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Properties

Label 2340.2.bi.c
Level $2340$
Weight $2$
Character orbit 2340.bi
Analytic conductor $18.685$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2340,2,Mod(161,2340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2340, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2340.161"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2340.bi (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.6849940730\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
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: \( x^{12} + 10 x^{10} - 16 x^{9} + 50 x^{8} - 32 x^{7} - 110 x^{6} + 40 x^{5} + 417 x^{4} + 712 x^{3} + \cdots + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + (\beta_{11} - \beta_{5} - \beta_{4} + \cdots - 1) q^{7} + ( - \beta_{10} + \beta_{3}) q^{11} + (\beta_{11} + \beta_{8} + \beta_{5} + 1) q^{13} + ( - \beta_{10} + \beta_{9} + \beta_{6}) q^{17}+ \cdots + ( - 2 \beta_{11} + 3 \beta_{8} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8 q^{7} + 12 q^{13} - 24 q^{31} - 12 q^{37} - 8 q^{55} - 32 q^{61} + 40 q^{67} - 12 q^{73} + 8 q^{79} + 4 q^{85} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 10 x^{10} - 16 x^{9} + 50 x^{8} - 32 x^{7} - 110 x^{6} + 40 x^{5} + 417 x^{4} + 712 x^{3} + \cdots + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2531383 \nu^{11} + 6370735 \nu^{10} + 16368115 \nu^{9} + 30395407 \nu^{8} - 68962297 \nu^{7} + \cdots + 881548768 ) / 252631392 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8003349 \nu^{11} + 3019691 \nu^{10} - 82583149 \nu^{9} + 160389851 \nu^{8} - 474664177 \nu^{7} + \cdots - 982328448 ) / 252631392 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9877427 \nu^{11} + 4018065 \nu^{10} - 100097135 \nu^{9} + 198492281 \nu^{8} + \cdots - 639172800 ) / 252631392 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2726533 \nu^{11} - 2168717 \nu^{10} + 27969673 \nu^{9} - 65297821 \nu^{8} + 178050285 \nu^{7} + \cdots - 116031512 ) / 63157848 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6765 \nu^{11} - 2010 \nu^{10} + 68487 \nu^{9} - 129388 \nu^{8} + 380069 \nu^{7} - 342022 \nu^{6} + \cdots + 625296 ) / 133104 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6861369 \nu^{11} + 5745144 \nu^{10} - 71950207 \nu^{9} + 168540570 \nu^{8} - 469088145 \nu^{7} + \cdots - 69706944 ) / 126315696 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1770535 \nu^{11} - 430121 \nu^{10} + 17477065 \nu^{9} - 31961173 \nu^{8} + 92189437 \nu^{7} + \cdots + 160290560 ) / 31578924 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18075315 \nu^{11} + 7811821 \nu^{10} - 183758495 \nu^{9} + 367937581 \nu^{8} + \cdots - 1049485664 ) / 252631392 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10693734 \nu^{11} + 1055053 \nu^{10} - 105939942 \nu^{9} + 180921483 \nu^{8} + \cdots - 1423470544 ) / 126315696 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16383048 \nu^{11} + 6095859 \nu^{10} - 166055858 \nu^{9} + 324642465 \nu^{8} + \cdots - 1079754000 ) / 126315696 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18928493 \nu^{11} - 6358549 \nu^{10} + 191366733 \nu^{9} - 366558165 \nu^{8} + 1068678901 \nu^{7} + \cdots + 1726100080 ) / 126315696 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 2\beta_{8} + \beta_{7} - \beta_{6} + 3\beta_{5} + \beta_{4} + 8\beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} + \beta_{10} - 6 \beta_{9} - 3 \beta_{8} + 3 \beta_{7} + 10 \beta_{6} + \beta_{5} + \cdots + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{10} - 2\beta_{9} - 8\beta_{8} - 13\beta_{7} - 32\beta_{5} - 18\beta_{3} - 18\beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13 \beta_{11} - 46 \beta_{10} + 13 \beta_{9} + 104 \beta_{8} + 39 \beta_{7} - 104 \beta_{6} + \cdots - 122 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - \beta_{11} - 24 \beta_{9} + 209 \beta_{7} + 209 \beta_{6} + 405 \beta_{5} + \beta_{4} + 624 \beta_{2} + \cdots + 405 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 397 \beta_{11} + 397 \beta_{10} + 140 \beta_{9} - 1125 \beta_{8} - 1125 \beta_{7} + 450 \beta_{6} + \cdots + 560 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 144 \beta_{10} + 144 \beta_{9} + 1269 \beta_{8} - 1824 \beta_{6} - 249 \beta_{4} + 2256 \beta_{3} + \cdots - 3293 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3743 \beta_{11} - 1452 \beta_{10} - 3743 \beta_{9} + 5100 \beta_{8} + 12467 \beta_{7} + 5100 \beta_{6} + \cdots + 7104 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4969 \beta_{11} + 7392 \beta_{10} - 42526 \beta_{8} - 29903 \beta_{7} + 29903 \beta_{6} - 50493 \beta_{5} + \cdots + 50493 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15239 \beta_{11} - 15239 \beta_{10} + 37746 \beta_{9} + 57765 \beta_{8} - 57765 \beta_{7} + \cdots - 207326 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).

\(n\) \(937\) \(1081\) \(1171\) \(2081\)
\(\chi(n)\) \(1\) \(-\beta_{5}\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.814709 + 0.337464i
1.85301 − 0.767544i
−0.331199 + 0.137187i
−1.29438 − 3.12492i
−0.393753 − 0.950605i
0.981031 + 2.36842i
−0.814709 − 0.337464i
1.85301 + 0.767544i
−0.331199 − 0.137187i
−1.29438 + 3.12492i
−0.393753 + 0.950605i
0.981031 − 2.36842i
0 0 0 −0.707107 − 0.707107i 0 −2.78746 − 2.78746i 0 0 0
161.2 0 0 0 −0.707107 − 0.707107i 0 −1.71815 − 1.71815i 0 0 0
161.3 0 0 0 −0.707107 − 0.707107i 0 2.50560 + 2.50560i 0 0 0
161.4 0 0 0 0.707107 + 0.707107i 0 −2.78746 − 2.78746i 0 0 0
161.5 0 0 0 0.707107 + 0.707107i 0 −1.71815 − 1.71815i 0 0 0
161.6 0 0 0 0.707107 + 0.707107i 0 2.50560 + 2.50560i 0 0 0
1061.1 0 0 0 −0.707107 + 0.707107i 0 −2.78746 + 2.78746i 0 0 0
1061.2 0 0 0 −0.707107 + 0.707107i 0 −1.71815 + 1.71815i 0 0 0
1061.3 0 0 0 −0.707107 + 0.707107i 0 2.50560 − 2.50560i 0 0 0
1061.4 0 0 0 0.707107 − 0.707107i 0 −2.78746 + 2.78746i 0 0 0
1061.5 0 0 0 0.707107 − 0.707107i 0 −1.71815 + 1.71815i 0 0 0
1061.6 0 0 0 0.707107 − 0.707107i 0 2.50560 − 2.50560i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.d odd 4 1 inner
39.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2340.2.bi.c ✓ 12
3.b odd 2 1 inner 2340.2.bi.c ✓ 12
13.d odd 4 1 inner 2340.2.bi.c ✓ 12
39.f even 4 1 inner 2340.2.bi.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2340.2.bi.c ✓ 12 1.a even 1 1 trivial
2340.2.bi.c ✓ 12 3.b odd 2 1 inner
2340.2.bi.c ✓ 12 13.d odd 4 1 inner
2340.2.bi.c ✓ 12 39.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 4T_{7}^{5} + 8T_{7}^{4} - 4T_{7}^{3} + 169T_{7}^{2} + 624T_{7} + 1152 \) acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{6} + 4 T^{5} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 114 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{6} - 6 T^{5} + \cdots + 2197)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 48 T^{4} + \cdots - 3042)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( (T^{6} - 38 T^{4} + 337 T^{2} - 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 26 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 12 T^{5} + \cdots + 10368)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots + 57122)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 10226 T^{8} + \cdots + 7311616 \) Copy content Toggle raw display
$43$ \( (T^{6} + 136 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 4616 T^{8} + \cdots + 331776 \) Copy content Toggle raw display
$53$ \( (T^{6} + 368 T^{4} + \cdots + 1801202)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 4294967296 \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} + \cdots - 562)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} - 20 T^{5} + \cdots + 123008)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 268435456 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + \cdots + 294912)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 2 T^{2} + \cdots - 428)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 116985856 \) Copy content Toggle raw display
$89$ \( T^{12} + 52322 T^{8} + \cdots + 37015056 \) Copy content Toggle raw display
$97$ \( (T^{6} + 14 T^{5} + \cdots + 116162)^{2} \) Copy content Toggle raw display
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