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Properties

Label 1155.2.q.i
Level $1155$
Weight $2$
Character orbit 1155.q
Analytic conductor $9.223$
Analytic rank $0$
Dimension $14$
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(331,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.331");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.q (of order \(3\), 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: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 9x^{12} + 58x^{10} + 179x^{8} - 2x^{7} + 403x^{6} + 18x^{5} + 322x^{4} + 92x^{3} + 196x^{2} + 28x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{9} q^{3} + (\beta_{11} + \beta_{9}) q^{4} + ( - \beta_{9} - 1) q^{5} - \beta_{4} q^{6} + ( - \beta_{12} + \beta_{6}) q^{7} + (\beta_{6} - \beta_{5}) q^{8} + ( - \beta_{9} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{9} q^{3} + (\beta_{11} + \beta_{9}) q^{4} + ( - \beta_{9} - 1) q^{5} - \beta_{4} q^{6} + ( - \beta_{12} + \beta_{6}) q^{7} + (\beta_{6} - \beta_{5}) q^{8} + ( - \beta_{9} - 1) q^{9} + ( - \beta_{4} + \beta_1) q^{10} + \beta_{9} q^{11} + (\beta_{11} + \beta_{9} + \beta_{2} + 1) q^{12} + (\beta_{6} - \beta_{5} + \beta_{4} + \cdots - 1) q^{13}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 7 q^{3} - 4 q^{4} - 7 q^{5} - q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 7 q^{3} - 4 q^{4} - 7 q^{5} - q^{7} - 7 q^{9} - 7 q^{11} + 4 q^{12} - 14 q^{13} + 2 q^{14} - 14 q^{15} + 10 q^{16} + 6 q^{17} + 8 q^{19} + 8 q^{20} + q^{21} + q^{23} - 7 q^{25} - 12 q^{26} - 14 q^{27} - 2 q^{28} - 18 q^{29} + 20 q^{31} + 7 q^{33} - 24 q^{34} + 2 q^{35} + 8 q^{36} - 6 q^{37} - 7 q^{39} - 10 q^{41} + 4 q^{42} + 30 q^{43} - 4 q^{44} - 7 q^{45} + 6 q^{46} + q^{47} + 20 q^{48} - 7 q^{49} - 6 q^{51} + 14 q^{52} - 13 q^{53} + 14 q^{55} + 42 q^{56} + 16 q^{57} + 10 q^{58} + 38 q^{59} + 4 q^{60} + 4 q^{61} - 8 q^{62} + 2 q^{63} - 48 q^{64} + 7 q^{65} - 22 q^{67} - 16 q^{68} + 2 q^{69} + 2 q^{70} - 52 q^{71} + 24 q^{73} - 14 q^{74} + 7 q^{75} - 104 q^{76} - q^{77} - 24 q^{78} + 10 q^{80} - 7 q^{81} + 18 q^{82} - 40 q^{83} - 4 q^{84} - 12 q^{85} + 8 q^{86} - 9 q^{87} + 34 q^{89} + 39 q^{91} - 48 q^{92} - 20 q^{93} + 28 q^{94} + 8 q^{95} - 20 q^{97} + 108 q^{98} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 9x^{12} + 58x^{10} + 179x^{8} - 2x^{7} + 403x^{6} + 18x^{5} + 322x^{4} + 92x^{3} + 196x^{2} + 28x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29394648 \nu^{13} - 335056307 \nu^{12} + 232810137 \nu^{11} - 2732811543 \nu^{10} + \cdots - 218724480750 ) / 72735276298 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 86717844 \nu^{13} + 3128733800 \nu^{12} - 1026079167 \nu^{11} + 29435542857 \nu^{10} + \cdots + 278193656902 ) / 72735276298 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 129662964 \nu^{13} + 29394648 \nu^{12} + 831910369 \nu^{11} + 232810137 \nu^{10} + \cdots - 10324076162 ) / 72735276298 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 183595549 \nu^{13} + 85836897 \nu^{12} + 594829933 \nu^{11} + 697868966 \nu^{10} + \cdots - 41413883240 ) / 72735276298 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 335056307 \nu^{13} - 31741695 \nu^{12} - 2732811543 \nu^{11} - 233371582 \nu^{10} + \cdots - 117578592 ) / 72735276298 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 403420311 \nu^{13} + 1510164907 \nu^{12} + 2396962373 \nu^{11} + 12780016767 \nu^{10} + \cdots + 178162558298 ) / 72735276298 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 595550880 \nu^{13} + 488515170 \nu^{12} + 3775280950 \nu^{11} + 3950957430 \nu^{10} + \cdots + 146328980836 ) / 72735276298 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5162038081 \nu^{13} - 259325928 \nu^{12} - 46517132025 \nu^{11} - 1663820738 \nu^{10} + \cdots - 147032159192 ) / 145470552596 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12089618295 \nu^{13} + 168427824 \nu^{12} - 110853840825 \nu^{11} + 2451569712 \nu^{10} + \cdots - 2490506532 ) / 145470552596 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15486114243 \nu^{13} + 777977784 \nu^{12} + 139551396075 \nu^{11} + 4991462214 \nu^{10} + \cdots + 441096477576 ) / 145470552596 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15916788731 \nu^{13} + 384261138 \nu^{12} + 141207799105 \nu^{11} + 2756637154 \nu^{10} + \cdots + 355588260640 ) / 145470552596 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 10742086857 \nu^{13} - 4113273572 \nu^{12} + 96347701170 \nu^{11} - 35706554914 \nu^{10} + \cdots - 4044435624 ) / 72735276298 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + 3\beta_{9} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + \beta_{5} - 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - 5\beta_{11} - 12\beta_{9} - \beta_{7} - \beta_{6} - 5\beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{11} - \beta_{10} + \beta_{8} - 6\beta_{5} + 17\beta_{4} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{7} + 8\beta_{6} - 8\beta_{5} + \beta_{4} - \beta_{3} + 24\beta_{2} + 52 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{11} + 9\beta_{10} + 2\beta_{9} + 31\beta_{6} + 9\beta_{2} + 75\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9\beta_{13} - 49\beta_{12} + 115\beta_{11} + 234\beta_{9} + 49\beta_{5} - 11\beta_{4} + 9\beta_{3} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -58\beta_{8} - 155\beta_{6} + 155\beta_{5} - 340\beta_{4} - 60\beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 58 \beta_{13} + 271 \beta_{12} - 553 \beta_{11} - 1078 \beta_{9} - 271 \beta_{7} - 273 \beta_{6} + \cdots - 1078 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2 \beta_{12} - 357 \beta_{11} - 329 \beta_{10} - 194 \beta_{9} + 329 \beta_{8} - 768 \beta_{5} + \cdots - 1573 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2\beta_{8} + 1424\beta_{7} + 1454\beta_{6} - 1454\beta_{5} + 551\beta_{4} - 329\beta_{3} + 2670\beta_{2} + 5048 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2 \beta_{13} - 34 \beta_{12} + 2007 \beta_{11} + 1753 \beta_{10} + 1326 \beta_{9} + 34 \beta_{7} + \cdots + 1326 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1
1.11656 − 1.93394i
0.860135 − 1.48980i
0.529611 − 0.917312i
−0.0740769 + 0.128305i
−0.405942 + 0.703112i
−0.943217 + 1.63370i
−1.08307 + 1.87593i
1.11656 + 1.93394i
0.860135 + 1.48980i
0.529611 + 0.917312i
−0.0740769 − 0.128305i
−0.405942 − 0.703112i
−0.943217 − 1.63370i
−1.08307 − 1.87593i
−1.11656 + 1.93394i 0.500000 + 0.866025i −1.49341 − 2.58666i −0.500000 + 0.866025i −2.23312 0.736549 − 2.54116i 2.20369 −0.500000 + 0.866025i −1.11656 − 1.93394i
331.2 −0.860135 + 1.48980i 0.500000 + 0.866025i −0.479666 − 0.830805i −0.500000 + 0.866025i −1.72027 −0.270725 + 2.63186i −1.79023 −0.500000 + 0.866025i −0.860135 − 1.48980i
331.3 −0.529611 + 0.917312i 0.500000 + 0.866025i 0.439025 + 0.760414i −0.500000 + 0.866025i −1.05922 −2.37300 + 1.16998i −3.04849 −0.500000 + 0.866025i −0.529611 − 0.917312i
331.4 0.0740769 − 0.128305i 0.500000 + 0.866025i 0.989025 + 1.71304i −0.500000 + 0.866025i 0.148154 −0.856004 − 2.50345i 0.589363 −0.500000 + 0.866025i 0.0740769 + 0.128305i
331.5 0.405942 − 0.703112i 0.500000 + 0.866025i 0.670422 + 1.16121i −0.500000 + 0.866025i 0.811884 2.64302 − 0.120126i 2.71238 −0.500000 + 0.866025i 0.405942 + 0.703112i
331.6 0.943217 − 1.63370i 0.500000 + 0.866025i −0.779317 − 1.34982i −0.500000 + 0.866025i 1.88643 1.89726 + 1.84402i 0.832608 −0.500000 + 0.866025i 0.943217 + 1.63370i
331.7 1.08307 − 1.87593i 0.500000 + 0.866025i −1.34608 − 2.33148i −0.500000 + 0.866025i 2.16614 −2.27710 − 1.34716i −1.49931 −0.500000 + 0.866025i 1.08307 + 1.87593i
991.1 −1.11656 − 1.93394i 0.500000 − 0.866025i −1.49341 + 2.58666i −0.500000 − 0.866025i −2.23312 0.736549 + 2.54116i 2.20369 −0.500000 − 0.866025i −1.11656 + 1.93394i
991.2 −0.860135 − 1.48980i 0.500000 − 0.866025i −0.479666 + 0.830805i −0.500000 − 0.866025i −1.72027 −0.270725 − 2.63186i −1.79023 −0.500000 − 0.866025i −0.860135 + 1.48980i
991.3 −0.529611 − 0.917312i 0.500000 − 0.866025i 0.439025 − 0.760414i −0.500000 − 0.866025i −1.05922 −2.37300 − 1.16998i −3.04849 −0.500000 − 0.866025i −0.529611 + 0.917312i
991.4 0.0740769 + 0.128305i 0.500000 − 0.866025i 0.989025 − 1.71304i −0.500000 − 0.866025i 0.148154 −0.856004 + 2.50345i 0.589363 −0.500000 − 0.866025i 0.0740769 − 0.128305i
991.5 0.405942 + 0.703112i 0.500000 − 0.866025i 0.670422 − 1.16121i −0.500000 − 0.866025i 0.811884 2.64302 + 0.120126i 2.71238 −0.500000 − 0.866025i 0.405942 − 0.703112i
991.6 0.943217 + 1.63370i 0.500000 − 0.866025i −0.779317 + 1.34982i −0.500000 − 0.866025i 1.88643 1.89726 − 1.84402i 0.832608 −0.500000 − 0.866025i 0.943217 − 1.63370i
991.7 1.08307 + 1.87593i 0.500000 − 0.866025i −1.34608 + 2.33148i −0.500000 − 0.866025i 2.16614 −2.27710 + 1.34716i −1.49931 −0.500000 − 0.866025i 1.08307 − 1.87593i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.q.i ✓ 14
7.c even 3 1 inner 1155.2.q.i ✓ 14
7.c even 3 1 8085.2.a.ca 7
7.d odd 6 1 8085.2.a.cc 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.i ✓ 14 1.a even 1 1 trivial
1155.2.q.i ✓ 14 7.c even 3 1 inner
8085.2.a.ca 7 7.c even 3 1
8085.2.a.cc 7 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{14} + 9 T_{2}^{12} + 58 T_{2}^{10} + 179 T_{2}^{8} + 2 T_{2}^{7} + 403 T_{2}^{6} - 18 T_{2}^{5} + \cdots + 4 \) Copy content Toggle raw display
\( T_{13}^{7} + 7T_{13}^{6} - 21T_{13}^{5} - 231T_{13}^{4} - 165T_{13}^{3} + 1405T_{13}^{2} + 1225T_{13} - 2797 \) Copy content Toggle raw display
\( T_{17}^{14} - 6 T_{17}^{13} + 124 T_{17}^{12} - 688 T_{17}^{11} + 9824 T_{17}^{10} - 50688 T_{17}^{9} + \cdots + 1015824384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 9 T^{12} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{7} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$13$ \( (T^{7} + 7 T^{6} + \cdots - 2797)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 1015824384 \) Copy content Toggle raw display
$19$ \( T^{14} - 8 T^{13} + \cdots + 12845056 \) Copy content Toggle raw display
$23$ \( T^{14} - T^{13} + \cdots + 29073664 \) Copy content Toggle raw display
$29$ \( (T^{7} + 9 T^{6} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} - 20 T^{13} + \cdots + 29637136 \) Copy content Toggle raw display
$37$ \( T^{14} + 6 T^{13} + \cdots + 9216 \) Copy content Toggle raw display
$41$ \( (T^{7} + 5 T^{6} + \cdots + 30768)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} - 15 T^{6} + \cdots + 1690793)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 201412864 \) Copy content Toggle raw display
$53$ \( T^{14} + 13 T^{13} + \cdots + 20736 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 111497223744 \) Copy content Toggle raw display
$61$ \( T^{14} - 4 T^{13} + \cdots + 16777216 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 186375250944 \) Copy content Toggle raw display
$71$ \( (T^{7} + 26 T^{6} + \cdots - 338)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 33361575298704 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 153814564864 \) Copy content Toggle raw display
$83$ \( (T^{7} + 20 T^{6} + \cdots + 2145038)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 166759523044 \) Copy content Toggle raw display
$97$ \( (T^{7} + 10 T^{6} + \cdots - 1952)^{2} \) Copy content Toggle raw display
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