LMFDB - Embedded newform 768.2.c.d.767.1

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:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(1.00000 - 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} -6.00000 q^{11} -5.65685i q^{17} -8.48528i q^{19} +5.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} +(-6.00000 + 8.48528i) q^{33} +11.3137i q^{41} -8.48528i q^{43} +7.00000 q^{49} +(-8.00000 - 5.65685i) q^{51} +(-12.0000 - 8.48528i) q^{57} +6.00000 q^{59} +8.48528i q^{67} +2.00000 q^{73} +(5.00000 - 7.07107i) q^{75} +(-7.00000 + 5.65685i) q^{81} -18.0000 q^{83} +5.65685i q^{89} +10.0000 q^{97} +(6.00000 + 16.9706i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 12 q^{11} + 10 q^{25} - 10 q^{27} - 12 q^{33} + 14 q^{49} - 16 q^{51} - 24 q^{57} + 12 q^{59} + 4 q^{73} + 10 q^{75} - 14 q^{81} - 36 q^{83} + 20 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^9 - 12 * q^11 + 10 * q^25 - 10 * q^27 - 12 * q^33 + 14 * q^49 - 16 * q^51 - 24 * q^57 + 12 * q^59 + 4 * q^73 + 10 * q^75 - 14 * q^81 - 36 * q^83 + 20 * q^97 + 12 * 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
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000	−	2.82843i	−0.333333	−	0.942809i
$10$		0			0
$11$	−6.00000			−1.80907			−0.904534	−	0.426401i	$-0.859781\pi$
$11$	−6.00000			−1.80907			−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		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
$19$		−	8.48528i		−	1.94666i	−0.229416	−	0.973329i	$-0.573682\pi$
$19$		−	8.48528i		−	1.94666i	0.229416	−	0.973329i	$-0.426318\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	−6.00000	+	8.48528i	−1.04447	+	1.47710i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			11.3137i			1.76690i	0.468521	+	0.883452i	$0.344787\pi$
$41$			11.3137i			1.76690i	−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	8.48528i		−	1.29399i	−0.762493	−	0.646997i	$-0.776025\pi$
$43$		−	8.48528i		−	1.29399i	0.762493	−	0.646997i	$-0.223975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−8.00000	−	5.65685i	−1.12022	−	0.792118i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−12.0000	−	8.48528i	−1.58944	−	1.12390i
$58$		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
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			8.48528i			1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$67$			8.48528i			1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	5.00000	−	7.07107i	0.577350	−	0.816497i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	−18.0000			−1.97576			−0.987878	−	0.155230i	$-0.950388\pi$
$83$	−18.0000			−1.97576			−0.987878	+	0.155230i	$0.950388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			5.65685i			0.599625i	0.953998	+	0.299813i	$0.0969242\pi$
$89$			5.65685i			0.599625i	−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		0			0
$99$	6.00000	+	16.9706i	0.603023	+	1.70561i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		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
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.3137i		−	1.06430i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$		−	11.3137i		−	1.06430i	0.846649	−	0.532152i	$-0.178617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$	16.0000	+	11.3137i	1.44267	+	1.02012i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−12.0000	−	8.48528i	−1.05654	−	0.747087i
$130$		0			0
$131$	18.0000			1.57267			0.786334	−	0.617802i	$-0.211977\pi$
$131$	18.0000			1.57267			0.786334	+	0.617802i	$0.211977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	22.6274i		−	1.93319i	−0.256307	−	0.966595i	$-0.582506\pi$
$137$		−	22.6274i		−	1.93319i	0.256307	−	0.966595i	$-0.417494\pi$
$138$		0			0
$139$			8.48528i			0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$			8.48528i			0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	7.00000	−	9.89949i	0.577350	−	0.816497i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−16.0000	+	5.65685i	−1.29352	+	0.457330i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			25.4558i			1.99386i	0.0783260	+	0.996928i	$0.475042\pi$
$163$			25.4558i			1.99386i	−0.0783260	+	0.996928i	$0.524958\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	−24.0000	+	8.48528i	−1.83533	+	0.648886i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		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
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			33.9411i			2.48202i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	22.0000			1.58359			0.791797	−	0.610784i	$-0.209146\pi$
$193$	22.0000			1.58359			0.791797	+	0.610784i	$0.209146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	12.0000	+	8.48528i	0.846415	+	0.598506i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			50.9117i			3.52164i
$210$		0			0
$211$		−	25.4558i		−	1.75245i	−0.481900	−	0.876226i	$-0.660053\pi$
$211$		−	25.4558i		−	1.75245i	0.481900	−	0.876226i	$-0.339947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.00000	−	2.82843i	0.135147	−	0.191127i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−5.00000	−	14.1421i	−0.333333	−	0.942809i
$226$		0			0
$227$	30.0000			1.99117			0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	30.0000			1.99117			0.995585	+	0.0938647i	$0.0299221\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			5.65685i			0.370593i	0.982683	+	0.185296i	$0.0593245\pi$
$233$			5.65685i			0.370593i	−0.982683	+	0.185296i	$0.940675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−26.0000			−1.67481			−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000			−1.67481			−0.837404	+	0.546585i	$0.815928\pi$
$242$		0			0
$243$	1.00000	+	15.5563i	0.0641500	+	0.997940i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−18.0000	+	25.4558i	−1.14070	+	1.61320i
$250$		0			0
$251$	−6.00000			−0.378717			−0.189358	−	0.981908i	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		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
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	8.00000	+	5.65685i	0.489592	+	0.346194i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−30.0000			−1.80907
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	28.2843i		−	1.68730i	−0.536895	−	0.843649i	$-0.680403\pi$
$281$		−	28.2843i		−	1.68730i	0.536895	−	0.843649i	$-0.319597\pi$
$282$		0			0
$283$		−	25.4558i		−	1.51319i	−0.653882	−	0.756596i	$-0.726861\pi$
$283$		−	25.4558i		−	1.51319i	0.653882	−	0.756596i	$-0.273139\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−15.0000			−0.882353
$290$		0			0
$291$	10.0000	−	14.1421i	0.586210	−	0.829027i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	30.0000	+	8.48528i	1.74078	+	0.492366i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			8.48528i			0.484281i	0.970241	+	0.242140i	$0.0778494\pi$
$307$			8.48528i			0.484281i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	10.0000			0.565233			0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000			0.565233			0.282617	+	0.959233i	$0.408798\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	6.00000	−	8.48528i	0.334887	−	0.473602i
$322$		0			0
$323$	−48.0000			−2.67079
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	25.4558i		−	1.39918i	−0.714545	−	0.699590i	$-0.753366\pi$
$331$		−	25.4558i		−	1.39918i	0.714545	−	0.699590i	$-0.246634\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$	−16.0000	−	11.3137i	−0.869001	−	0.614476i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−6.00000			−0.322097			−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000			−0.322097			−0.161048	+	0.986947i	$0.551488\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		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
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−53.0000			−2.78947
$362$		0			0
$363$	25.0000	−	35.3553i	1.31216	−	1.85567i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	32.0000	−	11.3137i	1.66585	−	0.588968i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	8.48528i		−	0.435860i	−0.975964	−	0.217930i	$-0.930070\pi$
$379$		−	8.48528i		−	0.435860i	0.975964	−	0.217930i	$-0.0699304\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−24.0000	+	8.48528i	−1.21999	+	0.431331i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	18.0000	−	25.4558i	0.907980	−	1.28408i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	39.5980i		−	1.97743i	−0.149813	−	0.988714i	$-0.547867\pi$
$401$		−	39.5980i		−	1.97743i	0.149813	−	0.988714i	$-0.452133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	22.0000			1.08783			0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000			1.08783			0.543915	+	0.839140i	$0.316941\pi$
$410$		0			0
$411$	−32.0000	−	22.6274i	−1.57844	−	1.11613i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	12.0000	+	8.48528i	0.587643	+	0.415526i
$418$		0			0
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	28.2843i		−	1.37199i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−38.0000			−1.82616			−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000			−1.82616			−0.913082	+	0.407777i	$0.866304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−7.00000	−	19.7990i	−0.333333	−	0.942809i
$442$		0			0
$443$	42.0000			1.99548			0.997740	−	0.0671913i	$-0.0214038\pi$
$443$	42.0000			1.99548			0.997740	+	0.0671913i	$0.0214038\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$		0			0
$451$		−	67.8823i		−	3.19645i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−26.0000			−1.21623			−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000			−1.21623			−0.608114	+	0.793849i	$0.708074\pi$
$458$		0			0
$459$	−8.00000	+	28.2843i	−0.373408	+	1.32020i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	30.0000			1.38823			0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000			1.38823			0.694117	+	0.719862i	$0.255795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$			50.9117i			2.34092i
$474$		0			0
$475$		−	42.4264i		−	1.94666i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	36.0000	+	25.4558i	1.62798	+	1.15115i
$490$		0			0
$491$	−42.0000			−1.89543			−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000			−1.89543			−0.947717	+	0.319113i	$0.896615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$			42.4264i			1.89927i	0.313363	+	0.949633i	$0.398544\pi$
$499$			42.4264i			1.89927i	−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13.0000	+	18.3848i	−0.577350	+	0.816497i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−12.0000	+	42.4264i	−0.529813	+	1.87317i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			45.2548i			1.98265i	0.131432	+	0.991325i	$0.458042\pi$
$521$			45.2548i			1.98265i	−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$			25.4558i			1.11311i	0.830812	+	0.556553i	$0.187876\pi$
$523$			25.4558i			1.11311i	−0.830812	+	0.556553i	$0.812124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	−6.00000	−	16.9706i	−0.260378	−	0.736460i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	18.0000	−	25.4558i	0.776757	−	1.09850i
$538$		0			0
$539$	−42.0000			−1.80907
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	8.48528i		−	0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$		−	8.48528i		−	0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	48.0000	+	33.9411i	2.02656	+	1.43300i
$562$		0			0
$563$	30.0000			1.26435			0.632175	−	0.774826i	$-0.282163\pi$
$563$	30.0000			1.26435			0.632175	+	0.774826i	$0.282163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	22.6274i		−	0.948591i	−0.880366	−	0.474295i	$-0.842703\pi$
$569$		−	22.6274i		−	0.948591i	0.880366	−	0.474295i	$-0.157297\pi$
$570$		0			0
$571$			42.4264i			1.77549i	0.460336	+	0.887745i	$0.347729\pi$
$571$			42.4264i			1.77549i	−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		0			0
$579$	22.0000	−	31.1127i	0.914289	−	1.29300i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.00000			0.247647			0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000			0.247647			0.123823	+	0.992304i	$0.460484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	45.2548i		−	1.85839i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$		−	45.2548i		−	1.85839i	0.369586	−	0.929197i	$-0.379500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	−46.0000			−1.87638			−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000			−1.87638			−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	24.0000	−	8.48528i	0.977356	−	0.345547i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			39.5980i			1.59415i	0.603877	+	0.797077i	$0.293622\pi$
$617$			39.5980i			1.59415i	−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$			42.4264i			1.70526i	0.522514	+	0.852631i	$0.324994\pi$
$619$			42.4264i			1.70526i	−0.522514	+	0.852631i	$0.675006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	72.0000	+	50.9117i	2.87540	+	2.03322i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−36.0000	−	25.4558i	−1.43087	−	1.01178i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			28.2843i			1.11716i	0.829450	+	0.558581i	$0.188654\pi$
$641$			28.2843i			1.11716i	−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$		−	8.48528i		−	0.334627i	−0.985904	−	0.167313i	$-0.946491\pi$
$643$		−	8.48528i		−	0.334627i	0.985904	−	0.167313i	$-0.0535092\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−36.0000			−1.41312
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−2.00000	−	5.65685i	−0.0780274	−	0.220695i
$658$		0			0
$659$	18.0000			0.701180			0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000			0.701180			0.350590	+	0.936529i	$0.385981\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	10.0000			0.385472			0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000			0.385472			0.192736	+	0.981251i	$0.438264\pi$
$674$		0			0
$675$	−25.0000	−	7.07107i	−0.962250	−	0.272166i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	30.0000	−	42.4264i	1.14960	−	1.62578i
$682$		0			0
$683$	42.0000			1.60709			0.803543	−	0.595247i	$-0.202946\pi$
$683$	42.0000			1.60709			0.803543	+	0.595247i	$0.202946\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$			25.4558i			0.968386i	0.874961	+	0.484193i	$0.160887\pi$
$691$			25.4558i			0.968386i	−0.874961	+	0.484193i	$0.839113\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	64.0000			2.42417
$698$		0			0
$699$	8.00000	+	5.65685i	0.302588	+	0.213962i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−26.0000	+	36.7696i	−0.966950	+	1.36747i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	23.0000	+	14.1421i	0.851852	+	0.523783i
$730$		0			0
$731$	−48.0000			−1.77534
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	50.9117i		−	1.87536i
$738$		0			0
$739$			42.4264i			1.56068i	0.625355	+	0.780340i	$0.284954\pi$
$739$			42.4264i			1.56068i	−0.625355	+	0.780340i	$0.715046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	18.0000	+	50.9117i	0.658586	+	1.86276i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−6.00000	+	8.48528i	−0.218652	+	0.309221i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			11.3137i			0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$			11.3137i			0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	22.0000			0.793340			0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000			0.793340			0.396670	+	0.917961i	$0.370166\pi$
$770$		0			0
$771$	−16.0000	−	11.3137i	−0.576226	−	0.407453i
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	96.0000			3.43956
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			25.4558i			0.907403i	0.891154	+	0.453701i	$0.149897\pi$
$787$			25.4558i			0.907403i	−0.891154	+	0.453701i	$0.850103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	16.0000	−	5.65685i	0.565332	−	0.199875i
$802$		0			0
$803$	−12.0000			−0.423471
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.41421i) q^{3} +(-1.00000 - 2.82843i) q^{9} -6.00000 q^{11} -5.65685i q^{17} -8.48528i q^{19} +5.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} +(-6.00000 + 8.48528i) q^{33} +11.3137i q^{41} -8.48528i q^{43} +7.00000 q^{49} +(-8.00000 - 5.65685i) q^{51} +(-12.0000 - 8.48528i) q^{57} +6.00000 q^{59} +8.48528i q^{67} +2.00000 q^{73} +(5.00000 - 7.07107i) q^{75} +(-7.00000 + 5.65685i) q^{81} -18.0000 q^{83} +5.65685i q^{89} +10.0000 q^{97} +(6.00000 + 16.9706i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{9} - 12 q^{11} + 10 q^{25} - 10 q^{27} - 12 q^{33} + 14 q^{49} - 16 q^{51} - 24 q^{57} + 12 q^{59} + 4 q^{73} + 10 q^{75} - 14 q^{81} - 36 q^{83} + 20 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^9 - 12 * q^11 + 10 * q^25 - 10 * q^27 - 12 * q^33 + 14 * q^49 - 16 * q^51 - 24 * q^57 + 12 * q^59 + 4 * q^73 + 10 * q^75 - 14 * q^81 - 36 * q^83 + 20 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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