Properties

Label 7056.2.h.o
Level $7056$
Weight $2$
Character orbit 7056.h
Analytic conductor $56.342$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7056,2,Mod(4607,7056)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7056, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("7056.4607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7056 = 2^{4} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7056.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: \(56.3424436662\)
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: \( x^{12} + 28x^{10} + 276x^{8} + 1178x^{6} + 2292x^{4} + 1888x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1008)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} + \beta_{3} q^{11} - \beta_{5} q^{13} + ( - \beta_{9} + \beta_{4}) q^{17} + ( - \beta_{10} + \beta_{8}) q^{19} + (\beta_{3} - \beta_{2}) q^{23} + ( - \beta_{5} + \beta_1 - 1) q^{25} + (\beta_{11} - \beta_{9} - 2 \beta_{4}) q^{29} + ( - 2 \beta_{10} - \beta_{8} + \beta_{7}) q^{31} + (\beta_{5} - 2 \beta_1) q^{37} + ( - \beta_{11} + \beta_{9} + \beta_{4}) q^{41} + (3 \beta_{10} + \beta_{8} - 2 \beta_{7}) q^{43} + ( - \beta_{6} + \beta_{3}) q^{47} + ( - \beta_{11} + \beta_{9} - 2 \beta_{4}) q^{53} + \beta_{8} q^{55} + ( - \beta_{6} - \beta_{3} + \beta_{2}) q^{59} + (\beta_1 - 3) q^{61} + ( - \beta_{11} + 2 \beta_{4}) q^{65} + ( - 2 \beta_{10} + 2 \beta_{8} + \beta_{7}) q^{67} + (\beta_{6} - 3 \beta_{3} - 2 \beta_{2}) q^{71} + (\beta_{5} + \beta_1 - 3) q^{73} + (\beta_{8} - \beta_{7}) q^{79} + (\beta_{3} - 2 \beta_{2}) q^{83} + (2 \beta_{5} - 3 \beta_1 + 7) q^{85} + ( - 2 \beta_{11} - 2 \beta_{4}) q^{89} + ( - \beta_{6} - \beta_{3}) q^{95} + (\beta_{5} - \beta_1 - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{13} - 4 q^{25} - 12 q^{37} - 32 q^{61} - 36 q^{73} + 64 q^{85} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 28x^{10} + 276x^{8} + 1178x^{6} + 2292x^{4} + 1888x^{2} + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{10} - 16\nu^{8} + 35\nu^{6} + 1377\nu^{4} + 4746\nu^{2} + 3397 ) / 238 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 23\nu^{10} + 487\nu^{8} + 2527\nu^{6} - 2635\nu^{4} - 27643\nu^{2} - 21011 ) / 1428 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{10} + 197\nu^{8} + 1931\nu^{6} + 7837\nu^{4} + 12355\nu^{2} + 5189 ) / 204 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\nu^{11} + 121\nu^{9} - 3381\nu^{7} - 43027\nu^{5} - 134141\nu^{3} - 99031\nu ) / 10948 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 23\nu^{10} + 606\nu^{8} + 5383\nu^{6} + 19023\nu^{4} + 26978\nu^{2} + 11357 ) / 238 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\nu^{10} + 311\nu^{8} + 2674\nu^{6} + 8823\nu^{4} + 11354\nu^{2} + 5170 ) / 119 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -227\nu^{11} - 8035\nu^{9} - 103201\nu^{7} - 575399\nu^{5} - 1301111\nu^{3} - 931057\nu ) / 49266 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 127\nu^{11} + 3579\nu^{9} + 35420\nu^{7} + 148801\nu^{5} + 259413\nu^{3} + 125144\nu ) / 5474 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 41\nu^{11} + 1217\nu^{9} + 12811\nu^{7} + 56831\nu^{5} + 95927\nu^{3} + 36537\nu ) / 1564 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -16\nu^{11} - 425\nu^{9} - 3818\nu^{7} - 13696\nu^{5} - 19675\nu^{3} - 8588\nu ) / 414 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 807\nu^{11} + 21837\nu^{9} + 202377\nu^{7} + 763725\nu^{5} + 1154055\nu^{3} + 519459\nu ) / 10948 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - 3\beta_{8} - 3\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - \beta_{5} + \beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + 9\beta_{10} - 3\beta_{9} + 21\beta_{8} - 9\beta_{7} + 33\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{6} + 13\beta_{5} - 4\beta_{3} - 4\beta_{2} - 13\beta _1 + 86 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -17\beta_{11} - 153\beta_{10} + 39\beta_{9} - 204\beta_{8} + 99\beta_{7} - 360\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{6} - 73\beta_{5} + 31\beta_{3} + 37\beta_{2} + 84\beta _1 - 436 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 457\beta_{11} + 2112\beta_{10} - 504\beta_{9} + 2217\beta_{8} - 960\beta_{7} + 3861\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1193\beta_{6} + 1601\beta_{5} - 760\beta_{3} - 1072\beta_{2} - 2125\beta _1 + 9322 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -7328\beta_{11} - 27135\beta_{10} + 6441\beta_{9} - 24933\beta_{8} + 9099\beta_{7} - 41601\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12677\beta_{6} - 17571\beta_{5} + 8822\beta_{3} + 14234\beta_{2} + 26249\beta _1 - 101916 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 102073\beta_{11} + 336543\beta_{10} - 80517\beta_{9} + 283662\beta_{8} - 86289\beta_{7} + 452886\beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1765\) \(4609\) \(6175\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4607.1
3.39892i
1.62198i
2.98196i
0.776357i
0.984122i
1.83117i
0.984122i
1.83117i
2.98196i
0.776357i
3.39892i
1.62198i
0 0 0 3.19116i 0 0 0 0 0
4607.2 0 0 0 3.19116i 0 0 0 0 0
4607.3 0 0 0 2.34411i 0 0 0 0 0
4607.4 0 0 0 2.34411i 0 0 0 0 0
4607.5 0 0 0 0.567166i 0 0 0 0 0
4607.6 0 0 0 0.567166i 0 0 0 0 0
4607.7 0 0 0 0.567166i 0 0 0 0 0
4607.8 0 0 0 0.567166i 0 0 0 0 0
4607.9 0 0 0 2.34411i 0 0 0 0 0
4607.10 0 0 0 2.34411i 0 0 0 0 0
4607.11 0 0 0 3.19116i 0 0 0 0 0
4607.12 0 0 0 3.19116i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4607.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.h.o 12
3.b odd 2 1 inner 7056.2.h.o 12
4.b odd 2 1 inner 7056.2.h.o 12
7.b odd 2 1 7056.2.h.n 12
7.d odd 6 1 1008.2.cq.b 12
7.d odd 6 1 1008.2.cq.c yes 12
12.b even 2 1 inner 7056.2.h.o 12
21.c even 2 1 7056.2.h.n 12
21.g even 6 1 1008.2.cq.b 12
21.g even 6 1 1008.2.cq.c yes 12
28.d even 2 1 7056.2.h.n 12
28.f even 6 1 1008.2.cq.b 12
28.f even 6 1 1008.2.cq.c yes 12
84.h odd 2 1 7056.2.h.n 12
84.j odd 6 1 1008.2.cq.b 12
84.j odd 6 1 1008.2.cq.c yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cq.b 12 7.d odd 6 1
1008.2.cq.b 12 21.g even 6 1
1008.2.cq.b 12 28.f even 6 1
1008.2.cq.b 12 84.j odd 6 1
1008.2.cq.c yes 12 7.d odd 6 1
1008.2.cq.c yes 12 21.g even 6 1
1008.2.cq.c yes 12 28.f even 6 1
1008.2.cq.c yes 12 84.j odd 6 1
7056.2.h.n 12 7.b odd 2 1
7056.2.h.n 12 21.c even 2 1
7056.2.h.n 12 28.d even 2 1
7056.2.h.n 12 84.h odd 2 1
7056.2.h.o 12 1.a even 1 1 trivial
7056.2.h.o 12 3.b odd 2 1 inner
7056.2.h.o 12 4.b odd 2 1 inner
7056.2.h.o 12 12.b 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 \(S_{2}^{\mathrm{new}}(7056, [\chi])\):

\( T_{5}^{6} + 16T_{5}^{4} + 61T_{5}^{2} + 18 \) Copy content Toggle raw display
\( T_{11}^{6} - 28T_{11}^{4} + 85T_{11}^{2} - 54 \) Copy content Toggle raw display
\( T_{13}^{3} - T_{13}^{2} - 22T_{13} - 14 \) Copy content Toggle raw display
\( T_{61}^{3} + 8T_{61}^{2} + 6T_{61} - 24 \) Copy content Toggle raw display
\( T_{83}^{6} - 220T_{83}^{4} + 15637T_{83}^{2} - 354294 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 16 T^{4} + 61 T^{2} + 18)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} - 28 T^{4} + 85 T^{2} - 54)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - T^{2} - 22 T - 14)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 100 T^{4} + 2308 T^{2} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 47 T^{4} + 712 T^{2} + 3468)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 76 T^{4} + 1828 T^{2} + \cdots - 13824)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 142 T^{4} + 3409 T^{2} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 117 T^{4} + 2511 T^{2} + \cdots + 14283)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 3 T^{2} - 54 T - 218)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 106 T^{4} + 2944 T^{2} + \cdots + 23328)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 263 T^{4} + 19912 T^{2} + \cdots + 465708)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 54)^{6} \) Copy content Toggle raw display
$53$ \( (T^{6} + 190 T^{4} + 2113 T^{2} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 214 T^{4} + 6913 T^{2} + \cdots - 55296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 8 T^{2} + 6 T - 24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 171 T^{4} + 5184 T^{2} + \cdots + 27648)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 466 T^{4} + 62956 T^{2} + \cdots - 2032344)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 9 T^{2} - 24 T - 248)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 93 T^{4} + 2079 T^{2} + 243)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 220 T^{4} + 15637 T^{2} + \cdots - 354294)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 328 T^{4} + 24976 T^{2} + \cdots + 294912)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 14 T^{2} + 41 T - 8)^{4} \) Copy content Toggle raw display
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