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Properties

Label 140.6.i.c
Level $140$
Weight $6$
Character orbit 140.i
Analytic conductor $22.454$
Analytic rank $0$
Dimension $8$
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] = [140,6,Mod(81,140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(140, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.81");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 140.i (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: \(22.4537347738\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 641x^{6} + 6298x^{5} + 385109x^{4} + 1662688x^{3} + 16270276x^{2} - 39277632x + 244484496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 8 \beta_{2} - \beta_1 + 8) q^{3} - 25 \beta_{2} q^{5} + (\beta_{6} + 2 \beta_{5} - 5 \beta_{3} + \cdots + 9) q^{7}+ \cdots + (5 \beta_{7} + 10 \beta_{3} - 135 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 8 \beta_{2} - \beta_1 + 8) q^{3} - 25 \beta_{2} q^{5} + (\beta_{6} + 2 \beta_{5} - 5 \beta_{3} + \cdots + 9) q^{7}+ \cdots + ( - 195 \beta_{6} + 390 \beta_{5} + \cdots - 17255) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 30 q^{3} - 100 q^{5} + 50 q^{7} - 530 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 30 q^{3} - 100 q^{5} + 50 q^{7} - 530 q^{9} + 776 q^{11} - 1180 q^{13} - 1500 q^{15} - 440 q^{17} - 1508 q^{19} + 7890 q^{21} - 1450 q^{23} - 2500 q^{25} - 12780 q^{27} + 9684 q^{29} - 7588 q^{31} + 12160 q^{33} - 1000 q^{35} + 19550 q^{37} - 5548 q^{39} + 27400 q^{41} - 34860 q^{43} - 13250 q^{45} - 5560 q^{47} + 34910 q^{49} - 16828 q^{51} + 27630 q^{53} - 38800 q^{55} - 150120 q^{57} + 85328 q^{59} + 374 q^{61} - 14660 q^{63} + 14750 q^{65} - 100830 q^{67} + 185932 q^{69} - 94256 q^{71} + 107060 q^{73} + 18750 q^{75} - 123610 q^{77} - 63984 q^{79} - 59864 q^{81} + 278580 q^{83} + 22000 q^{85} - 252390 q^{87} + 107578 q^{89} + 113420 q^{91} + 140060 q^{93} - 37700 q^{95} + 278480 q^{97} - 146200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 641x^{6} + 6298x^{5} + 385109x^{4} + 1662688x^{3} + 16270276x^{2} - 39277632x + 244484496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 222156724 \nu^{7} - 903031551 \nu^{6} - 134358704910 \nu^{5} - 2240345312787 \nu^{4} + \cdots + 83\!\cdots\!04 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1034033 \nu^{7} + 6173258 \nu^{6} - 645588845 \nu^{5} - 3949301554 \nu^{4} + \cdots + 41683710437568 ) / 16598069344500 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1030939 \nu^{7} - 1981324 \nu^{6} + 643657135 \nu^{5} + 3937484582 \nu^{4} + \cdots - 520909099994604 ) / 8299034672250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39255136507 \nu^{7} + 359782701582 \nu^{6} - 22418836582555 \nu^{5} + \cdots + 38\!\cdots\!72 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67036960007 \nu^{7} - 201243291172 \nu^{6} + 46007325640655 \nu^{5} + 366681492336416 \nu^{4} + \cdots - 14\!\cdots\!12 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7784128133 \nu^{7} - 23528937597 \nu^{6} - 4723584693195 \nu^{5} - 73434406776429 \nu^{4} + \cdots + 29\!\cdots\!88 ) / 10\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_{7} - 6\beta_{3} - 314\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -30\beta_{6} + 60\beta_{5} - 30\beta_{4} - 595\beta_{3} - 30\beta_{2} - 595\beta _1 - 2514 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3245\beta_{7} + 60\beta_{6} + 60\beta_{5} - 3245\beta_{4} + 189410\beta_{2} - 7524\beta _1 - 189470 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -38160\beta_{7} + 38460\beta_{6} - 19230\beta_{5} + 393499\beta_{3} + 2788236\beta_{2} - 19230 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 152040 \beta_{6} - 304080 \beta_{5} + 2140565 \beta_{4} + 6980610 \beta_{3} + 152040 \beta_{2} + \cdots + 127597646 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 36271410 \beta_{7} - 12235230 \beta_{6} - 12235230 \beta_{5} + 36271410 \beta_{4} - 2455541130 \beta_{2} + \cdots + 2467776360 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/140\mathbb{Z}\right)^\times\).

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
13.8084 − 23.9169i
1.68416 − 2.91704i
−4.00800 + 6.94206i
−10.4846 + 18.1598i
13.8084 + 23.9169i
1.68416 + 2.91704i
−4.00800 − 6.94206i
−10.4846 − 18.1598i
0 −9.80844 + 16.9887i 0 −12.5000 − 21.6506i 0 −129.248 − 10.0923i 0 −70.9108 − 122.821i 0
81.2 0 2.31584 − 4.01116i 0 −12.5000 − 21.6506i 0 78.6007 − 103.097i 0 110.774 + 191.866i 0
81.3 0 8.00800 − 13.8703i 0 −12.5000 − 21.6506i 0 −53.2723 + 118.191i 0 −6.75607 − 11.7018i 0
81.4 0 14.4846 − 25.0881i 0 −12.5000 − 21.6506i 0 128.920 − 13.6621i 0 −298.107 − 516.336i 0
121.1 0 −9.80844 − 16.9887i 0 −12.5000 + 21.6506i 0 −129.248 + 10.0923i 0 −70.9108 + 122.821i 0
121.2 0 2.31584 + 4.01116i 0 −12.5000 + 21.6506i 0 78.6007 + 103.097i 0 110.774 − 191.866i 0
121.3 0 8.00800 + 13.8703i 0 −12.5000 + 21.6506i 0 −53.2723 − 118.191i 0 −6.75607 + 11.7018i 0
121.4 0 14.4846 + 25.0881i 0 −12.5000 + 21.6506i 0 128.920 + 13.6621i 0 −298.107 + 516.336i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.6.i.c ✓ 8
7.c even 3 1 inner 140.6.i.c ✓ 8
7.c even 3 1 980.6.a.j 4
7.d odd 6 1 980.6.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.i.c ✓ 8 1.a even 1 1 trivial
140.6.i.c ✓ 8 7.c even 3 1 inner
980.6.a.j 4 7.c even 3 1
980.6.a.k 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 30 T_{3}^{7} + 1201 T_{3}^{6} - 13050 T_{3}^{5} + 463957 T_{3}^{4} - 5852400 T_{3}^{3} + \cdots + 1777128336 \) acting on \(S_{6}^{\mathrm{new}}(140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 1777128336 \) Copy content Toggle raw display
$5$ \( (T^{2} + 25 T + 625)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} + 590 T^{3} + \cdots + 617640956)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 17\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 40\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 10704301699716)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 50\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 97\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 58859927331471)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 22\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 34\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 96\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 86\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots - 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 56\!\cdots\!56)^{2} \) Copy content Toggle raw display
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