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 $4$

Related objects

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: Complex embeddings Normalized embeddings Satake parameters Satake angles

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

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