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Properties

Label 114.4.b.b
Level $114$
Weight $4$
Character orbit 114.b
Analytic conductor $6.726$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [114,4,Mod(113,114)]
 
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([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("114.113");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 114.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: \(6.72621774065\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 15 x^{8} + 30 x^{7} - 366 x^{6} - 2046 x^{5} + 15498 x^{4} + 29808 x^{3} + \cdots + 11925920 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} - \beta_{3} q^{3} + 4 q^{4} + (\beta_{4} + \beta_{3}) q^{5} - 2 \beta_{3} q^{6} + ( - \beta_{6} - \beta_{3} + 1) q^{7} + 8 q^{8} + ( - \beta_{8} + \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} - \beta_{3} q^{3} + 4 q^{4} + (\beta_{4} + \beta_{3}) q^{5} - 2 \beta_{3} q^{6} + ( - \beta_{6} - \beta_{3} + 1) q^{7} + 8 q^{8} + ( - \beta_{8} + \beta_{3} - 1) q^{9} + (2 \beta_{4} + 2 \beta_{3}) q^{10} + (\beta_{8} + \beta_{7} + \beta_{4} + \cdots + 1) q^{11}+ \cdots + (14 \beta_{8} + 32 \beta_{7} + \cdots + 285) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 20 q^{2} - 5 q^{3} + 40 q^{4} - 10 q^{6} + 10 q^{7} + 80 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 20 q^{2} - 5 q^{3} + 40 q^{4} - 10 q^{6} + 10 q^{7} + 80 q^{8} - 5 q^{9} - 20 q^{12} + 20 q^{14} + 146 q^{15} + 160 q^{16} - 10 q^{18} + 14 q^{19} + 191 q^{21} - 40 q^{24} - 382 q^{25} - 170 q^{27} + 40 q^{28} + 30 q^{29} + 292 q^{30} + 320 q^{32} - 214 q^{33} - 20 q^{36} + 28 q^{38} + 225 q^{39} - 444 q^{41} + 382 q^{42} - 892 q^{43} - 490 q^{45} - 80 q^{48} + 1488 q^{49} - 764 q^{50} - 859 q^{51} + 18 q^{53} - 340 q^{54} - 780 q^{55} + 80 q^{56} - 1331 q^{57} + 60 q^{58} - 582 q^{59} + 584 q^{60} + 28 q^{61} - 631 q^{63} + 640 q^{64} - 864 q^{65} - 428 q^{66} + 959 q^{69} + 1368 q^{71} - 40 q^{72} - 682 q^{73} - 669 q^{75} + 56 q^{76} + 450 q^{78} + 1159 q^{81} - 888 q^{82} + 764 q^{84} + 1608 q^{85} - 1784 q^{86} + 915 q^{87} + 348 q^{89} - 980 q^{90} + 2310 q^{93} + 3528 q^{95} - 160 q^{96} + 2976 q^{98} + 2990 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 15 x^{8} + 30 x^{7} - 366 x^{6} - 2046 x^{5} + 15498 x^{4} + 29808 x^{3} + \cdots + 11925920 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{9} + 4 \nu^{8} - 11 \nu^{7} - 41 \nu^{6} + 325 \nu^{5} + 2371 \nu^{4} - 13127 \nu^{3} + \cdots - 1360105 ) / 531441 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 4 \nu^{8} + 11 \nu^{7} + 41 \nu^{6} - 325 \nu^{5} - 2371 \nu^{4} + 13127 \nu^{3} + \cdots + 2422987 ) / 531441 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 389 \nu^{9} + 2285 \nu^{8} + 5441 \nu^{7} - 14248 \nu^{6} - 553732 \nu^{5} + \cdots + 1070326444 ) / 176438412 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 583 \nu^{9} + 8137 \nu^{8} - 61979 \nu^{7} + 13546 \nu^{6} + 389356 \nu^{5} + \cdots - 2346538684 ) / 176438412 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4 \nu^{9} + 70 \nu^{8} - 449 \nu^{7} + 754 \nu^{6} + 2488 \nu^{5} - 18785 \nu^{4} + \cdots - 6382180 ) / 1062882 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 259 \nu^{9} - 1675 \nu^{8} - 6169 \nu^{7} + 90800 \nu^{6} - 539764 \nu^{5} + 990386 \nu^{4} + \cdots + 423928480 ) / 58812804 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4 \nu^{9} + 43 \nu^{8} - 125 \nu^{7} + 52 \nu^{6} + 2623 \nu^{5} + 2032 \nu^{4} - 123977 \nu^{3} + \cdots - 3899503 ) / 531441 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1417 \nu^{9} + 5155 \nu^{8} - 27656 \nu^{7} + 253699 \nu^{6} + 242203 \nu^{5} + \cdots - 2607623884 ) / 88219206 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{8} - 3\beta_{7} + 9\beta_{6} - 6\beta_{5} + 3\beta_{4} + 6\beta_{2} + 4\beta _1 - 47 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{9} + 11 \beta_{8} + 13 \beta_{7} + 3 \beta_{6} - 16 \beta_{5} - 19 \beta_{4} - 72 \beta_{3} + \cdots + 83 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 210 \beta_{8} - 3 \beta_{7} - 45 \beta_{6} - 168 \beta_{5} - 699 \beta_{4} - 402 \beta_{3} - 45 \beta_{2} + \cdots + 5050 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 192 \beta_{9} + 621 \beta_{8} + 318 \beta_{7} - 372 \beta_{6} - 180 \beta_{5} - 522 \beta_{4} + \cdots - 1283 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2142 \beta_{9} + 8073 \beta_{8} - 3069 \beta_{7} - 7029 \beta_{6} - 3132 \beta_{5} + 7047 \beta_{4} + \cdots - 208889 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1050 \beta_{9} + 17100 \beta_{8} - 4623 \beta_{7} + 5451 \beta_{6} - 8658 \beta_{5} + 20043 \beta_{4} + \cdots - 521530 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 102456 \beta_{9} + 265269 \beta_{8} + 70044 \beta_{7} + 77058 \beta_{6} - 136950 \beta_{5} + \cdots - 8275133 ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1
−3.94474 + 1.59674i
−3.94474 − 1.59674i
−3.03930 + 3.26864i
−3.03930 − 3.26864i
1.15425 + 5.19386i
1.15425 − 5.19386i
2.51768 + 4.96957i
2.51768 − 4.96957i
5.81212 + 1.96049i
5.81212 − 1.96049i
2.00000 −4.94474 − 1.59674i 4.00000 − 0.432619i −9.88947 − 3.19348i 17.1594 8.00000 21.9009 + 15.7909i − 0.865237i
113.2 2.00000 −4.94474 + 1.59674i 4.00000 0.432619i −9.88947 + 3.19348i 17.1594 8.00000 21.9009 − 15.7909i 0.865237i
113.3 2.00000 −4.03930 − 3.26864i 4.00000 14.8595i −8.07861 − 6.53729i −34.4283 8.00000 5.63193 + 26.4061i 29.7191i
113.4 2.00000 −4.03930 + 3.26864i 4.00000 − 14.8595i −8.07861 + 6.53729i −34.4283 8.00000 5.63193 − 26.4061i − 29.7191i
113.5 2.00000 0.154247 − 5.19386i 4.00000 − 11.5116i 0.308495 − 10.3877i −7.36161 8.00000 −26.9524 − 1.60228i − 23.0232i
113.6 2.00000 0.154247 + 5.19386i 4.00000 11.5116i 0.308495 + 10.3877i −7.36161 8.00000 −26.9524 + 1.60228i 23.0232i
113.7 2.00000 1.51768 − 4.96957i 4.00000 20.1014i 3.03535 − 9.93915i 30.4049 8.00000 −22.3933 − 15.0844i 40.2027i
113.8 2.00000 1.51768 + 4.96957i 4.00000 − 20.1014i 3.03535 + 9.93915i 30.4049 8.00000 −22.3933 + 15.0844i − 40.2027i
113.9 2.00000 4.81212 − 1.96049i 4.00000 − 7.64361i 9.62424 − 3.92098i −0.774402 8.00000 19.3130 − 18.8682i − 15.2872i
113.10 2.00000 4.81212 + 1.96049i 4.00000 7.64361i 9.62424 + 3.92098i −0.774402 8.00000 19.3130 + 18.8682i 15.2872i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.b.b yes 10
3.b odd 2 1 114.4.b.a ✓ 10
19.b odd 2 1 114.4.b.a ✓ 10
57.d even 2 1 inner 114.4.b.b yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.b.a ✓ 10 3.b odd 2 1
114.4.b.a ✓ 10 19.b odd 2 1
114.4.b.b yes 10 1.a even 1 1 trivial
114.4.b.b yes 10 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{29}^{5} - 15T_{29}^{4} - 37908T_{29}^{3} - 360612T_{29}^{2} + 360067680T_{29} + 11179944000 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 5 T^{9} + \cdots + 14348907 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 129283200 \) Copy content Toggle raw display
$7$ \( (T^{5} - 5 T^{4} + \cdots + 102400)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 1015483507200 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 15\!\cdots\!99 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} - 15 T^{4} + \cdots + 11179944000)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( (T^{5} + 222 T^{4} + \cdots + 48844166400)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 446 T^{4} + \cdots + 4670227600)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 1526127827976)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 8884056088800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 1387964356000)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 25715101466880)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 341 T^{4} + \cdots - 492268510520)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 258564364974000)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
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