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 = [6,6,2,6,4,2,6] 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: \(6\)
Coefficient field: 6.6.84770496.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 10x^{4} + 24x^{3} + 6x^{2} - 28x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.36985\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.36985 q^{3} +1.00000 q^{4} +2.28223 q^{5} +2.36985 q^{6} +1.00000 q^{7} +1.00000 q^{8} +2.61619 q^{9} +2.28223 q^{10} +0.630151 q^{11} +2.36985 q^{12} +2.95108 q^{13} +1.00000 q^{14} +5.40854 q^{15} +1.00000 q^{16} +3.59074 q^{17} +2.61619 q^{18} +3.81687 q^{19} +2.28223 q^{20} +2.36985 q^{21} +0.630151 q^{22} +2.36985 q^{24} +0.208577 q^{25} +2.95108 q^{26} -0.909582 q^{27} +1.00000 q^{28} +0.667910 q^{29} +5.40854 q^{30} -6.23611 q^{31} +1.00000 q^{32} +1.49336 q^{33} +3.59074 q^{34} +2.28223 q^{35} +2.61619 q^{36} +3.40294 q^{37} +3.81687 q^{38} +6.99360 q^{39} +2.28223 q^{40} -7.02659 q^{41} +2.36985 q^{42} -1.00765 q^{43} +0.630151 q^{44} +5.97074 q^{45} -13.4621 q^{47} +2.36985 q^{48} +1.00000 q^{49} +0.208577 q^{50} +8.50951 q^{51} +2.95108 q^{52} +1.94911 q^{53} -0.909582 q^{54} +1.43815 q^{55} +1.00000 q^{56} +9.04541 q^{57} +0.667910 q^{58} +9.39644 q^{59} +5.40854 q^{60} -1.46970 q^{61} -6.23611 q^{62} +2.61619 q^{63} +1.00000 q^{64} +6.73503 q^{65} +1.49336 q^{66} -1.80290 q^{67} +3.59074 q^{68} +2.28223 q^{70} -1.19677 q^{71} +2.61619 q^{72} -2.90958 q^{73} +3.40294 q^{74} +0.494297 q^{75} +3.81687 q^{76} +0.630151 q^{77} +6.99360 q^{78} -10.0485 q^{79} +2.28223 q^{80} -10.0041 q^{81} -7.02659 q^{82} -3.35663 q^{83} +2.36985 q^{84} +8.19489 q^{85} -1.00765 q^{86} +1.58285 q^{87} +0.630151 q^{88} -17.4557 q^{89} +5.97074 q^{90} +2.95108 q^{91} -14.7786 q^{93} -13.4621 q^{94} +8.71098 q^{95} +2.36985 q^{96} +15.0386 q^{97} +1.00000 q^{98} +1.64859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 4 q^{5} + 2 q^{6} + 6 q^{7} + 6 q^{8} + 6 q^{9} + 4 q^{10} + 16 q^{11} + 2 q^{12} + 6 q^{14} + 6 q^{16} + 2 q^{17} + 6 q^{18} + 18 q^{19} + 4 q^{20} + 2 q^{21} + 16 q^{22}+ \cdots + 28 q^{99}+O(q^{100}) \) 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
\(3\) 2.36985 1.36823 0.684117 − 0.729373i \(-0.260188\pi\)
0.684117 + 0.729373i \(0.260188\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.28223 1.02064 0.510322 − 0.859983i \(-0.329526\pi\)
0.510322 + 0.859983i \(0.329526\pi\)
\(6\) 2.36985 0.967487
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 2.61619 0.872062
\(10\) 2.28223 0.721705
\(11\) 0.630151 0.189998 0.0949988 − 0.995477i \(-0.469715\pi\)
0.0949988 + 0.995477i \(0.469715\pi\)
\(12\) 2.36985 0.684117
\(13\) 2.95108 0.818481 0.409241 − 0.912427i \(-0.365794\pi\)
0.409241 + 0.912427i \(0.365794\pi\)
\(14\) 1.00000 0.267261
\(15\) 5.40854 1.39648
\(16\) 1.00000 0.250000
\(17\) 3.59074 0.870882 0.435441 − 0.900217i \(-0.356592\pi\)
0.435441 + 0.900217i \(0.356592\pi\)
\(18\) 2.61619 0.616641
\(19\) 3.81687 0.875650 0.437825 − 0.899060i \(-0.355749\pi\)
0.437825 + 0.899060i \(0.355749\pi\)
\(20\) 2.28223 0.510322
\(21\) 2.36985 0.517144
\(22\) 0.630151 0.134349
\(23\) 0 0
\(24\) 2.36985 0.483743
\(25\) 0.208577 0.0417155
\(26\) 2.95108 0.578753
\(27\) −0.909582 −0.175049
\(28\) 1.00000 0.188982
\(29\) 0.667910 0.124028 0.0620139 − 0.998075i \(-0.480248\pi\)
0.0620139 + 0.998075i \(0.480248\pi\)
\(30\) 5.40854 0.987460
\(31\) −6.23611 −1.12004 −0.560019 − 0.828480i \(-0.689206\pi\)
−0.560019 + 0.828480i \(0.689206\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.49336 0.259961
\(34\) 3.59074 0.615807
\(35\) 2.28223 0.385767
\(36\) 2.61619 0.436031
\(37\) 3.40294 0.559441 0.279720 − 0.960082i \(-0.409758\pi\)
0.279720 + 0.960082i \(0.409758\pi\)
\(38\) 3.81687 0.619178
\(39\) 6.99360 1.11987
\(40\) 2.28223 0.360852
\(41\) −7.02659 −1.09737 −0.548685 − 0.836029i \(-0.684871\pi\)
−0.548685 + 0.836029i \(0.684871\pi\)
\(42\) 2.36985 0.365676
\(43\) −1.00765 −0.153665 −0.0768324 − 0.997044i \(-0.524481\pi\)
−0.0768324 + 0.997044i \(0.524481\pi\)
\(44\) 0.630151 0.0949988
\(45\) 5.97074 0.890065
\(46\) 0 0
\(47\) −13.4621 −1.96365 −0.981826 − 0.189782i \(-0.939222\pi\)
−0.981826 + 0.189782i \(0.939222\pi\)
\(48\) 2.36985 0.342058
\(49\) 1.00000 0.142857
\(50\) 0.208577 0.0294973
\(51\) 8.50951 1.19157
\(52\) 2.95108 0.409241
\(53\) 1.94911 0.267730 0.133865 − 0.991000i \(-0.457261\pi\)
0.133865 + 0.991000i \(0.457261\pi\)
\(54\) −0.909582 −0.123778
\(55\) 1.43815 0.193920
\(56\) 1.00000 0.133631
\(57\) 9.04541 1.19809
\(58\) 0.667910 0.0877008
\(59\) 9.39644 1.22331 0.611656 − 0.791124i \(-0.290504\pi\)
0.611656 + 0.791124i \(0.290504\pi\)
\(60\) 5.40854 0.698240
\(61\) −1.46970 −0.188176 −0.0940879 − 0.995564i \(-0.529993\pi\)
−0.0940879 + 0.995564i \(0.529993\pi\)
\(62\) −6.23611 −0.791986
\(63\) 2.61619 0.329608
\(64\) 1.00000 0.125000
\(65\) 6.73503 0.835378
\(66\) 1.49336 0.183820
\(67\) −1.80290 −0.220260 −0.110130 − 0.993917i \(-0.535127\pi\)
−0.110130 + 0.993917i \(0.535127\pi\)
\(68\) 3.59074 0.435441
\(69\) 0 0
\(70\) 2.28223 0.272779
\(71\) −1.19677 −0.142031 −0.0710153 − 0.997475i \(-0.522624\pi\)
−0.0710153 + 0.997475i \(0.522624\pi\)
\(72\) 2.61619 0.308320
\(73\) −2.90958 −0.340541 −0.170270 − 0.985397i \(-0.554464\pi\)
−0.170270 + 0.985397i \(0.554464\pi\)
\(74\) 3.40294 0.395584
\(75\) 0.494297 0.0570765
\(76\) 3.81687 0.437825
\(77\) 0.630151 0.0718123
\(78\) 6.99360 0.791870
\(79\) −10.0485 −1.13055 −0.565274 − 0.824904i \(-0.691229\pi\)
−0.565274 + 0.824904i \(0.691229\pi\)
\(80\) 2.28223 0.255161
\(81\) −10.0041 −1.11157
\(82\) −7.02659 −0.775957
\(83\) −3.35663 −0.368438 −0.184219 − 0.982885i \(-0.558976\pi\)
−0.184219 + 0.982885i \(0.558976\pi\)
\(84\) 2.36985 0.258572
\(85\) 8.19489 0.888861
\(86\) −1.00765 −0.108657
\(87\) 1.58285 0.169699
\(88\) 0.630151 0.0671743
\(89\) −17.4557 −1.85030 −0.925152 − 0.379597i \(-0.876063\pi\)
−0.925152 + 0.379597i \(0.876063\pi\)
\(90\) 5.97074 0.629371
\(91\) 2.95108 0.309357
\(92\) 0 0
\(93\) −14.7786 −1.53247
\(94\) −13.4621 −1.38851
\(95\) 8.71098 0.893728
\(96\) 2.36985 0.241872
\(97\) 15.0386 1.52694 0.763468 − 0.645845i \(-0.223495\pi\)
0.763468 + 0.645845i \(0.223495\pi\)
\(98\) 1.00000 0.101015
\(99\) 1.64859 0.165690
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.bl.1.6 yes 6
23.22 odd 2 7406.2.a.bk.1.6 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bk.1.6 ✓ 6 23.22 odd 2
7406.2.a.bl.1.6 yes 6 1.1 even 1 trivial