LMFDB - Embedded newform 4680.2.l.e.2809.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4680,2,Mod(2809,4680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4680.2809");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4680.l (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$37.3699881460$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2809.3
Root			$0.432320 + 0.432320i$ of defining polynomial
Character	$\chi$	$=$	4680.2809
Dual form			4680.2.l.e.2809.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.432320 - 2.19388i) q^{5} +O(q^{10})$	$q+(-0.432320 - 2.19388i) q^{5} +2.86464 q^{11} +1.00000i q^{13} -5.52311i q^{17} -3.52311 q^{19} +7.52311i q^{23} +(-4.62620 + 1.89692i) q^{25} +6.77551 q^{29} +5.72928 q^{31} -3.72928i q^{37} +10.1170 q^{41} +5.52311i q^{43} -8.65847i q^{47} +7.00000 q^{49} -6.77551i q^{53} +(-1.23844 - 6.28467i) q^{55} +0.593923 q^{59} -5.25240 q^{61} +(2.19388 - 0.432320i) q^{65} -10.5048i q^{67} +2.38776 q^{71} -5.45856i q^{73} -2.47689 q^{79} +8.11704i q^{83} +(-12.1170 + 2.38776i) q^{85} -14.1170 q^{89} +(1.52311 + 7.72928i) q^{95} +6.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100}) $ 6 * q + 12 * q^11 + 4 * q^19 - 10 * q^25 - 20 * q^29 + 24 * q^31 + 20 * q^41 + 42 * q^49 - 20 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 16 * q^71 - 40 * q^79 - 32 * q^85 - 44 * q^89 - 16 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4680\mathbb{Z}\right)^\times$.

$n$	$937$	$1081$	$2081$	$2341$	$3511$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.432320	−	2.19388i	−0.193340	−	0.981132i
$5$	−0.432320	−	2.19388i	−0.193340	−	0.981132i
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.86464			0.863722			0.431861	−	0.901940i	$-0.357857\pi$
$11$	2.86464			0.863722			0.431861	+	0.901940i	$0.357857\pi$
$12$		0			0
$13$			1.00000i			0.277350i
$13$			1.00000i			0.277350i
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.52311i		−	1.33955i	−0.742563	−	0.669776i	$-0.766390\pi$
$17$		−	5.52311i		−	1.33955i	0.742563	−	0.669776i	$-0.233610\pi$
$18$		0			0
$19$	−3.52311			−0.808258			−0.404129	−	0.914702i	$-0.632425\pi$
$19$	−3.52311			−0.808258			−0.404129	+	0.914702i	$0.632425\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.52311i			1.56868i	0.620333	+	0.784339i	$0.286998\pi$
$23$			7.52311i			1.56868i	−0.620333	+	0.784339i	$0.713002\pi$
$24$		0			0
$25$	−4.62620	+	1.89692i	−0.925240	+	0.379383i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.77551			1.25818			0.629090	−	0.777332i	$-0.283428\pi$
$29$	6.77551			1.25818			0.629090	+	0.777332i	$0.283428\pi$
$30$		0			0
$31$	5.72928			1.02901			0.514505	−	0.857488i	$-0.327976\pi$
$31$	5.72928			1.02901			0.514505	+	0.857488i	$0.327976\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	3.72928i		−	0.613090i	−0.951856	−	0.306545i	$-0.900827\pi$
$37$		−	3.72928i		−	0.613090i	0.951856	−	0.306545i	$-0.0991730\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	10.1170			1.58002			0.790008	−	0.613097i	$-0.210076\pi$
$41$	10.1170			1.58002			0.790008	+	0.613097i	$0.210076\pi$
$42$		0			0
$43$			5.52311i			0.842267i	0.906999	+	0.421134i	$0.138368\pi$
$43$			5.52311i			0.842267i	−0.906999	+	0.421134i	$0.861632\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.65847i		−	1.26297i	−0.775389	−	0.631484i	$-0.782446\pi$
$47$		−	8.65847i		−	1.26297i	0.775389	−	0.631484i	$-0.217554\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	6.77551i		−	0.930688i	−0.885130	−	0.465344i	$-0.845931\pi$
$53$		−	6.77551i		−	0.930688i	0.885130	−	0.465344i	$-0.154069\pi$
$54$		0			0
$55$	−1.23844	−	6.28467i	−0.166992	−	0.847425i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.593923			0.0773221			0.0386611	−	0.999252i	$-0.487691\pi$
$59$	0.593923			0.0773221			0.0386611	+	0.999252i	$0.487691\pi$
$60$		0			0
$61$	−5.25240			−0.672500			−0.336250	−	0.941773i	$-0.609159\pi$
$61$	−5.25240			−0.672500			−0.336250	+	0.941773i	$0.609159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.19388	−	0.432320i	0.272117	−	0.0536228i
$66$		0			0
$67$		−	10.5048i		−	1.28336i	−0.766971	−	0.641682i	$-0.778237\pi$
$67$		−	10.5048i		−	1.28336i	0.766971	−	0.641682i	$-0.221763\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.38776			0.283374			0.141687	−	0.989911i	$-0.454747\pi$
$71$	2.38776			0.283374			0.141687	+	0.989911i	$0.454747\pi$
$72$		0			0
$73$		−	5.45856i		−	0.638877i	−0.947607	−	0.319438i	$-0.896506\pi$
$73$		−	5.45856i		−	0.638877i	0.947607	−	0.319438i	$-0.103494\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.47689			−0.278671			−0.139336	−	0.990245i	$-0.544497\pi$
$79$	−2.47689			−0.278671			−0.139336	+	0.990245i	$0.544497\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			8.11704i			0.890961i	0.895292	+	0.445480i	$0.146967\pi$
$83$			8.11704i			0.890961i	−0.895292	+	0.445480i	$0.853033\pi$
$84$		0			0
$85$	−12.1170	+	2.38776i	−1.31428	+	0.258988i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−14.1170			−1.49640			−0.748201	−	0.663472i	$-0.769082\pi$
$89$	−14.1170			−1.49640			−0.748201	+	0.663472i	$0.769082\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.52311	+	7.72928i	0.156268	+	0.793008i
$96$		0			0
$97$			6.00000i			0.609208i	0.952479	+	0.304604i	$0.0985241\pi$
$97$			6.00000i			0.609208i	−0.952479	+	0.304604i	$0.901476\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.7293			1.16711			0.583554	−	0.812075i	$-0.301662\pi$
$101$	11.7293			1.16711			0.583554	+	0.812075i	$0.301662\pi$
$102$		0			0
$103$		−	14.7755i		−	1.45587i	−0.685644	−	0.727937i	$-0.740479\pi$
$103$		−	14.7755i		−	1.45587i	0.685644	−	0.727937i	$-0.259521\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1.72928i		−	0.167176i	−0.996500	−	0.0835880i	$-0.973362\pi$
$107$		−	1.72928i		−	0.167176i	0.996500	−	0.0835880i	$-0.0266380\pi$
$108$		0			0
$109$	−12.0279			−1.15206			−0.576032	−	0.817427i	$-0.695400\pi$
$109$	−12.0279			−1.15206			−0.576032	+	0.817427i	$0.695400\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	7.25240i		−	0.682248i	−0.940018	−	0.341124i	$-0.889192\pi$
$113$		−	7.25240i		−	0.682248i	0.940018	−	0.341124i	$-0.110808\pi$
$114$		0			0
$115$	16.5048	−	3.25240i	1.53908	−	0.303288i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.79383			−0.253985
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.16160	+	9.32924i	0.551110	+	0.834432i
$126$		0			0
$127$			17.2524i			1.53090i	0.643494	+	0.765451i	$0.277484\pi$
$127$			17.2524i			1.53090i	−0.643494	+	0.765451i	$0.722516\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	8.77551			0.766720			0.383360	−	0.923599i	$-0.374767\pi$
$131$	8.77551			0.766720			0.383360	+	0.923599i	$0.374767\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	19.4340i		−	1.66036i	−0.557497	−	0.830179i	$-0.688238\pi$
$137$		−	19.4340i		−	1.66036i	0.557497	−	0.830179i	$-0.311762\pi$
$138$		0			0
$139$	−7.58767			−0.643577			−0.321789	−	0.946812i	$-0.604284\pi$
$139$	−7.58767			−0.643577			−0.321789	+	0.946812i	$0.604284\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			2.86464i			0.239553i
$144$		0			0
$145$	−2.92919	−	14.8646i	−0.243256	−	1.23444i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.40608			−0.115190			−0.0575952	−	0.998340i	$-0.518343\pi$
$149$	−1.40608			−0.115190			−0.0575952	+	0.998340i	$0.518343\pi$
$150$		0			0
$151$	−1.31695			−0.107172			−0.0535858	−	0.998563i	$-0.517065\pi$
$151$	−1.31695			−0.107172			−0.0535858	+	0.998563i	$0.517065\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.47689	−	12.5693i	−0.198948	−	1.00959i
$156$		0			0
$157$			20.7110i			1.65291i	0.562999	+	0.826457i	$0.309647\pi$
$157$			20.7110i			1.65291i	−0.562999	+	0.826457i	$0.690353\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	2.27072i		−	0.177856i	−0.996038	−	0.0889282i	$-0.971656\pi$
$163$		−	2.27072i		−	0.177856i	0.996038	−	0.0889282i	$-0.0283442\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	7.34153i		−	0.568104i	−0.958809	−	0.284052i	$-0.908321\pi$
$167$		−	7.34153i		−	0.568104i	0.958809	−	0.284052i	$-0.0916789\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	16.0925i		−	1.22349i	−0.791056	−	0.611743i	$-0.790468\pi$
$173$		−	16.0925i		−	1.22349i	0.791056	−	0.611743i	$-0.209532\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.0462			1.12461			0.562304	−	0.826931i	$-0.309915\pi$
$179$	15.0462			1.12461			0.562304	+	0.826931i	$0.309915\pi$
$180$		0			0
$181$	5.58767			0.415328			0.207664	−	0.978200i	$-0.433414\pi$
$181$	5.58767			0.415328			0.207664	+	0.978200i	$0.433414\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.18159	+	1.61224i	−0.601522	+	0.118535i
$186$		0			0
$187$		−	15.8217i		−	1.15700i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−17.5510			−1.26995			−0.634974	−	0.772534i	$-0.718989\pi$
$191$	−17.5510			−1.26995			−0.634974	+	0.772534i	$0.718989\pi$
$192$		0			0
$193$		−	13.8217i		−	0.994911i	−0.867490	−	0.497455i	$-0.834268\pi$
$193$		−	13.8217i		−	0.994911i	0.867490	−	0.497455i	$-0.165732\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	26.3511i		−	1.87744i	−0.344682	−	0.938719i	$-0.612013\pi$
$197$		−	26.3511i		−	1.87744i	0.344682	−	0.938719i	$-0.387987\pi$
$198$		0			0
$199$	23.0741			1.63568			0.817841	−	0.575444i	$-0.195171\pi$
$199$	23.0741			1.63568			0.817841	+	0.575444i	$0.195171\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−4.37380	−	22.1955i	−0.305480	−	1.55020i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−10.0925			−0.698110
$210$		0			0
$211$	1.52311			0.104856			0.0524278	−	0.998625i	$-0.483304\pi$
$211$	1.52311			0.104856			0.0524278	+	0.998625i	$0.483304\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	12.1170	−	2.38776i	0.826375	−	0.162844i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.52311			0.371525
$222$		0			0
$223$		−	13.3169i		−	0.891769i	−0.895091	−	0.445884i	$-0.852889\pi$
$223$		−	13.3169i		−	0.891769i	0.895091	−	0.445884i	$-0.147111\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.80009i			0.451338i	0.974204	+	0.225669i	$0.0724568\pi$
$227$			6.80009i			0.451338i	−0.974204	+	0.225669i	$0.927543\pi$
$228$		0			0
$229$	−1.52311			−0.100650			−0.0503251	−	0.998733i	$-0.516026\pi$
$229$	−1.52311			−0.100650			−0.0503251	+	0.998733i	$0.516026\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	10.7110i		−	0.701698i	−0.936432	−	0.350849i	$-0.885893\pi$
$233$		−	10.7110i		−	0.701698i	0.936432	−	0.350849i	$-0.114107\pi$
$234$		0			0
$235$	−18.9956	+	3.74324i	−1.23914	+	0.244182i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.65847			0.301332			0.150666	−	0.988585i	$-0.451858\pi$
$239$	4.65847			0.301332			0.150666	+	0.988585i	$0.451858\pi$
$240$		0			0
$241$	5.22449			0.336539			0.168269	−	0.985741i	$-0.446182\pi$
$241$	5.22449			0.336539			0.168269	+	0.985741i	$0.446182\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.02624	−	15.3571i	−0.193340	−	0.981132i
$246$		0			0
$247$		−	3.52311i		−	0.224170i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−5.85838			−0.369778			−0.184889	−	0.982759i	$-0.559193\pi$
$251$	−5.85838			−0.369778			−0.184889	+	0.982759i	$0.559193\pi$
$252$		0			0
$253$			21.5510i			1.35490i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			2.29862i			0.143384i	0.997427	+	0.0716921i	$0.0228399\pi$
$257$			2.29862i			0.143384i	−0.997427	+	0.0716921i	$0.977160\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	31.5789i		−	1.94724i	−0.228175	−	0.973620i	$-0.573276\pi$
$263$		−	31.5789i		−	1.94724i	0.228175	−	0.973620i	$-0.426724\pi$
$264$		0			0
$265$	−14.8646	+	2.92919i	−0.913128	+	0.179939i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−30.3632			−1.85128			−0.925638	−	0.378411i	$-0.876471\pi$
$269$	−30.3632			−1.85128			−0.925638	+	0.378411i	$0.876471\pi$
$270$		0			0
$271$	28.5972			1.73716			0.868580	−	0.495550i	$-0.165033\pi$
$271$	28.5972			1.73716			0.868580	+	0.495550i	$0.165033\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−13.2524	+	5.43398i	−0.799150	+	0.327682i
$276$		0			0
$277$			1.79383i			0.107781i	0.998547	+	0.0538905i	$0.0171622\pi$
$277$			1.79383i			0.107781i	−0.998547	+	0.0538905i	$0.982838\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−32.2095			−1.92146			−0.960729	−	0.277489i	$-0.910498\pi$
$281$	−32.2095			−1.92146			−0.960729	+	0.277489i	$0.910498\pi$
$282$		0			0
$283$		−	18.0925i		−	1.07548i	−0.843109	−	0.537742i	$-0.819277\pi$
$283$		−	18.0925i		−	1.07548i	0.843109	−	0.537742i	$-0.180723\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.5048			−0.794400
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	12.6218i		−	0.737375i	−0.929553	−	0.368688i	$-0.879807\pi$
$293$		−	12.6218i		−	0.737375i	0.929553	−	0.368688i	$-0.120193\pi$
$294$		0			0
$295$	−0.256765	−	1.30299i	−0.0149494	−	0.0758632i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.52311			−0.435073
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.27072	+	11.5231i	0.130021	+	0.659812i
$306$		0			0
$307$			20.0000i			1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$			20.0000i			1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	31.8217			1.80445			0.902223	−	0.431271i	$-0.141935\pi$
$311$	31.8217			1.80445			0.902223	+	0.431271i	$0.141935\pi$
$312$		0			0
$313$		−	11.7938i		−	0.666627i	−0.942816	−	0.333313i	$-0.891833\pi$
$313$		−	11.7938i		−	0.666627i	0.942816	−	0.333313i	$-0.108167\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11.4340i		−	0.642197i	−0.947046	−	0.321098i	$-0.895948\pi$
$317$		−	11.4340i		−	0.642197i	0.947046	−	0.321098i	$-0.104052\pi$
$318$		0			0
$319$	19.4094			1.08672
$320$		0			0
$321$		0			0
$322$		0			0
$323$			19.4586i			1.08270i
$324$		0			0
$325$	−1.89692	−	4.62620i	−0.105222	−	0.256615i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.20617			0.121262			0.0606310	−	0.998160i	$-0.480689\pi$
$331$	2.20617			0.121262			0.0606310	+	0.998160i	$0.480689\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−23.0462	+	4.54144i	−1.25915	+	0.248125i
$336$		0			0
$337$		−	16.8034i		−	0.915340i	−0.889122	−	0.457670i	$-0.848684\pi$
$337$		−	16.8034i		−	0.915340i	0.889122	−	0.457670i	$-0.151316\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	16.4123			0.888778
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	9.96336i		−	0.534861i	−0.963577	−	0.267430i	$-0.913825\pi$
$347$		−	9.96336i		−	0.534861i	0.963577	−	0.267430i	$-0.0861746\pi$
$348$		0			0
$349$	21.3449			1.14256			0.571282	−	0.820754i	$-0.306446\pi$
$349$	21.3449			1.14256			0.571282	+	0.820754i	$0.306446\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			33.3973i			1.77756i	0.458333	+	0.888781i	$0.348447\pi$
$353$			33.3973i			1.77756i	−0.458333	+	0.888781i	$0.651553\pi$
$354$		0			0
$355$	−1.03228	−	5.23844i	−0.0547875	−	0.278028i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	21.9754			1.15982			0.579909	−	0.814681i	$-0.303088\pi$
$359$	21.9754			1.15982			0.579909	+	0.814681i	$0.303088\pi$
$360$		0			0
$361$	−6.58767			−0.346719
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−11.9754	+	2.35985i	−0.626822	+	0.123520i
$366$		0			0
$367$			29.2803i			1.52842i	0.644968	+	0.764210i	$0.276871\pi$
$367$			29.2803i			1.52842i	−0.644968	+	0.764210i	$0.723129\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	2.41233i		−	0.124906i	−0.998048	−	0.0624530i	$-0.980108\pi$
$373$		−	2.41233i		−	0.124906i	0.998048	−	0.0624530i	$-0.0198923\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			6.77551i			0.348957i
$378$		0			0
$379$	−19.5231			−1.00284			−0.501418	−	0.865205i	$-0.667188\pi$
$379$	−19.5231			−1.00284			−0.501418	+	0.865205i	$0.667188\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	23.7047i		−	1.21125i	−0.795749	−	0.605627i	$-0.792922\pi$
$383$		−	23.7047i		−	1.21125i	0.795749	−	0.605627i	$-0.207078\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−14.0000			−0.709828			−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000			−0.709828			−0.354914	+	0.934899i	$0.615490\pi$
$390$		0			0
$391$	41.5510			2.10133
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.07081	+	5.43398i	0.0538782	+	0.273413i
$396$		0			0
$397$			28.8680i			1.44884i	0.689358	+	0.724421i	$0.257893\pi$
$397$			28.8680i			1.44884i	−0.689358	+	0.724421i	$0.742107\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−20.0804			−1.00277			−0.501383	−	0.865225i	$-0.667175\pi$
$401$	−20.0804			−1.00277			−0.501383	+	0.865225i	$0.667175\pi$
$402$		0			0
$403$			5.72928i			0.285396i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	10.6831i		−	0.529539i
$408$		0			0
$409$	13.6926			0.677057			0.338529	−	0.940956i	$-0.390071\pi$
$409$	13.6926			0.677057			0.338529	+	0.940956i	$0.390071\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	17.8078	−	3.50916i	0.874150	−	0.172258i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−16.3632			−0.799393			−0.399697	−	0.916647i	$-0.630885\pi$
$419$	−16.3632			−0.799393			−0.399697	+	0.916647i	$0.630885\pi$
$420$		0			0
$421$	24.9817			1.21753			0.608766	−	0.793350i	$-0.291665\pi$
$421$	24.9817			1.21753			0.608766	+	0.793350i	$0.291665\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	10.4769	+	25.5510i	0.508204	+	1.23941i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	27.9388			1.34576			0.672882	−	0.739750i	$-0.265056\pi$
$431$	27.9388			1.34576			0.672882	+	0.739750i	$0.265056\pi$
$432$		0			0
$433$			26.7110i			1.28365i	0.766852	+	0.641823i	$0.221822\pi$
$433$			26.7110i			1.28365i	−0.766852	+	0.641823i	$0.778178\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	26.5048i		−	1.26790i
$438$		0			0
$439$	5.00958			0.239094			0.119547	−	0.992829i	$-0.461856\pi$
$439$	5.00958			0.239094			0.119547	+	0.992829i	$0.461856\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	19.6926i		−	0.935625i	−0.883828	−	0.467813i	$-0.845042\pi$
$443$		−	19.6926i		−	0.935625i	0.883828	−	0.467813i	$-0.154958\pi$
$444$		0			0
$445$	6.10308	+	30.9711i	0.289314	+	1.46817i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	38.7143			1.82704			0.913520	−	0.406794i	$-0.133353\pi$
$449$	38.7143			1.82704			0.913520	+	0.406794i	$0.133353\pi$
$450$		0			0
$451$	28.9817			1.36469
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			10.9538i			0.512396i	0.966624	+	0.256198i	$0.0824699\pi$
$457$			10.9538i			0.512396i	−0.966624	+	0.256198i	$0.917530\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.864641			0.0402703			0.0201352	−	0.999797i	$-0.493590\pi$
$461$	0.864641			0.0402703			0.0201352	+	0.999797i	$0.493590\pi$
$462$		0			0
$463$		−	16.0000i		−	0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$		−	16.0000i		−	0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	20.7755i		−	0.961376i	−0.876892	−	0.480688i	$-0.840387\pi$
$467$		−	20.7755i		−	0.961376i	0.876892	−	0.480688i	$-0.159613\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			15.8217i			0.727484i
$474$		0			0
$475$	16.2986	−	6.68305i	0.747832	−	0.306640i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	20.6585			0.943910			0.471955	−	0.881623i	$-0.343549\pi$
$479$	20.6585			0.943910			0.471955	+	0.881623i	$0.343549\pi$
$480$		0			0
$481$	3.72928			0.170041
$482$		0			0
$483$		0			0
$484$		0			0
$485$	13.1633	−	2.59392i	0.597713	−	0.117784i
$486$		0			0
$487$		−	16.3632i		−	0.741486i	−0.928735	−	0.370743i	$-0.879103\pi$
$487$		−	16.3632i		−	0.741486i	0.928735	−	0.370743i	$-0.120897\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−26.7389			−1.20671			−0.603354	−	0.797473i	$-0.706169\pi$
$491$	−26.7389			−1.20671			−0.603354	+	0.797473i	$0.706169\pi$
$492$		0			0
$493$		−	37.4219i		−	1.68540i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4.24281			−0.189934			−0.0949672	−	0.995480i	$-0.530275\pi$
$499$	−4.24281			−0.189934			−0.0949672	+	0.995480i	$0.530275\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	0.889220i		−	0.0396484i	−0.999803	−	0.0198242i	$-0.993689\pi$
$503$		−	0.889220i		−	0.0396484i	0.999803	−	0.0198242i	$-0.00631065\pi$
$504$		0			0
$505$	−5.07081	−	25.7326i	−0.225648	−	1.14509i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−11.3694			−0.503941			−0.251971	−	0.967735i	$-0.581079\pi$
$509$	−11.3694			−0.503941			−0.251971	+	0.967735i	$0.581079\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−32.4157	+	6.38776i	−1.42840	+	0.281478i
$516$		0			0
$517$		−	24.8034i		−	1.09085i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.18785			−0.139662			−0.0698310	−	0.997559i	$-0.522246\pi$
$521$	−3.18785			−0.139662			−0.0698310	+	0.997559i	$0.522246\pi$
$522$		0			0
$523$			37.1666i			1.62518i	0.582835	+	0.812591i	$0.301944\pi$
$523$			37.1666i			1.62518i	−0.582835	+	0.812591i	$0.698056\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	31.6435i		−	1.37841i
$528$		0			0
$529$	−33.5972			−1.46075
$530$		0			0
$531$		0			0
$532$		0			0
$533$			10.1170i			0.438218i
$534$		0			0
$535$	−3.79383	+	0.747604i	−0.164022	+	0.0323217i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	20.0525			0.863722
$540$		0			0
$541$	−8.02791			−0.345147			−0.172573	−	0.984997i	$-0.555208\pi$
$541$	−8.02791			−0.345147			−0.172573	+	0.984997i	$0.555208\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.19991	+	26.3878i	0.222740	+	1.13033i
$546$		0			0
$547$			31.2032i			1.33415i	0.744989	+	0.667077i	$0.232455\pi$
$547$			31.2032i			1.33415i	−0.744989	+	0.667077i	$0.767545\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−23.8709			−1.01693
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$			29.2191i			1.23805i	0.785370	+	0.619026i	$0.212472\pi$
$557$			29.2191i			1.23805i	−0.785370	+	0.619026i	$0.787528\pi$
$558$		0			0
$559$	−5.52311			−0.233603
$560$		0			0
$561$		0			0
$562$		0			0
$563$			30.5048i			1.28562i	0.766024	+	0.642812i	$0.222232\pi$
$563$			30.5048i			1.28562i	−0.766024	+	0.642812i	$0.777768\pi$
$564$		0			0
$565$	−15.9109	+	3.13536i	−0.669375	+	0.131906i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−12.1416			−0.509003			−0.254502	−	0.967072i	$-0.581911\pi$
$569$	−12.1416			−0.509003			−0.254502	+	0.967072i	$0.581911\pi$
$570$		0			0
$571$	−8.85258			−0.370469			−0.185234	−	0.982694i	$-0.559304\pi$
$571$	−8.85258			−0.370469			−0.185234	+	0.982694i	$0.559304\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−14.2707	−	34.8034i	−0.595130	−	1.45140i
$576$		0			0
$577$		−	30.2341i		−	1.25866i	−0.777138	−	0.629330i	$-0.783329\pi$
$577$		−	30.2341i		−	1.25866i	0.777138	−	0.629330i	$-0.216671\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	19.4094i		−	0.803855i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			23.0342i			0.950722i	0.879791	+	0.475361i	$0.157682\pi$
$587$			23.0342i			0.950722i	−0.879791	+	0.475361i	$0.842318\pi$
$588$		0			0
$589$	−20.1849			−0.831705
$590$		0			0
$591$		0			0
$592$		0			0
$593$			27.9754i			1.14881i	0.818570	+	0.574406i	$0.194767\pi$
$593$			27.9754i			1.14881i	−0.818570	+	0.574406i	$0.805233\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	22.7389			0.929085			0.464542	−	0.885551i	$-0.346219\pi$
$599$	22.7389			0.929085			0.464542	+	0.885551i	$0.346219\pi$
$600$		0			0
$601$	−11.2158			−0.457500			−0.228750	−	0.973485i	$-0.573464\pi$
$601$	−11.2158			−0.457500			−0.228750	+	0.973485i	$0.573464\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.20783	+	6.12933i	0.0491053	+	0.249193i
$606$		0			0
$607$			23.7293i			0.963142i	0.876407	+	0.481571i	$0.159934\pi$
$607$			23.7293i			0.963142i	−0.876407	+	0.481571i	$0.840066\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.65847			0.350284
$612$		0			0
$613$			18.2341i			0.736467i	0.929733	+	0.368234i	$0.120037\pi$
$613$			18.2341i			0.736467i	−0.929733	+	0.368234i	$0.879963\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	0.0245796i		−	0.000989536i	−1.00000		0.000494768i	$-0.999843\pi$
$617$		−	0.0245796i		−	0.000989536i	1.00000		0.000494768i	$-0.000157490\pi$
$618$		0			0
$619$	−31.2158			−1.25467			−0.627334	−	0.778751i	$-0.715854\pi$
$619$	−31.2158			−1.25467			−0.627334	+	0.778751i	$0.715854\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	17.8034	−	17.5510i	0.712137	−	0.702041i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−20.5972			−0.821266
$630$		0			0
$631$	13.4460			0.535279			0.267639	−	0.963519i	$-0.413756\pi$
$631$	13.4460			0.535279			0.267639	+	0.963519i	$0.413756\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	37.8496	−	7.45856i	1.50202	−	0.295984i
$636$		0			0
$637$			7.00000i			0.277350i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.54144			−0.100381			−0.0501904	−	0.998740i	$-0.515983\pi$
$641$	−2.54144			−0.100381			−0.0501904	+	0.998740i	$0.515983\pi$
$642$		0			0
$643$		−	13.9634i		−	0.550661i	−0.961350	−	0.275330i	$-0.911213\pi$
$643$		−	13.9634i		−	0.550661i	0.961350	−	0.275330i	$-0.0887873\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			30.8034i			1.21101i	0.795843	+	0.605504i	$0.207028\pi$
$647$			30.8034i			1.21101i	−0.795843	+	0.605504i	$0.792972\pi$
$648$		0			0
$649$	1.70138			0.0667848
$650$		0			0
$651$		0			0
$652$		0			0
$653$			2.00000i			0.0782660i	0.999234	+	0.0391330i	$0.0124596\pi$
$653$			2.00000i			0.0782660i	−0.999234	+	0.0391330i	$0.987540\pi$
$654$		0			0
$655$	−3.79383	−	19.2524i	−0.148237	−	0.752253i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−4.05581			−0.157992			−0.0789960	−	0.996875i	$-0.525171\pi$
$659$	−4.05581			−0.157992			−0.0789960	+	0.996875i	$0.525171\pi$
$660$		0			0
$661$	−32.9325			−1.28093			−0.640463	−	0.767989i	$-0.721258\pi$
$661$	−32.9325			−1.28093			−0.640463	+	0.767989i	$0.721258\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			50.9729i			1.97368i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−15.0462			−0.580853
$672$		0			0
$673$		−	9.42192i		−	0.363188i	−0.983374	−	0.181594i	$-0.941874\pi$
$673$		−	9.42192i		−	0.363188i	0.983374	−	0.181594i	$-0.0581257\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	42.2341i		−	1.62319i	−0.584222	−	0.811594i	$-0.698600\pi$
$677$		−	42.2341i		−	1.62319i	0.584222	−	0.811594i	$-0.301400\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	18.7509i		−	0.717484i	−0.933437	−	0.358742i	$-0.883206\pi$
$683$		−	18.7509i		−	0.717484i	0.933437	−	0.358742i	$-0.116794\pi$
$684$		0			0
$685$	−42.6358	+	8.40171i	−1.62903	+	0.321013i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	6.77551			0.258126
$690$		0			0
$691$	45.2524			1.72148			0.860741	−	0.509043i	$-0.170001\pi$
$691$	45.2524			1.72148			0.860741	+	0.509043i	$0.170001\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.28030	+	16.6464i	0.124429	+	0.631434i
$696$		0			0
$697$		−	55.8776i		−	2.11651i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	37.4586			1.41479			0.707395	−	0.706818i	$-0.249870\pi$
$701$	37.4586			1.41479			0.707395	+	0.706818i	$0.249870\pi$
$702$		0			0
$703$			13.1387i			0.495535i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−11.8988			−0.446869			−0.223435	−	0.974719i	$-0.571727\pi$
$709$	−11.8988			−0.446869			−0.223435	+	0.974719i	$0.571727\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			43.1020i			1.61418i
$714$		0			0
$715$	6.28467	−	1.23844i	0.235033	−	0.0463151i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	25.7851			0.961622			0.480811	−	0.876824i	$-0.340342\pi$
$719$	25.7851			0.961622			0.480811	+	0.876824i	$0.340342\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−31.3449	+	12.8526i	−1.16412	+	0.477333i
$726$		0			0
$727$			34.8034i			1.29079i	0.763850	+	0.645394i	$0.223307\pi$
$727$			34.8034i			1.29079i	−0.763850	+	0.645394i	$0.776693\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	30.5048			1.12826
$732$		0			0
$733$		−	19.5510i		−	0.722133i	−0.932540	−	0.361067i	$-0.882413\pi$
$733$		−	19.5510i		−	0.722133i	0.932540	−	0.361067i	$-0.117587\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	30.0925i		−	1.10847i
$738$		0			0
$739$	−12.2986			−0.452412			−0.226206	−	0.974079i	$-0.572632\pi$
$739$	−12.2986			−0.452412			−0.226206	+	0.974079i	$0.572632\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			40.9975i			1.50405i	0.659133	+	0.752027i	$0.270924\pi$
$743$			40.9975i			1.50405i	−0.659133	+	0.752027i	$0.729076\pi$
$744$		0			0
$745$	0.607876	+	3.08476i	0.0222709	+	0.113017i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	33.5510			1.22429			0.612147	−	0.790744i	$-0.290306\pi$
$751$	33.5510			1.22429			0.612147	+	0.790744i	$0.290306\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.569343	+	2.88922i	0.0207205	+	0.105149i
$756$		0			0
$757$			25.1020i			0.912349i	0.889890	+	0.456175i	$0.150781\pi$
$757$			25.1020i			0.912349i	−0.889890	+	0.456175i	$0.849219\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	39.9021			1.44645			0.723226	−	0.690612i	$-0.242659\pi$
$761$	39.9021			1.44645			0.723226	+	0.690612i	$0.242659\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			0.593923i			0.0214453i
$768$		0			0
$769$	15.5510			0.560784			0.280392	−	0.959886i	$-0.409536\pi$
$769$	15.5510			0.560784			0.280392	+	0.959886i	$0.409536\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			27.7972i			0.999794i	0.866085	+	0.499897i	$0.166629\pi$
$773$			27.7972i			0.999794i	−0.866085	+	0.499897i	$0.833371\pi$
$774$		0			0
$775$	−26.5048	+	10.8680i	−0.952080	+	0.390389i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−35.6435			−1.27706
$780$		0			0
$781$	6.84006			0.244757
$782$		0			0
$783$		0			0
$784$		0			0
$785$	45.4373	−	8.95377i	1.62173	−	0.319574i
$786$		0			0
$787$		−	30.6339i		−	1.09198i	−0.837792	−	0.545990i	$-0.816154\pi$
$787$		−	30.6339i		−	1.09198i	0.837792	−	0.545990i	$-0.183846\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	5.25240i		−	0.186518i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			18.6464i			0.660490i	0.943895	+	0.330245i	$0.107131\pi$
$797$			18.6464i			0.660490i	−0.943895	+	0.330245i	$0.892869\pi$
$798$		0			0
$799$	−47.8217			−1.69181
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	15.6368i		−	0.551812i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−54.8313			−1.92777			−0.963883	−	0.266325i	$-0.914191\pi$
$809$	−54.8313			−1.92777			−0.963883	+	0.266325i	$0.914191\pi$
$810$		0			0
$811$	5.97209			0.209709			0.104854	−	0.994488i	$-0.466562\pi$
$811$	5.97209			0.209709			0.104854	+	0.994488i	$0.466562\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.98168	+	0.981678i	−0.174501	+	0.0343867i
$816$		0			0
$817$		−	19.4586i		−	0.680769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−30.2866			−1.05701			−0.528504	−	0.848931i	$-0.677247\pi$
$821$	−30.2866			−1.05701			−0.528504	+	0.848931i	$0.677247\pi$
$822$		0			0
$823$			31.2437i			1.08909i	0.838733	+	0.544543i	$0.183297\pi$
$823$			31.2437i			1.08909i	−0.838733	+	0.544543i	$0.816703\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			21.9754i			0.764160i	0.924129	+	0.382080i	$0.124792\pi$
$827$			21.9754i			0.764160i	−0.924129	+	0.382080i	$0.875208\pi$
$828$		0			0
$829$	11.7014			0.406406			0.203203	−	0.979137i	$-0.434865\pi$
$829$	11.7014			0.406406			0.203203	+	0.979137i	$0.434865\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	38.6618i		−	1.33955i
$834$		0			0
$835$	−16.1064	+	3.17389i	−0.557385	+	0.109837i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	11.7047			0.404091			0.202046	−	0.979376i	$-0.435241\pi$
$839$	11.7047			0.404091			0.202046	+	0.979376i	$0.435241\pi$
$840$		0			0
$841$	16.9075			0.583019
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.432320	+	2.19388i	0.0148723	+	0.0754717i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	28.0558			0.961741
$852$		0			0
$853$			15.3169i			0.524442i	0.965008	+	0.262221i	$0.0844550\pi$
$853$			15.3169i			0.524442i	−0.965008	+	0.262221i	$0.915545\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	20.7476i		−	0.708725i	−0.935108	−	0.354362i	$-0.884698\pi$
$857$		−	20.7476i		−	0.708725i	0.935108	−	0.354362i	$-0.115302\pi$
$858$		0			0
$859$	−53.0375			−1.80962			−0.904808	−	0.425820i	$-0.859986\pi$
$859$	−53.0375			−1.80962			−0.904808	+	0.425820i	$0.859986\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			10.2095i			0.347535i	0.984787	+	0.173768i	$0.0555942\pi$
$863$			10.2095i			0.347535i	−0.984787	+	0.173768i	$0.944406\pi$
$864$		0			0
$865$	−35.3049	+	6.95710i	−1.20040	+	0.236548i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.09539			−0.240695
$870$		0			0
$871$	10.5048			0.355941
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			9.28030i			0.313374i	0.987648	+	0.156687i	$0.0500813\pi$
$877$			9.28030i			0.313374i	−0.987648	+	0.156687i	$0.949919\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	22.4923			0.757784			0.378892	−	0.925441i	$-0.376305\pi$
$881$	22.4923			0.757784			0.378892	+	0.925441i	$0.376305\pi$
$882$		0			0
$883$			28.5972i			0.962374i	0.876618	+	0.481187i	$0.159794\pi$
$883$			28.5972i			0.962374i	−0.876618	+	0.481187i	$0.840206\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			34.2062i			1.14853i	0.818669	+	0.574265i	$0.194712\pi$
$887$			34.2062i			1.14853i	−0.818669	+	0.574265i	$0.805288\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			30.5048i			1.02080i
$894$		0			0
$895$	−6.50479	−	33.0096i	−0.217431	−	1.10339i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	38.8188			1.29468
$900$		0			0
$901$	−37.4219			−1.24670
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.41566	−	12.2587i	−0.0802993	−	0.407491i
$906$		0			0
$907$		−	25.9634i		−	0.862099i	−0.902328	−	0.431050i	$-0.858143\pi$
$907$		−	25.9634i		−	0.862099i	0.902328	−	0.431050i	$-0.141857\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	51.9267			1.72041			0.860204	−	0.509949i	$-0.170336\pi$
$911$	51.9267			1.72041			0.860204	+	0.509949i	$0.170336\pi$
$912$		0			0
$913$			23.2524i			0.769542i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	11.6156			0.383162			0.191581	−	0.981477i	$-0.438638\pi$
$919$	11.6156			0.383162			0.191581	+	0.981477i	$0.438638\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			2.38776i			0.0785939i
$924$		0			0
$925$	7.07414	+	17.2524i	0.232596	+	0.567255i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5.83380			0.191401			0.0957004	−	0.995410i	$-0.469491\pi$
$929$	5.83380			0.191401			0.0957004	+	0.995410i	$0.469491\pi$
$930$		0			0
$931$	−24.6618			−0.808258
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−34.7110	+	6.84006i	−1.13517	+	0.223694i
$936$		0			0
$937$			27.4373i			0.896338i	0.893949	+	0.448169i	$0.147924\pi$
$937$			27.4373i			0.896338i	−0.893949	+	0.448169i	$0.852076\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	34.9571			1.13957			0.569784	−	0.821794i	$-0.307027\pi$
$941$	34.9571			1.13957			0.569784	+	0.821794i	$0.307027\pi$
$942$		0			0
$943$			76.1116i			2.47854i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			53.3848i			1.73477i	0.497634	+	0.867387i	$0.334202\pi$
$947$			53.3848i			1.73477i	−0.497634	+	0.867387i	$0.665798\pi$
$948$		0			0
$949$	5.45856			0.177192
$950$		0			0
$951$		0			0
$952$		0			0
$953$			22.7110i			0.735680i	0.929889	+	0.367840i	$0.119903\pi$
$953$			22.7110i			0.735680i	−0.929889	+	0.367840i	$0.880097\pi$
$954$		0			0
$955$	7.58767	+	38.5048i	0.245531	+	1.24599i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.82467			0.0588603
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−30.3232	+	5.97542i	−0.976138	+	0.192356i
$966$		0			0
$967$		−	28.8313i		−	0.927153i	−0.886057	−	0.463576i	$-0.846566\pi$
$967$		−	28.8313i		−	0.927153i	0.886057	−	0.463576i	$-0.153434\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19.0462			−0.611223			−0.305611	−	0.952156i	$-0.598861\pi$
$971$	−19.0462			−0.611223			−0.305611	+	0.952156i	$0.598861\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			30.1729i			0.965315i	0.875809	+	0.482658i	$0.160328\pi$
$977$			30.1729i			0.965315i	−0.875809	+	0.482658i	$0.839672\pi$
$978$		0			0
$979$	−40.4402			−1.29248
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	18.9292i		−	0.603747i	−0.953348	−	0.301874i	$-0.902388\pi$
$983$		−	18.9292i		−	0.603747i	0.953348	−	0.301874i	$-0.0976121\pi$
$984$		0			0
$985$	−57.8111	+	11.3921i	−1.84202	+	0.362983i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−41.5510			−1.32125
$990$		0			0
$991$	−25.5510			−0.811655			−0.405827	−	0.913950i	$-0.633017\pi$
$991$	−25.5510			−0.811655			−0.405827	+	0.913950i	$0.633017\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−9.97542	−	50.6218i	−0.316242	−	1.60482i
$996$		0			0
$997$		−	4.16952i		−	0.132050i	−0.997818	−	0.0660251i	$-0.978968\pi$
$997$		−	4.16952i		−	0.132050i	0.997818	−	0.0660251i	$-0.0210317\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4680.2.l.e.2809.3		6
3.2	odd	2		1560.2.l.c.1249.5	yes	6
5.4	even	2	inner	4680.2.l.e.2809.4		6
12.11	even	2		3120.2.l.m.1249.2		6
15.2	even	4		7800.2.a.bp.1.2		3
15.8	even	4		7800.2.a.bj.1.2		3
15.14	odd	2		1560.2.l.c.1249.2	✓	6
60.59	even	2		3120.2.l.m.1249.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.c.1249.2	✓	6	15.14	odd	2
1560.2.l.c.1249.5	yes	6	3.2	odd	2
3120.2.l.m.1249.2		6	12.11	even	2
3120.2.l.m.1249.5		6	60.59	even	2
4680.2.l.e.2809.3		6	1.1	even	1	trivial
4680.2.l.e.2809.4		6	5.4	even	2	inner
7800.2.a.bj.1.2		3	15.8	even	4
7800.2.a.bp.1.2		3	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4680.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.432320 - 2.19388i) q^{5} +O(q^{10})\)	\(q+(-0.432320 - 2.19388i) q^{5} +2.86464 q^{11} +1.00000i q^{13} -5.52311i q^{17} -3.52311 q^{19} +7.52311i q^{23} +(-4.62620 + 1.89692i) q^{25} +6.77551 q^{29} +5.72928 q^{31} -3.72928i q^{37} +10.1170 q^{41} +5.52311i q^{43} -8.65847i q^{47} +7.00000 q^{49} -6.77551i q^{53} +(-1.23844 - 6.28467i) q^{55} +0.593923 q^{59} -5.25240 q^{61} +(2.19388 - 0.432320i) q^{65} -10.5048i q^{67} +2.38776 q^{71} -5.45856i q^{73} -2.47689 q^{79} +8.11704i q^{83} +(-12.1170 + 2.38776i) q^{85} -14.1170 q^{89} +(1.52311 + 7.72928i) q^{95} +6.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 12 q^{11} + 4 q^{19} - 10 q^{25} - 20 q^{29} + 24 q^{31} + 20 q^{41} + 42 q^{49} - 20 q^{55} - 12 q^{59} + 4 q^{61} - 2 q^{65} - 16 q^{71} - 40 q^{79} - 32 q^{85} - 44 q^{89} - 16 q^{95}+O(q^{100}) \) 6 * q + 12 * q^11 + 4 * q^19 - 10 * q^25 - 20 * q^29 + 24 * q^31 + 20 * q^41 + 42 * q^49 - 20 * q^55 - 12 * q^59 + 4 * q^61 - 2 * q^65 - 16 * q^71 - 40 * q^79 - 32 * q^85 - 44 * q^89 - 16 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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