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Properties

Label 945.2.d.f
Level $945$
Weight $2$
Character orbit 945.d
Analytic conductor $7.546$
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] = [945,2,Mod(379,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, 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("945.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.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: \(7.54586299101\)
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: \( x^{16} + 38x^{14} + 539x^{12} + 3626x^{10} + 12441x^{8} + 21432x^{6} + 16040x^{4} + 2944x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} + ( - \beta_{7} - 2) q^{4} - \beta_{9} q^{5} - \beta_{11} q^{7} + ( - \beta_{15} + 2 \beta_{10}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} + ( - \beta_{7} - 2) q^{4} - \beta_{9} q^{5} - \beta_{11} q^{7} + ( - \beta_{15} + 2 \beta_{10}) q^{8} + ( - \beta_{14} - \beta_{12} - \beta_{11} + \cdots + 1) q^{10}+ \cdots + \beta_{10} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 28 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 28 q^{4} + 8 q^{10} + 60 q^{16} - 36 q^{19} - 4 q^{25} + 40 q^{31} - 44 q^{34} - 4 q^{40} + 40 q^{46} - 16 q^{49} + 20 q^{55} + 76 q^{61} - 104 q^{64} - 8 q^{70} + 20 q^{76} - 52 q^{79} - 16 q^{85} + 20 q^{91} - 140 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 38x^{14} + 539x^{12} + 3626x^{10} + 12441x^{8} + 21432x^{6} + 16040x^{4} + 2944x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6931 \nu^{15} - 19635 \nu^{14} + 235187 \nu^{13} - 697320 \nu^{12} + 2731727 \nu^{11} + \cdots - 22815240 ) / 7716000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3927 \nu^{14} - 126604 \nu^{12} - 1317909 \nu^{10} - 4200328 \nu^{8} + 5989501 \nu^{6} + \cdots + 2432792 ) / 643000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6931 \nu^{15} - 19635 \nu^{14} - 235187 \nu^{13} - 697320 \nu^{12} - 2731727 \nu^{11} + \cdots - 22815240 ) / 7716000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10057 \nu^{14} - 359214 \nu^{12} - 4609719 \nu^{10} - 26250498 \nu^{8} - 68906109 \nu^{6} + \cdots - 3384128 ) / 1286000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 25349 \nu^{15} + 52572 \nu^{14} - 743398 \nu^{13} + 2088744 \nu^{12} - 5825683 \nu^{11} + \cdots + 114278688 ) / 15432000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25349 \nu^{15} + 52572 \nu^{14} + 743398 \nu^{13} + 2088744 \nu^{12} + 5825683 \nu^{11} + \cdots + 114278688 ) / 15432000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10057 \nu^{14} + 359214 \nu^{12} + 4609719 \nu^{10} + 26250498 \nu^{8} + 68906109 \nu^{6} + \cdots + 169128 ) / 643000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 144281 \nu^{15} - 3810 \nu^{14} + 5496262 \nu^{13} + 80580 \nu^{12} + 78256927 \nu^{11} + \cdots + 96850560 ) / 15432000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 144281 \nu^{15} - 3810 \nu^{14} - 5496262 \nu^{13} + 80580 \nu^{12} - 78256927 \nu^{11} + \cdots + 96850560 ) / 15432000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43711 \nu^{15} + 1630847 \nu^{13} + 22482587 \nu^{11} + 144666929 \nu^{9} + 465057057 \nu^{7} + \cdots + 50819044 \nu ) / 3858000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 43711 \nu^{15} + 1630847 \nu^{13} + 22482587 \nu^{11} + 144666929 \nu^{9} + 465057057 \nu^{7} + \cdots + 46961044 \nu ) / 3858000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 193603 \nu^{15} - 78258 \nu^{14} + 7441856 \nu^{13} - 2993916 \nu^{12} + 107392001 \nu^{11} + \cdots - 165253632 ) / 15432000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 193603 \nu^{15} - 78258 \nu^{14} - 7441856 \nu^{13} - 2993916 \nu^{12} - 107392001 \nu^{11} + \cdots - 165253632 ) / 15432000 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 33654 \nu^{15} - 1271633 \nu^{13} - 17872868 \nu^{11} - 118416431 \nu^{9} - 396150948 \nu^{7} + \cdots - 42290916 \nu ) / 1286000 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 128042 \nu^{15} + 4921309 \nu^{13} + 70925464 \nu^{11} + 487080163 \nu^{9} + 1705130304 \nu^{7} + \cdots + 227496068 \nu ) / 3858000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{11} + \beta_{10} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - 2\beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 3\beta_{14} + 13\beta_{11} - 9\beta_{10} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{9} + 4\beta_{8} + 13\beta_{7} - \beta_{6} - \beta_{5} + 28\beta_{4} - 4\beta_{2} + 50 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 19 \beta_{15} - 55 \beta_{14} + 5 \beta_{13} - 5 \beta_{12} - 166 \beta_{11} + 104 \beta_{10} + \cdots - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{13} + 2 \beta_{12} - 80 \beta_{9} - 80 \beta_{8} - 168 \beta_{7} + 27 \beta_{6} + 27 \beta_{5} + \cdots - 607 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 296 \beta_{15} + 840 \beta_{14} - 133 \beta_{13} + 133 \beta_{12} + 2157 \beta_{11} - 1299 \beta_{10} + \cdots + 31 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 86 \beta_{13} - 86 \beta_{12} + 1272 \beta_{9} + 1272 \beta_{8} + 2197 \beta_{7} - 518 \beta_{6} + \cdots + 7855 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4369 \beta_{15} - 12249 \beta_{14} + 2580 \beta_{13} - 2580 \beta_{12} - 28495 \beta_{11} + \cdots - 568 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2084 \beta_{13} + 2084 \beta_{12} - 18890 \beta_{9} - 18890 \beta_{8} - 29043 \beta_{7} + 8725 \beta_{6} + \cdots - 104128 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 63111 \beta_{15} + 175637 \beta_{14} - 44187 \beta_{13} + 44187 \beta_{12} + 380810 \beta_{11} + \cdots + 8913 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 40274 \beta_{13} - 40274 \beta_{12} + 273192 \beta_{9} + 273192 \beta_{8} + 387334 \beta_{7} + \cdots + 1396855 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 901740 \beta_{15} - 2497716 \beta_{14} + 708591 \beta_{13} - 708591 \beta_{12} - 5130971 \beta_{11} + \cdots - 131999 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 694050 \beta_{13} + 694050 \beta_{12} - 3902202 \beta_{9} - 3902202 \beta_{8} - 5203389 \beta_{7} + \cdots - 18880049 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 12799917 \beta_{15} + 35339939 \beta_{14} - 10917252 \beta_{13} + 10917252 \beta_{12} + \cdots + 1907232 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
379.1
3.73099i
1.73099i
1.35037i
3.35037i
0.284387i
2.28439i
1.60648i
− 0.393517i
− 1.60648i
0.393517i
− 0.284387i
− 2.28439i
− 1.35037i
− 3.35037i
− 3.73099i
− 1.73099i
− 2.73099i 0 −5.45830 −1.54123 + 1.62006i 0 1.00000i 9.44459i 0 4.42438 + 4.20908i
379.2 − 2.73099i 0 −5.45830 1.54123 + 1.62006i 0 − 1.00000i 9.44459i 0 4.42438 − 4.20908i
379.3 − 2.35037i 0 −3.52423 −0.670641 − 2.13313i 0 − 1.00000i 3.58249i 0 −5.01364 + 1.57625i
379.4 − 2.35037i 0 −3.52423 0.670641 − 2.13313i 0 1.00000i 3.58249i 0 −5.01364 − 1.57625i
379.5 − 1.28439i 0 0.350351 −1.48951 + 1.66775i 0 − 1.00000i − 3.01876i 0 2.14203 + 1.91310i
379.6 − 1.28439i 0 0.350351 1.48951 + 1.66775i 0 1.00000i − 3.01876i 0 2.14203 − 1.91310i
379.7 − 0.606483i 0 1.63218 −2.11098 + 0.737411i 0 1.00000i − 2.20285i 0 0.447227 + 1.28027i
379.8 − 0.606483i 0 1.63218 2.11098 + 0.737411i 0 − 1.00000i − 2.20285i 0 0.447227 − 1.28027i
379.9 0.606483i 0 1.63218 −2.11098 − 0.737411i 0 − 1.00000i 2.20285i 0 0.447227 − 1.28027i
379.10 0.606483i 0 1.63218 2.11098 − 0.737411i 0 1.00000i 2.20285i 0 0.447227 + 1.28027i
379.11 1.28439i 0 0.350351 −1.48951 − 1.66775i 0 1.00000i 3.01876i 0 2.14203 − 1.91310i
379.12 1.28439i 0 0.350351 1.48951 − 1.66775i 0 − 1.00000i 3.01876i 0 2.14203 + 1.91310i
379.13 2.35037i 0 −3.52423 −0.670641 + 2.13313i 0 1.00000i − 3.58249i 0 −5.01364 − 1.57625i
379.14 2.35037i 0 −3.52423 0.670641 + 2.13313i 0 − 1.00000i − 3.58249i 0 −5.01364 + 1.57625i
379.15 2.73099i 0 −5.45830 −1.54123 − 1.62006i 0 − 1.00000i − 9.44459i 0 4.42438 − 4.20908i
379.16 2.73099i 0 −5.45830 1.54123 − 1.62006i 0 1.00000i − 9.44459i 0 4.42438 + 4.20908i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 379.16
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 945.2.d.f ✓ 16
3.b odd 2 1 inner 945.2.d.f ✓ 16
5.b even 2 1 inner 945.2.d.f ✓ 16
5.c odd 4 1 4725.2.a.ce 8
5.c odd 4 1 4725.2.a.cf 8
15.d odd 2 1 inner 945.2.d.f ✓ 16
15.e even 4 1 4725.2.a.ce 8
15.e even 4 1 4725.2.a.cf 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.d.f ✓ 16 1.a even 1 1 trivial
945.2.d.f ✓ 16 3.b odd 2 1 inner
945.2.d.f ✓ 16 5.b even 2 1 inner
945.2.d.f ✓ 16 15.d odd 2 1 inner
4725.2.a.ce 8 5.c odd 4 1
4725.2.a.ce 8 15.e even 4 1
4725.2.a.cf 8 5.c odd 4 1
4725.2.a.cf 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\):

\( T_{2}^{8} + 15T_{2}^{6} + 68T_{2}^{4} + 91T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{8} - 72T_{11}^{6} + 1590T_{11}^{4} - 11464T_{11}^{2} + 5625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 15 T^{6} + \cdots + 25)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 2 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$11$ \( (T^{8} - 72 T^{6} + \cdots + 5625)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 87 T^{6} + \cdots + 44944)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 45 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 9 T^{3} + \cdots + 381)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 71 T^{6} + \cdots + 7569)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 73 T^{6} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} + \cdots + 325)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 120 T^{6} + \cdots + 7225)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 167 T^{6} + \cdots + 265225)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 98 T^{6} + \cdots + 53824)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 146 T^{6} + \cdots + 876096)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 61 T^{6} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 106 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 19 T^{3} + \cdots - 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 39 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 439 T^{6} + \cdots + 994009)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 458 T^{6} + \cdots + 1065024)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 13 T^{3} + \cdots - 7880)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 202 T^{6} + \cdots + 115600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 308 T^{6} + \cdots + 4601025)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 515 T^{6} + \cdots + 144384256)^{2} \) Copy content Toggle raw display
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