LMFDB - Embedded newform 1368.2.cu.a.521.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(449,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1368.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.cu (of order $6$, degree $2$, 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:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 18 x^{18} + 232 x^{17} + 208 x^{16} - 3152 x^{15} - 2732 x^{14} + 24552 x^{13} + \cdots + 414864 $ x^20 - 8x^19 - 18x^18 + 232x^17 + 208x^16 - 3152x^15 - 2732x^14 + 24552x^13 + 30728x^12 - 110512x^11 - 211368x^10 + 231024x^9 + 827916x^8 + 179664x^7 - 1551960x^6 - 1894880x^5 + 305564x^4 + 2772832x^3 + 3072904x^2 + 1645728*x + 414864
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			521.3
Root			$3.16384 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1368.521
Dual form			1368.2.cu.a.449.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.76516 - 1.01912i) q^{5} +3.69902 q^{7} +O(q^{10})$	$q+(-1.76516 - 1.01912i) q^{5} +3.69902 q^{7} -0.605557i q^{11} +(2.65015 - 1.53006i) q^{13} +(-2.40852 - 1.39056i) q^{17} +(0.780721 + 4.28841i) q^{19} +(2.23644 - 1.29121i) q^{23} +(-0.422804 - 0.732318i) q^{25} +(1.70567 + 2.95430i) q^{29} -4.42738i q^{31} +(-6.52937 - 3.76973i) q^{35} -3.34720i q^{37} +(1.50349 - 2.60412i) q^{41} +(0.562914 - 0.974996i) q^{43} +(3.05695 - 1.76493i) q^{47} +6.68277 q^{49} +(0.984893 + 1.70588i) q^{53} +(-0.617133 + 1.06891i) q^{55} +(3.48434 - 6.03505i) q^{59} +(4.98180 + 8.62874i) q^{61} -6.23725 q^{65} +(0.324291 - 0.187230i) q^{67} +(8.03969 - 13.9252i) q^{71} +(8.09850 - 14.0270i) q^{73} -2.23997i q^{77} +(-7.44995 - 4.30123i) q^{79} -7.17251i q^{83} +(2.83429 + 4.90913i) q^{85} +(3.91340 + 6.77822i) q^{89} +(9.80296 - 5.65974i) q^{91} +(2.99229 - 8.36538i) q^{95} +(8.57756 + 4.95226i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{7}+O(q^{10}) $ 20 * q - 4 * q^7	$ 20 q - 4 q^{7} + 6 q^{13} - 12 q^{17} + 4 q^{19} + 14 q^{25} + 12 q^{35} - 8 q^{41} - 2 q^{43} + 36 q^{47} + 32 q^{49} + 8 q^{53} + 12 q^{55} - 8 q^{59} - 2 q^{61} - 8 q^{65} + 30 q^{67} + 4 q^{71} - 22 q^{73} + 54 q^{79} - 4 q^{85} + 32 q^{89} + 18 q^{91} + 32 q^{95} + 12 q^{97}+O(q^{100}) $ 20 * q - 4 * q^7 + 6 * q^13 - 12 * q^17 + 4 * q^19 + 14 * q^25 + 12 * q^35 - 8 * q^41 - 2 * q^43 + 36 * q^47 + 32 * q^49 + 8 * q^53 + 12 * q^55 - 8 * q^59 - 2 * q^61 - 8 * q^65 + 30 * q^67 + 4 * q^71 - 22 * q^73 + 54 * q^79 - 4 * q^85 + 32 * q^89 + 18 * q^91 + 32 * q^95 + 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$	$-1$

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
$5$	−1.76516	−	1.01912i	−0.789404	−	0.455763i	0.0503486	−	0.998732i	$-0.483967\pi$
$5$	−1.76516	−	1.01912i	−0.789404	−	0.455763i	−0.839753	+	0.542969i	$0.817300\pi$
$6$		0			0
$7$	3.69902			1.39810			0.699050	−	0.715073i	$-0.253607\pi$
$7$	3.69902			1.39810			0.699050	+	0.715073i	$0.253607\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.605557i		−	0.182582i	−0.995824	−	0.0912912i	$-0.970901\pi$
$11$		−	0.605557i		−	0.182582i	0.995824	−	0.0912912i	$-0.0290994\pi$
$12$		0			0
$13$	2.65015	−	1.53006i	0.735019	−	0.424364i	−0.0852364	−	0.996361i	$-0.527165\pi$
$13$	2.65015	−	1.53006i	0.735019	−	0.424364i	0.820256	+	0.571997i	$0.193831\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.40852	−	1.39056i	−0.584153	−	0.337261i	0.178629	−	0.983916i	$-0.442834\pi$
$17$	−2.40852	−	1.39056i	−0.584153	−	0.337261i	−0.762782	+	0.646656i	$0.776167\pi$
$18$		0			0
$19$	0.780721	+	4.28841i	0.179110	+	0.983829i
$19$	0.780721	+	4.28841i	0.179110	+	0.983829i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.23644	−	1.29121i	0.466330	−	0.269236i	−0.248372	−	0.968665i	$-0.579896\pi$
$23$	2.23644	−	1.29121i	0.466330	−	0.269236i	0.714702	+	0.699429i	$0.246562\pi$
$24$		0			0
$25$	−0.422804	−	0.732318i	−0.0845607	−	0.146464i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.70567	+	2.95430i	0.316734	+	0.548600i	0.979805	−	0.199957i	$-0.0640803\pi$
$29$	1.70567	+	2.95430i	0.316734	+	0.548600i	−0.663070	+	0.748557i	$0.730747\pi$
$30$		0			0
$31$		−	4.42738i		−	0.795181i	−0.917563	−	0.397591i	$-0.869846\pi$
$31$		−	4.42738i		−	0.795181i	0.917563	−	0.397591i	$-0.130154\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.52937	−	3.76973i	−1.10367	−	0.637201i
$36$		0			0
$37$		−	3.34720i		−	0.550276i	−0.961405	−	0.275138i	$-0.911276\pi$
$37$		−	3.34720i		−	0.550276i	0.961405	−	0.275138i	$-0.0887236\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.50349	−	2.60412i	0.234806	−	0.406696i	−0.724410	−	0.689369i	$-0.757888\pi$
$41$	1.50349	−	2.60412i	0.234806	−	0.406696i	0.959216	+	0.282673i	$0.0912212\pi$
$42$		0			0
$43$	0.562914	−	0.974996i	0.0858436	−	0.148686i	−0.819907	−	0.572497i	$-0.805975\pi$
$43$	0.562914	−	0.974996i	0.0858436	−	0.148686i	0.905750	+	0.423812i	$0.139308\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.05695	−	1.76493i	0.445903	−	0.257442i	−0.260196	−	0.965556i	$-0.583787\pi$
$47$	3.05695	−	1.76493i	0.445903	−	0.257442i	0.706098	+	0.708114i	$0.250454\pi$
$48$		0			0
$49$	6.68277			0.954681
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.984893	+	1.70588i	0.135285	+	0.234321i	0.925706	−	0.378243i	$-0.123472\pi$
$53$	0.984893	+	1.70588i	0.135285	+	0.234321i	−0.790421	+	0.612564i	$0.790138\pi$
$54$		0			0
$55$	−0.617133	+	1.06891i	−0.0832142	+	0.144131i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.48434	−	6.03505i	0.453622	−	0.785696i	−0.544986	−	0.838445i	$-0.683465\pi$
$59$	3.48434	−	6.03505i	0.453622	−	0.785696i	0.998608	+	0.0527492i	$0.0167984\pi$
$60$		0			0
$61$	4.98180	+	8.62874i	0.637855	+	1.10480i	0.985903	+	0.167320i	$0.0535113\pi$
$61$	4.98180	+	8.62874i	0.637855	+	1.10480i	−0.348048	+	0.937477i	$0.613155\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−6.23725			−0.773636
$66$		0			0
$67$	0.324291	−	0.187230i	0.0396185	−	0.0228738i	−0.480060	−	0.877236i	$-0.659385\pi$
$67$	0.324291	−	0.187230i	0.0396185	−	0.0228738i	0.519678	+	0.854362i	$0.326052\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	8.03969	−	13.9252i	0.954136	−	1.65261i	0.217802	−	0.975993i	$-0.430111\pi$
$71$	8.03969	−	13.9252i	0.954136	−	1.65261i	0.736334	−	0.676618i	$-0.236555\pi$
$72$		0			0
$73$	8.09850	−	14.0270i	0.947858	−	1.64174i	0.197934	−	0.980215i	$-0.436577\pi$
$73$	8.09850	−	14.0270i	0.947858	−	1.64174i	0.749925	−	0.661523i	$-0.230090\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	2.23997i		−	0.255268i
$78$		0			0
$79$	−7.44995	−	4.30123i	−0.838185	−	0.483926i	0.0184618	−	0.999830i	$-0.494123\pi$
$79$	−7.44995	−	4.30123i	−0.838185	−	0.483926i	−0.856647	+	0.515903i	$0.827456\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	7.17251i		−	0.787285i	−0.919264	−	0.393643i	$-0.871215\pi$
$83$		−	7.17251i		−	0.787285i	0.919264	−	0.393643i	$-0.128785\pi$
$84$		0			0
$85$	2.83429	+	4.90913i	0.307422	+	0.532470i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	3.91340	+	6.77822i	0.414820	+	0.718489i	0.995410	−	0.0957070i	$-0.0305112\pi$
$89$	3.91340	+	6.77822i	0.414820	+	0.718489i	−0.580590	+	0.814196i	$0.697178\pi$
$90$		0			0
$91$	9.80296	−	5.65974i	1.02763	−	0.593302i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.99229	−	8.36538i	0.307003	−	0.858270i
$96$		0			0
$97$	8.57756	+	4.95226i	0.870919	+	0.502826i	0.867654	−	0.497169i	$-0.165627\pi$
$97$	8.57756	+	4.95226i	0.870919	+	0.502826i	0.00326577	+	0.999995i	$0.498960\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.3741	+	7.72154i	−1.33077	+	0.768322i	−0.985418	−	0.170150i	$-0.945575\pi$
$101$	−13.3741	+	7.72154i	−1.33077	+	0.768322i	−0.345355	+	0.938472i	$0.612241\pi$
$102$		0			0
$103$		−	13.3067i		−	1.31114i	−0.755133	−	0.655572i	$-0.772428\pi$
$103$		−	13.3067i		−	1.31114i	0.755133	−	0.655572i	$-0.227572\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.95982			−0.769505			−0.384753	−	0.923020i	$-0.625713\pi$
$107$	−7.95982			−0.769505			−0.384753	+	0.923020i	$0.625713\pi$
$108$		0			0
$109$	7.06803	+	4.08073i	0.676995	+	0.390863i	0.798722	−	0.601700i	$-0.205510\pi$
$109$	7.06803	+	4.08073i	0.676995	+	0.390863i	−0.121727	+	0.992564i	$0.538843\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.63294			−0.341759			−0.170879	−	0.985292i	$-0.554661\pi$
$113$	−3.63294			−0.341759			−0.170879	+	0.985292i	$0.554661\pi$
$114$		0			0
$115$	−5.26357			−0.490831
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.90918	−	5.14372i	−0.816704	−	0.471524i
$120$		0			0
$121$	10.6333			0.966664
$122$		0			0
$123$		0			0
$124$		0			0
$125$			11.9147i			1.06568i
$126$		0			0
$127$	−5.69581	+	3.28848i	−0.505421	+	0.291805i	−0.730950	−	0.682431i	$-0.760923\pi$
$127$	−5.69581	+	3.28848i	−0.505421	+	0.291805i	0.225528	+	0.974237i	$0.427589\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.27444	−	1.89050i	−0.286089	−	0.165174i	0.350088	−	0.936717i	$-0.386152\pi$
$131$	−3.27444	−	1.89050i	−0.286089	−	0.165174i	−0.636177	+	0.771543i	$0.719485\pi$
$132$		0			0
$133$	2.88790	+	15.8629i	0.250413	+	1.37549i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.5831	−	7.26483i	1.07504	−	0.620676i	0.145488	−	0.989360i	$-0.453525\pi$
$137$	12.5831	−	7.26483i	1.07504	−	0.620676i	0.929555	+	0.368684i	$0.120192\pi$
$138$		0			0
$139$	3.79838	+	6.57899i	0.322175	+	0.558023i	0.980936	−	0.194329i	$-0.0622529\pi$
$139$	3.79838	+	6.57899i	0.322175	+	0.558023i	−0.658762	+	0.752352i	$0.728920\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.926541	−	1.60482i	−0.0774813	−	0.134202i
$144$		0			0
$145$		−	6.95309i		−	0.577423i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.06792	+	1.77127i	0.251334	+	0.145108i	0.620375	−	0.784305i	$-0.286980\pi$
$149$	3.06792	+	1.77127i	0.251334	+	0.145108i	−0.369041	+	0.929413i	$0.620314\pi$
$150$		0			0
$151$		−	1.10333i		−	0.0897877i	−0.998992	−	0.0448938i	$-0.985705\pi$
$151$		−	1.10333i		−	0.0897877i	0.998992	−	0.0448938i	$-0.0142950\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.51202	+	7.81504i	−0.362414	+	0.627719i
$156$		0			0
$157$	−0.691234	+	1.19725i	−0.0551665	+	0.0955511i	−0.892290	−	0.451463i	$-0.850902\pi$
$157$	−0.691234	+	1.19725i	−0.0551665	+	0.0955511i	0.837123	+	0.547014i	$0.184236\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	8.27265	−	4.77622i	0.651976	−	0.376419i
$162$		0			0
$163$	−15.3481			−1.20216			−0.601080	−	0.799189i	$-0.705263\pi$
$163$	−15.3481			−1.20216			−0.601080	+	0.799189i	$0.705263\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.45424	+	4.25087i	0.189915	+	0.328942i	0.945222	−	0.326429i	$-0.105845\pi$
$167$	2.45424	+	4.25087i	0.189915	+	0.328942i	−0.755307	+	0.655371i	$0.772512\pi$
$168$		0			0
$169$	−1.81781	+	3.14853i	−0.139831	+	0.242195i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.2798	+	21.2693i	−0.933619	+	1.61708i	−0.156541	+	0.987671i	$0.550034\pi$
$173$	−12.2798	+	21.2693i	−0.933619	+	1.61708i	−0.777078	+	0.629404i	$0.783299\pi$
$174$		0			0
$175$	−1.56396	−	2.70886i	−0.118224	−	0.204770i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.5239			1.16031			0.580155	−	0.814506i	$-0.302992\pi$
$179$	15.5239			1.16031			0.580155	+	0.814506i	$0.302992\pi$
$180$		0			0
$181$	−11.6668	+	6.73585i	−0.867189	+	0.500672i	−0.866413	−	0.499328i	$-0.833580\pi$
$181$	−11.6668	+	6.73585i	−0.867189	+	0.500672i	−0.000775855		1.00000i	$0.500247\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.41119	+	5.90835i	−0.250795	+	0.434390i
$186$		0			0
$187$	−0.842065	+	1.45850i	−0.0615779	+	0.106656i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			14.3381i			1.03747i	0.854935	+	0.518734i	$0.173597\pi$
$191$			14.3381i			1.03747i	−0.854935	+	0.518734i	$0.826403\pi$
$192$		0			0
$193$	−16.7819	−	9.68904i	−1.20799	−	0.697432i	−0.245668	−	0.969354i	$-0.579007\pi$
$193$	−16.7819	−	9.68904i	−1.20799	−	0.697432i	−0.962319	+	0.271922i	$0.912341\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.2122i		−	1.01257i	−0.862365	−	0.506287i	$-0.831018\pi$
$197$		−	14.2122i		−	1.01257i	0.862365	−	0.506287i	$-0.168982\pi$
$198$		0			0
$199$	11.3046	+	19.5801i	0.801360	+	1.38800i	0.918721	+	0.394906i	$0.129223\pi$
$199$	11.3046	+	19.5801i	0.801360	+	1.38800i	−0.117362	+	0.993089i	$0.537444\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.30930	+	10.9280i	0.442826	+	0.766997i
$204$		0			0
$205$	−5.30781	+	3.06447i	−0.370714	+	0.214032i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.59688	−	0.472771i	0.179630	−	0.0327022i
$210$		0			0
$211$	−10.1117	−	5.83798i	−0.696116	−	0.401903i	0.109783	−	0.993956i	$-0.464984\pi$
$211$	−10.1117	−	5.83798i	−0.696116	−	0.401903i	−0.805899	+	0.592053i	$0.798318\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−1.98727	+	1.14735i	−0.135531	+	0.0782487i
$216$		0			0
$217$		−	16.3770i		−	1.11174i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.51060			−0.572485
$222$		0			0
$223$	−14.6238	−	8.44308i	−0.979285	−	0.565391i	−0.0772310	−	0.997013i	$-0.524608\pi$
$223$	−14.6238	−	8.44308i	−0.979285	−	0.565391i	−0.902054	+	0.431623i	$0.857941\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.3183			−1.01671			−0.508354	−	0.861148i	$-0.669746\pi$
$227$	−15.3183			−1.01671			−0.508354	+	0.861148i	$0.669746\pi$
$228$		0			0
$229$	−17.5798			−1.16171			−0.580854	−	0.814008i	$-0.697281\pi$
$229$	−17.5798			−1.16171			−0.580854	+	0.814008i	$0.697281\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	18.4561	+	10.6556i	1.20910	+	0.698074i	0.962563	−	0.271059i	$-0.0873738\pi$
$233$	18.4561	+	10.6556i	1.20910	+	0.698074i	0.246538	+	0.969133i	$0.420707\pi$
$234$		0			0
$235$	−7.19469			−0.469330
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5.70885i			0.369275i	0.982807	+	0.184638i	$0.0591111\pi$
$239$			5.70885i			0.369275i	−0.982807	+	0.184638i	$0.940889\pi$
$240$		0			0
$241$	−15.3638	+	8.87029i	−0.989668	+	0.571385i	−0.905175	−	0.425039i	$-0.860260\pi$
$241$	−15.3638	+	8.87029i	−0.989668	+	0.571385i	−0.0844932	+	0.996424i	$0.526927\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−11.7962	−	6.81052i	−0.753629	−	0.435108i
$246$		0			0
$247$	8.63057	+	10.1704i	0.549150	+	0.647126i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	24.8199	−	14.3298i	1.56662	−	0.904489i	0.570062	−	0.821602i	$-0.306919\pi$
$251$	24.8199	−	14.3298i	1.56662	−	0.904489i	0.996559	−	0.0828875i	$-0.0264142\pi$
$252$		0			0
$253$	−0.781902	−	1.35429i	−0.0491577	−	0.0851437i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.27016	+	10.8602i	0.391122	+	0.677443i	0.992598	−	0.121448i	$-0.0387538\pi$
$257$	6.27016	+	10.8602i	0.391122	+	0.677443i	−0.601476	+	0.798891i	$0.705420\pi$
$258$		0			0
$259$		−	12.3814i		−	0.769341i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.42476	+	0.822587i	0.0878546	+	0.0507229i	0.543284	−	0.839549i	$-0.317181\pi$
$263$	1.42476	+	0.822587i	0.0878546	+	0.0507229i	−0.455429	+	0.890272i	$0.650514\pi$
$264$		0			0
$265$		−	4.01488i		−	0.246632i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.25176	−	9.09632i	0.320206	−	0.554612i	−0.660325	−	0.750980i	$-0.729581\pi$
$269$	5.25176	−	9.09632i	0.320206	−	0.554612i	0.980530	+	0.196368i	$0.0629146\pi$
$270$		0			0
$271$	−6.37145	+	11.0357i	−0.387038	+	0.670370i	−0.992050	−	0.125847i	$-0.959835\pi$
$271$	−6.37145	+	11.0357i	−0.387038	+	0.670370i	0.605012	+	0.796217i	$0.293168\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.443460	+	0.256032i	−0.0267416	+	0.0154393i
$276$		0			0
$277$	19.4101			1.16624			0.583119	−	0.812387i	$-0.301832\pi$
$277$	19.4101			1.16624			0.583119	+	0.812387i	$0.301832\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.78424	+	13.4827i	0.464369	+	0.804311i	0.999173	−	0.0406655i	$-0.0129478\pi$
$281$	7.78424	+	13.4827i	0.464369	+	0.804311i	−0.534804	+	0.844976i	$0.679614\pi$
$282$		0			0
$283$	−5.22827	+	9.05563i	−0.310788	+	0.538301i	−0.978533	−	0.206089i	$-0.933926\pi$
$283$	−5.22827	+	9.05563i	−0.310788	+	0.538301i	0.667745	+	0.744390i	$0.267260\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.56145	−	9.63271i	0.328282	−	0.568601i
$288$		0			0
$289$	−4.63267	−	8.02403i	−0.272510	−	0.472002i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−28.1390			−1.64390			−0.821949	−	0.569561i	$-0.807113\pi$
$293$	−28.1390			−1.64390			−0.821949	+	0.569561i	$0.807113\pi$
$294$		0			0
$295$	−12.3008	+	7.10189i	−0.716182	+	0.413488i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.95127	−	6.84380i	0.228508	−	0.395787i
$300$		0			0
$301$	2.08223	−	3.60653i	0.120018	−	0.207877i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	20.3082i		−	1.16284i
$306$		0			0
$307$	11.1659	+	6.44663i	0.637271	+	0.367929i	0.783563	−	0.621313i	$-0.213400\pi$
$307$	11.1659	+	6.44663i	0.637271	+	0.367929i	−0.146292	+	0.989242i	$0.546734\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			13.6008i			0.771228i	0.922660	+	0.385614i	$0.126010\pi$
$311$			13.6008i			0.771228i	−0.922660	+	0.385614i	$0.873990\pi$
$312$		0			0
$313$	1.12148	+	1.94246i	0.0633899	+	0.109794i	0.895979	−	0.444097i	$-0.146476\pi$
$313$	1.12148	+	1.94246i	0.0633899	+	0.109794i	−0.832589	+	0.553892i	$0.813142\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	16.3488	+	28.3170i	0.918242	+	1.59044i	0.802084	+	0.597212i	$0.203725\pi$
$317$	16.3488	+	28.3170i	0.918242	+	1.59044i	0.116159	+	0.993231i	$0.462942\pi$
$318$		0			0
$319$	1.78900	−	1.03288i	0.100165	−	0.0578301i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.08292	−	11.4144i	0.227180	−	0.635113i
$324$		0			0
$325$	−2.24099	−	1.29383i	−0.124308	−	0.0717690i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	11.3077	−	6.52853i	0.623416	−	0.359929i
$330$		0			0
$331$		−	18.7734i		−	1.03188i	−0.856625	−	0.515940i	$-0.827443\pi$
$331$		−	18.7734i		−	1.03188i	0.856625	−	0.515940i	$-0.172557\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.763236			−0.0417000
$336$		0			0
$337$	−8.38982	−	4.84387i	−0.457023	−	0.263862i	0.253769	−	0.967265i	$-0.418330\pi$
$337$	−8.38982	−	4.84387i	−0.457023	−	0.263862i	−0.710792	+	0.703403i	$0.751663\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.68103			−0.145186
$342$		0			0
$343$	−1.17346			−0.0633606
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.2531	+	6.49698i	0.604098	+	0.348776i	0.770652	−	0.637256i	$-0.219931\pi$
$347$	11.2531	+	6.49698i	0.604098	+	0.348776i	−0.166554	+	0.986032i	$0.553264\pi$
$348$		0			0
$349$	6.98962			0.374146			0.187073	−	0.982346i	$-0.440100\pi$
$349$	6.98962			0.374146			0.187073	+	0.982346i	$0.440100\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			1.60313i			0.0853258i	0.999090	+	0.0426629i	$0.0135842\pi$
$353$			1.60313i			0.0853258i	−0.999090	+	0.0426629i	$0.986416\pi$
$354$		0			0
$355$	−28.3827	+	16.3868i	−1.50640	+	0.869719i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−19.1453	−	11.0535i	−1.01045	−	0.583383i	−0.0991256	−	0.995075i	$-0.531605\pi$
$359$	−19.1453	−	11.0535i	−1.01045	−	0.583383i	−0.911323	+	0.411692i	$0.864938\pi$
$360$		0			0
$361$	−17.7810	+	6.69610i	−0.935840	+	0.352426i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−28.5903	+	16.5066i	−1.49649	+	0.863997i
$366$		0			0
$367$	4.97781	+	8.62182i	0.259840	+	0.450055i	0.966199	−	0.257798i	$-0.0829970\pi$
$367$	4.97781	+	8.62182i	0.259840	+	0.450055i	−0.706359	+	0.707854i	$0.749664\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.64314	+	6.31011i	0.189142	+	0.327604i
$372$		0			0
$373$			16.4257i			0.850489i	0.905078	+	0.425245i	$0.139812\pi$
$373$			16.4257i			0.850489i	−0.905078	+	0.425245i	$0.860188\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.04054	+	5.21956i	0.465612	+	0.268821i
$378$		0			0
$379$		−	25.2429i		−	1.29664i	−0.761367	−	0.648322i	$-0.775471\pi$
$379$		−	25.2429i		−	1.29664i	0.761367	−	0.648322i	$-0.224529\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−17.5496	+	30.3968i	−0.896742	+	1.55320i	−0.0651092	+	0.997878i	$0.520740\pi$
$383$	−17.5496	+	30.3968i	−0.896742	+	1.55320i	−0.831633	+	0.555325i	$0.812594\pi$
$384$		0			0
$385$	−2.28279	+	3.95391i	−0.116342	+	0.201510i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−16.8623	+	9.73543i	−0.854950	+	0.493606i	−0.862318	−	0.506367i	$-0.830988\pi$
$389$	−16.8623	+	9.73543i	−0.854950	+	0.493606i	0.00736774	+	0.999973i	$0.497655\pi$
$390$		0			0
$391$	−7.18203			−0.363211
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.76691	+	15.1847i	0.441111	+	0.764027i
$396$		0			0
$397$	−8.78901	+	15.2230i	−0.441108	+	0.764021i	−0.997772	−	0.0667165i	$-0.978748\pi$
$397$	−8.78901	+	15.2230i	−0.441108	+	0.764021i	0.556664	+	0.830738i	$0.312081\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.54092	+	14.7933i	−0.426513	+	0.738743i	−0.996560	−	0.0828693i	$-0.973592\pi$
$401$	−8.54092	+	14.7933i	−0.426513	+	0.738743i	0.570047	+	0.821612i	$0.306925\pi$
$402$		0			0
$403$	−6.77418	−	11.7332i	−0.337446	−	0.584473i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.02692			−0.100471
$408$		0			0
$409$	13.3939	−	7.73296i	0.662284	−	0.382370i	−0.130862	−	0.991401i	$-0.541775\pi$
$409$	13.3939	−	7.73296i	0.662284	−	0.382370i	0.793147	+	0.609030i	$0.208441\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	12.8886	−	22.3238i	0.634208	−	1.09848i
$414$		0			0
$415$	−7.30962	+	12.6606i	−0.358815	+	0.621486i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			25.5092i			1.24621i	0.782140	+	0.623103i	$0.214128\pi$
$419$			25.5092i			1.24621i	−0.782140	+	0.623103i	$0.785872\pi$
$420$		0			0
$421$	6.76334	+	3.90482i	0.329625	+	0.190309i	0.655675	−	0.755044i	$-0.272384\pi$
$421$	6.76334	+	3.90482i	0.329625	+	0.190309i	−0.326050	+	0.945353i	$0.605718\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			2.35174i			0.114076i
$426$		0			0
$427$	18.4278	+	31.9179i	0.891784	+	1.54461i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	9.26660	+	16.0502i	0.446356	+	0.773112i	0.998146	−	0.0608716i	$-0.0193880\pi$
$431$	9.26660	+	16.0502i	0.446356	+	0.773112i	−0.551789	+	0.833984i	$0.686055\pi$
$432$		0			0
$433$	−28.2020	+	16.2824i	−1.35530	+	0.782484i	−0.988986	−	0.148007i	$-0.952714\pi$
$433$	−28.2020	+	16.2824i	−1.35530	+	0.782484i	−0.366316	+	0.930491i	$0.619381\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.28328	+	8.58271i	0.348406	+	0.410567i
$438$		0			0
$439$	−10.5290	−	6.07891i	−0.502521	−	0.290131i	0.227233	−	0.973840i	$-0.427032\pi$
$439$	−10.5290	−	6.07891i	−0.502521	−	0.290131i	−0.729754	+	0.683710i	$0.760365\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.7476	+	6.20513i	−0.510634	+	0.294815i	−0.733094	−	0.680127i	$-0.761925\pi$
$443$	−10.7476	+	6.20513i	−0.510634	+	0.294815i	0.222460	+	0.974942i	$0.428591\pi$
$444$		0			0
$445$		−	15.9529i		−	0.756238i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	38.8022			1.83119			0.915596	−	0.402100i	$-0.131720\pi$
$449$	38.8022			1.83119			0.915596	+	0.402100i	$0.131720\pi$
$450$		0			0
$451$	−1.57695	−	0.910450i	−0.0742555	−	0.0428714i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−23.0717			−1.08162
$456$		0			0
$457$	−31.4392			−1.47066			−0.735331	−	0.677708i	$-0.762973\pi$
$457$	−31.4392			−1.47066			−0.735331	+	0.677708i	$0.762973\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.76189	−	1.59458i	−0.128634	−	0.0742669i	0.434302	−	0.900767i	$-0.356995\pi$
$461$	−2.76189	−	1.59458i	−0.128634	−	0.0742669i	−0.562936	+	0.826500i	$0.690328\pi$
$462$		0			0
$463$	21.6707			1.00712			0.503562	−	0.863959i	$-0.332022\pi$
$463$	21.6707			1.00712			0.503562	+	0.863959i	$0.332022\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	19.0633i		−	0.882144i	−0.897472	−	0.441072i	$-0.854598\pi$
$467$		−	19.0633i		−	0.882144i	0.897472	−	0.441072i	$-0.145402\pi$
$468$		0			0
$469$	1.19956	−	0.692567i	0.0553906	−	0.0319798i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.590416	−	0.340877i	−0.0271474	−	0.0156735i
$474$		0			0
$475$	2.81039	−	2.38489i	0.128949	−	0.109426i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.51906	+	4.34113i	−0.343554	+	0.198351i	−0.661843	−	0.749643i	$-0.730225\pi$
$479$	−7.51906	+	4.34113i	−0.343554	+	0.198351i	0.318288	+	0.947994i	$0.396892\pi$
$480$		0			0
$481$	−5.12143	−	8.87058i	−0.233517	−	0.404464i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.0939	−	17.4831i	−0.458338	−	0.793865i
$486$		0			0
$487$			17.7742i			0.805425i	0.915326	+	0.402713i	$0.131933\pi$
$487$			17.7742i			0.805425i	−0.915326	+	0.402713i	$0.868067\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.13906	−	0.657635i	−0.0514050	−	0.0296787i	0.474077	−	0.880483i	$-0.342782\pi$
$491$	−1.13906	−	0.657635i	−0.0514050	−	0.0296787i	−0.525482	+	0.850805i	$0.676115\pi$
$492$		0			0
$493$		−	9.48734i		−	0.427288i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	29.7390	−	51.5094i	1.33398	−	2.31051i
$498$		0			0
$499$	−10.7952	+	18.6979i	−0.483261	+	0.837033i	−0.999815	−	0.0192214i	$-0.993881\pi$
$499$	−10.7952	+	18.6979i	−0.483261	+	0.837033i	0.516554	+	0.856255i	$0.327215\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	28.2076	−	16.2857i	1.25772	−	0.726143i	0.285086	−	0.958502i	$-0.407978\pi$
$503$	28.2076	−	16.2857i	1.25772	−	0.726143i	0.972630	+	0.232359i	$0.0746444\pi$
$504$		0			0
$505$	31.4766			1.40069
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.4800	−	19.8839i	−0.508842	−	0.881339i	−0.999948	−	0.0102396i	$-0.996741\pi$
$509$	−11.4800	−	19.8839i	−0.508842	−	0.881339i	0.491106	−	0.871100i	$-0.336593\pi$
$510$		0			0
$511$	29.9565	−	51.8863i	1.32520	−	2.29531i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−13.5610	+	23.4884i	−0.597570	+	1.03502i
$516$		0			0
$517$	−1.06877	−	1.85116i	−0.0470044	−	0.0814139i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14.0841			−0.617035			−0.308517	−	0.951219i	$-0.599833\pi$
$521$	−14.0841			−0.617035			−0.308517	+	0.951219i	$0.599833\pi$
$522$		0			0
$523$	33.1343	−	19.1301i	1.44886	−	0.836501i	0.450448	−	0.892802i	$-0.351264\pi$
$523$	33.1343	−	19.1301i	1.44886	−	0.836501i	0.998414	+	0.0563014i	$0.0179308\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6.15655	+	10.6635i	−0.268183	+	0.464507i
$528$		0			0
$529$	−8.16555	+	14.1431i	−0.355024	+	0.614920i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	9.20176i		−	0.398572i
$534$		0			0
$535$	14.0504	+	8.11199i	0.607450	+	0.350712i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	4.04680i		−	0.174308i
$540$		0			0
$541$	2.22232	+	3.84916i	0.0955448	+	0.165489i	0.909836	−	0.414968i	$-0.136207\pi$
$541$	2.22232	+	3.84916i	0.0955448	+	0.165489i	−0.814291	+	0.580457i	$0.802874\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.31748	−	14.4063i	−0.356282	−	0.617098i
$546$		0			0
$547$	18.2229	−	10.5210i	0.779156	−	0.449846i	−0.0569750	−	0.998376i	$-0.518146\pi$
$547$	18.2229	−	10.5210i	0.779156	−	0.449846i	0.836131	+	0.548530i	$0.184812\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11.3376	+	9.62108i	−0.482998	+	0.409872i
$552$		0			0
$553$	−27.5575	−	15.9104i	−1.17187	−	0.676577i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.4094	−	12.3607i	0.907145	−	0.523740i	0.0276336	−	0.999618i	$-0.491203\pi$
$557$	21.4094	−	12.3607i	0.907145	−	0.523740i	0.879511	+	0.475878i	$0.157869\pi$
$558$		0			0
$559$		−	3.44518i		−	0.145716i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−20.1514			−0.849281			−0.424640	−	0.905362i	$-0.639599\pi$
$563$	−20.1514			−0.849281			−0.424640	+	0.905362i	$0.639599\pi$
$564$		0			0
$565$	6.41273	+	3.70239i	0.269786	+	0.155761i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.91264			−0.122104			−0.0610520	−	0.998135i	$-0.519446\pi$
$569$	−2.91264			−0.122104			−0.0610520	+	0.998135i	$0.519446\pi$
$570$		0			0
$571$	5.30300			0.221924			0.110962	−	0.993825i	$-0.464607\pi$
$571$	5.30300			0.221924			0.110962	+	0.993825i	$0.464607\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.89115	−	1.09186i	−0.0788665	−	0.0455336i
$576$		0			0
$577$	29.0613			1.20984			0.604919	−	0.796287i	$-0.293205\pi$
$577$	29.0613			1.20984			0.604919	+	0.796287i	$0.293205\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	26.5313i		−	1.10070i
$582$		0			0
$583$	1.03301	−	0.596409i	0.0427829	−	0.0247007i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.60901	+	3.81571i	0.272783	+	0.157491i	0.630152	−	0.776472i	$-0.282993\pi$
$587$	6.60901	+	3.81571i	0.272783	+	0.157491i	−0.357369	+	0.933963i	$0.616326\pi$
$588$		0			0
$589$	18.9864	−	3.45655i	0.782322	−	0.142425i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	23.1987	−	13.3938i	0.952657	−	0.550017i	0.0587516	−	0.998273i	$-0.481288\pi$
$593$	23.1987	−	13.3938i	0.952657	−	0.550017i	0.893905	+	0.448256i	$0.147955\pi$
$594$		0			0
$595$	10.4841	+	18.1590i	0.429806	+	0.744446i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.4393	−	33.6699i	−0.794270	−	1.37572i	−0.923302	−	0.384075i	$-0.874520\pi$
$599$	−19.4393	−	33.6699i	−0.794270	−	1.37572i	0.129032	−	0.991640i	$-0.458813\pi$
$600$		0			0
$601$		−	20.4550i		−	0.834377i	−0.908820	−	0.417188i	$-0.863015\pi$
$601$		−	20.4550i		−	0.834377i	0.908820	−	0.417188i	$-0.136985\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−18.7695	−	10.8366i	−0.763088	−	0.440569i
$606$		0			0
$607$			4.65988i			0.189139i	0.995518	+	0.0945694i	$0.0301474\pi$
$607$			4.65988i			0.189139i	−0.995518	+	0.0945694i	$0.969853\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.40092	−	9.35467i	0.218498	−	0.378450i
$612$		0			0
$613$	−5.90269	+	10.2238i	−0.238408	+	0.412934i	−0.960258	−	0.279116i	$-0.909959\pi$
$613$	−5.90269	+	10.2238i	−0.238408	+	0.412934i	0.721850	+	0.692050i	$0.243292\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	42.1238	−	24.3202i	1.69584	−	0.979094i	0.746215	−	0.665705i	$-0.231869\pi$
$617$	42.1238	−	24.3202i	1.69584	−	0.979094i	0.949625	−	0.313389i	$-0.101464\pi$
$618$		0			0
$619$	42.7915			1.71993			0.859967	−	0.510350i	$-0.170484\pi$
$619$	42.7915			1.71993			0.859967	+	0.510350i	$0.170484\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	14.4758	+	25.0728i	0.579959	+	1.00452i
$624$		0			0
$625$	10.0285	−	17.3698i	0.401138	−	0.694792i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.65449	+	8.06181i	−0.185587	+	0.321445i
$630$		0			0
$631$	−18.6092	−	32.2321i	−0.740820	−	1.28314i	−0.952122	−	0.305717i	$-0.901104\pi$
$631$	−18.6092	−	32.2321i	−0.740820	−	1.28314i	0.211302	−	0.977421i	$-0.432230\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	13.4054			0.531976
$636$		0			0
$637$	17.7103	−	10.2251i	0.701709	−	0.405132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.29142	+	5.70091i	−0.130003	+	0.225172i	−0.923678	−	0.383171i	$-0.874832\pi$
$641$	−3.29142	+	5.70091i	−0.130003	+	0.225172i	0.793674	+	0.608343i	$0.208165\pi$
$642$		0			0
$643$	6.24623	−	10.8188i	0.246327	−	0.426651i	−0.716177	−	0.697919i	$-0.754110\pi$
$643$	6.24623	−	10.8188i	0.246327	−	0.426651i	0.962504	+	0.271268i	$0.0874429\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			4.23117i			0.166344i	0.996535	+	0.0831722i	$0.0265052\pi$
$647$			4.23117i			0.166344i	−0.996535	+	0.0831722i	$0.973495\pi$
$648$		0			0
$649$	−3.65456	−	2.10996i	−0.143454	−	0.0828233i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	2.95004i		−	0.115444i	−0.998333	−	0.0577219i	$-0.981616\pi$
$653$		−	2.95004i		−	0.115444i	0.998333	−	0.0577219i	$-0.0183837\pi$
$654$		0			0
$655$	3.85328	+	6.67408i	0.150560	+	0.260778i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.557166	+	0.965039i	0.0217041	+	0.0375926i	0.876673	−	0.481086i	$-0.159757\pi$
$659$	0.557166	+	0.965039i	0.0217041	+	0.0375926i	−0.854969	+	0.518679i	$0.826424\pi$
$660$		0			0
$661$	−21.5103	+	12.4190i	−0.836653	+	0.483042i	−0.856125	−	0.516768i	$-0.827135\pi$
$661$	−21.5103	+	12.4190i	−0.836653	+	0.483042i	0.0194720	+	0.999810i	$0.493801\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	11.0686	−	30.9437i	0.429220	−	1.19995i
$666$		0			0
$667$	7.62925	+	4.40475i	0.295406	+	0.170553i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.22519	−	3.01677i	0.201716	−	0.116461i
$672$		0			0
$673$			4.51767i			0.174144i	0.996202	+	0.0870718i	$0.0277509\pi$
$673$			4.51767i			0.174144i	−0.996202	+	0.0870718i	$0.972249\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−11.8210			−0.454319			−0.227159	−	0.973858i	$-0.572944\pi$
$677$	−11.8210			−0.454319			−0.227159	+	0.973858i	$0.572944\pi$
$678$		0			0
$679$	31.7286	+	18.3185i	1.21763	+	0.703000i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21.2681			−0.813803			−0.406901	−	0.913472i	$-0.633391\pi$
$683$	−21.2681			−0.813803			−0.406901	+	0.913472i	$0.633391\pi$
$684$		0			0
$685$	−29.6148			−1.13152
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.22023	+	3.01390i	0.198875	+	0.114820i
$690$		0			0
$691$	−34.9630			−1.33006			−0.665028	−	0.746819i	$-0.731580\pi$
$691$	−34.9630			−1.33006			−0.665028	+	0.746819i	$0.731580\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	15.4840i		−	0.587341i
$696$		0			0
$697$	−7.24239	+	4.18140i	−0.274325	+	0.158382i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.1442	−	22.0226i	−1.44069	−	0.831781i	−0.442791	−	0.896625i	$-0.646012\pi$
$701$	−38.1442	−	22.0226i	−1.44069	−	0.831781i	−0.997895	+	0.0648442i	$0.979345\pi$
$702$		0			0
$703$	14.3542	−	2.61323i	0.541378	−	0.0985598i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−49.4711	+	28.5622i	−1.86055	+	1.07419i
$708$		0			0
$709$	16.9260	+	29.3166i	0.635668	+	1.10101i	0.986373	+	0.164523i	$0.0526085\pi$
$709$	16.9260	+	29.3166i	0.635668	+	1.10101i	−0.350705	+	0.936486i	$0.614058\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−5.71668	−	9.90158i	−0.214091	−	0.370817i
$714$		0			0
$715$			3.77701i			0.141252i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	45.9929	+	26.5540i	1.71525	+	0.990298i	0.927103	+	0.374807i	$0.122291\pi$
$719$	45.9929	+	26.5540i	1.71525	+	0.990298i	0.788144	+	0.615491i	$0.211042\pi$
$720$		0			0
$721$		−	49.2216i		−	1.83311i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.44232	−	2.49818i	0.0535666	−	0.0927800i
$726$		0			0
$727$	−3.59081	+	6.21946i	−0.133176	+	0.230667i	−0.924899	−	0.380213i	$-0.875851\pi$
$727$	−3.59081	+	6.21946i	−0.133176	+	0.230667i	0.791723	+	0.610880i	$0.209184\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.71159	+	1.56553i	−0.100292	+	0.0579034i
$732$		0			0
$733$	28.8680			1.06626			0.533132	−	0.846032i	$-0.321015\pi$
$733$	28.8680			1.06626			0.533132	+	0.846032i	$0.321015\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.113378	−	0.196377i	−0.00417634	−	0.00723364i
$738$		0			0
$739$	16.1981	−	28.0559i	0.595855	−	1.03205i	−0.397570	−	0.917572i	$-0.630146\pi$
$739$	16.1981	−	28.0559i	0.595855	−	1.03205i	0.993426	−	0.114480i	$-0.0365202\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17.2449	−	29.8690i	0.632654	−	1.09579i	−0.354353	−	0.935112i	$-0.615299\pi$
$743$	17.2449	−	29.8690i	0.632654	−	1.09579i	0.987007	−	0.160677i	$-0.0513678\pi$
$744$		0			0
$745$	−3.61025	−	6.25314i	−0.132269	−	0.229097i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−29.4436			−1.07584
$750$		0			0
$751$	19.7822	−	11.4212i	0.721861	−	0.416767i	−0.0935759	−	0.995612i	$-0.529830\pi$
$751$	19.7822	−	11.4212i	0.721861	−	0.416767i	0.815437	+	0.578845i	$0.196496\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.12442	+	1.94755i	−0.0409219	+	0.0708788i
$756$		0			0
$757$	2.87620	−	4.98173i	0.104537	−	0.181064i	−0.809012	−	0.587792i	$-0.799997\pi$
$757$	2.87620	−	4.98173i	0.104537	−	0.181064i	0.913549	+	0.406728i	$0.133330\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			48.7434i			1.76695i	0.468481	+	0.883474i	$0.344801\pi$
$761$			48.7434i			1.76695i	−0.468481	+	0.883474i	$0.655199\pi$
$762$		0			0
$763$	26.1448	+	15.0947i	0.946506	+	0.546465i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	21.3250i		−	0.770002i
$768$		0			0
$769$	−3.50200	−	6.06565i	−0.126285	−	0.218733i	0.795949	−	0.605363i	$-0.206972\pi$
$769$	−3.50200	−	6.06565i	−0.126285	−	0.218733i	−0.922235	+	0.386631i	$0.873639\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5.00414	+	8.66743i	0.179987	+	0.311746i	0.941876	−	0.335962i	$-0.109061\pi$
$773$	5.00414	+	8.66743i	0.179987	+	0.311746i	−0.761889	+	0.647707i	$0.775728\pi$
$774$		0			0
$775$	−3.24225	+	1.87191i	−0.116465	+	0.0672411i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.3414	+	4.41450i	0.442175	+	0.158166i
$780$		0			0
$781$	−8.43247	−	4.86849i	−0.301738	−	0.174208i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.44028	−	1.40890i	0.0870973	−	0.0502856i
$786$		0			0
$787$			34.2689i			1.22155i	0.791803	+	0.610777i	$0.209143\pi$
$787$			34.2689i			1.22155i	−0.791803	+	0.610777i	$0.790857\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−13.4383			−0.477812
$792$		0			0
$793$	26.4050	+	15.2450i	0.937671	+	0.541364i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	27.8026			0.984817			0.492409	−	0.870364i	$-0.336117\pi$
$797$	27.8026			0.984817			0.492409	+	0.870364i	$0.336117\pi$
$798$		0			0
$799$	−9.81700			−0.347300
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−8.49416	−	4.90411i	−0.299752	−	0.173062i
$804$		0			0
$805$	−19.4701			−0.686230
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	38.6452i		−	1.35869i	−0.733817	−	0.679347i	$-0.762263\pi$
$809$		−	38.6452i		−	1.35869i	0.733817	−	0.679347i	$-0.237737\pi$
$810$		0			0
$811$	−37.9998	+	21.9392i	−1.33435	+	0.770390i	−0.985964	−	0.166960i	$-0.946605\pi$
$811$	−37.9998	+	21.9392i	−1.33435	+	0.770390i	−0.348390	+	0.937350i	$0.613272\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	27.0919	+	15.6415i	0.948989	+	0.547899i
$816$		0			0
$817$	4.62066	+	1.65281i	0.161657	+	0.0578245i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−45.8864	+	26.4925i	−1.60145	+	0.924596i	−0.610250	+	0.792209i	$0.708931\pi$
$821$	−45.8864	+	26.4925i	−1.60145	+	0.924596i	−0.991198	+	0.132387i	$0.957736\pi$
$822$		0			0
$823$	14.0908	+	24.4059i	0.491174	+	0.850737i	0.999948	−	0.0101621i	$-0.00323476\pi$
$823$	14.0908	+	24.4059i	0.491174	+	0.850737i	−0.508775	+	0.860900i	$0.669901\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−17.9263	−	31.0492i	−0.623358	−	1.07969i	−0.988856	−	0.148876i	$-0.952435\pi$
$827$	−17.9263	−	31.0492i	−0.623358	−	1.07969i	0.365498	−	0.930812i	$-0.380899\pi$
$828$		0			0
$829$			37.7685i			1.31175i	0.754868	+	0.655877i	$0.227701\pi$
$829$			37.7685i			1.31175i	−0.754868	+	0.655877i	$0.772299\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−16.0956	−	9.29280i	−0.557680	−	0.321976i
$834$		0			0
$835$		−	10.0046i		−	0.346225i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−11.3542	+	19.6660i	−0.391989	+	0.678945i	−0.992712	−	0.120513i	$-0.961546\pi$
$839$	−11.3542	+	19.6660i	−0.391989	+	0.678945i	0.600723	+	0.799457i	$0.294880\pi$
$840$		0			0
$841$	8.68140	−	15.0366i	0.299359	−	0.518505i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.41744	−	3.70511i	0.220767	−	0.127460i
$846$		0			0
$847$	39.3328			1.35149
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.32194	−	7.48582i	−0.148154	−	0.256611i
$852$		0			0
$853$	18.3713	−	31.8200i	0.629021	−	1.08950i	−0.358728	−	0.933442i	$-0.616789\pi$
$853$	18.3713	−	31.8200i	0.629021	−	1.08950i	0.987749	−	0.156054i	$-0.0498772\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.42058	−	5.92462i	0.116845	−	0.202381i	−0.801671	−	0.597766i	$-0.796055\pi$
$857$	3.42058	−	5.92462i	0.116845	−	0.202381i	0.918516	+	0.395385i	$0.129389\pi$
$858$		0			0
$859$	−11.3794	−	19.7097i	−0.388261	−	0.672488i	0.603955	−	0.797019i	$-0.293591\pi$
$859$	−11.3794	−	19.7097i	−0.388261	−	0.672488i	−0.992216	+	0.124531i	$0.960257\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.6907			0.670279			0.335139	−	0.942169i	$-0.391217\pi$
$863$	19.6907			0.670279			0.335139	+	0.942169i	$0.391217\pi$
$864$		0			0
$865$	43.3518	−	25.0292i	1.47401	−	0.851017i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.60464	+	4.51137i	−0.0883564	+	0.153038i
$870$		0			0
$871$	0.572947	−	0.992374i	0.0194136	−	0.0336253i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			44.0728i			1.48993i
$876$		0			0
$877$	13.0986	+	7.56250i	0.442309	+	0.255367i	0.704577	−	0.709628i	$-0.251137\pi$
$877$	13.0986	+	7.56250i	0.442309	+	0.255367i	−0.262267	+	0.964995i	$0.584470\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			41.3230i			1.39221i	0.717942	+	0.696103i	$0.245084\pi$
$881$			41.3230i			1.39221i	−0.717942	+	0.696103i	$0.754916\pi$
$882$		0			0
$883$	23.1270	+	40.0572i	0.778286	+	1.34803i	0.932929	+	0.360061i	$0.117244\pi$
$883$	23.1270	+	40.0572i	0.778286	+	1.34803i	−0.154642	+	0.987971i	$0.549422\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12.7167	−	22.0260i	−0.426986	−	0.739561i	0.569618	−	0.821910i	$-0.307091\pi$
$887$	−12.7167	−	22.0260i	−0.426986	−	0.739561i	−0.996604	+	0.0823488i	$0.973758\pi$
$888$		0			0
$889$	−21.0689	+	12.1641i	−0.706629	+	0.407972i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9.95539	+	11.7316i	0.333144	+	0.392582i
$894$		0			0
$895$	−27.4021	−	15.8206i	−0.915953	−	0.528826i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	13.0798	−	7.55164i	0.436236	−	0.251861i
$900$		0			0
$901$		−	5.47822i		−	0.182506i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	27.4585			0.912750
$906$		0			0
$907$	19.5518	+	11.2882i	0.649205	+	0.374819i	0.788152	−	0.615481i	$-0.211038\pi$
$907$	19.5518	+	11.2882i	0.649205	+	0.374819i	−0.138946	+	0.990300i	$0.544372\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−50.6150			−1.67695			−0.838475	−	0.544940i	$-0.816552\pi$
$911$	−50.6150			−1.67695			−0.838475	+	0.544940i	$0.816552\pi$
$912$		0			0
$913$	−4.34336			−0.143744
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.1122	−	6.99300i	−0.399981	−	0.230929i
$918$		0			0
$919$	−30.3240			−1.00030			−0.500148	−	0.865940i	$-0.666721\pi$
$919$	−30.3240			−1.00030			−0.500148	+	0.865940i	$0.666721\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	49.2050i		−	1.61960i
$924$		0			0
$925$	−2.45121	+	1.41521i	−0.0805954	+	0.0465318i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25.7976	−	14.8943i	−0.846393	−	0.488665i	0.0130391	−	0.999915i	$-0.495849\pi$
$929$	−25.7976	−	14.8943i	−0.846393	−	0.488665i	−0.859432	+	0.511250i	$0.829183\pi$
$930$		0			0
$931$	5.21737	+	28.6585i	0.170993	+	0.939243i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2.97276	−	1.71632i	0.0972196	−	0.0561298i
$936$		0			0
$937$	−13.1927	−	22.8504i	−0.430985	−	0.746488i	0.565973	−	0.824424i	$-0.308501\pi$
$937$	−13.1927	−	22.8504i	−0.430985	−	0.746488i	−0.996958	+	0.0779353i	$0.975167\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.47207	−	2.54971i	−0.0479883	−	0.0831181i	0.841034	−	0.540983i	$-0.181948\pi$
$941$	−1.47207	−	2.54971i	−0.0479883	−	0.0831181i	−0.889022	+	0.457865i	$0.848614\pi$
$942$		0			0
$943$		−	7.76530i		−	0.252873i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−2.65210	−	1.53119i	−0.0861815	−	0.0497569i	0.456290	−	0.889831i	$-0.349178\pi$
$947$	−2.65210	−	1.53119i	−0.0861815	−	0.0497569i	−0.542471	+	0.840074i	$0.682511\pi$
$948$		0			0
$949$		−	49.5649i		−	1.60895i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.84291	+	3.19201i	−0.0596977	+	0.103399i	−0.894330	−	0.447409i	$-0.852347\pi$
$953$	−1.84291	+	3.19201i	−0.0596977	+	0.103399i	0.834632	+	0.550808i	$0.185680\pi$
$954$		0			0
$955$	14.6122	−	25.3091i	0.472840	−	0.818982i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	46.5450	−	26.8728i	1.50302	−	0.867767i
$960$		0			0
$961$	11.3983			0.367687
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.7485	+	34.2054i	0.635727	+	1.10111i
$966$		0			0
$967$	16.0399	−	27.7819i	0.515808	−	0.893405i	−0.484024	−	0.875055i	$-0.660825\pi$
$967$	16.0399	−	27.7819i	0.515808	−	0.893405i	0.999832	−	0.0183505i	$-0.00584147\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7.26555	+	12.5843i	−0.233163	+	0.403849i	−0.958737	−	0.284294i	$-0.908241\pi$
$971$	−7.26555	+	12.5843i	−0.233163	+	0.403849i	0.725575	+	0.688144i	$0.241574\pi$
$972$		0			0
$973$	14.0503	+	24.3358i	0.450432	+	0.780171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.611504			−0.0195637			−0.00978187	−	0.999952i	$-0.503114\pi$
$977$	−0.611504			−0.0195637			−0.00978187	+	0.999952i	$0.503114\pi$
$978$		0			0
$979$	4.10460	−	2.36979i	0.131183	−	0.0757388i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.61045	+	2.78937i	−0.0513652	+	0.0889672i	−0.890565	−	0.454856i	$-0.849691\pi$
$983$	−1.61045	+	2.78937i	−0.0513652	+	0.0889672i	0.839200	+	0.543824i	$0.183024\pi$
$984$		0			0
$985$	−14.4838	+	25.0867i	−0.461493	+	0.799330i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	2.90736i		−	0.0924488i
$990$		0			0
$991$	−34.9740	−	20.1923i	−1.11099	−	0.641428i	−0.171902	−	0.985114i	$-0.554991\pi$
$991$	−34.9740	−	20.1923i	−1.11099	−	0.641428i	−0.939085	+	0.343686i	$0.888325\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	46.0827i		−	1.46092i
$996$		0			0
$997$	27.5964	+	47.7983i	0.873986	+	1.51379i	0.857839	+	0.513918i	$0.171806\pi$
$997$	27.5964	+	47.7983i	0.873986	+	1.51379i	0.0161464	+	0.999870i	$0.494860\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.cu.a.521.3	yes	20
3.2	odd	2		1368.2.cu.b.521.8	yes	20
4.3	odd	2		2736.2.dc.e.1889.3		20
12.11	even	2		2736.2.dc.f.1889.8		20
19.12	odd	6		1368.2.cu.b.449.8	yes	20
57.50	even	6	inner	1368.2.cu.a.449.3	✓	20
76.31	even	6		2736.2.dc.f.449.8		20
228.107	odd	6		2736.2.dc.e.449.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.2.cu.a.449.3	✓	20	57.50	even	6	inner
1368.2.cu.a.521.3	yes	20	1.1	even	1	trivial
1368.2.cu.b.449.8	yes	20	19.12	odd	6
1368.2.cu.b.521.8	yes	20	3.2	odd	2
2736.2.dc.e.449.3		20	228.107	odd	6
2736.2.dc.e.1889.3		20	4.3	odd	2
2736.2.dc.f.449.8		20	76.31	even	6
2736.2.dc.f.1889.8		20	12.11	even	2

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.cu (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.76516 - 1.01912i) q^{5} +3.69902 q^{7} +O(q^{10})\)	\(q+(-1.76516 - 1.01912i) q^{5} +3.69902 q^{7} -0.605557i q^{11} +(2.65015 - 1.53006i) q^{13} +(-2.40852 - 1.39056i) q^{17} +(0.780721 + 4.28841i) q^{19} +(2.23644 - 1.29121i) q^{23} +(-0.422804 - 0.732318i) q^{25} +(1.70567 + 2.95430i) q^{29} -4.42738i q^{31} +(-6.52937 - 3.76973i) q^{35} -3.34720i q^{37} +(1.50349 - 2.60412i) q^{41} +(0.562914 - 0.974996i) q^{43} +(3.05695 - 1.76493i) q^{47} +6.68277 q^{49} +(0.984893 + 1.70588i) q^{53} +(-0.617133 + 1.06891i) q^{55} +(3.48434 - 6.03505i) q^{59} +(4.98180 + 8.62874i) q^{61} -6.23725 q^{65} +(0.324291 - 0.187230i) q^{67} +(8.03969 - 13.9252i) q^{71} +(8.09850 - 14.0270i) q^{73} -2.23997i q^{77} +(-7.44995 - 4.30123i) q^{79} -7.17251i q^{83} +(2.83429 + 4.90913i) q^{85} +(3.91340 + 6.77822i) q^{89} +(9.80296 - 5.65974i) q^{91} +(2.99229 - 8.36538i) q^{95} +(8.57756 + 4.95226i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{7}+O(q^{10}) \) 20 * q - 4 * q^7	\( 20 q - 4 q^{7} + 6 q^{13} - 12 q^{17} + 4 q^{19} + 14 q^{25} + 12 q^{35} - 8 q^{41} - 2 q^{43} + 36 q^{47} + 32 q^{49} + 8 q^{53} + 12 q^{55} - 8 q^{59} - 2 q^{61} - 8 q^{65} + 30 q^{67} + 4 q^{71} - 22 q^{73} + 54 q^{79} - 4 q^{85} + 32 q^{89} + 18 q^{91} + 32 q^{95} + 12 q^{97}+O(q^{100}) \) 20 * q - 4 * q^7 + 6 * q^13 - 12 * q^17 + 4 * q^19 + 14 * q^25 + 12 * q^35 - 8 * q^41 - 2 * q^43 + 36 * q^47 + 32 * q^49 + 8 * q^53 + 12 * q^55 - 8 * q^59 - 2 * q^61 - 8 * q^65 + 30 * q^67 + 4 * q^71 - 22 * q^73 + 54 * q^79 - 4 * q^85 + 32 * q^89 + 18 * q^91 + 32 * q^95 + 12 * q^97

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