Properties

Label 99.4.d.c
Level $99$
Weight $4$
Character orbit 99.d
Analytic conductor $5.841$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,4,Mod(98,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.98");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 99.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: \(5.84118909057\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12261951429820416.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 62x^{6} + 1113x^{4} + 5786x^{2} + 5776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 5) q^{4} - 5 \beta_{6} q^{5} + ( - \beta_{5} - 2 \beta_{4}) q^{7} + (4 \beta_{2} + 7 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} + 5) q^{4} - 5 \beta_{6} q^{5} + ( - \beta_{5} - 2 \beta_{4}) q^{7} + (4 \beta_{2} + 7 \beta_1) q^{8} - 5 \beta_{4} q^{10} + (\beta_{7} + 12 \beta_{6} + 5 \beta_{2}) q^{11} + ( - 8 \beta_{5} + 2 \beta_{4}) q^{13} + ( - \beta_{7} - 25 \beta_{6}) q^{14} + (3 \beta_{3} + 55) q^{16} + ( - 2 \beta_{2} - 22 \beta_1) q^{17} + ( - 5 \beta_{5} + 10 \beta_{4}) q^{19} + ( - 5 \beta_{7} - 25 \beta_{6}) q^{20} + ( - 4 \beta_{5} + 22 \beta_{4} + 5 \beta_{3} + 5) q^{22} + ( - 10 \beta_{7} - 30 \beta_{6}) q^{23} + 75 q^{25} + (10 \beta_{7} + 34 \beta_{6}) q^{26} + (12 \beta_{5} - 19 \beta_{4}) q^{28} + (10 \beta_{2} - 30 \beta_1) q^{29} + ( - 12 \beta_{3} - 4) q^{31} + ( - 20 \beta_{2} + 29 \beta_1) q^{32} + ( - 24 \beta_{3} - 288) q^{34} + (10 \beta_{2} - 20 \beta_1) q^{35} + (20 \beta_{3} - 80) q^{37} + (15 \beta_{7} + 135 \beta_{6}) q^{38} + (20 \beta_{5} - 35 \beta_{4}) q^{40} + ( - 30 \beta_{2} + 50 \beta_1) q^{41} + (3 \beta_{5} + 38 \beta_{4}) q^{43} + (18 \beta_{7} + 194 \beta_{6} - 20 \beta_{2} + 55 \beta_1) q^{44} + (40 \beta_{5} - 130 \beta_{4}) q^{46} - 170 \beta_{6} q^{47} + (4 \beta_{3} + 185) q^{49} + 75 \beta_1 q^{50} + (24 \beta_{5} + 118 \beta_{4}) q^{52} + ( - 20 \beta_{7} + 5 \beta_{6}) q^{53} + (25 \beta_{5} + 10 \beta_{3} + 120) q^{55} + ( - 23 \beta_{7} - 59 \beta_{6}) q^{56} + ( - 20 \beta_{3} - 380) q^{58} + (30 \beta_{7} - 92 \beta_{6}) q^{59} + ( - 50 \beta_{5} + 60 \beta_{4}) q^{61} + ( - 48 \beta_{2} - 124 \beta_1) q^{62} + ( - 15 \beta_{3} - 83) q^{64} + (80 \beta_{2} + 20 \beta_1) q^{65} + 260 q^{67} + ( - 80 \beta_{2} - 352 \beta_1) q^{68} + ( - 10 \beta_{3} - 250) q^{70} + ( - 20 \beta_{7} + 254 \beta_{6}) q^{71} + ( - 82 \beta_{5} - 154 \beta_{4}) q^{73} + (80 \beta_{2} + 120 \beta_1) q^{74} + ( - 20 \beta_{5} + 205 \beta_{4}) q^{76} + ( - 20 \beta_{7} + 145 \beta_{6} + 10 \beta_{2} + 66 \beta_1) q^{77} + ( - 55 \beta_{5} - 170 \beta_{4}) q^{79} + ( - 15 \beta_{7} - 275 \beta_{6}) q^{80} + (20 \beta_{3} + 620) q^{82} + (60 \beta_{2} - 68 \beta_1) q^{83} + ( - 10 \beta_{5} + 110 \beta_{4}) q^{85} + (35 \beta_{7} + 491 \beta_{6}) q^{86} + ( - 40 \beta_{5} + 198 \beta_{4} - 5 \beta_{3} + 655) q^{88} + (36 \beta_{7} - 443 \beta_{6}) q^{89} + (68 \beta_{3} - 364) q^{91} + ( - 90 \beta_{7} - 1490 \beta_{6}) q^{92} - 170 \beta_{4} q^{94} + (50 \beta_{2} + 100 \beta_1) q^{95} + (60 \beta_{3} + 1070) q^{97} + (16 \beta_{2} + 225 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 44 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 44 q^{4} + 452 q^{16} + 60 q^{22} + 600 q^{25} - 80 q^{31} - 2400 q^{34} - 560 q^{37} + 1496 q^{49} + 1000 q^{55} - 3120 q^{58} - 724 q^{64} + 2080 q^{67} - 2040 q^{70} + 5040 q^{82} + 5220 q^{88} - 2640 q^{91} + 8800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 62x^{6} + 1113x^{4} + 5786x^{2} + 5776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -17\nu^{6} - 1018\nu^{4} - 15948\nu^{2} - 42252 ) / 4631 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} - 95\nu^{4} - 1059\nu^{2} - 2321 ) / 421 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} - 95\nu^{4} - 638\nu^{2} + 4415 ) / 421 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -36\nu^{7} - 2973\nu^{5} - 79265\nu^{3} - 644650\nu ) / 87989 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\nu^{7} + 1398\nu^{5} + 20605\nu^{3} + 128158\nu ) / 31996 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 25\nu^{7} + 1398\nu^{5} + 20605\nu^{3} + 64166\nu ) / 31996 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 223\nu^{7} + 13750\nu^{5} + 228591\nu^{3} + 754098\nu ) / 31996 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 47\beta_{6} - 26\beta_{5} - 11\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -33\beta_{3} + 50\beta_{2} - 22\beta _1 + 421 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 155\beta_{7} - 1949\beta_{6} + 768\beta_{5} + 385\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1038\beta_{3} - 2056\beta_{2} + 1045\beta _1 - 12686 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -6195\beta_{7} + 75377\beta_{6} - 24084\beta_{5} - 12463\beta_{4} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
98.1
1.14042i
1.14042i
5.86333i
5.86333i
4.44912i
4.44912i
2.55464i
2.55464i
−5.00866 0 17.0866 7.07107i 0 10.4716i −45.5118 0 35.4165i
98.2 −5.00866 0 17.0866 7.07107i 0 10.4716i −45.5118 0 35.4165i
98.3 −1.38325 0 −6.08663 7.07107i 0 14.2249i 19.4853 0 9.78103i
98.4 −1.38325 0 −6.08663 7.07107i 0 14.2249i 19.4853 0 9.78103i
98.5 1.38325 0 −6.08663 7.07107i 0 14.2249i −19.4853 0 9.78103i
98.6 1.38325 0 −6.08663 7.07107i 0 14.2249i −19.4853 0 9.78103i
98.7 5.00866 0 17.0866 7.07107i 0 10.4716i 45.5118 0 35.4165i
98.8 5.00866 0 17.0866 7.07107i 0 10.4716i 45.5118 0 35.4165i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 98.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.b odd 2 1 inner
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 99.4.d.c 8
3.b odd 2 1 inner 99.4.d.c 8
4.b odd 2 1 1584.4.b.g 8
11.b odd 2 1 inner 99.4.d.c 8
12.b even 2 1 1584.4.b.g 8
33.d even 2 1 inner 99.4.d.c 8
44.c even 2 1 1584.4.b.g 8
132.d odd 2 1 1584.4.b.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.4.d.c 8 1.a even 1 1 trivial
99.4.d.c 8 3.b odd 2 1 inner
99.4.d.c 8 11.b odd 2 1 inner
99.4.d.c 8 33.d even 2 1 inner
1584.4.b.g 8 4.b odd 2 1
1584.4.b.g 8 12.b even 2 1
1584.4.b.g 8 44.c even 2 1
1584.4.b.g 8 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 27T_{2}^{2} + 48 \) acting on \(S_{4}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 27 T^{2} + 48)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 312 T^{2} + 22188)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 676 T^{6} + \cdots + 3138428376721 \) Copy content Toggle raw display
$13$ \( (T^{4} + 8088 T^{2} + 11808768)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 13572 T^{2} + 3345408)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 9000 T^{2} + 8167500)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 58600 T^{2} + \cdots + 595360000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 28500 T^{2} + \cdots + 201720000)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 20 T - 19232)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 140 T - 48800)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 112500 T^{2} + \cdots + 2453880000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 77688 T^{2} + \cdots + 738151788)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 57800)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 214900 T^{2} + \cdots + 11524022500)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 507016 T^{2} + \cdots + 52804363264)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 530400 T^{2} + \cdots + 58968120000)^{2} \) Copy content Toggle raw display
$67$ \( (T - 260)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 452944 T^{2} + \cdots + 136235584)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1936008 T^{2} + \cdots + 816399986688)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1811400 T^{2} + \cdots + 811044007500)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 316368 T^{2} + \cdots + 9548295168)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 1418452 T^{2} + \cdots + 176199076)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2200 T + 726700)^{4} \) Copy content Toggle raw display
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