LMFDB - Embedded newform 252.9.c.a.197.12

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			197.12
Root			$640.282i$ of defining polynomial
Character	$\chi$	$=$	252.197
Dual form			252.9.c.a.197.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+640.282i q^{5} -907.493 q^{7} +O(q^{10})$	$q+640.282i q^{5} -907.493 q^{7} +8116.74i q^{11} +47941.1 q^{13} -29809.0i q^{17} -7034.96 q^{19} -351363. i q^{23} -19336.0 q^{25} -387709. i q^{29} +63727.1 q^{31} -581051. i q^{35} +2.45976e6 q^{37} +3.12120e6i q^{41} +2.12121e6 q^{43} -2.22184e6i q^{47} +823543. q^{49} +1.76942e6i q^{53} -5.19700e6 q^{55} -1.12039e7i q^{59} -4.08608e6 q^{61} +3.06958e7i q^{65} -2.48165e7 q^{67} +1.73411e7i q^{71} -3.58546e6 q^{73} -7.36589e6i q^{77} +5.14806e7 q^{79} +4.80626e7i q^{83} +1.90862e7 q^{85} -2.69599e7i q^{89} -4.35062e7 q^{91} -4.50436e6i q^{95} +1.47587e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	640.282i	1.02445i	0.858851	+	0.512226i	$0.171179\pi$
$5$	640.282i	1.02445i	−0.858851	+	0.512226i	$0.828821\pi$
$6$	0	0
$7$	−907.493	−0.377964
$7$	−907.493	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8116.74i	0.554385i	0.960814	+	0.277192i	$0.0894039\pi$
$11$	8116.74i	0.554385i	−0.960814	+	0.277192i	$0.910596\pi$
$12$	0	0
$13$	47941.1	1.67855	0.839275	−	0.543707i	$-0.182980\pi$
$13$	47941.1	1.67855	0.839275	+	0.543707i	$0.182980\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 29809.0i	− 0.356904i	−0.983949	−	0.178452i	$-0.942891\pi$
$17$	− 29809.0i	− 0.356904i	0.983949	−	0.178452i	$-0.0571090\pi$
$18$	0	0
$19$	−7034.96	−0.0539818	−0.0269909	−	0.999636i	$-0.508593\pi$
$19$	−7034.96	−0.0539818	−0.0269909	+	0.999636i	$0.508593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 351363.i	− 1.25558i	−0.778382	−	0.627791i	$-0.783959\pi$
$23$	− 351363.i	− 1.25558i	0.778382	−	0.627791i	$-0.216041\pi$
$24$	0	0
$25$	−19336.0	−0.0495001
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 387709.i	− 0.548168i	−0.961706	−	0.274084i	$-0.911625\pi$
$29$	− 387709.i	− 0.548168i	0.961706	−	0.274084i	$-0.0883746\pi$
$30$	0	0
$31$	63727.1	0.0690045	0.0345022	−	0.999405i	$-0.489015\pi$
$31$	63727.1	0.0690045	0.0345022	+	0.999405i	$0.489015\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 581051.i	− 0.387206i
$36$	0	0
$37$	2.45976e6	1.31246	0.656230	−	0.754561i	$-0.272150\pi$
$37$	2.45976e6	1.31246	0.656230	+	0.754561i	$0.272150\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.12120e6i	1.10455i	0.833662	+	0.552276i	$0.186240\pi$
$41$	3.12120e6i	1.10455i	−0.833662	+	0.552276i	$0.813760\pi$
$42$	0	0
$43$	2.12121e6	0.620454	0.310227	−	0.950663i	$-0.399595\pi$
$43$	2.12121e6	0.620454	0.310227	+	0.950663i	$0.399595\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.22184e6i	− 0.455325i	−0.973740	−	0.227663i	$-0.926892\pi$
$47$	− 2.22184e6i	− 0.455325i	0.973740	−	0.227663i	$-0.0731083\pi$
$48$	0	0
$49$	823543.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.76942e6i	0.224248i	0.993694	+	0.112124i	$0.0357653\pi$
$53$	1.76942e6i	0.224248i	−0.993694	+	0.112124i	$0.964235\pi$
$54$	0	0
$55$	−5.19700e6	−0.567940
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.12039e7i	− 0.924617i	−0.886719	−	0.462308i	$-0.847021\pi$
$59$	− 1.12039e7i	− 0.924617i	0.886719	−	0.462308i	$-0.152979\pi$
$60$	0	0
$61$	−4.08608e6	−0.295112	−0.147556	−	0.989054i	$-0.547141\pi$
$61$	−4.08608e6	−0.295112	−0.147556	+	0.989054i	$0.547141\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.06958e7i	1.71959i
$66$	0	0
$67$	−2.48165e7	−1.23152	−0.615759	−	0.787935i	$-0.711150\pi$
$67$	−2.48165e7	−1.23152	−0.615759	+	0.787935i	$0.711150\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.73411e7i	0.682406i	0.939990	+	0.341203i	$0.110834\pi$
$71$	1.73411e7i	0.682406i	−0.939990	+	0.341203i	$0.889166\pi$
$72$	0	0
$73$	−3.58546e6	−0.126257	−0.0631283	−	0.998005i	$-0.520108\pi$
$73$	−3.58546e6	−0.126257	−0.0631283	+	0.998005i	$0.520108\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7.36589e6i	− 0.209538i
$78$	0	0
$79$	5.14806e7	1.32171	0.660853	−	0.750515i	$-0.270195\pi$
$79$	5.14806e7	1.32171	0.660853	+	0.750515i	$0.270195\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.80626e7i	1.01273i	0.862319	+	0.506366i	$0.169012\pi$
$83$	4.80626e7i	1.01273i	−0.862319	+	0.506366i	$0.830988\pi$
$84$	0	0
$85$	1.90862e7	0.365631
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 2.69599e7i	− 0.429693i	−0.976648	−	0.214847i	$-0.931075\pi$
$89$	− 2.69599e7i	− 0.429693i	0.976648	−	0.214847i	$-0.0689252\pi$
$90$	0	0
$91$	−4.35062e7	−0.634433
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 4.50436e6i	− 0.0553017i
$96$	0	0
$97$	1.47587e8	1.66709	0.833547	−	0.552449i	$-0.186307\pi$
$97$	1.47587e8	1.66709	0.833547	+	0.552449i	$0.186307\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.04528e8i	− 1.00450i	−0.864723	−	0.502249i	$-0.832506\pi$
$101$	− 1.04528e8i	− 1.00450i	0.864723	−	0.502249i	$-0.167494\pi$
$102$	0	0
$103$	−4.86644e7	−0.432377	−0.216188	−	0.976352i	$-0.569363\pi$
$103$	−4.86644e7	−0.432377	−0.216188	+	0.976352i	$0.569363\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.18081e8i	1.66373i	0.554979	+	0.831865i	$0.312726\pi$
$107$	2.18081e8i	1.66373i	−0.554979	+	0.831865i	$0.687274\pi$
$108$	0	0
$109$	1.04183e8	0.738061	0.369031	−	0.929417i	$-0.379690\pi$
$109$	1.04183e8	0.738061	0.369031	+	0.929417i	$0.379690\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.22264e8i	0.749867i	0.927052	+	0.374934i	$0.122334\pi$
$113$	1.22264e8i	0.749867i	−0.927052	+	0.374934i	$0.877666\pi$
$114$	0	0
$115$	2.24972e8	1.28628
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.70515e7i	0.134897i
$120$	0	0
$121$	1.48477e8	0.692658
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.37730e8i	0.973741i
$126$	0	0
$127$	1.59142e8	0.611746	0.305873	−	0.952072i	$-0.401052\pi$
$127$	1.59142e8	0.611746	0.305873	+	0.952072i	$0.401052\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.75418e8i	0.935206i	0.883939	+	0.467603i	$0.154882\pi$
$131$	2.75418e8i	0.935206i	−0.883939	+	0.467603i	$0.845118\pi$
$132$	0	0
$133$	6.38417e6	0.0204032
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.92582e8i	0.546682i	0.961917	+	0.273341i	$0.0881287\pi$
$137$	1.92582e8i	0.546682i	−0.961917	+	0.273341i	$0.911871\pi$
$138$	0	0
$139$	−5.26604e8	−1.41067	−0.705334	−	0.708875i	$-0.749203\pi$
$139$	−5.26604e8	−1.41067	−0.705334	+	0.708875i	$0.749203\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.89126e8i	0.930563i
$144$	0	0
$145$	2.48243e8	0.561571
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.53599e7i	− 0.0514519i	−0.999669	−	0.0257260i	$-0.991810\pi$
$149$	− 2.53599e7i	− 0.0514519i	0.999669	−	0.0257260i	$-0.00818973\pi$
$150$	0	0
$151$	2.06642e8	0.397475	0.198738	−	0.980053i	$-0.436316\pi$
$151$	2.06642e8	0.397475	0.198738	+	0.980053i	$0.436316\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.08033e7i	0.0706917i
$156$	0	0
$157$	2.42458e8	0.399060	0.199530	−	0.979892i	$-0.436059\pi$
$157$	2.42458e8	0.399060	0.199530	+	0.979892i	$0.436059\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.18860e8i	0.474566i
$162$	0	0
$163$	−2.62352e8	−0.371650	−0.185825	−	0.982583i	$-0.559496\pi$
$163$	−2.62352e8	−0.371650	−0.185825	+	0.982583i	$0.559496\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.62142e8i	0.979873i	0.871758	+	0.489937i	$0.162980\pi$
$167$	7.62142e8i	0.979873i	−0.871758	+	0.489937i	$0.837020\pi$
$168$	0	0
$169$	1.48262e9	1.81753
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.60297e9i	1.78954i	0.446529	+	0.894769i	$0.352660\pi$
$173$	1.60297e9i	1.78954i	−0.446529	+	0.894769i	$0.647340\pi$
$174$	0	0
$175$	1.75472e7	0.0187093
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.28649e9i	1.25312i	0.779373	+	0.626560i	$0.215538\pi$
$179$	1.28649e9i	1.25312i	−0.779373	+	0.626560i	$0.784462\pi$
$180$	0	0
$181$	2.94339e8	0.274242	0.137121	−	0.990554i	$-0.456215\pi$
$181$	2.94339e8	0.274242	0.137121	+	0.990554i	$0.456215\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.57494e9i	1.34455i
$186$	0	0
$187$	2.41952e8	0.197862
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 6.83713e8i	− 0.513736i	−0.966446	−	0.256868i	$-0.917309\pi$
$191$	− 6.83713e8i	− 0.513736i	0.966446	−	0.256868i	$-0.0826906\pi$
$192$	0	0
$193$	1.74277e9	1.25606	0.628032	−	0.778187i	$-0.283861\pi$
$193$	1.74277e9	1.25606	0.628032	+	0.778187i	$0.283861\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.35657e9i	− 1.56464i	−0.622876	−	0.782321i	$-0.714036\pi$
$197$	− 2.35657e9i	− 1.56464i	0.622876	−	0.782321i	$-0.285964\pi$
$198$	0	0
$199$	2.71407e9	1.73065	0.865324	−	0.501212i	$-0.167112\pi$
$199$	2.71407e9	1.73065	0.865324	+	0.501212i	$0.167112\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.51843e8i	0.207188i
$204$	0	0
$205$	−1.99845e9	−1.13156
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 5.71009e7i	− 0.0299267i
$210$	0	0
$211$	2.44112e9	1.23157	0.615786	−	0.787913i	$-0.288839\pi$
$211$	2.44112e9	1.23157	0.615786	+	0.787913i	$0.288839\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.35817e9i	0.635625i
$216$	0	0
$217$	−5.78319e7	−0.0260812
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.42908e9i	− 0.599082i
$222$	0	0
$223$	−3.86582e8	−0.156323	−0.0781614	−	0.996941i	$-0.524905\pi$
$223$	−3.86582e8	−0.156323	−0.0781614	+	0.996941i	$0.524905\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.81490e9i	− 0.683516i	−0.939788	−	0.341758i	$-0.888978\pi$
$227$	− 1.81490e9i	− 0.683516i	0.939788	−	0.341758i	$-0.111022\pi$
$228$	0	0
$229$	−2.65732e9	−0.966276	−0.483138	−	0.875544i	$-0.660503\pi$
$229$	−2.65732e9	−0.966276	−0.483138	+	0.875544i	$0.660503\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.24408e9i	1.77929i	0.456657	+	0.889643i	$0.349047\pi$
$233$	5.24408e9i	1.77929i	−0.456657	+	0.889643i	$0.650953\pi$
$234$	0	0
$235$	1.42261e9	0.466458
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.39464e9i	0.733919i	0.930237	+	0.366959i	$0.119601\pi$
$239$	2.39464e9i	0.733919i	−0.930237	+	0.366959i	$0.880399\pi$
$240$	0	0
$241$	−2.18983e9	−0.649145	−0.324572	−	0.945861i	$-0.605220\pi$
$241$	−2.18983e9	−0.649145	−0.324572	+	0.945861i	$0.605220\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.27300e8i	0.146350i
$246$	0	0
$247$	−3.37263e8	−0.0906111
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.22517e8i	0.0308676i	0.999881	+	0.0154338i	$0.00491292\pi$
$251$	1.22517e8i	0.0308676i	−0.999881	+	0.0154338i	$0.995087\pi$
$252$	0	0
$253$	2.85193e9	0.696075
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.54143e9i	0.811795i	0.913919	+	0.405897i	$0.133041\pi$
$257$	3.54143e9i	0.811795i	−0.913919	+	0.405897i	$0.866959\pi$
$258$	0	0
$259$	−2.23221e9	−0.496063
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 3.24479e9i	− 0.678209i	−0.940749	−	0.339105i	$-0.889876\pi$
$263$	− 3.24479e9i	− 0.678209i	0.940749	−	0.339105i	$-0.110124\pi$
$264$	0	0
$265$	−1.13293e9	−0.229731
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.16580e9i	− 0.795590i	−0.917474	−	0.397795i	$-0.869776\pi$
$269$	− 4.16580e9i	− 0.795590i	0.917474	−	0.397795i	$-0.130224\pi$
$270$	0	0
$271$	−7.78226e9	−1.44287	−0.721437	−	0.692480i	$-0.756518\pi$
$271$	−7.78226e9	−1.44287	−0.721437	+	0.692480i	$0.756518\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.56945e8i	− 0.0274421i
$276$	0	0
$277$	−6.42407e9	−1.09117	−0.545583	−	0.838057i	$-0.683692\pi$
$277$	−6.42407e9	−1.09117	−0.545583	+	0.838057i	$0.683692\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.04039e10i	− 1.66867i	−0.551261	−	0.834333i	$-0.685853\pi$
$281$	− 1.04039e10i	− 1.66867i	0.551261	−	0.834333i	$-0.314147\pi$
$282$	0	0
$283$	−7.82909e8	−0.122058	−0.0610289	−	0.998136i	$-0.519438\pi$
$283$	−7.82909e8	−0.122058	−0.0610289	+	0.998136i	$0.519438\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.83246e9i	− 0.417481i
$288$	0	0
$289$	6.08718e9	0.872619
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.18853e10i	1.61264i	0.591477	+	0.806322i	$0.298545\pi$
$293$	1.18853e10i	1.61264i	−0.591477	+	0.806322i	$0.701455\pi$
$294$	0	0
$295$	7.17366e9	0.947225
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.68447e10i	− 2.10756i
$300$	0	0
$301$	−1.92498e9	−0.234510
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 2.61624e9i	− 0.302328i
$306$	0	0
$307$	−1.09190e10	−1.22922	−0.614611	−	0.788831i	$-0.710687\pi$
$307$	−1.09190e10	−1.22922	−0.614611	+	0.788831i	$0.710687\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 1.01076e10i	− 1.08046i	−0.841519	−	0.540228i	$-0.818338\pi$
$311$	− 1.01076e10i	− 1.08046i	0.841519	−	0.540228i	$-0.181662\pi$
$312$	0	0
$313$	7.14089e9	0.744003	0.372002	−	0.928232i	$-0.378672\pi$
$313$	7.14089e9	0.744003	0.372002	+	0.928232i	$0.378672\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.17626e9i	− 0.215514i	−0.994177	−	0.107757i	$-0.965633\pi$
$317$	− 2.17626e9i	− 0.215514i	0.994177	−	0.107757i	$-0.0343668\pi$
$318$	0	0
$319$	3.14693e9	0.303896
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.09705e8i	0.0192663i
$324$	0	0
$325$	−9.26987e8	−0.0830884
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.01631e9i	0.172097i
$330$	0	0
$331$	−8.58963e9	−0.715587	−0.357794	−	0.933801i	$-0.616471\pi$
$331$	−8.58963e9	−0.715587	−0.357794	+	0.933801i	$0.616471\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.58895e10i	− 1.26163i
$336$	0	0
$337$	−1.44960e10	−1.12390	−0.561952	−	0.827170i	$-0.689949\pi$
$337$	−1.44960e10	−1.12390	−0.561952	+	0.827170i	$0.689949\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.17257e8i	0.0382550i
$342$	0	0
$343$	−7.47359e8	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.48761e10i	− 1.02605i	−0.858373	−	0.513027i	$-0.828524\pi$
$347$	− 1.48761e10i	− 1.02605i	0.858373	−	0.513027i	$-0.171476\pi$
$348$	0	0
$349$	2.99091e9	0.201605	0.100803	−	0.994906i	$-0.467859\pi$
$349$	2.99091e9	0.201605	0.100803	+	0.994906i	$0.467859\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.22457e10i	− 0.788653i	−0.918970	−	0.394327i	$-0.870978\pi$
$353$	− 1.22457e10i	− 0.788653i	0.918970	−	0.394327i	$-0.129022\pi$
$354$	0	0
$355$	−1.11032e10	−0.699092
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.69094e10i	− 1.01800i	−0.860765	−	0.509002i	$-0.830015\pi$
$359$	− 1.69094e10i	− 1.01800i	0.860765	−	0.509002i	$-0.169985\pi$
$360$	0	0
$361$	−1.69341e10	−0.997086
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.29571e9i	− 0.129344i
$366$	0	0
$367$	1.25574e9	0.0692204	0.0346102	−	0.999401i	$-0.488981\pi$
$367$	1.25574e9	0.0692204	0.0346102	+	0.999401i	$0.488981\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.60574e9i	− 0.0847576i
$372$	0	0
$373$	−7.27627e9	−0.375901	−0.187950	−	0.982179i	$-0.560184\pi$
$373$	−7.27627e9	−0.375901	−0.187950	+	0.982179i	$0.560184\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1.85872e10i	− 0.920127i
$378$	0	0
$379$	−1.54709e10	−0.749824	−0.374912	−	0.927060i	$-0.622327\pi$
$379$	−1.54709e10	−0.749824	−0.374912	+	0.927060i	$0.622327\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.53417e10i	1.17772i	0.808237	+	0.588858i	$0.200422\pi$
$383$	2.53417e10i	1.17772i	−0.808237	+	0.588858i	$0.799578\pi$
$384$	0	0
$385$	4.71624e9	0.214661
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.80420e10i	0.787929i	0.919126	+	0.393964i	$0.128897\pi$
$389$	1.80420e10i	0.787929i	−0.919126	+	0.393964i	$0.871103\pi$
$390$	0	0
$391$	−1.04738e10	−0.448123
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.29621e10i	1.35402i
$396$	0	0
$397$	2.15193e10	0.866296	0.433148	−	0.901323i	$-0.357403\pi$
$397$	2.15193e10	0.866296	0.433148	+	0.901323i	$0.357403\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2.66386e10i	− 1.03023i	−0.857121	−	0.515115i	$-0.827749\pi$
$401$	− 2.66386e10i	− 1.03023i	0.857121	−	0.515115i	$-0.172251\pi$
$402$	0	0
$403$	3.05515e9	0.115828
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.99652e10i	0.727607i
$408$	0	0
$409$	3.17363e8	0.0113413	0.00567066	−	0.999984i	$-0.498195\pi$
$409$	3.17363e8	0.0113413	0.00567066	+	0.999984i	$0.498195\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.01675e10i	0.349472i
$414$	0	0
$415$	−3.07736e10	−1.03749
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.22236e10i	− 0.721039i	−0.932752	−	0.360519i	$-0.882599\pi$
$419$	− 2.22236e10i	− 0.721039i	0.932752	−	0.360519i	$-0.117401\pi$
$420$	0	0
$421$	−2.61883e10	−0.833642	−0.416821	−	0.908989i	$-0.636856\pi$
$421$	−2.61883e10	−0.833642	−0.416821	+	0.908989i	$0.636856\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.76386e8i	0.0176668i
$426$	0	0
$427$	3.70808e9	0.111542
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.43747e8i	0.0273493i	0.999906	+	0.0136747i	$0.00435292\pi$
$431$	9.43747e8i	0.0273493i	−0.999906	+	0.0136747i	$0.995647\pi$
$432$	0	0
$433$	5.68198e10	1.61640	0.808198	−	0.588910i	$-0.200443\pi$
$433$	5.68198e10	1.61640	0.808198	+	0.588910i	$0.200443\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.47183e9i	0.0677785i
$438$	0	0
$439$	7.47760e9	0.201328	0.100664	−	0.994920i	$-0.467903\pi$
$439$	7.47760e9	0.201328	0.100664	+	0.994920i	$0.467903\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.70177e10i	− 0.961157i	−0.876952	−	0.480579i	$-0.840427\pi$
$443$	− 3.70177e10i	− 0.961157i	0.876952	−	0.480579i	$-0.159573\pi$
$444$	0	0
$445$	1.72620e10	0.440200
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.00631e9i	− 0.0247597i	−0.999923	−	0.0123798i	$-0.996059\pi$
$449$	− 1.00631e9i	− 0.0247597i	0.999923	−	0.0123798i	$-0.00394072\pi$
$450$	0	0
$451$	−2.53340e10	−0.612346
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 2.78562e10i	− 0.649945i
$456$	0	0
$457$	2.29431e10	0.526003	0.263001	−	0.964795i	$-0.415288\pi$
$457$	2.29431e10	0.526003	0.263001	+	0.964795i	$0.415288\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.78011e10i	0.836953i	0.908228	+	0.418477i	$0.137436\pi$
$461$	3.78011e10i	0.836953i	−0.908228	+	0.418477i	$0.862564\pi$
$462$	0	0
$463$	−1.70770e9	−0.0371610	−0.0185805	−	0.999827i	$-0.505915\pi$
$463$	−1.70770e9	−0.0371610	−0.0185805	+	0.999827i	$0.505915\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.86432e10i	− 1.02271i	−0.859368	−	0.511357i	$-0.829143\pi$
$467$	− 4.86432e10i	− 1.02271i	0.859368	−	0.511357i	$-0.170857\pi$
$468$	0	0
$469$	2.25208e10	0.465470
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.72173e10i	0.343970i
$474$	0	0
$475$	1.36028e8	0.00267210
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3.55978e10i	0.676210i	0.941108	+	0.338105i	$0.109786\pi$
$479$	3.55978e10i	0.676210i	−0.941108	+	0.338105i	$0.890214\pi$
$480$	0	0
$481$	1.17924e11	2.20303
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.44970e10i	1.70786i
$486$	0	0
$487$	−2.70179e10	−0.480325	−0.240162	−	0.970733i	$-0.577201\pi$
$487$	−2.70179e10	−0.480325	−0.240162	+	0.970733i	$0.577201\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6.67674e10i	1.14878i	0.818580	+	0.574392i	$0.194762\pi$
$491$	6.67674e10i	1.14878i	−0.818580	+	0.574392i	$0.805238\pi$
$492$	0	0
$493$	−1.15572e10	−0.195643
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.57369e10i	− 0.257925i
$498$	0	0
$499$	9.30718e10	1.50112	0.750561	−	0.660802i	$-0.229784\pi$
$499$	9.30718e10	1.50112	0.750561	+	0.660802i	$0.229784\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.79146e10i	0.279856i	0.990162	+	0.139928i	$0.0446871\pi$
$503$	1.79146e10i	0.279856i	−0.990162	+	0.139928i	$0.955313\pi$
$504$	0	0
$505$	6.69277e10	1.02906
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.00413e11i	1.49596i	0.663724	+	0.747978i	$0.268975\pi$
$509$	1.00413e11i	1.49596i	−0.663724	+	0.747978i	$0.731025\pi$
$510$	0	0
$511$	3.25378e9	0.0477205
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.11589e10i	− 0.442949i
$516$	0	0
$517$	1.80341e10	0.252425
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.66810e10i	0.497841i	0.968524	+	0.248921i	$0.0800758\pi$
$521$	3.66810e10i	0.497841i	−0.968524	+	0.248921i	$0.919924\pi$
$522$	0	0
$523$	−2.96773e9	−0.0396659	−0.0198329	−	0.999803i	$-0.506313\pi$
$523$	−2.96773e9	−0.0396659	−0.0198329	+	0.999803i	$0.506313\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.89964e9i	− 0.0246280i
$528$	0	0
$529$	−4.51453e10	−0.576487
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.49634e11i	1.85405i
$534$	0	0
$535$	−1.39633e11	−1.70441
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.68449e9i	0.0791978i
$540$	0	0
$541$	4.71697e10	0.550648	0.275324	−	0.961351i	$-0.411215\pi$
$541$	4.71697e10	0.550648	0.275324	+	0.961351i	$0.411215\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.67067e10i	0.756107i
$546$	0	0
$547$	−7.66653e10	−0.856347	−0.428173	−	0.903697i	$-0.640843\pi$
$547$	−7.66653e10	−0.856347	−0.428173	+	0.903697i	$0.640843\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.72751e9i	0.0295911i
$552$	0	0
$553$	−4.67182e10	−0.499558
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.34358e10i	0.243478i	0.992562	+	0.121739i	$0.0388471\pi$
$557$	2.34358e10i	0.243478i	−0.992562	+	0.121739i	$0.961153\pi$
$558$	0	0
$559$	1.01693e11	1.04146
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1.18235e11i	− 1.17682i	−0.808561	−	0.588412i	$-0.799753\pi$
$563$	− 1.18235e11i	− 1.17682i	0.808561	−	0.588412i	$-0.200247\pi$
$564$	0	0
$565$	−7.82833e10	−0.768202
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.28655e11i	1.22738i	0.789549	+	0.613688i	$0.210315\pi$
$569$	1.28655e11i	1.22738i	−0.789549	+	0.613688i	$0.789685\pi$
$570$	0	0
$571$	1.13973e11	1.07216	0.536078	−	0.844168i	$-0.319905\pi$
$571$	1.13973e11	1.07216	0.536078	+	0.844168i	$0.319905\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.79395e9i	0.0621514i
$576$	0	0
$577$	1.31966e11	1.19058	0.595291	−	0.803510i	$-0.297037\pi$
$577$	1.31966e11	1.19058	0.595291	+	0.803510i	$0.297037\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.36164e10i	− 0.382777i
$582$	0	0
$583$	−1.43619e10	−0.124319
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.91244e10i	0.582209i	0.956691	+	0.291105i	$0.0940228\pi$
$587$	6.91244e10i	0.582209i	−0.956691	+	0.291105i	$0.905977\pi$
$588$	0	0
$589$	−4.48317e8	−0.00372498
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.96912e10i	0.482716i	0.970436	+	0.241358i	$0.0775928\pi$
$593$	5.96912e10i	0.482716i	−0.970436	+	0.241358i	$0.922407\pi$
$594$	0	0
$595$	−1.73206e10	−0.138196
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.24032e11i	− 0.963443i	−0.876324	−	0.481721i	$-0.840012\pi$
$599$	− 1.24032e11i	− 0.963443i	0.876324	−	0.481721i	$-0.159988\pi$
$600$	0	0
$601$	1.51053e11	1.15779	0.578896	−	0.815401i	$-0.303484\pi$
$601$	1.51053e11	1.15779	0.578896	+	0.815401i	$0.303484\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.50674e10i	0.709594i
$606$	0	0
$607$	1.77767e11	1.30947	0.654735	−	0.755859i	$-0.272780\pi$
$607$	1.77767e11	1.30947	0.654735	+	0.755859i	$0.272780\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.06518e11i	− 0.764286i
$612$	0	0
$613$	−9.29216e10	−0.658074	−0.329037	−	0.944317i	$-0.606724\pi$
$613$	−9.29216e10	−0.658074	−0.329037	+	0.944317i	$0.606724\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.33802e10i	0.161327i	0.996741	+	0.0806637i	$0.0257040\pi$
$617$	2.33802e10i	0.161327i	−0.996741	+	0.0806637i	$0.974296\pi$
$618$	0	0
$619$	1.29256e11	0.880418	0.440209	−	0.897895i	$-0.354904\pi$
$619$	1.29256e11	0.880418	0.440209	+	0.897895i	$0.354904\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.44659e10i	0.162409i
$624$	0	0
$625$	−1.59767e11	−1.04705
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 7.33230e10i	− 0.468422i
$630$	0	0
$631$	4.31032e10	0.271889	0.135945	−	0.990716i	$-0.456593\pi$
$631$	4.31032e10	0.271889	0.135945	+	0.990716i	$0.456593\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.01896e11i	0.626703i
$636$	0	0
$637$	3.94815e10	0.239793
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.47970e11i	0.876478i	0.898858	+	0.438239i	$0.144398\pi$
$641$	1.47970e11i	0.876478i	−0.898858	+	0.438239i	$0.855602\pi$
$642$	0	0
$643$	−2.78103e11	−1.62690	−0.813451	−	0.581634i	$-0.802414\pi$
$643$	−2.78103e11	−1.62690	−0.813451	+	0.581634i	$0.802414\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.85426e11i	− 1.62883i	−0.580282	−	0.814416i	$-0.697058\pi$
$647$	− 2.85426e11i	− 1.62883i	0.580282	−	0.814416i	$-0.302942\pi$
$648$	0	0
$649$	9.09393e10	0.512593
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.60195e11i	− 1.43102i	−0.698600	−	0.715512i	$-0.746193\pi$
$653$	− 2.60195e11i	− 1.43102i	0.698600	−	0.715512i	$-0.253807\pi$
$654$	0	0
$655$	−1.76345e11	−0.958073
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.08589e11i	− 1.10599i	−0.833185	−	0.552994i	$-0.813485\pi$
$659$	− 2.08589e11i	− 1.10599i	0.833185	−	0.552994i	$-0.186515\pi$
$660$	0	0
$661$	−2.70845e10	−0.141878	−0.0709390	−	0.997481i	$-0.522600\pi$
$661$	−2.70845e10	−0.141878	−0.0709390	+	0.997481i	$0.522600\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4.08767e9i	0.0209021i
$666$	0	0
$667$	−1.36227e11	−0.688270
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 3.31656e10i	− 0.163606i
$672$	0	0
$673$	4.08197e11	1.98980	0.994900	−	0.100862i	$-0.0321601\pi$
$673$	4.08197e11	1.98980	0.994900	+	0.100862i	$0.0321601\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.84230e11i	1.35305i	0.736418	+	0.676527i	$0.236516\pi$
$677$	2.84230e11i	1.35305i	−0.736418	+	0.676527i	$0.763484\pi$
$678$	0	0
$679$	−1.33934e11	−0.630102
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.10241e11i	− 1.42566i	−0.701335	−	0.712831i	$-0.747412\pi$
$683$	− 3.10241e11i	− 1.42566i	0.701335	−	0.712831i	$-0.252588\pi$
$684$	0	0
$685$	−1.23307e11	−0.560049
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8.48280e10i	0.376411i
$690$	0	0
$691$	2.06970e10	0.0907810	0.0453905	−	0.998969i	$-0.485547\pi$
$691$	2.06970e10	0.0907810	0.0453905	+	0.998969i	$0.485547\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 3.37175e11i	− 1.44516i
$696$	0	0
$697$	9.30398e10	0.394219
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.29778e11i	0.537440i	0.963218	+	0.268720i	$0.0866007\pi$
$701$	1.29778e11i	0.537440i	−0.963218	+	0.268720i	$0.913399\pi$
$702$	0	0
$703$	−1.73043e10	−0.0708489
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.48588e10i	0.379664i
$708$	0	0
$709$	−2.37263e11	−0.938956	−0.469478	−	0.882944i	$-0.655558\pi$
$709$	−2.37263e11	−0.938956	−0.469478	+	0.882944i	$0.655558\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.23914e10i	− 0.0866408i
$714$	0	0
$715$	−2.49150e11	−0.953316
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.40426e11i	1.27382i	0.770940	+	0.636908i	$0.219787\pi$
$719$	3.40426e11i	1.27382i	−0.770940	+	0.636908i	$0.780213\pi$
$720$	0	0
$721$	4.41626e10	0.163423
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.49672e9i	0.0271343i
$726$	0	0
$727$	−3.88892e11	−1.39217	−0.696084	−	0.717960i	$-0.745076\pi$
$727$	−3.88892e11	−1.39217	−0.696084	+	0.717960i	$0.745076\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 6.32311e10i	− 0.221443i
$732$	0	0
$733$	−1.53399e11	−0.531383	−0.265691	−	0.964058i	$-0.585600\pi$
$733$	−1.53399e11	−0.531383	−0.265691	+	0.964058i	$0.585600\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.01429e11i	− 0.682734i
$738$	0	0
$739$	−3.54937e11	−1.19007	−0.595036	−	0.803699i	$-0.702862\pi$
$739$	−3.54937e11	−1.19007	−0.595036	+	0.803699i	$0.702862\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 6.04592e11i	− 1.98384i	−0.126858	−	0.991921i	$-0.540489\pi$
$743$	− 6.04592e11i	− 1.98384i	0.126858	−	0.991921i	$-0.459511\pi$
$744$	0	0
$745$	1.62375e10	0.0527100
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.97907e11i	− 0.628831i
$750$	0	0
$751$	−2.76854e11	−0.870345	−0.435172	−	0.900347i	$-0.643313\pi$
$751$	−2.76854e11	−0.870345	−0.435172	+	0.900347i	$0.643313\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.32309e11i	0.407194i
$756$	0	0
$757$	5.28734e11	1.61010	0.805052	−	0.593204i	$-0.202137\pi$
$757$	5.28734e11	1.61010	0.805052	+	0.593204i	$0.202137\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.83174e10i	0.173884i	0.996213	+	0.0869419i	$0.0277095\pi$
$761$	5.83174e10i	0.173884i	−0.996213	+	0.0869419i	$0.972291\pi$
$762$	0	0
$763$	−9.45456e10	−0.278961
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.37128e11i	− 1.55202i
$768$	0	0
$769$	−1.31873e11	−0.377095	−0.188547	−	0.982064i	$-0.560378\pi$
$769$	−1.31873e11	−0.377095	−0.188547	+	0.982064i	$0.560378\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.91940e11i	− 1.37783i	−0.724844	−	0.688913i	$-0.758088\pi$
$773$	− 4.91940e11i	− 1.37783i	0.724844	−	0.688913i	$-0.241912\pi$
$774$	0	0
$775$	−1.23222e9	−0.00341573
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.19575e10i	− 0.0596256i
$780$	0	0
$781$	−1.40753e11	−0.378316
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.55241e11i	0.408817i
$786$	0	0
$787$	−6.56249e11	−1.71068	−0.855342	−	0.518065i	$-0.826653\pi$
$787$	−6.56249e11	−1.71068	−0.855342	+	0.518065i	$0.826653\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.10954e11i	− 0.283423i
$792$	0	0
$793$	−1.95891e11	−0.495361
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.56799e11i	0.388607i	0.980941	+	0.194303i	$0.0622446\pi$
$797$	1.56799e11i	0.388607i	−0.980941	+	0.194303i	$0.937755\pi$
$798$	0	0
$799$	−6.62309e10	−0.162508
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2.91023e10i	− 0.0699947i
$804$	0	0
$805$	−2.04160e11	−0.486169
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.63793e11i	− 1.08275i	−0.840780	−	0.541377i	$-0.817903\pi$
$809$	− 4.63793e11i	− 1.08275i	0.840780	−	0.541377i	$-0.182097\pi$
$810$	0	0
$811$	7.29900e11	1.68725	0.843626	−	0.536931i	$-0.180417\pi$
$811$	7.29900e11	1.68725	0.843626	+	0.536931i	$0.180417\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1.67979e11i	− 0.380737i
$816$	0	0
$817$	−1.49226e10	−0.0334932
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.81580e10i	− 0.0399664i	−0.999800	−	0.0199832i	$-0.993639\pi$
$821$	− 1.81580e10i	− 0.0399664i	0.999800	−	0.0199832i	$-0.00636127\pi$
$822$	0	0
$823$	9.12249e11	1.98845	0.994224	−	0.107329i	$-0.0342297\pi$
$823$	9.12249e11	1.98845	0.994224	+	0.107329i	$0.0342297\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.47368e11i	0.742623i	0.928508	+	0.371311i	$0.121092\pi$
$827$	3.47368e11i	0.742623i	−0.928508	+	0.371311i	$0.878908\pi$
$828$	0	0
$829$	6.72798e11	1.42451	0.712257	−	0.701919i	$-0.247673\pi$
$829$	6.72798e11	1.42451	0.712257	+	0.701919i	$0.247673\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.45490e10i	− 0.0509863i
$834$	0	0
$835$	−4.87986e11	−1.00383
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.22680e11i	0.247587i	0.992308	+	0.123793i	$0.0395060\pi$
$839$	1.22680e11i	0.247587i	−0.992308	+	0.123793i	$0.960494\pi$
$840$	0	0
$841$	3.49928e11	0.699512
$842$	0	0
$843$	0	0
$844$	0	0
$845$	9.49293e11i	1.86197i
$846$	0	0
$847$	−1.34742e11	−0.261800
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 8.64270e11i	− 1.64790i
$852$	0	0
$853$	5.53404e11	1.04531	0.522656	−	0.852544i	$-0.324941\pi$
$853$	5.53404e11	1.04531	0.522656	+	0.852544i	$0.324941\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6.47734e11i	− 1.20081i	−0.799697	−	0.600403i	$-0.795007\pi$
$857$	− 6.47734e11i	− 1.20081i	0.799697	−	0.600403i	$-0.204993\pi$
$858$	0	0
$859$	7.66128e11	1.40711	0.703556	−	0.710640i	$-0.251594\pi$
$859$	7.66128e11	1.40711	0.703556	+	0.710640i	$0.251594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.95413e11i	0.893150i	0.894746	+	0.446575i	$0.147356\pi$
$863$	4.95413e11i	0.893150i	−0.894746	+	0.446575i	$0.852644\pi$
$864$	0	0
$865$	−1.02635e12	−1.83329
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.17855e11i	0.732733i
$870$	0	0
$871$	−1.18973e12	−2.06716
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 2.15738e11i	− 0.368039i
$876$	0	0
$877$	8.33732e11	1.40938	0.704690	−	0.709516i	$-0.251086\pi$
$877$	8.33732e11	1.40938	0.704690	+	0.709516i	$0.251086\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 2.38563e11i	− 0.396005i	−0.980202	−	0.198002i	$-0.936555\pi$
$881$	− 2.38563e11i	− 0.396005i	0.980202	−	0.198002i	$-0.0634453\pi$
$882$	0	0
$883$	6.91704e11	1.13783	0.568915	−	0.822396i	$-0.307363\pi$
$883$	6.91704e11	1.13783	0.568915	+	0.822396i	$0.307363\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.58558e11i	0.256150i	0.991764	+	0.128075i	$0.0408798\pi$
$887$	1.58558e11i	0.256150i	−0.991764	+	0.128075i	$0.959120\pi$
$888$	0	0
$889$	−1.44421e11	−0.231218
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.56306e10i	0.0245793i
$894$	0	0
$895$	−8.23714e11	−1.28376
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.47075e10i	− 0.0378260i
$900$	0	0
$901$	5.27447e10	0.0800349
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.88460e11i	0.280947i
$906$	0	0
$907$	−1.20962e12	−1.78739	−0.893693	−	0.448678i	$-0.851895\pi$
$907$	−1.20962e12	−1.78739	−0.893693	+	0.448678i	$0.851895\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.19638e11i	0.609258i	0.952471	+	0.304629i	$0.0985325\pi$
$911$	4.19638e11i	0.609258i	−0.952471	+	0.304629i	$0.901467\pi$
$912$	0	0
$913$	−3.90112e11	−0.561443
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.49940e11i	− 0.353475i
$918$	0	0
$919$	2.65442e11	0.372140	0.186070	−	0.982536i	$-0.440425\pi$
$919$	2.65442e11	0.372140	0.186070	+	0.982536i	$0.440425\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.31351e11i	1.14545i
$924$	0	0
$925$	−4.75618e10	−0.0649668
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.02149e12i	1.37143i	0.727872	+	0.685713i	$0.240510\pi$
$929$	1.02149e12i	1.37143i	−0.727872	+	0.685713i	$0.759490\pi$
$930$	0	0
$931$	−5.79359e9	−0.00771168
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.54918e11i	0.202700i
$936$	0	0
$937$	9.49784e11	1.23216	0.616079	−	0.787684i	$-0.288720\pi$
$937$	9.49784e11	1.23216	0.616079	+	0.787684i	$0.288720\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.52549e11i	0.704713i	0.935866	+	0.352356i	$0.114620\pi$
$941$	5.52549e11i	0.704713i	−0.935866	+	0.352356i	$0.885380\pi$
$942$	0	0
$943$	1.09667e12	1.38686
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.31274e12i	− 1.63223i	−0.577893	−	0.816113i	$-0.696125\pi$
$947$	− 1.31274e12i	− 1.63223i	0.577893	−	0.816113i	$-0.303875\pi$
$948$	0	0
$949$	−1.71891e11	−0.211928
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.34394e11i	1.13281i	0.824126	+	0.566407i	$0.191667\pi$
$953$	9.34394e11i	1.13281i	−0.824126	+	0.566407i	$0.808333\pi$
$954$	0	0
$955$	4.37769e11	0.526298
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.74767e11i	− 0.206626i
$960$	0	0
$961$	−8.48830e11	−0.995238
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.11587e12i	1.28678i
$966$	0	0
$967$	−4.78441e11	−0.547170	−0.273585	−	0.961848i	$-0.588209\pi$
$967$	−4.78441e11	−0.547170	−0.273585	+	0.961848i	$0.588209\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	8.53487e11i	0.960108i	0.877239	+	0.480054i	$0.159383\pi$
$971$	8.53487e11i	0.960108i	−0.877239	+	0.480054i	$0.840617\pi$
$972$	0	0
$973$	4.77889e11	0.533182
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.25702e12i	1.37964i	0.723983	+	0.689818i	$0.242309\pi$
$977$	1.25702e12i	1.37964i	−0.723983	+	0.689818i	$0.757691\pi$
$978$	0	0
$979$	2.18827e11	0.238215
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.32959e10i	− 0.0142398i	−0.999975	−	0.00711989i	$-0.997734\pi$
$983$	− 1.32959e10i	− 0.0142398i	0.999975	−	0.00711989i	$-0.00226635\pi$
$984$	0	0
$985$	1.50887e12	1.60290
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 7.45315e11i	− 0.779031i
$990$	0	0
$991$	1.32471e12	1.37349	0.686746	−	0.726898i	$-0.259038\pi$
$991$	1.32471e12	1.37349	0.686746	+	0.726898i	$0.259038\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.73777e12i	1.77296i
$996$	0	0
$997$	9.09754e11	0.920753	0.460377	−	0.887724i	$-0.347714\pi$
$997$	9.09754e11	0.920753	0.460377	+	0.887724i	$0.347714\pi$
$998$	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.197.12	yes	16
3.2	odd	2	inner	252.9.c.a.197.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.9.c.a.197.5	✓	16	3.2	odd	2	inner
252.9.c.a.197.12	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)

\(f(q)\)	\(=\)	\(q+640.282i q^{5} -907.493 q^{7} +O(q^{10})\)	\(q+640.282i q^{5} -907.493 q^{7} +8116.74i q^{11} +47941.1 q^{13} -29809.0i q^{17} -7034.96 q^{19} -351363. i q^{23} -19336.0 q^{25} -387709. i q^{29} +63727.1 q^{31} -581051. i q^{35} +2.45976e6 q^{37} +3.12120e6i q^{41} +2.12121e6 q^{43} -2.22184e6i q^{47} +823543. q^{49} +1.76942e6i q^{53} -5.19700e6 q^{55} -1.12039e7i q^{59} -4.08608e6 q^{61} +3.06958e7i q^{65} -2.48165e7 q^{67} +1.73411e7i q^{71} -3.58546e6 q^{73} -7.36589e6i q^{77} +5.14806e7 q^{79} +4.80626e7i q^{83} +1.90862e7 q^{85} -2.69599e7i q^{89} -4.35062e7 q^{91} -4.50436e6i q^{95} +1.47587e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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