LMFDB - Embedded newform 3006.2.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3006,2,Mod(1,3006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3006, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3006 = 2 \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3006.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:	$24.0030308476$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 20x^{5} + 2x^{4} + 87x^{3} + 46x^{2} - 48x - 2 $ x^7 - 2x^6 - 20x^5 + 2x^4 + 87x^3 + 46x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.29679$ of defining polynomial
Character	$\chi$	$=$	3006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.29679 q^{5} -1.34851 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.29679 q^{5} -1.34851 q^{7} -1.00000 q^{8} +3.29679 q^{10} -5.12251 q^{11} -1.72691 q^{13} +1.34851 q^{14} +1.00000 q^{16} +0.817380 q^{17} -8.07208 q^{19} -3.29679 q^{20} +5.12251 q^{22} +4.72957 q^{23} +5.86881 q^{25} +1.72691 q^{26} -1.34851 q^{28} -7.99732 q^{29} -6.07808 q^{31} -1.00000 q^{32} -0.817380 q^{34} +4.44575 q^{35} -6.06922 q^{37} +8.07208 q^{38} +3.29679 q^{40} +4.77243 q^{41} +5.73329 q^{43} -5.12251 q^{44} -4.72957 q^{46} -9.77133 q^{47} -5.18152 q^{49} -5.86881 q^{50} -1.72691 q^{52} +1.53622 q^{53} +16.8878 q^{55} +1.34851 q^{56} +7.99732 q^{58} -3.30317 q^{59} +13.7653 q^{61} +6.07808 q^{62} +1.00000 q^{64} +5.69325 q^{65} -14.9508 q^{67} +0.817380 q^{68} -4.44575 q^{70} +12.2868 q^{71} +7.41202 q^{73} +6.06922 q^{74} -8.07208 q^{76} +6.90776 q^{77} -10.3474 q^{79} -3.29679 q^{80} -4.77243 q^{82} -5.92397 q^{83} -2.69473 q^{85} -5.73329 q^{86} +5.12251 q^{88} +5.19016 q^{89} +2.32875 q^{91} +4.72957 q^{92} +9.77133 q^{94} +26.6119 q^{95} +1.05901 q^{97} +5.18152 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} - 7 q^{8} + 5 q^{10} + 6 q^{13} - 7 q^{14} + 7 q^{16} - 6 q^{17} - 2 q^{19} - 5 q^{20} + 12 q^{25} - 6 q^{26} + 7 q^{28} + 4 q^{29} + 7 q^{31} - 7 q^{32} + 6 q^{34} + 13 q^{35} - 3 q^{37} + 2 q^{38} + 5 q^{40} + 12 q^{41} - 2 q^{43} + 11 q^{47} + 10 q^{49} - 12 q^{50} + 6 q^{52} - q^{53} - 2 q^{55} - 7 q^{56} - 4 q^{58} + 19 q^{59} + 12 q^{61} - 7 q^{62} + 7 q^{64} + 10 q^{65} - 17 q^{67} - 6 q^{68} - 13 q^{70} + 20 q^{71} + 10 q^{73} + 3 q^{74} - 2 q^{76} + 24 q^{77} + 2 q^{79} - 5 q^{80} - 12 q^{82} + 7 q^{83} - 18 q^{85} + 2 q^{86} + 3 q^{89} + 4 q^{91} - 11 q^{94} + 24 q^{95} - 3 q^{97} - 10 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 - 7 * q^8 + 5 * q^10 + 6 * q^13 - 7 * q^14 + 7 * q^16 - 6 * q^17 - 2 * q^19 - 5 * q^20 + 12 * q^25 - 6 * q^26 + 7 * q^28 + 4 * q^29 + 7 * q^31 - 7 * q^32 + 6 * q^34 + 13 * q^35 - 3 * q^37 + 2 * q^38 + 5 * q^40 + 12 * q^41 - 2 * q^43 + 11 * q^47 + 10 * q^49 - 12 * q^50 + 6 * q^52 - q^53 - 2 * q^55 - 7 * q^56 - 4 * q^58 + 19 * q^59 + 12 * q^61 - 7 * q^62 + 7 * q^64 + 10 * q^65 - 17 * q^67 - 6 * q^68 - 13 * q^70 + 20 * q^71 + 10 * q^73 + 3 * q^74 - 2 * q^76 + 24 * q^77 + 2 * q^79 - 5 * q^80 - 12 * q^82 + 7 * q^83 - 18 * q^85 + 2 * q^86 + 3 * q^89 + 4 * q^91 - 11 * q^94 + 24 * q^95 - 3 * q^97 - 10 * 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
$5$	−3.29679	−1.47437	−0.737184	−	0.675692i	$-0.763845\pi$
$5$	−3.29679	−1.47437	−0.737184	+	0.675692i	$0.763845\pi$
$6$	0	0
$7$	−1.34851	−0.509689	−0.254844	−	0.966982i	$-0.582024\pi$
$7$	−1.34851	−0.509689	−0.254844	+	0.966982i	$0.582024\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.29679	1.04254
$11$	−5.12251	−1.54450	−0.772248	−	0.635321i	$-0.780868\pi$
$11$	−5.12251	−1.54450	−0.772248	+	0.635321i	$0.780868\pi$
$12$	0	0
$13$	−1.72691	−0.478958	−0.239479	−	0.970901i	$-0.576977\pi$
$13$	−1.72691	−0.478958	−0.239479	+	0.970901i	$0.576977\pi$
$14$	1.34851	0.360404
$15$	0	0
$16$	1.00000	0.250000
$17$	0.817380	0.198244	0.0991219	−	0.995075i	$-0.468397\pi$
$17$	0.817380	0.198244	0.0991219	+	0.995075i	$0.468397\pi$
$18$	0	0
$19$	−8.07208	−1.85186	−0.925931	−	0.377693i	$-0.876717\pi$
$19$	−8.07208	−1.85186	−0.925931	+	0.377693i	$0.876717\pi$
$20$	−3.29679	−0.737184
$21$	0	0
$22$	5.12251	1.09212
$23$	4.72957	0.986183	0.493091	−	0.869978i	$-0.335867\pi$
$23$	4.72957	0.986183	0.493091	+	0.869978i	$0.335867\pi$
$24$	0	0
$25$	5.86881	1.17376
$26$	1.72691	0.338675
$27$	0	0
$28$	−1.34851	−0.254844
$29$	−7.99732	−1.48507	−0.742533	−	0.669810i	$-0.766376\pi$
$29$	−7.99732	−1.48507	−0.742533	+	0.669810i	$0.766376\pi$
$30$	0	0
$31$	−6.07808	−1.09165	−0.545827	−	0.837898i	$-0.683785\pi$
$31$	−6.07808	−1.09165	−0.545827	+	0.837898i	$0.683785\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.817380	−0.140180
$35$	4.44575	0.751469
$36$	0	0
$37$	−6.06922	−0.997773	−0.498887	−	0.866667i	$-0.666258\pi$
$37$	−6.06922	−0.997773	−0.498887	+	0.866667i	$0.666258\pi$
$38$	8.07208	1.30946
$39$	0	0
$40$	3.29679	0.521268
$41$	4.77243	0.745328	0.372664	−	0.927966i	$-0.378444\pi$
$41$	4.77243	0.745328	0.372664	+	0.927966i	$0.378444\pi$
$42$	0	0
$43$	5.73329	0.874318	0.437159	−	0.899384i	$-0.355985\pi$
$43$	5.73329	0.874318	0.437159	+	0.899384i	$0.355985\pi$
$44$	−5.12251	−0.772248
$45$	0	0
$46$	−4.72957	−0.697337
$47$	−9.77133	−1.42529	−0.712647	−	0.701522i	$-0.752504\pi$
$47$	−9.77133	−1.42529	−0.712647	+	0.701522i	$0.752504\pi$
$48$	0	0
$49$	−5.18152	−0.740217
$50$	−5.86881	−0.829975
$51$	0	0
$52$	−1.72691	−0.239479
$53$	1.53622	0.211016	0.105508	−	0.994418i	$-0.466353\pi$
$53$	1.53622	0.211016	0.105508	+	0.994418i	$0.466353\pi$
$54$	0	0
$55$	16.8878	2.27716
$56$	1.34851	0.180202
$57$	0	0
$58$	7.99732	1.05010
$59$	−3.30317	−0.430035	−0.215018	−	0.976610i	$-0.568981\pi$
$59$	−3.30317	−0.430035	−0.215018	+	0.976610i	$0.568981\pi$
$60$	0	0
$61$	13.7653	1.76247	0.881235	−	0.472679i	$-0.156713\pi$
$61$	13.7653	1.76247	0.881235	+	0.472679i	$0.156713\pi$
$62$	6.07808	0.771916
$63$	0	0
$64$	1.00000	0.125000
$65$	5.69325	0.706161
$66$	0	0
$67$	−14.9508	−1.82653	−0.913266	−	0.407364i	$-0.866448\pi$
$67$	−14.9508	−1.82653	−0.913266	+	0.407364i	$0.866448\pi$
$68$	0.817380	0.0991219
$69$	0	0
$70$	−4.44575	−0.531369
$71$	12.2868	1.45818	0.729089	−	0.684419i	$-0.239944\pi$
$71$	12.2868	1.45818	0.729089	+	0.684419i	$0.239944\pi$
$72$	0	0
$73$	7.41202	0.867511	0.433756	−	0.901031i	$-0.357188\pi$
$73$	7.41202	0.867511	0.433756	+	0.901031i	$0.357188\pi$
$74$	6.06922	0.705532
$75$	0	0
$76$	−8.07208	−0.925931
$77$	6.90776	0.787213
$78$	0	0
$79$	−10.3474	−1.16417	−0.582087	−	0.813126i	$-0.697764\pi$
$79$	−10.3474	−1.16417	−0.582087	+	0.813126i	$0.697764\pi$
$80$	−3.29679	−0.368592
$81$	0	0
$82$	−4.77243	−0.527027
$83$	−5.92397	−0.650240	−0.325120	−	0.945673i	$-0.605405\pi$
$83$	−5.92397	−0.650240	−0.325120	+	0.945673i	$0.605405\pi$
$84$	0	0
$85$	−2.69473	−0.292284
$86$	−5.73329	−0.618236
$87$	0	0
$88$	5.12251	0.546062
$89$	5.19016	0.550156	0.275078	−	0.961422i	$-0.411296\pi$
$89$	5.19016	0.550156	0.275078	+	0.961422i	$0.411296\pi$
$90$	0	0
$91$	2.32875	0.244120
$92$	4.72957	0.493091
$93$	0	0
$94$	9.77133	1.00784
$95$	26.6119	2.73033
$96$	0	0
$97$	1.05901	0.107526	0.0537629	−	0.998554i	$-0.482878\pi$
$97$	1.05901	0.107526	0.0537629	+	0.998554i	$0.482878\pi$
$98$	5.18152	0.523413
$99$	0	0
$100$	5.86881	0.586881
$101$	−8.60536	−0.856266	−0.428133	−	0.903716i	$-0.640828\pi$
$101$	−8.60536	−0.856266	−0.428133	+	0.903716i	$0.640828\pi$
$102$	0	0
$103$	14.3156	1.41055	0.705277	−	0.708932i	$-0.250822\pi$
$103$	14.3156	1.41055	0.705277	+	0.708932i	$0.250822\pi$
$104$	1.72691	0.169337
$105$	0	0
$106$	−1.53622	−0.149211
$107$	−4.20807	−0.406809	−0.203405	−	0.979095i	$-0.565201\pi$
$107$	−4.20807	−0.406809	−0.203405	+	0.979095i	$0.565201\pi$
$108$	0	0
$109$	−5.45271	−0.522275	−0.261137	−	0.965302i	$-0.584098\pi$
$109$	−5.45271	−0.522275	−0.261137	+	0.965302i	$0.584098\pi$
$110$	−16.8878	−1.61019
$111$	0	0
$112$	−1.34851	−0.127422
$113$	−4.52520	−0.425695	−0.212848	−	0.977085i	$-0.568274\pi$
$113$	−4.52520	−0.425695	−0.212848	+	0.977085i	$0.568274\pi$
$114$	0	0
$115$	−15.5924	−1.45400
$116$	−7.99732	−0.742533
$117$	0	0
$118$	3.30317	0.304081
$119$	−1.10225	−0.101043
$120$	0	0
$121$	15.2402	1.38547
$122$	−13.7653	−1.24625
$123$	0	0
$124$	−6.07808	−0.545827
$125$	−2.86428	−0.256189
$126$	0	0
$127$	9.79684	0.869329	0.434665	−	0.900592i	$-0.356867\pi$
$127$	9.79684	0.869329	0.434665	+	0.900592i	$0.356867\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−5.69325	−0.499331
$131$	10.4983	0.917239	0.458619	−	0.888633i	$-0.348344\pi$
$131$	10.4983	0.917239	0.458619	+	0.888633i	$0.348344\pi$
$132$	0	0
$133$	10.8853	0.943873
$134$	14.9508	1.29155
$135$	0	0
$136$	−0.817380	−0.0700898
$137$	15.0021	1.28172	0.640859	−	0.767659i	$-0.278578\pi$
$137$	15.0021	1.28172	0.640859	+	0.767659i	$0.278578\pi$
$138$	0	0
$139$	−6.00399	−0.509252	−0.254626	−	0.967040i	$-0.581952\pi$
$139$	−6.00399	−0.509252	−0.254626	+	0.967040i	$0.581952\pi$
$140$	4.44575	0.375735
$141$	0	0
$142$	−12.2868	−1.03109
$143$	8.84612	0.739750
$144$	0	0
$145$	26.3655	2.18953
$146$	−7.41202	−0.613423
$147$	0	0
$148$	−6.06922	−0.498887
$149$	−4.39903	−0.360383	−0.180191	−	0.983632i	$-0.557672\pi$
$149$	−4.39903	−0.360383	−0.180191	+	0.983632i	$0.557672\pi$
$150$	0	0
$151$	−6.58539	−0.535911	−0.267956	−	0.963431i	$-0.586348\pi$
$151$	−6.58539	−0.535911	−0.267956	+	0.963431i	$0.586348\pi$
$152$	8.07208	0.654732
$153$	0	0
$154$	−6.90776	−0.556643
$155$	20.0381	1.60950
$156$	0	0
$157$	23.5431	1.87894	0.939471	−	0.342630i	$-0.111318\pi$
$157$	23.5431	1.87894	0.939471	+	0.342630i	$0.111318\pi$
$158$	10.3474	0.823196
$159$	0	0
$160$	3.29679	0.260634
$161$	−6.37787	−0.502646
$162$	0	0
$163$	−2.57466	−0.201663	−0.100832	−	0.994904i	$-0.532150\pi$
$163$	−2.57466	−0.201663	−0.100832	+	0.994904i	$0.532150\pi$
$164$	4.77243	0.372664
$165$	0	0
$166$	5.92397	0.459789
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−10.0178	−0.770599
$170$	2.69473	0.206676
$171$	0	0
$172$	5.73329	0.437159
$173$	−15.9801	−1.21494	−0.607472	−	0.794341i	$-0.707816\pi$
$173$	−15.9801	−1.21494	−0.607472	+	0.794341i	$0.707816\pi$
$174$	0	0
$175$	−7.91415	−0.598253
$176$	−5.12251	−0.386124
$177$	0	0
$178$	−5.19016	−0.389019
$179$	16.0998	1.20335	0.601677	−	0.798739i	$-0.294499\pi$
$179$	16.0998	1.20335	0.601677	+	0.798739i	$0.294499\pi$
$180$	0	0
$181$	0.401921	0.0298746	0.0149373	−	0.999888i	$-0.495245\pi$
$181$	0.401921	0.0298746	0.0149373	+	0.999888i	$0.495245\pi$
$182$	−2.32875	−0.172619
$183$	0	0
$184$	−4.72957	−0.348668
$185$	20.0089	1.47108
$186$	0	0
$187$	−4.18704	−0.306187
$188$	−9.77133	−0.712647
$189$	0	0
$190$	−26.6119	−1.93063
$191$	7.29514	0.527857	0.263929	−	0.964542i	$-0.414982\pi$
$191$	7.29514	0.527857	0.263929	+	0.964542i	$0.414982\pi$
$192$	0	0
$193$	−7.26549	−0.522981	−0.261491	−	0.965206i	$-0.584214\pi$
$193$	−7.26549	−0.522981	−0.261491	+	0.965206i	$0.584214\pi$
$194$	−1.05901	−0.0760322
$195$	0	0
$196$	−5.18152	−0.370109
$197$	1.73072	0.123309	0.0616543	−	0.998098i	$-0.480362\pi$
$197$	1.73072	0.123309	0.0616543	+	0.998098i	$0.480362\pi$
$198$	0	0
$199$	−24.9415	−1.76805	−0.884027	−	0.467436i	$-0.845178\pi$
$199$	−24.9415	−1.76805	−0.884027	+	0.467436i	$0.845178\pi$
$200$	−5.86881	−0.414988
$201$	0	0
$202$	8.60536	0.605471
$203$	10.7845	0.756922
$204$	0	0
$205$	−15.7337	−1.09889
$206$	−14.3156	−0.997413
$207$	0	0
$208$	−1.72691	−0.119740
$209$	41.3493	2.86019
$210$	0	0
$211$	19.4045	1.33586	0.667929	−	0.744225i	$-0.267181\pi$
$211$	19.4045	1.33586	0.667929	+	0.744225i	$0.267181\pi$
$212$	1.53622	0.105508
$213$	0	0
$214$	4.20807	0.287658
$215$	−18.9014	−1.28907
$216$	0	0
$217$	8.19635	0.556404
$218$	5.45271	0.369304
$219$	0	0
$220$	16.8878	1.13858
$221$	−1.41154	−0.0949506
$222$	0	0
$223$	−18.2857	−1.22450	−0.612250	−	0.790664i	$-0.709735\pi$
$223$	−18.2857	−1.22450	−0.612250	+	0.790664i	$0.709735\pi$
$224$	1.34851	0.0901011
$225$	0	0
$226$	4.52520	0.301012
$227$	3.37535	0.224030	0.112015	−	0.993707i	$-0.464270\pi$
$227$	3.37535	0.224030	0.112015	+	0.993707i	$0.464270\pi$
$228$	0	0
$229$	18.4505	1.21925	0.609623	−	0.792692i	$-0.291321\pi$
$229$	18.4505	1.21925	0.609623	+	0.792692i	$0.291321\pi$
$230$	15.5924	1.02813
$231$	0	0
$232$	7.99732	0.525050
$233$	5.56810	0.364778	0.182389	−	0.983226i	$-0.441617\pi$
$233$	5.56810	0.364778	0.182389	+	0.983226i	$0.441617\pi$
$234$	0	0
$235$	32.2140	2.10141
$236$	−3.30317	−0.215018
$237$	0	0
$238$	1.10225	0.0714480
$239$	24.9446	1.61353	0.806767	−	0.590870i	$-0.201215\pi$
$239$	24.9446	1.61353	0.806767	+	0.590870i	$0.201215\pi$
$240$	0	0
$241$	−20.0196	−1.28957	−0.644787	−	0.764362i	$-0.723054\pi$
$241$	−20.0196	−1.28957	−0.644787	+	0.764362i	$0.723054\pi$
$242$	−15.2402	−0.979674
$243$	0	0
$244$	13.7653	0.881235
$245$	17.0824	1.09135
$246$	0	0
$247$	13.9397	0.886965
$248$	6.07808	0.385958
$249$	0	0
$250$	2.86428	0.181153
$251$	−5.41418	−0.341740	−0.170870	−	0.985294i	$-0.554658\pi$
$251$	−5.41418	−0.341740	−0.170870	+	0.985294i	$0.554658\pi$
$252$	0	0
$253$	−24.2273	−1.52316
$254$	−9.79684	−0.614709
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.80813	0.299923	0.149961	−	0.988692i	$-0.452085\pi$
$257$	4.80813	0.299923	0.149961	+	0.988692i	$0.452085\pi$
$258$	0	0
$259$	8.18440	0.508554
$260$	5.69325	0.353081
$261$	0	0
$262$	−10.4983	−0.648586
$263$	−24.0805	−1.48487	−0.742434	−	0.669919i	$-0.766329\pi$
$263$	−24.0805	−1.48487	−0.742434	+	0.669919i	$0.766329\pi$
$264$	0	0
$265$	−5.06460	−0.311116
$266$	−10.8853	−0.667419
$267$	0	0
$268$	−14.9508	−0.913266
$269$	−25.8709	−1.57738	−0.788688	−	0.614794i	$-0.789239\pi$
$269$	−25.8709	−1.57738	−0.788688	+	0.614794i	$0.789239\pi$
$270$	0	0
$271$	−12.9853	−0.788798	−0.394399	−	0.918939i	$-0.629047\pi$
$271$	−12.9853	−0.788798	−0.394399	+	0.918939i	$0.629047\pi$
$272$	0.817380	0.0495610
$273$	0	0
$274$	−15.0021	−0.906312
$275$	−30.0631	−1.81287
$276$	0	0
$277$	−7.33571	−0.440760	−0.220380	−	0.975414i	$-0.570730\pi$
$277$	−7.33571	−0.440760	−0.220380	+	0.975414i	$0.570730\pi$
$278$	6.00399	0.360096
$279$	0	0
$280$	−4.44575	−0.265684
$281$	11.4575	0.683500	0.341750	−	0.939791i	$-0.388980\pi$
$281$	11.4575	0.683500	0.341750	+	0.939791i	$0.388980\pi$
$282$	0	0
$283$	19.4159	1.15415	0.577076	−	0.816690i	$-0.304194\pi$
$283$	19.4159	1.15415	0.577076	+	0.816690i	$0.304194\pi$
$284$	12.2868	0.729089
$285$	0	0
$286$	−8.84612	−0.523082
$287$	−6.43567	−0.379885
$288$	0	0
$289$	−16.3319	−0.960699
$290$	−26.3655	−1.54823
$291$	0	0
$292$	7.41202	0.433756
$293$	−6.23247	−0.364105	−0.182052	−	0.983289i	$-0.558274\pi$
$293$	−6.23247	−0.364105	−0.182052	+	0.983289i	$0.558274\pi$
$294$	0	0
$295$	10.8898	0.634031
$296$	6.06922	0.352766
$297$	0	0
$298$	4.39903	0.254829
$299$	−8.16753	−0.472341
$300$	0	0
$301$	−7.73139	−0.445630
$302$	6.58539	0.378946
$303$	0	0
$304$	−8.07208	−0.462965
$305$	−45.3814	−2.59853
$306$	0	0
$307$	26.7588	1.52720	0.763602	−	0.645687i	$-0.223429\pi$
$307$	26.7588	1.52720	0.763602	+	0.645687i	$0.223429\pi$
$308$	6.90776	0.393606
$309$	0	0
$310$	−20.0381	−1.13809
$311$	−4.94659	−0.280495	−0.140248	−	0.990116i	$-0.544790\pi$
$311$	−4.94659	−0.280495	−0.140248	+	0.990116i	$0.544790\pi$
$312$	0	0
$313$	28.8861	1.63274	0.816370	−	0.577529i	$-0.195983\pi$
$313$	28.8861	1.63274	0.816370	+	0.577529i	$0.195983\pi$
$314$	−23.5431	−1.32861
$315$	0	0
$316$	−10.3474	−0.582087
$317$	−12.2872	−0.690120	−0.345060	−	0.938581i	$-0.612141\pi$
$317$	−12.2872	−0.690120	−0.345060	+	0.938581i	$0.612141\pi$
$318$	0	0
$319$	40.9664	2.29368
$320$	−3.29679	−0.184296
$321$	0	0
$322$	6.37787	0.355425
$323$	−6.59796	−0.367120
$324$	0	0
$325$	−10.1349	−0.562183
$326$	2.57466	0.142597
$327$	0	0
$328$	−4.77243	−0.263513
$329$	13.1767	0.726457
$330$	0	0
$331$	−12.8823	−0.708075	−0.354038	−	0.935231i	$-0.615191\pi$
$331$	−12.8823	−0.708075	−0.354038	+	0.935231i	$0.615191\pi$
$332$	−5.92397	−0.325120
$333$	0	0
$334$	1.00000	0.0547176
$335$	49.2896	2.69298
$336$	0	0
$337$	19.3811	1.05575	0.527877	−	0.849321i	$-0.322988\pi$
$337$	19.3811	1.05575	0.527877	+	0.849321i	$0.322988\pi$
$338$	10.0178	0.544896
$339$	0	0
$340$	−2.69473	−0.146142
$341$	31.1350	1.68606
$342$	0	0
$343$	16.4269	0.886969
$344$	−5.73329	−0.309118
$345$	0	0
$346$	15.9801	0.859095
$347$	12.1296	0.651150	0.325575	−	0.945516i	$-0.394442\pi$
$347$	12.1296	0.651150	0.325575	+	0.945516i	$0.394442\pi$
$348$	0	0
$349$	−14.7159	−0.787724	−0.393862	−	0.919169i	$-0.628861\pi$
$349$	−14.7159	−0.787724	−0.393862	+	0.919169i	$0.628861\pi$
$350$	7.91415	0.423029
$351$	0	0
$352$	5.12251	0.273031
$353$	−8.11247	−0.431783	−0.215892	−	0.976417i	$-0.569266\pi$
$353$	−8.11247	−0.431783	−0.215892	+	0.976417i	$0.569266\pi$
$354$	0	0
$355$	−40.5071	−2.14989
$356$	5.19016	0.275078
$357$	0	0
$358$	−16.0998	−0.850900
$359$	16.3977	0.865436	0.432718	−	0.901529i	$-0.357555\pi$
$359$	16.3977	0.865436	0.432718	+	0.901529i	$0.357555\pi$
$360$	0	0
$361$	46.1584	2.42939
$362$	−0.401921	−0.0211245
$363$	0	0
$364$	2.32875	0.122060
$365$	−24.4359	−1.27903
$366$	0	0
$367$	−30.4926	−1.59170	−0.795851	−	0.605492i	$-0.792976\pi$
$367$	−30.4926	−1.59170	−0.795851	+	0.605492i	$0.792976\pi$
$368$	4.72957	0.246546
$369$	0	0
$370$	−20.0089	−1.04021
$371$	−2.07161	−0.107553
$372$	0	0
$373$	−10.5100	−0.544189	−0.272095	−	0.962271i	$-0.587716\pi$
$373$	−10.5100	−0.544189	−0.272095	+	0.962271i	$0.587716\pi$
$374$	4.18704	0.216507
$375$	0	0
$376$	9.77133	0.503918
$377$	13.8107	0.711285
$378$	0	0
$379$	36.8309	1.89188	0.945939	−	0.324346i	$-0.105144\pi$
$379$	36.8309	1.89188	0.945939	+	0.324346i	$0.105144\pi$
$380$	26.6119	1.36516
$381$	0	0
$382$	−7.29514	−0.373252
$383$	2.34615	0.119883	0.0599414	−	0.998202i	$-0.480909\pi$
$383$	2.34615	0.119883	0.0599414	+	0.998202i	$0.480909\pi$
$384$	0	0
$385$	−22.7734	−1.16064
$386$	7.26549	0.369804
$387$	0	0
$388$	1.05901	0.0537629
$389$	−25.8027	−1.30825	−0.654124	−	0.756387i	$-0.726963\pi$
$389$	−25.8027	−1.30825	−0.654124	+	0.756387i	$0.726963\pi$
$390$	0	0
$391$	3.86585	0.195505
$392$	5.18152	0.261706
$393$	0	0
$394$	−1.73072	−0.0871923
$395$	34.1132	1.71642
$396$	0	0
$397$	14.5834	0.731918	0.365959	−	0.930631i	$-0.380741\pi$
$397$	14.5834	0.731918	0.365959	+	0.930631i	$0.380741\pi$
$398$	24.9415	1.25020
$399$	0	0
$400$	5.86881	0.293441
$401$	10.2580	0.512260	0.256130	−	0.966642i	$-0.417552\pi$
$401$	10.2580	0.512260	0.256130	+	0.966642i	$0.417552\pi$
$402$	0	0
$403$	10.4963	0.522857
$404$	−8.60536	−0.428133
$405$	0	0
$406$	−10.7845	−0.535224
$407$	31.0897	1.54106
$408$	0	0
$409$	12.1426	0.600411	0.300206	−	0.953874i	$-0.402945\pi$
$409$	12.1426	0.600411	0.300206	+	0.953874i	$0.402945\pi$
$410$	15.7337	0.777031
$411$	0	0
$412$	14.3156	0.705277
$413$	4.45435	0.219184
$414$	0	0
$415$	19.5301	0.958694
$416$	1.72691	0.0846687
$417$	0	0
$418$	−41.3493	−2.02246
$419$	−19.5126	−0.953253	−0.476627	−	0.879106i	$-0.658141\pi$
$419$	−19.5126	−0.953253	−0.476627	+	0.879106i	$0.658141\pi$
$420$	0	0
$421$	23.2244	1.13189	0.565944	−	0.824444i	$-0.308512\pi$
$421$	23.2244	1.13189	0.565944	+	0.824444i	$0.308512\pi$
$422$	−19.4045	−0.944595
$423$	0	0
$424$	−1.53622	−0.0746056
$425$	4.79705	0.232691
$426$	0	0
$427$	−18.5627	−0.898311
$428$	−4.20807	−0.203405
$429$	0	0
$430$	18.9014	0.911508
$431$	14.8917	0.717309	0.358654	−	0.933470i	$-0.383236\pi$
$431$	14.8917	0.717309	0.358654	+	0.933470i	$0.383236\pi$
$432$	0	0
$433$	−25.3991	−1.22060	−0.610301	−	0.792169i	$-0.708952\pi$
$433$	−25.3991	−1.22060	−0.610301	+	0.792169i	$0.708952\pi$
$434$	−8.19635	−0.393437
$435$	0	0
$436$	−5.45271	−0.261137
$437$	−38.1774	−1.82627
$438$	0	0
$439$	35.5080	1.69470	0.847351	−	0.531033i	$-0.178196\pi$
$439$	35.5080	1.69470	0.847351	+	0.531033i	$0.178196\pi$
$440$	−16.8878	−0.805096
$441$	0	0
$442$	1.41154	0.0671402
$443$	−27.0187	−1.28370	−0.641849	−	0.766831i	$-0.721832\pi$
$443$	−27.0187	−1.28370	−0.641849	+	0.766831i	$0.721832\pi$
$444$	0	0
$445$	−17.1109	−0.811132
$446$	18.2857	0.865852
$447$	0	0
$448$	−1.34851	−0.0637111
$449$	17.0796	0.806037	0.403019	−	0.915192i	$-0.367961\pi$
$449$	17.0796	0.806037	0.403019	+	0.915192i	$0.367961\pi$
$450$	0	0
$451$	−24.4468	−1.15116
$452$	−4.52520	−0.212848
$453$	0	0
$454$	−3.37535	−0.158413
$455$	−7.67741	−0.359922
$456$	0	0
$457$	30.3473	1.41958	0.709792	−	0.704411i	$-0.248789\pi$
$457$	30.3473	1.41958	0.709792	+	0.704411i	$0.248789\pi$
$458$	−18.4505	−0.862137
$459$	0	0
$460$	−15.5924	−0.726998
$461$	−38.3444	−1.78588	−0.892939	−	0.450178i	$-0.851360\pi$
$461$	−38.3444	−1.78588	−0.892939	+	0.450178i	$0.851360\pi$
$462$	0	0
$463$	1.88423	0.0875676	0.0437838	−	0.999041i	$-0.486059\pi$
$463$	1.88423	0.0875676	0.0437838	+	0.999041i	$0.486059\pi$
$464$	−7.99732	−0.371266
$465$	0	0
$466$	−5.56810	−0.257937
$467$	−1.93135	−0.0893724	−0.0446862	−	0.999001i	$-0.514229\pi$
$467$	−1.93135	−0.0893724	−0.0446862	+	0.999001i	$0.514229\pi$
$468$	0	0
$469$	20.1613	0.930963
$470$	−32.2140	−1.48592
$471$	0	0
$472$	3.30317	0.152040
$473$	−29.3688	−1.35038
$474$	0	0
$475$	−47.3735	−2.17364
$476$	−1.10225	−0.0505213
$477$	0	0
$478$	−24.9446	−1.14094
$479$	−13.3579	−0.610338	−0.305169	−	0.952298i	$-0.598713\pi$
$479$	−13.3579	−0.610338	−0.305169	+	0.952298i	$0.598713\pi$
$480$	0	0
$481$	10.4810	0.477892
$482$	20.0196	0.911867
$483$	0	0
$484$	15.2402	0.692734
$485$	−3.49132	−0.158533
$486$	0	0
$487$	−25.1680	−1.14047	−0.570236	−	0.821481i	$-0.693148\pi$
$487$	−25.1680	−1.14047	−0.570236	+	0.821481i	$0.693148\pi$
$488$	−13.7653	−0.623127
$489$	0	0
$490$	−17.0824	−0.771703
$491$	−13.1669	−0.594213	−0.297107	−	0.954844i	$-0.596022\pi$
$491$	−13.1669	−0.594213	−0.297107	+	0.954844i	$0.596022\pi$
$492$	0	0
$493$	−6.53686	−0.294405
$494$	−13.9397	−0.627179
$495$	0	0
$496$	−6.07808	−0.272914
$497$	−16.5689	−0.743217
$498$	0	0
$499$	−31.3895	−1.40519	−0.702593	−	0.711592i	$-0.747974\pi$
$499$	−31.3895	−1.40519	−0.702593	+	0.711592i	$0.747974\pi$
$500$	−2.86428	−0.128095
$501$	0	0
$502$	5.41418	0.241647
$503$	−35.8853	−1.60004	−0.800022	−	0.599970i	$-0.795179\pi$
$503$	−35.8853	−1.60004	−0.800022	+	0.599970i	$0.795179\pi$
$504$	0	0
$505$	28.3701	1.26245
$506$	24.2273	1.07703
$507$	0	0
$508$	9.79684	0.434665
$509$	35.1304	1.55713	0.778565	−	0.627564i	$-0.215948\pi$
$509$	35.1304	1.55713	0.778565	+	0.627564i	$0.215948\pi$
$510$	0	0
$511$	−9.99518	−0.442161
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−4.80813	−0.212077
$515$	−47.1954	−2.07968
$516$	0	0
$517$	50.0538	2.20136
$518$	−8.18440	−0.359602
$519$	0	0
$520$	−5.69325	−0.249666
$521$	−14.7898	−0.647955	−0.323977	−	0.946065i	$-0.605020\pi$
$521$	−14.7898	−0.647955	−0.323977	+	0.946065i	$0.605020\pi$
$522$	0	0
$523$	−14.6841	−0.642089	−0.321045	−	0.947064i	$-0.604034\pi$
$523$	−14.6841	−0.642089	−0.321045	+	0.947064i	$0.604034\pi$
$524$	10.4983	0.458619
$525$	0	0
$526$	24.0805	1.04996
$527$	−4.96810	−0.216414
$528$	0	0
$529$	−0.631202	−0.0274436
$530$	5.06460	0.219992
$531$	0	0
$532$	10.8853	0.471937
$533$	−8.24155	−0.356981
$534$	0	0
$535$	13.8731	0.599787
$536$	14.9508	0.645777
$537$	0	0
$538$	25.8709	1.11537
$539$	26.5424	1.14326
$540$	0	0
$541$	41.9287	1.80265	0.901327	−	0.433140i	$-0.142594\pi$
$541$	41.9287	1.80265	0.901327	+	0.433140i	$0.142594\pi$
$542$	12.9853	0.557765
$543$	0	0
$544$	−0.817380	−0.0350449
$545$	17.9764	0.770025
$546$	0	0
$547$	−42.7988	−1.82994	−0.914971	−	0.403519i	$-0.867787\pi$
$547$	−42.7988	−1.82994	−0.914971	+	0.403519i	$0.867787\pi$
$548$	15.0021	0.640859
$549$	0	0
$550$	30.0631	1.28189
$551$	64.5550	2.75014
$552$	0	0
$553$	13.9536	0.593367
$554$	7.33571	0.311665
$555$	0	0
$556$	−6.00399	−0.254626
$557$	−19.5772	−0.829512	−0.414756	−	0.909933i	$-0.636133\pi$
$557$	−19.5772	−0.829512	−0.414756	+	0.909933i	$0.636133\pi$
$558$	0	0
$559$	−9.90086	−0.418762
$560$	4.44575	0.187867
$561$	0	0
$562$	−11.4575	−0.483308
$563$	−9.38845	−0.395676	−0.197838	−	0.980235i	$-0.563392\pi$
$563$	−9.38845	−0.395676	−0.197838	+	0.980235i	$0.563392\pi$
$564$	0	0
$565$	14.9186	0.627631
$566$	−19.4159	−0.816109
$567$	0	0
$568$	−12.2868	−0.515544
$569$	9.59122	0.402085	0.201042	−	0.979583i	$-0.435567\pi$
$569$	9.59122	0.402085	0.201042	+	0.979583i	$0.435567\pi$
$570$	0	0
$571$	33.5875	1.40559	0.702797	−	0.711390i	$-0.251934\pi$
$571$	33.5875	1.40559	0.702797	+	0.711390i	$0.251934\pi$
$572$	8.84612	0.369875
$573$	0	0
$574$	6.43567	0.268620
$575$	27.7569	1.15754
$576$	0	0
$577$	−18.9190	−0.787606	−0.393803	−	0.919195i	$-0.628841\pi$
$577$	−18.9190	−0.787606	−0.393803	+	0.919195i	$0.628841\pi$
$578$	16.3319	0.679317
$579$	0	0
$580$	26.3655	1.09477
$581$	7.98854	0.331420
$582$	0	0
$583$	−7.86932	−0.325914
$584$	−7.41202	−0.306712
$585$	0	0
$586$	6.23247	0.257461
$587$	−33.5815	−1.38606	−0.693028	−	0.720910i	$-0.743724\pi$
$587$	−33.5815	−1.38606	−0.693028	+	0.720910i	$0.743724\pi$
$588$	0	0
$589$	49.0627	2.02159
$590$	−10.8898	−0.448327
$591$	0	0
$592$	−6.06922	−0.249443
$593$	−10.8213	−0.444378	−0.222189	−	0.975004i	$-0.571320\pi$
$593$	−10.8213	−0.444378	−0.222189	+	0.975004i	$0.571320\pi$
$594$	0	0
$595$	3.63387	0.148974
$596$	−4.39903	−0.180191
$597$	0	0
$598$	8.16753	0.333995
$599$	4.08937	0.167087	0.0835436	−	0.996504i	$-0.473376\pi$
$599$	4.08937	0.167087	0.0835436	+	0.996504i	$0.473376\pi$
$600$	0	0
$601$	7.25979	0.296133	0.148067	−	0.988977i	$-0.452695\pi$
$601$	7.25979	0.296133	0.148067	+	0.988977i	$0.452695\pi$
$602$	7.73139	0.315108
$603$	0	0
$604$	−6.58539	−0.267956
$605$	−50.2436	−2.04269
$606$	0	0
$607$	−28.3999	−1.15272	−0.576359	−	0.817197i	$-0.695527\pi$
$607$	−28.3999	−1.15272	−0.576359	+	0.817197i	$0.695527\pi$
$608$	8.07208	0.327366
$609$	0	0
$610$	45.3814	1.83744
$611$	16.8742	0.682657
$612$	0	0
$613$	−8.26220	−0.333707	−0.166854	−	0.985982i	$-0.553361\pi$
$613$	−8.26220	−0.333707	−0.166854	+	0.985982i	$0.553361\pi$
$614$	−26.7588	−1.07990
$615$	0	0
$616$	−6.90776	−0.278322
$617$	1.08807	0.0438039	0.0219020	−	0.999760i	$-0.493028\pi$
$617$	1.08807	0.0438039	0.0219020	+	0.999760i	$0.493028\pi$
$618$	0	0
$619$	−15.8966	−0.638938	−0.319469	−	0.947597i	$-0.603505\pi$
$619$	−15.8966	−0.638938	−0.319469	+	0.947597i	$0.603505\pi$
$620$	20.0381	0.804751
$621$	0	0
$622$	4.94659	0.198340
$623$	−6.99898	−0.280408
$624$	0	0
$625$	−19.9011	−0.796045
$626$	−28.8861	−1.15452
$627$	0	0
$628$	23.5431	0.939471
$629$	−4.96086	−0.197802
$630$	0	0
$631$	34.6392	1.37897	0.689483	−	0.724302i	$-0.257838\pi$
$631$	34.6392	1.37897	0.689483	+	0.724302i	$0.257838\pi$
$632$	10.3474	0.411598
$633$	0	0
$634$	12.2872	0.487988
$635$	−32.2981	−1.28171
$636$	0	0
$637$	8.94802	0.354533
$638$	−40.9664	−1.62188
$639$	0	0
$640$	3.29679	0.130317
$641$	−5.80967	−0.229468	−0.114734	−	0.993396i	$-0.536602\pi$
$641$	−5.80967	−0.229468	−0.114734	+	0.993396i	$0.536602\pi$
$642$	0	0
$643$	−20.8903	−0.823834	−0.411917	−	0.911221i	$-0.635141\pi$
$643$	−20.8903	−0.823834	−0.411917	+	0.911221i	$0.635141\pi$
$644$	−6.37787	−0.251323
$645$	0	0
$646$	6.59796	0.259593
$647$	−43.9369	−1.72734	−0.863668	−	0.504060i	$-0.831839\pi$
$647$	−43.9369	−1.72734	−0.863668	+	0.504060i	$0.831839\pi$
$648$	0	0
$649$	16.9205	0.664188
$650$	10.1349	0.397524
$651$	0	0
$652$	−2.57466	−0.100832
$653$	12.8105	0.501313	0.250656	−	0.968076i	$-0.419354\pi$
$653$	12.8105	0.501313	0.250656	+	0.968076i	$0.419354\pi$
$654$	0	0
$655$	−34.6106	−1.35235
$656$	4.77243	0.186332
$657$	0	0
$658$	−13.1767	−0.513683
$659$	39.1994	1.52699	0.763497	−	0.645812i	$-0.223481\pi$
$659$	39.1994	1.52699	0.763497	+	0.645812i	$0.223481\pi$
$660$	0	0
$661$	13.2802	0.516540	0.258270	−	0.966073i	$-0.416848\pi$
$661$	13.2802	0.516540	0.258270	+	0.966073i	$0.416848\pi$
$662$	12.8823	0.500685
$663$	0	0
$664$	5.92397	0.229895
$665$	−35.8864	−1.39162
$666$	0	0
$667$	−37.8239	−1.46455
$668$	−1.00000	−0.0386912
$669$	0	0
$670$	−49.2896	−1.90423
$671$	−70.5131	−2.72213
$672$	0	0
$673$	34.4859	1.32933	0.664667	−	0.747140i	$-0.268573\pi$
$673$	34.4859	1.32933	0.664667	+	0.747140i	$0.268573\pi$
$674$	−19.3811	−0.746531
$675$	0	0
$676$	−10.0178	−0.385299
$677$	−41.8714	−1.60925	−0.804625	−	0.593784i	$-0.797633\pi$
$677$	−41.8714	−1.60925	−0.804625	+	0.593784i	$0.797633\pi$
$678$	0	0
$679$	−1.42808	−0.0548047
$680$	2.69473	0.103338
$681$	0	0
$682$	−31.1350	−1.19222
$683$	21.3800	0.818082	0.409041	−	0.912516i	$-0.365863\pi$
$683$	21.3800	0.818082	0.409041	+	0.912516i	$0.365863\pi$
$684$	0	0
$685$	−49.4588	−1.88972
$686$	−16.4269	−0.627182
$687$	0	0
$688$	5.73329	0.218579
$689$	−2.65292	−0.101068
$690$	0	0
$691$	48.0267	1.82702	0.913510	−	0.406816i	$-0.133361\pi$
$691$	48.0267	1.82702	0.913510	+	0.406816i	$0.133361\pi$
$692$	−15.9801	−0.607472
$693$	0	0
$694$	−12.1296	−0.460432
$695$	19.7939	0.750825
$696$	0	0
$697$	3.90089	0.147757
$698$	14.7159	0.557005
$699$	0	0
$700$	−7.91415	−0.299127
$701$	4.81753	0.181956	0.0909778	−	0.995853i	$-0.471001\pi$
$701$	4.81753	0.181956	0.0909778	+	0.995853i	$0.471001\pi$
$702$	0	0
$703$	48.9912	1.84774
$704$	−5.12251	−0.193062
$705$	0	0
$706$	8.11247	0.305317
$707$	11.6044	0.436429
$708$	0	0
$709$	−10.7130	−0.402336	−0.201168	−	0.979557i	$-0.564474\pi$
$709$	−10.7130	−0.402336	−0.201168	+	0.979557i	$0.564474\pi$
$710$	40.5071	1.52020
$711$	0	0
$712$	−5.19016	−0.194509
$713$	−28.7467	−1.07657
$714$	0	0
$715$	−29.1638	−1.09066
$716$	16.0998	0.601677
$717$	0	0
$718$	−16.3977	−0.611956
$719$	−46.8098	−1.74571	−0.872856	−	0.487979i	$-0.837734\pi$
$719$	−46.8098	−1.74571	−0.872856	+	0.487979i	$0.837734\pi$
$720$	0	0
$721$	−19.3047	−0.718944
$722$	−46.1584	−1.71784
$723$	0	0
$724$	0.401921	0.0149373
$725$	−46.9348	−1.74311
$726$	0	0
$727$	−37.2837	−1.38277	−0.691387	−	0.722484i	$-0.743000\pi$
$727$	−37.2837	−1.38277	−0.691387	+	0.722484i	$0.743000\pi$
$728$	−2.32875	−0.0863094
$729$	0	0
$730$	24.4359	0.904412
$731$	4.68628	0.173328
$732$	0	0
$733$	−27.0063	−0.997500	−0.498750	−	0.866746i	$-0.666207\pi$
$733$	−27.0063	−0.997500	−0.498750	+	0.866746i	$0.666207\pi$
$734$	30.4926	1.12550
$735$	0	0
$736$	−4.72957	−0.174334
$737$	76.5857	2.82107
$738$	0	0
$739$	−7.54393	−0.277508	−0.138754	−	0.990327i	$-0.544310\pi$
$739$	−7.54393	−0.277508	−0.138754	+	0.990327i	$0.544310\pi$
$740$	20.0089	0.735542
$741$	0	0
$742$	2.07161	0.0760513
$743$	−4.24737	−0.155821	−0.0779105	−	0.996960i	$-0.524825\pi$
$743$	−4.24737	−0.155821	−0.0779105	+	0.996960i	$0.524825\pi$
$744$	0	0
$745$	14.5027	0.531337
$746$	10.5100	0.384800
$747$	0	0
$748$	−4.18704	−0.153093
$749$	5.67462	0.207346
$750$	0	0
$751$	−17.3220	−0.632089	−0.316045	−	0.948744i	$-0.602355\pi$
$751$	−17.3220	−0.632089	−0.316045	+	0.948744i	$0.602355\pi$
$752$	−9.77133	−0.356324
$753$	0	0
$754$	−13.8107	−0.502954
$755$	21.7106	0.790130
$756$	0	0
$757$	−1.02359	−0.0372030	−0.0186015	−	0.999827i	$-0.505921\pi$
$757$	−1.02359	−0.0372030	−0.0186015	+	0.999827i	$0.505921\pi$
$758$	−36.8309	−1.33776
$759$	0	0
$760$	−26.6119	−0.965316
$761$	−24.8520	−0.900885	−0.450442	−	0.892806i	$-0.648734\pi$
$761$	−24.8520	−0.900885	−0.450442	+	0.892806i	$0.648734\pi$
$762$	0	0
$763$	7.35303	0.266198
$764$	7.29514	0.263929
$765$	0	0
$766$	−2.34615	−0.0847699
$767$	5.70427	0.205969
$768$	0	0
$769$	−11.6985	−0.421859	−0.210929	−	0.977501i	$-0.567649\pi$
$769$	−11.6985	−0.421859	−0.210929	+	0.977501i	$0.567649\pi$
$770$	22.7734	0.820697
$771$	0	0
$772$	−7.26549	−0.261491
$773$	8.50666	0.305963	0.152982	−	0.988229i	$-0.451112\pi$
$773$	8.50666	0.305963	0.152982	+	0.988229i	$0.451112\pi$
$774$	0	0
$775$	−35.6711	−1.28134
$776$	−1.05901	−0.0380161
$777$	0	0
$778$	25.8027	0.925071
$779$	−38.5234	−1.38024
$780$	0	0
$781$	−62.9395	−2.25215
$782$	−3.86585	−0.138243
$783$	0	0
$784$	−5.18152	−0.185054
$785$	−77.6165	−2.77025
$786$	0	0
$787$	−45.8326	−1.63375	−0.816877	−	0.576812i	$-0.804297\pi$
$787$	−45.8326	−1.63375	−0.816877	+	0.576812i	$0.804297\pi$
$788$	1.73072	0.0616543
$789$	0	0
$790$	−34.1132	−1.21369
$791$	6.10228	0.216972
$792$	0	0
$793$	−23.7715	−0.844150
$794$	−14.5834	−0.517544
$795$	0	0
$796$	−24.9415	−0.884027
$797$	33.6790	1.19297	0.596486	−	0.802624i	$-0.296563\pi$
$797$	33.6790	1.19297	0.596486	+	0.802624i	$0.296563\pi$
$798$	0	0
$799$	−7.98689	−0.282556
$800$	−5.86881	−0.207494
$801$	0	0
$802$	−10.2580	−0.362223
$803$	−37.9682	−1.33987
$804$	0	0
$805$	21.0265	0.741086
$806$	−10.4963	−0.369716
$807$	0	0
$808$	8.60536	0.302736
$809$	−16.1836	−0.568984	−0.284492	−	0.958678i	$-0.591825\pi$
$809$	−16.1836	−0.568984	−0.284492	+	0.958678i	$0.591825\pi$
$810$	0	0
$811$	29.3132	1.02933	0.514663	−	0.857392i	$-0.327917\pi$
$811$	29.3132	1.02933	0.514663	+	0.857392i	$0.327917\pi$
$812$	10.7845	0.378461
$813$	0	0
$814$	−31.0897	−1.08969
$815$	8.48811	0.297326
$816$	0	0
$817$	−46.2795	−1.61912
$818$	−12.1426	−0.424555
$819$	0	0
$820$	−15.7337	−0.549444
$821$	20.4798	0.714751	0.357375	−	0.933961i	$-0.383672\pi$
$821$	20.4798	0.714751	0.357375	+	0.933961i	$0.383672\pi$
$822$	0	0
$823$	−16.5160	−0.575712	−0.287856	−	0.957674i	$-0.592942\pi$
$823$	−16.5160	−0.575712	−0.287856	+	0.957674i	$0.592942\pi$
$824$	−14.3156	−0.498706
$825$	0	0
$826$	−4.45435	−0.154987
$827$	5.82347	0.202502	0.101251	−	0.994861i	$-0.467715\pi$
$827$	5.82347	0.202502	0.101251	+	0.994861i	$0.467715\pi$
$828$	0	0
$829$	−36.3106	−1.26112	−0.630559	−	0.776141i	$-0.717174\pi$
$829$	−36.3106	−1.26112	−0.630559	+	0.776141i	$0.717174\pi$
$830$	−19.5301	−0.677899
$831$	0	0
$832$	−1.72691	−0.0598698
$833$	−4.23527	−0.146744
$834$	0	0
$835$	3.29679	0.114090
$836$	41.3493	1.43010
$837$	0	0
$838$	19.5126	0.674052
$839$	−24.4644	−0.844606	−0.422303	−	0.906455i	$-0.638778\pi$
$839$	−24.4644	−0.844606	−0.422303	+	0.906455i	$0.638778\pi$
$840$	0	0
$841$	34.9572	1.20542
$842$	−23.2244	−0.800365
$843$	0	0
$844$	19.4045	0.667929
$845$	33.0265	1.13615
$846$	0	0
$847$	−20.5515	−0.706158
$848$	1.53622	0.0527541
$849$	0	0
$850$	−4.79705	−0.164537
$851$	−28.7048	−0.983987
$852$	0	0
$853$	33.9073	1.16096	0.580481	−	0.814274i	$-0.302864\pi$
$853$	33.9073	1.16096	0.580481	+	0.814274i	$0.302864\pi$
$854$	18.5627	0.635202
$855$	0	0
$856$	4.20807	0.143829
$857$	−1.88438	−0.0643692	−0.0321846	−	0.999482i	$-0.510246\pi$
$857$	−1.88438	−0.0643692	−0.0321846	+	0.999482i	$0.510246\pi$
$858$	0	0
$859$	−37.4840	−1.27894	−0.639469	−	0.768817i	$-0.720846\pi$
$859$	−37.4840	−1.27894	−0.639469	+	0.768817i	$0.720846\pi$
$860$	−18.9014	−0.644533
$861$	0	0
$862$	−14.8917	−0.507214
$863$	−31.9368	−1.08714	−0.543571	−	0.839363i	$-0.682928\pi$
$863$	−31.9368	−1.08714	−0.543571	+	0.839363i	$0.682928\pi$
$864$	0	0
$865$	52.6829	1.79127
$866$	25.3991	0.863097
$867$	0	0
$868$	8.19635	0.278202
$869$	53.0048	1.79806
$870$	0	0
$871$	25.8187	0.874833
$872$	5.45271	0.184652
$873$	0	0
$874$	38.1774	1.29137
$875$	3.86252	0.130577
$876$	0	0
$877$	2.45965	0.0830566	0.0415283	−	0.999137i	$-0.486777\pi$
$877$	2.45965	0.0830566	0.0415283	+	0.999137i	$0.486777\pi$
$878$	−35.5080	−1.19834
$879$	0	0
$880$	16.8878	0.569289
$881$	−6.33929	−0.213576	−0.106788	−	0.994282i	$-0.534057\pi$
$881$	−6.33929	−0.213576	−0.106788	+	0.994282i	$0.534057\pi$
$882$	0	0
$883$	−37.3797	−1.25793	−0.628964	−	0.777435i	$-0.716521\pi$
$883$	−37.3797	−1.25793	−0.628964	+	0.777435i	$0.716521\pi$
$884$	−1.41154	−0.0474753
$885$	0	0
$886$	27.0187	0.907711
$887$	−35.3939	−1.18841	−0.594206	−	0.804313i	$-0.702534\pi$
$887$	−35.3939	−1.18841	−0.594206	+	0.804313i	$0.702534\pi$
$888$	0	0
$889$	−13.2111	−0.443087
$890$	17.1109	0.573557
$891$	0	0
$892$	−18.2857	−0.612250
$893$	78.8749	2.63945
$894$	0	0
$895$	−53.0776	−1.77419
$896$	1.34851	0.0450506
$897$	0	0
$898$	−17.0796	−0.569954
$899$	48.6084	1.62118
$900$	0	0
$901$	1.25568	0.0418327
$902$	24.4468	0.813991
$903$	0	0
$904$	4.52520	0.150506
$905$	−1.32505	−0.0440461
$906$	0	0
$907$	−19.9019	−0.660832	−0.330416	−	0.943835i	$-0.607189\pi$
$907$	−19.9019	−0.660832	−0.330416	+	0.943835i	$0.607189\pi$
$908$	3.37535	0.112015
$909$	0	0
$910$	7.67741	0.254504
$911$	14.4396	0.478405	0.239202	−	0.970970i	$-0.423114\pi$
$911$	14.4396	0.478405	0.239202	+	0.970970i	$0.423114\pi$
$912$	0	0
$913$	30.3456	1.00429
$914$	−30.3473	−1.00380
$915$	0	0
$916$	18.4505	0.609623
$917$	−14.1570	−0.467506
$918$	0	0
$919$	−21.9004	−0.722428	−0.361214	−	0.932483i	$-0.617638\pi$
$919$	−21.9004	−0.722428	−0.361214	+	0.932483i	$0.617638\pi$
$920$	15.5924	0.514065
$921$	0	0
$922$	38.3444	1.26281
$923$	−21.2182	−0.698407
$924$	0	0
$925$	−35.6191	−1.17115
$926$	−1.88423	−0.0619197
$927$	0	0
$928$	7.99732	0.262525
$929$	55.9746	1.83647	0.918234	−	0.396039i	$-0.129615\pi$
$929$	55.9746	1.83647	0.918234	+	0.396039i	$0.129615\pi$
$930$	0	0
$931$	41.8256	1.37078
$932$	5.56810	0.182389
$933$	0	0
$934$	1.93135	0.0631958
$935$	13.8038	0.451432
$936$	0	0
$937$	−29.6188	−0.967603	−0.483802	−	0.875178i	$-0.660744\pi$
$937$	−29.6188	−0.967603	−0.483802	+	0.875178i	$0.660744\pi$
$938$	−20.1613	−0.658290
$939$	0	0
$940$	32.2140	1.05070
$941$	−17.8076	−0.580510	−0.290255	−	0.956949i	$-0.593740\pi$
$941$	−17.8076	−0.580510	−0.290255	+	0.956949i	$0.593740\pi$
$942$	0	0
$943$	22.5715	0.735030
$944$	−3.30317	−0.107509
$945$	0	0
$946$	29.3688	0.954863
$947$	−38.4959	−1.25095	−0.625475	−	0.780244i	$-0.715095\pi$
$947$	−38.4959	−1.25095	−0.625475	+	0.780244i	$0.715095\pi$
$948$	0	0
$949$	−12.7999	−0.415502
$950$	47.3735	1.53700
$951$	0	0
$952$	1.10225	0.0357240
$953$	1.48750	0.0481848	0.0240924	−	0.999710i	$-0.492330\pi$
$953$	1.48750	0.0481848	0.0240924	+	0.999710i	$0.492330\pi$
$954$	0	0
$955$	−24.0505	−0.778256
$956$	24.9446	0.806767
$957$	0	0
$958$	13.3579	0.431574
$959$	−20.2305	−0.653277
$960$	0	0
$961$	5.94301	0.191710
$962$	−10.4810	−0.337921
$963$	0	0
$964$	−20.0196	−0.644787
$965$	23.9528	0.771067
$966$	0	0
$967$	−17.6109	−0.566327	−0.283163	−	0.959072i	$-0.591384\pi$
$967$	−17.6109	−0.566327	−0.283163	+	0.959072i	$0.591384\pi$
$968$	−15.2402	−0.489837
$969$	0	0
$970$	3.49132	0.112099
$971$	30.3532	0.974079	0.487040	−	0.873380i	$-0.338077\pi$
$971$	30.3532	0.974079	0.487040	+	0.873380i	$0.338077\pi$
$972$	0	0
$973$	8.09645	0.259560
$974$	25.1680	0.806435
$975$	0	0
$976$	13.7653	0.440617
$977$	35.5976	1.13887	0.569434	−	0.822037i	$-0.307163\pi$
$977$	35.5976	1.13887	0.569434	+	0.822037i	$0.307163\pi$
$978$	0	0
$979$	−26.5867	−0.849714
$980$	17.0824	0.545676
$981$	0	0
$982$	13.1669	0.420172
$983$	−24.2401	−0.773138	−0.386569	−	0.922260i	$-0.626340\pi$
$983$	−24.2401	−0.773138	−0.386569	+	0.922260i	$0.626340\pi$
$984$	0	0
$985$	−5.70581	−0.181802
$986$	6.53686	0.208176
$987$	0	0
$988$	13.9397	0.443482
$989$	27.1160	0.862237
$990$	0	0
$991$	30.4607	0.967617	0.483809	−	0.875174i	$-0.339253\pi$
$991$	30.4607	0.967617	0.483809	+	0.875174i	$0.339253\pi$
$992$	6.07808	0.192979
$993$	0	0
$994$	16.5689	0.525534
$995$	82.2267	2.60676
$996$	0	0
$997$	−10.3030	−0.326299	−0.163150	−	0.986601i	$-0.552165\pi$
$997$	−10.3030	−0.326299	−0.163150	+	0.986601i	$0.552165\pi$
$998$	31.3895	0.993616
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3006.2.a.u.1.1		7
3.2	odd	2		1002.2.a.k.1.7	✓	7
12.11	even	2		8016.2.a.v.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.k.1.7	✓	7	3.2	odd	2
3006.2.a.u.1.1		7	1.1	even	1	trivial
8016.2.a.v.1.7		7	12.11	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:	\( N \)	\(=\)	\( 3006 = 2 \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.29679 q^{5} -1.34851 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.29679 q^{5} -1.34851 q^{7} -1.00000 q^{8} +3.29679 q^{10} -5.12251 q^{11} -1.72691 q^{13} +1.34851 q^{14} +1.00000 q^{16} +0.817380 q^{17} -8.07208 q^{19} -3.29679 q^{20} +5.12251 q^{22} +4.72957 q^{23} +5.86881 q^{25} +1.72691 q^{26} -1.34851 q^{28} -7.99732 q^{29} -6.07808 q^{31} -1.00000 q^{32} -0.817380 q^{34} +4.44575 q^{35} -6.06922 q^{37} +8.07208 q^{38} +3.29679 q^{40} +4.77243 q^{41} +5.73329 q^{43} -5.12251 q^{44} -4.72957 q^{46} -9.77133 q^{47} -5.18152 q^{49} -5.86881 q^{50} -1.72691 q^{52} +1.53622 q^{53} +16.8878 q^{55} +1.34851 q^{56} +7.99732 q^{58} -3.30317 q^{59} +13.7653 q^{61} +6.07808 q^{62} +1.00000 q^{64} +5.69325 q^{65} -14.9508 q^{67} +0.817380 q^{68} -4.44575 q^{70} +12.2868 q^{71} +7.41202 q^{73} +6.06922 q^{74} -8.07208 q^{76} +6.90776 q^{77} -10.3474 q^{79} -3.29679 q^{80} -4.77243 q^{82} -5.92397 q^{83} -2.69473 q^{85} -5.73329 q^{86} +5.12251 q^{88} +5.19016 q^{89} +2.32875 q^{91} +4.72957 q^{92} +9.77133 q^{94} +26.6119 q^{95} +1.05901 q^{97} +5.18152 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 5 q^{5} + 7 q^{7} - 7 q^{8} + 5 q^{10} + 6 q^{13} - 7 q^{14} + 7 q^{16} - 6 q^{17} - 2 q^{19} - 5 q^{20} + 12 q^{25} - 6 q^{26} + 7 q^{28} + 4 q^{29} + 7 q^{31} - 7 q^{32} + 6 q^{34} + 13 q^{35} - 3 q^{37} + 2 q^{38} + 5 q^{40} + 12 q^{41} - 2 q^{43} + 11 q^{47} + 10 q^{49} - 12 q^{50} + 6 q^{52} - q^{53} - 2 q^{55} - 7 q^{56} - 4 q^{58} + 19 q^{59} + 12 q^{61} - 7 q^{62} + 7 q^{64} + 10 q^{65} - 17 q^{67} - 6 q^{68} - 13 q^{70} + 20 q^{71} + 10 q^{73} + 3 q^{74} - 2 q^{76} + 24 q^{77} + 2 q^{79} - 5 q^{80} - 12 q^{82} + 7 q^{83} - 18 q^{85} + 2 q^{86} + 3 q^{89} + 4 q^{91} - 11 q^{94} + 24 q^{95} - 3 q^{97} - 10 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 5 * q^5 + 7 * q^7 - 7 * q^8 + 5 * q^10 + 6 * q^13 - 7 * q^14 + 7 * q^16 - 6 * q^17 - 2 * q^19 - 5 * q^20 + 12 * q^25 - 6 * q^26 + 7 * q^28 + 4 * q^29 + 7 * q^31 - 7 * q^32 + 6 * q^34 + 13 * q^35 - 3 * q^37 + 2 * q^38 + 5 * q^40 + 12 * q^41 - 2 * q^43 + 11 * q^47 + 10 * q^49 - 12 * q^50 + 6 * q^52 - q^53 - 2 * q^55 - 7 * q^56 - 4 * q^58 + 19 * q^59 + 12 * q^61 - 7 * q^62 + 7 * q^64 + 10 * q^65 - 17 * q^67 - 6 * q^68 - 13 * q^70 + 20 * q^71 + 10 * q^73 + 3 * q^74 - 2 * q^76 + 24 * q^77 + 2 * q^79 - 5 * q^80 - 12 * q^82 + 7 * q^83 - 18 * q^85 + 2 * q^86 + 3 * q^89 + 4 * q^91 - 11 * q^94 + 24 * q^95 - 3 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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