Properties

Label 7406.2.a.t.1.1
Level $7406$
Weight $2$
Character 7406.1
Self dual yes
Analytic conductor $59.137$
Analytic rank $1$
Dimension $3$
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 = [3,3,-2,3,-2,-2,-3] 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: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.90321 q^{3} +1.00000 q^{4} +1.52543 q^{5} -2.90321 q^{6} -1.00000 q^{7} +1.00000 q^{8} +5.42864 q^{9} +1.52543 q^{10} +5.80642 q^{11} -2.90321 q^{12} -1.00000 q^{14} -4.42864 q^{15} +1.00000 q^{16} +0.903212 q^{17} +5.42864 q^{18} -6.42864 q^{19} +1.52543 q^{20} +2.90321 q^{21} +5.80642 q^{22} -2.90321 q^{24} -2.67307 q^{25} -7.05086 q^{27} -1.00000 q^{28} +0.622216 q^{29} -4.42864 q^{30} -3.95407 q^{31} +1.00000 q^{32} -16.8573 q^{33} +0.903212 q^{34} -1.52543 q^{35} +5.42864 q^{36} -2.62222 q^{37} -6.42864 q^{38} +1.52543 q^{40} -10.8573 q^{41} +2.90321 q^{42} -9.67307 q^{43} +5.80642 q^{44} +8.28100 q^{45} -8.90321 q^{47} -2.90321 q^{48} +1.00000 q^{49} -2.67307 q^{50} -2.62222 q^{51} +5.37778 q^{53} -7.05086 q^{54} +8.85728 q^{55} -1.00000 q^{56} +18.6637 q^{57} +0.622216 q^{58} -7.19850 q^{59} -4.42864 q^{60} -3.03657 q^{61} -3.95407 q^{62} -5.42864 q^{63} +1.00000 q^{64} -16.8573 q^{66} +10.6637 q^{67} +0.903212 q^{68} -1.52543 q^{70} +9.80642 q^{71} +5.42864 q^{72} +12.1017 q^{73} -2.62222 q^{74} +7.76049 q^{75} -6.42864 q^{76} -5.80642 q^{77} -15.1842 q^{79} +1.52543 q^{80} +4.18421 q^{81} -10.8573 q^{82} -1.57136 q^{83} +2.90321 q^{84} +1.37778 q^{85} -9.67307 q^{86} -1.80642 q^{87} +5.80642 q^{88} +18.6178 q^{89} +8.28100 q^{90} +11.4795 q^{93} -8.90321 q^{94} -9.80642 q^{95} -2.90321 q^{96} -3.65878 q^{97} +1.00000 q^{98} +31.5210 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 2 q^{10} + 4 q^{11} - 2 q^{12} - 3 q^{14} + 3 q^{16} - 4 q^{17} + 3 q^{18} - 6 q^{19} - 2 q^{20} + 2 q^{21} + 4 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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
\(3\) −2.90321 −1.67617 −0.838085 − 0.545540i \(-0.816325\pi\)
−0.838085 + 0.545540i \(0.816325\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.52543 0.682192 0.341096 − 0.940028i \(-0.389202\pi\)
0.341096 + 0.940028i \(0.389202\pi\)
\(6\) −2.90321 −1.18523
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 5.42864 1.80955
\(10\) 1.52543 0.482383
\(11\) 5.80642 1.75070 0.875351 − 0.483487i \(-0.160630\pi\)
0.875351 + 0.483487i \(0.160630\pi\)
\(12\) −2.90321 −0.838085
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) −1.00000 −0.267261
\(15\) −4.42864 −1.14347
\(16\) 1.00000 0.250000
\(17\) 0.903212 0.219061 0.109531 − 0.993983i \(-0.465065\pi\)
0.109531 + 0.993983i \(0.465065\pi\)
\(18\) 5.42864 1.27954
\(19\) −6.42864 −1.47483 −0.737416 − 0.675439i \(-0.763954\pi\)
−0.737416 + 0.675439i \(0.763954\pi\)
\(20\) 1.52543 0.341096
\(21\) 2.90321 0.633533
\(22\) 5.80642 1.23793
\(23\) 0 0
\(24\) −2.90321 −0.592616
\(25\) −2.67307 −0.534614
\(26\) 0 0
\(27\) −7.05086 −1.35694
\(28\) −1.00000 −0.188982
\(29\) 0.622216 0.115543 0.0577713 − 0.998330i \(-0.481601\pi\)
0.0577713 + 0.998330i \(0.481601\pi\)
\(30\) −4.42864 −0.808555
\(31\) −3.95407 −0.710171 −0.355086 − 0.934834i \(-0.615548\pi\)
−0.355086 + 0.934834i \(0.615548\pi\)
\(32\) 1.00000 0.176777
\(33\) −16.8573 −2.93448
\(34\) 0.903212 0.154900
\(35\) −1.52543 −0.257844
\(36\) 5.42864 0.904773
\(37\) −2.62222 −0.431090 −0.215545 − 0.976494i \(-0.569153\pi\)
−0.215545 + 0.976494i \(0.569153\pi\)
\(38\) −6.42864 −1.04286
\(39\) 0 0
\(40\) 1.52543 0.241191
\(41\) −10.8573 −1.69562 −0.847811 − 0.530298i \(-0.822080\pi\)
−0.847811 + 0.530298i \(0.822080\pi\)
\(42\) 2.90321 0.447975
\(43\) −9.67307 −1.47513 −0.737565 − 0.675276i \(-0.764024\pi\)
−0.737565 + 0.675276i \(0.764024\pi\)
\(44\) 5.80642 0.875351
\(45\) 8.28100 1.23446
\(46\) 0 0
\(47\) −8.90321 −1.29867 −0.649333 − 0.760504i \(-0.724952\pi\)
−0.649333 + 0.760504i \(0.724952\pi\)
\(48\) −2.90321 −0.419043
\(49\) 1.00000 0.142857
\(50\) −2.67307 −0.378029
\(51\) −2.62222 −0.367184
\(52\) 0 0
\(53\) 5.37778 0.738695 0.369348 − 0.929291i \(-0.379581\pi\)
0.369348 + 0.929291i \(0.379581\pi\)
\(54\) −7.05086 −0.959500
\(55\) 8.85728 1.19432
\(56\) −1.00000 −0.133631
\(57\) 18.6637 2.47207
\(58\) 0.622216 0.0817009
\(59\) −7.19850 −0.937165 −0.468582 − 0.883420i \(-0.655235\pi\)
−0.468582 + 0.883420i \(0.655235\pi\)
\(60\) −4.42864 −0.571735
\(61\) −3.03657 −0.388792 −0.194396 − 0.980923i \(-0.562275\pi\)
−0.194396 + 0.980923i \(0.562275\pi\)
\(62\) −3.95407 −0.502167
\(63\) −5.42864 −0.683944
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −16.8573 −2.07499
\(67\) 10.6637 1.30278 0.651389 − 0.758744i \(-0.274186\pi\)
0.651389 + 0.758744i \(0.274186\pi\)
\(68\) 0.903212 0.109531
\(69\) 0 0
\(70\) −1.52543 −0.182323
\(71\) 9.80642 1.16381 0.581904 − 0.813257i \(-0.302308\pi\)
0.581904 + 0.813257i \(0.302308\pi\)
\(72\) 5.42864 0.639771
\(73\) 12.1017 1.41640 0.708199 − 0.706013i \(-0.249508\pi\)
0.708199 + 0.706013i \(0.249508\pi\)
\(74\) −2.62222 −0.304826
\(75\) 7.76049 0.896104
\(76\) −6.42864 −0.737416
\(77\) −5.80642 −0.661703
\(78\) 0 0
\(79\) −15.1842 −1.70836 −0.854178 − 0.519980i \(-0.825939\pi\)
−0.854178 + 0.519980i \(0.825939\pi\)
\(80\) 1.52543 0.170548
\(81\) 4.18421 0.464912
\(82\) −10.8573 −1.19899
\(83\) −1.57136 −0.172479 −0.0862396 − 0.996274i \(-0.527485\pi\)
−0.0862396 + 0.996274i \(0.527485\pi\)
\(84\) 2.90321 0.316766
\(85\) 1.37778 0.149442
\(86\) −9.67307 −1.04307
\(87\) −1.80642 −0.193669
\(88\) 5.80642 0.618967
\(89\) 18.6178 1.97348 0.986740 − 0.162310i \(-0.0518944\pi\)
0.986740 + 0.162310i \(0.0518944\pi\)
\(90\) 8.28100 0.872894
\(91\) 0 0
\(92\) 0 0
\(93\) 11.4795 1.19037
\(94\) −8.90321 −0.918296
\(95\) −9.80642 −1.00612
\(96\) −2.90321 −0.296308
\(97\) −3.65878 −0.371493 −0.185746 − 0.982598i \(-0.559470\pi\)
−0.185746 + 0.982598i \(0.559470\pi\)
\(98\) 1.00000 0.101015
\(99\) 31.5210 3.16798
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.t.1.1 ✓ 3
23.22 odd 2 7406.2.a.u.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.t.1.1 ✓ 3 1.1 even 1 trivial
7406.2.a.u.1.1 yes 3 23.22 odd 2