LMFDB - Embedded newform 1200.5.j.b.799.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,5,Mod(799,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.799");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1200.j (of order $2$, degree $1$, not 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:	$124.043955701$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1200.799
Dual form			1200.5.j.b.799.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} +76.2102 q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} +76.2102 q^{7} +27.0000 q^{9} +20.7846i q^{11} +182.000i q^{13} -246.000i q^{17} -117.779i q^{19} +396.000 q^{21} +748.246 q^{23} +140.296 q^{27} -78.0000 q^{29} -1475.71i q^{31} +108.000i q^{33} +530.000i q^{37} +945.700i q^{39} -918.000 q^{41} +852.169 q^{43} +3782.80 q^{47} +3407.00 q^{49} -1278.25i q^{51} +4626.00i q^{53} -612.000i q^{57} -228.631i q^{59} +1346.00 q^{61} +2057.68 q^{63} +1087.73 q^{67} +3888.00 q^{69} +1829.05i q^{71} +926.000i q^{73} +1584.00i q^{77} -4399.41i q^{79} +729.000 q^{81} -11992.7 q^{83} -405.300 q^{87} -11586.0 q^{89} +13870.3i q^{91} -7668.00i q^{93} -13118.0i q^{97} +561.184i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{9}+O(q^{10}) $ 4 * q + 108 * q^9	$ 4 q + 108 q^{9} + 1584 q^{21} - 312 q^{29} - 3672 q^{41} + 13628 q^{49} + 5384 q^{61} + 15552 q^{69} + 2916 q^{81} - 46344 q^{89}+O(q^{100}) $ 4 * q + 108 * q^9 + 1584 * q^21 - 312 * q^29 - 3672 * q^41 + 13628 * q^49 + 5384 * q^61 + 15552 * q^69 + 2916 * q^81 - 46344 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	76.2102	1.55531	0.777655	−	0.628691i	$-0.216409\pi$
$7$	76.2102	1.55531	0.777655	+	0.628691i	$0.216409\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	20.7846i	0.171774i	0.996305	+	0.0858868i	$0.0273723\pi$
$11$	20.7846i	0.171774i	−0.996305	+	0.0858868i	$0.972628\pi$
$12$	0	0
$13$	182.000i	1.07692i	0.842650	+	0.538462i	$0.180994\pi$
$13$	182.000i	1.07692i	−0.842650	+	0.538462i	$0.819006\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 246.000i	− 0.851211i	−0.904909	−	0.425606i	$-0.860061\pi$
$17$	− 246.000i	− 0.851211i	0.904909	−	0.425606i	$-0.139939\pi$
$18$	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
$23$	748.246	1.41445	0.707227	−	0.706987i	$-0.249946\pi$
$23$	748.246	1.41445	0.707227	+	0.706987i	$0.249946\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	−78.0000	−0.0927467	−0.0463734	−	0.998924i	$-0.514766\pi$
$29$	−78.0000	−0.0927467	−0.0463734	+	0.998924i	$0.514766\pi$
$30$	0	0
$31$	− 1475.71i	− 1.53560i	−0.640692	−	0.767798i	$-0.721353\pi$
$31$	− 1475.71i	− 1.53560i	0.640692	−	0.767798i	$-0.278647\pi$
$32$	0	0
$33$	108.000i	0.0991736i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	530.000i	0.387144i	0.981086	+	0.193572i	$0.0620073\pi$
$37$	530.000i	0.387144i	−0.981086	+	0.193572i	$0.937993\pi$
$38$	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
$43$	852.169	0.460881	0.230441	−	0.973086i	$-0.425983\pi$
$43$	852.169	0.460881	0.230441	+	0.973086i	$0.425983\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3782.80	1.71245	0.856224	−	0.516604i	$-0.172804\pi$
$47$	3782.80	1.71245	0.856224	+	0.516604i	$0.172804\pi$
$48$	0	0
$49$	3407.00	1.41899
$50$	0	0
$51$	− 1278.25i	− 0.491447i
$52$	0	0
$53$	4626.00i	1.64685i	0.567426	+	0.823425i	$0.307939\pi$
$53$	4626.00i	1.64685i	−0.567426	+	0.823425i	$0.692061\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 612.000i	− 0.188366i
$58$	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
$63$	2057.68	0.518437
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1087.73	0.242310	0.121155	−	0.992634i	$-0.461340\pi$
$67$	1087.73	0.242310	0.121155	+	0.992634i	$0.461340\pi$
$68$	0	0
$69$	3888.00	0.816635
$70$	0	0
$71$	1829.05i	0.362834i	0.983406	+	0.181417i	$0.0580684\pi$
$71$	1829.05i	0.362834i	−0.983406	+	0.181417i	$0.941932\pi$
$72$	0	0
$73$	926.000i	0.173766i	0.996219	+	0.0868831i	$0.0276907\pi$
$73$	926.000i	0.173766i	−0.996219	+	0.0868831i	$0.972309\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1584.00i	0.267161i
$78$	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
$83$	−11992.7	−1.74085	−0.870425	−	0.492301i	$-0.836156\pi$
$83$	−11992.7	−1.74085	−0.870425	+	0.492301i	$0.836156\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−405.300	−0.0535473
$88$	0	0
$89$	−11586.0	−1.46269	−0.731347	−	0.682005i	$-0.761108\pi$
$89$	−11586.0	−1.46269	−0.731347	+	0.682005i	$0.761108\pi$
$90$	0	0
$91$	13870.3i	1.67495i
$92$	0	0
$93$	− 7668.00i	− 0.886576i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 13118.0i	− 1.39420i	−0.716975	−	0.697099i	$-0.754474\pi$
$97$	− 13118.0i	− 1.39420i	0.716975	−	0.697099i	$-0.245526\pi$
$98$	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
$103$	5701.91	0.537460	0.268730	−	0.963216i	$-0.413396\pi$
$103$	5701.91	0.537460	0.268730	+	0.963216i	$0.413396\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10080.5	0.880473	0.440237	−	0.897882i	$-0.354895\pi$
$107$	10080.5	0.880473	0.440237	+	0.897882i	$0.354895\pi$
$108$	0	0
$109$	16166.0	1.36066	0.680330	−	0.732906i	$-0.261836\pi$
$109$	16166.0	1.36066	0.680330	+	0.732906i	$0.261836\pi$
$110$	0	0
$111$	2753.96i	0.223518i
$112$	0	0
$113$	− 1842.00i	− 0.144256i	−0.997395	−	0.0721278i	$-0.977021\pi$
$113$	− 1842.00i	− 0.144256i	0.997395	−	0.0721278i	$-0.0229789\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4914.00i	0.358974i
$118$	0	0
$119$	− 18747.7i	− 1.32390i
$120$	0	0
$121$	14209.0	0.970494
$122$	0	0
$123$	−4770.07	−0.315293
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−394.908	−0.0244843	−0.0122422	−	0.999925i	$-0.503897\pi$
$127$	−394.908	−0.0244843	−0.0122422	+	0.999925i	$0.503897\pi$
$128$	0	0
$129$	4428.00	0.266090
$130$	0	0
$131$	353.338i	0.0205896i	0.999947	+	0.0102948i	$0.00327700\pi$
$131$	353.338i	0.0205896i	−0.999947	+	0.0102948i	$0.996723\pi$
$132$	0	0
$133$	− 8976.00i	− 0.507434i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 13254.0i	− 0.706164i	−0.935592	−	0.353082i	$-0.885134\pi$
$137$	− 13254.0i	− 0.706164i	0.935592	−	0.353082i	$-0.114866\pi$
$138$	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
$143$	−3782.80	−0.184987
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17703.3	0.819255
$148$	0	0
$149$	−438.000	−0.0197288	−0.00986442	−	0.999951i	$-0.503140\pi$
$149$	−438.000	−0.0197288	−0.00986442	+	0.999951i	$0.503140\pi$
$150$	0	0
$151$	28052.3i	1.23031i	0.788406	+	0.615155i	$0.210907\pi$
$151$	28052.3i	1.23031i	−0.788406	+	0.615155i	$0.789093\pi$
$152$	0	0
$153$	− 6642.00i	− 0.283737i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	19346.0i	0.784859i	0.919782	+	0.392430i	$0.128365\pi$
$157$	19346.0i	0.784859i	−0.919782	+	0.392430i	$0.871635\pi$
$158$	0	0
$159$	24037.4i	0.950809i
$160$	0	0
$161$	57024.0	2.19992
$162$	0	0
$163$	36255.3	1.36457	0.682286	−	0.731086i	$-0.260986\pi$
$163$	36255.3	1.36457	0.682286	+	0.731086i	$0.260986\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18747.7	−0.672226	−0.336113	−	0.941822i	$-0.609112\pi$
$167$	−18747.7	−0.672226	−0.336113	+	0.941822i	$0.609112\pi$
$168$	0	0
$169$	−4563.00	−0.159763
$170$	0	0
$171$	− 3180.05i	− 0.108753i
$172$	0	0
$173$	34410.0i	1.14972i	0.818251	+	0.574861i	$0.194944\pi$
$173$	34410.0i	1.14972i	−0.818251	+	0.574861i	$0.805056\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1188.00i	− 0.0379201i
$178$	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
$183$	6994.02	0.208845
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5113.01	0.146216
$188$	0	0
$189$	10692.0	0.299320
$190$	0	0
$191$	− 2660.43i	− 0.0729265i	−0.999335	−	0.0364632i	$-0.988391\pi$
$191$	− 2660.43i	− 0.0729265i	0.999335	−	0.0364632i	$-0.0116092\pi$
$192$	0	0
$193$	26782.0i	0.718999i	0.933145	+	0.359500i	$0.117053\pi$
$193$	26782.0i	0.718999i	−0.933145	+	0.359500i	$0.882947\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 52482.0i	− 1.35232i	−0.736757	−	0.676158i	$-0.763644\pi$
$197$	− 52482.0i	− 1.35232i	0.736757	−	0.676158i	$-0.236356\pi$
$198$	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
$203$	−5944.40	−0.144250
$204$	0	0
$205$	0	0
$206$	0	0
$207$	20202.6	0.471485
$208$	0	0
$209$	2448.00	0.0560427
$210$	0	0
$211$	− 23895.4i	− 0.536721i	−0.963319	−	0.268361i	$-0.913518\pi$
$211$	− 23895.4i	− 0.536721i	0.963319	−	0.268361i	$-0.0864819\pi$
$212$	0	0
$213$	9504.00i	0.209482i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 112464.i	− 2.38833i
$218$	0	0
$219$	4811.64i	0.100324i
$220$	0	0
$221$	44772.0	0.916689
$222$	0	0
$223$	−852.169	−0.0171363	−0.00856813	−	0.999963i	$-0.502727\pi$
$223$	−852.169	−0.0171363	−0.00856813	+	0.999963i	$0.502727\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	76175.6	1.47831	0.739153	−	0.673538i	$-0.235226\pi$
$227$	76175.6	1.47831	0.739153	+	0.673538i	$0.235226\pi$
$228$	0	0
$229$	48470.0	0.924277	0.462138	−	0.886808i	$-0.347082\pi$
$229$	48470.0	0.924277	0.462138	+	0.886808i	$0.347082\pi$
$230$	0	0
$231$	8230.71i	0.154246i
$232$	0	0
$233$	− 48738.0i	− 0.897751i	−0.893594	−	0.448875i	$-0.851825\pi$
$233$	− 48738.0i	− 0.897751i	0.893594	−	0.448875i	$-0.148175\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 22860.0i	− 0.406986i
$238$	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
$243$	3788.00	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	21435.9	0.351356
$248$	0	0
$249$	−62316.0	−1.00508
$250$	0	0
$251$	91888.8i	1.45853i	0.684232	+	0.729264i	$0.260138\pi$
$251$	91888.8i	1.45853i	−0.684232	+	0.729264i	$0.739862\pi$
$252$	0	0
$253$	15552.0i	0.242966i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 48894.0i	− 0.740269i	−0.928978	−	0.370134i	$-0.879312\pi$
$257$	− 48894.0i	− 0.740269i	0.928978	−	0.370134i	$-0.120688\pi$
$258$	0	0
$259$	40391.4i	0.602129i
$260$	0	0
$261$	−2106.00	−0.0309156
$262$	0	0
$263$	−78191.7	−1.13044	−0.565222	−	0.824939i	$-0.691210\pi$
$263$	−78191.7	−1.13044	−0.565222	+	0.824939i	$0.691210\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−60202.6	−0.844487
$268$	0	0
$269$	71538.0	0.988626	0.494313	−	0.869284i	$-0.335420\pi$
$269$	71538.0	0.988626	0.494313	+	0.869284i	$0.335420\pi$
$270$	0	0
$271$	108198.i	1.47326i	0.676296	+	0.736630i	$0.263584\pi$
$271$	108198.i	1.47326i	−0.676296	+	0.736630i	$0.736416\pi$
$272$	0	0
$273$	72072.0i	0.967033i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 120518.i	− 1.57070i	−0.619054	−	0.785348i	$-0.712484\pi$
$277$	− 120518.i	− 1.57070i	0.619054	−	0.785348i	$-0.287516\pi$
$278$	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.	−1.66014	−0.830071	−	0.557657i	$-0.811700\pi$
$283$	−132959.	−1.66014	−0.830071	+	0.557657i	$0.811700\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−69961.0	−0.849361
$288$	0	0
$289$	23005.0	0.275440
$290$	0	0
$291$	− 68163.1i	− 0.804940i
$292$	0	0
$293$	− 151662.i	− 1.76661i	−0.468795	−	0.883307i	$-0.655312\pi$
$293$	− 151662.i	− 1.76661i	0.468795	−	0.883307i	$-0.344688\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2916.00i	0.0330579i
$298$	0	0
$299$	136181.i	1.52326i
$300$	0	0
$301$	64944.0	0.716813
$302$	0	0
$303$	−28526.9	−0.310720
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−5424.78	−0.0575580	−0.0287790	−	0.999586i	$-0.509162\pi$
$307$	−5424.78	−0.0575580	−0.0287790	+	0.999586i	$0.509162\pi$
$308$	0	0
$309$	29628.0	0.310303
$310$	0	0
$311$	141127.i	1.45912i	0.683917	+	0.729560i	$0.260275\pi$
$311$	141127.i	1.45912i	−0.683917	+	0.729560i	$0.739725\pi$
$312$	0	0
$313$	128686.i	1.31354i	0.754092	+	0.656769i	$0.228077\pi$
$313$	128686.i	1.31354i	−0.754092	+	0.656769i	$0.771923\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 73986.0i	− 0.736260i	−0.929774	−	0.368130i	$-0.879998\pi$
$317$	− 73986.0i	− 0.736260i	0.929774	−	0.368130i	$-0.120002\pi$
$318$	0	0
$319$	− 1621.20i	− 0.0159314i
$320$	0	0
$321$	52380.0	0.508341
$322$	0	0
$323$	−28973.7	−0.277715
$324$	0	0
$325$	0	0
$326$	0	0
$327$	84001.0	0.785577
$328$	0	0
$329$	288288.	2.66339
$330$	0	0
$331$	− 57026.0i	− 0.520496i	−0.965542	−	0.260248i	$-0.916196\pi$
$331$	− 57026.0i	− 0.520496i	0.965542	−	0.260248i	$-0.0838043\pi$
$332$	0	0
$333$	14310.0i	0.129048i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	98674.0i	0.868846i	0.900709	+	0.434423i	$0.143048\pi$
$337$	98674.0i	0.868846i	−0.900709	+	0.434423i	$0.856952\pi$
$338$	0	0
$339$	− 9571.31i	− 0.0832860i
$340$	0	0
$341$	30672.0	0.263775
$342$	0	0
$343$	76667.5	0.651663
$344$	0	0
$345$	0	0
$346$	0	0
$347$	56929.0	0.472797	0.236399	−	0.971656i	$-0.424033\pi$
$347$	56929.0	0.472797	0.236399	+	0.971656i	$0.424033\pi$
$348$	0	0
$349$	−181346.	−1.48887	−0.744436	−	0.667694i	$-0.767281\pi$
$349$	−181346.	−1.48887	−0.744436	+	0.667694i	$0.767281\pi$
$350$	0	0
$351$	25533.9i	0.207254i
$352$	0	0
$353$	4302.00i	0.0345240i	0.999851	+	0.0172620i	$0.00549494\pi$
$353$	4302.00i	0.0345240i	−0.999851	+	0.0172620i	$0.994505\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 97416.0i	− 0.764353i
$358$	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
$363$	73832.1	0.560315
$364$	0	0
$365$	0	0
$366$	0	0
$367$	182690.	1.35638	0.678191	−	0.734885i	$-0.262764\pi$
$367$	182690.	1.35638	0.678191	+	0.734885i	$0.262764\pi$
$368$	0	0
$369$	−24786.0	−0.182035
$370$	0	0
$371$	352549.i	2.56136i
$372$	0	0
$373$	− 151778.i	− 1.09092i	−0.838138	−	0.545458i	$-0.816356\pi$
$373$	− 151778.i	− 1.09092i	0.838138	−	0.545458i	$-0.183644\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 14196.0i	− 0.0998811i
$378$	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
$383$	65346.8	0.445479	0.222739	−	0.974878i	$-0.428500\pi$
$383$	65346.8	0.445479	0.222739	+	0.974878i	$0.428500\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	23008.6	0.153627
$388$	0	0
$389$	−105750.	−0.698846	−0.349423	−	0.936965i	$-0.613622\pi$
$389$	−105750.	−0.698846	−0.349423	+	0.936965i	$0.613622\pi$
$390$	0	0
$391$	− 184069.i	− 1.20400i
$392$	0	0
$393$	1836.00i	0.0118874i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 27934.0i	− 0.177236i	−0.996066	−	0.0886180i	$-0.971755\pi$
$397$	− 27934.0i	− 0.177236i	0.996066	−	0.0886180i	$-0.0282450\pi$
$398$	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.	1.65372
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11015.8	−0.0665011
$408$	0	0
$409$	20270.0	0.121173	0.0605867	−	0.998163i	$-0.480703\pi$
$409$	20270.0	0.121173	0.0605867	+	0.998163i	$0.480703\pi$
$410$	0	0
$411$	− 68869.8i	− 0.407704i
$412$	0	0
$413$	− 17424.0i	− 0.102152i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	68652.0i	0.394804i
$418$	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.	0.570816
$424$	0	0
$425$	0	0
$426$	0	0
$427$	102579.	0.562604
$428$	0	0
$429$	−19656.0	−0.106802
$430$	0	0
$431$	− 355542.i	− 1.91397i	−0.290132	−	0.956986i	$-0.593699\pi$
$431$	− 355542.i	− 1.91397i	0.290132	−	0.956986i	$-0.406301\pi$
$432$	0	0
$433$	199726.i	1.06527i	0.846346	+	0.532634i	$0.178798\pi$
$433$	199726.i	1.06527i	−0.846346	+	0.532634i	$0.821202\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 88128.0i	− 0.461478i
$438$	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
$443$	−50444.2	−0.257042	−0.128521	−	0.991707i	$-0.541023\pi$
$443$	−50444.2	−0.257042	−0.128521	+	0.991707i	$0.541023\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−2275.91	−0.0113905
$448$	0	0
$449$	−149994.	−0.744014	−0.372007	−	0.928230i	$-0.621330\pi$
$449$	−149994.	−0.744014	−0.372007	+	0.928230i	$0.621330\pi$
$450$	0	0
$451$	− 19080.3i	− 0.0938062i
$452$	0	0
$453$	145764.i	0.710320i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	284338.i	1.36145i	0.732538	+	0.680726i	$0.238336\pi$
$457$	284338.i	1.36145i	−0.732538	+	0.680726i	$0.761664\pi$
$458$	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.	−0.804324	−0.402162	−	0.915568i	$-0.631741\pi$
$463$	−172422.	−0.804324	−0.402162	+	0.915568i	$0.631741\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−68734.7	−0.315168	−0.157584	−	0.987506i	$-0.550371\pi$
$467$	−68734.7	−0.315168	−0.157584	+	0.987506i	$0.550371\pi$
$468$	0	0
$469$	82896.0	0.376867
$470$	0	0
$471$	100525.i	0.453139i
$472$	0	0
$473$	17712.0i	0.0791672i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	124902.i	0.548950i
$478$	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.	1.27012
$484$	0	0
$485$	0	0
$486$	0	0
$487$	271108.	1.14310	0.571549	−	0.820568i	$-0.306343\pi$
$487$	271108.	1.14310	0.571549	+	0.820568i	$0.306343\pi$
$488$	0	0
$489$	188388.	0.787835
$490$	0	0
$491$	227862.i	0.945166i	0.881286	+	0.472583i	$0.156678\pi$
$491$	227862.i	0.945166i	−0.881286	+	0.472583i	$0.843322\pi$
$492$	0	0
$493$	19188.0i	0.0789470i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	139392.i	0.564320i
$498$	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.	−1.76490	−0.882452	−	0.470403i	$-0.844109\pi$
$503$	−446537.	−1.76490	−0.882452	+	0.470403i	$0.844109\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−23710.0	−0.0922394
$508$	0	0
$509$	39330.0	0.151806	0.0759029	−	0.997115i	$-0.475816\pi$
$509$	39330.0	0.151806	0.0759029	+	0.997115i	$0.475816\pi$
$510$	0	0
$511$	70570.7i	0.270260i
$512$	0	0
$513$	− 16524.0i	− 0.0627886i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	78624.0i	0.294154i
$518$	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.	0.495409	0.247704	−	0.968836i	$-0.420324\pi$
$523$	135509.	0.495409	0.247704	+	0.968836i	$0.420324\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−363024.	−1.30712
$528$	0	0
$529$	280031.	1.00068
$530$	0	0
$531$	− 6173.03i	− 0.0218932i
$532$	0	0
$533$	− 167076.i	− 0.588111i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	87588.0i	0.303736i
$538$	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
$543$	81610.8	0.276788
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−261644.	−0.874451	−0.437225	−	0.899352i	$-0.644039\pi$
$547$	−261644.	−0.874451	−0.437225	+	0.899352i	$0.644039\pi$
$548$	0	0
$549$	36342.0	0.120577
$550$	0	0
$551$	9186.80i	0.0302594i
$552$	0	0
$553$	− 335280.i	− 1.09637i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 233274.i	− 0.751893i	−0.926641	−	0.375946i	$-0.877318\pi$
$557$	− 233274.i	− 0.751893i	0.926641	−	0.375946i	$-0.122682\pi$
$558$	0	0
$559$	155095.i	0.496333i
$560$	0	0
$561$	26568.0	0.0844176
$562$	0	0
$563$	−419704.	−1.32412	−0.662058	−	0.749453i	$-0.730317\pi$
$563$	−419704.	−1.32412	−0.662058	+	0.749453i	$0.730317\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	55557.3	0.172812
$568$	0	0
$569$	−470058.	−1.45187	−0.725934	−	0.687765i	$-0.758592\pi$
$569$	−470058.	−1.45187	−0.725934	+	0.687765i	$0.758592\pi$
$570$	0	0
$571$	− 320381.i	− 0.982640i	−0.870979	−	0.491320i	$-0.836515\pi$
$571$	− 320381.i	− 0.982640i	0.870979	−	0.491320i	$-0.163485\pi$
$572$	0	0
$573$	− 13824.0i	− 0.0421041i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 341038.i	− 1.02436i	−0.858879	−	0.512178i	$-0.828839\pi$
$577$	− 341038.i	− 1.02436i	0.858879	−	0.512178i	$-0.171161\pi$
$578$	0	0
$579$	139163.i	0.415114i
$580$	0	0
$581$	−913968.	−2.70756
$582$	0	0
$583$	−96149.6	−0.282885
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−114128.	−0.331220	−0.165610	−	0.986191i	$-0.552959\pi$
$587$	−114128.	−0.331220	−0.165610	+	0.986191i	$0.552959\pi$
$588$	0	0
$589$	−173808.	−0.501002
$590$	0	0
$591$	− 272704.i	− 0.780760i
$592$	0	0
$593$	96846.0i	0.275405i	0.990474	+	0.137703i	$0.0439718\pi$
$593$	96846.0i	0.275405i	−0.990474	+	0.137703i	$0.956028\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 119916.i	− 0.336456i
$598$	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
$603$	29368.7	0.0807699
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−133195.	−0.361501	−0.180751	−	0.983529i	$-0.557853\pi$
$607$	−133195.	−0.361501	−0.180751	+	0.983529i	$0.557853\pi$
$608$	0	0
$609$	−30888.0	−0.0832828
$610$	0	0
$611$	688469.i	1.84418i
$612$	0	0
$613$	− 247202.i	− 0.657856i	−0.944355	−	0.328928i	$-0.893313\pi$
$613$	− 247202.i	− 0.657856i	0.944355	−	0.328928i	$-0.106687\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 31758.0i	− 0.0834224i	−0.999130	−	0.0417112i	$-0.986719\pi$
$617$	− 31758.0i	− 0.0834224i	0.999130	−	0.0417112i	$-0.0132809\pi$
$618$	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.	−2.27494
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12720.2	0.0323563
$628$	0	0
$629$	130380.	0.329541
$630$	0	0
$631$	− 417736.i	− 1.04916i	−0.851360	−	0.524582i	$-0.824222\pi$
$631$	− 417736.i	− 1.04916i	0.851360	−	0.524582i	$-0.175778\pi$
$632$	0	0
$633$	− 124164.i	− 0.309876i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	620074.i	1.52815i
$638$	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.	1.72727	0.863635	−	0.504117i	$-0.168182\pi$
$643$	714138.	1.72727	0.863635	+	0.504117i	$0.168182\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−259558.	−0.620049	−0.310025	−	0.950729i	$-0.600337\pi$
$647$	−259558.	−0.620049	−0.310025	+	0.950729i	$0.600337\pi$
$648$	0	0
$649$	4752.00	0.0112820
$650$	0	0
$651$	− 584380.i	− 1.37890i
$652$	0	0
$653$	330714.i	0.775579i	0.921748	+	0.387790i	$0.126761\pi$
$653$	330714.i	0.775579i	−0.921748	+	0.387790i	$0.873239\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	25002.0i	0.0579221i
$658$	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.	0.529251
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−58363.2	−0.131186
$668$	0	0
$669$	−4428.00	−0.00989362
$670$	0	0
$671$	27976.1i	0.0621358i
$672$	0	0
$673$	552910.i	1.22074i	0.792115	+	0.610372i	$0.208980\pi$
$673$	552910.i	1.22074i	−0.792115	+	0.610372i	$0.791020\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	609030.i	1.32881i	0.747375	+	0.664403i	$0.231314\pi$
$677$	609030.i	1.32881i	−0.747375	+	0.664403i	$0.768686\pi$
$678$	0	0
$679$	− 999726.i	− 2.16841i
$680$	0	0
$681$	395820.	0.853500
$682$	0	0
$683$	23715.2	0.0508377	0.0254189	−	0.999677i	$-0.491908\pi$
$683$	23715.2	0.0508377	0.0254189	+	0.999677i	$0.491908\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	251858.	0.533631
$688$	0	0
$689$	−841932.	−1.77353
$690$	0	0
$691$	− 431842.i	− 0.904417i	−0.891912	−	0.452208i	$-0.850636\pi$
$691$	− 431842.i	− 0.904417i	0.891912	−	0.452208i	$-0.149364\pi$
$692$	0	0
$693$	42768.0i	0.0890538i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	225828.i	0.464849i
$698$	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
$703$	62423.1	0.126309
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−418394.	−0.837041
$708$	0	0
$709$	−533002.	−1.06032	−0.530159	−	0.847898i	$-0.677868\pi$
$709$	−533002.	−1.06032	−0.530159	+	0.847898i	$0.677868\pi$
$710$	0	0
$711$	− 118784.i	− 0.234974i
$712$	0	0
$713$	− 1.10419e6i	− 2.17203i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	368928.i	0.717634i
$718$	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.	0.727023
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−755791.	−1.42999	−0.714995	−	0.699130i	$-0.753571\pi$
$727$	−755791.	−1.42999	−0.714995	+	0.699130i	$0.753571\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	− 209634.i	− 0.392307i
$732$	0	0
$733$	832982.i	1.55034i	0.631751	+	0.775171i	$0.282336\pi$
$733$	832982.i	1.55034i	−0.631751	+	0.775171i	$0.717664\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22608.0i	0.0416224i
$738$	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.	−0.835151	−0.417575	−	0.908642i	$-0.637120\pi$
$743$	−461044.	−0.835151	−0.417575	+	0.908642i	$0.637120\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−323803.	−0.580284
$748$	0	0
$749$	768240.	1.36941
$750$	0	0
$751$	− 937060.i	− 1.66145i	−0.556682	−	0.830726i	$-0.687926\pi$
$751$	− 937060.i	− 1.66145i	0.556682	−	0.830726i	$-0.312074\pi$
$752$	0	0
$753$	477468.i	0.842082i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	295786.i	0.516162i	0.966123	+	0.258081i	$0.0830901\pi$
$757$	295786.i	0.516162i	−0.966123	+	0.258081i	$0.916910\pi$
$758$	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
$763$	1.23201e6	2.11625
$764$	0	0
$765$	0	0
$766$	0	0
$767$	41610.8	0.0707319
$768$	0	0
$769$	−362306.	−0.612665	−0.306332	−	0.951925i	$-0.599102\pi$
$769$	−362306.	−0.612665	−0.306332	+	0.951925i	$0.599102\pi$
$770$	0	0
$771$	− 254061.i	− 0.427394i
$772$	0	0
$773$	− 1.02608e6i	− 1.71720i	−0.512644	−	0.858601i	$-0.671334\pi$
$773$	− 1.02608e6i	− 1.71720i	0.512644	−	0.858601i	$-0.328666\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	209880.i	0.347639i
$778$	0	0
$779$	108122.i	0.178171i
$780$	0	0
$781$	−38016.0	−0.0623253
$782$	0	0
$783$	−10943.1	−0.0178491
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−850042.	−1.37243	−0.686216	−	0.727398i	$-0.740730\pi$
$787$	−850042.	−1.37243	−0.686216	+	0.727398i	$0.740730\pi$
$788$	0	0
$789$	−406296.	−0.652662
$790$	0	0
$791$	− 140379.i	− 0.224362i
$792$	0	0
$793$	244972.i	0.389556i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	761478.i	1.19878i	0.800456	+	0.599392i	$0.204591\pi$
$797$	761478.i	1.19878i	−0.800456	+	0.599392i	$0.795409\pi$
$798$	0	0
$799$	− 930569.i	− 1.45766i
$800$	0	0
$801$	−312822.	−0.487565
$802$	0	0
$803$	−19246.5	−0.0298484
$804$	0	0
$805$	0	0
$806$	0	0
$807$	371722.	0.570784
$808$	0	0
$809$	−247674.	−0.378428	−0.189214	−	0.981936i	$-0.560594\pi$
$809$	−247674.	−0.378428	−0.189214	+	0.981936i	$0.560594\pi$
$810$	0	0
$811$	920197.i	1.39907i	0.714599	+	0.699534i	$0.246609\pi$
$811$	920197.i	1.39907i	−0.714599	+	0.699534i	$0.753391\pi$
$812$	0	0
$813$	562212.i	0.850588i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 100368.i	− 0.150367i
$818$	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.	0.591680	0.295840	−	0.955238i	$-0.404401\pi$
$823$	400762.	0.591680	0.295840	+	0.955238i	$0.404401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17272.0	−0.0252541	−0.0126270	−	0.999920i	$-0.504019\pi$
$827$	−17272.0	−0.0252541	−0.0126270	+	0.999920i	$0.504019\pi$
$828$	0	0
$829$	15686.0	0.0228246	0.0114123	−	0.999935i	$-0.496367\pi$
$829$	15686.0	0.0228246	0.0114123	+	0.999935i	$0.496367\pi$
$830$	0	0
$831$	− 626230.i	− 0.906842i
$832$	0	0
$833$	− 838122.i	− 1.20786i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 207036.i	− 0.295525i
$838$	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
$843$	−15869.0	−0.0223304
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.08287e6	1.50942
$848$	0	0
$849$	−690876.	−0.958484
$850$	0	0
$851$	396570.i	0.547597i
$852$	0	0
$853$	− 345938.i	− 0.475445i	−0.971333	−	0.237722i	$-0.923599\pi$
$853$	− 345938.i	− 0.475445i	0.971333	−	0.237722i	$-0.0764009\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 267990.i	− 0.364886i	−0.983216	−	0.182443i	$-0.941600\pi$
$857$	− 267990.i	− 0.364886i	0.983216	−	0.182443i	$-0.0584005\pi$
$858$	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.	−1.11027	−0.555135	−	0.831760i	$-0.687333\pi$
$863$	−826895.	−1.11027	−0.555135	+	0.831760i	$0.687333\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	119537.	0.159025
$868$	0	0
$869$	91440.0	0.121087
$870$	0	0
$871$	197966.i	0.260949i
$872$	0	0
$873$	− 354186.i	− 0.464732i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.11629e6i	1.45137i	0.688028	+	0.725685i	$0.258477\pi$
$877$	1.11629e6i	1.45137i	−0.688028	+	0.725685i	$0.741523\pi$
$878$	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.	−0.729662	−0.364831	−	0.931074i	$-0.618873\pi$
$883$	−568909.	−0.729662	−0.364831	+	0.931074i	$0.618873\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.09015e6	−1.38561	−0.692804	−	0.721126i	$-0.743625\pi$
$887$	−1.09015e6	−1.38561	−0.692804	+	0.721126i	$0.743625\pi$
$888$	0	0
$889$	−30096.0	−0.0380807
$890$	0	0
$891$	15152.0i	0.0190860i
$892$	0	0
$893$	− 445536.i	− 0.558702i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	707616.i	0.879453i
$898$	0	0
$899$	115105.i	0.142421i
$900$	0	0
$901$	1.13800e6	1.40182
$902$	0	0
$903$	337459.	0.413852
$904$	0	0
$905$	0	0
$906$	0	0
$907$	916193.	1.11371	0.556855	−	0.830610i	$-0.312008\pi$
$907$	916193.	1.11371	0.556855	+	0.830610i	$0.312008\pi$
$908$	0	0
$909$	−148230.	−0.179394
$910$	0	0
$911$	995500.i	1.19951i	0.800183	+	0.599756i	$0.204736\pi$
$911$	995500.i	1.19951i	−0.800183	+	0.599756i	$0.795264\pi$
$912$	0	0
$913$	− 249264.i	− 0.299032i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26928.0i	0.0320233i
$918$	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.	−0.390744
$924$	0	0
$925$	0	0
$926$	0	0
$927$	153952.	0.179153
$928$	0	0
$929$	1.27882e6	1.48176	0.740881	−	0.671636i	$-0.234408\pi$
$929$	1.27882e6	1.48176	0.740881	+	0.671636i	$0.234408\pi$
$930$	0	0
$931$	− 401275.i	− 0.462959i
$932$	0	0
$933$	733320.i	0.842423i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 981262.i	− 1.11765i	−0.829286	−	0.558825i	$-0.811252\pi$
$937$	− 981262.i	− 1.11765i	0.829286	−	0.558825i	$-0.188748\pi$
$938$	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.	−0.772438
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−993109.	−1.10738	−0.553691	−	0.832722i	$-0.686781\pi$
$947$	−993109.	−1.10738	−0.553691	+	0.832722i	$0.686781\pi$
$948$	0	0
$949$	−168532.	−0.187133
$950$	0	0
$951$	− 384443.i	− 0.425080i
$952$	0	0
$953$	− 602922.i	− 0.663858i	−0.943304	−	0.331929i	$-0.892301\pi$
$953$	− 602922.i	− 0.663858i	0.943304	−	0.331929i	$-0.107699\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 8424.00i	− 0.00919802i
$958$	0	0
$959$	− 1.01009e6i	− 1.09831i
$960$	0	0
$961$	−1.25419e6	−1.35805
$962$	0	0
$963$	272174.	0.293491
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−575810.	−0.615781	−0.307890	−	0.951422i	$-0.599623\pi$
$967$	−575810.	−0.615781	−0.307890	+	0.951422i	$0.599623\pi$
$968$	0	0
$969$	−150552.	−0.160339
$970$	0	0
$971$	1.23920e6i	1.31432i	0.753749	+	0.657162i	$0.228243\pi$
$971$	1.23920e6i	1.31432i	−0.753749	+	0.657162i	$0.771757\pi$
$972$	0	0
$973$	1.00690e6i	1.06355i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.04074e6i	− 1.09032i	−0.838332	−	0.545160i	$-0.816469\pi$
$977$	− 1.04074e6i	− 1.09032i	0.838332	−	0.545160i	$-0.183531\pi$
$978$	0	0
$979$	− 240810.i	− 0.251252i
$980$	0	0
$981$	436482.	0.453553
$982$	0	0
$983$	−948734.	−0.981833	−0.490916	−	0.871207i	$-0.663338\pi$
$983$	−948734.	−0.981833	−0.490916	+	0.871207i	$0.663338\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.49799e6	1.53771
$988$	0	0
$989$	637632.	0.651895
$990$	0	0
$991$	− 616007.i	− 0.627247i	−0.949547	−	0.313623i	$-0.898457\pi$
$991$	− 616007.i	− 0.627247i	0.949547	−	0.313623i	$-0.101543\pi$
$992$	0	0
$993$	− 296316.i	− 0.300508i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 535870.i	− 0.539100i	−0.962986	−	0.269550i	$-0.913125\pi$
$997$	− 535870.i	− 0.539100i	0.962986	−	0.269550i	$-0.0868749\pi$
$998$	0	0
$999$	74356.9i	0.0745059i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.5.j.b.799.4		4
4.3	odd	2	inner	1200.5.j.b.799.1		4
5.2	odd	4		1200.5.e.b.751.1		2
5.3	odd	4		48.5.g.a.31.2	yes	2
5.4	even	2	inner	1200.5.j.b.799.2		4
15.8	even	4		144.5.g.f.127.1		2
20.3	even	4		48.5.g.a.31.1	✓	2
20.7	even	4		1200.5.e.b.751.2		2
20.19	odd	2	inner	1200.5.j.b.799.3		4
40.3	even	4		192.5.g.b.127.2		2
40.13	odd	4		192.5.g.b.127.1		2
60.23	odd	4		144.5.g.f.127.2		2
80.3	even	4		768.5.b.c.127.4		4
80.13	odd	4		768.5.b.c.127.2		4
80.43	even	4		768.5.b.c.127.1		4
80.53	odd	4		768.5.b.c.127.3		4
120.53	even	4		576.5.g.d.127.1		2
120.83	odd	4		576.5.g.d.127.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.g.a.31.1	✓	2	20.3	even	4
48.5.g.a.31.2	yes	2	5.3	odd	4
144.5.g.f.127.1		2	15.8	even	4
144.5.g.f.127.2		2	60.23	odd	4
192.5.g.b.127.1		2	40.13	odd	4
192.5.g.b.127.2		2	40.3	even	4
576.5.g.d.127.1		2	120.53	even	4
576.5.g.d.127.2		2	120.83	odd	4
768.5.b.c.127.1		4	80.43	even	4
768.5.b.c.127.2		4	80.13	odd	4
768.5.b.c.127.3		4	80.53	odd	4
768.5.b.c.127.4		4	80.3	even	4
1200.5.e.b.751.1		2	5.2	odd	4
1200.5.e.b.751.2		2	20.7	even	4
1200.5.j.b.799.1		4	4.3	odd	2	inner
1200.5.j.b.799.2		4	5.4	even	2	inner
1200.5.j.b.799.3		4	20.19	odd	2	inner
1200.5.j.b.799.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1200.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} +76.2102 q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} +76.2102 q^{7} +27.0000 q^{9} +20.7846i q^{11} +182.000i q^{13} -246.000i q^{17} -117.779i q^{19} +396.000 q^{21} +748.246 q^{23} +140.296 q^{27} -78.0000 q^{29} -1475.71i q^{31} +108.000i q^{33} +530.000i q^{37} +945.700i q^{39} -918.000 q^{41} +852.169 q^{43} +3782.80 q^{47} +3407.00 q^{49} -1278.25i q^{51} +4626.00i q^{53} -612.000i q^{57} -228.631i q^{59} +1346.00 q^{61} +2057.68 q^{63} +1087.73 q^{67} +3888.00 q^{69} +1829.05i q^{71} +926.000i q^{73} +1584.00i q^{77} -4399.41i q^{79} +729.000 q^{81} -11992.7 q^{83} -405.300 q^{87} -11586.0 q^{89} +13870.3i q^{91} -7668.00i q^{93} -13118.0i q^{97} +561.184i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{9}+O(q^{10}) \) 4 * q + 108 * q^9	\( 4 q + 108 q^{9} + 1584 q^{21} - 312 q^{29} - 3672 q^{41} + 13628 q^{49} + 5384 q^{61} + 15552 q^{69} + 2916 q^{81} - 46344 q^{89}+O(q^{100}) \) 4 * q + 108 * q^9 + 1584 * q^21 - 312 * q^29 - 3672 * q^41 + 13628 * q^49 + 5384 * q^61 + 15552 * q^69 + 2916 * q^81 - 46344 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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