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Properties

Label 315.3.e.e
Level $315$
Weight $3$
Character orbit 315.e
Analytic conductor $8.583$
Analytic rank $0$
Dimension $16$
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] = [315,3,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, 3); 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, 3, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 315.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-32,0,0,0,0,0,0,56] 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: \(8.58312832735\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 72 x^{14} - 292 x^{13} + 1148 x^{12} - 2304 x^{11} + 4996 x^{10} - 4490 x^{9} + \cdots + 1849 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 105)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{4} - 2) q^{4} + (\beta_{10} - \beta_{8}) q^{5} + ( - \beta_{13} - \beta_{2}) q^{7} + (\beta_{5} - 3 \beta_{2}) q^{8} + ( - \beta_{13} + \beta_{11} + \cdots - 2 \beta_1) q^{10}+ \cdots + ( - 12 \beta_{13} + 4 \beta_{12} + \cdots + 3 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} + 56 q^{11} + 84 q^{14} + 112 q^{16} + 16 q^{25} + 32 q^{29} + 4 q^{35} - 568 q^{44} - 96 q^{46} - 152 q^{49} + 96 q^{50} - 444 q^{56} - 992 q^{64} + 296 q^{65} + 504 q^{70} + 56 q^{71}+ \cdots + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 72 x^{14} - 292 x^{13} + 1148 x^{12} - 2304 x^{11} + 4996 x^{10} - 4490 x^{9} + \cdots + 1849 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!43 \nu^{15} + \cdots + 24\!\cdots\!38 ) / 69\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 50\!\cdots\!03 \nu^{15} + \cdots + 16\!\cdots\!07 ) / 16\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 34\!\cdots\!72 \nu^{15} + \cdots + 24\!\cdots\!74 ) / 69\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 92\!\cdots\!48 \nu^{15} + \cdots + 15\!\cdots\!65 ) / 22\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 60\!\cdots\!14 \nu^{15} + \cdots + 19\!\cdots\!04 ) / 16\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26\!\cdots\!56 \nu^{15} + \cdots - 16\!\cdots\!46 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15\!\cdots\!71 \nu^{15} + \cdots + 14\!\cdots\!46 ) / 97\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41\!\cdots\!66 \nu^{15} + \cdots - 37\!\cdots\!92 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62\!\cdots\!57 \nu^{15} + \cdots + 27\!\cdots\!78 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!74 \nu^{15} + \cdots - 39\!\cdots\!02 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 38\!\cdots\!91 \nu^{15} + \cdots + 55\!\cdots\!80 ) / 69\!\cdots\!37 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 91\!\cdots\!21 \nu^{15} + \cdots - 47\!\cdots\!73 ) / 16\!\cdots\!59 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!53 \nu^{15} + \cdots + 10\!\cdots\!98 ) / 20\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 18\!\cdots\!24 \nu^{15} + \cdots - 15\!\cdots\!95 ) / 22\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 69\!\cdots\!88 \nu^{15} + \cdots - 23\!\cdots\!50 ) / 75\!\cdots\!43 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{13} - \beta_{10} + \beta_{9} + 2\beta_{8} + 2\beta_{6} + 2\beta_{4} + \beta_{3} - 4\beta_{2} + 2\beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{15} + 9 \beta_{13} + 3 \beta_{11} - 9 \beta_{10} + 9 \beta_{9} + 18 \beta_{8} - 6 \beta_{7} + \cdots - 82 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 15 \beta_{15} + 7 \beta_{14} - 24 \beta_{13} + 8 \beta_{11} + 24 \beta_{10} - 24 \beta_{9} + \cdots + 106 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 51 \beta_{15} + 95 \beta_{14} - 440 \beta_{13} + 19 \beta_{12} - 44 \beta_{11} + 440 \beta_{10} + \cdots + 2388 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 382 \beta_{15} - 78 \beta_{14} + 90 \beta_{13} + 78 \beta_{12} - 330 \beta_{11} - 45 \beta_{10} + \cdots - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4135 \beta_{15} - 3976 \beta_{14} + 13847 \beta_{13} - 779 \beta_{11} - 12915 \beta_{10} + 13705 \beta_{9} + \cdots - 64130 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4886 \beta_{15} - 4268 \beta_{14} + 17468 \beta_{13} - 2716 \beta_{12} + 8288 \beta_{11} + \cdots - 78049 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 149055 \beta_{15} + 95400 \beta_{14} - 305481 \beta_{13} - 19080 \beta_{12} + 68391 \beta_{11} + \cdots + 1419200 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 70179 \beta_{15} + 272586 \beta_{14} - 937566 \beta_{13} + 52128 \beta_{12} - 142329 \beta_{11} + \cdots + 4084397 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3918528 \beta_{15} - 1207569 \beta_{14} + 3619903 \beta_{13} + 887133 \beta_{12} - 2725383 \beta_{11} + \cdots - 19194622 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 7452756 \beta_{15} - 9674511 \beta_{14} + 32096940 \beta_{13} - 218862 \beta_{12} + 399360 \beta_{11} + \cdots - 143703808 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 72149628 \beta_{15} - 19529081 \beta_{14} + 72135045 \beta_{13} - 27029193 \beta_{12} + \cdots - 222089510 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 314292918 \beta_{15} + 252488234 \beta_{14} - 827687867 \beta_{13} - 31799894 \beta_{12} + \cdots + 3854381958 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 181008146 \beta_{15} + 980270325 \beta_{14} - 3307866177 \beta_{13} + 301465875 \beta_{12} + \cdots + 13705362729 ) / 2 \) 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
−0.366025 − 1.39311i
1.36603 + 5.19914i
−0.366025 − 0.842173i
1.36603 + 3.14303i
1.36603 + 2.63709i
−0.366025 − 0.706606i
−0.366025 − 0.257094i
1.36603 + 0.959486i
−0.366025 + 0.257094i
1.36603 − 0.959486i
1.36603 − 2.63709i
−0.366025 + 0.706606i
−0.366025 + 0.842173i
1.36603 − 3.14303i
−0.366025 + 1.39311i
1.36603 − 5.19914i
− 3.80604i 0 −10.4859 −3.71318 − 3.34847i 0 −5.08005 + 4.81592i 24.6856i 0 −12.7444 + 14.1325i
244.2 − 3.80604i 0 −10.4859 3.71318 + 3.34847i 0 5.08005 + 4.81592i 24.6856i 0 12.7444 − 14.1325i
244.3 − 2.30086i 0 −1.29396 −3.65761 + 3.40909i 0 6.39480 − 2.84720i − 6.22623i 0 7.84383 + 8.41565i
244.4 − 2.30086i 0 −1.29396 3.65761 − 3.40909i 0 −6.39480 − 2.84720i − 6.22623i 0 −7.84383 − 8.41565i
244.5 − 1.93048i 0 0.273228 −4.88618 + 1.06077i 0 −0.433408 + 6.98657i − 8.24940i 0 2.04780 + 9.43270i
244.6 − 1.93048i 0 0.273228 4.88618 − 1.06077i 0 0.433408 + 6.98657i − 8.24940i 0 −2.04780 − 9.43270i
244.7 − 0.702393i 0 3.50664 −0.979490 − 4.90312i 0 −3.48021 − 6.07356i − 5.27261i 0 −3.44392 + 0.687987i
244.8 − 0.702393i 0 3.50664 0.979490 + 4.90312i 0 3.48021 − 6.07356i − 5.27261i 0 3.44392 − 0.687987i
244.9 0.702393i 0 3.50664 −0.979490 + 4.90312i 0 −3.48021 + 6.07356i 5.27261i 0 −3.44392 − 0.687987i
244.10 0.702393i 0 3.50664 0.979490 − 4.90312i 0 3.48021 + 6.07356i 5.27261i 0 3.44392 + 0.687987i
244.11 1.93048i 0 0.273228 −4.88618 − 1.06077i 0 −0.433408 − 6.98657i 8.24940i 0 2.04780 − 9.43270i
244.12 1.93048i 0 0.273228 4.88618 + 1.06077i 0 0.433408 − 6.98657i 8.24940i 0 −2.04780 + 9.43270i
244.13 2.30086i 0 −1.29396 −3.65761 − 3.40909i 0 6.39480 + 2.84720i 6.22623i 0 7.84383 − 8.41565i
244.14 2.30086i 0 −1.29396 3.65761 + 3.40909i 0 −6.39480 + 2.84720i 6.22623i 0 −7.84383 + 8.41565i
244.15 3.80604i 0 −10.4859 −3.71318 + 3.34847i 0 −5.08005 − 4.81592i − 24.6856i 0 −12.7444 − 14.1325i
244.16 3.80604i 0 −10.4859 3.71318 − 3.34847i 0 5.08005 − 4.81592i − 24.6856i 0 12.7444 + 14.1325i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.16
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 315.3.e.e 16
3.b odd 2 1 105.3.e.a ✓ 16
5.b even 2 1 inner 315.3.e.e 16
7.b odd 2 1 inner 315.3.e.e 16
12.b even 2 1 1680.3.bd.c 16
15.d odd 2 1 105.3.e.a ✓ 16
15.e even 4 2 525.3.h.e 16
21.c even 2 1 105.3.e.a ✓ 16
35.c odd 2 1 inner 315.3.e.e 16
60.h even 2 1 1680.3.bd.c 16
84.h odd 2 1 1680.3.bd.c 16
105.g even 2 1 105.3.e.a ✓ 16
105.k odd 4 2 525.3.h.e 16
420.o odd 2 1 1680.3.bd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.e.a ✓ 16 3.b odd 2 1
105.3.e.a ✓ 16 15.d odd 2 1
105.3.e.a ✓ 16 21.c even 2 1
105.3.e.a ✓ 16 105.g even 2 1
315.3.e.e 16 1.a even 1 1 trivial
315.3.e.e 16 5.b even 2 1 inner
315.3.e.e 16 7.b odd 2 1 inner
315.3.e.e 16 35.c odd 2 1 inner
525.3.h.e 16 15.e even 4 2
525.3.h.e 16 105.k odd 4 2
1680.3.bd.c 16 12.b even 2 1
1680.3.bd.c 16 60.h even 2 1
1680.3.bd.c 16 84.h odd 2 1
1680.3.bd.c 16 420.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 \(S_{3}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{8} + 24T_{2}^{6} + 162T_{2}^{4} + 360T_{2}^{2} + 141 \) Copy content Toggle raw display
\( T_{13}^{8} - 556T_{13}^{6} + 95136T_{13}^{4} - 5488384T_{13}^{2} + 60715264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 24 T^{6} + \cdots + 141)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} - 14 T^{3} + \cdots + 3280)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 556 T^{6} + \cdots + 60715264)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1468 T^{6} + \cdots + 4251040000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 396 T^{6} + \cdots + 324864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 2904 T^{6} + \cdots + 382942464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots - 26384)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3840 T^{6} + \cdots + 1892910336)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 3497148290304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 26090305209600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 2508 T^{6} + \cdots + 86666496)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 1152376486144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 55452408710400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 2149592204544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 9868871097600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 674847686079744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 14 T^{3} + \cdots - 3270032)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 36785098365184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 116 T^{3} + \cdots + 28622512)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 162581970541824)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 338426355591424)^{2} \) Copy content Toggle raw display
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