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Properties

Label 140.6.i.d
Level $140$
Weight $6$
Character orbit 140.i
Analytic conductor $22.454$
Analytic rank $0$
Dimension $14$
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: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 955 x^{12} + 2982 x^{11} + 679713 x^{10} + 1284921 x^{9} + 189493777 x^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{5}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (4 \beta_{2} - \beta_1 + 4) q^{3} - 25 \beta_{2} q^{5} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 16) q^{7} + (\beta_{6} - \beta_{4} + \cdots - 5 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (4 \beta_{2} - \beta_1 + 4) q^{3} - 25 \beta_{2} q^{5} + ( - \beta_{5} - \beta_{3} - \beta_{2} - 16) q^{7} + (\beta_{6} - \beta_{4} + \cdots - 5 \beta_1) q^{9}+ \cdots + ( - 134 \beta_{13} - 158 \beta_{12} + \cdots + 54100) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 27 q^{3} + 175 q^{5} - 225 q^{7} - 312 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 27 q^{3} + 175 q^{5} - 225 q^{7} - 312 q^{9} - 590 q^{11} - 1708 q^{13} + 1350 q^{15} + 1328 q^{17} + 2990 q^{19} + 993 q^{21} - 4089 q^{23} - 4375 q^{25} - 8562 q^{27} - 6126 q^{29} + 5740 q^{31} - 1470 q^{33} - 3000 q^{35} - 12218 q^{37} + 1560 q^{39} - 16954 q^{41} - 18390 q^{43} + 7800 q^{45} + 28652 q^{47} + 20435 q^{49} - 41244 q^{51} - 24932 q^{53} - 29500 q^{55} - 125868 q^{57} + 38242 q^{59} + 61299 q^{61} + 145548 q^{63} - 21350 q^{65} - 24913 q^{67} - 172302 q^{69} - 72340 q^{71} + 94780 q^{73} + 16875 q^{75} + 198344 q^{77} + 594 q^{79} + 53265 q^{81} - 227170 q^{83} + 66400 q^{85} + 43875 q^{87} + 36823 q^{89} + 138176 q^{91} + 21828 q^{93} - 74750 q^{95} - 515900 q^{97} + 756864 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 955 x^{12} + 2982 x^{11} + 679713 x^{10} + 1284921 x^{9} + 189493777 x^{8} + \cdots + 29\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 35\!\cdots\!69 \nu^{13} + \cdots - 30\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!33 \nu^{13} + \cdots + 53\!\cdots\!00 ) / 15\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!41 \nu^{13} + \cdots + 23\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!91 \nu^{13} + \cdots - 54\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 21\!\cdots\!51 \nu^{13} + \cdots + 55\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 61\!\cdots\!19 \nu^{13} + \cdots - 20\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 40\!\cdots\!43 \nu^{13} + \cdots - 52\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 46\!\cdots\!17 \nu^{13} + \cdots - 96\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!67 \nu^{13} + \cdots + 13\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 32\!\cdots\!97 \nu^{13} + \cdots + 42\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!57 \nu^{13} + \cdots + 37\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!91 \nu^{13} + \cdots + 46\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{4} - 3\beta_{3} + 273\beta_{2} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 2 \beta_{13} + \beta_{12} + 14 \beta_{10} + 3 \beta_{9} - 3 \beta_{7} - 6 \beta_{6} - 11 \beta_{4} + \cdots - 766 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{13} - 18 \beta_{12} - 36 \beta_{11} + 90 \beta_{10} + 3 \beta_{8} - 15 \beta_{7} + \cdots - 129141 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 866 \beta_{13} - 2850 \beta_{12} + 408 \beta_{11} + 2342 \beta_{10} + 1364 \beta_{9} - 710 \beta_{8} + \cdots + 5438 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9708 \beta_{13} - 48246 \beta_{12} + 62280 \beta_{11} - 115104 \beta_{10} + 7182 \beta_{9} + \cdots + 75842548 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1256625 \beta_{13} + 1084410 \beta_{12} + 456876 \beta_{11} - 8845938 \beta_{10} - 1711944 \beta_{9} + \cdots + 1186818657 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14526717 \beta_{13} + 56128887 \beta_{12} - 22320144 \beta_{11} - 17692422 \beta_{10} - 15855531 \beta_{9} + \cdots - 62601387 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 345547532 \beta_{13} + 851856256 \beta_{12} - 802497528 \beta_{11} + 5084438594 \beta_{10} + \cdots - 994693088932 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 11413584672 \beta_{13} - 15711951288 \beta_{12} - 15616525464 \beta_{11} + 110174510664 \beta_{10} + \cdots - 33943076273556 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 326317689029 \beta_{13} - 1176012095487 \beta_{12} + 324861218040 \beta_{11} + 559789373036 \beta_{10} + \cdots + 1773015064643 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1217198462118 \beta_{13} - 19572284047719 \beta_{12} + 22046197675608 \beta_{11} - 75476161044276 \beta_{10} + \cdots + 24\!\cdots\!18 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 361008377791470 \beta_{13} + 375514411285080 \beta_{12} + 253112242520592 \beta_{11} + \cdots + 59\!\cdots\!14 \) 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
13.5944 − 23.5462i
6.80925 − 11.7940i
3.20706 − 5.55479i
2.48906 − 4.31118i
−5.89368 + 10.2082i
−9.26395 + 16.0456i
−10.4422 + 18.0864i
13.5944 + 23.5462i
6.80925 + 11.7940i
3.20706 + 5.55479i
2.48906 + 4.31118i
−5.89368 − 10.2082i
−9.26395 − 16.0456i
−10.4422 − 18.0864i
0 −11.5944 + 20.0821i 0 12.5000 + 21.6506i 0 −85.8544 + 97.1391i 0 −147.361 − 255.237i 0
81.2 0 −4.80925 + 8.32987i 0 12.5000 + 21.6506i 0 −22.9057 − 127.602i 0 75.2421 + 130.323i 0
81.3 0 −1.20706 + 2.09068i 0 12.5000 + 21.6506i 0 −128.644 + 16.0515i 0 118.586 + 205.397i 0
81.4 0 −0.489062 + 0.847081i 0 12.5000 + 21.6506i 0 124.053 − 37.6545i 0 121.022 + 209.616i 0
81.5 0 7.89368 − 13.6723i 0 12.5000 + 21.6506i 0 124.976 + 34.4692i 0 −3.12031 − 5.40454i 0
81.6 0 11.2640 − 19.5097i 0 12.5000 + 21.6506i 0 −42.3536 + 122.528i 0 −132.253 − 229.069i 0
81.7 0 12.4422 − 21.5505i 0 12.5000 + 21.6506i 0 −81.7704 − 100.601i 0 −188.115 − 325.825i 0
121.1 0 −11.5944 − 20.0821i 0 12.5000 − 21.6506i 0 −85.8544 − 97.1391i 0 −147.361 + 255.237i 0
121.2 0 −4.80925 − 8.32987i 0 12.5000 − 21.6506i 0 −22.9057 + 127.602i 0 75.2421 − 130.323i 0
121.3 0 −1.20706 − 2.09068i 0 12.5000 − 21.6506i 0 −128.644 − 16.0515i 0 118.586 − 205.397i 0
121.4 0 −0.489062 − 0.847081i 0 12.5000 − 21.6506i 0 124.053 + 37.6545i 0 121.022 − 209.616i 0
121.5 0 7.89368 + 13.6723i 0 12.5000 − 21.6506i 0 124.976 − 34.4692i 0 −3.12031 + 5.40454i 0
121.6 0 11.2640 + 19.5097i 0 12.5000 − 21.6506i 0 −42.3536 − 122.528i 0 −132.253 + 229.069i 0
121.7 0 12.4422 + 21.5505i 0 12.5000 − 21.6506i 0 −81.7704 + 100.601i 0 −188.115 + 325.825i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.7
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.d ✓ 14
7.c even 3 1 inner 140.6.i.d ✓ 14
7.c even 3 1 980.6.a.n 7
7.d odd 6 1 980.6.a.o 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.6.i.d ✓ 14 1.a even 1 1 trivial
140.6.i.d ✓ 14 7.c even 3 1 inner
980.6.a.n 7 7.c even 3 1
980.6.a.o 7 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} - 27 T_{3}^{13} + 1371 T_{3}^{12} - 18854 T_{3}^{11} + 832497 T_{3}^{10} + \cdots + 21726524413584 \) acting on \(S_{6}^{\mathrm{new}}(140, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 21726524413584 \) Copy content Toggle raw display
$5$ \( (T^{2} - 25 T + 625)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 37\!\cdots\!43 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots - 43\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 12\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 17\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 13\!\cdots\!85)^{2} \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots + 15\!\cdots\!20)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 93\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 90\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 73\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 35\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( (T^{7} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
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