LMFDB - Embedded newform 1368.2.e.a.379.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(379,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1368.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.e (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:	$10.9235349965$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			379.1
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1368.379
Dual form			1368.2.e.a.379.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +O(q^{10})$	$q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} -6.00000 q^{11} +4.00000 q^{16} +6.00000 q^{17} +(-1.00000 + 4.24264i) q^{19} +8.48528i q^{22} +5.00000 q^{25} -5.65685i q^{32} -8.48528i q^{34} +(6.00000 + 1.41421i) q^{38} +11.3137i q^{41} +10.0000 q^{43} +12.0000 q^{44} +7.00000 q^{49} -7.07107i q^{50} +14.1421i q^{59} -8.00000 q^{64} +8.48528i q^{67} -12.0000 q^{68} -2.00000 q^{73} +(2.00000 - 8.48528i) q^{76} +16.0000 q^{82} +18.0000 q^{83} -14.1421i q^{86} -16.9706i q^{88} +5.65685i q^{89} -16.9706i q^{97} -9.89949i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4}+O(q^{10}) $ 2 * q - 4 * q^4	$ 2 q - 4 q^{4} - 12 q^{11} + 8 q^{16} + 12 q^{17} - 2 q^{19} + 10 q^{25} + 12 q^{38} + 20 q^{43} + 24 q^{44} + 14 q^{49} - 16 q^{64} - 24 q^{68} - 4 q^{73} + 4 q^{76} + 32 q^{82} + 36 q^{83}+O(q^{100}) $ 2 * q - 4 * q^4 - 12 * q^11 + 8 * q^16 + 12 * q^17 - 2 * q^19 + 10 * q^25 + 12 * q^38 + 20 * q^43 + 24 * q^44 + 14 * q^49 - 16 * q^64 - 24 * q^68 - 4 * q^73 + 4 * q^76 + 32 * q^82 + 36 * q^83

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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 $

$2$		−	1.41421i		−	1.00000i
$2$		−	1.41421i		−	1.00000i
$3$		0			0
$3$		0			0
$4$	−2.00000			−1.00000
$5$		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			$\pi$
$8$			2.82843i			1.00000i
$9$		0			0
$10$		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
$16$	4.00000			1.00000
$17$	6.00000			1.45521			0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000			1.45521			0.727607	+	0.685994i	$0.240633\pi$
$18$		0			0
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$19$	−1.00000	+	4.24264i	−0.229416	+	0.973329i
$20$		0			0
$21$		0			0
$22$			8.48528i			1.80907i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$		−	5.65685i		−	1.00000i
$33$		0			0
$34$		−	8.48528i		−	1.45521i
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	6.00000	+	1.41421i	0.973329	+	0.229416i
$39$		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
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$	12.0000			1.80907
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		−	7.07107i		−	1.00000i
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			14.1421i			1.84115i	0.390567	+	0.920575i	$0.372279\pi$
$59$			14.1421i			1.84115i	−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		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	$0.826557\pi$
$68$	−12.0000			−1.45521
$69$		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	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$		0			0
$76$	2.00000	−	8.48528i	0.229416	−	0.973329i
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$		0			0
$82$	16.0000			1.76690
$83$	18.0000			1.97576			0.987878	−	0.155230i	$-0.0496119\pi$
$83$	18.0000			1.97576			0.987878	+	0.155230i	$0.0496119\pi$
$84$		0			0
$85$		0			0
$86$		−	14.1421i		−	1.52499i
$87$		0			0
$88$		−	16.9706i		−	1.80907i
$89$			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
$97$		−	16.9706i		−	1.72310i	−0.507673	−	0.861550i	$-0.669494\pi$
$97$		−	16.9706i		−	1.72310i	0.507673	−	0.861550i	$-0.330506\pi$
$98$		−	9.89949i		−	1.00000i
$99$		0			0
$100$	−10.0000			−1.00000
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	19.7990i		−	1.91404i	−0.290021	−	0.957020i	$-0.593662\pi$
$107$		−	19.7990i		−	1.91404i	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	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	20.0000			1.84115
$119$		0			0
$120$		0			0
$121$	25.0000			2.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$			11.3137i			1.00000i
$129$		0			0
$130$		0			0
$131$	−18.0000			−1.57267			−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000			−1.57267			−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$		0			0
$134$	12.0000			1.03664
$135$		0			0
$136$			16.9706i			1.45521i
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$	−22.0000			−1.86602			−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000			−1.86602			−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
$146$			2.82843i			0.234082i
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−12.0000	−	2.82843i	−0.973329	−	0.229416i
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$		−	22.6274i		−	1.76690i
$165$		0			0
$166$		−	25.4558i		−	1.97576i
$167$		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
$171$		0			0
$172$	−20.0000			−1.52499
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−24.0000			−1.80907
$177$		0			0
$178$	8.00000			0.599625
$179$			19.7990i			1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$			19.7990i			1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		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
$187$	−36.0000			−2.63258
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$			16.9706i			1.22157i	0.791797	+	0.610784i	$0.209146\pi$
$193$			16.9706i			1.22157i	−0.791797	+	0.610784i	$0.790854\pi$
$194$	−24.0000			−1.72310
$195$		0			0
$196$	−14.0000			−1.00000
$197$		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			$\pi$
$200$			14.1421i			1.00000i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.00000	−	25.4558i	0.415029	−	1.76082i
$210$		0			0
$211$			25.4558i			1.75245i	0.481900	+	0.876226i	$0.339947\pi$
$211$			25.4558i			1.75245i	−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$		0			0
$214$	−28.0000			−1.91404
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	16.0000			1.06430
$227$			2.82843i			0.187729i	0.995585	+	0.0938647i	$0.0299221\pi$
$227$			2.82843i			0.187729i	−0.995585	+	0.0938647i	$0.970078\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	30.0000			1.96537			0.982683	−	0.185296i	$-0.0593245\pi$
$233$	30.0000			1.96537			0.982683	+	0.185296i	$0.0593245\pi$
$234$		0			0
$235$		0			0
$236$		−	28.2843i		−	1.84115i
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			16.9706i			1.09317i	0.837404	+	0.546585i	$0.184072\pi$
$241$			16.9706i			1.09317i	−0.837404	+	0.546585i	$0.815928\pi$
$242$		−	35.3553i		−	2.27273i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$		−	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
$262$			25.4558i			1.57267i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		−	16.9706i		−	1.03664i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	24.0000			1.45521
$273$		0			0
$274$		−	8.48528i		−	0.512615i
$275$	−30.0000			−1.80907
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$			31.1127i			1.86602i
$279$		0			0
$280$		0			0
$281$			28.2843i			1.68730i	0.536895	+	0.843649i	$0.319597\pi$
$281$			28.2843i			1.68730i	−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−22.0000			−1.30776			−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000			−1.30776			−0.653882	+	0.756596i	$0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	19.0000			1.11765
$290$		0			0
$291$		0			0
$292$	4.00000			0.234082
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−4.00000	+	16.9706i	−0.229416	+	0.973329i
$305$		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		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		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			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.00000	+	25.4558i	−0.333849	+	1.41640i
$324$		0			0
$325$		0			0
$326$		−	2.82843i		−	0.156652i
$327$		0			0
$328$	−32.0000			−1.76690
$329$		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	$-0.246634\pi$
$332$	−36.0000			−1.97576
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	33.9411i		−	1.84889i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$		−	33.9411i		−	1.84889i	0.381314	−	0.924445i	$-0.375472\pi$
$338$			18.3848i			1.00000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$			28.2843i			1.52499i
$345$		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		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$			33.9411i			1.80907i
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		0			0
$356$		−	11.3137i		−	0.599625i
$357$		0			0
$358$	28.0000			1.47985
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−17.0000	−	8.48528i	−0.894737	−	0.446594i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$			50.9117i			2.63258i
$375$		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
$386$	24.0000			1.22157
$387$		0			0
$388$			33.9411i			1.72310i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$			19.7990i			1.00000i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	20.0000			1.00000
$401$		−	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
$409$			33.9411i			1.67828i	0.543915	+	0.839140i	$0.316941\pi$
$409$			33.9411i			1.67828i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$	−36.0000	−	8.48528i	−1.76082	−	0.415029i
$419$	−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	$0.5\pi$
$422$	36.0000			1.75245
$423$		0			0
$424$		0			0
$425$	30.0000			1.45521
$426$		0			0
$427$		0			0
$428$			39.5980i			1.91404i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$			16.9706i			0.815553i	0.913082	+	0.407777i	$0.133696\pi$
$433$			16.9706i			0.815553i	−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			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		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	$0.0426157\pi$
$449$			5.65685i			0.266963i	−0.991051	+	0.133482i	$0.957384\pi$
$450$		0			0
$451$		−	67.8823i		−	3.19645i
$452$		−	22.6274i		−	1.06430i
$453$		0			0
$454$	4.00000			0.187729
$455$		0			0
$456$		0			0
$457$	26.0000			1.21623			0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000			1.21623			0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$		0			0
$460$		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
$466$		−	42.4264i		−	1.96537i
$467$	−30.0000			−1.38823			−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000			−1.38823			−0.694117	+	0.719862i	$0.744205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−40.0000			−1.84115
$473$	−60.0000			−2.75880
$474$		0			0
$475$	−5.00000	+	21.2132i	−0.229416	+	0.973329i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$	24.0000			1.09317
$483$		0			0
$484$	−50.0000			−2.27273
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	42.0000			1.89543			0.947717	−	0.319113i	$-0.103385\pi$
$491$	42.0000			1.89543			0.947717	+	0.319113i	$0.103385\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−14.0000			−0.626726			−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000			−0.626726			−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$		−	8.48528i		−	0.378717i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		−	22.6274i		−	1.00000i
$513$		0			0
$514$	−16.0000			−0.705730
$515$		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	$-0.541958\pi$
$521$		−	45.2548i		−	1.98265i	0.131432	−	0.991325i	$-0.458042\pi$
$522$		0			0
$523$			25.4558i			1.11311i	0.830812	+	0.556553i	$0.187876\pi$
$523$			25.4558i			1.11311i	−0.830812	+	0.556553i	$0.812124\pi$
$524$	36.0000			1.57267
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−24.0000			−1.03664
$537$		0			0
$538$		0			0
$539$	−42.0000			−1.80907
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		−	33.9411i		−	1.45521i
$545$		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	$-0.0580636\pi$
$548$	−12.0000			−0.512615
$549$		0			0
$550$			42.4264i			1.80907i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	44.0000			1.86602
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	40.0000			1.68730
$563$		−	36.7696i		−	1.54965i	−0.632175	−	0.774826i	$-0.717837\pi$
$563$		−	36.7696i		−	1.54965i	0.632175	−	0.774826i	$-0.282163\pi$
$564$		0			0
$565$		0			0
$566$			31.1127i			1.30776i
$567$		0			0
$568$		0			0
$569$			22.6274i			0.948591i	0.880366	+	0.474295i	$0.157297\pi$
$569$			22.6274i			0.948591i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$	22.0000			0.920671			0.460336	−	0.887745i	$-0.347729\pi$
$571$	22.0000			0.920671			0.460336	+	0.887745i	$0.347729\pi$
$572$		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	$0.750276\pi$
$578$		−	26.8701i		−	1.11765i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		−	5.65685i		−	0.234082i
$585$		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
$593$	18.0000			0.739171			0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000			0.739171			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
$601$		−	16.9706i		−	0.692244i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$		−	16.9706i		−	0.692244i	0.938190	−	0.346122i	$-0.112502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	24.0000	+	5.65685i	0.973329	+	0.229416i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	12.0000			0.484281
$615$		0			0
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	−26.0000			−1.04503			−0.522514	−	0.852631i	$-0.675006\pi$
$619$	−26.0000			−1.04503			−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
$626$		−	14.1421i		−	0.565233i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		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	$-0.811346\pi$
$641$		−	28.2843i		−	1.11716i	0.829450	−	0.558581i	$-0.188654\pi$
$642$		0			0
$643$	50.0000			1.97181			0.985904	−	0.167313i	$-0.0535092\pi$
$643$	50.0000			1.97181			0.985904	+	0.167313i	$0.0535092\pi$
$644$		0			0
$645$		0			0
$646$	36.0000	+	8.48528i	1.41640	+	0.333849i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		−	84.8528i		−	3.33076i
$650$		0			0
$651$		0			0
$652$	−4.00000			−0.156652
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$			45.2548i			1.76690i
$657$		0			0
$658$		0			0
$659$		−	48.0833i		−	1.87306i	−0.350590	−	0.936529i	$-0.614019\pi$
$659$		−	48.0833i		−	1.87306i	0.350590	−	0.936529i	$-0.385981\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−36.0000			−1.39918
$663$		0			0
$664$			50.9117i			1.97576i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	50.9117i		−	1.96250i	−0.192736	−	0.981251i	$-0.561736\pi$
$673$		−	50.9117i		−	1.96250i	0.192736	−	0.981251i	$-0.438264\pi$
$674$	−48.0000			−1.84889
$675$		0			0
$676$	26.0000			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	31.1127i		−	1.19049i	−0.803543	−	0.595247i	$-0.797054\pi$
$683$		−	31.1127i		−	1.19049i	0.803543	−	0.595247i	$-0.202946\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	40.0000			1.52499
$689$		0			0
$690$		0			0
$691$	46.0000			1.74992			0.874961	−	0.484193i	$-0.160887\pi$
$691$	46.0000			1.74992			0.874961	+	0.484193i	$0.160887\pi$
$692$		0			0
$693$		0			0
$694$			8.48528i			0.322097i
$695$		0			0
$696$		0			0
$697$			67.8823i			2.57122i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	48.0000			1.80907
$705$		0			0
$706$			42.4264i			1.59674i
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	−16.0000			−0.599625
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	39.5980i		−	1.47985i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	−12.0000	+	24.0416i	−0.446594	+	0.894737i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	60.0000			2.21918
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	50.9117i		−	1.87536i
$738$		0			0
$739$	−34.0000			−1.25071			−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000			−1.25071			−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
$747$		0			0
$748$	72.0000			2.63258
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−12.0000			−0.435860
$759$		0			0
$760$		0			0
$761$	54.0000			1.95750			0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000			1.95750			0.978749	+	0.205061i	$0.0657392\pi$
$762$		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
$771$		0			0
$772$		−	33.9411i		−	1.22157i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$	48.0000			1.72310
$777$		0			0
$778$		0			0
$779$	−48.0000	−	11.3137i	−1.71978	−	0.405356i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$		0			0
$787$		−	25.4558i		−	0.907403i	−0.891154	−	0.453701i	$-0.850103\pi$
$787$		−	25.4558i		−	0.907403i	0.891154	−	0.453701i	$-0.149897\pi$
$788$		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			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	28.2843i		−	1.00000i
$801$		0			0
$802$	−56.0000			−1.97743
$803$	12.0000			0.423471
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$		−	42.4264i		−	1.48979i	−0.667180	−	0.744896i	$-0.732499\pi$
$811$		−	42.4264i		−	1.48979i	0.667180	−	0.744896i	$-0.267501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−10.0000	+	42.4264i	−0.349856	+	1.48431i
$818$	48.0000			1.67828
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			19.7990i			0.688478i	0.938882	+	0.344239i	$0.111863\pi$
$827$			19.7990i			0.688478i	−0.938882	+	0.344239i	$0.888137\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	42.0000			1.45521
$834$		0			0
$835$		0			0
$836$	−12.0000	+	50.9117i	−0.415029	+	1.76082i
$837$		0			0
$838$			25.4558i			0.879358i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		0			0
$844$		−	50.9117i		−	1.75245i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		−	42.4264i		−	1.45521i
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	56.0000			1.91404
$857$			22.6274i			0.772938i	0.922302	+	0.386469i	$0.126305\pi$
$857$			22.6274i			0.772938i	−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$	58.0000			1.97893			0.989467	−	0.144757i	$-0.0462401\pi$
$859$	58.0000			1.97893			0.989467	+	0.144757i	$0.0462401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$	24.0000			0.815553
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$		0			0
$885$		0			0
$886$		−	59.3970i		−	1.99548i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	8.00000			0.266963
$899$		0			0
$900$		0			0
$901$		0			0
$902$	−96.0000			−3.19645
$903$		0			0
$904$	−32.0000			−1.06430
$905$		0			0
$906$		0			0
$907$			59.3970i			1.97224i	0.166022	+	0.986122i	$0.446908\pi$
$907$			59.3970i			1.97224i	−0.166022	+	0.986122i	$0.553092\pi$
$908$		−	5.65685i		−	0.187729i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−108.000			−3.57428
$914$		−	36.7696i		−	1.21623i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−54.0000			−1.77168			−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000			−1.77168			−0.885841	+	0.463988i	$0.846418\pi$
$930$		0			0
$931$	−7.00000	+	29.6985i	−0.229416	+	0.973329i
$932$	−60.0000			−1.96537
$933$		0			0
$934$			42.4264i			1.38823i
$935$		0			0
$936$		0			0
$937$	34.0000			1.11073			0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000			1.11073			0.555366	+	0.831606i	$0.312578\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			56.5685i			1.84115i
$945$		0			0
$946$			84.8528i			2.75880i
$947$	30.0000			0.974869			0.487435	−	0.873160i	$-0.337933\pi$
$947$	30.0000			0.974869			0.487435	+	0.873160i	$0.337933\pi$
$948$		0			0
$949$		0			0
$950$	30.0000	+	7.07107i	0.973329	+	0.229416i
$951$		0			0
$952$		0			0
$953$			45.2548i			1.46595i	0.680257	+	0.732974i	$0.261868\pi$
$953$			45.2548i			1.46595i	−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−31.0000			−1.00000
$962$		0			0
$963$		0			0
$964$		−	33.9411i		−	1.09317i
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$			70.7107i			2.27273i
$969$		0			0
$970$		0			0
$971$		−	31.1127i		−	0.998454i	−0.866471	−	0.499227i	$-0.833617\pi$
$971$		−	31.1127i		−	0.998454i	0.866471	−	0.499227i	$-0.166383\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			62.2254i			1.99077i	0.0959785	+	0.995383i	$0.469402\pi$
$977$			62.2254i			1.99077i	−0.0959785	+	0.995383i	$0.530598\pi$
$978$		0			0
$979$		−	33.9411i		−	1.08476i
$980$		0			0
$981$		0			0
$982$		−	59.3970i		−	1.89543i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$			19.7990i			0.626726i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.e.a.379.1		2
3.2	odd	2		152.2.b.a.75.2	yes	2
4.3	odd	2		5472.2.e.a.5167.1		2
8.3	odd	2	CM	1368.2.e.a.379.1		2
8.5	even	2		5472.2.e.a.5167.1		2
12.11	even	2		608.2.b.a.303.1		2
19.18	odd	2	inner	1368.2.e.a.379.2		2
24.5	odd	2		608.2.b.a.303.1		2
24.11	even	2		152.2.b.a.75.2	yes	2
57.56	even	2		152.2.b.a.75.1	✓	2
76.75	even	2		5472.2.e.a.5167.2		2
152.37	odd	2		5472.2.e.a.5167.2		2
152.75	even	2	inner	1368.2.e.a.379.2		2
228.227	odd	2		608.2.b.a.303.2		2
456.227	odd	2		152.2.b.a.75.1	✓	2
456.341	even	2		608.2.b.a.303.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.b.a.75.1	✓	2	57.56	even	2
152.2.b.a.75.1	✓	2	456.227	odd	2
152.2.b.a.75.2	yes	2	3.2	odd	2
152.2.b.a.75.2	yes	2	24.11	even	2
608.2.b.a.303.1		2	12.11	even	2
608.2.b.a.303.1		2	24.5	odd	2
608.2.b.a.303.2		2	228.227	odd	2
608.2.b.a.303.2		2	456.341	even	2
1368.2.e.a.379.1		2	1.1	even	1	trivial
1368.2.e.a.379.1		2	8.3	odd	2	CM
1368.2.e.a.379.2		2	19.18	odd	2	inner
1368.2.e.a.379.2		2	152.75	even	2	inner
5472.2.e.a.5167.1		2	4.3	odd	2
5472.2.e.a.5167.1		2	8.5	even	2
5472.2.e.a.5167.2		2	76.75	even	2
5472.2.e.a.5167.2		2	152.37	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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +2.82843i q^{8} -6.00000 q^{11} +4.00000 q^{16} +6.00000 q^{17} +(-1.00000 + 4.24264i) q^{19} +8.48528i q^{22} +5.00000 q^{25} -5.65685i q^{32} -8.48528i q^{34} +(6.00000 + 1.41421i) q^{38} +11.3137i q^{41} +10.0000 q^{43} +12.0000 q^{44} +7.00000 q^{49} -7.07107i q^{50} +14.1421i q^{59} -8.00000 q^{64} +8.48528i q^{67} -12.0000 q^{68} -2.00000 q^{73} +(2.00000 - 8.48528i) q^{76} +16.0000 q^{82} +18.0000 q^{83} -14.1421i q^{86} -16.9706i q^{88} +5.65685i q^{89} -16.9706i q^{97} -9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4}+O(q^{10}) \) 2 * q - 4 * q^4	\( 2 q - 4 q^{4} - 12 q^{11} + 8 q^{16} + 12 q^{17} - 2 q^{19} + 10 q^{25} + 12 q^{38} + 20 q^{43} + 24 q^{44} + 14 q^{49} - 16 q^{64} - 24 q^{68} - 4 q^{73} + 4 q^{76} + 32 q^{82} + 36 q^{83}+O(q^{100}) \) 2 * q - 4 * q^4 - 12 * q^11 + 8 * q^16 + 12 * q^17 - 2 * q^19 + 10 * q^25 + 12 * q^38 + 20 * q^43 + 24 * q^44 + 14 * q^49 - 16 * q^64 - 24 * q^68 - 4 * q^73 + 4 * q^76 + 32 * q^82 + 36 * q^83

Introduction

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