Properties

Label 15.5.c.a
Level $15$
Weight $5$
Character orbit 15.c
Analytic conductor $1.551$
Analytic rank $0$
Dimension $6$
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] = [15,5,Mod(11,15)]
 
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([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("15.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 15.c (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: \(1.55054944626\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 73x^{4} + 1096x^{2} + 180 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{4} + \beta_{2} - 8) q^{4} + \beta_{3} q^{5} + (\beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 13) q^{7} + ( - \beta_{5} + 6 \beta_{3} - 2 \beta_{2} - 9 \beta_1 - 1) q^{8} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 9 \beta_1 + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + 1) q^{3} + (\beta_{4} + \beta_{2} - 8) q^{4} + \beta_{3} q^{5} + (\beta_{5} - 3 \beta_{3} + 3 \beta_1) q^{6} + ( - \beta_{5} - 2 \beta_{4} + 2 \beta_{2} + 13) q^{7} + ( - \beta_{5} + 6 \beta_{3} - 2 \beta_{2} - 9 \beta_1 - 1) q^{8} + ( - \beta_{5} - 3 \beta_{4} - 3 \beta_{3} - \beta_{2} - 9 \beta_1 + 19) q^{9} + (\beta_{5} - \beta_{4} - 5 \beta_{2} + 7) q^{10} + (4 \beta_{5} + 8 \beta_{2} - 2 \beta_1 + 4) q^{11} + ( - 3 \beta_{5} + 6 \beta_{4} - 6 \beta_{3} + 10 \beta_{2} + 3 \beta_1 - 73) q^{12} + (2 \beta_{4} + 2 \beta_{2} - 70) q^{13} + ( - 2 \beta_{5} + 6 \beta_{3} - 4 \beta_{2} + 36 \beta_1 - 2) q^{14} + (2 \beta_{5} + 3 \beta_{4} - 15 \beta_1 + 9) q^{15} + (8 \beta_{5} + \beta_{4} - 31 \beta_{2} + 126) q^{16} + ( - 8 \beta_{5} + 6 \beta_{3} - 16 \beta_{2} - 20 \beta_1 - 8) q^{17} + ( - 2 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} + 63 \beta_1 + 194) q^{18} + ( - 6 \beta_{5} + 2 \beta_{4} + 26 \beta_{2} - 34) q^{19} + (5 \beta_{5} - 8 \beta_{3} + 10 \beta_{2} + 35 \beta_1 + 5) q^{20} + (9 \beta_{5} - 3 \beta_{4} + 21 \beta_{3} - 21 \beta_{2} + 21 \beta_1 - 150) q^{21} + ( - 8 \beta_{5} - 2 \beta_{4} + 30 \beta_{2} + 64) q^{22} + (7 \beta_{5} - 48 \beta_{3} + 14 \beta_{2} - 72 \beta_1 + 7) q^{23} + (9 \beta_{4} + 18 \beta_{3} - 3 \beta_{2} - 144 \beta_1 - 132) q^{24} - 125 q^{25} + ( - 2 \beta_{5} + 12 \beta_{3} - 4 \beta_{2} - 104 \beta_1 - 2) q^{26} + ( - 5 \beta_{5} - 24 \beta_{4} + 48 \beta_{3} - 29 \beta_{2} + 18 \beta_1 - 70) q^{27} + ( - 6 \beta_{5} - 2 \beta_{4} + 22 \beta_{2} - 622) q^{28} + ( - 2 \beta_{5} - 48 \beta_{3} - 4 \beta_{2} + 146 \beta_1 - 2) q^{29} + ( - 15 \beta_{4} - 3 \beta_{3} - 5 \beta_{2} - 30 \beta_1 + 365) q^{30} + ( - 8 \beta_{5} - 2 \beta_{4} + 30 \beta_{2} + 636) q^{31} + (15 \beta_{5} - 42 \beta_{3} + 30 \beta_{2} + 53 \beta_1 + 15) q^{32} + (10 \beta_{5} + 36 \beta_{4} + 42 \beta_{3} + 12 \beta_{2} + 102 \beta_1 + 744) q^{33} + (22 \beta_{5} - 26 \beta_{4} - 114 \beta_{2} + 490) q^{34} + ( - 15 \beta_{5} + 20 \beta_{3} - 30 \beta_{2} + 20 \beta_1 - 15) q^{35} + ( - 26 \beta_{5} + 21 \beta_{4} - 42 \beta_{3} + 73 \beta_{2} + \cdots - 1252) q^{36}+ \cdots + (64 \beta_{5} + 300 \beta_{4} - 564 \beta_{3} - 632 \beta_{2} + \cdots + 1424) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{3} - 50 q^{4} - 2 q^{6} + 76 q^{7} + 118 q^{9} + 50 q^{10} - 452 q^{12} - 424 q^{13} + 50 q^{15} + 802 q^{16} + 1160 q^{18} - 244 q^{19} - 876 q^{21} + 340 q^{22} - 786 q^{24} - 750 q^{25} - 352 q^{27} - 3764 q^{28} + 2200 q^{30} + 3772 q^{31} + 4420 q^{33} + 3124 q^{34} - 7606 q^{36} + 1896 q^{37} - 1336 q^{39} - 4650 q^{40} - 1980 q^{42} - 7384 q^{43} + 1900 q^{45} + 8196 q^{46} + 14668 q^{48} - 1318 q^{49} - 8492 q^{51} + 8976 q^{52} - 278 q^{54} - 1300 q^{55} - 11584 q^{57} - 23740 q^{58} + 5050 q^{60} + 6452 q^{61} + 14796 q^{63} + 3174 q^{64} - 12760 q^{66} + 13816 q^{67} + 5472 q^{69} - 2100 q^{70} - 2040 q^{72} + 596 q^{73} - 1000 q^{75} + 21348 q^{76} - 1400 q^{78} - 16124 q^{79} + 5086 q^{81} - 31240 q^{82} - 14736 q^{84} - 3100 q^{85} - 4900 q^{87} + 15660 q^{88} + 7550 q^{90} - 11632 q^{91} - 8184 q^{93} + 34924 q^{94} + 14354 q^{96} + 9756 q^{97} + 9680 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 73x^{4} + 1096x^{2} + 180 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 71\nu^{3} - 47\nu^{2} + 1002\nu - 114 ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 79\nu^{3} + 1330\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 71\nu^{3} + 95\nu^{2} - 1002\nu + 1266 ) / 48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} + \nu^{4} + 142\nu^{3} + 47\nu^{2} + 2004\nu + 90 ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + 6\beta_{3} - 2\beta_{2} - 41\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} - 47\beta_{4} - 79\beta_{2} + 1022 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 79\beta_{5} - 426\beta_{3} + 158\beta_{2} + 1909\beta _1 + 79 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
7.20990i
4.56632i
0.407512i
0.407512i
4.56632i
7.20990i
7.20990i 8.77108 + 2.01697i −35.9827 11.1803i 14.5422 63.2386i 23.3388 144.073i 72.8637 + 35.3820i 80.6091
11.2 4.56632i −7.98405 4.15391i −4.85128 11.1803i −18.9681 + 36.4577i 61.6068 50.9086i 46.4900 + 66.3301i −51.0530
11.3 0.407512i 3.21297 + 8.40695i 15.8339 11.1803i 3.42594 1.30932i −46.9457 12.9727i −60.3537 + 54.0225i −4.55613
11.4 0.407512i 3.21297 8.40695i 15.8339 11.1803i 3.42594 + 1.30932i −46.9457 12.9727i −60.3537 54.0225i −4.55613
11.5 4.56632i −7.98405 + 4.15391i −4.85128 11.1803i −18.9681 36.4577i 61.6068 50.9086i 46.4900 66.3301i −51.0530
11.6 7.20990i 8.77108 2.01697i −35.9827 11.1803i 14.5422 + 63.2386i 23.3388 144.073i 72.8637 35.3820i 80.6091
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 15.5.c.a 6
3.b odd 2 1 inner 15.5.c.a 6
4.b odd 2 1 240.5.l.d 6
5.b even 2 1 75.5.c.i 6
5.c odd 4 2 75.5.d.d 12
12.b even 2 1 240.5.l.d 6
15.d odd 2 1 75.5.c.i 6
15.e even 4 2 75.5.d.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.5.c.a 6 1.a even 1 1 trivial
15.5.c.a 6 3.b odd 2 1 inner
75.5.c.i 6 5.b even 2 1
75.5.c.i 6 15.d odd 2 1
75.5.d.d 12 5.c odd 4 2
75.5.d.d 12 15.e even 4 2
240.5.l.d 6 4.b odd 2 1
240.5.l.d 6 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(15, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 73 T^{4} + 1096 T^{2} + \cdots + 180 \) Copy content Toggle raw display
$3$ \( T^{6} - 8 T^{5} - 27 T^{4} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{2} + 125)^{3} \) Copy content Toggle raw display
$7$ \( (T^{3} - 38 T^{2} - 2550 T + 67500)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 53380 T^{4} + \cdots + 552448800000 \) Copy content Toggle raw display
$13$ \( (T^{3} + 212 T^{2} + 12260 T + 179200)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 345773292505920 \) Copy content Toggle raw display
$19$ \( (T^{3} + 122 T^{2} - 106024 T - 6584528)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 512545320524880 \) Copy content Toggle raw display
$29$ \( T^{6} + 2816020 T^{4} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{3} - 1886 T^{2} + 1033632 T - 171289728)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 948 T^{2} - 1992060 T + 148979200)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 7332280 T^{4} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{3} + 3692 T^{2} + \cdots - 9836480000)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 9919408 T^{4} + \cdots + 17\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{6} + 15871708 T^{4} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{6} + 52482580 T^{4} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{3} - 3226 T^{2} + \cdots + 8122222912)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 6908 T^{2} + \cdots + 28105138800)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 30787120 T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{3} - 298 T^{2} + \cdots - 62842083800)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 8062 T^{2} + \cdots - 94953979728)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 73171728 T^{4} + \cdots + 77\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{6} + 121099680 T^{4} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} - 4878 T^{2} + \cdots + 501103547800)^{2} \) Copy content Toggle raw display
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