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Properties

Label 115.3.d.b
Level $115$
Weight $3$
Character orbit 115.d
Analytic conductor $3.134$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,3,Mod(91,115)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.91");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 115.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(3.13352304014\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{7} q^{3} + (\beta_{6} + 3) q^{4} + \beta_{5} q^{5} + (\beta_{9} + \beta_{6} + 2) q^{6} + (\beta_{5} + \beta_{3}) q^{7} + ( - \beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{8}+ \cdots + (\beta_{9} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{7} q^{3} + (\beta_{6} + 3) q^{4} + \beta_{5} q^{5} + (\beta_{9} + \beta_{6} + 2) q^{6} + (\beta_{5} + \beta_{3}) q^{7} + ( - \beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{8}+ \cdots + ( - 5 \beta_{8} + 4 \beta_{5} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} - 2 q^{3} + 34 q^{4} + 28 q^{6} - 20 q^{8} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{2} - 2 q^{3} + 34 q^{4} + 28 q^{6} - 20 q^{8} - 16 q^{9} - 24 q^{12} - 2 q^{13} - 38 q^{16} - 22 q^{18} + 44 q^{23} + 70 q^{24} - 50 q^{25} - 72 q^{26} + 40 q^{27} - 46 q^{29} + 16 q^{31} + 142 q^{32} - 60 q^{35} - 110 q^{36} + 72 q^{39} - 84 q^{41} + 86 q^{46} + 112 q^{47} - 236 q^{48} + 50 q^{49} - 10 q^{50} + 52 q^{52} + 30 q^{54} - 10 q^{55} + 246 q^{58} - 262 q^{59} - 136 q^{62} - 224 q^{64} + 124 q^{69} + 130 q^{70} + 236 q^{71} - 406 q^{72} + 168 q^{73} + 10 q^{75} + 300 q^{77} - 254 q^{78} - 258 q^{81} + 140 q^{82} + 540 q^{87} - 182 q^{92} + 100 q^{93} + 262 q^{94} - 90 q^{95} - 332 q^{96} + 388 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} - 10 x^{8} + 34 x^{7} + 346 x^{6} - 968 x^{5} + 165 x^{4} + 6972 x^{3} + 19344 x^{2} + \cdots + 225444 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 18413999298 \nu^{9} - 6940583162233 \nu^{8} - 2184040960024 \nu^{7} + \cdots - 64\!\cdots\!66 ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17276778863 \nu^{9} - 63973457098 \nu^{8} + 259306204096 \nu^{7} - 287663549294 \nu^{6} + \cdots - 11\!\cdots\!66 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3178911074 \nu^{9} + 78987068911 \nu^{8} + 430137087328 \nu^{7} - 1072934217052 \nu^{6} + \cdots + 34\!\cdots\!42 ) / 251783039496280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27126561061 \nu^{9} + 36976343259 \nu^{8} + 207292153512 \nu^{7} - 662996871978 \nu^{6} + \cdots + 16\!\cdots\!88 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27126561061 \nu^{9} + 36976343259 \nu^{8} + 207292153512 \nu^{7} - 662996871978 \nu^{6} + \cdots + 16\!\cdots\!88 ) / 14\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 17276778863 \nu^{9} - 63973457098 \nu^{8} + 259306204096 \nu^{7} + \cdots - 25\!\cdots\!76 ) / 723876238551805 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 468602611879 \nu^{9} + 217704046159 \nu^{8} - 6293663580928 \nu^{7} - 12414624463218 \nu^{6} + \cdots - 20\!\cdots\!62 ) / 57\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 113495858954 \nu^{9} + 178588851821 \nu^{8} + 1093879455048 \nu^{7} + \cdots + 82\!\cdots\!42 ) / 723876238551805 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41025221057 \nu^{9} + 13161849303 \nu^{8} + 712152635904 \nu^{7} + \cdots + 15\!\cdots\!06 ) / 251783039496280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + 3\beta_{8} - \beta_{7} + \beta_{6} - 16\beta_{5} - 6\beta_{4} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{8} + 4\beta_{7} - 16\beta_{6} + 8\beta_{5} + 3\beta_{4} + 4\beta_{3} - 16\beta_{2} + 4\beta _1 - 123 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 32\beta_{9} + 20\beta_{8} + 36\beta_{7} - 37\beta_{6} + 15\beta_{5} - 235\beta_{4} - 20\beta_{3} - 15\beta_{2} + 85 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} - 22 \beta_{8} - 223 \beta_{7} - 504 \beta_{6} - 110 \beta_{5} - 160 \beta_{4} - 16 \beta_{3} + \cdots - 1536 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 767 \beta_{9} - 728 \beta_{8} + 419 \beta_{7} - 334 \beta_{6} + 3002 \beta_{5} - 1497 \beta_{4} + \cdots - 215 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 120 \beta_{9} - 992 \beta_{8} - 2696 \beta_{7} - 911 \beta_{6} - 1080 \beta_{5} - 2834 \beta_{4} + \cdots + 5128 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 4673 \beta_{9} - 13959 \beta_{8} - 4049 \beta_{7} + 4271 \beta_{6} + 35288 \beta_{5} + 33546 \beta_{4} + \cdots - 18144 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
91.1
−3.34025 + 2.23607i
−3.34025 − 2.23607i
−2.67869 + 2.23607i
−2.67869 − 2.23607i
1.32878 + 2.23607i
1.32878 − 2.23607i
2.24810 + 2.23607i
2.24810 − 2.23607i
3.44206 + 2.23607i
3.44206 − 2.23607i
−3.34025 −3.57592 7.15727 − 2.23607i 11.9445 − 10.2321i −10.5461 3.78719 7.46903i
91.2 −3.34025 −3.57592 7.15727 2.23607i 11.9445 10.2321i −10.5461 3.78719 − 7.46903i
91.3 −2.67869 1.23330 3.17536 − 2.23607i −3.30362 0.521669i 2.20895 −7.47898 5.98972i
91.4 −2.67869 1.23330 3.17536 2.23607i −3.30362 − 0.521669i 2.20895 −7.47898 − 5.98972i
91.5 1.32878 −2.60299 −2.23434 − 2.23607i −3.45880 − 8.05652i −8.28408 −2.22446 − 2.97125i
91.6 1.32878 −2.60299 −2.23434 2.23607i −3.45880 8.05652i −8.28408 −2.22446 2.97125i
91.7 2.24810 3.98928 1.05393 − 2.23607i 8.96828 6.68423i −6.62304 6.91435 − 5.02689i
91.8 2.24810 3.98928 1.05393 2.23607i 8.96828 − 6.68423i −6.62304 6.91435 5.02689i
91.9 3.44206 −0.0436725 7.84777 − 2.23607i −0.150323 − 2.33372i 13.2442 −8.99809 − 7.69668i
91.10 3.44206 −0.0436725 7.84777 2.23607i −0.150323 2.33372i 13.2442 −8.99809 7.69668i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 91.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.3.d.b ✓ 10
3.b odd 2 1 1035.3.g.b 10
4.b odd 2 1 1840.3.k.b 10
5.b even 2 1 575.3.d.g 10
5.c odd 4 2 575.3.c.d 20
23.b odd 2 1 inner 115.3.d.b ✓ 10
69.c even 2 1 1035.3.g.b 10
92.b even 2 1 1840.3.k.b 10
115.c odd 2 1 575.3.d.g 10
115.e even 4 2 575.3.c.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.3.d.b ✓ 10 1.a even 1 1 trivial
115.3.d.b ✓ 10 23.b odd 2 1 inner
575.3.c.d 20 5.c odd 4 2
575.3.c.d 20 115.e even 4 2
575.3.d.g 10 5.b even 2 1
575.3.d.g 10 115.c odd 2 1
1035.3.g.b 10 3.b odd 2 1
1035.3.g.b 10 69.c even 2 1
1840.3.k.b 10 4.b odd 2 1
1840.3.k.b 10 92.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - T_{2}^{4} - 18T_{2}^{3} + 19T_{2}^{2} + 75T_{2} - 92 \) acting on \(S_{3}^{\mathrm{new}}(115, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{5} - T^{4} - 18 T^{3} + \cdots - 92)^{2} \) Copy content Toggle raw display
$3$ \( (T^{5} + T^{4} - 18 T^{3} + \cdots + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + 220 T^{8} + \cdots + 450000 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 5056200000 \) Copy content Toggle raw display
$13$ \( (T^{5} + T^{4} - 148 T^{3} + \cdots + 2272)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 32080050000 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 1154881800000 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( (T^{5} + 23 T^{4} + \cdots + 482044)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 8 T^{4} + \cdots + 4585671)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 632002759200000 \) Copy content Toggle raw display
$41$ \( (T^{5} + 42 T^{4} + \cdots + 137382991)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{5} - 56 T^{4} + \cdots - 75512)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 550998028800000 \) Copy content Toggle raw display
$59$ \( (T^{5} + 131 T^{4} + \cdots + 547216)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} - 118 T^{4} + \cdots - 2383066649)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} - 84 T^{4} + \cdots + 181129672)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 92\!\cdots\!00 \) Copy content Toggle raw display
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