Properties

Label 140.3.r.a
Level $140$
Weight $3$
Character orbit 140.r
Analytic conductor $3.815$
Analytic rank $0$
Dimension $12$
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] = [140,3,Mod(61,140)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("140.61");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 140.r (of order \(6\), 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: \(3.81472370104\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 99 x^{10} - 440 x^{9} + 3356 x^{8} - 10850 x^{7} + 46499 x^{6} - 103304 x^{5} + 238074 x^{4} - 315786 x^{3} + 343805 x^{2} - 201448 x + 54481 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_{11} - \beta_{8} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{9} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 5 \beta_{4} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{7} q^{5} + ( - \beta_{11} - \beta_{8} - \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{7} + ( - \beta_{9} - 2 \beta_{7} - \beta_{6} - \beta_{5} - 5 \beta_{4} - \beta_1) q^{9} + ( - \beta_{7} - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 1) q^{11} + (\beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 3) q^{13} + (\beta_{10} + \beta_{9} + \beta_{6} - 2) q^{15} + ( - \beta_{11} + 2 \beta_{9} - 3 \beta_{8} - \beta_{7} + \beta_{6} + 3 \beta_{4} - 2 \beta_{3} + \cdots - 4) q^{17}+ \cdots + (2 \beta_{11} + 4 \beta_{9} - 8 \beta_{8} - 12 \beta_{7} + 14 \beta_{6} + 2 \beta_{5} + \cdots - 46) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} + 14 q^{7} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} + 14 q^{7} + 30 q^{9} + 10 q^{11} - 20 q^{15} - 48 q^{17} - 6 q^{19} + 28 q^{21} + 18 q^{23} + 30 q^{25} - 60 q^{31} + 192 q^{33} - 10 q^{35} - 20 q^{37} + 24 q^{39} - 228 q^{43} + 60 q^{45} - 228 q^{47} - 200 q^{49} - 52 q^{51} - 44 q^{53} + 144 q^{57} - 24 q^{59} + 72 q^{61} - 152 q^{63} - 50 q^{65} - 154 q^{67} + 376 q^{71} - 48 q^{73} - 30 q^{75} + 240 q^{77} + 116 q^{79} - 38 q^{81} + 666 q^{87} + 300 q^{89} + 16 q^{91} + 164 q^{93} - 20 q^{95} - 540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 99 x^{10} - 440 x^{9} + 3356 x^{8} - 10850 x^{7} + 46499 x^{6} - 103304 x^{5} + 238074 x^{4} - 315786 x^{3} + 343805 x^{2} - 201448 x + 54481 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3 \nu^{10} + 15 \nu^{9} - 302 \nu^{8} + 1118 \nu^{7} - 10163 \nu^{6} + 26639 \nu^{5} - 126989 \nu^{4} + 210854 \nu^{3} - 379157 \nu^{2} + 277988 \nu + 162694 ) / 31213 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 694 \nu^{11} + 3350 \nu^{10} + 27379 \nu^{9} + 256605 \nu^{8} + 110174 \nu^{7} + 4959932 \nu^{6} - 1378575 \nu^{5} + 15352567 \nu^{4} + 26760926 \nu^{3} + \cdots - 3928127 ) / 63915306 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 694 \nu^{11} - 10984 \nu^{10} + 99049 \nu^{9} - 768276 \nu^{8} + 3779678 \nu^{7} - 16240054 \nu^{6} + 49679133 \nu^{5} - 109059386 \nu^{4} + \cdots - 75619628 ) / 63915306 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1388 \nu^{11} + 7634 \nu^{10} - 126428 \nu^{9} + 511671 \nu^{8} - 3889852 \nu^{7} + 11280122 \nu^{6} - 48300558 \nu^{5} + 93706819 \nu^{4} + \cdots + 79547755 ) / 63915306 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 96781 \nu^{11} + 829726 \nu^{10} - 7833394 \nu^{9} + 53043945 \nu^{8} - 188136191 \nu^{7} + 1091938660 \nu^{6} - 1275132618 \nu^{5} + \cdots - 24644038489 ) / 2684442852 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 113269 \nu^{11} - 454555 \nu^{10} + 10316239 \nu^{9} - 30048810 \nu^{8} + 316729109 \nu^{7} - 624777277 \nu^{6} + 3733905573 \nu^{5} + \cdots + 2867919628 ) / 2684442852 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 113269 \nu^{11} - 791404 \nu^{10} + 12000484 \nu^{9} - 61031751 \nu^{8} + 430555403 \nu^{7} - 1574397610 \nu^{6} + 6191448372 \nu^{5} + \cdots - 17020928225 ) / 2684442852 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 55792 \nu^{11} - 149182 \nu^{10} + 4574443 \nu^{9} - 8124351 \nu^{8} + 125649851 \nu^{7} - 117203410 \nu^{6} + 1304157726 \nu^{5} + \cdots + 5637912553 ) / 671110713 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 228382 \nu^{11} - 1428109 \nu^{10} + 21813343 \nu^{9} - 93405003 \nu^{8} + 672176528 \nu^{7} - 1872083389 \nu^{6} + 7669985193 \nu^{5} + \cdots - 511003031 ) / 2684442852 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 341651 \nu^{11} + 1538648 \nu^{10} - 30409502 \nu^{9} + 106382019 \nu^{8} - 930938941 \nu^{7} + 2480519906 \nu^{6} - 11550527586 \nu^{5} + \cdots + 18221905513 ) / 2684442852 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 448568 \nu^{11} - 2517293 \nu^{10} + 41992115 \nu^{9} - 178362705 \nu^{8} + 1351376590 \nu^{7} - 4281464033 \nu^{6} + 17909729109 \nu^{5} + \cdots - 53261887861 ) / 2684442852 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} + \beta_{6} + \beta_{4} + 2\beta_{2} + \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 3 \beta_{11} + \beta_{10} - 2 \beta_{8} + 2 \beta_{7} + 5 \beta_{6} - 48 \beta_{4} - 27 \beta_{3} - 23 \beta_{2} + \beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 6 \beta_{11} - 34 \beta_{10} - 36 \beta_{9} - 4 \beta_{8} - 8 \beta_{7} - 14 \beta_{6} - 97 \beta_{4} - 17 \beta_{3} - 85 \beta_{2} - 23 \beta _1 + 318 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 96 \beta_{11} - 94 \beta_{10} - 12 \beta_{9} + 64 \beta_{8} - 113 \beta_{7} - 226 \beta_{6} + 15 \beta_{5} + 1443 \beta_{4} + 745 \beta_{3} + 533 \beta_{2} - 34 \beta _1 + 961 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 278 \beta_{11} + 976 \beta_{10} + 1177 \beta_{9} + 302 \beta_{8} + 407 \beta_{7} - 221 \beta_{6} + 20 \beta_{5} + 4572 \beta_{4} + 1061 \beta_{3} + 3054 \beta_{2} + 550 \beta _1 - 7536 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2750 \beta_{11} + 4687 \beta_{10} + 965 \beta_{9} - 1425 \beta_{8} + 5109 \beta_{7} + 7293 \beta_{6} - 769 \beta_{5} - 41222 \beta_{4} - 20717 \beta_{3} - 11867 \beta_{2} + 1088 \beta _1 - 36287 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 10792 \beta_{11} - 25086 \beta_{10} - 37878 \beta_{9} - 13196 \beta_{8} - 13554 \beta_{7} + 24562 \beta_{6} - 1650 \beta_{5} - 186451 \beta_{4} - 47841 \beta_{3} - 103457 \beta_{2} + \cdots + 173049 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 76854 \beta_{11} - 189718 \beta_{10} - 55640 \beta_{9} + 21780 \beta_{8} - 206481 \beta_{7} - 201956 \beta_{6} + 30013 \beta_{5} + 1133056 \beta_{4} + 578062 \beta_{3} + 237622 \beta_{2} + \cdots + 1266297 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 399632 \beta_{11} + 565479 \beta_{10} + 1198118 \beta_{9} + 480220 \beta_{8} + 333009 \beta_{7} - 1204974 \beta_{6} + 95026 \beta_{5} + 7096153 \beta_{4} + 1921387 \beta_{3} + \cdots - 3662102 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2081863 \beta_{11} + 6931510 \beta_{10} + 2718051 \beta_{9} + 62489 \beta_{8} + 7705527 \beta_{7} + 4885422 \beta_{6} - 1023719 \beta_{5} - 29422728 \beta_{4} + \cdots - 42265590 \) Copy content Toggle raw display

Character values

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

\(n\) \(57\) \(71\) \(101\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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 4.71642i
0.500000 2.25410i
0.500000 1.26614i
0.500000 0.317877i
0.500000 + 4.61251i
0.500000 + 5.67407i
0.500000 + 4.71642i
0.500000 + 2.25410i
0.500000 + 1.26614i
0.500000 + 0.317877i
0.500000 4.61251i
0.500000 5.67407i
0 −4.83454 + 2.79122i 0 1.93649 + 1.11803i 0 −5.45772 + 4.38330i 0 11.0818 19.1943i 0
61.2 0 −2.70210 + 1.56006i 0 −1.93649 1.11803i 0 2.68333 6.46527i 0 0.367578 0.636664i 0
61.3 0 −1.84651 + 1.06608i 0 1.93649 + 1.11803i 0 5.58449 4.22061i 0 −2.22693 + 3.85716i 0
61.4 0 −1.02529 + 0.591951i 0 −1.93649 1.11803i 0 −0.503367 + 6.98188i 0 −3.79919 + 6.58039i 0
61.5 0 3.24455 1.87324i 0 1.93649 + 1.11803i 0 5.30972 + 4.56145i 0 2.51809 4.36146i 0
61.6 0 4.16389 2.40402i 0 −1.93649 1.11803i 0 −0.616458 6.97280i 0 7.05863 12.2259i 0
101.1 0 −4.83454 2.79122i 0 1.93649 1.11803i 0 −5.45772 4.38330i 0 11.0818 + 19.1943i 0
101.2 0 −2.70210 1.56006i 0 −1.93649 + 1.11803i 0 2.68333 + 6.46527i 0 0.367578 + 0.636664i 0
101.3 0 −1.84651 1.06608i 0 1.93649 1.11803i 0 5.58449 + 4.22061i 0 −2.22693 3.85716i 0
101.4 0 −1.02529 0.591951i 0 −1.93649 + 1.11803i 0 −0.503367 6.98188i 0 −3.79919 6.58039i 0
101.5 0 3.24455 + 1.87324i 0 1.93649 1.11803i 0 5.30972 4.56145i 0 2.51809 + 4.36146i 0
101.6 0 4.16389 + 2.40402i 0 −1.93649 + 1.11803i 0 −0.616458 + 6.97280i 0 7.05863 + 12.2259i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 140.3.r.a 12
3.b odd 2 1 1260.3.bn.c 12
4.b odd 2 1 560.3.bx.b 12
5.b even 2 1 700.3.s.d 12
5.c odd 4 2 700.3.o.c 24
7.b odd 2 1 980.3.r.b 12
7.c even 3 1 980.3.d.a 12
7.c even 3 1 980.3.r.b 12
7.d odd 6 1 inner 140.3.r.a 12
7.d odd 6 1 980.3.d.a 12
21.g even 6 1 1260.3.bn.c 12
28.f even 6 1 560.3.bx.b 12
35.i odd 6 1 700.3.s.d 12
35.k even 12 2 700.3.o.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.3.r.a 12 1.a even 1 1 trivial
140.3.r.a 12 7.d odd 6 1 inner
560.3.bx.b 12 4.b odd 2 1
560.3.bx.b 12 28.f even 6 1
700.3.o.c 24 5.c odd 4 2
700.3.o.c 24 35.k even 12 2
700.3.s.d 12 5.b even 2 1
700.3.s.d 12 35.i odd 6 1
980.3.d.a 12 7.c even 3 1
980.3.d.a 12 7.d odd 6 1
980.3.r.b 12 7.b odd 2 1
980.3.r.b 12 7.c even 3 1
1260.3.bn.c 12 3.b odd 2 1
1260.3.bn.c 12 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(140, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{11} - 24 T^{10} + \cdots + 627264 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} - 14 T^{11} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} - 10 T^{11} + \cdots + 9758278656 \) Copy content Toggle raw display
$13$ \( T^{12} + 1314 T^{10} + \cdots + 1132462045584 \) Copy content Toggle raw display
$17$ \( T^{12} + 48 T^{11} + \cdots + 4938153984 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} + \cdots + 44050990216464 \) Copy content Toggle raw display
$23$ \( T^{12} - 18 T^{11} + \cdots + 16\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{6} - 2422 T^{4} - 21936 T^{3} + \cdots - 32456556)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + 60 T^{11} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{12} + 20 T^{11} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{12} + 15306 T^{10} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( (T^{6} + 114 T^{5} - 3292 T^{4} + \cdots - 2416949216)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 228 T^{11} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{12} + 44 T^{11} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{12} + 24 T^{11} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{12} - 72 T^{11} + \cdots + 82\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{12} + 154 T^{11} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{6} - 188 T^{5} + \cdots + 22870147104)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 48 T^{11} + \cdots + 17\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{12} - 116 T^{11} + \cdots + 73\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{12} + 50172 T^{10} + \cdots + 67\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{12} - 300 T^{11} + \cdots + 74\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{12} + 47984 T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
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