LMFDB - Embedded newform 3024.2.k.l.1889.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3024,2,Mod(1889,3024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3024.1889");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3024 = 2^{4} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3024.k (of order $2$, degree $1$, not 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:	$24.1467615712$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 18 x^{14} - 36 x^{13} - 45 x^{12} + 306 x^{11} - 378 x^{10} + 1704 x^{9} + \cdots + 256 $ x^16 - 6x^15 + 18x^14 - 36x^13 - 45x^12 + 306x^11 - 378x^10 + 1704x^9 + 917x^8 - 12522x^7 + 27090x^6 - 35292x^5 + 30948x^4 - 18000x^3 + 7200x^2 - 1920*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.13
Root			$-2.43091 - 0.651359i$ of defining polynomial
Character	$\chi$	$=$	3024.1889
Dual form			3024.2.k.l.1889.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.85593 q^{5} +(2.30652 - 1.29613i) q^{7} +O(q^{10})$	$q+2.85593 q^{5} +(2.30652 - 1.29613i) q^{7} +3.55149i q^{11} -2.43521i q^{13} -0.860208 q^{17} +3.55909i q^{19} +5.94662i q^{23} +3.15635 q^{25} +2.88506i q^{29} +9.01892i q^{31} +(6.58727 - 3.70166i) q^{35} +7.43654 q^{37} +6.85451 q^{41} -3.48992 q^{43} +0.263674 q^{47} +(3.64010 - 5.97910i) q^{49} +7.76940i q^{53} +10.1428i q^{55} +9.72208 q^{59} +1.62542i q^{61} -6.95480i q^{65} -15.0496 q^{67} -15.1727i q^{71} -9.20637i q^{73} +(4.60318 + 8.19158i) q^{77} +7.55967 q^{79} -0.922263 q^{83} -2.45670 q^{85} -4.90208 q^{89} +(-3.15635 - 5.61687i) q^{91} +10.1645i q^{95} -10.1428i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{7}+O(q^{10}) $ 16 * q + 2 * q^7	$ 16 q + 2 q^{7} + 36 q^{25} - 8 q^{37} - 20 q^{43} + 2 q^{49} - 44 q^{67} - 40 q^{79} + 16 q^{85} - 36 q^{91}+O(q^{100}) $ 16 * q + 2 * q^7 + 36 * q^25 - 8 * q^37 - 20 * q^43 + 2 * q^49 - 44 * q^67 - 40 * q^79 + 16 * q^85 - 36 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$.

$n$	$757$	$785$	$1135$	$2593$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.85593			1.27721			0.638606	−	0.769534i	$-0.279511\pi$
$5$	2.85593			1.27721			0.638606	+	0.769534i	$0.279511\pi$
$6$		0			0
$7$	2.30652	−	1.29613i	0.871784	−	0.489891i
$7$	2.30652	−	1.29613i	0.871784	−	0.489891i
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.55149i			1.07081i	0.844594	+	0.535407i	$0.179842\pi$
$11$			3.55149i			1.07081i	−0.844594	+	0.535407i	$0.820158\pi$
$12$		0			0
$13$		−	2.43521i		−	0.675406i	−0.941253	−	0.337703i	$-0.890350\pi$
$13$		−	2.43521i		−	0.675406i	0.941253	−	0.337703i	$-0.109650\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.860208			−0.208631			−0.104316	−	0.994544i	$-0.533265\pi$
$17$	−0.860208			−0.208631			−0.104316	+	0.994544i	$0.533265\pi$
$18$		0			0
$19$			3.55909i			0.816512i	0.912867	+	0.408256i	$0.133863\pi$
$19$			3.55909i			0.816512i	−0.912867	+	0.408256i	$0.866137\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			5.94662i			1.23996i	0.784619	+	0.619978i	$0.212858\pi$
$23$			5.94662i			1.23996i	−0.784619	+	0.619978i	$0.787142\pi$
$24$		0			0
$25$	3.15635			0.631270
$26$		0			0
$27$		0			0
$28$		0			0
$29$			2.88506i			0.535742i	0.963455	+	0.267871i	$0.0863201\pi$
$29$			2.88506i			0.535742i	−0.963455	+	0.267871i	$0.913680\pi$
$30$		0			0
$31$			9.01892i			1.61985i	0.586536	+	0.809923i	$0.300491\pi$
$31$			9.01892i			1.61985i	−0.586536	+	0.809923i	$0.699509\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.58727	−	3.70166i	1.11345	−	0.625694i
$36$		0			0
$37$	7.43654			1.22256			0.611280	−	0.791414i	$-0.290655\pi$
$37$	7.43654			1.22256			0.611280	+	0.791414i	$0.290655\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.85451			1.07050			0.535248	−	0.844695i	$-0.320218\pi$
$41$	6.85451			1.07050			0.535248	+	0.844695i	$0.320218\pi$
$42$		0			0
$43$	−3.48992			−0.532208			−0.266104	−	0.963944i	$-0.585737\pi$
$43$	−3.48992			−0.532208			−0.266104	+	0.963944i	$0.585737\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	0.263674			0.0384607			0.0192304	−	0.999815i	$-0.493878\pi$
$47$	0.263674			0.0384607			0.0192304	+	0.999815i	$0.493878\pi$
$48$		0			0
$49$	3.64010	−	5.97910i	0.520014	−	0.854158i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			7.76940i			1.06721i	0.845734	+	0.533604i	$0.179163\pi$
$53$			7.76940i			1.06721i	−0.845734	+	0.533604i	$0.820837\pi$
$54$		0			0
$55$			10.1428i			1.36766i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.72208			1.26571			0.632854	−	0.774272i	$-0.281884\pi$
$59$	9.72208			1.26571			0.632854	+	0.774272i	$0.281884\pi$
$60$		0			0
$61$			1.62542i			0.208114i	0.994571	+	0.104057i	$0.0331825\pi$
$61$			1.62542i			0.208114i	−0.994571	+	0.104057i	$0.966818\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	6.95480i		−	0.862637i
$66$		0			0
$67$	−15.0496			−1.83860			−0.919300	−	0.393557i	$-0.871244\pi$
$67$	−15.0496			−1.83860			−0.919300	+	0.393557i	$0.871244\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	15.1727i		−	1.80067i	−0.435198	−	0.900335i	$-0.643322\pi$
$71$		−	15.1727i		−	1.80067i	0.435198	−	0.900335i	$-0.356678\pi$
$72$		0			0
$73$		−	9.20637i		−	1.07752i	−0.842458	−	0.538762i	$-0.818892\pi$
$73$		−	9.20637i		−	1.07752i	0.842458	−	0.538762i	$-0.181108\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.60318	+	8.19158i	0.524582	+	0.933517i
$78$		0			0
$79$	7.55967			0.850529			0.425264	−	0.905069i	$-0.360181\pi$
$79$	7.55967			0.850529			0.425264	+	0.905069i	$0.360181\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−0.922263			−0.101231			−0.0506157	−	0.998718i	$-0.516118\pi$
$83$	−0.922263			−0.101231			−0.0506157	+	0.998718i	$0.516118\pi$
$84$		0			0
$85$	−2.45670			−0.266466
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.90208			−0.519619			−0.259810	−	0.965660i	$-0.583660\pi$
$89$	−4.90208			−0.519619			−0.259810	+	0.965660i	$0.583660\pi$
$90$		0			0
$91$	−3.15635	−	5.61687i	−0.330875	−	0.588808i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			10.1645i			1.04286i
$96$		0			0
$97$		−	10.1428i		−	1.02985i	−0.857237	−	0.514923i	$-0.827821\pi$
$97$		−	10.1428i		−	1.02985i	0.857237	−	0.514923i	$-0.172179\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	13.1745			1.31092			0.655458	−	0.755231i	$-0.272476\pi$
$101$	13.1745			1.31092			0.655458	+	0.755231i	$0.272476\pi$
$102$		0			0
$103$		−	0.933897i		−	0.0920196i	−0.998941	−	0.0460098i	$-0.985349\pi$
$103$		−	0.933897i		−	0.0920196i	0.998941	−	0.0460098i	$-0.0146506\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			7.15707i			0.691900i	0.938253	+	0.345950i	$0.112443\pi$
$107$			7.15707i			0.691900i	−0.938253	+	0.345950i	$0.887557\pi$
$108$		0			0
$109$	−14.6626			−1.40443			−0.702213	−	0.711967i	$-0.747805\pi$
$109$	−14.6626			−1.40443			−0.702213	+	0.711967i	$0.747805\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			5.72871i			0.538912i	0.963013	+	0.269456i	$0.0868438\pi$
$113$			5.72871i			0.538912i	−0.963013	+	0.269456i	$0.913156\pi$
$114$		0			0
$115$			16.9831i			1.58369i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.98409	+	1.11494i	−0.181881	+	0.102206i
$120$		0			0
$121$	−1.61305			−0.146641
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.26534			−0.470946
$126$		0			0
$127$	−4.94662			−0.438942			−0.219471	−	0.975619i	$-0.570433\pi$
$127$	−4.94662			−0.438942			−0.219471	+	0.975619i	$0.570433\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13.9843			1.22182			0.610909	−	0.791701i	$-0.290804\pi$
$131$	13.9843			1.22182			0.610909	+	0.791701i	$0.290804\pi$
$132$		0			0
$133$	4.61305	+	8.20913i	0.400002	+	0.711822i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.28765i		−	0.708062i	−0.935234	−	0.354031i	$-0.884811\pi$
$137$		−	8.28765i		−	0.708062i	0.935234	−	0.354031i	$-0.115189\pi$
$138$		0			0
$139$		−	22.4151i		−	1.90122i	−0.310381	−	0.950612i	$-0.600457\pi$
$139$		−	22.4151i		−	1.90122i	0.310381	−	0.950612i	$-0.399543\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.64862			0.723234
$144$		0			0
$145$			8.23953i			0.684256i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	15.6212i		−	1.27974i	−0.768483	−	0.639870i	$-0.778988\pi$
$149$		−	15.6212i		−	1.27974i	0.768483	−	0.639870i	$-0.221012\pi$
$150$		0			0
$151$	−3.61377			−0.294084			−0.147042	−	0.989130i	$-0.546975\pi$
$151$	−3.61377			−0.294084			−0.147042	+	0.989130i	$0.546975\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			25.7574i			2.06889i
$156$		0			0
$157$		−	22.8449i		−	1.82322i	−0.411052	−	0.911612i	$-0.634839\pi$
$157$		−	22.8449i		−	1.82322i	0.411052	−	0.911612i	$-0.365161\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	7.70759	+	13.7160i	0.607443	+	1.08097i
$162$		0			0
$163$	−4.82278			−0.377749			−0.188874	−	0.982001i	$-0.560484\pi$
$163$	−4.82278			−0.377749			−0.188874	+	0.982001i	$0.560484\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.94411			−0.382587			−0.191293	−	0.981533i	$-0.561268\pi$
$167$	−4.94411			−0.382587			−0.191293	+	0.981533i	$0.561268\pi$
$168$		0			0
$169$	7.06974			0.543826
$170$		0			0
$171$		0			0
$172$		0			0
$173$	18.8864			1.43591			0.717954	−	0.696091i	$-0.245079\pi$
$173$	18.8864			1.43591			0.717954	+	0.696091i	$0.245079\pi$
$174$		0			0
$175$	7.28019	−	4.09104i	0.550331	−	0.309253i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	3.79773i		−	0.283856i	−0.989877	−	0.141928i	$-0.954670\pi$
$179$		−	3.79773i		−	0.283856i	0.989877	−	0.141928i	$-0.0453301\pi$
$180$		0			0
$181$		−	15.4844i		−	1.15094i	−0.817822	−	0.575472i	$-0.804818\pi$
$181$		−	15.4844i		−	1.15094i	0.817822	−	0.575472i	$-0.195182\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	21.2383			1.56147
$186$		0			0
$187$		−	3.05502i		−	0.223405i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			4.32906i			0.313240i	0.987659	+	0.156620i	$0.0500598\pi$
$191$			4.32906i			0.313240i	−0.987659	+	0.156620i	$0.949940\pi$
$192$		0			0
$193$	−26.1193			−1.88011			−0.940055	−	0.341022i	$-0.889227\pi$
$193$	−26.1193			−1.88011			−0.940055	+	0.341022i	$0.889227\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			16.7776i			1.19535i	0.801737	+	0.597676i	$0.203909\pi$
$197$			16.7776i			1.19535i	−0.801737	+	0.597676i	$0.796091\pi$
$198$		0			0
$199$		−	11.0243i		−	0.781493i	−0.920498	−	0.390746i	$-0.872217\pi$
$199$		−	11.0243i		−	0.781493i	0.920498	−	0.390746i	$-0.127783\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	3.73941	+	6.65446i	0.262455	+	0.467051i
$204$		0			0
$205$	19.5760			1.36725
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.6401			−0.874332
$210$		0			0
$211$	−8.06974			−0.555544			−0.277772	−	0.960647i	$-0.589596\pi$
$211$	−8.06974			−0.555544			−0.277772	+	0.960647i	$0.589596\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.96699			−0.679743
$216$		0			0
$217$	11.6897	+	20.8023i	0.793548	+	1.41216i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2.09479i			0.140911i
$222$		0			0
$223$			23.1041i			1.54716i	0.633696	+	0.773582i	$0.281537\pi$
$223$			23.1041i			1.54716i	−0.633696	+	0.773582i	$0.718463\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.7294			0.778510			0.389255	−	0.921130i	$-0.372732\pi$
$227$	11.7294			0.778510			0.389255	+	0.921130i	$0.372732\pi$
$228$		0			0
$229$			27.2138i			1.79834i	0.437601	+	0.899169i	$0.355828\pi$
$229$			27.2138i			1.79834i	−0.437601	+	0.899169i	$0.644172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			22.4936i			1.47360i	0.676108	+	0.736802i	$0.263665\pi$
$233$			22.4936i			1.47360i	−0.676108	+	0.736802i	$0.736335\pi$
$234$		0			0
$235$	0.753034			0.0491225
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	12.0414i		−	0.778894i	−0.921049	−	0.389447i	$-0.872666\pi$
$239$		−	12.0414i		−	0.778894i	0.921049	−	0.389447i	$-0.127334\pi$
$240$		0			0
$241$			2.00539i			0.129179i	0.997912	+	0.0645893i	$0.0205737\pi$
$241$			2.00539i			0.129179i	−0.997912	+	0.0645893i	$0.979426\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	10.3959	−	17.0759i	0.664168	−	1.09094i
$246$		0			0
$247$	8.66715			0.551477
$248$		0			0
$249$		0			0
$250$		0			0
$251$	27.6442			1.74489			0.872443	−	0.488716i	$-0.162535\pi$
$251$	27.6442			1.74489			0.872443	+	0.488716i	$0.162535\pi$
$252$		0			0
$253$	−21.1193			−1.32776
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.96818			0.247528			0.123764	−	0.992312i	$-0.460503\pi$
$257$	3.96818			0.247528			0.123764	+	0.992312i	$0.460503\pi$
$258$		0			0
$259$	17.1526	−	9.63872i	1.06581	−	0.598921i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	13.1978i		−	0.813809i	−0.913471	−	0.406904i	$-0.866608\pi$
$263$		−	13.1978i		−	0.813809i	0.913471	−	0.406904i	$-0.133392\pi$
$264$		0			0
$265$			22.1889i			1.36305i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.577767			−0.0352271			−0.0176135	−	0.999845i	$-0.505607\pi$
$269$	−0.577767			−0.0352271			−0.0176135	+	0.999845i	$0.505607\pi$
$270$		0			0
$271$		−	24.6907i		−	1.49986i	−0.661520	−	0.749928i	$-0.730088\pi$
$271$		−	24.6907i		−	1.49986i	0.661520	−	0.749928i	$-0.269912\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			11.2097i			0.675972i
$276$		0			0
$277$	13.0156			0.782034			0.391017	−	0.920383i	$-0.372123\pi$
$277$	13.0156			0.782034			0.391017	+	0.920383i	$0.372123\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	7.81042i		−	0.465931i	−0.972485	−	0.232965i	$-0.925157\pi$
$281$		−	7.81042i		−	0.465931i	0.972485	−	0.232965i	$-0.0748429\pi$
$282$		0			0
$283$			21.7210i			1.29118i	0.763683	+	0.645591i	$0.223389\pi$
$283$			21.7210i			1.29118i	−0.763683	+	0.645591i	$0.776611\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	15.8101	−	8.88434i	0.933240	−	0.524426i
$288$		0			0
$289$	−16.2600			−0.956473
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−24.7024			−1.44313			−0.721564	−	0.692348i	$-0.756576\pi$
$293$	−24.7024			−1.44313			−0.721564	+	0.692348i	$0.756576\pi$
$294$		0			0
$295$	27.7656			1.61658
$296$		0			0
$297$		0			0
$298$		0			0
$299$	14.4813			0.837474
$300$		0			0
$301$	−8.04959	+	4.52339i	−0.463971	+	0.260724i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			4.64210i			0.265806i
$306$		0			0
$307$		−	0.564882i		−	0.0322395i	−0.999870	−	0.0161198i	$-0.994869\pi$
$307$		−	0.564882i		−	0.0322395i	0.999870	−	0.0161198i	$-0.00513130\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−17.9409			−1.01733			−0.508667	−	0.860963i	$-0.669862\pi$
$311$	−17.9409			−1.01733			−0.508667	+	0.860963i	$0.669862\pi$
$312$		0			0
$313$			20.4401i			1.15534i	0.816269	+	0.577672i	$0.196039\pi$
$313$			20.4401i			1.15534i	−0.816269	+	0.577672i	$0.803961\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			33.1734i			1.86321i	0.363478	+	0.931603i	$0.381589\pi$
$317$			33.1734i			1.86321i	−0.363478	+	0.931603i	$0.618411\pi$
$318$		0			0
$319$	−10.2462			−0.573680
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	3.06156i		−	0.170350i
$324$		0			0
$325$		−	7.68638i		−	0.426364i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.608169	−	0.341755i	0.0335295	−	0.0188416i
$330$		0			0
$331$	25.9421			1.42591			0.712954	−	0.701211i	$-0.247357\pi$
$331$	25.9421			1.42591			0.712954	+	0.701211i	$0.247357\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−42.9806			−2.34828
$336$		0			0
$337$	−11.9257			−0.649637			−0.324818	−	0.945776i	$-0.605303\pi$
$337$	−11.9257			−0.649637			−0.324818	+	0.945776i	$0.605303\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−32.0306			−1.73455
$342$		0			0
$343$	0.646274	−	18.5090i	0.0348955	−	0.999391i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			5.42836i			0.291410i	0.989328	+	0.145705i	$0.0465450\pi$
$347$			5.42836i			0.291410i	−0.989328	+	0.145705i	$0.953455\pi$
$348$		0			0
$349$			30.5997i			1.63797i	0.573818	+	0.818983i	$0.305462\pi$
$349$			30.5997i			1.63797i	−0.573818	+	0.818983i	$0.694538\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.19210			0.329572			0.164786	−	0.986329i	$-0.447307\pi$
$353$	6.19210			0.329572			0.164786	+	0.986329i	$0.447307\pi$
$354$		0			0
$355$		−	43.3322i		−	2.29984i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	21.9227i		−	1.15703i	−0.815670	−	0.578517i	$-0.803632\pi$
$359$		−	21.9227i		−	1.15703i	0.815670	−	0.578517i	$-0.196368\pi$
$360$		0			0
$361$	6.33285			0.333308
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	26.2928i		−	1.37623i
$366$		0			0
$367$			5.95883i			0.311049i	0.987832	+	0.155524i	$0.0497067\pi$
$367$			5.95883i			0.311049i	−0.987832	+	0.155524i	$0.950293\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.0701	+	17.9203i	0.522816	+	0.930375i
$372$		0			0
$373$	19.0171			0.984667			0.492334	−	0.870407i	$-0.336144\pi$
$373$	19.0171			0.984667			0.492334	+	0.870407i	$0.336144\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.02573			0.361844
$378$		0			0
$379$	11.3134			0.581131			0.290566	−	0.956855i	$-0.406157\pi$
$379$	11.3134			0.581131			0.290566	+	0.956855i	$0.406157\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−23.2095			−1.18595			−0.592974	−	0.805221i	$-0.702046\pi$
$383$	−23.2095			−1.18595			−0.592974	+	0.805221i	$0.702046\pi$
$384$		0			0
$385$	13.1464	+	23.3946i	0.670002	+	1.19230i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	7.63430i		−	0.387075i	−0.981093	−	0.193537i	$-0.938004\pi$
$389$		−	7.63430i		−	0.387075i	0.981093	−	0.193537i	$-0.0619960\pi$
$390$		0			0
$391$		−	5.11533i		−	0.258693i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	21.5899			1.08631
$396$		0			0
$397$		−	7.93636i		−	0.398314i	−0.979968	−	0.199157i	$-0.936180\pi$
$397$		−	7.93636i		−	0.398314i	0.979968	−	0.199157i	$-0.0638204\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	1.98364i		−	0.0990581i	−0.998773	−	0.0495291i	$-0.984228\pi$
$401$		−	1.98364i		−	0.0990581i	0.998773	−	0.0495291i	$-0.0157720\pi$
$402$		0			0
$403$	21.9630			1.09405
$404$		0			0
$405$		0			0
$406$		0			0
$407$			26.4108i			1.30913i
$408$		0			0
$409$		−	29.3796i		−	1.45273i	−0.687311	−	0.726363i	$-0.741209\pi$
$409$		−	29.3796i		−	1.45273i	0.687311	−	0.726363i	$-0.258791\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	22.4242	−	12.6011i	1.10342	−	0.620058i
$414$		0			0
$415$	−2.63392			−0.129294
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−35.6304			−1.74066			−0.870330	−	0.492470i	$-0.836094\pi$
$419$	−35.6304			−1.74066			−0.870330	+	0.492470i	$0.836094\pi$
$420$		0			0
$421$	12.8716			0.627326			0.313663	−	0.949534i	$-0.398444\pi$
$421$	12.8716			0.627326			0.313663	+	0.949534i	$0.398444\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.71512			−0.131703
$426$		0			0
$427$	2.10676	+	3.74908i	0.101953	+	0.181431i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	8.97985i		−	0.432544i	−0.976333	−	0.216272i	$-0.930610\pi$
$431$		−	8.97985i		−	0.432544i	0.976333	−	0.216272i	$-0.0693898\pi$
$432$		0			0
$433$		−	16.1920i		−	0.778139i	−0.921208	−	0.389070i	$-0.872797\pi$
$433$		−	16.1920i		−	0.778139i	0.921208	−	0.389070i	$-0.127203\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−21.1646			−1.01244
$438$		0			0
$439$		−	20.9358i		−	0.999212i	−0.866253	−	0.499606i	$-0.833478\pi$
$439$		−	20.9358i		−	0.999212i	0.866253	−	0.499606i	$-0.166522\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	30.9548i		−	1.47071i	−0.677684	−	0.735353i	$-0.737016\pi$
$443$		−	30.9548i		−	1.47071i	0.677684	−	0.735353i	$-0.262984\pi$
$444$		0			0
$445$	−14.0000			−0.663664
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	9.81153i		−	0.463035i	−0.972831	−	0.231517i	$-0.925631\pi$
$449$		−	9.81153i		−	0.463035i	0.972831	−	0.231517i	$-0.0743690\pi$
$450$		0			0
$451$			24.3437i			1.14630i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−9.01432	−	16.0414i	−0.422598	−	0.752033i
$456$		0			0
$457$	−16.0171			−0.749248			−0.374624	−	0.927177i	$-0.622228\pi$
$457$	−16.0171			−0.749248			−0.374624	+	0.927177i	$0.622228\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6.99476			−0.325778			−0.162889	−	0.986644i	$-0.552081\pi$
$461$	−6.99476			−0.325778			−0.162889	+	0.986644i	$0.552081\pi$
$462$		0			0
$463$	−34.3787			−1.59771			−0.798856	−	0.601523i	$-0.794561\pi$
$463$	−34.3787			−1.59771			−0.798856	+	0.601523i	$0.794561\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.5778			−0.767131			−0.383566	−	0.923514i	$-0.625304\pi$
$467$	−16.5778			−0.767131			−0.383566	+	0.923514i	$0.625304\pi$
$468$		0			0
$469$	−34.7122	+	19.5062i	−1.60286	+	0.900713i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	12.3944i		−	0.569896i
$474$		0			0
$475$			11.2337i			0.515440i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	9.01313			0.411821			0.205910	−	0.978571i	$-0.433984\pi$
$479$	9.01313			0.411821			0.205910	+	0.978571i	$0.433984\pi$
$480$		0			0
$481$		−	18.1096i		−	0.825725i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	28.9672i		−	1.31533i
$486$		0			0
$487$	−19.7447			−0.894719			−0.447360	−	0.894354i	$-0.647636\pi$
$487$	−19.7447			−0.894719			−0.447360	+	0.894354i	$0.647636\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	3.46867i		−	0.156539i	−0.996932	−	0.0782694i	$-0.975061\pi$
$491$		−	3.46867i		−	0.156539i	0.996932	−	0.0782694i	$-0.0249394\pi$
$492$		0			0
$493$		−	2.48175i		−	0.111772i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−19.6658	−	34.9962i	−0.882132	−	1.56979i
$498$		0			0
$499$	1.42018			0.0635761			0.0317880	−	0.999495i	$-0.489880\pi$
$499$	1.42018			0.0635761			0.0317880	+	0.999495i	$0.489880\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−43.3051			−1.93088			−0.965439	−	0.260628i	$-0.916070\pi$
$503$	−43.3051			−1.93088			−0.965439	+	0.260628i	$0.916070\pi$
$504$		0			0
$505$	37.6256			1.67432
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.11225			−0.0492995			−0.0246497	−	0.999696i	$-0.507847\pi$
$509$	−1.11225			−0.0492995			−0.0246497	+	0.999696i	$0.507847\pi$
$510$		0			0
$511$	−11.9326	−	21.2347i	−0.527869	−	0.939368i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	2.66715i		−	0.117529i
$516$		0			0
$517$			0.936433i			0.0411843i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	28.5730			1.25181			0.625903	−	0.779901i	$-0.284730\pi$
$521$	28.5730			1.25181			0.625903	+	0.779901i	$0.284730\pi$
$522$		0			0
$523$		−	10.6114i		−	0.464004i	−0.972715	−	0.232002i	$-0.925472\pi$
$523$		−	10.6114i		−	0.464004i	0.972715	−	0.232002i	$-0.0745277\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	7.75815i		−	0.337950i
$528$		0			0
$529$	−12.3623			−0.537491
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	16.6922i		−	0.723019i
$534$		0			0
$535$			20.4401i			0.883703i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	21.2347	+	12.9278i	0.914643	+	0.556838i
$540$		0			0
$541$	−22.6612			−0.974281			−0.487140	−	0.873324i	$-0.661960\pi$
$541$	−22.6612			−0.974281			−0.487140	+	0.873324i	$0.661960\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−41.8755			−1.79375
$546$		0			0
$547$	6.22537			0.266178			0.133089	−	0.991104i	$-0.457510\pi$
$547$	6.22537			0.266178			0.133089	+	0.991104i	$0.457510\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10.2682			−0.437440
$552$		0			0
$553$	17.4365	−	9.79831i	0.741477	−	0.416666i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			9.18507i			0.389184i	0.980884	+	0.194592i	$0.0623382\pi$
$557$			9.18507i			0.389184i	−0.980884	+	0.194592i	$0.937662\pi$
$558$		0			0
$559$			8.49871i			0.359457i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	43.8608			1.84851			0.924257	−	0.381771i	$-0.124686\pi$
$563$	43.8608			1.84851			0.924257	+	0.381771i	$0.124686\pi$
$564$		0			0
$565$			16.3608i			0.688304i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			39.8932i			1.67241i	0.548416	+	0.836206i	$0.315231\pi$
$569$			39.8932i			1.67241i	−0.548416	+	0.836206i	$0.684769\pi$
$570$		0			0
$571$	−18.5101			−0.774623			−0.387311	−	0.921949i	$-0.626596\pi$
$571$	−18.5101			−0.774623			−0.387311	+	0.921949i	$0.626596\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			18.7696i			0.782747i
$576$		0			0
$577$		−	12.6633i		−	0.527182i	−0.964635	−	0.263591i	$-0.915093\pi$
$577$		−	12.6633i		−	0.527182i	0.964635	−	0.263591i	$-0.0849069\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2.12722	+	1.19537i	−0.0882520	+	0.0495924i
$582$		0			0
$583$	−27.5929			−1.14278
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.1403			−0.418534			−0.209267	−	0.977859i	$-0.567108\pi$
$587$	−10.1403			−0.418534			−0.209267	+	0.977859i	$0.567108\pi$
$588$		0			0
$589$	−32.0992			−1.32262
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−39.4894			−1.62163			−0.810817	−	0.585300i	$-0.800977\pi$
$593$	−39.4894			−1.62163			−0.810817	+	0.585300i	$0.800977\pi$
$594$		0			0
$595$	−5.66643	+	3.18420i	−0.232301	+	0.130539i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	25.0910i		−	1.02519i	−0.858630	−	0.512595i	$-0.828684\pi$
$599$		−	25.0910i		−	1.02519i	0.858630	−	0.512595i	$-0.171316\pi$
$600$		0			0
$601$		−	20.3877i		−	0.831633i	−0.909449	−	0.415816i	$-0.863496\pi$
$601$		−	20.3877i		−	0.831633i	0.909449	−	0.415816i	$-0.136504\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4.60675			−0.187291
$606$		0			0
$607$		−	11.3604i		−	0.461104i	−0.973060	−	0.230552i	$-0.925947\pi$
$607$		−	11.3604i		−	0.461104i	0.973060	−	0.230552i	$-0.0740532\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	0.642101i		−	0.0259766i
$612$		0			0
$613$	32.5201			1.31347			0.656737	−	0.754120i	$-0.271936\pi$
$613$	32.5201			1.31347			0.656737	+	0.754120i	$0.271936\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			35.2425i			1.41881i	0.704802	+	0.709404i	$0.251036\pi$
$617$			35.2425i			1.41881i	−0.704802	+	0.709404i	$0.748964\pi$
$618$		0			0
$619$			17.7296i			0.712613i	0.934369	+	0.356306i	$0.115964\pi$
$619$			17.7296i			0.712613i	−0.934369	+	0.356306i	$0.884036\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.3068	+	6.35373i	−0.452995	+	0.254557i
$624$		0			0
$625$	−30.8192			−1.23277
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.39698			−0.255064
$630$		0			0
$631$	−17.5720			−0.699531			−0.349766	−	0.936837i	$-0.613739\pi$
$631$	−17.5720			−0.699531			−0.349766	+	0.936837i	$0.613739\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−14.1272			−0.560621
$636$		0			0
$637$	−14.5604	−	8.86441i	−0.576904	−	0.351221i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			28.6167i			1.13029i	0.824991	+	0.565146i	$0.191180\pi$
$641$			28.6167i			1.13029i	−0.824991	+	0.565146i	$0.808820\pi$
$642$		0			0
$643$			22.5942i			0.891027i	0.895275	+	0.445514i	$0.146979\pi$
$643$			22.5942i			0.891027i	−0.895275	+	0.445514i	$0.853021\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−29.6658			−1.16628			−0.583142	−	0.812370i	$-0.698177\pi$
$647$	−29.6658			−1.16628			−0.583142	+	0.812370i	$0.698177\pi$
$648$		0			0
$649$			34.5278i			1.35534i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	12.3788i		−	0.484419i	−0.970224	−	0.242209i	$-0.922128\pi$
$653$		−	12.3788i		−	0.484419i	0.970224	−	0.242209i	$-0.0778721\pi$
$654$		0			0
$655$	39.9383			1.56052
$656$		0			0
$657$		0			0
$658$		0			0
$659$			2.82801i			0.110164i	0.998482	+	0.0550818i	$0.0175420\pi$
$659$			2.82801i			0.110164i	−0.998482	+	0.0550818i	$0.982458\pi$
$660$		0			0
$661$		−	23.7155i		−	0.922427i	−0.887289	−	0.461213i	$-0.847414\pi$
$661$		−	23.7155i		−	0.922427i	0.887289	−	0.461213i	$-0.152586\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	13.1745	+	23.4447i	0.510887	+	0.909147i
$666$		0			0
$667$	−17.1564			−0.664297
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5.77267			−0.222851
$672$		0			0
$673$	13.1037			0.505110			0.252555	−	0.967583i	$-0.418729\pi$
$673$	13.1037			0.505110			0.252555	+	0.967583i	$0.418729\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	5.83597			0.224295			0.112147	−	0.993692i	$-0.464227\pi$
$677$	5.83597			0.224295			0.112147	+	0.993692i	$0.464227\pi$
$678$		0			0
$679$	−13.1464	−	23.3946i	−0.504512	−	0.897803i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			29.4070i			1.12523i	0.826720	+	0.562614i	$0.190204\pi$
$683$			29.4070i			1.12523i	−0.826720	+	0.562614i	$0.809796\pi$
$684$		0			0
$685$		−	23.6690i		−	0.904345i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	18.9201			0.720799
$690$		0			0
$691$			12.9858i			0.494005i	0.969015	+	0.247002i	$0.0794456\pi$
$691$			12.9858i			0.494005i	−0.969015	+	0.247002i	$0.920554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	64.0160i		−	2.42827i
$696$		0			0
$697$	−5.89631			−0.223339
$698$		0			0
$699$		0			0
$700$		0			0
$701$			3.65969i			0.138225i	0.997609	+	0.0691123i	$0.0220167\pi$
$701$			3.65969i			0.138225i	−0.997609	+	0.0691123i	$0.977983\pi$
$702$		0			0
$703$			26.4674i			0.998235i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	30.3874	−	17.0759i	1.14284	−	0.642206i
$708$		0			0
$709$	45.3208			1.70206			0.851028	−	0.525120i	$-0.175979\pi$
$709$	45.3208			1.70206			0.851028	+	0.525120i	$0.175979\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−53.6321			−2.00854
$714$		0			0
$715$	24.6999			0.923723
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−52.9476			−1.97461			−0.987306	−	0.158829i	$-0.949228\pi$
$719$	−52.9476			−1.97461			−0.987306	+	0.158829i	$0.949228\pi$
$720$		0			0
$721$	−1.21045	−	2.15406i	−0.0450796	−	0.0802212i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			9.10626i			0.338198i
$726$		0			0
$727$			7.33604i			0.272079i	0.990703	+	0.136039i	$0.0434373\pi$
$727$			7.33604i			0.272079i	−0.990703	+	0.136039i	$0.956563\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.00206			0.111035
$732$		0			0
$733$		−	17.7980i		−	0.657384i	−0.944437	−	0.328692i	$-0.893392\pi$
$733$		−	17.7980i		−	0.657384i	0.944437	−	0.328692i	$-0.106608\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	53.4484i		−	1.96880i
$738$		0			0
$739$	35.1193			1.29189			0.645943	−	0.763386i	$-0.276464\pi$
$739$	35.1193			1.29189			0.645943	+	0.763386i	$0.276464\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	25.7418i		−	0.944375i	−0.881498	−	0.472187i	$-0.843465\pi$
$743$		−	25.7418i		−	0.944375i	0.881498	−	0.472187i	$-0.156535\pi$
$744$		0			0
$745$		−	44.6132i		−	1.63450i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.27649	+	16.5080i	0.338956	+	0.603187i
$750$		0			0
$751$	−15.6921			−0.572612			−0.286306	−	0.958138i	$-0.592427\pi$
$751$	−15.6921			−0.572612			−0.286306	+	0.958138i	$0.592427\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.3207			−0.375608
$756$		0			0
$757$	−32.3409			−1.17545			−0.587725	−	0.809061i	$-0.699976\pi$
$757$	−32.3409			−1.17545			−0.587725	+	0.809061i	$0.699976\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.20066			−0.0797737			−0.0398869	−	0.999204i	$-0.512700\pi$
$761$	−2.20066			−0.0797737			−0.0398869	+	0.999204i	$0.512700\pi$
$762$		0			0
$763$	−33.8197	+	19.0047i	−1.22436	+	0.688015i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	23.6753i		−	0.854867i
$768$		0			0
$769$		−	22.6271i		−	0.815953i	−0.912992	−	0.407977i	$-0.866235\pi$
$769$		−	22.6271i		−	0.815953i	0.912992	−	0.407977i	$-0.133765\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−41.7514			−1.50169			−0.750847	−	0.660477i	$-0.770354\pi$
$773$	−41.7514			−1.50169			−0.750847	+	0.660477i	$0.770354\pi$
$774$		0			0
$775$			28.4669i			1.02256i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			24.3959i			0.874072i
$780$		0			0
$781$	53.8857			1.92818
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	65.2436i		−	2.32864i
$786$		0			0
$787$			1.42959i			0.0509595i	0.999675	+	0.0254797i	$0.00811133\pi$
$787$			1.42959i			0.0509595i	−0.999675	+	0.0254797i	$0.991889\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	7.42515	+	13.2134i	0.264008	+	0.469814i
$792$		0			0
$793$	3.95825			0.140562
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−20.2269			−0.716472			−0.358236	−	0.933631i	$-0.616622\pi$
$797$	−20.2269			−0.716472			−0.358236	+	0.933631i	$0.616622\pi$
$798$		0			0
$799$	−0.226814			−0.00802411
$800$		0			0
$801$		0			0
$802$		0			0
$803$	32.6963			1.15383
$804$		0			0
$805$	22.0124	+	39.1720i	0.775834	+	1.38063i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	55.5476i		−	1.95295i	−0.215634	−	0.976474i	$-0.569182\pi$
$809$		−	55.5476i		−	1.95295i	0.215634	−	0.976474i	$-0.430818\pi$
$810$		0			0
$811$		−	1.07481i		−	0.0377416i	−0.999822	−	0.0188708i	$-0.993993\pi$
$811$		−	1.07481i		−	0.0377416i	0.999822	−	0.0188708i	$-0.00600712\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−13.7735			−0.482465
$816$		0			0
$817$		−	12.4210i		−	0.434555i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			28.6123i			0.998577i	0.866436	+	0.499288i	$0.166405\pi$
$821$			28.6123i			0.998577i	−0.866436	+	0.499288i	$0.833595\pi$
$822$		0			0
$823$	−15.4743			−0.539400			−0.269700	−	0.962944i	$-0.586924\pi$
$823$	−15.4743			−0.539400			−0.269700	+	0.962944i	$0.586924\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	20.7249i		−	0.720676i	−0.932822	−	0.360338i	$-0.882661\pi$
$827$		−	20.7249i		−	0.720676i	0.932822	−	0.360338i	$-0.117339\pi$
$828$		0			0
$829$		−	11.9802i		−	0.416089i	−0.978119	−	0.208045i	$-0.933290\pi$
$829$		−	11.9802i		−	0.416089i	0.978119	−	0.208045i	$-0.0667099\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.13124	+	5.14327i	−0.108491	+	0.178204i
$834$		0			0
$835$	−14.1201			−0.488645
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−33.0226			−1.14007			−0.570034	−	0.821621i	$-0.693070\pi$
$839$	−33.0226			−1.14007			−0.570034	+	0.821621i	$0.693070\pi$
$840$		0			0
$841$	20.6764			0.712980
$842$		0			0
$843$		0			0
$844$		0			0
$845$	20.1907			0.694581
$846$		0			0
$847$	−3.72053	+	2.09072i	−0.127839	+	0.0718379i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			44.2223i			1.51592i
$852$		0			0
$853$		−	32.8940i		−	1.12627i	−0.826365	−	0.563135i	$-0.809595\pi$
$853$		−	32.8940i		−	1.12627i	0.826365	−	0.563135i	$-0.190405\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	39.9334			1.36410			0.682050	−	0.731305i	$-0.261089\pi$
$857$	39.9334			1.36410			0.682050	+	0.731305i	$0.261089\pi$
$858$		0			0
$859$			7.37148i			0.251512i	0.992061	+	0.125756i	$0.0401356\pi$
$859$			7.37148i			0.251512i	−0.992061	+	0.125756i	$0.959864\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			47.7480i			1.62536i	0.582709	+	0.812681i	$0.301993\pi$
$863$			47.7480i			1.62536i	−0.582709	+	0.812681i	$0.698007\pi$
$864$		0			0
$865$	53.9383			1.83396
$866$		0			0
$867$		0			0
$868$		0			0
$869$			26.8480i			0.910757i
$870$		0			0
$871$			36.6489i			1.24180i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−12.1446	+	6.82456i	−0.410563	+	0.230712i
$876$		0			0
$877$	52.2014			1.76272			0.881358	−	0.472448i	$-0.156630\pi$
$877$	52.2014			1.76272			0.881358	+	0.472448i	$0.156630\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23.8797			−0.804527			−0.402264	−	0.915524i	$-0.631776\pi$
$881$	−23.8797			−0.804527			−0.402264	+	0.915524i	$0.631776\pi$
$882$		0			0
$883$	−15.0302			−0.505805			−0.252903	−	0.967492i	$-0.581385\pi$
$883$	−15.0302			−0.505805			−0.252903	+	0.967492i	$0.581385\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	36.4545			1.22402			0.612010	−	0.790850i	$-0.290361\pi$
$887$	36.4545			1.22402			0.612010	+	0.790850i	$0.290361\pi$
$888$		0			0
$889$	−11.4095	+	6.41146i	−0.382662	+	0.215033i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			0.938439i			0.0314037i
$894$		0			0
$895$		−	10.8461i		−	0.362544i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−26.0201			−0.867819
$900$		0			0
$901$		−	6.68330i		−	0.222653i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	44.2223i		−	1.47000i
$906$		0			0
$907$	10.3336			0.343121			0.171560	−	0.985174i	$-0.445119\pi$
$907$	10.3336			0.343121			0.171560	+	0.985174i	$0.445119\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	38.6091i		−	1.27918i	−0.768717	−	0.639589i	$-0.779105\pi$
$911$		−	38.6091i		−	1.27918i	0.768717	−	0.639589i	$-0.220895\pi$
$912$		0			0
$913$		−	3.27540i		−	0.108400i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	32.2552	−	18.1255i	1.06516	−	0.598557i
$918$		0			0
$919$	−5.29962			−0.174818			−0.0874092	−	0.996172i	$-0.527859\pi$
$919$	−5.29962			−0.174818			−0.0874092	+	0.996172i	$0.527859\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−36.9488			−1.21618
$924$		0			0
$925$	23.4723			0.771766
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−44.2474			−1.45171			−0.725855	−	0.687848i	$-0.758555\pi$
$929$	−44.2474			−1.45171			−0.725855	+	0.687848i	$0.758555\pi$
$930$		0			0
$931$	21.2802	+	12.9554i	0.697430	+	0.424598i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	8.72492i		−	0.285335i
$936$		0			0
$937$			46.2961i			1.51243i	0.654326	+	0.756213i	$0.272953\pi$
$937$			46.2961i			1.51243i	−0.654326	+	0.756213i	$0.727047\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	44.4961			1.45053			0.725266	−	0.688469i	$-0.241717\pi$
$941$	44.4961			1.45053			0.725266	+	0.688469i	$0.241717\pi$
$942$		0			0
$943$			40.7612i			1.32737i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1.93832i			0.0629869i	0.999504	+	0.0314934i	$0.0100263\pi$
$947$			1.93832i			0.0629869i	−0.999504	+	0.0314934i	$0.989974\pi$
$948$		0			0
$949$	−22.4195			−0.727766
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	6.80519i		−	0.220442i	−0.993907	−	0.110221i	$-0.964844\pi$
$953$		−	6.80519i		−	0.220442i	0.993907	−	0.110221i	$-0.0351558\pi$
$954$		0			0
$955$			12.3635i			0.400074i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−10.7419	−	19.1157i	−0.346873	−	0.617277i
$960$		0			0
$961$	−50.3409			−1.62390
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−74.5950			−2.40130
$966$		0			0
$967$	24.9182			0.801314			0.400657	−	0.916228i	$-0.368782\pi$
$967$	24.9182			0.801314			0.400657	+	0.916228i	$0.368782\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−4.79291			−0.153812			−0.0769060	−	0.997038i	$-0.524504\pi$
$971$	−4.79291			−0.153812			−0.0769060	+	0.997038i	$0.524504\pi$
$972$		0			0
$973$	−29.0529	−	51.7010i	−0.931392	−	1.65746i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	5.77156i		−	0.184649i	−0.995729	−	0.0923243i	$-0.970570\pi$
$977$		−	5.77156i		−	0.184649i	0.995729	−	0.0923243i	$-0.0294297\pi$
$978$		0			0
$979$		−	17.4097i		−	0.556415i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	27.3651			0.872810			0.436405	−	0.899750i	$-0.356251\pi$
$983$	27.3651			0.872810			0.436405	+	0.899750i	$0.356251\pi$
$984$		0			0
$985$			47.9156i			1.52672i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	20.7533i		−	0.659915i
$990$		0			0
$991$	59.8524			1.90128			0.950638	−	0.310303i	$-0.100431\pi$
$991$	59.8524			1.90128			0.950638	+	0.310303i	$0.100431\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	31.4847i		−	0.998132i
$996$		0			0
$997$		−	20.1124i		−	0.636967i	−0.947928	−	0.318483i	$-0.896827\pi$
$997$		−	20.1124i		−	0.636967i	0.947928	−	0.318483i	$-0.103173\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3024.2.k.l.1889.13		16
3.2	odd	2	inner	3024.2.k.l.1889.3		16
4.3	odd	2		1512.2.k.b.377.14	yes	16
7.6	odd	2	inner	3024.2.k.l.1889.4		16
12.11	even	2		1512.2.k.b.377.4	yes	16
21.20	even	2	inner	3024.2.k.l.1889.14		16
28.27	even	2		1512.2.k.b.377.3	✓	16
84.83	odd	2		1512.2.k.b.377.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.k.b.377.3	✓	16	28.27	even	2
1512.2.k.b.377.4	yes	16	12.11	even	2
1512.2.k.b.377.13	yes	16	84.83	odd	2
1512.2.k.b.377.14	yes	16	4.3	odd	2
3024.2.k.l.1889.3		16	3.2	odd	2	inner
3024.2.k.l.1889.4		16	7.6	odd	2	inner
3024.2.k.l.1889.13		16	1.1	even	1	trivial
3024.2.k.l.1889.14		16	21.20	even	2	inner

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:	\( N \)	\(=\)	\( 3024 = 2^{4} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3024.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.85593 q^{5} +(2.30652 - 1.29613i) q^{7} +O(q^{10})\)	\(q+2.85593 q^{5} +(2.30652 - 1.29613i) q^{7} +3.55149i q^{11} -2.43521i q^{13} -0.860208 q^{17} +3.55909i q^{19} +5.94662i q^{23} +3.15635 q^{25} +2.88506i q^{29} +9.01892i q^{31} +(6.58727 - 3.70166i) q^{35} +7.43654 q^{37} +6.85451 q^{41} -3.48992 q^{43} +0.263674 q^{47} +(3.64010 - 5.97910i) q^{49} +7.76940i q^{53} +10.1428i q^{55} +9.72208 q^{59} +1.62542i q^{61} -6.95480i q^{65} -15.0496 q^{67} -15.1727i q^{71} -9.20637i q^{73} +(4.60318 + 8.19158i) q^{77} +7.55967 q^{79} -0.922263 q^{83} -2.45670 q^{85} -4.90208 q^{89} +(-3.15635 - 5.61687i) q^{91} +10.1645i q^{95} -10.1428i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{7}+O(q^{10}) \) 16 * q + 2 * q^7	\( 16 q + 2 q^{7} + 36 q^{25} - 8 q^{37} - 20 q^{43} + 2 q^{49} - 44 q^{67} - 40 q^{79} + 16 q^{85} - 36 q^{91}+O(q^{100}) \) 16 * q + 2 * q^7 + 36 * q^25 - 8 * q^37 - 20 * q^43 + 2 * q^49 - 44 * q^67 - 40 * q^79 + 16 * q^85 - 36 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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