LMFDB - Embedded newform 768.3.g.g.511.2

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.2
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.3.g.g.511.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +8.89898 q^{5} +2.82843i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +8.89898 q^{5} +2.82843i q^{7} -3.00000 q^{9} -18.2419i q^{11} -5.79796 q^{13} -15.4135i q^{15} -21.5959 q^{17} -18.2419i q^{19} +4.89898 q^{21} -33.3697i q^{23} +54.1918 q^{25} +5.19615i q^{27} -4.49490 q^{29} -2.25697i q^{31} -31.5959 q^{33} +25.1701i q^{35} +43.1918 q^{37} +10.0424i q^{39} +1.59592 q^{41} +63.4967i q^{43} -26.6969 q^{45} -72.3962i q^{47} +41.0000 q^{49} +37.4052i q^{51} +70.2929 q^{53} -162.334i q^{55} -31.5959 q^{57} +34.6410i q^{59} -63.5959 q^{61} -8.48528i q^{63} -51.5959 q^{65} +3.24259i q^{67} -57.7980 q^{69} +68.4537i q^{71} +10.0000 q^{73} -93.8630i q^{75} +51.5959 q^{77} -35.0552i q^{79} +9.00000 q^{81} -42.2691i q^{83} -192.182 q^{85} +7.78539i q^{87} -5.19184 q^{89} -16.3991i q^{91} -3.90918 q^{93} -162.334i q^{95} -26.8082 q^{97} +54.7257i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 16 * q^5 - 12 * q^9	$ 4 q + 16 q^{5} - 12 q^{9} + 16 q^{13} - 8 q^{17} + 60 q^{25} + 80 q^{29} - 48 q^{33} + 16 q^{37} - 72 q^{41} - 48 q^{45} + 164 q^{49} + 144 q^{53} - 48 q^{57} - 176 q^{61} - 128 q^{65} - 192 q^{69} + 40 q^{73} + 128 q^{77} + 36 q^{81} - 416 q^{85} + 136 q^{89} - 192 q^{93} - 264 q^{97}+O(q^{100}) $ 4 * q + 16 * q^5 - 12 * q^9 + 16 * q^13 - 8 * q^17 + 60 * q^25 + 80 * q^29 - 48 * q^33 + 16 * q^37 - 72 * q^41 - 48 * q^45 + 164 * q^49 + 144 * q^53 - 48 * q^57 - 176 * q^61 - 128 * q^65 - 192 * q^69 + 40 * q^73 + 128 * q^77 + 36 * q^81 - 416 * q^85 + 136 * q^89 - 192 * q^93 - 264 * 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
$5$	8.89898	1.77980	0.889898	−	0.456160i	$-0.150775\pi$
$5$	8.89898	1.77980	0.889898	+	0.456160i	$0.150775\pi$
$6$	0	0
$7$	2.82843i	0.404061i	0.979379	+	0.202031i	$0.0647540\pi$
$7$	2.82843i	0.404061i	−0.979379	+	0.202031i	$0.935246\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 18.2419i	− 1.65836i	−0.558985	−	0.829178i	$-0.688809\pi$
$11$	− 18.2419i	− 1.65836i	0.558985	−	0.829178i	$-0.311191\pi$
$12$	0	0
$13$	−5.79796	−0.445997	−0.222998	−	0.974819i	$-0.571584\pi$
$13$	−5.79796	−0.445997	−0.222998	+	0.974819i	$0.571584\pi$
$14$	0	0
$15$	− 15.4135i	− 1.02757i
$16$	0	0
$17$	−21.5959	−1.27035	−0.635174	−	0.772369i	$-0.719072\pi$
$17$	−21.5959	−1.27035	−0.635174	+	0.772369i	$0.719072\pi$
$18$	0	0
$19$	− 18.2419i	− 0.960101i	−0.877241	−	0.480050i	$-0.840618\pi$
$19$	− 18.2419i	− 0.960101i	0.877241	−	0.480050i	$-0.159382\pi$
$20$	0	0
$21$	4.89898	0.233285
$22$	0	0
$23$	− 33.3697i	− 1.45086i	−0.688299	−	0.725428i	$-0.741642\pi$
$23$	− 33.3697i	− 1.45086i	0.688299	−	0.725428i	$-0.258358\pi$
$24$	0	0
$25$	54.1918	2.16767
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−4.49490	−0.154996	−0.0774982	−	0.996992i	$-0.524693\pi$
$29$	−4.49490	−0.154996	−0.0774982	+	0.996992i	$0.524693\pi$
$30$	0	0
$31$	− 2.25697i	− 0.0728054i	−0.999337	−	0.0364027i	$-0.988410\pi$
$31$	− 2.25697i	− 0.0728054i	0.999337	−	0.0364027i	$-0.0115899\pi$
$32$	0	0
$33$	−31.5959	−0.957452
$34$	0	0
$35$	25.1701i	0.719146i
$36$	0	0
$37$	43.1918	1.16735	0.583673	−	0.811988i	$-0.301615\pi$
$37$	43.1918	1.16735	0.583673	+	0.811988i	$0.301615\pi$
$38$	0	0
$39$	10.0424i	0.257496i
$40$	0	0
$41$	1.59592	0.0389248	0.0194624	−	0.999811i	$-0.493805\pi$
$41$	1.59592	0.0389248	0.0194624	+	0.999811i	$0.493805\pi$
$42$	0	0
$43$	63.4967i	1.47667i	0.674435	+	0.738334i	$0.264387\pi$
$43$	63.4967i	1.47667i	−0.674435	+	0.738334i	$0.735613\pi$
$44$	0	0
$45$	−26.6969	−0.593265
$46$	0	0
$47$	− 72.3962i	− 1.54034i	−0.637836	−	0.770172i	$-0.720170\pi$
$47$	− 72.3962i	− 1.54034i	0.637836	−	0.770172i	$-0.279830\pi$
$48$	0	0
$49$	41.0000	0.836735
$50$	0	0
$51$	37.4052i	0.733436i
$52$	0	0
$53$	70.2929	1.32628	0.663140	−	0.748495i	$-0.269223\pi$
$53$	70.2929	1.32628	0.663140	+	0.748495i	$0.269223\pi$
$54$	0	0
$55$	− 162.334i	− 2.95153i
$56$	0	0
$57$	−31.5959	−0.554314
$58$	0	0
$59$	34.6410i	0.587136i	0.955938	+	0.293568i	$0.0948427\pi$
$59$	34.6410i	0.587136i	−0.955938	+	0.293568i	$0.905157\pi$
$60$	0	0
$61$	−63.5959	−1.04256	−0.521278	−	0.853387i	$-0.674545\pi$
$61$	−63.5959	−1.04256	−0.521278	+	0.853387i	$0.674545\pi$
$62$	0	0
$63$	− 8.48528i	− 0.134687i
$64$	0	0
$65$	−51.5959	−0.793783
$66$	0	0
$67$	3.24259i	0.0483968i	0.999707	+	0.0241984i	$0.00770335\pi$
$67$	3.24259i	0.0483968i	−0.999707	+	0.0241984i	$0.992297\pi$
$68$	0	0
$69$	−57.7980	−0.837652
$70$	0	0
$71$	68.4537i	0.964137i	0.876134	+	0.482068i	$0.160114\pi$
$71$	68.4537i	0.964137i	−0.876134	+	0.482068i	$0.839886\pi$
$72$	0	0
$73$	10.0000	0.136986	0.0684932	−	0.997652i	$-0.478181\pi$
$73$	10.0000	0.136986	0.0684932	+	0.997652i	$0.478181\pi$
$74$	0	0
$75$	− 93.8630i	− 1.25151i
$76$	0	0
$77$	51.5959	0.670077
$78$	0	0
$79$	− 35.0552i	− 0.443736i	−0.975077	−	0.221868i	$-0.928785\pi$
$79$	− 35.0552i	− 0.443736i	0.975077	−	0.221868i	$-0.0712155\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 42.2691i	− 0.509266i	−0.967038	−	0.254633i	$-0.918045\pi$
$83$	− 42.2691i	− 0.509266i	0.967038	−	0.254633i	$-0.0819547\pi$
$84$	0	0
$85$	−192.182	−2.26096
$86$	0	0
$87$	7.78539i	0.0894872i
$88$	0	0
$89$	−5.19184	−0.0583352	−0.0291676	−	0.999575i	$-0.509286\pi$
$89$	−5.19184	−0.0583352	−0.0291676	+	0.999575i	$0.509286\pi$
$90$	0	0
$91$	− 16.3991i	− 0.180210i
$92$	0	0
$93$	−3.90918	−0.0420342
$94$	0	0
$95$	− 162.334i	− 1.70878i
$96$	0	0
$97$	−26.8082	−0.276373	−0.138186	−	0.990406i	$-0.544127\pi$
$97$	−26.8082	−0.276373	−0.138186	+	0.990406i	$0.544127\pi$
$98$	0	0
$99$	54.7257i	0.552785i
$100$	0	0
$101$	−50.8786	−0.503748	−0.251874	−	0.967760i	$-0.581047\pi$
$101$	−50.8786	−0.503748	−0.251874	+	0.967760i	$0.581047\pi$
$102$	0	0
$103$	− 148.764i	− 1.44431i	−0.691732	−	0.722154i	$-0.743152\pi$
$103$	− 148.764i	− 1.44431i	0.691732	−	0.722154i	$-0.256848\pi$
$104$	0	0
$105$	43.5959	0.415199
$106$	0	0
$107$	− 116.380i	− 1.08766i	−0.839195	−	0.543830i	$-0.816974\pi$
$107$	− 116.380i	− 1.08766i	0.839195	−	0.543830i	$-0.183026\pi$
$108$	0	0
$109$	44.1816	0.405336	0.202668	−	0.979248i	$-0.435039\pi$
$109$	44.1816	0.405336	0.202668	+	0.979248i	$0.435039\pi$
$110$	0	0
$111$	− 74.8105i	− 0.673968i
$112$	0	0
$113$	199.576	1.76615	0.883077	−	0.469227i	$-0.155467\pi$
$113$	199.576	1.76615	0.883077	+	0.469227i	$0.155467\pi$
$114$	0	0
$115$	− 296.956i	− 2.58223i
$116$	0	0
$117$	17.3939	0.148666
$118$	0	0
$119$	− 61.0825i	− 0.513298i
$120$	0	0
$121$	−211.767	−1.75014
$122$	0	0
$123$	− 2.76421i	− 0.0224733i
$124$	0	0
$125$	259.778	2.07822
$126$	0	0
$127$	183.276i	1.44312i	0.692352	+	0.721560i	$0.256575\pi$
$127$	183.276i	1.44312i	−0.692352	+	0.721560i	$0.743425\pi$
$128$	0	0
$129$	109.980	0.852555
$130$	0	0
$131$	176.891i	1.35031i	0.737676	+	0.675155i	$0.235923\pi$
$131$	176.891i	1.35031i	−0.737676	+	0.675155i	$0.764077\pi$
$132$	0	0
$133$	51.5959	0.387939
$134$	0	0
$135$	46.2405i	0.342522i
$136$	0	0
$137$	−147.980	−1.08014	−0.540071	−	0.841619i	$-0.681603\pi$
$137$	−147.980	−1.08014	−0.540071	+	0.841619i	$0.681603\pi$
$138$	0	0
$139$	114.980i	0.827193i	0.910460	+	0.413597i	$0.135728\pi$
$139$	114.980i	0.827193i	−0.910460	+	0.413597i	$0.864272\pi$
$140$	0	0
$141$	−125.394	−0.889318
$142$	0	0
$143$	105.766i	0.739621i
$144$	0	0
$145$	−40.0000	−0.275862
$146$	0	0
$147$	− 71.0141i	− 0.483089i
$148$	0	0
$149$	229.303	1.53895	0.769473	−	0.638679i	$-0.220519\pi$
$149$	229.303	1.53895	0.769473	+	0.638679i	$0.220519\pi$
$150$	0	0
$151$	225.674i	1.49453i	0.664527	+	0.747264i	$0.268633\pi$
$151$	225.674i	1.49453i	−0.664527	+	0.747264i	$0.731367\pi$
$152$	0	0
$153$	64.7878	0.423449
$154$	0	0
$155$	− 20.0847i	− 0.129579i
$156$	0	0
$157$	−36.8082	−0.234447	−0.117223	−	0.993106i	$-0.537399\pi$
$157$	−36.8082	−0.234447	−0.117223	+	0.993106i	$0.537399\pi$
$158$	0	0
$159$	− 121.751i	− 0.765728i
$160$	0	0
$161$	94.3837	0.586234
$162$	0	0
$163$	180.833i	1.10941i	0.832048	+	0.554703i	$0.187168\pi$
$163$	180.833i	1.10941i	−0.832048	+	0.554703i	$0.812832\pi$
$164$	0	0
$165$	−281.171	−1.70407
$166$	0	0
$167$	− 174.791i	− 1.04665i	−0.852132	−	0.523326i	$-0.824691\pi$
$167$	− 174.791i	− 1.04665i	0.852132	−	0.523326i	$-0.175309\pi$
$168$	0	0
$169$	−135.384	−0.801087
$170$	0	0
$171$	54.7257i	0.320034i
$172$	0	0
$173$	158.111	0.913938	0.456969	−	0.889483i	$-0.348935\pi$
$173$	158.111	0.913938	0.456969	+	0.889483i	$0.348935\pi$
$174$	0	0
$175$	153.278i	0.875872i
$176$	0	0
$177$	60.0000	0.338983
$178$	0	0
$179$	45.6979i	0.255295i	0.991820	+	0.127648i	$0.0407427\pi$
$179$	45.6979i	0.255295i	−0.991820	+	0.127648i	$0.959257\pi$
$180$	0	0
$181$	−263.778	−1.45733	−0.728667	−	0.684868i	$-0.759860\pi$
$181$	−263.778	−1.45733	−0.728667	+	0.684868i	$0.759860\pi$
$182$	0	0
$183$	110.151i	0.601920i
$184$	0	0
$185$	384.363	2.07764
$186$	0	0
$187$	393.951i	2.10669i
$188$	0	0
$189$	−14.6969	−0.0777616
$190$	0	0
$191$	154.963i	0.811325i	0.914023	+	0.405663i	$0.132959\pi$
$191$	154.963i	0.811325i	−0.914023	+	0.405663i	$0.867041\pi$
$192$	0	0
$193$	−182.767	−0.946981	−0.473491	−	0.880799i	$-0.657006\pi$
$193$	−182.767	−0.946981	−0.473491	+	0.880799i	$0.657006\pi$
$194$	0	0
$195$	89.3668i	0.458291i
$196$	0	0
$197$	178.697	0.907091	0.453546	−	0.891233i	$-0.350159\pi$
$197$	178.697	0.907091	0.453546	+	0.891233i	$0.350159\pi$
$198$	0	0
$199$	− 37.9125i	− 0.190515i	−0.995453	−	0.0952575i	$-0.969633\pi$
$199$	− 37.9125i	− 0.190515i	0.995453	−	0.0952575i	$-0.0303674\pi$
$200$	0	0
$201$	5.61633	0.0279419
$202$	0	0
$203$	− 12.7135i	− 0.0626280i
$204$	0	0
$205$	14.2020	0.0692782
$206$	0	0
$207$	100.109i	0.483618i
$208$	0	0
$209$	−332.767	−1.59219
$210$	0	0
$211$	− 19.8986i	− 0.0943060i	−0.998888	−	0.0471530i	$-0.984985\pi$
$211$	− 19.8986i	− 0.0943060i	0.998888	−	0.0471530i	$-0.0150148\pi$
$212$	0	0
$213$	118.565	0.556645
$214$	0	0
$215$	565.056i	2.62817i
$216$	0	0
$217$	6.38367	0.0294178
$218$	0	0
$219$	− 17.3205i	− 0.0790891i
$220$	0	0
$221$	125.212	0.566571
$222$	0	0
$223$	− 212.703i	− 0.953827i	−0.878950	−	0.476914i	$-0.841755\pi$
$223$	− 212.703i	− 0.953827i	0.878950	−	0.476914i	$-0.158245\pi$
$224$	0	0
$225$	−162.576	−0.722558
$226$	0	0
$227$	104.180i	0.458942i	0.973315	+	0.229471i	$0.0736997\pi$
$227$	104.180i	0.458942i	−0.973315	+	0.229471i	$0.926300\pi$
$228$	0	0
$229$	316.545	1.38229	0.691146	−	0.722715i	$-0.257106\pi$
$229$	316.545	1.38229	0.691146	+	0.722715i	$0.257106\pi$
$230$	0	0
$231$	− 89.3668i	− 0.386869i
$232$	0	0
$233$	−120.424	−0.516843	−0.258422	−	0.966032i	$-0.583202\pi$
$233$	−120.424	−0.516843	−0.258422	+	0.966032i	$0.583202\pi$
$234$	0	0
$235$	− 644.252i	− 2.74150i
$236$	0	0
$237$	−60.7173	−0.256191
$238$	0	0
$239$	178.734i	0.747839i	0.927461	+	0.373919i	$0.121986\pi$
$239$	178.734i	0.747839i	−0.927461	+	0.373919i	$0.878014\pi$
$240$	0	0
$241$	46.8082	0.194225	0.0971124	−	0.995273i	$-0.469039\pi$
$241$	46.8082	0.194225	0.0971124	+	0.995273i	$0.469039\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	364.858	1.48922
$246$	0	0
$247$	105.766i	0.428202i
$248$	0	0
$249$	−73.2122	−0.294025
$250$	0	0
$251$	39.9833i	0.159296i	0.996823	+	0.0796480i	$0.0253796\pi$
$251$	39.9833i	0.159296i	−0.996823	+	0.0796480i	$0.974620\pi$
$252$	0	0
$253$	−608.727	−2.40603
$254$	0	0
$255$	332.868i	1.30537i
$256$	0	0
$257$	290.000	1.12840	0.564202	−	0.825637i	$-0.309184\pi$
$257$	290.000	1.12840	0.564202	+	0.825637i	$0.309184\pi$
$258$	0	0
$259$	122.165i	0.471679i
$260$	0	0
$261$	13.4847	0.0516655
$262$	0	0
$263$	249.415i	0.948347i	0.880431	+	0.474174i	$0.157253\pi$
$263$	249.415i	0.948347i	−0.880431	+	0.474174i	$0.842747\pi$
$264$	0	0
$265$	625.535	2.36051
$266$	0	0
$267$	8.99252i	0.0336799i
$268$	0	0
$269$	−100.858	−0.374937	−0.187469	−	0.982271i	$-0.560028\pi$
$269$	−100.858	−0.374937	−0.187469	+	0.982271i	$0.560028\pi$
$270$	0	0
$271$	222.817i	0.822201i	0.911590	+	0.411101i	$0.134856\pi$
$271$	222.817i	0.822201i	−0.911590	+	0.411101i	$0.865144\pi$
$272$	0	0
$273$	−28.4041	−0.104044
$274$	0	0
$275$	− 988.563i	− 3.59477i
$276$	0	0
$277$	−194.202	−0.701090	−0.350545	−	0.936546i	$-0.614004\pi$
$277$	−194.202	−0.701090	−0.350545	+	0.936546i	$0.614004\pi$
$278$	0	0
$279$	6.77091i	0.0242685i
$280$	0	0
$281$	−202.767	−0.721592	−0.360796	−	0.932645i	$-0.617495\pi$
$281$	−202.767	−0.721592	−0.360796	+	0.932645i	$0.617495\pi$
$282$	0	0
$283$	− 156.549i	− 0.553177i	−0.960988	−	0.276589i	$-0.910796\pi$
$283$	− 156.549i	− 0.553177i	0.960988	−	0.276589i	$-0.0892039\pi$
$284$	0	0
$285$	−281.171	−0.986566
$286$	0	0
$287$	4.51394i	0.0157280i
$288$	0	0
$289$	177.384	0.613784
$290$	0	0
$291$	46.4331i	0.159564i
$292$	0	0
$293$	116.677	0.398213	0.199107	−	0.979978i	$-0.436196\pi$
$293$	116.677	0.398213	0.199107	+	0.979978i	$0.436196\pi$
$294$	0	0
$295$	308.270i	1.04498i
$296$	0	0
$297$	94.7878	0.319151
$298$	0	0
$299$	193.476i	0.647077i
$300$	0	0
$301$	−179.596	−0.596664
$302$	0	0
$303$	88.1243i	0.290839i
$304$	0	0
$305$	−565.939	−1.85554
$306$	0	0
$307$	17.9850i	0.0585832i	0.999571	+	0.0292916i	$0.00932514\pi$
$307$	17.9850i	0.0585832i	−0.999571	+	0.0292916i	$0.990675\pi$
$308$	0	0
$309$	−257.666	−0.833872
$310$	0	0
$311$	− 290.214i	− 0.933164i	−0.884478	−	0.466582i	$-0.845485\pi$
$311$	− 290.214i	− 0.933164i	0.884478	−	0.466582i	$-0.154515\pi$
$312$	0	0
$313$	255.535	0.816405	0.408202	−	0.912891i	$-0.366156\pi$
$313$	255.535	0.816405	0.408202	+	0.912891i	$0.366156\pi$
$314$	0	0
$315$	− 75.5103i	− 0.239715i
$316$	0	0
$317$	93.4847	0.294904	0.147452	−	0.989069i	$-0.452893\pi$
$317$	93.4847	0.294904	0.147452	+	0.989069i	$0.452893\pi$
$318$	0	0
$319$	81.9955i	0.257039i
$320$	0	0
$321$	−201.576	−0.627961
$322$	0	0
$323$	393.951i	1.21966i
$324$	0	0
$325$	−314.202	−0.966776
$326$	0	0
$327$	− 76.5248i	− 0.234021i
$328$	0	0
$329$	204.767	0.622393
$330$	0	0
$331$	− 151.464i	− 0.457594i	−0.973474	−	0.228797i	$-0.926521\pi$
$331$	− 151.464i	− 0.457594i	0.973474	−	0.228797i	$-0.0734793\pi$
$332$	0	0
$333$	−129.576	−0.389116
$334$	0	0
$335$	28.8557i	0.0861365i
$336$	0	0
$337$	102.767	0.304948	0.152474	−	0.988308i	$-0.451276\pi$
$337$	102.767	0.304948	0.152474	+	0.988308i	$0.451276\pi$
$338$	0	0
$339$	− 345.675i	− 1.01969i
$340$	0	0
$341$	−41.1714	−0.120737
$342$	0	0
$343$	254.558i	0.742153i
$344$	0	0
$345$	−514.343	−1.49085
$346$	0	0
$347$	333.626i	0.961458i	0.876869	+	0.480729i	$0.159628\pi$
$347$	333.626i	0.961458i	−0.876869	+	0.480729i	$0.840372\pi$
$348$	0	0
$349$	553.939	1.58722	0.793609	−	0.608429i	$-0.208200\pi$
$349$	553.939	1.58722	0.793609	+	0.608429i	$0.208200\pi$
$350$	0	0
$351$	− 30.1271i	− 0.0858321i
$352$	0	0
$353$	−398.727	−1.12954	−0.564768	−	0.825249i	$-0.691035\pi$
$353$	−398.727	−1.12954	−0.564768	+	0.825249i	$0.691035\pi$
$354$	0	0
$355$	609.168i	1.71597i
$356$	0	0
$357$	−105.798	−0.296353
$358$	0	0
$359$	− 110.280i	− 0.307186i	−0.988134	−	0.153593i	$-0.950916\pi$
$359$	− 110.280i	− 0.307186i	0.988134	−	0.153593i	$-0.0490845\pi$
$360$	0	0
$361$	28.2327	0.0782068
$362$	0	0
$363$	366.792i	1.01045i
$364$	0	0
$365$	88.9898	0.243808
$366$	0	0
$367$	− 372.181i	− 1.01412i	−0.861912	−	0.507058i	$-0.830733\pi$
$367$	− 372.181i	− 1.01412i	0.861912	−	0.507058i	$-0.169267\pi$
$368$	0	0
$369$	−4.78775	−0.0129749
$370$	0	0
$371$	198.818i	0.535898i
$372$	0	0
$373$	277.980	0.745254	0.372627	−	0.927981i	$-0.378457\pi$
$373$	277.980	0.745254	0.372627	+	0.927981i	$0.378457\pi$
$374$	0	0
$375$	− 449.948i	− 1.19986i
$376$	0	0
$377$	26.0612	0.0691279
$378$	0	0
$379$	320.797i	0.846430i	0.906029	+	0.423215i	$0.139099\pi$
$379$	320.797i	0.846430i	−0.906029	+	0.423215i	$0.860901\pi$
$380$	0	0
$381$	317.444	0.833186
$382$	0	0
$383$	609.797i	1.59216i	0.605191	+	0.796080i	$0.293097\pi$
$383$	609.797i	1.59216i	−0.605191	+	0.796080i	$0.706903\pi$
$384$	0	0
$385$	459.151	1.19260
$386$	0	0
$387$	− 190.490i	− 0.492223i
$388$	0	0
$389$	126.111	0.324193	0.162097	−	0.986775i	$-0.448174\pi$
$389$	126.111	0.324193	0.162097	+	0.986775i	$0.448174\pi$
$390$	0	0
$391$	720.649i	1.84309i
$392$	0	0
$393$	306.384	0.779602
$394$	0	0
$395$	− 311.955i	− 0.789760i
$396$	0	0
$397$	4.36326	0.0109906	0.00549529	−	0.999985i	$-0.498251\pi$
$397$	4.36326	0.0109906	0.00549529	+	0.999985i	$0.498251\pi$
$398$	0	0
$399$	− 89.3668i	− 0.223977i
$400$	0	0
$401$	691.049	1.72331	0.861657	−	0.507491i	$-0.169427\pi$
$401$	691.049	1.72331	0.861657	+	0.507491i	$0.169427\pi$
$402$	0	0
$403$	13.0858i	0.0324710i
$404$	0	0
$405$	80.0908	0.197755
$406$	0	0
$407$	− 787.902i	− 1.93588i
$408$	0	0
$409$	−485.959	−1.18816	−0.594082	−	0.804404i	$-0.702485\pi$
$409$	−485.959	−1.18816	−0.594082	+	0.804404i	$0.702485\pi$
$410$	0	0
$411$	256.308i	0.623621i
$412$	0	0
$413$	−97.9796	−0.237239
$414$	0	0
$415$	− 376.152i	− 0.906390i
$416$	0	0
$417$	199.151	0.477580
$418$	0	0
$419$	499.531i	1.19220i	0.802911	+	0.596098i	$0.203283\pi$
$419$	499.531i	1.19220i	−0.802911	+	0.596098i	$0.796717\pi$
$420$	0	0
$421$	21.8796	0.0519705	0.0259853	−	0.999662i	$-0.491728\pi$
$421$	21.8796	0.0519705	0.0259853	+	0.999662i	$0.491728\pi$
$422$	0	0
$423$	217.189i	0.513448i
$424$	0	0
$425$	−1170.32	−2.75370
$426$	0	0
$427$	− 179.876i	− 0.421256i
$428$	0	0
$429$	183.192	0.427021
$430$	0	0
$431$	− 513.631i	− 1.19172i	−0.803089	−	0.595859i	$-0.796812\pi$
$431$	− 513.631i	− 1.19172i	0.803089	−	0.595859i	$-0.203188\pi$
$432$	0	0
$433$	−16.3837	−0.0378376	−0.0189188	−	0.999821i	$-0.506022\pi$
$433$	−16.3837	−0.0378376	−0.0189188	+	0.999821i	$0.506022\pi$
$434$	0	0
$435$	69.2820i	0.159269i
$436$	0	0
$437$	−608.727	−1.39297
$438$	0	0
$439$	− 324.068i	− 0.738197i	−0.929390	−	0.369098i	$-0.879667\pi$
$439$	− 324.068i	− 0.738197i	0.929390	−	0.369098i	$-0.120333\pi$
$440$	0	0
$441$	−123.000	−0.278912
$442$	0	0
$443$	221.003i	0.498877i	0.968391	+	0.249439i	$0.0802461\pi$
$443$	221.003i	0.498877i	−0.968391	+	0.249439i	$0.919754\pi$
$444$	0	0
$445$	−46.2020	−0.103825
$446$	0	0
$447$	− 397.165i	− 0.888511i
$448$	0	0
$449$	690.322	1.53747	0.768733	−	0.639570i	$-0.220887\pi$
$449$	690.322	1.53747	0.768733	+	0.639570i	$0.220887\pi$
$450$	0	0
$451$	− 29.1126i	− 0.0645512i
$452$	0	0
$453$	390.879	0.862867
$454$	0	0
$455$	− 145.935i	− 0.320737i
$456$	0	0
$457$	−397.878	−0.870629	−0.435315	−	0.900278i	$-0.643363\pi$
$457$	−397.878	−0.870629	−0.435315	+	0.900278i	$0.643363\pi$
$458$	0	0
$459$	− 112.216i	− 0.244479i
$460$	0	0
$461$	707.160	1.53397	0.766985	−	0.641665i	$-0.221756\pi$
$461$	707.160	1.53397	0.766985	+	0.641665i	$0.221756\pi$
$462$	0	0
$463$	− 869.926i	− 1.87889i	−0.342700	−	0.939445i	$-0.611341\pi$
$463$	− 869.926i	− 1.87889i	0.342700	−	0.939445i	$-0.388659\pi$
$464$	0	0
$465$	−34.7878	−0.0748124
$466$	0	0
$467$	75.4396i	0.161541i	0.996733	+	0.0807705i	$0.0257381\pi$
$467$	75.4396i	0.161541i	−0.996733	+	0.0807705i	$0.974262\pi$
$468$	0	0
$469$	−9.17143	−0.0195553
$470$	0	0
$471$	63.7536i	0.135358i
$472$	0	0
$473$	1158.30	2.44884
$474$	0	0
$475$	− 988.563i	− 2.08118i
$476$	0	0
$477$	−210.879	−0.442093
$478$	0	0
$479$	90.5674i	0.189076i	0.995521	+	0.0945380i	$0.0301374\pi$
$479$	90.5674i	0.189076i	−0.995521	+	0.0945380i	$0.969863\pi$
$480$	0	0
$481$	−250.424	−0.520633
$482$	0	0
$483$	− 163.477i	− 0.338462i
$484$	0	0
$485$	−238.565	−0.491887
$486$	0	0
$487$	− 137.392i	− 0.282120i	−0.990001	−	0.141060i	$-0.954949\pi$
$487$	− 137.392i	− 0.282120i	0.990001	−	0.141060i	$-0.0450510\pi$
$488$	0	0
$489$	313.212	0.640516
$490$	0	0
$491$	89.5529i	0.182389i	0.995833	+	0.0911944i	$0.0290685\pi$
$491$	89.5529i	0.182389i	−0.995833	+	0.0911944i	$0.970932\pi$
$492$	0	0
$493$	97.0714	0.196899
$494$	0	0
$495$	487.003i	0.983845i
$496$	0	0
$497$	−193.616	−0.389570
$498$	0	0
$499$	− 268.286i	− 0.537648i	−0.963189	−	0.268824i	$-0.913365\pi$
$499$	− 268.286i	− 0.537648i	0.963189	−	0.268824i	$-0.0866350\pi$
$500$	0	0
$501$	−302.747	−0.604285
$502$	0	0
$503$	− 93.2515i	− 0.185391i	−0.995695	−	0.0926953i	$-0.970452\pi$
$503$	− 93.2515i	− 0.185391i	0.995695	−	0.0926953i	$-0.0295483\pi$
$504$	0	0
$505$	−452.767	−0.896569
$506$	0	0
$507$	234.491i	0.462508i
$508$	0	0
$509$	168.272	0.330594	0.165297	−	0.986244i	$-0.447142\pi$
$509$	168.272	0.330594	0.165297	+	0.986244i	$0.447142\pi$
$510$	0	0
$511$	28.2843i	0.0553508i
$512$	0	0
$513$	94.7878	0.184771
$514$	0	0
$515$	− 1323.85i	− 2.57057i
$516$	0	0
$517$	−1320.64	−2.55444
$518$	0	0
$519$	− 273.857i	− 0.527662i
$520$	0	0
$521$	−316.788	−0.608038	−0.304019	−	0.952666i	$-0.598329\pi$
$521$	−316.788	−0.608038	−0.304019	+	0.952666i	$0.598329\pi$
$522$	0	0
$523$	443.591i	0.848167i	0.905623	+	0.424083i	$0.139404\pi$
$523$	443.591i	0.848167i	−0.905623	+	0.424083i	$0.860596\pi$
$524$	0	0
$525$	265.485	0.505685
$526$	0	0
$527$	48.7413i	0.0924883i
$528$	0	0
$529$	−584.535	−1.10498
$530$	0	0
$531$	− 103.923i	− 0.195712i
$532$	0	0
$533$	−9.25307	−0.0173604
$534$	0	0
$535$	− 1035.66i	− 1.93581i
$536$	0	0
$537$	79.1510	0.147395
$538$	0	0
$539$	− 747.918i	− 1.38760i
$540$	0	0
$541$	966.443	1.78640	0.893200	−	0.449659i	$-0.148454\pi$
$541$	966.443	1.78640	0.893200	+	0.449659i	$0.148454\pi$
$542$	0	0
$543$	456.876i	0.841392i
$544$	0	0
$545$	393.171	0.721415
$546$	0	0
$547$	578.470i	1.05753i	0.848768	+	0.528766i	$0.177345\pi$
$547$	578.470i	1.05753i	−0.848768	+	0.528766i	$0.822655\pi$
$548$	0	0
$549$	190.788	0.347519
$550$	0	0
$551$	81.9955i	0.148812i
$552$	0	0
$553$	99.1510	0.179297
$554$	0	0
$555$	− 665.737i	− 1.19953i
$556$	0	0
$557$	−590.293	−1.05977	−0.529886	−	0.848069i	$-0.677765\pi$
$557$	−590.293	−1.05977	−0.529886	+	0.848069i	$0.677765\pi$
$558$	0	0
$559$	− 368.152i	− 0.658589i
$560$	0	0
$561$	682.343	1.21630
$562$	0	0
$563$	− 468.877i	− 0.832818i	−0.909177	−	0.416409i	$-0.863288\pi$
$563$	− 468.877i	− 0.832818i	0.909177	−	0.416409i	$-0.136712\pi$
$564$	0	0
$565$	1776.02	3.14340
$566$	0	0
$567$	25.4558i	0.0448957i
$568$	0	0
$569$	548.829	0.964549	0.482275	−	0.876020i	$-0.339811\pi$
$569$	548.829	0.964549	0.482275	+	0.876020i	$0.339811\pi$
$570$	0	0
$571$	133.036i	0.232987i	0.993191	+	0.116494i	$0.0371654\pi$
$571$	133.036i	0.232987i	−0.993191	+	0.116494i	$0.962835\pi$
$572$	0	0
$573$	268.404	0.468419
$574$	0	0
$575$	− 1808.36i	− 3.14498i
$576$	0	0
$577$	−735.535	−1.27476	−0.637378	−	0.770551i	$-0.719981\pi$
$577$	−735.535	−1.27476	−0.637378	+	0.770551i	$0.719981\pi$
$578$	0	0
$579$	316.562i	0.546740i
$580$	0	0
$581$	119.555	0.205775
$582$	0	0
$583$	− 1282.28i	− 2.19944i
$584$	0	0
$585$	154.788	0.264594
$586$	0	0
$587$	178.290i	0.303732i	0.988401	+	0.151866i	$0.0485282\pi$
$587$	178.290i	0.303732i	−0.988401	+	0.151866i	$0.951472\pi$
$588$	0	0
$589$	−41.1714	−0.0699006
$590$	0	0
$591$	− 309.512i	− 0.523709i
$592$	0	0
$593$	−584.261	−0.985263	−0.492632	−	0.870238i	$-0.663965\pi$
$593$	−584.261	−0.985263	−0.492632	+	0.870238i	$0.663965\pi$
$594$	0	0
$595$	− 543.572i	− 0.913566i
$596$	0	0
$597$	−65.6663	−0.109994
$598$	0	0
$599$	134.050i	0.223790i	0.993720	+	0.111895i	$0.0356920\pi$
$599$	134.050i	0.223790i	−0.993720	+	0.111895i	$0.964308\pi$
$600$	0	0
$601$	−621.918	−1.03481	−0.517403	−	0.855742i	$-0.673101\pi$
$601$	−621.918	−1.03481	−0.517403	+	0.855742i	$0.673101\pi$
$602$	0	0
$603$	− 9.72777i	− 0.0161323i
$604$	0	0
$605$	−1884.51	−3.11490
$606$	0	0
$607$	− 36.1404i	− 0.0595393i	−0.999557	−	0.0297697i	$-0.990523\pi$
$607$	− 36.1404i	− 0.0595393i	0.999557	−	0.0297697i	$-0.00947738\pi$
$608$	0	0
$609$	−22.0204	−0.0361583
$610$	0	0
$611$	419.750i	0.686989i
$612$	0	0
$613$	385.857	0.629457	0.314728	−	0.949182i	$-0.398087\pi$
$613$	385.857	0.629457	0.314728	+	0.949182i	$0.398087\pi$
$614$	0	0
$615$	− 24.5987i	− 0.0399978i
$616$	0	0
$617$	−290.849	−0.471392	−0.235696	−	0.971827i	$-0.575737\pi$
$617$	−290.849	−0.471392	−0.235696	+	0.971827i	$0.575737\pi$
$618$	0	0
$619$	275.658i	0.445327i	0.974895	+	0.222664i	$0.0714752\pi$
$619$	275.658i	0.445327i	−0.974895	+	0.222664i	$0.928525\pi$
$620$	0	0
$621$	173.394	0.279217
$622$	0	0
$623$	− 14.6847i	− 0.0235710i
$624$	0	0
$625$	956.959	1.53113
$626$	0	0
$627$	576.370i	0.919250i
$628$	0	0
$629$	−932.767	−1.48294
$630$	0	0
$631$	940.608i	1.49066i	0.666694	+	0.745331i	$0.267709\pi$
$631$	940.608i	1.49066i	−0.666694	+	0.745331i	$0.732291\pi$
$632$	0	0
$633$	−34.4653	−0.0544476
$634$	0	0
$635$	1630.97i	2.56846i
$636$	0	0
$637$	−237.716	−0.373181
$638$	0	0
$639$	− 205.361i	− 0.321379i
$640$	0	0
$641$	−34.3633	−0.0536088	−0.0268044	−	0.999641i	$-0.508533\pi$
$641$	−34.3633	−0.0536088	−0.0268044	+	0.999641i	$0.508533\pi$
$642$	0	0
$643$	− 308.340i	− 0.479534i	−0.970830	−	0.239767i	$-0.922929\pi$
$643$	− 308.340i	− 0.479534i	0.970830	−	0.239767i	$-0.0770711\pi$
$644$	0	0
$645$	978.706	1.51737
$646$	0	0
$647$	− 190.503i	− 0.294441i	−0.989104	−	0.147220i	$-0.952967\pi$
$647$	− 190.503i	− 0.294441i	0.989104	−	0.147220i	$-0.0470327\pi$
$648$	0	0
$649$	631.918	0.973680
$650$	0	0
$651$	− 11.0568i	− 0.0169844i
$652$	0	0
$653$	−1103.38	−1.68971	−0.844857	−	0.534993i	$-0.820314\pi$
$653$	−1103.38	−1.68971	−0.844857	+	0.534993i	$0.820314\pi$
$654$	0	0
$655$	1574.15i	2.40328i
$656$	0	0
$657$	−30.0000	−0.0456621
$658$	0	0
$659$	815.544i	1.23755i	0.785569	+	0.618774i	$0.212370\pi$
$659$	815.544i	1.23755i	−0.785569	+	0.618774i	$0.787630\pi$
$660$	0	0
$661$	121.576	0.183927	0.0919633	−	0.995762i	$-0.470686\pi$
$661$	121.576	0.183927	0.0919633	+	0.995762i	$0.470686\pi$
$662$	0	0
$663$	− 216.874i	− 0.327110i
$664$	0	0
$665$	459.151	0.690453
$666$	0	0
$667$	149.993i	0.224877i
$668$	0	0
$669$	−368.413	−0.550692
$670$	0	0
$671$	1160.11i	1.72893i
$672$	0	0
$673$	−1326.60	−1.97118	−0.985590	−	0.169152i	$-0.945897\pi$
$673$	−1326.60	−1.97118	−0.985590	+	0.169152i	$0.945897\pi$
$674$	0	0
$675$	281.589i	0.417169i
$676$	0	0
$677$	−657.526	−0.971234	−0.485617	−	0.874172i	$-0.661405\pi$
$677$	−657.526	−0.971234	−0.485617	+	0.874172i	$0.661405\pi$
$678$	0	0
$679$	− 75.8249i	− 0.111671i
$680$	0	0
$681$	180.445	0.264970
$682$	0	0
$683$	− 772.318i	− 1.13077i	−0.824826	−	0.565386i	$-0.808727\pi$
$683$	− 772.318i	− 1.13077i	0.824826	−	0.565386i	$-0.191273\pi$
$684$	0	0
$685$	−1316.87	−1.92243
$686$	0	0
$687$	− 548.272i	− 0.798067i
$688$	0	0
$689$	−407.555	−0.591517
$690$	0	0
$691$	1269.00i	1.83647i	0.396030	+	0.918237i	$0.370387\pi$
$691$	1269.00i	1.83647i	−0.396030	+	0.918237i	$0.629613\pi$
$692$	0	0
$693$	−154.788	−0.223359
$694$	0	0
$695$	1023.20i	1.47224i
$696$	0	0
$697$	−34.4653	−0.0494481
$698$	0	0
$699$	208.581i	0.298400i
$700$	0	0
$701$	535.383	0.763741	0.381871	−	0.924216i	$-0.375280\pi$
$701$	535.383	0.763741	0.381871	+	0.924216i	$0.375280\pi$
$702$	0	0
$703$	− 787.902i	− 1.12077i
$704$	0	0
$705$	−1115.88	−1.58281
$706$	0	0
$707$	− 143.906i	− 0.203545i
$708$	0	0
$709$	−56.8673	−0.0802078	−0.0401039	−	0.999196i	$-0.512769\pi$
$709$	−56.8673	−0.0802078	−0.0401039	+	0.999196i	$0.512769\pi$
$710$	0	0
$711$	105.166i	0.147912i
$712$	0	0
$713$	−75.3143	−0.105630
$714$	0	0
$715$	941.208i	1.31638i
$716$	0	0
$717$	309.576	0.431765
$718$	0	0
$719$	− 980.922i	− 1.36429i	−0.731219	−	0.682143i	$-0.761048\pi$
$719$	− 980.922i	− 1.36429i	0.731219	−	0.682143i	$-0.238952\pi$
$720$	0	0
$721$	420.767	0.583589
$722$	0	0
$723$	− 81.0741i	− 0.112136i
$724$	0	0
$725$	−243.587	−0.335982
$726$	0	0
$727$	83.1673i	0.114398i	0.998363	+	0.0571990i	$0.0182169\pi$
$727$	83.1673i	0.114398i	−0.998363	+	0.0571990i	$0.981783\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 1371.27i	− 1.87588i
$732$	0	0
$733$	345.476	0.471317	0.235659	−	0.971836i	$-0.424275\pi$
$733$	345.476	0.471317	0.235659	+	0.971836i	$0.424275\pi$
$734$	0	0
$735$	− 631.953i	− 0.859800i
$736$	0	0
$737$	59.1510	0.0802592
$738$	0	0
$739$	− 1388.23i	− 1.87852i	−0.343203	−	0.939261i	$-0.611512\pi$
$739$	− 1388.23i	− 1.87852i	0.343203	−	0.939261i	$-0.388488\pi$
$740$	0	0
$741$	183.192	0.247222
$742$	0	0
$743$	− 885.269i	− 1.19148i	−0.803178	−	0.595739i	$-0.796859\pi$
$743$	− 885.269i	− 1.19148i	0.803178	−	0.595739i	$-0.203141\pi$
$744$	0	0
$745$	2040.56	2.73901
$746$	0	0
$747$	126.807i	0.169755i
$748$	0	0
$749$	329.171	0.439481
$750$	0	0
$751$	1225.11i	1.63130i	0.578545	+	0.815651i	$0.303621\pi$
$751$	1225.11i	1.63130i	−0.578545	+	0.815651i	$0.696379\pi$
$752$	0	0
$753$	69.2531	0.0919695
$754$	0	0
$755$	2008.27i	2.65996i
$756$	0	0
$757$	86.9694	0.114887	0.0574435	−	0.998349i	$-0.481705\pi$
$757$	86.9694	0.114887	0.0574435	+	0.998349i	$0.481705\pi$
$758$	0	0
$759$	1054.35i	1.38912i
$760$	0	0
$761$	764.020	1.00397	0.501985	−	0.864877i	$-0.332603\pi$
$761$	764.020	1.00397	0.501985	+	0.864877i	$0.332603\pi$
$762$	0	0
$763$	124.965i	0.163781i
$764$	0	0
$765$	576.545	0.753653
$766$	0	0
$767$	− 200.847i	− 0.261861i
$768$	0	0
$769$	269.918	0.350999	0.175500	−	0.984480i	$-0.443846\pi$
$769$	269.918	0.350999	0.175500	+	0.984480i	$0.443846\pi$
$770$	0	0
$771$	− 502.295i	− 0.651485i
$772$	0	0
$773$	1109.24	1.43498	0.717490	−	0.696569i	$-0.245291\pi$
$773$	1109.24	1.43498	0.717490	+	0.696569i	$0.245291\pi$
$774$	0	0
$775$	− 122.309i	− 0.157818i
$776$	0	0
$777$	211.596	0.272324
$778$	0	0
$779$	− 29.1126i	− 0.0373718i
$780$	0	0
$781$	1248.73	1.59888
$782$	0	0
$783$	− 23.3562i	− 0.0298291i
$784$	0	0
$785$	−327.555	−0.417268
$786$	0	0
$787$	− 1260.98i	− 1.60226i	−0.598491	−	0.801129i	$-0.704233\pi$
$787$	− 1260.98i	− 1.60226i	0.598491	−	0.801129i	$-0.295767\pi$
$788$	0	0
$789$	432.000	0.547529
$790$	0	0
$791$	564.485i	0.713634i
$792$	0	0
$793$	368.727	0.464977
$794$	0	0
$795$	− 1083.46i	− 1.36284i
$796$	0	0
$797$	−386.779	−0.485293	−0.242647	−	0.970115i	$-0.578016\pi$
$797$	−386.779	−0.485293	−0.242647	+	0.970115i	$0.578016\pi$
$798$	0	0
$799$	1563.46i	1.95677i
$800$	0	0
$801$	15.5755	0.0194451
$802$	0	0
$803$	− 182.419i	− 0.227172i
$804$	0	0
$805$	839.918	1.04338
$806$	0	0
$807$	174.691i	0.216470i
$808$	0	0
$809$	−1139.98	−1.40912	−0.704561	−	0.709643i	$-0.748856\pi$
$809$	−1139.98	−1.40912	−0.704561	+	0.709643i	$0.748856\pi$
$810$	0	0
$811$	959.450i	1.18305i	0.806288	+	0.591523i	$0.201473\pi$
$811$	959.450i	1.18305i	−0.806288	+	0.591523i	$0.798527\pi$
$812$	0	0
$813$	385.930	0.474698
$814$	0	0
$815$	1609.23i	1.97452i
$816$	0	0
$817$	1158.30	1.41775
$818$	0	0
$819$	49.1973i	0.0600700i
$820$	0	0
$821$	−918.674	−1.11897	−0.559485	−	0.828840i	$-0.689001\pi$
$821$	−918.674	−1.11897	−0.559485	+	0.828840i	$0.689001\pi$
$822$	0	0
$823$	678.107i	0.823945i	0.911196	+	0.411972i	$0.135160\pi$
$823$	678.107i	0.823945i	−0.911196	+	0.411972i	$0.864840\pi$
$824$	0	0
$825$	−1712.24	−2.07544
$826$	0	0
$827$	− 1237.58i	− 1.49647i	−0.663434	−	0.748234i	$-0.730902\pi$
$827$	− 1237.58i	− 1.49647i	0.663434	−	0.748234i	$-0.269098\pi$
$828$	0	0
$829$	−778.120	−0.938625	−0.469313	−	0.883032i	$-0.655498\pi$
$829$	−778.120	−0.938625	−0.469313	+	0.883032i	$0.655498\pi$
$830$	0	0
$831$	336.368i	0.404775i
$832$	0	0
$833$	−885.433	−1.06294
$834$	0	0
$835$	− 1555.46i	− 1.86283i
$836$	0	0
$837$	11.7276	0.0140114
$838$	0	0
$839$	− 345.582i	− 0.411897i	−0.978563	−	0.205949i	$-0.933972\pi$
$839$	− 345.582i	− 0.411897i	0.978563	−	0.205949i	$-0.0660280\pi$
$840$	0	0
$841$	−820.796	−0.975976
$842$	0	0
$843$	351.203i	0.416611i
$844$	0	0
$845$	−1204.78	−1.42577
$846$	0	0
$847$	− 598.968i	− 0.707165i
$848$	0	0
$849$	−271.151	−0.319377
$850$	0	0
$851$	− 1441.30i	− 1.69365i
$852$	0	0
$853$	1026.38	1.20326	0.601632	−	0.798774i	$-0.294518\pi$
$853$	1026.38	1.20326	0.601632	+	0.798774i	$0.294518\pi$
$854$	0	0
$855$	487.003i	0.569594i
$856$	0	0
$857$	296.061	0.345462	0.172731	−	0.984969i	$-0.444741\pi$
$857$	296.061	0.345462	0.172731	+	0.984969i	$0.444741\pi$
$858$	0	0
$859$	− 167.118i	− 0.194550i	−0.995258	−	0.0972748i	$-0.968987\pi$
$859$	− 167.118i	− 0.194550i	0.995258	−	0.0972748i	$-0.0310126\pi$
$860$	0	0
$861$	7.81837	0.00908057
$862$	0	0
$863$	10.0553i	0.0116516i	0.999983	+	0.00582580i	$0.00185442\pi$
$863$	10.0553i	0.0116516i	−0.999983	+	0.00582580i	$0.998146\pi$
$864$	0	0
$865$	1407.03	1.62662
$866$	0	0
$867$	− 307.238i	− 0.354369i
$868$	0	0
$869$	−639.473	−0.735873
$870$	0	0
$871$	− 18.8004i	− 0.0215848i
$872$	0	0
$873$	80.4245	0.0921243
$874$	0	0
$875$	734.762i	0.839728i
$876$	0	0
$877$	1154.58	1.31651	0.658257	−	0.752793i	$-0.271294\pi$
$877$	1154.58	1.31651	0.658257	+	0.752793i	$0.271294\pi$
$878$	0	0
$879$	− 202.090i	− 0.229909i
$880$	0	0
$881$	160.220	0.181862	0.0909310	−	0.995857i	$-0.471016\pi$
$881$	160.220	0.181862	0.0909310	+	0.995857i	$0.471016\pi$
$882$	0	0
$883$	1311.46i	1.48523i	0.669718	+	0.742616i	$0.266415\pi$
$883$	1311.46i	1.48523i	−0.669718	+	0.742616i	$0.733585\pi$
$884$	0	0
$885$	533.939	0.603321
$886$	0	0
$887$	234.673i	0.264569i	0.991212	+	0.132285i	$0.0422313\pi$
$887$	234.673i	0.264569i	−0.991212	+	0.132285i	$0.957769\pi$
$888$	0	0
$889$	−518.384	−0.583109
$890$	0	0
$891$	− 164.177i	− 0.184262i
$892$	0	0
$893$	−1320.64	−1.47889
$894$	0	0
$895$	406.664i	0.454374i
$896$	0	0
$897$	335.110	0.373590
$898$	0	0
$899$	10.1448i	0.0112846i
$900$	0	0
$901$	−1518.04	−1.68484
$902$	0	0
$903$	311.069i	0.344484i
$904$	0	0
$905$	−2347.35	−2.59376
$906$	0	0
$907$	− 1161.70i	− 1.28081i	−0.768036	−	0.640406i	$-0.778766\pi$
$907$	− 1161.70i	− 1.28081i	0.768036	−	0.640406i	$-0.221234\pi$
$908$	0	0
$909$	152.636	0.167916
$910$	0	0
$911$	1382.33i	1.51737i	0.651456	+	0.758687i	$0.274159\pi$
$911$	1382.33i	1.51737i	−0.651456	+	0.758687i	$0.725841\pi$
$912$	0	0
$913$	−771.069	−0.844545
$914$	0	0
$915$	980.235i	1.07129i
$916$	0	0
$917$	−500.322	−0.545608
$918$	0	0
$919$	− 1206.60i	− 1.31294i	−0.754350	−	0.656472i	$-0.772048\pi$
$919$	− 1206.60i	− 1.31294i	0.754350	−	0.656472i	$-0.227952\pi$
$920$	0	0
$921$	31.1510	0.0338230
$922$	0	0
$923$	− 396.892i	− 0.430002i
$924$	0	0
$925$	2340.64	2.53043
$926$	0	0
$927$	446.291i	0.481436i
$928$	0	0
$929$	1159.90	1.24854	0.624272	−	0.781207i	$-0.285396\pi$
$929$	1159.90	1.24854	0.624272	+	0.781207i	$0.285396\pi$
$930$	0	0
$931$	− 747.918i	− 0.803349i
$932$	0	0
$933$	−502.665	−0.538762
$934$	0	0
$935$	3505.76i	3.74948i
$936$	0	0
$937$	−173.837	−0.185525	−0.0927624	−	0.995688i	$-0.529570\pi$
$937$	−173.837	−0.185525	−0.0927624	+	0.995688i	$0.529570\pi$
$938$	0	0
$939$	− 442.599i	− 0.471352i
$940$	0	0
$941$	603.968	0.641837	0.320918	−	0.947107i	$-0.396008\pi$
$941$	603.968	0.641837	0.320918	+	0.947107i	$0.396008\pi$
$942$	0	0
$943$	− 53.2553i	− 0.0564743i
$944$	0	0
$945$	−130.788	−0.138400
$946$	0	0
$947$	951.077i	1.00431i	0.864779	+	0.502153i	$0.167458\pi$
$947$	951.077i	1.00431i	−0.864779	+	0.502153i	$0.832542\pi$
$948$	0	0
$949$	−57.9796	−0.0610955
$950$	0	0
$951$	− 161.920i	− 0.170263i
$952$	0	0
$953$	1407.98	1.47742	0.738709	−	0.674024i	$-0.235436\pi$
$953$	1407.98	1.47742	0.738709	+	0.674024i	$0.235436\pi$
$954$	0	0
$955$	1379.01i	1.44399i
$956$	0	0
$957$	142.020	0.148402
$958$	0	0
$959$	− 418.549i	− 0.436444i
$960$	0	0
$961$	955.906	0.994699
$962$	0	0
$963$	349.139i	0.362554i
$964$	0	0
$965$	−1626.44	−1.68543
$966$	0	0
$967$	− 467.718i	− 0.483679i	−0.970316	−	0.241840i	$-0.922249\pi$
$967$	− 467.718i	− 0.483679i	0.970316	−	0.241840i	$-0.0777508\pi$
$968$	0	0
$969$	682.343	0.704172
$970$	0	0
$971$	1115.70i	1.14902i	0.818498	+	0.574510i	$0.194807\pi$
$971$	1115.70i	1.14902i	−0.818498	+	0.574510i	$0.805193\pi$
$972$	0	0
$973$	−325.212	−0.334237
$974$	0	0
$975$	544.214i	0.558168i
$976$	0	0
$977$	494.363	0.506001	0.253001	−	0.967466i	$-0.418583\pi$
$977$	494.363	0.506001	0.253001	+	0.967466i	$0.418583\pi$
$978$	0	0
$979$	94.7090i	0.0967406i
$980$	0	0
$981$	−132.545	−0.135112
$982$	0	0
$983$	− 188.391i	− 0.191649i	−0.995398	−	0.0958243i	$-0.969451\pi$
$983$	− 188.391i	− 0.191649i	0.995398	−	0.0958243i	$-0.0305487\pi$
$984$	0	0
$985$	1590.22	1.61444
$986$	0	0
$987$	− 354.667i	− 0.359339i
$988$	0	0
$989$	2118.87	2.14243
$990$	0	0
$991$	1895.81i	1.91303i	0.291681	+	0.956516i	$0.405785\pi$
$991$	1895.81i	1.91303i	−0.291681	+	0.956516i	$0.594215\pi$
$992$	0	0
$993$	−262.343	−0.264192
$994$	0	0
$995$	− 337.382i	− 0.339078i
$996$	0	0
$997$	1317.98	1.32195	0.660973	−	0.750410i	$-0.270144\pi$
$997$	1317.98	1.32195	0.660973	+	0.750410i	$0.270144\pi$
$998$	0	0
$999$	224.431i	0.224656i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.g.g.511.2		4
3.2	odd	2		2304.3.g.o.1279.2		4
4.3	odd	2	inner	768.3.g.g.511.4		4
8.3	odd	2		768.3.g.c.511.1		4
8.5	even	2		768.3.g.c.511.3		4
12.11	even	2		2304.3.g.o.1279.1		4
16.3	odd	4		384.3.b.c.319.1	✓	8
16.5	even	4		384.3.b.c.319.4	yes	8
16.11	odd	4		384.3.b.c.319.8	yes	8
16.13	even	4		384.3.b.c.319.5	yes	8
24.5	odd	2		2304.3.g.x.1279.4		4
24.11	even	2		2304.3.g.x.1279.3		4
48.5	odd	4		1152.3.b.j.703.1		8
48.11	even	4		1152.3.b.j.703.2		8
48.29	odd	4		1152.3.b.j.703.7		8
48.35	even	4		1152.3.b.j.703.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.b.c.319.1	✓	8	16.3	odd	4
384.3.b.c.319.4	yes	8	16.5	even	4
384.3.b.c.319.5	yes	8	16.13	even	4
384.3.b.c.319.8	yes	8	16.11	odd	4
768.3.g.c.511.1		4	8.3	odd	2
768.3.g.c.511.3		4	8.5	even	2
768.3.g.g.511.2		4	1.1	even	1	trivial
768.3.g.g.511.4		4	4.3	odd	2	inner
1152.3.b.j.703.1		8	48.5	odd	4
1152.3.b.j.703.2		8	48.11	even	4
1152.3.b.j.703.7		8	48.29	odd	4
1152.3.b.j.703.8		8	48.35	even	4
2304.3.g.o.1279.1		4	12.11	even	2
2304.3.g.o.1279.2		4	3.2	odd	2
2304.3.g.x.1279.3		4	24.11	even	2
2304.3.g.x.1279.4		4	24.5	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +8.89898 q^{5} +2.82843i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +8.89898 q^{5} +2.82843i q^{7} -3.00000 q^{9} -18.2419i q^{11} -5.79796 q^{13} -15.4135i q^{15} -21.5959 q^{17} -18.2419i q^{19} +4.89898 q^{21} -33.3697i q^{23} +54.1918 q^{25} +5.19615i q^{27} -4.49490 q^{29} -2.25697i q^{31} -31.5959 q^{33} +25.1701i q^{35} +43.1918 q^{37} +10.0424i q^{39} +1.59592 q^{41} +63.4967i q^{43} -26.6969 q^{45} -72.3962i q^{47} +41.0000 q^{49} +37.4052i q^{51} +70.2929 q^{53} -162.334i q^{55} -31.5959 q^{57} +34.6410i q^{59} -63.5959 q^{61} -8.48528i q^{63} -51.5959 q^{65} +3.24259i q^{67} -57.7980 q^{69} +68.4537i q^{71} +10.0000 q^{73} -93.8630i q^{75} +51.5959 q^{77} -35.0552i q^{79} +9.00000 q^{81} -42.2691i q^{83} -192.182 q^{85} +7.78539i q^{87} -5.19184 q^{89} -16.3991i q^{91} -3.90918 q^{93} -162.334i q^{95} -26.8082 q^{97} +54.7257i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 16 * q^5 - 12 * q^9	\( 4 q + 16 q^{5} - 12 q^{9} + 16 q^{13} - 8 q^{17} + 60 q^{25} + 80 q^{29} - 48 q^{33} + 16 q^{37} - 72 q^{41} - 48 q^{45} + 164 q^{49} + 144 q^{53} - 48 q^{57} - 176 q^{61} - 128 q^{65} - 192 q^{69} + 40 q^{73} + 128 q^{77} + 36 q^{81} - 416 q^{85} + 136 q^{89} - 192 q^{93} - 264 q^{97}+O(q^{100}) \) 4 * q + 16 * q^5 - 12 * q^9 + 16 * q^13 - 8 * q^17 + 60 * q^25 + 80 * q^29 - 48 * q^33 + 16 * q^37 - 72 * q^41 - 48 * q^45 + 164 * q^49 + 144 * q^53 - 48 * q^57 - 176 * q^61 - 128 * q^65 - 192 * q^69 + 40 * q^73 + 128 * q^77 + 36 * q^81 - 416 * q^85 + 136 * q^89 - 192 * q^93 - 264 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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