Properties

Label 114.3.d.a
Level $114$
Weight $3$
Character orbit 114.d
Analytic conductor $3.106$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [114,3,Mod(37,114)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(114, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("114.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 114.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: \(3.10627501371\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.184143974400.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + \beta_{4} q^{3} - 2 q^{4} + ( - \beta_{6} - \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{6} + \beta_1 - 2) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + \beta_{4} q^{3} - 2 q^{4} + ( - \beta_{6} - \beta_1) q^{5} + \beta_1 q^{6} + ( - \beta_{6} + \beta_1 - 2) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9} + (2 \beta_{4} - \beta_{2}) q^{10} - \beta_{7} q^{11} - 2 \beta_{4} q^{12} + (6 \beta_{5} + 4 \beta_{4}) q^{13} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{2}) q^{14} + (3 \beta_{5} + \beta_{3}) q^{15} + 4 q^{16} + ( - \beta_{6} + 3 \beta_1) q^{17} - 3 \beta_{5} q^{18} + (\beta_{7} - 3 \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1 - 4) q^{19} + (2 \beta_{6} + 2 \beta_1) q^{20} + ( - 3 \beta_{5} - 2 \beta_{4} + \beta_{3}) q^{21} + ( - 2 \beta_{3} - \beta_{2}) q^{22} + ( - \beta_{7} + 3 \beta_{6} - 5 \beta_1 - 6) q^{23} - 2 \beta_1 q^{24} + (2 \beta_{7} - 3 \beta_{6} - 5 \beta_1 + 17) q^{25} + (4 \beta_1 - 12) q^{26} - 3 \beta_{4} q^{27} + (2 \beta_{6} - 2 \beta_1 + 4) q^{28} + ( - 9 \beta_{5} + 6 \beta_{4} + 3 \beta_{2}) q^{29} + ( - \beta_{7} + \beta_{6} - 6) q^{30} + (6 \beta_{5} - 6 \beta_{4} - 2 \beta_{3}) q^{31} + 4 \beta_{5} q^{32} + (\beta_{3} + 3 \beta_{2}) q^{33} + ( - 6 \beta_{4} - \beta_{2}) q^{34} + (\beta_{6} - 3 \beta_1 + 30) q^{35} + 6 q^{36} + (12 \beta_{5} + 20 \beta_{4}) q^{37} + (\beta_{7} - \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \beta_{2} - \beta_1 + 6) q^{38} + (6 \beta_1 - 12) q^{39} + ( - 4 \beta_{4} + 2 \beta_{2}) q^{40} + ( - 3 \beta_{5} - 18 \beta_{4} + 4 \beta_{3} + 3 \beta_{2}) q^{41} + ( - \beta_{7} + \beta_{6} - 2 \beta_1 + 6) q^{42} + (2 \beta_{7} + \beta_{6} - 9 \beta_1 + 14) q^{43} + 2 \beta_{7} q^{44} + (3 \beta_{6} + 3 \beta_1) q^{45} + ( - 6 \beta_{5} + 10 \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{46} + ( - \beta_{7} - 4 \beta_1 - 24) q^{47} + 4 \beta_{4} q^{48} + ( - 2 \beta_{7} + 5 \beta_{6} - 9 \beta_1 - 3) q^{49} + (17 \beta_{5} + 10 \beta_{4} + 4 \beta_{3} - \beta_{2}) q^{50} + ( - 9 \beta_{5} + \beta_{3}) q^{51} + ( - 12 \beta_{5} - 8 \beta_{4}) q^{52} + (21 \beta_{5} - 18 \beta_{4} - 4 \beta_{3} + 3 \beta_{2}) q^{53} - 3 \beta_1 q^{54} + (5 \beta_{6} + 31 \beta_1 + 6) q^{55} + (4 \beta_{5} + 4 \beta_{4} + 2 \beta_{2}) q^{56} + ( - 3 \beta_{6} - 6 \beta_{5} - 4 \beta_{4} - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{57} + ( - 6 \beta_{6} + 6 \beta_1 + 18) q^{58} + ( - 36 \beta_{5} - 12 \beta_{4} - 4 \beta_{3} - 6 \beta_{2}) q^{59} + ( - 6 \beta_{5} - 2 \beta_{3}) q^{60} + ( - 2 \beta_{7} - 5 \beta_{6} - 11 \beta_1 - 26) q^{61} + (2 \beta_{7} - 2 \beta_{6} - 6 \beta_1 - 12) q^{62} + (3 \beta_{6} - 3 \beta_1 + 6) q^{63} - 8 q^{64} + (12 \beta_{5} + 12 \beta_{4} + 4 \beta_{3} - 6 \beta_{2}) q^{65} + ( - \beta_{7} - 5 \beta_{6}) q^{66} + ( - 42 \beta_{5} + 10 \beta_{4} - 2 \beta_{3} - 6 \beta_{2}) q^{67} + (2 \beta_{6} - 6 \beta_1) q^{68} + (15 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + 3 \beta_{2}) q^{69} + (30 \beta_{5} + 6 \beta_{4} + \beta_{2}) q^{70} + (36 \beta_{4} - 4 \beta_{3} - 6 \beta_{2}) q^{71} + 6 \beta_{5} q^{72} + (2 \beta_{7} - 5 \beta_{6} + 9 \beta_1 - 46) q^{73} + (20 \beta_1 - 24) q^{74} + (15 \beta_{5} + 17 \beta_{4} + \beta_{3} - 6 \beta_{2}) q^{75} + ( - 2 \beta_{7} + 6 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 4 \beta_1 + 8) q^{76} + ( - 5 \beta_{6} + 31 \beta_1 + 6) q^{77} + ( - 12 \beta_{5} - 12 \beta_{4}) q^{78} + (36 \beta_{5} - 40 \beta_{4} + 4 \beta_{3} - 6 \beta_{2}) q^{79} + ( - 4 \beta_{6} - 4 \beta_1) q^{80} + 9 q^{81} + ( - 4 \beta_{7} - 2 \beta_{6} - 18 \beta_1 + 6) q^{82} + ( - \beta_{7} + 3 \beta_{6} + 19 \beta_1 + 18) q^{83} + (6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3}) q^{84} + ( - 2 \beta_{7} + \beta_{6} - 5 \beta_1 + 18) q^{85} + (14 \beta_{5} + 18 \beta_{4} + 4 \beta_{3} + 3 \beta_{2}) q^{86} + (3 \beta_{7} - 3 \beta_{6} - 9 \beta_1 - 18) q^{87} + (4 \beta_{3} + 2 \beta_{2}) q^{88} + ( - 9 \beta_{5} + 42 \beta_{4} - 4 \beta_{3} - 3 \beta_{2}) q^{89} + ( - 6 \beta_{4} + 3 \beta_{2}) q^{90} + ( - 24 \beta_{5} - 20 \beta_{4} + 4 \beta_{3} - 6 \beta_{2}) q^{91} + (2 \beta_{7} - 6 \beta_{6} + 10 \beta_1 + 12) q^{92} + ( - 6 \beta_{6} + 6 \beta_1 + 18) q^{93} + ( - 24 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{94} + ( - 2 \beta_{7} + \beta_{6} - 18 \beta_{5} - 42 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} + \cdots - 18) q^{95}+ \cdots + 3 \beta_{7} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 4 q^{5} - 12 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 4 q^{5} - 12 q^{7} - 24 q^{9} + 4 q^{11} + 32 q^{16} + 4 q^{17} - 36 q^{19} - 8 q^{20} - 56 q^{23} + 140 q^{25} - 96 q^{26} + 24 q^{28} - 48 q^{30} + 236 q^{35} + 48 q^{36} + 48 q^{38} - 96 q^{39} + 48 q^{42} + 100 q^{43} - 8 q^{44} - 12 q^{45} - 188 q^{47} - 36 q^{49} + 28 q^{55} + 36 q^{57} + 168 q^{58} - 180 q^{61} - 96 q^{62} + 36 q^{63} - 64 q^{64} + 24 q^{66} - 8 q^{68} - 356 q^{73} - 192 q^{74} + 72 q^{76} + 68 q^{77} + 16 q^{80} + 72 q^{81} + 72 q^{82} + 136 q^{83} + 148 q^{85} - 144 q^{87} + 112 q^{92} + 168 q^{93} - 140 q^{95} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 17\nu^{4} - 39\nu^{3} - 88\nu^{2} + 108\nu + 36 ) / 90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -41\nu^{7} - 584\nu^{6} + 2938\nu^{5} + 18850\nu^{4} - 52061\nu^{3} - 192038\nu^{2} + 242412\nu + 645012 ) / 26190 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{7} + 263\nu^{6} - 1063\nu^{5} - 5275\nu^{4} + 14651\nu^{3} + 49778\nu^{2} - 63312\nu - 162522 ) / 5238 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} - 190\nu^{5} + 545\nu^{4} + 1556\nu^{3} - 2893\nu^{2} - 3948\nu + 2475 ) / 2619 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 82\nu^{7} - 287\nu^{6} - 1511\nu^{5} + 4495\nu^{4} + 12457\nu^{3} - 23324\nu^{2} - 30864\nu + 19476 ) / 26190 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -34\nu^{7} + 119\nu^{6} + 371\nu^{5} - 1225\nu^{4} + 371\nu^{3} + 728\nu^{2} - 17268\nu + 5850 ) / 5238 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -205\nu^{7} + 863\nu^{6} + 3341\nu^{5} - 13711\nu^{4} - 25468\nu^{3} + 71114\nu^{2} + 192396\nu - 132498 ) / 26190 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{7} + 5\beta_{5} + \beta _1 + 6 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 3\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 8\beta _1 + 38 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17\beta_{7} + 15\beta_{6} + 119\beta_{5} - 48\beta_{4} + 6\beta_{3} + 3\beta_{2} + 31\beta _1 + 126 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{7} + 15\beta_{6} + 132\beta_{5} - 64\beta_{4} + 38\beta_{3} + 34\beta_{2} + 113\beta _1 + 323 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 143\beta_{7} + 195\beta_{6} + 2015\beta_{5} - 1290\beta_{4} + 180\beta_{3} + 165\beta_{2} + 559\beta _1 + 1524 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 175\beta_{7} + 255\beta_{6} + 2952\beta_{5} - 1999\beta_{4} + 623\beta_{3} + 679\beta_{2} + 1055\beta _1 + 2480 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 845 \beta_{7} + 1455 \beta_{6} + 32027 \beta_{5} - 24024 \beta_{4} + 3738 \beta_{3} + 4179 \beta_{2} + 5515 \beta _1 + 12390 ) / 10 \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.51885 0.707107i
2.51885 0.707107i
3.87998 0.707107i
−2.87998 0.707107i
3.87998 + 0.707107i
−2.87998 + 0.707107i
−1.51885 + 0.707107i
2.51885 + 0.707107i
1.41421i 1.73205i −2.00000 −4.01452 −2.44949 −10.9135 2.82843i −3.00000 5.67739i
37.2 1.41421i 1.73205i −2.00000 9.91350 −2.44949 3.01452 2.82843i −3.00000 14.0198i
37.3 1.41421i 1.73205i −2.00000 −6.84873 2.44949 −3.94975 2.82843i −3.00000 9.68557i
37.4 1.41421i 1.73205i −2.00000 2.94975 2.44949 5.84873 2.82843i −3.00000 4.17158i
37.5 1.41421i 1.73205i −2.00000 −6.84873 2.44949 −3.94975 2.82843i −3.00000 9.68557i
37.6 1.41421i 1.73205i −2.00000 2.94975 2.44949 5.84873 2.82843i −3.00000 4.17158i
37.7 1.41421i 1.73205i −2.00000 −4.01452 −2.44949 −10.9135 2.82843i −3.00000 5.67739i
37.8 1.41421i 1.73205i −2.00000 9.91350 −2.44949 3.01452 2.82843i −3.00000 14.0198i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.3.d.a 8
3.b odd 2 1 342.3.d.b 8
4.b odd 2 1 912.3.o.d 8
12.b even 2 1 2736.3.o.n 8
19.b odd 2 1 inner 114.3.d.a 8
57.d even 2 1 342.3.d.b 8
76.d even 2 1 912.3.o.d 8
228.b odd 2 1 2736.3.o.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.d.a 8 1.a even 1 1 trivial
114.3.d.a 8 19.b odd 2 1 inner
342.3.d.b 8 3.b odd 2 1
342.3.d.b 8 57.d even 2 1
912.3.o.d 8 4.b odd 2 1
912.3.o.d 8 76.d even 2 1
2736.3.o.n 8 12.b even 2 1
2736.3.o.n 8 228.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(114, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{3} - 83 T^{2} - 36 T + 804)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 6 T^{3} - 71 T^{2} - 120 T + 760)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 389 T^{2} - 60 T + 28650)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 240 T^{2} + 576)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} - 179 T^{2} + 540 T - 60)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 36 T^{7} + \cdots + 16983563041 \) Copy content Toggle raw display
$23$ \( (T^{4} + 28 T^{3} - 914 T^{2} + \cdots - 302040)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 3924 T^{6} + \cdots + 30611001600 \) Copy content Toggle raw display
$31$ \( T^{8} + 2616 T^{6} + \cdots + 6046617600 \) Copy content Toggle raw display
$37$ \( (T^{4} + 2976 T^{2} + 831744)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 9924 T^{6} + \cdots + 4246237209600 \) Copy content Toggle raw display
$43$ \( (T^{4} - 50 T^{3} - 1935 T^{2} + \cdots - 1616000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 94 T^{3} + 2779 T^{2} + \cdots + 89850)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 492631164527616 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 652283427225600 \) Copy content Toggle raw display
$61$ \( (T^{4} + 90 T^{3} - 1775 T^{2} + \cdots + 250000)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 22440 T^{6} + \cdots + 292222411776 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 110381078937600 \) Copy content Toggle raw display
$73$ \( (T^{4} + 178 T^{3} + 8001 T^{2} + \cdots + 577600)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 52656 T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{4} - 68 T^{3} - 3794 T^{2} + \cdots - 57240)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 109746576000000 \) Copy content Toggle raw display
$97$ \( T^{8} + 39552 T^{6} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
show more
show less