Properties

Label 7406.2.a.bb.1.2
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 = [4,4,0,4,-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: \(1\)
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 1.2
Root \(2.68554\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.27133 q^{3} +1.00000 q^{4} -0.887611 q^{5} -1.27133 q^{6} +1.00000 q^{7} +1.00000 q^{8} -1.38372 q^{9} -0.887611 q^{10} -3.41421 q^{11} -1.27133 q^{12} +1.57316 q^{13} +1.00000 q^{14} +1.12845 q^{15} +1.00000 q^{16} -1.11239 q^{17} -1.38372 q^{18} +0.542661 q^{19} -0.887611 q^{20} -1.27133 q^{21} -3.41421 q^{22} -1.27133 q^{24} -4.21215 q^{25} +1.57316 q^{26} +5.57316 q^{27} +1.00000 q^{28} +8.58323 q^{29} +1.12845 q^{30} +1.49611 q^{31} +1.00000 q^{32} +4.34059 q^{33} -1.11239 q^{34} -0.887611 q^{35} -1.38372 q^{36} +9.01008 q^{37} +0.542661 q^{38} -2.00000 q^{39} -0.887611 q^{40} +8.13853 q^{41} -1.27133 q^{42} -6.86054 q^{43} -3.41421 q^{44} +1.22820 q^{45} -12.5240 q^{47} -1.27133 q^{48} +1.00000 q^{49} -4.21215 q^{50} +1.41421 q^{51} +1.57316 q^{52} +6.27476 q^{53} +5.57316 q^{54} +3.03049 q^{55} +1.00000 q^{56} -0.689901 q^{57} +8.58323 q^{58} +0.503892 q^{59} +1.12845 q^{60} -6.22658 q^{61} +1.49611 q^{62} -1.38372 q^{63} +1.00000 q^{64} -1.39635 q^{65} +4.34059 q^{66} -0.989921 q^{67} -1.11239 q^{68} -0.887611 q^{70} -3.88163 q^{71} -1.38372 q^{72} -4.79015 q^{73} +9.01008 q^{74} +5.35503 q^{75} +0.542661 q^{76} -3.41421 q^{77} -2.00000 q^{78} -17.1165 q^{79} -0.887611 q^{80} -2.93416 q^{81} +8.13853 q^{82} +2.42684 q^{83} -1.27133 q^{84} +0.987369 q^{85} -6.86054 q^{86} -10.9121 q^{87} -3.41421 q^{88} -14.9651 q^{89} +1.22820 q^{90} +1.57316 q^{91} -1.90205 q^{93} -12.5240 q^{94} -0.481672 q^{95} -1.27133 q^{96} +1.37545 q^{97} +1.00000 q^{98} +4.72431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8} - 4 q^{10} - 8 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{17} - 8 q^{19} - 4 q^{20} - 8 q^{22} - 4 q^{26} + 12 q^{27} + 4 q^{28} - 4 q^{29} + 4 q^{32}+ \cdots - 8 q^{99}+O(q^{100}) \) 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
\(3\) −1.27133 −0.734003 −0.367001 − 0.930220i \(-0.619616\pi\)
−0.367001 + 0.930220i \(0.619616\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.887611 −0.396952 −0.198476 − 0.980106i \(-0.563599\pi\)
−0.198476 + 0.980106i \(0.563599\pi\)
\(6\) −1.27133 −0.519018
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) −1.38372 −0.461240
\(10\) −0.887611 −0.280687
\(11\) −3.41421 −1.02942 −0.514712 − 0.857363i \(-0.672101\pi\)
−0.514712 + 0.857363i \(0.672101\pi\)
\(12\) −1.27133 −0.367001
\(13\) 1.57316 0.436315 0.218157 − 0.975914i \(-0.429995\pi\)
0.218157 + 0.975914i \(0.429995\pi\)
\(14\) 1.00000 0.267261
\(15\) 1.12845 0.291364
\(16\) 1.00000 0.250000
\(17\) −1.11239 −0.269794 −0.134897 − 0.990860i \(-0.543070\pi\)
−0.134897 + 0.990860i \(0.543070\pi\)
\(18\) −1.38372 −0.326146
\(19\) 0.542661 0.124495 0.0622475 − 0.998061i \(-0.480173\pi\)
0.0622475 + 0.998061i \(0.480173\pi\)
\(20\) −0.887611 −0.198476
\(21\) −1.27133 −0.277427
\(22\) −3.41421 −0.727913
\(23\) 0 0
\(24\) −1.27133 −0.259509
\(25\) −4.21215 −0.842429
\(26\) 1.57316 0.308521
\(27\) 5.57316 1.07255
\(28\) 1.00000 0.188982
\(29\) 8.58323 1.59387 0.796933 − 0.604067i \(-0.206454\pi\)
0.796933 + 0.604067i \(0.206454\pi\)
\(30\) 1.12845 0.206025
\(31\) 1.49611 0.268709 0.134354 − 0.990933i \(-0.457104\pi\)
0.134354 + 0.990933i \(0.457104\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.34059 0.755600
\(34\) −1.11239 −0.190773
\(35\) −0.887611 −0.150034
\(36\) −1.38372 −0.230620
\(37\) 9.01008 1.48125 0.740624 − 0.671920i \(-0.234530\pi\)
0.740624 + 0.671920i \(0.234530\pi\)
\(38\) 0.542661 0.0880312
\(39\) −2.00000 −0.320256
\(40\) −0.887611 −0.140344
\(41\) 8.13853 1.27102 0.635512 − 0.772091i \(-0.280789\pi\)
0.635512 + 0.772091i \(0.280789\pi\)
\(42\) −1.27133 −0.196171
\(43\) −6.86054 −1.04622 −0.523112 − 0.852264i \(-0.675229\pi\)
−0.523112 + 0.852264i \(0.675229\pi\)
\(44\) −3.41421 −0.514712
\(45\) 1.22820 0.183090
\(46\) 0 0
\(47\) −12.5240 −1.82682 −0.913410 − 0.407040i \(-0.866561\pi\)
−0.913410 + 0.407040i \(0.866561\pi\)
\(48\) −1.27133 −0.183501
\(49\) 1.00000 0.142857
\(50\) −4.21215 −0.595687
\(51\) 1.41421 0.198030
\(52\) 1.57316 0.218157
\(53\) 6.27476 0.861904 0.430952 − 0.902375i \(-0.358178\pi\)
0.430952 + 0.902375i \(0.358178\pi\)
\(54\) 5.57316 0.758410
\(55\) 3.03049 0.408632
\(56\) 1.00000 0.133631
\(57\) −0.689901 −0.0913796
\(58\) 8.58323 1.12703
\(59\) 0.503892 0.0656012 0.0328006 − 0.999462i \(-0.489557\pi\)
0.0328006 + 0.999462i \(0.489557\pi\)
\(60\) 1.12845 0.145682
\(61\) −6.22658 −0.797232 −0.398616 − 0.917118i \(-0.630509\pi\)
−0.398616 + 0.917118i \(0.630509\pi\)
\(62\) 1.49611 0.190006
\(63\) −1.38372 −0.174332
\(64\) 1.00000 0.125000
\(65\) −1.39635 −0.173196
\(66\) 4.34059 0.534290
\(67\) −0.989921 −0.120938 −0.0604691 − 0.998170i \(-0.519260\pi\)
−0.0604691 + 0.998170i \(0.519260\pi\)
\(68\) −1.11239 −0.134897
\(69\) 0 0
\(70\) −0.887611 −0.106090
\(71\) −3.88163 −0.460665 −0.230332 − 0.973112i \(-0.573981\pi\)
−0.230332 + 0.973112i \(0.573981\pi\)
\(72\) −1.38372 −0.163073
\(73\) −4.79015 −0.560644 −0.280322 − 0.959906i \(-0.590441\pi\)
−0.280322 + 0.959906i \(0.590441\pi\)
\(74\) 9.01008 1.04740
\(75\) 5.35503 0.618346
\(76\) 0.542661 0.0622475
\(77\) −3.41421 −0.389086
\(78\) −2.00000 −0.226455
\(79\) −17.1165 −1.92576 −0.962878 − 0.269938i \(-0.912997\pi\)
−0.962878 + 0.269938i \(0.912997\pi\)
\(80\) −0.887611 −0.0992380
\(81\) −2.93416 −0.326018
\(82\) 8.13853 0.898750
\(83\) 2.42684 0.266381 0.133190 − 0.991090i \(-0.457478\pi\)
0.133190 + 0.991090i \(0.457478\pi\)
\(84\) −1.27133 −0.138714
\(85\) 0.987369 0.107095
\(86\) −6.86054 −0.739791
\(87\) −10.9121 −1.16990
\(88\) −3.41421 −0.363956
\(89\) −14.9651 −1.58630 −0.793151 − 0.609025i \(-0.791561\pi\)
−0.793151 + 0.609025i \(0.791561\pi\)
\(90\) 1.22820 0.129464
\(91\) 1.57316 0.164911
\(92\) 0 0
\(93\) −1.90205 −0.197233
\(94\) −12.5240 −1.29176
\(95\) −0.481672 −0.0494185
\(96\) −1.27133 −0.129755
\(97\) 1.37545 0.139655 0.0698276 − 0.997559i \(-0.477755\pi\)
0.0698276 + 0.997559i \(0.477755\pi\)
\(98\) 1.00000 0.101015
\(99\) 4.72431 0.474811
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.bb.1.2 ✓ 4
23.22 odd 2 7406.2.a.bc.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.bb.1.2 ✓ 4 1.1 even 1 trivial
7406.2.a.bc.1.2 yes 4 23.22 odd 2