Properties

Label 276.6.a.b
Level $276$
Weight $6$
Character orbit 276.a
Self dual yes
Analytic conductor $44.266$
Analytic rank $1$
Dimension $4$
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] = [276,6,Mod(1,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 276.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: \(44.2659342684\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 223x^{2} - 168x + 7047 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
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 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.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( - \beta_{2} - \beta_1 - 13) q^{5} + (\beta_{3} + 3 \beta_1 - 15) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + ( - \beta_{2} - \beta_1 - 13) q^{5} + (\beta_{3} + 3 \beta_1 - 15) q^{7} + 81 q^{9} + ( - 7 \beta_{3} + 7 \beta_{2} - 98) q^{11} + (9 \beta_{3} + 5 \beta_{2} - 7 \beta_1 - 92) q^{13} + ( - 9 \beta_{2} - 9 \beta_1 - 117) q^{15} + ( - 25 \beta_{3} + 8 \beta_{2} - 3 \beta_1 - 207) q^{17} + (12 \beta_{3} + 29 \beta_{2} + 13 \beta_1 - 73) q^{19} + (9 \beta_{3} + 27 \beta_1 - 135) q^{21} - 529 q^{23} + (5 \beta_{3} + 25 \beta_{2} + 75 \beta_1 - 919) q^{25} + 729 q^{27} + ( - 2 \beta_{3} + 16 \beta_{2} + 34 \beta_1 - 2500) q^{29} + (44 \beta_{3} - 134 \beta_{2} - 24 \beta_1 - 1818) q^{31} + ( - 63 \beta_{3} + 63 \beta_{2} - 882) q^{33} + (\beta_{3} - 101 \beta_{2} - 163 \beta_1 - 2924) q^{35} + (13 \beta_{3} - 87 \beta_{2} + 120 \beta_1 - 4146) q^{37} + (81 \beta_{3} + 45 \beta_{2} - 63 \beta_1 - 828) q^{39} + (82 \beta_{3} - 78 \beta_{2} - 116 \beta_1 - 5094) q^{41} + ( - 46 \beta_{3} + 107 \beta_{2} + 5 \beta_1 - 4521) q^{43} + ( - 81 \beta_{2} - 81 \beta_1 - 1053) q^{45} + (97 \beta_{3} - 189 \beta_{2} - 275 \beta_1 - 7764) q^{47} + ( - 199 \beta_{3} + 273 \beta_{2} + 131 \beta_1 - 201) q^{49} + ( - 225 \beta_{3} + 72 \beta_{2} - 27 \beta_1 - 1863) q^{51} + (16 \beta_{3} - 191 \beta_{2} - 185 \beta_1 - 2875) q^{53} + ( - 14 \beta_{3} + 266 \beta_{2} + 70 \beta_1 - 4340) q^{55} + (108 \beta_{3} + 261 \beta_{2} + 117 \beta_1 - 657) q^{57} + ( - 277 \beta_{3} - 339 \beta_{2} - 359 \beta_1 - 4328) q^{59} + ( - 63 \beta_{3} + 53 \beta_{2} + 350 \beta_1 + 34) q^{61} + (81 \beta_{3} + 243 \beta_1 - 1215) q^{63} + ( - 94 \beta_{3} + 548 \beta_{2} + 520 \beta_1 - 322) q^{65} + ( - 142 \beta_{3} - 507 \beta_{2} - 409 \beta_1 - 3355) q^{67} - 4761 q^{69} + (58 \beta_{3} - 666 \beta_{2} - 64 \beta_1 - 17764) q^{71} + (892 \beta_{3} + 304 \beta_{2} + 452 \beta_1 + 1350) q^{73} + (45 \beta_{3} + 225 \beta_{2} + 675 \beta_1 - 8271) q^{75} + (868 \beta_{3} - 112 \beta_{2} + 266 \beta_1 - 21336) q^{77} + (387 \beta_{3} + 208 \beta_{2} - 975 \beta_1 - 26333) q^{79} + 6561 q^{81} + ( - 447 \beta_{3} + 775 \beta_{2} + 1758 \beta_1 - 34318) q^{83} + (63 \beta_{3} + 451 \beta_{2} + 321 \beta_1 + 4082) q^{85} + ( - 18 \beta_{3} + 144 \beta_{2} + 306 \beta_1 - 22500) q^{87} + (205 \beta_{3} + 1308 \beta_{2} + 245 \beta_1 - 29713) q^{89} + ( - 1194 \beta_{3} - 80 \beta_{2} - 646 \beta_1 - 30674) q^{91} + (396 \beta_{3} - 1206 \beta_{2} - 216 \beta_1 - 16362) q^{93} + ( - 237 \beta_{3} + 413 \beta_{2} - 741 \beta_1 - 46568) q^{95} + ( - 1724 \beta_{3} - 686 \beta_{2} - 326 \beta_1 - 26444) q^{97} + ( - 567 \beta_{3} + 567 \beta_{2} - 7938) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{3} - 50 q^{5} - 62 q^{7} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{3} - 50 q^{5} - 62 q^{7} + 324 q^{9} - 392 q^{11} - 396 q^{13} - 450 q^{15} - 794 q^{17} - 374 q^{19} - 558 q^{21} - 2116 q^{23} - 3736 q^{25} + 2916 q^{27} - 10028 q^{29} - 7092 q^{31} - 3528 q^{33} - 11496 q^{35} - 16436 q^{37} - 3564 q^{39} - 20384 q^{41} - 18206 q^{43} - 4050 q^{45} - 30872 q^{47} - 952 q^{49} - 7146 q^{51} - 11150 q^{53} - 17864 q^{55} - 3366 q^{57} - 16080 q^{59} + 156 q^{61} - 5022 q^{63} - 2196 q^{65} - 12122 q^{67} - 19044 q^{69} - 69840 q^{71} + 3008 q^{73} - 33624 q^{75} - 86856 q^{77} - 106522 q^{79} + 26244 q^{81} - 137928 q^{83} + 15300 q^{85} - 90252 q^{87} - 121878 q^{89} - 120148 q^{91} - 63828 q^{93} - 186624 q^{95} - 100956 q^{97} - 31752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 223x^{2} - 168x + 7047 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu^{2} - 268\nu - 1470 ) / 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 12\nu^{2} + 114\nu - 950 ) / 11 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 454 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{3} + 36\beta_{2} + 99\beta _1 + 938 ) / 2 \) 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
15.3152
−11.4900
−7.30627
5.48106
0 9.00000 0 −83.2716 0 164.449 0 81.0000 0
1.2 0 9.00000 0 −7.84635 0 −82.3982 0 81.0000 0
1.3 0 9.00000 0 −0.922658 0 −177.068 0 81.0000 0
1.4 0 9.00000 0 42.0407 0 33.0168 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(1\)

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 276.6.a.b 4
3.b odd 2 1 828.6.a.d 4
4.b odd 2 1 1104.6.a.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.6.a.b 4 1.a even 1 1 trivial
828.6.a.d 4 3.b odd 2 1
1104.6.a.l 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 50T_{5}^{3} - 3132T_{5}^{2} - 30400T_{5} - 25344 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(276))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 50 T^{3} - 3132 T^{2} + \cdots - 25344 \) Copy content Toggle raw display
$7$ \( T^{4} + 62 T^{3} - 31216 T^{2} + \cdots + 79217776 \) Copy content Toggle raw display
$11$ \( T^{4} + 392 T^{3} + \cdots - 3025951488 \) Copy content Toggle raw display
$13$ \( T^{4} + 396 T^{3} + \cdots - 140168104144 \) Copy content Toggle raw display
$17$ \( T^{4} + 794 T^{3} + \cdots - 553461499200 \) Copy content Toggle raw display
$19$ \( T^{4} + 374 T^{3} + \cdots + 2403073414064 \) Copy content Toggle raw display
$23$ \( (T + 529)^{4} \) Copy content Toggle raw display
$29$ \( T^{4} + 10028 T^{3} + \cdots + 10508867384496 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 677576433888000 \) Copy content Toggle raw display
$37$ \( T^{4} + 16436 T^{3} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 933488207082288 \) Copy content Toggle raw display
$43$ \( T^{4} + 18206 T^{3} + \cdots - 71181372294800 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 250538124570624 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 212742156308160 \) Copy content Toggle raw display
$59$ \( T^{4} + 16080 T^{3} + \cdots + 22\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} - 156 T^{3} + \cdots + 86\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + 12122 T^{3} + \cdots + 14\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{4} + 69840 T^{3} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{4} - 3008 T^{3} + \cdots + 31\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{4} + 106522 T^{3} + \cdots - 57\!\cdots\!40 \) Copy content Toggle raw display
$83$ \( T^{4} + 137928 T^{3} + \cdots - 21\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{4} + 121878 T^{3} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + 100956 T^{3} + \cdots + 33\!\cdots\!52 \) Copy content Toggle raw display
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