↑

Properties

Label 1610.2.e.c
Level $1610$
Weight $2$
Character orbit 1610.e
Analytic conductor $12.856$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1610,2,Mod(1289,1610)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1610, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1610.1289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1610 = 2 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1610.e (of order \(2\), degree \(1\), 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: \(12.8559147254\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 36 x^{16} + 524 x^{14} + 3948 x^{12} + 16316 x^{10} + 35812 x^{8} + 36192 x^{6} + 10688 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} + \beta_1 q^{3} - q^{4} - \beta_{11} q^{5} + \beta_{3} q^{6} - \beta_{9} q^{7} - \beta_{9} q^{8} + (\beta_{4} - \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} + \beta_1 q^{3} - q^{4} - \beta_{11} q^{5} + \beta_{3} q^{6} - \beta_{9} q^{7} - \beta_{9} q^{8} + (\beta_{4} - \beta_{3} - 1) q^{9} - \beta_{7} q^{10} + (\beta_{15} - \beta_{14} - \beta_{4} + \cdots + 1) q^{11}+ \cdots + (\beta_{17} + \beta_{15} + \beta_{13} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 18 q^{4} + 4 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 18 q^{4} + 4 q^{6} - 18 q^{9} - 2 q^{10} - 8 q^{11} + 18 q^{14} + 8 q^{15} + 18 q^{16} - 24 q^{19} - 4 q^{21} - 4 q^{24} - 2 q^{25} + 8 q^{26} + 8 q^{29} + 12 q^{30} - 24 q^{34} + 2 q^{35} + 18 q^{36} + 16 q^{39} + 2 q^{40} + 16 q^{41} + 8 q^{44} + 24 q^{45} + 18 q^{46} - 18 q^{49} + 8 q^{51} - 28 q^{54} - 12 q^{55} - 18 q^{56} + 8 q^{59} - 8 q^{60} + 28 q^{61} - 18 q^{64} - 20 q^{65} + 8 q^{66} - 4 q^{69} - 44 q^{71} - 8 q^{74} + 20 q^{75} + 24 q^{76} - 16 q^{79} + 10 q^{81} + 4 q^{84} - 36 q^{85} + 24 q^{86} - 20 q^{89} - 18 q^{90} - 8 q^{91} - 36 q^{94} - 32 q^{95} + 4 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 36 x^{16} + 524 x^{14} + 3948 x^{12} + 16316 x^{10} + 35812 x^{8} + 36192 x^{6} + 10688 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 372 \nu^{16} - 11971 \nu^{14} - 150784 \nu^{12} - 934238 \nu^{10} - 2872628 \nu^{8} + \cdots + 374104 ) / 160880 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 492 \nu^{16} + 17649 \nu^{14} + 255732 \nu^{12} + 1912858 \nu^{10} + 7792220 \nu^{8} + 16540544 \nu^{6} + \cdots + 72968 ) / 160880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 492 \nu^{16} + 17649 \nu^{14} + 255732 \nu^{12} + 1912858 \nu^{10} + 7792220 \nu^{8} + \cdots + 716488 ) / 160880 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3617 \nu^{17} + 1814 \nu^{16} - 129148 \nu^{15} + 64851 \nu^{14} - 1858906 \nu^{13} + \cdots - 655112 ) / 321760 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5147 \nu^{17} + 182438 \nu^{15} + 2599726 \nu^{13} + 18989248 \nu^{11} + 74619056 \nu^{9} + \cdots - 4166416 \nu ) / 321760 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3617 \nu^{17} - 1814 \nu^{16} - 129148 \nu^{15} - 64851 \nu^{14} - 1858906 \nu^{13} + \cdots + 655112 ) / 321760 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6442 \nu^{17} - 1447 \nu^{16} - 230474 \nu^{15} - 50938 \nu^{14} - 3324392 \nu^{13} + \cdots + 415216 ) / 321760 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9121 \nu^{17} - 327372 \nu^{15} - 4744106 \nu^{13} - 35498244 \nu^{11} - 144992520 \nu^{9} + \cdots - 3499264 \nu ) / 321760 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1814 \nu^{17} - 6842 \nu^{16} + 64851 \nu^{15} - 244038 \nu^{14} + 931012 \nu^{13} - 3507976 \nu^{12} + \cdots - 1561264 ) / 321760 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 7471 \nu^{17} - 3973 \nu^{16} + 268404 \nu^{15} - 142507 \nu^{14} + 3898338 \nu^{13} + \cdots - 674936 ) / 321760 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 7471 \nu^{17} - 3973 \nu^{16} - 268404 \nu^{15} - 142507 \nu^{14} - 3898338 \nu^{13} + \cdots - 674936 ) / 321760 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5620 \nu^{16} + 200227 \nu^{14} + 2878592 \nu^{12} + 21364870 \nu^{10} + 86606348 \nu^{8} + \cdots + 1584008 ) / 160880 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 14269 \nu^{17} + 552 \nu^{16} + 512237 \nu^{15} + 16466 \nu^{14} + 7426018 \nu^{13} + 183524 \nu^{12} + \cdots + 119536 ) / 321760 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 14269 \nu^{17} - 552 \nu^{16} + 512237 \nu^{15} - 16466 \nu^{14} + 7426018 \nu^{13} - 183524 \nu^{12} + \cdots - 119536 ) / 321760 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 25205 \nu^{17} + 552 \nu^{16} + 906887 \nu^{15} + 16466 \nu^{14} + 13183322 \nu^{13} + \cdots + 119536 ) / 321760 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 3550 \nu^{17} - 127419 \nu^{15} - 1846496 \nu^{13} - 13816726 \nu^{11} - 56438564 \nu^{9} + \cdots - 1417288 \nu ) / 32176 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{15} + \beta_{9} + \beta_{6} - 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{17} - \beta_{14} + \beta_{13} + 3 \beta_{12} + 2 \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 13 \beta_{17} - 14 \beta_{16} - 11 \beta_{15} + 2 \beta_{14} + \beta_{13} + \beta_{12} + \beta_{10} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10 \beta_{17} - 2 \beta_{15} + 12 \beta_{14} - 16 \beta_{13} - 42 \beta_{12} - 32 \beta_{11} + \cdots - 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 132 \beta_{17} + 148 \beta_{16} + 102 \beta_{15} - 32 \beta_{14} - 14 \beta_{13} - 16 \beta_{12} + \cdots - 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 76 \beta_{17} + 34 \beta_{15} - 110 \beta_{14} + 190 \beta_{13} + 460 \beta_{12} + 384 \beta_{11} + \cdots + 1746 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1234 \beta_{17} - 1444 \beta_{16} - 904 \beta_{15} + 382 \beta_{14} + 158 \beta_{13} + 194 \beta_{12} + \cdots + 158 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 498 \beta_{17} - 418 \beta_{15} + 916 \beta_{14} - 2022 \beta_{13} - 4646 \beta_{12} - 4148 \beta_{11} + \cdots - 14538 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 11120 \beta_{17} + 13672 \beta_{16} + 7920 \beta_{15} - 4096 \beta_{14} - 1656 \beta_{13} - 2120 \beta_{12} + \cdots - 1656 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2692 \beta_{17} + 4560 \beta_{15} - 7252 \beta_{14} + 20444 \beta_{13} + 45308 \beta_{12} + \cdots + 123684 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 98376 \beta_{17} - 127876 \beta_{16} - 69376 \beta_{15} + 41784 \beta_{14} + 16716 \beta_{13} + \cdots + 16716 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 8232 \beta_{17} - 47044 \beta_{15} + 55276 \beta_{14} - 201304 \beta_{13} - 434424 \beta_{12} + \cdots - 1070172 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 862180 \beta_{17} + 1190600 \beta_{16} + 610148 \beta_{15} - 415392 \beta_{14} - 165060 \beta_{13} + \cdots - 165060 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 66200 \beta_{17} + 471264 \beta_{15} - 405064 \beta_{14} + 1953920 \beta_{13} + 4130456 \beta_{12} + \cdots + 9387936 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 7524648 \beta_{17} - 11075504 \beta_{16} - 5396192 \beta_{15} + 4071200 \beta_{14} + 1608112 \beta_{13} + \cdots + 1608112 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1610\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(967\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1289.1
− 2.85646i
− 1.79450i
− 1.61134i
− 0.373014i
0.142304i
0.489328i
2.42161i
2.49010i
3.09198i
− 3.09198i
− 2.49010i
− 2.42161i
− 0.489328i
− 0.142304i
0.373014i
1.61134i
1.79450i
2.85646i
− 1.00000i − 2.85646i −1.00000 −1.88425 + 1.20400i −2.85646 1.00000i 1.00000i −5.15937 1.20400 + 1.88425i
1289.2 − 1.00000i − 1.79450i −1.00000 1.37039 + 1.76693i −1.79450 1.00000i 1.00000i −0.220238 1.76693 − 1.37039i
1289.3 − 1.00000i − 1.61134i −1.00000 −0.149342 − 2.23108i −1.61134 1.00000i 1.00000i 0.403572 −2.23108 + 0.149342i
1289.4 − 1.00000i − 0.373014i −1.00000 −2.09564 − 0.779927i −0.373014 1.00000i 1.00000i 2.86086 −0.779927 + 2.09564i
1289.5 − 1.00000i 0.142304i −1.00000 0.709991 − 2.12036i 0.142304 1.00000i 1.00000i 2.97975 −2.12036 − 0.709991i
1289.6 − 1.00000i 0.489328i −1.00000 0.844210 + 2.07058i 0.489328 1.00000i 1.00000i 2.76056 2.07058 − 0.844210i
1289.7 − 1.00000i 2.42161i −1.00000 2.18723 + 0.464772i 2.42161 1.00000i 1.00000i −2.86421 0.464772 − 2.18723i
1289.8 − 1.00000i 2.49010i −1.00000 1.19307 − 1.89119i 2.49010 1.00000i 1.00000i −3.20059 −1.89119 − 1.19307i
1289.9 − 1.00000i 3.09198i −1.00000 −2.17565 + 0.516263i 3.09198 1.00000i 1.00000i −6.56033 0.516263 + 2.17565i
1289.10 1.00000i − 3.09198i −1.00000 −2.17565 − 0.516263i 3.09198 − 1.00000i − 1.00000i −6.56033 0.516263 − 2.17565i
1289.11 1.00000i − 2.49010i −1.00000 1.19307 + 1.89119i 2.49010 − 1.00000i − 1.00000i −3.20059 −1.89119 + 1.19307i
1289.12 1.00000i − 2.42161i −1.00000 2.18723 − 0.464772i 2.42161 − 1.00000i − 1.00000i −2.86421 0.464772 + 2.18723i
1289.13 1.00000i − 0.489328i −1.00000 0.844210 − 2.07058i 0.489328 − 1.00000i − 1.00000i 2.76056 2.07058 + 0.844210i
1289.14 1.00000i − 0.142304i −1.00000 0.709991 + 2.12036i 0.142304 − 1.00000i − 1.00000i 2.97975 −2.12036 + 0.709991i
1289.15 1.00000i 0.373014i −1.00000 −2.09564 + 0.779927i −0.373014 − 1.00000i − 1.00000i 2.86086 −0.779927 − 2.09564i
1289.16 1.00000i 1.61134i −1.00000 −0.149342 + 2.23108i −1.61134 − 1.00000i − 1.00000i 0.403572 −2.23108 − 0.149342i
1289.17 1.00000i 1.79450i −1.00000 1.37039 − 1.76693i −1.79450 − 1.00000i − 1.00000i −0.220238 1.76693 + 1.37039i
1289.18 1.00000i 2.85646i −1.00000 −1.88425 − 1.20400i −2.85646 − 1.00000i − 1.00000i −5.15937 1.20400 − 1.88425i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1289.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1610.2.e.c ✓ 18
5.b even 2 1 inner 1610.2.e.c ✓ 18
5.c odd 4 1 8050.2.a.cg 9
5.c odd 4 1 8050.2.a.ch 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1610.2.e.c ✓ 18 1.a even 1 1 trivial
1610.2.e.c ✓ 18 5.b even 2 1 inner
8050.2.a.cg 9 5.c odd 4 1
8050.2.a.ch 9 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} + 36 T_{3}^{16} + 524 T_{3}^{14} + 3948 T_{3}^{12} + 16316 T_{3}^{10} + 35812 T_{3}^{8} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(1610, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$3$ \( T^{18} + 36 T^{16} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{18} + T^{16} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$11$ \( (T^{9} + 4 T^{8} + \cdots + 176)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + 120 T^{16} + \cdots + 16384 \) Copy content Toggle raw display
$17$ \( T^{18} + 152 T^{16} + \cdots + 69488896 \) Copy content Toggle raw display
$19$ \( (T^{9} + 12 T^{8} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$29$ \( (T^{9} - 4 T^{8} - 68 T^{7} + \cdots + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} - 130 T^{7} + \cdots + 153124)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 172134400 \) Copy content Toggle raw display
$41$ \( (T^{9} - 8 T^{8} + \cdots - 1125120)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + 264 T^{16} + \cdots + 3097600 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 513191392297984 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 67660495360000 \) Copy content Toggle raw display
$59$ \( (T^{9} - 4 T^{8} + \cdots - 1321400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 14 T^{8} + \cdots + 2355952)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 26454371904 \) Copy content Toggle raw display
$71$ \( (T^{9} + 22 T^{8} + \cdots + 329623296)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 26857601434624 \) Copy content Toggle raw display
$79$ \( (T^{9} + 8 T^{8} + \cdots + 1307448)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 321934311030784 \) Copy content Toggle raw display
$89$ \( (T^{9} + 10 T^{8} + \cdots + 688488)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 138122235904 \) Copy content Toggle raw display
show more
show less