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Properties

Label 600.3.c.d
Level $600$
Weight $3$
Character orbit 600.c
Analytic conductor $16.349$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,3,Mod(449,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.c (of order \(2\), degree \(1\), 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: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{14} + \beta_{12} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + (\beta_{14} + \beta_{12} - 3) q^{9} + ( - 2 \beta_{13} - 2 \beta_{11} + \cdots - 1) q^{11}+ \cdots + (4 \beta_{14} - 6 \beta_{13} + \cdots - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -763\nu^{14} - 105618\nu^{10} - 2636235\nu^{6} - 13306756\nu^{2} ) / 1927656 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11905 \nu^{15} + 35058 \nu^{14} - 9198 \nu^{13} + 1627734 \nu^{11} + 4519404 \nu^{10} + \cdots - 41323320 \nu ) / 92527488 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11905 \nu^{15} + 71346 \nu^{14} + 9198 \nu^{13} - 1627734 \nu^{11} + 9825516 \nu^{10} + \cdots + 41323320 \nu ) / 92527488 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11905 \nu^{15} + 9198 \nu^{13} - 32652 \nu^{12} + 1627734 \nu^{11} + 1237860 \nu^{9} + \cdots - 332141040 ) / 46263744 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11905 \nu^{15} + 73248 \nu^{14} - 9198 \nu^{13} + 1627734 \nu^{11} + 10139328 \nu^{10} + \cdots - 41323320 \nu ) / 46263744 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 103381 \nu^{15} - 71346 \nu^{14} - 5562 \nu^{13} - 14361006 \nu^{11} - 9825516 \nu^{10} + \cdots + 518165784 \nu ) / 92527488 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30217 \nu^{15} + 9198 \nu^{13} + 18648 \nu^{12} + 4162566 \nu^{11} + 1237860 \nu^{9} + \cdots - 95466816 ) / 23131872 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32945 \nu^{15} + 12762 \nu^{13} + 4537662 \nu^{11} + 1758996 \nu^{9} + 110503233 \nu^{7} + \cdots + 138394728 \nu ) / 23131872 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 32945 \nu^{15} + 12762 \nu^{13} - 4537662 \nu^{11} + 1758996 \nu^{9} - 110503233 \nu^{7} + \cdots + 138394728 \nu ) / 23131872 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 156583 \nu^{15} + 71346 \nu^{14} - 64386 \nu^{13} + 21533466 \nu^{11} + 9825516 \nu^{10} + \cdots - 659373192 \nu ) / 92527488 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21061 \nu^{15} - 9198 \nu^{13} - 2895150 \nu^{11} - 1237860 \nu^{9} - 70026525 \nu^{7} + \cdots - 87587064 \nu ) / 11565936 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 9559 \nu^{15} - 2030 \nu^{13} - 2936 \nu^{12} - 1320394 \nu^{11} - 284932 \nu^{9} + \cdots - 30449184 ) / 5140416 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24484 \nu^{15} + 6867 \nu^{13} + 12825 \nu^{12} + 3377820 \nu^{11} + 950562 \nu^{9} + \cdots + 53385588 ) / 11565936 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26543 \nu^{15} + 2754 \nu^{13} + 3672942 \nu^{11} + 402696 \nu^{9} + 91416699 \nu^{7} + \cdots + 126530712 \nu ) / 11565936 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 224249 \nu^{15} - 71346 \nu^{14} - 31230 \nu^{13} + 31011270 \nu^{11} - 9825516 \nu^{10} + \cdots - 1053569016 \nu ) / 92527488 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 4 \beta_{15} + 2 \beta_{14} - 4 \beta_{13} - \beta_{11} + \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + \cdots - 2 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{15} + \beta_{10} - \beta_{6} - 3\beta_{5} - 3\beta_{3} - 12\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{15} + 2 \beta_{14} - 12 \beta_{13} - \beta_{11} - 5 \beta_{10} - 4 \beta_{9} + 10 \beta_{8} + \cdots - 6 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{14} - 14\beta_{13} - 36\beta_{12} - 8\beta_{11} - 11\beta_{8} - 8\beta_{7} - 2\beta_{4} - 196 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 200 \beta_{15} - 34 \beta_{14} + 332 \beta_{13} - 19 \beta_{11} - 185 \beta_{10} - 204 \beta_{9} + \cdots + 166 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 69\beta_{15} - 69\beta_{10} + 69\beta_{6} + 131\beta_{5} + 75\beta_{3} + 20\beta_{2} + 286\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1888 \beta_{15} - 278 \beta_{14} + 3220 \beta_{13} - 509 \beta_{11} + 2119 \beta_{10} + 2484 \beta_{9} + \cdots + 1610 ) / 24 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1339 \beta_{14} + 1492 \beta_{13} + 4170 \beta_{12} + 538 \beta_{11} + 1339 \beta_{8} + 538 \beta_{7} + \cdots + 17048 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6264 \beta_{15} + 886 \beta_{14} - 10756 \beta_{13} + 2329 \beta_{11} + 7707 \beta_{10} + 9172 \beta_{9} + \cdots - 5378 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 24503 \beta_{15} + 24503 \beta_{10} - 24503 \beta_{6} - 43731 \beta_{5} - 23235 \beta_{3} + \cdots - 85872 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 191408 \beta_{15} + 26806 \beta_{14} - 329204 \beta_{13} + 81349 \beta_{11} - 245951 \beta_{10} + \cdots - 164602 ) / 24 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 49957 \beta_{14} - 52158 \beta_{13} - 152072 \beta_{12} - 16172 \beta_{11} - 49957 \beta_{8} + \cdots - 584540 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1969912 \beta_{15} - 275234 \beta_{14} + 3389356 \beta_{13} - 889139 \beta_{11} - 2583817 \beta_{10} + \cdots + 1694678 ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2694055 \beta_{15} - 2694055 \beta_{10} + 2694055 \beta_{6} + 4747917 \beta_{5} + 2491221 \beta_{3} + \cdots + 9116454 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 6790240 \beta_{15} - 948210 \beta_{14} + 11684060 \beta_{13} - 3151599 \beta_{11} + 8993629 \beta_{10} + \cdots + 5842030 ) / 8 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−2.27869 − 2.27869i
−2.27869 + 2.27869i
−1.09102 + 1.09102i
−1.09102 − 1.09102i
−1.57158 + 1.57158i
−1.57158 − 1.57158i
0.383916 + 0.383916i
0.383916 − 0.383916i
−0.383916 − 0.383916i
−0.383916 + 0.383916i
1.57158 − 1.57158i
1.57158 + 1.57158i
1.09102 − 1.09102i
1.09102 + 1.09102i
2.27869 + 2.27869i
2.27869 − 2.27869i
0 −2.98579 − 0.291610i 0 0 0 4.46268i 0 8.82993 + 1.74137i 0
449.2 0 −2.98579 + 0.291610i 0 0 0 − 4.46268i 0 8.82993 − 1.74137i 0
449.3 0 −1.79813 − 2.40140i 0 0 0 10.2132i 0 −2.53346 + 8.63606i 0
449.4 0 −1.79813 + 2.40140i 0 0 0 − 10.2132i 0 −2.53346 − 8.63606i 0
449.5 0 −0.864473 − 2.87275i 0 0 0 − 9.02416i 0 −7.50537 + 4.96683i 0
449.6 0 −0.864473 + 2.87275i 0 0 0 9.02416i 0 −7.50537 − 4.96683i 0
449.7 0 −0.323191 − 2.98254i 0 0 0 4.72640i 0 −8.79110 + 1.92786i 0
449.8 0 −0.323191 + 2.98254i 0 0 0 − 4.72640i 0 −8.79110 − 1.92786i 0
449.9 0 0.323191 − 2.98254i 0 0 0 4.72640i 0 −8.79110 − 1.92786i 0
449.10 0 0.323191 + 2.98254i 0 0 0 − 4.72640i 0 −8.79110 + 1.92786i 0
449.11 0 0.864473 − 2.87275i 0 0 0 − 9.02416i 0 −7.50537 − 4.96683i 0
449.12 0 0.864473 + 2.87275i 0 0 0 9.02416i 0 −7.50537 + 4.96683i 0
449.13 0 1.79813 − 2.40140i 0 0 0 10.2132i 0 −2.53346 − 8.63606i 0
449.14 0 1.79813 + 2.40140i 0 0 0 − 10.2132i 0 −2.53346 + 8.63606i 0
449.15 0 2.98579 − 0.291610i 0 0 0 4.46268i 0 8.82993 − 1.74137i 0
449.16 0 2.98579 + 0.291610i 0 0 0 − 4.46268i 0 8.82993 + 1.74137i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.c.d 16
3.b odd 2 1 inner 600.3.c.d 16
4.b odd 2 1 1200.3.c.m 16
5.b even 2 1 inner 600.3.c.d 16
5.c odd 4 1 120.3.l.a ✓ 8
5.c odd 4 1 600.3.l.f 8
12.b even 2 1 1200.3.c.m 16
15.d odd 2 1 inner 600.3.c.d 16
15.e even 4 1 120.3.l.a ✓ 8
15.e even 4 1 600.3.l.f 8
20.d odd 2 1 1200.3.c.m 16
20.e even 4 1 240.3.l.d 8
20.e even 4 1 1200.3.l.x 8
40.i odd 4 1 960.3.l.h 8
40.k even 4 1 960.3.l.g 8
60.h even 2 1 1200.3.c.m 16
60.l odd 4 1 240.3.l.d 8
60.l odd 4 1 1200.3.l.x 8
120.q odd 4 1 960.3.l.g 8
120.w even 4 1 960.3.l.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.l.a ✓ 8 5.c odd 4 1
120.3.l.a ✓ 8 15.e even 4 1
240.3.l.d 8 20.e even 4 1
240.3.l.d 8 60.l odd 4 1
600.3.c.d 16 1.a even 1 1 trivial
600.3.c.d 16 3.b odd 2 1 inner
600.3.c.d 16 5.b even 2 1 inner
600.3.c.d 16 15.d odd 2 1 inner
600.3.l.f 8 5.c odd 4 1
600.3.l.f 8 15.e even 4 1
960.3.l.g 8 40.k even 4 1
960.3.l.g 8 120.q odd 4 1
960.3.l.h 8 40.i odd 4 1
960.3.l.h 8 120.w even 4 1
1200.3.c.m 16 4.b odd 2 1
1200.3.c.m 16 12.b even 2 1
1200.3.c.m 16 20.d odd 2 1
1200.3.c.m 16 60.h even 2 1
1200.3.l.x 8 20.e even 4 1
1200.3.l.x 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 228T_{7}^{6} + 16788T_{7}^{4} + 441568T_{7}^{2} + 3779136 \) acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 20 T^{14} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 228 T^{6} + \cdots + 3779136)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 888 T^{6} + \cdots + 232989696)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 776 T^{6} + \cdots + 11943936)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 1104 T^{6} + \cdots + 15872256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 4 T^{3} + \cdots + 16736)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2964 T^{6} + \cdots + 93650688576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 2376 T^{6} + \cdots + 3474395136)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 60 T^{3} + \cdots - 151296)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 6472 T^{6} + \cdots + 967458816)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 43961355472896)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 2504800014336)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 2612 T^{6} + \cdots + 13517317696)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 6801580544256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 15563214360576)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 30631296)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 15484 T^{6} + \cdots + 446523314176)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 35499479924736)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 8637109698816)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 44 T^{3} + \cdots - 12384)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 336130569170496)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 105474871849216)^{2} \) Copy content Toggle raw display
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