LMFDB - Embedded newform 370.2.b.a.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [370,2,Mod(149,370)]

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([1, 0]))

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

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

chi := DirichletCharacter("370.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 370 = 2 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	370.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:	$2.95446487479$
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			149.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	370.149
Dual form			370.2.b.a.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +2.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} +(1.00000 + 2.00000i) q^{10} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -3.00000i q^{18} +6.00000 q^{19} +(2.00000 - 1.00000i) q^{20} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +2.00000 q^{26} -2.00000i q^{28} -4.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} +(-2.00000 - 4.00000i) q^{35} -3.00000 q^{36} +1.00000i q^{37} -6.00000i q^{38} +(-1.00000 - 2.00000i) q^{40} -10.0000 q^{41} -4.00000i q^{43} +(-6.00000 + 3.00000i) q^{45} +4.00000 q^{46} +2.00000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -2.00000i q^{52} +2.00000i q^{53} -2.00000 q^{56} +6.00000 q^{59} +4.00000i q^{62} +6.00000i q^{63} -1.00000 q^{64} +(-2.00000 - 4.00000i) q^{65} -8.00000i q^{67} -6.00000i q^{68} +(-4.00000 + 2.00000i) q^{70} +3.00000i q^{72} +8.00000i q^{73} +1.00000 q^{74} -6.00000 q^{76} -4.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} +9.00000 q^{81} +10.0000i q^{82} -12.0000i q^{83} +(-6.00000 - 12.0000i) q^{85} -4.00000 q^{86} -6.00000 q^{89} +(3.00000 + 6.00000i) q^{90} -4.00000 q^{91} -4.00000i q^{92} +2.00000 q^{94} +(-12.0000 + 6.00000i) q^{95} +10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 4 q^{5} + 6 q^{9} + 2 q^{10} + 4 q^{14} + 2 q^{16} + 12 q^{19} + 4 q^{20} + 6 q^{25} + 4 q^{26} - 8 q^{31} + 12 q^{34} - 4 q^{35} - 6 q^{36} - 2 q^{40} - 20 q^{41} - 12 q^{45} + 8 q^{46}+ \cdots - 24 q^{95}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^5 + 6 * q^9 + 2 * q^10 + 4 * q^14 + 2 * q^16 + 12 * q^19 + 4 * q^20 + 6 * q^25 + 4 * q^26 - 8 * q^31 + 12 * q^34 - 4 * q^35 - 6 * q^36 - 2 * q^40 - 20 * q^41 - 12 * q^45 + 8 * q^46 + 6 * q^49 - 8 * q^50 - 4 * q^56 + 12 * q^59 - 2 * q^64 - 4 * q^65 - 8 * q^70 + 2 * q^74 - 12 * q^76 - 8 * q^79 - 4 * q^80 + 18 * q^81 - 12 * q^85 - 8 * q^86 - 12 * q^89 + 6 * q^90 - 8 * q^91 + 4 * q^94 - 24 * q^95

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		1.00000			$0$
$3$		0			0		−1.00000			$\pi$
$4$	−1.00000			−0.500000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$			2.00000i			0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$			2.00000i			0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$			1.00000i			0.353553i
$9$	3.00000			1.00000
$10$	1.00000	+	2.00000i	0.316228	+	0.632456i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$			2.00000i			0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$			2.00000i			0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	2.00000			0.534522
$15$		0			0
$16$	1.00000			0.250000
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		−	3.00000i		−	0.707107i
$19$	6.00000			1.37649			0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000			1.37649			0.688247	+	0.725476i	$0.258380\pi$
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$		0			0
$22$		0			0
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$	2.00000			0.392232
$27$		0			0
$28$		−	2.00000i		−	0.377964i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	−4.00000			−0.718421			−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000			−0.718421			−0.359211	+	0.933257i	$0.616954\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	6.00000			1.02899
$35$	−2.00000	−	4.00000i	−0.338062	−	0.676123i
$36$	−3.00000			−0.500000
$37$			1.00000i			0.164399i
$37$			1.00000i			0.164399i
$38$		−	6.00000i		−	0.973329i
$39$		0			0
$40$	−1.00000	−	2.00000i	−0.158114	−	0.316228i
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$		−	4.00000i		−	0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$		−	4.00000i		−	0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$		0			0
$45$	−6.00000	+	3.00000i	−0.894427	+	0.447214i
$46$	4.00000			0.589768
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	−4.00000	−	3.00000i	−0.565685	−	0.424264i
$51$		0			0
$52$		−	2.00000i		−	0.277350i
$53$			2.00000i			0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$			2.00000i			0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$		0			0
$55$		0			0
$56$	−2.00000			−0.267261
$57$		0			0
$58$		0			0
$59$	6.00000			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$			4.00000i			0.508001i
$63$			6.00000i			0.755929i
$64$	−1.00000			−0.125000
$65$	−2.00000	−	4.00000i	−0.248069	−	0.496139i
$66$		0			0
$67$		−	8.00000i		−	0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$		−	8.00000i		−	0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$		−	6.00000i		−	0.727607i
$69$		0			0
$70$	−4.00000	+	2.00000i	−0.478091	+	0.239046i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$			3.00000i			0.353553i
$73$			8.00000i			0.936329i	0.883641	+	0.468165i	$0.155085\pi$
$73$			8.00000i			0.936329i	−0.883641	+	0.468165i	$0.844915\pi$
$74$	1.00000			0.116248
$75$		0			0
$76$	−6.00000			−0.688247
$77$		0			0
$78$		0			0
$79$	−4.00000			−0.450035			−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000			−0.450035			−0.225018	+	0.974355i	$0.572244\pi$
$80$	−2.00000	+	1.00000i	−0.223607	+	0.111803i
$81$	9.00000			1.00000
$82$			10.0000i			1.10432i
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$	−6.00000	−	12.0000i	−0.650791	−	1.30158i
$86$	−4.00000			−0.431331
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$	3.00000	+	6.00000i	0.316228	+	0.632456i
$91$	−4.00000			−0.419314
$92$		−	4.00000i		−	0.417029i
$93$		0			0
$94$	2.00000			0.206284
$95$	−12.0000	+	6.00000i	−1.23117	+	0.615587i
$96$		0			0
$97$			10.0000i			1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$			10.0000i			1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$		−	3.00000i		−	0.303046i
$99$		0			0
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−2.00000			−0.196116
$105$		0			0
$106$	2.00000			0.194257
$107$		−	8.00000i		−	0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$		−	8.00000i		−	0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$		0			0
$109$	20.0000			1.91565			0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000			1.91565			0.957826	+	0.287348i	$0.0927736\pi$
$110$		0			0
$111$		0			0
$112$			2.00000i			0.188982i
$113$			2.00000i			0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$			2.00000i			0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$		0			0
$115$	−4.00000	−	8.00000i	−0.373002	−	0.746004i
$116$		0			0
$117$			6.00000i			0.554700i
$118$		−	6.00000i		−	0.552345i
$119$	−12.0000			−1.10004
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$		0			0
$124$	4.00000			0.359211
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$	6.00000			0.534522
$127$		−	22.0000i		−	1.95218i	−0.217357	−	0.976092i	$-0.569744\pi$
$127$		−	22.0000i		−	1.95218i	0.217357	−	0.976092i	$-0.430256\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−4.00000	+	2.00000i	−0.350823	+	0.175412i
$131$	14.0000			1.22319			0.611593	−	0.791173i	$-0.290529\pi$
$131$	14.0000			1.22319			0.611593	+	0.791173i	$0.290529\pi$
$132$		0			0
$133$			12.0000i			1.04053i
$134$	−8.00000			−0.691095
$135$		0			0
$136$	−6.00000			−0.514496
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$	2.00000	+	4.00000i	0.169031	+	0.338062i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	3.00000			0.250000
$145$		0			0
$146$	8.00000			0.662085
$147$		0			0
$148$		−	1.00000i		−	0.0821995i
$149$	−14.0000			−1.14692			−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000			−1.14692			−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$			6.00000i			0.486664i
$153$			18.0000i			1.45521i
$154$		0			0
$155$	8.00000	−	4.00000i	0.642575	−	0.321288i
$156$		0			0
$157$		−	22.0000i		−	1.75579i	−0.478852	−	0.877896i	$-0.658947\pi$
$157$		−	22.0000i		−	1.75579i	0.478852	−	0.877896i	$-0.341053\pi$
$158$			4.00000i			0.318223i
$159$		0			0
$160$	1.00000	+	2.00000i	0.0790569	+	0.158114i
$161$	−8.00000			−0.630488
$162$		−	9.00000i		−	0.707107i
$163$		−	12.0000i		−	0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$		−	12.0000i		−	0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	10.0000			0.780869
$165$		0			0
$166$	−12.0000			−0.931381
$167$		−	16.0000i		−	1.23812i	−0.785345	−	0.619059i	$-0.787514\pi$
$167$		−	16.0000i		−	1.23812i	0.785345	−	0.619059i	$-0.212486\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$	−12.0000	+	6.00000i	−0.920358	+	0.460179i
$171$	18.0000			1.37649
$172$			4.00000i			0.304997i
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$	8.00000	+	6.00000i	0.604743	+	0.453557i
$176$		0			0
$177$		0			0
$178$			6.00000i			0.449719i
$179$	14.0000			1.04641			0.523205	−	0.852207i	$-0.324736\pi$
$179$	14.0000			1.04641			0.523205	+	0.852207i	$0.324736\pi$
$180$	6.00000	−	3.00000i	0.447214	−	0.223607i
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$			4.00000i			0.296500i
$183$		0			0
$184$	−4.00000			−0.294884
$185$	−1.00000	−	2.00000i	−0.0735215	−	0.147043i
$186$		0			0
$187$		0			0
$188$		−	2.00000i		−	0.145865i
$189$		0			0
$190$	6.00000	+	12.0000i	0.435286	+	0.870572i
$191$	4.00000			0.289430			0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000			0.289430			0.144715	+	0.989473i	$0.453773\pi$
$192$		0			0
$193$			10.0000i			0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$			10.0000i			0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	10.0000			0.717958
$195$		0			0
$196$	−3.00000			−0.214286
$197$			10.0000i			0.712470i	0.934396	+	0.356235i	$0.115940\pi$
$197$			10.0000i			0.712470i	−0.934396	+	0.356235i	$0.884060\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$	4.00000	+	3.00000i	0.282843	+	0.212132i
$201$		0			0
$202$			10.0000i			0.703598i
$203$		0			0
$204$		0			0
$205$	20.0000	−	10.0000i	1.39686	−	0.698430i
$206$		0			0
$207$			12.0000i			0.834058i
$208$			2.00000i			0.138675i
$209$		0			0
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		−	2.00000i		−	0.137361i
$213$		0			0
$214$	−8.00000			−0.546869
$215$	4.00000	+	8.00000i	0.272798	+	0.545595i
$216$		0			0
$217$		−	8.00000i		−	0.543075i
$218$		−	20.0000i		−	1.35457i
$219$		0			0
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$			14.0000i			0.937509i	0.883328	+	0.468755i	$0.155297\pi$
$223$			14.0000i			0.937509i	−0.883328	+	0.468755i	$0.844703\pi$
$224$	2.00000			0.133631
$225$	9.00000	−	12.0000i	0.600000	−	0.800000i
$226$	2.00000			0.133038
$227$		−	20.0000i		−	1.32745i	−0.747978	−	0.663723i	$-0.768975\pi$
$227$		−	20.0000i		−	1.32745i	0.747978	−	0.663723i	$-0.231025\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−8.00000	+	4.00000i	−0.527504	+	0.263752i
$231$		0			0
$232$		0			0
$233$		−	24.0000i		−	1.57229i	−0.618041	−	0.786146i	$-0.712073\pi$
$233$		−	24.0000i		−	1.57229i	0.618041	−	0.786146i	$-0.287927\pi$
$234$	6.00000			0.392232
$235$	−2.00000	−	4.00000i	−0.130466	−	0.260931i
$236$	−6.00000			−0.390567
$237$		0			0
$238$			12.0000i			0.777844i
$239$	−28.0000			−1.81117			−0.905585	−	0.424165i	$-0.860568\pi$
$239$	−28.0000			−1.81117			−0.905585	+	0.424165i	$0.860568\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$			11.0000i			0.707107i
$243$		0			0
$244$		0			0
$245$	−6.00000	+	3.00000i	−0.383326	+	0.191663i
$246$		0			0
$247$			12.0000i			0.763542i
$248$		−	4.00000i		−	0.254000i
$249$		0			0
$250$	11.0000	+	2.00000i	0.695701	+	0.126491i
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		−	6.00000i		−	0.377964i
$253$		0			0
$254$	−22.0000			−1.38040
$255$		0			0
$256$	1.00000			0.0625000
$257$			30.0000i			1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$			30.0000i			1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$		0			0
$259$	−2.00000			−0.124274
$260$	2.00000	+	4.00000i	0.124035	+	0.248069i
$261$		0			0
$262$		−	14.0000i		−	0.864923i
$263$		−	2.00000i		−	0.123325i	−0.998097	−	0.0616626i	$-0.980360\pi$
$263$		−	2.00000i		−	0.123325i	0.998097	−	0.0616626i	$-0.0196403\pi$
$264$		0			0
$265$	−2.00000	−	4.00000i	−0.122859	−	0.245718i
$266$	12.0000			0.735767
$267$		0			0
$268$			8.00000i			0.488678i
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$			6.00000i			0.363803i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$			18.0000i			1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$			18.0000i			1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$		−	20.0000i		−	1.19952i
$279$	−12.0000			−0.718421
$280$	4.00000	−	2.00000i	0.239046	−	0.119523i
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$			28.0000i			1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$			28.0000i			1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	20.0000i		−	1.18056i
$288$		−	3.00000i		−	0.176777i
$289$	−19.0000			−1.11765
$290$		0			0
$291$		0			0
$292$		−	8.00000i		−	0.468165i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$	−12.0000	+	6.00000i	−0.698667	+	0.349334i
$296$	−1.00000			−0.0581238
$297$		0			0
$298$			14.0000i			0.810998i
$299$	−8.00000			−0.462652
$300$		0			0
$301$	8.00000			0.461112
$302$		−	16.0000i		−	0.920697i
$303$		0			0
$304$	6.00000			0.344124
$305$		0			0
$306$	18.0000			1.02899
$307$		−	4.00000i		−	0.228292i	−0.993464	−	0.114146i	$-0.963587\pi$
$307$		−	4.00000i		−	0.228292i	0.993464	−	0.114146i	$-0.0364132\pi$
$308$		0			0
$309$		0			0
$310$	−4.00000	−	8.00000i	−0.227185	−	0.454369i
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	14.0000i		−	0.791327i	−0.918396	−	0.395663i	$-0.870515\pi$
$313$		−	14.0000i		−	0.791327i	0.918396	−	0.395663i	$-0.129485\pi$
$314$	−22.0000			−1.24153
$315$	−6.00000	−	12.0000i	−0.338062	−	0.676123i
$316$	4.00000			0.225018
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$		0			0
$320$	2.00000	−	1.00000i	0.111803	−	0.0559017i
$321$		0			0
$322$			8.00000i			0.445823i
$323$			36.0000i			2.00309i
$324$	−9.00000			−0.500000
$325$	8.00000	+	6.00000i	0.443760	+	0.332820i
$326$	−12.0000			−0.664619
$327$		0			0
$328$		−	10.0000i		−	0.552158i
$329$	−4.00000			−0.220527
$330$		0			0
$331$	−34.0000			−1.86881			−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000			−1.86881			−0.934405	+	0.356214i	$0.884068\pi$
$332$			12.0000i			0.658586i
$333$			3.00000i			0.164399i
$334$	−16.0000			−0.875481
$335$	8.00000	+	16.0000i	0.437087	+	0.874173i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		−	9.00000i		−	0.489535i
$339$		0			0
$340$	6.00000	+	12.0000i	0.325396	+	0.650791i
$341$		0			0
$342$		−	18.0000i		−	0.973329i
$343$			20.0000i			1.07990i
$344$	4.00000			0.215666
$345$		0			0
$346$	2.00000			0.107521
$347$		−	36.0000i		−	1.93258i	−0.257454	−	0.966291i	$-0.582883\pi$
$347$		−	36.0000i		−	1.93258i	0.257454	−	0.966291i	$-0.417117\pi$
$348$		0			0
$349$	−30.0000			−1.60586			−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000			−1.60586			−0.802932	+	0.596071i	$0.796728\pi$
$350$	6.00000	−	8.00000i	0.320713	−	0.427618i
$351$		0			0
$352$		0			0
$353$		−	18.0000i		−	0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$		−	18.0000i		−	0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$		0			0
$355$		0			0
$356$	6.00000			0.317999
$357$		0			0
$358$		−	14.0000i		−	0.739923i
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$	−3.00000	−	6.00000i	−0.158114	−	0.316228i
$361$	17.0000			0.894737
$362$		−	2.00000i		−	0.105118i
$363$		0			0
$364$	4.00000			0.209657
$365$	−8.00000	−	16.0000i	−0.418739	−	0.837478i
$366$		0			0
$367$			26.0000i			1.35719i	0.734513	+	0.678594i	$0.237411\pi$
$367$			26.0000i			1.35719i	−0.734513	+	0.678594i	$0.762589\pi$
$368$			4.00000i			0.208514i
$369$	−30.0000			−1.56174
$370$	−2.00000	+	1.00000i	−0.103975	+	0.0519875i
$371$	−4.00000			−0.207670
$372$		0			0
$373$		−	34.0000i		−	1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$		−	34.0000i		−	1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$		0			0
$375$		0			0
$376$	−2.00000			−0.103142
$377$		0			0
$378$		0			0
$379$	−12.0000			−0.616399			−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000			−0.616399			−0.308199	+	0.951322i	$0.599726\pi$
$380$	12.0000	−	6.00000i	0.615587	−	0.307794i
$381$		0			0
$382$		−	4.00000i		−	0.204658i
$383$		−	28.0000i		−	1.43073i	−0.698749	−	0.715367i	$-0.746260\pi$
$383$		−	28.0000i		−	1.43073i	0.698749	−	0.715367i	$-0.253740\pi$
$384$		0			0
$385$		0			0
$386$	10.0000			0.508987
$387$		−	12.0000i		−	0.609994i
$388$		−	10.0000i		−	0.507673i
$389$	16.0000			0.811232			0.405616	−	0.914044i	$-0.367057\pi$
$389$	16.0000			0.811232			0.405616	+	0.914044i	$0.367057\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$			3.00000i			0.151523i
$393$		0			0
$394$	10.0000			0.503793
$395$	8.00000	−	4.00000i	0.402524	−	0.201262i
$396$		0			0
$397$		−	2.00000i		−	0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$		−	2.00000i		−	0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$		−	20.0000i		−	1.00251i
$399$		0			0
$400$	3.00000	−	4.00000i	0.150000	−	0.200000i
$401$	−2.00000			−0.0998752			−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000			−0.0998752			−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$		−	8.00000i		−	0.398508i
$404$	10.0000			0.497519
$405$	−18.0000	+	9.00000i	−0.894427	+	0.447214i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$	−10.0000	−	20.0000i	−0.493865	−	0.987730i
$411$		0			0
$412$		0			0
$413$			12.0000i			0.590481i
$414$	12.0000			0.589768
$415$	12.0000	+	24.0000i	0.589057	+	1.17811i
$416$	2.00000			0.0980581
$417$		0			0
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	36.0000			1.75453			0.877266	−	0.480004i	$-0.159365\pi$
$421$	36.0000			1.75453			0.877266	+	0.480004i	$0.159365\pi$
$422$			16.0000i			0.778868i
$423$			6.00000i			0.291730i
$424$	−2.00000			−0.0971286
$425$	24.0000	+	18.0000i	1.16417	+	0.873128i
$426$		0			0
$427$		0			0
$428$			8.00000i			0.386695i
$429$		0			0
$430$	8.00000	−	4.00000i	0.385794	−	0.192897i
$431$	20.0000			0.963366			0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000			0.963366			0.481683	+	0.876346i	$0.340026\pi$
$432$		0			0
$433$		−	16.0000i		−	0.768911i	−0.923144	−	0.384455i	$-0.874389\pi$
$433$		−	16.0000i		−	0.768911i	0.923144	−	0.384455i	$-0.125611\pi$
$434$	−8.00000			−0.384012
$435$		0			0
$436$	−20.0000			−0.957826
$437$			24.0000i			1.14808i
$438$		0			0
$439$	−4.00000			−0.190910			−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000			−0.190910			−0.0954548	+	0.995434i	$0.530431\pi$
$440$		0			0
$441$	9.00000			0.428571
$442$			12.0000i			0.570782i
$443$			16.0000i			0.760183i	0.924949	+	0.380091i	$0.124107\pi$
$443$			16.0000i			0.760183i	−0.924949	+	0.380091i	$0.875893\pi$
$444$		0			0
$445$	12.0000	−	6.00000i	0.568855	−	0.284427i
$446$	14.0000			0.662919
$447$		0			0
$448$		−	2.00000i		−	0.0944911i
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$	−12.0000	−	9.00000i	−0.565685	−	0.424264i
$451$		0			0
$452$		−	2.00000i		−	0.0940721i
$453$		0			0
$454$	−20.0000			−0.938647
$455$	8.00000	−	4.00000i	0.375046	−	0.187523i
$456$		0			0
$457$			10.0000i			0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$			10.0000i			0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$			2.00000i			0.0934539i
$459$		0			0
$460$	4.00000	+	8.00000i	0.186501	+	0.373002i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	4.00000i		−	0.185896i	−0.995671	−	0.0929479i	$-0.970371\pi$
$463$		−	4.00000i		−	0.185896i	0.995671	−	0.0929479i	$-0.0296290\pi$
$464$		0			0
$465$		0			0
$466$	−24.0000			−1.11178
$467$			36.0000i			1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$			36.0000i			1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$		−	6.00000i		−	0.277350i
$469$	16.0000			0.738811
$470$	−4.00000	+	2.00000i	−0.184506	+	0.0922531i
$471$		0			0
$472$			6.00000i			0.276172i
$473$		0			0
$474$		0			0
$475$	18.0000	−	24.0000i	0.825897	−	1.10120i
$476$	12.0000			0.550019
$477$			6.00000i			0.274721i
$478$			28.0000i			1.28069i
$479$	−12.0000			−0.548294			−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000			−0.548294			−0.274147	+	0.961688i	$0.588395\pi$
$480$		0			0
$481$	−2.00000			−0.0911922
$482$		−	10.0000i		−	0.455488i
$483$		0			0
$484$	11.0000			0.500000
$485$	−10.0000	−	20.0000i	−0.454077	−	0.908153i
$486$		0			0
$487$		−	24.0000i		−	1.08754i	−0.839233	−	0.543772i	$-0.816996\pi$
$487$		−	24.0000i		−	1.08754i	0.839233	−	0.543772i	$-0.183004\pi$
$488$		0			0
$489$		0			0
$490$	3.00000	+	6.00000i	0.135526	+	0.271052i
$491$	24.0000			1.08310			0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000			1.08310			0.541552	+	0.840667i	$0.317837\pi$
$492$		0			0
$493$		0			0
$494$	12.0000			0.539906
$495$		0			0
$496$	−4.00000			−0.179605
$497$		0			0
$498$		0			0
$499$	−30.0000			−1.34298			−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000			−1.34298			−0.671492	+	0.741012i	$0.734346\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$		0			0
$502$		−	2.00000i		−	0.0892644i
$503$			40.0000i			1.78351i	0.452517	+	0.891756i	$0.350526\pi$
$503$			40.0000i			1.78351i	−0.452517	+	0.891756i	$0.649474\pi$
$504$	−6.00000			−0.267261
$505$	20.0000	−	10.0000i	0.889988	−	0.444994i
$506$		0			0
$507$		0			0
$508$			22.0000i			0.976092i
$509$	18.0000			0.797836			0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000			0.797836			0.398918	+	0.916987i	$0.369386\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	30.0000			1.32324
$515$		0			0
$516$		0			0
$517$		0			0
$518$			2.00000i			0.0878750i
$519$		0			0
$520$	4.00000	−	2.00000i	0.175412	−	0.0877058i
$521$	26.0000			1.13908			0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000			1.13908			0.569540	+	0.821963i	$0.307121\pi$
$522$		0			0
$523$			4.00000i			0.174908i	0.996169	+	0.0874539i	$0.0278730\pi$
$523$			4.00000i			0.174908i	−0.996169	+	0.0874539i	$0.972127\pi$
$524$	−14.0000			−0.611593
$525$		0			0
$526$	−2.00000			−0.0872041
$527$		−	24.0000i		−	1.04546i
$528$		0			0
$529$	7.00000			0.304348
$530$	−4.00000	+	2.00000i	−0.173749	+	0.0868744i
$531$	18.0000			0.781133
$532$		−	12.0000i		−	0.520266i
$533$		−	20.0000i		−	0.866296i
$534$		0			0
$535$	8.00000	+	16.0000i	0.345870	+	0.691740i
$536$	8.00000			0.345547
$537$		0			0
$538$		−	10.0000i		−	0.431131i
$539$		0			0
$540$		0			0
$541$	−16.0000			−0.687894			−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000			−0.687894			−0.343947	+	0.938989i	$0.611764\pi$
$542$			8.00000i			0.343629i
$543$		0			0
$544$	6.00000			0.257248
$545$	−40.0000	+	20.0000i	−1.71341	+	0.856706i
$546$		0			0
$547$			28.0000i			1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$			28.0000i			1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	8.00000i		−	0.340195i
$554$	18.0000			0.764747
$555$		0			0
$556$	−20.0000			−0.848189
$557$			30.0000i			1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$			30.0000i			1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$			12.0000i			0.508001i
$559$	8.00000			0.338364
$560$	−2.00000	−	4.00000i	−0.0845154	−	0.169031i
$561$		0			0
$562$		−	6.00000i		−	0.253095i
$563$		−	4.00000i		−	0.168580i	−0.996441	−	0.0842900i	$-0.973138\pi$
$563$		−	4.00000i		−	0.168580i	0.996441	−	0.0842900i	$-0.0268622\pi$
$564$		0			0
$565$	−2.00000	−	4.00000i	−0.0841406	−	0.168281i
$566$	28.0000			1.17693
$567$			18.0000i			0.755929i
$568$		0			0
$569$	34.0000			1.42535			0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000			1.42535			0.712677	+	0.701492i	$0.247483\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$		0			0
$574$	−20.0000			−0.834784
$575$	16.0000	+	12.0000i	0.667246	+	0.500435i
$576$	−3.00000			−0.125000
$577$			38.0000i			1.58196i	0.611842	+	0.790980i	$0.290429\pi$
$577$			38.0000i			1.58196i	−0.611842	+	0.790980i	$0.709571\pi$
$578$			19.0000i			0.790296i
$579$		0			0
$580$		0			0
$581$	24.0000			0.995688
$582$		0			0
$583$		0			0
$584$	−8.00000			−0.331042
$585$	−6.00000	−	12.0000i	−0.248069	−	0.496139i
$586$	6.00000			0.247858
$587$		−	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
$589$	−24.0000			−0.988903
$590$	6.00000	+	12.0000i	0.247016	+	0.494032i
$591$		0			0
$592$			1.00000i			0.0410997i
$593$			24.0000i			0.985562i	0.870153	+	0.492781i	$0.164020\pi$
$593$			24.0000i			0.985562i	−0.870153	+	0.492781i	$0.835980\pi$
$594$		0			0
$595$	24.0000	−	12.0000i	0.983904	−	0.491952i
$596$	14.0000			0.573462
$597$		0			0
$598$			8.00000i			0.327144i
$599$	8.00000			0.326871			0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000			0.326871			0.163436	+	0.986554i	$0.447742\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		−	8.00000i		−	0.326056i
$603$		−	24.0000i		−	0.977356i
$604$	−16.0000			−0.651031
$605$	22.0000	−	11.0000i	0.894427	−	0.447214i
$606$		0			0
$607$			36.0000i			1.46119i	0.682808	+	0.730597i	$0.260758\pi$
$607$			36.0000i			1.46119i	−0.682808	+	0.730597i	$0.739242\pi$
$608$		−	6.00000i		−	0.243332i
$609$		0			0
$610$		0			0
$611$	−4.00000			−0.161823
$612$		−	18.0000i		−	0.727607i
$613$		−	30.0000i		−	1.21169i	−0.795583	−	0.605844i	$-0.792835\pi$
$613$		−	30.0000i		−	1.21169i	0.795583	−	0.605844i	$-0.207165\pi$
$614$	−4.00000			−0.161427
$615$		0			0
$616$		0			0
$617$		−	12.0000i		−	0.483102i	−0.970388	−	0.241551i	$-0.922344\pi$
$617$		−	12.0000i		−	0.483102i	0.970388	−	0.241551i	$-0.0776561\pi$
$618$		0			0
$619$	32.0000			1.28619			0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000			1.28619			0.643094	+	0.765787i	$0.277650\pi$
$620$	−8.00000	+	4.00000i	−0.321288	+	0.160644i
$621$		0			0
$622$		0			0
$623$		−	12.0000i		−	0.480770i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−14.0000			−0.559553
$627$		0			0
$628$			22.0000i			0.877896i
$629$	−6.00000			−0.239236
$630$	−12.0000	+	6.00000i	−0.478091	+	0.239046i
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		−	4.00000i		−	0.159111i
$633$		0			0
$634$	6.00000			0.238290
$635$	22.0000	+	44.0000i	0.873043	+	1.74609i
$636$		0			0
$637$			6.00000i			0.237729i
$638$		0			0
$639$		0			0
$640$	−1.00000	−	2.00000i	−0.0395285	−	0.0790569i
$641$	14.0000			0.552967			0.276483	−	0.961019i	$-0.410831\pi$
$641$	14.0000			0.552967			0.276483	+	0.961019i	$0.410831\pi$
$642$		0			0
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$	8.00000			0.315244
$645$		0			0
$646$	36.0000			1.41640
$647$			16.0000i			0.629025i	0.949253	+	0.314512i	$0.101841\pi$
$647$			16.0000i			0.629025i	−0.949253	+	0.314512i	$0.898159\pi$
$648$			9.00000i			0.353553i
$649$		0			0
$650$	6.00000	−	8.00000i	0.235339	−	0.313786i
$651$		0			0
$652$			12.0000i			0.469956i
$653$			14.0000i			0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$			14.0000i			0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$		0			0
$655$	−28.0000	+	14.0000i	−1.09405	+	0.547025i
$656$	−10.0000			−0.390434
$657$			24.0000i			0.936329i
$658$			4.00000i			0.155936i
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	−28.0000			−1.08907			−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000			−1.08907			−0.544537	+	0.838737i	$0.683295\pi$
$662$			34.0000i			1.32145i
$663$		0			0
$664$	12.0000			0.465690
$665$	−12.0000	−	24.0000i	−0.465340	−	0.930680i
$666$	3.00000			0.116248
$667$		0			0
$668$			16.0000i			0.619059i
$669$		0			0
$670$	16.0000	−	8.00000i	0.618134	−	0.309067i
$671$		0			0
$672$		0			0
$673$		−	36.0000i		−	1.38770i	−0.720121	−	0.693849i	$-0.755914\pi$
$673$		−	36.0000i		−	1.38770i	0.720121	−	0.693849i	$-0.244086\pi$
$674$	−8.00000			−0.308148
$675$		0			0
$676$	−9.00000			−0.346154
$677$		−	30.0000i		−	1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$		−	30.0000i		−	1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$		0			0
$679$	−20.0000			−0.767530
$680$	12.0000	−	6.00000i	0.460179	−	0.230089i
$681$		0			0
$682$		0			0
$683$		−	20.0000i		−	0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$		−	20.0000i		−	0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	−18.0000			−0.688247
$685$		0			0
$686$	20.0000			0.763604
$687$		0			0
$688$		−	4.00000i		−	0.152499i
$689$	−4.00000			−0.152388
$690$		0			0
$691$	4.00000			0.152167			0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000			0.152167			0.0760836	+	0.997101i	$0.475758\pi$
$692$		−	2.00000i		−	0.0760286i
$693$		0			0
$694$	−36.0000			−1.36654
$695$	−40.0000	+	20.0000i	−1.51729	+	0.758643i
$696$		0			0
$697$		−	60.0000i		−	2.27266i
$698$			30.0000i			1.13552i
$699$		0			0
$700$	−8.00000	−	6.00000i	−0.302372	−	0.226779i
$701$	−4.00000			−0.151078			−0.0755390	−	0.997143i	$-0.524068\pi$
$701$	−4.00000			−0.151078			−0.0755390	+	0.997143i	$0.524068\pi$
$702$		0			0
$703$			6.00000i			0.226294i
$704$		0			0
$705$		0			0
$706$	−18.0000			−0.677439
$707$		−	20.0000i		−	0.752177i
$708$		0			0
$709$	40.0000			1.50223			0.751116	−	0.660171i	$-0.229516\pi$
$709$	40.0000			1.50223			0.751116	+	0.660171i	$0.229516\pi$
$710$		0			0
$711$	−12.0000			−0.450035
$712$		−	6.00000i		−	0.224860i
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$		0			0
$716$	−14.0000			−0.523205
$717$		0			0
$718$		−	8.00000i		−	0.298557i
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$	−6.00000	+	3.00000i	−0.223607	+	0.111803i
$721$		0			0
$722$		−	17.0000i		−	0.632674i
$723$		0			0
$724$	−2.00000			−0.0743294
$725$		0			0
$726$		0			0
$727$		−	28.0000i		−	1.03846i	−0.854634	−	0.519231i	$-0.826218\pi$
$727$		−	28.0000i		−	1.03846i	0.854634	−	0.519231i	$-0.173782\pi$
$728$		−	4.00000i		−	0.148250i
$729$	27.0000			1.00000
$730$	−16.0000	+	8.00000i	−0.592187	+	0.296093i
$731$	24.0000			0.887672
$732$		0			0
$733$		−	38.0000i		−	1.40356i	−0.712393	−	0.701781i	$-0.752388\pi$
$733$		−	38.0000i		−	1.40356i	0.712393	−	0.701781i	$-0.247612\pi$
$734$	26.0000			0.959678
$735$		0			0
$736$	4.00000			0.147442
$737$		0			0
$738$			30.0000i			1.10432i
$739$	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	+	2.00000i	0.0367607	+	0.0735215i
$741$		0			0
$742$			4.00000i			0.146845i
$743$			26.0000i			0.953847i	0.878945	+	0.476924i	$0.158248\pi$
$743$			26.0000i			0.953847i	−0.878945	+	0.476924i	$0.841752\pi$
$744$		0			0
$745$	28.0000	−	14.0000i	1.02584	−	0.512920i
$746$	−34.0000			−1.24483
$747$		−	36.0000i		−	1.31717i
$748$		0			0
$749$	16.0000			0.584627
$750$		0			0
$751$	−32.0000			−1.16770			−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000			−1.16770			−0.583848	+	0.811863i	$0.698454\pi$
$752$			2.00000i			0.0729325i
$753$		0			0
$754$		0			0
$755$	−32.0000	+	16.0000i	−1.16460	+	0.582300i
$756$		0			0
$757$		−	14.0000i		−	0.508839i	−0.967094	−	0.254419i	$-0.918116\pi$
$757$		−	14.0000i		−	0.508839i	0.967094	−	0.254419i	$-0.0818843\pi$
$758$			12.0000i			0.435860i
$759$		0			0
$760$	−6.00000	−	12.0000i	−0.217643	−	0.435286i
$761$	34.0000			1.23250			0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000			1.23250			0.616250	+	0.787551i	$0.288651\pi$
$762$		0			0
$763$			40.0000i			1.44810i
$764$	−4.00000			−0.144715
$765$	−18.0000	−	36.0000i	−0.650791	−	1.30158i
$766$	−28.0000			−1.01168
$767$			12.0000i			0.433295i
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$		0			0
$772$		−	10.0000i		−	0.359908i
$773$		−	54.0000i		−	1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$		−	54.0000i		−	1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	−12.0000			−0.431331
$775$	−12.0000	+	16.0000i	−0.431053	+	0.574737i
$776$	−10.0000			−0.358979
$777$		0			0
$778$		−	16.0000i		−	0.573628i
$779$	−60.0000			−2.14972
$780$		0			0
$781$		0			0
$782$			24.0000i			0.858238i
$783$		0			0
$784$	3.00000			0.107143
$785$	22.0000	+	44.0000i	0.785214	+	1.57043i
$786$		0			0
$787$		−	40.0000i		−	1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$		−	40.0000i		−	1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$		−	10.0000i		−	0.356235i
$789$		0			0
$790$	−4.00000	−	8.00000i	−0.142314	−	0.284627i
$791$	−4.00000			−0.142224
$792$		0			0
$793$		0			0
$794$	−2.00000			−0.0709773
$795$		0			0
$796$	−20.0000			−0.708881
$797$		−	6.00000i		−	0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$		−	6.00000i		−	0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$		0			0
$799$	−12.0000			−0.424529
$800$	−4.00000	−	3.00000i	−0.141421	−	0.106066i
$801$	−18.0000			−0.635999
$802$			2.00000i			0.0706225i
$803$		0			0
$804$		0			0
$805$	16.0000	−	8.00000i	0.563926	−	0.281963i
$806$	−8.00000			−0.281788
$807$		0			0
$808$		−	10.0000i		−	0.351799i
$809$	6.00000			0.210949			0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000			0.210949			0.105474	+	0.994422i	$0.466364\pi$
$810$	9.00000	+	18.0000i	0.316228	+	0.632456i
$811$	−36.0000			−1.26413			−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000			−1.26413			−0.632065	+	0.774915i	$0.717793\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.0000	+	24.0000i	0.420342	+	0.840683i
$816$		0			0
$817$		−	24.0000i		−	0.839654i
$818$			14.0000i			0.489499i
$819$	−12.0000			−0.419314
$820$	−20.0000	+	10.0000i	−0.698430	+	0.349215i
$821$	−50.0000			−1.74501			−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000			−1.74501			−0.872506	+	0.488603i	$0.837507\pi$
$822$		0			0
$823$		−	26.0000i		−	0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$		−	26.0000i		−	0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$		0			0
$825$		0			0
$826$	12.0000			0.417533
$827$		−	36.0000i		−	1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$		−	36.0000i		−	1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$		−	12.0000i		−	0.417029i
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$	24.0000	−	12.0000i	0.833052	−	0.416526i
$831$		0			0
$832$		−	2.00000i		−	0.0693375i
$833$			18.0000i			0.623663i
$834$		0			0
$835$	16.0000	+	32.0000i	0.553703	+	1.10741i
$836$		0			0
$837$		0			0
$838$			20.0000i			0.690889i
$839$	48.0000			1.65714			0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000			1.65714			0.828572	+	0.559883i	$0.189154\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		−	36.0000i		−	1.24064i
$843$		0			0
$844$	16.0000			0.550743
$845$	−18.0000	+	9.00000i	−0.619219	+	0.309609i
$846$	6.00000			0.206284
$847$		−	22.0000i		−	0.755929i
$848$			2.00000i			0.0686803i
$849$		0			0
$850$	18.0000	−	24.0000i	0.617395	−	0.823193i
$851$	−4.00000			−0.137118
$852$		0			0
$853$			30.0000i			1.02718i	0.858036	+	0.513590i	$0.171685\pi$
$853$			30.0000i			1.02718i	−0.858036	+	0.513590i	$0.828315\pi$
$854$		0			0
$855$	−36.0000	+	18.0000i	−1.23117	+	0.615587i
$856$	8.00000			0.273434
$857$			2.00000i			0.0683187i	0.999416	+	0.0341593i	$0.0108754\pi$
$857$			2.00000i			0.0683187i	−0.999416	+	0.0341593i	$0.989125\pi$
$858$		0			0
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$	−4.00000	−	8.00000i	−0.136399	−	0.272798i
$861$		0			0
$862$		−	20.0000i		−	0.681203i
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$	−2.00000	−	4.00000i	−0.0680020	−	0.136004i
$866$	−16.0000			−0.543702
$867$		0			0
$868$			8.00000i			0.271538i
$869$		0			0
$870$		0			0
$871$	16.0000			0.542139
$872$			20.0000i			0.677285i
$873$			30.0000i			1.01535i
$874$	24.0000			0.811812
$875$	−22.0000	−	4.00000i	−0.743736	−	0.135225i
$876$		0			0
$877$			34.0000i			1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$			34.0000i			1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$			4.00000i			0.134993i
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		−	9.00000i		−	0.303046i
$883$			20.0000i			0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$			20.0000i			0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	12.0000			0.403604
$885$		0			0
$886$	16.0000			0.537531
$887$			50.0000i			1.67884i	0.543487	+	0.839418i	$0.317104\pi$
$887$			50.0000i			1.67884i	−0.543487	+	0.839418i	$0.682896\pi$
$888$		0			0
$889$	44.0000			1.47571
$890$	−6.00000	−	12.0000i	−0.201120	−	0.402241i
$891$		0			0
$892$		−	14.0000i		−	0.468755i
$893$			12.0000i			0.401565i
$894$		0			0
$895$	−28.0000	+	14.0000i	−0.935937	+	0.467968i
$896$	−2.00000			−0.0668153
$897$		0			0
$898$			30.0000i			1.00111i
$899$		0			0
$900$	−9.00000	+	12.0000i	−0.300000	+	0.400000i
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$	−2.00000			−0.0665190
$905$	−4.00000	+	2.00000i	−0.132964	+	0.0664822i
$906$		0			0
$907$			20.0000i			0.664089i	0.943264	+	0.332045i	$0.107738\pi$
$907$			20.0000i			0.664089i	−0.943264	+	0.332045i	$0.892262\pi$
$908$			20.0000i			0.663723i
$909$	−30.0000			−0.995037
$910$	−4.00000	−	8.00000i	−0.132599	−	0.265197i
$911$	−48.0000			−1.59031			−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000			−1.59031			−0.795155	+	0.606406i	$0.792611\pi$
$912$		0			0
$913$		0			0
$914$	10.0000			0.330771
$915$		0			0
$916$	2.00000			0.0660819
$917$			28.0000i			0.924641i
$918$		0			0
$919$	−8.00000			−0.263896			−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000			−0.263896			−0.131948	+	0.991257i	$0.542123\pi$
$920$	8.00000	−	4.00000i	0.263752	−	0.131876i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	4.00000	+	3.00000i	0.131519	+	0.0986394i
$926$	−4.00000			−0.131448
$927$		0			0
$928$		0			0
$929$	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
$931$	18.0000			0.589926
$932$			24.0000i			0.786146i
$933$		0			0
$934$	36.0000			1.17796
$935$		0			0
$936$	−6.00000			−0.196116
$937$		−	4.00000i		−	0.130674i	−0.997863	−	0.0653372i	$-0.979188\pi$
$937$		−	4.00000i		−	0.130674i	0.997863	−	0.0653372i	$-0.0208123\pi$
$938$		−	16.0000i		−	0.522419i
$939$		0			0
$940$	2.00000	+	4.00000i	0.0652328	+	0.130466i
$941$	−22.0000			−0.717180			−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000			−0.717180			−0.358590	+	0.933495i	$0.616742\pi$
$942$		0			0
$943$		−	40.0000i		−	1.30258i
$944$	6.00000			0.195283
$945$		0			0
$946$		0			0
$947$		−	28.0000i		−	0.909878i	−0.890523	−	0.454939i	$-0.849661\pi$
$947$		−	28.0000i		−	0.909878i	0.890523	−	0.454939i	$-0.150339\pi$
$948$		0			0
$949$	−16.0000			−0.519382
$950$	−24.0000	−	18.0000i	−0.778663	−	0.583997i
$951$		0			0
$952$		−	12.0000i		−	0.388922i
$953$		−	24.0000i		−	0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$		−	24.0000i		−	0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	6.00000			0.194257
$955$	−8.00000	+	4.00000i	−0.258874	+	0.129437i
$956$	28.0000			0.905585
$957$		0			0
$958$			12.0000i			0.387702i
$959$		0			0
$960$		0			0
$961$	−15.0000			−0.483871
$962$			2.00000i			0.0644826i
$963$		−	24.0000i		−	0.773389i
$964$	−10.0000			−0.322078
$965$	−10.0000	−	20.0000i	−0.321911	−	0.643823i
$966$		0			0
$967$			28.0000i			0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$			28.0000i			0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$		−	11.0000i		−	0.353553i
$969$		0			0
$970$	−20.0000	+	10.0000i	−0.642161	+	0.321081i
$971$	32.0000			1.02693			0.513464	−	0.858111i	$-0.328362\pi$
$971$	32.0000			1.02693			0.513464	+	0.858111i	$0.328362\pi$
$972$		0			0
$973$			40.0000i			1.28234i
$974$	−24.0000			−0.769010
$975$		0			0
$976$		0			0
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$		0			0
$980$	6.00000	−	3.00000i	0.191663	−	0.0958315i
$981$	60.0000			1.91565
$982$		−	24.0000i		−	0.765871i
$983$		−	38.0000i		−	1.21201i	−0.795460	−	0.606006i	$-0.792771\pi$
$983$		−	38.0000i		−	1.21201i	0.795460	−	0.606006i	$-0.207229\pi$
$984$		0			0
$985$	−10.0000	−	20.0000i	−0.318626	−	0.637253i
$986$		0			0
$987$		0			0
$988$		−	12.0000i		−	0.381771i
$989$	16.0000			0.508770
$990$		0			0
$991$	48.0000			1.52477			0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000			1.52477			0.762385	+	0.647124i	$0.224028\pi$
$992$			4.00000i			0.127000i
$993$		0			0
$994$		0			0
$995$	−40.0000	+	20.0000i	−1.26809	+	0.634043i
$996$		0			0
$997$			26.0000i			0.823428i	0.911313	+	0.411714i	$0.135070\pi$
$997$			26.0000i			0.823428i	−0.911313	+	0.411714i	$0.864930\pi$
$998$			30.0000i			0.949633i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	370.2.b.a.149.1	✓	2
3.2	odd	2		3330.2.d.f.1999.2		2
5.2	odd	4		1850.2.a.j.1.1		1
5.3	odd	4		1850.2.a.e.1.1		1
5.4	even	2	inner	370.2.b.a.149.2	yes	2
15.14	odd	2		3330.2.d.f.1999.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.b.a.149.1	✓	2	1.1	even	1	trivial
370.2.b.a.149.2	yes	2	5.4	even	2	inner
1850.2.a.e.1.1		1	5.3	odd	4
1850.2.a.j.1.1		1	5.2	odd	4
3330.2.d.f.1999.1		2	15.14	odd	2
3330.2.d.f.1999.2		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +2.00000i q^{7} +1.00000i q^{8} +3.00000 q^{9} +(1.00000 + 2.00000i) q^{10} +2.00000i q^{13} +2.00000 q^{14} +1.00000 q^{16} +6.00000i q^{17} -3.00000i q^{18} +6.00000 q^{19} +(2.00000 - 1.00000i) q^{20} +4.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +2.00000 q^{26} -2.00000i q^{28} -4.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} +(-2.00000 - 4.00000i) q^{35} -3.00000 q^{36} +1.00000i q^{37} -6.00000i q^{38} +(-1.00000 - 2.00000i) q^{40} -10.0000 q^{41} -4.00000i q^{43} +(-6.00000 + 3.00000i) q^{45} +4.00000 q^{46} +2.00000i q^{47} +3.00000 q^{49} +(-4.00000 - 3.00000i) q^{50} -2.00000i q^{52} +2.00000i q^{53} -2.00000 q^{56} +6.00000 q^{59} +4.00000i q^{62} +6.00000i q^{63} -1.00000 q^{64} +(-2.00000 - 4.00000i) q^{65} -8.00000i q^{67} -6.00000i q^{68} +(-4.00000 + 2.00000i) q^{70} +3.00000i q^{72} +8.00000i q^{73} +1.00000 q^{74} -6.00000 q^{76} -4.00000 q^{79} +(-2.00000 + 1.00000i) q^{80} +9.00000 q^{81} +10.0000i q^{82} -12.0000i q^{83} +(-6.00000 - 12.0000i) q^{85} -4.00000 q^{86} -6.00000 q^{89} +(3.00000 + 6.00000i) q^{90} -4.00000 q^{91} -4.00000i q^{92} +2.00000 q^{94} +(-12.0000 + 6.00000i) q^{95} +10.0000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 4 q^{5} + 6 q^{9} + 2 q^{10} + 4 q^{14} + 2 q^{16} + 12 q^{19} + 4 q^{20} + 6 q^{25} + 4 q^{26} - 8 q^{31} + 12 q^{34} - 4 q^{35} - 6 q^{36} - 2 q^{40} - 20 q^{41} - 12 q^{45} + 8 q^{46}+ \cdots - 24 q^{95}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^5 + 6 * q^9 + 2 * q^10 + 4 * q^14 + 2 * q^16 + 12 * q^19 + 4 * q^20 + 6 * q^25 + 4 * q^26 - 8 * q^31 + 12 * q^34 - 4 * q^35 - 6 * q^36 - 2 * q^40 - 20 * q^41 - 12 * q^45 + 8 * q^46 + 6 * q^49 - 8 * q^50 - 4 * q^56 + 12 * q^59 - 2 * q^64 - 4 * q^65 - 8 * q^70 + 2 * q^74 - 12 * q^76 - 8 * q^79 - 4 * q^80 + 18 * q^81 - 12 * q^85 - 8 * q^86 - 12 * q^89 + 6 * q^90 - 8 * q^91 + 4 * q^94 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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