Properties

Label 507.4.b.h
Level $507$
Weight $4$
Character orbit 507.b
Analytic conductor $29.914$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [507,4,Mod(337,507)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(507, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("507.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 507.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: \(29.9139683729\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 13^{2} \)
Twist minimal: no (minimal twist has level 39)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - 3 q^{3} + (\beta_1 - 6) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + 3 \beta_{2} q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + 2 \beta_{5} + \cdots + 5 \beta_{2}) q^{8}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - 3 q^{3} + (\beta_1 - 6) q^{4} + (\beta_{5} + \beta_{2}) q^{5} + 3 \beta_{2} q^{6} + (\beta_{3} + \beta_{2}) q^{7} + ( - \beta_{7} + 2 \beta_{5} + \cdots + 5 \beta_{2}) q^{8}+ \cdots + ( - 9 \beta_{7} - 9 \beta_{5} + \cdots + 36 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{3} - 44 q^{4} + 72 q^{9} + 124 q^{10} + 132 q^{12} + 80 q^{14} + 244 q^{16} - 196 q^{17} + 440 q^{22} - 208 q^{23} + 116 q^{25} - 216 q^{27} + 388 q^{29} - 372 q^{30} + 176 q^{35} - 396 q^{36} - 664 q^{38} - 1996 q^{40} - 240 q^{42} - 900 q^{43} - 732 q^{48} - 2140 q^{49} + 588 q^{51} + 524 q^{53} + 408 q^{55} - 4328 q^{56} - 1856 q^{61} - 5560 q^{62} - 2052 q^{64} - 1320 q^{66} + 3572 q^{68} + 624 q^{69} + 2316 q^{74} - 348 q^{75} - 5016 q^{77} - 1492 q^{79} + 648 q^{81} - 3468 q^{82} - 1164 q^{87} - 6120 q^{88} + 1116 q^{90} + 664 q^{92} - 1544 q^{94} - 4408 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 54x^{6} + 889x^{4} + 4584x^{2} + 5776 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 40\nu^{4} - 329\nu^{2} + 178 ) / 78 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{7} - 518\nu^{5} - 6739\nu^{3} - 31348\nu ) / 11856 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\nu^{7} + 1622\nu^{5} + 41575\nu^{3} + 349012\nu ) / 11856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 118\nu^{4} + 2357\nu^{2} + 5048 ) / 156 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} + 397\nu^{5} + 6242\nu^{3} + 19814\nu ) / 1482 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} - 239\nu^{4} - 3166\nu^{2} - 8704 ) / 78 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\nu^{7} + 518\nu^{5} + 6739\nu^{3} + 19492\nu ) / 912 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - 13\beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 2\beta_{4} + 9\beta _1 - 179 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 24\beta_{7} - 13\beta_{5} + 13\beta_{3} + 273\beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} + 6\beta_{4} - 17\beta _1 + 291 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -698\beta_{7} + 663\beta_{5} - 377\beta_{3} - 6487\beta_{2} ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1422\beta_{6} - 2462\beta_{4} + 4865\beta _1 - 90115 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 21016\beta_{7} - 23257\beta_{5} + 9789\beta_{3} + 161265\beta_{2} ) / 13 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
4.33039i
5.22605i
1.36176i
2.46610i
2.46610i
1.36176i
5.22605i
4.33039i
5.33039i −3.00000 −20.4131 16.4131i 15.9912i 9.67968i 66.1667i 9.00000 87.4882
337.2 4.22605i −3.00000 −9.85953 5.85953i 12.6782i 24.1254i 7.85849i 9.00000 −24.7627
337.3 2.36176i −3.00000 2.42208 6.42208i 7.08529i 29.4938i 24.6145i 9.00000 −15.1674
337.4 1.46610i −3.00000 5.85055 9.85055i 4.39830i 29.9396i 20.3063i 9.00000 14.4419
337.5 1.46610i −3.00000 5.85055 9.85055i 4.39830i 29.9396i 20.3063i 9.00000 14.4419
337.6 2.36176i −3.00000 2.42208 6.42208i 7.08529i 29.4938i 24.6145i 9.00000 −15.1674
337.7 4.22605i −3.00000 −9.85953 5.85953i 12.6782i 24.1254i 7.85849i 9.00000 −24.7627
337.8 5.33039i −3.00000 −20.4131 16.4131i 15.9912i 9.67968i 66.1667i 9.00000 87.4882
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.4.b.h 8
13.b even 2 1 inner 507.4.b.h 8
13.d odd 4 1 507.4.a.i 4
13.d odd 4 1 507.4.a.m 4
13.f odd 12 2 39.4.e.c 8
39.f even 4 1 1521.4.a.v 4
39.f even 4 1 1521.4.a.bb 4
39.k even 12 2 117.4.g.e 8
52.l even 12 2 624.4.q.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 13.f odd 12 2
117.4.g.e 8 39.k even 12 2
507.4.a.i 4 13.d odd 4 1
507.4.a.m 4 13.d odd 4 1
507.4.b.h 8 1.a even 1 1 trivial
507.4.b.h 8 13.b even 2 1 inner
624.4.q.i 8 52.l even 12 2
1521.4.a.v 4 39.f even 4 1
1521.4.a.bb 4 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{8} + 54T_{2}^{6} + 877T_{2}^{4} + 4476T_{2}^{2} + 6084 \) Copy content Toggle raw display
\( T_{5}^{8} + 442T_{5}^{6} + 55249T_{5}^{4} + 2494440T_{5}^{2} + 37015056 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 54 T^{6} + \cdots + 6084 \) Copy content Toggle raw display
$3$ \( (T + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 442 T^{6} + \cdots + 37015056 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 42523388944 \) Copy content Toggle raw display
$11$ \( T^{8} + 4112 T^{6} + \cdots + 751198464 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 98 T^{3} + \cdots + 22571952)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 2959721107456 \) Copy content Toggle raw display
$23$ \( (T^{4} + 104 T^{3} + \cdots - 2571504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 194 T^{3} + \cdots - 274591068)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 738573457717264 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{4} + 450 T^{3} + \cdots - 2362804828)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{4} - 262 T^{3} + \cdots + 744728256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{4} + 928 T^{3} + \cdots - 5230543711)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 14\!\cdots\!09 \) Copy content Toggle raw display
$79$ \( (T^{4} + 746 T^{3} + \cdots + 680937616)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 72\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
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