LMFDB - Embedded newform 1650.2.c.g.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1650,2,Mod(199,1650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1650.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1650.c (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:	$13.1753163335$
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:	no (minimal twist has level 330)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1650.199
Dual form			1650.2.c.g.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -1.00000i q^{22} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +10.0000 q^{29} -1.00000i q^{32} -1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} -1.00000 q^{44} +8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} -4.00000i q^{57} -10.0000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -1.00000 q^{64} -1.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} -8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -6.00000 q^{74} -4.00000 q^{76} -6.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} -10.0000i q^{87} +1.00000i q^{88} +6.00000 q^{89} +8.00000 q^{94} -1.00000 q^{96} -18.0000i q^{97} -7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{16} + 8 q^{19} + 2 q^{24} + 12 q^{26} + 20 q^{29} - 4 q^{34} + 2 q^{36} + 12 q^{39} + 4 q^{41} - 2 q^{44} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 2 q^{66} - 16 q^{71} - 12 q^{74} - 8 q^{76} + 16 q^{79} + 2 q^{81} + 8 q^{86} + 12 q^{89} + 16 q^{94} - 2 q^{96} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^16 + 8 * q^19 + 2 * q^24 + 12 * q^26 + 20 * q^29 - 4 * q^34 + 2 * q^36 + 12 * q^39 + 4 * q^41 - 2 * q^44 + 14 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 2 * q^66 - 16 * q^71 - 12 * q^74 - 8 * q^76 + 16 * q^79 + 2 * q^81 + 8 * q^86 + 12 * q^89 + 16 * q^94 - 2 * q^96 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times$.

$n$	$551$	$727$	$1201$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	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	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	1.00000i	0.235702i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 0.213201i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000i	0.192450i
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 1.00000i	− 0.174078i
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	− 4.00000i	− 0.648886i
$39$	6.00000	0.960769
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	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
$46$	0	0
$47$	8.00000i	1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$	8.00000i	1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	− 1.00000i	− 0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	−2.00000	−0.280056
$52$	− 6.00000i	− 0.832050i
$53$	− 10.0000i	− 1.37361i	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	− 10.0000i	− 1.37361i	0.726844	−	0.686803i	$-0.240986\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	0	0
$57$	− 4.00000i	− 0.529813i
$58$	− 10.0000i	− 1.31306i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−1.00000	−0.123091
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	2.00000i	0.242536i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	− 1.00000i	− 0.117851i
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−4.00000	−0.458831
$77$	0	0
$78$	− 6.00000i	− 0.679366i
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.00000i	− 0.220863i
$83$	− 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
$85$	0	0
$86$	4.00000	0.431331
$87$	− 10.0000i	− 1.07211i
$88$	1.00000i	0.106600i
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	8.00000	0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	− 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
$99$	−1.00000	−0.100504
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	2.00000i	0.198030i
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−10.0000	−0.971286
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	− 1.00000i	− 0.0962250i
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	0	0
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	−4.00000	−0.374634
$115$	0	0
$116$	−10.0000	−0.928477
$117$	− 6.00000i	− 0.554700i
$118$	− 4.00000i	− 0.368230i
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000i	0.181071i
$123$	− 2.00000i	− 0.180334i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 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
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	1.00000i	0.0870388i
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	8.00000i	0.671345i
$143$	6.00000i	0.501745i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	2.00000	0.165521
$147$	− 7.00000i	− 0.577350i
$148$	6.00000i	0.493197i
$149$	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
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	4.00000i	0.324443i
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	−6.00000	−0.480384
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	− 8.00000i	− 0.636446i
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−12.0000	−0.931381
$167$	− 24.0000i	− 1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$	− 24.0000i	− 1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−4.00000	−0.305888
$172$	− 4.00000i	− 0.304997i
$173$	14.0000i	1.06440i	0.846619	+	0.532200i	$0.178635\pi$
$173$	14.0000i	1.06440i	−0.846619	+	0.532200i	$0.821365\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	1.00000	0.0753778
$177$	− 4.00000i	− 0.300658i
$178$	− 6.00000i	− 0.449719i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	− 8.00000i	− 0.583460i
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	1.00000i	0.0721688i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	1.00000i	0.0710669i
$199$	−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
$201$	4.00000	0.282138
$202$	− 14.0000i	− 0.985037i
$203$	0	0
$204$	2.00000	0.140028
$205$	0	0
$206$	16.0000	1.11477
$207$	0	0
$208$	6.00000i	0.416025i
$209$	4.00000	0.276686
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	10.0000i	0.686803i
$213$	8.00000i	0.548151i
$214$	4.00000	0.273434
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.0000i	0.948200i
$219$	2.00000	0.135147
$220$	0	0
$221$	12.0000	0.807207
$222$	6.00000i	0.402694i
$223$	24.0000i	1.60716i	0.595198	+	0.803579i	$0.297074\pi$
$223$	24.0000i	1.60716i	−0.595198	+	0.803579i	$0.702926\pi$
$224$	0	0
$225$	0	0
$226$	2.00000	0.133038
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	4.00000i	0.264906i
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	0	0
$231$	0	0
$232$	10.0000i	0.656532i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−4.00000	−0.260378
$237$	− 8.00000i	− 0.519656i
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	− 1.00000i	− 0.0642824i
$243$	− 1.00000i	− 0.0641500i
$244$	2.00000	0.128037
$245$	0	0
$246$	−2.00000	−0.127515
$247$	24.0000i	1.52708i
$248$	0	0
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	− 4.00000i	− 0.249029i
$259$	0	0
$260$	0	0
$261$	−10.0000	−0.618984
$262$	− 20.0000i	− 1.23560i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	0	0
$267$	− 6.00000i	− 0.367194i
$268$	− 4.00000i	− 0.244339i
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	− 2.00000i	− 0.121268i
$273$	0	0
$274$	6.00000	0.362473
$275$	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	$0.818138\pi$
$278$	4.00000i	0.239904i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	− 8.00000i	− 0.476393i
$283$	− 12.0000i	− 0.713326i	−0.934233	−	0.356663i	$-0.883914\pi$
$283$	− 12.0000i	− 0.713326i	0.934233	−	0.356663i	$-0.116086\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	6.00000	0.354787
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	13.0000	0.764706
$290$	0	0
$291$	−18.0000	−1.05518
$292$	− 2.00000i	− 0.117041i
$293$	6.00000i	0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$	6.00000i	0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	6.00000	0.348743
$297$	1.00000i	0.0580259i
$298$	− 18.0000i	− 1.04271i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	16.0000i	0.920697i
$303$	− 14.0000i	− 0.804279i
$304$	4.00000	0.229416
$305$	0	0
$306$	2.00000	0.114332
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	16.0000	0.910208
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	6.00000i	0.339683i
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−8.00000	−0.450035
$317$	34.0000i	1.90963i	0.297200	+	0.954815i	$0.403947\pi$
$317$	34.0000i	1.90963i	−0.297200	+	0.954815i	$0.596053\pi$
$318$	10.0000i	0.560772i
$319$	10.0000	0.559893
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−20.0000	−1.10770
$327$	14.0000i	0.774202i
$328$	2.00000i	0.110432i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000i	0.658586i
$333$	6.00000i	0.328798i
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	− 10.0000i	− 0.544735i	−0.962193	−	0.272367i	$-0.912193\pi$
$337$	− 10.0000i	− 0.544735i	0.962193	−	0.272367i	$-0.0878066\pi$
$338$	23.0000i	1.25104i
$339$	2.00000	0.108625
$340$	0	0
$341$	0	0
$342$	4.00000i	0.216295i
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	14.0000	0.752645
$347$	− 12.0000i	− 0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$	− 12.0000i	− 0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$	10.0000i	0.536056i
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	− 1.00000i	− 0.0533002i
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	−4.00000	−0.212598
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	10.0000i	0.525588i
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 18.0000i	− 0.932005i	−0.884783	−	0.466002i	$-0.845694\pi$
$373$	− 18.0000i	− 0.932005i	0.884783	−	0.466002i	$-0.154306\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−8.00000	−0.412568
$377$	60.0000i	3.09016i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	0	0
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	10.0000	0.508987
$387$	− 4.00000i	− 0.203331i
$388$	18.0000i	0.913812i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	0	0
$392$	7.00000i	0.353553i
$393$	− 20.0000i	− 1.00887i
$394$	−6.00000	−0.302276
$395$	0	0
$396$	1.00000	0.0502519
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	− 4.00000i	− 0.199502i
$403$	0	0
$404$	−14.0000	−0.696526
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	− 2.00000i	− 0.0990148i
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	− 16.0000i	− 0.788263i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	4.00000i	0.195881i
$418$	− 4.00000i	− 0.195646i
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	20.0000i	0.973585i
$423$	− 8.00000i	− 0.388973i
$424$	10.0000	0.485643
$425$	0	0
$426$	8.00000	0.387601
$427$	0	0
$428$	− 4.00000i	− 0.193347i
$429$	6.00000	0.289683
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	1.00000i	0.0481125i
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	− 2.00000i	− 0.0955637i
$439$	−32.0000	−1.52728	−0.763638	−	0.645644i	$-0.776589\pi$
$439$	−32.0000	−1.52728	−0.763638	+	0.645644i	$0.776589\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 12.0000i	− 0.570782i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	6.00000	0.284747
$445$	0	0
$446$	24.0000	1.13643
$447$	− 18.0000i	− 0.851371i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	− 2.00000i	− 0.0940721i
$453$	16.0000i	0.751746i
$454$	−4.00000	−0.187729
$455$	0	0
$456$	4.00000	0.187317
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	− 26.0000i	− 1.21490i
$459$	2.00000	0.0933520
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	40.0000i	1.85896i	0.368875	+	0.929479i	$0.379743\pi$
$463$	40.0000i	1.85896i	−0.368875	+	0.929479i	$0.620257\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	−6.00000	−0.277945
$467$	4.00000i	0.185098i	0.995708	+	0.0925490i	$0.0295015\pi$
$467$	4.00000i	0.185098i	−0.995708	+	0.0925490i	$0.970499\pi$
$468$	6.00000i	0.277350i
$469$	0	0
$470$	0	0
$471$	18.0000	0.829396
$472$	4.00000i	0.184115i
$473$	4.00000i	0.183920i
$474$	−8.00000	−0.367452
$475$	0	0
$476$	0	0
$477$	10.0000i	0.457869i
$478$	16.0000i	0.731823i
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	2.00000i	0.0901670i
$493$	− 20.0000i	− 0.900755i
$494$	24.0000	1.07981
$495$	0	0
$496$	0	0
$497$	0	0
$498$	12.0000i	0.537733i
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	−24.0000	−1.07224
$502$	4.00000i	0.178529i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0000i	1.02147i
$508$	8.00000i	0.354943i
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	4.00000i	0.176604i
$514$	14.0000	0.617514
$515$	0	0
$516$	−4.00000	−0.176090
$517$	8.00000i	0.351840i
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	10.0000i	0.437688i
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	− 1.00000i	− 0.0435194i
$529$	23.0000	1.00000
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	12.0000i	0.519778i
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	− 12.0000i	− 0.517838i
$538$	− 2.00000i	− 0.0862261i
$539$	7.00000	0.301511
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	8.00000i	0.343629i
$543$	10.0000i	0.429141i
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	− 6.00000i	− 0.256307i
$549$	2.00000	0.0853579
$550$	0	0
$551$	40.0000	1.70406
$552$	0	0
$553$	0	0
$554$	18.0000	0.764747
$555$	0	0
$556$	4.00000	0.169638
$557$	− 30.0000i	− 1.27114i	−0.772043	−	0.635570i	$-0.780765\pi$
$557$	− 30.0000i	− 1.27114i	0.772043	−	0.635570i	$-0.219235\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	0	0
$561$	−2.00000	−0.0844401
$562$	− 18.0000i	− 0.759284i
$563$	− 12.0000i	− 0.505740i	−0.967500	−	0.252870i	$-0.918626\pi$
$563$	− 12.0000i	− 0.505740i	0.967500	−	0.252870i	$-0.0813744\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−12.0000	−0.504398
$567$	0	0
$568$	− 8.00000i	− 0.335673i
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	− 6.00000i	− 0.250873i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	46.0000i	1.91501i	0.288425	+	0.957503i	$0.406868\pi$
$577$	46.0000i	1.91501i	−0.288425	+	0.957503i	$0.593132\pi$
$578$	− 13.0000i	− 0.540729i
$579$	10.0000	0.415586
$580$	0	0
$581$	0	0
$582$	18.0000i	0.746124i
$583$	− 10.0000i	− 0.414158i
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	6.00000	0.247858
$587$	12.0000i	0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$	12.0000i	0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$	7.00000i	0.288675i
$589$	0	0
$590$	0	0
$591$	−6.00000	−0.246807
$592$	− 6.00000i	− 0.246598i
$593$	− 46.0000i	− 1.88899i	−0.328521	−	0.944497i	$-0.606550\pi$
$593$	− 46.0000i	− 1.88899i	0.328521	−	0.944497i	$-0.393450\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	−18.0000	−0.737309
$597$	8.00000i	0.327418i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	42.0000	1.71322	0.856608	−	0.515968i	$-0.172568\pi$
$601$	42.0000	1.71322	0.856608	+	0.515968i	$0.172568\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	16.0000	0.651031
$605$	0	0
$606$	−14.0000	−0.568711
$607$	− 40.0000i	− 1.62355i	−0.583970	−	0.811775i	$-0.698502\pi$
$607$	− 40.0000i	− 1.62355i	0.583970	−	0.811775i	$-0.301498\pi$
$608$	− 4.00000i	− 0.162221i
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	− 2.00000i	− 0.0808452i
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	0	0
$617$	− 26.0000i	− 1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	− 26.0000i	− 1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$	− 16.0000i	− 0.643614i
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	− 24.0000i	− 0.962312i
$623$	0	0
$624$	6.00000	0.240192
$625$	0	0
$626$	−6.00000	−0.239808
$627$	− 4.00000i	− 0.159745i
$628$	− 18.0000i	− 0.718278i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	8.00000i	0.318223i
$633$	20.0000i	0.794929i
$634$	34.0000	1.35031
$635$	0	0
$636$	10.0000	0.396526
$637$	42.0000i	1.66410i
$638$	− 10.0000i	− 0.395904i
$639$	8.00000	0.316475
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	− 4.00000i	− 0.157867i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	−8.00000	−0.314756
$647$	− 48.0000i	− 1.88707i	−0.331266	−	0.943537i	$-0.607476\pi$
$647$	− 48.0000i	− 1.88707i	0.331266	−	0.943537i	$-0.392524\pi$
$648$	1.00000i	0.0392837i
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	20.0000i	0.783260i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	2.00000	0.0780869
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	4.00000i	0.155464i
$663$	− 12.0000i	− 0.466041i
$664$	12.0000	0.465690
$665$	0	0
$666$	6.00000	0.232495
$667$	0	0
$668$	24.0000i	0.928588i
$669$	24.0000	0.927894
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	− 22.0000i	− 0.848038i	−0.905653	−	0.424019i	$-0.860619\pi$
$673$	− 22.0000i	− 0.848038i	0.905653	−	0.424019i	$-0.139381\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	23.0000	0.884615
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	− 2.00000i	− 0.0768095i
$679$	0	0
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	20.0000i	0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$	20.0000i	0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	0	0
$687$	− 26.0000i	− 0.991962i
$688$	4.00000i	0.152499i
$689$	60.0000	2.28582
$690$	0	0
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	− 14.0000i	− 0.532200i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	10.0000	0.379049
$697$	− 4.00000i	− 0.151511i
$698$	30.0000i	1.13552i
$699$	−6.00000	−0.226941
$700$	0	0
$701$	−26.0000	−0.982006	−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000	−0.982006	−0.491003	+	0.871158i	$0.663370\pi$
$702$	6.00000i	0.226455i
$703$	− 24.0000i	− 0.905177i
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−14.0000	−0.526897
$707$	0	0
$708$	4.00000i	0.150329i
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000i	0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	16.0000i	0.597531i
$718$	− 8.00000i	− 0.298557i
$719$	32.0000	1.19340	0.596699	−	0.802465i	$-0.296479\pi$
$719$	32.0000	1.19340	0.596699	+	0.802465i	$0.296479\pi$
$720$	0	0
$721$	0	0
$722$	3.00000i	0.111648i
$723$	− 18.0000i	− 0.669427i
$724$	10.0000	0.371647
$725$	0	0
$726$	−1.00000	−0.0371135
$727$	− 32.0000i	− 1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$	− 32.0000i	− 1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	− 2.00000i	− 0.0739221i
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	4.00000i	0.147342i
$738$	2.00000i	0.0736210i
$739$	20.0000	0.735712	0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000	0.735712	0.367856	+	0.929883i	$0.380092\pi$
$740$	0	0
$741$	24.0000	0.881662
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	−18.0000	−0.659027
$747$	12.0000i	0.439057i
$748$	2.00000i	0.0731272i
$749$	0	0
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000i	0.291730i
$753$	4.00000i	0.145768i
$754$	60.0000	2.18507
$755$	0	0
$756$	0	0
$757$	− 38.0000i	− 1.38113i	−0.723269	−	0.690567i	$-0.757361\pi$
$757$	− 38.0000i	− 1.38113i	0.723269	−	0.690567i	$-0.242639\pi$
$758$	12.0000i	0.435860i
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	8.00000i	0.289809i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	−8.00000	−0.289052
$767$	24.0000i	0.866590i
$768$	− 1.00000i	− 0.0360844i
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	− 10.0000i	− 0.359908i
$773$	− 26.0000i	− 0.935155i	−0.883952	−	0.467578i	$-0.845127\pi$
$773$	− 26.0000i	− 0.935155i	0.883952	−	0.467578i	$-0.154873\pi$
$774$	−4.00000	−0.143777
$775$	0	0
$776$	18.0000	0.646162
$777$	0	0
$778$	6.00000i	0.215110i
$779$	8.00000	0.286630
$780$	0	0
$781$	−8.00000	−0.286263
$782$	0	0
$783$	10.0000i	0.357371i
$784$	7.00000	0.250000
$785$	0	0
$786$	−20.0000	−0.713376
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	6.00000i	0.213741i
$789$	24.0000	0.854423
$790$	0	0
$791$	0	0
$792$	− 1.00000i	− 0.0355335i
$793$	− 12.0000i	− 0.426132i
$794$	2.00000	0.0709773
$795$	0	0
$796$	8.00000	0.283552
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 2.00000i	− 0.0706225i
$803$	2.00000i	0.0705785i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	0	0
$807$	− 2.00000i	− 0.0704033i
$808$	14.0000i	0.492518i
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	0	0
$813$	8.00000i	0.280572i
$814$	−6.00000	−0.210300
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	16.0000i	0.559769i
$818$	10.0000i	0.349642i
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	− 6.00000i	− 0.209274i
$823$	32.0000i	1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$	32.0000i	1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	0	0
$827$	− 44.0000i	− 1.53003i	−0.644013	−	0.765015i	$-0.722732\pi$
$827$	− 44.0000i	− 1.53003i	0.644013	−	0.765015i	$-0.277268\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	18.0000	0.624413
$832$	− 6.00000i	− 0.208013i
$833$	− 14.0000i	− 0.485071i
$834$	4.00000	0.138509
$835$	0	0
$836$	−4.00000	−0.138343
$837$	0	0
$838$	36.0000i	1.24360i
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	− 38.0000i	− 1.30957i
$843$	− 18.0000i	− 0.619953i
$844$	20.0000	0.688428
$845$	0	0
$846$	−8.00000	−0.275046
$847$	0	0
$848$	− 10.0000i	− 0.343401i
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0	0
$852$	− 8.00000i	− 0.274075i
$853$	− 50.0000i	− 1.71197i	−0.517003	−	0.855984i	$-0.672952\pi$
$853$	− 50.0000i	− 1.71197i	0.517003	−	0.855984i	$-0.327048\pi$
$854$	0	0
$855$	0	0
$856$	−4.00000	−0.136717
$857$	22.0000i	0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$	22.0000i	0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$	− 6.00000i	− 0.204837i
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	0	0
$861$	0	0
$862$	− 16.0000i	− 0.544962i
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	2.00000	0.0679628
$867$	− 13.0000i	− 0.441503i
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−24.0000	−0.813209
$872$	− 14.0000i	− 0.474100i
$873$	18.0000i	0.609208i
$874$	0	0
$875$	0	0
$876$	−2.00000	−0.0675737
$877$	58.0000i	1.95852i	0.202606	+	0.979260i	$0.435059\pi$
$877$	58.0000i	1.95852i	−0.202606	+	0.979260i	$0.564941\pi$
$878$	32.0000i	1.07995i
$879$	6.00000	0.202375
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	7.00000i	0.235702i
$883$	12.0000i	0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$	12.0000i	0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	4.00000	0.134383
$887$	− 56.0000i	− 1.88030i	−0.340766	−	0.940148i	$-0.610687\pi$
$887$	− 56.0000i	− 1.88030i	0.340766	−	0.940148i	$-0.389313\pi$
$888$	− 6.00000i	− 0.201347i
$889$	0	0
$890$	0	0
$891$	1.00000	0.0335013
$892$	− 24.0000i	− 0.803579i
$893$	32.0000i	1.07084i
$894$	−18.0000	−0.602010
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 30.0000i	− 1.00111i
$899$	0	0
$900$	0	0
$901$	−20.0000	−0.666297
$902$	− 2.00000i	− 0.0665927i
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	0	0
$906$	16.0000	0.531564
$907$	12.0000i	0.398453i	0.979953	+	0.199227i	$0.0638430\pi$
$907$	12.0000i	0.398453i	−0.979953	+	0.199227i	$0.936157\pi$
$908$	4.00000i	0.132745i
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	− 4.00000i	− 0.132453i
$913$	− 12.0000i	− 0.397142i
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−26.0000	−0.859064
$917$	0	0
$918$	− 2.00000i	− 0.0660098i
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	− 6.00000i	− 0.197599i
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	0	0
$926$	40.0000	1.31448
$927$	− 16.0000i	− 0.525509i
$928$	− 10.0000i	− 0.328266i
$929$	−50.0000	−1.64045	−0.820223	−	0.572043i	$-0.806151\pi$
$929$	−50.0000	−1.64045	−0.820223	+	0.572043i	$0.806151\pi$
$930$	0	0
$931$	28.0000	0.917663
$932$	6.00000i	0.196537i
$933$	− 24.0000i	− 0.785725i
$934$	4.00000	0.130884
$935$	0	0
$936$	6.00000	0.196116
$937$	− 2.00000i	− 0.0653372i	−0.999466	−	0.0326686i	$-0.989599\pi$
$937$	− 2.00000i	− 0.0653372i	0.999466	−	0.0326686i	$-0.0104006\pi$
$938$	0	0
$939$	−6.00000	−0.195803
$940$	0	0
$941$	−10.0000	−0.325991	−0.162995	−	0.986627i	$-0.552116\pi$
$941$	−10.0000	−0.325991	−0.162995	+	0.986627i	$0.552116\pi$
$942$	− 18.0000i	− 0.586472i
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	4.00000	0.130051
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	8.00000i	0.259828i
$949$	−12.0000	−0.389536
$950$	0	0
$951$	34.0000	1.10253
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	16.0000	0.517477
$957$	− 10.0000i	− 0.323254i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 36.0000i	− 1.16069i
$963$	− 4.00000i	− 0.128898i
$964$	−18.0000	−0.579741
$965$	0	0
$966$	0	0
$967$	− 48.0000i	− 1.54358i	−0.635880	−	0.771788i	$-0.719363\pi$
$967$	− 48.0000i	− 1.54358i	0.635880	−	0.771788i	$-0.280637\pi$
$968$	1.00000i	0.0321412i
$969$	−8.00000	−0.256997
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000i	0.0320750i
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	20.0000i	0.639529i
$979$	6.00000	0.191761
$980$	0	0
$981$	14.0000	0.446986
$982$	36.0000i	1.14881i
$983$	− 48.0000i	− 1.53096i	−0.643458	−	0.765481i	$-0.722501\pi$
$983$	− 48.0000i	− 1.53096i	0.643458	−	0.765481i	$-0.277499\pi$
$984$	2.00000	0.0637577
$985$	0	0
$986$	−20.0000	−0.636930
$987$	0	0
$988$	− 24.0000i	− 0.763542i
$989$	0	0
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	4.00000i	0.126936i
$994$	0	0
$995$	0	0
$996$	12.0000	0.380235
$997$	2.00000i	0.0633406i	0.999498	+	0.0316703i	$0.0100827\pi$
$997$	2.00000i	0.0633406i	−0.999498	+	0.0316703i	$0.989917\pi$
$998$	20.0000i	0.633089i
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1650.2.c.g.199.1		2
3.2	odd	2		4950.2.c.j.199.2		2
5.2	odd	4		330.2.a.d.1.1	✓	1
5.3	odd	4		1650.2.a.h.1.1		1
5.4	even	2	inner	1650.2.c.g.199.2		2
15.2	even	4		990.2.a.b.1.1		1
15.8	even	4		4950.2.a.bg.1.1		1
15.14	odd	2		4950.2.c.j.199.1		2
20.7	even	4		2640.2.a.t.1.1		1
55.32	even	4		3630.2.a.f.1.1		1
60.47	odd	4		7920.2.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
330.2.a.d.1.1	✓	1	5.2	odd	4
990.2.a.b.1.1		1	15.2	even	4
1650.2.a.h.1.1		1	5.3	odd	4
1650.2.c.g.199.1		2	1.1	even	1	trivial
1650.2.c.g.199.2		2	5.4	even	2	inner
2640.2.a.t.1.1		1	20.7	even	4
3630.2.a.f.1.1		1	55.32	even	4
4950.2.a.bg.1.1		1	15.8	even	4
4950.2.c.j.199.1		2	15.14	odd	2
4950.2.c.j.199.2		2	3.2	odd	2
7920.2.a.m.1.1		1	60.47	odd	4

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:	\( N \)	\(=\)	\( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1650.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +1.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} -1.00000i q^{22} +1.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} +10.0000 q^{29} -1.00000i q^{32} -1.00000i q^{33} -2.00000 q^{34} +1.00000 q^{36} -6.00000i q^{37} -4.00000i q^{38} +6.00000 q^{39} +2.00000 q^{41} +4.00000i q^{43} -1.00000 q^{44} +8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} -10.0000i q^{53} +1.00000 q^{54} -4.00000i q^{57} -10.0000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -1.00000 q^{64} -1.00000 q^{66} +4.00000i q^{67} +2.00000i q^{68} -8.00000 q^{71} -1.00000i q^{72} +2.00000i q^{73} -6.00000 q^{74} -4.00000 q^{76} -6.00000i q^{78} +8.00000 q^{79} +1.00000 q^{81} -2.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} -10.0000i q^{87} +1.00000i q^{88} +6.00000 q^{89} +8.00000 q^{94} -1.00000 q^{96} -18.0000i q^{97} -7.00000i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} - 2 q^{6} - 2 q^{9} + 2 q^{11} + 2 q^{16} + 8 q^{19} + 2 q^{24} + 12 q^{26} + 20 q^{29} - 4 q^{34} + 2 q^{36} + 12 q^{39} + 4 q^{41} - 2 q^{44} + 14 q^{49} - 4 q^{51} + 2 q^{54} + 8 q^{59} - 4 q^{61} - 2 q^{64} - 2 q^{66} - 16 q^{71} - 12 q^{74} - 8 q^{76} + 16 q^{79} + 2 q^{81} + 8 q^{86} + 12 q^{89} + 16 q^{94} - 2 q^{96} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^6 - 2 * q^9 + 2 * q^11 + 2 * q^16 + 8 * q^19 + 2 * q^24 + 12 * q^26 + 20 * q^29 - 4 * q^34 + 2 * q^36 + 12 * q^39 + 4 * q^41 - 2 * q^44 + 14 * q^49 - 4 * q^51 + 2 * q^54 + 8 * q^59 - 4 * q^61 - 2 * q^64 - 2 * q^66 - 16 * q^71 - 12 * q^74 - 8 * q^76 + 16 * q^79 + 2 * q^81 + 8 * q^86 + 12 * q^89 + 16 * q^94 - 2 * q^96 - 2 * q^99

Introduction

L-functions

Modular forms

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