LMFDB - Embedded newform 7406.2.a.bd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7406,2,Mod(1,7406)] 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("7406.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7406, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,4,0,4,4,0,4] 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:	$59.1372077370$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.449797.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - x + 4 $ x^4 - 10*x^2 - x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.03941$ of defining polynomial
Character	$\chi$	$=$	7406.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} -3.03941 q^{3} +1.00000 q^{4} -2.03941 q^{5} -3.03941 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.23799 q^{9} -2.03941 q^{10} +0.723361 q^{11} -3.03941 q^{12} -6.23799 q^{13} +1.00000 q^{14} +6.19859 q^{15} +1.00000 q^{16} -5.76277 q^{17} +6.23799 q^{18} +2.31605 q^{19} -2.03941 q^{20} -3.03941 q^{21} +0.723361 q^{22} -3.03941 q^{24} -0.840820 q^{25} -6.23799 q^{26} -9.84158 q^{27} +1.00000 q^{28} +6.76277 q^{29} +6.19859 q^{30} -8.27740 q^{31} +1.00000 q^{32} -2.19859 q^{33} -5.76277 q^{34} -2.03941 q^{35} +6.23799 q^{36} -6.27740 q^{37} +2.31605 q^{38} +18.9598 q^{39} -2.03941 q^{40} +10.3168 q^{41} -3.03941 q^{42} -4.39486 q^{43} +0.723361 q^{44} -12.7218 q^{45} -2.92195 q^{47} -3.03941 q^{48} +1.00000 q^{49} -0.840820 q^{50} +17.5154 q^{51} -6.23799 q^{52} +8.76277 q^{53} -9.84158 q^{54} -1.47523 q^{55} +1.00000 q^{56} -7.03941 q^{57} +6.76277 q^{58} -4.51463 q^{59} +6.19859 q^{60} +12.7628 q^{61} -8.27740 q^{62} +6.23799 q^{63} +1.00000 q^{64} +12.7218 q^{65} -2.19859 q^{66} +5.96135 q^{67} -5.76277 q^{68} -2.03941 q^{70} -13.3555 q^{71} +6.23799 q^{72} -5.63209 q^{73} -6.27740 q^{74} +2.55559 q^{75} +2.31605 q^{76} +0.723361 q^{77} +18.9598 q^{78} -10.3562 q^{79} -2.03941 q^{80} +11.1986 q^{81} +10.3168 q^{82} -1.23799 q^{83} -3.03941 q^{84} +11.7526 q^{85} -4.39486 q^{86} -20.5548 q^{87} +0.723361 q^{88} +1.11746 q^{89} -12.7218 q^{90} -6.23799 q^{91} +25.1584 q^{93} -2.92195 q^{94} -4.72336 q^{95} -3.03941 q^{96} -15.4373 q^{97} +1.00000 q^{98} +4.51232 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 4 q^{7} + 4 q^{8} + 8 q^{9} + 4 q^{10} - 3 q^{11} - 8 q^{13} + 4 q^{14} + 20 q^{15} + 4 q^{16} - 5 q^{17} + 8 q^{18} + 3 q^{19} + 4 q^{20} - 3 q^{22} + 4 q^{25} - 8 q^{26}+ \cdots - 14 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 4 * q^7 + 4 * q^8 + 8 * q^9 + 4 * q^10 - 3 * q^11 - 8 * q^13 + 4 * q^14 + 20 * q^15 + 4 * q^16 - 5 * q^17 + 8 * q^18 + 3 * q^19 + 4 * q^20 - 3 * q^22 + 4 * q^25 - 8 * q^26 + 3 * q^27 + 4 * q^28 + 9 * q^29 + 20 * q^30 - 4 * q^31 + 4 * q^32 - 4 * q^33 - 5 * q^34 + 4 * q^35 + 8 * q^36 + 4 * q^37 + 3 * q^38 - 3 * q^39 + 4 * q^40 + 13 * q^43 - 3 * q^44 + 11 * q^45 - q^47 + 4 * q^49 + 4 * q^50 + 24 * q^51 - 8 * q^52 + 17 * q^53 + 3 * q^54 - 7 * q^55 + 4 * q^56 - 16 * q^57 + 9 * q^58 - 7 * q^59 + 20 * q^60 + 33 * q^61 - 4 * q^62 + 8 * q^63 + 4 * q^64 - 11 * q^65 - 4 * q^66 + q^67 - 5 * q^68 + 4 * q^70 - 35 * q^71 + 8 * q^72 - 10 * q^73 + 4 * q^74 + 43 * q^75 + 3 * q^76 - 3 * q^77 - 3 * q^78 + 12 * q^79 + 4 * q^80 + 40 * q^81 + 12 * q^83 + 19 * q^85 + 13 * q^86 - 24 * q^87 - 3 * q^88 + 3 * q^89 + 11 * q^90 - 8 * q^91 + 17 * q^93 - q^94 - 13 * q^95 - 5 * q^97 + 4 * q^98 - 14 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.03941	−1.75480	−0.877401	−	0.479757i	$-0.840725\pi$
$3$	−3.03941	−1.75480	−0.877401	+	0.479757i	$0.840725\pi$
$4$	1.00000	0.500000
$5$	−2.03941	−0.912050	−0.456025	−	0.889967i	$-0.650727\pi$
$5$	−2.03941	−0.912050	−0.456025	+	0.889967i	$0.650727\pi$
$6$	−3.03941	−1.24083
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	6.23799	2.07933
$10$	−2.03941	−0.644917
$11$	0.723361	0.218101	0.109051	−	0.994036i	$-0.465219\pi$
$11$	0.723361	0.218101	0.109051	+	0.994036i	$0.465219\pi$
$12$	−3.03941	−0.877401
$13$	−6.23799	−1.73011	−0.865054	−	0.501679i	$-0.832716\pi$
$13$	−6.23799	−1.73011	−0.865054	+	0.501679i	$0.832716\pi$
$14$	1.00000	0.267261
$15$	6.19859	1.60047
$16$	1.00000	0.250000
$17$	−5.76277	−1.39768	−0.698838	−	0.715280i	$-0.746299\pi$
$17$	−5.76277	−1.39768	−0.698838	+	0.715280i	$0.746299\pi$
$18$	6.23799	1.47031
$19$	2.31605	0.531337	0.265669	−	0.964064i	$-0.414407\pi$
$19$	2.31605	0.531337	0.265669	+	0.964064i	$0.414407\pi$
$20$	−2.03941	−0.456025
$21$	−3.03941	−0.663253
$22$	0.723361	0.154221
$23$	0	0
$23$	0	0
$24$	−3.03941	−0.620416
$25$	−0.840820	−0.168164
$26$	−6.23799	−1.22337
$27$	−9.84158	−1.89401
$28$	1.00000	0.188982
$29$	6.76277	1.25581	0.627907	−	0.778288i	$-0.283912\pi$
$29$	6.76277	1.25581	0.627907	+	0.778288i	$0.283912\pi$
$30$	6.19859	1.13170
$31$	−8.27740	−1.48667	−0.743333	−	0.668922i	$-0.766756\pi$
$31$	−8.27740	−1.48667	−0.743333	+	0.668922i	$0.766756\pi$
$32$	1.00000	0.176777
$33$	−2.19859	−0.382725
$34$	−5.76277	−0.988306
$35$	−2.03941	−0.344723
$36$	6.23799	1.03967
$37$	−6.27740	−1.03200	−0.515999	−	0.856589i	$-0.672579\pi$
$37$	−6.27740	−1.03200	−0.515999	+	0.856589i	$0.672579\pi$
$38$	2.31605	0.375712
$39$	18.9598	3.03600
$40$	−2.03941	−0.322459
$41$	10.3168	1.61121	0.805607	−	0.592450i	$-0.201839\pi$
$41$	10.3168	1.61121	0.805607	+	0.592450i	$0.201839\pi$
$42$	−3.03941	−0.468991
$43$	−4.39486	−0.670210	−0.335105	−	0.942181i	$-0.608772\pi$
$43$	−4.39486	−0.670210	−0.335105	+	0.942181i	$0.608772\pi$
$44$	0.723361	0.109051
$45$	−12.7218	−1.89646
$46$	0	0
$47$	−2.92195	−0.426210	−0.213105	−	0.977029i	$-0.568358\pi$
$47$	−2.92195	−0.426210	−0.213105	+	0.977029i	$0.568358\pi$
$48$	−3.03941	−0.438701
$49$	1.00000	0.142857
$50$	−0.840820	−0.118910
$51$	17.5154	2.45265
$52$	−6.23799	−0.865054
$53$	8.76277	1.20366	0.601829	−	0.798625i	$-0.294439\pi$
$53$	8.76277	1.20366	0.601829	+	0.798625i	$0.294439\pi$
$54$	−9.84158	−1.33927
$55$	−1.47523	−0.198920
$56$	1.00000	0.133631
$57$	−7.03941	−0.932392
$58$	6.76277	0.887995
$59$	−4.51463	−0.587755	−0.293878	−	0.955843i	$-0.594946\pi$
$59$	−4.51463	−0.587755	−0.293878	+	0.955843i	$0.594946\pi$
$60$	6.19859	0.800234
$61$	12.7628	1.63410	0.817052	−	0.576563i	$-0.195607\pi$
$61$	12.7628	1.63410	0.817052	+	0.576563i	$0.195607\pi$
$62$	−8.27740	−1.05123
$63$	6.23799	0.785913
$64$	1.00000	0.125000
$65$	12.7218	1.57795
$66$	−2.19859	−0.270627
$67$	5.96135	0.728295	0.364148	−	0.931341i	$-0.381360\pi$
$67$	5.96135	0.728295	0.364148	+	0.931341i	$0.381360\pi$
$68$	−5.76277	−0.698838
$69$	0	0
$70$	−2.03941	−0.243756
$71$	−13.3555	−1.58500	−0.792500	−	0.609871i	$-0.791221\pi$
$71$	−13.3555	−1.58500	−0.792500	+	0.609871i	$0.791221\pi$
$72$	6.23799	0.735155
$73$	−5.63209	−0.659187	−0.329593	−	0.944123i	$-0.606912\pi$
$73$	−5.63209	−0.659187	−0.329593	+	0.944123i	$0.606912\pi$
$74$	−6.27740	−0.729733
$75$	2.55559	0.295094
$76$	2.31605	0.265669
$77$	0.723361	0.0824346
$78$	18.9598	2.14677
$79$	−10.3562	−1.16517	−0.582583	−	0.812772i	$-0.697958\pi$
$79$	−10.3562	−1.16517	−0.582583	+	0.812772i	$0.697958\pi$
$80$	−2.03941	−0.228013
$81$	11.1986	1.24429
$82$	10.3168	1.13930
$83$	−1.23799	−0.135887	−0.0679437	−	0.997689i	$-0.521644\pi$
$83$	−1.23799	−0.135887	−0.0679437	+	0.997689i	$0.521644\pi$
$84$	−3.03941	−0.331626
$85$	11.7526	1.27475
$86$	−4.39486	−0.473910
$87$	−20.5548	−2.20371
$88$	0.723361	0.0771105
$89$	1.11746	0.118450	0.0592252	−	0.998245i	$-0.481137\pi$
$89$	1.11746	0.118450	0.0592252	+	0.998245i	$0.481137\pi$
$90$	−12.7218	−1.34100
$91$	−6.23799	−0.653919
$92$	0	0
$93$	25.1584	2.60880
$94$	−2.92195	−0.301376
$95$	−4.72336	−0.484607
$96$	−3.03941	−0.310208
$97$	−15.4373	−1.56742	−0.783712	−	0.621124i	$-0.786676\pi$
$97$	−15.4373	−1.56742	−0.783712	+	0.621124i	$0.786676\pi$
$98$	1.00000	0.101015
$99$	4.51232	0.453505

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	7406.2.a.bd.1.1	yes	4
23.22	odd	2		7406.2.a.ba.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7406.2.a.ba.1.1	✓	4	23.22	odd	2
7406.2.a.bd.1.1	yes	4	1.1	even	1	trivial

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Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.03941 q^{3} +1.00000 q^{4} -2.03941 q^{5} -3.03941 q^{6} +1.00000 q^{7} +1.00000 q^{8} +6.23799 q^{9} -2.03941 q^{10} +0.723361 q^{11} -3.03941 q^{12} -6.23799 q^{13} +1.00000 q^{14} +6.19859 q^{15} +1.00000 q^{16} -5.76277 q^{17} +6.23799 q^{18} +2.31605 q^{19} -2.03941 q^{20} -3.03941 q^{21} +0.723361 q^{22} -3.03941 q^{24} -0.840820 q^{25} -6.23799 q^{26} -9.84158 q^{27} +1.00000 q^{28} +6.76277 q^{29} +6.19859 q^{30} -8.27740 q^{31} +1.00000 q^{32} -2.19859 q^{33} -5.76277 q^{34} -2.03941 q^{35} +6.23799 q^{36} -6.27740 q^{37} +2.31605 q^{38} +18.9598 q^{39} -2.03941 q^{40} +10.3168 q^{41} -3.03941 q^{42} -4.39486 q^{43} +0.723361 q^{44} -12.7218 q^{45} -2.92195 q^{47} -3.03941 q^{48} +1.00000 q^{49} -0.840820 q^{50} +17.5154 q^{51} -6.23799 q^{52} +8.76277 q^{53} -9.84158 q^{54} -1.47523 q^{55} +1.00000 q^{56} -7.03941 q^{57} +6.76277 q^{58} -4.51463 q^{59} +6.19859 q^{60} +12.7628 q^{61} -8.27740 q^{62} +6.23799 q^{63} +1.00000 q^{64} +12.7218 q^{65} -2.19859 q^{66} +5.96135 q^{67} -5.76277 q^{68} -2.03941 q^{70} -13.3555 q^{71} +6.23799 q^{72} -5.63209 q^{73} -6.27740 q^{74} +2.55559 q^{75} +2.31605 q^{76} +0.723361 q^{77} +18.9598 q^{78} -10.3562 q^{79} -2.03941 q^{80} +11.1986 q^{81} +10.3168 q^{82} -1.23799 q^{83} -3.03941 q^{84} +11.7526 q^{85} -4.39486 q^{86} -20.5548 q^{87} +0.723361 q^{88} +1.11746 q^{89} -12.7218 q^{90} -6.23799 q^{91} +25.1584 q^{93} -2.92195 q^{94} -4.72336 q^{95} -3.03941 q^{96} -15.4373 q^{97} +1.00000 q^{98} +4.51232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 4 q^{7} + 4 q^{8} + 8 q^{9} + 4 q^{10} - 3 q^{11} - 8 q^{13} + 4 q^{14} + 20 q^{15} + 4 q^{16} - 5 q^{17} + 8 q^{18} + 3 q^{19} + 4 q^{20} - 3 q^{22} + 4 q^{25} - 8 q^{26}+ \cdots - 14 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^5 + 4 * q^7 + 4 * q^8 + 8 * q^9 + 4 * q^10 - 3 * q^11 - 8 * q^13 + 4 * q^14 + 20 * q^15 + 4 * q^16 - 5 * q^17 + 8 * q^18 + 3 * q^19 + 4 * q^20 - 3 * q^22 + 4 * q^25 - 8 * q^26 + 3 * q^27 + 4 * q^28 + 9 * q^29 + 20 * q^30 - 4 * q^31 + 4 * q^32 - 4 * q^33 - 5 * q^34 + 4 * q^35 + 8 * q^36 + 4 * q^37 + 3 * q^38 - 3 * q^39 + 4 * q^40 + 13 * q^43 - 3 * q^44 + 11 * q^45 - q^47 + 4 * q^49 + 4 * q^50 + 24 * q^51 - 8 * q^52 + 17 * q^53 + 3 * q^54 - 7 * q^55 + 4 * q^56 - 16 * q^57 + 9 * q^58 - 7 * q^59 + 20 * q^60 + 33 * q^61 - 4 * q^62 + 8 * q^63 + 4 * q^64 - 11 * q^65 - 4 * q^66 + q^67 - 5 * q^68 + 4 * q^70 - 35 * q^71 + 8 * q^72 - 10 * q^73 + 4 * q^74 + 43 * q^75 + 3 * q^76 - 3 * q^77 - 3 * q^78 + 12 * q^79 + 4 * q^80 + 40 * q^81 + 12 * q^83 + 19 * q^85 + 13 * q^86 - 24 * q^87 - 3 * q^88 + 3 * q^89 + 11 * q^90 - 8 * q^91 + 17 * q^93 - q^94 - 13 * q^95 - 5 * q^97 + 4 * q^98 - 14 * q^99

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