LMFDB - Embedded newform 3330.2.d.d.1999.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1999,3330)]

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, 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("3330.1999");

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.d (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			1999.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3330.1999
Dual form			3330.2.d.d.1999.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} -5.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +(1.00000 - 2.00000i) q^{20} +5.00000i q^{22} +4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{28} -3.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -1.00000 q^{34} +(-2.00000 - 1.00000i) q^{35} -1.00000i q^{37} +(-2.00000 - 1.00000i) q^{40} +1.00000 q^{41} -7.00000i q^{43} +5.00000 q^{44} +4.00000 q^{46} +4.00000i q^{47} +6.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} +3.00000i q^{53} +(5.00000 - 10.0000i) q^{55} -1.00000 q^{56} +3.00000i q^{58} -8.00000 q^{59} +5.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +1.00000i q^{68} +(-1.00000 + 2.00000i) q^{70} +6.00000 q^{71} -10.0000i q^{73} -1.00000 q^{74} -5.00000i q^{77} +(-1.00000 + 2.00000i) q^{80} -1.00000i q^{82} -8.00000i q^{83} +(2.00000 + 1.00000i) q^{85} -7.00000 q^{86} -5.00000i q^{88} +8.00000 q^{89} -4.00000i q^{92} +4.00000 q^{94} -1.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^4 - 2 * q^5	$ 2 q - 2 q^{4} - 2 q^{5} + 4 q^{10} - 10 q^{11} + 2 q^{14} + 2 q^{16} + 2 q^{20} - 6 q^{25} - 6 q^{29} + 2 q^{31} - 2 q^{34} - 4 q^{35} - 4 q^{40} + 2 q^{41} + 10 q^{44} + 8 q^{46} + 12 q^{49} - 8 q^{50} + 10 q^{55} - 2 q^{56} - 16 q^{59} + 10 q^{61} - 2 q^{64} - 2 q^{70} + 12 q^{71} - 2 q^{74} - 2 q^{80} + 4 q^{85} - 14 q^{86} + 16 q^{89} + 8 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^10 - 10 * q^11 + 2 * q^14 + 2 * q^16 + 2 * q^20 - 6 * q^25 - 6 * q^29 + 2 * q^31 - 2 * q^34 - 4 * q^35 - 4 * q^40 + 2 * q^41 + 10 * q^44 + 8 * q^46 + 12 * q^49 - 8 * q^50 + 10 * q^55 - 2 * q^56 - 16 * q^59 + 10 * q^61 - 2 * q^64 - 2 * q^70 + 12 * q^71 - 2 * q^74 - 2 * q^80 + 4 * q^85 - 14 * q^86 + 16 * q^89 + 8 * q^94

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
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$5$	−1.00000	+	2.00000i	−0.447214	+	0.894427i
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	−5.00000			−1.50756			−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000			−1.50756			−0.753778	+	0.657129i	$0.771771\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	1.00000			0.267261
$15$		0			0
$16$	1.00000			0.250000
$17$		−	1.00000i		−	0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$		−	1.00000i		−	0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	1.00000	−	2.00000i	0.223607	−	0.447214i
$21$		0			0
$22$			5.00000i			1.06600i
$23$			4.00000i			0.834058i	0.908893	+	0.417029i	$0.136929\pi$
$23$			4.00000i			0.834058i	−0.908893	+	0.417029i	$0.863071\pi$
$24$		0			0
$25$	−3.00000	−	4.00000i	−0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	−3.00000			−0.557086			−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000			−0.557086			−0.278543	+	0.960424i	$0.589851\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−1.00000			−0.171499
$35$	−2.00000	−	1.00000i	−0.338062	−	0.169031i
$36$		0			0
$37$		−	1.00000i		−	0.164399i
$37$		−	1.00000i		−	0.164399i
$38$		0			0
$39$		0			0
$40$	−2.00000	−	1.00000i	−0.316228	−	0.158114i
$41$	1.00000			0.156174			0.0780869	−	0.996947i	$-0.475119\pi$
$41$	1.00000			0.156174			0.0780869	+	0.996947i	$0.475119\pi$
$42$		0			0
$43$		−	7.00000i		−	1.06749i	−0.845645	−	0.533745i	$-0.820784\pi$
$43$		−	7.00000i		−	1.06749i	0.845645	−	0.533745i	$-0.179216\pi$
$44$	5.00000			0.753778
$45$		0			0
$46$	4.00000			0.589768
$47$			4.00000i			0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$			4.00000i			0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$	−4.00000	+	3.00000i	−0.565685	+	0.424264i
$51$		0			0
$52$		0			0
$53$			3.00000i			0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$			3.00000i			0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$		0			0
$55$	5.00000	−	10.0000i	0.674200	−	1.34840i
$56$	−1.00000			−0.133631
$57$		0			0
$58$			3.00000i			0.393919i
$59$	−8.00000			−1.04151			−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000			−1.04151			−0.520756	+	0.853706i	$0.674350\pi$
$60$		0			0
$61$	5.00000			0.640184			0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000			0.640184			0.320092	+	0.947386i	$0.396286\pi$
$62$		−	1.00000i		−	0.127000i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$			1.00000i			0.121268i
$69$		0			0
$70$	−1.00000	+	2.00000i	−0.119523	+	0.239046i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$		−	10.0000i		−	1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$		−	10.0000i		−	1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	−1.00000			−0.116248
$75$		0			0
$76$		0			0
$77$		−	5.00000i		−	0.569803i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−1.00000	+	2.00000i	−0.111803	+	0.223607i
$81$		0			0
$82$		−	1.00000i		−	0.110432i
$83$		−	8.00000i		−	0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$		−	8.00000i		−	0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$		0			0
$85$	2.00000	+	1.00000i	0.216930	+	0.108465i
$86$	−7.00000			−0.754829
$87$		0			0
$88$		−	5.00000i		−	0.533002i
$89$	8.00000			0.847998			0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000			0.847998			0.423999	+	0.905663i	$0.360626\pi$
$90$		0			0
$91$		0			0
$92$		−	4.00000i		−	0.417029i
$93$		0			0
$94$	4.00000			0.412568
$95$		0			0
$96$		0			0
$97$		−	1.00000i		−	0.101535i	−0.998711	−	0.0507673i	$-0.983833\pi$
$97$		−	1.00000i		−	0.101535i	0.998711	−	0.0507673i	$-0.0161667\pi$
$98$		−	6.00000i		−	0.606092i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$		0			0
$106$	3.00000			0.291386
$107$		−	6.00000i		−	0.580042i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$		−	6.00000i		−	0.580042i	0.957020	−	0.290021i	$-0.0936623\pi$
$108$		0			0
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$	−10.0000	−	5.00000i	−0.953463	−	0.476731i
$111$		0			0
$112$			1.00000i			0.0944911i
$113$		−	21.0000i		−	1.97551i	−0.156001	−	0.987757i	$-0.549860\pi$
$113$		−	21.0000i		−	1.97551i	0.156001	−	0.987757i	$-0.450140\pi$
$114$		0			0
$115$	−8.00000	−	4.00000i	−0.746004	−	0.373002i
$116$	3.00000			0.278543
$117$		0			0
$118$			8.00000i			0.736460i
$119$	1.00000			0.0916698
$120$		0			0
$121$	14.0000			1.27273
$122$		−	5.00000i		−	0.452679i
$123$		0			0
$124$	−1.00000			−0.0898027
$125$	11.0000	−	2.00000i	0.983870	−	0.178885i
$126$		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
$129$		0			0
$130$		0			0
$131$	6.00000			0.524222			0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000			0.524222			0.262111	+	0.965038i	$0.415581\pi$
$132$		0			0
$133$		0			0
$134$	−4.00000			−0.345547
$135$		0			0
$136$	1.00000			0.0857493
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	11.0000			0.933008			0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000			0.933008			0.466504	+	0.884519i	$0.345513\pi$
$140$	2.00000	+	1.00000i	0.169031	+	0.0845154i
$141$		0			0
$142$		−	6.00000i		−	0.503509i
$143$		0			0
$144$		0			0
$145$	3.00000	−	6.00000i	0.249136	−	0.498273i
$146$	−10.0000			−0.827606
$147$		0			0
$148$			1.00000i			0.0821995i
$149$	4.00000			0.327693			0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000			0.327693			0.163846	+	0.986486i	$0.447610\pi$
$150$		0			0
$151$	−2.00000			−0.162758			−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000			−0.162758			−0.0813788	+	0.996683i	$0.525932\pi$
$152$		0			0
$153$		0			0
$154$	−5.00000			−0.402911
$155$	−1.00000	+	2.00000i	−0.0803219	+	0.160644i
$156$		0			0
$157$			23.0000i			1.83560i	0.397043	+	0.917800i	$0.370036\pi$
$157$			23.0000i			1.83560i	−0.397043	+	0.917800i	$0.629964\pi$
$158$		0			0
$159$		0			0
$160$	2.00000	+	1.00000i	0.158114	+	0.0790569i
$161$	−4.00000			−0.315244
$162$		0			0
$163$			1.00000i			0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$			1.00000i			0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$	−1.00000			−0.0780869
$165$		0			0
$166$	−8.00000			−0.620920
$167$			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
$169$	13.0000			1.00000
$170$	1.00000	−	2.00000i	0.0766965	−	0.153393i
$171$		0			0
$172$			7.00000i			0.533745i
$173$		−	13.0000i		−	0.988372i	−0.869356	−	0.494186i	$-0.835466\pi$
$173$		−	13.0000i		−	0.988372i	0.869356	−	0.494186i	$-0.164534\pi$
$174$		0			0
$175$	4.00000	−	3.00000i	0.302372	−	0.226779i
$176$	−5.00000			−0.376889
$177$		0			0
$178$		−	8.00000i		−	0.599625i
$179$	10.0000			0.747435			0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000			0.747435			0.373718	+	0.927543i	$0.378083\pi$
$180$		0			0
$181$	−16.0000			−1.18927			−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000			−1.18927			−0.594635	+	0.803996i	$0.702704\pi$
$182$		0			0
$183$		0			0
$184$	−4.00000			−0.294884
$185$	2.00000	+	1.00000i	0.147043	+	0.0735215i
$186$		0			0
$187$			5.00000i			0.365636i
$188$		−	4.00000i		−	0.291730i
$189$		0			0
$190$		0			0
$191$	−3.00000			−0.217072			−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000			−0.217072			−0.108536	+	0.994092i	$0.534616\pi$
$192$		0			0
$193$		−	10.0000i		−	0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$		−	10.0000i		−	0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	−1.00000			−0.0717958
$195$		0			0
$196$	−6.00000			−0.428571
$197$		−	26.0000i		−	1.85242i	−0.377004	−	0.926212i	$-0.623046\pi$
$197$		−	26.0000i		−	1.85242i	0.377004	−	0.926212i	$-0.376954\pi$
$198$		0			0
$199$	−24.0000			−1.70131			−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000			−1.70131			−0.850657	+	0.525720i	$0.823796\pi$
$200$	4.00000	−	3.00000i	0.282843	−	0.212132i
$201$		0			0
$202$			10.0000i			0.703598i
$203$		−	3.00000i		−	0.210559i
$204$		0			0
$205$	−1.00000	+	2.00000i	−0.0698430	+	0.139686i
$206$	−8.00000			−0.557386
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−21.0000			−1.44570			−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000			−1.44570			−0.722850	+	0.691005i	$0.757168\pi$
$212$		−	3.00000i		−	0.206041i
$213$		0			0
$214$	−6.00000			−0.410152
$215$	14.0000	+	7.00000i	0.954792	+	0.477396i
$216$		0			0
$217$			1.00000i			0.0678844i
$218$			1.00000i			0.0677285i
$219$		0			0
$220$	−5.00000	+	10.0000i	−0.337100	+	0.674200i
$221$		0			0
$222$		0			0
$223$			5.00000i			0.334825i	0.985887	+	0.167412i	$0.0535411\pi$
$223$			5.00000i			0.334825i	−0.985887	+	0.167412i	$0.946459\pi$
$224$	1.00000			0.0668153
$225$		0			0
$226$	−21.0000			−1.39690
$227$		−	1.00000i		−	0.0663723i	−0.999449	−	0.0331862i	$-0.989435\pi$
$227$		−	1.00000i		−	0.0663723i	0.999449	−	0.0331862i	$-0.0105654\pi$
$228$		0			0
$229$	16.0000			1.05731			0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000			1.05731			0.528655	+	0.848837i	$0.322697\pi$
$230$	−4.00000	+	8.00000i	−0.263752	+	0.527504i
$231$		0			0
$232$		−	3.00000i		−	0.196960i
$233$		−	10.0000i		−	0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$		−	10.0000i		−	0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$		0			0
$235$	−8.00000	−	4.00000i	−0.521862	−	0.260931i
$236$	8.00000			0.520756
$237$		0			0
$238$		−	1.00000i		−	0.0648204i
$239$	11.0000			0.711531			0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000			0.711531			0.355765	+	0.934575i	$0.384220\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		−	14.0000i		−	0.899954i
$243$		0			0
$244$	−5.00000			−0.320092
$245$	−6.00000	+	12.0000i	−0.383326	+	0.766652i
$246$		0			0
$247$		0			0
$248$			1.00000i			0.0635001i
$249$		0			0
$250$	−2.00000	−	11.0000i	−0.126491	−	0.695701i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		−	20.0000i		−	1.25739i
$254$	−8.00000			−0.501965
$255$		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			0.0621370
$260$		0			0
$261$		0			0
$262$		−	6.00000i		−	0.370681i
$263$			19.0000i			1.17159i	0.810459	+	0.585795i	$0.199218\pi$
$263$			19.0000i			1.17159i	−0.810459	+	0.585795i	$0.800782\pi$
$264$		0			0
$265$	−6.00000	−	3.00000i	−0.368577	−	0.184289i
$266$		0			0
$267$		0			0
$268$			4.00000i			0.244339i
$269$	−12.0000			−0.731653			−0.365826	−	0.930683i	$-0.619214\pi$
$269$	−12.0000			−0.731653			−0.365826	+	0.930683i	$0.619214\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$		−	1.00000i		−	0.0606339i
$273$		0			0
$274$	−18.0000			−1.08742
$275$	15.0000	+	20.0000i	0.904534	+	1.20605i
$276$		0			0
$277$		−	8.00000i		−	0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$		−	8.00000i		−	0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$		−	11.0000i		−	0.659736i
$279$		0			0
$280$	1.00000	−	2.00000i	0.0597614	−	0.119523i
$281$	−18.0000			−1.07379			−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000			−1.07379			−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$			16.0000i			0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$			16.0000i			0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	−6.00000			−0.356034
$285$		0			0
$286$		0			0
$287$			1.00000i			0.0590281i
$288$		0			0
$289$	16.0000			0.941176
$290$	−6.00000	−	3.00000i	−0.352332	−	0.176166i
$291$		0			0
$292$			10.0000i			0.585206i
$293$			9.00000i			0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$			9.00000i			0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$		0			0
$295$	8.00000	−	16.0000i	0.465778	−	0.931556i
$296$	1.00000			0.0581238
$297$		0			0
$298$		−	4.00000i		−	0.231714i
$299$		0			0
$300$		0			0
$301$	7.00000			0.403473
$302$			2.00000i			0.115087i
$303$		0			0
$304$		0			0
$305$	−5.00000	+	10.0000i	−0.286299	+	0.572598i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$			5.00000i			0.284901i
$309$		0			0
$310$	2.00000	+	1.00000i	0.113592	+	0.0567962i
$311$	−15.0000			−0.850572			−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000			−0.850572			−0.425286	+	0.905059i	$0.639826\pi$
$312$		0			0
$313$		−	10.0000i		−	0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$		−	10.0000i		−	0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	23.0000			1.29797
$315$		0			0
$316$		0			0
$317$			5.00000i			0.280828i	0.990093	+	0.140414i	$0.0448433\pi$
$317$			5.00000i			0.280828i	−0.990093	+	0.140414i	$0.955157\pi$
$318$		0			0
$319$	15.0000			0.839839
$320$	1.00000	−	2.00000i	0.0559017	−	0.111803i
$321$		0			0
$322$			4.00000i			0.222911i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	1.00000			0.0553849
$327$		0			0
$328$			1.00000i			0.0552158i
$329$	−4.00000			−0.220527
$330$		0			0
$331$	−26.0000			−1.42909			−0.714545	−	0.699590i	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$			8.00000i			0.439057i
$333$		0			0
$334$	6.00000			0.328305
$335$	8.00000	+	4.00000i	0.437087	+	0.218543i
$336$		0			0
$337$			16.0000i			0.871576i	0.900049	+	0.435788i	$0.143530\pi$
$337$			16.0000i			0.871576i	−0.900049	+	0.435788i	$0.856470\pi$
$338$		−	13.0000i		−	0.707107i
$339$		0			0
$340$	−2.00000	−	1.00000i	−0.108465	−	0.0542326i
$341$	−5.00000			−0.270765
$342$		0			0
$343$			13.0000i			0.701934i
$344$	7.00000			0.377415
$345$		0			0
$346$	−13.0000			−0.698884
$347$		−	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
$349$	2.00000			0.107058			0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000			0.107058			0.0535288	+	0.998566i	$0.482953\pi$
$350$	−3.00000	−	4.00000i	−0.160357	−	0.213809i
$351$		0			0
$352$			5.00000i			0.266501i
$353$		−	31.0000i		−	1.64996i	−0.565159	−	0.824982i	$-0.691185\pi$
$353$		−	31.0000i		−	1.64996i	0.565159	−	0.824982i	$-0.308815\pi$
$354$		0			0
$355$	−6.00000	+	12.0000i	−0.318447	+	0.636894i
$356$	−8.00000			−0.423999
$357$		0			0
$358$		−	10.0000i		−	0.528516i
$359$	6.00000			0.316668			0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000			0.316668			0.158334	+	0.987386i	$0.449388\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$			16.0000i			0.840941i
$363$		0			0
$364$		0			0
$365$	20.0000	+	10.0000i	1.04685	+	0.523424i
$366$		0			0
$367$			13.0000i			0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$			13.0000i			0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$			4.00000i			0.208514i
$369$		0			0
$370$	1.00000	−	2.00000i	0.0519875	−	0.103975i
$371$	−3.00000			−0.155752
$372$		0			0
$373$		−	6.00000i		−	0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$		−	6.00000i		−	0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	5.00000			0.258544
$375$		0			0
$376$	−4.00000			−0.206284
$377$		0			0
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$		0			0
$381$		0			0
$382$			3.00000i			0.153493i
$383$			8.00000i			0.408781i	0.978889	+	0.204390i	$0.0655212\pi$
$383$			8.00000i			0.408781i	−0.978889	+	0.204390i	$0.934479\pi$
$384$		0			0
$385$	10.0000	+	5.00000i	0.509647	+	0.254824i
$386$	−10.0000			−0.508987
$387$		0			0
$388$			1.00000i			0.0507673i
$389$	23.0000			1.16615			0.583073	−	0.812420i	$-0.301850\pi$
$389$	23.0000			1.16615			0.583073	+	0.812420i	$0.301850\pi$
$390$		0			0
$391$	4.00000			0.202289
$392$			6.00000i			0.303046i
$393$		0			0
$394$	−26.0000			−1.30986
$395$		0			0
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$			24.0000i			1.20301i
$399$		0			0
$400$	−3.00000	−	4.00000i	−0.150000	−	0.200000i
$401$	−10.0000			−0.499376			−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000			−0.499376			−0.249688	+	0.968326i	$0.580328\pi$
$402$		0			0
$403$		0			0
$404$	10.0000			0.497519
$405$		0			0
$406$	−3.00000			−0.148888
$407$			5.00000i			0.247841i
$408$		0			0
$409$	40.0000			1.97787			0.988936	−	0.148340i	$-0.0473931\pi$
$409$	40.0000			1.97787			0.988936	+	0.148340i	$0.0473931\pi$
$410$	2.00000	+	1.00000i	0.0987730	+	0.0493865i
$411$		0			0
$412$			8.00000i			0.394132i
$413$		−	8.00000i		−	0.393654i
$414$		0			0
$415$	16.0000	+	8.00000i	0.785409	+	0.392705i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$		0			0
$421$	−38.0000			−1.85201			−0.926003	−	0.377515i	$-0.876779\pi$
$421$	−38.0000			−1.85201			−0.926003	+	0.377515i	$0.876779\pi$
$422$			21.0000i			1.02226i
$423$		0			0
$424$	−3.00000			−0.145693
$425$	−4.00000	+	3.00000i	−0.194029	+	0.145521i
$426$		0			0
$427$			5.00000i			0.241967i
$428$			6.00000i			0.290021i
$429$		0			0
$430$	7.00000	−	14.0000i	0.337570	−	0.675140i
$431$	13.0000			0.626188			0.313094	−	0.949722i	$-0.398635\pi$
$431$	13.0000			0.626188			0.313094	+	0.949722i	$0.398635\pi$
$432$		0			0
$433$			20.0000i			0.961139i	0.876957	+	0.480569i	$0.159570\pi$
$433$			20.0000i			0.961139i	−0.876957	+	0.480569i	$0.840430\pi$
$434$	1.00000			0.0480015
$435$		0			0
$436$	1.00000			0.0478913
$437$		0			0
$438$		0			0
$439$	33.0000			1.57500			0.787502	−	0.616312i	$-0.211374\pi$
$439$	33.0000			1.57500			0.787502	+	0.616312i	$0.211374\pi$
$440$	10.0000	+	5.00000i	0.476731	+	0.238366i
$441$		0			0
$442$		0			0
$443$		−	6.00000i		−	0.285069i	−0.989790	−	0.142534i	$-0.954475\pi$
$443$		−	6.00000i		−	0.285069i	0.989790	−	0.142534i	$-0.0455251\pi$
$444$		0			0
$445$	−8.00000	+	16.0000i	−0.379236	+	0.758473i
$446$	5.00000			0.236757
$447$		0			0
$448$		−	1.00000i		−	0.0472456i
$449$	−24.0000			−1.13263			−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000			−1.13263			−0.566315	+	0.824189i	$0.691631\pi$
$450$		0			0
$451$	−5.00000			−0.235441
$452$			21.0000i			0.987757i
$453$		0			0
$454$	−1.00000			−0.0469323
$455$		0			0
$456$		0			0
$457$			19.0000i			0.888783i	0.895833	+	0.444391i	$0.146580\pi$
$457$			19.0000i			0.888783i	−0.895833	+	0.444391i	$0.853420\pi$
$458$		−	16.0000i		−	0.747631i
$459$		0			0
$460$	8.00000	+	4.00000i	0.373002	+	0.186501i
$461$	−3.00000			−0.139724			−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000			−0.139724			−0.0698620	+	0.997557i	$0.522256\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	−3.00000			−0.139272
$465$		0			0
$466$	−10.0000			−0.463241
$467$		−	27.0000i		−	1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$		−	27.0000i		−	1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$		0			0
$469$	4.00000			0.184703
$470$	−4.00000	+	8.00000i	−0.184506	+	0.369012i
$471$		0			0
$472$		−	8.00000i		−	0.368230i
$473$			35.0000i			1.60930i
$474$		0			0
$475$		0			0
$476$	−1.00000			−0.0458349
$477$		0			0
$478$		−	11.0000i		−	0.503128i
$479$	−28.0000			−1.27935			−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000			−1.27935			−0.639676	+	0.768644i	$0.720932\pi$
$480$		0			0
$481$		0			0
$482$			2.00000i			0.0910975i
$483$		0			0
$484$	−14.0000			−0.636364
$485$	2.00000	+	1.00000i	0.0908153	+	0.0454077i
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$			5.00000i			0.226339i
$489$		0			0
$490$	12.0000	+	6.00000i	0.542105	+	0.271052i
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$			3.00000i			0.135113i
$494$		0			0
$495$		0			0
$496$	1.00000			0.0449013
$497$			6.00000i			0.269137i
$498$		0			0
$499$	30.0000			1.34298			0.671492	−	0.741012i	$-0.265654\pi$
$499$	30.0000			1.34298			0.671492	+	0.741012i	$0.265654\pi$
$500$	−11.0000	+	2.00000i	−0.491935	+	0.0894427i
$501$		0			0
$502$		0			0
$503$			6.00000i			0.267527i	0.991013	+	0.133763i	$0.0427062\pi$
$503$			6.00000i			0.267527i	−0.991013	+	0.133763i	$0.957294\pi$
$504$		0			0
$505$	10.0000	−	20.0000i	0.444994	−	0.889988i
$506$	−20.0000			−0.889108
$507$		0			0
$508$			8.00000i			0.354943i
$509$	−42.0000			−1.86162			−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000			−1.86162			−0.930809	+	0.365507i	$0.880896\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−6.00000			−0.264649
$515$	16.0000	+	8.00000i	0.705044	+	0.352522i
$516$		0			0
$517$		−	20.0000i		−	0.879599i
$518$		−	1.00000i		−	0.0439375i
$519$		0			0
$520$		0			0
$521$	35.0000			1.53338			0.766689	−	0.642019i	$-0.221903\pi$
$521$	35.0000			1.53338			0.766689	+	0.642019i	$0.221903\pi$
$522$		0			0
$523$		−	20.0000i		−	0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$		−	20.0000i		−	0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	−6.00000			−0.262111
$525$		0			0
$526$	19.0000			0.828439
$527$		−	1.00000i		−	0.0435607i
$528$		0			0
$529$	7.00000			0.304348
$530$	−3.00000	+	6.00000i	−0.130312	+	0.260623i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	12.0000	+	6.00000i	0.518805	+	0.259403i
$536$	4.00000			0.172774
$537$		0			0
$538$			12.0000i			0.517357i
$539$	−30.0000			−1.29219
$540$		0			0
$541$	34.0000			1.46177			0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000			1.46177			0.730887	+	0.682498i	$0.239107\pi$
$542$			14.0000i			0.601351i
$543$		0			0
$544$	−1.00000			−0.0428746
$545$	1.00000	−	2.00000i	0.0428353	−	0.0856706i
$546$		0			0
$547$		−	41.0000i		−	1.75303i	−0.481371	−	0.876517i	$-0.659861\pi$
$547$		−	41.0000i		−	1.75303i	0.481371	−	0.876517i	$-0.340139\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$	20.0000	−	15.0000i	0.852803	−	0.639602i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−8.00000			−0.339887
$555$		0			0
$556$	−11.0000			−0.466504
$557$		−	6.00000i		−	0.254228i	−0.991888	−	0.127114i	$-0.959429\pi$
$557$		−	6.00000i		−	0.254228i	0.991888	−	0.127114i	$-0.0405714\pi$
$558$		0			0
$559$		0			0
$560$	−2.00000	−	1.00000i	−0.0845154	−	0.0422577i
$561$		0			0
$562$			18.0000i			0.759284i
$563$		−	19.0000i		−	0.800755i	−0.916350	−	0.400377i	$-0.868879\pi$
$563$		−	19.0000i		−	0.800755i	0.916350	−	0.400377i	$-0.131121\pi$
$564$		0			0
$565$	42.0000	+	21.0000i	1.76695	+	0.883477i
$566$	16.0000			0.672530
$567$		0			0
$568$			6.00000i			0.251754i
$569$	−10.0000			−0.419222			−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000			−0.419222			−0.209611	+	0.977785i	$0.567220\pi$
$570$		0			0
$571$	−29.0000			−1.21361			−0.606806	−	0.794850i	$-0.707550\pi$
$571$	−29.0000			−1.21361			−0.606806	+	0.794850i	$0.707550\pi$
$572$		0			0
$573$		0			0
$574$	1.00000			0.0417392
$575$	16.0000	−	12.0000i	0.667246	−	0.500435i
$576$		0			0
$577$			22.0000i			0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$			22.0000i			0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$		−	16.0000i		−	0.665512i
$579$		0			0
$580$	−3.00000	+	6.00000i	−0.124568	+	0.249136i
$581$	8.00000			0.331896
$582$		0			0
$583$		−	15.0000i		−	0.621237i
$584$	10.0000			0.413803
$585$		0			0
$586$	9.00000			0.371787
$587$			33.0000i			1.36206i	0.732257	+	0.681028i	$0.238467\pi$
$587$			33.0000i			1.36206i	−0.732257	+	0.681028i	$0.761533\pi$
$588$		0			0
$589$		0			0
$590$	−16.0000	−	8.00000i	−0.658710	−	0.329355i
$591$		0			0
$592$		−	1.00000i		−	0.0410997i
$593$		−	24.0000i		−	0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$		−	24.0000i		−	0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$		0			0
$595$	−1.00000	+	2.00000i	−0.0409960	+	0.0819920i
$596$	−4.00000			−0.163846
$597$		0			0
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	−7.00000			−0.285536			−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000			−0.285536			−0.142768	+	0.989756i	$0.545600\pi$
$602$		−	7.00000i		−	0.285299i
$603$		0			0
$604$	2.00000			0.0813788
$605$	−14.0000	+	28.0000i	−0.569181	+	1.13836i
$606$		0			0
$607$		−	10.0000i		−	0.405887i	−0.979190	−	0.202944i	$-0.934949\pi$
$607$		−	10.0000i		−	0.405887i	0.979190	−	0.202944i	$-0.0650509\pi$
$608$		0			0
$609$		0			0
$610$	10.0000	+	5.00000i	0.404888	+	0.202444i
$611$		0			0
$612$		0			0
$613$		−	37.0000i		−	1.49442i	−0.664590	−	0.747208i	$-0.731394\pi$
$613$		−	37.0000i		−	1.49442i	0.664590	−	0.747208i	$-0.268606\pi$
$614$		0			0
$615$		0			0
$616$	5.00000			0.201456
$617$		−	34.0000i		−	1.36879i	−0.729112	−	0.684394i	$-0.760067\pi$
$617$		−	34.0000i		−	1.36879i	0.729112	−	0.684394i	$-0.239933\pi$
$618$		0			0
$619$	31.0000			1.24600			0.622998	−	0.782224i	$-0.285915\pi$
$619$	31.0000			1.24600			0.622998	+	0.782224i	$0.285915\pi$
$620$	1.00000	−	2.00000i	0.0401610	−	0.0803219i
$621$		0			0
$622$			15.0000i			0.601445i
$623$			8.00000i			0.320513i
$624$		0			0
$625$	−7.00000	+	24.0000i	−0.280000	+	0.960000i
$626$	−10.0000			−0.399680
$627$		0			0
$628$		−	23.0000i		−	0.917800i
$629$	−1.00000			−0.0398726
$630$		0			0
$631$	5.00000			0.199047			0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000			0.199047			0.0995234	+	0.995035i	$0.468268\pi$
$632$		0			0
$633$		0			0
$634$	5.00000			0.198575
$635$	16.0000	+	8.00000i	0.634941	+	0.317470i
$636$		0			0
$637$		0			0
$638$		−	15.0000i		−	0.593856i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	37.0000			1.46141			0.730706	−	0.682692i	$-0.239191\pi$
$641$	37.0000			1.46141			0.730706	+	0.682692i	$0.239191\pi$
$642$		0			0
$643$			21.0000i			0.828159i	0.910241	+	0.414080i	$0.135896\pi$
$643$			21.0000i			0.828159i	−0.910241	+	0.414080i	$0.864104\pi$
$644$	4.00000			0.157622
$645$		0			0
$646$		0			0
$647$		−	42.0000i		−	1.65119i	−0.564263	−	0.825595i	$-0.690840\pi$
$647$		−	42.0000i		−	1.65119i	0.564263	−	0.825595i	$-0.309160\pi$
$648$		0			0
$649$	40.0000			1.57014
$650$		0			0
$651$		0			0
$652$		−	1.00000i		−	0.0391630i
$653$			10.0000i			0.391330i	0.980671	+	0.195665i	$0.0626866\pi$
$653$			10.0000i			0.391330i	−0.980671	+	0.195665i	$0.937313\pi$
$654$		0			0
$655$	−6.00000	+	12.0000i	−0.234439	+	0.468879i
$656$	1.00000			0.0390434
$657$		0			0
$658$			4.00000i			0.155936i
$659$	−32.0000			−1.24654			−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000			−1.24654			−0.623272	+	0.782006i	$0.714197\pi$
$660$		0			0
$661$	−13.0000			−0.505641			−0.252821	−	0.967513i	$-0.581358\pi$
$661$	−13.0000			−0.505641			−0.252821	+	0.967513i	$0.581358\pi$
$662$			26.0000i			1.01052i
$663$		0			0
$664$	8.00000			0.310460
$665$		0			0
$666$		0			0
$667$		−	12.0000i		−	0.464642i
$668$		−	6.00000i		−	0.232147i
$669$		0			0
$670$	4.00000	−	8.00000i	0.154533	−	0.309067i
$671$	−25.0000			−0.965114
$672$		0			0
$673$		−	36.0000i		−	1.38770i	−0.720121	−	0.693849i	$-0.755914\pi$
$673$		−	36.0000i		−	1.38770i	0.720121	−	0.693849i	$-0.244086\pi$
$674$	16.0000			0.616297
$675$		0			0
$676$	−13.0000			−0.500000
$677$			2.00000i			0.0768662i	0.999261	+	0.0384331i	$0.0122367\pi$
$677$			2.00000i			0.0768662i	−0.999261	+	0.0384331i	$0.987763\pi$
$678$		0			0
$679$	1.00000			0.0383765
$680$	−1.00000	+	2.00000i	−0.0383482	+	0.0766965i
$681$		0			0
$682$			5.00000i			0.191460i
$683$			37.0000i			1.41577i	0.706330	+	0.707883i	$0.250350\pi$
$683$			37.0000i			1.41577i	−0.706330	+	0.707883i	$0.749650\pi$
$684$		0			0
$685$	36.0000	+	18.0000i	1.37549	+	0.687745i
$686$	13.0000			0.496342
$687$		0			0
$688$		−	7.00000i		−	0.266872i
$689$		0			0
$690$		0			0
$691$	−5.00000			−0.190209			−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000			−0.190209			−0.0951045	+	0.995467i	$0.530319\pi$
$692$			13.0000i			0.494186i
$693$		0			0
$694$	−12.0000			−0.455514
$695$	−11.0000	+	22.0000i	−0.417254	+	0.834508i
$696$		0			0
$697$		−	1.00000i		−	0.0378777i
$698$		−	2.00000i		−	0.0757011i
$699$		0			0
$700$	−4.00000	+	3.00000i	−0.151186	+	0.113389i
$701$	−26.0000			−0.982006			−0.491003	−	0.871158i	$-0.663370\pi$
$701$	−26.0000			−0.982006			−0.491003	+	0.871158i	$0.663370\pi$
$702$		0			0
$703$		0			0
$704$	5.00000			0.188445
$705$		0			0
$706$	−31.0000			−1.16670
$707$		−	10.0000i		−	0.376089i
$708$		0			0
$709$	−9.00000			−0.338002			−0.169001	−	0.985616i	$-0.554054\pi$
$709$	−9.00000			−0.338002			−0.169001	+	0.985616i	$0.554054\pi$
$710$	12.0000	+	6.00000i	0.450352	+	0.225176i
$711$		0			0
$712$			8.00000i			0.299813i
$713$			4.00000i			0.149801i
$714$		0			0
$715$		0			0
$716$	−10.0000			−0.373718
$717$		0			0
$718$		−	6.00000i		−	0.223918i
$719$	−28.0000			−1.04422			−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000			−1.04422			−0.522112	+	0.852877i	$0.674856\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$			19.0000i			0.707107i
$723$		0			0
$724$	16.0000			0.594635
$725$	9.00000	+	12.0000i	0.334252	+	0.445669i
$726$		0			0
$727$		−	42.0000i		−	1.55769i	−0.627214	−	0.778847i	$-0.715805\pi$
$727$		−	42.0000i		−	1.55769i	0.627214	−	0.778847i	$-0.284195\pi$
$728$		0			0
$729$		0			0
$730$	10.0000	−	20.0000i	0.370117	−	0.740233i
$731$	−7.00000			−0.258904
$732$		0			0
$733$			19.0000i			0.701781i	0.936416	+	0.350891i	$0.114121\pi$
$733$			19.0000i			0.701781i	−0.936416	+	0.350891i	$0.885879\pi$
$734$	13.0000			0.479839
$735$		0			0
$736$	4.00000			0.147442
$737$			20.0000i			0.736709i
$738$		0			0
$739$	−37.0000			−1.36107			−0.680534	−	0.732717i	$-0.738252\pi$
$739$	−37.0000			−1.36107			−0.680534	+	0.732717i	$0.738252\pi$
$740$	−2.00000	−	1.00000i	−0.0735215	−	0.0367607i
$741$		0			0
$742$			3.00000i			0.110133i
$743$			15.0000i			0.550297i	0.961402	+	0.275148i	$0.0887270\pi$
$743$			15.0000i			0.550297i	−0.961402	+	0.275148i	$0.911273\pi$
$744$		0			0
$745$	−4.00000	+	8.00000i	−0.146549	+	0.293097i
$746$	−6.00000			−0.219676
$747$		0			0
$748$		−	5.00000i		−	0.182818i
$749$	6.00000			0.219235
$750$		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
$753$		0			0
$754$		0			0
$755$	2.00000	−	4.00000i	0.0727875	−	0.145575i
$756$		0			0
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$			20.0000i			0.726433i
$759$		0			0
$760$		0			0
$761$	21.0000			0.761249			0.380625	−	0.924730i	$-0.375709\pi$
$761$	21.0000			0.761249			0.380625	+	0.924730i	$0.375709\pi$
$762$		0			0
$763$		−	1.00000i		−	0.0362024i
$764$	3.00000			0.108536
$765$		0			0
$766$	8.00000			0.289052
$767$		0			0
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$	5.00000	−	10.0000i	0.180187	−	0.360375i
$771$		0			0
$772$			10.0000i			0.359908i
$773$		−	27.0000i		−	0.971123i	−0.874203	−	0.485561i	$-0.838615\pi$
$773$		−	27.0000i		−	0.971123i	0.874203	−	0.485561i	$-0.161385\pi$
$774$		0			0
$775$	−3.00000	−	4.00000i	−0.107763	−	0.143684i
$776$	1.00000			0.0358979
$777$		0			0
$778$		−	23.0000i		−	0.824590i
$779$		0			0
$780$		0			0
$781$	−30.0000			−1.07348
$782$		−	4.00000i		−	0.143040i
$783$		0			0
$784$	6.00000			0.214286
$785$	−46.0000	−	23.0000i	−1.64181	−	0.820905i
$786$		0			0
$787$			10.0000i			0.356462i	0.983989	+	0.178231i	$0.0570374\pi$
$787$			10.0000i			0.356462i	−0.983989	+	0.178231i	$0.942963\pi$
$788$			26.0000i			0.926212i
$789$		0			0
$790$		0			0
$791$	21.0000			0.746674
$792$		0			0
$793$		0			0
$794$	22.0000			0.780751
$795$		0			0
$796$	24.0000			0.850657
$797$			8.00000i			0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$			8.00000i			0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$		0			0
$799$	4.00000			0.141510
$800$	−4.00000	+	3.00000i	−0.141421	+	0.106066i
$801$		0			0
$802$			10.0000i			0.353112i
$803$			50.0000i			1.76446i
$804$		0			0
$805$	4.00000	−	8.00000i	0.140981	−	0.281963i
$806$		0			0
$807$		0			0
$808$		−	10.0000i		−	0.351799i
$809$	48.0000			1.68759			0.843795	−	0.536666i	$-0.180316\pi$
$809$	48.0000			1.68759			0.843795	+	0.536666i	$0.180316\pi$
$810$		0			0
$811$	−8.00000			−0.280918			−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000			−0.280918			−0.140459	+	0.990086i	$0.544858\pi$
$812$			3.00000i			0.105279i
$813$		0			0
$814$	5.00000			0.175250
$815$	−2.00000	−	1.00000i	−0.0700569	−	0.0350285i
$816$		0			0
$817$		0			0
$818$		−	40.0000i		−	1.39857i
$819$		0			0
$820$	1.00000	−	2.00000i	0.0349215	−	0.0698430i
$821$	50.0000			1.74501			0.872506	−	0.488603i	$-0.162493\pi$
$821$	50.0000			1.74501			0.872506	+	0.488603i	$0.162493\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$	8.00000			0.278693
$825$		0			0
$826$	−8.00000			−0.278356
$827$			35.0000i			1.21707i	0.793527	+	0.608535i	$0.208242\pi$
$827$			35.0000i			1.21707i	−0.793527	+	0.608535i	$0.791758\pi$
$828$		0			0
$829$	−21.0000			−0.729360			−0.364680	−	0.931133i	$-0.618822\pi$
$829$	−21.0000			−0.729360			−0.364680	+	0.931133i	$0.618822\pi$
$830$	8.00000	−	16.0000i	0.277684	−	0.555368i
$831$		0			0
$832$		0			0
$833$		−	6.00000i		−	0.207888i
$834$		0			0
$835$	−12.0000	−	6.00000i	−0.415277	−	0.207639i
$836$		0			0
$837$		0			0
$838$		−	20.0000i		−	0.690889i
$839$	40.0000			1.38095			0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000			1.38095			0.690477	+	0.723355i	$0.257401\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$			38.0000i			1.30957i
$843$		0			0
$844$	21.0000			0.722850
$845$	−13.0000	+	26.0000i	−0.447214	+	0.894427i
$846$		0			0
$847$			14.0000i			0.481046i
$848$			3.00000i			0.103020i
$849$		0			0
$850$	3.00000	+	4.00000i	0.102899	+	0.137199i
$851$	4.00000			0.137118
$852$		0			0
$853$		−	2.00000i		−	0.0684787i	−0.999414	−	0.0342393i	$-0.989099\pi$
$853$		−	2.00000i		−	0.0684787i	0.999414	−	0.0342393i	$-0.0109009\pi$
$854$	5.00000			0.171096
$855$		0			0
$856$	6.00000			0.205076
$857$		−	27.0000i		−	0.922302i	−0.887322	−	0.461151i	$-0.847437\pi$
$857$		−	27.0000i		−	0.922302i	0.887322	−	0.461151i	$-0.152563\pi$
$858$		0			0
$859$	−8.00000			−0.272956			−0.136478	−	0.990643i	$-0.543578\pi$
$859$	−8.00000			−0.272956			−0.136478	+	0.990643i	$0.543578\pi$
$860$	−14.0000	−	7.00000i	−0.477396	−	0.238698i
$861$		0			0
$862$		−	13.0000i		−	0.442782i
$863$			21.0000i			0.714848i	0.933942	+	0.357424i	$0.116345\pi$
$863$			21.0000i			0.714848i	−0.933942	+	0.357424i	$0.883655\pi$
$864$		0			0
$865$	26.0000	+	13.0000i	0.884027	+	0.442013i
$866$	20.0000			0.679628
$867$		0			0
$868$		−	1.00000i		−	0.0339422i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	1.00000i		−	0.0338643i
$873$		0			0
$874$		0			0
$875$	2.00000	+	11.0000i	0.0676123	+	0.371868i
$876$		0			0
$877$		−	13.0000i		−	0.438979i	−0.975615	−	0.219489i	$-0.929561\pi$
$877$		−	13.0000i		−	0.438979i	0.975615	−	0.219489i	$-0.0704391\pi$
$878$		−	33.0000i		−	1.11370i
$879$		0			0
$880$	5.00000	−	10.0000i	0.168550	−	0.337100i
$881$	45.0000			1.51609			0.758044	−	0.652203i	$-0.226155\pi$
$881$	45.0000			1.51609			0.758044	+	0.652203i	$0.226155\pi$
$882$		0			0
$883$		−	33.0000i		−	1.11054i	−0.831671	−	0.555269i	$-0.812615\pi$
$883$		−	33.0000i		−	1.11054i	0.831671	−	0.555269i	$-0.187385\pi$
$884$		0			0
$885$		0			0
$886$	−6.00000			−0.201574
$887$			45.0000i			1.51095i	0.655176	+	0.755476i	$0.272594\pi$
$887$			45.0000i			1.51095i	−0.655176	+	0.755476i	$0.727406\pi$
$888$		0			0
$889$	8.00000			0.268311
$890$	16.0000	+	8.00000i	0.536321	+	0.268161i
$891$		0			0
$892$		−	5.00000i		−	0.167412i
$893$		0			0
$894$		0			0
$895$	−10.0000	+	20.0000i	−0.334263	+	0.668526i
$896$	−1.00000			−0.0334077
$897$		0			0
$898$			24.0000i			0.800890i
$899$	−3.00000			−0.100056
$900$		0			0
$901$	3.00000			0.0999445
$902$			5.00000i			0.166482i
$903$		0			0
$904$	21.0000			0.698450
$905$	16.0000	−	32.0000i	0.531858	−	1.06372i
$906$		0			0
$907$			36.0000i			1.19536i	0.801735	+	0.597680i	$0.203911\pi$
$907$			36.0000i			1.19536i	−0.801735	+	0.597680i	$0.796089\pi$
$908$			1.00000i			0.0331862i
$909$		0			0
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$			40.0000i			1.32381i
$914$	19.0000			0.628464
$915$		0			0
$916$	−16.0000			−0.528655
$917$			6.00000i			0.198137i
$918$		0			0
$919$	−48.0000			−1.58337			−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000			−1.58337			−0.791687	+	0.610927i	$0.790797\pi$
$920$	4.00000	−	8.00000i	0.131876	−	0.263752i
$921$		0			0
$922$			3.00000i			0.0987997i
$923$		0			0
$924$		0			0
$925$	−4.00000	+	3.00000i	−0.131519	+	0.0986394i
$926$	24.0000			0.788689
$927$		0			0
$928$			3.00000i			0.0984798i
$929$	−3.00000			−0.0984268			−0.0492134	−	0.998788i	$-0.515671\pi$
$929$	−3.00000			−0.0984268			−0.0492134	+	0.998788i	$0.515671\pi$
$930$		0			0
$931$		0			0
$932$			10.0000i			0.327561i
$933$		0			0
$934$	−27.0000			−0.883467
$935$	−10.0000	−	5.00000i	−0.327035	−	0.163517i
$936$		0			0
$937$		−	16.0000i		−	0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$		−	16.0000i		−	0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$		−	4.00000i		−	0.130605i
$939$		0			0
$940$	8.00000	+	4.00000i	0.260931	+	0.130466i
$941$	22.0000			0.717180			0.358590	−	0.933495i	$-0.383258\pi$
$941$	22.0000			0.717180			0.358590	+	0.933495i	$0.383258\pi$
$942$		0			0
$943$			4.00000i			0.130258i
$944$	−8.00000			−0.260378
$945$		0			0
$946$	35.0000			1.13795
$947$			27.0000i			0.877382i	0.898638	+	0.438691i	$0.144558\pi$
$947$			27.0000i			0.877382i	−0.898638	+	0.438691i	$0.855442\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$			1.00000i			0.0324102i
$953$			14.0000i			0.453504i	0.973952	+	0.226752i	$0.0728108\pi$
$953$			14.0000i			0.453504i	−0.973952	+	0.226752i	$0.927189\pi$
$954$		0			0
$955$	3.00000	−	6.00000i	0.0970777	−	0.194155i
$956$	−11.0000			−0.355765
$957$		0			0
$958$			28.0000i			0.904639i
$959$	18.0000			0.581250
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$	2.00000			0.0644157
$965$	20.0000	+	10.0000i	0.643823	+	0.321911i
$966$		0			0
$967$		−	10.0000i		−	0.321578i	−0.986989	−	0.160789i	$-0.948596\pi$
$967$		−	10.0000i		−	0.321578i	0.986989	−	0.160789i	$-0.0514039\pi$
$968$			14.0000i			0.449977i
$969$		0			0
$970$	1.00000	−	2.00000i	0.0321081	−	0.0642161i
$971$	−39.0000			−1.25157			−0.625785	−	0.779996i	$-0.715221\pi$
$971$	−39.0000			−1.25157			−0.625785	+	0.779996i	$0.715221\pi$
$972$		0			0
$973$			11.0000i			0.352644i
$974$	−12.0000			−0.384505
$975$		0			0
$976$	5.00000			0.160046
$977$			3.00000i			0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$			3.00000i			0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$		0			0
$979$	−40.0000			−1.27841
$980$	6.00000	−	12.0000i	0.191663	−	0.383326i
$981$		0			0
$982$			12.0000i			0.382935i
$983$		−	7.00000i		−	0.223265i	−0.993750	−	0.111633i	$-0.964392\pi$
$983$		−	7.00000i		−	0.223265i	0.993750	−	0.111633i	$-0.0356080\pi$
$984$		0			0
$985$	52.0000	+	26.0000i	1.65686	+	0.828429i
$986$	3.00000			0.0955395
$987$		0			0
$988$		0			0
$989$	28.0000			0.890348
$990$		0			0
$991$	−5.00000			−0.158830			−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000			−0.158830			−0.0794151	+	0.996842i	$0.525305\pi$
$992$		−	1.00000i		−	0.0317500i
$993$		0			0
$994$	6.00000			0.190308
$995$	24.0000	−	48.0000i	0.760851	−	1.52170i
$996$		0			0
$997$		−	24.0000i		−	0.760088i	−0.924968	−	0.380044i	$-0.875909\pi$
$997$		−	24.0000i		−	0.760088i	0.924968	−	0.380044i	$-0.124091\pi$
$998$		−	30.0000i		−	0.949633i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.d.d.1999.1		2
3.2	odd	2		1110.2.d.c.889.2	yes	2
5.4	even	2	inner	3330.2.d.d.1999.2		2
15.2	even	4		5550.2.a.d.1.1		1
15.8	even	4		5550.2.a.bo.1.1		1
15.14	odd	2		1110.2.d.c.889.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.d.c.889.1	✓	2	15.14	odd	2
1110.2.d.c.889.2	yes	2	3.2	odd	2
3330.2.d.d.1999.1		2	1.1	even	1	trivial
3330.2.d.d.1999.2		2	5.4	even	2	inner
5550.2.a.d.1.1		1	15.2	even	4
5550.2.a.bo.1.1		1	15.8	even	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +1.00000i q^{7} +1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} -5.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +(1.00000 - 2.00000i) q^{20} +5.00000i q^{22} +4.00000i q^{23} +(-3.00000 - 4.00000i) q^{25} -1.00000i q^{28} -3.00000 q^{29} +1.00000 q^{31} -1.00000i q^{32} -1.00000 q^{34} +(-2.00000 - 1.00000i) q^{35} -1.00000i q^{37} +(-2.00000 - 1.00000i) q^{40} +1.00000 q^{41} -7.00000i q^{43} +5.00000 q^{44} +4.00000 q^{46} +4.00000i q^{47} +6.00000 q^{49} +(-4.00000 + 3.00000i) q^{50} +3.00000i q^{53} +(5.00000 - 10.0000i) q^{55} -1.00000 q^{56} +3.00000i q^{58} -8.00000 q^{59} +5.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -4.00000i q^{67} +1.00000i q^{68} +(-1.00000 + 2.00000i) q^{70} +6.00000 q^{71} -10.0000i q^{73} -1.00000 q^{74} -5.00000i q^{77} +(-1.00000 + 2.00000i) q^{80} -1.00000i q^{82} -8.00000i q^{83} +(2.00000 + 1.00000i) q^{85} -7.00000 q^{86} -5.00000i q^{88} +8.00000 q^{89} -4.00000i q^{92} +4.00000 q^{94} -1.00000i q^{97} -6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^4 - 2 * q^5	\( 2 q - 2 q^{4} - 2 q^{5} + 4 q^{10} - 10 q^{11} + 2 q^{14} + 2 q^{16} + 2 q^{20} - 6 q^{25} - 6 q^{29} + 2 q^{31} - 2 q^{34} - 4 q^{35} - 4 q^{40} + 2 q^{41} + 10 q^{44} + 8 q^{46} + 12 q^{49} - 8 q^{50} + 10 q^{55} - 2 q^{56} - 16 q^{59} + 10 q^{61} - 2 q^{64} - 2 q^{70} + 12 q^{71} - 2 q^{74} - 2 q^{80} + 4 q^{85} - 14 q^{86} + 16 q^{89} + 8 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 2 * q^5 + 4 * q^10 - 10 * q^11 + 2 * q^14 + 2 * q^16 + 2 * q^20 - 6 * q^25 - 6 * q^29 + 2 * q^31 - 2 * q^34 - 4 * q^35 - 4 * q^40 + 2 * q^41 + 10 * q^44 + 8 * q^46 + 12 * q^49 - 8 * q^50 + 10 * q^55 - 2 * q^56 - 16 * q^59 + 10 * q^61 - 2 * q^64 - 2 * q^70 + 12 * q^71 - 2 * q^74 - 2 * q^80 + 4 * q^85 - 14 * q^86 + 16 * q^89 + 8 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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