LMFDB - Embedded newform 1368.3.cs.a.881.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,3,Mod(809,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1368.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1368.cs (of order $6$, degree $2$, 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:	$37.2753001645$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			881.19
Character	$\chi$	$=$	1368.881
Dual form			1368.3.cs.a.809.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(7.55555 - 4.36220i) q^{5} +6.97290 q^{7} +O(q^{10})$	$q+(7.55555 - 4.36220i) q^{5} +6.97290 q^{7} -1.54709i q^{11} +(2.76688 - 4.79238i) q^{13} +(26.4832 - 15.2901i) q^{17} +(-13.0307 + 13.8275i) q^{19} +(-15.3299 - 8.85070i) q^{23} +(25.5576 - 44.2671i) q^{25} +(22.8626 + 13.1997i) q^{29} +35.6349 q^{31} +(52.6841 - 30.4172i) q^{35} -25.7157 q^{37} +(-23.5273 + 13.5835i) q^{41} +(1.57062 + 2.72040i) q^{43} +(-23.2860 - 13.4442i) q^{47} -0.378685 q^{49} +(32.9300 + 19.0121i) q^{53} +(-6.74873 - 11.6891i) q^{55} +(-4.73454 + 2.73349i) q^{59} +(-9.10711 + 15.7740i) q^{61} -48.2788i q^{65} +(-22.9840 + 39.8095i) q^{67} +(-13.2880 + 7.67183i) q^{71} +(-28.2274 - 48.8913i) q^{73} -10.7877i q^{77} +(-39.4035 - 68.2489i) q^{79} +141.290i q^{83} +(133.397 - 231.050i) q^{85} +(80.0444 + 46.2137i) q^{89} +(19.2932 - 33.4168i) q^{91} +(-38.1360 + 161.317i) q^{95} +(-40.5021 - 70.1516i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 16 q^{7}+O(q^{10}) $ 40 * q - 16 * q^7	$ 40 q - 16 q^{7} - 44 q^{13} - 36 q^{19} + 100 q^{25} + 56 q^{31} - 72 q^{37} - 44 q^{43} + 48 q^{49} - 116 q^{55} - 52 q^{61} - 176 q^{67} - 180 q^{73} + 76 q^{79} + 472 q^{85} + 484 q^{91} + 72 q^{97}+O(q^{100}) $ 40 * q - 16 * q^7 - 44 * q^13 - 36 * q^19 + 100 * q^25 + 56 * q^31 - 72 * q^37 - 44 * q^43 + 48 * q^49 - 116 * q^55 - 52 * q^61 - 176 * q^67 - 180 * q^73 + 76 * q^79 + 472 * q^85 + 484 * q^91 + 72 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$	$-1$

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
$5$	7.55555	−	4.36220i	1.51111	−	0.872440i	0.511195	−	0.859465i	$-0.329203\pi$
$5$	7.55555	−	4.36220i	1.51111	−	0.872440i	0.999916	−	0.0129755i	$-0.00413035\pi$
$6$		0			0
$7$	6.97290			0.996128			0.498064	−	0.867140i	$-0.334044\pi$
$7$	6.97290			0.996128			0.498064	+	0.867140i	$0.334044\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.54709i		−	0.140645i	−0.997524	−	0.0703224i	$-0.977597\pi$
$11$		−	1.54709i		−	0.140645i	0.997524	−	0.0703224i	$-0.0224028\pi$
$12$		0			0
$13$	2.76688	−	4.79238i	0.212837	−	0.368645i	−0.739764	−	0.672866i	$-0.765063\pi$
$13$	2.76688	−	4.79238i	0.212837	−	0.368645i	0.952601	+	0.304221i	$0.0983963\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	26.4832	−	15.2901i	1.55783	−	0.899416i	0.560371	−	0.828242i	$-0.310659\pi$
$17$	26.4832	−	15.2901i	1.55783	−	0.899416i	0.997464	−	0.0711744i	$-0.0226747\pi$
$18$		0			0
$19$	−13.0307	+	13.8275i	−0.685828	+	0.727764i
$19$	−13.0307	+	13.8275i	−0.685828	+	0.727764i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−15.3299	−	8.85070i	−0.666516	−	0.384813i	0.128239	−	0.991743i	$-0.459067\pi$
$23$	−15.3299	−	8.85070i	−0.666516	−	0.384813i	−0.794755	+	0.606930i	$0.792401\pi$
$24$		0			0
$25$	25.5576	−	44.2671i	1.02230	−	1.77068i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	22.8626	+	13.1997i	0.788365	+	0.455163i	0.839387	−	0.543535i	$-0.182914\pi$
$29$	22.8626	+	13.1997i	0.788365	+	0.455163i	−0.0510214	+	0.998698i	$0.516248\pi$
$30$		0			0
$31$	35.6349			1.14951			0.574757	−	0.818324i	$-0.305097\pi$
$31$	35.6349			1.14951			0.574757	+	0.818324i	$0.305097\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	52.6841	−	30.4172i	1.50526	−	0.869062i
$36$		0			0
$37$	−25.7157			−0.695019			−0.347509	−	0.937676i	$-0.612973\pi$
$37$	−25.7157			−0.695019			−0.347509	+	0.937676i	$0.612973\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−23.5273	+	13.5835i	−0.573837	+	0.331305i	−0.758680	−	0.651463i	$-0.774156\pi$
$41$	−23.5273	+	13.5835i	−0.573837	+	0.331305i	0.184843	+	0.982768i	$0.440822\pi$
$42$		0			0
$43$	1.57062	+	2.72040i	0.0365261	+	0.0632651i	0.883711	−	0.468034i	$-0.155037\pi$
$43$	1.57062	+	2.72040i	0.0365261	+	0.0632651i	−0.847184	+	0.531299i	$0.821704\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−23.2860	−	13.4442i	−0.495447	−	0.286047i	0.231384	−	0.972862i	$-0.425675\pi$
$47$	−23.2860	−	13.4442i	−0.495447	−	0.286047i	−0.726832	+	0.686816i	$0.759008\pi$
$48$		0			0
$49$	−0.378685			−0.00772827
$50$		0			0
$51$		0			0
$52$		0			0
$53$	32.9300	+	19.0121i	0.621320	+	0.358719i	0.777383	−	0.629028i	$-0.216547\pi$
$53$	32.9300	+	19.0121i	0.621320	+	0.358719i	−0.156063	+	0.987747i	$0.549880\pi$
$54$		0			0
$55$	−6.74873	−	11.6891i	−0.122704	−	0.212530i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−4.73454	+	2.73349i	−0.0802464	+	0.0463303i	−0.539586	−	0.841930i	$-0.681419\pi$
$59$	−4.73454	+	2.73349i	−0.0802464	+	0.0463303i	0.459340	+	0.888261i	$0.348086\pi$
$60$		0			0
$61$	−9.10711	+	15.7740i	−0.149297	+	0.258590i	−0.930968	−	0.365101i	$-0.881034\pi$
$61$	−9.10711	+	15.7740i	−0.149297	+	0.258590i	0.781671	+	0.623691i	$0.214368\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	48.2788i		−	0.742751i
$66$		0			0
$67$	−22.9840	+	39.8095i	−0.343045	+	0.594172i	−0.984997	−	0.172574i	$-0.944792\pi$
$67$	−22.9840	+	39.8095i	−0.343045	+	0.594172i	0.641952	+	0.766745i	$0.278125\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−13.2880	+	7.67183i	−0.187155	+	0.108054i	−0.590650	−	0.806928i	$-0.701129\pi$
$71$	−13.2880	+	7.67183i	−0.187155	+	0.108054i	0.403495	+	0.914982i	$0.367795\pi$
$72$		0			0
$73$	−28.2274	−	48.8913i	−0.386677	−	0.669744i	0.605324	−	0.795979i	$-0.293044\pi$
$73$	−28.2274	−	48.8913i	−0.386677	−	0.669744i	−0.992000	+	0.126236i	$0.959710\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	10.7877i		−	0.140100i
$78$		0			0
$79$	−39.4035	−	68.2489i	−0.498778	−	0.863910i	0.501221	−	0.865320i	$-0.332885\pi$
$79$	−39.4035	−	68.2489i	−0.498778	−	0.863910i	−0.999999	+	0.00140999i	$0.999551\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			141.290i			1.70229i	0.524928	+	0.851147i	$0.324092\pi$
$83$			141.290i			1.70229i	−0.524928	+	0.851147i	$0.675908\pi$
$84$		0			0
$85$	133.397	−	231.050i	1.56937	−	2.71824i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	80.0444	+	46.2137i	0.899375	+	0.519254i	0.876997	−	0.480495i	$-0.159543\pi$
$89$	80.0444	+	46.2137i	0.899375	+	0.519254i	0.0223778	+	0.999750i	$0.492876\pi$
$90$		0			0
$91$	19.2932	−	33.4168i	0.212013	−	0.367218i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−38.1360	+	161.317i	−0.401431	+	1.69808i
$96$		0			0
$97$	−40.5021	−	70.1516i	−0.417547	−	0.723213i	0.578145	−	0.815934i	$-0.303777\pi$
$97$	−40.5021	−	70.1516i	−0.417547	−	0.723213i	−0.995692	+	0.0927214i	$0.970443\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−50.2450	−	29.0090i	−0.497475	−	0.287217i	0.230195	−	0.973145i	$-0.426064\pi$
$101$	−50.2450	−	29.0090i	−0.497475	−	0.287217i	−0.727670	+	0.685927i	$0.759397\pi$
$102$		0			0
$103$	−139.157			−1.35104			−0.675522	−	0.737340i	$-0.736082\pi$
$103$	−139.157			−1.35104			−0.675522	+	0.737340i	$0.736082\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			163.414i			1.52723i	0.645670	+	0.763617i	$0.276578\pi$
$107$			163.414i			1.52723i	−0.645670	+	0.763617i	$0.723422\pi$
$108$		0			0
$109$	−46.4582	−	80.4679i	−0.426222	−	0.738238i	0.570312	−	0.821428i	$-0.306822\pi$
$109$	−46.4582	−	80.4679i	−0.426222	−	0.738238i	−0.996534	+	0.0831906i	$0.973489\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			116.692i			1.03267i	0.856386	+	0.516336i	$0.172705\pi$
$113$			116.692i			1.03267i	−0.856386	+	0.516336i	$0.827295\pi$
$114$		0			0
$115$	−154.434			−1.34291
$116$		0			0
$117$		0			0
$118$		0			0
$119$	184.665	−	106.616i	1.55180	−	0.895934i
$120$		0			0
$121$	118.607			0.980219
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	227.840i		−	1.82272i
$126$		0			0
$127$	59.4345	−	102.944i	0.467989	−	0.810580i	−0.531342	−	0.847157i	$-0.678312\pi$
$127$	59.4345	−	102.944i	0.467989	−	0.810580i	0.999331	+	0.0365773i	$0.0116455\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	168.162	−	97.0885i	1.28368	−	0.741133i	0.306161	−	0.951980i	$-0.400955\pi$
$131$	168.162	−	97.0885i	1.28368	−	0.741133i	0.977519	+	0.210846i	$0.0676220\pi$
$132$		0			0
$133$	−90.8620	+	96.4178i	−0.683173	+	0.724946i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−117.778	−	67.9991i	−0.859692	−	0.496344i	0.00421682	−	0.999991i	$-0.498658\pi$
$137$	−117.778	−	67.9991i	−0.859692	−	0.496344i	−0.863909	+	0.503647i	$0.831991\pi$
$138$		0			0
$139$	104.924	−	181.734i	0.754852	−	1.30744i	−0.190596	−	0.981668i	$-0.561042\pi$
$139$	104.924	−	181.734i	0.754852	−	1.30744i	0.945448	−	0.325773i	$-0.105624\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.41426	−	4.28062i	−0.0518480	−	0.0299344i
$144$		0			0
$145$	230.319			1.58841
$146$		0			0
$147$		0			0
$148$		0			0
$149$	161.799	−	93.4147i	1.08590	−	0.626944i	0.153417	−	0.988161i	$-0.450972\pi$
$149$	161.799	−	93.4147i	1.08590	−	0.626944i	0.932482	+	0.361217i	$0.117639\pi$
$150$		0			0
$151$	−136.997			−0.907264			−0.453632	−	0.891189i	$-0.649872\pi$
$151$	−136.997			−0.907264			−0.453632	+	0.891189i	$0.649872\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	269.242	−	155.447i	1.73704	−	1.00288i
$156$		0			0
$157$	−72.4091	−	125.416i	−0.461204	−	0.798829i	0.537817	−	0.843062i	$-0.319249\pi$
$157$	−72.4091	−	125.416i	−0.461204	−	0.798829i	−0.999021	+	0.0442322i	$0.985916\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−106.894	−	61.7150i	−0.663935	−	0.383323i
$162$		0			0
$163$	117.323			0.719775			0.359887	−	0.932996i	$-0.382815\pi$
$163$	117.323			0.719775			0.359887	+	0.932996i	$0.382815\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	162.015	+	93.5392i	0.970147	+	0.560115i	0.899281	−	0.437371i	$-0.144090\pi$
$167$	162.015	+	93.5392i	0.970147	+	0.560115i	0.0708661	+	0.997486i	$0.477424\pi$
$168$		0			0
$169$	69.1887	+	119.838i	0.409401	+	0.709103i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	163.090	−	94.1603i	0.942719	−	0.544279i	0.0519073	−	0.998652i	$-0.483470\pi$
$173$	163.090	−	94.1603i	0.942719	−	0.544279i	0.890812	+	0.454373i	$0.150137\pi$
$174$		0			0
$175$	178.211	−	308.670i	1.01835	−	1.76383i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			233.023i			1.30181i	0.759161	+	0.650903i	$0.225609\pi$
$179$			233.023i			1.30181i	−0.759161	+	0.650903i	$0.774391\pi$
$180$		0			0
$181$	126.922	−	219.836i	0.701227	−	1.21456i	−0.266808	−	0.963750i	$-0.585969\pi$
$181$	126.922	−	219.836i	0.701227	−	1.21456i	0.968036	−	0.250812i	$-0.0806976\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−194.296	+	112.177i	−1.05025	+	0.606362i
$186$		0			0
$187$	−23.6552	−	40.9719i	−0.126498	−	0.219101i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			228.873i			1.19829i	0.800642	+	0.599143i	$0.204492\pi$
$191$			228.873i			1.19829i	−0.800642	+	0.599143i	$0.795508\pi$
$192$		0			0
$193$	151.550	+	262.493i	0.785234	+	1.36006i	0.928859	+	0.370433i	$0.120791\pi$
$193$	151.550	+	262.493i	0.785234	+	1.36006i	−0.143625	+	0.989632i	$0.545876\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	340.115i		−	1.72647i	−0.504802	−	0.863235i	$-0.668434\pi$
$197$		−	340.115i		−	1.72647i	0.504802	−	0.863235i	$-0.331566\pi$
$198$		0			0
$199$	−110.680	+	191.703i	−0.556180	+	0.963332i	0.441631	+	0.897197i	$0.354400\pi$
$199$	−110.680	+	191.703i	−0.556180	+	0.963332i	−0.997811	+	0.0661350i	$0.978933\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	159.419	+	92.0403i	0.785313	+	0.453401i
$204$		0			0
$205$	−118.508	+	205.262i	−0.578087	+	1.00128i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	21.3924	+	20.1597i	0.102356	+	0.0964581i
$210$		0			0
$211$	−18.1991	−	31.5218i	−0.0862518	−	0.149393i	0.819672	−	0.572833i	$-0.194156\pi$
$211$	−18.1991	−	31.5218i	−0.0862518	−	0.149393i	−0.905924	+	0.423440i	$0.860822\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	23.7339	+	13.7028i	0.110390	+	0.0637337i
$216$		0			0
$217$	248.479			1.14506
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	169.223i		−	0.765717i
$222$		0			0
$223$	5.87009	+	10.1673i	0.0263233	+	0.0455933i	0.878887	−	0.477030i	$-0.158287\pi$
$223$	5.87009	+	10.1673i	0.0263233	+	0.0455933i	−0.852564	+	0.522623i	$0.824953\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	308.513i		−	1.35909i	−0.733634	−	0.679545i	$-0.762177\pi$
$227$		−	308.513i		−	1.35909i	0.733634	−	0.679545i	$-0.237823\pi$
$228$		0			0
$229$	−199.914			−0.872985			−0.436493	−	0.899708i	$-0.643780\pi$
$229$	−199.914			−0.872985			−0.436493	+	0.899708i	$0.643780\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−150.150	+	86.6894i	−0.644422	+	0.372057i	−0.786316	−	0.617825i	$-0.788014\pi$
$233$	−150.150	+	86.6894i	−0.644422	+	0.372057i	0.141894	+	0.989882i	$0.454681\pi$
$234$		0			0
$235$	−234.585			−0.998234
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	92.3221i		−	0.386285i	−0.981171	−	0.193142i	$-0.938132\pi$
$239$		−	92.3221i		−	0.386285i	0.981171	−	0.193142i	$-0.0618679\pi$
$240$		0			0
$241$	−73.8527	+	127.917i	−0.306443	+	0.530775i	−0.977582	−	0.210557i	$-0.932472\pi$
$241$	−73.8527	+	127.917i	−0.306443	+	0.530775i	0.671139	+	0.741332i	$0.265806\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.86118	+	1.65190i	−0.0116783	+	0.00674246i
$246$		0			0
$247$	30.2122	+	100.707i	0.122317	+	0.407722i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	72.3166	+	41.7520i	0.288114	+	0.166343i	0.637091	−	0.770789i	$-0.280138\pi$
$251$	72.3166	+	41.7520i	0.288114	+	0.166343i	−0.348977	+	0.937131i	$0.613471\pi$
$252$		0			0
$253$	−13.6928	+	23.7167i	−0.0541219	+	0.0937419i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−237.979	−	137.397i	−0.925990	−	0.534620i	−0.0404487	−	0.999182i	$-0.512879\pi$
$257$	−237.979	−	137.397i	−0.925990	−	0.534620i	−0.885541	+	0.464561i	$0.846212\pi$
$258$		0			0
$259$	−179.313			−0.692328
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−155.944	+	90.0343i	−0.592943	+	0.342336i	−0.766260	−	0.642530i	$-0.777885\pi$
$263$	−155.944	+	90.0343i	−0.592943	+	0.342336i	0.173318	+	0.984866i	$0.444551\pi$
$264$		0			0
$265$	331.739			1.25185
$266$		0			0
$267$		0			0
$268$		0			0
$269$	131.944	−	76.1780i	0.490499	−	0.283190i	−0.234283	−	0.972169i	$-0.575274\pi$
$269$	131.944	−	76.1780i	0.490499	−	0.283190i	0.724781	+	0.688979i	$0.241941\pi$
$270$		0			0
$271$	−3.27548	−	5.67330i	−0.0120867	−	0.0209347i	0.859919	−	0.510431i	$-0.170514\pi$
$271$	−3.27548	−	5.67330i	−0.0120867	−	0.0209347i	−0.872005	+	0.489496i	$0.837181\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−68.4852	−	39.5400i	−0.249037	−	0.143782i
$276$		0			0
$277$	−486.332			−1.75571			−0.877856	−	0.478924i	$-0.841027\pi$
$277$	−486.332			−1.75571			−0.877856	+	0.478924i	$0.841027\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−235.113	−	135.743i	−0.836703	−	0.483070i	0.0194394	−	0.999811i	$-0.493812\pi$
$281$	−235.113	−	135.743i	−0.836703	−	0.483070i	−0.856142	+	0.516741i	$0.827145\pi$
$282$		0			0
$283$	196.799	+	340.866i	0.695404	+	1.20448i	0.970044	+	0.242928i	$0.0781079\pi$
$283$	196.799	+	340.866i	0.695404	+	1.20448i	−0.274640	+	0.961547i	$0.588559\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−164.054	+	94.7164i	−0.571615	+	0.330022i
$288$		0			0
$289$	323.073	−	559.579i	1.11790	−	1.93626i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			529.445i			1.80698i	0.428610	+	0.903489i	$0.359003\pi$
$293$			529.445i			1.80698i	−0.428610	+	0.903489i	$0.640997\pi$
$294$		0			0
$295$	−23.8480	+	41.3060i	−0.0808408	+	0.140020i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−84.8319	+	48.9777i	−0.283719	+	0.163805i
$300$		0			0
$301$	10.9518	+	18.9691i	0.0363847	+	0.0630202i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			158.908i			0.521010i
$306$		0			0
$307$	192.412	+	333.267i	0.626749	+	1.08556i	0.988200	+	0.153170i	$0.0489483\pi$
$307$	192.412	+	333.267i	0.626749	+	1.08556i	−0.361451	+	0.932391i	$0.617718\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	329.752i		−	1.06030i	−0.847905	−	0.530148i	$-0.822136\pi$
$311$		−	329.752i		−	1.06030i	0.847905	−	0.530148i	$-0.177864\pi$
$312$		0			0
$313$	−14.8195	+	25.6681i	−0.0473467	+	0.0820068i	−0.888728	−	0.458436i	$-0.848410\pi$
$313$	−14.8195	+	25.6681i	−0.0473467	+	0.0820068i	0.841381	+	0.540443i	$0.181743\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	541.900	+	312.866i	1.70946	+	0.986959i	0.935217	+	0.354076i	$0.115204\pi$
$317$	541.900	+	312.866i	1.70946	+	0.986959i	0.774247	+	0.632884i	$0.218129\pi$
$318$		0			0
$319$	20.4212	−	35.3705i	0.0640163	−	0.110879i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−133.672	+	565.437i	−0.413844	+	1.75058i
$324$		0			0
$325$	−141.430	−	244.964i	−0.435169	−	0.753734i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−162.371	−	93.7450i	−0.493529	−	0.284939i
$330$		0			0
$331$	−504.353			−1.52373			−0.761863	−	0.647738i	$-0.775715\pi$
$331$	−504.353			−1.52373			−0.761863	+	0.647738i	$0.775715\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			401.044i			1.19715i
$336$		0			0
$337$	−118.223	−	204.769i	−0.350811	−	0.607622i	0.635581	−	0.772034i	$-0.280761\pi$
$337$	−118.223	−	204.769i	−0.350811	−	0.607622i	−0.986392	+	0.164412i	$0.947427\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	55.1305i		−	0.161673i
$342$		0			0
$343$	−344.313			−1.00383
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−534.925	+	308.839i	−1.54157	+	0.890027i	−0.542832	+	0.839842i	$0.682648\pi$
$347$	−534.925	+	308.839i	−1.54157	+	0.890027i	−0.998740	+	0.0501853i	$0.984019\pi$
$348$		0			0
$349$	123.801			0.354730			0.177365	−	0.984145i	$-0.443243\pi$
$349$	123.801			0.354730			0.177365	+	0.984145i	$0.443243\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	153.890i		−	0.435949i	−0.975954	−	0.217975i	$-0.930055\pi$
$353$		−	153.890i		−	0.435949i	0.975954	−	0.217975i	$-0.0699450\pi$
$354$		0			0
$355$	−66.9321	+	115.930i	−0.188541	+	0.326563i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−405.104	+	233.887i	−1.12842	+	0.651495i	−0.943538	−	0.331265i	$-0.892524\pi$
$359$	−405.104	+	233.887i	−1.12842	+	0.651495i	−0.184885	+	0.982760i	$0.559191\pi$
$360$		0			0
$361$	−21.4002	−	360.365i	−0.0592803	−	0.998241i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−426.547	−	246.267i	−1.16862	−	0.674704i
$366$		0			0
$367$	−65.3268	+	113.149i	−0.178002	+	0.308309i	−0.941196	−	0.337861i	$-0.890297\pi$
$367$	−65.3268	+	113.149i	−0.178002	+	0.308309i	0.763194	+	0.646169i	$0.223630\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	229.617	+	132.570i	0.618915	+	0.357331i
$372$		0			0
$373$	7.77753			0.0208513			0.0104256	−	0.999946i	$-0.496681\pi$
$373$	7.77753			0.0208513			0.0104256	+	0.999946i	$0.496681\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	126.516	−	73.0442i	0.335587	−	0.193751i
$378$		0			0
$379$	−77.1360			−0.203525			−0.101763	−	0.994809i	$-0.532448\pi$
$379$	−77.1360			−0.203525			−0.101763	+	0.994809i	$0.532448\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	379.325	−	219.003i	0.990404	−	0.571810i	0.0850086	−	0.996380i	$-0.472908\pi$
$383$	379.325	−	219.003i	0.990404	−	0.571810i	0.905395	+	0.424570i	$0.139575\pi$
$384$		0			0
$385$	−47.0582	−	81.5072i	−0.122229	−	0.211707i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−145.895	−	84.2326i	−0.375052	−	0.216536i	0.300611	−	0.953747i	$-0.402809\pi$
$389$	−145.895	−	84.2326i	−0.375052	−	0.216536i	−0.675663	+	0.737210i	$0.736143\pi$
$390$		0			0
$391$	−541.311			−1.38443
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−595.430	−	343.772i	−1.50742	−	0.870309i
$396$		0			0
$397$	342.696	+	593.567i	0.863214	+	1.49513i	0.868809	+	0.495147i	$0.164886\pi$
$397$	342.696	+	593.567i	0.863214	+	1.49513i	−0.00559518	+	0.999984i	$0.501781\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−502.251	+	289.975i	−1.25250	+	0.723129i	−0.971605	−	0.236610i	$-0.923964\pi$
$401$	−502.251	+	289.975i	−1.25250	+	0.723129i	−0.280892	+	0.959739i	$0.590630\pi$
$402$		0			0
$403$	98.5977	−	170.776i	0.244659	−	0.423762i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			39.7845i			0.0977507i
$408$		0			0
$409$	−230.293	+	398.878i	−0.563062	+	0.975253i	0.434165	+	0.900834i	$0.357044\pi$
$409$	−230.293	+	398.878i	−0.563062	+	0.975253i	−0.997227	+	0.0744192i	$0.976290\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−33.0135	+	19.0603i	−0.0799357	+	0.0461509i
$414$		0			0
$415$	616.337	+	1067.53i	1.48515	+	2.57235i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	231.953i		−	0.553587i	−0.960929	−	0.276793i	$-0.910728\pi$
$419$		−	231.953i		−	0.553587i	0.960929	−	0.276793i	$-0.0892717\pi$
$420$		0			0
$421$	235.613	+	408.094i	0.559651	+	0.969345i	0.997525	+	0.0703082i	$0.0223983\pi$
$421$	235.613	+	408.094i	0.559651	+	0.969345i	−0.437874	+	0.899036i	$0.644268\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	1563.11i		−	3.67791i
$426$		0			0
$427$	−63.5030	+	109.990i	−0.148719	+	0.257589i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	709.700	+	409.745i	1.64664	+	0.950685i	0.978396	+	0.206738i	$0.0662849\pi$
$431$	709.700	+	409.745i	1.64664	+	0.950685i	0.668239	+	0.743947i	$0.267048\pi$
$432$		0			0
$433$	−68.9423	+	119.412i	−0.159220	+	0.275777i	−0.934588	−	0.355733i	$-0.884231\pi$
$433$	−68.9423	+	119.412i	−0.159220	+	0.275777i	0.775368	+	0.631510i	$0.217565\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	322.142	−	96.6428i	0.737168	−	0.221150i
$438$		0			0
$439$	107.874	+	186.843i	0.245727	+	0.425611i	0.962336	−	0.271864i	$-0.0876402\pi$
$439$	107.874	+	186.843i	0.245727	+	0.425611i	−0.716609	+	0.697475i	$0.754307\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−156.973	−	90.6285i	−0.354341	−	0.204579i	0.312254	−	0.949998i	$-0.398916\pi$
$443$	−156.973	−	90.6285i	−0.354341	−	0.204579i	−0.666596	+	0.745420i	$0.732249\pi$
$444$		0			0
$445$	806.373			1.81207
$446$		0			0
$447$		0			0
$448$		0			0
$449$			748.052i			1.66604i	0.553244	+	0.833019i	$0.313390\pi$
$449$			748.052i			1.66604i	−0.553244	+	0.833019i	$0.686610\pi$
$450$		0			0
$451$	21.0149	+	36.3989i	0.0465963	+	0.0807071i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	336.643i		−	0.739875i
$456$		0			0
$457$	−227.449			−0.497701			−0.248850	−	0.968542i	$-0.580053\pi$
$457$	−227.449			−0.497701			−0.248850	+	0.968542i	$0.580053\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	362.235	−	209.136i	0.785759	−	0.453658i	−0.0527086	−	0.998610i	$-0.516785\pi$
$461$	362.235	−	209.136i	0.785759	−	0.453658i	0.838467	+	0.544952i	$0.183452\pi$
$462$		0			0
$463$	689.238			1.48864			0.744318	−	0.667826i	$-0.232775\pi$
$463$	689.238			1.48864			0.744318	+	0.667826i	$0.232775\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			336.790i			0.721178i	0.932725	+	0.360589i	$0.117424\pi$
$467$			336.790i			0.721178i	−0.932725	+	0.360589i	$0.882576\pi$
$468$		0			0
$469$	−160.265	+	277.588i	−0.341717	+	0.591871i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.20871	−	2.42990i	0.00889790	−	0.00513721i
$474$		0			0
$475$	279.069	+	930.230i	0.587514	+	1.95838i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−382.931	−	221.085i	−0.799438	−	0.461556i	0.0438364	−	0.999039i	$-0.486042\pi$
$479$	−382.931	−	221.085i	−0.799438	−	0.461556i	−0.843275	+	0.537483i	$0.819375\pi$
$480$		0			0
$481$	−71.1524	+	123.239i	−0.147926	+	0.256215i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−612.031	−	353.356i	−1.26192	−	0.728570i
$486$		0			0
$487$	−933.143			−1.91610			−0.958052	−	0.286593i	$-0.907477\pi$
$487$	−933.143			−1.91610			−0.958052	+	0.286593i	$0.907477\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	736.877	−	425.436i	1.50077	−	0.866469i	0.500768	−	0.865581i	$-0.333051\pi$
$491$	736.877	−	425.436i	1.50077	−	0.866469i	1.00000		0.000887369i	$-0.000282458\pi$
$492$		0			0
$493$	807.299			1.63752
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−92.6559	+	53.4949i	−0.186430	+	0.107636i
$498$		0			0
$499$	−68.6506	−	118.906i	−0.137576	−	0.238289i	0.789002	−	0.614390i	$-0.210598\pi$
$499$	−68.6506	−	118.906i	−0.137576	−	0.238289i	−0.926579	+	0.376101i	$0.877265\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−526.517	−	303.985i	−1.04675	−	0.604344i	−0.125015	−	0.992155i	$-0.539898\pi$
$503$	−526.517	−	303.985i	−1.04675	−	0.604344i	−0.921739	+	0.387811i	$0.873231\pi$
$504$		0			0
$505$	−506.172			−1.00232
$506$		0			0
$507$		0			0
$508$		0			0
$509$	53.0407	+	30.6231i	0.104206	+	0.0601632i	0.551197	−	0.834375i	$-0.314171\pi$
$509$	53.0407	+	30.6231i	0.104206	+	0.0601632i	−0.446991	+	0.894538i	$0.647505\pi$
$510$		0			0
$511$	−196.827	−	340.914i	−0.385180	−	0.667151i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1051.41	+	607.033i	−2.04158	+	1.17870i
$516$		0			0
$517$	−20.7994	+	36.0256i	−0.0402309	+	0.0696820i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			210.458i			0.403950i	0.979391	+	0.201975i	$0.0647359\pi$
$521$			210.458i			0.403950i	−0.979391	+	0.201975i	$0.935264\pi$
$522$		0			0
$523$	−389.363	+	674.397i	−0.744480	+	1.28948i	0.205957	+	0.978561i	$0.433969\pi$
$523$	−389.363	+	674.397i	−0.744480	+	1.28948i	−0.950437	+	0.310917i	$0.899364\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	943.726	−	544.861i	1.79075	−	1.03389i
$528$		0			0
$529$	−107.830	−	186.768i	−0.203838	−	0.353058i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			150.336i			0.282056i
$534$		0			0
$535$	712.845	+	1234.68i	1.33242	+	2.30782i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			0.585861i			0.00108694i
$540$		0			0
$541$	−287.444	+	497.868i	−0.531320	+	0.920273i	0.468012	+	0.883722i	$0.344970\pi$
$541$	−287.444	+	497.868i	−0.531320	+	0.920273i	−0.999332	+	0.0365507i	$0.988363\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−702.034	−	405.320i	−1.28814	−	0.743706i
$546$		0			0
$547$	338.221	−	585.817i	0.618321	−	1.07096i	−0.371472	−	0.928444i	$-0.621147\pi$
$547$	338.221	−	585.817i	0.618321	−	1.07096i	0.989792	−	0.142518i	$-0.0455200\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−480.436	+	144.131i	−0.871934	+	0.261580i
$552$		0			0
$553$	−274.757	−	475.892i	−0.496847	−	0.860565i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	26.6883	+	15.4085i	0.0479144	+	0.0276634i	0.523766	−	0.851862i	$-0.324527\pi$
$557$	26.6883	+	15.4085i	0.0479144	+	0.0276634i	−0.475851	+	0.879526i	$0.657860\pi$
$558$		0			0
$559$	17.3829			0.0310965
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	566.760i		−	1.00668i	−0.864089	−	0.503339i	$-0.832105\pi$
$563$		−	566.760i		−	1.00668i	0.864089	−	0.503339i	$-0.167895\pi$
$564$		0			0
$565$	509.034	+	881.673i	0.900945	+	1.56048i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	490.796i		−	0.862559i	−0.902218	−	0.431280i	$-0.858062\pi$
$569$		−	490.796i		−	0.862559i	0.902218	−	0.431280i	$-0.141938\pi$
$570$		0			0
$571$	−446.171			−0.781385			−0.390692	−	0.920521i	$-0.627764\pi$
$571$	−446.171			−0.781385			−0.390692	+	0.920521i	$0.627764\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−783.589	+	452.405i	−1.36276	+	0.786792i
$576$		0			0
$577$	−309.678			−0.536704			−0.268352	−	0.963321i	$-0.586479\pi$
$577$	−309.678			−0.536704			−0.268352	+	0.963321i	$0.586479\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			985.204i			1.69570i
$582$		0			0
$583$	29.4135	−	50.9457i	0.0504520	−	0.0873854i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	267.910	−	154.678i	0.456406	−	0.263506i	−0.254126	−	0.967171i	$-0.581788\pi$
$587$	267.910	−	154.678i	0.456406	−	0.263506i	0.710532	+	0.703665i	$0.248454\pi$
$588$		0			0
$589$	−464.349	+	492.742i	−0.788368	+	0.836574i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−301.852	−	174.275i	−0.509026	−	0.293886i	0.223407	−	0.974725i	$-0.428282\pi$
$593$	−301.852	−	174.275i	−0.509026	−	0.293886i	−0.732433	+	0.680839i	$0.761615\pi$
$594$		0			0
$595$	930.162	−	1611.09i	1.56330	−	2.70771i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	370.252	+	213.765i	0.618117	+	0.356870i	0.776136	−	0.630566i	$-0.217177\pi$
$599$	370.252	+	213.765i	0.618117	+	0.356870i	−0.158019	+	0.987436i	$0.550511\pi$
$600$		0			0
$601$	697.294			1.16022			0.580111	−	0.814537i	$-0.303009\pi$
$601$	697.294			1.16022			0.580111	+	0.814537i	$0.303009\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	896.138	−	517.385i	1.48122	−	0.855183i
$606$		0			0
$607$	−521.986			−0.859944			−0.429972	−	0.902842i	$-0.641476\pi$
$607$	−521.986			−0.859944			−0.429972	+	0.902842i	$0.641476\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−128.859	+	74.3970i	−0.210899	+	0.121763i
$612$		0			0
$613$	357.609	+	619.398i	0.583376	+	1.01044i	0.995076	+	0.0991173i	$0.0316019\pi$
$613$	357.609	+	619.398i	0.583376	+	1.01044i	−0.411700	+	0.911320i	$0.635065\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	810.906	+	468.177i	1.31427	+	0.758795i	0.982801	−	0.184670i	$-0.0591216\pi$
$617$	810.906	+	468.177i	1.31427	+	0.758795i	0.331471	+	0.943465i	$0.392455\pi$
$618$		0			0
$619$	93.3680			0.150837			0.0754184	−	0.997152i	$-0.475971\pi$
$619$	93.3680			0.150837			0.0754184	+	0.997152i	$0.475971\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	558.141	+	322.243i	0.895893	+	0.517244i
$624$		0			0
$625$	−354.942	−	614.777i	−0.567907	−	0.983644i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−681.034	+	393.195i	−1.08272	+	0.625111i
$630$		0			0
$631$	−267.657	+	463.595i	−0.424178	+	0.734698i	−0.996343	−	0.0854402i	$-0.972770\pi$
$631$	−267.657	+	463.595i	−0.424178	+	0.734698i	0.572165	+	0.820139i	$0.306104\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	1037.06i		−	1.63317i
$636$		0			0
$637$	−1.04778	+	1.81481i	−0.00164486	+	0.00284899i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−586.854	+	338.820i	−0.915528	+	0.528580i	−0.882206	−	0.470864i	$-0.843942\pi$
$641$	−586.854	+	338.820i	−0.915528	+	0.528580i	−0.0333225	+	0.999445i	$0.510609\pi$
$642$		0			0
$643$	−433.732	−	751.246i	−0.674545	−	1.16835i	−0.976602	−	0.215056i	$-0.931007\pi$
$643$	−433.732	−	751.246i	−0.674545	−	1.16835i	0.302057	−	0.953290i	$-0.402327\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			599.147i			0.926039i	0.886348	+	0.463020i	$0.153234\pi$
$647$			599.147i			0.926039i	−0.886348	+	0.463020i	$0.846766\pi$
$648$		0			0
$649$	4.22896	+	7.32477i	0.00651611	+	0.0112862i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			83.4862i			0.127850i	0.997955	+	0.0639251i	$0.0203619\pi$
$653$			83.4862i			0.127850i	−0.997955	+	0.0639251i	$0.979638\pi$
$654$		0			0
$655$	847.039	−	1467.11i	1.29319	−	2.23987i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	811.780	+	468.681i	1.23184	+	0.711201i	0.967412	−	0.253206i	$-0.0814851\pi$
$659$	811.780	+	468.681i	1.23184	+	0.711201i	0.264423	+	0.964407i	$0.414818\pi$
$660$		0			0
$661$	175.330	−	303.680i	0.265250	−	0.459426i	−0.702379	−	0.711803i	$-0.747879\pi$
$661$	175.330	−	303.680i	0.265250	−	0.459426i	0.967629	+	0.252377i	$0.0812123\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−265.918	+	1124.85i	−0.399877	+	1.69150i
$666$		0			0
$667$	−233.654	−	404.700i	−0.350305	−	0.606746i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	24.4038	+	14.0895i	0.0363693	+	0.0209978i
$672$		0			0
$673$	−843.178			−1.25287			−0.626433	−	0.779476i	$-0.715486\pi$
$673$	−843.178			−1.25287			−0.626433	+	0.779476i	$0.715486\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			923.246i			1.36373i	0.731478	+	0.681865i	$0.238831\pi$
$677$			923.246i			1.36373i	−0.731478	+	0.681865i	$0.761169\pi$
$678$		0			0
$679$	−282.417	−	489.160i	−0.415930	−	0.720413i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			86.0356i			0.125967i	0.998015	+	0.0629836i	$0.0200616\pi$
$683$			86.0356i			0.125967i	−0.998015	+	0.0629836i	$0.979938\pi$
$684$		0			0
$685$	−1186.50			−1.73212
$686$		0			0
$687$		0			0
$688$		0			0
$689$	182.227	−	105.209i	0.264480	−	0.152698i
$690$		0			0
$691$	1240.37			1.79504			0.897518	−	0.440977i	$-0.145368\pi$
$691$	1240.37			1.79504			0.897518	+	0.440977i	$0.145368\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	1830.80i		−	2.63425i
$696$		0			0
$697$	−415.385	+	719.469i	−0.595962	+	1.03224i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−412.894	+	238.384i	−0.589007	+	0.340063i	−0.764705	−	0.644381i	$-0.777115\pi$
$701$	−412.894	+	238.384i	−0.589007	+	0.340063i	0.175698	+	0.984444i	$0.443782\pi$
$702$		0			0
$703$	335.094	−	355.584i	0.476663	−	0.505810i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−350.353	−	202.277i	−0.495549	−	0.286105i
$708$		0			0
$709$	−662.983	+	1148.32i	−0.935096	+	1.61963i	−0.160633	+	0.987014i	$0.551353\pi$
$709$	−662.983	+	1148.32i	−0.935096	+	1.61963i	−0.774463	+	0.632619i	$0.781980\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−546.278	−	315.394i	−0.766169	−	0.442348i
$714$		0			0
$715$	−74.6918			−0.104464
$716$		0			0
$717$		0			0
$718$		0			0
$719$	216.967	−	125.266i	0.301762	−	0.174222i	−0.341472	−	0.939892i	$-0.610926\pi$
$719$	216.967	−	125.266i	0.301762	−	0.174222i	0.643234	+	0.765670i	$0.277592\pi$
$720$		0			0
$721$	−970.331			−1.34581
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1168.63	−	674.707i	1.61190	−	0.930630i
$726$		0			0
$727$	−369.591	−	640.150i	−0.508378	−	0.880536i	−0.999953	−	0.00970078i	$-0.996912\pi$
$727$	−369.591	−	640.150i	−0.508378	−	0.880536i	0.491575	−	0.870835i	$-0.336421\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	83.1902	+	48.0299i	0.113803	+	0.0657044i
$732$		0			0
$733$	575.654			0.785340			0.392670	−	0.919680i	$-0.371552\pi$
$733$	575.654			0.785340			0.392670	+	0.919680i	$0.371552\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	61.5889	+	35.5584i	0.0835671	+	0.0482475i
$738$		0			0
$739$	296.205	+	513.043i	0.400819	+	0.694239i	0.993825	−	0.110959i	$-0.0353923\pi$
$739$	296.205	+	513.043i	0.400819	+	0.694239i	−0.593006	+	0.805198i	$0.702059\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−296.361	+	171.104i	−0.398871	+	0.230289i	−0.685997	−	0.727604i	$-0.740634\pi$
$743$	−296.361	+	171.104i	−0.398871	+	0.230289i	0.287125	+	0.957893i	$0.407300\pi$
$744$		0			0
$745$	814.987	−	1411.60i	1.09394	−	1.89476i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			1139.47i			1.52132i
$750$		0			0
$751$	−366.642	+	635.042i	−0.488204	+	0.845595i	−0.999908	−	0.0135672i	$-0.995681\pi$
$751$	−366.642	+	635.042i	−0.488204	+	0.845595i	0.511704	+	0.859162i	$0.329015\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1035.09	+	597.608i	−1.37098	+	0.791534i
$756$		0			0
$757$	−151.497	−	262.400i	−0.200128	−	0.346631i	0.748442	−	0.663200i	$-0.230802\pi$
$757$	−151.497	−	262.400i	−0.200128	−	0.346631i	−0.948569	+	0.316569i	$0.897469\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			876.517i			1.15180i	0.817521	+	0.575898i	$0.195348\pi$
$761$			876.517i			1.15180i	−0.817521	+	0.575898i	$0.804652\pi$
$762$		0			0
$763$	−323.948	−	561.094i	−0.424571	−	0.735379i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			30.2530i			0.0394432i
$768$		0			0
$769$	196.099	−	339.653i	0.255005	−	0.441681i	−0.709892	−	0.704310i	$-0.751256\pi$
$769$	196.099	−	339.653i	0.255005	−	0.441681i	0.964897	+	0.262629i	$0.0845896\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	490.012	+	282.908i	0.633909	+	0.365988i	0.782264	−	0.622946i	$-0.214065\pi$
$773$	490.012	+	282.908i	0.633909	+	0.365988i	−0.148355	+	0.988934i	$0.547398\pi$
$774$		0			0
$775$	910.743	−	1577.45i	1.17515	−	2.03542i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	118.752	−	502.327i	0.152442	−	0.644836i
$780$		0			0
$781$	11.8690	+	20.5578i	0.0151972	+	0.0263224i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1094.18	−	631.726i	−1.39386	−	0.804747i
$786$		0			0
$787$	−432.589			−0.549668			−0.274834	−	0.961492i	$-0.588623\pi$
$787$	−432.589			−0.549668			−0.274834	+	0.961492i	$0.588623\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			813.681i			1.02867i
$792$		0			0
$793$	50.3966	+	87.2895i	0.0635519	+	0.110075i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			62.7709i			0.0787590i	0.999224	+	0.0393795i	$0.0125381\pi$
$797$			62.7709i			0.0787590i	−0.999224	+	0.0393795i	$0.987462\pi$
$798$		0			0
$799$	−822.251			−1.02910
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−75.6393	+	43.6704i	−0.0941959	+	0.0543840i
$804$		0			0
$805$	−1076.85			−1.33771
$806$		0			0
$807$		0			0
$808$		0			0
$809$			246.241i			0.304377i	0.988351	+	0.152189i	$0.0486321\pi$
$809$			246.241i			0.304377i	−0.988351	+	0.152189i	$0.951368\pi$
$810$		0			0
$811$	−147.909	+	256.186i	−0.182378	+	0.315889i	−0.942690	−	0.333670i	$-0.891713\pi$
$811$	−147.909	+	256.186i	−0.182378	+	0.315889i	0.760312	+	0.649559i	$0.225046\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	886.442	−	511.788i	1.08766	−	0.627960i
$816$		0			0
$817$	−58.0827	−	13.7310i	−0.0710927	−	0.0168066i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1211.82	+	699.644i	1.47603	+	0.852185i	0.999634	−	0.0270480i	$-0.00861070\pi$
$821$	1211.82	+	699.644i	1.47603	+	0.852185i	0.476393	+	0.879233i	$0.341944\pi$
$822$		0			0
$823$	14.3206	−	24.8039i	0.0174004	−	0.0301384i	−0.857194	−	0.514994i	$-0.827794\pi$
$823$	14.3206	−	24.8039i	0.0174004	−	0.0301384i	0.874595	+	0.484855i	$0.161128\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−943.226	−	544.572i	−1.14054	−	0.658491i	−0.193975	−	0.981006i	$-0.562138\pi$
$827$	−943.226	−	544.572i	−1.14054	−	0.658491i	−0.946564	+	0.322516i	$0.895471\pi$
$828$		0			0
$829$	243.575			0.293817			0.146909	−	0.989150i	$-0.453068\pi$
$829$	243.575			0.293817			0.146909	+	0.989150i	$0.453068\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−10.0288	+	5.79013i	−0.0120394	+	0.00695093i
$834$		0			0
$835$	1632.15			1.95467
$836$		0			0
$837$		0			0
$838$		0			0
$839$	363.154	−	209.667i	0.432842	−	0.249901i	−0.267715	−	0.963498i	$-0.586268\pi$
$839$	363.154	−	209.667i	0.432842	−	0.249901i	0.700557	+	0.713597i	$0.252935\pi$
$840$		0			0
$841$	−72.0345	−	124.767i	−0.0856534	−	0.148356i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1045.52	+	603.630i	1.23730	+	0.714355i
$846$		0			0
$847$	827.031			0.976424
$848$		0			0
$849$		0			0
$850$		0			0
$851$	394.218	+	227.602i	0.463241	+	0.267452i
$852$		0			0
$853$	199.991	+	346.394i	0.234456	+	0.406089i	0.959114	−	0.283019i	$-0.0913359\pi$
$853$	199.991	+	346.394i	0.234456	+	0.406089i	−0.724659	+	0.689108i	$0.758003\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1158.17	−	668.667i	1.35142	−	0.780241i	0.362970	−	0.931801i	$-0.381763\pi$
$857$	1158.17	−	668.667i	1.35142	−	0.780241i	0.988448	+	0.151559i	$0.0484295\pi$
$858$		0			0
$859$	−523.987	+	907.572i	−0.609996	+	1.05654i	0.381244	+	0.924474i	$0.375496\pi$
$859$	−523.987	+	907.572i	−0.609996	+	1.05654i	−0.991240	+	0.132070i	$0.957838\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			588.373i			0.681777i	0.940104	+	0.340888i	$0.110728\pi$
$863$			588.373i			0.681777i	−0.940104	+	0.340888i	$0.889272\pi$
$864$		0			0
$865$	821.492	−	1422.87i	0.949702	−	1.64493i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−105.587	+	60.9608i	−0.121504	+	0.0701505i
$870$		0			0
$871$	127.188	+	220.297i	0.146026	+	0.252924i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	1588.70i		−	1.81566i
$876$		0			0
$877$	−3.13264	−	5.42590i	−0.00357200	−	0.00618689i	0.864234	−	0.503090i	$-0.167804\pi$
$877$	−3.13264	−	5.42590i	−0.00357200	−	0.00618689i	−0.867806	+	0.496903i	$0.834470\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	428.968i		−	0.486910i	−0.969912	−	0.243455i	$-0.921719\pi$
$881$		−	428.968i		−	0.486910i	0.969912	−	0.243455i	$-0.0782809\pi$
$882$		0			0
$883$	130.002	−	225.170i	0.147227	−	0.255005i	−0.782974	−	0.622054i	$-0.786298\pi$
$883$	130.002	−	225.170i	0.147227	−	0.255005i	0.930202	+	0.367049i	$0.119632\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1231.71	−	711.128i	−1.38862	−	0.801723i	−0.395464	−	0.918481i	$-0.629416\pi$
$887$	−1231.71	−	711.128i	−1.38862	−	0.801723i	−0.993160	+	0.116759i	$0.962750\pi$
$888$		0			0
$889$	414.431	−	717.816i	0.466177	−	0.807442i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	489.334	−	146.800i	0.547966	−	0.164390i
$894$		0			0
$895$	1016.49	+	1760.62i	1.13575	+	1.96717i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	814.707	+	470.371i	0.906237	+	0.523216i
$900$		0			0
$901$	1162.79			1.29055
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	2214.64i		−	2.44712i
$906$		0			0
$907$	−828.365	−	1434.77i	−0.913302	−	1.58189i	−0.809369	−	0.587301i	$-0.800191\pi$
$907$	−828.365	−	1434.77i	−0.913302	−	1.58189i	−0.103933	−	0.994584i	$-0.533143\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	893.788i		−	0.981106i	−0.871411	−	0.490553i	$-0.836795\pi$
$911$		−	893.788i		−	0.981106i	0.871411	−	0.490553i	$-0.163205\pi$
$912$		0			0
$913$	218.589			0.239419
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1172.58	−	676.988i	1.27871	−	0.738264i
$918$		0			0
$919$	−343.060			−0.373297			−0.186648	−	0.982427i	$-0.559763\pi$
$919$	−343.060			−0.373297			−0.186648	+	0.982427i	$0.559763\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			84.9083i			0.0919916i
$924$		0			0
$925$	−657.231	+	1138.36i	−0.710520	+	1.23066i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	709.290	−	409.509i	0.763498	−	0.440806i	−0.0670522	−	0.997749i	$-0.521359\pi$
$929$	709.290	−	409.509i	0.763498	−	0.440806i	0.830550	+	0.556944i	$0.188026\pi$
$930$		0			0
$931$	4.93455	−	5.23628i	0.00530026	−	0.00562436i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−357.456	−	206.377i	−0.382305	−	0.220724i
$936$		0			0
$937$	−222.612	+	385.576i	−0.237580	+	0.411500i	−0.960019	−	0.279934i	$-0.909687\pi$
$937$	−222.612	+	385.576i	−0.237580	+	0.411500i	0.722440	+	0.691434i	$0.243021\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	122.681	+	70.8301i	0.130373	+	0.0752711i	0.563768	−	0.825933i	$-0.309351\pi$
$941$	122.681	+	70.8301i	0.130373	+	0.0752711i	−0.433395	+	0.901204i	$0.642684\pi$
$942$		0			0
$943$	480.894			0.509962
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1406.28	−	811.919i	1.48499	−	0.857359i	0.485134	−	0.874440i	$-0.338771\pi$
$947$	1406.28	−	811.919i	1.48499	−	0.857359i	0.999854	+	0.0170810i	$0.00543732\pi$
$948$		0			0
$949$	−312.408			−0.329197
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−602.610	+	347.917i	−0.632330	+	0.365076i	−0.781654	−	0.623713i	$-0.785624\pi$
$953$	−602.610	+	347.917i	−0.632330	+	0.365076i	0.149324	+	0.988788i	$0.452290\pi$
$954$		0			0
$955$	998.388	+	1729.26i	1.04543	+	1.81074i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−821.253	−	474.151i	−0.856364	−	0.494422i
$960$		0			0
$961$	308.848			0.321381
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2290.09	+	1322.18i	2.37315	+	1.37014i
$966$		0			0
$967$	−584.211	−	1011.88i	−0.604147	−	1.04641i	−0.992186	−	0.124771i	$-0.960180\pi$
$967$	−584.211	−	1011.88i	−0.604147	−	1.04641i	0.388038	−	0.921643i	$-0.373153\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	113.347	−	65.4408i	0.116732	−	0.0673953i	−0.440497	−	0.897754i	$-0.645198\pi$
$971$	113.347	−	65.4408i	0.116732	−	0.0673953i	0.557229	+	0.830359i	$0.311864\pi$
$972$		0			0
$973$	731.627	−	1267.22i	0.751929	−	1.30238i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1191.76i			1.21982i	0.792472	+	0.609908i	$0.208794\pi$
$977$			1191.76i			1.21982i	−0.792472	+	0.609908i	$0.791206\pi$
$978$		0			0
$979$	71.4968	−	123.836i	0.0730304	−	0.126492i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1383.54	−	798.789i	1.40747	−	0.812603i	0.412326	−	0.911037i	$-0.364717\pi$
$983$	1383.54	−	798.789i	1.40747	−	0.812603i	0.995144	+	0.0984338i	$0.0313833\pi$
$984$		0			0
$985$	−1483.65	−	2569.76i	−1.50624	−	2.60889i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	55.6045i		−	0.0562229i
$990$		0			0
$991$	311.165	+	538.954i	0.313991	+	0.543848i	0.979222	−	0.202789i	$-0.0650005\pi$
$991$	311.165	+	538.954i	0.313991	+	0.543848i	−0.665232	+	0.746637i	$0.731667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			1931.23i			1.94094i
$996$		0			0
$997$	793.782	−	1374.87i	0.796171	−	1.37901i	−0.125922	−	0.992040i	$-0.540189\pi$
$997$	793.782	−	1374.87i	0.796171	−	1.37901i	0.922093	−	0.386968i	$-0.126478\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.3.cs.a.881.19	yes	40
3.2	odd	2	inner	1368.3.cs.a.881.2	yes	40
19.11	even	3	inner	1368.3.cs.a.809.2	✓	40
57.11	odd	6	inner	1368.3.cs.a.809.19	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.3.cs.a.809.2	✓	40	19.11	even	3	inner
1368.3.cs.a.809.19	yes	40	57.11	odd	6	inner
1368.3.cs.a.881.2	yes	40	3.2	odd	2	inner
1368.3.cs.a.881.19	yes	40	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1368.cs (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(7.55555 - 4.36220i) q^{5} +6.97290 q^{7} +O(q^{10})\)	\(q+(7.55555 - 4.36220i) q^{5} +6.97290 q^{7} -1.54709i q^{11} +(2.76688 - 4.79238i) q^{13} +(26.4832 - 15.2901i) q^{17} +(-13.0307 + 13.8275i) q^{19} +(-15.3299 - 8.85070i) q^{23} +(25.5576 - 44.2671i) q^{25} +(22.8626 + 13.1997i) q^{29} +35.6349 q^{31} +(52.6841 - 30.4172i) q^{35} -25.7157 q^{37} +(-23.5273 + 13.5835i) q^{41} +(1.57062 + 2.72040i) q^{43} +(-23.2860 - 13.4442i) q^{47} -0.378685 q^{49} +(32.9300 + 19.0121i) q^{53} +(-6.74873 - 11.6891i) q^{55} +(-4.73454 + 2.73349i) q^{59} +(-9.10711 + 15.7740i) q^{61} -48.2788i q^{65} +(-22.9840 + 39.8095i) q^{67} +(-13.2880 + 7.67183i) q^{71} +(-28.2274 - 48.8913i) q^{73} -10.7877i q^{77} +(-39.4035 - 68.2489i) q^{79} +141.290i q^{83} +(133.397 - 231.050i) q^{85} +(80.0444 + 46.2137i) q^{89} +(19.2932 - 33.4168i) q^{91} +(-38.1360 + 161.317i) q^{95} +(-40.5021 - 70.1516i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 16 q^{7}+O(q^{10}) \) 40 * q - 16 * q^7	\( 40 q - 16 q^{7} - 44 q^{13} - 36 q^{19} + 100 q^{25} + 56 q^{31} - 72 q^{37} - 44 q^{43} + 48 q^{49} - 116 q^{55} - 52 q^{61} - 176 q^{67} - 180 q^{73} + 76 q^{79} + 472 q^{85} + 484 q^{91} + 72 q^{97}+O(q^{100}) \) 40 * q - 16 * q^7 - 44 * q^13 - 36 * q^19 + 100 * q^25 + 56 * q^31 - 72 * q^37 - 44 * q^43 + 48 * q^49 - 116 * q^55 - 52 * q^61 - 176 * q^67 - 180 * q^73 + 76 * q^79 + 472 * q^85 + 484 * q^91 + 72 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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