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Properties

Label 1449.2.h.f
Level $1449$
Weight $2$
Character orbit 1449.h
Analytic conductor $11.570$
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] = [1449,2,Mod(1126,1449)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1449, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1449.1126");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1449.h (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: \(11.5703232529\)
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: \( x^{12} + 22x^{10} + 144x^{8} + 276x^{6} + 184x^{4} + 30x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - 2) q^{5} + (\beta_{6} + \beta_{5}) q^{7} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{2} + 1) q^{4} + (\beta_{7} - 2) q^{5} + (\beta_{6} + \beta_{5}) q^{7} + \beta_1 q^{8} + (2 \beta_{5} - \beta_{3} - \beta_1) q^{10} + ( - \beta_{9} + \beta_{8} + \beta_{4}) q^{11} + ( - \beta_{9} + 2 \beta_{8} - \beta_{4}) q^{13} + ( - \beta_{8} - \beta_{4} + \beta_{2} - 3) q^{14} + ( - \beta_{7} - 1) q^{16} + (\beta_{7} + \beta_{2} + 2) q^{17} + ( - \beta_{3} + \beta_1) q^{19} + (3 \beta_{2} - 2) q^{20} + (\beta_{11} - \beta_{10} - 2 \beta_{6}) q^{22} + ( - \beta_{9} - \beta_{5} + \cdots + \beta_1) q^{23}+ \cdots + ( - 2 \beta_{10} - 4 \beta_{6} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 20 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 20 q^{5} - 32 q^{14} - 16 q^{16} + 32 q^{17} - 12 q^{20} + 32 q^{25} + 12 q^{38} + 20 q^{46} - 20 q^{49} - 76 q^{58} - 12 q^{64} - 20 q^{68} + 52 q^{70} - 32 q^{80} + 56 q^{83} + 4 q^{85} - 52 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 22x^{10} + 144x^{8} + 276x^{6} + 184x^{4} + 30x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9\nu^{10} + 193\nu^{8} + 1184\nu^{6} + 1745\nu^{4} + 333\nu^{2} - 216 ) / 86 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\nu^{10} + 243\nu^{8} + 1292\nu^{6} + 836\nu^{4} - 416\nu^{2} + 13 ) / 43 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{10} + 19\nu^{8} - 229\nu^{6} - 2369\nu^{4} - 2678\nu^{2} - 220 ) / 43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31\nu^{11} + 660\nu^{9} + 3997\nu^{7} + 5743\nu^{5} + 1706\nu^{3} - 529\nu ) / 86 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46\nu^{10} + 953\nu^{8} + 5397\nu^{6} + 5670\nu^{4} + 541\nu^{2} - 115 ) / 86 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 57\nu^{11} + 1208\nu^{9} + 7255\nu^{7} + 10335\nu^{5} + 4818\nu^{3} + 1083\nu ) / 86 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 44\nu^{10} + 934\nu^{8} + 5626\nu^{6} + 8039\nu^{4} + 3262\nu^{2} + 277 ) / 43 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -75\nu^{11} - 1637\nu^{9} - 10526\nu^{7} - 19071\nu^{5} - 11547\nu^{3} - 1210\nu ) / 86 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -29\nu^{11} - 684\nu^{9} - 5129\nu^{7} - 13401\nu^{5} - 11006\nu^{3} - 1411\nu ) / 43 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 130\nu^{11} + 2783\nu^{9} + 17107\nu^{7} + 26486\nu^{5} + 12421\nu^{3} + 707\nu ) / 86 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 117\nu^{11} + 2552\nu^{9} + 16381\nu^{7} + 29479\nu^{5} + 17530\nu^{3} + 2137\nu ) / 43 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{9} - 2\beta_{6} - 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{5} + \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{11} - 4\beta_{10} - 9\beta_{9} + 4\beta_{8} + 22\beta_{6} + 22\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -11\beta_{7} + 20\beta_{5} - 12\beta_{3} + 4\beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 67\beta_{11} + 64\beta_{10} + 83\beta_{9} - 16\beta_{8} - 226\beta_{6} - 242\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 117\beta_{7} - 196\beta_{5} + 133\beta_{3} - 71\beta_{2} - 12\beta _1 - 344 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -685\beta_{11} - 864\beta_{10} - 787\beta_{9} - 128\beta_{8} + 2370\beta_{6} + 2654\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1256\beta_{7} + 1960\beta_{5} - 1472\beta_{3} + 1001\beta_{2} + 248\beta _1 + 3501 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 7181\beta_{11} + 10884\beta_{10} + 7661\beta_{9} + 4036\beta_{8} - 25274\beta_{6} - 29278\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13638\beta_{7} - 20050\beta_{5} + 16359\beta_{3} - 12901\beta_{2} - 3730\beta _1 - 36436 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -76575\beta_{11} - 131960\beta_{10} - 76497\beta_{9} - 66680\beta_{8} + 273138\beta_{6} + 324742\beta_{4} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1449\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(829\) \(1289\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1126.1
2.85264i
− 2.85264i
− 1.07227i
1.07227i
3.35642i
− 3.35642i
− 0.414543i
0.414543i
− 0.211825i
0.211825i
1.10924i
− 1.10924i
−2.27250 0 3.16425 −2.68397 0 2.27250 − 1.35490i −2.64575 0 6.09931
1126.2 −2.27250 0 3.16425 −2.68397 0 2.27250 + 1.35490i −2.64575 0 6.09931
1126.3 −1.49236 0 0.227134 1.40268 0 1.49236 − 2.18469i 2.64575 0 −2.09330
1126.4 −1.49236 0 0.227134 1.40268 0 1.49236 + 2.18469i 2.64575 0 −2.09330
1126.5 −0.780139 0 −1.39138 −3.71871 0 0.780139 − 2.52812i 2.64575 0 2.90111
1126.6 −0.780139 0 −1.39138 −3.71871 0 0.780139 + 2.52812i 2.64575 0 2.90111
1126.7 0.780139 0 −1.39138 −3.71871 0 −0.780139 − 2.52812i −2.64575 0 −2.90111
1126.8 0.780139 0 −1.39138 −3.71871 0 −0.780139 + 2.52812i −2.64575 0 −2.90111
1126.9 1.49236 0 0.227134 1.40268 0 −1.49236 − 2.18469i −2.64575 0 2.09330
1126.10 1.49236 0 0.227134 1.40268 0 −1.49236 + 2.18469i −2.64575 0 2.09330
1126.11 2.27250 0 3.16425 −2.68397 0 −2.27250 − 1.35490i 2.64575 0 −6.09931
1126.12 2.27250 0 3.16425 −2.68397 0 −2.27250 + 1.35490i 2.64575 0 −6.09931
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1126.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner
69.c even 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 1449.2.h.f ✓ 12
3.b odd 2 1 1449.2.h.i yes 12
7.b odd 2 1 1449.2.h.i yes 12
21.c even 2 1 inner 1449.2.h.f ✓ 12
23.b odd 2 1 1449.2.h.i yes 12
69.c even 2 1 inner 1449.2.h.f ✓ 12
161.c even 2 1 inner 1449.2.h.f ✓ 12
483.c odd 2 1 1449.2.h.i yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1449.2.h.f ✓ 12 1.a even 1 1 trivial
1449.2.h.f ✓ 12 21.c even 2 1 inner
1449.2.h.f ✓ 12 69.c even 2 1 inner
1449.2.h.f ✓ 12 161.c even 2 1 inner
1449.2.h.i yes 12 3.b odd 2 1
1449.2.h.i yes 12 7.b odd 2 1
1449.2.h.i yes 12 23.b odd 2 1
1449.2.h.i yes 12 483.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1449, [\chi])\):

\( T_{2}^{6} - 8T_{2}^{4} + 16T_{2}^{2} - 7 \) Copy content Toggle raw display
\( T_{5}^{3} + 5T_{5}^{2} + T_{5} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 8 T^{4} + 16 T^{2} - 7)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{3} + 5 T^{2} + T - 14)^{4} \) Copy content Toggle raw display
$7$ \( T^{12} + 10 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} + 40 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 73 T^{4} + \cdots + 6272)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 8 T^{2} + 9 T + 14)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} - 58 T^{4} + \cdots - 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 148035889 \) Copy content Toggle raw display
$29$ \( (T^{6} - 102 T^{4} + \cdots - 13552)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 160 T^{4} + \cdots + 100352)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 48 T^{4} + \cdots + 896)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 48 T^{4} + \cdots + 896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 259 T^{4} + \cdots + 537824)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 208 T^{4} + \cdots + 229376)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 175 T^{4} + \cdots + 119072)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 265 T^{4} + \cdots + 896)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 169 T^{4} + \cdots - 13552)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 153 T^{4} + \cdots + 64736)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 225 T^{4} + \cdots - 20412)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 264 T^{4} + \cdots + 25088)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 40 T^{4} + \cdots + 224)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 14 T^{2} + \cdots + 196)^{4} \) Copy content Toggle raw display
$89$ \( (T^{3} + 13 T^{2} + \cdots - 98)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} - 182 T^{4} + \cdots - 87808)^{2} \) Copy content Toggle raw display
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