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Properties

Label 722.6.a.q
Level $722$
Weight $6$
Character orbit 722.a
Self dual yes
Analytic conductor $115.797$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [722,6,Mod(1,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 722.a (trivial)

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: yes
Analytic conductor: \(115.797117905\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2871 x^{13} - 4674 x^{12} + 3170019 x^{11} + 9081402 x^{10} - 1680307373 x^{9} + \cdots - 34\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 19^{6} \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + \beta_1 q^{3} + 16 q^{4} + (\beta_{5} + 7) q^{5} - 4 \beta_1 q^{6} + ( - \beta_{10} + \beta_{4} - \beta_1 + 6) q^{7} - 64 q^{8} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 142) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + \beta_1 q^{3} + 16 q^{4} + (\beta_{5} + 7) q^{5} - 4 \beta_1 q^{6} + ( - \beta_{10} + \beta_{4} - \beta_1 + 6) q^{7} - 64 q^{8} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 142) q^{9} + ( - 4 \beta_{5} - 28) q^{10} + ( - \beta_{13} + \beta_{6} + \beta_{4} + \cdots + 8) q^{11}+ \cdots + (173 \beta_{14} - 84 \beta_{13} + \cdots - 10062) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 60 q^{2} + 240 q^{4} + 108 q^{5} + 84 q^{7} - 960 q^{8} + 2127 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 60 q^{2} + 240 q^{4} + 108 q^{5} + 84 q^{7} - 960 q^{8} + 2127 q^{9} - 432 q^{10} + 126 q^{11} + 114 q^{13} - 336 q^{14} + 3840 q^{16} + 4119 q^{17} - 8508 q^{18} + 1728 q^{20} - 3408 q^{21} - 504 q^{22} + 3936 q^{23} + 26895 q^{25} - 456 q^{26} + 13017 q^{27} + 1344 q^{28} - 14658 q^{29} - 6840 q^{31} - 15360 q^{32} + 3945 q^{33} - 16476 q^{34} + 12636 q^{35} + 34032 q^{36} + 4278 q^{37} + 4956 q^{39} - 6912 q^{40} - 5112 q^{41} + 13632 q^{42} + 94191 q^{43} + 2016 q^{44} + 31770 q^{45} - 15744 q^{46} + 702 q^{47} + 63777 q^{49} - 107580 q^{50} + 108 q^{51} + 1824 q^{52} - 47544 q^{53} - 52068 q^{54} + 16848 q^{55} - 5376 q^{56} + 58632 q^{58} + 8832 q^{59} + 119196 q^{61} + 27360 q^{62} - 88068 q^{63} + 61440 q^{64} - 80646 q^{65} - 15780 q^{66} - 64248 q^{67} + 65904 q^{68} - 124224 q^{69} - 50544 q^{70} + 53364 q^{71} - 136128 q^{72} - 4908 q^{73} - 17112 q^{74} + 87480 q^{75} + 121218 q^{77} - 19824 q^{78} + 115500 q^{79} + 27648 q^{80} + 481659 q^{81} + 20448 q^{82} + 201630 q^{83} - 54528 q^{84} - 150282 q^{85} - 376764 q^{86} + 376512 q^{87} - 8064 q^{88} + 101505 q^{89} - 127080 q^{90} - 414918 q^{91} + 62976 q^{92} + 165960 q^{93} - 2808 q^{94} - 297114 q^{97} - 255108 q^{98} - 149895 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2871 x^{13} - 4674 x^{12} + 3170019 x^{11} + 9081402 x^{10} - 1680307373 x^{9} + \cdots - 34\!\cdots\!72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 40\!\cdots\!68 \nu^{14} + \cdots - 12\!\cdots\!60 ) / 31\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 46\!\cdots\!99 \nu^{14} + \cdots + 34\!\cdots\!66 ) / 10\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 33\!\cdots\!43 \nu^{14} + \cdots + 16\!\cdots\!32 ) / 33\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40\!\cdots\!68 \nu^{14} + \cdots - 12\!\cdots\!60 ) / 16\!\cdots\!74 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12\!\cdots\!29 \nu^{14} + \cdots - 15\!\cdots\!75 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67\!\cdots\!99 \nu^{14} + \cdots + 12\!\cdots\!64 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14\!\cdots\!80 \nu^{14} + \cdots - 81\!\cdots\!22 ) / 20\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21\!\cdots\!75 \nu^{14} + \cdots + 48\!\cdots\!08 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 92\!\cdots\!91 \nu^{14} + \cdots - 23\!\cdots\!28 ) / 41\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!83 \nu^{14} + \cdots - 10\!\cdots\!72 ) / 61\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 76\!\cdots\!06 \nu^{14} + \cdots - 82\!\cdots\!32 ) / 20\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!99 \nu^{14} + \cdots + 17\!\cdots\!37 ) / 30\!\cdots\!67 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 78\!\cdots\!03 \nu^{14} + \cdots + 51\!\cdots\!60 ) / 12\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!47 \nu^{14} + \cdots - 16\!\cdots\!00 ) / 20\!\cdots\!78 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{4} + 19\beta_1 ) / 19 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{12} - 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \cdots + 7273 ) / 19 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 97 \beta_{14} - 38 \beta_{13} + 96 \beta_{12} + 30 \beta_{11} - 186 \beta_{10} - 85 \beta_{9} + \cdots + 17922 ) / 19 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1360 \beta_{14} + 190 \beta_{13} + 4214 \beta_{12} - 3954 \beta_{11} + 5642 \beta_{10} + \cdots + 4817020 ) / 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 108532 \beta_{14} - 34010 \beta_{13} + 131916 \beta_{12} + 27512 \beta_{11} - 132077 \beta_{10} + \cdots + 27567731 ) / 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1819163 \beta_{14} + 348536 \beta_{13} + 5433359 \beta_{12} - 4224877 \beta_{11} + 6529751 \beta_{10} + \cdots + 3629933860 ) / 19 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 106782040 \beta_{14} - 25659101 \beta_{13} + 160647482 \beta_{12} + 20459193 \beta_{11} + \cdots + 32796875225 ) / 19 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2101820106 \beta_{14} + 486960082 \beta_{13} + 6208915457 \beta_{12} - 3915486941 \beta_{11} + \cdots + 2910272012244 ) / 19 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 103318575512 \beta_{14} - 17505847961 \beta_{13} + 184253758157 \beta_{12} + 12758261518 \beta_{11} + \cdots + 35585553569167 ) / 19 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2314506788037 \beta_{14} + 611868793642 \beta_{13} + 6718510062981 \beta_{12} + \cdots + 24\!\cdots\!60 ) / 19 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 100447494063224 \beta_{14} - 9761809753832 \beta_{13} + 202526285762246 \beta_{12} + \cdots + 37\!\cdots\!67 ) / 19 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 24\!\cdots\!15 \beta_{14} + 721423171444289 \beta_{13} + \cdots + 21\!\cdots\!99 ) / 19 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 98\!\cdots\!03 \beta_{14} + \cdots + 37\!\cdots\!94 ) / 19 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 26\!\cdots\!83 \beta_{14} + \cdots + 18\!\cdots\!36 ) / 19 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−27.8974
−26.2549
−23.5180
−20.7673
−12.5784
−10.7498
−3.66933
−1.51331
3.01062
9.74479
10.3629
18.3407
25.8045
28.1551
31.5298
−4.00000 −26.3653 16.0000 84.5221 105.461 15.2607 −64.0000 452.132 −338.089
1.2 −4.00000 −25.9076 16.0000 −18.4597 103.630 −122.093 −64.0000 428.205 73.8387
1.3 −4.00000 −25.3974 16.0000 −93.4251 101.589 −182.176 −64.0000 402.026 373.700
1.4 −4.00000 −22.6466 16.0000 79.6824 90.5866 191.779 −64.0000 269.871 −318.730
1.5 −4.00000 −11.0464 16.0000 −52.2115 44.1854 215.645 −64.0000 −120.978 208.846
1.6 −4.00000 −10.4025 16.0000 95.5276 41.6100 232.219 −64.0000 −134.788 −382.111
1.7 −4.00000 −2.13724 16.0000 −10.7945 8.54896 −195.108 −64.0000 −238.432 43.1779
1.8 −4.00000 −1.16601 16.0000 −65.1319 4.66404 94.5243 −64.0000 −241.640 260.528
1.9 −4.00000 1.13124 16.0000 11.4222 −4.52494 15.4828 −64.0000 −241.720 −45.6887
1.10 −4.00000 7.86541 16.0000 −51.1536 −31.4616 18.3720 −64.0000 −181.135 204.615
1.11 −4.00000 10.7102 16.0000 59.4057 −42.8408 −230.434 −64.0000 −128.291 −237.623
1.12 −4.00000 19.8728 16.0000 105.633 −79.4911 36.4788 −64.0000 151.927 −422.533
1.13 −4.00000 27.3366 16.0000 −91.1494 −109.346 45.5647 −64.0000 504.290 364.597
1.14 −4.00000 28.5024 16.0000 −35.3418 −114.010 85.8987 −64.0000 569.386 141.367
1.15 −4.00000 29.6504 16.0000 89.4742 −118.602 −137.413 −64.0000 636.149 −357.897
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(-1\)

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 722.6.a.q 15
19.b odd 2 1 722.6.a.r 15
19.f odd 18 2 38.6.e.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.6.e.b ✓ 30 19.f odd 18 2
722.6.a.q 15 1.a even 1 1 trivial
722.6.a.r 15 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 2886 T_{3}^{13} - 4339 T_{3}^{12} + 3213570 T_{3}^{11} + 8713875 T_{3}^{10} + \cdots - 49\!\cdots\!71 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 49\!\cdots\!71 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 43\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 77\!\cdots\!12 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 66\!\cdots\!17 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 17\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 45\!\cdots\!11 \) Copy content Toggle raw display
$19$ \( T^{15} \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 50\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 53\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 42\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 71\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 92\!\cdots\!87 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 60\!\cdots\!87 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 33\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 31\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 11\!\cdots\!77 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 15\!\cdots\!32 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 19\!\cdots\!12 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 34\!\cdots\!44 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 54\!\cdots\!23 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 81\!\cdots\!39 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 55\!\cdots\!19 \) Copy content Toggle raw display
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