LMFDB - Embedded newform 370.2.d.a.221.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(221,370)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(370, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("370.221");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.d (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:	$2.95446487479$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			221.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	370.221
Dual form			370.2.d.a.221.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000 q^{7} +1.00000i q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000 q^{7} +1.00000i q^{8} -3.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -2.00000i q^{13} +2.00000i q^{14} +1.00000 q^{16} -6.00000i q^{17} +3.00000i q^{18} -2.00000i q^{19} +1.00000i q^{20} +4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} +2.00000i q^{29} -8.00000i q^{31} -1.00000i q^{32} -6.00000 q^{34} +2.00000i q^{35} +3.00000 q^{36} +(-6.00000 - 1.00000i) q^{37} -2.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +3.00000i q^{45} +8.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +1.00000i q^{50} +2.00000i q^{52} +12.0000 q^{53} +4.00000i q^{55} -2.00000i q^{56} +2.00000 q^{58} -2.00000i q^{59} -14.0000i q^{61} -8.00000 q^{62} +6.00000 q^{63} -1.00000 q^{64} -2.00000 q^{65} +16.0000 q^{67} +6.00000i q^{68} +2.00000 q^{70} -12.0000 q^{71} -3.00000i q^{72} +6.00000 q^{73} +(-1.00000 + 6.00000i) q^{74} +2.00000i q^{76} +8.00000 q^{77} -1.00000i q^{80} +9.00000 q^{81} +2.00000i q^{82} -12.0000 q^{83} -6.00000 q^{85} +4.00000 q^{86} -4.00000i q^{88} -4.00000i q^{89} +3.00000 q^{90} +4.00000i q^{91} -8.00000i q^{92} -6.00000i q^{94} -2.00000 q^{95} -10.0000i q^{97} +3.00000i q^{98} +12.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{7} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 4 * q^7 - 6 * q^9	$ 2 q - 2 q^{4} - 4 q^{7} - 6 q^{9} - 2 q^{10} - 8 q^{11} + 2 q^{16} - 2 q^{25} - 4 q^{26} + 4 q^{28} - 12 q^{34} + 6 q^{36} - 12 q^{37} - 4 q^{38} + 2 q^{40} - 4 q^{41} + 8 q^{44} + 16 q^{46} + 12 q^{47} - 6 q^{49} + 24 q^{53} + 4 q^{58} - 16 q^{62} + 12 q^{63} - 2 q^{64} - 4 q^{65} + 32 q^{67} + 4 q^{70} - 24 q^{71} + 12 q^{73} - 2 q^{74} + 16 q^{77} + 18 q^{81} - 24 q^{83} - 12 q^{85} + 8 q^{86} + 6 q^{90} - 4 q^{95} + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^7 - 6 * q^9 - 2 * q^10 - 8 * q^11 + 2 * q^16 - 2 * q^25 - 4 * q^26 + 4 * q^28 - 12 * q^34 + 6 * q^36 - 12 * q^37 - 4 * q^38 + 2 * q^40 - 4 * q^41 + 8 * q^44 + 16 * q^46 + 12 * q^47 - 6 * q^49 + 24 * q^53 + 4 * q^58 - 16 * q^62 + 12 * q^63 - 2 * q^64 - 4 * q^65 + 32 * q^67 + 4 * q^70 - 24 * q^71 + 12 * q^73 - 2 * q^74 + 16 * q^77 + 18 * q^81 - 24 * q^83 - 12 * q^85 + 8 * q^86 + 6 * q^90 - 4 * q^95 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/370\mathbb{Z}\right)^\times$.

$n$	$261$	$297$
$\chi(n)$	$-1$	$1$

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			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	−1.00000			−0.500000
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$	−2.00000			−0.755929			−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000			−0.755929			−0.377964	+	0.925820i	$0.623376\pi$
$8$			1.00000i			0.353553i
$9$	−3.00000			−1.00000
$10$	−1.00000			−0.316228
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$			2.00000i			0.534522i
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.00000i		−	1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$		−	6.00000i		−	1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$			3.00000i			0.707107i
$19$		−	2.00000i		−	0.458831i	−0.973329	−	0.229416i	$-0.926318\pi$
$19$		−	2.00000i		−	0.458831i	0.973329	−	0.229416i	$-0.0736815\pi$
$20$			1.00000i			0.223607i
$21$		0			0
$22$			4.00000i			0.852803i
$23$			8.00000i			1.66812i	0.551677	+	0.834058i	$0.313988\pi$
$23$			8.00000i			1.66812i	−0.551677	+	0.834058i	$0.686012\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	−2.00000			−0.392232
$27$		0			0
$28$	2.00000			0.377964
$29$			2.00000i			0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$			2.00000i			0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$		0			0
$31$		−	8.00000i		−	1.43684i	−0.695608	−	0.718421i	$-0.744865\pi$
$31$		−	8.00000i		−	1.43684i	0.695608	−	0.718421i	$-0.255135\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−6.00000			−1.02899
$35$			2.00000i			0.338062i
$36$	3.00000			0.500000
$37$	−6.00000	−	1.00000i	−0.986394	−	0.164399i
$37$	−6.00000	−	1.00000i	−0.986394	−	0.164399i
$38$	−2.00000			−0.324443
$39$		0			0
$40$	1.00000			0.158114
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	4.00000			0.603023
$45$			3.00000i			0.447214i
$46$	8.00000			1.17954
$47$	6.00000			0.875190			0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000			0.875190			0.437595	+	0.899172i	$0.355830\pi$
$48$		0			0
$49$	−3.00000			−0.428571
$50$			1.00000i			0.141421i
$51$		0			0
$52$			2.00000i			0.277350i
$53$	12.0000			1.64833			0.824163	−	0.566352i	$-0.191646\pi$
$53$	12.0000			1.64833			0.824163	+	0.566352i	$0.191646\pi$
$54$		0			0
$55$			4.00000i			0.539360i
$56$		−	2.00000i		−	0.267261i
$57$		0			0
$58$	2.00000			0.262613
$59$		−	2.00000i		−	0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$		−	2.00000i		−	0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$		0			0
$61$		−	14.0000i		−	1.79252i	−0.443533	−	0.896258i	$-0.646275\pi$
$61$		−	14.0000i		−	1.79252i	0.443533	−	0.896258i	$-0.353725\pi$
$62$	−8.00000			−1.01600
$63$	6.00000			0.755929
$64$	−1.00000			−0.125000
$65$	−2.00000			−0.248069
$66$		0			0
$67$	16.0000			1.95471			0.977356	−	0.211604i	$-0.0678686\pi$
$67$	16.0000			1.95471			0.977356	+	0.211604i	$0.0678686\pi$
$68$			6.00000i			0.727607i
$69$		0			0
$70$	2.00000			0.239046
$71$	−12.0000			−1.42414			−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		−	3.00000i		−	0.353553i
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$	−1.00000	+	6.00000i	−0.116248	+	0.697486i
$75$		0			0
$76$			2.00000i			0.229416i
$77$	8.00000			0.911685
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		−	1.00000i		−	0.111803i
$81$	9.00000			1.00000
$82$			2.00000i			0.220863i
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$	−6.00000			−0.650791
$86$	4.00000			0.431331
$87$		0			0
$88$		−	4.00000i		−	0.426401i
$89$		−	4.00000i		−	0.423999i	−0.977270	−	0.212000i	$-0.932002\pi$
$89$		−	4.00000i		−	0.423999i	0.977270	−	0.212000i	$-0.0679975\pi$
$90$	3.00000			0.316228
$91$			4.00000i			0.419314i
$92$		−	8.00000i		−	0.834058i
$93$		0			0
$94$		−	6.00000i		−	0.618853i
$95$	−2.00000			−0.205196
$96$		0			0
$97$		−	10.0000i		−	1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$		−	10.0000i		−	1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$			3.00000i			0.303046i
$99$	12.0000			1.20605
$100$	1.00000			0.100000
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	2.00000			0.196116
$105$		0			0
$106$		−	12.0000i		−	1.16554i
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		−	14.0000i		−	1.34096i	−0.741929	−	0.670478i	$-0.766089\pi$
$109$		−	14.0000i		−	1.34096i	0.741929	−	0.670478i	$-0.233911\pi$
$110$	4.00000			0.381385
$111$		0			0
$112$	−2.00000			−0.188982
$113$			6.00000i			0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$			6.00000i			0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		−	2.00000i		−	0.185695i
$117$			6.00000i			0.554700i
$118$	−2.00000			−0.184115
$119$			12.0000i			1.10004i
$120$		0			0
$121$	5.00000			0.454545
$122$	−14.0000			−1.26750
$123$		0			0
$124$			8.00000i			0.718421i
$125$			1.00000i			0.0894427i
$126$		−	6.00000i		−	0.534522i
$127$	−2.00000			−0.177471			−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000			−0.177471			−0.0887357	+	0.996055i	$0.528283\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$			2.00000i			0.175412i
$131$			6.00000i			0.524222i	0.965038	+	0.262111i	$0.0844187\pi$
$131$			6.00000i			0.524222i	−0.965038	+	0.262111i	$0.915581\pi$
$132$		0			0
$133$			4.00000i			0.346844i
$134$		−	16.0000i		−	1.38219i
$135$		0			0
$136$	6.00000			0.514496
$137$	−14.0000			−1.19610			−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000			−1.19610			−0.598050	+	0.801459i	$0.704058\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		−	2.00000i		−	0.169031i
$141$		0			0
$142$			12.0000i			1.00702i
$143$			8.00000i			0.668994i
$144$	−3.00000			−0.250000
$145$	2.00000			0.166091
$146$		−	6.00000i		−	0.496564i
$147$		0			0
$148$	6.00000	+	1.00000i	0.493197	+	0.0821995i
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$	2.00000			0.162221
$153$			18.0000i			1.45521i
$154$		−	8.00000i		−	0.644658i
$155$	−8.00000			−0.642575
$156$		0			0
$157$	8.00000			0.638470			0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000			0.638470			0.319235	+	0.947676i	$0.396574\pi$
$158$		0			0
$159$		0			0
$160$	−1.00000			−0.0790569
$161$		−	16.0000i		−	1.26098i
$162$		−	9.00000i		−	0.707107i
$163$			16.0000i			1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$			16.0000i			1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	2.00000			0.156174
$165$		0			0
$166$			12.0000i			0.931381i
$167$		−	8.00000i		−	0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$		−	8.00000i		−	0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$			6.00000i			0.460179i
$171$			6.00000i			0.458831i
$172$		−	4.00000i		−	0.304997i
$173$	−12.0000			−0.912343			−0.456172	−	0.889892i	$-0.650780\pi$
$173$	−12.0000			−0.912343			−0.456172	+	0.889892i	$0.650780\pi$
$174$		0			0
$175$	2.00000			0.151186
$176$	−4.00000			−0.301511
$177$		0			0
$178$	−4.00000			−0.299813
$179$			18.0000i			1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$			18.0000i			1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$		−	3.00000i		−	0.223607i
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$	4.00000			0.296500
$183$		0			0
$184$	−8.00000			−0.589768
$185$	−1.00000	+	6.00000i	−0.0735215	+	0.441129i
$186$		0			0
$187$			24.0000i			1.75505i
$188$	−6.00000			−0.437595
$189$		0			0
$190$			2.00000i			0.145095i
$191$			8.00000i			0.578860i	0.957199	+	0.289430i	$0.0934657\pi$
$191$			8.00000i			0.578860i	−0.957199	+	0.289430i	$0.906534\pi$
$192$		0			0
$193$		−	14.0000i		−	1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$		−	14.0000i		−	1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	−10.0000			−0.717958
$195$		0			0
$196$	3.00000			0.214286
$197$	8.00000			0.569976			0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000			0.569976			0.284988	+	0.958531i	$0.408010\pi$
$198$		−	12.0000i		−	0.852803i
$199$		−	8.00000i		−	0.567105i	−0.958957	−	0.283552i	$-0.908487\pi$
$199$		−	8.00000i		−	0.567105i	0.958957	−	0.283552i	$-0.0915130\pi$
$200$		−	1.00000i		−	0.0707107i
$201$		0			0
$202$			6.00000i			0.422159i
$203$		−	4.00000i		−	0.280745i
$204$		0			0
$205$			2.00000i			0.139686i
$206$		0			0
$207$		−	24.0000i		−	1.66812i
$208$		−	2.00000i		−	0.138675i
$209$			8.00000i			0.553372i
$210$		0			0
$211$	−20.0000			−1.37686			−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000			−1.37686			−0.688428	+	0.725304i	$0.741699\pi$
$212$	−12.0000			−0.824163
$213$		0			0
$214$		0			0
$215$	4.00000			0.272798
$216$		0			0
$217$			16.0000i			1.08615i
$218$	−14.0000			−0.948200
$219$		0			0
$220$		−	4.00000i		−	0.269680i
$221$	−12.0000			−0.807207
$222$		0			0
$223$	−26.0000			−1.74109			−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000			−1.74109			−0.870544	+	0.492090i	$0.836233\pi$
$224$			2.00000i			0.133631i
$225$	3.00000			0.200000
$226$	6.00000			0.399114
$227$			12.0000i			0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$			12.0000i			0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$		0			0
$229$	26.0000			1.71813			0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000			1.71813			0.859064	+	0.511868i	$0.171046\pi$
$230$		−	8.00000i		−	0.527504i
$231$		0			0
$232$	−2.00000			−0.131306
$233$	−10.0000			−0.655122			−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000			−0.655122			−0.327561	+	0.944830i	$0.606227\pi$
$234$	6.00000			0.392232
$235$		−	6.00000i		−	0.391397i
$236$			2.00000i			0.130189i
$237$		0			0
$238$	12.0000			0.777844
$239$		−	24.0000i		−	1.55243i	−0.630468	−	0.776215i	$-0.717137\pi$
$239$		−	24.0000i		−	1.55243i	0.630468	−	0.776215i	$-0.282863\pi$
$240$		0			0
$241$			8.00000i			0.515325i	0.966235	+	0.257663i	$0.0829523\pi$
$241$			8.00000i			0.515325i	−0.966235	+	0.257663i	$0.917048\pi$
$242$		−	5.00000i		−	0.321412i
$243$		0			0
$244$			14.0000i			0.896258i
$245$			3.00000i			0.191663i
$246$		0			0
$247$	−4.00000			−0.254514
$248$	8.00000			0.508001
$249$		0			0
$250$	1.00000			0.0632456
$251$			18.0000i			1.13615i	0.822977	+	0.568075i	$0.192312\pi$
$251$			18.0000i			1.13615i	−0.822977	+	0.568075i	$0.807688\pi$
$252$	−6.00000			−0.377964
$253$		−	32.0000i		−	2.01182i
$254$			2.00000i			0.125491i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	14.0000i		−	0.873296i	−0.899632	−	0.436648i	$-0.856166\pi$
$257$		−	14.0000i		−	0.873296i	0.899632	−	0.436648i	$-0.143834\pi$
$258$		0			0
$259$	12.0000	+	2.00000i	0.745644	+	0.124274i
$260$	2.00000			0.124035
$261$		−	6.00000i		−	0.371391i
$262$	6.00000			0.370681
$263$	10.0000			0.616626			0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000			0.616626			0.308313	+	0.951285i	$0.400236\pi$
$264$		0			0
$265$		−	12.0000i		−	0.737154i
$266$	4.00000			0.245256
$267$		0			0
$268$	−16.0000			−0.977356
$269$	6.00000			0.365826			0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000			0.365826			0.182913	+	0.983129i	$0.441447\pi$
$270$		0			0
$271$	12.0000			0.728948			0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000			0.728948			0.364474	+	0.931214i	$0.381249\pi$
$272$		−	6.00000i		−	0.363803i
$273$		0			0
$274$			14.0000i			0.845771i
$275$	4.00000			0.241209
$276$		0			0
$277$		−	10.0000i		−	0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$		−	10.0000i		−	0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$		−	12.0000i		−	0.719712i
$279$			24.0000i			1.43684i
$280$	−2.00000			−0.119523
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$		−	16.0000i		−	0.951101i	−0.879688	−	0.475551i	$-0.842249\pi$
$283$		−	16.0000i		−	0.951101i	0.879688	−	0.475551i	$-0.157751\pi$
$284$	12.0000			0.712069
$285$		0			0
$286$	8.00000			0.473050
$287$	4.00000			0.236113
$288$			3.00000i			0.176777i
$289$	−19.0000			−1.11765
$290$		−	2.00000i		−	0.117444i
$291$		0			0
$292$	−6.00000			−0.351123
$293$	24.0000			1.40209			0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000			1.40209			0.701047	+	0.713115i	$0.252716\pi$
$294$		0			0
$295$	−2.00000			−0.116445
$296$	1.00000	−	6.00000i	0.0581238	−	0.348743i
$297$		0			0
$298$			18.0000i			1.04271i
$299$	16.0000			0.925304
$300$		0			0
$301$		−	8.00000i		−	0.461112i
$302$			4.00000i			0.230174i
$303$		0			0
$304$		−	2.00000i		−	0.114708i
$305$	−14.0000			−0.801638
$306$	18.0000			1.02899
$307$	−4.00000			−0.228292			−0.114146	−	0.993464i	$-0.536413\pi$
$307$	−4.00000			−0.228292			−0.114146	+	0.993464i	$0.536413\pi$
$308$	−8.00000			−0.455842
$309$		0			0
$310$			8.00000i			0.454369i
$311$		−	28.0000i		−	1.58773i	−0.608091	−	0.793867i	$-0.708065\pi$
$311$		−	28.0000i		−	1.58773i	0.608091	−	0.793867i	$-0.291935\pi$
$312$		0			0
$313$		−	22.0000i		−	1.24351i	−0.783210	−	0.621757i	$-0.786419\pi$
$313$		−	22.0000i		−	1.24351i	0.783210	−	0.621757i	$-0.213581\pi$
$314$		−	8.00000i		−	0.451466i
$315$		−	6.00000i		−	0.338062i
$316$		0			0
$317$	12.0000			0.673987			0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000			0.673987			0.336994	+	0.941507i	$0.390590\pi$
$318$		0			0
$319$		−	8.00000i		−	0.447914i
$320$			1.00000i			0.0559017i
$321$		0			0
$322$	−16.0000			−0.891645
$323$	−12.0000			−0.667698
$324$	−9.00000			−0.500000
$325$			2.00000i			0.110940i
$326$	16.0000			0.886158
$327$		0			0
$328$		−	2.00000i		−	0.110432i
$329$	−12.0000			−0.661581
$330$		0			0
$331$			18.0000i			0.989369i	0.869072	+	0.494685i	$0.164716\pi$
$331$			18.0000i			0.989369i	−0.869072	+	0.494685i	$0.835284\pi$
$332$	12.0000			0.658586
$333$	18.0000	+	3.00000i	0.986394	+	0.164399i
$334$	−8.00000			−0.437741
$335$		−	16.0000i		−	0.874173i
$336$		0			0
$337$	22.0000			1.19842			0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000			1.19842			0.599208	+	0.800593i	$0.295482\pi$
$338$		−	9.00000i		−	0.489535i
$339$		0			0
$340$	6.00000			0.325396
$341$			32.0000i			1.73290i
$342$	6.00000			0.324443
$343$	20.0000			1.07990
$344$	−4.00000			−0.215666
$345$		0			0
$346$			12.0000i			0.645124i
$347$			4.00000i			0.214731i	0.994220	+	0.107366i	$0.0342415\pi$
$347$			4.00000i			0.214731i	−0.994220	+	0.107366i	$0.965758\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		−	2.00000i		−	0.106904i
$351$		0			0
$352$			4.00000i			0.213201i
$353$			2.00000i			0.106449i	0.998583	+	0.0532246i	$0.0169499\pi$
$353$			2.00000i			0.106449i	−0.998583	+	0.0532246i	$0.983050\pi$
$354$		0			0
$355$			12.0000i			0.636894i
$356$			4.00000i			0.212000i
$357$		0			0
$358$	18.0000			0.951330
$359$	−16.0000			−0.844448			−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000			−0.844448			−0.422224	+	0.906492i	$0.638750\pi$
$360$	−3.00000			−0.158114
$361$	15.0000			0.789474
$362$			22.0000i			1.15629i
$363$		0			0
$364$		−	4.00000i		−	0.209657i
$365$		−	6.00000i		−	0.314054i
$366$		0			0
$367$	−26.0000			−1.35719			−0.678594	−	0.734513i	$-0.737411\pi$
$367$	−26.0000			−1.35719			−0.678594	+	0.734513i	$0.737411\pi$
$368$			8.00000i			0.417029i
$369$	6.00000			0.312348
$370$	6.00000	+	1.00000i	0.311925	+	0.0519875i
$371$	−24.0000			−1.24602
$372$		0			0
$373$	8.00000			0.414224			0.207112	−	0.978317i	$-0.433593\pi$
$373$	8.00000			0.414224			0.207112	+	0.978317i	$0.433593\pi$
$374$	24.0000			1.24101
$375$		0			0
$376$			6.00000i			0.309426i
$377$	4.00000			0.206010
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$	2.00000			0.102598
$381$		0			0
$382$	8.00000			0.409316
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		0			0
$385$		−	8.00000i		−	0.407718i
$386$	−14.0000			−0.712581
$387$		−	12.0000i		−	0.609994i
$388$			10.0000i			0.507673i
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$	48.0000			2.42746
$392$		−	3.00000i		−	0.151523i
$393$		0			0
$394$		−	8.00000i		−	0.403034i
$395$		0			0
$396$	−12.0000			−0.603023
$397$	4.00000			0.200754			0.100377	−	0.994949i	$-0.467995\pi$
$397$	4.00000			0.200754			0.100377	+	0.994949i	$0.467995\pi$
$398$	−8.00000			−0.401004
$399$		0			0
$400$	−1.00000			−0.0500000
$401$		−	36.0000i		−	1.79775i	−0.438201	−	0.898877i	$-0.644384\pi$
$401$		−	36.0000i		−	1.79775i	0.438201	−	0.898877i	$-0.355616\pi$
$402$		0			0
$403$	−16.0000			−0.797017
$404$	6.00000			0.298511
$405$		−	9.00000i		−	0.447214i
$406$	−4.00000			−0.198517
$407$	24.0000	+	4.00000i	1.18964	+	0.198273i
$408$		0			0
$409$		−	20.0000i		−	0.988936i	−0.869196	−	0.494468i	$-0.835363\pi$
$409$		−	20.0000i		−	0.988936i	0.869196	−	0.494468i	$-0.164637\pi$
$410$	2.00000			0.0987730
$411$		0			0
$412$		0			0
$413$			4.00000i			0.196827i
$414$	−24.0000			−1.17954
$415$			12.0000i			0.589057i
$416$	−2.00000			−0.0980581
$417$		0			0
$418$	8.00000			0.391293
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$			30.0000i			1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$			30.0000i			1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$			20.0000i			0.973585i
$423$	−18.0000			−0.875190
$424$			12.0000i			0.582772i
$425$			6.00000i			0.291043i
$426$		0			0
$427$			28.0000i			1.35501i
$428$		0			0
$429$		0			0
$430$		−	4.00000i		−	0.192897i
$431$		−	16.0000i		−	0.770693i	−0.922772	−	0.385346i	$-0.874082\pi$
$431$		−	16.0000i		−	0.770693i	0.922772	−	0.385346i	$-0.125918\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$	16.0000			0.768025
$435$		0			0
$436$			14.0000i			0.670478i
$437$	16.0000			0.765384
$438$		0			0
$439$		−	32.0000i		−	1.52728i	−0.645644	−	0.763638i	$-0.723411\pi$
$439$		−	32.0000i		−	1.52728i	0.645644	−	0.763638i	$-0.276589\pi$
$440$	−4.00000			−0.190693
$441$	9.00000			0.428571
$442$			12.0000i			0.570782i
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$	−4.00000			−0.189618
$446$			26.0000i			1.23114i
$447$		0			0
$448$	2.00000			0.0944911
$449$			20.0000i			0.943858i	0.881636	+	0.471929i	$0.156442\pi$
$449$			20.0000i			0.943858i	−0.881636	+	0.471929i	$0.843558\pi$
$450$		−	3.00000i		−	0.141421i
$451$	8.00000			0.376705
$452$		−	6.00000i		−	0.282216i
$453$		0			0
$454$	12.0000			0.563188
$455$	4.00000			0.187523
$456$		0			0
$457$			30.0000i			1.40334i	0.712502	+	0.701670i	$0.247562\pi$
$457$			30.0000i			1.40334i	−0.712502	+	0.701670i	$0.752438\pi$
$458$		−	26.0000i		−	1.21490i
$459$		0			0
$460$	−8.00000			−0.373002
$461$			38.0000i			1.76984i	0.465746	+	0.884918i	$0.345786\pi$
$461$			38.0000i			1.76984i	−0.465746	+	0.884918i	$0.654214\pi$
$462$		0			0
$463$		−	40.0000i		−	1.85896i	−0.368875	−	0.929479i	$-0.620257\pi$
$463$		−	40.0000i		−	1.85896i	0.368875	−	0.929479i	$-0.379743\pi$
$464$			2.00000i			0.0928477i
$465$		0			0
$466$			10.0000i			0.463241i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		−	6.00000i		−	0.277350i
$469$	−32.0000			−1.47762
$470$	−6.00000			−0.276759
$471$		0			0
$472$	2.00000			0.0920575
$473$		−	16.0000i		−	0.735681i
$474$		0			0
$475$			2.00000i			0.0917663i
$476$		−	12.0000i		−	0.550019i
$477$	−36.0000			−1.64833
$478$	−24.0000			−1.09773
$479$		−	8.00000i		−	0.365529i	−0.983157	−	0.182765i	$-0.941495\pi$
$479$		−	8.00000i		−	0.365529i	0.983157	−	0.182765i	$-0.0585046\pi$
$480$		0			0
$481$	−2.00000	+	12.0000i	−0.0911922	+	0.547153i
$482$	8.00000			0.364390
$483$		0			0
$484$	−5.00000			−0.227273
$485$	−10.0000			−0.454077
$486$		0			0
$487$			8.00000i			0.362515i	0.983436	+	0.181257i	$0.0580167\pi$
$487$			8.00000i			0.362515i	−0.983436	+	0.181257i	$0.941983\pi$
$488$	14.0000			0.633750
$489$		0			0
$490$	3.00000			0.135526
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$	12.0000			0.540453
$494$			4.00000i			0.179969i
$495$		−	12.0000i		−	0.539360i
$496$		−	8.00000i		−	0.359211i
$497$	24.0000			1.07655
$498$		0			0
$499$		−	10.0000i		−	0.447661i	−0.974628	−	0.223831i	$-0.928144\pi$
$499$		−	10.0000i		−	0.447661i	0.974628	−	0.223831i	$-0.0718563\pi$
$500$		−	1.00000i		−	0.0447214i
$501$		0			0
$502$	18.0000			0.803379
$503$		−	24.0000i		−	1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$		−	24.0000i		−	1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$			6.00000i			0.267261i
$505$			6.00000i			0.266996i
$506$	−32.0000			−1.42257
$507$		0			0
$508$	2.00000			0.0887357
$509$	−22.0000			−0.975133			−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000			−0.975133			−0.487566	+	0.873086i	$0.662115\pi$
$510$		0			0
$511$	−12.0000			−0.530849
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−14.0000			−0.617514
$515$		0			0
$516$		0			0
$517$	−24.0000			−1.05552
$518$	2.00000	−	12.0000i	0.0878750	−	0.527250i
$519$		0			0
$520$		−	2.00000i		−	0.0877058i
$521$	38.0000			1.66481			0.832405	−	0.554168i	$-0.186963\pi$
$521$	38.0000			1.66481			0.832405	+	0.554168i	$0.186963\pi$
$522$	−6.00000			−0.262613
$523$			8.00000i			0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$			8.00000i			0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$		−	6.00000i		−	0.262111i
$525$		0			0
$526$		−	10.0000i		−	0.436021i
$527$	−48.0000			−2.09091
$528$		0			0
$529$	−41.0000			−1.78261
$530$	−12.0000			−0.521247
$531$			6.00000i			0.260378i
$532$		−	4.00000i		−	0.173422i
$533$			4.00000i			0.173259i
$534$		0			0
$535$		0			0
$536$			16.0000i			0.691095i
$537$		0			0
$538$		−	6.00000i		−	0.258678i
$539$	12.0000			0.516877
$540$		0			0
$541$		−	26.0000i		−	1.11783i	−0.829226	−	0.558914i	$-0.811218\pi$
$541$		−	26.0000i		−	1.11783i	0.829226	−	0.558914i	$-0.188782\pi$
$542$		−	12.0000i		−	0.515444i
$543$		0			0
$544$	−6.00000			−0.257248
$545$	−14.0000			−0.599694
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$	14.0000			0.598050
$549$			42.0000i			1.79252i
$550$		−	4.00000i		−	0.170561i
$551$	4.00000			0.170406
$552$		0			0
$553$		0			0
$554$	−10.0000			−0.424859
$555$		0			0
$556$	−12.0000			−0.508913
$557$		−	14.0000i		−	0.593199i	−0.955002	−	0.296600i	$-0.904147\pi$
$557$		−	14.0000i		−	0.593199i	0.955002	−	0.296600i	$-0.0958526\pi$
$558$	24.0000			1.01600
$559$	8.00000			0.338364
$560$			2.00000i			0.0845154i
$561$		0			0
$562$	20.0000			0.843649
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	6.00000			0.252422
$566$	−16.0000			−0.672530
$567$	−18.0000			−0.755929
$568$		−	12.0000i		−	0.503509i
$569$			12.0000i			0.503066i	0.967849	+	0.251533i	$0.0809347\pi$
$569$			12.0000i			0.503066i	−0.967849	+	0.251533i	$0.919065\pi$
$570$		0			0
$571$	20.0000			0.836974			0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000			0.836974			0.418487	+	0.908223i	$0.362561\pi$
$572$		−	8.00000i		−	0.334497i
$573$		0			0
$574$		−	4.00000i		−	0.166957i
$575$		−	8.00000i		−	0.333623i
$576$	3.00000			0.125000
$577$			10.0000i			0.416305i	0.978096	+	0.208153i	$0.0667451\pi$
$577$			10.0000i			0.416305i	−0.978096	+	0.208153i	$0.933255\pi$
$578$			19.0000i			0.790296i
$579$		0			0
$580$	−2.00000			−0.0830455
$581$	24.0000			0.995688
$582$		0			0
$583$	−48.0000			−1.98796
$584$			6.00000i			0.248282i
$585$	6.00000			0.248069
$586$		−	24.0000i		−	0.991431i
$587$		−	28.0000i		−	1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$		−	28.0000i		−	1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$			2.00000i			0.0823387i
$591$		0			0
$592$	−6.00000	−	1.00000i	−0.246598	−	0.0410997i
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			0.369586	+	0.929197i	$0.379500\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$	18.0000			0.737309
$597$		0			0
$598$		−	16.0000i		−	0.654289i
$599$	−4.00000			−0.163436			−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000			−0.163436			−0.0817178	+	0.996656i	$0.526041\pi$
$600$		0			0
$601$	−22.0000			−0.897399			−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000			−0.897399			−0.448699	+	0.893683i	$0.648113\pi$
$602$	−8.00000			−0.326056
$603$	−48.0000			−1.95471
$604$	4.00000			0.162758
$605$		−	5.00000i		−	0.203279i
$606$		0			0
$607$			24.0000i			0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$			24.0000i			0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	−2.00000			−0.0811107
$609$		0			0
$610$			14.0000i			0.566843i
$611$		−	12.0000i		−	0.485468i
$612$		−	18.0000i		−	0.727607i
$613$	12.0000			0.484675			0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000			0.484675			0.242338	+	0.970192i	$0.422086\pi$
$614$			4.00000i			0.161427i
$615$		0			0
$616$			8.00000i			0.322329i
$617$	18.0000			0.724653			0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000			0.724653			0.362326	+	0.932051i	$0.381983\pi$
$618$		0			0
$619$	−4.00000			−0.160774			−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000			−0.160774			−0.0803868	+	0.996764i	$0.525616\pi$
$620$	8.00000			0.321288
$621$		0			0
$622$	−28.0000			−1.12270
$623$			8.00000i			0.320513i
$624$		0			0
$625$	1.00000			0.0400000
$626$	−22.0000			−0.879297
$627$		0			0
$628$	−8.00000			−0.319235
$629$	−6.00000	+	36.0000i	−0.239236	+	1.43541i
$630$	−6.00000			−0.239046
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$		0			0
$634$		−	12.0000i		−	0.476581i
$635$			2.00000i			0.0793676i
$636$		0			0
$637$			6.00000i			0.237729i
$638$	−8.00000			−0.316723
$639$	36.0000			1.42414
$640$	1.00000			0.0395285
$641$	−6.00000			−0.236986			−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000			−0.236986			−0.118493	+	0.992955i	$0.537806\pi$
$642$		0			0
$643$			12.0000i			0.473234i	0.971603	+	0.236617i	$0.0760386\pi$
$643$			12.0000i			0.473234i	−0.971603	+	0.236617i	$0.923961\pi$
$644$			16.0000i			0.630488i
$645$		0			0
$646$			12.0000i			0.472134i
$647$		−	32.0000i		−	1.25805i	−0.777385	−	0.629025i	$-0.783454\pi$
$647$		−	32.0000i		−	1.25805i	0.777385	−	0.629025i	$-0.216546\pi$
$648$			9.00000i			0.353553i
$649$			8.00000i			0.314027i
$650$	2.00000			0.0784465
$651$		0			0
$652$		−	16.0000i		−	0.626608i
$653$			46.0000i			1.80012i	0.435767	+	0.900060i	$0.356477\pi$
$653$			46.0000i			1.80012i	−0.435767	+	0.900060i	$0.643523\pi$
$654$		0			0
$655$	6.00000			0.234439
$656$	−2.00000			−0.0780869
$657$	−18.0000			−0.702247
$658$			12.0000i			0.467809i
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$			14.0000i			0.544537i	0.962221	+	0.272268i	$0.0877739\pi$
$661$			14.0000i			0.544537i	−0.962221	+	0.272268i	$0.912226\pi$
$662$	18.0000			0.699590
$663$		0			0
$664$		−	12.0000i		−	0.465690i
$665$	4.00000			0.155113
$666$	3.00000	−	18.0000i	0.116248	−	0.697486i
$667$	−16.0000			−0.619522
$668$			8.00000i			0.309529i
$669$		0			0
$670$	−16.0000			−0.618134
$671$			56.0000i			2.16186i
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$		−	22.0000i		−	0.847408i
$675$		0			0
$676$	−9.00000			−0.346154
$677$	32.0000			1.22986			0.614930	−	0.788582i	$-0.289184\pi$
$677$	32.0000			1.22986			0.614930	+	0.788582i	$0.289184\pi$
$678$		0			0
$679$			20.0000i			0.767530i
$680$		−	6.00000i		−	0.230089i
$681$		0			0
$682$	32.0000			1.22534
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		−	6.00000i		−	0.229416i
$685$			14.0000i			0.534913i
$686$		−	20.0000i		−	0.763604i
$687$		0			0
$688$			4.00000i			0.152499i
$689$		−	24.0000i		−	0.914327i
$690$		0			0
$691$	−36.0000			−1.36950			−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000			−1.36950			−0.684752	+	0.728776i	$0.740090\pi$
$692$	12.0000			0.456172
$693$	−24.0000			−0.911685
$694$	4.00000			0.151838
$695$		−	12.0000i		−	0.455186i
$696$		0			0
$697$			12.0000i			0.454532i
$698$			10.0000i			0.378506i
$699$		0			0
$700$	−2.00000			−0.0755929
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$		0			0
$703$	−2.00000	+	12.0000i	−0.0754314	+	0.452589i
$704$	4.00000			0.150756
$705$		0			0
$706$	2.00000			0.0752710
$707$	12.0000			0.451306
$708$		0			0
$709$		−	10.0000i		−	0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$		−	10.0000i		−	0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	12.0000			0.450352
$711$		0			0
$712$	4.00000			0.149906
$713$	64.0000			2.39682
$714$		0			0
$715$	8.00000			0.299183
$716$		−	18.0000i		−	0.672692i
$717$		0			0
$718$			16.0000i			0.597115i
$719$	36.0000			1.34257			0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000			1.34257			0.671287	+	0.741198i	$0.265742\pi$
$720$			3.00000i			0.111803i
$721$		0			0
$722$		−	15.0000i		−	0.558242i
$723$		0			0
$724$	22.0000			0.817624
$725$		−	2.00000i		−	0.0742781i
$726$		0			0
$727$			8.00000i			0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$			8.00000i			0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	−4.00000			−0.148250
$729$	−27.0000			−1.00000
$730$	−6.00000			−0.222070
$731$	24.0000			0.887672
$732$		0			0
$733$	36.0000			1.32969			0.664845	−	0.746981i	$-0.268498\pi$
$733$	36.0000			1.32969			0.664845	+	0.746981i	$0.268498\pi$
$734$			26.0000i			0.959678i
$735$		0			0
$736$	8.00000			0.294884
$737$	−64.0000			−2.35747
$738$		−	6.00000i		−	0.220863i
$739$	36.0000			1.32428			0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000			1.32428			0.662141	+	0.749380i	$0.269648\pi$
$740$	1.00000	−	6.00000i	0.0367607	−	0.220564i
$741$		0			0
$742$			24.0000i			0.881068i
$743$	38.0000			1.39408			0.697042	−	0.717030i	$-0.254499\pi$
$743$	38.0000			1.39408			0.697042	+	0.717030i	$0.254499\pi$
$744$		0			0
$745$			18.0000i			0.659469i
$746$		−	8.00000i		−	0.292901i
$747$	36.0000			1.31717
$748$		−	24.0000i		−	0.877527i
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	6.00000			0.218797
$753$		0			0
$754$		−	4.00000i		−	0.145671i
$755$			4.00000i			0.145575i
$756$		0			0
$757$		−	22.0000i		−	0.799604i	−0.916602	−	0.399802i	$-0.869079\pi$
$757$		−	22.0000i		−	0.799604i	0.916602	−	0.399802i	$-0.130921\pi$
$758$		−	4.00000i		−	0.145287i
$759$		0			0
$760$		−	2.00000i		−	0.0725476i
$761$	26.0000			0.942499			0.471250	−	0.882000i	$-0.343803\pi$
$761$	26.0000			0.942499			0.471250	+	0.882000i	$0.343803\pi$
$762$		0			0
$763$			28.0000i			1.01367i
$764$		−	8.00000i		−	0.289430i
$765$	18.0000			0.650791
$766$	−24.0000			−0.867155
$767$	−4.00000			−0.144432
$768$		0			0
$769$		−	16.0000i		−	0.576975i	−0.957484	−	0.288487i	$-0.906848\pi$
$769$		−	16.0000i		−	0.576975i	0.957484	−	0.288487i	$-0.0931523\pi$
$770$	−8.00000			−0.288300
$771$		0			0
$772$			14.0000i			0.503871i
$773$	4.00000			0.143870			0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000			0.143870			0.0719350	+	0.997409i	$0.477083\pi$
$774$	−12.0000			−0.431331
$775$			8.00000i			0.287368i
$776$	10.0000			0.358979
$777$		0			0
$778$	6.00000			0.215110
$779$			4.00000i			0.143315i
$780$		0			0
$781$	48.0000			1.71758
$782$		−	48.0000i		−	1.71648i
$783$		0			0
$784$	−3.00000			−0.107143
$785$		−	8.00000i		−	0.285532i
$786$		0			0
$787$	8.00000			0.285169			0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000			0.285169			0.142585	+	0.989783i	$0.454459\pi$
$788$	−8.00000			−0.284988
$789$		0			0
$790$		0			0
$791$		−	12.0000i		−	0.426671i
$792$			12.0000i			0.426401i
$793$	−28.0000			−0.994309
$794$		−	4.00000i		−	0.141955i
$795$		0			0
$796$			8.00000i			0.283552i
$797$			30.0000i			1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$			30.0000i			1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$		0			0
$799$		−	36.0000i		−	1.27359i
$800$			1.00000i			0.0353553i
$801$			12.0000i			0.423999i
$802$	−36.0000			−1.27120
$803$	−24.0000			−0.846942
$804$		0			0
$805$	−16.0000			−0.563926
$806$			16.0000i			0.563576i
$807$		0			0
$808$		−	6.00000i		−	0.211079i
$809$			32.0000i			1.12506i	0.826777	+	0.562530i	$0.190172\pi$
$809$			32.0000i			1.12506i	−0.826777	+	0.562530i	$0.809828\pi$
$810$	−9.00000			−0.316228
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$			4.00000i			0.140372i
$813$		0			0
$814$	4.00000	−	24.0000i	0.140200	−	0.841200i
$815$	16.0000			0.560456
$816$		0			0
$817$	8.00000			0.279885
$818$	−20.0000			−0.699284
$819$		−	12.0000i		−	0.419314i
$820$		−	2.00000i		−	0.0698430i
$821$	6.00000			0.209401			0.104701	−	0.994504i	$-0.466612\pi$
$821$	6.00000			0.209401			0.104701	+	0.994504i	$0.466612\pi$
$822$		0			0
$823$	14.0000			0.488009			0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000			0.488009			0.244005	+	0.969774i	$0.421539\pi$
$824$		0			0
$825$		0			0
$826$	4.00000			0.139178
$827$		−	4.00000i		−	0.139094i	−0.997579	−	0.0695468i	$-0.977845\pi$
$827$		−	4.00000i		−	0.139094i	0.997579	−	0.0695468i	$-0.0221553\pi$
$828$			24.0000i			0.834058i
$829$		−	34.0000i		−	1.18087i	−0.807086	−	0.590434i	$-0.798956\pi$
$829$		−	34.0000i		−	1.18087i	0.807086	−	0.590434i	$-0.201044\pi$
$830$	12.0000			0.416526
$831$		0			0
$832$			2.00000i			0.0693375i
$833$			18.0000i			0.623663i
$834$		0			0
$835$	−8.00000			−0.276851
$836$		−	8.00000i		−	0.276686i
$837$		0			0
$838$		−	4.00000i		−	0.138178i
$839$	−48.0000			−1.65714			−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000			−1.65714			−0.828572	+	0.559883i	$0.810846\pi$
$840$		0			0
$841$	25.0000			0.862069
$842$	30.0000			1.03387
$843$		0			0
$844$	20.0000			0.688428
$845$		−	9.00000i		−	0.309609i
$846$			18.0000i			0.618853i
$847$	−10.0000			−0.343604
$848$	12.0000			0.412082
$849$		0			0
$850$	6.00000			0.205798
$851$	8.00000	−	48.0000i	0.274236	−	1.64542i
$852$		0			0
$853$		−	10.0000i		−	0.342393i	−0.985237	−	0.171197i	$-0.945237\pi$
$853$		−	10.0000i		−	0.342393i	0.985237	−	0.171197i	$-0.0547634\pi$
$854$	28.0000			0.958140
$855$	6.00000			0.205196
$856$		0			0
$857$			46.0000i			1.57133i	0.618652	+	0.785665i	$0.287679\pi$
$857$			46.0000i			1.57133i	−0.618652	+	0.785665i	$0.712321\pi$
$858$		0			0
$859$			30.0000i			1.02359i	0.859109	+	0.511793i	$0.171019\pi$
$859$			30.0000i			1.02359i	−0.859109	+	0.511793i	$0.828981\pi$
$860$	−4.00000			−0.136399
$861$		0			0
$862$	−16.0000			−0.544962
$863$	−30.0000			−1.02121			−0.510606	−	0.859815i	$-0.670579\pi$
$863$	−30.0000			−1.02121			−0.510606	+	0.859815i	$0.670579\pi$
$864$		0			0
$865$			12.0000i			0.408012i
$866$			14.0000i			0.475739i
$867$		0			0
$868$		−	16.0000i		−	0.543075i
$869$		0			0
$870$		0			0
$871$		−	32.0000i		−	1.08428i
$872$	14.0000			0.474100
$873$			30.0000i			1.01535i
$874$		−	16.0000i		−	0.541208i
$875$		−	2.00000i		−	0.0676123i
$876$		0			0
$877$	−8.00000			−0.270141			−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000			−0.270141			−0.135070	+	0.990836i	$0.543126\pi$
$878$	−32.0000			−1.07995
$879$		0			0
$880$			4.00000i			0.134840i
$881$	−2.00000			−0.0673817			−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000			−0.0673817			−0.0336909	+	0.999432i	$0.510726\pi$
$882$		−	9.00000i		−	0.303046i
$883$			44.0000i			1.48072i	0.672212	+	0.740359i	$0.265344\pi$
$883$			44.0000i			1.48072i	−0.672212	+	0.740359i	$0.734656\pi$
$884$	12.0000			0.403604
$885$		0			0
$886$		−	24.0000i		−	0.806296i
$887$	−22.0000			−0.738688			−0.369344	−	0.929293i	$-0.620418\pi$
$887$	−22.0000			−0.738688			−0.369344	+	0.929293i	$0.620418\pi$
$888$		0			0
$889$	4.00000			0.134156
$890$			4.00000i			0.134080i
$891$	−36.0000			−1.20605
$892$	26.0000			0.870544
$893$		−	12.0000i		−	0.401565i
$894$		0			0
$895$	18.0000			0.601674
$896$		−	2.00000i		−	0.0668153i
$897$		0			0
$898$	20.0000			0.667409
$899$	16.0000			0.533630
$900$	−3.00000			−0.100000
$901$		−	72.0000i		−	2.39867i
$902$		−	8.00000i		−	0.266371i
$903$		0			0
$904$	−6.00000			−0.199557
$905$			22.0000i			0.731305i
$906$		0			0
$907$		−	36.0000i		−	1.19536i	−0.801735	−	0.597680i	$-0.796089\pi$
$907$		−	36.0000i		−	1.19536i	0.801735	−	0.597680i	$-0.203911\pi$
$908$		−	12.0000i		−	0.398234i
$909$	18.0000			0.597022
$910$		−	4.00000i		−	0.132599i
$911$			12.0000i			0.397578i	0.980042	+	0.198789i	$0.0637008\pi$
$911$			12.0000i			0.397578i	−0.980042	+	0.198789i	$0.936299\pi$
$912$		0			0
$913$	48.0000			1.58857
$914$	30.0000			0.992312
$915$		0			0
$916$	−26.0000			−0.859064
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$		−	44.0000i		−	1.45143i	−0.687998	−	0.725713i	$-0.741510\pi$
$919$		−	44.0000i		−	1.45143i	0.687998	−	0.725713i	$-0.258490\pi$
$920$			8.00000i			0.263752i
$921$		0			0
$922$	38.0000			1.25146
$923$			24.0000i			0.789970i
$924$		0			0
$925$	6.00000	+	1.00000i	0.197279	+	0.0328798i
$926$	−40.0000			−1.31448
$927$		0			0
$928$	2.00000			0.0656532
$929$	18.0000			0.590561			0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000			0.590561			0.295280	+	0.955411i	$0.404587\pi$
$930$		0			0
$931$			6.00000i			0.196642i
$932$	10.0000			0.327561
$933$		0			0
$934$		0			0
$935$	24.0000			0.784884
$936$	−6.00000			−0.196116
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$			32.0000i			1.04484i
$939$		0			0
$940$			6.00000i			0.195698i
$941$	50.0000			1.62995			0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000			1.62995			0.814977	+	0.579494i	$0.196750\pi$
$942$		0			0
$943$		−	16.0000i		−	0.521032i
$944$		−	2.00000i		−	0.0650945i
$945$		0			0
$946$	−16.0000			−0.520205
$947$		−	32.0000i		−	1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$		−	32.0000i		−	1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$		0			0
$949$		−	12.0000i		−	0.389536i
$950$	2.00000			0.0648886
$951$		0			0
$952$	−12.0000			−0.388922
$953$	6.00000			0.194359			0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000			0.194359			0.0971795	+	0.995267i	$0.469018\pi$
$954$			36.0000i			1.16554i
$955$	8.00000			0.258874
$956$			24.0000i			0.776215i
$957$		0			0
$958$	−8.00000			−0.258468
$959$	28.0000			0.904167
$960$		0			0
$961$	−33.0000			−1.06452
$962$	12.0000	+	2.00000i	0.386896	+	0.0644826i
$963$		0			0
$964$		−	8.00000i		−	0.257663i
$965$	−14.0000			−0.450676
$966$		0			0
$967$		−	48.0000i		−	1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$		−	48.0000i		−	1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$			5.00000i			0.160706i
$969$		0			0
$970$			10.0000i			0.321081i
$971$	−20.0000			−0.641831			−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000			−0.641831			−0.320915	+	0.947108i	$0.603990\pi$
$972$		0			0
$973$	−24.0000			−0.769405
$974$	8.00000			0.256337
$975$		0			0
$976$		−	14.0000i		−	0.448129i
$977$		−	34.0000i		−	1.08776i	−0.839164	−	0.543878i	$-0.816955\pi$
$977$		−	34.0000i		−	1.08776i	0.839164	−	0.543878i	$-0.183045\pi$
$978$		0			0
$979$			16.0000i			0.511362i
$980$		−	3.00000i		−	0.0958315i
$981$			42.0000i			1.34096i
$982$			20.0000i			0.638226i
$983$	−34.0000			−1.08443			−0.542216	−	0.840239i	$-0.682414\pi$
$983$	−34.0000			−1.08443			−0.542216	+	0.840239i	$0.682414\pi$
$984$		0			0
$985$		−	8.00000i		−	0.254901i
$986$		−	12.0000i		−	0.382158i
$987$		0			0
$988$	4.00000			0.127257
$989$	−32.0000			−1.01754
$990$	−12.0000			−0.381385
$991$		−	20.0000i		−	0.635321i	−0.948205	−	0.317660i	$-0.897103\pi$
$991$		−	20.0000i		−	0.635321i	0.948205	−	0.317660i	$-0.102897\pi$
$992$	−8.00000			−0.254000
$993$		0			0
$994$		−	24.0000i		−	0.761234i
$995$	−8.00000			−0.253617
$996$		0			0
$997$			42.0000i			1.33015i	0.746775	+	0.665077i	$0.231601\pi$
$997$			42.0000i			1.33015i	−0.746775	+	0.665077i	$0.768399\pi$
$998$	−10.0000			−0.316544
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.d.a.221.1	✓	2
3.2	odd	2		3330.2.h.c.2071.2		2
4.3	odd	2		2960.2.p.e.961.1		2
5.2	odd	4		1850.2.c.d.1849.1		2
5.3	odd	4		1850.2.c.a.1849.2		2
5.4	even	2		1850.2.d.c.1701.2		2
37.36	even	2	inner	370.2.d.a.221.2	yes	2
111.110	odd	2		3330.2.h.c.2071.1		2
148.147	odd	2		2960.2.p.e.961.2		2
185.73	odd	4		1850.2.c.d.1849.2		2
185.147	odd	4		1850.2.c.a.1849.1		2
185.184	even	2		1850.2.d.c.1701.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.d.a.221.1	✓	2	1.1	even	1	trivial
370.2.d.a.221.2	yes	2	37.36	even	2	inner
1850.2.c.a.1849.1		2	185.147	odd	4
1850.2.c.a.1849.2		2	5.3	odd	4
1850.2.c.d.1849.1		2	5.2	odd	4
1850.2.c.d.1849.2		2	185.73	odd	4
1850.2.d.c.1701.1		2	185.184	even	2
1850.2.d.c.1701.2		2	5.4	even	2
2960.2.p.e.961.1		2	4.3	odd	2
2960.2.p.e.961.2		2	148.147	odd	2
3330.2.h.c.2071.1		2	111.110	odd	2
3330.2.h.c.2071.2		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 370 = 2 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	370.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000 q^{7} +1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{5} -2.00000 q^{7} +1.00000i q^{8} -3.00000 q^{9} -1.00000 q^{10} -4.00000 q^{11} -2.00000i q^{13} +2.00000i q^{14} +1.00000 q^{16} -6.00000i q^{17} +3.00000i q^{18} -2.00000i q^{19} +1.00000i q^{20} +4.00000i q^{22} +8.00000i q^{23} -1.00000 q^{25} -2.00000 q^{26} +2.00000 q^{28} +2.00000i q^{29} -8.00000i q^{31} -1.00000i q^{32} -6.00000 q^{34} +2.00000i q^{35} +3.00000 q^{36} +(-6.00000 - 1.00000i) q^{37} -2.00000 q^{38} +1.00000 q^{40} -2.00000 q^{41} +4.00000i q^{43} +4.00000 q^{44} +3.00000i q^{45} +8.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} +1.00000i q^{50} +2.00000i q^{52} +12.0000 q^{53} +4.00000i q^{55} -2.00000i q^{56} +2.00000 q^{58} -2.00000i q^{59} -14.0000i q^{61} -8.00000 q^{62} +6.00000 q^{63} -1.00000 q^{64} -2.00000 q^{65} +16.0000 q^{67} +6.00000i q^{68} +2.00000 q^{70} -12.0000 q^{71} -3.00000i q^{72} +6.00000 q^{73} +(-1.00000 + 6.00000i) q^{74} +2.00000i q^{76} +8.00000 q^{77} -1.00000i q^{80} +9.00000 q^{81} +2.00000i q^{82} -12.0000 q^{83} -6.00000 q^{85} +4.00000 q^{86} -4.00000i q^{88} -4.00000i q^{89} +3.00000 q^{90} +4.00000i q^{91} -8.00000i q^{92} -6.00000i q^{94} -2.00000 q^{95} -10.0000i q^{97} +3.00000i q^{98} +12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 4 * q^7 - 6 * q^9	\( 2 q - 2 q^{4} - 4 q^{7} - 6 q^{9} - 2 q^{10} - 8 q^{11} + 2 q^{16} - 2 q^{25} - 4 q^{26} + 4 q^{28} - 12 q^{34} + 6 q^{36} - 12 q^{37} - 4 q^{38} + 2 q^{40} - 4 q^{41} + 8 q^{44} + 16 q^{46} + 12 q^{47} - 6 q^{49} + 24 q^{53} + 4 q^{58} - 16 q^{62} + 12 q^{63} - 2 q^{64} - 4 q^{65} + 32 q^{67} + 4 q^{70} - 24 q^{71} + 12 q^{73} - 2 q^{74} + 16 q^{77} + 18 q^{81} - 24 q^{83} - 12 q^{85} + 8 q^{86} + 6 q^{90} - 4 q^{95} + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^7 - 6 * q^9 - 2 * q^10 - 8 * q^11 + 2 * q^16 - 2 * q^25 - 4 * q^26 + 4 * q^28 - 12 * q^34 + 6 * q^36 - 12 * q^37 - 4 * q^38 + 2 * q^40 - 4 * q^41 + 8 * q^44 + 16 * q^46 + 12 * q^47 - 6 * q^49 + 24 * q^53 + 4 * q^58 - 16 * q^62 + 12 * q^63 - 2 * q^64 - 4 * q^65 + 32 * q^67 + 4 * q^70 - 24 * q^71 + 12 * q^73 - 2 * q^74 + 16 * q^77 + 18 * q^81 - 24 * q^83 - 12 * q^85 + 8 * q^86 + 6 * q^90 - 4 * q^95 + 24 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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