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: \(0\)
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 1.3
Root \(0.517638\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.482362 q^{3} +1.00000 q^{4} +1.21441 q^{5} +0.482362 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.76733 q^{9} -1.21441 q^{10} -0.449490 q^{11} -0.482362 q^{12} +3.18154 q^{13} -1.00000 q^{14} -0.585786 q^{15} +1.00000 q^{16} -0.785587 q^{17} +2.76733 q^{18} +3.67095 q^{19} +1.21441 q^{20} -0.482362 q^{21} +0.449490 q^{22} +0.482362 q^{24} -3.52520 q^{25} -3.18154 q^{26} +2.78194 q^{27} +1.00000 q^{28} +4.19615 q^{29} +0.585786 q^{30} +2.58214 q^{31} -1.00000 q^{32} +0.216817 q^{33} +0.785587 q^{34} +1.21441 q^{35} -2.76733 q^{36} -0.515234 q^{37} -3.67095 q^{38} -1.53465 q^{39} -1.21441 q^{40} -1.03528 q^{41} +0.482362 q^{42} -1.86370 q^{43} -0.449490 q^{44} -3.36068 q^{45} +4.13922 q^{47} -0.482362 q^{48} +1.00000 q^{49} +3.52520 q^{50} +0.378937 q^{51} +3.18154 q^{52} +5.88437 q^{53} -2.78194 q^{54} -0.545866 q^{55} -1.00000 q^{56} -1.77073 q^{57} -4.19615 q^{58} +3.87591 q^{59} -0.585786 q^{60} -1.90654 q^{61} -2.58214 q^{62} -2.76733 q^{63} +1.00000 q^{64} +3.86370 q^{65} -0.216817 q^{66} +4.74202 q^{67} -0.785587 q^{68} -1.21441 q^{70} -11.8131 q^{71} +2.76733 q^{72} +0.575090 q^{73} +0.515234 q^{74} +1.70042 q^{75} +3.67095 q^{76} -0.449490 q^{77} +1.53465 q^{78} -12.5493 q^{79} +1.21441 q^{80} +6.96008 q^{81} +1.03528 q^{82} -9.11099 q^{83} -0.482362 q^{84} -0.954027 q^{85} +1.86370 q^{86} -2.02406 q^{87} +0.449490 q^{88} -0.320241 q^{89} +3.36068 q^{90} +3.18154 q^{91} -1.24553 q^{93} -4.13922 q^{94} +4.45805 q^{95} +0.482362 q^{96} -1.76028 q^{97} -1.00000 q^{98} +1.24389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 8 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} - 8 q^{15} + 4 q^{16} - 8 q^{17} + 8 q^{19} - 4 q^{21} - 8 q^{22} + 4 q^{24} + 4 q^{26} - 4 q^{27}+ \cdots + 24 q^{99}+O(q^{100}) \) 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
\(3\) −0.482362 −0.278492 −0.139246 − 0.990258i \(-0.544468\pi\)
−0.139246 + 0.990258i \(0.544468\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.21441 0.543102 0.271551 − 0.962424i \(-0.412463\pi\)
0.271551 + 0.962424i \(0.412463\pi\)
\(6\) 0.482362 0.196923
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.76733 −0.922442
\(10\) −1.21441 −0.384031
\(11\) −0.449490 −0.135526 −0.0677631 − 0.997701i \(-0.521586\pi\)
−0.0677631 + 0.997701i \(0.521586\pi\)
\(12\) −0.482362 −0.139246
\(13\) 3.18154 0.882401 0.441200 − 0.897409i \(-0.354553\pi\)
0.441200 + 0.897409i \(0.354553\pi\)
\(14\) −1.00000 −0.267261
\(15\) −0.585786 −0.151249
\(16\) 1.00000 0.250000
\(17\) −0.785587 −0.190533 −0.0952665 − 0.995452i \(-0.530370\pi\)
−0.0952665 + 0.995452i \(0.530370\pi\)
\(18\) 2.76733 0.652265
\(19\) 3.67095 0.842174 0.421087 − 0.907020i \(-0.361649\pi\)
0.421087 + 0.907020i \(0.361649\pi\)
\(20\) 1.21441 0.271551
\(21\) −0.482362 −0.105260
\(22\) 0.449490 0.0958315
\(23\) 0 0
\(24\) 0.482362 0.0984617
\(25\) −3.52520 −0.705040
\(26\) −3.18154 −0.623951
\(27\) 2.78194 0.535384
\(28\) 1.00000 0.188982
\(29\) 4.19615 0.779206 0.389603 − 0.920983i \(-0.372612\pi\)
0.389603 + 0.920983i \(0.372612\pi\)
\(30\) 0.585786 0.106949
\(31\) 2.58214 0.463766 0.231883 − 0.972744i \(-0.425511\pi\)
0.231883 + 0.972744i \(0.425511\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.216817 0.0377429
\(34\) 0.785587 0.134727
\(35\) 1.21441 0.205273
\(36\) −2.76733 −0.461221
\(37\) −0.515234 −0.0847040 −0.0423520 − 0.999103i \(-0.513485\pi\)
−0.0423520 + 0.999103i \(0.513485\pi\)
\(38\) −3.67095 −0.595507
\(39\) −1.53465 −0.245741
\(40\) −1.21441 −0.192016
\(41\) −1.03528 −0.161683 −0.0808415 − 0.996727i \(-0.525761\pi\)
−0.0808415 + 0.996727i \(0.525761\pi\)
\(42\) 0.482362 0.0744301
\(43\) −1.86370 −0.284212 −0.142106 − 0.989851i \(-0.545387\pi\)
−0.142106 + 0.989851i \(0.545387\pi\)
\(44\) −0.449490 −0.0677631
\(45\) −3.36068 −0.500980
\(46\) 0 0
\(47\) 4.13922 0.603767 0.301883 − 0.953345i \(-0.402385\pi\)
0.301883 + 0.953345i \(0.402385\pi\)
\(48\) −0.482362 −0.0696229
\(49\) 1.00000 0.142857
\(50\) 3.52520 0.498539
\(51\) 0.378937 0.0530618
\(52\) 3.18154 0.441200
\(53\) 5.88437 0.808280 0.404140 − 0.914697i \(-0.367571\pi\)
0.404140 + 0.914697i \(0.367571\pi\)
\(54\) −2.78194 −0.378574
\(55\) −0.545866 −0.0736046
\(56\) −1.00000 −0.133631
\(57\) −1.77073 −0.234538
\(58\) −4.19615 −0.550982
\(59\) 3.87591 0.504601 0.252300 − 0.967649i \(-0.418813\pi\)
0.252300 + 0.967649i \(0.418813\pi\)
\(60\) −0.585786 −0.0756247
\(61\) −1.90654 −0.244108 −0.122054 − 0.992523i \(-0.538948\pi\)
−0.122054 + 0.992523i \(0.538948\pi\)
\(62\) −2.58214 −0.327932
\(63\) −2.76733 −0.348650
\(64\) 1.00000 0.125000
\(65\) 3.86370 0.479233
\(66\) −0.216817 −0.0266883
\(67\) 4.74202 0.579330 0.289665 − 0.957128i \(-0.406456\pi\)
0.289665 + 0.957128i \(0.406456\pi\)
\(68\) −0.785587 −0.0952665
\(69\) 0 0
\(70\) −1.21441 −0.145150
\(71\) −11.8131 −1.40196 −0.700978 − 0.713183i \(-0.747253\pi\)
−0.700978 + 0.713183i \(0.747253\pi\)
\(72\) 2.76733 0.326133
\(73\) 0.575090 0.0673092 0.0336546 − 0.999434i \(-0.489285\pi\)
0.0336546 + 0.999434i \(0.489285\pi\)
\(74\) 0.515234 0.0598947
\(75\) 1.70042 0.196348
\(76\) 3.67095 0.421087
\(77\) −0.449490 −0.0512241
\(78\) 1.53465 0.173765
\(79\) −12.5493 −1.41190 −0.705951 − 0.708260i \(-0.749480\pi\)
−0.705951 + 0.708260i \(0.749480\pi\)
\(80\) 1.21441 0.135775
\(81\) 6.96008 0.773342
\(82\) 1.03528 0.114327
\(83\) −9.11099 −1.00006 −0.500030 − 0.866008i \(-0.666678\pi\)
−0.500030 + 0.866008i \(0.666678\pi\)
\(84\) −0.482362 −0.0526300
\(85\) −0.954027 −0.103479
\(86\) 1.86370 0.200968
\(87\) −2.02406 −0.217002
\(88\) 0.449490 0.0479158
\(89\) −0.320241 −0.0339455 −0.0169728 − 0.999856i \(-0.505403\pi\)
−0.0169728 + 0.999856i \(0.505403\pi\)
\(90\) 3.36068 0.354246
\(91\) 3.18154 0.333516
\(92\) 0 0
\(93\) −1.24553 −0.129155
\(94\) −4.13922 −0.426928
\(95\) 4.45805 0.457386
\(96\) 0.482362 0.0492309
\(97\) −1.76028 −0.178729 −0.0893646 − 0.995999i \(-0.528484\pi\)
−0.0893646 + 0.995999i \(0.528484\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.24389 0.125015
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.z.1.3 yes 4
23.22 odd 2 7406.2.a.y.1.3 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.y.1.3 ✓ 4 23.22 odd 2
7406.2.a.z.1.3 yes 4 1.1 even 1 trivial