LMFDB - Embedded newform 768.3.g.h.511.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(511,768)]

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, 0, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("768.511");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.g (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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.5
Root			$1.20036 - 0.747754i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.3.g.h.511.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -7.98203 q^{5} +2.13878i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -7.98203 q^{5} +2.13878i q^{7} -3.00000 q^{9} -8.00000i q^{11} -11.6865 q^{13} -13.8253i q^{15} +11.8564 q^{17} -14.9282i q^{19} -3.70447 q^{21} -4.27756i q^{23} +38.7128 q^{25} -5.19615i q^{27} -0.573084 q^{29} +57.4399i q^{31} +13.8564 q^{33} -17.0718i q^{35} +27.6506 q^{37} -20.2416i q^{39} +31.5692 q^{41} +28.7846i q^{43} +23.9461 q^{45} -59.5787i q^{47} +44.4256 q^{49} +20.5359i q^{51} +31.3550 q^{53} +63.8562i q^{55} +25.8564 q^{57} -52.7846i q^{59} +59.5787 q^{61} -6.41634i q^{63} +93.2820 q^{65} +84.7846i q^{67} +7.40895 q^{69} -42.4685i q^{71} +5.42563 q^{73} +67.0526i q^{75} +17.1102 q^{77} -44.6072i q^{79} +9.00000 q^{81} -67.7128i q^{83} -94.6382 q^{85} -0.992611i q^{87} +133.138 q^{89} -24.9948i q^{91} -99.4888 q^{93} +119.157i q^{95} +97.1384 q^{97} +24.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{9}+O(q^{10}) $ 8 * q - 24 * q^9	$ 8 q - 24 q^{9} - 16 q^{17} + 88 q^{25} - 80 q^{41} - 88 q^{49} + 96 q^{57} + 192 q^{65} - 400 q^{73} + 72 q^{81} + 400 q^{89} + 112 q^{97}+O(q^{100}) $ 8 * q - 24 * q^9 - 16 * q^17 + 88 * q^25 - 80 * q^41 - 88 * q^49 + 96 * q^57 + 192 * q^65 - 400 * q^73 + 72 * q^81 + 400 * q^89 + 112 * 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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−7.98203	−1.59641	−0.798203	−	0.602388i	$-0.794216\pi$
$5$	−7.98203	−1.59641	−0.798203	+	0.602388i	$0.794216\pi$
$6$	0	0
$7$	2.13878i	0.305540i	0.988262	+	0.152770i	$0.0488193\pi$
$7$	2.13878i	0.305540i	−0.988262	+	0.152770i	$0.951181\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 8.00000i	− 0.727273i	−0.931541	−	0.363636i	$-0.881535\pi$
$11$	− 8.00000i	− 0.727273i	0.931541	−	0.363636i	$-0.118465\pi$
$12$	0	0
$13$	−11.6865	−0.898962	−0.449481	−	0.893290i	$-0.648391\pi$
$13$	−11.6865	−0.898962	−0.449481	+	0.893290i	$0.648391\pi$
$14$	0	0
$15$	− 13.8253i	− 0.921686i
$16$	0	0
$17$	11.8564	0.697436	0.348718	−	0.937228i	$-0.386617\pi$
$17$	11.8564	0.697436	0.348718	+	0.937228i	$0.386617\pi$
$18$	0	0
$19$	− 14.9282i	− 0.785695i	−0.919604	−	0.392847i	$-0.871490\pi$
$19$	− 14.9282i	− 0.785695i	0.919604	−	0.392847i	$-0.128510\pi$
$20$	0	0
$21$	−3.70447	−0.176403
$22$	0	0
$23$	− 4.27756i	− 0.185981i	−0.995667	−	0.0929904i	$-0.970357\pi$
$23$	− 4.27756i	− 0.185981i	0.995667	−	0.0929904i	$-0.0296426\pi$
$24$	0	0
$25$	38.7128	1.54851
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−0.573084	−0.0197615	−0.00988076	−	0.999951i	$-0.503145\pi$
$29$	−0.573084	−0.0197615	−0.00988076	+	0.999951i	$0.503145\pi$
$30$	0	0
$31$	57.4399i	1.85290i	0.376417	+	0.926450i	$0.377156\pi$
$31$	57.4399i	1.85290i	−0.376417	+	0.926450i	$0.622844\pi$
$32$	0	0
$33$	13.8564	0.419891
$34$	0	0
$35$	− 17.0718i	− 0.487766i
$36$	0	0
$37$	27.6506	0.747313	0.373656	−	0.927567i	$-0.378104\pi$
$37$	27.6506	0.747313	0.373656	+	0.927567i	$0.378104\pi$
$38$	0	0
$39$	− 20.2416i	− 0.519016i
$40$	0	0
$41$	31.5692	0.769981	0.384990	−	0.922921i	$-0.374205\pi$
$41$	31.5692	0.769981	0.384990	+	0.922921i	$0.374205\pi$
$42$	0	0
$43$	28.7846i	0.669410i	0.942323	+	0.334705i	$0.108637\pi$
$43$	28.7846i	0.669410i	−0.942323	+	0.334705i	$0.891363\pi$
$44$	0	0
$45$	23.9461	0.532135
$46$	0	0
$47$	− 59.5787i	− 1.26763i	−0.773484	−	0.633816i	$-0.781488\pi$
$47$	− 59.5787i	− 1.26763i	0.773484	−	0.633816i	$-0.218512\pi$
$48$	0	0
$49$	44.4256	0.906645
$50$	0	0
$51$	20.5359i	0.402665i
$52$	0	0
$53$	31.3550	0.591604	0.295802	−	0.955249i	$-0.404413\pi$
$53$	31.3550	0.591604	0.295802	+	0.955249i	$0.404413\pi$
$54$	0	0
$55$	63.8562i	1.16102i
$56$	0	0
$57$	25.8564	0.453621
$58$	0	0
$59$	− 52.7846i	− 0.894654i	−0.894370	−	0.447327i	$-0.852376\pi$
$59$	− 52.7846i	− 0.894654i	0.894370	−	0.447327i	$-0.147624\pi$
$60$	0	0
$61$	59.5787	0.976700	0.488350	−	0.872648i	$-0.337599\pi$
$61$	59.5787	0.976700	0.488350	+	0.872648i	$0.337599\pi$
$62$	0	0
$63$	− 6.41634i	− 0.101847i
$64$	0	0
$65$	93.2820	1.43511
$66$	0	0
$67$	84.7846i	1.26544i	0.774380	+	0.632721i	$0.218062\pi$
$67$	84.7846i	1.26544i	−0.774380	+	0.632721i	$0.781938\pi$
$68$	0	0
$69$	7.40895	0.107376
$70$	0	0
$71$	− 42.4685i	− 0.598147i	−0.954230	−	0.299074i	$-0.903322\pi$
$71$	− 42.4685i	− 0.598147i	0.954230	−	0.299074i	$-0.0966776\pi$
$72$	0	0
$73$	5.42563	0.0743236	0.0371618	−	0.999309i	$-0.488168\pi$
$73$	5.42563	0.0743236	0.0371618	+	0.999309i	$0.488168\pi$
$74$	0	0
$75$	67.0526i	0.894034i
$76$	0	0
$77$	17.1102	0.222211
$78$	0	0
$79$	− 44.6072i	− 0.564649i	−0.959319	−	0.282324i	$-0.908895\pi$
$79$	− 44.6072i	− 0.564649i	0.959319	−	0.282324i	$-0.0911054\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 67.7128i	− 0.815817i	−0.913023	−	0.407909i	$-0.866258\pi$
$83$	− 67.7128i	− 0.815817i	0.913023	−	0.407909i	$-0.133742\pi$
$84$	0	0
$85$	−94.6382	−1.11339
$86$	0	0
$87$	− 0.992611i	− 0.0114093i
$88$	0	0
$89$	133.138	1.49594	0.747969	−	0.663734i	$-0.231029\pi$
$89$	133.138	1.49594	0.747969	+	0.663734i	$0.231029\pi$
$90$	0	0
$91$	− 24.9948i	− 0.274669i
$92$	0	0
$93$	−99.4888	−1.06977
$94$	0	0
$95$	119.157i	1.25429i
$96$	0	0
$97$	97.1384	1.00143	0.500714	−	0.865613i	$-0.333071\pi$
$97$	97.1384	1.00143	0.500714	+	0.865613i	$0.333071\pi$
$98$	0	0
$99$	24.0000i	0.242424i
$100$	0	0
$101$	−62.1370	−0.615218	−0.307609	−	0.951513i	$-0.599529\pi$
$101$	−62.1370	−0.615218	−0.307609	+	0.951513i	$0.599529\pi$
$102$	0	0
$103$	− 27.8041i	− 0.269943i	−0.990849	−	0.134971i	$-0.956906\pi$
$103$	− 27.8041i	− 0.269943i	0.990849	−	0.134971i	$-0.0430943\pi$
$104$	0	0
$105$	29.5692	0.281612
$106$	0	0
$107$	37.7795i	0.353079i	0.984294	+	0.176540i	$0.0564903\pi$
$107$	37.7795i	0.353079i	−0.984294	+	0.176540i	$0.943510\pi$
$108$	0	0
$109$	−141.691	−1.29992	−0.649960	−	0.759968i	$-0.725214\pi$
$109$	−141.691	−1.29992	−0.649960	+	0.759968i	$0.725214\pi$
$110$	0	0
$111$	47.8922i	0.431461i
$112$	0	0
$113$	−58.2872	−0.515816	−0.257908	−	0.966170i	$-0.583033\pi$
$113$	−58.2872	−0.515816	−0.257908	+	0.966170i	$0.583033\pi$
$114$	0	0
$115$	34.1436i	0.296901i
$116$	0	0
$117$	35.0595	0.299654
$118$	0	0
$119$	25.3582i	0.213094i
$120$	0	0
$121$	57.0000	0.471074
$122$	0	0
$123$	54.6795i	0.444549i
$124$	0	0
$125$	−109.456	−0.875649
$126$	0	0
$127$	− 185.152i	− 1.45789i	−0.684571	−	0.728946i	$-0.740010\pi$
$127$	− 185.152i	− 1.45789i	0.684571	−	0.728946i	$-0.259990\pi$
$128$	0	0
$129$	−49.8564	−0.386484
$130$	0	0
$131$	− 125.359i	− 0.956939i	−0.878104	−	0.478469i	$-0.841192\pi$
$131$	− 125.359i	− 0.956939i	0.878104	−	0.478469i	$-0.158808\pi$
$132$	0	0
$133$	31.9281	0.240061
$134$	0	0
$135$	41.4758i	0.307229i
$136$	0	0
$137$	−99.5692	−0.726783	−0.363391	−	0.931637i	$-0.618381\pi$
$137$	−99.5692	−0.726783	−0.363391	+	0.931637i	$0.618381\pi$
$138$	0	0
$139$	177.492i	1.27692i	0.769654	+	0.638461i	$0.220429\pi$
$139$	177.492i	1.27692i	−0.769654	+	0.638461i	$0.779571\pi$
$140$	0	0
$141$	103.193	0.731867
$142$	0	0
$143$	93.4920i	0.653790i
$144$	0	0
$145$	4.57437	0.0315474
$146$	0	0
$147$	76.9474i	0.523452i
$148$	0	0
$149$	87.8023	0.589277	0.294639	−	0.955609i	$-0.404801\pi$
$149$	87.8023	0.589277	0.294639	+	0.955609i	$0.404801\pi$
$150$	0	0
$151$	− 219.066i	− 1.45077i	−0.688345	−	0.725383i	$-0.741663\pi$
$151$	− 219.066i	− 1.45077i	0.688345	−	0.725383i	$-0.258337\pi$
$152$	0	0
$153$	−35.5692	−0.232479
$154$	0	0
$155$	− 458.487i	− 2.95798i
$156$	0	0
$157$	253.440	1.61427	0.807133	−	0.590370i	$-0.201018\pi$
$157$	253.440	1.61427	0.807133	+	0.590370i	$0.201018\pi$
$158$	0	0
$159$	54.3085i	0.341563i
$160$	0	0
$161$	9.14875	0.0568245
$162$	0	0
$163$	102.354i	0.627938i	0.949433	+	0.313969i	$0.101659\pi$
$163$	102.354i	0.627938i	−0.949433	+	0.313969i	$0.898341\pi$
$164$	0	0
$165$	−110.602	−0.670317
$166$	0	0
$167$	281.090i	1.68318i	0.540120	+	0.841588i	$0.318379\pi$
$167$	281.090i	1.68318i	−0.540120	+	0.841588i	$0.681621\pi$
$168$	0	0
$169$	−32.4256	−0.191868
$170$	0	0
$171$	44.7846i	0.261898i
$172$	0	0
$173$	242.858	1.40381	0.701903	−	0.712273i	$-0.252334\pi$
$173$	242.858	1.40381	0.701903	+	0.712273i	$0.252334\pi$
$174$	0	0
$175$	82.7981i	0.473132i
$176$	0	0
$177$	91.4256	0.516529
$178$	0	0
$179$	− 318.354i	− 1.77851i	−0.457409	−	0.889257i	$-0.651222\pi$
$179$	− 318.354i	− 1.77851i	0.457409	−	0.889257i	$-0.348778\pi$
$180$	0	0
$181$	79.5132	0.439299	0.219650	−	0.975579i	$-0.429509\pi$
$181$	79.5132	0.439299	0.219650	+	0.975579i	$0.429509\pi$
$182$	0	0
$183$	103.193i	0.563898i
$184$	0	0
$185$	−220.708	−1.19301
$186$	0	0
$187$	− 94.8513i	− 0.507226i
$188$	0	0
$189$	11.1134	0.0588012
$190$	0	0
$191$	− 352.887i	− 1.84758i	−0.382902	−	0.923789i	$-0.625075\pi$
$191$	− 352.887i	− 1.84758i	0.382902	−	0.923789i	$-0.374925\pi$
$192$	0	0
$193$	−284.277	−1.47294	−0.736469	−	0.676472i	$-0.763508\pi$
$193$	−284.277	−1.47294	−0.736469	+	0.676472i	$0.763508\pi$
$194$	0	0
$195$	161.569i	0.828560i
$196$	0	0
$197$	75.8087	0.384816	0.192408	−	0.981315i	$-0.438370\pi$
$197$	75.8087	0.384816	0.192408	+	0.981315i	$0.438370\pi$
$198$	0	0
$199$	− 104.186i	− 0.523547i	−0.965129	−	0.261774i	$-0.915693\pi$
$199$	− 104.186i	− 0.523547i	0.965129	−	0.261774i	$-0.0843074\pi$
$200$	0	0
$201$	−146.851	−0.730603
$202$	0	0
$203$	− 1.22570i	− 0.00603793i
$204$	0	0
$205$	−251.986	−1.22920
$206$	0	0
$207$	12.8327i	0.0619936i
$208$	0	0
$209$	−119.426	−0.571414
$210$	0	0
$211$	− 136.918i	− 0.648900i	−0.945903	−	0.324450i	$-0.894821\pi$
$211$	− 136.918i	− 0.648900i	0.945903	−	0.324450i	$-0.105179\pi$
$212$	0	0
$213$	73.5575	0.345341
$214$	0	0
$215$	− 229.760i	− 1.06865i
$216$	0	0
$217$	−122.851	−0.566135
$218$	0	0
$219$	9.39746i	0.0429108i
$220$	0	0
$221$	−138.560	−0.626968
$222$	0	0
$223$	53.1624i	0.238396i	0.992870	+	0.119198i	$0.0380324\pi$
$223$	53.1624i	0.238396i	−0.992870	+	0.119198i	$0.961968\pi$
$224$	0	0
$225$	−116.138	−0.516171
$226$	0	0
$227$	119.846i	0.527956i	0.964529	+	0.263978i	$0.0850347\pi$
$227$	119.846i	0.527956i	−0.964529	+	0.263978i	$0.914965\pi$
$228$	0	0
$229$	214.103	0.934946	0.467473	−	0.884007i	$-0.345164\pi$
$229$	214.103	0.934946	0.467473	+	0.884007i	$0.345164\pi$
$230$	0	0
$231$	29.6358i	0.128293i
$232$	0	0
$233$	127.436	0.546935	0.273468	−	0.961881i	$-0.411829\pi$
$233$	127.436	0.546935	0.273468	+	0.961881i	$0.411829\pi$
$234$	0	0
$235$	475.559i	2.02365i
$236$	0	0
$237$	77.2620	0.326000
$238$	0	0
$239$	319.281i	1.33590i	0.744204	+	0.667952i	$0.232829\pi$
$239$	319.281i	1.33590i	−0.744204	+	0.667952i	$0.767171\pi$
$240$	0	0
$241$	−247.415	−1.02662	−0.513310	−	0.858203i	$-0.671581\pi$
$241$	−247.415	−1.02662	−0.513310	+	0.858203i	$0.671581\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−354.607	−1.44737
$246$	0	0
$247$	174.459i	0.706310i
$248$	0	0
$249$	117.282	0.471012
$250$	0	0
$251$	− 214.851i	− 0.855981i	−0.903783	−	0.427991i	$-0.859222\pi$
$251$	− 214.851i	− 0.855981i	0.903783	−	0.427991i	$-0.140778\pi$
$252$	0	0
$253$	−34.2205	−0.135259
$254$	0	0
$255$	− 163.918i	− 0.642816i
$256$	0	0
$257$	−84.2769	−0.327926	−0.163963	−	0.986467i	$-0.552428\pi$
$257$	−84.2769	−0.327926	−0.163963	+	0.986467i	$0.552428\pi$
$258$	0	0
$259$	59.1384i	0.228334i
$260$	0	0
$261$	1.71925	0.00658717
$262$	0	0
$263$	− 277.120i	− 1.05369i	−0.849962	−	0.526844i	$-0.823375\pi$
$263$	− 277.120i	− 1.05369i	0.849962	−	0.526844i	$-0.176625\pi$
$264$	0	0
$265$	−250.277	−0.944441
$266$	0	0
$267$	230.603i	0.863680i
$268$	0	0
$269$	−123.701	−0.459855	−0.229927	−	0.973208i	$-0.573849\pi$
$269$	−123.701	−0.459855	−0.229927	+	0.973208i	$0.573849\pi$
$270$	0	0
$271$	197.985i	0.730572i	0.930895	+	0.365286i	$0.119029\pi$
$271$	197.985i	0.730572i	−0.930895	+	0.365286i	$0.880971\pi$
$272$	0	0
$273$	43.2923	0.158580
$274$	0	0
$275$	− 309.703i	− 1.12619i
$276$	0	0
$277$	−247.709	−0.894256	−0.447128	−	0.894470i	$-0.647553\pi$
$277$	−247.709	−0.894256	−0.447128	+	0.894470i	$0.647553\pi$
$278$	0	0
$279$	− 172.320i	− 0.617633i
$280$	0	0
$281$	−443.128	−1.57697	−0.788484	−	0.615055i	$-0.789134\pi$
$281$	−443.128	−1.57697	−0.788484	+	0.615055i	$0.789134\pi$
$282$	0	0
$283$	− 294.620i	− 1.04106i	−0.853843	−	0.520531i	$-0.825734\pi$
$283$	− 294.620i	− 1.04106i	0.853843	−	0.520531i	$-0.174266\pi$
$284$	0	0
$285$	−206.387	−0.724164
$286$	0	0
$287$	67.5196i	0.235260i
$288$	0	0
$289$	−148.426	−0.513583
$290$	0	0
$291$	168.249i	0.578174i
$292$	0	0
$293$	−66.7217	−0.227719	−0.113859	−	0.993497i	$-0.536321\pi$
$293$	−66.7217	−0.227719	−0.113859	+	0.993497i	$0.536321\pi$
$294$	0	0
$295$	421.328i	1.42823i
$296$	0	0
$297$	−41.5692	−0.139964
$298$	0	0
$299$	49.9897i	0.167190i
$300$	0	0
$301$	−61.5639	−0.204531
$302$	0	0
$303$	− 107.624i	− 0.355196i
$304$	0	0
$305$	−475.559	−1.55921
$306$	0	0
$307$	524.210i	1.70753i	0.520662	+	0.853763i	$0.325685\pi$
$307$	524.210i	1.70753i	−0.520662	+	0.853763i	$0.674315\pi$
$308$	0	0
$309$	48.1582	0.155852
$310$	0	0
$311$	362.057i	1.16417i	0.813128	+	0.582085i	$0.197763\pi$
$311$	362.057i	1.16417i	−0.813128	+	0.582085i	$0.802237\pi$
$312$	0	0
$313$	−252.277	−0.805996	−0.402998	−	0.915201i	$-0.632032\pi$
$313$	−252.277	−0.805996	−0.402998	+	0.915201i	$0.632032\pi$
$314$	0	0
$315$	51.2154i	0.162589i
$316$	0	0
$317$	80.3934	0.253607	0.126803	−	0.991928i	$-0.459528\pi$
$317$	80.3934	0.253607	0.126803	+	0.991928i	$0.459528\pi$
$318$	0	0
$319$	4.58467i	0.0143720i
$320$	0	0
$321$	−65.4359	−0.203850
$322$	0	0
$323$	− 176.995i	− 0.547972i
$324$	0	0
$325$	−452.417	−1.39205
$326$	0	0
$327$	− 245.417i	− 0.750509i
$328$	0	0
$329$	127.426	0.387312
$330$	0	0
$331$	172.056i	0.519808i	0.965635	+	0.259904i	$0.0836909\pi$
$331$	172.056i	0.519808i	−0.965635	+	0.259904i	$0.916309\pi$
$332$	0	0
$333$	−82.9517	−0.249104
$334$	0	0
$335$	− 676.753i	− 2.02016i
$336$	0	0
$337$	564.277	1.67441	0.837206	−	0.546888i	$-0.184187\pi$
$337$	564.277	1.67441	0.837206	+	0.546888i	$0.184187\pi$
$338$	0	0
$339$	− 100.956i	− 0.297806i
$340$	0	0
$341$	459.519	1.34756
$342$	0	0
$343$	199.817i	0.582556i
$344$	0	0
$345$	−59.1384	−0.171416
$346$	0	0
$347$	− 286.123i	− 0.824562i	−0.911057	−	0.412281i	$-0.864732\pi$
$347$	− 286.123i	− 0.824562i	0.911057	−	0.412281i	$-0.135268\pi$
$348$	0	0
$349$	421.021	1.20636	0.603182	−	0.797603i	$-0.293899\pi$
$349$	421.021	1.20636	0.603182	+	0.797603i	$0.293899\pi$
$350$	0	0
$351$	60.7249i	0.173005i
$352$	0	0
$353$	429.138	1.21569	0.607845	−	0.794056i	$-0.292034\pi$
$353$	429.138	1.21569	0.607845	+	0.794056i	$0.292034\pi$
$354$	0	0
$355$	338.985i	0.954886i
$356$	0	0
$357$	−43.9217	−0.123030
$358$	0	0
$359$	− 263.673i	− 0.734465i	−0.930129	−	0.367233i	$-0.880305\pi$
$359$	− 263.673i	− 0.734465i	0.930129	−	0.367233i	$-0.119695\pi$
$360$	0	0
$361$	138.149	0.382684
$362$	0	0
$363$	98.7269i	0.271975i
$364$	0	0
$365$	−43.3075	−0.118651
$366$	0	0
$367$	− 129.544i	− 0.352981i	−0.984302	−	0.176491i	$-0.943525\pi$
$367$	− 129.544i	− 0.352981i	0.984302	−	0.176491i	$-0.0564745\pi$
$368$	0	0
$369$	−94.7077	−0.256660
$370$	0	0
$371$	67.0615i	0.180759i
$372$	0	0
$373$	302.478	0.810933	0.405467	−	0.914110i	$-0.367109\pi$
$373$	302.478	0.810933	0.405467	+	0.914110i	$0.367109\pi$
$374$	0	0
$375$	− 189.584i	− 0.505556i
$376$	0	0
$377$	6.69735	0.0177649
$378$	0	0
$379$	− 116.210i	− 0.306623i	−0.988178	−	0.153312i	$-0.951006\pi$
$379$	− 116.210i	− 0.306623i	0.988178	−	0.153312i	$-0.0489938\pi$
$380$	0	0
$381$	320.693	0.841715
$382$	0	0
$383$	566.151i	1.47820i	0.673595	+	0.739101i	$0.264749\pi$
$383$	566.151i	1.47820i	−0.673595	+	0.739101i	$0.735251\pi$
$384$	0	0
$385$	−136.574	−0.354739
$386$	0	0
$387$	− 86.3538i	− 0.223137i
$388$	0	0
$389$	350.104	0.900011	0.450006	−	0.893026i	$-0.351422\pi$
$389$	350.104	0.900011	0.450006	+	0.893026i	$0.351422\pi$
$390$	0	0
$391$	− 50.7165i	− 0.129710i
$392$	0	0
$393$	217.128	0.552489
$394$	0	0
$395$	356.056i	0.901408i
$396$	0	0
$397$	544.149	1.37065	0.685326	−	0.728236i	$-0.259660\pi$
$397$	544.149	1.37065	0.685326	+	0.728236i	$0.259660\pi$
$398$	0	0
$399$	55.3011i	0.138599i
$400$	0	0
$401$	−296.431	−0.739229	−0.369614	−	0.929185i	$-0.620510\pi$
$401$	−296.431	−0.739229	−0.369614	+	0.929185i	$0.620510\pi$
$402$	0	0
$403$	− 671.272i	− 1.66569i
$404$	0	0
$405$	−71.8383	−0.177378
$406$	0	0
$407$	− 221.205i	− 0.543500i
$408$	0	0
$409$	247.415	0.604927	0.302464	−	0.953161i	$-0.402191\pi$
$409$	247.415	0.604927	0.302464	+	0.953161i	$0.402191\pi$
$410$	0	0
$411$	− 172.459i	− 0.419608i
$412$	0	0
$413$	112.895	0.273353
$414$	0	0
$415$	540.486i	1.30238i
$416$	0	0
$417$	−307.426	−0.737232
$418$	0	0
$419$	92.1333i	0.219889i	0.993938	+	0.109944i	$0.0350672\pi$
$419$	92.1333i	0.219889i	−0.993938	+	0.109944i	$0.964933\pi$
$420$	0	0
$421$	445.540	1.05829	0.529145	−	0.848531i	$-0.322513\pi$
$421$	445.540	1.05829	0.529145	+	0.848531i	$0.322513\pi$
$422$	0	0
$423$	178.736i	0.422544i
$424$	0	0
$425$	458.995	1.07999
$426$	0	0
$427$	127.426i	0.298421i
$428$	0	0
$429$	−161.933	−0.377466
$430$	0	0
$431$	− 186.677i	− 0.433125i	−0.976269	−	0.216563i	$-0.930515\pi$
$431$	− 186.677i	− 0.433125i	0.976269	−	0.216563i	$-0.0694845\pi$
$432$	0	0
$433$	291.128	0.672351	0.336176	−	0.941799i	$-0.390866\pi$
$433$	291.128	0.672351	0.336176	+	0.941799i	$0.390866\pi$
$434$	0	0
$435$	7.92305i	0.0182139i
$436$	0	0
$437$	−63.8562	−0.146124
$438$	0	0
$439$	− 87.6899i	− 0.199749i	−0.995000	−	0.0998746i	$-0.968156\pi$
$439$	− 87.6899i	− 0.199749i	0.995000	−	0.0998746i	$-0.0318442\pi$
$440$	0	0
$441$	−133.277	−0.302215
$442$	0	0
$443$	20.7077i	0.0467441i	0.999727	+	0.0233721i	$0.00744024\pi$
$443$	20.7077i	0.0467441i	−0.999727	+	0.0233721i	$0.992560\pi$
$444$	0	0
$445$	−1062.72	−2.38812
$446$	0	0
$447$	152.078i	0.340219i
$448$	0	0
$449$	584.410	1.30158	0.650791	−	0.759257i	$-0.274437\pi$
$449$	584.410	1.30158	0.650791	+	0.759257i	$0.274437\pi$
$450$	0	0
$451$	− 252.554i	− 0.559986i
$452$	0	0
$453$	379.433	0.837600
$454$	0	0
$455$	199.510i	0.438483i
$456$	0	0
$457$	269.692	0.590136	0.295068	−	0.955476i	$-0.404658\pi$
$457$	269.692	0.590136	0.295068	+	0.955476i	$0.404658\pi$
$458$	0	0
$459$	− 61.6077i	− 0.134222i
$460$	0	0
$461$	−44.4948	−0.0965181	−0.0482590	−	0.998835i	$-0.515367\pi$
$461$	−44.4948	−0.0965181	−0.0482590	+	0.998835i	$0.515367\pi$
$462$	0	0
$463$	− 611.065i	− 1.31980i	−0.751355	−	0.659898i	$-0.770600\pi$
$463$	− 611.065i	− 1.31980i	0.751355	−	0.659898i	$-0.229400\pi$
$464$	0	0
$465$	794.123	1.70779
$466$	0	0
$467$	− 146.410i	− 0.313512i	−0.987637	−	0.156756i	$-0.949896\pi$
$467$	− 146.410i	− 0.313512i	0.987637	−	0.156756i	$-0.0501036\pi$
$468$	0	0
$469$	−181.336	−0.386643
$470$	0	0
$471$	438.971i	0.931997i
$472$	0	0
$473$	230.277	0.486843
$474$	0	0
$475$	− 577.913i	− 1.21666i
$476$	0	0
$477$	−94.0651	−0.197201
$478$	0	0
$479$	− 191.876i	− 0.400576i	−0.979737	−	0.200288i	$-0.935812\pi$
$479$	− 191.876i	− 0.400576i	0.979737	−	0.200288i	$-0.0641878\pi$
$480$	0	0
$481$	−323.138	−0.671805
$482$	0	0
$483$	15.8461i	0.0328077i
$484$	0	0
$485$	−775.362	−1.59868
$486$	0	0
$487$	− 610.758i	− 1.25412i	−0.778969	−	0.627062i	$-0.784257\pi$
$487$	− 610.758i	− 1.25412i	0.778969	−	0.627062i	$-0.215743\pi$
$488$	0	0
$489$	−177.282	−0.362540
$490$	0	0
$491$	− 142.354i	− 0.289926i	−0.989437	−	0.144963i	$-0.953694\pi$
$491$	− 142.354i	− 0.289926i	0.989437	−	0.144963i	$-0.0463064\pi$
$492$	0	0
$493$	−6.79472	−0.0137824
$494$	0	0
$495$	− 191.569i	− 0.387008i
$496$	0	0
$497$	90.8306	0.182758
$498$	0	0
$499$	− 91.3693i	− 0.183105i	−0.995800	−	0.0915524i	$-0.970817\pi$
$499$	− 91.3693i	− 0.183105i	0.995800	−	0.0915524i	$-0.0291829\pi$
$500$	0	0
$501$	−486.863	−0.971782
$502$	0	0
$503$	230.067i	0.457389i	0.973498	+	0.228695i	$0.0734457\pi$
$503$	230.067i	0.457389i	−0.973498	+	0.228695i	$0.926554\pi$
$504$	0	0
$505$	495.979	0.982137
$506$	0	0
$507$	− 56.1628i	− 0.110775i
$508$	0	0
$509$	527.387	1.03612	0.518062	−	0.855343i	$-0.326654\pi$
$509$	527.387	1.03612	0.518062	+	0.855343i	$0.326654\pi$
$510$	0	0
$511$	11.6042i	0.0227088i
$512$	0	0
$513$	−77.5692	−0.151207
$514$	0	0
$515$	221.933i	0.430939i
$516$	0	0
$517$	−476.630	−0.921914
$518$	0	0
$519$	420.643i	0.810487i
$520$	0	0
$521$	−191.856	−0.368246	−0.184123	−	0.982903i	$-0.558945\pi$
$521$	−191.856	−0.368246	−0.184123	+	0.982903i	$0.558945\pi$
$522$	0	0
$523$	105.492i	0.201706i	0.994901	+	0.100853i	$0.0321572\pi$
$523$	105.492i	0.201706i	−0.994901	+	0.100853i	$0.967843\pi$
$524$	0	0
$525$	−143.411	−0.273163
$526$	0	0
$527$	681.031i	1.29228i
$528$	0	0
$529$	510.703	0.965411
$530$	0	0
$531$	158.354i	0.298218i
$532$	0	0
$533$	−368.934	−0.692184
$534$	0	0
$535$	− 301.557i	− 0.563658i
$536$	0	0
$537$	551.405	1.02682
$538$	0	0
$539$	− 355.405i	− 0.659378i
$540$	0	0
$541$	−459.744	−0.849804	−0.424902	−	0.905239i	$-0.639691\pi$
$541$	−459.744	−0.849804	−0.424902	+	0.905239i	$0.639691\pi$
$542$	0	0
$543$	137.721i	0.253630i
$544$	0	0
$545$	1130.98	2.07520
$546$	0	0
$547$	− 67.3693i	− 0.123161i	−0.998102	−	0.0615807i	$-0.980386\pi$
$547$	− 67.3693i	− 0.123161i	0.998102	−	0.0615807i	$-0.0196142\pi$
$548$	0	0
$549$	−178.736	−0.325567
$550$	0	0
$551$	8.55511i	0.0155265i
$552$	0	0
$553$	95.4050	0.172523
$554$	0	0
$555$	− 382.277i	− 0.688787i
$556$	0	0
$557$	−476.671	−0.855782	−0.427891	−	0.903830i	$-0.640743\pi$
$557$	−476.671	−0.855782	−0.427891	+	0.903830i	$0.640743\pi$
$558$	0	0
$559$	− 336.391i	− 0.601774i
$560$	0	0
$561$	164.287	0.292847
$562$	0	0
$563$	910.123i	1.61656i	0.588799	+	0.808280i	$0.299601\pi$
$563$	910.123i	1.61656i	−0.588799	+	0.808280i	$0.700399\pi$
$564$	0	0
$565$	465.250	0.823452
$566$	0	0
$567$	19.2490i	0.0339489i
$568$	0	0
$569$	124.123	0.218142	0.109071	−	0.994034i	$-0.465212\pi$
$569$	124.123	0.218142	0.109071	+	0.994034i	$0.465212\pi$
$570$	0	0
$571$	945.031i	1.65504i	0.561433	+	0.827522i	$0.310250\pi$
$571$	945.031i	1.65504i	−0.561433	+	0.827522i	$0.689750\pi$
$572$	0	0
$573$	611.219	1.06670
$574$	0	0
$575$	− 165.596i	− 0.287994i
$576$	0	0
$577$	215.682	0.373799	0.186899	−	0.982379i	$-0.440156\pi$
$577$	215.682	0.373799	0.186899	+	0.982379i	$0.440156\pi$
$578$	0	0
$579$	− 492.382i	− 0.850401i
$580$	0	0
$581$	144.823	0.249265
$582$	0	0
$583$	− 250.840i	− 0.430258i
$584$	0	0
$585$	−279.846	−0.478369
$586$	0	0
$587$	900.785i	1.53456i	0.641314	+	0.767278i	$0.278389\pi$
$587$	900.785i	1.53456i	−0.641314	+	0.767278i	$0.721611\pi$
$588$	0	0
$589$	857.475	1.45581
$590$	0	0
$591$	131.305i	0.222174i
$592$	0	0
$593$	−508.585	−0.857647	−0.428824	−	0.903388i	$-0.641072\pi$
$593$	−508.585	−0.857647	−0.428824	+	0.903388i	$0.641072\pi$
$594$	0	0
$595$	− 202.410i	− 0.340185i
$596$	0	0
$597$	180.455	0.302270
$598$	0	0
$599$	846.934i	1.41391i	0.707257	+	0.706957i	$0.249933\pi$
$599$	846.934i	1.41391i	−0.707257	+	0.706957i	$0.750067\pi$
$600$	0	0
$601$	406.000	0.675541	0.337770	−	0.941229i	$-0.390327\pi$
$601$	406.000	0.675541	0.337770	+	0.941229i	$0.390327\pi$
$602$	0	0
$603$	− 254.354i	− 0.421814i
$604$	0	0
$605$	−454.976	−0.752026
$606$	0	0
$607$	− 771.156i	− 1.27044i	−0.772332	−	0.635219i	$-0.780910\pi$
$607$	− 771.156i	− 1.27044i	0.772332	−	0.635219i	$-0.219090\pi$
$608$	0	0
$609$	2.12297	0.00348600
$610$	0	0
$611$	696.267i	1.13955i
$612$	0	0
$613$	336.699	0.549264	0.274632	−	0.961549i	$-0.411444\pi$
$613$	336.699	0.549264	0.274632	+	0.961549i	$0.411444\pi$
$614$	0	0
$615$	− 436.453i	− 0.709680i
$616$	0	0
$617$	908.831	1.47298	0.736492	−	0.676447i	$-0.236481\pi$
$617$	908.831	1.47298	0.736492	+	0.676447i	$0.236481\pi$
$618$	0	0
$619$	− 1047.77i	− 1.69268i	−0.532643	−	0.846340i	$-0.678801\pi$
$619$	− 1047.77i	− 1.69268i	0.532643	−	0.846340i	$-0.321199\pi$
$620$	0	0
$621$	−22.2268	−0.0357920
$622$	0	0
$623$	284.754i	0.457068i
$624$	0	0
$625$	−94.1384	−0.150622
$626$	0	0
$627$	− 206.851i	− 0.329906i
$628$	0	0
$629$	327.836	0.521202
$630$	0	0
$631$	− 610.758i	− 0.967921i	−0.875090	−	0.483961i	$-0.839198\pi$
$631$	− 610.758i	− 0.967921i	0.875090	−	0.483961i	$-0.160802\pi$
$632$	0	0
$633$	237.149	0.374643
$634$	0	0
$635$	1477.89i	2.32739i
$636$	0	0
$637$	−519.180	−0.815040
$638$	0	0
$639$	127.405i	0.199382i
$640$	0	0
$641$	9.60015	0.0149768	0.00748842	−	0.999972i	$-0.497616\pi$
$641$	9.60015	0.0149768	0.00748842	+	0.999972i	$0.497616\pi$
$642$	0	0
$643$	− 86.1999i	− 0.134059i	−0.997751	−	0.0670295i	$-0.978648\pi$
$643$	− 86.1999i	− 0.134059i	0.997751	−	0.0670295i	$-0.0213522\pi$
$644$	0	0
$645$	397.955	0.616985
$646$	0	0
$647$	352.580i	0.544946i	0.962163	+	0.272473i	$0.0878416\pi$
$647$	352.580i	0.544946i	−0.962163	+	0.272473i	$0.912158\pi$
$648$	0	0
$649$	−422.277	−0.650658
$650$	0	0
$651$	− 212.785i	− 0.326858i
$652$	0	0
$653$	−319.322	−0.489008	−0.244504	−	0.969648i	$-0.578625\pi$
$653$	−319.322	−0.489008	−0.244504	+	0.969648i	$0.578625\pi$
$654$	0	0
$655$	1000.62i	1.52766i
$656$	0	0
$657$	−16.2769	−0.0247745
$658$	0	0
$659$	− 275.328i	− 0.417797i	−0.977937	−	0.208898i	$-0.933012\pi$
$659$	− 275.328i	− 0.417797i	0.977937	−	0.208898i	$-0.0669878\pi$
$660$	0	0
$661$	133.668	0.202221	0.101111	−	0.994875i	$-0.467760\pi$
$661$	133.668	0.202221	0.101111	+	0.994875i	$0.467760\pi$
$662$	0	0
$663$	− 239.993i	− 0.361980i
$664$	0	0
$665$	−254.851	−0.383235
$666$	0	0
$667$	2.45140i	0.00367526i
$668$	0	0
$669$	−92.0799	−0.137638
$670$	0	0
$671$	− 476.630i	− 0.710327i
$672$	0	0
$673$	187.703	0.278904	0.139452	−	0.990229i	$-0.455466\pi$
$673$	187.703	0.278904	0.139452	+	0.990229i	$0.455466\pi$
$674$	0	0
$675$	− 201.158i	− 0.298011i
$676$	0	0
$677$	1169.84	1.72797	0.863986	−	0.503515i	$-0.167960\pi$
$677$	1169.84	1.72797	0.863986	+	0.503515i	$0.167960\pi$
$678$	0	0
$679$	207.758i	0.305976i
$680$	0	0
$681$	−207.580	−0.304816
$682$	0	0
$683$	89.4566i	0.130976i	0.997853	+	0.0654880i	$0.0208604\pi$
$683$	89.4566i	0.130976i	−0.997853	+	0.0654880i	$0.979140\pi$
$684$	0	0
$685$	794.765	1.16024
$686$	0	0
$687$	370.837i	0.539791i
$688$	0	0
$689$	−366.431	−0.531830
$690$	0	0
$691$	− 139.103i	− 0.201306i	−0.994922	−	0.100653i	$-0.967907\pi$
$691$	− 139.103i	− 0.201306i	0.994922	−	0.100653i	$-0.0320933\pi$
$692$	0	0
$693$	−51.3307	−0.0740703
$694$	0	0
$695$	− 1416.75i	− 2.03849i
$696$	0	0
$697$	374.297	0.537012
$698$	0	0
$699$	220.726i	0.315773i
$700$	0	0
$701$	1218.34	1.73801	0.869004	−	0.494804i	$-0.164760\pi$
$701$	1218.34	1.73801	0.869004	+	0.494804i	$0.164760\pi$
$702$	0	0
$703$	− 412.773i	− 0.587160i
$704$	0	0
$705$	−823.692	−1.16836
$706$	0	0
$707$	− 132.897i	− 0.187974i
$708$	0	0
$709$	−1080.13	−1.52346	−0.761730	−	0.647895i	$-0.775649\pi$
$709$	−1080.13	−1.52346	−0.761730	+	0.647895i	$0.775649\pi$
$710$	0	0
$711$	133.822i	0.188216i
$712$	0	0
$713$	245.703	0.344604
$714$	0	0
$715$	− 746.256i	− 1.04372i
$716$	0	0
$717$	−553.011	−0.771285
$718$	0	0
$719$	20.7736i	0.0288923i	0.999896	+	0.0144461i	$0.00459851\pi$
$719$	20.7736i	0.0288923i	−0.999896	+	0.0144461i	$0.995401\pi$
$720$	0	0
$721$	59.4669	0.0824783
$722$	0	0
$723$	− 428.536i	− 0.592719i
$724$	0	0
$725$	−22.1857	−0.0306010
$726$	0	0
$727$	1031.17i	1.41838i	0.705015	+	0.709192i	$0.250940\pi$
$727$	1031.17i	1.41838i	−0.705015	+	0.709192i	$0.749060\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	341.282i	0.466870i
$732$	0	0
$733$	881.072	1.20201	0.601004	−	0.799246i	$-0.294767\pi$
$733$	881.072	1.20201	0.601004	+	0.799246i	$0.294767\pi$
$734$	0	0
$735$	− 614.197i	− 0.835642i
$736$	0	0
$737$	678.277	0.920321
$738$	0	0
$739$	671.195i	0.908247i	0.890939	+	0.454124i	$0.150048\pi$
$739$	671.195i	0.908247i	−0.890939	+	0.454124i	$0.849952\pi$
$740$	0	0
$741$	−302.171	−0.407788
$742$	0	0
$743$	− 254.197i	− 0.342122i	−0.985260	−	0.171061i	$-0.945281\pi$
$743$	− 254.197i	− 0.342122i	0.985260	−	0.171061i	$-0.0547195\pi$
$744$	0	0
$745$	−700.841	−0.940726
$746$	0	0
$747$	203.138i	0.271939i
$748$	0	0
$749$	−80.8019	−0.107880
$750$	0	0
$751$	728.994i	0.970698i	0.874320	+	0.485349i	$0.161307\pi$
$751$	728.994i	0.970698i	−0.874320	+	0.485349i	$0.838693\pi$
$752$	0	0
$753$	372.133	0.494201
$754$	0	0
$755$	1748.59i	2.31601i
$756$	0	0
$757$	372.679	0.492311	0.246155	−	0.969230i	$-0.420833\pi$
$757$	372.679	0.492311	0.246155	+	0.969230i	$0.420833\pi$
$758$	0	0
$759$	− 59.2716i	− 0.0780917i
$760$	0	0
$761$	−1257.80	−1.65283	−0.826416	−	0.563060i	$-0.809624\pi$
$761$	−1257.80	−1.65283	−0.826416	+	0.563060i	$0.809624\pi$
$762$	0	0
$763$	− 303.046i	− 0.397177i
$764$	0	0
$765$	283.915	0.371130
$766$	0	0
$767$	616.868i	0.804260i
$768$	0	0
$769$	247.703	0.322110	0.161055	−	0.986945i	$-0.448510\pi$
$769$	247.703	0.322110	0.161055	+	0.986945i	$0.448510\pi$
$770$	0	0
$771$	− 145.972i	− 0.189328i
$772$	0	0
$773$	−587.805	−0.760420	−0.380210	−	0.924900i	$-0.624148\pi$
$773$	−587.805	−0.760420	−0.380210	+	0.924900i	$0.624148\pi$
$774$	0	0
$775$	2223.66i	2.86924i
$776$	0	0
$777$	−102.431	−0.131829
$778$	0	0
$779$	− 471.272i	− 0.604970i
$780$	0	0
$781$	−339.748	−0.435016
$782$	0	0
$783$	2.97783i	0.00380311i
$784$	0	0
$785$	−2022.96	−2.57702
$786$	0	0
$787$	31.0821i	0.0394944i	0.999805	+	0.0197472i	$0.00628614\pi$
$787$	31.0821i	0.0394944i	−0.999805	+	0.0197472i	$0.993714\pi$
$788$	0	0
$789$	479.986	0.608347
$790$	0	0
$791$	− 124.663i	− 0.157602i
$792$	0	0
$793$	−696.267	−0.878016
$794$	0	0
$795$	− 433.492i	− 0.545273i
$796$	0	0
$797$	−490.260	−0.615132	−0.307566	−	0.951527i	$-0.599514\pi$
$797$	−490.260	−0.615132	−0.307566	+	0.951527i	$0.599514\pi$
$798$	0	0
$799$	− 706.389i	− 0.884092i
$800$	0	0
$801$	−399.415	−0.498646
$802$	0	0
$803$	− 43.4050i	− 0.0540536i
$804$	0	0
$805$	−73.0256	−0.0907150
$806$	0	0
$807$	− 214.256i	− 0.265497i
$808$	0	0
$809$	−676.102	−0.835726	−0.417863	−	0.908510i	$-0.637221\pi$
$809$	−676.102	−0.835726	−0.417863	+	0.908510i	$0.637221\pi$
$810$	0	0
$811$	− 74.1793i	− 0.0914665i	−0.998954	−	0.0457332i	$-0.985438\pi$
$811$	− 74.1793i	− 0.0914665i	0.998954	−	0.0457332i	$-0.0145624\pi$
$812$	0	0
$813$	−342.920	−0.421796
$814$	0	0
$815$	− 816.991i	− 1.00244i
$816$	0	0
$817$	429.703	0.525952
$818$	0	0
$819$	74.9845i	0.0915562i
$820$	0	0
$821$	−1130.58	−1.37708	−0.688540	−	0.725198i	$-0.741748\pi$
$821$	−1130.58	−1.37708	−0.688540	+	0.725198i	$0.741748\pi$
$822$	0	0
$823$	− 82.1839i	− 0.0998589i	−0.998753	−	0.0499295i	$-0.984100\pi$
$823$	− 82.1839i	− 0.0998589i	0.998753	−	0.0499295i	$-0.0158997\pi$
$824$	0	0
$825$	536.420	0.650207
$826$	0	0
$827$	1504.57i	1.81931i	0.415365	+	0.909655i	$0.363654\pi$
$827$	1504.57i	1.81931i	−0.415365	+	0.909655i	$0.636346\pi$
$828$	0	0
$829$	−1409.65	−1.70042	−0.850209	−	0.526445i	$-0.823525\pi$
$829$	−1409.65	−1.70042	−0.850209	+	0.526445i	$0.823525\pi$
$830$	0	0
$831$	− 429.044i	− 0.516299i
$832$	0	0
$833$	526.728	0.632327
$834$	0	0
$835$	− 2243.67i	− 2.68703i
$836$	0	0
$837$	298.467	0.356591
$838$	0	0
$839$	− 288.110i	− 0.343397i	−0.985150	−	0.171698i	$-0.945075\pi$
$839$	− 288.110i	− 0.343397i	0.985150	−	0.171698i	$-0.0549254\pi$
$840$	0	0
$841$	−840.672	−0.999609
$842$	0	0
$843$	− 767.520i	− 0.910463i
$844$	0	0
$845$	258.822	0.306299
$846$	0	0
$847$	121.910i	0.143932i
$848$	0	0
$849$	510.297	0.601057
$850$	0	0
$851$	− 118.277i	− 0.138986i
$852$	0	0
$853$	−645.132	−0.756310	−0.378155	−	0.925742i	$-0.623441\pi$
$853$	−645.132	−0.756310	−0.378155	+	0.925742i	$0.623441\pi$
$854$	0	0
$855$	− 357.472i	− 0.418096i
$856$	0	0
$857$	995.549	1.16167	0.580833	−	0.814022i	$-0.302727\pi$
$857$	995.549	1.16167	0.580833	+	0.814022i	$0.302727\pi$
$858$	0	0
$859$	− 774.354i	− 0.901460i	−0.892660	−	0.450730i	$-0.851164\pi$
$859$	− 774.354i	− 0.901460i	0.892660	−	0.450730i	$-0.148836\pi$
$860$	0	0
$861$	−116.947	−0.135827
$862$	0	0
$863$	− 1007.33i	− 1.16724i	−0.812025	−	0.583622i	$-0.801635\pi$
$863$	− 1007.33i	− 1.16724i	0.812025	−	0.583622i	$-0.198365\pi$
$864$	0	0
$865$	−1938.50	−2.24104
$866$	0	0
$867$	− 257.081i	− 0.296518i
$868$	0	0
$869$	−356.858	−0.410654
$870$	0	0
$871$	− 990.836i	− 1.13758i
$872$	0	0
$873$	−291.415	−0.333809
$874$	0	0
$875$	− 234.102i	− 0.267546i
$876$	0	0
$877$	−681.645	−0.777246	−0.388623	−	0.921397i	$-0.627049\pi$
$877$	−681.645	−0.777246	−0.388623	+	0.921397i	$0.627049\pi$
$878$	0	0
$879$	− 115.565i	− 0.131474i
$880$	0	0
$881$	−679.108	−0.770837	−0.385419	−	0.922742i	$-0.625943\pi$
$881$	−679.108	−0.770837	−0.385419	+	0.922742i	$0.625943\pi$
$882$	0	0
$883$	1059.44i	1.19982i	0.800068	+	0.599910i	$0.204797\pi$
$883$	1059.44i	1.19982i	−0.800068	+	0.599910i	$0.795203\pi$
$884$	0	0
$885$	−729.762	−0.824590
$886$	0	0
$887$	− 856.411i	− 0.965514i	−0.875754	−	0.482757i	$-0.839635\pi$
$887$	− 856.411i	− 0.965514i	0.875754	−	0.482757i	$-0.160365\pi$
$888$	0	0
$889$	396.000	0.445444
$890$	0	0
$891$	− 72.0000i	− 0.0808081i
$892$	0	0
$893$	−889.403	−0.995972
$894$	0	0
$895$	2541.11i	2.83923i
$896$	0	0
$897$	−86.5847	−0.0965270
$898$	0	0
$899$	− 32.9179i	− 0.0366161i
$900$	0	0
$901$	371.758	0.412606
$902$	0	0
$903$	− 106.632i	− 0.118086i
$904$	0	0
$905$	−634.677	−0.701300
$906$	0	0
$907$	− 761.492i	− 0.839573i	−0.907623	−	0.419786i	$-0.862105\pi$
$907$	− 761.492i	− 0.839573i	0.907623	−	0.419786i	$-0.137895\pi$
$908$	0	0
$909$	186.411	0.205073
$910$	0	0
$911$	897.344i	0.985009i	0.870310	+	0.492505i	$0.163919\pi$
$911$	897.344i	0.985009i	−0.870310	+	0.492505i	$0.836081\pi$
$912$	0	0
$913$	−541.703	−0.593321
$914$	0	0
$915$	− 823.692i	− 0.900210i
$916$	0	0
$917$	268.115	0.292383
$918$	0	0
$919$	62.6388i	0.0681598i	0.999419	+	0.0340799i	$0.0108501\pi$
$919$	62.6388i	0.0681598i	−0.999419	+	0.0340799i	$0.989150\pi$
$920$	0	0
$921$	−907.959	−0.985840
$922$	0	0
$923$	496.308i	0.537712i
$924$	0	0
$925$	1070.43	1.15722
$926$	0	0
$927$	83.4124i	0.0899810i
$928$	0	0
$929$	253.313	0.272673	0.136336	−	0.990663i	$-0.456467\pi$
$929$	253.313	0.272673	0.136336	+	0.990663i	$0.456467\pi$
$930$	0	0
$931$	− 663.195i	− 0.712347i
$932$	0	0
$933$	−627.101	−0.672134
$934$	0	0
$935$	757.106i	0.809739i
$936$	0	0
$937$	30.5538	0.0326081	0.0163040	−	0.999867i	$-0.494810\pi$
$937$	30.5538	0.0326081	0.0163040	+	0.999867i	$0.494810\pi$
$938$	0	0
$939$	− 436.956i	− 0.465342i
$940$	0	0
$941$	388.745	0.413119	0.206559	−	0.978434i	$-0.433773\pi$
$941$	388.745	0.413119	0.206559	+	0.978434i	$0.433773\pi$
$942$	0	0
$943$	− 135.039i	− 0.143202i
$944$	0	0
$945$	−88.7077	−0.0938705
$946$	0	0
$947$	263.615i	0.278369i	0.990266	+	0.139184i	$0.0444481\pi$
$947$	263.615i	0.278369i	−0.990266	+	0.139184i	$0.955552\pi$
$948$	0	0
$949$	−63.4066	−0.0668141
$950$	0	0
$951$	139.245i	0.146420i
$952$	0	0
$953$	−1292.41	−1.35615	−0.678075	−	0.734993i	$-0.737185\pi$
$953$	−1292.41	−1.35615	−0.678075	+	0.734993i	$0.737185\pi$
$954$	0	0
$955$	2816.76i	2.94949i
$956$	0	0
$957$	−7.94089	−0.00829769
$958$	0	0
$959$	− 212.957i	− 0.222061i
$960$	0	0
$961$	−2338.34	−2.43324
$962$	0	0
$963$	− 113.338i	− 0.117693i
$964$	0	0
$965$	2269.11	2.35141
$966$	0	0
$967$	− 1110.00i	− 1.14788i	−0.818896	−	0.573942i	$-0.805413\pi$
$967$	− 1110.00i	− 1.14788i	0.818896	−	0.573942i	$-0.194587\pi$
$968$	0	0
$969$	306.564	0.316372
$970$	0	0
$971$	− 1341.66i	− 1.38173i	−0.722983	−	0.690866i	$-0.757230\pi$
$971$	− 1341.66i	− 1.38173i	0.722983	−	0.690866i	$-0.242770\pi$
$972$	0	0
$973$	−379.617	−0.390151
$974$	0	0
$975$	− 783.610i	− 0.803703i
$976$	0	0
$977$	−920.431	−0.942099	−0.471050	−	0.882107i	$-0.656125\pi$
$977$	−920.431	−0.942099	−0.471050	+	0.882107i	$0.656125\pi$
$978$	0	0
$979$	− 1065.11i	− 1.08795i
$980$	0	0
$981$	425.074	0.433307
$982$	0	0
$983$	1697.84i	1.72720i	0.504176	+	0.863601i	$0.331796\pi$
$983$	1697.84i	1.72720i	−0.504176	+	0.863601i	$0.668204\pi$
$984$	0	0
$985$	−605.108	−0.614322
$986$	0	0
$987$	220.708i	0.223615i
$988$	0	0
$989$	123.128	0.124497
$990$	0	0
$991$	− 284.765i	− 0.287351i	−0.989625	−	0.143675i	$-0.954108\pi$
$991$	− 284.765i	− 0.287351i	0.989625	−	0.143675i	$-0.0458921\pi$
$992$	0	0
$993$	−298.010	−0.300111
$994$	0	0
$995$	831.615i	0.835794i
$996$	0	0
$997$	1093.80	1.09710	0.548548	−	0.836119i	$-0.315181\pi$
$997$	1093.80	1.09710	0.548548	+	0.836119i	$0.315181\pi$
$998$	0	0
$999$	− 143.677i	− 0.143820i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.g.h.511.5		8
3.2	odd	2		2304.3.g.z.1279.8		8
4.3	odd	2	inner	768.3.g.h.511.1		8
8.3	odd	2	inner	768.3.g.h.511.8		8
8.5	even	2	inner	768.3.g.h.511.4		8
12.11	even	2		2304.3.g.z.1279.7		8
16.3	odd	4		96.3.b.a.79.4		4
16.5	even	4		96.3.b.a.79.3		4
16.11	odd	4		24.3.b.a.19.4	yes	4
16.13	even	4		24.3.b.a.19.3	✓	4
24.5	odd	2		2304.3.g.z.1279.2		8
24.11	even	2		2304.3.g.z.1279.1		8
48.5	odd	4		288.3.b.b.271.4		4
48.11	even	4		72.3.b.b.19.1		4
48.29	odd	4		72.3.b.b.19.2		4
48.35	even	4		288.3.b.b.271.1		4
80.3	even	4		2400.3.p.a.1999.7		8
80.13	odd	4		600.3.p.a.499.3		8
80.19	odd	4		2400.3.g.a.751.1		4
80.27	even	4		600.3.p.a.499.4		8
80.29	even	4		600.3.g.a.451.2		4
80.37	odd	4		2400.3.p.a.1999.3		8
80.43	even	4		600.3.p.a.499.5		8
80.53	odd	4		2400.3.p.a.1999.6		8
80.59	odd	4		600.3.g.a.451.1		4
80.67	even	4		2400.3.p.a.1999.2		8
80.69	even	4		2400.3.g.a.751.2		4
80.77	odd	4		600.3.p.a.499.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.3.b.a.19.3	✓	4	16.13	even	4
24.3.b.a.19.4	yes	4	16.11	odd	4
72.3.b.b.19.1		4	48.11	even	4
72.3.b.b.19.2		4	48.29	odd	4
96.3.b.a.79.3		4	16.5	even	4
96.3.b.a.79.4		4	16.3	odd	4
288.3.b.b.271.1		4	48.35	even	4
288.3.b.b.271.4		4	48.5	odd	4
600.3.g.a.451.1		4	80.59	odd	4
600.3.g.a.451.2		4	80.29	even	4
600.3.p.a.499.3		8	80.13	odd	4
600.3.p.a.499.4		8	80.27	even	4
600.3.p.a.499.5		8	80.43	even	4
600.3.p.a.499.6		8	80.77	odd	4
768.3.g.h.511.1		8	4.3	odd	2	inner
768.3.g.h.511.4		8	8.5	even	2	inner
768.3.g.h.511.5		8	1.1	even	1	trivial
768.3.g.h.511.8		8	8.3	odd	2	inner
2304.3.g.z.1279.1		8	24.11	even	2
2304.3.g.z.1279.2		8	24.5	odd	2
2304.3.g.z.1279.7		8	12.11	even	2
2304.3.g.z.1279.8		8	3.2	odd	2
2400.3.g.a.751.1		4	80.19	odd	4
2400.3.g.a.751.2		4	80.69	even	4
2400.3.p.a.1999.2		8	80.67	even	4
2400.3.p.a.1999.3		8	80.37	odd	4
2400.3.p.a.1999.6		8	80.53	odd	4
2400.3.p.a.1999.7		8	80.3	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -7.98203 q^{5} +2.13878i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -7.98203 q^{5} +2.13878i q^{7} -3.00000 q^{9} -8.00000i q^{11} -11.6865 q^{13} -13.8253i q^{15} +11.8564 q^{17} -14.9282i q^{19} -3.70447 q^{21} -4.27756i q^{23} +38.7128 q^{25} -5.19615i q^{27} -0.573084 q^{29} +57.4399i q^{31} +13.8564 q^{33} -17.0718i q^{35} +27.6506 q^{37} -20.2416i q^{39} +31.5692 q^{41} +28.7846i q^{43} +23.9461 q^{45} -59.5787i q^{47} +44.4256 q^{49} +20.5359i q^{51} +31.3550 q^{53} +63.8562i q^{55} +25.8564 q^{57} -52.7846i q^{59} +59.5787 q^{61} -6.41634i q^{63} +93.2820 q^{65} +84.7846i q^{67} +7.40895 q^{69} -42.4685i q^{71} +5.42563 q^{73} +67.0526i q^{75} +17.1102 q^{77} -44.6072i q^{79} +9.00000 q^{81} -67.7128i q^{83} -94.6382 q^{85} -0.992611i q^{87} +133.138 q^{89} -24.9948i q^{91} -99.4888 q^{93} +119.157i q^{95} +97.1384 q^{97} +24.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{9}+O(q^{10}) \) 8 * q - 24 * q^9	\( 8 q - 24 q^{9} - 16 q^{17} + 88 q^{25} - 80 q^{41} - 88 q^{49} + 96 q^{57} + 192 q^{65} - 400 q^{73} + 72 q^{81} + 400 q^{89} + 112 q^{97}+O(q^{100}) \) 8 * q - 24 * q^9 - 16 * q^17 + 88 * q^25 - 80 * q^41 - 88 * q^49 + 96 * q^57 + 192 * q^65 - 400 * q^73 + 72 * q^81 + 400 * q^89 + 112 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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