LMFDB - Embedded newform 684.2.k.a.505.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,2,Mod(505,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("684.505");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	684.k (of order $3$, 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:	$5.46176749826$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			505.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	684.505
Dual form			684.2.k.a.577.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 3.46410i) q^{5} -3.00000 q^{7} +O(q^{10})$	$q+(-2.00000 + 3.46410i) q^{5} -3.00000 q^{7} +4.00000 q^{11} +(-2.50000 - 4.33013i) q^{13} +(-4.00000 + 1.73205i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(-4.00000 - 6.92820i) q^{29} +1.00000 q^{31} +(6.00000 - 10.3923i) q^{35} -5.00000 q^{37} +(-4.00000 + 6.92820i) q^{41} +(2.50000 - 4.33013i) q^{43} +(4.00000 + 6.92820i) q^{47} +2.00000 q^{49} +(-2.00000 - 3.46410i) q^{53} +(-8.00000 + 13.8564i) q^{55} +(-6.00000 + 10.3923i) q^{59} +(0.500000 + 0.866025i) q^{61} +20.0000 q^{65} +(-1.50000 - 2.59808i) q^{67} +(-8.00000 + 13.8564i) q^{71} +(7.50000 - 12.9904i) q^{73} -12.0000 q^{77} +(3.50000 - 6.06218i) q^{79} +(6.00000 + 10.3923i) q^{89} +(7.50000 + 12.9904i) q^{91} +(2.00000 - 17.3205i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q - 4 * q^5 - 6 * q^7	$ 2 q - 4 q^{5} - 6 q^{7} + 8 q^{11} - 5 q^{13} - 8 q^{19} - 4 q^{23} - 11 q^{25} - 8 q^{29} + 2 q^{31} + 12 q^{35} - 10 q^{37} - 8 q^{41} + 5 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{53} - 16 q^{55} - 12 q^{59} + q^{61} + 40 q^{65} - 3 q^{67} - 16 q^{71} + 15 q^{73} - 24 q^{77} + 7 q^{79} + 12 q^{89} + 15 q^{91} + 4 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 6 * q^7 + 8 * q^11 - 5 * q^13 - 8 * q^19 - 4 * q^23 - 11 * q^25 - 8 * q^29 + 2 * q^31 + 12 * q^35 - 10 * q^37 - 8 * q^41 + 5 * q^43 + 8 * q^47 + 4 * q^49 - 4 * q^53 - 16 * q^55 - 12 * q^59 + q^61 + 40 * q^65 - 3 * q^67 - 16 * q^71 + 15 * q^73 - 24 * q^77 + 7 * q^79 + 12 * q^89 + 15 * q^91 + 4 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.0599153	+	0.998203i	$0.519083\pi$
$5$	−2.00000	+	3.46410i	−0.894427	+	1.54919i	−0.834512	+	0.550990i	$0.814250\pi$
$6$		0			0
$7$	−3.00000			−1.13389			−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000			−1.13389			−0.566947	+	0.823754i	$0.691875\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.00000			1.20605			0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000			1.20605			0.603023	+	0.797724i	$0.293963\pi$
$12$		0			0
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	0.277350	−	0.960769i	$-0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$19$	−4.00000	+	1.73205i	−0.917663	+	0.397360i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−5.50000	−	9.52628i	−1.10000	−	1.90526i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	−0.951224	−	0.308500i	$-0.900173\pi$
$29$	−4.00000	−	6.92820i	−0.742781	−	1.28654i	0.208443	−	0.978035i	$-0.433160\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	6.00000	−	10.3923i	1.01419	−	1.75662i
$36$		0			0
$37$	−5.00000			−0.821995			−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000			−0.821995			−0.410997	+	0.911636i	$0.634819\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.00000	+	6.92820i	−0.624695	+	1.08200i	0.363905	+	0.931436i	$0.381443\pi$
$41$	−4.00000	+	6.92820i	−0.624695	+	1.08200i	−0.988600	+	0.150567i	$0.951890\pi$
$42$		0			0
$43$	2.50000	−	4.33013i	0.381246	−	0.660338i	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	2.50000	−	4.33013i	0.381246	−	0.660338i	0.991241	+	0.132068i	$0.0421616\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	0.995066	+	0.0992202i	$0.0316348\pi$
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	−0.411606	+	0.911362i	$0.635032\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	0.695344	−	0.718677i	$-0.255252\pi$
$53$	−2.00000	−	3.46410i	−0.274721	−	0.475831i	−0.970065	+	0.242846i	$0.921919\pi$
$54$		0			0
$55$	−8.00000	+	13.8564i	−1.07872	+	1.86840i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	0.150148	+	0.988663i	$0.452025\pi$
$59$	−6.00000	+	10.3923i	−0.781133	+	1.35296i	−0.931282	+	0.364299i	$0.881308\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	20.0000			2.48069
$66$		0			0
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	0.759733	−	0.650236i	$-0.225330\pi$
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	−0.942987	+	0.332830i	$0.891996\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−8.00000	+	13.8564i	−0.949425	+	1.64445i	−0.202787	+	0.979223i	$0.565000\pi$
$71$	−8.00000	+	13.8564i	−0.949425	+	1.64445i	−0.746639	+	0.665230i	$0.768333\pi$
$72$		0			0
$73$	7.50000	−	12.9904i	0.877809	−	1.52041i	0.0240681	−	0.999710i	$-0.492338\pi$
$73$	7.50000	−	12.9904i	0.877809	−	1.52041i	0.853740	−	0.520699i	$-0.174329\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0000			−1.36753
$78$		0			0
$79$	3.50000	−	6.06218i	0.393781	−	0.682048i	−0.599164	−	0.800626i	$-0.704500\pi$
$79$	3.50000	−	6.06218i	0.393781	−	0.682048i	0.992945	+	0.118578i	$0.0378336\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000	+	10.3923i	0.635999	+	1.10158i	0.986303	+	0.164946i	$0.0527450\pi$
$89$	6.00000	+	10.3923i	0.635999	+	1.10158i	−0.350304	+	0.936636i	$0.613922\pi$
$90$		0			0
$91$	7.50000	+	12.9904i	0.786214	+	1.36176i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	−	17.3205i	0.205196	−	1.77705i
$96$		0			0
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	−0.810782	−	0.585348i	$-0.800958\pi$
$97$	1.00000	−	1.73205i	0.101535	−	0.175863i	0.912317	+	0.409484i	$0.134291\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−8.00000	−	13.8564i	−0.796030	−	1.37876i	−0.922183	−	0.386753i	$-0.873597\pi$
$101$	−8.00000	−	13.8564i	−0.796030	−	1.37876i	0.126153	−	0.992011i	$-0.459737\pi$
$102$		0			0
$103$	−15.0000			−1.47799			−0.738997	−	0.673709i	$-0.764700\pi$
$103$	−15.0000			−1.47799			−0.738997	+	0.673709i	$0.764700\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−8.00000			−0.773389			−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000			−0.773389			−0.386695	+	0.922208i	$0.626383\pi$
$108$		0			0
$109$	−9.00000	+	15.5885i	−0.862044	+	1.49310i	0.00790932	+	0.999969i	$0.497482\pi$
$109$	−9.00000	+	15.5885i	−0.862044	+	1.49310i	−0.869953	+	0.493135i	$0.835851\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.00000			0.376288			0.188144	−	0.982141i	$-0.439753\pi$
$113$	4.00000			0.376288			0.188144	+	0.982141i	$0.439753\pi$
$114$		0			0
$115$	16.0000			1.49201
$116$		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
$125$	24.0000			2.14663
$126$		0			0
$127$	6.00000	+	10.3923i	0.532414	+	0.922168i	0.999284	+	0.0378419i	$0.0120483\pi$
$127$	6.00000	+	10.3923i	0.532414	+	0.922168i	−0.466870	+	0.884326i	$0.654618\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	12.0000	−	5.19615i	1.04053	−	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−4.00000	−	6.92820i	−0.341743	−	0.591916i	0.643013	−	0.765855i	$-0.277684\pi$
$137$	−4.00000	−	6.92820i	−0.341743	−	0.591916i	−0.984757	+	0.173939i	$0.944351\pi$
$138$		0			0
$139$	7.50000	+	12.9904i	0.636142	+	1.10183i	0.986272	+	0.165129i	$0.0528040\pi$
$139$	7.50000	+	12.9904i	0.636142	+	1.10183i	−0.350130	+	0.936701i	$0.613863\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.0000	−	17.3205i	−0.836242	−	1.44841i
$144$		0			0
$145$	32.0000			2.65746
$146$		0			0
$147$		0			0
$148$		0			0
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	−0.508413	−	0.861113i	$-0.669768\pi$
$149$	6.00000	−	10.3923i	0.491539	−	0.851371i	0.999953	+	0.00974235i	$0.00310113\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.00000	+	3.46410i	−0.160644	+	0.278243i
$156$		0			0
$157$	2.50000	−	4.33013i	0.199522	−	0.345582i	−0.748852	−	0.662738i	$-0.769394\pi$
$157$	2.50000	−	4.33013i	0.199522	−	0.345582i	0.948373	+	0.317156i	$0.102728\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	6.00000	+	10.3923i	0.472866	+	0.819028i
$162$		0			0
$163$	−1.00000			−0.0783260			−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000			−0.0783260			−0.0391630	+	0.999233i	$0.512469\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−12.0000	+	20.7846i	−0.912343	+	1.58022i	−0.101598	+	0.994826i	$0.532395\pi$
$173$	−12.0000	+	20.7846i	−0.912343	+	1.58022i	−0.810745	+	0.585399i	$0.800938\pi$
$174$		0			0
$175$	16.5000	+	28.5788i	1.24728	+	2.16036i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.0000			−0.896922			−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000			−0.896922			−0.448461	+	0.893802i	$0.648028\pi$
$180$		0			0
$181$	−5.00000	−	8.66025i	−0.371647	−	0.643712i	0.618172	−	0.786043i	$-0.287874\pi$
$181$	−5.00000	−	8.66025i	−0.371647	−	0.643712i	−0.989819	+	0.142331i	$0.954540\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.0000	−	17.3205i	0.735215	−	1.27343i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.00000			−0.289430			−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000			−0.289430			−0.144715	+	0.989473i	$0.546227\pi$
$192$		0			0
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	−0.657376	−	0.753563i	$-0.728333\pi$
$193$	4.50000	−	7.79423i	0.323917	−	0.561041i	0.981293	+	0.192522i	$0.0616668\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−20.0000			−1.42494			−0.712470	−	0.701702i	$-0.752424\pi$
$197$	−20.0000			−1.42494			−0.712470	+	0.701702i	$0.752424\pi$
$198$		0			0
$199$	3.50000	+	6.06218i	0.248108	+	0.429736i	0.963001	−	0.269498i	$-0.0868577\pi$
$199$	3.50000	+	6.06218i	0.248108	+	0.429736i	−0.714893	+	0.699234i	$0.753524\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.0000	+	20.7846i	0.842235	+	1.45879i
$204$		0			0
$205$	−16.0000	−	27.7128i	−1.11749	−	1.93555i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−16.0000	+	6.92820i	−1.10674	+	0.479234i
$210$		0			0
$211$	−10.5000	+	18.1865i	−0.722850	+	1.25201i	0.237003	+	0.971509i	$0.423835\pi$
$211$	−10.5000	+	18.1865i	−0.722850	+	1.25201i	−0.959853	+	0.280504i	$0.909498\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	10.0000	+	17.3205i	0.681994	+	1.18125i
$216$		0			0
$217$	−3.00000			−0.203653
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	9.50000	−	16.4545i	0.636167	−	1.10187i	−0.350100	−	0.936713i	$-0.613852\pi$
$223$	9.50000	−	16.4545i	0.636167	−	1.10187i	0.986267	−	0.165161i	$-0.0528144\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.0000			1.32745			0.663723	−	0.747978i	$-0.268975\pi$
$227$	20.0000			1.32745			0.663723	+	0.747978i	$0.268975\pi$
$228$		0			0
$229$	−7.00000			−0.462573			−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000			−0.462573			−0.231287	+	0.972886i	$0.574293\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$	−32.0000			−2.08745
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	0.849472	−	0.527633i	$-0.176921\pi$
$241$	−0.500000	−	0.866025i	−0.0322078	−	0.0557856i	−0.881680	+	0.471848i	$0.843587\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.00000	+	6.92820i	−0.255551	+	0.442627i
$246$		0			0
$247$	17.5000	+	12.9904i	1.11350	+	0.826558i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.00000	−	13.8564i	−0.504956	−	0.874609i	−0.999984	−	0.00573163i	$-0.998176\pi$
$251$	−8.00000	−	13.8564i	−0.504956	−	0.874609i	0.495028	−	0.868877i	$-0.335158\pi$
$252$		0			0
$253$	−8.00000	−	13.8564i	−0.502956	−	0.871145i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.0000	+	17.3205i	0.623783	+	1.08042i	0.988775	+	0.149413i	$0.0477384\pi$
$257$	10.0000	+	17.3205i	0.623783	+	1.08042i	−0.364992	+	0.931011i	$0.618928\pi$
$258$		0			0
$259$	15.0000			0.932055
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	−0.623082	−	0.782157i	$-0.714120\pi$
$269$	6.00000	−	10.3923i	0.365826	−	0.633630i	0.988908	+	0.148527i	$0.0474530\pi$
$270$		0			0
$271$	−2.00000	+	3.46410i	−0.121491	+	0.210429i	−0.920356	−	0.391082i	$-0.872101\pi$
$271$	−2.00000	+	3.46410i	−0.121491	+	0.210429i	0.798865	+	0.601511i	$0.205434\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−22.0000	−	38.1051i	−1.32665	−	2.29783i
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−10.0000	−	17.3205i	−0.596550	−	1.03325i	−0.993326	−	0.115339i	$-0.963204\pi$
$281$	−10.0000	−	17.3205i	−0.596550	−	1.03325i	0.396776	−	0.917915i	$-0.370129\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	12.0000	−	20.7846i	0.708338	−	1.22688i
$288$		0			0
$289$	8.50000	+	14.7224i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	−24.0000	−	41.5692i	−1.39733	−	2.42025i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.0000	+	17.3205i	−0.578315	+	1.00167i
$300$		0			0
$301$	−7.50000	+	12.9904i	−0.432293	+	0.748753i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−4.00000			−0.229039
$306$		0			0
$307$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$307$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	20.0000			1.13410			0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000			1.13410			0.567048	+	0.823685i	$0.308085\pi$
$312$		0			0
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	−0.954810	−	0.297218i	$-0.903941\pi$
$313$	−13.0000	−	22.5167i	−0.734803	−	1.27272i	0.220006	−	0.975499i	$-0.429392\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	0.646872	−	0.762598i	$-0.276077\pi$
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	−0.983866	+	0.178908i	$0.942743\pi$
$318$		0			0
$319$	−16.0000	−	27.7128i	−0.895828	−	1.55162i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−27.5000	+	47.6314i	−1.52543	+	2.64211i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000	−	20.7846i	−0.661581	−	1.14589i
$330$		0			0
$331$	−25.0000			−1.37412			−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000			−1.37412			−0.687062	+	0.726599i	$0.741100\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.0000			0.655630
$336$		0			0
$337$	−6.50000	+	11.2583i	−0.354078	+	0.613280i	−0.986960	−	0.160968i	$-0.948538\pi$
$337$	−6.50000	+	11.2583i	−0.354078	+	0.613280i	0.632882	+	0.774248i	$0.281872\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.00000			0.216612
$342$		0			0
$343$	15.0000			0.809924
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.0000	−	24.2487i	0.751559	−	1.30174i	−0.195507	−	0.980702i	$-0.562635\pi$
$347$	14.0000	−	24.2487i	0.751559	−	1.30174i	0.947067	−	0.321037i	$-0.104031\pi$
$348$		0			0
$349$	−5.00000			−0.267644			−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000			−0.267644			−0.133822	+	0.991005i	$0.542725\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−36.0000			−1.91609			−0.958043	−	0.286623i	$-0.907467\pi$
$353$	−36.0000			−1.91609			−0.958043	+	0.286623i	$0.907467\pi$
$354$		0			0
$355$	−32.0000	−	55.4256i	−1.69838	−	2.94169i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	13.0000	−	13.8564i	0.684211	−	0.729285i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	30.0000	+	51.9615i	1.57027	+	2.71979i
$366$		0			0
$367$	2.50000	+	4.33013i	0.130499	+	0.226031i	0.923869	−	0.382709i	$-0.125009\pi$
$367$	2.50000	+	4.33013i	0.130499	+	0.226031i	−0.793370	+	0.608740i	$0.791675\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.00000	+	10.3923i	0.311504	+	0.539542i
$372$		0			0
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.0000	+	34.6410i	−1.03005	+	1.78410i
$378$		0			0
$379$	3.00000			0.154100			0.0770498	−	0.997027i	$-0.475450\pi$
$379$	3.00000			0.154100			0.0770498	+	0.997027i	$0.475450\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.0000	+	17.3205i	−0.510976	+	0.885037i	0.488943	+	0.872316i	$0.337383\pi$
$383$	−10.0000	+	17.3205i	−0.510976	+	0.885037i	−0.999919	+	0.0127209i	$0.995951\pi$
$384$		0			0
$385$	24.0000	−	41.5692i	1.22315	−	2.11856i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.00000	+	10.3923i	0.304212	+	0.526911i	0.977086	−	0.212847i	$-0.0682735\pi$
$389$	6.00000	+	10.3923i	0.304212	+	0.526911i	−0.672874	+	0.739758i	$0.734940\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.0000	+	24.2487i	0.704416	+	1.22009i
$396$		0			0
$397$	−17.5000	+	30.3109i	−0.878300	+	1.52126i	−0.0250943	+	0.999685i	$0.507989\pi$
$397$	−17.5000	+	30.3109i	−0.878300	+	1.52126i	−0.853206	+	0.521575i	$0.825345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.00000	−	3.46410i	0.0998752	−	0.172989i	−0.811758	−	0.583994i	$-0.801489\pi$
$401$	2.00000	−	3.46410i	0.0998752	−	0.172989i	0.911633	+	0.411005i	$0.134822\pi$
$402$		0			0
$403$	−2.50000	−	4.33013i	−0.124534	−	0.215699i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.0000			−0.991363
$408$		0			0
$409$	−11.0000	−	19.0526i	−0.543915	−	0.942088i	−0.998674	−	0.0514740i	$-0.983608\pi$
$409$	−11.0000	−	19.0526i	−0.543915	−	0.942088i	0.454759	−	0.890614i	$-0.349725\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	18.0000	−	31.1769i	0.885722	−	1.53412i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	12.0000			0.586238			0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000			0.586238			0.293119	+	0.956076i	$0.405307\pi$
$420$		0			0
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	0.353233	+	0.935536i	$0.385082\pi$
$421$	−13.0000	+	22.5167i	−0.633581	+	1.09739i	−0.986814	+	0.161859i	$0.948251\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.50000	−	2.59808i	−0.0725901	−	0.125730i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−12.0000	−	20.7846i	−0.578020	−	1.00116i	−0.995706	−	0.0925683i	$-0.970492\pi$
$431$	−12.0000	−	20.7846i	−0.578020	−	1.00116i	0.417687	−	0.908591i	$-0.362841\pi$
$432$		0			0
$433$	−5.50000	−	9.52628i	−0.264313	−	0.457804i	0.703070	−	0.711120i	$-0.251812\pi$
$433$	−5.50000	−	9.52628i	−0.264313	−	0.457804i	−0.967383	+	0.253317i	$0.918479\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.0000	+	10.3923i	0.669711	+	0.497131i
$438$		0			0
$439$	18.5000	−	32.0429i	0.882957	−	1.52933i	0.0349192	−	0.999390i	$-0.488883\pi$
$439$	18.5000	−	32.0429i	0.882957	−	1.52933i	0.848038	−	0.529936i	$-0.177784\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.0000	+	34.6410i	0.950229	+	1.64584i	0.744927	+	0.667146i	$0.232484\pi$
$443$	20.0000	+	34.6410i	0.950229	+	1.64584i	0.205301	+	0.978699i	$0.434183\pi$
$444$		0			0
$445$	−48.0000			−2.27542
$446$		0			0
$447$		0			0
$448$		0			0
$449$	8.00000			0.377543			0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000			0.377543			0.188772	+	0.982021i	$0.439549\pi$
$450$		0			0
$451$	−16.0000	+	27.7128i	−0.753411	+	1.30495i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−60.0000			−2.81284
$456$		0			0
$457$	−7.00000			−0.327446			−0.163723	−	0.986506i	$-0.552350\pi$
$457$	−7.00000			−0.327446			−0.163723	+	0.986506i	$0.552350\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.00000	+	3.46410i	−0.0931493	+	0.161339i	−0.908835	−	0.417156i	$-0.863027\pi$
$461$	−2.00000	+	3.46410i	−0.0931493	+	0.161339i	0.815685	+	0.578496i	$0.196360\pi$
$462$		0			0
$463$	31.0000			1.44069			0.720346	−	0.693615i	$-0.243983\pi$
$463$	31.0000			1.44069			0.720346	+	0.693615i	$0.243983\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−40.0000			−1.85098			−0.925490	−	0.378773i	$-0.876346\pi$
$467$	−40.0000			−1.85098			−0.925490	+	0.378773i	$0.876346\pi$
$468$		0			0
$469$	4.50000	+	7.79423i	0.207791	+	0.359904i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	10.0000	−	17.3205i	0.459800	−	0.796398i
$474$		0			0
$475$	38.5000	+	28.5788i	1.76650	+	1.31129i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−16.0000	−	27.7128i	−0.731059	−	1.26623i	−0.956431	−	0.291958i	$-0.905693\pi$
$479$	−16.0000	−	27.7128i	−0.731059	−	1.26623i	0.225372	−	0.974273i	$-0.427640\pi$
$480$		0			0
$481$	12.5000	+	21.6506i	0.569951	+	0.987184i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.00000	+	6.92820i	0.181631	+	0.314594i
$486$		0			0
$487$	−20.0000			−0.906287			−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000			−0.906287			−0.453143	+	0.891438i	$0.649697\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.00000	+	13.8564i	−0.361035	+	0.625331i	−0.988131	−	0.153611i	$-0.950910\pi$
$491$	−8.00000	+	13.8564i	−0.361035	+	0.625331i	0.627096	+	0.778942i	$0.284243\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	24.0000	−	41.5692i	1.07655	−	1.86463i
$498$		0			0
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	−0.146538	−	0.989205i	$-0.546813\pi$
$499$	17.5000	−	30.3109i	0.783408	−	1.35690i	0.929946	−	0.367697i	$-0.119854\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.0000	+	20.7846i	0.535054	+	0.926740i	0.999161	+	0.0409609i	$0.0130419\pi$
$503$	12.0000	+	20.7846i	0.535054	+	0.926740i	−0.464107	+	0.885779i	$0.653625\pi$
$504$		0			0
$505$	64.0000			2.84796
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−22.5000	+	38.9711i	−0.995341	+	1.72398i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	30.0000	−	51.9615i	1.32196	−	2.28970i
$516$		0			0
$517$	16.0000	+	27.7128i	0.703679	+	1.21881i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	4.00000			0.175243			0.0876216	−	0.996154i	$-0.472073\pi$
$521$	4.00000			0.175243			0.0876216	+	0.996154i	$0.472073\pi$
$522$		0			0
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	−0.986718	−	0.162446i	$-0.948062\pi$
$523$	−14.5000	−	25.1147i	−0.634041	−	1.09819i	0.352677	−	0.935745i	$-0.385272\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	40.0000			1.73259
$534$		0			0
$535$	16.0000	−	27.7128i	0.691740	−	1.19813i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8.00000			0.344584
$540$		0			0
$541$	9.50000	+	16.4545i	0.408437	+	0.707433i	0.994715	−	0.102677i	$-0.0327407\pi$
$541$	9.50000	+	16.4545i	0.408437	+	0.707433i	−0.586278	+	0.810110i	$0.699407\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−36.0000	−	62.3538i	−1.54207	−	2.67094i
$546$		0			0
$547$	−3.50000	−	6.06218i	−0.149649	−	0.259200i	0.781449	−	0.623970i	$-0.214481\pi$
$547$	−3.50000	−	6.06218i	−0.149649	−	0.259200i	−0.931098	+	0.364770i	$0.881148\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	28.0000	+	20.7846i	1.19284	+	0.885454i
$552$		0			0
$553$	−10.5000	+	18.1865i	−0.446505	+	0.773370i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4.00000	−	6.92820i	−0.169485	−	0.293557i	0.768754	−	0.639545i	$-0.220877\pi$
$557$	−4.00000	−	6.92820i	−0.169485	−	0.293557i	−0.938239	+	0.345988i	$0.887544\pi$
$558$		0			0
$559$	−25.0000			−1.05739
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−24.0000			−1.01148			−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000			−1.01148			−0.505740	+	0.862686i	$0.668780\pi$
$564$		0			0
$565$	−8.00000	+	13.8564i	−0.336563	+	0.582943i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	8.00000			0.335377			0.167689	−	0.985840i	$-0.446370\pi$
$569$	8.00000			0.335377			0.167689	+	0.985840i	$0.446370\pi$
$570$		0			0
$571$	41.0000			1.71580			0.857898	−	0.513820i	$-0.171770\pi$
$571$	41.0000			1.71580			0.857898	+	0.513820i	$0.171770\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−22.0000	+	38.1051i	−0.917463	+	1.58909i
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−8.00000	−	13.8564i	−0.331326	−	0.573874i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.00000	+	10.3923i	−0.247647	+	0.428936i	−0.962872	−	0.269957i	$-0.912990\pi$
$587$	−6.00000	+	10.3923i	−0.247647	+	0.428936i	0.715226	+	0.698893i	$0.246324\pi$
$588$		0			0
$589$	−4.00000	+	1.73205i	−0.164817	+	0.0713679i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	2.00000	+	3.46410i	0.0821302	+	0.142254i	0.904165	−	0.427184i	$-0.140494\pi$
$593$	2.00000	+	3.46410i	0.0821302	+	0.142254i	−0.822035	+	0.569438i	$0.807161\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	0.954521	+	0.298143i	$0.0963673\pi$
$599$	18.0000	+	31.1769i	0.735460	+	1.27385i	−0.219061	+	0.975711i	$0.570299\pi$
$600$		0			0
$601$	31.0000			1.26452			0.632258	−	0.774758i	$-0.282128\pi$
$601$	31.0000			1.26452			0.632258	+	0.774758i	$0.282128\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−10.0000	+	17.3205i	−0.406558	+	0.704179i
$606$		0			0
$607$	5.00000			0.202944			0.101472	−	0.994838i	$-0.467645\pi$
$607$	5.00000			0.202944			0.101472	+	0.994838i	$0.467645\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.0000	−	34.6410i	0.809113	−	1.40143i
$612$		0			0
$613$	13.0000	−	22.5167i	0.525065	−	0.909439i	−0.474509	−	0.880251i	$-0.657374\pi$
$613$	13.0000	−	22.5167i	0.525065	−	0.909439i	0.999574	−	0.0291886i	$-0.00929235\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.00000	−	10.3923i	−0.241551	−	0.418378i	0.719605	−	0.694383i	$-0.244323\pi$
$617$	−6.00000	−	10.3923i	−0.241551	−	0.418378i	−0.961156	+	0.276005i	$0.910989\pi$
$618$		0			0
$619$	25.0000			1.00483			0.502417	−	0.864625i	$-0.332444\pi$
$619$	25.0000			1.00483			0.502417	+	0.864625i	$0.332444\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−18.0000	−	31.1769i	−0.721155	−	1.24908i
$624$		0			0
$625$	−20.5000	+	35.5070i	−0.820000	+	1.42028i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	9.50000	+	16.4545i	0.378189	+	0.655043i	0.990799	−	0.135343i	$-0.0432136\pi$
$631$	9.50000	+	16.4545i	0.378189	+	0.655043i	−0.612610	+	0.790386i	$0.709880\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−48.0000			−1.90482
$636$		0			0
$637$	−5.00000	−	8.66025i	−0.198107	−	0.343132i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000	+	20.7846i	−0.473972	+	0.820943i	−0.999556	−	0.0297987i	$-0.990513\pi$
$641$	−12.0000	+	20.7846i	−0.473972	+	0.820943i	0.525584	+	0.850741i	$0.323847\pi$
$642$		0			0
$643$	−0.500000	+	0.866025i	−0.0197181	+	0.0341527i	−0.875716	−	0.482826i	$-0.839610\pi$
$643$	−0.500000	+	0.866025i	−0.0197181	+	0.0341527i	0.855998	+	0.516979i	$0.172944\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	40.0000			1.57256			0.786281	−	0.617869i	$-0.212004\pi$
$647$	40.0000			1.57256			0.786281	+	0.617869i	$0.212004\pi$
$648$		0			0
$649$	−24.0000	+	41.5692i	−0.942082	+	1.63173i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	10.0000	+	17.3205i	0.389545	+	0.674711i	0.992388	−	0.123148i	$-0.0392990\pi$
$659$	10.0000	+	17.3205i	0.389545	+	0.674711i	−0.602844	+	0.797859i	$0.705966\pi$
$660$		0			0
$661$	13.0000	+	22.5167i	0.505641	+	0.875797i	0.999979	+	0.00652642i	$0.00207744\pi$
$661$	13.0000	+	22.5167i	0.505641	+	0.875797i	−0.494337	+	0.869270i	$0.664589\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.00000	+	51.9615i	−0.232670	+	2.01498i
$666$		0			0
$667$	−16.0000	+	27.7128i	−0.619522	+	1.07304i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	2.00000	+	3.46410i	0.0772091	+	0.133730i
$672$		0			0
$673$	35.0000			1.34915			0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000			1.34915			0.674575	+	0.738206i	$0.264327\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	32.0000			1.22986			0.614930	−	0.788582i	$-0.289184\pi$
$677$	32.0000			1.22986			0.614930	+	0.788582i	$0.289184\pi$
$678$		0			0
$679$	−3.00000	+	5.19615i	−0.115129	+	0.199410i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.00000			−0.153056			−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000			−0.153056			−0.0765279	+	0.997067i	$0.524383\pi$
$684$		0			0
$685$	32.0000			1.22266
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−10.0000	+	17.3205i	−0.380970	+	0.659859i
$690$		0			0
$691$	−24.0000			−0.913003			−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000			−0.913003			−0.456502	+	0.889723i	$0.650898\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−60.0000			−2.27593
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.0000	−	17.3205i	0.377695	−	0.654187i	−0.613032	−	0.790058i	$-0.710050\pi$
$701$	10.0000	−	17.3205i	0.377695	−	0.654187i	0.990726	+	0.135872i	$0.0433835\pi$
$702$		0			0
$703$	20.0000	−	8.66025i	0.754314	−	0.326628i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.0000	+	41.5692i	0.902613	+	1.56337i
$708$		0			0
$709$	−16.5000	−	28.5788i	−0.619671	−	1.07330i	−0.989546	−	0.144219i	$-0.953933\pi$
$709$	−16.5000	−	28.5788i	−0.619671	−	1.07330i	0.369875	−	0.929081i	$-0.379400\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.00000	−	3.46410i	−0.0749006	−	0.129732i
$714$		0			0
$715$	80.0000			2.99183
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.0000	−	24.2487i	0.522112	−	0.904324i	−0.477557	−	0.878601i	$-0.658478\pi$
$719$	14.0000	−	24.2487i	0.522112	−	0.904324i	0.999669	−	0.0257237i	$-0.00818900\pi$
$720$		0			0
$721$	45.0000			1.67589
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−44.0000	+	76.2102i	−1.63412	+	2.83038i
$726$		0			0
$727$	−6.50000	+	11.2583i	−0.241072	+	0.417548i	−0.961020	−	0.276479i	$-0.910832\pi$
$727$	−6.50000	+	11.2583i	−0.241072	+	0.417548i	0.719948	+	0.694028i	$0.244166\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−30.0000			−1.10808			−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000			−1.10808			−0.554038	+	0.832492i	$0.686914\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−6.00000	−	10.3923i	−0.221013	−	0.382805i
$738$		0			0
$739$	6.50000	−	11.2583i	0.239106	−	0.414144i	−0.721352	−	0.692569i	$-0.756479\pi$
$739$	6.50000	−	11.2583i	0.239106	−	0.414144i	0.960458	+	0.278425i	$0.0898122\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	10.0000	−	17.3205i	0.366864	−	0.635428i	−0.622209	−	0.782851i	$-0.713765\pi$
$743$	10.0000	−	17.3205i	0.366864	−	0.635428i	0.989073	+	0.147423i	$0.0470980\pi$
$744$		0			0
$745$	24.0000	+	41.5692i	0.879292	+	1.52298i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24.0000			0.876941
$750$		0			0
$751$	15.5000	+	26.8468i	0.565603	+	0.979653i	0.996993	+	0.0774878i	$0.0246899\pi$
$751$	15.5000	+	26.8468i	0.565603	+	0.979653i	−0.431390	+	0.902165i	$0.641977\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.00000	−	13.8564i	0.291150	−	0.504286i
$756$		0			0
$757$	17.5000	−	30.3109i	0.636048	−	1.10167i	−0.350244	−	0.936659i	$-0.613901\pi$
$757$	17.5000	−	30.3109i	0.636048	−	1.10167i	0.986292	−	0.165009i	$-0.0527654\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	12.0000			0.435000			0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000			0.435000			0.217500	+	0.976060i	$0.430210\pi$
$762$		0			0
$763$	27.0000	−	46.7654i	0.977466	−	1.69302i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	60.0000			2.16647
$768$		0			0
$769$	8.50000	+	14.7224i	0.306518	+	0.530904i	0.977598	−	0.210480i	$-0.0675028\pi$
$769$	8.50000	+	14.7224i	0.306518	+	0.530904i	−0.671080	+	0.741385i	$0.734169\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−12.0000	−	20.7846i	−0.431610	−	0.747570i	0.565402	−	0.824815i	$-0.308721\pi$
$773$	−12.0000	−	20.7846i	−0.431610	−	0.747570i	−0.997012	+	0.0772449i	$0.975388\pi$
$774$		0			0
$775$	−5.50000	−	9.52628i	−0.197566	−	0.342194i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.00000	−	34.6410i	0.143315	−	1.24114i
$780$		0			0
$781$	−32.0000	+	55.4256i	−1.14505	+	1.98328i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.0000	+	17.3205i	0.356915	+	0.618195i
$786$		0			0
$787$	−23.0000			−0.819861			−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000			−0.819861			−0.409931	+	0.912117i	$0.634447\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−12.0000			−0.426671
$792$		0			0
$793$	2.50000	−	4.33013i	0.0887776	−	0.153767i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−48.0000			−1.70025			−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000			−1.70025			−0.850124	+	0.526583i	$0.823473\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	30.0000	−	51.9615i	1.05868	−	1.83368i
$804$		0			0
$805$	−48.0000			−1.69178
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−12.0000			−0.421898			−0.210949	−	0.977497i	$-0.567655\pi$
$809$	−12.0000			−0.421898			−0.210949	+	0.977497i	$0.567655\pi$
$810$		0			0
$811$	−12.0000	−	20.7846i	−0.421377	−	0.729846i	0.574697	−	0.818366i	$-0.305120\pi$
$811$	−12.0000	−	20.7846i	−0.421377	−	0.729846i	−0.996074	+	0.0885196i	$0.971786\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.00000	−	3.46410i	0.0700569	−	0.121342i
$816$		0			0
$817$	−2.50000	+	21.6506i	−0.0874639	+	0.757460i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12.0000	−	20.7846i	−0.418803	−	0.725388i	0.577016	−	0.816733i	$-0.304217\pi$
$821$	−12.0000	−	20.7846i	−0.418803	−	0.725388i	−0.995819	+	0.0913446i	$0.970884\pi$
$822$		0			0
$823$	−22.0000	−	38.1051i	−0.766872	−	1.32826i	−0.939251	−	0.343230i	$-0.888479\pi$
$823$	−22.0000	−	38.1051i	−0.766872	−	1.32826i	0.172379	−	0.985031i	$-0.444854\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.00000	+	6.92820i	0.139094	+	0.240917i	0.927154	−	0.374681i	$-0.122248\pi$
$827$	4.00000	+	6.92820i	0.139094	+	0.240917i	−0.788060	+	0.615598i	$0.788914\pi$
$828$		0			0
$829$	21.0000			0.729360			0.364680	−	0.931133i	$-0.381178\pi$
$829$	21.0000			0.729360			0.364680	+	0.931133i	$0.381178\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−48.0000			−1.66111
$836$		0			0
$837$		0			0
$838$		0			0
$839$	10.0000	−	17.3205i	0.345238	−	0.597970i	−0.640159	−	0.768243i	$-0.721131\pi$
$839$	10.0000	−	17.3205i	0.345238	−	0.597970i	0.985397	+	0.170272i	$0.0544647\pi$
$840$		0			0
$841$	−17.5000	+	30.3109i	−0.603448	+	1.04520i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−24.0000	−	41.5692i	−0.825625	−	1.43002i
$846$		0			0
$847$	−15.0000			−0.515406
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.0000	+	17.3205i	0.342796	+	0.593739i
$852$		0			0
$853$	−19.5000	+	33.7750i	−0.667667	+	1.15643i	0.310887	+	0.950447i	$0.399374\pi$
$853$	−19.5000	+	33.7750i	−0.667667	+	1.15643i	−0.978555	+	0.205987i	$0.933960\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	26.5000	+	45.8993i	0.904168	+	1.56607i	0.822030	+	0.569445i	$0.192842\pi$
$859$	26.5000	+	45.8993i	0.904168	+	1.56607i	0.0821386	+	0.996621i	$0.473825\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$	−48.0000	−	83.1384i	−1.63205	−	2.82679i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14.0000	−	24.2487i	0.474917	−	0.822581i
$870$		0			0
$871$	−7.50000	+	12.9904i	−0.254128	+	0.440162i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−72.0000			−2.43404
$876$		0			0
$877$	26.5000	−	45.8993i	0.894841	−	1.54991i	0.0608407	−	0.998147i	$-0.480622\pi$
$877$	26.5000	−	45.8993i	0.894841	−	1.54991i	0.834001	−	0.551763i	$-0.186045\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−36.0000			−1.21287			−0.606435	−	0.795133i	$-0.707401\pi$
$881$	−36.0000			−1.21287			−0.606435	+	0.795133i	$0.707401\pi$
$882$		0			0
$883$	−27.5000	−	47.6314i	−0.925449	−	1.60292i	−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−27.5000	−	47.6314i	−0.925449	−	1.60292i	−0.134611	−	0.990899i	$-0.542978\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.0000	−	17.3205i	−0.335767	−	0.581566i	0.647865	−	0.761755i	$-0.275662\pi$
$887$	−10.0000	−	17.3205i	−0.335767	−	0.581566i	−0.983632	+	0.180190i	$0.942329\pi$
$888$		0			0
$889$	−18.0000	−	31.1769i	−0.603701	−	1.04564i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−28.0000	−	20.7846i	−0.936984	−	0.695530i
$894$		0			0
$895$	24.0000	−	41.5692i	0.802232	−	1.38951i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.00000	−	6.92820i	−0.133407	−	0.231069i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	40.0000			1.32964
$906$		0			0
$907$	20.0000	−	34.6410i	0.664089	−	1.15024i	−0.315442	−	0.948945i	$-0.602153\pi$
$907$	20.0000	−	34.6410i	0.664089	−	1.15024i	0.979531	−	0.201291i	$-0.0645138\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24.0000			0.795155			0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000			0.795155			0.397578	+	0.917568i	$0.369851\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−25.0000			−0.824674			−0.412337	−	0.911031i	$-0.635287\pi$
$919$	−25.0000			−0.824674			−0.412337	+	0.911031i	$0.635287\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	80.0000			2.63323
$924$		0			0
$925$	27.5000	+	47.6314i	0.904194	+	1.56611i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−14.0000	+	24.2487i	−0.459325	+	0.795574i	−0.998925	−	0.0463469i	$-0.985242\pi$
$929$	−14.0000	+	24.2487i	−0.459325	+	0.795574i	0.539600	+	0.841921i	$0.318575\pi$
$930$		0			0
$931$	−8.00000	+	3.46410i	−0.262189	+	0.113531i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	6.50000	+	11.2583i	0.212346	+	0.367794i	0.952448	−	0.304700i	$-0.0985564\pi$
$937$	6.50000	+	11.2583i	0.212346	+	0.367794i	−0.740102	+	0.672494i	$0.765223\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8.00000	+	13.8564i	0.260793	+	0.451706i	0.966453	−	0.256844i	$-0.0826828\pi$
$941$	8.00000	+	13.8564i	0.260793	+	0.451706i	−0.705660	+	0.708550i	$0.749349\pi$
$942$		0			0
$943$	32.0000			1.04206
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−24.0000	+	41.5692i	−0.779895	+	1.35082i	0.152106	+	0.988364i	$0.451394\pi$
$947$	−24.0000	+	41.5692i	−0.779895	+	1.35082i	−0.932002	+	0.362454i	$0.881939\pi$
$948$		0			0
$949$	−75.0000			−2.43460
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.0000	−	17.3205i	0.323932	−	0.561066i	−0.657364	−	0.753573i	$-0.728329\pi$
$953$	10.0000	−	17.3205i	0.323932	−	0.561066i	0.981296	+	0.192507i	$0.0616618\pi$
$954$		0			0
$955$	8.00000	−	13.8564i	0.258874	−	0.448383i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	12.0000	+	20.7846i	0.387500	+	0.671170i
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$	18.0000	+	31.1769i	0.579441	+	1.00362i
$966$		0			0
$967$	−13.5000	+	23.3827i	−0.434131	+	0.751936i	−0.997224	−	0.0744567i	$-0.976278\pi$
$967$	−13.5000	+	23.3827i	−0.434131	+	0.751936i	0.563094	+	0.826393i	$0.309611\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20.0000	+	34.6410i	−0.641831	+	1.11168i	0.343193	+	0.939265i	$0.388491\pi$
$971$	−20.0000	+	34.6410i	−0.641831	+	1.11168i	−0.985024	+	0.172418i	$0.944842\pi$
$972$		0			0
$973$	−22.5000	−	38.9711i	−0.721317	−	1.24936i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−8.00000			−0.255943			−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000			−0.255943			−0.127971	+	0.991778i	$0.540847\pi$
$978$		0			0
$979$	24.0000	+	41.5692i	0.767043	+	1.32856i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	0.422027	+	0.906583i	$0.361319\pi$
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	−0.996138	+	0.0878058i	$0.972015\pi$
$984$		0			0
$985$	40.0000	−	69.2820i	1.27451	−	2.20751i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−20.0000			−0.635963
$990$		0			0
$991$	2.50000	−	4.33013i	0.0794151	−	0.137551i	−0.823583	−	0.567196i	$-0.808028\pi$
$991$	2.50000	−	4.33013i	0.0794151	−	0.137551i	0.902998	+	0.429645i	$0.141361\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−28.0000			−0.887660
$996$		0			0
$997$	26.5000	+	45.8993i	0.839263	+	1.45365i	0.890511	+	0.454961i	$0.150347\pi$
$997$	26.5000	+	45.8993i	0.839263	+	1.45365i	−0.0512480	+	0.998686i	$0.516320\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.2.k.a.505.1	✓	2
3.2	odd	2		684.2.k.e.505.1	yes	2
4.3	odd	2		2736.2.s.b.1873.1		2
12.11	even	2		2736.2.s.r.1873.1		2
19.7	even	3	inner	684.2.k.a.577.1	yes	2
57.26	odd	6		684.2.k.e.577.1	yes	2
76.7	odd	6		2736.2.s.b.577.1		2
228.83	even	6		2736.2.s.r.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.2.k.a.505.1	✓	2	1.1	even	1	trivial
684.2.k.a.577.1	yes	2	19.7	even	3	inner
684.2.k.e.505.1	yes	2	3.2	odd	2
684.2.k.e.577.1	yes	2	57.26	odd	6
2736.2.s.b.577.1		2	76.7	odd	6
2736.2.s.b.1873.1		2	4.3	odd	2
2736.2.s.r.577.1		2	228.83	even	6
2736.2.s.r.1873.1		2	12.11	even	2

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 + 3.46410i) q^{5} -3.00000 q^{7} +O(q^{10})\)	\(q+(-2.00000 + 3.46410i) q^{5} -3.00000 q^{7} +4.00000 q^{11} +(-2.50000 - 4.33013i) q^{13} +(-4.00000 + 1.73205i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-5.50000 - 9.52628i) q^{25} +(-4.00000 - 6.92820i) q^{29} +1.00000 q^{31} +(6.00000 - 10.3923i) q^{35} -5.00000 q^{37} +(-4.00000 + 6.92820i) q^{41} +(2.50000 - 4.33013i) q^{43} +(4.00000 + 6.92820i) q^{47} +2.00000 q^{49} +(-2.00000 - 3.46410i) q^{53} +(-8.00000 + 13.8564i) q^{55} +(-6.00000 + 10.3923i) q^{59} +(0.500000 + 0.866025i) q^{61} +20.0000 q^{65} +(-1.50000 - 2.59808i) q^{67} +(-8.00000 + 13.8564i) q^{71} +(7.50000 - 12.9904i) q^{73} -12.0000 q^{77} +(3.50000 - 6.06218i) q^{79} +(6.00000 + 10.3923i) q^{89} +(7.50000 + 12.9904i) q^{91} +(2.00000 - 17.3205i) q^{95} +(1.00000 - 1.73205i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q - 4 * q^5 - 6 * q^7	\( 2 q - 4 q^{5} - 6 q^{7} + 8 q^{11} - 5 q^{13} - 8 q^{19} - 4 q^{23} - 11 q^{25} - 8 q^{29} + 2 q^{31} + 12 q^{35} - 10 q^{37} - 8 q^{41} + 5 q^{43} + 8 q^{47} + 4 q^{49} - 4 q^{53} - 16 q^{55} - 12 q^{59} + q^{61} + 40 q^{65} - 3 q^{67} - 16 q^{71} + 15 q^{73} - 24 q^{77} + 7 q^{79} + 12 q^{89} + 15 q^{91} + 4 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 6 * q^7 + 8 * q^11 - 5 * q^13 - 8 * q^19 - 4 * q^23 - 11 * q^25 - 8 * q^29 + 2 * q^31 + 12 * q^35 - 10 * q^37 - 8 * q^41 + 5 * q^43 + 8 * q^47 + 4 * q^49 - 4 * q^53 - 16 * q^55 - 12 * q^59 + q^61 + 40 * q^65 - 3 * q^67 - 16 * q^71 + 15 * q^73 - 24 * q^77 + 7 * q^79 + 12 * q^89 + 15 * q^91 + 4 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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