LMFDB - Embedded newform 768.6.d.m.385.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,6,Mod(385,768)]

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([0, 1, 0]))

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

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

chi := DirichletCharacter("768.385");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.d (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:	$123.174773616$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			385.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	768.385
Dual form			768.6.d.m.385.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} -38.0000i q^{5} +120.000 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} -38.0000i q^{5} +120.000 q^{7} -81.0000 q^{9} +524.000i q^{11} -962.000i q^{13} -342.000 q^{15} -1358.00 q^{17} +2284.00i q^{19} -1080.00i q^{21} +2552.00 q^{23} +1681.00 q^{25} +729.000i q^{27} +3966.00i q^{29} +2992.00 q^{31} +4716.00 q^{33} -4560.00i q^{35} -13206.0i q^{37} -8658.00 q^{39} +15126.0 q^{41} -7316.00i q^{43} +3078.00i q^{45} +6960.00 q^{47} -2407.00 q^{49} +12222.0i q^{51} +17482.0i q^{53} +19912.0 q^{55} +20556.0 q^{57} +33884.0i q^{59} +39118.0i q^{61} -9720.00 q^{63} -36556.0 q^{65} -32996.0i q^{67} -22968.0i q^{69} +14248.0 q^{71} +35990.0 q^{73} -15129.0i q^{75} +62880.0i q^{77} +29888.0 q^{79} +6561.00 q^{81} +51884.0i q^{83} +51604.0i q^{85} +35694.0 q^{87} -30714.0 q^{89} -115440. i q^{91} -26928.0i q^{93} +86792.0 q^{95} -48478.0 q^{97} -42444.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 240 q^{7} - 162 q^{9}+O(q^{10}) $ 2 * q + 240 * q^7 - 162 * q^9	$ 2 q + 240 q^{7} - 162 q^{9} - 684 q^{15} - 2716 q^{17} + 5104 q^{23} + 3362 q^{25} + 5984 q^{31} + 9432 q^{33} - 17316 q^{39} + 30252 q^{41} + 13920 q^{47} - 4814 q^{49} + 39824 q^{55} + 41112 q^{57} - 19440 q^{63} - 73112 q^{65} + 28496 q^{71} + 71980 q^{73} + 59776 q^{79} + 13122 q^{81} + 71388 q^{87} - 61428 q^{89} + 173584 q^{95} - 96956 q^{97}+O(q^{100}) $ 2 * q + 240 * q^7 - 162 * q^9 - 684 * q^15 - 2716 * q^17 + 5104 * q^23 + 3362 * q^25 + 5984 * q^31 + 9432 * q^33 - 17316 * q^39 + 30252 * q^41 + 13920 * q^47 - 4814 * q^49 + 39824 * q^55 + 41112 * q^57 - 19440 * q^63 - 73112 * q^65 + 28496 * q^71 + 71980 * q^73 + 59776 * q^79 + 13122 * q^81 + 71388 * q^87 - 61428 * q^89 + 173584 * q^95 - 96956 * 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
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	0	0
$5$	− 38.0000i	− 0.679765i	−0.940468	−	0.339882i	$-0.889613\pi$
$5$	− 38.0000i	− 0.679765i	0.940468	−	0.339882i	$-0.110387\pi$
$6$	0	0
$7$	120.000	0.925627	0.462814	−	0.886456i	$-0.346840\pi$
$7$	120.000	0.925627	0.462814	+	0.886456i	$0.346840\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	524.000i	1.30572i	0.757479	+	0.652859i	$0.226431\pi$
$11$	524.000i	1.30572i	−0.757479	+	0.652859i	$0.773569\pi$
$12$	0	0
$13$	− 962.000i	− 1.57876i	−0.613904	−	0.789381i	$-0.710402\pi$
$13$	− 962.000i	− 1.57876i	0.613904	−	0.789381i	$-0.289598\pi$
$14$	0	0
$15$	−342.000	−0.392462
$16$	0	0
$17$	−1358.00	−1.13967	−0.569833	−	0.821761i	$-0.692992\pi$
$17$	−1358.00	−1.13967	−0.569833	+	0.821761i	$0.692992\pi$
$18$	0	0
$19$	2284.00i	1.45148i	0.687967	+	0.725742i	$0.258503\pi$
$19$	2284.00i	1.45148i	−0.687967	+	0.725742i	$0.741497\pi$
$20$	0	0
$21$	− 1080.00i	− 0.534411i
$22$	0	0
$23$	2552.00	1.00591	0.502957	−	0.864311i	$-0.332245\pi$
$23$	2552.00	1.00591	0.502957	+	0.864311i	$0.332245\pi$
$24$	0	0
$25$	1681.00	0.537920
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	3966.00i	0.875705i	0.899047	+	0.437852i	$0.144261\pi$
$29$	3966.00i	0.875705i	−0.899047	+	0.437852i	$0.855739\pi$
$30$	0	0
$31$	2992.00	0.559187	0.279594	−	0.960118i	$-0.409800\pi$
$31$	2992.00	0.559187	0.279594	+	0.960118i	$0.409800\pi$
$32$	0	0
$33$	4716.00	0.753857
$34$	0	0
$35$	− 4560.00i	− 0.629209i
$36$	0	0
$37$	− 13206.0i	− 1.58587i	−0.609308	−	0.792934i	$-0.708553\pi$
$37$	− 13206.0i	− 1.58587i	0.609308	−	0.792934i	$-0.291447\pi$
$38$	0	0
$39$	−8658.00	−0.911499
$40$	0	0
$41$	15126.0	1.40529	0.702643	−	0.711543i	$-0.252003\pi$
$41$	15126.0	1.40529	0.702643	+	0.711543i	$0.252003\pi$
$42$	0	0
$43$	− 7316.00i	− 0.603396i	−0.953404	−	0.301698i	$-0.902447\pi$
$43$	− 7316.00i	− 0.603396i	0.953404	−	0.301698i	$-0.0975535\pi$
$44$	0	0
$45$	3078.00i	0.226588i
$46$	0	0
$47$	6960.00	0.459584	0.229792	−	0.973240i	$-0.426195\pi$
$47$	6960.00	0.459584	0.229792	+	0.973240i	$0.426195\pi$
$48$	0	0
$49$	−2407.00	−0.143214
$50$	0	0
$51$	12222.0i	0.657986i
$52$	0	0
$53$	17482.0i	0.854873i	0.904045	+	0.427436i	$0.140583\pi$
$53$	17482.0i	0.854873i	−0.904045	+	0.427436i	$0.859417\pi$
$54$	0	0
$55$	19912.0	0.887581
$56$	0	0
$57$	20556.0	0.838014
$58$	0	0
$59$	33884.0i	1.26726i	0.773638	+	0.633628i	$0.218435\pi$
$59$	33884.0i	1.26726i	−0.773638	+	0.633628i	$0.781565\pi$
$60$	0	0
$61$	39118.0i	1.34602i	0.739633	+	0.673011i	$0.234999\pi$
$61$	39118.0i	1.34602i	−0.739633	+	0.673011i	$0.765001\pi$
$62$	0	0
$63$	−9720.00	−0.308542
$64$	0	0
$65$	−36556.0	−1.07319
$66$	0	0
$67$	− 32996.0i	− 0.897996i	−0.893533	−	0.448998i	$-0.851781\pi$
$67$	− 32996.0i	− 0.897996i	0.893533	−	0.448998i	$-0.148219\pi$
$68$	0	0
$69$	− 22968.0i	− 0.580765i
$70$	0	0
$71$	14248.0	0.335435	0.167717	−	0.985835i	$-0.446360\pi$
$71$	14248.0	0.335435	0.167717	+	0.985835i	$0.446360\pi$
$72$	0	0
$73$	35990.0	0.790451	0.395225	−	0.918584i	$-0.370667\pi$
$73$	35990.0	0.790451	0.395225	+	0.918584i	$0.370667\pi$
$74$	0	0
$75$	− 15129.0i	− 0.310568i
$76$	0	0
$77$	62880.0i	1.20861i
$78$	0	0
$79$	29888.0	0.538802	0.269401	−	0.963028i	$-0.413174\pi$
$79$	29888.0	0.538802	0.269401	+	0.963028i	$0.413174\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	51884.0i	0.826682i	0.910576	+	0.413341i	$0.135638\pi$
$83$	51884.0i	0.826682i	−0.910576	+	0.413341i	$0.864362\pi$
$84$	0	0
$85$	51604.0i	0.774704i
$86$	0	0
$87$	35694.0	0.505588
$88$	0	0
$89$	−30714.0	−0.411018	−0.205509	−	0.978655i	$-0.565885\pi$
$89$	−30714.0	−0.411018	−0.205509	+	0.978655i	$0.565885\pi$
$90$	0	0
$91$	− 115440.i	− 1.46135i
$92$	0	0
$93$	− 26928.0i	− 0.322847i
$94$	0	0
$95$	86792.0	0.986667
$96$	0	0
$97$	−48478.0	−0.523137	−0.261568	−	0.965185i	$-0.584240\pi$
$97$	−48478.0	−0.523137	−0.261568	+	0.965185i	$0.584240\pi$
$98$	0	0
$99$	− 42444.0i	− 0.435240i
$100$	0	0
$101$	− 93222.0i	− 0.909316i	−0.890666	−	0.454658i	$-0.849761\pi$
$101$	− 93222.0i	− 0.909316i	0.890666	−	0.454658i	$-0.150239\pi$
$102$	0	0
$103$	2296.00	0.0213245	0.0106622	−	0.999943i	$-0.496606\pi$
$103$	2296.00	0.0213245	0.0106622	+	0.999943i	$0.496606\pi$
$104$	0	0
$105$	−41040.0	−0.363274
$106$	0	0
$107$	38988.0i	0.329209i	0.986360	+	0.164604i	$0.0526348\pi$
$107$	38988.0i	0.329209i	−0.986360	+	0.164604i	$0.947365\pi$
$108$	0	0
$109$	6238.00i	0.0502897i	0.999684	+	0.0251449i	$0.00800470\pi$
$109$	6238.00i	0.0502897i	−0.999684	+	0.0251449i	$0.991995\pi$
$110$	0	0
$111$	−118854.	−0.915601
$112$	0	0
$113$	213618.	1.57377	0.786886	−	0.617099i	$-0.211692\pi$
$113$	213618.	1.57377	0.786886	+	0.617099i	$0.211692\pi$
$114$	0	0
$115$	− 96976.0i	− 0.683785i
$116$	0	0
$117$	77922.0i	0.526254i
$118$	0	0
$119$	−162960.	−1.05491
$120$	0	0
$121$	−113525.	−0.704901
$122$	0	0
$123$	− 136134.i	− 0.811342i
$124$	0	0
$125$	− 182628.i	− 1.04542i
$126$	0	0
$127$	−205072.	−1.12823	−0.564114	−	0.825697i	$-0.690782\pi$
$127$	−205072.	−1.12823	−0.564114	+	0.825697i	$0.690782\pi$
$128$	0	0
$129$	−65844.0	−0.348371
$130$	0	0
$131$	− 350116.i	− 1.78252i	−0.453495	−	0.891259i	$-0.649823\pi$
$131$	− 350116.i	− 1.78252i	0.453495	−	0.891259i	$-0.350177\pi$
$132$	0	0
$133$	274080.i	1.34353i
$134$	0	0
$135$	27702.0	0.130821
$136$	0	0
$137$	234486.	1.06737	0.533686	−	0.845683i	$-0.320807\pi$
$137$	234486.	1.06737	0.533686	+	0.845683i	$0.320807\pi$
$138$	0	0
$139$	16428.0i	0.0721187i	0.999350	+	0.0360593i	$0.0114805\pi$
$139$	16428.0i	0.0721187i	−0.999350	+	0.0360593i	$0.988519\pi$
$140$	0	0
$141$	− 62640.0i	− 0.265341i
$142$	0	0
$143$	504088.	2.06142
$144$	0	0
$145$	150708.	0.595273
$146$	0	0
$147$	21663.0i	0.0826847i
$148$	0	0
$149$	92426.0i	0.341058i	0.985353	+	0.170529i	$0.0545477\pi$
$149$	92426.0i	0.341058i	−0.985353	+	0.170529i	$0.945452\pi$
$150$	0	0
$151$	350984.	1.25269	0.626347	−	0.779544i	$-0.284549\pi$
$151$	350984.	1.25269	0.626347	+	0.779544i	$0.284549\pi$
$152$	0	0
$153$	109998.	0.379889
$154$	0	0
$155$	− 113696.i	− 0.380116i
$156$	0	0
$157$	46318.0i	0.149969i	0.997185	+	0.0749844i	$0.0238907\pi$
$157$	46318.0i	0.149969i	−0.997185	+	0.0749844i	$0.976109\pi$
$158$	0	0
$159$	157338.	0.493561
$160$	0	0
$161$	306240.	0.931102
$162$	0	0
$163$	394908.i	1.16420i	0.813118	+	0.582099i	$0.197768\pi$
$163$	394908.i	1.16420i	−0.813118	+	0.582099i	$0.802232\pi$
$164$	0	0
$165$	− 179208.i	− 0.512445i
$166$	0	0
$167$	514344.	1.42713	0.713563	−	0.700591i	$-0.247080\pi$
$167$	514344.	1.42713	0.713563	+	0.700591i	$0.247080\pi$
$168$	0	0
$169$	−554151.	−1.49249
$170$	0	0
$171$	− 185004.i	− 0.483828i
$172$	0	0
$173$	− 497874.i	− 1.26475i	−0.774663	−	0.632374i	$-0.782080\pi$
$173$	− 497874.i	− 1.26475i	0.774663	−	0.632374i	$-0.217920\pi$
$174$	0	0
$175$	201720.	0.497913
$176$	0	0
$177$	304956.	0.731651
$178$	0	0
$179$	− 711252.i	− 1.65917i	−0.558380	−	0.829585i	$-0.688577\pi$
$179$	− 711252.i	− 1.65917i	0.558380	−	0.829585i	$-0.311423\pi$
$180$	0	0
$181$	− 471366.i	− 1.06945i	−0.845025	−	0.534727i	$-0.820415\pi$
$181$	− 471366.i	− 1.06945i	0.845025	−	0.534727i	$-0.179585\pi$
$182$	0	0
$183$	352062.	0.777126
$184$	0	0
$185$	−501828.	−1.07802
$186$	0	0
$187$	− 711592.i	− 1.48808i
$188$	0	0
$189$	87480.0i	0.178137i
$190$	0	0
$191$	646080.	1.28145	0.640727	−	0.767769i	$-0.278633\pi$
$191$	646080.	1.28145	0.640727	+	0.767769i	$0.278633\pi$
$192$	0	0
$193$	−826558.	−1.59728	−0.798638	−	0.601811i	$-0.794446\pi$
$193$	−826558.	−1.59728	−0.798638	+	0.601811i	$0.794446\pi$
$194$	0	0
$195$	329004.i	0.619605i
$196$	0	0
$197$	126138.i	0.231569i	0.993274	+	0.115784i	$0.0369382\pi$
$197$	126138.i	0.231569i	−0.993274	+	0.115784i	$0.963062\pi$
$198$	0	0
$199$	−119144.	−0.213275	−0.106637	−	0.994298i	$-0.534008\pi$
$199$	−119144.	−0.213275	−0.106637	+	0.994298i	$0.534008\pi$
$200$	0	0
$201$	−296964.	−0.518458
$202$	0	0
$203$	475920.i	0.810576i
$204$	0	0
$205$	− 574788.i	− 0.955263i
$206$	0	0
$207$	−206712.	−0.335305
$208$	0	0
$209$	−1.19682e6	−1.89523
$210$	0	0
$211$	− 341620.i	− 0.528247i	−0.964489	−	0.264124i	$-0.914917\pi$
$211$	− 341620.i	− 0.528247i	0.964489	−	0.264124i	$-0.0850827\pi$
$212$	0	0
$213$	− 128232.i	− 0.193663i
$214$	0	0
$215$	−278008.	−0.410167
$216$	0	0
$217$	359040.	0.517599
$218$	0	0
$219$	− 323910.i	− 0.456367i
$220$	0	0
$221$	1.30640e6i	1.79926i
$222$	0	0
$223$	523088.	0.704389	0.352195	−	0.935927i	$-0.385436\pi$
$223$	523088.	0.704389	0.352195	+	0.935927i	$0.385436\pi$
$224$	0	0
$225$	−136161.	−0.179307
$226$	0	0
$227$	− 1.03261e6i	− 1.33006i	−0.746815	−	0.665032i	$-0.768418\pi$
$227$	− 1.03261e6i	− 1.33006i	0.746815	−	0.665032i	$-0.231582\pi$
$228$	0	0
$229$	− 50422.0i	− 0.0635377i	−0.999495	−	0.0317688i	$-0.989886\pi$
$229$	− 50422.0i	− 0.0635377i	0.999495	−	0.0317688i	$-0.0101140\pi$
$230$	0	0
$231$	565920.	0.697791
$232$	0	0
$233$	991030.	1.19591	0.597953	−	0.801531i	$-0.295981\pi$
$233$	991030.	1.19591	0.597953	+	0.801531i	$0.295981\pi$
$234$	0	0
$235$	− 264480.i	− 0.312409i
$236$	0	0
$237$	− 268992.i	− 0.311077i
$238$	0	0
$239$	−514864.	−0.583039	−0.291520	−	0.956565i	$-0.594161\pi$
$239$	−514864.	−0.583039	−0.291520	+	0.956565i	$0.594161\pi$
$240$	0	0
$241$	480498.	0.532904	0.266452	−	0.963848i	$-0.414149\pi$
$241$	480498.	0.532904	0.266452	+	0.963848i	$0.414149\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	91466.0i	0.0973519i
$246$	0	0
$247$	2.19721e6	2.29155
$248$	0	0
$249$	466956.	0.477285
$250$	0	0
$251$	− 768260.i	− 0.769704i	−0.922978	−	0.384852i	$-0.874252\pi$
$251$	− 768260.i	− 0.769704i	0.922978	−	0.384852i	$-0.125748\pi$
$252$	0	0
$253$	1.33725e6i	1.31344i
$254$	0	0
$255$	464436.	0.447276
$256$	0	0
$257$	316162.	0.298591	0.149296	−	0.988793i	$-0.452299\pi$
$257$	316162.	0.298591	0.149296	+	0.988793i	$0.452299\pi$
$258$	0	0
$259$	− 1.58472e6i	− 1.46792i
$260$	0	0
$261$	− 321246.i	− 0.291902i
$262$	0	0
$263$	−1.33017e6	−1.18582	−0.592908	−	0.805270i	$-0.702020\pi$
$263$	−1.33017e6	−1.18582	−0.592908	+	0.805270i	$0.702020\pi$
$264$	0	0
$265$	664316.	0.581112
$266$	0	0
$267$	276426.i	0.237302i
$268$	0	0
$269$	812590.i	0.684685i	0.939575	+	0.342342i	$0.111220\pi$
$269$	812590.i	0.684685i	−0.939575	+	0.342342i	$0.888780\pi$
$270$	0	0
$271$	1.99235e6	1.64795	0.823973	−	0.566629i	$-0.191753\pi$
$271$	1.99235e6	1.64795	0.823973	+	0.566629i	$0.191753\pi$
$272$	0	0
$273$	−1.03896e6	−0.843708
$274$	0	0
$275$	880844.i	0.702372i
$276$	0	0
$277$	− 356134.i	− 0.278878i	−0.990231	−	0.139439i	$-0.955470\pi$
$277$	− 356134.i	− 0.278878i	0.990231	−	0.139439i	$-0.0445299\pi$
$278$	0	0
$279$	−242352.	−0.186396
$280$	0	0
$281$	−644986.	−0.487287	−0.243643	−	0.969865i	$-0.578343\pi$
$281$	−644986.	−0.487287	−0.243643	+	0.969865i	$0.578343\pi$
$282$	0	0
$283$	− 677188.i	− 0.502624i	−0.967906	−	0.251312i	$-0.919138\pi$
$283$	− 677188.i	− 0.502624i	0.967906	−	0.251312i	$-0.0808620\pi$
$284$	0	0
$285$	− 781128.i	− 0.569653i
$286$	0	0
$287$	1.81512e6	1.30077
$288$	0	0
$289$	424307.	0.298838
$290$	0	0
$291$	436302.i	0.302033i
$292$	0	0
$293$	− 414822.i	− 0.282288i	−0.989989	−	0.141144i	$-0.954922\pi$
$293$	− 414822.i	− 0.282288i	0.989989	−	0.141144i	$-0.0450781\pi$
$294$	0	0
$295$	1.28759e6	0.861436
$296$	0	0
$297$	−381996.	−0.251286
$298$	0	0
$299$	− 2.45502e6i	− 1.58810i
$300$	0	0
$301$	− 877920.i	− 0.558520i
$302$	0	0
$303$	−838998.	−0.524994
$304$	0	0
$305$	1.48648e6	0.914978
$306$	0	0
$307$	362028.i	0.219228i	0.993974	+	0.109614i	$0.0349615\pi$
$307$	362028.i	0.219228i	−0.993974	+	0.109614i	$0.965039\pi$
$308$	0	0
$309$	− 20664.0i	− 0.0123117i
$310$	0	0
$311$	1.54342e6	0.904861	0.452431	−	0.891800i	$-0.350557\pi$
$311$	1.54342e6	0.904861	0.452431	+	0.891800i	$0.350557\pi$
$312$	0	0
$313$	1.54225e6	0.889801	0.444900	−	0.895580i	$-0.353239\pi$
$313$	1.54225e6	0.889801	0.444900	+	0.895580i	$0.353239\pi$
$314$	0	0
$315$	369360.i	0.209736i
$316$	0	0
$317$	33246.0i	0.0185819i	0.999957	+	0.00929097i	$0.00295745\pi$
$317$	33246.0i	0.0185819i	−0.999957	+	0.00929097i	$0.997043\pi$
$318$	0	0
$319$	−2.07818e6	−1.14342
$320$	0	0
$321$	350892.	0.190069
$322$	0	0
$323$	− 3.10167e6i	− 1.65421i
$324$	0	0
$325$	− 1.61712e6i	− 0.849248i
$326$	0	0
$327$	56142.0	0.0290348
$328$	0	0
$329$	835200.	0.425403
$330$	0	0
$331$	− 1.60738e6i	− 0.806396i	−0.915113	−	0.403198i	$-0.867899\pi$
$331$	− 1.60738e6i	− 0.806396i	0.915113	−	0.403198i	$-0.132101\pi$
$332$	0	0
$333$	1.06969e6i	0.528623i
$334$	0	0
$335$	−1.25385e6	−0.610426
$336$	0	0
$337$	−1.44958e6	−0.695293	−0.347647	−	0.937626i	$-0.613019\pi$
$337$	−1.44958e6	−0.695293	−0.347647	+	0.937626i	$0.613019\pi$
$338$	0	0
$339$	− 1.92256e6i	− 0.908618i
$340$	0	0
$341$	1.56781e6i	0.730141i
$342$	0	0
$343$	−2.30568e6	−1.05819
$344$	0	0
$345$	−872784.	−0.394784
$346$	0	0
$347$	474588.i	0.211589i	0.994388	+	0.105794i	$0.0337386\pi$
$347$	474588.i	0.211589i	−0.994388	+	0.105794i	$0.966261\pi$
$348$	0	0
$349$	− 2.98869e6i	− 1.31346i	−0.754125	−	0.656731i	$-0.771939\pi$
$349$	− 2.98869e6i	− 1.31346i	0.754125	−	0.656731i	$-0.228061\pi$
$350$	0	0
$351$	701298.	0.303833
$352$	0	0
$353$	3.26480e6	1.39451	0.697253	−	0.716826i	$-0.254406\pi$
$353$	3.26480e6	1.39451	0.697253	+	0.716826i	$0.254406\pi$
$354$	0	0
$355$	− 541424.i	− 0.228017i
$356$	0	0
$357$	1.46664e6i	0.609050i
$358$	0	0
$359$	1.92430e6	0.788017	0.394009	−	0.919107i	$-0.371088\pi$
$359$	1.92430e6	0.788017	0.394009	+	0.919107i	$0.371088\pi$
$360$	0	0
$361$	−2.74056e6	−1.10680
$362$	0	0
$363$	1.02172e6i	0.406975i
$364$	0	0
$365$	− 1.36762e6i	− 0.537320i
$366$	0	0
$367$	1.50013e6	0.581384	0.290692	−	0.956817i	$-0.406114\pi$
$367$	1.50013e6	0.581384	0.290692	+	0.956817i	$0.406114\pi$
$368$	0	0
$369$	−1.22521e6	−0.468428
$370$	0	0
$371$	2.09784e6i	0.791293i
$372$	0	0
$373$	4.70185e6i	1.74983i	0.484273	+	0.874917i	$0.339084\pi$
$373$	4.70185e6i	1.74983i	−0.484273	+	0.874917i	$0.660916\pi$
$374$	0	0
$375$	−1.64365e6	−0.603576
$376$	0	0
$377$	3.81529e6	1.38253
$378$	0	0
$379$	1.51526e6i	0.541863i	0.962599	+	0.270931i	$0.0873316\pi$
$379$	1.51526e6i	0.541863i	−0.962599	+	0.270931i	$0.912668\pi$
$380$	0	0
$381$	1.84565e6i	0.651383i
$382$	0	0
$383$	−155520.	−0.0541738	−0.0270869	−	0.999633i	$-0.508623\pi$
$383$	−155520.	−0.0541738	−0.0270869	+	0.999633i	$0.508623\pi$
$384$	0	0
$385$	2.38944e6	0.821569
$386$	0	0
$387$	592596.i	0.201132i
$388$	0	0
$389$	3.05084e6i	1.02222i	0.859514	+	0.511112i	$0.170766\pi$
$389$	3.05084e6i	1.02222i	−0.859514	+	0.511112i	$0.829234\pi$
$390$	0	0
$391$	−3.46562e6	−1.14641
$392$	0	0
$393$	−3.15104e6	−1.02914
$394$	0	0
$395$	− 1.13574e6i	− 0.366258i
$396$	0	0
$397$	196574.i	0.0625965i	0.999510	+	0.0312982i	$0.00996416\pi$
$397$	196574.i	0.0625965i	−0.999510	+	0.0312982i	$0.990036\pi$
$398$	0	0
$399$	2.46672e6	0.775689
$400$	0	0
$401$	−752910.	−0.233820	−0.116910	−	0.993142i	$-0.537299\pi$
$401$	−752910.	−0.233820	−0.116910	+	0.993142i	$0.537299\pi$
$402$	0	0
$403$	− 2.87830e6i	− 0.882824i
$404$	0	0
$405$	− 249318.i	− 0.0755294i
$406$	0	0
$407$	6.91994e6	2.07070
$408$	0	0
$409$	−5.61695e6	−1.66032	−0.830162	−	0.557523i	$-0.811752\pi$
$409$	−5.61695e6	−1.66032	−0.830162	+	0.557523i	$0.811752\pi$
$410$	0	0
$411$	− 2.11037e6i	− 0.616247i
$412$	0	0
$413$	4.06608e6i	1.17301i
$414$	0	0
$415$	1.97159e6	0.561949
$416$	0	0
$417$	147852.	0.0416377
$418$	0	0
$419$	6.35267e6i	1.76775i	0.467722	+	0.883876i	$0.345075\pi$
$419$	6.35267e6i	1.76775i	−0.467722	+	0.883876i	$0.654925\pi$
$420$	0	0
$421$	− 5.80991e6i	− 1.59759i	−0.601606	−	0.798793i	$-0.705472\pi$
$421$	− 5.80991e6i	− 1.59759i	0.601606	−	0.798793i	$-0.294528\pi$
$422$	0	0
$423$	−563760.	−0.153195
$424$	0	0
$425$	−2.28280e6	−0.613049
$426$	0	0
$427$	4.69416e6i	1.24591i
$428$	0	0
$429$	− 4.53679e6i	− 1.19016i
$430$	0	0
$431$	−1.31099e6	−0.339944	−0.169972	−	0.985449i	$-0.554368\pi$
$431$	−1.31099e6	−0.339944	−0.169972	+	0.985449i	$0.554368\pi$
$432$	0	0
$433$	3.76127e6	0.964083	0.482041	−	0.876148i	$-0.339895\pi$
$433$	3.76127e6	0.964083	0.482041	+	0.876148i	$0.339895\pi$
$434$	0	0
$435$	− 1.35637e6i	− 0.343681i
$436$	0	0
$437$	5.82877e6i	1.46007i
$438$	0	0
$439$	1.00116e6	0.247937	0.123969	−	0.992286i	$-0.460438\pi$
$439$	1.00116e6	0.247937	0.123969	+	0.992286i	$0.460438\pi$
$440$	0	0
$441$	194967.	0.0477380
$442$	0	0
$443$	4.01542e6i	0.972124i	0.873924	+	0.486062i	$0.161567\pi$
$443$	4.01542e6i	0.972124i	−0.873924	+	0.486062i	$0.838433\pi$
$444$	0	0
$445$	1.16713e6i	0.279396i
$446$	0	0
$447$	831834.	0.196910
$448$	0	0
$449$	322466.	0.0754863	0.0377431	−	0.999287i	$-0.487983\pi$
$449$	322466.	0.0754863	0.0377431	+	0.999287i	$0.487983\pi$
$450$	0	0
$451$	7.92602e6i	1.83491i
$452$	0	0
$453$	− 3.15886e6i	− 0.723243i
$454$	0	0
$455$	−4.38672e6	−0.993371
$456$	0	0
$457$	6.64994e6	1.48945	0.744727	−	0.667369i	$-0.232579\pi$
$457$	6.64994e6	1.48945	0.744727	+	0.667369i	$0.232579\pi$
$458$	0	0
$459$	− 989982.i	− 0.219329i
$460$	0	0
$461$	1.17793e6i	0.258148i	0.991635	+	0.129074i	$0.0412005\pi$
$461$	1.17793e6i	0.258148i	−0.991635	+	0.129074i	$0.958800\pi$
$462$	0	0
$463$	−5.85949e6	−1.27030	−0.635151	−	0.772388i	$-0.719062\pi$
$463$	−5.85949e6	−1.27030	−0.635151	+	0.772388i	$0.719062\pi$
$464$	0	0
$465$	−1.02326e6	−0.219460
$466$	0	0
$467$	4.28056e6i	0.908255i	0.890937	+	0.454128i	$0.150049\pi$
$467$	4.28056e6i	0.908255i	−0.890937	+	0.454128i	$0.849951\pi$
$468$	0	0
$469$	− 3.95952e6i	− 0.831209i
$470$	0	0
$471$	416862.	0.0865845
$472$	0	0
$473$	3.83358e6	0.787866
$474$	0	0
$475$	3.83940e6i	0.780782i
$476$	0	0
$477$	− 1.41604e6i	− 0.284958i
$478$	0	0
$479$	−5.75622e6	−1.14630	−0.573151	−	0.819450i	$-0.694279\pi$
$479$	−5.75622e6	−1.14630	−0.573151	+	0.819450i	$0.694279\pi$
$480$	0	0
$481$	−1.27042e7	−2.50371
$482$	0	0
$483$	− 2.75616e6i	− 0.537572i
$484$	0	0
$485$	1.84216e6i	0.355610i
$486$	0	0
$487$	−5.63127e6	−1.07593	−0.537965	−	0.842967i	$-0.680807\pi$
$487$	−5.63127e6	−1.07593	−0.537965	+	0.842967i	$0.680807\pi$
$488$	0	0
$489$	3.55417e6	0.672150
$490$	0	0
$491$	− 3.46885e6i	− 0.649355i	−0.945825	−	0.324677i	$-0.894744\pi$
$491$	− 3.46885e6i	− 0.649355i	0.945825	−	0.324677i	$-0.105256\pi$
$492$	0	0
$493$	− 5.38583e6i	− 0.998011i
$494$	0	0
$495$	−1.61287e6	−0.295860
$496$	0	0
$497$	1.70976e6	0.310488
$498$	0	0
$499$	− 5.98837e6i	− 1.07661i	−0.842751	−	0.538304i	$-0.819065\pi$
$499$	− 5.98837e6i	− 1.07661i	0.842751	−	0.538304i	$-0.180935\pi$
$500$	0	0
$501$	− 4.62910e6i	− 0.823952i
$502$	0	0
$503$	3.13058e6	0.551703	0.275852	−	0.961200i	$-0.411040\pi$
$503$	3.13058e6	0.551703	0.275852	+	0.961200i	$0.411040\pi$
$504$	0	0
$505$	−3.54244e6	−0.618121
$506$	0	0
$507$	4.98736e6i	0.861689i
$508$	0	0
$509$	3.11965e6i	0.533717i	0.963736	+	0.266858i	$0.0859856\pi$
$509$	3.11965e6i	0.533717i	−0.963736	+	0.266858i	$0.914014\pi$
$510$	0	0
$511$	4.31880e6	0.731663
$512$	0	0
$513$	−1.66504e6	−0.279338
$514$	0	0
$515$	− 87248.0i	− 0.0144956i
$516$	0	0
$517$	3.64704e6i	0.600087i
$518$	0	0
$519$	−4.48087e6	−0.730203
$520$	0	0
$521$	−4.34939e6	−0.701994	−0.350997	−	0.936377i	$-0.614157\pi$
$521$	−4.34939e6	−0.701994	−0.350997	+	0.936377i	$0.614157\pi$
$522$	0	0
$523$	− 2.08524e6i	− 0.333350i	−0.986012	−	0.166675i	$-0.946697\pi$
$523$	− 2.08524e6i	− 0.333350i	0.986012	−	0.166675i	$-0.0533031\pi$
$524$	0	0
$525$	− 1.81548e6i	− 0.287470i
$526$	0	0
$527$	−4.06314e6	−0.637287
$528$	0	0
$529$	76361.0	0.0118640
$530$	0	0
$531$	− 2.74460e6i	− 0.422419i
$532$	0	0
$533$	− 1.45512e7i	− 2.21861i
$534$	0	0
$535$	1.48154e6	0.223785
$536$	0	0
$537$	−6.40127e6	−0.957922
$538$	0	0
$539$	− 1.26127e6i	− 0.186997i
$540$	0	0
$541$	− 2.16722e6i	− 0.318353i	−0.987250	−	0.159177i	$-0.949116\pi$
$541$	− 2.16722e6i	− 0.318353i	0.987250	−	0.159177i	$-0.0508839\pi$
$542$	0	0
$543$	−4.24229e6	−0.617449
$544$	0	0
$545$	237044.	0.0341852
$546$	0	0
$547$	1.02512e6i	0.146489i	0.997314	+	0.0732444i	$0.0233353\pi$
$547$	1.02512e6i	0.146489i	−0.997314	+	0.0732444i	$0.976665\pi$
$548$	0	0
$549$	− 3.16856e6i	− 0.448674i
$550$	0	0
$551$	−9.05834e6	−1.27107
$552$	0	0
$553$	3.58656e6	0.498730
$554$	0	0
$555$	4.51645e6i	0.622393i
$556$	0	0
$557$	9.08401e6i	1.24062i	0.784356	+	0.620311i	$0.212994\pi$
$557$	9.08401e6i	1.24062i	−0.784356	+	0.620311i	$0.787006\pi$
$558$	0	0
$559$	−7.03799e6	−0.952619
$560$	0	0
$561$	−6.40433e6	−0.859145
$562$	0	0
$563$	8.45921e6i	1.12476i	0.826880	+	0.562379i	$0.190114\pi$
$563$	8.45921e6i	1.12476i	−0.826880	+	0.562379i	$0.809886\pi$
$564$	0	0
$565$	− 8.11748e6i	− 1.06979i
$566$	0	0
$567$	787320.	0.102847
$568$	0	0
$569$	1.16334e7	1.50634	0.753172	−	0.657824i	$-0.228523\pi$
$569$	1.16334e7	1.50634	0.753172	+	0.657824i	$0.228523\pi$
$570$	0	0
$571$	− 4.01840e6i	− 0.515779i	−0.966174	−	0.257889i	$-0.916973\pi$
$571$	− 4.01840e6i	− 0.515779i	0.966174	−	0.257889i	$-0.0830270\pi$
$572$	0	0
$573$	− 5.81472e6i	− 0.739848i
$574$	0	0
$575$	4.28991e6	0.541102
$576$	0	0
$577$	−3.55296e6	−0.444274	−0.222137	−	0.975016i	$-0.571303\pi$
$577$	−3.55296e6	−0.444274	−0.222137	+	0.975016i	$0.571303\pi$
$578$	0	0
$579$	7.43902e6i	0.922188i
$580$	0	0
$581$	6.22608e6i	0.765199i
$582$	0	0
$583$	−9.16057e6	−1.11622
$584$	0	0
$585$	2.96104e6	0.357729
$586$	0	0
$587$	6.29496e6i	0.754045i	0.926204	+	0.377023i	$0.123052\pi$
$587$	6.29496e6i	0.754045i	−0.926204	+	0.377023i	$0.876948\pi$
$588$	0	0
$589$	6.83373e6i	0.811651i
$590$	0	0
$591$	1.13524e6	0.133696
$592$	0	0
$593$	9.01935e6	1.05327	0.526633	−	0.850093i	$-0.323454\pi$
$593$	9.01935e6	1.05327	0.526633	+	0.850093i	$0.323454\pi$
$594$	0	0
$595$	6.19248e6i	0.717088i
$596$	0	0
$597$	1.07230e6i	0.123134i
$598$	0	0
$599$	−1.24315e7	−1.41565	−0.707826	−	0.706387i	$-0.750324\pi$
$599$	−1.24315e7	−1.41565	−0.707826	+	0.706387i	$0.750324\pi$
$600$	0	0
$601$	−4.74476e6	−0.535832	−0.267916	−	0.963442i	$-0.586335\pi$
$601$	−4.74476e6	−0.535832	−0.267916	+	0.963442i	$0.586335\pi$
$602$	0	0
$603$	2.67268e6i	0.299332i
$604$	0	0
$605$	4.31395e6i	0.479167i
$606$	0	0
$607$	−3.49784e6	−0.385326	−0.192663	−	0.981265i	$-0.561712\pi$
$607$	−3.49784e6	−0.385326	−0.192663	+	0.981265i	$0.561712\pi$
$608$	0	0
$609$	4.28328e6	0.467986
$610$	0	0
$611$	− 6.69552e6i	− 0.725573i
$612$	0	0
$613$	358762.i	0.0385616i	0.999814	+	0.0192808i	$0.00613765\pi$
$613$	358762.i	0.0385616i	−0.999814	+	0.0192808i	$0.993862\pi$
$614$	0	0
$615$	−5.17309e6	−0.551521
$616$	0	0
$617$	1.26388e7	1.33658	0.668289	−	0.743902i	$-0.267027\pi$
$617$	1.26388e7	1.33658	0.668289	+	0.743902i	$0.267027\pi$
$618$	0	0
$619$	1.06705e7i	1.11933i	0.828720	+	0.559664i	$0.189070\pi$
$619$	1.06705e7i	1.11933i	−0.828720	+	0.559664i	$0.810930\pi$
$620$	0	0
$621$	1.86041e6i	0.193588i
$622$	0	0
$623$	−3.68568e6	−0.380450
$624$	0	0
$625$	−1.68674e6	−0.172722
$626$	0	0
$627$	1.07713e7i	1.09421i
$628$	0	0
$629$	1.79337e7i	1.80736i
$630$	0	0
$631$	1.32621e7	1.32598	0.662991	−	0.748628i	$-0.269287\pi$
$631$	1.32621e7	1.32598	0.662991	+	0.748628i	$0.269287\pi$
$632$	0	0
$633$	−3.07458e6	−0.304984
$634$	0	0
$635$	7.79274e6i	0.766930i
$636$	0	0
$637$	2.31553e6i	0.226101i
$638$	0	0
$639$	−1.15409e6	−0.111812
$640$	0	0
$641$	7.29879e6	0.701626	0.350813	−	0.936446i	$-0.385905\pi$
$641$	7.29879e6	0.701626	0.350813	+	0.936446i	$0.385905\pi$
$642$	0	0
$643$	1.97747e6i	0.188618i	0.995543	+	0.0943088i	$0.0300641\pi$
$643$	1.97747e6i	0.188618i	−0.995543	+	0.0943088i	$0.969936\pi$
$644$	0	0
$645$	2.50207e6i	0.236810i
$646$	0	0
$647$	−272792.	−0.0256195	−0.0128098	−	0.999918i	$-0.504078\pi$
$647$	−272792.	−0.0256195	−0.0128098	+	0.999918i	$0.504078\pi$
$648$	0	0
$649$	−1.77552e7	−1.65468
$650$	0	0
$651$	− 3.23136e6i	− 0.298836i
$652$	0	0
$653$	4.13845e6i	0.379799i	0.981803	+	0.189900i	$0.0608163\pi$
$653$	4.13845e6i	0.379799i	−0.981803	+	0.189900i	$0.939184\pi$
$654$	0	0
$655$	−1.33044e7	−1.21169
$656$	0	0
$657$	−2.91519e6	−0.263484
$658$	0	0
$659$	− 445812.i	− 0.0399888i	−0.999800	−	0.0199944i	$-0.993635\pi$
$659$	− 445812.i	− 0.0399888i	0.999800	−	0.0199944i	$-0.00636484\pi$
$660$	0	0
$661$	3.65881e6i	0.325714i	0.986650	+	0.162857i	$0.0520709\pi$
$661$	3.65881e6i	0.325714i	−0.986650	+	0.162857i	$0.947929\pi$
$662$	0	0
$663$	1.17576e7	1.03880
$664$	0	0
$665$	1.04150e7	0.913286
$666$	0	0
$667$	1.01212e7i	0.880884i
$668$	0	0
$669$	− 4.70779e6i	− 0.406679i
$670$	0	0
$671$	−2.04978e7	−1.75753
$672$	0	0
$673$	−1.42444e7	−1.21229	−0.606147	−	0.795353i	$-0.707286\pi$
$673$	−1.42444e7	−1.21229	−0.606147	+	0.795353i	$0.707286\pi$
$674$	0	0
$675$	1.22545e6i	0.103523i
$676$	0	0
$677$	1.33128e7i	1.11634i	0.829727	+	0.558170i	$0.188496\pi$
$677$	1.33128e7i	1.11634i	−0.829727	+	0.558170i	$0.811504\pi$
$678$	0	0
$679$	−5.81736e6	−0.484230
$680$	0	0
$681$	−9.29351e6	−0.767913
$682$	0	0
$683$	− 1.49468e6i	− 0.122601i	−0.998119	−	0.0613007i	$-0.980475\pi$
$683$	− 1.49468e6i	− 0.122601i	0.998119	−	0.0613007i	$-0.0195249\pi$
$684$	0	0
$685$	− 8.91047e6i	− 0.725561i
$686$	0	0
$687$	−453798.	−0.0366835
$688$	0	0
$689$	1.68177e7	1.34964
$690$	0	0
$691$	1.11320e7i	0.886910i	0.896297	+	0.443455i	$0.146247\pi$
$691$	1.11320e7i	0.886910i	−0.896297	+	0.443455i	$0.853753\pi$
$692$	0	0
$693$	− 5.09328e6i	− 0.402870i
$694$	0	0
$695$	624264.	0.0490237
$696$	0	0
$697$	−2.05411e7	−1.60156
$698$	0	0
$699$	− 8.91927e6i	− 0.690457i
$700$	0	0
$701$	− 1.52327e7i	− 1.17080i	−0.810746	−	0.585398i	$-0.800938\pi$
$701$	− 1.52327e7i	− 1.17080i	0.810746	−	0.585398i	$-0.199062\pi$
$702$	0	0
$703$	3.01625e7	2.30186
$704$	0	0
$705$	−2.38032e6	−0.180369
$706$	0	0
$707$	− 1.11866e7i	− 0.841688i
$708$	0	0
$709$	8.59921e6i	0.642455i	0.947002	+	0.321228i	$0.104095\pi$
$709$	8.59921e6i	0.642455i	−0.947002	+	0.321228i	$0.895905\pi$
$710$	0	0
$711$	−2.42093e6	−0.179601
$712$	0	0
$713$	7.63558e6	0.562495
$714$	0	0
$715$	− 1.91553e7i	− 1.40128i
$716$	0	0
$717$	4.63378e6i	0.336618i
$718$	0	0
$719$	2.84891e6	0.205521	0.102761	−	0.994706i	$-0.467232\pi$
$719$	2.84891e6	0.205521	0.102761	+	0.994706i	$0.467232\pi$
$720$	0	0
$721$	275520.	0.0197385
$722$	0	0
$723$	− 4.32448e6i	− 0.307672i
$724$	0	0
$725$	6.66685e6i	0.471059i
$726$	0	0
$727$	−8.11615e6	−0.569527	−0.284763	−	0.958598i	$-0.591915\pi$
$727$	−8.11615e6	−0.569527	−0.284763	+	0.958598i	$0.591915\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	9.93513e6i	0.687670i
$732$	0	0
$733$	− 1.14038e7i	− 0.783954i	−0.919975	−	0.391977i	$-0.871791\pi$
$733$	− 1.14038e7i	− 0.783954i	0.919975	−	0.391977i	$-0.128209\pi$
$734$	0	0
$735$	823194.	0.0562062
$736$	0	0
$737$	1.72899e7	1.17253
$738$	0	0
$739$	2.28780e6i	0.154102i	0.997027	+	0.0770509i	$0.0245504\pi$
$739$	2.28780e6i	0.154102i	−0.997027	+	0.0770509i	$0.975450\pi$
$740$	0	0
$741$	− 1.97749e7i	− 1.32303i
$742$	0	0
$743$	−2.92359e7	−1.94287	−0.971437	−	0.237296i	$-0.923739\pi$
$743$	−2.92359e7	−1.94287	−0.971437	+	0.237296i	$0.923739\pi$
$744$	0	0
$745$	3.51219e6	0.231839
$746$	0	0
$747$	− 4.20260e6i	− 0.275561i
$748$	0	0
$749$	4.67856e6i	0.304725i
$750$	0	0
$751$	−1.71311e7	−1.10837	−0.554186	−	0.832393i	$-0.686970\pi$
$751$	−1.71311e7	−1.10837	−0.554186	+	0.832393i	$0.686970\pi$
$752$	0	0
$753$	−6.91434e6	−0.444389
$754$	0	0
$755$	− 1.33374e7i	− 0.851537i
$756$	0	0
$757$	− 3.31732e6i	− 0.210401i	−0.994451	−	0.105200i	$-0.966452\pi$
$757$	− 3.31732e6i	− 0.210401i	0.994451	−	0.105200i	$-0.0335484\pi$
$758$	0	0
$759$	1.20352e7	0.758316
$760$	0	0
$761$	−1.28948e7	−0.807146	−0.403573	−	0.914947i	$-0.632232\pi$
$761$	−1.28948e7	−0.807146	−0.403573	+	0.914947i	$0.632232\pi$
$762$	0	0
$763$	748560.i	0.0465495i
$764$	0	0
$765$	− 4.17992e6i	− 0.258235i
$766$	0	0
$767$	3.25964e7	2.00070
$768$	0	0
$769$	1.87622e7	1.14411	0.572056	−	0.820214i	$-0.306146\pi$
$769$	1.87622e7	1.14411	0.572056	+	0.820214i	$0.306146\pi$
$770$	0	0
$771$	− 2.84546e6i	− 0.172392i
$772$	0	0
$773$	− 6.84442e6i	− 0.411991i	−0.978553	−	0.205996i	$-0.933957\pi$
$773$	− 6.84442e6i	− 0.411991i	0.978553	−	0.205996i	$-0.0660433\pi$
$774$	0	0
$775$	5.02955e6	0.300798
$776$	0	0
$777$	−1.42625e7	−0.847505
$778$	0	0
$779$	3.45478e7i	2.03975i
$780$	0	0
$781$	7.46595e6i	0.437983i
$782$	0	0
$783$	−2.89121e6	−0.168529
$784$	0	0
$785$	1.76008e6	0.101943
$786$	0	0
$787$	2.91272e7i	1.67634i	0.545409	+	0.838170i	$0.316374\pi$
$787$	2.91272e7i	1.67634i	−0.545409	+	0.838170i	$0.683626\pi$
$788$	0	0
$789$	1.19715e7i	0.684631i
$790$	0	0
$791$	2.56342e7	1.45673
$792$	0	0
$793$	3.76315e7	2.12505
$794$	0	0
$795$	− 5.97884e6i	− 0.335505i
$796$	0	0
$797$	3.12485e7i	1.74254i	0.490800	+	0.871272i	$0.336705\pi$
$797$	3.12485e7i	1.74254i	−0.490800	+	0.871272i	$0.663295\pi$
$798$	0	0
$799$	−9.45168e6	−0.523772
$800$	0	0
$801$	2.48783e6	0.137006
$802$	0	0
$803$	1.88588e7i	1.03211i
$804$	0	0
$805$	− 1.16371e7i	− 0.632930i
$806$	0	0
$807$	7.31331e6	0.395303
$808$	0	0
$809$	1.27324e7	0.683972	0.341986	−	0.939705i	$-0.388901\pi$
$809$	1.27324e7	0.683972	0.341986	+	0.939705i	$0.388901\pi$
$810$	0	0
$811$	2.65194e7i	1.41583i	0.706297	+	0.707916i	$0.250364\pi$
$811$	2.65194e7i	1.41583i	−0.706297	+	0.707916i	$0.749636\pi$
$812$	0	0
$813$	− 1.79312e7i	− 0.951442i
$814$	0	0
$815$	1.50065e7	0.791381
$816$	0	0
$817$	1.67097e7	0.875820
$818$	0	0
$819$	9.35064e6i	0.487115i
$820$	0	0
$821$	2.84134e7i	1.47118i	0.677427	+	0.735590i	$0.263095\pi$
$821$	2.84134e7i	1.47118i	−0.677427	+	0.735590i	$0.736905\pi$
$822$	0	0
$823$	3.21639e7	1.65527	0.827637	−	0.561264i	$-0.189685\pi$
$823$	3.21639e7	1.65527	0.827637	+	0.561264i	$0.189685\pi$
$824$	0	0
$825$	7.92760e6	0.405515
$826$	0	0
$827$	− 3.01436e7i	− 1.53261i	−0.642478	−	0.766304i	$-0.722094\pi$
$827$	− 3.01436e7i	− 1.53261i	0.642478	−	0.766304i	$-0.277906\pi$
$828$	0	0
$829$	− 2.01164e7i	− 1.01663i	−0.861171	−	0.508315i	$-0.830268\pi$
$829$	− 2.01164e7i	− 1.01663i	0.861171	−	0.508315i	$-0.169732\pi$
$830$	0	0
$831$	−3.20521e6	−0.161010
$832$	0	0
$833$	3.26871e6	0.163216
$834$	0	0
$835$	− 1.95451e7i	− 0.970110i
$836$	0	0
$837$	2.18117e6i	0.107616i
$838$	0	0
$839$	−2.20685e7	−1.08235	−0.541175	−	0.840910i	$-0.682020\pi$
$839$	−2.20685e7	−1.08235	−0.541175	+	0.840910i	$0.682020\pi$
$840$	0	0
$841$	4.78199e6	0.233141
$842$	0	0
$843$	5.80487e6i	0.281335i
$844$	0	0
$845$	2.10577e7i	1.01454i
$846$	0	0
$847$	−1.36230e7	−0.652476
$848$	0	0
$849$	−6.09469e6	−0.290190
$850$	0	0
$851$	− 3.37017e7i	− 1.59525i
$852$	0	0
$853$	− 1.12740e7i	− 0.530524i	−0.964176	−	0.265262i	$-0.914542\pi$
$853$	− 1.12740e7i	− 0.530524i	0.964176	−	0.265262i	$-0.0854585\pi$
$854$	0	0
$855$	−7.03015e6	−0.328889
$856$	0	0
$857$	1.67412e7	0.778634	0.389317	−	0.921104i	$-0.372711\pi$
$857$	1.67412e7	0.778634	0.389317	+	0.921104i	$0.372711\pi$
$858$	0	0
$859$	− 3.26435e7i	− 1.50943i	−0.656051	−	0.754716i	$-0.727775\pi$
$859$	− 3.26435e7i	− 1.50943i	0.656051	−	0.754716i	$-0.272225\pi$
$860$	0	0
$861$	− 1.63361e7i	− 0.751000i
$862$	0	0
$863$	2.54029e7	1.16106	0.580532	−	0.814237i	$-0.302844\pi$
$863$	2.54029e7	1.16106	0.580532	+	0.814237i	$0.302844\pi$
$864$	0	0
$865$	−1.89192e7	−0.859731
$866$	0	0
$867$	− 3.81876e6i	− 0.172534i
$868$	0	0
$869$	1.56613e7i	0.703524i
$870$	0	0
$871$	−3.17422e7	−1.41772
$872$	0	0
$873$	3.92672e6	0.174379
$874$	0	0
$875$	− 2.19154e7i	− 0.967673i
$876$	0	0
$877$	3.55846e6i	0.156230i	0.996944	+	0.0781148i	$0.0248901\pi$
$877$	3.55846e6i	0.156230i	−0.996944	+	0.0781148i	$0.975110\pi$
$878$	0	0
$879$	−3.73340e6	−0.162979
$880$	0	0
$881$	−4.10156e7	−1.78037	−0.890184	−	0.455602i	$-0.849424\pi$
$881$	−4.10156e7	−1.78037	−0.890184	+	0.455602i	$0.849424\pi$
$882$	0	0
$883$	− 3.51736e7i	− 1.51815i	−0.651004	−	0.759075i	$-0.725652\pi$
$883$	− 3.51736e7i	− 1.51815i	0.651004	−	0.759075i	$-0.274348\pi$
$884$	0	0
$885$	− 1.15883e7i	− 0.497351i
$886$	0	0
$887$	4.33071e7	1.84821	0.924103	−	0.382144i	$-0.124814\pi$
$887$	4.33071e7	1.84821	0.924103	+	0.382144i	$0.124814\pi$
$888$	0	0
$889$	−2.46086e7	−1.04432
$890$	0	0
$891$	3.43796e6i	0.145080i
$892$	0	0
$893$	1.58966e7i	0.667078i
$894$	0	0
$895$	−2.70276e7	−1.12785
$896$	0	0
$897$	−2.20952e7	−0.916890
$898$	0	0
$899$	1.18663e7i	0.489683i
$900$	0	0
$901$	− 2.37406e7i	− 0.974269i
$902$	0	0
$903$	−7.90128e6	−0.322462
$904$	0	0
$905$	−1.79119e7	−0.726977
$906$	0	0
$907$	− 1.33192e7i	− 0.537599i	−0.963196	−	0.268800i	$-0.913373\pi$
$907$	− 1.33192e7i	− 0.537599i	0.963196	−	0.268800i	$-0.0866269\pi$
$908$	0	0
$909$	7.55098e6i	0.303105i
$910$	0	0
$911$	−8.92578e6	−0.356328	−0.178164	−	0.984001i	$-0.557016\pi$
$911$	−8.92578e6	−0.356328	−0.178164	+	0.984001i	$0.557016\pi$
$912$	0	0
$913$	−2.71872e7	−1.07941
$914$	0	0
$915$	− 1.33784e7i	− 0.528263i
$916$	0	0
$917$	− 4.20139e7i	− 1.64995i
$918$	0	0
$919$	4.13982e7	1.61694	0.808468	−	0.588540i	$-0.200297\pi$
$919$	4.13982e7	1.61694	0.808468	+	0.588540i	$0.200297\pi$
$920$	0	0
$921$	3.25825e6	0.126571
$922$	0	0
$923$	− 1.37066e7i	− 0.529572i
$924$	0	0
$925$	− 2.21993e7i	− 0.853070i
$926$	0	0
$927$	−185976.	−0.00710817
$928$	0	0
$929$	3.42481e7	1.30196	0.650980	−	0.759095i	$-0.274358\pi$
$929$	3.42481e7	1.30196	0.650980	+	0.759095i	$0.274358\pi$
$930$	0	0
$931$	− 5.49759e6i	− 0.207873i
$932$	0	0
$933$	− 1.38907e7i	− 0.522422i
$934$	0	0
$935$	−2.70405e7	−1.01155
$936$	0	0
$937$	−5.44468e6	−0.202593	−0.101296	−	0.994856i	$-0.532299\pi$
$937$	−5.44468e6	−0.202593	−0.101296	+	0.994856i	$0.532299\pi$
$938$	0	0
$939$	− 1.38802e7i	− 0.513727i
$940$	0	0
$941$	3.54752e7i	1.30602i	0.757347	+	0.653012i	$0.226495\pi$
$941$	3.54752e7i	1.30602i	−0.757347	+	0.653012i	$0.773505\pi$
$942$	0	0
$943$	3.86016e7	1.41360
$944$	0	0
$945$	3.32424e6	0.121091
$946$	0	0
$947$	− 1.22505e7i	− 0.443892i	−0.975059	−	0.221946i	$-0.928759\pi$
$947$	− 1.22505e7i	− 0.443892i	0.975059	−	0.221946i	$-0.0712408\pi$
$948$	0	0
$949$	− 3.46224e7i	− 1.24793i
$950$	0	0
$951$	299214.	0.0107283
$952$	0	0
$953$	−4.77139e6	−0.170181	−0.0850907	−	0.996373i	$-0.527118\pi$
$953$	−4.77139e6	−0.170181	−0.0850907	+	0.996373i	$0.527118\pi$
$954$	0	0
$955$	− 2.45510e7i	− 0.871087i
$956$	0	0
$957$	1.87037e7i	0.660156i
$958$	0	0
$959$	2.81383e7	0.987988
$960$	0	0
$961$	−1.96771e7	−0.687309
$962$	0	0
$963$	− 3.15803e6i	− 0.109736i
$964$	0	0
$965$	3.14092e7i	1.08577i
$966$	0	0
$967$	−4.31939e7	−1.48544	−0.742722	−	0.669600i	$-0.766466\pi$
$967$	−4.31939e7	−1.48544	−0.742722	+	0.669600i	$0.766466\pi$
$968$	0	0
$969$	−2.79150e7	−0.955056
$970$	0	0
$971$	1.73630e7i	0.590984i	0.955345	+	0.295492i	$0.0954835\pi$
$971$	1.73630e7i	0.590984i	−0.955345	+	0.295492i	$0.904516\pi$
$972$	0	0
$973$	1.97136e6i	0.0667550i
$974$	0	0
$975$	−1.45541e7	−0.490313
$976$	0	0
$977$	−1.71680e7	−0.575416	−0.287708	−	0.957718i	$-0.592893\pi$
$977$	−1.71680e7	−0.575416	−0.287708	+	0.957718i	$0.592893\pi$
$978$	0	0
$979$	− 1.60941e7i	− 0.536674i
$980$	0	0
$981$	− 505278.i	− 0.0167632i
$982$	0	0
$983$	3.53993e7	1.16845	0.584225	−	0.811592i	$-0.301398\pi$
$983$	3.53993e7	1.16845	0.584225	+	0.811592i	$0.301398\pi$
$984$	0	0
$985$	4.79324e6	0.157412
$986$	0	0
$987$	− 7.51680e6i	− 0.245607i
$988$	0	0
$989$	− 1.86704e7i	− 0.606965i
$990$	0	0
$991$	5.23340e7	1.69278	0.846389	−	0.532566i	$-0.178772\pi$
$991$	5.23340e7	1.69278	0.846389	+	0.532566i	$0.178772\pi$
$992$	0	0
$993$	−1.44664e7	−0.465573
$994$	0	0
$995$	4.52747e6i	0.144977i
$996$	0	0
$997$	7.21035e6i	0.229730i	0.993381	+	0.114865i	$0.0366436\pi$
$997$	7.21035e6i	0.229730i	−0.993381	+	0.114865i	$0.963356\pi$
$998$	0	0
$999$	9.62717e6	0.305200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.d.m.385.1		2
4.3	odd	2		768.6.d.f.385.2		2
8.3	odd	2		768.6.d.f.385.1		2
8.5	even	2	inner	768.6.d.m.385.2		2
16.3	odd	4		192.6.a.b.1.1		1
16.5	even	4		48.6.a.b.1.1		1
16.11	odd	4		24.6.a.c.1.1	✓	1
16.13	even	4		192.6.a.j.1.1		1
48.5	odd	4		144.6.a.d.1.1		1
48.11	even	4		72.6.a.b.1.1		1
48.29	odd	4		576.6.a.ba.1.1		1
48.35	even	4		576.6.a.bb.1.1		1
80.27	even	4		600.6.f.h.49.1		2
80.43	even	4		600.6.f.h.49.2		2
80.59	odd	4		600.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.a.c.1.1	✓	1	16.11	odd	4
48.6.a.b.1.1		1	16.5	even	4
72.6.a.b.1.1		1	48.11	even	4
144.6.a.d.1.1		1	48.5	odd	4
192.6.a.b.1.1		1	16.3	odd	4
192.6.a.j.1.1		1	16.13	even	4
576.6.a.ba.1.1		1	48.29	odd	4
576.6.a.bb.1.1		1	48.35	even	4
600.6.a.a.1.1		1	80.59	odd	4
600.6.f.h.49.1		2	80.27	even	4
600.6.f.h.49.2		2	80.43	even	4
768.6.d.f.385.1		2	8.3	odd	2
768.6.d.f.385.2		2	4.3	odd	2
768.6.d.m.385.1		2	1.1	even	1	trivial
768.6.d.m.385.2		2	8.5	even	2	inner

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} -38.0000i q^{5} +120.000 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} -38.0000i q^{5} +120.000 q^{7} -81.0000 q^{9} +524.000i q^{11} -962.000i q^{13} -342.000 q^{15} -1358.00 q^{17} +2284.00i q^{19} -1080.00i q^{21} +2552.00 q^{23} +1681.00 q^{25} +729.000i q^{27} +3966.00i q^{29} +2992.00 q^{31} +4716.00 q^{33} -4560.00i q^{35} -13206.0i q^{37} -8658.00 q^{39} +15126.0 q^{41} -7316.00i q^{43} +3078.00i q^{45} +6960.00 q^{47} -2407.00 q^{49} +12222.0i q^{51} +17482.0i q^{53} +19912.0 q^{55} +20556.0 q^{57} +33884.0i q^{59} +39118.0i q^{61} -9720.00 q^{63} -36556.0 q^{65} -32996.0i q^{67} -22968.0i q^{69} +14248.0 q^{71} +35990.0 q^{73} -15129.0i q^{75} +62880.0i q^{77} +29888.0 q^{79} +6561.00 q^{81} +51884.0i q^{83} +51604.0i q^{85} +35694.0 q^{87} -30714.0 q^{89} -115440. i q^{91} -26928.0i q^{93} +86792.0 q^{95} -48478.0 q^{97} -42444.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 240 q^{7} - 162 q^{9}+O(q^{10}) \) 2 * q + 240 * q^7 - 162 * q^9	\( 2 q + 240 q^{7} - 162 q^{9} - 684 q^{15} - 2716 q^{17} + 5104 q^{23} + 3362 q^{25} + 5984 q^{31} + 9432 q^{33} - 17316 q^{39} + 30252 q^{41} + 13920 q^{47} - 4814 q^{49} + 39824 q^{55} + 41112 q^{57} - 19440 q^{63} - 73112 q^{65} + 28496 q^{71} + 71980 q^{73} + 59776 q^{79} + 13122 q^{81} + 71388 q^{87} - 61428 q^{89} + 173584 q^{95} - 96956 q^{97}+O(q^{100}) \) 2 * q + 240 * q^7 - 162 * q^9 - 684 * q^15 - 2716 * q^17 + 5104 * q^23 + 3362 * q^25 + 5984 * q^31 + 9432 * q^33 - 17316 * q^39 + 30252 * q^41 + 13920 * q^47 - 4814 * q^49 + 39824 * q^55 + 41112 * q^57 - 19440 * q^63 - 73112 * q^65 + 28496 * q^71 + 71980 * q^73 + 59776 * q^79 + 13122 * q^81 + 71388 * q^87 - 61428 * q^89 + 173584 * q^95 - 96956 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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