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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{10 +4 \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} - 4x + 2 \) 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 1.1
Root \(0.334904\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.74912 q^{3} +1.00000 q^{4} +2.80853 q^{5} -1.74912 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.0594122 q^{9} +2.80853 q^{10} +0.585786 q^{11} -1.74912 q^{12} +1.14343 q^{13} -1.00000 q^{14} -4.91245 q^{15} +1.00000 q^{16} -0.808530 q^{17} +0.0594122 q^{18} -1.49824 q^{19} +2.80853 q^{20} +1.74912 q^{21} +0.585786 q^{22} -1.74912 q^{24} +2.88784 q^{25} +1.14343 q^{26} +5.14343 q^{27} -1.00000 q^{28} -3.21803 q^{29} -4.91245 q^{30} -1.86794 q^{31} +1.00000 q^{32} -1.02461 q^{33} -0.808530 q^{34} -2.80853 q^{35} +0.0594122 q^{36} +2.36147 q^{37} -1.49824 q^{38} -2.00000 q^{39} +2.80853 q^{40} +0.550984 q^{41} +1.74912 q^{42} +12.6135 q^{43} +0.585786 q^{44} +0.166861 q^{45} +6.85499 q^{47} -1.74912 q^{48} +1.00000 q^{49} +2.88784 q^{50} +1.41421 q^{51} +1.14343 q^{52} -9.19932 q^{53} +5.14343 q^{54} +1.64520 q^{55} -1.00000 q^{56} +2.62059 q^{57} -3.21803 q^{58} +3.86794 q^{59} -4.91245 q^{60} -7.96362 q^{61} -1.86794 q^{62} -0.0594122 q^{63} +1.00000 q^{64} +3.21137 q^{65} -1.02461 q^{66} +12.3615 q^{67} -0.808530 q^{68} -2.80853 q^{70} +11.2739 q^{71} +0.0594122 q^{72} +6.20951 q^{73} +2.36147 q^{74} -5.05117 q^{75} -1.49824 q^{76} -0.585786 q^{77} -2.00000 q^{78} -13.2524 q^{79} +2.80853 q^{80} -9.17471 q^{81} +0.550984 q^{82} -2.85657 q^{83} +1.74912 q^{84} -2.27078 q^{85} +12.6135 q^{86} +5.62872 q^{87} +0.585786 q^{88} +12.0691 q^{89} +0.166861 q^{90} -1.14343 q^{91} +3.26725 q^{93} +6.85499 q^{94} -4.20784 q^{95} -1.74912 q^{96} -0.955493 q^{97} +1.00000 q^{98} +0.0348029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{8} + 4 q^{10} + 8 q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} + 4 q^{17} + 8 q^{19} + 4 q^{20} + 8 q^{22} - 4 q^{26} + 12 q^{27} - 4 q^{28} - 4 q^{29} + 4 q^{32}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(3\) −1.74912 −1.00985 −0.504927 − 0.863162i \(-0.668480\pi\)
−0.504927 + 0.863162i \(0.668480\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.80853 1.25601 0.628006 − 0.778208i \(-0.283871\pi\)
0.628006 + 0.778208i \(0.283871\pi\)
\(6\) −1.74912 −0.714074
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0.0594122 0.0198041
\(10\) 2.80853 0.888135
\(11\) 0.585786 0.176621 0.0883106 − 0.996093i \(-0.471853\pi\)
0.0883106 + 0.996093i \(0.471853\pi\)
\(12\) −1.74912 −0.504927
\(13\) 1.14343 0.317131 0.158566 − 0.987348i \(-0.449313\pi\)
0.158566 + 0.987348i \(0.449313\pi\)
\(14\) −1.00000 −0.267261
\(15\) −4.91245 −1.26839
\(16\) 1.00000 0.250000
\(17\) −0.808530 −0.196097 −0.0980486 − 0.995182i \(-0.531260\pi\)
−0.0980486 + 0.995182i \(0.531260\pi\)
\(18\) 0.0594122 0.0140036
\(19\) −1.49824 −0.343719 −0.171859 − 0.985121i \(-0.554977\pi\)
−0.171859 + 0.985121i \(0.554977\pi\)
\(20\) 2.80853 0.628006
\(21\) 1.74912 0.381689
\(22\) 0.585786 0.124890
\(23\) 0 0
\(24\) −1.74912 −0.357037
\(25\) 2.88784 0.577568
\(26\) 1.14343 0.224246
\(27\) 5.14343 0.989854
\(28\) −1.00000 −0.188982
\(29\) −3.21803 −0.597573 −0.298787 − 0.954320i \(-0.596582\pi\)
−0.298787 + 0.954320i \(0.596582\pi\)
\(30\) −4.91245 −0.896886
\(31\) −1.86794 −0.335492 −0.167746 − 0.985830i \(-0.553649\pi\)
−0.167746 + 0.985830i \(0.553649\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.02461 −0.178362
\(34\) −0.808530 −0.138662
\(35\) −2.80853 −0.474728
\(36\) 0.0594122 0.00990203
\(37\) 2.36147 0.388222 0.194111 − 0.980980i \(-0.437818\pi\)
0.194111 + 0.980980i \(0.437818\pi\)
\(38\) −1.49824 −0.243046
\(39\) −2.00000 −0.320256
\(40\) 2.80853 0.444068
\(41\) 0.550984 0.0860492 0.0430246 − 0.999074i \(-0.486301\pi\)
0.0430246 + 0.999074i \(0.486301\pi\)
\(42\) 1.74912 0.269895
\(43\) 12.6135 1.92355 0.961773 − 0.273849i \(-0.0882969\pi\)
0.961773 + 0.273849i \(0.0882969\pi\)
\(44\) 0.585786 0.0883106
\(45\) 0.166861 0.0248742
\(46\) 0 0
\(47\) 6.85499 0.999903 0.499951 − 0.866053i \(-0.333351\pi\)
0.499951 + 0.866053i \(0.333351\pi\)
\(48\) −1.74912 −0.252463
\(49\) 1.00000 0.142857
\(50\) 2.88784 0.408402
\(51\) 1.41421 0.198030
\(52\) 1.14343 0.158566
\(53\) −9.19932 −1.26362 −0.631812 − 0.775122i \(-0.717688\pi\)
−0.631812 + 0.775122i \(0.717688\pi\)
\(54\) 5.14343 0.699933
\(55\) 1.64520 0.221839
\(56\) −1.00000 −0.133631
\(57\) 2.62059 0.347106
\(58\) −3.21803 −0.422548
\(59\) 3.86794 0.503563 0.251782 − 0.967784i \(-0.418984\pi\)
0.251782 + 0.967784i \(0.418984\pi\)
\(60\) −4.91245 −0.634194
\(61\) −7.96362 −1.01964 −0.509818 − 0.860282i \(-0.670287\pi\)
−0.509818 + 0.860282i \(0.670287\pi\)
\(62\) −1.86794 −0.237229
\(63\) −0.0594122 −0.00748523
\(64\) 1.00000 0.125000
\(65\) 3.21137 0.398321
\(66\) −1.02461 −0.126121
\(67\) 12.3615 1.51019 0.755097 − 0.655613i \(-0.227590\pi\)
0.755097 + 0.655613i \(0.227590\pi\)
\(68\) −0.808530 −0.0980486
\(69\) 0 0
\(70\) −2.80853 −0.335684
\(71\) 11.2739 1.33797 0.668984 − 0.743277i \(-0.266730\pi\)
0.668984 + 0.743277i \(0.266730\pi\)
\(72\) 0.0594122 0.00700179
\(73\) 6.20951 0.726768 0.363384 − 0.931639i \(-0.381621\pi\)
0.363384 + 0.931639i \(0.381621\pi\)
\(74\) 2.36147 0.274515
\(75\) −5.05117 −0.583259
\(76\) −1.49824 −0.171859
\(77\) −0.585786 −0.0667566
\(78\) −2.00000 −0.226455
\(79\) −13.2524 −1.49102 −0.745508 − 0.666497i \(-0.767793\pi\)
−0.745508 + 0.666497i \(0.767793\pi\)
\(80\) 2.80853 0.314003
\(81\) −9.17471 −1.01941
\(82\) 0.550984 0.0608460
\(83\) −2.85657 −0.313549 −0.156774 − 0.987634i \(-0.550110\pi\)
−0.156774 + 0.987634i \(0.550110\pi\)
\(84\) 1.74912 0.190844
\(85\) −2.27078 −0.246301
\(86\) 12.6135 1.36015
\(87\) 5.62872 0.603462
\(88\) 0.585786 0.0624450
\(89\) 12.0691 1.27932 0.639662 − 0.768656i \(-0.279074\pi\)
0.639662 + 0.768656i \(0.279074\pi\)
\(90\) 0.166861 0.0175887
\(91\) −1.14343 −0.119864
\(92\) 0 0
\(93\) 3.26725 0.338798
\(94\) 6.85499 0.707038
\(95\) −4.20784 −0.431715
\(96\) −1.74912 −0.178519
\(97\) −0.955493 −0.0970156 −0.0485078 − 0.998823i \(-0.515447\pi\)
−0.0485078 + 0.998823i \(0.515447\pi\)
\(98\) 1.00000 0.101015
\(99\) 0.0348029 0.00349782
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.bc.1.1 yes 4
23.22 odd 2 7406.2.a.bb.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bb.1.1 ✓ 4 23.22 odd 2
7406.2.a.bc.1.1 yes 4 1.1 even 1 trivial