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Properties

Label 109.2.e.b
Level $109$
Weight $2$
Character orbit 109.e
Analytic conductor $0.870$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [109,2,Mod(46,109)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(109, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("109.46");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 109 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 109.e (of order \(6\), degree \(2\), 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: \(0.870369382032\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 18x^{12} + 117x^{10} + 346x^{8} + 486x^{6} + 300x^{4} + 61x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{13} + \beta_{10} + \beta_{5}) q^{3} + ( - \beta_{12} + \beta_{11} + \cdots + \beta_1) q^{4}+ \cdots + ( - \beta_{11} - \beta_{10} - \beta_{8} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{13} + \beta_{10} + \beta_{5}) q^{3} + ( - \beta_{12} + \beta_{11} + \cdots + \beta_1) q^{4}+ \cdots + (3 \beta_{13} - 2 \beta_{12} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 12 q^{6} - 7 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{3} - 8 q^{4} - 2 q^{5} - 12 q^{6} - 7 q^{7} - 9 q^{9} + 9 q^{10} - 9 q^{11} + 5 q^{12} + 9 q^{13} + 12 q^{14} + 20 q^{16} + 3 q^{18} - 9 q^{20} + 13 q^{21} - 8 q^{22} + 9 q^{24} + 7 q^{25} - 14 q^{26} - 10 q^{27} + 8 q^{28} + 3 q^{29} - 21 q^{30} - 3 q^{31} - 16 q^{34} + q^{35} - 26 q^{36} + 24 q^{37} + 32 q^{38} - 39 q^{39} + 21 q^{40} - 30 q^{42} - 24 q^{43} - 30 q^{44} + 78 q^{45} + 48 q^{46} + 30 q^{47} + 18 q^{48} - 30 q^{49} - 18 q^{50} + 33 q^{51} - 54 q^{52} - 6 q^{53} + 48 q^{56} - 18 q^{57} - 27 q^{58} - 18 q^{59} + 21 q^{60} + 10 q^{61} - 12 q^{62} + 24 q^{63} - 56 q^{64} - 45 q^{65} + 52 q^{66} - 9 q^{67} - 39 q^{69} - 39 q^{70} - 64 q^{71} + 33 q^{72} + 5 q^{73} + 47 q^{74} + 8 q^{75} - 12 q^{78} + 30 q^{79} + 26 q^{80} - 27 q^{81} + 78 q^{82} + 28 q^{83} + 7 q^{84} + 15 q^{85} + 18 q^{87} - 4 q^{88} - 28 q^{89} - 24 q^{93} + 38 q^{94} - 15 q^{95} - 18 q^{96} - 19 q^{97} + 15 q^{98} + 39 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 18x^{12} + 117x^{10} + 346x^{8} + 486x^{6} + 300x^{4} + 61x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{12} + 44\nu^{10} + 353\nu^{8} + 1249\nu^{6} + 1864\nu^{4} + 931\nu^{2} - 57\nu + 84 ) / 114 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9 \nu^{13} + 11 \nu^{12} - 141 \nu^{11} + 204 \nu^{10} - 705 \nu^{9} + 1381 \nu^{8} - 1146 \nu^{7} + \cdots + 348 ) / 114 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{12} - 204\nu^{10} - 1381\nu^{8} - 4276\nu^{6} - 6129\nu^{4} - 3401\nu^{2} - 348 ) / 57 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 17\nu^{12} + 298\nu^{10} + 1851\nu^{8} + 5059\nu^{6} + 6230\nu^{4} + 3135\nu^{2} + 429 ) / 57 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 11 \nu^{13} - 21 \nu^{12} + 204 \nu^{11} - 348 \nu^{10} + 1381 \nu^{9} - 1968 \nu^{8} + 4276 \nu^{7} + \cdots - 141 ) / 114 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11 \nu^{13} + 21 \nu^{12} + 204 \nu^{11} + 348 \nu^{10} + 1381 \nu^{9} + 1968 \nu^{8} + 4276 \nu^{7} + \cdots + 141 ) / 114 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11 \nu^{13} - 37 \nu^{12} + 204 \nu^{11} - 643 \nu^{10} + 1381 \nu^{9} - 3937 \nu^{8} + 4276 \nu^{7} + \cdots - 585 ) / 114 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11 \nu^{13} - 37 \nu^{12} - 204 \nu^{11} - 643 \nu^{10} - 1381 \nu^{9} - 3937 \nu^{8} - 4276 \nu^{7} + \cdots - 585 ) / 114 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 28\nu^{13} + 502\nu^{11} + 3232\nu^{9} + 9335\nu^{7} + 12359\nu^{5} + 6536\nu^{3} + 777\nu - 57 ) / 114 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34 \nu^{13} + 15 \nu^{12} + 615 \nu^{11} + 254 \nu^{10} + 4025 \nu^{9} + 1498 \nu^{8} + 11999 \nu^{7} + \cdots + 459 ) / 114 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 34 \nu^{13} - 15 \nu^{12} + 615 \nu^{11} - 254 \nu^{10} + 4025 \nu^{9} - 1498 \nu^{8} + 11999 \nu^{7} + \cdots - 459 ) / 114 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 66 \nu^{13} + 17 \nu^{12} + 1167 \nu^{11} + 298 \nu^{10} + 7374 \nu^{9} + 1851 \nu^{8} + 20868 \nu^{7} + \cdots + 429 ) / 114 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{11} - \beta_{5} + 2\beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{12} - 7\beta_{11} - \beta_{7} + \beta_{6} + 9\beta_{5} + \beta_{4} - 16\beta_{2} - 8\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} - 2 \beta_{12} - 2 \beta_{11} - 11 \beta_{9} + 11 \beta_{8} - 10 \beta_{7} + \cdots + 19 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 49 \beta_{12} + 49 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} + 11 \beta_{7} - 11 \beta_{6} - 66 \beta_{5} + \cdots - 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 26 \beta_{13} + 26 \beta_{12} + 26 \beta_{11} + 6 \beta_{10} + 97 \beta_{9} - 97 \beta_{8} + 78 \beta_{7} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 348 \beta_{12} - 348 \beta_{11} - 26 \beta_{9} - 26 \beta_{8} - 95 \beta_{7} + 95 \beta_{6} + 471 \beta_{5} + \cdots + 362 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 252 \beta_{13} - 247 \beta_{12} - 247 \beta_{11} - 102 \beta_{10} - 790 \beta_{9} + 790 \beta_{8} + \cdots - 51 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2503 \beta_{12} + 2503 \beta_{11} + 247 \beta_{9} + 247 \beta_{8} + 759 \beta_{7} - 759 \beta_{6} + \cdots - 2543 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2176 \beta_{13} + 2094 \beta_{12} + 2094 \beta_{11} + 1128 \beta_{10} + 6197 \beta_{9} - 6197 \beta_{8} + \cdots + 564 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 18186 \beta_{12} - 18186 \beta_{11} - 2094 \beta_{9} - 2094 \beta_{8} - 5868 \beta_{7} + 5868 \beta_{6} + \cdots + 18277 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 17710 \beta_{13} - 16817 \beta_{12} - 16817 \beta_{11} - 10446 \beta_{10} - 47632 \beta_{9} + \cdots - 5223 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/109\mathbb{Z}\right)^\times\).

\(n\) \(6\)
\(\chi(n)\) \(1 + \beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
2.72773i
1.60541i
0.491162i
− 0.267645i
− 1.05205i
− 1.29332i
− 2.21128i
2.21128i
1.29332i
1.05205i
0.267645i
− 0.491162i
− 1.60541i
− 2.72773i
− 2.72773i −0.375846 − 0.650984i −5.44051 0.441643 + 0.764949i −1.77571 + 1.02521i −1.23451 − 2.13823i 9.38478i 1.21748 − 2.10874i 2.08657 − 1.20468i
46.2 − 1.60541i −1.47379 − 2.55267i −0.577347 −0.434345 − 0.752308i −4.09809 + 2.36603i 2.30780 + 3.99723i − 2.28394i −2.84409 + 4.92612i −1.20776 + 0.697303i
46.3 − 0.491162i −0.760683 − 1.31754i 1.75876 1.50253 + 2.60247i −0.647126 + 0.373619i −1.37516 − 2.38184i − 1.84616i 0.342721 − 0.593611i 1.27823 − 0.737988i
46.4 0.267645i 1.55776 + 2.69812i 1.92837 −1.17796 − 2.04029i −0.722139 + 0.416927i −1.86875 − 3.23678i 1.05141i −3.35323 + 5.80797i 0.546073 − 0.315275i
46.5 1.05205i −0.0984404 − 0.170504i 0.893187 −0.421618 − 0.730264i 0.179379 − 0.103564i 1.68951 + 2.92632i 3.04378i 1.48062 − 2.56451i 0.768275 − 0.443564i
46.6 1.29332i −1.37617 − 2.38360i 0.327314 −1.54578 − 2.67737i 3.08276 − 1.77983i −1.73906 − 3.01213i 3.00997i −2.28768 + 3.96239i 3.46271 − 1.99920i
46.7 2.21128i 0.527166 + 0.913077i −2.88977 0.635526 + 1.10076i −2.01907 + 1.16571i −1.27984 − 2.21674i − 1.96753i 0.944193 − 1.63539i −2.43410 + 1.40533i
64.1 − 2.21128i 0.527166 − 0.913077i −2.88977 0.635526 − 1.10076i −2.01907 − 1.16571i −1.27984 + 2.21674i 1.96753i 0.944193 + 1.63539i −2.43410 − 1.40533i
64.2 − 1.29332i −1.37617 + 2.38360i 0.327314 −1.54578 + 2.67737i 3.08276 + 1.77983i −1.73906 + 3.01213i − 3.00997i −2.28768 − 3.96239i 3.46271 + 1.99920i
64.3 − 1.05205i −0.0984404 + 0.170504i 0.893187 −0.421618 + 0.730264i 0.179379 + 0.103564i 1.68951 − 2.92632i − 3.04378i 1.48062 + 2.56451i 0.768275 + 0.443564i
64.4 − 0.267645i 1.55776 − 2.69812i 1.92837 −1.17796 + 2.04029i −0.722139 − 0.416927i −1.86875 + 3.23678i − 1.05141i −3.35323 − 5.80797i 0.546073 + 0.315275i
64.5 0.491162i −0.760683 + 1.31754i 1.75876 1.50253 − 2.60247i −0.647126 − 0.373619i −1.37516 + 2.38184i 1.84616i 0.342721 + 0.593611i 1.27823 + 0.737988i
64.6 1.60541i −1.47379 + 2.55267i −0.577347 −0.434345 + 0.752308i −4.09809 − 2.36603i 2.30780 − 3.99723i 2.28394i −2.84409 − 4.92612i −1.20776 − 0.697303i
64.7 2.72773i −0.375846 + 0.650984i −5.44051 0.441643 − 0.764949i −1.77571 − 1.02521i −1.23451 + 2.13823i − 9.38478i 1.21748 + 2.10874i 2.08657 + 1.20468i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
109.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 109.2.e.b ✓ 14
3.b odd 2 1 981.2.k.c 14
109.e even 6 1 inner 109.2.e.b ✓ 14
327.j odd 6 1 981.2.k.c 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
109.2.e.b ✓ 14 1.a even 1 1 trivial
109.2.e.b ✓ 14 109.e even 6 1 inner
981.2.k.c 14 3.b odd 2 1
981.2.k.c 14 327.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 18T_{2}^{12} + 117T_{2}^{10} + 346T_{2}^{8} + 486T_{2}^{6} + 300T_{2}^{4} + 61T_{2}^{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(109, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 18 T^{12} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{14} + 4 T^{13} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( T^{14} + 2 T^{13} + \cdots + 324 \) Copy content Toggle raw display
$7$ \( T^{14} + 7 T^{13} + \cdots + 12418576 \) Copy content Toggle raw display
$11$ \( T^{14} + 9 T^{13} + \cdots + 3888 \) Copy content Toggle raw display
$13$ \( T^{14} - 9 T^{13} + \cdots + 243 \) Copy content Toggle raw display
$17$ \( T^{14} + 84 T^{12} + \cdots + 2883 \) Copy content Toggle raw display
$19$ \( T^{14} + 95 T^{12} + \cdots + 972 \) Copy content Toggle raw display
$23$ \( T^{14} + 174 T^{12} + \cdots + 5746368 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 104714289 \) Copy content Toggle raw display
$31$ \( T^{14} + 3 T^{13} + \cdots + 20164 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 221337162387 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 54166696923 \) Copy content Toggle raw display
$43$ \( (T^{7} + 12 T^{6} + \cdots - 381994)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 221055168 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 91946413872 \) Copy content Toggle raw display
$59$ \( T^{14} + 18 T^{13} + \cdots + 5354688 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 1614914596 \) Copy content Toggle raw display
$67$ \( T^{14} + 9 T^{13} + \cdots + 12632112 \) Copy content Toggle raw display
$71$ \( (T^{7} + 32 T^{6} + \cdots - 1782)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} - 5 T^{13} + \cdots + 8898289 \) Copy content Toggle raw display
$79$ \( T^{14} - 30 T^{13} + \cdots + 108 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 18694545984 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 126360081 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 1965090901489 \) Copy content Toggle raw display
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