LMFDB - Embedded newform 180.7.l.b.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [180,7,Mod(37,180)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(180, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("180.37");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	180.l (of order $4$, 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:	$41.4097350516$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 66x^{10} + 1601x^{8} + 17520x^{6} + 84208x^{4} + 136704x^{2} + 14400 $ x^12 + 66x^10 + 1601x^8 + 17520x^6 + 84208x^4 + 136704*x^2 + 14400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{10}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			37.4
Root			$3.51909i$ of defining polynomial
Character	$\chi$	$=$	180.37
Dual form			180.7.l.b.73.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(23.5531 + 122.761i) q^{5} +(312.733 + 312.733i) q^{7} +O(q^{10})$	$q+(23.5531 + 122.761i) q^{5} +(312.733 + 312.733i) q^{7} +524.296 q^{11} +(-2815.91 + 2815.91i) q^{13} +(6552.59 + 6552.59i) q^{17} -7554.63i q^{19} +(8761.99 - 8761.99i) q^{23} +(-14515.5 + 5782.80i) q^{25} -23363.5i q^{29} -48026.9 q^{31} +(-31025.5 + 45757.2i) q^{35} +(34199.0 + 34199.0i) q^{37} +30870.8 q^{41} +(-21790.4 + 21790.4i) q^{43} +(-37819.6 - 37819.6i) q^{47} +77954.4i q^{49} +(-179394. + 179394. i) q^{53} +(12348.8 + 64363.1i) q^{55} +200563. i q^{59} -19690.2 q^{61} +(-412008. - 279361. i) q^{65} +(94194.5 + 94194.5i) q^{67} +281712. q^{71} +(-454330. + 454330. i) q^{73} +(163965. + 163965. i) q^{77} -134576. i q^{79} +(-535168. + 535168. i) q^{83} +(-650068. + 958736. i) q^{85} -78207.3i q^{89} -1.76126e6 q^{91} +(927413. - 177935. i) q^{95} +(-94504.5 - 94504.5i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 312 q^{5} + 120 q^{7}+O(q^{10}) $ 12 * q - 312 * q^5 + 120 * q^7	$ 12 q - 312 q^{5} + 120 q^{7} + 3248 q^{11} - 2100 q^{13} + 5540 q^{17} + 23840 q^{23} + 10044 q^{25} - 127152 q^{31} - 102976 q^{35} + 282900 q^{37} + 320720 q^{41} - 62880 q^{43} - 381600 q^{47} + 400300 q^{53} + 502152 q^{55} + 807024 q^{61} - 124500 q^{65} + 752160 q^{67} - 202400 q^{71} - 322020 q^{73} + 2448400 q^{77} - 1894560 q^{83} - 857124 q^{85} + 2294400 q^{91} + 2620000 q^{95} - 3161700 q^{97}+O(q^{100}) $ 12 * q - 312 * q^5 + 120 * q^7 + 3248 * q^11 - 2100 * q^13 + 5540 * q^17 + 23840 * q^23 + 10044 * q^25 - 127152 * q^31 - 102976 * q^35 + 282900 * q^37 + 320720 * q^41 - 62880 * q^43 - 381600 * q^47 + 400300 * q^53 + 502152 * q^55 + 807024 * q^61 - 124500 * q^65 + 752160 * q^67 - 202400 * q^71 - 322020 * q^73 + 2448400 * q^77 - 1894560 * q^83 - 857124 * q^85 + 2294400 * q^91 + 2620000 * q^95 - 3161700 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	23.5531	+	122.761i	0.188425	+	0.982088i
$5$	23.5531	+	122.761i	0.188425	+	0.982088i
$6$		0			0
$7$	312.733	+	312.733i	0.911757	+	0.911757i	0.996410	−	0.0846534i	$-0.0269783\pi$
$7$	312.733	+	312.733i	0.911757	+	0.911757i	−0.0846534	+	0.996410i	$0.526978\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	524.296			0.393911			0.196956	−	0.980412i	$-0.436894\pi$
$11$	524.296			0.393911			0.196956	+	0.980412i	$0.436894\pi$
$12$		0			0
$13$	−2815.91	+	2815.91i	−1.28171	+	1.28171i	−0.342014	+	0.939695i	$0.611109\pi$
$13$	−2815.91	+	2815.91i	−1.28171	+	1.28171i	−0.939695	+	0.342014i	$0.888891\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6552.59	+	6552.59i	1.33372	+	1.33372i	0.902011	+	0.431714i	$0.142091\pi$
$17$	6552.59	+	6552.59i	1.33372	+	1.33372i	0.431714	+	0.902011i	$0.357909\pi$
$18$		0			0
$19$		−	7554.63i		−	1.10142i	−0.834697	−	0.550709i	$-0.814357\pi$
$19$		−	7554.63i		−	1.10142i	0.834697	−	0.550709i	$-0.185643\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8761.99	−	8761.99i	0.720143	−	0.720143i	−0.248491	−	0.968634i	$-0.579935\pi$
$23$	8761.99	−	8761.99i	0.720143	−	0.720143i	0.968634	+	0.248491i	$0.0799346\pi$
$24$		0			0
$25$	−14515.5	+	5782.80i	−0.928992	+	0.370099i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	23363.5i		−	0.957952i	−0.877828	−	0.478976i	$-0.841008\pi$
$29$		−	23363.5i		−	0.957952i	0.877828	−	0.478976i	$-0.158992\pi$
$30$		0			0
$31$	−48026.9			−1.61213			−0.806064	−	0.591828i	$-0.798406\pi$
$31$	−48026.9			−1.61213			−0.806064	+	0.591828i	$0.798406\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−31025.5	+	45757.2i	−0.723628	+	1.06722i
$36$		0			0
$37$	34199.0	+	34199.0i	0.675163	+	0.675163i	0.958902	−	0.283739i	$-0.0915749\pi$
$37$	34199.0	+	34199.0i	0.675163	+	0.675163i	−0.283739	+	0.958902i	$0.591575\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	30870.8			0.447916			0.223958	−	0.974599i	$-0.428102\pi$
$41$	30870.8			0.447916			0.223958	+	0.974599i	$0.428102\pi$
$42$		0			0
$43$	−21790.4	+	21790.4i	−0.274068	+	0.274068i	−0.830736	−	0.556667i	$-0.812080\pi$
$43$	−21790.4	+	21790.4i	−0.274068	+	0.274068i	0.556667	+	0.830736i	$0.312080\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−37819.6	−	37819.6i	−0.364270	−	0.364270i	0.501113	−	0.865382i	$-0.332924\pi$
$47$	−37819.6	−	37819.6i	−0.364270	−	0.364270i	−0.865382	+	0.501113i	$0.832924\pi$
$48$		0			0
$49$			77954.4i			0.662602i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−179394.	+	179394.i	−1.20498	+	1.20498i	−0.232349	+	0.972632i	$0.574641\pi$
$53$	−179394.	+	179394.i	−1.20498	+	1.20498i	−0.972632	+	0.232349i	$0.925359\pi$
$54$		0			0
$55$	12348.8	+	64363.1i	0.0742226	+	0.386856i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			200563.i			0.976552i	0.872689	+	0.488276i	$0.162374\pi$
$59$			200563.i			0.976552i	−0.872689	+	0.488276i	$0.837626\pi$
$60$		0			0
$61$	−19690.2			−0.0867481			−0.0433741	−	0.999059i	$-0.513811\pi$
$61$	−19690.2			−0.0867481			−0.0433741	+	0.999059i	$0.513811\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−412008.	−	279361.i	−1.50026	−	1.01724i
$66$		0			0
$67$	94194.5	+	94194.5i	0.313185	+	0.313185i	0.846142	−	0.532957i	$-0.178919\pi$
$67$	94194.5	+	94194.5i	0.313185	+	0.313185i	−0.532957	+	0.846142i	$0.678919\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	281712.			0.787100			0.393550	−	0.919303i	$-0.371247\pi$
$71$	281712.			0.787100			0.393550	+	0.919303i	$0.371247\pi$
$72$		0			0
$73$	−454330.	+	454330.i	−1.16789	+	1.16789i	−0.185189	+	0.982703i	$0.559290\pi$
$73$	−454330.	+	454330.i	−1.16789	+	1.16789i	−0.982703	+	0.185189i	$0.940710\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	163965.	+	163965.i	0.359152	+	0.359152i
$78$		0			0
$79$		−	134576.i		−	0.272951i	−0.990643	−	0.136476i	$-0.956422\pi$
$79$		−	134576.i		−	0.272951i	0.990643	−	0.136476i	$-0.0435776\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−535168.	+	535168.i	−0.935957	+	0.935957i	−0.998069	−	0.0621126i	$-0.980216\pi$
$83$	−535168.	+	535168.i	−0.935957	+	0.935957i	0.0621126	+	0.998069i	$0.480216\pi$
$84$		0			0
$85$	−650068.	+	958736.i	−1.05853	+	1.56114i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	78207.3i		−	0.110937i	−0.998460	−	0.0554686i	$-0.982335\pi$
$89$		−	78207.3i		−	0.110937i	0.998460	−	0.0554686i	$-0.0176653\pi$
$90$		0			0
$91$	−1.76126e6			−2.33721
$92$		0			0
$93$		0			0
$94$		0			0
$95$	927413.	−	177935.i	1.08169	−	0.207534i
$96$		0			0
$97$	−94504.5	−	94504.5i	−0.103547	−	0.103547i	0.653435	−	0.756982i	$-0.273327\pi$
$97$	−94504.5	−	94504.5i	−0.103547	−	0.103547i	−0.756982	+	0.653435i	$0.773327\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	139520.			0.135417			0.0677085	−	0.997705i	$-0.478431\pi$
$101$	139520.			0.135417			0.0677085	+	0.997705i	$0.478431\pi$
$102$		0			0
$103$	1.00305e6	−	1.00305e6i	0.917935	−	0.917935i	−0.0789444	−	0.996879i	$-0.525155\pi$
$103$	1.00305e6	−	1.00305e6i	0.917935	−	0.917935i	0.996879	+	0.0789444i	$0.0251550\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−444305.	−	444305.i	−0.362686	−	0.362686i	0.502115	−	0.864801i	$-0.332555\pi$
$107$	−444305.	−	444305.i	−0.362686	−	0.362686i	−0.864801	+	0.502115i	$0.832555\pi$
$108$		0			0
$109$			670544.i			0.517783i	0.965906	+	0.258892i	$0.0833572\pi$
$109$			670544.i			0.517783i	−0.965906	+	0.258892i	$0.916643\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	912083.	−	912083.i	0.632119	−	0.632119i	−0.316480	−	0.948599i	$-0.602501\pi$
$113$	912083.	−	912083.i	0.632119	−	0.632119i	0.948599	+	0.316480i	$0.102501\pi$
$114$		0			0
$115$	1.28200e6	+	869258.i	0.842937	+	0.571551i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			4.09842e6i			2.43207i
$120$		0			0
$121$	−1.49667e6			−0.844834
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.05179e6	−	1.64573e6i	−0.538515	−	0.842616i
$126$		0			0
$127$	600626.	+	600626.i	0.293220	+	0.293220i	0.838351	−	0.545131i	$-0.183520\pi$
$127$	600626.	+	600626.i	0.293220	+	0.293220i	−0.545131	+	0.838351i	$0.683520\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−132740.			−0.0590455			−0.0295227	−	0.999564i	$-0.509399\pi$
$131$	−132740.			−0.0590455			−0.0295227	+	0.999564i	$0.509399\pi$
$132$		0			0
$133$	2.36258e6	−	2.36258e6i	1.00423	−	1.00423i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.77153e6	−	1.77153e6i	−0.688948	−	0.688948i	0.273051	−	0.961999i	$-0.411967\pi$
$137$	−1.77153e6	−	1.77153e6i	−0.688948	−	0.688948i	−0.961999	+	0.273051i	$0.911967\pi$
$138$		0			0
$139$		−	1.48696e6i		−	0.553675i	−0.960917	−	0.276837i	$-0.910714\pi$
$139$		−	1.48696e6i		−	0.553675i	0.960917	−	0.276837i	$-0.0892863\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.47637e6	+	1.47637e6i	−0.504880	+	0.504880i
$144$		0			0
$145$	2.86813e6	−	550282.i	0.940793	−	0.180502i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	982054.i		−	0.296877i	−0.988922	−	0.148439i	$-0.952575\pi$
$149$		−	982054.i		−	0.296877i	0.988922	−	0.148439i	$-0.0474247\pi$
$150$		0			0
$151$	2.61844e6			0.760521			0.380261	−	0.924879i	$-0.375834\pi$
$151$	2.61844e6			0.760521			0.380261	+	0.924879i	$0.375834\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.13118e6	−	5.89583e6i	−0.303765	−	1.58325i
$156$		0			0
$157$	2.03939e6	+	2.03939e6i	0.526989	+	0.526989i	0.919673	−	0.392685i	$-0.128454\pi$
$157$	2.03939e6	+	2.03939e6i	0.526989	+	0.526989i	−0.392685	+	0.919673i	$0.628454\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.48032e6			1.31319
$162$		0			0
$163$	2.34500e6	−	2.34500e6i	0.541477	−	0.541477i	−0.382485	−	0.923962i	$-0.624932\pi$
$163$	2.34500e6	−	2.34500e6i	0.541477	−	0.541477i	0.923962	+	0.382485i	$0.124932\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−107293.	−	107293.i	−0.0230369	−	0.0230369i	0.695495	−	0.718531i	$-0.255185\pi$
$167$	−107293.	−	107293.i	−0.0230369	−	0.0230369i	−0.718531	+	0.695495i	$0.755185\pi$
$168$		0			0
$169$		−	1.10319e7i		−	2.28556i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2.16801e6	−	2.16801e6i	0.418718	−	0.418718i	−0.466043	−	0.884762i	$-0.654321\pi$
$173$	2.16801e6	−	2.16801e6i	0.418718	−	0.418718i	0.884762	+	0.466043i	$0.154321\pi$
$174$		0			0
$175$	−6.34794e6	−	2.73100e6i	−1.18446	−	0.509575i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			22081.4i			0.00385006i	0.999998	+	0.00192503i	$0.000612757\pi$
$179$			22081.4i			0.00385006i	−0.999998	+	0.00192503i	$0.999387\pi$
$180$		0			0
$181$	−4.79204e6			−0.808137			−0.404069	−	0.914729i	$-0.632404\pi$
$181$	−4.79204e6			−0.808137			−0.404069	+	0.914729i	$0.632404\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.39281e6	+	5.00380e6i	−0.535852	+	0.790286i
$186$		0			0
$187$	3.43550e6	+	3.43550e6i	0.525369	+	0.525369i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.35557e7			1.94546			0.972730	−	0.231942i	$-0.0745079\pi$
$191$	1.35557e7			1.94546			0.972730	+	0.231942i	$0.0745079\pi$
$192$		0			0
$193$	2.38486e6	−	2.38486e6i	0.331735	−	0.331735i	−0.521510	−	0.853245i	$-0.674631\pi$
$193$	2.38486e6	−	2.38486e6i	0.331735	−	0.331735i	0.853245	+	0.521510i	$0.174631\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.12238e6	+	9.12238e6i	1.19319	+	1.19319i	0.976166	+	0.217024i	$0.0696348\pi$
$197$	9.12238e6	+	9.12238e6i	1.19319	+	1.19319i	0.217024	+	0.976166i	$0.430365\pi$
$198$		0			0
$199$			1.23227e7i			1.56367i	0.623484	+	0.781836i	$0.285717\pi$
$199$			1.23227e7i			1.56367i	−0.623484	+	0.781836i	$0.714283\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	7.30653e6	−	7.30653e6i	0.873420	−	0.873420i
$204$		0			0
$205$	727102.	+	3.78973e6i	0.0843983	+	0.439892i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	3.96086e6i		−	0.433861i
$210$		0			0
$211$	1.27619e7			1.35852			0.679262	−	0.733896i	$-0.262300\pi$
$211$	1.27619e7			1.35852			0.679262	+	0.733896i	$0.262300\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.18824e6	−	2.16178e6i	−0.320801	−	0.217518i
$216$		0			0
$217$	−1.50196e7	−	1.50196e7i	−1.46987	−	1.46987i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−3.69031e7			−3.41889
$222$		0			0
$223$	−2.06607e6	+	2.06607e6i	−0.186307	+	0.186307i	−0.794098	−	0.607790i	$-0.792056\pi$
$223$	−2.06607e6	+	2.06607e6i	−0.186307	+	0.186307i	0.607790	+	0.794098i	$0.292056\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.69282e6	+	3.69282e6i	0.315705	+	0.315705i	0.847115	−	0.531410i	$-0.178338\pi$
$227$	3.69282e6	+	3.69282e6i	0.315705	+	0.315705i	−0.531410	+	0.847115i	$0.678338\pi$
$228$		0			0
$229$		−	2.50689e6i		−	0.208751i	−0.994538	−	0.104376i	$-0.966716\pi$
$229$		−	2.50689e6i		−	0.208751i	0.994538	−	0.104376i	$-0.0332844\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.79869e6	−	6.79869e6i	0.537474	−	0.537474i	−0.385312	−	0.922786i	$-0.625906\pi$
$233$	6.79869e6	−	6.79869e6i	0.537474	−	0.537474i	0.922786	+	0.385312i	$0.125906\pi$
$234$		0			0
$235$	3.75200e6	−	5.53353e6i	0.289107	−	0.426382i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1.99520e6i		−	0.146148i	−0.997327	−	0.0730739i	$-0.976719\pi$
$239$		−	1.99520e6i		−	0.146148i	0.997327	−	0.0730739i	$-0.0232809\pi$
$240$		0			0
$241$	−1.49921e7			−1.07106			−0.535528	−	0.844517i	$-0.679887\pi$
$241$	−1.49921e7			−1.07106			−0.535528	+	0.844517i	$0.679887\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−9.56976e6	+	1.83607e6i	−0.650733	+	0.124851i
$246$		0			0
$247$	2.12732e7	+	2.12732e7i	1.41170	+	1.41170i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.81622e7			−1.14854			−0.574272	−	0.818665i	$-0.694715\pi$
$251$	−1.81622e7			−1.14854			−0.574272	+	0.818665i	$0.694715\pi$
$252$		0			0
$253$	4.59388e6	−	4.59388e6i	0.283673	−	0.283673i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6.05915e6	+	6.05915e6i	0.356954	+	0.356954i	0.862689	−	0.505735i	$-0.168779\pi$
$257$	6.05915e6	+	6.05915e6i	0.356954	+	0.356954i	−0.505735	+	0.862689i	$0.668779\pi$
$258$		0			0
$259$			2.13903e7i			1.23117i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	230685.	−	230685.i	0.0126810	−	0.0126810i	−0.700738	−	0.713419i	$-0.747146\pi$
$263$	230685.	−	230685.i	0.0126810	−	0.0126810i	0.713419	+	0.700738i	$0.247146\pi$
$264$		0			0
$265$	−2.62479e7	−	1.77973e7i	−1.41045	−	0.956349i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			2.30197e7i			1.18261i	0.806447	+	0.591307i	$0.201388\pi$
$269$			2.30197e7i			1.18261i	−0.806447	+	0.591307i	$0.798612\pi$
$270$		0			0
$271$	2.58489e7			1.29878			0.649389	−	0.760456i	$-0.275025\pi$
$271$	2.58489e7			1.29878			0.649389	+	0.760456i	$0.275025\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.61042e6	+	3.03190e6i	−0.365941	+	0.145786i
$276$		0			0
$277$	−3.76177e6	−	3.76177e6i	−0.176992	−	0.176992i	0.613051	−	0.790043i	$-0.289942\pi$
$277$	−3.76177e6	−	3.76177e6i	−0.176992	−	0.176992i	−0.790043	+	0.613051i	$0.789942\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−9.53259e6			−0.429627			−0.214814	−	0.976655i	$-0.568914\pi$
$281$	−9.53259e6			−0.429627			−0.214814	+	0.976655i	$0.568914\pi$
$282$		0			0
$283$	−2.62576e7	+	2.62576e7i	−1.15850	+	1.15850i	−0.173699	+	0.984799i	$0.555572\pi$
$283$	−2.62576e7	+	2.62576e7i	−1.15850	+	1.15850i	−0.984799	+	0.173699i	$0.944428\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	9.65430e6	+	9.65430e6i	0.408390	+	0.408390i
$288$		0			0
$289$			6.17352e7i			2.55764i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.82060e7	−	1.82060e7i	0.723790	−	0.723790i	−0.245585	−	0.969375i	$-0.578980\pi$
$293$	1.82060e7	−	1.82060e7i	0.723790	−	0.723790i	0.969375	+	0.245585i	$0.0789800\pi$
$294$		0			0
$295$	−2.46213e7	+	4.72388e6i	−0.959060	+	0.184006i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			4.93460e7i			1.84603i
$300$		0			0
$301$	−1.36291e7			−0.499768
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−463764.	−	2.41719e6i	−0.0163455	−	0.0851943i
$306$		0			0
$307$	−1.23071e7	−	1.23071e7i	−0.425346	−	0.425346i	0.461694	−	0.887039i	$-0.347242\pi$
$307$	−1.23071e7	−	1.23071e7i	−0.425346	−	0.425346i	−0.887039	+	0.461694i	$0.847242\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.48687e6			−0.0494301			−0.0247151	−	0.999695i	$-0.507868\pi$
$311$	−1.48687e6			−0.0494301			−0.0247151	+	0.999695i	$0.507868\pi$
$312$		0			0
$313$	−1.23460e7	+	1.23460e7i	−0.402618	+	0.402618i	−0.879155	−	0.476537i	$-0.841892\pi$
$313$	−1.23460e7	+	1.23460e7i	−0.402618	+	0.402618i	0.476537	+	0.879155i	$0.341892\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−8.48737e6	−	8.48737e6i	−0.266437	−	0.266437i	0.561225	−	0.827663i	$-0.310330\pi$
$317$	−8.48737e6	−	8.48737e6i	−0.266437	−	0.266437i	−0.827663	+	0.561225i	$0.810330\pi$
$318$		0			0
$319$		−	1.22494e7i		−	0.377348i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	4.95024e7	−	4.95024e7i	1.46899	−	1.46899i
$324$		0			0
$325$	2.45906e7	−	5.71583e7i	0.716338	−	1.66506i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	2.36548e7i		−	0.664251i
$330$		0			0
$331$	−4.20247e7			−1.15883			−0.579417	−	0.815031i	$-0.696720\pi$
$331$	−4.20247e7			−1.15883			−0.579417	+	0.815031i	$0.696720\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−9.34484e6	+	1.37820e7i	−0.248563	+	0.366587i
$336$		0			0
$337$	−2.18920e7	−	2.18920e7i	−0.572000	−	0.572000i	0.360687	−	0.932687i	$-0.382542\pi$
$337$	−2.18920e7	−	2.18920e7i	−0.572000	−	0.572000i	−0.932687	+	0.360687i	$0.882542\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.51803e7			−0.635036
$342$		0			0
$343$	1.24138e7	−	1.24138e7i	0.307625	−	0.307625i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.92252e7	+	1.92252e7i	0.460132	+	0.460132i	0.898699	−	0.438567i	$-0.144514\pi$
$347$	1.92252e7	+	1.92252e7i	0.460132	+	0.460132i	−0.438567	+	0.898699i	$0.644514\pi$
$348$		0			0
$349$			3.16328e6i			0.0744152i	0.999308	+	0.0372076i	$0.0118463\pi$
$349$			3.16328e6i			0.0744152i	−0.999308	+	0.0372076i	$0.988154\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.03601e7	−	4.03601e7i	0.917546	−	0.917546i	−0.0793047	−	0.996850i	$-0.525270\pi$
$353$	4.03601e7	−	4.03601e7i	0.917546	−	0.917546i	0.996850	+	0.0793047i	$0.0252700\pi$
$354$		0			0
$355$	6.63518e6	+	3.45832e7i	0.148309	+	0.773002i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6.51722e7i			1.40857i	0.709916	+	0.704286i	$0.248733\pi$
$359$			6.51722e7i			1.40857i	−0.709916	+	0.704286i	$0.751267\pi$
$360$		0			0
$361$	−1.00265e7			−0.213122
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.64748e7	−	4.50731e7i	−1.36703	−	0.926913i
$366$		0			0
$367$	6.30942e7	+	6.30942e7i	1.27641	+	1.27641i	0.942661	+	0.333752i	$0.108315\pi$
$367$	6.30942e7	+	6.30942e7i	1.27641	+	1.27641i	0.333752	+	0.942661i	$0.391685\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.12205e8			−2.19730
$372$		0			0
$373$	5.23236e7	−	5.23236e7i	1.00826	−	1.00826i	0.00829139	−	0.999966i	$-0.497361\pi$
$373$	5.23236e7	−	5.23236e7i	1.00826	−	1.00826i	0.999966	−	0.00829139i	$-0.00263926\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6.57896e7	+	6.57896e7i	1.22782	+	1.22782i
$378$		0			0
$379$		−	2.16294e7i		−	0.397308i	−0.980070	−	0.198654i	$-0.936343\pi$
$379$		−	2.16294e7i		−	0.397308i	0.980070	−	0.198654i	$-0.0636571\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	3.45802e6	−	3.45802e6i	0.0615505	−	0.0615505i	−0.675662	−	0.737212i	$-0.736142\pi$
$383$	3.45802e6	−	3.45802e6i	0.0615505	−	0.0615505i	0.737212	+	0.675662i	$0.236142\pi$
$384$		0			0
$385$	−1.62666e7	+	2.39903e7i	−0.285045	+	0.420391i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	9.91167e7i		−	1.68383i	−0.539611	−	0.841915i	$-0.681428\pi$
$389$		−	9.91167e7i		−	1.68383i	0.539611	−	0.841915i	$-0.318572\pi$
$390$		0			0
$391$	1.14827e8			1.92095
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.65206e7	−	3.16967e6i	0.268062	−	0.0514308i
$396$		0			0
$397$	3.80721e7	+	3.80721e7i	0.608464	+	0.608464i	0.942545	−	0.334080i	$-0.108426\pi$
$397$	3.80721e7	+	3.80721e7i	0.608464	+	0.608464i	−0.334080	+	0.942545i	$0.608426\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.01890e8			1.58014			0.790072	−	0.613014i	$-0.210043\pi$
$401$	1.01890e8			1.58014			0.790072	+	0.613014i	$0.210043\pi$
$402$		0			0
$403$	1.35240e8	−	1.35240e8i	2.06628	−	2.06628i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.79304e7	+	1.79304e7i	0.265954	+	0.265954i
$408$		0			0
$409$		−	2.97072e7i		−	0.434201i	−0.976149	−	0.217101i	$-0.930340\pi$
$409$		−	2.97072e7i		−	0.434201i	0.976149	−	0.217101i	$-0.0696600\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.27227e7	+	6.27227e7i	−0.890378	+	0.890378i
$414$		0			0
$415$	−7.83026e7	−	5.30929e7i	−1.09555	−	0.742834i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			5.47424e7i			0.744187i	0.928195	+	0.372093i	$0.121360\pi$
$419$			5.47424e7i			0.744187i	−0.928195	+	0.372093i	$0.878640\pi$
$420$		0			0
$421$	−4.56931e7			−0.612357			−0.306178	−	0.951974i	$-0.599050\pi$
$421$	−4.56931e7			−0.612357			−0.306178	+	0.951974i	$0.599050\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.33006e8	−	5.72218e7i	−1.73263	−	0.745410i
$426$		0			0
$427$	−6.15776e6	−	6.15776e6i	−0.0790932	−	0.0790932i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.36803e8			1.70870			0.854348	−	0.519701i	$-0.173957\pi$
$431$	1.36803e8			1.70870			0.854348	+	0.519701i	$0.173957\pi$
$432$		0			0
$433$	−1.42469e7	+	1.42469e7i	−0.175492	+	0.175492i	−0.789388	−	0.613895i	$-0.789602\pi$
$433$	−1.42469e7	+	1.42469e7i	−0.175492	+	0.175492i	0.613895	+	0.789388i	$0.289602\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.61935e7	−	6.61935e7i	−0.793179	−	0.793179i
$438$		0			0
$439$		−	1.22508e8i		−	1.44801i	−0.689793	−	0.724007i	$-0.742298\pi$
$439$		−	1.22508e8i		−	1.44801i	0.689793	−	0.724007i	$-0.257702\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.09099e8	−	1.09099e8i	1.25490	−	1.25490i	0.301401	−	0.953497i	$-0.402546\pi$
$443$	1.09099e8	−	1.09099e8i	1.25490	−	1.25490i	0.953497	−	0.301401i	$-0.0974543\pi$
$444$		0			0
$445$	9.60080e6	−	1.84202e6i	0.108950	−	0.0209033i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5.72987e7i			0.633003i	0.948592	+	0.316502i	$0.102508\pi$
$449$			5.72987e7i			0.633003i	−0.948592	+	0.316502i	$0.897492\pi$
$450$		0			0
$451$	1.61854e7			0.176439
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.14830e7	−	2.16214e8i	−0.440389	−	2.29535i
$456$		0			0
$457$	−1.62737e7	−	1.62737e7i	−0.170505	−	0.170505i	0.616696	−	0.787201i	$-0.288471\pi$
$457$	−1.62737e7	−	1.62737e7i	−0.170505	−	0.170505i	−0.787201	+	0.616696i	$0.788471\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	9.23525e7			0.942640			0.471320	−	0.881962i	$-0.343778\pi$
$461$	9.23525e7			0.942640			0.471320	+	0.881962i	$0.343778\pi$
$462$		0			0
$463$	−3.18877e7	+	3.18877e7i	−0.321278	+	0.321278i	−0.849257	−	0.527979i	$-0.822950\pi$
$463$	−3.18877e7	+	3.18877e7i	−0.321278	+	0.321278i	0.527979	+	0.849257i	$0.322950\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	6.91385e7	+	6.91385e7i	0.678843	+	0.678843i	0.959738	−	0.280895i	$-0.0906313\pi$
$467$	6.91385e7	+	6.91385e7i	0.678843	+	0.678843i	−0.280895	+	0.959738i	$0.590631\pi$
$468$		0			0
$469$			5.89154e7i			0.571097i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.14246e7	+	1.14246e7i	−0.107959	+	0.107959i
$474$		0			0
$475$	4.36869e7	+	1.09659e8i	0.407634	+	1.02321i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			1.78196e8i			1.62140i	0.585461	+	0.810701i	$0.300914\pi$
$479$			1.78196e8i			1.62140i	−0.585461	+	0.810701i	$0.699086\pi$
$480$		0			0
$481$	−1.92603e8			−1.73072
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9.37559e6	−	1.38273e7i	0.0821814	−	0.121203i
$486$		0			0
$487$	1.09058e7	+	1.09058e7i	0.0944212	+	0.0944212i	0.752740	−	0.658318i	$-0.228732\pi$
$487$	1.09058e7	+	1.09058e7i	0.0944212	+	0.0944212i	−0.658318	+	0.752740i	$0.728732\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.74098e8			−1.47079			−0.735395	−	0.677639i	$-0.763003\pi$
$491$	−1.74098e8			−1.47079			−0.735395	+	0.677639i	$0.763003\pi$
$492$		0			0
$493$	1.53091e8	−	1.53091e8i	1.27764	−	1.27764i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8.81005e7	+	8.81005e7i	0.717644	+	0.717644i
$498$		0			0
$499$			1.39200e8i			1.12031i	0.828387	+	0.560156i	$0.189259\pi$
$499$			1.39200e8i			1.12031i	−0.828387	+	0.560156i	$0.810741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.04234e8	−	1.04234e8i	0.819044	−	0.819044i	−0.166926	−	0.985969i	$-0.553384\pi$
$503$	1.04234e8	−	1.04234e8i	0.819044	−	0.819044i	0.985969	+	0.166926i	$0.0533840\pi$
$504$		0			0
$505$	3.28613e6	+	1.71276e7i	0.0255159	+	0.132991i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			1.18629e8i			0.899576i	0.893135	+	0.449788i	$0.148500\pi$
$509$			1.18629e8i			0.899576i	−0.893135	+	0.449788i	$0.851500\pi$
$510$		0			0
$511$	−2.84168e8			−2.12967
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.46761e8	+	9.95107e7i	1.07445	+	0.728531i
$516$		0			0
$517$	−1.98286e7	−	1.98286e7i	−0.143490	−	0.143490i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	9.44367e7			0.667771			0.333886	−	0.942614i	$-0.391640\pi$
$521$	9.44367e7			0.667771			0.333886	+	0.942614i	$0.391640\pi$
$522$		0			0
$523$	−1.29920e8	+	1.29920e8i	−0.908177	+	0.908177i	−0.996125	−	0.0879478i	$-0.971969\pi$
$523$	−1.29920e8	+	1.29920e8i	−0.908177	+	0.908177i	0.0879478	+	0.996125i	$0.471969\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.14701e8	−	3.14701e8i	−2.15013	−	2.15013i
$528$		0			0
$529$		−	5.50890e6i		−	0.0372133i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−8.69295e7	+	8.69295e7i	−0.574097	+	0.574097i
$534$		0			0
$535$	4.40786e7	−	6.50081e7i	0.287850	−	0.424528i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			4.08712e7i			0.261006i
$540$		0			0
$541$	1.48167e8			0.935750			0.467875	−	0.883795i	$-0.345020\pi$
$541$	1.48167e8			0.935750			0.467875	+	0.883795i	$0.345020\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.23167e7	+	1.57934e7i	−0.508509	+	0.0975631i
$546$		0			0
$547$	1.81777e8	+	1.81777e8i	1.11065	+	1.11065i	0.993063	+	0.117583i	$0.0375147\pi$
$547$	1.81777e8	+	1.81777e8i	1.11065	+	1.11065i	0.117583	+	0.993063i	$0.462485\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.76503e8			−1.05511
$552$		0			0
$553$	4.20862e7	−	4.20862e7i	0.248865	−	0.248865i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	2.04488e8	+	2.04488e8i	1.18332	+	1.18332i	0.978879	+	0.204442i	$0.0655381\pi$
$557$	2.04488e8	+	2.04488e8i	1.18332	+	1.18332i	0.204442	+	0.978879i	$0.434462\pi$
$558$		0			0
$559$		−	1.22720e8i		−	0.702552i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.28944e8	+	1.28944e8i	−0.722565	+	0.722565i	−0.969127	−	0.246562i	$-0.920699\pi$
$563$	−1.28944e8	+	1.28944e8i	−0.722565	+	0.722565i	0.246562	+	0.969127i	$0.420699\pi$
$564$		0			0
$565$	1.33451e8	+	9.04858e7i	0.739903	+	0.501690i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	7.27289e7i		−	0.394794i	−0.980324	−	0.197397i	$-0.936751\pi$
$569$		−	7.27289e7i		−	0.394794i	0.980324	−	0.197397i	$-0.0632487\pi$
$570$		0			0
$571$	−6.23414e7			−0.334864			−0.167432	−	0.985884i	$-0.553547\pi$
$571$	−6.23414e7			−0.334864			−0.167432	+	0.985884i	$0.553547\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−7.65159e7	+	1.77853e8i	−0.402483	+	0.935532i
$576$		0			0
$577$	1.65224e7	+	1.65224e7i	0.0860096	+	0.0860096i	0.748803	−	0.662793i	$-0.230629\pi$
$577$	1.65224e7	+	1.65224e7i	0.0860096	+	0.0860096i	−0.662793	+	0.748803i	$0.730629\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.34729e8			−1.70673
$582$		0			0
$583$	−9.40556e7	+	9.40556e7i	−0.474656	+	0.474656i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.40171e8	−	1.40171e8i	−0.693016	−	0.693016i	0.269879	−	0.962894i	$-0.413016\pi$
$587$	−1.40171e8	−	1.40171e8i	−0.693016	−	0.693016i	−0.962894	+	0.269879i	$0.913016\pi$
$588$		0			0
$589$			3.62826e8i			1.77563i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.73100e7	−	4.73100e7i	0.226876	−	0.226876i	−0.584510	−	0.811386i	$-0.698713\pi$
$593$	4.73100e7	−	4.73100e7i	0.226876	−	0.226876i	0.811386	+	0.584510i	$0.198713\pi$
$594$		0			0
$595$	−5.03126e8	+	9.65303e7i	−2.38850	+	0.458261i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	2.08410e8i		−	0.969701i	−0.874597	−	0.484851i	$-0.838874\pi$
$599$		−	2.08410e8i		−	0.969701i	0.874597	−	0.484851i	$-0.161126\pi$
$600$		0			0
$601$	7.41018e7			0.341354			0.170677	−	0.985327i	$-0.445404\pi$
$601$	7.41018e7			0.341354			0.170677	+	0.985327i	$0.445404\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.52513e7	−	1.83733e8i	−0.159187	−	0.829701i
$606$		0			0
$607$	−1.12886e8	−	1.12886e8i	−0.504749	−	0.504749i	0.408161	−	0.912910i	$-0.366170\pi$
$607$	−1.12886e8	−	1.12886e8i	−0.504749	−	0.504749i	−0.912910	+	0.408161i	$0.866170\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.12993e8			0.933775
$612$		0			0
$613$	−2.41064e8	+	2.41064e8i	−1.04653	+	1.04653i	−0.0476639	+	0.998863i	$0.515178\pi$
$613$	−2.41064e8	+	2.41064e8i	−1.04653	+	1.04653i	−0.998863	+	0.0476639i	$0.984822\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	6.28924e7	+	6.28924e7i	0.267758	+	0.267758i	0.828196	−	0.560438i	$-0.189367\pi$
$617$	6.28924e7	+	6.28924e7i	0.267758	+	0.267758i	−0.560438	+	0.828196i	$0.689367\pi$
$618$		0			0
$619$		−	1.20452e8i		−	0.507859i	−0.967223	−	0.253930i	$-0.918277\pi$
$619$		−	1.20452e8i		−	0.507859i	0.967223	−	0.253930i	$-0.0817232\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	2.44580e7	−	2.44580e7i	0.101148	−	0.101148i
$624$		0			0
$625$	1.77259e8	−	1.67880e8i	0.726053	−	0.687638i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			4.48184e8i			1.80096i
$630$		0			0
$631$	3.28657e8			1.30814			0.654071	−	0.756433i	$-0.273060\pi$
$631$	3.28657e8			1.30814			0.654071	+	0.756433i	$0.273060\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.95868e7	+	8.78800e7i	−0.232718	+	0.343217i
$636$		0			0
$637$	−2.19513e8	−	2.19513e8i	−0.849263	−	0.849263i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.45954e8			0.554169			0.277085	−	0.960845i	$-0.410632\pi$
$641$	1.45954e8			0.554169			0.277085	+	0.960845i	$0.410632\pi$
$642$		0			0
$643$	9.18706e6	−	9.18706e6i	0.0345576	−	0.0345576i	−0.689617	−	0.724174i	$-0.742221\pi$
$643$	9.18706e6	−	9.18706e6i	0.0345576	−	0.0345576i	0.724174	+	0.689617i	$0.242221\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.48972e8	+	2.48972e8i	0.919260	+	0.919260i	0.996976	−	0.0777156i	$-0.0247626\pi$
$647$	2.48972e8	+	2.48972e8i	0.919260	+	0.919260i	−0.0777156	+	0.996976i	$0.524763\pi$
$648$		0			0
$649$			1.05155e8i			0.384675i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	1.70989e8	−	1.70989e8i	0.614084	−	0.614084i	−0.329924	−	0.944008i	$-0.607023\pi$
$653$	1.70989e8	−	1.70989e8i	0.614084	−	0.614084i	0.944008	+	0.329924i	$0.107023\pi$
$654$		0			0
$655$	−3.12643e6	−	1.62952e7i	−0.0111256	−	0.0579878i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	3.67582e8i		−	1.28439i	−0.766540	−	0.642197i	$-0.778023\pi$
$659$		−	3.67582e8i		−	1.28439i	0.766540	−	0.642197i	$-0.221977\pi$
$660$		0			0
$661$	−9.82639e7			−0.340243			−0.170122	−	0.985423i	$-0.554416\pi$
$661$	−9.82639e7			−0.340243			−0.170122	+	0.985423i	$0.554416\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.45678e8	+	2.34386e8i	1.17546	+	0.797017i
$666$		0			0
$667$	−2.04711e8	−	2.04711e8i	−0.689863	−	0.689863i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.03235e7			−0.0341711
$672$		0			0
$673$	−2.33286e8	+	2.33286e8i	−0.765320	+	0.765320i	−0.977279	−	0.211959i	$-0.932016\pi$
$673$	−2.33286e8	+	2.33286e8i	−0.765320	+	0.765320i	0.211959	+	0.977279i	$0.432016\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.14487e7	−	2.14487e7i	−0.0691251	−	0.0691251i	0.671699	−	0.740824i	$-0.265565\pi$
$677$	−2.14487e7	−	2.14487e7i	−0.0691251	−	0.0691251i	−0.740824	+	0.671699i	$0.765565\pi$
$678$		0			0
$679$		−	5.91093e7i		−	0.188819i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.59602e8	+	2.59602e8i	−0.814790	+	0.814790i	−0.985348	−	0.170558i	$-0.945443\pi$
$683$	−2.59602e8	+	2.59602e8i	−0.814790	+	0.814790i	0.170558	+	0.985348i	$0.445443\pi$
$684$		0			0
$685$	1.75750e8	−	2.59199e8i	0.546793	−	0.806422i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	1.01032e9i		−	3.08887i
$690$		0			0
$691$	1.02213e8			0.309792			0.154896	−	0.987931i	$-0.450496\pi$
$691$	1.02213e8			0.309792			0.154896	+	0.987931i	$0.450496\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1.82541e8	−	3.50225e7i	0.543757	−	0.104326i
$696$		0			0
$697$	2.02284e8	+	2.02284e8i	0.597396	+	0.597396i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.00343e8			0.871894			0.435947	−	0.899972i	$-0.356413\pi$
$701$	3.00343e8			0.871894			0.435947	+	0.899972i	$0.356413\pi$
$702$		0			0
$703$	2.58361e8	−	2.58361e8i	0.743637	−	0.743637i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.36325e7	+	4.36325e7i	0.123467	+	0.123467i
$708$		0			0
$709$			2.55379e8i			0.716550i	0.933616	+	0.358275i	$0.116635\pi$
$709$			2.55379e8i			0.716550i	−0.933616	+	0.358275i	$0.883365\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.20811e8	+	4.20811e8i	−1.16096	+	1.16096i
$714$		0			0
$715$	−2.16014e8	−	1.46468e8i	−0.590968	−	0.400704i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			3.42719e6i			0.00922044i	0.999989	+	0.00461022i	$0.00146748\pi$
$719$			3.42719e6i			0.00922044i	−0.999989	+	0.00461022i	$0.998533\pi$
$720$		0			0
$721$	6.27374e8			1.67387
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.35106e8	+	3.39133e8i	0.354537	+	0.889930i
$726$		0			0
$727$	−1.94321e8	−	1.94321e8i	−0.505729	−	0.505729i	0.407484	−	0.913212i	$-0.366406\pi$
$727$	−1.94321e8	−	1.94321e8i	−0.505729	−	0.505729i	−0.913212	+	0.407484i	$0.866406\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.85567e8			−0.731064
$732$		0			0
$733$	1.01995e8	−	1.01995e8i	0.258980	−	0.258980i	−0.565659	−	0.824639i	$-0.691378\pi$
$733$	1.01995e8	−	1.01995e8i	0.258980	−	0.258980i	0.824639	+	0.565659i	$0.191378\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.93858e7	+	4.93858e7i	0.123367	+	0.123367i
$738$		0			0
$739$		−	5.45928e8i		−	1.35270i	−0.736579	−	0.676351i	$-0.763560\pi$
$739$		−	5.45928e8i		−	1.35270i	0.736579	−	0.676351i	$-0.236440\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	3.99074e8	−	3.99074e8i	0.972942	−	0.972942i	−0.0267014	−	0.999643i	$-0.508500\pi$
$743$	3.99074e8	−	3.99074e8i	0.972942	−	0.972942i	0.999643	+	0.0267014i	$0.00850032\pi$
$744$		0			0
$745$	1.20558e8	−	2.31304e7i	0.291559	−	0.0559389i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	2.77898e8i		−	0.661362i
$750$		0			0
$751$	5.75259e8			1.35814			0.679069	−	0.734074i	$-0.262384\pi$
$751$	5.75259e8			1.35814			0.679069	+	0.734074i	$0.262384\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.16723e7	+	3.21442e8i	0.143301	+	0.746898i
$756$		0			0
$757$	4.27701e8	+	4.27701e8i	0.985946	+	0.985946i	0.999903	−	0.0139566i	$-0.00444267\pi$
$757$	4.27701e8	+	4.27701e8i	0.985946	+	0.985946i	−0.0139566	+	0.999903i	$0.504443\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−8.12040e8			−1.84257			−0.921284	−	0.388891i	$-0.872858\pi$
$761$	−8.12040e8			−1.84257			−0.921284	+	0.388891i	$0.872858\pi$
$762$		0			0
$763$	−2.09701e8	+	2.09701e8i	−0.472093	+	0.472093i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−5.64769e8	−	5.64769e8i	−1.25166	−	1.25166i
$768$		0			0
$769$		−	6.57679e8i		−	1.44622i	−0.690731	−	0.723111i	$-0.742711\pi$
$769$		−	6.57679e8i		−	1.44622i	0.690731	−	0.723111i	$-0.257289\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	3.20098e8	−	3.20098e8i	0.693018	−	0.693018i	−0.269877	−	0.962895i	$-0.586983\pi$
$773$	3.20098e8	−	3.20098e8i	0.693018	−	0.693018i	0.962895	+	0.269877i	$0.0869830\pi$
$774$		0			0
$775$	6.97135e8	−	2.77730e8i	1.49765	−	0.596647i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	2.33217e8i		−	0.493342i
$780$		0			0
$781$	1.47700e8			0.310048
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.02324e8	+	2.98391e8i	−0.418251	+	0.616847i
$786$		0			0
$787$	−4.19160e8	−	4.19160e8i	−0.859916	−	0.859916i	0.131412	−	0.991328i	$-0.458049\pi$
$787$	−4.19160e8	−	4.19160e8i	−0.859916	−	0.859916i	−0.991328	+	0.131412i	$0.958049\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5.70476e8			1.15268
$792$		0			0
$793$	5.54459e7	−	5.54459e7i	0.111186	−	0.111186i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.71446e8	−	4.71446e8i	−0.931230	−	0.931230i	0.0665526	−	0.997783i	$-0.478800\pi$
$797$	−4.71446e8	−	4.71446e8i	−0.931230	−	0.931230i	−0.997783	+	0.0665526i	$0.978800\pi$
$798$		0			0
$799$		−	4.95632e8i		−	0.971670i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2.38203e8	+	2.38203e8i	−0.460046	+	0.460046i
$804$		0			0
$805$	1.29078e8	+	6.72769e8i	0.247438	+	1.28967i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	3.01131e8i		−	0.568735i	−0.958715	−	0.284368i	$-0.908216\pi$
$809$		−	3.01131e8i		−	0.568735i	0.958715	−	0.284368i	$-0.0917836\pi$
$810$		0			0
$811$	−3.82474e8			−0.717034			−0.358517	−	0.933523i	$-0.616718\pi$
$811$	−3.82474e8			−0.717034			−0.358517	+	0.933523i	$0.616718\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.43106e8	+	2.32642e8i	0.633805	+	0.429750i
$816$		0			0
$817$	1.64618e8	+	1.64618e8i	0.301864	+	0.301864i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.20534e8			0.940633			0.470316	−	0.882498i	$-0.344140\pi$
$821$	5.20534e8			0.940633			0.470316	+	0.882498i	$0.344140\pi$
$822$		0			0
$823$	5.86832e7	−	5.86832e7i	0.105272	−	0.105272i	−0.652509	−	0.757781i	$-0.726283\pi$
$823$	5.86832e7	−	5.86832e7i	0.105272	−	0.105272i	0.757781	+	0.652509i	$0.226283\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.82617e8	−	1.82617e8i	−0.322867	−	0.322867i	0.526999	−	0.849866i	$-0.323317\pi$
$827$	−1.82617e8	−	1.82617e8i	−0.322867	−	0.322867i	−0.849866	+	0.526999i	$0.823317\pi$
$828$		0			0
$829$		−	6.42030e7i		−	0.112692i	−0.998411	−	0.0563458i	$-0.982055\pi$
$829$		−	6.42030e7i		−	0.112692i	0.998411	−	0.0563458i	$-0.0179449\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5.10803e8	+	5.10803e8i	−0.883728	+	0.883728i
$834$		0			0
$835$	1.06444e7	−	1.56985e7i	0.0182835	−	0.0269650i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			8.94555e8i			1.51468i	0.653020	+	0.757341i	$0.273502\pi$
$839$			8.94555e8i			1.51468i	−0.653020	+	0.757341i	$0.726498\pi$
$840$		0			0
$841$	4.89702e7			0.0823272
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.35429e9	−	2.59836e8i	2.24462	−	0.430655i
$846$		0			0
$847$	−4.68059e8	−	4.68059e8i	−0.770283	−	0.770283i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.99303e8			0.972428
$852$		0			0
$853$	−2.04315e8	+	2.04315e8i	−0.329195	+	0.329195i	−0.852280	−	0.523085i	$-0.824781\pi$
$853$	−2.04315e8	+	2.04315e8i	−0.329195	+	0.329195i	0.523085	+	0.852280i	$0.324781\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.92490e8	+	6.92490e8i	1.10020	+	1.10020i	0.994386	+	0.105812i	$0.0337441\pi$
$857$	6.92490e8	+	6.92490e8i	1.10020	+	1.10020i	0.105812	+	0.994386i	$0.466256\pi$
$858$		0			0
$859$			1.16995e9i			1.84581i	0.385025	+	0.922906i	$0.374193\pi$
$859$			1.16995e9i			1.84581i	−0.385025	+	0.922906i	$0.625807\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4.85739e8	+	4.85739e8i	−0.755737	+	0.755737i	−0.975543	−	0.219807i	$-0.929457\pi$
$863$	−4.85739e8	+	4.85739e8i	−0.755737	+	0.755737i	0.219807	+	0.975543i	$0.429457\pi$
$864$		0			0
$865$	3.17210e8	+	2.15083e8i	0.490115	+	0.332321i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	7.05575e7i		−	0.107519i
$870$		0			0
$871$	−5.30487e8			−0.802824
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.85747e8	−	8.43603e8i	0.277267	−	1.25926i
$876$		0			0
$877$	−1.41359e8	−	1.41359e8i	−0.209568	−	0.209568i	0.594516	−	0.804084i	$-0.297344\pi$
$877$	−1.41359e8	−	1.41359e8i	−0.209568	−	0.209568i	−0.804084	+	0.594516i	$0.797344\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.09131e8			0.159595			0.0797975	−	0.996811i	$-0.474573\pi$
$881$	1.09131e8			0.159595			0.0797975	+	0.996811i	$0.474573\pi$
$882$		0			0
$883$	−3.96518e8	+	3.96518e8i	−0.575944	+	0.575944i	−0.933783	−	0.357839i	$-0.883514\pi$
$883$	−3.96518e8	+	3.96518e8i	−0.575944	+	0.575944i	0.357839	+	0.933783i	$0.383514\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−8.71187e8	−	8.71187e8i	−1.24836	−	1.24836i	−0.956443	−	0.291920i	$-0.905706\pi$
$887$	−8.71187e8	−	8.71187e8i	−1.24836	−	1.24836i	−0.291920	−	0.956443i	$-0.594294\pi$
$888$		0			0
$889$			3.75671e8i			0.534690i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.85713e8	+	2.85713e8i	−0.401213	+	0.401213i
$894$		0			0
$895$	−2.71074e6	+	520086.i	−0.00378110	+	0.000725447i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1.12208e9i			1.54434i
$900$		0			0
$901$	−2.35099e9			−3.21423
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.12867e8	−	5.88276e8i	−0.152273	−	0.793661i
$906$		0			0
$907$	1.20008e8	+	1.20008e8i	0.160838	+	0.160838i	0.782938	−	0.622100i	$-0.213720\pi$
$907$	1.20008e8	+	1.20008e8i	0.160838	+	0.160838i	−0.622100	+	0.782938i	$0.713720\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.44658e8			1.11719			0.558593	−	0.829442i	$-0.311341\pi$
$911$	8.44658e8			1.11719			0.558593	+	0.829442i	$0.311341\pi$
$912$		0			0
$913$	−2.80586e8	+	2.80586e8i	−0.368684	+	0.368684i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.15120e7	−	4.15120e7i	−0.0538351	−	0.0538351i
$918$		0			0
$919$			8.94371e8i			1.15232i	0.817339	+	0.576158i	$0.195449\pi$
$919$			8.94371e8i			1.15232i	−0.817339	+	0.576158i	$0.804551\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7.93277e8	+	7.93277e8i	−1.00883	+	1.00883i
$924$		0			0
$925$	−6.94182e8	−	2.98650e8i	−0.877098	−	0.377344i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	9.36762e8i		−	1.16837i	−0.811619	−	0.584187i	$-0.801413\pi$
$929$		−	9.36762e8i		−	1.16837i	0.811619	−	0.584187i	$-0.198587\pi$
$930$		0			0
$931$	5.88917e8			0.729802
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.40828e8	+	5.02661e8i	−0.416966	+	0.614951i
$936$		0			0
$937$	−4.41572e7	−	4.41572e7i	−0.0536763	−	0.0536763i	0.679759	−	0.733435i	$-0.262084\pi$
$937$	−4.41572e7	−	4.41572e7i	−0.0536763	−	0.0536763i	−0.733435	+	0.679759i	$0.762084\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−3.95331e8			−0.474452			−0.237226	−	0.971455i	$-0.576238\pi$
$941$	−3.95331e8			−0.474452			−0.237226	+	0.971455i	$0.576238\pi$
$942$		0			0
$943$	2.70489e8	−	2.70489e8i	0.322563	−	0.322563i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−8.00979e8	−	8.00979e8i	−0.943129	−	0.943129i	0.0553386	−	0.998468i	$-0.482376\pi$
$947$	−8.00979e8	−	8.00979e8i	−0.943129	−	0.943129i	−0.998468	+	0.0553386i	$0.982376\pi$
$948$		0			0
$949$		−	2.55871e9i		−	2.99379i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−9.59999e8	+	9.59999e8i	−1.10916	+	1.10916i	−0.115893	+	0.993262i	$0.536973\pi$
$953$	−9.59999e8	+	9.59999e8i	−1.10916	+	1.10916i	−0.993262	+	0.115893i	$0.963027\pi$
$954$		0			0
$955$	3.19279e8	+	1.66411e9i	0.366572	+	1.91061i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1.10803e9i		−	1.25631i
$960$		0			0
$961$	1.41908e9			1.59896
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3.48939e8	+	2.36597e8i	0.388300	+	0.263286i
$966$		0			0
$967$	4.20606e8	+	4.20606e8i	0.465153	+	0.465153i	0.900340	−	0.435187i	$-0.143318\pi$
$967$	4.20606e8	+	4.20606e8i	0.465153	+	0.465153i	−0.435187	+	0.900340i	$0.643318\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	6.58106e8			0.718850			0.359425	−	0.933174i	$-0.382973\pi$
$971$	6.58106e8			0.718850			0.359425	+	0.933174i	$0.382973\pi$
$972$		0			0
$973$	4.65021e8	−	4.65021e8i	0.504817	−	0.504817i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.52427e7	+	7.52427e7i	0.0806828	+	0.0806828i	0.746296	−	0.665614i	$-0.231830\pi$
$977$	7.52427e7	+	7.52427e7i	0.0806828	+	0.0806828i	−0.665614	+	0.746296i	$0.731830\pi$
$978$		0			0
$979$		−	4.10038e7i		−	0.0436994i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	3.89903e8	−	3.89903e8i	0.410484	−	0.410484i	−0.471423	−	0.881907i	$-0.656260\pi$
$983$	3.89903e8	−	3.89903e8i	0.410484	−	0.410484i	0.881907	+	0.471423i	$0.156260\pi$
$984$		0			0
$985$	−9.05012e8	+	1.33473e9i	−0.946991	+	1.39664i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			3.81854e8i			0.394737i
$990$		0			0
$991$	−2.98613e8			−0.306823			−0.153411	−	0.988162i	$-0.549026\pi$
$991$	−2.98613e8			−0.306823			−0.153411	+	0.988162i	$0.549026\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.51274e9	+	2.90237e8i	−1.53566	+	0.294634i
$996$		0			0
$997$	−2.31492e7	−	2.31492e7i	−0.0233588	−	0.0233588i	0.695331	−	0.718690i	$-0.255258\pi$
$997$	−2.31492e7	−	2.31492e7i	−0.0233588	−	0.0233588i	−0.718690	+	0.695331i	$0.755258\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.7.l.b.37.4		12
3.2	odd	2		60.7.k.a.37.2	yes	12
5.3	odd	4	inner	180.7.l.b.73.4		12
12.11	even	2		240.7.bg.d.97.5		12
15.2	even	4		300.7.k.d.193.4		12
15.8	even	4		60.7.k.a.13.2	✓	12
15.14	odd	2		300.7.k.d.157.4		12
60.23	odd	4		240.7.bg.d.193.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.7.k.a.13.2	✓	12	15.8	even	4
60.7.k.a.37.2	yes	12	3.2	odd	2
180.7.l.b.37.4		12	1.1	even	1	trivial
180.7.l.b.73.4		12	5.3	odd	4	inner
240.7.bg.d.97.5		12	12.11	even	2
240.7.bg.d.193.5		12	60.23	odd	4
300.7.k.d.157.4		12	15.14	odd	2
300.7.k.d.193.4		12	15.2	even	4

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:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	180.l (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(23.5531 + 122.761i) q^{5} +(312.733 + 312.733i) q^{7} +O(q^{10})\)	\(q+(23.5531 + 122.761i) q^{5} +(312.733 + 312.733i) q^{7} +524.296 q^{11} +(-2815.91 + 2815.91i) q^{13} +(6552.59 + 6552.59i) q^{17} -7554.63i q^{19} +(8761.99 - 8761.99i) q^{23} +(-14515.5 + 5782.80i) q^{25} -23363.5i q^{29} -48026.9 q^{31} +(-31025.5 + 45757.2i) q^{35} +(34199.0 + 34199.0i) q^{37} +30870.8 q^{41} +(-21790.4 + 21790.4i) q^{43} +(-37819.6 - 37819.6i) q^{47} +77954.4i q^{49} +(-179394. + 179394. i) q^{53} +(12348.8 + 64363.1i) q^{55} +200563. i q^{59} -19690.2 q^{61} +(-412008. - 279361. i) q^{65} +(94194.5 + 94194.5i) q^{67} +281712. q^{71} +(-454330. + 454330. i) q^{73} +(163965. + 163965. i) q^{77} -134576. i q^{79} +(-535168. + 535168. i) q^{83} +(-650068. + 958736. i) q^{85} -78207.3i q^{89} -1.76126e6 q^{91} +(927413. - 177935. i) q^{95} +(-94504.5 - 94504.5i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 312 q^{5} + 120 q^{7}+O(q^{10}) \) 12 * q - 312 * q^5 + 120 * q^7	\( 12 q - 312 q^{5} + 120 q^{7} + 3248 q^{11} - 2100 q^{13} + 5540 q^{17} + 23840 q^{23} + 10044 q^{25} - 127152 q^{31} - 102976 q^{35} + 282900 q^{37} + 320720 q^{41} - 62880 q^{43} - 381600 q^{47} + 400300 q^{53} + 502152 q^{55} + 807024 q^{61} - 124500 q^{65} + 752160 q^{67} - 202400 q^{71} - 322020 q^{73} + 2448400 q^{77} - 1894560 q^{83} - 857124 q^{85} + 2294400 q^{91} + 2620000 q^{95} - 3161700 q^{97}+O(q^{100}) \) 12 * q - 312 * q^5 + 120 * q^7 + 3248 * q^11 - 2100 * q^13 + 5540 * q^17 + 23840 * q^23 + 10044 * q^25 - 127152 * q^31 - 102976 * q^35 + 282900 * q^37 + 320720 * q^41 - 62880 * q^43 - 381600 * q^47 + 400300 * q^53 + 502152 * q^55 + 807024 * q^61 - 124500 * q^65 + 752160 * q^67 - 202400 * q^71 - 322020 * q^73 + 2448400 * q^77 - 1894560 * q^83 - 857124 * q^85 + 2294400 * q^91 + 2620000 * q^95 - 3161700 * q^97

Introduction

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