Properties

Label 114.4.e.c
Level $114$
Weight $4$
Character orbit 114.e
Analytic conductor $6.726$
Analytic rank $0$
Dimension $4$
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] = [114,4,Mod(7,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("114.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 114.e (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: \(6.72621774065\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} - 3 \beta_1 q^{3} + (4 \beta_1 - 4) q^{4} + ( - \beta_{2} - 4 \beta_1) q^{5} + (6 \beta_1 - 6) q^{6} + ( - \beta_{3} + 9) q^{7} + 8 q^{8} + (9 \beta_1 - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} - 3 \beta_1 q^{3} + (4 \beta_1 - 4) q^{4} + ( - \beta_{2} - 4 \beta_1) q^{5} + (6 \beta_1 - 6) q^{6} + ( - \beta_{3} + 9) q^{7} + 8 q^{8} + (9 \beta_1 - 9) q^{9} + ( - 2 \beta_{3} + 2 \beta_{2} + 8 \beta_1 - 8) q^{10} + ( - \beta_{3} + 10) q^{11} + 12 q^{12} + ( - 4 \beta_{3} + 4 \beta_{2} + 5 \beta_1 - 5) q^{13} + (2 \beta_{2} - 18 \beta_1) q^{14} + ( - 3 \beta_{3} + 3 \beta_{2} + 12 \beta_1 - 12) q^{15} - 16 \beta_1 q^{16} - 12 \beta_1 q^{17} + 18 q^{18} + ( - 7 \beta_{3} + 8 \beta_{2} - 2 \beta_1 - 3) q^{19} + (4 \beta_{3} + 16) q^{20} + (3 \beta_{2} - 27 \beta_1) q^{21} + (2 \beta_{2} - 20 \beta_1) q^{22} + ( - \beta_{3} + \beta_{2} + 40 \beta_1 - 40) q^{23} - 24 \beta_1 q^{24} + ( - 8 \beta_{3} + 8 \beta_{2} + 11 \beta_1 - 11) q^{25} + (8 \beta_{3} + 10) q^{26} + 27 q^{27} + (4 \beta_{3} - 4 \beta_{2} + 36 \beta_1 - 36) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + 32 \beta_1 - 32) q^{29} + (6 \beta_{3} + 24) q^{30} + ( - 11 \beta_{3} + 109) q^{31} + (32 \beta_1 - 32) q^{32} + (3 \beta_{2} - 30 \beta_1) q^{33} + (24 \beta_1 - 24) q^{34} + ( - 5 \beta_{2} + 84 \beta_1) q^{35} - 36 \beta_1 q^{36} + ( - 20 \beta_{3} + 43) q^{37} + (16 \beta_{3} - 2 \beta_{2} + 10 \beta_1 - 4) q^{38} + (12 \beta_{3} + 15) q^{39} + ( - 8 \beta_{2} - 32 \beta_1) q^{40} + ( - 6 \beta_{2} - 188 \beta_1) q^{41} + (6 \beta_{3} - 6 \beta_{2} + 54 \beta_1 - 54) q^{42} + ( - 33 \beta_{2} - 127 \beta_1) q^{43} + (4 \beta_{3} - 4 \beta_{2} + 40 \beta_1 - 40) q^{44} + (9 \beta_{3} + 36) q^{45} + (2 \beta_{3} + 80) q^{46} + (44 \beta_{3} - 44 \beta_{2} + 130 \beta_1 - 130) q^{47} + (48 \beta_1 - 48) q^{48} + ( - 18 \beta_{3} - 142) q^{49} + (16 \beta_{3} + 22) q^{50} + (36 \beta_1 - 36) q^{51} + ( - 16 \beta_{2} - 20 \beta_1) q^{52} + (3 \beta_{3} - 3 \beta_{2} - 152 \beta_1 + 152) q^{53} - 54 \beta_1 q^{54} + ( - 6 \beta_{2} + 80 \beta_1) q^{55} + ( - 8 \beta_{3} + 72) q^{56} + (24 \beta_{3} - 3 \beta_{2} + 15 \beta_1 - 6) q^{57} + (4 \beta_{3} + 64) q^{58} + ( - \beta_{2} - 270 \beta_1) q^{59} + ( - 12 \beta_{2} - 48 \beta_1) q^{60} + ( - 22 \beta_{3} + 22 \beta_{2} + 323 \beta_1 - 323) q^{61} + (22 \beta_{2} - 218 \beta_1) q^{62} + (9 \beta_{3} - 9 \beta_{2} + 81 \beta_1 - 81) q^{63} + 64 q^{64} + (21 \beta_{3} + 500) q^{65} + (6 \beta_{3} - 6 \beta_{2} + 60 \beta_1 - 60) q^{66} + (35 \beta_{3} - 35 \beta_{2} + 195 \beta_1 - 195) q^{67} + 48 q^{68} + (3 \beta_{3} + 120) q^{69} + ( - 10 \beta_{3} + 10 \beta_{2} - 168 \beta_1 + 168) q^{70} + ( - 34 \beta_{2} + 266 \beta_1) q^{71} + (72 \beta_1 - 72) q^{72} + (50 \beta_{2} + 435 \beta_1) q^{73} + (40 \beta_{2} - 86 \beta_1) q^{74} + (24 \beta_{3} + 33) q^{75} + ( - 4 \beta_{3} - 28 \beta_{2} - 12 \beta_1 + 20) q^{76} + ( - 19 \beta_{3} + 210) q^{77} + ( - 24 \beta_{2} - 30 \beta_1) q^{78} + (7 \beta_{2} - 881 \beta_1) q^{79} + ( - 16 \beta_{3} + 16 \beta_{2} + 64 \beta_1 - 64) q^{80} - 81 \beta_1 q^{81} + ( - 12 \beta_{3} + 12 \beta_{2} + 376 \beta_1 - 376) q^{82} + ( - 12 \beta_{3} - 912) q^{83} + ( - 12 \beta_{3} + 108) q^{84} + ( - 12 \beta_{3} + 12 \beta_{2} + 48 \beta_1 - 48) q^{85} + ( - 66 \beta_{3} + 66 \beta_{2} + 254 \beta_1 - 254) q^{86} + (6 \beta_{3} + 96) q^{87} + ( - 8 \beta_{3} + 80) q^{88} + ( - 77 \beta_{3} + 77 \beta_{2} + 30 \beta_1 - 30) q^{89} + ( - 18 \beta_{2} - 72 \beta_1) q^{90} + ( - 31 \beta_{3} + 31 \beta_{2} - 435 \beta_1 + 435) q^{91} + ( - 4 \beta_{2} - 160 \beta_1) q^{92} + (33 \beta_{2} - 327 \beta_1) q^{93} + ( - 88 \beta_{3} + 260) q^{94} + (30 \beta_{3} + \beta_{2} - 100 \beta_1 + 952) q^{95} + 96 q^{96} + (62 \beta_{2} - 802 \beta_1) q^{97} + (36 \beta_{2} + 284 \beta_1) q^{98} + (9 \beta_{3} - 9 \beta_{2} + 90 \beta_1 - 90) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} - 8 q^{5} - 12 q^{6} + 36 q^{7} + 32 q^{8} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} - 8 q^{5} - 12 q^{6} + 36 q^{7} + 32 q^{8} - 18 q^{9} - 16 q^{10} + 40 q^{11} + 48 q^{12} - 10 q^{13} - 36 q^{14} - 24 q^{15} - 32 q^{16} - 24 q^{17} + 72 q^{18} - 16 q^{19} + 64 q^{20} - 54 q^{21} - 40 q^{22} - 80 q^{23} - 48 q^{24} - 22 q^{25} + 40 q^{26} + 108 q^{27} - 72 q^{28} - 64 q^{29} + 96 q^{30} + 436 q^{31} - 64 q^{32} - 60 q^{33} - 48 q^{34} + 168 q^{35} - 72 q^{36} + 172 q^{37} + 4 q^{38} + 60 q^{39} - 64 q^{40} - 376 q^{41} - 108 q^{42} - 254 q^{43} - 80 q^{44} + 144 q^{45} + 320 q^{46} - 260 q^{47} - 96 q^{48} - 568 q^{49} + 88 q^{50} - 72 q^{51} - 40 q^{52} + 304 q^{53} - 108 q^{54} + 160 q^{55} + 288 q^{56} + 6 q^{57} + 256 q^{58} - 540 q^{59} - 96 q^{60} - 646 q^{61} - 436 q^{62} - 162 q^{63} + 256 q^{64} + 2000 q^{65} - 120 q^{66} - 390 q^{67} + 192 q^{68} + 480 q^{69} + 336 q^{70} + 532 q^{71} - 144 q^{72} + 870 q^{73} - 172 q^{74} + 132 q^{75} + 56 q^{76} + 840 q^{77} - 60 q^{78} - 1762 q^{79} - 128 q^{80} - 162 q^{81} - 752 q^{82} - 3648 q^{83} + 432 q^{84} - 96 q^{85} - 508 q^{86} + 384 q^{87} + 320 q^{88} - 60 q^{89} - 144 q^{90} + 870 q^{91} - 320 q^{92} - 654 q^{93} + 1040 q^{94} + 3608 q^{95} + 384 q^{96} - 1604 q^{97} + 568 q^{98} - 180 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 20\nu ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{3} + 10\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.73861 + 1.58114i
−2.73861 1.58114i
2.73861 1.58114i
−2.73861 + 1.58114i
−1.00000 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i −7.47723 12.9509i −3.00000 + 5.19615i −1.95445 8.00000 −4.50000 + 7.79423i −14.9545 + 25.9019i
7.2 −1.00000 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i 3.47723 + 6.02273i −3.00000 + 5.19615i 19.9545 8.00000 −4.50000 + 7.79423i 6.95445 12.0455i
49.1 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i −7.47723 + 12.9509i −3.00000 5.19615i −1.95445 8.00000 −4.50000 7.79423i −14.9545 25.9019i
49.2 −1.00000 + 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i 3.47723 6.02273i −3.00000 5.19615i 19.9545 8.00000 −4.50000 7.79423i 6.95445 + 12.0455i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.e.c 4
3.b odd 2 1 342.4.g.e 4
19.c even 3 1 inner 114.4.e.c 4
19.c even 3 1 2166.4.a.q 2
19.d odd 6 1 2166.4.a.k 2
57.h odd 6 1 342.4.g.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.c 4 1.a even 1 1 trivial
114.4.e.c 4 19.c even 3 1 inner
342.4.g.e 4 3.b odd 2 1
342.4.g.e 4 57.h odd 6 1
2166.4.a.k 2 19.d odd 6 1
2166.4.a.q 2 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8T_{5}^{3} + 168T_{5}^{2} - 832T_{5} + 10816 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 8 T^{3} + 168 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$7$ \( (T^{2} - 18 T - 39)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 20 T - 20)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 10 T^{3} + 1995 T^{2} + \cdots + 3591025 \) Copy content Toggle raw display
$17$ \( (T^{2} + 12 T + 144)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 16 T^{3} + 9462 T^{2} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{4} + 80 T^{3} + 4920 T^{2} + \cdots + 2190400 \) Copy content Toggle raw display
$29$ \( T^{4} + 64 T^{3} + 3552 T^{2} + \cdots + 295936 \) Copy content Toggle raw display
$31$ \( (T^{2} - 218 T - 2639)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 86 T - 46151)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 376 T^{3} + \cdots + 962488576 \) Copy content Toggle raw display
$43$ \( T^{4} + 254 T^{3} + \cdots + 13121931601 \) Copy content Toggle raw display
$47$ \( T^{4} + 260 T^{3} + \cdots + 46405776400 \) Copy content Toggle raw display
$53$ \( T^{4} - 304 T^{3} + \cdots + 485056576 \) Copy content Toggle raw display
$59$ \( T^{4} + 540 T^{3} + \cdots + 5296928400 \) Copy content Toggle raw display
$61$ \( T^{4} + 646 T^{3} + \cdots + 2138970001 \) Copy content Toggle raw display
$67$ \( T^{4} + 390 T^{3} + \cdots + 11875550625 \) Copy content Toggle raw display
$71$ \( T^{4} - 532 T^{3} + \cdots + 4619105296 \) Copy content Toggle raw display
$73$ \( T^{4} - 870 T^{3} + \cdots + 12271100625 \) Copy content Toggle raw display
$79$ \( T^{4} + 1762 T^{3} + \cdots + 593332818961 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1824 T + 814464)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 60 T^{3} + \cdots + 504923936400 \) Copy content Toggle raw display
$97$ \( T^{4} + 1604 T^{3} + \cdots + 33096341776 \) Copy content Toggle raw display
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