LMFDB - Embedded newform 3330.2.h.f.2071.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(2071,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("3330.2071");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.5901838731$
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 1110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2071.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3330.2071
Dual form			3330.2.h.f.2071.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	1.00000			0.377964			0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000			0.377964			0.188982	+	0.981981i	$0.439481\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−1.00000			−0.316228
$11$	−3.00000			−0.904534			−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000			−0.904534			−0.452267	+	0.891883i	$0.649385\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$			1.00000i			0.267261i
$15$		0			0
$16$	1.00000			0.250000
$17$			3.00000i			0.727607i	0.931476	+	0.363803i	$0.118522\pi$
$17$			3.00000i			0.727607i	−0.931476	+	0.363803i	$0.881478\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		−	1.00000i		−	0.223607i
$21$		0			0
$22$		−	3.00000i		−	0.639602i
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$	6.00000			1.17670
$27$		0			0
$28$	−1.00000			−0.188982
$29$			9.00000i			1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$			9.00000i			1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$		0			0
$31$			3.00000i			0.538816i	0.963026	+	0.269408i	$0.0868280\pi$
$31$			3.00000i			0.538816i	−0.963026	+	0.269408i	$0.913172\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−3.00000			−0.514496
$35$			1.00000i			0.169031i
$36$		0			0
$37$	1.00000	−	6.00000i	0.164399	−	0.986394i
$37$	1.00000	−	6.00000i	0.164399	−	0.986394i
$38$	−6.00000			−0.973329
$39$		0			0
$40$	1.00000			0.158114
$41$	−9.00000			−1.40556			−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000			−1.40556			−0.702782	+	0.711405i	$0.748059\pi$
$42$		0			0
$43$		−	9.00000i		−	1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$		−	9.00000i		−	1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	3.00000			0.452267
$45$		0			0
$46$	6.00000			0.884652
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−6.00000			−0.857143
$50$		−	1.00000i		−	0.141421i
$51$		0			0
$52$			6.00000i			0.832050i
$53$	−3.00000			−0.412082			−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000			−0.412082			−0.206041	+	0.978543i	$0.566058\pi$
$54$		0			0
$55$		−	3.00000i		−	0.404520i
$56$		−	1.00000i		−	0.133631i
$57$		0			0
$58$	−9.00000			−1.18176
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$		−	9.00000i		−	1.15233i	−0.817333	−	0.576166i	$-0.804548\pi$
$61$		−	9.00000i		−	1.15233i	0.817333	−	0.576166i	$-0.195452\pi$
$62$	−3.00000			−0.381000
$63$		0			0
$64$	−1.00000			−0.125000
$65$	6.00000			0.744208
$66$		0			0
$67$	−14.0000			−1.71037			−0.855186	−	0.518321i	$-0.826557\pi$
$67$	−14.0000			−1.71037			−0.855186	+	0.518321i	$0.826557\pi$
$68$		−	3.00000i		−	0.363803i
$69$		0			0
$70$	−1.00000			−0.119523
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$	6.00000	+	1.00000i	0.697486	+	0.116248i
$75$		0			0
$76$		−	6.00000i		−	0.688247i
$77$	−3.00000			−0.341882
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$			1.00000i			0.111803i
$81$		0			0
$82$		−	9.00000i		−	0.993884i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$	9.00000			0.970495
$87$		0			0
$88$			3.00000i			0.319801i
$89$		−	12.0000i		−	1.27200i	−0.771690	−	0.635999i	$-0.780588\pi$
$89$		−	12.0000i		−	1.27200i	0.771690	−	0.635999i	$-0.219412\pi$
$90$		0			0
$91$		−	6.00000i		−	0.628971i
$92$			6.00000i			0.625543i
$93$		0			0
$94$		0			0
$95$	−6.00000			−0.615587
$96$		0			0
$97$			3.00000i			0.304604i	0.988334	+	0.152302i	$0.0486686\pi$
$97$			3.00000i			0.304604i	−0.988334	+	0.152302i	$0.951331\pi$
$98$		−	6.00000i		−	0.606092i
$99$		0			0
$100$	1.00000			0.100000
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	−6.00000			−0.588348
$105$		0			0
$106$		−	3.00000i		−	0.291386i
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$			9.00000i			0.862044i	0.902342	+	0.431022i	$0.141847\pi$
$109$			9.00000i			0.862044i	−0.902342	+	0.431022i	$0.858153\pi$
$110$	3.00000			0.286039
$111$		0			0
$112$	1.00000			0.0944911
$113$		−	9.00000i		−	0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$		−	9.00000i		−	0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$		0			0
$115$	6.00000			0.559503
$116$		−	9.00000i		−	0.835629i
$117$		0			0
$118$		0			0
$119$			3.00000i			0.275010i
$120$		0			0
$121$	−2.00000			−0.181818
$122$	9.00000			0.814822
$123$		0			0
$124$		−	3.00000i		−	0.269408i
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$	20.0000			1.77471			0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000			1.77471			0.887357	+	0.461084i	$0.152539\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$			6.00000i			0.526235i
$131$		−	12.0000i		−	1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$		−	12.0000i		−	1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$		0			0
$133$			6.00000i			0.520266i
$134$		−	14.0000i		−	1.20942i
$135$		0			0
$136$	3.00000			0.257248
$137$	−12.0000			−1.02523			−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000			−1.02523			−0.512615	+	0.858619i	$0.671323\pi$
$138$		0			0
$139$	−5.00000			−0.424094			−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000			−0.424094			−0.212047	+	0.977259i	$0.568013\pi$
$140$		−	1.00000i		−	0.0845154i
$141$		0			0
$142$		−	6.00000i		−	0.503509i
$143$			18.0000i			1.50524i
$144$		0			0
$145$	−9.00000			−0.747409
$146$		−	2.00000i		−	0.165521i
$147$		0			0
$148$	−1.00000	+	6.00000i	−0.0821995	+	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
$151$	−10.0000			−0.813788			−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000			−0.813788			−0.406894	+	0.913475i	$0.633388\pi$
$152$	6.00000			0.486664
$153$		0			0
$154$		−	3.00000i		−	0.241747i
$155$	−3.00000			−0.240966
$156$		0			0
$157$	−5.00000			−0.399043			−0.199522	−	0.979893i	$-0.563939\pi$
$157$	−5.00000			−0.399043			−0.199522	+	0.979893i	$0.563939\pi$
$158$		0			0
$159$		0			0
$160$	−1.00000			−0.0790569
$161$		−	6.00000i		−	0.472866i
$162$		0			0
$163$			9.00000i			0.704934i	0.935824	+	0.352467i	$0.114657\pi$
$163$			9.00000i			0.704934i	−0.935824	+	0.352467i	$0.885343\pi$
$164$	9.00000			0.702782
$165$		0			0
$166$		0			0
$167$			6.00000i			0.464294i	0.972681	+	0.232147i	$0.0745750\pi$
$167$			6.00000i			0.464294i	−0.972681	+	0.232147i	$0.925425\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		−	3.00000i		−	0.230089i
$171$		0			0
$172$			9.00000i			0.686244i
$173$	3.00000			0.228086			0.114043	−	0.993476i	$-0.463620\pi$
$173$	3.00000			0.228086			0.114043	+	0.993476i	$0.463620\pi$
$174$		0			0
$175$	−1.00000			−0.0755929
$176$	−3.00000			−0.226134
$177$		0			0
$178$	12.0000			0.899438
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$	16.0000			1.18927			0.594635	−	0.803996i	$-0.297296\pi$
$181$	16.0000			1.18927			0.594635	+	0.803996i	$0.297296\pi$
$182$	6.00000			0.444750
$183$		0			0
$184$	−6.00000			−0.442326
$185$	6.00000	+	1.00000i	0.441129	+	0.0735215i
$186$		0			0
$187$		−	9.00000i		−	0.658145i
$188$		0			0
$189$		0			0
$190$		−	6.00000i		−	0.435286i
$191$		−	27.0000i		−	1.95365i	−0.214036	−	0.976826i	$-0.568661\pi$
$191$		−	27.0000i		−	1.95365i	0.214036	−	0.976826i	$-0.431339\pi$
$192$		0			0
$193$		−	18.0000i		−	1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$		−	18.0000i		−	1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$	−3.00000			−0.215387
$195$		0			0
$196$	6.00000			0.428571
$197$	−18.0000			−1.28245			−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000			−1.28245			−0.641223	+	0.767354i	$0.721573\pi$
$198$		0			0
$199$			12.0000i			0.850657i	0.905039	+	0.425329i	$0.139842\pi$
$199$			12.0000i			0.850657i	−0.905039	+	0.425329i	$0.860158\pi$
$200$			1.00000i			0.0707107i
$201$		0			0
$202$		−	12.0000i		−	0.844317i
$203$			9.00000i			0.631676i
$204$		0			0
$205$		−	9.00000i		−	0.628587i
$206$	−6.00000			−0.418040
$207$		0			0
$208$		−	6.00000i		−	0.416025i
$209$		−	18.0000i		−	1.24509i
$210$		0			0
$211$	23.0000			1.58339			0.791693	−	0.610920i	$-0.209200\pi$
$211$	23.0000			1.58339			0.791693	+	0.610920i	$0.209200\pi$
$212$	3.00000			0.206041
$213$		0			0
$214$			6.00000i			0.410152i
$215$	9.00000			0.613795
$216$		0			0
$217$			3.00000i			0.203653i
$218$	−9.00000			−0.609557
$219$		0			0
$220$			3.00000i			0.202260i
$221$	18.0000			1.21081
$222$		0			0
$223$	−1.00000			−0.0669650			−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000			−0.0669650			−0.0334825	+	0.999439i	$0.510660\pi$
$224$			1.00000i			0.0668153i
$225$		0			0
$226$	9.00000			0.598671
$227$		−	21.0000i		−	1.39382i	−0.717159	−	0.696909i	$-0.754558\pi$
$227$		−	21.0000i		−	1.39382i	0.717159	−	0.696909i	$-0.245442\pi$
$228$		0			0
$229$	−22.0000			−1.45380			−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000			−1.45380			−0.726900	+	0.686743i	$0.759040\pi$
$230$			6.00000i			0.395628i
$231$		0			0
$232$	9.00000			0.590879
$233$	−24.0000			−1.57229			−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000			−1.57229			−0.786146	+	0.618041i	$0.787927\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	−3.00000			−0.194461
$239$		−	15.0000i		−	0.970269i	−0.874439	−	0.485135i	$-0.838771\pi$
$239$		−	15.0000i		−	0.970269i	0.874439	−	0.485135i	$-0.161229\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		−	2.00000i		−	0.128565i
$243$		0			0
$244$			9.00000i			0.576166i
$245$		−	6.00000i		−	0.383326i
$246$		0			0
$247$	36.0000			2.29063
$248$	3.00000			0.190500
$249$		0			0
$250$	1.00000			0.0632456
$251$			12.0000i			0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$			12.0000i			0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$		0			0
$253$			18.0000i			1.13165i
$254$			20.0000i			1.25491i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	6.00000i		−	0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$		−	6.00000i		−	0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$		0			0
$259$	1.00000	−	6.00000i	0.0621370	−	0.372822i
$260$	−6.00000			−0.372104
$261$		0			0
$262$	12.0000			0.741362
$263$	−9.00000			−0.554964			−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000			−0.554964			−0.277482	+	0.960731i	$0.589500\pi$
$264$		0			0
$265$		−	3.00000i		−	0.184289i
$266$	−6.00000			−0.367884
$267$		0			0
$268$	14.0000			0.855186
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−20.0000			−1.21491			−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000			−1.21491			−0.607457	+	0.794353i	$0.707810\pi$
$272$			3.00000i			0.181902i
$273$		0			0
$274$		−	12.0000i		−	0.724947i
$275$	3.00000			0.180907
$276$		0			0
$277$			12.0000i			0.721010i	0.932757	+	0.360505i	$0.117396\pi$
$277$			12.0000i			0.721010i	−0.932757	+	0.360505i	$0.882604\pi$
$278$		−	5.00000i		−	0.299880i
$279$		0			0
$280$	1.00000			0.0597614
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$		−	24.0000i		−	1.42665i	−0.700832	−	0.713326i	$-0.747188\pi$
$283$		−	24.0000i		−	1.42665i	0.700832	−	0.713326i	$-0.252812\pi$
$284$	6.00000			0.356034
$285$		0			0
$286$	−18.0000			−1.06436
$287$	−9.00000			−0.531253
$288$		0			0
$289$	8.00000			0.470588
$290$		−	9.00000i		−	0.528498i
$291$		0			0
$292$	2.00000			0.117041
$293$	−21.0000			−1.22683			−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000			−1.22683			−0.613417	+	0.789760i	$0.710205\pi$
$294$		0			0
$295$		0			0
$296$	−6.00000	−	1.00000i	−0.348743	−	0.0581238i
$297$		0			0
$298$			18.0000i			1.04271i
$299$	−36.0000			−2.08193
$300$		0			0
$301$		−	9.00000i		−	0.518751i
$302$		−	10.0000i		−	0.575435i
$303$		0			0
$304$			6.00000i			0.344124i
$305$	9.00000			0.515339
$306$		0			0
$307$	−16.0000			−0.913168			−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000			−0.913168			−0.456584	+	0.889680i	$0.650927\pi$
$308$	3.00000			0.170941
$309$		0			0
$310$		−	3.00000i		−	0.170389i
$311$			27.0000i			1.53103i	0.643418	+	0.765515i	$0.277516\pi$
$311$			27.0000i			1.53103i	−0.643418	+	0.765515i	$0.722484\pi$
$312$		0			0
$313$			18.0000i			1.01742i	0.860938	+	0.508710i	$0.169877\pi$
$313$			18.0000i			1.01742i	−0.860938	+	0.508710i	$0.830123\pi$
$314$		−	5.00000i		−	0.282166i
$315$		0			0
$316$		0			0
$317$	27.0000			1.51647			0.758236	−	0.651981i	$-0.226062\pi$
$317$	27.0000			1.51647			0.758236	+	0.651981i	$0.226062\pi$
$318$		0			0
$319$		−	27.0000i		−	1.51171i
$320$		−	1.00000i		−	0.0559017i
$321$		0			0
$322$	6.00000			0.334367
$323$	−18.0000			−1.00155
$324$		0			0
$325$			6.00000i			0.332820i
$326$	−9.00000			−0.498464
$327$		0			0
$328$			9.00000i			0.496942i
$329$		0			0
$330$		0			0
$331$			18.0000i			0.989369i	0.869072	+	0.494685i	$0.164716\pi$
$331$			18.0000i			0.989369i	−0.869072	+	0.494685i	$0.835284\pi$
$332$		0			0
$333$		0			0
$334$	−6.00000			−0.328305
$335$		−	14.0000i		−	0.764902i
$336$		0			0
$337$	−4.00000			−0.217894			−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000			−0.217894			−0.108947	+	0.994048i	$0.534748\pi$
$338$		−	23.0000i		−	1.25104i
$339$		0			0
$340$	3.00000			0.162698
$341$		−	9.00000i		−	0.487377i
$342$		0			0
$343$	−13.0000			−0.701934
$344$	−9.00000			−0.485247
$345$		0			0
$346$			3.00000i			0.161281i
$347$		−	12.0000i		−	0.644194i	−0.946707	−	0.322097i	$-0.895612\pi$
$347$		−	12.0000i		−	0.644194i	0.946707	−	0.322097i	$-0.104388\pi$
$348$		0			0
$349$	−26.0000			−1.39175			−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000			−1.39175			−0.695874	+	0.718164i	$0.744983\pi$
$350$		−	1.00000i		−	0.0534522i
$351$		0			0
$352$		−	3.00000i		−	0.159901i
$353$			21.0000i			1.11772i	0.829263	+	0.558859i	$0.188761\pi$
$353$			21.0000i			1.11772i	−0.829263	+	0.558859i	$0.811239\pi$
$354$		0			0
$355$		−	6.00000i		−	0.318447i
$356$			12.0000i			0.635999i
$357$		0			0
$358$	18.0000			0.951330
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$			16.0000i			0.840941i
$363$		0			0
$364$			6.00000i			0.314485i
$365$		−	2.00000i		−	0.104685i
$366$		0			0
$367$	17.0000			0.887393			0.443696	−	0.896177i	$-0.353667\pi$
$367$	17.0000			0.887393			0.443696	+	0.896177i	$0.353667\pi$
$368$		−	6.00000i		−	0.312772i
$369$		0			0
$370$	−1.00000	+	6.00000i	−0.0519875	+	0.311925i
$371$	−3.00000			−0.155752
$372$		0			0
$373$	22.0000			1.13912			0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000			1.13912			0.569558	+	0.821951i	$0.307114\pi$
$374$	9.00000			0.465379
$375$		0			0
$376$		0			0
$377$	54.0000			2.78114
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$	6.00000			0.307794
$381$		0			0
$382$	27.0000			1.38144
$383$		−	18.0000i		−	0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$		−	18.0000i		−	0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$		0			0
$385$		−	3.00000i		−	0.152894i
$386$	18.0000			0.916176
$387$		0			0
$388$		−	3.00000i		−	0.152302i
$389$			21.0000i			1.06474i	0.846511	+	0.532371i	$0.178699\pi$
$389$			21.0000i			1.06474i	−0.846511	+	0.532371i	$0.821301\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$			6.00000i			0.303046i
$393$		0			0
$394$		−	18.0000i		−	0.906827i
$395$		0			0
$396$		0			0
$397$	34.0000			1.70641			0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000			1.70641			0.853206	+	0.521575i	$0.174655\pi$
$398$	−12.0000			−0.601506
$399$		0			0
$400$	−1.00000			−0.0500000
$401$		−	18.0000i		−	0.898877i	−0.893311	−	0.449439i	$-0.851624\pi$
$401$		−	18.0000i		−	0.898877i	0.893311	−	0.449439i	$-0.148376\pi$
$402$		0			0
$403$	18.0000			0.896644
$404$	12.0000			0.597022
$405$		0			0
$406$	−9.00000			−0.446663
$407$	−3.00000	+	18.0000i	−0.148704	+	0.892227i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$	9.00000			0.444478
$411$		0			0
$412$		−	6.00000i		−	0.295599i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	6.00000			0.294174
$417$		0			0
$418$	18.0000			0.880409
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$			30.0000i			1.46211i	0.682318	+	0.731055i	$0.260972\pi$
$421$			30.0000i			1.46211i	−0.682318	+	0.731055i	$0.739028\pi$
$422$			23.0000i			1.11962i
$423$		0			0
$424$			3.00000i			0.145693i
$425$		−	3.00000i		−	0.145521i
$426$		0			0
$427$		−	9.00000i		−	0.435541i
$428$	−6.00000			−0.290021
$429$		0			0
$430$			9.00000i			0.434019i
$431$		−	9.00000i		−	0.433515i	−0.976226	−	0.216757i	$-0.930452\pi$
$431$		−	9.00000i		−	0.433515i	0.976226	−	0.216757i	$-0.0695480\pi$
$432$		0			0
$433$	−38.0000			−1.82616			−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000			−1.82616			−0.913082	+	0.407777i	$0.866304\pi$
$434$	−3.00000			−0.144005
$435$		0			0
$436$		−	9.00000i		−	0.431022i
$437$	36.0000			1.72211
$438$		0			0
$439$			33.0000i			1.57500i	0.616312	+	0.787502i	$0.288626\pi$
$439$			33.0000i			1.57500i	−0.616312	+	0.787502i	$0.711374\pi$
$440$	−3.00000			−0.143019
$441$		0			0
$442$			18.0000i			0.856173i
$443$	12.0000			0.570137			0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000			0.570137			0.285069	+	0.958507i	$0.407984\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$		−	1.00000i		−	0.0473514i
$447$		0			0
$448$	−1.00000			−0.0472456
$449$			30.0000i			1.41579i	0.706319	+	0.707894i	$0.250354\pi$
$449$			30.0000i			1.41579i	−0.706319	+	0.707894i	$0.749646\pi$
$450$		0			0
$451$	27.0000			1.27138
$452$			9.00000i			0.423324i
$453$		0			0
$454$	21.0000			0.985579
$455$	6.00000			0.281284
$456$		0			0
$457$			39.0000i			1.82434i	0.409809	+	0.912172i	$0.365595\pi$
$457$			39.0000i			1.82434i	−0.409809	+	0.912172i	$0.634405\pi$
$458$		−	22.0000i		−	1.02799i
$459$		0			0
$460$	−6.00000			−0.279751
$461$			9.00000i			0.419172i	0.977790	+	0.209586i	$0.0672116\pi$
$461$			9.00000i			0.419172i	−0.977790	+	0.209586i	$0.932788\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$			9.00000i			0.417815i
$465$		0			0
$466$		−	24.0000i		−	1.11178i
$467$			33.0000i			1.52706i	0.645774	+	0.763529i	$0.276535\pi$
$467$			33.0000i			1.52706i	−0.645774	+	0.763529i	$0.723465\pi$
$468$		0			0
$469$	−14.0000			−0.646460
$470$		0			0
$471$		0			0
$472$		0			0
$473$			27.0000i			1.24146i
$474$		0			0
$475$		−	6.00000i		−	0.275299i
$476$		−	3.00000i		−	0.137505i
$477$		0			0
$478$	15.0000			0.686084
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$	−36.0000	−	6.00000i	−1.64146	−	0.273576i
$482$		0			0
$483$		0			0
$484$	2.00000			0.0909091
$485$	−3.00000			−0.136223
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$	−9.00000			−0.407411
$489$		0			0
$490$	6.00000			0.271052
$491$	36.0000			1.62466			0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000			1.62466			0.812329	+	0.583200i	$0.198200\pi$
$492$		0			0
$493$	−27.0000			−1.21602
$494$			36.0000i			1.61972i
$495$		0			0
$496$			3.00000i			0.134704i
$497$	−6.00000			−0.269137
$498$		0			0
$499$		−	6.00000i		−	0.268597i	−0.990941	−	0.134298i	$-0.957122\pi$
$499$		−	6.00000i		−	0.268597i	0.990941	−	0.134298i	$-0.0428781\pi$
$500$			1.00000i			0.0447214i
$501$		0			0
$502$	−12.0000			−0.535586
$503$			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
$505$		−	12.0000i		−	0.533993i
$506$	−18.0000			−0.800198
$507$		0			0
$508$	−20.0000			−0.887357
$509$	−36.0000			−1.59567			−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000			−1.59567			−0.797836	+	0.602875i	$0.794022\pi$
$510$		0			0
$511$	−2.00000			−0.0884748
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	6.00000			0.264649
$515$	−6.00000			−0.264392
$516$		0			0
$517$		0			0
$518$	6.00000	+	1.00000i	0.263625	+	0.0439375i
$519$		0			0
$520$		−	6.00000i		−	0.263117i
$521$	−3.00000			−0.131432			−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000			−0.131432			−0.0657162	+	0.997838i	$0.520933\pi$
$522$		0			0
$523$		−	36.0000i		−	1.57417i	−0.616844	−	0.787085i	$-0.711589\pi$
$523$		−	36.0000i		−	1.57417i	0.616844	−	0.787085i	$-0.288411\pi$
$524$			12.0000i			0.524222i
$525$		0			0
$526$		−	9.00000i		−	0.392419i
$527$	−9.00000			−0.392046
$528$		0			0
$529$	−13.0000			−0.565217
$530$	3.00000			0.130312
$531$		0			0
$532$		−	6.00000i		−	0.260133i
$533$			54.0000i			2.33900i
$534$		0			0
$535$			6.00000i			0.259403i
$536$			14.0000i			0.604708i
$537$		0			0
$538$		0			0
$539$	18.0000			0.775315
$540$		0			0
$541$		−	6.00000i		−	0.257960i	−0.991647	−	0.128980i	$-0.958830\pi$
$541$		−	6.00000i		−	0.257960i	0.991647	−	0.128980i	$-0.0411703\pi$
$542$		−	20.0000i		−	0.859074i
$543$		0			0
$544$	−3.00000			−0.128624
$545$	−9.00000			−0.385518
$546$		0			0
$547$			21.0000i			0.897895i	0.893558	+	0.448948i	$0.148201\pi$
$547$			21.0000i			0.897895i	−0.893558	+	0.448948i	$0.851799\pi$
$548$	12.0000			0.512615
$549$		0			0
$550$			3.00000i			0.127920i
$551$	−54.0000			−2.30048
$552$		0			0
$553$		0			0
$554$	−12.0000			−0.509831
$555$		0			0
$556$	5.00000			0.212047
$557$		−	12.0000i		−	0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$		−	12.0000i		−	0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$		0			0
$559$	−54.0000			−2.28396
$560$			1.00000i			0.0422577i
$561$		0			0
$562$	18.0000			0.759284
$563$			27.0000i			1.13791i	0.822367	+	0.568957i	$0.192653\pi$
$563$			27.0000i			1.13791i	−0.822367	+	0.568957i	$0.807347\pi$
$564$		0			0
$565$	9.00000			0.378633
$566$	24.0000			1.00880
$567$		0			0
$568$			6.00000i			0.251754i
$569$			12.0000i			0.503066i	0.967849	+	0.251533i	$0.0809347\pi$
$569$			12.0000i			0.503066i	−0.967849	+	0.251533i	$0.919065\pi$
$570$		0			0
$571$	−5.00000			−0.209243			−0.104622	−	0.994512i	$-0.533363\pi$
$571$	−5.00000			−0.209243			−0.104622	+	0.994512i	$0.533363\pi$
$572$		−	18.0000i		−	0.752618i
$573$		0			0
$574$		−	9.00000i		−	0.375653i
$575$			6.00000i			0.250217i
$576$		0			0
$577$			18.0000i			0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$			18.0000i			0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$			8.00000i			0.332756i
$579$		0			0
$580$	9.00000			0.373705
$581$		0			0
$582$		0			0
$583$	9.00000			0.372742
$584$			2.00000i			0.0827606i
$585$		0			0
$586$		−	21.0000i		−	0.867502i
$587$			3.00000i			0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$			3.00000i			0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$		0			0
$589$	−18.0000			−0.741677
$590$		0			0
$591$		0			0
$592$	1.00000	−	6.00000i	0.0410997	−	0.246598i
$593$	12.0000			0.492781			0.246390	−	0.969171i	$-0.420755\pi$
$593$	12.0000			0.492781			0.246390	+	0.969171i	$0.420755\pi$
$594$		0			0
$595$	−3.00000			−0.122988
$596$	−18.0000			−0.737309
$597$		0			0
$598$		−	36.0000i		−	1.47215i
$599$	6.00000			0.245153			0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000			0.245153			0.122577	+	0.992459i	$0.460884\pi$
$600$		0			0
$601$	−35.0000			−1.42768			−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000			−1.42768			−0.713840	+	0.700309i	$0.753046\pi$
$602$	9.00000			0.366813
$603$		0			0
$604$	10.0000			0.406894
$605$		−	2.00000i		−	0.0813116i
$606$		0			0
$607$		−	24.0000i		−	0.974130i	−0.873366	−	0.487065i	$-0.838067\pi$
$607$		−	24.0000i		−	0.974130i	0.873366	−	0.487065i	$-0.161933\pi$
$608$	−6.00000			−0.243332
$609$		0			0
$610$			9.00000i			0.364399i
$611$		0			0
$612$		0			0
$613$	−25.0000			−1.00974			−0.504870	−	0.863195i	$-0.668460\pi$
$613$	−25.0000			−1.00974			−0.504870	+	0.863195i	$0.668460\pi$
$614$		−	16.0000i		−	0.645707i
$615$		0			0
$616$			3.00000i			0.120873i
$617$	−12.0000			−0.483102			−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000			−0.483102			−0.241551	+	0.970388i	$0.577656\pi$
$618$		0			0
$619$	35.0000			1.40677			0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000			1.40677			0.703384	+	0.710810i	$0.251671\pi$
$620$	3.00000			0.120483
$621$		0			0
$622$	−27.0000			−1.08260
$623$		−	12.0000i		−	0.480770i
$624$		0			0
$625$	1.00000			0.0400000
$626$	−18.0000			−0.719425
$627$		0			0
$628$	5.00000			0.199522
$629$	18.0000	+	3.00000i	0.717707	+	0.119618i
$630$		0			0
$631$			27.0000i			1.07485i	0.843311	+	0.537427i	$0.180603\pi$
$631$			27.0000i			1.07485i	−0.843311	+	0.537427i	$0.819397\pi$
$632$		0			0
$633$		0			0
$634$			27.0000i			1.07231i
$635$			20.0000i			0.793676i
$636$		0			0
$637$			36.0000i			1.42637i
$638$	27.0000			1.06894
$639$		0			0
$640$	1.00000			0.0395285
$641$	33.0000			1.30342			0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000			1.30342			0.651711	+	0.758468i	$0.274052\pi$
$642$		0			0
$643$		−	27.0000i		−	1.06478i	−0.846500	−	0.532388i	$-0.821295\pi$
$643$		−	27.0000i		−	1.06478i	0.846500	−	0.532388i	$-0.178705\pi$
$644$			6.00000i			0.236433i
$645$		0			0
$646$		−	18.0000i		−	0.708201i
$647$			18.0000i			0.707653i	0.935311	+	0.353827i	$0.115120\pi$
$647$			18.0000i			0.707653i	−0.935311	+	0.353827i	$0.884880\pi$
$648$		0			0
$649$		0			0
$650$	−6.00000			−0.235339
$651$		0			0
$652$		−	9.00000i		−	0.352467i
$653$			42.0000i			1.64359i	0.569785	+	0.821794i	$0.307026\pi$
$653$			42.0000i			1.64359i	−0.569785	+	0.821794i	$0.692974\pi$
$654$		0			0
$655$	12.0000			0.468879
$656$	−9.00000			−0.351391
$657$		0			0
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$		−	15.0000i		−	0.583432i	−0.956505	−	0.291716i	$-0.905774\pi$
$661$		−	15.0000i		−	0.583432i	0.956505	−	0.291716i	$-0.0942263\pi$
$662$	−18.0000			−0.699590
$663$		0			0
$664$		0			0
$665$	−6.00000			−0.232670
$666$		0			0
$667$	54.0000			2.09089
$668$		−	6.00000i		−	0.232147i
$669$		0			0
$670$	14.0000			0.540867
$671$			27.0000i			1.04232i
$672$		0			0
$673$	−26.0000			−1.00223			−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000			−1.00223			−0.501113	+	0.865382i	$0.667076\pi$
$674$		−	4.00000i		−	0.154074i
$675$		0			0
$676$	23.0000			0.884615
$677$	−6.00000			−0.230599			−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000			−0.230599			−0.115299	+	0.993331i	$0.536783\pi$
$678$		0			0
$679$			3.00000i			0.115129i
$680$			3.00000i			0.115045i
$681$		0			0
$682$	9.00000			0.344628
$683$		−	9.00000i		−	0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$		−	9.00000i		−	0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$		0			0
$685$		−	12.0000i		−	0.458496i
$686$		−	13.0000i		−	0.496342i
$687$		0			0
$688$		−	9.00000i		−	0.343122i
$689$			18.0000i			0.685745i
$690$		0			0
$691$	37.0000			1.40755			0.703773	−	0.710425i	$-0.251497\pi$
$691$	37.0000			1.40755			0.703773	+	0.710425i	$0.251497\pi$
$692$	−3.00000			−0.114043
$693$		0			0
$694$	12.0000			0.455514
$695$		−	5.00000i		−	0.189661i
$696$		0			0
$697$		−	27.0000i		−	1.02270i
$698$		−	26.0000i		−	0.984115i
$699$		0			0
$700$	1.00000			0.0377964
$701$		−	30.0000i		−	1.13308i	−0.824033	−	0.566542i	$-0.808281\pi$
$701$		−	30.0000i		−	1.13308i	0.824033	−	0.566542i	$-0.191719\pi$
$702$		0			0
$703$	36.0000	+	6.00000i	1.35777	+	0.226294i
$704$	3.00000			0.113067
$705$		0			0
$706$	−21.0000			−0.790345
$707$	−12.0000			−0.451306
$708$		0			0
$709$		−	39.0000i		−	1.46468i	−0.680941	−	0.732338i	$-0.738429\pi$
$709$		−	39.0000i		−	1.46468i	0.680941	−	0.732338i	$-0.261571\pi$
$710$	6.00000			0.225176
$711$		0			0
$712$	−12.0000			−0.449719
$713$	18.0000			0.674105
$714$		0			0
$715$	−18.0000			−0.673162
$716$			18.0000i			0.672692i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$			6.00000i			0.223452i
$722$		−	17.0000i		−	0.632674i
$723$		0			0
$724$	−16.0000			−0.594635
$725$		−	9.00000i		−	0.334252i
$726$		0			0
$727$		−	30.0000i		−	1.11264i	−0.830969	−	0.556319i	$-0.812213\pi$
$727$		−	30.0000i		−	1.11264i	0.830969	−	0.556319i	$-0.187787\pi$
$728$	−6.00000			−0.222375
$729$		0			0
$730$	2.00000			0.0740233
$731$	27.0000			0.998631
$732$		0			0
$733$	23.0000			0.849524			0.424762	−	0.905305i	$-0.360358\pi$
$733$	23.0000			0.849524			0.424762	+	0.905305i	$0.360358\pi$
$734$			17.0000i			0.627481i
$735$		0			0
$736$	6.00000			0.221163
$737$	42.0000			1.54709
$738$		0			0
$739$	11.0000			0.404642			0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000			0.404642			0.202321	+	0.979319i	$0.435152\pi$
$740$	−6.00000	−	1.00000i	−0.220564	−	0.0367607i
$741$		0			0
$742$		−	3.00000i		−	0.110133i
$743$	15.0000			0.550297			0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000			0.550297			0.275148	+	0.961402i	$0.411273\pi$
$744$		0			0
$745$			18.0000i			0.659469i
$746$			22.0000i			0.805477i
$747$		0			0
$748$			9.00000i			0.329073i
$749$	6.00000			0.219235
$750$		0			0
$751$	−50.0000			−1.82453			−0.912263	−	0.409605i	$-0.865667\pi$
$751$	−50.0000			−1.82453			−0.912263	+	0.409605i	$0.865667\pi$
$752$		0			0
$753$		0			0
$754$			54.0000i			1.96656i
$755$		−	10.0000i		−	0.363937i
$756$		0			0
$757$			36.0000i			1.30844i	0.756303	+	0.654221i	$0.227003\pi$
$757$			36.0000i			1.30844i	−0.756303	+	0.654221i	$0.772997\pi$
$758$		−	20.0000i		−	0.726433i
$759$		0			0
$760$			6.00000i			0.217643i
$761$	3.00000			0.108750			0.0543750	−	0.998521i	$-0.482683\pi$
$761$	3.00000			0.108750			0.0543750	+	0.998521i	$0.482683\pi$
$762$		0			0
$763$			9.00000i			0.325822i
$764$			27.0000i			0.976826i
$765$		0			0
$766$	18.0000			0.650366
$767$		0			0
$768$		0			0
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$	3.00000			0.108112
$771$		0			0
$772$			18.0000i			0.647834i
$773$	3.00000			0.107903			0.0539513	−	0.998544i	$-0.482818\pi$
$773$	3.00000			0.107903			0.0539513	+	0.998544i	$0.482818\pi$
$774$		0			0
$775$		−	3.00000i		−	0.107763i
$776$	3.00000			0.107694
$777$		0			0
$778$	−21.0000			−0.752886
$779$		−	54.0000i		−	1.93475i
$780$		0			0
$781$	18.0000			0.644091
$782$			18.0000i			0.643679i
$783$		0			0
$784$	−6.00000			−0.214286
$785$		−	5.00000i		−	0.178458i
$786$		0			0
$787$	−22.0000			−0.784215			−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000			−0.784215			−0.392108	+	0.919919i	$0.628254\pi$
$788$	18.0000			0.641223
$789$		0			0
$790$		0			0
$791$		−	9.00000i		−	0.320003i
$792$		0			0
$793$	−54.0000			−1.91760
$794$			34.0000i			1.20661i
$795$		0			0
$796$		−	12.0000i		−	0.425329i
$797$		−	12.0000i		−	0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$		−	12.0000i		−	0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$		0			0
$799$		0			0
$800$		−	1.00000i		−	0.0353553i
$801$		0			0
$802$	18.0000			0.635602
$803$	6.00000			0.211735
$804$		0			0
$805$	6.00000			0.211472
$806$			18.0000i			0.634023i
$807$		0			0
$808$			12.0000i			0.422159i
$809$		−	18.0000i		−	0.632846i	−0.948618	−	0.316423i	$-0.897518\pi$
$809$		−	18.0000i		−	0.632846i	0.948618	−	0.316423i	$-0.102482\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		−	9.00000i		−	0.315838i
$813$		0			0
$814$	−18.0000	−	3.00000i	−0.630900	−	0.105150i
$815$	−9.00000			−0.315256
$816$		0			0
$817$	54.0000			1.88922
$818$		0			0
$819$		0			0
$820$			9.00000i			0.314294i
$821$	−36.0000			−1.25641			−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000			−1.25641			−0.628204	+	0.778048i	$0.716210\pi$
$822$		0			0
$823$	32.0000			1.11545			0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000			1.11545			0.557725	+	0.830026i	$0.311674\pi$
$824$	6.00000			0.209020
$825$		0			0
$826$		0			0
$827$			27.0000i			0.938882i	0.882964	+	0.469441i	$0.155545\pi$
$827$			27.0000i			0.938882i	−0.882964	+	0.469441i	$0.844455\pi$
$828$		0			0
$829$		−	9.00000i		−	0.312583i	−0.987711	−	0.156291i	$-0.950046\pi$
$829$		−	9.00000i		−	0.312583i	0.987711	−	0.156291i	$-0.0499539\pi$
$830$		0			0
$831$		0			0
$832$			6.00000i			0.208013i
$833$		−	18.0000i		−	0.623663i
$834$		0			0
$835$	−6.00000			−0.207639
$836$			18.0000i			0.622543i
$837$		0			0
$838$			24.0000i			0.829066i
$839$	−30.0000			−1.03572			−0.517858	−	0.855467i	$-0.673270\pi$
$839$	−30.0000			−1.03572			−0.517858	+	0.855467i	$0.673270\pi$
$840$		0			0
$841$	−52.0000			−1.79310
$842$	−30.0000			−1.03387
$843$		0			0
$844$	−23.0000			−0.791693
$845$		−	23.0000i		−	0.791224i
$846$		0			0
$847$	−2.00000			−0.0687208
$848$	−3.00000			−0.103020
$849$		0			0
$850$	3.00000			0.102899
$851$	−36.0000	−	6.00000i	−1.23406	−	0.205677i
$852$		0			0
$853$			6.00000i			0.205436i	0.994711	+	0.102718i	$0.0327539\pi$
$853$			6.00000i			0.205436i	−0.994711	+	0.102718i	$0.967246\pi$
$854$	9.00000			0.307974
$855$		0			0
$856$		−	6.00000i		−	0.205076i
$857$			21.0000i			0.717346i	0.933463	+	0.358673i	$0.116771\pi$
$857$			21.0000i			0.717346i	−0.933463	+	0.358673i	$0.883229\pi$
$858$		0			0
$859$			42.0000i			1.43302i	0.697576	+	0.716511i	$0.254262\pi$
$859$			42.0000i			1.43302i	−0.697576	+	0.716511i	$0.745738\pi$
$860$	−9.00000			−0.306897
$861$		0			0
$862$	9.00000			0.306541
$863$	3.00000			0.102121			0.0510606	−	0.998696i	$-0.483740\pi$
$863$	3.00000			0.102121			0.0510606	+	0.998696i	$0.483740\pi$
$864$		0			0
$865$			3.00000i			0.102003i
$866$		−	38.0000i		−	1.29129i
$867$		0			0
$868$		−	3.00000i		−	0.101827i
$869$		0			0
$870$		0			0
$871$			84.0000i			2.84623i
$872$	9.00000			0.304778
$873$		0			0
$874$			36.0000i			1.21772i
$875$		−	1.00000i		−	0.0338062i
$876$		0			0
$877$	−41.0000			−1.38447			−0.692236	−	0.721671i	$-0.743374\pi$
$877$	−41.0000			−1.38447			−0.692236	+	0.721671i	$0.743374\pi$
$878$	−33.0000			−1.11370
$879$		0			0
$880$		−	3.00000i		−	0.101130i
$881$	−57.0000			−1.92038			−0.960189	−	0.279350i	$-0.909881\pi$
$881$	−57.0000			−1.92038			−0.960189	+	0.279350i	$0.909881\pi$
$882$		0			0
$883$			39.0000i			1.31245i	0.754563	+	0.656227i	$0.227849\pi$
$883$			39.0000i			1.31245i	−0.754563	+	0.656227i	$0.772151\pi$
$884$	−18.0000			−0.605406
$885$		0			0
$886$			12.0000i			0.403148i
$887$	−9.00000			−0.302190			−0.151095	−	0.988519i	$-0.548280\pi$
$887$	−9.00000			−0.302190			−0.151095	+	0.988519i	$0.548280\pi$
$888$		0			0
$889$	20.0000			0.670778
$890$			12.0000i			0.402241i
$891$		0			0
$892$	1.00000			0.0334825
$893$		0			0
$894$		0			0
$895$	18.0000			0.601674
$896$		−	1.00000i		−	0.0334077i
$897$		0			0
$898$	−30.0000			−1.00111
$899$	−27.0000			−0.900500
$900$		0			0
$901$		−	9.00000i		−	0.299833i
$902$			27.0000i			0.899002i
$903$		0			0
$904$	−9.00000			−0.299336
$905$			16.0000i			0.531858i
$906$		0			0
$907$		−	36.0000i		−	1.19536i	−0.801735	−	0.597680i	$-0.796089\pi$
$907$		−	36.0000i		−	1.19536i	0.801735	−	0.597680i	$-0.203911\pi$
$908$			21.0000i			0.696909i
$909$		0			0
$910$			6.00000i			0.198898i
$911$			24.0000i			0.795155i	0.917568	+	0.397578i	$0.130149\pi$
$911$			24.0000i			0.795155i	−0.917568	+	0.397578i	$0.869851\pi$
$912$		0			0
$913$		0			0
$914$	−39.0000			−1.29001
$915$		0			0
$916$	22.0000			0.726900
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$		−	48.0000i		−	1.58337i	−0.610927	−	0.791687i	$-0.709203\pi$
$919$		−	48.0000i		−	1.58337i	0.610927	−	0.791687i	$-0.290797\pi$
$920$		−	6.00000i		−	0.197814i
$921$		0			0
$922$	−9.00000			−0.296399
$923$			36.0000i			1.18495i
$924$		0			0
$925$	−1.00000	+	6.00000i	−0.0328798	+	0.197279i
$926$		0			0
$927$		0			0
$928$	−9.00000			−0.295439
$929$	−39.0000			−1.27955			−0.639774	−	0.768563i	$-0.720972\pi$
$929$	−39.0000			−1.27955			−0.639774	+	0.768563i	$0.720972\pi$
$930$		0			0
$931$		−	36.0000i		−	1.17985i
$932$	24.0000			0.786146
$933$		0			0
$934$	−33.0000			−1.07979
$935$	9.00000			0.294331
$936$		0			0
$937$	20.0000			0.653372			0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000			0.653372			0.326686	+	0.945133i	$0.394068\pi$
$938$		−	14.0000i		−	0.457116i
$939$		0			0
$940$		0			0
$941$	36.0000			1.17357			0.586783	−	0.809744i	$-0.300394\pi$
$941$	36.0000			1.17357			0.586783	+	0.809744i	$0.300394\pi$
$942$		0			0
$943$			54.0000i			1.75848i
$944$		0			0
$945$		0			0
$946$	−27.0000			−0.877846
$947$		−	21.0000i		−	0.682408i	−0.939989	−	0.341204i	$-0.889165\pi$
$947$		−	21.0000i		−	0.682408i	0.939989	−	0.341204i	$-0.110835\pi$
$948$		0			0
$949$			12.0000i			0.389536i
$950$	6.00000			0.194666
$951$		0			0
$952$	3.00000			0.0972306
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$	27.0000			0.873699
$956$			15.0000i			0.485135i
$957$		0			0
$958$		0			0
$959$	−12.0000			−0.387500
$960$		0			0
$961$	22.0000			0.709677
$962$	6.00000	−	36.0000i	0.193448	−	1.16069i
$963$		0			0
$964$		0			0
$965$	18.0000			0.579441
$966$		0			0
$967$			42.0000i			1.35063i	0.737530	+	0.675314i	$0.235992\pi$
$967$			42.0000i			1.35063i	−0.737530	+	0.675314i	$0.764008\pi$
$968$			2.00000i			0.0642824i
$969$		0			0
$970$		−	3.00000i		−	0.0963242i
$971$	57.0000			1.82922			0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000			1.82922			0.914609	+	0.404341i	$0.132499\pi$
$972$		0			0
$973$	−5.00000			−0.160293
$974$	12.0000			0.384505
$975$		0			0
$976$		−	9.00000i		−	0.288083i
$977$		−	3.00000i		−	0.0959785i	−0.998848	−	0.0479893i	$-0.984719\pi$
$977$		−	3.00000i		−	0.0959785i	0.998848	−	0.0479893i	$-0.0152813\pi$
$978$		0			0
$979$			36.0000i			1.15056i
$980$			6.00000i			0.191663i
$981$		0			0
$982$			36.0000i			1.14881i
$983$	9.00000			0.287055			0.143528	−	0.989646i	$-0.454155\pi$
$983$	9.00000			0.287055			0.143528	+	0.989646i	$0.454155\pi$
$984$		0			0
$985$		−	18.0000i		−	0.573528i
$986$		−	27.0000i		−	0.859855i
$987$		0			0
$988$	−36.0000			−1.14531
$989$	−54.0000			−1.71710
$990$		0			0
$991$		−	27.0000i		−	0.857683i	−0.903380	−	0.428842i	$-0.858922\pi$
$991$		−	27.0000i		−	0.857683i	0.903380	−	0.428842i	$-0.141078\pi$
$992$	−3.00000			−0.0952501
$993$		0			0
$994$		−	6.00000i		−	0.190308i
$995$	−12.0000			−0.380426
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	6.00000			0.189927
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.h.f.2071.2		2
3.2	odd	2		1110.2.h.c.961.1	✓	2
37.36	even	2	inner	3330.2.h.f.2071.1		2
111.110	odd	2		1110.2.h.c.961.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.h.c.961.1	✓	2	3.2	odd	2
1110.2.h.c.961.2	yes	2	111.110	odd	2
3330.2.h.f.2071.1		2	37.36	even	2	inner
3330.2.h.f.2071.2		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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.h (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more