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Properties

Label 2205.2.b.b
Level $2205$
Weight $2$
Character orbit 2205.b
Analytic conductor $17.607$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,2,Mod(881,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, 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("2205.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.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: \(17.6070136457\)
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: \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 312x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 315)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} + ( - \beta_{6} + \beta_{5} - \beta_1) q^{8} + \beta_1 q^{10} + (\beta_{7} - \beta_{4} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{4}) q^{13} + ( - \beta_{10} - \beta_{9} + \beta_{3} + \cdots + 2) q^{16}+ \cdots + ( - \beta_{7} - 2 \beta_{6} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16 q^{4} + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16 q^{4} + 12 q^{5} + 32 q^{16} - 16 q^{20} + 32 q^{22} + 12 q^{25} + 20 q^{37} + 72 q^{38} + 24 q^{41} - 4 q^{43} + 8 q^{46} - 40 q^{58} + 48 q^{59} + 24 q^{62} - 56 q^{64} - 12 q^{67} + 120 q^{68} - 36 q^{79} + 32 q^{80} - 24 q^{83} - 8 q^{88} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 20x^{10} + 144x^{8} + 452x^{6} + 604x^{4} + 312x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 54\nu^{2} + 38 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 20\nu^{9} + 144\nu^{7} + 446\nu^{5} + 544\nu^{3} + 180\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{11} + 94\nu^{9} + 612\nu^{7} + 1594\nu^{5} + 1412\nu^{3} + 324\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{11} + 94\nu^{9} + 612\nu^{7} + 1594\nu^{5} + 1388\nu^{3} + 204\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{11} + 94\nu^{9} + 606\nu^{7} + 1510\nu^{5} + 1064\nu^{3} - 48\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 18\nu^{8} - 110\nu^{6} - 258\nu^{4} - 180\nu^{2} - 16 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{10} + 19\nu^{8} + 125\nu^{6} + 326\nu^{4} + 272\nu^{2} + 42 ) / 4 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{10} - 19\nu^{8} - 124\nu^{6} - 316\nu^{4} - 250\nu^{2} - 36 ) / 4 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4\nu^{11} + 77\nu^{9} + 516\nu^{7} + 1394\nu^{5} + 1294\nu^{3} + 288\nu ) / 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} - \beta_{9} + \beta_{3} - 8\beta_{2} + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - 2\beta_{7} + 10\beta_{6} - 9\beta_{5} - 3\beta_{4} + 31\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{10} + 14\beta_{9} - 10\beta_{3} + 58\beta_{2} - 100 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -14\beta_{11} + 24\beta_{7} - 82\beta_{6} + 72\beta_{5} + 42\beta_{4} - 206\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -142\beta_{10} - 138\beta_{9} + 4\beta_{8} + 82\beta_{3} - 418\beta_{2} + 662 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 146\beta_{11} - 220\beta_{7} + 634\beta_{6} - 564\beta_{5} - 418\beta_{4} + 1416\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1274\beta_{10} + 1202\beta_{9} - 76\beta_{8} - 634\beta_{3} + 3028\beta_{2} - 4520 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1350\beta_{11} + 1836\beta_{7} - 4788\beta_{6} + 4382\beta_{5} + 3674\beta_{4} - 9942\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
− 2.74137i
− 2.46680i
− 1.99567i
− 1.13965i
− 0.979668i
− 0.398211i
0.398211i
0.979668i
1.13965i
1.99567i
2.46680i
2.74137i
− 2.74137i 0 −5.51509 1.00000 0 0 9.63615i 0 − 2.74137i
881.2 − 2.46680i 0 −4.08510 1.00000 0 0 5.14351i 0 − 2.46680i
881.3 − 1.99567i 0 −1.98270 1.00000 0 0 − 0.0345244i 0 − 1.99567i
881.4 − 1.13965i 0 0.701207 1.00000 0 0 − 3.07842i 0 − 1.13965i
881.5 − 0.979668i 0 1.04025 1.00000 0 0 − 2.97844i 0 − 0.979668i
881.6 − 0.398211i 0 1.84143 1.00000 0 0 − 1.52970i 0 − 0.398211i
881.7 0.398211i 0 1.84143 1.00000 0 0 1.52970i 0 0.398211i
881.8 0.979668i 0 1.04025 1.00000 0 0 2.97844i 0 0.979668i
881.9 1.13965i 0 0.701207 1.00000 0 0 3.07842i 0 1.13965i
881.10 1.99567i 0 −1.98270 1.00000 0 0 0.0345244i 0 1.99567i
881.11 2.46680i 0 −4.08510 1.00000 0 0 − 5.14351i 0 2.46680i
881.12 2.74137i 0 −5.51509 1.00000 0 0 − 9.63615i 0 2.74137i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.12
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 2205.2.b.b 12
3.b odd 2 1 2205.2.b.a 12
7.b odd 2 1 2205.2.b.a 12
7.c even 3 1 315.2.bj.a ✓ 12
7.d odd 6 1 315.2.bj.b yes 12
21.c even 2 1 inner 2205.2.b.b 12
21.g even 6 1 315.2.bj.a ✓ 12
21.h odd 6 1 315.2.bj.b yes 12
35.i odd 6 1 1575.2.bk.e 12
35.j even 6 1 1575.2.bk.f 12
35.k even 12 2 1575.2.bc.c 24
35.l odd 12 2 1575.2.bc.d 24
105.o odd 6 1 1575.2.bk.e 12
105.p even 6 1 1575.2.bk.f 12
105.w odd 12 2 1575.2.bc.d 24
105.x even 12 2 1575.2.bc.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.bj.a ✓ 12 7.c even 3 1
315.2.bj.a ✓ 12 21.g even 6 1
315.2.bj.b yes 12 7.d odd 6 1
315.2.bj.b yes 12 21.h odd 6 1
1575.2.bc.c 24 35.k even 12 2
1575.2.bc.c 24 105.x even 12 2
1575.2.bc.d 24 35.l odd 12 2
1575.2.bc.d 24 105.w odd 12 2
1575.2.bk.e 12 35.i odd 6 1
1575.2.bk.e 12 105.o odd 6 1
1575.2.bk.f 12 35.j even 6 1
1575.2.bk.f 12 105.p even 6 1
2205.2.b.a 12 3.b odd 2 1
2205.2.b.a 12 7.b odd 2 1
2205.2.b.b 12 1.a even 1 1 trivial
2205.2.b.b 12 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\):

\( T_{2}^{12} + 20T_{2}^{10} + 144T_{2}^{8} + 452T_{2}^{6} + 604T_{2}^{4} + 312T_{2}^{2} + 36 \) Copy content Toggle raw display
\( T_{17}^{6} - 60T_{17}^{4} + 60T_{17}^{3} + 678T_{17}^{2} - 324T_{17} - 1458 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 20 T^{10} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T - 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 68 T^{10} + \cdots + 19044 \) Copy content Toggle raw display
$13$ \( T^{12} + 102 T^{10} + \cdots + 2025 \) Copy content Toggle raw display
$17$ \( (T^{6} - 60 T^{4} + \cdots - 1458)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 114 T^{10} + \cdots + 227529 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 113422500 \) Copy content Toggle raw display
$29$ \( T^{12} + 164 T^{10} + \cdots + 4088484 \) Copy content Toggle raw display
$31$ \( T^{12} + 258 T^{10} + \cdots + 1750329 \) Copy content Toggle raw display
$37$ \( (T^{6} - 10 T^{5} + \cdots + 110137)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 12 T^{5} + \cdots - 36450)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} + \cdots - 114167)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 120 T^{4} + \cdots - 21600)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 200 T^{10} + \cdots + 43877376 \) Copy content Toggle raw display
$59$ \( (T^{6} - 24 T^{5} + \cdots + 270)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 593994384 \) Copy content Toggle raw display
$67$ \( (T^{6} + 6 T^{5} + \cdots - 15359)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 491508900 \) Copy content Toggle raw display
$73$ \( T^{12} + 402 T^{10} + \cdots + 1565001 \) Copy content Toggle raw display
$79$ \( (T^{6} + 18 T^{5} + \cdots + 44425)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 12 T^{5} + \cdots - 287874)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 12 T^{5} + \cdots - 29538)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 732 T^{10} + \cdots + 71571600 \) Copy content Toggle raw display
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