LMFDB - Embedded newform 768.3.g.g.511.3

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			511.3
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.3.g.g.511.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -0.898979 q^{5} +2.82843i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} -0.898979 q^{5} +2.82843i q^{7} -3.00000 q^{9} -4.38551i q^{11} +13.7980 q^{13} -1.55708i q^{15} +17.5959 q^{17} -4.38551i q^{19} -4.89898 q^{21} +22.0560i q^{23} -24.1918 q^{25} -5.19615i q^{27} +44.4949 q^{29} +53.1687i q^{31} +7.59592 q^{33} -2.54270i q^{35} -35.1918 q^{37} +23.8988i q^{39} -37.5959 q^{41} +49.6403i q^{43} +2.69694 q^{45} +38.4551i q^{47} +41.0000 q^{49} +30.4770i q^{51} +1.70714 q^{53} +3.94248i q^{55} +7.59592 q^{57} -34.6410i q^{59} -24.4041 q^{61} -8.48528i q^{63} -12.4041 q^{65} -93.7523i q^{67} -38.2020 q^{69} +123.879i q^{71} +10.0000 q^{73} -41.9015i q^{75} +12.4041 q^{77} +131.222i q^{79} +9.00000 q^{81} +110.151i q^{83} -15.8184 q^{85} +77.0674i q^{87} +73.1918 q^{89} +39.0265i q^{91} -92.0908 q^{93} +3.94248i q^{95} -105.192 q^{97} +13.1565i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.898979	−0.179796	−0.0898979	−	0.995951i	$-0.528654\pi$
$5$	−0.898979	−0.179796	−0.0898979	+	0.995951i	$0.528654\pi$
$6$	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
$11$	− 4.38551i	− 0.398682i	−0.979930	−	0.199341i	$-0.936120\pi$
$11$	− 4.38551i	− 0.398682i	0.979930	−	0.199341i	$-0.0638802\pi$
$12$	0	0
$13$	13.7980	1.06138	0.530691	−	0.847566i	$-0.321933\pi$
$13$	13.7980	1.06138	0.530691	+	0.847566i	$0.321933\pi$
$14$	0	0
$15$	− 1.55708i	− 0.103805i
$16$	0	0
$17$	17.5959	1.03505	0.517527	−	0.855667i	$-0.326853\pi$
$17$	17.5959	1.03505	0.517527	+	0.855667i	$0.326853\pi$
$18$	0	0
$19$	− 4.38551i	− 0.230816i	−0.993318	−	0.115408i	$-0.963182\pi$
$19$	− 4.38551i	− 0.230816i	0.993318	−	0.115408i	$-0.0368176\pi$
$20$	0	0
$21$	−4.89898	−0.233285
$22$	0	0
$23$	22.0560i	0.958955i	0.877554	+	0.479477i	$0.159174\pi$
$23$	22.0560i	0.958955i	−0.877554	+	0.479477i	$0.840826\pi$
$24$	0	0
$25$	−24.1918	−0.967673
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	44.4949	1.53431	0.767153	−	0.641464i	$-0.221672\pi$
$29$	44.4949	1.53431	0.767153	+	0.641464i	$0.221672\pi$
$30$	0	0
$31$	53.1687i	1.71512i	0.514386	+	0.857559i	$0.328020\pi$
$31$	53.1687i	1.71512i	−0.514386	+	0.857559i	$0.671980\pi$
$32$	0	0
$33$	7.59592	0.230179
$34$	0	0
$35$	− 2.54270i	− 0.0726485i
$36$	0	0
$37$	−35.1918	−0.951131	−0.475565	−	0.879680i	$-0.657756\pi$
$37$	−35.1918	−0.951131	−0.475565	+	0.879680i	$0.657756\pi$
$38$	0	0
$39$	23.8988i	0.612789i
$40$	0	0
$41$	−37.5959	−0.916974	−0.458487	−	0.888701i	$-0.651608\pi$
$41$	−37.5959	−0.916974	−0.458487	+	0.888701i	$0.651608\pi$
$42$	0	0
$43$	49.6403i	1.15443i	0.816593	+	0.577213i	$0.195860\pi$
$43$	49.6403i	1.15443i	−0.816593	+	0.577213i	$0.804140\pi$
$44$	0	0
$45$	2.69694	0.0599320
$46$	0	0
$47$	38.4551i	0.818193i	0.912491	+	0.409096i	$0.134156\pi$
$47$	38.4551i	0.818193i	−0.912491	+	0.409096i	$0.865844\pi$
$48$	0	0
$49$	41.0000	0.836735
$50$	0	0
$51$	30.4770i	0.597589i
$52$	0	0
$53$	1.70714	0.0322103	0.0161051	−	0.999870i	$-0.494873\pi$
$53$	1.70714	0.0322103	0.0161051	+	0.999870i	$0.494873\pi$
$54$	0	0
$55$	3.94248i	0.0716814i
$56$	0	0
$57$	7.59592	0.133262
$58$	0	0
$59$	− 34.6410i	− 0.587136i	−0.955938	−	0.293568i	$-0.905157\pi$
$59$	− 34.6410i	− 0.587136i	0.955938	−	0.293568i	$-0.0948427\pi$
$60$	0	0
$61$	−24.4041	−0.400067	−0.200033	−	0.979789i	$-0.564105\pi$
$61$	−24.4041	−0.400067	−0.200033	+	0.979789i	$0.564105\pi$
$62$	0	0
$63$	− 8.48528i	− 0.134687i
$64$	0	0
$65$	−12.4041	−0.190832
$66$	0	0
$67$	− 93.7523i	− 1.39929i	−0.714492	−	0.699644i	$-0.753342\pi$
$67$	− 93.7523i	− 1.39929i	0.714492	−	0.699644i	$-0.246658\pi$
$68$	0	0
$69$	−38.2020	−0.553653
$70$	0	0
$71$	123.879i	1.74478i	0.488811	+	0.872390i	$0.337431\pi$
$71$	123.879i	1.74478i	−0.488811	+	0.872390i	$0.662569\pi$
$72$	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
$75$	− 41.9015i	− 0.558687i
$76$	0	0
$77$	12.4041	0.161092
$78$	0	0
$79$	131.222i	1.66103i	0.556993	+	0.830517i	$0.311955\pi$
$79$	131.222i	1.66103i	−0.556993	+	0.830517i	$0.688045\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	110.151i	1.32712i	0.748121	+	0.663562i	$0.230956\pi$
$83$	110.151i	1.32712i	−0.748121	+	0.663562i	$0.769044\pi$
$84$	0	0
$85$	−15.8184	−0.186098
$86$	0	0
$87$	77.0674i	0.885832i
$88$	0	0
$89$	73.1918	0.822380	0.411190	−	0.911550i	$-0.365113\pi$
$89$	73.1918	0.822380	0.411190	+	0.911550i	$0.365113\pi$
$90$	0	0
$91$	39.0265i	0.428863i
$92$	0	0
$93$	−92.0908	−0.990224
$94$	0	0
$95$	3.94248i	0.0414998i
$96$	0	0
$97$	−105.192	−1.08445	−0.542226	−	0.840233i	$-0.682418\pi$
$97$	−105.192	−1.08445	−0.542226	+	0.840233i	$0.682418\pi$
$98$	0	0
$99$	13.1565i	0.132894i
$100$	0	0
$101$	154.879	1.53345	0.766726	−	0.641975i	$-0.221885\pi$
$101$	154.879	1.53345	0.766726	+	0.641975i	$0.221885\pi$
$102$	0	0
$103$	− 37.9125i	− 0.368082i	−0.982919	−	0.184041i	$-0.941082\pi$
$103$	− 37.9125i	− 0.368082i	0.982919	−	0.184041i	$-0.0589180\pi$
$104$	0	0
$105$	4.40408	0.0419436
$106$	0	0
$107$	− 19.3848i	− 0.181167i	−0.995889	−	0.0905833i	$-0.971127\pi$
$107$	− 19.3848i	− 0.181167i	0.995889	−	0.0905833i	$-0.0288731\pi$
$108$	0	0
$109$	−132.182	−1.21268	−0.606338	−	0.795207i	$-0.707362\pi$
$109$	−132.182	−1.21268	−0.606338	+	0.795207i	$0.707362\pi$
$110$	0	0
$111$	− 60.9540i	− 0.549136i
$112$	0	0
$113$	−35.5755	−0.314828	−0.157414	−	0.987533i	$-0.550316\pi$
$113$	−35.5755	−0.314828	−0.157414	+	0.987533i	$0.550316\pi$
$114$	0	0
$115$	− 19.8279i	− 0.172416i
$116$	0	0
$117$	−41.3939	−0.353794
$118$	0	0
$119$	49.7688i	0.418225i
$120$	0	0
$121$	101.767	0.841052
$122$	0	0
$123$	− 65.1180i	− 0.529415i
$124$	0	0
$125$	44.2225	0.353780
$126$	0	0
$127$	127.851i	1.00670i	0.864083	+	0.503349i	$0.167899\pi$
$127$	127.851i	1.00670i	−0.864083	+	0.503349i	$0.832101\pi$
$128$	0	0
$129$	−85.9796	−0.666508
$130$	0	0
$131$	− 86.3810i	− 0.659397i	−0.944086	−	0.329699i	$-0.893053\pi$
$131$	− 86.3810i	− 0.659397i	0.944086	−	0.329699i	$-0.106947\pi$
$132$	0	0
$133$	12.4041	0.0932638
$134$	0	0
$135$	4.67123i	0.0346017i
$136$	0	0
$137$	47.9796	0.350216	0.175108	−	0.984549i	$-0.443973\pi$
$137$	47.9796	0.350216	0.175108	+	0.984549i	$0.443973\pi$
$138$	0	0
$139$	156.549i	1.12625i	0.826371	+	0.563126i	$0.190402\pi$
$139$	156.549i	1.12625i	−0.826371	+	0.563126i	$0.809598\pi$
$140$	0	0
$141$	−66.6061	−0.472384
$142$	0	0
$143$	− 60.5110i	− 0.423154i
$144$	0	0
$145$	−40.0000	−0.275862
$146$	0	0
$147$	71.0141i	0.483089i
$148$	0	0
$149$	258.697	1.73622	0.868111	−	0.496371i	$-0.165334\pi$
$149$	258.697	1.73622	0.868111	+	0.496371i	$0.165334\pi$
$150$	0	0
$151$	− 106.880i	− 0.707814i	−0.935281	−	0.353907i	$-0.884853\pi$
$151$	− 106.880i	− 0.707814i	0.935281	−	0.353907i	$-0.115147\pi$
$152$	0	0
$153$	−52.7878	−0.345018
$154$	0	0
$155$	− 47.7975i	− 0.308371i
$156$	0	0
$157$	−115.192	−0.733706	−0.366853	−	0.930279i	$-0.619565\pi$
$157$	−115.192	−0.733706	−0.366853	+	0.930279i	$0.619565\pi$
$158$	0	0
$159$	2.95686i	0.0185966i
$160$	0	0
$161$	−62.3837	−0.387476
$162$	0	0
$163$	− 248.715i	− 1.52586i	−0.646480	−	0.762931i	$-0.723760\pi$
$163$	− 248.715i	− 1.52586i	0.646480	−	0.762931i	$-0.276240\pi$
$164$	0	0
$165$	−6.82857	−0.0413853
$166$	0	0
$167$	− 119.365i	− 0.714763i	−0.933959	−	0.357381i	$-0.883670\pi$
$167$	− 119.365i	− 0.714763i	0.933959	−	0.357381i	$-0.116330\pi$
$168$	0	0
$169$	21.3837	0.126531
$170$	0	0
$171$	13.1565i	0.0769387i
$172$	0	0
$173$	265.889	1.53693	0.768465	−	0.639892i	$-0.221021\pi$
$173$	265.889	1.53693	0.768465	+	0.639892i	$0.221021\pi$
$174$	0	0
$175$	− 68.4248i	− 0.390999i
$176$	0	0
$177$	60.0000	0.338983
$178$	0	0
$179$	225.831i	1.26163i	0.775935	+	0.630813i	$0.217279\pi$
$179$	225.831i	1.26163i	−0.775935	+	0.630813i	$0.782721\pi$
$180$	0	0
$181$	−48.2225	−0.266422	−0.133211	−	0.991088i	$-0.542529\pi$
$181$	−48.2225	−0.266422	−0.133211	+	0.991088i	$0.542529\pi$
$182$	0	0
$183$	− 42.2691i	− 0.230979i
$184$	0	0
$185$	31.6367	0.171009
$186$	0	0
$187$	− 77.1670i	− 0.412658i
$188$	0	0
$189$	14.6969	0.0777616
$190$	0	0
$191$	− 177.591i	− 0.929794i	−0.885365	−	0.464897i	$-0.846091\pi$
$191$	− 177.591i	− 0.929794i	0.885365	−	0.464897i	$-0.153909\pi$
$192$	0	0
$193$	130.767	0.677551	0.338776	−	0.940867i	$-0.389987\pi$
$193$	130.767	0.677551	0.338776	+	0.940867i	$0.389987\pi$
$194$	0	0
$195$	− 21.4845i	− 0.110177i
$196$	0	0
$197$	149.303	0.757884	0.378942	−	0.925421i	$-0.376288\pi$
$197$	149.303	0.757884	0.378942	+	0.925421i	$0.376288\pi$
$198$	0	0
$199$	− 148.764i	− 0.747556i	−0.927518	−	0.373778i	$-0.878062\pi$
$199$	− 148.764i	− 0.747556i	0.927518	−	0.373778i	$-0.121938\pi$
$200$	0	0
$201$	162.384	0.807879
$202$	0	0
$203$	125.851i	0.619954i
$204$	0	0
$205$	33.7980	0.164868
$206$	0	0
$207$	− 66.1679i	− 0.319652i
$208$	0	0
$209$	−19.2327	−0.0920223
$210$	0	0
$211$	381.937i	1.81013i	0.425274	+	0.905065i	$0.360178\pi$
$211$	381.937i	1.81013i	−0.425274	+	0.905065i	$0.639822\pi$
$212$	0	0
$213$	−214.565	−1.00735
$214$	0	0
$215$	− 44.6256i	− 0.207561i
$216$	0	0
$217$	−150.384	−0.693012
$218$	0	0
$219$	17.3205i	0.0790891i
$220$	0	0
$221$	242.788	1.09859
$222$	0	0
$223$	− 268.129i	− 1.20237i	−0.799109	−	0.601186i	$-0.794695\pi$
$223$	− 268.129i	− 1.20237i	0.799109	−	0.601186i	$-0.205305\pi$
$224$	0	0
$225$	72.5755	0.322558
$226$	0	0
$227$	− 353.082i	− 1.55543i	−0.628620	−	0.777713i	$-0.716380\pi$
$227$	− 353.082i	− 1.55543i	0.628620	−	0.777713i	$-0.283620\pi$
$228$	0	0
$229$	−212.545	−0.928144	−0.464072	−	0.885798i	$-0.653612\pi$
$229$	−212.545	−0.928144	−0.464072	+	0.885798i	$0.653612\pi$
$230$	0	0
$231$	21.4845i	0.0930065i
$232$	0	0
$233$	−355.576	−1.52608	−0.763038	−	0.646354i	$-0.776293\pi$
$233$	−355.576	−1.52608	−0.763038	+	0.646354i	$0.776293\pi$
$234$	0	0
$235$	− 34.5703i	− 0.147108i
$236$	0	0
$237$	−227.283	−0.958999
$238$	0	0
$239$	− 42.9690i	− 0.179787i	−0.995951	−	0.0898933i	$-0.971347\pi$
$239$	− 42.9690i	− 0.179787i	0.995951	−	0.0898933i	$-0.0286526\pi$
$240$	0	0
$241$	125.192	0.519468	0.259734	−	0.965680i	$-0.416365\pi$
$241$	125.192	0.519468	0.259734	+	0.965680i	$0.416365\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−36.8582	−0.150441
$246$	0	0
$247$	− 60.5110i	− 0.244984i
$248$	0	0
$249$	−190.788	−0.766216
$250$	0	0
$251$	− 334.140i	− 1.33123i	−0.746294	−	0.665617i	$-0.768168\pi$
$251$	− 334.140i	− 1.33123i	0.746294	−	0.665617i	$-0.231832\pi$
$252$	0	0
$253$	96.7265	0.382318
$254$	0	0
$255$	− 27.3982i	− 0.107444i
$256$	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
$259$	− 99.5375i	− 0.384315i
$260$	0	0
$261$	−133.485	−0.511436
$262$	0	0
$263$	− 249.415i	− 0.948347i	−0.880431	−	0.474174i	$-0.842747\pi$
$263$	− 249.415i	− 0.948347i	0.880431	−	0.474174i	$-0.157253\pi$
$264$	0	0
$265$	−1.53469	−0.00579127
$266$	0	0
$267$	126.772i	0.474801i
$268$	0	0
$269$	300.858	1.11843	0.559216	−	0.829022i	$-0.311102\pi$
$269$	300.858	1.11843	0.559216	+	0.829022i	$0.311102\pi$
$270$	0	0
$271$	− 386.865i	− 1.42755i	−0.700376	−	0.713774i	$-0.746985\pi$
$271$	− 386.865i	− 1.42755i	0.700376	−	0.713774i	$-0.253015\pi$
$272$	0	0
$273$	−67.5959	−0.247604
$274$	0	0
$275$	106.093i	0.385794i
$276$	0	0
$277$	−213.798	−0.771834	−0.385917	−	0.922534i	$-0.626115\pi$
$277$	−213.798	−0.771834	−0.385917	+	0.922534i	$0.626115\pi$
$278$	0	0
$279$	− 159.506i	− 0.571706i
$280$	0	0
$281$	110.767	0.394190	0.197095	−	0.980384i	$-0.436849\pi$
$281$	110.767	0.394190	0.197095	+	0.980384i	$0.436849\pi$
$282$	0	0
$283$	− 114.980i	− 0.406289i	−0.979149	−	0.203145i	$-0.934884\pi$
$283$	− 114.980i	− 0.406289i	0.979149	−	0.203145i	$-0.0651162\pi$
$284$	0	0
$285$	−6.82857	−0.0239599
$286$	0	0
$287$	− 106.337i	− 0.370513i
$288$	0	0
$289$	20.6163	0.0713368
$290$	0	0
$291$	− 182.198i	− 0.626109i
$292$	0	0
$293$	−108.677	−0.370910	−0.185455	−	0.982653i	$-0.559376\pi$
$293$	−108.677	−0.370910	−0.185455	+	0.982653i	$0.559376\pi$
$294$	0	0
$295$	31.1416i	0.105565i
$296$	0	0
$297$	−22.7878	−0.0767264
$298$	0	0
$299$	304.327i	1.01782i
$300$	0	0
$301$	−140.404	−0.466459
$302$	0	0
$303$	268.258i	0.885338i
$304$	0	0
$305$	21.9388	0.0719304
$306$	0	0
$307$	253.544i	0.825876i	0.910759	+	0.412938i	$0.135497\pi$
$307$	253.544i	0.825876i	−0.910759	+	0.412938i	$0.864503\pi$
$308$	0	0
$309$	65.6663	0.212512
$310$	0	0
$311$	− 456.491i	− 1.46782i	−0.679249	−	0.733908i	$-0.737694\pi$
$311$	− 456.491i	− 1.46782i	0.679249	−	0.733908i	$-0.262306\pi$
$312$	0	0
$313$	−371.535	−1.18701	−0.593506	−	0.804830i	$-0.702257\pi$
$313$	−371.535	−1.18701	−0.593506	+	0.804830i	$0.702257\pi$
$314$	0	0
$315$	7.62809i	0.0242162i
$316$	0	0
$317$	−53.4847	−0.168721	−0.0843607	−	0.996435i	$-0.526885\pi$
$317$	−53.4847	−0.168721	−0.0843607	+	0.996435i	$0.526885\pi$
$318$	0	0
$319$	− 195.133i	− 0.611701i
$320$	0	0
$321$	33.5755	0.104597
$322$	0	0
$323$	− 77.1670i	− 0.238907i
$324$	0	0
$325$	−333.798	−1.02707
$326$	0	0
$327$	− 228.945i	− 0.700139i
$328$	0	0
$329$	−108.767	−0.330600
$330$	0	0
$331$	− 165.320i	− 0.499457i	−0.968316	−	0.249728i	$-0.919659\pi$
$331$	− 165.320i	− 0.499457i	0.968316	−	0.249728i	$-0.0803413\pi$
$332$	0	0
$333$	105.576	0.317044
$334$	0	0
$335$	84.2814i	0.251586i
$336$	0	0
$337$	−210.767	−0.625422	−0.312711	−	0.949848i	$-0.601237\pi$
$337$	−210.767	−0.625422	−0.312711	+	0.949848i	$0.601237\pi$
$338$	0	0
$339$	− 61.6186i	− 0.181766i
$340$	0	0
$341$	233.171	0.683787
$342$	0	0
$343$	254.558i	0.742153i
$344$	0	0
$345$	34.3429	0.0995445
$346$	0	0
$347$	458.334i	1.32085i	0.750894	+	0.660423i	$0.229623\pi$
$347$	458.334i	1.32085i	−0.750894	+	0.660423i	$0.770377\pi$
$348$	0	0
$349$	−33.9388	−0.0972458	−0.0486229	−	0.998817i	$-0.515483\pi$
$349$	−33.9388	−0.0972458	−0.0486229	+	0.998817i	$0.515483\pi$
$350$	0	0
$351$	− 71.6963i	− 0.204263i
$352$	0	0
$353$	306.727	0.868914	0.434457	−	0.900693i	$-0.356940\pi$
$353$	306.727	0.868914	0.434457	+	0.900693i	$0.356940\pi$
$354$	0	0
$355$	− 111.365i	− 0.313704i
$356$	0	0
$357$	−86.2020	−0.241462
$358$	0	0
$359$	166.848i	0.464759i	0.972625	+	0.232379i	$0.0746511\pi$
$359$	166.848i	0.464759i	−0.972625	+	0.232379i	$0.925349\pi$
$360$	0	0
$361$	341.767	0.946724
$362$	0	0
$363$	176.266i	0.485582i
$364$	0	0
$365$	−8.98979	−0.0246296
$366$	0	0
$367$	15.7988i	0.0430485i	0.999768	+	0.0215242i	$0.00685190\pi$
$367$	15.7988i	0.0430485i	−0.999768	+	0.0215242i	$0.993148\pi$
$368$	0	0
$369$	112.788	0.305658
$370$	0	0
$371$	4.82853i	0.0130149i
$372$	0	0
$373$	82.0204	0.219894	0.109947	−	0.993937i	$-0.464932\pi$
$373$	82.0204	0.219894	0.109947	+	0.993937i	$0.464932\pi$
$374$	0	0
$375$	76.5955i	0.204255i
$376$	0	0
$377$	613.939	1.62848
$378$	0	0
$379$	− 524.444i	− 1.38376i	−0.722014	−	0.691878i	$-0.756783\pi$
$379$	− 524.444i	− 1.38376i	0.722014	−	0.691878i	$-0.243217\pi$
$380$	0	0
$381$	−221.444	−0.581218
$382$	0	0
$383$	498.946i	1.30273i	0.758764	+	0.651366i	$0.225804\pi$
$383$	498.946i	1.30273i	−0.758764	+	0.651366i	$0.774196\pi$
$384$	0	0
$385$	−11.1510	−0.0289637
$386$	0	0
$387$	− 148.921i	− 0.384809i
$388$	0	0
$389$	233.889	0.601256	0.300628	−	0.953741i	$-0.402804\pi$
$389$	233.889	0.601256	0.300628	+	0.953741i	$0.402804\pi$
$390$	0	0
$391$	388.095i	0.992570i
$392$	0	0
$393$	149.616	0.380703
$394$	0	0
$395$	− 117.966i	− 0.298647i
$396$	0	0
$397$	−348.363	−0.877489	−0.438745	−	0.898612i	$-0.644577\pi$
$397$	−348.363	−0.877489	−0.438745	+	0.898612i	$0.644577\pi$
$398$	0	0
$399$	21.4845i	0.0538459i
$400$	0	0
$401$	−759.049	−1.89289	−0.946445	−	0.322865i	$-0.895354\pi$
$401$	−759.049	−1.89289	−0.946445	+	0.322865i	$0.895354\pi$
$402$	0	0
$403$	733.619i	1.82039i
$404$	0	0
$405$	−8.09082	−0.0199773
$406$	0	0
$407$	154.334i	0.379199i
$408$	0	0
$409$	−94.0408	−0.229929	−0.114964	−	0.993370i	$-0.536675\pi$
$409$	−94.0408	−0.229929	−0.114964	+	0.993370i	$0.536675\pi$
$410$	0	0
$411$	83.1031i	0.202197i
$412$	0	0
$413$	97.9796	0.237239
$414$	0	0
$415$	− 99.0238i	− 0.238612i
$416$	0	0
$417$	−271.151	−0.650242
$418$	0	0
$419$	− 567.413i	− 1.35421i	−0.735888	−	0.677104i	$-0.763235\pi$
$419$	− 567.413i	− 1.35421i	0.735888	−	0.677104i	$-0.236765\pi$
$420$	0	0
$421$	786.120	1.86727	0.933635	−	0.358227i	$-0.116618\pi$
$421$	786.120	1.86727	0.933635	+	0.358227i	$0.116618\pi$
$422$	0	0
$423$	− 115.365i	− 0.272731i
$424$	0	0
$425$	−425.678	−1.00159
$426$	0	0
$427$	− 69.0252i	− 0.161651i
$428$	0	0
$429$	104.808	0.244308
$430$	0	0
$431$	− 402.780i	− 0.934523i	−0.884119	−	0.467262i	$-0.845241\pi$
$431$	− 402.780i	− 0.934523i	0.884119	−	0.467262i	$-0.154759\pi$
$432$	0	0
$433$	140.384	0.324212	0.162106	−	0.986773i	$-0.448171\pi$
$433$	140.384	0.324212	0.162106	+	0.986773i	$0.448171\pi$
$434$	0	0
$435$	− 69.2820i	− 0.159269i
$436$	0	0
$437$	96.7265	0.221342
$438$	0	0
$439$	341.039i	0.776854i	0.921479	+	0.388427i	$0.126981\pi$
$439$	341.039i	0.776854i	−0.921479	+	0.388427i	$0.873019\pi$
$440$	0	0
$441$	−123.000	−0.278912
$442$	0	0
$443$	− 153.120i	− 0.345644i	−0.984953	−	0.172822i	$-0.944711\pi$
$443$	− 153.120i	− 0.345644i	0.984953	−	0.172822i	$-0.0552886\pi$
$444$	0	0
$445$	−65.7980	−0.147861
$446$	0	0
$447$	448.076i	1.00241i
$448$	0	0
$449$	−54.3224	−0.120985	−0.0604927	−	0.998169i	$-0.519267\pi$
$449$	−54.3224	−0.120985	−0.0604927	+	0.998169i	$0.519267\pi$
$450$	0	0
$451$	164.877i	0.365581i
$452$	0	0
$453$	185.121	0.408657
$454$	0	0
$455$	− 35.0840i	− 0.0771078i
$456$	0	0
$457$	777.878	1.70214	0.851070	−	0.525053i	$-0.175955\pi$
$457$	777.878	1.70214	0.851070	+	0.525053i	$0.175955\pi$
$458$	0	0
$459$	− 91.4311i	− 0.199196i
$460$	0	0
$461$	−635.160	−1.37779	−0.688894	−	0.724862i	$-0.741903\pi$
$461$	−635.160	−1.37779	−0.688894	+	0.724862i	$0.741903\pi$
$462$	0	0
$463$	72.3096i	0.156176i	0.996946	+	0.0780881i	$0.0248815\pi$
$463$	72.3096i	0.156176i	−0.996946	+	0.0780881i	$0.975118\pi$
$464$	0	0
$465$	82.7878	0.178038
$466$	0	0
$467$	671.265i	1.43740i	0.695321	+	0.718699i	$0.255262\pi$
$467$	671.265i	1.43740i	−0.695321	+	0.718699i	$0.744738\pi$
$468$	0	0
$469$	265.171	0.565397
$470$	0	0
$471$	− 199.518i	− 0.423605i
$472$	0	0
$473$	217.698	0.460249
$474$	0	0
$475$	106.093i	0.223355i
$476$	0	0
$477$	−5.12143	−0.0107368
$478$	0	0
$479$	644.824i	1.34619i	0.739557	+	0.673094i	$0.235035\pi$
$479$	644.824i	1.34619i	−0.739557	+	0.673094i	$0.764965\pi$
$480$	0	0
$481$	−485.576	−1.00951
$482$	0	0
$483$	− 108.052i	− 0.223710i
$484$	0	0
$485$	94.5653	0.194980
$486$	0	0
$487$	527.715i	1.08360i	0.840506	+	0.541802i	$0.182258\pi$
$487$	527.715i	1.08360i	−0.840506	+	0.541802i	$0.817742\pi$
$488$	0	0
$489$	430.788	0.880957
$490$	0	0
$491$	408.250i	0.831467i	0.909486	+	0.415733i	$0.136475\pi$
$491$	408.250i	0.831467i	−0.909486	+	0.415733i	$0.863525\pi$
$492$	0	0
$493$	782.929	1.58809
$494$	0	0
$495$	− 11.8274i	− 0.0238938i
$496$	0	0
$497$	−350.384	−0.704997
$498$	0	0
$499$	− 365.281i	− 0.732027i	−0.930610	−	0.366013i	$-0.880722\pi$
$499$	− 365.281i	− 0.732027i	0.930610	−	0.366013i	$-0.119278\pi$
$500$	0	0
$501$	206.747	0.412669
$502$	0	0
$503$	738.133i	1.46746i	0.679441	+	0.733731i	$0.262223\pi$
$503$	738.133i	1.46746i	−0.679441	+	0.733731i	$0.737777\pi$
$504$	0	0
$505$	−139.233	−0.275708
$506$	0	0
$507$	37.0376i	0.0730525i
$508$	0	0
$509$	−96.2724	−0.189140	−0.0945702	−	0.995518i	$-0.530148\pi$
$509$	−96.2724	−0.189140	−0.0945702	+	0.995518i	$0.530148\pi$
$510$	0	0
$511$	28.2843i	0.0553508i
$512$	0	0
$513$	−22.7878	−0.0444206
$514$	0	0
$515$	34.0825i	0.0661797i
$516$	0	0
$517$	168.645	0.326199
$518$	0	0
$519$	460.533i	0.887347i
$520$	0	0
$521$	−199.212	−0.382365	−0.191183	−	0.981554i	$-0.561232\pi$
$521$	−199.212	−0.382365	−0.191183	+	0.981554i	$0.561232\pi$
$522$	0	0
$523$	− 13.6702i	− 0.0261381i	−0.999915	−	0.0130691i	$-0.995840\pi$
$523$	− 13.6702i	− 0.0261381i	0.999915	−	0.0130691i	$-0.00416013\pi$
$524$	0	0
$525$	118.515	0.225743
$526$	0	0
$527$	935.551i	1.77524i
$528$	0	0
$529$	42.5347	0.0804058
$530$	0	0
$531$	103.923i	0.195712i
$532$	0	0
$533$	−518.747	−0.973259
$534$	0	0
$535$	17.4266i	0.0325730i
$536$	0	0
$537$	−391.151	−0.728400
$538$	0	0
$539$	− 179.806i	− 0.333591i
$540$	0	0
$541$	−542.443	−1.00267	−0.501333	−	0.865254i	$-0.667157\pi$
$541$	−542.443	−1.00267	−0.501333	+	0.865254i	$0.667157\pi$
$542$	0	0
$543$	− 83.5237i	− 0.153819i
$544$	0	0
$545$	118.829	0.218034
$546$	0	0
$547$	− 239.058i	− 0.437035i	−0.975833	−	0.218518i	$-0.929878\pi$
$547$	− 239.058i	− 0.437035i	0.975833	−	0.218518i	$-0.0701221\pi$
$548$	0	0
$549$	73.2122	0.133356
$550$	0	0
$551$	− 195.133i	− 0.354143i
$552$	0	0
$553$	−371.151	−0.671159
$554$	0	0
$555$	54.7964i	0.0987323i
$556$	0	0
$557$	−521.707	−0.936638	−0.468319	−	0.883560i	$-0.655140\pi$
$557$	−521.707	−0.936638	−0.468319	+	0.883560i	$0.655140\pi$
$558$	0	0
$559$	684.935i	1.22529i
$560$	0	0
$561$	133.657	0.238248
$562$	0	0
$563$	− 1092.42i	− 1.94035i	−0.242413	−	0.970173i	$-0.577939\pi$
$563$	− 1092.42i	− 1.94035i	0.242413	−	0.970173i	$-0.422061\pi$
$564$	0	0
$565$	31.9817	0.0566047
$566$	0	0
$567$	25.4558i	0.0448957i
$568$	0	0
$569$	823.171	1.44670	0.723349	−	0.690482i	$-0.242602\pi$
$569$	823.171	1.44670	0.723349	+	0.690482i	$0.242602\pi$
$570$	0	0
$571$	− 268.800i	− 0.470753i	−0.971904	−	0.235377i	$-0.924368\pi$
$571$	− 268.800i	− 0.470753i	0.971904	−	0.235377i	$-0.0756324\pi$
$572$	0	0
$573$	307.596	0.536817
$574$	0	0
$575$	− 533.574i	− 0.927955i
$576$	0	0
$577$	−108.465	−0.187981	−0.0939907	−	0.995573i	$-0.529962\pi$
$577$	−108.465	−0.187981	−0.0939907	+	0.995573i	$0.529962\pi$
$578$	0	0
$579$	226.496i	0.391184i
$580$	0	0
$581$	−311.555	−0.536239
$582$	0	0
$583$	− 7.48669i	− 0.0128417i
$584$	0	0
$585$	37.2122	0.0636107
$586$	0	0
$587$	− 223.545i	− 0.380827i	−0.981704	−	0.190413i	$-0.939017\pi$
$587$	− 223.545i	− 0.380827i	0.981704	−	0.190413i	$-0.0609828\pi$
$588$	0	0
$589$	233.171	0.395877
$590$	0	0
$591$	258.600i	0.437564i
$592$	0	0
$593$	748.261	1.26182	0.630912	−	0.775855i	$-0.282681\pi$
$593$	748.261	1.26182	0.630912	+	0.775855i	$0.282681\pi$
$594$	0	0
$595$	− 44.7411i	− 0.0751951i
$596$	0	0
$597$	257.666	0.431602
$598$	0	0
$599$	− 32.2268i	− 0.0538009i	−0.999638	−	0.0269005i	$-0.991436\pi$
$599$	− 32.2268i	− 0.0538009i	0.999638	−	0.0269005i	$-0.00856372\pi$
$600$	0	0
$601$	161.918	0.269415	0.134707	−	0.990885i	$-0.456991\pi$
$601$	161.918	0.269415	0.134707	+	0.990885i	$0.456991\pi$
$602$	0	0
$603$	281.257i	0.466429i
$604$	0	0
$605$	−91.4868	−0.151218
$606$	0	0
$607$	573.542i	0.944879i	0.881363	+	0.472439i	$0.156626\pi$
$607$	573.542i	0.944879i	−0.881363	+	0.472439i	$0.843374\pi$
$608$	0	0
$609$	−217.980	−0.357930
$610$	0	0
$611$	530.601i	0.868415i
$612$	0	0
$613$	−985.857	−1.60825	−0.804125	−	0.594460i	$-0.797366\pi$
$613$	−985.857	−1.60825	−0.804125	+	0.594460i	$0.797366\pi$
$614$	0	0
$615$	58.5398i	0.0951866i
$616$	0	0
$617$	−761.151	−1.23363	−0.616816	−	0.787107i	$-0.711578\pi$
$617$	−761.151	−1.23363	−0.616816	+	0.787107i	$0.711578\pi$
$618$	0	0
$619$	538.929i	0.870645i	0.900275	+	0.435323i	$0.143366\pi$
$619$	538.929i	0.870645i	−0.900275	+	0.435323i	$0.856634\pi$
$620$	0	0
$621$	114.606	0.184551
$622$	0	0
$623$	207.018i	0.332292i
$624$	0	0
$625$	565.041	0.904065
$626$	0	0
$627$	− 33.3119i	− 0.0531291i
$628$	0	0
$629$	−619.233	−0.984472
$630$	0	0
$631$	− 278.756i	− 0.441768i	−0.975300	−	0.220884i	$-0.929106\pi$
$631$	− 278.756i	− 0.441768i	0.975300	−	0.220884i	$-0.0708943\pi$
$632$	0	0
$633$	−661.535	−1.04508
$634$	0	0
$635$	− 114.935i	− 0.181000i
$636$	0	0
$637$	565.716	0.888095
$638$	0	0
$639$	− 371.638i	− 0.581593i
$640$	0	0
$641$	318.363	0.496667	0.248333	−	0.968675i	$-0.420117\pi$
$641$	318.363	0.496667	0.248333	+	0.968675i	$0.420117\pi$
$642$	0	0
$643$	647.752i	1.00739i	0.863882	+	0.503695i	$0.168026\pi$
$643$	647.752i	1.00739i	−0.863882	+	0.503695i	$0.831974\pi$
$644$	0	0
$645$	77.2939	0.119835
$646$	0	0
$647$	1084.29i	1.67587i	0.545772	+	0.837934i	$0.316236\pi$
$647$	1084.29i	1.67587i	−0.545772	+	0.837934i	$0.683764\pi$
$648$	0	0
$649$	−151.918	−0.234081
$650$	0	0
$651$	− 260.472i	− 0.400111i
$652$	0	0
$653$	23.3826	0.0358080	0.0179040	−	0.999840i	$-0.494301\pi$
$653$	23.3826	0.0358080	0.0179040	+	0.999840i	$0.494301\pi$
$654$	0	0
$655$	77.6548i	0.118557i
$656$	0	0
$657$	−30.0000	−0.0456621
$658$	0	0
$659$	496.846i	0.753940i	0.926225	+	0.376970i	$0.123034\pi$
$659$	496.846i	0.753940i	−0.926225	+	0.376970i	$0.876966\pi$
$660$	0	0
$661$	−113.576	−0.171824	−0.0859119	−	0.996303i	$-0.527380\pi$
$661$	−113.576	−0.171824	−0.0859119	+	0.996303i	$0.527380\pi$
$662$	0	0
$663$	420.521i	0.634270i
$664$	0	0
$665$	−11.1510	−0.0167684
$666$	0	0
$667$	981.378i	1.47133i
$668$	0	0
$669$	464.413	0.694190
$670$	0	0
$671$	107.024i	0.159500i
$672$	0	0
$673$	554.604	0.824077	0.412039	−	0.911166i	$-0.364817\pi$
$673$	554.604	0.824077	0.412039	+	0.911166i	$0.364817\pi$
$674$	0	0
$675$	125.704i	0.186229i
$676$	0	0
$677$	−902.474	−1.33305	−0.666525	−	0.745483i	$-0.732219\pi$
$677$	−902.474	−1.33305	−0.666525	+	0.745483i	$0.732219\pi$
$678$	0	0
$679$	− 297.527i	− 0.438185i
$680$	0	0
$681$	611.555	0.898025
$682$	0	0
$683$	− 924.738i	− 1.35394i	−0.736012	−	0.676968i	$-0.763294\pi$
$683$	− 924.738i	− 1.35394i	0.736012	−	0.676968i	$-0.236706\pi$
$684$	0	0
$685$	−43.1327	−0.0629674
$686$	0	0
$687$	− 368.139i	− 0.535864i
$688$	0	0
$689$	23.5551	0.0341874
$690$	0	0
$691$	− 1155.87i	− 1.67275i	−0.548161	−	0.836373i	$-0.684672\pi$
$691$	− 1155.87i	− 1.67275i	0.548161	−	0.836373i	$-0.315328\pi$
$692$	0	0
$693$	−37.2122	−0.0536973
$694$	0	0
$695$	− 140.734i	− 0.202496i
$696$	0	0
$697$	−661.535	−0.949117
$698$	0	0
$699$	− 615.875i	− 0.881080i
$700$	0	0
$701$	−591.383	−0.843627	−0.421814	−	0.906683i	$-0.638606\pi$
$701$	−591.383	−0.843627	−0.421814	+	0.906683i	$0.638606\pi$
$702$	0	0
$703$	154.334i	0.219536i
$704$	0	0
$705$	59.8775	0.0849327
$706$	0	0
$707$	438.063i	0.619608i
$708$	0	0
$709$	1216.87	1.71631	0.858157	−	0.513387i	$-0.171609\pi$
$709$	1216.87	1.71631	0.858157	+	0.513387i	$0.171609\pi$
$710$	0	0
$711$	− 393.665i	− 0.553678i
$712$	0	0
$713$	−1172.69	−1.64472
$714$	0	0
$715$	54.3982i	0.0760814i
$716$	0	0
$717$	74.4245	0.103800
$718$	0	0
$719$	− 1202.62i	− 1.67263i	−0.548246	−	0.836317i	$-0.684704\pi$
$719$	− 1202.62i	− 1.67263i	0.548246	−	0.836317i	$-0.315296\pi$
$720$	0	0
$721$	107.233	0.148728
$722$	0	0
$723$	216.839i	0.299915i
$724$	0	0
$725$	−1076.41	−1.48471
$726$	0	0
$727$	194.019i	0.266876i	0.991057	+	0.133438i	$0.0426016\pi$
$727$	194.019i	0.266876i	−0.991057	+	0.133438i	$0.957398\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	873.467i	1.19489i
$732$	0	0
$733$	1070.52	1.46047	0.730235	−	0.683196i	$-0.239411\pi$
$733$	1070.52	1.46047	0.730235	+	0.683196i	$0.239411\pi$
$734$	0	0
$735$	− 63.8402i	− 0.0868574i
$736$	0	0
$737$	−411.151	−0.557871
$738$	0	0
$739$	− 376.710i	− 0.509757i	−0.966973	−	0.254879i	$-0.917965\pi$
$739$	− 376.710i	− 0.509757i	0.966973	−	0.254879i	$-0.0820355\pi$
$740$	0	0
$741$	104.808	0.141442
$742$	0	0
$743$	− 608.141i	− 0.818494i	−0.912424	−	0.409247i	$-0.865792\pi$
$743$	− 608.141i	− 0.818494i	0.912424	−	0.409247i	$-0.134208\pi$
$744$	0	0
$745$	−232.563	−0.312165
$746$	0	0
$747$	− 330.454i	− 0.442375i
$748$	0	0
$749$	54.8286	0.0732024
$750$	0	0
$751$	− 382.236i	− 0.508969i	−0.967077	−	0.254485i	$-0.918094\pi$
$751$	− 382.236i	− 0.508969i	0.967077	−	0.254485i	$-0.0819058\pi$
$752$	0	0
$753$	578.747	0.768588
$754$	0	0
$755$	96.0828i	0.127262i
$756$	0	0
$757$	−206.969	−0.273407	−0.136704	−	0.990612i	$-0.543651\pi$
$757$	−206.969	−0.273407	−0.136704	+	0.990612i	$0.543651\pi$
$758$	0	0
$759$	167.535i	0.220732i
$760$	0	0
$761$	959.980	1.26147	0.630736	−	0.775998i	$-0.282753\pi$
$761$	959.980	1.26147	0.630736	+	0.775998i	$0.282753\pi$
$762$	0	0
$763$	− 373.866i	− 0.489995i
$764$	0	0
$765$	47.4551	0.0620328
$766$	0	0
$767$	− 477.975i	− 0.623175i
$768$	0	0
$769$	−513.918	−0.668294	−0.334147	−	0.942521i	$-0.608448\pi$
$769$	−513.918	−0.668294	−0.334147	+	0.942521i	$0.608448\pi$
$770$	0	0
$771$	502.295i	0.651485i
$772$	0	0
$773$	−1389.24	−1.79721	−0.898603	−	0.438763i	$-0.855417\pi$
$773$	−1389.24	−1.79721	−0.898603	+	0.438763i	$0.855417\pi$
$774$	0	0
$775$	− 1286.25i	− 1.65967i
$776$	0	0
$777$	172.404	0.221884
$778$	0	0
$779$	164.877i	0.211652i
$780$	0	0
$781$	543.273	0.695613
$782$	0	0
$783$	− 231.202i	− 0.295277i
$784$	0	0
$785$	103.555	0.131917
$786$	0	0
$787$	− 526.588i	− 0.669108i	−0.942377	−	0.334554i	$-0.891414\pi$
$787$	− 526.588i	− 0.669108i	0.942377	−	0.334554i	$-0.108586\pi$
$788$	0	0
$789$	432.000	0.547529
$790$	0	0
$791$	− 100.623i	− 0.127210i
$792$	0	0
$793$	−336.727	−0.424624
$794$	0	0
$795$	− 2.65816i	− 0.00334359i
$796$	0	0
$797$	−1141.22	−1.43190	−0.715948	−	0.698153i	$-0.754005\pi$
$797$	−1141.22	−1.43190	−0.715948	+	0.698153i	$0.754005\pi$
$798$	0	0
$799$	676.652i	0.846874i
$800$	0	0
$801$	−219.576	−0.274127
$802$	0	0
$803$	− 43.8551i	− 0.0546140i
$804$	0	0
$805$	56.0816	0.0696666
$806$	0	0
$807$	521.102i	0.645727i
$808$	0	0
$809$	−944.020	−1.16690	−0.583449	−	0.812150i	$-0.698297\pi$
$809$	−944.020	−1.16690	−0.583449	+	0.812150i	$0.698297\pi$
$810$	0	0
$811$	58.7837i	0.0724830i	0.999343	+	0.0362415i	$0.0115385\pi$
$811$	58.7837i	0.0724830i	−0.999343	+	0.0362415i	$0.988461\pi$
$812$	0	0
$813$	670.070	0.824195
$814$	0	0
$815$	223.590i	0.274344i
$816$	0	0
$817$	217.698	0.266460
$818$	0	0
$819$	− 117.080i	− 0.142954i
$820$	0	0
$821$	1246.67	1.51848	0.759241	−	0.650809i	$-0.225570\pi$
$821$	1246.67	1.51848	0.759241	+	0.650809i	$0.225570\pi$
$822$	0	0
$823$	− 762.960i	− 0.927047i	−0.886085	−	0.463523i	$-0.846585\pi$
$823$	− 762.960i	− 0.927047i	0.886085	−	0.463523i	$-0.153415\pi$
$824$	0	0
$825$	−183.759	−0.222738
$826$	0	0
$827$	− 1251.44i	− 1.51322i	−0.653864	−	0.756612i	$-0.726853\pi$
$827$	− 1251.44i	− 1.51322i	0.653864	−	0.756612i	$-0.273147\pi$
$828$	0	0
$829$	−13.8796	−0.0167426	−0.00837129	−	0.999965i	$-0.502665\pi$
$829$	−13.8796	−0.0167426	−0.00837129	+	0.999965i	$0.502665\pi$
$830$	0	0
$831$	− 370.309i	− 0.445618i
$832$	0	0
$833$	721.433	0.866066
$834$	0	0
$835$	107.307i	0.128511i
$836$	0	0
$837$	276.272	0.330075
$838$	0	0
$839$	153.249i	0.182656i	0.995821	+	0.0913282i	$0.0291112\pi$
$839$	153.249i	0.182656i	−0.995821	+	0.0913282i	$0.970889\pi$
$840$	0	0
$841$	1138.80	1.35410
$842$	0	0
$843$	191.855i	0.227586i
$844$	0	0
$845$	−19.2235	−0.0227497
$846$	0	0
$847$	287.842i	0.339836i
$848$	0	0
$849$	199.151	0.234571
$850$	0	0
$851$	− 776.190i	− 0.912091i
$852$	0	0
$853$	869.616	1.01948	0.509740	−	0.860329i	$-0.329742\pi$
$853$	869.616	1.01948	0.509740	+	0.860329i	$0.329742\pi$
$854$	0	0
$855$	− 11.8274i	− 0.0138333i
$856$	0	0
$857$	883.939	1.03143	0.515717	−	0.856759i	$-0.327526\pi$
$857$	883.939	1.03143	0.515717	+	0.856759i	$0.327526\pi$
$858$	0	0
$859$	1592.65i	1.85407i	0.374976	+	0.927035i	$0.377651\pi$
$859$	1592.65i	1.85407i	−0.374976	+	0.927035i	$0.622349\pi$
$860$	0	0
$861$	184.182	0.213916
$862$	0	0
$863$	− 1209.31i	− 1.40128i	−0.713513	−	0.700642i	$-0.752897\pi$
$863$	− 1209.31i	− 1.40128i	0.713513	−	0.700642i	$-0.247103\pi$
$864$	0	0
$865$	−239.029	−0.276334
$866$	0	0
$867$	35.7085i	0.0411863i
$868$	0	0
$869$	575.473	0.662225
$870$	0	0
$871$	− 1293.59i	− 1.48518i
$872$	0	0
$873$	315.576	0.361484
$874$	0	0
$875$	125.080i	0.142949i
$876$	0	0
$877$	−922.584	−1.05198	−0.525988	−	0.850492i	$-0.676304\pi$
$877$	−922.584	−1.05198	−0.525988	+	0.850492i	$0.676304\pi$
$878$	0	0
$879$	− 188.233i	− 0.214145i
$880$	0	0
$881$	−1564.22	−1.77551	−0.887753	−	0.460321i	$-0.847734\pi$
$881$	−1564.22	−1.77551	−0.887753	+	0.460321i	$0.847734\pi$
$882$	0	0
$883$	− 836.284i	− 0.947094i	−0.880769	−	0.473547i	$-0.842973\pi$
$883$	− 836.284i	− 0.947094i	0.880769	−	0.473547i	$-0.157027\pi$
$884$	0	0
$885$	−53.9388	−0.0609478
$886$	0	0
$887$	− 596.712i	− 0.672730i	−0.941732	−	0.336365i	$-0.890802\pi$
$887$	− 596.712i	− 0.672730i	0.941732	−	0.336365i	$-0.109198\pi$
$888$	0	0
$889$	−361.616	−0.406768
$890$	0	0
$891$	− 39.4695i	− 0.0442980i
$892$	0	0
$893$	168.645	0.188852
$894$	0	0
$895$	− 203.018i	− 0.226835i
$896$	0	0
$897$	−527.110	−0.587637
$898$	0	0
$899$	2365.73i	2.63152i
$900$	0	0
$901$	30.0388	0.0333394
$902$	0	0
$903$	− 243.187i	− 0.269310i
$904$	0	0
$905$	43.3510	0.0479017
$906$	0	0
$907$	− 399.595i	− 0.440567i	−0.975436	−	0.220284i	$-0.929302\pi$
$907$	− 399.595i	− 0.440567i	0.975436	−	0.220284i	$-0.0706983\pi$
$908$	0	0
$909$	−464.636	−0.511150
$910$	0	0
$911$	− 612.995i	− 0.672882i	−0.941705	−	0.336441i	$-0.890777\pi$
$911$	− 612.995i	− 0.672882i	0.941705	−	0.336441i	$-0.109223\pi$
$912$	0	0
$913$	483.069	0.529101
$914$	0	0
$915$	37.9991i	0.0415290i
$916$	0	0
$917$	244.322	0.266437
$918$	0	0
$919$	− 1095.74i	− 1.19232i	−0.802865	−	0.596161i	$-0.796692\pi$
$919$	− 1095.74i	− 1.19232i	0.802865	−	0.596161i	$-0.203308\pi$
$920$	0	0
$921$	−439.151	−0.476820
$922$	0	0
$923$	1709.28i	1.85188i
$924$	0	0
$925$	851.355	0.920384
$926$	0	0
$927$	113.737i	0.122694i
$928$	0	0
$929$	180.102	0.193867	0.0969333	−	0.995291i	$-0.469097\pi$
$929$	180.102	0.193867	0.0969333	+	0.995291i	$0.469097\pi$
$930$	0	0
$931$	− 179.806i	− 0.193132i
$932$	0	0
$933$	790.665	0.847444
$934$	0	0
$935$	69.3715i	0.0741942i
$936$	0	0
$937$	1393.84	1.48755	0.743776	−	0.668429i	$-0.233033\pi$
$937$	1393.84	1.48755	0.743776	+	0.668429i	$0.233033\pi$
$938$	0	0
$939$	− 643.517i	− 0.685322i
$940$	0	0
$941$	−659.968	−0.701348	−0.350674	−	0.936498i	$-0.614047\pi$
$941$	−659.968	−0.701348	−0.350674	+	0.936498i	$0.614047\pi$
$942$	0	0
$943$	− 829.214i	− 0.879336i
$944$	0	0
$945$	−13.2122	−0.0139812
$946$	0	0
$947$	− 1584.65i	− 1.67333i	−0.547714	−	0.836666i	$-0.684502\pi$
$947$	− 1584.65i	− 1.67333i	0.547714	−	0.836666i	$-0.315498\pi$
$948$	0	0
$949$	137.980	0.145395
$950$	0	0
$951$	− 92.6382i	− 0.0974114i
$952$	0	0
$953$	1212.02	1.27179	0.635897	−	0.771774i	$-0.280630\pi$
$953$	1212.02	1.27179	0.635897	+	0.771774i	$0.280630\pi$
$954$	0	0
$955$	159.650i	0.167173i
$956$	0	0
$957$	337.980	0.353166
$958$	0	0
$959$	135.707i	0.141509i
$960$	0	0
$961$	−1865.91	−1.94163
$962$	0	0
$963$	58.1545i	0.0603889i
$964$	0	0
$965$	−117.557	−0.121821
$966$	0	0
$967$	529.943i	0.548028i	0.961726	+	0.274014i	$0.0883515\pi$
$967$	529.943i	0.548028i	−0.961726	+	0.274014i	$0.911648\pi$
$968$	0	0
$969$	133.657	0.137933
$970$	0	0
$971$	− 1364.60i	− 1.40535i	−0.711509	−	0.702677i	$-0.751988\pi$
$971$	− 1364.60i	− 1.40535i	0.711509	−	0.702677i	$-0.248012\pi$
$972$	0	0
$973$	−442.788	−0.455075
$974$	0	0
$975$	− 578.155i	− 0.592980i
$976$	0	0
$977$	141.637	0.144971	0.0724855	−	0.997369i	$-0.476907\pi$
$977$	141.637	0.144971	0.0724855	+	0.997369i	$0.476907\pi$
$978$	0	0
$979$	− 320.983i	− 0.327868i
$980$	0	0
$981$	396.545	0.404225
$982$	0	0
$983$	− 354.667i	− 0.360801i	−0.983593	−	0.180401i	$-0.942261\pi$
$983$	− 354.667i	− 0.360801i	0.983593	−	0.180401i	$-0.0577394\pi$
$984$	0	0
$985$	−134.220	−0.136264
$986$	0	0
$987$	− 188.391i	− 0.190872i
$988$	0	0
$989$	−1094.87	−1.10704
$990$	0	0
$991$	− 1596.00i	− 1.61050i	−0.592939	−	0.805248i	$-0.702032\pi$
$991$	− 1596.00i	− 1.61050i	0.592939	−	0.805248i	$-0.297968\pi$
$992$	0	0
$993$	286.343	0.288361
$994$	0	0
$995$	133.736i	0.134408i
$996$	0	0
$997$	1122.02	1.12540	0.562698	−	0.826662i	$-0.309763\pi$
$997$	1122.02	1.12540	0.562698	+	0.826662i	$0.309763\pi$
$998$	0	0
$999$	182.862i	0.183045i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.g.g.511.3		4
3.2	odd	2		2304.3.g.o.1279.4		4
4.3	odd	2	inner	768.3.g.g.511.1		4
8.3	odd	2		768.3.g.c.511.4		4
8.5	even	2		768.3.g.c.511.2		4
12.11	even	2		2304.3.g.o.1279.3		4
16.3	odd	4		384.3.b.c.319.7	yes	8
16.5	even	4		384.3.b.c.319.6	yes	8
16.11	odd	4		384.3.b.c.319.2	✓	8
16.13	even	4		384.3.b.c.319.3	yes	8
24.5	odd	2		2304.3.g.x.1279.2		4
24.11	even	2		2304.3.g.x.1279.1		4
48.5	odd	4		1152.3.b.j.703.5		8
48.11	even	4		1152.3.b.j.703.6		8
48.29	odd	4		1152.3.b.j.703.3		8
48.35	even	4		1152.3.b.j.703.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.b.c.319.2	✓	8	16.11	odd	4
384.3.b.c.319.3	yes	8	16.13	even	4
384.3.b.c.319.6	yes	8	16.5	even	4
384.3.b.c.319.7	yes	8	16.3	odd	4
768.3.g.c.511.2		4	8.5	even	2
768.3.g.c.511.4		4	8.3	odd	2
768.3.g.g.511.1		4	4.3	odd	2	inner
768.3.g.g.511.3		4	1.1	even	1	trivial
1152.3.b.j.703.3		8	48.29	odd	4
1152.3.b.j.703.4		8	48.35	even	4
1152.3.b.j.703.5		8	48.5	odd	4
1152.3.b.j.703.6		8	48.11	even	4
2304.3.g.o.1279.3		4	12.11	even	2
2304.3.g.o.1279.4		4	3.2	odd	2
2304.3.g.x.1279.1		4	24.11	even	2
2304.3.g.x.1279.2		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)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -0.898979 q^{5} +2.82843i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} -0.898979 q^{5} +2.82843i q^{7} -3.00000 q^{9} -4.38551i q^{11} +13.7980 q^{13} -1.55708i q^{15} +17.5959 q^{17} -4.38551i q^{19} -4.89898 q^{21} +22.0560i q^{23} -24.1918 q^{25} -5.19615i q^{27} +44.4949 q^{29} +53.1687i q^{31} +7.59592 q^{33} -2.54270i q^{35} -35.1918 q^{37} +23.8988i q^{39} -37.5959 q^{41} +49.6403i q^{43} +2.69694 q^{45} +38.4551i q^{47} +41.0000 q^{49} +30.4770i q^{51} +1.70714 q^{53} +3.94248i q^{55} +7.59592 q^{57} -34.6410i q^{59} -24.4041 q^{61} -8.48528i q^{63} -12.4041 q^{65} -93.7523i q^{67} -38.2020 q^{69} +123.879i q^{71} +10.0000 q^{73} -41.9015i q^{75} +12.4041 q^{77} +131.222i q^{79} +9.00000 q^{81} +110.151i q^{83} -15.8184 q^{85} +77.0674i q^{87} +73.1918 q^{89} +39.0265i q^{91} -92.0908 q^{93} +3.94248i q^{95} -105.192 q^{97} +13.1565i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{5} - 12 q^{9}+O(q^{10}) \) 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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