LMFDB - Embedded newform 384.3.b.c.319.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(319,384)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	384.b (of order $2$, degree $1$, 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:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.7
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.3.b.c.319.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +0.898979i q^{5} +2.82843i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} +0.898979i q^{5} +2.82843i q^{7} +3.00000 q^{9} +4.38551 q^{11} +13.7980i q^{13} +1.55708i q^{15} +17.5959 q^{17} -4.38551 q^{19} +4.89898i q^{21} +22.0560i q^{23} +24.1918 q^{25} +5.19615 q^{27} +44.4949i q^{29} -53.1687i q^{31} +7.59592 q^{33} -2.54270 q^{35} +35.1918i q^{37} +23.8988i q^{39} +37.5959 q^{41} -49.6403 q^{43} +2.69694i q^{45} -38.4551i q^{47} +41.0000 q^{49} +30.4770 q^{51} -1.70714i q^{53} +3.94248i q^{55} -7.59592 q^{57} +34.6410 q^{59} -24.4041i q^{61} +8.48528i q^{63} -12.4041 q^{65} -93.7523 q^{67} +38.2020i q^{69} +123.879i q^{71} -10.0000 q^{73} +41.9015 q^{75} +12.4041i q^{77} -131.222i q^{79} +9.00000 q^{81} +110.151 q^{83} +15.8184i q^{85} +77.0674i q^{87} -73.1918 q^{89} -39.0265 q^{91} -92.0908i q^{93} -3.94248i q^{95} -105.192 q^{97} +13.1565 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{9}+O(q^{10}) $ 8 * q + 24 * q^9	$ 8 q + 24 q^{9} - 16 q^{17} - 120 q^{25} - 96 q^{33} + 144 q^{41} + 328 q^{49} + 96 q^{57} - 256 q^{65} - 80 q^{73} + 72 q^{81} - 272 q^{89} - 528 q^{97}+O(q^{100}) $ 8 * q + 24 * q^9 - 16 * q^17 - 120 * q^25 - 96 * q^33 + 144 * q^41 + 328 * q^49 + 96 * q^57 - 256 * q^65 - 80 * q^73 + 72 * q^81 - 272 * q^89 - 528 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	1.73205	0.577350
$3$	1.73205	0.577350
$4$	0	0
$5$	0.898979i	0.179796i	0.995951	+	0.0898979i	$0.0286541\pi$
$5$	0.898979i	0.179796i	−0.995951	+	0.0898979i	$0.971346\pi$
$6$	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
$11$	4.38551	0.398682	0.199341	−	0.979930i	$-0.436120\pi$
$11$	4.38551	0.398682	0.199341	+	0.979930i	$0.436120\pi$
$12$	0	0
$13$	13.7980i	1.06138i	0.847566	+	0.530691i	$0.178067\pi$
$13$	13.7980i	1.06138i	−0.847566	+	0.530691i	$0.821933\pi$
$14$	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
$19$	−4.38551	−0.230816	−0.115408	−	0.993318i	$-0.536818\pi$
$19$	−4.38551	−0.230816	−0.115408	+	0.993318i	$0.536818\pi$
$20$	0	0
$21$	4.89898i	0.233285i
$22$	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
$27$	5.19615	0.192450
$28$	0	0
$29$	44.4949i	1.53431i	0.641464	+	0.767153i	$0.278328\pi$
$29$	44.4949i	1.53431i	−0.641464	+	0.767153i	$0.721672\pi$
$30$	0	0
$31$	− 53.1687i	− 1.71512i	−0.514386	−	0.857559i	$-0.671980\pi$
$31$	− 53.1687i	− 1.71512i	0.514386	−	0.857559i	$-0.328020\pi$
$32$	0	0
$33$	7.59592	0.230179
$34$	0	0
$35$	−2.54270	−0.0726485
$36$	0	0
$37$	35.1918i	0.951131i	0.879680	+	0.475565i	$0.157756\pi$
$37$	35.1918i	0.951131i	−0.879680	+	0.475565i	$0.842244\pi$
$38$	0	0
$39$	23.8988i	0.612789i
$40$	0	0
$41$	37.5959	0.916974	0.458487	−	0.888701i	$-0.348392\pi$
$41$	37.5959	0.916974	0.458487	+	0.888701i	$0.348392\pi$
$42$	0	0
$43$	−49.6403	−1.15443	−0.577213	−	0.816593i	$-0.695860\pi$
$43$	−49.6403	−1.15443	−0.577213	+	0.816593i	$0.695860\pi$
$44$	0	0
$45$	2.69694i	0.0599320i
$46$	0	0
$47$	− 38.4551i	− 0.818193i	−0.912491	−	0.409096i	$-0.865844\pi$
$47$	− 38.4551i	− 0.818193i	0.912491	−	0.409096i	$-0.134156\pi$
$48$	0	0
$49$	41.0000	0.836735
$50$	0	0
$51$	30.4770	0.597589
$52$	0	0
$53$	− 1.70714i	− 0.0322103i	−0.999870	−	0.0161051i	$-0.994873\pi$
$53$	− 1.70714i	− 0.0322103i	0.999870	−	0.0161051i	$-0.00512664\pi$
$54$	0	0
$55$	3.94248i	0.0716814i
$56$	0	0
$57$	−7.59592	−0.133262
$58$	0	0
$59$	34.6410	0.587136	0.293568	−	0.955938i	$-0.405157\pi$
$59$	34.6410	0.587136	0.293568	+	0.955938i	$0.405157\pi$
$60$	0	0
$61$	− 24.4041i	− 0.400067i	−0.979789	−	0.200033i	$-0.935895\pi$
$61$	− 24.4041i	− 0.400067i	0.979789	−	0.200033i	$-0.0641051\pi$
$62$	0	0
$63$	8.48528i	0.134687i
$64$	0	0
$65$	−12.4041	−0.190832
$66$	0	0
$67$	−93.7523	−1.39929	−0.699644	−	0.714492i	$-0.746658\pi$
$67$	−93.7523	−1.39929	−0.699644	+	0.714492i	$0.746658\pi$
$68$	0	0
$69$	38.2020i	0.553653i
$70$	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	$-0.521819\pi$
$73$	−10.0000	−0.136986	−0.0684932	+	0.997652i	$0.521819\pi$
$74$	0	0
$75$	41.9015	0.558687
$76$	0	0
$77$	12.4041i	0.161092i
$78$	0	0
$79$	− 131.222i	− 1.66103i	−0.556993	−	0.830517i	$-0.688045\pi$
$79$	− 131.222i	− 1.66103i	0.556993	−	0.830517i	$-0.311955\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	110.151	1.32712	0.663562	−	0.748121i	$-0.269044\pi$
$83$	110.151	1.32712	0.663562	+	0.748121i	$0.269044\pi$
$84$	0	0
$85$	15.8184i	0.186098i
$86$	0	0
$87$	77.0674i	0.885832i
$88$	0	0
$89$	−73.1918	−0.822380	−0.411190	−	0.911550i	$-0.634887\pi$
$89$	−73.1918	−0.822380	−0.411190	+	0.911550i	$0.634887\pi$
$90$	0	0
$91$	−39.0265	−0.428863
$92$	0	0
$93$	− 92.0908i	− 0.990224i
$94$	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
$99$	13.1565	0.132894
$100$	0	0
$101$	− 154.879i	− 1.53345i	−0.641975	−	0.766726i	$-0.721885\pi$
$101$	− 154.879i	− 1.53345i	0.641975	−	0.766726i	$-0.278115\pi$
$102$	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
$107$	19.3848	0.181167	0.0905833	−	0.995889i	$-0.471127\pi$
$107$	19.3848	0.181167	0.0905833	+	0.995889i	$0.471127\pi$
$108$	0	0
$109$	− 132.182i	− 1.21268i	−0.795207	−	0.606338i	$-0.792638\pi$
$109$	− 132.182i	− 1.21268i	0.795207	−	0.606338i	$-0.207362\pi$
$110$	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
$115$	−19.8279	−0.172416
$116$	0	0
$117$	41.3939i	0.353794i
$118$	0	0
$119$	49.7688i	0.418225i
$120$	0	0
$121$	−101.767	−0.841052
$122$	0	0
$123$	65.1180	0.529415
$124$	0	0
$125$	44.2225i	0.353780i
$126$	0	0
$127$	− 127.851i	− 1.00670i	−0.864083	−	0.503349i	$-0.832101\pi$
$127$	− 127.851i	− 1.00670i	0.864083	−	0.503349i	$-0.167899\pi$
$128$	0	0
$129$	−85.9796	−0.666508
$130$	0	0
$131$	−86.3810	−0.659397	−0.329699	−	0.944086i	$-0.606947\pi$
$131$	−86.3810	−0.659397	−0.329699	+	0.944086i	$0.606947\pi$
$132$	0	0
$133$	− 12.4041i	− 0.0932638i
$134$	0	0
$135$	4.67123i	0.0346017i
$136$	0	0
$137$	−47.9796	−0.350216	−0.175108	−	0.984549i	$-0.556027\pi$
$137$	−47.9796	−0.350216	−0.175108	+	0.984549i	$0.556027\pi$
$138$	0	0
$139$	−156.549	−1.12625	−0.563126	−	0.826371i	$-0.690402\pi$
$139$	−156.549	−1.12625	−0.563126	+	0.826371i	$0.690402\pi$
$140$	0	0
$141$	− 66.6061i	− 0.472384i
$142$	0	0
$143$	60.5110i	0.423154i
$144$	0	0
$145$	−40.0000	−0.275862
$146$	0	0
$147$	71.0141	0.483089
$148$	0	0
$149$	− 258.697i	− 1.73622i	−0.496371	−	0.868111i	$-0.665334\pi$
$149$	− 258.697i	− 1.73622i	0.496371	−	0.868111i	$-0.334666\pi$
$150$	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
$155$	47.7975	0.308371
$156$	0	0
$157$	− 115.192i	− 0.733706i	−0.930279	−	0.366853i	$-0.880435\pi$
$157$	− 115.192i	− 0.733706i	0.930279	−	0.366853i	$-0.119565\pi$
$158$	0	0
$159$	− 2.95686i	− 0.0185966i
$160$	0	0
$161$	−62.3837	−0.387476
$162$	0	0
$163$	−248.715	−1.52586	−0.762931	−	0.646480i	$-0.776240\pi$
$163$	−248.715	−1.52586	−0.762931	+	0.646480i	$0.776240\pi$
$164$	0	0
$165$	6.82857i	0.0413853i
$166$	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
$171$	−13.1565	−0.0769387
$172$	0	0
$173$	265.889i	1.53693i	0.639892	+	0.768465i	$0.278979\pi$
$173$	265.889i	1.53693i	−0.639892	+	0.768465i	$0.721021\pi$
$174$	0	0
$175$	68.4248i	0.390999i
$176$	0	0
$177$	60.0000	0.338983
$178$	0	0
$179$	225.831	1.26163	0.630813	−	0.775935i	$-0.282721\pi$
$179$	225.831	1.26163	0.630813	+	0.775935i	$0.282721\pi$
$180$	0	0
$181$	48.2225i	0.266422i	0.991088	+	0.133211i	$0.0425289\pi$
$181$	48.2225i	0.266422i	−0.991088	+	0.133211i	$0.957471\pi$
$182$	0	0
$183$	− 42.2691i	− 0.230979i
$184$	0	0
$185$	−31.6367	−0.171009
$186$	0	0
$187$	77.1670	0.412658
$188$	0	0
$189$	14.6969i	0.0777616i
$190$	0	0
$191$	177.591i	0.929794i	0.885365	+	0.464897i	$0.153909\pi$
$191$	177.591i	0.929794i	−0.885365	+	0.464897i	$0.846091\pi$
$192$	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
$195$	−21.4845	−0.110177
$196$	0	0
$197$	− 149.303i	− 0.757884i	−0.925421	−	0.378942i	$-0.876288\pi$
$197$	− 149.303i	− 0.757884i	0.925421	−	0.378942i	$-0.123712\pi$
$198$	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
$203$	−125.851	−0.619954
$204$	0	0
$205$	33.7980i	0.164868i
$206$	0	0
$207$	66.1679i	0.319652i
$208$	0	0
$209$	−19.2327	−0.0920223
$210$	0	0
$211$	381.937	1.81013	0.905065	−	0.425274i	$-0.139822\pi$
$211$	381.937	1.81013	0.905065	+	0.425274i	$0.139822\pi$
$212$	0	0
$213$	214.565i	1.00735i
$214$	0	0
$215$	− 44.6256i	− 0.207561i
$216$	0	0
$217$	150.384	0.693012
$218$	0	0
$219$	−17.3205	−0.0790891
$220$	0	0
$221$	242.788i	1.09859i
$222$	0	0
$223$	268.129i	1.20237i	0.799109	+	0.601186i	$0.205305\pi$
$223$	268.129i	1.20237i	−0.799109	+	0.601186i	$0.794695\pi$
$224$	0	0
$225$	72.5755	0.322558
$226$	0	0
$227$	−353.082	−1.55543	−0.777713	−	0.628620i	$-0.783620\pi$
$227$	−353.082	−1.55543	−0.777713	+	0.628620i	$0.783620\pi$
$228$	0	0
$229$	212.545i	0.928144i	0.885798	+	0.464072i	$0.153612\pi$
$229$	212.545i	0.928144i	−0.885798	+	0.464072i	$0.846388\pi$
$230$	0	0
$231$	21.4845i	0.0930065i
$232$	0	0
$233$	355.576	1.52608	0.763038	−	0.646354i	$-0.223707\pi$
$233$	355.576	1.52608	0.763038	+	0.646354i	$0.223707\pi$
$234$	0	0
$235$	34.5703	0.147108
$236$	0	0
$237$	− 227.283i	− 0.958999i
$238$	0	0
$239$	42.9690i	0.179787i	0.995951	+	0.0898933i	$0.0286526\pi$
$239$	42.9690i	0.179787i	−0.995951	+	0.0898933i	$0.971347\pi$
$240$	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
$243$	15.5885	0.0641500
$244$	0	0
$245$	36.8582i	0.150441i
$246$	0	0
$247$	− 60.5110i	− 0.244984i
$248$	0	0
$249$	190.788	0.766216
$250$	0	0
$251$	334.140	1.33123	0.665617	−	0.746294i	$-0.268168\pi$
$251$	334.140	1.33123	0.665617	+	0.746294i	$0.268168\pi$
$252$	0	0
$253$	96.7265i	0.382318i
$254$	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
$259$	−99.5375	−0.384315
$260$	0	0
$261$	133.485i	0.511436i
$262$	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
$267$	−126.772	−0.474801
$268$	0	0
$269$	300.858i	1.11843i	0.829022	+	0.559216i	$0.188898\pi$
$269$	300.858i	1.11843i	−0.829022	+	0.559216i	$0.811102\pi$
$270$	0	0
$271$	386.865i	1.42755i	0.700376	+	0.713774i	$0.253015\pi$
$271$	386.865i	1.42755i	−0.700376	+	0.713774i	$0.746985\pi$
$272$	0	0
$273$	−67.5959	−0.247604
$274$	0	0
$275$	106.093	0.385794
$276$	0	0
$277$	213.798i	0.771834i	0.922534	+	0.385917i	$0.126115\pi$
$277$	213.798i	0.771834i	−0.922534	+	0.385917i	$0.873885\pi$
$278$	0	0
$279$	− 159.506i	− 0.571706i
$280$	0	0
$281$	−110.767	−0.394190	−0.197095	−	0.980384i	$-0.563151\pi$
$281$	−110.767	−0.394190	−0.197095	+	0.980384i	$0.563151\pi$
$282$	0	0
$283$	114.980	0.406289	0.203145	−	0.979149i	$-0.434884\pi$
$283$	114.980	0.406289	0.203145	+	0.979149i	$0.434884\pi$
$284$	0	0
$285$	− 6.82857i	− 0.0239599i
$286$	0	0
$287$	106.337i	0.370513i
$288$	0	0
$289$	20.6163	0.0713368
$290$	0	0
$291$	−182.198	−0.626109
$292$	0	0
$293$	108.677i	0.370910i	0.982653	+	0.185455i	$0.0593758\pi$
$293$	108.677i	0.370910i	−0.982653	+	0.185455i	$0.940624\pi$
$294$	0	0
$295$	31.1416i	0.105565i
$296$	0	0
$297$	22.7878	0.0767264
$298$	0	0
$299$	−304.327	−1.01782
$300$	0	0
$301$	− 140.404i	− 0.466459i
$302$	0	0
$303$	− 268.258i	− 0.885338i
$304$	0	0
$305$	21.9388	0.0719304
$306$	0	0
$307$	253.544	0.825876	0.412938	−	0.910759i	$-0.364503\pi$
$307$	253.544	0.825876	0.412938	+	0.910759i	$0.364503\pi$
$308$	0	0
$309$	− 65.6663i	− 0.212512i
$310$	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	$-0.297743\pi$
$313$	371.535	1.18701	0.593506	+	0.804830i	$0.297743\pi$
$314$	0	0
$315$	−7.62809	−0.0242162
$316$	0	0
$317$	− 53.4847i	− 0.168721i	−0.996435	−	0.0843607i	$-0.973115\pi$
$317$	− 53.4847i	− 0.168721i	0.996435	−	0.0843607i	$-0.0268848\pi$
$318$	0	0
$319$	195.133i	0.611701i
$320$	0	0
$321$	33.5755	0.104597
$322$	0	0
$323$	−77.1670	−0.238907
$324$	0	0
$325$	333.798i	1.02707i
$326$	0	0
$327$	− 228.945i	− 0.700139i
$328$	0	0
$329$	108.767	0.330600
$330$	0	0
$331$	165.320	0.499457	0.249728	−	0.968316i	$-0.419659\pi$
$331$	165.320	0.499457	0.249728	+	0.968316i	$0.419659\pi$
$332$	0	0
$333$	105.576i	0.317044i
$334$	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
$339$	−61.6186	−0.181766
$340$	0	0
$341$	− 233.171i	− 0.683787i
$342$	0	0
$343$	254.558i	0.742153i
$344$	0	0
$345$	−34.3429	−0.0995445
$346$	0	0
$347$	−458.334	−1.32085	−0.660423	−	0.750894i	$-0.729623\pi$
$347$	−458.334	−1.32085	−0.660423	+	0.750894i	$0.729623\pi$
$348$	0	0
$349$	− 33.9388i	− 0.0972458i	−0.998817	−	0.0486229i	$-0.984517\pi$
$349$	− 33.9388i	− 0.0972458i	0.998817	−	0.0486229i	$-0.0154832\pi$
$350$	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
$355$	−111.365	−0.313704
$356$	0	0
$357$	86.2020i	0.241462i
$358$	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
$363$	−176.266	−0.485582
$364$	0	0
$365$	− 8.98979i	− 0.0246296i
$366$	0	0
$367$	− 15.7988i	− 0.0430485i	−0.999768	−	0.0215242i	$-0.993148\pi$
$367$	− 15.7988i	− 0.0430485i	0.999768	−	0.0215242i	$-0.00685190\pi$
$368$	0	0
$369$	112.788	0.305658
$370$	0	0
$371$	4.82853	0.0130149
$372$	0	0
$373$	− 82.0204i	− 0.219894i	−0.993937	−	0.109947i	$-0.964932\pi$
$373$	− 82.0204i	− 0.219894i	0.993937	−	0.109947i	$-0.0350681\pi$
$374$	0	0
$375$	76.5955i	0.204255i
$376$	0	0
$377$	−613.939	−1.62848
$378$	0	0
$379$	524.444	1.38376	0.691878	−	0.722014i	$-0.256783\pi$
$379$	524.444	1.38376	0.691878	+	0.722014i	$0.256783\pi$
$380$	0	0
$381$	− 221.444i	− 0.581218i
$382$	0	0
$383$	− 498.946i	− 1.30273i	−0.758764	−	0.651366i	$-0.774196\pi$
$383$	− 498.946i	− 1.30273i	0.758764	−	0.651366i	$-0.225804\pi$
$384$	0	0
$385$	−11.1510	−0.0289637
$386$	0	0
$387$	−148.921	−0.384809
$388$	0	0
$389$	− 233.889i	− 0.601256i	−0.953741	−	0.300628i	$-0.902804\pi$
$389$	− 233.889i	− 0.601256i	0.953741	−	0.300628i	$-0.0971963\pi$
$390$	0	0
$391$	388.095i	0.992570i
$392$	0	0
$393$	−149.616	−0.380703
$394$	0	0
$395$	117.966	0.298647
$396$	0	0
$397$	− 348.363i	− 0.877489i	−0.898612	−	0.438745i	$-0.855423\pi$
$397$	− 348.363i	− 0.877489i	0.898612	−	0.438745i	$-0.144577\pi$
$398$	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
$403$	733.619	1.82039
$404$	0	0
$405$	8.09082i	0.0199773i
$406$	0	0
$407$	154.334i	0.379199i
$408$	0	0
$409$	94.0408	0.229929	0.114964	−	0.993370i	$-0.463325\pi$
$409$	94.0408	0.229929	0.114964	+	0.993370i	$0.463325\pi$
$410$	0	0
$411$	−83.1031	−0.202197
$412$	0	0
$413$	97.9796i	0.237239i
$414$	0	0
$415$	99.0238i	0.238612i
$416$	0	0
$417$	−271.151	−0.650242
$418$	0	0
$419$	−567.413	−1.35421	−0.677104	−	0.735888i	$-0.736765\pi$
$419$	−567.413	−1.35421	−0.677104	+	0.735888i	$0.736765\pi$
$420$	0	0
$421$	− 786.120i	− 1.86727i	−0.358227	−	0.933635i	$-0.616618\pi$
$421$	− 786.120i	− 1.86727i	0.358227	−	0.933635i	$-0.383382\pi$
$422$	0	0
$423$	− 115.365i	− 0.272731i
$424$	0	0
$425$	425.678	1.00159
$426$	0	0
$427$	69.0252	0.161651
$428$	0	0
$429$	104.808i	0.244308i
$430$	0	0
$431$	402.780i	0.934523i	0.884119	+	0.467262i	$0.154759\pi$
$431$	402.780i	0.934523i	−0.884119	+	0.467262i	$0.845241\pi$
$432$	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
$435$	−69.2820	−0.159269
$436$	0	0
$437$	− 96.7265i	− 0.221342i
$438$	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
$443$	153.120	0.345644	0.172822	−	0.984953i	$-0.444711\pi$
$443$	153.120	0.345644	0.172822	+	0.984953i	$0.444711\pi$
$444$	0	0
$445$	− 65.7980i	− 0.147861i
$446$	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
$451$	164.877	0.365581
$452$	0	0
$453$	− 185.121i	− 0.408657i
$454$	0	0
$455$	− 35.0840i	− 0.0771078i
$456$	0	0
$457$	−777.878	−1.70214	−0.851070	−	0.525053i	$-0.824045\pi$
$457$	−777.878	−1.70214	−0.851070	+	0.525053i	$0.824045\pi$
$458$	0	0
$459$	91.4311	0.199196
$460$	0	0
$461$	− 635.160i	− 1.37779i	−0.724862	−	0.688894i	$-0.758097\pi$
$461$	− 635.160i	− 1.37779i	0.724862	−	0.688894i	$-0.241903\pi$
$462$	0	0
$463$	− 72.3096i	− 0.156176i	−0.996946	−	0.0780881i	$-0.975118\pi$
$463$	− 72.3096i	− 0.156176i	0.996946	−	0.0780881i	$-0.0248815\pi$
$464$	0	0
$465$	82.7878	0.178038
$466$	0	0
$467$	671.265	1.43740	0.718699	−	0.695321i	$-0.244738\pi$
$467$	671.265	1.43740	0.718699	+	0.695321i	$0.244738\pi$
$468$	0	0
$469$	− 265.171i	− 0.565397i
$470$	0	0
$471$	− 199.518i	− 0.423605i
$472$	0	0
$473$	−217.698	−0.460249
$474$	0	0
$475$	−106.093	−0.223355
$476$	0	0
$477$	− 5.12143i	− 0.0107368i
$478$	0	0
$479$	− 644.824i	− 1.34619i	−0.739557	−	0.673094i	$-0.764965\pi$
$479$	− 644.824i	− 1.34619i	0.739557	−	0.673094i	$-0.235035\pi$
$480$	0	0
$481$	−485.576	−1.00951
$482$	0	0
$483$	−108.052	−0.223710
$484$	0	0
$485$	− 94.5653i	− 0.194980i
$486$	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
$491$	−408.250	−0.831467	−0.415733	−	0.909486i	$-0.636475\pi$
$491$	−408.250	−0.831467	−0.415733	+	0.909486i	$0.636475\pi$
$492$	0	0
$493$	782.929i	1.58809i
$494$	0	0
$495$	11.8274i	0.0238938i
$496$	0	0
$497$	−350.384	−0.704997
$498$	0	0
$499$	−365.281	−0.732027	−0.366013	−	0.930610i	$-0.619278\pi$
$499$	−365.281	−0.732027	−0.366013	+	0.930610i	$0.619278\pi$
$500$	0	0
$501$	− 206.747i	− 0.412669i
$502$	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
$507$	−37.0376	−0.0730525
$508$	0	0
$509$	− 96.2724i	− 0.189140i	−0.995518	−	0.0945702i	$-0.969852\pi$
$509$	− 96.2724i	− 0.189140i	0.995518	−	0.0945702i	$-0.0301477\pi$
$510$	0	0
$511$	− 28.2843i	− 0.0553508i
$512$	0	0
$513$	−22.7878	−0.0444206
$514$	0	0
$515$	34.0825	0.0661797
$516$	0	0
$517$	− 168.645i	− 0.326199i
$518$	0	0
$519$	460.533i	0.887347i
$520$	0	0
$521$	199.212	0.382365	0.191183	−	0.981554i	$-0.438768\pi$
$521$	199.212	0.382365	0.191183	+	0.981554i	$0.438768\pi$
$522$	0	0
$523$	13.6702	0.0261381	0.0130691	−	0.999915i	$-0.495840\pi$
$523$	13.6702	0.0261381	0.0130691	+	0.999915i	$0.495840\pi$
$524$	0	0
$525$	118.515i	0.225743i
$526$	0	0
$527$	− 935.551i	− 1.77524i
$528$	0	0
$529$	42.5347	0.0804058
$530$	0	0
$531$	103.923	0.195712
$532$	0	0
$533$	518.747i	0.973259i
$534$	0	0
$535$	17.4266i	0.0325730i
$536$	0	0
$537$	391.151	0.728400
$538$	0	0
$539$	179.806	0.333591
$540$	0	0
$541$	− 542.443i	− 1.00267i	−0.865254	−	0.501333i	$-0.832843\pi$
$541$	− 542.443i	− 1.00267i	0.865254	−	0.501333i	$-0.167157\pi$
$542$	0	0
$543$	83.5237i	0.153819i
$544$	0	0
$545$	118.829	0.218034
$546$	0	0
$547$	−239.058	−0.437035	−0.218518	−	0.975833i	$-0.570122\pi$
$547$	−239.058	−0.437035	−0.218518	+	0.975833i	$0.570122\pi$
$548$	0	0
$549$	− 73.2122i	− 0.133356i
$550$	0	0
$551$	− 195.133i	− 0.354143i
$552$	0	0
$553$	371.151	0.671159
$554$	0	0
$555$	−54.7964	−0.0987323
$556$	0	0
$557$	− 521.707i	− 0.936638i	−0.883560	−	0.468319i	$-0.844860\pi$
$557$	− 521.707i	− 0.936638i	0.883560	−	0.468319i	$-0.155140\pi$
$558$	0	0
$559$	− 684.935i	− 1.22529i
$560$	0	0
$561$	133.657	0.238248
$562$	0	0
$563$	−1092.42	−1.94035	−0.970173	−	0.242413i	$-0.922061\pi$
$563$	−1092.42	−1.94035	−0.970173	+	0.242413i	$0.922061\pi$
$564$	0	0
$565$	− 31.9817i	− 0.0566047i
$566$	0	0
$567$	25.4558i	0.0448957i
$568$	0	0
$569$	−823.171	−1.44670	−0.723349	−	0.690482i	$-0.757398\pi$
$569$	−823.171	−1.44670	−0.723349	+	0.690482i	$0.757398\pi$
$570$	0	0
$571$	268.800	0.470753	0.235377	−	0.971904i	$-0.424368\pi$
$571$	268.800	0.470753	0.235377	+	0.971904i	$0.424368\pi$
$572$	0	0
$573$	307.596i	0.536817i
$574$	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
$579$	226.496	0.391184
$580$	0	0
$581$	311.555i	0.536239i
$582$	0	0
$583$	− 7.48669i	− 0.0128417i
$584$	0	0
$585$	−37.2122	−0.0636107
$586$	0	0
$587$	223.545	0.380827	0.190413	−	0.981704i	$-0.439017\pi$
$587$	223.545	0.380827	0.190413	+	0.981704i	$0.439017\pi$
$588$	0	0
$589$	233.171i	0.395877i
$590$	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
$595$	−44.7411	−0.0751951
$596$	0	0
$597$	− 257.666i	− 0.431602i
$598$	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	$-0.543009\pi$
$601$	−161.918	−0.269415	−0.134707	+	0.990885i	$0.543009\pi$
$602$	0	0
$603$	−281.257	−0.466429
$604$	0	0
$605$	− 91.4868i	− 0.151218i
$606$	0	0
$607$	− 573.542i	− 0.944879i	−0.881363	−	0.472439i	$-0.843374\pi$
$607$	− 573.542i	− 0.944879i	0.881363	−	0.472439i	$-0.156626\pi$
$608$	0	0
$609$	−217.980	−0.357930
$610$	0	0
$611$	530.601	0.868415
$612$	0	0
$613$	985.857i	1.60825i	0.594460	+	0.804125i	$0.297366\pi$
$613$	985.857i	1.60825i	−0.594460	+	0.804125i	$0.702634\pi$
$614$	0	0
$615$	58.5398i	0.0951866i
$616$	0	0
$617$	761.151	1.23363	0.616816	−	0.787107i	$-0.288422\pi$
$617$	761.151	1.23363	0.616816	+	0.787107i	$0.288422\pi$
$618$	0	0
$619$	−538.929	−0.870645	−0.435323	−	0.900275i	$-0.643366\pi$
$619$	−538.929	−0.870645	−0.435323	+	0.900275i	$0.643366\pi$
$620$	0	0
$621$	114.606i	0.184551i
$622$	0	0
$623$	− 207.018i	− 0.332292i
$624$	0	0
$625$	565.041	0.904065
$626$	0	0
$627$	−33.3119	−0.0531291
$628$	0	0
$629$	619.233i	0.984472i
$630$	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
$635$	114.935	0.181000
$636$	0	0
$637$	565.716i	0.888095i
$638$	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
$643$	647.752	1.00739	0.503695	−	0.863882i	$-0.331974\pi$
$643$	647.752	1.00739	0.503695	+	0.863882i	$0.331974\pi$
$644$	0	0
$645$	− 77.2939i	− 0.119835i
$646$	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
$651$	260.472	0.400111
$652$	0	0
$653$	23.3826i	0.0358080i	0.999840	+	0.0179040i	$0.00569933\pi$
$653$	23.3826i	0.0358080i	−0.999840	+	0.0179040i	$0.994301\pi$
$654$	0	0
$655$	− 77.6548i	− 0.118557i
$656$	0	0
$657$	−30.0000	−0.0456621
$658$	0	0
$659$	496.846	0.753940	0.376970	−	0.926225i	$-0.376966\pi$
$659$	496.846	0.753940	0.376970	+	0.926225i	$0.376966\pi$
$660$	0	0
$661$	113.576i	0.171824i	0.996303	+	0.0859119i	$0.0273804\pi$
$661$	113.576i	0.171824i	−0.996303	+	0.0859119i	$0.972620\pi$
$662$	0	0
$663$	420.521i	0.634270i
$664$	0	0
$665$	11.1510	0.0167684
$666$	0	0
$667$	−981.378	−1.47133
$668$	0	0
$669$	464.413i	0.694190i
$670$	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
$675$	125.704	0.186229
$676$	0	0
$677$	902.474i	1.33305i	0.745483	+	0.666525i	$0.232219\pi$
$677$	902.474i	1.33305i	−0.745483	+	0.666525i	$0.767781\pi$
$678$	0	0
$679$	− 297.527i	− 0.438185i
$680$	0	0
$681$	−611.555	−0.898025
$682$	0	0
$683$	924.738	1.35394	0.676968	−	0.736012i	$-0.263294\pi$
$683$	924.738	1.35394	0.676968	+	0.736012i	$0.263294\pi$
$684$	0	0
$685$	− 43.1327i	− 0.0629674i
$686$	0	0
$687$	368.139i	0.535864i
$688$	0	0
$689$	23.5551	0.0341874
$690$	0	0
$691$	−1155.87	−1.67275	−0.836373	−	0.548161i	$-0.815328\pi$
$691$	−1155.87	−1.67275	−0.836373	+	0.548161i	$0.815328\pi$
$692$	0	0
$693$	37.2122i	0.0536973i
$694$	0	0
$695$	− 140.734i	− 0.202496i
$696$	0	0
$697$	661.535	0.949117
$698$	0	0
$699$	615.875	0.881080
$700$	0	0
$701$	− 591.383i	− 0.843627i	−0.906683	−	0.421814i	$-0.861394\pi$
$701$	− 591.383i	− 0.843627i	0.906683	−	0.421814i	$-0.138606\pi$
$702$	0	0
$703$	− 154.334i	− 0.219536i
$704$	0	0
$705$	59.8775	0.0849327
$706$	0	0
$707$	438.063	0.619608
$708$	0	0
$709$	− 1216.87i	− 1.71631i	−0.513387	−	0.858157i	$-0.671609\pi$
$709$	− 1216.87i	− 1.71631i	0.513387	−	0.858157i	$-0.328391\pi$
$710$	0	0
$711$	− 393.665i	− 0.553678i
$712$	0	0
$713$	1172.69	1.64472
$714$	0	0
$715$	−54.3982	−0.0760814
$716$	0	0
$717$	74.4245i	0.103800i
$718$	0	0
$719$	1202.62i	1.67263i	0.548246	+	0.836317i	$0.315296\pi$
$719$	1202.62i	1.67263i	−0.548246	+	0.836317i	$0.684704\pi$
$720$	0	0
$721$	107.233	0.148728
$722$	0	0
$723$	216.839	0.299915
$724$	0	0
$725$	1076.41i	1.48471i
$726$	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
$731$	−873.467	−1.19489
$732$	0	0
$733$	1070.52i	1.46047i	0.683196	+	0.730235i	$0.260589\pi$
$733$	1070.52i	1.46047i	−0.683196	+	0.730235i	$0.739411\pi$
$734$	0	0
$735$	63.8402i	0.0868574i
$736$	0	0
$737$	−411.151	−0.557871
$738$	0	0
$739$	−376.710	−0.509757	−0.254879	−	0.966973i	$-0.582035\pi$
$739$	−376.710	−0.509757	−0.254879	+	0.966973i	$0.582035\pi$
$740$	0	0
$741$	− 104.808i	− 0.141442i
$742$	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
$747$	330.454	0.442375
$748$	0	0
$749$	54.8286i	0.0732024i
$750$	0	0
$751$	382.236i	0.508969i	0.967077	+	0.254485i	$0.0819058\pi$
$751$	382.236i	0.508969i	−0.967077	+	0.254485i	$0.918094\pi$
$752$	0	0
$753$	578.747	0.768588
$754$	0	0
$755$	96.0828	0.127262
$756$	0	0
$757$	206.969i	0.273407i	0.990612	+	0.136704i	$0.0436508\pi$
$757$	206.969i	0.273407i	−0.990612	+	0.136704i	$0.956349\pi$
$758$	0	0
$759$	167.535i	0.220732i
$760$	0	0
$761$	−959.980	−1.26147	−0.630736	−	0.775998i	$-0.717247\pi$
$761$	−959.980	−1.26147	−0.630736	+	0.775998i	$0.717247\pi$
$762$	0	0
$763$	373.866	0.489995
$764$	0	0
$765$	47.4551i	0.0620328i
$766$	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
$771$	502.295	0.651485
$772$	0	0
$773$	1389.24i	1.79721i	0.438763	+	0.898603i	$0.355417\pi$
$773$	1389.24i	1.79721i	−0.438763	+	0.898603i	$0.644583\pi$
$774$	0	0
$775$	− 1286.25i	− 1.65967i
$776$	0	0
$777$	−172.404	−0.221884
$778$	0	0
$779$	−164.877	−0.211652
$780$	0	0
$781$	543.273i	0.695613i
$782$	0	0
$783$	231.202i	0.295277i
$784$	0	0
$785$	103.555	0.131917
$786$	0	0
$787$	−526.588	−0.669108	−0.334554	−	0.942377i	$-0.608586\pi$
$787$	−526.588	−0.669108	−0.334554	+	0.942377i	$0.608586\pi$
$788$	0	0
$789$	− 432.000i	− 0.547529i
$790$	0	0
$791$	− 100.623i	− 0.127210i
$792$	0	0
$793$	336.727	0.424624
$794$	0	0
$795$	2.65816	0.00334359
$796$	0	0
$797$	− 1141.22i	− 1.43190i	−0.698153	−	0.715948i	$-0.745995\pi$
$797$	− 1141.22i	− 1.43190i	0.698153	−	0.715948i	$-0.254005\pi$
$798$	0	0
$799$	− 676.652i	− 0.846874i
$800$	0	0
$801$	−219.576	−0.274127
$802$	0	0
$803$	−43.8551	−0.0546140
$804$	0	0
$805$	− 56.0816i	− 0.0696666i
$806$	0	0
$807$	521.102i	0.645727i
$808$	0	0
$809$	944.020	1.16690	0.583449	−	0.812150i	$-0.301703\pi$
$809$	944.020	1.16690	0.583449	+	0.812150i	$0.301703\pi$
$810$	0	0
$811$	−58.7837	−0.0724830	−0.0362415	−	0.999343i	$-0.511539\pi$
$811$	−58.7837	−0.0724830	−0.0362415	+	0.999343i	$0.511539\pi$
$812$	0	0
$813$	670.070i	0.824195i
$814$	0	0
$815$	− 223.590i	− 0.274344i
$816$	0	0
$817$	217.698	0.266460
$818$	0	0
$819$	−117.080	−0.142954
$820$	0	0
$821$	− 1246.67i	− 1.51848i	−0.650809	−	0.759241i	$-0.725570\pi$
$821$	− 1246.67i	− 1.51848i	0.650809	−	0.759241i	$-0.274430\pi$
$822$	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
$827$	1251.44	1.51322	0.756612	−	0.653864i	$-0.226853\pi$
$827$	1251.44	1.51322	0.756612	+	0.653864i	$0.226853\pi$
$828$	0	0
$829$	− 13.8796i	− 0.0167426i	−0.999965	−	0.00837129i	$-0.997335\pi$
$829$	− 13.8796i	− 0.0167426i	0.999965	−	0.00837129i	$-0.00266470\pi$
$830$	0	0
$831$	370.309i	0.445618i
$832$	0	0
$833$	721.433	0.866066
$834$	0	0
$835$	107.307	0.128511
$836$	0	0
$837$	− 276.272i	− 0.330075i
$838$	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
$843$	−191.855	−0.227586
$844$	0	0
$845$	− 19.2235i	− 0.0227497i
$846$	0	0
$847$	− 287.842i	− 0.339836i
$848$	0	0
$849$	199.151	0.234571
$850$	0	0
$851$	−776.190	−0.912091
$852$	0	0
$853$	− 869.616i	− 1.01948i	−0.860329	−	0.509740i	$-0.829742\pi$
$853$	− 869.616i	− 1.01948i	0.860329	−	0.509740i	$-0.170258\pi$
$854$	0	0
$855$	− 11.8274i	− 0.0138333i
$856$	0	0
$857$	−883.939	−1.03143	−0.515717	−	0.856759i	$-0.672474\pi$
$857$	−883.939	−1.03143	−0.515717	+	0.856759i	$0.672474\pi$
$858$	0	0
$859$	−1592.65	−1.85407	−0.927035	−	0.374976i	$-0.877651\pi$
$859$	−1592.65	−1.85407	−0.927035	+	0.374976i	$0.877651\pi$
$860$	0	0
$861$	184.182i	0.213916i
$862$	0	0
$863$	1209.31i	1.40128i	0.713513	+	0.700642i	$0.247103\pi$
$863$	1209.31i	1.40128i	−0.713513	+	0.700642i	$0.752897\pi$
$864$	0	0
$865$	−239.029	−0.276334
$866$	0	0
$867$	35.7085	0.0411863
$868$	0	0
$869$	− 575.473i	− 0.662225i
$870$	0	0
$871$	− 1293.59i	− 1.48518i
$872$	0	0
$873$	−315.576	−0.361484
$874$	0	0
$875$	−125.080	−0.142949
$876$	0	0
$877$	− 922.584i	− 1.05198i	−0.850492	−	0.525988i	$-0.823696\pi$
$877$	− 922.584i	− 1.05198i	0.850492	−	0.525988i	$-0.176304\pi$
$878$	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
$883$	−836.284	−0.947094	−0.473547	−	0.880769i	$-0.657027\pi$
$883$	−836.284	−0.947094	−0.473547	+	0.880769i	$0.657027\pi$
$884$	0	0
$885$	53.9388i	0.0609478i
$886$	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
$891$	39.4695	0.0442980
$892$	0	0
$893$	168.645i	0.188852i
$894$	0	0
$895$	203.018i	0.226835i
$896$	0	0
$897$	−527.110	−0.587637
$898$	0	0
$899$	2365.73	2.63152
$900$	0	0
$901$	− 30.0388i	− 0.0333394i
$902$	0	0
$903$	− 243.187i	− 0.269310i
$904$	0	0
$905$	−43.3510	−0.0479017
$906$	0	0
$907$	399.595	0.440567	0.220284	−	0.975436i	$-0.429302\pi$
$907$	399.595	0.440567	0.220284	+	0.975436i	$0.429302\pi$
$908$	0	0
$909$	− 464.636i	− 0.511150i
$910$	0	0
$911$	612.995i	0.672882i	0.941705	+	0.336441i	$0.109223\pi$
$911$	612.995i	0.672882i	−0.941705	+	0.336441i	$0.890777\pi$
$912$	0	0
$913$	483.069	0.529101
$914$	0	0
$915$	37.9991	0.0415290
$916$	0	0
$917$	− 244.322i	− 0.266437i
$918$	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
$923$	−1709.28	−1.85188
$924$	0	0
$925$	851.355i	0.920384i
$926$	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
$931$	−179.806	−0.193132
$932$	0	0
$933$	− 790.665i	− 0.847444i
$934$	0	0
$935$	69.3715i	0.0741942i
$936$	0	0
$937$	−1393.84	−1.48755	−0.743776	−	0.668429i	$-0.766967\pi$
$937$	−1393.84	−1.48755	−0.743776	+	0.668429i	$0.766967\pi$
$938$	0	0
$939$	643.517	0.685322
$940$	0	0
$941$	− 659.968i	− 0.701348i	−0.936498	−	0.350674i	$-0.885953\pi$
$941$	− 659.968i	− 0.701348i	0.936498	−	0.350674i	$-0.114047\pi$
$942$	0	0
$943$	829.214i	0.879336i
$944$	0	0
$945$	−13.2122	−0.0139812
$946$	0	0
$947$	−1584.65	−1.67333	−0.836666	−	0.547714i	$-0.815498\pi$
$947$	−1584.65	−1.67333	−0.836666	+	0.547714i	$0.815498\pi$
$948$	0	0
$949$	− 137.980i	− 0.145395i
$950$	0	0
$951$	− 92.6382i	− 0.0974114i
$952$	0	0
$953$	−1212.02	−1.27179	−0.635897	−	0.771774i	$-0.719370\pi$
$953$	−1212.02	−1.27179	−0.635897	+	0.771774i	$0.719370\pi$
$954$	0	0
$955$	−159.650	−0.167173
$956$	0	0
$957$	337.980i	0.353166i
$958$	0	0
$959$	− 135.707i	− 0.141509i
$960$	0	0
$961$	−1865.91	−1.94163
$962$	0	0
$963$	58.1545	0.0603889
$964$	0	0
$965$	117.557i	0.121821i
$966$	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
$971$	1364.60	1.40535	0.702677	−	0.711509i	$-0.251988\pi$
$971$	1364.60	1.40535	0.702677	+	0.711509i	$0.251988\pi$
$972$	0	0
$973$	− 442.788i	− 0.455075i
$974$	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
$979$	−320.983	−0.327868
$980$	0	0
$981$	− 396.545i	− 0.404225i
$982$	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
$987$	188.391	0.190872
$988$	0	0
$989$	− 1094.87i	− 1.10704i
$990$	0	0
$991$	1596.00i	1.61050i	0.592939	+	0.805248i	$0.297968\pi$
$991$	1596.00i	1.61050i	−0.592939	+	0.805248i	$0.702032\pi$
$992$	0	0
$993$	286.343	0.288361
$994$	0	0
$995$	133.736	0.134408
$996$	0	0
$997$	− 1122.02i	− 1.12540i	−0.826662	−	0.562698i	$-0.809763\pi$
$997$	− 1122.02i	− 1.12540i	0.826662	−	0.562698i	$-0.190237\pi$
$998$	0	0
$999$	182.862i	0.183045i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.b.c.319.2	✓	8	8.5	even	2	inner
384.3.b.c.319.3	yes	8	4.3	odd	2	inner
384.3.b.c.319.6	yes	8	8.3	odd	2	inner
384.3.b.c.319.7	yes	8	1.1	even	1	trivial
768.3.g.c.511.2		4	16.3	odd	4
768.3.g.c.511.4		4	16.13	even	4
768.3.g.g.511.1		4	16.5	even	4
768.3.g.g.511.3		4	16.11	odd	4
1152.3.b.j.703.3		8	12.11	even	2
1152.3.b.j.703.4		8	3.2	odd	2
1152.3.b.j.703.5		8	24.11	even	2
1152.3.b.j.703.6		8	24.5	odd	2
2304.3.g.o.1279.3		4	48.5	odd	4
2304.3.g.o.1279.4		4	48.11	even	4
2304.3.g.x.1279.1		4	48.29	odd	4
2304.3.g.x.1279.2		4	48.35	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +0.898979i q^{5} +2.82843i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} +0.898979i q^{5} +2.82843i q^{7} +3.00000 q^{9} +4.38551 q^{11} +13.7980i q^{13} +1.55708i q^{15} +17.5959 q^{17} -4.38551 q^{19} +4.89898i q^{21} +22.0560i q^{23} +24.1918 q^{25} +5.19615 q^{27} +44.4949i q^{29} -53.1687i q^{31} +7.59592 q^{33} -2.54270 q^{35} +35.1918i q^{37} +23.8988i q^{39} +37.5959 q^{41} -49.6403 q^{43} +2.69694i q^{45} -38.4551i q^{47} +41.0000 q^{49} +30.4770 q^{51} -1.70714i q^{53} +3.94248i q^{55} -7.59592 q^{57} +34.6410 q^{59} -24.4041i q^{61} +8.48528i q^{63} -12.4041 q^{65} -93.7523 q^{67} +38.2020i q^{69} +123.879i q^{71} -10.0000 q^{73} +41.9015 q^{75} +12.4041i q^{77} -131.222i q^{79} +9.00000 q^{81} +110.151 q^{83} +15.8184i q^{85} +77.0674i q^{87} -73.1918 q^{89} -39.0265 q^{91} -92.0908i q^{93} -3.94248i q^{95} -105.192 q^{97} +13.1565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{9}+O(q^{10}) \) 8 * q + 24 * q^9	\( 8 q + 24 q^{9} - 16 q^{17} - 120 q^{25} - 96 q^{33} + 144 q^{41} + 328 q^{49} + 96 q^{57} - 256 q^{65} - 80 q^{73} + 72 q^{81} - 272 q^{89} - 528 q^{97}+O(q^{100}) \) 8 * q + 24 * q^9 - 16 * q^17 - 120 * q^25 - 96 * q^33 + 144 * q^41 + 328 * q^49 + 96 * q^57 - 256 * q^65 - 80 * q^73 + 72 * q^81 - 272 * q^89 - 528 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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