Properties

Label 7406.2.a.ba.1.2
Level $7406$
Weight $2$
Character 7406.1
Self dual yes
Analytic conductor $59.137$
Analytic rank $1$
Dimension $4$
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 = [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)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(59.1372077370\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.449797.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} - x + 4 \) Copy content Toggle raw display
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 1.2
Root \(-0.703441\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.703441 q^{3} +1.00000 q^{4} -0.296559 q^{5} -0.703441 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.50517 q^{9} -0.296559 q^{10} +5.98289 q^{11} -0.703441 q^{12} +2.50517 q^{13} -1.00000 q^{14} +0.208612 q^{15} +1.00000 q^{16} -3.27945 q^{17} -2.50517 q^{18} -6.68633 q^{19} -0.296559 q^{20} +0.703441 q^{21} +5.98289 q^{22} -0.703441 q^{24} -4.91205 q^{25} +2.50517 q^{26} +3.87256 q^{27} -1.00000 q^{28} -2.27945 q^{29} +0.208612 q^{30} +2.80173 q^{31} +1.00000 q^{32} -4.20861 q^{33} -3.27945 q^{34} +0.296559 q^{35} -2.50517 q^{36} -4.80173 q^{37} -6.68633 q^{38} -1.76224 q^{39} -0.296559 q^{40} -3.09829 q^{41} +0.703441 q^{42} +4.09321 q^{43} +5.98289 q^{44} +0.742930 q^{45} +10.1915 q^{47} -0.703441 q^{48} +1.00000 q^{49} -4.91205 q^{50} +2.30690 q^{51} +2.50517 q^{52} +0.279447 q^{53} +3.87256 q^{54} -1.77428 q^{55} -1.00000 q^{56} +4.70344 q^{57} -2.27945 q^{58} -2.47772 q^{59} +0.208612 q^{60} -3.72055 q^{61} +2.80173 q^{62} +2.50517 q^{63} +1.00000 q^{64} -0.742930 q^{65} -4.20861 q^{66} +9.48806 q^{67} -3.27945 q^{68} +0.296559 q^{70} -15.3898 q^{71} -2.50517 q^{72} -14.3727 q^{73} -4.80173 q^{74} +3.45534 q^{75} -6.68633 q^{76} -5.98289 q^{77} -1.76224 q^{78} -5.39485 q^{79} -0.296559 q^{80} +4.79139 q^{81} -3.09829 q^{82} -7.50517 q^{83} +0.703441 q^{84} +0.972548 q^{85} +4.09321 q^{86} +1.60346 q^{87} +5.98289 q^{88} -11.8949 q^{89} +0.742930 q^{90} -2.50517 q^{91} -1.97085 q^{93} +10.1915 q^{94} +1.98289 q^{95} -0.703441 q^{96} -17.4984 q^{97} +1.00000 q^{98} -14.9882 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{7} + 4 q^{8} + 8 q^{9} - 4 q^{10} + 3 q^{11} - 8 q^{13} - 4 q^{14} - 20 q^{15} + 4 q^{16} + 5 q^{17} + 8 q^{18} - 3 q^{19} - 4 q^{20} + 3 q^{22} + 4 q^{25} - 8 q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.703441 −0.406132 −0.203066 − 0.979165i \(-0.565091\pi\)
−0.203066 + 0.979165i \(0.565091\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.296559 −0.132625 −0.0663125 − 0.997799i \(-0.521123\pi\)
−0.0663125 + 0.997799i \(0.521123\pi\)
\(6\) −0.703441 −0.287179
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) −2.50517 −0.835057
\(10\) −0.296559 −0.0937801
\(11\) 5.98289 1.80391 0.901954 − 0.431831i \(-0.142132\pi\)
0.901954 + 0.431831i \(0.142132\pi\)
\(12\) −0.703441 −0.203066
\(13\) 2.50517 0.694809 0.347405 − 0.937715i \(-0.387063\pi\)
0.347405 + 0.937715i \(0.387063\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.208612 0.0538633
\(16\) 1.00000 0.250000
\(17\) −3.27945 −0.795383 −0.397691 − 0.917519i \(-0.630189\pi\)
−0.397691 + 0.917519i \(0.630189\pi\)
\(18\) −2.50517 −0.590474
\(19\) −6.68633 −1.53395 −0.766975 − 0.641677i \(-0.778239\pi\)
−0.766975 + 0.641677i \(0.778239\pi\)
\(20\) −0.296559 −0.0663125
\(21\) 0.703441 0.153503
\(22\) 5.98289 1.27556
\(23\) 0 0
\(24\) −0.703441 −0.143589
\(25\) −4.91205 −0.982411
\(26\) 2.50517 0.491304
\(27\) 3.87256 0.745275
\(28\) −1.00000 −0.188982
\(29\) −2.27945 −0.423283 −0.211641 − 0.977347i \(-0.567881\pi\)
−0.211641 + 0.977347i \(0.567881\pi\)
\(30\) 0.208612 0.0380871
\(31\) 2.80173 0.503205 0.251603 − 0.967831i \(-0.419042\pi\)
0.251603 + 0.967831i \(0.419042\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.20861 −0.732625
\(34\) −3.27945 −0.562421
\(35\) 0.296559 0.0501276
\(36\) −2.50517 −0.417528
\(37\) −4.80173 −0.789399 −0.394700 − 0.918810i \(-0.629151\pi\)
−0.394700 + 0.918810i \(0.629151\pi\)
\(38\) −6.68633 −1.08467
\(39\) −1.76224 −0.282184
\(40\) −0.296559 −0.0468900
\(41\) −3.09829 −0.483871 −0.241936 − 0.970292i \(-0.577782\pi\)
−0.241936 + 0.970292i \(0.577782\pi\)
\(42\) 0.703441 0.108543
\(43\) 4.09321 0.624209 0.312105 − 0.950048i \(-0.398966\pi\)
0.312105 + 0.950048i \(0.398966\pi\)
\(44\) 5.98289 0.901954
\(45\) 0.742930 0.110749
\(46\) 0 0
\(47\) 10.1915 1.48658 0.743292 − 0.668968i \(-0.233264\pi\)
0.743292 + 0.668968i \(0.233264\pi\)
\(48\) −0.703441 −0.101533
\(49\) 1.00000 0.142857
\(50\) −4.91205 −0.694669
\(51\) 2.30690 0.323030
\(52\) 2.50517 0.347405
\(53\) 0.279447 0.0383850 0.0191925 − 0.999816i \(-0.493890\pi\)
0.0191925 + 0.999816i \(0.493890\pi\)
\(54\) 3.87256 0.526989
\(55\) −1.77428 −0.239243
\(56\) −1.00000 −0.133631
\(57\) 4.70344 0.622986
\(58\) −2.27945 −0.299306
\(59\) −2.47772 −0.322571 −0.161286 − 0.986908i \(-0.551564\pi\)
−0.161286 + 0.986908i \(0.551564\pi\)
\(60\) 0.208612 0.0269316
\(61\) −3.72055 −0.476368 −0.238184 − 0.971220i \(-0.576552\pi\)
−0.238184 + 0.971220i \(0.576552\pi\)
\(62\) 2.80173 0.355820
\(63\) 2.50517 0.315622
\(64\) 1.00000 0.125000
\(65\) −0.742930 −0.0921491
\(66\) −4.20861 −0.518044
\(67\) 9.48806 1.15915 0.579575 − 0.814919i \(-0.303218\pi\)
0.579575 + 0.814919i \(0.303218\pi\)
\(68\) −3.27945 −0.397691
\(69\) 0 0
\(70\) 0.296559 0.0354455
\(71\) −15.3898 −1.82643 −0.913215 − 0.407478i \(-0.866408\pi\)
−0.913215 + 0.407478i \(0.866408\pi\)
\(72\) −2.50517 −0.295237
\(73\) −14.3727 −1.68219 −0.841096 − 0.540885i \(-0.818089\pi\)
−0.841096 + 0.540885i \(0.818089\pi\)
\(74\) −4.80173 −0.558190
\(75\) 3.45534 0.398988
\(76\) −6.68633 −0.766975
\(77\) −5.98289 −0.681813
\(78\) −1.76224 −0.199534
\(79\) −5.39485 −0.606968 −0.303484 − 0.952837i \(-0.598150\pi\)
−0.303484 + 0.952837i \(0.598150\pi\)
\(80\) −0.296559 −0.0331563
\(81\) 4.79139 0.532376
\(82\) −3.09829 −0.342149
\(83\) −7.50517 −0.823799 −0.411900 − 0.911229i \(-0.635135\pi\)
−0.411900 + 0.911229i \(0.635135\pi\)
\(84\) 0.703441 0.0767517
\(85\) 0.972548 0.105488
\(86\) 4.09321 0.441383
\(87\) 1.60346 0.171909
\(88\) 5.98289 0.637778
\(89\) −11.8949 −1.26086 −0.630431 − 0.776246i \(-0.717122\pi\)
−0.630431 + 0.776246i \(0.717122\pi\)
\(90\) 0.742930 0.0783117
\(91\) −2.50517 −0.262613
\(92\) 0 0
\(93\) −1.97085 −0.204368
\(94\) 10.1915 1.05117
\(95\) 1.98289 0.203440
\(96\) −0.703441 −0.0717947
\(97\) −17.4984 −1.77669 −0.888347 − 0.459173i \(-0.848146\pi\)
−0.888347 + 0.459173i \(0.848146\pi\)
\(98\) 1.00000 0.101015
\(99\) −14.9882 −1.50637
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.ba.1.2 ✓ 4
23.22 odd 2 7406.2.a.bd.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.ba.1.2 ✓ 4 1.1 even 1 trivial
7406.2.a.bd.1.2 yes 4 23.22 odd 2