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Properties

Label 1155.2.q.h
Level $1155$
Weight $2$
Character orbit 1155.q
Analytic conductor $9.223$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 9x^{10} + 61x^{8} - 2x^{7} + 164x^{6} - 36x^{5} + 328x^{4} - 40x^{3} + 164x^{2} + 16x + 64 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 9x^{10} + 61x^{8} - 2x^{7} + 164x^{6} - 36x^{5} + 328x^{4} - 40x^{3} + 164x^{2} + 16x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 157070 \nu^{11} + 761011 \nu^{10} - 1380620 \nu^{9} + 5123059 \nu^{8} - 9188422 \nu^{7} + \cdots - 446661052 ) / 179226324 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1618579 \nu^{11} - 5688620 \nu^{10} + 13938931 \nu^{9} - 48153536 \nu^{8} + 93210839 \nu^{7} + \cdots - 138040624 ) / 716905296 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 676159 \nu^{11} - 1122554 \nu^{10} + 5655211 \nu^{9} - 17415182 \nu^{8} + 38080307 \nu^{7} + \cdots - 308838400 ) / 179226324 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1618579 \nu^{11} + 5688620 \nu^{10} - 13938931 \nu^{9} + 48153536 \nu^{8} - 93210839 \nu^{7} + \cdots + 138040624 ) / 238968432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1861165 \nu^{11} - 2114092 \nu^{10} - 11467045 \nu^{9} - 18061444 \nu^{8} - 64914857 \nu^{7} + \cdots - 330485120 ) / 238968432 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1422155 \nu^{11} + 157070 \nu^{10} + 12038384 \nu^{9} + 1380620 \nu^{8} + 81628396 \nu^{7} + \cdots + 25897264 ) / 179226324 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8627539 \nu^{11} - 6474316 \nu^{10} - 54893371 \nu^{9} - 55755724 \nu^{8} - 333665735 \nu^{7} + \cdots - 1031665280 ) / 716905296 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2331565 \nu^{11} - 2682956 \nu^{10} + 19929256 \nu^{9} - 17666354 \nu^{8} + 134479856 \nu^{7} + \cdots + 245399744 ) / 179226324 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13718905 \nu^{11} - 3637640 \nu^{10} + 134830249 \nu^{9} - 31563488 \nu^{8} + 913041101 \nu^{7} + \cdots + 135580880 ) / 716905296 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 39117961 \nu^{11} + 628280 \nu^{10} - 355105693 \nu^{9} + 5522480 \nu^{8} - 2406687857 \nu^{7} + \cdots - 613316240 ) / 716905296 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} - 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{8} + 5\beta_{7} - \beta_{6} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{5} + \beta_{4} + 14\beta_{3} - 6\beta_{2} - \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{11} - 9\beta_{10} - 27\beta_{7} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{9} - 11\beta_{7} + 34\beta_{2} + 11\beta _1 + 74 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -43\beta_{8} + 61\beta_{6} - 2\beta_{5} - 24\beta_{3} + 2\beta_{2} + 151\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{11} + 2\beta_{10} - 61\beta_{9} + 89\beta_{7} - 194\beta_{5} - 61\beta_{4} - 410\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 255 \beta_{11} + 377 \beta_{10} - 2 \beta_{9} + 255 \beta_{8} + 861 \beta_{7} - 377 \beta_{6} + \cdots - 204 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32\beta_{8} - 36\beta_{6} + 1114\beta_{5} + 377\beta_{4} + 2326\beta_{3} - 1114\beta_{2} - 643\beta _1 - 2326 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -1491\beta_{11} - 2245\beta_{10} + 36\beta_{9} - 4967\beta_{7} + 298\beta_{5} + 36\beta_{4} + 1520\beta_{3} \) 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
−1.16512 + 2.01805i
−0.898535 + 1.55631i
−0.319182 + 0.552839i
0.408375 − 0.707327i
0.745141 − 1.29062i
1.22932 − 2.12925i
−1.16512 − 2.01805i
−0.898535 − 1.55631i
−0.319182 − 0.552839i
0.408375 + 0.707327i
0.745141 + 1.29062i
1.22932 + 2.12925i
−1.16512 + 2.01805i −0.500000 − 0.866025i −1.71503 − 2.97052i 0.500000 − 0.866025i 2.33025 −1.60347 − 2.10449i 3.33238 −0.500000 + 0.866025i 1.16512 + 2.01805i
331.2 −0.898535 + 1.55631i −0.500000 − 0.866025i −0.614730 − 1.06474i 0.500000 − 0.866025i 1.79707 2.63828 − 0.198718i −1.38471 −0.500000 + 0.866025i 0.898535 + 1.55631i
331.3 −0.319182 + 0.552839i −0.500000 − 0.866025i 0.796246 + 1.37914i 0.500000 − 0.866025i 0.638364 −1.39973 + 2.24516i −2.29332 −0.500000 + 0.866025i 0.319182 + 0.552839i
331.4 0.408375 − 0.707327i −0.500000 − 0.866025i 0.666459 + 1.15434i 0.500000 − 0.866025i −0.816751 1.70825 + 2.02037i 2.72216 −0.500000 + 0.866025i −0.408375 − 0.707327i
331.5 0.745141 − 1.29062i −0.500000 − 0.866025i −0.110472 − 0.191342i 0.500000 − 0.866025i −1.49028 −2.55828 + 0.674683i 2.65130 −0.500000 + 0.866025i −0.745141 − 1.29062i
331.6 1.22932 − 2.12925i −0.500000 − 0.866025i −2.02248 − 3.50303i 0.500000 − 0.866025i −2.45865 0.214953 − 2.63700i −5.02782 −0.500000 + 0.866025i −1.22932 − 2.12925i
991.1 −1.16512 − 2.01805i −0.500000 + 0.866025i −1.71503 + 2.97052i 0.500000 + 0.866025i 2.33025 −1.60347 + 2.10449i 3.33238 −0.500000 − 0.866025i 1.16512 − 2.01805i
991.2 −0.898535 − 1.55631i −0.500000 + 0.866025i −0.614730 + 1.06474i 0.500000 + 0.866025i 1.79707 2.63828 + 0.198718i −1.38471 −0.500000 − 0.866025i 0.898535 − 1.55631i
991.3 −0.319182 − 0.552839i −0.500000 + 0.866025i 0.796246 − 1.37914i 0.500000 + 0.866025i 0.638364 −1.39973 − 2.24516i −2.29332 −0.500000 − 0.866025i 0.319182 − 0.552839i
991.4 0.408375 + 0.707327i −0.500000 + 0.866025i 0.666459 − 1.15434i 0.500000 + 0.866025i −0.816751 1.70825 − 2.02037i 2.72216 −0.500000 − 0.866025i −0.408375 + 0.707327i
991.5 0.745141 + 1.29062i −0.500000 + 0.866025i −0.110472 + 0.191342i 0.500000 + 0.866025i −1.49028 −2.55828 − 0.674683i 2.65130 −0.500000 − 0.866025i −0.745141 + 1.29062i
991.6 1.22932 + 2.12925i −0.500000 + 0.866025i −2.02248 + 3.50303i 0.500000 + 0.866025i −2.45865 0.214953 + 2.63700i −5.02782 −0.500000 − 0.866025i −1.22932 + 2.12925i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.6
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.h ✓ 12
7.c even 3 1 inner 1155.2.q.h ✓ 12
7.c even 3 1 8085.2.a.bz 6
7.d odd 6 1 8085.2.a.bx 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.q.h ✓ 12 1.a even 1 1 trivial
1155.2.q.h ✓ 12 7.c even 3 1 inner
8085.2.a.bx 6 7.d odd 6 1
8085.2.a.bz 6 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 \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\( T_{2}^{12} + 9T_{2}^{10} + 61T_{2}^{8} - 2T_{2}^{7} + 164T_{2}^{6} - 36T_{2}^{5} + 328T_{2}^{4} - 40T_{2}^{3} + 164T_{2}^{2} + 16T_{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{6} + 4T_{13}^{5} - 42T_{13}^{4} - 160T_{13}^{3} + 37T_{13}^{2} + 20T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{12} + 2 T_{17}^{11} + 57 T_{17}^{10} - 122 T_{17}^{9} + 2193 T_{17}^{8} - 3112 T_{17}^{7} + \cdots + 2408704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 9 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{12} + 2 T^{11} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T^{6} + 4 T^{5} - 42 T^{4} + \cdots - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + \cdots + 2408704 \) Copy content Toggle raw display
$19$ \( T^{12} - 13 T^{11} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{12} - 7 T^{11} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( (T^{6} - 3 T^{5} + \cdots + 5238)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 15 T^{11} + \cdots + 495616 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 1574978596 \) Copy content Toggle raw display
$41$ \( (T^{6} - 3 T^{5} + \cdots - 384)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 4 T^{5} + \cdots + 293)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 145781476 \) Copy content Toggle raw display
$53$ \( T^{12} - T^{11} + \cdots + 66064384 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 29578496256 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 17994612736 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1660073536 \) Copy content Toggle raw display
$71$ \( (T^{6} - 8 T^{5} - 139 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 11708971264 \) Copy content Toggle raw display
$79$ \( T^{12} - 13 T^{11} + \cdots + 65536 \) Copy content Toggle raw display
$83$ \( (T^{6} + 10 T^{5} + \cdots + 6912)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2368963584 \) Copy content Toggle raw display
$97$ \( (T^{6} + 6 T^{5} + \cdots + 556)^{2} \) Copy content Toggle raw display
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