↑

Properties

Label 525.3.h.e
Level $525$
Weight $3$
Character orbit 525.h
Analytic conductor $14.305$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,3,Mod(76,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.76");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 525.h (of order \(2\), degree \(1\), 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: \(14.3052138789\)
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: \( x^{16} + 24x^{14} + 414x^{12} + 3168x^{10} + 17463x^{8} + 51552x^{6} + 106758x^{4} + 50760x^{2} + 19881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: no (minimal twist has level 105)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} - \beta_{2} q^{3} + (\beta_1 + 2) q^{4} + \beta_{4} q^{6} + ( - \beta_{14} + \beta_{5}) q^{7} + ( - \beta_{12} + 2 \beta_{5}) q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} - \beta_{2} q^{3} + (\beta_1 + 2) q^{4} + \beta_{4} q^{6} + ( - \beta_{14} + \beta_{5}) q^{7} + ( - \beta_{12} + 2 \beta_{5}) q^{8} - 3 q^{9} + ( - \beta_{6} - \beta_1 - 3) q^{11} + ( - \beta_{14} - \beta_{13} + \cdots + 2 \beta_{7}) q^{12}+ \cdots + (3 \beta_{6} + 3 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 48 q^{9} - 56 q^{11} + 84 q^{14} + 112 q^{16} - 12 q^{21} + 32 q^{29} - 96 q^{36} - 72 q^{39} - 568 q^{44} - 96 q^{46} + 152 q^{49} + 24 q^{51} + 444 q^{56} + 992 q^{64} - 56 q^{71} + 48 q^{74} - 464 q^{79} + 144 q^{81} - 228 q^{84} - 456 q^{86} - 88 q^{91} + 168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 414x^{12} + 3168x^{10} + 17463x^{8} + 51552x^{6} + 106758x^{4} + 50760x^{2} + 19881 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 150606 \nu^{14} + 3240000 \nu^{12} + 54449647 \nu^{10} + 341708280 \nu^{8} + 1780233858 \nu^{6} + \cdots - 50073186330 ) / 9157890147 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 23045648 \nu^{14} - 538938588 \nu^{12} - 9236338272 \nu^{10} - 67890346046 \nu^{8} + \cdots - 634653260373 ) / 430420836909 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 116525215 \nu^{15} + 1219186695 \nu^{13} + 10866988062 \nu^{11} - 256938610557 \nu^{9} + \cdots - 30168209004228 \nu ) / 5165050042908 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11522824 \nu^{15} + 269469294 \nu^{13} + 4618169136 \nu^{11} + 33945173023 \nu^{9} + \cdots + 962957885550 \nu ) / 430420836909 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11522824 \nu^{15} - 269469294 \nu^{13} - 4618169136 \nu^{11} - 33945173023 \nu^{9} + \cdots - 102116211732 \nu ) / 430420836909 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2974266 \nu^{14} - 66956497 \nu^{12} - 1075307317 \nu^{10} - 6748279080 \nu^{8} + \cdots - 113994462651 ) / 36631560588 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 747294116 \nu^{14} - 16419189733 \nu^{12} - 278201786919 \nu^{10} - 1869866410434 \nu^{8} + \cdots - 18310197050973 ) / 5165050042908 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 760233940 \nu^{14} + 19393223853 \nu^{12} + 337375111941 \nu^{10} + 2767529091714 \nu^{8} + \cdots + 22664233154853 ) / 5165050042908 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2348190 \nu^{14} + 51611341 \nu^{12} + 848957655 \nu^{10} + 5327782200 \nu^{8} + \cdots - 101270058857 ) / 12210520196 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1096330097 \nu^{15} - 1078226154 \nu^{14} + 27462408585 \nu^{13} - 23658957243 \nu^{12} + \cdots + 76243562029539 ) / 10330100085816 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 350442240 \nu^{15} - 8632212133 \nu^{13} - 149762165028 \nu^{11} + \cdots - 41004509961708 \nu ) / 2582525021454 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 115228240 \nu^{15} - 2694692940 \nu^{13} - 46181691360 \nu^{11} - 339451730230 \nu^{9} + \cdots - 1021162117320 \nu ) / 430420836909 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3296279751 \nu^{15} - 1233572852 \nu^{14} + 78809212737 \nu^{13} - 28189911095 \nu^{12} + \cdots - 31570293926223 ) / 10330100085816 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 3296279751 \nu^{15} - 1233572852 \nu^{14} - 78809212737 \nu^{13} - 28189911095 \nu^{12} + \cdots - 31570293926223 ) / 10330100085816 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 616151752 \nu^{15} - 14449453031 \nu^{13} - 247634960664 \nu^{11} + \cdots - 5475664344618 \nu ) / 860841673818 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{14} - \beta_{13} - \beta_{8} + 2\beta_{7} - 4\beta_{2} - \beta _1 - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - 10\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 18\beta_{14} + 18\beta_{13} + 2\beta_{9} + 12\beta_{8} - 30\beta_{7} + 2\beta_{6} + 33\beta_{2} - 16\beta _1 - 65 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2 \beta_{15} - 18 \beta_{12} + 6 \beta_{11} + 36 \beta_{10} + 18 \beta_{9} + 125 \beta_{5} - 129 \beta_{4} + \cdots - 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -38\beta_{9} - 42\beta_{6} + 237\beta _1 + 850 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 42 \beta_{15} - 4 \beta_{14} + 4 \beta_{13} - 275 \beta_{12} - 126 \beta_{11} - 558 \beta_{10} + \cdots + 279 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 4140 \beta_{14} - 4140 \beta_{13} + 588 \beta_{9} - 2376 \beta_{8} + 6228 \beta_{7} + 684 \beta_{6} + \cdots - 11889 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -684\beta_{15} + 96\beta_{14} - 96\beta_{13} + 4044\beta_{12} - 24441\beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 60501 \beta_{14} + 60501 \beta_{13} + 8676 \beta_{9} + 34473 \beta_{8} - 90006 \beta_{7} + 10332 \beta_{6} + \cdots - 170190 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10332 \beta_{15} - 1656 \beta_{14} + 1656 \beta_{13} - 58845 \beta_{12} + 30996 \beta_{11} + \cdots - 60501 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -126366\beta_{9} - 151998\beta_{6} + 727248\beta _1 + 2455377 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 151998 \beta_{15} - 25632 \beta_{14} + 25632 \beta_{13} - 853614 \beta_{12} - 455994 \beta_{11} + \cdots + 879246 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 12751803 \beta_{14} - 12751803 \beta_{13} + 1833594 \beta_{9} - 7251021 \beta_{8} + 18860148 \beta_{7} + \cdots - 35518458 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -2214486\beta_{15} + 380892\beta_{14} - 380892\beta_{13} + 12370911\beta_{12} - 73619646\beta_{5} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−1.90302 + 3.29612i
−1.90302 − 3.29612i
−1.15043 + 1.99260i
−1.15043 − 1.99260i
−0.965242 + 1.67185i
−0.965242 − 1.67185i
−0.351196 + 0.608290i
−0.351196 − 0.608290i
0.351196 − 0.608290i
0.351196 + 0.608290i
0.965242 − 1.67185i
0.965242 + 1.67185i
1.15043 − 1.99260i
1.15043 + 1.99260i
1.90302 − 3.29612i
1.90302 + 3.29612i
−3.80604 − 1.73205i 10.4859 0 6.59225i −4.81592 − 5.08005i −24.6856 −3.00000 0
76.2 −3.80604 1.73205i 10.4859 0 − 6.59225i −4.81592 + 5.08005i −24.6856 −3.00000 0
76.3 −2.30086 − 1.73205i 1.29396 0 3.98521i 2.84720 + 6.39480i 6.22623 −3.00000 0
76.4 −2.30086 1.73205i 1.29396 0 − 3.98521i 2.84720 − 6.39480i 6.22623 −3.00000 0
76.5 −1.93048 − 1.73205i −0.273228 0 3.34370i −6.98657 + 0.433408i 8.24940 −3.00000 0
76.6 −1.93048 1.73205i −0.273228 0 − 3.34370i −6.98657 − 0.433408i 8.24940 −3.00000 0
76.7 −0.702393 − 1.73205i −3.50664 0 1.21658i 6.07356 − 3.48021i 5.27261 −3.00000 0
76.8 −0.702393 1.73205i −3.50664 0 − 1.21658i 6.07356 + 3.48021i 5.27261 −3.00000 0
76.9 0.702393 − 1.73205i −3.50664 0 − 1.21658i −6.07356 − 3.48021i −5.27261 −3.00000 0
76.10 0.702393 1.73205i −3.50664 0 1.21658i −6.07356 + 3.48021i −5.27261 −3.00000 0
76.11 1.93048 − 1.73205i −0.273228 0 − 3.34370i 6.98657 + 0.433408i −8.24940 −3.00000 0
76.12 1.93048 1.73205i −0.273228 0 3.34370i 6.98657 − 0.433408i −8.24940 −3.00000 0
76.13 2.30086 − 1.73205i 1.29396 0 − 3.98521i −2.84720 + 6.39480i −6.22623 −3.00000 0
76.14 2.30086 1.73205i 1.29396 0 3.98521i −2.84720 − 6.39480i −6.22623 −3.00000 0
76.15 3.80604 − 1.73205i 10.4859 0 − 6.59225i 4.81592 − 5.08005i 24.6856 −3.00000 0
76.16 3.80604 1.73205i 10.4859 0 6.59225i 4.81592 + 5.08005i 24.6856 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.3.h.e 16
5.b even 2 1 inner 525.3.h.e 16
5.c odd 4 2 105.3.e.a ✓ 16
7.b odd 2 1 inner 525.3.h.e 16
15.e even 4 2 315.3.e.e 16
20.e even 4 2 1680.3.bd.c 16
35.c odd 2 1 inner 525.3.h.e 16
35.f even 4 2 105.3.e.a ✓ 16
105.k odd 4 2 315.3.e.e 16
140.j odd 4 2 1680.3.bd.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.e.a ✓ 16 5.c odd 4 2
105.3.e.a ✓ 16 35.f even 4 2
315.3.e.e 16 15.e even 4 2
315.3.e.e 16 105.k odd 4 2
525.3.h.e 16 1.a even 1 1 trivial
525.3.h.e 16 5.b even 2 1 inner
525.3.h.e 16 7.b odd 2 1 inner
525.3.h.e 16 35.c odd 2 1 inner
1680.3.bd.c 16 20.e even 4 2
1680.3.bd.c 16 140.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 24T_{2}^{6} + 162T_{2}^{4} - 360T_{2}^{2} + 141 \) acting on \(S_{3}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 24 T^{6} + \cdots + 141)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{4} + 14 T^{3} + \cdots + 3280)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + 556 T^{6} + \cdots + 60715264)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1468 T^{6} + \cdots + 4251040000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 396 T^{6} + \cdots + 324864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2904 T^{6} + \cdots + 382942464)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots - 26384)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3840 T^{6} + \cdots + 1892910336)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 3497148290304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 26090305209600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 2508 T^{6} + \cdots + 86666496)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 1152376486144)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 55452408710400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 2149592204544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 9868871097600)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 674847686079744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 14 T^{3} + \cdots - 3270032)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 36785098365184)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 116 T^{3} + \cdots + 28622512)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 26\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 162581970541824)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 338426355591424)^{2} \) Copy content Toggle raw display
show more
show less