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Properties

Label 2254.2.c.b
Level $2254$
Weight $2$
Character orbit 2254.c
Analytic conductor $17.998$
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] = [2254,2,Mod(2253,2254)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2254, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2254.2253");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2254 = 2 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2254.c (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: \(17.9982806156\)
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: \( x^{16} + 12x^{14} + 73x^{12} + 312x^{10} + 1045x^{8} + 2808x^{6} + 5913x^{4} + 8748x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 322)
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.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{13} q^{3} + q^{4} + (\beta_{10} - \beta_{2}) q^{5} + \beta_{13} q^{6} - q^{8} + (\beta_{12} - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{13} q^{3} + q^{4} + (\beta_{10} - \beta_{2}) q^{5} + \beta_{13} q^{6} - q^{8} + (\beta_{12} - \beta_{4} - 1) q^{9} + ( - \beta_{10} + \beta_{2}) q^{10} + (\beta_{3} + \beta_1) q^{11} - \beta_{13} q^{12} + (\beta_{11} + \beta_{7}) q^{13} + (\beta_{14} + \beta_{6} - \beta_{3}) q^{15} + q^{16} + ( - \beta_{15} + \beta_{10}) q^{17} + ( - \beta_{12} + \beta_{4} + 1) q^{18} + (2 \beta_{15} + \beta_{10} - \beta_{2}) q^{19} + (\beta_{10} - \beta_{2}) q^{20} + ( - \beta_{3} - \beta_1) q^{22} + (\beta_{12} + \beta_{8} - \beta_{3} + \cdots - 1) q^{23}+ \cdots + (5 \beta_{14} + 6 \beta_{6} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 16 q^{4} - 16 q^{8} - 20 q^{9} + 16 q^{16} + 20 q^{18} - 16 q^{23} - 4 q^{25} - 16 q^{29} - 16 q^{32} - 20 q^{36} + 4 q^{39} + 16 q^{46} + 4 q^{50} + 16 q^{58} + 16 q^{64} - 52 q^{71} + 20 q^{72} - 4 q^{78} + 40 q^{81} + 80 q^{85} - 16 q^{92} + 52 q^{93} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 12x^{14} + 73x^{12} + 312x^{10} + 1045x^{8} + 2808x^{6} + 5913x^{4} + 8748x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 8 \nu^{15} - 306 \nu^{13} - 2890 \nu^{11} - 12351 \nu^{9} - 44029 \nu^{7} - 125385 \nu^{5} + \cdots - 379809 \nu ) / 103518 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 12\nu^{13} + 73\nu^{11} + 312\nu^{9} + 1045\nu^{7} + 2808\nu^{5} + 5913\nu^{3} + 6561\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 169 \nu^{15} + 1470 \nu^{13} + 5884 \nu^{11} + 15639 \nu^{9} + 31183 \nu^{7} + 53280 \nu^{5} + \cdots - 473121 \nu ) / 310554 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} + 12\nu^{12} + 73\nu^{10} + 312\nu^{8} + 1045\nu^{6} + 2808\nu^{4} + 5184\nu^{2} + 6561 ) / 729 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{15} - 66\nu^{13} - 346\nu^{11} - 1347\nu^{9} - 4531\nu^{7} - 10845\nu^{5} - 20817\nu^{3} - 24057\nu ) / 4374 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 311 \nu^{15} + 3174 \nu^{13} + 16250 \nu^{11} + 59943 \nu^{9} + 179573 \nu^{7} + 452016 \nu^{5} + \cdots + 1079649 \nu ) / 310554 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -28\nu^{14} - 136\nu^{12} - 535\nu^{10} - 1750\nu^{8} - 2986\nu^{6} - 2950\nu^{4} + 13878\nu^{2} + 43983 ) / 11502 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{14} - 75\nu^{12} - 403\nu^{10} - 1527\nu^{8} - 4507\nu^{6} - 10980\nu^{4} - 19764\nu^{2} - 18954 ) / 1458 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 386 \nu^{14} - 3660 \nu^{12} - 19349 \nu^{10} - 74748 \nu^{8} - 227600 \nu^{6} - 541512 \nu^{4} + \cdots - 1019142 ) / 51759 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -17\nu^{15} - 132\nu^{13} - 620\nu^{11} - 2235\nu^{9} - 6479\nu^{7} - 14778\nu^{5} - 23247\nu^{3} - 19683\nu ) / 4374 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 383 \nu^{14} - 3189 \nu^{12} - 15800 \nu^{10} - 58905 \nu^{8} - 173741 \nu^{6} - 411342 \nu^{4} + \cdots - 680643 ) / 34506 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -17\nu^{14} - 159\nu^{12} - 782\nu^{10} - 2991\nu^{8} - 8909\nu^{6} - 20880\nu^{4} - 36936\nu^{2} - 35721 ) / 1458 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 509 \nu^{14} - 4653 \nu^{12} - 23639 \nu^{10} - 88719 \nu^{8} - 267053 \nu^{6} - 630588 \nu^{4} + \cdots - 1077948 ) / 34506 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 752 \nu^{15} + 7020 \nu^{13} + 35060 \nu^{11} + 134259 \nu^{9} + 398813 \nu^{7} + \cdots + 1512675 \nu ) / 103518 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} + 9\nu^{13} + 46\nu^{11} + 174\nu^{9} + 523\nu^{7} + 1239\nu^{5} + 2196\nu^{3} + 2160\nu ) / 81 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{3} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{11} - \beta_{9} + \beta_{7} - \beta_{4} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{14} + 2\beta_{10} + \beta_{6} + \beta_{3} + 2\beta_{2} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{13} - \beta_{11} + 6\beta_{9} - 2\beta_{8} - \beta_{7} + 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{15} + 2\beta_{14} - 5\beta_{10} + \beta_{6} + \beta_{5} + 2\beta_{3} + 5\beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -11\beta_{13} + 2\beta_{12} + 11\beta_{11} - 8\beta_{9} + 14\beta_{8} + \beta_{7} - 5\beta_{4} - 12 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9\beta_{15} - 13\beta_{14} - 2\beta_{10} - 33\beta_{6} - 18\beta_{5} - 9\beta_{3} - 8\beta_{2} + 9\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 51\beta_{13} - 24\beta_{12} - 9\beta_{11} - 27\beta_{9} - 22\beta_{8} - 23\beta_{7} + 7\beta_{4} + 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -12\beta_{15} + 62\beta_{14} + 58\beta_{10} + 21\beta_{6} + 52\beta_{5} - 51\beta_{3} - 51\beta_{2} + 10\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -46\beta_{13} + 62\beta_{12} - 2\beta_{11} - 22\beta_{8} + 14\beta_{7} - 36\beta_{4} + 27 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 152 \beta_{15} - 214 \beta_{14} + 102 \beta_{10} + 29 \beta_{6} + 12 \beta_{5} + 245 \beta_{3} + \cdots - 90 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -59\beta_{13} - 360\beta_{12} + 73\beta_{11} + 519\beta_{9} + 18\beta_{8} + 159\beta_{7} + 63\beta_{4} + 55 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 537 \beta_{15} + 483 \beta_{14} - 486 \beta_{10} + 763 \beta_{6} - 414 \beta_{5} - 157 \beta_{3} + \cdots + 297 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 631 \beta_{13} + 666 \beta_{12} - 1279 \beta_{11} - 1108 \beta_{9} + 846 \beta_{8} - 197 \beta_{7} + \cdots - 552 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 311 \beta_{15} - 626 \beta_{14} - 683 \beta_{10} - 715 \beta_{6} + 531 \beta_{5} - 634 \beta_{3} + \cdots + 411 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2254\mathbb{Z}\right)^\times\).

\(n\) \(1473\) \(1569\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2253.1
−0.105715 + 1.72882i
0.105715 − 1.72882i
0.956239 + 1.44416i
−0.956239 − 1.44416i
0.452119 − 1.67200i
−0.452119 + 1.67200i
−1.36749 + 1.06300i
1.36749 − 1.06300i
−1.36749 − 1.06300i
1.36749 + 1.06300i
0.452119 + 1.67200i
−0.452119 − 1.67200i
0.956239 − 1.44416i
−0.956239 + 1.44416i
−0.105715 − 1.72882i
0.105715 + 1.72882i
−1.00000 − 3.05986i 1.00000 −1.84099 3.05986i 0 −1.00000 −6.36275 1.84099
2253.2 −1.00000 − 3.05986i 1.00000 1.84099 3.05986i 0 −1.00000 −6.36275 −1.84099
2253.3 −1.00000 − 2.46537i 1.00000 −1.02224 2.46537i 0 −1.00000 −3.07807 1.02224
2253.4 −1.00000 − 2.46537i 1.00000 1.02224 2.46537i 0 −1.00000 −3.07807 −1.02224
2253.5 −1.00000 − 1.24411i 1.00000 −1.33797 1.24411i 0 −1.00000 1.45218 1.33797
2253.6 −1.00000 − 1.24411i 1.00000 1.33797 1.24411i 0 −1.00000 1.45218 −1.33797
2253.7 −1.00000 − 0.106550i 1.00000 −3.57430 0.106550i 0 −1.00000 2.98865 3.57430
2253.8 −1.00000 − 0.106550i 1.00000 3.57430 0.106550i 0 −1.00000 2.98865 −3.57430
2253.9 −1.00000 0.106550i 1.00000 −3.57430 − 0.106550i 0 −1.00000 2.98865 3.57430
2253.10 −1.00000 0.106550i 1.00000 3.57430 − 0.106550i 0 −1.00000 2.98865 −3.57430
2253.11 −1.00000 1.24411i 1.00000 −1.33797 − 1.24411i 0 −1.00000 1.45218 1.33797
2253.12 −1.00000 1.24411i 1.00000 1.33797 − 1.24411i 0 −1.00000 1.45218 −1.33797
2253.13 −1.00000 2.46537i 1.00000 −1.02224 − 2.46537i 0 −1.00000 −3.07807 1.02224
2253.14 −1.00000 2.46537i 1.00000 1.02224 − 2.46537i 0 −1.00000 −3.07807 −1.02224
2253.15 −1.00000 3.05986i 1.00000 −1.84099 − 3.05986i 0 −1.00000 −6.36275 1.84099
2253.16 −1.00000 3.05986i 1.00000 1.84099 − 3.05986i 0 −1.00000 −6.36275 −1.84099
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2253.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2254.2.c.b 16
7.b odd 2 1 inner 2254.2.c.b 16
7.c even 3 1 322.2.g.b ✓ 16
7.d odd 6 1 322.2.g.b ✓ 16
23.b odd 2 1 inner 2254.2.c.b 16
161.c even 2 1 inner 2254.2.c.b 16
161.f odd 6 1 322.2.g.b ✓ 16
161.g even 6 1 322.2.g.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
322.2.g.b ✓ 16 7.c even 3 1
322.2.g.b ✓ 16 7.d odd 6 1
322.2.g.b ✓ 16 161.f odd 6 1
322.2.g.b ✓ 16 161.g even 6 1
2254.2.c.b 16 1.a even 1 1 trivial
2254.2.c.b 16 7.b odd 2 1 inner
2254.2.c.b 16 23.b odd 2 1 inner
2254.2.c.b 16 161.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 17T_{3}^{6} + 81T_{3}^{4} + 89T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(2254, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 17 T^{6} + 81 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 19 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + 64 T^{6} + \cdots + 8649)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 55 T^{6} + 562 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 49 T^{6} + \cdots + 441)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 83 T^{6} + \cdots + 25281)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 8 T^{7} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 11 T^{2} + \cdots - 15)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 28 T^{6} + \cdots + 225)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 62 T^{6} + 958 T^{4} + \cdots + 9)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 131 T^{6} + \cdots + 961)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 80 T^{6} + \cdots + 729)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 233 T^{6} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 315 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 236 T^{6} + \cdots + 73441)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 345 T^{6} + \cdots + 3279721)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 405 T^{6} + \cdots + 1520289)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 13 T^{3} + \cdots - 15)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 304 T^{6} + \cdots + 10595025)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 323 T^{6} + \cdots + 20385225)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 196 T^{6} + \cdots + 1896129)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 351 T^{6} + \cdots + 11029041)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 900 T^{6} + \cdots + 2486119321)^{2} \) Copy content Toggle raw display
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