Properties

Label 945.2.b.c
Level $945$
Weight $2$
Character orbit 945.b
Analytic conductor $7.546$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(566,945)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.566");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.b (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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 20x^{10} + 144x^{8} + 446x^{6} + 532x^{4} + 132x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} + \beta_{6} q^{7} + (\beta_{9} + \beta_{7} + \beta_{6} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} + \beta_{6} q^{7} + (\beta_{9} + \beta_{7} + \beta_{6} - \beta_1) q^{8} - \beta_1 q^{10} + ( - \beta_{5} - \beta_{4}) q^{11} + ( - \beta_{5} - \beta_1) q^{13} + (\beta_{11} - \beta_{7} - \beta_{2} - \beta_1 + 1) q^{14} + (\beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} - 2 \beta_{2} + 3) q^{16} + ( - \beta_{8} + \beta_{7} - \beta_{2}) q^{17} + ( - \beta_{9} + \beta_{4}) q^{19} + ( - \beta_{2} + 1) q^{20} + ( - \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{22} + (\beta_{9} - \beta_{5}) q^{23} + q^{25} + ( - \beta_{8} + \beta_{7} - \beta_{2} + 3) q^{26} + ( - \beta_{10} - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} + \beta_1 + 1) q^{28} + (\beta_{11} - \beta_{10} + \beta_{7} + \beta_{6}) q^{29} + (\beta_{11} - \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5}) q^{31} + ( - \beta_{9} - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_1) q^{32} + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_1) q^{34} - \beta_{6} q^{35} + (\beta_{11} + \beta_{10} - 2) q^{37} + (\beta_{11} + \beta_{10} - \beta_{7} + \beta_{6} - \beta_{3} + 1) q^{38} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \beta_1) q^{40} + (\beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{2} + 1) q^{41} + (\beta_{3} - \beta_{2} - 1) q^{43} + ( - \beta_{11} + \beta_{10} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + 3 \beta_{4}) q^{44} + ( - 2 \beta_{8} + 2 \beta_{7} - \beta_{2}) q^{46} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - 1) q^{47} + ( - \beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} - 2 \beta_1 + 1) q^{49} + \beta_1 q^{50} + ( - \beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_1) q^{52} + (\beta_{9} + \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_1) q^{53} + (\beta_{5} + \beta_{4}) q^{55} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 3 \beta_{2} + \cdots - 5) q^{56}+ \cdots + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + 2 \beta_{7} + \beta_{6} + 2 \beta_{4} + \beta_1 + 4) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4} - 12 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{4} - 12 q^{5} - 2 q^{7} + 12 q^{14} + 32 q^{16} + 16 q^{20} - 4 q^{22} + 12 q^{25} + 36 q^{26} + 10 q^{28} + 2 q^{35} - 28 q^{37} - 4 q^{43} - 4 q^{46} + 12 q^{49} - 54 q^{56} - 4 q^{58} + 24 q^{59} - 48 q^{62} - 20 q^{64} + 24 q^{67} - 72 q^{68} - 12 q^{70} + 24 q^{77} - 32 q^{80} + 48 q^{83} + 64 q^{88} + 36 q^{89} + 60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 144x^{8} + 446x^{6} + 532x^{4} + 132x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 12\nu^{4} + 36\nu^{2} + 13 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 20\nu^{9} + 144\nu^{7} + 443\nu^{5} + 502\nu^{3} + 66\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} - 20\nu^{9} - 143\nu^{7} - 431\nu^{5} - 464\nu^{3} - 41\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{11} - 3 \nu^{10} - 100 \nu^{9} - 57 \nu^{8} - 717 \nu^{7} - 384 \nu^{6} - 2185 \nu^{5} - 1089 \nu^{4} - 2450 \nu^{3} - 1125 \nu^{2} - 351 \nu - 138 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{11} + 3 \nu^{10} - 100 \nu^{9} + 57 \nu^{8} - 717 \nu^{7} + 384 \nu^{6} - 2185 \nu^{5} + 1089 \nu^{4} - 2450 \nu^{3} + 1125 \nu^{2} - 351 \nu + 138 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5 \nu^{11} + 3 \nu^{10} - 100 \nu^{9} + 63 \nu^{8} - 717 \nu^{7} + 474 \nu^{6} - 2185 \nu^{5} + 1497 \nu^{4} - 2450 \nu^{3} + 1671 \nu^{2} - 351 \nu + 192 ) / 12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5\nu^{11} + 100\nu^{9} + 717\nu^{7} + 2185\nu^{5} + 2456\nu^{3} + 381\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 8 \nu^{11} + 3 \nu^{10} + 157 \nu^{9} + 60 \nu^{8} + 1101 \nu^{7} + 429 \nu^{6} + 3274 \nu^{5} + 1299 \nu^{4} + 3575 \nu^{3} + 1446 \nu^{2} + 477 \nu + 207 ) / 12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8 \nu^{11} + 3 \nu^{10} - 157 \nu^{9} + 60 \nu^{8} - 1101 \nu^{7} + 429 \nu^{6} - 3274 \nu^{5} + 1299 \nu^{4} - 3575 \nu^{3} + 1446 \nu^{2} - 477 \nu + 207 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} + \beta_{6} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} - 8\beta_{2} + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{9} - 10\beta_{7} - 10\beta_{6} + \beta_{5} - 2\beta_{4} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{11} - 12\beta_{10} + 12\beta_{8} - 12\beta_{6} + 2\beta_{3} + 60\beta_{2} - 109 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 70\beta_{9} + 82\beta_{7} + 82\beta_{6} - 10\beta_{5} + 30\beta_{4} - 183\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 112\beta_{11} + 112\beta_{10} - 110\beta_{8} - 2\beta_{7} + 112\beta_{6} - 30\beta_{3} - 447\beta_{2} + 743 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2\beta_{11} - 2\beta_{10} - 527\beta_{9} - 641\beta_{7} - 641\beta_{6} + 80\beta_{5} - 314\beta_{4} + 1219\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -955\beta_{11} - 955\beta_{10} + 917\beta_{8} + 40\beta_{7} - 957\beta_{6} + 314\beta_{3} + 3342\beta_{2} - 5257 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 40 \beta_{11} + 40 \beta_{10} + 3945 \beta_{9} + 4940 \beta_{7} + 4940 \beta_{6} - 603 \beta_{5} + 2852 \beta_{4} - 8431 \beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
566.1
2.76021i
2.30576i
2.13535i
1.48407i
0.422829i
0.351785i
0.351785i
0.422829i
1.48407i
2.13535i
2.30576i
2.76021i
2.76021i 0 −5.61875 −1.00000 0 −0.00652987 + 2.64574i 9.98851i 0 2.76021i
566.2 2.30576i 0 −3.31651 −1.00000 0 −2.45075 0.996917i 3.03555i 0 2.30576i
566.3 2.13535i 0 −2.55974 −1.00000 0 2.31317 + 1.28422i 1.19524i 0 2.13535i
566.4 1.48407i 0 −0.202463 −1.00000 0 −2.16580 1.51964i 2.66767i 0 1.48407i
566.5 0.422829i 0 1.82122 −1.00000 0 2.53835 0.746160i 1.61572i 0 0.422829i
566.6 0.351785i 0 1.87625 −1.00000 0 −1.22845 + 2.34327i 1.36361i 0 0.351785i
566.7 0.351785i 0 1.87625 −1.00000 0 −1.22845 2.34327i 1.36361i 0 0.351785i
566.8 0.422829i 0 1.82122 −1.00000 0 2.53835 + 0.746160i 1.61572i 0 0.422829i
566.9 1.48407i 0 −0.202463 −1.00000 0 −2.16580 + 1.51964i 2.66767i 0 1.48407i
566.10 2.13535i 0 −2.55974 −1.00000 0 2.31317 1.28422i 1.19524i 0 2.13535i
566.11 2.30576i 0 −3.31651 −1.00000 0 −2.45075 + 0.996917i 3.03555i 0 2.30576i
566.12 2.76021i 0 −5.61875 −1.00000 0 −0.00652987 2.64574i 9.98851i 0 2.76021i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 566.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.b.c 12
3.b odd 2 1 945.2.b.d yes 12
7.b odd 2 1 945.2.b.d yes 12
21.c even 2 1 inner 945.2.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.b.c 12 1.a even 1 1 trivial
945.2.b.c 12 21.c even 2 1 inner
945.2.b.d yes 12 3.b odd 2 1
945.2.b.d yes 12 7.b odd 2 1

Hecke kernels

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

\( T_{2}^{12} + 20T_{2}^{10} + 144T_{2}^{8} + 446T_{2}^{6} + 532T_{2}^{4} + 132T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{17}^{6} - 66T_{17}^{4} + 12T_{17}^{3} + 789T_{17}^{2} + 1116T_{17} + 216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 20 T^{10} + 144 T^{8} + 446 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} - 4 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{12} + 98 T^{10} + 3573 T^{8} + \cdots + 318096 \) Copy content Toggle raw display
$13$ \( T^{12} + 78 T^{10} + 1929 T^{8} + \cdots + 104976 \) Copy content Toggle raw display
$17$ \( (T^{6} - 66 T^{4} + 12 T^{3} + 789 T^{2} + \cdots + 216)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 102 T^{10} + 3495 T^{8} + \cdots + 9801 \) Copy content Toggle raw display
$23$ \( T^{12} + 140 T^{10} + \cdots + 10890000 \) Copy content Toggle raw display
$29$ \( T^{12} + 248 T^{10} + \cdots + 343286784 \) Copy content Toggle raw display
$31$ \( T^{12} + 228 T^{10} + \cdots + 545876496 \) Copy content Toggle raw display
$37$ \( (T^{6} + 14 T^{5} + 14 T^{4} - 484 T^{3} + \cdots + 4432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 177 T^{4} + 18 T^{3} + \cdots - 19116)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} - 85 T^{4} + 194 T^{3} + \cdots + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 111 T^{4} - 348 T^{3} + 1032 T^{2} + \cdots - 432)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 218 T^{10} + \cdots + 25674489 \) Copy content Toggle raw display
$59$ \( (T^{6} - 12 T^{5} - 72 T^{4} + 936 T^{3} + \cdots - 6912)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 228 T^{10} + \cdots + 545876496 \) Copy content Toggle raw display
$67$ \( (T^{6} - 12 T^{5} - 123 T^{4} + \cdots + 86992)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 200 T^{10} + 11628 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$73$ \( T^{12} + 414 T^{10} + 54849 T^{8} + \cdots + 156816 \) Copy content Toggle raw display
$79$ \( (T^{6} - 150 T^{4} - 448 T^{3} + \cdots + 9844)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 24 T^{5} - 105 T^{4} + \cdots + 507141)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 18 T^{5} - 147 T^{4} + \cdots + 147636)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 600 T^{10} + \cdots + 303595776 \) Copy content Toggle raw display
show more
show less