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Properties

Label 2790.2.d.n
Level $2790$
Weight $2$
Character orbit 2790.d
Analytic conductor $22.278$
Analytic rank $0$
Dimension $12$
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] = [2790,2,Mod(559,2790)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2790, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2790.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.d (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: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 9 x^{10} + 10 x^{9} + 35 x^{8} - 266 x^{7} + 171 x^{6} + 526 x^{5} + 365 x^{4} + \cdots + 1810 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{11} + \beta_{7} + \beta_{6}) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} - \beta_{2} q^{5} + (\beta_{11} + \beta_{7} + \beta_{6}) q^{7} + \beta_1 q^{8} + \beta_{6} q^{10} - \beta_{8} q^{11} + \beta_{9} q^{13} + (\beta_{8} - \beta_{3} + \beta_{2}) q^{14} + q^{16} + (\beta_{3} + \beta_{2} + 2 \beta_1) q^{17} + (\beta_{7} + \beta_{5} + 1) q^{19} + \beta_{2} q^{20} + \beta_{11} q^{22} + (\beta_{4} + \beta_{2} + 3 \beta_1) q^{23} + ( - \beta_{11} + \beta_{5} - 1) q^{25} - \beta_{10} q^{26} + ( - \beta_{11} - \beta_{7} - \beta_{6}) q^{28} + 2 \beta_{8} q^{29} + q^{31} - \beta_1 q^{32} + (\beta_{7} - \beta_{6} + 2) q^{34} + (\beta_{10} + \beta_{4} + \cdots - 3 \beta_1) q^{35}+ \cdots + ( - \beta_{4} - \beta_{2} + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} + 2 q^{10} + 12 q^{16} + 8 q^{19} - 14 q^{25} + 12 q^{31} + 20 q^{34} - 2 q^{40} + 36 q^{46} - 48 q^{49} - 10 q^{55} - 28 q^{61} - 12 q^{64} - 36 q^{70} - 8 q^{76} - 64 q^{79} + 70 q^{85} + 8 q^{91} - 28 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 9 x^{10} + 10 x^{9} + 35 x^{8} - 266 x^{7} + 171 x^{6} + 526 x^{5} + 365 x^{4} + \cdots + 1810 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5 \nu^{10} + 25 \nu^{9} - 25 \nu^{8} - 50 \nu^{7} - 218 \nu^{6} + 934 \nu^{5} - 75 \nu^{4} + \cdots + 2458 ) / 2176 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 169 \nu^{11} + 296 \nu^{10} + 2428 \nu^{9} - 6811 \nu^{8} - 11764 \nu^{7} + 5044 \nu^{6} + \cdots + 562186 ) / 393856 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 169 \nu^{11} - 1563 \nu^{10} + 3907 \nu^{9} + 476 \nu^{8} - 906 \nu^{7} - 79978 \nu^{6} + \cdots + 966540 ) / 393856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 169 \nu^{11} - 3373 \nu^{10} + 12957 \nu^{9} - 8574 \nu^{8} - 19006 \nu^{7} - 158894 \nu^{6} + \cdots + 1462480 ) / 393856 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 114 \nu^{11} - 627 \nu^{10} + 803 \nu^{9} + 2175 \nu^{8} + 1186 \nu^{7} - 26654 \nu^{6} + \cdots - 461550 ) / 196928 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 114 \nu^{11} + 627 \nu^{10} - 803 \nu^{9} - 2899 \nu^{8} + 1710 \nu^{7} + 25206 \nu^{6} + \cdots - 721466 ) / 196928 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 114 \nu^{11} + 627 \nu^{10} - 803 \nu^{9} + 721 \nu^{8} - 12770 \nu^{7} + 32446 \nu^{6} + \cdots + 664270 ) / 196928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 530 \nu^{11} - 2915 \nu^{10} + 2679 \nu^{9} + 9807 \nu^{8} + 19562 \nu^{7} - 134638 \nu^{6} + \cdots + 887986 ) / 393856 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 210 \nu^{11} + 1155 \nu^{10} - 1403 \nu^{9} - 2349 \nu^{8} - 9882 \nu^{7} + 53634 \nu^{6} + \cdots - 73486 ) / 98464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1118 \nu^{11} - 6149 \nu^{10} + 4001 \nu^{9} + 28113 \nu^{8} + 25222 \nu^{7} - 262514 \nu^{6} + \cdots + 1063918 ) / 393856 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 938 \nu^{11} - 5159 \nu^{10} + 7763 \nu^{9} + 3759 \nu^{8} + 42402 \nu^{7} - 202062 \nu^{6} + \cdots + 114754 ) / 196928 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} + 2\beta_{3} - 2\beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{6} + 4\beta_{4} - 2\beta_{3} - 2\beta_{2} - 8\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{10} - 8\beta_{9} + 12\beta_{7} + 12\beta_{6} + 6\beta_{4} - 13\beta_{3} + 7\beta_{2} - 12\beta _1 + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{10} - 15 \beta_{9} + 2 \beta_{8} + 19 \beta_{7} + 7 \beta_{6} - 20 \beta_{5} + 16 \beta_{4} + \cdots - 74 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4 \beta_{11} + 19 \beta_{10} - 10 \beta_{9} + 34 \beta_{8} - 22 \beta_{7} - 52 \beta_{6} - 50 \beta_{5} + \cdots - 198 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 12 \beta_{11} + 52 \beta_{10} + 7 \beta_{9} + 96 \beta_{8} - 123 \beta_{7} - 213 \beta_{6} + \cdots - 624 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 92 \beta_{11} + 33 \beta_{10} + 282 \beta_{9} + 170 \beta_{8} - 1046 \beta_{7} - 1256 \beta_{6} + \cdots - 1482 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 312 \beta_{11} - 106 \beta_{10} + 1061 \beta_{9} + 238 \beta_{8} - 3125 \beta_{7} - 3337 \beta_{6} + \cdots + 438 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 536 \beta_{11} - 1835 \beta_{10} + 2442 \beta_{9} - 720 \beta_{8} - 5582 \beta_{7} - 5402 \beta_{6} + \cdots + 10016 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 424 \beta_{11} - 8026 \beta_{10} + 4375 \beta_{9} - 4726 \beta_{8} - 4307 \beta_{7} - 1747 \beta_{6} + \cdots + 52246 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6380 \beta_{11} - 22503 \beta_{10} - 4926 \beta_{9} - 15214 \beta_{8} + 51678 \beta_{7} + 62128 \beta_{6} + \cdots + 181458 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2790\mathbb{Z}\right)^\times\).

\(n\) \(1117\) \(1801\) \(2171\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
559.1
−1.56466 + 1.73575i
1.73429 − 0.522947i
−1.06962 − 1.10168i
2.06962 + 1.10168i
−0.734291 + 0.522947i
2.56466 − 1.73575i
−1.56466 − 1.73575i
1.73429 + 0.522947i
−1.06962 + 1.10168i
2.06962 − 1.10168i
−0.734291 − 0.522947i
2.56466 + 1.73575i
− 1.00000i 0 −1.00000 −2.19196 + 0.441924i 0 4.12933i 1.00000i 0 0.441924 + 2.19196i
559.2 − 1.00000i 0 −1.00000 −0.772145 − 2.09852i 0 − 2.46858i 1.00000i 0 −2.09852 + 0.772145i
559.3 − 1.00000i 0 −1.00000 −0.590837 + 2.15660i 0 3.13923i 1.00000i 0 2.15660 + 0.590837i
559.4 − 1.00000i 0 −1.00000 0.590837 + 2.15660i 0 − 3.13923i 1.00000i 0 2.15660 − 0.590837i
559.5 − 1.00000i 0 −1.00000 0.772145 − 2.09852i 0 2.46858i 1.00000i 0 −2.09852 − 0.772145i
559.6 − 1.00000i 0 −1.00000 2.19196 + 0.441924i 0 − 4.12933i 1.00000i 0 0.441924 − 2.19196i
559.7 1.00000i 0 −1.00000 −2.19196 − 0.441924i 0 − 4.12933i − 1.00000i 0 0.441924 − 2.19196i
559.8 1.00000i 0 −1.00000 −0.772145 + 2.09852i 0 2.46858i − 1.00000i 0 −2.09852 − 0.772145i
559.9 1.00000i 0 −1.00000 −0.590837 − 2.15660i 0 − 3.13923i − 1.00000i 0 2.15660 − 0.590837i
559.10 1.00000i 0 −1.00000 0.590837 − 2.15660i 0 3.13923i − 1.00000i 0 2.15660 + 0.590837i
559.11 1.00000i 0 −1.00000 0.772145 + 2.09852i 0 − 2.46858i − 1.00000i 0 −2.09852 + 0.772145i
559.12 1.00000i 0 −1.00000 2.19196 − 0.441924i 0 4.12933i − 1.00000i 0 0.441924 + 2.19196i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 559.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2790.2.d.n ✓ 12
3.b odd 2 1 inner 2790.2.d.n ✓ 12
5.b even 2 1 inner 2790.2.d.n ✓ 12
15.d odd 2 1 inner 2790.2.d.n ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2790.2.d.n ✓ 12 1.a even 1 1 trivial
2790.2.d.n ✓ 12 3.b odd 2 1 inner
2790.2.d.n ✓ 12 5.b even 2 1 inner
2790.2.d.n ✓ 12 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2790, [\chi])\):

\( T_{7}^{6} + 33T_{7}^{4} + 332T_{7}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11}^{6} - 20T_{11}^{4} + 63T_{11}^{2} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{10} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + 33 T^{4} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 20 T^{4} + 63 T^{2} - 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 51 T^{4} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 45 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 2 T^{2} - 27 T + 8)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + 89 T^{4} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 80 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{12} \) Copy content Toggle raw display
$37$ \( (T^{6} + 76 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 212 T^{4} + \cdots - 25600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 173 T^{4} + \cdots + 43264)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 53 T^{4} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 181 T^{4} + \cdots + 141376)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 72 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 7 T^{2} - 2 T - 40)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 51 T^{4} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 300 T^{4} + \cdots - 586756)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 173 T^{4} + \cdots + 43264)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 16 T^{2} + \cdots - 400)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 329 T^{4} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 273 T^{4} + \cdots - 4096)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 83 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
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