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Properties

Label 2401.4.a.c.1.3
Level $2401$
Weight $4$
Character 2401.1
Self dual yes
Analytic conductor $141.664$
Analytic rank $1$
Dimension $39$
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] = [2401,4,Mod(1,2401)] 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("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2401 = 7^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2401.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [39,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(141.663585924\)
Analytic rank: \(1\)
Dimension: \(39\)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 2401.1

$q$-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)
 
\(f(q)\) \(=\) \(q-5.14785 q^{2} -6.96013 q^{3} +18.5003 q^{4} -3.24701 q^{5} +35.8297 q^{6} -54.0542 q^{8} +21.4435 q^{9} +16.7151 q^{10} +58.8488 q^{11} -128.765 q^{12} -47.9754 q^{13} +22.5996 q^{15} +130.260 q^{16} +84.9507 q^{17} -110.388 q^{18} -54.8763 q^{19} -60.0707 q^{20} -302.944 q^{22} +150.540 q^{23} +376.225 q^{24} -114.457 q^{25} +246.970 q^{26} +38.6742 q^{27} +44.2844 q^{29} -116.339 q^{30} -95.7503 q^{31} -238.126 q^{32} -409.595 q^{33} -437.313 q^{34} +396.712 q^{36} +209.605 q^{37} +282.495 q^{38} +333.915 q^{39} +175.514 q^{40} -354.943 q^{41} +221.030 q^{43} +1088.72 q^{44} -69.6271 q^{45} -774.956 q^{46} +235.194 q^{47} -906.628 q^{48} +589.207 q^{50} -591.268 q^{51} -887.561 q^{52} -508.784 q^{53} -199.089 q^{54} -191.082 q^{55} +381.947 q^{57} -227.969 q^{58} -439.523 q^{59} +418.100 q^{60} +51.4628 q^{61} +492.908 q^{62} +183.755 q^{64} +155.776 q^{65} +2108.53 q^{66} -263.337 q^{67} +1571.62 q^{68} -1047.78 q^{69} +203.727 q^{71} -1159.11 q^{72} +679.617 q^{73} -1079.02 q^{74} +796.636 q^{75} -1015.23 q^{76} -1718.94 q^{78} -189.020 q^{79} -422.955 q^{80} -848.151 q^{81} +1827.19 q^{82} -520.046 q^{83} -275.835 q^{85} -1137.83 q^{86} -308.225 q^{87} -3181.02 q^{88} -1494.87 q^{89} +358.430 q^{90} +2785.04 q^{92} +666.435 q^{93} -1210.74 q^{94} +178.184 q^{95} +1657.39 q^{96} -712.036 q^{97} +1261.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 39 q + q^{2} - q^{3} + 145 q^{4} - 27 q^{5} - 41 q^{6} - 12 q^{8} + 312 q^{9} - 78 q^{10} + q^{11} + 91 q^{12} - 77 q^{13} - 161 q^{15} + 461 q^{16} - 211 q^{17} + 8 q^{18} - 314 q^{19} - 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.14785 −1.82004 −0.910020 − 0.414565i \(-0.863934\pi\)
−0.910020 + 0.414565i \(0.863934\pi\)
\(3\) −6.96013 −1.33948 −0.669739 − 0.742596i \(-0.733594\pi\)
−0.669739 + 0.742596i \(0.733594\pi\)
\(4\) 18.5003 2.31254
\(5\) −3.24701 −0.290421 −0.145210 − 0.989401i \(-0.546386\pi\)
−0.145210 + 0.989401i \(0.546386\pi\)
\(6\) 35.8297 2.43790
\(7\) 0 0
\(8\) −54.0542 −2.38888
\(9\) 21.4435 0.794203
\(10\) 16.7151 0.528578
\(11\) 58.8488 1.61305 0.806526 − 0.591199i \(-0.201345\pi\)
0.806526 + 0.591199i \(0.201345\pi\)
\(12\) −128.765 −3.09760
\(13\) −47.9754 −1.02354 −0.511768 − 0.859123i \(-0.671009\pi\)
−0.511768 + 0.859123i \(0.671009\pi\)
\(14\) 0 0
\(15\) 22.5996 0.389013
\(16\) 130.260 2.03531
\(17\) 84.9507 1.21197 0.605987 − 0.795474i \(-0.292778\pi\)
0.605987 + 0.795474i \(0.292778\pi\)
\(18\) −110.388 −1.44548
\(19\) −54.8763 −0.662605 −0.331302 − 0.943525i \(-0.607488\pi\)
−0.331302 + 0.943525i \(0.607488\pi\)
\(20\) −60.0707 −0.671611
\(21\) 0 0
\(22\) −302.944 −2.93582
\(23\) 150.540 1.36477 0.682385 − 0.730993i \(-0.260943\pi\)
0.682385 + 0.730993i \(0.260943\pi\)
\(24\) 376.225 3.19986
\(25\) −114.457 −0.915656
\(26\) 246.970 1.86288
\(27\) 38.6742 0.275661
\(28\) 0 0
\(29\) 44.2844 0.283566 0.141783 − 0.989898i \(-0.454717\pi\)
0.141783 + 0.989898i \(0.454717\pi\)
\(30\) −116.339 −0.708018
\(31\) −95.7503 −0.554750 −0.277375 − 0.960762i \(-0.589464\pi\)
−0.277375 + 0.960762i \(0.589464\pi\)
\(32\) −238.126 −1.31547
\(33\) −409.595 −2.16065
\(34\) −437.313 −2.20584
\(35\) 0 0
\(36\) 396.712 1.83663
\(37\) 209.605 0.931321 0.465661 − 0.884963i \(-0.345817\pi\)
0.465661 + 0.884963i \(0.345817\pi\)
\(38\) 282.495 1.20597
\(39\) 333.915 1.37101
\(40\) 175.514 0.693781
\(41\) −354.943 −1.35202 −0.676010 − 0.736893i \(-0.736292\pi\)
−0.676010 + 0.736893i \(0.736292\pi\)
\(42\) 0 0
\(43\) 221.030 0.783877 0.391939 − 0.919991i \(-0.371805\pi\)
0.391939 + 0.919991i \(0.371805\pi\)
\(44\) 1088.72 3.73025
\(45\) −69.6271 −0.230653
\(46\) −774.956 −2.48394
\(47\) 235.194 0.729928 0.364964 − 0.931022i \(-0.381081\pi\)
0.364964 + 0.931022i \(0.381081\pi\)
\(48\) −906.628 −2.72626
\(49\) 0 0
\(50\) 589.207 1.66653
\(51\) −591.268 −1.62341
\(52\) −887.561 −2.36697
\(53\) −508.784 −1.31862 −0.659310 − 0.751871i \(-0.729151\pi\)
−0.659310 + 0.751871i \(0.729151\pi\)
\(54\) −199.089 −0.501714
\(55\) −191.082 −0.468464
\(56\) 0 0
\(57\) 381.947 0.887545
\(58\) −227.969 −0.516101
\(59\) −439.523 −0.969847 −0.484924 − 0.874556i \(-0.661153\pi\)
−0.484924 + 0.874556i \(0.661153\pi\)
\(60\) 418.100 0.899609
\(61\) 51.4628 0.108019 0.0540093 − 0.998540i \(-0.482800\pi\)
0.0540093 + 0.998540i \(0.482800\pi\)
\(62\) 492.908 1.00967
\(63\) 0 0
\(64\) 183.755 0.358896
\(65\) 155.776 0.297257
\(66\) 2108.53 3.93246
\(67\) −263.337 −0.480175 −0.240087 − 0.970751i \(-0.577176\pi\)
−0.240087 + 0.970751i \(0.577176\pi\)
\(68\) 1571.62 2.80274
\(69\) −1047.78 −1.82808
\(70\) 0 0
\(71\) 203.727 0.340534 0.170267 − 0.985398i \(-0.445537\pi\)
0.170267 + 0.985398i \(0.445537\pi\)
\(72\) −1159.11 −1.89726
\(73\) 679.617 1.08963 0.544816 − 0.838556i \(-0.316600\pi\)
0.544816 + 0.838556i \(0.316600\pi\)
\(74\) −1079.02 −1.69504
\(75\) 796.636 1.22650
\(76\) −1015.23 −1.53230
\(77\) 0 0
\(78\) −1718.94 −2.49528
\(79\) −189.020 −0.269196 −0.134598 − 0.990900i \(-0.542974\pi\)
−0.134598 + 0.990900i \(0.542974\pi\)
\(80\) −422.955 −0.591098
\(81\) −848.151 −1.16344
\(82\) 1827.19 2.46073
\(83\) −520.046 −0.687740 −0.343870 − 0.939017i \(-0.611738\pi\)
−0.343870 + 0.939017i \(0.611738\pi\)
\(84\) 0 0
\(85\) −275.835 −0.351983
\(86\) −1137.83 −1.42669
\(87\) −308.225 −0.379830
\(88\) −3181.02 −3.85339
\(89\) −1494.87 −1.78041 −0.890203 − 0.455565i \(-0.849437\pi\)
−0.890203 + 0.455565i \(0.849437\pi\)
\(90\) 358.430 0.419798
\(91\) 0 0
\(92\) 2785.04 3.15609
\(93\) 666.435 0.743076
\(94\) −1210.74 −1.32850
\(95\) 178.184 0.192434
\(96\) 1657.39 1.76205
\(97\) −712.036 −0.745322 −0.372661 − 0.927967i \(-0.621555\pi\)
−0.372661 + 0.927967i \(0.621555\pi\)
\(98\) 0 0
\(99\) 1261.92 1.28109
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 2401.4.a.c.1.3 39
7.6 odd 2 2401.4.a.d.1.3 39
49.13 odd 14 49.4.e.a.22.13 ✓ 78
49.34 odd 14 49.4.e.a.29.13 yes 78
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.4.e.a.22.13 ✓ 78 49.13 odd 14
49.4.e.a.29.13 yes 78 49.34 odd 14
2401.4.a.c.1.3 39 1.1 even 1 trivial
2401.4.a.d.1.3 39 7.6 odd 2