Properties

Label 588.4.k.d
Level $588$
Weight $4$
Character orbit 588.k
Analytic conductor $34.693$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,4,Mod(509,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.509");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.k (of order \(6\), degree \(2\), 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: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 2x^{14} - 94x^{12} - 128x^{10} + 2719x^{8} - 10368x^{6} - 616734x^{4} + 1062882x^{2} + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{6}) q^{3} - \beta_{12} q^{5} + ( - \beta_{5} - \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{6}) q^{3} - \beta_{12} q^{5} + ( - \beta_{5} - \beta_{4} + 1) q^{9} - \beta_{13} q^{11} + (\beta_{8} + 3 \beta_{6} - \beta_{3}) q^{13} + (\beta_{14} + \beta_{13} - \beta_{9} + \beta_{5} - \beta_{4} - \beta_{2} + \beta_1 - 8) q^{15} + (\beta_{11} + \beta_{10} - 3 \beta_{7} - 3 \beta_{6}) q^{17} + ( - 2 \beta_{15} + \beta_{10} - \beta_{8} + 3 \beta_{7} + 2 \beta_{3}) q^{19} + (\beta_{14} - 3 \beta_{9} + 3 \beta_{5} - 6 \beta_{4} - 3) q^{23} + (5 \beta_{9} - 40 \beta_{5} - 5 \beta_{2}) q^{25} + (6 \beta_{12} - 6 \beta_{11} + 5 \beta_{8} + 3 \beta_{3}) q^{27} + ( - 2 \beta_{14} - 2 \beta_{13} - 6 \beta_{9} + 6 \beta_{5} - 6 \beta_{4} + 3 \beta_{2} + \cdots - 3) q^{29}+ \cdots + ( - 3 \beta_{14} - 3 \beta_{13} + 24 \beta_{9} - 24 \beta_{5} + 24 \beta_{4} + \cdots + 84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 12 q^{9} - 120 q^{15} - 320 q^{25} - 80 q^{37} + 732 q^{39} + 640 q^{43} - 552 q^{51} - 1560 q^{57} - 1840 q^{67} + 3176 q^{79} + 3456 q^{81} + 1920 q^{85} - 3960 q^{93} + 1152 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 2x^{14} - 94x^{12} - 128x^{10} + 2719x^{8} - 10368x^{6} - 616734x^{4} + 1062882x^{2} + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 367 \nu^{14} - 20012 \nu^{12} + 179650 \nu^{10} + 632201 \nu^{8} - 4107139 \nu^{6} - 18982350 \nu^{4} + 678066228 \nu^{2} + 7659659133 ) / 465010875 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 98 \nu^{14} + 34 \nu^{12} - 2975 \nu^{10} + 15806 \nu^{8} - 860977 \nu^{6} - 2296350 \nu^{4} - 40920957 \nu^{2} + 111071169 ) / 31000725 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4657 \nu^{15} - 154547 \nu^{13} - 325100 \nu^{11} + 1080071 \nu^{9} - 84122284 \nu^{7} + 298446525 \nu^{5} + 5734202463 \nu^{3} + \cdots + 63309503448 \nu ) / 8370195750 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2258 \nu^{14} + 14762 \nu^{12} + 67100 \nu^{10} - 296201 \nu^{8} - 3030236 \nu^{6} + 46198350 \nu^{4} + 2335893147 \nu^{2} - 9984713508 ) / 465010875 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2258 \nu^{14} - 14762 \nu^{12} - 67100 \nu^{10} + 296201 \nu^{8} + 3030236 \nu^{6} - 46198350 \nu^{4} - 940860522 \nu^{2} + 10449724383 ) / 465010875 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2258 \nu^{15} + 14762 \nu^{13} + 67100 \nu^{11} - 296201 \nu^{9} - 3030236 \nu^{7} + 46198350 \nu^{5} + 940860522 \nu^{3} - 10914735258 \nu ) / 465010875 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2258 \nu^{15} + 14762 \nu^{13} + 67100 \nu^{11} - 296201 \nu^{9} - 3030236 \nu^{7} + 46198350 \nu^{5} + 940860522 \nu^{3} - 9519702633 \nu ) / 465010875 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5971 \nu^{15} - 49024 \nu^{13} - 162700 \nu^{11} + 1047412 \nu^{9} + 8237722 \nu^{7} - 143723700 \nu^{5} - 2435163264 \nu^{3} + 31749702516 \nu ) / 465010875 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3481 \nu^{14} + 25762 \nu^{12} + 117100 \nu^{10} - 524407 \nu^{8} - 5288236 \nu^{6} + 80623350 \nu^{4} + 1395780579 \nu^{2} - 17424887508 ) / 31000725 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 2282 \nu^{15} + 15758 \nu^{13} + 81650 \nu^{11} - 311804 \nu^{9} - 3538949 \nu^{7} + 50931900 \nu^{5} + 991601838 \nu^{3} - 10893241422 \nu ) / 155003625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 50440 \nu^{15} + 355879 \nu^{13} + 1567375 \nu^{11} - 6390280 \nu^{9} - 64883587 \nu^{7} + 1211922000 \nu^{5} + 19926642735 \nu^{3} + \cdots - 245536902261 \nu ) / 3348078300 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 53467 \nu^{15} - 364162 \nu^{13} - 1605025 \nu^{11} + 9495949 \nu^{9} + 66583861 \nu^{7} - 1237844025 \nu^{5} - 22893284178 \nu^{3} + \cdots + 251139353283 \nu ) / 3348078300 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 30163 \nu^{14} - 193312 \nu^{12} - 873475 \nu^{10} + 3838861 \nu^{8} + 41105011 \nu^{6} - 755977725 \nu^{4} - 12659265792 \nu^{2} + \cdots + 137968992333 ) / 206671500 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 92729 \nu^{14} - 647141 \nu^{12} - 2941550 \nu^{10} + 18033863 \nu^{8} + 132840398 \nu^{6} - 2025257175 \nu^{4} - 38526854661 \nu^{2} + \cdots + 437712876594 ) / 620014500 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 300184 \nu^{15} - 2251561 \nu^{13} - 10045675 \nu^{11} + 52077748 \nu^{9} + 404260783 \nu^{7} - 6655308300 \nu^{5} - 117765823131 \nu^{3} + \cdots + 1502298144999 \nu ) / 8370195750 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{4} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{15} - 6\beta_{12} - 6\beta_{11} + 10\beta_{10} - 5\beta_{8} - 3\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{13} + \beta_{9} + 144\beta_{5} - \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21 \beta_{15} - 66 \beta_{12} + 132 \beta_{11} + 8 \beta_{10} + 8 \beta_{8} - 159 \beta_{7} - 318 \beta_{6} + 21 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8\beta_{14} + 8\beta_{13} + 17\beta_{9} - 17\beta_{5} + 17\beta_{4} - 96\beta_{2} - 17\beta _1 - 62 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 336 \beta_{15} - 912 \beta_{12} + 456 \beta_{11} - 223 \beta_{10} + 446 \beta_{8} + 168 \beta_{7} - 168 \beta_{6} - 672 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 376\beta_{14} + 256\beta_{9} - 5533\beta_{5} + 128\beta_{4} + 5533 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1248 \beta_{15} + 2136 \beta_{12} + 2136 \beta_{11} + 1360 \beta_{10} - 680 \beta_{8} + 2510 \beta_{7} + 1255 \beta_{6} - 624 \beta_{3} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 888\beta_{13} + 2753\beta_{9} + 39518\beta_{5} - 2753\beta_{2} + 4639\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 13587 \beta_{15} - 3198 \beta_{12} + 6396 \beta_{11} + 51875 \beta_{10} + 51875 \beta_{8} - 49752 \beta_{7} - 99504 \beta_{6} - 13587 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 10124 \beta_{14} + 10124 \beta_{13} + 24576 \beta_{9} - 24576 \beta_{5} + 24576 \beta_{4} - 4063 \beta_{2} - 24576 \beta _1 + 1407216 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 72933 \beta_{15} + 254964 \beta_{12} - 127482 \beta_{11} + 215056 \beta_{10} - 430112 \beta_{8} + 1339881 \beta_{7} - 1339881 \beta_{6} - 145866 \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 6128\beta_{14} + 206543\beta_{9} + 5743010\beta_{5} + 537743\beta_{4} - 5743010 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 2060736 \beta_{15} - 1895280 \beta_{12} - 1895280 \beta_{11} + 7042030 \beta_{10} - 3521015 \beta_{8} - 11686560 \beta_{7} - 5843280 \beta_{6} - 1030368 \beta_{3} ) / 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
509.1
0.00599769 + 2.99999i
−1.34095 + 2.68363i
2.59507 + 1.50519i
2.99456 + 0.180515i
−2.99456 0.180515i
−2.59507 1.50519i
1.34095 2.68363i
−0.00599769 2.99999i
0.00599769 2.99999i
−1.34095 2.68363i
2.59507 1.50519i
2.99456 0.180515i
−2.99456 + 0.180515i
−2.59507 + 1.50519i
1.34095 + 2.68363i
−0.00599769 + 2.99999i
0 −5.19614 + 0.0103883i 0 8.45479 14.6441i 0 0 0 26.9998 0.107958i 0
509.2 0 −4.64818 2.32260i 0 −3.31912 + 5.74888i 0 0 0 16.2111 + 21.5917i 0
509.3 0 −2.60707 + 4.49480i 0 −8.45479 + 14.6441i 0 0 0 −13.4064 23.4365i 0
509.4 0 −0.312662 + 5.18674i 0 3.31912 5.74888i 0 0 0 −26.8045 3.24339i 0
509.5 0 0.312662 5.18674i 0 −3.31912 + 5.74888i 0 0 0 −26.8045 3.24339i 0
509.6 0 2.60707 4.49480i 0 8.45479 14.6441i 0 0 0 −13.4064 23.4365i 0
509.7 0 4.64818 + 2.32260i 0 3.31912 5.74888i 0 0 0 16.2111 + 21.5917i 0
509.8 0 5.19614 0.0103883i 0 −8.45479 + 14.6441i 0 0 0 26.9998 0.107958i 0
521.1 0 −5.19614 0.0103883i 0 8.45479 + 14.6441i 0 0 0 26.9998 + 0.107958i 0
521.2 0 −4.64818 + 2.32260i 0 −3.31912 5.74888i 0 0 0 16.2111 21.5917i 0
521.3 0 −2.60707 4.49480i 0 −8.45479 14.6441i 0 0 0 −13.4064 + 23.4365i 0
521.4 0 −0.312662 5.18674i 0 3.31912 + 5.74888i 0 0 0 −26.8045 + 3.24339i 0
521.5 0 0.312662 + 5.18674i 0 −3.31912 5.74888i 0 0 0 −26.8045 + 3.24339i 0
521.6 0 2.60707 + 4.49480i 0 8.45479 + 14.6441i 0 0 0 −13.4064 + 23.4365i 0
521.7 0 4.64818 2.32260i 0 3.31912 + 5.74888i 0 0 0 16.2111 21.5917i 0
521.8 0 5.19614 + 0.0103883i 0 −8.45479 14.6441i 0 0 0 26.9998 + 0.107958i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 509.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.4.k.d 16
3.b odd 2 1 inner 588.4.k.d 16
7.b odd 2 1 inner 588.4.k.d 16
7.c even 3 1 84.4.f.a 8
7.c even 3 1 inner 588.4.k.d 16
7.d odd 6 1 84.4.f.a 8
7.d odd 6 1 inner 588.4.k.d 16
21.c even 2 1 inner 588.4.k.d 16
21.g even 6 1 84.4.f.a 8
21.g even 6 1 inner 588.4.k.d 16
21.h odd 6 1 84.4.f.a 8
21.h odd 6 1 inner 588.4.k.d 16
28.f even 6 1 336.4.k.d 8
28.g odd 6 1 336.4.k.d 8
84.j odd 6 1 336.4.k.d 8
84.n even 6 1 336.4.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.f.a 8 7.c even 3 1
84.4.f.a 8 7.d odd 6 1
84.4.f.a 8 21.g even 6 1
84.4.f.a 8 21.h odd 6 1
336.4.k.d 8 28.f even 6 1
336.4.k.d 8 28.g odd 6 1
336.4.k.d 8 84.j odd 6 1
336.4.k.d 8 84.n even 6 1
588.4.k.d 16 1.a even 1 1 trivial
588.4.k.d 16 3.b odd 2 1 inner
588.4.k.d 16 7.b odd 2 1 inner
588.4.k.d 16 7.c even 3 1 inner
588.4.k.d 16 7.d odd 6 1 inner
588.4.k.d 16 21.c even 2 1 inner
588.4.k.d 16 21.g even 6 1 inner
588.4.k.d 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{8} + 330T_{5}^{6} + 96300T_{5}^{4} + 4158000T_{5}^{2} + 158760000 \) Copy content Toggle raw display
\( T_{13}^{4} + 2694T_{13}^{2} + 522144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - 6 T^{14} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( (T^{8} + 330 T^{6} + 96300 T^{4} + \cdots + 158760000)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 4716 T^{6} + \cdots + 28843602149376)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2694 T^{2} + 522144)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1068 T^{6} + 1138608 T^{4} + \cdots + 4064256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 7410 T^{6} + \cdots + 126684029160000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 48240 T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 57204 T^{2} + 65790144)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 48060 T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 20 T^{3} + 84540 T^{2} + \cdots + 7079539600)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 274380 T^{2} + \cdots + 2076933600)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 80 T - 7760)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 322008 T^{6} + \cdots + 81\!\cdots\!56)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 221940 T^{6} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 558870 T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 67350 T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 460 T^{3} + 273360 T^{2} + \cdots + 3814297600)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1163340 T^{2} + \cdots + 537062400)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 484896 T^{6} + \cdots + 20\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 794 T^{3} + \cdots + 229878055936)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1944870 T^{2} + \cdots + 869269413600)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 57780 T^{6} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 3181656 T^{2} + \cdots + 2446437556224)^{4} \) Copy content Toggle raw display
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