Properties

Label 115.6.a.c
Level $115$
Weight $6$
Character orbit 115.a
Self dual yes
Analytic conductor $18.444$
Analytic rank $0$
Dimension $7$
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] = [115,6,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.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: \(18.4441392785\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - \beta_1 + 26) q^{4} - 25 q^{5} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \cdots - 54) q^{6}+ \cdots + ( - 4 \beta_{5} + 2 \beta_{4} + \cdots + 70) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - \beta_1 + 26) q^{4} - 25 q^{5} + (3 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + \cdots - 54) q^{6}+ \cdots + ( - 519 \beta_{6} - 552 \beta_{5} + \cdots + 32707) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 4 q^{2} - 3 q^{3} + 178 q^{4} - 175 q^{5} - 381 q^{6} + 33 q^{7} + 546 q^{8} + 440 q^{9} - 100 q^{10} + 1373 q^{11} - 285 q^{12} + 605 q^{13} + 1317 q^{14} + 75 q^{15} + 3770 q^{16} + 2505 q^{17} + 7971 q^{18} - 115 q^{19} - 4450 q^{20} + 608 q^{21} + 2977 q^{22} + 3703 q^{23} - 12447 q^{24} + 4375 q^{25} + 9379 q^{26} - 12276 q^{27} + 5777 q^{28} + 2440 q^{29} + 9525 q^{30} + 13565 q^{31} + 14086 q^{32} + 10519 q^{33} + 26997 q^{34} - 825 q^{35} + 79889 q^{36} + 9414 q^{37} + 28717 q^{38} - 21738 q^{39} - 13650 q^{40} + 13725 q^{41} + 12426 q^{42} + 76694 q^{43} + 55203 q^{44} - 11000 q^{45} + 2116 q^{46} + 59692 q^{47} - 32985 q^{48} - 53608 q^{49} + 2500 q^{50} - 24725 q^{51} + 61195 q^{52} + 49536 q^{53} - 156168 q^{54} - 34325 q^{55} - 54461 q^{56} - 7580 q^{57} - 95562 q^{58} + 44536 q^{59} + 7125 q^{60} - 49097 q^{61} - 25763 q^{62} - 3578 q^{63} - 18654 q^{64} - 15125 q^{65} - 201873 q^{66} + 788 q^{67} + 163845 q^{68} - 1587 q^{69} - 32925 q^{70} + 49521 q^{71} + 328503 q^{72} - 3760 q^{73} + 88170 q^{74} - 1875 q^{75} - 411465 q^{76} + 77728 q^{77} - 389832 q^{78} + 918 q^{79} - 94250 q^{80} + 121235 q^{81} - 227459 q^{82} + 99202 q^{83} + 336602 q^{84} - 62625 q^{85} + 24584 q^{86} - 38666 q^{87} - 201275 q^{88} - 141676 q^{89} - 199275 q^{90} - 223605 q^{91} + 94162 q^{92} + 51412 q^{93} - 354292 q^{94} + 2875 q^{95} - 592095 q^{96} + 28731 q^{97} - 149557 q^{98} + 237333 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 196x^{5} + 464x^{4} + 11003x^{3} - 21041x^{2} - 142416x + 243340 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{6} - 631\nu^{5} + 1662\nu^{4} + 73276\nu^{3} - 253039\nu^{2} - 1466045\nu + 3713902 ) / 178848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 57 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{6} - 71\nu^{5} - 1986\nu^{4} + 13772\nu^{3} + 23512\nu^{2} - 708589\nu + 1154738 ) / 44712 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -13\nu^{6} - 7\nu^{5} + 1950\nu^{4} + 5836\nu^{3} - 68887\nu^{2} - 377597\nu + 320206 ) / 19872 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 343\nu^{6} - 707\nu^{5} - 64698\nu^{4} + 62828\nu^{3} + 3331141\nu^{2} - 191713\nu - 29553850 ) / 178848 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 57 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + 4\beta_{4} + 2\beta_{3} - 2\beta_{2} + 87\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -10\beta_{6} - 16\beta_{5} + 24\beta_{4} + 121\beta_{3} + 2\beta_{2} + 203\beta _1 + 4841 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -52\beta_{6} + 136\beta_{5} + 616\beta_{4} + 386\beta_{3} - 516\beta_{2} + 8447\beta _1 + 6024 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1472\beta_{6} - 3104\beta_{5} + 5064\beta_{4} + 13541\beta_{3} - 320\beta_{2} + 30613\beta _1 + 454473 \) 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.

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
10.7957
8.76815
4.33855
1.66253
−4.26829
−8.71037
−9.58627
−9.79570 20.6902 63.9557 −25.0000 −202.675 61.8409 −313.028 185.085 244.892
1.2 −7.76815 −8.56566 28.3442 −25.0000 66.5393 −61.0525 28.3987 −169.630 194.204
1.3 −3.33855 13.0297 −20.8541 −25.0000 −43.5004 −186.649 176.456 −73.2260 83.4637
1.4 −0.662532 −4.47473 −31.5611 −25.0000 2.96465 91.5985 42.1112 −222.977 16.5633
1.5 5.26829 −11.8410 −4.24511 −25.0000 −62.3818 72.8708 −190.950 −102.791 −131.707
1.6 9.71037 18.9751 62.2913 −25.0000 184.255 84.0805 294.140 117.055 −242.759
1.7 10.5863 −30.8137 80.0691 −25.0000 −326.202 −29.6892 508.872 706.483 −264.657
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)
\(23\) \(-1\)

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 115.6.a.c 7
3.b odd 2 1 1035.6.a.b 7
5.b even 2 1 575.6.a.d 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.6.a.c 7 1.a even 1 1 trivial
575.6.a.d 7 5.b even 2 1
1035.6.a.b 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 4T_{2}^{6} - 193T_{2}^{5} + 526T_{2}^{4} + 10874T_{2}^{3} - 12768T_{2}^{2} - 150624T_{2} - 91152 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(115))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 4 T^{6} + \cdots - 91152 \) Copy content Toggle raw display
$3$ \( T^{7} + 3 T^{6} + \cdots - 71539200 \) Copy content Toggle raw display
$5$ \( (T + 25)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots + 11741977345536 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 12\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots - 25\!\cdots\!28 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 25\!\cdots\!20 \) Copy content Toggle raw display
$19$ \( T^{7} + \cdots + 53\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( (T - 529)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots - 11\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 23\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots + 67\!\cdots\!90 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 14\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 14\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots - 17\!\cdots\!52 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 20\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 72\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 77\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 16\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
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