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Properties

Label 960.3.bg.o
Level $960$
Weight $3$
Character orbit 960.bg
Analytic conductor $26.158$
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] = [960,3,Mod(193,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.bg (of order \(4\), degree \(2\), not 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: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 50x^{10} + 947x^{8} + 8580x^{6} + 38883x^{4} + 84370x^{2} + 69169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5 \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{7} + 1) q^{5} + ( - \beta_{10} - \beta_{7} - \beta_{6} + 1) q^{7} + 3 \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{7} + 1) q^{5} + ( - \beta_{10} - \beta_{7} - \beta_{6} + 1) q^{7} + 3 \beta_{6} q^{9} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots + 3) q^{11}+ \cdots + (3 \beta_{10} + 3 \beta_{9} - 3 \beta_{7} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{5} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{5} + 16 q^{7} + 32 q^{11} + 4 q^{13} + 28 q^{17} + 80 q^{23} - 4 q^{25} - 128 q^{31} - 24 q^{33} - 48 q^{35} + 92 q^{37} - 144 q^{41} - 64 q^{43} - 12 q^{45} + 80 q^{47} + 192 q^{51} - 100 q^{53} - 48 q^{55} + 96 q^{57} - 240 q^{61} + 48 q^{63} + 212 q^{65} - 160 q^{67} - 288 q^{71} - 4 q^{73} - 144 q^{75} + 144 q^{77} - 108 q^{81} - 256 q^{83} + 308 q^{85} - 96 q^{87} + 320 q^{91} + 48 q^{93} + 432 q^{95} + 220 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 50x^{10} + 947x^{8} + 8580x^{6} + 38883x^{4} + 84370x^{2} + 69169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 123 \nu^{11} - 789 \nu^{10} - 5887 \nu^{9} - 36031 \nu^{8} - 103594 \nu^{7} - 592802 \nu^{6} + \cdots - 12928291 ) / 210400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 76 \nu^{11} + 526 \nu^{10} + 3274 \nu^{9} + 24459 \nu^{8} + 48828 \nu^{7} + 411858 \nu^{6} + \cdots + 11296639 ) / 105200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 123 \nu^{11} - 789 \nu^{10} + 5887 \nu^{9} - 36031 \nu^{8} + 103594 \nu^{7} - 592802 \nu^{6} + \cdots - 12928291 ) / 210400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 76 \nu^{11} + 263 \nu^{10} - 3274 \nu^{9} + 11572 \nu^{8} - 48828 \nu^{7} + 175684 \nu^{6} + \cdots + 258792 ) / 105200 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 389 \nu^{11} + 263 \nu^{10} + 18661 \nu^{9} + 12887 \nu^{8} + 329722 \nu^{7} + 236174 \nu^{6} + \cdots + 11037847 ) / 210400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 209\nu^{11} + 9661\nu^{9} + 161892\nu^{7} + 1197788\nu^{5} + 3788099\nu^{3} + 4132751\nu ) / 105200 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 299 \nu^{11} - 1578 \nu^{10} + 14161 \nu^{9} - 73377 \nu^{8} + 244492 \nu^{7} - 1235574 \nu^{6} + \cdots - 30365717 ) / 105200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 338 \nu^{11} - 1578 \nu^{10} + 15322 \nu^{9} - 73377 \nu^{8} + 248024 \nu^{7} - 1240834 \nu^{6} + \cdots - 32580177 ) / 105200 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 799 \nu^{11} - 2367 \nu^{10} + 36531 \nu^{9} - 110723 \nu^{8} + 599642 \nu^{7} - 1873086 \nu^{6} + \cdots - 44536683 ) / 210400 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 561 \nu^{11} - 526 \nu^{10} - 26209 \nu^{9} - 24459 \nu^{8} - 443688 \nu^{7} - 417118 \nu^{6} + \cdots - 13511099 ) / 105200 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1098 \nu^{11} - 1578 \nu^{10} - 50692 \nu^{9} - 73377 \nu^{8} - 844134 \nu^{7} - 1235574 \nu^{6} + \cdots - 30365717 ) / 105200 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - 4 \beta_{6} + 4 \beta_{5} + \cdots - 2 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - 2 \beta_{10} + 4 \beta_{9} - 4 \beta_{8} - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 84 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 26 \beta_{11} - 52 \beta_{10} - 36 \beta_{9} + 16 \beta_{8} - 6 \beta_{7} + 82 \beta_{6} - 62 \beta_{5} + \cdots + 36 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{11} + 3 \beta_{10} - 8 \beta_{9} + 9 \beta_{8} + 2 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} + \cdots + 101 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 392 \beta_{11} + 794 \beta_{10} + 602 \beta_{9} - 192 \beta_{8} - 18 \beta_{7} - 1464 \beta_{6} + \cdots - 602 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 81 \beta_{11} - 452 \beta_{10} + 1484 \beta_{9} - 1774 \beta_{8} - 371 \beta_{7} + 1322 \beta_{6} + \cdots - 14344 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 6456 \beta_{11} - 12802 \beta_{10} - 10056 \beta_{9} + 2746 \beta_{8} + 854 \beta_{7} + 26762 \beta_{6} + \cdots + 10056 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 72 \beta_{11} + 736 \beta_{10} - 2776 \beta_{9} + 3368 \beta_{8} + 664 \beta_{7} - 2632 \beta_{6} + \cdots + 21961 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 111082 \beta_{11} + 211684 \beta_{10} + 169202 \beta_{9} - 42482 \beta_{8} - 15638 \beta_{7} + \cdots - 169202 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9761 \beta_{11} - 126002 \beta_{10} + 519764 \beta_{9} - 626244 \beta_{8} - 116241 \beta_{7} + \cdots - 3504004 ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1948786 \beta_{11} - 3555652 \beta_{10} - 2869076 \beta_{9} + 686576 \beta_{8} + 233714 \beta_{7} + \cdots + 2869076 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
2.01000i
3.63131i
− 4.19183i
− 3.07927i
1.49590i
− 1.86613i
− 2.01000i
− 3.63131i
4.19183i
3.07927i
− 1.49590i
1.86613i
0 −1.22474 − 1.22474i 0 −4.46869 + 2.24295i 0 3.87277 − 3.87277i 0 3.00000i 0
193.2 0 −1.22474 − 1.22474i 0 2.52446 − 4.31591i 0 2.70972 − 2.70972i 0 3.00000i 0
193.3 0 −1.22474 − 1.22474i 0 3.94423 + 3.07296i 0 −2.58249 + 2.58249i 0 3.00000i 0
193.4 0 1.22474 + 1.22474i 0 −2.65477 − 4.23700i 0 0.640101 − 0.640101i 0 3.00000i 0
193.5 0 1.22474 + 1.22474i 0 −0.339047 + 4.98849i 0 9.45605 − 9.45605i 0 3.00000i 0
193.6 0 1.22474 + 1.22474i 0 4.99382 + 0.248509i 0 −6.09615 + 6.09615i 0 3.00000i 0
577.1 0 −1.22474 + 1.22474i 0 −4.46869 − 2.24295i 0 3.87277 + 3.87277i 0 − 3.00000i 0
577.2 0 −1.22474 + 1.22474i 0 2.52446 + 4.31591i 0 2.70972 + 2.70972i 0 − 3.00000i 0
577.3 0 −1.22474 + 1.22474i 0 3.94423 − 3.07296i 0 −2.58249 − 2.58249i 0 − 3.00000i 0
577.4 0 1.22474 − 1.22474i 0 −2.65477 + 4.23700i 0 0.640101 + 0.640101i 0 − 3.00000i 0
577.5 0 1.22474 − 1.22474i 0 −0.339047 − 4.98849i 0 9.45605 + 9.45605i 0 − 3.00000i 0
577.6 0 1.22474 − 1.22474i 0 4.99382 − 0.248509i 0 −6.09615 − 6.09615i 0 − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.bg.o 12
4.b odd 2 1 960.3.bg.n 12
5.c odd 4 1 inner 960.3.bg.o 12
8.b even 2 1 480.3.bg.d yes 12
8.d odd 2 1 480.3.bg.c ✓ 12
20.e even 4 1 960.3.bg.n 12
40.i odd 4 1 480.3.bg.d yes 12
40.k even 4 1 480.3.bg.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.bg.c ✓ 12 8.d odd 2 1
480.3.bg.c ✓ 12 40.k even 4 1
480.3.bg.d yes 12 8.b even 2 1
480.3.bg.d yes 12 40.i odd 4 1
960.3.bg.n 12 4.b odd 2 1
960.3.bg.n 12 20.e even 4 1
960.3.bg.o 12 1.a even 1 1 trivial
960.3.bg.o 12 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 16 T_{7}^{11} + 128 T_{7}^{10} + 224 T_{7}^{9} + 7624 T_{7}^{8} - 94144 T_{7}^{7} + \cdots + 64000000 \) acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} - 16 T^{11} + \cdots + 64000000 \) Copy content Toggle raw display
$11$ \( (T^{6} - 16 T^{5} + \cdots - 3200)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1852441600 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 216730217497600 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 304376872960000 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 640000000000 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + 64 T^{5} + \cdots + 1884160)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + 72 T^{5} + \cdots - 270464000)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 81\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + 120 T^{5} + \cdots - 103808000)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + 144 T^{5} + \cdots - 507084800)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 38\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
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