LMFDB - Embedded newform 1850.2.b.e.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.e.149.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +2.00000i q^{18} +3.00000 q^{19} -4.00000 q^{21} +3.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} -5.00000i q^{27} +4.00000i q^{28} +1.00000i q^{32} -3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} +1.00000i q^{37} +3.00000i q^{38} +6.00000 q^{39} -3.00000 q^{41} -4.00000i q^{42} -4.00000i q^{43} -3.00000 q^{44} -2.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -3.00000 q^{51} -6.00000i q^{52} -2.00000i q^{53} +5.00000 q^{54} -4.00000 q^{56} -3.00000i q^{57} +12.0000 q^{59} +12.0000 q^{61} -8.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -9.00000i q^{67} +3.00000i q^{68} +2.00000 q^{69} -2.00000 q^{71} -2.00000i q^{72} -9.00000i q^{73} -1.00000 q^{74} -3.00000 q^{76} -12.0000i q^{77} +6.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} -3.00000i q^{82} -7.00000i q^{83} +4.00000 q^{84} +4.00000 q^{86} -3.00000i q^{88} +3.00000 q^{89} +24.0000 q^{91} -2.00000i q^{92} +4.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -9.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9	$ 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 6 q^{19} - 8 q^{21} - 2 q^{24} - 12 q^{26} + 6 q^{34} - 4 q^{36} + 12 q^{39} - 6 q^{41} - 6 q^{44} - 4 q^{46} - 18 q^{49} - 6 q^{51} + 10 q^{54} - 8 q^{56} + 24 q^{59} + 24 q^{61} - 2 q^{64} + 6 q^{66} + 4 q^{69} - 4 q^{71} - 2 q^{74} - 6 q^{76} + 4 q^{79} + 2 q^{81} + 8 q^{84} + 8 q^{86} + 6 q^{89} + 48 q^{91} + 8 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 + 6 * q^11 + 8 * q^14 + 2 * q^16 + 6 * q^19 - 8 * q^21 - 2 * q^24 - 12 * q^26 + 6 * q^34 - 4 * q^36 + 12 * q^39 - 6 * q^41 - 6 * q^44 - 4 * q^46 - 18 * q^49 - 6 * q^51 + 10 * q^54 - 8 * q^56 + 24 * q^59 + 24 * q^61 - 2 * q^64 + 6 * q^66 + 4 * q^69 - 4 * q^71 - 2 * q^74 - 6 * q^76 + 4 * q^79 + 2 * q^81 + 8 * q^84 + 8 * q^86 + 6 * q^89 + 48 * q^91 + 8 * q^94 + 2 * 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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	− 1.00000i	− 0.353553i
$9$	2.00000	0.666667
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	1.00000i	0.288675i
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	2.00000i	0.471405i
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	3.00000i	0.639602i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.00000	−1.17670
$27$	− 5.00000i	− 0.962250i
$28$	4.00000i	0.755929i
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	− 3.00000i	− 0.522233i
$34$	3.00000	0.514496
$35$	0	0
$36$	−2.00000	−0.333333
$37$	1.00000i	0.164399i
$37$	1.00000i	0.164399i
$38$	3.00000i	0.486664i
$39$	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	$0.575267\pi$
$42$	− 4.00000i	− 0.617213i
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−2.00000	−0.294884
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	− 1.00000i	− 0.144338i
$49$	−9.00000	−1.28571
$50$	0	0
$51$	−3.00000	−0.420084
$52$	− 6.00000i	− 0.832050i
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	5.00000	0.680414
$55$	0	0
$56$	−4.00000	−0.534522
$57$	− 3.00000i	− 0.397360i
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	12.0000	1.53644	0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000	1.53644	0.768221	+	0.640184i	$0.221142\pi$
$62$	0	0
$63$	− 8.00000i	− 1.00791i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	3.00000	0.369274
$67$	− 9.00000i	− 1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$	− 9.00000i	− 1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$	3.00000i	0.363803i
$69$	2.00000	0.240772
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	− 2.00000i	− 0.235702i
$73$	− 9.00000i	− 1.05337i	−0.850060	−	0.526685i	$-0.823435\pi$
$73$	− 9.00000i	− 1.05337i	0.850060	−	0.526685i	$-0.176565\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−3.00000	−0.344124
$77$	− 12.0000i	− 1.36753i
$78$	6.00000i	0.679366i
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 3.00000i	− 0.331295i
$83$	− 7.00000i	− 0.768350i	−0.923260	−	0.384175i	$-0.874486\pi$
$83$	− 7.00000i	− 0.768350i	0.923260	−	0.384175i	$-0.125514\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	24.0000	2.51588
$92$	− 2.00000i	− 0.208514i
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	1.00000	0.102062
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 9.00000i	− 0.909137i
$99$	6.00000	0.603023
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 3.00000i	− 0.297044i
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	2.00000	0.194257
$107$	1.00000i	0.0966736i	0.998831	+	0.0483368i	$0.0153921\pi$
$107$	1.00000i	0.0966736i	−0.998831	+	0.0483368i	$0.984608\pi$
$108$	5.00000i	0.481125i
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	0	0
$111$	1.00000	0.0949158
$112$	− 4.00000i	− 0.377964i
$113$	5.00000i	0.470360i	0.971952	+	0.235180i	$0.0755680\pi$
$113$	5.00000i	0.470360i	−0.971952	+	0.235180i	$0.924432\pi$
$114$	3.00000	0.280976
$115$	0	0
$116$	0	0
$117$	12.0000i	1.10940i
$118$	12.0000i	1.10469i
$119$	−12.0000	−1.10004
$120$	0	0
$121$	−2.00000	−0.181818
$122$	12.0000i	1.08643i
$123$	3.00000i	0.270501i
$124$	0	0
$125$	0	0
$126$	8.00000	0.712697
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	−4.00000	−0.352180
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	3.00000i	0.261116i
$133$	− 12.0000i	− 1.04053i
$134$	9.00000	0.777482
$135$	0	0
$136$	−3.00000	−0.257248
$137$	7.00000i	0.598050i	0.954245	+	0.299025i	$0.0966615\pi$
$137$	7.00000i	0.598050i	−0.954245	+	0.299025i	$0.903339\pi$
$138$	2.00000i	0.170251i
$139$	−3.00000	−0.254457	−0.127228	−	0.991873i	$-0.540608\pi$
$139$	−3.00000	−0.254457	−0.127228	+	0.991873i	$0.540608\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	− 2.00000i	− 0.167836i
$143$	18.0000i	1.50524i
$144$	2.00000	0.166667
$145$	0	0
$146$	9.00000	0.744845
$147$	9.00000i	0.742307i
$148$	− 1.00000i	− 0.0821995i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	− 3.00000i	− 0.243332i
$153$	− 6.00000i	− 0.485071i
$154$	12.0000	0.966988
$155$	0	0
$156$	−6.00000	−0.480384
$157$	− 24.0000i	− 1.91541i	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	− 24.0000i	− 1.91541i	0.287754	−	0.957704i	$-0.407091\pi$
$158$	2.00000i	0.159111i
$159$	−2.00000	−0.158610
$160$	0	0
$161$	8.00000	0.630488
$162$	1.00000i	0.0785674i
$163$	− 17.0000i	− 1.33154i	−0.746156	−	0.665771i	$-0.768103\pi$
$163$	− 17.0000i	− 1.33154i	0.746156	−	0.665771i	$-0.231897\pi$
$164$	3.00000	0.234261
$165$	0	0
$166$	7.00000	0.543305
$167$	18.0000i	1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$167$	18.0000i	1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$	4.00000i	0.308607i
$169$	−23.0000	−1.76923
$170$	0	0
$171$	6.00000	0.458831
$172$	4.00000i	0.304997i
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	− 12.0000i	− 0.901975i
$178$	3.00000i	0.224860i
$179$	−21.0000	−1.56961	−0.784807	−	0.619740i	$-0.787238\pi$
$179$	−21.0000	−1.56961	−0.784807	+	0.619740i	$0.787238\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	24.0000i	1.77900i
$183$	− 12.0000i	− 0.887066i
$184$	2.00000	0.147442
$185$	0	0
$186$	0	0
$187$	− 9.00000i	− 0.658145i
$188$	4.00000i	0.291730i
$189$	−20.0000	−1.45479
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	1.00000i	0.0721688i
$193$	11.0000i	0.791797i	0.918294	+	0.395899i	$0.129567\pi$
$193$	11.0000i	0.791797i	−0.918294	+	0.395899i	$0.870433\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	6.00000i	0.426401i
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−9.00000	−0.634811
$202$	− 10.0000i	− 0.703598i
$203$	0	0
$204$	3.00000	0.210042
$205$	0	0
$206$	−14.0000	−0.975426
$207$	4.00000i	0.278019i
$208$	6.00000i	0.416025i
$209$	9.00000	0.622543
$210$	0	0
$211$	9.00000	0.619586	0.309793	−	0.950804i	$-0.399740\pi$
$211$	9.00000	0.619586	0.309793	+	0.950804i	$0.399740\pi$
$212$	2.00000i	0.137361i
$213$	2.00000i	0.137038i
$214$	−1.00000	−0.0683586
$215$	0	0
$216$	−5.00000	−0.340207
$217$	0	0
$218$	− 18.0000i	− 1.21911i
$219$	−9.00000	−0.608164
$220$	0	0
$221$	18.0000	1.21081
$222$	1.00000i	0.0671156i
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	4.00000	0.267261
$225$	0	0
$226$	−5.00000	−0.332595
$227$	− 28.0000i	− 1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$	− 28.0000i	− 1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$	3.00000i	0.198680i
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	−12.0000	−0.789542
$232$	0	0
$233$	− 26.0000i	− 1.70332i	−0.524097	−	0.851658i	$-0.675597\pi$
$233$	− 26.0000i	− 1.70332i	0.524097	−	0.851658i	$-0.324403\pi$
$234$	−12.0000	−0.784465
$235$	0	0
$236$	−12.0000	−0.781133
$237$	− 2.00000i	− 0.129914i
$238$	− 12.0000i	− 0.777844i
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−21.0000	−1.35273	−0.676364	−	0.736567i	$-0.736446\pi$
$241$	−21.0000	−1.35273	−0.676364	+	0.736567i	$0.736446\pi$
$242$	− 2.00000i	− 0.128565i
$243$	− 16.0000i	− 1.02640i
$244$	−12.0000	−0.768221
$245$	0	0
$246$	−3.00000	−0.191273
$247$	18.0000i	1.14531i
$248$	0	0
$249$	−7.00000	−0.443607
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	8.00000i	0.503953i
$253$	6.00000i	0.377217i
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000i	1.37232i	0.727450	+	0.686161i	$0.240706\pi$
$257$	22.0000i	1.37232i	−0.727450	+	0.686161i	$0.759294\pi$
$258$	− 4.00000i	− 0.249029i
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	12.0000i	0.741362i
$263$	26.0000i	1.60323i	0.597841	+	0.801614i	$0.296025\pi$
$263$	26.0000i	1.60323i	−0.597841	+	0.801614i	$0.703975\pi$
$264$	−3.00000	−0.184637
$265$	0	0
$266$	12.0000	0.735767
$267$	− 3.00000i	− 0.183597i
$268$	9.00000i	0.549762i
$269$	28.0000	1.70719	0.853595	−	0.520937i	$-0.174417\pi$
$269$	28.0000	1.70719	0.853595	+	0.520937i	$0.174417\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	− 3.00000i	− 0.181902i
$273$	− 24.0000i	− 1.45255i
$274$	−7.00000	−0.422885
$275$	0	0
$276$	−2.00000	−0.120386
$277$	− 28.0000i	− 1.68236i	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	− 28.0000i	− 1.68236i	0.540758	−	0.841178i	$-0.318138\pi$
$278$	− 3.00000i	− 0.179928i
$279$	0	0
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	− 4.00000i	− 0.238197i
$283$	7.00000i	0.416107i	0.978117	+	0.208053i	$0.0667128\pi$
$283$	7.00000i	0.416107i	−0.978117	+	0.208053i	$0.933287\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	−18.0000	−1.06436
$287$	12.0000i	0.708338i
$288$	2.00000i	0.117851i
$289$	8.00000	0.470588
$290$	0	0
$291$	−2.00000	−0.117242
$292$	9.00000i	0.526685i
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	−9.00000	−0.524891
$295$	0	0
$296$	1.00000	0.0581238
$297$	− 15.0000i	− 0.870388i
$298$	18.0000i	1.04271i
$299$	−12.0000	−0.693978
$300$	0	0
$301$	−16.0000	−0.922225
$302$	22.0000i	1.26596i
$303$	10.0000i	0.574485i
$304$	3.00000	0.172062
$305$	0	0
$306$	6.00000	0.342997
$307$	29.0000i	1.65512i	0.561379	+	0.827559i	$0.310271\pi$
$307$	29.0000i	1.65512i	−0.561379	+	0.827559i	$0.689729\pi$
$308$	12.0000i	0.683763i
$309$	14.0000	0.796432
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	− 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	$-0.0912024\pi$
$314$	24.0000	1.35440
$315$	0	0
$316$	−2.00000	−0.112509
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	− 2.00000i	− 0.112154i
$319$	0	0
$320$	0	0
$321$	1.00000	0.0558146
$322$	8.00000i	0.445823i
$323$	− 9.00000i	− 0.500773i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	17.0000	0.941543
$327$	18.0000i	0.995402i
$328$	3.00000i	0.165647i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	7.00000i	0.384175i
$333$	2.00000i	0.109599i
$334$	−18.0000	−0.984916
$335$	0	0
$336$	−4.00000	−0.218218
$337$	9.00000i	0.490261i	0.969490	+	0.245131i	$0.0788309\pi$
$337$	9.00000i	0.490261i	−0.969490	+	0.245131i	$0.921169\pi$
$338$	− 23.0000i	− 1.25104i
$339$	5.00000	0.271563
$340$	0	0
$341$	0	0
$342$	6.00000i	0.324443i
$343$	8.00000i	0.431959i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−24.0000	−1.29025
$347$	31.0000i	1.66417i	0.554650	+	0.832084i	$0.312852\pi$
$347$	31.0000i	1.66417i	−0.554650	+	0.832084i	$0.687148\pi$
$348$	0	0
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	0	0
$351$	30.0000	1.60128
$352$	3.00000i	0.159901i
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−3.00000	−0.159000
$357$	12.0000i	0.635107i
$358$	− 21.0000i	− 1.10988i
$359$	−34.0000	−1.79445	−0.897226	−	0.441572i	$-0.854421\pi$
$359$	−34.0000	−1.79445	−0.897226	+	0.441572i	$0.854421\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	− 10.0000i	− 0.525588i
$363$	2.00000i	0.104973i
$364$	−24.0000	−1.25794
$365$	0	0
$366$	12.0000	0.627250
$367$	2.00000i	0.104399i	0.998637	+	0.0521996i	$0.0166232\pi$
$367$	2.00000i	0.104399i	−0.998637	+	0.0521996i	$0.983377\pi$
$368$	2.00000i	0.104257i
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	28.0000i	1.44979i	0.688862	+	0.724893i	$0.258111\pi$
$373$	28.0000i	1.44979i	−0.688862	+	0.724893i	$0.741889\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−4.00000	−0.206284
$377$	0	0
$378$	− 20.0000i	− 1.02869i
$379$	−1.00000	−0.0513665	−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000	−0.0513665	−0.0256833	+	0.999670i	$0.508176\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	6.00000i	0.306987i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−11.0000	−0.559885
$387$	− 8.00000i	− 0.406663i
$388$	2.00000i	0.101535i
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	9.00000i	0.454569i
$393$	− 12.0000i	− 0.605320i
$394$	12.0000	0.604551
$395$	0	0
$396$	−6.00000	−0.301511
$397$	30.0000i	1.50566i	0.658217	+	0.752828i	$0.271311\pi$
$397$	30.0000i	1.50566i	−0.658217	+	0.752828i	$0.728689\pi$
$398$	− 8.00000i	− 0.401004i
$399$	−12.0000	−0.600751
$400$	0	0
$401$	−5.00000	−0.249688	−0.124844	−	0.992176i	$-0.539843\pi$
$401$	−5.00000	−0.249688	−0.124844	+	0.992176i	$0.539843\pi$
$402$	− 9.00000i	− 0.448879i
$403$	0	0
$404$	10.0000	0.497519
$405$	0	0
$406$	0	0
$407$	3.00000i	0.148704i
$408$	3.00000i	0.148522i
$409$	−15.0000	−0.741702	−0.370851	−	0.928692i	$-0.620934\pi$
$409$	−15.0000	−0.741702	−0.370851	+	0.928692i	$0.620934\pi$
$410$	0	0
$411$	7.00000	0.345285
$412$	− 14.0000i	− 0.689730i
$413$	− 48.0000i	− 2.36193i
$414$	−4.00000	−0.196589
$415$	0	0
$416$	−6.00000	−0.294174
$417$	3.00000i	0.146911i
$418$	9.00000i	0.440204i
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	0	0
$421$	−32.0000	−1.55958	−0.779792	−	0.626038i	$-0.784675\pi$
$421$	−32.0000	−1.55958	−0.779792	+	0.626038i	$0.784675\pi$
$422$	9.00000i	0.438113i
$423$	− 8.00000i	− 0.388973i
$424$	−2.00000	−0.0971286
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	− 48.0000i	− 2.32288i
$428$	− 1.00000i	− 0.0483368i
$429$	18.0000	0.869048
$430$	0	0
$431$	2.00000	0.0963366	0.0481683	−	0.998839i	$-0.484662\pi$
$431$	2.00000	0.0963366	0.0481683	+	0.998839i	$0.484662\pi$
$432$	− 5.00000i	− 0.240563i
$433$	5.00000i	0.240285i	0.992757	+	0.120142i	$0.0383351\pi$
$433$	5.00000i	0.240285i	−0.992757	+	0.120142i	$0.961665\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	6.00000i	0.287019i
$438$	− 9.00000i	− 0.430037i
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−18.0000	−0.857143
$442$	18.0000i	0.856173i
$443$	5.00000i	0.237557i	0.992921	+	0.118779i	$0.0378979\pi$
$443$	5.00000i	0.237557i	−0.992921	+	0.118779i	$0.962102\pi$
$444$	−1.00000	−0.0474579
$445$	0	0
$446$	−8.00000	−0.378811
$447$	− 18.0000i	− 0.851371i
$448$	4.00000i	0.188982i
$449$	37.0000	1.74614	0.873069	−	0.487597i	$-0.162126\pi$
$449$	37.0000	1.74614	0.873069	+	0.487597i	$0.162126\pi$
$450$	0	0
$451$	−9.00000	−0.423793
$452$	− 5.00000i	− 0.235180i
$453$	− 22.0000i	− 1.03365i
$454$	28.0000	1.31411
$455$	0	0
$456$	−3.00000	−0.140488
$457$	33.0000i	1.54367i	0.635820	+	0.771837i	$0.280662\pi$
$457$	33.0000i	1.54367i	−0.635820	+	0.771837i	$0.719338\pi$
$458$	0	0
$459$	−15.0000	−0.700140
$460$	0	0
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	− 12.0000i	− 0.558291i
$463$	6.00000i	0.278844i	0.990233	+	0.139422i	$0.0445244\pi$
$463$	6.00000i	0.278844i	−0.990233	+	0.139422i	$0.955476\pi$
$464$	0	0
$465$	0	0
$466$	26.0000	1.20443
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	− 12.0000i	− 0.554700i
$469$	−36.0000	−1.66233
$470$	0	0
$471$	−24.0000	−1.10586
$472$	− 12.0000i	− 0.552345i
$473$	− 12.0000i	− 0.551761i
$474$	2.00000	0.0918630
$475$	0	0
$476$	12.0000	0.550019
$477$	− 4.00000i	− 0.183147i
$478$	12.0000i	0.548867i
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	− 21.0000i	− 0.956524i
$483$	− 8.00000i	− 0.364013i
$484$	2.00000	0.0909091
$485$	0	0
$486$	16.0000	0.725775
$487$	− 28.0000i	− 1.26880i	−0.773004	−	0.634401i	$-0.781247\pi$
$487$	− 28.0000i	− 1.26880i	0.773004	−	0.634401i	$-0.218753\pi$
$488$	− 12.0000i	− 0.543214i
$489$	−17.0000	−0.768767
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	− 3.00000i	− 0.135250i
$493$	0	0
$494$	−18.0000	−0.809858
$495$	0	0
$496$	0	0
$497$	8.00000i	0.358849i
$498$	− 7.00000i	− 0.313678i
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	− 9.00000i	− 0.401690i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	−8.00000	−0.356348
$505$	0	0
$506$	−6.00000	−0.266733
$507$	23.0000i	1.02147i
$508$	8.00000i	0.354943i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−36.0000	−1.59255
$512$	1.00000i	0.0441942i
$513$	− 15.0000i	− 0.662266i
$514$	−22.0000	−0.970378
$515$	0	0
$516$	4.00000	0.176090
$517$	− 12.0000i	− 0.527759i
$518$	4.00000i	0.175750i
$519$	24.0000	1.05348
$520$	0	0
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	0	0
$523$	37.0000i	1.61790i	0.587879	+	0.808949i	$0.299963\pi$
$523$	37.0000i	1.61790i	−0.587879	+	0.808949i	$0.700037\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−26.0000	−1.13365
$527$	0	0
$528$	− 3.00000i	− 0.130558i
$529$	19.0000	0.826087
$530$	0	0
$531$	24.0000	1.04151
$532$	12.0000i	0.520266i
$533$	− 18.0000i	− 0.779667i
$534$	3.00000	0.129823
$535$	0	0
$536$	−9.00000	−0.388741
$537$	21.0000i	0.906217i
$538$	28.0000i	1.20717i
$539$	−27.0000	−1.16297
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	− 28.0000i	− 1.20270i
$543$	10.0000i	0.429141i
$544$	3.00000	0.128624
$545$	0	0
$546$	24.0000	1.02711
$547$	− 5.00000i	− 0.213785i	−0.994271	−	0.106892i	$-0.965910\pi$
$547$	− 5.00000i	− 0.213785i	0.994271	−	0.106892i	$-0.0340900\pi$
$548$	− 7.00000i	− 0.299025i
$549$	24.0000	1.02430
$550$	0	0
$551$	0	0
$552$	− 2.00000i	− 0.0851257i
$553$	− 8.00000i	− 0.340195i
$554$	28.0000	1.18961
$555$	0	0
$556$	3.00000	0.127228
$557$	− 32.0000i	− 1.35588i	−0.735116	−	0.677942i	$-0.762872\pi$
$557$	− 32.0000i	− 1.35588i	0.735116	−	0.677942i	$-0.237128\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	−9.00000	−0.379980
$562$	− 6.00000i	− 0.253095i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	4.00000	0.168430
$565$	0	0
$566$	−7.00000	−0.294232
$567$	− 4.00000i	− 0.167984i
$568$	2.00000i	0.0839181i
$569$	−19.0000	−0.796521	−0.398261	−	0.917272i	$-0.630386\pi$
$569$	−19.0000	−0.796521	−0.398261	+	0.917272i	$0.630386\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 18.0000i	− 0.752618i
$573$	− 6.00000i	− 0.250654i
$574$	−12.0000	−0.500870
$575$	0	0
$576$	−2.00000	−0.0833333
$577$	15.0000i	0.624458i	0.950007	+	0.312229i	$0.101076\pi$
$577$	15.0000i	0.624458i	−0.950007	+	0.312229i	$0.898924\pi$
$578$	8.00000i	0.332756i
$579$	11.0000	0.457144
$580$	0	0
$581$	−28.0000	−1.16164
$582$	− 2.00000i	− 0.0829027i
$583$	− 6.00000i	− 0.248495i
$584$	−9.00000	−0.372423
$585$	0	0
$586$	−12.0000	−0.495715
$587$	13.0000i	0.536567i	0.963340	+	0.268284i	$0.0864565\pi$
$587$	13.0000i	0.536567i	−0.963340	+	0.268284i	$0.913544\pi$
$588$	− 9.00000i	− 0.371154i
$589$	0	0
$590$	0	0
$591$	−12.0000	−0.493614
$592$	1.00000i	0.0410997i
$593$	− 9.00000i	− 0.369586i	−0.982777	−	0.184793i	$-0.940839\pi$
$593$	− 9.00000i	− 0.369586i	0.982777	−	0.184793i	$-0.0591614\pi$
$594$	15.0000	0.615457
$595$	0	0
$596$	−18.0000	−0.737309
$597$	8.00000i	0.327418i
$598$	− 12.0000i	− 0.490716i
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	−33.0000	−1.34610	−0.673049	−	0.739598i	$-0.735016\pi$
$601$	−33.0000	−1.34610	−0.673049	+	0.739598i	$0.735016\pi$
$602$	− 16.0000i	− 0.652111i
$603$	− 18.0000i	− 0.733017i
$604$	−22.0000	−0.895167
$605$	0	0
$606$	−10.0000	−0.406222
$607$	26.0000i	1.05531i	0.849460	+	0.527654i	$0.176928\pi$
$607$	26.0000i	1.05531i	−0.849460	+	0.527654i	$0.823072\pi$
$608$	3.00000i	0.121666i
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	6.00000i	0.242536i
$613$	− 46.0000i	− 1.85792i	−0.370177	−	0.928961i	$-0.620703\pi$
$613$	− 46.0000i	− 1.85792i	0.370177	−	0.928961i	$-0.379297\pi$
$614$	−29.0000	−1.17034
$615$	0	0
$616$	−12.0000	−0.483494
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	14.0000i	0.563163i
$619$	16.0000	0.643094	0.321547	−	0.946894i	$-0.395797\pi$
$619$	16.0000	0.643094	0.321547	+	0.946894i	$0.395797\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	28.0000i	1.12270i
$623$	− 12.0000i	− 0.480770i
$624$	6.00000	0.240192
$625$	0	0
$626$	10.0000	0.399680
$627$	− 9.00000i	− 0.359425i
$628$	24.0000i	0.957704i
$629$	3.00000	0.119618
$630$	0	0
$631$	22.0000	0.875806	0.437903	−	0.899022i	$-0.355721\pi$
$631$	22.0000	0.875806	0.437903	+	0.899022i	$0.355721\pi$
$632$	− 2.00000i	− 0.0795557i
$633$	− 9.00000i	− 0.357718i
$634$	−18.0000	−0.714871
$635$	0	0
$636$	2.00000	0.0793052
$637$	− 54.0000i	− 2.13956i
$638$	0	0
$639$	−4.00000	−0.158238
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	1.00000i	0.0394669i
$643$	20.0000i	0.788723i	0.918955	+	0.394362i	$0.129034\pi$
$643$	20.0000i	0.788723i	−0.918955	+	0.394362i	$0.870966\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	9.00000	0.354100
$647$	− 42.0000i	− 1.65119i	−0.564263	−	0.825595i	$-0.690840\pi$
$647$	− 42.0000i	− 1.65119i	0.564263	−	0.825595i	$-0.309160\pi$
$648$	− 1.00000i	− 0.0392837i
$649$	36.0000	1.41312
$650$	0	0
$651$	0	0
$652$	17.0000i	0.665771i
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	−18.0000	−0.703856
$655$	0	0
$656$	−3.00000	−0.117130
$657$	− 18.0000i	− 0.702247i
$658$	− 16.0000i	− 0.623745i
$659$	−9.00000	−0.350590	−0.175295	−	0.984516i	$-0.556088\pi$
$659$	−9.00000	−0.350590	−0.175295	+	0.984516i	$0.556088\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	− 3.00000i	− 0.116598i
$663$	− 18.0000i	− 0.699062i
$664$	−7.00000	−0.271653
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	0	0
$668$	− 18.0000i	− 0.696441i
$669$	8.00000	0.309298
$670$	0	0
$671$	36.0000	1.38976
$672$	− 4.00000i	− 0.154303i
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	−9.00000	−0.346667
$675$	0	0
$676$	23.0000	0.884615
$677$	8.00000i	0.307465i	0.988113	+	0.153732i	$0.0491294\pi$
$677$	8.00000i	0.307465i	−0.988113	+	0.153732i	$0.950871\pi$
$678$	5.00000i	0.192024i
$679$	−8.00000	−0.307012
$680$	0	0
$681$	−28.0000	−1.07296
$682$	0	0
$683$	17.0000i	0.650487i	0.945630	+	0.325243i	$0.105446\pi$
$683$	17.0000i	0.650487i	−0.945630	+	0.325243i	$0.894554\pi$
$684$	−6.00000	−0.229416
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	12.0000	0.457164
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	− 24.0000i	− 0.912343i
$693$	− 24.0000i	− 0.911685i
$694$	−31.0000	−1.17674
$695$	0	0
$696$	0	0
$697$	9.00000i	0.340899i
$698$	16.0000i	0.605609i
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	30.0000i	1.13228i
$703$	3.00000i	0.113147i
$704$	−3.00000	−0.113067
$705$	0	0
$706$	14.0000	0.526897
$707$	40.0000i	1.50435i
$708$	12.0000i	0.450988i
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	− 3.00000i	− 0.112430i
$713$	0	0
$714$	−12.0000	−0.449089
$715$	0	0
$716$	21.0000	0.784807
$717$	− 12.0000i	− 0.448148i
$718$	− 34.0000i	− 1.26887i
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	0	0
$721$	56.0000	2.08555
$722$	− 10.0000i	− 0.372161i
$723$	21.0000i	0.780998i
$724$	10.0000	0.371647
$725$	0	0
$726$	−2.00000	−0.0742270
$727$	44.0000i	1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$	44.0000i	1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$	− 24.0000i	− 0.889499i
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−12.0000	−0.443836
$732$	12.0000i	0.443533i
$733$	40.0000i	1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$	40.0000i	1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$	−2.00000	−0.0738213
$735$	0	0
$736$	−2.00000	−0.0737210
$737$	− 27.0000i	− 0.994558i
$738$	− 6.00000i	− 0.220863i
$739$	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
$741$	18.0000	0.661247
$742$	− 8.00000i	− 0.293689i
$743$	6.00000i	0.220119i	0.993925	+	0.110059i	$0.0351041\pi$
$743$	6.00000i	0.220119i	−0.993925	+	0.110059i	$0.964896\pi$
$744$	0	0
$745$	0	0
$746$	−28.0000	−1.02515
$747$	− 14.0000i	− 0.512233i
$748$	9.00000i	0.329073i
$749$	4.00000	0.146157
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	− 4.00000i	− 0.145865i
$753$	9.00000i	0.327978i
$754$	0	0
$755$	0	0
$756$	20.0000	0.727393
$757$	− 16.0000i	− 0.581530i	−0.956795	−	0.290765i	$-0.906090\pi$
$757$	− 16.0000i	− 0.581530i	0.956795	−	0.290765i	$-0.0939098\pi$
$758$	− 1.00000i	− 0.0363216i
$759$	6.00000	0.217786
$760$	0	0
$761$	45.0000	1.63125	0.815624	−	0.578582i	$-0.196394\pi$
$761$	45.0000	1.63125	0.815624	+	0.578582i	$0.196394\pi$
$762$	− 8.00000i	− 0.289809i
$763$	72.0000i	2.60658i
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−24.0000	−0.867155
$767$	72.0000i	2.59977i
$768$	− 1.00000i	− 0.0360844i
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	− 11.0000i	− 0.395899i
$773$	46.0000i	1.65451i	0.561830	+	0.827253i	$0.310097\pi$
$773$	46.0000i	1.65451i	−0.561830	+	0.827253i	$0.689903\pi$
$774$	8.00000	0.287554
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	− 4.00000i	− 0.143499i
$778$	12.0000i	0.430221i
$779$	−9.00000	−0.322458
$780$	0	0
$781$	−6.00000	−0.214697
$782$	6.00000i	0.214560i
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	12.0000	0.428026
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	12.0000i	0.427482i
$789$	26.0000	0.925625
$790$	0	0
$791$	20.0000	0.711118
$792$	− 6.00000i	− 0.213201i
$793$	72.0000i	2.55679i
$794$	−30.0000	−1.06466
$795$	0	0
$796$	8.00000	0.283552
$797$	− 16.0000i	− 0.566749i	−0.959009	−	0.283375i	$-0.908546\pi$
$797$	− 16.0000i	− 0.566749i	0.959009	−	0.283375i	$-0.0914540\pi$
$798$	− 12.0000i	− 0.424795i
$799$	−12.0000	−0.424529
$800$	0	0
$801$	6.00000	0.212000
$802$	− 5.00000i	− 0.176556i
$803$	− 27.0000i	− 0.952809i
$804$	9.00000	0.317406
$805$	0	0
$806$	0	0
$807$	− 28.0000i	− 0.985647i
$808$	10.0000i	0.351799i
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	28.0000i	0.982003i
$814$	−3.00000	−0.105150
$815$	0	0
$816$	−3.00000	−0.105021
$817$	− 12.0000i	− 0.419827i
$818$	− 15.0000i	− 0.524463i
$819$	48.0000	1.67726
$820$	0	0
$821$	−4.00000	−0.139601	−0.0698005	−	0.997561i	$-0.522236\pi$
$821$	−4.00000	−0.139601	−0.0698005	+	0.997561i	$0.522236\pi$
$822$	7.00000i	0.244153i
$823$	42.0000i	1.46403i	0.681290	+	0.732014i	$0.261419\pi$
$823$	42.0000i	1.46403i	−0.681290	+	0.732014i	$0.738581\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	48.0000	1.67013
$827$	− 7.00000i	− 0.243414i	−0.992566	−	0.121707i	$-0.961163\pi$
$827$	− 7.00000i	− 0.243414i	0.992566	−	0.121707i	$-0.0388368\pi$
$828$	− 4.00000i	− 0.139010i
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	0	0
$831$	−28.0000	−0.971309
$832$	− 6.00000i	− 0.208013i
$833$	27.0000i	0.935495i
$834$	−3.00000	−0.103882
$835$	0	0
$836$	−9.00000	−0.311272
$837$	0	0
$838$	− 27.0000i	− 0.932700i
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	− 32.0000i	− 1.10279i
$843$	6.00000i	0.206651i
$844$	−9.00000	−0.309793
$845$	0	0
$846$	8.00000	0.275046
$847$	8.00000i	0.274883i
$848$	− 2.00000i	− 0.0686803i
$849$	7.00000	0.240239
$850$	0	0
$851$	−2.00000	−0.0685591
$852$	− 2.00000i	− 0.0685189i
$853$	40.0000i	1.36957i	0.728743	+	0.684787i	$0.240105\pi$
$853$	40.0000i	1.36957i	−0.728743	+	0.684787i	$0.759895\pi$
$854$	48.0000	1.64253
$855$	0	0
$856$	1.00000	0.0341793
$857$	− 3.00000i	− 0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$	− 3.00000i	− 0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$	18.0000i	0.614510i
$859$	−9.00000	−0.307076	−0.153538	−	0.988143i	$-0.549067\pi$
$859$	−9.00000	−0.307076	−0.153538	+	0.988143i	$0.549067\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	2.00000i	0.0681203i
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	5.00000	0.170103
$865$	0	0
$866$	−5.00000	−0.169907
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	6.00000	0.203536
$870$	0	0
$871$	54.0000	1.82972
$872$	18.0000i	0.609557i
$873$	− 4.00000i	− 0.135379i
$874$	−6.00000	−0.202953
$875$	0	0
$876$	9.00000	0.304082
$877$	− 26.0000i	− 0.877958i	−0.898497	−	0.438979i	$-0.855340\pi$
$877$	− 26.0000i	− 0.877958i	0.898497	−	0.438979i	$-0.144660\pi$
$878$	− 16.0000i	− 0.539974i
$879$	12.0000	0.404750
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	− 18.0000i	− 0.606092i
$883$	− 7.00000i	− 0.235569i	−0.993039	−	0.117784i	$-0.962421\pi$
$883$	− 7.00000i	− 0.235569i	0.993039	−	0.117784i	$-0.0375792\pi$
$884$	−18.0000	−0.605406
$885$	0	0
$886$	−5.00000	−0.167978
$887$	36.0000i	1.20876i	0.796696	+	0.604381i	$0.206579\pi$
$887$	36.0000i	1.20876i	−0.796696	+	0.604381i	$0.793421\pi$
$888$	− 1.00000i	− 0.0335578i
$889$	−32.0000	−1.07325
$890$	0	0
$891$	3.00000	0.100504
$892$	− 8.00000i	− 0.267860i
$893$	− 12.0000i	− 0.401565i
$894$	18.0000	0.602010
$895$	0	0
$896$	−4.00000	−0.133631
$897$	12.0000i	0.400668i
$898$	37.0000i	1.23471i
$899$	0	0
$900$	0	0
$901$	−6.00000	−0.199889
$902$	− 9.00000i	− 0.299667i
$903$	16.0000i	0.532447i
$904$	5.00000	0.166298
$905$	0	0
$906$	22.0000	0.730901
$907$	− 36.0000i	− 1.19536i	−0.801735	−	0.597680i	$-0.796089\pi$
$907$	− 36.0000i	− 1.19536i	0.801735	−	0.597680i	$-0.203911\pi$
$908$	28.0000i	0.929213i
$909$	−20.0000	−0.663358
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	− 3.00000i	− 0.0993399i
$913$	− 21.0000i	− 0.694999i
$914$	−33.0000	−1.09154
$915$	0	0
$916$	0	0
$917$	− 48.0000i	− 1.58510i
$918$	− 15.0000i	− 0.495074i
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	29.0000	0.955582
$922$	20.0000i	0.658665i
$923$	− 12.0000i	− 0.394985i
$924$	12.0000	0.394771
$925$	0	0
$926$	−6.00000	−0.197172
$927$	28.0000i	0.919641i
$928$	0	0
$929$	54.0000	1.77168	0.885841	−	0.463988i	$-0.153582\pi$
$929$	54.0000	1.77168	0.885841	+	0.463988i	$0.153582\pi$
$930$	0	0
$931$	−27.0000	−0.884889
$932$	26.0000i	0.851658i
$933$	− 28.0000i	− 0.916679i
$934$	−28.0000	−0.916188
$935$	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	$-0.0364753\pi$
$938$	− 36.0000i	− 1.17544i
$939$	−10.0000	−0.326338
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	− 24.0000i	− 0.781962i
$943$	− 6.00000i	− 0.195387i
$944$	12.0000	0.390567
$945$	0	0
$946$	12.0000	0.390154
$947$	24.0000i	0.779895i	0.920837	+	0.389948i	$0.127507\pi$
$947$	24.0000i	0.779895i	−0.920837	+	0.389948i	$0.872493\pi$
$948$	2.00000i	0.0649570i
$949$	54.0000	1.75291
$950$	0	0
$951$	18.0000	0.583690
$952$	12.0000i	0.388922i
$953$	33.0000i	1.06897i	0.845176	+	0.534487i	$0.179495\pi$
$953$	33.0000i	1.06897i	−0.845176	+	0.534487i	$0.820505\pi$
$954$	4.00000	0.129505
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	− 16.0000i	− 0.516937i
$959$	28.0000	0.904167
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 6.00000i	− 0.193448i
$963$	2.00000i	0.0644491i
$964$	21.0000	0.676364
$965$	0	0
$966$	8.00000	0.257396
$967$	48.0000i	1.54358i	0.635880	+	0.771788i	$0.280637\pi$
$967$	48.0000i	1.54358i	−0.635880	+	0.771788i	$0.719363\pi$
$968$	2.00000i	0.0642824i
$969$	−9.00000	−0.289122
$970$	0	0
$971$	3.00000	0.0962746	0.0481373	−	0.998841i	$-0.484672\pi$
$971$	3.00000	0.0962746	0.0481373	+	0.998841i	$0.484672\pi$
$972$	16.0000i	0.513200i
$973$	12.0000i	0.384702i
$974$	28.0000	0.897178
$975$	0	0
$976$	12.0000	0.384111
$977$	− 23.0000i	− 0.735835i	−0.929858	−	0.367918i	$-0.880071\pi$
$977$	− 23.0000i	− 0.735835i	0.929858	−	0.367918i	$-0.119929\pi$
$978$	− 17.0000i	− 0.543600i
$979$	9.00000	0.287641
$980$	0	0
$981$	−36.0000	−1.14939
$982$	− 20.0000i	− 0.638226i
$983$	18.0000i	0.574111i	0.957914	+	0.287055i	$0.0926764\pi$
$983$	18.0000i	0.574111i	−0.957914	+	0.287055i	$0.907324\pi$
$984$	3.00000	0.0956365
$985$	0	0
$986$	0	0
$987$	16.0000i	0.509286i
$988$	− 18.0000i	− 0.572656i
$989$	8.00000	0.254385
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	0	0
$993$	3.00000i	0.0952021i
$994$	−8.00000	−0.253745
$995$	0	0
$996$	7.00000	0.221803
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	16.0000i	0.506471i
$999$	5.00000	0.158193

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.c.1.1	✓	1	5.2	odd	4
1850.2.a.m.1.1	yes	1	5.3	odd	4
1850.2.b.e.149.1		2	5.4	even	2	inner
1850.2.b.e.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)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} +2.00000 q^{9} +3.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +2.00000i q^{18} +3.00000 q^{19} -4.00000 q^{21} +3.00000i q^{22} +2.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} -5.00000i q^{27} +4.00000i q^{28} +1.00000i q^{32} -3.00000i q^{33} +3.00000 q^{34} -2.00000 q^{36} +1.00000i q^{37} +3.00000i q^{38} +6.00000 q^{39} -3.00000 q^{41} -4.00000i q^{42} -4.00000i q^{43} -3.00000 q^{44} -2.00000 q^{46} -4.00000i q^{47} -1.00000i q^{48} -9.00000 q^{49} -3.00000 q^{51} -6.00000i q^{52} -2.00000i q^{53} +5.00000 q^{54} -4.00000 q^{56} -3.00000i q^{57} +12.0000 q^{59} +12.0000 q^{61} -8.00000i q^{63} -1.00000 q^{64} +3.00000 q^{66} -9.00000i q^{67} +3.00000i q^{68} +2.00000 q^{69} -2.00000 q^{71} -2.00000i q^{72} -9.00000i q^{73} -1.00000 q^{74} -3.00000 q^{76} -12.0000i q^{77} +6.00000i q^{78} +2.00000 q^{79} +1.00000 q^{81} -3.00000i q^{82} -7.00000i q^{83} +4.00000 q^{84} +4.00000 q^{86} -3.00000i q^{88} +3.00000 q^{89} +24.0000 q^{91} -2.00000i q^{92} +4.00000 q^{94} +1.00000 q^{96} -2.00000i q^{97} -9.00000i q^{98} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} + 2 q^{6} + 4 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 6 q^{19} - 8 q^{21} - 2 q^{24} - 12 q^{26} + 6 q^{34} - 4 q^{36} + 12 q^{39} - 6 q^{41} - 6 q^{44} - 4 q^{46} - 18 q^{49} - 6 q^{51} + 10 q^{54} - 8 q^{56} + 24 q^{59} + 24 q^{61} - 2 q^{64} + 6 q^{66} + 4 q^{69} - 4 q^{71} - 2 q^{74} - 6 q^{76} + 4 q^{79} + 2 q^{81} + 8 q^{84} + 8 q^{86} + 6 q^{89} + 48 q^{91} + 8 q^{94} + 2 q^{96} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^6 + 4 * q^9 + 6 * q^11 + 8 * q^14 + 2 * q^16 + 6 * q^19 - 8 * q^21 - 2 * q^24 - 12 * q^26 + 6 * q^34 - 4 * q^36 + 12 * q^39 - 6 * q^41 - 6 * q^44 - 4 * q^46 - 18 * q^49 - 6 * q^51 + 10 * q^54 - 8 * q^56 + 24 * q^59 + 24 * q^61 - 2 * q^64 + 6 * q^66 + 4 * q^69 - 4 * q^71 - 2 * q^74 - 6 * q^76 + 4 * q^79 + 2 * q^81 + 8 * q^84 + 8 * q^86 + 6 * q^89 + 48 * q^91 + 8 * q^94 + 2 * q^96 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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