LMFDB - Embedded newform 768.5.b.c.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.b (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:	$79.3881316484$
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			127.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.5.b.c.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615 q^{3} +42.0000i q^{5} +76.2102i q^{7} +27.0000 q^{9} +O(q^{10})$	$q-5.19615 q^{3} +42.0000i q^{5} +76.2102i q^{7} +27.0000 q^{9} +20.7846 q^{11} -182.000i q^{13} -218.238i q^{15} -246.000 q^{17} -117.779 q^{19} -396.000i q^{21} -748.246i q^{23} -1139.00 q^{25} -140.296 q^{27} +78.0000i q^{29} -1475.71i q^{31} -108.000 q^{33} -3200.83 q^{35} -530.000i q^{37} +945.700i q^{39} +918.000 q^{41} +852.169 q^{43} +1134.00i q^{45} -3782.80i q^{47} -3407.00 q^{49} +1278.25 q^{51} +4626.00i q^{53} +872.954i q^{55} +612.000 q^{57} +228.631 q^{59} +1346.00i q^{61} +2057.68i q^{63} +7644.00 q^{65} +1087.73 q^{67} +3888.00i q^{69} -1829.05i q^{71} +926.000 q^{73} +5918.42 q^{75} +1584.00i q^{77} +4399.41i q^{79} +729.000 q^{81} +11992.7 q^{83} -10332.0i q^{85} -405.300i q^{87} -11586.0 q^{89} +13870.3 q^{91} +7668.00i q^{93} -4946.74i q^{95} -13118.0 q^{97} +561.184 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 108 q^{9}+O(q^{10}) $ 4 * q + 108 * q^9	$ 4 q + 108 q^{9} - 984 q^{17} - 4556 q^{25} - 432 q^{33} + 3672 q^{41} - 13628 q^{49} + 2448 q^{57} + 30576 q^{65} + 3704 q^{73} + 2916 q^{81} - 46344 q^{89} - 52472 q^{97}+O(q^{100}) $ 4 * q + 108 * q^9 - 984 * q^17 - 4556 * q^25 - 432 * q^33 + 3672 * q^41 - 13628 * q^49 + 2448 * q^57 + 30576 * q^65 + 3704 * q^73 + 2916 * q^81 - 46344 * q^89 - 52472 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−5.19615	−0.577350
$3$	−5.19615	−0.577350
$4$	0	0
$5$	42.0000i	1.68000i	0.542586	+	0.840000i	$0.317445\pi$
$5$	42.0000i	1.68000i	−0.542586	+	0.840000i	$0.682555\pi$
$6$	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
$11$	20.7846	0.171774	0.0858868	−	0.996305i	$-0.472628\pi$
$11$	20.7846	0.171774	0.0858868	+	0.996305i	$0.472628\pi$
$12$	0	0
$13$	− 182.000i	− 1.07692i	−0.842650	−	0.538462i	$-0.819006\pi$
$13$	− 182.000i	− 1.07692i	0.842650	−	0.538462i	$-0.180994\pi$
$14$	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
$19$	−117.779	−0.326259	−0.163129	−	0.986605i	$-0.552159\pi$
$19$	−117.779	−0.326259	−0.163129	+	0.986605i	$0.552159\pi$
$20$	0	0
$21$	− 396.000i	− 0.897959i
$22$	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
$27$	−140.296	−0.192450
$28$	0	0
$29$	78.0000i	0.0927467i	0.998924	+	0.0463734i	$0.0147664\pi$
$29$	78.0000i	0.0927467i	−0.998924	+	0.0463734i	$0.985234\pi$
$30$	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
$33$	−108.000	−0.0991736
$34$	0	0
$35$	−3200.83	−2.61292
$36$	0	0
$37$	− 530.000i	− 0.387144i	−0.981086	−	0.193572i	$-0.937993\pi$
$37$	− 530.000i	− 0.387144i	0.981086	−	0.193572i	$-0.0620073\pi$
$38$	0	0
$39$	945.700i	0.621762i
$40$	0	0
$41$	918.000	0.546104	0.273052	−	0.961999i	$-0.411967\pi$
$41$	918.000	0.546104	0.273052	+	0.961999i	$0.411967\pi$
$42$	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
$45$	1134.00i	0.560000i
$46$	0	0
$47$	− 3782.80i	− 1.71245i	−0.516604	−	0.856224i	$-0.672804\pi$
$47$	− 3782.80i	− 1.71245i	0.516604	−	0.856224i	$-0.327196\pi$
$48$	0	0
$49$	−3407.00	−1.41899
$50$	0	0
$51$	1278.25	0.491447
$52$	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
$55$	872.954i	0.288580i
$56$	0	0
$57$	612.000	0.188366
$58$	0	0
$59$	228.631	0.0656796	0.0328398	−	0.999461i	$-0.489545\pi$
$59$	228.631	0.0656796	0.0328398	+	0.999461i	$0.489545\pi$
$60$	0	0
$61$	1346.00i	0.361731i	0.983508	+	0.180865i	$0.0578898\pi$
$61$	1346.00i	0.361731i	−0.983508	+	0.180865i	$0.942110\pi$
$62$	0	0
$63$	2057.68i	0.518437i
$64$	0	0
$65$	7644.00	1.80923
$66$	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
$69$	3888.00i	0.816635i
$70$	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	$-0.472309\pi$
$73$	926.000	0.173766	0.0868831	+	0.996219i	$0.472309\pi$
$74$	0	0
$75$	5918.42	1.05216
$76$	0	0
$77$	1584.00i	0.267161i
$78$	0	0
$79$	4399.41i	0.704921i	0.935827	+	0.352460i	$0.114655\pi$
$79$	4399.41i	0.704921i	−0.935827	+	0.352460i	$0.885345\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	11992.7	1.74085	0.870425	−	0.492301i	$-0.163844\pi$
$83$	11992.7	1.74085	0.870425	+	0.492301i	$0.163844\pi$
$84$	0	0
$85$	− 10332.0i	− 1.43003i
$86$	0	0
$87$	− 405.300i	− 0.0535473i
$88$	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
$91$	13870.3	1.67495
$92$	0	0
$93$	7668.00i	0.886576i
$94$	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
$99$	561.184	0.0572579
$100$	0	0
$101$	5490.00i	0.538183i	0.963115	+	0.269091i	$0.0867233\pi$
$101$	5490.00i	0.538183i	−0.963115	+	0.269091i	$0.913277\pi$
$102$	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
$107$	−10080.5	−0.880473	−0.440237	−	0.897882i	$-0.645105\pi$
$107$	−10080.5	−0.880473	−0.440237	+	0.897882i	$0.645105\pi$
$108$	0	0
$109$	− 16166.0i	− 1.36066i	−0.732906	−	0.680330i	$-0.761836\pi$
$109$	− 16166.0i	− 1.36066i	0.732906	−	0.680330i	$-0.238164\pi$
$110$	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
$115$	31426.3	2.37628
$116$	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
$125$	− 21588.0i	− 1.38163i
$126$	0	0
$127$	394.908i	0.0244843i	0.999925	+	0.0122422i	$0.00389690\pi$
$127$	394.908i	0.0244843i	−0.999925	+	0.0122422i	$0.996103\pi$
$128$	0	0
$129$	−4428.00	−0.266090
$130$	0	0
$131$	−353.338	−0.0205896	−0.0102948	−	0.999947i	$-0.503277\pi$
$131$	−353.338	−0.0205896	−0.0102948	+	0.999947i	$0.503277\pi$
$132$	0	0
$133$	− 8976.00i	− 0.507434i
$134$	0	0
$135$	− 5892.44i	− 0.323316i
$136$	0	0
$137$	13254.0	0.706164	0.353082	−	0.935592i	$-0.385134\pi$
$137$	13254.0	0.706164	0.353082	+	0.935592i	$0.385134\pi$
$138$	0	0
$139$	−13212.1	−0.683820	−0.341910	−	0.939733i	$-0.611074\pi$
$139$	−13212.1	−0.683820	−0.341910	+	0.939733i	$0.611074\pi$
$140$	0	0
$141$	19656.0i	0.988683i
$142$	0	0
$143$	− 3782.80i	− 0.184987i
$144$	0	0
$145$	−3276.00	−0.155815
$146$	0	0
$147$	17703.3	0.819255
$148$	0	0
$149$	− 438.000i	− 0.0197288i	−0.999951	−	0.00986442i	$-0.996860\pi$
$149$	− 438.000i	− 0.0197288i	0.999951	−	0.00986442i	$-0.00313999\pi$
$150$	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
$155$	61979.7	2.57980
$156$	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	$-0.739014\pi$
$163$	−36255.3	−1.36457	−0.682286	+	0.731086i	$0.739014\pi$
$164$	0	0
$165$	− 4536.00i	− 0.166612i
$166$	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
$171$	−3180.05	−0.108753
$172$	0	0
$173$	− 34410.0i	− 1.14972i	−0.818251	−	0.574861i	$-0.805056\pi$
$173$	− 34410.0i	− 1.14972i	0.818251	−	0.574861i	$-0.194944\pi$
$174$	0	0
$175$	− 86803.5i	− 2.83440i
$176$	0	0
$177$	−1188.00	−0.0379201
$178$	0	0
$179$	16856.3	0.526086	0.263043	−	0.964784i	$-0.415274\pi$
$179$	16856.3	0.526086	0.263043	+	0.964784i	$0.415274\pi$
$180$	0	0
$181$	− 15706.0i	− 0.479411i	−0.970846	−	0.239706i	$-0.922949\pi$
$181$	− 15706.0i	− 0.479411i	0.970846	−	0.239706i	$-0.0770510\pi$
$182$	0	0
$183$	− 6994.02i	− 0.208845i
$184$	0	0
$185$	22260.0	0.650402
$186$	0	0
$187$	−5113.01	−0.146216
$188$	0	0
$189$	− 10692.0i	− 0.299320i
$190$	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
$193$	−26782.0	−0.718999	−0.359500	−	0.933145i	$-0.617053\pi$
$193$	−26782.0	−0.718999	−0.359500	+	0.933145i	$0.617053\pi$
$194$	0	0
$195$	−39719.4	−1.04456
$196$	0	0
$197$	52482.0i	1.35232i	0.736757	+	0.676158i	$0.236356\pi$
$197$	52482.0i	1.35232i	−0.736757	+	0.676158i	$0.763644\pi$
$198$	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
$205$	38556.0i	0.917454i
$206$	0	0
$207$	− 20202.6i	− 0.471485i
$208$	0	0
$209$	−2448.00	−0.0560427
$210$	0	0
$211$	23895.4	0.536721	0.268361	−	0.963319i	$-0.413518\pi$
$211$	23895.4	0.536721	0.268361	+	0.963319i	$0.413518\pi$
$212$	0	0
$213$	9504.00i	0.209482i
$214$	0	0
$215$	35791.1i	0.774280i
$216$	0	0
$217$	112464.	2.38833
$218$	0	0
$219$	−4811.64	−0.100324
$220$	0	0
$221$	44772.0i	0.916689i
$222$	0	0
$223$	− 852.169i	− 0.0171363i	−0.999963	−	0.00856813i	$-0.997273\pi$
$223$	− 852.169i	− 0.0171363i	0.999963	−	0.00856813i	$-0.00272735\pi$
$224$	0	0
$225$	−30753.0	−0.607467
$226$	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
$229$	48470.0i	0.924277i	0.886808	+	0.462138i	$0.152918\pi$
$229$	48470.0i	0.924277i	−0.886808	+	0.462138i	$0.847082\pi$
$230$	0	0
$231$	− 8230.71i	− 0.154246i
$232$	0	0
$233$	−48738.0	−0.897751	−0.448875	−	0.893594i	$-0.648175\pi$
$233$	−48738.0	−0.897751	−0.448875	+	0.893594i	$0.648175\pi$
$234$	0	0
$235$	158878.	2.87691
$236$	0	0
$237$	− 22860.0i	− 0.406986i
$238$	0	0
$239$	− 71000.2i	− 1.24298i	−0.783422	−	0.621490i	$-0.786528\pi$
$239$	− 71000.2i	− 1.24298i	0.783422	−	0.621490i	$-0.213472\pi$
$240$	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
$245$	− 143094.i	− 2.38391i
$246$	0	0
$247$	21435.9i	0.351356i
$248$	0	0
$249$	−62316.0	−1.00508
$250$	0	0
$251$	91888.8	1.45853	0.729264	−	0.684232i	$-0.239862\pi$
$251$	91888.8	1.45853	0.729264	+	0.684232i	$0.239862\pi$
$252$	0	0
$253$	− 15552.0i	− 0.242966i
$254$	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
$259$	40391.4	0.602129
$260$	0	0
$261$	2106.00i	0.0309156i
$262$	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
$267$	60202.6	0.844487
$268$	0	0
$269$	− 71538.0i	− 0.988626i	−0.869284	−	0.494313i	$-0.835420\pi$
$269$	− 71538.0i	− 0.988626i	0.869284	−	0.494313i	$-0.164580\pi$
$270$	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
$273$	−72072.0	−0.967033
$274$	0	0
$275$	−23673.7	−0.313040
$276$	0	0
$277$	120518.i	1.57070i	0.619054	+	0.785348i	$0.287516\pi$
$277$	120518.i	1.57070i	−0.619054	+	0.785348i	$0.712484\pi$
$278$	0	0
$279$	− 39844.1i	− 0.511865i
$280$	0	0
$281$	3054.00	0.0386773	0.0193387	−	0.999813i	$-0.493844\pi$
$281$	3054.00	0.0386773	0.0193387	+	0.999813i	$0.493844\pi$
$282$	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
$285$	25704.0i	0.316454i
$286$	0	0
$287$	69961.0i	0.849361i
$288$	0	0
$289$	−23005.0	−0.275440
$290$	0	0
$291$	68163.1	0.804940
$292$	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
$295$	9602.49i	0.110342i
$296$	0	0
$297$	−2916.00	−0.0330579
$298$	0	0
$299$	−136181.	−1.52326
$300$	0	0
$301$	64944.0i	0.716813i
$302$	0	0
$303$	− 28526.9i	− 0.310720i
$304$	0	0
$305$	−56532.0	−0.607708
$306$	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
$309$	29628.0i	0.310303i
$310$	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	$-0.271923\pi$
$313$	128686.	1.31354	0.656769	+	0.754092i	$0.271923\pi$
$314$	0	0
$315$	−86422.4	−0.870974
$316$	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
$325$	207298.i	1.96258i
$326$	0	0
$327$	84001.0i	0.785577i
$328$	0	0
$329$	288288.	2.66339
$330$	0	0
$331$	−57026.0	−0.520496	−0.260248	−	0.965542i	$-0.583804\pi$
$331$	−57026.0	−0.520496	−0.260248	+	0.965542i	$0.583804\pi$
$332$	0	0
$333$	− 14310.0i	− 0.129048i
$334$	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
$339$	−9571.31	−0.0832860
$340$	0	0
$341$	− 30672.0i	− 0.263775i
$342$	0	0
$343$	− 76667.5i	− 0.651663i
$344$	0	0
$345$	−163296.	−1.37195
$346$	0	0
$347$	−56929.0	−0.472797	−0.236399	−	0.971656i	$-0.575967\pi$
$347$	−56929.0	−0.472797	−0.236399	+	0.971656i	$0.575967\pi$
$348$	0	0
$349$	181346.i	1.48887i	0.667694	+	0.744436i	$0.267281\pi$
$349$	181346.i	1.48887i	−0.667694	+	0.744436i	$0.732719\pi$
$350$	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
$355$	76819.9	0.609561
$356$	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
$365$	38892.0i	0.291927i
$366$	0	0
$367$	− 182690.i	− 1.35638i	−0.734885	−	0.678191i	$-0.762764\pi$
$367$	− 182690.i	− 1.35638i	0.734885	−	0.678191i	$-0.237236\pi$
$368$	0	0
$369$	24786.0	0.182035
$370$	0	0
$371$	−352549.	−2.56136
$372$	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
$375$	112175.i	0.797686i
$376$	0	0
$377$	14196.0	0.0998811
$378$	0	0
$379$	−36005.9	−0.250666	−0.125333	−	0.992115i	$-0.540000\pi$
$379$	−36005.9	−0.250666	−0.125333	+	0.992115i	$0.540000\pi$
$380$	0	0
$381$	− 2052.00i	− 0.0141360i
$382$	0	0
$383$	65346.8i	0.445479i	0.974878	+	0.222739i	$0.0714999\pi$
$383$	65346.8i	0.445479i	−0.974878	+	0.222739i	$0.928500\pi$
$384$	0	0
$385$	−66528.0	−0.448831
$386$	0	0
$387$	23008.6	0.153627
$388$	0	0
$389$	− 105750.i	− 0.698846i	−0.936965	−	0.349423i	$-0.886378\pi$
$389$	− 105750.i	− 0.698846i	0.936965	−	0.349423i	$-0.113622\pi$
$390$	0	0
$391$	184069.i	1.20400i
$392$	0	0
$393$	1836.00	0.0118874
$394$	0	0
$395$	−184775.	−1.18427
$396$	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
$405$	30618.0i	0.186667i
$406$	0	0
$407$	− 11015.8i	− 0.0665011i
$408$	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
$411$	−68869.8	−0.407704
$412$	0	0
$413$	17424.0i	0.102152i
$414$	0	0
$415$	503694.i	2.92463i
$416$	0	0
$417$	68652.0	0.394804
$418$	0	0
$419$	−24089.4	−0.137214	−0.0686068	−	0.997644i	$-0.521855\pi$
$419$	−24089.4	−0.137214	−0.0686068	+	0.997644i	$0.521855\pi$
$420$	0	0
$421$	− 116698.i	− 0.658414i	−0.944258	−	0.329207i	$-0.893219\pi$
$421$	− 116698.i	− 0.658414i	0.944258	−	0.329207i	$-0.106781\pi$
$422$	0	0
$423$	− 102136.i	− 0.570816i
$424$	0	0
$425$	280194.	1.55125
$426$	0	0
$427$	−102579.	−0.562604
$428$	0	0
$429$	19656.0i	0.106802i
$430$	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.	−1.06527	−0.532634	−	0.846346i	$-0.678798\pi$
$433$	−199726.	−1.06527	−0.532634	+	0.846346i	$0.678798\pi$
$434$	0	0
$435$	17022.6	0.0899595
$436$	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
$445$	− 486612.i	− 2.45733i
$446$	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
$451$	19080.3	0.0938062
$452$	0	0
$453$	145764.i	0.710320i
$454$	0	0
$455$	582551.i	2.81392i
$456$	0	0
$457$	−284338.	−1.36145	−0.680726	−	0.732538i	$-0.738336\pi$
$457$	−284338.	−1.36145	−0.680726	+	0.732538i	$0.738336\pi$
$458$	0	0
$459$	34512.8	0.163816
$460$	0	0
$461$	− 183402.i	− 0.862983i	−0.902117	−	0.431491i	$-0.857987\pi$
$461$	− 183402.i	− 0.862983i	0.902117	−	0.431491i	$-0.142013\pi$
$462$	0	0
$463$	− 172422.i	− 0.804324i	−0.915568	−	0.402162i	$-0.868259\pi$
$463$	− 172422.i	− 0.804324i	0.915568	−	0.402162i	$-0.131741\pi$
$464$	0	0
$465$	−322056.	−1.48945
$466$	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
$469$	82896.0i	0.376867i
$470$	0	0
$471$	− 100525.i	− 0.453139i
$472$	0	0
$473$	17712.0	0.0791672
$474$	0	0
$475$	134151.	0.594574
$476$	0	0
$477$	124902.i	0.548950i
$478$	0	0
$479$	249956.i	1.08941i	0.838627	+	0.544706i	$0.183359\pi$
$479$	249956.i	1.08941i	−0.838627	+	0.544706i	$0.816641\pi$
$480$	0	0
$481$	−96460.0	−0.416924
$482$	0	0
$483$	−296305.	−1.27012
$484$	0	0
$485$	− 550956.i	− 2.34225i
$486$	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.	0.945166	0.472583	−	0.881286i	$-0.343322\pi$
$491$	227862.	0.945166	0.472583	+	0.881286i	$0.343322\pi$
$492$	0	0
$493$	− 19188.0i	− 0.0789470i
$494$	0	0
$495$	23569.7i	0.0961932i
$496$	0	0
$497$	139392.	0.564320
$498$	0	0
$499$	−248854.	−0.999410	−0.499705	−	0.866196i	$-0.666558\pi$
$499$	−248854.	−0.999410	−0.499705	+	0.866196i	$0.666558\pi$
$500$	0	0
$501$	97416.0i	0.388110i
$502$	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
$507$	23710.0	0.0922394
$508$	0	0
$509$	− 39330.0i	− 0.151806i	−0.997115	−	0.0759029i	$-0.975816\pi$
$509$	− 39330.0i	− 0.151806i	0.997115	−	0.0759029i	$-0.0241839\pi$
$510$	0	0
$511$	70570.7i	0.270260i
$512$	0	0
$513$	16524.0	0.0627886
$514$	0	0
$515$	239480.	0.902933
$516$	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	$-0.173066\pi$
$521$	464598.	1.71160	0.855799	+	0.517308i	$0.173066\pi$
$522$	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
$525$	451044.i	1.63644i
$526$	0	0
$527$	363024.i	1.30712i
$528$	0	0
$529$	−280031.	−1.00068
$530$	0	0
$531$	6173.03	0.0218932
$532$	0	0
$533$	− 167076.i	− 0.588111i
$534$	0	0
$535$	− 423382.i	− 1.47919i
$536$	0	0
$537$	−87588.0	−0.303736
$538$	0	0
$539$	−70813.2	−0.243745
$540$	0	0
$541$	360442.i	1.23152i	0.787934	+	0.615759i	$0.211151\pi$
$541$	360442.i	1.23152i	−0.787934	+	0.615759i	$0.788849\pi$
$542$	0	0
$543$	81610.8i	0.276788i
$544$	0	0
$545$	678972.	2.28591
$546$	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
$549$	36342.0i	0.120577i
$550$	0	0
$551$	− 9186.80i	− 0.0302594i
$552$	0	0
$553$	−335280.	−1.09637
$554$	0	0
$555$	−115666.	−0.375510
$556$	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	$-0.269683\pi$
$563$	419704.	1.32412	0.662058	+	0.749453i	$0.269683\pi$
$564$	0	0
$565$	77364.0i	0.242349i
$566$	0	0
$567$	55557.3i	0.172812i
$568$	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.	−0.982640	−0.491320	−	0.870979i	$-0.663485\pi$
$571$	−320381.	−0.982640	−0.491320	+	0.870979i	$0.663485\pi$
$572$	0	0
$573$	13824.0i	0.0421041i
$574$	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.	0.415114
$580$	0	0
$581$	913968.i	2.70756i
$582$	0	0
$583$	96149.6i	0.282885i
$584$	0	0
$585$	206388.	0.603077
$586$	0	0
$587$	114128.	0.331220	0.165610	−	0.986191i	$-0.447041\pi$
$587$	114128.	0.331220	0.165610	+	0.986191i	$0.447041\pi$
$588$	0	0
$589$	173808.i	0.501002i
$590$	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.	2.22415
$596$	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	$-0.164786\pi$
$601$	627742.	1.73793	0.868965	+	0.494874i	$0.164786\pi$
$602$	0	0
$603$	29368.7	0.0807699
$604$	0	0
$605$	− 596778.i	− 1.63043i
$606$	0	0
$607$	133195.i	0.361501i	0.983529	+	0.180751i	$0.0578527\pi$
$607$	133195.i	0.361501i	−0.983529	+	0.180751i	$0.942147\pi$
$608$	0	0
$609$	30888.0	0.0832828
$610$	0	0
$611$	−688469.	−1.84418
$612$	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
$615$	− 200343.i	− 0.529692i
$616$	0	0
$617$	31758.0	0.0834224	0.0417112	−	0.999130i	$-0.486719\pi$
$617$	31758.0	0.0834224	0.0417112	+	0.999130i	$0.486719\pi$
$618$	0	0
$619$	−656094.	−1.71232	−0.856160	−	0.516712i	$-0.827156\pi$
$619$	−656094.	−1.71232	−0.856160	+	0.516712i	$0.827156\pi$
$620$	0	0
$621$	104976.i	0.272212i
$622$	0	0
$623$	− 882972.i	− 2.27494i
$624$	0	0
$625$	194821.	0.498742
$626$	0	0
$627$	12720.2	0.0323563
$628$	0	0
$629$	130380.i	0.329541i
$630$	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
$635$	−16586.1	−0.0411337
$636$	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	$-0.831818\pi$
$643$	−714138.	−1.72727	−0.863635	+	0.504117i	$0.831818\pi$
$644$	0	0
$645$	− 185976.i	− 0.447031i
$646$	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.	−1.37890
$652$	0	0
$653$	− 330714.i	− 0.775579i	−0.921748	−	0.387790i	$-0.873239\pi$
$653$	− 330714.i	− 0.775579i	0.921748	−	0.387790i	$-0.126761\pi$
$654$	0	0
$655$	− 14840.2i	− 0.0345906i
$656$	0	0
$657$	25002.0	0.0579221
$658$	0	0
$659$	253884.	0.584608	0.292304	−	0.956326i	$-0.405578\pi$
$659$	253884.	0.584608	0.292304	+	0.956326i	$0.405578\pi$
$660$	0	0
$661$	722158.i	1.65283i	0.563058	+	0.826417i	$0.309625\pi$
$661$	722158.i	1.65283i	−0.563058	+	0.826417i	$0.690375\pi$
$662$	0	0
$663$	− 232642.i	− 0.529251i
$664$	0	0
$665$	376992.	0.852489
$666$	0	0
$667$	58363.2	0.131186
$668$	0	0
$669$	4428.00i	0.00989362i
$670$	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.	0.350721
$676$	0	0
$677$	− 609030.i	− 1.32881i	−0.747375	−	0.664403i	$-0.768686\pi$
$677$	− 609030.i	− 1.32881i	0.747375	−	0.664403i	$-0.231314\pi$
$678$	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
$685$	556668.i	1.18636i
$686$	0	0
$687$	− 251858.i	− 0.533631i
$688$	0	0
$689$	841932.	1.77353
$690$	0	0
$691$	431842.	0.904417	0.452208	−	0.891912i	$-0.350636\pi$
$691$	431842.	0.904417	0.452208	+	0.891912i	$0.350636\pi$
$692$	0	0
$693$	42768.0i	0.0890538i
$694$	0	0
$695$	− 554908.i	− 1.14882i
$696$	0	0
$697$	−225828.	−0.464849
$698$	0	0
$699$	253250.	0.518317
$700$	0	0
$701$	44958.0i	0.0914894i	0.998953	+	0.0457447i	$0.0145661\pi$
$701$	44958.0i	0.0914894i	−0.998953	+	0.0457447i	$0.985434\pi$
$702$	0	0
$703$	62423.1i	0.126309i
$704$	0	0
$705$	−825552.	−1.66099
$706$	0	0
$707$	−418394.	−0.837041
$708$	0	0
$709$	− 533002.i	− 1.06032i	−0.847898	−	0.530159i	$-0.822132\pi$
$709$	− 533002.i	− 1.06032i	0.847898	−	0.530159i	$-0.177868\pi$
$710$	0	0
$711$	118784.i	0.234974i
$712$	0	0
$713$	−1.10419e6	−2.17203
$714$	0	0
$715$	158878.	0.310778
$716$	0	0
$717$	368928.i	0.717634i
$718$	0	0
$719$	− 292107.i	− 0.565046i	−0.959260	−	0.282523i	$-0.908829\pi$
$719$	− 292107.i	− 0.565046i	0.959260	−	0.282523i	$-0.0911714\pi$
$720$	0	0
$721$	434544.	0.835917
$722$	0	0
$723$	−380036.	−0.727023
$724$	0	0
$725$	− 88842.0i	− 0.169022i
$726$	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.	−0.392307
$732$	0	0
$733$	− 832982.i	− 1.55034i	−0.631751	−	0.775171i	$-0.717664\pi$
$733$	− 832982.i	− 1.55034i	0.631751	−	0.775171i	$-0.282336\pi$
$734$	0	0
$735$	743538.i	1.37635i
$736$	0	0
$737$	22608.0	0.0416224
$738$	0	0
$739$	698093.	1.27827	0.639137	−	0.769093i	$-0.279292\pi$
$739$	698093.	1.27827	0.639137	+	0.769093i	$0.279292\pi$
$740$	0	0
$741$	− 111384.i	− 0.202855i
$742$	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.	0.580284
$748$	0	0
$749$	− 768240.i	− 1.36941i
$750$	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.	−0.842082
$754$	0	0
$755$	1.17820e6	2.06692
$756$	0	0
$757$	− 295786.i	− 0.516162i	−0.966123	−	0.258081i	$-0.916910\pi$
$757$	− 295786.i	− 0.516162i	0.966123	−	0.258081i	$-0.0830901\pi$
$758$	0	0
$759$	80810.6i	0.140276i
$760$	0	0
$761$	1.02615e6	1.77191	0.885955	−	0.463772i	$-0.153504\pi$
$761$	1.02615e6	1.77191	0.885955	+	0.463772i	$0.153504\pi$
$762$	0	0
$763$	1.23201e6	2.11625
$764$	0	0
$765$	− 278964.i	− 0.476678i
$766$	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.	0.427394
$772$	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
$775$	1.68083e6i	2.79847i
$776$	0	0
$777$	−209880.	−0.347639
$778$	0	0
$779$	−108122.	−0.178171
$780$	0	0
$781$	− 38016.0i	− 0.0623253i
$782$	0	0
$783$	− 10943.1i	− 0.0178491i
$784$	0	0
$785$	−812532.	−1.31856
$786$	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.i	− 0.652662i
$790$	0	0
$791$	140379.i	0.224362i
$792$	0	0
$793$	244972.	0.389556
$794$	0	0
$795$	1.00957e6	1.59736
$796$	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
$805$	2.39501e6i	3.69586i
$806$	0	0
$807$	371722.i	0.570784i
$808$	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.	1.39907	0.699534	−	0.714599i	$-0.253391\pi$
$811$	920197.	1.39907	0.699534	+	0.714599i	$0.253391\pi$
$812$	0	0
$813$	− 562212.i	− 0.850588i
$814$	0	0
$815$	− 1.52272e6i	− 2.29248i
$816$	0	0
$817$	−100368.	−0.150367
$818$	0	0
$819$	374497.	0.558317
$820$	0	0
$821$	250242.i	0.371256i	0.982620	+	0.185628i	$0.0594320\pi$
$821$	250242.i	0.371256i	−0.982620	+	0.185628i	$0.940568\pi$
$822$	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
$827$	17272.0	0.0252541	0.0126270	−	0.999920i	$-0.495981\pi$
$827$	17272.0	0.0252541	0.0126270	+	0.999920i	$0.495981\pi$
$828$	0	0
$829$	− 15686.0i	− 0.0228246i	−0.999935	−	0.0114123i	$-0.996367\pi$
$829$	− 15686.0i	− 0.0228246i	0.999935	−	0.0114123i	$-0.00363273\pi$
$830$	0	0
$831$	− 626230.i	− 0.906842i
$832$	0	0
$833$	838122.	1.20786
$834$	0	0
$835$	787404.	1.12934
$836$	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
$845$	− 191646.i	− 0.268402i
$846$	0	0
$847$	− 1.08287e6i	− 1.50942i
$848$	0	0
$849$	690876.	0.958484
$850$	0	0
$851$	−396570.	−0.547597
$852$	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
$855$	− 133562.i	− 0.182705i
$856$	0	0
$857$	267990.	0.364886	0.182443	−	0.983216i	$-0.441600\pi$
$857$	267990.	0.364886	0.182443	+	0.983216i	$0.441600\pi$
$858$	0	0
$859$	522407.	0.707983	0.353992	−	0.935249i	$-0.384824\pi$
$859$	522407.	0.707983	0.353992	+	0.935249i	$0.384824\pi$
$860$	0	0
$861$	− 363528.i	− 0.490379i
$862$	0	0
$863$	− 826895.i	− 1.11027i	−0.831760	−	0.555135i	$-0.812667\pi$
$863$	− 826895.i	− 1.11027i	0.831760	−	0.555135i	$-0.187333\pi$
$864$	0	0
$865$	1.44522e6	1.93153
$866$	0	0
$867$	119537.	0.159025
$868$	0	0
$869$	91440.0i	0.121087i
$870$	0	0
$871$	− 197966.i	− 0.260949i
$872$	0	0
$873$	−354186.	−0.464732
$874$	0	0
$875$	1.64523e6	2.14887
$876$	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	$-0.381127\pi$
$883$	568909.	0.729662	0.364831	+	0.931074i	$0.381127\pi$
$884$	0	0
$885$	− 49896.0i	− 0.0637058i
$886$	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
$891$	15152.0	0.0190860
$892$	0	0
$893$	445536.i	0.558702i
$894$	0	0
$895$	707965.i	0.883824i
$896$	0	0
$897$	707616.	0.879453
$898$	0	0
$899$	115105.	0.142421
$900$	0	0
$901$	− 1.13800e6i	− 1.40182i
$902$	0	0
$903$	− 337459.i	− 0.413852i
$904$	0	0
$905$	659652.	0.805411
$906$	0	0
$907$	−916193.	−1.11371	−0.556855	−	0.830610i	$-0.687992\pi$
$907$	−916193.	−1.11371	−0.556855	+	0.830610i	$0.687992\pi$
$908$	0	0
$909$	148230.i	0.179394i
$910$	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.	0.299032
$914$	0	0
$915$	293749.	0.350860
$916$	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
$925$	603670.i	0.705531i
$926$	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.	0.462959
$932$	0	0
$933$	733320.i	0.842423i
$934$	0	0
$935$	− 214747.i	− 0.245642i
$936$	0	0
$937$	981262.	1.11765	0.558825	−	0.829286i	$-0.311252\pi$
$937$	981262.	1.11765	0.558825	+	0.829286i	$0.311252\pi$
$938$	0	0
$939$	−668672.	−0.758371
$940$	0	0
$941$	284406.i	0.321188i	0.987021	+	0.160594i	$0.0513410\pi$
$941$	284406.i	0.321188i	−0.987021	+	0.160594i	$0.948659\pi$
$942$	0	0
$943$	− 686890.i	− 0.772438i
$944$	0	0
$945$	449064.	0.502857
$946$	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.i	− 0.187133i
$950$	0	0
$951$	384443.i	0.425080i
$952$	0	0
$953$	−602922.	−0.663858	−0.331929	−	0.943304i	$-0.607699\pi$
$953$	−602922.	−0.663858	−0.331929	+	0.943304i	$0.607699\pi$
$954$	0	0
$955$	111738.	0.122516
$956$	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
$965$	− 1.12484e6i	− 1.20792i
$966$	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
$971$	1.23920e6	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$971$	1.23920e6	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$972$	0	0
$973$	− 1.00690e6i	− 1.06355i
$974$	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.	−0.251252
$980$	0	0
$981$	− 436482.i	− 0.453553i
$982$	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
$987$	−1.49799e6	−1.53771
$988$	0	0
$989$	− 637632.i	− 0.651895i
$990$	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.	0.300508
$994$	0	0
$995$	969269.	0.979035
$996$	0	0
$997$	535870.i	0.539100i	0.962986	+	0.269550i	$0.0868749\pi$
$997$	535870.i	0.539100i	−0.962986	+	0.269550i	$0.913125\pi$
$998$	0	0
$999$	74356.9i	0.0745059i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.b.c.127.2		4
4.3	odd	2	inner	768.5.b.c.127.4		4
8.3	odd	2	inner	768.5.b.c.127.1		4
8.5	even	2	inner	768.5.b.c.127.3		4
16.3	odd	4		192.5.g.b.127.2		2
16.5	even	4		48.5.g.a.31.2	yes	2
16.11	odd	4		48.5.g.a.31.1	✓	2
16.13	even	4		192.5.g.b.127.1		2
48.5	odd	4		144.5.g.f.127.1		2
48.11	even	4		144.5.g.f.127.2		2
48.29	odd	4		576.5.g.d.127.1		2
48.35	even	4		576.5.g.d.127.2		2
80.27	even	4		1200.5.j.b.799.1		4
80.37	odd	4		1200.5.j.b.799.4		4
80.43	even	4		1200.5.j.b.799.3		4
80.53	odd	4		1200.5.j.b.799.2		4
80.59	odd	4		1200.5.e.b.751.2		2
80.69	even	4		1200.5.e.b.751.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.5.g.a.31.1	✓	2	16.11	odd	4
48.5.g.a.31.2	yes	2	16.5	even	4
144.5.g.f.127.1		2	48.5	odd	4
144.5.g.f.127.2		2	48.11	even	4
192.5.g.b.127.1		2	16.13	even	4
192.5.g.b.127.2		2	16.3	odd	4
576.5.g.d.127.1		2	48.29	odd	4
576.5.g.d.127.2		2	48.35	even	4
768.5.b.c.127.1		4	8.3	odd	2	inner
768.5.b.c.127.2		4	1.1	even	1	trivial
768.5.b.c.127.3		4	8.5	even	2	inner
768.5.b.c.127.4		4	4.3	odd	2	inner
1200.5.e.b.751.1		2	80.69	even	4
1200.5.e.b.751.2		2	80.59	odd	4
1200.5.j.b.799.1		4	80.27	even	4
1200.5.j.b.799.2		4	80.53	odd	4
1200.5.j.b.799.3		4	80.43	even	4
1200.5.j.b.799.4		4	80.37	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.19615 q^{3} +42.0000i q^{5} +76.2102i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q-5.19615 q^{3} +42.0000i q^{5} +76.2102i q^{7} +27.0000 q^{9} +20.7846 q^{11} -182.000i q^{13} -218.238i q^{15} -246.000 q^{17} -117.779 q^{19} -396.000i q^{21} -748.246i q^{23} -1139.00 q^{25} -140.296 q^{27} +78.0000i q^{29} -1475.71i q^{31} -108.000 q^{33} -3200.83 q^{35} -530.000i q^{37} +945.700i q^{39} +918.000 q^{41} +852.169 q^{43} +1134.00i q^{45} -3782.80i q^{47} -3407.00 q^{49} +1278.25 q^{51} +4626.00i q^{53} +872.954i q^{55} +612.000 q^{57} +228.631 q^{59} +1346.00i q^{61} +2057.68i q^{63} +7644.00 q^{65} +1087.73 q^{67} +3888.00i q^{69} -1829.05i q^{71} +926.000 q^{73} +5918.42 q^{75} +1584.00i q^{77} +4399.41i q^{79} +729.000 q^{81} +11992.7 q^{83} -10332.0i q^{85} -405.300i q^{87} -11586.0 q^{89} +13870.3 q^{91} +7668.00i q^{93} -4946.74i q^{95} -13118.0 q^{97} +561.184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 108 q^{9}+O(q^{10}) \) 4 * q + 108 * q^9	\( 4 q + 108 q^{9} - 984 q^{17} - 4556 q^{25} - 432 q^{33} + 3672 q^{41} - 13628 q^{49} + 2448 q^{57} + 30576 q^{65} + 3704 q^{73} + 2916 q^{81} - 46344 q^{89} - 52472 q^{97}+O(q^{100}) \) 4 * q + 108 * q^9 - 984 * q^17 - 4556 * q^25 - 432 * q^33 + 3672 * q^41 - 13628 * q^49 + 2448 * q^57 + 30576 * q^65 + 3704 * q^73 + 2916 * q^81 - 46344 * q^89 - 52472 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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