Properties

Label 7406.2.a.ba.1.4
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 1.4
Root \(3.14867\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.14867 q^{3} +1.00000 q^{4} -4.14867 q^{5} +3.14867 q^{6} -1.00000 q^{7} +1.00000 q^{8} +6.91413 q^{9} -4.14867 q^{10} +2.87829 q^{11} +3.14867 q^{12} -6.91413 q^{13} -1.00000 q^{14} -13.0628 q^{15} +1.00000 q^{16} -4.02697 q^{17} +6.91413 q^{18} +0.270377 q^{19} -4.14867 q^{20} -3.14867 q^{21} +2.87829 q^{22} +3.14867 q^{24} +12.2115 q^{25} -6.91413 q^{26} +12.3243 q^{27} -1.00000 q^{28} -3.02697 q^{29} -13.0628 q^{30} -2.76546 q^{31} +1.00000 q^{32} +9.06280 q^{33} -4.02697 q^{34} +4.14867 q^{35} +6.91413 q^{36} +0.765459 q^{37} +0.270377 q^{38} -21.7703 q^{39} -4.14867 q^{40} -1.38321 q^{41} -3.14867 q^{42} -10.5677 q^{43} +2.87829 q^{44} -28.6845 q^{45} -6.18451 q^{47} +3.14867 q^{48} +1.00000 q^{49} +12.2115 q^{50} -12.6796 q^{51} -6.91413 q^{52} +1.02697 q^{53} +12.3243 q^{54} -11.9411 q^{55} -1.00000 q^{56} +0.851329 q^{57} -3.02697 q^{58} -8.79242 q^{59} -13.0628 q^{60} -2.97303 q^{61} -2.76546 q^{62} -6.91413 q^{63} +1.00000 q^{64} +28.6845 q^{65} +9.06280 q^{66} -3.03584 q^{67} -4.02697 q^{68} +4.14867 q^{70} -4.58095 q^{71} +6.91413 q^{72} -0.459246 q^{73} +0.765459 q^{74} +38.4499 q^{75} +0.270377 q^{76} -2.87829 q^{77} -21.7703 q^{78} -7.53188 q^{79} -4.14867 q^{80} +18.0628 q^{81} -1.38321 q^{82} +1.91413 q^{83} -3.14867 q^{84} +16.7066 q^{85} -10.5677 q^{86} -9.53092 q^{87} +2.87829 q^{88} +8.33318 q^{89} -28.6845 q^{90} +6.91413 q^{91} -8.70752 q^{93} -6.18451 q^{94} -1.12171 q^{95} +3.14867 q^{96} +13.8641 q^{97} +1.00000 q^{98} +19.9009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} - 4 q^{7} + 4 q^{8} + 8 q^{9} - 4 q^{10} + 3 q^{11} - 8 q^{13} - 4 q^{14} - 20 q^{15} + 4 q^{16} + 5 q^{17} + 8 q^{18} - 3 q^{19} - 4 q^{20} + 3 q^{22} + 4 q^{25} - 8 q^{26}+ \cdots + 14 q^{99}+O(q^{100}) \) 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
\(3\) 3.14867 1.81789 0.908943 − 0.416920i \(-0.136891\pi\)
0.908943 + 0.416920i \(0.136891\pi\)
\(4\) 1.00000 0.500000
\(5\) −4.14867 −1.85534 −0.927671 − 0.373398i \(-0.878192\pi\)
−0.927671 + 0.373398i \(0.878192\pi\)
\(6\) 3.14867 1.28544
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 6.91413 2.30471
\(10\) −4.14867 −1.31192
\(11\) 2.87829 0.867838 0.433919 − 0.900952i \(-0.357130\pi\)
0.433919 + 0.900952i \(0.357130\pi\)
\(12\) 3.14867 0.908943
\(13\) −6.91413 −1.91763 −0.958817 − 0.284024i \(-0.908331\pi\)
−0.958817 + 0.284024i \(0.908331\pi\)
\(14\) −1.00000 −0.267261
\(15\) −13.0628 −3.37280
\(16\) 1.00000 0.250000
\(17\) −4.02697 −0.976682 −0.488341 − 0.872653i \(-0.662398\pi\)
−0.488341 + 0.872653i \(0.662398\pi\)
\(18\) 6.91413 1.62968
\(19\) 0.270377 0.0620288 0.0310144 − 0.999519i \(-0.490126\pi\)
0.0310144 + 0.999519i \(0.490126\pi\)
\(20\) −4.14867 −0.927671
\(21\) −3.14867 −0.687096
\(22\) 2.87829 0.613654
\(23\) 0 0
\(24\) 3.14867 0.642720
\(25\) 12.2115 2.44229
\(26\) −6.91413 −1.35597
\(27\) 12.3243 2.37181
\(28\) −1.00000 −0.188982
\(29\) −3.02697 −0.562093 −0.281047 − 0.959694i \(-0.590682\pi\)
−0.281047 + 0.959694i \(0.590682\pi\)
\(30\) −13.0628 −2.38493
\(31\) −2.76546 −0.496691 −0.248346 − 0.968672i \(-0.579887\pi\)
−0.248346 + 0.968672i \(0.579887\pi\)
\(32\) 1.00000 0.176777
\(33\) 9.06280 1.57763
\(34\) −4.02697 −0.690619
\(35\) 4.14867 0.701253
\(36\) 6.91413 1.15235
\(37\) 0.765459 0.125841 0.0629203 − 0.998019i \(-0.479959\pi\)
0.0629203 + 0.998019i \(0.479959\pi\)
\(38\) 0.270377 0.0438610
\(39\) −21.7703 −3.48604
\(40\) −4.14867 −0.655962
\(41\) −1.38321 −0.216021 −0.108011 − 0.994150i \(-0.534448\pi\)
−0.108011 + 0.994150i \(0.534448\pi\)
\(42\) −3.14867 −0.485850
\(43\) −10.5677 −1.61156 −0.805781 − 0.592214i \(-0.798254\pi\)
−0.805781 + 0.592214i \(0.798254\pi\)
\(44\) 2.87829 0.433919
\(45\) −28.6845 −4.27603
\(46\) 0 0
\(47\) −6.18451 −0.902103 −0.451052 − 0.892498i \(-0.648951\pi\)
−0.451052 + 0.892498i \(0.648951\pi\)
\(48\) 3.14867 0.454472
\(49\) 1.00000 0.142857
\(50\) 12.2115 1.72696
\(51\) −12.6796 −1.77550
\(52\) −6.91413 −0.958817
\(53\) 1.02697 0.141064 0.0705322 − 0.997509i \(-0.477530\pi\)
0.0705322 + 0.997509i \(0.477530\pi\)
\(54\) 12.3243 1.67713
\(55\) −11.9411 −1.61014
\(56\) −1.00000 −0.133631
\(57\) 0.851329 0.112761
\(58\) −3.02697 −0.397460
\(59\) −8.79242 −1.14468 −0.572338 − 0.820018i \(-0.693963\pi\)
−0.572338 + 0.820018i \(0.693963\pi\)
\(60\) −13.0628 −1.68640
\(61\) −2.97303 −0.380658 −0.190329 − 0.981720i \(-0.560955\pi\)
−0.190329 + 0.981720i \(0.560955\pi\)
\(62\) −2.76546 −0.351214
\(63\) −6.91413 −0.871098
\(64\) 1.00000 0.125000
\(65\) 28.6845 3.55787
\(66\) 9.06280 1.11555
\(67\) −3.03584 −0.370886 −0.185443 − 0.982655i \(-0.559372\pi\)
−0.185443 + 0.982655i \(0.559372\pi\)
\(68\) −4.02697 −0.488341
\(69\) 0 0
\(70\) 4.14867 0.495861
\(71\) −4.58095 −0.543659 −0.271829 − 0.962345i \(-0.587629\pi\)
−0.271829 + 0.962345i \(0.587629\pi\)
\(72\) 6.91413 0.814838
\(73\) −0.459246 −0.0537506 −0.0268753 − 0.999639i \(-0.508556\pi\)
−0.0268753 + 0.999639i \(0.508556\pi\)
\(74\) 0.765459 0.0889828
\(75\) 38.4499 4.43981
\(76\) 0.270377 0.0310144
\(77\) −2.87829 −0.328012
\(78\) −21.7703 −2.46500
\(79\) −7.53188 −0.847403 −0.423702 − 0.905802i \(-0.639269\pi\)
−0.423702 + 0.905802i \(0.639269\pi\)
\(80\) −4.14867 −0.463836
\(81\) 18.0628 2.00698
\(82\) −1.38321 −0.152750
\(83\) 1.91413 0.210103 0.105052 − 0.994467i \(-0.466499\pi\)
0.105052 + 0.994467i \(0.466499\pi\)
\(84\) −3.14867 −0.343548
\(85\) 16.7066 1.81208
\(86\) −10.5677 −1.13955
\(87\) −9.53092 −1.02182
\(88\) 2.87829 0.306827
\(89\) 8.33318 0.883315 0.441658 − 0.897184i \(-0.354391\pi\)
0.441658 + 0.897184i \(0.354391\pi\)
\(90\) −28.6845 −3.02361
\(91\) 6.91413 0.724798
\(92\) 0 0
\(93\) −8.70752 −0.902928
\(94\) −6.18451 −0.637883
\(95\) −1.12171 −0.115085
\(96\) 3.14867 0.321360
\(97\) 13.8641 1.40769 0.703843 − 0.710356i \(-0.251466\pi\)
0.703843 + 0.710356i \(0.251466\pi\)
\(98\) 1.00000 0.101015
\(99\) 19.9009 2.00012
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.1.4 ✓ 4
23.22 odd 2 7406.2.a.bd.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.ba.1.4 ✓ 4 1.1 even 1 trivial
7406.2.a.bd.1.4 yes 4 23.22 odd 2