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Properties

Label 1155.2.q.j
Level $1155$
Weight $2$
Character orbit 1155.q
Analytic conductor $9.223$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 13 x^{14} + 116 x^{12} + 545 x^{10} - 6 x^{9} + 1849 x^{8} + 78 x^{7} + 3192 x^{6} + 636 x^{5} + \cdots + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 13 x^{14} + 116 x^{12} + 545 x^{10} - 6 x^{9} + 1849 x^{8} + 78 x^{7} + 3192 x^{6} + 636 x^{5} + \cdots + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1052323422108 \nu^{15} + 6789491580751 \nu^{14} + 13752657090600 \nu^{13} + \cdots - 35\!\cdots\!08 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14301649856387 \nu^{15} - 51169199919132 \nu^{14} - 181712154444599 \nu^{13} + \cdots - 77\!\cdots\!12 ) / 57\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38695655880729 \nu^{15} - 126349633434392 \nu^{14} - 490125834971397 \nu^{13} + \cdots - 20\!\cdots\!04 ) / 57\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12792299979783 \nu^{15} + 1052323422108 \nu^{14} - 159510408156428 \nu^{13} + \cdots + 20\!\cdots\!28 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 168828305445689 \nu^{15} - 255109958373564 \nu^{14} + \cdots - 74\!\cdots\!40 ) / 17\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19975964629135 \nu^{15} - 106977830653263 \nu^{14} + 338914482978299 \nu^{13} + \cdots - 52\!\cdots\!96 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 258092904141113 \nu^{15} - 698629967463864 \nu^{14} + \cdots - 10\!\cdots\!52 ) / 17\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 87236550584191 \nu^{15} + 92337059649456 \nu^{14} + \cdots + 60\!\cdots\!68 ) / 57\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 225737113727417 \nu^{15} - 6968957788736 \nu^{14} + \cdots - 66\!\cdots\!40 ) / 57\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 186689425750364 \nu^{15} - 241742927215932 \nu^{14} + \cdots - 17\!\cdots\!16 ) / 43\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 81351716361131 \nu^{15} - 10564133477445 \nu^{14} - 963194108937821 \nu^{13} + \cdots - 65\!\cdots\!40 ) / 14\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 76696915955089 \nu^{15} + 54691716865524 \nu^{14} - 940644573385813 \nu^{13} + \cdots - 17\!\cdots\!60 ) / 13\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 304197521667406 \nu^{15} - 58973541029706 \nu^{14} + \cdots - 97\!\cdots\!92 ) / 43\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 36593349740146 \nu^{15} - 3305487068346 \nu^{14} - 456723671279011 \nu^{13} + \cdots - 52\!\cdots\!92 ) / 33\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 3\beta_{3} - \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} - \beta_{6} - 5\beta_{5} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + \beta_{11} - \beta_{8} - 7\beta_{4} + 14\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 7 \beta_{15} + 9 \beta_{14} + \beta_{11} - \beta_{9} - \beta_{8} - 2 \beta_{6} + 27 \beta_{5} + \cdots - 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{15} + \beta_{14} - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} - \beta_{9} + 10 \beta_{8} + \cdots + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 11 \beta_{14} - 44 \beta_{13} - 66 \beta_{11} - 13 \beta_{10} + 22 \beta_{8} + 55 \beta_{6} + \cdots + 152 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{15} - 90 \beta_{14} - 77 \beta_{11} + 13 \beta_{9} + 13 \beta_{7} + 77 \beta_{6} - 19 \beta_{5} + \cdots - 425 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 273 \beta_{15} - 363 \beta_{14} + 273 \beta_{13} + 363 \beta_{11} + 116 \beta_{10} + 116 \beta_{9} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 543 \beta_{14} + 122 \beta_{13} - 116 \beta_{12} + 665 \beta_{11} + 116 \beta_{10} - 543 \beta_{8} + \cdots - 224 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1697 \beta_{15} + 3021 \beta_{14} + 665 \beta_{11} - 891 \beta_{9} - 659 \beta_{8} - 1324 \beta_{6} + \cdots + 123 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1011 \beta_{15} + 1017 \beta_{14} - 1011 \beta_{13} + 891 \beta_{12} - 1017 \beta_{11} - 897 \beta_{10} + \cdots + 14851 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4703 \beta_{14} - 10598 \beta_{13} + 6 \beta_{12} - 19866 \beta_{11} - 6353 \beta_{10} + \cdots + 31482 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 7853 \beta_{15} - 32428 \beta_{14} - 24431 \beta_{11} + 6503 \beta_{9} + 144 \beta_{8} + 6347 \beta_{7} + \cdots - 90063 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 66511 \beta_{15} - 97145 \beta_{14} + 66511 \beta_{13} - 156 \beta_{12} + 97145 \beta_{11} + \cdots - 28936 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1 - \beta_{3}\)

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
−1.21150 − 2.09838i
−1.13705 − 1.96943i
−0.544976 − 0.943925i
−0.428252 − 0.741755i
0.307276 + 0.532217i
0.765367 + 1.32565i
0.967294 + 1.67540i
1.28184 + 2.22022i
−1.21150 + 2.09838i
−1.13705 + 1.96943i
−0.544976 + 0.943925i
−0.428252 + 0.741755i
0.307276 − 0.532217i
0.765367 − 1.32565i
0.967294 − 1.67540i
1.28184 − 2.22022i
−1.21150 + 2.09838i −0.500000 − 0.866025i −1.93546 − 3.35232i 0.500000 − 0.866025i 2.42300 −0.185653 − 2.63923i 4.53326 −0.500000 + 0.866025i 1.21150 + 2.09838i
331.2 −1.13705 + 1.96943i −0.500000 − 0.866025i −1.58577 − 2.74664i 0.500000 − 0.866025i 2.27410 1.22797 + 2.34352i 2.66421 −0.500000 + 0.866025i 1.13705 + 1.96943i
331.3 −0.544976 + 0.943925i −0.500000 − 0.866025i 0.406003 + 0.703218i 0.500000 − 0.866025i 1.08995 −0.0476864 + 2.64532i −3.06495 −0.500000 + 0.866025i 0.544976 + 0.943925i
331.4 −0.428252 + 0.741755i −0.500000 − 0.866025i 0.633200 + 1.09673i 0.500000 − 0.866025i 0.856504 −1.20148 − 2.35721i −2.79769 −0.500000 + 0.866025i 0.428252 + 0.741755i
331.5 0.307276 − 0.532217i −0.500000 − 0.866025i 0.811163 + 1.40498i 0.500000 − 0.866025i −0.614552 −2.24946 + 1.39281i 2.22611 −0.500000 + 0.866025i −0.307276 − 0.532217i
331.6 0.765367 − 1.32565i −0.500000 − 0.866025i −0.171572 − 0.297171i 0.500000 − 0.866025i −1.53073 1.95584 − 1.78176i 2.53620 −0.500000 + 0.866025i −0.765367 − 1.32565i
331.7 0.967294 − 1.67540i −0.500000 − 0.866025i −0.871314 − 1.50916i 0.500000 − 0.866025i −1.93459 2.59557 + 0.512871i 0.497910 −0.500000 + 0.866025i −0.967294 − 1.67540i
331.8 1.28184 − 2.22022i −0.500000 − 0.866025i −2.28624 − 3.95989i 0.500000 − 0.866025i −2.56369 −2.09509 + 1.61573i −6.59506 −0.500000 + 0.866025i −1.28184 − 2.22022i
991.1 −1.21150 − 2.09838i −0.500000 + 0.866025i −1.93546 + 3.35232i 0.500000 + 0.866025i 2.42300 −0.185653 + 2.63923i 4.53326 −0.500000 − 0.866025i 1.21150 − 2.09838i
991.2 −1.13705 − 1.96943i −0.500000 + 0.866025i −1.58577 + 2.74664i 0.500000 + 0.866025i 2.27410 1.22797 − 2.34352i 2.66421 −0.500000 − 0.866025i 1.13705 − 1.96943i
991.3 −0.544976 − 0.943925i −0.500000 + 0.866025i 0.406003 − 0.703218i 0.500000 + 0.866025i 1.08995 −0.0476864 − 2.64532i −3.06495 −0.500000 − 0.866025i 0.544976 − 0.943925i
991.4 −0.428252 − 0.741755i −0.500000 + 0.866025i 0.633200 − 1.09673i 0.500000 + 0.866025i 0.856504 −1.20148 + 2.35721i −2.79769 −0.500000 − 0.866025i 0.428252 − 0.741755i
991.5 0.307276 + 0.532217i −0.500000 + 0.866025i 0.811163 − 1.40498i 0.500000 + 0.866025i −0.614552 −2.24946 − 1.39281i 2.22611 −0.500000 − 0.866025i −0.307276 + 0.532217i
991.6 0.765367 + 1.32565i −0.500000 + 0.866025i −0.171572 + 0.297171i 0.500000 + 0.866025i −1.53073 1.95584 + 1.78176i 2.53620 −0.500000 − 0.866025i −0.765367 + 1.32565i
991.7 0.967294 + 1.67540i −0.500000 + 0.866025i −0.871314 + 1.50916i 0.500000 + 0.866025i −1.93459 2.59557 − 0.512871i 0.497910 −0.500000 − 0.866025i −0.967294 + 1.67540i
991.8 1.28184 + 2.22022i −0.500000 + 0.866025i −2.28624 + 3.95989i 0.500000 + 0.866025i −2.56369 −2.09509 − 1.61573i −6.59506 −0.500000 − 0.866025i −1.28184 + 2.22022i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.8
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.j ✓ 16
7.c even 3 1 inner 1155.2.q.j ✓ 16
7.c even 3 1 8085.2.a.cf 8
7.d odd 6 1 8085.2.a.ce 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.j ✓ 16 1.a even 1 1 trivial
1155.2.q.j ✓ 16 7.c even 3 1 inner
8085.2.a.ce 8 7.d odd 6 1
8085.2.a.cf 8 7.c even 3 1

Hecke kernels

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

\( T_{2}^{16} + 13 T_{2}^{14} + 116 T_{2}^{12} + 545 T_{2}^{10} - 6 T_{2}^{9} + 1849 T_{2}^{8} + 78 T_{2}^{7} + \cdots + 576 \) Copy content Toggle raw display
\( T_{13}^{8} - 4T_{13}^{7} - 70T_{13}^{6} + 236T_{13}^{5} + 1320T_{13}^{4} - 2436T_{13}^{3} - 8810T_{13}^{2} - 3428T_{13} + 7 \) Copy content Toggle raw display
\( T_{17}^{16} + 4 T_{17}^{15} + 92 T_{17}^{14} + 112 T_{17}^{13} + 4624 T_{17}^{12} + 2816 T_{17}^{11} + \cdots + 897122304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 13 T^{14} + \cdots + 576 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + 10 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} - 4 T^{7} - 70 T^{6} + \cdots + 7)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 897122304 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 31655526400 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 144000000 \) Copy content Toggle raw display
$29$ \( (T^{8} + 5 T^{7} + \cdots - 2048736)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 5 T^{15} + \cdots + 3168400 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 5738056139776 \) Copy content Toggle raw display
$41$ \( (T^{8} - 9 T^{7} + \cdots + 777600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 14 T^{7} + \cdots + 84133)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 10282847929344 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 890894352384 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1055990016 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 579076096 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 674605966336 \) Copy content Toggle raw display
$71$ \( (T^{8} + 30 T^{7} + \cdots + 2088384)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 5186880400 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 334731673600 \) Copy content Toggle raw display
$83$ \( (T^{8} - 12 T^{7} + \cdots - 35281392)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 2634152976 \) Copy content Toggle raw display
$97$ \( (T^{8} - 44 T^{7} + \cdots + 7517120)^{2} \) Copy content Toggle raw display
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