# Properties

 Label 1911.1.ea.a Level $1911$ Weight $1$ Character orbit 1911.ea 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.ea (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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{84}^{33} q^{3} + \zeta_{84} q^{4} + \zeta_{84}^{32} q^{7} -\zeta_{84}^{24} q^{9} +O(q^{10})$$ $$q -\zeta_{84}^{33} q^{3} + \zeta_{84} q^{4} + \zeta_{84}^{32} q^{7} -\zeta_{84}^{24} q^{9} -\zeta_{84}^{34} q^{12} -\zeta_{84}^{37} q^{13} + \zeta_{84}^{2} q^{16} + ( -\zeta_{84}^{4} + \zeta_{84}^{17} ) q^{19} + \zeta_{84}^{23} q^{21} -\zeta_{84}^{17} q^{25} -\zeta_{84}^{15} q^{27} + \zeta_{84}^{33} q^{28} + ( -\zeta_{84}^{20} + \zeta_{84}^{29} ) q^{31} -\zeta_{84}^{25} q^{36} + ( \zeta_{84}^{21} - \zeta_{84}^{40} ) q^{37} -\zeta_{84}^{28} q^{39} + ( \zeta_{84}^{6} + \zeta_{84}^{40} ) q^{43} -\zeta_{84}^{35} q^{48} -\zeta_{84}^{22} q^{49} -\zeta_{84}^{38} q^{52} + ( \zeta_{84}^{8} + \zeta_{84}^{37} ) q^{57} + ( \zeta_{84}^{19} + \zeta_{84}^{35} ) q^{61} + \zeta_{84}^{14} q^{63} + \zeta_{84}^{3} q^{64} + ( -\zeta_{84}^{9} - \zeta_{84}^{12} ) q^{67} + ( \zeta_{84}^{25} - \zeta_{84}^{30} ) q^{73} -\zeta_{84}^{8} q^{75} + ( -\zeta_{84}^{5} + \zeta_{84}^{18} ) q^{76} + ( -\zeta_{84} + \zeta_{84}^{13} ) q^{79} -\zeta_{84}^{6} q^{81} + \zeta_{84}^{24} q^{84} + \zeta_{84}^{27} q^{91} + ( -\zeta_{84}^{11} + \zeta_{84}^{20} ) q^{93} + ( -\zeta_{84}^{10} + \zeta_{84}^{39} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$24 q + 2 q^{7} + 4 q^{9} + O(q^{10})$$ $$24 q + 2 q^{7} + 4 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{19} - 2 q^{31} - 2 q^{37} + 12 q^{39} + 6 q^{43} + 2 q^{49} + 2 q^{52} + 2 q^{57} + 12 q^{63} + 4 q^{67} - 4 q^{73} - 2 q^{75} + 4 q^{76} - 4 q^{81} - 4 q^{84} + 2 q^{93} + 2 q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$638$$ $$1471$$ $$1522$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{84}^{35}$$ $$\zeta_{84}^{10}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
110.1
 0.149042 + 0.988831i −0.680173 − 0.733052i 0.680173 + 0.733052i −0.563320 − 0.826239i −0.563320 + 0.826239i 0.997204 + 0.0747301i 0.563320 − 0.826239i 0.930874 + 0.365341i −0.930874 + 0.365341i −0.930874 − 0.365341i −0.997204 − 0.0747301i 0.149042 − 0.988831i 0.680173 − 0.733052i −0.149042 + 0.988831i 0.930874 − 0.365341i −0.997204 + 0.0747301i −0.294755 + 0.955573i 0.997204 − 0.0747301i −0.680173 + 0.733052i 0.294755 + 0.955573i
0 0.974928 0.222521i 0.149042 + 0.988831i 0 0 0.0747301 + 0.997204i 0 0.900969 0.433884i 0
206.1 0 −0.433884 + 0.900969i −0.680173 0.733052i 0 0 0.365341 + 0.930874i 0 −0.623490 0.781831i 0
353.1 0 0.433884 0.900969i 0.680173 + 0.733052i 0 0 0.365341 + 0.930874i 0 −0.623490 0.781831i 0
383.1 0 0.781831 + 0.623490i −0.563320 0.826239i 0 0 0.955573 0.294755i 0 0.222521 + 0.974928i 0
479.1 0 0.781831 0.623490i −0.563320 + 0.826239i 0 0 0.955573 + 0.294755i 0 0.222521 0.974928i 0
488.1 0 0.781831 0.623490i 0.997204 + 0.0747301i 0 0 −0.733052 + 0.680173i 0 0.222521 0.974928i 0
626.1 0 −0.781831 + 0.623490i 0.563320 0.826239i 0 0 0.955573 + 0.294755i 0 0.222521 0.974928i 0
752.1 0 −0.974928 + 0.222521i 0.930874 + 0.365341i 0 0 0.826239 0.563320i 0 0.900969 0.433884i 0
761.1 0 0.974928 + 0.222521i −0.930874 + 0.365341i 0 0 0.826239 + 0.563320i 0 0.900969 + 0.433884i 0
899.1 0 0.974928 0.222521i −0.930874 0.365341i 0 0 0.826239 0.563320i 0 0.900969 0.433884i 0
929.1 0 −0.781831 + 0.623490i −0.997204 0.0747301i 0 0 −0.733052 + 0.680173i 0 0.222521 0.974928i 0
1025.1 0 0.974928 + 0.222521i 0.149042 0.988831i 0 0 0.0747301 0.997204i 0 0.900969 + 0.433884i 0
1034.1 0 0.433884 + 0.900969i 0.680173 0.733052i 0 0 0.365341 0.930874i 0 −0.623490 + 0.781831i 0
1172.1 0 −0.974928 0.222521i −0.149042 + 0.988831i 0 0 0.0747301 0.997204i 0 0.900969 + 0.433884i 0
1202.1 0 −0.974928 0.222521i 0.930874 0.365341i 0 0 0.826239 + 0.563320i 0 0.900969 + 0.433884i 0
1298.1 0 −0.781831 0.623490i −0.997204 + 0.0747301i 0 0 −0.733052 0.680173i 0 0.222521 + 0.974928i 0
1307.1 0 −0.433884 + 0.900969i −0.294755 + 0.955573i 0 0 −0.988831 0.149042i 0 −0.623490 0.781831i 0
1445.1 0 0.781831 + 0.623490i 0.997204 0.0747301i 0 0 −0.733052 0.680173i 0 0.222521 + 0.974928i 0
1475.1 0 −0.433884 0.900969i −0.680173 + 0.733052i 0 0 0.365341 0.930874i 0 −0.623490 + 0.781831i 0
1571.1 0 0.433884 + 0.900969i 0.294755 + 0.955573i 0 0 −0.988831 + 0.149042i 0 −0.623490 + 0.781831i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1853.1 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.ch even 84 1 inner
1911.ea 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.ea.a 24
3.b odd 2 1 CM 1911.1.ea.a 24
13.f odd 12 1 1911.1.ep.a yes 24
39.k even 12 1 1911.1.ep.a yes 24
49.h odd 42 1 1911.1.ep.a yes 24
147.o even 42 1 1911.1.ep.a yes 24
637.ch even 84 1 inner 1911.1.ea.a 24
1911.ea odd 84 1 inner 1911.1.ea.a 24

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{24}$$
$3$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$
$5$ $$T^{24}$$
$7$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )^{2}$$
$11$ $$T^{24}$$
$13$ $$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$
$17$ $$T^{24}$$
$19$ $$1 - 16 T + 128 T^{2} + 436 T^{3} + 848 T^{4} - 910 T^{5} + 1064 T^{6} + 1640 T^{7} + 1948 T^{8} - 1492 T^{9} + 1346 T^{10} + 1470 T^{11} + 1070 T^{12} - 504 T^{13} + 464 T^{14} + 470 T^{15} + 268 T^{16} - 58 T^{17} + 56 T^{18} + 56 T^{19} + 29 T^{20} - 2 T^{21} + 2 T^{22} + 2 T^{23} + T^{24}$$
$23$ $$T^{24}$$
$29$ $$T^{24}$$
$31$ $$1 + 8 T + 32 T^{2} + 172 T^{3} + 662 T^{4} + 926 T^{5} + 1016 T^{6} + 1526 T^{7} + 199 T^{8} - 1684 T^{9} - 1222 T^{10} - 1152 T^{11} + 704 T^{12} + 1188 T^{13} + 566 T^{14} + 542 T^{15} + 187 T^{16} - 70 T^{17} - 34 T^{18} - 34 T^{19} - 13 T^{20} + 4 T^{21} + 2 T^{22} + 2 T^{23} + T^{24}$$
$37$ $$1 + 32 T + 326 T^{2} + 1104 T^{3} + 959 T^{4} - 1176 T^{5} - 1212 T^{6} + 1156 T^{7} + 2403 T^{8} + 1492 T^{9} + 1502 T^{10} + 1470 T^{11} + 1779 T^{12} + 1306 T^{13} + 1202 T^{14} + 702 T^{15} + 634 T^{16} + 306 T^{17} + 258 T^{18} + 96 T^{19} + 73 T^{20} + 20 T^{21} + 13 T^{22} + 2 T^{23} + T^{24}$$
$41$ $$T^{24}$$
$43$ $$( 1 - 8 T + 24 T^{2} - 47 T^{3} + 88 T^{4} - 105 T^{5} + 92 T^{6} - 56 T^{7} + 23 T^{8} - 9 T^{9} + 6 T^{10} - 3 T^{11} + T^{12} )^{2}$$
$47$ $$T^{24}$$
$53$ $$T^{24}$$
$59$ $$T^{24}$$
$61$ $$1 + 54 T^{2} + 1907 T^{4} - 3392 T^{6} + 9259 T^{8} - 7748 T^{10} + 3045 T^{12} - 888 T^{14} + 404 T^{16} - 158 T^{18} + 45 T^{20} - 9 T^{22} + T^{24}$$
$67$ $$( 1 - 8 T + 32 T^{2} - 42 T^{3} + 26 T^{4} + 2 T^{5} + 34 T^{6} - 34 T^{7} + 17 T^{8} + 2 T^{10} - 2 T^{11} + T^{12} )^{2}$$
$71$ $$T^{24}$$
$73$ $$1 - 12 T + 188 T^{2} - 176 T^{3} - 900 T^{4} + 922 T^{5} + 3639 T^{6} + 2672 T^{7} + 284 T^{8} + 226 T^{9} + 523 T^{10} - 248 T^{11} - 482 T^{12} - 798 T^{13} - 696 T^{14} - 440 T^{15} - 75 T^{16} + 50 T^{17} + 97 T^{18} + 84 T^{19} + 56 T^{20} + 26 T^{21} + 11 T^{22} + 4 T^{23} + T^{24}$$
$79$ $$( 49 + 98 T^{2} + 147 T^{4} + 84 T^{6} + 35 T^{8} + 7 T^{10} + T^{12} )^{2}$$
$83$ $$T^{24}$$
$89$ $$T^{24}$$
$97$ $$1 + 10 T + 200 T^{2} + 548 T^{3} - 28 T^{4} - 266 T^{5} + 692 T^{6} - 2612 T^{7} + 3166 T^{8} - 3368 T^{9} + 2447 T^{10} + 294 T^{11} - 1210 T^{12} + 864 T^{13} - 436 T^{14} - 128 T^{15} + 214 T^{16} - 110 T^{17} + 62 T^{18} + 16 T^{19} - 25 T^{20} + 8 T^{21} - T^{22} - 2 T^{23} + T^{24}$$