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Properties

Label 124.3.h.a
Level $124$
Weight $3$
Character orbit 124.h
Analytic conductor $3.379$
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] = [124,3,Mod(37,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 124.h (of order \(6\), degree \(2\), 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: \(3.37875527807\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 16x^{8} + 37x^{7} + 196x^{6} + 211x^{5} + 401x^{4} + 102x^{3} + 434x^{2} + 180x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{6} - \beta_1) q^{5} + ( - \beta_{9} + \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + (\beta_{9} - \beta_{8} + \beta_{5} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{3} + ( - \beta_{6} - \beta_1) q^{5} + ( - \beta_{9} + \beta_{7} + \beta_{6} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{9} - \beta_{7} + \cdots - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{3} + 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{3} + 3 q^{5} + q^{7} + 2 q^{9} + 3 q^{11} + 3 q^{13} + 15 q^{17} + 11 q^{19} - 9 q^{21} + 56 q^{31} + 18 q^{33} - 34 q^{35} - 93 q^{37} + 34 q^{39} - 41 q^{41} - 117 q^{43} - 74 q^{45} - 40 q^{47} - 112 q^{49} + 25 q^{51} - 171 q^{53} + 81 q^{55} - 159 q^{57} + 97 q^{59} + 248 q^{63} + 153 q^{65} + 61 q^{67} + 10 q^{69} + 71 q^{71} - 123 q^{73} + 24 q^{75} + 405 q^{79} + 171 q^{81} + 15 q^{83} + 44 q^{87} + 377 q^{93} + 426 q^{95} - 208 q^{97} - 504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 16x^{8} + 37x^{7} + 196x^{6} + 211x^{5} + 401x^{4} + 102x^{3} + 434x^{2} + 180x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 116935 \nu^{9} - 2235378 \nu^{8} + 6576137 \nu^{7} - 53752351 \nu^{6} - 10836443 \nu^{5} + \cdots - 937271060 ) / 133012554 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1453741 \nu^{9} + 8052416 \nu^{8} - 35601681 \nu^{7} + 33636353 \nu^{6} - 76244111 \nu^{5} + \cdots - 253679730 ) / 665062770 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2639151 \nu^{9} + 7093066 \nu^{8} - 55885081 \nu^{7} - 38065577 \nu^{6} - 441071211 \nu^{5} + \cdots - 1621228300 ) / 665062770 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 541135 \nu^{9} + 2103386 \nu^{8} - 9366825 \nu^{7} + 2998389 \nu^{6} - 39358229 \nu^{5} + \cdots + 462788320 ) / 133012554 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3512429 \nu^{9} - 4105134 \nu^{8} + 55719189 \nu^{7} + 119818173 \nu^{6} + 652585119 \nu^{5} + \cdots - 210990550 ) / 665062770 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2650478 \nu^{9} + 1021637 \nu^{8} + 37435153 \nu^{7} + 167118041 \nu^{6} + 619702443 \nu^{5} + \cdots + 926209365 ) / 332531385 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2132891 \nu^{9} - 940854 \nu^{8} + 33946073 \nu^{7} + 94109177 \nu^{6} + 481452541 \nu^{5} + \cdots + 728700790 ) / 133012554 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12713097 \nu^{9} + 26273002 \nu^{8} - 220420237 \nu^{7} - 259985269 \nu^{6} + \cdots + 1860209325 ) / 332531385 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{9} + 2\beta_{8} - \beta_{7} - 13\beta_{6} - \beta_{5} - 2\beta_{4} + 3\beta_{3} - \beta_{2} + 2\beta _1 - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 6\beta_{7} - 3\beta_{6} - 2\beta_{5} - 17\beta_{4} + 14\beta_{3} - 3\beta_{2} - 3\beta _1 - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17\beta_{9} - 13\beta_{8} - 19\beta_{7} + 176\beta_{6} + 9\beta_{5} - 19\beta_{4} - 19\beta_{3} - 71\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 73 \beta_{9} - 78 \beta_{8} + 75 \beta_{7} + 753 \beta_{6} + 39 \beta_{5} + 266 \beta_{4} - 341 \beta_{3} + \cdots + 946 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 227 \beta_{8} + 750 \beta_{7} + 375 \beta_{6} + 116 \beta_{5} + 1529 \beta_{4} - 1154 \beta_{3} + \cdots + 4574 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1561 \beta_{9} + 927 \beta_{8} + 1645 \beta_{7} - 14906 \beta_{6} - 293 \beta_{5} + 1645 \beta_{4} + \cdots + 7089 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7173 \beta_{9} + 9034 \beta_{8} - 7723 \beta_{7} - 77479 \beta_{6} - 4517 \beta_{5} - 24604 \beta_{4} + \cdots - 94910 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20087 \beta_{8} - 69966 \beta_{7} - 34983 \beta_{6} - 12790 \beta_{5} - 148633 \beta_{4} + 113650 \beta_{3} + \cdots - 432406 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.519964 − 0.900604i
−1.16362 + 2.01545i
2.29262 − 3.97093i
−0.898118 + 1.55559i
−0.250842 + 0.434472i
0.519964 + 0.900604i
−1.16362 − 2.01545i
2.29262 + 3.97093i
−0.898118 − 1.55559i
−0.250842 − 0.434472i
0 −2.71957 + 1.57015i 0 −0.539928 + 0.935182i 0 −2.37748 − 4.11792i 0 0.430712 − 0.746016i 0
37.2 0 −2.61732 + 1.51111i 0 2.82724 − 4.89692i 0 4.49442 + 7.78456i 0 0.0669081 − 0.115888i 0
37.3 0 0.823315 − 0.475341i 0 −4.08523 + 7.07583i 0 1.51961 + 2.63203i 0 −4.04810 + 7.01152i 0
37.4 0 1.94750 − 1.12439i 0 2.29624 − 3.97720i 0 −6.86778 − 11.8953i 0 −1.97149 + 3.41472i 0
37.5 0 4.06607 − 2.34755i 0 1.00168 − 1.73497i 0 3.73124 + 6.46270i 0 6.52197 − 11.2964i 0
57.1 0 −2.71957 − 1.57015i 0 −0.539928 − 0.935182i 0 −2.37748 + 4.11792i 0 0.430712 + 0.746016i 0
57.2 0 −2.61732 − 1.51111i 0 2.82724 + 4.89692i 0 4.49442 − 7.78456i 0 0.0669081 + 0.115888i 0
57.3 0 0.823315 + 0.475341i 0 −4.08523 − 7.07583i 0 1.51961 − 2.63203i 0 −4.04810 − 7.01152i 0
57.4 0 1.94750 + 1.12439i 0 2.29624 + 3.97720i 0 −6.86778 + 11.8953i 0 −1.97149 − 3.41472i 0
57.5 0 4.06607 + 2.34755i 0 1.00168 + 1.73497i 0 3.73124 − 6.46270i 0 6.52197 + 11.2964i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.e odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 124.3.h.a ✓ 10
3.b odd 2 1 1116.3.t.b 10
4.b odd 2 1 496.3.r.c 10
31.e odd 6 1 inner 124.3.h.a ✓ 10
93.g even 6 1 1116.3.t.b 10
124.g even 6 1 496.3.r.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.3.h.a ✓ 10 1.a even 1 1 trivial
124.3.h.a ✓ 10 31.e odd 6 1 inner
496.3.r.c 10 4.b odd 2 1
496.3.r.c 10 124.g even 6 1
1116.3.t.b 10 3.b odd 2 1
1116.3.t.b 10 93.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(124, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 3 T^{9} + \cdots + 9075 \) Copy content Toggle raw display
$5$ \( T^{10} - 3 T^{9} + \cdots + 210681 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 177289225 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 124443184683 \) Copy content Toggle raw display
$13$ \( T^{10} - 3 T^{9} + \cdots + 541875 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 5270601675 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10658291121 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 622932787200 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 3216109780992 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 819628286980801 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 25584018226875 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 742089351894489 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 25\!\cdots\!75 \) Copy content Toggle raw display
$47$ \( (T^{5} + 20 T^{4} + \cdots + 629847360)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 627926970825675 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 17\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 25\!\cdots\!08 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 41332876774225 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 19\!\cdots\!21 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 27\!\cdots\!75 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 34\!\cdots\!07 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 55\!\cdots\!75 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 706421069709312 \) Copy content Toggle raw display
$97$ \( (T^{5} + 104 T^{4} + \cdots + 87055120)^{2} \) Copy content Toggle raw display
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