Properties

Label 1911.1.ep.a
Level $1911$
Weight $1$
Character orbit 1911.ep
Analytic conductor $0.954$
Analytic rank $0$
Dimension $24$
Projective image $D_{84}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1911.ep (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{84})\)
Defining polynomial: \(x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{84}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{84} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{84}^{13} q^{3} + \zeta_{84}^{39} q^{4} + \zeta_{84}^{37} q^{7} + \zeta_{84}^{26} q^{9} +O(q^{10})\) \( q -\zeta_{84}^{13} q^{3} + \zeta_{84}^{39} q^{4} + \zeta_{84}^{37} q^{7} + \zeta_{84}^{26} q^{9} + \zeta_{84}^{10} q^{12} -\zeta_{84}^{17} q^{13} -\zeta_{84}^{36} q^{16} + ( \zeta_{84}^{9} + \zeta_{84}^{40} ) q^{19} + \zeta_{84}^{8} q^{21} + \zeta_{84}^{5} q^{25} -\zeta_{84}^{39} q^{27} -\zeta_{84}^{34} q^{28} + ( \zeta_{84}^{17} + \zeta_{84}^{32} ) q^{31} -\zeta_{84}^{23} q^{36} + ( -\zeta_{84}^{28} - \zeta_{84}^{41} ) q^{37} + \zeta_{84}^{30} q^{39} + ( \zeta_{84}^{6} - \zeta_{84}^{38} ) q^{43} -\zeta_{84}^{7} q^{48} -\zeta_{84}^{32} q^{49} + \zeta_{84}^{14} q^{52} + ( \zeta_{84}^{11} - \zeta_{84}^{22} ) q^{57} + ( \zeta_{84}^{7} + \zeta_{84}^{27} ) q^{61} -\zeta_{84}^{21} q^{63} + \zeta_{84}^{33} q^{64} + ( \zeta_{84}^{13} + \zeta_{84}^{22} ) q^{67} + ( -\zeta_{84}^{15} - \zeta_{84}^{16} ) q^{73} -\zeta_{84}^{18} q^{75} + ( -\zeta_{84}^{6} - \zeta_{84}^{37} ) q^{76} + ( -\zeta_{84}^{25} - \zeta_{84}^{31} ) q^{79} -\zeta_{84}^{10} q^{81} -\zeta_{84}^{5} q^{84} + \zeta_{84}^{12} q^{91} + ( \zeta_{84}^{3} - \zeta_{84}^{30} ) q^{93} + ( \zeta_{84}^{2} - \zeta_{84}^{33} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{9} + O(q^{10}) \) \( 24 q - 2 q^{9} - 2 q^{12} + 4 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{28} + 2 q^{31} + 12 q^{37} + 4 q^{39} + 6 q^{43} - 2 q^{49} + 12 q^{52} + 2 q^{57} - 2 q^{67} - 2 q^{73} - 4 q^{75} - 4 q^{76} + 2 q^{81} - 4 q^{91} - 4 q^{93} - 2 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(\zeta_{84}^{7}\) \(-\zeta_{84}^{40}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1
0.563320 0.826239i
0.149042 + 0.988831i
−0.149042 0.988831i
−0.149042 + 0.988831i
−0.563320 + 0.826239i
−0.294755 0.955573i
0.930874 0.365341i
0.149042 0.988831i
−0.930874 + 0.365341i
0.680173 + 0.733052i
0.294755 0.955573i
0.294755 + 0.955573i
−0.294755 + 0.955573i
−0.930874 0.365341i
−0.563320 0.826239i
−0.680173 0.733052i
0.563320 + 0.826239i
0.997204 0.0747301i
−0.997204 0.0747301i
0.930874 + 0.365341i
0 −0.997204 + 0.0747301i 0.974928 0.222521i 0 0 −0.149042 + 0.988831i 0 0.988831 0.149042i 0
89.1 0 −0.930874 + 0.365341i 0.433884 0.900969i 0 0 −0.680173 + 0.733052i 0 0.733052 0.680173i 0
236.1 0 0.930874 0.365341i −0.433884 + 0.900969i 0 0 0.680173 0.733052i 0 0.733052 0.680173i 0
332.1 0 0.930874 + 0.365341i −0.433884 0.900969i 0 0 0.680173 + 0.733052i 0 0.733052 + 0.680173i 0
500.1 0 0.997204 0.0747301i −0.974928 + 0.222521i 0 0 0.149042 0.988831i 0 0.988831 0.149042i 0
605.1 0 −0.680173 0.733052i −0.781831 + 0.623490i 0 0 0.997204 0.0747301i 0 −0.0747301 + 0.997204i 0
635.1 0 −0.149042 0.988831i −0.433884 0.900969i 0 0 0.294755 0.955573i 0 −0.955573 + 0.294755i 0
773.1 0 −0.930874 0.365341i 0.433884 + 0.900969i 0 0 −0.680173 0.733052i 0 0.733052 + 0.680173i 0
782.1 0 0.149042 + 0.988831i 0.433884 + 0.900969i 0 0 −0.294755 + 0.955573i 0 −0.955573 + 0.294755i 0
878.1 0 0.294755 + 0.955573i 0.781831 + 0.623490i 0 0 0.563320 0.826239i 0 −0.826239 + 0.563320i 0
908.1 0 0.680173 0.733052i 0.781831 + 0.623490i 0 0 −0.997204 0.0747301i 0 −0.0747301 0.997204i 0
1046.1 0 0.680173 + 0.733052i 0.781831 0.623490i 0 0 −0.997204 + 0.0747301i 0 −0.0747301 + 0.997204i 0
1055.1 0 −0.680173 + 0.733052i −0.781831 0.623490i 0 0 0.997204 + 0.0747301i 0 −0.0747301 0.997204i 0
1151.1 0 0.149042 0.988831i 0.433884 0.900969i 0 0 −0.294755 0.955573i 0 −0.955573 0.294755i 0
1181.1 0 0.997204 + 0.0747301i −0.974928 0.222521i 0 0 0.149042 + 0.988831i 0 0.988831 + 0.149042i 0
1319.1 0 −0.294755 0.955573i −0.781831 0.623490i 0 0 −0.563320 + 0.826239i 0 −0.826239 + 0.563320i 0
1328.1 0 −0.997204 0.0747301i 0.974928 + 0.222521i 0 0 −0.149042 0.988831i 0 0.988831 + 0.149042i 0
1424.1 0 −0.563320 + 0.826239i −0.974928 0.222521i 0 0 −0.930874 0.365341i 0 −0.365341 0.930874i 0
1454.1 0 0.563320 + 0.826239i 0.974928 0.222521i 0 0 0.930874 0.365341i 0 −0.365341 + 0.930874i 0
1592.1 0 −0.149042 + 0.988831i −0.433884 + 0.900969i 0 0 0.294755 + 0.955573i 0 −0.955573 0.294755i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1874.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
637.cd even 84 1 inner
1911.ep odd 84 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1911.1.ep.a yes 24
3.b odd 2 1 CM 1911.1.ep.a yes 24
13.f odd 12 1 1911.1.ea.a 24
39.k even 12 1 1911.1.ea.a 24
49.h odd 42 1 1911.1.ea.a 24
147.o even 42 1 1911.1.ea.a 24
637.cd even 84 1 inner 1911.1.ep.a yes 24
1911.ep odd 84 1 inner 1911.1.ep.a yes 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.1.ea.a 24 13.f odd 12 1
1911.1.ea.a 24 39.k even 12 1
1911.1.ea.a 24 49.h odd 42 1
1911.1.ea.a 24 147.o even 42 1
1911.1.ep.a yes 24 1.a even 1 1 trivial
1911.1.ep.a yes 24 3.b odd 2 1 CM
1911.1.ep.a yes 24 637.cd even 84 1 inner
1911.1.ep.a yes 24 1911.ep odd 84 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1911, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{24} \)
$3$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$5$ \( T^{24} \)
$7$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$11$ \( T^{24} \)
$13$ \( 1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24} \)
$17$ \( T^{24} \)
$19$ \( 1 + 10 T + 200 T^{2} + 548 T^{3} - 28 T^{4} - 266 T^{5} + 692 T^{6} - 2612 T^{7} + 3166 T^{8} - 3368 T^{9} + 2447 T^{10} + 294 T^{11} - 1210 T^{12} + 864 T^{13} - 436 T^{14} - 128 T^{15} + 214 T^{16} - 110 T^{17} + 62 T^{18} + 16 T^{19} - 25 T^{20} + 8 T^{21} - T^{22} - 2 T^{23} + T^{24} \)
$23$ \( T^{24} \)
$29$ \( T^{24} \)
$31$ \( 1 - 8 T + 32 T^{2} - 172 T^{3} + 662 T^{4} - 926 T^{5} + 1016 T^{6} - 1526 T^{7} + 199 T^{8} + 1684 T^{9} - 1222 T^{10} + 1152 T^{11} + 704 T^{12} - 1188 T^{13} + 566 T^{14} - 542 T^{15} + 187 T^{16} + 70 T^{17} - 34 T^{18} + 34 T^{19} - 13 T^{20} - 4 T^{21} + 2 T^{22} - 2 T^{23} + T^{24} \)
$37$ \( 1 + 12 T - 12 T^{2} - 908 T^{3} + 6049 T^{4} - 19418 T^{5} + 41314 T^{6} - 69396 T^{7} + 101369 T^{8} - 132070 T^{9} + 153568 T^{10} - 159978 T^{11} + 149813 T^{12} - 126126 T^{13} + 95328 T^{14} - 64490 T^{15} + 38852 T^{16} - 20694 T^{17} + 9646 T^{18} - 3878 T^{19} + 1317 T^{20} - 366 T^{21} + 79 T^{22} - 12 T^{23} + T^{24} \)
$41$ \( T^{24} \)
$43$ \( ( 1 - T + 3 T^{2} + 23 T^{3} + 25 T^{4} + 8 T^{6} - 7 T^{7} + 2 T^{8} - 9 T^{9} + 6 T^{10} - 3 T^{11} + T^{12} )^{2} \)
$47$ \( T^{24} \)
$53$ \( T^{24} \)
$59$ \( T^{24} \)
$61$ \( 1 + 36 T^{2} + 2114 T^{4} + 6670 T^{6} + 4717 T^{8} - 2744 T^{10} - 657 T^{12} + 126 T^{14} + 104 T^{16} - 50 T^{18} + 21 T^{20} - 6 T^{22} + T^{24} \)
$67$ \( 1 + 8 T + 32 T^{2} + 172 T^{3} + 662 T^{4} + 926 T^{5} + 1016 T^{6} + 1526 T^{7} + 199 T^{8} - 1684 T^{9} - 1222 T^{10} - 1152 T^{11} + 704 T^{12} + 1188 T^{13} + 566 T^{14} + 542 T^{15} + 187 T^{16} - 70 T^{17} - 34 T^{18} - 34 T^{19} - 13 T^{20} + 4 T^{21} + 2 T^{22} + 2 T^{23} + T^{24} \)
$71$ \( T^{24} \)
$73$ \( 1 - 24 T + 172 T^{2} - 58 T^{3} + 1260 T^{4} - 2044 T^{5} + 4003 T^{6} - 3436 T^{7} + 2452 T^{8} - 160 T^{9} - 185 T^{10} - 28 T^{11} + 50 T^{12} + 32 T^{13} + 12 T^{14} + 2 T^{15} - 115 T^{16} - 58 T^{17} + 41 T^{18} + 26 T^{19} - 4 T^{20} - 8 T^{21} - T^{22} + 2 T^{23} + T^{24} \)
$79$ \( ( 49 + 98 T^{2} + 147 T^{4} + 84 T^{6} + 35 T^{8} + 7 T^{10} + T^{12} )^{2} \)
$83$ \( T^{24} \)
$89$ \( T^{24} \)
$97$ \( 1 - 10 T + 200 T^{2} - 548 T^{3} - 28 T^{4} + 266 T^{5} + 692 T^{6} + 2612 T^{7} + 3166 T^{8} + 3368 T^{9} + 2447 T^{10} - 294 T^{11} - 1210 T^{12} - 864 T^{13} - 436 T^{14} + 128 T^{15} + 214 T^{16} + 110 T^{17} + 62 T^{18} - 16 T^{19} - 25 T^{20} - 8 T^{21} - T^{22} + 2 T^{23} + T^{24} \)
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