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Properties

Label 1170.2.cu.f
Level $1170$
Weight $2$
Character orbit 1170.cu
Analytic conductor $9.342$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,2,Mod(71,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.cu (of order \(12\), degree \(4\), 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: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{14} q^{2} + \beta_{5} q^{4} - \beta_{13} q^{5} + ( - \beta_{8} - \beta_{4} - \beta_{2} + 1) q^{7} + \beta_{15} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{14} q^{2} + \beta_{5} q^{4} - \beta_{13} q^{5} + ( - \beta_{8} - \beta_{4} - \beta_{2} + 1) q^{7} + \beta_{15} q^{8} - \beta_{4} q^{10} + (\beta_{15} + 2 \beta_{13} + \cdots + \beta_{6}) q^{11}+ \cdots + (\beta_{14} + 4 \beta_{13} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{7} - 8 q^{13} + 8 q^{16} - 52 q^{19} - 8 q^{22} - 12 q^{28} + 36 q^{31} + 16 q^{34} + 8 q^{37} - 16 q^{40} + 72 q^{43} + 32 q^{46} + 60 q^{49} + 12 q^{52} - 8 q^{55} - 8 q^{58} - 12 q^{61} - 12 q^{67} + 12 q^{70} + 8 q^{73} - 52 q^{76} + 8 q^{79} + 4 q^{85} + 24 q^{88} - 84 q^{91} - 16 q^{94} - 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 12x^{14} + 49x^{12} - 12x^{10} - 600x^{8} + 108x^{6} + 4057x^{4} + 18252x^{2} + 28561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97553 \nu^{14} - 59673197 \nu^{12} + 497612320 \nu^{10} - 1386197292 \nu^{8} + \cdots + 33039693843 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5554765 \nu^{15} + 146362481 \nu^{13} + 1743430336 \nu^{11} - 27938665412 \nu^{9} + \cdots - 2063244529359 \nu ) / 2964568772464 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4506447 \nu^{14} - 121241175 \nu^{12} + 1034556480 \nu^{10} - 3947923892 \nu^{8} + \cdots - 42326241815 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13282279 \nu^{14} + 200884791 \nu^{12} - 1177246560 \nu^{10} + 3219861076 \nu^{8} + \cdots - 296484914921 ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19681995 \nu^{15} + 459692017 \nu^{13} - 3195349744 \nu^{11} + 7267451924 \nu^{9} + \cdots - 67134153359 \nu ) / 2964568772464 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7844936 \nu^{14} + 136368445 \nu^{12} - 773884672 \nu^{10} + 1974780696 \nu^{8} + \cdots - 181212319067 ) / 114021875864 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1322533 \nu^{14} - 13200781 \nu^{12} + 41605136 \nu^{10} + 115043140 \nu^{8} + \cdots + 3326257155 ) / 17541827056 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 30695 \nu^{14} - 419209 \nu^{12} + 2520928 \nu^{10} - 7255428 \nu^{8} + 1602740 \nu^{6} + \cdots + 379906423 ) / 289763344 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 32743733 \nu^{15} - 1359249727 \nu^{13} + 12996379200 \nu^{11} - 61356708508 \nu^{9} + \cdots + 1574354566993 \nu ) / 2964568772464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2640659 \nu^{14} - 35680975 \nu^{12} + 174215264 \nu^{10} - 156875412 \nu^{8} + \cdots + 33548140145 ) / 17541827056 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4506447 \nu^{15} + 121241175 \nu^{13} - 1034556480 \nu^{11} + 3947923892 \nu^{9} + \cdots + 42326241815 \nu ) / 228043751728 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 599257 \nu^{15} - 8739462 \nu^{13} + 46755552 \nu^{11} - 102847788 \nu^{9} + \cdots + 9359839554 \nu ) / 14676083032 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 198509705 \nu^{15} + 2828387831 \nu^{13} - 15757060320 \nu^{11} + \cdots - 1729708183385 \nu ) / 2964568772464 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1582155 \nu^{15} - 21956880 \nu^{13} + 119050416 \nu^{11} - 229166428 \nu^{9} + \cdots + 14779694904 \nu ) / 14676083032 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{7} + 3\beta_{5} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{14} + 4\beta_{13} - 3\beta_{12} - 3\beta_{10} - 2\beta_{6} + \beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{9} - 2\beta_{8} - 2\beta_{7} + 12\beta_{5} - 7\beta_{4} + 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{15} + 25\beta_{14} + 15\beta_{13} - 5\beta_{12} - 10\beta_{10} - 34\beta_{6} + 8\beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{11} + 36\beta_{9} - 42\beta_{8} + 47\beta_{5} - 47\beta_{4} + 20\beta_{2} - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 138\beta_{15} + 233\beta_{14} - 5\beta_{10} - 122\beta_{6} + 16\beta_{3} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 233\beta_{11} + 44\beta_{9} - 260\beta_{8} + 130\beta_{7} + 167\beta_{5} - 74\beta_{4} - 44 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1133\beta_{15} + 1413\beta_{14} - 753\beta_{13} - 93\beta_{12} + 93\beta_{10} - 189\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1320\beta_{11} + 99\beta_{9} - 1133\beta_{8} + 1133\beta_{7} + 1133\beta_{5} + 1035\beta_{4} - 660\beta_{2} + 1122 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6325\beta_{15} + 6512\beta_{14} - 6512\beta_{13} - 2168\beta_{12} + 3773\beta_{6} - 1221\beta_{3} + 1221\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4344 \beta_{11} + 4344 \beta_{9} - 2552 \beta_{8} + 5104 \beta_{7} + 12156 \beta_{5} + 7052 \beta_{4} + \cdots + 12335 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 25032 \beta_{15} + 24156 \beta_{14} - 29104 \beta_{13} - 19208 \beta_{12} - 9604 \beta_{10} + \cdots + 16679 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 4948\beta_{11} + 64991\beta_{9} + 8353\beta_{7} + 103881\beta_{5} - 9896\beta_{2} + 55095 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 88188 \beta_{15} + 103881 \beta_{14} - 48256 \beta_{13} - 103881 \beta_{12} - 103881 \beta_{10} + \cdots + 120086 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
2.13219 − 1.08896i
−2.39101 + 0.123030i
−2.13219 + 1.08896i
2.39101 − 0.123030i
−0.115299 − 1.50155i
−0.850627 + 1.24273i
0.115299 + 1.50155i
0.850627 − 1.24273i
2.13219 + 1.08896i
−2.39101 − 0.123030i
−2.13219 − 1.08896i
2.39101 + 0.123030i
−0.115299 + 1.50155i
−0.850627 − 1.24273i
0.115299 − 1.50155i
0.850627 + 1.24273i
−0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.707107 + 0.707107i 0 −0.429711 + 1.60370i −0.707107 + 0.707107i 0 −0.866025 − 0.500000i
71.2 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.707107 + 0.707107i 0 0.197660 − 0.737676i −0.707107 + 0.707107i 0 −0.866025 − 0.500000i
71.3 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.707107 − 0.707107i 0 −0.429711 + 1.60370i 0.707107 − 0.707107i 0 −0.866025 − 0.500000i
71.4 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.707107 − 0.707107i 0 0.197660 − 0.737676i 0.707107 − 0.707107i 0 −0.866025 − 0.500000i
431.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.707107 − 0.707107i 0 −1.03475 + 0.277260i 0.707107 − 0.707107i 0 0.866025 − 0.500000i
431.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i −0.707107 − 0.707107i 0 4.26680 − 1.14329i 0.707107 − 0.707107i 0 0.866025 − 0.500000i
431.3 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.707107 + 0.707107i 0 −1.03475 + 0.277260i −0.707107 + 0.707107i 0 0.866025 − 0.500000i
431.4 0.258819 − 0.965926i 0 −0.866025 − 0.500000i 0.707107 + 0.707107i 0 4.26680 − 1.14329i −0.707107 + 0.707107i 0 0.866025 − 0.500000i
791.1 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.707107 − 0.707107i 0 −0.429711 − 1.60370i −0.707107 − 0.707107i 0 −0.866025 + 0.500000i
791.2 −0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.707107 − 0.707107i 0 0.197660 + 0.737676i −0.707107 − 0.707107i 0 −0.866025 + 0.500000i
791.3 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 −0.429711 − 1.60370i 0.707107 + 0.707107i 0 −0.866025 + 0.500000i
791.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 0.197660 + 0.737676i 0.707107 + 0.707107i 0 −0.866025 + 0.500000i
1151.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 −1.03475 − 0.277260i 0.707107 + 0.707107i 0 0.866025 + 0.500000i
1151.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 4.26680 + 1.14329i 0.707107 + 0.707107i 0 0.866025 + 0.500000i
1151.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.707107 − 0.707107i 0 −1.03475 − 0.277260i −0.707107 − 0.707107i 0 0.866025 + 0.500000i
1151.4 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0.707107 − 0.707107i 0 4.26680 + 1.14329i −0.707107 − 0.707107i 0 0.866025 + 0.500000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.cu.f ✓ 16
3.b odd 2 1 inner 1170.2.cu.f ✓ 16
13.f odd 12 1 inner 1170.2.cu.f ✓ 16
39.k even 12 1 inner 1170.2.cu.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.cu.f ✓ 16 1.a even 1 1 trivial
1170.2.cu.f ✓ 16 3.b odd 2 1 inner
1170.2.cu.f ✓ 16 13.f odd 12 1 inner
1170.2.cu.f ✓ 16 39.k even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{8} - 6T_{7}^{7} + 3T_{7}^{6} + 12T_{7}^{5} + 51T_{7}^{4} + 90T_{7}^{3} + 54T_{7}^{2} + 36T_{7} + 36 \) Copy content Toggle raw display
\( T_{17}^{16} + 44 T_{17}^{14} + 1612 T_{17}^{12} + 12848 T_{17}^{10} + 73744 T_{17}^{8} + 205568 T_{17}^{6} + \cdots + 65536 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{8} - 6 T^{7} + 3 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 24 T^{14} + \cdots + 28561 \) Copy content Toggle raw display
$13$ \( (T^{8} + 4 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 44 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{8} + 26 T^{7} + \cdots + 274576)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 454371856 \) Copy content Toggle raw display
$29$ \( T^{16} - 44 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$31$ \( (T^{8} - 18 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 4 T^{7} + \cdots + 1907161)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 256 T^{4} + 65536)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 36 T^{7} + \cdots + 1971216)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 4784350561 \) Copy content Toggle raw display
$53$ \( (T^{8} + 80 T^{6} + \cdots + 131044)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 429981696 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6 T^{7} + \cdots + 1272384)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 6 T^{7} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 38317882740736 \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{7} + \cdots + 173056)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots - 1052)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + 60 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$97$ \( (T^{8} + 48 T^{7} + \cdots + 20358144)^{2} \) Copy content Toggle raw display
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