LMFDB - Embedded newform 1080.2.q.c.721.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(361,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

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

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

chi := DirichletCharacter("1080.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, 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:	$8.62384341830$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.2.q.c.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{5} +(-1.72474 + 2.98735i) q^{7} +2.00000 q^{17} -6.89898 q^{19} +(-3.72474 - 6.45145i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.949490 + 1.64456i) q^{29} +(-0.550510 - 0.953512i) q^{31} +3.44949 q^{35} -6.00000 q^{37} +(-4.94949 - 8.57277i) q^{41} +(-5.89898 + 10.2173i) q^{43} +(-4.72474 + 8.18350i) q^{47} +(-2.44949 - 4.24264i) q^{49} -7.79796 q^{53} +(-0.550510 - 0.953512i) q^{59} +(1.50000 - 2.59808i) q^{61} +(6.62372 + 11.4726i) q^{67} -9.79796 q^{71} +13.7980 q^{73} +(3.44949 - 5.97469i) q^{79} +(-2.72474 + 4.71940i) q^{83} +(-1.00000 - 1.73205i) q^{85} -2.79796 q^{89} +(3.44949 + 5.97469i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{5} - 2 q^{7} + 8 q^{17} - 8 q^{19} - 10 q^{23} - 2 q^{25} + 6 q^{29} - 12 q^{31} + 4 q^{35} - 24 q^{37} - 10 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{53} - 12 q^{59} + 6 q^{61} + 2 q^{67} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q - 2 * q^5 - 2 * q^7 + 8 * q^17 - 8 * q^19 - 10 * q^23 - 2 * q^25 + 6 * q^29 - 12 * q^31 + 4 * q^35 - 24 * q^37 - 10 * q^41 - 4 * q^43 - 14 * q^47 + 8 * q^53 - 12 * q^59 + 6 * q^61 + 2 * q^67 + 16 * q^73 + 4 * q^79 - 6 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$	−1.72474	+	2.98735i	−0.651892	+	1.12911i	0.330771	+	0.943711i	$0.392691\pi$
$7$	−1.72474	+	2.98735i	−0.651892	+	1.12911i	−0.982663	+	0.185399i	$0.940642\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	−6.89898			−1.58273			−0.791367	−	0.611341i	$-0.790630\pi$
$19$	−6.89898			−1.58273			−0.791367	+	0.611341i	$0.790630\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.72474	−	6.45145i	−0.776663	−	1.34522i	−0.933855	−	0.357652i	$-0.883578\pi$
$23$	−3.72474	−	6.45145i	−0.776663	−	1.34522i	0.157192	−	0.987568i	$-0.449756\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.949490	+	1.64456i	−0.176316	+	0.305388i	−0.940616	−	0.339473i	$-0.889751\pi$
$29$	−0.949490	+	1.64456i	−0.176316	+	0.305388i	0.764300	+	0.644861i	$0.223085\pi$
$30$		0			0
$31$	−0.550510	−	0.953512i	−0.0988746	−	0.171256i	0.812345	−	0.583178i	$-0.198191\pi$
$31$	−0.550510	−	0.953512i	−0.0988746	−	0.171256i	−0.911219	+	0.411922i	$0.864858\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.44949			0.583070
$36$		0			0
$37$	−6.00000			−0.986394			−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000			−0.986394			−0.493197	+	0.869918i	$0.664172\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.94949	−	8.57277i	−0.772980	−	1.33884i	−0.935923	−	0.352206i	$-0.885432\pi$
$41$	−4.94949	−	8.57277i	−0.772980	−	1.33884i	0.162942	−	0.986636i	$-0.447902\pi$
$42$		0			0
$43$	−5.89898	+	10.2173i	−0.899586	+	1.55813i	−0.0715617	+	0.997436i	$0.522798\pi$
$43$	−5.89898	+	10.2173i	−0.899586	+	1.55813i	−0.828024	+	0.560692i	$0.810535\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.72474	+	8.18350i	−0.689175	+	1.19369i	0.282930	+	0.959140i	$0.408693\pi$
$47$	−4.72474	+	8.18350i	−0.689175	+	1.19369i	−0.972105	+	0.234545i	$0.924640\pi$
$48$		0			0
$49$	−2.44949	−	4.24264i	−0.349927	−	0.606092i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.79796			−1.07113			−0.535566	−	0.844493i	$-0.679902\pi$
$53$	−7.79796			−1.07113			−0.535566	+	0.844493i	$0.679902\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.550510	−	0.953512i	−0.0716703	−	0.124137i	0.827963	−	0.560783i	$-0.189500\pi$
$59$	−0.550510	−	0.953512i	−0.0716703	−	0.124137i	−0.899633	+	0.436646i	$0.856166\pi$
$60$		0			0
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	−0.753876	−	0.657017i	$-0.771818\pi$
$61$	1.50000	−	2.59808i	0.192055	−	0.332650i	0.945931	+	0.324367i	$0.105151\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	6.62372	+	11.4726i	0.809217	+	1.40160i	0.913407	+	0.407047i	$0.133442\pi$
$67$	6.62372	+	11.4726i	0.809217	+	1.40160i	−0.104190	+	0.994557i	$0.533225\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.79796			−1.16280			−0.581402	−	0.813617i	$-0.697496\pi$
$71$	−9.79796			−1.16280			−0.581402	+	0.813617i	$0.697496\pi$
$72$		0			0
$73$	13.7980			1.61493			0.807464	−	0.589916i	$-0.200839\pi$
$73$	13.7980			1.61493			0.807464	+	0.589916i	$0.200839\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	3.44949	−	5.97469i	0.388098	−	0.672205i	−0.604096	−	0.796912i	$-0.706466\pi$
$79$	3.44949	−	5.97469i	0.388098	−	0.672205i	0.992194	+	0.124706i	$0.0397989\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.72474	+	4.71940i	−0.299080	+	0.518021i	−0.975926	−	0.218104i	$-0.930013\pi$
$83$	−2.72474	+	4.71940i	−0.299080	+	0.518021i	0.676846	+	0.736125i	$0.263346\pi$
$84$		0			0
$85$	−1.00000	−	1.73205i	−0.108465	−	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.79796			−0.296583			−0.148292	−	0.988944i	$-0.547377\pi$
$89$	−2.79796			−0.296583			−0.148292	+	0.988944i	$0.547377\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.44949	+	5.97469i	0.353910	+	0.612990i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	−0.911479	−	0.411346i	$-0.865059\pi$
$101$	−1.00000	+	1.73205i	−0.0995037	+	0.172345i	0.811976	+	0.583691i	$0.198392\pi$
$102$		0			0
$103$	−5.00000	−	8.66025i	−0.492665	−	0.853320i	0.507300	−	0.861770i	$-0.330644\pi$
$103$	−5.00000	−	8.66025i	−0.492665	−	0.853320i	−0.999964	+	0.00844953i	$0.997310\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.3485			1.19377			0.596886	−	0.802326i	$-0.296405\pi$
$107$	12.3485			1.19377			0.596886	+	0.802326i	$0.296405\pi$
$108$		0			0
$109$	−10.7980			−1.03426			−0.517128	−	0.855908i	$-0.672999\pi$
$109$	−10.7980			−1.03426			−0.517128	+	0.855908i	$0.672999\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.89898	−	8.48528i	−0.460857	−	0.798228i	0.538147	−	0.842851i	$-0.319125\pi$
$113$	−4.89898	−	8.48528i	−0.460857	−	0.798228i	−0.999004	+	0.0446231i	$0.985791\pi$
$114$		0			0
$115$	−3.72474	+	6.45145i	−0.347334	+	0.601601i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.44949	+	5.97469i	−0.316214	+	0.547699i
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	0.348469			0.0309216			0.0154608	−	0.999880i	$-0.495078\pi$
$127$	0.348469			0.0309216			0.0154608	+	0.999880i	$0.495078\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.44949	+	2.51059i	0.126643	+	0.219351i	0.922374	−	0.386299i	$-0.126247\pi$
$131$	1.44949	+	2.51059i	0.126643	+	0.219351i	−0.795731	+	0.605650i	$0.792913\pi$
$132$		0			0
$133$	11.8990	−	20.6096i	1.03177	−	1.78708i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	9.79796	−	16.9706i	0.837096	−	1.44989i	−0.0552162	−	0.998474i	$-0.517585\pi$
$137$	9.79796	−	16.9706i	0.837096	−	1.44989i	0.892312	−	0.451419i	$-0.149082\pi$
$138$		0			0
$139$	−9.79796	−	16.9706i	−0.831052	−	1.43942i	−0.897205	−	0.441615i	$-0.854406\pi$
$139$	−9.79796	−	16.9706i	−0.831052	−	1.43942i	0.0661527	−	0.997810i	$-0.478928\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.89898			0.157702
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.5000	−	18.1865i	−0.860194	−	1.48990i	−0.871742	−	0.489966i	$-0.837009\pi$
$149$	−10.5000	−	18.1865i	−0.860194	−	1.48990i	0.0115483	−	0.999933i	$-0.496324\pi$
$150$		0			0
$151$	−6.00000	+	10.3923i	−0.488273	+	0.845714i	−0.999909	−	0.0134886i	$-0.995706\pi$
$151$	−6.00000	+	10.3923i	−0.488273	+	0.845714i	0.511636	+	0.859202i	$0.329040\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.550510	+	0.953512i	−0.0442180	+	0.0765879i
$156$		0			0
$157$	11.8990	+	20.6096i	0.949642	+	1.64483i	0.746179	+	0.665745i	$0.231886\pi$
$157$	11.8990	+	20.6096i	0.949642	+	1.64483i	0.203463	+	0.979083i	$0.434780\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	25.6969			2.02520
$162$		0			0
$163$	−7.79796			−0.610783			−0.305392	−	0.952227i	$-0.598787\pi$
$163$	−7.79796			−0.610783			−0.305392	+	0.952227i	$0.598787\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.17423	+	7.22999i	0.323012	+	0.559473i	0.981108	−	0.193461i	$-0.0619713\pi$
$167$	4.17423	+	7.22999i	0.323012	+	0.559473i	−0.658096	+	0.752934i	$0.728638\pi$
$168$		0			0
$169$	6.50000	−	11.2583i	0.500000	−	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	−0.998755	−	0.0498898i	$-0.984113\pi$
$173$	−6.00000	+	10.3923i	−0.456172	+	0.790112i	0.542583	+	0.840002i	$0.317446\pi$
$174$		0			0
$175$	−1.72474	−	2.98735i	−0.130378	−	0.225822i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	13.7980			1.03131			0.515654	−	0.856797i	$-0.327549\pi$
$179$	13.7980			1.03131			0.515654	+	0.856797i	$0.327549\pi$
$180$		0			0
$181$	−9.69694			−0.720768			−0.360384	−	0.932804i	$-0.617354\pi$
$181$	−9.69694			−0.720768			−0.360384	+	0.932804i	$0.617354\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.00000	+	5.19615i	0.220564	+	0.382029i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−7.44949	+	12.9029i	−0.539026	+	0.933621i	0.459931	+	0.887955i	$0.347874\pi$
$191$	−7.44949	+	12.9029i	−0.539026	+	0.933621i	−0.998957	+	0.0456658i	$0.985459\pi$
$192$		0			0
$193$	1.10102	+	1.90702i	0.0792532	+	0.137271i	0.902928	−	0.429792i	$-0.141413\pi$
$193$	1.10102	+	1.90702i	0.0792532	+	0.137271i	−0.823675	+	0.567063i	$0.808080\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.5959			1.11116			0.555582	−	0.831462i	$-0.312496\pi$
$197$	15.5959			1.11116			0.555582	+	0.831462i	$0.312496\pi$
$198$		0			0
$199$	22.8990			1.62327			0.811633	−	0.584168i	$-0.198579\pi$
$199$	22.8990			1.62327			0.811633	+	0.584168i	$0.198579\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3.27526	−	5.67291i	−0.229878	−	0.398160i
$204$		0			0
$205$	−4.94949	+	8.57277i	−0.345687	+	0.598748i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−6.00000	−	10.3923i	−0.413057	−	0.715436i	0.582165	−	0.813070i	$-0.302206\pi$
$211$	−6.00000	−	10.3923i	−0.413057	−	0.715436i	−0.995222	+	0.0976347i	$0.968872\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.7980			0.804614
$216$		0			0
$217$	3.79796			0.257822
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.72474	+	4.71940i	−0.182462	+	0.316034i	−0.942718	−	0.333589i	$-0.891740\pi$
$223$	−2.72474	+	4.71940i	−0.182462	+	0.316034i	0.760256	+	0.649623i	$0.225073\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−4.10102	+	7.10318i	−0.272194	+	0.471454i	−0.969423	−	0.245394i	$-0.921083\pi$
$227$	−4.10102	+	7.10318i	−0.272194	+	0.471454i	0.697229	+	0.716848i	$0.254416\pi$
$228$		0			0
$229$	2.05051	+	3.55159i	0.135502	+	0.234696i	0.925789	−	0.378041i	$-0.123402\pi$
$229$	2.05051	+	3.55159i	0.135502	+	0.234696i	−0.790287	+	0.612736i	$0.790069\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	19.7980			1.29701			0.648504	−	0.761211i	$-0.275395\pi$
$233$	19.7980			1.29701			0.648504	+	0.761211i	$0.275395\pi$
$234$		0			0
$235$	9.44949			0.616417
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.10102	+	1.90702i	0.0712191	+	0.123355i	0.899436	−	0.437053i	$-0.143978\pi$
$239$	1.10102	+	1.90702i	0.0712191	+	0.123355i	−0.828217	+	0.560408i	$0.810644\pi$
$240$		0			0
$241$	1.84847	−	3.20164i	0.119070	−	0.206236i	−0.800329	−	0.599561i	$-0.795342\pi$
$241$	1.84847	−	3.20164i	0.119070	−	0.206236i	0.919400	+	0.393325i	$0.128675\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2.44949	+	4.24264i	−0.156492	+	0.271052i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.89898			−0.182982			−0.0914910	−	0.995806i	$-0.529163\pi$
$251$	−2.89898			−0.182982			−0.0914910	+	0.995806i	$0.529163\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.8990	+	24.0737i	0.866995	+	1.50168i	0.865053	+	0.501680i	$0.167285\pi$
$257$	13.8990	+	24.0737i	0.866995	+	1.50168i	0.00194150	+	0.999998i	$0.499382\pi$
$258$		0			0
$259$	10.3485	−	17.9241i	0.643023	−	1.11375i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	−0.442943	−	0.896550i	$-0.646065\pi$
$263$	9.00000	−	15.5885i	0.554964	−	0.961225i	0.997906	−	0.0646755i	$-0.0206012\pi$
$264$		0			0
$265$	3.89898	+	6.75323i	0.239512	+	0.414848i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−22.5959			−1.37770			−0.688849	−	0.724905i	$-0.741884\pi$
$269$	−22.5959			−1.37770			−0.688849	+	0.724905i	$0.741884\pi$
$270$		0			0
$271$	30.8990			1.87698			0.938490	−	0.345307i	$-0.112225\pi$
$271$	30.8990			1.87698			0.938490	+	0.345307i	$0.112225\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	−0.575408	−	0.817867i	$-0.695157\pi$
$277$	7.00000	−	12.1244i	0.420589	−	0.728482i	0.995997	+	0.0893846i	$0.0284900\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−4.05051	+	7.01569i	−0.241633	+	0.418521i	−0.961180	−	0.275923i	$-0.911016\pi$
$281$	−4.05051	+	7.01569i	−0.241633	+	0.418521i	0.719546	+	0.694444i	$0.244350\pi$
$282$		0			0
$283$	1.82577	+	3.16232i	0.108530	+	0.187980i	0.915175	−	0.403056i	$-0.132052\pi$
$283$	1.82577	+	3.16232i	0.108530	+	0.187980i	−0.806645	+	0.591037i	$0.798719\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	34.1464			2.01560
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8.79796	+	15.2385i	0.513982	+	0.890243i	0.999868	+	0.0162213i	$0.00516362\pi$
$293$	8.79796	+	15.2385i	0.513982	+	0.890243i	−0.485886	+	0.874022i	$0.661503\pi$
$294$		0			0
$295$	−0.550510	+	0.953512i	−0.0320519	+	0.0555156i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−20.3485	−	35.2446i	−1.17287	−	2.03146i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.00000			−0.171780
$306$		0			0
$307$	−4.75255			−0.271242			−0.135621	−	0.990761i	$-0.543303\pi$
$307$	−4.75255			−0.271242			−0.135621	+	0.990761i	$0.543303\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.44949	−	5.97469i	−0.195603	−	0.338794i	0.751495	−	0.659738i	$-0.229333\pi$
$311$	−3.44949	−	5.97469i	−0.195603	−	0.338794i	−0.947098	+	0.320945i	$0.896000\pi$
$312$		0			0
$313$	7.79796	−	13.5065i	0.440767	−	0.763430i	−0.556980	−	0.830526i	$-0.688040\pi$
$313$	7.79796	−	13.5065i	0.440767	−	0.763430i	0.997747	+	0.0670957i	$0.0213733\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.8990	+	22.3417i	−0.724479	+	1.25483i	0.234709	+	0.972066i	$0.424586\pi$
$317$	−12.8990	+	22.3417i	−0.724479	+	1.25483i	−0.959188	+	0.282769i	$0.908747\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−13.7980			−0.767739
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−16.2980	−	28.2289i	−0.898536	−	1.55631i
$330$		0			0
$331$	3.44949	−	5.97469i	0.189601	−	0.328399i	−0.755516	−	0.655130i	$-0.772614\pi$
$331$	3.44949	−	5.97469i	0.189601	−	0.328399i	0.945117	+	0.326731i	$0.105947\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	6.62372	−	11.4726i	0.361893	−	0.626817i
$336$		0			0
$337$	14.8990	+	25.8058i	0.811599	+	1.40573i	0.911744	+	0.410758i	$0.134736\pi$
$337$	14.8990	+	25.8058i	0.811599	+	1.40573i	−0.100145	+	0.994973i	$0.531931\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−7.24745			−0.391325
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.7980	−	22.1667i	−0.687030	−	1.18997i	−0.972794	−	0.231671i	$-0.925581\pi$
$347$	−12.7980	−	22.1667i	−0.687030	−	1.18997i	0.285764	−	0.958300i	$-0.407753\pi$
$348$		0			0
$349$	1.15153	−	1.99451i	0.0616400	−	0.106764i	−0.833559	−	0.552431i	$-0.813700\pi$
$349$	1.15153	−	1.99451i	0.0616400	−	0.106764i	0.895199	+	0.445667i	$0.147034\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$	4.89898	+	8.48528i	0.260011	+	0.450352i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−28.6969			−1.51457			−0.757283	−	0.653087i	$-0.773474\pi$
$359$	−28.6969			−1.51457			−0.757283	+	0.653087i	$0.773474\pi$
$360$		0			0
$361$	28.5959			1.50505
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.89898	−	11.9494i	−0.361109	−	0.625459i
$366$		0			0
$367$	−11.0000	+	19.0526i	−0.574195	+	0.994535i	0.421933	+	0.906627i	$0.361352\pi$
$367$	−11.0000	+	19.0526i	−0.574195	+	0.994535i	−0.996129	+	0.0879086i	$0.971982\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	13.4495	−	23.2952i	0.698263	−	1.20943i
$372$		0			0
$373$	−10.7980	−	18.7026i	−0.559097	−	0.968385i	−0.997572	−	0.0696412i	$-0.977815\pi$
$373$	−10.7980	−	18.7026i	−0.559097	−	0.968385i	0.438475	−	0.898743i	$-0.355519\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	26.8990			1.38171			0.690854	−	0.722994i	$-0.257235\pi$
$379$	26.8990			1.38171			0.690854	+	0.722994i	$0.257235\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.00000	+	1.73205i	0.0510976	+	0.0885037i	0.890443	−	0.455095i	$-0.150395\pi$
$383$	1.00000	+	1.73205i	0.0510976	+	0.0885037i	−0.839345	+	0.543599i	$0.817061\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.29796	+	14.3725i	−0.420723	+	0.728714i	−0.996010	−	0.0892375i	$-0.971557\pi$
$389$	−8.29796	+	14.3725i	−0.420723	+	0.728714i	0.575287	+	0.817952i	$0.304890\pi$
$390$		0			0
$391$	−7.44949	−	12.9029i	−0.376737	−	0.652527i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.89898			−0.347125
$396$		0			0
$397$	−22.0000			−1.10415			−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000			−1.10415			−0.552074	+	0.833795i	$0.686163\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	12.7980	+	22.1667i	0.639100	+	1.10695i	0.985631	+	0.168914i	$0.0540262\pi$
$401$	12.7980	+	22.1667i	0.639100	+	1.10695i	−0.346531	+	0.938038i	$0.612641\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.79796	+	8.31031i	0.237244	+	0.410918i	0.959922	−	0.280266i	$-0.0904226\pi$
$409$	4.79796	+	8.31031i	0.237244	+	0.410918i	−0.722679	+	0.691184i	$0.757089\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.79796			0.186885
$414$		0			0
$415$	5.44949			0.267505
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	0.999913	−	0.0131919i	$-0.00419923\pi$
$419$	10.0000	+	17.3205i	0.488532	+	0.846162i	−0.511381	+	0.859354i	$0.670866\pi$
$420$		0			0
$421$	−12.7980	+	22.1667i	−0.623734	+	1.08034i	0.365050	+	0.930988i	$0.381052\pi$
$421$	−12.7980	+	22.1667i	−0.623734	+	1.08034i	−0.988784	+	0.149352i	$0.952281\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.00000	+	1.73205i	−0.0485071	+	0.0840168i
$426$		0			0
$427$	5.17423	+	8.96204i	0.250399	+	0.433703i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.8990			−0.717659			−0.358829	−	0.933403i	$-0.616824\pi$
$431$	−14.8990			−0.717659			−0.358829	+	0.933403i	$0.616824\pi$
$432$		0			0
$433$	−11.7980			−0.566974			−0.283487	−	0.958976i	$-0.591491\pi$
$433$	−11.7980			−0.566974			−0.283487	+	0.958976i	$0.591491\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	25.6969	+	44.5084i	1.22925	+	2.12913i
$438$		0			0
$439$	8.89898	−	15.4135i	0.424725	−	0.735645i	−0.571670	−	0.820484i	$-0.693704\pi$
$439$	8.89898	−	15.4135i	0.424725	−	0.735645i	0.996395	+	0.0848384i	$0.0270374\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.07321	+	7.05501i	−0.193524	+	0.335194i	−0.946416	−	0.322951i	$-0.895325\pi$
$443$	−4.07321	+	7.05501i	−0.193524	+	0.335194i	0.752892	+	0.658145i	$0.228658\pi$
$444$		0			0
$445$	1.39898	+	2.42310i	0.0663180	+	0.114866i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.5959			−1.01917			−0.509587	−	0.860419i	$-0.670202\pi$
$449$	−21.5959			−1.01917			−0.509587	+	0.860419i	$0.670202\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8.79796	+	15.2385i	−0.411551	+	0.712828i	−0.995060	−	0.0992796i	$-0.968346\pi$
$457$	−8.79796	+	15.2385i	−0.411551	+	0.712828i	0.583508	+	0.812107i	$0.301680\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.05051	−	10.4798i	0.281800	−	0.488093i	−0.690028	−	0.723783i	$-0.742402\pi$
$461$	6.05051	−	10.4798i	0.281800	−	0.488093i	0.971828	+	0.235690i	$0.0757351\pi$
$462$		0			0
$463$	−6.10102	−	10.5673i	−0.283538	−	0.491103i	0.688715	−	0.725032i	$-0.258175\pi$
$463$	−6.10102	−	10.5673i	−0.283538	−	0.491103i	−0.972254	+	0.233929i	$0.924842\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.79796			−0.360847			−0.180423	−	0.983589i	$-0.557747\pi$
$467$	−7.79796			−0.360847			−0.180423	+	0.983589i	$0.557747\pi$
$468$		0			0
$469$	−45.6969			−2.11009
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	3.44949	−	5.97469i	0.158273	−	0.274138i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	15.2474	−	26.4094i	0.696674	−	1.20667i	−0.272939	−	0.962031i	$-0.587996\pi$
$479$	15.2474	−	26.4094i	0.696674	−	1.20667i	0.969613	−	0.244643i	$-0.0786708\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.00000			0.0908153
$486$		0			0
$487$	−13.5959			−0.616090			−0.308045	−	0.951372i	$-0.599675\pi$
$487$	−13.5959			−0.616090			−0.308045	+	0.951372i	$0.599675\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6.89898	−	11.9494i	−0.311347	−	0.539268i	0.667308	−	0.744782i	$-0.267447\pi$
$491$	−6.89898	−	11.9494i	−0.311347	−	0.539268i	−0.978654	+	0.205514i	$0.934113\pi$
$492$		0			0
$493$	−1.89898	+	3.28913i	−0.0855257	+	0.148135i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.8990	−	29.2699i	0.758023	−	1.31293i
$498$		0			0
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	0.988204	+	0.153141i	$0.0489388\pi$
$499$	14.0000	+	24.2487i	0.626726	+	1.08552i	−0.361478	+	0.932381i	$0.617728\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−25.0454			−1.11672			−0.558360	−	0.829599i	$-0.688569\pi$
$503$	−25.0454			−1.11672			−0.558360	+	0.829599i	$0.688569\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.05051	−	3.55159i	−0.0908873	−	0.157421i	0.816997	−	0.576641i	$-0.195637\pi$
$509$	−2.05051	−	3.55159i	−0.0908873	−	0.157421i	−0.907885	+	0.419220i	$0.862304\pi$
$510$		0			0
$511$	−23.7980	+	41.2193i	−1.05276	+	1.82343i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−5.00000	+	8.66025i	−0.220326	+	0.381616i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9.20204			−0.403149			−0.201574	−	0.979473i	$-0.564606\pi$
$521$	−9.20204			−0.403149			−0.201574	+	0.979473i	$0.564606\pi$
$522$		0			0
$523$	−34.3485			−1.50195			−0.750977	−	0.660329i	$-0.770417\pi$
$523$	−34.3485			−1.50195			−0.750977	+	0.660329i	$0.770417\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.10102	−	1.90702i	−0.0479612	−	0.0830712i
$528$		0			0
$529$	−16.2474	+	28.1414i	−0.706411	+	1.22354i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−6.17423	−	10.6941i	−0.266935	−	0.462346i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	12.1010			0.520264			0.260132	−	0.965573i	$-0.416234\pi$
$541$	12.1010			0.520264			0.260132	+	0.965573i	$0.416234\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	5.39898	+	9.35131i	0.231267	+	0.400566i
$546$		0			0
$547$	8.37628	−	14.5081i	0.358144	−	0.620323i	−0.629507	−	0.776995i	$-0.716743\pi$
$547$	8.37628	−	14.5081i	0.358144	−	0.620323i	0.987651	+	0.156672i	$0.0500765\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.55051	−	11.3458i	0.279061	−	0.483348i
$552$		0			0
$553$	11.8990	+	20.6096i	0.505996	+	0.876411i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−43.7980			−1.85578			−0.927890	−	0.372855i	$-0.878379\pi$
$557$	−43.7980			−1.85578			−0.927890	+	0.372855i	$0.878379\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5.17423	−	8.96204i	−0.218068	−	0.377705i	0.736149	−	0.676819i	$-0.236642\pi$
$563$	−5.17423	−	8.96204i	−0.218068	−	0.377705i	−0.954217	+	0.299114i	$0.903309\pi$
$564$		0			0
$565$	−4.89898	+	8.48528i	−0.206102	+	0.356978i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5.00000	−	8.66025i	0.209611	−	0.363057i	−0.741981	−	0.670421i	$-0.766114\pi$
$569$	5.00000	−	8.66025i	0.209611	−	0.363057i	0.951592	+	0.307364i	$0.0994469\pi$
$570$		0			0
$571$	−12.8990	−	22.3417i	−0.539805	−	0.934971i	−0.998914	−	0.0465904i	$-0.985164\pi$
$571$	−12.8990	−	22.3417i	−0.539805	−	0.934971i	0.459109	−	0.888380i	$-0.348169\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7.44949			0.310665
$576$		0			0
$577$	−26.0000			−1.08239			−0.541197	−	0.840896i	$-0.682029\pi$
$577$	−26.0000			−1.08239			−0.541197	+	0.840896i	$0.682029\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.39898	−	16.2795i	−0.389935	−	0.675388i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.82577	−	6.62642i	0.157906	−	0.273502i	−0.776207	−	0.630478i	$-0.782859\pi$
$587$	3.82577	−	6.62642i	0.157906	−	0.273502i	0.934113	+	0.356976i	$0.116192\pi$
$588$		0			0
$589$	3.79796	+	6.57826i	0.156492	+	0.271052i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	31.3939			1.28919			0.644596	−	0.764523i	$-0.277026\pi$
$593$	31.3939			1.28919			0.644596	+	0.764523i	$0.277026\pi$
$594$		0			0
$595$	6.89898			0.282831
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.3485	−	21.3882i	−0.504545	−	0.873897i	−0.999986	−	0.00525583i	$-0.998327\pi$
$599$	−12.3485	−	21.3882i	−0.504545	−	0.873897i	0.495441	−	0.868641i	$-0.335006\pi$
$600$		0			0
$601$	−7.00000	+	12.1244i	−0.285536	+	0.494563i	−0.972739	−	0.231903i	$-0.925505\pi$
$601$	−7.00000	+	12.1244i	−0.285536	+	0.494563i	0.687203	+	0.726465i	$0.258838\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.50000	−	9.52628i	0.223607	−	0.387298i
$606$		0			0
$607$	−1.07321	−	1.85886i	−0.0435604	−	0.0754489i	0.843423	−	0.537250i	$-0.180537\pi$
$607$	−1.07321	−	1.85886i	−0.0435604	−	0.0754489i	−0.886984	+	0.461801i	$0.847203\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−29.5959			−1.19537			−0.597684	−	0.801732i	$-0.703912\pi$
$613$	−29.5959			−1.19537			−0.597684	+	0.801732i	$0.703912\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17.5959	−	30.4770i	−0.708385	−	1.22696i	−0.965456	−	0.260566i	$-0.916091\pi$
$617$	−17.5959	−	30.4770i	−0.708385	−	1.22696i	0.257071	−	0.966393i	$-0.417243\pi$
$618$		0			0
$619$	12.3485	−	21.3882i	0.496327	−	0.859663i	−0.503664	−	0.863900i	$-0.668015\pi$
$619$	12.3485	−	21.3882i	0.496327	−	0.859663i	0.999991	+	0.00423617i	$0.00134842\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.82577	−	8.35847i	0.193340	−	0.334875i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	−15.5959			−0.620864			−0.310432	−	0.950596i	$-0.600474\pi$
$631$	−15.5959			−0.620864			−0.310432	+	0.950596i	$0.600474\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−0.174235	−	0.301783i	−0.00691429	−	0.0119759i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−6.94949	+	12.0369i	−0.274488	+	0.475428i	−0.970006	−	0.243081i	$-0.921842\pi$
$641$	−6.94949	+	12.0369i	−0.274488	+	0.475428i	0.695518	+	0.718509i	$0.255175\pi$
$642$		0			0
$643$	10.0732	+	17.4473i	0.397249	+	0.688055i	0.993385	−	0.114828i	$-0.0366317\pi$
$643$	10.0732	+	17.4473i	0.397249	+	0.688055i	−0.596137	+	0.802883i	$0.703298\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−6.55051			−0.257527			−0.128764	−	0.991675i	$-0.541101\pi$
$647$	−6.55051			−0.257527			−0.128764	+	0.991675i	$0.541101\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.8990	+	24.0737i	0.543909	+	0.942078i	0.998675	+	0.0514670i	$0.0163897\pi$
$653$	13.8990	+	24.0737i	0.543909	+	0.942078i	−0.454766	+	0.890611i	$0.650277\pi$
$654$		0			0
$655$	1.44949	−	2.51059i	0.0566363	−	0.0980969i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5.10102	−	8.83523i	0.198708	−	0.344172i	−0.749402	−	0.662115i	$-0.769659\pi$
$659$	5.10102	−	8.83523i	0.198708	−	0.344172i	0.948110	+	0.317944i	$0.102992\pi$
$660$		0			0
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	0.752252	−	0.658876i	$-0.228968\pi$
$661$	−5.00000	−	8.66025i	−0.194477	−	0.336845i	−0.946729	+	0.322031i	$0.895634\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−23.7980			−0.922845
$666$		0			0
$667$	14.1464			0.547752
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	8.00000	−	13.8564i	0.308377	−	0.534125i	−0.669630	−	0.742695i	$-0.733547\pi$
$673$	8.00000	−	13.8564i	0.308377	−	0.534125i	0.978008	+	0.208569i	$0.0668807\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.10102	−	12.2993i	0.272914	−	0.472702i	−0.696692	−	0.717370i	$-0.745346\pi$
$677$	7.10102	−	12.2993i	0.272914	−	0.472702i	0.969607	+	0.244668i	$0.0786791\pi$
$678$		0			0
$679$	−3.44949	−	5.97469i	−0.132379	−	0.229288i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−23.3939			−0.895142			−0.447571	−	0.894248i	$-0.647711\pi$
$683$	−23.3939			−0.895142			−0.447571	+	0.894248i	$0.647711\pi$
$684$		0			0
$685$	−19.5959			−0.748722
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−19.4495	+	33.6875i	−0.739893	+	1.28153i	0.212649	+	0.977129i	$0.431791\pi$
$691$	−19.4495	+	33.6875i	−0.739893	+	1.28153i	−0.952543	+	0.304404i	$0.901543\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−9.79796	+	16.9706i	−0.371658	+	0.643730i
$696$		0			0
$697$	−9.89898	−	17.1455i	−0.374951	−	0.649433i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−30.3939			−1.14796			−0.573980	−	0.818869i	$-0.694601\pi$
$701$	−30.3939			−1.14796			−0.573980	+	0.818869i	$0.694601\pi$
$702$		0			0
$703$	41.3939			1.56120
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.44949	−	5.97469i	−0.129731	−	0.224701i
$708$		0			0
$709$	−4.84847	+	8.39780i	−0.182088	+	0.315386i	−0.942591	−	0.333948i	$-0.891619\pi$
$709$	−4.84847	+	8.39780i	−0.182088	+	0.315386i	0.760503	+	0.649334i	$0.224952\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4.10102	+	7.10318i	−0.153584	+	0.266016i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	28.2929			1.05515			0.527573	−	0.849510i	$-0.323102\pi$
$719$	28.2929			1.05515			0.527573	+	0.849510i	$0.323102\pi$
$720$		0			0
$721$	34.4949			1.28466
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−0.949490	−	1.64456i	−0.0352632	−	0.0610776i
$726$		0			0
$727$	15.6237	−	27.0611i	0.579452	−	1.00364i	−0.416090	−	0.909323i	$-0.636600\pi$
$727$	15.6237	−	27.0611i	0.579452	−	1.00364i	0.995542	−	0.0943168i	$-0.0300667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−11.7980	+	20.4347i	−0.436363	+	0.755803i
$732$		0			0
$733$	14.7980	+	25.6308i	0.546575	+	0.946696i	0.998506	+	0.0546429i	$0.0174020\pi$
$733$	14.7980	+	25.6308i	0.546575	+	0.946696i	−0.451931	+	0.892053i	$0.649265\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−33.1010			−1.21764			−0.608820	−	0.793308i	$-0.708357\pi$
$739$	−33.1010			−1.21764			−0.608820	+	0.793308i	$0.708357\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.52270	−	6.10150i	−0.129235	−	0.223842i	0.794145	−	0.607728i	$-0.207919\pi$
$743$	−3.52270	−	6.10150i	−0.129235	−	0.223842i	−0.923381	+	0.383886i	$0.874586\pi$
$744$		0			0
$745$	−10.5000	+	18.1865i	−0.384690	+	0.666303i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−21.2980	+	36.8891i	−0.778210	+	1.34790i
$750$		0			0
$751$	−13.1010	−	22.6916i	−0.478063	−	0.828029i	0.521621	−	0.853177i	$-0.325328\pi$
$751$	−13.1010	−	22.6916i	−0.478063	−	0.828029i	−0.999684	+	0.0251480i	$0.991994\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	12.0000			0.436725
$756$		0			0
$757$	31.5959			1.14837			0.574187	−	0.818724i	$-0.305318\pi$
$757$	31.5959			1.14837			0.574187	+	0.818724i	$0.305318\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	17.5000	+	30.3109i	0.634375	+	1.09877i	0.986647	+	0.162872i	$0.0520756\pi$
$761$	17.5000	+	30.3109i	0.634375	+	1.09877i	−0.352273	+	0.935897i	$0.614591\pi$
$762$		0			0
$763$	18.6237	−	32.2572i	0.674224	−	1.16779i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	2.60102	+	4.50510i	0.0937952	+	0.162458i	0.909105	−	0.416567i	$-0.136767\pi$
$769$	2.60102	+	4.50510i	0.0937952	+	0.162458i	−0.815310	+	0.579025i	$0.803433\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6.20204			−0.223072			−0.111536	−	0.993760i	$-0.535577\pi$
$773$	−6.20204			−0.223072			−0.111536	+	0.993760i	$0.535577\pi$
$774$		0			0
$775$	1.10102			0.0395498
$776$		0			0
$777$		0			0
$778$		0			0
$779$	34.1464	+	59.1433i	1.22342	+	2.11903i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	11.8990	−	20.6096i	0.424693	−	0.735589i
$786$		0			0
$787$	24.7980	+	42.9513i	0.883952	+	1.53105i	0.846910	+	0.531736i	$0.178460\pi$
$787$	24.7980	+	42.9513i	0.883952	+	1.53105i	0.0370414	+	0.999314i	$0.488207\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	33.7980			1.20172
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.00000	−	6.92820i	−0.141687	−	0.245410i	0.786445	−	0.617661i	$-0.211919\pi$
$797$	−4.00000	−	6.92820i	−0.141687	−	0.245410i	−0.928132	+	0.372251i	$0.878586\pi$
$798$		0			0
$799$	−9.44949	+	16.3670i	−0.334299	+	0.579023i
$800$		0

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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{5} +(-1.72474 + 2.98735i) q^{7} +2.00000 q^{17} -6.89898 q^{19} +(-3.72474 - 6.45145i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.949490 + 1.64456i) q^{29} +(-0.550510 - 0.953512i) q^{31} +3.44949 q^{35} -6.00000 q^{37} +(-4.94949 - 8.57277i) q^{41} +(-5.89898 + 10.2173i) q^{43} +(-4.72474 + 8.18350i) q^{47} +(-2.44949 - 4.24264i) q^{49} -7.79796 q^{53} +(-0.550510 - 0.953512i) q^{59} +(1.50000 - 2.59808i) q^{61} +(6.62372 + 11.4726i) q^{67} -9.79796 q^{71} +13.7980 q^{73} +(3.44949 - 5.97469i) q^{79} +(-2.72474 + 4.71940i) q^{83} +(-1.00000 - 1.73205i) q^{85} -2.79796 q^{89} +(3.44949 + 5.97469i) q^{95} +(-1.00000 + 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{5} - 2 q^{7} + 8 q^{17} - 8 q^{19} - 10 q^{23} - 2 q^{25} + 6 q^{29} - 12 q^{31} + 4 q^{35} - 24 q^{37} - 10 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{53} - 12 q^{59} + 6 q^{61} + 2 q^{67} + 16 q^{73}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q - 2 * q^5 - 2 * q^7 + 8 * q^17 - 8 * q^19 - 10 * q^23 - 2 * q^25 + 6 * q^29 - 12 * q^31 + 4 * q^35 - 24 * q^37 - 10 * q^41 - 4 * q^43 - 14 * q^47 + 8 * q^53 - 12 * q^59 + 6 * q^61 + 2 * q^67 + 16 * q^73 + 4 * q^79 - 6 * q^83 - 4 * q^85 + 28 * q^89 + 4 * q^95 - 4 * q^97

Introduction

L-functions

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