Properties

Label 740.2.i.a
Level $740$
Weight $2$
Character orbit 740.i
Analytic conductor $5.909$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(121,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("740.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.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: \(5.90892974957\)
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: \( x^{14} + 24x^{12} + 204x^{10} + 727x^{8} + 1008x^{6} + 426x^{4} + 64x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + (\beta_{10} - 1) q^{5} + ( - \beta_{10} + \beta_{6} + \beta_{5} + 1) q^{7} + ( - \beta_{13} + \beta_{11} + \cdots + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + (\beta_{10} - 1) q^{5} + ( - \beta_{10} + \beta_{6} + \beta_{5} + 1) q^{7} + ( - \beta_{13} + \beta_{11} + \cdots + \beta_{6}) q^{9}+ \cdots + ( - 2 \beta_{13} + 2 \beta_{11} + \cdots - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 7 q^{5} + 4 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 7 q^{5} + 4 q^{7} - 13 q^{9} + 14 q^{11} + 4 q^{13} + q^{17} + 4 q^{19} - 3 q^{21} + 12 q^{23} - 7 q^{25} - 6 q^{27} - 4 q^{29} - 24 q^{31} + 13 q^{33} + 4 q^{35} + 10 q^{37} + 21 q^{39} + 5 q^{41} - 6 q^{43} + 26 q^{45} - 16 q^{47} - 11 q^{49} + 26 q^{51} + 14 q^{53} - 7 q^{55} + 24 q^{57} - 16 q^{59} + 12 q^{61} - 98 q^{63} + 4 q^{65} + 21 q^{69} - 9 q^{71} - 40 q^{73} + 3 q^{77} + 3 q^{79} - 43 q^{81} + 5 q^{83} - 2 q^{85} + 31 q^{87} + 20 q^{89} - 2 q^{91} + 37 q^{93} + 4 q^{95} - 58 q^{97} - 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 24x^{12} + 204x^{10} + 727x^{8} + 1008x^{6} + 426x^{4} + 64x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{12} + 25\nu^{10} + 226\nu^{8} + 893\nu^{6} + 1496\nu^{4} + 863\nu^{2} + 60 ) / 27 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{12} + 25\nu^{10} + 226\nu^{8} + 893\nu^{6} + 1496\nu^{4} + 890\nu^{2} + 141 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{12} + 24\nu^{10} + 203\nu^{8} + 713\nu^{6} + 954\nu^{4} + 361\nu^{2} + 27\nu + 30 ) / 18 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{12} - 24\nu^{10} - 203\nu^{8} - 713\nu^{6} - 954\nu^{4} - 361\nu^{2} - 30 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{12} + 310\nu^{10} + 2602\nu^{8} + 9017\nu^{6} + 11588\nu^{4} + 3680\nu^{2} + 303 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9 \nu^{13} - 13 \nu^{12} + 219 \nu^{11} - 310 \nu^{10} + 1905 \nu^{9} - 2602 \nu^{8} + 7092 \nu^{7} + \cdots - 303 ) / 54 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18 \nu^{13} + \nu^{12} - 432 \nu^{11} + 25 \nu^{10} - 3672 \nu^{9} + 226 \nu^{8} - 13077 \nu^{7} + \cdots + 60 ) / 54 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -23\nu^{12} - 548\nu^{10} - 4595\nu^{8} - 15895\nu^{6} - 20287\nu^{4} - 6034\nu^{2} - 354 ) / 27 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\nu^{12} + 239\nu^{10} + 2016\nu^{8} + 7067\nu^{6} + 9367\nu^{4} + 3306\nu^{2} + 279 ) / 9 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -10\nu^{13} - 239\nu^{11} - 2016\nu^{9} - 7067\nu^{7} - 9367\nu^{5} - 3306\nu^{3} - 279\nu + 9 ) / 18 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9 \nu^{13} - 10 \nu^{12} - 216 \nu^{11} - 239 \nu^{10} - 1836 \nu^{9} - 2016 \nu^{8} - 6543 \nu^{7} + \cdots - 279 ) / 18 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 69 \nu^{13} - \nu^{12} - 1647 \nu^{11} - 25 \nu^{10} - 13863 \nu^{9} - 226 \nu^{8} - 48387 \nu^{7} + \cdots - 141 ) / 54 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 135 \nu^{13} + 23 \nu^{12} + 3219 \nu^{11} + 548 \nu^{10} + 27039 \nu^{9} + 4595 \nu^{8} + 93933 \nu^{7} + \cdots + 354 ) / 54 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{13} + 4 \beta_{12} - 4 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + \beta_{8} - 2 \beta_{7} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + 2\beta_{5} + \beta_{4} - 9\beta_{2} + 9\beta _1 + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20 \beta_{13} - 46 \beta_{12} + 40 \beta_{11} - 28 \beta_{10} + 20 \beta_{9} - 10 \beta_{8} + \cdots + 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} - 11\beta_{8} - 25\beta_{5} - 15\beta_{4} + 76\beta_{2} - 79\beta _1 - 166 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 174 \beta_{13} + 438 \beta_{12} - 360 \beta_{11} + 168 \beta_{10} - 180 \beta_{9} + 87 \beta_{8} + \cdots - 84 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -12\beta_{9} + 102\beta_{8} + 243\beta_{5} + 165\beta_{4} - 641\beta_{2} + 683\beta _1 + 1362 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1486 \beta_{13} - 3962 \beta_{12} + 3134 \beta_{11} - 938 \beta_{10} + 1567 \beta_{9} - 743 \beta_{8} + \cdots + 469 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 96\beta_{9} - 908\beta_{8} - 2173\beta_{5} - 1628\beta_{4} + 5439\beta_{2} - 5838\beta _1 - 11352 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 12694 \beta_{13} + 35156 \beta_{12} - 26900 \beta_{11} + 4610 \beta_{10} - 13450 \beta_{9} + 6347 \beta_{8} + \cdots - 2305 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -581\beta_{9} + 7975\beta_{8} + 18740\beta_{5} + 15309\beta_{4} - 46376\beta_{2} + 49565\beta _1 + 95339 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 108702 \beta_{13} - 309258 \beta_{12} + 229362 \beta_{11} - 14904 \beta_{10} + 114681 \beta_{9} + \cdots + 7452 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(-1 + \beta_{10}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.569716i
0.401547i
2.12787i
0.300390i
2.93925i
1.39878i
2.88103i
0.569716i
0.401547i
2.12787i
0.300390i
2.93925i
1.39878i
2.88103i
0 −1.67818 2.90670i 0 −0.500000 0.866025i 0 1.40373 + 2.43134i 0 −4.13261 + 7.15788i 0
121.2 0 −0.812038 1.40649i 0 −0.500000 0.866025i 0 0.0339191 + 0.0587497i 0 0.181190 0.313830i 0
121.3 0 −0.795627 1.37807i 0 −0.500000 0.866025i 0 −1.53626 2.66088i 0 0.233956 0.405224i 0
121.4 0 −0.117270 0.203117i 0 −0.500000 0.866025i 0 1.58956 + 2.75319i 0 1.47250 2.55044i 0
121.5 0 0.627876 + 1.08751i 0 −0.500000 0.866025i 0 −0.536763 0.929701i 0 0.711543 1.23243i 0
121.6 0 1.13042 + 1.95795i 0 −0.500000 0.866025i 0 −1.38845 2.40487i 0 −1.05570 + 1.82853i 0
121.7 0 1.64482 + 2.84891i 0 −0.500000 0.866025i 0 2.43427 + 4.21628i 0 −3.91088 + 6.77384i 0
581.1 0 −1.67818 + 2.90670i 0 −0.500000 + 0.866025i 0 1.40373 2.43134i 0 −4.13261 7.15788i 0
581.2 0 −0.812038 + 1.40649i 0 −0.500000 + 0.866025i 0 0.0339191 0.0587497i 0 0.181190 + 0.313830i 0
581.3 0 −0.795627 + 1.37807i 0 −0.500000 + 0.866025i 0 −1.53626 + 2.66088i 0 0.233956 + 0.405224i 0
581.4 0 −0.117270 + 0.203117i 0 −0.500000 + 0.866025i 0 1.58956 2.75319i 0 1.47250 + 2.55044i 0
581.5 0 0.627876 1.08751i 0 −0.500000 + 0.866025i 0 −0.536763 + 0.929701i 0 0.711543 + 1.23243i 0
581.6 0 1.13042 1.95795i 0 −0.500000 + 0.866025i 0 −1.38845 + 2.40487i 0 −1.05570 1.82853i 0
581.7 0 1.64482 2.84891i 0 −0.500000 + 0.866025i 0 2.43427 4.21628i 0 −3.91088 6.77384i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 740.2.i.a 14
37.c even 3 1 inner 740.2.i.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.i.a 14 1.a even 1 1 trivial
740.2.i.a 14 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + 17 T_{3}^{12} + 2 T_{3}^{11} + 216 T_{3}^{10} + 34 T_{3}^{9} + 1080 T_{3}^{8} + 486 T_{3}^{7} + \cdots + 361 \) acting on \(S_{2}^{\mathrm{new}}(740, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 17 T^{12} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} - 4 T^{13} + \cdots + 729 \) Copy content Toggle raw display
$11$ \( (T^{7} - 7 T^{6} + \cdots + 171)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} - 4 T^{13} + \cdots + 6365529 \) Copy content Toggle raw display
$17$ \( T^{14} - T^{13} + \cdots + 50936769 \) Copy content Toggle raw display
$19$ \( T^{14} - 4 T^{13} + \cdots + 3651921 \) Copy content Toggle raw display
$23$ \( (T^{7} - 6 T^{6} + \cdots - 17403)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + 2 T^{6} + \cdots + 2439)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + 12 T^{6} + \cdots + 10623)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 94931877133 \) Copy content Toggle raw display
$41$ \( T^{14} - 5 T^{13} + \cdots + 99620361 \) Copy content Toggle raw display
$43$ \( (T^{7} + 3 T^{6} + \cdots - 56313)^{2} \) Copy content Toggle raw display
$47$ \( (T^{7} + 8 T^{6} + \cdots - 834219)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} - 14 T^{13} + \cdots + 729 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 307686681 \) Copy content Toggle raw display
$61$ \( T^{14} - 12 T^{13} + \cdots + 1521 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 1259819921889 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 13021320321 \) Copy content Toggle raw display
$73$ \( (T^{7} + 20 T^{6} + \cdots - 8451)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 581726161 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 1719106455609 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 2708430524361 \) Copy content Toggle raw display
$97$ \( (T^{7} + 29 T^{6} + \cdots + 5722633)^{2} \) Copy content Toggle raw display
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