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Properties

Label 525.2.r.h
Level $525$
Weight $2$
Character orbit 525.r
Analytic conductor $4.192$
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] = [525,2,Mod(424,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.424");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.r (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: \(4.19214610612\)
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: \( x^{16} - 15x^{14} + 158x^{12} - 843x^{10} + 3258x^{8} - 4947x^{6} + 5489x^{4} - 1296x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 1) q^{4}+ \cdots - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{12} q^{3} + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots + 1) q^{4}+ \cdots + (\beta_{9} - \beta_{7} + \beta_{6} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 14 q^{4} - 4 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 14 q^{4} - 4 q^{6} + 8 q^{9} - 16 q^{11} + 24 q^{14} - 34 q^{16} + 6 q^{19} + 4 q^{21} - 6 q^{24} + 38 q^{26} - 64 q^{29} - 18 q^{31} - 56 q^{34} + 28 q^{36} + 14 q^{39} + 16 q^{41} + 52 q^{44} + 12 q^{46} + 42 q^{49} - 12 q^{51} - 2 q^{54} - 42 q^{56} + 20 q^{59} + 40 q^{61} - 84 q^{64} - 24 q^{66} + 8 q^{69} + 88 q^{71} - 42 q^{74} - 92 q^{76} + 16 q^{79} - 8 q^{81} + 36 q^{84} - 24 q^{86} - 20 q^{89} + 42 q^{91} + 44 q^{94} + 34 q^{96} - 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 15x^{14} + 158x^{12} - 843x^{10} + 3258x^{8} - 4947x^{6} + 5489x^{4} - 1296x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 62937 \nu^{14} + 791239 \nu^{12} - 7673262 \nu^{10} + 29204099 \nu^{8} - 68243802 \nu^{6} + \cdots - 741055184 ) / 245379408 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 288167 \nu^{14} + 4051357 \nu^{12} - 40373487 \nu^{10} + 188912823 \nu^{8} - 628102215 \nu^{6} + \cdots - 306634312 ) / 644120946 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1037621 \nu^{14} - 15868765 \nu^{12} + 166205130 \nu^{10} - 885323493 \nu^{8} + \cdots - 227006720 ) / 1288241892 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1708867 \nu^{14} - 25153821 \nu^{12} + 263454282 \nu^{10} - 1373439057 \nu^{8} + \cdots - 2077487664 ) / 1717655856 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2751857 \nu^{14} + 38259711 \nu^{12} - 393558942 \nu^{10} + 1888414411 \nu^{8} + \cdots - 3459116192 ) / 1717655856 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 9945497 \nu^{14} + 154655863 \nu^{12} - 1657786662 \nu^{10} + 9304722867 \nu^{8} + \cdots + 22301372336 ) / 5152967568 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 886745 \nu^{15} - 12263554 \nu^{13} + 124236945 \nu^{11} - 581320905 \nu^{9} + \cdots + 2571974473 \nu ) / 1288241892 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8142721 \nu^{14} + 123758843 \nu^{12} - 1302663054 \nu^{10} + 7014096807 \nu^{8} + \cdots + 9670329352 ) / 2576483784 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2706149 \nu^{15} - 45046027 \nu^{13} + 488786070 \nu^{11} - 2912604567 \nu^{9} + \cdots - 18897225584 \nu ) / 2944552896 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 1708867 \nu^{15} + 25153821 \nu^{13} - 263454282 \nu^{11} + 1373439057 \nu^{9} + \cdots + 2077487664 \nu ) / 1717655856 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 38329289 \nu^{15} + 584160679 \nu^{13} - 6185671086 \nu^{11} + 33603542211 \nu^{9} + \cdots + 52315090832 \nu ) / 20611870272 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58035667 \nu^{15} + 852889805 \nu^{13} - 8898554154 \nu^{11} + 46016072625 \nu^{9} + \cdots - 79171609568 \nu ) / 20611870272 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 126993521 \nu^{15} + 1897063711 \nu^{13} - 19953851358 \nu^{11} + 105905971323 \nu^{9} + \cdots + 42923508464 \nu ) / 20611870272 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 68497297 \nu^{15} - 1000621895 \nu^{13} + 10437980742 \nu^{11} - 53765247483 \nu^{9} + \cdots - 6421385992 \nu ) / 10305935136 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{6} + 4\beta_{5} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} + \beta_{12} - 5\beta_{11} + \beta_{10} - \beta_{8} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{9} - \beta_{7} + 8\beta_{6} + 25\beta_{5} + 7\beta_{4} - \beta_{3} - 8\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{15} - 2\beta_{14} - 8\beta_{13} + 12\beta_{12} - 39\beta_{11} + 2\beta_{10} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 57\beta_{9} - 57\beta_{7} + 10\beta_{6} + 57\beta_{4} - 63\beta_{3} - 47\beta_{2} - 109 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -83\beta_{15} - 83\beta_{14} + 26\beta_{13} - 83\beta_{11} - 57\beta_{10} + 121\beta_{8} - 286\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 83\beta_{9} - 317\beta_{7} - 317\beta_{6} - 1123\beta_{5} + 173\beta_{4} - 317\beta_{3} + 83\beta_{2} - 806 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 256 \beta_{15} - 400 \beta_{14} + 656 \beta_{13} - 1092 \beta_{12} + 1101 \beta_{11} + \cdots - 1757 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2157 \beta_{9} + 656 \beta_{7} - 2813 \beta_{6} - 7684 \beta_{5} - 1209 \beta_{4} + 1604 \beta_{3} + \cdots - 656 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2813\beta_{15} + 2260\beta_{14} + 2813\beta_{13} - 9229\beta_{12} + 11998\beta_{11} - 2260\beta_{10} + 2260\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -19884\beta_{9} + 19884\beta_{7} - 5073\beta_{6} - 19884\beta_{4} + 28560\beta_{3} + 14811\beta_{2} + 33181 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 38706 \beta_{15} + 38706 \beta_{14} - 18822 \beta_{13} + 38706 \beta_{11} + 19884 \beta_{10} + \cdots + 101509 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 38706 \beta_{9} + 102571 \beta_{7} + 102571 \beta_{6} + 369454 \beta_{5} - 112524 \beta_{4} + \cdots + 266883 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 151230 \beta_{15} + 141277 \beta_{14} - 292507 \beta_{13} + 591373 \beta_{12} - 282089 \beta_{11} + \cdots + 574596 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
424.1
−2.34074 + 1.35143i
−2.11082 + 1.21868i
−1.06407 + 0.614340i
−0.427967 + 0.247087i
0.427967 − 0.247087i
1.06407 − 0.614340i
2.11082 − 1.21868i
2.34074 − 1.35143i
−2.34074 − 1.35143i
−2.11082 − 1.21868i
−1.06407 − 0.614340i
−0.427967 − 0.247087i
0.427967 + 0.247087i
1.06407 + 0.614340i
2.11082 + 1.21868i
2.34074 + 1.35143i
−2.34074 + 1.35143i 0.866025 + 0.500000i 2.65271 − 4.59463i 0 −2.70285 −2.06605 + 1.65271i 8.93406i 0.500000 + 0.866025i 0
424.2 −2.11082 + 1.21868i −0.866025 − 0.500000i 1.97036 − 3.41277i 0 2.43736 −2.46138 − 0.970361i 4.73024i 0.500000 + 0.866025i 0
424.3 −1.06407 + 0.614340i 0.866025 + 0.500000i −0.245174 + 0.424653i 0 −1.22868 2.33443 − 1.24517i − 3.05984i 0.500000 + 0.866025i 0
424.4 −0.427967 + 0.247087i −0.866025 − 0.500000i −0.877896 + 1.52056i 0 0.494173 1.86373 + 1.87790i − 1.85601i 0.500000 + 0.866025i 0
424.5 0.427967 − 0.247087i 0.866025 + 0.500000i −0.877896 + 1.52056i 0 0.494173 −1.86373 − 1.87790i 1.85601i 0.500000 + 0.866025i 0
424.6 1.06407 − 0.614340i −0.866025 − 0.500000i −0.245174 + 0.424653i 0 −1.22868 −2.33443 + 1.24517i 3.05984i 0.500000 + 0.866025i 0
424.7 2.11082 − 1.21868i 0.866025 + 0.500000i 1.97036 − 3.41277i 0 2.43736 2.46138 + 0.970361i − 4.73024i 0.500000 + 0.866025i 0
424.8 2.34074 − 1.35143i −0.866025 − 0.500000i 2.65271 − 4.59463i 0 −2.70285 2.06605 − 1.65271i − 8.93406i 0.500000 + 0.866025i 0
499.1 −2.34074 − 1.35143i 0.866025 − 0.500000i 2.65271 + 4.59463i 0 −2.70285 −2.06605 − 1.65271i − 8.93406i 0.500000 − 0.866025i 0
499.2 −2.11082 − 1.21868i −0.866025 + 0.500000i 1.97036 + 3.41277i 0 2.43736 −2.46138 + 0.970361i − 4.73024i 0.500000 − 0.866025i 0
499.3 −1.06407 − 0.614340i 0.866025 − 0.500000i −0.245174 − 0.424653i 0 −1.22868 2.33443 + 1.24517i 3.05984i 0.500000 − 0.866025i 0
499.4 −0.427967 − 0.247087i −0.866025 + 0.500000i −0.877896 − 1.52056i 0 0.494173 1.86373 − 1.87790i 1.85601i 0.500000 − 0.866025i 0
499.5 0.427967 + 0.247087i 0.866025 − 0.500000i −0.877896 − 1.52056i 0 0.494173 −1.86373 + 1.87790i − 1.85601i 0.500000 − 0.866025i 0
499.6 1.06407 + 0.614340i −0.866025 + 0.500000i −0.245174 − 0.424653i 0 −1.22868 −2.33443 − 1.24517i − 3.05984i 0.500000 − 0.866025i 0
499.7 2.11082 + 1.21868i 0.866025 − 0.500000i 1.97036 + 3.41277i 0 2.43736 2.46138 − 0.970361i 4.73024i 0.500000 − 0.866025i 0
499.8 2.34074 + 1.35143i −0.866025 + 0.500000i 2.65271 + 4.59463i 0 −2.70285 2.06605 + 1.65271i 8.93406i 0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 424.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.r.h 16
5.b even 2 1 inner 525.2.r.h 16
5.c odd 4 1 525.2.i.i ✓ 8
5.c odd 4 1 525.2.i.j yes 8
7.c even 3 1 inner 525.2.r.h 16
35.j even 6 1 inner 525.2.r.h 16
35.k even 12 1 3675.2.a.bq 4
35.k even 12 1 3675.2.a.bx 4
35.l odd 12 1 525.2.i.i ✓ 8
35.l odd 12 1 525.2.i.j yes 8
35.l odd 12 1 3675.2.a.br 4
35.l odd 12 1 3675.2.a.bw 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.i.i ✓ 8 5.c odd 4 1
525.2.i.i ✓ 8 35.l odd 12 1
525.2.i.j yes 8 5.c odd 4 1
525.2.i.j yes 8 35.l odd 12 1
525.2.r.h 16 1.a even 1 1 trivial
525.2.r.h 16 5.b even 2 1 inner
525.2.r.h 16 7.c even 3 1 inner
525.2.r.h 16 35.j even 6 1 inner
3675.2.a.bq 4 35.k even 12 1
3675.2.a.br 4 35.l odd 12 1
3675.2.a.bw 4 35.l odd 12 1
3675.2.a.bx 4 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\):

\( T_{2}^{16} - 15T_{2}^{14} + 158T_{2}^{12} - 843T_{2}^{10} + 3258T_{2}^{8} - 4947T_{2}^{6} + 5489T_{2}^{4} - 1296T_{2}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{8} + 8T_{11}^{7} + 72T_{11}^{6} + 296T_{11}^{5} + 1832T_{11}^{4} + 6688T_{11}^{3} + 29776T_{11}^{2} + 59040T_{11} + 107584 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 15 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 21 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 8 T^{7} + \cdots + 107584)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 87 T^{6} + \cdots + 37636)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 76 T^{14} + \cdots + 2560000 \) Copy content Toggle raw display
$19$ \( (T^{8} - 3 T^{7} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 849346560000 \) Copy content Toggle raw display
$29$ \( (T + 4)^{16} \) Copy content Toggle raw display
$31$ \( (T^{8} + 9 T^{7} + \cdots + 589824)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 357040905841 \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots - 200)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 129 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 536902045696 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 1529041063936 \) Copy content Toggle raw display
$59$ \( (T^{8} - 10 T^{7} + \cdots + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 25)^{8} \) Copy content Toggle raw display
$67$ \( T^{16} - 35 T^{14} + \cdots + 12960000 \) Copy content Toggle raw display
$71$ \( (T^{4} - 22 T^{3} + \cdots - 12736)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 6029426229121 \) Copy content Toggle raw display
$79$ \( (T^{8} - 8 T^{7} + \cdots + 23409)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 228 T^{6} + \cdots + 97344)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 10 T^{7} + \cdots + 576)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 260 T^{6} + \cdots + 6241)^{2} \) Copy content Toggle raw display
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