Properties

Label 7406.2.a.g.1.1
Level $7406$
Weight $2$
Character 7406.1
Self dual yes
Analytic conductor $59.137$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7406,2,Mod(1,7406)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content 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);
 
Copy content 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

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-1,1,-4,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.1372077370\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7406.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{9} -4.00000 q^{10} -1.00000 q^{12} +2.00000 q^{13} -1.00000 q^{14} +4.00000 q^{15} +1.00000 q^{16} +5.00000 q^{17} -2.00000 q^{18} -7.00000 q^{19} -4.00000 q^{20} +1.00000 q^{21} -1.00000 q^{24} +11.0000 q^{25} +2.00000 q^{26} +5.00000 q^{27} -1.00000 q^{28} +8.00000 q^{29} +4.00000 q^{30} -2.00000 q^{31} +1.00000 q^{32} +5.00000 q^{34} +4.00000 q^{35} -2.00000 q^{36} +8.00000 q^{37} -7.00000 q^{38} -2.00000 q^{39} -4.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} -3.00000 q^{43} +8.00000 q^{45} -1.00000 q^{48} +1.00000 q^{49} +11.0000 q^{50} -5.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +5.00000 q^{54} -1.00000 q^{56} +7.00000 q^{57} +8.00000 q^{58} -9.00000 q^{59} +4.00000 q^{60} +6.00000 q^{61} -2.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -8.00000 q^{65} +5.00000 q^{67} +5.00000 q^{68} +4.00000 q^{70} -2.00000 q^{71} -2.00000 q^{72} +17.0000 q^{73} +8.00000 q^{74} -11.0000 q^{75} -7.00000 q^{76} -2.00000 q^{78} -4.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -13.0000 q^{83} +1.00000 q^{84} -20.0000 q^{85} -3.00000 q^{86} -8.00000 q^{87} +6.00000 q^{89} +8.00000 q^{90} -2.00000 q^{91} +2.00000 q^{93} +28.0000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)

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 \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −1.00000 −0.577350 −0.288675 − 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.00000 −1.78885 −0.894427 − 0.447214i \(-0.852416\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.00000 −0.666667
\(10\) −4.00000 −1.26491
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) −2.00000 −0.471405
\(19\) −7.00000 −1.60591 −0.802955 − 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −4.00000 −0.894427
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 0 0
\(24\) −1.00000 −0.204124
\(25\) 11.0000 2.20000
\(26\) 2.00000 0.392232
\(27\) 5.00000 0.962250
\(28\) −1.00000 −0.188982
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 4.00000 0.730297
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 5.00000 0.857493
\(35\) 4.00000 0.676123
\(36\) −2.00000 −0.333333
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −7.00000 −1.13555
\(39\) −2.00000 −0.320256
\(40\) −4.00000 −0.632456
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 1.00000 0.154303
\(43\) −3.00000 −0.457496 −0.228748 − 0.973486i \(-0.573463\pi\)
−0.228748 + 0.973486i \(0.573463\pi\)
\(44\) 0 0
\(45\) 8.00000 1.19257
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 11.0000 1.55563
\(51\) −5.00000 −0.700140
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 − 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 7.00000 0.927173
\(58\) 8.00000 1.05045
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 4.00000 0.516398
\(61\) 6.00000 0.768221 0.384111 − 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −2.00000 −0.254000
\(63\) 2.00000 0.251976
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 5.00000 0.606339
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) −2.00000 −0.237356 −0.118678 − 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −2.00000 −0.235702
\(73\) 17.0000 1.98970 0.994850 − 0.101361i \(-0.0323196\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 8.00000 0.929981
\(75\) −11.0000 −1.27017
\(76\) −7.00000 −0.802955
\(77\) 0 0
\(78\) −2.00000 −0.226455
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −13.0000 −1.42694 −0.713468 − 0.700688i \(-0.752876\pi\)
−0.713468 + 0.700688i \(0.752876\pi\)
\(84\) 1.00000 0.109109
\(85\) −20.0000 −2.16930
\(86\) −3.00000 −0.323498
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 8.00000 0.843274
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 28.0000 2.87274
\(96\) −1.00000 −0.102062
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 1.00000 0.101015
\(99\) 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 7406.2.a.g.1.1 ✓ 1
23.22 odd 2 7406.2.a.h.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.g.1.1 ✓ 1 1.1 even 1 trivial
7406.2.a.h.1.1 yes 1 23.22 odd 2