LMFDB - Embedded newform 1440.2.x.d.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,2,Mod(127,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))

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

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

chi := DirichletCharacter("1440.127");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1440.x (of order $4$, 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:	$11.4984578911$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			127.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1440.127
Dual form			1440.2.x.d.703.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 2.00000i) q^{5} +O(q^{10})$	$q+(-1.00000 - 2.00000i) q^{5} +4.00000i q^{11} +(-3.00000 + 3.00000i) q^{13} +(-1.00000 - 1.00000i) q^{17} +8.00000 q^{19} +(-4.00000 - 4.00000i) q^{23} +(-3.00000 + 4.00000i) q^{25} +6.00000i q^{29} +4.00000i q^{31} +(5.00000 + 5.00000i) q^{37} -2.00000 q^{41} +(4.00000 + 4.00000i) q^{43} +7.00000i q^{49} +(5.00000 - 5.00000i) q^{53} +(8.00000 - 4.00000i) q^{55} +12.0000 q^{59} -4.00000 q^{61} +(9.00000 + 3.00000i) q^{65} +(4.00000 - 4.00000i) q^{67} -8.00000i q^{71} +(-5.00000 + 5.00000i) q^{73} +4.00000 q^{79} +(8.00000 + 8.00000i) q^{83} +(-1.00000 + 3.00000i) q^{85} +2.00000i q^{89} +(-8.00000 - 16.0000i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} - 6 q^{13} - 2 q^{17} + 16 q^{19} - 8 q^{23} - 6 q^{25} + 10 q^{37} - 4 q^{41} + 8 q^{43} + 10 q^{53} + 16 q^{55} + 24 q^{59} - 8 q^{61} + 18 q^{65} + 8 q^{67} - 10 q^{73} + 8 q^{79} + 16 q^{83} - 2 q^{85} - 16 q^{95} + 6 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 6 * q^13 - 2 * q^17 + 16 * q^19 - 8 * q^23 - 6 * q^25 + 10 * q^37 - 4 * q^41 + 8 * q^43 + 10 * q^53 + 16 * q^55 + 24 * q^59 - 8 * q^61 + 18 * q^65 + 8 * q^67 - 10 * q^73 + 8 * q^79 + 16 * q^83 - 2 * q^85 - 16 * q^95 + 6 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$991$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$-1$

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
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$5$	−1.00000	−	2.00000i	−0.447214	−	0.894427i
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.00000i			1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$			4.00000i			1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$		0			0
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	−0.987797	−	0.155747i	$-0.950222\pi$
$13$	−3.00000	+	3.00000i	−0.832050	+	0.832050i	0.155747	+	0.987797i	$0.450222\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	−	1.00000i	−0.242536	−	0.242536i	0.575363	−	0.817898i	$-0.304861\pi$
$17$	−1.00000	−	1.00000i	−0.242536	−	0.242536i	−0.817898	+	0.575363i	$0.804861\pi$
$18$		0			0
$19$	8.00000			1.83533			0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000			1.83533			0.917663	+	0.397360i	$0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	0.154011	−	0.988069i	$-0.450781\pi$
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	−0.988069	+	0.154011i	$0.950781\pi$
$24$		0			0
$25$	−3.00000	+	4.00000i	−0.600000	+	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	0.986394	−	0.164399i	$-0.0525685\pi$
$37$	5.00000	+	5.00000i	0.821995	+	0.821995i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.00000			−0.312348			−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000			−0.312348			−0.156174	+	0.987730i	$0.549916\pi$
$42$		0			0
$43$	4.00000	+	4.00000i	0.609994	+	0.609994i	0.942944	−	0.332950i	$-0.108044\pi$
$43$	4.00000	+	4.00000i	0.609994	+	0.609994i	−0.332950	+	0.942944i	$0.608044\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.00000	−	5.00000i	0.686803	−	0.686803i	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	5.00000	−	5.00000i	0.686803	−	0.686803i	0.961524	+	0.274721i	$0.0885855\pi$
$54$		0			0
$55$	8.00000	−	4.00000i	1.07872	−	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.0000			1.56227			0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000			1.56227			0.781133	+	0.624364i	$0.214642\pi$
$60$		0			0
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.00000	+	3.00000i	1.11631	+	0.372104i
$66$		0			0
$67$	4.00000	−	4.00000i	0.488678	−	0.488678i	−0.419211	−	0.907889i	$-0.637693\pi$
$67$	4.00000	−	4.00000i	0.488678	−	0.488678i	0.907889	+	0.419211i	$0.137693\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	8.00000i		−	0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00000i		−	0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$		0			0
$73$	−5.00000	+	5.00000i	−0.585206	+	0.585206i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−5.00000	+	5.00000i	−0.585206	+	0.585206i	0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.00000			0.450035			0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000			0.450035			0.225018	+	0.974355i	$0.427756\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	8.00000	+	8.00000i	0.878114	+	0.878114i	0.993339	−	0.115225i	$-0.0367590\pi$
$83$	8.00000	+	8.00000i	0.878114	+	0.878114i	−0.115225	+	0.993339i	$0.536759\pi$
$84$		0			0
$85$	−1.00000	+	3.00000i	−0.108465	+	0.325396i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			2.00000i			0.212000i	0.994366	+	0.106000i	$0.0338043\pi$
$89$			2.00000i			0.212000i	−0.994366	+	0.106000i	$0.966196\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.00000	−	16.0000i	−0.820783	−	1.64157i
$96$		0			0
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	0.842812	−	0.538208i	$-0.180899\pi$
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	−0.538208	+	0.842812i	$0.680899\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$	12.0000	+	12.0000i	1.18240	+	1.18240i	0.979122	+	0.203273i	$0.0651579\pi$
$103$	12.0000	+	12.0000i	1.18240	+	1.18240i	0.203273	+	0.979122i	$0.434842\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000	+	12.0000i	−1.16008	+	1.16008i	−0.175627	+	0.984457i	$0.556195\pi$
$107$	−12.0000	+	12.0000i	−1.16008	+	1.16008i	−0.984457	+	0.175627i	$0.943805\pi$
$108$		0			0
$109$			14.0000i			1.34096i	0.741929	+	0.670478i	$0.233911\pi$
$109$			14.0000i			1.34096i	−0.741929	+	0.670478i	$0.766089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.00000	+	3.00000i	−0.282216	+	0.282216i	−0.833992	−	0.551776i	$-0.813950\pi$
$113$	−3.00000	+	3.00000i	−0.282216	+	0.282216i	0.551776	+	0.833992i	$0.313950\pi$
$114$		0			0
$115$	−4.00000	+	12.0000i	−0.373002	+	1.11901i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$	4.00000	−	4.00000i	0.354943	−	0.354943i	−0.507002	−	0.861945i	$-0.669246\pi$
$127$	4.00000	−	4.00000i	0.354943	−	0.354943i	0.861945	+	0.507002i	$0.169246\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			20.0000i			1.74741i	0.486458	+	0.873704i	$0.338289\pi$
$131$			20.0000i			1.74741i	−0.486458	+	0.873704i	$0.661711\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−13.0000	−	13.0000i	−1.11066	−	1.11066i	−0.993061	−	0.117604i	$-0.962479\pi$
$137$	−13.0000	−	13.0000i	−1.11066	−	1.11066i	−0.117604	−	0.993061i	$-0.537521\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−12.0000	−	12.0000i	−1.00349	−	1.00349i
$144$		0			0
$145$	12.0000	−	6.00000i	0.996546	−	0.498273i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			4.00000i			0.327693i	0.986486	+	0.163846i	$0.0523901\pi$
$149$			4.00000i			0.327693i	−0.986486	+	0.163846i	$0.947610\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	8.00000	−	4.00000i	0.642575	−	0.321288i
$156$		0			0
$157$	−13.0000	−	13.0000i	−1.03751	−	1.03751i	−0.999268	−	0.0382445i	$-0.987823\pi$
$157$	−13.0000	−	13.0000i	−1.03751	−	1.03751i	−0.0382445	−	0.999268i	$-0.512177\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−16.0000	−	16.0000i	−1.25322	−	1.25322i	−0.954270	−	0.298947i	$-0.903365\pi$
$163$	−16.0000	−	16.0000i	−1.25322	−	1.25322i	−0.298947	−	0.954270i	$-0.596635\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.00000	+	8.00000i	−0.619059	+	0.619059i	−0.945290	−	0.326231i	$-0.894221\pi$
$167$	−8.00000	+	8.00000i	−0.619059	+	0.619059i	0.326231	+	0.945290i	$0.394221\pi$
$168$		0			0
$169$		−	5.00000i		−	0.384615i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000	+	1.00000i	−0.0760286	+	0.0760286i	−0.744099	−	0.668070i	$-0.767121\pi$
$173$	−1.00000	+	1.00000i	−0.0760286	+	0.0760286i	0.668070	+	0.744099i	$0.267121\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.00000	−	15.0000i	0.367607	−	1.10282i
$186$		0			0
$187$	4.00000	−	4.00000i	0.292509	−	0.292509i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			8.00000i			0.578860i	0.957199	+	0.289430i	$0.0934657\pi$
$191$			8.00000i			0.578860i	−0.957199	+	0.289430i	$0.906534\pi$
$192$		0			0
$193$	3.00000	−	3.00000i	0.215945	−	0.215945i	−0.590842	−	0.806787i	$-0.701204\pi$
$193$	3.00000	−	3.00000i	0.215945	−	0.215945i	0.806787	+	0.590842i	$0.201204\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.0000	−	17.0000i	−1.21120	−	1.21120i	−0.970632	−	0.240567i	$-0.922666\pi$
$197$	−17.0000	−	17.0000i	−1.21120	−	1.21120i	−0.240567	−	0.970632i	$-0.577334\pi$
$198$		0			0
$199$	−20.0000			−1.41776			−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000			−1.41776			−0.708881	+	0.705328i	$0.750800\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.00000	+	4.00000i	0.139686	+	0.279372i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			32.0000i			2.21349i
$210$		0			0
$211$			16.0000i			1.10149i	0.834675	+	0.550743i	$0.185655\pi$
$211$			16.0000i			1.10149i	−0.834675	+	0.550743i	$0.814345\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.00000	−	12.0000i	0.272798	−	0.818393i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000			0.403604
$222$		0			0
$223$	4.00000	+	4.00000i	0.267860	+	0.267860i	0.828237	−	0.560378i	$-0.189344\pi$
$223$	4.00000	+	4.00000i	0.267860	+	0.267860i	−0.560378	+	0.828237i	$0.689344\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.00000	−	4.00000i	0.265489	−	0.265489i	−0.561790	−	0.827280i	$-0.689887\pi$
$227$	4.00000	−	4.00000i	0.265489	−	0.265489i	0.827280	+	0.561790i	$0.189887\pi$
$228$		0			0
$229$		−	20.0000i		−	1.32164i	−0.750546	−	0.660819i	$-0.770209\pi$
$229$		−	20.0000i		−	1.32164i	0.750546	−	0.660819i	$-0.229791\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	15.0000	−	15.0000i	0.982683	−	0.982683i	−0.0171699	−	0.999853i	$-0.505466\pi$
$233$	15.0000	−	15.0000i	0.982683	−	0.982683i	0.999853	+	0.0171699i	$0.00546562\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	16.0000			1.03495			0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000			1.03495			0.517477	+	0.855697i	$0.326871\pi$
$240$		0			0
$241$	−8.00000			−0.515325			−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000			−0.515325			−0.257663	+	0.966235i	$0.582952\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	14.0000	−	7.00000i	0.894427	−	0.447214i
$246$		0			0
$247$	−24.0000	+	24.0000i	−1.52708	+	1.52708i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			12.0000i			0.757433i	0.925513	+	0.378717i	$0.123635\pi$
$251$			12.0000i			0.757433i	−0.925513	+	0.378717i	$0.876365\pi$
$252$		0			0
$253$	16.0000	−	16.0000i	1.00591	−	1.00591i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.0000	+	13.0000i	0.810918	+	0.810918i	0.984771	−	0.173854i	$-0.0556220\pi$
$257$	13.0000	+	13.0000i	0.810918	+	0.810918i	−0.173854	+	0.984771i	$0.555622\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−16.0000	−	16.0000i	−0.986602	−	0.986602i	0.0133092	−	0.999911i	$-0.495763\pi$
$263$	−16.0000	−	16.0000i	−0.986602	−	0.986602i	−0.999911	+	0.0133092i	$0.995763\pi$
$264$		0			0
$265$	−15.0000	−	5.00000i	−0.921443	−	0.307148i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			28.0000i			1.70719i	0.520937	+	0.853595i	$0.325583\pi$
$269$			28.0000i			1.70719i	−0.520937	+	0.853595i	$0.674417\pi$
$270$		0			0
$271$		−	20.0000i		−	1.21491i	−0.794353	−	0.607457i	$-0.792190\pi$
$271$		−	20.0000i		−	1.21491i	0.794353	−	0.607457i	$-0.207810\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.0000	−	12.0000i	−0.964836	−	0.723627i
$276$		0			0
$277$	5.00000	+	5.00000i	0.300421	+	0.300421i	0.841178	−	0.540758i	$-0.181862\pi$
$277$	5.00000	+	5.00000i	0.300421	+	0.300421i	−0.540758	+	0.841178i	$0.681862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−30.0000			−1.78965			−0.894825	−	0.446417i	$-0.852700\pi$
$281$	−30.0000			−1.78965			−0.894825	+	0.446417i	$0.852700\pi$
$282$		0			0
$283$	−20.0000	−	20.0000i	−1.18888	−	1.18888i	−0.977378	−	0.211498i	$-0.932166\pi$
$283$	−20.0000	−	20.0000i	−1.18888	−	1.18888i	−0.211498	−	0.977378i	$-0.567834\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		−	15.0000i		−	0.882353i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.00000	−	1.00000i	0.0584206	−	0.0584206i	−0.677293	−	0.735714i	$-0.736847\pi$
$293$	1.00000	−	1.00000i	0.0584206	−	0.0584206i	0.735714	+	0.677293i	$0.236847\pi$
$294$		0			0
$295$	−12.0000	−	24.0000i	−0.698667	−	1.39733i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	24.0000			1.38796
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.00000	+	8.00000i	0.229039	+	0.458079i
$306$		0			0
$307$	8.00000	−	8.00000i	0.456584	−	0.456584i	−0.440948	−	0.897532i	$-0.645358\pi$
$307$	8.00000	−	8.00000i	0.456584	−	0.456584i	0.897532	+	0.440948i	$0.145358\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	15.0000	−	15.0000i	0.847850	−	0.847850i	−0.142014	−	0.989865i	$-0.545358\pi$
$313$	15.0000	−	15.0000i	0.847850	−	0.847850i	0.989865	+	0.142014i	$0.0453579\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	7.00000	+	7.00000i	0.393159	+	0.393159i	0.875812	−	0.482653i	$-0.160327\pi$
$317$	7.00000	+	7.00000i	0.393159	+	0.393159i	−0.482653	+	0.875812i	$0.660327\pi$
$318$		0			0
$319$	−24.0000			−1.34374
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.00000	−	8.00000i	−0.445132	−	0.445132i
$324$		0			0
$325$	−3.00000	−	21.0000i	−0.166410	−	1.16487i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			8.00000i			0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$			8.00000i			0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−12.0000	−	4.00000i	−0.655630	−	0.218543i
$336$		0			0
$337$	−9.00000	−	9.00000i	−0.490261	−	0.490261i	0.418127	−	0.908388i	$-0.362687\pi$
$337$	−9.00000	−	9.00000i	−0.490261	−	0.490261i	−0.908388	+	0.418127i	$0.862687\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000	+	12.0000i	−0.644194	+	0.644194i	−0.951584	−	0.307390i	$-0.900544\pi$
$347$	−12.0000	+	12.0000i	−0.644194	+	0.644194i	0.307390	+	0.951584i	$0.400544\pi$
$348$		0			0
$349$		−	20.0000i		−	1.07058i	−0.844670	−	0.535288i	$-0.820203\pi$
$349$		−	20.0000i		−	1.07058i	0.844670	−	0.535288i	$-0.179797\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	21.0000	−	21.0000i	1.11772	−	1.11772i	0.125642	−	0.992076i	$-0.459901\pi$
$353$	21.0000	−	21.0000i	1.11772	−	1.11772i	0.992076	−	0.125642i	$-0.0400989\pi$
$354$		0			0
$355$	−16.0000	+	8.00000i	−0.849192	+	0.424596i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	32.0000			1.68890			0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000			1.68890			0.844448	+	0.535638i	$0.179929\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$		0			0
$363$		0			0
$364$		0			0
$365$	15.0000	+	5.00000i	0.785136	+	0.261712i
$366$		0			0
$367$	−24.0000	+	24.0000i	−1.25279	+	1.25279i	−0.298326	+	0.954464i	$0.596428\pi$
$367$	−24.0000	+	24.0000i	−1.25279	+	1.25279i	−0.954464	+	0.298326i	$0.903572\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−15.0000	+	15.0000i	−0.776671	+	0.776671i	−0.979263	−	0.202593i	$-0.935063\pi$
$373$	−15.0000	+	15.0000i	−0.776671	+	0.776671i	0.202593	+	0.979263i	$0.435063\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−18.0000	−	18.0000i	−0.927047	−	0.927047i
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	20.0000	+	20.0000i	1.02195	+	1.02195i	0.999754	+	0.0221987i	$0.00706664\pi$
$383$	20.0000	+	20.0000i	1.02195	+	1.02195i	0.0221987	+	0.999754i	$0.492933\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	20.0000i		−	1.01404i	−0.861934	−	0.507020i	$-0.830747\pi$
$389$		−	20.0000i		−	1.01404i	0.861934	−	0.507020i	$-0.169253\pi$
$390$		0			0
$391$			8.00000i			0.404577i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.00000	−	8.00000i	−0.201262	−	0.402524i
$396$		0			0
$397$	9.00000	+	9.00000i	0.451697	+	0.451697i	0.895918	−	0.444220i	$-0.146519\pi$
$397$	9.00000	+	9.00000i	0.451697	+	0.451697i	−0.444220	+	0.895918i	$0.646519\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.00000			−0.399501			−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000			−0.399501			−0.199750	+	0.979847i	$0.564013\pi$
$402$		0			0
$403$	−12.0000	−	12.0000i	−0.597763	−	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.0000	+	20.0000i	−0.991363	+	0.991363i
$408$		0			0
$409$		−	10.0000i		−	0.494468i	−0.968956	−	0.247234i	$-0.920478\pi$
$409$		−	10.0000i		−	0.494468i	0.968956	−	0.247234i	$-0.0795217\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	8.00000	−	24.0000i	0.392705	−	1.17811i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	4.00000			0.195413			0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000			0.195413			0.0977064	+	0.995215i	$0.468849\pi$
$420$		0			0
$421$	−20.0000			−0.974740			−0.487370	−	0.873195i	$-0.662044\pi$
$421$	−20.0000			−0.974740			−0.487370	+	0.873195i	$0.662044\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	7.00000	−	1.00000i	0.339550	−	0.0485071i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			24.0000i			1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$			24.0000i			1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$		0			0
$433$	−3.00000	+	3.00000i	−0.144171	+	0.144171i	−0.775508	−	0.631337i	$-0.782506\pi$
$433$	−3.00000	+	3.00000i	−0.144171	+	0.144171i	0.631337	+	0.775508i	$0.282506\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−32.0000	−	32.0000i	−1.53077	−	1.53077i
$438$		0			0
$439$	−4.00000			−0.190910			−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000			−0.190910			−0.0954548	+	0.995434i	$0.530431\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.0000	−	20.0000i	−0.950229	−	0.950229i	0.0485901	−	0.998819i	$-0.484527\pi$
$443$	−20.0000	−	20.0000i	−0.950229	−	0.950229i	−0.998819	+	0.0485901i	$0.984527\pi$
$444$		0			0
$445$	4.00000	−	2.00000i	0.189618	−	0.0948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			24.0000i			1.13263i	0.824189	+	0.566315i	$0.191631\pi$
$449$			24.0000i			1.13263i	−0.824189	+	0.566315i	$0.808369\pi$
$450$		0			0
$451$		−	8.00000i		−	0.376705i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−15.0000	−	15.0000i	−0.701670	−	0.701670i	0.263099	−	0.964769i	$-0.415256\pi$
$457$	−15.0000	−	15.0000i	−0.701670	−	0.701670i	−0.964769	+	0.263099i	$0.915256\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.00000			−0.186299			−0.0931493	−	0.995652i	$-0.529693\pi$
$461$	−4.00000			−0.186299			−0.0931493	+	0.995652i	$0.529693\pi$
$462$		0			0
$463$	16.0000	+	16.0000i	0.743583	+	0.743583i	0.973266	−	0.229683i	$-0.0737688\pi$
$463$	16.0000	+	16.0000i	0.743583	+	0.743583i	−0.229683	+	0.973266i	$0.573769\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−16.0000	+	16.0000i	−0.740392	+	0.740392i	−0.972653	−	0.232262i	$-0.925387\pi$
$467$	−16.0000	+	16.0000i	−0.740392	+	0.740392i	0.232262	+	0.972653i	$0.425387\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16.0000	+	16.0000i	−0.735681	+	0.735681i
$474$		0			0
$475$	−24.0000	+	32.0000i	−1.10120	+	1.46826i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.0000			−0.731059			−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000			−0.731059			−0.365529	+	0.930800i	$0.619112\pi$
$480$		0			0
$481$	−30.0000			−1.36788
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.00000	−	9.00000i	0.136223	−	0.408669i
$486$		0			0
$487$	28.0000	−	28.0000i	1.26880	−	1.26880i	0.322093	−	0.946708i	$-0.395614\pi$
$487$	28.0000	−	28.0000i	1.26880	−	1.26880i	0.946708	−	0.322093i	$-0.104386\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	28.0000i		−	1.26362i	−0.775122	−	0.631811i	$-0.782312\pi$
$491$		−	28.0000i		−	1.26362i	0.775122	−	0.631811i	$-0.217688\pi$
$492$		0			0
$493$	6.00000	−	6.00000i	0.270226	−	0.270226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	8.00000			0.358129			0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000			0.358129			0.179065	+	0.983837i	$0.442693\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16.0000	+	16.0000i	0.713405	+	0.713405i	0.967246	−	0.253841i	$-0.0816941\pi$
$503$	16.0000	+	16.0000i	0.713405	+	0.713405i	−0.253841	+	0.967246i	$0.581694\pi$
$504$		0			0
$505$	12.0000	+	24.0000i	0.533993	+	1.06799i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			42.0000i			1.86162i	0.365507	+	0.930809i	$0.380896\pi$
$509$			42.0000i			1.86162i	−0.365507	+	0.930809i	$0.619104\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	12.0000	−	36.0000i	0.528783	−	1.58635i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8.00000			0.350486			0.175243	−	0.984525i	$-0.443929\pi$
$521$	8.00000			0.350486			0.175243	+	0.984525i	$0.443929\pi$
$522$		0			0
$523$	−24.0000	−	24.0000i	−1.04945	−	1.04945i	−0.998712	−	0.0507346i	$-0.983844\pi$
$523$	−24.0000	−	24.0000i	−1.04945	−	1.04945i	−0.0507346	−	0.998712i	$-0.516156\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.00000	−	4.00000i	0.174243	−	0.174243i
$528$		0			0
$529$			9.00000i			0.391304i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.00000	−	6.00000i	0.259889	−	0.259889i
$534$		0			0
$535$	36.0000	+	12.0000i	1.55642	+	0.518805i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−28.0000			−1.20605
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	28.0000	−	14.0000i	1.19939	−	0.599694i
$546$		0			0
$547$	4.00000	−	4.00000i	0.171028	−	0.171028i	−0.616403	−	0.787431i	$-0.711411\pi$
$547$	4.00000	−	4.00000i	0.171028	−	0.171028i	0.787431	+	0.616403i	$0.211411\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			48.0000i			2.04487i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	21.0000	+	21.0000i	0.889799	+	0.889799i	0.994503	−	0.104705i	$-0.0333898\pi$
$557$	21.0000	+	21.0000i	0.889799	+	0.889799i	−0.104705	+	0.994503i	$0.533390\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−12.0000	−	12.0000i	−0.505740	−	0.505740i	0.407476	−	0.913216i	$-0.366409\pi$
$563$	−12.0000	−	12.0000i	−0.505740	−	0.505740i	−0.913216	+	0.407476i	$0.866409\pi$
$564$		0			0
$565$	9.00000	+	3.00000i	0.378633	+	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	24.0000i		−	1.00613i	−0.864248	−	0.503066i	$-0.832205\pi$
$569$		−	24.0000i		−	1.00613i	0.864248	−	0.503066i	$-0.167795\pi$
$570$		0			0
$571$			16.0000i			0.669579i	0.942293	+	0.334790i	$0.108665\pi$
$571$			16.0000i			0.669579i	−0.942293	+	0.334790i	$0.891335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	28.0000	−	4.00000i	1.16768	−	0.166812i
$576$		0			0
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	0.999133	−	0.0416305i	$-0.0132552\pi$
$577$	23.0000	+	23.0000i	0.957503	+	0.957503i	−0.0416305	+	0.999133i	$0.513255\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	20.0000	+	20.0000i	0.828315	+	0.828315i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	20.0000	−	20.0000i	0.825488	−	0.825488i	−0.161401	−	0.986889i	$-0.551601\pi$
$587$	20.0000	−	20.0000i	0.825488	−	0.825488i	0.986889	+	0.161401i	$0.0516011\pi$
$588$		0			0
$589$			32.0000i			1.31854i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−29.0000	+	29.0000i	−1.19089	+	1.19089i	−0.214069	+	0.976819i	$0.568672\pi$
$593$	−29.0000	+	29.0000i	−1.19089	+	1.19089i	−0.976819	+	0.214069i	$0.931328\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$		0			0
$601$	−16.0000			−0.652654			−0.326327	−	0.945257i	$-0.605811\pi$
$601$	−16.0000			−0.652654			−0.326327	+	0.945257i	$0.605811\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.00000	+	10.0000i	0.203279	+	0.406558i
$606$		0			0
$607$	−8.00000	+	8.00000i	−0.324710	+	0.324710i	−0.850571	−	0.525861i	$-0.823743\pi$
$607$	−8.00000	+	8.00000i	−0.324710	+	0.324710i	0.525861	+	0.850571i	$0.323743\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	3.00000	−	3.00000i	0.121169	−	0.121169i	−0.643922	−	0.765091i	$-0.722694\pi$
$613$	3.00000	−	3.00000i	0.121169	−	0.121169i	0.765091	+	0.643922i	$0.222694\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−25.0000	−	25.0000i	−1.00646	−	1.00646i	−0.999979	−	0.00648312i	$-0.997936\pi$
$617$	−25.0000	−	25.0000i	−1.00646	−	1.00646i	−0.00648312	−	0.999979i	$-0.502064\pi$
$618$		0			0
$619$	24.0000			0.964641			0.482321	−	0.875995i	$-0.339794\pi$
$619$	24.0000			0.964641			0.482321	+	0.875995i	$0.339794\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	10.0000i		−	0.398726i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0000	−	4.00000i	−0.476205	−	0.158735i
$636$		0			0
$637$	−21.0000	−	21.0000i	−0.832050	−	0.832050i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	42.0000			1.65890			0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000			1.65890			0.829450	+	0.558581i	$0.188654\pi$
$642$		0			0
$643$	24.0000	+	24.0000i	0.946468	+	0.946468i	0.998638	−	0.0521706i	$-0.0166140\pi$
$643$	24.0000	+	24.0000i	0.946468	+	0.946468i	−0.0521706	+	0.998638i	$0.516614\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	8.00000	−	8.00000i	0.314512	−	0.314512i	−0.532142	−	0.846655i	$-0.678613\pi$
$647$	8.00000	−	8.00000i	0.314512	−	0.314512i	0.846655	+	0.532142i	$0.178613\pi$
$648$		0			0
$649$			48.0000i			1.88416i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.0000	−	13.0000i	0.508729	−	0.508729i	−0.405407	−	0.914136i	$-0.632870\pi$
$653$	13.0000	−	13.0000i	0.508729	−	0.508729i	0.914136	+	0.405407i	$0.132870\pi$
$654$		0			0
$655$	40.0000	−	20.0000i	1.56293	−	0.781465i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	−36.0000			−1.40024			−0.700119	−	0.714026i	$-0.746870\pi$
$661$	−36.0000			−1.40024			−0.700119	+	0.714026i	$0.746870\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	24.0000	−	24.0000i	0.929284	−	0.929284i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	16.0000i		−	0.617673i
$672$		0			0
$673$	−27.0000	+	27.0000i	−1.04077	+	1.04077i	−0.0416409	+	0.999133i	$0.513259\pi$
$673$	−27.0000	+	27.0000i	−1.04077	+	1.04077i	−0.999133	+	0.0416409i	$0.986741\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	9.00000	+	9.00000i	0.345898	+	0.345898i	0.858579	−	0.512681i	$-0.171348\pi$
$677$	9.00000	+	9.00000i	0.345898	+	0.345898i	−0.512681	+	0.858579i	$0.671348\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.00000	−	8.00000i	−0.306111	−	0.306111i	0.537288	−	0.843399i	$-0.319449\pi$
$683$	−8.00000	−	8.00000i	−0.306111	−	0.306111i	−0.843399	+	0.537288i	$0.819449\pi$
$684$		0			0
$685$	−13.0000	+	39.0000i	−0.496704	+	1.49011i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			30.0000i			1.14291i
$690$		0			0
$691$		−	8.00000i		−	0.304334i	−0.988355	−	0.152167i	$-0.951375\pi$
$691$		−	8.00000i		−	0.304334i	0.988355	−	0.152167i	$-0.0486252\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.00000	+	2.00000i	0.0757554	+	0.0757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−10.0000			−0.377695			−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000			−0.377695			−0.188847	+	0.982006i	$0.560475\pi$
$702$		0			0
$703$	40.0000	+	40.0000i	1.50863	+	1.50863i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$			36.0000i			1.35201i	0.736898	+	0.676004i	$0.236290\pi$
$709$			36.0000i			1.35201i	−0.736898	+	0.676004i	$0.763710\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	16.0000	−	16.0000i	0.599205	−	0.599205i
$714$		0			0
$715$	−12.0000	+	36.0000i	−0.448775	+	1.34632i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	40.0000			1.49175			0.745874	−	0.666087i	$-0.232032\pi$
$719$	40.0000			1.49175			0.745874	+	0.666087i	$0.232032\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−24.0000	−	18.0000i	−0.891338	−	0.668503i
$726$		0			0
$727$	−12.0000	+	12.0000i	−0.445055	+	0.445055i	−0.893707	−	0.448651i	$-0.851904\pi$
$727$	−12.0000	+	12.0000i	−0.445055	+	0.445055i	0.448651	+	0.893707i	$0.351904\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	8.00000i		−	0.295891i
$732$		0			0
$733$	−9.00000	+	9.00000i	−0.332423	+	0.332423i	−0.853506	−	0.521083i	$-0.825528\pi$
$733$	−9.00000	+	9.00000i	−0.332423	+	0.332423i	0.521083	+	0.853506i	$0.325528\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	16.0000	+	16.0000i	0.589368	+	0.589368i
$738$		0			0
$739$	40.0000			1.47142			0.735712	−	0.677295i	$-0.236848\pi$
$739$	40.0000			1.47142			0.735712	+	0.677295i	$0.236848\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.0000	+	12.0000i	0.440237	+	0.440237i	0.892092	−	0.451854i	$-0.149237\pi$
$743$	12.0000	+	12.0000i	0.440237	+	0.440237i	−0.451854	+	0.892092i	$0.649237\pi$
$744$		0			0
$745$	8.00000	−	4.00000i	0.293097	−	0.146549i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	4.00000i		−	0.145962i	−0.997333	−	0.0729810i	$-0.976749\pi$
$751$		−	4.00000i		−	0.145962i	0.997333	−	0.0729810i	$-0.0232513\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−24.0000	+	12.0000i	−0.873449	+	0.436725i
$756$		0			0
$757$	13.0000	+	13.0000i	0.472493	+	0.472493i	0.902721	−	0.430227i	$-0.141567\pi$
$757$	13.0000	+	13.0000i	0.472493	+	0.472493i	−0.430227	+	0.902721i	$0.641567\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−24.0000			−0.869999			−0.435000	−	0.900431i	$-0.643252\pi$
$761$	−24.0000			−0.869999			−0.435000	+	0.900431i	$0.643252\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−36.0000	+	36.0000i	−1.29988	+	1.29988i
$768$		0			0
$769$		−	8.00000i		−	0.288487i	−0.989542	−	0.144244i	$-0.953925\pi$
$769$		−	8.00000i		−	0.288487i	0.989542	−	0.144244i	$-0.0460749\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	11.0000	−	11.0000i	0.395643	−	0.395643i	−0.481050	−	0.876693i	$-0.659745\pi$
$773$	11.0000	−	11.0000i	0.395643	−	0.395643i	0.876693	+	0.481050i	$0.159745\pi$
$774$		0			0
$775$	−16.0000	−	12.0000i	−0.574737	−	0.431053i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−16.0000			−0.573259
$780$		0			0
$781$	32.0000			1.14505
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−13.0000	+	39.0000i	−0.463990	+	1.39197i
$786$		0			0
$787$	8.00000	−	8.00000i	0.285169	−	0.285169i	−0.549997	−	0.835166i	$-0.685371\pi$
$787$	8.00000	−	8.00000i	0.285169	−	0.285169i	0.835166	+	0.549997i	$0.185371\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	12.0000	−	12.0000i	0.426132	−	0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.00000	+	1.00000i	0.0354218	+	0.0354218i	0.724596	−	0.689174i	$-0.242026\pi$
$797$	1.00000	+	1.00000i	0.0354218	+	0.0354218i	−0.689174	+	0.724596i	$0.742026\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−20.0000	−	20.0000i	−0.705785	−	0.705785i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	34.0000i		−	1.19538i	−0.801729	−	0.597688i	$-0.796086\pi$
$809$		−	34.0000i		−	1.19538i	0.801729	−	0.597688i	$-0.203914\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−16.0000	+	48.0000i	−0.560456	+	1.68137i
$816$		0			0
$817$	32.0000	+	32.0000i	1.11954	+	1.11954i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−34.0000			−1.18661			−0.593304	−	0.804978i	$-0.702177\pi$
$821$	−34.0000			−1.18661			−0.593304	+	0.804978i	$0.702177\pi$
$822$		0			0
$823$	−4.00000	−	4.00000i	−0.139431	−	0.139431i	0.633946	−	0.773377i	$-0.281434\pi$
$823$	−4.00000	−	4.00000i	−0.139431	−	0.139431i	−0.773377	+	0.633946i	$0.781434\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.00000	−	8.00000i	0.278187	−	0.278187i	−0.554198	−	0.832385i	$-0.686975\pi$
$827$	8.00000	−	8.00000i	0.278187	−	0.278187i	0.832385	+	0.554198i	$0.186975\pi$
$828$		0			0
$829$			2.00000i			0.0694629i	0.999397	+	0.0347314i	$0.0110576\pi$
$829$			2.00000i			0.0694629i	−0.999397	+	0.0347314i	$0.988942\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.00000	−	7.00000i	0.242536	−	0.242536i
$834$		0			0
$835$	24.0000	+	8.00000i	0.830554	+	0.276851i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−24.0000			−0.828572			−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000			−0.828572			−0.414286	+	0.910147i	$0.635969\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10.0000	+	5.00000i	−0.344010	+	0.172005i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	40.0000i		−	1.37118i
$852$		0			0
$853$	−21.0000	+	21.0000i	−0.719026	+	0.719026i	−0.968406	−	0.249380i	$-0.919773\pi$
$853$	−21.0000	+	21.0000i	−0.719026	+	0.719026i	0.249380	+	0.968406i	$0.419773\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	5.00000	+	5.00000i	0.170797	+	0.170797i	0.787329	−	0.616533i	$-0.211463\pi$
$857$	5.00000	+	5.00000i	0.170797	+	0.170797i	−0.616533	+	0.787329i	$0.711463\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	8.00000	+	8.00000i	0.272323	+	0.272323i	0.830035	−	0.557712i	$-0.188320\pi$
$863$	8.00000	+	8.00000i	0.272323	+	0.272323i	−0.557712	+	0.830035i	$0.688320\pi$
$864$		0			0
$865$	3.00000	+	1.00000i	0.102003	+	0.0340010i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			16.0000i			0.542763i
$870$		0			0
$871$			24.0000i			0.813209i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	0.617630	−	0.786468i	$-0.288093\pi$
$877$	−5.00000	−	5.00000i	−0.168838	−	0.168838i	−0.786468	+	0.617630i	$0.788093\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−42.0000			−1.41502			−0.707508	−	0.706705i	$-0.750181\pi$
$881$	−42.0000			−1.41502			−0.707508	+	0.706705i	$0.750181\pi$
$882$		0			0
$883$	−24.0000	−	24.0000i	−0.807664	−	0.807664i	0.176616	−	0.984280i	$-0.443485\pi$
$883$	−24.0000	−	24.0000i	−0.807664	−	0.807664i	−0.984280	+	0.176616i	$0.943485\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.00000	−	8.00000i	0.268614	−	0.268614i	−0.559928	−	0.828541i	$-0.689171\pi$
$887$	8.00000	−	8.00000i	0.268614	−	0.268614i	0.828541	+	0.559928i	$0.189171\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−12.0000	−	24.0000i	−0.401116	−	0.802232i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0000			−0.800445
$900$		0			0
$901$	−10.0000			−0.333148
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	−	20.0000i	−0.332411	−	0.664822i
$906$		0			0
$907$	36.0000	−	36.0000i	1.19536	−	1.19536i	0.219820	−	0.975540i	$-0.429453\pi$
$907$	36.0000	−	36.0000i	1.19536	−	1.19536i	0.975540	−	0.219820i	$-0.0705470\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			24.0000i			0.795155i	0.917568	+	0.397578i	$0.130149\pi$
$911$			24.0000i			0.795155i	−0.917568	+	0.397578i	$0.869851\pi$
$912$		0			0
$913$	−32.0000	+	32.0000i	−1.05905	+	1.05905i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	12.0000			0.395843			0.197922	−	0.980218i	$-0.436581\pi$
$919$	12.0000			0.395843			0.197922	+	0.980218i	$0.436581\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	24.0000	+	24.0000i	0.789970	+	0.789970i
$924$		0			0
$925$	−35.0000	+	5.00000i	−1.15079	+	0.164399i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	40.0000i		−	1.31236i	−0.754606	−	0.656179i	$-0.772172\pi$
$929$		−	40.0000i		−	1.31236i	0.754606	−	0.656179i	$-0.227828\pi$
$930$		0			0
$931$			56.0000i			1.83533i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.0000	−	4.00000i	−0.392442	−	0.130814i
$936$		0			0
$937$	−21.0000	−	21.0000i	−0.686040	−	0.686040i	0.275314	−	0.961354i	$-0.411218\pi$
$937$	−21.0000	−	21.0000i	−0.686040	−	0.686040i	−0.961354	+	0.275314i	$0.911218\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	52.0000			1.69515			0.847576	−	0.530674i	$-0.178061\pi$
$941$	52.0000			1.69515			0.847576	+	0.530674i	$0.178061\pi$
$942$		0			0
$943$	8.00000	+	8.00000i	0.260516	+	0.260516i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	16.0000	−	16.0000i	0.519930	−	0.519930i	−0.397620	−	0.917550i	$-0.630164\pi$
$947$	16.0000	−	16.0000i	0.519930	−	0.519930i	0.917550	+	0.397620i	$0.130164\pi$
$948$		0			0
$949$		−	30.0000i		−	0.973841i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.0000	−	13.0000i	0.421111	−	0.421111i	−0.464475	−	0.885586i	$-0.653757\pi$
$953$	13.0000	−	13.0000i	0.421111	−	0.421111i	0.885586	+	0.464475i	$0.153757\pi$
$954$		0			0
$955$	16.0000	−	8.00000i	0.517748	−	0.258874i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.00000	−	3.00000i	−0.289720	−	0.0965734i
$966$		0			0
$967$	−28.0000	+	28.0000i	−0.900419	+	0.900419i	−0.995472	−	0.0950529i	$-0.969698\pi$
$967$	−28.0000	+	28.0000i	−0.900419	+	0.900419i	0.0950529	+	0.995472i	$0.469698\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			28.0000i			0.898563i	0.893390	+	0.449281i	$0.148320\pi$
$971$			28.0000i			0.898563i	−0.893390	+	0.449281i	$0.851680\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	9.00000	+	9.00000i	0.287936	+	0.287936i	0.836263	−	0.548328i	$-0.184735\pi$
$977$	9.00000	+	9.00000i	0.287936	+	0.287936i	−0.548328	+	0.836263i	$0.684735\pi$
$978$		0			0
$979$	−8.00000			−0.255681
$980$		0			0
$981$		0			0
$982$		0			0
$983$	4.00000	+	4.00000i	0.127580	+	0.127580i	0.768014	−	0.640433i	$-0.221245\pi$
$983$	4.00000	+	4.00000i	0.127580	+	0.127580i	−0.640433	+	0.768014i	$0.721245\pi$
$984$		0			0
$985$	−17.0000	+	51.0000i	−0.541665	+	1.62500i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	32.0000i		−	1.01754i
$990$		0			0
$991$		−	20.0000i		−	0.635321i	−0.948205	−	0.317660i	$-0.897103\pi$
$991$		−	20.0000i		−	0.635321i	0.948205	−	0.317660i	$-0.102897\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	20.0000	+	40.0000i	0.634043	+	1.26809i
$996$		0			0
$997$	31.0000	+	31.0000i	0.981780	+	0.981780i	0.999837	−	0.0180571i	$-0.00574807\pi$
$997$	31.0000	+	31.0000i	0.981780	+	0.981780i	−0.0180571	+	0.999837i	$0.505748\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.2.x.d.127.1	yes	2
3.2	odd	2		1440.2.x.h.127.1	yes	2
4.3	odd	2		1440.2.x.c.127.1	✓	2
5.3	odd	4		1440.2.x.c.703.1	yes	2
12.11	even	2		1440.2.x.g.127.1	yes	2
15.8	even	4		1440.2.x.g.703.1	yes	2
20.3	even	4	inner	1440.2.x.d.703.1	yes	2
60.23	odd	4		1440.2.x.h.703.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.2.x.c.127.1	✓	2	4.3	odd	2
1440.2.x.c.703.1	yes	2	5.3	odd	4
1440.2.x.d.127.1	yes	2	1.1	even	1	trivial
1440.2.x.d.703.1	yes	2	20.3	even	4	inner
1440.2.x.g.127.1	yes	2	12.11	even	2
1440.2.x.g.703.1	yes	2	15.8	even	4
1440.2.x.h.127.1	yes	2	3.2	odd	2
1440.2.x.h.703.1	yes	2	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1440.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 2.00000i) q^{5} +O(q^{10})\)	\(q+(-1.00000 - 2.00000i) q^{5} +4.00000i q^{11} +(-3.00000 + 3.00000i) q^{13} +(-1.00000 - 1.00000i) q^{17} +8.00000 q^{19} +(-4.00000 - 4.00000i) q^{23} +(-3.00000 + 4.00000i) q^{25} +6.00000i q^{29} +4.00000i q^{31} +(5.00000 + 5.00000i) q^{37} -2.00000 q^{41} +(4.00000 + 4.00000i) q^{43} +7.00000i q^{49} +(5.00000 - 5.00000i) q^{53} +(8.00000 - 4.00000i) q^{55} +12.0000 q^{59} -4.00000 q^{61} +(9.00000 + 3.00000i) q^{65} +(4.00000 - 4.00000i) q^{67} -8.00000i q^{71} +(-5.00000 + 5.00000i) q^{73} +4.00000 q^{79} +(8.00000 + 8.00000i) q^{83} +(-1.00000 + 3.00000i) q^{85} +2.00000i q^{89} +(-8.00000 - 16.0000i) q^{95} +(3.00000 + 3.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} - 6 q^{13} - 2 q^{17} + 16 q^{19} - 8 q^{23} - 6 q^{25} + 10 q^{37} - 4 q^{41} + 8 q^{43} + 10 q^{53} + 16 q^{55} + 24 q^{59} - 8 q^{61} + 18 q^{65} + 8 q^{67} - 10 q^{73} + 8 q^{79} + 16 q^{83} - 2 q^{85} - 16 q^{95} + 6 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 6 * q^13 - 2 * q^17 + 16 * q^19 - 8 * q^23 - 6 * q^25 + 10 * q^37 - 4 * q^41 + 8 * q^43 + 10 * q^53 + 16 * q^55 + 24 * q^59 - 8 * q^61 + 18 * q^65 + 8 * q^67 - 10 * q^73 + 8 * q^79 + 16 * q^83 - 2 * q^85 - 16 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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