Properties

Label 45.6.a.e
Level $45$
Weight $6$
Character orbit 45.a
Self dual yes
Analytic conductor $7.217$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,6,Mod(1,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 45.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: \(7.21727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{409}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 102 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
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 \(\beta = \frac{1}{2}(1 + \sqrt{409})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 70) q^{4} - 25 q^{5} + (16 \beta - 64) q^{7} + (39 \beta + 102) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 70) q^{4} - 25 q^{5} + (16 \beta - 64) q^{7} + (39 \beta + 102) q^{8} - 25 \beta q^{10} + ( - 32 \beta - 108) q^{11} + ( - 16 \beta + 446) q^{13} + ( - 48 \beta + 1632) q^{14} + (109 \beta + 1738) q^{16} + ( - 80 \beta - 978) q^{17} + ( - 208 \beta + 836) q^{19} + ( - 25 \beta - 1750) q^{20} + ( - 140 \beta - 3264) q^{22} + ( - 48 \beta + 1632) q^{23} + 625 q^{25} + (430 \beta - 1632) q^{26} + (1072 \beta - 2848) q^{28} + ( - 64 \beta - 942) q^{29} + ( - 176 \beta + 1424) q^{31} + (599 \beta + 7854) q^{32} + ( - 1058 \beta - 8160) q^{34} + ( - 400 \beta + 1600) q^{35} + (816 \beta + 3926) q^{37} + (628 \beta - 21216) q^{38} + ( - 975 \beta - 2550) q^{40} + ( - 544 \beta + 4086) q^{41} + ( - 64 \beta - 8188) q^{43} + ( - 2380 \beta - 10824) q^{44} + (1584 \beta - 4896) q^{46} + ( - 1232 \beta + 10296) q^{47} + ( - 1792 \beta + 13401) q^{49} + 625 \beta q^{50} + ( - 690 \beta + 29588) q^{52} + (2272 \beta + 6042) q^{53} + (800 \beta + 2700) q^{55} + ( - 240 \beta + 57120) q^{56} + ( - 1006 \beta - 6528) q^{58} + (3232 \beta - 1164) q^{59} + (1568 \beta + 9326) q^{61} + (1248 \beta - 17952) q^{62} + (4965 \beta + 5482) q^{64} + (400 \beta - 11150) q^{65} + ( - 1280 \beta - 5812) q^{67} + ( - 6658 \beta - 76620) q^{68} + (1200 \beta - 40800) q^{70} + (3200 \beta + 18888) q^{71} + (608 \beta + 29258) q^{73} + (4742 \beta + 83232) q^{74} + ( - 13932 \beta + 37304) q^{76} + ( - 192 \beta - 45312) q^{77} + (3760 \beta + 51920) q^{79} + ( - 2725 \beta - 43450) q^{80} + (3542 \beta - 55488) q^{82} + ( - 4032 \beta + 63060) q^{83} + (2000 \beta + 24450) q^{85} + ( - 8252 \beta - 6528) q^{86} + ( - 8724 \beta - 138312) q^{88} + ( - 7392 \beta - 48186) q^{89} + (7904 \beta - 54656) q^{91} + ( - 1776 \beta + 109344) q^{92} + (9064 \beta - 125664) q^{94} + (5200 \beta - 20900) q^{95} + ( - 12480 \beta - 6142) q^{97} + (11609 \beta - 182784) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 141 q^{4} - 50 q^{5} - 112 q^{7} + 243 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 141 q^{4} - 50 q^{5} - 112 q^{7} + 243 q^{8} - 25 q^{10} - 248 q^{11} + 876 q^{13} + 3216 q^{14} + 3585 q^{16} - 2036 q^{17} + 1464 q^{19} - 3525 q^{20} - 6668 q^{22} + 3216 q^{23} + 1250 q^{25} - 2834 q^{26} - 4624 q^{28} - 1948 q^{29} + 2672 q^{31} + 16307 q^{32} - 17378 q^{34} + 2800 q^{35} + 8668 q^{37} - 41804 q^{38} - 6075 q^{40} + 7628 q^{41} - 16440 q^{43} - 24028 q^{44} - 8208 q^{46} + 19360 q^{47} + 25010 q^{49} + 625 q^{50} + 58486 q^{52} + 14356 q^{53} + 6200 q^{55} + 114000 q^{56} - 14062 q^{58} + 904 q^{59} + 20220 q^{61} - 34656 q^{62} + 15929 q^{64} - 21900 q^{65} - 12904 q^{67} - 159898 q^{68} - 80400 q^{70} + 40976 q^{71} + 59124 q^{73} + 171206 q^{74} + 60676 q^{76} - 90816 q^{77} + 107600 q^{79} - 89625 q^{80} - 107434 q^{82} + 122088 q^{83} + 50900 q^{85} - 21308 q^{86} - 285348 q^{88} - 103764 q^{89} - 101408 q^{91} + 216912 q^{92} - 242264 q^{94} - 36600 q^{95} - 24764 q^{97} - 353959 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−9.61187
10.6119
−9.61187 0 60.3881 −25.0000 0 −217.790 −272.863 0 240.297
1.2 10.6119 0 80.6119 −25.0000 0 105.790 515.863 0 −265.297
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

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 45.6.a.e 2
3.b odd 2 1 15.6.a.c 2
4.b odd 2 1 720.6.a.bd 2
5.b even 2 1 225.6.a.m 2
5.c odd 4 2 225.6.b.g 4
12.b even 2 1 240.6.a.q 2
15.d odd 2 1 75.6.a.h 2
15.e even 4 2 75.6.b.e 4
21.c even 2 1 735.6.a.g 2
24.f even 2 1 960.6.a.bf 2
24.h odd 2 1 960.6.a.bj 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.c 2 3.b odd 2 1
45.6.a.e 2 1.a even 1 1 trivial
75.6.a.h 2 15.d odd 2 1
75.6.b.e 4 15.e even 4 2
225.6.a.m 2 5.b even 2 1
225.6.b.g 4 5.c odd 4 2
240.6.a.q 2 12.b even 2 1
720.6.a.bd 2 4.b odd 2 1
735.6.a.g 2 21.c even 2 1
960.6.a.bf 2 24.f even 2 1
960.6.a.bj 2 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 102 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(45))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 102 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 112T - 23040 \) Copy content Toggle raw display
$11$ \( T^{2} + 248T - 89328 \) Copy content Toggle raw display
$13$ \( T^{2} - 876T + 165668 \) Copy content Toggle raw display
$17$ \( T^{2} + 2036 T + 381924 \) Copy content Toggle raw display
$19$ \( T^{2} - 1464 T - 3887920 \) Copy content Toggle raw display
$23$ \( T^{2} - 3216 T + 2350080 \) Copy content Toggle raw display
$29$ \( T^{2} + 1948 T + 529860 \) Copy content Toggle raw display
$31$ \( T^{2} - 2672 T - 1382400 \) Copy content Toggle raw display
$37$ \( T^{2} - 8668 T - 49300220 \) Copy content Toggle raw display
$41$ \( T^{2} - 7628 T - 15712860 \) Copy content Toggle raw display
$43$ \( T^{2} + 16440 T + 67149584 \) Copy content Toggle raw display
$47$ \( T^{2} - 19360 T - 61495104 \) Copy content Toggle raw display
$53$ \( T^{2} - 14356 T - 476289180 \) Copy content Toggle raw display
$59$ \( T^{2} - 904 T - 1067881200 \) Copy content Toggle raw display
$61$ \( T^{2} - 20220 T - 149182204 \) Copy content Toggle raw display
$67$ \( T^{2} + 12904 T - 125898096 \) Copy content Toggle raw display
$71$ \( T^{2} - 40976 T - 627281856 \) Copy content Toggle raw display
$73$ \( T^{2} - 59124 T + 836113700 \) Copy content Toggle raw display
$79$ \( T^{2} - 107600 T + 1448870400 \) Copy content Toggle raw display
$83$ \( T^{2} - 122088 T + 2064089232 \) Copy content Toggle raw display
$89$ \( T^{2} + 103764 T - 2895368220 \) Copy content Toggle raw display
$97$ \( T^{2} + 24764 T - 15772164476 \) Copy content Toggle raw display
show more
show less