LMFDB - Embedded newform 546.2.c.b.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [546,2,Mod(337,546)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(546, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("546.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$546 = 2 \cdot 3 \cdot 7 \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	546.c (of order $2$ , degree $1$ , 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:	$4.35983195036$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	546.337
Dual form			546.2.c.b.337.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} +1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +(2.00000 + 3.00000i) q^{13} -1.00000 q^{14} +2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000i q^{18} +4.00000i q^{19} -2.00000i q^{20} +1.00000i q^{21} -6.00000 q^{23} -1.00000i q^{24} +1.00000 q^{25} +(-3.00000 + 2.00000i) q^{26} +1.00000 q^{27} -1.00000i q^{28} -2.00000 q^{30} +1.00000i q^{32} -2.00000i q^{34} -2.00000 q^{35} -1.00000 q^{36} -2.00000i q^{37} -4.00000 q^{38} +(2.00000 + 3.00000i) q^{39} +2.00000 q^{40} -1.00000 q^{42} +4.00000 q^{43} +2.00000i q^{45} -6.00000i q^{46} +8.00000i q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000i q^{50} -2.00000 q^{51} +(-2.00000 - 3.00000i) q^{52} +4.00000 q^{53} +1.00000i q^{54} +1.00000 q^{56} +4.00000i q^{57} -6.00000i q^{59} -2.00000i q^{60} +12.0000 q^{61} +1.00000i q^{63} -1.00000 q^{64} +(-6.00000 + 4.00000i) q^{65} -2.00000i q^{67} +2.00000 q^{68} -6.00000 q^{69} -2.00000i q^{70} -1.00000i q^{72} -14.0000i q^{73} +2.00000 q^{74} +1.00000 q^{75} -4.00000i q^{76} +(-3.00000 + 2.00000i) q^{78} +2.00000i q^{80} +1.00000 q^{81} -14.0000i q^{83} -1.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} +4.00000i q^{89} -2.00000 q^{90} +(-3.00000 + 2.00000i) q^{91} +6.00000 q^{92} -8.00000 q^{94} -8.00000 q^{95} +1.00000i q^{96} -2.00000i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 4 q^{10} - 2 q^{12} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 4 q^{17} - 12 q^{23} + 2 q^{25} - 6 q^{26} + 2 q^{27} - 4 q^{30} - 4 q^{35} - 2 q^{36} - 8 q^{38} + 4 q^{39} + 4 q^{40}+ \cdots - 16 q^{95}+O(q^{100})$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 4 * q^10 - 2 * q^12 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 4 * q^17 - 12 * q^23 + 2 * q^25 - 6 * q^26 + 2 * q^27 - 4 * q^30 - 4 * q^35 - 2 * q^36 - 8 * q^38 + 4 * q^39 + 4 * q^40 - 2 * q^42 + 8 * q^43 + 2 * q^48 - 2 * q^49 - 4 * q^51 - 4 * q^52 + 8 * q^53 + 2 * q^56 + 24 * q^61 - 2 * q^64 - 12 * q^65 + 4 * q^68 - 12 * q^69 + 4 * q^74 + 2 * q^75 - 6 * q^78 + 2 * q^81 - 4 * q^90 - 6 * q^91 + 12 * q^92 - 16 * q^94 - 16 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/546\mathbb{Z}\right)^\times$ .

$n$	$157$	$365$	$379$
$\chi(n)$	$1$	$1$	$-1$

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

)

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

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$	−1.00000			−0.500000
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$			1.00000i			0.408248i
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$	−2.00000			−0.632456
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−1.00000			−0.288675
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$13$	2.00000	+	3.00000i	0.554700	+	0.832050i
$14$	−1.00000			−0.267261
$15$			2.00000i			0.516398i
$16$	1.00000			0.250000
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$			1.00000i			0.235702i
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		−	2.00000i		−	0.447214i
$21$			1.00000i			0.218218i
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		−	1.00000i		−	0.204124i
$25$	1.00000			0.200000
$26$	−3.00000	+	2.00000i	−0.588348	+	0.392232i
$27$	1.00000			0.192450
$28$		−	1.00000i		−	0.188982i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$	−2.00000			−0.365148
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	2.00000i		−	0.342997i
$35$	−2.00000			−0.338062
$36$	−1.00000			−0.166667
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	−4.00000			−0.648886
$39$	2.00000	+	3.00000i	0.320256	+	0.480384i
$40$	2.00000			0.316228
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−1.00000			−0.154303
$43$	4.00000			0.609994			0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000			0.609994			0.304997	+	0.952353i	$0.401344\pi$
$44$		0			0
$45$			2.00000i			0.298142i
$46$		−	6.00000i		−	0.884652i
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$	1.00000			0.144338
$49$	−1.00000			−0.142857
$50$			1.00000i			0.141421i
$51$	−2.00000			−0.280056
$52$	−2.00000	−	3.00000i	−0.277350	−	0.416025i
$53$	4.00000			0.549442			0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000			0.549442			0.274721	+	0.961524i	$0.411414\pi$
$54$			1.00000i			0.136083i
$55$		0			0
$56$	1.00000			0.133631
$57$			4.00000i			0.529813i
$58$		0			0
$59$		−	6.00000i		−	0.781133i	−0.920575	−	0.390567i	$-0.872279\pi$
$59$		−	6.00000i		−	0.781133i	0.920575	−	0.390567i	$-0.127721\pi$
$60$		−	2.00000i		−	0.258199i
$61$	12.0000			1.53644			0.768221	−	0.640184i	$-0.221142\pi$
$61$	12.0000			1.53644			0.768221	+	0.640184i	$0.221142\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$	−1.00000			−0.125000
$65$	−6.00000	+	4.00000i	−0.744208	+	0.496139i
$66$		0			0
$67$		−	2.00000i		−	0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$		−	2.00000i		−	0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	2.00000			0.242536
$69$	−6.00000			−0.722315
$70$		−	2.00000i		−	0.239046i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		−	1.00000i		−	0.117851i
$73$		−	14.0000i		−	1.63858i	−0.573382	−	0.819288i	$-0.694369\pi$
$73$		−	14.0000i		−	1.63858i	0.573382	−	0.819288i	$-0.305631\pi$
$74$	2.00000			0.232495
$75$	1.00000			0.115470
$76$		−	4.00000i		−	0.458831i
$77$		0			0
$78$	−3.00000	+	2.00000i	−0.339683	+	0.226455i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$			2.00000i			0.223607i
$81$	1.00000			0.111111
$82$		0			0
$83$		−	14.0000i		−	1.53670i	−0.640030	−	0.768350i	$-0.721078\pi$
$83$		−	14.0000i		−	1.53670i	0.640030	−	0.768350i	$-0.278922\pi$
$84$		−	1.00000i		−	0.109109i
$85$		−	4.00000i		−	0.433861i
$86$			4.00000i			0.431331i
$87$		0			0
$88$		0			0
$89$			4.00000i			0.423999i	0.977270	+	0.212000i	$0.0679975\pi$
$89$			4.00000i			0.423999i	−0.977270	+	0.212000i	$0.932002\pi$
$90$	−2.00000			−0.210819
$91$	−3.00000	+	2.00000i	−0.314485	+	0.209657i
$92$	6.00000			0.625543
$93$		0			0
$94$	−8.00000			−0.825137
$95$	−8.00000			−0.820783
$96$			1.00000i			0.102062i
$97$		−	2.00000i		−	0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$		−	2.00000i		−	0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$		−	1.00000i		−	0.101015i
$99$		0			0
$100$	−1.00000			−0.100000
$101$	2.00000			0.199007			0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000			0.199007			0.0995037	+	0.995037i	$0.468274\pi$
$102$		−	2.00000i		−	0.198030i
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$	3.00000	−	2.00000i	0.294174	−	0.196116i
$105$	−2.00000			−0.195180
$106$			4.00000i			0.388514i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000			−0.0962250
$109$		−	6.00000i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$109$		−	6.00000i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$110$		0			0
$111$		−	2.00000i		−	0.189832i
$112$			1.00000i			0.0944911i
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$	−4.00000			−0.374634
$115$		−	12.0000i		−	1.11901i
$116$		0			0
$117$	2.00000	+	3.00000i	0.184900	+	0.277350i
$118$	6.00000			0.552345
$119$		−	2.00000i		−	0.183340i
$120$	2.00000			0.182574
$121$	11.0000			1.00000
$122$			12.0000i			1.08643i
$123$		0			0
$124$		0			0
$125$			12.0000i			1.07331i
$126$	−1.00000			−0.0890871
$127$	−12.0000			−1.06483			−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000			−1.06483			−0.532414	+	0.846484i	$0.678715\pi$
$128$		−	1.00000i		−	0.0883883i
$129$	4.00000			0.352180
$130$	−4.00000	−	6.00000i	−0.350823	−	0.526235i
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−4.00000			−0.346844
$134$	2.00000			0.172774
$135$			2.00000i			0.172133i
$136$			2.00000i			0.171499i
$137$		−	2.00000i		−	0.170872i	−0.996344	−	0.0854358i	$-0.972772\pi$
$137$		−	2.00000i		−	0.170872i	0.996344	−	0.0854358i	$-0.0272282\pi$
$138$		−	6.00000i		−	0.510754i
$139$	20.0000			1.69638			0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000			1.69638			0.848189	+	0.529694i	$0.177693\pi$
$140$	2.00000			0.169031
$141$			8.00000i			0.673722i
$142$		0			0
$143$		0			0
$144$	1.00000			0.0833333
$145$		0			0
$146$	14.0000			1.15865
$147$	−1.00000			−0.0824786
$148$			2.00000i			0.164399i
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$			1.00000i			0.0816497i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	4.00000			0.324443
$153$	−2.00000			−0.161690
$154$		0			0
$155$		0			0
$156$	−2.00000	−	3.00000i	−0.160128	−	0.240192i
$157$	8.00000			0.638470			0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000			0.638470			0.319235	+	0.947676i	$0.396574\pi$
$158$		0			0
$159$	4.00000			0.317221
$160$	−2.00000			−0.158114
$161$		−	6.00000i		−	0.472866i
$162$			1.00000i			0.0785674i
$163$		−	14.0000i		−	1.09656i	−0.836293	−	0.548282i	$-0.815282\pi$
$163$		−	14.0000i		−	1.09656i	0.836293	−	0.548282i	$-0.184718\pi$
$164$		0			0
$165$		0			0
$166$	14.0000			1.08661
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	1.00000			0.0771517
$169$	−5.00000	+	12.0000i	−0.384615	+	0.923077i
$170$	4.00000			0.306786
$171$			4.00000i			0.305888i
$172$	−4.00000			−0.304997
$173$	−6.00000			−0.456172			−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000			−0.456172			−0.228086	+	0.973641i	$0.573247\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$		−	6.00000i		−	0.450988i
$178$	−4.00000			−0.299813
$179$	−20.0000			−1.49487			−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000			−1.49487			−0.747435	+	0.664335i	$0.768715\pi$
$180$		−	2.00000i		−	0.149071i
$181$	12.0000			0.891953			0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000			0.891953			0.445976	+	0.895045i	$0.352856\pi$
$182$	−2.00000	−	3.00000i	−0.148250	−	0.222375i
$183$	12.0000			0.887066
$184$			6.00000i			0.442326i
$185$	4.00000			0.294086
$186$		0			0
$187$		0			0
$188$		−	8.00000i		−	0.583460i
$189$			1.00000i			0.0727393i
$190$		−	8.00000i		−	0.580381i
$191$	2.00000			0.144715			0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000			0.144715			0.0723575	+	0.997379i	$0.476948\pi$
$192$	−1.00000			−0.0721688
$193$		−	4.00000i		−	0.287926i	−0.989583	−	0.143963i	$-0.954015\pi$
$193$		−	4.00000i		−	0.287926i	0.989583	−	0.143963i	$-0.0459847\pi$
$194$	2.00000			0.143592
$195$	−6.00000	+	4.00000i	−0.429669	+	0.286446i
$196$	1.00000			0.0714286
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$	−10.0000			−0.708881			−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000			−0.708881			−0.354441	+	0.935079i	$0.615329\pi$
$200$		−	1.00000i		−	0.0707107i
$201$		−	2.00000i		−	0.141069i
$202$			2.00000i			0.140720i
$203$		0			0
$204$	2.00000			0.140028
$205$		0			0
$206$			14.0000i			0.975426i
$207$	−6.00000			−0.417029
$208$	2.00000	+	3.00000i	0.138675	+	0.208013i
$209$		0			0
$210$		−	2.00000i		−	0.138013i
$211$	−28.0000			−1.92760			−0.963800	−	0.266627i	$-0.914091\pi$
$211$	−28.0000			−1.92760			−0.963800	+	0.266627i	$0.914091\pi$
$212$	−4.00000			−0.274721
$213$		0			0
$214$		−	12.0000i		−	0.820303i
$215$			8.00000i			0.545595i
$216$		−	1.00000i		−	0.0680414i
$217$		0			0
$218$	6.00000			0.406371
$219$		−	14.0000i		−	0.946032i
$220$		0			0
$221$	−4.00000	−	6.00000i	−0.269069	−	0.403604i
$222$	2.00000			0.134231
$223$			16.0000i			1.07144i	0.844396	+	0.535720i	$0.179960\pi$
$223$			16.0000i			1.07144i	−0.844396	+	0.535720i	$0.820040\pi$
$224$	−1.00000			−0.0668153
$225$	1.00000			0.0666667
$226$			14.0000i			0.931266i
$227$		−	22.0000i		−	1.46019i	−0.683345	−	0.730096i	$-0.739475\pi$
$227$		−	22.0000i		−	1.46019i	0.683345	−	0.730096i	$-0.260525\pi$
$228$		−	4.00000i		−	0.264906i
$229$			14.0000i			0.925146i	0.886581	+	0.462573i	$0.153074\pi$
$229$			14.0000i			0.925146i	−0.886581	+	0.462573i	$0.846926\pi$
$230$	12.0000			0.791257
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$	−3.00000	+	2.00000i	−0.196116	+	0.130744i
$235$	−16.0000			−1.04372
$236$			6.00000i			0.390567i
$237$		0			0
$238$	2.00000			0.129641
$239$		−	16.0000i		−	1.03495i	−0.855697	−	0.517477i	$-0.826871\pi$
$239$		−	16.0000i		−	1.03495i	0.855697	−	0.517477i	$-0.173129\pi$
$240$			2.00000i			0.129099i
$241$			10.0000i			0.644157i	0.946713	+	0.322078i	$0.104381\pi$
$241$			10.0000i			0.644157i	−0.946713	+	0.322078i	$0.895619\pi$
$242$			11.0000i			0.707107i
$243$	1.00000			0.0641500
$244$	−12.0000			−0.768221
$245$		−	2.00000i		−	0.127775i
$246$		0			0
$247$	−12.0000	+	8.00000i	−0.763542	+	0.509028i
$248$		0			0
$249$		−	14.0000i		−	0.887214i
$250$	−12.0000			−0.758947
$251$	−28.0000			−1.76734			−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000			−1.76734			−0.883672	+	0.468106i	$0.844936\pi$
$252$		−	1.00000i		−	0.0629941i
$253$		0			0
$254$		−	12.0000i		−	0.752947i
$255$		−	4.00000i		−	0.250490i
$256$	1.00000			0.0625000
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$			4.00000i			0.249029i
$259$	2.00000			0.124274
$260$	6.00000	−	4.00000i	0.372104	−	0.248069i
$261$		0			0
$262$			12.0000i			0.741362i
$263$	−6.00000			−0.369976			−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000			−0.369976			−0.184988	+	0.982741i	$0.559225\pi$
$264$		0			0
$265$			8.00000i			0.491436i
$266$		−	4.00000i		−	0.245256i
$267$			4.00000i			0.244796i
$268$			2.00000i			0.122169i
$269$	10.0000			0.609711			0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000			0.609711			0.304855	+	0.952399i	$0.401392\pi$
$270$	−2.00000			−0.121716
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−2.00000			−0.121268
$273$	−3.00000	+	2.00000i	−0.181568	+	0.121046i
$274$	2.00000			0.120824
$275$		0			0
$276$	6.00000			0.361158
$277$	−22.0000			−1.32185			−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000			−1.32185			−0.660926	+	0.750451i	$0.729836\pi$
$278$			20.0000i			1.19952i
$279$		0			0
$280$			2.00000i			0.119523i
$281$			30.0000i			1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$			30.0000i			1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$	−8.00000			−0.476393
$283$	−16.0000			−0.951101			−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000			−0.951101			−0.475551	+	0.879688i	$0.657751\pi$
$284$		0			0
$285$	−8.00000			−0.473879
$286$		0			0
$287$		0			0
$288$			1.00000i			0.0589256i
$289$	−13.0000			−0.764706
$290$		0			0
$291$		−	2.00000i		−	0.117242i
$292$			14.0000i			0.819288i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		−	1.00000i		−	0.0583212i
$295$	12.0000			0.698667
$296$	−2.00000			−0.116248
$297$		0			0
$298$	6.00000			0.347571
$299$	−12.0000	−	18.0000i	−0.693978	−	1.04097i
$300$	−1.00000			−0.0577350
$301$			4.00000i			0.230556i
$302$		0			0
$303$	2.00000			0.114897
$304$			4.00000i			0.229416i
$305$			24.0000i			1.37424i
$306$		−	2.00000i		−	0.114332i
$307$			28.0000i			1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$			28.0000i			1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$		0			0
$309$	14.0000			0.796432
$310$		0			0
$311$	12.0000			0.680458			0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000			0.680458			0.340229	+	0.940343i	$0.389495\pi$
$312$	3.00000	−	2.00000i	0.169842	−	0.113228i
$313$	−6.00000			−0.339140			−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000			−0.339140			−0.169570	+	0.985518i	$0.554238\pi$
$314$			8.00000i			0.451466i
$315$	−2.00000			−0.112687
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$			4.00000i			0.224309i
$319$		0			0
$320$		−	2.00000i		−	0.111803i
$321$	−12.0000			−0.669775
$322$	6.00000			0.334367
$323$		−	8.00000i		−	0.445132i
$324$	−1.00000			−0.0555556
$325$	2.00000	+	3.00000i	0.110940	+	0.166410i
$326$	14.0000			0.775388
$327$		−	6.00000i		−	0.331801i
$328$		0			0
$329$	−8.00000			−0.441054
$330$		0			0
$331$		−	10.0000i		−	0.549650i	−0.961494	−	0.274825i	$-0.911380\pi$
$331$		−	10.0000i		−	0.549650i	0.961494	−	0.274825i	$-0.0886199\pi$
$332$			14.0000i			0.768350i
$333$		−	2.00000i		−	0.109599i
$334$	12.0000			0.656611
$335$	4.00000			0.218543
$336$			1.00000i			0.0545545i
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$	−12.0000	−	5.00000i	−0.652714	−	0.271964i
$339$	14.0000			0.760376
$340$			4.00000i			0.216930i
$341$		0			0
$342$	−4.00000			−0.216295
$343$		−	1.00000i		−	0.0539949i
$344$		−	4.00000i		−	0.215666i
$345$		−	12.0000i		−	0.646058i
$346$		−	6.00000i		−	0.322562i
$347$	−32.0000			−1.71785			−0.858925	−	0.512101i	$-0.828867\pi$
$347$	−32.0000			−1.71785			−0.858925	+	0.512101i	$0.828867\pi$
$348$		0			0
$349$			14.0000i			0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$			14.0000i			0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$	−1.00000			−0.0534522
$351$	2.00000	+	3.00000i	0.106752	+	0.160128i
$352$		0			0
$353$		−	24.0000i		−	1.27739i	−0.769460	−	0.638696i	$-0.779474\pi$
$353$		−	24.0000i		−	1.27739i	0.769460	−	0.638696i	$-0.220526\pi$
$354$	6.00000			0.318896
$355$		0			0
$356$		−	4.00000i		−	0.212000i
$357$		−	2.00000i		−	0.105851i
$358$		−	20.0000i		−	1.05703i
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$	2.00000			0.105409
$361$	3.00000			0.157895
$362$			12.0000i			0.630706i
$363$	11.0000			0.577350
$364$	3.00000	−	2.00000i	0.157243	−	0.104828i
$365$	28.0000			1.46559
$366$			12.0000i			0.627250i
$367$	−22.0000			−1.14839			−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000			−1.14839			−0.574195	+	0.818718i	$0.694685\pi$
$368$	−6.00000			−0.312772
$369$		0			0
$370$			4.00000i			0.207950i
$371$			4.00000i			0.207670i
$372$		0			0
$373$	34.0000			1.76045			0.880227	−	0.474554i	$-0.157390\pi$
$373$	34.0000			1.76045			0.880227	+	0.474554i	$0.157390\pi$
$374$		0			0
$375$			12.0000i			0.619677i
$376$	8.00000			0.412568
$377$		0			0
$378$	−1.00000			−0.0514344
$379$		−	26.0000i		−	1.33553i	−0.744372	−	0.667765i	$-0.767251\pi$
$379$		−	26.0000i		−	1.33553i	0.744372	−	0.667765i	$-0.232749\pi$
$380$	8.00000			0.410391
$381$	−12.0000			−0.614779
$382$			2.00000i			0.102329i
$383$		−	24.0000i		−	1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$		−	24.0000i		−	1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$		−	1.00000i		−	0.0510310i
$385$		0			0
$386$	4.00000			0.203595
$387$	4.00000			0.203331
$388$			2.00000i			0.101535i
$389$	20.0000			1.01404			0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000			1.01404			0.507020	+	0.861934i	$0.330747\pi$
$390$	−4.00000	−	6.00000i	−0.202548	−	0.303822i
$391$	12.0000			0.606866
$392$			1.00000i			0.0505076i
$393$	12.0000			0.605320
$394$	−18.0000			−0.906827
$395$		0			0
$396$		0			0
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$		−	10.0000i		−	0.501255i
$399$	−4.00000			−0.200250
$400$	1.00000			0.0500000
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$	2.00000			0.0997509
$403$		0			0
$404$	−2.00000			−0.0995037
$405$			2.00000i			0.0993808i
$406$		0			0
$407$		0			0
$408$			2.00000i			0.0990148i
$409$			14.0000i			0.692255i	0.938187	+	0.346128i	$0.112504\pi$
$409$			14.0000i			0.692255i	−0.938187	+	0.346128i	$0.887496\pi$
$410$		0			0
$411$		−	2.00000i		−	0.0986527i
$412$	−14.0000			−0.689730
$413$	6.00000			0.295241
$414$		−	6.00000i		−	0.294884i
$415$	28.0000			1.37447
$416$	−3.00000	+	2.00000i	−0.147087	+	0.0980581i
$417$	20.0000			0.979404
$418$		0			0
$419$	20.0000			0.977064			0.488532	−	0.872546i	$-0.337533\pi$
$419$	20.0000			0.977064			0.488532	+	0.872546i	$0.337533\pi$
$420$	2.00000			0.0975900
$421$		−	30.0000i		−	1.46211i	−0.682318	−	0.731055i	$-0.739028\pi$
$421$		−	30.0000i		−	1.46211i	0.682318	−	0.731055i	$-0.260972\pi$
$422$		−	28.0000i		−	1.36302i
$423$			8.00000i			0.388973i
$424$		−	4.00000i		−	0.194257i
$425$	−2.00000			−0.0970143
$426$		0			0
$427$			12.0000i			0.580721i
$428$	12.0000			0.580042
$429$		0			0
$430$	−8.00000			−0.385794
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	1.00000			0.0481125
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$			6.00000i			0.287348i
$437$		−	24.0000i		−	1.14808i
$438$	14.0000			0.668946
$439$	30.0000			1.43182			0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000			1.43182			0.715911	+	0.698192i	$0.246012\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$	6.00000	−	4.00000i	0.285391	−	0.190261i
$443$	4.00000			0.190046			0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000			0.190046			0.0950229	+	0.995475i	$0.469708\pi$
$444$			2.00000i			0.0949158i
$445$	−8.00000			−0.379236
$446$	−16.0000			−0.757622
$447$		−	6.00000i		−	0.283790i
$448$		−	1.00000i		−	0.0472456i
$449$			14.0000i			0.660701i	0.943858	+	0.330350i	$0.107167\pi$
$449$			14.0000i			0.660701i	−0.943858	+	0.330350i	$0.892833\pi$
$450$			1.00000i			0.0471405i
$451$		0			0
$452$	−14.0000			−0.658505
$453$		0			0
$454$	22.0000			1.03251
$455$	−4.00000	−	6.00000i	−0.187523	−	0.281284i
$456$	4.00000			0.187317
$457$			8.00000i			0.374224i	0.982339	+	0.187112i	$0.0599128\pi$
$457$			8.00000i			0.374224i	−0.982339	+	0.187112i	$0.940087\pi$
$458$	−14.0000			−0.654177
$459$	−2.00000			−0.0933520
$460$			12.0000i			0.559503i
$461$			10.0000i			0.465746i	0.972507	+	0.232873i	$0.0748127\pi$
$461$			10.0000i			0.465746i	−0.972507	+	0.232873i	$0.925187\pi$
$462$		0			0
$463$			16.0000i			0.743583i	0.928316	+	0.371792i	$0.121256\pi$
$463$			16.0000i			0.743583i	−0.928316	+	0.371792i	$0.878744\pi$
$464$		0			0
$465$		0			0
$466$		−	6.00000i		−	0.277945i
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$	−2.00000	−	3.00000i	−0.0924500	−	0.138675i
$469$	2.00000			0.0923514
$470$		−	16.0000i		−	0.738025i
$471$	8.00000			0.368621
$472$	−6.00000			−0.276172
$473$		0			0
$474$		0			0
$475$			4.00000i			0.183533i
$476$			2.00000i			0.0916698i
$477$	4.00000			0.183147
$478$	16.0000			0.731823
$479$		−	16.0000i		−	0.731059i	−0.930800	−	0.365529i	$-0.880888\pi$
$479$		−	16.0000i		−	0.731059i	0.930800	−	0.365529i	$-0.119112\pi$
$480$	−2.00000			−0.0912871
$481$	6.00000	−	4.00000i	0.273576	−	0.182384i
$482$	−10.0000			−0.455488
$483$		−	6.00000i		−	0.273009i
$484$	−11.0000			−0.500000
$485$	4.00000			0.181631
$486$			1.00000i			0.0453609i
$487$			28.0000i			1.26880i	0.773004	+	0.634401i	$0.218753\pi$
$487$			28.0000i			1.26880i	−0.773004	+	0.634401i	$0.781247\pi$
$488$		−	12.0000i		−	0.543214i
$489$		−	14.0000i		−	0.633102i
$490$	2.00000			0.0903508
$491$	−8.00000			−0.361035			−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000			−0.361035			−0.180517	+	0.983572i	$0.557777\pi$
$492$		0			0
$493$		0			0
$494$	−8.00000	−	12.0000i	−0.359937	−	0.539906i
$495$		0			0
$496$		0			0
$497$		0			0
$498$	14.0000			0.627355
$499$			14.0000i			0.626726i	0.949633	+	0.313363i	$0.101456\pi$
$499$			14.0000i			0.626726i	−0.949633	+	0.313363i	$0.898544\pi$
$500$		−	12.0000i		−	0.536656i
$501$		−	12.0000i		−	0.536120i
$502$		−	28.0000i		−	1.24970i
$503$	−36.0000			−1.60516			−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000			−1.60516			−0.802580	+	0.596544i	$0.796540\pi$
$504$	1.00000			0.0445435
$505$			4.00000i			0.177998i
$506$		0			0
$507$	−5.00000	+	12.0000i	−0.222058	+	0.532939i
$508$	12.0000			0.532414
$509$		−	26.0000i		−	1.15243i	−0.817298	−	0.576215i	$-0.804529\pi$
$509$		−	26.0000i		−	1.15243i	0.817298	−	0.576215i	$-0.195471\pi$
$510$	4.00000			0.177123
$511$	14.0000			0.619324
$512$			1.00000i			0.0441942i
$513$			4.00000i			0.176604i
$514$			18.0000i			0.793946i
$515$			28.0000i			1.23383i
$516$	−4.00000			−0.176090
$517$		0			0
$518$			2.00000i			0.0878750i
$519$	−6.00000			−0.263371
$520$	4.00000	+	6.00000i	0.175412	+	0.263117i
$521$	42.0000			1.84005			0.920027	−	0.391856i	$-0.128167\pi$
$521$	42.0000			1.84005			0.920027	+	0.391856i	$0.128167\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$	−12.0000			−0.524222
$525$			1.00000i			0.0436436i
$526$		−	6.00000i		−	0.261612i
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$	−8.00000			−0.347498
$531$		−	6.00000i		−	0.260378i
$532$	4.00000			0.173422
$533$		0			0
$534$	−4.00000			−0.173097
$535$		−	24.0000i		−	1.03761i
$536$	−2.00000			−0.0863868
$537$	−20.0000			−0.863064
$538$			10.0000i			0.431131i
$539$		0			0
$540$		−	2.00000i		−	0.0860663i
$541$		−	30.0000i		−	1.28980i	−0.764267	−	0.644900i	$-0.776899\pi$
$541$		−	30.0000i		−	1.28980i	0.764267	−	0.644900i	$-0.223101\pi$
$542$		0			0
$543$	12.0000			0.514969
$544$		−	2.00000i		−	0.0857493i
$545$	12.0000			0.514024
$546$	−2.00000	−	3.00000i	−0.0855921	−	0.128388i
$547$	−12.0000			−0.513083			−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000			−0.513083			−0.256541	+	0.966533i	$0.582583\pi$
$548$			2.00000i			0.0854358i
$549$	12.0000			0.512148
$550$		0			0
$551$		0			0
$552$			6.00000i			0.255377i
$553$		0			0
$554$		−	22.0000i		−	0.934690i
$555$	4.00000			0.169791
$556$	−20.0000			−0.848189
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$	8.00000	+	12.0000i	0.338364	+	0.507546i
$560$	−2.00000			−0.0845154
$561$		0			0
$562$	−30.0000			−1.26547
$563$	4.00000			0.168580			0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000			0.168580			0.0842900	+	0.996441i	$0.473138\pi$
$564$		−	8.00000i		−	0.336861i
$565$			28.0000i			1.17797i
$566$		−	16.0000i		−	0.672530i
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	30.0000			1.25767			0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000			1.25767			0.628833	+	0.777541i	$0.283533\pi$
$570$		−	8.00000i		−	0.335083i
$571$	12.0000			0.502184			0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000			0.502184			0.251092	+	0.967963i	$0.419210\pi$
$572$		0			0
$573$	2.00000			0.0835512
$574$		0			0
$575$	−6.00000			−0.250217
$576$	−1.00000			−0.0416667
$577$		−	22.0000i		−	0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$		−	22.0000i		−	0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$		−	13.0000i		−	0.540729i
$579$		−	4.00000i		−	0.166234i
$580$		0			0
$581$	14.0000			0.580818
$582$	2.00000			0.0829027
$583$		0			0
$584$	−14.0000			−0.579324
$585$	−6.00000	+	4.00000i	−0.248069	+	0.165380i
$586$	−6.00000			−0.247858
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$	1.00000			0.0412393
$589$		0			0
$590$			12.0000i			0.494032i
$591$			18.0000i			0.740421i
$592$		−	2.00000i		−	0.0821995i
$593$		−	24.0000i		−	0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$		−	24.0000i		−	0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$			6.00000i			0.245770i
$597$	−10.0000			−0.409273
$598$	18.0000	−	12.0000i	0.736075	−	0.490716i
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		−	1.00000i		−	0.0408248i
$601$	−18.0000			−0.734235			−0.367118	−	0.930175i	$-0.619655\pi$
$601$	−18.0000			−0.734235			−0.367118	+	0.930175i	$0.619655\pi$
$602$	−4.00000			−0.163028
$603$		−	2.00000i		−	0.0814463i
$604$		0			0
$605$			22.0000i			0.894427i
$606$			2.00000i			0.0812444i
$607$	18.0000			0.730597			0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000			0.730597			0.365299	+	0.930890i	$0.380967\pi$
$608$	−4.00000			−0.162221
$609$		0			0
$610$	−24.0000			−0.971732
$611$	−24.0000	+	16.0000i	−0.970936	+	0.647291i
$612$	2.00000			0.0808452
$613$		−	34.0000i		−	1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$		−	34.0000i		−	1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	−28.0000			−1.12999
$615$		0			0
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$			14.0000i			0.563163i
$619$		−	16.0000i		−	0.643094i	−0.946894	−	0.321547i	$-0.895797\pi$
$619$		−	16.0000i		−	0.643094i	0.946894	−	0.321547i	$-0.104203\pi$
$620$		0			0
$621$	−6.00000			−0.240772
$622$			12.0000i			0.481156i
$623$	−4.00000			−0.160257
$624$	2.00000	+	3.00000i	0.0800641	+	0.120096i
$625$	−19.0000			−0.760000
$626$		−	6.00000i		−	0.239808i
$627$		0			0
$628$	−8.00000			−0.319235
$629$			4.00000i			0.159490i
$630$		−	2.00000i		−	0.0796819i
$631$		−	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
$633$	−28.0000			−1.11290
$634$	2.00000			0.0794301
$635$		−	24.0000i		−	0.952411i
$636$	−4.00000			−0.158610
$637$	−2.00000	−	3.00000i	−0.0792429	−	0.118864i
$638$		0			0
$639$		0			0
$640$	2.00000			0.0790569
$641$	42.0000			1.65890			0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000			1.65890			0.829450	+	0.558581i	$0.188654\pi$
$642$		−	12.0000i		−	0.473602i
$643$		−	44.0000i		−	1.73519i	−0.497271	−	0.867595i	$-0.665665\pi$
$643$		−	44.0000i		−	1.73519i	0.497271	−	0.867595i	$-0.334335\pi$
$644$			6.00000i			0.236433i
$645$			8.00000i			0.315000i
$646$	8.00000			0.314756
$647$	28.0000			1.10079			0.550397	−	0.834903i	$-0.314476\pi$
$647$	28.0000			1.10079			0.550397	+	0.834903i	$0.314476\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		0			0
$650$	−3.00000	+	2.00000i	−0.117670	+	0.0784465i
$651$		0			0
$652$			14.0000i			0.548282i
$653$	4.00000			0.156532			0.0782660	−	0.996933i	$-0.475062\pi$
$653$	4.00000			0.156532			0.0782660	+	0.996933i	$0.475062\pi$
$654$	6.00000			0.234619
$655$			24.0000i			0.937758i
$656$		0			0
$657$		−	14.0000i		−	0.546192i
$658$		−	8.00000i		−	0.311872i
$659$	20.0000			0.779089			0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000			0.779089			0.389545	+	0.921008i	$0.372632\pi$
$660$		0			0
$661$			10.0000i			0.388955i	0.980907	+	0.194477i	$0.0623011\pi$
$661$			10.0000i			0.388955i	−0.980907	+	0.194477i	$0.937699\pi$
$662$	10.0000			0.388661
$663$	−4.00000	−	6.00000i	−0.155347	−	0.233021i
$664$	−14.0000			−0.543305
$665$		−	8.00000i		−	0.310227i
$666$	2.00000			0.0774984
$667$		0			0
$668$			12.0000i			0.464294i
$669$			16.0000i			0.618596i
$670$			4.00000i			0.154533i
$671$		0			0
$672$	−1.00000			−0.0385758
$673$	−46.0000			−1.77317			−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000			−1.77317			−0.886585	+	0.462566i	$0.846929\pi$
$674$		−	22.0000i		−	0.847408i
$675$	1.00000			0.0384900
$676$	5.00000	−	12.0000i	0.192308	−	0.461538i
$677$	−22.0000			−0.845529			−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000			−0.845529			−0.422764	+	0.906240i	$0.638940\pi$
$678$			14.0000i			0.537667i
$679$	2.00000			0.0767530
$680$	−4.00000			−0.153393
$681$		−	22.0000i		−	0.843042i
$682$		0			0
$683$			16.0000i			0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$			16.0000i			0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$		−	4.00000i		−	0.152944i
$685$	4.00000			0.152832
$686$	1.00000			0.0381802
$687$			14.0000i			0.534133i
$688$	4.00000			0.152499
$689$	8.00000	+	12.0000i	0.304776	+	0.457164i
$690$	12.0000			0.456832
$691$		−	40.0000i		−	1.52167i	−0.648944	−	0.760836i	$-0.724789\pi$
$691$		−	40.0000i		−	1.52167i	0.648944	−	0.760836i	$-0.275211\pi$
$692$	6.00000			0.228086
$693$		0			0
$694$		−	32.0000i		−	1.21470i
$695$			40.0000i			1.51729i
$696$		0			0
$697$		0			0
$698$	−14.0000			−0.529908
$699$	−6.00000			−0.226941
$700$		−	1.00000i		−	0.0377964i
$701$	−48.0000			−1.81293			−0.906467	−	0.422276i	$-0.861231\pi$
$701$	−48.0000			−1.81293			−0.906467	+	0.422276i	$0.861231\pi$
$702$	−3.00000	+	2.00000i	−0.113228	+	0.0754851i
$703$	8.00000			0.301726
$704$		0			0
$705$	−16.0000			−0.602595
$706$	24.0000			0.903252
$707$			2.00000i			0.0752177i
$708$			6.00000i			0.225494i
$709$		−	46.0000i		−	1.72757i	−0.503864	−	0.863783i	$-0.668089\pi$
$709$		−	46.0000i		−	1.72757i	0.503864	−	0.863783i	$-0.331911\pi$
$710$		0			0
$711$		0			0
$712$	4.00000			0.149906
$713$		0			0
$714$	2.00000			0.0748481
$715$		0			0
$716$	20.0000			0.747435
$717$		−	16.0000i		−	0.597531i
$718$	−24.0000			−0.895672
$719$	20.0000			0.745874			0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000			0.745874			0.372937	+	0.927857i	$0.378351\pi$
$720$			2.00000i			0.0745356i
$721$			14.0000i			0.521387i
$722$			3.00000i			0.111648i
$723$			10.0000i			0.371904i
$724$	−12.0000			−0.445976
$725$		0			0
$726$			11.0000i			0.408248i
$727$	38.0000			1.40934			0.704671	−	0.709534i	$-0.251095\pi$
$727$	38.0000			1.40934			0.704671	+	0.709534i	$0.251095\pi$
$728$	2.00000	+	3.00000i	0.0741249	+	0.111187i
$729$	1.00000			0.0370370
$730$			28.0000i			1.03633i
$731$	−8.00000			−0.295891
$732$	−12.0000			−0.443533
$733$		−	14.0000i		−	0.517102i	−0.965998	−	0.258551i	$-0.916755\pi$
$733$		−	14.0000i		−	0.517102i	0.965998	−	0.258551i	$-0.0832450\pi$
$734$		−	22.0000i		−	0.812035i
$735$		−	2.00000i		−	0.0737711i
$736$		−	6.00000i		−	0.221163i
$737$		0			0
$738$		0			0
$739$			34.0000i			1.25071i	0.780340	+	0.625355i	$0.215046\pi$
$739$			34.0000i			1.25071i	−0.780340	+	0.625355i	$0.784954\pi$
$740$	−4.00000			−0.147043
$741$	−12.0000	+	8.00000i	−0.440831	+	0.293887i
$742$	−4.00000			−0.146845
$743$			16.0000i			0.586983i	0.955962	+	0.293492i	$0.0948173\pi$
$743$			16.0000i			0.586983i	−0.955962	+	0.293492i	$0.905183\pi$
$744$		0			0
$745$	12.0000			0.439646
$746$			34.0000i			1.24483i
$747$		−	14.0000i		−	0.512233i
$748$		0			0
$749$		−	12.0000i		−	0.438470i
$750$	−12.0000			−0.438178
$751$	12.0000			0.437886			0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000			0.437886			0.218943	+	0.975738i	$0.429739\pi$
$752$			8.00000i			0.291730i
$753$	−28.0000			−1.02038
$754$		0			0
$755$		0			0
$756$		−	1.00000i		−	0.0363696i
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$	26.0000			0.944363
$759$		0			0
$760$			8.00000i			0.290191i
$761$			40.0000i			1.45000i	0.688749	+	0.724999i	$0.258160\pi$
$761$			40.0000i			1.45000i	−0.688749	+	0.724999i	$0.741840\pi$
$762$		−	12.0000i		−	0.434714i
$763$	6.00000			0.217215
$764$	−2.00000			−0.0723575
$765$		−	4.00000i		−	0.144620i
$766$	24.0000			0.867155
$767$	18.0000	−	12.0000i	0.649942	−	0.433295i
$768$	1.00000			0.0360844
$769$		−	26.0000i		−	0.937584i	−0.883309	−	0.468792i	$-0.844689\pi$
$769$		−	26.0000i		−	0.937584i	0.883309	−	0.468792i	$-0.155311\pi$
$770$		0			0
$771$	18.0000			0.648254
$772$			4.00000i			0.143963i
$773$			26.0000i			0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$			26.0000i			0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$			4.00000i			0.143777i
$775$		0			0
$776$	−2.00000			−0.0717958
$777$	2.00000			0.0717496
$778$			20.0000i			0.717035i
$779$		0			0
$780$	6.00000	−	4.00000i	0.214834	−	0.143223i
$781$		0			0
$782$			12.0000i			0.429119i
$783$		0			0
$784$	−1.00000			−0.0357143
$785$			16.0000i			0.571064i
$786$			12.0000i			0.428026i
$787$			48.0000i			1.71102i	0.517790	+	0.855508i	$0.326755\pi$
$787$			48.0000i			1.71102i	−0.517790	+	0.855508i	$0.673245\pi$
$788$		−	18.0000i		−	0.641223i
$789$	−6.00000			−0.213606
$790$		0			0
$791$			14.0000i			0.497783i
$792$		0			0
$793$	24.0000	+	36.0000i	0.852265	+	1.27840i
$794$	−18.0000			−0.638796
$795$			8.00000i			0.283731i
$796$	10.0000			0.354441
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		−	4.00000i		−	0.141598i
$799$		−	16.0000i		−	0.566039i
$800$			1.00000i			0.0353553i
$801$			4.00000i			0.141333i
$802$	30.0000			1.05934
$803$		0			0
$804$			2.00000i			0.0705346i
$805$	12.0000			0.422944
$806$		0			0
$807$	10.0000			0.352017
$808$		−	2.00000i		−	0.0703598i
$809$	50.0000			1.75791			0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000			1.75791			0.878953	+	0.476908i	$0.158243\pi$
$810$	−2.00000			−0.0702728
$811$		−	40.0000i		−	1.40459i	−0.711886	−	0.702295i	$-0.752159\pi$
$811$		−	40.0000i		−	1.40459i	0.711886	−	0.702295i	$-0.247841\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	28.0000			0.980797
$816$	−2.00000			−0.0700140
$817$			16.0000i			0.559769i
$818$	−14.0000			−0.489499
$819$	−3.00000	+	2.00000i	−0.104828	+	0.0698857i
$820$		0			0
$821$			30.0000i			1.04701i	0.852023	+	0.523504i	$0.175375\pi$
$821$			30.0000i			1.04701i	−0.852023	+	0.523504i	$0.824625\pi$
$822$	2.00000			0.0697580
$823$	4.00000			0.139431			0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000			0.139431			0.0697156	+	0.997567i	$0.477791\pi$
$824$		−	14.0000i		−	0.487713i
$825$		0			0
$826$			6.00000i			0.208767i
$827$			28.0000i			0.973655i	0.873498	+	0.486828i	$0.161846\pi$
$827$			28.0000i			0.973655i	−0.873498	+	0.486828i	$0.838154\pi$
$828$	6.00000			0.208514
$829$	−40.0000			−1.38926			−0.694629	−	0.719368i	$-0.744431\pi$
$829$	−40.0000			−1.38926			−0.694629	+	0.719368i	$0.744431\pi$
$830$			28.0000i			0.971894i
$831$	−22.0000			−0.763172
$832$	−2.00000	−	3.00000i	−0.0693375	−	0.104006i
$833$	2.00000			0.0692959
$834$			20.0000i			0.692543i
$835$	24.0000			0.830554
$836$		0			0
$837$		0			0
$838$			20.0000i			0.690889i
$839$			4.00000i			0.138095i	0.997613	+	0.0690477i	$0.0219961\pi$
$839$			4.00000i			0.138095i	−0.997613	+	0.0690477i	$0.978004\pi$
$840$			2.00000i			0.0690066i
$841$	−29.0000			−1.00000
$842$	30.0000			1.03387
$843$			30.0000i			1.03325i
$844$	28.0000			0.963800
$845$	−24.0000	−	10.0000i	−0.825625	−	0.344010i
$846$	−8.00000			−0.275046
$847$			11.0000i			0.377964i
$848$	4.00000			0.137361
$849$	−16.0000			−0.549119
$850$		−	2.00000i		−	0.0685994i
$851$			12.0000i			0.411355i
$852$		0			0
$853$			26.0000i			0.890223i	0.895475	+	0.445112i	$0.146836\pi$
$853$			26.0000i			0.890223i	−0.895475	+	0.445112i	$0.853164\pi$
$854$	−12.0000			−0.410632
$855$	−8.00000			−0.273594
$856$			12.0000i			0.410152i
$857$	−42.0000			−1.43469			−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000			−1.43469			−0.717346	+	0.696717i	$0.754643\pi$
$858$		0			0
$859$	−40.0000			−1.36478			−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000			−1.36478			−0.682391	+	0.730987i	$0.739060\pi$
$860$		−	8.00000i		−	0.272798i
$861$		0			0
$862$		0			0
$863$			16.0000i			0.544646i	0.962206	+	0.272323i	$0.0877920\pi$
$863$			16.0000i			0.544646i	−0.962206	+	0.272323i	$0.912208\pi$
$864$			1.00000i			0.0340207i
$865$		−	12.0000i		−	0.408012i
$866$		−	26.0000i		−	0.883516i
$867$	−13.0000			−0.441503
$868$		0			0
$869$		0			0
$870$		0			0
$871$	6.00000	−	4.00000i	0.203302	−	0.135535i
$872$	−6.00000			−0.203186
$873$		−	2.00000i		−	0.0676897i
$874$	24.0000			0.811812
$875$	−12.0000			−0.405674
$876$			14.0000i			0.473016i
$877$		−	2.00000i		−	0.0675352i	−0.999430	−	0.0337676i	$-0.989249\pi$
$877$		−	2.00000i		−	0.0675352i	0.999430	−	0.0337676i	$-0.0107506\pi$
$878$			30.0000i			1.01245i
$879$			6.00000i			0.202375i
$880$		0			0
$881$	2.00000			0.0673817			0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000			0.0673817			0.0336909	+	0.999432i	$0.489274\pi$
$882$		−	1.00000i		−	0.0336718i
$883$	4.00000			0.134611			0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000			0.134611			0.0673054	+	0.997732i	$0.478560\pi$
$884$	4.00000	+	6.00000i	0.134535	+	0.201802i
$885$	12.0000			0.403376
$886$			4.00000i			0.134383i
$887$	8.00000			0.268614			0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000			0.268614			0.134307	+	0.990940i	$0.457119\pi$
$888$	−2.00000			−0.0671156
$889$		−	12.0000i		−	0.402467i
$890$		−	8.00000i		−	0.268161i
$891$		0			0
$892$		−	16.0000i		−	0.535720i
$893$	−32.0000			−1.07084
$894$	6.00000			0.200670
$895$		−	40.0000i		−	1.33705i
$896$	1.00000			0.0334077
$897$	−12.0000	−	18.0000i	−0.400668	−	0.601003i
$898$	−14.0000			−0.467186
$899$		0			0
$900$	−1.00000			−0.0333333
$901$	−8.00000			−0.266519
$902$		0			0
$903$			4.00000i			0.133112i
$904$		−	14.0000i		−	0.465633i
$905$			24.0000i			0.797787i
$906$		0			0
$907$	−12.0000			−0.398453			−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000			−0.398453			−0.199227	+	0.979953i	$0.563843\pi$
$908$			22.0000i			0.730096i
$909$	2.00000			0.0663358
$910$	6.00000	−	4.00000i	0.198898	−	0.132599i
$911$	22.0000			0.728893			0.364446	−	0.931224i	$-0.381258\pi$
$911$	22.0000			0.728893			0.364446	+	0.931224i	$0.381258\pi$
$912$			4.00000i			0.132453i
$913$		0			0
$914$	−8.00000			−0.264616
$915$			24.0000i			0.793416i
$916$		−	14.0000i		−	0.462573i
$917$			12.0000i			0.396275i
$918$		−	2.00000i		−	0.0660098i
$919$	−20.0000			−0.659739			−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000			−0.659739			−0.329870	+	0.944027i	$0.607005\pi$
$920$	−12.0000			−0.395628
$921$			28.0000i			0.922631i
$922$	−10.0000			−0.329332
$923$		0			0
$924$		0			0
$925$		−	2.00000i		−	0.0657596i
$926$	−16.0000			−0.525793
$927$	14.0000			0.459820
$928$		0			0
$929$			24.0000i			0.787414i	0.919236	+	0.393707i	$0.128808\pi$
$929$			24.0000i			0.787414i	−0.919236	+	0.393707i	$0.871192\pi$
$930$		0			0
$931$		−	4.00000i		−	0.131095i
$932$	6.00000			0.196537
$933$	12.0000			0.392862
$934$		−	12.0000i		−	0.392652i
$935$		0			0
$936$	3.00000	−	2.00000i	0.0980581	−	0.0653720i
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$			2.00000i			0.0653023i
$939$	−6.00000			−0.195803
$940$	16.0000			0.521862
$941$		−	10.0000i		−	0.325991i	−0.986627	−	0.162995i	$-0.947884\pi$
$941$		−	10.0000i		−	0.325991i	0.986627	−	0.162995i	$-0.0521156\pi$
$942$			8.00000i			0.260654i
$943$		0			0
$944$		−	6.00000i		−	0.195283i
$945$	−2.00000			−0.0650600
$946$		0			0
$947$		−	32.0000i		−	1.03986i	−0.854209	−	0.519930i	$-0.825958\pi$
$947$		−	32.0000i		−	1.03986i	0.854209	−	0.519930i	$-0.174042\pi$
$948$		0			0
$949$	42.0000	−	28.0000i	1.36338	−	0.908918i
$950$	−4.00000			−0.129777
$951$		−	2.00000i		−	0.0648544i
$952$	−2.00000			−0.0648204
$953$	−26.0000			−0.842223			−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000			−0.842223			−0.421111	+	0.907009i	$0.638360\pi$
$954$			4.00000i			0.129505i
$955$			4.00000i			0.129437i
$956$			16.0000i			0.517477i
$957$		0			0
$958$	16.0000			0.516937
$959$	2.00000			0.0645834
$960$		−	2.00000i		−	0.0645497i
$961$	31.0000			1.00000
$962$	4.00000	+	6.00000i	0.128965	+	0.193448i
$963$	−12.0000			−0.386695
$964$		−	10.0000i		−	0.322078i
$965$	8.00000			0.257529
$966$	6.00000			0.193047
$967$			28.0000i			0.900419i	0.892923	+	0.450210i	$0.148651\pi$
$967$			28.0000i			0.900419i	−0.892923	+	0.450210i	$0.851349\pi$
$968$		−	11.0000i		−	0.353553i
$969$		−	8.00000i		−	0.256997i
$970$			4.00000i			0.128432i
$971$	−28.0000			−0.898563			−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000			−0.898563			−0.449281	+	0.893390i	$0.648320\pi$
$972$	−1.00000			−0.0320750
$973$			20.0000i			0.641171i
$974$	−28.0000			−0.897178
$975$	2.00000	+	3.00000i	0.0640513	+	0.0960769i
$976$	12.0000			0.384111
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	14.0000			0.447671
$979$		0			0
$980$			2.00000i			0.0638877i
$981$		−	6.00000i		−	0.191565i
$982$		−	8.00000i		−	0.255290i
$983$		−	4.00000i		−	0.127580i	−0.997963	−	0.0637901i	$-0.979681\pi$
$983$		−	4.00000i		−	0.127580i	0.997963	−	0.0637901i	$-0.0203188\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$		0			0
$987$	−8.00000			−0.254643
$988$	12.0000	−	8.00000i	0.381771	−	0.254514i
$989$	−24.0000			−0.763156
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$		0			0
$993$		−	10.0000i		−	0.317340i
$994$		0			0
$995$		−	20.0000i		−	0.634043i
$996$			14.0000i			0.443607i
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$	−14.0000			−0.443162
$999$		−	2.00000i		−	0.0632772i

Display

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

with

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up to: 50 250 1000 (See only

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.2.c.b.337.2	yes	2
3.2	odd	2		1638.2.c.b.883.1		2
4.3	odd	2		4368.2.h.f.337.2		2
7.6	odd	2		3822.2.c.c.883.2		2
13.5	odd	4		7098.2.a.bc.1.1		1
13.8	odd	4		7098.2.a.k.1.1		1
13.12	even	2	inner	546.2.c.b.337.1	✓	2
39.38	odd	2		1638.2.c.b.883.2		2
52.51	odd	2		4368.2.h.f.337.1		2
91.90	odd	2		3822.2.c.c.883.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.b.337.1	✓	2	13.12	even	2	inner
546.2.c.b.337.2	yes	2	1.1	even	1	trivial
1638.2.c.b.883.1		2	3.2	odd	2
1638.2.c.b.883.2		2	39.38	odd	2
3822.2.c.c.883.1		2	91.90	odd	2
3822.2.c.c.883.2		2	7.6	odd	2
4368.2.h.f.337.1		2	52.51	odd	2
4368.2.h.f.337.2		2	4.3	odd	2
7098.2.a.k.1.1		1	13.8	odd	4
7098.2.a.bc.1.1		1	13.5	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	546.2.c.b.337.2

Level	$546$
Weight	$2$
Character	546.337
Analytic conductor	$4.360$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$