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Properties

Label 108.6.b.b
Level $108$
Weight $6$
Character orbit 108.b
Analytic conductor $17.321$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 30x^{14} + 619x^{12} + 5604x^{10} + 40971x^{8} - 4866x^{6} + 568069x^{4} - 7909632x^{2} + 20340100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{32}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + 6) q^{4} + (\beta_{3} - 3 \beta_1) q^{5} + ( - \beta_{10} + \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{3} + \cdots + 6 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{4} + 6) q^{4} + (\beta_{3} - 3 \beta_1) q^{5} + ( - \beta_{10} + \beta_{4}) q^{7} + ( - \beta_{7} + \beta_{3} + \cdots + 6 \beta_1) q^{8}+ \cdots + ( - 174 \beta_{13} - 700 \beta_{9} + \cdots - 3322 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 94 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 94 q^{4} + 1454 q^{10} + 896 q^{13} + 178 q^{16} + 30 q^{22} + 9888 q^{25} + 11454 q^{28} - 6172 q^{34} - 71008 q^{37} - 16618 q^{40} + 35304 q^{46} - 49376 q^{49} + 14876 q^{52} - 10492 q^{58} + 77888 q^{61} + 89206 q^{64} + 229398 q^{70} - 38032 q^{73} + 48960 q^{76} - 224488 q^{82} - 371264 q^{85} + 249102 q^{88} + 68772 q^{94} - 976 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 30x^{14} + 619x^{12} + 5604x^{10} + 40971x^{8} - 4866x^{6} + 568069x^{4} - 7909632x^{2} + 20340100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8580011824 \nu^{15} + 1068480910 \nu^{14} + 249341355770 \nu^{13} + 130620941625 \nu^{12} + \cdots - 46\!\cdots\!00 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10356042052 \nu^{15} - 512818688367 \nu^{14} - 559634572300 \nu^{13} + \cdots + 28\!\cdots\!20 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 102960141888 \nu^{15} + 993088055285 \nu^{14} + 2992096269240 \nu^{13} + \cdots - 44\!\cdots\!00 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9163365761 \nu^{15} + 19341597480 \nu^{14} + 350667454149 \nu^{13} + 550630309632 \nu^{12} + \cdots - 37\!\cdots\!88 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9163365761 \nu^{15} + 58012811010 \nu^{14} + 350667454149 \nu^{13} + 7239682944882 \nu^{12} + \cdots - 32\!\cdots\!40 ) / 54\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55695042027 \nu^{15} + 975870014470 \nu^{14} + 2694274061535 \nu^{13} + \cdots - 16\!\cdots\!00 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 71549127163 \nu^{15} + 1068480910 \nu^{14} + 2035013416385 \nu^{13} + 130620941625 \nu^{12} + \cdots - 46\!\cdots\!00 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 77070036758 \nu^{15} - 4413231616935 \nu^{14} + 4391182699990 \nu^{13} + \cdots + 20\!\cdots\!00 ) / 27\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 68640094592 \nu^{15} - 214858520889 \nu^{14} + 1994730846160 \nu^{13} + \cdots + 15\!\cdots\!80 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 197752231459 \nu^{15} + 77366389920 \nu^{14} + 5720648825457 \nu^{13} + \cdots - 14\!\cdots\!52 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 255661874717 \nu^{15} - 72011660196 \nu^{14} - 7569237235779 \nu^{13} + \cdots - 15\!\cdots\!64 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 376776329737 \nu^{15} - 397541409048 \nu^{14} - 11873706511995 \nu^{13} + \cdots + 41\!\cdots\!44 ) / 21\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2514369237472 \nu^{15} - 2572641289115 \nu^{14} - 85268253719840 \nu^{13} + \cdots + 14\!\cdots\!00 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 225638294345 \nu^{15} - 79151299828 \nu^{14} + 7443744491583 \nu^{13} + \cdots + 71\!\cdots\!24 ) / 72\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 842117283421 \nu^{15} + 1165850578524 \nu^{14} - 30011954301315 \nu^{13} + \cdots - 39\!\cdots\!00 ) / 21\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 19 \beta_{15} + 52 \beta_{14} - 24 \beta_{12} - 15 \beta_{11} - 64 \beta_{10} - 240 \beta_{7} + \cdots + 62 ) / 6480 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5 \beta_{15} + 20 \beta_{14} + 60 \beta_{12} - 25 \beta_{11} + 20 \beta_{10} + 66 \beta_{9} + \cdots - 8130 ) / 2160 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 52 \beta_{15} - 76 \beta_{14} - 48 \beta_{12} - 120 \beta_{11} + 232 \beta_{10} - 1215 \beta_{9} + \cdots - 56 ) / 6480 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 15 \beta_{15} + 120 \beta_{14} + 360 \beta_{12} - 105 \beta_{11} + 120 \beta_{10} - 648 \beta_{9} + \cdots - 91350 ) / 2160 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3334 \beta_{15} - 7732 \beta_{14} + 2400 \beta_{13} + 2304 \beta_{12} + 210 \beta_{11} + \cdots - 8972 ) / 6480 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2535 \beta_{15} - 5260 \beta_{14} - 15780 \beta_{12} + 7795 \beta_{11} - 5260 \beta_{10} + \cdots + 3222790 ) / 2160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 62722 \beta_{15} + 159316 \beta_{14} - 10080 \beta_{13} - 37152 \beta_{12} + 22290 \beta_{11} + \cdots + 162596 ) / 6480 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 44805 \beta_{15} + 68760 \beta_{14} + 206280 \beta_{12} - 113565 \beta_{11} + 68760 \beta_{10} + \cdots - 38898510 ) / 2160 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 379946 \beta_{15} - 1114628 \beta_{14} - 501120 \beta_{13} + 289536 \beta_{12} - 155610 \beta_{11} + \cdots - 1049428 ) / 6480 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 165935 \beta_{15} - 69900 \beta_{14} - 209700 \beta_{12} + 235835 \beta_{11} - 69900 \beta_{10} + \cdots + 23948470 ) / 2160 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6305902 \beta_{15} - 14271196 \beta_{14} + 11689920 \beta_{13} + 2874432 \beta_{12} + \cdots - 15486236 ) / 6480 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 2593905 \beta_{15} - 4597560 \beta_{14} - 13792680 \beta_{12} + 7191465 \beta_{11} + \cdots + 2693267670 ) / 720 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 183929966 \beta_{15} + 496673228 \beta_{14} - 93506400 \beta_{13} - 129359136 \beta_{12} + \cdots + 497219068 ) / 6480 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 188417785 \beta_{15} + 265125940 \beta_{14} + 795377820 \beta_{12} - 453543725 \beta_{11} + \cdots - 152142237690 ) / 2160 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1869056678 \beta_{15} - 5836123724 \beta_{14} - 1223903520 \beta_{13} + 1787937888 \beta_{12} + \cdots - 5526051244 ) / 6480 \) 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
−1.73205 + 3.53958i
−1.73205 − 3.53958i
1.73205 − 0.353400i
1.73205 + 0.353400i
−1.73205 + 3.22829i
−1.73205 − 3.22829i
1.73205 + 1.98106i
1.73205 − 1.98106i
−1.73205 − 1.98106i
−1.73205 + 1.98106i
1.73205 − 3.22829i
1.73205 + 3.22829i
−1.73205 + 0.353400i
−1.73205 − 0.353400i
1.73205 − 3.53958i
1.73205 + 3.53958i
−5.65395 − 0.181119i 0 31.9344 + 2.04808i 69.1814i 0 238.886i −180.185 − 17.3637i 0 12.5301 − 391.148i
107.2 −5.65395 + 0.181119i 0 31.9344 − 2.04808i − 69.1814i 0 − 238.886i −180.185 + 17.3637i 0 12.5301 + 391.148i
107.3 −5.08799 − 2.47231i 0 19.7753 + 25.1582i − 33.1536i 0 59.4073i −38.4178 − 176.896i 0 −81.9660 + 168.685i
107.4 −5.08799 + 2.47231i 0 19.7753 − 25.1582i 33.1536i 0 − 59.4073i −38.4178 + 176.896i 0 −81.9660 − 168.685i
107.5 −3.82945 − 4.16357i 0 −2.67058 + 31.8884i 46.0468i 0 − 134.772i 142.996 − 110.996i 0 191.719 − 176.334i
107.6 −3.82945 + 4.16357i 0 −2.67058 − 31.8884i − 46.0468i 0 134.772i 142.996 + 110.996i 0 191.719 + 176.334i
107.7 −1.79734 − 5.36373i 0 −25.5392 + 19.2809i 44.9719i 0 − 28.5094i 149.320 + 102.331i 0 241.217 − 80.8297i
107.8 −1.79734 + 5.36373i 0 −25.5392 − 19.2809i − 44.9719i 0 28.5094i 149.320 − 102.331i 0 241.217 + 80.8297i
107.9 1.79734 − 5.36373i 0 −25.5392 − 19.2809i 44.9719i 0 28.5094i −149.320 + 102.331i 0 241.217 + 80.8297i
107.10 1.79734 + 5.36373i 0 −25.5392 + 19.2809i − 44.9719i 0 − 28.5094i −149.320 − 102.331i 0 241.217 − 80.8297i
107.11 3.82945 − 4.16357i 0 −2.67058 − 31.8884i 46.0468i 0 134.772i −142.996 − 110.996i 0 191.719 + 176.334i
107.12 3.82945 + 4.16357i 0 −2.67058 + 31.8884i − 46.0468i 0 − 134.772i −142.996 + 110.996i 0 191.719 − 176.334i
107.13 5.08799 − 2.47231i 0 19.7753 − 25.1582i − 33.1536i 0 − 59.4073i 38.4178 − 176.896i 0 −81.9660 − 168.685i
107.14 5.08799 + 2.47231i 0 19.7753 + 25.1582i 33.1536i 0 59.4073i 38.4178 + 176.896i 0 −81.9660 + 168.685i
107.15 5.65395 − 0.181119i 0 31.9344 − 2.04808i 69.1814i 0 − 238.886i 180.185 − 17.3637i 0 12.5301 + 391.148i
107.16 5.65395 + 0.181119i 0 31.9344 + 2.04808i − 69.1814i 0 238.886i 180.185 + 17.3637i 0 12.5301 − 391.148i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.16
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.b ✓ 16
3.b odd 2 1 inner 108.6.b.b ✓ 16
4.b odd 2 1 inner 108.6.b.b ✓ 16
12.b even 2 1 inner 108.6.b.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.b.b ✓ 16 1.a even 1 1 trivial
108.6.b.b ✓ 16 3.b odd 2 1 inner
108.6.b.b ✓ 16 4.b odd 2 1 inner
108.6.b.b ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 10028T_{5}^{6} + 33930078T_{5}^{4} + 47031065276T_{5}^{2} + 22559032633225 \) acting on \(S_{6}^{\mathrm{new}}(108, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 1099511627776 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots + 22559032633225)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 29\!\cdots\!25)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 14\!\cdots\!09)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 224 T^{3} + \cdots + 64817879632)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 43\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 26\!\cdots\!25)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 57\!\cdots\!96)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 39\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 12\!\cdots\!25)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 97\!\cdots\!64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 12\!\cdots\!36)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 37\!\cdots\!25)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 51\!\cdots\!01)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 16\!\cdots\!17)^{4} \) Copy content Toggle raw display
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