LMFDB - Embedded newform 1850.2.b.a.149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, 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("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.b (of order $2$, degree $1$, not 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:	$14.7723243739$
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			149.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.a.149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -1.00000i q^{8} -6.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -1.00000i q^{8} -6.00000 q^{9} -1.00000 q^{11} -3.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -7.00000i q^{17} -6.00000i q^{18} -5.00000 q^{19} -1.00000i q^{22} -6.00000i q^{23} +3.00000 q^{24} -2.00000 q^{26} -9.00000i q^{27} -4.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +7.00000 q^{34} +6.00000 q^{36} +1.00000i q^{37} -5.00000i q^{38} -6.00000 q^{39} -3.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} +3.00000i q^{48} +7.00000 q^{49} +21.0000 q^{51} -2.00000i q^{52} -2.00000i q^{53} +9.00000 q^{54} -15.0000i q^{57} -4.00000 q^{59} -8.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +3.00000 q^{66} -13.0000i q^{67} +7.00000i q^{68} +18.0000 q^{69} -6.00000 q^{71} +6.00000i q^{72} +7.00000i q^{73} -1.00000 q^{74} +5.00000 q^{76} -6.00000i q^{78} -14.0000 q^{79} +9.00000 q^{81} -3.00000i q^{82} -3.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} +7.00000 q^{89} +6.00000i q^{92} -12.0000i q^{93} +4.00000 q^{94} -3.00000 q^{96} -18.0000i q^{97} +7.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9	$ 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 2 q^{11} + 2 q^{16} - 10 q^{19} + 6 q^{24} - 4 q^{26} - 8 q^{31} + 14 q^{34} + 12 q^{36} - 12 q^{39} - 6 q^{41} + 2 q^{44} + 12 q^{46} + 14 q^{49} + 42 q^{51} + 18 q^{54} - 8 q^{59} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 36 q^{69} - 12 q^{71} - 2 q^{74} + 10 q^{76} - 28 q^{79} + 18 q^{81} - 8 q^{86} + 14 q^{89} + 8 q^{94} - 6 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 - 2 * q^11 + 2 * q^16 - 10 * q^19 + 6 * q^24 - 4 * q^26 - 8 * q^31 + 14 * q^34 + 12 * q^36 - 12 * q^39 - 6 * q^41 + 2 * q^44 + 12 * q^46 + 14 * q^49 + 42 * q^51 + 18 * q^54 - 8 * q^59 - 16 * q^61 - 2 * q^64 + 6 * q^66 + 36 * q^69 - 12 * q^71 - 2 * q^74 + 10 * q^76 - 28 * q^79 + 18 * q^81 - 8 * q^86 + 14 * q^89 + 8 * q^94 - 6 * q^96 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

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
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−3.00000	−1.22474
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	− 1.00000i	− 0.353553i
$9$	−6.00000	−2.00000
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	− 3.00000i	− 0.866025i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 7.00000i	− 1.69775i	−0.528594	−	0.848875i	$-0.677281\pi$
$17$	− 7.00000i	− 1.69775i	0.528594	−	0.848875i	$-0.322719\pi$
$18$	− 6.00000i	− 1.41421i
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−2.00000	−0.392232
$27$	− 9.00000i	− 1.73205i
$28$	0	0
$29$	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
$33$	− 3.00000i	− 0.522233i
$34$	7.00000	1.20049
$35$	0	0
$36$	6.00000	1.00000
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	− 5.00000i	− 0.811107i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	6.00000	0.884652
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	3.00000i	0.433013i
$49$	7.00000	1.00000
$50$	0	0
$51$	21.0000	2.94059
$52$	− 2.00000i	− 0.277350i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	9.00000	1.22474
$55$	0	0
$56$	0	0
$57$	− 15.0000i	− 1.98680i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	− 13.0000i	− 1.58820i	−0.607785	−	0.794101i	$-0.707942\pi$
$67$	− 13.0000i	− 1.58820i	0.607785	−	0.794101i	$-0.292058\pi$
$68$	7.00000i	0.848875i
$69$	18.0000	2.16695
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	6.00000i	0.707107i
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	5.00000	0.573539
$77$	0	0
$78$	− 6.00000i	− 0.679366i
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 3.00000i	− 0.331295i
$83$	− 3.00000i	− 0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$	− 3.00000i	− 0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	1.00000i	0.106600i
$89$	7.00000	0.741999	0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000	0.741999	0.370999	+	0.928633i	$0.379015\pi$
$90$	0	0
$91$	0	0
$92$	6.00000i	0.625543i
$93$	− 12.0000i	− 1.24434i
$94$	4.00000	0.412568
$95$	0	0
$96$	−3.00000	−0.306186
$97$	− 18.0000i	− 1.82762i	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	− 18.0000i	− 1.82762i	0.406138	−	0.913812i	$-0.366875\pi$
$98$	7.00000i	0.707107i
$99$	6.00000	0.603023
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	21.0000i	2.07931i
$103$	18.0000i	1.77359i	0.462160	+	0.886796i	$0.347074\pi$
$103$	18.0000i	1.77359i	−0.462160	+	0.886796i	$0.652926\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	5.00000i	0.483368i	0.970355	+	0.241684i	$0.0776998\pi$
$107$	5.00000i	0.483368i	−0.970355	+	0.241684i	$0.922300\pi$
$108$	9.00000i	0.866025i
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	17.0000i	1.59923i	0.600516	+	0.799613i	$0.294962\pi$
$113$	17.0000i	1.59923i	−0.600516	+	0.799613i	$0.705038\pi$
$114$	15.0000	1.40488
$115$	0	0
$116$	0	0
$117$	− 12.0000i	− 1.10940i
$118$	− 4.00000i	− 0.368230i
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	− 8.00000i	− 0.724286i
$123$	− 9.00000i	− 0.811503i
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	4.00000i	0.354943i	0.984126	+	0.177471i	$0.0567917\pi$
$127$	4.00000i	0.354943i	−0.984126	+	0.177471i	$0.943208\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−12.0000	−1.05654
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	3.00000i	0.261116i
$133$	0	0
$134$	13.0000	1.12303
$135$	0	0
$136$	−7.00000	−0.600245
$137$	15.0000i	1.28154i	0.767734	+	0.640768i	$0.221384\pi$
$137$	15.0000i	1.28154i	−0.767734	+	0.640768i	$0.778616\pi$
$138$	18.0000i	1.53226i
$139$	1.00000	0.0848189	0.0424094	−	0.999100i	$-0.486497\pi$
$139$	1.00000	0.0848189	0.0424094	+	0.999100i	$0.486497\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	− 6.00000i	− 0.503509i
$143$	− 2.00000i	− 0.167248i
$144$	−6.00000	−0.500000
$145$	0	0
$146$	−7.00000	−0.579324
$147$	21.0000i	1.73205i
$148$	− 1.00000i	− 0.0821995i
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	5.00000i	0.405554i
$153$	42.0000i	3.39550i
$154$	0	0
$155$	0	0
$156$	6.00000	0.480384
$157$	12.0000i	0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$	12.0000i	0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$	− 14.0000i	− 1.11378i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	9.00000i	0.707107i
$163$	− 1.00000i	− 0.0783260i	−0.999233	−	0.0391630i	$-0.987531\pi$
$163$	− 1.00000i	− 0.0783260i	0.999233	−	0.0391630i	$-0.0124692\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	3.00000	0.232845
$167$	− 2.00000i	− 0.154765i	−0.997001	−	0.0773823i	$-0.975344\pi$
$167$	− 2.00000i	− 0.154765i	0.997001	−	0.0773823i	$-0.0246562\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	30.0000	2.29416
$172$	− 4.00000i	− 0.304997i
$173$	− 8.00000i	− 0.608229i	−0.952636	−	0.304114i	$-0.901639\pi$
$173$	− 8.00000i	− 0.608229i	0.952636	−	0.304114i	$-0.0983605\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	− 12.0000i	− 0.901975i
$178$	7.00000i	0.524672i
$179$	−13.0000	−0.971666	−0.485833	−	0.874052i	$-0.661484\pi$
$179$	−13.0000	−0.971666	−0.485833	+	0.874052i	$0.661484\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	− 24.0000i	− 1.77413i
$184$	−6.00000	−0.442326
$185$	0	0
$186$	12.0000	0.879883
$187$	7.00000i	0.511891i
$188$	4.00000i	0.291730i
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	− 3.00000i	− 0.216506i
$193$	− 25.0000i	− 1.79954i	−0.436365	−	0.899770i	$-0.643734\pi$
$193$	− 25.0000i	− 1.79954i	0.436365	−	0.899770i	$-0.356266\pi$
$194$	18.0000	1.29232
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 8.00000i	− 0.569976i	−0.958531	−	0.284988i	$-0.908010\pi$
$197$	− 8.00000i	− 0.569976i	0.958531	−	0.284988i	$-0.0919897\pi$
$198$	6.00000i	0.426401i
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	39.0000	2.75085
$202$	6.00000i	0.422159i
$203$	0	0
$204$	−21.0000	−1.47029
$205$	0	0
$206$	−18.0000	−1.25412
$207$	36.0000i	2.50217i
$208$	2.00000i	0.138675i
$209$	5.00000	0.345857
$210$	0	0
$211$	13.0000	0.894957	0.447478	−	0.894295i	$-0.352322\pi$
$211$	13.0000	0.894957	0.447478	+	0.894295i	$0.352322\pi$
$212$	2.00000i	0.137361i
$213$	− 18.0000i	− 1.23334i
$214$	−5.00000	−0.341793
$215$	0	0
$216$	−9.00000	−0.612372
$217$	0	0
$218$	6.00000i	0.406371i
$219$	−21.0000	−1.41905
$220$	0	0
$221$	14.0000	0.941742
$222$	− 3.00000i	− 0.201347i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	−17.0000	−1.13082
$227$	28.0000i	1.85843i	0.369546	+	0.929213i	$0.379513\pi$
$227$	28.0000i	1.85843i	−0.369546	+	0.929213i	$0.620487\pi$
$228$	15.0000i	0.993399i
$229$	−24.0000	−1.58596	−0.792982	−	0.609245i	$-0.791473\pi$
$229$	−24.0000	−1.58596	−0.792982	+	0.609245i	$0.791473\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.0000i	0.917170i	0.888650	+	0.458585i	$0.151644\pi$
$233$	14.0000i	0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$	12.0000	0.784465
$235$	0	0
$236$	4.00000	0.260378
$237$	− 42.0000i	− 2.72819i
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	− 10.0000i	− 0.642824i
$243$	0	0
$244$	8.00000	0.512148
$245$	0	0
$246$	9.00000	0.573819
$247$	− 10.0000i	− 0.636285i
$248$	4.00000i	0.254000i
$249$	9.00000	0.570352
$250$	0	0
$251$	−1.00000	−0.0631194	−0.0315597	−	0.999502i	$-0.510047\pi$
$251$	−1.00000	−0.0631194	−0.0315597	+	0.999502i	$0.510047\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	− 12.0000i	− 0.747087i
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 12.0000i	− 0.741362i
$263$	− 14.0000i	− 0.863277i	−0.902047	−	0.431638i	$-0.857936\pi$
$263$	− 14.0000i	− 0.863277i	0.902047	−	0.431638i	$-0.142064\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	0	0
$267$	21.0000i	1.28518i
$268$	13.0000i	0.794101i
$269$	4.00000	0.243884	0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000	0.243884	0.121942	+	0.992537i	$0.461088\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	− 7.00000i	− 0.424437i
$273$	0	0
$274$	−15.0000	−0.906183
$275$	0	0
$276$	−18.0000	−1.08347
$277$	− 20.0000i	− 1.20168i	−0.799368	−	0.600842i	$-0.794832\pi$
$277$	− 20.0000i	− 1.20168i	0.799368	−	0.600842i	$-0.205168\pi$
$278$	1.00000i	0.0599760i
$279$	24.0000	1.43684
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	12.0000i	0.714590i
$283$	7.00000i	0.416107i	0.978117	+	0.208053i	$0.0667128\pi$
$283$	7.00000i	0.416107i	−0.978117	+	0.208053i	$0.933287\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	− 6.00000i	− 0.353553i
$289$	−32.0000	−1.88235
$290$	0	0
$291$	54.0000	3.16554
$292$	− 7.00000i	− 0.409644i
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	−21.0000	−1.22474
$295$	0	0
$296$	1.00000	0.0581238
$297$	9.00000i	0.522233i
$298$	22.0000i	1.27443i
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	− 18.0000i	− 1.03578i
$303$	18.0000i	1.03407i
$304$	−5.00000	−0.286770
$305$	0	0
$306$	−42.0000	−2.40098
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	0	0
$309$	−54.0000	−3.07195
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	6.00000i	0.339683i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−12.0000	−0.677199
$315$	0	0
$316$	14.0000	0.787562
$317$	− 2.00000i	− 0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$	− 2.00000i	− 0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	0	0
$321$	−15.0000	−0.837218
$322$	0	0
$323$	35.0000i	1.94745i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	1.00000	0.0553849
$327$	18.0000i	0.995402i
$328$	3.00000i	0.165647i
$329$	0	0
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	3.00000i	0.164646i
$333$	− 6.00000i	− 0.328798i
$334$	2.00000	0.109435
$335$	0	0
$336$	0	0
$337$	− 31.0000i	− 1.68868i	−0.535810	−	0.844339i	$-0.679994\pi$
$337$	− 31.0000i	− 1.68868i	0.535810	−	0.844339i	$-0.320006\pi$
$338$	9.00000i	0.489535i
$339$	−51.0000	−2.76994
$340$	0	0
$341$	4.00000	0.216612
$342$	30.0000i	1.62221i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	8.00000	0.430083
$347$	23.0000i	1.23470i	0.786687	+	0.617352i	$0.211795\pi$
$347$	23.0000i	1.23470i	−0.786687	+	0.617352i	$0.788205\pi$
$348$	0	0
$349$	12.0000	0.642345	0.321173	−	0.947021i	$-0.395923\pi$
$349$	12.0000	0.642345	0.321173	+	0.947021i	$0.395923\pi$
$350$	0	0
$351$	18.0000	0.960769
$352$	− 1.00000i	− 0.0533002i
$353$	− 6.00000i	− 0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$	− 6.00000i	− 0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−7.00000	−0.370999
$357$	0	0
$358$	− 13.0000i	− 0.687071i
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	− 2.00000i	− 0.105118i
$363$	− 30.0000i	− 1.57459i
$364$	0	0
$365$	0	0
$366$	24.0000	1.25450
$367$	− 26.0000i	− 1.35719i	−0.734513	−	0.678594i	$-0.762589\pi$
$367$	− 26.0000i	− 1.35719i	0.734513	−	0.678594i	$-0.237411\pi$
$368$	− 6.00000i	− 0.312772i
$369$	18.0000	0.937043
$370$	0	0
$371$	0	0
$372$	12.0000i	0.622171i
$373$	− 32.0000i	− 1.65690i	−0.560065	−	0.828449i	$-0.689224\pi$
$373$	− 32.0000i	− 1.65690i	0.560065	−	0.828449i	$-0.310776\pi$
$374$	−7.00000	−0.361961
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	0	0
$379$	19.0000	0.975964	0.487982	−	0.872854i	$-0.337733\pi$
$379$	19.0000	0.975964	0.487982	+	0.872854i	$0.337733\pi$
$380$	0	0
$381$	−12.0000	−0.614779
$382$	6.00000i	0.306987i
$383$	− 36.0000i	− 1.83951i	−0.392488	−	0.919757i	$-0.628386\pi$
$383$	− 36.0000i	− 1.83951i	0.392488	−	0.919757i	$-0.371614\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	25.0000	1.27247
$387$	− 24.0000i	− 1.21999i
$388$	18.0000i	0.913812i
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	−42.0000	−2.12403
$392$	− 7.00000i	− 0.353553i
$393$	− 36.0000i	− 1.81596i
$394$	8.00000	0.403034
$395$	0	0
$396$	−6.00000	−0.301511
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	− 16.0000i	− 0.802008i
$399$	0	0
$400$	0	0
$401$	23.0000	1.14857	0.574283	−	0.818657i	$-0.305281\pi$
$401$	23.0000	1.14857	0.574283	+	0.818657i	$0.305281\pi$
$402$	39.0000i	1.94514i
$403$	− 8.00000i	− 0.398508i
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	− 1.00000i	− 0.0495682i
$408$	− 21.0000i	− 1.03965i
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	−45.0000	−2.21969
$412$	− 18.0000i	− 0.886796i
$413$	0	0
$414$	−36.0000	−1.76930
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	3.00000i	0.146911i
$418$	5.00000i	0.244558i
$419$	17.0000	0.830504	0.415252	−	0.909706i	$-0.363693\pi$
$419$	17.0000	0.830504	0.415252	+	0.909706i	$0.363693\pi$
$420$	0	0
$421$	24.0000	1.16969	0.584844	−	0.811146i	$-0.301156\pi$
$421$	24.0000	1.16969	0.584844	+	0.811146i	$0.301156\pi$
$422$	13.0000i	0.632830i
$423$	24.0000i	1.16692i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	18.0000	0.872103
$427$	0	0
$428$	− 5.00000i	− 0.241684i
$429$	6.00000	0.289683
$430$	0	0
$431$	−38.0000	−1.83040	−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000	−1.83040	−0.915198	+	0.403005i	$0.867966\pi$
$432$	− 9.00000i	− 0.433013i
$433$	− 11.0000i	− 0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$	− 11.0000i	− 0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	30.0000i	1.43509i
$438$	− 21.0000i	− 1.00342i
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	−42.0000	−2.00000
$442$	14.0000i	0.665912i
$443$	− 15.0000i	− 0.712672i	−0.934358	−	0.356336i	$-0.884026\pi$
$443$	− 15.0000i	− 0.712672i	0.934358	−	0.356336i	$-0.115974\pi$
$444$	3.00000	0.142374
$445$	0	0
$446$	16.0000	0.757622
$447$	66.0000i	3.12169i
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	3.00000	0.141264
$452$	− 17.0000i	− 0.799613i
$453$	− 54.0000i	− 2.53714i
$454$	−28.0000	−1.31411
$455$	0	0
$456$	−15.0000	−0.702439
$457$	21.0000i	0.982339i	0.871064	+	0.491169i	$0.163430\pi$
$457$	21.0000i	0.982339i	−0.871064	+	0.491169i	$0.836570\pi$
$458$	− 24.0000i	− 1.12145i
$459$	−63.0000	−2.94059
$460$	0	0
$461$	4.00000	0.186299	0.0931493	−	0.995652i	$-0.470307\pi$
$461$	4.00000	0.186299	0.0931493	+	0.995652i	$0.470307\pi$
$462$	0	0
$463$	26.0000i	1.20832i	0.796862	+	0.604161i	$0.206492\pi$
$463$	26.0000i	1.20832i	−0.796862	+	0.604161i	$0.793508\pi$
$464$	0	0
$465$	0	0
$466$	−14.0000	−0.648537
$467$	− 4.00000i	− 0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$	− 4.00000i	− 0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	12.0000i	0.554700i
$469$	0	0
$470$	0	0
$471$	−36.0000	−1.65879
$472$	4.00000i	0.184115i
$473$	− 4.00000i	− 0.183920i
$474$	42.0000	1.92912
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	− 12.0000i	− 0.548867i
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	− 25.0000i	− 1.13872i
$483$	0	0
$484$	10.0000	0.454545
$485$	0	0
$486$	0	0
$487$	16.0000i	0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$	16.0000i	0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$	8.00000i	0.362143i
$489$	3.00000	0.135665
$490$	0	0
$491$	44.0000	1.98569	0.992846	−	0.119401i	$-0.0380974\pi$
$491$	44.0000	1.98569	0.992846	+	0.119401i	$0.0380974\pi$
$492$	9.00000i	0.405751i
$493$	0	0
$494$	10.0000	0.449921
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	9.00000i	0.403300i
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	− 1.00000i	− 0.0446322i
$503$	36.0000i	1.60516i	0.596544	+	0.802580i	$0.296540\pi$
$503$	36.0000i	1.60516i	−0.596544	+	0.802580i	$0.703460\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	27.0000i	1.19911i
$508$	− 4.00000i	− 0.177471i
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	0.0441942i
$513$	45.0000i	1.98680i
$514$	2.00000	0.0882162
$515$	0	0
$516$	12.0000	0.528271
$517$	4.00000i	0.175920i
$518$	0	0
$519$	24.0000	1.05348
$520$	0	0
$521$	−35.0000	−1.53338	−0.766689	−	0.642019i	$-0.778097\pi$
$521$	−35.0000	−1.53338	−0.766689	+	0.642019i	$0.778097\pi$
$522$	0	0
$523$	45.0000i	1.96771i	0.178960	+	0.983856i	$0.442727\pi$
$523$	45.0000i	1.96771i	−0.178960	+	0.983856i	$0.557273\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	14.0000	0.610429
$527$	28.0000i	1.21970i
$528$	− 3.00000i	− 0.130558i
$529$	−13.0000	−0.565217
$530$	0	0
$531$	24.0000	1.04151
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	−21.0000	−0.908759
$535$	0	0
$536$	−13.0000	−0.561514
$537$	− 39.0000i	− 1.68297i
$538$	4.00000i	0.172452i
$539$	−7.00000	−0.301511
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	16.0000i	0.687259i
$543$	− 6.00000i	− 0.257485i
$544$	7.00000	0.300123
$545$	0	0
$546$	0	0
$547$	11.0000i	0.470326i	0.971956	+	0.235163i	$0.0755624\pi$
$547$	11.0000i	0.470326i	−0.971956	+	0.235163i	$0.924438\pi$
$548$	− 15.0000i	− 0.640768i
$549$	48.0000	2.04859
$550$	0	0
$551$	0	0
$552$	− 18.0000i	− 0.766131i
$553$	0	0
$554$	20.0000	0.849719
$555$	0	0
$556$	−1.00000	−0.0424094
$557$	40.0000i	1.69485i	0.530912	+	0.847427i	$0.321850\pi$
$557$	40.0000i	1.69485i	−0.530912	+	0.847427i	$0.678150\pi$
$558$	24.0000i	1.01600i
$559$	−8.00000	−0.338364
$560$	0	0
$561$	−21.0000	−0.886621
$562$	− 22.0000i	− 0.928014i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	−7.00000	−0.294232
$567$	0	0
$568$	6.00000i	0.251754i
$569$	−39.0000	−1.63497	−0.817483	−	0.575953i	$-0.804631\pi$
$569$	−39.0000	−1.63497	−0.817483	+	0.575953i	$0.804631\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	2.00000i	0.0836242i
$573$	18.0000i	0.751961i
$574$	0	0
$575$	0	0
$576$	6.00000	0.250000
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	− 32.0000i	− 1.33102i
$579$	75.0000	3.11689
$580$	0	0
$581$	0	0
$582$	54.0000i	2.23837i
$583$	2.00000i	0.0828315i
$584$	7.00000	0.289662
$585$	0	0
$586$	24.0000	0.991431
$587$	− 3.00000i	− 0.123823i	−0.998082	−	0.0619116i	$-0.980280\pi$
$587$	− 3.00000i	− 0.123823i	0.998082	−	0.0619116i	$-0.0197197\pi$
$588$	− 21.0000i	− 0.866025i
$589$	20.0000	0.824086
$590$	0	0
$591$	24.0000	0.987228
$592$	1.00000i	0.0410997i
$593$	− 41.0000i	− 1.68367i	−0.539736	−	0.841834i	$-0.681476\pi$
$593$	− 41.0000i	− 1.68367i	0.539736	−	0.841834i	$-0.318524\pi$
$594$	−9.00000	−0.369274
$595$	0	0
$596$	−22.0000	−0.901155
$597$	− 48.0000i	− 1.96451i
$598$	12.0000i	0.490716i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−17.0000	−0.693444	−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000	−0.693444	−0.346722	+	0.937968i	$0.612705\pi$
$602$	0	0
$603$	78.0000i	3.17641i
$604$	18.0000	0.732410
$605$	0	0
$606$	−18.0000	−0.731200
$607$	− 18.0000i	− 0.730597i	−0.930890	−	0.365299i	$-0.880967\pi$
$607$	− 18.0000i	− 0.730597i	0.930890	−	0.365299i	$-0.119033\pi$
$608$	− 5.00000i	− 0.202777i
$609$	0	0
$610$	0	0
$611$	8.00000	0.323645
$612$	− 42.0000i	− 1.69775i
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	0	0
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	− 54.0000i	− 2.17220i
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	−54.0000	−2.16695
$622$	− 24.0000i	− 0.962312i
$623$	0	0
$624$	−6.00000	−0.240192
$625$	0	0
$626$	10.0000	0.399680
$627$	15.0000i	0.599042i
$628$	− 12.0000i	− 0.478852i
$629$	7.00000	0.279108
$630$	0	0
$631$	30.0000	1.19428	0.597141	−	0.802137i	$-0.296303\pi$
$631$	30.0000	1.19428	0.597141	+	0.802137i	$0.296303\pi$
$632$	14.0000i	0.556890i
$633$	39.0000i	1.55011i
$634$	2.00000	0.0794301
$635$	0	0
$636$	−6.00000	−0.237915
$637$	14.0000i	0.554700i
$638$	0	0
$639$	36.0000	1.42414
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	− 15.0000i	− 0.592003i
$643$	28.0000i	1.10421i	0.833774	+	0.552106i	$0.186176\pi$
$643$	28.0000i	1.10421i	−0.833774	+	0.552106i	$0.813824\pi$
$644$	0	0
$645$	0	0
$646$	−35.0000	−1.37706
$647$	− 38.0000i	− 1.49393i	−0.664861	−	0.746967i	$-0.731509\pi$
$647$	− 38.0000i	− 1.49393i	0.664861	−	0.746967i	$-0.268491\pi$
$648$	− 9.00000i	− 0.353553i
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	1.00000i	0.0391630i
$653$	40.0000i	1.56532i	0.622449	+	0.782660i	$0.286138\pi$
$653$	40.0000i	1.56532i	−0.622449	+	0.782660i	$0.713862\pi$
$654$	−18.0000	−0.703856
$655$	0	0
$656$	−3.00000	−0.117130
$657$	− 42.0000i	− 1.63858i
$658$	0	0
$659$	−5.00000	−0.194772	−0.0973862	−	0.995247i	$-0.531048\pi$
$659$	−5.00000	−0.194772	−0.0973862	+	0.995247i	$0.531048\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	− 19.0000i	− 0.738456i
$663$	42.0000i	1.63114i
$664$	−3.00000	−0.116423
$665$	0	0
$666$	6.00000	0.232495
$667$	0	0
$668$	2.00000i	0.0773823i
$669$	48.0000	1.85579
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	31.0000	1.19408
$675$	0	0
$676$	−9.00000	−0.346154
$677$	− 8.00000i	− 0.307465i	−0.988113	−	0.153732i	$-0.950871\pi$
$677$	− 8.00000i	− 0.307465i	0.988113	−	0.153732i	$-0.0491294\pi$
$678$	− 51.0000i	− 1.95864i
$679$	0	0
$680$	0	0
$681$	−84.0000	−3.21889
$682$	4.00000i	0.153168i
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	−30.0000	−1.14708
$685$	0	0
$686$	0	0
$687$	− 72.0000i	− 2.74697i
$688$	4.00000i	0.152499i
$689$	4.00000	0.152388
$690$	0	0
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	8.00000i	0.304114i
$693$	0	0
$694$	−23.0000	−0.873068
$695$	0	0
$696$	0	0
$697$	21.0000i	0.795432i
$698$	12.0000i	0.454207i
$699$	−42.0000	−1.58859
$700$	0	0
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	18.0000i	0.679366i
$703$	− 5.00000i	− 0.188579i
$704$	1.00000	0.0376889
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	12.0000i	0.450988i
$709$	−24.0000	−0.901339	−0.450669	−	0.892691i	$-0.648815\pi$
$709$	−24.0000	−0.901339	−0.450669	+	0.892691i	$0.648815\pi$
$710$	0	0
$711$	84.0000	3.15025
$712$	− 7.00000i	− 0.262336i
$713$	24.0000i	0.898807i
$714$	0	0
$715$	0	0
$716$	13.0000	0.485833
$717$	− 36.0000i	− 1.34444i
$718$	6.00000i	0.223918i
$719$	−8.00000	−0.298350	−0.149175	−	0.988811i	$-0.547662\pi$
$719$	−8.00000	−0.298350	−0.149175	+	0.988811i	$0.547662\pi$
$720$	0	0
$721$	0	0
$722$	6.00000i	0.223297i
$723$	− 75.0000i	− 2.78928i
$724$	2.00000	0.0743294
$725$	0	0
$726$	30.0000	1.11340
$727$	20.0000i	0.741759i	0.928681	+	0.370879i	$0.120944\pi$
$727$	20.0000i	0.741759i	−0.928681	+	0.370879i	$0.879056\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	28.0000	1.03562
$732$	24.0000i	0.887066i
$733$	− 32.0000i	− 1.18195i	−0.806691	−	0.590973i	$-0.798744\pi$
$733$	− 32.0000i	− 1.18195i	0.806691	−	0.590973i	$-0.201256\pi$
$734$	26.0000	0.959678
$735$	0	0
$736$	6.00000	0.221163
$737$	13.0000i	0.478861i
$738$	18.0000i	0.662589i
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	30.0000	1.10208
$742$	0	0
$743$	54.0000i	1.98107i	0.137268	+	0.990534i	$0.456168\pi$
$743$	54.0000i	1.98107i	−0.137268	+	0.990534i	$0.543832\pi$
$744$	−12.0000	−0.439941
$745$	0	0
$746$	32.0000	1.17160
$747$	18.0000i	0.658586i
$748$	− 7.00000i	− 0.255945i
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	− 4.00000i	− 0.145865i
$753$	− 3.00000i	− 0.109326i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.0000i	1.16306i	0.813525	+	0.581530i	$0.197546\pi$
$757$	32.0000i	1.16306i	−0.813525	+	0.581530i	$0.802454\pi$
$758$	19.0000i	0.690111i
$759$	−18.0000	−0.653359
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	− 12.0000i	− 0.434714i
$763$	0	0
$764$	−6.00000	−0.217072
$765$	0	0
$766$	36.0000	1.30073
$767$	− 8.00000i	− 0.288863i
$768$	3.00000i	0.108253i
$769$	21.0000	0.757279	0.378640	−	0.925544i	$-0.376392\pi$
$769$	21.0000	0.757279	0.378640	+	0.925544i	$0.376392\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	25.0000i	0.899770i
$773$	− 22.0000i	− 0.791285i	−0.918405	−	0.395643i	$-0.870522\pi$
$773$	− 22.0000i	− 0.791285i	0.918405	−	0.395643i	$-0.129478\pi$
$774$	24.0000	0.862662
$775$	0	0
$776$	−18.0000	−0.646162
$777$	0	0
$778$	8.00000i	0.286814i
$779$	15.0000	0.537431
$780$	0	0
$781$	6.00000	0.214697
$782$	− 42.0000i	− 1.50192i
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	36.0000	1.28408
$787$	− 24.0000i	− 0.855508i	−0.903895	−	0.427754i	$-0.859305\pi$
$787$	− 24.0000i	− 0.855508i	0.903895	−	0.427754i	$-0.140695\pi$
$788$	8.00000i	0.284988i
$789$	42.0000	1.49524
$790$	0	0
$791$	0	0
$792$	− 6.00000i	− 0.213201i
$793$	− 16.0000i	− 0.568177i
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	16.0000	0.567105
$797$	12.0000i	0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$	12.0000i	0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$	0	0
$799$	−28.0000	−0.990569
$800$	0	0
$801$	−42.0000	−1.48400
$802$	23.0000i	0.812158i
$803$	− 7.00000i	− 0.247025i
$804$	−39.0000	−1.37542
$805$	0	0
$806$	8.00000	0.281788
$807$	12.0000i	0.422420i
$808$	− 6.00000i	− 0.211079i
$809$	22.0000	0.773479	0.386739	−	0.922189i	$-0.373601\pi$
$809$	22.0000	0.773479	0.386739	+	0.922189i	$0.373601\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	48.0000i	1.68343i
$814$	1.00000	0.0350500
$815$	0	0
$816$	21.0000	0.735147
$817$	− 20.0000i	− 0.699711i
$818$	− 11.0000i	− 0.384606i
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	− 45.0000i	− 1.56956i
$823$	− 34.0000i	− 1.18517i	−0.805510	−	0.592583i	$-0.798108\pi$
$823$	− 34.0000i	− 1.18517i	0.805510	−	0.592583i	$-0.201892\pi$
$824$	18.0000	0.627060
$825$	0	0
$826$	0	0
$827$	33.0000i	1.14752i	0.819023	+	0.573761i	$0.194516\pi$
$827$	33.0000i	1.14752i	−0.819023	+	0.573761i	$0.805484\pi$
$828$	− 36.0000i	− 1.25109i
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	60.0000	2.08138
$832$	− 2.00000i	− 0.0693375i
$833$	− 49.0000i	− 1.69775i
$834$	−3.00000	−0.103882
$835$	0	0
$836$	−5.00000	−0.172929
$837$	36.0000i	1.24434i
$838$	17.0000i	0.587255i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	24.0000i	0.827095i
$843$	− 66.0000i	− 2.27316i
$844$	−13.0000	−0.447478
$845$	0	0
$846$	−24.0000	−0.825137
$847$	0	0
$848$	− 2.00000i	− 0.0686803i
$849$	−21.0000	−0.720718
$850$	0	0
$851$	6.00000	0.205677
$852$	18.0000i	0.616670i
$853$	− 20.0000i	− 0.684787i	−0.939557	−	0.342393i	$-0.888762\pi$
$853$	− 20.0000i	− 0.684787i	0.939557	−	0.342393i	$-0.111238\pi$
$854$	0	0
$855$	0	0
$856$	5.00000	0.170896
$857$	9.00000i	0.307434i	0.988115	+	0.153717i	$0.0491244\pi$
$857$	9.00000i	0.307434i	−0.988115	+	0.153717i	$0.950876\pi$
$858$	6.00000i	0.204837i
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	0	0
$862$	− 38.0000i	− 1.29429i
$863$	− 6.00000i	− 0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$	− 6.00000i	− 0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$	9.00000	0.306186
$865$	0	0
$866$	11.0000	0.373795
$867$	− 96.0000i	− 3.26033i
$868$	0	0
$869$	14.0000	0.474917
$870$	0	0
$871$	26.0000	0.880976
$872$	− 6.00000i	− 0.203186i
$873$	108.000i	3.65525i
$874$	−30.0000	−1.01477
$875$	0	0
$876$	21.0000	0.709524
$877$	42.0000i	1.41824i	0.705088	+	0.709120i	$0.250907\pi$
$877$	42.0000i	1.41824i	−0.705088	+	0.709120i	$0.749093\pi$
$878$	32.0000i	1.07995i
$879$	72.0000	2.42850
$880$	0	0
$881$	30.0000	1.01073	0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000	1.01073	0.505363	+	0.862907i	$0.331359\pi$
$882$	− 42.0000i	− 1.41421i
$883$	− 7.00000i	− 0.235569i	−0.993039	−	0.117784i	$-0.962421\pi$
$883$	− 7.00000i	− 0.235569i	0.993039	−	0.117784i	$-0.0375792\pi$
$884$	−14.0000	−0.470871
$885$	0	0
$886$	15.0000	0.503935
$887$	52.0000i	1.74599i	0.487730	+	0.872995i	$0.337825\pi$
$887$	52.0000i	1.74599i	−0.487730	+	0.872995i	$0.662175\pi$
$888$	3.00000i	0.100673i
$889$	0	0
$890$	0	0
$891$	−9.00000	−0.301511
$892$	16.0000i	0.535720i
$893$	20.0000i	0.669274i
$894$	−66.0000	−2.20737
$895$	0	0
$896$	0	0
$897$	36.0000i	1.20201i
$898$	− 15.0000i	− 0.500556i
$899$	0	0
$900$	0	0
$901$	−14.0000	−0.466408
$902$	3.00000i	0.0998891i
$903$	0	0
$904$	17.0000	0.565412
$905$	0	0
$906$	54.0000	1.79403
$907$	− 28.0000i	− 0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$	− 28.0000i	− 0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$	− 28.0000i	− 0.929213i
$909$	−36.0000	−1.19404
$910$	0	0
$911$	30.0000	0.993944	0.496972	−	0.867766i	$-0.334445\pi$
$911$	30.0000	0.993944	0.496972	+	0.867766i	$0.334445\pi$
$912$	− 15.0000i	− 0.496700i
$913$	3.00000i	0.0992855i
$914$	−21.0000	−0.694618
$915$	0	0
$916$	24.0000	0.792982
$917$	0	0
$918$	− 63.0000i	− 2.07931i
$919$	−14.0000	−0.461817	−0.230909	−	0.972975i	$-0.574170\pi$
$919$	−14.0000	−0.461817	−0.230909	+	0.972975i	$0.574170\pi$
$920$	0	0
$921$	21.0000	0.691974
$922$	4.00000i	0.131733i
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	−26.0000	−0.854413
$927$	− 108.000i	− 3.54719i
$928$	0	0
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−35.0000	−1.14708
$932$	− 14.0000i	− 0.458585i
$933$	− 72.0000i	− 2.35717i
$934$	4.00000	0.130884
$935$	0	0
$936$	−12.0000	−0.392232
$937$	− 7.00000i	− 0.228680i	−0.993442	−	0.114340i	$-0.963525\pi$
$937$	− 7.00000i	− 0.228680i	0.993442	−	0.114340i	$-0.0364753\pi$
$938$	0	0
$939$	30.0000	0.979013
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	− 36.0000i	− 1.17294i
$943$	18.0000i	0.586161i
$944$	−4.00000	−0.130189
$945$	0	0
$946$	4.00000	0.130051
$947$	− 56.0000i	− 1.81976i	−0.414876	−	0.909878i	$-0.636175\pi$
$947$	− 56.0000i	− 1.81976i	0.414876	−	0.909878i	$-0.363825\pi$
$948$	42.0000i	1.36410i
$949$	−14.0000	−0.454459
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	9.00000i	0.291539i	0.989319	+	0.145769i	$0.0465657\pi$
$953$	9.00000i	0.291539i	−0.989319	+	0.145769i	$0.953434\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 4.00000i	− 0.129234i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 2.00000i	− 0.0644826i
$963$	− 30.0000i	− 0.966736i
$964$	25.0000	0.805196
$965$	0	0
$966$	0	0
$967$	32.0000i	1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$	32.0000i	1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	10.0000i	0.321412i
$969$	−105.000	−3.37309
$970$	0	0
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	0	0
$974$	−16.0000	−0.512673
$975$	0	0
$976$	−8.00000	−0.256074
$977$	− 43.0000i	− 1.37569i	−0.725857	−	0.687846i	$-0.758556\pi$
$977$	− 43.0000i	− 1.37569i	0.725857	−	0.687846i	$-0.241444\pi$
$978$	3.00000i	0.0959294i
$979$	−7.00000	−0.223721
$980$	0	0
$981$	−36.0000	−1.14939
$982$	44.0000i	1.40410i
$983$	42.0000i	1.33959i	0.742545	+	0.669796i	$0.233618\pi$
$983$	42.0000i	1.33959i	−0.742545	+	0.669796i	$0.766382\pi$
$984$	−9.00000	−0.286910
$985$	0	0
$986$	0	0
$987$	0	0
$988$	10.0000i	0.318142i
$989$	24.0000	0.763156
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	− 4.00000i	− 0.127000i
$993$	− 57.0000i	− 1.80884i
$994$	0	0
$995$	0	0
$996$	−9.00000	−0.285176
$997$	− 10.0000i	− 0.316703i	−0.987383	−	0.158352i	$-0.949382\pi$
$997$	− 10.0000i	− 0.316703i	0.987383	−	0.158352i	$-0.0506179\pi$
$998$	− 16.0000i	− 0.506471i
$999$	9.00000	0.284747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.a.149.2		2
5.2	odd	4		1850.2.a.g.1.1	✓	1
5.3	odd	4		1850.2.a.h.1.1	yes	1
5.4	even	2	inner	1850.2.b.a.149.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.g.1.1	✓	1	5.2	odd	4
1850.2.a.h.1.1	yes	1	5.3	odd	4
1850.2.b.a.149.1		2	5.4	even	2	inner
1850.2.b.a.149.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -1.00000i q^{8} -6.00000 q^{9} -1.00000 q^{11} -3.00000i q^{12} +2.00000i q^{13} +1.00000 q^{16} -7.00000i q^{17} -6.00000i q^{18} -5.00000 q^{19} -1.00000i q^{22} -6.00000i q^{23} +3.00000 q^{24} -2.00000 q^{26} -9.00000i q^{27} -4.00000 q^{31} +1.00000i q^{32} -3.00000i q^{33} +7.00000 q^{34} +6.00000 q^{36} +1.00000i q^{37} -5.00000i q^{38} -6.00000 q^{39} -3.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} +3.00000i q^{48} +7.00000 q^{49} +21.0000 q^{51} -2.00000i q^{52} -2.00000i q^{53} +9.00000 q^{54} -15.0000i q^{57} -4.00000 q^{59} -8.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} +3.00000 q^{66} -13.0000i q^{67} +7.00000i q^{68} +18.0000 q^{69} -6.00000 q^{71} +6.00000i q^{72} +7.00000i q^{73} -1.00000 q^{74} +5.00000 q^{76} -6.00000i q^{78} -14.0000 q^{79} +9.00000 q^{81} -3.00000i q^{82} -3.00000i q^{83} -4.00000 q^{86} +1.00000i q^{88} +7.00000 q^{89} +6.00000i q^{92} -12.0000i q^{93} +4.00000 q^{94} -3.00000 q^{96} -18.0000i q^{97} +7.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9	\( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} - 2 q^{11} + 2 q^{16} - 10 q^{19} + 6 q^{24} - 4 q^{26} - 8 q^{31} + 14 q^{34} + 12 q^{36} - 12 q^{39} - 6 q^{41} + 2 q^{44} + 12 q^{46} + 14 q^{49} + 42 q^{51} + 18 q^{54} - 8 q^{59} - 16 q^{61} - 2 q^{64} + 6 q^{66} + 36 q^{69} - 12 q^{71} - 2 q^{74} + 10 q^{76} - 28 q^{79} + 18 q^{81} - 8 q^{86} + 14 q^{89} + 8 q^{94} - 6 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^6 - 12 * q^9 - 2 * q^11 + 2 * q^16 - 10 * q^19 + 6 * q^24 - 4 * q^26 - 8 * q^31 + 14 * q^34 + 12 * q^36 - 12 * q^39 - 6 * q^41 + 2 * q^44 + 12 * q^46 + 14 * q^49 + 42 * q^51 + 18 * q^54 - 8 * q^59 - 16 * q^61 - 2 * q^64 + 6 * q^66 + 36 * q^69 - 12 * q^71 - 2 * q^74 + 10 * q^76 - 28 * q^79 + 18 * q^81 - 8 * q^86 + 14 * q^89 + 8 * q^94 - 6 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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