LMFDB - Embedded newform 384.6.d.j.193.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,6,Mod(193,384)]

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([0, 1, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$61.5873868082$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 338x^{10} + 43555x^{8} + 2692222x^{6} + 81680965x^{4} + 1098257588x^{2} + 4742525956 $ x^12 + 338x^10 + 43555x^8 + 2692222x^6 + 81680965x^4 + 1098257588*x^2 + 4742525956
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{57}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.4
Root			$8.37346i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.6.d.j.193.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +21.1198i q^{5} +241.194 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} +21.1198i q^{5} +241.194 q^{7} -81.0000 q^{9} -641.936i q^{11} +316.614i q^{13} +190.078 q^{15} -901.822 q^{17} -2167.92i q^{19} -2170.75i q^{21} -397.284 q^{23} +2678.95 q^{25} +729.000i q^{27} +5175.25i q^{29} +2390.97 q^{31} -5777.42 q^{33} +5093.97i q^{35} -13524.8i q^{37} +2849.53 q^{39} -11053.7 q^{41} +16930.3i q^{43} -1710.70i q^{45} +22654.9 q^{47} +41367.6 q^{49} +8116.40i q^{51} -7717.44i q^{53} +13557.6 q^{55} -19511.3 q^{57} -28603.9i q^{59} -10732.3i q^{61} -19536.7 q^{63} -6686.83 q^{65} -35933.3i q^{67} +3575.56i q^{69} -64951.4 q^{71} +14484.6 q^{73} -24110.6i q^{75} -154831. i q^{77} -37956.2 q^{79} +6561.00 q^{81} -50726.1i q^{83} -19046.3i q^{85} +46577.2 q^{87} -17057.3 q^{89} +76365.5i q^{91} -21518.8i q^{93} +45786.1 q^{95} -95957.4 q^{97} +51996.8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 972 q^{9}+O(q^{10}) $ 12 * q - 972 * q^9	$ 12 q - 972 q^{9} - 4888 q^{17} - 18660 q^{25} - 720 q^{33} - 25096 q^{41} + 135564 q^{49} - 13104 q^{57} + 254272 q^{65} - 187032 q^{73} + 78732 q^{81} - 575544 q^{89} + 93864 q^{97}+O(q^{100}) $ 12 * q - 972 * q^9 - 4888 * q^17 - 18660 * q^25 - 720 * q^33 - 25096 * q^41 + 135564 * q^49 - 13104 * q^57 + 254272 * q^65 - 187032 * q^73 + 78732 * q^81 - 575544 * q^89 + 93864 * 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
$3$	− 9.00000i	− 0.577350i
$3$	− 9.00000i	− 0.577350i
$4$	0	0
$5$	21.1198i	0.377803i	0.981996	+	0.188901i	$0.0604926\pi$
$5$	21.1198i	0.377803i	−0.981996	+	0.188901i	$0.939507\pi$
$6$	0	0
$7$	241.194	1.86046	0.930232	−	0.366971i	$-0.119605\pi$
$7$	241.194	1.86046	0.930232	+	0.366971i	$0.119605\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	− 641.936i	− 1.59959i	−0.600271	−	0.799797i	$-0.704940\pi$
$11$	− 641.936i	− 1.59959i	0.600271	−	0.799797i	$-0.295060\pi$
$12$	0	0
$13$	316.614i	0.519604i	0.965662	+	0.259802i	$0.0836572\pi$
$13$	316.614i	0.519604i	−0.965662	+	0.259802i	$0.916343\pi$
$14$	0	0
$15$	190.078	0.218124
$16$	0	0
$17$	−901.822	−0.756831	−0.378415	−	0.925636i	$-0.623531\pi$
$17$	−901.822	−0.756831	−0.378415	+	0.925636i	$0.623531\pi$
$18$	0	0
$19$	− 2167.92i	− 1.37771i	−0.724897	−	0.688857i	$-0.758113\pi$
$19$	− 2167.92i	− 1.37771i	0.724897	−	0.688857i	$-0.241887\pi$
$20$	0	0
$21$	− 2170.75i	− 1.07414i
$22$	0	0
$23$	−397.284	−0.156596	−0.0782981	−	0.996930i	$-0.524949\pi$
$23$	−397.284	−0.156596	−0.0782981	+	0.996930i	$0.524949\pi$
$24$	0	0
$25$	2678.95	0.857265
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	5175.25i	1.14271i	0.820703	+	0.571355i	$0.193582\pi$
$29$	5175.25i	1.14271i	−0.820703	+	0.571355i	$0.806418\pi$
$30$	0	0
$31$	2390.97	0.446859	0.223429	−	0.974720i	$-0.428275\pi$
$31$	2390.97	0.446859	0.223429	+	0.974720i	$0.428275\pi$
$32$	0	0
$33$	−5777.42	−0.923526
$34$	0	0
$35$	5093.97i	0.702888i
$36$	0	0
$37$	− 13524.8i	− 1.62415i	−0.583554	−	0.812074i	$-0.698339\pi$
$37$	− 13524.8i	− 1.62415i	0.583554	−	0.812074i	$-0.301661\pi$
$38$	0	0
$39$	2849.53	0.299993
$40$	0	0
$41$	−11053.7	−1.02695	−0.513475	−	0.858105i	$-0.671642\pi$
$41$	−11053.7	−1.02695	−0.513475	+	0.858105i	$0.671642\pi$
$42$	0	0
$43$	16930.3i	1.39635i	0.715928	+	0.698174i	$0.246004\pi$
$43$	16930.3i	1.39635i	−0.715928	+	0.698174i	$0.753996\pi$
$44$	0	0
$45$	− 1710.70i	− 0.125934i
$46$	0	0
$47$	22654.9	1.49595	0.747976	−	0.663726i	$-0.231026\pi$
$47$	22654.9	1.49595	0.747976	+	0.663726i	$0.231026\pi$
$48$	0	0
$49$	41367.6	2.46133
$50$	0	0
$51$	8116.40i	0.436956i
$52$	0	0
$53$	− 7717.44i	− 0.377384i	−0.982036	−	0.188692i	$-0.939575\pi$
$53$	− 7717.44i	− 0.377384i	0.982036	−	0.188692i	$-0.0604248\pi$
$54$	0	0
$55$	13557.6	0.604331
$56$	0	0
$57$	−19511.3	−0.795424
$58$	0	0
$59$	− 28603.9i	− 1.06978i	−0.844921	−	0.534891i	$-0.820353\pi$
$59$	− 28603.9i	− 1.06978i	0.844921	−	0.534891i	$-0.179647\pi$
$60$	0	0
$61$	− 10732.3i	− 0.369292i	−0.982805	−	0.184646i	$-0.940886\pi$
$61$	− 10732.3i	− 0.369292i	0.982805	−	0.184646i	$-0.0591138\pi$
$62$	0	0
$63$	−19536.7	−0.620155
$64$	0	0
$65$	−6686.83	−0.196308
$66$	0	0
$67$	− 35933.3i	− 0.977934i	−0.872302	−	0.488967i	$-0.837374\pi$
$67$	− 35933.3i	− 0.977934i	0.872302	−	0.488967i	$-0.162626\pi$
$68$	0	0
$69$	3575.56i	0.0904109i
$70$	0	0
$71$	−64951.4	−1.52912	−0.764562	−	0.644550i	$-0.777045\pi$
$71$	−64951.4	−1.52912	−0.764562	+	0.644550i	$0.777045\pi$
$72$	0	0
$73$	14484.6	0.318125	0.159063	−	0.987268i	$-0.449153\pi$
$73$	14484.6	0.318125	0.159063	+	0.987268i	$0.449153\pi$
$74$	0	0
$75$	− 24110.6i	− 0.494942i
$76$	0	0
$77$	− 154831.i	− 2.97599i
$78$	0	0
$79$	−37956.2	−0.684251	−0.342125	−	0.939654i	$-0.611147\pi$
$79$	−37956.2	−0.684251	−0.342125	+	0.939654i	$0.611147\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	− 50726.1i	− 0.808233i	−0.914707	−	0.404117i	$-0.867579\pi$
$83$	− 50726.1i	− 0.808233i	0.914707	−	0.404117i	$-0.132421\pi$
$84$	0	0
$85$	− 19046.3i	− 0.285933i
$86$	0	0
$87$	46577.2	0.659744
$88$	0	0
$89$	−17057.3	−0.228263	−0.114131	−	0.993466i	$-0.536409\pi$
$89$	−17057.3	−0.228263	−0.114131	+	0.993466i	$0.536409\pi$
$90$	0	0
$91$	76365.5i	0.966704i
$92$	0	0
$93$	− 21518.8i	− 0.257994i
$94$	0	0
$95$	45786.1	0.520504
$96$	0	0
$97$	−95957.4	−1.03550	−0.517749	−	0.855533i	$-0.673230\pi$
$97$	−95957.4	−1.03550	−0.517749	+	0.855533i	$0.673230\pi$
$98$	0	0
$99$	51996.8i	0.533198i
$100$	0	0
$101$	− 124044.i	− 1.20996i	−0.796239	−	0.604982i	$-0.793180\pi$
$101$	− 124044.i	− 1.20996i	0.796239	−	0.604982i	$-0.206820\pi$
$102$	0	0
$103$	76641.5	0.711821	0.355911	−	0.934520i	$-0.384171\pi$
$103$	76641.5	0.711821	0.355911	+	0.934520i	$0.384171\pi$
$104$	0	0
$105$	45845.7	0.405813
$106$	0	0
$107$	165790.i	1.39991i	0.714189	+	0.699953i	$0.246796\pi$
$107$	165790.i	1.39991i	−0.714189	+	0.699953i	$0.753204\pi$
$108$	0	0
$109$	− 215641.i	− 1.73847i	−0.494403	−	0.869233i	$-0.664613\pi$
$109$	− 215641.i	− 1.73847i	0.494403	−	0.869233i	$-0.335387\pi$
$110$	0	0
$111$	−121723.	−0.937702
$112$	0	0
$113$	131695.	0.970223	0.485112	−	0.874452i	$-0.338779\pi$
$113$	131695.	0.970223	0.485112	+	0.874452i	$0.338779\pi$
$114$	0	0
$115$	− 8390.56i	− 0.0591625i
$116$	0	0
$117$	− 25645.8i	− 0.173201i
$118$	0	0
$119$	−217514.	−1.40806
$120$	0	0
$121$	−251030.	−1.55870
$122$	0	0
$123$	99483.6i	0.592910i
$124$	0	0
$125$	122578.i	0.701680i
$126$	0	0
$127$	−67988.9	−0.374049	−0.187025	−	0.982355i	$-0.559884\pi$
$127$	−67988.9	−0.374049	−0.187025	+	0.982355i	$0.559884\pi$
$128$	0	0
$129$	152373.	0.806182
$130$	0	0
$131$	− 122344.i	− 0.622883i	−0.950265	−	0.311441i	$-0.899188\pi$
$131$	− 122344.i	− 0.622883i	0.950265	−	0.311441i	$-0.100812\pi$
$132$	0	0
$133$	− 522889.i	− 2.56319i
$134$	0	0
$135$	−15396.3	−0.0727081
$136$	0	0
$137$	107765.	0.490544	0.245272	−	0.969454i	$-0.421123\pi$
$137$	107765.	0.490544	0.245272	+	0.969454i	$0.421123\pi$
$138$	0	0
$139$	− 112298.i	− 0.492987i	−0.969144	−	0.246493i	$-0.920722\pi$
$139$	− 112298.i	− 0.492987i	0.969144	−	0.246493i	$-0.0792783\pi$
$140$	0	0
$141$	− 203894.i	− 0.863688i
$142$	0	0
$143$	203246.	0.831155
$144$	0	0
$145$	−109300.	−0.431719
$146$	0	0
$147$	− 372308.i	− 1.42105i
$148$	0	0
$149$	− 233896.i	− 0.863094i	−0.902091	−	0.431547i	$-0.857968\pi$
$149$	− 233896.i	− 0.863094i	0.902091	−	0.431547i	$-0.142032\pi$
$150$	0	0
$151$	372017.	1.32776	0.663881	−	0.747838i	$-0.268908\pi$
$151$	372017.	1.32776	0.663881	+	0.747838i	$0.268908\pi$
$152$	0	0
$153$	73047.6	0.252277
$154$	0	0
$155$	50496.9i	0.168824i
$156$	0	0
$157$	221812.i	0.718185i	0.933302	+	0.359093i	$0.116914\pi$
$157$	221812.i	0.718185i	−0.933302	+	0.359093i	$0.883086\pi$
$158$	0	0
$159$	−69457.0	−0.217883
$160$	0	0
$161$	−95822.5	−0.291342
$162$	0	0
$163$	− 290409.i	− 0.856132i	−0.903747	−	0.428066i	$-0.859195\pi$
$163$	− 290409.i	− 0.856132i	0.903747	−	0.428066i	$-0.140805\pi$
$164$	0	0
$165$	− 122018.i	− 0.348911i
$166$	0	0
$167$	173938.	0.482618	0.241309	−	0.970448i	$-0.422423\pi$
$167$	173938.	0.482618	0.241309	+	0.970448i	$0.422423\pi$
$168$	0	0
$169$	271048.	0.730012
$170$	0	0
$171$	175602.i	0.459238i
$172$	0	0
$173$	− 468959.i	− 1.19129i	−0.803246	−	0.595647i	$-0.796896\pi$
$173$	− 468959.i	− 1.19129i	0.803246	−	0.595647i	$-0.203104\pi$
$174$	0	0
$175$	646148.	1.59491
$176$	0	0
$177$	−257435.	−0.617639
$178$	0	0
$179$	− 283666.i	− 0.661721i	−0.943680	−	0.330860i	$-0.892661\pi$
$179$	− 283666.i	− 0.661721i	0.943680	−	0.330860i	$-0.107339\pi$
$180$	0	0
$181$	96082.7i	0.217996i	0.994042	+	0.108998i	$0.0347642\pi$
$181$	96082.7i	0.217996i	−0.994042	+	0.108998i	$0.965236\pi$
$182$	0	0
$183$	−96591.0	−0.213211
$184$	0	0
$185$	285641.	0.613607
$186$	0	0
$187$	578912.i	1.21062i
$188$	0	0
$189$	175830.i	0.358047i
$190$	0	0
$191$	−691835.	−1.37220	−0.686102	−	0.727505i	$-0.740680\pi$
$191$	−691835.	−1.37220	−0.686102	+	0.727505i	$0.740680\pi$
$192$	0	0
$193$	448634.	0.866960	0.433480	−	0.901163i	$-0.357285\pi$
$193$	448634.	0.866960	0.433480	+	0.901163i	$0.357285\pi$
$194$	0	0
$195$	60181.5i	0.113338i
$196$	0	0
$197$	188295.i	0.345680i	0.984950	+	0.172840i	$0.0552943\pi$
$197$	188295.i	0.345680i	−0.984950	+	0.172840i	$0.944706\pi$
$198$	0	0
$199$	−44526.8	−0.0797056	−0.0398528	−	0.999206i	$-0.512689\pi$
$199$	−44526.8	−0.0797056	−0.0398528	+	0.999206i	$0.512689\pi$
$200$	0	0
$201$	−323399.	−0.564611
$202$	0	0
$203$	1.24824e6i	2.12597i
$204$	0	0
$205$	− 233453.i	− 0.387984i
$206$	0	0
$207$	32180.0	0.0521988
$208$	0	0
$209$	−1.39167e6	−2.20378
$210$	0	0
$211$	57184.3i	0.0884241i	0.999022	+	0.0442121i	$0.0140777\pi$
$211$	57184.3i	0.0884241i	−0.999022	+	0.0442121i	$0.985922\pi$
$212$	0	0
$213$	584563.i	0.882841i
$214$	0	0
$215$	−357565.	−0.527544
$216$	0	0
$217$	576688.	0.831365
$218$	0	0
$219$	− 130361.i	− 0.183670i
$220$	0	0
$221$	− 285530.i	− 0.393252i
$222$	0	0
$223$	1.04768e6	1.41081	0.705404	−	0.708806i	$-0.250766\pi$
$223$	1.04768e6	1.41081	0.705404	+	0.708806i	$0.250766\pi$
$224$	0	0
$225$	−216995.	−0.285755
$226$	0	0
$227$	− 694916.i	− 0.895092i	−0.894261	−	0.447546i	$-0.852298\pi$
$227$	− 694916.i	− 0.895092i	0.894261	−	0.447546i	$-0.147702\pi$
$228$	0	0
$229$	1.29816e6i	1.63584i	0.575334	+	0.817919i	$0.304872\pi$
$229$	1.29816e6i	1.63584i	−0.575334	+	0.817919i	$0.695128\pi$
$230$	0	0
$231$	−1.39348e6	−1.71819
$232$	0	0
$233$	−85333.4	−0.102974	−0.0514872	−	0.998674i	$-0.516396\pi$
$233$	−85333.4	−0.102974	−0.0514872	+	0.998674i	$0.516396\pi$
$234$	0	0
$235$	478467.i	0.565174i
$236$	0	0
$237$	341606.i	0.395052i
$238$	0	0
$239$	−994203.	−1.12585	−0.562925	−	0.826508i	$-0.690324\pi$
$239$	−994203.	−1.12585	−0.562925	+	0.826508i	$0.690324\pi$
$240$	0	0
$241$	424680.	0.470998	0.235499	−	0.971875i	$-0.424328\pi$
$241$	424680.	0.470998	0.235499	+	0.971875i	$0.424328\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	873675.i	0.929897i
$246$	0	0
$247$	686395.	0.715866
$248$	0	0
$249$	−456535.	−0.466634
$250$	0	0
$251$	64258.2i	0.0643790i	0.999482	+	0.0321895i	$0.0102480\pi$
$251$	64258.2i	0.0643790i	−0.999482	+	0.0321895i	$0.989752\pi$
$252$	0	0
$253$	255031.i	0.250491i
$254$	0	0
$255$	−171417.	−0.165083
$256$	0	0
$257$	1.34112e6	1.26659	0.633295	−	0.773911i	$-0.281702\pi$
$257$	1.34112e6	1.26659	0.633295	+	0.773911i	$0.281702\pi$
$258$	0	0
$259$	− 3.26209e6i	− 3.02167i
$260$	0	0
$261$	− 419195.i	− 0.380903i
$262$	0	0
$263$	1.40129e6	1.24922	0.624612	−	0.780936i	$-0.285257\pi$
$263$	1.40129e6	1.24922	0.624612	+	0.780936i	$0.285257\pi$
$264$	0	0
$265$	162991.	0.142577
$266$	0	0
$267$	153516.i	0.131788i
$268$	0	0
$269$	2.10516e6i	1.77380i	0.461964	+	0.886899i	$0.347145\pi$
$269$	2.10516e6i	1.77380i	−0.461964	+	0.886899i	$0.652855\pi$
$270$	0	0
$271$	668046.	0.552564	0.276282	−	0.961077i	$-0.410898\pi$
$271$	668046.	0.552564	0.276282	+	0.961077i	$0.410898\pi$
$272$	0	0
$273$	687289.	0.558127
$274$	0	0
$275$	− 1.71972e6i	− 1.37128i
$276$	0	0
$277$	− 380201.i	− 0.297724i	−0.988858	−	0.148862i	$-0.952439\pi$
$277$	− 380201.i	− 0.297724i	0.988858	−	0.148862i	$-0.0475611\pi$
$278$	0	0
$279$	−193669.	−0.148953
$280$	0	0
$281$	−1.01955e6	−0.770272	−0.385136	−	0.922860i	$-0.625845\pi$
$281$	−1.01955e6	−0.770272	−0.385136	+	0.922860i	$0.625845\pi$
$282$	0	0
$283$	1.97634e6i	1.46688i	0.679752	+	0.733442i	$0.262087\pi$
$283$	1.97634e6i	1.46688i	−0.679752	+	0.733442i	$0.737913\pi$
$284$	0	0
$285$	− 412075.i	− 0.300513i
$286$	0	0
$287$	−2.66609e6	−1.91060
$288$	0	0
$289$	−606574.	−0.427207
$290$	0	0
$291$	863616.i	0.597845i
$292$	0	0
$293$	1.78662e6i	1.21580i	0.794013	+	0.607901i	$0.207988\pi$
$293$	1.78662e6i	1.21580i	−0.794013	+	0.607901i	$0.792012\pi$
$294$	0	0
$295$	604109.	0.404166
$296$	0	0
$297$	467971.	0.307842
$298$	0	0
$299$	− 125786.i	− 0.0813680i
$300$	0	0
$301$	4.08349e6i	2.59786i
$302$	0	0
$303$	−1.11640e6	−0.698573
$304$	0	0
$305$	226665.	0.139519
$306$	0	0
$307$	1.55113e6i	0.939293i	0.882855	+	0.469646i	$0.155619\pi$
$307$	1.55113e6i	0.939293i	−0.882855	+	0.469646i	$0.844381\pi$
$308$	0	0
$309$	− 689773.i	− 0.410970i
$310$	0	0
$311$	−2.78547e6	−1.63304	−0.816522	−	0.577314i	$-0.804101\pi$
$311$	−2.78547e6	−1.63304	−0.816522	+	0.577314i	$0.804101\pi$
$312$	0	0
$313$	−3.24609e6	−1.87284	−0.936419	−	0.350884i	$-0.885881\pi$
$313$	−3.24609e6	−1.87284	−0.936419	+	0.350884i	$0.885881\pi$
$314$	0	0
$315$	− 412612.i	− 0.234296i
$316$	0	0
$317$	2.51001e6i	1.40290i	0.712717	+	0.701452i	$0.247464\pi$
$317$	2.51001e6i	1.40290i	−0.712717	+	0.701452i	$0.752536\pi$
$318$	0	0
$319$	3.32217e6	1.82787
$320$	0	0
$321$	1.49211e6	0.808236
$322$	0	0
$323$	1.95508e6i	1.04270i
$324$	0	0
$325$	848195.i	0.445438i
$326$	0	0
$327$	−1.94077e6	−1.00370
$328$	0	0
$329$	5.46423e6	2.78317
$330$	0	0
$331$	2.63607e6i	1.32247i	0.750178	+	0.661236i	$0.229968\pi$
$331$	2.63607e6i	1.32247i	−0.750178	+	0.661236i	$0.770032\pi$
$332$	0	0
$333$	1.09551e6i	0.541383i
$334$	0	0
$335$	758904.	0.369466
$336$	0	0
$337$	−1.01687e6	−0.487745	−0.243872	−	0.969807i	$-0.578418\pi$
$337$	−1.01687e6	−0.487745	−0.243872	+	0.969807i	$0.578418\pi$
$338$	0	0
$339$	− 1.18525e6i	− 0.560159i
$340$	0	0
$341$	− 1.53485e6i	− 0.714793i
$342$	0	0
$343$	5.92386e6	2.71875
$344$	0	0
$345$	−75515.1	−0.0341575
$346$	0	0
$347$	584840.i	0.260744i	0.991465	+	0.130372i	$0.0416171\pi$
$347$	584840.i	0.260744i	−0.991465	+	0.130372i	$0.958383\pi$
$348$	0	0
$349$	− 4.07200e6i	− 1.78955i	−0.446514	−	0.894777i	$-0.647335\pi$
$349$	− 4.07200e6i	− 1.78955i	0.446514	−	0.894777i	$-0.352665\pi$
$350$	0	0
$351$	−230812.	−0.0999978
$352$	0	0
$353$	−4.12456e6	−1.76174	−0.880869	−	0.473360i	$-0.843041\pi$
$353$	−4.12456e6	−1.76174	−0.880869	+	0.473360i	$0.843041\pi$
$354$	0	0
$355$	− 1.37176e6i	− 0.577707i
$356$	0	0
$357$	1.95763e6i	0.812942i
$358$	0	0
$359$	1.62705e6	0.666292	0.333146	−	0.942875i	$-0.391890\pi$
$359$	1.62705e6	0.666292	0.333146	+	0.942875i	$0.391890\pi$
$360$	0	0
$361$	−2.22378e6	−0.898098
$362$	0	0
$363$	2.25927e6i	0.899916i
$364$	0	0
$365$	305911.i	0.120189i
$366$	0	0
$367$	1.91820e6	0.743412	0.371706	−	0.928351i	$-0.378773\pi$
$367$	1.91820e6	0.743412	0.371706	+	0.928351i	$0.378773\pi$
$368$	0	0
$369$	895352.	0.342317
$370$	0	0
$371$	− 1.86140e6i	− 0.702110i
$372$	0	0
$373$	3.12704e6i	1.16375i	0.813277	+	0.581876i	$0.197681\pi$
$373$	3.12704e6i	1.16375i	−0.813277	+	0.581876i	$0.802319\pi$
$374$	0	0
$375$	1.10321e6	0.405115
$376$	0	0
$377$	−1.63856e6	−0.593756
$378$	0	0
$379$	− 4.61578e6i	− 1.65062i	−0.564681	−	0.825310i	$-0.691001\pi$
$379$	− 4.61578e6i	− 1.65062i	0.564681	−	0.825310i	$-0.308999\pi$
$380$	0	0
$381$	611900.i	0.215957i
$382$	0	0
$383$	805082.	0.280442	0.140221	−	0.990120i	$-0.455219\pi$
$383$	805082.	0.280442	0.140221	+	0.990120i	$0.455219\pi$
$384$	0	0
$385$	3.27000e6	1.12434
$386$	0	0
$387$	− 1.37135e6i	− 0.465449i
$388$	0	0
$389$	− 1.51690e6i	− 0.508258i	−0.967170	−	0.254129i	$-0.918211\pi$
$389$	− 1.51690e6i	− 0.508258i	0.967170	−	0.254129i	$-0.0817887\pi$
$390$	0	0
$391$	358280.	0.118517
$392$	0	0
$393$	−1.10110e6	−0.359621
$394$	0	0
$395$	− 801628.i	− 0.258512i
$396$	0	0
$397$	− 852385.i	− 0.271431i	−0.990748	−	0.135716i	$-0.956667\pi$
$397$	− 852385.i	− 0.271431i	0.990748	−	0.135716i	$-0.0433333\pi$
$398$	0	0
$399$	−4.70601e6	−1.47986
$400$	0	0
$401$	3.26307e6	1.01336	0.506682	−	0.862133i	$-0.330872\pi$
$401$	3.26307e6	1.01336	0.506682	+	0.862133i	$0.330872\pi$
$402$	0	0
$403$	757016.i	0.232190i
$404$	0	0
$405$	138567.i	0.0419781i
$406$	0	0
$407$	−8.68203e6	−2.59798
$408$	0	0
$409$	−357757.	−0.105750	−0.0528750	−	0.998601i	$-0.516838\pi$
$409$	−357757.	−0.105750	−0.0528750	+	0.998601i	$0.516838\pi$
$410$	0	0
$411$	− 969888.i	− 0.283215i
$412$	0	0
$413$	− 6.89909e6i	− 1.99029i
$414$	0	0
$415$	1.07133e6	0.305353
$416$	0	0
$417$	−1.01068e6	−0.284626
$418$	0	0
$419$	− 3.18860e6i	− 0.887288i	−0.896203	−	0.443644i	$-0.853685\pi$
$419$	− 3.18860e6i	− 0.887288i	0.896203	−	0.443644i	$-0.146315\pi$
$420$	0	0
$421$	1.56580e6i	0.430557i	0.976553	+	0.215278i	$0.0690659\pi$
$421$	1.56580e6i	0.430557i	−0.976553	+	0.215278i	$0.930934\pi$
$422$	0	0
$423$	−1.83505e6	−0.498650
$424$	0	0
$425$	−2.41594e6	−0.648805
$426$	0	0
$427$	− 2.58858e6i	− 0.687055i
$428$	0	0
$429$	− 1.82921e6i	− 0.479868i
$430$	0	0
$431$	3.21418e6	0.833445	0.416722	−	0.909034i	$-0.363179\pi$
$431$	3.21418e6	0.833445	0.416722	+	0.909034i	$0.363179\pi$
$432$	0	0
$433$	3.60956e6	0.925199	0.462599	−	0.886567i	$-0.346917\pi$
$433$	3.60956e6	0.925199	0.462599	+	0.886567i	$0.346917\pi$
$434$	0	0
$435$	983702.i	0.249253i
$436$	0	0
$437$	861280.i	0.215745i
$438$	0	0
$439$	4.38465e6	1.08586	0.542930	−	0.839778i	$-0.317315\pi$
$439$	4.38465e6	1.08586	0.542930	+	0.839778i	$0.317315\pi$
$440$	0	0
$441$	−3.35077e6	−0.820443
$442$	0	0
$443$	3.38752e6i	0.820111i	0.912061	+	0.410055i	$0.134491\pi$
$443$	3.38752e6i	0.820111i	−0.912061	+	0.410055i	$0.865509\pi$
$444$	0	0
$445$	− 360247.i	− 0.0862383i
$446$	0	0
$447$	−2.10507e6	−0.498307
$448$	0	0
$449$	−5.62999e6	−1.31793	−0.658964	−	0.752174i	$-0.729005\pi$
$449$	−5.62999e6	−1.31793	−0.658964	+	0.752174i	$0.729005\pi$
$450$	0	0
$451$	7.09578e6i	1.64270i
$452$	0	0
$453$	− 3.34815e6i	− 0.766584i
$454$	0	0
$455$	−1.61282e6	−0.365223
$456$	0	0
$457$	6.44361e6	1.44324	0.721621	−	0.692289i	$-0.243398\pi$
$457$	6.44361e6	1.44324	0.721621	+	0.692289i	$0.243398\pi$
$458$	0	0
$459$	− 657428.i	− 0.145652i
$460$	0	0
$461$	− 3.24074e6i	− 0.710219i	−0.934825	−	0.355110i	$-0.884444\pi$
$461$	− 3.24074e6i	− 0.710219i	0.934825	−	0.355110i	$-0.115556\pi$
$462$	0	0
$463$	315049.	0.0683009	0.0341504	−	0.999417i	$-0.489127\pi$
$463$	315049.	0.0683009	0.0341504	+	0.999417i	$0.489127\pi$
$464$	0	0
$465$	454472.	0.0974708
$466$	0	0
$467$	8.41249e6i	1.78498i	0.451071	+	0.892488i	$0.351042\pi$
$467$	8.41249e6i	1.78498i	−0.451071	+	0.892488i	$0.648958\pi$
$468$	0	0
$469$	− 8.66689e6i	− 1.81941i
$470$	0	0
$471$	1.99631e6	0.414645
$472$	0	0
$473$	1.08682e7	2.23359
$474$	0	0
$475$	− 5.80776e6i	− 1.18107i
$476$	0	0
$477$	625113.i	0.125795i
$478$	0	0
$479$	−2.90994e6	−0.579489	−0.289745	−	0.957104i	$-0.593570\pi$
$479$	−2.90994e6	−0.579489	−0.289745	+	0.957104i	$0.593570\pi$
$480$	0	0
$481$	4.28214e6	0.843913
$482$	0	0
$483$	862403.i	0.168206i
$484$	0	0
$485$	− 2.02660e6i	− 0.391214i
$486$	0	0
$487$	5.68765e6	1.08670	0.543351	−	0.839506i	$-0.317155\pi$
$487$	5.68765e6	1.08670	0.543351	+	0.839506i	$0.317155\pi$
$488$	0	0
$489$	−2.61368e6	−0.494288
$490$	0	0
$491$	4.68878e6i	0.877721i	0.898555	+	0.438860i	$0.144618\pi$
$491$	4.68878e6i	0.877721i	−0.898555	+	0.438860i	$0.855382\pi$
$492$	0	0
$493$	− 4.66715e6i	− 0.864838i
$494$	0	0
$495$	−1.09816e6	−0.201444
$496$	0	0
$497$	−1.56659e7	−2.84488
$498$	0	0
$499$	1.85099e6i	0.332776i	0.986060	+	0.166388i	$0.0532104\pi$
$499$	1.85099e6i	0.332776i	−0.986060	+	0.166388i	$0.946790\pi$
$500$	0	0
$501$	− 1.56544e6i	− 0.278640i
$502$	0	0
$503$	−9.26198e6	−1.63224	−0.816120	−	0.577883i	$-0.803879\pi$
$503$	−9.26198e6	−1.63224	−0.816120	+	0.577883i	$0.803879\pi$
$504$	0	0
$505$	2.61979e6	0.457128
$506$	0	0
$507$	− 2.43944e6i	− 0.421473i
$508$	0	0
$509$	6.96980e6i	1.19241i	0.802832	+	0.596206i	$0.203326\pi$
$509$	6.96980e6i	1.19241i	−0.802832	+	0.596206i	$0.796674\pi$
$510$	0	0
$511$	3.49359e6	0.591861
$512$	0	0
$513$	1.58041e6	0.265141
$514$	0	0
$515$	1.61865e6i	0.268928i
$516$	0	0
$517$	− 1.45430e7i	− 2.39291i
$518$	0	0
$519$	−4.22063e6	−0.687794
$520$	0	0
$521$	3.50414e6	0.565571	0.282785	−	0.959183i	$-0.408742\pi$
$521$	3.50414e6	0.565571	0.282785	+	0.959183i	$0.408742\pi$
$522$	0	0
$523$	6.15562e6i	0.984050i	0.870581	+	0.492025i	$0.163743\pi$
$523$	6.15562e6i	0.984050i	−0.870581	+	0.492025i	$0.836257\pi$
$524$	0	0
$525$	− 5.81533e6i	− 0.920823i
$526$	0	0
$527$	−2.15623e6	−0.338196
$528$	0	0
$529$	−6.27851e6	−0.975478
$530$	0	0
$531$	2.31692e6i	0.356594i
$532$	0	0
$533$	− 3.49977e6i	− 0.533607i
$534$	0	0
$535$	−3.50145e6	−0.528888
$536$	0	0
$537$	−2.55299e6	−0.382045
$538$	0	0
$539$	− 2.65553e7i	− 3.93713i
$540$	0	0
$541$	675255.i	0.0991915i	0.998769	+	0.0495957i	$0.0157933\pi$
$541$	675255.i	0.0991915i	−0.998769	+	0.0495957i	$0.984207\pi$
$542$	0	0
$543$	864744.	0.125860
$544$	0	0
$545$	4.55431e6	0.656797
$546$	0	0
$547$	− 1.97692e6i	− 0.282501i	−0.989974	−	0.141251i	$-0.954888\pi$
$547$	− 1.97692e6i	− 0.282501i	0.989974	−	0.141251i	$-0.0451123\pi$
$548$	0	0
$549$	869319.i	0.123097i
$550$	0	0
$551$	1.12195e7	1.57433
$552$	0	0
$553$	−9.15481e6	−1.27302
$554$	0	0
$555$	− 2.57077e6i	− 0.354266i
$556$	0	0
$557$	− 5.06259e6i	− 0.691408i	−0.938344	−	0.345704i	$-0.887640\pi$
$557$	− 5.06259e6i	− 0.691408i	0.938344	−	0.345704i	$-0.112360\pi$
$558$	0	0
$559$	−5.36038e6	−0.725548
$560$	0	0
$561$	5.21021e6	0.698953
$562$	0	0
$563$	6.19735e6i	0.824015i	0.911181	+	0.412007i	$0.135172\pi$
$563$	6.19735e6i	0.824015i	−0.911181	+	0.412007i	$0.864828\pi$
$564$	0	0
$565$	2.78136e6i	0.366553i
$566$	0	0
$567$	1.58247e6	0.206718
$568$	0	0
$569$	3.46075e6	0.448115	0.224058	−	0.974576i	$-0.428070\pi$
$569$	3.46075e6	0.448115	0.224058	+	0.974576i	$0.428070\pi$
$570$	0	0
$571$	− 7.09829e6i	− 0.911094i	−0.890212	−	0.455547i	$-0.849444\pi$
$571$	− 7.09829e6i	− 0.911094i	0.890212	−	0.455547i	$-0.150556\pi$
$572$	0	0
$573$	6.22651e6i	0.792243i
$574$	0	0
$575$	−1.06431e6	−0.134245
$576$	0	0
$577$	−873055.	−0.109170	−0.0545849	−	0.998509i	$-0.517384\pi$
$577$	−873055.	−0.109170	−0.0545849	+	0.998509i	$0.517384\pi$
$578$	0	0
$579$	− 4.03771e6i	− 0.500540i
$580$	0	0
$581$	− 1.22348e7i	− 1.50369i
$582$	0	0
$583$	−4.95410e6	−0.603661
$584$	0	0
$585$	541634.	0.0654359
$586$	0	0
$587$	− 55569.1i	− 0.00665638i	−0.999994	−	0.00332819i	$-0.998941\pi$
$587$	− 55569.1i	− 0.00665638i	0.999994	−	0.00332819i	$-0.00105940\pi$
$588$	0	0
$589$	− 5.18344e6i	− 0.615644i
$590$	0	0
$591$	1.69466e6	0.199578
$592$	0	0
$593$	−2.43242e6	−0.284055	−0.142028	−	0.989863i	$-0.545362\pi$
$593$	−2.43242e6	−0.284055	−0.142028	+	0.989863i	$0.545362\pi$
$594$	0	0
$595$	− 4.59386e6i	− 0.531968i
$596$	0	0
$597$	400741.i	0.0460180i
$598$	0	0
$599$	1.20790e7	1.37551	0.687756	−	0.725942i	$-0.258596\pi$
$599$	1.20790e7	1.37551	0.687756	+	0.725942i	$0.258596\pi$
$600$	0	0
$601$	−1.33462e7	−1.50720	−0.753598	−	0.657335i	$-0.771684\pi$
$601$	−1.33462e7	−1.50720	−0.753598	+	0.657335i	$0.771684\pi$
$602$	0	0
$603$	2.91059e6i	0.325978i
$604$	0	0
$605$	− 5.30171e6i	− 0.588881i
$606$	0	0
$607$	1.16077e7	1.27871	0.639357	−	0.768910i	$-0.279201\pi$
$607$	1.16077e7	1.27871	0.639357	+	0.768910i	$0.279201\pi$
$608$	0	0
$609$	1.12341e7	1.22743
$610$	0	0
$611$	7.17287e6i	0.777302i
$612$	0	0
$613$	1.58382e7i	1.70237i	0.524866	+	0.851185i	$0.324116\pi$
$613$	1.58382e7i	1.70237i	−0.524866	+	0.851185i	$0.675884\pi$
$614$	0	0
$615$	−2.10107e6	−0.224003
$616$	0	0
$617$	1.55010e7	1.63925	0.819626	−	0.572900i	$-0.194182\pi$
$617$	1.55010e7	1.63925	0.819626	+	0.572900i	$0.194182\pi$
$618$	0	0
$619$	1.69642e6i	0.177953i	0.996034	+	0.0889766i	$0.0283596\pi$
$619$	1.69642e6i	0.177953i	−0.996034	+	0.0889766i	$0.971640\pi$
$620$	0	0
$621$	− 289620.i	− 0.0301370i
$622$	0	0
$623$	−4.11412e6	−0.424675
$624$	0	0
$625$	5.78290e6	0.592169
$626$	0	0
$627$	1.25250e7i	1.27236i
$628$	0	0
$629$	1.21969e7i	1.22920i
$630$	0	0
$631$	1.02431e7	1.02413	0.512067	−	0.858945i	$-0.328880\pi$
$631$	1.02431e7	1.02413	0.512067	+	0.858945i	$0.328880\pi$
$632$	0	0
$633$	514659.	0.0510517
$634$	0	0
$635$	− 1.43591e6i	− 0.141317i
$636$	0	0
$637$	1.30976e7i	1.27892i
$638$	0	0
$639$	5.26107e6	0.509708
$640$	0	0
$641$	−1.16350e7	−1.11846	−0.559229	−	0.829013i	$-0.688903\pi$
$641$	−1.16350e7	−1.11846	−0.559229	+	0.829013i	$0.688903\pi$
$642$	0	0
$643$	1.40073e7i	1.33607i	0.744131	+	0.668034i	$0.232864\pi$
$643$	1.40073e7i	1.33607i	−0.744131	+	0.668034i	$0.767136\pi$
$644$	0	0
$645$	3.21808e6i	0.304578i
$646$	0	0
$647$	7.52725e6	0.706929	0.353464	−	0.935448i	$-0.385004\pi$
$647$	7.52725e6	0.706929	0.353464	+	0.935448i	$0.385004\pi$
$648$	0	0
$649$	−1.83619e7	−1.71122
$650$	0	0
$651$	− 5.19020e6i	− 0.479989i
$652$	0	0
$653$	− 1.92615e7i	− 1.76769i	−0.467779	−	0.883845i	$-0.654946\pi$
$653$	− 1.92615e7i	− 1.76769i	0.467779	−	0.883845i	$-0.345054\pi$
$654$	0	0
$655$	2.58389e6	0.235327
$656$	0	0
$657$	−1.17325e6	−0.106042
$658$	0	0
$659$	1.43325e7i	1.28561i	0.766031	+	0.642803i	$0.222229\pi$
$659$	1.43325e7i	1.28561i	−0.766031	+	0.642803i	$0.777771\pi$
$660$	0	0
$661$	− 4.02140e6i	− 0.357992i	−0.983850	−	0.178996i	$-0.942715\pi$
$661$	− 4.02140e6i	− 0.357992i	0.983850	−	0.178996i	$-0.0572850\pi$
$662$	0	0
$663$	−2.56977e6	−0.227044
$664$	0	0
$665$	1.10433e7	0.968380
$666$	0	0
$667$	− 2.05604e6i	− 0.178944i
$668$	0	0
$669$	− 9.42915e6i	− 0.814530i
$670$	0	0
$671$	−6.88947e6	−0.590717
$672$	0	0
$673$	4.33436e6	0.368882	0.184441	−	0.982844i	$-0.440953\pi$
$673$	4.33436e6	0.368882	0.184441	+	0.982844i	$0.440953\pi$
$674$	0	0
$675$	1.95296e6i	0.164981i
$676$	0	0
$677$	7.53072e6i	0.631488i	0.948844	+	0.315744i	$0.102254\pi$
$677$	7.53072e6i	0.631488i	−0.948844	+	0.315744i	$0.897746\pi$
$678$	0	0
$679$	−2.31443e7	−1.92651
$680$	0	0
$681$	−6.25424e6	−0.516781
$682$	0	0
$683$	3.41843e6i	0.280398i	0.990123	+	0.140199i	$0.0447742\pi$
$683$	3.41843e6i	0.280398i	−0.990123	+	0.140199i	$0.955226\pi$
$684$	0	0
$685$	2.27598e6i	0.185329i
$686$	0	0
$687$	1.16835e7	0.944451
$688$	0	0
$689$	2.44345e6	0.196090
$690$	0	0
$691$	5.56351e6i	0.443255i	0.975131	+	0.221628i	$0.0711370\pi$
$691$	5.56351e6i	0.443255i	−0.975131	+	0.221628i	$0.928863\pi$
$692$	0	0
$693$	1.25413e7i	0.991996i
$694$	0	0
$695$	2.37171e6	0.186252
$696$	0	0
$697$	9.96850e6	0.777227
$698$	0	0
$699$	768001.i	0.0594523i
$700$	0	0
$701$	2.50136e7i	1.92256i	0.275566	+	0.961282i	$0.411135\pi$
$701$	2.50136e7i	1.92256i	−0.275566	+	0.961282i	$0.588865\pi$
$702$	0	0
$703$	−2.93206e7	−2.23761
$704$	0	0
$705$	4.30620e6	0.326304
$706$	0	0
$707$	− 2.99187e7i	− 2.25110i
$708$	0	0
$709$	7.44336e6i	0.556100i	0.960567	+	0.278050i	$0.0896882\pi$
$709$	7.44336e6i	0.556100i	−0.960567	+	0.278050i	$0.910312\pi$
$710$	0	0
$711$	3.07445e6	0.228084
$712$	0	0
$713$	−949895.	−0.0699764
$714$	0	0
$715$	4.29252e6i	0.314012i
$716$	0	0
$717$	8.94783e6i	0.650009i
$718$	0	0
$719$	1.75536e6	0.126632	0.0633162	−	0.997994i	$-0.479832\pi$
$719$	1.75536e6	0.126632	0.0633162	+	0.997994i	$0.479832\pi$
$720$	0	0
$721$	1.84855e7	1.32432
$722$	0	0
$723$	− 3.82212e6i	− 0.271931i
$724$	0	0
$725$	1.38642e7i	0.979605i
$726$	0	0
$727$	−2.51518e7	−1.76495	−0.882477	−	0.470355i	$-0.844126\pi$
$727$	−2.51518e7	−1.76495	−0.882477	+	0.470355i	$0.844126\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	− 1.52681e7i	− 1.05680i
$732$	0	0
$733$	− 1.37042e7i	− 0.942094i	−0.882108	−	0.471047i	$-0.843876\pi$
$733$	− 1.37042e7i	− 0.942094i	0.882108	−	0.471047i	$-0.156124\pi$
$734$	0	0
$735$	7.86308e6	0.536876
$736$	0	0
$737$	−2.30668e7	−1.56430
$738$	0	0
$739$	1.66362e6i	0.112058i	0.998429	+	0.0560290i	$0.0178439\pi$
$739$	1.66362e6i	0.112058i	−0.998429	+	0.0560290i	$0.982156\pi$
$740$	0	0
$741$	− 6.17755e6i	− 0.413305i
$742$	0	0
$743$	−8.11218e6	−0.539095	−0.269548	−	0.962987i	$-0.586874\pi$
$743$	−8.11218e6	−0.539095	−0.269548	+	0.962987i	$0.586874\pi$
$744$	0	0
$745$	4.93985e6	0.326079
$746$	0	0
$747$	4.10882e6i	0.269411i
$748$	0	0
$749$	3.99875e7i	2.60447i
$750$	0	0
$751$	1.60115e7	1.03594	0.517968	−	0.855400i	$-0.326689\pi$
$751$	1.60115e7	1.03594	0.517968	+	0.855400i	$0.326689\pi$
$752$	0	0
$753$	578324.	0.0371692
$754$	0	0
$755$	7.85692e6i	0.501632i
$756$	0	0
$757$	3.03674e6i	0.192605i	0.995352	+	0.0963025i	$0.0307016\pi$
$757$	3.03674e6i	0.192605i	−0.995352	+	0.0963025i	$0.969298\pi$
$758$	0	0
$759$	2.29528e6	0.144621
$760$	0	0
$761$	−2.06516e6	−0.129268	−0.0646341	−	0.997909i	$-0.520588\pi$
$761$	−2.06516e6	−0.129268	−0.0646341	+	0.997909i	$0.520588\pi$
$762$	0	0
$763$	− 5.20114e7i	− 3.23435i
$764$	0	0
$765$	1.54275e6i	0.0953109i
$766$	0	0
$767$	9.05641e6	0.555863
$768$	0	0
$769$	−3.54420e6	−0.216124	−0.108062	−	0.994144i	$-0.534464\pi$
$769$	−3.54420e6	−0.216124	−0.108062	+	0.994144i	$0.534464\pi$
$770$	0	0
$771$	− 1.20701e7i	− 0.731266i
$772$	0	0
$773$	1.02157e7i	0.614920i	0.951561	+	0.307460i	$0.0994791\pi$
$773$	1.02157e7i	0.614920i	−0.951561	+	0.307460i	$0.900521\pi$
$774$	0	0
$775$	6.40531e6	0.383077
$776$	0	0
$777$	−2.93588e7	−1.74456
$778$	0	0
$779$	2.39636e7i	1.41484i
$780$	0	0
$781$	4.16946e7i	2.44598i
$782$	0	0
$783$	−3.77275e6	−0.219915
$784$	0	0
$785$	−4.68463e6	−0.271332
$786$	0	0
$787$	9.76102e6i	0.561770i	0.959741	+	0.280885i	$0.0906279\pi$
$787$	9.76102e6i	0.561770i	−0.959741	+	0.280885i	$0.909372\pi$
$788$	0	0
$789$	− 1.26117e7i	− 0.721239i
$790$	0	0
$791$	3.17639e7	1.80507
$792$	0	0
$793$	3.39801e6	0.191885
$794$	0	0
$795$	− 1.46692e6i	− 0.0823167i
$796$	0	0
$797$	1.56946e6i	0.0875196i	0.999042	+	0.0437598i	$0.0139336\pi$
$797$	1.56946e6i	0.0875196i	−0.999042	+	0.0437598i	$0.986066\pi$
$798$	0	0
$799$	−2.04307e7	−1.13218
$800$	0	0
$801$	1.38164e6	0.0760876
$802$	0	0
$803$	− 9.29816e6i	− 0.508872i
$804$	0	0
$805$	− 2.02375e6i	− 0.110070i
$806$	0	0
$807$	1.89464e7	1.02410
$808$	0	0
$809$	9.30648e6	0.499936	0.249968	−	0.968254i	$-0.419580\pi$
$809$	9.30648e6	0.499936	0.249968	+	0.968254i	$0.419580\pi$
$810$	0	0
$811$	3.06018e7i	1.63378i	0.576790	+	0.816892i	$0.304305\pi$
$811$	3.06018e7i	1.63378i	−0.576790	+	0.816892i	$0.695695\pi$
$812$	0	0
$813$	− 6.01241e6i	− 0.319023i
$814$	0	0
$815$	6.13338e6	0.323449
$816$	0	0
$817$	3.67036e7	1.92377
$818$	0	0
$819$	− 6.18561e6i	− 0.322235i
$820$	0	0
$821$	− 1.73699e7i	− 0.899370i	−0.893187	−	0.449685i	$-0.851536\pi$
$821$	− 1.73699e7i	− 0.899370i	0.893187	−	0.449685i	$-0.148464\pi$
$822$	0	0
$823$	−3.15925e7	−1.62586	−0.812931	−	0.582360i	$-0.802130\pi$
$823$	−3.15925e7	−1.62586	−0.812931	+	0.582360i	$0.802130\pi$
$824$	0	0
$825$	−1.54774e7	−0.791707
$826$	0	0
$827$	4.68014e6i	0.237955i	0.992897	+	0.118978i	$0.0379617\pi$
$827$	4.68014e6i	0.237955i	−0.992897	+	0.118978i	$0.962038\pi$
$828$	0	0
$829$	1.34506e6i	0.0679759i	0.999422	+	0.0339879i	$0.0108208\pi$
$829$	1.34506e6i	0.0679759i	−0.999422	+	0.0339879i	$0.989179\pi$
$830$	0	0
$831$	−3.42181e6	−0.171891
$832$	0	0
$833$	−3.73062e7	−1.86281
$834$	0	0
$835$	3.67354e6i	0.182334i
$836$	0	0
$837$	1.74302e6i	0.0859980i
$838$	0	0
$839$	−2.25209e7	−1.10454	−0.552268	−	0.833666i	$-0.686238\pi$
$839$	−2.25209e7	−1.10454	−0.552268	+	0.833666i	$0.686238\pi$
$840$	0	0
$841$	−6.27202e6	−0.305786
$842$	0	0
$843$	9.17597e6i	0.444716i
$844$	0	0
$845$	5.72449e6i	0.275800i
$846$	0	0
$847$	−6.05470e7	−2.89991
$848$	0	0
$849$	1.77871e7	0.846906
$850$	0	0
$851$	5.37318e6i	0.254336i
$852$	0	0
$853$	1.95759e7i	0.921190i	0.887610	+	0.460595i	$0.152364\pi$
$853$	1.95759e7i	0.921190i	−0.887610	+	0.460595i	$0.847636\pi$
$854$	0	0
$855$	−3.70867e6	−0.173501
$856$	0	0
$857$	2.33665e7	1.08678	0.543389	−	0.839481i	$-0.317141\pi$
$857$	2.33665e7	1.08678	0.543389	+	0.839481i	$0.317141\pi$
$858$	0	0
$859$	1.67915e7i	0.776438i	0.921567	+	0.388219i	$0.126910\pi$
$859$	1.67915e7i	0.776438i	−0.921567	+	0.388219i	$0.873090\pi$
$860$	0	0
$861$	2.39948e7i	1.10309i
$862$	0	0
$863$	2.87342e7	1.31332	0.656662	−	0.754185i	$-0.271968\pi$
$863$	2.87342e7	1.31332	0.656662	+	0.754185i	$0.271968\pi$
$864$	0	0
$865$	9.90432e6	0.450074
$866$	0	0
$867$	5.45916e6i	0.246648i
$868$	0	0
$869$	2.43654e7i	1.09452i
$870$	0	0
$871$	1.13770e7	0.508138
$872$	0	0
$873$	7.77255e6	0.345166
$874$	0	0
$875$	2.95652e7i	1.30545i
$876$	0	0
$877$	7.93487e6i	0.348370i	0.984713	+	0.174185i	$0.0557291\pi$
$877$	7.93487e6i	0.348370i	−0.984713	+	0.174185i	$0.944271\pi$
$878$	0	0
$879$	1.60796e7	0.701944
$880$	0	0
$881$	−515307.	−0.0223680	−0.0111840	−	0.999937i	$-0.503560\pi$
$881$	−515307.	−0.0223680	−0.0111840	+	0.999937i	$0.503560\pi$
$882$	0	0
$883$	− 1.31862e7i	− 0.569138i	−0.958656	−	0.284569i	$-0.908150\pi$
$883$	− 1.31862e7i	− 0.569138i	0.958656	−	0.284569i	$-0.0918505\pi$
$884$	0	0
$885$	− 5.43698e6i	− 0.233346i
$886$	0	0
$887$	−7.11666e6	−0.303716	−0.151858	−	0.988402i	$-0.548526\pi$
$887$	−7.11666e6	−0.303716	−0.151858	+	0.988402i	$0.548526\pi$
$888$	0	0
$889$	−1.63985e7	−0.695905
$890$	0	0
$891$	− 4.21174e6i	− 0.177733i
$892$	0	0
$893$	− 4.91140e7i	− 2.06099i
$894$	0	0
$895$	5.99097e6	0.250000
$896$	0	0
$897$	−1.13207e6	−0.0469778
$898$	0	0
$899$	1.23739e7i	0.510630i
$900$	0	0
$901$	6.95976e6i	0.285616i
$902$	0	0
$903$	3.67514e7	1.49987
$904$	0	0
$905$	−2.02925e6	−0.0823595
$906$	0	0
$907$	3.45645e7i	1.39512i	0.716526	+	0.697561i	$0.245731\pi$
$907$	3.45645e7i	1.39512i	−0.716526	+	0.697561i	$0.754269\pi$
$908$	0	0
$909$	1.00476e7i	0.403321i
$910$	0	0
$911$	3.63060e7	1.44938	0.724691	−	0.689074i	$-0.241983\pi$
$911$	3.63060e7	1.44938	0.724691	+	0.689074i	$0.241983\pi$
$912$	0	0
$913$	−3.25629e7	−1.29284
$914$	0	0
$915$	− 2.03998e6i	− 0.0805516i
$916$	0	0
$917$	− 2.95088e7i	− 1.15885i
$918$	0	0
$919$	1.09911e7	0.429291	0.214645	−	0.976692i	$-0.431140\pi$
$919$	1.09911e7	0.429291	0.214645	+	0.976692i	$0.431140\pi$
$920$	0	0
$921$	1.39601e7	0.542301
$922$	0	0
$923$	− 2.05646e7i	− 0.794539i
$924$	0	0
$925$	− 3.62322e7i	− 1.39233i
$926$	0	0
$927$	−6.20796e6	−0.237274
$928$	0	0
$929$	3.44048e7	1.30792	0.653958	−	0.756531i	$-0.273107\pi$
$929$	3.44048e7	1.30792	0.653958	+	0.756531i	$0.273107\pi$
$930$	0	0
$931$	− 8.96816e7i	− 3.39101i
$932$	0	0
$933$	2.50693e7i	0.942839i
$934$	0	0
$935$	−1.22265e7	−0.457376
$936$	0	0
$937$	−264943.	−0.00985833	−0.00492916	−	0.999988i	$-0.501569\pi$
$937$	−264943.	−0.00985833	−0.00492916	+	0.999988i	$0.501569\pi$
$938$	0	0
$939$	2.92148e7i	1.08128i
$940$	0	0
$941$	− 6.61265e6i	− 0.243445i	−0.992564	−	0.121723i	$-0.961158\pi$
$941$	− 6.61265e6i	− 0.243445i	0.992564	−	0.121723i	$-0.0388418\pi$
$942$	0	0
$943$	4.39147e6	0.160817
$944$	0	0
$945$	−3.71351e6	−0.135271
$946$	0	0
$947$	− 2.84346e7i	− 1.03032i	−0.857094	−	0.515160i	$-0.827732\pi$
$947$	− 2.84346e7i	− 1.03032i	0.857094	−	0.515160i	$-0.172268\pi$
$948$	0	0
$949$	4.58602e6i	0.165299i
$950$	0	0
$951$	2.25901e7	0.809967
$952$	0	0
$953$	4.59185e7	1.63778	0.818891	−	0.573950i	$-0.194589\pi$
$953$	4.59185e7	1.63778	0.818891	+	0.573950i	$0.194589\pi$
$954$	0	0
$955$	− 1.46114e7i	− 0.518422i
$956$	0	0
$957$	− 2.98996e7i	− 1.05532i
$958$	0	0
$959$	2.59924e7	0.912639
$960$	0	0
$961$	−2.29124e7	−0.800317
$962$	0	0
$963$	− 1.34290e7i	− 0.466635i
$964$	0	0
$965$	9.47507e6i	0.327540i
$966$	0	0
$967$	−2.35557e7	−0.810082	−0.405041	−	0.914299i	$-0.632743\pi$
$967$	−2.35557e7	−0.810082	−0.405041	+	0.914299i	$0.632743\pi$
$968$	0	0
$969$	1.75957e7	0.602001
$970$	0	0
$971$	6.54594e6i	0.222805i	0.993775	+	0.111402i	$0.0355342\pi$
$971$	6.54594e6i	0.222805i	−0.993775	+	0.111402i	$0.964466\pi$
$972$	0	0
$973$	− 2.70856e7i	− 0.917184i
$974$	0	0
$975$	7.63376e6	0.257174
$976$	0	0
$977$	−5.78429e7	−1.93871	−0.969357	−	0.245656i	$-0.920996\pi$
$977$	−5.78429e7	−1.93871	−0.969357	+	0.245656i	$0.920996\pi$
$978$	0	0
$979$	1.09497e7i	0.365128i
$980$	0	0
$981$	1.74670e7i	0.579488i
$982$	0	0
$983$	4.74656e7	1.56673	0.783367	−	0.621559i	$-0.213500\pi$
$983$	4.74656e7	1.56673	0.783367	+	0.621559i	$0.213500\pi$
$984$	0	0
$985$	−3.97676e6	−0.130599
$986$	0	0
$987$	− 4.91780e7i	− 1.60686i
$988$	0	0
$989$	− 6.72614e6i	− 0.218663i
$990$	0	0
$991$	−5.05840e7	−1.63617	−0.818086	−	0.575097i	$-0.804964\pi$
$991$	−5.05840e7	−1.63617	−0.818086	+	0.575097i	$0.804964\pi$
$992$	0	0
$993$	2.37246e7	0.763529
$994$	0	0
$995$	− 940397.i	− 0.0301130i
$996$	0	0
$997$	− 2.32042e7i	− 0.739314i	−0.929168	−	0.369657i	$-0.879475\pi$
$997$	− 2.32042e7i	− 0.739314i	0.929168	−	0.369657i	$-0.120525\pi$
$998$	0	0
$999$	9.85956e6	0.312567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.d.j.193.4	yes	12
4.3	odd	2	inner	384.6.d.j.193.10	yes	12
8.3	odd	2	inner	384.6.d.j.193.3	✓	12
8.5	even	2	inner	384.6.d.j.193.9	yes	12
16.3	odd	4		768.6.a.be.1.4		6
16.5	even	4		768.6.a.be.1.3		6
16.11	odd	4		768.6.a.bf.1.3		6
16.13	even	4		768.6.a.bf.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.j.193.3	✓	12	8.3	odd	2	inner
384.6.d.j.193.4	yes	12	1.1	even	1	trivial
384.6.d.j.193.9	yes	12	8.5	even	2	inner
384.6.d.j.193.10	yes	12	4.3	odd	2	inner
768.6.a.be.1.3		6	16.5	even	4
768.6.a.be.1.4		6	16.3	odd	4
768.6.a.bf.1.3		6	16.11	odd	4
768.6.a.bf.1.4		6	16.13	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} +21.1198i q^{5} +241.194 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} +21.1198i q^{5} +241.194 q^{7} -81.0000 q^{9} -641.936i q^{11} +316.614i q^{13} +190.078 q^{15} -901.822 q^{17} -2167.92i q^{19} -2170.75i q^{21} -397.284 q^{23} +2678.95 q^{25} +729.000i q^{27} +5175.25i q^{29} +2390.97 q^{31} -5777.42 q^{33} +5093.97i q^{35} -13524.8i q^{37} +2849.53 q^{39} -11053.7 q^{41} +16930.3i q^{43} -1710.70i q^{45} +22654.9 q^{47} +41367.6 q^{49} +8116.40i q^{51} -7717.44i q^{53} +13557.6 q^{55} -19511.3 q^{57} -28603.9i q^{59} -10732.3i q^{61} -19536.7 q^{63} -6686.83 q^{65} -35933.3i q^{67} +3575.56i q^{69} -64951.4 q^{71} +14484.6 q^{73} -24110.6i q^{75} -154831. i q^{77} -37956.2 q^{79} +6561.00 q^{81} -50726.1i q^{83} -19046.3i q^{85} +46577.2 q^{87} -17057.3 q^{89} +76365.5i q^{91} -21518.8i q^{93} +45786.1 q^{95} -95957.4 q^{97} +51996.8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 972 q^{9}+O(q^{10}) \) 12 * q - 972 * q^9	\( 12 q - 972 q^{9} - 4888 q^{17} - 18660 q^{25} - 720 q^{33} - 25096 q^{41} + 135564 q^{49} - 13104 q^{57} + 254272 q^{65} - 187032 q^{73} + 78732 q^{81} - 575544 q^{89} + 93864 q^{97}+O(q^{100}) \) 12 * q - 972 * q^9 - 4888 * q^17 - 18660 * q^25 - 720 * q^33 - 25096 * q^41 + 135564 * q^49 - 13104 * q^57 + 254272 * q^65 - 187032 * q^73 + 78732 * q^81 - 575544 * q^89 + 93864 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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