LMFDB - Embedded newform 48.5.g.a.31.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,5,Mod(31,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	48.g (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:	$4.96175822802$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.5.g.a.31.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} -42.0000 q^{5} +76.2102i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} -42.0000 q^{5} +76.2102i q^{7} -27.0000 q^{9} -20.7846i q^{11} -182.000 q^{13} +218.238i q^{15} -246.000 q^{17} -117.779i q^{19} +396.000 q^{21} -748.246i q^{23} +1139.00 q^{25} +140.296i q^{27} +78.0000 q^{29} +1475.71i q^{31} -108.000 q^{33} -3200.83i q^{35} +530.000 q^{37} +945.700i q^{39} -918.000 q^{41} -852.169i q^{43} +1134.00 q^{45} +3782.80i q^{47} -3407.00 q^{49} +1278.25i q^{51} -4626.00 q^{53} +872.954i q^{55} -612.000 q^{57} -228.631i q^{59} +1346.00 q^{61} -2057.68i q^{63} +7644.00 q^{65} +1087.73i q^{67} -3888.00 q^{69} -1829.05i q^{71} -926.000 q^{73} -5918.42i q^{75} +1584.00 q^{77} -4399.41i q^{79} +729.000 q^{81} +11992.7i q^{83} +10332.0 q^{85} -405.300i q^{87} +11586.0 q^{89} -13870.3i q^{91} +7668.00 q^{93} +4946.74i q^{95} -13118.0 q^{97} +561.184i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 84 q^{5} - 54 q^{9}+O(q^{10}) $ 2 * q - 84 * q^5 - 54 * q^9	$ 2 q - 84 q^{5} - 54 q^{9} - 364 q^{13} - 492 q^{17} + 792 q^{21} + 2278 q^{25} + 156 q^{29} - 216 q^{33} + 1060 q^{37} - 1836 q^{41} + 2268 q^{45} - 6814 q^{49} - 9252 q^{53} - 1224 q^{57} + 2692 q^{61} + 15288 q^{65} - 7776 q^{69} - 1852 q^{73} + 3168 q^{77} + 1458 q^{81} + 20664 q^{85} + 23172 q^{89} + 15336 q^{93} - 26236 q^{97}+O(q^{100}) $ 2 * q - 84 * q^5 - 54 * q^9 - 364 * q^13 - 492 * q^17 + 792 * q^21 + 2278 * q^25 + 156 * q^29 - 216 * q^33 + 1060 * q^37 - 1836 * q^41 + 2268 * q^45 - 6814 * q^49 - 9252 * q^53 - 1224 * q^57 + 2692 * q^61 + 15288 * q^65 - 7776 * q^69 - 1852 * q^73 + 3168 * q^77 + 1458 * q^81 + 20664 * q^85 + 23172 * q^89 + 15336 * q^93 - 26236 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	−42.0000	−1.68000	−0.840000	−	0.542586i	$-0.817445\pi$
$5$	−42.0000	−1.68000	−0.840000	+	0.542586i	$0.817445\pi$
$6$	0	0
$7$	76.2102i	1.55531i	0.628691	+	0.777655i	$0.283591\pi$
$7$	76.2102i	1.55531i	−0.628691	+	0.777655i	$0.716409\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	− 20.7846i	− 0.171774i	−0.996305	−	0.0858868i	$-0.972628\pi$
$11$	− 20.7846i	− 0.171774i	0.996305	−	0.0858868i	$-0.0273723\pi$
$12$	0	0
$13$	−182.000	−1.07692	−0.538462	−	0.842650i	$-0.680994\pi$
$13$	−182.000	−1.07692	−0.538462	+	0.842650i	$0.680994\pi$
$14$	0	0
$15$	218.238i	0.969948i
$16$	0	0
$17$	−246.000	−0.851211	−0.425606	−	0.904909i	$-0.639939\pi$
$17$	−246.000	−0.851211	−0.425606	+	0.904909i	$0.639939\pi$
$18$	0	0
$19$	− 117.779i	− 0.326259i	−0.986605	−	0.163129i	$-0.947841\pi$
$19$	− 117.779i	− 0.326259i	0.986605	−	0.163129i	$-0.0521588\pi$
$20$	0	0
$21$	396.000	0.897959
$22$	0	0
$23$	− 748.246i	− 1.41445i	−0.706987	−	0.707227i	$-0.749946\pi$
$23$	− 748.246i	− 1.41445i	0.706987	−	0.707227i	$-0.250054\pi$
$24$	0	0
$25$	1139.00	1.82240
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	78.0000	0.0927467	0.0463734	−	0.998924i	$-0.485234\pi$
$29$	78.0000	0.0927467	0.0463734	+	0.998924i	$0.485234\pi$
$30$	0	0
$31$	1475.71i	1.53560i	0.640692	+	0.767798i	$0.278647\pi$
$31$	1475.71i	1.53560i	−0.640692	+	0.767798i	$0.721353\pi$
$32$	0	0
$33$	−108.000	−0.0991736
$34$	0	0
$35$	− 3200.83i	− 2.61292i
$36$	0	0
$37$	530.000	0.387144	0.193572	−	0.981086i	$-0.437993\pi$
$37$	530.000	0.387144	0.193572	+	0.981086i	$0.437993\pi$
$38$	0	0
$39$	945.700i	0.621762i
$40$	0	0
$41$	−918.000	−0.546104	−0.273052	−	0.961999i	$-0.588033\pi$
$41$	−918.000	−0.546104	−0.273052	+	0.961999i	$0.588033\pi$
$42$	0	0
$43$	− 852.169i	− 0.460881i	−0.973086	−	0.230441i	$-0.925983\pi$
$43$	− 852.169i	− 0.460881i	0.973086	−	0.230441i	$-0.0740167\pi$
$44$	0	0
$45$	1134.00	0.560000
$46$	0	0
$47$	3782.80i	1.71245i	0.516604	+	0.856224i	$0.327196\pi$
$47$	3782.80i	1.71245i	−0.516604	+	0.856224i	$0.672804\pi$
$48$	0	0
$49$	−3407.00	−1.41899
$50$	0	0
$51$	1278.25i	0.491447i
$52$	0	0
$53$	−4626.00	−1.64685	−0.823425	−	0.567426i	$-0.807939\pi$
$53$	−4626.00	−1.64685	−0.823425	+	0.567426i	$0.807939\pi$
$54$	0	0
$55$	872.954i	0.288580i
$56$	0	0
$57$	−612.000	−0.188366
$58$	0	0
$59$	− 228.631i	− 0.0656796i	−0.999461	−	0.0328398i	$-0.989545\pi$
$59$	− 228.631i	− 0.0656796i	0.999461	−	0.0328398i	$-0.0104551\pi$
$60$	0	0
$61$	1346.00	0.361731	0.180865	−	0.983508i	$-0.442110\pi$
$61$	1346.00	0.361731	0.180865	+	0.983508i	$0.442110\pi$
$62$	0	0
$63$	− 2057.68i	− 0.518437i
$64$	0	0
$65$	7644.00	1.80923
$66$	0	0
$67$	1087.73i	0.242310i	0.992634	+	0.121155i	$0.0386597\pi$
$67$	1087.73i	0.242310i	−0.992634	+	0.121155i	$0.961340\pi$
$68$	0	0
$69$	−3888.00	−0.816635
$70$	0	0
$71$	− 1829.05i	− 0.362834i	−0.983406	−	0.181417i	$-0.941932\pi$
$71$	− 1829.05i	− 0.362834i	0.983406	−	0.181417i	$-0.0580684\pi$
$72$	0	0
$73$	−926.000	−0.173766	−0.0868831	−	0.996219i	$-0.527691\pi$
$73$	−926.000	−0.173766	−0.0868831	+	0.996219i	$0.527691\pi$
$74$	0	0
$75$	− 5918.42i	− 1.05216i
$76$	0	0
$77$	1584.00	0.267161
$78$	0	0
$79$	− 4399.41i	− 0.704921i	−0.935827	−	0.352460i	$-0.885345\pi$
$79$	− 4399.41i	− 0.704921i	0.935827	−	0.352460i	$-0.114655\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	11992.7i	1.74085i	0.492301	+	0.870425i	$0.336156\pi$
$83$	11992.7i	1.74085i	−0.492301	+	0.870425i	$0.663844\pi$
$84$	0	0
$85$	10332.0	1.43003
$86$	0	0
$87$	− 405.300i	− 0.0535473i
$88$	0	0
$89$	11586.0	1.46269	0.731347	−	0.682005i	$-0.238892\pi$
$89$	11586.0	1.46269	0.731347	+	0.682005i	$0.238892\pi$
$90$	0	0
$91$	− 13870.3i	− 1.67495i
$92$	0	0
$93$	7668.00	0.886576
$94$	0	0
$95$	4946.74i	0.548115i
$96$	0	0
$97$	−13118.0	−1.39420	−0.697099	−	0.716975i	$-0.745526\pi$
$97$	−13118.0	−1.39420	−0.697099	+	0.716975i	$0.745526\pi$
$98$	0	0
$99$	561.184i	0.0572579i
$100$	0	0
$101$	−5490.00	−0.538183	−0.269091	−	0.963115i	$-0.586723\pi$
$101$	−5490.00	−0.538183	−0.269091	+	0.963115i	$0.586723\pi$
$102$	0	0
$103$	− 5701.91i	− 0.537460i	−0.963216	−	0.268730i	$-0.913396\pi$
$103$	− 5701.91i	− 0.537460i	0.963216	−	0.268730i	$-0.0866039\pi$
$104$	0	0
$105$	−16632.0	−1.50857
$106$	0	0
$107$	10080.5i	0.880473i	0.897882	+	0.440237i	$0.145105\pi$
$107$	10080.5i	0.880473i	−0.897882	+	0.440237i	$0.854895\pi$
$108$	0	0
$109$	−16166.0	−1.36066	−0.680330	−	0.732906i	$-0.738164\pi$
$109$	−16166.0	−1.36066	−0.680330	+	0.732906i	$0.738164\pi$
$110$	0	0
$111$	− 2753.96i	− 0.223518i
$112$	0	0
$113$	1842.00	0.144256	0.0721278	−	0.997395i	$-0.477021\pi$
$113$	1842.00	0.144256	0.0721278	+	0.997395i	$0.477021\pi$
$114$	0	0
$115$	31426.3i	2.37628i
$116$	0	0
$117$	4914.00	0.358974
$118$	0	0
$119$	− 18747.7i	− 1.32390i
$120$	0	0
$121$	14209.0	0.970494
$122$	0	0
$123$	4770.07i	0.315293i
$124$	0	0
$125$	−21588.0	−1.38163
$126$	0	0
$127$	− 394.908i	− 0.0244843i	−0.999925	−	0.0122422i	$-0.996103\pi$
$127$	− 394.908i	− 0.0244843i	0.999925	−	0.0122422i	$-0.00389690\pi$
$128$	0	0
$129$	−4428.00	−0.266090
$130$	0	0
$131$	− 353.338i	− 0.0205896i	−0.999947	−	0.0102948i	$-0.996723\pi$
$131$	− 353.338i	− 0.0205896i	0.999947	−	0.0102948i	$-0.00327700\pi$
$132$	0	0
$133$	8976.00	0.507434
$134$	0	0
$135$	− 5892.44i	− 0.323316i
$136$	0	0
$137$	−13254.0	−0.706164	−0.353082	−	0.935592i	$-0.614866\pi$
$137$	−13254.0	−0.706164	−0.353082	+	0.935592i	$0.614866\pi$
$138$	0	0
$139$	13212.1i	0.683820i	0.939733	+	0.341910i	$0.111074\pi$
$139$	13212.1i	0.683820i	−0.939733	+	0.341910i	$0.888926\pi$
$140$	0	0
$141$	19656.0	0.988683
$142$	0	0
$143$	3782.80i	0.184987i
$144$	0	0
$145$	−3276.00	−0.155815
$146$	0	0
$147$	17703.3i	0.819255i
$148$	0	0
$149$	438.000	0.0197288	0.00986442	−	0.999951i	$-0.496860\pi$
$149$	438.000	0.0197288	0.00986442	+	0.999951i	$0.496860\pi$
$150$	0	0
$151$	− 28052.3i	− 1.23031i	−0.788406	−	0.615155i	$-0.789093\pi$
$151$	− 28052.3i	− 1.23031i	0.788406	−	0.615155i	$-0.210907\pi$
$152$	0	0
$153$	6642.00	0.283737
$154$	0	0
$155$	− 61979.7i	− 2.57980i
$156$	0	0
$157$	19346.0	0.784859	0.392430	−	0.919782i	$-0.371635\pi$
$157$	19346.0	0.784859	0.392430	+	0.919782i	$0.371635\pi$
$158$	0	0
$159$	24037.4i	0.950809i
$160$	0	0
$161$	57024.0	2.19992
$162$	0	0
$163$	− 36255.3i	− 1.36457i	−0.731086	−	0.682286i	$-0.760986\pi$
$163$	− 36255.3i	− 1.36457i	0.731086	−	0.682286i	$-0.239014\pi$
$164$	0	0
$165$	4536.00	0.166612
$166$	0	0
$167$	− 18747.7i	− 0.672226i	−0.941822	−	0.336113i	$-0.890888\pi$
$167$	− 18747.7i	− 0.672226i	0.941822	−	0.336113i	$-0.109112\pi$
$168$	0	0
$169$	4563.00	0.159763
$170$	0	0
$171$	3180.05i	0.108753i
$172$	0	0
$173$	−34410.0	−1.14972	−0.574861	−	0.818251i	$-0.694944\pi$
$173$	−34410.0	−1.14972	−0.574861	+	0.818251i	$0.694944\pi$
$174$	0	0
$175$	86803.5i	2.83440i
$176$	0	0
$177$	−1188.00	−0.0379201
$178$	0	0
$179$	16856.3i	0.526086i	0.964784	+	0.263043i	$0.0847261\pi$
$179$	16856.3i	0.526086i	−0.964784	+	0.263043i	$0.915274\pi$
$180$	0	0
$181$	15706.0	0.479411	0.239706	−	0.970846i	$-0.422949\pi$
$181$	15706.0	0.479411	0.239706	+	0.970846i	$0.422949\pi$
$182$	0	0
$183$	− 6994.02i	− 0.208845i
$184$	0	0
$185$	−22260.0	−0.650402
$186$	0	0
$187$	5113.01i	0.146216i
$188$	0	0
$189$	−10692.0	−0.299320
$190$	0	0
$191$	2660.43i	0.0729265i	0.999335	+	0.0364632i	$0.0116092\pi$
$191$	2660.43i	0.0729265i	−0.999335	+	0.0364632i	$0.988391\pi$
$192$	0	0
$193$	−26782.0	−0.718999	−0.359500	−	0.933145i	$-0.617053\pi$
$193$	−26782.0	−0.718999	−0.359500	+	0.933145i	$0.617053\pi$
$194$	0	0
$195$	− 39719.4i	− 1.04456i
$196$	0	0
$197$	−52482.0	−1.35232	−0.676158	−	0.736757i	$-0.736356\pi$
$197$	−52482.0	−1.35232	−0.676158	+	0.736757i	$0.736356\pi$
$198$	0	0
$199$	− 23077.8i	− 0.582759i	−0.956608	−	0.291380i	$-0.905886\pi$
$199$	− 23077.8i	− 0.582759i	0.956608	−	0.291380i	$-0.0941143\pi$
$200$	0	0
$201$	5652.00	0.139898
$202$	0	0
$203$	5944.40i	0.144250i
$204$	0	0
$205$	38556.0	0.917454
$206$	0	0
$207$	20202.6i	0.471485i
$208$	0	0
$209$	−2448.00	−0.0560427
$210$	0	0
$211$	23895.4i	0.536721i	0.963319	+	0.268361i	$0.0864819\pi$
$211$	23895.4i	0.536721i	−0.963319	+	0.268361i	$0.913518\pi$
$212$	0	0
$213$	−9504.00	−0.209482
$214$	0	0
$215$	35791.1i	0.774280i
$216$	0	0
$217$	−112464.	−2.38833
$218$	0	0
$219$	4811.64i	0.100324i
$220$	0	0
$221$	44772.0	0.916689
$222$	0	0
$223$	852.169i	0.0171363i	0.999963	+	0.00856813i	$0.00272735\pi$
$223$	852.169i	0.0171363i	−0.999963	+	0.00856813i	$0.997273\pi$
$224$	0	0
$225$	−30753.0	−0.607467
$226$	0	0
$227$	76175.6i	1.47831i	0.673538	+	0.739153i	$0.264774\pi$
$227$	76175.6i	1.47831i	−0.673538	+	0.739153i	$0.735226\pi$
$228$	0	0
$229$	−48470.0	−0.924277	−0.462138	−	0.886808i	$-0.652918\pi$
$229$	−48470.0	−0.924277	−0.462138	+	0.886808i	$0.652918\pi$
$230$	0	0
$231$	− 8230.71i	− 0.154246i
$232$	0	0
$233$	48738.0	0.897751	0.448875	−	0.893594i	$-0.351825\pi$
$233$	48738.0	0.897751	0.448875	+	0.893594i	$0.351825\pi$
$234$	0	0
$235$	− 158878.i	− 2.87691i
$236$	0	0
$237$	−22860.0	−0.406986
$238$	0	0
$239$	71000.2i	1.24298i	0.783422	+	0.621490i	$0.213472\pi$
$239$	71000.2i	1.24298i	−0.783422	+	0.621490i	$0.786528\pi$
$240$	0	0
$241$	73138.0	1.25924	0.629621	−	0.776903i	$-0.283210\pi$
$241$	73138.0	1.25924	0.629621	+	0.776903i	$0.283210\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	143094.	2.38391
$246$	0	0
$247$	21435.9i	0.351356i
$248$	0	0
$249$	62316.0	1.00508
$250$	0	0
$251$	− 91888.8i	− 1.45853i	−0.684232	−	0.729264i	$-0.739862\pi$
$251$	− 91888.8i	− 1.45853i	0.684232	−	0.729264i	$-0.260138\pi$
$252$	0	0
$253$	−15552.0	−0.242966
$254$	0	0
$255$	− 53686.6i	− 0.825631i
$256$	0	0
$257$	−48894.0	−0.740269	−0.370134	−	0.928978i	$-0.620688\pi$
$257$	−48894.0	−0.740269	−0.370134	+	0.928978i	$0.620688\pi$
$258$	0	0
$259$	40391.4i	0.602129i
$260$	0	0
$261$	−2106.00	−0.0309156
$262$	0	0
$263$	78191.7i	1.13044i	0.824939	+	0.565222i	$0.191210\pi$
$263$	78191.7i	1.13044i	−0.824939	+	0.565222i	$0.808790\pi$
$264$	0	0
$265$	194292.	2.76671
$266$	0	0
$267$	− 60202.6i	− 0.844487i
$268$	0	0
$269$	−71538.0	−0.988626	−0.494313	−	0.869284i	$-0.664580\pi$
$269$	−71538.0	−0.988626	−0.494313	+	0.869284i	$0.664580\pi$
$270$	0	0
$271$	− 108198.i	− 1.47326i	−0.676296	−	0.736630i	$-0.736416\pi$
$271$	− 108198.i	− 1.47326i	0.676296	−	0.736630i	$-0.263584\pi$
$272$	0	0
$273$	−72072.0	−0.967033
$274$	0	0
$275$	− 23673.7i	− 0.313040i
$276$	0	0
$277$	−120518.	−1.57070	−0.785348	−	0.619054i	$-0.787516\pi$
$277$	−120518.	−1.57070	−0.785348	+	0.619054i	$0.787516\pi$
$278$	0	0
$279$	− 39844.1i	− 0.511865i
$280$	0	0
$281$	−3054.00	−0.0386773	−0.0193387	−	0.999813i	$-0.506156\pi$
$281$	−3054.00	−0.0386773	−0.0193387	+	0.999813i	$0.506156\pi$
$282$	0	0
$283$	132959.i	1.66014i	0.557657	+	0.830071i	$0.311700\pi$
$283$	132959.i	1.66014i	−0.557657	+	0.830071i	$0.688300\pi$
$284$	0	0
$285$	25704.0	0.316454
$286$	0	0
$287$	− 69961.0i	− 0.849361i
$288$	0	0
$289$	−23005.0	−0.275440
$290$	0	0
$291$	68163.1i	0.804940i
$292$	0	0
$293$	151662.	1.76661	0.883307	−	0.468795i	$-0.155312\pi$
$293$	151662.	1.76661	0.883307	+	0.468795i	$0.155312\pi$
$294$	0	0
$295$	9602.49i	0.110342i
$296$	0	0
$297$	2916.00	0.0330579
$298$	0	0
$299$	136181.i	1.52326i
$300$	0	0
$301$	64944.0	0.716813
$302$	0	0
$303$	28526.9i	0.310720i
$304$	0	0
$305$	−56532.0	−0.607708
$306$	0	0
$307$	− 5424.78i	− 0.0575580i	−0.999586	−	0.0287790i	$-0.990838\pi$
$307$	− 5424.78i	− 0.0575580i	0.999586	−	0.0287790i	$-0.00916190\pi$
$308$	0	0
$309$	−29628.0	−0.310303
$310$	0	0
$311$	− 141127.i	− 1.45912i	−0.683917	−	0.729560i	$-0.739725\pi$
$311$	− 141127.i	− 1.45912i	0.683917	−	0.729560i	$-0.260275\pi$
$312$	0	0
$313$	−128686.	−1.31354	−0.656769	−	0.754092i	$-0.728077\pi$
$313$	−128686.	−1.31354	−0.656769	+	0.754092i	$0.728077\pi$
$314$	0	0
$315$	86422.4i	0.870974i
$316$	0	0
$317$	−73986.0	−0.736260	−0.368130	−	0.929774i	$-0.620002\pi$
$317$	−73986.0	−0.736260	−0.368130	+	0.929774i	$0.620002\pi$
$318$	0	0
$319$	− 1621.20i	− 0.0159314i
$320$	0	0
$321$	52380.0	0.508341
$322$	0	0
$323$	28973.7i	0.277715i
$324$	0	0
$325$	−207298.	−1.96258
$326$	0	0
$327$	84001.0i	0.785577i
$328$	0	0
$329$	−288288.	−2.66339
$330$	0	0
$331$	57026.0i	0.520496i	0.965542	+	0.260248i	$0.0838043\pi$
$331$	57026.0i	0.520496i	−0.965542	+	0.260248i	$0.916196\pi$
$332$	0	0
$333$	−14310.0	−0.129048
$334$	0	0
$335$	− 45684.6i	− 0.407080i
$336$	0	0
$337$	98674.0	0.868846	0.434423	−	0.900709i	$-0.356952\pi$
$337$	98674.0	0.868846	0.434423	+	0.900709i	$0.356952\pi$
$338$	0	0
$339$	− 9571.31i	− 0.0832860i
$340$	0	0
$341$	30672.0	0.263775
$342$	0	0
$343$	− 76667.5i	− 0.651663i
$344$	0	0
$345$	163296.	1.37195
$346$	0	0
$347$	56929.0i	0.472797i	0.971656	+	0.236399i	$0.0759671\pi$
$347$	56929.0i	0.472797i	−0.971656	+	0.236399i	$0.924033\pi$
$348$	0	0
$349$	181346.	1.48887	0.744436	−	0.667694i	$-0.232719\pi$
$349$	181346.	1.48887	0.744436	+	0.667694i	$0.232719\pi$
$350$	0	0
$351$	− 25533.9i	− 0.207254i
$352$	0	0
$353$	−4302.00	−0.0345240	−0.0172620	−	0.999851i	$-0.505495\pi$
$353$	−4302.00	−0.0345240	−0.0172620	+	0.999851i	$0.505495\pi$
$354$	0	0
$355$	76819.9i	0.609561i
$356$	0	0
$357$	−97416.0	−0.764353
$358$	0	0
$359$	− 185232.i	− 1.43724i	−0.695405	−	0.718618i	$-0.744775\pi$
$359$	− 185232.i	− 1.43724i	0.695405	−	0.718618i	$-0.255225\pi$
$360$	0	0
$361$	116449.	0.893555
$362$	0	0
$363$	− 73832.1i	− 0.560315i
$364$	0	0
$365$	38892.0	0.291927
$366$	0	0
$367$	182690.i	1.35638i	0.734885	+	0.678191i	$0.237236\pi$
$367$	182690.i	1.35638i	−0.734885	+	0.678191i	$0.762764\pi$
$368$	0	0
$369$	24786.0	0.182035
$370$	0	0
$371$	− 352549.i	− 2.56136i
$372$	0	0
$373$	151778.	1.09092	0.545458	−	0.838138i	$-0.316356\pi$
$373$	151778.	1.09092	0.545458	+	0.838138i	$0.316356\pi$
$374$	0	0
$375$	112175.i	0.797686i
$376$	0	0
$377$	−14196.0	−0.0998811
$378$	0	0
$379$	36005.9i	0.250666i	0.992115	+	0.125333i	$0.0399999\pi$
$379$	36005.9i	0.250666i	−0.992115	+	0.125333i	$0.960000\pi$
$380$	0	0
$381$	−2052.00	−0.0141360
$382$	0	0
$383$	− 65346.8i	− 0.445479i	−0.974878	−	0.222739i	$-0.928500\pi$
$383$	− 65346.8i	− 0.445479i	0.974878	−	0.222739i	$-0.0714999\pi$
$384$	0	0
$385$	−66528.0	−0.448831
$386$	0	0
$387$	23008.6i	0.153627i
$388$	0	0
$389$	105750.	0.698846	0.349423	−	0.936965i	$-0.386378\pi$
$389$	105750.	0.698846	0.349423	+	0.936965i	$0.386378\pi$
$390$	0	0
$391$	184069.i	1.20400i
$392$	0	0
$393$	−1836.00	−0.0118874
$394$	0	0
$395$	184775.i	1.18427i
$396$	0	0
$397$	−27934.0	−0.177236	−0.0886180	−	0.996066i	$-0.528245\pi$
$397$	−27934.0	−0.177236	−0.0886180	+	0.996066i	$0.528245\pi$
$398$	0	0
$399$	− 46640.7i	− 0.292967i
$400$	0	0
$401$	237882.	1.47936	0.739678	−	0.672961i	$-0.234978\pi$
$401$	237882.	1.47936	0.739678	+	0.672961i	$0.234978\pi$
$402$	0	0
$403$	− 268579.i	− 1.65372i
$404$	0	0
$405$	−30618.0	−0.186667
$406$	0	0
$407$	− 11015.8i	− 0.0665011i
$408$	0	0
$409$	−20270.0	−0.121173	−0.0605867	−	0.998163i	$-0.519297\pi$
$409$	−20270.0	−0.121173	−0.0605867	+	0.998163i	$0.519297\pi$
$410$	0	0
$411$	68869.8i	0.407704i
$412$	0	0
$413$	17424.0	0.102152
$414$	0	0
$415$	− 503694.i	− 2.92463i
$416$	0	0
$417$	68652.0	0.394804
$418$	0	0
$419$	− 24089.4i	− 0.137214i	−0.997644	−	0.0686068i	$-0.978145\pi$
$419$	− 24089.4i	− 0.137214i	0.997644	−	0.0686068i	$-0.0218554\pi$
$420$	0	0
$421$	116698.	0.658414	0.329207	−	0.944258i	$-0.393219\pi$
$421$	116698.	0.658414	0.329207	+	0.944258i	$0.393219\pi$
$422$	0	0
$423$	− 102136.i	− 0.570816i
$424$	0	0
$425$	−280194.	−1.55125
$426$	0	0
$427$	102579.i	0.562604i
$428$	0	0
$429$	19656.0	0.106802
$430$	0	0
$431$	355542.i	1.91397i	0.290132	+	0.956986i	$0.406301\pi$
$431$	355542.i	1.91397i	−0.290132	+	0.956986i	$0.593699\pi$
$432$	0	0
$433$	−199726.	−1.06527	−0.532634	−	0.846346i	$-0.678798\pi$
$433$	−199726.	−1.06527	−0.532634	+	0.846346i	$0.678798\pi$
$434$	0	0
$435$	17022.6i	0.0899595i
$436$	0	0
$437$	−88128.0	−0.461478
$438$	0	0
$439$	146469.i	0.760006i	0.924985	+	0.380003i	$0.124077\pi$
$439$	146469.i	0.760006i	−0.924985	+	0.380003i	$0.875923\pi$
$440$	0	0
$441$	91989.0	0.472997
$442$	0	0
$443$	50444.2i	0.257042i	0.991707	+	0.128521i	$0.0410230\pi$
$443$	50444.2i	0.257042i	−0.991707	+	0.128521i	$0.958977\pi$
$444$	0	0
$445$	−486612.	−2.45733
$446$	0	0
$447$	− 2275.91i	− 0.0113905i
$448$	0	0
$449$	149994.	0.744014	0.372007	−	0.928230i	$-0.378670\pi$
$449$	149994.	0.744014	0.372007	+	0.928230i	$0.378670\pi$
$450$	0	0
$451$	19080.3i	0.0938062i
$452$	0	0
$453$	−145764.	−0.710320
$454$	0	0
$455$	582551.i	2.81392i
$456$	0	0
$457$	284338.	1.36145	0.680726	−	0.732538i	$-0.261664\pi$
$457$	284338.	1.36145	0.680726	+	0.732538i	$0.261664\pi$
$458$	0	0
$459$	− 34512.8i	− 0.163816i
$460$	0	0
$461$	−183402.	−0.862983	−0.431491	−	0.902117i	$-0.642013\pi$
$461$	−183402.	−0.862983	−0.431491	+	0.902117i	$0.642013\pi$
$462$	0	0
$463$	172422.i	0.804324i	0.915568	+	0.402162i	$0.131741\pi$
$463$	172422.i	0.804324i	−0.915568	+	0.402162i	$0.868259\pi$
$464$	0	0
$465$	−322056.	−1.48945
$466$	0	0
$467$	− 68734.7i	− 0.315168i	−0.987506	−	0.157584i	$-0.949629\pi$
$467$	− 68734.7i	− 0.315168i	0.987506	−	0.157584i	$-0.0503705\pi$
$468$	0	0
$469$	−82896.0	−0.376867
$470$	0	0
$471$	− 100525.i	− 0.453139i
$472$	0	0
$473$	−17712.0	−0.0791672
$474$	0	0
$475$	− 134151.i	− 0.594574i
$476$	0	0
$477$	124902.	0.548950
$478$	0	0
$479$	− 249956.i	− 1.08941i	−0.838627	−	0.544706i	$-0.816641\pi$
$479$	− 249956.i	− 1.08941i	0.838627	−	0.544706i	$-0.183359\pi$
$480$	0	0
$481$	−96460.0	−0.416924
$482$	0	0
$483$	− 296305.i	− 1.27012i
$484$	0	0
$485$	550956.	2.34225
$486$	0	0
$487$	271108.i	1.14310i	0.820568	+	0.571549i	$0.193657\pi$
$487$	271108.i	1.14310i	−0.820568	+	0.571549i	$0.806343\pi$
$488$	0	0
$489$	−188388.	−0.787835
$490$	0	0
$491$	− 227862.i	− 0.945166i	−0.881286	−	0.472583i	$-0.843322\pi$
$491$	− 227862.i	− 0.945166i	0.881286	−	0.472583i	$-0.156678\pi$
$492$	0	0
$493$	−19188.0	−0.0789470
$494$	0	0
$495$	− 23569.7i	− 0.0961932i
$496$	0	0
$497$	139392.	0.564320
$498$	0	0
$499$	− 248854.i	− 0.999410i	−0.866196	−	0.499705i	$-0.833442\pi$
$499$	− 248854.i	− 0.999410i	0.866196	−	0.499705i	$-0.166558\pi$
$500$	0	0
$501$	−97416.0	−0.388110
$502$	0	0
$503$	446537.i	1.76490i	0.470403	+	0.882452i	$0.344109\pi$
$503$	446537.i	1.76490i	−0.470403	+	0.882452i	$0.655891\pi$
$504$	0	0
$505$	230580.	0.904147
$506$	0	0
$507$	− 23710.0i	− 0.0922394i
$508$	0	0
$509$	−39330.0	−0.151806	−0.0759029	−	0.997115i	$-0.524184\pi$
$509$	−39330.0	−0.151806	−0.0759029	+	0.997115i	$0.524184\pi$
$510$	0	0
$511$	− 70570.7i	− 0.270260i
$512$	0	0
$513$	16524.0	0.0627886
$514$	0	0
$515$	239480.i	0.902933i
$516$	0	0
$517$	78624.0	0.294154
$518$	0	0
$519$	178800.i	0.663792i
$520$	0	0
$521$	−464598.	−1.71160	−0.855799	−	0.517308i	$-0.826934\pi$
$521$	−464598.	−1.71160	−0.855799	+	0.517308i	$0.826934\pi$
$522$	0	0
$523$	− 135509.i	− 0.495409i	−0.968836	−	0.247704i	$-0.920324\pi$
$523$	− 135509.i	− 0.495409i	0.968836	−	0.247704i	$-0.0796762\pi$
$524$	0	0
$525$	451044.	1.63644
$526$	0	0
$527$	− 363024.i	− 1.30712i
$528$	0	0
$529$	−280031.	−1.00068
$530$	0	0
$531$	6173.03i	0.0218932i
$532$	0	0
$533$	167076.	0.588111
$534$	0	0
$535$	− 423382.i	− 1.47919i
$536$	0	0
$537$	87588.0	0.303736
$538$	0	0
$539$	70813.2i	0.243745i
$540$	0	0
$541$	360442.	1.23152	0.615759	−	0.787934i	$-0.288849\pi$
$541$	360442.	1.23152	0.615759	+	0.787934i	$0.288849\pi$
$542$	0	0
$543$	− 81610.8i	− 0.276788i
$544$	0	0
$545$	678972.	2.28591
$546$	0	0
$547$	− 261644.i	− 0.874451i	−0.899352	−	0.437225i	$-0.855961\pi$
$547$	− 261644.i	− 0.874451i	0.899352	−	0.437225i	$-0.144039\pi$
$548$	0	0
$549$	−36342.0	−0.120577
$550$	0	0
$551$	− 9186.80i	− 0.0302594i
$552$	0	0
$553$	335280.	1.09637
$554$	0	0
$555$	115666.i	0.375510i
$556$	0	0
$557$	−233274.	−0.751893	−0.375946	−	0.926641i	$-0.622682\pi$
$557$	−233274.	−0.751893	−0.375946	+	0.926641i	$0.622682\pi$
$558$	0	0
$559$	155095.i	0.496333i
$560$	0	0
$561$	26568.0	0.0844176
$562$	0	0
$563$	419704.i	1.32412i	0.749453	+	0.662058i	$0.230317\pi$
$563$	419704.i	1.32412i	−0.749453	+	0.662058i	$0.769683\pi$
$564$	0	0
$565$	−77364.0	−0.242349
$566$	0	0
$567$	55557.3i	0.172812i
$568$	0	0
$569$	470058.	1.45187	0.725934	−	0.687765i	$-0.241408\pi$
$569$	470058.	1.45187	0.725934	+	0.687765i	$0.241408\pi$
$570$	0	0
$571$	320381.i	0.982640i	0.870979	+	0.491320i	$0.163485\pi$
$571$	320381.i	0.982640i	−0.870979	+	0.491320i	$0.836515\pi$
$572$	0	0
$573$	13824.0	0.0421041
$574$	0	0
$575$	− 852252.i	− 2.57770i
$576$	0	0
$577$	−341038.	−1.02436	−0.512178	−	0.858879i	$-0.671161\pi$
$577$	−341038.	−1.02436	−0.512178	+	0.858879i	$0.671161\pi$
$578$	0	0
$579$	139163.i	0.415114i
$580$	0	0
$581$	−913968.	−2.70756
$582$	0	0
$583$	96149.6i	0.282885i
$584$	0	0
$585$	−206388.	−0.603077
$586$	0	0
$587$	− 114128.i	− 0.331220i	−0.986191	−	0.165610i	$-0.947041\pi$
$587$	− 114128.i	− 0.331220i	0.986191	−	0.165610i	$-0.0529594\pi$
$588$	0	0
$589$	173808.	0.501002
$590$	0	0
$591$	272704.i	0.780760i
$592$	0	0
$593$	−96846.0	−0.275405	−0.137703	−	0.990474i	$-0.543972\pi$
$593$	−96846.0	−0.275405	−0.137703	+	0.990474i	$0.543972\pi$
$594$	0	0
$595$	787404.i	2.22415i
$596$	0	0
$597$	−119916.	−0.336456
$598$	0	0
$599$	− 519782.i	− 1.44866i	−0.689452	−	0.724331i	$-0.742149\pi$
$599$	− 519782.i	− 1.44866i	0.689452	−	0.724331i	$-0.257851\pi$
$600$	0	0
$601$	−627742.	−1.73793	−0.868965	−	0.494874i	$-0.835214\pi$
$601$	−627742.	−1.73793	−0.868965	+	0.494874i	$0.835214\pi$
$602$	0	0
$603$	− 29368.7i	− 0.0807699i
$604$	0	0
$605$	−596778.	−1.63043
$606$	0	0
$607$	− 133195.i	− 0.361501i	−0.983529	−	0.180751i	$-0.942147\pi$
$607$	− 133195.i	− 0.361501i	0.983529	−	0.180751i	$-0.0578527\pi$
$608$	0	0
$609$	30888.0	0.0832828
$610$	0	0
$611$	− 688469.i	− 1.84418i
$612$	0	0
$613$	247202.	0.657856	0.328928	−	0.944355i	$-0.393313\pi$
$613$	247202.	0.657856	0.328928	+	0.944355i	$0.393313\pi$
$614$	0	0
$615$	− 200343.i	− 0.529692i
$616$	0	0
$617$	−31758.0	−0.0834224	−0.0417112	−	0.999130i	$-0.513281\pi$
$617$	−31758.0	−0.0834224	−0.0417112	+	0.999130i	$0.513281\pi$
$618$	0	0
$619$	656094.i	1.71232i	0.516712	+	0.856160i	$0.327156\pi$
$619$	656094.i	1.71232i	−0.516712	+	0.856160i	$0.672844\pi$
$620$	0	0
$621$	104976.	0.272212
$622$	0	0
$623$	882972.i	2.27494i
$624$	0	0
$625$	194821.	0.498742
$626$	0	0
$627$	12720.2i	0.0323563i
$628$	0	0
$629$	−130380.	−0.329541
$630$	0	0
$631$	417736.i	1.04916i	0.851360	+	0.524582i	$0.175778\pi$
$631$	417736.i	1.04916i	−0.851360	+	0.524582i	$0.824222\pi$
$632$	0	0
$633$	124164.	0.309876
$634$	0	0
$635$	16586.1i	0.0411337i
$636$	0	0
$637$	620074.	1.52815
$638$	0	0
$639$	49384.2i	0.120945i
$640$	0	0
$641$	−152214.	−0.370458	−0.185229	−	0.982695i	$-0.559303\pi$
$641$	−152214.	−0.370458	−0.185229	+	0.982695i	$0.559303\pi$
$642$	0	0
$643$	− 714138.i	− 1.72727i	−0.504117	−	0.863635i	$-0.668182\pi$
$643$	− 714138.i	− 1.72727i	0.504117	−	0.863635i	$-0.331818\pi$
$644$	0	0
$645$	185976.	0.447031
$646$	0	0
$647$	− 259558.i	− 0.620049i	−0.950729	−	0.310025i	$-0.899663\pi$
$647$	− 259558.i	− 0.620049i	0.950729	−	0.310025i	$-0.100337\pi$
$648$	0	0
$649$	−4752.00	−0.0112820
$650$	0	0
$651$	584380.i	1.37890i
$652$	0	0
$653$	−330714.	−0.775579	−0.387790	−	0.921748i	$-0.626761\pi$
$653$	−330714.	−0.775579	−0.387790	+	0.921748i	$0.626761\pi$
$654$	0	0
$655$	14840.2i	0.0345906i
$656$	0	0
$657$	25002.0	0.0579221
$658$	0	0
$659$	253884.i	0.584608i	0.956326	+	0.292304i	$0.0944219\pi$
$659$	253884.i	0.584608i	−0.956326	+	0.292304i	$0.905578\pi$
$660$	0	0
$661$	−722158.	−1.65283	−0.826417	−	0.563058i	$-0.809625\pi$
$661$	−722158.	−1.65283	−0.826417	+	0.563058i	$0.809625\pi$
$662$	0	0
$663$	− 232642.i	− 0.529251i
$664$	0	0
$665$	−376992.	−0.852489
$666$	0	0
$667$	− 58363.2i	− 0.131186i
$668$	0	0
$669$	4428.00	0.00989362
$670$	0	0
$671$	− 27976.1i	− 0.0621358i
$672$	0	0
$673$	−552910.	−1.22074	−0.610372	−	0.792115i	$-0.708980\pi$
$673$	−552910.	−1.22074	−0.610372	+	0.792115i	$0.708980\pi$
$674$	0	0
$675$	159797.i	0.350721i
$676$	0	0
$677$	609030.	1.32881	0.664403	−	0.747375i	$-0.268686\pi$
$677$	609030.	1.32881	0.664403	+	0.747375i	$0.268686\pi$
$678$	0	0
$679$	− 999726.i	− 2.16841i
$680$	0	0
$681$	395820.	0.853500
$682$	0	0
$683$	− 23715.2i	− 0.0508377i	−0.999677	−	0.0254189i	$-0.991908\pi$
$683$	− 23715.2i	− 0.0508377i	0.999677	−	0.0254189i	$-0.00809195\pi$
$684$	0	0
$685$	556668.	1.18636
$686$	0	0
$687$	251858.i	0.533631i
$688$	0	0
$689$	841932.	1.77353
$690$	0	0
$691$	431842.i	0.904417i	0.891912	+	0.452208i	$0.149364\pi$
$691$	431842.i	0.904417i	−0.891912	+	0.452208i	$0.850636\pi$
$692$	0	0
$693$	−42768.0	−0.0890538
$694$	0	0
$695$	− 554908.i	− 1.14882i
$696$	0	0
$697$	225828.	0.464849
$698$	0	0
$699$	− 253250.i	− 0.518317i
$700$	0	0
$701$	44958.0	0.0914894	0.0457447	−	0.998953i	$-0.485434\pi$
$701$	44958.0	0.0914894	0.0457447	+	0.998953i	$0.485434\pi$
$702$	0	0
$703$	− 62423.1i	− 0.126309i
$704$	0	0
$705$	−825552.	−1.66099
$706$	0	0
$707$	− 418394.i	− 0.837041i
$708$	0	0
$709$	533002.	1.06032	0.530159	−	0.847898i	$-0.322132\pi$
$709$	533002.	1.06032	0.530159	+	0.847898i	$0.322132\pi$
$710$	0	0
$711$	118784.i	0.234974i
$712$	0	0
$713$	1.10419e6	2.17203
$714$	0	0
$715$	− 158878.i	− 0.310778i
$716$	0	0
$717$	368928.	0.717634
$718$	0	0
$719$	292107.i	0.565046i	0.959260	+	0.282523i	$0.0911714\pi$
$719$	292107.i	0.565046i	−0.959260	+	0.282523i	$0.908829\pi$
$720$	0	0
$721$	434544.	0.835917
$722$	0	0
$723$	− 380036.i	− 0.727023i
$724$	0	0
$725$	88842.0	0.169022
$726$	0	0
$727$	− 755791.i	− 1.42999i	−0.699130	−	0.714995i	$-0.746429\pi$
$727$	− 755791.i	− 1.42999i	0.699130	−	0.714995i	$-0.253571\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	209634.i	0.392307i
$732$	0	0
$733$	−832982.	−1.55034	−0.775171	−	0.631751i	$-0.782336\pi$
$733$	−832982.	−1.55034	−0.775171	+	0.631751i	$0.782336\pi$
$734$	0	0
$735$	− 743538.i	− 1.37635i
$736$	0	0
$737$	22608.0	0.0416224
$738$	0	0
$739$	698093.i	1.27827i	0.769093	+	0.639137i	$0.220708\pi$
$739$	698093.i	1.27827i	−0.769093	+	0.639137i	$0.779292\pi$
$740$	0	0
$741$	111384.	0.202855
$742$	0	0
$743$	461044.i	0.835151i	0.908642	+	0.417575i	$0.137120\pi$
$743$	461044.i	0.835151i	−0.908642	+	0.417575i	$0.862880\pi$
$744$	0	0
$745$	−18396.0	−0.0331445
$746$	0	0
$747$	− 323803.i	− 0.580284i
$748$	0	0
$749$	−768240.	−1.36941
$750$	0	0
$751$	937060.i	1.66145i	0.556682	+	0.830726i	$0.312074\pi$
$751$	937060.i	1.66145i	−0.556682	+	0.830726i	$0.687926\pi$
$752$	0	0
$753$	−477468.	−0.842082
$754$	0	0
$755$	1.17820e6i	2.06692i
$756$	0	0
$757$	295786.	0.516162	0.258081	−	0.966123i	$-0.416910\pi$
$757$	295786.	0.516162	0.258081	+	0.966123i	$0.416910\pi$
$758$	0	0
$759$	80810.6i	0.140276i
$760$	0	0
$761$	−1.02615e6	−1.77191	−0.885955	−	0.463772i	$-0.846496\pi$
$761$	−1.02615e6	−1.77191	−0.885955	+	0.463772i	$0.846496\pi$
$762$	0	0
$763$	− 1.23201e6i	− 2.11625i
$764$	0	0
$765$	−278964.	−0.476678
$766$	0	0
$767$	41610.8i	0.0707319i
$768$	0	0
$769$	362306.	0.612665	0.306332	−	0.951925i	$-0.400898\pi$
$769$	362306.	0.612665	0.306332	+	0.951925i	$0.400898\pi$
$770$	0	0
$771$	254061.i	0.427394i
$772$	0	0
$773$	1.02608e6	1.71720	0.858601	−	0.512644i	$-0.171334\pi$
$773$	1.02608e6	1.71720	0.858601	+	0.512644i	$0.171334\pi$
$774$	0	0
$775$	1.68083e6i	2.79847i
$776$	0	0
$777$	209880.	0.347639
$778$	0	0
$779$	108122.i	0.178171i
$780$	0	0
$781$	−38016.0	−0.0623253
$782$	0	0
$783$	10943.1i	0.0178491i
$784$	0	0
$785$	−812532.	−1.31856
$786$	0	0
$787$	− 850042.i	− 1.37243i	−0.727398	−	0.686216i	$-0.759270\pi$
$787$	− 850042.i	− 1.37243i	0.727398	−	0.686216i	$-0.240730\pi$
$788$	0	0
$789$	406296.	0.652662
$790$	0	0
$791$	140379.i	0.224362i
$792$	0	0
$793$	−244972.	−0.389556
$794$	0	0
$795$	− 1.00957e6i	− 1.59736i
$796$	0	0
$797$	761478.	1.19878	0.599392	−	0.800456i	$-0.295409\pi$
$797$	761478.	1.19878	0.599392	+	0.800456i	$0.295409\pi$
$798$	0	0
$799$	− 930569.i	− 1.45766i
$800$	0	0
$801$	−312822.	−0.487565
$802$	0	0
$803$	19246.5i	0.0298484i
$804$	0	0
$805$	−2.39501e6	−3.69586
$806$	0	0
$807$	371722.i	0.570784i
$808$	0	0
$809$	247674.	0.378428	0.189214	−	0.981936i	$-0.439406\pi$
$809$	247674.	0.378428	0.189214	+	0.981936i	$0.439406\pi$
$810$	0	0
$811$	− 920197.i	− 1.39907i	−0.714599	−	0.699534i	$-0.753391\pi$
$811$	− 920197.i	− 1.39907i	0.714599	−	0.699534i	$-0.246609\pi$
$812$	0	0
$813$	−562212.	−0.850588
$814$	0	0
$815$	1.52272e6i	2.29248i
$816$	0	0
$817$	−100368.	−0.150367
$818$	0	0
$819$	374497.i	0.558317i
$820$	0	0
$821$	−250242.	−0.371256	−0.185628	−	0.982620i	$-0.559432\pi$
$821$	−250242.	−0.371256	−0.185628	+	0.982620i	$0.559432\pi$
$822$	0	0
$823$	− 400762.i	− 0.591680i	−0.955238	−	0.295840i	$-0.904401\pi$
$823$	− 400762.i	− 0.591680i	0.955238	−	0.295840i	$-0.0955995\pi$
$824$	0	0
$825$	−123012.	−0.180734
$826$	0	0
$827$	− 17272.0i	− 0.0252541i	−0.999920	−	0.0126270i	$-0.995981\pi$
$827$	− 17272.0i	− 0.0252541i	0.999920	−	0.0126270i	$-0.00401942\pi$
$828$	0	0
$829$	−15686.0	−0.0228246	−0.0114123	−	0.999935i	$-0.503633\pi$
$829$	−15686.0	−0.0228246	−0.0114123	+	0.999935i	$0.503633\pi$
$830$	0	0
$831$	626230.i	0.906842i
$832$	0	0
$833$	838122.	1.20786
$834$	0	0
$835$	787404.i	1.12934i
$836$	0	0
$837$	−207036.	−0.295525
$838$	0	0
$839$	− 115479.i	− 0.164051i	−0.996630	−	0.0820257i	$-0.973861\pi$
$839$	− 115479.i	− 0.164051i	0.996630	−	0.0820257i	$-0.0261390\pi$
$840$	0	0
$841$	−701197.	−0.991398
$842$	0	0
$843$	15869.0i	0.0223304i
$844$	0	0
$845$	−191646.	−0.268402
$846$	0	0
$847$	1.08287e6i	1.50942i
$848$	0	0
$849$	690876.	0.958484
$850$	0	0
$851$	− 396570.i	− 0.547597i
$852$	0	0
$853$	345938.	0.475445	0.237722	−	0.971333i	$-0.423599\pi$
$853$	345938.	0.475445	0.237722	+	0.971333i	$0.423599\pi$
$854$	0	0
$855$	− 133562.i	− 0.182705i
$856$	0	0
$857$	−267990.	−0.364886	−0.182443	−	0.983216i	$-0.558400\pi$
$857$	−267990.	−0.364886	−0.182443	+	0.983216i	$0.558400\pi$
$858$	0	0
$859$	− 522407.i	− 0.707983i	−0.935249	−	0.353992i	$-0.884824\pi$
$859$	− 522407.i	− 0.707983i	0.935249	−	0.353992i	$-0.115176\pi$
$860$	0	0
$861$	−363528.	−0.490379
$862$	0	0
$863$	826895.i	1.11027i	0.831760	+	0.555135i	$0.187333\pi$
$863$	826895.i	1.11027i	−0.831760	+	0.555135i	$0.812667\pi$
$864$	0	0
$865$	1.44522e6	1.93153
$866$	0	0
$867$	119537.i	0.159025i
$868$	0	0
$869$	−91440.0	−0.121087
$870$	0	0
$871$	− 197966.i	− 0.260949i
$872$	0	0
$873$	354186.	0.464732
$874$	0	0
$875$	− 1.64523e6i	− 2.14887i
$876$	0	0
$877$	1.11629e6	1.45137	0.725685	−	0.688028i	$-0.241523\pi$
$877$	1.11629e6	1.45137	0.725685	+	0.688028i	$0.241523\pi$
$878$	0	0
$879$	− 788059.i	− 1.01995i
$880$	0	0
$881$	19170.0	0.0246985	0.0123492	−	0.999924i	$-0.496069\pi$
$881$	19170.0	0.0246985	0.0123492	+	0.999924i	$0.496069\pi$
$882$	0	0
$883$	568909.i	0.729662i	0.931074	+	0.364831i	$0.118873\pi$
$883$	568909.i	0.729662i	−0.931074	+	0.364831i	$0.881127\pi$
$884$	0	0
$885$	49896.0	0.0637058
$886$	0	0
$887$	− 1.09015e6i	− 1.38561i	−0.721126	−	0.692804i	$-0.756375\pi$
$887$	− 1.09015e6i	− 1.38561i	0.721126	−	0.692804i	$-0.243625\pi$
$888$	0	0
$889$	30096.0	0.0380807
$890$	0	0
$891$	− 15152.0i	− 0.0190860i
$892$	0	0
$893$	445536.	0.558702
$894$	0	0
$895$	− 707965.i	− 0.883824i
$896$	0	0
$897$	707616.	0.879453
$898$	0	0
$899$	115105.i	0.142421i
$900$	0	0
$901$	1.13800e6	1.40182
$902$	0	0
$903$	− 337459.i	− 0.413852i
$904$	0	0
$905$	−659652.	−0.805411
$906$	0	0
$907$	916193.i	1.11371i	0.830610	+	0.556855i	$0.187992\pi$
$907$	916193.i	1.11371i	−0.830610	+	0.556855i	$0.812008\pi$
$908$	0	0
$909$	148230.	0.179394
$910$	0	0
$911$	− 995500.i	− 1.19951i	−0.800183	−	0.599756i	$-0.795264\pi$
$911$	− 995500.i	− 1.19951i	0.800183	−	0.599756i	$-0.204736\pi$
$912$	0	0
$913$	249264.	0.299032
$914$	0	0
$915$	293749.i	0.350860i
$916$	0	0
$917$	26928.0	0.0320233
$918$	0	0
$919$	− 97084.9i	− 0.114953i	−0.998347	−	0.0574766i	$-0.981695\pi$
$919$	− 97084.9i	− 0.114953i	0.998347	−	0.0574766i	$-0.0183054\pi$
$920$	0	0
$921$	−28188.0	−0.0332311
$922$	0	0
$923$	332886.i	0.390744i
$924$	0	0
$925$	603670.	0.705531
$926$	0	0
$927$	153952.i	0.179153i
$928$	0	0
$929$	−1.27882e6	−1.48176	−0.740881	−	0.671636i	$-0.765592\pi$
$929$	−1.27882e6	−1.48176	−0.740881	+	0.671636i	$0.765592\pi$
$930$	0	0
$931$	401275.i	0.462959i
$932$	0	0
$933$	−733320.	−0.842423
$934$	0	0
$935$	− 214747.i	− 0.245642i
$936$	0	0
$937$	−981262.	−1.11765	−0.558825	−	0.829286i	$-0.688748\pi$
$937$	−981262.	−1.11765	−0.558825	+	0.829286i	$0.688748\pi$
$938$	0	0
$939$	668672.i	0.758371i
$940$	0	0
$941$	284406.	0.321188	0.160594	−	0.987021i	$-0.448659\pi$
$941$	284406.	0.321188	0.160594	+	0.987021i	$0.448659\pi$
$942$	0	0
$943$	686890.i	0.772438i
$944$	0	0
$945$	449064.	0.502857
$946$	0	0
$947$	− 993109.i	− 1.10738i	−0.832722	−	0.553691i	$-0.813219\pi$
$947$	− 993109.i	− 1.10738i	0.832722	−	0.553691i	$-0.186781\pi$
$948$	0	0
$949$	168532.	0.187133
$950$	0	0
$951$	384443.i	0.425080i
$952$	0	0
$953$	602922.	0.663858	0.331929	−	0.943304i	$-0.392301\pi$
$953$	602922.	0.663858	0.331929	+	0.943304i	$0.392301\pi$
$954$	0	0
$955$	− 111738.i	− 0.122516i
$956$	0	0
$957$	−8424.00	−0.00919802
$958$	0	0
$959$	− 1.01009e6i	− 1.09831i
$960$	0	0
$961$	−1.25419e6	−1.35805
$962$	0	0
$963$	− 272174.i	− 0.293491i
$964$	0	0
$965$	1.12484e6	1.20792
$966$	0	0
$967$	− 575810.i	− 0.615781i	−0.951422	−	0.307890i	$-0.900377\pi$
$967$	− 575810.i	− 0.615781i	0.951422	−	0.307890i	$-0.0996230\pi$
$968$	0	0
$969$	150552.	0.160339
$970$	0	0
$971$	− 1.23920e6i	− 1.31432i	−0.753749	−	0.657162i	$-0.771757\pi$
$971$	− 1.23920e6i	− 1.31432i	0.753749	−	0.657162i	$-0.228243\pi$
$972$	0	0
$973$	−1.00690e6	−1.06355
$974$	0	0
$975$	1.07715e6i	1.13310i
$976$	0	0
$977$	−1.04074e6	−1.09032	−0.545160	−	0.838332i	$-0.683531\pi$
$977$	−1.04074e6	−1.09032	−0.545160	+	0.838332i	$0.683531\pi$
$978$	0	0
$979$	− 240810.i	− 0.251252i
$980$	0	0
$981$	436482.	0.453553
$982$	0	0
$983$	948734.i	0.981833i	0.871207	+	0.490916i	$0.163338\pi$
$983$	948734.i	0.981833i	−0.871207	+	0.490916i	$0.836662\pi$
$984$	0	0
$985$	2.20424e6	2.27189
$986$	0	0
$987$	1.49799e6i	1.53771i
$988$	0	0
$989$	−637632.	−0.651895
$990$	0	0
$991$	616007.i	0.627247i	0.949547	+	0.313623i	$0.101543\pi$
$991$	616007.i	0.627247i	−0.949547	+	0.313623i	$0.898457\pi$
$992$	0	0
$993$	296316.	0.300508
$994$	0	0
$995$	969269.i	0.979035i
$996$	0	0
$997$	−535870.	−0.539100	−0.269550	−	0.962986i	$-0.586875\pi$
$997$	−535870.	−0.539100	−0.269550	+	0.962986i	$0.586875\pi$
$998$	0	0
$999$	74356.9i	0.0745059i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.g.a.31.1	✓	2
3.2	odd	2		144.5.g.f.127.2		2
4.3	odd	2	inner	48.5.g.a.31.2	yes	2
5.2	odd	4		1200.5.j.b.799.1		4
5.3	odd	4		1200.5.j.b.799.3		4
5.4	even	2		1200.5.e.b.751.2		2
8.3	odd	2		192.5.g.b.127.1		2
8.5	even	2		192.5.g.b.127.2		2
12.11	even	2		144.5.g.f.127.1		2
16.3	odd	4		768.5.b.c.127.2		4
16.5	even	4		768.5.b.c.127.1		4
16.11	odd	4		768.5.b.c.127.3		4
16.13	even	4		768.5.b.c.127.4		4
20.3	even	4		1200.5.j.b.799.2		4
20.7	even	4		1200.5.j.b.799.4		4
20.19	odd	2		1200.5.e.b.751.1		2
24.5	odd	2		576.5.g.d.127.2		2
24.11	even	2		576.5.g.d.127.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.g.a.31.1	✓	2	1.1	even	1	trivial
48.5.g.a.31.2	yes	2	4.3	odd	2	inner
144.5.g.f.127.1		2	12.11	even	2
144.5.g.f.127.2		2	3.2	odd	2
192.5.g.b.127.1		2	8.3	odd	2
192.5.g.b.127.2		2	8.5	even	2
576.5.g.d.127.1		2	24.11	even	2
576.5.g.d.127.2		2	24.5	odd	2
768.5.b.c.127.1		4	16.5	even	4
768.5.b.c.127.2		4	16.3	odd	4
768.5.b.c.127.3		4	16.11	odd	4
768.5.b.c.127.4		4	16.13	even	4
1200.5.e.b.751.1		2	20.19	odd	2
1200.5.e.b.751.2		2	5.4	even	2
1200.5.j.b.799.1		4	5.2	odd	4
1200.5.j.b.799.2		4	20.3	even	4
1200.5.j.b.799.3		4	5.3	odd	4
1200.5.j.b.799.4		4	20.7	even	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} -42.0000 q^{5} +76.2102i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} -42.0000 q^{5} +76.2102i q^{7} -27.0000 q^{9} -20.7846i q^{11} -182.000 q^{13} +218.238i q^{15} -246.000 q^{17} -117.779i q^{19} +396.000 q^{21} -748.246i q^{23} +1139.00 q^{25} +140.296i q^{27} +78.0000 q^{29} +1475.71i q^{31} -108.000 q^{33} -3200.83i q^{35} +530.000 q^{37} +945.700i q^{39} -918.000 q^{41} -852.169i q^{43} +1134.00 q^{45} +3782.80i q^{47} -3407.00 q^{49} +1278.25i q^{51} -4626.00 q^{53} +872.954i q^{55} -612.000 q^{57} -228.631i q^{59} +1346.00 q^{61} -2057.68i q^{63} +7644.00 q^{65} +1087.73i q^{67} -3888.00 q^{69} -1829.05i q^{71} -926.000 q^{73} -5918.42i q^{75} +1584.00 q^{77} -4399.41i q^{79} +729.000 q^{81} +11992.7i q^{83} +10332.0 q^{85} -405.300i q^{87} +11586.0 q^{89} -13870.3i q^{91} +7668.00 q^{93} +4946.74i q^{95} -13118.0 q^{97} +561.184i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 84 q^{5} - 54 q^{9}+O(q^{10}) \) 2 * q - 84 * q^5 - 54 * q^9	\( 2 q - 84 q^{5} - 54 q^{9} - 364 q^{13} - 492 q^{17} + 792 q^{21} + 2278 q^{25} + 156 q^{29} - 216 q^{33} + 1060 q^{37} - 1836 q^{41} + 2268 q^{45} - 6814 q^{49} - 9252 q^{53} - 1224 q^{57} + 2692 q^{61} + 15288 q^{65} - 7776 q^{69} - 1852 q^{73} + 3168 q^{77} + 1458 q^{81} + 20664 q^{85} + 23172 q^{89} + 15336 q^{93} - 26236 q^{97}+O(q^{100}) \) 2 * q - 84 * q^5 - 54 * q^9 - 364 * q^13 - 492 * q^17 + 792 * q^21 + 2278 * q^25 + 156 * q^29 - 216 * q^33 + 1060 * q^37 - 1836 * q^41 + 2268 * q^45 - 6814 * q^49 - 9252 * q^53 - 1224 * q^57 + 2692 * q^61 + 15288 * q^65 - 7776 * q^69 - 1852 * q^73 + 3168 * q^77 + 1458 * q^81 + 20664 * q^85 + 23172 * q^89 + 15336 * q^93 - 26236 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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