Properties

Label 225.7.g.b
Level $225$
Weight $7$
Character orbit 225.g
Analytic conductor $51.762$
Analytic rank $0$
Dimension $2$
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] = [225,7,Mod(82,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), 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: \(51.7621688145\)
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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (3 i + 3) q^{2} - 46 i q^{4} + (207 i + 207) q^{7} + ( - 330 i + 330) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 i + 3) q^{2} - 46 i q^{4} + (207 i + 207) q^{7} + ( - 330 i + 330) q^{8} + 1188 q^{11} + ( - 1548 i + 1548) q^{13} + 1242 i q^{14} - 964 q^{16} + ( - 3252 i - 3252) q^{17} + 5060 i q^{19} + (3564 i + 3564) q^{22} + (5313 i - 5313) q^{23} + 9288 q^{26} + ( - 9522 i + 9522) q^{28} + 8910 i q^{29} + 25432 q^{31} + ( - 24012 i - 24012) q^{32} - 19512 i q^{34} + (20592 i + 20592) q^{37} + (15180 i - 15180) q^{38} + 19008 q^{41} + ( - 80343 i + 80343) q^{43} - 54648 i q^{44} - 31878 q^{46} + ( - 16137 i - 16137) q^{47} - 31951 i q^{49} + ( - 71208 i - 71208) q^{52} + ( - 155892 i + 155892) q^{53} + 136620 q^{56} + (26730 i - 26730) q^{58} - 360180 i q^{59} + 178112 q^{61} + (76296 i + 76296) q^{62} - 82376 i q^{64} + ( - 240273 i - 240273) q^{67} + (149592 i - 149592) q^{68} + 617328 q^{71} + (306612 i - 306612) q^{73} + 123552 i q^{74} + 232760 q^{76} + (245916 i + 245916) q^{77} - 232760 i q^{79} + (57024 i + 57024) q^{82} + ( - 134097 i + 134097) q^{83} + 482058 q^{86} + ( - 392040 i + 392040) q^{88} - 270270 i q^{89} + 640872 q^{91} + (244398 i + 244398) q^{92} - 96822 i q^{94} + (810612 i + 810612) q^{97} + ( - 95853 i + 95853) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{2} + 414 q^{7} + 660 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{2} + 414 q^{7} + 660 q^{8} + 2376 q^{11} + 3096 q^{13} - 1928 q^{16} - 6504 q^{17} + 7128 q^{22} - 10626 q^{23} + 18576 q^{26} + 19044 q^{28} + 50864 q^{31} - 48024 q^{32} + 41184 q^{37} - 30360 q^{38} + 38016 q^{41} + 160686 q^{43} - 63756 q^{46} - 32274 q^{47} - 142416 q^{52} + 311784 q^{53} + 273240 q^{56} - 53460 q^{58} + 356224 q^{61} + 152592 q^{62} - 480546 q^{67} - 299184 q^{68} + 1234656 q^{71} - 613224 q^{73} + 465520 q^{76} + 491832 q^{77} + 114048 q^{82} + 268194 q^{83} + 964116 q^{86} + 784080 q^{88} + 1281744 q^{91} + 488796 q^{92} + 1621224 q^{97} + 191706 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
1.00000i
1.00000i
3.00000 + 3.00000i 0 46.0000i 0 0 207.000 + 207.000i 330.000 330.000i 0 0
118.1 3.00000 3.00000i 0 46.0000i 0 0 207.000 207.000i 330.000 + 330.000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.7.g.b 2
3.b odd 2 1 25.7.c.a 2
5.b even 2 1 225.7.g.a 2
5.c odd 4 1 225.7.g.a 2
5.c odd 4 1 inner 225.7.g.b 2
15.d odd 2 1 25.7.c.b yes 2
15.e even 4 1 25.7.c.a 2
15.e even 4 1 25.7.c.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.7.c.a 2 3.b odd 2 1
25.7.c.a 2 15.e even 4 1
25.7.c.b yes 2 15.d odd 2 1
25.7.c.b yes 2 15.e even 4 1
225.7.g.a 2 5.b even 2 1
225.7.g.a 2 5.c odd 4 1
225.7.g.b 2 1.a even 1 1 trivial
225.7.g.b 2 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 6T_{2} + 18 \) acting on \(S_{7}^{\mathrm{new}}(225, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 6T + 18 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 414T + 85698 \) Copy content Toggle raw display
$11$ \( (T - 1188)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3096 T + 4792608 \) Copy content Toggle raw display
$17$ \( T^{2} + 6504 T + 21151008 \) Copy content Toggle raw display
$19$ \( T^{2} + 25603600 \) Copy content Toggle raw display
$23$ \( T^{2} + 10626 T + 56455938 \) Copy content Toggle raw display
$29$ \( T^{2} + 79388100 \) Copy content Toggle raw display
$31$ \( (T - 25432)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 41184 T + 848060928 \) Copy content Toggle raw display
$41$ \( (T - 19008)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 160686 T + 12909995298 \) Copy content Toggle raw display
$47$ \( T^{2} + 32274 T + 520805538 \) Copy content Toggle raw display
$53$ \( T^{2} - 311784 T + 48604631328 \) Copy content Toggle raw display
$59$ \( T^{2} + 129729632400 \) Copy content Toggle raw display
$61$ \( (T - 178112)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 480546 T + 115462229058 \) Copy content Toggle raw display
$71$ \( (T - 617328)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 613224 T + 188021837088 \) Copy content Toggle raw display
$79$ \( T^{2} + 54177217600 \) Copy content Toggle raw display
$83$ \( T^{2} - 268194 T + 35964010818 \) Copy content Toggle raw display
$89$ \( T^{2} + 73045872900 \) Copy content Toggle raw display
$97$ \( T^{2} - 1621224 T + 1314183629088 \) Copy content Toggle raw display
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