Properties

Label 231.2.be.a
Level 231
Weight 2
Character orbit 231.be
Analytic conductor 1.845
Analytic rank 0
Dimension 224
CM no
Inner twists 8

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Newspace parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.be (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(224\)
Relative dimension: \(28\) over \(\Q(\zeta_{30})\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 224q - 3q^{3} + 18q^{4} - 20q^{6} - 20q^{7} - 9q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 224q - 3q^{3} + 18q^{4} - 20q^{6} - 20q^{7} - 9q^{9} - 16q^{12} - 40q^{13} + 12q^{15} + 34q^{16} - 5q^{18} - 10q^{19} - 76q^{22} - 25q^{24} - 18q^{25} + 6q^{27} + 10q^{28} - 35q^{30} - 8q^{33} - 96q^{34} - 48q^{36} - 10q^{37} - 45q^{39} - 120q^{40} + 34q^{42} - 24q^{45} - 50q^{46} + 14q^{48} - 56q^{49} - 45q^{51} - 10q^{52} + 48q^{55} + 60q^{57} + 44q^{58} - 47q^{60} - 50q^{61} + 60q^{63} - 72q^{64} + 77q^{66} - 80q^{67} + 78q^{69} + 36q^{70} + 55q^{72} - 70q^{73} - 11q^{75} + 36q^{78} - 90q^{79} + 23q^{81} - 6q^{82} + 125q^{84} + 160q^{85} + 86q^{88} + 30q^{90} - 128q^{91} - 38q^{93} - 210q^{94} + 135q^{96} + 40q^{97} + 80q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.69754 1.88531i −1.50120 0.863943i −0.463691 + 4.41173i 0.0606689 0.285425i 0.919548 + 4.29680i 0.885230 + 2.49326i 4.99976 3.63254i 1.50720 + 2.59390i −0.641101 + 0.370140i
2.2 −1.65113 1.83377i 1.72117 0.193857i −0.427411 + 4.06654i −0.660086 + 3.10546i −3.19736 2.83614i −2.64526 0.0509795i 4.17018 3.02981i 2.92484 0.667320i 6.78458 3.91708i
2.3 −1.62295 1.80247i 0.590856 1.62816i −0.405869 + 3.86159i 0.153087 0.720219i −3.89363 + 1.57742i 1.64166 2.07484i 3.69461 2.68429i −2.30178 1.92401i −1.54662 + 0.892944i
2.4 −1.41473 1.57122i −1.63740 + 0.564721i −0.258206 + 2.45666i −0.705701 + 3.32006i 3.20379 + 1.77379i −0.181035 2.63955i 0.804266 0.584333i 2.36218 1.84935i 6.21493 3.58819i
2.5 −1.36136 1.51195i 0.270437 + 1.71081i −0.223616 + 2.12757i −0.0600850 + 0.282678i 2.21849 2.73792i −2.04626 + 1.67715i 0.229262 0.166568i −2.85373 + 0.925333i 0.509191 0.293981i
2.6 −1.06973 1.18806i −0.998701 + 1.41513i −0.0580986 + 0.552771i 0.161799 0.761203i 2.74960 0.327297i 1.95474 + 1.78298i −1.86786 + 1.35708i −1.00519 2.82659i −1.07744 + 0.622058i
2.7 −1.06933 1.18762i 1.50618 + 0.855229i −0.0578990 + 0.550872i 0.711086 3.34540i −0.594929 2.70329i −0.575806 2.58233i −1.86963 + 1.35837i 1.53717 + 2.57626i −4.73343 + 2.73285i
2.8 −1.03170 1.14582i 0.391245 1.68728i −0.0394378 + 0.375225i 0.582427 2.74010i −2.33696 + 1.29247i −2.49909 + 0.868634i −2.02413 + 1.47062i −2.69386 1.32028i −3.74054 + 2.15960i
2.9 −0.830242 0.922077i −0.464723 1.66854i 0.0481326 0.457951i −0.859357 + 4.04296i −1.15269 + 1.81380i −0.152025 + 2.64138i −2.46985 + 1.79445i −2.56807 + 1.55082i 4.44139 2.56424i
2.10 −0.675892 0.750655i 1.68081 0.418192i 0.102405 0.974320i −0.347823 + 1.63638i −1.44996 0.979053i 2.62714 0.313255i −2.43498 + 1.76911i 2.65023 1.40580i 1.46344 0.844920i
2.11 −0.590583 0.655909i −1.48384 0.893427i 0.127629 1.21431i 0.627181 2.95066i 0.290326 + 1.50091i 2.53863 0.745226i −2.29995 + 1.67101i 1.40358 + 2.65141i −2.30576 + 1.33123i
2.12 −0.475341 0.527920i −1.72047 + 0.199922i 0.156307 1.48716i −0.0305586 + 0.143767i 0.923355 + 0.813242i −2.63481 0.240370i −2.00883 + 1.45950i 2.92006 0.687921i 0.0904232 0.0522059i
2.13 −0.109613 0.121738i −0.460773 + 1.66964i 0.206252 1.96236i 0.280949 1.32176i 0.253764 0.126921i −0.638360 2.56759i −0.526557 + 0.382566i −2.57538 1.53865i −0.191704 + 0.110680i
2.14 −0.0572311 0.0635616i 1.72188 0.187403i 0.208292 1.98177i 0.496749 2.33702i −0.110457 0.0987204i −0.857429 + 2.50296i −0.276277 + 0.200727i 2.92976 0.645370i −0.176974 + 0.102176i
2.15 0.0572311 + 0.0635616i 1.01290 + 1.40501i 0.208292 1.98177i −0.496749 + 2.33702i −0.0313351 + 0.144791i −0.857429 + 2.50296i 0.276277 0.200727i −0.948078 + 2.84625i −0.176974 + 0.102176i
2.16 0.109613 + 0.121738i 0.932465 1.45963i 0.206252 1.96236i −0.280949 + 1.32176i 0.279902 0.0464781i −0.638360 2.56759i 0.526557 0.382566i −1.26102 2.72210i −0.191704 + 0.110680i
2.17 0.475341 + 0.527920i −1.00265 1.41234i 0.156307 1.48716i 0.0305586 0.143767i 0.268999 1.20066i −2.63481 0.240370i 2.00883 1.45950i −0.989382 + 2.83216i 0.0904232 0.0522059i
2.18 0.590583 + 0.655909i −1.65683 0.504891i 0.127629 1.21431i −0.627181 + 2.95066i −0.647333 1.38491i 2.53863 0.745226i 2.29995 1.67101i 2.49017 + 1.67304i −2.30576 + 1.33123i
2.19 0.675892 + 0.750655i 0.813902 + 1.52891i 0.102405 0.974320i 0.347823 1.63638i −0.597572 + 1.64434i 2.62714 0.313255i 2.43498 1.76911i −1.67513 + 2.48877i 1.46344 0.844920i
2.20 0.830242 + 0.922077i −1.55093 + 0.771117i 0.0481326 0.457951i 0.859357 4.04296i −1.99867 0.789862i −0.152025 + 2.64138i 2.46985 1.79445i 1.81076 2.39189i 4.44139 2.56424i
See next 80 embeddings (of 224 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 200.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
11.d odd 10 1 inner
21.h odd 6 1 inner
33.f even 10 1 inner
77.o odd 30 1 inner
231.be even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.be.a 224
3.b odd 2 1 inner 231.2.be.a 224
7.c even 3 1 inner 231.2.be.a 224
11.d odd 10 1 inner 231.2.be.a 224
21.h odd 6 1 inner 231.2.be.a 224
33.f even 10 1 inner 231.2.be.a 224
77.o odd 30 1 inner 231.2.be.a 224
231.be even 30 1 inner 231.2.be.a 224
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.be.a 224 1.a even 1 1 trivial
231.2.be.a 224 3.b odd 2 1 inner
231.2.be.a 224 7.c even 3 1 inner
231.2.be.a 224 11.d odd 10 1 inner
231.2.be.a 224 21.h odd 6 1 inner
231.2.be.a 224 33.f even 10 1 inner
231.2.be.a 224 77.o odd 30 1 inner
231.2.be.a 224 231.be even 30 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database