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Properties

Label 315.4.j.i
Level $315$
Weight $4$
Character orbit 315.j
Analytic conductor $18.586$
Analytic rank $0$
Dimension $16$
CM no
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 61 x^{14} - 88 x^{13} + 1920 x^{12} - 2032 x^{11} + 36527 x^{10} - 1714 x^{9} + \cdots + 13660416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_1) q^{2} + (\beta_{11} - 6 \beta_{6} - \beta_{5} + \cdots - 6) q^{4}+ \cdots + ( - \beta_{12} + \beta_{8} + \beta_{6} + \cdots + 9) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_1) q^{2} + (\beta_{11} - 6 \beta_{6} - \beta_{5} + \cdots - 6) q^{4}+ \cdots + (6 \beta_{15} + 21 \beta_{14} + \cdots - 96) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} - 42 q^{4} + 40 q^{5} + 22 q^{7} + 96 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} - 42 q^{4} + 40 q^{5} + 22 q^{7} + 96 q^{8} + 20 q^{10} - 100 q^{11} - 204 q^{13} + 178 q^{14} - 266 q^{16} - 56 q^{17} - 420 q^{20} - 140 q^{22} - 190 q^{23} - 200 q^{25} - 60 q^{26} - 4 q^{28} + 592 q^{29} - 42 q^{31} - 718 q^{32} - 976 q^{34} + 10 q^{35} - 314 q^{37} + 514 q^{38} + 240 q^{40} - 56 q^{41} + 1428 q^{43} - 410 q^{44} + 650 q^{46} - 326 q^{47} - 754 q^{49} + 200 q^{50} + 794 q^{52} - 1282 q^{53} - 1000 q^{55} + 1218 q^{56} - 942 q^{58} + 924 q^{59} + 536 q^{61} + 100 q^{62} + 3804 q^{64} - 510 q^{65} - 2 q^{67} - 2690 q^{68} + 970 q^{70} + 3032 q^{71} - 86 q^{73} - 4754 q^{74} + 60 q^{76} + 1358 q^{77} + 42 q^{79} + 1330 q^{80} + 3602 q^{82} + 1144 q^{83} - 560 q^{85} - 1758 q^{86} + 4312 q^{88} - 940 q^{89} - 1530 q^{91} + 7688 q^{92} - 2866 q^{94} - 3440 q^{97} - 1514 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 61 x^{14} - 88 x^{13} + 1920 x^{12} - 2032 x^{11} + 36527 x^{10} - 1714 x^{9} + \cdots + 13660416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 60\!\cdots\!59 \nu^{15} + \cdots - 71\!\cdots\!88 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 56\!\cdots\!51 \nu^{15} + \cdots - 20\!\cdots\!36 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!93 \nu^{15} + \cdots - 38\!\cdots\!72 ) / 38\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 86\!\cdots\!65 \nu^{15} + \cdots - 14\!\cdots\!92 ) / 12\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61\!\cdots\!99 \nu^{15} + \cdots - 61\!\cdots\!12 ) / 85\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!41 \nu^{15} + \cdots + 12\!\cdots\!80 ) / 33\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 22\!\cdots\!55 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 61\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!47 \nu^{15} + \cdots + 94\!\cdots\!12 ) / 24\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 83\!\cdots\!39 \nu^{15} + \cdots + 27\!\cdots\!48 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 70\!\cdots\!51 \nu^{15} + \cdots - 75\!\cdots\!20 ) / 85\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!19 \nu^{15} + \cdots - 12\!\cdots\!76 ) / 13\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 60\!\cdots\!65 \nu^{15} + \cdots + 33\!\cdots\!40 ) / 26\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!44 \nu^{15} + \cdots - 13\!\cdots\!64 ) / 33\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 56\!\cdots\!01 \nu^{15} + \cdots + 66\!\cdots\!04 ) / 13\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - 13\beta_{6} - \beta_{5} - \beta_{4} + \beta _1 - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} + \beta_{8} + \beta_{6} - 2\beta_{5} - 22\beta_{4} + \beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{14} - \beta_{13} - 2 \beta_{12} - 32 \beta_{11} + \beta_{10} + \beta_{9} + 3 \beta_{8} + \cdots - 50 \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 42 \beta_{14} - 42 \beta_{13} - 6 \beta_{12} - 96 \beta_{11} + 14 \beta_{10} + 28 \beta_{9} + \cdots + 722 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50 \beta_{15} - 74 \beta_{13} + 70 \beta_{12} + 24 \beta_{10} - 172 \beta_{8} + 50 \beta_{7} + \cdots + 7997 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1563 \beta_{14} + 1169 \beta_{13} + 1565 \beta_{12} + 3820 \beta_{11} - 396 \beta_{10} + \cdots + 17192 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 2013 \beta_{15} - 7571 \beta_{14} + 6960 \beta_{13} + 3469 \beta_{12} + 31546 \beta_{11} + \cdots - 238739 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8760 \beta_{15} + 19064 \beta_{13} - 38120 \beta_{12} - 19016 \beta_{10} + 56916 \beta_{8} + \cdots - 1029984 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 302172 \beta_{14} - 149032 \beta_{13} - 290552 \beta_{12} - 1038829 \beta_{11} + 141520 \beta_{10} + \cdots - 2543549 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 381240 \beta_{15} + 2057257 \beta_{14} - 2074373 \beta_{13} - 803576 \beta_{12} - 5234054 \beta_{11} + \cdots + 36710499 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2814613 \beta_{15} - 5445677 \beta_{13} + 5869457 \beta_{12} + 3894592 \beta_{10} - 11488431 \beta_{8} + \cdots + 246606449 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 74008386 \beta_{14} + 43252592 \beta_{13} + 74858386 \beta_{12} + 188363376 \beta_{11} + \cdots + 575973043 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 102637418 \beta_{15} - 425070592 \beta_{14} + 424470416 \beta_{13} + 203311002 \beta_{12} + \cdots - 8285372429 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 578940360 \beta_{15} + 1195029020 \beta_{13} - 1494652213 \beta_{12} - 1022446512 \beta_{10} + \cdots - 45663266503 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(1\)

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
−2.29598 − 3.97675i
−1.71036 − 2.96243i
−1.01204 − 1.75290i
−0.475331 − 0.823298i
0.906849 + 1.57071i
1.15131 + 1.99413i
2.46040 + 4.26153i
2.97515 + 5.15311i
−2.29598 + 3.97675i
−1.71036 + 2.96243i
−1.01204 + 1.75290i
−0.475331 + 0.823298i
0.906849 − 1.57071i
1.15131 − 1.99413i
2.46040 − 4.26153i
2.97515 − 5.15311i
−2.79598 − 4.84278i 0 −11.6350 + 20.1524i 2.50000 + 4.33013i 0 8.61636 + 16.3939i 85.3894 0 13.9799 − 24.2139i
46.2 −2.21036 − 3.82846i 0 −5.77139 + 9.99634i 2.50000 + 4.33013i 0 0.502371 − 18.5134i 15.6616 0 11.0518 − 19.1423i
46.3 −1.51204 − 2.61893i 0 −0.572522 + 0.991637i 2.50000 + 4.33013i 0 −16.1159 + 9.12560i −20.7299 0 7.56019 − 13.0946i
46.4 −0.975331 − 1.68932i 0 2.09746 − 3.63290i 2.50000 + 4.33013i 0 18.2579 + 3.10650i −23.7882 0 4.87666 − 8.44662i
46.5 0.406849 + 0.704684i 0 3.66895 − 6.35480i 2.50000 + 4.33013i 0 0.736536 + 18.5056i 12.4804 0 −2.03425 + 3.52342i
46.6 0.651312 + 1.12811i 0 3.15158 − 5.45870i 2.50000 + 4.33013i 0 −14.0975 − 12.0108i 18.6317 0 −3.25656 + 5.64053i
46.7 1.96040 + 3.39551i 0 −3.68632 + 6.38489i 2.50000 + 4.33013i 0 17.2936 + 6.62818i 2.45977 0 −9.80199 + 16.9775i
46.8 2.47515 + 4.28709i 0 −8.25275 + 14.2942i 2.50000 + 4.33013i 0 −4.19324 − 18.0393i −42.1048 0 −12.3758 + 21.4354i
226.1 −2.79598 + 4.84278i 0 −11.6350 − 20.1524i 2.50000 − 4.33013i 0 8.61636 − 16.3939i 85.3894 0 13.9799 + 24.2139i
226.2 −2.21036 + 3.82846i 0 −5.77139 − 9.99634i 2.50000 − 4.33013i 0 0.502371 + 18.5134i 15.6616 0 11.0518 + 19.1423i
226.3 −1.51204 + 2.61893i 0 −0.572522 − 0.991637i 2.50000 − 4.33013i 0 −16.1159 − 9.12560i −20.7299 0 7.56019 + 13.0946i
226.4 −0.975331 + 1.68932i 0 2.09746 + 3.63290i 2.50000 − 4.33013i 0 18.2579 − 3.10650i −23.7882 0 4.87666 + 8.44662i
226.5 0.406849 − 0.704684i 0 3.66895 + 6.35480i 2.50000 − 4.33013i 0 0.736536 − 18.5056i 12.4804 0 −2.03425 − 3.52342i
226.6 0.651312 − 1.12811i 0 3.15158 + 5.45870i 2.50000 − 4.33013i 0 −14.0975 + 12.0108i 18.6317 0 −3.25656 − 5.64053i
226.7 1.96040 − 3.39551i 0 −3.68632 − 6.38489i 2.50000 − 4.33013i 0 17.2936 − 6.62818i 2.45977 0 −9.80199 − 16.9775i
226.8 2.47515 − 4.28709i 0 −8.25275 − 14.2942i 2.50000 − 4.33013i 0 −4.19324 + 18.0393i −42.1048 0 −12.3758 − 21.4354i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.8
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.i ✓ 16
3.b odd 2 1 315.4.j.j yes 16
7.c even 3 1 inner 315.4.j.i ✓ 16
7.c even 3 1 2205.4.a.cf 8
7.d odd 6 1 2205.4.a.cg 8
21.g even 6 1 2205.4.a.cb 8
21.h odd 6 1 315.4.j.j yes 16
21.h odd 6 1 2205.4.a.cc 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.j.i ✓ 16 1.a even 1 1 trivial
315.4.j.i ✓ 16 7.c even 3 1 inner
315.4.j.j yes 16 3.b odd 2 1
315.4.j.j yes 16 21.h odd 6 1
2205.4.a.cb 8 21.g even 6 1
2205.4.a.cc 8 21.h odd 6 1
2205.4.a.cf 8 7.c even 3 1
2205.4.a.cg 8 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 \(S_{4}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{16} + 4 T_{2}^{15} + 61 T_{2}^{14} + 148 T_{2}^{13} + 2115 T_{2}^{12} + 4642 T_{2}^{11} + \cdots + 9000000 \) Copy content Toggle raw display
\( T_{13}^{8} + 102 T_{13}^{7} - 6020 T_{13}^{6} - 866738 T_{13}^{5} - 9517198 T_{13}^{4} + \cdots - 1519211972487 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 4 T^{15} + \cdots + 9000000 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} - 5 T + 25)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 1519211972487)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 32\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 54\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 19\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 47\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 24\!\cdots\!80)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 90\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 31\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 69\!\cdots\!88)^{2} \) Copy content Toggle raw display
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