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Properties

Label 600.3.u.h
Level $600$
Weight $3$
Character orbit 600.u
Analytic conductor $16.349$
Analytic rank $0$
Dimension $8$
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] = [600,3,Mod(193,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.193");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.u (of order \(4\), 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: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{6} - \beta_{3}) q^{7} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{6} - \beta_{3}) q^{7} - 3 \beta_{2} q^{9} + ( - \beta_{4} - 2 \beta_{3} + 3 \beta_1 + 4) q^{11} + (\beta_{7} - \beta_{2} - 3 \beta_1) q^{13} + ( - \beta_{6} - 7 \beta_{3} - 7 \beta_{2} - 7) q^{17} + (\beta_{7} - \beta_{6} + \beta_{3} + \cdots - 1) q^{19}+ \cdots + (3 \beta_{5} + 9 \beta_{3} + \cdots + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} + 32 q^{11} + 4 q^{13} - 52 q^{17} - 24 q^{21} + 40 q^{23} + 96 q^{31} - 60 q^{33} + 60 q^{37} - 152 q^{41} + 88 q^{43} + 16 q^{47} - 168 q^{51} - 108 q^{53} + 24 q^{57} + 264 q^{61} - 12 q^{63} + 216 q^{67} - 240 q^{71} + 208 q^{73} - 168 q^{77} - 72 q^{81} - 336 q^{83} - 252 q^{87} + 592 q^{91} - 264 q^{93} + 208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3096 \nu^{7} - 17905 \nu^{6} + 34598 \nu^{5} + 15150 \nu^{4} + 876904 \nu^{3} - 4737945 \nu^{2} + \cdots + 2940830 ) / 1277880 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2465 \nu^{7} + 10455 \nu^{6} - 21142 \nu^{5} + 2900 \nu^{4} - 663785 \nu^{3} + 2966245 \nu^{2} + \cdots - 838660 ) / 958410 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4775 \nu^{7} - 20987 \nu^{6} + 31148 \nu^{5} - 24410 \nu^{4} + 1286695 \nu^{3} - 5631403 \nu^{2} + \cdots - 227650 ) / 1277880 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 40034 \nu^{7} - 107007 \nu^{6} + 1294 \nu^{5} - 216230 \nu^{4} + 9466346 \nu^{3} - 29697223 \nu^{2} + \cdots + 17361370 ) / 3833640 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 17425 \nu^{7} + 50772 \nu^{6} - 167078 \nu^{5} + 20500 \nu^{4} - 4251625 \nu^{3} + \cdots - 5575940 ) / 1277880 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 71513 \nu^{7} - 305517 \nu^{6} + 737284 \nu^{5} - 140510 \nu^{4} + 19259897 \nu^{3} + \cdots - 3461150 ) / 3833640 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33016 \nu^{7} + 119837 \nu^{6} - 241278 \nu^{5} + 20050 \nu^{4} - 8904504 \nu^{3} + \cdots - 19230750 ) / 1277880 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{6} + \beta_{5} + \beta_{4} - 5\beta_{3} + 6\beta_{2} + 6 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} - 2\beta_{5} + 19\beta_{3} + 108\beta_{2} + 19\beta _1 + 3 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -24\beta_{7} + 17\beta_{5} - 17\beta_{4} + 17\beta_{3} + 52\beta_{2} - 58\beta _1 - 28 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -31\beta_{7} - 31\beta_{6} + 34\beta_{4} - 377\beta_{3} + 343\beta _1 - 1505 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -308\beta_{6} - 279\beta_{5} - 279\beta_{4} + 1145\beta_{3} - 224\beta_{2} - 224 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 297\beta_{7} - 297\beta_{6} + 598\beta_{5} - 5881\beta_{3} - 22392\beta_{2} - 5881\beta _1 - 297 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4076\beta_{7} - 4533\beta_{5} + 4533\beta_{4} - 4533\beta_{3} + 4152\beta_{2} + 13942\beta _1 - 8228 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−0.137883 + 0.137883i
2.36263 − 2.36263i
−2.88489 + 2.88489i
2.66015 − 2.66015i
−0.137883 − 0.137883i
2.36263 + 2.36263i
−2.88489 − 2.88489i
2.66015 + 2.66015i
0 −1.22474 − 1.22474i 0 0 0 −7.33876 + 7.33876i 0 3.00000i 0
193.2 0 −1.22474 − 1.22474i 0 0 0 8.78825 − 8.78825i 0 3.00000i 0
193.3 0 1.22474 + 1.22474i 0 0 0 −6.02356 + 6.02356i 0 3.00000i 0
193.4 0 1.22474 + 1.22474i 0 0 0 2.57407 − 2.57407i 0 3.00000i 0
457.1 0 −1.22474 + 1.22474i 0 0 0 −7.33876 − 7.33876i 0 − 3.00000i 0
457.2 0 −1.22474 + 1.22474i 0 0 0 8.78825 + 8.78825i 0 − 3.00000i 0
457.3 0 1.22474 − 1.22474i 0 0 0 −6.02356 − 6.02356i 0 − 3.00000i 0
457.4 0 1.22474 − 1.22474i 0 0 0 2.57407 + 2.57407i 0 − 3.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.u.h 8
3.b odd 2 1 1800.3.v.t 8
4.b odd 2 1 1200.3.bg.q 8
5.b even 2 1 120.3.u.b ✓ 8
5.c odd 4 1 120.3.u.b ✓ 8
5.c odd 4 1 inner 600.3.u.h 8
15.d odd 2 1 360.3.v.f 8
15.e even 4 1 360.3.v.f 8
15.e even 4 1 1800.3.v.t 8
20.d odd 2 1 240.3.bg.e 8
20.e even 4 1 240.3.bg.e 8
20.e even 4 1 1200.3.bg.q 8
40.e odd 2 1 960.3.bg.k 8
40.f even 2 1 960.3.bg.l 8
40.i odd 4 1 960.3.bg.l 8
40.k even 4 1 960.3.bg.k 8
60.h even 2 1 720.3.bh.o 8
60.l odd 4 1 720.3.bh.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.b ✓ 8 5.b even 2 1
120.3.u.b ✓ 8 5.c odd 4 1
240.3.bg.e 8 20.d odd 2 1
240.3.bg.e 8 20.e even 4 1
360.3.v.f 8 15.d odd 2 1
360.3.v.f 8 15.e even 4 1
600.3.u.h 8 1.a even 1 1 trivial
600.3.u.h 8 5.c odd 4 1 inner
720.3.bh.o 8 60.h even 2 1
720.3.bh.o 8 60.l odd 4 1
960.3.bg.k 8 40.e odd 2 1
960.3.bg.k 8 40.k even 4 1
960.3.bg.l 8 40.f even 2 1
960.3.bg.l 8 40.i odd 4 1
1200.3.bg.q 8 4.b odd 2 1
1200.3.bg.q 8 20.e even 4 1
1800.3.v.t 8 3.b odd 2 1
1800.3.v.t 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 4T_{7}^{7} + 8T_{7}^{6} + 120T_{7}^{5} + 20900T_{7}^{4} + 120000T_{7}^{3} + 320000T_{7}^{2} - 3200000T_{7} + 16000000 \) acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} + \cdots + 16000000 \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{3} + \cdots - 32288)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{7} + \cdots + 53824 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 2570084416 \) Copy content Toggle raw display
$19$ \( T^{8} + 888 T^{6} + \cdots + 82591744 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 3064729600 \) Copy content Toggle raw display
$29$ \( T^{8} + 4500 T^{6} + \cdots + 619810816 \) Copy content Toggle raw display
$31$ \( (T^{4} - 48 T^{3} + \cdots - 1815680)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 16133718222400 \) Copy content Toggle raw display
$41$ \( (T^{4} + 76 T^{3} + \cdots - 63872)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 949580087296 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 287296000000 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 5939943840000 \) Copy content Toggle raw display
$59$ \( T^{8} + 2348 T^{6} + \cdots + 861184 \) Copy content Toggle raw display
$61$ \( (T^{4} - 132 T^{3} + \cdots - 13929728)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 3555366682624 \) Copy content Toggle raw display
$71$ \( (T^{4} + 120 T^{3} + \cdots - 4371968)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 1421016580096 \) Copy content Toggle raw display
$83$ \( T^{8} + 336 T^{7} + \cdots + 2166784 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 693289369600 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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