LMFDB - Embedded newform 768.2.c.e.767.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(767,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("768.767");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			767.2
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	768.767
Dual form			768.2.c.e.767.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 1.41421i) q^{3} -2.82843i q^{5} -2.82843i q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})$	$q+(1.00000 + 1.41421i) q^{3} -2.82843i q^{5} -2.82843i q^{7} +(-1.00000 + 2.82843i) q^{9} +2.00000 q^{11} -4.00000 q^{13} +(4.00000 - 2.82843i) q^{15} -5.65685i q^{17} -2.82843i q^{19} +(4.00000 - 2.82843i) q^{21} +8.00000 q^{23} -3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} +2.82843i q^{29} -8.48528i q^{31} +(2.00000 + 2.82843i) q^{33} -8.00000 q^{35} -4.00000 q^{37} +(-4.00000 - 5.65685i) q^{39} -2.82843i q^{43} +(8.00000 + 2.82843i) q^{45} -1.00000 q^{49} +(8.00000 - 5.65685i) q^{51} +8.48528i q^{53} -5.65685i q^{55} +(4.00000 - 2.82843i) q^{57} +6.00000 q^{59} -4.00000 q^{61} +(8.00000 + 2.82843i) q^{63} +11.3137i q^{65} +14.1421i q^{67} +(8.00000 + 11.3137i) q^{69} +8.00000 q^{71} +10.0000 q^{73} +(-3.00000 - 4.24264i) q^{75} -5.65685i q^{77} +2.82843i q^{79} +(-7.00000 - 5.65685i) q^{81} +6.00000 q^{83} -16.0000 q^{85} +(-4.00000 + 2.82843i) q^{87} -5.65685i q^{89} +11.3137i q^{91} +(12.0000 - 8.48528i) q^{93} -8.00000 q^{95} -6.00000 q^{97} +(-2.00000 + 5.65685i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{9} + 4 q^{11} - 8 q^{13} + 8 q^{15} + 8 q^{21} + 16 q^{23} - 6 q^{25} - 10 q^{27} + 4 q^{33} - 16 q^{35} - 8 q^{37} - 8 q^{39} + 16 q^{45} - 2 q^{49} + 16 q^{51} + 8 q^{57} + 12 q^{59} - 8 q^{61} + 16 q^{63} + 16 q^{69} + 16 q^{71} + 20 q^{73} - 6 q^{75} - 14 q^{81} + 12 q^{83} - 32 q^{85} - 8 q^{87} + 24 q^{93} - 16 q^{95} - 12 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^9 + 4 * q^11 - 8 * q^13 + 8 * q^15 + 8 * q^21 + 16 * q^23 - 6 * q^25 - 10 * q^27 + 4 * q^33 - 16 * q^35 - 8 * q^37 - 8 * q^39 + 16 * q^45 - 2 * q^49 + 16 * q^51 + 8 * q^57 + 12 * q^59 - 8 * q^61 + 16 * q^63 + 16 * q^69 + 16 * q^71 + 20 * q^73 - 6 * q^75 - 14 * q^81 + 12 * q^83 - 32 * q^85 - 8 * q^87 + 24 * q^93 - 16 * q^95 - 12 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$3$	1.00000	+	1.41421i	0.577350	+	0.816497i
$4$		0			0
$5$		−	2.82843i		−	1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$		−	2.82843i		−	1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$		0			0
$7$		−	2.82843i		−	1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$		−	2.82843i		−	1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$	4.00000	−	2.82843i	1.03280	−	0.730297i
$16$		0			0
$17$		−	5.65685i		−	1.37199i	−0.727607	−	0.685994i	$-0.759367\pi$
$17$		−	5.65685i		−	1.37199i	0.727607	−	0.685994i	$-0.240633\pi$
$18$		0			0
$19$		−	2.82843i		−	0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$		−	2.82843i		−	0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$		0			0
$21$	4.00000	−	2.82843i	0.872872	−	0.617213i
$22$		0			0
$23$	8.00000			1.66812			0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000			1.66812			0.834058	+	0.551677i	$0.186012\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$	−5.00000	+	1.41421i	−0.962250	+	0.272166i
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		−	8.48528i		−	1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$		−	8.48528i		−	1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$		0			0
$33$	2.00000	+	2.82843i	0.348155	+	0.492366i
$34$		0			0
$35$	−8.00000			−1.35225
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$	−4.00000	−	5.65685i	−0.640513	−	0.905822i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	2.82843i		−	0.431331i	−0.976467	−	0.215666i	$-0.930808\pi$
$43$		−	2.82843i		−	0.431331i	0.976467	−	0.215666i	$-0.0691921\pi$
$44$		0			0
$45$	8.00000	+	2.82843i	1.19257	+	0.421637i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	8.00000	−	5.65685i	1.12022	−	0.792118i
$52$		0			0
$53$			8.48528i			1.16554i	0.812636	+	0.582772i	$0.198032\pi$
$53$			8.48528i			1.16554i	−0.812636	+	0.582772i	$0.801968\pi$
$54$		0			0
$55$		−	5.65685i		−	0.762770i
$56$		0			0
$57$	4.00000	−	2.82843i	0.529813	−	0.374634i
$58$		0			0
$59$	6.00000			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$		0			0
$63$	8.00000	+	2.82843i	1.00791	+	0.356348i
$64$		0			0
$65$			11.3137i			1.40329i
$66$		0			0
$67$			14.1421i			1.72774i	0.503718	+	0.863868i	$0.331965\pi$
$67$			14.1421i			1.72774i	−0.503718	+	0.863868i	$0.668035\pi$
$68$		0			0
$69$	8.00000	+	11.3137i	0.963087	+	1.36201i
$70$		0			0
$71$	8.00000			0.949425			0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000			0.949425			0.474713	+	0.880141i	$0.342552\pi$
$72$		0			0
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	−3.00000	−	4.24264i	−0.346410	−	0.489898i
$76$		0			0
$77$		−	5.65685i		−	0.644658i
$78$		0			0
$79$			2.82843i			0.318223i	0.987261	+	0.159111i	$0.0508629\pi$
$79$			2.82843i			0.318223i	−0.987261	+	0.159111i	$0.949137\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$	−16.0000			−1.73544
$86$		0			0
$87$	−4.00000	+	2.82843i	−0.428845	+	0.303239i
$88$		0			0
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		0			0
$91$			11.3137i			1.18600i
$92$		0			0
$93$	12.0000	−	8.48528i	1.24434	−	0.879883i
$94$		0			0
$95$	−8.00000			−0.820783
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$	−2.00000	+	5.65685i	−0.201008	+	0.568535i
$100$		0			0
$101$		−	2.82843i		−	0.281439i	−0.990050	−	0.140720i	$-0.955058\pi$
$101$		−	2.82843i		−	0.281439i	0.990050	−	0.140720i	$-0.0449416\pi$
$102$		0			0
$103$			8.48528i			0.836080i	0.908429	+	0.418040i	$0.137283\pi$
$103$			8.48528i			0.836080i	−0.908429	+	0.418040i	$0.862717\pi$
$104$		0			0
$105$	−8.00000	−	11.3137i	−0.780720	−	1.10410i
$106$		0			0
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$	−4.00000			−0.383131			−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000			−0.383131			−0.191565	+	0.981480i	$0.561356\pi$
$110$		0			0
$111$	−4.00000	−	5.65685i	−0.379663	−	0.536925i
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		−	22.6274i		−	2.11002i
$116$		0			0
$117$	4.00000	−	11.3137i	0.369800	−	1.04595i
$118$		0			0
$119$	−16.0000			−1.46672
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	5.65685i		−	0.505964i
$126$		0			0
$127$			14.1421i			1.25491i	0.778652	+	0.627456i	$0.215904\pi$
$127$			14.1421i			1.25491i	−0.778652	+	0.627456i	$0.784096\pi$
$128$		0			0
$129$	4.00000	−	2.82843i	0.352180	−	0.249029i
$130$		0			0
$131$	−14.0000			−1.22319			−0.611593	−	0.791173i	$-0.709471\pi$
$131$	−14.0000			−1.22319			−0.611593	+	0.791173i	$0.709471\pi$
$132$		0			0
$133$	−8.00000			−0.693688
$134$		0			0
$135$	4.00000	+	14.1421i	0.344265	+	1.21716i
$136$		0			0
$137$			11.3137i			0.966595i	0.875456	+	0.483298i	$0.160561\pi$
$137$			11.3137i			0.966595i	−0.875456	+	0.483298i	$0.839439\pi$
$138$		0			0
$139$		−	8.48528i		−	0.719712i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$		−	8.48528i		−	0.719712i	0.933008	−	0.359856i	$-0.117174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−8.00000			−0.668994
$144$		0			0
$145$	8.00000			0.664364
$146$		0			0
$147$	−1.00000	−	1.41421i	−0.0824786	−	0.116642i
$148$		0			0
$149$		−	2.82843i		−	0.231714i	−0.993266	−	0.115857i	$-0.963039\pi$
$149$		−	2.82843i		−	0.231714i	0.993266	−	0.115857i	$-0.0369614\pi$
$150$		0			0
$151$			8.48528i			0.690522i	0.938507	+	0.345261i	$0.112210\pi$
$151$			8.48528i			0.690522i	−0.938507	+	0.345261i	$0.887790\pi$
$152$		0			0
$153$	16.0000	+	5.65685i	1.29352	+	0.457330i
$154$		0			0
$155$	−24.0000			−1.92773
$156$		0			0
$157$	−20.0000			−1.59617			−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000			−1.59617			−0.798087	+	0.602542i	$0.794154\pi$
$158$		0			0
$159$	−12.0000	+	8.48528i	−0.951662	+	0.672927i
$160$		0			0
$161$		−	22.6274i		−	1.78329i
$162$		0			0
$163$			8.48528i			0.664619i	0.943170	+	0.332309i	$0.107828\pi$
$163$			8.48528i			0.664619i	−0.943170	+	0.332309i	$0.892172\pi$
$164$		0			0
$165$	8.00000	−	5.65685i	0.622799	−	0.440386i
$166$		0			0
$167$	−8.00000			−0.619059			−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000			−0.619059			−0.309529	+	0.950890i	$0.600171\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	8.00000	+	2.82843i	0.611775	+	0.216295i
$172$		0			0
$173$			2.82843i			0.215041i	0.994203	+	0.107521i	$0.0342912\pi$
$173$			2.82843i			0.215041i	−0.994203	+	0.107521i	$0.965709\pi$
$174$		0			0
$175$			8.48528i			0.641427i
$176$		0			0
$177$	6.00000	+	8.48528i	0.450988	+	0.637793i
$178$		0			0
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$	12.0000			0.891953			0.445976	−	0.895045i	$-0.352856\pi$
$181$	12.0000			0.891953			0.445976	+	0.895045i	$0.352856\pi$
$182$		0			0
$183$	−4.00000	−	5.65685i	−0.295689	−	0.418167i
$184$		0			0
$185$			11.3137i			0.831800i
$186$		0			0
$187$		−	11.3137i		−	0.827340i
$188$		0			0
$189$	4.00000	+	14.1421i	0.290957	+	1.02869i
$190$		0			0
$191$	16.0000			1.15772			0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000			1.15772			0.578860	+	0.815427i	$0.303498\pi$
$192$		0			0
$193$	−2.00000			−0.143963			−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000			−0.143963			−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$	−16.0000	+	11.3137i	−1.14578	+	0.810191i
$196$		0			0
$197$		−	14.1421i		−	1.00759i	−0.863825	−	0.503793i	$-0.831938\pi$
$197$		−	14.1421i		−	1.00759i	0.863825	−	0.503793i	$-0.168062\pi$
$198$		0			0
$199$			8.48528i			0.601506i	0.953702	+	0.300753i	$0.0972379\pi$
$199$			8.48528i			0.601506i	−0.953702	+	0.300753i	$0.902762\pi$
$200$		0			0
$201$	−20.0000	+	14.1421i	−1.41069	+	0.997509i
$202$		0			0
$203$	8.00000			0.561490
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−8.00000	+	22.6274i	−0.556038	+	1.57271i
$208$		0			0
$209$		−	5.65685i		−	0.391293i
$210$		0			0
$211$			2.82843i			0.194717i	0.995249	+	0.0973585i	$0.0310393\pi$
$211$			2.82843i			0.194717i	−0.995249	+	0.0973585i	$0.968961\pi$
$212$		0			0
$213$	8.00000	+	11.3137i	0.548151	+	0.775203i
$214$		0			0
$215$	−8.00000			−0.545595
$216$		0			0
$217$	−24.0000			−1.62923
$218$		0			0
$219$	10.0000	+	14.1421i	0.675737	+	0.955637i
$220$		0			0
$221$			22.6274i			1.52208i
$222$		0			0
$223$			2.82843i			0.189405i	0.995506	+	0.0947027i	$0.0301901\pi$
$223$			2.82843i			0.189405i	−0.995506	+	0.0947027i	$0.969810\pi$
$224$		0			0
$225$	3.00000	−	8.48528i	0.200000	−	0.565685i
$226$		0			0
$227$	22.0000			1.46019			0.730096	−	0.683345i	$-0.239475\pi$
$227$	22.0000			1.46019			0.730096	+	0.683345i	$0.239475\pi$
$228$		0			0
$229$	−4.00000			−0.264327			−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000			−0.264327			−0.132164	+	0.991228i	$0.542192\pi$
$230$		0			0
$231$	8.00000	−	5.65685i	0.526361	−	0.372194i
$232$		0			0
$233$		−	28.2843i		−	1.85296i	−0.376339	−	0.926482i	$-0.622817\pi$
$233$		−	28.2843i		−	1.85296i	0.376339	−	0.926482i	$-0.377183\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−4.00000	+	2.82843i	−0.259828	+	0.183726i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	14.0000			0.901819			0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000			0.901819			0.450910	+	0.892570i	$0.351100\pi$
$242$		0			0
$243$	1.00000	−	15.5563i	0.0641500	−	0.997940i
$244$		0			0
$245$			2.82843i			0.180702i
$246$		0			0
$247$			11.3137i			0.719874i
$248$		0			0
$249$	6.00000	+	8.48528i	0.380235	+	0.537733i
$250$		0			0
$251$	2.00000			0.126239			0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000			0.126239			0.0631194	+	0.998006i	$0.479895\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$	−16.0000	−	22.6274i	−1.00196	−	1.41698i
$256$		0			0
$257$		−	11.3137i		−	0.705730i	−0.935674	−	0.352865i	$-0.885208\pi$
$257$		−	11.3137i		−	0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$		0			0
$259$			11.3137i			0.703000i
$260$		0			0
$261$	−8.00000	−	2.82843i	−0.495188	−	0.175075i
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$	24.0000			1.47431
$266$		0			0
$267$	8.00000	−	5.65685i	0.489592	−	0.346194i
$268$		0			0
$269$			14.1421i			0.862261i	0.902290	+	0.431131i	$0.141885\pi$
$269$			14.1421i			0.862261i	−0.902290	+	0.431131i	$0.858115\pi$
$270$		0			0
$271$		−	19.7990i		−	1.20270i	−0.798985	−	0.601351i	$-0.794629\pi$
$271$		−	19.7990i		−	1.20270i	0.798985	−	0.601351i	$-0.205371\pi$
$272$		0			0
$273$	−16.0000	+	11.3137i	−0.968364	+	0.684737i
$274$		0			0
$275$	−6.00000			−0.361814
$276$		0			0
$277$	12.0000			0.721010			0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000			0.721010			0.360505	+	0.932757i	$0.382604\pi$
$278$		0			0
$279$	24.0000	+	8.48528i	1.43684	+	0.508001i
$280$		0			0
$281$			5.65685i			0.337460i	0.985662	+	0.168730i	$0.0539665\pi$
$281$			5.65685i			0.337460i	−0.985662	+	0.168730i	$0.946033\pi$
$282$		0			0
$283$			25.4558i			1.51319i	0.653882	+	0.756596i	$0.273139\pi$
$283$			25.4558i			1.51319i	−0.653882	+	0.756596i	$0.726861\pi$
$284$		0			0
$285$	−8.00000	−	11.3137i	−0.473879	−	0.670166i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$	−6.00000	−	8.48528i	−0.351726	−	0.497416i
$292$		0			0
$293$			31.1127i			1.81762i	0.417207	+	0.908812i	$0.363009\pi$
$293$			31.1127i			1.81762i	−0.417207	+	0.908812i	$0.636991\pi$
$294$		0			0
$295$		−	16.9706i		−	0.988064i
$296$		0			0
$297$	−10.0000	+	2.82843i	−0.580259	+	0.164122i
$298$		0			0
$299$	−32.0000			−1.85061
$300$		0			0
$301$	−8.00000			−0.461112
$302$		0			0
$303$	4.00000	−	2.82843i	0.229794	−	0.162489i
$304$		0			0
$305$			11.3137i			0.647821i
$306$		0			0
$307$		−	8.48528i		−	0.484281i	−0.970241	−	0.242140i	$-0.922151\pi$
$307$		−	8.48528i		−	0.484281i	0.970241	−	0.242140i	$-0.0778494\pi$
$308$		0			0
$309$	−12.0000	+	8.48528i	−0.682656	+	0.482711i
$310$		0			0
$311$	24.0000			1.36092			0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000			1.36092			0.680458	+	0.732787i	$0.261781\pi$
$312$		0			0
$313$	2.00000			0.113047			0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000			0.113047			0.0565233	+	0.998401i	$0.481998\pi$
$314$		0			0
$315$	8.00000	−	22.6274i	0.450749	−	1.27491i
$316$		0			0
$317$		−	31.1127i		−	1.74746i	−0.486408	−	0.873732i	$-0.661693\pi$
$317$		−	31.1127i		−	1.74746i	0.486408	−	0.873732i	$-0.338307\pi$
$318$		0			0
$319$			5.65685i			0.316723i
$320$		0			0
$321$	6.00000	+	8.48528i	0.334887	+	0.473602i
$322$		0			0
$323$	−16.0000			−0.890264
$324$		0			0
$325$	12.0000			0.665640
$326$		0			0
$327$	−4.00000	−	5.65685i	−0.221201	−	0.312825i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			2.82843i			0.155464i	0.996974	+	0.0777322i	$0.0247679\pi$
$331$			2.82843i			0.155464i	−0.996974	+	0.0777322i	$0.975232\pi$
$332$		0			0
$333$	4.00000	−	11.3137i	0.219199	−	0.619987i
$334$		0			0
$335$	40.0000			2.18543
$336$		0			0
$337$	6.00000			0.326841			0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000			0.326841			0.163420	+	0.986557i	$0.447747\pi$
$338$		0			0
$339$	−16.0000	+	11.3137i	−0.869001	+	0.614476i
$340$		0			0
$341$		−	16.9706i		−	0.919007i
$342$		0			0
$343$		−	16.9706i		−	0.916324i
$344$		0			0
$345$	32.0000	−	22.6274i	1.72282	−	1.21822i
$346$		0			0
$347$	2.00000			0.107366			0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000			0.107366			0.0536828	+	0.998558i	$0.482904\pi$
$348$		0			0
$349$	28.0000			1.49881			0.749403	−	0.662114i	$-0.230341\pi$
$349$	28.0000			1.49881			0.749403	+	0.662114i	$0.230341\pi$
$350$		0			0
$351$	20.0000	−	5.65685i	1.06752	−	0.301941i
$352$		0			0
$353$			22.6274i			1.20434i	0.798369	+	0.602168i	$0.205696\pi$
$353$			22.6274i			1.20434i	−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		−	22.6274i		−	1.20094i
$356$		0			0
$357$	−16.0000	−	22.6274i	−0.846810	−	1.19757i
$358$		0			0
$359$	8.00000			0.422224			0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000			0.422224			0.211112	+	0.977462i	$0.432292\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$	−7.00000	−	9.89949i	−0.367405	−	0.519589i
$364$		0			0
$365$		−	28.2843i		−	1.48047i
$366$		0			0
$367$		−	8.48528i		−	0.442928i	−0.975169	−	0.221464i	$-0.928916\pi$
$367$		−	8.48528i		−	0.442928i	0.975169	−	0.221464i	$-0.0710835\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	24.0000			1.24602
$372$		0			0
$373$	−36.0000			−1.86401			−0.932005	−	0.362446i	$-0.881942\pi$
$373$	−36.0000			−1.86401			−0.932005	+	0.362446i	$0.881942\pi$
$374$		0			0
$375$	8.00000	−	5.65685i	0.413118	−	0.292119i
$376$		0			0
$377$		−	11.3137i		−	0.582686i
$378$		0			0
$379$		−	25.4558i		−	1.30758i	−0.756677	−	0.653789i	$-0.773178\pi$
$379$		−	25.4558i		−	1.30758i	0.756677	−	0.653789i	$-0.226822\pi$
$380$		0			0
$381$	−20.0000	+	14.1421i	−1.02463	+	0.724524i
$382$		0			0
$383$	−16.0000			−0.817562			−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000			−0.817562			−0.408781	+	0.912633i	$0.634046\pi$
$384$		0			0
$385$	−16.0000			−0.815436
$386$		0			0
$387$	8.00000	+	2.82843i	0.406663	+	0.143777i
$388$		0			0
$389$			8.48528i			0.430221i	0.976590	+	0.215110i	$0.0690111\pi$
$389$			8.48528i			0.430221i	−0.976590	+	0.215110i	$0.930989\pi$
$390$		0			0
$391$		−	45.2548i		−	2.28864i
$392$		0			0
$393$	−14.0000	−	19.7990i	−0.706207	−	0.998727i
$394$		0			0
$395$	8.00000			0.402524
$396$		0			0
$397$	28.0000			1.40528			0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000			1.40528			0.702640	+	0.711546i	$0.252005\pi$
$398$		0			0
$399$	−8.00000	−	11.3137i	−0.400501	−	0.566394i
$400$		0			0
$401$			5.65685i			0.282490i	0.989975	+	0.141245i	$0.0451105\pi$
$401$			5.65685i			0.282490i	−0.989975	+	0.141245i	$0.954889\pi$
$402$		0			0
$403$			33.9411i			1.69073i
$404$		0			0
$405$	−16.0000	+	19.7990i	−0.795046	+	0.983820i
$406$		0			0
$407$	−8.00000			−0.396545
$408$		0			0
$409$	−26.0000			−1.28562			−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000			−1.28562			−0.642809	+	0.766027i	$0.722231\pi$
$410$		0			0
$411$	−16.0000	+	11.3137i	−0.789222	+	0.558064i
$412$		0			0
$413$		−	16.9706i		−	0.835067i
$414$		0			0
$415$		−	16.9706i		−	0.833052i
$416$		0			0
$417$	12.0000	−	8.48528i	0.587643	−	0.415526i
$418$		0			0
$419$	6.00000			0.293119			0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000			0.293119			0.146560	+	0.989202i	$0.453180\pi$
$420$		0			0
$421$	28.0000			1.36464			0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000			1.36464			0.682318	+	0.731055i	$0.260972\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			16.9706i			0.823193i
$426$		0			0
$427$			11.3137i			0.547509i
$428$		0			0
$429$	−8.00000	−	11.3137i	−0.386244	−	0.546231i
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$		0			0
$435$	8.00000	+	11.3137i	0.383571	+	0.542451i
$436$		0			0
$437$		−	22.6274i		−	1.08242i
$438$		0			0
$439$			8.48528i			0.404980i	0.979284	+	0.202490i	$0.0649034\pi$
$439$			8.48528i			0.404980i	−0.979284	+	0.202490i	$0.935097\pi$
$440$		0			0
$441$	1.00000	−	2.82843i	0.0476190	−	0.134687i
$442$		0			0
$443$	18.0000			0.855206			0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000			0.855206			0.427603	+	0.903967i	$0.359358\pi$
$444$		0			0
$445$	−16.0000			−0.758473
$446$		0			0
$447$	4.00000	−	2.82843i	0.189194	−	0.133780i
$448$		0			0
$449$			16.9706i			0.800890i	0.916321	+	0.400445i	$0.131145\pi$
$449$			16.9706i			0.800890i	−0.916321	+	0.400445i	$0.868855\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−12.0000	+	8.48528i	−0.563809	+	0.398673i
$454$		0			0
$455$	32.0000			1.50018
$456$		0			0
$457$	22.0000			1.02912			0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000			1.02912			0.514558	+	0.857455i	$0.327956\pi$
$458$		0			0
$459$	8.00000	+	28.2843i	0.373408	+	1.32020i
$460$		0			0
$461$			25.4558i			1.18560i	0.805351	+	0.592798i	$0.201977\pi$
$461$			25.4558i			1.18560i	−0.805351	+	0.592798i	$0.798023\pi$
$462$		0			0
$463$		−	8.48528i		−	0.394344i	−0.980369	−	0.197172i	$-0.936824\pi$
$463$		−	8.48528i		−	0.394344i	0.980369	−	0.197172i	$-0.0631758\pi$
$464$		0			0
$465$	−24.0000	−	33.9411i	−1.11297	−	1.57398i
$466$		0			0
$467$	−42.0000			−1.94353			−0.971764	−	0.235954i	$-0.924178\pi$
$467$	−42.0000			−1.94353			−0.971764	+	0.235954i	$0.924178\pi$
$468$		0			0
$469$	40.0000			1.84703
$470$		0			0
$471$	−20.0000	−	28.2843i	−0.921551	−	1.30327i
$472$		0			0
$473$		−	5.65685i		−	0.260102i
$474$		0			0
$475$			8.48528i			0.389331i
$476$		0			0
$477$	−24.0000	−	8.48528i	−1.09888	−	0.388514i
$478$		0			0
$479$	32.0000			1.46212			0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000			1.46212			0.731059	+	0.682315i	$0.239027\pi$
$480$		0			0
$481$	16.0000			0.729537
$482$		0			0
$483$	32.0000	−	22.6274i	1.45605	−	1.02958i
$484$		0			0
$485$			16.9706i			0.770594i
$486$		0			0
$487$			42.4264i			1.92252i	0.275636	+	0.961262i	$0.411111\pi$
$487$			42.4264i			1.92252i	−0.275636	+	0.961262i	$0.588889\pi$
$488$		0			0
$489$	−12.0000	+	8.48528i	−0.542659	+	0.383718i
$490$		0			0
$491$	22.0000			0.992846			0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000			0.992846			0.496423	+	0.868081i	$0.334646\pi$
$492$		0			0
$493$	16.0000			0.720604
$494$		0			0
$495$	16.0000	+	5.65685i	0.719147	+	0.254257i
$496$		0			0
$497$		−	22.6274i		−	1.01498i
$498$		0			0
$499$		−	19.7990i		−	0.886325i	−0.896441	−	0.443162i	$-0.853857\pi$
$499$		−	19.7990i		−	0.886325i	0.896441	−	0.443162i	$-0.146143\pi$
$500$		0			0
$501$	−8.00000	−	11.3137i	−0.357414	−	0.505459i
$502$		0			0
$503$	24.0000			1.07011			0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$	3.00000	+	4.24264i	0.133235	+	0.188422i
$508$		0			0
$509$			25.4558i			1.12831i	0.825669	+	0.564155i	$0.190798\pi$
$509$			25.4558i			1.12831i	−0.825669	+	0.564155i	$0.809202\pi$
$510$		0			0
$511$		−	28.2843i		−	1.25122i
$512$		0			0
$513$	4.00000	+	14.1421i	0.176604	+	0.624391i
$514$		0			0
$515$	24.0000			1.05757
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−4.00000	+	2.82843i	−0.175581	+	0.124154i
$520$		0			0
$521$			33.9411i			1.48699i	0.668743	+	0.743494i	$0.266833\pi$
$521$			33.9411i			1.48699i	−0.668743	+	0.743494i	$0.733167\pi$
$522$		0			0
$523$		−	14.1421i		−	0.618392i	−0.950998	−	0.309196i	$-0.899940\pi$
$523$		−	14.1421i		−	0.618392i	0.950998	−	0.309196i	$-0.100060\pi$
$524$		0			0
$525$	−12.0000	+	8.48528i	−0.523723	+	0.370328i
$526$		0			0
$527$	−48.0000			−2.09091
$528$		0			0
$529$	41.0000			1.78261
$530$		0			0
$531$	−6.00000	+	16.9706i	−0.260378	+	0.736460i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	16.9706i		−	0.733701i
$536$		0			0
$537$	18.0000	+	25.4558i	0.776757	+	1.09850i
$538$		0			0
$539$	−2.00000			−0.0861461
$540$		0			0
$541$	28.0000			1.20381			0.601907	−	0.798566i	$-0.294408\pi$
$541$	28.0000			1.20381			0.601907	+	0.798566i	$0.294408\pi$
$542$		0			0
$543$	12.0000	+	16.9706i	0.514969	+	0.728277i
$544$		0			0
$545$			11.3137i			0.484626i
$546$		0			0
$547$		−	25.4558i		−	1.08841i	−0.838951	−	0.544207i	$-0.816831\pi$
$547$		−	25.4558i		−	1.08841i	0.838951	−	0.544207i	$-0.183169\pi$
$548$		0			0
$549$	4.00000	−	11.3137i	0.170716	−	0.482857i
$550$		0			0
$551$	8.00000			0.340811
$552$		0			0
$553$	8.00000			0.340195
$554$		0			0
$555$	−16.0000	+	11.3137i	−0.679162	+	0.480240i
$556$		0			0
$557$			36.7696i			1.55798i	0.627039	+	0.778988i	$0.284267\pi$
$557$			36.7696i			1.55798i	−0.627039	+	0.778988i	$0.715733\pi$
$558$		0			0
$559$			11.3137i			0.478519i
$560$		0			0
$561$	16.0000	−	11.3137i	0.675521	−	0.477665i
$562$		0			0
$563$	−42.0000			−1.77009			−0.885044	−	0.465506i	$-0.845872\pi$
$563$	−42.0000			−1.77009			−0.885044	+	0.465506i	$0.845872\pi$
$564$		0			0
$565$	32.0000			1.34625
$566$		0			0
$567$	−16.0000	+	19.7990i	−0.671937	+	0.831479i
$568$		0			0
$569$		−	11.3137i		−	0.474295i	−0.971474	−	0.237148i	$-0.923787\pi$
$569$		−	11.3137i		−	0.474295i	0.971474	−	0.237148i	$-0.0762125\pi$
$570$		0			0
$571$		−	42.4264i		−	1.77549i	−0.460336	−	0.887745i	$-0.652271\pi$
$571$		−	42.4264i		−	1.77549i	0.460336	−	0.887745i	$-0.347729\pi$
$572$		0			0
$573$	16.0000	+	22.6274i	0.668410	+	0.945274i
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$	22.0000			0.915872			0.457936	−	0.888985i	$-0.348589\pi$
$577$	22.0000			0.915872			0.457936	+	0.888985i	$0.348589\pi$
$578$		0			0
$579$	−2.00000	−	2.82843i	−0.0831172	−	0.117545i
$580$		0			0
$581$		−	16.9706i		−	0.704058i
$582$		0			0
$583$			16.9706i			0.702849i
$584$		0			0
$585$	−32.0000	−	11.3137i	−1.32304	−	0.467764i
$586$		0			0
$587$	−26.0000			−1.07313			−0.536567	−	0.843857i	$-0.680279\pi$
$587$	−26.0000			−1.07313			−0.536567	+	0.843857i	$0.680279\pi$
$588$		0			0
$589$	−24.0000			−0.988903
$590$		0			0
$591$	20.0000	−	14.1421i	0.822690	−	0.581730i
$592$		0			0
$593$		−	22.6274i		−	0.929197i	−0.885522	−	0.464598i	$-0.846199\pi$
$593$		−	22.6274i		−	0.929197i	0.885522	−	0.464598i	$-0.153801\pi$
$594$		0			0
$595$			45.2548i			1.85527i
$596$		0			0
$597$	−12.0000	+	8.48528i	−0.491127	+	0.347279i
$598$		0			0
$599$	−24.0000			−0.980613			−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000			−0.980613			−0.490307	+	0.871550i	$0.663115\pi$
$600$		0			0
$601$	−38.0000			−1.55005			−0.775026	−	0.631929i	$-0.782263\pi$
$601$	−38.0000			−1.55005			−0.775026	+	0.631929i	$0.782263\pi$
$602$		0			0
$603$	−40.0000	−	14.1421i	−1.62893	−	0.575912i
$604$		0			0
$605$			19.7990i			0.804943i
$606$		0			0
$607$			36.7696i			1.49243i	0.665705	+	0.746215i	$0.268131\pi$
$607$			36.7696i			1.49243i	−0.665705	+	0.746215i	$0.731869\pi$
$608$		0			0
$609$	8.00000	+	11.3137i	0.324176	+	0.458455i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	12.0000			0.484675			0.242338	−	0.970192i	$-0.422086\pi$
$613$	12.0000			0.484675			0.242338	+	0.970192i	$0.422086\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			5.65685i			0.227736i	0.993496	+	0.113868i	$0.0363242\pi$
$617$			5.65685i			0.227736i	−0.993496	+	0.113868i	$0.963676\pi$
$618$		0			0
$619$			25.4558i			1.02316i	0.859237	+	0.511578i	$0.170939\pi$
$619$			25.4558i			1.02316i	−0.859237	+	0.511578i	$0.829061\pi$
$620$		0			0
$621$	−40.0000	+	11.3137i	−1.60514	+	0.454003i
$622$		0			0
$623$	−16.0000			−0.641026
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	8.00000	−	5.65685i	0.319489	−	0.225913i
$628$		0			0
$629$			22.6274i			0.902214i
$630$		0			0
$631$		−	25.4558i		−	1.01338i	−0.862128	−	0.506691i	$-0.830869\pi$
$631$		−	25.4558i		−	1.01338i	0.862128	−	0.506691i	$-0.169131\pi$
$632$		0			0
$633$	−4.00000	+	2.82843i	−0.158986	+	0.112420i
$634$		0			0
$635$	40.0000			1.58735
$636$		0			0
$637$	4.00000			0.158486
$638$		0			0
$639$	−8.00000	+	22.6274i	−0.316475	+	0.895127i
$640$		0			0
$641$		−	39.5980i		−	1.56403i	−0.623262	−	0.782013i	$-0.714193\pi$
$641$		−	39.5980i		−	1.56403i	0.623262	−	0.782013i	$-0.285807\pi$
$642$		0			0
$643$		−	25.4558i		−	1.00388i	−0.864902	−	0.501940i	$-0.832620\pi$
$643$		−	25.4558i		−	1.00388i	0.864902	−	0.501940i	$-0.167380\pi$
$644$		0			0
$645$	−8.00000	−	11.3137i	−0.315000	−	0.445477i
$646$		0			0
$647$	−8.00000			−0.314512			−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000			−0.314512			−0.157256	+	0.987558i	$0.550265\pi$
$648$		0			0
$649$	12.0000			0.471041
$650$		0			0
$651$	−24.0000	−	33.9411i	−0.940634	−	1.33026i
$652$		0			0
$653$		−	42.4264i		−	1.66027i	−0.557560	−	0.830137i	$-0.688262\pi$
$653$		−	42.4264i		−	1.66027i	0.557560	−	0.830137i	$-0.311738\pi$
$654$		0			0
$655$			39.5980i			1.54722i
$656$		0			0
$657$	−10.0000	+	28.2843i	−0.390137	+	1.10347i
$658$		0			0
$659$	−14.0000			−0.545363			−0.272681	−	0.962104i	$-0.587910\pi$
$659$	−14.0000			−0.545363			−0.272681	+	0.962104i	$0.587910\pi$
$660$		0			0
$661$	−4.00000			−0.155582			−0.0777910	−	0.996970i	$-0.524787\pi$
$661$	−4.00000			−0.155582			−0.0777910	+	0.996970i	$0.524787\pi$
$662$		0			0
$663$	−32.0000	+	22.6274i	−1.24278	+	0.878776i
$664$		0			0
$665$			22.6274i			0.877454i
$666$		0			0
$667$			22.6274i			0.876137i
$668$		0			0
$669$	−4.00000	+	2.82843i	−0.154649	+	0.109353i
$670$		0			0
$671$	−8.00000			−0.308837
$672$		0			0
$673$	−38.0000			−1.46479			−0.732396	−	0.680879i	$-0.761598\pi$
$673$	−38.0000			−1.46479			−0.732396	+	0.680879i	$0.761598\pi$
$674$		0			0
$675$	15.0000	−	4.24264i	0.577350	−	0.163299i
$676$		0			0
$677$		−	25.4558i		−	0.978348i	−0.872186	−	0.489174i	$-0.837298\pi$
$677$		−	25.4558i		−	0.978348i	0.872186	−	0.489174i	$-0.162702\pi$
$678$		0			0
$679$			16.9706i			0.651270i
$680$		0			0
$681$	22.0000	+	31.1127i	0.843042	+	1.19224i
$682$		0			0
$683$	18.0000			0.688751			0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000			0.688751			0.344375	+	0.938832i	$0.388091\pi$
$684$		0			0
$685$	32.0000			1.22266
$686$		0			0
$687$	−4.00000	−	5.65685i	−0.152610	−	0.215822i
$688$		0			0
$689$		−	33.9411i		−	1.29305i
$690$		0			0
$691$			31.1127i			1.18358i	0.806091	+	0.591791i	$0.201579\pi$
$691$			31.1127i			1.18358i	−0.806091	+	0.591791i	$0.798421\pi$
$692$		0			0
$693$	16.0000	+	5.65685i	0.607790	+	0.214886i
$694$		0			0
$695$	−24.0000			−0.910372
$696$		0			0
$697$		0			0
$698$		0			0
$699$	40.0000	−	28.2843i	1.51294	−	1.06981i
$700$		0			0
$701$		−	31.1127i		−	1.17511i	−0.809184	−	0.587555i	$-0.800091\pi$
$701$		−	31.1127i		−	1.17511i	0.809184	−	0.587555i	$-0.199909\pi$
$702$		0			0
$703$			11.3137i			0.426705i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−8.00000			−0.300871
$708$		0			0
$709$	−4.00000			−0.150223			−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000			−0.150223			−0.0751116	+	0.997175i	$0.523931\pi$
$710$		0			0
$711$	−8.00000	−	2.82843i	−0.300023	−	0.106074i
$712$		0			0
$713$		−	67.8823i		−	2.54221i
$714$		0			0
$715$			22.6274i			0.846217i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.0000			−0.596699			−0.298350	−	0.954457i	$-0.596436\pi$
$719$	−16.0000			−0.596699			−0.298350	+	0.954457i	$0.596436\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$		0			0
$723$	14.0000	+	19.7990i	0.520666	+	0.736332i
$724$		0			0
$725$		−	8.48528i		−	0.315135i
$726$		0			0
$727$			31.1127i			1.15391i	0.816777	+	0.576953i	$0.195758\pi$
$727$			31.1127i			1.15391i	−0.816777	+	0.576953i	$0.804242\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$	−16.0000			−0.591781
$732$		0			0
$733$	−36.0000			−1.32969			−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000			−1.32969			−0.664845	+	0.746981i	$0.731502\pi$
$734$		0			0
$735$	−4.00000	+	2.82843i	−0.147542	+	0.104328i
$736$		0			0
$737$			28.2843i			1.04186i
$738$		0			0
$739$		−	42.4264i		−	1.56068i	−0.625355	−	0.780340i	$-0.715046\pi$
$739$		−	42.4264i		−	1.56068i	0.625355	−	0.780340i	$-0.284954\pi$
$740$		0			0
$741$	−16.0000	+	11.3137i	−0.587775	+	0.415619i
$742$		0			0
$743$	24.0000			0.880475			0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000			0.880475			0.440237	+	0.897881i	$0.354894\pi$
$744$		0			0
$745$	−8.00000			−0.293097
$746$		0			0
$747$	−6.00000	+	16.9706i	−0.219529	+	0.620920i
$748$		0			0
$749$		−	16.9706i		−	0.620091i
$750$		0			0
$751$			2.82843i			0.103211i	0.998668	+	0.0516054i	$0.0164338\pi$
$751$			2.82843i			0.103211i	−0.998668	+	0.0516054i	$0.983566\pi$
$752$		0			0
$753$	2.00000	+	2.82843i	0.0728841	+	0.103074i
$754$		0			0
$755$	24.0000			0.873449
$756$		0			0
$757$	12.0000			0.436147			0.218074	−	0.975932i	$-0.430023\pi$
$757$	12.0000			0.436147			0.218074	+	0.975932i	$0.430023\pi$
$758$		0			0
$759$	16.0000	+	22.6274i	0.580763	+	0.821323i
$760$		0			0
$761$		−	22.6274i		−	0.820243i	−0.912031	−	0.410122i	$-0.865486\pi$
$761$		−	22.6274i		−	0.820243i	0.912031	−	0.410122i	$-0.134514\pi$
$762$		0			0
$763$			11.3137i			0.409584i
$764$		0			0
$765$	16.0000	−	45.2548i	0.578481	−	1.63619i
$766$		0			0
$767$	−24.0000			−0.866590
$768$		0			0
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$	16.0000	−	11.3137i	0.576226	−	0.407453i
$772$		0			0
$773$		−	2.82843i		−	0.101731i	−0.998706	−	0.0508657i	$-0.983802\pi$
$773$		−	2.82843i		−	0.101731i	0.998706	−	0.0508657i	$-0.0161981\pi$
$774$		0			0
$775$			25.4558i			0.914401i
$776$		0			0
$777$	−16.0000	+	11.3137i	−0.573997	+	0.405877i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	16.0000			0.572525
$782$		0			0
$783$	−4.00000	−	14.1421i	−0.142948	−	0.505399i
$784$		0			0
$785$			56.5685i			2.01902i
$786$		0			0
$787$			31.1127i			1.10905i	0.832168	+	0.554524i	$0.187100\pi$
$787$			31.1127i			1.10905i	−0.832168	+	0.554524i	$0.812900\pi$
$788$		0			0
$789$	−24.0000	−	33.9411i	−0.854423	−	1.20834i
$790$		0			0
$791$	32.0000			1.13779
$792$		0			0
$793$	16.0000			0.568177
$794$		0			0
$795$	24.0000	+	33.9411i	0.851192	+	1.20377i
$796$		0			0
$797$			14.1421i			0.500940i	0.968124	+	0.250470i	$0.0805852\pi$
$797$			14.1421i			0.500940i	−0.968124	+	0.250470i	$0.919415\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	16.0000	+	5.65685i	0.565332	+	0.199875i
$802$		0			0
$803$	20.0000			0.705785
$804$		0			0
$805$	−64.0000			−2.25570
$806$		0			0
$807$	−20.0000	+	14.1421i	−0.704033	+	0.497827i
$808$		0			0
$809$			22.6274i			0.795538i	0.917486	+	0.397769i	$0.130215\pi$
$809$			22.6274i			0.795538i	−0.917486	+	0.397769i	$0.869785\pi$
$810$		0			0
$811$			8.48528i			0.297959i	0.988840	+	0.148979i	$0.0475988\pi$
$811$			8.48528i			0.297959i	−0.988840	+	0.148979i	$0.952401\pi$
$812$		0			0
$813$	28.0000	−	19.7990i	0.982003	−	0.694381i
$814$		0			0
$815$	24.0000			0.840683
$816$		0			0
$817$	−8.00000			−0.279885
$818$		0			0
$819$	−32.0000	−	11.3137i	−1.11817	−	0.395333i
$820$		0			0
$821$			19.7990i			0.690990i	0.938421	+	0.345495i	$0.112289\pi$
$821$			19.7990i			0.690990i	−0.938421	+	0.345495i	$0.887711\pi$
$822$		0			0
$823$			42.4264i			1.47889i	0.673216	+	0.739446i	$0.264912\pi$
$823$			42.4264i			1.47889i	−0.673216	+	0.739446i	$0.735088\pi$
$824$		0			0
$825$	−6.00000	−	8.48528i	−0.208893	−	0.295420i
$826$		0			0
$827$	22.0000			0.765015			0.382507	−	0.923952i	$-0.375061\pi$
$827$	22.0000			0.765015			0.382507	+	0.923952i	$0.375061\pi$
$828$		0			0
$829$	12.0000			0.416777			0.208389	−	0.978046i	$-0.433178\pi$
$829$	12.0000			0.416777			0.208389	+	0.978046i	$0.433178\pi$
$830$		0			0
$831$	12.0000	+	16.9706i	0.416275	+	0.588702i
$832$		0			0
$833$			5.65685i			0.195998i
$834$		0			0
$835$			22.6274i			0.783054i
$836$		0			0
$837$	12.0000	+	42.4264i	0.414781	+	1.46647i
$838$		0			0
$839$	−56.0000			−1.93333			−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000			−1.93333			−0.966667	+	0.256036i	$0.917584\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	−8.00000	+	5.65685i	−0.275535	+	0.194832i
$844$		0			0
$845$		−	8.48528i		−	0.291903i
$846$		0			0
$847$			19.7990i			0.680301i
$848$		0			0
$849$	−36.0000	+	25.4558i	−1.23552	+	0.873642i
$850$		0			0
$851$	−32.0000			−1.09695
$852$		0			0
$853$	−36.0000			−1.23262			−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000			−1.23262			−0.616308	+	0.787505i	$0.711372\pi$
$854$		0			0
$855$	8.00000	−	22.6274i	0.273594	−	0.773841i
$856$		0			0
$857$		−	11.3137i		−	0.386469i	−0.981153	−	0.193234i	$-0.938102\pi$
$857$		−	11.3137i		−	0.386469i	0.981153	−	0.193234i	$-0.0618978\pi$
$858$		0			0
$859$		−	25.4558i		−	0.868542i	−0.900782	−	0.434271i	$-0.857006\pi$
$859$		−	25.4558i		−	0.868542i	0.900782	−	0.434271i	$-0.142994\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−16.0000			−0.544646			−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000			−0.544646			−0.272323	+	0.962206i	$0.587792\pi$
$864$		0			0
$865$	8.00000			0.272008
$866$		0			0
$867$	−15.0000	−	21.2132i	−0.509427	−	0.720438i
$868$		0			0
$869$			5.65685i			0.191896i
$870$		0			0
$871$		−	56.5685i		−	1.91675i
$872$		0			0
$873$	6.00000	−	16.9706i	0.203069	−	0.574367i
$874$		0			0
$875$	−16.0000			−0.540899
$876$		0			0
$877$	−20.0000			−0.675352			−0.337676	−	0.941262i	$-0.609641\pi$
$877$	−20.0000			−0.675352			−0.337676	+	0.941262i	$0.609641\pi$
$878$		0			0
$879$	−44.0000	+	31.1127i	−1.48408	+	1.04941i
$880$		0			0
$881$			33.9411i			1.14351i	0.820426	+	0.571753i	$0.193736\pi$
$881$			33.9411i			1.14351i	−0.820426	+	0.571753i	$0.806264\pi$
$882$		0			0
$883$		−	25.4558i		−	0.856657i	−0.903623	−	0.428329i	$-0.859103\pi$
$883$		−	25.4558i		−	0.856657i	0.903623	−	0.428329i	$-0.140897\pi$
$884$		0			0
$885$	24.0000	−	16.9706i	0.806751	−	0.570459i
$886$		0			0
$887$	−8.00000			−0.268614			−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000			−0.268614			−0.134307	+	0.990940i	$0.542881\pi$
$888$		0			0
$889$	40.0000			1.34156
$890$		0			0
$891$	−14.0000	−	11.3137i	−0.469018	−	0.379023i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	50.9117i		−	1.70179i
$896$		0			0
$897$	−32.0000	−	45.2548i	−1.06845	−	1.51101i
$898$		0			0
$899$	24.0000			0.800445
$900$		0			0
$901$	48.0000			1.59911
$902$		0			0
$903$	−8.00000	−	11.3137i	−0.266223	−	0.376497i
$904$		0			0
$905$		−	33.9411i		−	1.12824i
$906$		0			0
$907$		−	25.4558i		−	0.845247i	−0.906305	−	0.422624i	$-0.861109\pi$
$907$		−	25.4558i		−	0.845247i	0.906305	−	0.422624i	$-0.138891\pi$
$908$		0			0
$909$	8.00000	+	2.82843i	0.265343	+	0.0938130i
$910$		0			0
$911$	32.0000			1.06021			0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000			1.06021			0.530104	+	0.847933i	$0.322153\pi$
$912$		0			0
$913$	12.0000			0.397142
$914$		0			0
$915$	−16.0000	+	11.3137i	−0.528944	+	0.374020i
$916$		0			0
$917$			39.5980i			1.30764i
$918$		0			0
$919$		−	2.82843i		−	0.0933012i	−0.998911	−	0.0466506i	$-0.985145\pi$
$919$		−	2.82843i		−	0.0933012i	0.998911	−	0.0466506i	$-0.0148547\pi$
$920$		0			0
$921$	12.0000	−	8.48528i	0.395413	−	0.279600i
$922$		0			0
$923$	−32.0000			−1.05329
$924$		0			0
$925$	12.0000			0.394558
$926$		0			0
$927$	−24.0000	−	8.48528i	−0.788263	−	0.278693i
$928$		0			0
$929$		−	39.5980i		−	1.29917i	−0.760290	−	0.649584i	$-0.774943\pi$
$929$		−	39.5980i		−	1.29917i	0.760290	−	0.649584i	$-0.225057\pi$
$930$		0			0
$931$			2.82843i			0.0926980i
$932$		0			0
$933$	24.0000	+	33.9411i	0.785725	+	1.11118i
$934$		0			0
$935$	−32.0000			−1.04651
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	2.00000	+	2.82843i	0.0652675	+	0.0923022i
$940$		0			0
$941$			2.82843i			0.0922041i	0.998937	+	0.0461020i	$0.0146799\pi$
$941$			2.82843i			0.0922041i	−0.998937	+	0.0461020i	$0.985320\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	40.0000	−	11.3137i	1.30120	−	0.368035i
$946$		0			0
$947$	2.00000			0.0649913			0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000			0.0649913			0.0324956	+	0.999472i	$0.489654\pi$
$948$		0			0
$949$	−40.0000			−1.29845
$950$		0			0
$951$	44.0000	−	31.1127i	1.42680	−	1.00890i
$952$		0			0
$953$			56.5685i			1.83243i	0.400681	+	0.916217i	$0.368773\pi$
$953$			56.5685i			1.83243i	−0.400681	+	0.916217i	$0.631227\pi$
$954$		0			0
$955$		−	45.2548i		−	1.46441i
$956$		0			0
$957$	−8.00000	+	5.65685i	−0.258603	+	0.182860i
$958$		0			0
$959$	32.0000			1.03333
$960$		0			0
$961$	−41.0000			−1.32258
$962$		0			0
$963$	−6.00000	+	16.9706i	−0.193347	+	0.546869i
$964$		0			0
$965$			5.65685i			0.182101i
$966$		0			0
$967$			19.7990i			0.636693i	0.947974	+	0.318346i	$0.103127\pi$
$967$			19.7990i			0.636693i	−0.947974	+	0.318346i	$0.896873\pi$
$968$		0			0
$969$	−16.0000	−	22.6274i	−0.513994	−	0.726897i
$970$		0			0
$971$	−14.0000			−0.449281			−0.224641	−	0.974442i	$-0.572121\pi$
$971$	−14.0000			−0.449281			−0.224641	+	0.974442i	$0.572121\pi$
$972$		0			0
$973$	−24.0000			−0.769405
$974$		0			0
$975$	12.0000	+	16.9706i	0.384308	+	0.543493i
$976$		0			0
$977$			16.9706i			0.542936i	0.962447	+	0.271468i	$0.0875092\pi$
$977$			16.9706i			0.542936i	−0.962447	+	0.271468i	$0.912491\pi$
$978$		0			0
$979$		−	11.3137i		−	0.361588i
$980$		0			0
$981$	4.00000	−	11.3137i	0.127710	−	0.361219i
$982$		0			0
$983$	24.0000			0.765481			0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000			0.765481			0.382741	+	0.923856i	$0.374980\pi$
$984$		0			0
$985$	−40.0000			−1.27451
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	22.6274i		−	0.719510i
$990$		0			0
$991$			2.82843i			0.0898479i	0.998990	+	0.0449240i	$0.0143046\pi$
$991$			2.82843i			0.0898479i	−0.998990	+	0.0449240i	$0.985695\pi$
$992$		0			0
$993$	−4.00000	+	2.82843i	−0.126936	+	0.0897574i
$994$		0			0
$995$	24.0000			0.760851
$996$		0			0
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$		0			0
$999$	20.0000	−	5.65685i	0.632772	−	0.178975i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.c.e.767.2		2
3.2	odd	2		768.2.c.a.767.2		2
4.3	odd	2		768.2.c.a.767.1		2
8.3	odd	2		768.2.c.f.767.2		2
8.5	even	2		768.2.c.b.767.1		2
12.11	even	2	inner	768.2.c.e.767.1		2
16.3	odd	4		384.2.f.b.191.3	yes	4
16.5	even	4		384.2.f.d.191.3	yes	4
16.11	odd	4		384.2.f.b.191.2	yes	4
16.13	even	4		384.2.f.d.191.2	yes	4
24.5	odd	2		768.2.c.f.767.1		2
24.11	even	2		768.2.c.b.767.2		2
48.5	odd	4		384.2.f.b.191.4	yes	4
48.11	even	4		384.2.f.d.191.1	yes	4
48.29	odd	4		384.2.f.b.191.1	✓	4
48.35	even	4		384.2.f.d.191.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.f.b.191.1	✓	4	48.29	odd	4
384.2.f.b.191.2	yes	4	16.11	odd	4
384.2.f.b.191.3	yes	4	16.3	odd	4
384.2.f.b.191.4	yes	4	48.5	odd	4
384.2.f.d.191.1	yes	4	48.11	even	4
384.2.f.d.191.2	yes	4	16.13	even	4
384.2.f.d.191.3	yes	4	16.5	even	4
384.2.f.d.191.4	yes	4	48.35	even	4
768.2.c.a.767.1		2	4.3	odd	2
768.2.c.a.767.2		2	3.2	odd	2
768.2.c.b.767.1		2	8.5	even	2
768.2.c.b.767.2		2	24.11	even	2
768.2.c.e.767.1		2	12.11	even	2	inner
768.2.c.e.767.2		2	1.1	even	1	trivial
768.2.c.f.767.1		2	24.5	odd	2
768.2.c.f.767.2		2	8.3	odd	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.41421i) q^{3} -2.82843i q^{5} -2.82843i q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.00000 + 1.41421i) q^{3} -2.82843i q^{5} -2.82843i q^{7} +(-1.00000 + 2.82843i) q^{9} +2.00000 q^{11} -4.00000 q^{13} +(4.00000 - 2.82843i) q^{15} -5.65685i q^{17} -2.82843i q^{19} +(4.00000 - 2.82843i) q^{21} +8.00000 q^{23} -3.00000 q^{25} +(-5.00000 + 1.41421i) q^{27} +2.82843i q^{29} -8.48528i q^{31} +(2.00000 + 2.82843i) q^{33} -8.00000 q^{35} -4.00000 q^{37} +(-4.00000 - 5.65685i) q^{39} -2.82843i q^{43} +(8.00000 + 2.82843i) q^{45} -1.00000 q^{49} +(8.00000 - 5.65685i) q^{51} +8.48528i q^{53} -5.65685i q^{55} +(4.00000 - 2.82843i) q^{57} +6.00000 q^{59} -4.00000 q^{61} +(8.00000 + 2.82843i) q^{63} +11.3137i q^{65} +14.1421i q^{67} +(8.00000 + 11.3137i) q^{69} +8.00000 q^{71} +10.0000 q^{73} +(-3.00000 - 4.24264i) q^{75} -5.65685i q^{77} +2.82843i q^{79} +(-7.00000 - 5.65685i) q^{81} +6.00000 q^{83} -16.0000 q^{85} +(-4.00000 + 2.82843i) q^{87} -5.65685i q^{89} +11.3137i q^{91} +(12.0000 - 8.48528i) q^{93} -8.00000 q^{95} -6.00000 q^{97} +(-2.00000 + 5.65685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{9} + 4 q^{11} - 8 q^{13} + 8 q^{15} + 8 q^{21} + 16 q^{23} - 6 q^{25} - 10 q^{27} + 4 q^{33} - 16 q^{35} - 8 q^{37} - 8 q^{39} + 16 q^{45} - 2 q^{49} + 16 q^{51} + 8 q^{57} + 12 q^{59} - 8 q^{61} + 16 q^{63} + 16 q^{69} + 16 q^{71} + 20 q^{73} - 6 q^{75} - 14 q^{81} + 12 q^{83} - 32 q^{85} - 8 q^{87} + 24 q^{93} - 16 q^{95} - 12 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^9 + 4 * q^11 - 8 * q^13 + 8 * q^15 + 8 * q^21 + 16 * q^23 - 6 * q^25 - 10 * q^27 + 4 * q^33 - 16 * q^35 - 8 * q^37 - 8 * q^39 + 16 * q^45 - 2 * q^49 + 16 * q^51 + 8 * q^57 + 12 * q^59 - 8 * q^61 + 16 * q^63 + 16 * q^69 + 16 * q^71 + 20 * q^73 - 6 * q^75 - 14 * q^81 + 12 * q^83 - 32 * q^85 - 8 * q^87 + 24 * q^93 - 16 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

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Database

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