Properties

Label 825.2.bi.h
Level $825$
Weight $2$
Character orbit 825.bi
Analytic conductor $6.588$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.bi (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80q - 4q^{4} - 10q^{6} + 10q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q - 4q^{4} - 10q^{6} + 10q^{9} - 60q^{16} + 20q^{19} + 30q^{24} - 20q^{31} + 56q^{34} + 2q^{36} + 50q^{39} - 40q^{46} - 72q^{49} + 30q^{51} - 96q^{64} - 42q^{66} - 30q^{69} - 66q^{81} - 140q^{84} + 48q^{91} - 60q^{94} - 70q^{96} - 60q^{99} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −0.694544 + 2.13759i 0.571289 + 1.63512i −2.46885 1.79372i 0 −3.89200 + 0.0855139i −2.41366 + 3.32212i 1.91228 1.38935i −2.34726 + 1.86826i 0
101.2 −0.694544 + 2.13759i 1.37856 + 1.04861i −2.46885 1.79372i 0 −3.19896 + 2.21848i 2.41366 3.32212i 1.91228 1.38935i 0.800839 + 2.89113i 0
101.3 −0.664572 + 2.04534i −1.37038 + 1.05927i −2.12373 1.54298i 0 −1.25585 3.50686i −1.14456 + 1.57535i 1.08756 0.790156i 0.755895 2.90321i 0
101.4 −0.664572 + 2.04534i 1.43090 0.975979i −2.12373 1.54298i 0 1.04528 + 3.57528i 1.14456 1.57535i 1.08756 0.790156i 1.09493 2.79305i 0
101.5 −0.428747 + 1.31955i −1.68532 0.399638i 0.0606486 + 0.0440638i 0 1.24992 2.05251i −0.527114 + 0.725510i −2.32910 + 1.69219i 2.68058 + 1.34703i 0
101.6 −0.428747 + 1.31955i 0.140712 1.72633i 0.0606486 + 0.0440638i 0 2.21764 + 0.925835i 0.527114 0.725510i −2.32910 + 1.69219i −2.96040 0.485831i 0
101.7 −0.270252 + 0.831751i −1.17137 + 1.27589i 0.999260 + 0.726005i 0 −0.744657 1.31910i 1.92808 2.65378i −2.28897 + 1.66303i −0.255787 2.98908i 0
101.8 −0.270252 + 0.831751i 1.57542 0.719767i 0.999260 + 0.726005i 0 0.172907 + 1.50487i −1.92808 + 2.65378i −2.28897 + 1.66303i 1.96387 2.26786i 0
101.9 −0.0382790 + 0.117811i −1.55555 0.761757i 1.60562 + 1.16655i 0 0.149288 0.154101i −1.86875 + 2.57211i −0.399325 + 0.290127i 1.83945 + 2.36990i 0
101.10 −0.0382790 + 0.117811i −0.243783 1.71481i 1.60562 + 1.16655i 0 0.211354 + 0.0369209i 1.86875 2.57211i −0.399325 + 0.290127i −2.88114 + 0.836083i 0
101.11 0.0382790 0.117811i 0.243783 + 1.71481i 1.60562 + 1.16655i 0 0.211354 + 0.0369209i −1.86875 + 2.57211i 0.399325 0.290127i −2.88114 + 0.836083i 0
101.12 0.0382790 0.117811i 1.55555 + 0.761757i 1.60562 + 1.16655i 0 0.149288 0.154101i 1.86875 2.57211i 0.399325 0.290127i 1.83945 + 2.36990i 0
101.13 0.270252 0.831751i −1.57542 + 0.719767i 0.999260 + 0.726005i 0 0.172907 + 1.50487i 1.92808 2.65378i 2.28897 1.66303i 1.96387 2.26786i 0
101.14 0.270252 0.831751i 1.17137 1.27589i 0.999260 + 0.726005i 0 −0.744657 1.31910i −1.92808 + 2.65378i 2.28897 1.66303i −0.255787 2.98908i 0
101.15 0.428747 1.31955i −0.140712 + 1.72633i 0.0606486 + 0.0440638i 0 2.21764 + 0.925835i −0.527114 + 0.725510i 2.32910 1.69219i −2.96040 0.485831i 0
101.16 0.428747 1.31955i 1.68532 + 0.399638i 0.0606486 + 0.0440638i 0 1.24992 2.05251i 0.527114 0.725510i 2.32910 1.69219i 2.68058 + 1.34703i 0
101.17 0.664572 2.04534i −1.43090 + 0.975979i −2.12373 1.54298i 0 1.04528 + 3.57528i −1.14456 + 1.57535i −1.08756 + 0.790156i 1.09493 2.79305i 0
101.18 0.664572 2.04534i 1.37038 1.05927i −2.12373 1.54298i 0 −1.25585 3.50686i 1.14456 1.57535i −1.08756 + 0.790156i 0.755895 2.90321i 0
101.19 0.694544 2.13759i −1.37856 1.04861i −2.46885 1.79372i 0 −3.19896 + 2.21848i −2.41366 + 3.32212i −1.91228 + 1.38935i 0.800839 + 2.89113i 0
101.20 0.694544 2.13759i −0.571289 1.63512i −2.46885 1.79372i 0 −3.89200 + 0.0855139i 2.41366 3.32212i −1.91228 + 1.38935i −2.34726 + 1.86826i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 776.20
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
11.d odd 10 1 inner
15.d odd 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
165.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.2.bi.h 80
3.b odd 2 1 inner 825.2.bi.h 80
5.b even 2 1 inner 825.2.bi.h 80
5.c odd 4 2 165.2.r.a 80
11.d odd 10 1 inner 825.2.bi.h 80
15.d odd 2 1 inner 825.2.bi.h 80
15.e even 4 2 165.2.r.a 80
33.f even 10 1 inner 825.2.bi.h 80
55.h odd 10 1 inner 825.2.bi.h 80
55.l even 20 2 165.2.r.a 80
165.r even 10 1 inner 825.2.bi.h 80
165.u odd 20 2 165.2.r.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.2.r.a 80 5.c odd 4 2
165.2.r.a 80 15.e even 4 2
165.2.r.a 80 55.l even 20 2
165.2.r.a 80 165.u odd 20 2
825.2.bi.h 80 1.a even 1 1 trivial
825.2.bi.h 80 3.b odd 2 1 inner
825.2.bi.h 80 5.b even 2 1 inner
825.2.bi.h 80 11.d odd 10 1 inner
825.2.bi.h 80 15.d odd 2 1 inner
825.2.bi.h 80 33.f even 10 1 inner
825.2.bi.h 80 55.h odd 10 1 inner
825.2.bi.h 80 165.r even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(825, [\chi])\):

\(T_{2}^{40} + \cdots\)
\(21\!\cdots\!88\)\( T_{7}^{16} - \)\(73\!\cdots\!84\)\( T_{7}^{14} + \)\(20\!\cdots\!81\)\( T_{7}^{12} - \)\(37\!\cdots\!56\)\( T_{7}^{10} + \)\(43\!\cdots\!71\)\( T_{7}^{8} - \)\(33\!\cdots\!05\)\( T_{7}^{6} + \)\(25\!\cdots\!60\)\( T_{7}^{4} - \)\(15\!\cdots\!00\)\( T_{7}^{2} + \)\(50\!\cdots\!25\)\( \)">\(T_{7}^{40} - \cdots\)