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Properties

Label 968.4.a.r
Level $968$
Weight $4$
Character orbit 968.a
Self dual yes
Analytic conductor $57.114$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [968,4,Mod(1,968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(968, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("968.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 968 = 2^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 968.a (trivial)

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: yes
Analytic conductor: \(57.1138488856\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4 x^{9} - 193 x^{8} + 670 x^{7} + 10959 x^{6} - 33408 x^{5} - 177207 x^{4} + 365822 x^{3} + \cdots - 781744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 11^{2} \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{4} + 1) q^{5} + \beta_{3} q^{7} + (\beta_{2} - \beta_1 + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{4} + 1) q^{5} + \beta_{3} q^{7} + (\beta_{2} - \beta_1 + 14) q^{9} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \cdots - 5) q^{13}+ \cdots + ( - 4 \beta_{6} + 27 \beta_{5} + \cdots + 352) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37} + 524 q^{39} - 967 q^{41} + 380 q^{43} - 44 q^{45} + 769 q^{47} + 503 q^{49} + 956 q^{51} + 701 q^{53} - 1293 q^{57} + 1291 q^{59} + 1359 q^{61} - 929 q^{63} - 173 q^{65} + 2260 q^{67} + 1988 q^{69} + 465 q^{71} - 111 q^{73} + 4584 q^{75} - 1827 q^{79} + 6874 q^{81} + 4947 q^{83} - 2609 q^{85} - 1303 q^{87} + 446 q^{89} + 2176 q^{91} + 4204 q^{93} - 108 q^{95} + 3511 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 4 x^{9} - 193 x^{8} + 670 x^{7} + 10959 x^{6} - 33408 x^{5} - 177207 x^{4} + 365822 x^{3} + \cdots - 781744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 45881774 \nu^{9} - 308254005 \nu^{8} - 8199887677 \nu^{7} + 52771607612 \nu^{6} + \cdots - 12264499903016 ) / 16765066978000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 124726909 \nu^{9} - 655294705 \nu^{8} - 22030819032 \nu^{7} + 105853227992 \nu^{6} + \cdots - 362786607604856 ) / 8382533489000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 886103049301 \nu^{9} + 5110637218145 \nu^{8} + 157023230871898 \nu^{7} + \cdots - 10\!\cdots\!16 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5500012093053 \nu^{9} - 33421412056235 \nu^{8} - 943912252076344 \nu^{7} + \cdots - 13\!\cdots\!52 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45881774 \nu^{9} + 308254005 \nu^{8} + 8199887677 \nu^{7} - 52771607612 \nu^{6} + \cdots + 19884984893016 ) / 1524096998000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6115871966796 \nu^{9} - 20258898060520 \nu^{8} + \cdots - 16\!\cdots\!64 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14378178257272 \nu^{9} - 36675098233515 \nu^{8} + \cdots + 96\!\cdots\!52 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 3104915030216 \nu^{9} + 19632238269445 \nu^{8} + 545824901023943 \nu^{7} + \cdots + 14\!\cdots\!44 ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22691812274787 \nu^{9} - 138059936943815 \nu^{8} + \cdots - 13\!\cdots\!08 ) / 22\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - 11\beta _1 + 5 ) / 11 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{8} - 2\beta_{7} + 2\beta_{6} - 3\beta_{5} + 2\beta_{4} + 11\beta_{2} - 3\beta _1 + 438 ) / 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 33 \beta_{9} + \beta_{8} - 39 \beta_{7} + 61 \beta_{6} - 113 \beta_{5} + 50 \beta_{4} - 11 \beta_{3} + \cdots + 448 ) / 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 121 \beta_{9} + 646 \beta_{8} - 301 \beta_{7} + 367 \beta_{6} - 764 \beta_{5} + 587 \beta_{4} + \cdots + 35092 ) / 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4642 \beta_{9} + 2215 \beta_{8} - 4600 \beta_{7} + 7394 \beta_{6} - 15574 \beta_{5} + 8197 \beta_{4} + \cdots + 61694 ) / 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 21560 \beta_{9} + 82952 \beta_{8} - 39364 \beta_{7} + 52498 \beta_{6} - 116219 \beta_{5} + \cdots + 3263570 ) / 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 565719 \beta_{9} + 426733 \beta_{8} - 507149 \beta_{7} + 824499 \beta_{6} - 1945263 \beta_{5} + \cdots + 9243884 ) / 11 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3092067 \beta_{9} + 9967658 \beta_{8} - 4811239 \beta_{7} + 6780041 \beta_{6} - 15417334 \beta_{5} + \cdots + 322922600 ) / 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 66053768 \beta_{9} + 62722195 \beta_{8} - 56493166 \beta_{7} + 92430760 \beta_{6} + \cdots + 1307910274 ) / 11 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.75981
−9.47148
−2.61151
−3.06402
1.28082
−0.614830
3.51662
4.92098
7.99327
10.8099
0 −9.37785 0 −8.73196 0 12.2405 0 60.9440 0
1.2 0 −7.85344 0 1.64930 0 −8.79751 0 34.6766 0
1.3 0 −3.22954 0 20.4651 0 −6.65561 0 −16.5701 0
1.4 0 −1.44598 0 −9.31187 0 22.5481 0 −24.9091 0
1.5 0 0.662787 0 −9.39213 0 −19.7277 0 −26.5607 0
1.6 0 1.00320 0 6.07450 0 −26.7097 0 −25.9936 0
1.7 0 2.89859 0 −6.53312 0 24.8165 0 −18.5982 0
1.8 0 6.53902 0 21.4485 0 30.2596 0 15.7588 0
1.9 0 9.61131 0 −16.7145 0 −19.9185 0 65.3772 0
1.10 0 10.1919 0 14.0462 0 −11.0557 0 76.8751 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 968.4.a.r 10
4.b odd 2 1 1936.4.a.by 10
11.b odd 2 1 968.4.a.s 10
11.d odd 10 2 88.4.i.b ✓ 20
44.c even 2 1 1936.4.a.bx 10
44.g even 10 2 176.4.m.f 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.i.b ✓ 20 11.d odd 10 2
176.4.m.f 20 44.g even 10 2
968.4.a.r 10 1.a even 1 1 trivial
968.4.a.s 10 11.b odd 2 1
1936.4.a.bx 10 44.c even 2 1
1936.4.a.by 10 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(968))\):

\( T_{3}^{10} - 9 T_{3}^{9} - 165 T_{3}^{8} + 1426 T_{3}^{7} + 7565 T_{3}^{6} - 62039 T_{3}^{5} + \cdots + 424589 \) Copy content Toggle raw display
\( T_{7}^{10} + 3 T_{7}^{9} - 1962 T_{7}^{8} - 12337 T_{7}^{7} + 1320558 T_{7}^{6} + 12489139 T_{7}^{5} + \cdots + 1408143313424 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 9 T^{9} + \cdots + 424589 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 5151191824 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 1408143313424 \) Copy content Toggle raw display
$11$ \( T^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 36\!\cdots\!20 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 19107414793525 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 14\!\cdots\!59 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 65\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots - 29\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 67\!\cdots\!80 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 84\!\cdots\!05 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 45\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 23\!\cdots\!99 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 65\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 16\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 40\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 14\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 52\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 65\!\cdots\!81 \) Copy content Toggle raw display
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