↑

Properties

Label 525.2.bg.b
Level $525$
Weight $2$
Character orbit 525.bg
Analytic conductor $4.192$
Analytic rank $0$
Dimension $160$
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] = [525,2,Mod(16,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([0, 6, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.bg (of order \(15\), degree \(8\), 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: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{15})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{2} + 20 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 30 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 2 q^{2} + 20 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{7} - 30 q^{8} + 20 q^{9} - 6 q^{10} + 12 q^{11} + 20 q^{12} + 12 q^{14} + 4 q^{15} + 12 q^{16} - 6 q^{17} - 8 q^{18} + 12 q^{19} - 7 q^{21} + 14 q^{23} - 30 q^{24} - 4 q^{25} - 56 q^{26} - 40 q^{27} - 27 q^{28} - 2 q^{29} + 4 q^{30} + 27 q^{31} - 78 q^{32} - 8 q^{33} + 76 q^{34} + 21 q^{35} - 40 q^{36} + 14 q^{37} - 17 q^{38} + 10 q^{40} - 6 q^{41} - 2 q^{42} + 28 q^{43} - 22 q^{44} + 3 q^{45} - 22 q^{46} + 7 q^{47} - 24 q^{48} + 152 q^{49} - 62 q^{50} + 4 q^{51} + 13 q^{52} - 26 q^{53} + 2 q^{54} + 64 q^{55} - 33 q^{56} + 56 q^{57} + 48 q^{59} + 15 q^{60} + 36 q^{61} - 88 q^{62} - 7 q^{63} - 2 q^{64} + 46 q^{65} + 15 q^{66} + 42 q^{67} - 28 q^{68} - 8 q^{69} - 32 q^{70} + 36 q^{71} + 15 q^{72} + 10 q^{73} - 16 q^{74} + q^{75} - 76 q^{76} + 12 q^{77} - 8 q^{78} - 10 q^{79} + 100 q^{80} + 20 q^{81} + 14 q^{82} - 74 q^{83} - 6 q^{84} + 90 q^{85} + 6 q^{86} - 4 q^{87} + 102 q^{88} + 33 q^{89} - 18 q^{90} - 31 q^{91} - 94 q^{92} - 98 q^{93} - 30 q^{94} - 106 q^{95} + 47 q^{96} - 26 q^{97} + 26 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.43210 + 1.08284i 0.669131 − 0.743145i 3.40432 − 3.78088i −1.23217 + 1.86595i −0.822687 + 2.53197i 1.91069 − 1.83010i −2.54019 + 7.81792i −0.104528 − 0.994522i 0.976241 − 5.87242i
16.2 −2.19023 + 0.975155i 0.669131 − 0.743145i 2.50794 − 2.78535i −0.992732 − 2.00362i −0.740871 + 2.28017i 2.55323 + 0.693547i −1.29508 + 3.98585i −0.104528 − 0.994522i 4.12815 + 3.42032i
16.3 −2.04004 + 0.908286i 0.669131 − 0.743145i 1.99854 − 2.21960i 1.97582 − 1.04697i −0.690068 + 2.12381i −0.0626701 − 2.64501i −0.680936 + 2.09571i −0.104528 − 0.994522i −3.07982 + 3.93047i
16.4 −1.72477 + 0.767919i 0.669131 − 0.743145i 1.04689 − 1.16268i −2.14015 + 0.647903i −0.583424 + 1.79560i −2.28147 + 1.33974i 0.254054 − 0.781898i −0.104528 − 0.994522i 3.19373 − 2.76095i
16.5 −1.53240 + 0.682267i 0.669131 − 0.743145i 0.544489 − 0.604717i 1.08380 + 1.95586i −0.518350 + 1.59532i −2.12083 − 1.58180i 0.614905 − 1.89248i −0.104528 − 0.994522i −2.99523 − 2.25771i
16.6 −1.19605 + 0.532515i 0.669131 − 0.743145i −0.191302 + 0.212462i 1.53481 + 1.62614i −0.404577 + 1.24516i 2.55996 + 0.668275i 0.924820 − 2.84630i −0.104528 − 0.994522i −2.70166 − 1.12764i
16.7 −0.989117 + 0.440383i 0.669131 − 0.743145i −0.553846 + 0.615109i −0.228741 − 2.22434i −0.334580 + 1.02973i −1.49334 + 2.18402i 0.946095 − 2.91178i −0.104528 − 0.994522i 1.20581 + 2.09940i
16.8 −0.504477 + 0.224608i 0.669131 − 0.743145i −1.13421 + 1.25967i −1.89312 + 1.19000i −0.170645 + 0.525192i 0.160877 − 2.64086i 0.630543 − 1.94061i −0.104528 − 0.994522i 0.687754 − 1.02554i
16.9 −0.308545 + 0.137373i 0.669131 − 0.743145i −1.26193 + 1.40152i −2.19353 − 0.434088i −0.104369 + 0.321214i 2.54788 + 0.712971i 0.405570 − 1.24822i −0.104528 − 0.994522i 0.736435 − 0.167396i
16.10 −0.195464 + 0.0870263i 0.669131 − 0.743145i −1.30763 + 1.45227i 2.23437 − 0.0871077i −0.0661180 + 0.203490i −2.62302 − 0.346096i 0.261445 − 0.804645i −0.104528 − 0.994522i −0.429159 + 0.211475i
16.11 0.340400 − 0.151556i 0.669131 − 0.743145i −1.24536 + 1.38311i 2.23604 − 0.0110432i 0.115144 − 0.354377i 0.0639980 + 2.64498i −0.444590 + 1.36831i −0.104528 − 0.994522i 0.759475 − 0.342644i
16.12 0.811470 − 0.361290i 0.669131 − 0.743145i −0.810308 + 0.899939i −1.71756 − 1.43178i 0.274489 − 0.844789i −1.96006 − 1.77712i −0.881380 + 2.71261i −0.104528 − 0.994522i −1.91103 − 0.541305i
16.13 0.845870 − 0.376606i 0.669131 − 0.743145i −0.764597 + 0.849171i 0.707816 − 2.12108i 0.286125 − 0.880602i 2.58582 − 0.559924i −0.899197 + 2.76744i −0.104528 − 0.994522i −0.200092 − 2.06073i
16.14 1.05172 − 0.468254i 0.669131 − 0.743145i −0.451416 + 0.501348i −1.26593 + 1.84321i 0.355755 − 1.09490i −2.58874 + 0.546295i −0.951513 + 2.92846i −0.104528 − 0.994522i −0.468306 + 2.53131i
16.15 1.23117 − 0.548154i 0.669131 − 0.743145i −0.122944 + 0.136543i −0.120990 + 2.23279i 0.416458 − 1.28173i 2.20566 − 1.46119i −0.909436 + 2.79896i −0.104528 − 0.994522i 1.07495 + 2.81528i
16.16 1.88139 − 0.837651i 0.669131 − 0.743145i 1.49973 − 1.66561i 2.15442 − 0.598733i 0.636403 − 1.95865i −0.705516 − 2.54995i 0.153568 − 0.472633i −0.104528 − 0.994522i 3.55178 − 2.93110i
16.17 1.90891 − 0.849901i 0.669131 − 0.743145i 1.58334 − 1.75848i 1.75701 + 1.38308i 0.645710 − 1.98729i −1.00807 + 2.44618i 0.236500 − 0.727873i −0.104528 − 0.994522i 4.52946 + 1.14688i
16.18 2.10352 − 0.936548i 0.669131 − 0.743145i 2.20942 − 2.45381i −0.282304 − 2.21818i 0.711540 − 2.18990i 0.729324 + 2.54324i 0.926375 − 2.85109i −0.104528 − 0.994522i −2.67126 − 4.40159i
16.19 2.34261 − 1.04299i 0.669131 − 0.743145i 3.06170 − 3.40036i −1.80799 − 1.31574i 0.792412 − 2.43879i −2.61898 − 0.375455i 2.04097 − 6.28145i −0.104528 − 0.994522i −5.60772 − 1.19654i
16.20 2.42324 − 1.07889i 0.669131 − 0.743145i 3.36980 − 3.74254i −0.617899 + 2.14900i 0.819687 − 2.52274i 2.64524 + 0.0517719i 2.48864 − 7.65924i −0.104528 − 0.994522i 0.821228 + 5.87418i
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
25.d even 5 1 inner
175.q even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.2.bg.b ✓ 160
7.c even 3 1 inner 525.2.bg.b ✓ 160
25.d even 5 1 inner 525.2.bg.b ✓ 160
175.q even 15 1 inner 525.2.bg.b ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
525.2.bg.b ✓ 160 1.a even 1 1 trivial
525.2.bg.b ✓ 160 7.c even 3 1 inner
525.2.bg.b ✓ 160 25.d even 5 1 inner
525.2.bg.b ✓ 160 175.q even 15 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 2 T_{2}^{159} - 28 T_{2}^{158} + 74 T_{2}^{157} + 287 T_{2}^{156} - 1048 T_{2}^{155} + \cdots + 28228234650625 \) acting on \(S_{2}^{\mathrm{new}}(525, [\chi])\). Copy content Toggle raw display