Properties

Label 225.8.a.t
Level $225$
Weight $8$
Character orbit 225.a
Self dual yes
Analytic conductor $70.287$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,8,Mod(1,225)]
 
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, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(70.2866307339\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{601}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 150 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{601})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 4) q^{2} + ( - 7 \beta + 38) q^{4} + ( - 56 \beta - 624) q^{7} + (69 \beta + 690) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 4) q^{2} + ( - 7 \beta + 38) q^{4} + ( - 56 \beta - 624) q^{7} + (69 \beta + 690) q^{8} + ( - 464 \beta - 1492) q^{11} + (824 \beta + 4082) q^{13} + (456 \beta + 5904) q^{14} + (413 \beta - 12454) q^{16} + (1400 \beta - 3446) q^{17} + (2056 \beta - 25820) q^{19} + (100 \beta + 63632) q^{22} + ( - 5208 \beta + 48528) q^{23} + ( - 1610 \beta - 107272) q^{26} + (2632 \beta + 35088) q^{28} + (8288 \beta - 95030) q^{29} + (2168 \beta + 151032) q^{31} + (4861 \beta - 200086) q^{32} + (7646 \beta - 223784) q^{34} + (14424 \beta + 243946) q^{37} + (31988 \beta - 411680) q^{38} + (21392 \beta - 326282) q^{41} + (7136 \beta - 180388) q^{43} + ( - 3940 \beta + 430504) q^{44} + ( - 64152 \beta + 975312) q^{46} + (5912 \beta - 236696) q^{47} + (73024 \beta + 36233) q^{49} + ( - 3030 \beta - 710084) q^{52} + (13232 \beta - 290642) q^{53} + ( - 85560 \beta - 1010160) q^{56} + (119894 \beta - 1623320) q^{58} + (149776 \beta - 218500) q^{59} + (1456 \beta - 1257818) q^{61} + ( - 144528 \beta + 278928) q^{62} + (161805 \beta + 64618) q^{64} + ( - 129920 \beta + 2601876) q^{67} + (67522 \beta - 1600948) q^{68} + ( - 76480 \beta + 1912648) q^{71} + (388208 \beta + 544502) q^{73} + ( - 200674 \beta - 1187816) q^{74} + (244476 \beta - 3139960) q^{76} + (399072 \beta + 4828608) q^{77} + ( - 80440 \beta - 2273640) q^{79} + (390458 \beta - 4513928) q^{82} + ( - 91872 \beta - 2990532) q^{83} + (201796 \beta - 1791952) q^{86} + ( - 455124 \beta - 5831880) q^{88} + ( - 20496 \beta - 8247930) q^{89} + ( - 788912 \beta - 9468768) q^{91} + ( - 501144 \beta + 7312464) q^{92} + (254432 \beta - 1833584) q^{94} + ( - 428640 \beta - 1147394) q^{97} + (182839 \beta - 10808668) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{2} + 69 q^{4} - 1304 q^{7} + 1449 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 7 q^{2} + 69 q^{4} - 1304 q^{7} + 1449 q^{8} - 3448 q^{11} + 8988 q^{13} + 12264 q^{14} - 24495 q^{16} - 5492 q^{17} - 49584 q^{19} + 127364 q^{22} + 91848 q^{23} - 216154 q^{26} + 72808 q^{28} - 181772 q^{29} + 304232 q^{31} - 395311 q^{32} - 439922 q^{34} + 502316 q^{37} - 791372 q^{38} - 631172 q^{41} - 353640 q^{43} + 857068 q^{44} + 1886472 q^{46} - 467480 q^{47} + 145490 q^{49} - 1423198 q^{52} - 568052 q^{53} - 2105880 q^{56} - 3126746 q^{58} - 287224 q^{59} - 2514180 q^{61} + 413328 q^{62} + 291041 q^{64} + 5073832 q^{67} - 3134374 q^{68} + 3748816 q^{71} + 1477212 q^{73} - 2576306 q^{74} - 6035444 q^{76} + 10056288 q^{77} - 4627720 q^{79} - 8637398 q^{82} - 6072936 q^{83} - 3382108 q^{86} - 12118884 q^{88} - 16516356 q^{89} - 19726448 q^{91} + 14123784 q^{92} - 3412736 q^{94} - 2723428 q^{97} - 21434497 q^{98}+O(q^{100}) \) 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
12.7577
−11.7577
−8.75765 0 −51.3036 0 0 −1338.43 1570.28 0 0
1.2 15.7577 0 120.304 0 0 34.4284 −121.278 0 0
\(n\): 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 225.8.a.t 2
3.b odd 2 1 75.8.a.e 2
5.b even 2 1 45.8.a.i 2
5.c odd 4 2 225.8.b.n 4
15.d odd 2 1 15.8.a.c 2
15.e even 4 2 75.8.b.d 4
60.h even 2 1 240.8.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.8.a.c 2 15.d odd 2 1
45.8.a.i 2 5.b even 2 1
75.8.a.e 2 3.b odd 2 1
75.8.b.d 4 15.e even 4 2
225.8.a.t 2 1.a even 1 1 trivial
225.8.b.n 4 5.c odd 4 2
240.8.a.p 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{2} - 7T_{2} - 138 \) Copy content Toggle raw display
\( T_{7}^{2} + 1304T_{7} - 46080 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 7T - 138 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1304T - 46080 \) Copy content Toggle raw display
$11$ \( T^{2} + 3448 T - 29376048 \) Copy content Toggle raw display
$13$ \( T^{2} - 8988 T - 81820108 \) Copy content Toggle raw display
$17$ \( T^{2} + 5492 T - 286949484 \) Copy content Toggle raw display
$19$ \( T^{2} + 49584 T - 20483920 \) Copy content Toggle raw display
$23$ \( T^{2} - 91848 T - 1966256640 \) Copy content Toggle raw display
$29$ \( T^{2} + 181772 T - 2060549340 \) Copy content Toggle raw display
$31$ \( T^{2} - 304232 T + 22433068800 \) Copy content Toggle raw display
$37$ \( T^{2} - 502316 T + 31820561620 \) Copy content Toggle raw display
$41$ \( T^{2} + 631172 T + 30837469380 \) Copy content Toggle raw display
$43$ \( T^{2} + 353640 T + 23614207376 \) Copy content Toggle raw display
$47$ \( T^{2} + 467480 T + 49382888064 \) Copy content Toggle raw display
$53$ \( T^{2} + 568052 T + 54364123620 \) Copy content Toggle raw display
$59$ \( T^{2} + 287224 T - 3349911332400 \) Copy content Toggle raw display
$61$ \( T^{2} + 2514180 T + 1579956747716 \) Copy content Toggle raw display
$67$ \( T^{2} - 5073832 T + 3899842029456 \) Copy content Toggle raw display
$71$ \( T^{2} - 3748816 T + 2634564492864 \) Copy content Toggle raw display
$73$ \( T^{2} - 1477212 T - 22097955229180 \) Copy content Toggle raw display
$79$ \( T^{2} + 4627720 T + 4381741411200 \) Copy content Toggle raw display
$83$ \( T^{2} + 6072936 T + 7951958141328 \) Copy content Toggle raw display
$89$ \( T^{2} + 16516356 T + 68134385955780 \) Copy content Toggle raw display
$97$ \( T^{2} + 2723428 T - 25751505484604 \) Copy content Toggle raw display
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