Properties

Label 1815.2.c.e
Level $1815$
Weight $2$
Character orbit 1815.c
Analytic conductor $14.493$
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] = [1815,2,Mod(364,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{3} q^{3} + (\beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + \beta_{2} q^{6} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{3} q^{3} + (\beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + \beta_{2} q^{6} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{7} + ( - \beta_{5} + \beta_{3}) q^{8} - q^{9} + (\beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} + 1) q^{10} + ( - \beta_{5} + \beta_{4}) q^{12} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{13} + (3 \beta_{2} - \beta_1 - 1) q^{14} + ( - \beta_{4} - \beta_1) q^{15} + ( - 2 \beta_1 - 1) q^{16} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3}) q^{17} + \beta_{4} q^{18} + ( - 2 \beta_1 - 2) q^{19} + (\beta_{5} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{20} + (\beta_{2} - \beta_1 - 1) q^{21} - 4 \beta_{3} q^{23} + (\beta_1 + 1) q^{24} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{25} + (\beta_{2} - \beta_1 - 1) q^{26} + \beta_{3} q^{27} + ( - \beta_{5} + 3 \beta_{4} + 3 \beta_{3}) q^{28} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{30} + (2 \beta_{2} - 2) q^{31} + ( - 2 \beta_{5} + 3 \beta_{4}) q^{32} + ( - 3 \beta_{2} + 3 \beta_1 + 5) q^{34} + (\beta_{5} - 3 \beta_{4} - \beta_{3} - 2) q^{35} + ( - \beta_{2} + \beta_1) q^{36} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3}) q^{37} + (4 \beta_{4} - 2 \beta_{3}) q^{38} + (\beta_{2} - \beta_1 + 1) q^{39} + (2 \beta_{4} - \beta_{3} + \beta_{2} + 3) q^{40} + (2 \beta_{2} - 6 \beta_1 - 2) q^{41} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3}) q^{42} + (3 \beta_{5} - 3 \beta_{4} + 5 \beta_{3}) q^{43} + ( - \beta_{5} + \beta_{2}) q^{45} + 4 \beta_{2} q^{46} + ( - 2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3}) q^{47} + ( - 2 \beta_{5} + \beta_{3}) q^{48} + (4 \beta_{2} - 2 \beta_1 + 3) q^{49} + (2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 4) q^{50} + ( - 3 \beta_{2} + \beta_1 - 1) q^{51} + (\beta_{5} + \beta_{4} + 3 \beta_{3}) q^{52} + (2 \beta_{5} - 4 \beta_{4} - 4 \beta_{3}) q^{53} - \beta_{2} q^{54} + ( - \beta_{2} + \beta_1 + 3) q^{56} + ( - 2 \beta_{5} + 2 \beta_{3}) q^{57} + (2 \beta_{5} - 2 \beta_{4} - 6 \beta_{3}) q^{58} + ( - 4 \beta_{2} - 2 \beta_1 - 4) q^{59} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_1 + 2) q^{60} + ( - 4 \beta_{2} + 6) q^{61} + ( - 2 \beta_{5} + 4 \beta_{4} + 4 \beta_{3}) q^{62} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{63} + ( - 5 \beta_{2} - \beta_1 + 2) q^{64} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{65} + (4 \beta_{5} + 4 \beta_{3}) q^{67} + (\beta_{5} - 5 \beta_{4} - 5 \beta_{3}) q^{68} - 4 q^{69} + (2 \beta_{4} + 5 \beta_{2} - 3 \beta_1 - 5) q^{70} + ( - 6 \beta_1 - 4) q^{71} + (\beta_{5} - \beta_{3}) q^{72} + ( - \beta_{5} + 5 \beta_{4} - 5 \beta_{3}) q^{73} + (2 \beta_{2} - 2 \beta_1 - 2) q^{74} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{75} + ( - 2 \beta_{2} + 4) q^{76} + ( - \beta_{5} + \beta_{4} + \beta_{3}) q^{78} + (4 \beta_{2} + 6 \beta_1 - 2) q^{79} + (\beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 3 \beta_{2} + 2) q^{80} + q^{81} + ( - 2 \beta_{5} + 10 \beta_{4} - 2 \beta_{3}) q^{82} + (5 \beta_{5} - \beta_{4} - 3 \beta_{3}) q^{83} + ( - 3 \beta_{2} + \beta_1 + 3) q^{84} + ( - 3 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{85} + (\beta_{2} - 3 \beta_1 - 3) q^{86} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{87} + ( - 6 \beta_{2} - 8) q^{89} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} - 1) q^{90} + (2 \beta_{2} - 2) q^{91} + ( - 4 \beta_{5} + 4 \beta_{4}) q^{92} + (2 \beta_{4} + 2 \beta_{3}) q^{93} + ( - 6 \beta_{2} - 2 \beta_1 - 6) q^{94} + ( - 2 \beta_{4} + 6 \beta_{3} + 4 \beta_{2} + 2) q^{95} + ( - 3 \beta_{2} + 2 \beta_1) q^{96} + ( - 8 \beta_{5} + 4 \beta_{4} + 4 \beta_{3}) q^{97} + ( - 4 \beta_{5} + 3 \beta_{4} + 6 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} + 2 q^{5} - 2 q^{6} - 6 q^{9} + 4 q^{10} - 12 q^{14} - 6 q^{16} - 12 q^{19} - 10 q^{20} - 8 q^{21} + 6 q^{24} - 2 q^{25} - 8 q^{26} + 16 q^{29} - 14 q^{30} - 16 q^{31} + 36 q^{34} - 12 q^{35} + 2 q^{36} + 4 q^{39} + 16 q^{40} - 16 q^{41} - 2 q^{45} - 8 q^{46} + 10 q^{49} - 24 q^{50} + 2 q^{54} + 20 q^{56} - 16 q^{59} + 12 q^{60} + 44 q^{61} + 22 q^{64} - 12 q^{65} - 24 q^{69} - 40 q^{70} - 24 q^{71} - 16 q^{74} + 8 q^{75} + 28 q^{76} - 20 q^{79} + 6 q^{80} + 6 q^{81} + 24 q^{84} + 4 q^{85} - 20 q^{86} - 36 q^{89} - 4 q^{90} - 16 q^{91} - 24 q^{94} + 4 q^{95} + 6 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
−0.854638 0.854638i
0.403032 0.403032i
1.45161 + 1.45161i
1.45161 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
2.17009i 1.00000i −2.70928 2.17009 0.539189i −2.17009 3.70928i 1.53919i −1.00000 −1.17009 4.70928i
364.2 1.48119i 1.00000i −0.193937 −1.48119 + 1.67513i 1.48119 1.19394i 2.67513i −1.00000 2.48119 + 2.19394i
364.3 0.311108i 1.00000i 1.90321 0.311108 + 2.21432i −0.311108 0.903212i 1.21432i −1.00000 0.688892 0.0967881i
364.4 0.311108i 1.00000i 1.90321 0.311108 2.21432i −0.311108 0.903212i 1.21432i −1.00000 0.688892 + 0.0967881i
364.5 1.48119i 1.00000i −0.193937 −1.48119 1.67513i 1.48119 1.19394i 2.67513i −1.00000 2.48119 2.19394i
364.6 2.17009i 1.00000i −2.70928 2.17009 + 0.539189i −2.17009 3.70928i 1.53919i −1.00000 −1.17009 + 4.70928i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.6
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 1815.2.c.e 6
5.b even 2 1 inner 1815.2.c.e 6
5.c odd 4 1 9075.2.a.cg 3
5.c odd 4 1 9075.2.a.ch 3
11.b odd 2 1 165.2.c.b 6
33.d even 2 1 495.2.c.e 6
44.c even 2 1 2640.2.d.h 6
55.d odd 2 1 165.2.c.b 6
55.e even 4 1 825.2.a.j 3
55.e even 4 1 825.2.a.l 3
165.d even 2 1 495.2.c.e 6
165.l odd 4 1 2475.2.a.ba 3
165.l odd 4 1 2475.2.a.bc 3
220.g even 2 1 2640.2.d.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.c.b 6 11.b odd 2 1
165.2.c.b 6 55.d odd 2 1
495.2.c.e 6 33.d even 2 1
495.2.c.e 6 165.d even 2 1
825.2.a.j 3 55.e even 4 1
825.2.a.l 3 55.e even 4 1
1815.2.c.e 6 1.a even 1 1 trivial
1815.2.c.e 6 5.b even 2 1 inner
2475.2.a.ba 3 165.l odd 4 1
2475.2.a.bc 3 165.l odd 4 1
2640.2.d.h 6 44.c even 2 1
2640.2.d.h 6 220.g even 2 1
9075.2.a.cg 3 5.c odd 4 1
9075.2.a.ch 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1815, [\chi])\):

\( T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{3} + 6T_{19}^{2} - 4T_{19} - 40 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 7 T^{4} + 11 T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 2 T^{5} + 3 T^{4} - 12 T^{3} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 16 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 12 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{6} + 56 T^{4} + 784 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$19$ \( (T^{3} + 6 T^{2} - 4 T - 40)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 16)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} - 8 T^{2} - 16 T + 160)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 8 T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 48 T^{4} + 512 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$41$ \( (T^{3} + 8 T^{2} - 112 T - 928)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 144 T^{4} + 3552 T^{2} + \cdots + 21904 \) Copy content Toggle raw display
$47$ \( T^{6} + 160 T^{4} + 1280 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{6} + 192 T^{4} + 512 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{3} + 8 T^{2} - 64 T + 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 22 T^{2} + 108 T - 8)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 176 T^{4} + 1792 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( (T^{3} + 12 T^{2} - 96 T - 944)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 188 T^{4} + 9696 T^{2} + \cdots + 150544 \) Copy content Toggle raw display
$79$ \( (T^{3} + 10 T^{2} - 212 T - 1720)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 220 T^{4} + 15680 T^{2} + \cdots + 364816 \) Copy content Toggle raw display
$89$ \( (T^{3} + 18 T^{2} - 12 T - 520)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 576 T^{4} + 103424 T^{2} + \cdots + 5914624 \) Copy content Toggle raw display
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