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Properties

Label 1340.2.i.c
Level $1340$
Weight $2$
Character orbit 1340.i
Analytic conductor $10.700$
Analytic rank $0$
Dimension $16$
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] = [1340,2,Mod(841,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1340.841");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1340.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: \(10.6999538709\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 20 x^{14} - 11 x^{13} + 268 x^{12} - 118 x^{11} + 1854 x^{10} - 42 x^{9} + 8772 x^{8} + \cdots + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{3} - q^{5} + \beta_{11} q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{3} - q^{5} + \beta_{11} q^{7} + (\beta_{2} + 2) q^{9} + (3 \beta_{3} + 3) q^{11} + ( - \beta_{10} - \beta_{8} + \cdots + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{7} + 6 \beta_{3} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} - 16 q^{5} + 30 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} - 16 q^{5} + 30 q^{9} + 24 q^{11} + 2 q^{13} + 2 q^{15} - 3 q^{17} - 5 q^{21} + 3 q^{23} + 16 q^{25} - 14 q^{27} - 7 q^{29} + 7 q^{31} - 3 q^{33} - 11 q^{37} - 13 q^{39} - 15 q^{41} - 14 q^{43} - 30 q^{45} - 11 q^{47} - 10 q^{49} - 5 q^{51} + 12 q^{53} - 24 q^{55} + 26 q^{57} + 24 q^{59} + 9 q^{61} + 23 q^{63} - 2 q^{65} - q^{67} - q^{69} - 9 q^{71} - 13 q^{73} - 2 q^{75} + 5 q^{79} + 56 q^{81} - 8 q^{83} + 3 q^{85} + 14 q^{87} - 34 q^{89} + 30 q^{91} - 40 q^{93} + 25 q^{97} + 45 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 20 x^{14} - 11 x^{13} + 268 x^{12} - 118 x^{11} + 1854 x^{10} - 42 x^{9} + 8772 x^{8} + \cdots + 2304 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 57\!\cdots\!80 \nu^{15} + \cdots - 11\!\cdots\!65 ) / 23\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\!\cdots\!47 \nu^{15} + \cdots - 44\!\cdots\!20 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!89 \nu^{15} + \cdots + 11\!\cdots\!64 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!67 \nu^{15} + \cdots + 43\!\cdots\!84 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31\!\cdots\!95 \nu^{15} + \cdots - 25\!\cdots\!68 ) / 23\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 78\!\cdots\!55 \nu^{15} + \cdots - 21\!\cdots\!20 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38\!\cdots\!27 \nu^{15} + \cdots - 68\!\cdots\!36 ) / 15\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 78\!\cdots\!21 \nu^{15} + \cdots - 10\!\cdots\!76 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 80\!\cdots\!55 \nu^{15} + \cdots - 28\!\cdots\!44 ) / 25\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16\!\cdots\!89 \nu^{15} + \cdots - 23\!\cdots\!28 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 66\!\cdots\!86 \nu^{15} + \cdots + 33\!\cdots\!44 ) / 64\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 42\!\cdots\!11 \nu^{15} + \cdots + 47\!\cdots\!12 ) / 30\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28\!\cdots\!02 \nu^{15} + \cdots - 18\!\cdots\!73 ) / 19\!\cdots\!09 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 34\!\cdots\!25 \nu^{15} + \cdots + 84\!\cdots\!22 ) / 19\!\cdots\!09 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 5\beta_{3} - \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{12} + \beta_{11} - 8\beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 10\beta_{7} - 2\beta_{5} - 2\beta_{4} + 39\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} - 13 \beta_{14} - 3 \beta_{13} + 15 \beta_{12} - 11 \beta_{11} + 15 \beta_{10} + 13 \beta_{9} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3 \beta_{15} + 18 \beta_{14} + 31 \beta_{13} - 4 \beta_{12} + 26 \beta_{11} - 6 \beta_{6} + \cdots + 342 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -171\beta_{10} - 141\beta_{9} - 37\beta_{8} - 5\beta_{7} - 114\beta_{5} - 53\beta_{4} + 65\beta_{3} + 675\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 50 \beta_{15} - 239 \beta_{14} - 365 \beta_{13} + 81 \beta_{12} - 280 \beta_{11} + 81 \beta_{10} + \cdots - 3153 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 482 \beta_{15} + 1444 \beta_{14} + 685 \beta_{13} - 1795 \beta_{12} + 1177 \beta_{11} - 6447 \beta_{6} + \cdots + 1042 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1155 \beta_{10} - 2831 \beta_{9} - 601 \beta_{8} - 8734 \beta_{7} - 2893 \beta_{5} + \cdots + 2266 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5538 \beta_{15} - 14476 \beta_{14} - 7910 \beta_{13} + 18258 \beta_{12} - 12119 \beta_{11} + \cdots - 14573 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 6485 \beta_{15} + 31739 \beta_{14} + 40644 \beta_{13} - 14428 \beta_{12} + 29586 \beta_{11} + \cdots + 285859 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 183279 \beta_{10} - 144084 \beta_{9} - 60108 \beta_{8} - 34779 \beta_{7} - 124362 \beta_{5} + \cdots + 604339 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 67338 \beta_{15} - 345255 \beta_{14} - 414174 \beta_{13} + 169206 \beta_{12} - 302034 \beta_{11} + \cdots - 2765816 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 633444 \beta_{15} + 1432666 \beta_{14} + 928635 \beta_{13} - 1829566 \beta_{12} + 1272292 \beta_{11} + \cdots + 2284656 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times\).

\(n\) \(537\) \(671\) \(1141\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
841.1
1.52184 + 2.63590i
1.47670 + 2.55772i
0.900610 + 1.55990i
0.276626 + 0.479131i
−0.279640 − 0.484351i
−0.656092 − 1.13638i
−1.14287 − 1.97951i
−1.59717 − 2.76638i
1.52184 − 2.63590i
1.47670 − 2.55772i
0.900610 − 1.55990i
0.276626 − 0.479131i
−0.279640 + 0.484351i
−0.656092 + 1.13638i
−1.14287 + 1.97951i
−1.59717 + 2.76638i
0 −3.04367 0 −1.00000 0 −1.18520 + 2.05282i 0 6.26395 0
841.2 0 −2.95340 0 −1.00000 0 2.02193 − 3.50208i 0 5.72255 0
841.3 0 −1.80122 0 −1.00000 0 −0.650435 + 1.12659i 0 0.244390 0
841.4 0 −0.553252 0 −1.00000 0 1.88582 − 3.26633i 0 −2.69391 0
841.5 0 0.559280 0 −1.00000 0 −0.701745 + 1.21546i 0 −2.68721 0
841.6 0 1.31218 0 −1.00000 0 −2.24497 + 3.88839i 0 −1.27817 0
841.7 0 2.28574 0 −1.00000 0 −0.308683 + 0.534654i 0 2.22460 0
841.8 0 3.19434 0 −1.00000 0 1.18328 − 2.04950i 0 7.20381 0
1101.1 0 −3.04367 0 −1.00000 0 −1.18520 − 2.05282i 0 6.26395 0
1101.2 0 −2.95340 0 −1.00000 0 2.02193 + 3.50208i 0 5.72255 0
1101.3 0 −1.80122 0 −1.00000 0 −0.650435 − 1.12659i 0 0.244390 0
1101.4 0 −0.553252 0 −1.00000 0 1.88582 + 3.26633i 0 −2.69391 0
1101.5 0 0.559280 0 −1.00000 0 −0.701745 − 1.21546i 0 −2.68721 0
1101.6 0 1.31218 0 −1.00000 0 −2.24497 − 3.88839i 0 −1.27817 0
1101.7 0 2.28574 0 −1.00000 0 −0.308683 − 0.534654i 0 2.22460 0
1101.8 0 3.19434 0 −1.00000 0 1.18328 + 2.04950i 0 7.20381 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 841.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.2.i.c ✓ 16
67.c even 3 1 inner 1340.2.i.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.2.i.c ✓ 16 1.a even 1 1 trivial
1340.2.i.c ✓ 16 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + T_{3}^{7} - 19T_{3}^{6} - 15T_{3}^{5} + 108T_{3}^{4} + 49T_{3}^{3} - 187T_{3}^{2} - 13T_{3} + 48 \) acting on \(S_{2}^{\mathrm{new}}(1340, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{7} - 19 T^{6} + \cdots + 48)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 33 T^{14} + \cdots + 187489 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T + 9)^{8} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 1322704161 \) Copy content Toggle raw display
$17$ \( T^{16} + 3 T^{15} + \cdots + 37564641 \) Copy content Toggle raw display
$19$ \( T^{16} + 93 T^{14} + \cdots + 954529 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 321213497049 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 453817809 \) Copy content Toggle raw display
$31$ \( T^{16} - 7 T^{15} + \cdots + 471969 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 818703769 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 532189217169 \) Copy content Toggle raw display
$43$ \( (T^{8} + 7 T^{7} + \cdots + 8112708)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 176914712263041 \) Copy content Toggle raw display
$53$ \( (T^{8} - 6 T^{7} + \cdots - 8586)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + \cdots + 18468)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 3641482943289 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 406067677556641 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 31487619172689 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 228665115292329 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 753335997499089 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 174319209 \) Copy content Toggle raw display
$89$ \( (T^{8} + 17 T^{7} + \cdots - 1118286)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 121934531505609 \) Copy content Toggle raw display
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