LMFDB - Embedded newform 7406.2.a.w.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 = [3,3,-1,3,4,-1,-3] 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:	$1$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*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			$2.92542$ 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)

$f(q)$	$=$	\(q+1.00000 q^{2} -2.92542 q^{3} +1.00000 q^{4} +3.92542 q^{5} -2.92542 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.55810 q^{9} +3.92542 q^{10} -4.55810 q^{11} -2.92542 q^{12} +3.55810 q^{13} -1.00000 q^{14} -11.4835 q^{15} +1.00000 q^{16} -3.63268 q^{17} +5.55810 q^{18} -7.48352 q^{19} +3.92542 q^{20} +2.92542 q^{21} -4.55810 q^{22} -2.92542 q^{24} +10.4089 q^{25} +3.55810 q^{26} -7.48352 q^{27} -1.00000 q^{28} +2.48352 q^{29} -11.4835 q^{30} +7.48352 q^{31} +1.00000 q^{32} +13.3344 q^{33} -3.63268 q^{34} -3.92542 q^{35} +5.55810 q^{36} -4.36732 q^{37} -7.48352 q^{38} -10.4089 q^{39} +3.92542 q^{40} +0.292747 q^{41} +2.92542 q^{42} +3.48352 q^{43} -4.55810 q^{44} +21.8179 q^{45} -8.77627 q^{47} -2.92542 q^{48} +1.00000 q^{49} +10.4089 q^{50} +10.6271 q^{51} +3.55810 q^{52} -7.21817 q^{53} -7.48352 q^{54} -17.8925 q^{55} -1.00000 q^{56} +21.8925 q^{57} +2.48352 q^{58} -9.11620 q^{59} -11.4835 q^{60} +0.483522 q^{61} +7.48352 q^{62} -5.55810 q^{63} +1.00000 q^{64} +13.9670 q^{65} +13.3344 q^{66} -11.7017 q^{67} -3.63268 q^{68} -3.92542 q^{70} +9.29275 q^{71} +5.55810 q^{72} -10.1162 q^{73} -4.36732 q^{74} -30.4506 q^{75} -7.48352 q^{76} +4.55810 q^{77} -10.4089 q^{78} -14.7489 q^{79} +3.92542 q^{80} +5.21817 q^{81} +0.292747 q^{82} +2.70725 q^{83} +2.92542 q^{84} -14.2598 q^{85} +3.48352 q^{86} -7.26535 q^{87} -4.55810 q^{88} -15.0000 q^{89} +21.8179 q^{90} -3.55810 q^{91} -21.8925 q^{93} -8.77627 q^{94} -29.3760 q^{95} -2.92542 q^{96} +9.43634 q^{97} +1.00000 q^{98} -25.3344 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - q^{3} + 3 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9} + 4 q^{10} - 3 q^{11} - q^{12} - 3 q^{14} - 16 q^{15} + 3 q^{16} - 8 q^{17} + 6 q^{18} - 4 q^{19} + 4 q^{20} + q^{21}+ \cdots - 42 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - q^3 + 3 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 3 * q^8 + 6 * q^9 + 4 * q^10 - 3 * q^11 - q^12 - 3 * q^14 - 16 * q^15 + 3 * q^16 - 8 * q^17 + 6 * q^18 - 4 * q^19 + 4 * q^20 + q^21 - 3 * q^22 - q^24 + 5 * q^25 - 4 * q^27 - 3 * q^28 - 11 * q^29 - 16 * q^30 + 4 * q^31 + 3 * q^32 + 6 * q^33 - 8 * q^34 - 4 * q^35 + 6 * q^36 - 16 * q^37 - 4 * q^38 - 5 * q^39 + 4 * q^40 - 4 * q^41 + q^42 - 8 * q^43 - 3 * q^44 + 13 * q^45 - 3 * q^47 - q^48 + 3 * q^49 + 5 * q^50 - 7 * q^51 - 9 * q^53 - 4 * q^54 - 9 * q^55 - 3 * q^56 + 21 * q^57 - 11 * q^58 - 6 * q^59 - 16 * q^60 - 17 * q^61 + 4 * q^62 - 6 * q^63 + 3 * q^64 + 5 * q^65 + 6 * q^66 - 4 * q^67 - 8 * q^68 - 4 * q^70 + 23 * q^71 + 6 * q^72 - 9 * q^73 - 16 * q^74 - 36 * q^75 - 4 * q^76 + 3 * q^77 - 5 * q^78 - 20 * q^79 + 4 * q^80 + 3 * q^81 - 4 * q^82 + 13 * q^83 + q^84 - q^85 - 8 * q^86 - 16 * q^87 - 3 * q^88 - 45 * q^89 + 13 * q^90 - 21 * q^93 - 3 * q^94 - 25 * q^95 - q^96 + 3 * q^97 + 3 * q^98 - 42 * 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
$3$	−2.92542	−1.68899	−0.844497	−	0.535561i	$-0.820100\pi$
$3$	−2.92542	−1.68899	−0.844497	+	0.535561i	$0.820100\pi$
$4$	1.00000	0.500000
$5$	3.92542	1.75550	0.877751	−	0.479117i	$-0.159043\pi$
$5$	3.92542	1.75550	0.877751	+	0.479117i	$0.159043\pi$
$6$	−2.92542	−1.19430
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	5.55810	1.85270
$10$	3.92542	1.24133
$11$	−4.55810	−1.37432	−0.687159	−	0.726507i	$-0.741143\pi$
$11$	−4.55810	−1.37432	−0.687159	+	0.726507i	$0.741143\pi$
$12$	−2.92542	−0.844497
$13$	3.55810	0.986839	0.493420	−	0.869791i	$-0.335747\pi$
$13$	3.55810	0.986839	0.493420	+	0.869791i	$0.335747\pi$
$14$	−1.00000	−0.267261
$15$	−11.4835	−2.96503
$16$	1.00000	0.250000
$17$	−3.63268	−0.881053	−0.440527	−	0.897740i	$-0.645208\pi$
$17$	−3.63268	−0.881053	−0.440527	+	0.897740i	$0.645208\pi$
$18$	5.55810	1.31006
$19$	−7.48352	−1.71684	−0.858419	−	0.512949i	$-0.828553\pi$
$19$	−7.48352	−1.71684	−0.858419	+	0.512949i	$0.828553\pi$
$20$	3.92542	0.877751
$21$	2.92542	0.638380
$22$	−4.55810	−0.971790
$23$	0	0
$23$	0	0
$24$	−2.92542	−0.597149
$25$	10.4089	2.08179
$26$	3.55810	0.697801
$27$	−7.48352	−1.44020
$28$	−1.00000	−0.188982
$29$	2.48352	0.461178	0.230589	−	0.973051i	$-0.425935\pi$
$29$	2.48352	0.461178	0.230589	+	0.973051i	$0.425935\pi$
$30$	−11.4835	−2.09659
$31$	7.48352	1.34408	0.672040	−	0.740515i	$-0.265418\pi$
$31$	7.48352	1.34408	0.672040	+	0.740515i	$0.265418\pi$
$32$	1.00000	0.176777
$33$	13.3344	2.32122
$34$	−3.63268	−0.622999
$35$	−3.92542	−0.663518
$36$	5.55810	0.926350
$37$	−4.36732	−0.717984	−0.358992	−	0.933341i	$-0.616879\pi$
$37$	−4.36732	−0.717984	−0.358992	+	0.933341i	$0.616879\pi$
$38$	−7.48352	−1.21399
$39$	−10.4089	−1.66677
$40$	3.92542	0.620664
$41$	0.292747	0.0457193	0.0228597	−	0.999739i	$-0.492723\pi$
$41$	0.292747	0.0457193	0.0228597	+	0.999739i	$0.492723\pi$
$42$	2.92542	0.451403
$43$	3.48352	0.531232	0.265616	−	0.964079i	$-0.414425\pi$
$43$	3.48352	0.531232	0.265616	+	0.964079i	$0.414425\pi$
$44$	−4.55810	−0.687159
$45$	21.8179	3.25242
$46$	0	0
$47$	−8.77627	−1.28015	−0.640075	−	0.768312i	$-0.721097\pi$
$47$	−8.77627	−1.28015	−0.640075	+	0.768312i	$0.721097\pi$
$48$	−2.92542	−0.422248
$49$	1.00000	0.142857
$50$	10.4089	1.47205
$51$	10.6271	1.48809
$52$	3.55810	0.493420
$53$	−7.21817	−0.991492	−0.495746	−	0.868468i	$-0.665105\pi$
$53$	−7.21817	−0.991492	−0.495746	+	0.868468i	$0.665105\pi$
$54$	−7.48352	−1.01838
$55$	−17.8925	−2.41262
$56$	−1.00000	−0.133631
$57$	21.8925	2.89973
$58$	2.48352	0.326102
$59$	−9.11620	−1.18683	−0.593414	−	0.804897i	$-0.702220\pi$
$59$	−9.11620	−1.18683	−0.593414	+	0.804897i	$0.702220\pi$
$60$	−11.4835	−1.48252
$61$	0.483522	0.0619087	0.0309543	−	0.999521i	$-0.490145\pi$
$61$	0.483522	0.0619087	0.0309543	+	0.999521i	$0.490145\pi$
$62$	7.48352	0.950408
$63$	−5.55810	−0.700255
$64$	1.00000	0.125000
$65$	13.9670	1.73240
$66$	13.3344	1.64135
$67$	−11.7017	−1.42959	−0.714795	−	0.699334i	$-0.753480\pi$
$67$	−11.7017	−1.42959	−0.714795	+	0.699334i	$0.753480\pi$
$68$	−3.63268	−0.440527
$69$	0	0
$70$	−3.92542	−0.469178
$71$	9.29275	1.10285	0.551423	−	0.834226i	$-0.314085\pi$
$71$	9.29275	1.10285	0.551423	+	0.834226i	$0.314085\pi$
$72$	5.55810	0.655028
$73$	−10.1162	−1.18401	−0.592006	−	0.805934i	$-0.701664\pi$
$73$	−10.1162	−1.18401	−0.592006	+	0.805934i	$0.701664\pi$
$74$	−4.36732	−0.507691
$75$	−30.4506	−3.51613
$76$	−7.48352	−0.858419
$77$	4.55810	0.519444
$78$	−10.4089	−1.17858
$79$	−14.7489	−1.65938	−0.829689	−	0.558226i	$-0.811482\pi$
$79$	−14.7489	−1.65938	−0.829689	+	0.558226i	$0.811482\pi$
$80$	3.92542	0.438876
$81$	5.21817	0.579797
$82$	0.292747	0.0323285
$83$	2.70725	0.297160	0.148580	−	0.988900i	$-0.452530\pi$
$83$	2.70725	0.297160	0.148580	+	0.988900i	$0.452530\pi$
$84$	2.92542	0.319190
$85$	−14.2598	−1.54669
$86$	3.48352	0.375638
$87$	−7.26535	−0.778928
$88$	−4.55810	−0.485895
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	21.8179	2.29981
$91$	−3.55810	−0.372990
$92$	0	0
$93$	−21.8925	−2.27014
$94$	−8.77627	−0.905203
$95$	−29.3760	−3.01391
$96$	−2.92542	−0.298575
$97$	9.43634	0.958115	0.479058	−	0.877784i	$-0.340979\pi$
$97$	9.43634	0.958115	0.479058	+	0.877784i	$0.340979\pi$
$98$	1.00000	0.101015
$99$	−25.3344	−2.54620

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.w.1.1	yes	3
23.22	odd	2		7406.2.a.v.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7406.2.a.v.1.1	✓	3	23.22	odd	2
7406.2.a.w.1.1	yes	3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.92542 q^{3} +1.00000 q^{4} +3.92542 q^{5} -2.92542 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.55810 q^{9} +3.92542 q^{10} -4.55810 q^{11} -2.92542 q^{12} +3.55810 q^{13} -1.00000 q^{14} -11.4835 q^{15} +1.00000 q^{16} -3.63268 q^{17} +5.55810 q^{18} -7.48352 q^{19} +3.92542 q^{20} +2.92542 q^{21} -4.55810 q^{22} -2.92542 q^{24} +10.4089 q^{25} +3.55810 q^{26} -7.48352 q^{27} -1.00000 q^{28} +2.48352 q^{29} -11.4835 q^{30} +7.48352 q^{31} +1.00000 q^{32} +13.3344 q^{33} -3.63268 q^{34} -3.92542 q^{35} +5.55810 q^{36} -4.36732 q^{37} -7.48352 q^{38} -10.4089 q^{39} +3.92542 q^{40} +0.292747 q^{41} +2.92542 q^{42} +3.48352 q^{43} -4.55810 q^{44} +21.8179 q^{45} -8.77627 q^{47} -2.92542 q^{48} +1.00000 q^{49} +10.4089 q^{50} +10.6271 q^{51} +3.55810 q^{52} -7.21817 q^{53} -7.48352 q^{54} -17.8925 q^{55} -1.00000 q^{56} +21.8925 q^{57} +2.48352 q^{58} -9.11620 q^{59} -11.4835 q^{60} +0.483522 q^{61} +7.48352 q^{62} -5.55810 q^{63} +1.00000 q^{64} +13.9670 q^{65} +13.3344 q^{66} -11.7017 q^{67} -3.63268 q^{68} -3.92542 q^{70} +9.29275 q^{71} +5.55810 q^{72} -10.1162 q^{73} -4.36732 q^{74} -30.4506 q^{75} -7.48352 q^{76} +4.55810 q^{77} -10.4089 q^{78} -14.7489 q^{79} +3.92542 q^{80} +5.21817 q^{81} +0.292747 q^{82} +2.70725 q^{83} +2.92542 q^{84} -14.2598 q^{85} +3.48352 q^{86} -7.26535 q^{87} -4.55810 q^{88} -15.0000 q^{89} +21.8179 q^{90} -3.55810 q^{91} -21.8925 q^{93} -8.77627 q^{94} -29.3760 q^{95} -2.92542 q^{96} +9.43634 q^{97} +1.00000 q^{98} -25.3344 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - q^{3} + 3 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 6 q^{9} + 4 q^{10} - 3 q^{11} - q^{12} - 3 q^{14} - 16 q^{15} + 3 q^{16} - 8 q^{17} + 6 q^{18} - 4 q^{19} + 4 q^{20} + q^{21}+ \cdots - 42 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - q^3 + 3 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 3 * q^8 + 6 * q^9 + 4 * q^10 - 3 * q^11 - q^12 - 3 * q^14 - 16 * q^15 + 3 * q^16 - 8 * q^17 + 6 * q^18 - 4 * q^19 + 4 * q^20 + q^21 - 3 * q^22 - q^24 + 5 * q^25 - 4 * q^27 - 3 * q^28 - 11 * q^29 - 16 * q^30 + 4 * q^31 + 3 * q^32 + 6 * q^33 - 8 * q^34 - 4 * q^35 + 6 * q^36 - 16 * q^37 - 4 * q^38 - 5 * q^39 + 4 * q^40 - 4 * q^41 + q^42 - 8 * q^43 - 3 * q^44 + 13 * q^45 - 3 * q^47 - q^48 + 3 * q^49 + 5 * q^50 - 7 * q^51 - 9 * q^53 - 4 * q^54 - 9 * q^55 - 3 * q^56 + 21 * q^57 - 11 * q^58 - 6 * q^59 - 16 * q^60 - 17 * q^61 + 4 * q^62 - 6 * q^63 + 3 * q^64 + 5 * q^65 + 6 * q^66 - 4 * q^67 - 8 * q^68 - 4 * q^70 + 23 * q^71 + 6 * q^72 - 9 * q^73 - 16 * q^74 - 36 * q^75 - 4 * q^76 + 3 * q^77 - 5 * q^78 - 20 * q^79 + 4 * q^80 + 3 * q^81 - 4 * q^82 + 13 * q^83 + q^84 - q^85 - 8 * q^86 - 16 * q^87 - 3 * q^88 - 45 * q^89 + 13 * q^90 - 21 * q^93 - 3 * q^94 - 25 * q^95 - q^96 + 3 * q^97 + 3 * q^98 - 42 * q^99

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