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Properties

Label 1650.2.d.f
Level $1650$
Weight $2$
Character orbit 1650.d
Analytic conductor $13.175$
Analytic rank $0$
Dimension $8$
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] = [1650,2,Mod(1451,1650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1650.1451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.d (of order \(2\), degree \(1\), 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: \(13.1753163335\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.2051727616.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 330)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} + q^{4} + ( - \beta_{4} - \beta_{3}) q^{6} + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{6} + \beta_{5} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{4} - \beta_{3}) q^{3} + q^{4} + ( - \beta_{4} - \beta_{3}) q^{6} + ( - \beta_{7} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{6} - 4 \beta_{5} - 4 \beta_{4} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{6} + 8 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 2 q^{3} + 8 q^{4} - 2 q^{6} + 8 q^{8} + 2 q^{9} - 6 q^{11} - 2 q^{12} + 8 q^{16} + 4 q^{17} + 2 q^{18} + 8 q^{21} - 6 q^{22} - 2 q^{24} + 22 q^{27} + 4 q^{29} - 4 q^{31} + 8 q^{32} - 18 q^{33} + 4 q^{34} + 2 q^{36} + 12 q^{37} + 8 q^{39} + 16 q^{41} + 8 q^{42} - 6 q^{44} - 2 q^{48} - 4 q^{49} + 28 q^{51} + 22 q^{54} + 14 q^{57} + 4 q^{58} - 4 q^{62} + 14 q^{63} + 8 q^{64} - 18 q^{66} + 12 q^{67} + 4 q^{68} + 8 q^{69} + 2 q^{72} + 12 q^{74} + 36 q^{77} + 8 q^{78} + 2 q^{81} + 16 q^{82} - 72 q^{83} + 8 q^{84} - 22 q^{87} - 6 q^{88} - 48 q^{91} - 8 q^{93} - 2 q^{96} + 8 q^{97} - 4 q^{98} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 37x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 9\nu^{5} + 14\nu^{4} + 27\nu^{3} + 18\nu^{2} + 30\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 4\nu^{4} + 47\nu^{3} + 20\nu^{2} + 34\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} - 9\nu^{5} - 14\nu^{4} - 19\nu^{3} - 18\nu^{2} + 2\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 9\nu^{5} + 23\nu^{3} + 14\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{5} + 9\nu^{4} + 14\nu^{3} + 23\nu^{2} + 22\nu + 10 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 13\nu^{5} + 4\nu^{4} - 47\nu^{3} + 20\nu^{2} - 34\nu + 4 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} - 9\nu^{4} + 14\nu^{3} - 23\nu^{2} + 22\nu - 10 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 2\beta_{6} - 2\beta_{5} - 2\beta_{4} + 3\beta_{3} + 2\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5\beta_{7} + 7\beta_{6} - 5\beta_{5} - 5\beta_{4} - 5\beta_{3} + 7\beta_{2} + 5\beta _1 + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19\beta_{7} + 17\beta_{6} + 19\beta_{5} + 17\beta_{4} - 31\beta_{3} - 17\beta_{2} - 31\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{7} - 20\beta_{6} + 13\beta_{5} + 11\beta_{4} + 11\beta_{3} - 20\beta_{2} - 11\beta _1 - 67 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -93\beta_{7} - 75\beta_{6} - 93\beta_{5} - 67\beta_{4} + 155\beta_{3} + 75\beta_{2} + 155\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1650\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(727\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1451.1
1.18994i
− 1.18994i
0.356500i
− 0.356500i
− 2.25619i
2.25619i
2.08963i
− 2.08963i
1.00000 −1.38699 − 1.03743i 1.00000 0 −1.38699 − 1.03743i 0.394100i 1.00000 0.847487 + 2.87781i 0
1451.2 1.00000 −1.38699 + 1.03743i 1.00000 0 −1.38699 + 1.03743i − 0.394100i 1.00000 0.847487 − 2.87781i 0
1451.3 1.00000 −1.25820 − 1.19035i 1.00000 0 −1.25820 − 1.19035i − 3.22941i 1.00000 0.166154 + 2.99540i 0
1451.4 1.00000 −1.25820 + 1.19035i 1.00000 0 −1.25820 + 1.19035i 3.22941i 1.00000 0.166154 − 2.99540i 0
1451.5 1.00000 −0.0828988 − 1.73007i 1.00000 0 −0.0828988 − 1.73007i 4.34658i 1.00000 −2.98626 + 0.286841i 0
1451.6 1.00000 −0.0828988 + 1.73007i 1.00000 0 −0.0828988 + 1.73007i − 4.34658i 1.00000 −2.98626 − 0.286841i 0
1451.7 1.00000 1.72809 − 0.117016i 1.00000 0 1.72809 − 0.117016i − 0.723074i 1.00000 2.97261 − 0.404430i 0
1451.8 1.00000 1.72809 + 0.117016i 1.00000 0 1.72809 + 0.117016i 0.723074i 1.00000 2.97261 + 0.404430i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1451.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
33.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1650.2.d.f 8
3.b odd 2 1 1650.2.d.c 8
5.b even 2 1 330.2.d.a ✓ 8
5.c odd 4 1 1650.2.f.c 8
5.c odd 4 1 1650.2.f.e 8
11.b odd 2 1 1650.2.d.c 8
15.d odd 2 1 330.2.d.b yes 8
15.e even 4 1 1650.2.f.d 8
15.e even 4 1 1650.2.f.f 8
20.d odd 2 1 2640.2.f.b 8
33.d even 2 1 inner 1650.2.d.f 8
55.d odd 2 1 330.2.d.b yes 8
55.e even 4 1 1650.2.f.d 8
55.e even 4 1 1650.2.f.f 8
60.h even 2 1 2640.2.f.a 8
165.d even 2 1 330.2.d.a ✓ 8
165.l odd 4 1 1650.2.f.c 8
165.l odd 4 1 1650.2.f.e 8
220.g even 2 1 2640.2.f.a 8
660.g odd 2 1 2640.2.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.d.a ✓ 8 5.b even 2 1
330.2.d.a ✓ 8 165.d even 2 1
330.2.d.b yes 8 15.d odd 2 1
330.2.d.b yes 8 55.d odd 2 1
1650.2.d.c 8 3.b odd 2 1
1650.2.d.c 8 11.b odd 2 1
1650.2.d.f 8 1.a even 1 1 trivial
1650.2.d.f 8 33.d even 2 1 inner
1650.2.f.c 8 5.c odd 4 1
1650.2.f.c 8 165.l odd 4 1
1650.2.f.d 8 15.e even 4 1
1650.2.f.d 8 55.e even 4 1
1650.2.f.e 8 5.c odd 4 1
1650.2.f.e 8 165.l odd 4 1
1650.2.f.f 8 15.e even 4 1
1650.2.f.f 8 55.e even 4 1
2640.2.f.a 8 60.h even 2 1
2640.2.f.a 8 220.g even 2 1
2640.2.f.b 8 20.d odd 2 1
2640.2.f.b 8 660.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1650, [\chi])\):

\( T_{7}^{8} + 30T_{7}^{6} + 217T_{7}^{4} + 136T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{17}^{4} - 2T_{17}^{3} - 31T_{17}^{2} + 104T_{17} - 68 \) Copy content Toggle raw display
\( T_{29}^{4} - 2T_{29}^{3} - 43T_{29}^{2} + 50T_{29} + 452 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 2 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 30 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 6 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} - 31 T^{2} + \cdots - 68)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 98 T^{6} + \cdots + 73984 \) Copy content Toggle raw display
$23$ \( T^{8} + 104 T^{6} + \cdots + 25600 \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{3} + \cdots + 452)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2 T^{3} + \cdots + 2000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 6 T^{3} + \cdots - 1388)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} - 12 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 160 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{8} + 256 T^{6} + \cdots + 2715904 \) Copy content Toggle raw display
$53$ \( T^{8} + 58 T^{6} + \cdots + 2704 \) Copy content Toggle raw display
$59$ \( T^{8} + 188 T^{6} + \cdots + 640000 \) Copy content Toggle raw display
$61$ \( T^{8} + 310 T^{6} + \cdots + 712336 \) Copy content Toggle raw display
$67$ \( (T^{4} - 6 T^{3} + \cdots + 3824)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 170 T^{6} + \cdots + 150544 \) Copy content Toggle raw display
$73$ \( T^{8} + 380 T^{6} + \cdots + 1183744 \) Copy content Toggle raw display
$79$ \( T^{8} + 316 T^{6} + \cdots + 173056 \) Copy content Toggle raw display
$83$ \( (T^{4} + 36 T^{3} + \cdots - 2944)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 262 T^{6} + \cdots + 135424 \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots + 11408)^{2} \) Copy content Toggle raw display
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