Properties

Label 928.4.a.h
Level $928$
Weight $4$
Character orbit 928.a
Self dual yes
Analytic conductor $54.754$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 928 = 2^{5} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 928.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(54.7537724853\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + 7106592 x^{4} - 7979328 x^{3} - 38912400 x^{2} + \cdots + 14751072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{4} - 1) q^{5} + (\beta_{6} - 4) q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1 + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + (\beta_{4} - 1) q^{5} + (\beta_{6} - 4) q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1 + 11) q^{9} + (\beta_{7} - \beta_{3} - 2 \beta_1 - 3) q^{11} + ( - \beta_{11} - \beta_{8} - \beta_{6} - 3) q^{13} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} - 3 \beta_{4} + 2 \beta_1 + 4) q^{15} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{17} + (\beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} + \cdots - 12) q^{19}+ \cdots + ( - 11 \beta_{11} + 23 \beta_{10} + 3 \beta_{9} - 3 \beta_{8} + 10 \beta_{7} + \cdots - 567) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 14 q^{3} - 10 q^{5} - 44 q^{7} + 130 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 14 q^{3} - 10 q^{5} - 44 q^{7} + 130 q^{9} - 46 q^{11} - 34 q^{13} + 50 q^{15} + 36 q^{17} - 148 q^{19} - 92 q^{21} - 328 q^{23} + 486 q^{25} - 326 q^{27} + 348 q^{29} - 374 q^{31} + 710 q^{33} - 204 q^{35} - 340 q^{37} + 122 q^{39} + 32 q^{41} - 462 q^{43} - 1132 q^{45} - 434 q^{47} + 1508 q^{49} - 440 q^{51} + 610 q^{53} - 46 q^{55} - 932 q^{57} - 1240 q^{59} - 1228 q^{61} - 4240 q^{63} + 730 q^{65} - 1672 q^{67} - 528 q^{69} - 3220 q^{71} + 564 q^{73} - 6032 q^{75} + 644 q^{77} - 1862 q^{79} + 3040 q^{81} - 3736 q^{83} - 808 q^{85} - 406 q^{87} + 584 q^{89} - 4844 q^{91} - 3226 q^{93} - 2844 q^{95} + 904 q^{97} - 6832 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 217 x^{10} + 520 x^{9} + 16022 x^{8} - 37368 x^{7} - 509640 x^{6} + 989168 x^{5} + 7106592 x^{4} - 7979328 x^{3} - 38912400 x^{2} + \cdots + 14751072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 8262965802853 \nu^{11} + 902081932283552 \nu^{10} + \cdots - 13\!\cdots\!68 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9923410960919 \nu^{11} - 37676872998341 \nu^{10} + \cdots - 24\!\cdots\!64 ) / 18\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 31658966007779 \nu^{11} + 107796402154664 \nu^{10} + \cdots - 11\!\cdots\!72 ) / 37\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 79714148233592 \nu^{11} + 244066653715127 \nu^{10} + \cdots - 37\!\cdots\!08 ) / 94\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17168595976539 \nu^{11} - 61050049383782 \nu^{10} + \cdots + 69\!\cdots\!20 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29213196068751 \nu^{11} + 128904048786731 \nu^{10} + \cdots - 13\!\cdots\!64 ) / 10\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 558861445914547 \nu^{11} + \cdots + 38\!\cdots\!28 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 717031054281463 \nu^{11} + \cdots + 26\!\cdots\!32 ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 520026572887418 \nu^{11} + \cdots - 12\!\cdots\!72 ) / 94\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 550701841585907 \nu^{11} + \cdots - 17\!\cdots\!48 ) / 62\!\cdots\!80 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} - \beta_{4} + \beta_{3} + 37 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3 \beta_{11} - 3 \beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{7} + 3 \beta_{6} + 6 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + 65 \beta _1 - 34 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 12 \beta_{11} + 4 \beta_{10} - 11 \beta_{9} - 24 \beta_{8} + \beta_{7} - 97 \beta_{6} + 7 \beta_{5} - 132 \beta_{4} + 113 \beta_{3} + 18 \beta_{2} - 84 \beta _1 + 2524 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 403 \beta_{11} - 335 \beta_{10} + 208 \beta_{9} + 215 \beta_{8} - 218 \beta_{7} + 447 \beta_{6} - 43 \beta_{5} + 819 \beta_{4} - 568 \beta_{3} - 304 \beta_{2} + 5316 \beta _1 - 6939 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2178 \beta_{11} + 908 \beta_{10} - 2191 \beta_{9} - 3382 \beta_{8} + 199 \beta_{7} - 9563 \beta_{6} + 1041 \beta_{5} - 14650 \beta_{4} + 12069 \beta_{3} + 2886 \beta_{2} - 16672 \beta _1 + 216826 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 45353 \beta_{11} - 34219 \beta_{10} + 27832 \beta_{9} + 30467 \beta_{8} - 20172 \beta_{7} + 58231 \beta_{6} - 8569 \beta_{5} + 102333 \beta_{4} - 81610 \beta_{3} - 38376 \beta_{2} + 492164 \beta _1 - 1017961 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 305376 \beta_{11} + 148482 \beta_{10} - 300869 \beta_{9} - 401240 \beta_{8} + 38209 \beta_{7} - 990991 \beta_{6} + 123045 \beta_{5} - 1595212 \beta_{4} + 1301983 \beta_{3} + \cdots + 20994312 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4965951 \beta_{11} - 3534029 \beta_{10} + 3361826 \beta_{9} + 3819385 \beta_{8} - 1875142 \beta_{7} + 7243335 \beta_{6} - 1203425 \beta_{5} + 12496687 \beta_{4} + \cdots - 134215891 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 38971046 \beta_{11} + 20764400 \beta_{10} - 36942863 \beta_{9} - 45988898 \beta_{8} + 6147163 \beta_{7} - 106364455 \beta_{6} + 13991589 \beta_{5} - 174987162 \beta_{4} + \cdots + 2172946166 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 546331693 \beta_{11} - 373781107 \beta_{10} + 393952724 \beta_{9} + 457049227 \beta_{8} - 181642712 \beta_{7} + 878676335 \beta_{6} - 149963201 \beta_{5} + \cdots - 16780997049 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
8.74783
8.22268
5.28045
5.24549
4.35906
0.613335
−0.676012
−2.71183
−3.34435
−6.04158
−6.97229
−10.7228
0 −9.74783 0 −20.5968 0 −29.4259 0 68.0201 0
1.2 0 −9.22268 0 9.91795 0 18.1458 0 58.0578 0
1.3 0 −6.28045 0 18.9942 0 −32.3083 0 12.4440 0
1.4 0 −6.24549 0 5.77776 0 −10.3273 0 12.0061 0
1.5 0 −5.35906 0 −20.7045 0 26.2041 0 1.71955 0
1.6 0 −1.61334 0 −10.5322 0 −13.7418 0 −24.3972 0
1.7 0 −0.323988 0 −5.78413 0 34.7780 0 −26.8950 0
1.8 0 1.71183 0 8.28780 0 9.40926 0 −24.0696 0
1.9 0 2.34435 0 11.5573 0 −8.54075 0 −21.5040 0
1.10 0 5.04158 0 10.9552 0 1.11655 0 −1.58244 0
1.11 0 5.97229 0 −11.6005 0 −8.29658 0 8.66827 0
1.12 0 9.72277 0 −6.27199 0 −31.0130 0 67.5323 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(29\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 928.4.a.h 12
4.b odd 2 1 928.4.a.j yes 12
8.b even 2 1 1856.4.a.bl 12
8.d odd 2 1 1856.4.a.bj 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.h 12 1.a even 1 1 trivial
928.4.a.j yes 12 4.b odd 2 1
1856.4.a.bj 12 8.d odd 2 1
1856.4.a.bl 12 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 14 T_{3}^{11} - 129 T_{3}^{10} - 2360 T_{3}^{9} + 2402 T_{3}^{8} + 122236 T_{3}^{7} + 113150 T_{3}^{6} - 2483536 T_{3}^{5} - 3164363 T_{3}^{4} + 18457158 T_{3}^{3} + 11333891 T_{3}^{2} + \cdots - 11605016 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(928))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 14 T^{11} - 129 T^{10} + \cdots - 11605016 \) Copy content Toggle raw display
$5$ \( T^{12} + 10 T^{11} + \cdots + 2158837575300 \) Copy content Toggle raw display
$7$ \( T^{12} + 44 T^{11} + \cdots - 51510514450432 \) Copy content Toggle raw display
$11$ \( T^{12} + 46 T^{11} + \cdots - 51\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{12} + 34 T^{11} + \cdots + 49\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{12} - 36 T^{11} + \cdots + 95\!\cdots\!48 \) Copy content Toggle raw display
$19$ \( T^{12} + 148 T^{11} + \cdots - 91\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( T^{12} + 328 T^{11} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T - 29)^{12} \) Copy content Toggle raw display
$31$ \( T^{12} + 374 T^{11} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + 340 T^{11} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{12} - 32 T^{11} + \cdots + 67\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{12} + 462 T^{11} + \cdots - 12\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{12} + 434 T^{11} + \cdots + 46\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{12} - 610 T^{11} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{12} + 1240 T^{11} + \cdots + 40\!\cdots\!72 \) Copy content Toggle raw display
$61$ \( T^{12} + 1228 T^{11} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$67$ \( T^{12} + 1672 T^{11} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{12} + 3220 T^{11} + \cdots - 50\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{12} - 564 T^{11} + \cdots - 76\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + 1862 T^{11} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + 3736 T^{11} + \cdots - 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} - 584 T^{11} + \cdots + 74\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} - 904 T^{11} + \cdots - 53\!\cdots\!72 \) Copy content Toggle raw display
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