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Properties

Label 15.8.b.a
Level $15$
Weight $8$
Character orbit 15.b
Analytic conductor $4.686$
Analytic rank $0$
Dimension $8$
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] = [15,8,Mod(4,15)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(15, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 15.b (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: \(4.68577538226\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{12}\cdot 5^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{4} - 83) q^{4} + ( - \beta_{3} + 3 \beta_{2} + \cdots - 55) q^{5}+ \cdots - 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{3} + ( - \beta_{4} - 83) q^{4} + ( - \beta_{3} + 3 \beta_{2} + \cdots - 55) q^{5}+ \cdots + ( - 729 \beta_{6} + 1458 \beta_{5} + \cdots - 985608) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9} - 6686 q^{10} + 10752 q^{11} - 13524 q^{14} - 17496 q^{15} + 86530 q^{16} - 30464 q^{19} + 87444 q^{20} + 64152 q^{21} - 110322 q^{24} + 127616 q^{25} - 793524 q^{26} + 240072 q^{29} - 172044 q^{30} + 233728 q^{31} + 184748 q^{34} + 593520 q^{35} + 485514 q^{36} - 454896 q^{39} - 1147102 q^{40} + 507648 q^{41} + 2578572 q^{44} + 323676 q^{45} + 662408 q^{46} - 3267160 q^{49} - 5117736 q^{50} + 264384 q^{51} - 354294 q^{54} + 3525696 q^{55} + 705660 q^{56} + 1091424 q^{59} + 5307606 q^{60} - 6433520 q^{61} - 568594 q^{64} - 2555592 q^{65} - 2382372 q^{66} - 5940864 q^{69} + 12097800 q^{70} - 1381824 q^{71} + 23961276 q^{74} + 7768224 q^{75} + 13115664 q^{76} - 14380160 q^{79} - 31251876 q^{80} + 4251528 q^{81} - 36423756 q^{84} - 1452008 q^{85} - 19837608 q^{86} + 45778896 q^{89} + 4874094 q^{90} + 24075648 q^{91} - 45728896 q^{94} - 25774656 q^{95} + 33586002 q^{96} - 7838208 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} - 486\nu^{5} - 22883\nu^{3} - 286700\nu ) / 20000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{7} + 1458\nu^{5} + 62649\nu^{3} + 494100\nu ) / 20000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} + 160\nu^{6} + 86\nu^{5} + 21920\nu^{4} - 29517\nu^{3} + 609760\nu^{2} - 1453300\nu - 996000 ) / 20000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 242\nu^{4} + 13441\nu^{2} + 81400 ) / 500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} + 20\nu^{6} + 1408\nu^{5} + 2740\nu^{4} + 60849\nu^{3} + 76220\nu^{2} + 791350\nu - 124500 ) / 2500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 189 \nu^{7} - 160 \nu^{6} + 27018 \nu^{5} - 21920 \nu^{4} + 932029 \nu^{3} - 609760 \nu^{2} + \cdots + 996000 ) / 20000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 152\nu^{4} + 6151\nu^{2} + 45300 ) / 50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{5} - 3\beta_{3} - 5\beta_{2} + 54\beta_1 ) / 270 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 14\beta_{7} + 3\beta_{6} - 6\beta_{5} - 20\beta_{4} - 21\beta_{3} - 9\beta_{2} - 3\beta _1 - 10922 ) / 270 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -183\beta_{5} + 183\beta_{3} - 595\beta_{2} - 5994\beta_1 ) / 270 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -428\beta_{7} - 81\beta_{6} + 162\beta_{5} + 1040\beta_{4} + 567\beta_{3} + 243\beta_{2} + 81\beta _1 + 258794 ) / 90 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1350\beta_{6} + 13623\beta_{5} - 14973\beta_{3} + 103745\beta_{2} + 538164\beta_1 ) / 270 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 122554 \beta_{7} + 18483 \beta_{6} - 36966 \beta_{5} - 351220 \beta_{4} - 129381 \beta_{3} + \cdots - 63059842 ) / 270 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 218700\beta_{6} - 1097763\beta_{5} + 1316463\beta_{3} - 11790395\beta_{2} - 48422934\beta_1 ) / 270 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
9.60626i
− 1.49423i
− 3.83609i
7.26440i
− 7.26440i
3.83609i
1.49423i
− 9.60626i
− 21.0486i 27.0000i −315.042 −238.301 + 146.073i 568.311 − 923.886i 3936.97i −729.000 3074.63 + 5015.90i
4.2 − 17.7812i − 27.0000i −188.170 57.4967 − 273.531i −480.092 1233.23i 1069.90i −729.000 −4863.70 − 1022.36i
4.3 − 8.75511i 27.0000i 51.3480 231.605 − 156.474i 236.388 − 536.160i − 1570.21i −729.000 −1369.95 − 2027.73i
4.4 − 3.02250i − 27.0000i 118.865 −272.800 − 60.8703i −81.6074 − 1505.28i − 746.147i −729.000 −183.980 + 824.537i
4.5 3.02250i 27.0000i 118.865 −272.800 + 60.8703i −81.6074 1505.28i 746.147i −729.000 −183.980 − 824.537i
4.6 8.75511i − 27.0000i 51.3480 231.605 + 156.474i 236.388 536.160i 1570.21i −729.000 −1369.95 + 2027.73i
4.7 17.7812i 27.0000i −188.170 57.4967 + 273.531i −480.092 − 1233.23i − 1069.90i −729.000 −4863.70 + 1022.36i
4.8 21.0486i − 27.0000i −315.042 −238.301 − 146.073i 568.311 923.886i − 3936.97i −729.000 3074.63 − 5015.90i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.8.b.a ✓ 8
3.b odd 2 1 45.8.b.d 8
4.b odd 2 1 240.8.f.e 8
5.b even 2 1 inner 15.8.b.a ✓ 8
5.c odd 4 1 75.8.a.i 4
5.c odd 4 1 75.8.a.j 4
15.d odd 2 1 45.8.b.d 8
15.e even 4 1 225.8.a.z 4
15.e even 4 1 225.8.a.bb 4
20.d odd 2 1 240.8.f.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.b.a ✓ 8 1.a even 1 1 trivial
15.8.b.a ✓ 8 5.b even 2 1 inner
45.8.b.d 8 3.b odd 2 1
45.8.b.d 8 15.d odd 2 1
75.8.a.i 4 5.c odd 4 1
75.8.a.j 4 5.c odd 4 1
225.8.a.z 4 15.e even 4 1
225.8.a.bb 4 15.e even 4 1
240.8.f.e 8 4.b odd 2 1
240.8.f.e 8 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 845 T^{6} + \cdots + 98089216 \) Copy content Toggle raw display
$3$ \( (T^{2} + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 37\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 36924996384576)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 56\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 76\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 64\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 87\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
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