897433
$293$	1.48286	0.0866294	0.0433147	−	0.999061i	$-0.486208\pi$
$293$	1.48286	0.0866294	0.0433147	+	0.999061i	$0.486208\pi$
$294$	0	0
$295$	45.7162	2.66170
$296$	−23.1069	−1.34306
$297$	0	0
$298$	18.0195	1.04384
$299$	9.55686	0.552688
$300$	0	0
$301$	0	0
$302$	6.38422	0.367371
$303$	0	0
$304$	0.933914	0.0535637
$305$	10.3672	0.593622
$306$	0	0
$307$	29.9624	1.71004	0.855022	−	0.518592i	$-0.173544\pi$
$307$	29.9624	1.71004	0.855022	+	0.518592i	$0.173544\pi$
$308$	0	0
$309$	0	0
$310$	−14.4748	−0.822111
$311$	4.38115	0.248432	0.124216	−	0.992255i	$-0.460358\pi$
$311$	4.38115	0.248432	0.124216	+	0.992255i	$0.460358\pi$
$312$	0	0
$313$	−20.4536	−1.15611	−0.578053	−	0.815999i	$-0.696187\pi$
$313$	−20.4536	−1.15611	−0.578053	+	0.815999i	$0.696187\pi$
$314$	−3.33362	−0.188127
$315$	0	0
$316$	−12.2972	−0.691774
$317$	8.81648	0.495183	0.247591	−	0.968865i	$-0.420361\pi$
$317$	8.81648	0.495183	0.247591	+	0.968865i	$0.420361\pi$
$318$	0	0
$319$	−30.1867	−1.69013
$320$	10.8127	0.604447
$321$	0	0
$322$	0	0
$323$	0.590042	0.0328308
$324$	0	0
$325$	28.8549	1.60058
$326$	−0.110330	−0.00611062
$327$	0	0
$328$	5.61885	0.310249
$329$	0	0
$330$	0	0
$331$	12.0580	0.662767	0.331383	−	0.943496i	$-0.392485\pi$
$331$	12.0580	0.662767	0.331383	+	0.943496i	$0.392485\pi$
$332$	6.11470	0.335588
$333$	0	0
$334$	0.665138	0.0363947
$335$	−24.1145	−1.31752
$336$	0	0
$337$	27.5871	1.50277	0.751383	−	0.659866i	$-0.229387\pi$
$337$	27.5871	1.50277	0.751383	+	0.659866i	$0.229387\pi$
$338$	−6.17123	−0.335671
$339$	0	0
$340$	−3.61401	−0.195997
$341$	18.5213	1.00299
$342$	0	0
$343$	0	0
$344$	22.4398	1.20987
$345$	0	0
$346$	18.2227	0.979658
$347$	−0.991536	−0.0532284	−0.0266142	−	0.999646i	$-0.508473\pi$
$347$	−0.991536	−0.0532284	−0.0266142	+	0.999646i	$0.508473\pi$
$348$	0	0
$349$	−31.6583	−1.69463	−0.847316	−	0.531090i	$-0.821783\pi$
$349$	−31.6583	−1.69463	−0.847316	+	0.531090i	$0.821783\pi$
$350$	0	0
$351$	0	0
$352$	−24.0298	−1.28079
$353$	8.90381	0.473902	0.236951	−	0.971522i	$-0.423852\pi$
$353$	8.90381	0.473902	0.236951	+	0.971522i	$0.423852\pi$
$354$	0	0
$355$	−35.5069	−1.88451
$356$	−5.44389	−0.288525
$357$	0	0
$358$	9.72323	0.513889
$359$	21.6998	1.14527	0.572635	−	0.819811i	$-0.305921\pi$
$359$	21.6998	1.14527	0.572635	+	0.819811i	$0.305921\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−13.9716	−0.734332
$363$	0	0
$364$	0	0
$365$	45.5526	2.38433
$366$	0	0
$367$	−12.7772	−0.666964	−0.333482	−	0.942757i	$-0.608224\pi$
$367$	−12.7772	−0.666964	−0.333482	+	0.942757i	$0.608224\pi$
$368$	4.08109	0.212742
$369$	0	0
$370$	28.6851	1.49127
$371$	0	0
$372$	0	0
$373$	−13.2182	−0.684415	−0.342207	−	0.939624i	$-0.611175\pi$
$373$	−13.2182	−0.684415	−0.342207	+	0.939624i	$0.611175\pi$
$374$	−1.81641	−0.0939244
$375$	0	0
$376$	30.5357	1.57476
$377$	16.1059	0.829496
$378$	0	0
$379$	−1.72767	−0.0887443	−0.0443722	−	0.999015i	$-0.514129\pi$
$379$	−1.72767	−0.0887443	−0.0443722	+	0.999015i	$0.514129\pi$
$380$	−6.12500	−0.314206
$381$	0	0
$382$	17.4904	0.894889
$383$	−4.85563	−0.248111	−0.124056	−	0.992275i	$-0.539590\pi$
$383$	−4.85563	−0.248111	−0.124056	+	0.992275i	$0.539590\pi$
$384$	0	0
$385$	0	0
$386$	2.39459	0.121881
$387$	0	0
$388$	3.03026	0.153838
$389$	−10.6299	−0.538955	−0.269477	−	0.963007i	$-0.586851\pi$
$389$	−10.6299	−0.538955	−0.269477	+	0.963007i	$0.586851\pi$
$390$	0	0
$391$	2.57841	0.130396
$392$	0	0
$393$	0	0
$394$	6.19729	0.312215
$395$	36.5281	1.83793
$396$	0	0
$397$	−12.7514	−0.639974	−0.319987	−	0.947422i	$-0.603678\pi$
$397$	−12.7514	−0.639974	−0.319987	+	0.947422i	$0.603678\pi$
$398$	14.7219	0.737941
$399$	0	0
$400$	12.3220	0.616098
$401$	−19.2514	−0.961367	−0.480683	−	0.876894i	$-0.659611\pi$
$401$	−19.2514	−0.961367	−0.480683	+	0.876894i	$0.659611\pi$
$402$	0	0
$403$	−9.88190	−0.492253
$404$	−21.6265	−1.07596
$405$	0	0
$406$	0	0
$407$	−36.7043	−1.81936
$408$	0	0
$409$	−3.13980	−0.155253	−0.0776266	−	0.996983i	$-0.524734\pi$
$409$	−3.13980	−0.155253	−0.0776266	+	0.996983i	$0.524734\pi$
$410$	−6.97529	−0.344485
$411$	0	0
$412$	7.02736	0.346213
$413$	0	0
$414$	0	0
$415$	−18.1633	−0.891601
$416$	12.8209	0.628598
$417$	0	0
$418$	−3.07844	−0.150572
$419$	12.4881	0.610085	0.305043	−	0.952339i	$-0.401329\pi$
$419$	12.4881	0.610085	0.305043	+	0.952339i	$0.401329\pi$
$420$	0	0
$421$	−14.5141	−0.707376	−0.353688	−	0.935364i	$-0.615072\pi$
$421$	−14.5141	−0.707376	−0.353688	+	0.935364i	$0.615072\pi$
$422$	7.12780	0.346976
$423$	0	0
$424$	11.3799	0.552655
$425$	7.78495	0.377626
$426$	0	0
$427$	0	0
$428$	1.27607	0.0616811
$429$	0	0
$430$	−27.8569	−1.34338
$431$	−23.6671	−1.14001	−0.570003	−	0.821643i	$-0.693058\pi$
$431$	−23.6671	−1.14001	−0.570003	+	0.821643i	$0.693058\pi$
$432$	0	0
$433$	−4.61912	−0.221981	−0.110990	−	0.993821i	$-0.535402\pi$
$433$	−4.61912	−0.221981	−0.110990	+	0.993821i	$0.535402\pi$
$434$	0	0
$435$	0	0
$436$	1.59586	0.0764277
$437$	4.36988	0.209040
$438$	0	0
$439$	−18.9603	−0.904925	−0.452462	−	0.891783i	$-0.649454\pi$
$439$	−18.9603	−0.904925	−0.452462	+	0.891783i	$0.649454\pi$
$440$	45.1172	2.15088
$441$	0	0
$442$	0.969132	0.0460969
$443$	22.4689	1.06753	0.533764	−	0.845633i	$-0.320777\pi$
$443$	22.4689	1.06753	0.533764	+	0.845633i	$0.320777\pi$
$444$	0	0
$445$	16.1707	0.766564
$446$	14.1815	0.671515
$447$	0	0
$448$	0	0
$449$	−15.7360	−0.742629	−0.371314	−	0.928507i	$-0.621093\pi$
$449$	−15.7360	−0.742629	−0.371314	+	0.928507i	$0.621093\pi$
$450$	0	0
$451$	8.92529	0.420276
$452$	−18.5470	−0.872379
$453$	0	0
$454$	−15.4264	−0.723997
$455$	0	0
$456$	0	0
$457$	22.8566	1.06919	0.534594	−	0.845109i	$-0.320464\pi$
$457$	22.8566	1.06919	0.534594	+	0.845109i	$0.320464\pi$
$458$	5.24590	0.245125
$459$	0	0
$460$	−26.7655	−1.24795
$461$	26.0021	1.21104	0.605519	−	0.795831i	$-0.292965\pi$
$461$	26.0021	1.21104	0.605519	+	0.795831i	$0.292965\pi$
$462$	0	0
$463$	20.3481	0.945654	0.472827	−	0.881155i	$-0.343233\pi$
$463$	20.3481	0.945654	0.472827	+	0.881155i	$0.343233\pi$
$464$	6.87774	0.319291
$465$	0	0
$466$	−14.6649	−0.679337
$467$	15.1302	0.700141	0.350071	−	0.936723i	$-0.386158\pi$
$467$	15.1302	0.700141	0.350071	+	0.936723i	$0.386158\pi$
$468$	0	0
$469$	0	0
$470$	−37.9073	−1.74853
$471$	0	0
$472$	−27.6573	−1.27303
$473$	35.6446	1.63894
$474$	0	0
$475$	13.1939	0.605377
$476$	0	0
$477$	0	0
$478$	13.3724	0.611640
$479$	27.9265	1.27599	0.637997	−	0.770039i	$-0.279763\pi$
$479$	27.9265	1.27599	0.637997	+	0.770039i	$0.279763\pi$
$480$	0	0
$481$	19.5833	0.892921
$482$	11.4357	0.520881
$483$	0	0
$484$	−8.33109	−0.378686
$485$	−9.00118	−0.408722
$486$	0	0
$487$	0.660652	0.0299370	0.0149685	−	0.999888i	$-0.495235\pi$
$487$	0.660652	0.0299370	0.0149685	+	0.999888i	$0.495235\pi$
$488$	−6.27190	−0.283916
$489$	0	0
$490$	0	0
$491$	5.09463	0.229917	0.114959	−	0.993370i	$-0.463326\pi$
$491$	5.09463	0.229917	0.114959	+	0.993370i	$0.463326\pi$
$492$	0	0
$493$	4.34532	0.195703
$494$	1.64248	0.0738987
$495$	0	0
$496$	−4.21989	−0.189479
$497$	0	0
$498$	0	0
$499$	−12.7990	−0.572963	−0.286482	−	0.958086i	$-0.592486\pi$
$499$	−12.7990	−0.572963	−0.286482	+	0.958086i	$0.592486\pi$
$500$	−50.1876	−2.24446
$501$	0	0
$502$	−5.03420	−0.224687
$503$	32.1398	1.43304	0.716522	−	0.697565i	$-0.245733\pi$
$503$	32.1398	1.43304	0.716522	+	0.697565i	$0.245733\pi$
$504$	0	0
$505$	64.2401	2.85865
$506$	−13.4524	−0.598033
$507$	0	0
$508$	4.47120	0.198377
$509$	36.4645	1.61626	0.808130	−	0.589004i	$-0.200480\pi$
$509$	36.4645	1.61626	0.808130	+	0.589004i	$0.200480\pi$
$510$	0	0
$511$	0	0
$512$	10.2949	0.454973
$513$	0	0
$514$	−6.19697	−0.273336
$515$	−20.8743	−0.919832
$516$	0	0
$517$	48.5046	2.13323
$518$	0	0
$519$	0	0
$520$	−24.0720	−1.05562
$521$	35.5212	1.55621	0.778107	−	0.628132i	$-0.216180\pi$
$521$	35.5212	1.55621	0.778107	+	0.628132i	$0.216180\pi$
$522$	0	0
$523$	−14.1623	−0.619273	−0.309636	−	0.950855i	$-0.600207\pi$
$523$	−14.1623	−0.619273	−0.309636	+	0.950855i	$0.600207\pi$
$524$	−5.32341	−0.232554
$525$	0	0
$526$	0.525946	0.0229323
$527$	−2.66611	−0.116137
$528$	0	0
$529$	−3.90417	−0.169747
$530$	−14.1270	−0.613639
$531$	0	0
$532$	0	0
$533$	−4.76202	−0.206266
$534$	0	0
$535$	−3.79047	−0.163876
$536$	14.5888	0.630138
$537$	0	0
$538$	−10.4882	−0.452178
$539$	0	0
$540$	0	0
$541$	−16.7454	−0.719940	−0.359970	−	0.932964i	$-0.617213\pi$
$541$	−16.7454	−0.719940	−0.359970	+	0.932964i	$0.617213\pi$
$542$	−6.25280	−0.268581
$543$	0	0
$544$	3.45905	0.148305
$545$	−4.74039	−0.203056
$546$	0	0
$547$	−41.3888	−1.76966	−0.884829	−	0.465917i	$-0.845725\pi$
$547$	−41.3888	−1.76966	−0.884829	+	0.465917i	$0.845725\pi$
$548$	1.66713	0.0712161
$549$	0	0
$550$	−40.6166	−1.73190
$551$	7.36442	0.313735
$552$	0	0
$553$	0	0
$554$	−11.3781	−0.483408
$555$	0	0
$556$	−6.09757	−0.258594
$557$	−37.6750	−1.59634	−0.798172	−	0.602430i	$-0.794199\pi$
$557$	−37.6750	−1.59634	−0.798172	+	0.602430i	$0.794199\pi$
$558$	0	0
$559$	−19.0179	−0.804372
$560$	0	0
$561$	0	0
$562$	7.66330	0.323257
$563$	−3.45814	−0.145743	−0.0728717	−	0.997341i	$-0.523216\pi$
$563$	−3.45814	−0.145743	−0.0728717	+	0.997341i	$0.523216\pi$
$564$	0	0
$565$	55.0927	2.31777
$566$	−4.72482	−0.198599
$567$	0	0
$568$	21.4809	0.901318
$569$	−7.69204	−0.322467	−0.161234	−	0.986916i	$-0.551547\pi$
$569$	−7.69204	−0.322467	−0.161234	+	0.986916i	$0.551547\pi$
$570$	0	0
$571$	−0.417131	−0.0174564	−0.00872819	−	0.999962i	$-0.502778\pi$
$571$	−0.417131	−0.0174564	−0.00872819	+	0.999962i	$0.502778\pi$
$572$	12.8726	0.538231
$573$	0	0
$574$	0	0
$575$	57.6557	2.40441
$576$	0	0
$577$	−23.2905	−0.969596	−0.484798	−	0.874626i	$-0.661107\pi$
$577$	−23.2905	−0.969596	−0.484798	+	0.874626i	$0.661107\pi$
$578$	−12.5059	−0.520179
$579$	0	0
$580$	−45.1071	−1.87297
$581$	0	0
$582$	0	0
$583$	18.0764	0.748647
$584$	−27.5583	−1.14037
$585$	0	0
$586$	1.11366	0.0460049
$587$	−30.5061	−1.25912	−0.629562	−	0.776951i	$-0.716766\pi$
$587$	−30.5061	−1.25912	−0.629562	+	0.776951i	$0.716766\pi$
$588$	0	0
$589$	−4.51850	−0.186182
$590$	34.3340	1.41351
$591$	0	0
$592$	8.36270	0.343705
$593$	−32.6861	−1.34226	−0.671129	−	0.741341i	$-0.734190\pi$
$593$	−32.6861	−1.34226	−0.671129	+	0.741341i	$0.734190\pi$
$594$	0	0
$595$	0	0
$596$	−34.4533	−1.41126
$597$	0	0
$598$	7.17744	0.293507
$599$	20.3738	0.832451	0.416225	−	0.909262i	$-0.363353\pi$
$599$	20.3738	0.832451	0.416225	+	0.909262i	$0.363353\pi$
$600$	0	0
$601$	−12.3891	−0.505363	−0.252682	−	0.967549i	$-0.581313\pi$
$601$	−12.3891	−0.505363	−0.252682	+	0.967549i	$0.581313\pi$
$602$	0	0
$603$	0	0
$604$	−12.2067	−0.496683
$605$	24.7469	1.00611
$606$	0	0
$607$	33.3480	1.35355	0.676776	−	0.736189i	$-0.263376\pi$
$607$	33.3480	1.35355	0.676776	+	0.736189i	$0.263376\pi$
$608$	5.86237	0.237751
$609$	0	0
$610$	7.78599	0.315245
$611$	−25.8793	−1.04696
$612$	0	0
$613$	−38.7985	−1.56706	−0.783528	−	0.621356i	$-0.786582\pi$
$613$	−38.7985	−1.56706	−0.783528	+	0.621356i	$0.786582\pi$
$614$	22.5025	0.908126
$615$	0	0
$616$	0	0
$617$	−1.98936	−0.0800887	−0.0400443	−	0.999198i	$-0.512750\pi$
$617$	−1.98936	−0.0800887	−0.0400443	+	0.999198i	$0.512750\pi$
$618$	0	0
$619$	−13.0014	−0.522572	−0.261286	−	0.965261i	$-0.584147\pi$
$619$	−13.0014	−0.522572	−0.261286	+	0.965261i	$0.584147\pi$
$620$	27.6758	1.11149
$621$	0	0
$622$	3.29035	0.131931
$623$	0	0
$624$	0	0
$625$	83.1093	3.32437
$626$	−15.3611	−0.613955
$627$	0	0
$628$	6.37391	0.254347
$629$	5.28351	0.210667
$630$	0	0
$631$	−14.9918	−0.596814	−0.298407	−	0.954439i	$-0.596455\pi$
$631$	−14.9918	−0.596814	−0.298407	+	0.954439i	$0.596455\pi$
$632$	−22.0987	−0.879040
$633$	0	0
$634$	6.62139	0.262969
$635$	−13.2814	−0.527056
$636$	0	0
$637$	0	0
$638$	−22.6710	−0.897552
$639$	0	0
$640$	−41.8905	−1.65587
$641$	−24.0547	−0.950105	−0.475053	−	0.879957i	$-0.657571\pi$
$641$	−24.0547	−0.950105	−0.475053	+	0.879957i	$0.657571\pi$
$642$	0	0
$643$	24.3676	0.960964	0.480482	−	0.877005i	$-0.340462\pi$
$643$	24.3676	0.960964	0.480482	+	0.877005i	$0.340462\pi$
$644$	0	0
$645$	0	0
$646$	0.443136	0.0174350
$647$	38.2402	1.50338	0.751688	−	0.659519i	$-0.229240\pi$
$647$	38.2402	1.50338	0.751688	+	0.659519i	$0.229240\pi$
$648$	0	0
$649$	−43.9324	−1.72450
$650$	21.6707	0.849995
$651$	0	0
$652$	0.210952	0.00826151
$653$	−21.1624	−0.828147	−0.414074	−	0.910243i	$-0.635894\pi$
$653$	−21.1624	−0.828147	−0.414074	+	0.910243i	$0.635894\pi$
$654$	0	0
$655$	15.8128	0.617858
$656$	−2.03354	−0.0793963
$657$	0	0
$658$	0	0
$659$	−24.6069	−0.958548	−0.479274	−	0.877665i	$-0.659100\pi$
$659$	−24.6069	−0.958548	−0.479274	+	0.877665i	$0.659100\pi$
$660$	0	0
$661$	−38.1953	−1.48562	−0.742812	−	0.669500i	$-0.766508\pi$
$661$	−38.1953	−1.48562	−0.742812	+	0.669500i	$0.766508\pi$
$662$	9.05584	0.351965
$663$	0	0
$664$	10.9884	0.426432
$665$	0	0
$666$	0	0
$667$	32.1816	1.24608
$668$	−1.27175	−0.0492054
$669$	0	0
$670$	−18.1106	−0.699673
$671$	−9.96264	−0.384603
$672$	0	0
$673$	−28.0333	−1.08060	−0.540302	−	0.841471i	$-0.681690\pi$
$673$	−28.0333	−1.08060	−0.540302	+	0.841471i	$0.681690\pi$
$674$	20.7186	0.798050
$675$	0	0
$676$	11.7994	0.453824
$677$	5.23635	0.201249	0.100625	−	0.994924i	$-0.467916\pi$
$677$	5.23635	0.201249	0.100625	+	0.994924i	$0.467916\pi$
$678$	0	0
$679$	0	0
$680$	−6.49454	−0.249054
$681$	0	0
$682$	13.9099	0.532639
$683$	1.25171	0.0478955	0.0239478	−	0.999713i	$-0.492376\pi$
$683$	1.25171	0.0478955	0.0239478	+	0.999713i	$0.492376\pi$
$684$	0	0
$685$	−4.95209	−0.189210
$686$	0	0
$687$	0	0
$688$	−8.12126	−0.309620
$689$	−9.64452	−0.367427
$690$	0	0
$691$	50.8050	1.93271	0.966357	−	0.257204i	$-0.0828012\pi$
$691$	50.8050	1.93271	0.966357	+	0.257204i	$0.0828012\pi$
$692$	−34.8419	−1.32449
$693$	0	0
$694$	−0.744668	−0.0282672
$695$	18.1124	0.687043
$696$	0	0
$697$	−1.28478	−0.0486645
$698$	−23.7762	−0.899941
$699$	0	0
$700$	0	0
$701$	−0.998799	−0.0377241	−0.0188621	−	0.999822i	$-0.506004\pi$
$701$	−0.998799	−0.0377241	−0.0188621	+	0.999822i	$0.506004\pi$
$702$	0	0
$703$	8.95446	0.337724
$704$	−10.3908	−0.391617
$705$	0	0
$706$	6.68698	0.251668
$707$	0	0
$708$	0	0
$709$	−13.9625	−0.524371	−0.262185	−	0.965017i	$-0.584443\pi$
$709$	−13.9625	−0.524371	−0.262185	+	0.965017i	$0.584443\pi$
$710$	−26.6665	−1.00078
$711$	0	0
$712$	−9.78291	−0.366630
$713$	−19.7453	−0.739467
$714$	0	0
$715$	−38.2372	−1.42999
$716$	−18.5909	−0.694774
$717$	0	0
$718$	16.2970	0.608200
$719$	18.5694	0.692521	0.346260	−	0.938138i	$-0.387451\pi$
$719$	18.5694	0.692521	0.346260	+	0.938138i	$0.387451\pi$
$720$	0	0
$721$	0	0
$722$	0.751024	0.0279502
$723$	0	0
$724$	26.7138	0.992811
$725$	97.1654	3.60863
$726$	0	0
$727$	32.1991	1.19420	0.597099	−	0.802168i	$-0.296320\pi$
$727$	32.1991	1.19420	0.597099	+	0.802168i	$0.296320\pi$
$728$	0	0
$729$	0	0
$730$	34.2111	1.26621
$731$	−5.13097	−0.189776
$732$	0	0
$733$	−9.55623	−0.352968	−0.176484	−	0.984304i	$-0.556472\pi$
$733$	−9.55623	−0.352968	−0.176484	+	0.984304i	$0.556472\pi$
$734$	−9.59598	−0.354194
$735$	0	0
$736$	25.6179	0.944287
$737$	23.1736	0.853609
$738$	0	0
$739$	−20.2850	−0.746194	−0.373097	−	0.927792i	$-0.621704\pi$
$739$	−20.2850	−0.746194	−0.373097	+	0.927792i	$0.621704\pi$
$740$	−54.8461	−2.01618
$741$	0	0
$742$	0	0
$743$	−0.965374	−0.0354161	−0.0177081	−	0.999843i	$-0.505637\pi$
$743$	−0.965374	−0.0354161	−0.0177081	+	0.999843i	$0.505637\pi$
$744$	0	0
$745$	102.341	3.74949
$746$	−9.92722	−0.363461
$747$	0	0
$748$	3.47299	0.126985
$749$	0	0
$750$	0	0
$751$	−0.657779	−0.0240027	−0.0120013	−	0.999928i	$-0.503820\pi$
$751$	−0.657779	−0.0240027	−0.0120013	+	0.999928i	$0.503820\pi$
$752$	−11.0513	−0.402999
$753$	0	0
$754$	12.0959	0.440507
$755$	36.2591	1.31960
$756$	0	0
$757$	−1.47679	−0.0536749	−0.0268375	−	0.999640i	$-0.508544\pi$
$757$	−1.47679	−0.0536749	−0.0268375	+	0.999640i	$0.508544\pi$
$758$	−1.29752	−0.0471281
$759$	0	0
$760$	−11.0069	−0.399262
$761$	18.4847	0.670070	0.335035	−	0.942206i	$-0.391252\pi$
$761$	18.4847	0.670070	0.335035	+	0.942206i	$0.391252\pi$
$762$	0	0
$763$	0	0
$764$	−33.4418	−1.20988
$765$	0	0
$766$	−3.64669	−0.131760
$767$	23.4398	0.846362
$768$	0	0
$769$	4.34052	0.156523	0.0782617	−	0.996933i	$-0.475063\pi$
$769$	4.34052	0.156523	0.0782617	+	0.996933i	$0.475063\pi$
$770$	0	0
$771$	0	0
$772$	−4.57847	−0.164783
$773$	33.5513	1.20676	0.603379	−	0.797455i	$-0.293821\pi$
$773$	33.5513	1.20676	0.603379	+	0.797455i	$0.293821\pi$
$774$	0	0
$775$	−59.6166	−2.14149
$776$	5.44551	0.195483
$777$	0	0
$778$	−7.98328	−0.286214
$779$	−2.17744	−0.0780147
$780$	0	0
$781$	34.1214	1.22096
$782$	1.93645	0.0692473
$783$	0	0
$784$	0	0
$785$	−18.9333	−0.675757
$786$	0	0
$787$	42.0571	1.49917	0.749587	−	0.661906i	$-0.230252\pi$
$787$	42.0571	1.49917	0.749587	+	0.661906i	$0.230252\pi$
$788$	−11.8493	−0.422112
$789$	0	0
$790$	27.4335	0.976041
$791$	0	0
$792$	0	0
$793$	5.31549	0.188759
$794$	−9.57660	−0.339861
$795$	0	0
$796$	−28.1483	−0.997691
$797$	2.71101	0.0960289	0.0480145	−	0.998847i	$-0.484711\pi$
$797$	2.71101	0.0960289	0.0480145	+	0.998847i	$0.484711\pi$
$798$	0	0
$799$	−6.98214	−0.247010
$800$	77.3475	2.73465
$801$	0	0
$802$	−14.4582	−0.510538
$803$	−43.7751	−1.54479
$804$	0	0
$805$	0	0
$806$	−7.42155	−0.261413
$807$	0	0
$808$	−38.8638	−1.36722
$809$	−13.3158	−0.468159	−0.234079	−	0.972217i	$-0.575208\pi$
$809$	−13.3158	−0.468159	−0.234079	+	0.972217i	$0.575208\pi$
$810$	0	0
$811$	15.1310	0.531320	0.265660	−	0.964067i	$-0.414410\pi$
$811$	15.1310	0.531320	0.265660	+	0.964067i	$0.414410\pi$
$812$	0	0
$813$	0	0
$814$	−27.5658	−0.966181
$815$	−0.626618	−0.0219495
$816$	0	0
$817$	−8.69594	−0.304232
$818$	−2.35807	−0.0824478
$819$	0	0
$820$	13.3368	0.465741
$821$	24.3734	0.850639	0.425319	−	0.905043i	$-0.360162\pi$
$821$	24.3734	0.850639	0.425319	+	0.905043i	$0.360162\pi$
$822$	0	0
$823$	11.7877	0.410892	0.205446	−	0.978668i	$-0.434135\pi$
$823$	11.7877	0.410892	0.205446	+	0.978668i	$0.434135\pi$
$824$	12.6285	0.439934
$825$	0	0
$826$	0	0
$827$	−17.5429	−0.610028	−0.305014	−	0.952348i	$-0.598661\pi$
$827$	−17.5429	−0.610028	−0.305014	+	0.952348i	$0.598661\pi$
$828$	0	0
$829$	−50.9728	−1.77036	−0.885180	−	0.465249i	$-0.845965\pi$
$829$	−50.9728	−1.77036	−0.885180	+	0.465249i	$0.845965\pi$
$830$	−13.6411	−0.473488
$831$	0	0
$832$	5.54391	0.192201
$833$	0	0
$834$	0	0
$835$	3.77764	0.130731
$836$	5.88600	0.203572
$837$	0	0
$838$	9.37889	0.323988
$839$	−5.84311	−0.201727	−0.100863	−	0.994900i	$-0.532160\pi$
$839$	−5.84311	−0.201727	−0.100863	+	0.994900i	$0.532160\pi$
$840$	0	0
$841$	25.2347	0.870163
$842$	−10.9005	−0.375655
$843$	0	0
$844$	−13.6284	−0.469109
$845$	−35.0494	−1.20574
$846$	0	0
$847$	0	0
$848$	−4.11852	−0.141431
$849$	0	0
$850$	5.84669	0.200540
$851$	39.1299	1.34136
$852$	0	0
$853$	37.0096	1.26719	0.633593	−	0.773667i	$-0.281579\pi$
$853$	37.0096	1.26719	0.633593	+	0.773667i	$0.281579\pi$
$854$	0	0
$855$	0	0
$856$	2.29315	0.0783784
$857$	40.1269	1.37071	0.685355	−	0.728209i	$-0.259647\pi$
$857$	40.1269	1.37071	0.685355	+	0.728209i	$0.259647\pi$
$858$	0	0
$859$	14.4388	0.492644	0.246322	−	0.969188i	$-0.420778\pi$
$859$	14.4388	0.492644	0.246322	+	0.969188i	$0.420778\pi$
$860$	53.2626	1.81624
$861$	0	0
$862$	−17.7746	−0.605405
$863$	−30.6183	−1.04226	−0.521130	−	0.853477i	$-0.674489\pi$
$863$	−30.6183	−1.04226	−0.521130	+	0.853477i	$0.674489\pi$
$864$	0	0
$865$	103.496	3.51895
$866$	−3.46907	−0.117884
$867$	0	0
$868$	0	0
$869$	−35.1028	−1.19078
$870$	0	0
$871$	−12.3641	−0.418941
$872$	2.86783	0.0971169
$873$	0	0
$874$	3.28188	0.111011
$875$	0	0
$876$	0	0
$877$	−16.0685	−0.542596	−0.271298	−	0.962495i	$-0.587453\pi$
$877$	−16.0685	−0.542596	−0.271298	+	0.962495i	$0.587453\pi$
$878$	−14.2396	−0.480564
$879$	0	0
$880$	−16.3285	−0.550435
$881$	−40.2468	−1.35595	−0.677975	−	0.735085i	$-0.737142\pi$
$881$	−40.2468	−1.35595	−0.677975	+	0.735085i	$0.737142\pi$
$882$	0	0
$883$	−0.564426	−0.0189944	−0.00949722	−	0.999955i	$-0.503023\pi$
$883$	−0.564426	−0.0189944	−0.00949722	+	0.999955i	$0.503023\pi$
$884$	−1.85299	−0.0623227
$885$	0	0
$886$	16.8747	0.566915
$887$	−35.5836	−1.19478	−0.597390	−	0.801951i	$-0.703796\pi$
$887$	−35.5836	−1.19478	−0.597390	+	0.801951i	$0.703796\pi$
$888$	0	0
$889$	0	0
$890$	12.1446	0.407087
$891$	0	0
$892$	−27.1152	−0.907883
$893$	−11.8333	−0.395986
$894$	0	0
$895$	55.2229	1.84590
$896$	0	0
$897$	0	0
$898$	−11.8181	−0.394376
$899$	−33.2762	−1.10982
$900$	0	0
$901$	−2.60206	−0.0866872
$902$	6.70311	0.223189
$903$	0	0
$904$	−33.3299	−1.10854
$905$	−79.3516	−2.63774
$906$	0	0
$907$	30.3682	1.00836	0.504179	−	0.863599i	$-0.331795\pi$
$907$	30.3682	1.00836	0.504179	+	0.863599i	$0.331795\pi$
$908$	29.4954	0.978839
$909$	0	0
$910$	0	0
$911$	−17.9947	−0.596191	−0.298096	−	0.954536i	$-0.596351\pi$
$911$	−17.9947	−0.596191	−0.298096	+	0.954536i	$0.596351\pi$
$912$	0	0
$913$	17.4546	0.577662
$914$	17.1659	0.567797
$915$	0	0
$916$	−10.0302	−0.331407
$917$	0	0
$918$	0	0
$919$	41.9568	1.38403	0.692014	−	0.721884i	$-0.256724\pi$
$919$	41.9568	1.38403	0.692014	+	0.721884i	$0.256724\pi$
$920$	−48.0988	−1.58577
$921$	0	0
$922$	19.5282	0.643127
$923$	−18.2052	−0.599232
$924$	0	0
$925$	118.144	3.88456
$926$	15.2819	0.502194
$927$	0	0
$928$	43.1730	1.41722
$929$	−19.1145	−0.627127	−0.313563	−	0.949567i	$-0.601523\pi$
$929$	−19.1145	−0.627127	−0.313563	+	0.949567i	$0.601523\pi$
$930$	0	0
$931$	0	0
$932$	28.0393	0.918458
$933$	0	0
$934$	11.3631	0.371813
$935$	−10.3163	−0.337378
$936$	0	0
$937$	−44.3779	−1.44976	−0.724881	−	0.688874i	$-0.758105\pi$
$937$	−44.3779	−1.44976	−0.724881	+	0.688874i	$0.758105\pi$
$938$	0	0
$939$	0	0
$940$	72.4789	2.36400
$941$	−37.6172	−1.22628	−0.613142	−	0.789973i	$-0.710095\pi$
$941$	−37.6172	−1.22628	−0.613142	+	0.789973i	$0.710095\pi$
$942$	0	0
$943$	−9.51513	−0.309855
$944$	10.0096	0.325783
$945$	0	0
$946$	26.7699	0.870366
$947$	−29.1696	−0.947883	−0.473942	−	0.880556i	$-0.657169\pi$
$947$	−29.1696	−0.947883	−0.473942	+	0.880556i	$0.657169\pi$
$948$	0	0
$949$	23.3559	0.758164
$950$	9.90893	0.321488
$951$	0	0
$952$	0	0
$953$	−25.0234	−0.810586	−0.405293	−	0.914187i	$-0.632830\pi$
$953$	−25.0234	−0.810586	−0.405293	+	0.914187i	$0.632830\pi$
$954$	0	0
$955$	99.3367	3.21446
$956$	−25.5681	−0.826933
$957$	0	0
$958$	20.9735	0.677622
$959$	0	0
$960$	0	0
$961$	−10.5831	−0.341392
$962$	14.7075	0.474190
$963$	0	0
$964$	−21.8651	−0.704228
$965$	13.6000	0.437801
$966$	0	0
$967$	−28.8469	−0.927655	−0.463828	−	0.885926i	$-0.653524\pi$
$967$	−28.8469	−0.927655	−0.463828	+	0.885926i	$0.653524\pi$
$968$	−14.9713	−0.481197
$969$	0	0
$970$	−6.76010	−0.217054
$971$	3.19928	0.102670	0.0513348	−	0.998681i	$-0.483652\pi$
$971$	3.19928	0.102670	0.0513348	+	0.998681i	$0.483652\pi$
$972$	0	0
$973$	0	0
$974$	0.496166	0.0158982
$975$	0	0
$976$	2.26989	0.0726573
$977$	54.4969	1.74351	0.871755	−	0.489943i	$-0.162982\pi$
$977$	54.4969	1.74351	0.871755	+	0.489943i	$0.162982\pi$
$978$	0	0
$979$	−15.5397	−0.496651
$980$	0	0
$981$	0	0
$982$	3.82619	0.122099
$983$	−5.45760	−0.174070	−0.0870352	−	0.996205i	$-0.527739\pi$
$983$	−5.45760	−0.174070	−0.0870352	+	0.996205i	$0.527739\pi$
$984$	0	0
$985$	35.1974	1.12148
$986$	3.26344	0.103929
$987$	0	0
$988$	−3.14043	−0.0999104
$989$	−38.0002	−1.20834
$990$	0	0
$991$	34.2911	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$991$	34.2911	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$992$	−26.4891	−0.841031
$993$	0	0
$994$	0	0
$995$	83.6127	2.65070
$996$	0	0
$997$	−23.2138	−0.735188	−0.367594	−	0.929986i	$-0.619818\pi$
$997$	−23.2138	−0.735188	−0.367594	+	0.929986i	$0.619818\pi$
$998$	−9.61238	−0.304275
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bv.1.3		4
3.2	odd	2		931.2.a.l.1.2	✓	4
7.6	odd	2		8379.2.a.bu.1.3		4
21.2	odd	6		931.2.f.o.704.3		8
21.5	even	6		931.2.f.n.704.3		8
21.11	odd	6		931.2.f.o.324.3		8
21.17	even	6		931.2.f.n.324.3		8
21.20	even	2		931.2.a.m.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
931.2.a.l.1.2	✓	4	3.2	odd	2
931.2.a.m.1.2	yes	4	21.20	even	2
931.2.f.n.324.3		8	21.17	even	6
931.2.f.n.704.3		8	21.5	even	6
931.2.f.o.324.3		8	21.11	odd	6
931.2.f.o.704.3		8	21.2	odd	6
8379.2.a.bu.1.3		4	7.6	odd	2
8379.2.a.bv.1.3		4	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+0.751024 q^{2} -1.43596 q^{4} +4.26543 q^{5} -2.58049 q^{8} +3.20344 q^{10} -4.09899 q^{11} +2.18699 q^{13} +0.933914 q^{16} +0.590042 q^{17} +1.00000 q^{19} -6.12500 q^{20} -3.07844 q^{22} +4.36988 q^{23} +13.1939 q^{25} +1.64248 q^{26} +7.36442 q^{29} -4.51850 q^{31} +5.86237 q^{32} +0.443136 q^{34} +8.95446 q^{37} +0.751024 q^{38} -11.0069 q^{40} -2.17744 q^{41} -8.69594 q^{43} +5.88600 q^{44} +3.28188 q^{46} -11.8333 q^{47} +9.90893 q^{50} -3.14043 q^{52} -4.40996 q^{53} -17.4840 q^{55} +5.53086 q^{58} +10.7178 q^{59} +2.43051 q^{61} -3.39350 q^{62} +2.53496 q^{64} +9.32844 q^{65} -5.65348 q^{67} -0.847279 q^{68} -8.32434 q^{71} +10.6795 q^{73} +6.72502 q^{74} -1.43596 q^{76} +8.56377 q^{79} +3.98355 q^{80} -1.63531 q^{82} -4.25826 q^{83} +2.51678 q^{85} -6.53086 q^{86} +10.5774 q^{88} +3.79110 q^{89} -6.27498 q^{92} -8.88709 q^{94} +4.26543 q^{95} -2.11026 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 8 q^{5} - 6 q^{8} - 10 q^{10} + 6 q^{11} - 2 q^{13} + 2 q^{16} + 8 q^{17} + 4 q^{19} - 14 q^{22} + 8 q^{23} + 20 q^{25} + 16 q^{26} - 2 q^{29} - 2 q^{32} + 22 q^{34} + 10 q^{37} - 22 q^{40}+ \cdots - 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 + 8 * q^5 - 6 * q^8 - 10 * q^10 + 6 * q^11 - 2 * q^13 + 2 * q^16 + 8 * q^17 + 4 * q^19 - 14 * q^22 + 8 * q^23 + 20 * q^25 + 16 * q^26 - 2 * q^29 - 2 * q^32 + 22 * q^34 + 10 * q^37 - 22 * q^40 + 12 * q^41 + 4 * q^43 + 14 * q^44 - 8 * q^46 + 16 * q^47 - 12 * q^50 - 28 * q^52 - 12 * q^53 + 8 * q^55 + 4 * q^58 + 14 * q^59 - 20 * q^61 - 20 * q^62 - 20 * q^64 - 10 * q^65 + 2 * q^67 - 24 * q^68 + 2 * q^71 + 16 * q^73 + 26 * q^74 + 2 * q^76 - 8 * q^79 + 28 * q^80 + 12 * q^82 + 20 * q^83 - 14 * q^85 - 8 * q^86 - 2 * q^88 + 16 * q^89 - 26 * q^92 - 10 * q^94 + 8 * q^95 - 2 * q^97

Introduction

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