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Properties

Label 108.6.b.c
Level $108$
Weight $6$
Character orbit 108.b
Analytic conductor $17.321$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [108,6,Mod(107,108)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(108, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("108.107");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 108 = 2^{2} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 108.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.3214525398\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 94 x^{18} + 5872 x^{16} - 207192 x^{14} + 5271952 x^{12} - 76648960 x^{10} + 792478720 x^{8} + \cdots + 41943040000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{50}\cdot 3^{40} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{2} + 1) q^{4} + \beta_{15} q^{5} - \beta_{5} q^{7} + (\beta_{17} + \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{2} + 1) q^{4} + \beta_{15} q^{5} - \beta_{5} q^{7} + (\beta_{17} + \beta_{4}) q^{8} + ( - \beta_1 + 9) q^{10} + (\beta_{11} + \beta_{4}) q^{11} + ( - \beta_{8} + \beta_1 - 6) q^{13} + ( - 2 \beta_{15} - \beta_{9} - \beta_{4}) q^{14} + (\beta_{8} - \beta_{7} + 2 \beta_{5} + \cdots - 208) q^{16}+ \cdots + ( - 4 \beta_{19} + \cdots - 2000 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 20 q^{4} + 184 q^{10} - 116 q^{13} - 4168 q^{16} + 696 q^{22} - 15228 q^{25} - 4764 q^{28} - 16520 q^{34} - 6452 q^{37} + 1504 q^{40} - 9336 q^{46} - 44464 q^{49} + 8236 q^{52} - 58736 q^{58} + 84604 q^{61} - 6496 q^{64} + 138696 q^{70} + 85420 q^{73} + 89172 q^{76} + 221200 q^{82} + 180320 q^{85} - 85824 q^{88} - 60936 q^{94} - 219908 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 94 x^{18} + 5872 x^{16} - 207192 x^{14} + 5271952 x^{12} - 76648960 x^{10} + 792478720 x^{8} + \cdots + 41943040000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 460704143658979 \nu^{18} + \cdots + 43\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20\!\cdots\!21 \nu^{18} + \cdots - 28\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 79\!\cdots\!41 \nu^{18} + \cdots + 15\!\cdots\!00 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 41\!\cdots\!79 \nu^{19} + \cdots + 89\!\cdots\!40 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 61\!\cdots\!17 \nu^{18} + \cdots - 13\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 13\!\cdots\!73 \nu^{18} + \cdots + 36\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!01 \nu^{18} + \cdots + 28\!\cdots\!40 ) / 19\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32\!\cdots\!29 \nu^{18} + \cdots + 28\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29\!\cdots\!63 \nu^{19} + \cdots - 10\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26\!\cdots\!89 \nu^{18} + \cdots + 16\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13\!\cdots\!79 \nu^{19} + \cdots - 16\!\cdots\!00 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 75\!\cdots\!39 \nu^{18} + \cdots - 35\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 75\!\cdots\!79 \nu^{19} + \cdots - 10\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 46\!\cdots\!41 \nu^{19} + \cdots - 10\!\cdots\!00 \nu ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 31\!\cdots\!79 \nu^{19} + \cdots - 30\!\cdots\!00 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 11\!\cdots\!51 \nu^{19} + \cdots - 51\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 60\!\cdots\!91 \nu^{19} + \cdots - 11\!\cdots\!00 \nu ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 80\!\cdots\!81 \nu^{19} + \cdots + 19\!\cdots\!00 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!03 \nu^{19} + \cdots - 10\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{19} + 12\beta_{18} + 2\beta_{16} - 8\beta_{15} + 29\beta_{14} - 2\beta_{13} + 51\beta_{4} ) / 2592 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{12} + 2 \beta_{10} - 3 \beta_{8} - 4 \beta_{7} + 6 \beta_{6} - 5 \beta_{5} - 83 \beta_{3} + \cdots + 6089 ) / 648 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 34 \beta_{19} + 56 \beta_{18} - 96 \beta_{17} - 82 \beta_{16} - 352 \beta_{15} + \cdots + 16965 \beta_{4} ) / 1296 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 31 \beta_{12} + 30 \beta_{10} + \beta_{8} + 24 \beta_{7} + 96 \beta_{6} - 53 \beta_{5} - 443 \beta_{3} + \cdots - 31415 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 170 \beta_{19} - 824 \beta_{18} - 2000 \beta_{17} - 362 \beta_{16} - 840 \beta_{15} + \cdots + 104361 \beta_{4} ) / 216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1759 \beta_{12} + 634 \beta_{10} + 2025 \beta_{8} + 2536 \beta_{7} + 3714 \beta_{6} + 491 \beta_{5} + \cdots - 3299855 ) / 162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1110 \beta_{19} - 60424 \beta_{18} - 77952 \beta_{17} + 11766 \beta_{16} + 91960 \beta_{15} + \cdots + 77229 \beta_{4} ) / 324 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1983 \beta_{12} - 21522 \beta_{10} + 37467 \beta_{8} + 22884 \beta_{7} - 31176 \beta_{6} + \cdots - 29846433 ) / 81 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 167546 \beta_{19} - 255784 \beta_{18} + 732960 \beta_{17} + 583274 \beta_{16} + \cdots - 101562429 \beta_{4} ) / 162 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1267466 \beta_{12} - 1224500 \beta_{10} - 42966 \beta_{8} - 836528 \beta_{7} - 3753900 \beta_{6} + \cdots + 1092980314 ) / 81 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2606246 \beta_{19} + 15694200 \beta_{18} + 40762128 \beta_{17} + 6613574 \beta_{16} + \cdots - 1867847859 \beta_{4} ) / 81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 14478644 \beta_{12} - 5122296 \beta_{10} - 17613356 \beta_{8} - 20489184 \beta_{7} + \cdots + 26727703156 ) / 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 5784084 \beta_{19} + 249255888 \beta_{18} + 332180544 \beta_{17} - 51139988 \beta_{16} + \cdots + 50202002 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 124343864 \beta_{12} + 1107810320 \beta_{10} - 1889345688 \beta_{8} - 1131258016 \beta_{7} + \cdots + 1472401899464 ) / 81 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1395291160 \beta_{19} + 2075597728 \beta_{18} - 6328900352 \beta_{17} - 4864253144 \beta_{16} + \cdots + 842754549724 \beta_{4} ) / 27 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 63228608976 \beta_{12} + 61786455456 \beta_{10} + 1442153520 \beta_{8} + 41667848832 \beta_{7} + \cdots - 54201041207376 ) / 81 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 131214426224 \beta_{19} - 783186283072 \beta_{18} - 2050461281664 \beta_{17} + \cdots + 92570416336344 \beta_{4} ) / 81 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 2150512607200 \beta_{12} + 767566959424 \beta_{10} + 2587457624352 \beta_{8} + 3070267837696 \beta_{7} + \cdots - 39\!\cdots\!12 ) / 81 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 870395542112 \beta_{19} - 37222217409152 \beta_{18} - 49312297651200 \beta_{17} + \cdots - 20574644736816 \beta_{4} ) / 81 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/108\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(55\)
\(\chi(n)\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
3.14287 − 1.81454i
3.14287 + 1.81454i
5.24560 − 3.02855i
5.24560 + 3.02855i
−1.31722 − 0.760496i
−1.31722 + 0.760496i
−1.93085 − 1.11477i
−1.93085 + 1.11477i
5.25764 − 3.03550i
5.25764 + 3.03550i
−5.25764 − 3.03550i
−5.25764 + 3.03550i
1.93085 − 1.11477i
1.93085 + 1.11477i
1.31722 − 0.760496i
1.31722 + 0.760496i
−5.24560 − 3.02855i
−5.24560 + 3.02855i
−3.14287 − 1.81454i
−3.14287 + 1.81454i
−5.64474 − 0.370087i 0 31.7261 + 4.17809i − 38.9183i 0 132.403i −177.539 − 35.3256i 0 −14.4032 + 219.683i
107.2 −5.64474 + 0.370087i 0 31.7261 − 4.17809i 38.9183i 0 − 132.403i −177.539 + 35.3256i 0 −14.4032 − 219.683i
107.3 −4.45658 − 3.48409i 0 7.72226 + 31.0543i 40.4806i 0 − 148.504i 73.7809 − 165.301i 0 141.038 − 180.405i
107.4 −4.45658 + 3.48409i 0 7.72226 − 31.0543i − 40.4806i 0 148.504i 73.7809 + 165.301i 0 141.038 + 180.405i
107.5 −4.10571 − 3.89142i 0 1.71366 + 31.9541i − 98.6190i 0 − 67.7266i 117.311 − 137.863i 0 −383.768 + 404.901i
107.6 −4.10571 + 3.89142i 0 1.71366 − 31.9541i 98.6190i 0 67.7266i 117.311 + 137.863i 0 −383.768 − 404.901i
107.7 −3.63934 − 4.33073i 0 −5.51036 + 31.5220i 80.5384i 0 216.819i 156.567 − 90.8555i 0 348.790 − 293.107i
107.8 −3.63934 + 4.33073i 0 −5.51036 − 31.5220i − 80.5384i 0 − 216.819i 156.567 + 90.8555i 0 348.790 + 293.107i
107.9 −0.821088 − 5.59695i 0 −30.6516 + 9.19117i − 8.15731i 0 63.0026i 76.6102 + 164.009i 0 −45.6560 + 6.69787i
107.10 −0.821088 + 5.59695i 0 −30.6516 − 9.19117i 8.15731i 0 − 63.0026i 76.6102 − 164.009i 0 −45.6560 − 6.69787i
107.11 0.821088 − 5.59695i 0 −30.6516 − 9.19117i − 8.15731i 0 − 63.0026i −76.6102 + 164.009i 0 −45.6560 − 6.69787i
107.12 0.821088 + 5.59695i 0 −30.6516 + 9.19117i 8.15731i 0 63.0026i −76.6102 − 164.009i 0 −45.6560 + 6.69787i
107.13 3.63934 − 4.33073i 0 −5.51036 − 31.5220i 80.5384i 0 − 216.819i −156.567 − 90.8555i 0 348.790 + 293.107i
107.14 3.63934 + 4.33073i 0 −5.51036 + 31.5220i − 80.5384i 0 216.819i −156.567 + 90.8555i 0 348.790 − 293.107i
107.15 4.10571 − 3.89142i 0 1.71366 − 31.9541i − 98.6190i 0 67.7266i −117.311 − 137.863i 0 −383.768 − 404.901i
107.16 4.10571 + 3.89142i 0 1.71366 + 31.9541i 98.6190i 0 − 67.7266i −117.311 + 137.863i 0 −383.768 + 404.901i
107.17 4.45658 − 3.48409i 0 7.72226 − 31.0543i 40.4806i 0 148.504i −73.7809 − 165.301i 0 141.038 + 180.405i
107.18 4.45658 + 3.48409i 0 7.72226 + 31.0543i − 40.4806i 0 − 148.504i −73.7809 + 165.301i 0 141.038 − 180.405i
107.19 5.64474 − 0.370087i 0 31.7261 − 4.17809i − 38.9183i 0 − 132.403i 177.539 − 35.3256i 0 −14.4032 − 219.683i
107.20 5.64474 + 0.370087i 0 31.7261 + 4.17809i 38.9183i 0 132.403i 177.539 + 35.3256i 0 −14.4032 + 219.683i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 108.6.b.c ✓ 20
3.b odd 2 1 inner 108.6.b.c ✓ 20
4.b odd 2 1 inner 108.6.b.c ✓ 20
12.b even 2 1 inner 108.6.b.c ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.b.c ✓ 20 1.a even 1 1 trivial
108.6.b.c ✓ 20 3.b odd 2 1 inner
108.6.b.c ✓ 20 4.b odd 2 1 inner
108.6.b.c ✓ 20 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 19432T_{5}^{8} + 117977728T_{5}^{6} + 246930301952T_{5}^{4} + 172491534061568T_{5}^{2} + 10418897768972288 \) acting on \(S_{6}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( (T^{10} + \cdots + 10\!\cdots\!88)^{2} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 33\!\cdots\!83)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 13976511398953)^{4} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 71\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 69\!\cdots\!75)^{2} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 22\!\cdots\!36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 14\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 48\!\cdots\!07)^{4} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 20\!\cdots\!72)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 35\!\cdots\!12)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 41\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 10\!\cdots\!25)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 88\!\cdots\!43)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots - 71\!\cdots\!03)^{4} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 97\!\cdots\!27)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 18\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 29\!\cdots\!61)^{4} \) Copy content Toggle raw display
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