LMFDB - Embedded newform 9248.2.a.z.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9248,2,Mod(1,9248)] 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("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-6,0,0,0,6,0,0,0,2,0,0,0,0,0,0,0,-4,0,0,0,2,0,0,0,-22, 0,0,0,14,0,0,0,-16,0,0,0,8,0,0,0,4,0,0,0,22,0,0,0,-30,0,0,0,-18,0,0,0, -28,0,0,0,36,0,0,0,-4,0,0,0,-20,0,0,0,46,0,0,0,-32,0,0,0,0,0,0,0,8,0,0, 0,-14,0,0,0,-30,0,0,0,12,0,0,0,-46,0,0,0,20,0,0,0,-30,0,0,0,42,0,0,0,16, 0,0,0,-12,0,0,0,58,0,0,0,4,0,0,0,-40,0,0,0,-86,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.8456517893$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.45968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 17 $ x^4 - 9*x^2 + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.64232$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.64232 q^{3} -3.30278 q^{5} +3.78190 q^{7} -0.302776 q^{9} +5.42422 q^{11} -4.90833 q^{13} -5.42422 q^{15} -1.64232 q^{19} +6.21110 q^{21} +3.78190 q^{23} +5.90833 q^{25} -5.42422 q^{27} -3.69722 q^{29} -5.42422 q^{31} +8.90833 q^{33} -12.4908 q^{35} -4.00000 q^{37} -8.06106 q^{39} +9.21110 q^{41} +1.14507 q^{43} +1.00000 q^{45} -11.9935 q^{47} +7.30278 q^{49} -12.9083 q^{53} -17.9150 q^{55} -2.69722 q^{57} -15.1276 q^{59} +0.211103 q^{61} -1.14507 q^{63} +16.2111 q^{65} +6.56929 q^{67} +6.21110 q^{69} +6.56929 q^{71} +2.21110 q^{73} +9.70338 q^{75} +20.5139 q^{77} -8.00000 q^{81} -8.21161 q^{83} -6.07204 q^{87} -5.21110 q^{89} -18.5628 q^{91} -8.90833 q^{93} +5.42422 q^{95} -16.5139 q^{97} -1.64232 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} + 6 q^{9} + 2 q^{13} - 4 q^{21} + 2 q^{25} - 22 q^{29} + 14 q^{33} - 16 q^{37} + 8 q^{41} + 4 q^{45} + 22 q^{49} - 30 q^{53} - 18 q^{57} - 28 q^{61} + 36 q^{65} - 4 q^{69} - 20 q^{73} + 46 q^{77}+ \cdots - 30 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 + 6 * q^9 + 2 * q^13 - 4 * q^21 + 2 * q^25 - 22 * q^29 + 14 * q^33 - 16 * q^37 + 8 * q^41 + 4 * q^45 + 22 * q^49 - 30 * q^53 - 18 * q^57 - 28 * q^61 + 36 * q^65 - 4 * q^69 - 20 * q^73 + 46 * q^77 - 32 * q^81 + 8 * q^89 - 14 * q^93 - 30 * 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	0
$2$	0	0
$3$	1.64232	0.948196	0.474098	−	0.880472i	$-0.342774\pi$
$3$	1.64232	0.948196	0.474098	+	0.880472i	$0.342774\pi$
$4$	0	0
$5$	−3.30278	−1.47705	−0.738523	−	0.674228i	$-0.764476\pi$
$5$	−3.30278	−1.47705	−0.738523	+	0.674228i	$0.764476\pi$
$6$	0	0
$7$	3.78190	1.42942	0.714712	−	0.699419i	$-0.246558\pi$
$7$	3.78190	1.42942	0.714712	+	0.699419i	$0.246558\pi$
$8$	0	0
$9$	−0.302776	−0.100925
$10$	0	0
$11$	5.42422	1.63547	0.817733	−	0.575598i	$-0.195231\pi$
$11$	5.42422	1.63547	0.817733	+	0.575598i	$0.195231\pi$
$12$	0	0
$13$	−4.90833	−1.36132	−0.680662	−	0.732597i	$-0.738308\pi$
$13$	−4.90833	−1.36132	−0.680662	+	0.732597i	$0.738308\pi$
$14$	0	0
$15$	−5.42422	−1.40053
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−1.64232	−0.376775	−0.188387	−	0.982095i	$-0.560326\pi$
$19$	−1.64232	−0.376775	−0.188387	+	0.982095i	$0.560326\pi$
$20$	0	0
$21$	6.21110	1.35537
$22$	0	0
$23$	3.78190	0.788581	0.394290	−	0.918986i	$-0.370990\pi$
$23$	3.78190	0.788581	0.394290	+	0.918986i	$0.370990\pi$
$24$	0	0
$25$	5.90833	1.18167
$26$	0	0
$27$	−5.42422	−1.04389
$28$	0	0
$29$	−3.69722	−0.686557	−0.343279	−	0.939234i	$-0.611538\pi$
$29$	−3.69722	−0.686557	−0.343279	+	0.939234i	$0.611538\pi$
$30$	0	0
$31$	−5.42422	−0.974219	−0.487110	−	0.873341i	$-0.661949\pi$
$31$	−5.42422	−0.974219	−0.487110	+	0.873341i	$0.661949\pi$
$32$	0	0
$33$	8.90833	1.55074
$34$	0	0
$35$	−12.4908	−2.11133
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−8.06106	−1.29080
$40$	0	0
$41$	9.21110	1.43853	0.719266	−	0.694735i	$-0.244478\pi$
$41$	9.21110	1.43853	0.719266	+	0.694735i	$0.244478\pi$
$42$	0	0
$43$	1.14507	0.174621	0.0873106	−	0.996181i	$-0.472173\pi$
$43$	1.14507	0.174621	0.0873106	+	0.996181i	$0.472173\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−11.9935	−1.74943	−0.874717	−	0.484634i	$-0.838953\pi$
$47$	−11.9935	−1.74943	−0.874717	+	0.484634i	$0.838953\pi$
$48$	0	0
$49$	7.30278	1.04325
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.9083	−1.77310	−0.886548	−	0.462638i	$-0.846903\pi$
$53$	−12.9083	−1.77310	−0.886548	+	0.462638i	$0.846903\pi$
$54$	0	0
$55$	−17.9150	−2.41566
$56$	0	0
$57$	−2.69722	−0.357256
$58$	0	0
$59$	−15.1276	−1.96945	−0.984723	−	0.174127i	$-0.944290\pi$
$59$	−15.1276	−1.96945	−0.984723	+	0.174127i	$0.944290\pi$
$60$	0	0
$61$	0.211103	0.0270289	0.0135145	−	0.999909i	$-0.495698\pi$
$61$	0.211103	0.0270289	0.0135145	+	0.999909i	$0.495698\pi$
$62$	0	0
$63$	−1.14507	−0.144265
$64$	0	0
$65$	16.2111	2.01074
$66$	0	0
$67$	6.56929	0.802567	0.401283	−	0.915954i	$-0.368564\pi$
$67$	6.56929	0.802567	0.401283	+	0.915954i	$0.368564\pi$
$68$	0	0
$69$	6.21110	0.747729
$70$	0	0
$71$	6.56929	0.779631	0.389816	−	0.920893i	$-0.372539\pi$
$71$	6.56929	0.779631	0.389816	+	0.920893i	$0.372539\pi$
$72$	0	0
$73$	2.21110	0.258790	0.129395	−	0.991593i	$-0.458696\pi$
$73$	2.21110	0.258790	0.129395	+	0.991593i	$0.458696\pi$
$74$	0	0
$75$	9.70338	1.12045
$76$	0	0
$77$	20.5139	2.33777
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−8.00000	−0.888889
$82$	0	0
$83$	−8.21161	−0.901342	−0.450671	−	0.892690i	$-0.648815\pi$
$83$	−8.21161	−0.901342	−0.450671	+	0.892690i	$0.648815\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.07204	−0.650991
$88$	0	0
$89$	−5.21110	−0.552376	−0.276188	−	0.961104i	$-0.589071\pi$
$89$	−5.21110	−0.552376	−0.276188	+	0.961104i	$0.589071\pi$
$90$	0	0
$91$	−18.5628	−1.94591
$92$	0	0
$93$	−8.90833	−0.923750
$94$	0	0
$95$	5.42422	0.556514
$96$	0	0
$97$	−16.5139	−1.67673	−0.838365	−	0.545109i	$-0.816488\pi$
$97$	−16.5139	−1.67673	−0.838365	+	0.545109i	$0.816488\pi$
$98$	0	0
$99$	−1.64232	−0.165060
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	15.1276	1.49057	0.745284	−	0.666748i	$-0.232314\pi$
$103$	15.1276	1.49057	0.745284	+	0.666748i	$0.232314\pi$
$104$	0	0
$105$	−20.5139	−2.00195
$106$	0	0
$107$	11.9935	1.15946	0.579728	−	0.814810i	$-0.303159\pi$
$107$	11.9935	1.15946	0.579728	+	0.814810i	$0.303159\pi$
$108$	0	0
$109$	19.4222	1.86031	0.930155	−	0.367167i	$-0.119672\pi$
$109$	19.4222	1.86031	0.930155	+	0.367167i	$0.119672\pi$
$110$	0	0
$111$	−6.56929	−0.623530
$112$	0	0
$113$	−2.09167	−0.196768	−0.0983840	−	0.995149i	$-0.531367\pi$
$113$	−2.09167	−0.196768	−0.0983840	+	0.995149i	$0.531367\pi$
$114$	0	0
$115$	−12.4908	−1.16477
$116$	0	0
$117$	1.48612	0.137392
$118$	0	0
$119$	0	0
$120$	0	0
$121$	18.4222	1.67475
$122$	0	0
$123$	15.1276	1.36401
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−1.14507	−0.101608	−0.0508042	−	0.998709i	$-0.516178\pi$
$127$	−1.14507	−0.101608	−0.0508042	+	0.998709i	$0.516178\pi$
$128$	0	0
$129$	1.88057	0.165575
$130$	0	0
$131$	6.56929	0.573962	0.286981	−	0.957936i	$-0.407348\pi$
$131$	6.56929	0.573962	0.286981	+	0.957936i	$0.407348\pi$
$132$	0	0
$133$	−6.21110	−0.538571
$134$	0	0
$135$	17.9150	1.54188
$136$	0	0
$137$	−2.78890	−0.238272	−0.119136	−	0.992878i	$-0.538012\pi$
$137$	−2.78890	−0.238272	−0.119136	+	0.992878i	$0.538012\pi$
$138$	0	0
$139$	−4.77641	−0.405130	−0.202565	−	0.979269i	$-0.564928\pi$
$139$	−4.77641	−0.405130	−0.202565	+	0.979269i	$0.564928\pi$
$140$	0	0
$141$	−19.6972	−1.65881
$142$	0	0
$143$	−26.6239	−2.22640
$144$	0	0
$145$	12.2111	1.01408
$146$	0	0
$147$	11.9935	0.989208
$148$	0	0
$149$	9.21110	0.754603	0.377301	−	0.926090i	$-0.376852\pi$
$149$	9.21110	0.754603	0.377301	+	0.926090i	$0.376852\pi$
$150$	0	0
$151$	−5.42422	−0.441417	−0.220709	−	0.975340i	$-0.570837\pi$
$151$	−5.42422	−0.441417	−0.220709	+	0.975340i	$0.570837\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.9150	1.43897
$156$	0	0
$157$	−13.9083	−1.11001	−0.555003	−	0.831849i	$-0.687283\pi$
$157$	−13.9083	−1.11001	−0.555003	+	0.831849i	$0.687283\pi$
$158$	0	0
$159$	−21.1996	−1.68124
$160$	0	0
$161$	14.3028	1.12722
$162$	0	0
$163$	−11.9935	−0.939405	−0.469702	−	0.882825i	$-0.655639\pi$
$163$	−11.9935	−0.939405	−0.469702	+	0.882825i	$0.655639\pi$
$164$	0	0
$165$	−29.4222	−2.29052
$166$	0	0
$167$	−4.92697	−0.381260	−0.190630	−	0.981662i	$-0.561053\pi$
$167$	−4.92697	−0.381260	−0.190630	+	0.981662i	$0.561053\pi$
$168$	0	0
$169$	11.0917	0.853206
$170$	0	0
$171$	0.497255	0.0380261
$172$	0	0
$173$	−8.21110	−0.624279	−0.312139	−	0.950036i	$-0.601046\pi$
$173$	−8.21110	−0.624279	−0.312139	+	0.950036i	$0.601046\pi$
$174$	0	0
$175$	22.3447	1.68910
$176$	0	0
$177$	−24.8444	−1.86742
$178$	0	0
$179$	9.35668	0.699351	0.349676	−	0.936871i	$-0.386292\pi$
$179$	9.35668	0.699351	0.349676	+	0.936871i	$0.386292\pi$
$180$	0	0
$181$	−18.2111	−1.35362	−0.676810	−	0.736157i	$-0.736638\pi$
$181$	−18.2111	−1.35362	−0.676810	+	0.736157i	$0.736638\pi$
$182$	0	0
$183$	0.346699	0.0256287
$184$	0	0
$185$	13.2111	0.971300
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−20.5139	−1.49217
$190$	0	0
$191$	−11.4963	−0.831840	−0.415920	−	0.909401i	$-0.636540\pi$
$191$	−11.4963	−0.831840	−0.415920	+	0.909401i	$0.636540\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	26.6239	1.90657
$196$	0	0
$197$	−22.9361	−1.63413	−0.817064	−	0.576547i	$-0.804400\pi$
$197$	−22.9361	−1.63413	−0.817064	+	0.576547i	$0.804400\pi$
$198$	0	0
$199$	−7.71436	−0.546856	−0.273428	−	0.961892i	$-0.588158\pi$
$199$	−7.71436	−0.546856	−0.273428	+	0.961892i	$0.588158\pi$
$200$	0	0
$201$	10.7889	0.760990
$202$	0	0
$203$	−13.9825	−0.981382
$204$	0	0
$205$	−30.4222	−2.12478
$206$	0	0
$207$	−1.14507	−0.0795877
$208$	0	0
$209$	−8.90833	−0.616202
$210$	0	0
$211$	−18.9095	−1.30178	−0.650892	−	0.759170i	$-0.725605\pi$
$211$	−18.9095	−1.30178	−0.650892	+	0.759170i	$0.725605\pi$
$212$	0	0
$213$	10.7889	0.739243
$214$	0	0
$215$	−3.78190	−0.257924
$216$	0	0
$217$	−20.5139	−1.39257
$218$	0	0
$219$	3.63134	0.245384
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−11.3457	−0.759764	−0.379882	−	0.925035i	$-0.624035\pi$
$223$	−11.3457	−0.759764	−0.379882	+	0.925035i	$0.624035\pi$
$224$	0	0
$225$	−1.78890	−0.119260
$226$	0	0
$227$	−22.8420	−1.51607	−0.758037	−	0.652211i	$-0.773841\pi$
$227$	−22.8420	−1.51607	−0.758037	+	0.652211i	$0.773841\pi$
$228$	0	0
$229$	−6.78890	−0.448623	−0.224311	−	0.974517i	$-0.572013\pi$
$229$	−6.78890	−0.448623	−0.224311	+	0.974517i	$0.572013\pi$
$230$	0	0
$231$	33.6904	2.21667
$232$	0	0
$233$	22.4222	1.46893	0.734464	−	0.678648i	$-0.237434\pi$
$233$	22.4222	1.46893	0.734464	+	0.678648i	$0.237434\pi$
$234$	0	0
$235$	39.6119	2.58399
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.5518	−1.32939	−0.664694	−	0.747116i	$-0.731438\pi$
$239$	−20.5518	−1.32939	−0.664694	+	0.747116i	$0.731438\pi$
$240$	0	0
$241$	−13.9083	−0.895914	−0.447957	−	0.894055i	$-0.647848\pi$
$241$	−13.9083	−0.895914	−0.447957	+	0.894055i	$0.647848\pi$
$242$	0	0
$243$	3.13409	0.201052
$244$	0	0
$245$	−24.1194	−1.54093
$246$	0	0
$247$	8.06106	0.512913
$248$	0	0
$249$	−13.4861	−0.854648
$250$	0	0
$251$	16.7699	1.05851	0.529254	−	0.848463i	$-0.322472\pi$
$251$	16.7699	1.05851	0.529254	+	0.848463i	$0.322472\pi$
$252$	0	0
$253$	20.5139	1.28970
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.4861	0.716485	0.358242	−	0.933629i	$-0.383376\pi$
$257$	11.4861	0.716485	0.358242	+	0.933629i	$0.383376\pi$
$258$	0	0
$259$	−15.1276	−0.939984
$260$	0	0
$261$	1.11943	0.0692909
$262$	0	0
$263$	3.63134	0.223918	0.111959	−	0.993713i	$-0.464287\pi$
$263$	3.63134	0.223918	0.111959	+	0.993713i	$0.464287\pi$
$264$	0	0
$265$	42.6333	2.61894
$266$	0	0
$267$	−8.55831	−0.523760
$268$	0	0
$269$	7.09167	0.432387	0.216193	−	0.976351i	$-0.430636\pi$
$269$	7.09167	0.432387	0.216193	+	0.976351i	$0.430636\pi$
$270$	0	0
$271$	10.3512	0.628790	0.314395	−	0.949292i	$-0.398198\pi$
$271$	10.3512	0.628790	0.314395	+	0.949292i	$0.398198\pi$
$272$	0	0
$273$	−30.4861	−1.84510
$274$	0	0
$275$	32.0481	1.93257
$276$	0	0
$277$	−25.2111	−1.51479	−0.757394	−	0.652958i	$-0.773528\pi$
$277$	−25.2111	−1.51479	−0.757394	+	0.652958i	$0.773528\pi$
$278$	0	0
$279$	1.64232	0.0983233
$280$	0	0
$281$	10.0917	0.602019	0.301009	−	0.953621i	$-0.402676\pi$
$281$	10.0917	0.602019	0.301009	+	0.953621i	$0.402676\pi$
$282$	0	0
$283$	−30.2552	−1.79849	−0.899243	−	0.437450i	$-0.855882\pi$
$283$	−30.2552	−1.79849	−0.899243	+	0.437450i	$0.855882\pi$
$284$	0	0
$285$	8.90833	0.527684
$286$	0	0
$287$	34.8355	2.05627
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−27.1211	−1.58987
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	49.9631	2.90896
$296$	0	0
$297$	−29.4222	−1.70725
$298$	0	0
$299$	−18.5628	−1.07351
$300$	0	0
$301$	4.33053	0.249608
$302$	0	0
$303$	4.92697	0.283047
$304$	0	0
$305$	−0.697224	−0.0399230
$306$	0	0
$307$	11.3457	0.647533	0.323767	−	0.946137i	$-0.395051\pi$
$307$	11.3457	0.647533	0.323767	+	0.946137i	$0.395051\pi$
$308$	0	0
$309$	24.8444	1.41335
$310$	0	0
$311$	13.4853	0.764680	0.382340	−	0.924022i	$-0.375118\pi$
$311$	13.4853	0.764680	0.382340	+	0.924022i	$0.375118\pi$
$312$	0	0
$313$	18.4222	1.04128	0.520642	−	0.853775i	$-0.325693\pi$
$313$	18.4222	1.04128	0.520642	+	0.853775i	$0.325693\pi$
$314$	0	0
$315$	3.78190	0.213086
$316$	0	0
$317$	15.4222	0.866197	0.433099	−	0.901347i	$-0.357420\pi$
$317$	15.4222	0.866197	0.433099	+	0.901347i	$0.357420\pi$
$318$	0	0
$319$	−20.0546	−1.12284
$320$	0	0
$321$	19.6972	1.09939
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−29.0000	−1.60863
$326$	0	0
$327$	31.8975	1.76394
$328$	0	0
$329$	−45.3583	−2.50068
$330$	0	0
$331$	33.0426	1.81619	0.908093	−	0.418769i	$-0.137538\pi$
$331$	33.0426	1.81619	0.908093	+	0.418769i	$0.137538\pi$
$332$	0	0
$333$	1.21110	0.0663680
$334$	0	0
$335$	−21.6969	−1.18543
$336$	0	0
$337$	18.1194	0.987028	0.493514	−	0.869738i	$-0.335712\pi$
$337$	18.1194	0.987028	0.493514	+	0.869738i	$0.335712\pi$
$338$	0	0
$339$	−3.43520	−0.186575
$340$	0	0
$341$	−29.4222	−1.59330
$342$	0	0
$343$	1.14507	0.0618278
$344$	0	0
$345$	−20.5139	−1.10443
$346$	0	0
$347$	−1.64232	−0.0881645	−0.0440823	−	0.999028i	$-0.514036\pi$
$347$	−1.64232	−0.0881645	−0.0440823	+	0.999028i	$0.514036\pi$
$348$	0	0
$349$	−31.4222	−1.68199	−0.840996	−	0.541041i	$-0.818030\pi$
$349$	−31.4222	−1.68199	−0.840996	+	0.541041i	$0.818030\pi$
$350$	0	0
$351$	26.6239	1.42108
$352$	0	0
$353$	18.6333	0.991751	0.495875	−	0.868394i	$-0.334847\pi$
$353$	18.6333	0.991751	0.495875	+	0.868394i	$0.334847\pi$
$354$	0	0
$355$	−21.6969	−1.15155
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.78739	−0.147113	−0.0735564	−	0.997291i	$-0.523435\pi$
$359$	−2.78739	−0.147113	−0.0735564	+	0.997291i	$0.523435\pi$
$360$	0	0
$361$	−16.3028	−0.858041
$362$	0	0
$363$	30.2552	1.58799
$364$	0	0
$365$	−7.30278	−0.382245
$366$	0	0
$367$	−30.2552	−1.57931	−0.789655	−	0.613552i	$-0.789740\pi$
$367$	−30.2552	−1.57931	−0.789655	+	0.613552i	$0.789740\pi$
$368$	0	0
$369$	−2.78890	−0.145184
$370$	0	0
$371$	−48.8180	−2.53451
$372$	0	0
$373$	−8.21110	−0.425155	−0.212577	−	0.977144i	$-0.568186\pi$
$373$	−8.21110	−0.425155	−0.212577	+	0.977144i	$0.568186\pi$
$374$	0	0
$375$	−4.92697	−0.254428
$376$	0	0
$377$	18.1472	0.934628
$378$	0	0
$379$	−5.42422	−0.278624	−0.139312	−	0.990249i	$-0.544489\pi$
$379$	−5.42422	−0.278624	−0.139312	+	0.990249i	$0.544489\pi$
$380$	0	0
$381$	−1.88057	−0.0963445
$382$	0	0
$383$	−18.0656	−0.923107	−0.461553	−	0.887112i	$-0.652708\pi$
$383$	−18.0656	−0.923107	−0.461553	+	0.887112i	$0.652708\pi$
$384$	0	0
$385$	−67.7527	−3.45300
$386$	0	0
$387$	−0.346699	−0.0176237
$388$	0	0
$389$	17.0000	0.861934	0.430967	−	0.902368i	$-0.358172\pi$
$389$	17.0000	0.861934	0.430967	+	0.902368i	$0.358172\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	10.7889	0.544228
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.7889	−0.692045	−0.346022	−	0.938226i	$-0.612468\pi$
$397$	−13.7889	−0.692045	−0.346022	+	0.938226i	$0.612468\pi$
$398$	0	0
$399$	−10.2006	−0.510671
$400$	0	0
$401$	28.3305	1.41476	0.707380	−	0.706834i	$-0.249877\pi$
$401$	28.3305	1.41476	0.707380	+	0.706834i	$0.249877\pi$
$402$	0	0
$403$	26.6239	1.32623
$404$	0	0
$405$	26.4222	1.31293
$406$	0	0
$407$	−21.6969	−1.07548
$408$	0	0
$409$	−34.2111	−1.69163	−0.845815	−	0.533476i	$-0.820885\pi$
$409$	−34.2111	−1.69163	−0.845815	+	0.533476i	$0.820885\pi$
$410$	0	0
$411$	−4.58027	−0.225928
$412$	0	0
$413$	−57.2111	−2.81517
$414$	0	0
$415$	27.1211	1.33132
$416$	0	0
$417$	−7.84441	−0.384142
$418$	0	0
$419$	−14.2837	−0.697802	−0.348901	−	0.937160i	$-0.613445\pi$
$419$	−14.2837	−0.697802	−0.348901	+	0.937160i	$0.613445\pi$
$420$	0	0
$421$	19.4861	0.949695	0.474848	−	0.880068i	$-0.342503\pi$
$421$	19.4861	0.949695	0.474848	+	0.880068i	$0.342503\pi$
$422$	0	0
$423$	3.63134	0.176562
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.798369	0.0386358
$428$	0	0
$429$	−43.7250	−2.11106
$430$	0	0
$431$	−5.77092	−0.277976	−0.138988	−	0.990294i	$-0.544385\pi$
$431$	−5.77092	−0.277976	−0.138988	+	0.990294i	$0.544385\pi$
$432$	0	0
$433$	10.7889	0.518481	0.259241	−	0.965813i	$-0.416528\pi$
$433$	10.7889	0.518481	0.259241	+	0.965813i	$0.416528\pi$
$434$	0	0
$435$	20.0546	0.961543
$436$	0	0
$437$	−6.21110	−0.297117
$438$	0	0
$439$	−26.2772	−1.25414	−0.627070	−	0.778963i	$-0.715746\pi$
$439$	−26.2772	−1.25414	−0.627070	+	0.778963i	$0.715746\pi$
$440$	0	0
$441$	−2.21110	−0.105291
$442$	0	0
$443$	−9.70338	−0.461022	−0.230511	−	0.973070i	$-0.574040\pi$
$443$	−9.70338	−0.461022	−0.230511	+	0.973070i	$0.574040\pi$
$444$	0	0
$445$	17.2111	0.815885
$446$	0	0
$447$	15.1276	0.715511
$448$	0	0
$449$	0.908327	0.0428666	0.0214333	−	0.999770i	$-0.493177\pi$
$449$	0.908327	0.0428666	0.0214333	+	0.999770i	$0.493177\pi$
$450$	0	0
$451$	49.9631	2.35267
$452$	0	0
$453$	−8.90833	−0.418550
$454$	0	0
$455$	61.3088	2.87420
$456$	0	0
$457$	−17.7250	−0.829140	−0.414570	−	0.910018i	$-0.636068\pi$
$457$	−17.7250	−0.829140	−0.414570	+	0.910018i	$0.636068\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15.6333	0.728116	0.364058	−	0.931376i	$-0.381391\pi$
$461$	15.6333	0.728116	0.364058	+	0.931376i	$0.381391\pi$
$462$	0	0
$463$	13.1386	0.610602	0.305301	−	0.952256i	$-0.401243\pi$
$463$	13.1386	0.610602	0.305301	+	0.952256i	$0.401243\pi$
$464$	0	0
$465$	29.4222	1.36442
$466$	0	0
$467$	−25.4788	−1.17902	−0.589509	−	0.807762i	$-0.700679\pi$
$467$	−25.4788	−1.17902	−0.589509	+	0.807762i	$0.700679\pi$
$468$	0	0
$469$	24.8444	1.14721
$470$	0	0
$471$	−22.8420	−1.05250
$472$	0	0
$473$	6.21110	0.285587
$474$	0	0
$475$	−9.70338	−0.445222
$476$	0	0
$477$	3.90833	0.178950
$478$	0	0
$479$	3.13409	0.143200	0.0716001	−	0.997433i	$-0.477189\pi$
$479$	3.13409	0.143200	0.0716001	+	0.997433i	$0.477189\pi$
$480$	0	0
$481$	19.6333	0.895202
$482$	0	0
$483$	23.4898	1.06882
$484$	0	0
$485$	54.5416	2.47661
$486$	0	0
$487$	−29.9085	−1.35528	−0.677642	−	0.735392i	$-0.736998\pi$
$487$	−29.9085	−1.35528	−0.677642	+	0.735392i	$0.736998\pi$
$488$	0	0
$489$	−19.6972	−0.890739
$490$	0	0
$491$	−5.77092	−0.260438	−0.130219	−	0.991485i	$-0.541568\pi$
$491$	−5.77092	−0.260438	−0.130219	+	0.991485i	$0.541568\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	5.42422	0.243801
$496$	0	0
$497$	24.8444	1.11442
$498$	0	0
$499$	15.9260	0.712944	0.356472	−	0.934306i	$-0.383979\pi$
$499$	15.9260	0.712944	0.356472	+	0.934306i	$0.383979\pi$
$500$	0	0
$501$	−8.09167	−0.361509
$502$	0	0
$503$	5.42422	0.241854	0.120927	−	0.992661i	$-0.461413\pi$
$503$	5.42422	0.241854	0.120927	+	0.992661i	$0.461413\pi$
$504$	0	0
$505$	−9.90833	−0.440915
$506$	0	0
$507$	18.2161	0.809006
$508$	0	0
$509$	3.78890	0.167940	0.0839700	−	0.996468i	$-0.473240\pi$
$509$	3.78890	0.167940	0.0839700	+	0.996468i	$0.473240\pi$
$510$	0	0
$511$	8.36217	0.369921
$512$	0	0
$513$	8.90833	0.393312
$514$	0	0
$515$	−49.9631	−2.20164
$516$	0	0
$517$	−65.0555	−2.86114
$518$	0	0
$519$	−13.4853	−0.591938
$520$	0	0
$521$	−9.21110	−0.403546	−0.201773	−	0.979432i	$-0.564670\pi$
$521$	−9.21110	−0.403546	−0.201773	+	0.979432i	$0.564670\pi$
$522$	0	0
$523$	11.9935	0.524440	0.262220	−	0.965008i	$-0.415545\pi$
$523$	11.9935	0.524440	0.262220	+	0.965008i	$0.415545\pi$
$524$	0	0
$525$	36.6972	1.60160
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−8.69722	−0.378140
$530$	0	0
$531$	4.58027	0.198767
$532$	0	0
$533$	−45.2111	−1.95831
$534$	0	0
$535$	−39.6119	−1.71257
$536$	0	0
$537$	15.3667	0.663122
$538$	0	0
$539$	39.6119	1.70620
$540$	0	0
$541$	3.69722	0.158956	0.0794780	−	0.996837i	$-0.474675\pi$
$541$	3.69722	0.158956	0.0794780	+	0.996837i	$0.474675\pi$
$542$	0	0
$543$	−29.9085	−1.28350
$544$	0	0
$545$	−64.1472	−2.74776
$546$	0	0
$547$	−27.1211	−1.15962	−0.579808	−	0.814753i	$-0.696872\pi$
$547$	−27.1211	−1.15962	−0.579808	+	0.814753i	$0.696872\pi$
$548$	0	0
$549$	−0.0639167	−0.00272790
$550$	0	0
$551$	6.07204	0.258677
$552$	0	0
$553$	0	0
$554$	0	0
$555$	21.6969	0.920982
$556$	0	0
$557$	17.2111	0.729258	0.364629	−	0.931153i	$-0.381196\pi$
$557$	17.2111	0.729258	0.364629	+	0.931153i	$0.381196\pi$
$558$	0	0
$559$	−5.62037	−0.237716
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.55831	0.360690	0.180345	−	0.983603i	$-0.442279\pi$
$563$	8.55831	0.360690	0.180345	+	0.983603i	$0.442279\pi$
$564$	0	0
$565$	6.90833	0.290635
$566$	0	0
$567$	−30.2552	−1.27060
$568$	0	0
$569$	−21.7889	−0.913438	−0.456719	−	0.889611i	$-0.650976\pi$
$569$	−21.7889	−0.913438	−0.456719	+	0.889611i	$0.650976\pi$
$570$	0	0
$571$	16.7699	0.701800	0.350900	−	0.936413i	$-0.385876\pi$
$571$	16.7699	0.701800	0.350900	+	0.936413i	$0.385876\pi$
$572$	0	0
$573$	−18.8806	−0.788747
$574$	0	0
$575$	22.3447	0.931839
$576$	0	0
$577$	32.6333	1.35854	0.679271	−	0.733887i	$-0.262296\pi$
$577$	32.6333	1.35854	0.679271	+	0.733887i	$0.262296\pi$
$578$	0	0
$579$	−8.21161	−0.341263
$580$	0	0
$581$	−31.0555	−1.28840
$582$	0	0
$583$	−70.0177	−2.89984
$584$	0	0
$585$	−4.90833	−0.202934
$586$	0	0
$587$	−29.9085	−1.23446	−0.617228	−	0.786784i	$-0.711744\pi$
$587$	−29.9085	−1.23446	−0.617228	+	0.786784i	$0.711744\pi$
$588$	0	0
$589$	8.90833	0.367061
$590$	0	0
$591$	−37.6685	−1.54947
$592$	0	0
$593$	−22.2111	−0.912101	−0.456050	−	0.889954i	$-0.650736\pi$
$593$	−22.2111	−0.912101	−0.456050	+	0.889954i	$0.650736\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.6695	−0.518527
$598$	0	0
$599$	22.8420	0.933297	0.466649	−	0.884443i	$-0.345461\pi$
$599$	22.8420	0.933297	0.466649	+	0.884443i	$0.345461\pi$
$600$	0	0
$601$	29.7527	1.21364	0.606820	−	0.794839i	$-0.292445\pi$
$601$	29.7527	1.21364	0.606820	+	0.794839i	$0.292445\pi$
$602$	0	0
$603$	−1.98902	−0.0809992
$604$	0	0
$605$	−60.8444	−2.47368
$606$	0	0
$607$	22.8420	0.927127	0.463563	−	0.886064i	$-0.346571\pi$
$607$	22.8420	0.927127	0.463563	+	0.886064i	$0.346571\pi$
$608$	0	0
$609$	−22.9638	−0.930542
$610$	0	0
$611$	58.8681	2.38155
$612$	0	0
$613$	10.7889	0.435759	0.217880	−	0.975976i	$-0.430086\pi$
$613$	10.7889	0.435759	0.217880	+	0.975976i	$0.430086\pi$
$614$	0	0
$615$	−49.9631	−2.01471
$616$	0	0
$617$	−47.0555	−1.89438	−0.947192	−	0.320668i	$-0.896093\pi$
$617$	−47.0555	−1.89438	−0.947192	+	0.320668i	$0.896093\pi$
$618$	0	0
$619$	−1.14507	−0.0460241	−0.0230121	−	0.999735i	$-0.507326\pi$
$619$	−1.14507	−0.0460241	−0.0230121	+	0.999735i	$0.507326\pi$
$620$	0	0
$621$	−20.5139	−0.823194
$622$	0	0
$623$	−19.7079	−0.789579
$624$	0	0
$625$	−19.6333	−0.785332
$626$	0	0
$627$	−14.6303	−0.584280
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1.64232	0.0653798	0.0326899	−	0.999466i	$-0.489593\pi$
$631$	1.64232	0.0653798	0.0326899	+	0.999466i	$0.489593\pi$
$632$	0	0
$633$	−31.0555	−1.23435
$634$	0	0
$635$	3.78190	0.150080
$636$	0	0
$637$	−35.8444	−1.42021
$638$	0	0
$639$	−1.98902	−0.0786845
$640$	0	0
$641$	22.7250	0.897583	0.448791	−	0.893637i	$-0.351855\pi$
$641$	22.7250	0.897583	0.448791	+	0.893637i	$0.351855\pi$
$642$	0	0
$643$	1.98902	0.0784393	0.0392197	−	0.999231i	$-0.487513\pi$
$643$	1.98902	0.0784393	0.0392197	+	0.999231i	$0.487513\pi$
$644$	0	0
$645$	−6.21110	−0.244562
$646$	0	0
$647$	7.71436	0.303283	0.151641	−	0.988436i	$-0.451544\pi$
$647$	7.71436	0.303283	0.151641	+	0.988436i	$0.451544\pi$
$648$	0	0
$649$	−82.0555	−3.22096
$650$	0	0
$651$	−33.6904	−1.32043
$652$	0	0
$653$	−1.42221	−0.0556552	−0.0278276	−	0.999613i	$-0.508859\pi$
$653$	−1.42221	−0.0556552	−0.0278276	+	0.999613i	$0.508859\pi$
$654$	0	0
$655$	−21.6969	−0.847768
$656$	0	0
$657$	−0.669468	−0.0261184
$658$	0	0
$659$	7.71436	0.300509	0.150254	−	0.988647i	$-0.451991\pi$
$659$	7.71436	0.300509	0.150254	+	0.988647i	$0.451991\pi$
$660$	0	0
$661$	32.5139	1.26464	0.632322	−	0.774706i	$-0.282102\pi$
$661$	32.5139	1.26464	0.632322	+	0.774706i	$0.282102\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	20.5139	0.795494
$666$	0	0
$667$	−13.9825	−0.541406
$668$	0	0
$669$	−18.6333	−0.720405
$670$	0	0
$671$	1.14507	0.0442048
$672$	0	0
$673$	−14.4222	−0.555935	−0.277968	−	0.960590i	$-0.589661\pi$
$673$	−14.4222	−0.555935	−0.277968	+	0.960590i	$0.589661\pi$
$674$	0	0
$675$	−32.0481	−1.23353
$676$	0	0
$677$	22.4861	0.864212	0.432106	−	0.901823i	$-0.357771\pi$
$677$	22.4861	0.864212	0.432106	+	0.901823i	$0.357771\pi$
$678$	0	0
$679$	−62.4539	−2.39676
$680$	0	0
$681$	−37.5139	−1.43753
$682$	0	0
$683$	−50.3098	−1.92505	−0.962525	−	0.271192i	$-0.912582\pi$
$683$	−50.3098	−1.92505	−0.962525	+	0.271192i	$0.912582\pi$
$684$	0	0
$685$	9.21110	0.351938
$686$	0	0
$687$	−11.1496	−0.425382
$688$	0	0
$689$	63.3583	2.41376
$690$	0	0
$691$	19.4068	0.738268	0.369134	−	0.929376i	$-0.379654\pi$
$691$	19.4068	0.738268	0.369134	+	0.929376i	$0.379654\pi$
$692$	0	0
$693$	−6.21110	−0.235940
$694$	0	0
$695$	15.7754	0.598396
$696$	0	0
$697$	0	0
$698$	0	0
$699$	36.8245	1.39283
$700$	0	0
$701$	−24.2111	−0.914441	−0.457220	−	0.889353i	$-0.651155\pi$
$701$	−24.2111	−0.914441	−0.457220	+	0.889353i	$0.651155\pi$
$702$	0	0
$703$	6.56929	0.247766
$704$	0	0
$705$	65.0555	2.45013
$706$	0	0
$707$	11.3457	0.426699
$708$	0	0
$709$	−16.9083	−0.635006	−0.317503	−	0.948257i	$-0.602844\pi$
$709$	−16.9083	−0.635006	−0.317503	+	0.948257i	$0.602844\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−20.5139	−0.768251
$714$	0	0
$715$	87.9327	3.28849
$716$	0	0
$717$	−33.7527	−1.26052
$718$	0	0
$719$	24.8310	0.926039	0.463020	−	0.886348i	$-0.346766\pi$
$719$	24.8310	0.926039	0.463020	+	0.886348i	$0.346766\pi$
$720$	0	0
$721$	57.2111	2.13065
$722$	0	0
$723$	−22.8420	−0.849502
$724$	0	0
$725$	−21.8444	−0.811281
$726$	0	0
$727$	−38.8135	−1.43951	−0.719757	−	0.694226i	$-0.755747\pi$
$727$	−38.8135	−1.43951	−0.719757	+	0.694226i	$0.755747\pi$
$728$	0	0
$729$	29.1472	1.07953
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−4.90833	−0.181293	−0.0906466	−	0.995883i	$-0.528893\pi$
$733$	−4.90833	−0.181293	−0.0906466	+	0.995883i	$0.528893\pi$
$734$	0	0
$735$	−39.6119	−1.46111
$736$	0	0
$737$	35.6333	1.31257
$738$	0	0
$739$	−20.8529	−0.767088	−0.383544	−	0.923523i	$-0.625297\pi$
$739$	−20.8529	−0.767088	−0.383544	+	0.923523i	$0.625297\pi$
$740$	0	0
$741$	13.2389	0.486342
$742$	0	0
$743$	29.4113	1.07899	0.539497	−	0.841987i	$-0.318614\pi$
$743$	29.4113	1.07899	0.539497	+	0.841987i	$0.318614\pi$
$744$	0	0
$745$	−30.4222	−1.11458
$746$	0	0
$747$	2.48628	0.0909681
$748$	0	0
$749$	45.3583	1.65736
$750$	0	0
$751$	9.70338	0.354081	0.177041	−	0.984204i	$-0.443348\pi$
$751$	9.70338	0.354081	0.177041	+	0.984204i	$0.443348\pi$
$752$	0	0
$753$	27.5416	1.00367
$754$	0	0
$755$	17.9150	0.651993
$756$	0	0
$757$	−22.2111	−0.807276	−0.403638	−	0.914919i	$-0.632255\pi$
$757$	−22.2111	−0.807276	−0.403638	+	0.914919i	$0.632255\pi$
$758$	0	0
$759$	33.6904	1.22288
$760$	0	0
$761$	−34.2111	−1.24015	−0.620076	−	0.784542i	$-0.712898\pi$
$761$	−34.2111	−1.24015	−0.620076	+	0.784542i	$0.712898\pi$
$762$	0	0
$763$	73.4529	2.65917
$764$	0	0
$765$	0	0
$766$	0	0
$767$	74.2512	2.68106
$768$	0	0
$769$	−11.3305	−0.408589	−0.204295	−	0.978909i	$-0.565490\pi$
$769$	−11.3305	−0.408589	−0.204295	+	0.978909i	$0.565490\pi$
$770$	0	0
$771$	18.8639	0.679367
$772$	0	0
$773$	18.6333	0.670193	0.335097	−	0.942184i	$-0.391231\pi$
$773$	18.6333	0.670193	0.335097	+	0.942184i	$0.391231\pi$
$774$	0	0
$775$	−32.0481	−1.15120
$776$	0	0
$777$	−24.8444	−0.891288
$778$	0	0
$779$	−15.1276	−0.542003
$780$	0	0
$781$	35.6333	1.27506
$782$	0	0
$783$	20.0546	0.716692
$784$	0	0
$785$	45.9361	1.63953
$786$	0	0
$787$	43.3938	1.54682	0.773411	−	0.633905i	$-0.218549\pi$
$787$	43.3938	1.54682	0.773411	+	0.633905i	$0.218549\pi$
$788$	0	0
$789$	5.96384	0.212318
$790$	0	0
$791$	−7.91050	−0.281265
$792$	0	0
$793$	−1.03616	−0.0367951
$794$	0	0
$795$	70.0177	2.48327
$796$	0	0
$797$	−23.8444	−0.844612	−0.422306	−	0.906453i	$-0.638779\pi$
$797$	−23.8444	−0.844612	−0.422306	+	0.906453i	$0.638779\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	1.57779	0.0557486
$802$	0	0
$803$	11.9935	0.423242
$804$	0	0
$805$	−47.2389	−1.66495
$806$	0	0
$807$	11.6468	0.409987
$808$	0	0
$809$	4.36669	0.153525	0.0767624	−	0.997049i	$-0.475542\pi$
$809$	4.36669	0.153525	0.0767624	+	0.997049i	$0.475542\pi$
$810$	0	0
$811$	15.1276	0.531202	0.265601	−	0.964083i	$-0.414430\pi$
$811$	15.1276	0.531202	0.265601	+	0.964083i	$0.414430\pi$
$812$	0	0
$813$	17.0000	0.596216
$814$	0	0
$815$	39.6119	1.38754
$816$	0	0
$817$	−1.88057	−0.0657928
$818$	0	0
$819$	5.62037	0.196391
$820$	0	0
$821$	−2.57779	−0.0899657	−0.0449828	−	0.998988i	$-0.514323\pi$
$821$	−2.57779	−0.0899657	−0.0449828	+	0.998988i	$0.514323\pi$
$822$	0	0
$823$	46.8290	1.63236	0.816178	−	0.577800i	$-0.196089\pi$
$823$	46.8290	1.63236	0.816178	+	0.577800i	$0.196089\pi$
$824$	0	0
$825$	52.6333	1.83246
$826$	0	0
$827$	35.6794	1.24070	0.620348	−	0.784327i	$-0.286991\pi$
$827$	35.6794	1.24070	0.620348	+	0.784327i	$0.286991\pi$
$828$	0	0
$829$	26.4222	0.917681	0.458841	−	0.888519i	$-0.348265\pi$
$829$	26.4222	0.917681	0.458841	+	0.888519i	$0.348265\pi$
$830$	0	0
$831$	−41.4048	−1.43632
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.2727	0.563139
$836$	0	0
$837$	29.4222	1.01698
$838$	0	0
$839$	47.9741	1.65625	0.828124	−	0.560545i	$-0.189408\pi$
$839$	47.9741	1.65625	0.828124	+	0.560545i	$0.189408\pi$
$840$	0	0
$841$	−15.3305	−0.528639
$842$	0	0
$843$	16.5738	0.570831
$844$	0	0
$845$	−36.6333	−1.26022
$846$	0	0
$847$	69.6710	2.39392
$848$	0	0
$849$	−49.6888	−1.70532
$850$	0	0
$851$	−15.1276	−0.518568
$852$	0	0
$853$	−19.7889	−0.677559	−0.338779	−	0.940866i	$-0.610014\pi$
$853$	−19.7889	−0.677559	−0.338779	+	0.940866i	$0.610014\pi$
$854$	0	0
$855$	−1.64232	−0.0561663
$856$	0	0
$857$	43.4222	1.48327	0.741637	−	0.670801i	$-0.234050\pi$
$857$	43.4222	1.48327	0.741637	+	0.670801i	$0.234050\pi$
$858$	0	0
$859$	−49.9631	−1.70472	−0.852359	−	0.522957i	$-0.824829\pi$
$859$	−49.9631	−1.70472	−0.852359	+	0.522957i	$0.824829\pi$
$860$	0	0
$861$	57.2111	1.94975
$862$	0	0
$863$	25.4788	0.867308	0.433654	−	0.901079i	$-0.357224\pi$
$863$	25.4788	0.867308	0.433654	+	0.901079i	$0.357224\pi$
$864$	0	0
$865$	27.1194	0.922088
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−32.2442	−1.09255
$872$	0	0
$873$	5.00000	0.169224
$874$	0	0
$875$	−11.3457	−0.383555
$876$	0	0
$877$	−32.3305	−1.09172	−0.545862	−	0.837875i	$-0.683798\pi$
$877$	−32.3305	−1.09172	−0.545862	+	0.837875i	$0.683798\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42.3583	1.42709	0.713543	−	0.700611i	$-0.247089\pi$
$881$	42.3583	1.42709	0.713543	+	0.700611i	$0.247089\pi$
$882$	0	0
$883$	−13.4853	−0.453816	−0.226908	−	0.973916i	$-0.572862\pi$
$883$	−13.4853	−0.453816	−0.226908	+	0.973916i	$0.572862\pi$
$884$	0	0
$885$	82.0555	2.75827
$886$	0	0
$887$	25.4788	0.855494	0.427747	−	0.903898i	$-0.359307\pi$
$887$	25.4788	0.855494	0.427747	+	0.903898i	$0.359307\pi$
$888$	0	0
$889$	−4.33053	−0.145241
$890$	0	0
$891$	−43.3938	−1.45375
$892$	0	0
$893$	19.6972	0.659142
$894$	0	0
$895$	−30.9030	−1.03297
$896$	0	0
$897$	−30.4861	−1.01790
$898$	0	0
$899$	20.0546	0.668857
$900$	0	0
$901$	0	0
$902$	0	0
$903$	7.11213	0.236677
$904$	0	0
$905$	60.1472	1.99936
$906$	0	0
$907$	−21.3502	−0.708922	−0.354461	−	0.935071i	$-0.615336\pi$
$907$	−21.3502	−0.708922	−0.354461	+	0.935071i	$0.615336\pi$
$908$	0	0
$909$	−0.908327	−0.0301273
$910$	0	0
$911$	12.3402	0.408850	0.204425	−	0.978882i	$-0.434468\pi$
$911$	12.3402	0.408850	0.204425	+	0.978882i	$0.434468\pi$
$912$	0	0
$913$	−44.5416	−1.47411
$914$	0	0
$915$	−1.14507	−0.0378548
$916$	0	0
$917$	24.8444	0.820435
$918$	0	0
$919$	−43.0471	−1.41999	−0.709996	−	0.704205i	$-0.751303\pi$
$919$	−43.0471	−1.41999	−0.709996	+	0.704205i	$0.751303\pi$
$920$	0	0
$921$	18.6333	0.613988
$922$	0	0
$923$	−32.2442	−1.06133
$924$	0	0
$925$	−23.6333	−0.777058
$926$	0	0
$927$	−4.58027	−0.150436
$928$	0	0
$929$	11.0917	0.363906	0.181953	−	0.983307i	$-0.441758\pi$
$929$	11.0917	0.363906	0.181953	+	0.983307i	$0.441758\pi$
$930$	0	0
$931$	−11.9935	−0.393072
$932$	0	0
$933$	22.1472	0.725066
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−36.1472	−1.18088	−0.590439	−	0.807083i	$-0.701045\pi$
$937$	−36.1472	−1.18088	−0.590439	+	0.807083i	$0.701045\pi$
$938$	0	0
$939$	30.2552	0.987341
$940$	0	0
$941$	−8.33053	−0.271568	−0.135784	−	0.990738i	$-0.543355\pi$
$941$	−8.33053	−0.271568	−0.135784	+	0.990738i	$0.543355\pi$
$942$	0	0
$943$	34.8355	1.13440
$944$	0	0
$945$	67.7527	2.20400
$946$	0	0
$947$	−44.5389	−1.44732	−0.723659	−	0.690157i	$-0.757541\pi$
$947$	−44.5389	−1.44732	−0.723659	+	0.690157i	$0.757541\pi$
$948$	0	0
$949$	−10.8528	−0.352297
$950$	0	0
$951$	25.3282	0.821324
$952$	0	0
$953$	32.5139	1.05323	0.526614	−	0.850105i	$-0.323461\pi$
$953$	32.5139	1.05323	0.526614	+	0.850105i	$0.323461\pi$
$954$	0	0
$955$	37.9696	1.22867
$956$	0	0
$957$	−32.9361	−1.06467
$958$	0	0
$959$	−10.5473	−0.340591
$960$	0	0
$961$	−1.57779	−0.0508966
$962$	0	0
$963$	−3.63134	−0.117018
$964$	0	0
$965$	16.5139	0.531601
$966$	0	0
$967$	31.0536	0.998616	0.499308	−	0.866425i	$-0.333588\pi$
$967$	31.0536	0.998616	0.499308	+	0.866425i	$0.333588\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.9205	0.543004	0.271502	−	0.962438i	$-0.412480\pi$
$971$	16.9205	0.543004	0.271502	+	0.962438i	$0.412480\pi$
$972$	0	0
$973$	−18.0639	−0.579103
$974$	0	0
$975$	−47.6274	−1.52530
$976$	0	0
$977$	27.6972	0.886113	0.443056	−	0.896494i	$-0.353894\pi$
$977$	27.6972	0.886113	0.443056	+	0.896494i	$0.353894\pi$
$978$	0	0
$979$	−28.2662	−0.903391
$980$	0	0
$981$	−5.88057	−0.187752
$982$	0	0
$983$	8.55831	0.272968	0.136484	−	0.990642i	$-0.456420\pi$
$983$	8.55831	0.272968	0.136484	+	0.990642i	$0.456420\pi$
$984$	0	0
$985$	75.7527	2.41368
$986$	0	0
$987$	−74.4930	−2.37114
$988$	0	0
$989$	4.33053	0.137703
$990$	0	0
$991$	−52.9010	−1.68046	−0.840228	−	0.542233i	$-0.817579\pi$
$991$	−52.9010	−1.68046	−0.840228	+	0.542233i	$0.817579\pi$
$992$	0	0
$993$	54.2666	1.72210
$994$	0	0
$995$	25.4788	0.807732
$996$	0	0
$997$	−32.3028	−1.02304	−0.511520	−	0.859272i	$-0.670917\pi$
$997$	−32.3028	−1.02304	−0.511520	+	0.859272i	$0.670917\pi$
$998$	0	0
$999$	21.6969	0.686459

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.z.1.3	yes	4
4.3	odd	2	inner	9248.2.a.z.1.2	✓	4
17.16	even	2		9248.2.a.bi.1.2	yes	4
68.67	odd	2		9248.2.a.bi.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9248.2.a.z.1.2	✓	4	4.3	odd	2	inner
9248.2.a.z.1.3	yes	4	1.1	even	1	trivial
9248.2.a.bi.1.2	yes	4	17.16	even	2
9248.2.a.bi.1.3	yes	4	68.67	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:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q+1.64232 q^{3} -3.30278 q^{5} +3.78190 q^{7} -0.302776 q^{9} +5.42422 q^{11} -4.90833 q^{13} -5.42422 q^{15} -1.64232 q^{19} +6.21110 q^{21} +3.78190 q^{23} +5.90833 q^{25} -5.42422 q^{27} -3.69722 q^{29} -5.42422 q^{31} +8.90833 q^{33} -12.4908 q^{35} -4.00000 q^{37} -8.06106 q^{39} +9.21110 q^{41} +1.14507 q^{43} +1.00000 q^{45} -11.9935 q^{47} +7.30278 q^{49} -12.9083 q^{53} -17.9150 q^{55} -2.69722 q^{57} -15.1276 q^{59} +0.211103 q^{61} -1.14507 q^{63} +16.2111 q^{65} +6.56929 q^{67} +6.21110 q^{69} +6.56929 q^{71} +2.21110 q^{73} +9.70338 q^{75} +20.5139 q^{77} -8.00000 q^{81} -8.21161 q^{83} -6.07204 q^{87} -5.21110 q^{89} -18.5628 q^{91} -8.90833 q^{93} +5.42422 q^{95} -16.5139 q^{97} -1.64232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} + 6 q^{9} + 2 q^{13} - 4 q^{21} + 2 q^{25} - 22 q^{29} + 14 q^{33} - 16 q^{37} + 8 q^{41} + 4 q^{45} + 22 q^{49} - 30 q^{53} - 18 q^{57} - 28 q^{61} + 36 q^{65} - 4 q^{69} - 20 q^{73} + 46 q^{77}+ \cdots - 30 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 + 6 * q^9 + 2 * q^13 - 4 * q^21 + 2 * q^25 - 22 * q^29 + 14 * q^33 - 16 * q^37 + 8 * q^41 + 4 * q^45 + 22 * q^49 - 30 * q^53 - 18 * q^57 - 28 * q^61 + 36 * q^65 - 4 * q^69 - 20 * q^73 + 46 * q^77 - 32 * q^81 + 8 * q^89 - 14 * q^93 - 30 * q^97

Introduction

L-functions

Modular forms

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Database

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