Properties

Label 225.8.b.o
Level $225$
Weight $8$
Character orbit 225.b
Analytic conductor $70.287$
Analytic rank $0$
Dimension $4$
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] = [225,8,Mod(199,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.199");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.b (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: \(70.2866307339\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{31})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 15x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 75)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 4 \beta_1) q^{2} + ( - 8 \beta_{2} - 12) q^{4} + (14 \beta_{3} + 43 \beta_1) q^{7} + ( - 84 \beta_{3} + 528 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 4 \beta_1) q^{2} + ( - 8 \beta_{2} - 12) q^{4} + (14 \beta_{3} + 43 \beta_1) q^{7} + ( - 84 \beta_{3} + 528 \beta_1) q^{8} + (266 \beta_{2} + 2306) q^{11} + (296 \beta_{3} - 4509 \beta_1) q^{13} + (99 \beta_{2} + 1908) q^{14} + ( - 832 \beta_{2} - 9840) q^{16} + (2210 \beta_{3} + 9974 \beta_1) q^{17} + ( - 2746 \beta_{2} + 11967) q^{19} + ( - 3370 \beta_{3} - 42208 \beta_1) q^{22} + ( - 3882 \beta_{3} + 9906 \beta_1) q^{23} + ( - 3325 \beta_{2} + 18668) q^{26} + ( - 512 \beta_{3} - 14404 \beta_1) q^{28} + ( - 478 \beta_{2} - 173254) q^{29} + ( - 15502 \beta_{2} - 64589) q^{31} + (2416 \beta_{3} + 210112 \beta_1) q^{32} + (18814 \beta_{2} + 313936) q^{34} + (40764 \beta_{3} + 29018 \beta_1) q^{37} + ( - 983 \beta_{3} + 292636 \beta_1) q^{38} + ( - 41678 \beta_{2} + 170464) q^{41} + (9014 \beta_{3} - 543675 \beta_1) q^{43} + ( - 21640 \beta_{2} - 291544) q^{44} + ( - 5622 \beta_{2} - 441744) q^{46} + ( - 15808 \beta_{3} + 224210 \beta_1) q^{47} + ( - 1204 \beta_{2} + 797390) q^{49} + (32520 \beta_{3} - 239524 \beta_1) q^{52} + (116638 \beta_{3} + 364276 \beta_1) q^{53} + ( - 3780 \beta_{2} + 123120) q^{56} + (175166 \beta_{3} + 752288 \beta_1) q^{58} + (116464 \beta_{2} + 453922) q^{59} + ( - 261524 \beta_{2} - 239205) q^{61} + (126597 \beta_{3} + 2180604 \beta_1) q^{62} + (113280 \beta_{2} - 119488) q^{64} + (3110 \beta_{3} + 2257351 \beta_1) q^{67} + ( - 106312 \beta_{3} - 2312008 \beta_1) q^{68} + (126490 \beta_{2} + 1303648) q^{71} + (57692 \beta_{3} + 2035134 \beta_1) q^{73} + (192074 \beta_{2} + 5170808) q^{74} + ( - 62784 \beta_{2} + 2580428) q^{76} + (43722 \beta_{3} + 560934 \beta_1) q^{77} + ( - 30080 \beta_{2} - 3113680) q^{79} + ( - 3752 \beta_{3} + 4486216 \beta_1) q^{82} + (413472 \beta_{3} - 2884542 \beta_1) q^{83} + ( - 507619 \beta_{2} - 1056964) q^{86} + ( - 53256 \beta_{3} - 1553088 \beta_1) q^{88} + ( - 418104 \beta_{2} + 8178048) q^{89} + (50398 \beta_{2} - 319969) q^{91} + ( - 32664 \beta_{3} + 3732072 \beta_1) q^{92} + (160978 \beta_{2} - 1063352) q^{94} + (516240 \beta_{3} + 8716721 \beta_1) q^{97} + ( - 792574 \beta_{3} - 3040264 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 48 q^{4} + 9224 q^{11} + 7632 q^{14} - 39360 q^{16} + 47868 q^{19} + 74672 q^{26} - 693016 q^{29} - 258356 q^{31} + 1255744 q^{34} + 681856 q^{41} - 1166176 q^{44} - 1766976 q^{46} + 3189560 q^{49} + 492480 q^{56} + 1815688 q^{59} - 956820 q^{61} - 477952 q^{64} + 5214592 q^{71} + 20683232 q^{74} + 10321712 q^{76} - 12454720 q^{79} - 4227856 q^{86} + 32712192 q^{89} - 1279876 q^{91} - 4253408 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 15x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 7\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 23\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 30 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 30 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{2} + 46\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

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

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

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
2.78388 + 0.500000i
−2.78388 0.500000i
−2.78388 + 0.500000i
2.78388 0.500000i
15.1355i 0 −101.084 0 0 198.897i 407.384i 0 0
199.2 7.13553i 0 77.0842 0 0 112.897i 1463.38i 0 0
199.3 7.13553i 0 77.0842 0 0 112.897i 1463.38i 0 0
199.4 15.1355i 0 −101.084 0 0 198.897i 407.384i 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 225.8.b.o 4
3.b odd 2 1 75.8.b.e 4
5.b even 2 1 inner 225.8.b.o 4
5.c odd 4 1 225.8.a.l 2
5.c odd 4 1 225.8.a.u 2
15.d odd 2 1 75.8.b.e 4
15.e even 4 1 75.8.a.d 2
15.e even 4 1 75.8.a.f yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.8.a.d 2 15.e even 4 1
75.8.a.f yes 2 15.e even 4 1
75.8.b.e 4 3.b odd 2 1
75.8.b.e 4 15.d odd 2 1
225.8.a.l 2 5.c odd 4 1
225.8.a.u 2 5.c odd 4 1
225.8.b.o 4 1.a even 1 1 trivial
225.8.b.o 4 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 \(S_{8}^{\mathrm{new}}(225, [\chi])\):

\( T_{2}^{4} + 280T_{2}^{2} + 11664 \) Copy content Toggle raw display
\( T_{11}^{2} - 4612T_{11} - 3456108 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 280 T^{2} + 11664 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 52306 T^{2} + \cdots + 504227025 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4612 T - 3456108)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 62390930 T^{2} + \cdots + 89618352089809 \) Copy content Toggle raw display
$17$ \( T^{4} + 1410218152 T^{2} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{2} - 23934 T - 791814895)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3933598824 T^{2} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + 346508 T + 29988616500)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 129178 T - 25626949575)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 413786605256 T^{2} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} - 340928 T - 186336929520)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 611315555858 T^{2} + \cdots + 81\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{4} + 162513678472 T^{2} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{4} + 3639290923264 T^{2} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{2} - 907844 T - 1475873866620)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 478410 T - 8423736487399)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 10193665755202 T^{2} + \cdots + 25\!\cdots\!01 \) Copy content Toggle raw display
$71$ \( (T^{2} - 2607296 T - 284467184496)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 9108975778184 T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} + 6227360 T + 9582807148800)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 59039020605960 T^{2} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( (T^{2} - 16356096 T + 45203910693120)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 218055376908482 T^{2} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
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