LMFDB - Embedded newform 2070.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, 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("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} +0.791288 q^{11} +5.79129 q^{13} -1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +5.79129 q^{19} +1.00000 q^{20} +0.791288 q^{22} -1.00000 q^{23} +1.00000 q^{25} +5.79129 q^{26} -1.79129 q^{28} -7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -0.791288 q^{34} -1.79129 q^{35} -4.00000 q^{37} +5.79129 q^{38} +1.00000 q^{40} +6.79129 q^{41} +11.1652 q^{43} +0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} -3.79129 q^{49} +1.00000 q^{50} +5.79129 q^{52} -6.00000 q^{53} +0.791288 q^{55} -1.79129 q^{56} -7.58258 q^{58} +13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +1.00000 q^{64} +5.79129 q^{65} +11.1652 q^{67} -0.791288 q^{68} -1.79129 q^{70} -8.37386 q^{71} +12.7477 q^{73} -4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} +8.00000 q^{79} +1.00000 q^{80} +6.79129 q^{82} +6.00000 q^{83} -0.791288 q^{85} +11.1652 q^{86} +0.791288 q^{88} -15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} +4.41742 q^{94} +5.79129 q^{95} -7.95644 q^{97} -3.79129 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 2 q^{20} - 3 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + q^{28} - 6 q^{29} + 7 q^{31} + 2 q^{32} + 3 q^{34} + q^{35} - 8 q^{37} + 7 q^{38} + 2 q^{40} + 9 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 7 q^{52} - 12 q^{53} - 3 q^{55} + q^{56} - 6 q^{58} + 18 q^{59} + 7 q^{61} + 7 q^{62} + 2 q^{64} + 7 q^{65} + 4 q^{67} + 3 q^{68} + q^{70} - 3 q^{71} - 2 q^{73} - 8 q^{74} + 7 q^{76} - 12 q^{77} + 16 q^{79} + 2 q^{80} + 9 q^{82} + 12 q^{83} + 3 q^{85} + 4 q^{86} - 3 q^{88} - 12 q^{89} - 7 q^{91} - 2 q^{92} + 18 q^{94} + 7 q^{95} + 7 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 2 * q^20 - 3 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + q^28 - 6 * q^29 + 7 * q^31 + 2 * q^32 + 3 * q^34 + q^35 - 8 * q^37 + 7 * q^38 + 2 * q^40 + 9 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 7 * q^52 - 12 * q^53 - 3 * q^55 + q^56 - 6 * q^58 + 18 * q^59 + 7 * q^61 + 7 * q^62 + 2 * q^64 + 7 * q^65 + 4 * q^67 + 3 * q^68 + q^70 - 3 * q^71 - 2 * q^73 - 8 * q^74 + 7 * q^76 - 12 * q^77 + 16 * q^79 + 2 * q^80 + 9 * q^82 + 12 * q^83 + 3 * q^85 + 4 * q^86 - 3 * q^88 - 12 * q^89 - 7 * q^91 - 2 * q^92 + 18 * q^94 + 7 * q^95 + 7 * q^97 - 3 * q^98

Display 100 coefficients 10 coefficients

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
$7$	−1.79129	−0.677043	−0.338522	−	0.940959i	$-0.609927\pi$
$7$	−1.79129	−0.677043	−0.338522	+	0.940959i	$0.609927\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0.791288	0.238582	0.119291	−	0.992859i	$-0.461938\pi$
$11$	0.791288	0.238582	0.119291	+	0.992859i	$0.461938\pi$
$12$	0	0
$13$	5.79129	1.60621	0.803107	−	0.595835i	$-0.203179\pi$
$13$	5.79129	1.60621	0.803107	+	0.595835i	$0.203179\pi$
$14$	−1.79129	−0.478742
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.791288	−0.191915	−0.0959577	−	0.995385i	$-0.530591\pi$
$17$	−0.791288	−0.191915	−0.0959577	+	0.995385i	$0.530591\pi$
$18$	0	0
$19$	5.79129	1.32861	0.664306	−	0.747460i	$-0.268727\pi$
$19$	5.79129	1.32861	0.664306	+	0.747460i	$0.268727\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0.791288	0.168703
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	5.79129	1.13576
$27$	0	0
$28$	−1.79129	−0.338522
$29$	−7.58258	−1.40805	−0.704024	−	0.710176i	$-0.748615\pi$
$29$	−7.58258	−1.40805	−0.704024	+	0.710176i	$0.748615\pi$
$30$	0	0
$31$	−3.37386	−0.605964	−0.302982	−	0.952996i	$-0.597982\pi$
$31$	−3.37386	−0.605964	−0.302982	+	0.952996i	$0.597982\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.791288	−0.135705
$35$	−1.79129	−0.302783
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	5.79129	0.939471
$39$	0	0
$40$	1.00000	0.158114
$41$	6.79129	1.06062	0.530310	−	0.847804i	$-0.322075\pi$
$41$	6.79129	1.06062	0.530310	+	0.847804i	$0.322075\pi$
$42$	0	0
$43$	11.1652	1.70267	0.851335	−	0.524623i	$-0.175794\pi$
$43$	11.1652	1.70267	0.851335	+	0.524623i	$0.175794\pi$
$44$	0.791288	0.119291
$45$	0	0
$46$	−1.00000	−0.147442
$47$	4.41742	0.644348	0.322174	−	0.946681i	$-0.395586\pi$
$47$	4.41742	0.644348	0.322174	+	0.946681i	$0.395586\pi$
$48$	0	0
$49$	−3.79129	−0.541613
$50$	1.00000	0.141421
$51$	0	0
$52$	5.79129	0.803107
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0.791288	0.106697
$56$	−1.79129	−0.239371
$57$	0	0
$58$	−7.58258	−0.995641
$59$	13.5826	1.76830	0.884150	−	0.467202i	$-0.154738\pi$
$59$	13.5826	1.76830	0.884150	+	0.467202i	$0.154738\pi$
$60$	0	0
$61$	10.3739	1.32824	0.664119	−	0.747627i	$-0.268807\pi$
$61$	10.3739	1.32824	0.664119	+	0.747627i	$0.268807\pi$
$62$	−3.37386	−0.428481
$63$	0	0
$64$	1.00000	0.125000
$65$	5.79129	0.718321
$66$	0	0
$67$	11.1652	1.36404	0.682020	−	0.731333i	$-0.261102\pi$
$67$	11.1652	1.36404	0.682020	+	0.731333i	$0.261102\pi$
$68$	−0.791288	−0.0959577
$69$	0	0
$70$	−1.79129	−0.214100
$71$	−8.37386	−0.993795	−0.496897	−	0.867809i	$-0.665527\pi$
$71$	−8.37386	−0.993795	−0.496897	+	0.867809i	$0.665527\pi$
$72$	0	0
$73$	12.7477	1.49201	0.746004	−	0.665941i	$-0.231970\pi$
$73$	12.7477	1.49201	0.746004	+	0.665941i	$0.231970\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	5.79129	0.664306
$77$	−1.41742	−0.161530
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	6.79129	0.749972
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−0.791288	−0.0858272
$86$	11.1652	1.20397
$87$	0	0
$88$	0.791288	0.0843516
$89$	−15.1652	−1.60750	−0.803751	−	0.594965i	$-0.797166\pi$
$89$	−15.1652	−1.60750	−0.803751	+	0.594965i	$0.797166\pi$
$90$	0	0
$91$	−10.3739	−1.08748
$92$	−1.00000	−0.104257
$93$	0	0
$94$	4.41742	0.455623
$95$	5.79129	0.594174
$96$	0	0
$97$	−7.95644	−0.807854	−0.403927	−	0.914791i	$-0.632355\pi$
$97$	−7.95644	−0.807854	−0.403927	+	0.914791i	$0.632355\pi$
$98$	−3.79129	−0.382978
$99$	0	0
$100$	1.00000	0.100000
$101$	−4.41742	−0.439550	−0.219775	−	0.975551i	$-0.570532\pi$
$101$	−4.41742	−0.439550	−0.219775	+	0.975551i	$0.570532\pi$
$102$	0	0
$103$	−6.37386	−0.628035	−0.314018	−	0.949417i	$-0.601675\pi$
$103$	−6.37386	−0.628035	−0.314018	+	0.949417i	$0.601675\pi$
$104$	5.79129	0.567882
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−4.41742	−0.427049	−0.213524	−	0.976938i	$-0.568494\pi$
$107$	−4.41742	−0.427049	−0.213524	+	0.976938i	$0.568494\pi$
$108$	0	0
$109$	−3.37386	−0.323158	−0.161579	−	0.986860i	$-0.551659\pi$
$109$	−3.37386	−0.323158	−0.161579	+	0.986860i	$0.551659\pi$
$110$	0.791288	0.0754463
$111$	0	0
$112$	−1.79129	−0.169261
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−7.58258	−0.704024
$117$	0	0
$118$	13.5826	1.25038
$119$	1.41742	0.129935
$120$	0	0
$121$	−10.3739	−0.943079
$122$	10.3739	0.939205
$123$	0	0
$124$	−3.37386	−0.302982
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.7477	1.13118	0.565589	−	0.824687i	$-0.308649\pi$
$127$	12.7477	1.13118	0.565589	+	0.824687i	$0.308649\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.79129	0.507930
$131$	9.16515	0.800763	0.400381	−	0.916349i	$-0.368878\pi$
$131$	9.16515	0.800763	0.400381	+	0.916349i	$0.368878\pi$
$132$	0	0
$133$	−10.3739	−0.899528
$134$	11.1652	0.964522
$135$	0	0
$136$	−0.791288	−0.0678524
$137$	−3.79129	−0.323912	−0.161956	−	0.986798i	$-0.551780\pi$
$137$	−3.79129	−0.323912	−0.161956	+	0.986798i	$0.551780\pi$
$138$	0	0
$139$	12.7477	1.08125	0.540624	−	0.841264i	$-0.318188\pi$
$139$	12.7477	1.08125	0.540624	+	0.841264i	$0.318188\pi$
$140$	−1.79129	−0.151391
$141$	0	0
$142$	−8.37386	−0.702719
$143$	4.58258	0.383214
$144$	0	0
$145$	−7.58258	−0.629699
$146$	12.7477	1.05501
$147$	0	0
$148$	−4.00000	−0.328798
$149$	−8.20871	−0.672484	−0.336242	−	0.941776i	$-0.609156\pi$
$149$	−8.20871	−0.672484	−0.336242	+	0.941776i	$0.609156\pi$
$150$	0	0
$151$	−10.7913	−0.878183	−0.439091	−	0.898442i	$-0.644700\pi$
$151$	−10.7913	−0.878183	−0.439091	+	0.898442i	$0.644700\pi$
$152$	5.79129	0.469735
$153$	0	0
$154$	−1.41742	−0.114219
$155$	−3.37386	−0.270995
$156$	0	0
$157$	−14.7477	−1.17700	−0.588498	−	0.808498i	$-0.700281\pi$
$157$	−14.7477	−1.17700	−0.588498	+	0.808498i	$0.700281\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	1.00000	0.0790569
$161$	1.79129	0.141173
$162$	0	0
$163$	8.62614	0.675651	0.337826	−	0.941209i	$-0.390309\pi$
$163$	8.62614	0.675651	0.337826	+	0.941209i	$0.390309\pi$
$164$	6.79129	0.530310
$165$	0	0
$166$	6.00000	0.465690
$167$	−18.3303	−1.41844	−0.709221	−	0.704987i	$-0.750953\pi$
$167$	−18.3303	−1.41844	−0.709221	+	0.704987i	$0.750953\pi$
$168$	0	0
$169$	20.5390	1.57992
$170$	−0.791288	−0.0606890
$171$	0	0
$172$	11.1652	0.851335
$173$	18.7913	1.42868	0.714338	−	0.699801i	$-0.246728\pi$
$173$	18.7913	1.42868	0.714338	+	0.699801i	$0.246728\pi$
$174$	0	0
$175$	−1.79129	−0.135409
$176$	0.791288	0.0596456
$177$	0	0
$178$	−15.1652	−1.13668
$179$	−10.7477	−0.803323	−0.401661	−	0.915788i	$-0.631567\pi$
$179$	−10.7477	−0.803323	−0.401661	+	0.915788i	$0.631567\pi$
$180$	0	0
$181$	−18.5390	−1.37799	−0.688997	−	0.724764i	$-0.741949\pi$
$181$	−18.5390	−1.37799	−0.688997	+	0.724764i	$0.741949\pi$
$182$	−10.3739	−0.768962
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−4.00000	−0.294086
$186$	0	0
$187$	−0.626136	−0.0457876
$188$	4.41742	0.322174
$189$	0	0
$190$	5.79129	0.420144
$191$	−25.5826	−1.85109	−0.925545	−	0.378637i	$-0.876393\pi$
$191$	−25.5826	−1.85109	−0.925545	+	0.378637i	$0.876393\pi$
$192$	0	0
$193$	−20.7477	−1.49345	−0.746727	−	0.665131i	$-0.768376\pi$
$193$	−20.7477	−1.49345	−0.746727	+	0.665131i	$0.768376\pi$
$194$	−7.95644	−0.571239
$195$	0	0
$196$	−3.79129	−0.270806
$197$	11.5390	0.822121	0.411060	−	0.911608i	$-0.365159\pi$
$197$	11.5390	0.822121	0.411060	+	0.911608i	$0.365159\pi$
$198$	0	0
$199$	−16.3303	−1.15762	−0.578812	−	0.815461i	$-0.696484\pi$
$199$	−16.3303	−1.15762	−0.578812	+	0.815461i	$0.696484\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−4.41742	−0.310809
$203$	13.5826	0.953310
$204$	0	0
$205$	6.79129	0.474324
$206$	−6.37386	−0.444088
$207$	0	0
$208$	5.79129	0.401554
$209$	4.58258	0.316983
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−4.41742	−0.301969
$215$	11.1652	0.761457
$216$	0	0
$217$	6.04356	0.410264
$218$	−3.37386	−0.228507
$219$	0	0
$220$	0.791288	0.0533486
$221$	−4.58258	−0.308257
$222$	0	0
$223$	−7.16515	−0.479814	−0.239907	−	0.970796i	$-0.577117\pi$
$223$	−7.16515	−0.479814	−0.239907	+	0.970796i	$0.577117\pi$
$224$	−1.79129	−0.119685
$225$	0	0
$226$	−6.00000	−0.399114
$227$	22.7477	1.50982	0.754910	−	0.655829i	$-0.227681\pi$
$227$	22.7477	1.50982	0.754910	+	0.655829i	$0.227681\pi$
$228$	0	0
$229$	20.3303	1.34346	0.671732	−	0.740794i	$-0.265551\pi$
$229$	20.3303	1.34346	0.671732	+	0.740794i	$0.265551\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−7.58258	−0.497820
$233$	1.58258	0.103678	0.0518390	−	0.998655i	$-0.483492\pi$
$233$	1.58258	0.103678	0.0518390	+	0.998655i	$0.483492\pi$
$234$	0	0
$235$	4.41742	0.288161
$236$	13.5826	0.884150
$237$	0	0
$238$	1.41742	0.0918780
$239$	15.1652	0.980952	0.490476	−	0.871455i	$-0.336823\pi$
$239$	15.1652	0.980952	0.490476	+	0.871455i	$0.336823\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	−10.3739	−0.666857
$243$	0	0
$244$	10.3739	0.664119
$245$	−3.79129	−0.242216
$246$	0	0
$247$	33.5390	2.13404
$248$	−3.37386	−0.214241
$249$	0	0
$250$	1.00000	0.0632456
$251$	−26.2087	−1.65428	−0.827140	−	0.561996i	$-0.810033\pi$
$251$	−26.2087	−1.65428	−0.827140	+	0.561996i	$0.810033\pi$
$252$	0	0
$253$	−0.791288	−0.0497478
$254$	12.7477	0.799864
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.74773	0.296155	0.148078	−	0.988976i	$-0.452691\pi$
$257$	4.74773	0.296155	0.148078	+	0.988976i	$0.452691\pi$
$258$	0	0
$259$	7.16515	0.445221
$260$	5.79129	0.359160
$261$	0	0
$262$	9.16515	0.566225
$263$	−11.2087	−0.691159	−0.345579	−	0.938390i	$-0.612318\pi$
$263$	−11.2087	−0.691159	−0.345579	+	0.938390i	$0.612318\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	−10.3739	−0.636062
$267$	0	0
$268$	11.1652	0.682020
$269$	10.7477	0.655300	0.327650	−	0.944799i	$-0.393743\pi$
$269$	10.7477	0.655300	0.327650	+	0.944799i	$0.393743\pi$
$270$	0	0
$271$	18.1216	1.10081	0.550404	−	0.834898i	$-0.314474\pi$
$271$	18.1216	1.10081	0.550404	+	0.834898i	$0.314474\pi$
$272$	−0.791288	−0.0479789
$273$	0	0
$274$	−3.79129	−0.229040
$275$	0.791288	0.0477165
$276$	0	0
$277$	17.1652	1.03135	0.515677	−	0.856783i	$-0.327540\pi$
$277$	17.1652	1.03135	0.515677	+	0.856783i	$0.327540\pi$
$278$	12.7477	0.764558
$279$	0	0
$280$	−1.79129	−0.107050
$281$	−10.7477	−0.641156	−0.320578	−	0.947222i	$-0.603877\pi$
$281$	−10.7477	−0.641156	−0.320578	+	0.947222i	$0.603877\pi$
$282$	0	0
$283$	8.33030	0.495185	0.247593	−	0.968864i	$-0.420361\pi$
$283$	8.33030	0.495185	0.247593	+	0.968864i	$0.420361\pi$
$284$	−8.37386	−0.496897
$285$	0	0
$286$	4.58258	0.270973
$287$	−12.1652	−0.718086
$288$	0	0
$289$	−16.3739	−0.963168
$290$	−7.58258	−0.445264
$291$	0	0
$292$	12.7477	0.746004
$293$	−27.4955	−1.60630	−0.803151	−	0.595776i	$-0.796845\pi$
$293$	−27.4955	−1.60630	−0.803151	+	0.595776i	$0.796845\pi$
$294$	0	0
$295$	13.5826	0.790808
$296$	−4.00000	−0.232495
$297$	0	0
$298$	−8.20871	−0.475518
$299$	−5.79129	−0.334919
$300$	0	0
$301$	−20.0000	−1.15278
$302$	−10.7913	−0.620969
$303$	0	0
$304$	5.79129	0.332153
$305$	10.3739	0.594006
$306$	0	0
$307$	−15.5390	−0.886858	−0.443429	−	0.896309i	$-0.646238\pi$
$307$	−15.5390	−0.886858	−0.443429	+	0.896309i	$0.646238\pi$
$308$	−1.41742	−0.0807652
$309$	0	0
$310$	−3.37386	−0.191623
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−4.62614	−0.261485	−0.130742	−	0.991416i	$-0.541736\pi$
$313$	−4.62614	−0.261485	−0.130742	+	0.991416i	$0.541736\pi$
$314$	−14.7477	−0.832262
$315$	0	0
$316$	8.00000	0.450035
$317$	−9.79129	−0.549934	−0.274967	−	0.961454i	$-0.588667\pi$
$317$	−9.79129	−0.549934	−0.274967	+	0.961454i	$0.588667\pi$
$318$	0	0
$319$	−6.00000	−0.335936
$320$	1.00000	0.0559017
$321$	0	0
$322$	1.79129	0.0998246
$323$	−4.58258	−0.254981
$324$	0	0
$325$	5.79129	0.321243
$326$	8.62614	0.477758
$327$	0	0
$328$	6.79129	0.374986
$329$	−7.91288	−0.436251
$330$	0	0
$331$	−20.7477	−1.14040	−0.570199	−	0.821507i	$-0.693134\pi$
$331$	−20.7477	−1.14040	−0.570199	+	0.821507i	$0.693134\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−18.3303	−1.00299
$335$	11.1652	0.610017
$336$	0	0
$337$	−12.2087	−0.665051	−0.332525	−	0.943094i	$-0.607901\pi$
$337$	−12.2087	−0.665051	−0.332525	+	0.943094i	$0.607901\pi$
$338$	20.5390	1.11718
$339$	0	0
$340$	−0.791288	−0.0429136
$341$	−2.66970	−0.144572
$342$	0	0
$343$	19.3303	1.04374
$344$	11.1652	0.601985
$345$	0	0
$346$	18.7913	1.01023
$347$	−5.20871	−0.279618	−0.139809	−	0.990178i	$-0.544649\pi$
$347$	−5.20871	−0.279618	−0.139809	+	0.990178i	$0.544649\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−1.79129	−0.0957484
$351$	0	0
$352$	0.791288	0.0421758
$353$	−3.16515	−0.168464	−0.0842320	−	0.996446i	$-0.526844\pi$
$353$	−3.16515	−0.168464	−0.0842320	+	0.996446i	$0.526844\pi$
$354$	0	0
$355$	−8.37386	−0.444439
$356$	−15.1652	−0.803751
$357$	0	0
$358$	−10.7477	−0.568035
$359$	−9.16515	−0.483718	−0.241859	−	0.970311i	$-0.577757\pi$
$359$	−9.16515	−0.483718	−0.241859	+	0.970311i	$0.577757\pi$
$360$	0	0
$361$	14.5390	0.765211
$362$	−18.5390	−0.974389
$363$	0	0
$364$	−10.3739	−0.543738
$365$	12.7477	0.667247
$366$	0	0
$367$	−19.1652	−1.00041	−0.500206	−	0.865906i	$-0.666743\pi$
$367$	−19.1652	−1.00041	−0.500206	+	0.865906i	$0.666743\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−4.00000	−0.207950
$371$	10.7477	0.557994
$372$	0	0
$373$	12.7477	0.660052	0.330026	−	0.943972i	$-0.392942\pi$
$373$	12.7477	0.660052	0.330026	+	0.943972i	$0.392942\pi$
$374$	−0.626136	−0.0323767
$375$	0	0
$376$	4.41742	0.227811
$377$	−43.9129	−2.26163
$378$	0	0
$379$	−6.37386	−0.327403	−0.163702	−	0.986510i	$-0.552343\pi$
$379$	−6.37386	−0.327403	−0.163702	+	0.986510i	$0.552343\pi$
$380$	5.79129	0.297087
$381$	0	0
$382$	−25.5826	−1.30892
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	−1.41742	−0.0722386
$386$	−20.7477	−1.05603
$387$	0	0
$388$	−7.95644	−0.403927
$389$	20.7042	1.04974	0.524871	−	0.851182i	$-0.324113\pi$
$389$	20.7042	1.04974	0.524871	+	0.851182i	$0.324113\pi$
$390$	0	0
$391$	0.791288	0.0400171
$392$	−3.79129	−0.191489
$393$	0	0
$394$	11.5390	0.581327
$395$	8.00000	0.402524
$396$	0	0
$397$	−15.5390	−0.779881	−0.389940	−	0.920840i	$-0.627504\pi$
$397$	−15.5390	−0.779881	−0.389940	+	0.920840i	$0.627504\pi$
$398$	−16.3303	−0.818564
$399$	0	0
$400$	1.00000	0.0500000
$401$	4.74773	0.237090	0.118545	−	0.992949i	$-0.462177\pi$
$401$	4.74773	0.237090	0.118545	+	0.992949i	$0.462177\pi$
$402$	0	0
$403$	−19.5390	−0.973308
$404$	−4.41742	−0.219775
$405$	0	0
$406$	13.5826	0.674092
$407$	−3.16515	−0.156891
$408$	0	0
$409$	−18.2087	−0.900363	−0.450181	−	0.892937i	$-0.648641\pi$
$409$	−18.2087	−0.900363	−0.450181	+	0.892937i	$0.648641\pi$
$410$	6.79129	0.335398
$411$	0	0
$412$	−6.37386	−0.314018
$413$	−24.3303	−1.19722
$414$	0	0
$415$	6.00000	0.294528
$416$	5.79129	0.283941
$417$	0	0
$418$	4.58258	0.224141
$419$	−20.8348	−1.01785	−0.508924	−	0.860811i	$-0.669957\pi$
$419$	−20.8348	−1.01785	−0.508924	+	0.860811i	$0.669957\pi$
$420$	0	0
$421$	18.1216	0.883192	0.441596	−	0.897214i	$-0.354412\pi$
$421$	18.1216	0.883192	0.441596	+	0.897214i	$0.354412\pi$
$422$	−10.0000	−0.486792
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−0.791288	−0.0383831
$426$	0	0
$427$	−18.5826	−0.899274
$428$	−4.41742	−0.213524
$429$	0	0
$430$	11.1652	0.538431
$431$	−25.9129	−1.24818	−0.624090	−	0.781353i	$-0.714530\pi$
$431$	−25.9129	−1.24818	−0.624090	+	0.781353i	$0.714530\pi$
$432$	0	0
$433$	−30.5390	−1.46761	−0.733806	−	0.679359i	$-0.762258\pi$
$433$	−30.5390	−1.46761	−0.733806	+	0.679359i	$0.762258\pi$
$434$	6.04356	0.290100
$435$	0	0
$436$	−3.37386	−0.161579
$437$	−5.79129	−0.277035
$438$	0	0
$439$	−6.53901	−0.312090	−0.156045	−	0.987750i	$-0.549875\pi$
$439$	−6.53901	−0.312090	−0.156045	+	0.987750i	$0.549875\pi$
$440$	0.791288	0.0377232
$441$	0	0
$442$	−4.58258	−0.217971
$443$	39.7913	1.89054	0.945271	−	0.326288i	$-0.105798\pi$
$443$	39.7913	1.89054	0.945271	+	0.326288i	$0.105798\pi$
$444$	0	0
$445$	−15.1652	−0.718897
$446$	−7.16515	−0.339280
$447$	0	0
$448$	−1.79129	−0.0846304
$449$	16.1216	0.760825	0.380412	−	0.924817i	$-0.375782\pi$
$449$	16.1216	0.760825	0.380412	+	0.924817i	$0.375782\pi$
$450$	0	0
$451$	5.37386	0.253045
$452$	−6.00000	−0.282216
$453$	0	0
$454$	22.7477	1.06760
$455$	−10.3739	−0.486334
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	20.3303	0.949973
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	28.7477	1.33892	0.669458	−	0.742850i	$-0.266527\pi$
$461$	28.7477	1.33892	0.669458	+	0.742850i	$0.266527\pi$
$462$	0	0
$463$	−10.0000	−0.464739	−0.232370	−	0.972628i	$-0.574648\pi$
$463$	−10.0000	−0.464739	−0.232370	+	0.972628i	$0.574648\pi$
$464$	−7.58258	−0.352012
$465$	0	0
$466$	1.58258	0.0733114
$467$	19.9129	0.921458	0.460729	−	0.887541i	$-0.347588\pi$
$467$	19.9129	0.921458	0.460729	+	0.887541i	$0.347588\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	4.41742	0.203761
$471$	0	0
$472$	13.5826	0.625189
$473$	8.83485	0.406227
$474$	0	0
$475$	5.79129	0.265723
$476$	1.41742	0.0649675
$477$	0	0
$478$	15.1652	0.693638
$479$	−15.4955	−0.708005	−0.354003	−	0.935244i	$-0.615180\pi$
$479$	−15.4955	−0.708005	−0.354003	+	0.935244i	$0.615180\pi$
$480$	0	0
$481$	−23.1652	−1.05624
$482$	−28.0000	−1.27537
$483$	0	0
$484$	−10.3739	−0.471539
$485$	−7.95644	−0.361283
$486$	0	0
$487$	6.41742	0.290801	0.145401	−	0.989373i	$-0.453553\pi$
$487$	6.41742	0.290801	0.145401	+	0.989373i	$0.453553\pi$
$488$	10.3739	0.469603
$489$	0	0
$490$	−3.79129	−0.171273
$491$	−10.7477	−0.485038	−0.242519	−	0.970147i	$-0.577974\pi$
$491$	−10.7477	−0.485038	−0.242519	+	0.970147i	$0.577974\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	33.5390	1.50899
$495$	0	0
$496$	−3.37386	−0.151491
$497$	15.0000	0.672842
$498$	0	0
$499$	4.83485	0.216438	0.108219	−	0.994127i	$-0.465485\pi$
$499$	4.83485	0.216438	0.108219	+	0.994127i	$0.465485\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−26.2087	−1.16975
$503$	−14.2087	−0.633535	−0.316768	−	0.948503i	$-0.602598\pi$
$503$	−14.2087	−0.633535	−0.316768	+	0.948503i	$0.602598\pi$
$504$	0	0
$505$	−4.41742	−0.196573
$506$	−0.791288	−0.0351770
$507$	0	0
$508$	12.7477	0.565589
$509$	−34.7477	−1.54017	−0.770083	−	0.637944i	$-0.779785\pi$
$509$	−34.7477	−1.54017	−0.770083	+	0.637944i	$0.779785\pi$
$510$	0	0
$511$	−22.8348	−1.01015
$512$	1.00000	0.0441942
$513$	0	0
$514$	4.74773	0.209413
$515$	−6.37386	−0.280866
$516$	0	0
$517$	3.49545	0.153730
$518$	7.16515	0.314819
$519$	0	0
$520$	5.79129	0.253965
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	17.1652	0.750580	0.375290	−	0.926908i	$-0.377543\pi$
$523$	17.1652	0.750580	0.375290	+	0.926908i	$0.377543\pi$
$524$	9.16515	0.400381
$525$	0	0
$526$	−11.2087	−0.488723
$527$	2.66970	0.116294
$528$	0	0
$529$	1.00000	0.0434783
$530$	−6.00000	−0.260623
$531$	0	0
$532$	−10.3739	−0.449764
$533$	39.3303	1.70358
$534$	0	0
$535$	−4.41742	−0.190982
$536$	11.1652	0.482261
$537$	0	0
$538$	10.7477	0.463367
$539$	−3.00000	−0.129219
$540$	0	0
$541$	1.66970	0.0717859	0.0358929	−	0.999356i	$-0.488572\pi$
$541$	1.66970	0.0717859	0.0358929	+	0.999356i	$0.488572\pi$
$542$	18.1216	0.778389
$543$	0	0
$544$	−0.791288	−0.0339262
$545$	−3.37386	−0.144520
$546$	0	0
$547$	−26.1216	−1.11688	−0.558439	−	0.829545i	$-0.688600\pi$
$547$	−26.1216	−1.11688	−0.558439	+	0.829545i	$0.688600\pi$
$548$	−3.79129	−0.161956
$549$	0	0
$550$	0.791288	0.0337406
$551$	−43.9129	−1.87075
$552$	0	0
$553$	−14.3303	−0.609386
$554$	17.1652	0.729277
$555$	0	0
$556$	12.7477	0.540624
$557$	6.33030	0.268224	0.134112	−	0.990966i	$-0.457182\pi$
$557$	6.33030	0.268224	0.134112	+	0.990966i	$0.457182\pi$
$558$	0	0
$559$	64.6606	2.73485
$560$	−1.79129	−0.0756957
$561$	0	0
$562$	−10.7477	−0.453366
$563$	15.1652	0.639135	0.319567	−	0.947564i	$-0.396462\pi$
$563$	15.1652	0.639135	0.319567	+	0.947564i	$0.396462\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	8.33030	0.350149
$567$	0	0
$568$	−8.37386	−0.351360
$569$	39.4955	1.65574	0.827868	−	0.560923i	$-0.189554\pi$
$569$	39.4955	1.65574	0.827868	+	0.560923i	$0.189554\pi$
$570$	0	0
$571$	−11.1216	−0.465424	−0.232712	−	0.972546i	$-0.574760\pi$
$571$	−11.1216	−0.465424	−0.232712	+	0.972546i	$0.574760\pi$
$572$	4.58258	0.191607
$573$	0	0
$574$	−12.1652	−0.507764
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	41.1652	1.71373	0.856864	−	0.515543i	$-0.172410\pi$
$577$	41.1652	1.71373	0.856864	+	0.515543i	$0.172410\pi$
$578$	−16.3739	−0.681063
$579$	0	0
$580$	−7.58258	−0.314849
$581$	−10.7477	−0.445891
$582$	0	0
$583$	−4.74773	−0.196631
$584$	12.7477	0.527505
$585$	0	0
$586$	−27.4955	−1.13583
$587$	30.7913	1.27089	0.635446	−	0.772145i	$-0.280816\pi$
$587$	30.7913	1.27089	0.635446	+	0.772145i	$0.280816\pi$
$588$	0	0
$589$	−19.5390	−0.805091
$590$	13.5826	0.559186
$591$	0	0
$592$	−4.00000	−0.164399
$593$	31.9129	1.31050	0.655252	−	0.755410i	$-0.272562\pi$
$593$	31.9129	1.31050	0.655252	+	0.755410i	$0.272562\pi$
$594$	0	0
$595$	1.41742	0.0581087
$596$	−8.20871	−0.336242
$597$	0	0
$598$	−5.79129	−0.236823
$599$	−1.12159	−0.0458270	−0.0229135	−	0.999737i	$-0.507294\pi$
$599$	−1.12159	−0.0458270	−0.0229135	+	0.999737i	$0.507294\pi$
$600$	0	0
$601$	−18.2087	−0.742749	−0.371374	−	0.928483i	$-0.621113\pi$
$601$	−18.2087	−0.742749	−0.371374	+	0.928483i	$0.621113\pi$
$602$	−20.0000	−0.815139
$603$	0	0
$604$	−10.7913	−0.439091
$605$	−10.3739	−0.421758
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	5.79129	0.234868
$609$	0	0
$610$	10.3739	0.420025
$611$	25.5826	1.03496
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−15.5390	−0.627104
$615$	0	0
$616$	−1.41742	−0.0571097
$617$	23.8693	0.960943	0.480471	−	0.877010i	$-0.340466\pi$
$617$	23.8693	0.960943	0.480471	+	0.877010i	$0.340466\pi$
$618$	0	0
$619$	−1.79129	−0.0719979	−0.0359990	−	0.999352i	$-0.511461\pi$
$619$	−1.79129	−0.0719979	−0.0359990	+	0.999352i	$0.511461\pi$
$620$	−3.37386	−0.135498
$621$	0	0
$622$	−12.0000	−0.481156
$623$	27.1652	1.08835
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.62614	−0.184898
$627$	0	0
$628$	−14.7477	−0.588498
$629$	3.16515	0.126203
$630$	0	0
$631$	27.9129	1.11119	0.555597	−	0.831452i	$-0.312490\pi$
$631$	27.9129	1.11119	0.555597	+	0.831452i	$0.312490\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	−9.79129	−0.388862
$635$	12.7477	0.505878
$636$	0	0
$637$	−21.9564	−0.869946
$638$	−6.00000	−0.237542
$639$	0	0
$640$	1.00000	0.0395285
$641$	−15.1652	−0.598987	−0.299494	−	0.954098i	$-0.596818\pi$
$641$	−15.1652	−0.598987	−0.299494	+	0.954098i	$0.596818\pi$
$642$	0	0
$643$	6.74773	0.266104	0.133052	−	0.991109i	$-0.457522\pi$
$643$	6.74773	0.266104	0.133052	+	0.991109i	$0.457522\pi$
$644$	1.79129	0.0705866
$645$	0	0
$646$	−4.58258	−0.180299
$647$	−21.1652	−0.832088	−0.416044	−	0.909344i	$-0.636584\pi$
$647$	−21.1652	−0.832088	−0.416044	+	0.909344i	$0.636584\pi$
$648$	0	0
$649$	10.7477	0.421885
$650$	5.79129	0.227153
$651$	0	0
$652$	8.62614	0.337826
$653$	−3.46099	−0.135439	−0.0677194	−	0.997704i	$-0.521572\pi$
$653$	−3.46099	−0.135439	−0.0677194	+	0.997704i	$0.521572\pi$
$654$	0	0
$655$	9.16515	0.358112
$656$	6.79129	0.265155
$657$	0	0
$658$	−7.91288	−0.308476
$659$	8.83485	0.344157	0.172078	−	0.985083i	$-0.444952\pi$
$659$	8.83485	0.344157	0.172078	+	0.985083i	$0.444952\pi$
$660$	0	0
$661$	−25.6261	−0.996741	−0.498371	−	0.866964i	$-0.666068\pi$
$661$	−25.6261	−0.996741	−0.498371	+	0.866964i	$0.666068\pi$
$662$	−20.7477	−0.806383
$663$	0	0
$664$	6.00000	0.232845
$665$	−10.3739	−0.402281
$666$	0	0
$667$	7.58258	0.293599
$668$	−18.3303	−0.709221
$669$	0	0
$670$	11.1652	0.431347
$671$	8.20871	0.316894
$672$	0	0
$673$	38.0000	1.46479	0.732396	−	0.680879i	$-0.238402\pi$
$673$	38.0000	1.46479	0.732396	+	0.680879i	$0.238402\pi$
$674$	−12.2087	−0.470262
$675$	0	0
$676$	20.5390	0.789962
$677$	−42.6606	−1.63958	−0.819790	−	0.572664i	$-0.805910\pi$
$677$	−42.6606	−1.63958	−0.819790	+	0.572664i	$0.805910\pi$
$678$	0	0
$679$	14.2523	0.546952
$680$	−0.791288	−0.0303445
$681$	0	0
$682$	−2.66970	−0.102228
$683$	11.3739	0.435209	0.217604	−	0.976037i	$-0.430176\pi$
$683$	11.3739	0.435209	0.217604	+	0.976037i	$0.430176\pi$
$684$	0	0
$685$	−3.79129	−0.144858
$686$	19.3303	0.738034
$687$	0	0
$688$	11.1652	0.425667
$689$	−34.7477	−1.32378
$690$	0	0
$691$	42.7477	1.62620	0.813100	−	0.582124i	$-0.197778\pi$
$691$	42.7477	1.62620	0.813100	+	0.582124i	$0.197778\pi$
$692$	18.7913	0.714338
$693$	0	0
$694$	−5.20871	−0.197720
$695$	12.7477	0.483549
$696$	0	0
$697$	−5.37386	−0.203550
$698$	26.0000	0.984115
$699$	0	0
$700$	−1.79129	−0.0677043
$701$	−23.3739	−0.882819	−0.441409	−	0.897306i	$-0.645521\pi$
$701$	−23.3739	−0.882819	−0.441409	+	0.897306i	$0.645521\pi$
$702$	0	0
$703$	−23.1652	−0.873690
$704$	0.791288	0.0298228
$705$	0	0
$706$	−3.16515	−0.119122
$707$	7.91288	0.297594
$708$	0	0
$709$	2.46099	0.0924242	0.0462121	−	0.998932i	$-0.485285\pi$
$709$	2.46099	0.0924242	0.0462121	+	0.998932i	$0.485285\pi$
$710$	−8.37386	−0.314265
$711$	0	0
$712$	−15.1652	−0.568338
$713$	3.37386	0.126352
$714$	0	0
$715$	4.58258	0.171379
$716$	−10.7477	−0.401661
$717$	0	0
$718$	−9.16515	−0.342040
$719$	−2.53901	−0.0946893	−0.0473446	−	0.998879i	$-0.515076\pi$
$719$	−2.53901	−0.0946893	−0.0473446	+	0.998879i	$0.515076\pi$
$720$	0	0
$721$	11.4174	0.425207
$722$	14.5390	0.541086
$723$	0	0
$724$	−18.5390	−0.688997
$725$	−7.58258	−0.281610
$726$	0	0
$727$	39.1216	1.45094	0.725470	−	0.688254i	$-0.241623\pi$
$727$	39.1216	1.45094	0.725470	+	0.688254i	$0.241623\pi$
$728$	−10.3739	−0.384481
$729$	0	0
$730$	12.7477	0.471815
$731$	−8.83485	−0.326769
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−19.1652	−0.707399
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	8.83485	0.325436
$738$	0	0
$739$	8.00000	0.294285	0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000	0.294285	0.147142	+	0.989115i	$0.452992\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	10.7477	0.394561
$743$	12.9564	0.475326	0.237663	−	0.971348i	$-0.423619\pi$
$743$	12.9564	0.475326	0.237663	+	0.971348i	$0.423619\pi$
$744$	0	0
$745$	−8.20871	−0.300744
$746$	12.7477	0.466727
$747$	0	0
$748$	−0.626136	−0.0228938
$749$	7.91288	0.289130
$750$	0	0
$751$	−8.74773	−0.319209	−0.159605	−	0.987181i	$-0.551022\pi$
$751$	−8.74773	−0.319209	−0.159605	+	0.987181i	$0.551022\pi$
$752$	4.41742	0.161087
$753$	0	0
$754$	−43.9129	−1.59921
$755$	−10.7913	−0.392735
$756$	0	0
$757$	−10.3303	−0.375461	−0.187731	−	0.982221i	$-0.560113\pi$
$757$	−10.3303	−0.375461	−0.187731	+	0.982221i	$0.560113\pi$
$758$	−6.37386	−0.231509
$759$	0	0
$760$	5.79129	0.210072
$761$	11.0436	0.400329	0.200164	−	0.979762i	$-0.435852\pi$
$761$	11.0436	0.400329	0.200164	+	0.979762i	$0.435852\pi$
$762$	0	0
$763$	6.04356	0.218792
$764$	−25.5826	−0.925545
$765$	0	0
$766$	24.0000	0.867155
$767$	78.6606	2.84027
$768$	0	0
$769$	−40.3303	−1.45435	−0.727174	−	0.686453i	$-0.759167\pi$
$769$	−40.3303	−1.45435	−0.727174	+	0.686453i	$0.759167\pi$
$770$	−1.41742	−0.0510804
$771$	0	0
$772$	−20.7477	−0.746727
$773$	−33.4955	−1.20475	−0.602374	−	0.798214i	$-0.705778\pi$
$773$	−33.4955	−1.20475	−0.602374	+	0.798214i	$0.705778\pi$
$774$	0	0
$775$	−3.37386	−0.121193
$776$	−7.95644	−0.285620
$777$	0	0
$778$	20.7042	0.742280
$779$	39.3303	1.40915
$780$	0	0
$781$	−6.62614	−0.237102
$782$	0.791288	0.0282964
$783$	0	0
$784$	−3.79129	−0.135403
$785$	−14.7477	−0.526369
$786$	0	0
$787$	−17.5826	−0.626751	−0.313376	−	0.949629i	$-0.601460\pi$
$787$	−17.5826	−0.626751	−0.313376	+	0.949629i	$0.601460\pi$
$788$	11.5390	0.411060
$789$	0	0
$790$	8.00000	0.284627
$791$	10.7477	0.382145
$792$	0	0
$793$	60.0780	2.13343
$794$	−15.5390	−0.551459
$795$	0	0
$796$	−16.3303	−0.578812
$797$	4.08712	0.144773	0.0723866	−	0.997377i	$-0.476938\pi$
$797$	4.08712	0.144773	0.0723866	+	0.997377i	$0.476938\pi$
$798$	0	0
$799$	−3.49545	−0.123660
$800$	1.00000	0.0353553
$801$	0	0
$802$	4.74773	0.167648
$803$	10.0871	0.355967
$804$	0	0
$805$	1.79129	0.0631346
$806$	−19.5390	−0.688232
$807$	0	0
$808$	−4.41742	−0.155404
$809$	−33.9564	−1.19384	−0.596922	−	0.802299i	$-0.703610\pi$
$809$	−33.9564	−1.19384	−0.596922	+	0.802299i	$0.703610\pi$
$810$	0	0
$811$	−2.08712	−0.0732887	−0.0366444	−	0.999328i	$-0.511667\pi$
$811$	−2.08712	−0.0732887	−0.0366444	+	0.999328i	$0.511667\pi$
$812$	13.5826	0.476655
$813$	0	0
$814$	−3.16515	−0.110938
$815$	8.62614	0.302160
$816$	0	0
$817$	64.6606	2.26219
$818$	−18.2087	−0.636653
$819$	0	0
$820$	6.79129	0.237162
$821$	−21.1652	−0.738669	−0.369334	−	0.929297i	$-0.620414\pi$
$821$	−21.1652	−0.738669	−0.369334	+	0.929297i	$0.620414\pi$
$822$	0	0
$823$	22.8348	0.795973	0.397986	−	0.917391i	$-0.369709\pi$
$823$	22.8348	0.795973	0.397986	+	0.917391i	$0.369709\pi$
$824$	−6.37386	−0.222044
$825$	0	0
$826$	−24.3303	−0.846560
$827$	23.0780	0.802502	0.401251	−	0.915968i	$-0.368576\pi$
$827$	23.0780	0.802502	0.401251	+	0.915968i	$0.368576\pi$
$828$	0	0
$829$	23.4955	0.816031	0.408015	−	0.912975i	$-0.366221\pi$
$829$	23.4955	0.816031	0.408015	+	0.912975i	$0.366221\pi$
$830$	6.00000	0.208263
$831$	0	0
$832$	5.79129	0.200777
$833$	3.00000	0.103944
$834$	0	0
$835$	−18.3303	−0.634346
$836$	4.58258	0.158492
$837$	0	0
$838$	−20.8348	−0.719728
$839$	31.5826	1.09035	0.545176	−	0.838322i	$-0.316463\pi$
$839$	31.5826	1.09035	0.545176	+	0.838322i	$0.316463\pi$
$840$	0	0
$841$	28.4955	0.982602
$842$	18.1216	0.624511
$843$	0	0
$844$	−10.0000	−0.344214
$845$	20.5390	0.706564
$846$	0	0
$847$	18.5826	0.638505
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−0.791288	−0.0271409
$851$	4.00000	0.137118
$852$	0	0
$853$	40.5390	1.38803	0.694015	−	0.719961i	$-0.255840\pi$
$853$	40.5390	1.38803	0.694015	+	0.719961i	$0.255840\pi$
$854$	−18.5826	−0.635883
$855$	0	0
$856$	−4.41742	−0.150984
$857$	9.16515	0.313076	0.156538	−	0.987672i	$-0.449967\pi$
$857$	9.16515	0.313076	0.156538	+	0.987672i	$0.449967\pi$
$858$	0	0
$859$	−26.7477	−0.912621	−0.456310	−	0.889821i	$-0.650829\pi$
$859$	−26.7477	−0.912621	−0.456310	+	0.889821i	$0.650829\pi$
$860$	11.1652	0.380729
$861$	0	0
$862$	−25.9129	−0.882596
$863$	22.4174	0.763098	0.381549	−	0.924349i	$-0.375391\pi$
$863$	22.4174	0.763098	0.381549	+	0.924349i	$0.375391\pi$
$864$	0	0
$865$	18.7913	0.638923
$866$	−30.5390	−1.03776
$867$	0	0
$868$	6.04356	0.205132
$869$	6.33030	0.214741
$870$	0	0
$871$	64.6606	2.19094
$872$	−3.37386	−0.114253
$873$	0	0
$874$	−5.79129	−0.195893
$875$	−1.79129	−0.0605566
$876$	0	0
$877$	−42.7042	−1.44202	−0.721009	−	0.692926i	$-0.756321\pi$
$877$	−42.7042	−1.44202	−0.721009	+	0.692926i	$0.756321\pi$
$878$	−6.53901	−0.220681
$879$	0	0
$880$	0.791288	0.0266743
$881$	−30.3303	−1.02185	−0.510927	−	0.859624i	$-0.670698\pi$
$881$	−30.3303	−1.02185	−0.510927	+	0.859624i	$0.670698\pi$
$882$	0	0
$883$	−34.9564	−1.17638	−0.588189	−	0.808724i	$-0.700159\pi$
$883$	−34.9564	−1.17638	−0.588189	+	0.808724i	$0.700159\pi$
$884$	−4.58258	−0.154129
$885$	0	0
$886$	39.7913	1.33681
$887$	−15.1652	−0.509196	−0.254598	−	0.967047i	$-0.581943\pi$
$887$	−15.1652	−0.509196	−0.254598	+	0.967047i	$0.581943\pi$
$888$	0	0
$889$	−22.8348	−0.765856
$890$	−15.1652	−0.508337
$891$	0	0
$892$	−7.16515	−0.239907
$893$	25.5826	0.856088
$894$	0	0
$895$	−10.7477	−0.359257
$896$	−1.79129	−0.0598427
$897$	0	0
$898$	16.1216	0.537984
$899$	25.5826	0.853227
$900$	0	0
$901$	4.74773	0.158170
$902$	5.37386	0.178930
$903$	0	0
$904$	−6.00000	−0.199557
$905$	−18.5390	−0.616258
$906$	0	0
$907$	6.74773	0.224055	0.112027	−	0.993705i	$-0.464266\pi$
$907$	6.74773	0.224055	0.112027	+	0.993705i	$0.464266\pi$
$908$	22.7477	0.754910
$909$	0	0
$910$	−10.3739	−0.343890
$911$	−4.41742	−0.146356	−0.0731779	−	0.997319i	$-0.523314\pi$
$911$	−4.41742	−0.146356	−0.0731779	+	0.997319i	$0.523314\pi$
$912$	0	0
$913$	4.74773	0.157127
$914$	−10.0000	−0.330771
$915$	0	0
$916$	20.3303	0.671732
$917$	−16.4174	−0.542151
$918$	0	0
$919$	−55.1652	−1.81973	−0.909865	−	0.414904i	$-0.863815\pi$
$919$	−55.1652	−1.81973	−0.909865	+	0.414904i	$0.863815\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	28.7477	0.946756
$923$	−48.4955	−1.59625
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−10.0000	−0.328620
$927$	0	0
$928$	−7.58258	−0.248910
$929$	−15.4955	−0.508389	−0.254195	−	0.967153i	$-0.581810\pi$
$929$	−15.4955	−0.508389	−0.254195	+	0.967153i	$0.581810\pi$
$930$	0	0
$931$	−21.9564	−0.719593
$932$	1.58258	0.0518390
$933$	0	0
$934$	19.9129	0.651569
$935$	−0.626136	−0.0204769
$936$	0	0
$937$	44.6261	1.45787	0.728936	−	0.684582i	$-0.240015\pi$
$937$	44.6261	1.45787	0.728936	+	0.684582i	$0.240015\pi$
$938$	−20.0000	−0.653023
$939$	0	0
$940$	4.41742	0.144080
$941$	−32.0436	−1.04459	−0.522295	−	0.852765i	$-0.674924\pi$
$941$	−32.0436	−1.04459	−0.522295	+	0.852765i	$0.674924\pi$
$942$	0	0
$943$	−6.79129	−0.221155
$944$	13.5826	0.442075
$945$	0	0
$946$	8.83485	0.287246
$947$	−2.53901	−0.0825069	−0.0412534	−	0.999149i	$-0.513135\pi$
$947$	−2.53901	−0.0825069	−0.0412534	+	0.999149i	$0.513135\pi$
$948$	0	0
$949$	73.8258	2.39649
$950$	5.79129	0.187894
$951$	0	0
$952$	1.41742	0.0459390
$953$	−5.53901	−0.179426	−0.0897131	−	0.995968i	$-0.528595\pi$
$953$	−5.53901	−0.179426	−0.0897131	+	0.995968i	$0.528595\pi$
$954$	0	0
$955$	−25.5826	−0.827833
$956$	15.1652	0.490476
$957$	0	0
$958$	−15.4955	−0.500635
$959$	6.79129	0.219302
$960$	0	0
$961$	−19.6170	−0.632808
$962$	−23.1652	−0.746874
$963$	0	0
$964$	−28.0000	−0.901819
$965$	−20.7477	−0.667893
$966$	0	0
$967$	−32.7477	−1.05310	−0.526548	−	0.850145i	$-0.676514\pi$
$967$	−32.7477	−1.05310	−0.526548	+	0.850145i	$0.676514\pi$
$968$	−10.3739	−0.333429
$969$	0	0
$970$	−7.95644	−0.255466
$971$	−15.9564	−0.512067	−0.256033	−	0.966668i	$-0.582416\pi$
$971$	−15.9564	−0.512067	−0.256033	+	0.966668i	$0.582416\pi$
$972$	0	0
$973$	−22.8348	−0.732052
$974$	6.41742	0.205628
$975$	0	0
$976$	10.3739	0.332059
$977$	34.1216	1.09165	0.545823	−	0.837900i	$-0.316217\pi$
$977$	34.1216	1.09165	0.545823	+	0.837900i	$0.316217\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	−3.79129	−0.121108
$981$	0	0
$982$	−10.7477	−0.342974
$983$	14.3739	0.458455	0.229228	−	0.973373i	$-0.426380\pi$
$983$	14.3739	0.458455	0.229228	+	0.973373i	$0.426380\pi$
$984$	0	0
$985$	11.5390	0.367664
$986$	6.00000	0.191079
$987$	0	0
$988$	33.5390	1.06702
$989$	−11.1652	−0.355031
$990$	0	0
$991$	−33.2087	−1.05491	−0.527455	−	0.849583i	$-0.676854\pi$
$991$	−33.2087	−1.05491	−0.527455	+	0.849583i	$0.676854\pi$
$992$	−3.37386	−0.107120
$993$	0	0
$994$	15.0000	0.475771
$995$	−16.3303	−0.517705
$996$	0	0
$997$	−43.4955	−1.37751	−0.688757	−	0.724992i	$-0.741843\pi$
$997$	−43.4955	−1.37751	−0.688757	+	0.724992i	$0.741843\pi$
$998$	4.83485	0.153044
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.x.1.1		2
3.2	odd	2		230.2.a.a.1.1	✓	2
12.11	even	2		1840.2.a.n.1.2		2
15.2	even	4		1150.2.b.g.599.2		4
15.8	even	4		1150.2.b.g.599.3		4
15.14	odd	2		1150.2.a.o.1.2		2
24.5	odd	2		7360.2.a.bq.1.2		2
24.11	even	2		7360.2.a.bk.1.1		2
60.59	even	2		9200.2.a.bs.1.1		2
69.68	even	2		5290.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.a.1.1	✓	2	3.2	odd	2
1150.2.a.o.1.2		2	15.14	odd	2
1150.2.b.g.599.2		4	15.2	even	4
1150.2.b.g.599.3		4	15.8	even	4
1840.2.a.n.1.2		2	12.11	even	2
2070.2.a.x.1.1		2	1.1	even	1	trivial
5290.2.a.e.1.1		2	69.68	even	2
7360.2.a.bk.1.1		2	24.11	even	2
7360.2.a.bq.1.2		2	24.5	odd	2
9200.2.a.bs.1.1		2	60.59	even	2

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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} +0.791288 q^{11} +5.79129 q^{13} -1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +5.79129 q^{19} +1.00000 q^{20} +0.791288 q^{22} -1.00000 q^{23} +1.00000 q^{25} +5.79129 q^{26} -1.79129 q^{28} -7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -0.791288 q^{34} -1.79129 q^{35} -4.00000 q^{37} +5.79129 q^{38} +1.00000 q^{40} +6.79129 q^{41} +11.1652 q^{43} +0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} -3.79129 q^{49} +1.00000 q^{50} +5.79129 q^{52} -6.00000 q^{53} +0.791288 q^{55} -1.79129 q^{56} -7.58258 q^{58} +13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +1.00000 q^{64} +5.79129 q^{65} +11.1652 q^{67} -0.791288 q^{68} -1.79129 q^{70} -8.37386 q^{71} +12.7477 q^{73} -4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} +8.00000 q^{79} +1.00000 q^{80} +6.79129 q^{82} +6.00000 q^{83} -0.791288 q^{85} +11.1652 q^{86} +0.791288 q^{88} -15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} +4.41742 q^{94} +5.79129 q^{95} -7.95644 q^{97} -3.79129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 2 q^{20} - 3 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + q^{28} - 6 q^{29} + 7 q^{31} + 2 q^{32} + 3 q^{34} + q^{35} - 8 q^{37} + 7 q^{38} + 2 q^{40} + 9 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 7 q^{52} - 12 q^{53} - 3 q^{55} + q^{56} - 6 q^{58} + 18 q^{59} + 7 q^{61} + 7 q^{62} + 2 q^{64} + 7 q^{65} + 4 q^{67} + 3 q^{68} + q^{70} - 3 q^{71} - 2 q^{73} - 8 q^{74} + 7 q^{76} - 12 q^{77} + 16 q^{79} + 2 q^{80} + 9 q^{82} + 12 q^{83} + 3 q^{85} + 4 q^{86} - 3 q^{88} - 12 q^{89} - 7 q^{91} - 2 q^{92} + 18 q^{94} + 7 q^{95} + 7 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 2 * q^20 - 3 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + q^28 - 6 * q^29 + 7 * q^31 + 2 * q^32 + 3 * q^34 + q^35 - 8 * q^37 + 7 * q^38 + 2 * q^40 + 9 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 7 * q^52 - 12 * q^53 - 3 * q^55 + q^56 - 6 * q^58 + 18 * q^59 + 7 * q^61 + 7 * q^62 + 2 * q^64 + 7 * q^65 + 4 * q^67 + 3 * q^68 + q^70 - 3 * q^71 - 2 * q^73 - 8 * q^74 + 7 * q^76 - 12 * q^77 + 16 * q^79 + 2 * q^80 + 9 * q^82 + 12 * q^83 + 3 * q^85 + 4 * q^86 - 3 * q^88 - 12 * q^89 - 7 * q^91 - 2 * q^92 + 18 * q^94 + 7 * q^95 + 7 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more