Properties

Label 7406.2.a.y.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.93185 q^{3} +1.00000 q^{4} -0.199801 q^{5} +2.93185 q^{6} -1.00000 q^{7} -1.00000 q^{8} +5.59575 q^{9} +0.199801 q^{10} -4.44949 q^{11} -2.93185 q^{12} -5.18154 q^{13} +1.00000 q^{14} +0.585786 q^{15} +1.00000 q^{16} +1.80020 q^{17} -5.59575 q^{18} +8.15623 q^{19} -0.199801 q^{20} +2.93185 q^{21} +4.44949 q^{22} +2.93185 q^{24} -4.96008 q^{25} +5.18154 q^{26} -7.61037 q^{27} -1.00000 q^{28} -6.19615 q^{29} -0.585786 q^{30} -8.82478 q^{31} -1.00000 q^{32} +13.0452 q^{33} -1.80020 q^{34} +0.199801 q^{35} +5.59575 q^{36} +10.3132 q^{37} -8.15623 q^{38} +15.1915 q^{39} +0.199801 q^{40} +3.86370 q^{41} -2.93185 q^{42} -3.03528 q^{43} -4.44949 q^{44} -1.11804 q^{45} +6.58871 q^{47} -2.93185 q^{48} +1.00000 q^{49} +4.96008 q^{50} -5.27792 q^{51} -5.18154 q^{52} +1.88437 q^{53} +7.61037 q^{54} +0.889012 q^{55} +1.00000 q^{56} -23.9129 q^{57} +6.19615 q^{58} +9.19516 q^{59} +0.585786 q^{60} -4.00705 q^{61} +8.82478 q^{62} -5.59575 q^{63} +1.00000 q^{64} +1.03528 q^{65} -13.0452 q^{66} +7.08516 q^{67} +1.80020 q^{68} -0.199801 q^{70} +0.0140964 q^{71} -5.59575 q^{72} -4.91824 q^{73} -10.3132 q^{74} +14.5422 q^{75} +8.15623 q^{76} +4.44949 q^{77} -15.1915 q^{78} -6.20612 q^{79} -0.199801 q^{80} +5.52520 q^{81} -3.86370 q^{82} +10.5459 q^{83} +2.93185 q^{84} -0.359681 q^{85} +3.03528 q^{86} +18.1662 q^{87} +4.44949 q^{88} -15.3913 q^{89} +1.11804 q^{90} +5.18154 q^{91} +25.8729 q^{93} -6.58871 q^{94} -1.62962 q^{95} +2.93185 q^{96} -0.689211 q^{97} -1.00000 q^{98} -24.8983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{7} - 4 q^{8} - 8 q^{11} - 4 q^{12} - 4 q^{13} + 4 q^{14} + 8 q^{15} + 4 q^{16} + 8 q^{17} - 8 q^{19} + 4 q^{21} + 8 q^{22} + 4 q^{24} + 4 q^{26} - 4 q^{27}+ \cdots - 24 q^{99}+O(q^{100}) \) 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
\(3\) −2.93185 −1.69271 −0.846353 − 0.532623i \(-0.821206\pi\)
−0.846353 + 0.532623i \(0.821206\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.199801 −0.0893537 −0.0446768 − 0.999001i \(-0.514226\pi\)
−0.0446768 + 0.999001i \(0.514226\pi\)
\(6\) 2.93185 1.19692
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 5.59575 1.86525
\(10\) 0.199801 0.0631826
\(11\) −4.44949 −1.34157 −0.670786 − 0.741651i \(-0.734043\pi\)
−0.670786 + 0.741651i \(0.734043\pi\)
\(12\) −2.93185 −0.846353
\(13\) −5.18154 −1.43710 −0.718550 − 0.695475i \(-0.755194\pi\)
−0.718550 + 0.695475i \(0.755194\pi\)
\(14\) 1.00000 0.267261
\(15\) 0.585786 0.151249
\(16\) 1.00000 0.250000
\(17\) 1.80020 0.436612 0.218306 − 0.975880i \(-0.429947\pi\)
0.218306 + 0.975880i \(0.429947\pi\)
\(18\) −5.59575 −1.31893
\(19\) 8.15623 1.87117 0.935584 − 0.353104i \(-0.114874\pi\)
0.935584 + 0.353104i \(0.114874\pi\)
\(20\) −0.199801 −0.0446768
\(21\) 2.93185 0.639782
\(22\) 4.44949 0.948634
\(23\) 0 0
\(24\) 2.93185 0.598462
\(25\) −4.96008 −0.992016
\(26\) 5.18154 1.01618
\(27\) −7.61037 −1.46462
\(28\) −1.00000 −0.188982
\(29\) −6.19615 −1.15060 −0.575298 − 0.817944i \(-0.695114\pi\)
−0.575298 + 0.817944i \(0.695114\pi\)
\(30\) −0.585786 −0.106949
\(31\) −8.82478 −1.58498 −0.792489 − 0.609887i \(-0.791215\pi\)
−0.792489 + 0.609887i \(0.791215\pi\)
\(32\) −1.00000 −0.176777
\(33\) 13.0452 2.27089
\(34\) −1.80020 −0.308732
\(35\) 0.199801 0.0337725
\(36\) 5.59575 0.932626
\(37\) 10.3132 1.69548 0.847739 − 0.530413i \(-0.177963\pi\)
0.847739 + 0.530413i \(0.177963\pi\)
\(38\) −8.15623 −1.32312
\(39\) 15.1915 2.43259
\(40\) 0.199801 0.0315913
\(41\) 3.86370 0.603409 0.301705 − 0.953401i \(-0.402444\pi\)
0.301705 + 0.953401i \(0.402444\pi\)
\(42\) −2.93185 −0.452395
\(43\) −3.03528 −0.462875 −0.231438 − 0.972850i \(-0.574343\pi\)
−0.231438 + 0.972850i \(0.574343\pi\)
\(44\) −4.44949 −0.670786
\(45\) −1.11804 −0.166667
\(46\) 0 0
\(47\) 6.58871 0.961062 0.480531 − 0.876978i \(-0.340444\pi\)
0.480531 + 0.876978i \(0.340444\pi\)
\(48\) −2.93185 −0.423176
\(49\) 1.00000 0.142857
\(50\) 4.96008 0.701461
\(51\) −5.27792 −0.739056
\(52\) −5.18154 −0.718550
\(53\) 1.88437 0.258838 0.129419 − 0.991590i \(-0.458689\pi\)
0.129419 + 0.991590i \(0.458689\pi\)
\(54\) 7.61037 1.03564
\(55\) 0.889012 0.119874
\(56\) 1.00000 0.133631
\(57\) −23.9129 −3.16734
\(58\) 6.19615 0.813595
\(59\) 9.19516 1.19711 0.598554 − 0.801083i \(-0.295742\pi\)
0.598554 + 0.801083i \(0.295742\pi\)
\(60\) 0.585786 0.0756247
\(61\) −4.00705 −0.513050 −0.256525 − 0.966538i \(-0.582578\pi\)
−0.256525 + 0.966538i \(0.582578\pi\)
\(62\) 8.82478 1.12075
\(63\) −5.59575 −0.704999
\(64\) 1.00000 0.125000
\(65\) 1.03528 0.128410
\(66\) −13.0452 −1.60576
\(67\) 7.08516 0.865591 0.432795 − 0.901492i \(-0.357527\pi\)
0.432795 + 0.901492i \(0.357527\pi\)
\(68\) 1.80020 0.218306
\(69\) 0 0
\(70\) −0.199801 −0.0238808
\(71\) 0.0140964 0.00167294 0.000836469 − 1.00000i \(-0.499734\pi\)
0.000836469 1.00000i \(0.499734\pi\)
\(72\) −5.59575 −0.659466
\(73\) −4.91824 −0.575636 −0.287818 − 0.957685i \(-0.592930\pi\)
−0.287818 + 0.957685i \(0.592930\pi\)
\(74\) −10.3132 −1.19888
\(75\) 14.5422 1.67919
\(76\) 8.15623 0.935584
\(77\) 4.44949 0.507066
\(78\) −15.1915 −1.72010
\(79\) −6.20612 −0.698243 −0.349122 − 0.937077i \(-0.613520\pi\)
−0.349122 + 0.937077i \(0.613520\pi\)
\(80\) −0.199801 −0.0223384
\(81\) 5.52520 0.613911
\(82\) −3.86370 −0.426675
\(83\) 10.5459 1.15756 0.578780 − 0.815484i \(-0.303529\pi\)
0.578780 + 0.815484i \(0.303529\pi\)
\(84\) 2.93185 0.319891
\(85\) −0.359681 −0.0390129
\(86\) 3.03528 0.327302
\(87\) 18.1662 1.94762
\(88\) 4.44949 0.474317
\(89\) −15.3913 −1.63148 −0.815738 − 0.578422i \(-0.803669\pi\)
−0.815738 + 0.578422i \(0.803669\pi\)
\(90\) 1.11804 0.117851
\(91\) 5.18154 0.543173
\(92\) 0 0
\(93\) 25.8729 2.68290
\(94\) −6.58871 −0.679573
\(95\) −1.62962 −0.167196
\(96\) 2.93185 0.299231
\(97\) −0.689211 −0.0699788 −0.0349894 − 0.999388i \(-0.511140\pi\)
−0.0349894 + 0.999388i \(0.511140\pi\)
\(98\) −1.00000 −0.101015
\(99\) −24.8983 −2.50237
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.1.1 ✓ 4
23.22 odd 2 7406.2.a.z.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.y.1.1 ✓ 4 1.1 even 1 trivial
7406.2.a.z.1.1 yes 4 23.22 odd 2