LMFDB - Embedded newform 2940.2.d.a.881.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(881,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2940.881"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.29471584693248.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 13x^{6} - 36x^{5} + 39x^{4} - 36x^{3} + 54x^{2} - 162x + 243 $ x^10 - 2x^9 + 2x^8 - 4x^7 + 13x^6 - 36x^5 + 39x^4 - 36x^3 + 54x^2 - 162*x + 243
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.3
Root			$1.15038 - 1.29484i$ of defining polynomial
Character	$\chi$	$=$	2940.881
Dual form			2940.2.d.a.881.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.15038 - 1.29484i) q^{3} -1.00000 q^{5} +(-0.353244 + 2.97913i) q^{9} -1.35200i q^{11} +4.94296i q^{13} +(1.15038 + 1.29484i) q^{15} -5.74210 q^{17} +3.28736i q^{19} -5.00540i q^{23} +1.00000 q^{25} +(4.26388 - 2.96974i) q^{27} -5.68630i q^{29} +2.83086i q^{31} +(-1.75063 + 1.55531i) q^{33} +3.85090 q^{37} +(6.40037 - 5.68630i) q^{39} -3.73802 q^{41} +4.06339 q^{43} +(0.353244 - 2.97913i) q^{45} +5.68596 q^{47} +(6.60560 + 7.43512i) q^{51} -1.46155i q^{53} +1.35200i q^{55} +(4.25662 - 3.78172i) q^{57} +8.68477 q^{59} +1.90879i q^{61} -4.94296i q^{65} +5.03878 q^{67} +(-6.48121 + 5.75812i) q^{69} +3.38259i q^{71} -16.3273i q^{73} +(-1.15038 - 1.29484i) q^{75} +4.83387 q^{79} +(-8.75044 - 2.10472i) q^{81} -16.6525 q^{83} +5.74210 q^{85} +(-7.36287 + 6.54141i) q^{87} +16.1630 q^{89} +(3.66552 - 3.25657i) q^{93} -3.28736i q^{95} -12.0577i q^{97} +(4.02778 + 0.477584i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 10 q^{5} + 2 q^{15} + 12 q^{17} + 10 q^{25} - 8 q^{27} - 16 q^{33} + 2 q^{37} + 6 q^{39} + 8 q^{41} - 26 q^{43} + 28 q^{47} + 4 q^{51} + 18 q^{57} - 14 q^{67} + 14 q^{69} - 2 q^{75} - 2 q^{79}+ \cdots + 36 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 10 * q^5 + 2 * q^15 + 12 * q^17 + 10 * q^25 - 8 * q^27 - 16 * q^33 + 2 * q^37 + 6 * q^39 + 8 * q^41 - 26 * q^43 + 28 * q^47 + 4 * q^51 + 18 * q^57 - 14 * q^67 + 14 * q^69 - 2 * q^75 - 2 * q^79 - 28 * q^81 + 8 * q^83 - 12 * q^85 - 34 * q^87 + 56 * q^89 + 22 * q^93 + 36 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.15038	−	1.29484i	−0.664173	−	0.747579i
$3$	−1.15038	−	1.29484i	−0.664173	−	0.747579i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.353244	+	2.97913i	−0.117748	+	0.993044i
$10$		0			0
$11$		−	1.35200i		−	0.407643i	−0.979008	−	0.203821i	$-0.934664\pi$
$11$		−	1.35200i		−	0.407643i	0.979008	−	0.203821i	$-0.0653361\pi$
$12$		0			0
$13$			4.94296i			1.37093i	0.728105	+	0.685466i	$0.240401\pi$
$13$			4.94296i			1.37093i	−0.728105	+	0.685466i	$0.759599\pi$
$14$		0			0
$15$	1.15038	+	1.29484i	0.297027	+	0.334327i
$16$		0			0
$17$	−5.74210			−1.39266			−0.696332	−	0.717720i	$-0.745186\pi$
$17$	−5.74210			−1.39266			−0.696332	+	0.717720i	$0.745186\pi$
$18$		0			0
$19$			3.28736i			0.754173i	0.926178	+	0.377087i	$0.123074\pi$
$19$			3.28736i			0.754173i	−0.926178	+	0.377087i	$0.876926\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	5.00540i		−	1.04370i	−0.853038	−	0.521849i	$-0.825243\pi$
$23$		−	5.00540i		−	1.04370i	0.853038	−	0.521849i	$-0.174757\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.26388	−	2.96974i	0.820583	−	0.571527i
$28$		0			0
$29$		−	5.68630i		−	1.05592i	−0.849270	−	0.527959i	$-0.822957\pi$
$29$		−	5.68630i		−	1.05592i	0.849270	−	0.527959i	$-0.177043\pi$
$30$		0			0
$31$			2.83086i			0.508437i	0.967147	+	0.254219i	$0.0818183\pi$
$31$			2.83086i			0.508437i	−0.967147	+	0.254219i	$0.918182\pi$
$32$		0			0
$33$	−1.75063	+	1.55531i	−0.304745	+	0.270745i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.85090			0.633084			0.316542	−	0.948578i	$-0.397478\pi$
$37$	3.85090			0.633084			0.316542	+	0.948578i	$0.397478\pi$
$38$		0			0
$39$	6.40037	−	5.68630i	1.02488	−	0.910536i
$40$		0			0
$41$	−3.73802			−0.583781			−0.291890	−	0.956452i	$-0.594284\pi$
$41$	−3.73802			−0.583781			−0.291890	+	0.956452i	$0.594284\pi$
$42$		0			0
$43$	4.06339			0.619661			0.309830	−	0.950792i	$-0.399728\pi$
$43$	4.06339			0.619661			0.309830	+	0.950792i	$0.399728\pi$
$44$		0			0
$45$	0.353244	−	2.97913i	0.0526584	−	0.444103i
$46$		0			0
$47$	5.68596			0.829383			0.414691	−	0.909962i	$-0.363890\pi$
$47$	5.68596			0.829383			0.414691	+	0.909962i	$0.363890\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	6.60560	+	7.43512i	0.924970	+	1.04113i
$52$		0			0
$53$		−	1.46155i		−	0.200759i	−0.994949	−	0.100379i	$-0.967994\pi$
$53$		−	1.46155i		−	0.200759i	0.994949	−	0.100379i	$-0.0320057\pi$
$54$		0			0
$55$			1.35200i			0.182303i
$56$		0			0
$57$	4.25662	−	3.78172i	0.563804	−	0.500902i
$58$		0			0
$59$	8.68477			1.13066			0.565330	−	0.824865i	$-0.308749\pi$
$59$	8.68477			1.13066			0.565330	+	0.824865i	$0.308749\pi$
$60$		0			0
$61$			1.90879i			0.244395i	0.992506	+	0.122198i	$0.0389942\pi$
$61$			1.90879i			0.244395i	−0.992506	+	0.122198i	$0.961006\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	4.94296i		−	0.613099i
$66$		0			0
$67$	5.03878			0.615585			0.307793	−	0.951453i	$-0.400410\pi$
$67$	5.03878			0.615585			0.307793	+	0.951453i	$0.400410\pi$
$68$		0			0
$69$	−6.48121	+	5.75812i	−0.780246	+	0.693196i
$70$		0			0
$71$			3.38259i			0.401440i	0.979649	+	0.200720i	$0.0643281\pi$
$71$			3.38259i			0.401440i	−0.979649	+	0.200720i	$0.935672\pi$
$72$		0			0
$73$		−	16.3273i		−	1.91096i	−0.295052	−	0.955481i	$-0.595337\pi$
$73$		−	16.3273i		−	1.91096i	0.295052	−	0.955481i	$-0.404663\pi$
$74$		0			0
$75$	−1.15038	−	1.29484i	−0.132835	−	0.149516i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.83387			0.543853			0.271926	−	0.962318i	$-0.412339\pi$
$79$	4.83387			0.543853			0.271926	+	0.962318i	$0.412339\pi$
$80$		0			0
$81$	−8.75044	−	2.10472i	−0.972271	−	0.233858i
$82$		0			0
$83$	−16.6525			−1.82785			−0.913927	−	0.405879i	$-0.866965\pi$
$83$	−16.6525			−1.82785			−0.913927	+	0.405879i	$0.866965\pi$
$84$		0			0
$85$	5.74210			0.622818
$86$		0			0
$87$	−7.36287	+	6.54141i	−0.789382	+	0.701313i
$88$		0			0
$89$	16.1630			1.71327			0.856637	−	0.515920i	$-0.172550\pi$
$89$	16.1630			1.71327			0.856637	+	0.515920i	$0.172550\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	3.66552	−	3.25657i	0.380097	−	0.337691i
$94$		0			0
$95$		−	3.28736i		−	0.337276i
$96$		0			0
$97$		−	12.0577i		−	1.22427i	−0.790751	−	0.612137i	$-0.790310\pi$
$97$		−	12.0577i		−	1.22427i	0.790751	−	0.612137i	$-0.209690\pi$
$98$		0			0
$99$	4.02778	+	0.477584i	0.404807	+	0.0479990i
$100$		0			0
$101$	−18.0337			−1.79442			−0.897208	−	0.441608i	$-0.854408\pi$
$101$	−18.0337			−1.79442			−0.897208	+	0.441608i	$0.854408\pi$
$102$		0			0
$103$			0.616166i			0.0607126i	0.999539	+	0.0303563i	$0.00966420\pi$
$103$			0.616166i			0.0607126i	−0.999539	+	0.0303563i	$0.990336\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	17.7101i		−	1.71210i	−0.516896	−	0.856048i	$-0.672913\pi$
$107$		−	17.7101i		−	1.71210i	0.516896	−	0.856048i	$-0.327087\pi$
$108$		0			0
$109$	−1.71056			−0.163842			−0.0819211	−	0.996639i	$-0.526106\pi$
$109$	−1.71056			−0.163842			−0.0819211	+	0.996639i	$0.526106\pi$
$110$		0			0
$111$	−4.43001	−	4.98632i	−0.420478	−	0.473280i
$112$		0			0
$113$			13.1214i			1.23436i	0.786822	+	0.617180i	$0.211725\pi$
$113$			13.1214i			1.23436i	−0.786822	+	0.617180i	$0.788275\pi$
$114$		0			0
$115$			5.00540i			0.466756i
$116$		0			0
$117$	−14.7257	−	1.74607i	−1.36139	−	0.161424i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.17210			0.833828
$122$		0			0
$123$	4.30015	+	4.84015i	0.387732	+	0.436422i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−15.5324			−1.37828			−0.689139	−	0.724629i	$-0.742011\pi$
$127$	−15.5324			−1.37828			−0.689139	+	0.724629i	$0.742011\pi$
$128$		0			0
$129$	−4.67445	−	5.26145i	−0.411562	−	0.463245i
$130$		0			0
$131$	8.95644			0.782528			0.391264	−	0.920278i	$-0.372038\pi$
$131$	8.95644			0.782528			0.391264	+	0.920278i	$0.372038\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−4.26388	+	2.96974i	−0.366976	+	0.255595i
$136$		0			0
$137$		−	15.4785i		−	1.32241i	−0.750204	−	0.661207i	$-0.770045\pi$
$137$		−	15.4785i		−	1.32241i	0.750204	−	0.661207i	$-0.229955\pi$
$138$		0			0
$139$		−	20.1547i		−	1.70950i	−0.519044	−	0.854748i	$-0.673712\pi$
$139$		−	20.1547i		−	1.70950i	0.519044	−	0.854748i	$-0.326288\pi$
$140$		0			0
$141$	−6.54103	−	7.36243i	−0.550854	−	0.620029i
$142$		0			0
$143$	6.68287			0.558850
$144$		0			0
$145$			5.68630i			0.472221i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	1.44277i		−	0.118196i	−0.998252	−	0.0590981i	$-0.981178\pi$
$149$		−	1.44277i		−	0.118196i	0.998252	−	0.0590981i	$-0.0188225\pi$
$150$		0			0
$151$	16.2422			1.32177			0.660884	−	0.750488i	$-0.270182\pi$
$151$	16.2422			1.32177			0.660884	+	0.750488i	$0.270182\pi$
$152$		0			0
$153$	2.02836	−	17.1065i	0.163983	−	1.38298i
$154$		0			0
$155$		−	2.83086i		−	0.227380i
$156$		0			0
$157$		−	18.2375i		−	1.45551i	−0.685836	−	0.727756i	$-0.740563\pi$
$157$		−	18.2375i		−	1.45551i	0.685836	−	0.727756i	$-0.259437\pi$
$158$		0			0
$159$	−1.89247	+	1.68134i	−0.150083	+	0.133339i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−24.3260			−1.90536			−0.952679	−	0.303978i	$-0.901685\pi$
$163$	−24.3260			−1.90536			−0.952679	+	0.303978i	$0.901685\pi$
$164$		0			0
$165$	1.75063	−	1.55531i	0.136286	−	0.121081i
$166$		0			0
$167$	12.5007			0.967336			0.483668	−	0.875252i	$-0.339304\pi$
$167$	12.5007			0.967336			0.483668	+	0.875252i	$0.339304\pi$
$168$		0			0
$169$	−11.4329			−0.879453
$170$		0			0
$171$	−9.79349	−	1.16124i	−0.748927	−	0.0888023i
$172$		0			0
$173$	14.8418			1.12840			0.564200	−	0.825638i	$-0.309185\pi$
$173$	14.8418			1.12840			0.564200	+	0.825638i	$0.309185\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.99080	−	11.2454i	−0.750955	−	0.845258i
$178$		0			0
$179$		−	11.4723i		−	0.857483i	−0.903427	−	0.428741i	$-0.858957\pi$
$179$		−	11.4723i		−	0.857483i	0.903427	−	0.428741i	$-0.141043\pi$
$180$		0			0
$181$			5.91099i			0.439361i	0.975572	+	0.219680i	$0.0705014\pi$
$181$			5.91099i			0.439361i	−0.975572	+	0.219680i	$0.929499\pi$
$182$		0			0
$183$	2.47158	−	2.19584i	0.182705	−	0.162321i
$184$		0			0
$185$	−3.85090			−0.283124
$186$		0			0
$187$			7.76330i			0.567709i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			10.5743i			0.765132i	0.923928	+	0.382566i	$0.124959\pi$
$191$			10.5743i			0.765132i	−0.923928	+	0.382566i	$0.875041\pi$
$192$		0			0
$193$	12.0646			0.868430			0.434215	−	0.900809i	$-0.357026\pi$
$193$	12.0646			0.868430			0.434215	+	0.900809i	$0.357026\pi$
$194$		0			0
$195$	−6.40037	+	5.68630i	−0.458340	+	0.407204i
$196$		0			0
$197$		−	17.2423i		−	1.22846i	−0.789126	−	0.614231i	$-0.789466\pi$
$197$		−	17.2423i		−	1.22846i	0.789126	−	0.614231i	$-0.210534\pi$
$198$		0			0
$199$		−	2.84518i		−	0.201689i	−0.994902	−	0.100845i	$-0.967845\pi$
$199$		−	2.84518i		−	0.201689i	0.994902	−	0.100845i	$-0.0321545\pi$
$200$		0			0
$201$	−5.79653	−	6.52444i	−0.408855	−	0.460199i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	3.73802			0.261075
$206$		0			0
$207$	14.9117	+	1.76812i	1.03644	+	0.122893i
$208$		0			0
$209$	4.44451			0.307433
$210$		0			0
$211$	22.4677			1.54674			0.773372	−	0.633953i	$-0.218569\pi$
$211$	22.4677			1.54674			0.773372	+	0.633953i	$0.218569\pi$
$212$		0			0
$213$	4.37993	−	3.89127i	0.300108	−	0.266626i
$214$		0			0
$215$	−4.06339			−0.277121
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−21.1413	+	18.7826i	−1.42859	+	1.26921i
$220$		0			0
$221$		−	28.3830i		−	1.90925i
$222$		0			0
$223$			5.27620i			0.353321i	0.984272	+	0.176660i	$0.0565294\pi$
$223$			5.27620i			0.353321i	−0.984272	+	0.176660i	$0.943471\pi$
$224$		0			0
$225$	−0.353244	+	2.97913i	−0.0235496	+	0.198609i
$226$		0			0
$227$	−12.4811			−0.828400			−0.414200	−	0.910186i	$-0.635939\pi$
$227$	−12.4811			−0.828400			−0.414200	+	0.910186i	$0.635939\pi$
$228$		0			0
$229$		−	16.2676i		−	1.07499i	−0.843266	−	0.537496i	$-0.819370\pi$
$229$		−	16.2676i		−	1.07499i	0.843266	−	0.537496i	$-0.180630\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			8.83101i			0.578539i	0.957248	+	0.289269i	$0.0934124\pi$
$233$			8.83101i			0.578539i	−0.957248	+	0.289269i	$0.906588\pi$
$234$		0			0
$235$	−5.68596			−0.370911
$236$		0			0
$237$	−5.56079	−	6.25911i	−0.361212	−	0.406573i
$238$		0			0
$239$			12.3333i			0.797773i	0.917000	+	0.398886i	$0.130603\pi$
$239$			12.3333i			0.797773i	−0.917000	+	0.398886i	$0.869397\pi$
$240$		0			0
$241$		−	7.95376i		−	0.512346i	−0.966631	−	0.256173i	$-0.917538\pi$
$241$		−	7.95376i		−	0.512346i	0.966631	−	0.256173i	$-0.0824618\pi$
$242$		0			0
$243$	7.34106	+	13.7517i	0.470929	+	0.882171i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−16.2493			−1.03392
$248$		0			0
$249$	19.1568	+	21.5624i	1.21401	+	1.36646i
$250$		0			0
$251$	−7.64756			−0.482710			−0.241355	−	0.970437i	$-0.577592\pi$
$251$	−7.64756			−0.482710			−0.241355	+	0.970437i	$0.577592\pi$
$252$		0			0
$253$	−6.76728			−0.425455
$254$		0			0
$255$	−6.60560	−	7.43512i	−0.413659	−	0.465605i
$256$		0			0
$257$	−27.4956			−1.71513			−0.857563	−	0.514379i	$-0.828022\pi$
$257$	−27.4956			−1.71513			−0.857563	+	0.514379i	$0.828022\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	16.9402	+	2.00865i	1.04857	+	0.124332i
$262$		0			0
$263$		−	19.0223i		−	1.17297i	−0.809962	−	0.586483i	$-0.800512\pi$
$263$		−	19.0223i		−	1.17297i	0.809962	−	0.586483i	$-0.199488\pi$
$264$		0			0
$265$			1.46155i			0.0897821i
$266$		0			0
$267$	−18.5936	−	20.9286i	−1.13791	−	1.28081i
$268$		0			0
$269$	−8.92020			−0.543874			−0.271937	−	0.962315i	$-0.587664\pi$
$269$	−8.92020			−0.543874			−0.271937	+	0.962315i	$0.587664\pi$
$270$		0			0
$271$			7.54161i			0.458120i	0.973412	+	0.229060i	$0.0735653\pi$
$271$			7.54161i			0.458120i	−0.973412	+	0.229060i	$0.926435\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	1.35200i		−	0.0815285i
$276$		0			0
$277$	−1.77712			−0.106777			−0.0533885	−	0.998574i	$-0.517002\pi$
$277$	−1.77712			−0.106777			−0.0533885	+	0.998574i	$0.517002\pi$
$278$		0			0
$279$	−8.43350	−	0.999983i	−0.504900	−	0.0598674i
$280$		0			0
$281$			25.3819i			1.51416i	0.653324	+	0.757079i	$0.273374\pi$
$281$			25.3819i			1.51416i	−0.653324	+	0.757079i	$0.726626\pi$
$282$		0			0
$283$			16.1397i			0.959408i	0.877430	+	0.479704i	$0.159256\pi$
$283$			16.1397i			0.959408i	−0.877430	+	0.479704i	$0.840744\pi$
$284$		0			0
$285$	−4.25662	+	3.78172i	−0.252141	+	0.224010i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	15.9717			0.939510
$290$		0			0
$291$	−15.6129	+	13.8710i	−0.915242	+	0.813130i
$292$		0			0
$293$	19.5542			1.14237			0.571186	−	0.820821i	$-0.306484\pi$
$293$	19.5542			1.14237			0.571186	+	0.820821i	$0.306484\pi$
$294$		0			0
$295$	−8.68477			−0.505647
$296$		0			0
$297$	−4.01508	−	5.76475i	−0.232979	−	0.334505i
$298$		0			0
$299$	24.7415			1.43084
$300$		0			0
$301$		0			0
$302$		0			0
$303$	20.7456	+	23.3508i	1.19180	+	1.34147i
$304$		0			0
$305$		−	1.90879i		−	0.109297i
$306$		0			0
$307$		−	11.8231i		−	0.674777i	−0.941365	−	0.337389i	$-0.890456\pi$
$307$		−	11.8231i		−	0.674777i	0.941365	−	0.337389i	$-0.109544\pi$
$308$		0			0
$309$	0.797839	−	0.708826i	0.0453875	−	0.0403237i
$310$		0			0
$311$	22.3378			1.26666			0.633331	−	0.773881i	$-0.281687\pi$
$311$	22.3378			1.26666			0.633331	+	0.773881i	$0.281687\pi$
$312$		0			0
$313$		−	13.3080i		−	0.752210i	−0.926577	−	0.376105i	$-0.877263\pi$
$313$		−	13.3080i		−	0.752210i	0.926577	−	0.376105i	$-0.122737\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	33.8784i		−	1.90280i	−0.307958	−	0.951400i	$-0.599645\pi$
$317$		−	33.8784i		−	1.90280i	0.307958	−	0.951400i	$-0.400355\pi$
$318$		0			0
$319$	−7.68786			−0.430437
$320$		0			0
$321$	−22.9318	+	20.3733i	−1.27993	+	1.13713i
$322$		0			0
$323$		−	18.8764i		−	1.05031i
$324$		0			0
$325$			4.94296i			0.274186i
$326$		0			0
$327$	1.96780	+	2.21491i	0.108820	+	0.122485i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.97965			−0.108811			−0.0544057	−	0.998519i	$-0.517326\pi$
$331$	−1.97965			−0.108811			−0.0544057	+	0.998519i	$0.517326\pi$
$332$		0			0
$333$	−1.36031	+	11.4723i	−0.0745443	+	0.628680i
$334$		0			0
$335$	−5.03878			−0.275298
$336$		0			0
$337$	−12.0217			−0.654862			−0.327431	−	0.944875i	$-0.606183\pi$
$337$	−12.0217			−0.654862			−0.327431	+	0.944875i	$0.606183\pi$
$338$		0			0
$339$	16.9902	−	15.0946i	0.922781	−	0.819828i
$340$		0			0
$341$	3.82731			0.207261
$342$		0			0
$343$		0			0
$344$		0			0
$345$	6.48121	−	5.75812i	0.348936	−	0.310007i
$346$		0			0
$347$			14.6637i			0.787190i	0.919284	+	0.393595i	$0.128769\pi$
$347$			14.6637i			0.787190i	−0.919284	+	0.393595i	$0.871231\pi$
$348$		0			0
$349$		−	25.0573i		−	1.34129i	−0.741780	−	0.670644i	$-0.766018\pi$
$349$		−	25.0573i		−	1.34129i	0.741780	−	0.670644i	$-0.233982\pi$
$350$		0			0
$351$	14.6793	+	21.0762i	0.783525	+	1.12496i
$352$		0			0
$353$	−7.76731			−0.413412			−0.206706	−	0.978403i	$-0.566274\pi$
$353$	−7.76731			−0.413412			−0.206706	+	0.978403i	$0.566274\pi$
$354$		0			0
$355$		−	3.38259i		−	0.179529i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	4.46814i		−	0.235819i	−0.993024	−	0.117910i	$-0.962381\pi$
$359$		−	4.46814i		−	0.235819i	0.993024	−	0.117910i	$-0.0376193\pi$
$360$		0			0
$361$	8.19324			0.431223
$362$		0			0
$363$	−10.5514	−	11.8764i	−0.553806	−	0.623352i
$364$		0			0
$365$			16.3273i			0.854608i
$366$		0			0
$367$		−	1.68448i		−	0.0879292i	−0.999033	−	0.0439646i	$-0.986001\pi$
$367$		−	1.68448i		−	0.0879292i	0.999033	−	0.0439646i	$-0.0139989\pi$
$368$		0			0
$369$	1.32043	−	11.1361i	0.0687389	−	0.579720i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	13.8464			0.716942			0.358471	−	0.933541i	$-0.383298\pi$
$373$	13.8464			0.716942			0.358471	+	0.933541i	$0.383298\pi$
$374$		0			0
$375$	1.15038	+	1.29484i	0.0594055	+	0.0668655i
$376$		0			0
$377$	28.1072			1.44759
$378$		0			0
$379$	−2.15603			−0.110748			−0.0553739	−	0.998466i	$-0.517635\pi$
$379$	−2.15603			−0.110748			−0.0553739	+	0.998466i	$0.517635\pi$
$380$		0			0
$381$	17.8682	+	20.1121i	0.915416	+	1.03037i
$382$		0			0
$383$	3.73196			0.190694			0.0953471	−	0.995444i	$-0.469604\pi$
$383$	3.73196			0.190694			0.0953471	+	0.995444i	$0.469604\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.43537	+	12.1054i	−0.0729637	+	0.615350i
$388$		0			0
$389$			13.8000i			0.699688i	0.936808	+	0.349844i	$0.113765\pi$
$389$			13.8000i			0.699688i	−0.936808	+	0.349844i	$0.886235\pi$
$390$		0			0
$391$			28.7415i			1.45352i
$392$		0			0
$393$	−10.3033	−	11.5972i	−0.519734	−	0.585001i
$394$		0			0
$395$	−4.83387			−0.243218
$396$		0			0
$397$		−	16.3513i		−	0.820649i	−0.911940	−	0.410324i	$-0.865416\pi$
$397$		−	16.3513i		−	0.820649i	0.911940	−	0.410324i	$-0.134584\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	13.8182i		−	0.690048i	−0.938594	−	0.345024i	$-0.887871\pi$
$401$		−	13.8182i		−	0.690048i	0.938594	−	0.345024i	$-0.112129\pi$
$402$		0			0
$403$	−13.9928			−0.697033
$404$		0			0
$405$	8.75044	+	2.10472i	0.434813	+	0.104584i
$406$		0			0
$407$		−	5.20641i		−	0.258072i
$408$		0			0
$409$			9.54923i			0.472179i	0.971731	+	0.236089i	$0.0758658\pi$
$409$			9.54923i			0.472179i	−0.971731	+	0.236089i	$0.924134\pi$
$410$		0			0
$411$	−20.0422	+	17.8061i	−0.988608	+	0.878312i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	16.6525			0.817441
$416$		0			0
$417$	−26.0971	+	23.1855i	−1.27798	+	1.13540i
$418$		0			0
$419$	9.71886			0.474797			0.237399	−	0.971412i	$-0.423705\pi$
$419$	9.71886			0.474797			0.237399	+	0.971412i	$0.423705\pi$
$420$		0			0
$421$	34.8713			1.69953			0.849763	−	0.527165i	$-0.176745\pi$
$421$	34.8713			1.69953			0.849763	+	0.527165i	$0.176745\pi$
$422$		0			0
$423$	−2.00853	+	16.9392i	−0.0976580	+	0.823613i
$424$		0			0
$425$	−5.74210			−0.278533
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−7.68786	−	8.65328i	−0.371173	−	0.417784i
$430$		0			0
$431$			1.81610i			0.0874786i	0.999043	+	0.0437393i	$0.0139271\pi$
$431$			1.81610i			0.0874786i	−0.999043	+	0.0437393i	$0.986073\pi$
$432$		0			0
$433$			10.6773i			0.513117i	0.966529	+	0.256558i	$0.0825886\pi$
$433$			10.6773i			0.513117i	−0.966529	+	0.256558i	$0.917411\pi$
$434$		0			0
$435$	7.36287	−	6.54141i	0.353022	−	0.313637i
$436$		0			0
$437$	16.4546			0.787128
$438$		0			0
$439$			30.8027i			1.47013i	0.677996	+	0.735066i	$0.262849\pi$
$439$			30.8027i			1.47013i	−0.677996	+	0.735066i	$0.737151\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	16.5328i		−	0.785497i	−0.919646	−	0.392749i	$-0.871524\pi$
$443$		−	16.5328i		−	0.785497i	0.919646	−	0.392749i	$-0.128476\pi$
$444$		0			0
$445$	−16.1630			−0.766199
$446$		0			0
$447$	−1.86816	+	1.65974i	−0.0883610	+	0.0785028i
$448$		0			0
$449$		−	2.59098i		−	0.122276i	−0.998129	−	0.0611380i	$-0.980527\pi$
$449$		−	2.59098i		−	0.122276i	0.998129	−	0.0611380i	$-0.0194730\pi$
$450$		0			0
$451$			5.05379i			0.237974i
$452$		0			0
$453$	−18.6847	−	21.0311i	−0.877883	−	0.988126i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−22.1748			−1.03729			−0.518647	−	0.854988i	$-0.673564\pi$
$457$	−22.1748			−1.03729			−0.518647	+	0.854988i	$0.673564\pi$
$458$		0			0
$459$	−24.4836	+	17.0525i	−1.14280	+	0.795945i
$460$		0			0
$461$	−30.5304			−1.42194			−0.710971	−	0.703221i	$-0.751744\pi$
$461$	−30.5304			−1.42194			−0.710971	+	0.703221i	$0.751744\pi$
$462$		0			0
$463$	3.54824			0.164901			0.0824504	−	0.996595i	$-0.473725\pi$
$463$	3.54824			0.164901			0.0824504	+	0.996595i	$0.473725\pi$
$464$		0			0
$465$	−3.66552	+	3.25657i	−0.169985	+	0.151020i
$466$		0			0
$467$	34.3127			1.58780			0.793901	−	0.608047i	$-0.208047\pi$
$467$	34.3127			1.58780			0.793901	+	0.608047i	$0.208047\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−23.6147	+	20.9801i	−1.08811	+	0.966712i
$472$		0			0
$473$		−	5.49369i		−	0.252600i
$474$		0			0
$475$			3.28736i			0.150835i
$476$		0			0
$477$	4.35414	+	0.516282i	0.199362	+	0.0236389i
$478$		0			0
$479$	19.5725			0.894289			0.447145	−	0.894462i	$-0.352441\pi$
$479$	19.5725			0.894289			0.447145	+	0.894462i	$0.352441\pi$
$480$		0			0
$481$			19.0349i			0.867915i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			12.0577i			0.547512i
$486$		0			0
$487$	25.2265			1.14312			0.571561	−	0.820559i	$-0.306338\pi$
$487$	25.2265			1.14312			0.571561	+	0.820559i	$0.306338\pi$
$488$		0			0
$489$	27.9842	+	31.4984i	1.26549	+	1.42441i
$490$		0			0
$491$			38.6556i			1.74450i	0.489057	+	0.872252i	$0.337341\pi$
$491$			38.6556i			1.74450i	−0.489057	+	0.872252i	$0.662659\pi$
$492$		0			0
$493$			32.6513i			1.47054i
$494$		0			0
$495$	−4.02778	−	0.477584i	−0.181035	−	0.0214658i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	5.61821			0.251506			0.125753	−	0.992062i	$-0.459865\pi$
$499$	5.61821			0.251506			0.125753	+	0.992062i	$0.459865\pi$
$500$		0			0
$501$	−14.3806	−	16.1865i	−0.642478	−	0.723160i
$502$		0			0
$503$	25.8655			1.15329			0.576644	−	0.816996i	$-0.304362\pi$
$503$	25.8655			1.15329			0.576644	+	0.816996i	$0.304362\pi$
$504$		0			0
$505$	18.0337			0.802487
$506$		0			0
$507$	13.1522	+	14.8038i	0.584109	+	0.657460i
$508$		0			0
$509$	−30.4611			−1.35016			−0.675082	−	0.737742i	$-0.735892\pi$
$509$	−30.4611			−1.35016			−0.675082	+	0.737742i	$0.735892\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	9.76262	+	14.0169i	0.431030	+	0.618862i
$514$		0			0
$515$		−	0.616166i		−	0.0271515i
$516$		0			0
$517$		−	7.68740i		−	0.338092i
$518$		0			0
$519$	−17.0737	−	19.2178i	−0.749453	−	0.843568i
$520$		0			0
$521$	21.2364			0.930384			0.465192	−	0.885210i	$-0.345985\pi$
$521$	21.2364			0.930384			0.465192	+	0.885210i	$0.345985\pi$
$522$		0			0
$523$		−	12.3815i		−	0.541407i	−0.962663	−	0.270704i	$-0.912744\pi$
$523$		−	12.3815i		−	0.541407i	0.962663	−	0.270704i	$-0.0872563\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	16.2551i		−	0.708082i
$528$		0			0
$529$	−2.05398			−0.0893036
$530$		0			0
$531$	−3.06784	+	25.8731i	−0.133133	+	1.12280i
$532$		0			0
$533$		−	18.4769i		−	0.800323i
$534$		0			0
$535$			17.7101i			0.765673i
$536$		0			0
$537$	−14.8549	+	13.1976i	−0.641036	+	0.569517i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	13.8600			0.595888			0.297944	−	0.954583i	$-0.403699\pi$
$541$	13.8600			0.595888			0.297944	+	0.954583i	$0.403699\pi$
$542$		0			0
$543$	7.65381	−	6.79990i	0.328457	−	0.291812i
$544$		0			0
$545$	1.71056			0.0732725
$546$		0			0
$547$	17.7100			0.757226			0.378613	−	0.925555i	$-0.376401\pi$
$547$	17.7100			0.757226			0.378613	+	0.925555i	$0.376401\pi$
$548$		0			0
$549$	−5.68653	−	0.674267i	−0.242695	−	0.0287770i
$550$		0			0
$551$	18.6929			0.796345
$552$		0			0
$553$		0			0
$554$		0			0
$555$	4.43001	+	4.98632i	0.188043	+	0.211657i
$556$		0			0
$557$		−	3.47714i		−	0.147331i	−0.997283	−	0.0736656i	$-0.976530\pi$
$557$		−	3.47714i		−	0.147331i	0.997283	−	0.0736656i	$-0.0234697\pi$
$558$		0			0
$559$			20.0852i			0.849512i
$560$		0			0
$561$	10.0523	−	8.93076i	0.424407	−	0.377057i
$562$		0			0
$563$	12.0579			0.508182			0.254091	−	0.967180i	$-0.418224\pi$
$563$	12.0579			0.508182			0.254091	+	0.967180i	$0.418224\pi$
$564$		0			0
$565$		−	13.1214i		−	0.552022i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	13.6330i		−	0.571525i	−0.958301	−	0.285762i	$-0.907753\pi$
$569$		−	13.6330i		−	0.571525i	0.958301	−	0.285762i	$-0.0922468\pi$
$570$		0			0
$571$	−44.0949			−1.84531			−0.922657	−	0.385623i	$-0.873987\pi$
$571$	−44.0949			−1.84531			−0.922657	+	0.385623i	$0.873987\pi$
$572$		0			0
$573$	13.6921	−	12.1645i	0.571996	−	0.508180i
$574$		0			0
$575$		−	5.00540i		−	0.208739i
$576$		0			0
$577$			6.99868i			0.291359i	0.989332	+	0.145679i	$0.0465368\pi$
$577$			6.99868i			0.291359i	−0.989332	+	0.145679i	$0.953463\pi$
$578$		0			0
$579$	−13.8789	−	15.6218i	−0.576788	−	0.649220i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.97601			−0.0818378
$584$		0			0
$585$	14.7257	+	1.74607i	0.608834	+	0.0721911i
$586$		0			0
$587$	29.9161			1.23477			0.617385	−	0.786661i	$-0.288192\pi$
$587$	29.9161			1.23477			0.617385	+	0.786661i	$0.288192\pi$
$588$		0			0
$589$	−9.30607			−0.383450
$590$		0			0
$591$	−22.3261	+	19.8352i	−0.918372	+	0.815911i
$592$		0			0
$593$	24.0851			0.989058			0.494529	−	0.869161i	$-0.335341\pi$
$593$	24.0851			0.989058			0.494529	+	0.869161i	$0.335341\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−3.68407	+	3.27304i	−0.150779	+	0.133957i
$598$		0			0
$599$			5.94409i			0.242869i	0.992599	+	0.121434i	$0.0387494\pi$
$599$			5.94409i			0.242869i	−0.992599	+	0.121434i	$0.961251\pi$
$600$		0			0
$601$			46.9992i			1.91714i	0.284863	+	0.958568i	$0.408052\pi$
$601$			46.9992i			1.91714i	−0.284863	+	0.958568i	$0.591948\pi$
$602$		0			0
$603$	−1.77992	+	15.0112i	−0.0724839	+	0.611303i
$604$		0			0
$605$	−9.17210			−0.372899
$606$		0			0
$607$		−	9.94645i		−	0.403714i	−0.979415	−	0.201857i	$-0.935302\pi$
$607$		−	9.94645i		−	0.403714i	0.979415	−	0.201857i	$-0.0646976\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			28.1055i			1.13703i
$612$		0			0
$613$	−32.5479			−1.31460			−0.657298	−	0.753631i	$-0.728301\pi$
$613$	−32.5479			−1.31460			−0.657298	+	0.753631i	$0.728301\pi$
$614$		0			0
$615$	−4.30015	−	4.84015i	−0.173399	−	0.195174i
$616$		0			0
$617$			7.69843i			0.309927i	0.987920	+	0.154964i	$0.0495260\pi$
$617$			7.69843i			0.309927i	−0.987920	+	0.154964i	$0.950474\pi$
$618$		0			0
$619$		−	3.18894i		−	0.128174i	−0.997944	−	0.0640872i	$-0.979586\pi$
$619$		−	3.18894i		−	0.128174i	0.997944	−	0.0640872i	$-0.0204136\pi$
$620$		0			0
$621$	−14.8647	−	21.3424i	−0.596501	−	0.856440i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−5.11288	−	5.75495i	−0.204189	−	0.229830i
$628$		0			0
$629$	−22.1122			−0.881673
$630$		0			0
$631$	7.97461			0.317464			0.158732	−	0.987322i	$-0.449259\pi$
$631$	7.97461			0.317464			0.158732	+	0.987322i	$0.449259\pi$
$632$		0			0
$633$	−25.8465	−	29.0922i	−1.02731	−	1.15631i
$634$		0			0
$635$	15.5324			0.616385
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−10.0772	−	1.19488i	−0.398647	−	0.0472687i
$640$		0			0
$641$			27.4558i			1.08444i	0.840237	+	0.542219i	$0.182416\pi$
$641$			27.4558i			1.08444i	−0.840237	+	0.542219i	$0.817584\pi$
$642$		0			0
$643$			29.0919i			1.14727i	0.819110	+	0.573637i	$0.194468\pi$
$643$			29.0919i			1.14727i	−0.819110	+	0.573637i	$0.805532\pi$
$644$		0			0
$645$	4.67445	+	5.26145i	0.184056	+	0.207170i
$646$		0			0
$647$	43.1999			1.69836			0.849181	−	0.528102i	$-0.177096\pi$
$647$	43.1999			1.69836			0.849181	+	0.528102i	$0.177096\pi$
$648$		0			0
$649$		−	11.7418i		−	0.460905i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	7.24161i		−	0.283386i	−0.989911	−	0.141693i	$-0.954745\pi$
$653$		−	7.24161i		−	0.283386i	0.989911	−	0.141693i	$-0.0452546\pi$
$654$		0			0
$655$	−8.95644			−0.349957
$656$		0			0
$657$	48.6411	+	5.76750i	1.89767	+	0.225012i
$658$		0			0
$659$			5.29324i			0.206196i	0.994671	+	0.103098i	$0.0328755\pi$
$659$			5.29324i			0.206196i	−0.994671	+	0.103098i	$0.967125\pi$
$660$		0			0
$661$		−	49.9567i		−	1.94309i	−0.236856	−	0.971545i	$-0.576117\pi$
$661$		−	49.9567i		−	1.94309i	0.236856	−	0.971545i	$-0.423883\pi$
$662$		0			0
$663$	−36.7515	+	32.6513i	−1.42731	+	1.26807i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−28.4622			−1.10206
$668$		0			0
$669$	6.83186	−	6.06965i	0.264135	−	0.234666i
$670$		0			0
$671$	2.58068			0.0996259
$672$		0			0
$673$	−21.8855			−0.843625			−0.421812	−	0.906683i	$-0.638606\pi$
$673$	−21.8855			−0.843625			−0.421812	+	0.906683i	$0.638606\pi$
$674$		0			0
$675$	4.26388	−	2.96974i	0.164117	−	0.114305i
$676$		0			0
$677$	13.7447			0.528253			0.264126	−	0.964488i	$-0.414916\pi$
$677$	13.7447			0.528253			0.264126	+	0.964488i	$0.414916\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	14.3580	+	16.1611i	0.550201	+	0.619294i
$682$		0			0
$683$			42.9026i			1.64162i	0.571200	+	0.820811i	$0.306478\pi$
$683$			42.9026i			1.64162i	−0.571200	+	0.820811i	$0.693522\pi$
$684$		0			0
$685$			15.4785i			0.591401i
$686$		0			0
$687$	−21.0640	+	18.7139i	−0.803642	+	0.713982i
$688$		0			0
$689$	7.22437			0.275227
$690$		0			0
$691$		−	23.7284i		−	0.902670i	−0.892355	−	0.451335i	$-0.850948\pi$
$691$		−	23.7284i		−	0.902670i	0.892355	−	0.451335i	$-0.149052\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			20.1547i			0.764509i
$696$		0			0
$697$	21.4641			0.813010
$698$		0			0
$699$	11.4348	−	10.1590i	0.432503	−	0.384250i
$700$		0			0
$701$		−	21.2721i		−	0.803436i	−0.915763	−	0.401718i	$-0.868413\pi$
$701$		−	21.2721i		−	0.803436i	0.915763	−	0.401718i	$-0.131587\pi$
$702$		0			0
$703$			12.6593i			0.477455i
$704$		0			0
$705$	6.54103	+	7.36243i	0.246349	+	0.277285i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−45.6137			−1.71306			−0.856529	−	0.516098i	$-0.827384\pi$
$709$	−45.6137			−1.71306			−0.856529	+	0.516098i	$0.827384\pi$
$710$		0			0
$711$	−1.70753	+	14.4007i	−0.0640375	+	0.540069i
$712$		0			0
$713$	14.1696			0.530655
$714$		0			0
$715$	−6.68287			−0.249925
$716$		0			0
$717$	15.9697	−	14.1880i	0.596398	−	0.529859i
$718$		0			0
$719$	19.0170			0.709215			0.354607	−	0.935015i	$-0.384615\pi$
$719$	19.0170			0.709215			0.354607	+	0.935015i	$0.384615\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−10.2989	+	9.14986i	−0.383019	+	0.340287i
$724$		0			0
$725$		−	5.68630i		−	0.211184i
$726$		0			0
$727$		−	34.0540i		−	1.26299i	−0.775379	−	0.631496i	$-0.782441\pi$
$727$		−	34.0540i		−	1.26299i	0.775379	−	0.631496i	$-0.217559\pi$
$728$		0			0
$729$	9.36126	−	25.3252i	0.346714	−	0.937971i
$730$		0			0
$731$	−23.3324			−0.862978
$732$		0			0
$733$		−	7.75770i		−	0.286537i	−0.989684	−	0.143269i	$-0.954239\pi$
$733$		−	7.75770i		−	0.286537i	0.989684	−	0.143269i	$-0.0457613\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	6.81242i		−	0.250939i
$738$		0			0
$739$	−8.39318			−0.308748			−0.154374	−	0.988012i	$-0.549336\pi$
$739$	−8.39318			−0.308748			−0.154374	+	0.988012i	$0.549336\pi$
$740$		0			0
$741$	18.6929	+	21.0403i	0.686702	+	0.772936i
$742$		0			0
$743$			14.7540i			0.541273i	0.962682	+	0.270636i	$0.0872341\pi$
$743$			14.7540i			0.541273i	−0.962682	+	0.270636i	$0.912766\pi$
$744$		0			0
$745$			1.44277i			0.0528590i
$746$		0			0
$747$	5.88240	−	49.6101i	0.215226	−	1.81514i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	41.2133			1.50390			0.751948	−	0.659222i	$-0.229114\pi$
$751$	41.2133			1.50390			0.751948	+	0.659222i	$0.229114\pi$
$752$		0			0
$753$	8.79762	+	9.90240i	0.320603	+	0.360864i
$754$		0			0
$755$	−16.2422			−0.591113
$756$		0			0
$757$	−11.8681			−0.431354			−0.215677	−	0.976465i	$-0.569196\pi$
$757$	−11.8681			−0.431354			−0.215677	+	0.976465i	$0.569196\pi$
$758$		0			0
$759$	7.78496	+	8.76258i	0.282576	+	0.318061i
$760$		0			0
$761$	−35.5411			−1.28836			−0.644181	−	0.764873i	$-0.722802\pi$
$761$	−35.5411			−1.28836			−0.644181	+	0.764873i	$0.722802\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.02836	+	17.1065i	−0.0733355	+	0.618485i
$766$		0			0
$767$			42.9285i			1.55006i
$768$		0			0
$769$			6.25608i			0.225600i	0.993618	+	0.112800i	$0.0359820\pi$
$769$			6.25608i			0.225600i	−0.993618	+	0.112800i	$0.964018\pi$
$770$		0			0
$771$	31.6304	+	35.6025i	1.13914	+	1.28219i
$772$		0			0
$773$	20.0321			0.720506			0.360253	−	0.932855i	$-0.382690\pi$
$773$	20.0321			0.720506			0.360253	+	0.932855i	$0.382690\pi$
$774$		0			0
$775$			2.83086i			0.101687i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	12.2882i		−	0.440272i
$780$		0			0
$781$	4.57326			0.163644
$782$		0			0
$783$	−16.8868	−	24.2457i	−0.603486	−	0.866469i
$784$		0			0
$785$			18.2375i			0.650925i
$786$		0			0
$787$			1.47611i			0.0526175i	0.999654	+	0.0263088i	$0.00837530\pi$
$787$			1.47611i			0.0526175i	−0.999654	+	0.0263088i	$0.991625\pi$
$788$		0			0
$789$	−24.6309	+	21.8829i	−0.876884	+	0.779052i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−9.43507			−0.335049
$794$		0			0
$795$	1.89247	−	1.68134i	0.0671192	−	0.0596308i
$796$		0			0
$797$	−22.9470			−0.812823			−0.406411	−	0.913690i	$-0.633220\pi$
$797$	−22.9470			−0.812823			−0.406411	+	0.913690i	$0.633220\pi$
$798$		0			0
$799$	−32.6493			−1.15505
$800$		0			0
$801$	−5.70947	+	48.1517i	−0.201734	+	1.70136i
$802$		0			0
$803$	−22.0744			−0.778990
$804$		0			0
$805$		0			0
$806$		0			0
$807$	10.2616	+	11.5503i	0.361227	+	0.406589i
$808$		0			0
$809$		−	22.5874i		−	0.794132i	−0.917790	−	0.397066i	$-0.870028\pi$
$809$		−	22.5874i		−	0.794132i	0.917790	−	0.397066i	$-0.129972\pi$
$810$		0			0
$811$		−	41.9366i		−	1.47259i	−0.676659	−	0.736296i	$-0.736573\pi$
$811$		−	41.9366i		−	1.47259i	0.676659	−	0.736296i	$-0.263427\pi$
$812$		0			0
$813$	9.76522	−	8.67573i	0.342481	−	0.304271i
$814$		0			0
$815$	24.3260			0.852102
$816$		0			0
$817$			13.3578i			0.467331i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			0.516180i			0.0180148i	0.999959	+	0.00900741i	$0.00286718\pi$
$821$			0.516180i			0.0180148i	−0.999959	+	0.00900741i	$0.997133\pi$
$822$		0			0
$823$	−20.5681			−0.716960			−0.358480	−	0.933537i	$-0.616705\pi$
$823$	−20.5681			−0.716960			−0.358480	+	0.933537i	$0.616705\pi$
$824$		0			0
$825$	−1.75063	+	1.55531i	−0.0609490	+	0.0541491i
$826$		0			0
$827$		−	24.0587i		−	0.836603i	−0.908308	−	0.418301i	$-0.862626\pi$
$827$		−	24.0587i		−	0.836603i	0.908308	−	0.418301i	$-0.137374\pi$
$828$		0			0
$829$			45.3883i			1.57640i	0.615420	+	0.788200i	$0.288987\pi$
$829$			45.3883i			1.57640i	−0.615420	+	0.788200i	$0.711013\pi$
$830$		0			0
$831$	2.04437	+	2.30110i	0.0709185	+	0.0798243i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−12.5007			−0.432606
$836$		0			0
$837$	8.40692	+	12.0704i	0.290586	+	0.417215i
$838$		0			0
$839$	39.8758			1.37666			0.688332	−	0.725395i	$-0.258343\pi$
$839$	39.8758			1.37666			0.688332	+	0.725395i	$0.258343\pi$
$840$		0			0
$841$	−3.33396			−0.114964
$842$		0			0
$843$	32.8656	−	29.1989i	1.13195	−	1.00566i
$844$		0			0
$845$	11.4329			0.393303
$846$		0			0
$847$		0			0
$848$		0			0
$849$	20.8984	−	18.5669i	0.717233	−	0.637213i
$850$		0			0
$851$		−	19.2753i		−	0.660748i
$852$		0			0
$853$			16.9504i			0.580372i	0.956970	+	0.290186i	$0.0937172\pi$
$853$			16.9504i			0.580372i	−0.956970	+	0.290186i	$0.906283\pi$
$854$		0			0
$855$	9.79349	+	1.16124i	0.334930	+	0.0397136i
$856$		0			0
$857$	0.631287			0.0215644			0.0107822	−	0.999942i	$-0.496568\pi$
$857$	0.631287			0.0215644			0.0107822	+	0.999942i	$0.496568\pi$
$858$		0			0
$859$		−	6.46218i		−	0.220487i	−0.993905	−	0.110243i	$-0.964837\pi$
$859$		−	6.46218i		−	0.220487i	0.993905	−	0.110243i	$-0.0351630\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	7.29564i		−	0.248346i	−0.992261	−	0.124173i	$-0.960372\pi$
$863$		−	7.29564i		−	0.248346i	0.992261	−	0.124173i	$-0.0396278\pi$
$864$		0			0
$865$	−14.8418			−0.504636
$866$		0			0
$867$	−18.3735	−	20.6808i	−0.623998	−	0.702358i
$868$		0			0
$869$		−	6.53538i		−	0.221698i
$870$		0			0
$871$			24.9065i			0.843925i
$872$		0			0
$873$	35.9215	+	4.25931i	1.21576	+	0.144156i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	29.6884			1.00251			0.501253	−	0.865301i	$-0.332873\pi$
$877$	29.6884			1.00251			0.501253	+	0.865301i	$0.332873\pi$
$878$		0			0
$879$	−22.4949	−	25.3197i	−0.758732	−	0.854013i
$880$		0			0
$881$	−21.6272			−0.728638			−0.364319	−	0.931274i	$-0.618698\pi$
$881$	−21.6272			−0.728638			−0.364319	+	0.931274i	$0.618698\pi$
$882$		0			0
$883$	−29.9462			−1.00777			−0.503884	−	0.863771i	$-0.668096\pi$
$883$	−29.9462			−1.00777			−0.503884	+	0.863771i	$0.668096\pi$
$884$		0			0
$885$	9.99080	+	11.2454i	0.335837	+	0.378011i
$886$		0			0
$887$	11.7964			0.396085			0.198043	−	0.980193i	$-0.436542\pi$
$887$	11.7964			0.396085			0.198043	+	0.980193i	$0.436542\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−2.84557	+	11.8306i	−0.0953303	+	0.396339i
$892$		0			0
$893$			18.6918i			0.625498i
$894$		0			0
$895$			11.4723i			0.383478i
$896$		0			0
$897$	−28.4622	−	32.0364i	−0.950324	−	1.06966i
$898$		0			0
$899$	16.0971			0.536868
$900$		0			0
$901$			8.39234i			0.279589i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	5.91099i		−	0.196488i
$906$		0			0
$907$	−28.4910			−0.946028			−0.473014	−	0.881055i	$-0.656834\pi$
$907$	−28.4910			−0.946028			−0.473014	+	0.881055i	$0.656834\pi$
$908$		0			0
$909$	6.37028	−	53.7246i	0.211289	−	1.78193i
$910$		0			0
$911$			14.4884i			0.480023i	0.970770	+	0.240011i	$0.0771511\pi$
$911$			14.4884i			0.480023i	−0.970770	+	0.240011i	$0.922849\pi$
$912$		0			0
$913$			22.5142i			0.745111i
$914$		0			0
$915$	−2.47158	+	2.19584i	−0.0817080	+	0.0725921i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−5.66806			−0.186972			−0.0934861	−	0.995621i	$-0.529801\pi$
$919$	−5.66806			−0.186972			−0.0934861	+	0.995621i	$0.529801\pi$
$920$		0			0
$921$	−15.3090	+	13.6010i	−0.504449	+	0.448169i
$922$		0			0
$923$	−16.7200			−0.550346
$924$		0			0
$925$	3.85090			0.126617
$926$		0			0
$927$	−1.83564	−	0.217657i	−0.0602903	−	0.00714878i
$928$		0			0
$929$	−17.0232			−0.558512			−0.279256	−	0.960217i	$-0.590088\pi$
$929$	−17.0232			−0.558512			−0.279256	+	0.960217i	$0.590088\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−25.6970	−	28.9240i	−0.841283	−	0.946930i
$934$		0			0
$935$		−	7.76330i		−	0.253887i
$936$		0			0
$937$		−	11.1936i		−	0.365678i	−0.983143	−	0.182839i	$-0.941471\pi$
$937$		−	11.1936i		−	0.365678i	0.983143	−	0.182839i	$-0.0585287\pi$
$938$		0			0
$939$	−17.2317	+	15.3092i	−0.562336	+	0.499598i
$940$		0			0
$941$	−22.0202			−0.717837			−0.358919	−	0.933369i	$-0.616854\pi$
$941$	−22.0202			−0.717837			−0.358919	+	0.933369i	$0.616854\pi$
$942$		0			0
$943$			18.7103i			0.609290i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			28.6069i			0.929600i	0.885416	+	0.464800i	$0.153874\pi$
$947$			28.6069i			0.929600i	−0.885416	+	0.464800i	$0.846126\pi$
$948$		0			0
$949$	80.7051			2.61980
$950$		0			0
$951$	−43.8672	+	38.9731i	−1.42249	+	1.26379i
$952$		0			0
$953$			45.2210i			1.46485i	0.680846	+	0.732426i	$0.261612\pi$
$953$			45.2210i			1.46485i	−0.680846	+	0.732426i	$0.738388\pi$
$954$		0			0
$955$		−	10.5743i		−	0.342177i
$956$		0			0
$957$	8.84397	+	9.95458i	0.285885	+	0.321786i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	22.9862			0.741491
$962$		0			0
$963$	52.7606	+	6.25597i	1.70019	+	0.201596i
$964$		0			0
$965$	−12.0646			−0.388374
$966$		0			0
$967$	−54.5961			−1.75569			−0.877846	−	0.478943i	$-0.841020\pi$
$967$	−54.5961			−1.75569			−0.877846	+	0.478943i	$0.841020\pi$
$968$		0			0
$969$	−24.4420	+	21.7150i	−0.785189	+	0.697587i
$970$		0			0
$971$	−24.3183			−0.780411			−0.390206	−	0.920728i	$-0.627596\pi$
$971$	−24.3183			−0.780411			−0.390206	+	0.920728i	$0.627596\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	6.40037	−	5.68630i	0.204976	−	0.182107i
$976$		0			0
$977$		−	9.17797i		−	0.293629i	−0.989164	−	0.146815i	$-0.953098\pi$
$977$		−	9.17797i		−	0.293629i	0.989164	−	0.146815i	$-0.0469021\pi$
$978$		0			0
$979$		−	21.8523i		−	0.698403i
$980$		0			0
$981$	0.604246	−	5.09599i	0.0192921	−	0.162703i
$982$		0			0
$983$	−16.5954			−0.529310			−0.264655	−	0.964343i	$-0.585258\pi$
$983$	−16.5954			−0.529310			−0.264655	+	0.964343i	$0.585258\pi$
$984$		0			0
$985$			17.2423i			0.549385i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	20.3389i		−	0.646738i
$990$		0			0
$991$	18.8983			0.600323			0.300162	−	0.953888i	$-0.402959\pi$
$991$	18.8983			0.600323			0.300162	+	0.953888i	$0.402959\pi$
$992$		0			0
$993$	2.27735	+	2.56334i	0.0722696	+	0.0813450i
$994$		0			0
$995$			2.84518i			0.0901983i
$996$		0			0
$997$		−	14.7421i		−	0.466886i	−0.972370	−	0.233443i	$-0.925001\pi$
$997$		−	14.7421i		−	0.466886i	0.972370	−	0.233443i	$-0.0749992\pi$
$998$		0			0
$999$	16.4198	−	11.4362i	0.519498	−	0.361825i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.d.a.881.3		10
3.2	odd	2		2940.2.d.b.881.7		10
7.2	even	3		420.2.bh.b.101.5	yes	10
7.3	odd	6		420.2.bh.a.341.4	yes	10
7.6	odd	2		2940.2.d.b.881.8		10
21.2	odd	6		420.2.bh.a.101.4	✓	10
21.17	even	6		420.2.bh.b.341.5	yes	10
21.20	even	2	inner	2940.2.d.a.881.4		10
35.2	odd	12		2100.2.bo.g.1949.5		20
35.3	even	12		2100.2.bo.h.1349.9		20
35.9	even	6		2100.2.bi.j.101.1		10
35.17	even	12		2100.2.bo.h.1349.2		20
35.23	odd	12		2100.2.bo.g.1949.6		20
35.24	odd	6		2100.2.bi.k.1601.2		10
105.2	even	12		2100.2.bo.h.1949.9		20
105.17	odd	12		2100.2.bo.g.1349.6		20
105.23	even	12		2100.2.bo.h.1949.2		20
105.38	odd	12		2100.2.bo.g.1349.5		20
105.44	odd	6		2100.2.bi.k.101.2		10
105.59	even	6		2100.2.bi.j.1601.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.bh.a.101.4	✓	10	21.2	odd	6
420.2.bh.a.341.4	yes	10	7.3	odd	6
420.2.bh.b.101.5	yes	10	7.2	even	3
420.2.bh.b.341.5	yes	10	21.17	even	6
2100.2.bi.j.101.1		10	35.9	even	6
2100.2.bi.j.1601.1		10	105.59	even	6
2100.2.bi.k.101.2		10	105.44	odd	6
2100.2.bi.k.1601.2		10	35.24	odd	6
2100.2.bo.g.1349.5		20	105.38	odd	12
2100.2.bo.g.1349.6		20	105.17	odd	12
2100.2.bo.g.1949.5		20	35.2	odd	12
2100.2.bo.g.1949.6		20	35.23	odd	12
2100.2.bo.h.1349.2		20	35.17	even	12
2100.2.bo.h.1349.9		20	35.3	even	12
2100.2.bo.h.1949.2		20	105.23	even	12
2100.2.bo.h.1949.9		20	105.2	even	12
2940.2.d.a.881.3		10	1.1	even	1	trivial
2940.2.d.a.881.4		10	21.20	even	2	inner
2940.2.d.b.881.7		10	3.2	odd	2
2940.2.d.b.881.8		10	7.6	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.15038 - 1.29484i) q^{3} -1.00000 q^{5} +(-0.353244 + 2.97913i) q^{9} -1.35200i q^{11} +4.94296i q^{13} +(1.15038 + 1.29484i) q^{15} -5.74210 q^{17} +3.28736i q^{19} -5.00540i q^{23} +1.00000 q^{25} +(4.26388 - 2.96974i) q^{27} -5.68630i q^{29} +2.83086i q^{31} +(-1.75063 + 1.55531i) q^{33} +3.85090 q^{37} +(6.40037 - 5.68630i) q^{39} -3.73802 q^{41} +4.06339 q^{43} +(0.353244 - 2.97913i) q^{45} +5.68596 q^{47} +(6.60560 + 7.43512i) q^{51} -1.46155i q^{53} +1.35200i q^{55} +(4.25662 - 3.78172i) q^{57} +8.68477 q^{59} +1.90879i q^{61} -4.94296i q^{65} +5.03878 q^{67} +(-6.48121 + 5.75812i) q^{69} +3.38259i q^{71} -16.3273i q^{73} +(-1.15038 - 1.29484i) q^{75} +4.83387 q^{79} +(-8.75044 - 2.10472i) q^{81} -16.6525 q^{83} +5.74210 q^{85} +(-7.36287 + 6.54141i) q^{87} +16.1630 q^{89} +(3.66552 - 3.25657i) q^{93} -3.28736i q^{95} -12.0577i q^{97} +(4.02778 + 0.477584i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 10 q^{5} + 2 q^{15} + 12 q^{17} + 10 q^{25} - 8 q^{27} - 16 q^{33} + 2 q^{37} + 6 q^{39} + 8 q^{41} - 26 q^{43} + 28 q^{47} + 4 q^{51} + 18 q^{57} - 14 q^{67} + 14 q^{69} - 2 q^{75} - 2 q^{79}+ \cdots + 36 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 10 * q^5 + 2 * q^15 + 12 * q^17 + 10 * q^25 - 8 * q^27 - 16 * q^33 + 2 * q^37 + 6 * q^39 + 8 * q^41 - 26 * q^43 + 28 * q^47 + 4 * q^51 + 18 * q^57 - 14 * q^67 + 14 * q^69 - 2 * q^75 - 2 * q^79 - 28 * q^81 + 8 * q^83 - 12 * q^85 - 34 * q^87 + 56 * q^89 + 22 * q^93 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more