LMFDB - Embedded newform 2475.2.a.bb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2475,2,Mod(1,2475)] 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("2475.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,5,0,0,0,-9,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} -1.00000 q^{11} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} +7.05086 q^{19} -1.90321 q^{22} +8.85728 q^{23} +1.18421 q^{26} +7.18421 q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} +2.00000 q^{37} +13.4193 q^{38} +0.193576 q^{41} -5.67307 q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557 q^{47} +12.6128 q^{49} +1.00937 q^{52} -10.8573 q^{53} -3.18421 q^{56} +14.8573 q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} +1.24443 q^{67} -8.40990 q^{68} -2.75557 q^{71} -4.23506 q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864 q^{77} +8.56199 q^{79} +0.368416 q^{82} +0.133353 q^{83} -10.7971 q^{86} +0.719004 q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684 q^{92} -5.24443 q^{94} -7.24443 q^{97} +24.0049 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41}+ \cdots + 39 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 22 * q^97 + 39 * q^98

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 $

$2$	1.90321	1.34577	0.672887	−	0.739745i	$-0.265054\pi$
$2$	1.90321	1.34577	0.672887	+	0.739745i	$0.265054\pi$
$3$	0	0
$3$	0	0
$4$	1.62222	0.811108
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.42864	1.67387	0.836934	−	0.547304i	$-0.184346\pi$
$7$	4.42864	1.67387	0.836934	+	0.547304i	$0.184346\pi$
$8$	−0.719004	−0.254206
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.622216	0.172572	0.0862858	−	0.996270i	$-0.472500\pi$
$13$	0.622216	0.172572	0.0862858	+	0.996270i	$0.472500\pi$
$14$	8.42864	2.25265
$15$	0	0
$16$	−4.61285	−1.15321
$17$	−5.18421	−1.25736	−0.628678	−	0.777666i	$-0.716403\pi$
$17$	−5.18421	−1.25736	−0.628678	+	0.777666i	$0.716403\pi$
$18$	0	0
$19$	7.05086	1.61758	0.808789	−	0.588100i	$-0.200124\pi$
$19$	7.05086	1.61758	0.808789	+	0.588100i	$0.200124\pi$
$20$	0	0
$21$	0	0
$22$	−1.90321	−0.405766
$23$	8.85728	1.84687	0.923435	−	0.383754i	$-0.125369\pi$
$23$	8.85728	1.84687	0.923435	+	0.383754i	$0.125369\pi$
$24$	0	0
$25$	0	0
$26$	1.18421	0.232242
$27$	0	0
$28$	7.18421	1.35769
$29$	7.80642	1.44962	0.724808	−	0.688951i	$-0.241928\pi$
$29$	7.80642	1.44962	0.724808	+	0.688951i	$0.241928\pi$
$30$	0	0
$31$	2.75557	0.494915	0.247457	−	0.968899i	$-0.420405\pi$
$31$	2.75557	0.494915	0.247457	+	0.968899i	$0.420405\pi$
$32$	−7.34122	−1.29776
$33$	0	0
$34$	−9.86665	−1.69212
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	13.4193	2.17689
$39$	0	0
$40$	0	0
$41$	0.193576	0.0302315	0.0151158	−	0.999886i	$-0.495188\pi$
$41$	0.193576	0.0302315	0.0151158	+	0.999886i	$0.495188\pi$
$42$	0	0
$43$	−5.67307	−0.865135	−0.432568	−	0.901602i	$-0.642392\pi$
$43$	−5.67307	−0.865135	−0.432568	+	0.901602i	$0.642392\pi$
$44$	−1.62222	−0.244558
$45$	0	0
$46$	16.8573	2.48547
$47$	−2.75557	−0.401941	−0.200971	−	0.979597i	$-0.564410\pi$
$47$	−2.75557	−0.401941	−0.200971	+	0.979597i	$0.564410\pi$
$48$	0	0
$49$	12.6128	1.80184
$50$	0	0
$51$	0	0
$52$	1.00937	0.139974
$53$	−10.8573	−1.49136	−0.745681	−	0.666303i	$-0.767876\pi$
$53$	−10.8573	−1.49136	−0.745681	+	0.666303i	$0.767876\pi$
$54$	0	0
$55$	0	0
$56$	−3.18421	−0.425508
$57$	0	0
$58$	14.8573	1.95086
$59$	4.85728	0.632364	0.316182	−	0.948699i	$-0.397599\pi$
$59$	4.85728	0.632364	0.316182	+	0.948699i	$0.397599\pi$
$60$	0	0
$61$	6.85728	0.877985	0.438992	−	0.898491i	$-0.355336\pi$
$61$	6.85728	0.877985	0.438992	+	0.898491i	$0.355336\pi$
$62$	5.24443	0.666043
$63$	0	0
$64$	−4.74620	−0.593275
$65$	0	0
$66$	0	0
$67$	1.24443	0.152031	0.0760157	−	0.997107i	$-0.475780\pi$
$67$	1.24443	0.152031	0.0760157	+	0.997107i	$0.475780\pi$
$68$	−8.40990	−1.01985
$69$	0	0
$70$	0	0
$71$	−2.75557	−0.327026	−0.163513	−	0.986541i	$-0.552283\pi$
$71$	−2.75557	−0.327026	−0.163513	+	0.986541i	$0.552283\pi$
$72$	0	0
$73$	−4.23506	−0.495677	−0.247838	−	0.968801i	$-0.579720\pi$
$73$	−4.23506	−0.495677	−0.247838	+	0.968801i	$0.579720\pi$
$74$	3.80642	0.442488
$75$	0	0
$76$	11.4380	1.31203
$77$	−4.42864	−0.504690
$78$	0	0
$79$	8.56199	0.963299	0.481650	−	0.876364i	$-0.340038\pi$
$79$	8.56199	0.963299	0.481650	+	0.876364i	$0.340038\pi$
$80$	0	0
$81$	0	0
$82$	0.368416	0.0406848
$83$	0.133353	0.0146374	0.00731870	−	0.999973i	$-0.497670\pi$
$83$	0.133353	0.0146374	0.00731870	+	0.999973i	$0.497670\pi$
$84$	0	0
$85$	0	0
$86$	−10.7971	−1.16428
$87$	0	0
$88$	0.719004	0.0766461
$89$	−5.61285	−0.594961	−0.297480	−	0.954728i	$-0.596146\pi$
$89$	−5.61285	−0.594961	−0.297480	+	0.954728i	$0.596146\pi$
$90$	0	0
$91$	2.75557	0.288862
$92$	14.3684	1.49801
$93$	0	0
$94$	−5.24443	−0.540922
$95$	0	0
$96$	0	0
$97$	−7.24443	−0.735561	−0.367780	−	0.929913i	$-0.619882\pi$
$97$	−7.24443	−0.735561	−0.367780	+	0.929913i	$0.619882\pi$
$98$	24.0049	2.42486
$99$	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	2475.2.a.bb.1.3		3
3.2	odd	2		825.2.a.k.1.1		3
5.2	odd	4		2475.2.c.r.199.5		6
5.3	odd	4		2475.2.c.r.199.2		6
5.4	even	2		495.2.a.e.1.1		3
15.2	even	4		825.2.c.g.199.2		6
15.8	even	4		825.2.c.g.199.5		6
15.14	odd	2		165.2.a.c.1.3	✓	3
20.19	odd	2		7920.2.a.cj.1.3		3
33.32	even	2		9075.2.a.cf.1.3		3
55.54	odd	2		5445.2.a.z.1.3		3
60.59	even	2		2640.2.a.be.1.3		3
105.104	even	2		8085.2.a.bk.1.3		3
165.164	even	2		1815.2.a.m.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.3	✓	3	15.14	odd	2
495.2.a.e.1.1		3	5.4	even	2
825.2.a.k.1.1		3	3.2	odd	2
825.2.c.g.199.2		6	15.2	even	4
825.2.c.g.199.5		6	15.8	even	4
1815.2.a.m.1.1		3	165.164	even	2
2475.2.a.bb.1.3		3	1.1	even	1	trivial
2475.2.c.r.199.2		6	5.3	odd	4
2475.2.c.r.199.5		6	5.2	odd	4
2640.2.a.be.1.3		3	60.59	even	2
5445.2.a.z.1.3		3	55.54	odd	2
7920.2.a.cj.1.3		3	20.19	odd	2
8085.2.a.bk.1.3		3	105.104	even	2
9075.2.a.cf.1.3		3	33.32	even	2

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Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+1.90321 q^{2} +1.62222 q^{4} +4.42864 q^{7} -0.719004 q^{8} -1.00000 q^{11} +0.622216 q^{13} +8.42864 q^{14} -4.61285 q^{16} -5.18421 q^{17} +7.05086 q^{19} -1.90321 q^{22} +8.85728 q^{23} +1.18421 q^{26} +7.18421 q^{28} +7.80642 q^{29} +2.75557 q^{31} -7.34122 q^{32} -9.86665 q^{34} +2.00000 q^{37} +13.4193 q^{38} +0.193576 q^{41} -5.67307 q^{43} -1.62222 q^{44} +16.8573 q^{46} -2.75557 q^{47} +12.6128 q^{49} +1.00937 q^{52} -10.8573 q^{53} -3.18421 q^{56} +14.8573 q^{58} +4.85728 q^{59} +6.85728 q^{61} +5.24443 q^{62} -4.74620 q^{64} +1.24443 q^{67} -8.40990 q^{68} -2.75557 q^{71} -4.23506 q^{73} +3.80642 q^{74} +11.4380 q^{76} -4.42864 q^{77} +8.56199 q^{79} +0.368416 q^{82} +0.133353 q^{83} -10.7971 q^{86} +0.719004 q^{88} -5.61285 q^{89} +2.75557 q^{91} +14.3684 q^{92} -5.24443 q^{94} -7.24443 q^{97} +24.0049 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 9 q^{8} - 3 q^{11} + 2 q^{13} + 12 q^{14} + 13 q^{16} - 2 q^{17} + 8 q^{19} + q^{22} - 10 q^{26} + 8 q^{28} + 10 q^{29} + 8 q^{31} - 29 q^{32} - 30 q^{34} + 6 q^{37} + 14 q^{41}+ \cdots + 39 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 9 * q^8 - 3 * q^11 + 2 * q^13 + 12 * q^14 + 13 * q^16 - 2 * q^17 + 8 * q^19 + q^22 - 10 * q^26 + 8 * q^28 + 10 * q^29 + 8 * q^31 - 29 * q^32 - 30 * q^34 + 6 * q^37 + 14 * q^41 - 4 * q^43 - 5 * q^44 + 24 * q^46 - 8 * q^47 + 11 * q^49 + 30 * q^52 - 6 * q^53 + 4 * q^56 + 18 * q^58 - 12 * q^59 - 6 * q^61 + 16 * q^62 + 13 * q^64 + 4 * q^67 + 42 * q^68 - 8 * q^71 + 14 * q^73 - 2 * q^74 + 48 * q^76 + 12 * q^79 - 26 * q^82 + 8 * q^86 + 9 * q^88 + 10 * q^89 + 8 * q^91 + 16 * q^92 - 16 * q^94 - 22 * q^97 + 39 * q^98

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