LMFDB - Embedded newform 3330.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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, 0]))

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

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

chi := DirichletCharacter("3330.1");

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.a (trivial)

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:	yes
Analytic conductor:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.23544108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 $ x^5 - x^4 - 20x^3 + 39x^2 + 9*x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.76376$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.19175 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.64179 q^{11} +4.86897 q^{13} -1.19175 q^{14} +1.00000 q^{16} +3.19175 q^{17} +2.27506 q^{19} +1.00000 q^{20} -3.64179 q^{22} -2.86897 q^{23} +1.00000 q^{25} +4.86897 q^{26} -1.19175 q^{28} +6.06072 q^{29} +5.36672 q^{31} +1.00000 q^{32} +3.19175 q^{34} -1.19175 q^{35} +1.00000 q^{37} +2.27506 q^{38} +1.00000 q^{40} -8.75471 q^{41} -6.75471 q^{43} -3.64179 q^{44} -2.86897 q^{46} +4.13955 q^{47} -5.57973 q^{49} +1.00000 q^{50} +4.86897 q^{52} +5.88574 q^{53} -3.64179 q^{55} -1.19175 q^{56} +6.06072 q^{58} +7.28357 q^{59} +14.5443 q^{61} +5.36672 q^{62} +1.00000 q^{64} +4.86897 q^{65} -4.90007 q^{67} +3.19175 q^{68} -1.19175 q^{70} +5.38798 q^{71} +11.8026 q^{73} +1.00000 q^{74} +2.27506 q^{76} +4.34010 q^{77} +7.38798 q^{79} +1.00000 q^{80} -8.75471 q^{82} +4.41461 q^{83} +3.19175 q^{85} -6.75471 q^{86} -3.64179 q^{88} +9.63597 q^{89} -5.80259 q^{91} -2.86897 q^{92} +4.13955 q^{94} +2.27506 q^{95} +14.8943 q^{97} -5.57973 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + 5 q^{10} + 5 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 7 q^{17} - q^{19} + 5 q^{20} + 5 q^{22} + 5 q^{23} + 5 q^{25} + 5 q^{26} + 3 q^{28} + 2 q^{29}+ \cdots + 16 q^{98}+O(q^{100}) $ 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8 + 5 * q^10 + 5 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 7 * q^17 - q^19 + 5 * q^20 + 5 * q^22 + 5 * q^23 + 5 * q^25 + 5 * q^26 + 3 * q^28 + 2 * q^29 + 16 * q^31 + 5 * q^32 + 7 * q^34 + 3 * q^35 + 5 * q^37 - q^38 + 5 * q^40 + 2 * q^41 + 12 * q^43 + 5 * q^44 + 5 * q^46 + 6 * q^47 + 16 * q^49 + 5 * q^50 + 5 * q^52 + 3 * q^53 + 5 * q^55 + 3 * q^56 + 2 * q^58 - 10 * q^59 + 16 * q^61 + 16 * q^62 + 5 * q^64 + 5 * q^65 + 4 * q^67 + 7 * q^68 + 3 * q^70 - 8 * q^71 - 3 * q^73 + 5 * q^74 - q^76 + 3 * q^77 + 2 * q^79 + 5 * q^80 + 2 * q^82 - 5 * q^83 + 7 * q^85 + 12 * q^86 + 5 * q^88 - 7 * q^89 + 33 * q^91 + 5 * q^92 + 6 * q^94 - q^95 + 14 * q^97 + 16 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.19175	−0.450439	−0.225220	−	0.974308i	$-0.572310\pi$
$7$	−1.19175	−0.450439	−0.225220	+	0.974308i	$0.572310\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.64179	−1.09804	−0.549020	−	0.835809i	$-0.684999\pi$
$11$	−3.64179	−1.09804	−0.549020	+	0.835809i	$0.684999\pi$
$12$	0	0
$13$	4.86897	1.35041	0.675204	−	0.737631i	$-0.264056\pi$
$13$	4.86897	1.35041	0.675204	+	0.737631i	$0.264056\pi$
$14$	−1.19175	−0.318509
$15$	0	0
$16$	1.00000	0.250000
$17$	3.19175	0.774113	0.387057	−	0.922056i	$-0.373492\pi$
$17$	3.19175	0.774113	0.387057	+	0.922056i	$0.373492\pi$
$18$	0	0
$19$	2.27506	0.521935	0.260968	−	0.965348i	$-0.415958\pi$
$19$	2.27506	0.521935	0.260968	+	0.965348i	$0.415958\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.64179	−0.776431
$23$	−2.86897	−0.598221	−0.299110	−	0.954219i	$-0.596690\pi$
$23$	−2.86897	−0.598221	−0.299110	+	0.954219i	$0.596690\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	4.86897	0.954883
$27$	0	0
$28$	−1.19175	−0.225220
$29$	6.06072	1.12545	0.562723	−	0.826645i	$-0.309754\pi$
$29$	6.06072	1.12545	0.562723	+	0.826645i	$0.309754\pi$
$30$	0	0
$31$	5.36672	0.963892	0.481946	−	0.876201i	$-0.339930\pi$
$31$	5.36672	0.963892	0.481946	+	0.876201i	$0.339930\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.19175	0.547381
$35$	−1.19175	−0.201443
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	2.27506	0.369064
$39$	0	0
$40$	1.00000	0.158114
$41$	−8.75471	−1.36726	−0.683628	−	0.729831i	$-0.739599\pi$
$41$	−8.75471	−1.36726	−0.683628	+	0.729831i	$0.739599\pi$
$42$	0	0
$43$	−6.75471	−1.03008	−0.515042	−	0.857165i	$-0.672224\pi$
$43$	−6.75471	−1.03008	−0.515042	+	0.857165i	$0.672224\pi$
$44$	−3.64179	−0.549020
$45$	0	0
$46$	−2.86897	−0.423006
$47$	4.13955	0.603815	0.301907	−	0.953337i	$-0.402377\pi$
$47$	4.13955	0.603815	0.301907	+	0.953337i	$0.402377\pi$
$48$	0	0
$49$	−5.57973	−0.797105
$50$	1.00000	0.141421
$51$	0	0
$52$	4.86897	0.675204
$53$	5.88574	0.808469	0.404234	−	0.914655i	$-0.367538\pi$
$53$	5.88574	0.808469	0.404234	+	0.914655i	$0.367538\pi$
$54$	0	0
$55$	−3.64179	−0.491058
$56$	−1.19175	−0.159254
$57$	0	0
$58$	6.06072	0.795811
$59$	7.28357	0.948240	0.474120	−	0.880460i	$-0.342766\pi$
$59$	7.28357	0.948240	0.474120	+	0.880460i	$0.342766\pi$
$60$	0	0
$61$	14.5443	1.86221	0.931104	−	0.364755i	$-0.118847\pi$
$61$	14.5443	1.86221	0.931104	+	0.364755i	$0.118847\pi$
$62$	5.36672	0.681575
$63$	0	0
$64$	1.00000	0.125000
$65$	4.86897	0.603921
$66$	0	0
$67$	−4.90007	−0.598639	−0.299320	−	0.954153i	$-0.596760\pi$
$67$	−4.90007	−0.598639	−0.299320	+	0.954153i	$0.596760\pi$
$68$	3.19175	0.387057
$69$	0	0
$70$	−1.19175	−0.142441
$71$	5.38798	0.639436	0.319718	−	0.947513i	$-0.396412\pi$
$71$	5.38798	0.639436	0.319718	+	0.947513i	$0.396412\pi$
$72$	0	0
$73$	11.8026	1.38139	0.690694	−	0.723147i	$-0.257305\pi$
$73$	11.8026	1.38139	0.690694	+	0.723147i	$0.257305\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.27506	0.260968
$77$	4.34010	0.494600
$78$	0	0
$79$	7.38798	0.831213	0.415606	−	0.909545i	$-0.363569\pi$
$79$	7.38798	0.831213	0.415606	+	0.909545i	$0.363569\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.75471	−0.966796
$83$	4.41461	0.484566	0.242283	−	0.970206i	$-0.422104\pi$
$83$	4.41461	0.484566	0.242283	+	0.970206i	$0.422104\pi$
$84$	0	0
$85$	3.19175	0.346194
$86$	−6.75471	−0.728379
$87$	0	0
$88$	−3.64179	−0.388216
$89$	9.63597	1.02141	0.510705	−	0.859756i	$-0.329384\pi$
$89$	9.63597	1.02141	0.510705	+	0.859756i	$0.329384\pi$
$90$	0	0
$91$	−5.80259	−0.608277
$92$	−2.86897	−0.299110
$93$	0	0
$94$	4.13955	0.426962
$95$	2.27506	0.233416
$96$	0	0
$97$	14.8943	1.51228	0.756141	−	0.654409i	$-0.227082\pi$
$97$	14.8943	1.51228	0.756141	+	0.654409i	$0.227082\pi$
$98$	−5.57973	−0.563638
$99$	0	0
$100$	1.00000	0.100000
$101$	−7.63194	−0.759406	−0.379703	−	0.925108i	$-0.623974\pi$
$101$	−7.63194	−0.759406	−0.379703	+	0.925108i	$0.623974\pi$
$102$	0	0
$103$	−2.13955	−0.210816	−0.105408	−	0.994429i	$-0.533615\pi$
$103$	−2.13955	−0.210816	−0.105408	+	0.994429i	$0.533615\pi$
$104$	4.86897	0.477441
$105$	0	0
$106$	5.88574	0.571674
$107$	−14.9904	−1.44918	−0.724588	−	0.689182i	$-0.757970\pi$
$107$	−14.9904	−1.44918	−0.724588	+	0.689182i	$0.757970\pi$
$108$	0	0
$109$	0.253804	0.0243101	0.0121550	−	0.999926i	$-0.496131\pi$
$109$	0.253804	0.0243101	0.0121550	+	0.999926i	$0.496131\pi$
$110$	−3.64179	−0.347231
$111$	0	0
$112$	−1.19175	−0.112610
$113$	−13.3443	−1.25533	−0.627663	−	0.778486i	$-0.715988\pi$
$113$	−13.3443	−1.25533	−0.627663	+	0.778486i	$0.715988\pi$
$114$	0	0
$115$	−2.86897	−0.267532
$116$	6.06072	0.562723
$117$	0	0
$118$	7.28357	0.670507
$119$	−3.80377	−0.348691
$120$	0	0
$121$	2.26261	0.205692
$122$	14.5443	1.31678
$123$	0	0
$124$	5.36672	0.481946
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.21891	−0.640575	−0.320288	−	0.947320i	$-0.603780\pi$
$127$	−7.21891	−0.640575	−0.320288	+	0.947320i	$0.603780\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	4.86897	0.427037
$131$	−20.0506	−1.75183	−0.875913	−	0.482468i	$-0.839740\pi$
$131$	−20.0506	−1.75183	−0.875913	+	0.482468i	$0.839740\pi$
$132$	0	0
$133$	−2.71131	−0.235100
$134$	−4.90007	−0.423302
$135$	0	0
$136$	3.19175	0.273690
$137$	9.52753	0.813992	0.406996	−	0.913430i	$-0.366576\pi$
$137$	9.52753	0.813992	0.406996	+	0.913430i	$0.366576\pi$
$138$	0	0
$139$	20.7323	1.75849	0.879244	−	0.476372i	$-0.158048\pi$
$139$	20.7323	1.75849	0.879244	+	0.476372i	$0.158048\pi$
$140$	−1.19175	−0.100721
$141$	0	0
$142$	5.38798	0.452149
$143$	−17.7317	−1.48280
$144$	0	0
$145$	6.06072	0.503315
$146$	11.8026	0.976789
$147$	0	0
$148$	1.00000	0.0821995
$149$	−9.17758	−0.751857	−0.375928	−	0.926649i	$-0.622676\pi$
$149$	−9.17758	−0.751857	−0.375928	+	0.926649i	$0.622676\pi$
$150$	0	0
$151$	17.6024	1.43246	0.716232	−	0.697862i	$-0.245865\pi$
$151$	17.6024	1.43246	0.716232	+	0.697862i	$0.245865\pi$
$152$	2.27506	0.184532
$153$	0	0
$154$	4.34010	0.349735
$155$	5.36672	0.431066
$156$	0	0
$157$	6.89857	0.550566	0.275283	−	0.961363i	$-0.411228\pi$
$157$	6.89857	0.550566	0.275283	+	0.961363i	$0.411228\pi$
$158$	7.38798	0.587756
$159$	0	0
$160$	1.00000	0.0790569
$161$	3.41909	0.269462
$162$	0	0
$163$	−10.8194	−0.847438	−0.423719	−	0.905794i	$-0.639276\pi$
$163$	−10.8194	−0.847438	−0.423719	+	0.905794i	$0.639276\pi$
$164$	−8.75471	−0.683628
$165$	0	0
$166$	4.41461	0.342640
$167$	2.86897	0.222007	0.111004	−	0.993820i	$-0.464593\pi$
$167$	2.86897	0.222007	0.111004	+	0.993820i	$0.464593\pi$
$168$	0	0
$169$	10.7068	0.823602
$170$	3.19175	0.244796
$171$	0	0
$172$	−6.75471	−0.515042
$173$	18.3571	1.39567	0.697833	−	0.716260i	$-0.254148\pi$
$173$	18.3571	1.39567	0.697833	+	0.716260i	$0.254148\pi$
$174$	0	0
$175$	−1.19175	−0.0900878
$176$	−3.64179	−0.274510
$177$	0	0
$178$	9.63597	0.722246
$179$	−23.1259	−1.72851	−0.864256	−	0.503052i	$-0.832210\pi$
$179$	−23.1259	−1.72851	−0.864256	+	0.503052i	$0.832210\pi$
$180$	0	0
$181$	14.9001	1.10751	0.553757	−	0.832678i	$-0.313194\pi$
$181$	14.9001	1.10751	0.553757	+	0.832678i	$0.313194\pi$
$182$	−5.80259	−0.430117
$183$	0	0
$184$	−2.86897	−0.211503
$185$	1.00000	0.0735215
$186$	0	0
$187$	−11.6237	−0.850007
$188$	4.13955	0.301907
$189$	0	0
$190$	2.27506	0.165050
$191$	8.31766	0.601845	0.300922	−	0.953649i	$-0.402705\pi$
$191$	8.31766	0.601845	0.300922	+	0.953649i	$0.402705\pi$
$192$	0	0
$193$	−21.5445	−1.55081	−0.775405	−	0.631464i	$-0.782454\pi$
$193$	−21.5445	−1.55081	−0.775405	+	0.631464i	$0.782454\pi$
$194$	14.8943	1.06934
$195$	0	0
$196$	−5.57973	−0.398552
$197$	−3.76904	−0.268533	−0.134266	−	0.990945i	$-0.542868\pi$
$197$	−3.76904	−0.268533	−0.134266	+	0.990945i	$0.542868\pi$
$198$	0	0
$199$	−5.92914	−0.420306	−0.210153	−	0.977669i	$-0.567396\pi$
$199$	−5.92914	−0.420306	−0.210153	+	0.977669i	$0.567396\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−7.63194	−0.536981
$203$	−7.22286	−0.506945
$204$	0	0
$205$	−8.75471	−0.611455
$206$	−2.13955	−0.149069
$207$	0	0
$208$	4.86897	0.337602
$209$	−8.28529	−0.573105
$210$	0	0
$211$	−3.51059	−0.241679	−0.120840	−	0.992672i	$-0.538559\pi$
$211$	−3.51059	−0.241679	−0.120840	+	0.992672i	$0.538559\pi$
$212$	5.88574	0.404234
$213$	0	0
$214$	−14.9904	−1.02472
$215$	−6.75471	−0.460667
$216$	0	0
$217$	−6.39579	−0.434175
$218$	0.253804	0.0171898
$219$	0	0
$220$	−3.64179	−0.245529
$221$	15.5405	1.04537
$222$	0	0
$223$	19.8322	1.32806	0.664031	−	0.747705i	$-0.268844\pi$
$223$	19.8322	1.32806	0.664031	+	0.747705i	$0.268844\pi$
$224$	−1.19175	−0.0796271
$225$	0	0
$226$	−13.3443	−0.887649
$227$	−12.9608	−0.860238	−0.430119	−	0.902772i	$-0.641528\pi$
$227$	−12.9608	−0.860238	−0.430119	+	0.902772i	$0.641528\pi$
$228$	0	0
$229$	−23.2384	−1.53564	−0.767818	−	0.640668i	$-0.778657\pi$
$229$	−23.2384	−1.53564	−0.767818	+	0.640668i	$0.778657\pi$
$230$	−2.86897	−0.189174
$231$	0	0
$232$	6.06072	0.397905
$233$	−8.88196	−0.581876	−0.290938	−	0.956742i	$-0.593967\pi$
$233$	−8.88196	−0.581876	−0.290938	+	0.956742i	$0.593967\pi$
$234$	0	0
$235$	4.13955	0.270034
$236$	7.28357	0.474120
$237$	0	0
$238$	−3.80377	−0.246562
$239$	−25.3778	−1.64156	−0.820778	−	0.571247i	$-0.806460\pi$
$239$	−25.3778	−1.64156	−0.820778	+	0.571247i	$0.806460\pi$
$240$	0	0
$241$	−10.2429	−0.659801	−0.329900	−	0.944016i	$-0.607015\pi$
$241$	−10.2429	−0.659801	−0.329900	+	0.944016i	$0.607015\pi$
$242$	2.26261	0.145446
$243$	0	0
$244$	14.5443	0.931104
$245$	−5.57973	−0.356476
$246$	0	0
$247$	11.0772	0.704825
$248$	5.36672	0.340787
$249$	0	0
$250$	1.00000	0.0632456
$251$	−10.2258	−0.645449	−0.322725	−	0.946493i	$-0.604599\pi$
$251$	−10.2258	−0.645449	−0.322725	+	0.946493i	$0.604599\pi$
$252$	0	0
$253$	10.4482	0.656870
$254$	−7.21891	−0.452955
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0481	1.12581	0.562906	−	0.826521i	$-0.309683\pi$
$257$	18.0481	1.12581	0.562906	+	0.826521i	$0.309683\pi$
$258$	0	0
$259$	−1.19175	−0.0740517
$260$	4.86897	0.301960
$261$	0	0
$262$	−20.0506	−1.23873
$263$	13.5658	0.836503	0.418252	−	0.908331i	$-0.362643\pi$
$263$	13.5658	0.836503	0.418252	+	0.908331i	$0.362643\pi$
$264$	0	0
$265$	5.88574	0.361558
$266$	−2.71131	−0.166241
$267$	0	0
$268$	−4.90007	−0.299320
$269$	−18.3629	−1.11961	−0.559804	−	0.828625i	$-0.689124\pi$
$269$	−18.3629	−1.11961	−0.559804	+	0.828625i	$0.689124\pi$
$270$	0	0
$271$	−0.279091	−0.0169536	−0.00847678	−	0.999964i	$-0.502698\pi$
$271$	−0.279091	−0.0169536	−0.00847678	+	0.999964i	$0.502698\pi$
$272$	3.19175	0.193528
$273$	0	0
$274$	9.52753	0.575579
$275$	−3.64179	−0.219608
$276$	0	0
$277$	24.9195	1.49727	0.748635	−	0.662982i	$-0.230710\pi$
$277$	24.9195	1.49727	0.748635	+	0.662982i	$0.230710\pi$
$278$	20.7323	1.24344
$279$	0	0
$280$	−1.19175	−0.0712207
$281$	2.25695	0.134638	0.0673191	−	0.997731i	$-0.478555\pi$
$281$	2.25695	0.134638	0.0673191	+	0.997731i	$0.478555\pi$
$282$	0	0
$283$	4.31884	0.256728	0.128364	−	0.991727i	$-0.459027\pi$
$283$	4.31884	0.256728	0.128364	+	0.991727i	$0.459027\pi$
$284$	5.38798	0.319718
$285$	0	0
$286$	−17.7317	−1.04850
$287$	10.4334	0.615865
$288$	0	0
$289$	−6.81273	−0.400749
$290$	6.06072	0.355897
$291$	0	0
$292$	11.8026	0.690694
$293$	7.43138	0.434146	0.217073	−	0.976155i	$-0.430349\pi$
$293$	7.43138	0.434146	0.217073	+	0.976155i	$0.430349\pi$
$294$	0	0
$295$	7.28357	0.424066
$296$	1.00000	0.0581238
$297$	0	0
$298$	−9.17758	−0.531643
$299$	−13.9689	−0.807842
$300$	0	0
$301$	8.04992	0.463990
$302$	17.6024	1.01291
$303$	0	0
$304$	2.27506	0.130484
$305$	14.5443	0.832804
$306$	0	0
$307$	3.28357	0.187403	0.0937017	−	0.995600i	$-0.470130\pi$
$307$	3.28357	0.187403	0.0937017	+	0.995600i	$0.470130\pi$
$308$	4.34010	0.247300
$309$	0	0
$310$	5.36672	0.304809
$311$	−11.8438	−0.671603	−0.335801	−	0.941933i	$-0.609007\pi$
$311$	−11.8438	−0.671603	−0.335801	+	0.941933i	$0.609007\pi$
$312$	0	0
$313$	15.4653	0.874151	0.437076	−	0.899425i	$-0.356014\pi$
$313$	15.4653	0.874151	0.437076	+	0.899425i	$0.356014\pi$
$314$	6.89857	0.389309
$315$	0	0
$316$	7.38798	0.415606
$317$	−16.4553	−0.924223	−0.462112	−	0.886822i	$-0.652908\pi$
$317$	−16.4553	−0.924223	−0.462112	+	0.886822i	$0.652908\pi$
$318$	0	0
$319$	−22.0718	−1.23579
$320$	1.00000	0.0559017
$321$	0	0
$322$	3.41909	0.190538
$323$	7.26143	0.404037
$324$	0	0
$325$	4.86897	0.270082
$326$	−10.8194	−0.599229
$327$	0	0
$328$	−8.75471	−0.483398
$329$	−4.93330	−0.271982
$330$	0	0
$331$	35.4204	1.94688	0.973442	−	0.228935i	$-0.0735243\pi$
$331$	35.4204	1.94688	0.973442	+	0.228935i	$0.0735243\pi$
$332$	4.41461	0.242283
$333$	0	0
$334$	2.86897	0.156983
$335$	−4.90007	−0.267720
$336$	0	0
$337$	−4.80675	−0.261840	−0.130920	−	0.991393i	$-0.541793\pi$
$337$	−4.80675	−0.261840	−0.130920	+	0.991393i	$0.541793\pi$
$338$	10.7068	0.582374
$339$	0	0
$340$	3.19175	0.173097
$341$	−19.5445	−1.05839
$342$	0	0
$343$	14.9919	0.809486
$344$	−6.75471	−0.364189
$345$	0	0
$346$	18.3571	0.986885
$347$	18.5049	0.993397	0.496698	−	0.867923i	$-0.334546\pi$
$347$	18.5049	0.993397	0.496698	+	0.867923i	$0.334546\pi$
$348$	0	0
$349$	−33.7975	−1.80914	−0.904568	−	0.426328i	$-0.859807\pi$
$349$	−33.7975	−1.80914	−0.904568	+	0.426328i	$0.859807\pi$
$350$	−1.19175	−0.0637017
$351$	0	0
$352$	−3.64179	−0.194108
$353$	−37.4657	−1.99410	−0.997049	−	0.0767613i	$-0.975542\pi$
$353$	−37.4657	−1.99410	−0.997049	+	0.0767613i	$0.975542\pi$
$354$	0	0
$355$	5.38798	0.285964
$356$	9.63597	0.510705
$357$	0	0
$358$	−23.1259	−1.22224
$359$	14.6886	0.775233	0.387617	−	0.921821i	$-0.373298\pi$
$359$	14.6886	0.775233	0.387617	+	0.921821i	$0.373298\pi$
$360$	0	0
$361$	−13.8241	−0.727584
$362$	14.9001	0.783130
$363$	0	0
$364$	−5.80259	−0.304138
$365$	11.8026	0.617776
$366$	0	0
$367$	16.1175	0.841326	0.420663	−	0.907217i	$-0.361797\pi$
$367$	16.1175	0.841326	0.420663	+	0.907217i	$0.361797\pi$
$368$	−2.86897	−0.149555
$369$	0	0
$370$	1.00000	0.0519875
$371$	−7.01433	−0.364166
$372$	0	0
$373$	3.25002	0.168280	0.0841399	−	0.996454i	$-0.473186\pi$
$373$	3.25002	0.168280	0.0841399	+	0.996454i	$0.473186\pi$
$374$	−11.6237	−0.601046
$375$	0	0
$376$	4.13955	0.213481
$377$	29.5094	1.51981
$378$	0	0
$379$	−33.7639	−1.73434	−0.867168	−	0.498016i	$-0.834062\pi$
$379$	−33.7639	−1.73434	−0.867168	+	0.498016i	$0.834062\pi$
$380$	2.27506	0.116708
$381$	0	0
$382$	8.31766	0.425569
$383$	−14.3859	−0.735087	−0.367544	−	0.930006i	$-0.619801\pi$
$383$	−14.3859	−0.735087	−0.367544	+	0.930006i	$0.619801\pi$
$384$	0	0
$385$	4.34010	0.221192
$386$	−21.5445	−1.09659
$387$	0	0
$388$	14.8943	0.756141
$389$	−25.5445	−1.29516	−0.647578	−	0.761999i	$-0.724218\pi$
$389$	−25.5445	−1.29516	−0.647578	+	0.761999i	$0.724218\pi$
$390$	0	0
$391$	−9.15702	−0.463090
$392$	−5.57973	−0.281819
$393$	0	0
$394$	−3.76904	−0.189881
$395$	7.38798	0.371730
$396$	0	0
$397$	−4.81219	−0.241517	−0.120759	−	0.992682i	$-0.538533\pi$
$397$	−4.81219	−0.241517	−0.120759	+	0.992682i	$0.538533\pi$
$398$	−5.92914	−0.297201
$399$	0	0
$400$	1.00000	0.0500000
$401$	−8.45191	−0.422068	−0.211034	−	0.977479i	$-0.567683\pi$
$401$	−8.45191	−0.422068	−0.211034	+	0.977479i	$0.567683\pi$
$402$	0	0
$403$	26.1304	1.30165
$404$	−7.63194	−0.379703
$405$	0	0
$406$	−7.22286	−0.358464
$407$	−3.64179	−0.180517
$408$	0	0
$409$	−5.51209	−0.272555	−0.136278	−	0.990671i	$-0.543514\pi$
$409$	−5.51209	−0.272555	−0.136278	+	0.990671i	$0.543514\pi$
$410$	−8.75471	−0.432364
$411$	0	0
$412$	−2.13955	−0.105408
$413$	−8.68020	−0.427125
$414$	0	0
$415$	4.41461	0.216705
$416$	4.86897	0.238721
$417$	0	0
$418$	−8.28529	−0.405247
$419$	−0.853529	−0.0416976	−0.0208488	−	0.999783i	$-0.506637\pi$
$419$	−0.853529	−0.0416976	−0.0208488	+	0.999783i	$0.506637\pi$
$420$	0	0
$421$	23.5275	1.14666	0.573331	−	0.819324i	$-0.305651\pi$
$421$	23.5275	1.14666	0.573331	+	0.819324i	$0.305651\pi$
$422$	−3.51059	−0.170893
$423$	0	0
$424$	5.88574	0.285837
$425$	3.19175	0.154823
$426$	0	0
$427$	−17.3332	−0.838811
$428$	−14.9904	−0.724588
$429$	0	0
$430$	−6.75471	−0.325741
$431$	−34.1849	−1.64663	−0.823315	−	0.567585i	$-0.807878\pi$
$431$	−34.1849	−1.64663	−0.823315	+	0.567585i	$0.807878\pi$
$432$	0	0
$433$	−0.747535	−0.0359242	−0.0179621	−	0.999839i	$-0.505718\pi$
$433$	−0.747535	−0.0359242	−0.0179621	+	0.999839i	$0.505718\pi$
$434$	−6.39579	−0.307008
$435$	0	0
$436$	0.253804	0.0121550
$437$	−6.52707	−0.312232
$438$	0	0
$439$	14.4504	0.689682	0.344841	−	0.938661i	$-0.387933\pi$
$439$	14.4504	0.689682	0.344841	+	0.938661i	$0.387933\pi$
$440$	−3.64179	−0.173615
$441$	0	0
$442$	15.5405	0.739187
$443$	−34.8974	−1.65803	−0.829013	−	0.559230i	$-0.811097\pi$
$443$	−34.8974	−1.65803	−0.829013	+	0.559230i	$0.811097\pi$
$444$	0	0
$445$	9.63597	0.456789
$446$	19.8322	0.939082
$447$	0	0
$448$	−1.19175	−0.0563049
$449$	−7.54564	−0.356101	−0.178050	−	0.984021i	$-0.556979\pi$
$449$	−7.54564	−0.356101	−0.178050	+	0.984021i	$0.556979\pi$
$450$	0	0
$451$	31.8828	1.50130
$452$	−13.3443	−0.627663
$453$	0	0
$454$	−12.9608	−0.608280
$455$	−5.80259	−0.272030
$456$	0	0
$457$	−5.70645	−0.266936	−0.133468	−	0.991053i	$-0.542611\pi$
$457$	−5.70645	−0.266936	−0.133468	+	0.991053i	$0.542611\pi$
$458$	−23.2384	−1.08586
$459$	0	0
$460$	−2.86897	−0.133766
$461$	18.0324	0.839851	0.419926	−	0.907559i	$-0.362056\pi$
$461$	18.0324	0.839851	0.419926	+	0.907559i	$0.362056\pi$
$462$	0	0
$463$	−25.3869	−1.17983	−0.589914	−	0.807466i	$-0.700839\pi$
$463$	−25.3869	−1.17983	−0.589914	+	0.807466i	$0.700839\pi$
$464$	6.06072	0.281362
$465$	0	0
$466$	−8.88196	−0.411449
$467$	23.6153	1.09279	0.546393	−	0.837529i	$-0.316000\pi$
$467$	23.6153	1.09279	0.546393	+	0.837529i	$0.316000\pi$
$468$	0	0
$469$	5.83966	0.269651
$470$	4.13955	0.190943
$471$	0	0
$472$	7.28357	0.335254
$473$	24.5992	1.13107
$474$	0	0
$475$	2.27506	0.104387
$476$	−3.80377	−0.174345
$477$	0	0
$478$	−25.3778	−1.16076
$479$	−34.7981	−1.58997	−0.794983	−	0.606632i	$-0.792520\pi$
$479$	−34.7981	−1.58997	−0.794983	+	0.606632i	$0.792520\pi$
$480$	0	0
$481$	4.86897	0.222006
$482$	−10.2429	−0.466550
$483$	0	0
$484$	2.26261	0.102846
$485$	14.8943	0.676313
$486$	0	0
$487$	−8.65612	−0.392246	−0.196123	−	0.980579i	$-0.562835\pi$
$487$	−8.65612	−0.392246	−0.196123	+	0.980579i	$0.562835\pi$
$488$	14.5443	0.658390
$489$	0	0
$490$	−5.57973	−0.252067
$491$	12.0632	0.544407	0.272203	−	0.962240i	$-0.412248\pi$
$491$	12.0632	0.544407	0.272203	+	0.962240i	$0.412248\pi$
$492$	0	0
$493$	19.3443	0.871223
$494$	11.0772	0.498387
$495$	0	0
$496$	5.36672	0.240973
$497$	−6.42113	−0.288027
$498$	0	0
$499$	1.14647	0.0513231	0.0256616	−	0.999671i	$-0.491831\pi$
$499$	1.14647	0.0513231	0.0256616	+	0.999671i	$0.491831\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−10.2258	−0.456402
$503$	6.98795	0.311577	0.155789	−	0.987790i	$-0.450208\pi$
$503$	6.98795	0.311577	0.155789	+	0.987790i	$0.450208\pi$
$504$	0	0
$505$	−7.63194	−0.339617
$506$	10.4482	0.464477
$507$	0	0
$508$	−7.21891	−0.320288
$509$	−27.9592	−1.23927	−0.619634	−	0.784891i	$-0.712719\pi$
$509$	−27.9592	−1.23927	−0.619634	+	0.784891i	$0.712719\pi$
$510$	0	0
$511$	−14.0657	−0.622232
$512$	1.00000	0.0441942
$513$	0	0
$514$	18.0481	0.796069
$515$	−2.13955	−0.0942796
$516$	0	0
$517$	−15.0753	−0.663013
$518$	−1.19175	−0.0523625
$519$	0	0
$520$	4.86897	0.213518
$521$	29.5052	1.29265	0.646323	−	0.763064i	$-0.276306\pi$
$521$	29.5052	1.29265	0.646323	+	0.763064i	$0.276306\pi$
$522$	0	0
$523$	13.0588	0.571022	0.285511	−	0.958375i	$-0.407837\pi$
$523$	13.0588	0.571022	0.285511	+	0.958375i	$0.407837\pi$
$524$	−20.0506	−0.875913
$525$	0	0
$526$	13.5658	0.591497
$527$	17.1292	0.746162
$528$	0	0
$529$	−14.7690	−0.642132
$530$	5.88574	0.255660
$531$	0	0
$532$	−2.71131	−0.117550
$533$	−42.6264	−1.84635
$534$	0	0
$535$	−14.9904	−0.648091
$536$	−4.90007	−0.211651
$537$	0	0
$538$	−18.3629	−0.791683
$539$	20.3202	0.875253
$540$	0	0
$541$	−23.5134	−1.01092	−0.505461	−	0.862850i	$-0.668678\pi$
$541$	−23.5134	−1.01092	−0.505461	+	0.862850i	$0.668678\pi$
$542$	−0.279091	−0.0119880
$543$	0	0
$544$	3.19175	0.136845
$545$	0.253804	0.0108718
$546$	0	0
$547$	−33.5166	−1.43307	−0.716533	−	0.697553i	$-0.754272\pi$
$547$	−33.5166	−1.43307	−0.716533	+	0.697553i	$0.754272\pi$
$548$	9.52753	0.406996
$549$	0	0
$550$	−3.64179	−0.155286
$551$	13.7885	0.587410
$552$	0	0
$553$	−8.80463	−0.374411
$554$	24.9195	1.05873
$555$	0	0
$556$	20.7323	0.879244
$557$	−11.6308	−0.492815	−0.246407	−	0.969166i	$-0.579250\pi$
$557$	−11.6308	−0.492815	−0.246407	+	0.969166i	$0.579250\pi$
$558$	0	0
$559$	−32.8884	−1.39103
$560$	−1.19175	−0.0503606
$561$	0	0
$562$	2.25695	0.0952036
$563$	2.37336	0.100025	0.0500125	−	0.998749i	$-0.484074\pi$
$563$	2.37336	0.100025	0.0500125	+	0.998749i	$0.484074\pi$
$564$	0	0
$565$	−13.3443	−0.561398
$566$	4.31884	0.181534
$567$	0	0
$568$	5.38798	0.226075
$569$	3.83093	0.160601	0.0803005	−	0.996771i	$-0.474412\pi$
$569$	3.83093	0.160601	0.0803005	+	0.996771i	$0.474412\pi$
$570$	0	0
$571$	3.85678	0.161401	0.0807007	−	0.996738i	$-0.474284\pi$
$571$	3.85678	0.161401	0.0807007	+	0.996738i	$0.474284\pi$
$572$	−17.7317	−0.741401
$573$	0	0
$574$	10.4334	0.435483
$575$	−2.86897	−0.119644
$576$	0	0
$577$	−15.2565	−0.635136	−0.317568	−	0.948235i	$-0.602866\pi$
$577$	−15.2565	−0.635136	−0.317568	+	0.948235i	$0.602866\pi$
$578$	−6.81273	−0.283372
$579$	0	0
$580$	6.06072	0.251658
$581$	−5.26111	−0.218268
$582$	0	0
$583$	−21.4346	−0.887731
$584$	11.8026	0.488395
$585$	0	0
$586$	7.43138	0.306988
$587$	20.0442	0.827312	0.413656	−	0.910433i	$-0.364252\pi$
$587$	20.0442	0.827312	0.413656	+	0.910433i	$0.364252\pi$
$588$	0	0
$589$	12.2096	0.503089
$590$	7.28357	0.299860
$591$	0	0
$592$	1.00000	0.0410997
$593$	38.0414	1.56217	0.781087	−	0.624422i	$-0.214666\pi$
$593$	38.0414	1.56217	0.781087	+	0.624422i	$0.214666\pi$
$594$	0	0
$595$	−3.80377	−0.155939
$596$	−9.17758	−0.375928
$597$	0	0
$598$	−13.9689	−0.571230
$599$	0.900073	0.0367760	0.0183880	−	0.999831i	$-0.494147\pi$
$599$	0.900073	0.0367760	0.0183880	+	0.999831i	$0.494147\pi$
$600$	0	0
$601$	17.8298	0.727291	0.363645	−	0.931537i	$-0.381532\pi$
$601$	17.8298	0.727291	0.363645	+	0.931537i	$0.381532\pi$
$602$	8.04992	0.328090
$603$	0	0
$604$	17.6024	0.716232
$605$	2.26261	0.0919881
$606$	0	0
$607$	−38.6109	−1.56717	−0.783585	−	0.621285i	$-0.786611\pi$
$607$	−38.6109	−1.56717	−0.783585	+	0.621285i	$0.786611\pi$
$608$	2.27506	0.0922659
$609$	0	0
$610$	14.5443	0.588882
$611$	20.1553	0.815396
$612$	0	0
$613$	13.7108	0.553773	0.276886	−	0.960903i	$-0.410697\pi$
$613$	13.7108	0.553773	0.276886	+	0.960903i	$0.410697\pi$
$614$	3.28357	0.132514
$615$	0	0
$616$	4.34010	0.174868
$617$	−23.0875	−0.929468	−0.464734	−	0.885450i	$-0.653850\pi$
$617$	−23.0875	−0.929468	−0.464734	+	0.885450i	$0.653850\pi$
$618$	0	0
$619$	28.4872	1.14500	0.572499	−	0.819905i	$-0.305974\pi$
$619$	28.4872	1.14500	0.572499	+	0.819905i	$0.305974\pi$
$620$	5.36672	0.215533
$621$	0	0
$622$	−11.8438	−0.474895
$623$	−11.4837	−0.460083
$624$	0	0
$625$	1.00000	0.0400000
$626$	15.4653	0.618118
$627$	0	0
$628$	6.89857	0.275283
$629$	3.19175	0.127263
$630$	0	0
$631$	5.84975	0.232875	0.116437	−	0.993198i	$-0.462853\pi$
$631$	5.84975	0.232875	0.116437	+	0.993198i	$0.462853\pi$
$632$	7.38798	0.293878
$633$	0	0
$634$	−16.4553	−0.653525
$635$	−7.21891	−0.286474
$636$	0	0
$637$	−27.1675	−1.07642
$638$	−22.0718	−0.873832
$639$	0	0
$640$	1.00000	0.0395285
$641$	−16.5884	−0.655203	−0.327601	−	0.944816i	$-0.606240\pi$
$641$	−16.5884	−0.655203	−0.327601	+	0.944816i	$0.606240\pi$
$642$	0	0
$643$	−23.3695	−0.921603	−0.460801	−	0.887503i	$-0.652438\pi$
$643$	−23.3695	−0.921603	−0.460801	+	0.887503i	$0.652438\pi$
$644$	3.41909	0.134731
$645$	0	0
$646$	7.26143	0.285697
$647$	25.8618	1.01673	0.508366	−	0.861141i	$-0.330250\pi$
$647$	25.8618	1.01673	0.508366	+	0.861141i	$0.330250\pi$
$648$	0	0
$649$	−26.5252	−1.04121
$650$	4.86897	0.190977
$651$	0	0
$652$	−10.8194	−0.423719
$653$	34.9853	1.36908	0.684540	−	0.728976i	$-0.260003\pi$
$653$	34.9853	1.36908	0.684540	+	0.728976i	$0.260003\pi$
$654$	0	0
$655$	−20.0506	−0.783441
$656$	−8.75471	−0.341814
$657$	0	0
$658$	−4.93330	−0.192320
$659$	−21.3274	−0.830796	−0.415398	−	0.909640i	$-0.636358\pi$
$659$	−21.3274	−0.830796	−0.415398	+	0.909640i	$0.636358\pi$
$660$	0	0
$661$	−13.8420	−0.538390	−0.269195	−	0.963086i	$-0.586758\pi$
$661$	−13.8420	−0.538390	−0.269195	+	0.963086i	$0.586758\pi$
$662$	35.4204	1.37665
$663$	0	0
$664$	4.41461	0.171320
$665$	−2.71131	−0.105140
$666$	0	0
$667$	−17.3880	−0.673265
$668$	2.86897	0.111004
$669$	0	0
$670$	−4.90007	−0.189306
$671$	−52.9672	−2.04478
$672$	0	0
$673$	−34.5866	−1.33322	−0.666608	−	0.745409i	$-0.732254\pi$
$673$	−34.5866	−1.33322	−0.666608	+	0.745409i	$0.732254\pi$
$674$	−4.80675	−0.185149
$675$	0	0
$676$	10.7068	0.411801
$677$	32.0401	1.23140	0.615700	−	0.787981i	$-0.288873\pi$
$677$	32.0401	1.23140	0.615700	+	0.787981i	$0.288873\pi$
$678$	0	0
$679$	−17.7502	−0.681191
$680$	3.19175	0.122398
$681$	0	0
$682$	−19.5445	−0.748396
$683$	−25.2036	−0.964391	−0.482195	−	0.876064i	$-0.660160\pi$
$683$	−25.2036	−0.964391	−0.482195	+	0.876064i	$0.660160\pi$
$684$	0	0
$685$	9.52753	0.364028
$686$	14.9919	0.572393
$687$	0	0
$688$	−6.75471	−0.257521
$689$	28.6575	1.09176
$690$	0	0
$691$	−21.5818	−0.821009	−0.410505	−	0.911859i	$-0.634647\pi$
$691$	−21.5818	−0.821009	−0.410505	+	0.911859i	$0.634647\pi$
$692$	18.3571	0.697833
$693$	0	0
$694$	18.5049	0.702438
$695$	20.7323	0.786420
$696$	0	0
$697$	−27.9428	−1.05841
$698$	−33.7975	−1.27925
$699$	0	0
$700$	−1.19175	−0.0450439
$701$	14.0622	0.531123	0.265561	−	0.964094i	$-0.414443\pi$
$701$	14.0622	0.531123	0.265561	+	0.964094i	$0.414443\pi$
$702$	0	0
$703$	2.27506	0.0858056
$704$	−3.64179	−0.137255
$705$	0	0
$706$	−37.4657	−1.41004
$707$	9.09536	0.342066
$708$	0	0
$709$	−3.83300	−0.143951	−0.0719756	−	0.997406i	$-0.522930\pi$
$709$	−3.83300	−0.143951	−0.0719756	+	0.997406i	$0.522930\pi$
$710$	5.38798	0.202207
$711$	0	0
$712$	9.63597	0.361123
$713$	−15.3969	−0.576620
$714$	0	0
$715$	−17.7317	−0.663129
$716$	−23.1259	−0.864256
$717$	0	0
$718$	14.6886	0.548173
$719$	−40.5298	−1.51151	−0.755754	−	0.654856i	$-0.772729\pi$
$719$	−40.5298	−1.51151	−0.755754	+	0.654856i	$0.772729\pi$
$720$	0	0
$721$	2.54980	0.0949596
$722$	−13.8241	−0.514479
$723$	0	0
$724$	14.9001	0.553757
$725$	6.06072	0.225089
$726$	0	0
$727$	29.5871	1.09732	0.548662	−	0.836044i	$-0.315137\pi$
$727$	29.5871	1.09732	0.548662	+	0.836044i	$0.315137\pi$
$728$	−5.80259	−0.215058
$729$	0	0
$730$	11.8026	0.436833
$731$	−21.5593	−0.797401
$732$	0	0
$733$	−30.5400	−1.12802	−0.564010	−	0.825768i	$-0.690742\pi$
$733$	−30.5400	−1.12802	−0.564010	+	0.825768i	$0.690742\pi$
$734$	16.1175	0.594907
$735$	0	0
$736$	−2.86897	−0.105751
$737$	17.8450	0.657330
$738$	0	0
$739$	−1.49357	−0.0549419	−0.0274709	−	0.999623i	$-0.508745\pi$
$739$	−1.49357	−0.0549419	−0.0274709	+	0.999623i	$0.508745\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−7.01433	−0.257504
$743$	−25.2442	−0.926120	−0.463060	−	0.886327i	$-0.653249\pi$
$743$	−25.2442	−0.926120	−0.463060	+	0.886327i	$0.653249\pi$
$744$	0	0
$745$	−9.17758	−0.336240
$746$	3.25002	0.118992
$747$	0	0
$748$	−11.6237	−0.425004
$749$	17.8648	0.652766
$750$	0	0
$751$	8.44304	0.308091	0.154045	−	0.988064i	$-0.450770\pi$
$751$	8.44304	0.308091	0.154045	+	0.988064i	$0.450770\pi$
$752$	4.13955	0.150954
$753$	0	0
$754$	29.5094	1.07467
$755$	17.6024	0.640617
$756$	0	0
$757$	35.8169	1.30179	0.650894	−	0.759168i	$-0.274394\pi$
$757$	35.8169	1.30179	0.650894	+	0.759168i	$0.274394\pi$
$758$	−33.7639	−1.22636
$759$	0	0
$760$	2.27506	0.0825252
$761$	27.0428	0.980299	0.490150	−	0.871638i	$-0.336942\pi$
$761$	27.0428	0.980299	0.490150	+	0.871638i	$0.336942\pi$
$762$	0	0
$763$	−0.302471	−0.0109502
$764$	8.31766	0.300922
$765$	0	0
$766$	−14.3859	−0.519785
$767$	35.4635	1.28051
$768$	0	0
$769$	−42.3051	−1.52556	−0.762780	−	0.646658i	$-0.776166\pi$
$769$	−42.3051	−1.52556	−0.762780	+	0.646658i	$0.776166\pi$
$770$	4.34010	0.156406
$771$	0	0
$772$	−21.5445	−0.775405
$773$	11.5566	0.415661	0.207830	−	0.978165i	$-0.433360\pi$
$773$	11.5566	0.415661	0.207830	+	0.978165i	$0.433360\pi$
$774$	0	0
$775$	5.36672	0.192778
$776$	14.8943	0.534672
$777$	0	0
$778$	−25.5445	−0.915813
$779$	−19.9175	−0.713618
$780$	0	0
$781$	−19.6219	−0.702126
$782$	−9.15702	−0.327454
$783$	0	0
$784$	−5.57973	−0.199276
$785$	6.89857	0.246221
$786$	0	0
$787$	33.9099	1.20876	0.604379	−	0.796697i	$-0.293421\pi$
$787$	33.9099	1.20876	0.604379	+	0.796697i	$0.293421\pi$
$788$	−3.76904	−0.134266
$789$	0	0
$790$	7.38798	0.262853
$791$	15.9031	0.565448
$792$	0	0
$793$	70.8157	2.51474
$794$	−4.81219	−0.170778
$795$	0	0
$796$	−5.92914	−0.210153
$797$	−20.6886	−0.732827	−0.366413	−	0.930452i	$-0.619414\pi$
$797$	−20.6886	−0.732827	−0.366413	+	0.930452i	$0.619414\pi$
$798$	0	0
$799$	13.2124	0.467421
$800$	1.00000	0.0353553
$801$	0	0
$802$	−8.45191	−0.298447
$803$	−42.9825	−1.51682
$804$	0	0
$805$	3.41909	0.120507
$806$	26.1304	0.920404
$807$	0	0
$808$	−7.63194	−0.268491
$809$	−44.3868	−1.56056	−0.780278	−	0.625433i	$-0.784922\pi$
$809$	−44.3868	−1.56056	−0.780278	+	0.625433i	$0.784922\pi$
$810$	0	0
$811$	−41.4929	−1.45701	−0.728506	−	0.685039i	$-0.759785\pi$
$811$	−41.4929	−1.45701	−0.728506	+	0.685039i	$0.759785\pi$
$812$	−7.22286	−0.253473
$813$	0	0
$814$	−3.64179	−0.127645
$815$	−10.8194	−0.378986
$816$	0	0
$817$	−15.3674	−0.537636
$818$	−5.51209	−0.192726
$819$	0	0
$820$	−8.75471	−0.305728
$821$	21.9624	0.766494	0.383247	−	0.923646i	$-0.374806\pi$
$821$	21.9624	0.766494	0.383247	+	0.923646i	$0.374806\pi$
$822$	0	0
$823$	31.5911	1.10120	0.550598	−	0.834770i	$-0.314400\pi$
$823$	31.5911	1.10120	0.550598	+	0.834770i	$0.314400\pi$
$824$	−2.13955	−0.0745346
$825$	0	0
$826$	−8.68020	−0.302023
$827$	5.42271	0.188566	0.0942831	−	0.995545i	$-0.469944\pi$
$827$	5.42271	0.188566	0.0942831	+	0.995545i	$0.469944\pi$
$828$	0	0
$829$	−54.0226	−1.87628	−0.938141	−	0.346252i	$-0.887454\pi$
$829$	−54.0226	−1.87628	−0.938141	+	0.346252i	$0.887454\pi$
$830$	4.41461	0.153233
$831$	0	0
$832$	4.86897	0.168801
$833$	−17.8091	−0.617049
$834$	0	0
$835$	2.86897	0.0992846
$836$	−8.28529	−0.286553
$837$	0	0
$838$	−0.853529	−0.0294847
$839$	−22.4803	−0.776108	−0.388054	−	0.921637i	$-0.626853\pi$
$839$	−22.4803	−0.776108	−0.388054	+	0.921637i	$0.626853\pi$
$840$	0	0
$841$	7.73227	0.266630
$842$	23.5275	0.810812
$843$	0	0
$844$	−3.51059	−0.120840
$845$	10.7068	0.368326
$846$	0	0
$847$	−2.69646	−0.0926516
$848$	5.88574	0.202117
$849$	0	0
$850$	3.19175	0.109476
$851$	−2.86897	−0.0983469
$852$	0	0
$853$	24.4482	0.837089	0.418545	−	0.908196i	$-0.362540\pi$
$853$	24.4482	0.837089	0.418545	+	0.908196i	$0.362540\pi$
$854$	−17.3332	−0.593129
$855$	0	0
$856$	−14.9904	−0.512361
$857$	19.1295	0.653452	0.326726	−	0.945119i	$-0.394054\pi$
$857$	19.1295	0.653452	0.326726	+	0.945119i	$0.394054\pi$
$858$	0	0
$859$	−11.5662	−0.394634	−0.197317	−	0.980340i	$-0.563223\pi$
$859$	−11.5662	−0.394634	−0.197317	+	0.980340i	$0.563223\pi$
$860$	−6.75471	−0.230334
$861$	0	0
$862$	−34.1849	−1.16434
$863$	14.2397	0.484726	0.242363	−	0.970186i	$-0.422078\pi$
$863$	14.2397	0.484726	0.242363	+	0.970186i	$0.422078\pi$
$864$	0	0
$865$	18.3571	0.624161
$866$	−0.747535	−0.0254023
$867$	0	0
$868$	−6.39579	−0.217087
$869$	−26.9055	−0.912705
$870$	0	0
$871$	−23.8583	−0.808407
$872$	0.253804	0.00859490
$873$	0	0
$874$	−6.52707	−0.220782
$875$	−1.19175	−0.0402885
$876$	0	0
$877$	−36.8021	−1.24272	−0.621358	−	0.783526i	$-0.713419\pi$
$877$	−36.8021	−1.24272	−0.621358	+	0.783526i	$0.713419\pi$
$878$	14.4504	0.487679
$879$	0	0
$880$	−3.64179	−0.122765
$881$	5.78645	0.194951	0.0974753	−	0.995238i	$-0.468923\pi$
$881$	5.78645	0.194951	0.0974753	+	0.995238i	$0.468923\pi$
$882$	0	0
$883$	8.30279	0.279411	0.139706	−	0.990193i	$-0.455384\pi$
$883$	8.30279	0.279411	0.139706	+	0.990193i	$0.455384\pi$
$884$	15.5405	0.522684
$885$	0	0
$886$	−34.8974	−1.17240
$887$	40.2125	1.35020	0.675101	−	0.737725i	$-0.264100\pi$
$887$	40.2125	1.35020	0.675101	+	0.737725i	$0.264100\pi$
$888$	0	0
$889$	8.60314	0.288540
$890$	9.63597	0.322998
$891$	0	0
$892$	19.8322	0.664031
$893$	9.41772	0.315152
$894$	0	0
$895$	−23.1259	−0.773014
$896$	−1.19175	−0.0398136
$897$	0	0
$898$	−7.54564	−0.251801
$899$	32.5262	1.08481
$900$	0	0
$901$	18.7858	0.625846
$902$	31.8828	1.06158
$903$	0	0
$904$	−13.3443	−0.443824
$905$	14.9001	0.495295
$906$	0	0
$907$	17.9432	0.595795	0.297898	−	0.954598i	$-0.403715\pi$
$907$	17.9432	0.595795	0.297898	+	0.954598i	$0.403715\pi$
$908$	−12.9608	−0.430119
$909$	0	0
$910$	−5.80259	−0.192354
$911$	10.0090	0.331612	0.165806	−	0.986158i	$-0.446977\pi$
$911$	10.0090	0.331612	0.165806	+	0.986158i	$0.446977\pi$
$912$	0	0
$913$	−16.0771	−0.532073
$914$	−5.70645	−0.188752
$915$	0	0
$916$	−23.2384	−0.767818
$917$	23.8953	0.789091
$918$	0	0
$919$	53.4439	1.76295	0.881476	−	0.472228i	$-0.156550\pi$
$919$	53.4439	1.76295	0.881476	+	0.472228i	$0.156550\pi$
$920$	−2.86897	−0.0945870
$921$	0	0
$922$	18.0324	0.593865
$923$	26.2339	0.863499
$924$	0	0
$925$	1.00000	0.0328798
$926$	−25.3869	−0.834265
$927$	0	0
$928$	6.06072	0.198953
$929$	17.5723	0.576528	0.288264	−	0.957551i	$-0.406922\pi$
$929$	17.5723	0.576528	0.288264	+	0.957551i	$0.406922\pi$
$930$	0	0
$931$	−12.6942	−0.416037
$932$	−8.88196	−0.290938
$933$	0	0
$934$	23.6153	0.772717
$935$	−11.6237	−0.380135
$936$	0	0
$937$	−32.5641	−1.06382	−0.531912	−	0.846800i	$-0.678526\pi$
$937$	−32.5641	−1.06382	−0.531912	+	0.846800i	$0.678526\pi$
$938$	5.83966	0.190672
$939$	0	0
$940$	4.13955	0.135017
$941$	17.0656	0.556323	0.278161	−	0.960534i	$-0.410275\pi$
$941$	17.0656	0.556323	0.278161	+	0.960534i	$0.410275\pi$
$942$	0	0
$943$	25.1169	0.817920
$944$	7.28357	0.237060
$945$	0	0
$946$	24.5992	0.799789
$947$	−25.2036	−0.819009	−0.409504	−	0.912308i	$-0.634298\pi$
$947$	−25.2036	−0.819009	−0.409504	+	0.912308i	$0.634298\pi$
$948$	0	0
$949$	57.4664	1.86544
$950$	2.27506	0.0738128
$951$	0	0
$952$	−3.80377	−0.123281
$953$	21.7560	0.704747	0.352374	−	0.935859i	$-0.385375\pi$
$953$	21.7560	0.704747	0.352374	+	0.935859i	$0.385375\pi$
$954$	0	0
$955$	8.31766	0.269153
$956$	−25.3778	−0.820778
$957$	0	0
$958$	−34.7981	−1.12428
$959$	−11.3544	−0.366654
$960$	0	0
$961$	−2.19827	−0.0709119
$962$	4.86897	0.156982
$963$	0	0
$964$	−10.2429	−0.329900
$965$	−21.5445	−0.693544
$966$	0	0
$967$	−12.3654	−0.397644	−0.198822	−	0.980036i	$-0.563712\pi$
$967$	−12.3654	−0.397644	−0.198822	+	0.980036i	$0.563712\pi$
$968$	2.26261	0.0727230
$969$	0	0
$970$	14.8943	0.478226
$971$	50.2114	1.61136	0.805680	−	0.592351i	$-0.201800\pi$
$971$	50.2114	1.61136	0.805680	+	0.592351i	$0.201800\pi$
$972$	0	0
$973$	−24.7077	−0.792092
$974$	−8.65612	−0.277360
$975$	0	0
$976$	14.5443	0.465552
$977$	8.06724	0.258094	0.129047	−	0.991638i	$-0.458808\pi$
$977$	8.06724	0.258094	0.129047	+	0.991638i	$0.458808\pi$
$978$	0	0
$979$	−35.0921	−1.12155
$980$	−5.57973	−0.178238
$981$	0	0
$982$	12.0632	0.384954
$983$	−42.2151	−1.34645	−0.673227	−	0.739436i	$-0.735092\pi$
$983$	−42.2151	−1.34645	−0.673227	+	0.739436i	$0.735092\pi$
$984$	0	0
$985$	−3.76904	−0.120092
$986$	19.3443	0.616048
$987$	0	0
$988$	11.0772	0.352413
$989$	19.3790	0.616217
$990$	0	0
$991$	−16.9294	−0.537780	−0.268890	−	0.963171i	$-0.586657\pi$
$991$	−16.9294	−0.537780	−0.268890	+	0.963171i	$0.586657\pi$
$992$	5.36672	0.170394
$993$	0	0
$994$	−6.42113	−0.203666
$995$	−5.92914	−0.187966
$996$	0	0
$997$	−8.75699	−0.277337	−0.138668	−	0.990339i	$-0.544282\pi$
$997$	−8.75699	−0.277337	−0.138668	+	0.990339i	$0.544282\pi$
$998$	1.14647	0.0362909
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bl.1.2	yes	5
3.2	odd	2		3330.2.a.bk.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.bk.1.2	✓	5	3.2	odd	2
3330.2.a.bl.1.2	yes	5	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.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.19175 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.64179 q^{11} +4.86897 q^{13} -1.19175 q^{14} +1.00000 q^{16} +3.19175 q^{17} +2.27506 q^{19} +1.00000 q^{20} -3.64179 q^{22} -2.86897 q^{23} +1.00000 q^{25} +4.86897 q^{26} -1.19175 q^{28} +6.06072 q^{29} +5.36672 q^{31} +1.00000 q^{32} +3.19175 q^{34} -1.19175 q^{35} +1.00000 q^{37} +2.27506 q^{38} +1.00000 q^{40} -8.75471 q^{41} -6.75471 q^{43} -3.64179 q^{44} -2.86897 q^{46} +4.13955 q^{47} -5.57973 q^{49} +1.00000 q^{50} +4.86897 q^{52} +5.88574 q^{53} -3.64179 q^{55} -1.19175 q^{56} +6.06072 q^{58} +7.28357 q^{59} +14.5443 q^{61} +5.36672 q^{62} +1.00000 q^{64} +4.86897 q^{65} -4.90007 q^{67} +3.19175 q^{68} -1.19175 q^{70} +5.38798 q^{71} +11.8026 q^{73} +1.00000 q^{74} +2.27506 q^{76} +4.34010 q^{77} +7.38798 q^{79} +1.00000 q^{80} -8.75471 q^{82} +4.41461 q^{83} +3.19175 q^{85} -6.75471 q^{86} -3.64179 q^{88} +9.63597 q^{89} -5.80259 q^{91} -2.86897 q^{92} +4.13955 q^{94} +2.27506 q^{95} +14.8943 q^{97} -5.57973 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{2} + 5 q^{4} + 5 q^{5} + 3 q^{7} + 5 q^{8} + 5 q^{10} + 5 q^{11} + 5 q^{13} + 3 q^{14} + 5 q^{16} + 7 q^{17} - q^{19} + 5 q^{20} + 5 q^{22} + 5 q^{23} + 5 q^{25} + 5 q^{26} + 3 q^{28} + 2 q^{29}+ \cdots + 16 q^{98}+O(q^{100}) \) 5 * q + 5 * q^2 + 5 * q^4 + 5 * q^5 + 3 * q^7 + 5 * q^8 + 5 * q^10 + 5 * q^11 + 5 * q^13 + 3 * q^14 + 5 * q^16 + 7 * q^17 - q^19 + 5 * q^20 + 5 * q^22 + 5 * q^23 + 5 * q^25 + 5 * q^26 + 3 * q^28 + 2 * q^29 + 16 * q^31 + 5 * q^32 + 7 * q^34 + 3 * q^35 + 5 * q^37 - q^38 + 5 * q^40 + 2 * q^41 + 12 * q^43 + 5 * q^44 + 5 * q^46 + 6 * q^47 + 16 * q^49 + 5 * q^50 + 5 * q^52 + 3 * q^53 + 5 * q^55 + 3 * q^56 + 2 * q^58 - 10 * q^59 + 16 * q^61 + 16 * q^62 + 5 * q^64 + 5 * q^65 + 4 * q^67 + 7 * q^68 + 3 * q^70 - 8 * q^71 - 3 * q^73 + 5 * q^74 - q^76 + 3 * q^77 + 2 * q^79 + 5 * q^80 + 2 * q^82 - 5 * q^83 + 7 * q^85 + 12 * q^86 + 5 * q^88 - 7 * q^89 + 33 * q^91 + 5 * q^92 + 6 * q^94 - q^95 + 14 * q^97 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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