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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [58,6,Mod(57,58)] 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("58.57"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(58, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 58.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.30226154915\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1393x^{10} + 654324x^{8} + 119183764x^{6} + 7919971024x^{4} + 86379720768x^{2} + 65958566976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{16}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{7} + \beta_{6}) q^{3} - 16 q^{4} + (\beta_{2} - 1) q^{5} + ( - \beta_1 + 13) q^{6} + ( - \beta_{4} - \beta_{2} + 6) q^{7} + 16 \beta_{7} q^{8} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{9}+ \cdots + (159 \beta_{11} + \cdots + 4702 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 192 q^{4} - 10 q^{5} + 152 q^{6} + 76 q^{7} + 10 q^{9} - 1766 q^{13} + 3072 q^{16} + 160 q^{20} + 8 q^{22} - 4764 q^{23} - 2432 q^{24} + 9942 q^{25} - 1216 q^{28} - 3808 q^{29} - 7864 q^{30} + 16986 q^{33}+ \cdots + 38912 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1393x^{10} + 654324x^{8} + 119183764x^{6} + 7919971024x^{4} + 86379720768x^{2} + 65958566976 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 6998 \nu^{10} - 8540457 \nu^{8} - 3116505009 \nu^{6} - 296217297926 \nu^{4} + \cdots + 11627702808858 ) / 2416236979050 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3005333224 \nu^{10} - 3943978629271 \nu^{8} + \cdots - 10\!\cdots\!16 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7468335331 \nu^{10} + 9254431606204 \nu^{8} + \cdots + 17\!\cdots\!24 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8919979457 \nu^{10} + 11026038384913 \nu^{8} + \cdots - 61\!\cdots\!72 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38230467044 \nu^{10} - 51139276608391 \nu^{8} + \cdots - 53\!\cdots\!56 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5977807 \nu^{11} + 8626627543 \nu^{9} + 4276988308896 \nu^{7} + 845856419130784 \nu^{5} + \cdots + 91\!\cdots\!48 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5977807 \nu^{11} - 8626627543 \nu^{9} - 4276988308896 \nu^{7} + \cdots - 50\!\cdots\!48 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 204012737093 \nu^{11} - 391439250773969 \nu^{9} + \cdots - 18\!\cdots\!60 \nu ) / 59\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 783548976421 \nu^{11} + \cdots + 55\!\cdots\!16 \nu ) / 29\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3434960491213 \nu^{11} + \cdots - 22\!\cdots\!52 \nu ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1250629934381 \nu^{11} + \cdots + 83\!\cdots\!44 \nu ) / 19\!\cdots\!60 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + 4\beta_{6} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{4} + 2\beta_{3} - \beta _1 - 465 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{11} + 21\beta_{10} + 44\beta_{9} - 2\beta_{8} - 14\beta_{7} - 912\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 23\beta_{5} + 592\beta_{4} - 560\beta_{3} - 357\beta_{2} + 445\beta _1 + 105515 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2836\beta_{11} - 16005\beta_{10} - 31450\beta_{9} + 880\beta_{8} - 52555\beta_{7} + 472495\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16483\beta_{5} - 343798\beta_{4} + 302082\beta_{3} + 246327\beta_{2} - 317145\beta _1 - 54475525 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1288182 \beta_{11} + 9905157 \beta_{10} + 18722646 \beta_{9} - 507300 \beta_{8} + \cdots - 254546465 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9624939 \beta_{5} + 198055306 \beta_{4} - 162098518 \beta_{3} - 141590331 \beta_{2} + 211347731 \beta _1 + 29272844871 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 519815666 \beta_{11} - 5844736533 \beta_{10} - 10710017006 \beta_{9} + 314025164 \beta_{8} + \cdots + 138900956541 \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 5379397463 \beta_{5} - 113700449158 \beta_{4} + 87041656310 \beta_{3} + 78210400047 \beta_{2} + \cdots - 15938721851603 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 179572354090 \beta_{11} + 3401509323765 \beta_{10} + 6068437289182 \beta_{9} + \cdots - 76195988670805 \beta_{6} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).

\(n\) \(31\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
− 23.8535i
− 12.4001i
− 0.908439i
3.52029i
11.9820i
22.6597i
− 22.6597i
− 11.9820i
− 3.52029i
0.908439i
12.4001i
23.8535i
− 4.00000i − 20.8535i −16.0000 −22.3112 −83.4139 −96.1657 64.0000i −191.868 89.2448i
57.2 − 4.00000i − 9.40010i −16.0000 96.6936 −37.6004 85.6921 64.0000i 154.638 − 386.774i
57.3 − 4.00000i 2.09156i −16.0000 −97.4975 8.36624 163.931 64.0000i 238.625 389.990i
57.4 − 4.00000i 6.52029i −16.0000 15.8925 26.0812 7.47869 64.0000i 200.486 − 63.5700i
57.5 − 4.00000i 14.9820i −16.0000 46.4586 59.9281 −163.799 64.0000i 18.5393 − 185.834i
57.6 − 4.00000i 25.6597i −16.0000 −44.2359 102.639 40.8625 64.0000i −415.421 176.944i
57.7 4.00000i − 25.6597i −16.0000 −44.2359 102.639 40.8625 − 64.0000i −415.421 − 176.944i
57.8 4.00000i − 14.9820i −16.0000 46.4586 59.9281 −163.799 − 64.0000i 18.5393 185.834i
57.9 4.00000i − 6.52029i −16.0000 15.8925 26.0812 7.47869 − 64.0000i 200.486 63.5700i
57.10 4.00000i − 2.09156i −16.0000 −97.4975 8.36624 163.931 − 64.0000i 238.625 − 389.990i
57.11 4.00000i 9.40010i −16.0000 96.6936 −37.6004 85.6921 − 64.0000i 154.638 386.774i
57.12 4.00000i 20.8535i −16.0000 −22.3112 −83.4139 −96.1657 − 64.0000i −191.868 − 89.2448i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 57.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.6.b.a ✓ 12
3.b odd 2 1 522.6.d.b 12
4.b odd 2 1 464.6.e.b 12
29.b even 2 1 inner 58.6.b.a ✓ 12
87.d odd 2 1 522.6.d.b 12
116.d odd 2 1 464.6.e.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.6.b.a ✓ 12 1.a even 1 1 trivial
58.6.b.a ✓ 12 29.b even 2 1 inner
464.6.e.b 12 4.b odd 2 1
464.6.e.b 12 116.d odd 2 1
522.6.d.b 12 3.b odd 2 1
522.6.d.b 12 87.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(58, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 1056187844100 \) Copy content Toggle raw display
$5$ \( (T^{6} + 5 T^{5} + \cdots - 6869852746)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 38 T^{5} + \cdots + 67621201152)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 50\!\cdots\!10)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 18\!\cdots\!60)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 74\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 43\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 42\!\cdots\!94)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 22\!\cdots\!40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 34\!\cdots\!96)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
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