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Properties

Label 1340.4.a.e
Level $1340$
Weight $4$
Character orbit 1340.a
Self dual yes
Analytic conductor $79.063$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1340,4,Mod(1,1340)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1340, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1340.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1340.a (trivial)

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: yes
Analytic conductor: \(79.0625594077\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 4 x^{16} - 315 x^{15} + 972 x^{14} + 40607 x^{13} - 89672 x^{12} - 2732228 x^{11} + \cdots - 19975750400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + 5 q^{5} + ( - \beta_{6} + 3) q^{7} + (\beta_{2} + \beta_1 + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + 5 q^{5} + ( - \beta_{6} + 3) q^{7} + (\beta_{2} + \beta_1 + 11) q^{9} + (\beta_{7} + 6) q^{11} + (\beta_{7} - \beta_{6} - \beta_{5}) q^{13} + 5 \beta_1 q^{15} + ( - \beta_{15} - \beta_{13} - \beta_{9} + \cdots + 3) q^{17}+ \cdots + (9 \beta_{16} + 6 \beta_{15} + \cdots + 273) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 4 q^{3} + 85 q^{5} + 55 q^{7} + 187 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 4 q^{3} + 85 q^{5} + 55 q^{7} + 187 q^{9} + 105 q^{11} + 14 q^{13} + 20 q^{15} + 35 q^{17} + 337 q^{19} + 160 q^{21} + 207 q^{23} + 425 q^{25} + 712 q^{27} + 391 q^{29} + 376 q^{31} + 298 q^{33} + 275 q^{35} + 452 q^{37} + 658 q^{39} + 176 q^{41} + 1018 q^{43} + 935 q^{45} + 145 q^{47} + 1308 q^{49} + 172 q^{51} + 722 q^{53} + 525 q^{55} - 88 q^{57} + 1699 q^{59} + 581 q^{61} + 891 q^{63} + 70 q^{65} - 1139 q^{67} - 108 q^{69} + 1811 q^{71} - 1245 q^{73} + 100 q^{75} - 1669 q^{77} + 3364 q^{79} + 953 q^{81} - 809 q^{83} + 175 q^{85} + 2918 q^{87} + 1318 q^{89} + 3326 q^{91} - 726 q^{93} + 1685 q^{95} + 407 q^{97} + 4407 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 4 x^{16} - 315 x^{15} + 972 x^{14} + 40607 x^{13} - 89672 x^{12} - 2732228 x^{11} + \cdots - 19975750400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 38 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47\!\cdots\!87 \nu^{16} + \cdots + 65\!\cdots\!80 ) / 82\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!59 \nu^{16} + \cdots + 28\!\cdots\!20 ) / 50\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 51\!\cdots\!73 \nu^{16} + \cdots - 29\!\cdots\!80 ) / 16\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\!\cdots\!13 \nu^{16} + \cdots + 10\!\cdots\!00 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 43\!\cdots\!49 \nu^{16} + \cdots - 10\!\cdots\!60 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53\!\cdots\!31 \nu^{16} + \cdots + 24\!\cdots\!44 ) / 11\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!77 \nu^{16} + \cdots + 31\!\cdots\!60 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!17 \nu^{16} + \cdots + 17\!\cdots\!40 ) / 55\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 22\!\cdots\!77 \nu^{16} + \cdots - 48\!\cdots\!80 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 29\!\cdots\!89 \nu^{16} + \cdots + 82\!\cdots\!20 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 59\!\cdots\!25 \nu^{16} + \cdots - 19\!\cdots\!28 ) / 44\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 50\!\cdots\!57 \nu^{16} + \cdots + 16\!\cdots\!40 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!93 \nu^{16} + \cdots - 29\!\cdots\!60 ) / 66\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 35\!\cdots\!11 \nu^{16} + \cdots + 11\!\cdots\!20 ) / 16\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{16} + \beta_{15} - \beta_{14} - \beta_{13} - \beta_{10} - \beta_{5} + 2\beta_{2} + 62\beta _1 + 41 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{16} + 12 \beta_{15} - 5 \beta_{14} - 2 \beta_{13} - 6 \beta_{11} + \beta_{10} + 4 \beta_{9} + \cdots + 2397 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 120 \beta_{16} + 157 \beta_{15} - 136 \beta_{14} - 119 \beta_{13} + 22 \beta_{12} - 58 \beta_{11} + \cdots + 5858 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 911 \beta_{16} + 1819 \beta_{15} - 823 \beta_{14} - 320 \beta_{13} + 240 \beta_{12} - 882 \beta_{11} + \cdots + 172337 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12409 \beta_{16} + 19243 \beta_{15} - 14343 \beta_{14} - 11367 \beta_{13} + 5073 \beta_{12} + \cdots + 672748 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 105510 \beta_{16} + 215917 \beta_{15} - 100506 \beta_{14} - 41668 \beta_{13} + 53449 \beta_{12} + \cdots + 13562456 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1234574 \beta_{16} + 2170727 \beta_{15} - 1415552 \beta_{14} - 1029457 \beta_{13} + 786418 \beta_{12} + \cdots + 72450986 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11191763 \beta_{16} + 23758252 \beta_{15} - 11092344 \beta_{14} - 4912618 \beta_{13} + 8168727 \beta_{12} + \cdots + 1150200313 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 121782466 \beta_{16} + 236394389 \beta_{15} - 138042160 \beta_{14} - 93165197 \beta_{13} + \cdots + 7625069754 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1148584487 \beta_{16} + 2538536045 \beta_{15} - 1174869661 \beta_{14} - 548395491 \beta_{13} + \cdots + 103606535354 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12050132252 \beta_{16} + 25308128681 \beta_{15} - 13550825404 \beta_{14} - 8593113777 \beta_{13} + \cdots + 795993354820 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 116617887095 \beta_{16} + 267942843644 \beta_{15} - 122351113600 \beta_{14} - 59385457209 \beta_{13} + \cdots + 9771667503150 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1201505910807 \beta_{16} + 2685163040433 \beta_{15} - 1346174972417 \beta_{14} - 812493094351 \beta_{13} + \cdots + 82900579594098 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 11830671031250 \beta_{16} + 28136393498474 \beta_{15} - 12664859410431 \beta_{14} + \cdots + 952905828419050 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.24041
−8.09584
−7.22323
−6.57344
−6.22788
−3.32603
−2.54513
−1.33937
0.578125
0.698673
1.72848
3.72711
6.46814
7.51144
8.03879
8.58633
10.2343
0 −8.24041 0 5.00000 0 −16.1529 0 40.9043 0
1.2 0 −8.09584 0 5.00000 0 −13.4382 0 38.5426 0
1.3 0 −7.22323 0 5.00000 0 11.1976 0 25.1751 0
1.4 0 −6.57344 0 5.00000 0 32.8259 0 16.2101 0
1.5 0 −6.22788 0 5.00000 0 20.2271 0 11.7865 0
1.6 0 −3.32603 0 5.00000 0 −0.417962 0 −15.9375 0
1.7 0 −2.54513 0 5.00000 0 −33.8779 0 −20.5223 0
1.8 0 −1.33937 0 5.00000 0 14.7862 0 −25.2061 0
1.9 0 0.578125 0 5.00000 0 −11.0005 0 −26.6658 0
1.10 0 0.698673 0 5.00000 0 9.07308 0 −26.5119 0
1.11 0 1.72848 0 5.00000 0 −15.8641 0 −24.0124 0
1.12 0 3.72711 0 5.00000 0 34.0709 0 −13.1087 0
1.13 0 6.46814 0 5.00000 0 26.2225 0 14.8369 0
1.14 0 7.51144 0 5.00000 0 −30.5782 0 29.4217 0
1.15 0 8.03879 0 5.00000 0 21.4324 0 37.6221 0
1.16 0 8.58633 0 5.00000 0 3.21268 0 46.7251 0
1.17 0 10.2343 0 5.00000 0 3.28123 0 77.7400 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(67\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1340.4.a.e ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1340.4.a.e ✓ 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{17} - 4 T_{3}^{16} - 315 T_{3}^{15} + 972 T_{3}^{14} + 40607 T_{3}^{13} - 89672 T_{3}^{12} + \cdots - 19975750400 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1340))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} \) Copy content Toggle raw display
$3$ \( T^{17} + \cdots - 19975750400 \) Copy content Toggle raw display
$5$ \( (T - 5)^{17} \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 33\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{17} + \cdots - 27\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 31\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 58\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots - 10\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 44\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots + 36\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 24\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 30\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T + 67)^{17} \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots - 41\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 56\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 11\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots + 24\!\cdots\!50 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots - 27\!\cdots\!20 \) Copy content Toggle raw display
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