Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,5,0] 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: \(44.1055504486\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.50110\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-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)
 
\(f(q)\) \(=\) \(q+6.96278 q^{2} -22.6701 q^{3} +16.4803 q^{4} -157.847 q^{6} +169.118 q^{7} -108.060 q^{8} +270.932 q^{9} -121.000 q^{11} -373.610 q^{12} -25.3182 q^{13} +1177.53 q^{14} -1279.77 q^{16} +2016.26 q^{17} +1886.44 q^{18} -773.486 q^{19} -3833.91 q^{21} -842.497 q^{22} +541.643 q^{23} +2449.73 q^{24} -176.285 q^{26} -633.231 q^{27} +2787.11 q^{28} -5882.28 q^{29} -915.584 q^{31} -5452.83 q^{32} +2743.08 q^{33} +14038.8 q^{34} +4465.06 q^{36} -11360.3 q^{37} -5385.62 q^{38} +573.964 q^{39} -15477.7 q^{41} -26694.7 q^{42} -6097.77 q^{43} -1994.12 q^{44} +3771.34 q^{46} -15131.8 q^{47} +29012.5 q^{48} +11793.8 q^{49} -45708.8 q^{51} -417.252 q^{52} -10443.0 q^{53} -4409.05 q^{54} -18274.9 q^{56} +17535.0 q^{57} -40957.0 q^{58} -50295.3 q^{59} +45523.0 q^{61} -6375.01 q^{62} +45819.4 q^{63} +2985.73 q^{64} +19099.5 q^{66} -11285.0 q^{67} +33228.7 q^{68} -12279.1 q^{69} +64741.3 q^{71} -29277.0 q^{72} -77769.4 q^{73} -79099.6 q^{74} -12747.3 q^{76} -20463.2 q^{77} +3996.39 q^{78} -87890.2 q^{79} -51481.2 q^{81} -107768. q^{82} -18403.3 q^{83} -63184.1 q^{84} -42457.4 q^{86} +133352. q^{87} +13075.3 q^{88} +52660.1 q^{89} -4281.74 q^{91} +8926.45 q^{92} +20756.4 q^{93} -105359. q^{94} +123616. q^{96} +38745.0 q^{97} +82117.3 q^{98} -32782.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} - 157 q^{6} + 90 q^{7} + 135 q^{8} + 22 q^{9} - 484 q^{11} + 795 q^{12} - 820 q^{13} - 1687 q^{14} - 2671 q^{16} + 3800 q^{17} + 1610 q^{18} - 3394 q^{19} - 4708 q^{21} - 605 q^{22}+ \cdots - 2662 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.96278 1.23086 0.615429 − 0.788192i \(-0.288983\pi\)
0.615429 + 0.788192i \(0.288983\pi\)
\(3\) −22.6701 −1.45429 −0.727143 − 0.686486i \(-0.759152\pi\)
−0.727143 + 0.686486i \(0.759152\pi\)
\(4\) 16.4803 0.515010
\(5\) 0 0
\(6\) −157.847 −1.79002
\(7\) 169.118 1.30450 0.652249 − 0.758004i \(-0.273826\pi\)
0.652249 + 0.758004i \(0.273826\pi\)
\(8\) −108.060 −0.596953
\(9\) 270.932 1.11495
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −373.610 −0.748973
\(13\) −25.3182 −0.0415502 −0.0207751 − 0.999784i \(-0.506613\pi\)
−0.0207751 + 0.999784i \(0.506613\pi\)
\(14\) 1177.53 1.60565
\(15\) 0 0
\(16\) −1279.77 −1.24977
\(17\) 2016.26 1.69209 0.846047 − 0.533108i \(-0.178976\pi\)
0.846047 + 0.533108i \(0.178976\pi\)
\(18\) 1886.44 1.37234
\(19\) −773.486 −0.491551 −0.245775 − 0.969327i \(-0.579043\pi\)
−0.245775 + 0.969327i \(0.579043\pi\)
\(20\) 0 0
\(21\) −3833.91 −1.89711
\(22\) −842.497 −0.371118
\(23\) 541.643 0.213498 0.106749 − 0.994286i \(-0.465956\pi\)
0.106749 + 0.994286i \(0.465956\pi\)
\(24\) 2449.73 0.868141
\(25\) 0 0
\(26\) −176.285 −0.0511424
\(27\) −633.231 −0.167168
\(28\) 2787.11 0.671830
\(29\) −5882.28 −1.29882 −0.649412 − 0.760437i \(-0.724985\pi\)
−0.649412 + 0.760437i \(0.724985\pi\)
\(30\) 0 0
\(31\) −915.584 −0.171117 −0.0855587 − 0.996333i \(-0.527268\pi\)
−0.0855587 + 0.996333i \(0.527268\pi\)
\(32\) −5452.83 −0.941342
\(33\) 2743.08 0.438484
\(34\) 14038.8 2.08273
\(35\) 0 0
\(36\) 4465.06 0.574210
\(37\) −11360.3 −1.36423 −0.682115 − 0.731245i \(-0.738939\pi\)
−0.682115 + 0.731245i \(0.738939\pi\)
\(38\) −5385.62 −0.605029
\(39\) 573.964 0.0604259
\(40\) 0 0
\(41\) −15477.7 −1.43796 −0.718979 − 0.695031i \(-0.755390\pi\)
−0.718979 + 0.695031i \(0.755390\pi\)
\(42\) −26694.7 −2.33508
\(43\) −6097.77 −0.502921 −0.251460 − 0.967868i \(-0.580911\pi\)
−0.251460 + 0.967868i \(0.580911\pi\)
\(44\) −1994.12 −0.155281
\(45\) 0 0
\(46\) 3771.34 0.262785
\(47\) −15131.8 −0.999185 −0.499593 − 0.866260i \(-0.666517\pi\)
−0.499593 + 0.866260i \(0.666517\pi\)
\(48\) 29012.5 1.81753
\(49\) 11793.8 0.701717
\(50\) 0 0
\(51\) −45708.8 −2.46079
\(52\) −417.252 −0.0213988
\(53\) −10443.0 −0.510666 −0.255333 − 0.966853i \(-0.582185\pi\)
−0.255333 + 0.966853i \(0.582185\pi\)
\(54\) −4409.05 −0.205760
\(55\) 0 0
\(56\) −18274.9 −0.778724
\(57\) 17535.0 0.714856
\(58\) −40957.0 −1.59867
\(59\) −50295.3 −1.88104 −0.940519 − 0.339741i \(-0.889661\pi\)
−0.940519 + 0.339741i \(0.889661\pi\)
\(60\) 0 0
\(61\) 45523.0 1.56641 0.783206 − 0.621762i \(-0.213583\pi\)
0.783206 + 0.621762i \(0.213583\pi\)
\(62\) −6375.01 −0.210621
\(63\) 45819.4 1.45445
\(64\) 2985.73 0.0911172
\(65\) 0 0
\(66\) 19099.5 0.539711
\(67\) −11285.0 −0.307124 −0.153562 − 0.988139i \(-0.549074\pi\)
−0.153562 + 0.988139i \(0.549074\pi\)
\(68\) 33228.7 0.871446
\(69\) −12279.1 −0.310487
\(70\) 0 0
\(71\) 64741.3 1.52418 0.762089 − 0.647473i \(-0.224174\pi\)
0.762089 + 0.647473i \(0.224174\pi\)
\(72\) −29277.0 −0.665572
\(73\) −77769.4 −1.70805 −0.854027 − 0.520229i \(-0.825847\pi\)
−0.854027 + 0.520229i \(0.825847\pi\)
\(74\) −79099.6 −1.67917
\(75\) 0 0
\(76\) −12747.3 −0.253154
\(77\) −20463.2 −0.393321
\(78\) 3996.39 0.0743757
\(79\) −87890.2 −1.58443 −0.792214 − 0.610243i \(-0.791072\pi\)
−0.792214 + 0.610243i \(0.791072\pi\)
\(80\) 0 0
\(81\) −51481.2 −0.871838
\(82\) −107768. −1.76992
\(83\) −18403.3 −0.293225 −0.146613 − 0.989194i \(-0.546837\pi\)
−0.146613 + 0.989194i \(0.546837\pi\)
\(84\) −63184.1 −0.977034
\(85\) 0 0
\(86\) −42457.4 −0.619024
\(87\) 133352. 1.88886
\(88\) 13075.3 0.179988
\(89\) 52660.1 0.704704 0.352352 − 0.935868i \(-0.385382\pi\)
0.352352 + 0.935868i \(0.385382\pi\)
\(90\) 0 0
\(91\) −4281.74 −0.0542022
\(92\) 8926.45 0.109954
\(93\) 20756.4 0.248854
\(94\) −105359. −1.22985
\(95\) 0 0
\(96\) 123616. 1.36898
\(97\) 38745.0 0.418106 0.209053 − 0.977904i \(-0.432962\pi\)
0.209053 + 0.977904i \(0.432962\pi\)
\(98\) 82117.3 0.863714
\(99\) −32782.8 −0.336170
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 275.6.a.d.1.3 4
5.2 odd 4 275.6.b.d.199.6 8
5.3 odd 4 275.6.b.d.199.3 8
5.4 even 2 55.6.a.b.1.2 ✓ 4
15.14 odd 2 495.6.a.g.1.3 4
20.19 odd 2 880.6.a.n.1.1 4
55.54 odd 2 605.6.a.c.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.2 ✓ 4 5.4 even 2
275.6.a.d.1.3 4 1.1 even 1 trivial
275.6.b.d.199.3 8 5.3 odd 4
275.6.b.d.199.6 8 5.2 odd 4
495.6.a.g.1.3 4 15.14 odd 2
605.6.a.c.1.3 4 55.54 odd 2
880.6.a.n.1.1 4 20.19 odd 2