Properties

Label 117.4.f.a
Level $117$
Weight $4$
Character orbit 117.f
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(61,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.61");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.f (of order \(3\), degree \(2\), 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: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 5 \zeta_{6} q^{2} + (3 \zeta_{6} - 6) q^{3} + (17 \zeta_{6} - 17) q^{4} + 19 \zeta_{6} q^{5} + ( - 15 \zeta_{6} - 15) q^{6} + 11 q^{7} - 45 q^{8} + ( - 27 \zeta_{6} + 27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 \zeta_{6} q^{2} + (3 \zeta_{6} - 6) q^{3} + (17 \zeta_{6} - 17) q^{4} + 19 \zeta_{6} q^{5} + ( - 15 \zeta_{6} - 15) q^{6} + 11 q^{7} - 45 q^{8} + ( - 27 \zeta_{6} + 27) q^{9} + (95 \zeta_{6} - 95) q^{10} - 16 \zeta_{6} q^{11} + ( - 102 \zeta_{6} + 51) q^{12} + ( - 17 \zeta_{6} + 53) q^{13} + 55 \zeta_{6} q^{14} + ( - 57 \zeta_{6} - 57) q^{15} - 89 \zeta_{6} q^{16} - 131 \zeta_{6} q^{17} + 135 q^{18} + 93 \zeta_{6} q^{19} - 323 q^{20} + (33 \zeta_{6} - 66) q^{21} + ( - 80 \zeta_{6} + 80) q^{22} + 69 q^{23} + ( - 135 \zeta_{6} + 270) q^{24} + (236 \zeta_{6} - 236) q^{25} + (180 \zeta_{6} + 85) q^{26} + (162 \zeta_{6} - 81) q^{27} + (187 \zeta_{6} - 187) q^{28} - 86 \zeta_{6} q^{29} + ( - 570 \zeta_{6} + 285) q^{30} + 173 \zeta_{6} q^{31} + ( - 85 \zeta_{6} + 85) q^{32} + (48 \zeta_{6} + 48) q^{33} + ( - 655 \zeta_{6} + 655) q^{34} + 209 \zeta_{6} q^{35} + 459 \zeta_{6} q^{36} + ( - 323 \zeta_{6} + 323) q^{37} + (465 \zeta_{6} - 465) q^{38} + (210 \zeta_{6} - 267) q^{39} - 855 \zeta_{6} q^{40} - 9 q^{41} + ( - 165 \zeta_{6} - 165) q^{42} + 69 q^{43} + 272 q^{44} + 513 q^{45} + 345 \zeta_{6} q^{46} + (369 \zeta_{6} - 369) q^{47} + (267 \zeta_{6} + 267) q^{48} - 222 q^{49} - 1180 q^{50} + (393 \zeta_{6} + 393) q^{51} + (901 \zeta_{6} - 612) q^{52} - 306 q^{53} + (405 \zeta_{6} - 810) q^{54} + ( - 304 \zeta_{6} + 304) q^{55} - 495 q^{56} + ( - 279 \zeta_{6} - 279) q^{57} + ( - 430 \zeta_{6} + 430) q^{58} + ( - 420 \zeta_{6} + 420) q^{59} + ( - 969 \zeta_{6} + 1938) q^{60} - 383 q^{61} + (865 \zeta_{6} - 865) q^{62} + ( - 297 \zeta_{6} + 297) q^{63} - 287 q^{64} + (684 \zeta_{6} + 323) q^{65} + (480 \zeta_{6} - 240) q^{66} + 139 q^{67} + 2227 q^{68} + (207 \zeta_{6} - 414) q^{69} + (1045 \zeta_{6} - 1045) q^{70} - 237 \zeta_{6} q^{71} + (1215 \zeta_{6} - 1215) q^{72} + 518 q^{73} + 1615 q^{74} + ( - 1416 \zeta_{6} + 708) q^{75} - 1581 q^{76} - 176 \zeta_{6} q^{77} + ( - 285 \zeta_{6} - 1050) q^{78} + (41 \zeta_{6} - 41) q^{79} + ( - 1691 \zeta_{6} + 1691) q^{80} - 729 \zeta_{6} q^{81} - 45 \zeta_{6} q^{82} + ( - 905 \zeta_{6} + 905) q^{83} + ( - 1122 \zeta_{6} + 561) q^{84} + ( - 2489 \zeta_{6} + 2489) q^{85} + 345 \zeta_{6} q^{86} + (258 \zeta_{6} + 258) q^{87} + 720 \zeta_{6} q^{88} + ( - 929 \zeta_{6} + 929) q^{89} + 2565 \zeta_{6} q^{90} + ( - 187 \zeta_{6} + 583) q^{91} + (1173 \zeta_{6} - 1173) q^{92} + ( - 519 \zeta_{6} - 519) q^{93} - 1845 q^{94} + (1767 \zeta_{6} - 1767) q^{95} + (510 \zeta_{6} - 255) q^{96} - 5 q^{97} - 1110 \zeta_{6} q^{98} - 432 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} - 9 q^{3} - 17 q^{4} + 19 q^{5} - 45 q^{6} + 22 q^{7} - 90 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} - 9 q^{3} - 17 q^{4} + 19 q^{5} - 45 q^{6} + 22 q^{7} - 90 q^{8} + 27 q^{9} - 95 q^{10} - 16 q^{11} + 89 q^{13} + 55 q^{14} - 171 q^{15} - 89 q^{16} - 131 q^{17} + 270 q^{18} + 93 q^{19} - 646 q^{20} - 99 q^{21} + 80 q^{22} + 138 q^{23} + 405 q^{24} - 236 q^{25} + 350 q^{26} - 187 q^{28} - 86 q^{29} + 173 q^{31} + 85 q^{32} + 144 q^{33} + 655 q^{34} + 209 q^{35} + 459 q^{36} + 323 q^{37} - 465 q^{38} - 324 q^{39} - 855 q^{40} - 18 q^{41} - 495 q^{42} + 138 q^{43} + 544 q^{44} + 1026 q^{45} + 345 q^{46} - 369 q^{47} + 801 q^{48} - 444 q^{49} - 2360 q^{50} + 1179 q^{51} - 323 q^{52} - 612 q^{53} - 1215 q^{54} + 304 q^{55} - 990 q^{56} - 837 q^{57} + 430 q^{58} + 420 q^{59} + 2907 q^{60} - 766 q^{61} - 865 q^{62} + 297 q^{63} - 574 q^{64} + 1330 q^{65} + 278 q^{67} + 4454 q^{68} - 621 q^{69} - 1045 q^{70} - 237 q^{71} - 1215 q^{72} + 1036 q^{73} + 3230 q^{74} - 3162 q^{76} - 176 q^{77} - 2385 q^{78} - 41 q^{79} + 1691 q^{80} - 729 q^{81} - 45 q^{82} + 905 q^{83} + 2489 q^{85} + 345 q^{86} + 774 q^{87} + 720 q^{88} + 929 q^{89} + 2565 q^{90} + 979 q^{91} - 1173 q^{92} - 1557 q^{93} - 3690 q^{94} - 1767 q^{95} - 10 q^{97} - 1110 q^{98} - 864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
0.500000 + 0.866025i
0.500000 0.866025i
2.50000 + 4.33013i −4.50000 + 2.59808i −8.50000 + 14.7224i 9.50000 + 16.4545i −22.5000 12.9904i 11.0000 −45.0000 13.5000 23.3827i −47.5000 + 82.2724i
94.1 2.50000 4.33013i −4.50000 2.59808i −8.50000 14.7224i 9.50000 16.4545i −22.5000 + 12.9904i 11.0000 −45.0000 13.5000 + 23.3827i −47.5000 82.2724i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.4.f.a 2
3.b odd 2 1 351.4.f.a 2
9.c even 3 1 117.4.h.a yes 2
9.d odd 6 1 351.4.h.a 2
13.c even 3 1 117.4.h.a yes 2
39.i odd 6 1 351.4.h.a 2
117.f even 3 1 inner 117.4.f.a 2
117.u odd 6 1 351.4.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.f.a 2 1.a even 1 1 trivial
117.4.f.a 2 117.f even 3 1 inner
117.4.h.a yes 2 9.c even 3 1
117.4.h.a yes 2 13.c even 3 1
351.4.f.a 2 3.b odd 2 1
351.4.f.a 2 117.u odd 6 1
351.4.h.a 2 9.d odd 6 1
351.4.h.a 2 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5T_{2} + 25 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$5$ \( T^{2} - 19T + 361 \) Copy content Toggle raw display
$7$ \( (T - 11)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$13$ \( T^{2} - 89T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 131T + 17161 \) Copy content Toggle raw display
$19$ \( T^{2} - 93T + 8649 \) Copy content Toggle raw display
$23$ \( (T - 69)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 86T + 7396 \) Copy content Toggle raw display
$31$ \( T^{2} - 173T + 29929 \) Copy content Toggle raw display
$37$ \( T^{2} - 323T + 104329 \) Copy content Toggle raw display
$41$ \( (T + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T - 69)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 369T + 136161 \) Copy content Toggle raw display
$53$ \( (T + 306)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 420T + 176400 \) Copy content Toggle raw display
$61$ \( (T + 383)^{2} \) Copy content Toggle raw display
$67$ \( (T - 139)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 237T + 56169 \) Copy content Toggle raw display
$73$ \( (T - 518)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 41T + 1681 \) Copy content Toggle raw display
$83$ \( T^{2} - 905T + 819025 \) Copy content Toggle raw display
$89$ \( T^{2} - 929T + 863041 \) Copy content Toggle raw display
$97$ \( (T + 5)^{2} \) Copy content Toggle raw display
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