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Properties

Label 315.4.j.h
Level $315$
Weight $4$
Character orbit 315.j
Analytic conductor $18.586$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,4,Mod(46,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.46");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.j (of order \(3\), degree \(2\), 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: \(18.5856016518\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} + 34 x^{8} + 16 x^{7} + 791 x^{6} - 132 x^{5} + 4906 x^{4} - 1674 x^{3} + 25257 x^{2} + \cdots + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - \beta_1) q^{2} + (\beta_{7} - 5 \beta_{4} - 5) q^{4} - 5 \beta_{4} q^{5} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} - 4) q^{7} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - \beta_1) q^{2} + (\beta_{7} - 5 \beta_{4} - 5) q^{4} - 5 \beta_{4} q^{5} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2} - 4) q^{7} + (\beta_{9} + \beta_{8} + \beta_{7} + \cdots - 6) q^{8}+ \cdots + (10 \beta_{9} - 5 \beta_{8} + \cdots + 323) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 3 q^{2} - 25 q^{4} + 25 q^{5} - 32 q^{7} - 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 3 q^{2} - 25 q^{4} + 25 q^{5} - 32 q^{7} - 42 q^{8} - 15 q^{10} + 43 q^{11} + 246 q^{13} + 23 q^{14} - 161 q^{16} + 124 q^{17} - 37 q^{19} - 250 q^{20} - 442 q^{22} + 77 q^{23} - 125 q^{25} - 79 q^{26} - 71 q^{28} - 720 q^{29} - 314 q^{31} - 59 q^{32} + 352 q^{34} - 155 q^{35} - 225 q^{37} + 759 q^{38} - 105 q^{40} - 682 q^{41} + 64 q^{43} + 679 q^{44} + 331 q^{46} + 25 q^{47} + 710 q^{49} - 150 q^{50} - 2299 q^{52} - 317 q^{53} + 430 q^{55} - 1884 q^{56} - 8 q^{58} + 676 q^{59} + 188 q^{61} + 696 q^{62} - 2206 q^{64} + 615 q^{65} + 1776 q^{67} + 1280 q^{68} - 475 q^{70} + 12 q^{71} - 2006 q^{73} - 2729 q^{74} + 2834 q^{76} - 3731 q^{77} - 200 q^{79} + 805 q^{80} + 539 q^{82} + 664 q^{83} + 1240 q^{85} + 4262 q^{86} + 4529 q^{88} + 894 q^{89} + 2016 q^{91} + 7374 q^{92} - 4233 q^{94} + 185 q^{95} - 1152 q^{97} - 2539 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2 x^{9} + 34 x^{8} + 16 x^{7} + 791 x^{6} - 132 x^{5} + 4906 x^{4} - 1674 x^{3} + 25257 x^{2} + \cdots + 7056 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12943327 \nu^{9} + 75909616 \nu^{8} - 373768160 \nu^{7} + 4214332012 \nu^{6} + \cdots + 1484766936324 ) / 2726355561837 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 75909616 \nu^{9} - 965660510 \nu^{8} + 4754775100 \nu^{7} - 23328533825 \nu^{6} + \cdots - 35777128730004 ) / 2726355561837 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5891932287 \nu^{9} + 12146277730 \nu^{8} - 198200228510 \nu^{7} - 104736425072 \nu^{6} + \cdots - 4926976726560 ) / 76337955731436 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 16488755029 \nu^{9} + 119035582802 \nu^{8} - 655311132625 \nu^{7} + \cdots + 20\!\cdots\!30 ) / 114506933597154 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 30058669202 \nu^{9} + 236260514137 \nu^{8} - 1094116349765 \nu^{7} + \cdots + 16\!\cdots\!86 ) / 114506933597154 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17494590283 \nu^{9} + 37501567814 \nu^{8} - 599833439770 \nu^{7} + \cdots + 235019674123164 ) / 19084488932859 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 279383812109 \nu^{9} + 132041218924 \nu^{8} - 8344840988516 \nu^{7} + \cdots - 10\!\cdots\!92 ) / 229013867194308 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 477073435099 \nu^{9} - 729818535632 \nu^{8} + 16408777816366 \nu^{7} + \cdots - 32\!\cdots\!32 ) / 229013867194308 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 12\beta_{4} - 2\beta_{2} + 2\beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} - 3\beta_{3} - 24\beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{9} + 2\beta_{8} - 33\beta_{7} + \beta_{6} - \beta_{5} + 267\beta_{4} - 30\beta_{3} - 89\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{9} + 36\beta_{8} - 132\beta_{7} + 36\beta_{6} + 4\beta_{5} + 804\beta_{4} + 701\beta_{2} - 701\beta _1 + 804 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 72\beta_{9} + 72\beta_{8} + 72\beta_{7} + 68\beta_{6} + 140\beta_{5} + 937\beta_{3} + 3322\beta_{2} + 7452 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1141 \beta_{9} - 212 \beta_{8} + 5820 \beta_{7} - 929 \beta_{6} + 929 \beta_{5} - 31863 \beta_{4} + \cdots + 22212 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2070 \beta_{9} - 5315 \beta_{8} + 32384 \beta_{7} - 5315 \beta_{6} - 2070 \beta_{5} - 230007 \beta_{4} + \cdots - 230007 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 27964 \beta_{9} - 27964 \beta_{8} - 27964 \beta_{7} - 8560 \beta_{6} - 36524 \beta_{5} + \cdots - 1151376 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(\beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
2.90324 + 5.02855i
1.22021 + 2.11347i
0.269375 + 0.466571i
−1.33997 − 2.32090i
−2.05285 − 3.55565i
2.90324 − 5.02855i
1.22021 − 2.11347i
0.269375 − 0.466571i
−1.33997 + 2.32090i
−2.05285 + 3.55565i
−2.40324 − 4.16253i 0 −7.55109 + 13.0789i 2.50000 + 4.33013i 0 −18.4976 + 0.916253i 34.1365 0 12.0162 − 20.8126i
46.2 −0.720214 − 1.24745i 0 2.96258 − 5.13135i 2.50000 + 4.33013i 0 18.2377 + 3.22290i −20.0582 0 3.60107 − 6.23723i
46.3 0.230625 + 0.399454i 0 3.89362 − 6.74396i 2.50000 + 4.33013i 0 −18.0800 + 4.01437i 7.28187 0 −1.15312 + 1.99727i
46.4 1.83997 + 3.18692i 0 −2.77099 + 4.79950i 2.50000 + 4.33013i 0 5.08172 − 17.8094i 9.04535 0 −9.19986 + 15.9346i
46.5 2.55285 + 4.42167i 0 −9.03412 + 15.6476i 2.50000 + 4.33013i 0 −2.74187 + 18.3162i −51.4055 0 −12.7643 + 22.1084i
226.1 −2.40324 + 4.16253i 0 −7.55109 − 13.0789i 2.50000 − 4.33013i 0 −18.4976 − 0.916253i 34.1365 0 12.0162 + 20.8126i
226.2 −0.720214 + 1.24745i 0 2.96258 + 5.13135i 2.50000 − 4.33013i 0 18.2377 − 3.22290i −20.0582 0 3.60107 + 6.23723i
226.3 0.230625 − 0.399454i 0 3.89362 + 6.74396i 2.50000 − 4.33013i 0 −18.0800 − 4.01437i 7.28187 0 −1.15312 − 1.99727i
226.4 1.83997 − 3.18692i 0 −2.77099 − 4.79950i 2.50000 − 4.33013i 0 5.08172 + 17.8094i 9.04535 0 −9.19986 − 15.9346i
226.5 2.55285 − 4.42167i 0 −9.03412 − 15.6476i 2.50000 − 4.33013i 0 −2.74187 − 18.3162i −51.4055 0 −12.7643 − 22.1084i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.j.h 10
3.b odd 2 1 105.4.i.d ✓ 10
7.c even 3 1 inner 315.4.j.h 10
7.c even 3 1 2205.4.a.br 5
7.d odd 6 1 2205.4.a.bs 5
21.g even 6 1 735.4.a.z 5
21.h odd 6 1 105.4.i.d ✓ 10
21.h odd 6 1 735.4.a.ba 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.i.d ✓ 10 3.b odd 2 1
105.4.i.d ✓ 10 21.h odd 6 1
315.4.j.h 10 1.a even 1 1 trivial
315.4.j.h 10 7.c even 3 1 inner
735.4.a.z 5 21.g even 6 1
735.4.a.ba 5 21.h odd 6 1
2205.4.a.br 5 7.c even 3 1
2205.4.a.bs 5 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{10} - 3 T_{2}^{9} + 37 T_{2}^{8} - 56 T_{2}^{7} + 890 T_{2}^{6} - 1396 T_{2}^{5} + 7632 T_{2}^{4} + \cdots + 3600 \) Copy content Toggle raw display
\( T_{13}^{5} - 123T_{13}^{4} - 1074T_{13}^{3} + 566590T_{13}^{2} - 14427235T_{13} - 125988247 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 3 T^{9} + \cdots + 3600 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 4747561509943 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} - 123 T^{4} + \cdots - 125988247)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 420843889670400 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 55\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{5} + 360 T^{4} + \cdots + 15483733056)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 45\!\cdots\!69 \) Copy content Toggle raw display
$41$ \( (T^{5} + 341 T^{4} + \cdots + 914820763500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 32 T^{4} + \cdots + 45414054020)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 12244368636072)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 217219935694608)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 207081920604160)^{2} \) Copy content Toggle raw display
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