Properties

Label 7406.2.a.y.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,-4,4,0,4,-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: \(\Q(\zeta_{24})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) 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 \(1.93185\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.931852 q^{3} +1.00000 q^{4} +3.66390 q^{5} -0.931852 q^{6} -1.00000 q^{7} -1.00000 q^{8} -2.13165 q^{9} -3.66390 q^{10} +0.449490 q^{11} +0.931852 q^{12} -0.282561 q^{13} +1.00000 q^{14} +3.41421 q^{15} +1.00000 q^{16} +5.66390 q^{17} +2.13165 q^{18} -5.22803 q^{19} +3.66390 q^{20} -0.931852 q^{21} -0.449490 q^{22} -0.931852 q^{24} +8.42418 q^{25} +0.282561 q^{26} -4.78194 q^{27} -1.00000 q^{28} -6.19615 q^{29} -3.41421 q^{30} -7.03163 q^{31} -1.00000 q^{32} +0.418858 q^{33} -5.66390 q^{34} -3.66390 q^{35} -2.13165 q^{36} -2.31319 q^{37} +5.22803 q^{38} -0.263305 q^{39} -3.66390 q^{40} -3.86370 q^{41} +0.931852 q^{42} -0.964724 q^{43} +0.449490 q^{44} -7.81017 q^{45} -8.58871 q^{47} +0.931852 q^{48} +1.00000 q^{49} -8.42418 q^{50} +5.27792 q^{51} -0.282561 q^{52} -12.8126 q^{53} +4.78194 q^{54} +1.64689 q^{55} +1.00000 q^{56} -4.87175 q^{57} +6.19615 q^{58} -10.1234 q^{59} +3.41421 q^{60} -11.4571 q^{61} +7.03163 q^{62} +2.13165 q^{63} +1.00000 q^{64} -1.03528 q^{65} -0.418858 q^{66} +7.84304 q^{67} +5.66390 q^{68} +3.66390 q^{70} +14.9141 q^{71} +2.13165 q^{72} -15.4741 q^{73} +2.31319 q^{74} +7.85009 q^{75} -5.22803 q^{76} -0.449490 q^{77} +0.263305 q^{78} +4.34971 q^{79} +3.66390 q^{80} +1.93890 q^{81} +3.86370 q^{82} -0.00996768 q^{83} -0.931852 q^{84} +20.7520 q^{85} +0.964724 q^{86} -5.77389 q^{87} -0.449490 q^{88} +3.92721 q^{89} +7.81017 q^{90} +0.282561 q^{91} -6.55243 q^{93} +8.58871 q^{94} -19.1550 q^{95} -0.931852 q^{96} -5.31079 q^{97} -1.00000 q^{98} -0.958156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{7} - 4 q^{8} - 8 q^{11} - 4 q^{12} - 4 q^{13} + 4 q^{14} + 8 q^{15} + 4 q^{16} + 8 q^{17} - 8 q^{19} + 4 q^{21} + 8 q^{22} + 4 q^{24} + 4 q^{26} - 4 q^{27}+ \cdots - 24 q^{99}+O(q^{100}) \) 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
\(3\) 0.931852 0.538005 0.269002 − 0.963140i \(-0.413306\pi\)
0.269002 + 0.963140i \(0.413306\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.66390 1.63855 0.819273 − 0.573403i \(-0.194377\pi\)
0.819273 + 0.573403i \(0.194377\pi\)
\(6\) −0.931852 −0.380427
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.13165 −0.710551
\(10\) −3.66390 −1.15863
\(11\) 0.449490 0.135526 0.0677631 − 0.997701i \(-0.478414\pi\)
0.0677631 + 0.997701i \(0.478414\pi\)
\(12\) 0.931852 0.269002
\(13\) −0.282561 −0.0783683 −0.0391842 − 0.999232i \(-0.512476\pi\)
−0.0391842 + 0.999232i \(0.512476\pi\)
\(14\) 1.00000 0.267261
\(15\) 3.41421 0.881546
\(16\) 1.00000 0.250000
\(17\) 5.66390 1.37370 0.686849 − 0.726800i \(-0.258993\pi\)
0.686849 + 0.726800i \(0.258993\pi\)
\(18\) 2.13165 0.502435
\(19\) −5.22803 −1.19939 −0.599696 − 0.800228i \(-0.704712\pi\)
−0.599696 + 0.800228i \(0.704712\pi\)
\(20\) 3.66390 0.819273
\(21\) −0.931852 −0.203347
\(22\) −0.449490 −0.0958315
\(23\) 0 0
\(24\) −0.931852 −0.190213
\(25\) 8.42418 1.68484
\(26\) 0.282561 0.0554148
\(27\) −4.78194 −0.920285
\(28\) −1.00000 −0.188982
\(29\) −6.19615 −1.15060 −0.575298 − 0.817944i \(-0.695114\pi\)
−0.575298 + 0.817944i \(0.695114\pi\)
\(30\) −3.41421 −0.623347
\(31\) −7.03163 −1.26292 −0.631459 − 0.775409i \(-0.717544\pi\)
−0.631459 + 0.775409i \(0.717544\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.418858 0.0729138
\(34\) −5.66390 −0.971351
\(35\) −3.66390 −0.619313
\(36\) −2.13165 −0.355275
\(37\) −2.31319 −0.380287 −0.190143 − 0.981756i \(-0.560895\pi\)
−0.190143 + 0.981756i \(0.560895\pi\)
\(38\) 5.22803 0.848098
\(39\) −0.263305 −0.0421625
\(40\) −3.66390 −0.579314
\(41\) −3.86370 −0.603409 −0.301705 − 0.953401i \(-0.597556\pi\)
−0.301705 + 0.953401i \(0.597556\pi\)
\(42\) 0.931852 0.143788
\(43\) −0.964724 −0.147119 −0.0735595 − 0.997291i \(-0.523436\pi\)
−0.0735595 + 0.997291i \(0.523436\pi\)
\(44\) 0.449490 0.0677631
\(45\) −7.81017 −1.16427
\(46\) 0 0
\(47\) −8.58871 −1.25279 −0.626396 − 0.779505i \(-0.715471\pi\)
−0.626396 + 0.779505i \(0.715471\pi\)
\(48\) 0.931852 0.134501
\(49\) 1.00000 0.142857
\(50\) −8.42418 −1.19136
\(51\) 5.27792 0.739056
\(52\) −0.282561 −0.0391842
\(53\) −12.8126 −1.75994 −0.879971 − 0.475027i \(-0.842438\pi\)
−0.879971 + 0.475027i \(0.842438\pi\)
\(54\) 4.78194 0.650739
\(55\) 1.64689 0.222066
\(56\) 1.00000 0.133631
\(57\) −4.87175 −0.645279
\(58\) 6.19615 0.813595
\(59\) −10.1234 −1.31795 −0.658975 − 0.752165i \(-0.729010\pi\)
−0.658975 + 0.752165i \(0.729010\pi\)
\(60\) 3.41421 0.440773
\(61\) −11.4571 −1.46693 −0.733463 − 0.679730i \(-0.762097\pi\)
−0.733463 + 0.679730i \(0.762097\pi\)
\(62\) 7.03163 0.893018
\(63\) 2.13165 0.268563
\(64\) 1.00000 0.125000
\(65\) −1.03528 −0.128410
\(66\) −0.418858 −0.0515578
\(67\) 7.84304 0.958180 0.479090 − 0.877766i \(-0.340967\pi\)
0.479090 + 0.877766i \(0.340967\pi\)
\(68\) 5.66390 0.686849
\(69\) 0 0
\(70\) 3.66390 0.437920
\(71\) 14.9141 1.76998 0.884989 − 0.465611i \(-0.154165\pi\)
0.884989 + 0.465611i \(0.154165\pi\)
\(72\) 2.13165 0.251218
\(73\) −15.4741 −1.81110 −0.905551 − 0.424237i \(-0.860543\pi\)
−0.905551 + 0.424237i \(0.860543\pi\)
\(74\) 2.31319 0.268903
\(75\) 7.85009 0.906450
\(76\) −5.22803 −0.599696
\(77\) −0.449490 −0.0512241
\(78\) 0.263305 0.0298134
\(79\) 4.34971 0.489381 0.244691 − 0.969601i \(-0.421314\pi\)
0.244691 + 0.969601i \(0.421314\pi\)
\(80\) 3.66390 0.409637
\(81\) 1.93890 0.215433
\(82\) 3.86370 0.426675
\(83\) −0.00996768 −0.00109409 −0.000547047 − 1.00000i \(-0.500174\pi\)
−0.000547047 1.00000i \(0.500174\pi\)
\(84\) −0.931852 −0.101673
\(85\) 20.7520 2.25087
\(86\) 0.964724 0.104029
\(87\) −5.77389 −0.619027
\(88\) −0.449490 −0.0479158
\(89\) 3.92721 0.416283 0.208142 − 0.978099i \(-0.433259\pi\)
0.208142 + 0.978099i \(0.433259\pi\)
\(90\) 7.81017 0.823264
\(91\) 0.282561 0.0296204
\(92\) 0 0
\(93\) −6.55243 −0.679456
\(94\) 8.58871 0.885857
\(95\) −19.1550 −1.96526
\(96\) −0.931852 −0.0951067
\(97\) −5.31079 −0.539229 −0.269614 − 0.962968i \(-0.586896\pi\)
−0.269614 + 0.962968i \(0.586896\pi\)
\(98\) −1.00000 −0.101015
\(99\) −0.958156 −0.0962983
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.y.1.4 ✓ 4
23.22 odd 2 7406.2.a.z.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.y.1.4 ✓ 4 1.1 even 1 trivial
7406.2.a.z.1.4 yes 4 23.22 odd 2