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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Newspace parameters

Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Defining polynomial: \(x^{12} + 18 x^{10} + 207 x^{8} + 1014 x^{6} + 1065 x^{4} - 5508 x^{2} + 8464\)
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

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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{7} q^{7} + ( -3 + \beta_{2} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + \beta_{7} q^{7} + ( -3 + \beta_{2} ) q^{9} + \beta_{5} q^{13} -\beta_{8} q^{17} -\beta_{10} q^{19} + ( -\beta_{5} - \beta_{6} - \beta_{11} ) q^{21} -\beta_{1} q^{23} + 5 q^{25} + ( -3 \beta_{4} - 2 \beta_{10} ) q^{27} + ( -\beta_{5} + \beta_{6} ) q^{29} -\beta_{11} q^{37} + ( \beta_{1} + \beta_{3} + 2 \beta_{7} ) q^{39} -3 \beta_{3} q^{47} + ( -7 + 2 \beta_{2} + \beta_{8} ) q^{49} + ( \beta_{9} - 2 \beta_{10} ) q^{51} + ( \beta_{5} + 2 \beta_{6} ) q^{53} + ( -2 \beta_{2} + \beta_{8} ) q^{57} + ( -2 \beta_{4} - \beta_{9} ) q^{59} + ( \beta_{1} + 5 \beta_{3} - 5 \beta_{7} ) q^{63} + ( 2 \beta_{4} - \beta_{9} ) q^{67} + ( \beta_{5} - 2 \beta_{6} - \beta_{11} ) q^{69} + ( -\beta_{2} - 2 \beta_{8} ) q^{73} + 5 \beta_{4} q^{75} + ( 9 - 4 \beta_{2} + 2 \beta_{8} ) q^{81} + ( -2 \beta_{1} - 7 \beta_{3} - \beta_{7} ) q^{87} + ( 4 \beta_{4} + \beta_{9} - 2 \beta_{10} ) q^{91} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 36q^{9} + O(q^{10}) \) \( 12q - 36q^{9} + 60q^{25} - 84q^{49} + 108q^{81} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -857 \nu^{11} - 108754 \nu^{9} - 1882257 \nu^{7} - 19921624 \nu^{5} - 91847935 \nu^{3} + 56141760 \nu \)\()/60769588\)
\(\beta_{2}\)\(=\)\((\)\( 4848 \nu^{10} + 67487 \nu^{8} + 612280 \nu^{6} + 1941799 \nu^{4} - 6158396 \nu^{2} + 20243266 \)\()/15192397\)
\(\beta_{3}\)\(=\)\((\)\( -171 \nu^{11} - 3142 \nu^{9} - 37173 \nu^{7} - 203958 \nu^{5} - 397511 \nu^{3} + 101088 \nu \)\()/589996\)
\(\beta_{4}\)\(=\)\((\)\( 6496 \nu^{10} + 113219 \nu^{8} + 1399303 \nu^{6} + 7488246 \nu^{4} + 15491689 \nu^{2} - 17025842 \)\()/15192397\)
\(\beta_{5}\)\(=\)\((\)\( 26777 \nu^{11} + 555190 \nu^{9} + 6273577 \nu^{7} + 34391608 \nu^{5} + 16535587 \nu^{3} - 306111640 \nu \)\()/60769588\)
\(\beta_{6}\)\(=\)\((\)\( -14647 \nu^{11} - 250356 \nu^{9} - 2790517 \nu^{7} - 12120654 \nu^{5} - 6158189 \nu^{3} + 112303816 \nu \)\()/30384794\)
\(\beta_{7}\)\(=\)\((\)\( -2741 \nu^{11} - 55406 \nu^{9} - 678481 \nu^{7} - 4014812 \nu^{5} - 9469483 \nu^{3} + 3424640 \nu \)\()/5524508\)
\(\beta_{8}\)\(=\)\((\)\( 11493 \nu^{10} + 188193 \nu^{8} + 1977982 \nu^{6} + 6662232 \nu^{4} - 20729120 \nu^{2} - 100079394 \)\()/15192397\)
\(\beta_{9}\)\(=\)\((\)\( 11527 \nu^{10} + 356749 \nu^{8} + 4352527 \nu^{6} + 33217533 \nu^{4} + 72677113 \nu^{2} - 83718298 \)\()/15192397\)
\(\beta_{10}\)\(=\)\((\)\( -1347 \nu^{10} - 22950 \nu^{8} - 268517 \nu^{6} - 1266390 \nu^{4} - 2440467 \nu^{2} + 2608806 \)\()/1321078\)
\(\beta_{11}\)\(=\)\((\)\( 9 \nu^{11} + 146 \nu^{9} + 1695 \nu^{7} + 7074 \nu^{5} + 4013 \nu^{3} - 67464 \nu \)\()/9476\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{10} - \beta_{8} - 3 \beta_{4} - 6\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-4 \beta_{11} - 3 \beta_{7} - 15 \beta_{6} - 6 \beta_{5} + 8 \beta_{3} - 3 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(18 \beta_{10} + \beta_{9} - 2 \beta_{8} + 32 \beta_{4} + 17 \beta_{2} - 30\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(5 \beta_{11} + 66 \beta_{7} + 35 \beta_{6} + 19 \beta_{5} - 125 \beta_{3} - 19 \beta_{1}\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-4 \beta_{10} - 21 \beta_{9} + 69 \beta_{8} + 87 \beta_{4} - 243 \beta_{2} + 768\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(462 \beta_{11} - 1141 \beta_{7} + 889 \beta_{6} + 4 \beta_{5} + 1974 \beta_{3} + 425 \beta_{1}\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(-1674 \beta_{10} + 319 \beta_{9} - 282 \beta_{8} - 4612 \beta_{4} + 741 \beta_{2} - 2950\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-6592 \beta_{11} - 3717 \beta_{7} - 15345 \beta_{6} - 1998 \beta_{5} + 7204 \beta_{3} + 855 \beta_{1}\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(14058 \beta_{10} - 2189 \beta_{9} - 5258 \beta_{8} + 36586 \beta_{4} + 19833 \beta_{2} - 59886\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(9031 \beta_{11} + 82638 \beta_{7} + 20333 \beta_{6} + 2233 \beta_{5} - 147851 \beta_{3} - 27605 \beta_{1}\)\()/2\)

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.

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 \(S_{2}^{\mathrm{new}}(1216, [\chi])\):

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

Hecke characteristic polynomials

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