LMFDB - Embedded newform 1170.2.b.g.181.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(181,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1170.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3057647616.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 30x^{4} + 9 $ x^8 + 30*x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.5
Root			$0.524648 - 0.524648i$ of defining polynomial
Character	$\chi$	$=$	1170.181
Dual form			1170.2.b.g.181.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -4.66883i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -4.66883i q^{7} -1.00000i q^{8} -1.00000 q^{10} +4.89898i q^{11} +(-3.44949 - 1.04930i) q^{13} +4.66883 q^{14} +1.00000 q^{16} -2.57024 q^{17} +6.76742i q^{19} -1.00000i q^{20} -4.89898 q^{22} -4.66883 q^{23} -1.00000 q^{25} +(1.04930 - 3.44949i) q^{26} +4.66883i q^{28} +6.76742 q^{29} +2.09859i q^{31} +1.00000i q^{32} -2.57024i q^{34} +4.66883 q^{35} +11.4362i q^{37} -6.76742 q^{38} +1.00000 q^{40} +10.8990i q^{41} -4.00000 q^{43} -4.89898i q^{44} -4.66883i q^{46} +9.79796i q^{47} -14.7980 q^{49} -1.00000i q^{50} +(3.44949 + 1.04930i) q^{52} -2.09859 q^{53} -4.89898 q^{55} -4.66883 q^{56} +6.76742i q^{58} -4.89898i q^{59} -2.00000 q^{61} -2.09859 q^{62} -1.00000 q^{64} +(1.04930 - 3.44949i) q^{65} -7.23907i q^{67} +2.57024 q^{68} +4.66883i q^{70} -14.0065i q^{73} -11.4362 q^{74} -6.76742i q^{76} +22.8725 q^{77} -5.79796 q^{79} +1.00000i q^{80} -10.8990 q^{82} -9.79796i q^{83} -2.57024i q^{85} -4.00000i q^{86} +4.89898 q^{88} -1.10102i q^{89} +(-4.89898 + 16.1051i) q^{91} +4.66883 q^{92} -9.79796 q^{94} -6.76742 q^{95} +9.80930i q^{97} -14.7980i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} - 8 q^{10} - 8 q^{13} + 8 q^{16} - 8 q^{25} + 8 q^{40} - 32 q^{43} - 40 q^{49} + 8 q^{52} - 16 q^{61} - 8 q^{64} + 32 q^{79} - 48 q^{82}+O(q^{100}) $ 8 * q - 8 * q^4 - 8 * q^10 - 8 * q^13 + 8 * q^16 - 8 * q^25 + 8 * q^40 - 32 * q^43 - 40 * q^49 + 8 * q^52 - 16 * q^61 - 8 * q^64 + 32 * q^79 - 48 * q^82

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times$.

$n$	$911$	$937$	$1081$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	4.66883i		−	1.76465i	−0.470639	−	0.882326i	$-0.655977\pi$
$7$		−	4.66883i		−	1.76465i	0.470639	−	0.882326i	$-0.344023\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.00000			−0.316228
$11$			4.89898i			1.47710i	0.674200	+	0.738549i	$0.264489\pi$
$11$			4.89898i			1.47710i	−0.674200	+	0.738549i	$0.735511\pi$
$12$		0			0
$13$	−3.44949	−	1.04930i	−0.956716	−	0.291022i
$13$	−3.44949	−	1.04930i	−0.956716	−	0.291022i
$14$	4.66883			1.24780
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.57024			−0.623374			−0.311687	−	0.950185i	$-0.600894\pi$
$17$	−2.57024			−0.623374			−0.311687	+	0.950185i	$0.600894\pi$
$18$		0			0
$19$			6.76742i			1.55255i	0.630393	+	0.776276i	$0.282894\pi$
$19$			6.76742i			1.55255i	−0.630393	+	0.776276i	$0.717106\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$	−4.89898			−1.04447
$23$	−4.66883			−0.973518			−0.486759	−	0.873536i	$-0.661821\pi$
$23$	−4.66883			−0.973518			−0.486759	+	0.873536i	$0.661821\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	1.04930	−	3.44949i	0.205784	−	0.676501i
$27$		0			0
$28$			4.66883i			0.882326i
$29$	6.76742			1.25668			0.628339	−	0.777940i	$-0.283735\pi$
$29$	6.76742			1.25668			0.628339	+	0.777940i	$0.283735\pi$
$30$		0			0
$31$			2.09859i			0.376918i	0.982081	+	0.188459i	$0.0603493\pi$
$31$			2.09859i			0.376918i	−0.982081	+	0.188459i	$0.939651\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	2.57024i		−	0.440792i
$35$	4.66883			0.789176
$36$		0			0
$37$			11.4362i			1.88011i	0.341026	+	0.940054i	$0.389225\pi$
$37$			11.4362i			1.88011i	−0.341026	+	0.940054i	$0.610775\pi$
$38$	−6.76742			−1.09782
$39$		0			0
$40$	1.00000			0.158114
$41$			10.8990i			1.70213i	0.525057	+	0.851067i	$0.324044\pi$
$41$			10.8990i			1.70213i	−0.525057	+	0.851067i	$0.675956\pi$
$42$		0			0
$43$	−4.00000			−0.609994			−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000			−0.609994			−0.304997	+	0.952353i	$0.598656\pi$
$44$		−	4.89898i		−	0.738549i
$45$		0			0
$46$		−	4.66883i		−	0.688381i
$47$			9.79796i			1.42918i	0.699544	+	0.714590i	$0.253387\pi$
$47$			9.79796i			1.42918i	−0.699544	+	0.714590i	$0.746613\pi$
$48$		0			0
$49$	−14.7980			−2.11399
$50$		−	1.00000i		−	0.141421i
$51$		0			0
$52$	3.44949	+	1.04930i	0.478358	+	0.145511i
$53$	−2.09859			−0.288264			−0.144132	−	0.989559i	$-0.546039\pi$
$53$	−2.09859			−0.288264			−0.144132	+	0.989559i	$0.546039\pi$
$54$		0			0
$55$	−4.89898			−0.660578
$56$	−4.66883			−0.623898
$57$		0			0
$58$			6.76742i			0.888606i
$59$		−	4.89898i		−	0.637793i	−0.947790	−	0.318896i	$-0.896688\pi$
$59$		−	4.89898i		−	0.637793i	0.947790	−	0.318896i	$-0.103312\pi$
$60$		0			0
$61$	−2.00000			−0.256074			−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000			−0.256074			−0.128037	+	0.991769i	$0.540868\pi$
$62$	−2.09859			−0.266521
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.04930	−	3.44949i	0.130149	−	0.427857i
$66$		0			0
$67$		−	7.23907i		−	0.884393i	−0.896918	−	0.442196i	$-0.854199\pi$
$67$		−	7.23907i		−	0.884393i	0.896918	−	0.442196i	$-0.145801\pi$
$68$	2.57024			0.311687
$69$		0			0
$70$			4.66883i			0.558032i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	14.0065i		−	1.63934i	−0.572840	−	0.819668i	$-0.694158\pi$
$73$		−	14.0065i		−	1.63934i	0.572840	−	0.819668i	$-0.305842\pi$
$74$	−11.4362			−1.32944
$75$		0			0
$76$		−	6.76742i		−	0.776276i
$77$	22.8725			2.60656
$78$		0			0
$79$	−5.79796			−0.652321			−0.326161	−	0.945314i	$-0.605755\pi$
$79$	−5.79796			−0.652321			−0.326161	+	0.945314i	$0.605755\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$	−10.8990			−1.20359
$83$		−	9.79796i		−	1.07547i	−0.843115	−	0.537733i	$-0.819281\pi$
$83$		−	9.79796i		−	1.07547i	0.843115	−	0.537733i	$-0.180719\pi$
$84$		0			0
$85$		−	2.57024i		−	0.278781i
$86$		−	4.00000i		−	0.431331i
$87$		0			0
$88$	4.89898			0.522233
$89$		−	1.10102i		−	0.116708i	−0.998296	−	0.0583540i	$-0.981415\pi$
$89$		−	1.10102i		−	0.116708i	0.998296	−	0.0583540i	$-0.0185852\pi$
$90$		0			0
$91$	−4.89898	+	16.1051i	−0.513553	+	1.68827i
$92$	4.66883			0.486759
$93$		0			0
$94$	−9.79796			−1.01058
$95$	−6.76742			−0.694323
$96$		0			0
$97$			9.80930i			0.995984i	0.867182	+	0.497992i	$0.165929\pi$
$97$			9.80930i			0.995984i	−0.867182	+	0.497992i	$0.834071\pi$
$98$		−	14.7980i		−	1.49482i
$99$		0			0
$100$	1.00000			0.100000
$101$	10.9646			1.09102			0.545509	−	0.838105i	$-0.316336\pi$
$101$	10.9646			1.09102			0.545509	+	0.838105i	$0.316336\pi$
$102$		0			0
$103$	3.10102			0.305553			0.152776	−	0.988261i	$-0.451179\pi$
$103$	3.10102			0.305553			0.152776	+	0.988261i	$0.451179\pi$
$104$	−1.04930	+	3.44949i	−0.102892	+	0.338250i
$105$		0			0
$106$		−	2.09859i		−	0.203833i
$107$	−18.6753			−1.80541			−0.902705	−	0.430259i	$-0.858422\pi$
$107$	−18.6753			−1.80541			−0.902705	+	0.430259i	$0.858422\pi$
$108$		0			0
$109$		−	9.80930i		−	0.939561i	−0.882783	−	0.469780i	$-0.844333\pi$
$109$		−	9.80930i		−	0.939561i	0.882783	−	0.469780i	$-0.155667\pi$
$110$		−	4.89898i		−	0.467099i
$111$		0			0
$112$		−	4.66883i		−	0.441163i
$113$	−6.76742			−0.636625			−0.318313	−	0.947986i	$-0.603116\pi$
$113$	−6.76742			−0.636625			−0.318313	+	0.947986i	$0.603116\pi$
$114$		0			0
$115$		−	4.66883i		−	0.435370i
$116$	−6.76742			−0.628339
$117$		0			0
$118$	4.89898			0.450988
$119$			12.0000i			1.10004i
$120$		0			0
$121$	−13.0000			−1.18182
$122$		−	2.00000i		−	0.181071i
$123$		0			0
$124$		−	2.09859i		−	0.188459i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	−3.10102			−0.275171			−0.137586	−	0.990490i	$-0.543934\pi$
$127$	−3.10102			−0.275171			−0.137586	+	0.990490i	$0.543934\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	3.44949	+	1.04930i	0.302540	+	0.0920293i
$131$	−11.9079			−1.04040			−0.520199	−	0.854045i	$-0.674142\pi$
$131$	−11.9079			−1.04040			−0.520199	+	0.854045i	$0.674142\pi$
$132$		0			0
$133$	31.5959			2.73971
$134$	7.23907			0.625360
$135$		0			0
$136$			2.57024i			0.220396i
$137$			15.7980i			1.34971i	0.737950	+	0.674855i	$0.235794\pi$
$137$			15.7980i			1.34971i	−0.737950	+	0.674855i	$0.764206\pi$
$138$		0			0
$139$	5.79796			0.491776			0.245888	−	0.969298i	$-0.420920\pi$
$139$	5.79796			0.491776			0.245888	+	0.969298i	$0.420920\pi$
$140$	−4.66883			−0.394588
$141$		0			0
$142$		0			0
$143$	5.14048	−	16.8990i	0.429868	−	1.41316i
$144$		0			0
$145$			6.76742i			0.562004i
$146$	14.0065			1.15918
$147$		0			0
$148$		−	11.4362i		−	0.940054i
$149$			3.79796i			0.311141i	0.987825	+	0.155570i	$0.0497216\pi$
$149$			3.79796i			0.311141i	−0.987825	+	0.155570i	$0.950278\pi$
$150$		0			0
$151$		−	2.09859i		−	0.170781i	−0.996348	−	0.0853904i	$-0.972786\pi$
$151$		−	2.09859i		−	0.170781i	0.996348	−	0.0853904i	$-0.0272138\pi$
$152$	6.76742			0.548910
$153$		0			0
$154$			22.8725i			1.84312i
$155$	−2.09859			−0.168563
$156$		0			0
$157$	18.8990			1.50830			0.754151	−	0.656701i	$-0.228048\pi$
$157$	18.8990			1.50830			0.754151	+	0.656701i	$0.228048\pi$
$158$		−	5.79796i		−	0.461261i
$159$		0			0
$160$	−1.00000			−0.0790569
$161$			21.7980i			1.71792i
$162$		0			0
$163$		−	2.09859i		−	0.164374i	−0.996617	−	0.0821871i	$-0.973809\pi$
$163$		−	2.09859i		−	0.164374i	0.996617	−	0.0821871i	$-0.0261905\pi$
$164$		−	10.8990i		−	0.851067i
$165$		0			0
$166$	9.79796			0.760469
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	10.7980	+	7.23907i	0.830612	+	0.556851i
$170$	2.57024			0.197128
$171$		0			0
$172$	4.00000			0.304997
$173$	16.5767			1.26030			0.630152	−	0.776471i	$-0.282992\pi$
$173$	16.5767			1.26030			0.630152	+	0.776471i	$0.282992\pi$
$174$		0			0
$175$			4.66883i			0.352930i
$176$			4.89898i			0.369274i
$177$		0			0
$178$	1.10102			0.0825250
$179$	−1.62694			−0.121603			−0.0608017	−	0.998150i	$-0.519366\pi$
$179$	−1.62694			−0.121603			−0.0608017	+	0.998150i	$0.519366\pi$
$180$		0			0
$181$	4.20204			0.312335			0.156168	−	0.987731i	$-0.450086\pi$
$181$	4.20204			0.312335			0.156168	+	0.987731i	$0.450086\pi$
$182$	−16.1051	−	4.89898i	−1.19379	−	0.363137i
$183$		0			0
$184$			4.66883i			0.344191i
$185$	−11.4362			−0.840810
$186$		0			0
$187$		−	12.5915i		−	0.920785i
$188$		−	9.79796i		−	0.714590i
$189$		0			0
$190$		−	6.76742i		−	0.490960i
$191$	27.0697			1.95869			0.979346	−	0.202189i	$-0.0648056\pi$
$191$	27.0697			1.95869			0.979346	+	0.202189i	$0.0648056\pi$
$192$		0			0
$193$			9.80930i			0.706089i	0.935606	+	0.353045i	$0.114854\pi$
$193$			9.80930i			0.706089i	−0.935606	+	0.353045i	$0.885146\pi$
$194$	−9.80930			−0.704267
$195$		0			0
$196$	14.7980			1.05700
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$			1.00000i			0.0707107i
$201$		0			0
$202$			10.9646i			0.771467i
$203$		−	31.5959i		−	2.21760i
$204$		0			0
$205$	−10.8990			−0.761218
$206$			3.10102i			0.216058i
$207$		0			0
$208$	−3.44949	−	1.04930i	−0.239179	−	0.0727555i
$209$	−33.1534			−2.29327
$210$		0			0
$211$	13.7980			0.949891			0.474945	−	0.880015i	$-0.342468\pi$
$211$	13.7980			0.949891			0.474945	+	0.880015i	$0.342468\pi$
$212$	2.09859			0.144132
$213$		0			0
$214$		−	18.6753i		−	1.27662i
$215$		−	4.00000i		−	0.272798i
$216$		0			0
$217$	9.79796			0.665129
$218$	9.80930			0.664370
$219$		0			0
$220$	4.89898			0.330289
$221$	8.86601	+	2.69694i	0.596392	+	0.181416i
$222$		0			0
$223$			9.80930i			0.656880i	0.944525	+	0.328440i	$0.106523\pi$
$223$			9.80930i			0.656880i	−0.944525	+	0.328440i	$0.893477\pi$
$224$	4.66883			0.311949
$225$		0			0
$226$		−	6.76742i		−	0.450162i
$227$		−	21.7980i		−	1.44678i	−0.690439	−	0.723391i	$-0.742583\pi$
$227$		−	21.7980i		−	1.44678i	0.690439	−	0.723391i	$-0.257417\pi$
$228$		0			0
$229$		−	4.66883i		−	0.308525i	−0.988030	−	0.154262i	$-0.950700\pi$
$229$		−	4.66883i		−	0.308525i	0.988030	−	0.154262i	$-0.0493001\pi$
$230$	4.66883			0.307853
$231$		0			0
$232$		−	6.76742i		−	0.444303i
$233$	−6.76742			−0.443348			−0.221674	−	0.975121i	$-0.571152\pi$
$233$	−6.76742			−0.443348			−0.221674	+	0.975121i	$0.571152\pi$
$234$		0			0
$235$	−9.79796			−0.639148
$236$			4.89898i			0.318896i
$237$		0			0
$238$	−12.0000			−0.777844
$239$			21.7980i			1.40999i	0.709211	+	0.704996i	$0.249051\pi$
$239$			21.7980i			1.40999i	−0.709211	+	0.704996i	$0.750949\pi$
$240$		0			0
$241$		−	18.6753i		−	1.20298i	−0.798879	−	0.601491i	$-0.794573\pi$
$241$		−	18.6753i		−	1.20298i	0.798879	−	0.601491i	$-0.205427\pi$
$242$		−	13.0000i		−	0.835672i
$243$		0			0
$244$	2.00000			0.128037
$245$		−	14.7980i		−	0.945407i
$246$		0			0
$247$	7.10102	−	23.3441i	0.451827	−	1.48535i
$248$	2.09859			0.133261
$249$		0			0
$250$	1.00000			0.0632456
$251$	2.57024			0.162232			0.0811160	−	0.996705i	$-0.474152\pi$
$251$	2.57024			0.162232			0.0811160	+	0.996705i	$0.474152\pi$
$252$		0			0
$253$		−	22.8725i		−	1.43798i
$254$		−	3.10102i		−	0.194575i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−11.9079			−0.742794			−0.371397	−	0.928474i	$-0.621121\pi$
$257$	−11.9079			−0.742794			−0.371397	+	0.928474i	$0.621121\pi$
$258$		0			0
$259$	53.3939			3.31773
$260$	−1.04930	+	3.44949i	−0.0650745	+	0.213928i
$261$		0			0
$262$		−	11.9079i		−	0.735672i
$263$	−8.86601			−0.546702			−0.273351	−	0.961914i	$-0.588132\pi$
$263$	−8.86601			−0.546702			−0.273351	+	0.961914i	$0.588132\pi$
$264$		0			0
$265$		−	2.09859i		−	0.128915i
$266$			31.5959i			1.93727i
$267$		0			0
$268$			7.23907i			0.442196i
$269$	−21.2456			−1.29536			−0.647682	−	0.761911i	$-0.724261\pi$
$269$	−21.2456			−1.29536			−0.647682	+	0.761911i	$0.724261\pi$
$270$		0			0
$271$		−	16.5767i		−	1.00696i	−0.864006	−	0.503482i	$-0.832052\pi$
$271$		−	16.5767i		−	1.00696i	0.864006	−	0.503482i	$-0.167948\pi$
$272$	−2.57024			−0.155844
$273$		0			0
$274$	−15.7980			−0.954390
$275$		−	4.89898i		−	0.295420i
$276$		0			0
$277$	4.69694			0.282212			0.141106	−	0.989995i	$-0.454934\pi$
$277$	4.69694			0.282212			0.141106	+	0.989995i	$0.454934\pi$
$278$			5.79796i			0.347738i
$279$		0			0
$280$		−	4.66883i		−	0.279016i
$281$			20.6969i			1.23468i	0.786698	+	0.617338i	$0.211789\pi$
$281$			20.6969i			1.23468i	−0.786698	+	0.617338i	$0.788211\pi$
$282$		0			0
$283$	−10.2020			−0.606448			−0.303224	−	0.952919i	$-0.598063\pi$
$283$	−10.2020			−0.606448			−0.303224	+	0.952919i	$0.598063\pi$
$284$		0			0
$285$		0			0
$286$	16.8990	+	5.14048i	0.999258	+	0.303963i
$287$	50.8855			3.00367
$288$		0			0
$289$	−10.3939			−0.611405
$290$	−6.76742			−0.397397
$291$		0			0
$292$			14.0065i			0.819668i
$293$			18.0000i			1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$			18.0000i			1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$		0			0
$295$	4.89898			0.285230
$296$	11.4362			0.664718
$297$		0			0
$298$	−3.79796			−0.220010
$299$	16.1051	+	4.89898i	0.931381	+	0.283315i
$300$		0			0
$301$			18.6753i			1.07643i
$302$	2.09859			0.120760
$303$		0			0
$304$			6.76742i			0.388138i
$305$		−	2.00000i		−	0.114520i
$306$		0			0
$307$			7.23907i			0.413155i	0.978430	+	0.206578i	$0.0662326\pi$
$307$			7.23907i			0.413155i	−0.978430	+	0.206578i	$0.933767\pi$
$308$	−22.8725			−1.30328
$309$		0			0
$310$		−	2.09859i		−	0.119192i
$311$	−18.6753			−1.05898			−0.529490	−	0.848316i	$-0.677617\pi$
$311$	−18.6753			−1.05898			−0.529490	+	0.848316i	$0.677617\pi$
$312$		0			0
$313$	27.3939			1.54839			0.774197	−	0.632945i	$-0.218154\pi$
$313$	27.3939			1.54839			0.774197	+	0.632945i	$0.218154\pi$
$314$			18.8990i			1.06653i
$315$		0			0
$316$	5.79796			0.326161
$317$			25.5959i			1.43761i	0.695212	+	0.718805i	$0.255311\pi$
$317$			25.5959i			1.43761i	−0.695212	+	0.718805i	$0.744689\pi$
$318$		0			0
$319$			33.1534i			1.85624i
$320$		−	1.00000i		−	0.0559017i
$321$		0			0
$322$	−21.7980			−1.21475
$323$		−	17.3939i		−	0.967821i
$324$		0			0
$325$	3.44949	+	1.04930i	0.191343	+	0.0582044i
$326$	2.09859			0.116230
$327$		0			0
$328$	10.8990			0.601795
$329$	45.7450			2.52200
$330$		0			0
$331$		−	7.71071i		−	0.423819i	−0.977289	−	0.211910i	$-0.932032\pi$
$331$		−	7.71071i		−	0.423819i	0.977289	−	0.211910i	$-0.0679682\pi$
$332$			9.79796i			0.537733i
$333$		0			0
$334$		0			0
$335$	7.23907			0.395512
$336$		0			0
$337$	10.0000			0.544735			0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000			0.544735			0.272367	+	0.962193i	$0.412193\pi$
$338$	−7.23907	+	10.7980i	−0.393753	+	0.587332i
$339$		0			0
$340$			2.57024i			0.139391i
$341$	−10.2810			−0.556745
$342$		0			0
$343$			36.4073i			1.96581i
$344$			4.00000i			0.215666i
$345$		0			0
$346$			16.5767i			0.891170i
$347$	19.6186			1.05318			0.526591	−	0.850119i	$-0.323470\pi$
$347$	19.6186			1.05318			0.526591	+	0.850119i	$0.323470\pi$
$348$		0			0
$349$		−	31.7385i		−	1.69892i	−0.527650	−	0.849462i	$-0.676927\pi$
$349$		−	31.7385i		−	1.69892i	0.527650	−	0.849462i	$-0.323073\pi$
$350$	−4.66883			−0.249559
$351$		0			0
$352$	−4.89898			−0.261116
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$		0			0
$356$			1.10102i			0.0583540i
$357$		0			0
$358$		−	1.62694i		−	0.0859866i
$359$		−	9.79796i		−	0.517116i	−0.965996	−	0.258558i	$-0.916753\pi$
$359$		−	9.79796i		−	0.517116i	0.965996	−	0.258558i	$-0.0832474\pi$
$360$		0			0
$361$	−26.7980			−1.41042
$362$			4.20204i			0.220854i
$363$		0			0
$364$	4.89898	−	16.1051i	0.256776	−	0.844135i
$365$	14.0065			0.733133
$366$		0			0
$367$	22.6969			1.18477			0.592385	−	0.805655i	$-0.298186\pi$
$367$	22.6969			1.18477			0.592385	+	0.805655i	$0.298186\pi$
$368$	−4.66883			−0.243380
$369$		0			0
$370$		−	11.4362i		−	0.594542i
$371$			9.79796i			0.508685i
$372$		0			0
$373$	−16.6969			−0.864535			−0.432267	−	0.901745i	$-0.642286\pi$
$373$	−16.6969			−0.864535			−0.432267	+	0.901745i	$0.642286\pi$
$374$	12.5915			0.651093
$375$		0			0
$376$	9.79796			0.505291
$377$	−23.3441	−	7.10102i	−1.20228	−	0.365721i
$378$		0			0
$379$		−	29.6399i		−	1.52250i	−0.648458	−	0.761250i	$-0.724586\pi$
$379$		−	29.6399i		−	1.52250i	0.648458	−	0.761250i	$-0.275414\pi$
$380$	6.76742			0.347161
$381$		0			0
$382$			27.0697i			1.38501i
$383$		−	14.2020i		−	0.725690i	−0.931849	−	0.362845i	$-0.881805\pi$
$383$		−	14.2020i		−	0.725690i	0.931849	−	0.362845i	$-0.118195\pi$
$384$		0			0
$385$			22.8725i			1.16569i
$386$	−9.80930			−0.499280
$387$		0			0
$388$		−	9.80930i		−	0.497992i
$389$	21.2456			1.07719			0.538596	−	0.842564i	$-0.318955\pi$
$389$	21.2456			1.07719			0.538596	+	0.842564i	$0.318955\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$			14.7980i			0.747410i
$393$		0			0
$394$	−18.0000			−0.906827
$395$		−	5.79796i		−	0.291727i
$396$		0			0
$397$			7.23907i			0.363318i	0.983362	+	0.181659i	$0.0581468\pi$
$397$			7.23907i			0.363318i	−0.983362	+	0.181659i	$0.941853\pi$
$398$		−	20.0000i		−	1.00251i
$399$		0			0
$400$	−1.00000			−0.0500000
$401$			20.6969i			1.03356i	0.856120	+	0.516778i	$0.172869\pi$
$401$			20.6969i			1.03356i	−0.856120	+	0.516778i	$0.827131\pi$
$402$		0			0
$403$	2.20204	−	7.23907i	0.109691	−	0.360604i
$404$	−10.9646			−0.545509
$405$		0			0
$406$	31.5959			1.56808
$407$	−56.0259			−2.77710
$408$		0			0
$409$			13.5348i			0.669255i	0.942351	+	0.334627i	$0.108610\pi$
$409$			13.5348i			0.669255i	−0.942351	+	0.334627i	$0.891390\pi$
$410$		−	10.8990i		−	0.538262i
$411$		0			0
$412$	−3.10102			−0.152776
$413$	−22.8725			−1.12548
$414$		0			0
$415$	9.79796			0.480963
$416$	1.04930	−	3.44949i	0.0514459	−	0.169125i
$417$		0			0
$418$		−	33.1534i		−	1.62159i
$419$	17.0484			0.832867			0.416434	−	0.909166i	$-0.363280\pi$
$419$	17.0484			0.832867			0.416434	+	0.909166i	$0.363280\pi$
$420$		0			0
$421$			14.0065i			0.682634i	0.939948	+	0.341317i	$0.110873\pi$
$421$			14.0065i			0.682634i	−0.939948	+	0.341317i	$0.889127\pi$
$422$			13.7980i			0.671674i
$423$		0			0
$424$			2.09859i			0.101917i
$425$	2.57024			0.124675
$426$		0			0
$427$			9.33766i			0.451881i
$428$	18.6753			0.902705
$429$		0			0
$430$	4.00000			0.192897
$431$		−	31.5959i		−	1.52192i	−0.648798	−	0.760961i	$-0.724728\pi$
$431$		−	31.5959i		−	1.52192i	0.648798	−	0.760961i	$-0.275272\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$			9.79796i			0.470317i
$435$		0			0
$436$			9.80930i			0.469780i
$437$		−	31.5959i		−	1.51144i
$438$		0			0
$439$	27.5959			1.31708			0.658541	−	0.752545i	$-0.271174\pi$
$439$	27.5959			1.31708			0.658541	+	0.752545i	$0.271174\pi$
$440$			4.89898i			0.233550i
$441$		0			0
$442$	−2.69694	+	8.86601i	−0.128280	+	0.421713i
$443$	−40.6045			−1.92918			−0.964589	−	0.263756i	$-0.915039\pi$
$443$	−40.6045			−1.92918			−0.964589	+	0.263756i	$0.915039\pi$
$444$		0			0
$445$	1.10102			0.0521934
$446$	−9.80930			−0.464484
$447$		0			0
$448$			4.66883i			0.220581i
$449$		−	1.10102i		−	0.0519604i	−0.999662	−	0.0259802i	$-0.991729\pi$
$449$		−	1.10102i		−	0.0519604i	0.999662	−	0.0259802i	$-0.00827068\pi$
$450$		0			0
$451$	−53.3939			−2.51422
$452$	6.76742			0.318313
$453$		0			0
$454$	21.7980			1.02303
$455$	−16.1051	−	4.89898i	−0.755018	−	0.229668i
$456$		0			0
$457$		−	8.86601i		−	0.414734i	−0.978263	−	0.207367i	$-0.933511\pi$
$457$		−	8.86601i		−	0.414734i	0.978263	−	0.207367i	$-0.0664895\pi$
$458$	4.66883			0.218160
$459$		0			0
$460$			4.66883i			0.217685i
$461$		−	3.79796i		−	0.176889i	−0.996081	−	0.0884443i	$-0.971810\pi$
$461$		−	3.79796i		−	0.176889i	0.996081	−	0.0884443i	$-0.0281895\pi$
$462$		0			0
$463$			8.86601i			0.412038i	0.978548	+	0.206019i	$0.0660509\pi$
$463$			8.86601i			0.412038i	−0.978548	+	0.206019i	$0.933949\pi$
$464$	6.76742			0.314170
$465$		0			0
$466$		−	6.76742i		−	0.313495i
$467$	−4.19718			−0.194222			−0.0971112	−	0.995274i	$-0.530960\pi$
$467$	−4.19718			−0.194222			−0.0971112	+	0.995274i	$0.530960\pi$
$468$		0			0
$469$	−33.7980			−1.56064
$470$		−	9.79796i		−	0.451946i
$471$		0			0
$472$	−4.89898			−0.225494
$473$		−	19.5959i		−	0.901021i
$474$		0			0
$475$		−	6.76742i		−	0.310510i
$476$		−	12.0000i		−	0.550019i
$477$		0			0
$478$	−21.7980			−0.997015
$479$		−	21.7980i		−	0.995974i	−0.867184	−	0.497987i	$-0.834073\pi$
$479$		−	21.7980i		−	0.995974i	0.867184	−	0.497987i	$-0.165927\pi$
$480$		0			0
$481$	12.0000	−	39.4492i	0.547153	−	1.79873i
$482$	18.6753			0.850637
$483$		0			0
$484$	13.0000			0.590909
$485$	−9.80930			−0.445418
$486$		0			0
$487$		−	32.6818i		−	1.48095i	−0.672082	−	0.740477i	$-0.734600\pi$
$487$		−	32.6818i		−	1.48095i	0.672082	−	0.740477i	$-0.265400\pi$
$488$			2.00000i			0.0905357i
$489$		0			0
$490$	14.7980			0.668504
$491$	−10.9646			−0.494825			−0.247413	−	0.968910i	$-0.579580\pi$
$491$	−10.9646			−0.494825			−0.247413	+	0.968910i	$0.579580\pi$
$492$		0			0
$493$	−17.3939			−0.783381
$494$	23.3441	+	7.10102i	1.05030	+	0.319490i
$495$		0			0
$496$			2.09859i			0.0942295i
$497$		0			0
$498$		0			0
$499$		−	6.76742i		−	0.302951i	−0.988461	−	0.151476i	$-0.951597\pi$
$499$		−	6.76742i		−	0.302951i	0.988461	−	0.151476i	$-0.0484025\pi$
$500$			1.00000i			0.0447214i
$501$		0			0
$502$			2.57024i			0.114715i
$503$	9.80930			0.437375			0.218688	−	0.975795i	$-0.429822\pi$
$503$	9.80930			0.437375			0.218688	+	0.975795i	$0.429822\pi$
$504$		0			0
$505$			10.9646i			0.487918i
$506$	22.8725			1.01681
$507$		0			0
$508$	3.10102			0.137586
$509$			30.0000i			1.32973i	0.746965	+	0.664863i	$0.231510\pi$
$509$			30.0000i			1.32973i	−0.746965	+	0.664863i	$0.768490\pi$
$510$		0			0
$511$	−65.3939			−2.89285
$512$			1.00000i			0.0441942i
$513$		0			0
$514$		−	11.9079i		−	0.525235i
$515$			3.10102i			0.136647i
$516$		0			0
$517$	−48.0000			−2.11104
$518$			53.3939i			2.34599i
$519$		0			0
$520$	−3.44949	−	1.04930i	−0.151270	−	0.0460146i
$521$	19.6186			0.859507			0.429753	−	0.902946i	$-0.358600\pi$
$521$	19.6186			0.859507			0.429753	+	0.902946i	$0.358600\pi$
$522$		0			0
$523$	22.2020			0.970827			0.485414	−	0.874285i	$-0.338669\pi$
$523$	22.2020			0.970827			0.485414	+	0.874285i	$0.338669\pi$
$524$	11.9079			0.520199
$525$		0			0
$526$		−	8.86601i		−	0.386576i
$527$		−	5.39388i		−	0.234961i
$528$		0			0
$529$	−1.20204			−0.0522627
$530$	2.09859			0.0911569
$531$		0			0
$532$	−31.5959			−1.36986
$533$	11.4362	−	37.5959i	0.495359	−	1.62846i
$534$		0			0
$535$		−	18.6753i		−	0.807404i
$536$	−7.23907			−0.312680
$537$		0			0
$538$		−	21.2456i		−	0.915961i
$539$		−	72.4949i		−	3.12258i
$540$		0			0
$541$			35.9357i			1.54500i	0.635017	+	0.772498i	$0.280993\pi$
$541$			35.9357i			1.54500i	−0.635017	+	0.772498i	$0.719007\pi$
$542$	16.5767			0.712031
$543$		0			0
$544$		−	2.57024i		−	0.110198i
$545$	9.80930			0.420184
$546$		0			0
$547$	32.0000			1.36822			0.684111	−	0.729378i	$-0.260191\pi$
$547$	32.0000			1.36822			0.684111	+	0.729378i	$0.260191\pi$
$548$		−	15.7980i		−	0.674855i
$549$		0			0
$550$	4.89898			0.208893
$551$			45.7980i			1.95106i
$552$		0			0
$553$			27.0697i			1.15112i
$554$			4.69694i			0.199554i
$555$		0			0
$556$	−5.79796			−0.245888
$557$			25.5959i			1.08453i	0.840206	+	0.542267i	$0.182434\pi$
$557$			25.5959i			1.08453i	−0.840206	+	0.542267i	$0.817566\pi$
$558$		0			0
$559$	13.7980	+	4.19718i	0.583591	+	0.177522i
$560$	4.66883			0.197294
$561$		0			0
$562$	−20.6969			−0.873048
$563$	−21.9292			−0.924206			−0.462103	−	0.886826i	$-0.652905\pi$
$563$	−21.9292			−0.924206			−0.462103	+	0.886826i	$0.652905\pi$
$564$		0			0
$565$		−	6.76742i		−	0.284707i
$566$		−	10.2020i		−	0.428824i
$567$		0			0
$568$		0			0
$569$	−9.33766			−0.391455			−0.195727	−	0.980658i	$-0.562707\pi$
$569$	−9.33766			−0.391455			−0.195727	+	0.980658i	$0.562707\pi$
$570$		0			0
$571$	−15.5959			−0.652669			−0.326334	−	0.945254i	$-0.605814\pi$
$571$	−15.5959			−0.652669			−0.326334	+	0.945254i	$0.605814\pi$
$572$	−5.14048	+	16.8990i	−0.214934	+	0.706582i
$573$		0			0
$574$			50.8855i			2.12392i
$575$	4.66883			0.194704
$576$		0			0
$577$			13.0632i			0.543828i	0.962322	+	0.271914i	$0.0876566\pi$
$577$			13.0632i			0.543828i	−0.962322	+	0.271914i	$0.912343\pi$
$578$		−	10.3939i		−	0.432328i
$579$		0			0
$580$		−	6.76742i		−	0.281002i
$581$	−45.7450			−1.89782
$582$		0			0
$583$		−	10.2810i		−	0.425794i
$584$	−14.0065			−0.579592
$585$		0			0
$586$	−18.0000			−0.743573
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$	−14.2020			−0.585185
$590$			4.89898i			0.201688i
$591$		0			0
$592$			11.4362i			0.470027i
$593$			13.5959i			0.558317i	0.960245	+	0.279159i	$0.0900556\pi$
$593$			13.5959i			0.558317i	−0.960245	+	0.279159i	$0.909944\pi$
$594$		0			0
$595$	−12.0000			−0.491952
$596$		−	3.79796i		−	0.155570i
$597$		0			0
$598$	−4.89898	+	16.1051i	−0.200334	+	0.658586i
$599$	−37.3506			−1.52611			−0.763053	−	0.646336i	$-0.776300\pi$
$599$	−37.3506			−1.52611			−0.763053	+	0.646336i	$0.776300\pi$
$600$		0			0
$601$	−19.3939			−0.791093			−0.395546	−	0.918446i	$-0.629445\pi$
$601$	−19.3939			−0.791093			−0.395546	+	0.918446i	$0.629445\pi$
$602$	−18.6753			−0.761149
$603$		0			0
$604$			2.09859i			0.0853904i
$605$		−	13.0000i		−	0.528525i
$606$		0			0
$607$	−30.6969			−1.24595			−0.622975	−	0.782242i	$-0.714076\pi$
$607$	−30.6969			−1.24595			−0.622975	+	0.782242i	$0.714076\pi$
$608$	−6.76742			−0.274455
$609$		0			0
$610$	2.00000			0.0809776
$611$	10.2810	−	33.7980i	0.415923	−	1.36732i
$612$		0			0
$613$			11.4362i			0.461906i	0.972965	+	0.230953i	$0.0741843\pi$
$613$			11.4362i			0.461906i	−0.972965	+	0.230953i	$0.925816\pi$
$614$	−7.23907			−0.292145
$615$		0			0
$616$		−	22.8725i		−	0.921559i
$617$		−	20.2020i		−	0.813304i	−0.913583	−	0.406652i	$-0.866696\pi$
$617$		−	20.2020i		−	0.813304i	0.913583	−	0.406652i	$-0.133304\pi$
$618$		0			0
$619$			31.5265i			1.26716i	0.773678	+	0.633579i	$0.218415\pi$
$619$			31.5265i			1.26716i	−0.773678	+	0.633579i	$0.781585\pi$
$620$	2.09859			0.0842814
$621$		0			0
$622$		−	18.6753i		−	0.748812i
$623$	−5.14048			−0.205949
$624$		0			0
$625$	1.00000			0.0400000
$626$			27.3939i			1.09488i
$627$		0			0
$628$	−18.8990			−0.754151
$629$		−	29.3939i		−	1.17201i
$630$		0			0
$631$			19.8306i			0.789444i	0.918801	+	0.394722i	$0.129159\pi$
$631$			19.8306i			0.789444i	−0.918801	+	0.394722i	$0.870841\pi$
$632$			5.79796i			0.230630i
$633$		0			0
$634$	−25.5959			−1.01654
$635$		−	3.10102i		−	0.123060i
$636$		0			0
$637$	51.0454	+	15.5274i	2.02249	+	0.615219i
$638$	−33.1534			−1.31256
$639$		0			0
$640$	1.00000			0.0395285
$641$	27.0697			1.06919			0.534594	−	0.845109i	$-0.320465\pi$
$641$	27.0697			1.06919			0.534594	+	0.845109i	$0.320465\pi$
$642$		0			0
$643$			1.15530i			0.0455604i	0.999740	+	0.0227802i	$0.00725179\pi$
$643$			1.15530i			0.0455604i	−0.999740	+	0.0227802i	$0.992748\pi$
$644$		−	21.7980i		−	0.858960i
$645$		0			0
$646$	17.3939			0.684353
$647$	8.86601			0.348559			0.174279	−	0.984696i	$-0.444240\pi$
$647$	8.86601			0.348559			0.174279	+	0.984696i	$0.444240\pi$
$648$		0			0
$649$	24.0000			0.942082
$650$	−1.04930	+	3.44949i	−0.0411567	+	0.135300i
$651$		0			0
$652$			2.09859i			0.0821871i
$653$	34.3087			1.34260			0.671302	−	0.741184i	$-0.265735\pi$
$653$	34.3087			1.34260			0.671302	+	0.741184i	$0.265735\pi$
$654$		0			0
$655$		−	11.9079i		−	0.465280i
$656$			10.8990i			0.425534i
$657$		0			0
$658$			45.7450i			1.78333i
$659$	−16.1051			−0.627365			−0.313682	−	0.949528i	$-0.601563\pi$
$659$	−16.1051			−0.627365			−0.313682	+	0.949528i	$0.601563\pi$
$660$		0			0
$661$			31.7385i			1.23448i	0.786773	+	0.617242i	$0.211750\pi$
$661$			31.7385i			1.23448i	−0.786773	+	0.617242i	$0.788250\pi$
$662$	7.71071			0.299685
$663$		0			0
$664$	−9.79796			−0.380235
$665$			31.5959i			1.22524i
$666$		0			0
$667$	−31.5959			−1.22340
$668$		0			0
$669$		0			0
$670$			7.23907i			0.279670i
$671$		−	9.79796i		−	0.378246i
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$			10.0000i			0.385186i
$675$		0			0
$676$	−10.7980	−	7.23907i	−0.415306	−	0.278426i
$677$	30.1116			1.15728			0.578641	−	0.815583i	$-0.303583\pi$
$677$	30.1116			1.15728			0.578641	+	0.815583i	$0.303583\pi$
$678$		0			0
$679$	45.7980			1.75756
$680$	−2.57024			−0.0985641
$681$		0			0
$682$		−	10.2810i		−	0.393678i
$683$			14.2020i			0.543426i	0.962378	+	0.271713i	$0.0875901\pi$
$683$			14.2020i			0.543426i	−0.962378	+	0.271713i	$0.912410\pi$
$684$		0			0
$685$	−15.7980			−0.603609
$686$	−36.4073			−1.39004
$687$		0			0
$688$	−4.00000			−0.152499
$689$	7.23907	+	2.20204i	0.275786	+	0.0838911i
$690$		0			0
$691$			38.0343i			1.44689i	0.690381	+	0.723446i	$0.257443\pi$
$691$			38.0343i			1.44689i	−0.690381	+	0.723446i	$0.742557\pi$
$692$	−16.5767			−0.630152
$693$		0			0
$694$			19.6186i			0.744712i
$695$			5.79796i			0.219929i
$696$		0			0
$697$		−	28.0130i		−	1.06107i
$698$	31.7385			1.20132
$699$		0			0
$700$		−	4.66883i		−	0.176465i
$701$	33.8371			1.27801			0.639005	−	0.769203i	$-0.279346\pi$
$701$	33.8371			1.27801			0.639005	+	0.769203i	$0.279346\pi$
$702$		0			0
$703$	−77.3939			−2.91897
$704$		−	4.89898i		−	0.184637i
$705$		0			0
$706$	6.00000			0.225813
$707$		−	51.1918i		−	1.92527i
$708$		0			0
$709$		−	14.9498i		−	0.561451i	−0.959788	−	0.280725i	$-0.909425\pi$
$709$		−	14.9498i		−	0.561451i	0.959788	−	0.280725i	$-0.0905750\pi$
$710$		0			0
$711$		0			0
$712$	−1.10102			−0.0412625
$713$		−	9.79796i		−	0.366936i
$714$		0			0
$715$	16.8990	+	5.14048i	0.631986	+	0.192243i
$716$	1.62694			0.0608017
$717$		0			0
$718$	9.79796			0.365657
$719$	−13.5348			−0.504764			−0.252382	−	0.967628i	$-0.581214\pi$
$719$	−13.5348			−0.504764			−0.252382	+	0.967628i	$0.581214\pi$
$720$		0			0
$721$		−	14.4781i		−	0.539194i
$722$		−	26.7980i		−	0.997317i
$723$		0			0
$724$	−4.20204			−0.156168
$725$	−6.76742			−0.251336
$726$		0			0
$727$	6.69694			0.248376			0.124188	−	0.992259i	$-0.460367\pi$
$727$	6.69694			0.248376			0.124188	+	0.992259i	$0.460367\pi$
$728$	16.1051	+	4.89898i	0.596894	+	0.181568i
$729$		0			0
$730$			14.0065i			0.518403i
$731$	10.2810			0.380255
$732$		0			0
$733$		−	24.0278i		−	0.887487i	−0.896154	−	0.443743i	$-0.853650\pi$
$733$		−	24.0278i		−	0.887487i	0.896154	−	0.443743i	$-0.146350\pi$
$734$			22.6969i			0.837759i
$735$		0			0
$736$		−	4.66883i		−	0.172095i
$737$	35.4640			1.30633
$738$		0			0
$739$			2.57024i			0.0945477i	0.998882	+	0.0472739i	$0.0150533\pi$
$739$			2.57024i			0.0945477i	−0.998882	+	0.0472739i	$0.984947\pi$
$740$	11.4362			0.420405
$741$		0			0
$742$	−9.79796			−0.359694
$743$		−	4.40408i		−	0.161570i	−0.996732	−	0.0807851i	$-0.974257\pi$
$743$		−	4.40408i		−	0.161570i	0.996732	−	0.0807851i	$-0.0257427\pi$
$744$		0			0
$745$	−3.79796			−0.139146
$746$		−	16.6969i		−	0.611318i
$747$		0			0
$748$			12.5915i			0.460392i
$749$			87.1918i			3.18592i
$750$		0			0
$751$	−6.20204			−0.226316			−0.113158	−	0.993577i	$-0.536097\pi$
$751$	−6.20204			−0.226316			−0.113158	+	0.993577i	$0.536097\pi$
$752$			9.79796i			0.357295i
$753$		0			0
$754$	7.10102	−	23.3441i	0.258604	−	0.850144i
$755$	2.09859			0.0763755
$756$		0			0
$757$	29.1010			1.05769			0.528847	−	0.848717i	$-0.322624\pi$
$757$	29.1010			1.05769			0.528847	+	0.848717i	$0.322624\pi$
$758$	29.6399			1.07657
$759$		0			0
$760$			6.76742i			0.245480i
$761$		−	16.2929i		−	0.590616i	−0.955402	−	0.295308i	$-0.904578\pi$
$761$		−	16.2929i		−	0.590616i	0.955402	−	0.295308i	$-0.0954222\pi$
$762$		0			0
$763$	−45.7980			−1.65800
$764$	−27.0697			−0.979346
$765$		0			0
$766$	14.2020			0.513141
$767$	−5.14048	+	16.8990i	−0.185612	+	0.610187i
$768$		0			0
$769$		−	51.8288i		−	1.86899i	−0.355972	−	0.934496i	$-0.615850\pi$
$769$		−	51.8288i		−	1.86899i	0.355972	−	0.934496i	$-0.384150\pi$
$770$	−22.8725			−0.824267
$771$		0			0
$772$		−	9.80930i		−	0.353045i
$773$			25.5959i			0.920621i	0.887758	+	0.460311i	$0.152262\pi$
$773$			25.5959i			0.920621i	−0.887758	+	0.460311i	$0.847738\pi$
$774$		0			0
$775$		−	2.09859i		−	0.0753836i
$776$	9.80930			0.352134
$777$		0			0
$778$			21.2456i			0.761690i
$779$	−73.7580			−2.64265
$780$		0			0
$781$		0			0
$782$			12.0000i			0.429119i
$783$		0			0
$784$	−14.7980			−0.528499
$785$			18.8990i			0.674534i
$786$		0			0
$787$			12.3795i			0.441283i	0.975355	+	0.220642i	$0.0708151\pi$
$787$			12.3795i			0.441283i	−0.975355	+	0.220642i	$0.929185\pi$
$788$		−	18.0000i		−	0.641223i
$789$		0			0
$790$	5.79796			0.206282
$791$			31.5959i			1.12342i
$792$		0			0
$793$	6.89898	+	2.09859i	0.244990	+	0.0745231i
$794$	−7.23907			−0.256905
$795$		0			0
$796$	20.0000			0.708881
$797$	−14.6901			−0.520351			−0.260176	−	0.965561i	$-0.583780\pi$
$797$	−14.6901			−0.520351			−0.260176	+	0.965561i	$0.583780\pi$
$798$		0			0
$799$		−	25.1831i		−	0.890914i
$800$		−	1.00000i		−	0.0353553i
$801$		0			0
$802$	−20.6969			−0.730834
$803$	68.6175			2.42146
$804$		0			0
$805$	−21.7980			−0.768277
$806$	7.23907	+	2.20204i	0.254985	+	0.0775636i
$807$		0			0
$808$		−	10.9646i		−	0.385733i
$809$	10.2810			0.361459			0.180730	−	0.983533i	$-0.442154\pi$
$809$	10.2810			0.361459			0.180730	+	0.983533i	$0.442154\pi$
$810$		0			0
$811$			6.76742i			0.237636i	0.992916	+	0.118818i	$0.0379105\pi$
$811$			6.76742i			0.237636i	−0.992916	+	0.118818i	$0.962089\pi$
$812$			31.5959i			1.10880i
$813$		0			0
$814$		−	56.0259i		−	1.96371i
$815$	2.09859			0.0735104
$816$		0			0
$817$		−	27.0697i		−	0.947048i
$818$	−13.5348			−0.473235
$819$		0			0
$820$	10.8990			0.380609
$821$		−	30.0000i		−	1.04701i	−0.852023	−	0.523504i	$-0.824625\pi$
$821$		−	30.0000i		−	1.04701i	0.852023	−	0.523504i	$-0.175375\pi$
$822$		0			0
$823$	−30.6969			−1.07003			−0.535014	−	0.844843i	$-0.679694\pi$
$823$	−30.6969			−1.07003			−0.535014	+	0.844843i	$0.679694\pi$
$824$		−	3.10102i		−	0.108029i
$825$		0			0
$826$		−	22.8725i		−	0.795836i
$827$			43.5959i			1.51598i	0.652267	+	0.757989i	$0.273818\pi$
$827$			43.5959i			1.51598i	−0.652267	+	0.757989i	$0.726182\pi$
$828$		0			0
$829$	0.202041			0.00701717			0.00350859	−	0.999994i	$-0.498883\pi$
$829$	0.202041			0.00701717			0.00350859	+	0.999994i	$0.498883\pi$
$830$			9.79796i			0.340092i
$831$		0			0
$832$	3.44949	+	1.04930i	0.119590	+	0.0363778i
$833$	38.0343			1.31781
$834$		0			0
$835$		0			0
$836$	33.1534			1.14664
$837$		0			0
$838$			17.0484i			0.588926i
$839$		−	16.4041i		−	0.566332i	−0.959071	−	0.283166i	$-0.908615\pi$
$839$		−	16.4041i		−	0.566332i	0.959071	−	0.283166i	$-0.0913847\pi$
$840$		0			0
$841$	16.7980			0.579240
$842$	−14.0065			−0.482695
$843$		0			0
$844$	−13.7980			−0.474945
$845$	−7.23907	+	10.7980i	−0.249031	+	0.371461i
$846$		0			0
$847$			60.6948i			2.08550i
$848$	−2.09859			−0.0720659
$849$		0			0
$850$			2.57024i			0.0881584i
$851$		−	53.3939i		−	1.83032i
$852$		0			0
$853$		−	42.7031i		−	1.46213i	−0.682310	−	0.731063i	$-0.739024\pi$
$853$		−	42.7031i		−	1.46213i	0.682310	−	0.731063i	$-0.260976\pi$
$854$	−9.33766			−0.319528
$855$		0			0
$856$			18.6753i			0.638309i
$857$	−16.1051			−0.550139			−0.275069	−	0.961424i	$-0.588701\pi$
$857$	−16.1051			−0.550139			−0.275069	+	0.961424i	$0.588701\pi$
$858$		0			0
$859$	33.3939			1.13938			0.569692	−	0.821858i	$-0.307062\pi$
$859$	33.3939			1.13938			0.569692	+	0.821858i	$0.307062\pi$
$860$			4.00000i			0.136399i
$861$		0			0
$862$	31.5959			1.07616
$863$			43.5959i			1.48402i	0.670388	+	0.742011i	$0.266128\pi$
$863$			43.5959i			1.48402i	−0.670388	+	0.742011i	$0.733872\pi$
$864$		0			0
$865$			16.5767i			0.563625i
$866$		−	14.0000i		−	0.475739i
$867$		0			0
$868$	−9.79796			−0.332564
$869$		−	28.4041i		−	0.963542i
$870$		0			0
$871$	−7.59592	+	24.9711i	−0.257378	+	0.846113i
$872$	−9.80930			−0.332185
$873$		0			0
$874$	31.5959			1.06875
$875$	−4.66883			−0.157835
$876$		0			0
$877$			20.7739i			0.701485i	0.936472	+	0.350742i	$0.114071\pi$
$877$			20.7739i			0.701485i	−0.936472	+	0.350742i	$0.885929\pi$
$878$			27.5959i			0.931317i
$879$		0			0
$880$	−4.89898			−0.165145
$881$	−9.33766			−0.314594			−0.157297	−	0.987551i	$-0.550278\pi$
$881$	−9.33766			−0.314594			−0.157297	+	0.987551i	$0.550278\pi$
$882$		0			0
$883$	45.3939			1.52763			0.763813	−	0.645438i	$-0.223325\pi$
$883$	45.3939			1.52763			0.763813	+	0.645438i	$0.223325\pi$
$884$	−8.86601	−	2.69694i	−0.298196	−	0.0907079i
$885$		0			0
$886$		−	40.6045i		−	1.36414i
$887$	−14.0065			−0.470292			−0.235146	−	0.971960i	$-0.575557\pi$
$887$	−14.0065			−0.470292			−0.235146	+	0.971960i	$0.575557\pi$
$888$		0			0
$889$			14.4781i			0.485581i
$890$			1.10102i			0.0369063i
$891$		0			0
$892$		−	9.80930i		−	0.328440i
$893$	−66.3069			−2.21888
$894$		0			0
$895$		−	1.62694i		−	0.0543827i
$896$	−4.66883			−0.155975
$897$		0			0
$898$	1.10102			0.0367415
$899$			14.2020i			0.473665i
$900$		0			0
$901$	5.39388			0.179696
$902$		−	53.3939i		−	1.77782i
$903$		0			0
$904$			6.76742i			0.225081i
$905$			4.20204i			0.139681i
$906$		0			0
$907$	−0.404082			−0.0134173			−0.00670866	−	0.999977i	$-0.502135\pi$
$907$	−0.404082			−0.0134173			−0.00670866	+	0.999977i	$0.502135\pi$
$908$			21.7980i			0.723391i
$909$		0			0
$910$	4.89898	−	16.1051i	0.162400	−	0.533878i
$911$	3.25389			0.107806			0.0539030	−	0.998546i	$-0.482834\pi$
$911$	3.25389			0.107806			0.0539030	+	0.998546i	$0.482834\pi$
$912$		0			0
$913$	48.0000			1.58857
$914$	8.86601			0.293262
$915$		0			0
$916$			4.66883i			0.154262i
$917$			55.5959i			1.83594i
$918$		0			0
$919$	1.79796			0.0593092			0.0296546	−	0.999560i	$-0.490559\pi$
$919$	1.79796			0.0593092			0.0296546	+	0.999560i	$0.490559\pi$
$920$	−4.66883			−0.153927
$921$		0			0
$922$	3.79796			0.125079
$923$		0			0
$924$		0			0
$925$		−	11.4362i		−	0.376021i
$926$	−8.86601			−0.291355
$927$		0			0
$928$			6.76742i			0.222151i
$929$		−	20.6969i		−	0.679045i	−0.940598	−	0.339522i	$-0.889735\pi$
$929$		−	20.6969i		−	0.679045i	0.940598	−	0.339522i	$-0.110265\pi$
$930$		0			0
$931$		−	100.144i		−	3.28209i
$932$	6.76742			0.221674
$933$		0			0
$934$		−	4.19718i		−	0.137336i
$935$	12.5915			0.411787
$936$		0			0
$937$	39.3939			1.28694			0.643471	−	0.765471i	$-0.277494\pi$
$937$	39.3939			1.28694			0.643471	+	0.765471i	$0.277494\pi$
$938$		−	33.7980i		−	1.10354i
$939$		0			0
$940$	9.79796			0.319574
$941$		−	6.00000i		−	0.195594i	−0.995206	−	0.0977972i	$-0.968820\pi$
$941$		−	6.00000i		−	0.195594i	0.995206	−	0.0977972i	$-0.0311797\pi$
$942$		0			0
$943$		−	50.8855i		−	1.65706i
$944$		−	4.89898i		−	0.159448i
$945$		0			0
$946$	19.5959			0.637118
$947$			19.5959i			0.636782i	0.947960	+	0.318391i	$0.103142\pi$
$947$			19.5959i			0.636782i	−0.947960	+	0.318391i	$0.896858\pi$
$948$		0			0
$949$	−14.6969	+	48.3152i	−0.477083	+	1.56838i
$950$	6.76742			0.219564
$951$		0			0
$952$	12.0000			0.388922
$953$	43.1748			1.39857			0.699284	−	0.714844i	$-0.253502\pi$
$953$	43.1748			1.39857			0.699284	+	0.714844i	$0.253502\pi$
$954$		0			0
$955$			27.0697i			0.875954i
$956$		−	21.7980i		−	0.704996i
$957$		0			0
$958$	21.7980			0.704260
$959$	73.7580			2.38177
$960$		0			0
$961$	26.5959			0.857933
$962$	39.4492	+	12.0000i	1.27189	+	0.386896i
$963$		0			0
$964$			18.6753i			0.601491i
$965$	−9.80930			−0.315773
$966$		0			0
$967$		−	24.2874i		−	0.781031i	−0.920596	−	0.390516i	$-0.872297\pi$
$967$		−	24.2874i		−	0.781031i	0.920596	−	0.390516i	$-0.127703\pi$
$968$			13.0000i			0.417836i
$969$		0			0
$970$		−	9.80930i		−	0.314958i
$971$	−38.0343			−1.22058			−0.610289	−	0.792179i	$-0.708947\pi$
$971$	−38.0343			−1.22058			−0.610289	+	0.792179i	$0.708947\pi$
$972$		0			0
$973$		−	27.0697i		−	0.867814i
$974$	32.6818			1.04719
$975$		0			0
$976$	−2.00000			−0.0640184
$977$		−	23.3939i		−	0.748436i	−0.927341	−	0.374218i	$-0.877911\pi$
$977$		−	23.3939i		−	0.748436i	0.927341	−	0.374218i	$-0.122089\pi$
$978$		0			0
$979$	5.39388			0.172389
$980$			14.7980i			0.472703i
$981$		0			0
$982$		−	10.9646i		−	0.349894i
$983$			43.5959i			1.39049i	0.718771	+	0.695247i	$0.244705\pi$
$983$			43.5959i			1.39049i	−0.718771	+	0.695247i	$0.755295\pi$
$984$		0			0
$985$	−18.0000			−0.573528
$986$		−	17.3939i		−	0.553934i
$987$		0			0
$988$	−7.10102	+	23.3441i	−0.225914	+	0.742676i
$989$	18.6753			0.593840
$990$		0			0
$991$	20.0000			0.635321			0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000			0.635321			0.317660	+	0.948205i	$0.397103\pi$
$992$	−2.09859			−0.0666303
$993$		0			0
$994$		0			0
$995$		−	20.0000i		−	0.634043i
$996$		0			0
$997$	−2.89898			−0.0918116			−0.0459058	−	0.998946i	$-0.514617\pi$
$997$	−2.89898			−0.0918116			−0.0459058	+	0.998946i	$0.514617\pi$
$998$	6.76742			0.214219
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.b.g.181.5	yes	8
3.2	odd	2	inner	1170.2.b.g.181.1	✓	8
13.12	even	2	inner	1170.2.b.g.181.4	yes	8
39.38	odd	2	inner	1170.2.b.g.181.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1170.2.b.g.181.1	✓	8	3.2	odd	2	inner
1170.2.b.g.181.4	yes	8	13.12	even	2	inner
1170.2.b.g.181.5	yes	8	1.1	even	1	trivial
1170.2.b.g.181.8	yes	8	39.38	odd	2	inner

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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -4.66883i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{5} -4.66883i q^{7} -1.00000i q^{8} -1.00000 q^{10} +4.89898i q^{11} +(-3.44949 - 1.04930i) q^{13} +4.66883 q^{14} +1.00000 q^{16} -2.57024 q^{17} +6.76742i q^{19} -1.00000i q^{20} -4.89898 q^{22} -4.66883 q^{23} -1.00000 q^{25} +(1.04930 - 3.44949i) q^{26} +4.66883i q^{28} +6.76742 q^{29} +2.09859i q^{31} +1.00000i q^{32} -2.57024i q^{34} +4.66883 q^{35} +11.4362i q^{37} -6.76742 q^{38} +1.00000 q^{40} +10.8990i q^{41} -4.00000 q^{43} -4.89898i q^{44} -4.66883i q^{46} +9.79796i q^{47} -14.7980 q^{49} -1.00000i q^{50} +(3.44949 + 1.04930i) q^{52} -2.09859 q^{53} -4.89898 q^{55} -4.66883 q^{56} +6.76742i q^{58} -4.89898i q^{59} -2.00000 q^{61} -2.09859 q^{62} -1.00000 q^{64} +(1.04930 - 3.44949i) q^{65} -7.23907i q^{67} +2.57024 q^{68} +4.66883i q^{70} -14.0065i q^{73} -11.4362 q^{74} -6.76742i q^{76} +22.8725 q^{77} -5.79796 q^{79} +1.00000i q^{80} -10.8990 q^{82} -9.79796i q^{83} -2.57024i q^{85} -4.00000i q^{86} +4.89898 q^{88} -1.10102i q^{89} +(-4.89898 + 16.1051i) q^{91} +4.66883 q^{92} -9.79796 q^{94} -6.76742 q^{95} +9.80930i q^{97} -14.7980i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} - 8 q^{10} - 8 q^{13} + 8 q^{16} - 8 q^{25} + 8 q^{40} - 32 q^{43} - 40 q^{49} + 8 q^{52} - 16 q^{61} - 8 q^{64} + 32 q^{79} - 48 q^{82}+O(q^{100}) \) 8 * q - 8 * q^4 - 8 * q^10 - 8 * q^13 + 8 * q^16 - 8 * q^25 + 8 * q^40 - 32 * q^43 - 40 * q^49 + 8 * q^52 - 16 * q^61 - 8 * q^64 + 32 * q^79 - 48 * q^82

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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