Properties

Label 450.6.c.n
Level $450$
Weight $6$
Character orbit 450.c
Analytic conductor $72.173$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [450,6,Mod(199,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.199");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not 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: \(72.1727189158\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 150)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 i q^{2} - 16 q^{4} + 233 i q^{7} - 64 i q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{2} - 16 q^{4} + 233 i q^{7} - 64 i q^{8} + 498 q^{11} - 809 i q^{13} - 932 q^{14} + 256 q^{16} + 1002 i q^{17} + 1705 q^{19} + 1992 i q^{22} + 1554 i q^{23} + 3236 q^{26} - 3728 i q^{28} + 7830 q^{29} + 977 q^{31} + 1024 i q^{32} - 4008 q^{34} - 4822 i q^{37} + 6820 i q^{38} + 8148 q^{41} - 19469 i q^{43} - 7968 q^{44} - 6216 q^{46} - 8418 i q^{47} - 37482 q^{49} + 12944 i q^{52} + 17664 i q^{53} + 14912 q^{56} + 31320 i q^{58} + 35910 q^{59} + 3527 q^{61} + 3908 i q^{62} - 4096 q^{64} + 57473 i q^{67} - 16032 i q^{68} + 7548 q^{71} + 646 i q^{73} + 19288 q^{74} - 27280 q^{76} + 116034 i q^{77} + 22720 q^{79} + 32592 i q^{82} + 11574 i q^{83} + 77876 q^{86} - 31872 i q^{88} - 78960 q^{89} + 188497 q^{91} - 24864 i q^{92} + 33672 q^{94} + 54593 i q^{97} - 149928 i q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 996 q^{11} - 1864 q^{14} + 512 q^{16} + 3410 q^{19} + 6472 q^{26} + 15660 q^{29} + 1954 q^{31} - 8016 q^{34} + 16296 q^{41} - 15936 q^{44} - 12432 q^{46} - 74964 q^{49} + 29824 q^{56} + 71820 q^{59} + 7054 q^{61} - 8192 q^{64} + 15096 q^{71} + 38576 q^{74} - 54560 q^{76} + 45440 q^{79} + 155752 q^{86} - 157920 q^{89} + 376994 q^{91} + 67344 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
4.00000i 0 −16.0000 0 0 233.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 233.000i 64.0000i 0 0
\(n\): e.g. 2-40 or 990-1000
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 450.6.c.n 2
3.b odd 2 1 150.6.c.e 2
5.b even 2 1 inner 450.6.c.n 2
5.c odd 4 1 450.6.a.a 1
5.c odd 4 1 450.6.a.x 1
15.d odd 2 1 150.6.c.e 2
15.e even 4 1 150.6.a.c 1
15.e even 4 1 150.6.a.l yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.c 1 15.e even 4 1
150.6.a.l yes 1 15.e even 4 1
150.6.c.e 2 3.b odd 2 1
150.6.c.e 2 15.d odd 2 1
450.6.a.a 1 5.c odd 4 1
450.6.a.x 1 5.c odd 4 1
450.6.c.n 2 1.a even 1 1 trivial
450.6.c.n 2 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(450, [\chi])\):

\( T_{7}^{2} + 54289 \) Copy content Toggle raw display
\( T_{11} - 498 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 54289 \) Copy content Toggle raw display
$11$ \( (T - 498)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 654481 \) Copy content Toggle raw display
$17$ \( T^{2} + 1004004 \) Copy content Toggle raw display
$19$ \( (T - 1705)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2414916 \) Copy content Toggle raw display
$29$ \( (T - 7830)^{2} \) Copy content Toggle raw display
$31$ \( (T - 977)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 23251684 \) Copy content Toggle raw display
$41$ \( (T - 8148)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 379041961 \) Copy content Toggle raw display
$47$ \( T^{2} + 70862724 \) Copy content Toggle raw display
$53$ \( T^{2} + 312016896 \) Copy content Toggle raw display
$59$ \( (T - 35910)^{2} \) Copy content Toggle raw display
$61$ \( (T - 3527)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 3303145729 \) Copy content Toggle raw display
$71$ \( (T - 7548)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 417316 \) Copy content Toggle raw display
$79$ \( (T - 22720)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 133957476 \) Copy content Toggle raw display
$89$ \( (T + 78960)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2980395649 \) Copy content Toggle raw display
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