LMFDB - Embedded newform 252.9.c.a.197.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,9,Mod(197,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("252.197");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	252.c (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:	$102.659409735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + \cdots + 19\!\cdots\!00 $ x^16 + 4002260x^14 + 6534459751956x^12 + 5613923146579405376x^10 + 2733728904154246859079616x^8 + 757873148017661341349205888000x^6 + 113644318422397913452531577312640000x^4 + 8098650340007618970326973663348480000000*x^2 + 199066230990417435753898292645889849600000000
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{37}\cdot 7^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.14
Root			$899.394i$ of defining polynomial
Character	$\chi$	$=$	252.197
Dual form			252.9.c.a.197.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+899.394i q^{5} +907.493 q^{7} +O(q^{10})$	$q+899.394i q^{5} +907.493 q^{7} -21934.5i q^{11} +20852.6 q^{13} -54914.5i q^{17} -112287. q^{19} +364864. i q^{23} -418285. q^{25} +493187. i q^{29} -947146. q^{31} +816194. i q^{35} -2.31637e6 q^{37} -336127. i q^{41} -164303. q^{43} +1.21616e6i q^{47} +823543. q^{49} -2.50757e6i q^{53} +1.97278e7 q^{55} -1.27740e7i q^{59} -1.01258e7 q^{61} +1.87547e7i q^{65} +7.53739e6 q^{67} -4.21448e7i q^{71} -3.47745e7 q^{73} -1.99054e7i q^{77} +1.95932e6 q^{79} -2.01258e7i q^{83} +4.93898e7 q^{85} +1.15659e7i q^{89} +1.89236e7 q^{91} -1.00990e8i q^{95} -9.91217e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) $ 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	899.394i	1.43903i	0.694477	+	0.719515i	$0.255636\pi$
$5$	899.394i	1.43903i	−0.694477	+	0.719515i	$0.744364\pi$
$6$	0	0
$7$	907.493	0.377964
$7$	907.493	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 21934.5i	− 1.49816i	−0.662481	−	0.749079i	$-0.730496\pi$
$11$	− 21934.5i	− 1.49816i	0.662481	−	0.749079i	$-0.269504\pi$
$12$	0	0
$13$	20852.6	0.730108	0.365054	−	0.930986i	$-0.381051\pi$
$13$	20852.6	0.730108	0.365054	+	0.930986i	$0.381051\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 54914.5i	− 0.657493i	−0.944418	−	0.328747i	$-0.893374\pi$
$17$	− 54914.5i	− 0.657493i	0.944418	−	0.328747i	$-0.106626\pi$
$18$	0	0
$19$	−112287.	−0.861620	−0.430810	−	0.902443i	$-0.641772\pi$
$19$	−112287.	−0.861620	−0.430810	+	0.902443i	$0.641772\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	364864.i	1.30383i	0.758294	+	0.651913i	$0.226033\pi$
$23$	364864.i	1.30383i	−0.758294	+	0.651913i	$0.773967\pi$
$24$	0	0
$25$	−418285.	−1.07081
$26$	0	0
$27$	0	0
$28$	0	0
$29$	493187.i	0.697299i	0.937253	+	0.348650i	$0.113360\pi$
$29$	493187.i	0.697299i	−0.937253	+	0.348650i	$0.886640\pi$
$30$	0	0
$31$	−947146.	−1.02558	−0.512791	−	0.858514i	$-0.671388\pi$
$31$	−947146.	−1.02558	−0.512791	+	0.858514i	$0.671388\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	816194.i	0.543902i
$36$	0	0
$37$	−2.31637e6	−1.23595	−0.617976	−	0.786197i	$-0.712047\pi$
$37$	−2.31637e6	−1.23595	−0.617976	+	0.786197i	$0.712047\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 336127.i	− 0.118951i	−0.998230	−	0.0594755i	$-0.981057\pi$
$41$	− 336127.i	− 0.118951i	0.998230	−	0.0594755i	$-0.0189428\pi$
$42$	0	0
$43$	−164303.	−0.0480587	−0.0240294	−	0.999711i	$-0.507650\pi$
$43$	−164303.	−0.0480587	−0.0240294	+	0.999711i	$0.507650\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.21616e6i	0.249229i	0.992205	+	0.124614i	$0.0397694\pi$
$47$	1.21616e6i	0.249229i	−0.992205	+	0.124614i	$0.960231\pi$
$48$	0	0
$49$	823543.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.50757e6i	− 0.317797i	−0.987295	−	0.158898i	$-0.949206\pi$
$53$	− 2.50757e6i	− 0.317797i	0.987295	−	0.158898i	$-0.0507943\pi$
$54$	0	0
$55$	1.97278e7	2.15589
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.27740e7i	− 1.05419i	−0.849807	−	0.527094i	$-0.823282\pi$
$59$	− 1.27740e7i	− 1.05419i	0.849807	−	0.527094i	$-0.176718\pi$
$60$	0	0
$61$	−1.01258e7	−0.731323	−0.365661	−	0.930748i	$-0.619157\pi$
$61$	−1.01258e7	−0.731323	−0.365661	+	0.930748i	$0.619157\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.87547e7i	1.05065i
$66$	0	0
$67$	7.53739e6	0.374043	0.187022	−	0.982356i	$-0.440117\pi$
$67$	7.53739e6	0.374043	0.187022	+	0.982356i	$0.440117\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 4.21448e7i	− 1.65848i	−0.558892	−	0.829240i	$-0.688774\pi$
$71$	− 4.21448e7i	− 1.65848i	0.558892	−	0.829240i	$-0.311226\pi$
$72$	0	0
$73$	−3.47745e7	−1.22453	−0.612266	−	0.790652i	$-0.709742\pi$
$73$	−3.47745e7	−1.22453	−0.612266	+	0.790652i	$0.709742\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.99054e7i	− 0.566250i
$78$	0	0
$79$	1.95932e6	0.0503033	0.0251517	−	0.999684i	$-0.491993\pi$
$79$	1.95932e6	0.0503033	0.0251517	+	0.999684i	$0.491993\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 2.01258e7i	− 0.424073i	−0.977262	−	0.212037i	$-0.931990\pi$
$83$	− 2.01258e7i	− 0.424073i	0.977262	−	0.212037i	$-0.0680096\pi$
$84$	0	0
$85$	4.93898e7	0.946153
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.15659e7i	0.184341i	0.995743	+	0.0921703i	$0.0293804\pi$
$89$	1.15659e7i	0.184341i	−0.995743	+	0.0921703i	$0.970620\pi$
$90$	0	0
$91$	1.89236e7	0.275955
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.00990e8i	− 1.23990i
$96$	0	0
$97$	−9.91217e7	−1.11965	−0.559825	−	0.828611i	$-0.689131\pi$
$97$	−9.91217e7	−1.11965	−0.559825	+	0.828611i	$0.689131\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.13987e8i	1.09539i	0.836679	+	0.547694i	$0.184494\pi$
$101$	1.13987e8i	1.09539i	−0.836679	+	0.547694i	$0.815506\pi$
$102$	0	0
$103$	−1.40209e8	−1.24574	−0.622868	−	0.782327i	$-0.714033\pi$
$103$	−1.40209e8	−1.24574	−0.622868	+	0.782327i	$0.714033\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.47465e7i	0.646527i	0.946309	+	0.323263i	$0.104780\pi$
$107$	8.47465e7i	0.646527i	−0.946309	+	0.323263i	$0.895220\pi$
$108$	0	0
$109$	−6.16339e7	−0.436630	−0.218315	−	0.975878i	$-0.570056\pi$
$109$	−6.16339e7	−0.436630	−0.218315	+	0.975878i	$0.570056\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1.48925e8i	− 0.913382i	−0.889625	−	0.456691i	$-0.849034\pi$
$113$	− 1.48925e8i	− 0.913382i	0.889625	−	0.456691i	$-0.150966\pi$
$114$	0	0
$115$	−3.28157e8	−1.87625
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 4.98345e7i	− 0.248509i
$120$	0	0
$121$	−2.66765e8	−1.24448
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.48770e7i	− 0.101896i
$126$	0	0
$127$	4.37067e8	1.68009	0.840046	−	0.542515i	$-0.182528\pi$
$127$	4.37067e8	1.68009	0.840046	+	0.542515i	$0.182528\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 3.99853e8i	− 1.35774i	−0.734261	−	0.678868i	$-0.762471\pi$
$131$	− 3.99853e8i	− 1.35774i	0.734261	−	0.678868i	$-0.237529\pi$
$132$	0	0
$133$	−1.01900e8	−0.325662
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 4.51026e8i	− 1.28032i	−0.768240	−	0.640161i	$-0.778867\pi$
$137$	− 4.51026e8i	− 1.28032i	0.768240	−	0.640161i	$-0.221133\pi$
$138$	0	0
$139$	−6.59839e7	−0.176758	−0.0883789	−	0.996087i	$-0.528169\pi$
$139$	−6.59839e7	−0.176758	−0.0883789	+	0.996087i	$0.528169\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.57392e8i	− 1.09382i
$144$	0	0
$145$	−4.43569e8	−1.00344
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.48851e8i	− 1.51932i	−0.650319	−	0.759662i	$-0.725365\pi$
$149$	− 7.48851e8i	− 1.51932i	0.650319	−	0.759662i	$-0.274635\pi$
$150$	0	0
$151$	8.69152e8	1.67181	0.835907	−	0.548871i	$-0.184942\pi$
$151$	8.69152e8	1.67181	0.835907	+	0.548871i	$0.184942\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8.51858e8i	− 1.47584i
$156$	0	0
$157$	3.43252e8	0.564956	0.282478	−	0.959274i	$-0.408844\pi$
$157$	3.43252e8	0.564956	0.282478	+	0.959274i	$0.408844\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.31112e8i	0.492800i
$162$	0	0
$163$	−1.03494e9	−1.46610	−0.733052	−	0.680173i	$-0.761905\pi$
$163$	−1.03494e9	−1.46610	−0.733052	+	0.680173i	$0.761905\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.55929e8i	− 1.10045i	−0.835015	−	0.550227i	$-0.814541\pi$
$167$	− 8.55929e8i	− 1.10045i	0.835015	−	0.550227i	$-0.185459\pi$
$168$	0	0
$169$	−3.80900e8	−0.466943
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.21109e9i	1.35204i	0.736882	+	0.676021i	$0.236297\pi$
$173$	1.21109e9i	1.35204i	−0.736882	+	0.676021i	$0.763703\pi$
$174$	0	0
$175$	−3.79590e8	−0.404728
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.65570e8i	− 0.161276i	−0.996743	−	0.0806381i	$-0.974304\pi$
$179$	− 1.65570e8i	− 0.161276i	0.996743	−	0.0806381i	$-0.0256958\pi$
$180$	0	0
$181$	−1.87197e9	−1.74416	−0.872078	−	0.489367i	$-0.837228\pi$
$181$	−1.87197e9	−1.74416	−0.872078	+	0.489367i	$0.837228\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 2.08333e9i	− 1.77857i
$186$	0	0
$187$	−1.20452e9	−0.985029
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.54193e8i	0.416416i	0.978085	+	0.208208i	$0.0667631\pi$
$191$	5.54193e8i	0.416416i	−0.978085	+	0.208208i	$0.933237\pi$
$192$	0	0
$193$	−2.65475e9	−1.91335	−0.956677	−	0.291152i	$-0.905961\pi$
$193$	−2.65475e9	−1.91335	−0.956677	+	0.291152i	$0.905961\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.36750e9i	− 1.57190i	−0.618292	−	0.785949i	$-0.712175\pi$
$197$	− 2.36750e9i	− 1.57190i	0.618292	−	0.785949i	$-0.287825\pi$
$198$	0	0
$199$	2.24978e9	1.43459	0.717294	−	0.696771i	$-0.245380\pi$
$199$	2.24978e9	1.43459	0.717294	+	0.696771i	$0.245380\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.47563e8i	0.263554i
$204$	0	0
$205$	3.02311e8	0.171174
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.46297e9i	1.29084i
$210$	0	0
$211$	1.61666e9	0.815624	0.407812	−	0.913066i	$-0.366292\pi$
$211$	1.61666e9	0.815624	0.407812	+	0.913066i	$0.366292\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.47773e8i	− 0.0691580i
$216$	0	0
$217$	−8.59528e8	−0.387633
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.14511e9i	− 0.480041i
$222$	0	0
$223$	−4.03660e9	−1.63229	−0.816143	−	0.577850i	$-0.803892\pi$
$223$	−4.03660e9	−1.63229	−0.816143	+	0.577850i	$0.803892\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.28153e8i	− 0.311894i	−0.987765	−	0.155947i	$-0.950157\pi$
$227$	− 8.28153e8i	− 0.311894i	0.987765	−	0.155947i	$-0.0498429\pi$
$228$	0	0
$229$	−2.05916e9	−0.748770	−0.374385	−	0.927273i	$-0.622146\pi$
$229$	−2.05916e9	−0.748770	−0.374385	+	0.927273i	$0.622146\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3.06700e9i	− 1.04062i	−0.853979	−	0.520308i	$-0.825817\pi$
$233$	− 3.06700e9i	− 1.04062i	0.853979	−	0.520308i	$-0.174183\pi$
$234$	0	0
$235$	−1.09380e9	−0.358648
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 3.90569e9i	− 1.19703i	−0.801111	−	0.598516i	$-0.795757\pi$
$239$	− 3.90569e9i	− 1.19703i	0.801111	−	0.598516i	$-0.204243\pi$
$240$	0	0
$241$	5.96315e9	1.76769	0.883847	−	0.467775i	$-0.154944\pi$
$241$	5.96315e9	1.76769	0.883847	+	0.467775i	$0.154944\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	7.40690e8i	0.205576i
$246$	0	0
$247$	−2.34148e9	−0.629075
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.52021e9i	0.634954i	0.948266	+	0.317477i	$0.102836\pi$
$251$	2.52021e9i	0.634954i	−0.948266	+	0.317477i	$0.897164\pi$
$252$	0	0
$253$	8.00312e9	1.95334
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.17920e8i	0.0499533i	0.999688	+	0.0249766i	$0.00795114\pi$
$257$	2.17920e8i	0.0499533i	−0.999688	+	0.0249766i	$0.992049\pi$
$258$	0	0
$259$	−2.10209e9	−0.467146
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.78536e9i	0.791196i	0.918424	+	0.395598i	$0.129463\pi$
$263$	3.78536e9i	0.791196i	−0.918424	+	0.395598i	$0.870537\pi$
$264$	0	0
$265$	2.25529e9	0.457320
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.04980e9i	− 1.34638i	−0.739469	−	0.673190i	$-0.764923\pi$
$269$	− 7.04980e9i	− 1.34638i	0.739469	−	0.673190i	$-0.235077\pi$
$270$	0	0
$271$	−1.60874e8	−0.0298270	−0.0149135	−	0.999889i	$-0.504747\pi$
$271$	−1.60874e8	−0.0298270	−0.0149135	+	0.999889i	$0.504747\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.17488e9i	1.60424i
$276$	0	0
$277$	9.44864e9	1.60491	0.802454	−	0.596714i	$-0.203527\pi$
$277$	9.44864e9	1.60491	0.802454	+	0.596714i	$0.203527\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.14862e9i	0.825783i	0.910780	+	0.412891i	$0.135481\pi$
$281$	5.14862e9i	0.825783i	−0.910780	+	0.412891i	$0.864519\pi$
$282$	0	0
$283$	−4.72901e9	−0.737266	−0.368633	−	0.929575i	$-0.620174\pi$
$283$	−4.72901e9	−0.737266	−0.368633	+	0.929575i	$0.620174\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.05033e8i	− 0.0449592i
$288$	0	0
$289$	3.96016e9	0.567703
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.31304e9i	0.178158i	0.996025	+	0.0890791i	$0.0283924\pi$
$293$	1.31304e9i	0.178158i	−0.996025	+	0.0890791i	$0.971608\pi$
$294$	0	0
$295$	1.14888e10	1.51701
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.60837e9i	0.951934i
$300$	0	0
$301$	−1.49104e8	−0.0181645
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 9.10707e9i	− 1.05240i
$306$	0	0
$307$	−2.29371e9	−0.258217	−0.129108	−	0.991630i	$-0.541212\pi$
$307$	−2.29371e9	−0.258217	−0.129108	+	0.991630i	$0.541212\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.29318e10i	1.38235i	0.722687	+	0.691176i	$0.242907\pi$
$311$	1.29318e10i	1.38235i	−0.722687	+	0.691176i	$0.757093\pi$
$312$	0	0
$313$	−1.63693e10	−1.70550	−0.852751	−	0.522317i	$-0.825068\pi$
$313$	−1.63693e10	−1.70550	−0.852751	+	0.522317i	$0.825068\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.30871e10i	1.29601i	0.761637	+	0.648004i	$0.224396\pi$
$317$	1.30871e10i	1.29601i	−0.761637	+	0.648004i	$0.775604\pi$
$318$	0	0
$319$	1.08178e10	1.04466
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.16619e9i	0.566509i
$324$	0	0
$325$	−8.72233e9	−0.781806
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.10365e9i	0.0941996i
$330$	0	0
$331$	−1.33867e10	−1.11522	−0.557611	−	0.830102i	$-0.688282\pi$
$331$	−1.33867e10	−1.11522	−0.557611	+	0.830102i	$0.688282\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.77908e9i	0.538260i
$336$	0	0
$337$	−6.92070e9	−0.536575	−0.268288	−	0.963339i	$-0.586458\pi$
$337$	−6.92070e9	−0.536575	−0.268288	+	0.963339i	$0.586458\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.07752e10i	1.53648i
$342$	0	0
$343$	7.47359e8	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.48176e9i	0.378096i	0.981968	+	0.189048i	$0.0605402\pi$
$347$	5.48176e9i	0.378096i	−0.981968	+	0.189048i	$0.939460\pi$
$348$	0	0
$349$	1.62728e10	1.09689	0.548443	−	0.836188i	$-0.315221\pi$
$349$	1.62728e10	1.09689	0.548443	+	0.836188i	$0.315221\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.57447e10i	1.65802i	0.559237	+	0.829008i	$0.311094\pi$
$353$	2.57447e10i	1.65802i	−0.559237	+	0.829008i	$0.688906\pi$
$354$	0	0
$355$	3.79048e10	2.38660
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 5.68498e9i	− 0.342256i	−0.985249	−	0.171128i	$-0.945259\pi$
$359$	− 5.68498e9i	− 0.342256i	0.985249	−	0.171128i	$-0.0547412\pi$
$360$	0	0
$361$	−4.37515e9	−0.257611
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.12760e10i	− 1.76214i
$366$	0	0
$367$	1.12442e10	0.619820	0.309910	−	0.950766i	$-0.399701\pi$
$367$	1.12442e10	0.619820	0.309910	+	0.950766i	$0.399701\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2.27560e9i	− 0.120116i
$372$	0	0
$373$	1.82711e10	0.943910	0.471955	−	0.881623i	$-0.343549\pi$
$373$	1.82711e10	0.943910	0.471955	+	0.881623i	$0.343549\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.02842e10i	0.509104i
$378$	0	0
$379$	2.30958e10	1.11938	0.559689	−	0.828703i	$-0.310921\pi$
$379$	2.30958e10	1.11938	0.559689	+	0.828703i	$0.310921\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.12755e10i	− 0.988745i	−0.869250	−	0.494372i	$-0.835398\pi$
$383$	− 2.12755e10i	− 0.988745i	0.869250	−	0.494372i	$-0.164602\pi$
$384$	0	0
$385$	1.79028e10	0.814852
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.68614e9i	0.160981i	0.996755	+	0.0804903i	$0.0256486\pi$
$389$	3.68614e9i	0.160981i	−0.996755	+	0.0804903i	$0.974351\pi$
$390$	0	0
$391$	2.00363e10	0.857257
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.76220e9i	0.0723880i
$396$	0	0
$397$	2.57692e9	0.103738	0.0518692	−	0.998654i	$-0.483482\pi$
$397$	2.57692e9	0.103738	0.0518692	+	0.998654i	$0.483482\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.27654e10i	− 0.493693i	−0.969055	−	0.246847i	$-0.920606\pi$
$401$	− 1.27654e10i	− 0.493693i	0.969055	−	0.246847i	$-0.0793944\pi$
$402$	0	0
$403$	−1.97505e10	−0.748785
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.08085e10i	1.85165i
$408$	0	0
$409$	−1.66538e10	−0.595142	−0.297571	−	0.954700i	$-0.596177\pi$
$409$	−1.66538e10	−0.595142	−0.297571	+	0.954700i	$0.596177\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1.15923e10i	− 0.398446i
$414$	0	0
$415$	1.81010e10	0.610254
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.06318e10i	− 0.669391i	−0.942326	−	0.334696i	$-0.891367\pi$
$419$	− 2.06318e10i	− 0.669391i	0.942326	−	0.334696i	$-0.108633\pi$
$420$	0	0
$421$	−5.94606e10	−1.89278	−0.946391	−	0.323023i	$-0.895301\pi$
$421$	−5.94606e10	−1.89278	−0.946391	+	0.323023i	$0.895301\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.29699e10i	0.704050i
$426$	0	0
$427$	−9.18907e9	−0.276414
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3.73736e10i	1.08307i	0.840679	+	0.541534i	$0.182156\pi$
$431$	3.73736e10i	1.08307i	−0.840679	+	0.541534i	$0.817844\pi$
$432$	0	0
$433$	−3.44971e10	−0.981365	−0.490683	−	0.871338i	$-0.663253\pi$
$433$	−3.44971e10	−0.981365	−0.490683	+	0.871338i	$0.663253\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.09696e10i	− 1.12340i
$438$	0	0
$439$	5.40944e10	1.45644	0.728222	−	0.685341i	$-0.240347\pi$
$439$	5.40944e10	1.45644	0.728222	+	0.685341i	$0.240347\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.18604e10i	− 0.827249i	−0.910448	−	0.413625i	$-0.864263\pi$
$443$	− 3.18604e10i	− 0.827249i	0.910448	−	0.413625i	$-0.135737\pi$
$444$	0	0
$445$	−1.04023e10	−0.265272
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.50863e10i	1.35537i	0.735352	+	0.677686i	$0.237017\pi$
$449$	5.50863e10i	1.35537i	−0.735352	+	0.677686i	$0.762983\pi$
$450$	0	0
$451$	−7.37279e9	−0.178207
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.70198e10i	0.397107i
$456$	0	0
$457$	−4.41618e10	−1.01247	−0.506234	−	0.862396i	$-0.668963\pi$
$457$	−4.41618e10	−1.01247	−0.506234	+	0.862396i	$0.668963\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 7.85244e10i	− 1.73861i	−0.494280	−	0.869303i	$-0.664568\pi$
$461$	− 7.85244e10i	− 1.73861i	0.494280	−	0.869303i	$-0.335432\pi$
$462$	0	0
$463$	−7.61686e10	−1.65750	−0.828748	−	0.559623i	$-0.810946\pi$
$463$	−7.61686e10	−1.65750	−0.828748	+	0.559623i	$0.810946\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 3.86924e10i	− 0.813502i	−0.913539	−	0.406751i	$-0.866662\pi$
$467$	− 3.86924e10i	− 0.813502i	0.913539	−	0.406751i	$-0.133338\pi$
$468$	0	0
$469$	6.84013e9	0.141375
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.60391e9i	0.0719996i
$474$	0	0
$475$	4.69680e10	0.922630
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 3.21577e9i	− 0.0610862i	−0.999533	−	0.0305431i	$-0.990276\pi$
$479$	− 3.21577e9i	− 0.0610862i	0.999533	−	0.0305431i	$-0.00972368\pi$
$480$	0	0
$481$	−4.83024e10	−0.902378
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 8.91495e10i	− 1.61121i
$486$	0	0
$487$	−6.36460e10	−1.13150	−0.565751	−	0.824576i	$-0.691414\pi$
$487$	−6.36460e10	−1.13150	−0.565751	+	0.824576i	$0.691414\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.53919e10i	1.29717i	0.761140	+	0.648587i	$0.224640\pi$
$491$	7.53919e10i	1.29717i	−0.761140	+	0.648587i	$0.775360\pi$
$492$	0	0
$493$	2.70831e10	0.458470
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.82461e10i	− 0.626847i
$498$	0	0
$499$	6.00879e10	0.969136	0.484568	−	0.874754i	$-0.338977\pi$
$499$	6.00879e10	0.969136	0.484568	+	0.874754i	$0.338977\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.02391e10i	1.09725i	0.836068	+	0.548626i	$0.184849\pi$
$503$	7.02391e10i	1.09725i	−0.836068	+	0.548626i	$0.815151\pi$
$504$	0	0
$505$	−1.02519e11	−1.57630
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 7.49035e10i	− 1.11591i	−0.829870	−	0.557957i	$-0.811585\pi$
$509$	− 7.49035e10i	− 1.11591i	0.829870	−	0.557957i	$-0.188415\pi$
$510$	0	0
$511$	−3.15576e10	−0.462829
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.26103e11i	− 1.79265i
$516$	0	0
$517$	2.66758e10	0.373384
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 8.53205e10i	− 1.15798i	−0.815333	−	0.578992i	$-0.803446\pi$
$521$	− 8.53205e10i	− 1.15798i	0.815333	−	0.578992i	$-0.196554\pi$
$522$	0	0
$523$	6.90127e10	0.922407	0.461203	−	0.887294i	$-0.347418\pi$
$523$	6.90127e10	0.922407	0.461203	+	0.887294i	$0.347418\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.20120e10i	0.674313i
$528$	0	0
$529$	−5.48148e10	−0.699963
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7.00912e9i	− 0.0868470i
$534$	0	0
$535$	−7.62205e10	−0.930372
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.80640e10i	− 0.214023i
$540$	0	0
$541$	2.81530e10	0.328652	0.164326	−	0.986406i	$-0.447455\pi$
$541$	2.81530e10	0.328652	0.164326	+	0.986406i	$0.447455\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 5.54332e10i	− 0.628324i
$546$	0	0
$547$	−1.27027e10	−0.141888	−0.0709440	−	0.997480i	$-0.522601\pi$
$547$	−1.27027e10	−0.141888	−0.0709440	+	0.997480i	$0.522601\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 5.53785e10i	− 0.600807i
$552$	0	0
$553$	1.77807e9	0.0190129
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 9.93534e10i	− 1.03220i	−0.856530	−	0.516098i	$-0.827384\pi$
$557$	− 9.93534e10i	− 1.03220i	0.856530	−	0.516098i	$-0.172616\pi$
$558$	0	0
$559$	−3.42615e9	−0.0350881
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.28318e11i	1.27719i	0.769544	+	0.638594i	$0.220484\pi$
$563$	1.28318e11i	1.27719i	−0.769544	+	0.638594i	$0.779516\pi$
$564$	0	0
$565$	1.33942e11	1.31438
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.69562e10i	− 0.924968i	−0.886628	−	0.462484i	$-0.846958\pi$
$569$	− 9.69562e10i	− 0.924968i	0.886628	−	0.462484i	$-0.153042\pi$
$570$	0	0
$571$	4.85788e10	0.456985	0.228493	−	0.973546i	$-0.426620\pi$
$571$	4.85788e10	0.456985	0.228493	+	0.973546i	$0.426620\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 1.52617e11i	− 1.39615i
$576$	0	0
$577$	−1.00425e11	−0.906024	−0.453012	−	0.891504i	$-0.649650\pi$
$577$	−1.00425e11	−0.906024	−0.453012	+	0.891504i	$0.649650\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 1.82640e10i	− 0.160285i
$582$	0	0
$583$	−5.50024e10	−0.476110
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 5.94122e10i	− 0.500407i	−0.968193	−	0.250203i	$-0.919502\pi$
$587$	− 5.94122e10i	− 0.500407i	0.968193	−	0.250203i	$-0.0804975\pi$
$588$	0	0
$589$	1.06352e11	0.883662
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.44264e11i	1.16665i	0.812240	+	0.583324i	$0.198248\pi$
$593$	1.44264e11i	1.16665i	−0.812240	+	0.583324i	$0.801752\pi$
$594$	0	0
$595$	4.48209e10	0.357612
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.20875e10i	0.326923i	0.986550	+	0.163462i	$0.0522660\pi$
$599$	4.20875e10i	0.326923i	−0.986550	+	0.163462i	$0.947734\pi$
$600$	0	0
$601$	−8.91047e10	−0.682972	−0.341486	−	0.939887i	$-0.610930\pi$
$601$	−8.91047e10	−0.682972	−0.341486	+	0.939887i	$0.610930\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.39927e11i	− 1.79084i
$606$	0	0
$607$	−2.58228e10	−0.190217	−0.0951083	−	0.995467i	$-0.530320\pi$
$607$	−2.58228e10	−0.190217	−0.0951083	+	0.995467i	$0.530320\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.53600e10i	0.181964i
$612$	0	0
$613$	−2.19898e11	−1.55733	−0.778663	−	0.627442i	$-0.784102\pi$
$613$	−2.19898e11	−1.55733	−0.778663	+	0.627442i	$0.784102\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 6.21752e10i	− 0.429019i	−0.976722	−	0.214510i	$-0.931185\pi$
$617$	− 6.21752e10i	− 0.429019i	0.976722	−	0.214510i	$-0.0688153\pi$
$618$	0	0
$619$	−2.05984e11	−1.40304	−0.701521	−	0.712649i	$-0.747495\pi$
$619$	−2.05984e11	−1.40304	−0.701521	+	0.712649i	$0.747495\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.04960e10i	0.0696742i
$624$	0	0
$625$	−1.41018e11	−0.924177
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.27202e11i	0.812630i
$630$	0	0
$631$	1.41366e11	0.891715	0.445858	−	0.895104i	$-0.352899\pi$
$631$	1.41366e11	0.891715	0.445858	+	0.895104i	$0.352899\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.93095e11i	2.41770i
$636$	0	0
$637$	1.71730e10	0.104301
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.65042e10i	− 0.0977605i	−0.998805	−	0.0488802i	$-0.984435\pi$
$641$	− 1.65042e10i	− 0.0977605i	0.998805	−	0.0488802i	$-0.0155653\pi$
$642$	0	0
$643$	−5.08388e10	−0.297407	−0.148704	−	0.988882i	$-0.547510\pi$
$643$	−5.08388e10	−0.297407	−0.148704	+	0.988882i	$0.547510\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.01779e11i	1.15149i	0.817630	+	0.575745i	$0.195288\pi$
$647$	2.01779e11i	1.15149i	−0.817630	+	0.575745i	$0.804712\pi$
$648$	0	0
$649$	−2.80191e11	−1.57934
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.45752e10i	0.245155i	0.992459	+	0.122577i	$0.0391160\pi$
$653$	4.45752e10i	0.245155i	−0.992459	+	0.122577i	$0.960884\pi$
$654$	0	0
$655$	3.59625e11	1.95382
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.28817e11i	1.21324i	0.794993	+	0.606619i	$0.207475\pi$
$659$	2.28817e11i	1.21324i	−0.794993	+	0.606619i	$0.792525\pi$
$660$	0	0
$661$	6.44329e10	0.337522	0.168761	−	0.985657i	$-0.446023\pi$
$661$	6.44329e10	0.337522	0.168761	+	0.985657i	$0.446023\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 9.16481e10i	− 0.468637i
$666$	0	0
$667$	−1.79946e11	−0.909157
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.22104e11i	1.09564i
$672$	0	0
$673$	3.53716e11	1.72423	0.862114	−	0.506715i	$-0.169140\pi$
$673$	3.53716e11	1.72423	0.862114	+	0.506715i	$0.169140\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.09182e10i	0.289996i	0.989432	+	0.144998i	$0.0463176\pi$
$677$	6.09182e10i	0.289996i	−0.989432	+	0.144998i	$0.953682\pi$
$678$	0	0
$679$	−8.99523e10	−0.423188
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.41129e10i	− 0.294620i	−0.989090	−	0.147310i	$-0.952938\pi$
$683$	− 6.41129e10i	− 0.294620i	0.989090	−	0.147310i	$-0.0470615\pi$
$684$	0	0
$685$	4.05650e11	1.84242
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 5.22894e10i	− 0.232026i
$690$	0	0
$691$	1.88610e11	0.827280	0.413640	−	0.910441i	$-0.364257\pi$
$691$	1.88610e11	0.827280	0.413640	+	0.910441i	$0.364257\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 5.93455e10i	− 0.254360i
$696$	0	0
$697$	−1.84582e10	−0.0782094
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.73439e11i	− 0.718248i	−0.933290	−	0.359124i	$-0.883076\pi$
$701$	− 1.73439e11i	− 0.718248i	0.933290	−	0.359124i	$-0.116924\pi$
$702$	0	0
$703$	2.60099e11	1.06492
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.03442e11i	0.414018i
$708$	0	0
$709$	−3.87397e11	−1.53310	−0.766551	−	0.642184i	$-0.778029\pi$
$709$	−3.87397e11	−1.53310	−0.766551	+	0.642184i	$0.778029\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 3.45580e11i	− 1.33718i
$714$	0	0
$715$	4.11376e11	1.57404
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.88408e11i	1.45336i	0.686976	+	0.726680i	$0.258938\pi$
$719$	3.88408e11i	1.45336i	−0.686976	+	0.726680i	$0.741062\pi$
$720$	0	0
$721$	−1.27238e11	−0.470844
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.06292e11i	− 0.746674i
$726$	0	0
$727$	−5.15997e10	−0.184718	−0.0923591	−	0.995726i	$-0.529441\pi$
$727$	−5.15997e10	−0.184718	−0.0923591	+	0.995726i	$0.529441\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.02263e9i	0.0315983i
$732$	0	0
$733$	1.68072e11	0.582209	0.291105	−	0.956691i	$-0.405977\pi$
$733$	1.68072e11	0.582209	0.291105	+	0.956691i	$0.405977\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.65329e11i	− 0.560376i
$738$	0	0
$739$	2.48752e11	0.834043	0.417021	−	0.908897i	$-0.363074\pi$
$739$	2.48752e11	0.834043	0.417021	+	0.908897i	$0.363074\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.00329e11i	0.329209i	0.986360	+	0.164605i	$0.0526348\pi$
$743$	1.00329e11i	0.329209i	−0.986360	+	0.164605i	$0.947365\pi$
$744$	0	0
$745$	6.73512e11	2.18635
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.69068e10i	0.244364i
$750$	0	0
$751$	3.60192e11	1.13233	0.566166	−	0.824291i	$-0.308426\pi$
$751$	3.60192e11	1.13233	0.566166	+	0.824291i	$0.308426\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7.81710e11i	2.40579i
$756$	0	0
$757$	3.05727e11	0.931000	0.465500	−	0.885048i	$-0.345874\pi$
$757$	3.05727e11	0.931000	0.465500	+	0.885048i	$0.345874\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.98951e11i	1.48771i	0.668339	+	0.743857i	$0.267005\pi$
$761$	4.98951e11i	1.48771i	−0.668339	+	0.743857i	$0.732995\pi$
$762$	0	0
$763$	−5.59323e10	−0.165031
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.66371e11i	− 0.769670i
$768$	0	0
$769$	−5.27348e11	−1.50797	−0.753983	−	0.656894i	$-0.771870\pi$
$769$	−5.27348e11	−1.50797	−0.753983	+	0.656894i	$0.771870\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.10274e10i	0.198933i	0.995041	+	0.0994667i	$0.0317137\pi$
$773$	7.10274e10i	0.198933i	−0.995041	+	0.0994667i	$0.968286\pi$
$774$	0	0
$775$	3.96177e11	1.09820
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.77427e10i	0.102491i
$780$	0	0
$781$	−9.24426e11	−2.48467
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.08719e11i	0.812989i
$786$	0	0
$787$	−4.06243e10	−0.105898	−0.0529489	−	0.998597i	$-0.516862\pi$
$787$	−4.06243e10	−0.105898	−0.0529489	+	0.998597i	$0.516862\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.35148e11i	− 0.345226i
$792$	0	0
$793$	−2.11149e11	−0.533944
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.67239e10i	0.0414482i	0.999785	+	0.0207241i	$0.00659715\pi$
$797$	1.67239e10i	0.0414482i	−0.999785	+	0.0207241i	$0.993403\pi$
$798$	0	0
$799$	6.67846e10	0.163866
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.62763e11i	1.83454i
$804$	0	0
$805$	−2.97800e11	−0.709154
$806$	0	0
$807$	0	0
$808$	0	0
$809$	5.87647e11i	1.37190i	0.727648	+	0.685950i	$0.240613\pi$
$809$	5.87647e11i	1.37190i	−0.727648	+	0.685950i	$0.759387\pi$
$810$	0	0
$811$	−4.74855e11	−1.09768	−0.548842	−	0.835926i	$-0.684931\pi$
$811$	−4.74855e11	−1.09768	−0.548842	+	0.835926i	$0.684931\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 9.30819e11i	− 2.10977i
$816$	0	0
$817$	1.84492e10	0.0414084
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.95199e11i	1.31005i	0.755605	+	0.655027i	$0.227343\pi$
$821$	5.95199e11i	1.31005i	−0.755605	+	0.655027i	$0.772657\pi$
$822$	0	0
$823$	6.88252e11	1.50020	0.750098	−	0.661326i	$-0.230006\pi$
$823$	6.88252e11	1.50020	0.750098	+	0.661326i	$0.230006\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.12237e11i	0.239946i	0.992777	+	0.119973i	$0.0382808\pi$
$827$	1.12237e11i	0.239946i	−0.992777	+	0.119973i	$0.961719\pi$
$828$	0	0
$829$	−4.65905e11	−0.986460	−0.493230	−	0.869899i	$-0.664184\pi$
$829$	−4.65905e11	−0.986460	−0.493230	+	0.869899i	$0.664184\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 4.52244e10i	− 0.0939276i
$834$	0	0
$835$	7.69818e11	1.58359
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.26729e11i	1.66846i	0.551418	+	0.834229i	$0.314087\pi$
$839$	8.26729e11i	1.66846i	−0.551418	+	0.834229i	$0.685913\pi$
$840$	0	0
$841$	2.57013e11	0.513774
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 3.42579e11i	− 0.671945i
$846$	0	0
$847$	−2.42087e11	−0.470368
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 8.45161e11i	− 1.61147i
$852$	0	0
$853$	4.95037e11	0.935064	0.467532	−	0.883976i	$-0.345143\pi$
$853$	4.95037e11	0.935064	0.467532	+	0.883976i	$0.345143\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.44761e11i	− 0.639138i	−0.947563	−	0.319569i	$-0.896462\pi$
$857$	− 3.44761e11i	− 0.639138i	0.947563	−	0.319569i	$-0.103538\pi$
$858$	0	0
$859$	8.07526e11	1.48315	0.741573	−	0.670872i	$-0.234080\pi$
$859$	8.07526e11	1.48315	0.741573	+	0.670872i	$0.234080\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 7.70866e11i	− 1.38975i	−0.719132	−	0.694873i	$-0.755461\pi$
$863$	− 7.70866e11i	− 1.38975i	0.719132	−	0.694873i	$-0.244539\pi$
$864$	0	0
$865$	−1.08924e12	−1.94563
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 4.29767e10i	− 0.0753623i
$870$	0	0
$871$	1.57174e11	0.273092
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 2.25757e10i	− 0.0385131i
$876$	0	0
$877$	−5.63983e11	−0.953384	−0.476692	−	0.879071i	$-0.658164\pi$
$877$	−5.63983e11	−0.953384	−0.476692	+	0.879071i	$0.658164\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.69781e11i	1.27780i	0.769289	+	0.638901i	$0.220611\pi$
$881$	7.69781e11i	1.27780i	−0.769289	+	0.638901i	$0.779389\pi$
$882$	0	0
$883$	−8.40219e11	−1.38213	−0.691066	−	0.722792i	$-0.742859\pi$
$883$	−8.40219e11	−1.38213	−0.691066	+	0.722792i	$0.742859\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.09144e12i	1.76322i	0.471983	+	0.881608i	$0.343539\pi$
$887$	1.09144e12i	1.76322i	−0.471983	+	0.881608i	$0.656461\pi$
$888$	0	0
$889$	3.96635e11	0.635015
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.36559e11i	− 0.214740i
$894$	0	0
$895$	1.48913e11	0.232081
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 4.67120e11i	− 0.715137i
$900$	0	0
$901$	−1.37702e11	−0.208949
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.68364e12i	− 2.50989i
$906$	0	0
$907$	1.73121e10	0.0255812	0.0127906	−	0.999918i	$-0.495929\pi$
$907$	1.73121e10	0.0255812	0.0127906	+	0.999918i	$0.495929\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.09440e12i	1.58892i	0.607319	+	0.794458i	$0.292245\pi$
$911$	1.09440e12i	1.58892i	−0.607319	+	0.794458i	$0.707755\pi$
$912$	0	0
$913$	−4.41450e11	−0.635329
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.62864e11i	− 0.513176i
$918$	0	0
$919$	−1.09637e12	−1.53707	−0.768537	−	0.639805i	$-0.779015\pi$
$919$	−1.09637e12	−1.53707	−0.768537	+	0.639805i	$0.779015\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 8.78828e11i	− 1.21087i
$924$	0	0
$925$	9.68903e11	1.32347
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.77872e11i	0.910090i	0.890468	+	0.455045i	$0.150377\pi$
$929$	6.77872e11i	0.910090i	−0.890468	+	0.455045i	$0.849623\pi$
$930$	0	0
$931$	−9.24733e10	−0.123089
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1.08334e12i	− 1.41749i
$936$	0	0
$937$	1.32341e12	1.71687	0.858434	−	0.512924i	$-0.171438\pi$
$937$	1.32341e12	1.71687	0.858434	+	0.512924i	$0.171438\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 5.32278e11i	− 0.678859i	−0.940631	−	0.339430i	$-0.889766\pi$
$941$	− 5.32278e11i	− 0.678859i	0.940631	−	0.339430i	$-0.110234\pi$
$942$	0	0
$943$	1.22641e11	0.155091
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.55370e11i	0.441856i	0.975290	+	0.220928i	$0.0709086\pi$
$947$	3.55370e11i	0.441856i	−0.975290	+	0.220928i	$0.929091\pi$
$948$	0	0
$949$	−7.25139e11	−0.894039
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 8.27352e11i	− 1.00304i	−0.865146	−	0.501520i	$-0.832774\pi$
$953$	− 8.27352e11i	− 1.00304i	0.865146	−	0.501520i	$-0.167226\pi$
$954$	0	0
$955$	−4.98438e11	−0.599236
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 4.09303e11i	− 0.483917i
$960$	0	0
$961$	4.41948e10	0.0518177
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 2.38767e12i	− 2.75337i
$966$	0	0
$967$	−5.09343e11	−0.582511	−0.291256	−	0.956645i	$-0.594073\pi$
$967$	−5.09343e11	−0.582511	−0.291256	+	0.956645i	$0.594073\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.25774e12i	− 1.41486i	−0.706784	−	0.707429i	$-0.749855\pi$
$971$	− 1.25774e12i	− 1.41486i	0.706784	−	0.707429i	$-0.250145\pi$
$972$	0	0
$973$	−5.98799e10	−0.0668082
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.03465e12i	− 1.13557i	−0.823177	−	0.567785i	$-0.807801\pi$
$977$	− 1.03465e12i	− 1.13557i	0.823177	−	0.567785i	$-0.192199\pi$
$978$	0	0
$979$	2.53694e11	0.276171
$980$	0	0
$981$	0	0
$982$	0	0
$983$	9.31373e11i	0.997492i	0.866748	+	0.498746i	$0.166206\pi$
$983$	9.31373e11i	0.997492i	−0.866748	+	0.498746i	$0.833794\pi$
$984$	0	0
$985$	2.12931e12	2.26201
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 5.99484e10i	− 0.0626603i
$990$	0	0
$991$	−7.90752e11	−0.819871	−0.409936	−	0.912114i	$-0.634449\pi$
$991$	−7.90752e11	−0.819871	−0.409936	+	0.912114i	$0.634449\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.02344e12i	2.06442i
$996$	0	0
$997$	−1.61160e12	−1.63109	−0.815543	−	0.578696i	$-0.803562\pi$
$997$	−1.61160e12	−1.63109	−0.815543	+	0.578696i	$0.803562\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.9.c.a.197.14	yes	16
3.2	odd	2	inner	252.9.c.a.197.3	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.c.a.197.3	✓	16	3.2	odd	2	inner
252.9.c.a.197.14	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	252.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+899.394i q^{5} +907.493 q^{7} +O(q^{10})\)	\(q+899.394i q^{5} +907.493 q^{7} -21934.5i q^{11} +20852.6 q^{13} -54914.5i q^{17} -112287. q^{19} +364864. i q^{23} -418285. q^{25} +493187. i q^{29} -947146. q^{31} +816194. i q^{35} -2.31637e6 q^{37} -336127. i q^{41} -164303. q^{43} +1.21616e6i q^{47} +823543. q^{49} -2.50757e6i q^{53} +1.97278e7 q^{55} -1.27740e7i q^{59} -1.01258e7 q^{61} +1.87547e7i q^{65} +7.53739e6 q^{67} -4.21448e7i q^{71} -3.47745e7 q^{73} -1.99054e7i q^{77} +1.95932e6 q^{79} -2.01258e7i q^{83} +4.93898e7 q^{85} +1.15659e7i q^{89} +1.89236e7 q^{91} -1.00990e8i q^{95} -9.91217e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 95480 q^{13} - 287560 q^{19} - 1754520 q^{25} - 3554264 q^{31} - 182920 q^{37} + 8472416 q^{43} + 13176688 q^{49} - 18692072 q^{55} + 34224568 q^{61} + 22683096 q^{67} + 2137296 q^{73} - 90245624 q^{79} - 56204456 q^{85} - 25661888 q^{91} + 134041152 q^{97}+O(q^{100}) \) 16 * q - 95480 * q^13 - 287560 * q^19 - 1754520 * q^25 - 3554264 * q^31 - 182920 * q^37 + 8472416 * q^43 + 13176688 * q^49 - 18692072 * q^55 + 34224568 * q^61 + 22683096 * q^67 + 2137296 * q^73 - 90245624 * q^79 - 56204456 * q^85 - 25661888 * q^91 + 134041152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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