Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,0,0,2,0,0,0,9,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.618034 q^{2} -1.61803 q^{4} +1.00000 q^{7} -2.23607 q^{8} +5.61803 q^{11} +4.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} +0.618034 q^{17} -1.85410 q^{19} +3.47214 q^{22} +1.47214 q^{23} +2.61803 q^{26} -1.61803 q^{28} +5.38197 q^{29} +6.70820 q^{31} +5.61803 q^{32} +0.381966 q^{34} -1.00000 q^{37} -1.14590 q^{38} +0.527864 q^{41} +11.5623 q^{43} -9.09017 q^{44} +0.909830 q^{46} -1.47214 q^{47} -6.00000 q^{49} -6.85410 q^{52} -10.4721 q^{53} -2.23607 q^{56} +3.32624 q^{58} -4.09017 q^{59} +6.70820 q^{61} +4.14590 q^{62} -0.236068 q^{64} -14.7984 q^{67} -1.00000 q^{68} -4.32624 q^{71} -8.56231 q^{73} -0.618034 q^{74} +3.00000 q^{76} +5.61803 q^{77} +6.09017 q^{79} +0.326238 q^{82} -11.3262 q^{83} +7.14590 q^{86} -12.5623 q^{88} +15.1803 q^{89} +4.23607 q^{91} -2.38197 q^{92} -0.909830 q^{94} -9.00000 q^{97} -3.70820 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 2 q^{7} + 9 q^{11} + 4 q^{13} - q^{14} - 3 q^{16} - q^{17} + 3 q^{19} - 2 q^{22} - 6 q^{23} + 3 q^{26} - q^{28} + 13 q^{29} + 9 q^{32} + 3 q^{34} - 2 q^{37} - 9 q^{38} + 10 q^{41}+ \cdots + 6 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.618034 0.437016 0.218508 − 0.975835i \(-0.429881\pi\)
0.218508 + 0.975835i \(0.429881\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 − 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) 5.61803 1.69390 0.846950 − 0.531672i \(-0.178436\pi\)
0.846950 + 0.531672i \(0.178436\pi\)
\(12\) 0 0
\(13\) 4.23607 1.17487 0.587437 − 0.809270i \(-0.300137\pi\)
0.587437 + 0.809270i \(0.300137\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 0.618034 0.149895 0.0749476 − 0.997187i \(-0.476121\pi\)
0.0749476 + 0.997187i \(0.476121\pi\)
\(18\) 0 0
\(19\) −1.85410 −0.425360 −0.212680 − 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.47214 0.740262
\(23\) 1.47214 0.306962 0.153481 − 0.988152i \(-0.450952\pi\)
0.153481 + 0.988152i \(0.450952\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.61803 0.513439
\(27\) 0 0
\(28\) −1.61803 −0.305780
\(29\) 5.38197 0.999406 0.499703 − 0.866197i \(-0.333442\pi\)
0.499703 + 0.866197i \(0.333442\pi\)
\(30\) 0 0
\(31\) 6.70820 1.20483 0.602414 − 0.798183i \(-0.294205\pi\)
0.602414 + 0.798183i \(0.294205\pi\)
\(32\) 5.61803 0.993137
\(33\) 0 0
\(34\) 0.381966 0.0655066
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) −1.14590 −0.185889
\(39\) 0 0
\(40\) 0 0
\(41\) 0.527864 0.0824385 0.0412193 − 0.999150i \(-0.486876\pi\)
0.0412193 + 0.999150i \(0.486876\pi\)
\(42\) 0 0
\(43\) 11.5623 1.76324 0.881618 − 0.471964i \(-0.156455\pi\)
0.881618 + 0.471964i \(0.156455\pi\)
\(44\) −9.09017 −1.37039
\(45\) 0 0
\(46\) 0.909830 0.134147
\(47\) −1.47214 −0.214733 −0.107367 − 0.994220i \(-0.534242\pi\)
−0.107367 + 0.994220i \(0.534242\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) −6.85410 −0.950493
\(53\) −10.4721 −1.43846 −0.719229 − 0.694773i \(-0.755505\pi\)
−0.719229 + 0.694773i \(0.755505\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) 3.32624 0.436756
\(59\) −4.09017 −0.532495 −0.266247 − 0.963905i \(-0.585784\pi\)
−0.266247 + 0.963905i \(0.585784\pi\)
\(60\) 0 0
\(61\) 6.70820 0.858898 0.429449 − 0.903091i \(-0.358708\pi\)
0.429449 + 0.903091i \(0.358708\pi\)
\(62\) 4.14590 0.526530
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) −14.7984 −1.80791 −0.903955 − 0.427629i \(-0.859349\pi\)
−0.903955 + 0.427629i \(0.859349\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) −4.32624 −0.513430 −0.256715 − 0.966487i \(-0.582640\pi\)
−0.256715 + 0.966487i \(0.582640\pi\)
\(72\) 0 0
\(73\) −8.56231 −1.00214 −0.501071 − 0.865406i \(-0.667060\pi\)
−0.501071 + 0.865406i \(0.667060\pi\)
\(74\) −0.618034 −0.0718450
\(75\) 0 0
\(76\) 3.00000 0.344124
\(77\) 5.61803 0.640234
\(78\) 0 0
\(79\) 6.09017 0.685198 0.342599 − 0.939482i \(-0.388693\pi\)
0.342599 + 0.939482i \(0.388693\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.326238 0.0360270
\(83\) −11.3262 −1.24322 −0.621608 − 0.783328i \(-0.713520\pi\)
−0.621608 + 0.783328i \(0.713520\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.14590 0.770562
\(87\) 0 0
\(88\) −12.5623 −1.33915
\(89\) 15.1803 1.60911 0.804556 − 0.593876i \(-0.202403\pi\)
0.804556 + 0.593876i \(0.202403\pi\)
\(90\) 0 0
\(91\) 4.23607 0.444061
\(92\) −2.38197 −0.248337
\(93\) 0 0
\(94\) −0.909830 −0.0938418
\(95\) 0 0
\(96\) 0 0
\(97\) −9.00000 −0.913812 −0.456906 − 0.889515i \(-0.651042\pi\)
−0.456906 + 0.889515i \(0.651042\pi\)
\(98\) −3.70820 −0.374585
\(99\) 0 0
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 8325.2.a.bh.1.2 2
3.2 odd 2 2775.2.a.n.1.1 2
5.4 even 2 1665.2.a.j.1.1 2
15.14 odd 2 555.2.a.e.1.2 ✓ 2
60.59 even 2 8880.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.a.e.1.2 ✓ 2 15.14 odd 2
1665.2.a.j.1.1 2 5.4 even 2
2775.2.a.n.1.1 2 3.2 odd 2
8325.2.a.bh.1.2 2 1.1 even 1 trivial
8880.2.a.bf.1.2 2 60.59 even 2