Properties

Label 231.2.j.e
Level 231
Weight 2
Character orbit 231.j
Analytic conductor 1.845
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 231 = 3 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 231.j (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.84454428669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} -3 \zeta_{10}^{3} q^{4} + ( -1 - \zeta_{10}^{2} ) q^{5} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} +O(q^{10})\) \( q + ( 1 + \zeta_{10} - \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{2} -\zeta_{10}^{2} q^{3} -3 \zeta_{10}^{3} q^{4} + ( -1 - \zeta_{10}^{2} ) q^{5} + ( -2 + \zeta_{10} - 2 \zeta_{10}^{2} ) q^{6} -\zeta_{10}^{3} q^{7} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{8} + ( -1 + \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{9} + ( -3 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{10} + ( 1 - \zeta_{10} + 3 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{11} -3 q^{12} + ( 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{13} + ( -2 \zeta_{10} + \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{14} + ( -1 + \zeta_{10} + \zeta_{10}^{3} ) q^{15} + \zeta_{10} q^{16} + ( 3 - 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{17} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{18} + ( -3 \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{19} + ( -3 + 3 \zeta_{10}^{3} ) q^{20} - q^{21} + ( 6 + 2 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{22} + ( 6 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{23} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{24} + ( \zeta_{10} - 4 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{25} + ( -2 + 2 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{26} + \zeta_{10} q^{27} -3 \zeta_{10} q^{28} + 6 \zeta_{10}^{3} q^{29} + ( \zeta_{10} + 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{30} + ( -5 + 5 \zeta_{10}^{3} ) q^{31} + ( 3 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{32} + ( 4 - 3 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{33} + ( 6 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{34} + ( -1 + \zeta_{10}^{3} ) q^{35} + 3 \zeta_{10}^{2} q^{36} + ( -1 + \zeta_{10} - 3 \zeta_{10}^{3} ) q^{37} + ( -1 - 7 \zeta_{10} - \zeta_{10}^{2} ) q^{38} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{39} + ( -1 + \zeta_{10} + 3 \zeta_{10}^{3} ) q^{40} + ( \zeta_{10} + 8 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{41} + ( -1 - \zeta_{10} + \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{42} + ( -6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{43} + ( 6 + 6 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{44} + ( 1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{45} + ( 8 + 3 \zeta_{10} - 3 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{46} + ( -4 \zeta_{10} + 2 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{47} -\zeta_{10}^{3} q^{48} -\zeta_{10} q^{49} + ( -7 + 6 \zeta_{10} - 7 \zeta_{10}^{2} ) q^{50} + ( 3 - 3 \zeta_{10} ) q^{51} + ( -6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{52} + ( -10 + 8 \zeta_{10} - 8 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{53} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{54} + ( 3 - 2 \zeta_{10} - \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{55} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{56} + ( -2 - \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{57} + ( 12 \zeta_{10} - 6 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{58} + ( -2 + 2 \zeta_{10} ) q^{59} + ( 3 + 3 \zeta_{10}^{2} ) q^{60} + ( -5 + 5 \zeta_{10} + 15 \zeta_{10}^{3} ) q^{62} + \zeta_{10}^{2} q^{63} + ( 13 - 13 \zeta_{10} + 13 \zeta_{10}^{2} - 13 \zeta_{10}^{3} ) q^{64} -2 q^{65} + ( 5 - 2 \zeta_{10} - 4 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{66} + ( 9 \zeta_{10} - 9 \zeta_{10}^{2} ) q^{68} + ( -\zeta_{10} - 5 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{69} + ( -1 + \zeta_{10} + 3 \zeta_{10}^{3} ) q^{70} + ( -8 - 8 \zeta_{10}^{2} ) q^{71} + ( 2 - \zeta_{10} + 2 \zeta_{10}^{2} ) q^{72} + ( -6 + 6 \zeta_{10} - 4 \zeta_{10}^{3} ) q^{73} + ( -7 \zeta_{10} + 6 \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{74} + ( -3 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{75} + ( -6 - 9 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{76} + ( 2 + 2 \zeta_{10} - \zeta_{10}^{2} ) q^{77} + ( -4 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{78} + ( 10 - 10 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{79} + ( -\zeta_{10} - \zeta_{10}^{3} ) q^{80} -\zeta_{10}^{3} q^{81} + ( 17 - 6 \zeta_{10} + 17 \zeta_{10}^{2} ) q^{82} + ( 4 - 14 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{83} + 3 \zeta_{10}^{3} q^{84} -3 \zeta_{10}^{2} q^{85} + ( -12 + 18 \zeta_{10} - 18 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{86} + 6 q^{87} + ( 2 + 3 \zeta_{10} - 6 \zeta_{10}^{3} ) q^{88} + ( 4 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{89} + ( 3 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{90} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{91} + ( 3 - 3 \zeta_{10} - 18 \zeta_{10}^{3} ) q^{92} + ( 5 + 5 \zeta_{10}^{2} ) q^{93} -10 \zeta_{10} q^{94} + ( -2 + 2 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{95} + ( -6 \zeta_{10} + 3 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{96} + ( 6 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{97} + ( -1 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{98} + ( -2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 5q^{2} + q^{3} - 3q^{4} - 3q^{5} - 5q^{6} - q^{7} - 5q^{8} - q^{9} + O(q^{10}) \) \( 4q + 5q^{2} + q^{3} - 3q^{4} - 3q^{5} - 5q^{6} - q^{7} - 5q^{8} - q^{9} - 10q^{10} + q^{11} - 12q^{12} + 4q^{13} - 5q^{14} - 2q^{15} + q^{16} + 6q^{17} - 5q^{18} - 7q^{19} - 9q^{20} - 4q^{21} + 25q^{22} + 22q^{23} - 5q^{24} + 6q^{25} - 10q^{26} + q^{27} - 3q^{28} + 6q^{29} - 15q^{31} + 9q^{33} + 30q^{34} - 3q^{35} - 3q^{36} - 6q^{37} - 10q^{38} - 4q^{39} - 6q^{41} - 5q^{42} + 12q^{43} + 33q^{44} + 2q^{45} + 30q^{46} - 10q^{47} - q^{48} - q^{49} - 15q^{50} + 9q^{51} - 18q^{52} - 14q^{53} + 8q^{55} - 8q^{57} + 30q^{58} - 6q^{59} + 9q^{60} - q^{63} + 13q^{64} - 8q^{65} + 20q^{66} + 18q^{68} + 3q^{69} - 24q^{71} + 5q^{72} - 22q^{73} - 20q^{74} - q^{75} - 6q^{76} + 11q^{77} - 20q^{78} + 10q^{79} - 2q^{80} - q^{81} + 45q^{82} - 2q^{83} + 3q^{84} + 3q^{85} + 24q^{87} + 5q^{88} + 18q^{89} + 5q^{90} - 6q^{91} - 9q^{92} + 15q^{93} - 10q^{94} - q^{95} - 15q^{96} + 6q^{97} + q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(211\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{10}^{3}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
64.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
0.690983 2.12663i 0.809017 0.587785i −2.42705 1.76336i −0.190983 0.587785i −0.690983 2.12663i −0.809017 0.587785i −1.80902 + 1.31433i 0.309017 0.951057i −1.38197
148.1 0.690983 + 2.12663i 0.809017 + 0.587785i −2.42705 + 1.76336i −0.190983 + 0.587785i −0.690983 + 2.12663i −0.809017 + 0.587785i −1.80902 1.31433i 0.309017 + 0.951057i −1.38197
169.1 1.80902 + 1.31433i −0.309017 + 0.951057i 0.927051 + 2.85317i −1.30902 + 0.951057i −1.80902 + 1.31433i 0.309017 + 0.951057i −0.690983 + 2.12663i −0.809017 0.587785i −3.61803
190.1 1.80902 1.31433i −0.309017 0.951057i 0.927051 2.85317i −1.30902 0.951057i −1.80902 1.31433i 0.309017 0.951057i −0.690983 2.12663i −0.809017 + 0.587785i −3.61803
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 231.2.j.e 4
3.b odd 2 1 693.2.m.a 4
11.c even 5 1 inner 231.2.j.e 4
11.c even 5 1 2541.2.a.w 2
11.d odd 10 1 2541.2.a.v 2
33.f even 10 1 7623.2.a.bj 2
33.h odd 10 1 693.2.m.a 4
33.h odd 10 1 7623.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.j.e 4 1.a even 1 1 trivial
231.2.j.e 4 11.c even 5 1 inner
693.2.m.a 4 3.b odd 2 1
693.2.m.a 4 33.h odd 10 1
2541.2.a.v 2 11.d odd 10 1
2541.2.a.w 2 11.c even 5 1
7623.2.a.bj 2 33.f even 10 1
7623.2.a.bk 2 33.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 5 T_{2}^{3} + 15 T_{2}^{2} - 25 T_{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(231, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 5 T + 13 T^{2} - 25 T^{3} + 39 T^{4} - 50 T^{5} + 52 T^{6} - 40 T^{7} + 16 T^{8} \)
$3$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
$5$ \( 1 + 3 T - T^{2} - 3 T^{3} + 16 T^{4} - 15 T^{5} - 25 T^{6} + 375 T^{7} + 625 T^{8} \)
$7$ \( 1 + T + T^{2} + T^{3} + T^{4} \)
$11$ \( 1 - T + 21 T^{2} - 11 T^{3} + 121 T^{4} \)
$13$ \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 650 T^{5} + 507 T^{6} - 8788 T^{7} + 28561 T^{8} \)
$17$ \( 1 - 6 T + 19 T^{2} - 132 T^{3} + 829 T^{4} - 2244 T^{5} + 5491 T^{6} - 29478 T^{7} + 83521 T^{8} \)
$19$ \( 1 + 7 T + 15 T^{2} + 107 T^{3} + 824 T^{4} + 2033 T^{5} + 5415 T^{6} + 48013 T^{7} + 130321 T^{8} \)
$23$ \( ( 1 - 11 T + 75 T^{2} - 253 T^{3} + 529 T^{4} )^{2} \)
$29$ \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 3828 T^{5} + 5887 T^{6} - 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 15 T + 69 T^{2} + 95 T^{3} + 36 T^{4} + 2945 T^{5} + 66309 T^{6} + 446865 T^{7} + 923521 T^{8} \)
$37$ \( 1 + 6 T - 21 T^{2} - 248 T^{3} - 471 T^{4} - 9176 T^{5} - 28749 T^{6} + 303918 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 6 T + 35 T^{2} + 84 T^{3} - 371 T^{4} + 3444 T^{5} + 58835 T^{6} + 413526 T^{7} + 2825761 T^{8} \)
$43$ \( ( 1 - 6 T + 50 T^{2} - 258 T^{3} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 10 T + 13 T^{2} + 200 T^{3} + 3549 T^{4} + 9400 T^{5} + 28717 T^{6} + 1038230 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 14 T + 43 T^{2} - 650 T^{3} - 8799 T^{4} - 34450 T^{5} + 120787 T^{6} + 2084278 T^{7} + 7890481 T^{8} \)
$59$ \( 1 + 6 T - 43 T^{2} - 102 T^{3} + 3025 T^{4} - 6018 T^{5} - 149683 T^{6} + 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 61 T^{2} + 3721 T^{4} - 226981 T^{6} + 13845841 T^{8} \)
$67$ \( ( 1 + 67 T^{2} )^{4} \)
$71$ \( 1 + 24 T + 185 T^{2} + 456 T^{3} + 49 T^{4} + 32376 T^{5} + 932585 T^{6} + 8589864 T^{7} + 25411681 T^{8} \)
$73$ \( 1 + 22 T + 111 T^{2} - 1054 T^{3} - 17431 T^{4} - 76942 T^{5} + 591519 T^{6} + 8558374 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 45820 T^{5} + 131061 T^{6} - 4930390 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 2 T + 121 T^{2} + 736 T^{3} + 7989 T^{4} + 61088 T^{5} + 833569 T^{6} + 1143574 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 9 T + 197 T^{2} - 801 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( 1 - 6 T - 61 T^{2} + 948 T^{3} + 229 T^{4} + 91956 T^{5} - 573949 T^{6} - 5476038 T^{7} + 88529281 T^{8} \)
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