Properties

Label 1856.4.a.bl
Level $1856$
Weight $4$
Character orbit 1856.a
Self dual yes
Analytic conductor $109.508$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(0\)
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: no (minimal twist has level 928)
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 \( q + (\beta_1 + 1) q^{3} + ( - \beta_{4} + 1) q^{5} + (\beta_{6} - 4) q^{7} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1 + 11) q^{9} + ( - \beta_{7} + \beta_{3} + 2 \beta_1 + 3) q^{11} + (\beta_{11} + \beta_{8} + \beta_{6} + 3) q^{13} + (\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - 2 \beta_{7} + \beta_{6} - 3 \beta_{4} + 2 \beta_1 + 4) q^{15} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{17} + ( - \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + 2 \beta_{3} + \cdots + 12) q^{19}+ \cdots + (11 \beta_{11} - 23 \beta_{10} - 3 \beta_{9} + 3 \beta_{8} - 10 \beta_{7} + \cdots + 567) q^{99}+O(q^{100}) \) 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
−10.7228
−6.97229
−6.04158
−3.34435
−2.71183
−0.676012
0.613335
4.35906
5.24549
5.28045
8.22268
8.74783
0 −9.72277 0 6.27199 0 −31.0130 0 67.5323 0
1.2 0 −5.97229 0 11.6005 0 −8.29658 0 8.66827 0
1.3 0 −5.04158 0 −10.9552 0 1.11655 0 −1.58244 0
1.4 0 −2.34435 0 −11.5573 0 −8.54075 0 −21.5040 0
1.5 0 −1.71183 0 −8.28780 0 9.40926 0 −24.0696 0
1.6 0 0.323988 0 5.78413 0 34.7780 0 −26.8950 0
1.7 0 1.61334 0 10.5322 0 −13.7418 0 −24.3972 0
1.8 0 5.35906 0 20.7045 0 26.2041 0 1.71955 0
1.9 0 6.24549 0 −5.77776 0 −10.3273 0 12.0061 0
1.10 0 6.28045 0 −18.9942 0 −32.3083 0 12.4440 0
1.11 0 9.22268 0 −9.91795 0 18.1458 0 58.0578 0
1.12 0 9.74783 0 20.5968 0 −29.4259 0 68.0201 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 1856.4.a.bl 12
4.b odd 2 1 1856.4.a.bj 12
8.b even 2 1 928.4.a.h 12
8.d odd 2 1 928.4.a.j yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.h 12 8.b even 2 1
928.4.a.j yes 12 8.d odd 2 1
1856.4.a.bj 12 4.b odd 2 1
1856.4.a.bl 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1856))\):

\( T_{3}^{12} - 14 T_{3}^{11} - 129 T_{3}^{10} + 2360 T_{3}^{9} + 2402 T_{3}^{8} - 122236 T_{3}^{7} + 113150 T_{3}^{6} + 2483536 T_{3}^{5} - 3164363 T_{3}^{4} - 18457158 T_{3}^{3} + 11333891 T_{3}^{2} + \cdots - 11605016 \) Copy content Toggle raw display
\( T_{5}^{12} - 10 T_{5}^{11} - 943 T_{5}^{10} + 6652 T_{5}^{9} + 331222 T_{5}^{8} - 1372684 T_{5}^{7} - 54452022 T_{5}^{6} + 112129888 T_{5}^{5} + 4379523877 T_{5}^{4} - 3397758554 T_{5}^{3} + \cdots + 2158837575300 \) Copy content Toggle raw display
\( T_{7}^{12} + 44 T_{7}^{11} - 1844 T_{7}^{10} - 100512 T_{7}^{9} + 566608 T_{7}^{8} + 69463360 T_{7}^{7} + 376753216 T_{7}^{6} - 15344836096 T_{7}^{5} - 169954399488 T_{7}^{4} + \cdots - 51510514450432 \) 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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