LMFDB - Embedded newform 768.6.d.c.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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
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.c.385.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +66.0000i q^{5} -176.000 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} +66.0000i q^{5} -176.000 q^{7} -81.0000 q^{9} +60.0000i q^{11} -658.000i q^{13} +594.000 q^{15} -414.000 q^{17} +956.000i q^{19} +1584.00i q^{21} -600.000 q^{23} -1231.00 q^{25} +729.000i q^{27} +5574.00i q^{29} -3592.00 q^{31} +540.000 q^{33} -11616.0i q^{35} +8458.00i q^{37} -5922.00 q^{39} -19194.0 q^{41} -13316.0i q^{43} -5346.00i q^{45} -19680.0 q^{47} +14169.0 q^{49} +3726.00i q^{51} +31266.0i q^{53} -3960.00 q^{55} +8604.00 q^{57} -26340.0i q^{59} -31090.0i q^{61} +14256.0 q^{63} +43428.0 q^{65} -16804.0i q^{67} +5400.00i q^{69} -6120.00 q^{71} +25558.0 q^{73} +11079.0i q^{75} -10560.0i q^{77} +74408.0 q^{79} +6561.00 q^{81} -6468.00i q^{83} -27324.0i q^{85} +50166.0 q^{87} +32742.0 q^{89} +115808. i q^{91} +32328.0i q^{93} -63096.0 q^{95} +166082. q^{97} -4860.00i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 352 q^{7} - 162 q^{9}+O(q^{10}) $ 2 * q - 352 * q^7 - 162 * q^9	$ 2 q - 352 q^{7} - 162 q^{9} + 1188 q^{15} - 828 q^{17} - 1200 q^{23} - 2462 q^{25} - 7184 q^{31} + 1080 q^{33} - 11844 q^{39} - 38388 q^{41} - 39360 q^{47} + 28338 q^{49} - 7920 q^{55} + 17208 q^{57} + 28512 q^{63} + 86856 q^{65} - 12240 q^{71} + 51116 q^{73} + 148816 q^{79} + 13122 q^{81} + 100332 q^{87} + 65484 q^{89} - 126192 q^{95} + 332164 q^{97}+O(q^{100}) $ 2 * q - 352 * q^7 - 162 * q^9 + 1188 * q^15 - 828 * q^17 - 1200 * q^23 - 2462 * q^25 - 7184 * q^31 + 1080 * q^33 - 11844 * q^39 - 38388 * q^41 - 39360 * q^47 + 28338 * q^49 - 7920 * q^55 + 17208 * q^57 + 28512 * q^63 + 86856 * q^65 - 12240 * q^71 + 51116 * q^73 + 148816 * q^79 + 13122 * q^81 + 100332 * q^87 + 65484 * q^89 - 126192 * q^95 + 332164 * 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
$5$	66.0000i	1.18064i	0.807168	+	0.590322i	$0.200999\pi$
$5$	66.0000i	1.18064i	−0.807168	+	0.590322i	$0.799001\pi$
$6$	0	0
$7$	−176.000	−1.35759	−0.678793	−	0.734329i	$-0.737497\pi$
$7$	−176.000	−1.35759	−0.678793	+	0.734329i	$0.737497\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	60.0000i	0.149510i	0.997202	+	0.0747549i	$0.0238174\pi$
$11$	60.0000i	0.149510i	−0.997202	+	0.0747549i	$0.976183\pi$
$12$	0	0
$13$	− 658.000i	− 1.07986i	−0.841710	−	0.539930i	$-0.818451\pi$
$13$	− 658.000i	− 1.07986i	0.841710	−	0.539930i	$-0.181549\pi$
$14$	0	0
$15$	594.000	0.681645
$16$	0	0
$17$	−414.000	−0.347439	−0.173719	−	0.984795i	$-0.555579\pi$
$17$	−414.000	−0.347439	−0.173719	+	0.984795i	$0.555579\pi$
$18$	0	0
$19$	956.000i	0.607539i	0.952746	+	0.303769i	$0.0982452\pi$
$19$	956.000i	0.607539i	−0.952746	+	0.303769i	$0.901755\pi$
$20$	0	0
$21$	1584.00i	0.783803i
$22$	0	0
$23$	−600.000	−0.236500	−0.118250	−	0.992984i	$-0.537728\pi$
$23$	−600.000	−0.236500	−0.118250	+	0.992984i	$0.537728\pi$
$24$	0	0
$25$	−1231.00	−0.393920
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	5574.00i	1.23076i	0.788232	+	0.615378i	$0.210997\pi$
$29$	5574.00i	1.23076i	−0.788232	+	0.615378i	$0.789003\pi$
$30$	0	0
$31$	−3592.00	−0.671324	−0.335662	−	0.941983i	$-0.608960\pi$
$31$	−3592.00	−0.671324	−0.335662	+	0.941983i	$0.608960\pi$
$32$	0	0
$33$	540.000	0.0863195
$34$	0	0
$35$	− 11616.0i	− 1.60283i
$36$	0	0
$37$	8458.00i	1.01570i	0.861447	+	0.507848i	$0.169559\pi$
$37$	8458.00i	1.01570i	−0.861447	+	0.507848i	$0.830441\pi$
$38$	0	0
$39$	−5922.00	−0.623458
$40$	0	0
$41$	−19194.0	−1.78322	−0.891612	−	0.452800i	$-0.850425\pi$
$41$	−19194.0	−1.78322	−0.891612	+	0.452800i	$0.850425\pi$
$42$	0	0
$43$	− 13316.0i	− 1.09825i	−0.835739	−	0.549127i	$-0.814960\pi$
$43$	− 13316.0i	− 1.09825i	0.835739	−	0.549127i	$-0.185040\pi$
$44$	0	0
$45$	− 5346.00i	− 0.393548i
$46$	0	0
$47$	−19680.0	−1.29951	−0.649756	−	0.760143i	$-0.725129\pi$
$47$	−19680.0	−1.29951	−0.649756	+	0.760143i	$0.725129\pi$
$48$	0	0
$49$	14169.0	0.843042
$50$	0	0
$51$	3726.00i	0.200594i
$52$	0	0
$53$	31266.0i	1.52891i	0.644676	+	0.764456i	$0.276992\pi$
$53$	31266.0i	1.52891i	−0.644676	+	0.764456i	$0.723008\pi$
$54$	0	0
$55$	−3960.00	−0.176518
$56$	0	0
$57$	8604.00	0.350763
$58$	0	0
$59$	− 26340.0i	− 0.985112i	−0.870281	−	0.492556i	$-0.836063\pi$
$59$	− 26340.0i	− 0.985112i	0.870281	−	0.492556i	$-0.163937\pi$
$60$	0	0
$61$	− 31090.0i	− 1.06978i	−0.844920	−	0.534892i	$-0.820352\pi$
$61$	− 31090.0i	− 1.06978i	0.844920	−	0.534892i	$-0.179648\pi$
$62$	0	0
$63$	14256.0	0.452529
$64$	0	0
$65$	43428.0	1.27493
$66$	0	0
$67$	− 16804.0i	− 0.457326i	−0.973506	−	0.228663i	$-0.926565\pi$
$67$	− 16804.0i	− 0.457326i	0.973506	−	0.228663i	$-0.0734353\pi$
$68$	0	0
$69$	5400.00i	0.136544i
$70$	0	0
$71$	−6120.00	−0.144081	−0.0720403	−	0.997402i	$-0.522951\pi$
$71$	−6120.00	−0.144081	−0.0720403	+	0.997402i	$0.522951\pi$
$72$	0	0
$73$	25558.0	0.561332	0.280666	−	0.959806i	$-0.409445\pi$
$73$	25558.0	0.561332	0.280666	+	0.959806i	$0.409445\pi$
$74$	0	0
$75$	11079.0i	0.227430i
$76$	0	0
$77$	− 10560.0i	− 0.202972i
$78$	0	0
$79$	74408.0	1.34138	0.670690	−	0.741738i	$-0.265998\pi$
$79$	74408.0	1.34138	0.670690	+	0.741738i	$0.265998\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	− 6468.00i	− 0.103056i	−0.998672	−	0.0515282i	$-0.983591\pi$
$83$	− 6468.00i	− 0.103056i	0.998672	−	0.0515282i	$-0.0164092\pi$
$84$	0	0
$85$	− 27324.0i	− 0.410201i
$86$	0	0
$87$	50166.0	0.710577
$88$	0	0
$89$	32742.0	0.438157	0.219079	−	0.975707i	$-0.429695\pi$
$89$	32742.0	0.438157	0.219079	+	0.975707i	$0.429695\pi$
$90$	0	0
$91$	115808.i	1.46600i
$92$	0	0
$93$	32328.0i	0.387589i
$94$	0	0
$95$	−63096.0	−0.717287
$96$	0	0
$97$	166082.	1.79223	0.896114	−	0.443824i	$-0.146378\pi$
$97$	166082.	1.79223	0.896114	+	0.443824i	$0.146378\pi$
$98$	0	0
$99$	− 4860.00i	− 0.0498366i
$100$	0	0
$101$	22002.0i	0.214614i	0.994226	+	0.107307i	$0.0342228\pi$
$101$	22002.0i	0.214614i	−0.994226	+	0.107307i	$0.965777\pi$
$102$	0	0
$103$	79264.0	0.736178	0.368089	−	0.929791i	$-0.380012\pi$
$103$	79264.0	0.736178	0.368089	+	0.929791i	$0.380012\pi$
$104$	0	0
$105$	−104544.	−0.925392
$106$	0	0
$107$	− 227988.i	− 1.92510i	−0.271110	−	0.962548i	$-0.587391\pi$
$107$	− 227988.i	− 1.92510i	0.271110	−	0.962548i	$-0.412609\pi$
$108$	0	0
$109$	− 8530.00i	− 0.0687674i	−0.999409	−	0.0343837i	$-0.989053\pi$
$109$	− 8530.00i	− 0.0687674i	0.999409	−	0.0343837i	$-0.0109468\pi$
$110$	0	0
$111$	76122.0	0.586412
$112$	0	0
$113$	−195438.	−1.43984	−0.719918	−	0.694059i	$-0.755821\pi$
$113$	−195438.	−1.43984	−0.719918	+	0.694059i	$0.755821\pi$
$114$	0	0
$115$	− 39600.0i	− 0.279223i
$116$	0	0
$117$	53298.0i	0.359953i
$118$	0	0
$119$	72864.0	0.471678
$120$	0	0
$121$	157451.	0.977647
$122$	0	0
$123$	172746.i	1.02954i
$124$	0	0
$125$	125004.i	0.715565i
$126$	0	0
$127$	173000.	0.951780	0.475890	−	0.879505i	$-0.342126\pi$
$127$	173000.	0.951780	0.475890	+	0.879505i	$0.342126\pi$
$128$	0	0
$129$	−119844.	−0.634077
$130$	0	0
$131$	151260.i	0.770098i	0.922896	+	0.385049i	$0.125815\pi$
$131$	151260.i	0.770098i	−0.922896	+	0.385049i	$0.874185\pi$
$132$	0	0
$133$	− 168256.i	− 0.824786i
$134$	0	0
$135$	−48114.0	−0.227215
$136$	0	0
$137$	128454.	0.584718	0.292359	−	0.956309i	$-0.405560\pi$
$137$	128454.	0.584718	0.292359	+	0.956309i	$0.405560\pi$
$138$	0	0
$139$	− 154196.i	− 0.676918i	−0.940981	−	0.338459i	$-0.890094\pi$
$139$	− 154196.i	− 0.676918i	0.940981	−	0.338459i	$-0.109906\pi$
$140$	0	0
$141$	177120.i	0.750274i
$142$	0	0
$143$	39480.0	0.161450
$144$	0	0
$145$	−367884.	−1.45308
$146$	0	0
$147$	− 127521.i	− 0.486730i
$148$	0	0
$149$	− 29454.0i	− 0.108687i	−0.998522	−	0.0543436i	$-0.982693\pi$
$149$	− 29454.0i	− 0.108687i	0.998522	−	0.0543436i	$-0.0173066\pi$
$150$	0	0
$151$	203872.	0.727638	0.363819	−	0.931470i	$-0.381473\pi$
$151$	203872.	0.727638	0.363819	+	0.931470i	$0.381473\pi$
$152$	0	0
$153$	33534.0	0.115813
$154$	0	0
$155$	− 237072.i	− 0.792594i
$156$	0	0
$157$	136142.i	0.440801i	0.975409	+	0.220401i	$0.0707365\pi$
$157$	136142.i	0.440801i	−0.975409	+	0.220401i	$0.929263\pi$
$158$	0	0
$159$	281394.	0.882718
$160$	0	0
$161$	105600.	0.321070
$162$	0	0
$163$	− 171124.i	− 0.504478i	−0.967665	−	0.252239i	$-0.918833\pi$
$163$	− 171124.i	− 0.504478i	0.967665	−	0.252239i	$-0.0811669\pi$
$164$	0	0
$165$	35640.0i	0.101913i
$166$	0	0
$167$	676200.	1.87622	0.938110	−	0.346336i	$-0.112574\pi$
$167$	676200.	1.87622	0.938110	+	0.346336i	$0.112574\pi$
$168$	0	0
$169$	−61671.0	−0.166098
$170$	0	0
$171$	− 77436.0i	− 0.202513i
$172$	0	0
$173$	133158.i	0.338261i	0.985594	+	0.169131i	$0.0540959\pi$
$173$	133158.i	0.338261i	−0.985594	+	0.169131i	$0.945904\pi$
$174$	0	0
$175$	216656.	0.534781
$176$	0	0
$177$	−237060.	−0.568755
$178$	0	0
$179$	− 693396.i	− 1.61752i	−0.588141	−	0.808758i	$-0.700140\pi$
$179$	− 693396.i	− 1.61752i	0.588141	−	0.808758i	$-0.299860\pi$
$180$	0	0
$181$	− 377174.i	− 0.855747i	−0.903839	−	0.427873i	$-0.859263\pi$
$181$	− 377174.i	− 0.855747i	0.903839	−	0.427873i	$-0.140737\pi$
$182$	0	0
$183$	−279810.	−0.617640
$184$	0	0
$185$	−558228.	−1.19917
$186$	0	0
$187$	− 24840.0i	− 0.0519455i
$188$	0	0
$189$	− 128304.i	− 0.261268i
$190$	0	0
$191$	−265344.	−0.526291	−0.263145	−	0.964756i	$-0.584760\pi$
$191$	−265344.	−0.526291	−0.263145	+	0.964756i	$0.584760\pi$
$192$	0	0
$193$	295298.	0.570647	0.285323	−	0.958431i	$-0.407899\pi$
$193$	295298.	0.570647	0.285323	+	0.958431i	$0.407899\pi$
$194$	0	0
$195$	− 390852.i	− 0.736081i
$196$	0	0
$197$	− 201294.i	− 0.369543i	−0.982781	−	0.184772i	$-0.940845\pi$
$197$	− 201294.i	− 0.369543i	0.982781	−	0.184772i	$-0.0591545\pi$
$198$	0	0
$199$	−652448.	−1.16792	−0.583960	−	0.811782i	$-0.698498\pi$
$199$	−652448.	−1.16792	−0.583960	+	0.811782i	$0.698498\pi$
$200$	0	0
$201$	−151236.	−0.264037
$202$	0	0
$203$	− 981024.i	− 1.67086i
$204$	0	0
$205$	− 1.26680e6i	− 2.10535i
$206$	0	0
$207$	48600.0	0.0788334
$208$	0	0
$209$	−57360.0	−0.0908330
$210$	0	0
$211$	− 1.14706e6i	− 1.77370i	−0.462058	−	0.886850i	$-0.652889\pi$
$211$	− 1.14706e6i	− 1.77370i	0.462058	−	0.886850i	$-0.347111\pi$
$212$	0	0
$213$	55080.0i	0.0831850i
$214$	0	0
$215$	878856.	1.29665
$216$	0	0
$217$	632192.	0.911380
$218$	0	0
$219$	− 230022.i	− 0.324085i
$220$	0	0
$221$	272412.i	0.375185i
$222$	0	0
$223$	701960.	0.945258	0.472629	−	0.881262i	$-0.343305\pi$
$223$	701960.	0.945258	0.472629	+	0.881262i	$0.343305\pi$
$224$	0	0
$225$	99711.0	0.131307
$226$	0	0
$227$	1.23611e6i	1.59218i	0.605179	+	0.796089i	$0.293101\pi$
$227$	1.23611e6i	1.59218i	−0.605179	+	0.796089i	$0.706899\pi$
$228$	0	0
$229$	− 105830.i	− 0.133358i	−0.997774	−	0.0666792i	$-0.978760\pi$
$229$	− 105830.i	− 0.133358i	0.997774	−	0.0666792i	$-0.0212404\pi$
$230$	0	0
$231$	−95040.0	−0.117186
$232$	0	0
$233$	438678.	0.529366	0.264683	−	0.964335i	$-0.414733\pi$
$233$	438678.	0.529366	0.264683	+	0.964335i	$0.414733\pi$
$234$	0	0
$235$	− 1.29888e6i	− 1.53426i
$236$	0	0
$237$	− 669672.i	− 0.774446i
$238$	0	0
$239$	28464.0	0.0322330	0.0161165	−	0.999870i	$-0.494870\pi$
$239$	28464.0	0.0322330	0.0161165	+	0.999870i	$0.494870\pi$
$240$	0	0
$241$	892562.	0.989910	0.494955	−	0.868919i	$-0.335185\pi$
$241$	892562.	0.989910	0.494955	+	0.868919i	$0.335185\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	935154.i	0.995332i
$246$	0	0
$247$	629048.	0.656057
$248$	0	0
$249$	−58212.0	−0.0594996
$250$	0	0
$251$	110124.i	0.110331i	0.998477	+	0.0551655i	$0.0175686\pi$
$251$	110124.i	0.110331i	−0.998477	+	0.0551655i	$0.982431\pi$
$252$	0	0
$253$	− 36000.0i	− 0.0353591i
$254$	0	0
$255$	−245916.	−0.236830
$256$	0	0
$257$	140802.	0.132977	0.0664884	−	0.997787i	$-0.478820\pi$
$257$	140802.	0.132977	0.0664884	+	0.997787i	$0.478820\pi$
$258$	0	0
$259$	− 1.48861e6i	− 1.37889i
$260$	0	0
$261$	− 451494.i	− 0.410252i
$262$	0	0
$263$	938760.	0.836884	0.418442	−	0.908244i	$-0.362576\pi$
$263$	938760.	0.836884	0.418442	+	0.908244i	$0.362576\pi$
$264$	0	0
$265$	−2.06356e6	−1.80510
$266$	0	0
$267$	− 294678.i	− 0.252970i
$268$	0	0
$269$	− 1.11451e6i	− 0.939078i	−0.882912	−	0.469539i	$-0.844420\pi$
$269$	− 1.11451e6i	− 0.939078i	0.882912	−	0.469539i	$-0.155580\pi$
$270$	0	0
$271$	567704.	0.469568	0.234784	−	0.972048i	$-0.424562\pi$
$271$	567704.	0.469568	0.234784	+	0.972048i	$0.424562\pi$
$272$	0	0
$273$	1.04227e6	0.846398
$274$	0	0
$275$	− 73860.0i	− 0.0588949i
$276$	0	0
$277$	1.21326e6i	0.950066i	0.879968	+	0.475033i	$0.157564\pi$
$277$	1.21326e6i	0.950066i	−0.879968	+	0.475033i	$0.842436\pi$
$278$	0	0
$279$	290952.	0.223775
$280$	0	0
$281$	−687738.	−0.519586	−0.259793	−	0.965664i	$-0.583654\pi$
$281$	−687738.	−0.519586	−0.259793	+	0.965664i	$0.583654\pi$
$282$	0	0
$283$	830908.i	0.616718i	0.951270	+	0.308359i	$0.0997799\pi$
$283$	830908.i	0.616718i	−0.951270	+	0.308359i	$0.900220\pi$
$284$	0	0
$285$	567864.i	0.414126i
$286$	0	0
$287$	3.37814e6	2.42088
$288$	0	0
$289$	−1.24846e6	−0.879286
$290$	0	0
$291$	− 1.49474e6i	− 1.03474i
$292$	0	0
$293$	1.31263e6i	0.893248i	0.894722	+	0.446624i	$0.147374\pi$
$293$	1.31263e6i	0.893248i	−0.894722	+	0.446624i	$0.852626\pi$
$294$	0	0
$295$	1.73844e6	1.16307
$296$	0	0
$297$	−43740.0	−0.0287732
$298$	0	0
$299$	394800.i	0.255387i
$300$	0	0
$301$	2.34362e6i	1.49097i
$302$	0	0
$303$	198018.	0.123908
$304$	0	0
$305$	2.05194e6	1.26303
$306$	0	0
$307$	1.69022e6i	1.02352i	0.859128	+	0.511761i	$0.171007\pi$
$307$	1.69022e6i	1.02352i	−0.859128	+	0.511761i	$0.828993\pi$
$308$	0	0
$309$	− 713376.i	− 0.425033i
$310$	0	0
$311$	1.50204e6	0.880604	0.440302	−	0.897850i	$-0.354871\pi$
$311$	1.50204e6	0.880604	0.440302	+	0.897850i	$0.354871\pi$
$312$	0	0
$313$	−810842.	−0.467816	−0.233908	−	0.972259i	$-0.575152\pi$
$313$	−810842.	−0.467816	−0.233908	+	0.972259i	$0.575152\pi$
$314$	0	0
$315$	940896.i	0.534275i
$316$	0	0
$317$	903558.i	0.505019i	0.967594	+	0.252510i	$0.0812559\pi$
$317$	903558.i	0.505019i	−0.967594	+	0.252510i	$0.918744\pi$
$318$	0	0
$319$	−334440.	−0.184010
$320$	0	0
$321$	−2.05189e6	−1.11146
$322$	0	0
$323$	− 395784.i	− 0.211082i
$324$	0	0
$325$	809998.i	0.425379i
$326$	0	0
$327$	−76770.0	−0.0397029
$328$	0	0
$329$	3.46368e6	1.76420
$330$	0	0
$331$	− 1.12197e6i	− 0.562875i	−0.959580	−	0.281438i	$-0.909189\pi$
$331$	− 1.12197e6i	− 0.562875i	0.959580	−	0.281438i	$-0.0908112\pi$
$332$	0	0
$333$	− 685098.i	− 0.338565i
$334$	0	0
$335$	1.10906e6	0.539939
$336$	0	0
$337$	−2.75217e6	−1.32008	−0.660041	−	0.751229i	$-0.729461\pi$
$337$	−2.75217e6	−1.32008	−0.660041	+	0.751229i	$0.729461\pi$
$338$	0	0
$339$	1.75894e6i	0.831289i
$340$	0	0
$341$	− 215520.i	− 0.100369i
$342$	0	0
$343$	464288.	0.213085
$344$	0	0
$345$	−356400.	−0.161209
$346$	0	0
$347$	− 1.91749e6i	− 0.854889i	−0.904042	−	0.427445i	$-0.859414\pi$
$347$	− 1.91749e6i	− 0.854889i	0.904042	−	0.427445i	$-0.140586\pi$
$348$	0	0
$349$	1.83659e6i	0.807140i	0.914949	+	0.403570i	$0.132231\pi$
$349$	1.83659e6i	0.807140i	−0.914949	+	0.403570i	$0.867769\pi$
$350$	0	0
$351$	479682.	0.207819
$352$	0	0
$353$	−622014.	−0.265683	−0.132841	−	0.991137i	$-0.542410\pi$
$353$	−622014.	−0.265683	−0.132841	+	0.991137i	$0.542410\pi$
$354$	0	0
$355$	− 403920.i	− 0.170108i
$356$	0	0
$357$	− 655776.i	− 0.272323i
$358$	0	0
$359$	−3.74062e6	−1.53182	−0.765909	−	0.642949i	$-0.777711\pi$
$359$	−3.74062e6	−1.53182	−0.765909	+	0.642949i	$0.777711\pi$
$360$	0	0
$361$	1.56216e6	0.630897
$362$	0	0
$363$	− 1.41706e6i	− 0.564445i
$364$	0	0
$365$	1.68683e6i	0.662733i
$366$	0	0
$367$	16232.0	0.00629081	0.00314541	−	0.999995i	$-0.498999\pi$
$367$	16232.0	0.00629081	0.00314541	+	0.999995i	$0.498999\pi$
$368$	0	0
$369$	1.55471e6	0.594408
$370$	0	0
$371$	− 5.50282e6i	− 2.07563i
$372$	0	0
$373$	− 293606.i	− 0.109268i	−0.998506	−	0.0546340i	$-0.982601\pi$
$373$	− 293606.i	− 0.109268i	0.998506	−	0.0546340i	$-0.0173992\pi$
$374$	0	0
$375$	1.12504e6	0.413131
$376$	0	0
$377$	3.66769e6	1.32904
$378$	0	0
$379$	− 3.18012e6i	− 1.13722i	−0.822607	−	0.568611i	$-0.807481\pi$
$379$	− 3.18012e6i	− 1.13722i	0.822607	−	0.568611i	$-0.192519\pi$
$380$	0	0
$381$	− 1.55700e6i	− 0.549511i
$382$	0	0
$383$	−2.97984e6	−1.03800	−0.518998	−	0.854775i	$-0.673695\pi$
$383$	−2.97984e6	−1.03800	−0.518998	+	0.854775i	$0.673695\pi$
$384$	0	0
$385$	696960.	0.239638
$386$	0	0
$387$	1.07860e6i	0.366085i
$388$	0	0
$389$	− 3.45977e6i	− 1.15924i	−0.814887	−	0.579620i	$-0.803201\pi$
$389$	− 3.45977e6i	− 1.15924i	0.814887	−	0.579620i	$-0.196799\pi$
$390$	0	0
$391$	248400.	0.0821693
$392$	0	0
$393$	1.36134e6	0.444616
$394$	0	0
$395$	4.91093e6i	1.58369i
$396$	0	0
$397$	− 3.90416e6i	− 1.24323i	−0.783323	−	0.621615i	$-0.786477\pi$
$397$	− 3.90416e6i	− 1.24323i	0.783323	−	0.621615i	$-0.213523\pi$
$398$	0	0
$399$	−1.51430e6	−0.476191
$400$	0	0
$401$	5.44115e6	1.68978	0.844890	−	0.534940i	$-0.179666\pi$
$401$	5.44115e6	1.68978	0.844890	+	0.534940i	$0.179666\pi$
$402$	0	0
$403$	2.36354e6i	0.724936i
$404$	0	0
$405$	433026.i	0.131183i
$406$	0	0
$407$	−507480.	−0.151856
$408$	0	0
$409$	−1.96995e6	−0.582299	−0.291150	−	0.956678i	$-0.594038\pi$
$409$	−1.96995e6	−0.582299	−0.291150	+	0.956678i	$0.594038\pi$
$410$	0	0
$411$	− 1.15609e6i	− 0.337587i
$412$	0	0
$413$	4.63584e6i	1.33738i
$414$	0	0
$415$	426888.	0.121673
$416$	0	0
$417$	−1.38776e6	−0.390819
$418$	0	0
$419$	139020.i	0.0386850i	0.999813	+	0.0193425i	$0.00615729\pi$
$419$	139020.i	0.0386850i	−0.999813	+	0.0193425i	$0.993843\pi$
$420$	0	0
$421$	− 4.32743e6i	− 1.18994i	−0.803748	−	0.594970i	$-0.797164\pi$
$421$	− 4.32743e6i	− 1.18994i	0.803748	−	0.594970i	$-0.202836\pi$
$422$	0	0
$423$	1.59408e6	0.433171
$424$	0	0
$425$	509634.	0.136863
$426$	0	0
$427$	5.47184e6i	1.45232i
$428$	0	0
$429$	− 355320.i	− 0.0932130i
$430$	0	0
$431$	−2.79936e6	−0.725881	−0.362941	−	0.931812i	$-0.618227\pi$
$431$	−2.79936e6	−0.725881	−0.362941	+	0.931812i	$0.618227\pi$
$432$	0	0
$433$	−5.90241e6	−1.51290	−0.756449	−	0.654052i	$-0.773068\pi$
$433$	−5.90241e6	−1.51290	−0.756449	+	0.654052i	$0.773068\pi$
$434$	0	0
$435$	3.31096e6i	0.838939i
$436$	0	0
$437$	− 573600.i	− 0.143683i
$438$	0	0
$439$	446512.	0.110579	0.0552894	−	0.998470i	$-0.482392\pi$
$439$	446512.	0.110579	0.0552894	+	0.998470i	$0.482392\pi$
$440$	0	0
$441$	−1.14769e6	−0.281014
$442$	0	0
$443$	− 3.49525e6i	− 0.846193i	−0.906085	−	0.423096i	$-0.860943\pi$
$443$	− 3.49525e6i	− 0.846193i	0.906085	−	0.423096i	$-0.139057\pi$
$444$	0	0
$445$	2.16097e6i	0.517308i
$446$	0	0
$447$	−265086.	−0.0627506
$448$	0	0
$449$	−1.20613e6	−0.282343	−0.141171	−	0.989985i	$-0.545087\pi$
$449$	−1.20613e6	−0.282343	−0.141171	+	0.989985i	$0.545087\pi$
$450$	0	0
$451$	− 1.15164e6i	− 0.266609i
$452$	0	0
$453$	− 1.83485e6i	− 0.420102i
$454$	0	0
$455$	−7.64333e6	−1.73083
$456$	0	0
$457$	−233546.	−0.0523097	−0.0261548	−	0.999658i	$-0.508326\pi$
$457$	−233546.	−0.0523097	−0.0261548	+	0.999658i	$0.508326\pi$
$458$	0	0
$459$	− 301806.i	− 0.0668646i
$460$	0	0
$461$	− 1.74489e6i	− 0.382398i	−0.981551	−	0.191199i	$-0.938762\pi$
$461$	− 1.74489e6i	− 0.382398i	0.981551	−	0.191199i	$-0.0612376\pi$
$462$	0	0
$463$	−2.91786e6	−0.632576	−0.316288	−	0.948663i	$-0.602437\pi$
$463$	−2.91786e6	−0.632576	−0.316288	+	0.948663i	$0.602437\pi$
$464$	0	0
$465$	−2.13365e6	−0.457605
$466$	0	0
$467$	− 5.31076e6i	− 1.12684i	−0.826169	−	0.563422i	$-0.809484\pi$
$467$	− 5.31076e6i	− 1.12684i	0.826169	−	0.563422i	$-0.190516\pi$
$468$	0	0
$469$	2.95750e6i	0.620859i
$470$	0	0
$471$	1.22528e6	0.254497
$472$	0	0
$473$	798960.	0.164200
$474$	0	0
$475$	− 1.17684e6i	− 0.239322i
$476$	0	0
$477$	− 2.53255e6i	− 0.509638i
$478$	0	0
$479$	2.34466e6	0.466918	0.233459	−	0.972367i	$-0.424996\pi$
$479$	2.34466e6	0.466918	0.233459	+	0.972367i	$0.424996\pi$
$480$	0	0
$481$	5.56536e6	1.09681
$482$	0	0
$483$	− 950400.i	− 0.185370i
$484$	0	0
$485$	1.09614e7i	2.11598i
$486$	0	0
$487$	−9.81531e6	−1.87535	−0.937674	−	0.347517i	$-0.887025\pi$
$487$	−9.81531e6	−1.87535	−0.937674	+	0.347517i	$0.887025\pi$
$488$	0	0
$489$	−1.54012e6	−0.291260
$490$	0	0
$491$	5.94520e6i	1.11292i	0.830876	+	0.556458i	$0.187840\pi$
$491$	5.94520e6i	1.11292i	−0.830876	+	0.556458i	$0.812160\pi$
$492$	0	0
$493$	− 2.30764e6i	− 0.427612i
$494$	0	0
$495$	320760.	0.0588393
$496$	0	0
$497$	1.07712e6	0.195602
$498$	0	0
$499$	6.47832e6i	1.16469i	0.812941	+	0.582346i	$0.197865\pi$
$499$	6.47832e6i	1.16469i	−0.812941	+	0.582346i	$0.802135\pi$
$500$	0	0
$501$	− 6.08580e6i	− 1.08324i
$502$	0	0
$503$	−4.71794e6	−0.831444	−0.415722	−	0.909492i	$-0.636471\pi$
$503$	−4.71794e6	−0.831444	−0.415722	+	0.909492i	$0.636471\pi$
$504$	0	0
$505$	−1.45213e6	−0.253383
$506$	0	0
$507$	555039.i	0.0958967i
$508$	0	0
$509$	− 1.90771e6i	− 0.326375i	−0.986595	−	0.163188i	$-0.947822\pi$
$509$	− 1.90771e6i	− 0.326375i	0.986595	−	0.163188i	$-0.0521776\pi$
$510$	0	0
$511$	−4.49821e6	−0.762057
$512$	0	0
$513$	−696924.	−0.116921
$514$	0	0
$515$	5.23142e6i	0.869164i
$516$	0	0
$517$	− 1.18080e6i	− 0.194290i
$518$	0	0
$519$	1.19842e6	0.195295
$520$	0	0
$521$	−8.01974e6	−1.29439	−0.647196	−	0.762324i	$-0.724059\pi$
$521$	−8.01974e6	−1.29439	−0.647196	+	0.762324i	$0.724059\pi$
$522$	0	0
$523$	− 1.91162e6i	− 0.305596i	−0.988257	−	0.152798i	$-0.951172\pi$
$523$	− 1.91162e6i	− 0.305596i	0.988257	−	0.152798i	$-0.0488284\pi$
$524$	0	0
$525$	− 1.94990e6i	− 0.308756i
$526$	0	0
$527$	1.48709e6	0.233244
$528$	0	0
$529$	−6.07634e6	−0.944068
$530$	0	0
$531$	2.13354e6i	0.328371i
$532$	0	0
$533$	1.26297e7i	1.92563i
$534$	0	0
$535$	1.50472e7	2.27285
$536$	0	0
$537$	−6.24056e6	−0.933874
$538$	0	0
$539$	850140.i	0.126043i
$540$	0	0
$541$	− 1.19900e7i	− 1.76128i	−0.473788	−	0.880639i	$-0.657114\pi$
$541$	− 1.19900e7i	− 1.76128i	0.473788	−	0.880639i	$-0.342886\pi$
$542$	0	0
$543$	−3.39457e6	−0.494066
$544$	0	0
$545$	562980.	0.0811898
$546$	0	0
$547$	4.45809e6i	0.637061i	0.947913	+	0.318530i	$0.103189\pi$
$547$	4.45809e6i	0.637061i	−0.947913	+	0.318530i	$0.896811\pi$
$548$	0	0
$549$	2.51829e6i	0.356595i
$550$	0	0
$551$	−5.32874e6	−0.747732
$552$	0	0
$553$	−1.30958e7	−1.82104
$554$	0	0
$555$	5.02405e6i	0.692344i
$556$	0	0
$557$	9.02612e6i	1.23272i	0.787466	+	0.616358i	$0.211393\pi$
$557$	9.02612e6i	1.23272i	−0.787466	+	0.616358i	$0.788607\pi$
$558$	0	0
$559$	−8.76193e6	−1.18596
$560$	0	0
$561$	−223560.	−0.0299907
$562$	0	0
$563$	6.84899e6i	0.910658i	0.890323	+	0.455329i	$0.150478\pi$
$563$	6.84899e6i	0.910658i	−0.890323	+	0.455329i	$0.849522\pi$
$564$	0	0
$565$	− 1.28989e7i	− 1.69993i
$566$	0	0
$567$	−1.15474e6	−0.150843
$568$	0	0
$569$	5.46322e6	0.707405	0.353703	−	0.935358i	$-0.384923\pi$
$569$	5.46322e6	0.707405	0.353703	+	0.935358i	$0.384923\pi$
$570$	0	0
$571$	1.02324e7i	1.31337i	0.754166	+	0.656684i	$0.228041\pi$
$571$	1.02324e7i	1.31337i	−0.754166	+	0.656684i	$0.771959\pi$
$572$	0	0
$573$	2.38810e6i	0.303854i
$574$	0	0
$575$	738600.	0.0931622
$576$	0	0
$577$	1.59437e7	1.99365	0.996825	−	0.0796186i	$-0.0253702\pi$
$577$	1.59437e7	1.99365	0.996825	+	0.0796186i	$0.0253702\pi$
$578$	0	0
$579$	− 2.65768e6i	− 0.329463i
$580$	0	0
$581$	1.13837e6i	0.139908i
$582$	0	0
$583$	−1.87596e6	−0.228587
$584$	0	0
$585$	−3.51767e6	−0.424977
$586$	0	0
$587$	9.47713e6i	1.13522i	0.823296	+	0.567612i	$0.192133\pi$
$587$	9.47713e6i	1.13522i	−0.823296	+	0.567612i	$0.807867\pi$
$588$	0	0
$589$	− 3.43395e6i	− 0.407855i
$590$	0	0
$591$	−1.81165e6	−0.213356
$592$	0	0
$593$	2.45349e6	0.286515	0.143258	−	0.989685i	$-0.454242\pi$
$593$	2.45349e6	0.286515	0.143258	+	0.989685i	$0.454242\pi$
$594$	0	0
$595$	4.80902e6i	0.556884i
$596$	0	0
$597$	5.87203e6i	0.674299i
$598$	0	0
$599$	9.29978e6	1.05902	0.529512	−	0.848302i	$-0.322375\pi$
$599$	9.29978e6	1.05902	0.529512	+	0.848302i	$0.322375\pi$
$600$	0	0
$601$	1.14617e7	1.29438	0.647192	−	0.762327i	$-0.275943\pi$
$601$	1.14617e7	1.29438	0.647192	+	0.762327i	$0.275943\pi$
$602$	0	0
$603$	1.36112e6i	0.152442i
$604$	0	0
$605$	1.03918e7i	1.15425i
$606$	0	0
$607$	1.12784e7	1.24244	0.621219	−	0.783637i	$-0.286638\pi$
$607$	1.12784e7	1.24244	0.621219	+	0.783637i	$0.286638\pi$
$608$	0	0
$609$	−8.82922e6	−0.964670
$610$	0	0
$611$	1.29494e7i	1.40329i
$612$	0	0
$613$	− 93782.0i	− 0.0100802i	−0.999987	−	0.00504009i	$-0.998396\pi$
$613$	− 93782.0i	− 0.0100802i	0.999987	−	0.00504009i	$-0.00160432\pi$
$614$	0	0
$615$	−1.14012e7	−1.21553
$616$	0	0
$617$	1.49642e7	1.58248	0.791242	−	0.611504i	$-0.209435\pi$
$617$	1.49642e7	1.58248	0.791242	+	0.611504i	$0.209435\pi$
$618$	0	0
$619$	5.06888e6i	0.531723i	0.964011	+	0.265861i	$0.0856563\pi$
$619$	5.06888e6i	0.531723i	−0.964011	+	0.265861i	$0.914344\pi$
$620$	0	0
$621$	− 437400.i	− 0.0455145i
$622$	0	0
$623$	−5.76259e6	−0.594837
$624$	0	0
$625$	−1.20971e7	−1.23875
$626$	0	0
$627$	516240.i	0.0524424i
$628$	0	0
$629$	− 3.50161e6i	− 0.352892i
$630$	0	0
$631$	−1.55919e7	−1.55892	−0.779462	−	0.626450i	$-0.784507\pi$
$631$	−1.55919e7	−1.55892	−0.779462	+	0.626450i	$0.784507\pi$
$632$	0	0
$633$	−1.03235e7	−1.02405
$634$	0	0
$635$	1.14180e7i	1.12371i
$636$	0	0
$637$	− 9.32320e6i	− 0.910367i
$638$	0	0
$639$	495720.	0.0480269
$640$	0	0
$641$	1.09701e7	1.05455	0.527274	−	0.849695i	$-0.323214\pi$
$641$	1.09701e7	1.05455	0.527274	+	0.849695i	$0.323214\pi$
$642$	0	0
$643$	− 2.83704e6i	− 0.270607i	−0.990804	−	0.135303i	$-0.956799\pi$
$643$	− 2.83704e6i	− 0.270607i	0.990804	−	0.135303i	$-0.0432009\pi$
$644$	0	0
$645$	− 7.90970e6i	− 0.748619i
$646$	0	0
$647$	6.05686e6	0.568835	0.284418	−	0.958700i	$-0.408200\pi$
$647$	6.05686e6	0.568835	0.284418	+	0.958700i	$0.408200\pi$
$648$	0	0
$649$	1.58040e6	0.147284
$650$	0	0
$651$	− 5.68973e6i	− 0.526186i
$652$	0	0
$653$	− 1.08892e6i	− 0.0999341i	−0.998751	−	0.0499671i	$-0.984088\pi$
$653$	− 1.08892e6i	− 0.0999341i	0.998751	−	0.0499671i	$-0.0159116\pi$
$654$	0	0
$655$	−9.98316e6	−0.909211
$656$	0	0
$657$	−2.07020e6	−0.187111
$658$	0	0
$659$	7.41803e6i	0.665388i	0.943035	+	0.332694i	$0.107958\pi$
$659$	7.41803e6i	0.665388i	−0.943035	+	0.332694i	$0.892042\pi$
$660$	0	0
$661$	− 767654.i	− 0.0683379i	−0.999416	−	0.0341690i	$-0.989122\pi$
$661$	− 767654.i	− 0.0683379i	0.999416	−	0.0341690i	$-0.0108784\pi$
$662$	0	0
$663$	2.45171e6	0.216613
$664$	0	0
$665$	1.11049e7	0.973779
$666$	0	0
$667$	− 3.34440e6i	− 0.291074i
$668$	0	0
$669$	− 6.31764e6i	− 0.545745i
$670$	0	0
$671$	1.86540e6	0.159943
$672$	0	0
$673$	1.42263e6	0.121075	0.0605373	−	0.998166i	$-0.480719\pi$
$673$	1.42263e6	0.121075	0.0605373	+	0.998166i	$0.480719\pi$
$674$	0	0
$675$	− 897399.i	− 0.0758099i
$676$	0	0
$677$	6.16231e6i	0.516739i	0.966046	+	0.258370i	$0.0831853\pi$
$677$	6.16231e6i	0.516739i	−0.966046	+	0.258370i	$0.916815\pi$
$678$	0	0
$679$	−2.92304e7	−2.43310
$680$	0	0
$681$	1.11250e7	0.919245
$682$	0	0
$683$	− 1.50621e7i	− 1.23548i	−0.786383	−	0.617739i	$-0.788049\pi$
$683$	− 1.50621e7i	− 1.23548i	0.786383	−	0.617739i	$-0.211951\pi$
$684$	0	0
$685$	8.47796e6i	0.690343i
$686$	0	0
$687$	−952470.	−0.0769945
$688$	0	0
$689$	2.05730e7	1.65101
$690$	0	0
$691$	− 5.87636e6i	− 0.468180i	−0.972215	−	0.234090i	$-0.924789\pi$
$691$	− 5.87636e6i	− 0.468180i	0.972215	−	0.234090i	$-0.0752111\pi$
$692$	0	0
$693$	855360.i	0.0676575i
$694$	0	0
$695$	1.01769e7	0.799199
$696$	0	0
$697$	7.94632e6	0.619561
$698$	0	0
$699$	− 3.94810e6i	− 0.305630i
$700$	0	0
$701$	3.60077e6i	0.276758i	0.990379	+	0.138379i	$0.0441893\pi$
$701$	3.60077e6i	0.276758i	−0.990379	+	0.138379i	$0.955811\pi$
$702$	0	0
$703$	−8.08585e6	−0.617074
$704$	0	0
$705$	−1.16899e7	−0.885806
$706$	0	0
$707$	− 3.87235e6i	− 0.291358i
$708$	0	0
$709$	− 9.22516e6i	− 0.689221i	−0.938746	−	0.344610i	$-0.888011\pi$
$709$	− 9.22516e6i	− 0.689221i	0.938746	−	0.344610i	$-0.111989\pi$
$710$	0	0
$711$	−6.02705e6	−0.447127
$712$	0	0
$713$	2.15520e6	0.158768
$714$	0	0
$715$	2.60568e6i	0.190615i
$716$	0	0
$717$	− 256176.i	− 0.0186098i
$718$	0	0
$719$	−2.63923e7	−1.90395	−0.951975	−	0.306177i	$-0.900950\pi$
$719$	−2.63923e7	−1.90395	−0.951975	+	0.306177i	$0.900950\pi$
$720$	0	0
$721$	−1.39505e7	−0.999426
$722$	0	0
$723$	− 8.03306e6i	− 0.571525i
$724$	0	0
$725$	− 6.86159e6i	− 0.484819i
$726$	0	0
$727$	9.79485e6	0.687324	0.343662	−	0.939093i	$-0.388333\pi$
$727$	9.79485e6	0.687324	0.343662	+	0.939093i	$0.388333\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	5.51282e6i	0.381576i
$732$	0	0
$733$	4.07584e6i	0.280193i	0.990138	+	0.140096i	$0.0447412\pi$
$733$	4.07584e6i	0.280193i	−0.990138	+	0.140096i	$0.955259\pi$
$734$	0	0
$735$	8.41639e6	0.574655
$736$	0	0
$737$	1.00824e6	0.0683747
$738$	0	0
$739$	− 1.65709e7i	− 1.11618i	−0.829781	−	0.558089i	$-0.811535\pi$
$739$	− 1.65709e7i	− 1.11618i	0.829781	−	0.558089i	$-0.188465\pi$
$740$	0	0
$741$	− 5.66143e6i	− 0.378775i
$742$	0	0
$743$	−1.44141e7	−0.957892	−0.478946	−	0.877844i	$-0.658981\pi$
$743$	−1.44141e7	−0.957892	−0.478946	+	0.877844i	$0.658981\pi$
$744$	0	0
$745$	1.94396e6	0.128321
$746$	0	0
$747$	523908.i	0.0343521i
$748$	0	0
$749$	4.01259e7i	2.61349i
$750$	0	0
$751$	1.67944e7	1.08659	0.543295	−	0.839542i	$-0.317177\pi$
$751$	1.67944e7	1.08659	0.543295	+	0.839542i	$0.317177\pi$
$752$	0	0
$753$	991116.	0.0636997
$754$	0	0
$755$	1.34556e7i	0.859081i
$756$	0	0
$757$	− 1.32943e7i	− 0.843188i	−0.906785	−	0.421594i	$-0.861471\pi$
$757$	− 1.32943e7i	− 0.843188i	0.906785	−	0.421594i	$-0.138529\pi$
$758$	0	0
$759$	−324000.	−0.0204146
$760$	0	0
$761$	2.14786e6	0.134445	0.0672225	−	0.997738i	$-0.478586\pi$
$761$	2.14786e6	0.134445	0.0672225	+	0.997738i	$0.478586\pi$
$762$	0	0
$763$	1.50128e6i	0.0933577i
$764$	0	0
$765$	2.21324e6i	0.136734i
$766$	0	0
$767$	−1.73317e7	−1.06378
$768$	0	0
$769$	−1.31059e7	−0.799193	−0.399596	−	0.916691i	$-0.630850\pi$
$769$	−1.31059e7	−0.799193	−0.399596	+	0.916691i	$0.630850\pi$
$770$	0	0
$771$	− 1.26722e6i	− 0.0767742i
$772$	0	0
$773$	2.37154e7i	1.42752i	0.700392	+	0.713759i	$0.253009\pi$
$773$	2.37154e7i	1.42752i	−0.700392	+	0.713759i	$0.746991\pi$
$774$	0	0
$775$	4.42175e6	0.264448
$776$	0	0
$777$	−1.33975e7	−0.796105
$778$	0	0
$779$	− 1.83495e7i	− 1.08338i
$780$	0	0
$781$	− 367200.i	− 0.0215415i
$782$	0	0
$783$	−4.06345e6	−0.236859
$784$	0	0
$785$	−8.98537e6	−0.520430
$786$	0	0
$787$	− 8.40048e6i	− 0.483468i	−0.970343	−	0.241734i	$-0.922284\pi$
$787$	− 8.40048e6i	− 0.483468i	0.970343	−	0.241734i	$-0.0777161\pi$
$788$	0	0
$789$	− 8.44884e6i	− 0.483175i
$790$	0	0
$791$	3.43971e7	1.95470
$792$	0	0
$793$	−2.04572e7	−1.15522
$794$	0	0
$795$	1.85720e7i	1.04218i
$796$	0	0
$797$	5.41023e6i	0.301696i	0.988557	+	0.150848i	$0.0482004\pi$
$797$	5.41023e6i	0.301696i	−0.988557	+	0.150848i	$0.951800\pi$
$798$	0	0
$799$	8.14752e6	0.451501
$800$	0	0
$801$	−2.65210e6	−0.146052
$802$	0	0
$803$	1.53348e6i	0.0839246i
$804$	0	0
$805$	6.96960e6i	0.379069i
$806$	0	0
$807$	−1.00306e7	−0.542177
$808$	0	0
$809$	2.60777e7	1.40087	0.700436	−	0.713715i	$-0.252989\pi$
$809$	2.60777e7	1.40087	0.700436	+	0.713715i	$0.252989\pi$
$810$	0	0
$811$	− 1.90021e7i	− 1.01449i	−0.861800	−	0.507247i	$-0.830663\pi$
$811$	− 1.90021e7i	− 1.01449i	0.861800	−	0.507247i	$-0.169337\pi$
$812$	0	0
$813$	− 5.10934e6i	− 0.271105i
$814$	0	0
$815$	1.12942e7	0.595608
$816$	0	0
$817$	1.27301e7	0.667231
$818$	0	0
$819$	− 9.38045e6i	− 0.488668i
$820$	0	0
$821$	3.10173e7i	1.60600i	0.595978	+	0.803001i	$0.296764\pi$
$821$	3.10173e7i	1.60600i	−0.595978	+	0.803001i	$0.703236\pi$
$822$	0	0
$823$	1.56290e7	0.804323	0.402162	−	0.915569i	$-0.368259\pi$
$823$	1.56290e7	0.804323	0.402162	+	0.915569i	$0.368259\pi$
$824$	0	0
$825$	−664740.	−0.0340030
$826$	0	0
$827$	− 1.58421e7i	− 0.805467i	−0.915317	−	0.402733i	$-0.868060\pi$
$827$	− 1.58421e7i	− 0.805467i	0.915317	−	0.402733i	$-0.131940\pi$
$828$	0	0
$829$	2.06176e6i	0.104196i	0.998642	+	0.0520980i	$0.0165908\pi$
$829$	2.06176e6i	0.104196i	−0.998642	+	0.0520980i	$0.983409\pi$
$830$	0	0
$831$	1.09193e7	0.548521
$832$	0	0
$833$	−5.86597e6	−0.292905
$834$	0	0
$835$	4.46292e7i	2.21515i
$836$	0	0
$837$	− 2.61857e6i	− 0.129196i
$838$	0	0
$839$	−3.03900e7	−1.49048	−0.745240	−	0.666796i	$-0.767665\pi$
$839$	−3.03900e7	−1.49048	−0.745240	+	0.666796i	$0.767665\pi$
$840$	0	0
$841$	−1.05583e7	−0.514760
$842$	0	0
$843$	6.18964e6i	0.299983i
$844$	0	0
$845$	− 4.07029e6i	− 0.196103i
$846$	0	0
$847$	−2.77114e7	−1.32724
$848$	0	0
$849$	7.47817e6	0.356062
$850$	0	0
$851$	− 5.07480e6i	− 0.240212i
$852$	0	0
$853$	2.97738e7i	1.40108i	0.713615	+	0.700538i	$0.247056\pi$
$853$	2.97738e7i	1.40108i	−0.713615	+	0.700538i	$0.752944\pi$
$854$	0	0
$855$	5.11078e6	0.239096
$856$	0	0
$857$	−8.64100e6	−0.401894	−0.200947	−	0.979602i	$-0.564402\pi$
$857$	−8.64100e6	−0.401894	−0.200947	+	0.979602i	$0.564402\pi$
$858$	0	0
$859$	3.35663e7i	1.55210i	0.630670	+	0.776051i	$0.282780\pi$
$859$	3.35663e7i	1.55210i	−0.630670	+	0.776051i	$0.717220\pi$
$860$	0	0
$861$	− 3.04033e7i	− 1.39770i
$862$	0	0
$863$	3.90191e7	1.78341	0.891703	−	0.452621i	$-0.149511\pi$
$863$	3.90191e7	1.78341	0.891703	+	0.452621i	$0.149511\pi$
$864$	0	0
$865$	−8.78843e6	−0.399366
$866$	0	0
$867$	1.12361e7i	0.507656i
$868$	0	0
$869$	4.46448e6i	0.200549i
$870$	0	0
$871$	−1.10570e7	−0.493848
$872$	0	0
$873$	−1.34526e7	−0.597409
$874$	0	0
$875$	− 2.20007e7i	− 0.971441i
$876$	0	0
$877$	− 1.81382e7i	− 0.796333i	−0.917313	−	0.398166i	$-0.869647\pi$
$877$	− 1.81382e7i	− 0.796333i	0.917313	−	0.398166i	$-0.130353\pi$
$878$	0	0
$879$	1.18136e7	0.515717
$880$	0	0
$881$	3.05312e7	1.32527	0.662634	−	0.748943i	$-0.269438\pi$
$881$	3.05312e7	1.32527	0.662634	+	0.748943i	$0.269438\pi$
$882$	0	0
$883$	− 4.35533e7i	− 1.87983i	−0.341405	−	0.939916i	$-0.610903\pi$
$883$	− 4.35533e7i	− 1.87983i	0.341405	−	0.939916i	$-0.389097\pi$
$884$	0	0
$885$	− 1.56460e7i	− 0.671497i
$886$	0	0
$887$	1.34152e7	0.572515	0.286257	−	0.958153i	$-0.407589\pi$
$887$	1.34152e7	0.572515	0.286257	+	0.958153i	$0.407589\pi$
$888$	0	0
$889$	−3.04480e7	−1.29212
$890$	0	0
$891$	393660.i	0.0166122i
$892$	0	0
$893$	− 1.88141e7i	− 0.789504i
$894$	0	0
$895$	4.57641e7	1.90971
$896$	0	0
$897$	3.55320e6	0.147448
$898$	0	0
$899$	− 2.00218e7i	− 0.826236i
$900$	0	0
$901$	− 1.29441e7i	− 0.531203i
$902$	0	0
$903$	2.10925e7	0.860815
$904$	0	0
$905$	2.48935e7	1.01033
$906$	0	0
$907$	− 3.10816e6i	− 0.125454i	−0.998031	−	0.0627272i	$-0.980020\pi$
$907$	− 3.10816e6i	− 0.125454i	0.998031	−	0.0627272i	$-0.0199798\pi$
$908$	0	0
$909$	− 1.78216e6i	− 0.0715381i
$910$	0	0
$911$	1.19035e6	0.0475203	0.0237602	−	0.999718i	$-0.492436\pi$
$911$	1.19035e6	0.0475203	0.0237602	+	0.999718i	$0.492436\pi$
$912$	0	0
$913$	388080.	0.0154079
$914$	0	0
$915$	− 1.84675e7i	− 0.729213i
$916$	0	0
$917$	− 2.66218e7i	− 1.04547i
$918$	0	0
$919$	4.71996e7	1.84353	0.921764	−	0.387752i	$-0.126748\pi$
$919$	4.71996e7	1.84353	0.921764	+	0.387752i	$0.126748\pi$
$920$	0	0
$921$	1.52120e7	0.590931
$922$	0	0
$923$	4.02696e6i	0.155587i
$924$	0	0
$925$	− 1.04118e7i	− 0.400103i
$926$	0	0
$927$	−6.42038e6	−0.245393
$928$	0	0
$929$	1.33595e6	0.0507870	0.0253935	−	0.999678i	$-0.491916\pi$
$929$	1.33595e6	0.0507870	0.0253935	+	0.999678i	$0.491916\pi$
$930$	0	0
$931$	1.35456e7i	0.512180i
$932$	0	0
$933$	− 1.35184e7i	− 0.508417i
$934$	0	0
$935$	1.63944e6	0.0613291
$936$	0	0
$937$	−1.47238e7	−0.547861	−0.273931	−	0.961749i	$-0.588324\pi$
$937$	−1.47238e7	−0.547861	−0.273931	+	0.961749i	$0.588324\pi$
$938$	0	0
$939$	7.29758e6i	0.270094i
$940$	0	0
$941$	− 2.69196e7i	− 0.991049i	−0.868594	−	0.495525i	$-0.834976\pi$
$941$	− 2.69196e7i	− 0.991049i	0.868594	−	0.495525i	$-0.165024\pi$
$942$	0	0
$943$	1.15164e7	0.421733
$944$	0	0
$945$	8.46806e6	0.308464
$946$	0	0
$947$	− 3.73160e6i	− 0.135214i	−0.997712	−	0.0676068i	$-0.978464\pi$
$947$	− 3.73160e6i	− 0.135214i	0.997712	−	0.0676068i	$-0.0215364\pi$
$948$	0	0
$949$	− 1.68172e7i	− 0.606160i
$950$	0	0
$951$	8.13202e6	0.291573
$952$	0	0
$953$	−2.18735e7	−0.780166	−0.390083	−	0.920780i	$-0.627554\pi$
$953$	−2.18735e7	−0.780166	−0.390083	+	0.920780i	$0.627554\pi$
$954$	0	0
$955$	− 1.75127e7i	− 0.621362i
$956$	0	0
$957$	3.00996e6i	0.106238i
$958$	0	0
$959$	−2.26079e7	−0.793805
$960$	0	0
$961$	−1.57267e7	−0.549324
$962$	0	0
$963$	1.84670e7i	0.641699i
$964$	0	0
$965$	1.94897e7i	0.673730i
$966$	0	0
$967$	−1.76025e7	−0.605352	−0.302676	−	0.953093i	$-0.597880\pi$
$967$	−1.76025e7	−0.605352	−0.302676	+	0.953093i	$0.597880\pi$
$968$	0	0
$969$	−3.56206e6	−0.121868
$970$	0	0
$971$	− 1.67317e7i	− 0.569497i	−0.958602	−	0.284749i	$-0.908090\pi$
$971$	− 1.67317e7i	− 0.569497i	0.958602	−	0.284749i	$-0.0919101\pi$
$972$	0	0
$973$	2.71385e7i	0.918975i
$974$	0	0
$975$	7.28998e6	0.245592
$976$	0	0
$977$	5.55382e7	1.86147	0.930733	−	0.365699i	$-0.119170\pi$
$977$	5.55382e7	1.86147	0.930733	+	0.365699i	$0.119170\pi$
$978$	0	0
$979$	1.96452e6i	0.0655088i
$980$	0	0
$981$	690930.i	0.0229225i
$982$	0	0
$983$	3.86784e7	1.27669	0.638344	−	0.769751i	$-0.279620\pi$
$983$	3.86784e7	1.27669	0.638344	+	0.769751i	$0.279620\pi$
$984$	0	0
$985$	1.32854e7	0.436299
$986$	0	0
$987$	− 3.11731e7i	− 1.01856i
$988$	0	0
$989$	7.98960e6i	0.259737i
$990$	0	0
$991$	9.58498e6	0.310033	0.155016	−	0.987912i	$-0.450457\pi$
$991$	9.58498e6	0.310033	0.155016	+	0.987912i	$0.450457\pi$
$992$	0	0
$993$	−1.00977e7	−0.324976
$994$	0	0
$995$	− 4.30616e7i	− 1.37890i
$996$	0	0
$997$	1.03650e7i	0.330242i	0.986273	+	0.165121i	$0.0528015\pi$
$997$	1.03650e7i	0.330242i	−0.986273	+	0.165121i	$0.947198\pi$
$998$	0	0
$999$	−6.16588e6	−0.195471

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.d.c.385.1		2
4.3	odd	2		768.6.d.p.385.2		2
8.3	odd	2		768.6.d.p.385.1		2
8.5	even	2	inner	768.6.d.c.385.2		2
16.3	odd	4		192.6.a.g.1.1		1
16.5	even	4		6.6.a.a.1.1	✓	1
16.11	odd	4		48.6.a.c.1.1		1
16.13	even	4		192.6.a.o.1.1		1
48.5	odd	4		18.6.a.b.1.1		1
48.11	even	4		144.6.a.j.1.1		1
48.29	odd	4		576.6.a.j.1.1		1
48.35	even	4		576.6.a.i.1.1		1
80.37	odd	4		150.6.c.b.49.2		2
80.53	odd	4		150.6.c.b.49.1		2
80.69	even	4		150.6.a.d.1.1		1
112.5	odd	12		294.6.e.a.67.1		2
112.37	even	12		294.6.e.g.67.1		2
112.53	even	12		294.6.e.g.79.1		2
112.69	odd	4		294.6.a.m.1.1		1
112.101	odd	12		294.6.e.a.79.1		2
144.5	odd	12		162.6.c.h.55.1		2
144.85	even	12		162.6.c.e.55.1		2
144.101	odd	12		162.6.c.h.109.1		2
144.133	even	12		162.6.c.e.109.1		2
176.21	odd	4		726.6.a.a.1.1		1
208.181	even	4		1014.6.a.c.1.1		1
240.53	even	4		450.6.c.j.199.2		2
240.149	odd	4		450.6.a.m.1.1		1
240.197	even	4		450.6.c.j.199.1		2
336.293	even	4		882.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.6.a.a.1.1	✓	1	16.5	even	4
18.6.a.b.1.1		1	48.5	odd	4
48.6.a.c.1.1		1	16.11	odd	4
144.6.a.j.1.1		1	48.11	even	4
150.6.a.d.1.1		1	80.69	even	4
150.6.c.b.49.1		2	80.53	odd	4
150.6.c.b.49.2		2	80.37	odd	4
162.6.c.e.55.1		2	144.85	even	12
162.6.c.e.109.1		2	144.133	even	12
162.6.c.h.55.1		2	144.5	odd	12
162.6.c.h.109.1		2	144.101	odd	12
192.6.a.g.1.1		1	16.3	odd	4
192.6.a.o.1.1		1	16.13	even	4
294.6.a.m.1.1		1	112.69	odd	4
294.6.e.a.67.1		2	112.5	odd	12
294.6.e.a.79.1		2	112.101	odd	12
294.6.e.g.67.1		2	112.37	even	12
294.6.e.g.79.1		2	112.53	even	12
450.6.a.m.1.1		1	240.149	odd	4
450.6.c.j.199.1		2	240.197	even	4
450.6.c.j.199.2		2	240.53	even	4
576.6.a.i.1.1		1	48.35	even	4
576.6.a.j.1.1		1	48.29	odd	4
726.6.a.a.1.1		1	176.21	odd	4
768.6.d.c.385.1		2	1.1	even	1	trivial
768.6.d.c.385.2		2	8.5	even	2	inner
768.6.d.p.385.1		2	8.3	odd	2
768.6.d.p.385.2		2	4.3	odd	2
882.6.a.a.1.1		1	336.293	even	4
1014.6.a.c.1.1		1	208.181	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} +66.0000i q^{5} -176.000 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} +66.0000i q^{5} -176.000 q^{7} -81.0000 q^{9} +60.0000i q^{11} -658.000i q^{13} +594.000 q^{15} -414.000 q^{17} +956.000i q^{19} +1584.00i q^{21} -600.000 q^{23} -1231.00 q^{25} +729.000i q^{27} +5574.00i q^{29} -3592.00 q^{31} +540.000 q^{33} -11616.0i q^{35} +8458.00i q^{37} -5922.00 q^{39} -19194.0 q^{41} -13316.0i q^{43} -5346.00i q^{45} -19680.0 q^{47} +14169.0 q^{49} +3726.00i q^{51} +31266.0i q^{53} -3960.00 q^{55} +8604.00 q^{57} -26340.0i q^{59} -31090.0i q^{61} +14256.0 q^{63} +43428.0 q^{65} -16804.0i q^{67} +5400.00i q^{69} -6120.00 q^{71} +25558.0 q^{73} +11079.0i q^{75} -10560.0i q^{77} +74408.0 q^{79} +6561.00 q^{81} -6468.00i q^{83} -27324.0i q^{85} +50166.0 q^{87} +32742.0 q^{89} +115808. i q^{91} +32328.0i q^{93} -63096.0 q^{95} +166082. q^{97} -4860.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 352 q^{7} - 162 q^{9}+O(q^{10}) \) 2 * q - 352 * q^7 - 162 * q^9	\( 2 q - 352 q^{7} - 162 q^{9} + 1188 q^{15} - 828 q^{17} - 1200 q^{23} - 2462 q^{25} - 7184 q^{31} + 1080 q^{33} - 11844 q^{39} - 38388 q^{41} - 39360 q^{47} + 28338 q^{49} - 7920 q^{55} + 17208 q^{57} + 28512 q^{63} + 86856 q^{65} - 12240 q^{71} + 51116 q^{73} + 148816 q^{79} + 13122 q^{81} + 100332 q^{87} + 65484 q^{89} - 126192 q^{95} + 332164 q^{97}+O(q^{100}) \) 2 * q - 352 * q^7 - 162 * q^9 + 1188 * q^15 - 828 * q^17 - 1200 * q^23 - 2462 * q^25 - 7184 * q^31 + 1080 * q^33 - 11844 * q^39 - 38388 * q^41 - 39360 * q^47 + 28338 * q^49 - 7920 * q^55 + 17208 * q^57 + 28512 * q^63 + 86856 * q^65 - 12240 * q^71 + 51116 * q^73 + 148816 * q^79 + 13122 * q^81 + 100332 * q^87 + 65484 * q^89 - 126192 * q^95 + 332164 * q^97

Introduction

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