LMFDB - Embedded newform 7938.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$63.3852491245$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 1134)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -4.24264 q^{11} -2.24264 q^{13} +1.00000 q^{16} -2.24264 q^{19} -4.24264 q^{22} -1.24264 q^{23} -5.00000 q^{25} -2.24264 q^{26} +4.24264 q^{29} +9.24264 q^{31} +1.00000 q^{32} -4.00000 q^{37} -2.24264 q^{38} +11.4853 q^{41} +10.4853 q^{43} -4.24264 q^{44} -1.24264 q^{46} +4.75736 q^{47} -5.00000 q^{50} -2.24264 q^{52} -4.24264 q^{53} +4.24264 q^{58} -2.24264 q^{61} +9.24264 q^{62} +1.00000 q^{64} +0.242641 q^{67} +1.24264 q^{71} -7.00000 q^{73} -4.00000 q^{74} -2.24264 q^{76} +0.757359 q^{79} +11.4853 q^{82} +16.2426 q^{83} +10.4853 q^{86} -4.24264 q^{88} +11.4853 q^{89} -1.24264 q^{92} +4.75736 q^{94} +4.48528 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 4 q^{13} + 2 q^{16} + 4 q^{19} + 6 q^{23} - 10 q^{25} + 4 q^{26} + 10 q^{31} + 2 q^{32} - 8 q^{37} + 4 q^{38} + 6 q^{41} + 4 q^{43} + 6 q^{46} + 18 q^{47} - 10 q^{50}+ \cdots - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 4 * q^13 + 2 * q^16 + 4 * q^19 + 6 * q^23 - 10 * q^25 + 4 * q^26 + 10 * q^31 + 2 * q^32 - 8 * q^37 + 4 * q^38 + 6 * q^41 + 4 * q^43 + 6 * q^46 + 18 * q^47 - 10 * q^50 + 4 * q^52 + 4 * q^61 + 10 * q^62 + 2 * q^64 - 8 * q^67 - 6 * q^71 - 14 * q^73 - 8 * q^74 + 4 * q^76 + 10 * q^79 + 6 * q^82 + 24 * q^83 + 4 * q^86 + 6 * q^89 + 6 * q^92 + 18 * q^94 - 8 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	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
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	−2.24264	−0.621997	−0.310998	−	0.950410i	$-0.600663\pi$
$13$	−2.24264	−0.621997	−0.310998	+	0.950410i	$0.600663\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−2.24264	−0.514497	−0.257249	−	0.966345i	$-0.582816\pi$
$19$	−2.24264	−0.514497	−0.257249	+	0.966345i	$0.582816\pi$
$20$	0	0
$21$	0	0
$22$	−4.24264	−0.904534
$23$	−1.24264	−0.259108	−0.129554	−	0.991572i	$-0.541355\pi$
$23$	−1.24264	−0.259108	−0.129554	+	0.991572i	$0.541355\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	−2.24264	−0.439818
$27$	0	0
$28$	0	0
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	9.24264	1.66003	0.830014	−	0.557743i	$-0.188333\pi$
$31$	9.24264	1.66003	0.830014	+	0.557743i	$0.188333\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−2.24264	−0.363804
$39$	0	0
$40$	0	0
$41$	11.4853	1.79370	0.896850	−	0.442335i	$-0.145850\pi$
$41$	11.4853	1.79370	0.896850	+	0.442335i	$0.145850\pi$
$42$	0	0
$43$	10.4853	1.59899	0.799495	−	0.600672i	$-0.205100\pi$
$43$	10.4853	1.59899	0.799495	+	0.600672i	$0.205100\pi$
$44$	−4.24264	−0.639602
$45$	0	0
$46$	−1.24264	−0.183217
$47$	4.75736	0.693932	0.346966	−	0.937878i	$-0.387212\pi$
$47$	4.75736	0.693932	0.346966	+	0.937878i	$0.387212\pi$
$48$	0	0
$49$	0	0
$50$	−5.00000	−0.707107
$51$	0	0
$52$	−2.24264	−0.310998
$53$	−4.24264	−0.582772	−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264	−0.582772	−0.291386	+	0.956606i	$0.594116\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	4.24264	0.557086
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.24264	−0.287141	−0.143570	−	0.989640i	$-0.545858\pi$
$61$	−2.24264	−0.287141	−0.143570	+	0.989640i	$0.545858\pi$
$62$	9.24264	1.17382
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0.242641	0.0296433	0.0148216	−	0.999890i	$-0.495282\pi$
$67$	0.242641	0.0296433	0.0148216	+	0.999890i	$0.495282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.24264	0.147474	0.0737372	−	0.997278i	$-0.476507\pi$
$71$	1.24264	0.147474	0.0737372	+	0.997278i	$0.476507\pi$
$72$	0	0
$73$	−7.00000	−0.819288	−0.409644	−	0.912245i	$-0.634347\pi$
$73$	−7.00000	−0.819288	−0.409644	+	0.912245i	$0.634347\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−2.24264	−0.257249
$77$	0	0
$78$	0	0
$79$	0.757359	0.0852096	0.0426048	−	0.999092i	$-0.486434\pi$
$79$	0.757359	0.0852096	0.0426048	+	0.999092i	$0.486434\pi$
$80$	0	0
$81$	0	0
$82$	11.4853	1.26834
$83$	16.2426	1.78286	0.891431	−	0.453157i	$-0.149702\pi$
$83$	16.2426	1.78286	0.891431	+	0.453157i	$0.149702\pi$
$84$	0	0
$85$	0	0
$86$	10.4853	1.13066
$87$	0	0
$88$	−4.24264	−0.452267
$89$	11.4853	1.21744	0.608719	−	0.793386i	$-0.291684\pi$
$89$	11.4853	1.21744	0.608719	+	0.793386i	$0.291684\pi$
$90$	0	0
$91$	0	0
$92$	−1.24264	−0.129554
$93$	0	0
$94$	4.75736	0.490684
$95$	0	0
$96$	0	0
$97$	4.48528	0.455411	0.227706	−	0.973730i	$-0.426878\pi$
$97$	4.48528	0.455411	0.227706	+	0.973730i	$0.426878\pi$
$98$	0	0
$99$	0	0
$100$	−5.00000	−0.500000
$101$	16.2426	1.61620	0.808102	−	0.589043i	$-0.200495\pi$
$101$	16.2426	1.61620	0.808102	+	0.589043i	$0.200495\pi$
$102$	0	0
$103$	9.24264	0.910704	0.455352	−	0.890311i	$-0.349513\pi$
$103$	9.24264	0.910704	0.455352	+	0.890311i	$0.349513\pi$
$104$	−2.24264	−0.219909
$105$	0	0
$106$	−4.24264	−0.412082
$107$	14.4853	1.40035	0.700173	−	0.713974i	$-0.253106\pi$
$107$	14.4853	1.40035	0.700173	+	0.713974i	$0.253106\pi$
$108$	0	0
$109$	6.24264	0.597937	0.298968	−	0.954263i	$-0.403358\pi$
$109$	6.24264	0.597937	0.298968	+	0.954263i	$0.403358\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.51472	−0.330637	−0.165318	−	0.986240i	$-0.552865\pi$
$113$	−3.51472	−0.330637	−0.165318	+	0.986240i	$0.552865\pi$
$114$	0	0
$115$	0	0
$116$	4.24264	0.393919
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	−2.24264	−0.203039
$123$	0	0
$124$	9.24264	0.830014
$125$	0	0
$126$	0	0
$127$	15.2426	1.35257	0.676283	−	0.736642i	$-0.263590\pi$
$127$	15.2426	1.35257	0.676283	+	0.736642i	$0.263590\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−16.2426	−1.41913	−0.709563	−	0.704642i	$-0.751108\pi$
$131$	−16.2426	−1.41913	−0.709563	+	0.704642i	$0.751108\pi$
$132$	0	0
$133$	0	0
$134$	0.242641	0.0209610
$135$	0	0
$136$	0	0
$137$	13.9706	1.19359	0.596793	−	0.802395i	$-0.296441\pi$
$137$	13.9706	1.19359	0.596793	+	0.802395i	$0.296441\pi$
$138$	0	0
$139$	20.7279	1.75812	0.879060	−	0.476712i	$-0.158171\pi$
$139$	20.7279	1.75812	0.879060	+	0.476712i	$0.158171\pi$
$140$	0	0
$141$	0	0
$142$	1.24264	0.104280
$143$	9.51472	0.795661
$144$	0	0
$145$	0	0
$146$	−7.00000	−0.579324
$147$	0	0
$148$	−4.00000	−0.328798
$149$	7.75736	0.635508	0.317754	−	0.948173i	$-0.397071\pi$
$149$	7.75736	0.635508	0.317754	+	0.948173i	$0.397071\pi$
$150$	0	0
$151$	−11.2426	−0.914913	−0.457457	−	0.889232i	$-0.651239\pi$
$151$	−11.2426	−0.914913	−0.457457	+	0.889232i	$0.651239\pi$
$152$	−2.24264	−0.181902
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	0.757359	0.0602523
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.2426	−1.58553	−0.792763	−	0.609530i	$-0.791358\pi$
$163$	−20.2426	−1.58553	−0.792763	+	0.609530i	$0.791358\pi$
$164$	11.4853	0.896850
$165$	0	0
$166$	16.2426	1.26067
$167$	−18.2132	−1.40938	−0.704690	−	0.709515i	$-0.748914\pi$
$167$	−18.2132	−1.40938	−0.704690	+	0.709515i	$0.748914\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	0	0
$171$	0	0
$172$	10.4853	0.799495
$173$	22.9706	1.74642	0.873210	−	0.487345i	$-0.162034\pi$
$173$	22.9706	1.74642	0.873210	+	0.487345i	$0.162034\pi$
$174$	0	0
$175$	0	0
$176$	−4.24264	−0.319801
$177$	0	0
$178$	11.4853	0.860858
$179$	−7.75736	−0.579812	−0.289906	−	0.957055i	$-0.593624\pi$
$179$	−7.75736	−0.579812	−0.289906	+	0.957055i	$0.593624\pi$
$180$	0	0
$181$	−11.7574	−0.873918	−0.436959	−	0.899482i	$-0.643944\pi$
$181$	−11.7574	−0.873918	−0.436959	+	0.899482i	$0.643944\pi$
$182$	0	0
$183$	0	0
$184$	−1.24264	−0.0916087
$185$	0	0
$186$	0	0
$187$	0	0
$188$	4.75736	0.346966
$189$	0	0
$190$	0	0
$191$	−8.48528	−0.613973	−0.306987	−	0.951714i	$-0.599321\pi$
$191$	−8.48528	−0.613973	−0.306987	+	0.951714i	$0.599321\pi$
$192$	0	0
$193$	−21.4853	−1.54654	−0.773272	−	0.634074i	$-0.781381\pi$
$193$	−21.4853	−1.54654	−0.773272	+	0.634074i	$0.781381\pi$
$194$	4.48528	0.322024
$195$	0	0
$196$	0	0
$197$	−16.9706	−1.20910	−0.604551	−	0.796566i	$-0.706648\pi$
$197$	−16.9706	−1.20910	−0.604551	+	0.796566i	$0.706648\pi$
$198$	0	0
$199$	−23.2426	−1.64763	−0.823814	−	0.566861i	$-0.808158\pi$
$199$	−23.2426	−1.64763	−0.823814	+	0.566861i	$0.808158\pi$
$200$	−5.00000	−0.353553
$201$	0	0
$202$	16.2426	1.14283
$203$	0	0
$204$	0	0
$205$	0	0
$206$	9.24264	0.643965
$207$	0	0
$208$	−2.24264	−0.155499
$209$	9.51472	0.658147
$210$	0	0
$211$	10.4853	0.721837	0.360918	−	0.932597i	$-0.382463\pi$
$211$	10.4853	0.721837	0.360918	+	0.932597i	$0.382463\pi$
$212$	−4.24264	−0.291386
$213$	0	0
$214$	14.4853	0.990193
$215$	0	0
$216$	0	0
$217$	0	0
$218$	6.24264	0.422805
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−13.7279	−0.919290	−0.459645	−	0.888103i	$-0.652023\pi$
$223$	−13.7279	−0.919290	−0.459645	+	0.888103i	$0.652023\pi$
$224$	0	0
$225$	0	0
$226$	−3.51472	−0.233796
$227$	9.51472	0.631514	0.315757	−	0.948840i	$-0.397742\pi$
$227$	9.51472	0.631514	0.315757	+	0.948840i	$0.397742\pi$
$228$	0	0
$229$	−8.97056	−0.592791	−0.296396	−	0.955065i	$-0.595785\pi$
$229$	−8.97056	−0.592791	−0.296396	+	0.955065i	$0.595785\pi$
$230$	0	0
$231$	0	0
$232$	4.24264	0.278543
$233$	3.51472	0.230257	0.115128	−	0.993351i	$-0.463272\pi$
$233$	3.51472	0.230257	0.115128	+	0.993351i	$0.463272\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−21.7279	−1.40546	−0.702731	−	0.711455i	$-0.748036\pi$
$239$	−21.7279	−1.40546	−0.702731	+	0.711455i	$0.748036\pi$
$240$	0	0
$241$	25.4853	1.64165	0.820826	−	0.571179i	$-0.193514\pi$
$241$	25.4853	1.64165	0.820826	+	0.571179i	$0.193514\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−2.24264	−0.143570
$245$	0	0
$246$	0	0
$247$	5.02944	0.320015
$248$	9.24264	0.586908
$249$	0	0
$250$	0	0
$251$	−6.72792	−0.424663	−0.212331	−	0.977198i	$-0.568106\pi$
$251$	−6.72792	−0.424663	−0.212331	+	0.977198i	$0.568106\pi$
$252$	0	0
$253$	5.27208	0.331453
$254$	15.2426	0.956408
$255$	0	0
$256$	1.00000	0.0625000
$257$	−11.4853	−0.716432	−0.358216	−	0.933639i	$-0.616615\pi$
$257$	−11.4853	−0.716432	−0.358216	+	0.933639i	$0.616615\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−16.2426	−1.00347
$263$	−10.9706	−0.676474	−0.338237	−	0.941061i	$-0.609831\pi$
$263$	−10.9706	−0.676474	−0.338237	+	0.941061i	$0.609831\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0.242641	0.0148216
$269$	22.9706	1.40054	0.700270	−	0.713878i	$-0.253063\pi$
$269$	22.9706	1.40054	0.700270	+	0.713878i	$0.253063\pi$
$270$	0	0
$271$	4.48528	0.272461	0.136231	−	0.990677i	$-0.456501\pi$
$271$	4.48528	0.272461	0.136231	+	0.990677i	$0.456501\pi$
$272$	0	0
$273$	0	0
$274$	13.9706	0.843993
$275$	21.2132	1.27920
$276$	0	0
$277$	23.2132	1.39475	0.697373	−	0.716708i	$-0.254352\pi$
$277$	23.2132	1.39475	0.697373	+	0.716708i	$0.254352\pi$
$278$	20.7279	1.24318
$279$	0	0
$280$	0	0
$281$	−28.4558	−1.69753	−0.848767	−	0.528768i	$-0.822654\pi$
$281$	−28.4558	−1.69753	−0.848767	+	0.528768i	$0.822654\pi$
$282$	0	0
$283$	−8.97056	−0.533245	−0.266622	−	0.963801i	$-0.585908\pi$
$283$	−8.97056	−0.533245	−0.266622	+	0.963801i	$0.585908\pi$
$284$	1.24264	0.0737372
$285$	0	0
$286$	9.51472	0.562617
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	−7.00000	−0.409644
$293$	22.9706	1.34195	0.670977	−	0.741478i	$-0.265875\pi$
$293$	22.9706	1.34195	0.670977	+	0.741478i	$0.265875\pi$
$294$	0	0
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	7.75736	0.449372
$299$	2.78680	0.161165
$300$	0	0
$301$	0	0
$302$	−11.2426	−0.646941
$303$	0	0
$304$	−2.24264	−0.128624
$305$	0	0
$306$	0	0
$307$	−28.0000	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$307$	−28.0000	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.9706	1.30254	0.651271	−	0.758846i	$-0.274236\pi$
$311$	22.9706	1.30254	0.651271	+	0.758846i	$0.274236\pi$
$312$	0	0
$313$	15.9706	0.902710	0.451355	−	0.892345i	$-0.350941\pi$
$313$	15.9706	0.902710	0.451355	+	0.892345i	$0.350941\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	0.757359	0.0426048
$317$	16.9706	0.953162	0.476581	−	0.879131i	$-0.341876\pi$
$317$	16.9706	0.953162	0.476581	+	0.879131i	$0.341876\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	11.2132	0.621997
$326$	−20.2426	−1.12114
$327$	0	0
$328$	11.4853	0.634169
$329$	0	0
$330$	0	0
$331$	−1.51472	−0.0832565	−0.0416282	−	0.999133i	$-0.513255\pi$
$331$	−1.51472	−0.0832565	−0.0416282	+	0.999133i	$0.513255\pi$
$332$	16.2426	0.891431
$333$	0	0
$334$	−18.2132	−0.996582
$335$	0	0
$336$	0	0
$337$	−12.4853	−0.680117	−0.340058	−	0.940404i	$-0.610447\pi$
$337$	−12.4853	−0.680117	−0.340058	+	0.940404i	$0.610447\pi$
$338$	−7.97056	−0.433541
$339$	0	0
$340$	0	0
$341$	−39.2132	−2.12351
$342$	0	0
$343$	0	0
$344$	10.4853	0.565328
$345$	0	0
$346$	22.9706	1.23491
$347$	18.7279	1.00537	0.502684	−	0.864470i	$-0.332346\pi$
$347$	18.7279	1.00537	0.502684	+	0.864470i	$0.332346\pi$
$348$	0	0
$349$	−8.97056	−0.480183	−0.240092	−	0.970750i	$-0.577177\pi$
$349$	−8.97056	−0.480183	−0.240092	+	0.970750i	$0.577177\pi$
$350$	0	0
$351$	0	0
$352$	−4.24264	−0.226134
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	0	0
$355$	0	0
$356$	11.4853	0.608719
$357$	0	0
$358$	−7.75736	−0.409989
$359$	19.2426	1.01559	0.507794	−	0.861479i	$-0.330461\pi$
$359$	19.2426	1.01559	0.507794	+	0.861479i	$0.330461\pi$
$360$	0	0
$361$	−13.9706	−0.735293
$362$	−11.7574	−0.617953
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.7279	−0.716592	−0.358296	−	0.933608i	$-0.616642\pi$
$367$	−13.7279	−0.716592	−0.358296	+	0.933608i	$0.616642\pi$
$368$	−1.24264	−0.0647771
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	35.2132	1.82327	0.911635	−	0.411000i	$-0.134820\pi$
$373$	35.2132	1.82327	0.911635	+	0.411000i	$0.134820\pi$
$374$	0	0
$375$	0	0
$376$	4.75736	0.245342
$377$	−9.51472	−0.490033
$378$	0	0
$379$	26.7279	1.37292	0.686461	−	0.727167i	$-0.259163\pi$
$379$	26.7279	1.37292	0.686461	+	0.727167i	$0.259163\pi$
$380$	0	0
$381$	0	0
$382$	−8.48528	−0.434145
$383$	18.2132	0.930651	0.465326	−	0.885140i	$-0.345937\pi$
$383$	18.2132	0.930651	0.465326	+	0.885140i	$0.345937\pi$
$384$	0	0
$385$	0	0
$386$	−21.4853	−1.09357
$387$	0	0
$388$	4.48528	0.227706
$389$	1.75736	0.0891017	0.0445508	−	0.999007i	$-0.485814\pi$
$389$	1.75736	0.0891017	0.0445508	+	0.999007i	$0.485814\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−16.9706	−0.854965
$395$	0	0
$396$	0	0
$397$	−11.7574	−0.590085	−0.295042	−	0.955484i	$-0.595334\pi$
$397$	−11.7574	−0.590085	−0.295042	+	0.955484i	$0.595334\pi$
$398$	−23.2426	−1.16505
$399$	0	0
$400$	−5.00000	−0.250000
$401$	12.5147	0.624955	0.312478	−	0.949925i	$-0.398841\pi$
$401$	12.5147	0.624955	0.312478	+	0.949925i	$0.398841\pi$
$402$	0	0
$403$	−20.7279	−1.03253
$404$	16.2426	0.808102
$405$	0	0
$406$	0	0
$407$	16.9706	0.841200
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	0	0
$412$	9.24264	0.455352
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.24264	−0.109955
$417$	0	0
$418$	9.51472	0.465380
$419$	−16.2426	−0.793505	−0.396752	−	0.917926i	$-0.629863\pi$
$419$	−16.2426	−0.793505	−0.396752	+	0.917926i	$0.629863\pi$
$420$	0	0
$421$	−5.75736	−0.280597	−0.140298	−	0.990109i	$-0.544806\pi$
$421$	−5.75736	−0.280597	−0.140298	+	0.990109i	$0.544806\pi$
$422$	10.4853	0.510416
$423$	0	0
$424$	−4.24264	−0.206041
$425$	0	0
$426$	0	0
$427$	0	0
$428$	14.4853	0.700173
$429$	0	0
$430$	0	0
$431$	−28.7574	−1.38519	−0.692597	−	0.721325i	$-0.743533\pi$
$431$	−28.7574	−1.38519	−0.692597	+	0.721325i	$0.743533\pi$
$432$	0	0
$433$	−29.9706	−1.44029	−0.720147	−	0.693822i	$-0.755926\pi$
$433$	−29.9706	−1.44029	−0.720147	+	0.693822i	$0.755926\pi$
$434$	0	0
$435$	0	0
$436$	6.24264	0.298968
$437$	2.78680	0.133311
$438$	0	0
$439$	−13.7279	−0.655198	−0.327599	−	0.944817i	$-0.606239\pi$
$439$	−13.7279	−0.655198	−0.327599	+	0.944817i	$0.606239\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.4853	1.25835	0.629177	−	0.777262i	$-0.283392\pi$
$443$	26.4853	1.25835	0.629177	+	0.777262i	$0.283392\pi$
$444$	0	0
$445$	0	0
$446$	−13.7279	−0.650036
$447$	0	0
$448$	0	0
$449$	28.9706	1.36721	0.683603	−	0.729854i	$-0.260412\pi$
$449$	28.9706	1.36721	0.683603	+	0.729854i	$0.260412\pi$
$450$	0	0
$451$	−48.7279	−2.29451
$452$	−3.51472	−0.165318
$453$	0	0
$454$	9.51472	0.446548
$455$	0	0
$456$	0	0
$457$	28.4853	1.33249	0.666243	−	0.745735i	$-0.267902\pi$
$457$	28.4853	1.33249	0.666243	+	0.745735i	$0.267902\pi$
$458$	−8.97056	−0.419167
$459$	0	0
$460$	0	0
$461$	25.7574	1.19964	0.599820	−	0.800135i	$-0.295239\pi$
$461$	25.7574	1.19964	0.599820	+	0.800135i	$0.295239\pi$
$462$	0	0
$463$	−17.2426	−0.801333	−0.400667	−	0.916224i	$-0.631221\pi$
$463$	−17.2426	−0.801333	−0.400667	+	0.916224i	$0.631221\pi$
$464$	4.24264	0.196960
$465$	0	0
$466$	3.51472	0.162816
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−44.4853	−2.04544
$474$	0	0
$475$	11.2132	0.514497
$476$	0	0
$477$	0	0
$478$	−21.7279	−0.993812
$479$	−4.75736	−0.217369	−0.108685	−	0.994076i	$-0.534664\pi$
$479$	−4.75736	−0.217369	−0.108685	+	0.994076i	$0.534664\pi$
$480$	0	0
$481$	8.97056	0.409022
$482$	25.4853	1.16082
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	0	0
$487$	−11.2426	−0.509453	−0.254726	−	0.967013i	$-0.581985\pi$
$487$	−11.2426	−0.509453	−0.254726	+	0.967013i	$0.581985\pi$
$488$	−2.24264	−0.101520
$489$	0	0
$490$	0	0
$491$	−18.7279	−0.845179	−0.422590	−	0.906321i	$-0.638879\pi$
$491$	−18.7279	−0.845179	−0.422590	+	0.906321i	$0.638879\pi$
$492$	0	0
$493$	0	0
$494$	5.02944	0.226285
$495$	0	0
$496$	9.24264	0.415007
$497$	0	0
$498$	0	0
$499$	14.7279	0.659312	0.329656	−	0.944101i	$-0.393067\pi$
$499$	14.7279	0.659312	0.329656	+	0.944101i	$0.393067\pi$
$500$	0	0
$501$	0	0
$502$	−6.72792	−0.300282
$503$	18.2132	0.812087	0.406043	−	0.913854i	$-0.366908\pi$
$503$	18.2132	0.812087	0.406043	+	0.913854i	$0.366908\pi$
$504$	0	0
$505$	0	0
$506$	5.27208	0.234372
$507$	0	0
$508$	15.2426	0.676283
$509$	16.2426	0.719942	0.359971	−	0.932963i	$-0.382787\pi$
$509$	16.2426	0.719942	0.359971	+	0.932963i	$0.382787\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−11.4853	−0.506594
$515$	0	0
$516$	0	0
$517$	−20.1838	−0.887681
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−34.4558	−1.50954	−0.754769	−	0.655991i	$-0.772251\pi$
$521$	−34.4558	−1.50954	−0.754769	+	0.655991i	$0.772251\pi$
$522$	0	0
$523$	−11.7574	−0.514113	−0.257057	−	0.966396i	$-0.582753\pi$
$523$	−11.7574	−0.514113	−0.257057	+	0.966396i	$0.582753\pi$
$524$	−16.2426	−0.709563
$525$	0	0
$526$	−10.9706	−0.478339
$527$	0	0
$528$	0	0
$529$	−21.4558	−0.932863
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−25.7574	−1.11568
$534$	0	0
$535$	0	0
$536$	0.242641	0.0104805
$537$	0	0
$538$	22.9706	0.990331
$539$	0	0
$540$	0	0
$541$	18.9706	0.815608	0.407804	−	0.913069i	$-0.366295\pi$
$541$	18.9706	0.815608	0.407804	+	0.913069i	$0.366295\pi$
$542$	4.48528	0.192659
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.4853	0.448318	0.224159	−	0.974553i	$-0.428036\pi$
$547$	10.4853	0.448318	0.224159	+	0.974553i	$0.428036\pi$
$548$	13.9706	0.596793
$549$	0	0
$550$	21.2132	0.904534
$551$	−9.51472	−0.405341
$552$	0	0
$553$	0	0
$554$	23.2132	0.986235
$555$	0	0
$556$	20.7279	0.879060
$557$	−7.02944	−0.297847	−0.148923	−	0.988849i	$-0.547581\pi$
$557$	−7.02944	−0.297847	−0.148923	+	0.988849i	$0.547581\pi$
$558$	0	0
$559$	−23.5147	−0.994567
$560$	0	0
$561$	0	0
$562$	−28.4558	−1.20034
$563$	−6.72792	−0.283548	−0.141774	−	0.989899i	$-0.545281\pi$
$563$	−6.72792	−0.283548	−0.141774	+	0.989899i	$0.545281\pi$
$564$	0	0
$565$	0	0
$566$	−8.97056	−0.377061
$567$	0	0
$568$	1.24264	0.0521400
$569$	−42.9411	−1.80019	−0.900093	−	0.435698i	$-0.856501\pi$
$569$	−42.9411	−1.80019	−0.900093	+	0.435698i	$0.856501\pi$
$570$	0	0
$571$	12.9706	0.542801	0.271401	−	0.962466i	$-0.412513\pi$
$571$	12.9706	0.542801	0.271401	+	0.962466i	$0.412513\pi$
$572$	9.51472	0.397830
$573$	0	0
$574$	0	0
$575$	6.21320	0.259108
$576$	0	0
$577$	−31.9411	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$577$	−31.9411	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	18.0000	0.745484
$584$	−7.00000	−0.289662
$585$	0	0
$586$	22.9706	0.948905
$587$	39.2132	1.61850	0.809251	−	0.587463i	$-0.199873\pi$
$587$	39.2132	1.61850	0.809251	+	0.587463i	$0.199873\pi$
$588$	0	0
$589$	−20.7279	−0.854079
$590$	0	0
$591$	0	0
$592$	−4.00000	−0.164399
$593$	11.4853	0.471644	0.235822	−	0.971796i	$-0.424222\pi$
$593$	11.4853	0.471644	0.235822	+	0.971796i	$0.424222\pi$
$594$	0	0
$595$	0	0
$596$	7.75736	0.317754
$597$	0	0
$598$	2.78680	0.113961
$599$	−10.9706	−0.448245	−0.224123	−	0.974561i	$-0.571952\pi$
$599$	−10.9706	−0.448245	−0.224123	+	0.974561i	$0.571952\pi$
$600$	0	0
$601$	15.9706	0.651453	0.325726	−	0.945464i	$-0.394391\pi$
$601$	15.9706	0.651453	0.325726	+	0.945464i	$0.394391\pi$
$602$	0	0
$603$	0	0
$604$	−11.2426	−0.457457
$605$	0	0
$606$	0	0
$607$	−5.02944	−0.204139	−0.102069	−	0.994777i	$-0.532546\pi$
$607$	−5.02944	−0.204139	−0.102069	+	0.994777i	$0.532546\pi$
$608$	−2.24264	−0.0909511
$609$	0	0
$610$	0	0
$611$	−10.6690	−0.431623
$612$	0	0
$613$	29.9411	1.20931	0.604655	−	0.796487i	$-0.293311\pi$
$613$	29.9411	1.20931	0.604655	+	0.796487i	$0.293311\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	0	0
$617$	22.4558	0.904038	0.452019	−	0.892008i	$-0.350704\pi$
$617$	22.4558	0.904038	0.452019	+	0.892008i	$0.350704\pi$
$618$	0	0
$619$	−18.4853	−0.742986	−0.371493	−	0.928436i	$-0.621154\pi$
$619$	−18.4853	−0.742986	−0.371493	+	0.928436i	$0.621154\pi$
$620$	0	0
$621$	0	0
$622$	22.9706	0.921036
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	15.9706	0.638312
$627$	0	0
$628$	14.0000	0.558661
$629$	0	0
$630$	0	0
$631$	−7.51472	−0.299156	−0.149578	−	0.988750i	$-0.547792\pi$
$631$	−7.51472	−0.299156	−0.149578	+	0.988750i	$0.547792\pi$
$632$	0.757359	0.0301261
$633$	0	0
$634$	16.9706	0.673987
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−18.0000	−0.712627
$639$	0	0
$640$	0	0
$641$	46.4558	1.83490	0.917448	−	0.397856i	$-0.130246\pi$
$641$	46.4558	1.83490	0.917448	+	0.397856i	$0.130246\pi$
$642$	0	0
$643$	−21.2721	−0.838889	−0.419444	−	0.907781i	$-0.637775\pi$
$643$	−21.2721	−0.838889	−0.419444	+	0.907781i	$0.637775\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.7279	1.09010	0.545049	−	0.838404i	$-0.316511\pi$
$647$	27.7279	1.09010	0.545049	+	0.838404i	$0.316511\pi$
$648$	0	0
$649$	0	0
$650$	11.2132	0.439818
$651$	0	0
$652$	−20.2426	−0.792763
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	0	0
$656$	11.4853	0.448425
$657$	0	0
$658$	0	0
$659$	3.21320	0.125169	0.0625843	−	0.998040i	$-0.480066\pi$
$659$	3.21320	0.125169	0.0625843	+	0.998040i	$0.480066\pi$
$660$	0	0
$661$	34.1838	1.32959	0.664797	−	0.747024i	$-0.268518\pi$
$661$	34.1838	1.32959	0.664797	+	0.747024i	$0.268518\pi$
$662$	−1.51472	−0.0588712
$663$	0	0
$664$	16.2426	0.630337
$665$	0	0
$666$	0	0
$667$	−5.27208	−0.204136
$668$	−18.2132	−0.704690
$669$	0	0
$670$	0	0
$671$	9.51472	0.367312
$672$	0	0
$673$	−10.5147	−0.405313	−0.202656	−	0.979250i	$-0.564957\pi$
$673$	−10.5147	−0.405313	−0.202656	+	0.979250i	$0.564957\pi$
$674$	−12.4853	−0.480915
$675$	0	0
$676$	−7.97056	−0.306560
$677$	25.7574	0.989936	0.494968	−	0.868911i	$-0.335180\pi$
$677$	25.7574	0.989936	0.494968	+	0.868911i	$0.335180\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	−39.2132	−1.50155
$683$	−17.6985	−0.677214	−0.338607	−	0.940928i	$-0.609956\pi$
$683$	−17.6985	−0.677214	−0.338607	+	0.940928i	$0.609956\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	10.4853	0.399748
$689$	9.51472	0.362482
$690$	0	0
$691$	−2.24264	−0.0853141	−0.0426570	−	0.999090i	$-0.513582\pi$
$691$	−2.24264	−0.0853141	−0.0426570	+	0.999090i	$0.513582\pi$
$692$	22.9706	0.873210
$693$	0	0
$694$	18.7279	0.710902
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−8.97056	−0.339541
$699$	0	0
$700$	0	0
$701$	22.2426	0.840093	0.420046	−	0.907503i	$-0.362014\pi$
$701$	22.2426	0.840093	0.420046	+	0.907503i	$0.362014\pi$
$702$	0	0
$703$	8.97056	0.338331
$704$	−4.24264	−0.159901
$705$	0	0
$706$	21.0000	0.790345
$707$	0	0
$708$	0	0
$709$	−33.6985	−1.26557	−0.632787	−	0.774326i	$-0.718089\pi$
$709$	−33.6985	−1.26557	−0.632787	+	0.774326i	$0.718089\pi$
$710$	0	0
$711$	0	0
$712$	11.4853	0.430429
$713$	−11.4853	−0.430127
$714$	0	0
$715$	0	0
$716$	−7.75736	−0.289906
$717$	0	0
$718$	19.2426	0.718129
$719$	−27.7279	−1.03408	−0.517039	−	0.855962i	$-0.672966\pi$
$719$	−27.7279	−1.03408	−0.517039	+	0.855962i	$0.672966\pi$
$720$	0	0
$721$	0	0
$722$	−13.9706	−0.519931
$723$	0	0
$724$	−11.7574	−0.436959
$725$	−21.2132	−0.787839
$726$	0	0
$727$	−0.272078	−0.0100908	−0.00504541	−	0.999987i	$-0.501606\pi$
$727$	−0.272078	−0.0100908	−0.00504541	+	0.999987i	$0.501606\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−5.02944	−0.185767	−0.0928833	−	0.995677i	$-0.529608\pi$
$733$	−5.02944	−0.185767	−0.0928833	+	0.995677i	$0.529608\pi$
$734$	−13.7279	−0.506707
$735$	0	0
$736$	−1.24264	−0.0458043
$737$	−1.02944	−0.0379198
$738$	0	0
$739$	−6.48528	−0.238565	−0.119282	−	0.992860i	$-0.538059\pi$
$739$	−6.48528	−0.238565	−0.119282	+	0.992860i	$0.538059\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	43.2426	1.58642	0.793209	−	0.608949i	$-0.208409\pi$
$743$	43.2426	1.58642	0.793209	+	0.608949i	$0.208409\pi$
$744$	0	0
$745$	0	0
$746$	35.2132	1.28925
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.75736	−0.319561	−0.159780	−	0.987153i	$-0.551079\pi$
$751$	−8.75736	−0.319561	−0.159780	+	0.987153i	$0.551079\pi$
$752$	4.75736	0.173483
$753$	0	0
$754$	−9.51472	−0.346506
$755$	0	0
$756$	0	0
$757$	−20.9706	−0.762188	−0.381094	−	0.924536i	$-0.624453\pi$
$757$	−20.9706	−0.762188	−0.381094	+	0.924536i	$0.624453\pi$
$758$	26.7279	0.970802
$759$	0	0
$760$	0	0
$761$	21.0000	0.761249	0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000	0.761249	0.380625	+	0.924730i	$0.375709\pi$
$762$	0	0
$763$	0	0
$764$	−8.48528	−0.306987
$765$	0	0
$766$	18.2132	0.658070
$767$	0	0
$768$	0	0
$769$	4.48528	0.161743	0.0808717	−	0.996725i	$-0.474230\pi$
$769$	4.48528	0.161743	0.0808717	+	0.996725i	$0.474230\pi$
$770$	0	0
$771$	0	0
$772$	−21.4853	−0.773272
$773$	−2.78680	−0.100234	−0.0501171	−	0.998743i	$-0.515959\pi$
$773$	−2.78680	−0.100234	−0.0501171	+	0.998743i	$0.515959\pi$
$774$	0	0
$775$	−46.2132	−1.66003
$776$	4.48528	0.161012
$777$	0	0
$778$	1.75736	0.0630044
$779$	−25.7574	−0.922853
$780$	0	0
$781$	−5.27208	−0.188650
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11.2132	0.399708	0.199854	−	0.979826i	$-0.435953\pi$
$787$	11.2132	0.399708	0.199854	+	0.979826i	$0.435953\pi$
$788$	−16.9706	−0.604551
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.02944	0.178601
$794$	−11.7574	−0.417253
$795$	0	0
$796$	−23.2426	−0.823814
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	0	0
$802$	12.5147	0.441910
$803$	29.6985	1.04804
$804$	0	0
$805$	0	0
$806$	−20.7279	−0.730110
$807$	0	0
$808$	16.2426	0.571414
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	27.4558	0.964105	0.482053	−	0.876142i	$-0.339891\pi$
$811$	27.4558	0.964105	0.482053	+	0.876142i	$0.339891\pi$
$812$	0	0
$813$	0	0
$814$	16.9706	0.594818
$815$	0	0
$816$	0	0
$817$	−23.5147	−0.822676
$818$	35.0000	1.22375
$819$	0	0
$820$	0	0
$821$	44.1838	1.54202	0.771012	−	0.636821i	$-0.219751\pi$
$821$	44.1838	1.54202	0.771012	+	0.636821i	$0.219751\pi$
$822$	0	0
$823$	−18.6985	−0.651788	−0.325894	−	0.945406i	$-0.605665\pi$
$823$	−18.6985	−0.651788	−0.325894	+	0.945406i	$0.605665\pi$
$824$	9.24264	0.321983
$825$	0	0
$826$	0	0
$827$	−16.9706	−0.590124	−0.295062	−	0.955478i	$-0.595340\pi$
$827$	−16.9706	−0.590124	−0.295062	+	0.955478i	$0.595340\pi$
$828$	0	0
$829$	17.9411	0.623121	0.311561	−	0.950226i	$-0.399148\pi$
$829$	17.9411	0.623121	0.311561	+	0.950226i	$0.399148\pi$
$830$	0	0
$831$	0	0
$832$	−2.24264	−0.0777496
$833$	0	0
$834$	0	0
$835$	0	0
$836$	9.51472	0.329073
$837$	0	0
$838$	−16.2426	−0.561093
$839$	32.4853	1.12152	0.560758	−	0.827980i	$-0.310510\pi$
$839$	32.4853	1.12152	0.560758	+	0.827980i	$0.310510\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	−5.75736	−0.198412
$843$	0	0
$844$	10.4853	0.360918
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−4.24264	−0.145693
$849$	0	0
$850$	0	0
$851$	4.97056	0.170389
$852$	0	0
$853$	−48.1838	−1.64978	−0.824890	−	0.565293i	$-0.808763\pi$
$853$	−48.1838	−1.64978	−0.824890	+	0.565293i	$0.808763\pi$
$854$	0	0
$855$	0	0
$856$	14.4853	0.495097
$857$	11.4853	0.392330	0.196165	−	0.980571i	$-0.437151\pi$
$857$	11.4853	0.392330	0.196165	+	0.980571i	$0.437151\pi$
$858$	0	0
$859$	−15.6985	−0.535625	−0.267813	−	0.963471i	$-0.586301\pi$
$859$	−15.6985	−0.535625	−0.267813	+	0.963471i	$0.586301\pi$
$860$	0	0
$861$	0	0
$862$	−28.7574	−0.979480
$863$	42.2132	1.43695	0.718477	−	0.695551i	$-0.244840\pi$
$863$	42.2132	1.43695	0.718477	+	0.695551i	$0.244840\pi$
$864$	0	0
$865$	0	0
$866$	−29.9706	−1.01844
$867$	0	0
$868$	0	0
$869$	−3.21320	−0.109000
$870$	0	0
$871$	−0.544156	−0.0184380
$872$	6.24264	0.211402
$873$	0	0
$874$	2.78680	0.0942648
$875$	0	0
$876$	0	0
$877$	−33.6985	−1.13792	−0.568958	−	0.822366i	$-0.692653\pi$
$877$	−33.6985	−1.13792	−0.568958	+	0.822366i	$0.692653\pi$
$878$	−13.7279	−0.463295
$879$	0	0
$880$	0	0
$881$	−53.4853	−1.80196	−0.900982	−	0.433856i	$-0.857153\pi$
$881$	−53.4853	−1.80196	−0.900982	+	0.433856i	$0.857153\pi$
$882$	0	0
$883$	−9.69848	−0.326380	−0.163190	−	0.986595i	$-0.552178\pi$
$883$	−9.69848	−0.326380	−0.163190	+	0.986595i	$0.552178\pi$
$884$	0	0
$885$	0	0
$886$	26.4853	0.889790
$887$	14.2721	0.479209	0.239605	−	0.970871i	$-0.422982\pi$
$887$	14.2721	0.479209	0.239605	+	0.970871i	$0.422982\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	−13.7279	−0.459645
$893$	−10.6690	−0.357026
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	28.9706	0.966760
$899$	39.2132	1.30783
$900$	0	0
$901$	0	0
$902$	−48.7279	−1.62246
$903$	0	0
$904$	−3.51472	−0.116898
$905$	0	0
$906$	0	0
$907$	29.9411	0.994179	0.497089	−	0.867699i	$-0.334402\pi$
$907$	29.9411	0.994179	0.497089	+	0.867699i	$0.334402\pi$
$908$	9.51472	0.315757
$909$	0	0
$910$	0	0
$911$	−39.7279	−1.31624	−0.658122	−	0.752911i	$-0.728649\pi$
$911$	−39.7279	−1.31624	−0.658122	+	0.752911i	$0.728649\pi$
$912$	0	0
$913$	−68.9117	−2.28064
$914$	28.4853	0.942209
$915$	0	0
$916$	−8.97056	−0.296396
$917$	0	0
$918$	0	0
$919$	−17.4558	−0.575815	−0.287908	−	0.957658i	$-0.592960\pi$
$919$	−17.4558	−0.575815	−0.287908	+	0.957658i	$0.592960\pi$
$920$	0	0
$921$	0	0
$922$	25.7574	0.848273
$923$	−2.78680	−0.0917285
$924$	0	0
$925$	20.0000	0.657596
$926$	−17.2426	−0.566628
$927$	0	0
$928$	4.24264	0.139272
$929$	−43.9706	−1.44263	−0.721314	−	0.692609i	$-0.756461\pi$
$929$	−43.9706	−1.44263	−0.721314	+	0.692609i	$0.756461\pi$
$930$	0	0
$931$	0	0
$932$	3.51472	0.115128
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20.4558	−0.668263	−0.334132	−	0.942526i	$-0.608443\pi$
$937$	−20.4558	−0.668263	−0.334132	+	0.942526i	$0.608443\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	52.6690	1.71696	0.858481	−	0.512845i	$-0.171409\pi$
$941$	52.6690	1.71696	0.858481	+	0.512845i	$0.171409\pi$
$942$	0	0
$943$	−14.2721	−0.464763
$944$	0	0
$945$	0	0
$946$	−44.4853	−1.44634
$947$	−38.4853	−1.25060	−0.625302	−	0.780383i	$-0.715024\pi$
$947$	−38.4853	−1.25060	−0.625302	+	0.780383i	$0.715024\pi$
$948$	0	0
$949$	15.6985	0.509594
$950$	11.2132	0.363804
$951$	0	0
$952$	0	0
$953$	−23.4853	−0.760763	−0.380381	−	0.924830i	$-0.624207\pi$
$953$	−23.4853	−0.760763	−0.380381	+	0.924830i	$0.624207\pi$
$954$	0	0
$955$	0	0
$956$	−21.7279	−0.702731
$957$	0	0
$958$	−4.75736	−0.153703
$959$	0	0
$960$	0	0
$961$	54.4264	1.75569
$962$	8.97056	0.289223
$963$	0	0
$964$	25.4853	0.820826
$965$	0	0
$966$	0	0
$967$	52.6985	1.69467	0.847335	−	0.531060i	$-0.178206\pi$
$967$	52.6985	1.69467	0.847335	+	0.531060i	$0.178206\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	−9.51472	−0.305342	−0.152671	−	0.988277i	$-0.548787\pi$
$971$	−9.51472	−0.305342	−0.152671	+	0.988277i	$0.548787\pi$
$972$	0	0
$973$	0	0
$974$	−11.2426	−0.360237
$975$	0	0
$976$	−2.24264	−0.0717852
$977$	19.9706	0.638915	0.319457	−	0.947601i	$-0.396499\pi$
$977$	19.9706	0.638915	0.319457	+	0.947601i	$0.396499\pi$
$978$	0	0
$979$	−48.7279	−1.55735
$980$	0	0
$981$	0	0
$982$	−18.7279	−0.597632
$983$	−32.4853	−1.03612	−0.518060	−	0.855344i	$-0.673346\pi$
$983$	−32.4853	−1.03612	−0.518060	+	0.855344i	$0.673346\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	5.02944	0.160008
$989$	−13.0294	−0.414312
$990$	0	0
$991$	7.78680	0.247356	0.123678	−	0.992322i	$-0.460531\pi$
$991$	7.78680	0.247356	0.123678	+	0.992322i	$0.460531\pi$
$992$	9.24264	0.293454
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	14.7279	0.466204
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bq.1.1		2
3.2	odd	2		7938.2.a.bk.1.2		2
7.2	even	3		1134.2.g.i.487.2	yes	4
7.4	even	3		1134.2.g.i.163.2	✓	4
7.6	odd	2		7938.2.a.bp.1.1		2
21.2	odd	6		1134.2.g.j.487.2	yes	4
21.11	odd	6		1134.2.g.j.163.2	yes	4
21.20	even	2		7938.2.a.bj.1.2		2
63.2	odd	6		1134.2.h.s.109.1		4
63.4	even	3		1134.2.h.r.541.2		4
63.11	odd	6		1134.2.e.r.919.1		4
63.16	even	3		1134.2.h.r.109.1		4
63.23	odd	6		1134.2.e.r.865.1		4
63.25	even	3		1134.2.e.s.919.1		4
63.32	odd	6		1134.2.h.s.541.2		4
63.58	even	3		1134.2.e.s.865.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.e.r.865.1		4	63.23	odd	6
1134.2.e.r.919.1		4	63.11	odd	6
1134.2.e.s.865.1		4	63.58	even	3
1134.2.e.s.919.1		4	63.25	even	3
1134.2.g.i.163.2	✓	4	7.4	even	3
1134.2.g.i.487.2	yes	4	7.2	even	3
1134.2.g.j.163.2	yes	4	21.11	odd	6
1134.2.g.j.487.2	yes	4	21.2	odd	6
1134.2.h.r.109.1		4	63.16	even	3
1134.2.h.r.541.2		4	63.4	even	3
1134.2.h.s.109.1		4	63.2	odd	6
1134.2.h.s.541.2		4	63.32	odd	6
7938.2.a.bj.1.2		2	21.20	even	2
7938.2.a.bk.1.2		2	3.2	odd	2
7938.2.a.bp.1.1		2	7.6	odd	2
7938.2.a.bq.1.1		2	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{8} -4.24264 q^{11} -2.24264 q^{13} +1.00000 q^{16} -2.24264 q^{19} -4.24264 q^{22} -1.24264 q^{23} -5.00000 q^{25} -2.24264 q^{26} +4.24264 q^{29} +9.24264 q^{31} +1.00000 q^{32} -4.00000 q^{37} -2.24264 q^{38} +11.4853 q^{41} +10.4853 q^{43} -4.24264 q^{44} -1.24264 q^{46} +4.75736 q^{47} -5.00000 q^{50} -2.24264 q^{52} -4.24264 q^{53} +4.24264 q^{58} -2.24264 q^{61} +9.24264 q^{62} +1.00000 q^{64} +0.242641 q^{67} +1.24264 q^{71} -7.00000 q^{73} -4.00000 q^{74} -2.24264 q^{76} +0.757359 q^{79} +11.4853 q^{82} +16.2426 q^{83} +10.4853 q^{86} -4.24264 q^{88} +11.4853 q^{89} -1.24264 q^{92} +4.75736 q^{94} +4.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 4 q^{13} + 2 q^{16} + 4 q^{19} + 6 q^{23} - 10 q^{25} + 4 q^{26} + 10 q^{31} + 2 q^{32} - 8 q^{37} + 4 q^{38} + 6 q^{41} + 4 q^{43} + 6 q^{46} + 18 q^{47} - 10 q^{50}+ \cdots - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^8 + 4 * q^13 + 2 * q^16 + 4 * q^19 + 6 * q^23 - 10 * q^25 + 4 * q^26 + 10 * q^31 + 2 * q^32 - 8 * q^37 + 4 * q^38 + 6 * q^41 + 4 * q^43 + 6 * q^46 + 18 * q^47 - 10 * q^50 + 4 * q^52 + 4 * q^61 + 10 * q^62 + 2 * q^64 - 8 * q^67 - 6 * q^71 - 14 * q^73 - 8 * q^74 + 4 * q^76 + 10 * q^79 + 6 * q^82 + 24 * q^83 + 4 * q^86 + 6 * q^89 + 6 * q^92 + 18 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more