LMFDB - Embedded newform 7935.2.a.bg.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7935 = 3 \cdot 5 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7935.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,6,4,6,0,-2] 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:	$63.3612940039$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4507648.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.146243$ of defining polynomial
Character	$\chi$	$=$	7935.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.83237 q^{2} +1.00000 q^{3} +1.35758 q^{4} +1.00000 q^{5} +1.83237 q^{6} -1.47931 q^{7} -1.17715 q^{8} +1.00000 q^{9} +1.83237 q^{10} -1.03419 q^{11} +1.35758 q^{12} +2.31168 q^{13} -2.71064 q^{14} +1.00000 q^{15} -4.87214 q^{16} -6.53907 q^{17} +1.83237 q^{18} -2.23773 q^{19} +1.35758 q^{20} -1.47931 q^{21} -1.89502 q^{22} -1.17715 q^{24} +1.00000 q^{25} +4.23585 q^{26} +1.00000 q^{27} -2.00828 q^{28} -0.725637 q^{29} +1.83237 q^{30} -3.74794 q^{31} -6.57326 q^{32} -1.03419 q^{33} -11.9820 q^{34} -1.47931 q^{35} +1.35758 q^{36} +2.04696 q^{37} -4.10034 q^{38} +2.31168 q^{39} -1.17715 q^{40} +12.5264 q^{41} -2.71064 q^{42} -3.10241 q^{43} -1.40400 q^{44} +1.00000 q^{45} -5.34307 q^{47} -4.87214 q^{48} -4.81164 q^{49} +1.83237 q^{50} -6.53907 q^{51} +3.13829 q^{52} -4.03148 q^{53} +1.83237 q^{54} -1.03419 q^{55} +1.74137 q^{56} -2.23773 q^{57} -1.32964 q^{58} -5.43666 q^{59} +1.35758 q^{60} -1.05908 q^{61} -6.86762 q^{62} -1.47931 q^{63} -2.30037 q^{64} +2.31168 q^{65} -1.89502 q^{66} -2.28137 q^{67} -8.87732 q^{68} -2.71064 q^{70} -12.0445 q^{71} -1.17715 q^{72} +1.97157 q^{73} +3.75079 q^{74} +1.00000 q^{75} -3.03789 q^{76} +1.52989 q^{77} +4.23585 q^{78} -5.74694 q^{79} -4.87214 q^{80} +1.00000 q^{81} +22.9529 q^{82} -3.13832 q^{83} -2.00828 q^{84} -6.53907 q^{85} -5.68477 q^{86} -0.725637 q^{87} +1.21739 q^{88} -9.77601 q^{89} +1.83237 q^{90} -3.41969 q^{91} -3.74794 q^{93} -9.79048 q^{94} -2.23773 q^{95} -6.57326 q^{96} -11.1959 q^{97} -8.81671 q^{98} -1.03419 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} - 12 q^{11} + 4 q^{12} - 4 q^{13} - 8 q^{14} + 6 q^{15} - 14 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{21} - 12 q^{22} + 6 q^{25} + 24 q^{26} + 6 q^{27} - 4 q^{28}+ \cdots - 12 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 4 * q^4 + 6 * q^5 - 2 * q^7 + 6 * q^9 - 12 * q^11 + 4 * q^12 - 4 * q^13 - 8 * q^14 + 6 * q^15 - 14 * q^17 - 8 * q^19 + 4 * q^20 - 2 * q^21 - 12 * q^22 + 6 * q^25 + 24 * q^26 + 6 * q^27 - 4 * q^28 - 14 * q^29 - 6 * q^31 - 20 * q^32 - 12 * q^33 - 8 * q^34 - 2 * q^35 + 4 * q^36 + 2 * q^37 - 8 * q^38 - 4 * q^39 - 2 * q^41 - 8 * q^42 - 12 * q^43 - 8 * q^44 + 6 * q^45 + 8 * q^47 - 14 * q^51 - 14 * q^53 - 12 * q^55 - 16 * q^56 - 8 * q^57 + 12 * q^58 - 14 * q^59 + 4 * q^60 + 8 * q^61 - 12 * q^62 - 2 * q^63 - 20 * q^64 - 4 * q^65 - 12 * q^66 + 2 * q^67 - 20 * q^68 - 8 * q^70 - 10 * q^71 - 4 * q^73 - 24 * q^74 + 6 * q^75 - 4 * q^76 + 40 * q^77 + 24 * q^78 + 16 * q^79 + 6 * q^81 + 4 * q^82 + 10 * q^83 - 4 * q^84 - 14 * q^85 - 20 * q^86 - 14 * q^87 - 36 * q^88 - 16 * q^89 - 48 * q^91 - 6 * q^93 - 44 * q^94 - 8 * q^95 - 20 * q^96 - 8 * q^97 - 28 * q^98 - 12 * q^99

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 $

$2$	1.83237	1.29568	0.647841	−	0.761776i	$-0.275672\pi$
$2$	1.83237	1.29568	0.647841	+	0.761776i	$0.275672\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.35758	0.678791
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.83237	0.748062
$7$	−1.47931	−0.559126	−0.279563	−	0.960127i	$-0.590190\pi$
$7$	−1.47931	−0.559126	−0.279563	+	0.960127i	$0.590190\pi$
$8$	−1.17715	−0.416185
$9$	1.00000	0.333333
$10$	1.83237	0.579446
$11$	−1.03419	−0.311820	−0.155910	−	0.987771i	$-0.549831\pi$
$11$	−1.03419	−0.311820	−0.155910	+	0.987771i	$0.549831\pi$
$12$	1.35758	0.391900
$13$	2.31168	0.641145	0.320572	−	0.947224i	$-0.396125\pi$
$13$	2.31168	0.641145	0.320572	+	0.947224i	$0.396125\pi$
$14$	−2.71064	−0.724450
$15$	1.00000	0.258199
$16$	−4.87214	−1.21803
$17$	−6.53907	−1.58596	−0.792979	−	0.609249i	$-0.791471\pi$
$17$	−6.53907	−1.58596	−0.792979	+	0.609249i	$0.791471\pi$
$18$	1.83237	0.431894
$19$	−2.23773	−0.513369	−0.256685	−	0.966495i	$-0.582630\pi$
$19$	−2.23773	−0.513369	−0.256685	+	0.966495i	$0.582630\pi$
$20$	1.35758	0.303564
$21$	−1.47931	−0.322812
$22$	−1.89502	−0.404019
$23$	0	0
$23$	0	0
$24$	−1.17715	−0.240285
$25$	1.00000	0.200000
$26$	4.23585	0.830719
$27$	1.00000	0.192450
$28$	−2.00828	−0.379530
$29$	−0.725637	−0.134747	−0.0673737	−	0.997728i	$-0.521462\pi$
$29$	−0.725637	−0.134747	−0.0673737	+	0.997728i	$0.521462\pi$
$30$	1.83237	0.334544
$31$	−3.74794	−0.673150	−0.336575	−	0.941657i	$-0.609269\pi$
$31$	−3.74794	−0.673150	−0.336575	+	0.941657i	$0.609269\pi$
$32$	−6.57326	−1.16200
$33$	−1.03419	−0.180029
$34$	−11.9820	−2.05490
$35$	−1.47931	−0.250049
$36$	1.35758	0.226264
$37$	2.04696	0.336518	0.168259	−	0.985743i	$-0.446186\pi$
$37$	2.04696	0.336518	0.168259	+	0.985743i	$0.446186\pi$
$38$	−4.10034	−0.665163
$39$	2.31168	0.370165
$40$	−1.17715	−0.186124
$41$	12.5264	1.95629	0.978145	−	0.207924i	$-0.0666705\pi$
$41$	12.5264	1.95629	0.978145	+	0.207924i	$0.0666705\pi$
$42$	−2.71064	−0.418261
$43$	−3.10241	−0.473113	−0.236557	−	0.971618i	$-0.576019\pi$
$43$	−3.10241	−0.473113	−0.236557	+	0.971618i	$0.576019\pi$
$44$	−1.40400	−0.211660
$45$	1.00000	0.149071
$46$	0	0
$47$	−5.34307	−0.779366	−0.389683	−	0.920949i	$-0.627415\pi$
$47$	−5.34307	−0.779366	−0.389683	+	0.920949i	$0.627415\pi$
$48$	−4.87214	−0.703232
$49$	−4.81164	−0.687378
$50$	1.83237	0.259136
$51$	−6.53907	−0.915653
$52$	3.13829	0.435203
$53$	−4.03148	−0.553767	−0.276883	−	0.960904i	$-0.589302\pi$
$53$	−4.03148	−0.553767	−0.276883	+	0.960904i	$0.589302\pi$
$54$	1.83237	0.249354
$55$	−1.03419	−0.139450
$56$	1.74137	0.232700
$57$	−2.23773	−0.296394
$58$	−1.32964	−0.174590
$59$	−5.43666	−0.707792	−0.353896	−	0.935285i	$-0.615143\pi$
$59$	−5.43666	−0.707792	−0.353896	+	0.935285i	$0.615143\pi$
$60$	1.35758	0.175263
$61$	−1.05908	−0.135601	−0.0678007	−	0.997699i	$-0.521598\pi$
$61$	−1.05908	−0.135601	−0.0678007	+	0.997699i	$0.521598\pi$
$62$	−6.86762	−0.872188
$63$	−1.47931	−0.186375
$64$	−2.30037	−0.287546
$65$	2.31168	0.286729
$66$	−1.89502	−0.233260
$67$	−2.28137	−0.278714	−0.139357	−	0.990242i	$-0.544504\pi$
$67$	−2.28137	−0.278714	−0.139357	+	0.990242i	$0.544504\pi$
$68$	−8.87732	−1.07653
$69$	0	0
$70$	−2.71064	−0.323984
$71$	−12.0445	−1.42942	−0.714711	−	0.699420i	$-0.753442\pi$
$71$	−12.0445	−1.42942	−0.714711	+	0.699420i	$0.753442\pi$
$72$	−1.17715	−0.138728
$73$	1.97157	0.230755	0.115377	−	0.993322i	$-0.463192\pi$
$73$	1.97157	0.230755	0.115377	+	0.993322i	$0.463192\pi$
$74$	3.75079	0.436020
$75$	1.00000	0.115470
$76$	−3.03789	−0.348470
$77$	1.52989	0.174347
$78$	4.23585	0.479616
$79$	−5.74694	−0.646582	−0.323291	−	0.946300i	$-0.604789\pi$
$79$	−5.74694	−0.646582	−0.323291	+	0.946300i	$0.604789\pi$
$80$	−4.87214	−0.544721
$81$	1.00000	0.111111
$82$	22.9529	2.53473
$83$	−3.13832	−0.344475	−0.172238	−	0.985055i	$-0.555100\pi$
$83$	−3.13832	−0.344475	−0.172238	+	0.985055i	$0.555100\pi$
$84$	−2.00828	−0.219122
$85$	−6.53907	−0.709262
$86$	−5.68477	−0.613004
$87$	−0.725637	−0.0777965
$88$	1.21739	0.129775
$89$	−9.77601	−1.03626	−0.518128	−	0.855303i	$-0.673371\pi$
$89$	−9.77601	−1.03626	−0.518128	+	0.855303i	$0.673371\pi$
$90$	1.83237	0.193149
$91$	−3.41969	−0.358481
$92$	0	0
$93$	−3.74794	−0.388643
$94$	−9.79048	−1.00981
$95$	−2.23773	−0.229586
$96$	−6.57326	−0.670880
$97$	−11.1959	−1.13677	−0.568386	−	0.822762i	$-0.692432\pi$
$97$	−11.1959	−1.13677	−0.568386	+	0.822762i	$0.692432\pi$
$98$	−8.81671	−0.890623
$99$	−1.03419	−0.103940

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	7935.2.a.bg.1.5	yes	6
23.22	odd	2		7935.2.a.bf.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7935.2.a.bf.1.5	✓	6	23.22	odd	2
7935.2.a.bg.1.5	yes	6	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7935.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83237 q^{2} +1.00000 q^{3} +1.35758 q^{4} +1.00000 q^{5} +1.83237 q^{6} -1.47931 q^{7} -1.17715 q^{8} +1.00000 q^{9} +1.83237 q^{10} -1.03419 q^{11} +1.35758 q^{12} +2.31168 q^{13} -2.71064 q^{14} +1.00000 q^{15} -4.87214 q^{16} -6.53907 q^{17} +1.83237 q^{18} -2.23773 q^{19} +1.35758 q^{20} -1.47931 q^{21} -1.89502 q^{22} -1.17715 q^{24} +1.00000 q^{25} +4.23585 q^{26} +1.00000 q^{27} -2.00828 q^{28} -0.725637 q^{29} +1.83237 q^{30} -3.74794 q^{31} -6.57326 q^{32} -1.03419 q^{33} -11.9820 q^{34} -1.47931 q^{35} +1.35758 q^{36} +2.04696 q^{37} -4.10034 q^{38} +2.31168 q^{39} -1.17715 q^{40} +12.5264 q^{41} -2.71064 q^{42} -3.10241 q^{43} -1.40400 q^{44} +1.00000 q^{45} -5.34307 q^{47} -4.87214 q^{48} -4.81164 q^{49} +1.83237 q^{50} -6.53907 q^{51} +3.13829 q^{52} -4.03148 q^{53} +1.83237 q^{54} -1.03419 q^{55} +1.74137 q^{56} -2.23773 q^{57} -1.32964 q^{58} -5.43666 q^{59} +1.35758 q^{60} -1.05908 q^{61} -6.86762 q^{62} -1.47931 q^{63} -2.30037 q^{64} +2.31168 q^{65} -1.89502 q^{66} -2.28137 q^{67} -8.87732 q^{68} -2.71064 q^{70} -12.0445 q^{71} -1.17715 q^{72} +1.97157 q^{73} +3.75079 q^{74} +1.00000 q^{75} -3.03789 q^{76} +1.52989 q^{77} +4.23585 q^{78} -5.74694 q^{79} -4.87214 q^{80} +1.00000 q^{81} +22.9529 q^{82} -3.13832 q^{83} -2.00828 q^{84} -6.53907 q^{85} -5.68477 q^{86} -0.725637 q^{87} +1.21739 q^{88} -9.77601 q^{89} +1.83237 q^{90} -3.41969 q^{91} -3.74794 q^{93} -9.79048 q^{94} -2.23773 q^{95} -6.57326 q^{96} -11.1959 q^{97} -8.81671 q^{98} -1.03419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} - 12 q^{11} + 4 q^{12} - 4 q^{13} - 8 q^{14} + 6 q^{15} - 14 q^{17} - 8 q^{19} + 4 q^{20} - 2 q^{21} - 12 q^{22} + 6 q^{25} + 24 q^{26} + 6 q^{27} - 4 q^{28}+ \cdots - 12 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 4 * q^4 + 6 * q^5 - 2 * q^7 + 6 * q^9 - 12 * q^11 + 4 * q^12 - 4 * q^13 - 8 * q^14 + 6 * q^15 - 14 * q^17 - 8 * q^19 + 4 * q^20 - 2 * q^21 - 12 * q^22 + 6 * q^25 + 24 * q^26 + 6 * q^27 - 4 * q^28 - 14 * q^29 - 6 * q^31 - 20 * q^32 - 12 * q^33 - 8 * q^34 - 2 * q^35 + 4 * q^36 + 2 * q^37 - 8 * q^38 - 4 * q^39 - 2 * q^41 - 8 * q^42 - 12 * q^43 - 8 * q^44 + 6 * q^45 + 8 * q^47 - 14 * q^51 - 14 * q^53 - 12 * q^55 - 16 * q^56 - 8 * q^57 + 12 * q^58 - 14 * q^59 + 4 * q^60 + 8 * q^61 - 12 * q^62 - 2 * q^63 - 20 * q^64 - 4 * q^65 - 12 * q^66 + 2 * q^67 - 20 * q^68 - 8 * q^70 - 10 * q^71 - 4 * q^73 - 24 * q^74 + 6 * q^75 - 4 * q^76 + 40 * q^77 + 24 * q^78 + 16 * q^79 + 6 * q^81 + 4 * q^82 + 10 * q^83 - 4 * q^84 - 14 * q^85 - 20 * q^86 - 14 * q^87 - 36 * q^88 - 16 * q^89 - 48 * q^91 - 6 * q^93 - 44 * q^94 - 8 * q^95 - 20 * q^96 - 8 * q^97 - 28 * q^98 - 12 * q^99

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