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Properties

Label 315.5.e.e
Level $315$
Weight $5$
Character orbit 315.e
Analytic conductor $32.562$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,5,Mod(244,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.244"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 315.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-172,0,0,0,0,0,0,376] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.5615383714\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 110x^{6} + 7113x^{4} + 190880x^{2} + 4177936 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{2} - 22) q^{4} + ( - \beta_{7} - \beta_1) q^{5} + ( - \beta_{7} + \beta_{6} - 5 \beta_{3}) q^{7} + (2 \beta_{6} - 11 \beta_{4} + \cdots + \beta_1) q^{8} + (\beta_{7} + \beta_{5} + \cdots - 2 \beta_1) q^{10}+ \cdots + ( - 278 \beta_{7} - 170 \beta_{6} + \cdots - 224 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 172 q^{4} + 376 q^{11} + 24 q^{14} + 596 q^{16} + 3500 q^{25} + 64 q^{29} + 4160 q^{35} + 2104 q^{44} - 28440 q^{46} - 7828 q^{49} - 7740 q^{50} - 27300 q^{56} - 5092 q^{64} + 24940 q^{65} - 10620 q^{70}+ \cdots - 25260 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 110x^{6} + 7113x^{4} + 190880x^{2} + 4177936 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} + 125\nu^{4} + 12513\nu^{2} + 188720 ) / 16898 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} - 125\nu^{4} - 4064\nu^{2} + 47852 ) / 8449 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{7} - 2166\nu^{5} - 141325\nu^{3} - 1118204\nu ) / 4934216 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{7} + 2166\nu^{5} + 141325\nu^{3} + 6052420\nu ) / 4934216 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} + 436\nu^{4} + 22497\nu^{2} + 517664 ) / 1988 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 113 \nu^{7} - 146 \nu^{6} + 783 \nu^{5} - 9125 \nu^{4} + 330432 \nu^{3} - 913449 \nu^{2} + \cdots - 13776560 ) / 2467108 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 154 \nu^{7} - 146 \nu^{6} + 16867 \nu^{5} - 9125 \nu^{4} + 782451 \nu^{3} - 913449 \nu^{2} + \cdots - 13776560 ) / 2467108 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 - 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{7} + 2\beta_{6} - 13\beta_{4} - 105\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{5} - 15\beta_{2} - 132\beta _1 - 524 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 550\beta_{7} + 50\beta_{6} - 1769\beta_{4} + 4555\beta_{3} + 300\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -500\beta_{5} - 5319\beta_{2} + 4186\beta _1 + 113572 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13734\beta_{7} - 15638\beta_{6} + 182027\beta_{4} - 43177\beta_{3} - 14686\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
3.16228 − 7.21587i
−3.16228 − 7.21587i
−3.16228 − 4.78865i
3.16228 − 4.78865i
−3.16228 + 4.78865i
3.16228 + 4.78865i
3.16228 + 7.21587i
−3.16228 + 7.21587i
− 7.21587i 0 −36.0688 −22.2447 + 11.4093i 0 −36.4750 + 32.7197i 144.814i 0 82.3280 + 160.515i
244.2 − 7.21587i 0 −36.0688 22.2447 − 11.4093i 0 36.4750 + 32.7197i 144.814i 0 −82.3280 − 160.515i
244.3 − 4.78865i 0 −6.93120 −23.8259 − 7.57153i 0 −9.59562 − 48.0513i − 43.4273i 0 −36.2574 + 114.094i
244.4 − 4.78865i 0 −6.93120 23.8259 + 7.57153i 0 9.59562 − 48.0513i − 43.4273i 0 36.2574 − 114.094i
244.5 4.78865i 0 −6.93120 −23.8259 + 7.57153i 0 −9.59562 + 48.0513i 43.4273i 0 −36.2574 − 114.094i
244.6 4.78865i 0 −6.93120 23.8259 − 7.57153i 0 9.59562 + 48.0513i 43.4273i 0 36.2574 + 114.094i
244.7 7.21587i 0 −36.0688 −22.2447 − 11.4093i 0 −36.4750 − 32.7197i − 144.814i 0 82.3280 − 160.515i
244.8 7.21587i 0 −36.0688 22.2447 + 11.4093i 0 36.4750 − 32.7197i − 144.814i 0 −82.3280 + 160.515i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.e.e 8
3.b odd 2 1 35.5.c.e ✓ 8
5.b even 2 1 inner 315.5.e.e 8
7.b odd 2 1 inner 315.5.e.e 8
12.b even 2 1 560.5.p.g 8
15.d odd 2 1 35.5.c.e ✓ 8
15.e even 4 2 175.5.d.g 8
21.c even 2 1 35.5.c.e ✓ 8
35.c odd 2 1 inner 315.5.e.e 8
60.h even 2 1 560.5.p.g 8
84.h odd 2 1 560.5.p.g 8
105.g even 2 1 35.5.c.e ✓ 8
105.k odd 4 2 175.5.d.g 8
420.o odd 2 1 560.5.p.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.e ✓ 8 3.b odd 2 1
35.5.c.e ✓ 8 15.d odd 2 1
35.5.c.e ✓ 8 21.c even 2 1
35.5.c.e ✓ 8 105.g even 2 1
175.5.d.g 8 15.e even 4 2
175.5.d.g 8 105.k odd 4 2
315.5.e.e 8 1.a even 1 1 trivial
315.5.e.e 8 5.b even 2 1 inner
315.5.e.e 8 7.b odd 2 1 inner
315.5.e.e 8 35.c odd 2 1 inner
560.5.p.g 8 12.b even 2 1
560.5.p.g 8 60.h even 2 1
560.5.p.g 8 84.h odd 2 1
560.5.p.g 8 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{4} + 75T_{2}^{2} + 1194 \) Copy content Toggle raw display
\( T_{13}^{4} - 52250T_{13}^{2} + 145926400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 75 T^{2} + 1194)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{2} - 94 T - 5432)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 52250 T^{2} + 145926400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 221000 T^{2} + 8745990400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 347850 T^{2} + 10320458400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1011930 T^{2} + 933058464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 16 T - 3332)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 1373776281600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2666340 T^{2} + 250900015104)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1115820 T^{2} + 61553565600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 18960156666024)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 899330 T^{2} + 169299331600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1107108 T^{2} + 23031858816)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 40047069962400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 499402073322600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 357890658714024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 4504 T - 2114432)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 268409242240000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2156 T - 7670912)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 403700 T^{2} + 2403940900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 17469797990400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 160123400 T^{2} + 608524806400)^{2} \) Copy content Toggle raw display
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