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 1.4
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.931852 q^{3} +1.00000 q^{4} -3.66390 q^{5} -0.931852 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.13165 q^{9} +3.66390 q^{10} -0.449490 q^{11} +0.931852 q^{12} -0.282561 q^{13} -1.00000 q^{14} -3.41421 q^{15} +1.00000 q^{16} -5.66390 q^{17} +2.13165 q^{18} +5.22803 q^{19} -3.66390 q^{20} +0.931852 q^{21} +0.449490 q^{22} -0.931852 q^{24} +8.42418 q^{25} +0.282561 q^{26} -4.78194 q^{27} +1.00000 q^{28} -6.19615 q^{29} +3.41421 q^{30} -7.03163 q^{31} -1.00000 q^{32} -0.418858 q^{33} +5.66390 q^{34} -3.66390 q^{35} -2.13165 q^{36} +2.31319 q^{37} -5.22803 q^{38} -0.263305 q^{39} +3.66390 q^{40} -3.86370 q^{41} -0.931852 q^{42} +0.964724 q^{43} -0.449490 q^{44} +7.81017 q^{45} -8.58871 q^{47} +0.931852 q^{48} +1.00000 q^{49} -8.42418 q^{50} -5.27792 q^{51} -0.282561 q^{52} +12.8126 q^{53} +4.78194 q^{54} +1.64689 q^{55} -1.00000 q^{56} +4.87175 q^{57} +6.19615 q^{58} -10.1234 q^{59} -3.41421 q^{60} +11.4571 q^{61} +7.03163 q^{62} -2.13165 q^{63} +1.00000 q^{64} +1.03528 q^{65} +0.418858 q^{66} -7.84304 q^{67} -5.66390 q^{68} +3.66390 q^{70} +14.9141 q^{71} +2.13165 q^{72} -15.4741 q^{73} -2.31319 q^{74} +7.85009 q^{75} +5.22803 q^{76} -0.449490 q^{77} +0.263305 q^{78} -4.34971 q^{79} -3.66390 q^{80} +1.93890 q^{81} +3.86370 q^{82} +0.00996768 q^{83} +0.931852 q^{84} +20.7520 q^{85} -0.964724 q^{86} -5.77389 q^{87} +0.449490 q^{88} -3.92721 q^{89} -7.81017 q^{90} -0.282561 q^{91} -6.55243 q^{93} +8.58871 q^{94} -19.1550 q^{95} -0.931852 q^{96} +5.31079 q^{97} -1.00000 q^{98} +0.958156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} + 4 q^{7} - 4 q^{8} + 8 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} - 8 q^{15} + 4 q^{16} - 8 q^{17} + 8 q^{19} - 4 q^{21} - 8 q^{22} + 4 q^{24} + 4 q^{26} - 4 q^{27}+ \cdots + 24 q^{99}+O(q^{100}) \) 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
\(3\) 0.931852 0.538005 0.269002 − 0.963140i \(-0.413306\pi\)
0.269002 + 0.963140i \(0.413306\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.66390 −1.63855 −0.819273 − 0.573403i \(-0.805623\pi\)
−0.819273 + 0.573403i \(0.805623\pi\)
\(6\) −0.931852 −0.380427
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.13165 −0.710551
\(10\) 3.66390 1.15863
\(11\) −0.449490 −0.135526 −0.0677631 − 0.997701i \(-0.521586\pi\)
−0.0677631 + 0.997701i \(0.521586\pi\)
\(12\) 0.931852 0.269002
\(13\) −0.282561 −0.0783683 −0.0391842 − 0.999232i \(-0.512476\pi\)
−0.0391842 + 0.999232i \(0.512476\pi\)
\(14\) −1.00000 −0.267261
\(15\) −3.41421 −0.881546
\(16\) 1.00000 0.250000
\(17\) −5.66390 −1.37370 −0.686849 − 0.726800i \(-0.741007\pi\)
−0.686849 + 0.726800i \(0.741007\pi\)
\(18\) 2.13165 0.502435
\(19\) 5.22803 1.19939 0.599696 − 0.800228i \(-0.295288\pi\)
0.599696 + 0.800228i \(0.295288\pi\)
\(20\) −3.66390 −0.819273
\(21\) 0.931852 0.203347
\(22\) 0.449490 0.0958315
\(23\) 0 0
\(24\) −0.931852 −0.190213
\(25\) 8.42418 1.68484
\(26\) 0.282561 0.0554148
\(27\) −4.78194 −0.920285
\(28\) 1.00000 0.188982
\(29\) −6.19615 −1.15060 −0.575298 − 0.817944i \(-0.695114\pi\)
−0.575298 + 0.817944i \(0.695114\pi\)
\(30\) 3.41421 0.623347
\(31\) −7.03163 −1.26292 −0.631459 − 0.775409i \(-0.717544\pi\)
−0.631459 + 0.775409i \(0.717544\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.418858 −0.0729138
\(34\) 5.66390 0.971351
\(35\) −3.66390 −0.619313
\(36\) −2.13165 −0.355275
\(37\) 2.31319 0.380287 0.190143 − 0.981756i \(-0.439105\pi\)
0.190143 + 0.981756i \(0.439105\pi\)
\(38\) −5.22803 −0.848098
\(39\) −0.263305 −0.0421625
\(40\) 3.66390 0.579314
\(41\) −3.86370 −0.603409 −0.301705 − 0.953401i \(-0.597556\pi\)
−0.301705 + 0.953401i \(0.597556\pi\)
\(42\) −0.931852 −0.143788
\(43\) 0.964724 0.147119 0.0735595 − 0.997291i \(-0.476564\pi\)
0.0735595 + 0.997291i \(0.476564\pi\)
\(44\) −0.449490 −0.0677631
\(45\) 7.81017 1.16427
\(46\) 0 0
\(47\) −8.58871 −1.25279 −0.626396 − 0.779505i \(-0.715471\pi\)
−0.626396 + 0.779505i \(0.715471\pi\)
\(48\) 0.931852 0.134501
\(49\) 1.00000 0.142857
\(50\) −8.42418 −1.19136
\(51\) −5.27792 −0.739056
\(52\) −0.282561 −0.0391842
\(53\) 12.8126 1.75994 0.879971 − 0.475027i \(-0.157562\pi\)
0.879971 + 0.475027i \(0.157562\pi\)
\(54\) 4.78194 0.650739
\(55\) 1.64689 0.222066
\(56\) −1.00000 −0.133631
\(57\) 4.87175 0.645279
\(58\) 6.19615 0.813595
\(59\) −10.1234 −1.31795 −0.658975 − 0.752165i \(-0.729010\pi\)
−0.658975 + 0.752165i \(0.729010\pi\)
\(60\) −3.41421 −0.440773
\(61\) 11.4571 1.46693 0.733463 − 0.679730i \(-0.237903\pi\)
0.733463 + 0.679730i \(0.237903\pi\)
\(62\) 7.03163 0.893018
\(63\) −2.13165 −0.268563
\(64\) 1.00000 0.125000
\(65\) 1.03528 0.128410
\(66\) 0.418858 0.0515578
\(67\) −7.84304 −0.958180 −0.479090 − 0.877766i \(-0.659033\pi\)
−0.479090 + 0.877766i \(0.659033\pi\)
\(68\) −5.66390 −0.686849
\(69\) 0 0
\(70\) 3.66390 0.437920
\(71\) 14.9141 1.76998 0.884989 − 0.465611i \(-0.154165\pi\)
0.884989 + 0.465611i \(0.154165\pi\)
\(72\) 2.13165 0.251218
\(73\) −15.4741 −1.81110 −0.905551 − 0.424237i \(-0.860543\pi\)
−0.905551 + 0.424237i \(0.860543\pi\)
\(74\) −2.31319 −0.268903
\(75\) 7.85009 0.906450
\(76\) 5.22803 0.599696
\(77\) −0.449490 −0.0512241
\(78\) 0.263305 0.0298134
\(79\) −4.34971 −0.489381 −0.244691 − 0.969601i \(-0.578686\pi\)
−0.244691 + 0.969601i \(0.578686\pi\)
\(80\) −3.66390 −0.409637
\(81\) 1.93890 0.215433
\(82\) 3.86370 0.426675
\(83\) 0.00996768 0.00109409 0.000547047 − 1.00000i \(-0.499826\pi\)
0.000547047 1.00000i \(0.499826\pi\)
\(84\) 0.931852 0.101673
\(85\) 20.7520 2.25087
\(86\) −0.964724 −0.104029
\(87\) −5.77389 −0.619027
\(88\) 0.449490 0.0479158
\(89\) −3.92721 −0.416283 −0.208142 − 0.978099i \(-0.566741\pi\)
−0.208142 + 0.978099i \(0.566741\pi\)
\(90\) −7.81017 −0.823264
\(91\) −0.282561 −0.0296204
\(92\) 0 0
\(93\) −6.55243 −0.679456
\(94\) 8.58871 0.885857
\(95\) −19.1550 −1.96526
\(96\) −0.931852 −0.0951067
\(97\) 5.31079 0.539229 0.269614 − 0.962968i \(-0.413104\pi\)
0.269614 + 0.962968i \(0.413104\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0.958156 0.0962983
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.1.4 yes 4
23.22 odd 2 7406.2.a.y.1.4 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7406.2.a.y.1.4 ✓ 4 23.22 odd 2
7406.2.a.z.1.4 yes 4 1.1 even 1 trivial