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Properties

Label 1216.2.b.f
Level $1216$
Weight $2$
Character orbit 1216.b
Analytic conductor $9.710$
Analytic rank $0$
Dimension $12$
CM discriminant -152
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1216,2,Mod(607,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1216.607");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.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: \(9.70980888579\)
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: \( x^{12} + 18x^{10} + 207x^{8} + 1014x^{6} + 1065x^{4} - 5508x^{2} + 8464 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{7} q^{7} + (\beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{3} + \beta_{7} q^{7} + (\beta_{2} - 3) q^{9} + \beta_{5} q^{13} - \beta_{8} q^{17} - \beta_{10} q^{19} + ( - \beta_{11} - \beta_{6} - \beta_{5}) q^{21} - \beta_1 q^{23} + 5 q^{25} + ( - 2 \beta_{10} - 3 \beta_{4}) q^{27} + (\beta_{6} - \beta_{5}) q^{29} - \beta_{11} q^{37} + (2 \beta_{7} + \beta_{3} + \beta_1) q^{39} - 3 \beta_{3} q^{47} + (\beta_{8} + 2 \beta_{2} - 7) q^{49} + ( - 2 \beta_{10} + \beta_{9}) q^{51} + (2 \beta_{6} + \beta_{5}) q^{53} + (\beta_{8} - 2 \beta_{2}) q^{57} + ( - \beta_{9} - 2 \beta_{4}) q^{59} + ( - 5 \beta_{7} + 5 \beta_{3} + \beta_1) q^{63} + ( - \beta_{9} + 2 \beta_{4}) q^{67} + ( - \beta_{11} - 2 \beta_{6} + \beta_{5}) q^{69} + ( - 2 \beta_{8} - \beta_{2}) q^{73} + 5 \beta_{4} q^{75} + (2 \beta_{8} - 4 \beta_{2} + 9) q^{81} + ( - \beta_{7} - 7 \beta_{3} - 2 \beta_1) q^{87} + ( - 2 \beta_{10} + \beta_{9} + 4 \beta_{4}) q^{91}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 36 q^{9} + 60 q^{25} - 84 q^{49} + 108 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18x^{10} + 207x^{8} + 1014x^{6} + 1065x^{4} - 5508x^{2} + 8464 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -857\nu^{11} - 108754\nu^{9} - 1882257\nu^{7} - 19921624\nu^{5} - 91847935\nu^{3} + 56141760\nu ) / 60769588 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4848\nu^{10} + 67487\nu^{8} + 612280\nu^{6} + 1941799\nu^{4} - 6158396\nu^{2} + 20243266 ) / 15192397 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -171\nu^{11} - 3142\nu^{9} - 37173\nu^{7} - 203958\nu^{5} - 397511\nu^{3} + 101088\nu ) / 589996 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6496\nu^{10} + 113219\nu^{8} + 1399303\nu^{6} + 7488246\nu^{4} + 15491689\nu^{2} - 17025842 ) / 15192397 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26777\nu^{11} + 555190\nu^{9} + 6273577\nu^{7} + 34391608\nu^{5} + 16535587\nu^{3} - 306111640\nu ) / 60769588 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -14647\nu^{11} - 250356\nu^{9} - 2790517\nu^{7} - 12120654\nu^{5} - 6158189\nu^{3} + 112303816\nu ) / 30384794 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2741\nu^{11} - 55406\nu^{9} - 678481\nu^{7} - 4014812\nu^{5} - 9469483\nu^{3} + 3424640\nu ) / 5524508 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11493\nu^{10} + 188193\nu^{8} + 1977982\nu^{6} + 6662232\nu^{4} - 20729120\nu^{2} - 100079394 ) / 15192397 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11527\nu^{10} + 356749\nu^{8} + 4352527\nu^{6} + 33217533\nu^{4} + 72677113\nu^{2} - 83718298 ) / 15192397 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -1347\nu^{10} - 22950\nu^{8} - 268517\nu^{6} - 1266390\nu^{4} - 2440467\nu^{2} + 2608806 ) / 1321078 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{11} + 146\nu^{9} + 1695\nu^{7} + 7074\nu^{5} + 4013\nu^{3} - 67464\nu ) / 9476 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{10} - \beta_{8} - 3\beta_{4} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{11} - 3\beta_{7} - 15\beta_{6} - 6\beta_{5} + 8\beta_{3} - 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18\beta_{10} + \beta_{9} - 2\beta_{8} + 32\beta_{4} + 17\beta_{2} - 30 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{11} + 66\beta_{7} + 35\beta_{6} + 19\beta_{5} - 125\beta_{3} - 19\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{10} - 21\beta_{9} + 69\beta_{8} + 87\beta_{4} - 243\beta_{2} + 768 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 462\beta_{11} - 1141\beta_{7} + 889\beta_{6} + 4\beta_{5} + 1974\beta_{3} + 425\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -1674\beta_{10} + 319\beta_{9} - 282\beta_{8} - 4612\beta_{4} + 741\beta_{2} - 2950 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -6592\beta_{11} - 3717\beta_{7} - 15345\beta_{6} - 1998\beta_{5} + 7204\beta_{3} + 855\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14058\beta_{10} - 2189\beta_{9} - 5258\beta_{8} + 36586\beta_{4} + 19833\beta_{2} - 59886 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9031\beta_{11} + 82638\beta_{7} + 20333\beta_{6} + 2233\beta_{5} - 147851\beta_{3} - 27605\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
607.1
1.70027 + 2.78183i
−1.70027 − 2.78183i
−1.13617 + 0.418247i
1.13617 − 0.418247i
−0.564101 + 2.36358i
0.564101 − 2.36358i
0.564101 + 2.36358i
−0.564101 − 2.36358i
1.13617 + 0.418247i
−1.13617 − 0.418247i
−1.70027 + 2.78183i
1.70027 − 2.78183i
0 − 3.40054i 0 0 0 − 4.62947i 0 −8.56365 0
607.2 0 − 3.40054i 0 0 0 4.62947i 0 −8.56365 0
607.3 0 − 2.27234i 0 0 0 − 4.53419i 0 −2.16351 0
607.4 0 − 2.27234i 0 0 0 4.53419i 0 −2.16351 0
607.5 0 − 1.12820i 0 0 0 − 0.0952793i 0 1.72716 0
607.6 0 − 1.12820i 0 0 0 0.0952793i 0 1.72716 0
607.7 0 1.12820i 0 0 0 − 0.0952793i 0 1.72716 0
607.8 0 1.12820i 0 0 0 0.0952793i 0 1.72716 0
607.9 0 2.27234i 0 0 0 − 4.53419i 0 −2.16351 0
607.10 0 2.27234i 0 0 0 4.53419i 0 −2.16351 0
607.11 0 3.40054i 0 0 0 − 4.62947i 0 −8.56365 0
607.12 0 3.40054i 0 0 0 4.62947i 0 −8.56365 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
152.g odd 2 1 CM by \(\Q(\sqrt{-38}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
19.b odd 2 1 inner
76.d even 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1216.2.b.f ✓ 12
4.b odd 2 1 inner 1216.2.b.f ✓ 12
8.b even 2 1 inner 1216.2.b.f ✓ 12
8.d odd 2 1 inner 1216.2.b.f ✓ 12
19.b odd 2 1 inner 1216.2.b.f ✓ 12
76.d even 2 1 inner 1216.2.b.f ✓ 12
152.b even 2 1 inner 1216.2.b.f ✓ 12
152.g odd 2 1 CM 1216.2.b.f ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1216.2.b.f ✓ 12 1.a even 1 1 trivial
1216.2.b.f ✓ 12 4.b odd 2 1 inner
1216.2.b.f ✓ 12 8.b even 2 1 inner
1216.2.b.f ✓ 12 8.d odd 2 1 inner
1216.2.b.f ✓ 12 19.b odd 2 1 inner
1216.2.b.f ✓ 12 76.d even 2 1 inner
1216.2.b.f ✓ 12 152.b even 2 1 inner
1216.2.b.f ✓ 12 152.g odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

\( T_{3}^{6} + 18T_{3}^{4} + 81T_{3}^{2} + 76 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{6} - 78T_{13}^{4} + 1521T_{13}^{2} - 76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 18 T^{4} + 81 T^{2} + 76)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 42 T^{4} + 441 T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( (T^{6} - 78 T^{4} + 1521 T^{2} - 76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 51 T - 138)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{6} \) Copy content Toggle raw display
$23$ \( (T^{6} + 138 T^{4} + 4761 T^{2} + \cdots + 30276)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 174 T^{4} + 7569 T^{2} + \cdots - 82764)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} \) Copy content Toggle raw display
$37$ \( (T^{2} - 76)^{6} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( (T^{2} + 36)^{6} \) Copy content Toggle raw display
$53$ \( (T^{6} - 318 T^{4} + 25281 T^{2} + \cdots - 197676)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 354 T^{4} + 31329 T^{2} + \cdots + 246924)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( (T^{6} + 402 T^{4} + 40401 T^{2} + \cdots + 870124)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{3} - 219 T - 394)^{4} \) Copy content Toggle raw display
$79$ \( T^{12} \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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