LMFDB - Embedded newform 363.4.a.l.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,4,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,38,18,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$21.4176933321$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.19615 q^{2} -3.00000 q^{3} +19.0000 q^{4} +9.00000 q^{5} -15.5885 q^{6} +24.2487 q^{7} +57.1577 q^{8} +9.00000 q^{9} +46.7654 q^{10} -57.0000 q^{12} -71.0141 q^{13} +126.000 q^{14} -27.0000 q^{15} +145.000 q^{16} -88.3346 q^{17} +46.7654 q^{18} +145.492 q^{19} +171.000 q^{20} -72.7461 q^{21} +90.0000 q^{23} -171.473 q^{24} -44.0000 q^{25} -369.000 q^{26} -27.0000 q^{27} +460.726 q^{28} +88.3346 q^{29} -140.296 q^{30} -188.000 q^{31} +296.181 q^{32} -459.000 q^{34} +218.238 q^{35} +171.000 q^{36} +133.000 q^{37} +756.000 q^{38} +213.042 q^{39} +514.419 q^{40} -36.3731 q^{41} -378.000 q^{42} +72.7461 q^{43} +81.0000 q^{45} +467.654 q^{46} +72.0000 q^{47} -435.000 q^{48} +245.000 q^{49} -228.631 q^{50} +265.004 q^{51} -1349.27 q^{52} -45.0000 q^{53} -140.296 q^{54} +1386.00 q^{56} -436.477 q^{57} +459.000 q^{58} +378.000 q^{59} -513.000 q^{60} -623.538 q^{61} -976.877 q^{62} +218.238 q^{63} +379.000 q^{64} -639.127 q^{65} -386.000 q^{67} -1678.36 q^{68} -270.000 q^{69} +1134.00 q^{70} -198.000 q^{71} +514.419 q^{72} -76.2102 q^{73} +691.088 q^{74} +132.000 q^{75} +2764.35 q^{76} +1107.00 q^{78} +152.420 q^{79} +1305.00 q^{80} +81.0000 q^{81} -189.000 q^{82} -1247.08 q^{83} -1382.18 q^{84} -795.011 q^{85} +378.000 q^{86} -265.004 q^{87} +45.0000 q^{89} +420.888 q^{90} -1722.00 q^{91} +1710.00 q^{92} +564.000 q^{93} +374.123 q^{94} +1309.43 q^{95} -888.542 q^{96} +89.0000 q^{97} +1273.06 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9} - 114 q^{12} + 252 q^{14} - 54 q^{15} + 290 q^{16} + 342 q^{20} + 180 q^{23} - 88 q^{25} - 738 q^{26} - 54 q^{27} - 376 q^{31} - 918 q^{34} + 342 q^{36} + 266 q^{37}+ \cdots + 178 q^{97}+O(q^{100}) $ 2 * q - 6 * q^3 + 38 * q^4 + 18 * q^5 + 18 * q^9 - 114 * q^12 + 252 * q^14 - 54 * q^15 + 290 * q^16 + 342 * q^20 + 180 * q^23 - 88 * q^25 - 738 * q^26 - 54 * q^27 - 376 * q^31 - 918 * q^34 + 342 * q^36 + 266 * q^37 + 1512 * q^38 - 756 * q^42 + 162 * q^45 + 144 * q^47 - 870 * q^48 + 490 * q^49 - 90 * q^53 + 2772 * q^56 + 918 * q^58 + 756 * q^59 - 1026 * q^60 + 758 * q^64 - 772 * q^67 - 540 * q^69 + 2268 * q^70 - 396 * q^71 + 264 * q^75 + 2214 * q^78 + 2610 * q^80 + 162 * q^81 - 378 * q^82 + 756 * q^86 + 90 * q^89 - 3444 * q^91 + 3420 * q^92 + 1128 * q^93 + 178 * q^97

Coefficient data

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

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display 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 $

$2$	5.19615	1.83712	0.918559	−	0.395285i	$-0.129354\pi$
$2$	5.19615	1.83712	0.918559	+	0.395285i	$0.129354\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	19.0000	2.37500
$5$	9.00000	0.804984	0.402492	−	0.915423i	$-0.368144\pi$
$5$	9.00000	0.804984	0.402492	+	0.915423i	$0.368144\pi$
$6$	−15.5885	−1.06066
$7$	24.2487	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	24.2487	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	57.1577	2.52604
$9$	9.00000	0.333333
$10$	46.7654	1.47885
$11$	0	0
$11$	0	0
$12$	−57.0000	−1.37121
$13$	−71.0141	−1.51506	−0.757529	−	0.652801i	$-0.773594\pi$
$13$	−71.0141	−1.51506	−0.757529	+	0.652801i	$0.773594\pi$
$14$	126.000	2.40535
$15$	−27.0000	−0.464758
$16$	145.000	2.26562
$17$	−88.3346	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−88.3346	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	46.7654	0.612372
$19$	145.492	1.75675	0.878374	−	0.477974i	$-0.158629\pi$
$19$	145.492	1.75675	0.878374	+	0.477974i	$0.158629\pi$
$20$	171.000	1.91184
$21$	−72.7461	−0.755929
$22$	0	0
$23$	90.0000	0.815926	0.407963	−	0.912998i	$-0.366239\pi$
$23$	90.0000	0.815926	0.407963	+	0.912998i	$0.366239\pi$
$24$	−171.473	−1.45841
$25$	−44.0000	−0.352000
$26$	−369.000	−2.78334
$27$	−27.0000	−0.192450
$28$	460.726	3.10960
$29$	88.3346	0.565632	0.282816	−	0.959174i	$-0.408731\pi$
$29$	88.3346	0.565632	0.282816	+	0.959174i	$0.408731\pi$
$30$	−140.296	−0.853815
$31$	−188.000	−1.08922	−0.544610	−	0.838690i	$-0.683322\pi$
$31$	−188.000	−1.08922	−0.544610	+	0.838690i	$0.683322\pi$
$32$	296.181	1.63618
$33$	0	0
$34$	−459.000	−2.31523
$35$	218.238	1.05397
$36$	171.000	0.791667
$37$	133.000	0.590948	0.295474	−	0.955351i	$-0.404522\pi$
$37$	133.000	0.590948	0.295474	+	0.955351i	$0.404522\pi$
$38$	756.000	3.22735
$39$	213.042	0.874720
$40$	514.419	2.03342
$41$	−36.3731	−0.138549	−0.0692746	−	0.997598i	$-0.522068\pi$
$41$	−36.3731	−0.138549	−0.0692746	+	0.997598i	$0.522068\pi$
$42$	−378.000	−1.38873
$43$	72.7461	0.257993	0.128996	−	0.991645i	$-0.458824\pi$
$43$	72.7461	0.257993	0.128996	+	0.991645i	$0.458824\pi$
$44$	0	0
$45$	81.0000	0.268328
$46$	467.654	1.49895
$47$	72.0000	0.223453	0.111726	−	0.993739i	$-0.464362\pi$
$47$	72.0000	0.223453	0.111726	+	0.993739i	$0.464362\pi$
$48$	−435.000	−1.30806
$49$	245.000	0.714286
$50$	−228.631	−0.646665
$51$	265.004	0.727607
$52$	−1349.27	−3.59826
$53$	−45.0000	−0.116627	−0.0583134	−	0.998298i	$-0.518572\pi$
$53$	−45.0000	−0.116627	−0.0583134	+	0.998298i	$0.518572\pi$
$54$	−140.296	−0.353553
$55$	0	0
$56$	1386.00	3.30736
$57$	−436.477	−1.01426
$58$	459.000	1.03913
$59$	378.000	0.834092	0.417046	−	0.908885i	$-0.363065\pi$
$59$	378.000	0.834092	0.417046	+	0.908885i	$0.363065\pi$
$60$	−513.000	−1.10380
$61$	−623.538	−1.30879	−0.654393	−	0.756155i	$-0.727076\pi$
$61$	−623.538	−1.30879	−0.654393	+	0.756155i	$0.727076\pi$
$62$	−976.877	−2.00102
$63$	218.238	0.436436
$64$	379.000	0.740234
$65$	−639.127	−1.21960
$66$	0	0
$67$	−386.000	−0.703842	−0.351921	−	0.936030i	$-0.614471\pi$
$67$	−386.000	−0.703842	−0.351921	+	0.936030i	$0.614471\pi$
$68$	−1678.36	−2.99310
$69$	−270.000	−0.471075
$70$	1134.00	1.93627
$71$	−198.000	−0.330962	−0.165481	−	0.986213i	$-0.552918\pi$
$71$	−198.000	−0.330962	−0.165481	+	0.986213i	$0.552918\pi$
$72$	514.419	0.842012
$73$	−76.2102	−0.122188	−0.0610941	−	0.998132i	$-0.519459\pi$
$73$	−76.2102	−0.122188	−0.0610941	+	0.998132i	$0.519459\pi$
$74$	691.088	1.08564
$75$	132.000	0.203227
$76$	2764.35	4.17228
$77$	0	0
$78$	1107.00	1.60696
$79$	152.420	0.217071	0.108536	−	0.994093i	$-0.465384\pi$
$79$	152.420	0.217071	0.108536	+	0.994093i	$0.465384\pi$
$80$	1305.00	1.82379
$81$	81.0000	0.111111
$82$	−189.000	−0.254531
$83$	−1247.08	−1.64921	−0.824605	−	0.565709i	$-0.808603\pi$
$83$	−1247.08	−1.64921	−0.824605	+	0.565709i	$0.808603\pi$
$84$	−1382.18	−1.79533
$85$	−795.011	−1.01448
$86$	378.000	0.473963
$87$	−265.004	−0.326568
$88$	0	0
$89$	45.0000	0.0535954	0.0267977	−	0.999641i	$-0.491469\pi$
$89$	45.0000	0.0535954	0.0267977	+	0.999641i	$0.491469\pi$
$90$	420.888	0.492950
$91$	−1722.00	−1.98368
$92$	1710.00	1.93782
$93$	564.000	0.628861
$94$	374.123	0.410509
$95$	1309.43	1.41416
$96$	−888.542	−0.944650
$97$	89.0000	0.0931606	0.0465803	−	0.998915i	$-0.485168\pi$
$97$	89.0000	0.0931606	0.0465803	+	0.998915i	$0.485168\pi$
$98$	1273.06	1.31223
$99$	0	0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.4.a.l.1.2	yes	2
3.2	odd	2		1089.4.a.s.1.1		2
11.10	odd	2	inner	363.4.a.l.1.1	✓	2
33.32	even	2		1089.4.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.4.a.l.1.1	✓	2	11.10	odd	2	inner
363.4.a.l.1.2	yes	2	1.1	even	1	trivial
1089.4.a.s.1.1		2	3.2	odd	2
1089.4.a.s.1.2		2	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+5.19615 q^{2} -3.00000 q^{3} +19.0000 q^{4} +9.00000 q^{5} -15.5885 q^{6} +24.2487 q^{7} +57.1577 q^{8} +9.00000 q^{9} +46.7654 q^{10} -57.0000 q^{12} -71.0141 q^{13} +126.000 q^{14} -27.0000 q^{15} +145.000 q^{16} -88.3346 q^{17} +46.7654 q^{18} +145.492 q^{19} +171.000 q^{20} -72.7461 q^{21} +90.0000 q^{23} -171.473 q^{24} -44.0000 q^{25} -369.000 q^{26} -27.0000 q^{27} +460.726 q^{28} +88.3346 q^{29} -140.296 q^{30} -188.000 q^{31} +296.181 q^{32} -459.000 q^{34} +218.238 q^{35} +171.000 q^{36} +133.000 q^{37} +756.000 q^{38} +213.042 q^{39} +514.419 q^{40} -36.3731 q^{41} -378.000 q^{42} +72.7461 q^{43} +81.0000 q^{45} +467.654 q^{46} +72.0000 q^{47} -435.000 q^{48} +245.000 q^{49} -228.631 q^{50} +265.004 q^{51} -1349.27 q^{52} -45.0000 q^{53} -140.296 q^{54} +1386.00 q^{56} -436.477 q^{57} +459.000 q^{58} +378.000 q^{59} -513.000 q^{60} -623.538 q^{61} -976.877 q^{62} +218.238 q^{63} +379.000 q^{64} -639.127 q^{65} -386.000 q^{67} -1678.36 q^{68} -270.000 q^{69} +1134.00 q^{70} -198.000 q^{71} +514.419 q^{72} -76.2102 q^{73} +691.088 q^{74} +132.000 q^{75} +2764.35 q^{76} +1107.00 q^{78} +152.420 q^{79} +1305.00 q^{80} +81.0000 q^{81} -189.000 q^{82} -1247.08 q^{83} -1382.18 q^{84} -795.011 q^{85} +378.000 q^{86} -265.004 q^{87} +45.0000 q^{89} +420.888 q^{90} -1722.00 q^{91} +1710.00 q^{92} +564.000 q^{93} +374.123 q^{94} +1309.43 q^{95} -888.542 q^{96} +89.0000 q^{97} +1273.06 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 38 q^{4} + 18 q^{5} + 18 q^{9} - 114 q^{12} + 252 q^{14} - 54 q^{15} + 290 q^{16} + 342 q^{20} + 180 q^{23} - 88 q^{25} - 738 q^{26} - 54 q^{27} - 376 q^{31} - 918 q^{34} + 342 q^{36} + 266 q^{37}+ \cdots + 178 q^{97}+O(q^{100}) \) 2 * q - 6 * q^3 + 38 * q^4 + 18 * q^5 + 18 * q^9 - 114 * q^12 + 252 * q^14 - 54 * q^15 + 290 * q^16 + 342 * q^20 + 180 * q^23 - 88 * q^25 - 738 * q^26 - 54 * q^27 - 376 * q^31 - 918 * q^34 + 342 * q^36 + 266 * q^37 + 1512 * q^38 - 756 * q^42 + 162 * q^45 + 144 * q^47 - 870 * q^48 + 490 * q^49 - 90 * q^53 + 2772 * q^56 + 918 * q^58 + 756 * q^59 - 1026 * q^60 + 758 * q^64 - 772 * q^67 - 540 * q^69 + 2268 * q^70 - 396 * q^71 + 264 * q^75 + 2214 * q^78 + 2610 * q^80 + 162 * q^81 - 378 * q^82 + 756 * q^86 + 90 * q^89 - 3444 * q^91 + 3420 * q^92 + 1128 * q^93 + 178 * q^97

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