LMFDB - Embedded newform 8379.2.a.bc.1.1

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 = [2,0,0,2,0,0,0,0,0,0,0,0,0,0,0,-10,-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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2793)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{8} +3.46410 q^{11} +3.46410 q^{13} -5.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} -6.00000 q^{22} +0.535898 q^{23} -5.00000 q^{25} -6.00000 q^{26} +4.00000 q^{29} -2.92820 q^{31} +5.19615 q^{32} +6.92820 q^{34} +4.92820 q^{37} -1.73205 q^{38} +10.9282 q^{41} -10.9282 q^{43} +3.46410 q^{44} -0.928203 q^{46} -4.92820 q^{47} +8.66025 q^{50} +3.46410 q^{52} +4.00000 q^{53} -6.92820 q^{58} +8.00000 q^{59} +1.07180 q^{61} +5.07180 q^{62} +1.00000 q^{64} -6.39230 q^{67} -4.00000 q^{68} +3.46410 q^{71} -8.00000 q^{73} -8.53590 q^{74} +1.00000 q^{76} +3.46410 q^{79} -18.9282 q^{82} +8.92820 q^{83} +18.9282 q^{86} +6.00000 q^{88} +12.0000 q^{89} +0.535898 q^{92} +8.53590 q^{94} +19.4641 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 10 q^{16} - 8 q^{17} + 2 q^{19} - 12 q^{22} + 8 q^{23} - 10 q^{25} - 12 q^{26} + 8 q^{29} + 8 q^{31} - 4 q^{37} + 8 q^{41} - 8 q^{43} + 12 q^{46} + 4 q^{47} + 8 q^{53} + 16 q^{59} + 16 q^{61}+ \cdots + 32 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 10 * q^16 - 8 * q^17 + 2 * q^19 - 12 * q^22 + 8 * q^23 - 10 * q^25 - 12 * q^26 + 8 * q^29 + 8 * q^31 - 4 * q^37 + 8 * q^41 - 8 * q^43 + 12 * q^46 + 4 * q^47 + 8 * q^53 + 16 * q^59 + 16 * q^61 + 24 * q^62 + 2 * q^64 + 8 * q^67 - 8 * q^68 - 16 * q^73 - 24 * q^74 + 2 * q^76 - 24 * q^82 + 4 * q^83 + 24 * q^86 + 12 * q^88 + 24 * q^89 + 8 * q^92 + 24 * q^94 + 32 * 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 $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.73205	0.612372
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	−6.00000	−1.27920
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−6.00000	−1.17670
$27$	0	0
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−2.92820	−0.525921	−0.262960	−	0.964807i	$-0.584699\pi$
$31$	−2.92820	−0.525921	−0.262960	+	0.964807i	$0.584699\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	6.92820	1.18818
$35$	0	0
$36$	0	0
$37$	4.92820	0.810192	0.405096	−	0.914274i	$-0.367238\pi$
$37$	4.92820	0.810192	0.405096	+	0.914274i	$0.367238\pi$
$38$	−1.73205	−0.280976
$39$	0	0
$40$	0	0
$41$	10.9282	1.70670	0.853349	−	0.521340i	$-0.174568\pi$
$41$	10.9282	1.70670	0.853349	+	0.521340i	$0.174568\pi$
$42$	0	0
$43$	−10.9282	−1.66654	−0.833268	−	0.552870i	$-0.813533\pi$
$43$	−10.9282	−1.66654	−0.833268	+	0.552870i	$0.813533\pi$
$44$	3.46410	0.522233
$45$	0	0
$46$	−0.928203	−0.136856
$47$	−4.92820	−0.718852	−0.359426	−	0.933174i	$-0.617028\pi$
$47$	−4.92820	−0.718852	−0.359426	+	0.933174i	$0.617028\pi$
$48$	0	0
$49$	0	0
$50$	8.66025	1.22474
$51$	0	0
$52$	3.46410	0.480384
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−6.92820	−0.909718
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	1.07180	0.137230	0.0686148	−	0.997643i	$-0.478142\pi$
$61$	1.07180	0.137230	0.0686148	+	0.997643i	$0.478142\pi$
$62$	5.07180	0.644119
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−6.39230	−0.780944	−0.390472	−	0.920615i	$-0.627688\pi$
$67$	−6.39230	−0.780944	−0.390472	+	0.920615i	$0.627688\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	0	0
$71$	3.46410	0.411113	0.205557	−	0.978645i	$-0.434100\pi$
$71$	3.46410	0.411113	0.205557	+	0.978645i	$0.434100\pi$
$72$	0	0
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	−8.53590	−0.992278
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	0	0
$79$	3.46410	0.389742	0.194871	−	0.980829i	$-0.437571\pi$
$79$	3.46410	0.389742	0.194871	+	0.980829i	$0.437571\pi$
$80$	0	0
$81$	0	0
$82$	−18.9282	−2.09027
$83$	8.92820	0.979998	0.489999	−	0.871723i	$-0.336997\pi$
$83$	8.92820	0.979998	0.489999	+	0.871723i	$0.336997\pi$
$84$	0	0
$85$	0	0
$86$	18.9282	2.04108
$87$	0	0
$88$	6.00000	0.639602
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	0.535898	0.0558713
$93$	0	0
$94$	8.53590	0.880411
$95$	0	0
$96$	0	0
$97$	19.4641	1.97628	0.988140	−	0.153555i	$-0.0490723\pi$
$97$	19.4641	1.97628	0.988140	+	0.153555i	$0.0490723\pi$
$98$	0	0
$99$	0	0
$100$	−5.00000	−0.500000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−6.92820	−0.672927
$107$	−2.39230	−0.231273	−0.115636	−	0.993292i	$-0.536891\pi$
$107$	−2.39230	−0.231273	−0.115636	+	0.993292i	$0.536891\pi$
$108$	0	0
$109$	4.92820	0.472036	0.236018	−	0.971749i	$-0.424158\pi$
$109$	4.92820	0.472036	0.236018	+	0.971749i	$0.424158\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.9282	1.78062	0.890308	−	0.455359i	$-0.150489\pi$
$113$	18.9282	1.78062	0.890308	+	0.455359i	$0.150489\pi$
$114$	0	0
$115$	0	0
$116$	4.00000	0.371391
$117$	0	0
$118$	−13.8564	−1.27559
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.85641	−0.168071
$123$	0	0
$124$	−2.92820	−0.262960
$125$	0	0
$126$	0	0
$127$	−19.4641	−1.72716	−0.863580	−	0.504212i	$-0.831783\pi$
$127$	−19.4641	−1.72716	−0.863580	+	0.504212i	$0.831783\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	0	0
$131$	4.92820	0.430579	0.215290	−	0.976550i	$-0.430930\pi$
$131$	4.92820	0.430579	0.215290	+	0.976550i	$0.430930\pi$
$132$	0	0
$133$	0	0
$134$	11.0718	0.956458
$135$	0	0
$136$	−6.92820	−0.594089
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	−6.00000	−0.503509
$143$	12.0000	1.00349
$144$	0	0
$145$	0	0
$146$	13.8564	1.14676
$147$	0	0
$148$	4.92820	0.405096
$149$	−7.07180	−0.579344	−0.289672	−	0.957126i	$-0.593546\pi$
$149$	−7.07180	−0.579344	−0.289672	+	0.957126i	$0.593546\pi$
$150$	0	0
$151$	10.3923	0.845714	0.422857	−	0.906196i	$-0.361027\pi$
$151$	10.3923	0.845714	0.422857	+	0.906196i	$0.361027\pi$
$152$	1.73205	0.140488
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.9282	0.855963	0.427981	−	0.903788i	$-0.359225\pi$
$163$	10.9282	0.855963	0.427981	+	0.903788i	$0.359225\pi$
$164$	10.9282	0.853349
$165$	0	0
$166$	−15.4641	−1.20025
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	−10.9282	−0.833268
$173$	−17.8564	−1.35760	−0.678799	−	0.734324i	$-0.737499\pi$
$173$	−17.8564	−1.35760	−0.678799	+	0.734324i	$0.737499\pi$
$174$	0	0
$175$	0	0
$176$	−17.3205	−1.30558
$177$	0	0
$178$	−20.7846	−1.55787
$179$	4.53590	0.339029	0.169514	−	0.985528i	$-0.445780\pi$
$179$	4.53590	0.339029	0.169514	+	0.985528i	$0.445780\pi$
$180$	0	0
$181$	9.32051	0.692788	0.346394	−	0.938089i	$-0.387406\pi$
$181$	9.32051	0.692788	0.346394	+	0.938089i	$0.387406\pi$
$182$	0	0
$183$	0	0
$184$	0.928203	0.0684280
$185$	0	0
$186$	0	0
$187$	−13.8564	−1.01328
$188$	−4.92820	−0.359426
$189$	0	0
$190$	0	0
$191$	7.46410	0.540083	0.270042	−	0.962849i	$-0.412963\pi$
$191$	7.46410	0.540083	0.270042	+	0.962849i	$0.412963\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−33.7128	−2.42044
$195$	0	0
$196$	0	0
$197$	−8.92820	−0.636108	−0.318054	−	0.948073i	$-0.603029\pi$
$197$	−8.92820	−0.636108	−0.318054	+	0.948073i	$0.603029\pi$
$198$	0	0
$199$	−21.8564	−1.54936	−0.774680	−	0.632354i	$-0.782089\pi$
$199$	−21.8564	−1.54936	−0.774680	+	0.632354i	$0.782089\pi$
$200$	−8.66025	−0.612372
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	5.07180	0.353369
$207$	0	0
$208$	−17.3205	−1.20096
$209$	3.46410	0.239617
$210$	0	0
$211$	−7.46410	−0.513850	−0.256925	−	0.966431i	$-0.582709\pi$
$211$	−7.46410	−0.513850	−0.256925	+	0.966431i	$0.582709\pi$
$212$	4.00000	0.274721
$213$	0	0
$214$	4.14359	0.283250
$215$	0	0
$216$	0	0
$217$	0	0
$218$	−8.53590	−0.578124
$219$	0	0
$220$	0	0
$221$	−13.8564	−0.932083
$222$	0	0
$223$	26.9282	1.80325	0.901623	−	0.432523i	$-0.142377\pi$
$223$	26.9282	1.80325	0.901623	+	0.432523i	$0.142377\pi$
$224$	0	0
$225$	0	0
$226$	−32.7846	−2.18080
$227$	−1.85641	−0.123214	−0.0616070	−	0.998100i	$-0.519623\pi$
$227$	−1.85641	−0.123214	−0.0616070	+	0.998100i	$0.519623\pi$
$228$	0	0
$229$	−6.92820	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$230$	0	0
$231$	0	0
$232$	6.92820	0.454859
$233$	19.8564	1.30084	0.650418	−	0.759576i	$-0.274594\pi$
$233$	19.8564	1.30084	0.650418	+	0.759576i	$0.274594\pi$
$234$	0	0
$235$	0	0
$236$	8.00000	0.520756
$237$	0	0
$238$	0	0
$239$	8.53590	0.552141	0.276071	−	0.961137i	$-0.410968\pi$
$239$	8.53590	0.552141	0.276071	+	0.961137i	$0.410968\pi$
$240$	0	0
$241$	−3.46410	−0.223142	−0.111571	−	0.993756i	$-0.535588\pi$
$241$	−3.46410	−0.223142	−0.111571	+	0.993756i	$0.535588\pi$
$242$	−1.73205	−0.111340
$243$	0	0
$244$	1.07180	0.0686148
$245$	0	0
$246$	0	0
$247$	3.46410	0.220416
$248$	−5.07180	−0.322059
$249$	0	0
$250$	0	0
$251$	−0.928203	−0.0585877	−0.0292938	−	0.999571i	$-0.509326\pi$
$251$	−0.928203	−0.0585877	−0.0292938	+	0.999571i	$0.509326\pi$
$252$	0	0
$253$	1.85641	0.116711
$254$	33.7128	2.11533
$255$	0	0
$256$	19.0000	1.18750
$257$	−17.8564	−1.11385	−0.556926	−	0.830562i	$-0.688019\pi$
$257$	−17.8564	−1.11385	−0.556926	+	0.830562i	$0.688019\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−8.53590	−0.527350
$263$	0.535898	0.0330449	0.0165225	−	0.999863i	$-0.494740\pi$
$263$	0.535898	0.0330449	0.0165225	+	0.999863i	$0.494740\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−6.39230	−0.390472
$269$	−2.92820	−0.178536	−0.0892679	−	0.996008i	$-0.528453\pi$
$269$	−2.92820	−0.178536	−0.0892679	+	0.996008i	$0.528453\pi$
$270$	0	0
$271$	1.85641	0.112769	0.0563843	−	0.998409i	$-0.482043\pi$
$271$	1.85641	0.112769	0.0563843	+	0.998409i	$0.482043\pi$
$272$	20.0000	1.21268
$273$	0	0
$274$	24.2487	1.46492
$275$	−17.3205	−1.04447
$276$	0	0
$277$	29.7128	1.78527	0.892635	−	0.450780i	$-0.148854\pi$
$277$	29.7128	1.78527	0.892635	+	0.450780i	$0.148854\pi$
$278$	6.92820	0.415526
$279$	0	0
$280$	0	0
$281$	6.92820	0.413302	0.206651	−	0.978415i	$-0.433744\pi$
$281$	6.92820	0.413302	0.206651	+	0.978415i	$0.433744\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	3.46410	0.205557
$285$	0	0
$286$	−20.7846	−1.22902
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	−8.00000	−0.468165
$293$	32.7846	1.91530	0.957649	−	0.287939i	$-0.0929701\pi$
$293$	32.7846	1.91530	0.957649	+	0.287939i	$0.0929701\pi$
$294$	0	0
$295$	0	0
$296$	8.53590	0.496139
$297$	0	0
$298$	12.2487	0.709549
$299$	1.85641	0.107359
$300$	0	0
$301$	0	0
$302$	−18.0000	−1.03578
$303$	0	0
$304$	−5.00000	−0.286770
$305$	0	0
$306$	0	0
$307$	1.85641	0.105951	0.0529754	−	0.998596i	$-0.483130\pi$
$307$	1.85641	0.105951	0.0529754	+	0.998596i	$0.483130\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.9282	1.41355	0.706774	−	0.707439i	$-0.250150\pi$
$311$	24.9282	1.41355	0.706774	+	0.707439i	$0.250150\pi$
$312$	0	0
$313$	−9.07180	−0.512768	−0.256384	−	0.966575i	$-0.582531\pi$
$313$	−9.07180	−0.512768	−0.256384	+	0.966575i	$0.582531\pi$
$314$	−27.7128	−1.56392
$315$	0	0
$316$	3.46410	0.194871
$317$	−28.0000	−1.57264	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	−28.0000	−1.57264	−0.786318	+	0.617822i	$0.788015\pi$
$318$	0	0
$319$	13.8564	0.775810
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	−17.3205	−0.960769
$326$	−18.9282	−1.04834
$327$	0	0
$328$	18.9282	1.04514
$329$	0	0
$330$	0	0
$331$	−36.2487	−1.99241	−0.996205	−	0.0870415i	$-0.972259\pi$
$331$	−36.2487	−1.99241	−0.996205	+	0.0870415i	$0.972259\pi$
$332$	8.92820	0.489999
$333$	0	0
$334$	24.0000	1.31322
$335$	0	0
$336$	0	0
$337$	−6.00000	−0.326841	−0.163420	−	0.986557i	$-0.552253\pi$
$337$	−6.00000	−0.326841	−0.163420	+	0.986557i	$0.552253\pi$
$338$	1.73205	0.0942111
$339$	0	0
$340$	0	0
$341$	−10.1436	−0.549306
$342$	0	0
$343$	0	0
$344$	−18.9282	−1.02054
$345$	0	0
$346$	30.9282	1.66271
$347$	20.5359	1.10242	0.551212	−	0.834365i	$-0.314165\pi$
$347$	20.5359	1.10242	0.551212	+	0.834365i	$0.314165\pi$
$348$	0	0
$349$	22.9282	1.22732	0.613659	−	0.789571i	$-0.289697\pi$
$349$	22.9282	1.22732	0.613659	+	0.789571i	$0.289697\pi$
$350$	0	0
$351$	0	0
$352$	18.0000	0.959403
$353$	−17.8564	−0.950401	−0.475200	−	0.879878i	$-0.657624\pi$
$353$	−17.8564	−0.950401	−0.475200	+	0.879878i	$0.657624\pi$
$354$	0	0
$355$	0	0
$356$	12.0000	0.635999
$357$	0	0
$358$	−7.85641	−0.415224
$359$	−14.3923	−0.759597	−0.379798	−	0.925069i	$-0.624007\pi$
$359$	−14.3923	−0.759597	−0.379798	+	0.925069i	$0.624007\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−16.1436	−0.848488
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	−2.67949	−0.139678
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	18.7846	0.972630	0.486315	−	0.873784i	$-0.338341\pi$
$373$	18.7846	0.972630	0.486315	+	0.873784i	$0.338341\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	−8.53590	−0.440205
$377$	13.8564	0.713641
$378$	0	0
$379$	9.60770	0.493514	0.246757	−	0.969077i	$-0.420635\pi$
$379$	9.60770	0.493514	0.246757	+	0.969077i	$0.420635\pi$
$380$	0	0
$381$	0	0
$382$	−12.9282	−0.661464
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	0	0
$386$	17.3205	0.881591
$387$	0	0
$388$	19.4641	0.988140
$389$	19.0718	0.966978	0.483489	−	0.875350i	$-0.339369\pi$
$389$	19.0718	0.966978	0.483489	+	0.875350i	$0.339369\pi$
$390$	0	0
$391$	−2.14359	−0.108406
$392$	0	0
$393$	0	0
$394$	15.4641	0.779070
$395$	0	0
$396$	0	0
$397$	−28.7846	−1.44466	−0.722329	−	0.691549i	$-0.756928\pi$
$397$	−28.7846	−1.44466	−0.722329	+	0.691549i	$0.756928\pi$
$398$	37.8564	1.89757
$399$	0	0
$400$	25.0000	1.25000
$401$	1.07180	0.0535230	0.0267615	−	0.999642i	$-0.491481\pi$
$401$	1.07180	0.0535230	0.0267615	+	0.999642i	$0.491481\pi$
$402$	0	0
$403$	−10.1436	−0.505288
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.0718	0.846218
$408$	0	0
$409$	−9.32051	−0.460869	−0.230435	−	0.973088i	$-0.574015\pi$
$409$	−9.32051	−0.460869	−0.230435	+	0.973088i	$0.574015\pi$
$410$	0	0
$411$	0	0
$412$	−2.92820	−0.144262
$413$	0	0
$414$	0	0
$415$	0	0
$416$	18.0000	0.882523
$417$	0	0
$418$	−6.00000	−0.293470
$419$	36.6410	1.79003	0.895015	−	0.446035i	$-0.147164\pi$
$419$	36.6410	1.79003	0.895015	+	0.446035i	$0.147164\pi$
$420$	0	0
$421$	4.92820	0.240186	0.120093	−	0.992763i	$-0.461681\pi$
$421$	4.92820	0.240186	0.120093	+	0.992763i	$0.461681\pi$
$422$	12.9282	0.629335
$423$	0	0
$424$	6.92820	0.336463
$425$	20.0000	0.970143
$426$	0	0
$427$	0	0
$428$	−2.39230	−0.115636
$429$	0	0
$430$	0	0
$431$	5.60770	0.270113	0.135057	−	0.990838i	$-0.456878\pi$
$431$	5.60770	0.270113	0.135057	+	0.990838i	$0.456878\pi$
$432$	0	0
$433$	25.3205	1.21683	0.608413	−	0.793621i	$-0.291806\pi$
$433$	25.3205	1.21683	0.608413	+	0.793621i	$0.291806\pi$
$434$	0	0
$435$	0	0
$436$	4.92820	0.236018
$437$	0.535898	0.0256355
$438$	0	0
$439$	30.6410	1.46242	0.731208	−	0.682155i	$-0.238957\pi$
$439$	30.6410	1.46242	0.731208	+	0.682155i	$0.238957\pi$
$440$	0	0
$441$	0	0
$442$	24.0000	1.14156
$443$	−1.32051	−0.0627392	−0.0313696	−	0.999508i	$-0.509987\pi$
$443$	−1.32051	−0.0627392	−0.0313696	+	0.999508i	$0.509987\pi$
$444$	0	0
$445$	0	0
$446$	−46.6410	−2.20852
$447$	0	0
$448$	0	0
$449$	−30.9282	−1.45959	−0.729796	−	0.683665i	$-0.760385\pi$
$449$	−30.9282	−1.45959	−0.729796	+	0.683665i	$0.760385\pi$
$450$	0	0
$451$	37.8564	1.78259
$452$	18.9282	0.890308
$453$	0	0
$454$	3.21539	0.150906
$455$	0	0
$456$	0	0
$457$	18.0000	0.842004	0.421002	−	0.907060i	$-0.361678\pi$
$457$	18.0000	0.842004	0.421002	+	0.907060i	$0.361678\pi$
$458$	12.0000	0.560723
$459$	0	0
$460$	0	0
$461$	−32.0000	−1.49039	−0.745194	−	0.666847i	$-0.767643\pi$
$461$	−32.0000	−1.49039	−0.745194	+	0.666847i	$0.767643\pi$
$462$	0	0
$463$	13.8564	0.643962	0.321981	−	0.946746i	$-0.395651\pi$
$463$	13.8564	0.643962	0.321981	+	0.946746i	$0.395651\pi$
$464$	−20.0000	−0.928477
$465$	0	0
$466$	−34.3923	−1.59319
$467$	4.92820	0.228050	0.114025	−	0.993478i	$-0.463626\pi$
$467$	4.92820	0.228050	0.114025	+	0.993478i	$0.463626\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	13.8564	0.637793
$473$	−37.8564	−1.74064
$474$	0	0
$475$	−5.00000	−0.229416
$476$	0	0
$477$	0	0
$478$	−14.7846	−0.676232
$479$	23.0718	1.05418	0.527089	−	0.849810i	$-0.323284\pi$
$479$	23.0718	1.05418	0.527089	+	0.849810i	$0.323284\pi$
$480$	0	0
$481$	17.0718	0.778407
$482$	6.00000	0.273293
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	−18.3923	−0.833435	−0.416717	−	0.909036i	$-0.636820\pi$
$487$	−18.3923	−0.833435	−0.416717	+	0.909036i	$0.636820\pi$
$488$	1.85641	0.0840356
$489$	0	0
$490$	0	0
$491$	−33.3205	−1.50373	−0.751867	−	0.659315i	$-0.770846\pi$
$491$	−33.3205	−1.50373	−0.751867	+	0.659315i	$0.770846\pi$
$492$	0	0
$493$	−16.0000	−0.720604
$494$	−6.00000	−0.269953
$495$	0	0
$496$	14.6410	0.657401
$497$	0	0
$498$	0	0
$499$	10.9282	0.489214	0.244607	−	0.969622i	$-0.421341\pi$
$499$	10.9282	0.489214	0.244607	+	0.969622i	$0.421341\pi$
$500$	0	0
$501$	0	0
$502$	1.60770	0.0717549
$503$	−32.6410	−1.45539	−0.727695	−	0.685900i	$-0.759409\pi$
$503$	−32.6410	−1.45539	−0.727695	+	0.685900i	$0.759409\pi$
$504$	0	0
$505$	0	0
$506$	−3.21539	−0.142942
$507$	0	0
$508$	−19.4641	−0.863580
$509$	−1.85641	−0.0822838	−0.0411419	−	0.999153i	$-0.513100\pi$
$509$	−1.85641	−0.0822838	−0.0411419	+	0.999153i	$0.513100\pi$
$510$	0	0
$511$	0	0
$512$	−8.66025	−0.382733
$513$	0	0
$514$	30.9282	1.36418
$515$	0	0
$516$	0	0
$517$	−17.0718	−0.750817
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.9282	1.17975	0.589873	−	0.807496i	$-0.299178\pi$
$521$	26.9282	1.17975	0.589873	+	0.807496i	$0.299178\pi$
$522$	0	0
$523$	33.8564	1.48044	0.740219	−	0.672366i	$-0.234722\pi$
$523$	33.8564	1.48044	0.740219	+	0.672366i	$0.234722\pi$
$524$	4.92820	0.215290
$525$	0	0
$526$	−0.928203	−0.0404716
$527$	11.7128	0.510218
$528$	0	0
$529$	−22.7128	−0.987514
$530$	0	0
$531$	0	0
$532$	0	0
$533$	37.8564	1.63974
$534$	0	0
$535$	0	0
$536$	−11.0718	−0.478229
$537$	0	0
$538$	5.07180	0.218661
$539$	0	0
$540$	0	0
$541$	10.0000	0.429934	0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000	0.429934	0.214967	+	0.976621i	$0.431036\pi$
$542$	−3.21539	−0.138113
$543$	0	0
$544$	−20.7846	−0.891133
$545$	0	0
$546$	0	0
$547$	7.46410	0.319142	0.159571	−	0.987186i	$-0.448989\pi$
$547$	7.46410	0.319142	0.159571	+	0.987186i	$0.448989\pi$
$548$	−14.0000	−0.598050
$549$	0	0
$550$	30.0000	1.27920
$551$	4.00000	0.170406
$552$	0	0
$553$	0	0
$554$	−51.4641	−2.18650
$555$	0	0
$556$	−4.00000	−0.169638
$557$	−3.07180	−0.130156	−0.0650781	−	0.997880i	$-0.520730\pi$
$557$	−3.07180	−0.130156	−0.0650781	+	0.997880i	$0.520730\pi$
$558$	0	0
$559$	−37.8564	−1.60116
$560$	0	0
$561$	0	0
$562$	−12.0000	−0.506189
$563$	17.8564	0.752558	0.376279	−	0.926506i	$-0.377203\pi$
$563$	17.8564	0.752558	0.376279	+	0.926506i	$0.377203\pi$
$564$	0	0
$565$	0	0
$566$	−13.8564	−0.582428
$567$	0	0
$568$	6.00000	0.251754
$569$	−24.7846	−1.03902	−0.519512	−	0.854463i	$-0.673886\pi$
$569$	−24.7846	−1.03902	−0.519512	+	0.854463i	$0.673886\pi$
$570$	0	0
$571$	−17.8564	−0.747267	−0.373634	−	0.927576i	$-0.621888\pi$
$571$	−17.8564	−0.747267	−0.373634	+	0.927576i	$0.621888\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	0	0
$575$	−2.67949	−0.111743
$576$	0	0
$577$	28.7846	1.19832	0.599159	−	0.800630i	$-0.295502\pi$
$577$	28.7846	1.19832	0.599159	+	0.800630i	$0.295502\pi$
$578$	1.73205	0.0720438
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	13.8564	0.573874
$584$	−13.8564	−0.573382
$585$	0	0
$586$	−56.7846	−2.34575
$587$	−11.0718	−0.456982	−0.228491	−	0.973546i	$-0.573379\pi$
$587$	−11.0718	−0.456982	−0.228491	+	0.973546i	$0.573379\pi$
$588$	0	0
$589$	−2.92820	−0.120655
$590$	0	0
$591$	0	0
$592$	−24.6410	−1.01274
$593$	15.7128	0.645248	0.322624	−	0.946527i	$-0.395435\pi$
$593$	15.7128	0.645248	0.322624	+	0.946527i	$0.395435\pi$
$594$	0	0
$595$	0	0
$596$	−7.07180	−0.289672
$597$	0	0
$598$	−3.21539	−0.131487
$599$	34.3923	1.40523	0.702616	−	0.711569i	$-0.252015\pi$
$599$	34.3923	1.40523	0.702616	+	0.711569i	$0.252015\pi$
$600$	0	0
$601$	−2.39230	−0.0975842	−0.0487921	−	0.998809i	$-0.515537\pi$
$601$	−2.39230	−0.0975842	−0.0487921	+	0.998809i	$0.515537\pi$
$602$	0	0
$603$	0	0
$604$	10.3923	0.422857
$605$	0	0
$606$	0	0
$607$	24.7846	1.00598	0.502988	−	0.864293i	$-0.332234\pi$
$607$	24.7846	1.00598	0.502988	+	0.864293i	$0.332234\pi$
$608$	5.19615	0.210732
$609$	0	0
$610$	0	0
$611$	−17.0718	−0.690651
$612$	0	0
$613$	−7.85641	−0.317317	−0.158659	−	0.987333i	$-0.550717\pi$
$613$	−7.85641	−0.317317	−0.158659	+	0.987333i	$0.550717\pi$
$614$	−3.21539	−0.129763
$615$	0	0
$616$	0	0
$617$	31.8564	1.28249	0.641245	−	0.767336i	$-0.278418\pi$
$617$	31.8564	1.28249	0.641245	+	0.767336i	$0.278418\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	0	0
$622$	−43.1769	−1.73124
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	15.7128	0.628010
$627$	0	0
$628$	16.0000	0.638470
$629$	−19.7128	−0.786001
$630$	0	0
$631$	−12.7846	−0.508947	−0.254474	−	0.967080i	$-0.581902\pi$
$631$	−12.7846	−0.508947	−0.254474	+	0.967080i	$0.581902\pi$
$632$	6.00000	0.238667
$633$	0	0
$634$	48.4974	1.92608
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−24.0000	−0.950169
$639$	0	0
$640$	0	0
$641$	26.9282	1.06360	0.531800	−	0.846870i	$-0.321516\pi$
$641$	26.9282	1.06360	0.531800	+	0.846870i	$0.321516\pi$
$642$	0	0
$643$	−37.5692	−1.48159	−0.740793	−	0.671734i	$-0.765550\pi$
$643$	−37.5692	−1.48159	−0.740793	+	0.671734i	$0.765550\pi$
$644$	0	0
$645$	0	0
$646$	6.92820	0.272587
$647$	−42.7846	−1.68204	−0.841018	−	0.541007i	$-0.818043\pi$
$647$	−42.7846	−1.68204	−0.841018	+	0.541007i	$0.818043\pi$
$648$	0	0
$649$	27.7128	1.08782
$650$	30.0000	1.17670
$651$	0	0
$652$	10.9282	0.427981
$653$	−32.6410	−1.27734	−0.638671	−	0.769480i	$-0.720515\pi$
$653$	−32.6410	−1.27734	−0.638671	+	0.769480i	$0.720515\pi$
$654$	0	0
$655$	0	0
$656$	−54.6410	−2.13337
$657$	0	0
$658$	0	0
$659$	−3.46410	−0.134942	−0.0674711	−	0.997721i	$-0.521493\pi$
$659$	−3.46410	−0.134942	−0.0674711	+	0.997721i	$0.521493\pi$
$660$	0	0
$661$	−23.1769	−0.901477	−0.450739	−	0.892656i	$-0.648839\pi$
$661$	−23.1769	−0.901477	−0.450739	+	0.892656i	$0.648839\pi$
$662$	62.7846	2.44019
$663$	0	0
$664$	15.4641	0.600124
$665$	0	0
$666$	0	0
$667$	2.14359	0.0830003
$668$	−13.8564	−0.536120
$669$	0	0
$670$	0	0
$671$	3.71281	0.143332
$672$	0	0
$673$	12.1436	0.468101	0.234051	−	0.972224i	$-0.424802\pi$
$673$	12.1436	0.468101	0.234051	+	0.972224i	$0.424802\pi$
$674$	10.3923	0.400297
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	36.0000	1.38359	0.691796	−	0.722093i	$-0.256820\pi$
$677$	36.0000	1.38359	0.691796	+	0.722093i	$0.256820\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	17.5692	0.672760
$683$	−8.24871	−0.315628	−0.157814	−	0.987469i	$-0.550445\pi$
$683$	−8.24871	−0.315628	−0.157814	+	0.987469i	$0.550445\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	54.6410	2.08317
$689$	13.8564	0.527887
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−17.8564	−0.678799
$693$	0	0
$694$	−35.5692	−1.35019
$695$	0	0
$696$	0	0
$697$	−43.7128	−1.65574
$698$	−39.7128	−1.50315
$699$	0	0
$700$	0	0
$701$	8.92820	0.337214	0.168607	−	0.985683i	$-0.446073\pi$
$701$	8.92820	0.337214	0.168607	+	0.985683i	$0.446073\pi$
$702$	0	0
$703$	4.92820	0.185871
$704$	3.46410	0.130558
$705$	0	0
$706$	30.9282	1.16400
$707$	0	0
$708$	0	0
$709$	12.1436	0.456062	0.228031	−	0.973654i	$-0.426771\pi$
$709$	12.1436	0.456062	0.228031	+	0.973654i	$0.426771\pi$
$710$	0	0
$711$	0	0
$712$	20.7846	0.778936
$713$	−1.56922	−0.0587677
$714$	0	0
$715$	0	0
$716$	4.53590	0.169514
$717$	0	0
$718$	24.9282	0.930312
$719$	40.6410	1.51565	0.757827	−	0.652455i	$-0.226261\pi$
$719$	40.6410	1.51565	0.757827	+	0.652455i	$0.226261\pi$
$720$	0	0
$721$	0	0
$722$	−1.73205	−0.0644603
$723$	0	0
$724$	9.32051	0.346394
$725$	−20.0000	−0.742781
$726$	0	0
$727$	47.7128	1.76957	0.884785	−	0.465999i	$-0.154305\pi$
$727$	47.7128	1.76957	0.884785	+	0.465999i	$0.154305\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	43.7128	1.61678
$732$	0	0
$733$	50.6410	1.87047	0.935234	−	0.354029i	$-0.115189\pi$
$733$	50.6410	1.87047	0.935234	+	0.354029i	$0.115189\pi$
$734$	−6.92820	−0.255725
$735$	0	0
$736$	2.78461	0.102642
$737$	−22.1436	−0.815670
$738$	0	0
$739$	−25.8564	−0.951143	−0.475572	−	0.879677i	$-0.657759\pi$
$739$	−25.8564	−0.951143	−0.475572	+	0.879677i	$0.657759\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.2487	0.889599	0.444799	−	0.895630i	$-0.353275\pi$
$743$	24.2487	0.889599	0.444799	+	0.895630i	$0.353275\pi$
$744$	0	0
$745$	0	0
$746$	−32.5359	−1.19122
$747$	0	0
$748$	−13.8564	−0.506640
$749$	0	0
$750$	0	0
$751$	11.4641	0.418331	0.209166	−	0.977880i	$-0.432925\pi$
$751$	11.4641	0.418331	0.209166	+	0.977880i	$0.432925\pi$
$752$	24.6410	0.898565
$753$	0	0
$754$	−24.0000	−0.874028
$755$	0	0
$756$	0	0
$757$	−51.8564	−1.88475	−0.942377	−	0.334554i	$-0.891414\pi$
$757$	−51.8564	−1.88475	−0.942377	+	0.334554i	$0.891414\pi$
$758$	−16.6410	−0.604429
$759$	0	0
$760$	0	0
$761$	31.7128	1.14959	0.574794	−	0.818298i	$-0.305082\pi$
$761$	31.7128	1.14959	0.574794	+	0.818298i	$0.305082\pi$
$762$	0	0
$763$	0	0
$764$	7.46410	0.270042
$765$	0	0
$766$	6.92820	0.250326
$767$	27.7128	1.00065
$768$	0	0
$769$	34.6410	1.24919	0.624593	−	0.780950i	$-0.285265\pi$
$769$	34.6410	1.24919	0.624593	+	0.780950i	$0.285265\pi$
$770$	0	0
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−7.71281	−0.277411	−0.138705	−	0.990334i	$-0.544294\pi$
$773$	−7.71281	−0.277411	−0.138705	+	0.990334i	$0.544294\pi$
$774$	0	0
$775$	14.6410	0.525921
$776$	33.7128	1.21022
$777$	0	0
$778$	−33.0333	−1.18430
$779$	10.9282	0.391544
$780$	0	0
$781$	12.0000	0.429394
$782$	3.71281	0.132770
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−6.14359	−0.218995	−0.109498	−	0.993987i	$-0.534924\pi$
$787$	−6.14359	−0.218995	−0.109498	+	0.993987i	$0.534924\pi$
$788$	−8.92820	−0.318054
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.71281	0.131846
$794$	49.8564	1.76934
$795$	0	0
$796$	−21.8564	−0.774680
$797$	24.7846	0.877916	0.438958	−	0.898508i	$-0.355348\pi$
$797$	24.7846	0.877916	0.438958	+	0.898508i	$0.355348\pi$
$798$	0	0
$799$	19.7128	0.697389
$800$	−25.9808	−0.918559
$801$	0	0
$802$	−1.85641	−0.0655520
$803$	−27.7128	−0.977964
$804$	0	0
$805$	0	0
$806$	17.5692	0.618849
$807$	0	0
$808$	0	0
$809$	37.7128	1.32591	0.662956	−	0.748658i	$-0.269302\pi$
$809$	37.7128	1.32591	0.662956	+	0.748658i	$0.269302\pi$
$810$	0	0
$811$	−7.71281	−0.270833	−0.135417	−	0.990789i	$-0.543237\pi$
$811$	−7.71281	−0.270833	−0.135417	+	0.990789i	$0.543237\pi$
$812$	0	0
$813$	0	0
$814$	−29.5692	−1.03640
$815$	0	0
$816$	0	0
$817$	−10.9282	−0.382329
$818$	16.1436	0.564448
$819$	0	0
$820$	0	0
$821$	2.78461	0.0971835	0.0485918	−	0.998819i	$-0.484527\pi$
$821$	2.78461	0.0971835	0.0485918	+	0.998819i	$0.484527\pi$
$822$	0	0
$823$	−18.1436	−0.632446	−0.316223	−	0.948685i	$-0.602415\pi$
$823$	−18.1436	−0.632446	−0.316223	+	0.948685i	$0.602415\pi$
$824$	−5.07180	−0.176684
$825$	0	0
$826$	0	0
$827$	−2.39230	−0.0831886	−0.0415943	−	0.999135i	$-0.513244\pi$
$827$	−2.39230	−0.0831886	−0.0415943	+	0.999135i	$0.513244\pi$
$828$	0	0
$829$	32.2487	1.12004	0.560022	−	0.828478i	$-0.310793\pi$
$829$	32.2487	1.12004	0.560022	+	0.828478i	$0.310793\pi$
$830$	0	0
$831$	0	0
$832$	3.46410	0.120096
$833$	0	0
$834$	0	0
$835$	0	0
$836$	3.46410	0.119808
$837$	0	0
$838$	−63.4641	−2.19233
$839$	33.8564	1.16885	0.584426	−	0.811447i	$-0.301320\pi$
$839$	33.8564	1.16885	0.584426	+	0.811447i	$0.301320\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−8.53590	−0.294166
$843$	0	0
$844$	−7.46410	−0.256925
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−20.0000	−0.686803
$849$	0	0
$850$	−34.6410	−1.18818
$851$	2.64102	0.0905329
$852$	0	0
$853$	−35.7128	−1.22278	−0.611392	−	0.791328i	$-0.709390\pi$
$853$	−35.7128	−1.22278	−0.611392	+	0.791328i	$0.709390\pi$
$854$	0	0
$855$	0	0
$856$	−4.14359	−0.141625
$857$	25.8564	0.883238	0.441619	−	0.897203i	$-0.354404\pi$
$857$	25.8564	0.883238	0.441619	+	0.897203i	$0.354404\pi$
$858$	0	0
$859$	53.8564	1.83756	0.918778	−	0.394774i	$-0.129177\pi$
$859$	53.8564	1.83756	0.918778	+	0.394774i	$0.129177\pi$
$860$	0	0
$861$	0	0
$862$	−9.71281	−0.330820
$863$	5.60770	0.190888	0.0954441	−	0.995435i	$-0.469573\pi$
$863$	5.60770	0.190888	0.0954441	+	0.995435i	$0.469573\pi$
$864$	0	0
$865$	0	0
$866$	−43.8564	−1.49030
$867$	0	0
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−22.1436	−0.750307
$872$	8.53590	0.289062
$873$	0	0
$874$	−0.928203	−0.0313969
$875$	0	0
$876$	0	0
$877$	−58.7846	−1.98502	−0.992508	−	0.122183i	$-0.961011\pi$
$877$	−58.7846	−1.98502	−0.992508	+	0.122183i	$0.961011\pi$
$878$	−53.0718	−1.79109
$879$	0	0
$880$	0	0
$881$	−45.5692	−1.53527	−0.767633	−	0.640890i	$-0.778566\pi$
$881$	−45.5692	−1.53527	−0.767633	+	0.640890i	$0.778566\pi$
$882$	0	0
$883$	24.7846	0.834069	0.417034	−	0.908891i	$-0.363070\pi$
$883$	24.7846	0.834069	0.417034	+	0.908891i	$0.363070\pi$
$884$	−13.8564	−0.466041
$885$	0	0
$886$	2.28719	0.0768396
$887$	53.8564	1.80832	0.904161	−	0.427193i	$-0.140497\pi$
$887$	53.8564	1.80832	0.904161	+	0.427193i	$0.140497\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	26.9282	0.901623
$893$	−4.92820	−0.164916
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	53.5692	1.78763
$899$	−11.7128	−0.390644
$900$	0	0
$901$	−16.0000	−0.533037
$902$	−65.5692	−2.18322
$903$	0	0
$904$	32.7846	1.09040
$905$	0	0
$906$	0	0
$907$	−10.6795	−0.354607	−0.177303	−	0.984156i	$-0.556737\pi$
$907$	−10.6795	−0.354607	−0.177303	+	0.984156i	$0.556737\pi$
$908$	−1.85641	−0.0616070
$909$	0	0
$910$	0	0
$911$	−55.1769	−1.82809	−0.914046	−	0.405610i	$-0.867059\pi$
$911$	−55.1769	−1.82809	−0.914046	+	0.405610i	$0.867059\pi$
$912$	0	0
$913$	30.9282	1.02357
$914$	−31.1769	−1.03124
$915$	0	0
$916$	−6.92820	−0.228914
$917$	0	0
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	0	0
$922$	55.4256	1.82535
$923$	12.0000	0.394985
$924$	0	0
$925$	−24.6410	−0.810192
$926$	−24.0000	−0.788689
$927$	0	0
$928$	20.7846	0.682288
$929$	44.0000	1.44359	0.721797	−	0.692105i	$-0.243317\pi$
$929$	44.0000	1.44359	0.721797	+	0.692105i	$0.243317\pi$
$930$	0	0
$931$	0	0
$932$	19.8564	0.650418
$933$	0	0
$934$	−8.53590	−0.279303
$935$	0	0
$936$	0	0
$937$	−12.7846	−0.417655	−0.208827	−	0.977952i	$-0.566965\pi$
$937$	−12.7846	−0.417655	−0.208827	+	0.977952i	$0.566965\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	39.7128	1.29460	0.647300	−	0.762235i	$-0.275898\pi$
$941$	39.7128	1.29460	0.647300	+	0.762235i	$0.275898\pi$
$942$	0	0
$943$	5.85641	0.190711
$944$	−40.0000	−1.30189
$945$	0	0
$946$	65.5692	2.13184
$947$	41.3205	1.34274	0.671368	−	0.741124i	$-0.265707\pi$
$947$	41.3205	1.34274	0.671368	+	0.741124i	$0.265707\pi$
$948$	0	0
$949$	−27.7128	−0.899596
$950$	8.66025	0.280976
$951$	0	0
$952$	0	0
$953$	17.0718	0.553010	0.276505	−	0.961013i	$-0.410824\pi$
$953$	17.0718	0.553010	0.276505	+	0.961013i	$0.410824\pi$
$954$	0	0
$955$	0	0
$956$	8.53590	0.276071
$957$	0	0
$958$	−39.9615	−1.29110
$959$	0	0
$960$	0	0
$961$	−22.4256	−0.723407
$962$	−29.5692	−0.953350
$963$	0	0
$964$	−3.46410	−0.111571
$965$	0	0
$966$	0	0
$967$	−12.7846	−0.411125	−0.205563	−	0.978644i	$-0.565902\pi$
$967$	−12.7846	−0.411125	−0.205563	+	0.978644i	$0.565902\pi$
$968$	1.73205	0.0556702
$969$	0	0
$970$	0	0
$971$	22.1436	0.710622	0.355311	−	0.934748i	$-0.384375\pi$
$971$	22.1436	0.710622	0.355311	+	0.934748i	$0.384375\pi$
$972$	0	0
$973$	0	0
$974$	31.8564	1.02075
$975$	0	0
$976$	−5.35898	−0.171537
$977$	58.6410	1.87609	0.938046	−	0.346510i	$-0.112633\pi$
$977$	58.6410	1.87609	0.938046	+	0.346510i	$0.112633\pi$
$978$	0	0
$979$	41.5692	1.32856
$980$	0	0
$981$	0	0
$982$	57.7128	1.84169
$983$	3.71281	0.118420	0.0592102	−	0.998246i	$-0.481142\pi$
$983$	3.71281	0.118420	0.0592102	+	0.998246i	$0.481142\pi$
$984$	0	0
$985$	0	0
$986$	27.7128	0.882556
$987$	0	0
$988$	3.46410	0.110208
$989$	−5.85641	−0.186223
$990$	0	0
$991$	−53.0333	−1.68466	−0.842329	−	0.538963i	$-0.818816\pi$
$991$	−53.0333	−1.68466	−0.842329	+	0.538963i	$0.818816\pi$
$992$	−15.2154	−0.483089
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.14359	0.0678883	0.0339441	−	0.999424i	$-0.489193\pi$
$997$	2.14359	0.0678883	0.0339441	+	0.999424i	$0.489193\pi$
$998$	−18.9282	−0.599162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bc.1.1		2
3.2	odd	2		2793.2.a.q.1.2	✓	2
7.6	odd	2		8379.2.a.bd.1.1		2
21.20	even	2		2793.2.a.r.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2793.2.a.q.1.2	✓	2	3.2	odd	2
2793.2.a.r.1.2	yes	2	21.20	even	2
8379.2.a.bc.1.1		2	1.1	even	1	trivial
8379.2.a.bd.1.1		2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} +1.73205 q^{8} +3.46410 q^{11} +3.46410 q^{13} -5.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} -6.00000 q^{22} +0.535898 q^{23} -5.00000 q^{25} -6.00000 q^{26} +4.00000 q^{29} -2.92820 q^{31} +5.19615 q^{32} +6.92820 q^{34} +4.92820 q^{37} -1.73205 q^{38} +10.9282 q^{41} -10.9282 q^{43} +3.46410 q^{44} -0.928203 q^{46} -4.92820 q^{47} +8.66025 q^{50} +3.46410 q^{52} +4.00000 q^{53} -6.92820 q^{58} +8.00000 q^{59} +1.07180 q^{61} +5.07180 q^{62} +1.00000 q^{64} -6.39230 q^{67} -4.00000 q^{68} +3.46410 q^{71} -8.00000 q^{73} -8.53590 q^{74} +1.00000 q^{76} +3.46410 q^{79} -18.9282 q^{82} +8.92820 q^{83} +18.9282 q^{86} +6.00000 q^{88} +12.0000 q^{89} +0.535898 q^{92} +8.53590 q^{94} +19.4641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 10 q^{16} - 8 q^{17} + 2 q^{19} - 12 q^{22} + 8 q^{23} - 10 q^{25} - 12 q^{26} + 8 q^{29} + 8 q^{31} - 4 q^{37} + 8 q^{41} - 8 q^{43} + 12 q^{46} + 4 q^{47} + 8 q^{53} + 16 q^{59} + 16 q^{61}+ \cdots + 32 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 10 * q^16 - 8 * q^17 + 2 * q^19 - 12 * q^22 + 8 * q^23 - 10 * q^25 - 12 * q^26 + 8 * q^29 + 8 * q^31 - 4 * q^37 + 8 * q^41 - 8 * q^43 + 12 * q^46 + 4 * q^47 + 8 * q^53 + 16 * q^59 + 16 * q^61 + 24 * q^62 + 2 * q^64 + 8 * q^67 - 8 * q^68 - 16 * q^73 - 24 * q^74 + 2 * q^76 - 24 * q^82 + 4 * q^83 + 24 * q^86 + 12 * q^88 + 24 * q^89 + 8 * q^92 + 24 * q^94 + 32 * q^97

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