Properties

Label 825.4.a.m
Level $825$
Weight $4$
Character orbit 825.a
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 3 q^{3} + (\beta - 4) q^{4} - 3 \beta q^{6} + ( - 4 \beta + 4) q^{7} + ( - 11 \beta + 4) q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 3 q^{3} + (\beta - 4) q^{4} - 3 \beta q^{6} + ( - 4 \beta + 4) q^{7} + ( - 11 \beta + 4) q^{8} + 9 q^{9} - 11 q^{11} + ( - 3 \beta + 12) q^{12} + (2 \beta + 44) q^{13} - 16 q^{14} + ( - 15 \beta - 12) q^{16} + ( - 44 \beta + 30) q^{17} + 9 \beta q^{18} + ( - 22 \beta - 74) q^{19} + (12 \beta - 12) q^{21} - 11 \beta q^{22} + (60 \beta + 32) q^{23} + (33 \beta - 12) q^{24} + (46 \beta + 8) q^{26} - 27 q^{27} + (16 \beta - 32) q^{28} + (34 \beta - 96) q^{29} + ( - 12 \beta + 36) q^{31} + (61 \beta - 92) q^{32} + 33 q^{33} + ( - 14 \beta - 176) q^{34} + (9 \beta - 36) q^{36} + (112 \beta + 130) q^{37} + ( - 96 \beta - 88) q^{38} + ( - 6 \beta - 132) q^{39} + ( - 154 \beta + 96) q^{41} + 48 q^{42} + (124 \beta + 196) q^{43} + ( - 11 \beta + 44) q^{44} + (92 \beta + 240) q^{46} + ( - 216 \beta - 4) q^{47} + (45 \beta + 36) q^{48} + ( - 16 \beta - 263) q^{49} + (132 \beta - 90) q^{51} + (38 \beta - 168) q^{52} + (196 \beta - 334) q^{53} - 27 \beta q^{54} + ( - 16 \beta + 192) q^{56} + (66 \beta + 222) q^{57} + ( - 62 \beta + 136) q^{58} + (240 \beta + 4) q^{59} + (364 \beta - 146) q^{61} + (24 \beta - 48) q^{62} + ( - 36 \beta + 36) q^{63} + (89 \beta + 340) q^{64} + 33 \beta q^{66} + ( - 16 \beta + 380) q^{67} + (162 \beta - 296) q^{68} + ( - 180 \beta - 96) q^{69} + (44 \beta + 1008) q^{71} + ( - 99 \beta + 36) q^{72} + ( - 58 \beta + 272) q^{73} + (242 \beta + 448) q^{74} + ( - 8 \beta + 208) q^{76} + (44 \beta - 44) q^{77} + ( - 138 \beta - 24) q^{78} + ( - 306 \beta + 474) q^{79} + 81 q^{81} + ( - 58 \beta - 616) q^{82} + (426 \beta - 70) q^{83} + ( - 48 \beta + 96) q^{84} + (320 \beta + 496) q^{86} + ( - 102 \beta + 288) q^{87} + (121 \beta - 44) q^{88} + ( - 128 \beta + 186) q^{89} + ( - 176 \beta + 144) q^{91} + ( - 148 \beta + 112) q^{92} + (36 \beta - 108) q^{93} + ( - 220 \beta - 864) q^{94} + ( - 183 \beta + 276) q^{96} + ( - 428 \beta + 298) q^{97} + ( - 279 \beta - 64) q^{98} - 99 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9} - 22 q^{11} + 21 q^{12} + 90 q^{13} - 32 q^{14} - 39 q^{16} + 16 q^{17} + 9 q^{18} - 170 q^{19} - 12 q^{21} - 11 q^{22} + 124 q^{23} + 9 q^{24} + 62 q^{26} - 54 q^{27} - 48 q^{28} - 158 q^{29} + 60 q^{31} - 123 q^{32} + 66 q^{33} - 366 q^{34} - 63 q^{36} + 372 q^{37} - 272 q^{38} - 270 q^{39} + 38 q^{41} + 96 q^{42} + 516 q^{43} + 77 q^{44} + 572 q^{46} - 224 q^{47} + 117 q^{48} - 542 q^{49} - 48 q^{51} - 298 q^{52} - 472 q^{53} - 27 q^{54} + 368 q^{56} + 510 q^{57} + 210 q^{58} + 248 q^{59} + 72 q^{61} - 72 q^{62} + 36 q^{63} + 769 q^{64} + 33 q^{66} + 744 q^{67} - 430 q^{68} - 372 q^{69} + 2060 q^{71} - 27 q^{72} + 486 q^{73} + 1138 q^{74} + 408 q^{76} - 44 q^{77} - 186 q^{78} + 642 q^{79} + 162 q^{81} - 1290 q^{82} + 286 q^{83} + 144 q^{84} + 1312 q^{86} + 474 q^{87} + 33 q^{88} + 244 q^{89} + 112 q^{91} + 76 q^{92} - 180 q^{93} - 1948 q^{94} + 369 q^{96} + 168 q^{97} - 407 q^{98} - 198 q^{99}+O(q^{100}) \) 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
−1.56155
2.56155
−1.56155 −3.00000 −5.56155 0 4.68466 10.2462 21.1771 9.00000 0
1.2 2.56155 −3.00000 −1.43845 0 −7.68466 −6.24621 −24.1771 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)

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 825.4.a.m 2
3.b odd 2 1 2475.4.a.n 2
5.b even 2 1 165.4.a.c 2
5.c odd 4 2 825.4.c.j 4
15.d odd 2 1 495.4.a.d 2
55.d odd 2 1 1815.4.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.c 2 5.b even 2 1
495.4.a.d 2 15.d odd 2 1
825.4.a.m 2 1.a even 1 1 trivial
825.4.c.j 4 5.c odd 4 2
1815.4.a.n 2 55.d odd 2 1
2475.4.a.n 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\):

\( T_{2}^{2} - T_{2} - 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} - 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 4 \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T - 64 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 90T + 2008 \) Copy content Toggle raw display
$17$ \( T^{2} - 16T - 8164 \) Copy content Toggle raw display
$19$ \( T^{2} + 170T + 5168 \) Copy content Toggle raw display
$23$ \( T^{2} - 124T - 11456 \) Copy content Toggle raw display
$29$ \( T^{2} + 158T + 1328 \) Copy content Toggle raw display
$31$ \( T^{2} - 60T + 288 \) Copy content Toggle raw display
$37$ \( T^{2} - 372T - 18716 \) Copy content Toggle raw display
$41$ \( T^{2} - 38T - 100432 \) Copy content Toggle raw display
$43$ \( T^{2} - 516T + 1216 \) Copy content Toggle raw display
$47$ \( T^{2} + 224T - 185744 \) Copy content Toggle raw display
$53$ \( T^{2} + 472T - 107572 \) Copy content Toggle raw display
$59$ \( T^{2} - 248T - 229424 \) Copy content Toggle raw display
$61$ \( T^{2} - 72T - 561812 \) Copy content Toggle raw display
$67$ \( T^{2} - 744T + 137296 \) Copy content Toggle raw display
$71$ \( T^{2} - 2060 T + 1052672 \) Copy content Toggle raw display
$73$ \( T^{2} - 486T + 44752 \) Copy content Toggle raw display
$79$ \( T^{2} - 642T - 294912 \) Copy content Toggle raw display
$83$ \( T^{2} - 286T - 750824 \) Copy content Toggle raw display
$89$ \( T^{2} - 244T - 54748 \) Copy content Toggle raw display
$97$ \( T^{2} - 168T - 771476 \) Copy content Toggle raw display
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