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Properties

Label 117.4.b.a
Level $117$
Weight $4$
Character orbit 117.b
Analytic conductor $6.903$
Analytic rank $0$
Dimension $2$
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] = [117,4,Mod(64,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.64");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.b (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: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 13)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} - 3 i q^{5} + 5 i q^{7} + 7 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - q^{4} - 3 i q^{5} + 5 i q^{7} + 7 i q^{8} + 27 q^{10} + 16 i q^{11} + (13 i + 26) q^{13} - 45 q^{14} - 71 q^{16} + 45 q^{17} - 2 i q^{19} + 3 i q^{20} - 144 q^{22} - 162 q^{23} + 44 q^{25} + (26 i - 117) q^{26} - 5 i q^{28} + 144 q^{29} - 88 i q^{31} - 15 i q^{32} + 45 i q^{34} + 135 q^{35} + 101 i q^{37} + 18 q^{38} + 189 q^{40} - 64 i q^{41} - 97 q^{43} - 16 i q^{44} - 162 i q^{46} - 37 i q^{47} + 118 q^{49} + 44 i q^{50} + ( - 13 i - 26) q^{52} + 414 q^{53} + 432 q^{55} - 315 q^{56} + 144 i q^{58} - 174 i q^{59} + 376 q^{61} + 792 q^{62} - 433 q^{64} + ( - 78 i + 351) q^{65} + 12 i q^{67} - 45 q^{68} + 135 i q^{70} + 119 i q^{71} - 366 i q^{73} - 909 q^{74} + 2 i q^{76} - 720 q^{77} - 830 q^{79} + 213 i q^{80} + 576 q^{82} - 146 i q^{83} - 135 i q^{85} - 97 i q^{86} - 1008 q^{88} + 146 i q^{89} + (130 i - 585) q^{91} + 162 q^{92} + 333 q^{94} - 54 q^{95} + 284 i q^{97} + 118 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 54 q^{10} + 52 q^{13} - 90 q^{14} - 142 q^{16} + 90 q^{17} - 288 q^{22} - 324 q^{23} + 88 q^{25} - 234 q^{26} + 288 q^{29} + 270 q^{35} + 36 q^{38} + 378 q^{40} - 194 q^{43} + 236 q^{49} - 52 q^{52} + 828 q^{53} + 864 q^{55} - 630 q^{56} + 752 q^{61} + 1584 q^{62} - 866 q^{64} + 702 q^{65} - 90 q^{68} - 1818 q^{74} - 1440 q^{77} - 1660 q^{79} + 1152 q^{82} - 2016 q^{88} - 1170 q^{91} + 324 q^{92} + 666 q^{94} - 108 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
− 1.00000i
1.00000i
− 3.00000i 0 −1.00000 9.00000i 0 − 15.0000i − 21.0000i 0 27.0000
64.2 3.00000i 0 −1.00000 − 9.00000i 0 15.0000i 21.0000i 0 27.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.b.a 2
3.b odd 2 1 13.4.b.a ✓ 2
12.b even 2 1 208.4.f.b 2
13.b even 2 1 inner 117.4.b.a 2
13.d odd 4 1 1521.4.a.d 1
13.d odd 4 1 1521.4.a.i 1
15.d odd 2 1 325.4.c.b 2
15.e even 4 1 325.4.d.a 2
15.e even 4 1 325.4.d.b 2
24.f even 2 1 832.4.f.c 2
24.h odd 2 1 832.4.f.e 2
39.d odd 2 1 13.4.b.a ✓ 2
39.f even 4 1 169.4.a.b 1
39.f even 4 1 169.4.a.c 1
39.h odd 6 2 169.4.e.d 4
39.i odd 6 2 169.4.e.d 4
39.k even 12 2 169.4.c.b 2
39.k even 12 2 169.4.c.c 2
156.h even 2 1 208.4.f.b 2
195.e odd 2 1 325.4.c.b 2
195.s even 4 1 325.4.d.a 2
195.s even 4 1 325.4.d.b 2
312.b odd 2 1 832.4.f.e 2
312.h even 2 1 832.4.f.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a ✓ 2 3.b odd 2 1
13.4.b.a ✓ 2 39.d odd 2 1
117.4.b.a 2 1.a even 1 1 trivial
117.4.b.a 2 13.b even 2 1 inner
169.4.a.b 1 39.f even 4 1
169.4.a.c 1 39.f even 4 1
169.4.c.b 2 39.k even 12 2
169.4.c.c 2 39.k even 12 2
169.4.e.d 4 39.h odd 6 2
169.4.e.d 4 39.i odd 6 2
208.4.f.b 2 12.b even 2 1
208.4.f.b 2 156.h even 2 1
325.4.c.b 2 15.d odd 2 1
325.4.c.b 2 195.e odd 2 1
325.4.d.a 2 15.e even 4 1
325.4.d.a 2 195.s even 4 1
325.4.d.b 2 15.e even 4 1
325.4.d.b 2 195.s even 4 1
832.4.f.c 2 24.f even 2 1
832.4.f.c 2 312.h even 2 1
832.4.f.e 2 24.h odd 2 1
832.4.f.e 2 312.b odd 2 1
1521.4.a.d 1 13.d odd 4 1
1521.4.a.i 1 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 225 \) Copy content Toggle raw display
$11$ \( T^{2} + 2304 \) Copy content Toggle raw display
$13$ \( T^{2} - 52T + 2197 \) Copy content Toggle raw display
$17$ \( (T - 45)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( (T + 162)^{2} \) Copy content Toggle raw display
$29$ \( (T - 144)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 69696 \) Copy content Toggle raw display
$37$ \( T^{2} + 91809 \) Copy content Toggle raw display
$41$ \( T^{2} + 36864 \) Copy content Toggle raw display
$43$ \( (T + 97)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12321 \) Copy content Toggle raw display
$53$ \( (T - 414)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 272484 \) Copy content Toggle raw display
$61$ \( (T - 376)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( T^{2} + 127449 \) Copy content Toggle raw display
$73$ \( T^{2} + 1205604 \) Copy content Toggle raw display
$79$ \( (T + 830)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 191844 \) Copy content Toggle raw display
$89$ \( T^{2} + 191844 \) Copy content Toggle raw display
$97$ \( T^{2} + 725904 \) Copy content Toggle raw display
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