Properties

Label 912.2.f.i
Level $912$
Weight $2$
Character orbit 912.f
Analytic conductor $7.282$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.20322144469993472.1
Defining polynomial: \( x^{10} - x^{9} - x^{8} - 2x^{7} - 2x^{6} + 22x^{5} - 6x^{4} - 18x^{3} - 27x^{2} - 81x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 456)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{6} q^{5} + \beta_{4} q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{6} q^{5} + \beta_{4} q^{7} + \beta_{2} q^{9} - \beta_{9} q^{11} + (\beta_{9} + \beta_{6} + \beta_{5} + \beta_1) q^{13} + ( - \beta_{8} - \beta_{6} - 1) q^{15} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} - \beta_1) q^{17} + ( - \beta_{8} + \beta_{3} + \beta_1) q^{19} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{21} + (\beta_{9} + \beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{4} - \beta_1 - 1) q^{25} + ( - \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + 1) q^{27} + (\beta_{3} - \beta_{2}) q^{29} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6}) q^{31} + (\beta_{9} - \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1 - 1) q^{33} + ( - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_1) q^{35} + ( - \beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{37} + ( - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{39} + ( - \beta_{5} + \beta_{3} - \beta_{2} + \beta_1) q^{41} + ( - \beta_{8} + \beta_{7} - \beta_{5} - \beta_{4} + \beta_1 - 2) q^{43} + ( - \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_1) q^{45} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{47} + ( - 2 \beta_{5} + \beta_{4} + 2 \beta_1 + 1) q^{49} + (2 \beta_{9} - \beta_{8} + 2 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{51} + ( - \beta_{8} + \beta_{7} + \beta_{3} - \beta_{2} - 2) q^{53} + ( - \beta_{4} + \beta_{3} - \beta_{2} + 2) q^{55} + ( - \beta_{9} - \beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{2} + 1) q^{57} + (\beta_{8} - \beta_{7} + 2 \beta_{4} + \beta_{3} - \beta_{2}) q^{59} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1) q^{61} + (\beta_{8} - \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{63} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{65} + ( - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + \beta_1) q^{67} + ( - 2 \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 3) q^{69} + (\beta_{8} - \beta_{7} - 2 \beta_{3} + 2 \beta_{2} - 2) q^{71} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{73} + ( - \beta_{9} - \beta_{8} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{75} + (\beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{77} + (\beta_{9} + \beta_{6} - 2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{79} + (\beta_{9} - 2 \beta_{7} + 4 \beta_{5} + \beta_{4} + \beta_{2} + 2 \beta_1 + 2) q^{81} + (\beta_{8} + \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{83} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{85} + (\beta_{9} + 4 \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{87} + ( - \beta_{8} + \beta_{7} + 2 \beta_{4}) q^{89} + (\beta_{8} + \beta_{7} + 4 \beta_{6} - \beta_{3} - \beta_{2}) q^{91} + (2 \beta_{9} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{93} + (\beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{95} + (\beta_{8} + \beta_{7} - 2 \beta_{6} - 2 \beta_{3} - 2 \beta_{2}) q^{97} + ( - 2 \beta_{9} + \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 2 q^{7} + 3 q^{9} - 10 q^{15} - 2 q^{19} + 5 q^{21} - 14 q^{25} + 10 q^{27} - 6 q^{29} - 10 q^{33} - 7 q^{39} - 4 q^{41} - 20 q^{43} - 2 q^{45} + 16 q^{49} - q^{51} - 26 q^{53} + 12 q^{55} + 9 q^{57} - 2 q^{59} - 4 q^{61} + 17 q^{63} + 32 q^{65} - 27 q^{69} - 8 q^{71} - 26 q^{73} - 39 q^{75} + 23 q^{81} - 8 q^{85} - 13 q^{87} + 4 q^{89} - 18 q^{93} + 8 q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - x^{8} - 2x^{7} - 2x^{6} + 22x^{5} - 6x^{4} - 18x^{3} - 27x^{2} - 81x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} + 2\nu^{8} - 4\nu^{7} - 5\nu^{6} - 8\nu^{5} + 16\nu^{4} + 60\nu^{3} - 36\nu^{2} - 81\nu - 162 ) / 81 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{9} - 4\nu^{8} - \nu^{7} + 19\nu^{6} + 25\nu^{5} - 14\nu^{4} - 21\nu^{3} - 45\nu^{2} + 135\nu + 324 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{9} - \nu^{8} - \nu^{7} - 2\nu^{6} - 2\nu^{5} + 22\nu^{4} - 6\nu^{3} - 18\nu^{2} - 27\nu - 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{9} - 4\nu^{8} + 5\nu^{7} + \nu^{6} - 17\nu^{5} - 47\nu^{4} - 15\nu^{3} + 81\nu^{2} + 81 ) / 324 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{9} - 5\nu^{8} - 8\nu^{7} + 8\nu^{6} + 11\nu^{5} - 13\nu^{4} - 60\nu^{3} - 54\nu^{2} + 135\nu + 162 ) / 162 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{9} + 2\nu^{8} - 7\nu^{7} - 14\nu^{6} + 40\nu^{5} + \nu^{4} + 3\nu^{3} - 108\nu^{2} - 162\nu + 405 ) / 162 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{9} + 2\nu^{8} - 7\nu^{7} + 13\nu^{6} + 13\nu^{5} - 26\nu^{4} - 51\nu^{3} - 81\nu^{2} + 27\nu + 324 ) / 162 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 2\beta_{7} + 4\beta_{5} + \beta_{4} + \beta_{2} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{9} + 2\beta_{8} - 4\beta_{6} - 2\beta_{5} + 2\beta_{4} - \beta_{3} + \beta_{2} - \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{9} - 4\beta_{8} - 2\beta_{7} - 4\beta_{6} + 5\beta_{4} + \beta_{3} + \beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -8\beta_{9} - 2\beta_{8} - 6\beta_{7} - 8\beta_{6} - 5\beta_{5} + 2\beta_{4} - 7\beta_{3} - 9\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 15 \beta_{9} - 2 \beta_{8} - 12 \beta_{7} - 20 \beta_{6} - 6 \beta_{5} - 9 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 8 \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 19 \beta_{9} - 8 \beta_{8} + 22 \beta_{7} - 44 \beta_{6} - 28 \beta_{5} - 9 \beta_{4} - 4 \beta_{3} - 7 \beta_{2} - 29 \beta _1 + 29 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
−1.69134 0.373340i
−1.69134 + 0.373340i
−0.729858 1.57077i
−0.729858 + 1.57077i
−0.171092 1.72358i
−0.171092 + 1.72358i
1.39399 1.02801i
1.39399 + 1.02801i
1.69830 0.340259i
1.69830 + 0.340259i
0 −1.69134 0.373340i 0 0.568907i 0 −0.718465 0 2.72123 + 1.26289i 0
113.2 0 −1.69134 + 0.373340i 0 0.568907i 0 −0.718465 0 2.72123 1.26289i 0
113.3 0 −0.729858 1.57077i 0 1.22502i 0 2.80880 0 −1.93462 + 2.29287i 0
113.4 0 −0.729858 + 1.57077i 0 1.22502i 0 2.80880 0 −1.93462 2.29287i 0
113.5 0 −0.171092 1.72358i 0 3.81594i 0 −2.25057 0 −2.94146 + 0.589781i 0
113.6 0 −0.171092 + 1.72358i 0 3.81594i 0 −2.25057 0 −2.94146 0.589781i 0
113.7 0 1.39399 1.02801i 0 1.12291i 0 −3.21832 0 0.886389 2.86606i 0
113.8 0 1.39399 + 1.02801i 0 1.12291i 0 −3.21832 0 0.886389 + 2.86606i 0
113.9 0 1.69830 0.340259i 0 3.78858i 0 4.37856 0 2.76845 1.15572i 0
113.10 0 1.69830 + 0.340259i 0 3.78858i 0 4.37856 0 2.76845 + 1.15572i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.f.i 10
3.b odd 2 1 912.2.f.h 10
4.b odd 2 1 456.2.f.a 10
12.b even 2 1 456.2.f.b yes 10
19.b odd 2 1 912.2.f.h 10
57.d even 2 1 inner 912.2.f.i 10
76.d even 2 1 456.2.f.b yes 10
228.b odd 2 1 456.2.f.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.2.f.a 10 4.b odd 2 1
456.2.f.a 10 228.b odd 2 1
456.2.f.b yes 10 12.b even 2 1
456.2.f.b yes 10 76.d even 2 1
912.2.f.h 10 3.b odd 2 1
912.2.f.h 10 19.b odd 2 1
912.2.f.i 10 1.a even 1 1 trivial
912.2.f.i 10 57.d 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 \(S_{2}^{\mathrm{new}}(912, [\chi])\):

\( T_{5}^{10} + 32T_{5}^{8} + 301T_{5}^{6} + 726T_{5}^{4} + 600T_{5}^{2} + 128 \) Copy content Toggle raw display
\( T_{7}^{5} - T_{7}^{4} - 21T_{7}^{3} + T_{7}^{2} + 100T_{7} + 64 \) Copy content Toggle raw display
\( T_{29}^{5} + 3T_{29}^{4} - 52T_{29}^{3} - 140T_{29}^{2} + 608T_{29} + 1216 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} - T^{8} - 2 T^{7} - 2 T^{6} + \cdots + 243 \) Copy content Toggle raw display
$5$ \( T^{10} + 32 T^{8} + 301 T^{6} + \cdots + 128 \) Copy content Toggle raw display
$7$ \( (T^{5} - T^{4} - 21 T^{3} + T^{2} + 100 T + 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 50 T^{8} + 889 T^{6} + \cdots + 41472 \) Copy content Toggle raw display
$13$ \( T^{10} + 57 T^{8} + 1014 T^{6} + \cdots + 128 \) Copy content Toggle raw display
$17$ \( T^{10} + 101 T^{8} + 3391 T^{6} + \cdots + 524288 \) Copy content Toggle raw display
$19$ \( T^{10} + 2 T^{9} + 3 T^{8} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} + 129 T^{8} + 5314 T^{6} + \cdots + 46208 \) Copy content Toggle raw display
$29$ \( (T^{5} + 3 T^{4} - 52 T^{3} - 140 T^{2} + \cdots + 1216)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 134 T^{8} + 6028 T^{6} + \cdots + 107648 \) Copy content Toggle raw display
$37$ \( T^{10} + 174 T^{8} + \cdots + 20891648 \) Copy content Toggle raw display
$41$ \( (T^{5} + 2 T^{4} - 58 T^{3} + 52 T^{2} + \cdots - 256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 10 T^{4} - 59 T^{3} - 616 T^{2} + \cdots - 432)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + 204 T^{8} + 12093 T^{6} + \cdots + 30752 \) Copy content Toggle raw display
$53$ \( (T^{5} + 13 T^{4} - 22 T^{3} - 798 T^{2} + \cdots + 1432)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + T^{4} - 206 T^{3} - 42 T^{2} + \cdots - 18336)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + 2 T^{4} - 123 T^{3} + 16 T^{2} + \cdots + 3488)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 471 T^{8} + \cdots + 757071872 \) Copy content Toggle raw display
$71$ \( (T^{5} + 4 T^{4} - 194 T^{3} + 48 T^{2} + \cdots + 7936)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 13 T^{4} - 75 T^{3} - 1837 T^{2} + \cdots - 13752)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 486 T^{8} + \cdots + 11829248 \) Copy content Toggle raw display
$83$ \( T^{10} + 240 T^{8} + 17552 T^{6} + \cdots + 2654208 \) Copy content Toggle raw display
$89$ \( (T^{5} - 2 T^{4} - 222 T^{3} + 1044 T^{2} + \cdots - 432)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 620 T^{8} + \cdots + 169869312 \) Copy content Toggle raw display
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