LMFDB - Embedded newform 384.6.d.j.193.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.d (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 338x^{10} + 43555x^{8} + 2692222x^{6} + 81680965x^{4} + 1098257588x^{2} + 4742525956 $ x^12 + 338x^10 + 43555x^8 + 2692222x^6 + 81680965x^4 + 1098257588*x^2 + 4742525956
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{57}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.2
Root			$2.80773i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.6.d.j.193.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} -46.1613i q^{5} -59.9278 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} -46.1613i q^{5} -59.9278 q^{7} -81.0000 q^{9} +400.065i q^{11} +351.898i q^{13} -415.451 q^{15} +1662.58 q^{17} -564.314i q^{19} +539.350i q^{21} +3831.57 q^{23} +994.139 q^{25} +729.000i q^{27} -5944.12i q^{29} +2678.95 q^{31} +3600.59 q^{33} +2766.34i q^{35} -771.803i q^{37} +3167.08 q^{39} -5119.70 q^{41} -9518.90i q^{43} +3739.06i q^{45} -8138.39 q^{47} -13215.7 q^{49} -14963.2i q^{51} -11051.9i q^{53} +18467.5 q^{55} -5078.83 q^{57} +32143.3i q^{59} -35454.8i q^{61} +4854.15 q^{63} +16244.0 q^{65} +11362.8i q^{67} -34484.1i q^{69} +10394.7 q^{71} -40098.7 q^{73} -8947.25i q^{75} -23975.0i q^{77} -25754.8 q^{79} +6561.00 q^{81} +2239.75i q^{83} -76746.5i q^{85} -53497.1 q^{87} -15149.4 q^{89} -21088.5i q^{91} -24110.6i q^{93} -26049.5 q^{95} +158426. q^{97} -32405.3i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 972 q^{9}+O(q^{10}) $ 12 * q - 972 * q^9	$ 12 q - 972 q^{9} - 4888 q^{17} - 18660 q^{25} - 720 q^{33} - 25096 q^{41} + 135564 q^{49} - 13104 q^{57} + 254272 q^{65} - 187032 q^{73} + 78732 q^{81} - 575544 q^{89} + 93864 q^{97}+O(q^{100}) $ 12 * q - 972 * q^9 - 4888 * q^17 - 18660 * q^25 - 720 * q^33 - 25096 * q^41 + 135564 * q^49 - 13104 * q^57 + 254272 * q^65 - 187032 * q^73 + 78732 * q^81 - 575544 * q^89 + 93864 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	0	0
$5$	− 46.1613i	− 0.825758i	−0.910786	−	0.412879i	$-0.864523\pi$
$5$	− 46.1613i	− 0.825758i	0.910786	−	0.412879i	$-0.135477\pi$
$6$	0	0
$7$	−59.9278	−0.462257	−0.231128	−	0.972923i	$-0.574242\pi$
$7$	−59.9278	−0.462257	−0.231128	+	0.972923i	$0.574242\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	400.065i	0.996894i	0.866920	+	0.498447i	$0.166096\pi$
$11$	400.065i	0.996894i	−0.866920	+	0.498447i	$0.833904\pi$
$12$	0	0
$13$	351.898i	0.577508i	0.957403	+	0.288754i	$0.0932410\pi$
$13$	351.898i	0.577508i	−0.957403	+	0.288754i	$0.906759\pi$
$14$	0	0
$15$	−415.451	−0.476751
$16$	0	0
$17$	1662.58	1.39527	0.697636	−	0.716452i	$-0.254235\pi$
$17$	1662.58	1.39527	0.697636	+	0.716452i	$0.254235\pi$
$18$	0	0
$19$	− 564.314i	− 0.358622i	−0.983792	−	0.179311i	$-0.942613\pi$
$19$	− 564.314i	− 0.358622i	0.983792	−	0.179311i	$-0.0573869\pi$
$20$	0	0
$21$	539.350i	0.266884i
$22$	0	0
$23$	3831.57	1.51028	0.755139	−	0.655565i	$-0.227570\pi$
$23$	3831.57	1.51028	0.755139	+	0.655565i	$0.227570\pi$
$24$	0	0
$25$	994.139	0.318124
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	− 5944.12i	− 1.31248i	−0.754552	−	0.656240i	$-0.772146\pi$
$29$	− 5944.12i	− 1.31248i	0.754552	−	0.656240i	$-0.227854\pi$
$30$	0	0
$31$	2678.95	0.500680	0.250340	−	0.968158i	$-0.419458\pi$
$31$	2678.95	0.500680	0.250340	+	0.968158i	$0.419458\pi$
$32$	0	0
$33$	3600.59	0.575557
$34$	0	0
$35$	2766.34i	0.381712i
$36$	0	0
$37$	− 771.803i	− 0.0926834i	−0.998926	−	0.0463417i	$-0.985244\pi$
$37$	− 771.803i	− 0.0926834i	0.998926	−	0.0463417i	$-0.0147563\pi$
$38$	0	0
$39$	3167.08	0.333425
$40$	0	0
$41$	−5119.70	−0.475647	−0.237824	−	0.971308i	$-0.576434\pi$
$41$	−5119.70	−0.475647	−0.237824	+	0.971308i	$0.576434\pi$
$42$	0	0
$43$	− 9518.90i	− 0.785083i	−0.919734	−	0.392541i	$-0.871596\pi$
$43$	− 9518.90i	− 0.785083i	0.919734	−	0.392541i	$-0.128404\pi$
$44$	0	0
$45$	3739.06i	0.275253i
$46$	0	0
$47$	−8138.39	−0.537395	−0.268698	−	0.963225i	$-0.586593\pi$
$47$	−8138.39	−0.537395	−0.268698	+	0.963225i	$0.586593\pi$
$48$	0	0
$49$	−13215.7	−0.786319
$50$	0	0
$51$	− 14963.2i	− 0.805561i
$52$	0	0
$53$	− 11051.9i	− 0.540441i	−0.962799	−	0.270220i	$-0.912903\pi$
$53$	− 11051.9i	− 0.540441i	0.962799	−	0.270220i	$-0.0870966\pi$
$54$	0	0
$55$	18467.5	0.823193
$56$	0	0
$57$	−5078.83	−0.207051
$58$	0	0
$59$	32143.3i	1.20215i	0.799191	+	0.601077i	$0.205261\pi$
$59$	32143.3i	1.20215i	−0.799191	+	0.601077i	$0.794739\pi$
$60$	0	0
$61$	− 35454.8i	− 1.21997i	−0.792412	−	0.609986i	$-0.791175\pi$
$61$	− 35454.8i	− 1.21997i	0.792412	−	0.609986i	$-0.208825\pi$
$62$	0	0
$63$	4854.15	0.154086
$64$	0	0
$65$	16244.0	0.476882
$66$	0	0
$67$	11362.8i	0.309242i	0.987974	+	0.154621i	$0.0494157\pi$
$67$	11362.8i	0.309242i	−0.987974	+	0.154621i	$0.950584\pi$
$68$	0	0
$69$	− 34484.1i	− 0.871959i
$70$	0	0
$71$	10394.7	0.244719	0.122360	−	0.992486i	$-0.460954\pi$
$71$	10394.7	0.244719	0.122360	+	0.992486i	$0.460954\pi$
$72$	0	0
$73$	−40098.7	−0.880689	−0.440345	−	0.897829i	$-0.645144\pi$
$73$	−40098.7	−0.880689	−0.440345	+	0.897829i	$0.645144\pi$
$74$	0	0
$75$	− 8947.25i	− 0.183669i
$76$	0	0
$77$	− 23975.0i	− 0.460821i
$78$	0	0
$79$	−25754.8	−0.464291	−0.232146	−	0.972681i	$-0.574575\pi$
$79$	−25754.8	−0.464291	−0.232146	+	0.972681i	$0.574575\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	2239.75i	0.0356866i	0.999841	+	0.0178433i	$0.00568000\pi$
$83$	2239.75i	0.0356866i	−0.999841	+	0.0178433i	$0.994320\pi$
$84$	0	0
$85$	− 76746.5i	− 1.15216i
$86$	0	0
$87$	−53497.1	−0.757761
$88$	0	0
$89$	−15149.4	−0.202732	−0.101366	−	0.994849i	$-0.532321\pi$
$89$	−15149.4	−0.202732	−0.101366	+	0.994849i	$0.532321\pi$
$90$	0	0
$91$	− 21088.5i	− 0.266957i
$92$	0	0
$93$	− 24110.6i	− 0.289068i
$94$	0	0
$95$	−26049.5	−0.296135
$96$	0	0
$97$	158426.	1.70961	0.854804	−	0.518951i	$-0.173677\pi$
$97$	158426.	1.70961	0.854804	+	0.518951i	$0.173677\pi$
$98$	0	0
$99$	− 32405.3i	− 0.332298i
$100$	0	0
$101$	− 154639.i	− 1.50840i	−0.656647	−	0.754198i	$-0.728026\pi$
$101$	− 154639.i	− 1.50840i	0.656647	−	0.754198i	$-0.271974\pi$
$102$	0	0
$103$	−89671.0	−0.832835	−0.416417	−	0.909174i	$-0.636714\pi$
$103$	−89671.0	−0.832835	−0.416417	+	0.909174i	$0.636714\pi$
$104$	0	0
$105$	24897.1	0.220382
$106$	0	0
$107$	113338.i	0.957006i	0.878086	+	0.478503i	$0.158820\pi$
$107$	113338.i	0.957006i	−0.878086	+	0.478503i	$0.841180\pi$
$108$	0	0
$109$	− 221403.i	− 1.78491i	−0.451133	−	0.892457i	$-0.648980\pi$
$109$	− 221403.i	− 1.78491i	0.451133	−	0.892457i	$-0.351020\pi$
$110$	0	0
$111$	−6946.23	−0.0535108
$112$	0	0
$113$	−6039.39	−0.0444936	−0.0222468	−	0.999753i	$-0.507082\pi$
$113$	−6039.39	−0.0444936	−0.0222468	+	0.999753i	$0.507082\pi$
$114$	0	0
$115$	− 176870.i	− 1.24712i
$116$	0	0
$117$	− 28503.7i	− 0.192503i
$118$	0	0
$119$	−99634.5	−0.644974
$120$	0	0
$121$	998.858	0.00620212
$122$	0	0
$123$	46077.3i	0.274615i
$124$	0	0
$125$	− 190145.i	− 1.08845i
$126$	0	0
$127$	328162.	1.80542	0.902711	−	0.430247i	$-0.141573\pi$
$127$	328162.	1.80542	0.902711	+	0.430247i	$0.141573\pi$
$128$	0	0
$129$	−85670.1	−0.453268
$130$	0	0
$131$	− 176176.i	− 0.896952i	−0.893795	−	0.448476i	$-0.851967\pi$
$131$	− 176176.i	− 0.896952i	0.893795	−	0.448476i	$-0.148033\pi$
$132$	0	0
$133$	33818.1i	0.165776i
$134$	0	0
$135$	33651.6	0.158917
$136$	0	0
$137$	50245.9	0.228718	0.114359	−	0.993440i	$-0.463519\pi$
$137$	50245.9	0.228718	0.114359	+	0.993440i	$0.463519\pi$
$138$	0	0
$139$	364925.i	1.60201i	0.598655	+	0.801007i	$0.295702\pi$
$139$	364925.i	1.60201i	−0.598655	+	0.801007i	$0.704298\pi$
$140$	0	0
$141$	73245.5i	0.310265i
$142$	0	0
$143$	−140782.	−0.575715
$144$	0	0
$145$	−274388.	−1.08379
$146$	0	0
$147$	118941.i	0.453981i
$148$	0	0
$149$	− 502438.i	− 1.85403i	−0.375023	−	0.927016i	$-0.622365\pi$
$149$	− 502438.i	− 1.85403i	0.375023	−	0.927016i	$-0.377635\pi$
$150$	0	0
$151$	225684.	0.805485	0.402743	−	0.915313i	$-0.368057\pi$
$151$	225684.	0.805485	0.402743	+	0.915313i	$0.368057\pi$
$152$	0	0
$153$	−134669.	−0.465091
$154$	0	0
$155$	− 123664.i	− 0.413441i
$156$	0	0
$157$	− 464228.i	− 1.50308i	−0.659688	−	0.751539i	$-0.729312\pi$
$157$	− 464228.i	− 1.50308i	0.659688	−	0.751539i	$-0.270688\pi$
$158$	0	0
$159$	−99467.3	−0.312024
$160$	0	0
$161$	−229617.	−0.698136
$162$	0	0
$163$	− 143142.i	− 0.421987i	−0.977487	−	0.210994i	$-0.932330\pi$
$163$	− 143142.i	− 0.421987i	0.977487	−	0.210994i	$-0.0676699\pi$
$164$	0	0
$165$	− 166208.i	− 0.475271i
$166$	0	0
$167$	170711.	0.473663	0.236831	−	0.971551i	$-0.423891\pi$
$167$	170711.	0.473663	0.236831	+	0.971551i	$0.423891\pi$
$168$	0	0
$169$	247461.	0.666484
$170$	0	0
$171$	45709.5i	0.119541i
$172$	0	0
$173$	38779.6i	0.0985118i	0.998786	+	0.0492559i	$0.0156850\pi$
$173$	38779.6i	0.0985118i	−0.998786	+	0.0492559i	$0.984315\pi$
$174$	0	0
$175$	−59576.6	−0.147055
$176$	0	0
$177$	289289.	0.694064
$178$	0	0
$179$	− 399489.i	− 0.931907i	−0.884809	−	0.465954i	$-0.845711\pi$
$179$	− 399489.i	− 0.931907i	0.884809	−	0.465954i	$-0.154289\pi$
$180$	0	0
$181$	168069.i	0.381322i	0.981656	+	0.190661i	$0.0610632\pi$
$181$	168069.i	0.381322i	−0.981656	+	0.190661i	$0.938937\pi$
$182$	0	0
$183$	−319093.	−0.704351
$184$	0	0
$185$	−35627.4	−0.0765340
$186$	0	0
$187$	665138.i	1.39094i
$188$	0	0
$189$	− 43687.4i	− 0.0889614i
$190$	0	0
$191$	230781.	0.457738	0.228869	−	0.973457i	$-0.426497\pi$
$191$	230781.	0.457738	0.228869	+	0.973457i	$0.426497\pi$
$192$	0	0
$193$	7236.58	0.0139843	0.00699214	−	0.999976i	$-0.497774\pi$
$193$	7236.58	0.0139843	0.00699214	+	0.999976i	$0.497774\pi$
$194$	0	0
$195$	− 146196.i	− 0.275328i
$196$	0	0
$197$	582795.i	1.06992i	0.844878	+	0.534959i	$0.179673\pi$
$197$	582795.i	1.06992i	−0.844878	+	0.534959i	$0.820327\pi$
$198$	0	0
$199$	1.07918e6	1.93180	0.965899	−	0.258920i	$-0.0833668\pi$
$199$	1.07918e6	1.93180	0.965899	+	0.258920i	$0.0833668\pi$
$200$	0	0
$201$	102265.	0.178541
$202$	0	0
$203$	356218.i	0.606703i
$204$	0	0
$205$	236332.i	0.392769i
$206$	0	0
$207$	−310357.	−0.503426
$208$	0	0
$209$	225763.	0.357508
$210$	0	0
$211$	− 860927.i	− 1.33125i	−0.746286	−	0.665626i	$-0.768165\pi$
$211$	− 860927.i	− 1.33125i	0.746286	−	0.665626i	$-0.231835\pi$
$212$	0	0
$213$	− 93552.7i	− 0.141289i
$214$	0	0
$215$	−439404.	−0.648288
$216$	0	0
$217$	−160544.	−0.231443
$218$	0	0
$219$	360888.i	0.508466i
$220$	0	0
$221$	585057.i	0.805781i
$222$	0	0
$223$	442484.	0.595848	0.297924	−	0.954590i	$-0.403706\pi$
$223$	442484.	0.595848	0.297924	+	0.954590i	$0.403706\pi$
$224$	0	0
$225$	−80525.2	−0.106041
$226$	0	0
$227$	− 1.25053e6i	− 1.61076i	−0.592761	−	0.805378i	$-0.701962\pi$
$227$	− 1.25053e6i	− 1.61076i	0.592761	−	0.805378i	$-0.298038\pi$
$228$	0	0
$229$	− 508471.i	− 0.640734i	−0.947293	−	0.320367i	$-0.896194\pi$
$229$	− 508471.i	− 0.640734i	0.947293	−	0.320367i	$-0.103806\pi$
$230$	0	0
$231$	−215775.	−0.266055
$232$	0	0
$233$	1.28856e6	1.55495	0.777474	−	0.628915i	$-0.216501\pi$
$233$	1.28856e6	1.55495	0.777474	+	0.628915i	$0.216501\pi$
$234$	0	0
$235$	375678.i	0.443758i
$236$	0	0
$237$	231793.i	0.268059i
$238$	0	0
$239$	−1.42040e6	−1.60848	−0.804240	−	0.594304i	$-0.797428\pi$
$239$	−1.42040e6	−1.60848	−0.804240	+	0.594304i	$0.797428\pi$
$240$	0	0
$241$	−196969.	−0.218452	−0.109226	−	0.994017i	$-0.534837\pi$
$241$	−196969.	−0.218452	−0.109226	+	0.994017i	$0.534837\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	610051.i	0.649309i
$246$	0	0
$247$	198581.	0.207107
$248$	0	0
$249$	20157.8	0.0206037
$250$	0	0
$251$	1.87291e6i	1.87643i	0.346052	+	0.938215i	$0.387522\pi$
$251$	1.87291e6i	1.87643i	−0.346052	+	0.938215i	$0.612478\pi$
$252$	0	0
$253$	1.53288e6i	1.50559i
$254$	0	0
$255$	−690719.	−0.665198
$256$	0	0
$257$	−854792.	−0.807287	−0.403643	−	0.914916i	$-0.632256\pi$
$257$	−854792.	−0.807287	−0.403643	+	0.914916i	$0.632256\pi$
$258$	0	0
$259$	46252.4i	0.0428435i
$260$	0	0
$261$	481474.i	0.437493i
$262$	0	0
$263$	−1.94301e6	−1.73215	−0.866076	−	0.499913i	$-0.833365\pi$
$263$	−1.94301e6	−1.73215	−0.866076	+	0.499913i	$0.833365\pi$
$264$	0	0
$265$	−510170.	−0.446273
$266$	0	0
$267$	136345.i	0.117047i
$268$	0	0
$269$	223635.i	0.188434i	0.995552	+	0.0942168i	$0.0300347\pi$
$269$	223635.i	0.188434i	−0.995552	+	0.0942168i	$0.969965\pi$
$270$	0	0
$271$	1.73900e6	1.43839	0.719196	−	0.694807i	$-0.244510\pi$
$271$	1.73900e6	1.43839	0.719196	+	0.694807i	$0.244510\pi$
$272$	0	0
$273$	−189796.	−0.154128
$274$	0	0
$275$	397720.i	0.317136i
$276$	0	0
$277$	2.35793e6i	1.84642i	0.384294	+	0.923211i	$0.374445\pi$
$277$	2.35793e6i	1.84642i	−0.384294	+	0.923211i	$0.625555\pi$
$278$	0	0
$279$	−216995.	−0.166893
$280$	0	0
$281$	−923073.	−0.697382	−0.348691	−	0.937238i	$-0.613374\pi$
$281$	−923073.	−0.697382	−0.348691	+	0.937238i	$0.613374\pi$
$282$	0	0
$283$	− 848669.i	− 0.629901i	−0.949108	−	0.314951i	$-0.898012\pi$
$283$	− 848669.i	− 0.629901i	0.949108	−	0.314951i	$-0.101988\pi$
$284$	0	0
$285$	234445.i	0.170974i
$286$	0	0
$287$	306813.	0.219871
$288$	0	0
$289$	1.34430e6	0.946785
$290$	0	0
$291$	− 1.42583e6i	− 0.987043i
$292$	0	0
$293$	1.09123e6i	0.742585i	0.928516	+	0.371293i	$0.121085\pi$
$293$	1.09123e6i	0.742585i	−0.928516	+	0.371293i	$0.878915\pi$
$294$	0	0
$295$	1.48377e6	0.992687
$296$	0	0
$297$	−291648.	−0.191852
$298$	0	0
$299$	1.34832e6i	0.872197i
$300$	0	0
$301$	570447.i	0.362910i
$302$	0	0
$303$	−1.39175e6	−0.870873
$304$	0	0
$305$	−1.63664e6	−1.00740
$306$	0	0
$307$	− 1.55616e6i	− 0.942340i	−0.882042	−	0.471170i	$-0.843832\pi$
$307$	− 1.55616e6i	− 0.942340i	0.882042	−	0.471170i	$-0.156168\pi$
$308$	0	0
$309$	807039.i	0.480837i
$310$	0	0
$311$	457231.	0.268062	0.134031	−	0.990977i	$-0.457208\pi$
$311$	457231.	0.268062	0.134031	+	0.990977i	$0.457208\pi$
$312$	0	0
$313$	−1.73593e6	−1.00154	−0.500772	−	0.865579i	$-0.666951\pi$
$313$	−1.73593e6	−1.00154	−0.500772	+	0.865579i	$0.666951\pi$
$314$	0	0
$315$	− 224074.i	− 0.127237i
$316$	0	0
$317$	450224.i	0.251640i	0.992053	+	0.125820i	$0.0401562\pi$
$317$	450224.i	0.251640i	−0.992053	+	0.125820i	$0.959844\pi$
$318$	0	0
$319$	2.37804e6	1.30840
$320$	0	0
$321$	1.02004e6	0.552528
$322$	0	0
$323$	− 938215.i	− 0.500376i
$324$	0	0
$325$	349835.i	0.183719i
$326$	0	0
$327$	−1.99263e6	−1.03052
$328$	0	0
$329$	487716.	0.248415
$330$	0	0
$331$	2.79853e6i	1.40397i	0.712189	+	0.701987i	$0.247704\pi$
$331$	2.79853e6i	1.40397i	−0.712189	+	0.701987i	$0.752296\pi$
$332$	0	0
$333$	62516.0i	0.0308945i
$334$	0	0
$335$	524522.	0.255359
$336$	0	0
$337$	1.88545e6	0.904359	0.452180	−	0.891927i	$-0.350647\pi$
$337$	1.88545e6	0.904359	0.452180	+	0.891927i	$0.350647\pi$
$338$	0	0
$339$	54354.5i	0.0256884i
$340$	0	0
$341$	1.07176e6i	0.499125i
$342$	0	0
$343$	1.79919e6	0.825738
$344$	0	0
$345$	−1.59183e6	−0.720027
$346$	0	0
$347$	− 841512.i	− 0.375178i	−0.982248	−	0.187589i	$-0.939933\pi$
$347$	− 841512.i	− 0.375178i	0.982248	−	0.187589i	$-0.0600672\pi$
$348$	0	0
$349$	3.05425e6i	1.34227i	0.741333	+	0.671137i	$0.234194\pi$
$349$	3.05425e6i	1.34227i	−0.741333	+	0.671137i	$0.765806\pi$
$350$	0	0
$351$	−256533.	−0.111142
$352$	0	0
$353$	3.20066e6	1.36711	0.683555	−	0.729899i	$-0.260433\pi$
$353$	3.20066e6	1.36711	0.683555	+	0.729899i	$0.260433\pi$
$354$	0	0
$355$	− 479834.i	− 0.202079i
$356$	0	0
$357$	896710.i	0.372376i
$358$	0	0
$359$	−1.95304e6	−0.799788	−0.399894	−	0.916561i	$-0.630953\pi$
$359$	−1.95304e6	−0.799788	−0.399894	+	0.916561i	$0.630953\pi$
$360$	0	0
$361$	2.15765e6	0.871390
$362$	0	0
$363$	− 8989.72i	− 0.00358080i
$364$	0	0
$365$	1.85100e6i	0.727236i
$366$	0	0
$367$	3.31321e6	1.28406	0.642028	−	0.766682i	$-0.278093\pi$
$367$	3.31321e6	1.28406	0.642028	+	0.766682i	$0.278093\pi$
$368$	0	0
$369$	414696.	0.158549
$370$	0	0
$371$	662317.i	0.249822i
$372$	0	0
$373$	− 3.65917e6i	− 1.36179i	−0.732381	−	0.680895i	$-0.761591\pi$
$373$	− 3.65917e6i	− 1.36179i	0.732381	−	0.680895i	$-0.238409\pi$
$374$	0	0
$375$	−1.71130e6	−0.628418
$376$	0	0
$377$	2.09172e6	0.757968
$378$	0	0
$379$	− 1.76629e6i	− 0.631631i	−0.948821	−	0.315816i	$-0.897722\pi$
$379$	− 1.76629e6i	− 0.631631i	0.948821	−	0.315816i	$-0.102278\pi$
$380$	0	0
$381$	− 2.95346e6i	− 1.04236i
$382$	0	0
$383$	−3.69691e6	−1.28778	−0.643890	−	0.765118i	$-0.722680\pi$
$383$	−3.69691e6	−1.28778	−0.643890	+	0.765118i	$0.722680\pi$
$384$	0	0
$385$	−1.10672e6	−0.380526
$386$	0	0
$387$	771031.i	0.261694i
$388$	0	0
$389$	669156.i	0.224209i	0.993696	+	0.112105i	$0.0357592\pi$
$389$	669156.i	0.224209i	−0.993696	+	0.112105i	$0.964241\pi$
$390$	0	0
$391$	6.37027e6	2.10725
$392$	0	0
$393$	−1.58559e6	−0.517855
$394$	0	0
$395$	1.18887e6i	0.383392i
$396$	0	0
$397$	− 2.19806e6i	− 0.699945i	−0.936760	−	0.349973i	$-0.886191\pi$
$397$	− 2.19806e6i	− 0.699945i	0.936760	−	0.349973i	$-0.113809\pi$
$398$	0	0
$399$	304363.	0.0957105
$400$	0	0
$401$	−4.54750e6	−1.41225	−0.706125	−	0.708087i	$-0.749558\pi$
$401$	−4.54750e6	−1.41225	−0.706125	+	0.708087i	$0.749558\pi$
$402$	0	0
$403$	942717.i	0.289147i
$404$	0	0
$405$	− 302864.i	− 0.0917508i
$406$	0	0
$407$	308771.	0.0923956
$408$	0	0
$409$	−1.46461e6	−0.432926	−0.216463	−	0.976291i	$-0.569452\pi$
$409$	−1.46461e6	−0.432926	−0.216463	+	0.976291i	$0.569452\pi$
$410$	0	0
$411$	− 452213.i	− 0.132050i
$412$	0	0
$413$	− 1.92628e6i	− 0.555704i
$414$	0	0
$415$	103390.	0.0294685
$416$	0	0
$417$	3.28432e6	0.924923
$418$	0	0
$419$	868371.i	0.241641i	0.992674	+	0.120820i	$0.0385525\pi$
$419$	868371.i	0.241641i	−0.992674	+	0.120820i	$0.961447\pi$
$420$	0	0
$421$	− 1.31760e6i	− 0.362308i	−0.983455	−	0.181154i	$-0.942017\pi$
$421$	− 1.31760e6i	− 0.362308i	0.983455	−	0.181154i	$-0.0579833\pi$
$422$	0	0
$423$	659210.	0.179132
$424$	0	0
$425$	1.65283e6	0.443870
$426$	0	0
$427$	2.12473e6i	0.563941i
$428$	0	0
$429$	1.26704e6i	0.332389i
$430$	0	0
$431$	−3.26185e6	−0.845805	−0.422902	−	0.906175i	$-0.638989\pi$
$431$	−3.26185e6	−0.845805	−0.422902	+	0.906175i	$0.638989\pi$
$432$	0	0
$433$	−278010.	−0.0712591	−0.0356295	−	0.999365i	$-0.511344\pi$
$433$	−278010.	−0.0712591	−0.0356295	+	0.999365i	$0.511344\pi$
$434$	0	0
$435$	2.46949e6i	0.625727i
$436$	0	0
$437$	− 2.16221e6i	− 0.541619i
$438$	0	0
$439$	2.77446e6	0.687096	0.343548	−	0.939135i	$-0.388371\pi$
$439$	2.77446e6	0.687096	0.343548	+	0.939135i	$0.388371\pi$
$440$	0	0
$441$	1.07047e6	0.262106
$442$	0	0
$443$	− 5.97809e6i	− 1.44728i	−0.690176	−	0.723641i	$-0.742467\pi$
$443$	− 5.97809e6i	− 1.44728i	0.690176	−	0.723641i	$-0.257533\pi$
$444$	0	0
$445$	699317.i	0.167407i
$446$	0	0
$447$	−4.52194e6	−1.07043
$448$	0	0
$449$	−5.68566e6	−1.33096	−0.665480	−	0.746416i	$-0.731773\pi$
$449$	−5.68566e6	−1.33096	−0.665480	+	0.746416i	$0.731773\pi$
$450$	0	0
$451$	− 2.04821e6i	− 0.474170i
$452$	0	0
$453$	− 2.03115e6i	− 0.465047i
$454$	0	0
$455$	−973470.	−0.220442
$456$	0	0
$457$	−2.65598e6	−0.594886	−0.297443	−	0.954740i	$-0.596134\pi$
$457$	−2.65598e6	−0.594886	−0.297443	+	0.954740i	$0.596134\pi$
$458$	0	0
$459$	1.21202e6i	0.268520i
$460$	0	0
$461$	− 3.88629e6i	− 0.851692i	−0.904796	−	0.425846i	$-0.859976\pi$
$461$	− 3.88629e6i	− 0.851692i	0.904796	−	0.425846i	$-0.140024\pi$
$462$	0	0
$463$	3.78966e6	0.821576	0.410788	−	0.911731i	$-0.365254\pi$
$463$	3.78966e6	0.821576	0.410788	+	0.911731i	$0.365254\pi$
$464$	0	0
$465$	−1.11297e6	−0.238700
$466$	0	0
$467$	− 2.31109e6i	− 0.490371i	−0.969476	−	0.245185i	$-0.921151\pi$
$467$	− 2.31109e6i	− 0.490371i	0.969476	−	0.245185i	$-0.0788488\pi$
$468$	0	0
$469$	− 680948.i	− 0.142949i
$470$	0	0
$471$	−4.17805e6	−0.867803
$472$	0	0
$473$	3.80818e6	0.782645
$474$	0	0
$475$	− 561007.i	− 0.114086i
$476$	0	0
$477$	895206.i	0.180147i
$478$	0	0
$479$	−4.62388e6	−0.920804	−0.460402	−	0.887710i	$-0.652295\pi$
$479$	−4.62388e6	−0.920804	−0.460402	+	0.887710i	$0.652295\pi$
$480$	0	0
$481$	271596.	0.0535254
$482$	0	0
$483$	2.06656e6i	0.403069i
$484$	0	0
$485$	− 7.31313e6i	− 1.41172i
$486$	0	0
$487$	−7.55618e6	−1.44371	−0.721855	−	0.692044i	$-0.756710\pi$
$487$	−7.55618e6	−1.44371	−0.721855	+	0.692044i	$0.756710\pi$
$488$	0	0
$489$	−1.28828e6	−0.243634
$490$	0	0
$491$	− 4.40919e6i	− 0.825382i	−0.910871	−	0.412691i	$-0.864589\pi$
$491$	− 4.40919e6i	− 0.825382i	0.910871	−	0.412691i	$-0.135411\pi$
$492$	0	0
$493$	− 9.88255e6i	− 1.83127i
$494$	0	0
$495$	−1.49587e6	−0.274398
$496$	0	0
$497$	−622934.	−0.113123
$498$	0	0
$499$	8.45546e6i	1.52015i	0.649837	+	0.760074i	$0.274837\pi$
$499$	8.45546e6i	1.52015i	−0.649837	+	0.760074i	$0.725163\pi$
$500$	0	0
$501$	− 1.53639e6i	− 0.273469i
$502$	0	0
$503$	1.17007e6	0.206201	0.103100	−	0.994671i	$-0.467124\pi$
$503$	1.17007e6	0.206201	0.103100	+	0.994671i	$0.467124\pi$
$504$	0	0
$505$	−7.13833e6	−1.24557
$506$	0	0
$507$	− 2.22715e6i	− 0.384795i
$508$	0	0
$509$	− 9.48209e6i	− 1.62222i	−0.584893	−	0.811110i	$-0.698864\pi$
$509$	− 9.48209e6i	− 1.62222i	0.584893	−	0.811110i	$-0.301136\pi$
$510$	0	0
$511$	2.40302e6	0.407105
$512$	0	0
$513$	411385.	0.0690169
$514$	0	0
$515$	4.13932e6i	0.687720i
$516$	0	0
$517$	− 3.25589e6i	− 0.535726i
$518$	0	0
$519$	349016.	0.0568758
$520$	0	0
$521$	1.08606e7	1.75290	0.876451	−	0.481491i	$-0.159905\pi$
$521$	1.08606e7	1.75290	0.876451	+	0.481491i	$0.159905\pi$
$522$	0	0
$523$	− 7.05571e6i	− 1.12794i	−0.825795	−	0.563971i	$-0.809273\pi$
$523$	− 7.05571e6i	− 1.12794i	0.825795	−	0.563971i	$-0.190727\pi$
$524$	0	0
$525$	536189.i	0.0849023i
$526$	0	0
$527$	4.45396e6	0.698585
$528$	0	0
$529$	8.24455e6	1.28094
$530$	0	0
$531$	− 2.60360e6i	− 0.400718i
$532$	0	0
$533$	− 1.80161e6i	− 0.274690i
$534$	0	0
$535$	5.23181e6	0.790255
$536$	0	0
$537$	−3.59541e6	−0.538037
$538$	0	0
$539$	− 5.28712e6i	− 0.783876i
$540$	0	0
$541$	− 2.11215e6i	− 0.310264i	−0.987894	−	0.155132i	$-0.950420\pi$
$541$	− 2.11215e6i	− 0.310264i	0.987894	−	0.155132i	$-0.0495804\pi$
$542$	0	0
$543$	1.51262e6	0.220157
$544$	0	0
$545$	−1.02202e7	−1.47391
$546$	0	0
$547$	9.37947e6i	1.34032i	0.742215	+	0.670162i	$0.233775\pi$
$547$	9.37947e6i	1.34032i	−0.742215	+	0.670162i	$0.766225\pi$
$548$	0	0
$549$	2.87184e6i	0.406657i
$550$	0	0
$551$	−3.35435e6	−0.470685
$552$	0	0
$553$	1.54343e6	0.214622
$554$	0	0
$555$	320646.i	0.0441869i
$556$	0	0
$557$	3.61461e6i	0.493655i	0.969059	+	0.246827i	$0.0793881\pi$
$557$	3.61461e6i	0.493655i	−0.969059	+	0.246827i	$0.920612\pi$
$558$	0	0
$559$	3.34968e6	0.453392
$560$	0	0
$561$	5.98625e6	0.803059
$562$	0	0
$563$	1.04180e7i	1.38520i	0.721321	+	0.692601i	$0.243535\pi$
$563$	1.04180e7i	1.38520i	−0.721321	+	0.692601i	$0.756465\pi$
$564$	0	0
$565$	278786.i	0.0367409i
$566$	0	0
$567$	−393186.	−0.0513619
$568$	0	0
$569$	7.80034e6	1.01003	0.505014	−	0.863111i	$-0.331487\pi$
$569$	7.80034e6	1.01003	0.505014	+	0.863111i	$0.331487\pi$
$570$	0	0
$571$	− 1.13367e7i	− 1.45511i	−0.686049	−	0.727555i	$-0.740657\pi$
$571$	− 1.13367e7i	− 1.45511i	0.686049	−	0.727555i	$-0.259343\pi$
$572$	0	0
$573$	− 2.07703e6i	− 0.264275i
$574$	0	0
$575$	3.80911e6	0.480456
$576$	0	0
$577$	3.41197e6	0.426644	0.213322	−	0.976982i	$-0.431572\pi$
$577$	3.41197e6	0.426644	0.213322	+	0.976982i	$0.431572\pi$
$578$	0	0
$579$	− 65129.2i	− 0.00807383i
$580$	0	0
$581$	− 134224.i	− 0.0164964i
$582$	0	0
$583$	4.42149e6	0.538762
$584$	0	0
$585$	−1.31577e6	−0.158961
$586$	0	0
$587$	9.10881e6i	1.09111i	0.838077	+	0.545553i	$0.183680\pi$
$587$	9.10881e6i	1.09111i	−0.838077	+	0.545553i	$0.816320\pi$
$588$	0	0
$589$	− 1.51177e6i	− 0.179555i
$590$	0	0
$591$	5.24516e6	0.617717
$592$	0	0
$593$	−5.87052e6	−0.685551	−0.342776	−	0.939417i	$-0.611367\pi$
$593$	−5.87052e6	−0.685551	−0.342776	+	0.939417i	$0.611367\pi$
$594$	0	0
$595$	4.59925e6i	0.532592i
$596$	0	0
$597$	− 9.71263e6i	− 1.11532i
$598$	0	0
$599$	−9.18159e6	−1.04556	−0.522782	−	0.852466i	$-0.675106\pi$
$599$	−9.18159e6	−1.04556	−0.522782	+	0.852466i	$0.675106\pi$
$600$	0	0
$601$	1.01852e7	1.15022	0.575112	−	0.818075i	$-0.304959\pi$
$601$	1.01852e7	1.15022	0.575112	+	0.818075i	$0.304959\pi$
$602$	0	0
$603$	− 920388.i	− 0.103081i
$604$	0	0
$605$	− 46108.5i	− 0.00512145i
$606$	0	0
$607$	−4.74894e6	−0.523148	−0.261574	−	0.965183i	$-0.584242\pi$
$607$	−4.74894e6	−0.523148	−0.261574	+	0.965183i	$0.584242\pi$
$608$	0	0
$609$	3.20596e6	0.350280
$610$	0	0
$611$	− 2.86388e6i	− 0.310350i
$612$	0	0
$613$	− 1.16623e6i	− 0.125352i	−0.998034	−	0.0626760i	$-0.980037\pi$
$613$	− 1.16623e6i	− 0.125352i	0.998034	−	0.0626760i	$-0.0199635\pi$
$614$	0	0
$615$	2.12699e6	0.226766
$616$	0	0
$617$	−1.29443e6	−0.136888	−0.0684440	−	0.997655i	$-0.521803\pi$
$617$	−1.29443e6	−0.136888	−0.0684440	+	0.997655i	$0.521803\pi$
$618$	0	0
$619$	4.23949e6i	0.444721i	0.974965	+	0.222360i	$0.0713762\pi$
$619$	4.23949e6i	0.444721i	−0.974965	+	0.222360i	$0.928624\pi$
$620$	0	0
$621$	2.79321e6i	0.290653i
$622$	0	0
$623$	907873.	0.0937141
$624$	0	0
$625$	−5.67063e6	−0.580672
$626$	0	0
$627$	− 2.03186e6i	− 0.206408i
$628$	0	0
$629$	− 1.28318e6i	− 0.129319i
$630$	0	0
$631$	1.24132e6	0.124111	0.0620553	−	0.998073i	$-0.480234\pi$
$631$	1.24132e6	0.124111	0.0620553	+	0.998073i	$0.480234\pi$
$632$	0	0
$633$	−7.74834e6	−0.768599
$634$	0	0
$635$	− 1.51484e7i	− 1.49084i
$636$	0	0
$637$	− 4.65056e6i	− 0.454105i
$638$	0	0
$639$	−841974.	−0.0815730
$640$	0	0
$641$	−1.09152e7	−1.04927	−0.524635	−	0.851327i	$-0.675798\pi$
$641$	−1.09152e7	−1.04927	−0.524635	+	0.851327i	$0.675798\pi$
$642$	0	0
$643$	− 1.48054e7i	− 1.41219i	−0.708118	−	0.706095i	$-0.750455\pi$
$643$	− 1.48054e7i	− 1.41219i	0.708118	−	0.706095i	$-0.249545\pi$
$644$	0	0
$645$	3.95464e6i	0.374289i
$646$	0	0
$647$	7.50071e6	0.704436	0.352218	−	0.935918i	$-0.385428\pi$
$647$	7.50071e6	0.704436	0.352218	+	0.935918i	$0.385428\pi$
$648$	0	0
$649$	−1.28594e7	−1.19842
$650$	0	0
$651$	1.44489e6i	0.133624i
$652$	0	0
$653$	1.84442e7i	1.69269i	0.532638	+	0.846343i	$0.321201\pi$
$653$	1.84442e7i	1.69269i	−0.532638	+	0.846343i	$0.678799\pi$
$654$	0	0
$655$	−8.13252e6	−0.740665
$656$	0	0
$657$	3.24799e6	0.293563
$658$	0	0
$659$	1.50202e7i	1.34729i	0.739055	+	0.673645i	$0.235272\pi$
$659$	1.50202e7i	1.34729i	−0.739055	+	0.673645i	$0.764728\pi$
$660$	0	0
$661$	1.62211e7i	1.44403i	0.691876	+	0.722017i	$0.256785\pi$
$661$	1.62211e7i	1.44403i	−0.691876	+	0.722017i	$0.743215\pi$
$662$	0	0
$663$	5.26551e6	0.465218
$664$	0	0
$665$	1.56109e6	0.136890
$666$	0	0
$667$	− 2.27753e7i	− 1.98221i
$668$	0	0
$669$	− 3.98236e6i	− 0.344013i
$670$	0	0
$671$	1.41842e7	1.21618
$672$	0	0
$673$	8.38719e6	0.713804	0.356902	−	0.934142i	$-0.383833\pi$
$673$	8.38719e6	0.713804	0.356902	+	0.934142i	$0.383833\pi$
$674$	0	0
$675$	724727.i	0.0612231i
$676$	0	0
$677$	5.22589e6i	0.438216i	0.975701	+	0.219108i	$0.0703147\pi$
$677$	5.22589e6i	0.438216i	−0.975701	+	0.219108i	$0.929685\pi$
$678$	0	0
$679$	−9.49411e6	−0.790278
$680$	0	0
$681$	−1.12548e7	−0.929970
$682$	0	0
$683$	1.61671e7i	1.32611i	0.748570	+	0.663056i	$0.230741\pi$
$683$	1.61671e7i	1.32611i	−0.748570	+	0.663056i	$0.769259\pi$
$684$	0	0
$685$	− 2.31942e6i	− 0.188865i
$686$	0	0
$687$	−4.57624e6	−0.369928
$688$	0	0
$689$	3.88915e6	0.312109
$690$	0	0
$691$	2.38372e7i	1.89915i	0.313533	+	0.949577i	$0.398487\pi$
$691$	2.38372e7i	1.89915i	−0.313533	+	0.949577i	$0.601513\pi$
$692$	0	0
$693$	1.94198e6i	0.153607i
$694$	0	0
$695$	1.68454e7	1.32287
$696$	0	0
$697$	−8.51189e6	−0.663658
$698$	0	0
$699$	− 1.15971e7i	− 0.897750i
$700$	0	0
$701$	9.30715e6i	0.715355i	0.933845	+	0.357678i	$0.116431\pi$
$701$	9.30715e6i	0.715355i	−0.933845	+	0.357678i	$0.883569\pi$
$702$	0	0
$703$	−435539.	−0.0332383
$704$	0	0
$705$	3.38110e6	0.256204
$706$	0	0
$707$	9.26717e6i	0.697266i
$708$	0	0
$709$	5.17752e6i	0.386818i	0.981118	+	0.193409i	$0.0619544\pi$
$709$	5.17752e6i	0.386818i	−0.981118	+	0.193409i	$0.938046\pi$
$710$	0	0
$711$	2.08614e6	0.154764
$712$	0	0
$713$	1.02646e7	0.756166
$714$	0	0
$715$	6.49868e6i	0.475401i
$716$	0	0
$717$	1.27836e7i	0.928657i
$718$	0	0
$719$	−3.77696e6	−0.272471	−0.136236	−	0.990676i	$-0.543500\pi$
$719$	−3.77696e6	−0.272471	−0.136236	+	0.990676i	$0.543500\pi$
$720$	0	0
$721$	5.37378e6	0.384983
$722$	0	0
$723$	1.77272e6i	0.126123i
$724$	0	0
$725$	− 5.90928e6i	− 0.417532i
$726$	0	0
$727$	−2.50517e7	−1.75793	−0.878966	−	0.476885i	$-0.841766\pi$
$727$	−2.50517e7	−1.75793	−0.878966	+	0.476885i	$0.841766\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	− 1.58259e7i	− 1.09540i
$732$	0	0
$733$	2.58312e7i	1.77576i	0.460074	+	0.887881i	$0.347823\pi$
$733$	2.58312e7i	1.77576i	−0.460074	+	0.887881i	$0.652177\pi$
$734$	0	0
$735$	5.49046e6	0.374879
$736$	0	0
$737$	−4.54587e6	−0.308282
$738$	0	0
$739$	− 1.40532e7i	− 0.946593i	−0.880903	−	0.473297i	$-0.843064\pi$
$739$	− 1.40532e7i	− 0.946593i	0.880903	−	0.473297i	$-0.156936\pi$
$740$	0	0
$741$	− 1.78723e6i	− 0.119573i
$742$	0	0
$743$	−3.60964e6	−0.239878	−0.119939	−	0.992781i	$-0.538270\pi$
$743$	−3.60964e6	−0.239878	−0.119939	+	0.992781i	$0.538270\pi$
$744$	0	0
$745$	−2.31932e7	−1.53098
$746$	0	0
$747$	− 181420.i	− 0.0118955i
$748$	0	0
$749$	− 6.79207e6i	− 0.442382i
$750$	0	0
$751$	−2.91748e7	−1.88759	−0.943795	−	0.330531i	$-0.892772\pi$
$751$	−2.91748e7	−1.88759	−0.943795	+	0.330531i	$0.892772\pi$
$752$	0	0
$753$	1.68562e7	1.08336
$754$	0	0
$755$	− 1.04178e7i	− 0.665136i
$756$	0	0
$757$	1.12305e7i	0.712294i	0.934430	+	0.356147i	$0.115910\pi$
$757$	1.12305e7i	0.712294i	−0.934430	+	0.356147i	$0.884090\pi$
$758$	0	0
$759$	1.37959e7	0.869251
$760$	0	0
$761$	1.69614e7	1.06170	0.530849	−	0.847466i	$-0.321873\pi$
$761$	1.69614e7	1.06170	0.530849	+	0.847466i	$0.321873\pi$
$762$	0	0
$763$	1.32682e7i	0.825089i
$764$	0	0
$765$	6.21647e6i	0.384052i
$766$	0	0
$767$	−1.13111e7	−0.694254
$768$	0	0
$769$	2.61093e7	1.59214	0.796068	−	0.605208i	$-0.206910\pi$
$769$	2.61093e7	1.59214	0.796068	+	0.605208i	$0.206910\pi$
$770$	0	0
$771$	7.69313e6i	0.466087i
$772$	0	0
$773$	2.28744e7i	1.37690i	0.725286	+	0.688448i	$0.241708\pi$
$773$	2.28744e7i	1.37690i	−0.725286	+	0.688448i	$0.758292\pi$
$774$	0	0
$775$	2.66325e6	0.159279
$776$	0	0
$777$	416272.	0.0247357
$778$	0	0
$779$	2.88912e6i	0.170578i
$780$	0	0
$781$	4.15857e6i	0.243959i
$782$	0	0
$783$	4.33327e6	0.252587
$784$	0	0
$785$	−2.14293e7	−1.24118
$786$	0	0
$787$	1.44464e6i	0.0831423i	0.999136	+	0.0415712i	$0.0132363\pi$
$787$	1.44464e6i	0.0831423i	−0.999136	+	0.0415712i	$0.986764\pi$
$788$	0	0
$789$	1.74871e7i	1.00006i
$790$	0	0
$791$	361927.	0.0205674
$792$	0	0
$793$	1.24765e7	0.704544
$794$	0	0
$795$	4.59153e6i	0.257656i
$796$	0	0
$797$	5.37136e6i	0.299529i	0.988722	+	0.149764i	$0.0478515\pi$
$797$	5.37136e6i	0.299529i	−0.988722	+	0.149764i	$0.952148\pi$
$798$	0	0
$799$	−1.35307e7	−0.749813
$800$	0	0
$801$	1.22710e6	0.0675772
$802$	0	0
$803$	− 1.60421e7i	− 0.877954i
$804$	0	0
$805$	1.05994e7i	0.576491i
$806$	0	0
$807$	2.01271e6	0.108792
$808$	0	0
$809$	8.86137e6	0.476025	0.238012	−	0.971262i	$-0.423504\pi$
$809$	8.86137e6	0.476025	0.238012	+	0.971262i	$0.423504\pi$
$810$	0	0
$811$	2.91251e7i	1.55495i	0.628915	+	0.777474i	$0.283499\pi$
$811$	2.91251e7i	1.55495i	−0.628915	+	0.777474i	$0.716501\pi$
$812$	0	0
$813$	− 1.56510e7i	− 0.830456i
$814$	0	0
$815$	−6.60763e6	−0.348459
$816$	0	0
$817$	−5.37165e6	−0.281548
$818$	0	0
$819$	1.70817e6i	0.0889857i
$820$	0	0
$821$	2.62586e7i	1.35961i	0.733394	+	0.679803i	$0.237935\pi$
$821$	2.62586e7i	1.35961i	−0.733394	+	0.679803i	$0.762065\pi$
$822$	0	0
$823$	−1.88150e7	−0.968287	−0.484143	−	0.874989i	$-0.660869\pi$
$823$	−1.88150e7	−0.968287	−0.484143	+	0.874989i	$0.660869\pi$
$824$	0	0
$825$	3.57948e6	0.183099
$826$	0	0
$827$	− 1.26962e7i	− 0.645520i	−0.946481	−	0.322760i	$-0.895389\pi$
$827$	− 1.26962e7i	− 0.645520i	0.946481	−	0.322760i	$-0.104611\pi$
$828$	0	0
$829$	2.01838e7i	1.02004i	0.860164	+	0.510018i	$0.170361\pi$
$829$	2.01838e7i	1.02004i	−0.860164	+	0.510018i	$0.829639\pi$
$830$	0	0
$831$	2.12213e7	1.06603
$832$	0	0
$833$	−2.19720e7	−1.09713
$834$	0	0
$835$	− 7.88021e6i	− 0.391130i
$836$	0	0
$837$	1.95296e6i	0.0963560i
$838$	0	0
$839$	1.27185e7	0.623781	0.311890	−	0.950118i	$-0.399038\pi$
$839$	1.27185e7	0.623781	0.311890	+	0.950118i	$0.399038\pi$
$840$	0	0
$841$	−1.48215e7	−0.722605
$842$	0	0
$843$	8.30766e6i	0.402633i
$844$	0	0
$845$	− 1.14231e7i	− 0.550354i
$846$	0	0
$847$	−59859.3	−0.00286697
$848$	0	0
$849$	−7.63802e6	−0.363674
$850$	0	0
$851$	− 2.95721e6i	− 0.139978i
$852$	0	0
$853$	− 1.95957e7i	− 0.922121i	−0.887369	−	0.461060i	$-0.847469\pi$
$853$	− 1.95957e7i	− 0.922121i	0.887369	−	0.461060i	$-0.152531\pi$
$854$	0	0
$855$	2.11001e6	0.0987117
$856$	0	0
$857$	3.68287e7	1.71291	0.856454	−	0.516224i	$-0.172663\pi$
$857$	3.68287e7	1.71291	0.856454	+	0.516224i	$0.172663\pi$
$858$	0	0
$859$	− 1.70593e7i	− 0.788823i	−0.918934	−	0.394411i	$-0.870949\pi$
$859$	− 1.70593e7i	− 0.788823i	0.918934	−	0.394411i	$-0.129051\pi$
$860$	0	0
$861$	− 2.76131e6i	− 0.126943i
$862$	0	0
$863$	3.56411e7	1.62901	0.814506	−	0.580155i	$-0.197008\pi$
$863$	3.56411e7	1.62901	0.814506	+	0.580155i	$0.197008\pi$
$864$	0	0
$865$	1.79012e6	0.0813468
$866$	0	0
$867$	− 1.20987e7i	− 0.546626i
$868$	0	0
$869$	− 1.03036e7i	− 0.462849i
$870$	0	0
$871$	−3.99855e6	−0.178590
$872$	0	0
$873$	−1.28325e7	−0.569869
$874$	0	0
$875$	1.13949e7i	0.503144i
$876$	0	0
$877$	6.78596e6i	0.297929i	0.988843	+	0.148964i	$0.0475939\pi$
$877$	6.78596e6i	0.297929i	−0.988843	+	0.148964i	$0.952406\pi$
$878$	0	0
$879$	9.82105e6	0.428732
$880$	0	0
$881$	6.27597e6	0.272421	0.136211	−	0.990680i	$-0.456508\pi$
$881$	6.27597e6	0.272421	0.136211	+	0.990680i	$0.456508\pi$
$882$	0	0
$883$	− 2.46975e7i	− 1.06599i	−0.846120	−	0.532993i	$-0.821067\pi$
$883$	− 2.46975e7i	− 1.06599i	0.846120	−	0.532993i	$-0.178933\pi$
$884$	0	0
$885$	− 1.33540e7i	− 0.573128i
$886$	0	0
$887$	1.04054e6	0.0444070	0.0222035	−	0.999753i	$-0.492932\pi$
$887$	1.04054e6	0.0444070	0.0222035	+	0.999753i	$0.492932\pi$
$888$	0	0
$889$	−1.96660e7	−0.834569
$890$	0	0
$891$	2.62483e6i	0.110766i
$892$	0	0
$893$	4.59261e6i	0.192722i
$894$	0	0
$895$	−1.84409e7	−0.769530
$896$	0	0
$897$	1.21349e7	0.503563
$898$	0	0
$899$	− 1.59240e7i	− 0.657133i
$900$	0	0
$901$	− 1.83746e7i	− 0.754062i
$902$	0	0
$903$	5.13402e6	0.209526
$904$	0	0
$905$	7.75829e6	0.314880
$906$	0	0
$907$	2.41754e7i	0.975789i	0.872903	+	0.487894i	$0.162235\pi$
$907$	2.41754e7i	0.975789i	−0.872903	+	0.487894i	$0.837765\pi$
$908$	0	0
$909$	1.25258e7i	0.502799i
$910$	0	0
$911$	3.64416e7	1.45479	0.727396	−	0.686217i	$-0.240730\pi$
$911$	3.64416e7	1.45479	0.727396	+	0.686217i	$0.240730\pi$
$912$	0	0
$913$	−896047.	−0.0355758
$914$	0	0
$915$	1.47297e7i	0.581624i
$916$	0	0
$917$	1.05579e7i	0.414622i
$918$	0	0
$919$	−3.29796e7	−1.28812	−0.644060	−	0.764975i	$-0.722751\pi$
$919$	−3.29796e7	−1.28812	−0.644060	+	0.764975i	$0.722751\pi$
$920$	0	0
$921$	−1.40054e7	−0.544060
$922$	0	0
$923$	3.65789e6i	0.141327i
$924$	0	0
$925$	− 767279.i	− 0.0294849i
$926$	0	0
$927$	7.26335e6	0.277612
$928$	0	0
$929$	−3.83111e7	−1.45642	−0.728209	−	0.685356i	$-0.759647\pi$
$929$	−3.83111e7	−1.45642	−0.728209	+	0.685356i	$0.759647\pi$
$930$	0	0
$931$	7.45779e6i	0.281991i
$932$	0	0
$933$	− 4.11508e6i	− 0.154765i
$934$	0	0
$935$	3.07036e7	1.14858
$936$	0	0
$937$	1.48708e7	0.553332	0.276666	−	0.960966i	$-0.410770\pi$
$937$	1.48708e7	0.553332	0.276666	+	0.960966i	$0.410770\pi$
$938$	0	0
$939$	1.56233e7i	0.578242i
$940$	0	0
$941$	2.67852e7i	0.986099i	0.870001	+	0.493049i	$0.164118\pi$
$941$	2.67852e7i	0.986099i	−0.870001	+	0.493049i	$0.835882\pi$
$942$	0	0
$943$	−1.96165e7	−0.718359
$944$	0	0
$945$	−2.01666e6	−0.0734605
$946$	0	0
$947$	− 1.24319e7i	− 0.450467i	−0.974305	−	0.225233i	$-0.927686\pi$
$947$	− 1.24319e7i	− 0.450467i	0.974305	−	0.225233i	$-0.0723145\pi$
$948$	0	0
$949$	− 1.41106e7i	− 0.508605i
$950$	0	0
$951$	4.05201e6	0.145285
$952$	0	0
$953$	−3.05330e7	−1.08902	−0.544511	−	0.838753i	$-0.683285\pi$
$953$	−3.05330e7	−1.08902	−0.544511	+	0.838753i	$0.683285\pi$
$954$	0	0
$955$	− 1.06532e7i	− 0.377981i
$956$	0	0
$957$	− 2.14023e7i	− 0.755407i
$958$	0	0
$959$	−3.01113e6	−0.105726
$960$	0	0
$961$	−2.14524e7	−0.749319
$962$	0	0
$963$	− 9.18035e6i	− 0.319002i
$964$	0	0
$965$	− 334050.i	− 0.0115476i
$966$	0	0
$967$	−3.23903e7	−1.11391	−0.556954	−	0.830543i	$-0.688030\pi$
$967$	−3.23903e7	−1.11391	−0.556954	+	0.830543i	$0.688030\pi$
$968$	0	0
$969$	−8.44394e6	−0.288892
$970$	0	0
$971$	5.00157e7i	1.70239i	0.524853	+	0.851193i	$0.324120\pi$
$971$	5.00157e7i	1.70239i	−0.524853	+	0.851193i	$0.675880\pi$
$972$	0	0
$973$	− 2.18691e7i	− 0.740542i
$974$	0	0
$975$	3.14852e6	0.106070
$976$	0	0
$977$	621156.	0.0208192	0.0104096	−	0.999946i	$-0.496686\pi$
$977$	621156.	0.0208192	0.0104096	+	0.999946i	$0.496686\pi$
$978$	0	0
$979$	− 6.06076e6i	− 0.202102i
$980$	0	0
$981$	1.79336e7i	0.594971i
$982$	0	0
$983$	−6.73840e6	−0.222420	−0.111210	−	0.993797i	$-0.535473\pi$
$983$	−6.73840e6	−0.222420	−0.111210	+	0.993797i	$0.535473\pi$
$984$	0	0
$985$	2.69026e7	0.883493
$986$	0	0
$987$	− 4.38944e6i	− 0.143422i
$988$	0	0
$989$	− 3.64723e7i	− 1.18569i
$990$	0	0
$991$	1.73933e7	0.562597	0.281298	−	0.959620i	$-0.409235\pi$
$991$	1.73933e7	0.562597	0.281298	+	0.959620i	$0.409235\pi$
$992$	0	0
$993$	2.51867e7	0.810585
$994$	0	0
$995$	− 4.98163e7i	− 1.59520i
$996$	0	0
$997$	− 9.67436e6i	− 0.308237i	−0.988052	−	0.154118i	$-0.950746\pi$
$997$	− 9.67436e6i	− 0.308237i	0.988052	−	0.154118i	$-0.0492537\pi$
$998$	0	0
$999$	562644.	0.0178369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.d.j.193.2	✓	12
4.3	odd	2	inner	384.6.d.j.193.8	yes	12
8.3	odd	2	inner	384.6.d.j.193.5	yes	12
8.5	even	2	inner	384.6.d.j.193.11	yes	12
16.3	odd	4		768.6.a.be.1.2		6
16.5	even	4		768.6.a.be.1.5		6
16.11	odd	4		768.6.a.bf.1.5		6
16.13	even	4		768.6.a.bf.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.j.193.2	✓	12	1.1	even	1	trivial
384.6.d.j.193.5	yes	12	8.3	odd	2	inner
384.6.d.j.193.8	yes	12	4.3	odd	2	inner
384.6.d.j.193.11	yes	12	8.5	even	2	inner
768.6.a.be.1.2		6	16.3	odd	4
768.6.a.be.1.5		6	16.5	even	4
768.6.a.bf.1.2		6	16.13	even	4
768.6.a.bf.1.5		6	16.11	odd	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} -46.1613i q^{5} -59.9278 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} -46.1613i q^{5} -59.9278 q^{7} -81.0000 q^{9} +400.065i q^{11} +351.898i q^{13} -415.451 q^{15} +1662.58 q^{17} -564.314i q^{19} +539.350i q^{21} +3831.57 q^{23} +994.139 q^{25} +729.000i q^{27} -5944.12i q^{29} +2678.95 q^{31} +3600.59 q^{33} +2766.34i q^{35} -771.803i q^{37} +3167.08 q^{39} -5119.70 q^{41} -9518.90i q^{43} +3739.06i q^{45} -8138.39 q^{47} -13215.7 q^{49} -14963.2i q^{51} -11051.9i q^{53} +18467.5 q^{55} -5078.83 q^{57} +32143.3i q^{59} -35454.8i q^{61} +4854.15 q^{63} +16244.0 q^{65} +11362.8i q^{67} -34484.1i q^{69} +10394.7 q^{71} -40098.7 q^{73} -8947.25i q^{75} -23975.0i q^{77} -25754.8 q^{79} +6561.00 q^{81} +2239.75i q^{83} -76746.5i q^{85} -53497.1 q^{87} -15149.4 q^{89} -21088.5i q^{91} -24110.6i q^{93} -26049.5 q^{95} +158426. q^{97} -32405.3i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 972 q^{9}+O(q^{10}) \) 12 * q - 972 * q^9	\( 12 q - 972 q^{9} - 4888 q^{17} - 18660 q^{25} - 720 q^{33} - 25096 q^{41} + 135564 q^{49} - 13104 q^{57} + 254272 q^{65} - 187032 q^{73} + 78732 q^{81} - 575544 q^{89} + 93864 q^{97}+O(q^{100}) \) 12 * q - 972 * q^9 - 4888 * q^17 - 18660 * q^25 - 720 * q^33 - 25096 * q^41 + 135564 * q^49 - 13104 * q^57 + 254272 * q^65 - 187032 * q^73 + 78732 * q^81 - 575544 * q^89 + 93864 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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