LMFDB - Embedded newform 324.9.g.h.53.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,9,Mod(53,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("324.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	324.g (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$131.990669660$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.10
Character	$\chi$	$=$	324.53
Dual form			324.9.g.h.269.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(341.792 + 197.334i) q^{5} +(-4.04023 - 6.99788i) q^{7} +(-15472.9 + 8933.30i) q^{11} +(-6367.34 + 11028.6i) q^{13} -65225.9i q^{17} -127296. q^{19} +(118747. + 68558.8i) q^{23} +(-117431. - 203397. i) q^{25} +(-131189. + 75741.9i) q^{29} +(343848. - 595563. i) q^{31} -3189.09i q^{35} +2.42635e6 q^{37} +(377176. + 217762. i) q^{41} +(56622.9 + 98073.7i) q^{43} +(-435276. + 251307. i) q^{47} +(2.88237e6 - 4.99241e6i) q^{49} -4.55661e6i q^{53} -7.05137e6 q^{55} +(3.15244e6 + 1.82006e6i) q^{59} +(-4.25161e6 - 7.36400e6i) q^{61} +(-4.35261e6 + 2.51298e6i) q^{65} +(-9.51606e6 + 1.64823e7i) q^{67} +3.34851e6i q^{71} +1.29864e6 q^{73} +(125028. + 72185.1i) q^{77} +(2.26931e7 + 3.93057e7i) q^{79} +(5.40909e7 - 3.12294e7i) q^{83} +(1.28713e7 - 2.22937e7i) q^{85} +1.81886e7i q^{89} +102902. q^{91} +(-4.35087e7 - 2.51198e7i) q^{95} +(1.09949e6 + 1.90438e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3692 q^{7} + 50380 q^{13} + 370216 q^{19} + 1958288 q^{25} + 285568 q^{31} - 5820104 q^{37} + 8122852 q^{43} - 14602560 q^{49} + 76640328 q^{55} + 5700244 q^{61} + 6328540 q^{67} - 103457408 q^{73}+ \cdots + 181751968 q^{97}+O(q^{100}) $ 32 * q - 3692 * q^7 + 50380 * q^13 + 370216 * q^19 + 1958288 * q^25 + 285568 * q^31 - 5820104 * q^37 + 8122852 * q^43 - 14602560 * q^49 + 76640328 * q^55 + 5700244 * q^61 + 6328540 * q^67 - 103457408 * q^73 - 168603092 * q^79 + 96322644 * q^85 - 399487432 * q^91 + 181751968 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	341.792	+	197.334i	0.546868	+	0.315734i	0.747858	−	0.663859i	$-0.231083\pi$
$5$	341.792	+	197.334i	0.546868	+	0.315734i	−0.200990	+	0.979593i	$0.564416\pi$
$6$		0			0
$7$	−4.04023	−	6.99788i	−0.00168273	−	0.00291457i	0.865183	−	0.501457i	$-0.167202\pi$
$7$	−4.04023	−	6.99788i	−0.00168273	−	0.00291457i	−0.866866	+	0.498542i	$0.833869\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−15472.9	+	8933.30i	−1.05682	+	0.610156i	−0.924551	−	0.381058i	$-0.875560\pi$
$11$	−15472.9	+	8933.30i	−1.05682	+	0.610156i	−0.132270	+	0.991214i	$0.542227\pi$
$12$		0			0
$13$	−6367.34	+	11028.6i	−0.222938	+	0.386140i	−0.955699	−	0.294346i	$-0.904898\pi$
$13$	−6367.34	+	11028.6i	−0.222938	+	0.386140i	0.732761	+	0.680486i	$0.238231\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	65225.9i		−	0.780952i	−0.920613	−	0.390476i	$-0.872310\pi$
$17$		−	65225.9i		−	0.780952i	0.920613	−	0.390476i	$-0.127690\pi$
$18$		0			0
$19$	−127296.			−0.976787			−0.488393	−	0.872624i	$-0.662417\pi$
$19$	−127296.			−0.976787			−0.488393	+	0.872624i	$0.662417\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	118747.	+	68558.8i	0.424339	+	0.244992i	0.696932	−	0.717137i	$-0.254548\pi$
$23$	118747.	+	68558.8i	0.424339	+	0.244992i	−0.272593	+	0.962129i	$0.587881\pi$
$24$		0			0
$25$	−117431.	−	203397.i	−0.300624	−	0.520696i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−131189.	+	75741.9i	−0.185483	+	0.107089i	−0.589866	−	0.807501i	$-0.700820\pi$
$29$	−131189.	+	75741.9i	−0.185483	+	0.107089i	0.404383	+	0.914590i	$0.367486\pi$
$30$		0			0
$31$	343848.	−	595563.i	0.372323	−	0.644882i	−0.617599	−	0.786493i	$-0.711895\pi$
$31$	343848.	−	595563.i	0.372323	−	0.644882i	0.989922	+	0.141610i	$0.0452280\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	3189.09i		−	0.00212518i
$36$		0			0
$37$	2.42635e6			1.29463			0.647316	−	0.762222i	$-0.275891\pi$
$37$	2.42635e6			1.29463			0.647316	+	0.762222i	$0.275891\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	377176.	+	217762.i	0.133478	+	0.0770633i	0.565252	−	0.824918i	$-0.308779\pi$
$41$	377176.	+	217762.i	0.133478	+	0.0770633i	−0.431774	+	0.901982i	$0.642112\pi$
$42$		0			0
$43$	56622.9	+	98073.7i	0.0165622	+	0.0286866i	0.874188	−	0.485588i	$-0.161395\pi$
$43$	56622.9	+	98073.7i	0.0165622	+	0.0286866i	−0.857626	+	0.514275i	$0.828061\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−435276.	+	251307.i	−0.0892018	+	0.0515007i	−0.543937	−	0.839126i	$-0.683067\pi$
$47$	−435276.	+	251307.i	−0.0892018	+	0.0515007i	0.454736	+	0.890626i	$0.349734\pi$
$48$		0			0
$49$	2.88237e6	−	4.99241e6i	0.499994	−	0.866016i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	4.55661e6i		−	0.577482i	−0.957407	−	0.288741i	$-0.906763\pi$
$53$		−	4.55661e6i		−	0.577482i	0.957407	−	0.288741i	$-0.0932366\pi$
$54$		0			0
$55$	−7.05137e6			−0.770588
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.15244e6	+	1.82006e6i	0.260159	+	0.150203i	0.624407	−	0.781099i	$-0.285341\pi$
$59$	3.15244e6	+	1.82006e6i	0.260159	+	0.150203i	−0.364248	+	0.931302i	$0.618674\pi$
$60$		0			0
$61$	−4.25161e6	−	7.36400e6i	−0.307068	−	0.531857i	0.670652	−	0.741772i	$-0.266014\pi$
$61$	−4.25161e6	−	7.36400e6i	−0.307068	−	0.531857i	−0.977720	+	0.209916i	$0.932681\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.35261e6	+	2.51298e6i	−0.243835	+	0.140778i
$66$		0			0
$67$	−9.51606e6	+	1.64823e7i	−0.472235	+	0.817935i	−0.999495	−	0.0317691i	$-0.989886\pi$
$67$	−9.51606e6	+	1.64823e7i	−0.472235	+	0.817935i	0.527260	+	0.849704i	$0.323219\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.34851e6i			0.131770i	0.997827	+	0.0658852i	$0.0209871\pi$
$71$			3.34851e6i			0.131770i	−0.997827	+	0.0658852i	$0.979013\pi$
$72$		0			0
$73$	1.29864e6			0.0457294			0.0228647	−	0.999739i	$-0.492721\pi$
$73$	1.29864e6			0.0457294			0.0228647	+	0.999739i	$0.492721\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	125028.	+	72185.1i	0.00355668	+	0.00205345i
$78$		0			0
$79$	2.26931e7	+	3.93057e7i	0.582621	+	1.00913i	0.995167	+	0.0981930i	$0.0313062\pi$
$79$	2.26931e7	+	3.93057e7i	0.582621	+	1.00913i	−0.412546	+	0.910937i	$0.635360\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.40909e7	−	3.12294e7i	1.13976	−	0.658039i	0.193386	−	0.981123i	$-0.438053\pi$
$83$	5.40909e7	−	3.12294e7i	1.13976	−	0.658039i	0.946370	+	0.323084i	$0.104720\pi$
$84$		0			0
$85$	1.28713e7	−	2.22937e7i	0.246573	−	0.427077i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.81886e7i			0.289894i	0.989439	+	0.144947i	$0.0463011\pi$
$89$			1.81886e7i			0.289894i	−0.989439	+	0.144947i	$0.953699\pi$
$90$		0			0
$91$	102902.			0.00150058
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.35087e7	−	2.51198e7i	−0.534173	−	0.308405i
$96$		0			0
$97$	1.09949e6	+	1.90438e6i	0.0124195	+	0.0215113i	0.872168	−	0.489206i	$-0.162713\pi$
$97$	1.09949e6	+	1.90438e6i	0.0124195	+	0.0215113i	−0.859749	+	0.510717i	$0.829380\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.21096e7	−	1.85385e7i	0.308567	−	0.178151i	−0.337718	−	0.941247i	$-0.609655\pi$
$101$	3.21096e7	−	1.85385e7i	0.308567	−	0.178151i	0.646285	+	0.763096i	$0.276322\pi$
$102$		0			0
$103$	−1.42038e7	+	2.46016e7i	−0.126199	+	0.218582i	−0.922201	−	0.386711i	$-0.873611\pi$
$103$	−1.42038e7	+	2.46016e7i	−0.126199	+	0.218582i	0.796002	+	0.605294i	$0.206944\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	7.17957e7i		−	0.547726i	−0.961769	−	0.273863i	$-0.911699\pi$
$107$		−	7.17957e7i		−	0.547726i	0.961769	−	0.273863i	$-0.0883015\pi$
$108$		0			0
$109$	−1.40547e8			−0.995674			−0.497837	−	0.867271i	$-0.665872\pi$
$109$	−1.40547e8			−0.995674			−0.497837	+	0.867271i	$0.665872\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.24914e8	+	7.21190e7i	0.766120	+	0.442319i	0.831489	−	0.555542i	$-0.187489\pi$
$113$	1.24914e8	+	7.21190e7i	0.766120	+	0.442319i	−0.0653689	+	0.997861i	$0.520822\pi$
$114$		0			0
$115$	2.70580e7	+	4.68657e7i	0.154705	+	0.267956i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−456443.	+	263527.i	−0.00227614	+	0.00131413i
$120$		0			0
$121$	5.24281e7	−	9.08081e7i	0.244581	−	0.423627i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	2.46860e8i		−	1.01114i
$126$		0			0
$127$	−1.90655e7			−0.0732882			−0.0366441	−	0.999328i	$-0.511667\pi$
$127$	−1.90655e7			−0.0732882			−0.0366441	+	0.999328i	$0.511667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	3.20736e8	+	1.85177e8i	1.08909	+	0.628785i	0.933334	−	0.359011i	$-0.116886\pi$
$131$	3.20736e8	+	1.85177e8i	1.08909	+	0.628785i	0.155755	+	0.987796i	$0.450219\pi$
$132$		0			0
$133$	514304.	+	890801.i	0.00164367	+	0.00284691i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.50436e8	−	3.17794e8i	1.56252	−	0.902119i	0.565514	−	0.824738i	$-0.308678\pi$
$137$	5.50436e8	−	3.17794e8i	1.56252	−	0.902119i	0.997002	−	0.0773805i	$-0.0246556\pi$
$138$		0			0
$139$	2.79213e8	−	4.83611e8i	0.747956	−	1.29550i	−0.200844	−	0.979623i	$-0.564369\pi$
$139$	2.79213e8	−	4.83611e8i	0.747956	−	1.29550i	0.948801	−	0.315875i	$-0.102298\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	2.27525e8i		−	0.544108i
$144$		0			0
$145$	−5.97857e7			−0.135246
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.64558e8	−	2.10478e8i	−0.739643	−	0.427033i	0.0822965	−	0.996608i	$-0.473775\pi$
$149$	−3.64558e8	−	2.10478e8i	−0.739643	−	0.427033i	−0.821939	+	0.569575i	$0.807108\pi$
$150$		0			0
$151$	1.96304e8	+	3.40009e8i	0.377591	+	0.654007i	0.990711	−	0.135983i	$-0.0434191\pi$
$151$	1.96304e8	+	3.40009e8i	0.377591	+	0.654007i	−0.613120	+	0.789990i	$0.710086\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.35049e8	−	1.35706e8i	0.407223	−	0.235110i
$156$		0			0
$157$	2.05058e8	−	3.55171e8i	0.337504	−	0.584574i	−0.646459	−	0.762949i	$-0.723751\pi$
$157$	2.05058e8	−	3.55171e8i	0.337504	−	0.584574i	0.983963	+	0.178375i	$0.0570841\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	1.10797e6i		−	0.00164902i
$162$		0			0
$163$	−1.91898e8			−0.271845			−0.135922	−	0.990719i	$-0.543400\pi$
$163$	−1.91898e8			−0.271845			−0.135922	+	0.990719i	$0.543400\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	7.97982e8	+	4.60715e8i	1.02595	+	0.592334i	0.915822	−	0.401583i	$-0.131540\pi$
$167$	7.97982e8	+	4.60715e8i	1.02595	+	0.592334i	0.110130	+	0.993917i	$0.464873\pi$
$168$		0			0
$169$	3.26779e8	+	5.65999e8i	0.400597	+	0.693855i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.26386e9	−	7.29690e8i	1.41096	−	0.814618i	0.415482	−	0.909602i	$-0.363613\pi$
$173$	1.26386e9	−	7.29690e8i	1.41096	−	0.814618i	0.995479	+	0.0949832i	$0.0302797\pi$
$174$		0			0
$175$	−948898.	+	1.64354e6i	−0.00101174	+	0.00175238i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1.92583e9i		−	1.87588i	−0.346799	−	0.937940i	$-0.612731\pi$
$179$		−	1.92583e9i		−	1.87588i	0.346799	−	0.937940i	$-0.387269\pi$
$180$		0			0
$181$	−1.94288e9			−1.81022			−0.905110	−	0.425178i	$-0.860211\pi$
$181$	−1.94288e9			−1.81022			−0.905110	+	0.425178i	$0.860211\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.29307e8	+	4.78801e8i	0.707992	+	0.408760i
$186$		0			0
$187$	5.82682e8	+	1.00923e9i	0.476503	+	0.825327i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.88923e9	−	1.09075e9i	1.41955	−	0.819580i	0.423295	−	0.905992i	$-0.360873\pi$
$191$	1.88923e9	−	1.09075e9i	1.41955	−	0.819580i	0.996259	+	0.0864120i	$0.0275402\pi$
$192$		0			0
$193$	−1.65137e7	+	2.86025e7i	−0.0119019	+	0.0206146i	−0.871915	−	0.489657i	$-0.837122\pi$
$193$	−1.65137e7	+	2.86025e7i	−0.0119019	+	0.0206146i	0.860013	+	0.510272i	$0.170455\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.06649e7i		−	0.0402784i	−0.999797	−	0.0201392i	$-0.993589\pi$
$197$		−	6.06649e7i		−	0.0402784i	0.999797	−	0.0201392i	$-0.00641095\pi$
$198$		0			0
$199$	−1.33086e9			−0.848633			−0.424316	−	0.905514i	$-0.639486\pi$
$199$	−1.33086e9			−0.848633			−0.424316	+	0.905514i	$0.639486\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.06007e6	+	612029.i	0.000624235	+	0.000360402i
$204$		0			0
$205$	8.59438e7	+	1.48859e8i	0.0486630	+	0.0842868i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.96964e9	−	1.13717e9i	1.03229	−	0.595992i
$210$		0			0
$211$	6.06460e8	−	1.05042e9i	0.305965	−	0.529948i	−0.671510	−	0.740995i	$-0.734354\pi$
$211$	6.06460e8	−	1.05042e9i	0.305965	−	0.529948i	0.977476	+	0.211048i	$0.0676875\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			4.46944e7i			0.0209170i
$216$		0			0
$217$	−5.55690e6			−0.00250607
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7.19347e8	+	4.15315e8i	0.301557	+	0.174104i
$222$		0			0
$223$	1.73268e9	+	3.00108e9i	0.700645	+	1.21355i	0.968240	+	0.250021i	$0.0804374\pi$
$223$	1.73268e9	+	3.00108e9i	0.700645	+	1.21355i	−0.267596	+	0.963531i	$0.586229\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.22439e9	−	1.86160e9i	1.21435	−	0.701105i	0.250645	−	0.968079i	$-0.419357\pi$
$227$	3.22439e9	−	1.86160e9i	1.21435	−	0.701105i	0.963704	+	0.266974i	$0.0860239\pi$
$228$		0			0
$229$	2.24392e9	−	3.88658e9i	0.815952	−	1.41327i	−0.0926902	−	0.995695i	$-0.529547\pi$
$229$	2.24392e9	−	3.88658e9i	0.815952	−	1.41327i	0.908642	−	0.417575i	$-0.137120\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.95736e9i		−	1.68200i	−0.541033	−	0.841001i	$-0.681967\pi$
$233$		−	4.95736e9i		−	1.68200i	0.541033	−	0.841001i	$-0.318033\pi$
$234$		0			0
$235$	−1.98365e8			−0.0650421
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.26997e9	+	7.33215e8i	0.389225	+	0.224719i	0.681824	−	0.731516i	$-0.261187\pi$
$239$	1.26997e9	+	7.33215e8i	0.389225	+	0.224719i	−0.292599	+	0.956235i	$0.594520\pi$
$240$		0			0
$241$	6.40342e8	+	1.10910e9i	0.189821	+	0.328779i	0.945190	−	0.326520i	$-0.105876\pi$
$241$	6.40342e8	+	1.10910e9i	0.189821	+	0.328779i	−0.755370	+	0.655299i	$0.772543\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.97034e9	−	1.13758e9i	0.546861	−	0.315731i
$246$		0			0
$247$	8.10535e8	−	1.40389e9i	0.217763	−	0.377177i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	2.01240e9i		−	0.507015i	−0.967333	−	0.253507i	$-0.918416\pi$
$251$		−	2.01240e9i		−	0.507015i	0.967333	−	0.253507i	$-0.0815842\pi$
$252$		0			0
$253$	−2.44983e9			−0.597934
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.89110e9	+	2.24652e9i	0.891948	+	0.514966i	0.874579	−	0.484883i	$-0.161138\pi$
$257$	3.89110e9	+	2.24652e9i	0.891948	+	0.514966i	0.0173686	+	0.999849i	$0.494471\pi$
$258$		0			0
$259$	−9.80301e6	−	1.69793e7i	−0.00217851	−	0.00377330i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2.91060e9	−	1.68044e9i	0.608358	−	0.351236i	−0.163964	−	0.986466i	$-0.552428\pi$
$263$	2.91060e9	−	1.68044e9i	0.608358	−	0.351236i	0.772323	+	0.635230i	$0.219095\pi$
$264$		0			0
$265$	8.99173e8	−	1.55741e9i	0.182331	−	0.315806i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	2.36691e9i		−	0.452035i	−0.974123	−	0.226018i	$-0.927429\pi$
$269$		−	2.36691e9i		−	0.452035i	0.974123	−	0.226018i	$-0.0725707\pi$
$270$		0			0
$271$	−6.17018e9			−1.14399			−0.571993	−	0.820259i	$-0.693829\pi$
$271$	−6.17018e9			−1.14399			−0.571993	+	0.820259i	$0.693829\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.63401e9	+	2.09810e9i	0.635412	+	0.366855i
$276$		0			0
$277$	−1.81714e9	−	3.14738e9i	−0.308652	−	0.534601i	0.669416	−	0.742888i	$-0.266545\pi$
$277$	−1.81714e9	−	3.14738e9i	−0.308652	−	0.534601i	−0.978068	+	0.208287i	$0.933211\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.56704e8	−	1.48208e8i	0.0411725	−	0.0237710i	−0.479272	−	0.877666i	$-0.659099\pi$
$281$	2.56704e8	−	1.48208e8i	0.0411725	−	0.0237710i	0.520445	+	0.853895i	$0.325766\pi$
$282$		0			0
$283$	1.47999e8	−	2.56341e8i	0.0230734	−	0.0399644i	−0.854258	−	0.519849i	$-0.825988\pi$
$283$	1.47999e8	−	2.56341e8i	0.0230734	−	0.0399644i	0.877332	+	0.479885i	$0.159322\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	3.51924e6i		−	0.000518706i
$288$		0			0
$289$	2.72134e9			0.390114
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.07208e9	−	2.35102e9i	−0.552517	−	0.318996i	0.197620	−	0.980279i	$-0.436679\pi$
$293$	−4.07208e9	−	2.35102e9i	−0.552517	−	0.318996i	−0.750137	+	0.661283i	$0.770012\pi$
$294$		0			0
$295$	7.18320e8	+	1.24417e9i	0.0948484	+	0.164282i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.51221e9	+	8.73074e8i	−0.189203	+	0.109236i
$300$		0			0
$301$	457539.	−	792480.i	5.57393e−5	−	9.65434e-5i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.35594e9i		−	0.387807i
$306$		0			0
$307$	1.51016e9			0.170008			0.0850042	−	0.996381i	$-0.472910\pi$
$307$	1.51016e9			0.170008			0.0850042	+	0.996381i	$0.472910\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2.88139e9	−	1.66357e9i	−0.308007	−	0.177828i	0.338027	−	0.941136i	$-0.390240\pi$
$311$	−2.88139e9	−	1.66357e9i	−0.308007	−	0.177828i	−0.646034	+	0.763308i	$0.723574\pi$
$312$		0			0
$313$	8.61062e9	+	1.49140e10i	0.897133	+	1.55388i	0.831142	+	0.556060i	$0.187688\pi$
$313$	8.61062e9	+	1.49140e10i	0.897133	+	1.55388i	0.0659916	+	0.997820i	$0.478979\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.26336e10	−	7.29404e9i	1.25110	−	0.722322i	0.279771	−	0.960067i	$-0.409742\pi$
$317$	1.26336e10	−	7.29404e9i	1.25110	−	0.722322i	0.971328	+	0.237745i	$0.0764082\pi$
$318$		0			0
$319$	1.35325e9	−	2.34390e9i	0.130682	−	0.226347i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.30298e9i			0.762823i
$324$		0			0
$325$	2.99090e9			0.268082
$326$		0			0
$327$		0			0
$328$		0			0
$329$	3.51723e6	+	2.03067e6i	0.000300205	+	0.000173323i
$330$		0			0
$331$	−3.74292e9	−	6.48294e9i	−0.311816	−	0.540082i	0.666939	−	0.745112i	$-0.267604\pi$
$331$	−3.74292e9	−	6.48294e9i	−0.311816	−	0.540082i	−0.978756	+	0.205030i	$0.934271\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.50503e9	+	3.75568e9i	−0.516500	+	0.298201i
$336$		0			0
$337$	−8.00476e9	+	1.38647e10i	−0.620624	+	1.07495i	0.368745	+	0.929531i	$0.379788\pi$
$337$	−8.00476e9	+	1.38647e10i	−0.620624	+	1.07495i	−0.989370	+	0.145423i	$0.953546\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			1.22868e10i			0.908701i
$342$		0			0
$343$	−9.31639e7			−0.00673087
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.04024e10	−	6.00580e9i	−0.717487	−	0.414241i	0.0963402	−	0.995348i	$-0.469286\pi$
$347$	−1.04024e10	−	6.00580e9i	−0.717487	−	0.414241i	−0.813827	+	0.581107i	$0.802620\pi$
$348$		0			0
$349$	−4.93157e9	−	8.54173e9i	−0.332417	−	0.575764i	0.650568	−	0.759448i	$-0.274531\pi$
$349$	−4.93157e9	−	8.54173e9i	−0.332417	−	0.575764i	−0.982985	+	0.183684i	$0.941198\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.38319e10	+	1.37594e10i	−1.53483	+	0.886134i	−0.535699	+	0.844409i	$0.679952\pi$
$353$	−2.38319e10	+	1.37594e10i	−1.53483	+	0.886134i	−0.999129	+	0.0417247i	$0.986715\pi$
$354$		0			0
$355$	−6.60774e8	+	1.14449e9i	−0.0416044	+	0.0720610i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	9.64206e8i		−	0.0580486i	−0.999579	−	0.0290243i	$-0.990760\pi$
$359$		−	9.64206e8i		−	0.0580486i	0.999579	−	0.0290243i	$-0.00924003\pi$
$360$		0			0
$361$	−7.79336e8			−0.0458877
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.43864e8	+	2.56265e8i	0.0250079	+	0.0144383i
$366$		0			0
$367$	−1.12446e10	−	1.94763e10i	−0.619843	−	1.07360i	−0.989514	−	0.144437i	$-0.953863\pi$
$367$	−1.12446e10	−	1.94763e10i	−0.619843	−	1.07360i	0.369671	−	0.929163i	$-0.379470\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.18866e7	+	1.84097e7i	−0.00168311	+	0.000971745i
$372$		0			0
$373$	−1.05932e10	+	1.83480e10i	−0.547260	+	0.947882i	0.451201	+	0.892422i	$0.350996\pi$
$373$	−1.05932e10	+	1.83480e10i	−0.547260	+	0.947882i	−0.998461	+	0.0554597i	$0.982338\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.92910e9i		−	0.0954967i
$378$		0			0
$379$	6.09727e9			0.295514			0.147757	−	0.989024i	$-0.452795\pi$
$379$	6.09727e9			0.295514			0.147757	+	0.989024i	$0.452795\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.45726e10	+	8.41349e9i	0.677239	+	0.391004i	0.798814	−	0.601578i	$-0.205461\pi$
$383$	1.45726e10	+	8.41349e9i	0.677239	+	0.391004i	−0.121575	+	0.992582i	$0.538795\pi$
$384$		0			0
$385$	2.84891e7	+	4.93446e7i	0.00129669	+	0.00224593i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.46519e10	+	1.42328e10i	−1.07660	+	0.621573i	−0.929976	−	0.367620i	$-0.880173\pi$
$389$	−2.46519e10	+	1.42328e10i	−1.07660	+	0.621573i	−0.146620	+	0.989193i	$0.546839\pi$
$390$		0			0
$391$	4.47181e9	−	7.74540e9i	0.191327	−	0.331388i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			1.79125e10i			0.735814i
$396$		0			0
$397$	1.86725e10			0.751694			0.375847	−	0.926682i	$-0.377352\pi$
$397$	1.86725e10			0.751694			0.375847	+	0.926682i	$0.377352\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2.38689e10	−	1.37807e10i	−0.923111	−	0.532959i	−0.0384850	−	0.999259i	$-0.512253\pi$
$401$	−2.38689e10	−	1.37807e10i	−0.923111	−	0.532959i	−0.884626	+	0.466301i	$0.845587\pi$
$402$		0			0
$403$	4.37879e9	+	7.58429e9i	0.166010	+	0.287538i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.75427e10	+	2.16753e10i	−1.36820	+	0.789928i
$408$		0			0
$409$	6.26759e8	−	1.08558e9i	0.0223979	−	0.0387943i	−0.854609	−	0.519272i	$-0.826203\pi$
$409$	6.26759e8	−	1.08558e9i	0.0223979	−	0.0387943i	0.877007	+	0.480477i	$0.159537\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	2.94139e7i		−	0.00101100i
$414$		0			0
$415$	2.46505e10			0.831061
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.01663e10	−	1.16430e10i	−0.654291	−	0.377755i	0.135807	−	0.990735i	$-0.456637\pi$
$419$	−2.01663e10	−	1.16430e10i	−0.654291	−	0.377755i	−0.790098	+	0.612980i	$0.789970\pi$
$420$		0			0
$421$	−2.26268e10	−	3.91908e10i	−0.720270	−	1.24754i	−0.960892	−	0.276925i	$-0.910685\pi$
$421$	−2.26268e10	−	3.91908e10i	−0.720270	−	1.24754i	0.240622	−	0.970619i	$-0.422649\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.32667e10	+	7.65955e9i	−0.406638	+	0.234773i
$426$		0			0
$427$	−3.43549e7	+	5.95045e7i	−0.00103342	+	0.00178994i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			5.72462e10i			1.65897i	0.558531	+	0.829484i	$0.311365\pi$
$431$			5.72462e10i			1.65897i	−0.558531	+	0.829484i	$0.688635\pi$
$432$		0			0
$433$	5.31445e10			1.51184			0.755922	−	0.654662i	$-0.227189\pi$
$433$	5.31445e10			1.51184			0.755922	+	0.654662i	$0.227189\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.51160e10	−	8.72725e9i	−0.414488	−	0.239305i
$438$		0			0
$439$	−6.31044e9	−	1.09300e10i	−0.169903	−	0.294281i	0.768482	−	0.639871i	$-0.221012\pi$
$439$	−6.31044e9	−	1.09300e10i	−0.169903	−	0.294281i	−0.938386	+	0.345590i	$0.887679\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.05623e10	+	1.76452e10i	−0.793545	+	0.458153i	−0.841209	−	0.540710i	$-0.818156\pi$
$443$	−3.05623e10	+	1.76452e10i	−0.793545	+	0.458153i	0.0476641	+	0.998863i	$0.484822\pi$
$444$		0			0
$445$	−3.58922e9	+	6.21671e9i	−0.0915293	+	0.158533i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.20870e10i		−	0.297395i	−0.988883	−	0.148698i	$-0.952492\pi$
$449$		−	1.20870e10i		−	0.297395i	0.988883	−	0.148698i	$-0.0475081\pi$
$450$		0			0
$451$	−7.78135e9			−0.188083
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3.51711e7	+	2.03060e7i	0.000820616	+	0.000473783i
$456$		0			0
$457$	−2.23708e10	−	3.87474e10i	−0.512882	−	0.888338i	−0.999888	−	0.0149392i	$-0.995245\pi$
$457$	−2.23708e10	−	3.87474e10i	−0.512882	−	0.888338i	0.487006	−	0.873398i	$-0.338089\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.84929e10	+	1.06769e10i	−0.409451	+	0.236397i	−0.690554	−	0.723281i	$-0.742633\pi$
$461$	−1.84929e10	+	1.06769e10i	−0.409451	+	0.236397i	0.281103	+	0.959678i	$0.409300\pi$
$462$		0			0
$463$	−3.52158e10	+	6.09955e10i	−0.766326	+	1.32732i	0.173217	+	0.984884i	$0.444584\pi$
$463$	−3.52158e10	+	6.09955e10i	−0.766326	+	1.32732i	−0.939543	+	0.342431i	$0.888750\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			4.76789e10i			1.00244i	0.865319	+	0.501221i	$0.167116\pi$
$467$			4.76789e10i			1.00244i	−0.865319	+	0.501221i	$0.832884\pi$
$468$		0			0
$469$	1.53788e8			0.00317857
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.75224e9	−	1.01166e9i	−0.0350066	−	0.0202111i
$474$		0			0
$475$	1.49485e10	+	2.58916e10i	0.293645	+	0.508609i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−4.23202e10	+	2.44336e10i	−0.803907	+	0.464136i	−0.844835	−	0.535026i	$-0.820302\pi$
$479$	−4.23202e10	+	2.44336e10i	−0.803907	+	0.464136i	0.0409287	+	0.999162i	$0.486968\pi$
$480$		0			0
$481$	−1.54494e10	+	2.67591e10i	−0.288623	+	0.499910i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			8.67868e8i			0.0156851i
$486$		0			0
$487$	−5.24487e10			−0.932435			−0.466218	−	0.884670i	$-0.654384\pi$
$487$	−5.24487e10			−0.932435			−0.466218	+	0.884670i	$0.654384\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.12422e9	+	1.22642e9i	0.0365488	+	0.0211015i	0.518163	−	0.855282i	$-0.326616\pi$
$491$	2.12422e9	+	1.22642e9i	0.0365488	+	0.0211015i	−0.481614	+	0.876383i	$0.659949\pi$
$492$		0			0
$493$	4.94033e9	+	8.55690e9i	0.0836312	+	0.144853i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2.34325e7	−	1.35287e7i	0.000384054	−	0.000221734i
$498$		0			0
$499$	−1.93400e10	+	3.34978e10i	−0.311927	+	0.540274i	−0.978780	−	0.204916i	$-0.934308\pi$
$499$	−1.93400e10	+	3.34978e10i	−0.311927	+	0.540274i	0.666852	+	0.745190i	$0.267641\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	4.00804e10i		−	0.626123i	−0.949733	−	0.313062i	$-0.898645\pi$
$503$		−	4.00804e10i		−	0.626123i	0.949733	−	0.313062i	$-0.101355\pi$
$504$		0			0
$505$	1.46331e10			0.224994
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.33363e10	−	1.92467e10i	−0.496645	−	0.286738i	0.230682	−	0.973029i	$-0.425904\pi$
$509$	−3.33363e10	−	1.92467e10i	−0.496645	−	0.286738i	−0.727327	+	0.686291i	$0.759238\pi$
$510$		0			0
$511$	−5.24679e6	−	9.08770e6i	−7.69502e−5	−	0.000133282i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−9.70947e9	+	5.60577e9i	−0.138028	+	0.0796904i
$516$		0			0
$517$	4.49000e9	−	7.77690e9i	0.0628469	−	0.108854i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			5.63244e10i			0.764443i	0.924071	+	0.382222i	$0.124841\pi$
$521$			5.63244e10i			0.764443i	−0.924071	+	0.382222i	$0.875159\pi$
$522$		0			0
$523$	6.62177e10			0.885049			0.442525	−	0.896756i	$-0.354083\pi$
$523$	6.62177e10			0.885049			0.442525	+	0.896756i	$0.354083\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.88461e10	−	2.24278e10i	−0.503622	−	0.290766i
$528$		0			0
$529$	−2.97549e10	−	5.15369e10i	−0.379958	−	0.658106i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.80321e9	+	2.77313e9i	−0.0595145	+	0.0343607i
$534$		0			0
$535$	1.41677e10	−	2.45392e10i	0.172936	−	0.299534i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1.02996e11i			1.22030i
$540$		0			0
$541$	1.36192e10			0.158987			0.0794934	−	0.996835i	$-0.474670\pi$
$541$	1.36192e10			0.158987			0.0794934	+	0.996835i	$0.474670\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.80380e10	−	2.77348e10i	−0.544502	−	0.314368i
$546$		0			0
$547$	1.79231e10	+	3.10436e10i	0.200199	+	0.346755i	0.948593	−	0.316500i	$-0.102508\pi$
$547$	1.79231e10	+	3.10436e10i	0.200199	+	0.346755i	−0.748393	+	0.663255i	$0.769174\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.66998e10	−	9.64162e9i	0.181178	−	0.104603i
$552$		0			0
$553$	1.83371e8	−	3.17608e8i	0.00196079	−	0.00339618i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	1.07472e11i		−	1.11654i	−0.829658	−	0.558272i	$-0.811465\pi$
$557$		−	1.07472e11i		−	1.11654i	0.829658	−	0.558272i	$-0.188535\pi$
$558$		0			0
$559$	−1.44215e9			−0.0147694
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.27547e11	+	7.36395e10i	1.26951	+	0.732955i	0.974896	−	0.222660i	$-0.0714741\pi$
$563$	1.27547e11	+	7.36395e10i	1.26951	+	0.732955i	0.294619	+	0.955615i	$0.404807\pi$
$564$		0			0
$565$	2.84630e10	+	4.92994e10i	0.279311	+	0.483780i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.48144e11	−	8.55309e10i	1.41330	−	0.815970i	0.417603	−	0.908630i	$-0.362870\pi$
$569$	1.48144e11	−	8.55309e10i	1.41330	−	0.815970i	0.995698	+	0.0926600i	$0.0295370\pi$
$570$		0			0
$571$	9.03252e10	−	1.56448e11i	0.849698	−	1.47172i	−0.0317799	−	0.999495i	$-0.510118\pi$
$571$	9.03252e10	−	1.56448e11i	0.849698	−	1.47172i	0.881478	−	0.472225i	$-0.156549\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.22038e10i		−	0.294602i
$576$		0			0
$577$	−7.69345e10			−0.694093			−0.347047	−	0.937848i	$-0.612815\pi$
$577$	−7.69345e10			−0.694093			−0.347047	+	0.937848i	$0.612815\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.37080e8	−	2.52348e8i	−0.00383580	−	0.00221460i
$582$		0			0
$583$	4.07055e10	+	7.05041e10i	0.352354	+	0.610295i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−4.45190e10	+	2.57031e10i	−0.374967	+	0.216487i	−0.675626	−	0.737244i	$-0.736127\pi$
$587$	−4.45190e10	+	2.57031e10i	−0.374967	+	0.216487i	0.300659	+	0.953732i	$0.402793\pi$
$588$		0			0
$589$	−4.37704e10	+	7.58126e10i	−0.363680	+	0.629913i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1.68949e11i		−	1.36627i	−0.730292	−	0.683135i	$-0.760616\pi$
$593$		−	1.68949e11i		−	1.36627i	0.730292	−	0.683135i	$-0.239384\pi$
$594$		0			0
$595$	−2.08011e8			−0.00165966
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1.77314e11	+	1.02372e11i	1.37732	+	0.795199i	0.991837	−	0.127514i	$-0.0406998\pi$
$599$	1.77314e11	+	1.02372e11i	1.37732	+	0.795199i	0.385488	+	0.922713i	$0.374033\pi$
$600$		0			0
$601$	−8.57845e10	−	1.48583e11i	−0.657523	−	1.13886i	−0.981255	−	0.192714i	$-0.938271\pi$
$601$	−8.57845e10	−	1.48583e11i	−0.657523	−	1.13886i	0.323732	−	0.946149i	$-0.395062\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.58390e10	−	2.06917e10i	0.267507	−	0.154445i
$606$		0			0
$607$	4.16767e10	−	7.21861e10i	0.307000	−	0.531740i	−0.670705	−	0.741725i	$-0.734008\pi$
$607$	4.16767e10	−	7.21861e10i	0.307000	−	0.531740i	0.977705	+	0.209985i	$0.0673415\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	6.40062e9i		−	0.0459259i
$612$		0			0
$613$	6.83395e9			0.0483983			0.0241991	−	0.999707i	$-0.492296\pi$
$613$	6.83395e9			0.0483983			0.0241991	+	0.999707i	$0.492296\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.76348e10	−	1.01815e10i	−0.121683	−	0.0702539i	0.437923	−	0.899013i	$-0.355714\pi$
$617$	−1.76348e10	−	1.01815e10i	−0.121683	−	0.0702539i	−0.559606	+	0.828759i	$0.689048\pi$
$618$		0			0
$619$	4.13501e10	+	7.16205e10i	0.281653	+	0.487837i	0.971792	−	0.235840i	$-0.0757840\pi$
$619$	4.13501e10	+	7.16205e10i	0.281653	+	0.487837i	−0.690139	+	0.723677i	$0.742451\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.27281e8	−	7.34860e7i	0.000844915	−	0.000487812i
$624$		0			0
$625$	2.84219e9	−	4.92281e9i	0.0186266	−	0.0322621i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.58261e11i		−	1.01105i
$630$		0			0
$631$	−2.84224e11			−1.79284			−0.896422	−	0.443201i	$-0.853843\pi$
$631$	−2.84224e11			−1.79284			−0.896422	+	0.443201i	$0.853843\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.51645e9	−	3.76227e9i	−0.0400789	−	0.0231396i
$636$		0			0
$637$	3.67060e10	+	6.35767e10i	0.222936	+	0.386136i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.64904e11	−	9.52074e10i	0.976785	−	0.563947i	0.0754874	−	0.997147i	$-0.475949\pi$
$641$	1.64904e11	−	9.52074e10i	0.976785	−	0.563947i	0.901298	+	0.433199i	$0.142615\pi$
$642$		0			0
$643$	1.10690e11	−	1.91720e11i	0.647534	−	1.12156i	−0.336176	−	0.941799i	$-0.609134\pi$
$643$	1.10690e11	−	1.91720e11i	0.647534	−	1.12156i	0.983710	−	0.179763i	$-0.0575331\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1.25057e10i			0.0713660i	0.999363	+	0.0356830i	$0.0113607\pi$
$647$			1.25057e10i			0.0713660i	−0.999363	+	0.0356830i	$0.988639\pi$
$648$		0			0
$649$	−6.50367e10			−0.366589
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−5.21783e10	−	3.01251e10i	−0.286970	−	0.165682i	0.349604	−	0.936897i	$-0.386316\pi$
$653$	−5.21783e10	−	3.01251e10i	−0.286970	−	0.165682i	−0.636575	+	0.771215i	$0.719649\pi$
$654$		0			0
$655$	7.30835e10	+	1.26584e11i	0.397058	+	0.687724i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.23470e11	−	1.29020e11i	1.18489	−	0.684096i	0.227748	−	0.973720i	$-0.426864\pi$
$659$	2.23470e11	−	1.29020e11i	1.18489	−	0.684096i	0.957140	+	0.289624i	$0.0935303\pi$
$660$		0			0
$661$	1.10021e10	−	1.90562e10i	0.0576327	−	0.0998228i	−0.835770	−	0.549080i	$-0.814978\pi$
$661$	1.10021e10	−	1.90562e10i	0.0576327	−	0.0998228i	0.893402	+	0.449258i	$0.148311\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			4.05958e8i			0.00207585i
$666$		0			0
$667$	−2.07711e10			−0.104944
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.31570e11	+	7.59617e10i	0.649031	+	0.374718i
$672$		0			0
$673$	6.11542e10	+	1.05922e11i	0.298103	+	0.516329i	0.975702	−	0.219102i	$-0.0703129\pi$
$673$	6.11542e10	+	1.05922e11i	0.298103	+	0.516329i	−0.677599	+	0.735431i	$0.736980\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	2.98067e11	−	1.72089e11i	1.41892	−	0.819216i	0.422719	−	0.906261i	$-0.361076\pi$
$677$	2.98067e11	−	1.72089e11i	1.41892	−	0.819216i	0.996204	+	0.0870453i	$0.0277425\pi$
$678$		0			0
$679$	8.88440e6	−	1.53882e7i	4.17974e−5	−	7.23952e-5i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			4.83943e10i			0.222388i	0.993799	+	0.111194i	$0.0354675\pi$
$683$			4.83943e10i			0.222388i	−0.993799	+	0.111194i	$0.964532\pi$
$684$		0			0
$685$	2.50846e11			1.13932
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.02528e10	+	2.90135e10i	0.222989	+	0.128743i
$690$		0			0
$691$	1.10913e11	+	1.92107e11i	0.486486	+	0.842619i	0.999879	−	0.0155347i	$-0.00494504\pi$
$691$	1.10913e11	+	1.92107e11i	0.486486	+	0.842619i	−0.513393	+	0.858154i	$0.671612\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.90866e11	−	1.10196e11i	0.818066	−	0.472311i
$696$		0			0
$697$	1.42037e10	−	2.46016e10i	0.0601827	−	0.104240i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	4.62432e11i		−	1.91503i	−0.288380	−	0.957516i	$-0.593117\pi$
$701$		−	4.62432e11i		−	1.91503i	0.288380	−	0.957516i	$-0.406883\pi$
$702$		0			0
$703$	−3.08864e11			−1.26458
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.59460e8	−	1.49799e8i	−0.00103847	−	0.000599560i
$708$		0			0
$709$	7.09038e10	+	1.22809e11i	0.280598	+	0.486010i	0.971532	−	0.236908i	$-0.0761340\pi$
$709$	7.09038e10	+	1.22809e11i	0.280598	+	0.486010i	−0.690934	+	0.722918i	$0.742801\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.16621e10	−	4.71477e10i	0.315982	−	0.182432i
$714$		0			0
$715$	4.48984e10	−	7.77663e10i	0.171794	−	0.297555i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			3.42588e11i			1.28191i	0.767579	+	0.640954i	$0.221461\pi$
$719$			3.42588e11i			1.28191i	−0.767579	+	0.640954i	$0.778539\pi$
$720$		0			0
$721$	2.29546e8			0.000849432
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.08113e10	+	1.77889e10i	0.111521	+	0.0643869i
$726$		0			0
$727$	9.30839e9	+	1.61226e10i	0.0333225	+	0.0577162i	0.882206	−	0.470864i	$-0.156058\pi$
$727$	9.30839e9	+	1.61226e10i	0.0333225	+	0.0577162i	−0.848883	+	0.528580i	$0.822725\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6.39694e9	−	3.69328e9i	0.0224028	−	0.0129343i
$732$		0			0
$733$	−3.76770e10	+	6.52585e10i	−0.130515	+	0.226059i	−0.923875	−	0.382694i	$-0.874996\pi$
$733$	−3.76770e10	+	6.52585e10i	−0.130515	+	0.226059i	0.793360	+	0.608753i	$0.208330\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3.40039e11i		−	1.15255i
$738$		0			0
$739$	−3.72658e11			−1.24949			−0.624746	−	0.780828i	$-0.714797\pi$
$739$	−3.72658e11			−1.24949			−0.624746	+	0.780828i	$0.714797\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.49606e11	+	1.44110e11i	0.819029	+	0.472867i	0.850082	−	0.526651i	$-0.176553\pi$
$743$	2.49606e11	+	1.44110e11i	0.819029	+	0.472867i	−0.0310525	+	0.999518i	$0.509886\pi$
$744$		0			0
$745$	−8.30688e10	−	1.43879e11i	−0.269658	−	0.467061i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−5.02418e8	+	2.90071e8i	−0.00159639	+	0.000921674i
$750$		0			0
$751$	−7.52005e10	+	1.30251e11i	−0.236407	+	0.409470i	−0.959681	−	0.281092i	$-0.909303\pi$
$751$	−7.52005e10	+	1.30251e11i	−0.236407	+	0.409470i	0.723273	+	0.690562i	$0.242637\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1.54950e11i			0.476874i
$756$		0			0
$757$	−7.41765e10			−0.225883			−0.112941	−	0.993602i	$-0.536027\pi$
$757$	−7.41765e10			−0.225883			−0.112941	+	0.993602i	$0.536027\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.79569e11	−	1.03674e11i	−0.535417	−	0.309123i	0.207803	−	0.978171i	$-0.433369\pi$
$761$	−1.79569e11	−	1.03674e11i	−0.535417	−	0.309123i	−0.743219	+	0.669048i	$0.766702\pi$
$762$		0			0
$763$	5.67844e8	+	9.83535e8i	0.00167545	+	0.00290196i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.01453e10	+	2.31779e10i	−0.115999	+	0.0669720i
$768$		0			0
$769$	8.44424e10	−	1.46259e11i	0.241466	−	0.418231i	−0.719666	−	0.694320i	$-0.755705\pi$
$769$	8.44424e10	−	1.46259e11i	0.241466	−	0.418231i	0.961132	+	0.276089i	$0.0890386\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	2.24840e11i		−	0.629731i	−0.949136	−	0.314866i	$-0.898041\pi$
$773$		−	2.24840e11i		−	0.629731i	0.949136	−	0.314866i	$-0.101959\pi$
$774$		0			0
$775$	−1.61514e11			−0.447717
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.80129e10	−	2.77203e10i	−0.130379	−	0.0752744i
$780$		0			0
$781$	−2.99132e10	−	5.18112e10i	−0.0804005	−	0.139258i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	1.40175e11	−	8.09298e10i	0.369140	−	0.213123i
$786$		0			0
$787$	1.89324e11	−	3.27919e11i	0.493523	−	0.854807i	−0.506449	−	0.862270i	$-0.669042\pi$
$787$	1.89324e11	−	3.27919e11i	0.493523	−	0.854807i	0.999972	+	0.00746264i	$0.00237545\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	1.16551e9i		−	0.00297721i
$792$		0			0
$793$	1.08286e11			0.273828
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.46999e11	+	2.00340e11i	0.859992	+	0.496517i	0.864010	−	0.503475i	$-0.167946\pi$
$797$	3.46999e11	+	2.00340e11i	0.859992	+	0.496517i	−0.00401764	+	0.999992i	$0.501279\pi$
$798$		0			0
$799$	1.63917e10	+	2.83913e10i	0.0402195	+	0.0696623i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.00937e10	+	1.16011e10i	−0.0483279	+	0.0279021i
$804$		0			0
$805$	2.18641e8	−	3.78697e8i	0.000520652	−	0.000901795i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.78516e11i		−	1.58404i	−0.610495	−	0.792020i	$-0.709030\pi$
$809$		−	6.78516e11i		−	1.58404i	0.610495	−	0.792020i	$-0.290970\pi$
$810$		0			0
$811$	4.45203e11			1.02914			0.514570	−	0.857448i	$-0.327951\pi$
$811$	4.45203e11			1.02914			0.514570	+	0.857448i	$0.327951\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−6.55894e10	−	3.78680e10i	−0.148663	−	0.0858306i
$816$		0			0
$817$	−7.20786e9	−	1.24844e10i	−0.0161777	−	0.0280207i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.69000e10	−	1.55307e10i	0.0592078	−	0.0341836i	−0.470104	−	0.882611i	$-0.655784\pi$
$821$	2.69000e10	−	1.55307e10i	0.0592078	−	0.0341836i	0.529312	+	0.848427i	$0.322450\pi$
$822$		0			0
$823$	2.01664e11	−	3.49292e11i	0.439570	−	0.761358i	−0.558086	−	0.829783i	$-0.688464\pi$
$823$	2.01664e11	−	3.49292e11i	0.439570	−	0.761358i	0.997656	+	0.0684249i	$0.0217974\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			8.30509e11i			1.77551i	0.460319	+	0.887754i	$0.347735\pi$
$827$			8.30509e11i			1.77551i	−0.460319	+	0.887754i	$0.652265\pi$
$828$		0			0
$829$	5.27954e11			1.11784			0.558918	−	0.829223i	$-0.311217\pi$
$829$	5.27954e11			1.11784			0.558918	+	0.829223i	$0.311217\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.25634e11	−	1.88005e11i	−0.676316	−	0.390471i
$834$		0			0
$835$	1.81829e11	+	3.14938e11i	0.374040	+	0.647856i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−6.75268e11	+	3.89866e11i	−1.36279	+	0.786806i	−0.989994	−	0.141109i	$-0.954933\pi$
$839$	−6.75268e11	+	3.89866e11i	−1.36279	+	0.786806i	−0.372793	+	0.927914i	$0.621600\pi$
$840$		0			0
$841$	−2.38650e11	+	4.13353e11i	−0.477064	+	0.826299i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			2.57939e11i			0.505929i
$846$		0			0
$847$	−8.47286e8			−0.00164625
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.88123e11	+	1.66348e11i	0.549363	+	0.317175i
$852$		0			0
$853$	−4.29865e11	−	7.44549e11i	−0.811963	−	1.40636i	−0.911488	−	0.411327i	$-0.865066\pi$
$853$	−4.29865e11	−	7.44549e11i	−0.811963	−	1.40636i	0.0995248	−	0.995035i	$-0.468268\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−2.57962e11	+	1.48934e11i	−0.478225	+	0.276103i	−0.719677	−	0.694309i	$-0.755710\pi$
$857$	−2.57962e11	+	1.48934e11i	−0.478225	+	0.276103i	0.241451	+	0.970413i	$0.422377\pi$
$858$		0			0
$859$	−4.28589e9	+	7.42339e9i	−0.00787170	+	0.0136342i	−0.869934	−	0.493167i	$-0.835839\pi$
$859$	−4.28589e9	+	7.42339e9i	−0.00787170	+	0.0136342i	0.862063	+	0.506802i	$0.169172\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			6.98002e11i			1.25838i	0.777250	+	0.629192i	$0.216614\pi$
$863$			6.98002e11i			1.25838i	−0.777250	+	0.629192i	$0.783386\pi$
$864$		0			0
$865$	5.75970e11			1.02881
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.02259e11	−	4.05449e11i	−1.23145	−	0.710980i
$870$		0			0
$871$	−1.21184e11	−	2.09897e11i	−0.210558	−	0.364698i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.72749e9	+	9.97369e8i	−0.00294703	+	0.00170147i
$876$		0			0
$877$	−4.92366e11	+	8.52802e11i	−0.832318	+	1.44162i	0.0638771	+	0.997958i	$0.479653\pi$
$877$	−4.92366e11	+	8.52802e11i	−0.832318	+	1.44162i	−0.896195	+	0.443660i	$0.853680\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	3.95599e11i		−	0.656677i	−0.944560	−	0.328338i	$-0.893511\pi$
$881$		−	3.95599e11i		−	0.656677i	0.944560	−	0.328338i	$-0.106489\pi$
$882$		0			0
$883$	−4.59500e11			−0.755863			−0.377932	−	0.925834i	$-0.623365\pi$
$883$	−4.59500e11			−0.755863			−0.377932	+	0.925834i	$0.623365\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.06335e11	+	2.34598e11i	0.656433	+	0.378992i	0.790916	−	0.611924i	$-0.209604\pi$
$887$	4.06335e11	+	2.34598e11i	0.656433	+	0.378992i	−0.134484	+	0.990916i	$0.542938\pi$
$888$		0			0
$889$	7.70291e7	+	1.33418e8i	0.000123324	+	0.000213603i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.54088e10	−	3.19903e10i	0.0871311	−	0.0503052i
$894$		0			0
$895$	3.80031e11	−	6.58232e11i	0.592279	−	1.02586i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1.04175e11i			0.159487i
$900$		0			0
$901$	−2.97209e11			−0.450986
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−6.64061e11	−	3.83396e11i	−0.989950	−	0.571548i
$906$		0			0
$907$	−2.37393e10	−	4.11177e10i	−0.0350783	−	0.0607574i	0.847953	−	0.530071i	$-0.177835\pi$
$907$	−2.37393e10	−	4.11177e10i	−0.0350783	−	0.0607574i	−0.883032	+	0.469314i	$0.844501\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−9.85584e11	+	5.69027e11i	−1.43094	+	0.826151i	−0.997192	−	0.0748848i	$-0.976141\pi$
$911$	−9.85584e11	+	5.69027e11i	−1.43094	+	0.826151i	−0.433744	+	0.901036i	$0.642808\pi$
$912$		0			0
$913$	−5.57963e11	+	9.66421e11i	−0.803013	+	1.39086i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2.99263e9i		−	0.00423230i
$918$		0			0
$919$	−6.52263e11			−0.914452			−0.457226	−	0.889350i	$-0.651157\pi$
$919$	−6.52263e11			−0.914452			−0.457226	+	0.889350i	$0.651157\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3.69292e10	−	2.13211e10i	−0.0508819	−	0.0293767i
$924$		0			0
$925$	−2.84929e11	−	4.93512e11i	−0.389197	−	0.674110i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.04709e11	+	2.91394e11i	−0.677608	+	0.391217i	−0.798953	−	0.601393i	$-0.794613\pi$
$929$	−5.04709e11	+	2.91394e11i	−0.677608	+	0.391217i	0.121345	+	0.992610i	$0.461279\pi$
$930$		0			0
$931$	−3.66913e11	+	6.35513e11i	−0.488388	+	0.845913i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			4.59931e11i			0.601792i
$936$		0			0
$937$	8.72906e11			1.13242			0.566212	−	0.824259i	$-0.308408\pi$
$937$	8.72906e11			1.13242			0.566212	+	0.824259i	$0.308408\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.09406e11	−	2.36370e11i	−0.522150	−	0.301463i	0.215664	−	0.976468i	$-0.430808\pi$
$941$	−4.09406e11	−	2.36370e11i	−0.522150	−	0.301463i	−0.737814	+	0.675004i	$0.764142\pi$
$942$		0			0
$943$	2.98591e10	+	5.17174e10i	0.0377598	+	0.0654019i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.41756e11	+	4.28253e11i	−0.922277	+	0.532477i	−0.884361	−	0.466804i	$-0.845405\pi$
$947$	−7.41756e11	+	4.28253e11i	−0.922277	+	0.532477i	−0.0379160	+	0.999281i	$0.512072\pi$
$948$		0			0
$949$	−8.26885e9	+	1.43221e10i	−0.0101948	+	0.0176580i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			7.57655e11i			0.918544i	0.888296	+	0.459272i	$0.151890\pi$
$953$			7.57655e11i			0.918544i	−0.888296	+	0.459272i	$0.848110\pi$
$954$		0			0
$955$	8.60967e11			1.03508
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.44777e9	−	2.56792e9i	−0.00525858	−	0.00303604i
$960$		0			0
$961$	1.89982e11	+	3.29059e11i	0.222751	+	0.385816i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.12885e10	+	6.51741e9i	−0.0130175	+	0.00751564i
$966$		0			0
$967$	−4.81667e11	+	8.34271e11i	−0.550859	+	0.954116i	0.447354	+	0.894357i	$0.352367\pi$
$967$	−4.81667e11	+	8.34271e11i	−0.550859	+	0.954116i	−0.998213	+	0.0597591i	$0.980967\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1.04836e12i			1.17932i	0.807650	+	0.589662i	$0.200739\pi$
$971$			1.04836e12i			1.17932i	−0.807650	+	0.589662i	$0.799261\pi$
$972$		0			0
$973$	−4.51233e9			−0.00503443
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−3.42076e10	−	1.97498e10i	−0.0375444	−	0.0216762i	0.481110	−	0.876660i	$-0.340234\pi$
$977$	−3.42076e10	−	1.97498e10i	−0.0375444	−	0.0216762i	−0.518655	+	0.854984i	$0.673567\pi$
$978$		0			0
$979$	−1.62484e11	−	2.81430e11i	−0.176880	−	0.306366i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4.01120e11	+	2.31587e11i	−0.429596	+	0.248027i	−0.699175	−	0.714951i	$-0.746449\pi$
$983$	−4.01120e11	+	2.31587e11i	−0.429596	+	0.248027i	0.269578	+	0.962978i	$0.413116\pi$
$984$		0			0
$985$	1.19712e10	−	2.07348e10i	0.0127173	−	0.0220270i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			1.55280e10i			0.0162304i
$990$		0			0
$991$	1.38294e12			1.43386			0.716932	−	0.697143i	$-0.245546\pi$
$991$	1.38294e12			1.43386			0.716932	+	0.697143i	$0.245546\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−4.54877e11	−	2.62624e11i	−0.464090	−	0.267942i
$996$		0			0
$997$	−4.12488e11	−	7.14450e11i	−0.417475	−	0.723088i	0.578209	−	0.815888i	$-0.303752\pi$
$997$	−4.12488e11	−	7.14450e11i	−0.417475	−	0.723088i	−0.995685	+	0.0927999i	$0.970418\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.9.g.h.53.10		32
3.2	odd	2	inner	324.9.g.h.53.7		32
9.2	odd	6	inner	324.9.g.h.269.10		32
9.4	even	3		324.9.c.b.161.7	✓	16
9.5	odd	6		324.9.c.b.161.10	yes	16
9.7	even	3	inner	324.9.g.h.269.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.9.c.b.161.7	✓	16	9.4	even	3
324.9.c.b.161.10	yes	16	9.5	odd	6
324.9.g.h.53.7		32	3.2	odd	2	inner
324.9.g.h.53.10		32	1.1	even	1	trivial
324.9.g.h.269.7		32	9.7	even	3	inner
324.9.g.h.269.10		32	9.2	odd	6	inner

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(341.792 + 197.334i) q^{5} +(-4.04023 - 6.99788i) q^{7} +(-15472.9 + 8933.30i) q^{11} +(-6367.34 + 11028.6i) q^{13} -65225.9i q^{17} -127296. q^{19} +(118747. + 68558.8i) q^{23} +(-117431. - 203397. i) q^{25} +(-131189. + 75741.9i) q^{29} +(343848. - 595563. i) q^{31} -3189.09i q^{35} +2.42635e6 q^{37} +(377176. + 217762. i) q^{41} +(56622.9 + 98073.7i) q^{43} +(-435276. + 251307. i) q^{47} +(2.88237e6 - 4.99241e6i) q^{49} -4.55661e6i q^{53} -7.05137e6 q^{55} +(3.15244e6 + 1.82006e6i) q^{59} +(-4.25161e6 - 7.36400e6i) q^{61} +(-4.35261e6 + 2.51298e6i) q^{65} +(-9.51606e6 + 1.64823e7i) q^{67} +3.34851e6i q^{71} +1.29864e6 q^{73} +(125028. + 72185.1i) q^{77} +(2.26931e7 + 3.93057e7i) q^{79} +(5.40909e7 - 3.12294e7i) q^{83} +(1.28713e7 - 2.22937e7i) q^{85} +1.81886e7i q^{89} +102902. q^{91} +(-4.35087e7 - 2.51198e7i) q^{95} +(1.09949e6 + 1.90438e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3692 q^{7} + 50380 q^{13} + 370216 q^{19} + 1958288 q^{25} + 285568 q^{31} - 5820104 q^{37} + 8122852 q^{43} - 14602560 q^{49} + 76640328 q^{55} + 5700244 q^{61} + 6328540 q^{67} - 103457408 q^{73}+ \cdots + 181751968 q^{97}+O(q^{100}) \) 32 * q - 3692 * q^7 + 50380 * q^13 + 370216 * q^19 + 1958288 * q^25 + 285568 * q^31 - 5820104 * q^37 + 8122852 * q^43 - 14602560 * q^49 + 76640328 * q^55 + 5700244 * q^61 + 6328540 * q^67 - 103457408 * q^73 - 168603092 * q^79 + 96322644 * q^85 - 399487432 * q^91 + 181751968 * q^97

Introduction

L-functions

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