LMFDB - Embedded newform 3420.2.t.w.3241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1261,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("3420.1261");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.t (of order $3$, 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:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 9x^{6} + 2x^{5} + 65x^{4} - 20x^{3} + 25x^{2} + 6x + 4 $ x^8 - x^7 + 9x^6 + 2x^5 + 65x^4 - 20x^3 + 25x^2 + 6x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3241.1
Root			$-0.176725 - 0.306096i$ of defining polynomial
Character	$\chi$	$=$	3420.3241
Dual form			3420.2.t.w.1261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} -4.30507 q^{7} +O(q^{10})$	$q+(0.500000 - 0.866025i) q^{5} -4.30507 q^{7} -6.01196 q^{11} +(2.97581 + 5.15425i) q^{13} +(1.93754 - 3.35591i) q^{17} +(4.19835 + 1.17212i) q^{19} +(0.391721 + 0.678480i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(3.98179 + 6.89666i) q^{29} -4.49034 q^{31} +(-2.15253 + 3.72830i) q^{35} -0.988035 q^{37} +(3.15253 - 5.46035i) q^{41} +(0.785004 - 1.35967i) q^{43} +(-0.630909 - 1.09277i) q^{47} +11.5336 q^{49} +(-4.07443 - 7.05712i) q^{53} +(-3.00598 + 5.20651i) q^{55} +(2.62834 - 4.55242i) q^{59} +(-2.80507 - 4.85852i) q^{61} +5.95162 q^{65} +(-3.52162 - 6.09963i) q^{67} +(-2.90736 + 5.03570i) q^{71} +(4.62024 - 8.00250i) q^{73} +25.8819 q^{77} +(6.99743 - 12.1199i) q^{79} -6.58197 q^{83} +(-1.93754 - 3.35591i) q^{85} +(-1.69237 - 2.93126i) q^{89} +(-12.8110 - 22.1894i) q^{91} +(3.11426 - 3.04982i) q^{95} +(3.69835 - 6.40573i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5}+O(q^{10}) $ 8 * q + 4 * q^5	$ 8 q + 4 q^{5} - 4 q^{11} + 9 q^{13} - q^{17} + 3 q^{19} - 4 q^{25} - 5 q^{29} - 20 q^{31} - 52 q^{37} + 8 q^{41} + 7 q^{43} - 16 q^{47} + 20 q^{49} - 5 q^{53} - 2 q^{55} - 11 q^{59} + 12 q^{61} + 18 q^{65} - 14 q^{71} - 4 q^{73} + 44 q^{77} + 13 q^{79} - 10 q^{83} + q^{85} - 5 q^{89} - 46 q^{91} + 6 q^{95} - q^{97}+O(q^{100}) $ 8 * q + 4 * q^5 - 4 * q^11 + 9 * q^13 - q^17 + 3 * q^19 - 4 * q^25 - 5 * q^29 - 20 * q^31 - 52 * q^37 + 8 * q^41 + 7 * q^43 - 16 * q^47 + 20 * q^49 - 5 * q^53 - 2 * q^55 - 11 * q^59 + 12 * q^61 + 18 * q^65 - 14 * q^71 - 4 * q^73 + 44 * q^77 + 13 * q^79 - 10 * q^83 + q^85 - 5 * q^89 - 46 * q^91 + 6 * q^95 - q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−4.30507			−1.62716			−0.813581	−	0.581452i	$-0.802485\pi$
$7$	−4.30507			−1.62716			−0.813581	+	0.581452i	$0.802485\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.01196			−1.81268			−0.906338	−	0.422554i	$-0.861134\pi$
$11$	−6.01196			−1.81268			−0.906338	+	0.422554i	$0.861134\pi$
$12$		0			0
$13$	2.97581	+	5.15425i	0.825341	+	1.42953i	0.901659	+	0.432448i	$0.142350\pi$
$13$	2.97581	+	5.15425i	0.825341	+	1.42953i	−0.0763181	+	0.997084i	$0.524316\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.93754	−	3.35591i	0.469922	−	0.813928i	−0.529487	−	0.848318i	$-0.677615\pi$
$17$	1.93754	−	3.35591i	0.469922	−	0.813928i	0.999408	+	0.0343900i	$0.0109488\pi$
$18$		0			0
$19$	4.19835	+	1.17212i	0.963167	+	0.268903i
$19$	4.19835	+	1.17212i	0.963167	+	0.268903i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.391721	+	0.678480i	0.0816794	+	0.141473i	0.903971	−	0.427593i	$-0.140638\pi$
$23$	0.391721	+	0.678480i	0.0816794	+	0.141473i	−0.822292	+	0.569066i	$0.807305\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.98179	+	6.89666i	0.739400	+	1.28068i	0.952766	+	0.303706i	$0.0982240\pi$
$29$	3.98179	+	6.89666i	0.739400	+	1.28068i	−0.213366	+	0.976972i	$0.568443\pi$
$30$		0			0
$31$	−4.49034			−0.806489			−0.403245	−	0.915092i	$-0.632118\pi$
$31$	−4.49034			−0.806489			−0.403245	+	0.915092i	$0.632118\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.15253	+	3.72830i	−0.363844	+	0.630197i
$36$		0			0
$37$	−0.988035			−0.162432			−0.0812160	−	0.996697i	$-0.525880\pi$
$37$	−0.988035			−0.162432			−0.0812160	+	0.996697i	$0.525880\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.15253	−	5.46035i	0.492343	−	0.852763i	−0.507618	−	0.861582i	$-0.669474\pi$
$41$	3.15253	−	5.46035i	0.492343	−	0.852763i	0.999961	+	0.00881921i	$0.00280728\pi$
$42$		0			0
$43$	0.785004	−	1.35967i	0.119712	−	0.207347i	−0.799942	−	0.600078i	$-0.795136\pi$
$43$	0.785004	−	1.35967i	0.119712	−	0.207347i	0.919654	+	0.392731i	$0.128470\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.630909	−	1.09277i	−0.0920275	−	0.159396i	0.816337	−	0.577576i	$-0.196001\pi$
$47$	−0.630909	−	1.09277i	−0.0920275	−	0.159396i	−0.908364	+	0.418180i	$0.862668\pi$
$48$		0			0
$49$	11.5336			1.64766
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−4.07443	−	7.05712i	−0.559666	−	0.969369i	−0.997524	−	0.0703255i	$-0.977596\pi$
$53$	−4.07443	−	7.05712i	−0.559666	−	0.969369i	0.437858	−	0.899044i	$-0.355737\pi$
$54$		0			0
$55$	−3.00598	+	5.20651i	−0.405327	+	0.702046i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.62834	−	4.55242i	0.342181	−	0.592675i	−0.642657	−	0.766154i	$-0.722168\pi$
$59$	2.62834	−	4.55242i	0.342181	−	0.592675i	0.984837	+	0.173480i	$0.0555011\pi$
$60$		0			0
$61$	−2.80507	−	4.85852i	−0.359152	−	0.622069i	0.628668	−	0.777674i	$-0.283601\pi$
$61$	−2.80507	−	4.85852i	−0.359152	−	0.622069i	−0.987819	+	0.155605i	$0.950267\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	5.95162			0.738207
$66$		0			0
$67$	−3.52162	−	6.09963i	−0.430235	−	0.745189i	0.566658	−	0.823953i	$-0.308236\pi$
$67$	−3.52162	−	6.09963i	−0.430235	−	0.745189i	−0.996893	+	0.0787642i	$0.974903\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.90736	+	5.03570i	−0.345040	+	0.597628i	−0.985361	−	0.170480i	$-0.945468\pi$
$71$	−2.90736	+	5.03570i	−0.345040	+	0.597628i	0.640321	+	0.768108i	$0.278801\pi$
$72$		0			0
$73$	4.62024	−	8.00250i	0.540759	−	0.936621i	−0.458102	−	0.888900i	$-0.651471\pi$
$73$	4.62024	−	8.00250i	0.540759	−	0.936621i	0.998861	−	0.0477218i	$-0.0151961\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	25.8819			2.94952
$78$		0			0
$79$	6.99743	−	12.1199i	0.787273	−	1.36360i	−0.140359	−	0.990101i	$-0.544826\pi$
$79$	6.99743	−	12.1199i	0.787273	−	1.36360i	0.927632	−	0.373495i	$-0.121841\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−6.58197			−0.722465			−0.361233	−	0.932476i	$-0.617644\pi$
$83$	−6.58197			−0.722465			−0.361233	+	0.932476i	$0.617644\pi$
$84$		0			0
$85$	−1.93754	−	3.35591i	−0.210155	−	0.364000i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.69237	−	2.93126i	−0.179390	−	0.310713i	0.762281	−	0.647246i	$-0.224079\pi$
$89$	−1.69237	−	2.93126i	−0.179390	−	0.310713i	−0.941672	+	0.336532i	$0.890746\pi$
$90$		0			0
$91$	−12.8110	−	22.1894i	−1.34296	−	2.32608i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.11426	−	3.04982i	0.319516	−	0.312904i
$96$		0			0
$97$	3.69835	−	6.40573i	0.375510	−	0.650403i	−0.614893	−	0.788611i	$-0.710801\pi$
$97$	3.69835	−	6.40573i	0.375510	−	0.650403i	0.990403	+	0.138208i	$0.0441341\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.90369	−	8.49343i	−0.487935	−	0.845128i	0.511969	−	0.859004i	$-0.328916\pi$
$101$	−4.90369	−	8.49343i	−0.487935	−	0.845128i	−0.999904	+	0.0138759i	$0.995583\pi$
$102$		0			0
$103$	14.4368			1.42250			0.711251	−	0.702938i	$-0.248129\pi$
$103$	14.4368			1.42250			0.711251	+	0.702938i	$0.248129\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	9.49034			0.917466			0.458733	−	0.888574i	$-0.348303\pi$
$107$	9.49034			0.917466			0.458733	+	0.888574i	$0.348303\pi$
$108$		0			0
$109$	1.30920	−	2.26759i	0.125398	−	0.217196i	−0.796490	−	0.604651i	$-0.793312\pi$
$109$	1.30920	−	2.26759i	0.125398	−	0.217196i	0.921889	+	0.387455i	$0.126646\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	13.6705			1.28601			0.643005	−	0.765862i	$-0.277687\pi$
$113$	13.6705			1.28601			0.643005	+	0.765862i	$0.277687\pi$
$114$		0			0
$115$	0.783442			0.0730563
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−8.34122	+	14.4474i	−0.764639	+	1.32439i
$120$		0			0
$121$	25.1437			2.28579
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−6.53359	−	11.3165i	−0.579762	−	1.00418i	−0.995506	−	0.0946960i	$-0.969812\pi$
$127$	−6.53359	−	11.3165i	−0.579762	−	1.00418i	0.415744	−	0.909482i	$-0.363521\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.75070	+	6.49640i	−0.327700	+	0.567593i	−0.982055	−	0.188594i	$-0.939607\pi$
$131$	−3.75070	+	6.49640i	−0.327700	+	0.567593i	0.654355	+	0.756188i	$0.272940\pi$
$132$		0			0
$133$	−18.0742	−	5.04606i	−1.56723	−	0.437549i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.21500	−	7.30059i	−0.360111	−	0.623731i	0.627867	−	0.778320i	$-0.283928\pi$
$137$	−4.21500	−	7.30059i	−0.360111	−	0.623731i	−0.987979	+	0.154589i	$0.950595\pi$
$138$		0			0
$139$	−4.38961	−	7.60302i	−0.372322	−	0.644880i	0.617601	−	0.786492i	$-0.288105\pi$
$139$	−4.38961	−	7.60302i	−0.372322	−	0.644880i	−0.989922	+	0.141612i	$0.954771\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−17.8905	−	30.9872i	−1.49607	−	2.59128i
$144$		0			0
$145$	7.96358			0.661339
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.915913	+	1.58641i	−0.0750345	+	0.129964i	−0.901101	−	0.433609i	$-0.857240\pi$
$149$	−0.915913	+	1.58641i	−0.0750345	+	0.129964i	0.826067	+	0.563572i	$0.190573\pi$
$150$		0			0
$151$	−0.389869			−0.0317271			−0.0158635	−	0.999874i	$-0.505050\pi$
$151$	−0.389869			−0.0317271			−0.0158635	+	0.999874i	$0.505050\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.24517	+	3.88875i	−0.180336	+	0.312352i
$156$		0			0
$157$	1.37608	−	2.38344i	0.109823	−	0.190219i	−0.805875	−	0.592085i	$-0.798305\pi$
$157$	1.37608	−	2.38344i	0.109823	−	0.190219i	0.915698	+	0.401866i	$0.131638\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.68638	−	2.92090i	−0.132906	−	0.230199i
$162$		0			0
$163$	−15.0953			−1.18236			−0.591179	−	0.806540i	$-0.701337\pi$
$163$	−15.0953			−1.18236			−0.591179	+	0.806540i	$0.701337\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.49402	+	2.58771i	0.115611	+	0.200243i	0.918024	−	0.396526i	$-0.129784\pi$
$167$	1.49402	+	2.58771i	0.115611	+	0.200243i	−0.802413	+	0.596769i	$0.796451\pi$
$168$		0			0
$169$	−11.2109	+	19.4178i	−0.862374	+	1.49368i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−11.5945	+	20.0822i	−0.881513	+	1.52682i	−0.0318535	+	0.999493i	$0.510141\pi$
$173$	−11.5945	+	20.0822i	−0.881513	+	1.52682i	−0.849659	+	0.527332i	$0.823192\pi$
$174$		0			0
$175$	2.15253	+	3.72830i	0.162716	+	0.281833i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.5091			1.00972			0.504860	−	0.863201i	$-0.331544\pi$
$179$	13.5091			1.00972			0.504860	+	0.863201i	$0.331544\pi$
$180$		0			0
$181$	10.1559	+	17.5906i	0.754886	+	1.30750i	0.945431	+	0.325822i	$0.105641\pi$
$181$	10.1559	+	17.5906i	0.754886	+	1.30750i	−0.190546	+	0.981678i	$0.561026\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.494018	+	0.855664i	−0.0363209	+	0.0629096i
$186$		0			0
$187$	−11.6484	+	20.1756i	−0.851816	+	1.47539i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.04838			0.510003			0.255002	−	0.966941i	$-0.417924\pi$
$191$	7.04838			0.510003			0.255002	+	0.966941i	$0.417924\pi$
$192$		0			0
$193$	11.8892	−	20.5926i	0.855800	−	1.48229i	−0.0201010	−	0.999798i	$-0.506399\pi$
$193$	11.8892	−	20.5926i	0.855800	−	1.48229i	0.875901	−	0.482491i	$-0.160268\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−23.7428			−1.69160			−0.845802	−	0.533497i	$-0.820878\pi$
$197$	−23.7428			−1.69160			−0.845802	+	0.533497i	$0.820878\pi$
$198$		0			0
$199$	11.4893	+	19.9001i	0.814457	+	1.41068i	0.909717	+	0.415229i	$0.136299\pi$
$199$	11.4893	+	19.9001i	0.814457	+	1.41068i	−0.0952595	+	0.995452i	$0.530368\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−17.1419	−	29.6906i	−1.20312	−	2.08387i
$204$		0			0
$205$	−3.15253	−	5.46035i	−0.220182	−	0.381367i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−25.2403	−	7.04675i	−1.74591	−	0.487434i
$210$		0			0
$211$	0.692366	−	1.19921i	0.0476645	−	0.0825573i	−0.841209	−	0.540710i	$-0.818156\pi$
$211$	0.692366	−	1.19921i	0.0476645	−	0.0825573i	0.888873	+	0.458153i	$0.151489\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.785004	−	1.35967i	−0.0535368	−	0.0927285i
$216$		0			0
$217$	19.3312			1.31229
$218$		0			0
$219$		0			0
$220$		0			0
$221$	23.0629			1.55138
$222$		0			0
$223$	11.6500	−	20.1783i	0.780139	−	1.35124i	−0.151721	−	0.988423i	$-0.548481\pi$
$223$	11.6500	−	20.1783i	0.780139	−	1.35124i	0.931860	−	0.362818i	$-0.118185\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.48943			0.297974			0.148987	−	0.988839i	$-0.452399\pi$
$227$	4.48943			0.297974			0.148987	+	0.988839i	$0.452399\pi$
$228$		0			0
$229$	9.20830			0.608501			0.304251	−	0.952592i	$-0.401594\pi$
$229$	9.20830			0.608501			0.304251	+	0.952592i	$0.401594\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.16265	+	3.74581i	−0.141680	+	0.245396i	−0.928129	−	0.372258i	$-0.878584\pi$
$233$	−2.16265	+	3.74581i	−0.141680	+	0.245396i	0.786450	+	0.617654i	$0.211917\pi$
$234$		0			0
$235$	−1.26182			−0.0823119
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.8267			0.959059			0.479529	−	0.877526i	$-0.340807\pi$
$239$	14.8267			0.959059			0.479529	+	0.877526i	$0.340807\pi$
$240$		0			0
$241$	−1.44453	−	2.50199i	−0.0930500	−	0.161167i	0.815743	−	0.578414i	$-0.196328\pi$
$241$	−1.44453	−	2.50199i	−0.0930500	−	0.161167i	−0.908793	+	0.417247i	$0.862995\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5.76679	−	9.98838i	0.368427	−	0.638134i
$246$		0			0
$247$	6.45207	+	25.1273i	0.410535	+	1.59881i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7.65253	+	13.2546i	0.483024	+	0.836621i	0.999810	−	0.0194930i	$-0.00620520\pi$
$251$	7.65253	+	13.2546i	0.483024	+	0.836621i	−0.516786	+	0.856114i	$0.672872\pi$
$252$		0			0
$253$	−2.35501	−	4.07900i	−0.148058	−	0.256445i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.54214	+	11.3313i	0.408087	+	0.706828i	0.994675	−	0.103057i	$-0.0328625\pi$
$257$	6.54214	+	11.3313i	0.408087	+	0.706828i	−0.586588	+	0.809886i	$0.699529\pi$
$258$		0			0
$259$	4.25356			0.264303
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.68980	−	16.7832i	0.597499	−	1.03490i	−0.395691	−	0.918384i	$-0.629495\pi$
$263$	9.68980	−	16.7832i	0.597499	−	1.03490i	0.993189	−	0.116514i	$-0.0371720\pi$
$264$		0			0
$265$	−8.14886			−0.500580
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.728523	+	1.26184i	−0.0444188	+	0.0769357i	−0.887380	−	0.461039i	$-0.847477\pi$
$269$	−0.728523	+	1.26184i	−0.0444188	+	0.0769357i	0.842961	+	0.537974i	$0.180810\pi$
$270$		0			0
$271$	−6.28133	+	10.8796i	−0.381563	+	0.660887i	−0.991286	−	0.131728i	$-0.957948\pi$
$271$	−6.28133	+	10.8796i	−0.381563	+	0.660887i	0.609722	+	0.792615i	$0.291281\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00598	+	5.20651i	0.181268	+	0.313965i
$276$		0			0
$277$	−4.39448			−0.264039			−0.132019	−	0.991247i	$-0.542146\pi$
$277$	−4.39448			−0.264039			−0.132019	+	0.991247i	$0.542146\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.16265	+	3.74581i	0.129013	+	0.223456i	0.923294	−	0.384093i	$-0.125486\pi$
$281$	2.16265	+	3.74581i	0.129013	+	0.223456i	−0.794282	+	0.607550i	$0.792153\pi$
$282$		0			0
$283$	−3.74885	+	6.49319i	−0.222846	+	0.385980i	−0.955671	−	0.294437i	$-0.904868\pi$
$283$	−3.74885	+	6.49319i	−0.222846	+	0.385980i	0.732825	+	0.680417i	$0.238201\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−13.5719	+	23.5072i	−0.801122	+	1.38758i
$288$		0			0
$289$	0.991903	+	1.71803i	0.0583472	+	0.101060i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	22.2837			1.30183			0.650915	−	0.759151i	$-0.274385\pi$
$293$	22.2837			1.30183			0.650915	+	0.759151i	$0.274385\pi$
$294$		0			0
$295$	−2.62834	−	4.55242i	−0.153028	−	0.265052i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.33137	+	4.03805i	−0.134827	+	0.233527i
$300$		0			0
$301$	−3.37949	+	5.85345i	−0.194791	+	0.337387i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.61013			−0.321235
$306$		0			0
$307$	15.1403	−	26.2238i	0.864103	−	1.49667i	−0.00383236	−	0.999993i	$-0.501220\pi$
$307$	15.1403	−	26.2238i	0.864103	−	1.49667i	0.867935	−	0.496677i	$-0.165447\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.37224			0.304632			0.152316	−	0.988332i	$-0.451327\pi$
$311$	5.37224			0.304632			0.152316	+	0.988332i	$0.451327\pi$
$312$		0			0
$313$	2.40369	+	4.16331i	0.135864	+	0.235324i	0.925927	−	0.377702i	$-0.123286\pi$
$313$	2.40369	+	4.16331i	0.135864	+	0.235324i	−0.790063	+	0.613026i	$0.789952\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−14.8855	−	25.7824i	−0.836052	−	1.44808i	−0.893171	−	0.449717i	$-0.851525\pi$
$317$	−14.8855	−	25.7824i	−0.836052	−	1.44808i	0.0571197	−	0.998367i	$-0.481808\pi$
$318$		0			0
$319$	−23.9384	−	41.4625i	−1.34029	−	2.32145i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	12.0680	−	11.8183i	0.671481	−	0.657586i
$324$		0			0
$325$	2.97581	−	5.15425i	0.165068	−	0.285906i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	2.71610	+	4.70443i	0.149744	+	0.259364i
$330$		0			0
$331$	30.8042			1.69315			0.846577	−	0.532266i	$-0.178659\pi$
$331$	30.8042			1.69315			0.846577	+	0.532266i	$0.178659\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.04325			−0.384814
$336$		0			0
$337$	3.32529	−	5.75957i	0.181140	−	0.313744i	−0.761129	−	0.648601i	$-0.775355\pi$
$337$	3.32529	−	5.75957i	0.181140	−	0.313744i	0.942269	+	0.334857i	$0.108688\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	26.9958			1.46190
$342$		0			0
$343$	−19.5174			−1.05384
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.70534	−	13.3460i	0.413644	−	0.716453i	−0.581641	−	0.813446i	$-0.697589\pi$
$347$	7.70534	−	13.3460i	0.413644	−	0.716453i	0.995285	+	0.0969930i	$0.0309224\pi$
$348$		0			0
$349$	−27.0157			−1.44612			−0.723058	−	0.690788i	$-0.757264\pi$
$349$	−27.0157			−1.44612			−0.723058	+	0.690788i	$0.757264\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−19.7783			−1.05269			−0.526346	−	0.850270i	$-0.676439\pi$
$353$	−19.7783			−1.05269			−0.526346	+	0.850270i	$0.676439\pi$
$354$		0			0
$355$	2.90736	+	5.03570i	0.154307	+	0.267267i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	17.8408	−	30.9011i	0.941600	−	1.63090i	0.179179	−	0.983816i	$-0.442656\pi$
$359$	17.8408	−	30.9011i	0.941600	−	1.63090i	0.762420	−	0.647082i	$-0.224011\pi$
$360$		0			0
$361$	16.2523	+	9.84195i	0.855382	+	0.517997i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.62024	−	8.00250i	−0.241835	−	0.418870i
$366$		0			0
$367$	1.05235	+	1.82272i	0.0549322	+	0.0951454i	0.892184	−	0.451672i	$-0.149172\pi$
$367$	1.05235	+	1.82272i	0.0549322	+	0.0951454i	−0.837252	+	0.546818i	$0.815839\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	17.5407	+	30.3813i	0.910667	+	1.57732i
$372$		0			0
$373$	−32.0208			−1.65797			−0.828987	−	0.559268i	$-0.811082\pi$
$373$	−32.0208			−1.65797			−0.828987	+	0.559268i	$0.811082\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−23.6981	+	41.0463i	−1.22051	+	2.11399i
$378$		0			0
$379$	−2.24784			−0.115464			−0.0577319	−	0.998332i	$-0.518387\pi$
$379$	−2.24784			−0.115464			−0.0577319	+	0.998332i	$0.518387\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.33479	−	2.31192i	0.0682044	−	0.118133i	−0.829907	−	0.557902i	$-0.811606\pi$
$383$	1.33479	−	2.31192i	0.0682044	−	0.118133i	0.898111	+	0.439769i	$0.144940\pi$
$384$		0			0
$385$	12.9410	−	22.4144i	0.659532	−	1.14234i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.76036	+	13.4413i	0.393466	+	0.681503i	0.992904	−	0.118918i	$-0.0379427\pi$
$389$	7.76036	+	13.4413i	0.393466	+	0.681503i	−0.599438	+	0.800421i	$0.704609\pi$
$390$		0			0
$391$	3.03589			0.153532
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.99743	−	12.1199i	−0.352079	−	0.609819i
$396$		0			0
$397$	−1.24839	+	2.16228i	−0.0626551	+	0.108522i	−0.895651	−	0.444757i	$-0.853290\pi$
$397$	−1.24839	+	2.16228i	−0.0626551	+	0.108522i	0.832996	+	0.553278i	$0.186623\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−10.8590	+	18.8083i	−0.542271	+	0.939242i	0.456502	+	0.889723i	$0.349102\pi$
$401$	−10.8590	+	18.8083i	−0.542271	+	0.939242i	−0.998773	+	0.0495192i	$0.984231\pi$
$402$		0			0
$403$	−13.3624	−	23.1443i	−0.665628	−	1.15290i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	5.94003			0.294437
$408$		0			0
$409$	−12.1200	−	20.9924i	−0.599294	−	1.03801i	−0.992925	−	0.118740i	$-0.962115\pi$
$409$	−12.1200	−	20.9924i	−0.599294	−	1.03801i	0.393631	−	0.919269i	$-0.371219\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−11.3152	+	19.5985i	−0.556784	+	0.964377i
$414$		0			0
$415$	−3.29099	+	5.70016i	−0.161548	+	0.279810i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	37.6543			1.83953			0.919766	−	0.392467i	$-0.128378\pi$
$419$	37.6543			1.83953			0.919766	+	0.392467i	$0.128378\pi$
$420$		0			0
$421$	18.6884	−	32.3693i	0.910818	−	1.57758i	0.0979071	−	0.995196i	$-0.468785\pi$
$421$	18.6884	−	32.3693i	0.910818	−	1.57758i	0.812911	−	0.582388i	$-0.197881\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.87507			−0.187969
$426$		0			0
$427$	12.0760	+	20.9162i	0.584398	+	1.01221i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	5.41491	+	9.37889i	0.260827	+	0.451765i	0.966462	−	0.256810i	$-0.0826715\pi$
$431$	5.41491	+	9.37889i	0.260827	+	0.451765i	−0.705635	+	0.708576i	$0.749338\pi$
$432$		0			0
$433$	−5.71031	−	9.89055i	−0.274420	−	0.475310i	0.695569	−	0.718460i	$-0.255153\pi$
$433$	−5.71031	−	9.89055i	−0.274420	−	0.475310i	−0.969989	+	0.243150i	$0.921819\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.849319	+	3.30764i	0.0406284	+	0.158226i
$438$		0			0
$439$	−15.0612	+	26.0868i	−0.718832	+	1.24505i	0.242631	+	0.970119i	$0.421989\pi$
$439$	−15.0612	+	26.0868i	−0.718832	+	1.24505i	−0.961463	+	0.274934i	$0.911344\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.3045	−	17.8479i	−0.489582	−	0.847981i	0.510346	−	0.859969i	$-0.329517\pi$
$443$	−10.3045	−	17.8479i	−0.489582	−	0.847981i	−0.999928	+	0.0119880i	$0.996184\pi$
$444$		0			0
$445$	−3.38473			−0.160452
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.61436			0.170572			0.0852861	−	0.996357i	$-0.472820\pi$
$449$	3.61436			0.170572			0.0852861	+	0.996357i	$0.472820\pi$
$450$		0			0
$451$	−18.9529	+	32.8274i	−0.892458	+	1.54578i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−25.6221			−1.20118
$456$		0			0
$457$	5.50452			0.257491			0.128745	−	0.991678i	$-0.458905\pi$
$457$	5.50452			0.257491			0.128745	+	0.991678i	$0.458905\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.66053	−	16.7325i	0.449936	−	0.779312i	−0.548446	−	0.836186i	$-0.684780\pi$
$461$	9.66053	−	16.7325i	0.449936	−	0.779312i	0.998381	+	0.0568746i	$0.0181135\pi$
$462$		0			0
$463$	1.05722			0.0491334			0.0245667	−	0.999698i	$-0.492179\pi$
$463$	1.05722			0.0491334			0.0245667	+	0.999698i	$0.492179\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−21.9413			−1.01532			−0.507662	−	0.861556i	$-0.669490\pi$
$467$	−21.9413			−1.01532			−0.507662	+	0.861556i	$0.669490\pi$
$468$		0			0
$469$	15.1608	+	26.2593i	0.700062	+	1.21254i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.71942	+	8.17427i	−0.216999	+	0.375853i
$474$		0			0
$475$	−1.08409	−	4.22194i	−0.0497413	−	0.193716i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−6.60203	−	11.4351i	−0.301655	−	0.522481i	0.674856	−	0.737949i	$-0.264206\pi$
$479$	−6.60203	−	11.4351i	−0.301655	−	0.522481i	−0.976511	+	0.215468i	$0.930872\pi$
$480$		0			0
$481$	−2.94020	−	5.09258i	−0.134062	−	0.232202i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.69835	−	6.40573i	−0.167933	−	0.290869i
$486$		0			0
$487$	15.5627			0.705211			0.352606	−	0.935772i	$-0.385296\pi$
$487$	15.5627			0.705211			0.352606	+	0.935772i	$0.385296\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.7978	+	25.6306i	−0.667816	+	1.15669i	0.310698	+	0.950509i	$0.399437\pi$
$491$	−14.7978	+	25.6306i	−0.667816	+	1.15669i	−0.978514	+	0.206182i	$0.933896\pi$
$492$		0			0
$493$	30.8595			1.38984
$494$		0			0
$495$		0			0
$496$		0			0
$497$	12.5164	−	21.6790i	0.561437	−	0.972437i
$498$		0			0
$499$	2.92190	−	5.06087i	0.130802	−	0.226556i	−0.793184	−	0.608982i	$-0.791578\pi$
$499$	2.92190	−	5.06087i	0.130802	−	0.226556i	0.923986	+	0.382426i	$0.124911\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.14544	+	5.44807i	0.140248	+	0.242917i	0.927590	−	0.373600i	$-0.121877\pi$
$503$	3.14544	+	5.44807i	0.140248	+	0.242917i	−0.787342	+	0.616517i	$0.788543\pi$
$504$		0			0
$505$	−9.80737			−0.436422
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.84591	−	3.19720i	−0.0818183	−	0.141713i	0.822213	−	0.569180i	$-0.192739\pi$
$509$	−1.84591	−	3.19720i	−0.0818183	−	0.141713i	−0.904031	+	0.427467i	$0.859406\pi$
$510$		0			0
$511$	−19.8905	+	34.4513i	−0.879902	+	1.52403i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.21841	−	12.5027i	0.318081	−	0.550933i
$516$		0			0
$517$	3.79300	+	6.56967i	0.166816	+	0.288934i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−29.0510			−1.27275			−0.636373	−	0.771381i	$-0.719566\pi$
$521$	−29.0510			−1.27275			−0.636373	+	0.771381i	$0.719566\pi$
$522$		0			0
$523$	−9.21685	−	15.9640i	−0.403025	−	0.698059i	0.591065	−	0.806624i	$-0.298708\pi$
$523$	−9.21685	−	15.9640i	−0.403025	−	0.698059i	−0.994089	+	0.108565i	$0.965374\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.70020	+	15.0692i	−0.378987	+	0.656424i
$528$		0			0
$529$	11.1931	−	19.3870i	0.486657	−	0.842915i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	37.5253			1.62540
$534$		0			0
$535$	4.74517	−	8.21888i	0.205152	−	0.355333i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−69.3395			−2.98666
$540$		0			0
$541$	21.0875	+	36.5246i	0.906622	+	1.57032i	0.818724	+	0.574187i	$0.194682\pi$
$541$	21.0875	+	36.5246i	0.906622	+	1.57032i	0.0878981	+	0.996129i	$0.471985\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.30920	−	2.26759i	−0.0560798	−	0.0971331i
$546$		0			0
$547$	−7.87719	−	13.6437i	−0.336804	−	0.583362i	0.647025	−	0.762468i	$-0.276013\pi$
$547$	−7.87719	−	13.6437i	−0.336804	−	0.583362i	−0.983830	+	0.179106i	$0.942679\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8.63321	+	33.6217i	0.367787	+	1.43233i
$552$		0			0
$553$	−30.1244	+	52.1770i	−1.28102	+	2.21879i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.5924	+	20.0786i	0.491185	+	0.850757i	0.999948	−	0.0101493i	$-0.00323068\pi$
$557$	11.5924	+	20.0786i	0.491185	+	0.850757i	−0.508764	+	0.860906i	$0.669897\pi$
$558$		0			0
$559$	9.34408			0.395213
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.970934			0.0409200			0.0204600	−	0.999791i	$-0.493487\pi$
$563$	0.970934			0.0409200			0.0204600	+	0.999791i	$0.493487\pi$
$564$		0			0
$565$	6.83524	−	11.8390i	0.287561	−	0.498070i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	30.1395			1.26351			0.631757	−	0.775167i	$-0.282334\pi$
$569$	30.1395			1.26351			0.631757	+	0.775167i	$0.282334\pi$
$570$		0			0
$571$	−31.8976			−1.33487			−0.667436	−	0.744667i	$-0.732608\pi$
$571$	−31.8976			−1.33487			−0.667436	+	0.744667i	$0.732608\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.391721	−	0.678480i	0.0163359	−	0.0282946i
$576$		0			0
$577$	23.2889			0.969528			0.484764	−	0.874645i	$-0.338905\pi$
$577$	23.2889			0.969528			0.484764	+	0.874645i	$0.338905\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	28.3358			1.17557
$582$		0			0
$583$	24.4953	+	42.4271i	1.01449	+	1.75715i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.74059	−	8.21094i	0.195665	−	0.338902i	−0.751453	−	0.659786i	$-0.770647\pi$
$587$	4.74059	−	8.21094i	0.195665	−	0.338902i	0.947118	+	0.320885i	$0.103980\pi$
$588$		0			0
$589$	−18.8520	−	5.26323i	−0.776784	−	0.216867i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.2234	−	36.7600i	−0.871540	−	1.50955i	−0.860403	−	0.509614i	$-0.829788\pi$
$593$	−21.2234	−	36.7600i	−0.871540	−	1.50955i	−0.0111366	−	0.999938i	$-0.503545\pi$
$594$		0			0
$595$	8.34122	+	14.4474i	0.341957	+	0.592287i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.8961	−	22.3368i	−0.526922	−	0.912656i	−0.999508	−	0.0313711i	$-0.990013\pi$
$599$	−12.8961	−	22.3368i	−0.526922	−	0.912656i	0.472586	−	0.881285i	$-0.343321\pi$
$600$		0			0
$601$	0.206080			0.00840619			0.00420310	−	0.999991i	$-0.498662\pi$
$601$	0.206080			0.00840619			0.00420310	+	0.999991i	$0.498662\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	12.5719	−	21.7751i	0.511119	−	0.885284i
$606$		0			0
$607$	0.100472			0.00407802			0.00203901	−	0.999998i	$-0.499351\pi$
$607$	0.100472			0.00407802			0.00203901	+	0.999998i	$0.499351\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3.75493	−	6.50373i	0.151908	−	0.263113i
$612$		0			0
$613$	19.7400	−	34.1907i	0.797292	−	1.38095i	−0.124081	−	0.992272i	$-0.539598\pi$
$613$	19.7400	−	34.1907i	0.797292	−	1.38095i	0.921373	−	0.388679i	$-0.127068\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.1658	−	24.5359i	−0.570294	−	0.987777i	−0.996536	−	0.0831682i	$-0.973496\pi$
$617$	−14.1658	−	24.5359i	−0.570294	−	0.987777i	0.426242	−	0.904609i	$-0.359837\pi$
$618$		0			0
$619$	−1.39670			−0.0561380			−0.0280690	−	0.999606i	$-0.508936\pi$
$619$	−1.39670			−0.0561380			−0.0280690	+	0.999606i	$0.508936\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	7.28575	+	12.6193i	0.291897	+	0.505581i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.91435	+	3.31576i	−0.0763303	+	0.132208i
$630$		0			0
$631$	−4.07397	−	7.05633i	−0.162182	−	0.280908i	0.773469	−	0.633834i	$-0.218520\pi$
$631$	−4.07397	−	7.05633i	−0.162182	−	0.280908i	−0.935651	+	0.352926i	$0.885187\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.0672			−0.518555
$636$		0			0
$637$	34.3217	+	59.4470i	1.35988	+	2.35538i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.76367	−	16.9112i	0.385642	−	0.667951i	−0.606216	−	0.795300i	$-0.707313\pi$
$641$	9.76367	−	16.9112i	0.385642	−	0.667951i	0.991858	+	0.127349i	$0.0406467\pi$
$642$		0			0
$643$	−21.5945	+	37.4028i	−0.851604	+	1.47502i	0.0281570	+	0.999604i	$0.491036\pi$
$643$	−21.5945	+	37.4028i	−0.851604	+	1.47502i	−0.879761	+	0.475417i	$0.842297\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−27.4390			−1.07874			−0.539370	−	0.842069i	$-0.681338\pi$
$647$	−27.4390			−1.07874			−0.539370	+	0.842069i	$0.681338\pi$
$648$		0			0
$649$	−15.8015	+	27.3690i	−0.620263	+	1.07433i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.40515			0.133254			0.0666270	−	0.997778i	$-0.478776\pi$
$653$	3.40515			0.133254			0.0666270	+	0.997778i	$0.478776\pi$
$654$		0			0
$655$	3.75070	+	6.49640i	0.146552	+	0.253835i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	6.09862	+	10.5631i	0.237569	+	0.411481i	0.960016	−	0.279945i	$-0.0903162\pi$
$659$	6.09862	+	10.5631i	0.237569	+	0.411481i	−0.722448	+	0.691426i	$0.756983\pi$
$660$		0			0
$661$	19.0683	+	33.0272i	0.741670	+	1.28461i	0.951734	+	0.306923i	$0.0992994\pi$
$661$	19.0683	+	33.0272i	0.741670	+	1.28461i	−0.210064	+	0.977688i	$0.567367\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−13.4071	+	13.1297i	−0.519905	+	0.509146i
$666$		0			0
$667$	−3.11950	+	5.40313i	−0.120788	+	0.209210i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	16.8640	+	29.2092i	0.651026	+	1.12761i
$672$		0			0
$673$	−37.9505			−1.46288			−0.731442	−	0.681903i	$-0.761152\pi$
$673$	−37.9505			−1.46288			−0.731442	+	0.681903i	$0.761152\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	32.4499			1.24715			0.623575	−	0.781763i	$-0.285679\pi$
$677$	32.4499			1.24715			0.623575	+	0.781763i	$0.285679\pi$
$678$		0			0
$679$	−15.9216	+	27.5771i	−0.611016	+	1.05831i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−40.5283			−1.55077			−0.775385	−	0.631488i	$-0.782444\pi$
$683$	−40.5283			−1.55077			−0.775385	+	0.631488i	$0.782444\pi$
$684$		0			0
$685$	−8.42999			−0.322093
$686$		0			0
$687$		0			0
$688$		0			0
$689$	24.2494	−	42.0012i	0.923830	−	1.60012i
$690$		0			0
$691$	1.78657			0.0679642			0.0339821	−	0.999422i	$-0.489181\pi$
$691$	1.78657			0.0679642			0.0339821	+	0.999422i	$0.489181\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.77921			−0.333015
$696$		0			0
$697$	−12.2163	−	21.1592i	−0.462725	−	0.801464i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.8614	+	37.8650i	−0.825693	+	1.43014i	0.0756952	+	0.997131i	$0.475882\pi$
$701$	−21.8614	+	37.8650i	−0.825693	+	1.43014i	−0.901388	+	0.433011i	$0.857451\pi$
$702$		0			0
$703$	−4.14812	−	1.15810i	−0.156449	−	0.0436785i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	21.1107	+	36.5648i	0.793949	+	1.37516i
$708$		0			0
$709$	−7.32015	−	12.6789i	−0.274914	−	0.476165i	0.695199	−	0.718817i	$-0.255316\pi$
$709$	−7.32015	−	12.6789i	−0.274914	−	0.476165i	−0.970113	+	0.242652i	$0.921983\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.75896	−	3.04661i	−0.0658736	−	0.114096i
$714$		0			0
$715$	−35.7809			−1.33813
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−11.1778	+	19.3606i	−0.416863	+	0.722028i	−0.995622	−	0.0934709i	$-0.970204\pi$
$719$	−11.1778	+	19.3606i	−0.416863	+	0.722028i	0.578759	+	0.815499i	$0.303537\pi$
$720$		0			0
$721$	−62.1515			−2.31464
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.98179	−	6.89666i	0.147880	−	0.256136i
$726$		0			0
$727$	−4.15042	+	7.18874i	−0.153930	+	0.266615i	−0.932669	−	0.360733i	$-0.882527\pi$
$727$	−4.15042	+	7.18874i	−0.153930	+	0.266615i	0.778739	+	0.627349i	$0.215860\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.04195	−	5.26881i	−0.112511	−	0.194874i
$732$		0			0
$733$	35.7912			1.32198			0.660989	−	0.750396i	$-0.270137\pi$
$733$	35.7912			1.32198			0.660989	+	0.750396i	$0.270137\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21.1719	+	36.6708i	0.779876	+	1.35079i
$738$		0			0
$739$	16.6216	−	28.7895i	0.611437	−	1.05904i	−0.379561	−	0.925167i	$-0.623925\pi$
$739$	16.6216	−	28.7895i	0.611437	−	1.05904i	0.990998	−	0.133873i	$-0.0427415\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−9.27444	+	16.0638i	−0.340246	+	0.589324i	−0.984478	−	0.175506i	$-0.943844\pi$
$743$	−9.27444	+	16.0638i	−0.340246	+	0.589324i	0.644232	+	0.764830i	$0.277177\pi$
$744$		0			0
$745$	0.915913	+	1.58641i	0.0335565	+	0.0581215i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−40.8565			−1.49287
$750$		0			0
$751$	−21.5748	−	37.3687i	−0.787276	−	1.36360i	−0.927630	−	0.373501i	$-0.878157\pi$
$751$	−21.5748	−	37.3687i	−0.787276	−	1.36360i	0.140353	−	0.990101i	$-0.455176\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.194935	+	0.337637i	−0.00709440	+	0.0122879i
$756$		0			0
$757$	−4.77434	+	8.26940i	−0.173526	+	0.300556i	−0.939650	−	0.342136i	$-0.888849\pi$
$757$	−4.77434	+	8.26940i	−0.173526	+	0.300556i	0.766124	+	0.642693i	$0.222183\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−35.8512			−1.29960			−0.649802	−	0.760103i	$-0.725148\pi$
$761$	−35.8512			−1.29960			−0.649802	+	0.760103i	$0.725148\pi$
$762$		0			0
$763$	−5.63617	+	9.76214i	−0.204043	+	0.353413i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	31.2857			1.12966
$768$		0			0
$769$	8.93698	+	15.4793i	0.322276	+	0.558198i	0.980957	−	0.194224i	$-0.0622188\pi$
$769$	8.93698	+	15.4793i	0.322276	+	0.558198i	−0.658681	+	0.752422i	$0.728886\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.926769	+	1.60521i	0.0333336	+	0.0577354i	0.882211	−	0.470854i	$-0.156054\pi$
$773$	0.926769	+	1.60521i	0.0333336	+	0.0577354i	−0.848877	+	0.528590i	$0.822721\pi$
$774$		0			0
$775$	2.24517	+	3.88875i	0.0806489	+	0.139688i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	19.6356	−	19.2293i	0.703519	−	0.688961i
$780$		0			0
$781$	17.4790	−	30.2744i	0.625446	−	1.08330i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.37608	−	2.38344i	−0.0491144	−	0.0850686i
$786$		0			0
$787$	−53.8501			−1.91955			−0.959774	−	0.280773i	$-0.909409\pi$
$787$	−53.8501			−1.91955			−0.959774	+	0.280773i	$0.909409\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−58.8523			−2.09255
$792$		0			0
$793$	16.6947	−	28.9160i	0.592845	−	1.02684i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	44.2766			1.56836			0.784179	−	0.620535i	$-0.213085\pi$
$797$	44.2766			1.56836			0.784179	+	0.620535i	$0.213085\pi$
$798$		0			0
$799$	−4.88964			−0.172983
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−27.7767	+	48.1107i	−0.980220	+	1.69779i
$804$		0			0
$805$	−3.37277			−0.118874
$806$		0			0
$807$		0			0
$808$		0			0
$809$	16.7641			0.589395			0.294698	−	0.955591i	$-0.404781\pi$
$809$	16.7641			0.589395			0.294698	+	0.955591i	$0.404781\pi$
$810$		0			0
$811$	13.7203	+	23.7642i	0.481784	+	0.834474i	0.999781	−	0.0209083i	$-0.00665581\pi$
$811$	13.7203	+	23.7642i	0.481784	+	0.834474i	−0.517998	+	0.855382i	$0.673322\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.54767	+	13.0729i	−0.264383	+	0.457925i
$816$		0			0
$817$	4.88942	−	4.78823i	0.171059	−	0.167519i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.9404	−	20.6814i	−0.416723	−	0.721785i	0.578885	−	0.815409i	$-0.303488\pi$
$821$	−11.9404	−	20.6814i	−0.416723	−	0.721785i	−0.995608	+	0.0936244i	$0.970155\pi$
$822$		0			0
$823$	11.3410	+	19.6431i	0.395321	+	0.684716i	0.993142	−	0.116913i	$-0.0373000\pi$
$823$	11.3410	+	19.6431i	0.395321	+	0.684716i	−0.597821	+	0.801630i	$0.703967\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	16.1121	+	27.9071i	0.560274	+	0.970423i	0.997472	+	0.0710581i	$0.0226376\pi$
$827$	16.1121	+	27.9071i	0.560274	+	0.970423i	−0.437198	+	0.899365i	$0.644029\pi$
$828$		0			0
$829$	−0.593350			−0.0206079			−0.0103040	−	0.999947i	$-0.503280\pi$
$829$	−0.593350			−0.0206079			−0.0103040	+	0.999947i	$0.503280\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	22.3468	−	38.7057i	0.774269	−	1.34107i
$834$		0			0
$835$	2.98804			0.103405
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−10.9777	+	19.0139i	−0.378991	+	0.656431i	−0.990916	−	0.134484i	$-0.957062\pi$
$839$	−10.9777	+	19.0139i	−0.378991	+	0.656431i	0.611925	+	0.790916i	$0.290396\pi$
$840$		0			0
$841$	−17.2093	+	29.8074i	−0.593424	+	1.02784i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.2109	+	19.4178i	0.385665	+	0.667992i
$846$		0			0
$847$	−108.245			−3.71935
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.387034	−	0.670363i	−0.0132674	−	0.0229797i
$852$		0			0
$853$	−9.76396	+	16.9117i	−0.334312	+	0.579045i	−0.983352	−	0.181709i	$-0.941837\pi$
$853$	−9.76396	+	16.9117i	−0.334312	+	0.579045i	0.649041	+	0.760754i	$0.275171\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.421704	+	0.730413i	−0.0144051	+	0.0249504i	−0.873138	−	0.487473i	$-0.837919\pi$
$857$	−0.421704	+	0.730413i	−0.0144051	+	0.0249504i	0.858733	+	0.512423i	$0.171252\pi$
$858$		0			0
$859$	−16.7147	−	28.9508i	−0.570299	−	0.987787i	−0.996535	−	0.0831752i	$-0.973494\pi$
$859$	−16.7147	−	28.9508i	−0.570299	−	0.987787i	0.426236	−	0.904612i	$-0.359839\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	7.13064			0.242730			0.121365	−	0.992608i	$-0.461273\pi$
$863$	7.13064			0.242730			0.121365	+	0.992608i	$0.461273\pi$
$864$		0			0
$865$	11.5945	+	20.0822i	0.394224	+	0.682817i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−42.0683	+	72.8645i	−1.42707	+	2.47176i
$870$		0			0
$871$	20.9594	−	36.3027i	0.710181	−	1.23007i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	4.30507			0.145538
$876$		0			0
$877$	−8.01564	+	13.8835i	−0.270669	+	0.468812i	−0.969033	−	0.246930i	$-0.920578\pi$
$877$	−8.01564	+	13.8835i	−0.270669	+	0.468812i	0.698364	+	0.715742i	$0.253912\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	23.0319			0.775963			0.387982	−	0.921667i	$-0.373172\pi$
$881$	23.0319			0.775963			0.387982	+	0.921667i	$0.373172\pi$
$882$		0			0
$883$	−20.5300	−	35.5590i	−0.690890	−	1.19666i	−0.971547	−	0.236848i	$-0.923886\pi$
$883$	−20.5300	−	35.5590i	−0.690890	−	1.19666i	0.280657	−	0.959808i	$-0.409448\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.06900	+	8.77977i	0.170200	+	0.294796i	0.938490	−	0.345307i	$-0.112225\pi$
$887$	5.06900	+	8.77977i	0.170200	+	0.294796i	−0.768289	+	0.640103i	$0.778892\pi$
$888$		0			0
$889$	28.1275	+	48.7183i	0.943367	+	1.63396i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.36792	−	5.32732i	−0.0457757	−	0.178272i
$894$		0			0
$895$	6.75457	−	11.6993i	0.225780	−	0.391063i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.8796	−	30.9684i	−0.596318	−	1.03285i
$900$		0			0
$901$	−31.5774			−1.05200
$902$		0			0
$903$		0			0
$904$		0			0
$905$	20.3119			0.675190
$906$		0			0
$907$	−19.0916	+	33.0677i	−0.633927	+	1.09799i	0.352814	+	0.935693i	$0.385225\pi$
$907$	−19.0916	+	33.0677i	−0.633927	+	1.09799i	−0.986741	+	0.162301i	$0.948109\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	14.4138			0.477550			0.238775	−	0.971075i	$-0.423254\pi$
$911$	14.4138			0.477550			0.238775	+	0.971075i	$0.423254\pi$
$912$		0			0
$913$	39.5706			1.30960
$914$		0			0
$915$		0			0
$916$		0			0
$917$	16.1470	−	27.9674i	0.533221	−	0.923566i
$918$		0			0
$919$	5.20920			0.171836			0.0859179	−	0.996302i	$-0.472618\pi$
$919$	5.20920			0.171836			0.0859179	+	0.996302i	$0.472618\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−34.6070			−1.13910
$924$		0			0
$925$	0.494018	+	0.855664i	0.0162432	+	0.0281340i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.1769	+	33.2154i	−0.629174	+	1.08976i	0.358544	+	0.933513i	$0.383273\pi$
$929$	−19.1769	+	33.2154i	−0.629174	+	1.08976i	−0.987718	+	0.156249i	$0.950060\pi$
$930$		0			0
$931$	48.4220	+	13.5188i	1.58697	+	0.443060i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	11.6484	+	20.1756i	0.380943	+	0.659813i
$936$		0			0
$937$	−30.2289	−	52.3580i	−0.987536	−	1.71046i	−0.630077	−	0.776533i	$-0.716977\pi$
$937$	−30.2289	−	52.3580i	−0.987536	−	1.71046i	−0.357459	−	0.933929i	$-0.616357\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12.0859	−	20.9335i	−0.393990	−	0.682411i	0.598981	−	0.800763i	$-0.295572\pi$
$941$	−12.0859	−	20.9335i	−0.393990	−	0.682411i	−0.992972	+	0.118352i	$0.962239\pi$
$942$		0			0
$943$	4.93965			0.160857
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.91390	−	8.51112i	0.159680	−	0.276574i	−0.775073	−	0.631872i	$-0.782287\pi$
$947$	4.91390	−	8.51112i	0.159680	−	0.276574i	0.934753	+	0.355297i	$0.115620\pi$
$948$		0			0
$949$	54.9958			1.78524
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.04867	+	1.81636i	−0.0339699	+	0.0588375i	−0.882511	−	0.470293i	$-0.844148\pi$
$953$	−1.04867	+	1.81636i	−0.0339699	+	0.0588375i	0.848541	+	0.529130i	$0.177482\pi$
$954$		0			0
$955$	3.52419	−	6.10408i	0.114040	−	0.197523i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	18.1458	+	31.4295i	0.585960	+	1.01491i
$960$		0			0
$961$	−10.8368			−0.349575
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11.8892	−	20.5926i	−0.382725	−	0.662900i
$966$		0			0
$967$	19.1770	−	33.2156i	0.616691	−	1.06814i	−0.373394	−	0.927673i	$-0.621806\pi$
$967$	19.1770	−	33.2156i	0.616691	−	1.06814i	0.990085	−	0.140468i	$-0.0448606\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	9.56818	−	16.5726i	0.307058	−	0.531839i	−0.670660	−	0.741765i	$-0.733989\pi$
$971$	9.56818	−	16.5726i	0.307058	−	0.531839i	0.977717	+	0.209926i	$0.0673222\pi$
$972$		0			0
$973$	18.8975	+	32.7315i	0.605827	+	1.04932i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19.8444			−0.634878			−0.317439	−	0.948279i	$-0.602823\pi$
$977$	−19.8444			−0.634878			−0.317439	+	0.948279i	$0.602823\pi$
$978$		0			0
$979$	10.1744	+	17.6227i	0.325177	+	0.563223i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	20.8080	−	36.0406i	0.663673	−	1.14952i	−0.315970	−	0.948769i	$-0.602330\pi$
$983$	20.8080	−	36.0406i	0.663673	−	1.14952i	0.979643	−	0.200746i	$-0.0643367\pi$
$984$		0			0
$985$	−11.8714	+	20.5619i	−0.378254	+	0.655155i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.23001			0.0391120
$990$		0			0
$991$	5.96872	−	10.3381i	0.189603	−	0.328401i	−0.755515	−	0.655131i	$-0.772613\pi$
$991$	5.96872	−	10.3381i	0.189603	−	0.328401i	0.945118	+	0.326730i	$0.105947\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	22.9787			0.728473
$996$		0			0
$997$	9.63745	+	16.6925i	0.305221	+	0.528658i	0.977311	−	0.211812i	$-0.0679363\pi$
$997$	9.63745	+	16.6925i	0.305221	+	0.528658i	−0.672089	+	0.740470i	$0.734603\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.t.w.3241.1		8
3.2	odd	2		380.2.i.c.201.3	yes	8
12.11	even	2		1520.2.q.m.961.2		8
15.2	even	4		1900.2.s.d.49.4		16
15.8	even	4		1900.2.s.d.49.5		16
15.14	odd	2		1900.2.i.d.201.2		8
19.7	even	3	inner	3420.2.t.w.1261.1		8
57.8	even	6		7220.2.a.p.1.3		4
57.11	odd	6		7220.2.a.r.1.2		4
57.26	odd	6		380.2.i.c.121.3	✓	8
228.83	even	6		1520.2.q.m.881.2		8
285.83	even	12		1900.2.s.d.349.4		16
285.197	even	12		1900.2.s.d.349.5		16
285.254	odd	6		1900.2.i.d.501.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.i.c.121.3	✓	8	57.26	odd	6
380.2.i.c.201.3	yes	8	3.2	odd	2
1520.2.q.m.881.2		8	228.83	even	6
1520.2.q.m.961.2		8	12.11	even	2
1900.2.i.d.201.2		8	15.14	odd	2
1900.2.i.d.501.2		8	285.254	odd	6
1900.2.s.d.49.4		16	15.2	even	4
1900.2.s.d.49.5		16	15.8	even	4
1900.2.s.d.349.4		16	285.83	even	12
1900.2.s.d.349.5		16	285.197	even	12
3420.2.t.w.1261.1		8	19.7	even	3	inner
3420.2.t.w.3241.1		8	1.1	even	1	trivial
7220.2.a.p.1.3		4	57.8	even	6
7220.2.a.r.1.2		4	57.11	odd	6

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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.t (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{5} -4.30507 q^{7} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{5} -4.30507 q^{7} -6.01196 q^{11} +(2.97581 + 5.15425i) q^{13} +(1.93754 - 3.35591i) q^{17} +(4.19835 + 1.17212i) q^{19} +(0.391721 + 0.678480i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(3.98179 + 6.89666i) q^{29} -4.49034 q^{31} +(-2.15253 + 3.72830i) q^{35} -0.988035 q^{37} +(3.15253 - 5.46035i) q^{41} +(0.785004 - 1.35967i) q^{43} +(-0.630909 - 1.09277i) q^{47} +11.5336 q^{49} +(-4.07443 - 7.05712i) q^{53} +(-3.00598 + 5.20651i) q^{55} +(2.62834 - 4.55242i) q^{59} +(-2.80507 - 4.85852i) q^{61} +5.95162 q^{65} +(-3.52162 - 6.09963i) q^{67} +(-2.90736 + 5.03570i) q^{71} +(4.62024 - 8.00250i) q^{73} +25.8819 q^{77} +(6.99743 - 12.1199i) q^{79} -6.58197 q^{83} +(-1.93754 - 3.35591i) q^{85} +(-1.69237 - 2.93126i) q^{89} +(-12.8110 - 22.1894i) q^{91} +(3.11426 - 3.04982i) q^{95} +(3.69835 - 6.40573i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5}+O(q^{10}) \) 8 * q + 4 * q^5	\( 8 q + 4 q^{5} - 4 q^{11} + 9 q^{13} - q^{17} + 3 q^{19} - 4 q^{25} - 5 q^{29} - 20 q^{31} - 52 q^{37} + 8 q^{41} + 7 q^{43} - 16 q^{47} + 20 q^{49} - 5 q^{53} - 2 q^{55} - 11 q^{59} + 12 q^{61} + 18 q^{65} - 14 q^{71} - 4 q^{73} + 44 q^{77} + 13 q^{79} - 10 q^{83} + q^{85} - 5 q^{89} - 46 q^{91} + 6 q^{95} - q^{97}+O(q^{100}) \) 8 * q + 4 * q^5 - 4 * q^11 + 9 * q^13 - q^17 + 3 * q^19 - 4 * q^25 - 5 * q^29 - 20 * q^31 - 52 * q^37 + 8 * q^41 + 7 * q^43 - 16 * q^47 + 20 * q^49 - 5 * q^53 - 2 * q^55 - 11 * q^59 + 12 * q^61 + 18 * q^65 - 14 * q^71 - 4 * q^73 + 44 * q^77 + 13 * q^79 - 10 * q^83 + q^85 - 5 * q^89 - 46 * q^91 + 6 * q^95 - q^97

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