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Properties

Label 570.2.k.b
Level $570$
Weight $2$
Character orbit 570.k
Analytic conductor $4.551$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 570.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.55147291521\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36q + 4q^{3} + 4q^{6} + 20q^{7} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36q + 4q^{3} + 4q^{6} + 20q^{7} - 4q^{10} - 4q^{12} + 8q^{13} + 4q^{15} - 36q^{16} + 16q^{21} - 4q^{22} + 16q^{25} - 44q^{27} + 20q^{28} + 32q^{30} - 24q^{31} - 4q^{33} + 4q^{36} - 8q^{40} + 12q^{42} - 8q^{43} + 28q^{45} - 16q^{46} - 4q^{48} + 40q^{51} - 8q^{52} - 36q^{55} - 4q^{57} + 44q^{58} + 16q^{60} - 120q^{61} - 12q^{63} + 80q^{67} - 36q^{70} + 44q^{73} + 4q^{75} - 36q^{76} - 64q^{78} + 36q^{81} + 8q^{82} - 24q^{85} - 28q^{87} - 4q^{88} + 44q^{90} - 72q^{93} - 4q^{96} + 92q^{97} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 −0.707107 + 0.707107i −1.69896 − 0.336929i − 1.00000i 2.14526 − 0.630765i 1.43959 − 0.963104i 0.804978 + 0.804978i 0.707107 + 0.707107i 2.77296 + 1.14486i −1.07091 + 1.96295i
77.2 −0.707107 + 0.707107i −1.63140 + 0.581841i − 1.00000i −1.36872 + 1.76822i 0.742149 − 1.56500i 0.102829 + 0.102829i 0.707107 + 0.707107i 2.32292 − 1.89843i −0.282491 − 2.21815i
77.3 −0.707107 + 0.707107i −1.32879 − 1.11099i − 1.00000i −1.28568 + 1.82949i 1.72519 − 0.154007i 1.30605 + 1.30605i 0.707107 + 0.707107i 0.531382 + 2.95256i −0.384532 − 2.20276i
77.4 −0.707107 + 0.707107i −0.0916341 − 1.72963i − 1.00000i −2.17182 − 0.532166i 1.28782 + 1.15823i 1.32508 + 1.32508i 0.707107 + 0.707107i −2.98321 + 0.316985i 1.91201 − 1.15941i
77.5 −0.707107 + 0.707107i 0.492817 + 1.66046i − 1.00000i −1.86278 + 1.23695i −1.52260 − 0.825649i −3.33681 − 3.33681i 0.707107 + 0.707107i −2.51426 + 1.63661i 0.442526 − 2.19184i
77.6 −0.707107 + 0.707107i 0.577604 + 1.63290i − 1.00000i 2.23603 − 0.0135681i −1.56306 − 0.746209i 3.38538 + 3.38538i 0.707107 + 0.707107i −2.33275 + 1.88634i −1.57152 + 1.59070i
77.7 −0.707107 + 0.707107i 0.901679 − 1.47884i − 1.00000i 1.61979 − 1.54152i 0.408116 + 1.68328i −1.95232 − 1.95232i 0.707107 + 0.707107i −1.37395 − 2.66688i −0.0553424 + 2.23538i
77.8 −0.707107 + 0.707107i 1.34719 − 1.08861i − 1.00000i −0.539236 − 2.17007i −0.182842 + 1.72237i 2.86820 + 2.86820i 0.707107 + 0.707107i 0.629843 − 2.93314i 1.91577 + 1.15318i
77.9 −0.707107 + 0.707107i 1.72439 + 0.162692i − 1.00000i 0.520052 + 2.17475i −1.33437 + 1.10429i 0.496606 + 0.496606i 0.707107 + 0.707107i 2.94706 + 0.561090i −1.90551 − 1.17005i
77.10 0.707107 − 0.707107i −1.66046 − 0.492817i − 1.00000i 1.86278 − 1.23695i −1.52260 + 0.825649i −3.33681 − 3.33681i −0.707107 − 0.707107i 2.51426 + 1.63661i 0.442526 − 2.19184i
77.11 0.707107 − 0.707107i −1.63290 − 0.577604i − 1.00000i −2.23603 + 0.0135681i −1.56306 + 0.746209i 3.38538 + 3.38538i −0.707107 − 0.707107i 2.33275 + 1.88634i −1.57152 + 1.59070i
77.12 0.707107 − 0.707107i −0.581841 + 1.63140i − 1.00000i 1.36872 − 1.76822i 0.742149 + 1.56500i 0.102829 + 0.102829i −0.707107 − 0.707107i −2.32292 − 1.89843i −0.282491 − 2.21815i
77.13 0.707107 − 0.707107i −0.162692 − 1.72439i − 1.00000i −0.520052 − 2.17475i −1.33437 − 1.10429i 0.496606 + 0.496606i −0.707107 − 0.707107i −2.94706 + 0.561090i −1.90551 − 1.17005i
77.14 0.707107 − 0.707107i 0.336929 + 1.69896i − 1.00000i −2.14526 + 0.630765i 1.43959 + 0.963104i 0.804978 + 0.804978i −0.707107 − 0.707107i −2.77296 + 1.14486i −1.07091 + 1.96295i
77.15 0.707107 − 0.707107i 1.08861 − 1.34719i − 1.00000i 0.539236 + 2.17007i −0.182842 − 1.72237i 2.86820 + 2.86820i −0.707107 − 0.707107i −0.629843 − 2.93314i 1.91577 + 1.15318i
77.16 0.707107 − 0.707107i 1.11099 + 1.32879i − 1.00000i 1.28568 − 1.82949i 1.72519 + 0.154007i 1.30605 + 1.30605i −0.707107 − 0.707107i −0.531382 + 2.95256i −0.384532 − 2.20276i
77.17 0.707107 − 0.707107i 1.47884 − 0.901679i − 1.00000i −1.61979 + 1.54152i 0.408116 − 1.68328i −1.95232 − 1.95232i −0.707107 − 0.707107i 1.37395 − 2.66688i −0.0553424 + 2.23538i
77.18 0.707107 − 0.707107i 1.72963 + 0.0916341i − 1.00000i 2.17182 + 0.532166i 1.28782 − 1.15823i 1.32508 + 1.32508i −0.707107 − 0.707107i 2.98321 + 0.316985i 1.91201 − 1.15941i
533.1 −0.707107 − 0.707107i −1.69896 + 0.336929i 1.00000i 2.14526 + 0.630765i 1.43959 + 0.963104i 0.804978 − 0.804978i 0.707107 − 0.707107i 2.77296 − 1.14486i −1.07091 − 1.96295i
533.2 −0.707107 − 0.707107i −1.63140 − 0.581841i 1.00000i −1.36872 − 1.76822i 0.742149 + 1.56500i 0.102829 − 0.102829i 0.707107 − 0.707107i 2.32292 + 1.89843i −0.282491 + 2.21815i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 533.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 570.2.k.b ✓ 36
3.b odd 2 1 inner 570.2.k.b ✓ 36
5.c odd 4 1 inner 570.2.k.b ✓ 36
15.e even 4 1 inner 570.2.k.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.k.b ✓ 36 1.a even 1 1 trivial
570.2.k.b ✓ 36 3.b odd 2 1 inner
570.2.k.b ✓ 36 5.c odd 4 1 inner
570.2.k.b ✓ 36 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{18} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).