Properties

Label 1155.2.c.d
Level $1155$
Weight $2$
Character orbit 1155.c
Analytic conductor $9.223$
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] = [1155,2,Mod(694,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.694");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.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: \(9.22272143346\)
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: 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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + \beta_1 q^{6} + \beta_{3} q^{7} + 2 \beta_{3} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_{3} q^{3} + ( - \beta_{2} + \beta_1 - 1) q^{4} + (\beta_{5} - \beta_{2}) q^{5} + \beta_1 q^{6} + \beta_{3} q^{7} + 2 \beta_{3} q^{8} - q^{9} + (\beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{10} + q^{11} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{12} + ( - \beta_{4} + 3 \beta_{3}) q^{13} - \beta_1 q^{14} + ( - \beta_{4} - \beta_1) q^{15} + ( - 2 \beta_{2} - 2) q^{16} + (\beta_{5} + \beta_{4}) q^{17} + \beta_{5} q^{18} + ( - 2 \beta_1 + 3) q^{19} + ( - 2 \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{20} + q^{21} - \beta_{5} q^{22} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{23} + 2 q^{24} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{25} + ( - \beta_{2} - 3 \beta_1 - 1) q^{26} + \beta_{3} q^{27} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{28} + (\beta_{2} + 8) q^{29} + ( - \beta_{5} + \beta_{4} - 3 \beta_{3} - \beta_{2} - 1) q^{30} + (2 \beta_{2} - 3 \beta_1 + 2) q^{31} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{32} - \beta_{3} q^{33} + (2 \beta_{2} - \beta_1 + 4) q^{34} + (\beta_{4} + \beta_1) q^{35} + (\beta_{2} - \beta_1 + 1) q^{36} + ( - 3 \beta_{4} + 5 \beta_{3}) q^{37} + ( - 5 \beta_{5} + 2 \beta_{4} - 6 \beta_{3}) q^{38} + (\beta_{2} + 3) q^{39} + (2 \beta_{4} + 2 \beta_1) q^{40} + (\beta_{2} - 2 \beta_1 - 5) q^{41} - \beta_{5} q^{42} + ( - 2 \beta_{5} - 3 \beta_{4}) q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{44} + ( - \beta_{5} + \beta_{2}) q^{45} + ( - \beta_1 - 2) q^{46} + ( - 3 \beta_{5} - 2 \beta_{4} - 4 \beta_{3}) q^{47} + ( - 2 \beta_{4} + 2 \beta_{3}) q^{48} - q^{49} + (3 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 2) q^{50} + ( - \beta_{2} - \beta_1) q^{51} + ( - 2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3}) q^{52} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{53} - \beta_1 q^{54} + (\beta_{5} - \beta_{2}) q^{55} - 2 q^{56} + ( - 2 \beta_{5} - 3 \beta_{3}) q^{57} + ( - 8 \beta_{5} - \beta_{4} + \beta_{3}) q^{58} + (4 \beta_{2} - \beta_1 + 1) q^{59} + (\beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 2) q^{60} + ( - 5 \beta_{2} + \beta_1 - 2) q^{61} + ( - 5 \beta_{5} + \beta_{4} - 7 \beta_{3}) q^{62} - \beta_{3} q^{63} + ( - 4 \beta_1 + 4) q^{64} + (\beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + 3 \beta_1 + 1) q^{65} + \beta_1 q^{66} + (4 \beta_{5} - 8 \beta_{4} + 6 \beta_{3}) q^{67} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3}) q^{68} + ( - \beta_{2} + \beta_1 + 2) q^{69} + (\beta_{5} - \beta_{4} + 3 \beta_{3} + \beta_{2} + 1) q^{70} + ( - \beta_{2} + 2 \beta_1 - 3) q^{71} - 2 \beta_{3} q^{72} + (2 \beta_{5} + 4 \beta_{3}) q^{73} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{74} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{75} + ( - 3 \beta_{2} + 7 \beta_1 - 7) q^{76} + \beta_{3} q^{77} + ( - 3 \beta_{5} - \beta_{4} + \beta_{3}) q^{78} + (2 \beta_{2} - 3 \beta_1 + 6) q^{79} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 4) q^{80} + q^{81} + (3 \beta_{5} + \beta_{4} - 5 \beta_{3}) q^{82} + ( - \beta_{5} + \beta_{4} + 4 \beta_{3}) q^{83} + ( - \beta_{2} + \beta_1 - 1) q^{84} + ( - \beta_{5} + 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{85} + ( - 5 \beta_{2} + 2 \beta_1 - 9) q^{86} + (\beta_{4} - 8 \beta_{3}) q^{87} + 2 \beta_{3} q^{88} + (5 \beta_{2} - 6 \beta_1 + 6) q^{89} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 3) q^{90} + ( - \beta_{2} - 3) q^{91} + ( - \beta_{5} + 3 \beta_{4} + \beta_{3}) q^{92} + ( - 3 \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{93} + ( - 5 \beta_{2} + 7 \beta_1 - 11) q^{94} + (5 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - \beta_{2} + 2) q^{95} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{96} + (6 \beta_{5} - 5 \beta_{4}) q^{97} + \beta_{5} q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{4} + 2 q^{5} - 6 q^{9} + 16 q^{10} + 6 q^{11} - 8 q^{16} + 18 q^{19} + 8 q^{20} + 6 q^{21} + 12 q^{24} - 2 q^{25} - 4 q^{26} + 46 q^{29} - 4 q^{30} + 8 q^{31} + 20 q^{34} + 4 q^{36} + 16 q^{39} - 32 q^{41} - 4 q^{44} - 2 q^{45} - 12 q^{46} - 6 q^{49} - 8 q^{50} + 2 q^{51} + 2 q^{55} - 12 q^{56} - 2 q^{59} - 12 q^{60} - 2 q^{61} + 24 q^{64} + 4 q^{65} + 14 q^{69} + 4 q^{70} - 16 q^{71} - 12 q^{74} + 8 q^{75} - 36 q^{76} + 32 q^{79} + 24 q^{80} + 6 q^{81} - 4 q^{84} - 20 q^{85} - 44 q^{86} + 26 q^{89} - 16 q^{90} - 16 q^{91} - 56 q^{94} + 14 q^{95} + 16 q^{96} - 6 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
694.1
1.45161 + 1.45161i
0.403032 0.403032i
−0.854638 + 0.854638i
−0.854638 0.854638i
0.403032 + 0.403032i
1.45161 1.45161i
2.21432i 1.00000i −2.90321 0.311108 + 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 0.688892i
694.2 1.67513i 1.00000i −0.806063 −1.48119 + 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 + 2.48119i
694.3 0.539189i 1.00000i 1.70928 2.17009 + 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 1.17009i
694.4 0.539189i 1.00000i 1.70928 2.17009 0.539189i 0.539189 1.00000i 2.00000i −1.00000 0.290725 + 1.17009i
694.5 1.67513i 1.00000i −0.806063 −1.48119 1.67513i 1.67513 1.00000i 2.00000i −1.00000 2.80606 2.48119i
694.6 2.21432i 1.00000i −2.90321 0.311108 2.21432i −2.21432 1.00000i 2.00000i −1.00000 4.90321 + 0.688892i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.6
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 1155.2.c.d 6
5.b even 2 1 inner 1155.2.c.d 6
5.c odd 4 1 5775.2.a.bs 3
5.c odd 4 1 5775.2.a.bv 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.c.d 6 1.a even 1 1 trivial
1155.2.c.d 6 5.b even 2 1 inner
5775.2.a.bs 3 5.c odd 4 1
5775.2.a.bv 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 \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{6} + 8T_{2}^{4} + 16T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{6} + 28T_{13}^{4} + 164T_{13}^{2} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 8 T^{4} + 16 T^{2} + 4 \) 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
$7$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 28 T^{4} + 164 T^{2} + \cdots + 100 \) Copy content Toggle raw display
$17$ \( T^{6} + 19 T^{4} + 107 T^{2} + \cdots + 169 \) Copy content Toggle raw display
$19$ \( (T^{3} - 9 T^{2} + 11 T + 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 27 T^{4} + 107 T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{3} - 23 T^{2} + 173 T - 425)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 32 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 108 T^{4} + 1716 T^{2} + \cdots + 3364 \) Copy content Toggle raw display
$41$ \( (T^{3} + 16 T^{2} + 70 T + 50)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 119 T^{4} + 4387 T^{2} + \cdots + 51529 \) Copy content Toggle raw display
$47$ \( T^{6} + 188 T^{4} + 8472 T^{2} + \cdots + 91204 \) Copy content Toggle raw display
$53$ \( T^{6} + 75 T^{4} + 1235 T^{2} + \cdots + 4489 \) Copy content Toggle raw display
$59$ \( (T^{3} + T^{2} - 49 T + 85)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + T^{2} - 77 T - 193)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 460 T^{4} + 69680 T^{2} + \cdots + 3474496 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} + 6 T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 80 T^{4} + 640 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$79$ \( (T^{3} - 16 T^{2} + 48 T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 67 T^{4} + 1067 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$89$ \( (T^{3} - 13 T^{2} - 111 T + 481)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 343 T^{4} + 29651 T^{2} + \cdots + 190969 \) Copy content Toggle raw display
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