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 1.4
Root \(-1.27133\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.68554 q^{3} +1.00000 q^{4} +2.52660 q^{5} +2.68554 q^{6} -1.00000 q^{7} +1.00000 q^{8} +4.21215 q^{9} +2.52660 q^{10} +3.41421 q^{11} +2.68554 q^{12} -0.744728 q^{13} -1.00000 q^{14} +6.78530 q^{15} +1.00000 q^{16} -0.526602 q^{17} +4.21215 q^{18} +7.37109 q^{19} +2.52660 q^{20} -2.68554 q^{21} +3.41421 q^{22} +2.68554 q^{24} +1.38372 q^{25} -0.744728 q^{26} +3.25527 q^{27} -1.00000 q^{28} -4.92638 q^{29} +6.78530 q^{30} -5.73875 q^{31} +1.00000 q^{32} +9.16902 q^{33} -0.526602 q^{34} -2.52660 q^{35} +4.21215 q^{36} +2.18165 q^{37} +7.37109 q^{38} -2.00000 q^{39} +2.52660 q^{40} -10.9670 q^{41} -2.68554 q^{42} -5.68897 q^{43} +3.41421 q^{44} +10.6424 q^{45} +2.62455 q^{47} +2.68554 q^{48} +1.00000 q^{49} +1.38372 q^{50} -1.41421 q^{51} -0.744728 q^{52} +6.27476 q^{53} +3.25527 q^{54} +8.62636 q^{55} -1.00000 q^{56} +19.7954 q^{57} -4.92638 q^{58} +7.73875 q^{59} +6.78530 q^{60} +12.5013 q^{61} -5.73875 q^{62} -4.21215 q^{63} +1.00000 q^{64} -1.88163 q^{65} +9.16902 q^{66} +12.1817 q^{67} -0.526602 q^{68} -2.52660 q^{70} -0.603650 q^{71} +4.21215 q^{72} +15.2754 q^{73} +2.18165 q^{74} +3.71604 q^{75} +7.37109 q^{76} -3.41421 q^{77} -2.00000 q^{78} -0.631207 q^{79} +2.52660 q^{80} -3.89426 q^{81} -10.9670 q^{82} -4.74473 q^{83} -2.68554 q^{84} -1.33051 q^{85} -5.68897 q^{86} -13.2300 q^{87} +3.41421 q^{88} -13.6931 q^{89} +10.6424 q^{90} +0.744728 q^{91} -15.4117 q^{93} +2.62455 q^{94} +18.6238 q^{95} +2.68554 q^{96} -16.5240 q^{97} +1.00000 q^{98} +14.3812 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{8} + 4 q^{10} + 8 q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} + 4 q^{17} + 8 q^{19} + 4 q^{20} + 8 q^{22} - 4 q^{26} + 12 q^{27} - 4 q^{28} - 4 q^{29} + 4 q^{32}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(3\) 2.68554 1.55050 0.775250 − 0.631655i \(-0.217624\pi\)
0.775250 + 0.631655i \(0.217624\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.52660 1.12993 0.564965 − 0.825115i \(-0.308889\pi\)
0.564965 + 0.825115i \(0.308889\pi\)
\(6\) 2.68554 1.09637
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 4.21215 1.40405
\(10\) 2.52660 0.798982
\(11\) 3.41421 1.02942 0.514712 − 0.857363i \(-0.327899\pi\)
0.514712 + 0.857363i \(0.327899\pi\)
\(12\) 2.68554 0.775250
\(13\) −0.744728 −0.206550 −0.103275 − 0.994653i \(-0.532932\pi\)
−0.103275 + 0.994653i \(0.532932\pi\)
\(14\) −1.00000 −0.267261
\(15\) 6.78530 1.75196
\(16\) 1.00000 0.250000
\(17\) −0.526602 −0.127720 −0.0638599 − 0.997959i \(-0.520341\pi\)
−0.0638599 + 0.997959i \(0.520341\pi\)
\(18\) 4.21215 0.992812
\(19\) 7.37109 1.69104 0.845522 − 0.533941i \(-0.179290\pi\)
0.845522 + 0.533941i \(0.179290\pi\)
\(20\) 2.52660 0.564965
\(21\) −2.68554 −0.586034
\(22\) 3.41421 0.727913
\(23\) 0 0
\(24\) 2.68554 0.548184
\(25\) 1.38372 0.276744
\(26\) −0.744728 −0.146053
\(27\) 3.25527 0.626477
\(28\) −1.00000 −0.188982
\(29\) −4.92638 −0.914806 −0.457403 − 0.889260i \(-0.651220\pi\)
−0.457403 + 0.889260i \(0.651220\pi\)
\(30\) 6.78530 1.23882
\(31\) −5.73875 −1.03071 −0.515355 − 0.856977i \(-0.672340\pi\)
−0.515355 + 0.856977i \(0.672340\pi\)
\(32\) 1.00000 0.176777
\(33\) 9.16902 1.59612
\(34\) −0.526602 −0.0903115
\(35\) −2.52660 −0.427074
\(36\) 4.21215 0.702024
\(37\) 2.18165 0.358661 0.179331 − 0.983789i \(-0.442607\pi\)
0.179331 + 0.983789i \(0.442607\pi\)
\(38\) 7.37109 1.19575
\(39\) −2.00000 −0.320256
\(40\) 2.52660 0.399491
\(41\) −10.9670 −1.71275 −0.856375 − 0.516354i \(-0.827289\pi\)
−0.856375 + 0.516354i \(0.827289\pi\)
\(42\) −2.68554 −0.414388
\(43\) −5.68897 −0.867560 −0.433780 − 0.901019i \(-0.642820\pi\)
−0.433780 + 0.901019i \(0.642820\pi\)
\(44\) 3.41421 0.514712
\(45\) 10.6424 1.58648
\(46\) 0 0
\(47\) 2.62455 0.382831 0.191415 − 0.981509i \(-0.438692\pi\)
0.191415 + 0.981509i \(0.438692\pi\)
\(48\) 2.68554 0.387625
\(49\) 1.00000 0.142857
\(50\) 1.38372 0.195687
\(51\) −1.41421 −0.198030
\(52\) −0.744728 −0.103275
\(53\) 6.27476 0.861904 0.430952 − 0.902375i \(-0.358178\pi\)
0.430952 + 0.902375i \(0.358178\pi\)
\(54\) 3.25527 0.442986
\(55\) 8.62636 1.16318
\(56\) −1.00000 −0.133631
\(57\) 19.7954 2.62196
\(58\) −4.92638 −0.646865
\(59\) 7.73875 1.00750 0.503750 − 0.863850i \(-0.331953\pi\)
0.503750 + 0.863850i \(0.331953\pi\)
\(60\) 6.78530 0.875979
\(61\) 12.5013 1.60063 0.800316 − 0.599578i \(-0.204665\pi\)
0.800316 + 0.599578i \(0.204665\pi\)
\(62\) −5.73875 −0.728822
\(63\) −4.21215 −0.530681
\(64\) 1.00000 0.125000
\(65\) −1.88163 −0.233388
\(66\) 9.16902 1.12863
\(67\) 12.1817 1.48823 0.744113 − 0.668054i \(-0.232872\pi\)
0.744113 + 0.668054i \(0.232872\pi\)
\(68\) −0.526602 −0.0638599
\(69\) 0 0
\(70\) −2.52660 −0.301987
\(71\) −0.603650 −0.0716400 −0.0358200 − 0.999358i \(-0.511404\pi\)
−0.0358200 + 0.999358i \(0.511404\pi\)
\(72\) 4.21215 0.496406
\(73\) 15.2754 1.78785 0.893927 − 0.448213i \(-0.147939\pi\)
0.893927 + 0.448213i \(0.147939\pi\)
\(74\) 2.18165 0.253612
\(75\) 3.71604 0.429091
\(76\) 7.37109 0.845522
\(77\) −3.41421 −0.389086
\(78\) −2.00000 −0.226455
\(79\) −0.631207 −0.0710163 −0.0355081 − 0.999369i \(-0.511305\pi\)
−0.0355081 + 0.999369i \(0.511305\pi\)
\(80\) 2.52660 0.282483
\(81\) −3.89426 −0.432696
\(82\) −10.9670 −1.21110
\(83\) −4.74473 −0.520802 −0.260401 − 0.965501i \(-0.583855\pi\)
−0.260401 + 0.965501i \(0.583855\pi\)
\(84\) −2.68554 −0.293017
\(85\) −1.33051 −0.144315
\(86\) −5.68897 −0.613458
\(87\) −13.2300 −1.41841
\(88\) 3.41421 0.363956
\(89\) −13.6931 −1.45146 −0.725731 − 0.687978i \(-0.758498\pi\)
−0.725731 + 0.687978i \(0.758498\pi\)
\(90\) 10.6424 1.12181
\(91\) 0.744728 0.0780687
\(92\) 0 0
\(93\) −15.4117 −1.59811
\(94\) 2.62455 0.270702
\(95\) 18.6238 1.91076
\(96\) 2.68554 0.274092
\(97\) −16.5240 −1.67776 −0.838882 − 0.544314i \(-0.816790\pi\)
−0.838882 + 0.544314i \(0.816790\pi\)
\(98\) 1.00000 0.101015
\(99\) 14.3812 1.44536
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.1.4 yes 4
23.22 odd 2 7406.2.a.bb.1.4 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bb.1.4 ✓ 4 23.22 odd 2
7406.2.a.bc.1.4 yes 4 1.1 even 1 trivial