LMFDB - Embedded newform 384.6.d.j.193.12

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			193.12
Root			$4.65977i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.6.d.j.193.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} +107.066i q^{5} +150.154 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} +107.066i q^{5} +150.154 q^{7} -81.0000 q^{9} -221.870i q^{11} -504.463i q^{13} -963.592 q^{15} -1982.75 q^{17} -2368.23i q^{19} +1351.38i q^{21} +1143.26 q^{23} -8338.09 q^{25} -729.000i q^{27} -3900.86i q^{29} -9007.16 q^{31} +1996.83 q^{33} +16076.3i q^{35} +448.595i q^{37} +4540.17 q^{39} +9899.43 q^{41} -4624.59i q^{43} -8672.33i q^{45} -20086.5 q^{47} +5739.09 q^{49} -17844.8i q^{51} -17426.0i q^{53} +23754.7 q^{55} +21314.1 q^{57} +35967.4i q^{59} -12519.7i q^{61} -12162.4 q^{63} +54010.8 q^{65} -35342.5i q^{67} +10289.3i q^{69} -1761.27 q^{71} -21143.9 q^{73} -75042.8i q^{75} -33314.6i q^{77} -46562.3 q^{79} +6561.00 q^{81} +103022. i q^{83} -212285. i q^{85} +35107.7 q^{87} -111679. q^{89} -75747.0i q^{91} -81064.4i q^{93} +253557. q^{95} -39002.5 q^{97} +17971.5i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	107.066i	1.91525i	0.288015	+	0.957626i	$0.407005\pi$
$5$	107.066i	1.91525i	−0.288015	+	0.957626i	$0.592995\pi$
$6$	0	0
$7$	150.154	1.15822	0.579109	−	0.815250i	$-0.303400\pi$
$7$	150.154	1.15822	0.579109	+	0.815250i	$0.303400\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	− 221.870i	− 0.552863i	−0.961034	−	0.276432i	$-0.910848\pi$
$11$	− 221.870i	− 0.552863i	0.961034	−	0.276432i	$-0.0891519\pi$
$12$	0	0
$13$	− 504.463i	− 0.827887i	−0.910302	−	0.413944i	$-0.864151\pi$
$13$	− 504.463i	− 0.827887i	0.910302	−	0.413944i	$-0.135849\pi$
$14$	0	0
$15$	−963.592	−1.10577
$16$	0	0
$17$	−1982.75	−1.66397	−0.831986	−	0.554796i	$-0.812796\pi$
$17$	−1982.75	−1.66397	−0.831986	+	0.554796i	$0.812796\pi$
$18$	0	0
$19$	− 2368.23i	− 1.50501i	−0.658584	−	0.752507i	$-0.728844\pi$
$19$	− 2368.23i	− 1.50501i	0.658584	−	0.752507i	$-0.271156\pi$
$20$	0	0
$21$	1351.38i	0.668698i
$22$	0	0
$23$	1143.26	0.450634	0.225317	−	0.974286i	$-0.427658\pi$
$23$	1143.26	0.450634	0.225317	+	0.974286i	$0.427658\pi$
$24$	0	0
$25$	−8338.09	−2.66819
$26$	0	0
$27$	− 729.000i	− 0.192450i
$28$	0	0
$29$	− 3900.86i	− 0.861321i	−0.902514	−	0.430660i	$-0.858281\pi$
$29$	− 3900.86i	− 0.861321i	0.902514	−	0.430660i	$-0.141719\pi$
$30$	0	0
$31$	−9007.16	−1.68339	−0.841693	−	0.539957i	$-0.818441\pi$
$31$	−9007.16	−1.68339	−0.841693	+	0.539957i	$0.818441\pi$
$32$	0	0
$33$	1996.83	0.319196
$34$	0	0
$35$	16076.3i	2.21828i
$36$	0	0
$37$	448.595i	0.0538704i	0.999637	+	0.0269352i	$0.00857477\pi$
$37$	448.595i	0.0538704i	−0.999637	+	0.0269352i	$0.991425\pi$
$38$	0	0
$39$	4540.17	0.477981
$40$	0	0
$41$	9899.43	0.919709	0.459855	−	0.887994i	$-0.347901\pi$
$41$	9899.43	0.919709	0.459855	+	0.887994i	$0.347901\pi$
$42$	0	0
$43$	− 4624.59i	− 0.381419i	−0.981647	−	0.190709i	$-0.938921\pi$
$43$	− 4624.59i	− 0.381419i	0.981647	−	0.190709i	$-0.0610788\pi$
$44$	0	0
$45$	− 8672.33i	− 0.638417i
$46$	0	0
$47$	−20086.5	−1.32635	−0.663176	−	0.748463i	$-0.730792\pi$
$47$	−20086.5	−1.32635	−0.663176	+	0.748463i	$0.730792\pi$
$48$	0	0
$49$	5739.09	0.341470
$50$	0	0
$51$	− 17844.8i	− 0.960695i
$52$	0	0
$53$	− 17426.0i	− 0.852135i	−0.904691	−	0.426068i	$-0.859899\pi$
$53$	− 17426.0i	− 0.852135i	0.904691	−	0.426068i	$-0.140101\pi$
$54$	0	0
$55$	23754.7	1.05887
$56$	0	0
$57$	21314.1	0.868921
$58$	0	0
$59$	35967.4i	1.34517i	0.740018	+	0.672587i	$0.234817\pi$
$59$	35967.4i	1.34517i	−0.740018	+	0.672587i	$0.765183\pi$
$60$	0	0
$61$	− 12519.7i	− 0.430795i	−0.976526	−	0.215397i	$-0.930895\pi$
$61$	− 12519.7i	− 0.430795i	0.976526	−	0.215397i	$-0.0691047\pi$
$62$	0	0
$63$	−12162.4	−0.386073
$64$	0	0
$65$	54010.8	1.58561
$66$	0	0
$67$	− 35342.5i	− 0.961855i	−0.876760	−	0.480928i	$-0.840300\pi$
$67$	− 35342.5i	− 0.961855i	0.876760	−	0.480928i	$-0.159700\pi$
$68$	0	0
$69$	10289.3i	0.260173i
$70$	0	0
$71$	−1761.27	−0.0414649	−0.0207325	−	0.999785i	$-0.506600\pi$
$71$	−1761.27	−0.0414649	−0.0207325	+	0.999785i	$0.506600\pi$
$72$	0	0
$73$	−21143.9	−0.464385	−0.232193	−	0.972670i	$-0.574590\pi$
$73$	−21143.9	−0.464385	−0.232193	+	0.972670i	$0.574590\pi$
$74$	0	0
$75$	− 75042.8i	− 1.54048i
$76$	0	0
$77$	− 33314.6i	− 0.640337i
$78$	0	0
$79$	−46562.3	−0.839395	−0.419697	−	0.907664i	$-0.637864\pi$
$79$	−46562.3	−0.839395	−0.419697	+	0.907664i	$0.637864\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	103022.i	1.64147i	0.571309	+	0.820735i	$0.306436\pi$
$83$	103022.i	1.64147i	−0.571309	+	0.820735i	$0.693564\pi$
$84$	0	0
$85$	− 212285.i	− 3.18693i
$86$	0	0
$87$	35107.7	0.497284
$88$	0	0
$89$	−111679.	−1.49451	−0.747253	−	0.664540i	$-0.768628\pi$
$89$	−111679.	−1.49451	−0.747253	+	0.664540i	$0.768628\pi$
$90$	0	0
$91$	− 75747.0i	− 0.958874i
$92$	0	0
$93$	− 81064.4i	− 0.971903i
$94$	0	0
$95$	253557.	2.88248
$96$	0	0
$97$	−39002.5	−0.420884	−0.210442	−	0.977606i	$-0.567490\pi$
$97$	−39002.5	−0.420884	−0.210442	+	0.977606i	$0.567490\pi$
$98$	0	0
$99$	17971.5i	0.184288i
$100$	0	0
$101$	− 61681.4i	− 0.601660i	−0.953678	−	0.300830i	$-0.902736\pi$
$101$	− 61681.4i	− 0.601660i	0.953678	−	0.300830i	$-0.0972636\pi$
$102$	0	0
$103$	−47754.1	−0.443524	−0.221762	−	0.975101i	$-0.571181\pi$
$103$	−47754.1	−0.443524	−0.221762	+	0.975101i	$0.571181\pi$
$104$	0	0
$105$	−144687.	−1.28072
$106$	0	0
$107$	70099.5i	0.591910i	0.955202	+	0.295955i	$0.0956378\pi$
$107$	70099.5i	0.591910i	−0.955202	+	0.295955i	$0.904362\pi$
$108$	0	0
$109$	113761.i	0.917121i	0.888663	+	0.458561i	$0.151635\pi$
$109$	113761.i	0.917121i	−0.888663	+	0.458561i	$0.848365\pi$
$110$	0	0
$111$	−4037.35	−0.0311021
$112$	0	0
$113$	−162065.	−1.19397	−0.596985	−	0.802252i	$-0.703635\pi$
$113$	−162065.	−1.19397	−0.596985	+	0.802252i	$0.703635\pi$
$114$	0	0
$115$	122404.i	0.863077i
$116$	0	0
$117$	40861.5i	0.275962i
$118$	0	0
$119$	−297717.	−1.92724
$120$	0	0
$121$	111824.	0.694342
$122$	0	0
$123$	89094.9i	0.530994i
$124$	0	0
$125$	− 558144.i	− 3.19500i
$126$	0	0
$127$	208540.	1.14731	0.573654	−	0.819097i	$-0.305525\pi$
$127$	208540.	1.14731	0.573654	+	0.819097i	$0.305525\pi$
$128$	0	0
$129$	41621.3	0.220212
$130$	0	0
$131$	− 25436.7i	− 0.129504i	−0.997901	−	0.0647520i	$-0.979374\pi$
$131$	− 25436.7i	− 0.129504i	0.997901	−	0.0647520i	$-0.0206256\pi$
$132$	0	0
$133$	− 355599.i	− 1.74314i
$134$	0	0
$135$	78051.0	0.368590
$136$	0	0
$137$	249283.	1.13473	0.567363	−	0.823468i	$-0.307964\pi$
$137$	249283.	1.13473	0.567363	+	0.823468i	$0.307964\pi$
$138$	0	0
$139$	− 199937.i	− 0.877722i	−0.898555	−	0.438861i	$-0.855382\pi$
$139$	− 199937.i	− 0.877722i	0.898555	−	0.438861i	$-0.144618\pi$
$140$	0	0
$141$	− 180778.i	− 0.765770i
$142$	0	0
$143$	−111926.	−0.457709
$144$	0	0
$145$	417648.	1.64965
$146$	0	0
$147$	51651.8i	0.197148i
$148$	0	0
$149$	183640.i	0.677645i	0.940850	+	0.338823i	$0.110029\pi$
$149$	183640.i	0.677645i	−0.940850	+	0.338823i	$0.889971\pi$
$150$	0	0
$151$	−319160.	−1.13911	−0.569555	−	0.821953i	$-0.692884\pi$
$151$	−319160.	−1.13911	−0.569555	+	0.821953i	$0.692884\pi$
$152$	0	0
$153$	160603.	0.554658
$154$	0	0
$155$	− 964359.i	− 3.22411i
$156$	0	0
$157$	− 105413.i	− 0.341308i	−0.985331	−	0.170654i	$-0.945412\pi$
$157$	− 105413.i	− 0.341308i	0.985331	−	0.170654i	$-0.0545881\pi$
$158$	0	0
$159$	156834.	0.491981
$160$	0	0
$161$	171664.	0.521932
$162$	0	0
$163$	511453.i	1.50777i	0.657004	+	0.753887i	$0.271824\pi$
$163$	511453.i	1.50777i	−0.657004	+	0.753887i	$0.728176\pi$
$164$	0	0
$165$	213793.i	0.611340i
$166$	0	0
$167$	585529.	1.62464	0.812320	−	0.583212i	$-0.198204\pi$
$167$	585529.	1.62464	0.812320	+	0.583212i	$0.198204\pi$
$168$	0	0
$169$	116810.	0.314602
$170$	0	0
$171$	191827.i	0.501672i
$172$	0	0
$173$	− 149097.i	− 0.378751i	−0.981905	−	0.189375i	$-0.939354\pi$
$173$	− 149097.i	− 0.378751i	0.981905	−	0.189375i	$-0.0606463\pi$
$174$	0	0
$175$	−1.25199e6	−3.09035
$176$	0	0
$177$	−323706.	−0.776637
$178$	0	0
$179$	− 101335.i	− 0.236390i	−0.992990	−	0.118195i	$-0.962289\pi$
$179$	− 101335.i	− 0.236390i	0.992990	−	0.118195i	$-0.0377108\pi$
$180$	0	0
$181$	− 709024.i	− 1.60866i	−0.594182	−	0.804330i	$-0.702524\pi$
$181$	− 709024.i	− 1.60866i	0.594182	−	0.804330i	$-0.297476\pi$
$182$	0	0
$183$	112678.	0.248719
$184$	0	0
$185$	−48029.2	−0.103175
$186$	0	0
$187$	439914.i	0.919950i
$188$	0	0
$189$	− 109462.i	− 0.222899i
$190$	0	0
$191$	54546.3	0.108189	0.0540943	−	0.998536i	$-0.482773\pi$
$191$	54546.3	0.108189	0.0540943	+	0.998536i	$0.482773\pi$
$192$	0	0
$193$	−831213.	−1.60627	−0.803136	−	0.595796i	$-0.796837\pi$
$193$	−831213.	−1.60627	−0.803136	+	0.595796i	$0.796837\pi$
$194$	0	0
$195$	486097.i	0.915454i
$196$	0	0
$197$	− 171308.i	− 0.314495i	−0.987559	−	0.157247i	$-0.949738\pi$
$197$	− 171308.i	− 0.314495i	0.987559	−	0.157247i	$-0.0502620\pi$
$198$	0	0
$199$	119632.	0.214149	0.107075	−	0.994251i	$-0.465852\pi$
$199$	119632.	0.214149	0.107075	+	0.994251i	$0.465852\pi$
$200$	0	0
$201$	318082.	0.555327
$202$	0	0
$203$	− 585727.i	− 0.997598i
$204$	0	0
$205$	1.05989e6i	1.76148i
$206$	0	0
$207$	−92603.7	−0.150211
$208$	0	0
$209$	−525441.	−0.832067
$210$	0	0
$211$	877085.i	1.35624i	0.734952	+	0.678119i	$0.237204\pi$
$211$	877085.i	1.35624i	−0.734952	+	0.678119i	$0.762796\pi$
$212$	0	0
$213$	− 15851.5i	− 0.0239398i
$214$	0	0
$215$	495136.	0.730513
$216$	0	0
$217$	−1.35246e6	−1.94973
$218$	0	0
$219$	− 190295.i	− 0.268113i
$220$	0	0
$221$	1.00023e6i	1.37758i
$222$	0	0
$223$	194513.	0.261931	0.130965	−	0.991387i	$-0.458192\pi$
$223$	194513.	0.261931	0.130965	+	0.991387i	$0.458192\pi$
$224$	0	0
$225$	675385.	0.889397
$226$	0	0
$227$	597533.i	0.769658i	0.922988	+	0.384829i	$0.125740\pi$
$227$	597533.i	0.769658i	−0.922988	+	0.384829i	$0.874260\pi$
$228$	0	0
$229$	794517.i	1.00119i	0.865683	+	0.500593i	$0.166885\pi$
$229$	794517.i	1.00119i	−0.865683	+	0.500593i	$0.833115\pi$
$230$	0	0
$231$	299832.	0.369698
$232$	0	0
$233$	437908.	0.528437	0.264218	−	0.964463i	$-0.414886\pi$
$233$	437908.	0.528437	0.264218	+	0.964463i	$0.414886\pi$
$234$	0	0
$235$	− 2.15057e6i	− 2.54030i
$236$	0	0
$237$	− 419060.i	− 0.484625i
$238$	0	0
$239$	−649053.	−0.734997	−0.367499	−	0.930024i	$-0.619786\pi$
$239$	−649053.	−0.734997	−0.367499	+	0.930024i	$0.619786\pi$
$240$	0	0
$241$	−427372.	−0.473984	−0.236992	−	0.971512i	$-0.576162\pi$
$241$	−427372.	−0.473984	−0.236992	+	0.971512i	$0.576162\pi$
$242$	0	0
$243$	59049.0i	0.0641500i
$244$	0	0
$245$	614460.i	0.654001i
$246$	0	0
$247$	−1.19469e6	−1.24598
$248$	0	0
$249$	−927194.	−0.947703
$250$	0	0
$251$	514388.i	0.515355i	0.966231	+	0.257678i	$0.0829572\pi$
$251$	514388.i	0.515355i	−0.966231	+	0.257678i	$0.917043\pi$
$252$	0	0
$253$	− 253655.i	− 0.249139i
$254$	0	0
$255$	1.91057e6	1.83997
$256$	0	0
$257$	−594709.	−0.561657	−0.280829	−	0.959758i	$-0.590609\pi$
$257$	−594709.	−0.561657	−0.280829	+	0.959758i	$0.590609\pi$
$258$	0	0
$259$	67358.1i	0.0623937i
$260$	0	0
$261$	315969.i	0.287107i
$262$	0	0
$263$	2.06496e6	1.84087	0.920434	−	0.390898i	$-0.127836\pi$
$263$	2.06496e6	1.84087	0.920434	+	0.390898i	$0.127836\pi$
$264$	0	0
$265$	1.86573e6	1.63205
$266$	0	0
$267$	− 1.00511e6i	− 0.862853i
$268$	0	0
$269$	667420.i	0.562365i	0.959654	+	0.281183i	$0.0907266\pi$
$269$	667420.i	0.562365i	−0.959654	+	0.281183i	$0.909273\pi$
$270$	0	0
$271$	−692916.	−0.573136	−0.286568	−	0.958060i	$-0.592514\pi$
$271$	−692916.	−0.573136	−0.286568	+	0.958060i	$0.592514\pi$
$272$	0	0
$273$	681723.	0.553606
$274$	0	0
$275$	1.84998e6i	1.47514i
$276$	0	0
$277$	2.11671e6i	1.65753i	0.559595	+	0.828767i	$0.310957\pi$
$277$	2.11671e6i	1.65753i	−0.559595	+	0.828767i	$0.689043\pi$
$278$	0	0
$279$	729580.	0.561128
$280$	0	0
$281$	1.82468e6	1.37854	0.689271	−	0.724504i	$-0.257931\pi$
$281$	1.82468e6	1.37854	0.689271	+	0.724504i	$0.257931\pi$
$282$	0	0
$283$	− 1.18653e6i	− 0.880665i	−0.897835	−	0.440333i	$-0.854861\pi$
$283$	− 1.18653e6i	− 0.880665i	0.897835	−	0.440333i	$-0.145139\pi$
$284$	0	0
$285$	2.28201e6i	1.66420i
$286$	0	0
$287$	1.48643e6	1.06522
$288$	0	0
$289$	2.51145e6	1.76881
$290$	0	0
$291$	− 351022.i	− 0.242998i
$292$	0	0
$293$	183945.i	0.125175i	0.998039	+	0.0625877i	$0.0199353\pi$
$293$	183945.i	0.125175i	−0.998039	+	0.0625877i	$0.980065\pi$
$294$	0	0
$295$	−3.85088e6	−2.57635
$296$	0	0
$297$	−161744.	−0.106399
$298$	0	0
$299$	− 576730.i	− 0.373074i
$300$	0	0
$301$	− 694399.i	− 0.441766i
$302$	0	0
$303$	555133.	0.347368
$304$	0	0
$305$	1.34044e6	0.825080
$306$	0	0
$307$	− 700476.i	− 0.424177i	−0.977250	−	0.212089i	$-0.931973\pi$
$307$	− 700476.i	− 0.424177i	0.977250	−	0.212089i	$-0.0680265\pi$
$308$	0	0
$309$	− 429787.i	− 0.256069i
$310$	0	0
$311$	−1.32274e6	−0.775483	−0.387741	−	0.921768i	$-0.626745\pi$
$311$	−1.32274e6	−0.775483	−0.387741	+	0.921768i	$0.626745\pi$
$312$	0	0
$313$	−118214.	−0.0682036	−0.0341018	−	0.999418i	$-0.510857\pi$
$313$	−118214.	−0.0682036	−0.0341018	+	0.999418i	$0.510857\pi$
$314$	0	0
$315$	− 1.30218e6i	− 0.739427i
$316$	0	0
$317$	2.55712e6i	1.42923i	0.699516	+	0.714617i	$0.253399\pi$
$317$	2.55712e6i	1.42923i	−0.699516	+	0.714617i	$0.746601\pi$
$318$	0	0
$319$	−865485.	−0.476193
$320$	0	0
$321$	−630896.	−0.341739
$322$	0	0
$323$	4.69562e6i	2.50430i
$324$	0	0
$325$	4.20626e6i	2.20896i
$326$	0	0
$327$	−1.02385e6	−0.529500
$328$	0	0
$329$	−3.01605e6	−1.53621
$330$	0	0
$331$	2.66796e6i	1.33847i	0.743049	+	0.669237i	$0.233379\pi$
$331$	2.66796e6i	1.33847i	−0.743049	+	0.669237i	$0.766621\pi$
$332$	0	0
$333$	− 36336.2i	− 0.0179568i
$334$	0	0
$335$	3.78397e6	1.84220
$336$	0	0
$337$	−352164.	−0.168916	−0.0844580	−	0.996427i	$-0.526916\pi$
$337$	−352164.	−0.168916	−0.0844580	+	0.996427i	$0.526916\pi$
$338$	0	0
$339$	− 1.45859e6i	− 0.689339i
$340$	0	0
$341$	1.99842e6i	0.930682i
$342$	0	0
$343$	−1.66189e6	−0.762722
$344$	0	0
$345$	−1.10163e6	−0.498298
$346$	0	0
$347$	1.65583e6i	0.738230i	0.929384	+	0.369115i	$0.120339\pi$
$347$	1.65583e6i	0.738230i	−0.929384	+	0.369115i	$0.879661\pi$
$348$	0	0
$349$	− 1.61505e6i	− 0.709778i	−0.934908	−	0.354889i	$-0.884519\pi$
$349$	− 1.61505e6i	− 0.709778i	0.934908	−	0.354889i	$-0.115481\pi$
$350$	0	0
$351$	−367754.	−0.159327
$352$	0	0
$353$	−2.16955e6	−0.926687	−0.463343	−	0.886179i	$-0.653350\pi$
$353$	−2.16955e6	−0.926687	−0.463343	+	0.886179i	$0.653350\pi$
$354$	0	0
$355$	− 188572.i	− 0.0794158i
$356$	0	0
$357$	− 2.67946e6i	− 1.11269i
$358$	0	0
$359$	285544.	0.116933	0.0584665	−	0.998289i	$-0.481379\pi$
$359$	285544.	0.116933	0.0584665	+	0.998289i	$0.481379\pi$
$360$	0	0
$361$	−3.13244e6	−1.26507
$362$	0	0
$363$	1.00642e6i	0.400879i
$364$	0	0
$365$	− 2.26379e6i	− 0.889414i
$366$	0	0
$367$	−3.77489e6	−1.46298	−0.731491	−	0.681851i	$-0.761175\pi$
$367$	−3.77489e6	−1.46298	−0.731491	+	0.681851i	$0.761175\pi$
$368$	0	0
$369$	−801854.	−0.306570
$370$	0	0
$371$	− 2.61658e6i	− 0.986959i
$372$	0	0
$373$	37031.8i	0.0137817i	0.999976	+	0.00689085i	$0.00219344\pi$
$373$	37031.8i	0.0137817i	−0.999976	+	0.00689085i	$0.997807\pi$
$374$	0	0
$375$	5.02330e6	1.84464
$376$	0	0
$377$	−1.96784e6	−0.713077
$378$	0	0
$379$	− 4.66986e6i	− 1.66996i	−0.550280	−	0.834980i	$-0.685479\pi$
$379$	− 4.66986e6i	− 1.66996i	0.550280	−	0.834980i	$-0.314521\pi$
$380$	0	0
$381$	1.87686e6i	0.662399i
$382$	0	0
$383$	−2.20726e6	−0.768878	−0.384439	−	0.923150i	$-0.625605\pi$
$383$	−2.20726e6	−0.768878	−0.384439	+	0.923150i	$0.625605\pi$
$384$	0	0
$385$	3.56686e6	1.22641
$386$	0	0
$387$	374592.i	0.127140i
$388$	0	0
$389$	− 3.43408e6i	− 1.15063i	−0.817931	−	0.575316i	$-0.804879\pi$
$389$	− 3.43408e6i	− 1.15063i	0.817931	−	0.575316i	$-0.195121\pi$
$390$	0	0
$391$	−2.26679e6	−0.749842
$392$	0	0
$393$	228931.	0.0747691
$394$	0	0
$395$	− 4.98523e6i	− 1.60765i
$396$	0	0
$397$	2.14935e6i	0.684432i	0.939621	+	0.342216i	$0.111177\pi$
$397$	2.14935e6i	0.684432i	−0.939621	+	0.342216i	$0.888823\pi$
$398$	0	0
$399$	3.20039e6	1.00640
$400$	0	0
$401$	−5.45334e6	−1.69356	−0.846782	−	0.531940i	$-0.821463\pi$
$401$	−5.45334e6	−1.69356	−0.846782	+	0.531940i	$0.821463\pi$
$402$	0	0
$403$	4.54378e6i	1.39365i
$404$	0	0
$405$	702459.i	0.212806i
$406$	0	0
$407$	99529.9	0.0297829
$408$	0	0
$409$	−2.77610e6	−0.820591	−0.410296	−	0.911953i	$-0.634574\pi$
$409$	−2.77610e6	−0.820591	−0.410296	+	0.911953i	$0.634574\pi$
$410$	0	0
$411$	2.24354e6i	0.655134i
$412$	0	0
$413$	5.40063e6i	1.55801i
$414$	0	0
$415$	−1.10301e7	−3.14383
$416$	0	0
$417$	1.79944e6	0.506753
$418$	0	0
$419$	− 5.88907e6i	− 1.63875i	−0.573261	−	0.819373i	$-0.694322\pi$
$419$	− 5.88907e6i	− 1.63875i	0.573261	−	0.819373i	$-0.305678\pi$
$420$	0	0
$421$	− 2.20288e6i	− 0.605740i	−0.953032	−	0.302870i	$-0.902055\pi$
$421$	− 2.20288e6i	− 0.605740i	0.953032	−	0.302870i	$-0.0979449\pi$
$422$	0	0
$423$	1.62700e6	0.442118
$424$	0	0
$425$	1.65324e7	4.43980
$426$	0	0
$427$	− 1.87988e6i	− 0.498954i
$428$	0	0
$429$	− 1.00733e6i	− 0.264258i
$430$	0	0
$431$	2.86779e6	0.743626	0.371813	−	0.928308i	$-0.378736\pi$
$431$	2.86779e6	0.743626	0.371813	+	0.928308i	$0.378736\pi$
$432$	0	0
$433$	−1.24865e6	−0.320052	−0.160026	−	0.987113i	$-0.551158\pi$
$433$	−1.24865e6	−0.320052	−0.160026	+	0.987113i	$0.551158\pi$
$434$	0	0
$435$	3.75884e6i	0.952424i
$436$	0	0
$437$	− 2.70750e6i	− 0.678210i
$438$	0	0
$439$	5.67269e6	1.40484	0.702422	−	0.711761i	$-0.252102\pi$
$439$	5.67269e6	1.40484	0.702422	+	0.711761i	$0.252102\pi$
$440$	0	0
$441$	−464866.	−0.113823
$442$	0	0
$443$	4.54963e6i	1.10145i	0.834685	+	0.550727i	$0.185649\pi$
$443$	4.54963e6i	1.10145i	−0.834685	+	0.550727i	$0.814351\pi$
$444$	0	0
$445$	− 1.19570e7i	− 2.86235i
$446$	0	0
$447$	−1.65276e6	−0.391239
$448$	0	0
$449$	913805.	0.213913	0.106957	−	0.994264i	$-0.465889\pi$
$449$	913805.	0.213913	0.106957	+	0.994264i	$0.465889\pi$
$450$	0	0
$451$	− 2.19639e6i	− 0.508474i
$452$	0	0
$453$	− 2.87244e6i	− 0.657665i
$454$	0	0
$455$	8.10991e6	1.83649
$456$	0	0
$457$	−1.33073e6	−0.298056	−0.149028	−	0.988833i	$-0.547614\pi$
$457$	−1.33073e6	−0.298056	−0.149028	+	0.988833i	$0.547614\pi$
$458$	0	0
$459$	1.44543e6i	0.320232i
$460$	0	0
$461$	− 4.20403e6i	− 0.921327i	−0.887575	−	0.460663i	$-0.847612\pi$
$461$	− 4.20403e6i	− 0.921327i	0.887575	−	0.460663i	$-0.152388\pi$
$462$	0	0
$463$	5.44968e6	1.18146	0.590729	−	0.806870i	$-0.298840\pi$
$463$	5.44968e6	1.18146	0.590729	+	0.806870i	$0.298840\pi$
$464$	0	0
$465$	8.67923e6	1.86144
$466$	0	0
$467$	− 6.16082e6i	− 1.30721i	−0.756835	−	0.653606i	$-0.773255\pi$
$467$	− 6.16082e6i	− 1.30721i	0.756835	−	0.653606i	$-0.226745\pi$
$468$	0	0
$469$	− 5.30680e6i	− 1.11404i
$470$	0	0
$471$	948721.	0.197055
$472$	0	0
$473$	−1.02606e6	−0.210872
$474$	0	0
$475$	1.97466e7i	4.01566i
$476$	0	0
$477$	1.41151e6i	0.284045i
$478$	0	0
$479$	−1.41659e6	−0.282102	−0.141051	−	0.990002i	$-0.545048\pi$
$479$	−1.41659e6	−0.282102	−0.141051	+	0.990002i	$0.545048\pi$
$480$	0	0
$481$	226300.	0.0445986
$482$	0	0
$483$	1.54497e6i	0.301338i
$484$	0	0
$485$	− 4.17583e6i	− 0.806099i
$486$	0	0
$487$	−5.72820e6	−1.09445	−0.547225	−	0.836985i	$-0.684316\pi$
$487$	−5.72820e6	−1.09445	−0.547225	+	0.836985i	$0.684316\pi$
$488$	0	0
$489$	−4.60307e6	−0.870514
$490$	0	0
$491$	− 1.89913e6i	− 0.355509i	−0.984075	−	0.177755i	$-0.943117\pi$
$491$	− 1.89913e6i	− 0.355509i	0.984075	−	0.177755i	$-0.0568833\pi$
$492$	0	0
$493$	7.73443e6i	1.43321i
$494$	0	0
$495$	−1.92413e6	−0.352957
$496$	0	0
$497$	−264461.	−0.0480254
$498$	0	0
$499$	− 6.03047e6i	− 1.08418i	−0.840322	−	0.542088i	$-0.817634\pi$
$499$	− 6.03047e6i	− 1.08418i	0.840322	−	0.542088i	$-0.182366\pi$
$500$	0	0
$501$	5.26976e6i	0.937986i
$502$	0	0
$503$	−1.73041e6	−0.304951	−0.152475	−	0.988307i	$-0.548724\pi$
$503$	−1.73041e6	−0.304951	−0.152475	+	0.988307i	$0.548724\pi$
$504$	0	0
$505$	6.60397e6	1.15233
$506$	0	0
$507$	1.05129e6i	0.181636i
$508$	0	0
$509$	− 1.10086e7i	− 1.88337i	−0.336496	−	0.941685i	$-0.609242\pi$
$509$	− 1.10086e7i	− 1.88337i	0.336496	−	0.941685i	$-0.390758\pi$
$510$	0	0
$511$	−3.17483e6	−0.537859
$512$	0	0
$513$	−1.72644e6	−0.289640
$514$	0	0
$515$	− 5.11283e6i	− 0.849461i
$516$	0	0
$517$	4.45659e6i	0.733292i
$518$	0	0
$519$	1.34187e6	0.218672
$520$	0	0
$521$	−816327.	−0.131756	−0.0658779	−	0.997828i	$-0.520985\pi$
$521$	−816327.	−0.131756	−0.0658779	+	0.997828i	$0.520985\pi$
$522$	0	0
$523$	− 8.59191e6i	− 1.37352i	−0.726883	−	0.686761i	$-0.759032\pi$
$523$	− 8.59191e6i	− 1.37352i	0.726883	−	0.686761i	$-0.240968\pi$
$524$	0	0
$525$	− 1.12679e7i	− 1.78421i
$526$	0	0
$527$	1.78590e7	2.80111
$528$	0	0
$529$	−5.12931e6	−0.796929
$530$	0	0
$531$	− 2.91336e6i	− 0.448391i
$532$	0	0
$533$	− 4.99390e6i	− 0.761416i
$534$	0	0
$535$	−7.50526e6	−1.13366
$536$	0	0
$537$	912019.	0.136480
$538$	0	0
$539$	− 1.27333e6i	− 0.188786i
$540$	0	0
$541$	− 9.94402e6i	− 1.46073i	−0.683059	−	0.730363i	$-0.739351\pi$
$541$	− 9.94402e6i	− 1.46073i	0.683059	−	0.730363i	$-0.260649\pi$
$542$	0	0
$543$	6.38122e6	0.928761
$544$	0	0
$545$	−1.21799e7	−1.75652
$546$	0	0
$547$	9.99790e6i	1.42870i	0.699789	+	0.714349i	$0.253277\pi$
$547$	9.99790e6i	1.42870i	−0.699789	+	0.714349i	$0.746723\pi$
$548$	0	0
$549$	1.01410e6i	0.143598i
$550$	0	0
$551$	−9.23814e6	−1.29630
$552$	0	0
$553$	−6.99149e6	−0.972203
$554$	0	0
$555$	− 432263.i	− 0.0595683i
$556$	0	0
$557$	− 9.35849e6i	− 1.27811i	−0.769161	−	0.639055i	$-0.779326\pi$
$557$	− 9.35849e6i	− 1.27811i	0.769161	−	0.639055i	$-0.220674\pi$
$558$	0	0
$559$	−2.33294e6	−0.315772
$560$	0	0
$561$	−3.95923e6	−0.531133
$562$	0	0
$563$	− 768517.i	− 0.102184i	−0.998694	−	0.0510920i	$-0.983730\pi$
$563$	− 768517.i	− 0.102184i	0.998694	−	0.0510920i	$-0.0162702\pi$
$564$	0	0
$565$	− 1.73516e7i	− 2.28675i
$566$	0	0
$567$	985157.	0.128691
$568$	0	0
$569$	2.54628e6	0.329704	0.164852	−	0.986318i	$-0.447285\pi$
$569$	2.54628e6	0.329704	0.164852	+	0.986318i	$0.447285\pi$
$570$	0	0
$571$	5.88507e6i	0.755373i	0.925934	+	0.377686i	$0.123280\pi$
$571$	5.88507e6i	0.755373i	−0.925934	+	0.377686i	$0.876720\pi$
$572$	0	0
$573$	490917.i	0.0624628i
$574$	0	0
$575$	−9.53257e6	−1.20238
$576$	0	0
$577$	−4.66115e6	−0.582846	−0.291423	−	0.956594i	$-0.594129\pi$
$577$	−4.66115e6	−0.582846	−0.291423	+	0.956594i	$0.594129\pi$
$578$	0	0
$579$	− 7.48092e6i	− 0.927381i
$580$	0	0
$581$	1.54691e7i	1.90118i
$582$	0	0
$583$	−3.86632e6	−0.471114
$584$	0	0
$585$	−4.37487e6	−0.528538
$586$	0	0
$587$	1.05639e7i	1.26541i	0.774395	+	0.632703i	$0.218054\pi$
$587$	1.05639e7i	1.26541i	−0.774395	+	0.632703i	$0.781946\pi$
$588$	0	0
$589$	2.13311e7i	2.53352i
$590$	0	0
$591$	1.54178e6	0.181573
$592$	0	0
$593$	−9.02809e6	−1.05429	−0.527144	−	0.849776i	$-0.676737\pi$
$593$	−9.02809e6	−1.05429	−0.527144	+	0.849776i	$0.676737\pi$
$594$	0	0
$595$	− 3.18754e7i	− 3.69116i
$596$	0	0
$597$	1.07669e6i	0.123639i
$598$	0	0
$599$	−5.98892e6	−0.681996	−0.340998	−	0.940064i	$-0.610765\pi$
$599$	−5.98892e6	−0.681996	−0.340998	+	0.940064i	$0.610765\pi$
$600$	0	0
$601$	−5.66239e6	−0.639460	−0.319730	−	0.947509i	$-0.603592\pi$
$601$	−5.66239e6	−0.639460	−0.319730	+	0.947509i	$0.603592\pi$
$602$	0	0
$603$	2.86274e6i	0.320618i
$604$	0	0
$605$	1.19726e7i	1.32984i
$606$	0	0
$607$	1.56734e7	1.72659	0.863297	−	0.504697i	$-0.168396\pi$
$607$	1.56734e7	1.72659	0.863297	+	0.504697i	$0.168396\pi$
$608$	0	0
$609$	5.27155e6	0.575963
$610$	0	0
$611$	1.01329e7i	1.09807i
$612$	0	0
$613$	− 1.52155e7i	− 1.63544i	−0.575615	−	0.817720i	$-0.695237\pi$
$613$	− 1.52155e7i	− 1.63544i	0.575615	−	0.817720i	$-0.304763\pi$
$614$	0	0
$615$	−9.53902e6	−1.01699
$616$	0	0
$617$	4.91925e6	0.520219	0.260110	−	0.965579i	$-0.416241\pi$
$617$	4.91925e6	0.520219	0.260110	+	0.965579i	$0.416241\pi$
$618$	0	0
$619$	8.16312e6i	0.856308i	0.903706	+	0.428154i	$0.140836\pi$
$619$	8.16312e6i	0.856308i	−0.903706	+	0.428154i	$0.859164\pi$
$620$	0	0
$621$	− 833433.i	− 0.0867245i
$622$	0	0
$623$	−1.67690e7	−1.73096
$624$	0	0
$625$	3.37016e7	3.45105
$626$	0	0
$627$	− 4.72897e6i	− 0.480394i
$628$	0	0
$629$	− 889453.i	− 0.0896388i
$630$	0	0
$631$	1.04567e7	1.04550	0.522749	−	0.852487i	$-0.324907\pi$
$631$	1.04567e7	1.04550	0.522749	+	0.852487i	$0.324907\pi$
$632$	0	0
$633$	−7.89377e6	−0.783024
$634$	0	0
$635$	2.23275e7i	2.19739i
$636$	0	0
$637$	− 2.89516e6i	− 0.282699i
$638$	0	0
$639$	142663.	0.0138216
$640$	0	0
$641$	−8.30291e6	−0.798152	−0.399076	−	0.916918i	$-0.630669\pi$
$641$	−8.30291e6	−0.798152	−0.399076	+	0.916918i	$0.630669\pi$
$642$	0	0
$643$	− 8.73090e6i	− 0.832782i	−0.909186	−	0.416391i	$-0.863295\pi$
$643$	− 8.73090e6i	− 0.832782i	0.909186	−	0.416391i	$-0.136705\pi$
$644$	0	0
$645$	4.45622e6i	0.421762i
$646$	0	0
$647$	1.10770e7	1.04030	0.520151	−	0.854074i	$-0.325876\pi$
$647$	1.10770e7	1.04030	0.520151	+	0.854074i	$0.325876\pi$
$648$	0	0
$649$	7.98010e6	0.743698
$650$	0	0
$651$	− 1.21721e7i	− 1.12568i
$652$	0	0
$653$	− 7.70900e6i	− 0.707481i	−0.935344	−	0.353741i	$-0.884910\pi$
$653$	− 7.70900e6i	− 0.707481i	0.935344	−	0.353741i	$-0.115090\pi$
$654$	0	0
$655$	2.72340e6	0.248033
$656$	0	0
$657$	1.71266e6	0.154795
$658$	0	0
$659$	1.26694e7i	1.13643i	0.822881	+	0.568214i	$0.192365\pi$
$659$	1.26694e7i	1.13643i	−0.822881	+	0.568214i	$0.807635\pi$
$660$	0	0
$661$	− 1.86962e6i	− 0.166437i	−0.996531	−	0.0832187i	$-0.973480\pi$
$661$	− 1.86962e6i	− 0.166437i	0.996531	−	0.0832187i	$-0.0265200\pi$
$662$	0	0
$663$	−9.00204e6	−0.795347
$664$	0	0
$665$	3.80725e7	3.33854
$666$	0	0
$667$	− 4.45967e6i	− 0.388140i
$668$	0	0
$669$	1.75062e6i	0.151226i
$670$	0	0
$671$	−2.77776e6	−0.238171
$672$	0	0
$673$	1.99120e7	1.69464	0.847318	−	0.531086i	$-0.178216\pi$
$673$	1.99120e7	1.69464	0.847318	+	0.531086i	$0.178216\pi$
$674$	0	0
$675$	6.07847e6i	0.513493i
$676$	0	0
$677$	− 652964.i	− 0.0547542i	−0.999625	−	0.0273771i	$-0.991285\pi$
$677$	− 652964.i	− 0.0547542i	0.999625	−	0.0273771i	$-0.00871550\pi$
$678$	0	0
$679$	−5.85636e6	−0.487476
$680$	0	0
$681$	−5.37780e6	−0.444362
$682$	0	0
$683$	3.88770e6i	0.318890i	0.987207	+	0.159445i	$0.0509705\pi$
$683$	3.88770e6i	0.318890i	−0.987207	+	0.159445i	$0.949030\pi$
$684$	0	0
$685$	2.66897e7i	2.17329i
$686$	0	0
$687$	−7.15066e6	−0.578035
$688$	0	0
$689$	−8.79079e6	−0.705472
$690$	0	0
$691$	1.60215e7i	1.27647i	0.769844	+	0.638233i	$0.220334\pi$
$691$	1.60215e7i	1.27647i	−0.769844	+	0.638233i	$0.779666\pi$
$692$	0	0
$693$	2.69849e6i	0.213446i
$694$	0	0
$695$	2.14065e7	1.68106
$696$	0	0
$697$	−1.96281e7	−1.53037
$698$	0	0
$699$	3.94117e6i	0.305093i
$700$	0	0
$701$	4.68614e6i	0.360180i	0.983650	+	0.180090i	$0.0576390\pi$
$701$	4.68614e6i	0.360180i	−0.983650	+	0.180090i	$0.942361\pi$
$702$	0	0
$703$	1.06238e6	0.0810757
$704$	0	0
$705$	1.93552e7	1.46664
$706$	0	0
$707$	− 9.26168e6i	− 0.696853i
$708$	0	0
$709$	1.74126e7i	1.30091i	0.759543	+	0.650457i	$0.225423\pi$
$709$	1.74126e7i	1.30091i	−0.759543	+	0.650457i	$0.774577\pi$
$710$	0	0
$711$	3.77154e6	0.279798
$712$	0	0
$713$	−1.02975e7	−0.758590
$714$	0	0
$715$	− 1.19834e7i	− 0.876627i
$716$	0	0
$717$	− 5.84148e6i	− 0.424351i
$718$	0	0
$719$	−1.20677e7	−0.870568	−0.435284	−	0.900293i	$-0.643352\pi$
$719$	−1.20677e7	−0.870568	−0.435284	+	0.900293i	$0.643352\pi$
$720$	0	0
$721$	−7.17045e6	−0.513698
$722$	0	0
$723$	− 3.84635e6i	− 0.273655i
$724$	0	0
$725$	3.25257e7i	2.29817i
$726$	0	0
$727$	1.76436e6	0.123809	0.0619044	−	0.998082i	$-0.480283\pi$
$727$	1.76436e6	0.123809	0.0619044	+	0.998082i	$0.480283\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	9.16942e6i	0.634671i
$732$	0	0
$733$	392726.i	0.0269979i	0.999909	+	0.0134989i	$0.00429698\pi$
$733$	392726.i	0.0269979i	−0.999909	+	0.0134989i	$0.995703\pi$
$734$	0	0
$735$	−5.53014e6	−0.377588
$736$	0	0
$737$	−7.84145e6	−0.531775
$738$	0	0
$739$	− 1.06652e7i	− 0.718386i	−0.933263	−	0.359193i	$-0.883052\pi$
$739$	− 1.06652e7i	− 0.718386i	0.933263	−	0.359193i	$-0.116948\pi$
$740$	0	0
$741$	− 1.07522e7i	− 0.719368i
$742$	0	0
$743$	−3.95018e6	−0.262509	−0.131255	−	0.991349i	$-0.541901\pi$
$743$	−3.95018e6	−0.262509	−0.131255	+	0.991349i	$0.541901\pi$
$744$	0	0
$745$	−1.96616e7	−1.29786
$746$	0	0
$747$	− 8.34475e6i	− 0.547157i
$748$	0	0
$749$	1.05257e7i	0.685561i
$750$	0	0
$751$	−8.88791e6	−0.575042	−0.287521	−	0.957774i	$-0.592831\pi$
$751$	−8.88791e6	−0.575042	−0.287521	+	0.957774i	$0.592831\pi$
$752$	0	0
$753$	−4.62949e6	−0.297540
$754$	0	0
$755$	− 3.41711e7i	− 2.18168i
$756$	0	0
$757$	2.27449e7i	1.44259i	0.692626	+	0.721297i	$0.256454\pi$
$757$	2.27449e7i	1.44259i	−0.692626	+	0.721297i	$0.743546\pi$
$758$	0	0
$759$	2.28289e6	0.143840
$760$	0	0
$761$	3.84679e6	0.240789	0.120395	−	0.992726i	$-0.461584\pi$
$761$	3.84679e6	0.240789	0.120395	+	0.992726i	$0.461584\pi$
$762$	0	0
$763$	1.70816e7i	1.06223i
$764$	0	0
$765$	1.71951e7i	1.06231i
$766$	0	0
$767$	1.81442e7	1.11365
$768$	0	0
$769$	−2.84393e7	−1.73421	−0.867107	−	0.498123i	$-0.834023\pi$
$769$	−2.84393e7	−1.73421	−0.867107	+	0.498123i	$0.834023\pi$
$770$	0	0
$771$	− 5.35238e6i	− 0.324273i
$772$	0	0
$773$	− 1.38002e7i	− 0.830686i	−0.909665	−	0.415343i	$-0.863662\pi$
$773$	− 1.38002e7i	− 0.830686i	0.909665	−	0.415343i	$-0.136338\pi$
$774$	0	0
$775$	7.51025e7	4.49159
$776$	0	0
$777$	−606223.	−0.0360230
$778$	0	0
$779$	− 2.34442e7i	− 1.38418i
$780$	0	0
$781$	390775.i	0.0229244i
$782$	0	0
$783$	−2.84372e6	−0.165761
$784$	0	0
$785$	1.12862e7	0.653692
$786$	0	0
$787$	6.48735e6i	0.373362i	0.982421	+	0.186681i	$0.0597732\pi$
$787$	6.48735e6i	0.373362i	−0.982421	+	0.186681i	$0.940227\pi$
$788$	0	0
$789$	1.85847e7i	1.06283i
$790$	0	0
$791$	−2.43347e7	−1.38288
$792$	0	0
$793$	−6.31575e6	−0.356650
$794$	0	0
$795$	1.67916e7i	0.942267i
$796$	0	0
$797$	8.51124e6i	0.474621i	0.971434	+	0.237311i	$0.0762659\pi$
$797$	8.51124e6i	0.474621i	−0.971434	+	0.237311i	$0.923734\pi$
$798$	0	0
$799$	3.98265e7	2.20701
$800$	0	0
$801$	9.04602e6	0.498168
$802$	0	0
$803$	4.69121e6i	0.256741i
$804$	0	0
$805$	1.83793e7i	0.999632i
$806$	0	0
$807$	−6.00678e6	−0.324682
$808$	0	0
$809$	−1.40776e7	−0.756236	−0.378118	−	0.925757i	$-0.623429\pi$
$809$	−1.40776e7	−0.756236	−0.378118	+	0.925757i	$0.623429\pi$
$810$	0	0
$811$	− 2.02736e7i	− 1.08238i	−0.840901	−	0.541189i	$-0.817974\pi$
$811$	− 2.02736e7i	− 1.08238i	0.840901	−	0.541189i	$-0.182026\pi$
$812$	0	0
$813$	− 6.23624e6i	− 0.330900i
$814$	0	0
$815$	−5.47591e7	−2.88777
$816$	0	0
$817$	−1.09521e7	−0.574041
$818$	0	0
$819$	6.13550e6i	0.319625i
$820$	0	0
$821$	− 4.15969e6i	− 0.215379i	−0.994185	−	0.107689i	$-0.965655\pi$
$821$	− 4.15969e6i	− 0.215379i	0.994185	−	0.107689i	$-0.0343452\pi$
$822$	0	0
$823$	−755688.	−0.0388904	−0.0194452	−	0.999811i	$-0.506190\pi$
$823$	−755688.	−0.0388904	−0.0194452	+	0.999811i	$0.506190\pi$
$824$	0	0
$825$	−1.66498e7	−0.851675
$826$	0	0
$827$	4.45268e6i	0.226390i	0.993573	+	0.113195i	$0.0361085\pi$
$827$	4.45268e6i	0.226390i	−0.993573	+	0.113195i	$0.963891\pi$
$828$	0	0
$829$	− 1.31726e7i	− 0.665712i	−0.942978	−	0.332856i	$-0.891988\pi$
$829$	− 1.31726e7i	− 0.665712i	0.942978	−	0.332856i	$-0.108012\pi$
$830$	0	0
$831$	−1.90504e7	−0.956977
$832$	0	0
$833$	−1.13792e7	−0.568197
$834$	0	0
$835$	6.26901e7i	3.11159i
$836$	0	0
$837$	6.56622e6i	0.323968i
$838$	0	0
$839$	−1.34089e7	−0.657640	−0.328820	−	0.944393i	$-0.606651\pi$
$839$	−1.34089e7	−0.657640	−0.328820	+	0.944393i	$0.606651\pi$
$840$	0	0
$841$	5.29447e6	0.258127
$842$	0	0
$843$	1.64221e7i	0.795902i
$844$	0	0
$845$	1.25063e7i	0.602543i
$846$	0	0
$847$	1.67908e7	0.804200
$848$	0	0
$849$	1.06787e7	0.508452
$850$	0	0
$851$	512858.i	0.0242758i
$852$	0	0
$853$	− 1.93105e6i	− 0.0908699i	−0.998967	−	0.0454349i	$-0.985533\pi$
$853$	− 1.93105e6i	− 0.0908699i	0.998967	−	0.0454349i	$-0.0144674\pi$
$854$	0	0
$855$	−2.05381e7	−0.960827
$856$	0	0
$857$	1.52761e7	0.710495	0.355248	−	0.934772i	$-0.384397\pi$
$857$	1.52761e7	0.710495	0.355248	+	0.934772i	$0.384397\pi$
$858$	0	0
$859$	2.73909e7i	1.26656i	0.773925	+	0.633278i	$0.218291\pi$
$859$	2.73909e7i	1.26656i	−0.773925	+	0.633278i	$0.781709\pi$
$860$	0	0
$861$	1.33779e7i	0.615008i
$862$	0	0
$863$	2.94149e7	1.34444	0.672219	−	0.740352i	$-0.265341\pi$
$863$	2.94149e7	1.34444	0.672219	+	0.740352i	$0.265341\pi$
$864$	0	0
$865$	1.59632e7	0.725403
$866$	0	0
$867$	2.26031e7i	1.02122i
$868$	0	0
$869$	1.03308e7i	0.464071i
$870$	0	0
$871$	−1.78290e7	−0.796308
$872$	0	0
$873$	3.15920e6	0.140295
$874$	0	0
$875$	− 8.38073e7i	− 3.70051i
$876$	0	0
$877$	− 3.48384e7i	− 1.52953i	−0.644308	−	0.764766i	$-0.722854\pi$
$877$	− 3.48384e7i	− 1.52953i	0.644308	−	0.764766i	$-0.277146\pi$
$878$	0	0
$879$	−1.65550e6	−0.0722700
$880$	0	0
$881$	−2.80051e7	−1.21562	−0.607809	−	0.794083i	$-0.707951\pi$
$881$	−2.80051e7	−1.21562	−0.607809	+	0.794083i	$0.707951\pi$
$882$	0	0
$883$	− 2.74027e7i	− 1.18275i	−0.806397	−	0.591374i	$-0.798586\pi$
$883$	− 2.74027e7i	− 1.18275i	0.806397	−	0.591374i	$-0.201414\pi$
$884$	0	0
$885$	− 3.46579e7i	− 1.48745i
$886$	0	0
$887$	−1.87337e7	−0.799493	−0.399747	−	0.916626i	$-0.630902\pi$
$887$	−1.87337e7	−0.799493	−0.399747	+	0.916626i	$0.630902\pi$
$888$	0	0
$889$	3.13130e7	1.32883
$890$	0	0
$891$	− 1.45569e6i	− 0.0614293i
$892$	0	0
$893$	4.75695e7i	1.99618i
$894$	0	0
$895$	1.08496e7	0.452746
$896$	0	0
$897$	5.19057e6	0.215394
$898$	0	0
$899$	3.51356e7i	1.44993i
$900$	0	0
$901$	3.45515e7i	1.41793i
$902$	0	0
$903$	6.24959e6	0.255054
$904$	0	0
$905$	7.59122e7	3.08099
$906$	0	0
$907$	− 1.80666e7i	− 0.729219i	−0.931161	−	0.364609i	$-0.881202\pi$
$907$	− 1.80666e7i	− 0.729219i	0.931161	−	0.364609i	$-0.118798\pi$
$908$	0	0
$909$	4.99620e6i	0.200553i
$910$	0	0
$911$	2.90330e7	1.15903	0.579517	−	0.814960i	$-0.303241\pi$
$911$	2.90330e7	1.15903	0.579517	+	0.814960i	$0.303241\pi$
$912$	0	0
$913$	2.28575e7	0.907509
$914$	0	0
$915$	1.20639e7i	0.476360i
$916$	0	0
$917$	− 3.81942e6i	− 0.149994i
$918$	0	0
$919$	−1.76004e7	−0.687437	−0.343718	−	0.939073i	$-0.611687\pi$
$919$	−1.76004e7	−0.687437	−0.343718	+	0.939073i	$0.611687\pi$
$920$	0	0
$921$	6.30428e6	0.244899
$922$	0	0
$923$	888498.i	0.0343283i
$924$	0	0
$925$	− 3.74042e6i	− 0.143736i
$926$	0	0
$927$	3.86808e6	0.147841
$928$	0	0
$929$	−2.19603e7	−0.834834	−0.417417	−	0.908715i	$-0.637064\pi$
$929$	−2.19603e7	−0.834834	−0.417417	+	0.908715i	$0.637064\pi$
$930$	0	0
$931$	− 1.35915e7i	− 0.513917i
$932$	0	0
$933$	− 1.19046e7i	− 0.447725i
$934$	0	0
$935$	−4.70998e7	−1.76194
$936$	0	0
$937$	−3.36644e7	−1.25263	−0.626314	−	0.779571i	$-0.715437\pi$
$937$	−3.36644e7	−1.25263	−0.626314	+	0.779571i	$0.715437\pi$
$938$	0	0
$939$	− 1.06392e6i	− 0.0393774i
$940$	0	0
$941$	− 3.50282e7i	− 1.28956i	−0.764366	−	0.644782i	$-0.776948\pi$
$941$	− 3.50282e7i	− 1.28956i	0.764366	−	0.644782i	$-0.223052\pi$
$942$	0	0
$943$	1.13176e7	0.414452
$944$	0	0
$945$	1.17196e7	0.426908
$946$	0	0
$947$	− 8.09345e6i	− 0.293264i	−0.989191	−	0.146632i	$-0.953157\pi$
$947$	− 8.09345e6i	− 0.293264i	0.989191	−	0.146632i	$-0.0468433\pi$
$948$	0	0
$949$	1.06663e7i	0.384458i
$950$	0	0
$951$	−2.30141e7	−0.825169
$952$	0	0
$953$	1.70686e7	0.608788	0.304394	−	0.952546i	$-0.401546\pi$
$953$	1.70686e7	0.608788	0.304394	+	0.952546i	$0.401546\pi$
$954$	0	0
$955$	5.84005e6i	0.207209i
$956$	0	0
$957$	− 7.78936e6i	− 0.274930i
$958$	0	0
$959$	3.74307e7	1.31426
$960$	0	0
$961$	5.24997e7	1.83379
$962$	0	0
$963$	− 5.67806e6i	− 0.197303i
$964$	0	0
$965$	− 8.89945e7i	− 3.07641i
$966$	0	0
$967$	2.24163e7	0.770901	0.385450	−	0.922729i	$-0.374046\pi$
$967$	2.24163e7	0.770901	0.385450	+	0.922729i	$0.374046\pi$
$968$	0	0
$969$	−4.22606e7	−1.44586
$970$	0	0
$971$	− 2.08206e7i	− 0.708671i	−0.935118	−	0.354336i	$-0.884707\pi$
$971$	− 2.08206e7i	− 0.708671i	0.935118	−	0.354336i	$-0.115293\pi$
$972$	0	0
$973$	− 3.00213e7i	− 1.01659i
$974$	0	0
$975$	−3.78564e7	−1.27534
$976$	0	0
$977$	−817996.	−0.0274167	−0.0137083	−	0.999906i	$-0.504364\pi$
$977$	−817996.	−0.0274167	−0.0137083	+	0.999906i	$0.504364\pi$
$978$	0	0
$979$	2.47783e7i	0.826257i
$980$	0	0
$981$	− 9.21463e6i	− 0.305707i
$982$	0	0
$983$	−1.17691e7	−0.388472	−0.194236	−	0.980955i	$-0.562223\pi$
$983$	−1.17691e7	−0.388472	−0.194236	+	0.980955i	$0.562223\pi$
$984$	0	0
$985$	1.83413e7	0.602336
$986$	0	0
$987$	− 2.71445e7i	− 0.886929i
$988$	0	0
$989$	− 5.28709e6i	− 0.171880i
$990$	0	0
$991$	−9.50426e6	−0.307421	−0.153711	−	0.988116i	$-0.549122\pi$
$991$	−9.50426e6	−0.307421	−0.153711	+	0.988116i	$0.549122\pi$
$992$	0	0
$993$	−2.40117e7	−0.772768
$994$	0	0
$995$	1.28085e7i	0.410150i
$996$	0	0
$997$	− 9.79258e6i	− 0.312003i	−0.987757	−	0.156002i	$-0.950139\pi$
$997$	− 9.79258e6i	− 0.312003i	0.987757	−	0.156002i	$-0.0498605\pi$
$998$	0	0
$999$	327026.	0.0103674

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.6.d.j.193.12	yes	12
4.3	odd	2	inner	384.6.d.j.193.6	yes	12
8.3	odd	2	inner	384.6.d.j.193.7	yes	12
8.5	even	2	inner	384.6.d.j.193.1	✓	12
16.3	odd	4		768.6.a.bf.1.6		6
16.5	even	4		768.6.a.bf.1.1		6
16.11	odd	4		768.6.a.be.1.1		6
16.13	even	4		768.6.a.be.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.j.193.1	✓	12	8.5	even	2	inner
384.6.d.j.193.6	yes	12	4.3	odd	2	inner
384.6.d.j.193.7	yes	12	8.3	odd	2	inner
384.6.d.j.193.12	yes	12	1.1	even	1	trivial
768.6.a.be.1.1		6	16.11	odd	4
768.6.a.be.1.6		6	16.13	even	4
768.6.a.bf.1.1		6	16.5	even	4
768.6.a.bf.1.6		6	16.3	odd	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)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} +107.066i q^{5} +150.154 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} +107.066i q^{5} +150.154 q^{7} -81.0000 q^{9} -221.870i q^{11} -504.463i q^{13} -963.592 q^{15} -1982.75 q^{17} -2368.23i q^{19} +1351.38i q^{21} +1143.26 q^{23} -8338.09 q^{25} -729.000i q^{27} -3900.86i q^{29} -9007.16 q^{31} +1996.83 q^{33} +16076.3i q^{35} +448.595i q^{37} +4540.17 q^{39} +9899.43 q^{41} -4624.59i q^{43} -8672.33i q^{45} -20086.5 q^{47} +5739.09 q^{49} -17844.8i q^{51} -17426.0i q^{53} +23754.7 q^{55} +21314.1 q^{57} +35967.4i q^{59} -12519.7i q^{61} -12162.4 q^{63} +54010.8 q^{65} -35342.5i q^{67} +10289.3i q^{69} -1761.27 q^{71} -21143.9 q^{73} -75042.8i q^{75} -33314.6i q^{77} -46562.3 q^{79} +6561.00 q^{81} +103022. i q^{83} -212285. i q^{85} +35107.7 q^{87} -111679. q^{89} -75747.0i q^{91} -81064.4i q^{93} +253557. q^{95} -39002.5 q^{97} +17971.5i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 972 q^{9}+O(q^{10}) \) 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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