# Properties

 Label 980.1.bq.a Level $980$ Weight $1$ Character orbit 980.bq Analytic conductor $0.489$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 980.bq (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.489083712380$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{21}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{21} + \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{42}^{4} q^{2} + ( \zeta_{42}^{3} + \zeta_{42}^{7} ) q^{3} + \zeta_{42}^{8} q^{4} -\zeta_{42}^{19} q^{5} + ( -\zeta_{42}^{7} - \zeta_{42}^{11} ) q^{6} -\zeta_{42}^{8} q^{7} -\zeta_{42}^{12} q^{8} + ( \zeta_{42}^{6} + \zeta_{42}^{10} + \zeta_{42}^{14} ) q^{9} +O(q^{10})$$ $$q -\zeta_{42}^{4} q^{2} + ( \zeta_{42}^{3} + \zeta_{42}^{7} ) q^{3} + \zeta_{42}^{8} q^{4} -\zeta_{42}^{19} q^{5} + ( -\zeta_{42}^{7} - \zeta_{42}^{11} ) q^{6} -\zeta_{42}^{8} q^{7} -\zeta_{42}^{12} q^{8} + ( \zeta_{42}^{6} + \zeta_{42}^{10} + \zeta_{42}^{14} ) q^{9} -\zeta_{42}^{2} q^{10} + ( \zeta_{42}^{11} + \zeta_{42}^{15} ) q^{12} + \zeta_{42}^{12} q^{14} + ( \zeta_{42} + \zeta_{42}^{5} ) q^{15} + \zeta_{42}^{16} q^{16} + ( -\zeta_{42}^{10} - \zeta_{42}^{14} - \zeta_{42}^{18} ) q^{18} + \zeta_{42}^{6} q^{20} + ( -\zeta_{42}^{11} - \zeta_{42}^{15} ) q^{21} + ( -\zeta_{42}^{6} + \zeta_{42}^{17} ) q^{23} + ( -\zeta_{42}^{15} - \zeta_{42}^{19} ) q^{24} -\zeta_{42}^{17} q^{25} + ( -1 + \zeta_{42}^{9} + \zeta_{42}^{13} + \zeta_{42}^{17} ) q^{27} -\zeta_{42}^{16} q^{28} + ( -\zeta_{42}^{5} - \zeta_{42}^{7} ) q^{29} + ( -\zeta_{42}^{5} - \zeta_{42}^{9} ) q^{30} -\zeta_{42}^{20} q^{32} -\zeta_{42}^{6} q^{35} + ( -\zeta_{42} + \zeta_{42}^{14} + \zeta_{42}^{18} ) q^{36} -\zeta_{42}^{10} q^{40} + ( -\zeta_{42} + \zeta_{42}^{2} ) q^{41} + ( \zeta_{42}^{15} + \zeta_{42}^{19} ) q^{42} + ( \zeta_{42}^{19} - \zeta_{42}^{20} ) q^{43} + ( \zeta_{42}^{4} + \zeta_{42}^{8} + \zeta_{42}^{12} ) q^{45} + ( 1 + \zeta_{42}^{10} ) q^{46} + ( \zeta_{42}^{13} - \zeta_{42}^{16} ) q^{47} + ( -\zeta_{42}^{2} + \zeta_{42}^{19} ) q^{48} + \zeta_{42}^{16} q^{49} - q^{50} + ( 1 + \zeta_{42}^{4} - \zeta_{42}^{13} - \zeta_{42}^{17} ) q^{54} + \zeta_{42}^{20} q^{56} + ( \zeta_{42}^{9} + \zeta_{42}^{11} ) q^{58} + ( \zeta_{42}^{9} + \zeta_{42}^{13} ) q^{60} + ( \zeta_{42}^{8} + \zeta_{42}^{18} ) q^{61} + ( \zeta_{42} - \zeta_{42}^{14} - \zeta_{42}^{18} ) q^{63} -\zeta_{42}^{3} q^{64} + \zeta_{42}^{14} q^{67} + ( -\zeta_{42}^{3} - \zeta_{42}^{9} - \zeta_{42}^{13} + \zeta_{42}^{20} ) q^{69} + \zeta_{42}^{10} q^{70} + ( \zeta_{42} + \zeta_{42}^{5} - \zeta_{42}^{18} ) q^{72} + ( \zeta_{42}^{3} - \zeta_{42}^{20} ) q^{75} + \zeta_{42}^{14} q^{80} + ( -\zeta_{42}^{3} - \zeta_{42}^{7} + \zeta_{42}^{12} + \zeta_{42}^{16} + \zeta_{42}^{20} ) q^{81} + ( \zeta_{42}^{5} - \zeta_{42}^{6} ) q^{82} + ( \zeta_{42}^{11} - \zeta_{42}^{16} ) q^{83} + ( \zeta_{42}^{2} - \zeta_{42}^{19} ) q^{84} + ( \zeta_{42}^{2} - \zeta_{42}^{3} ) q^{86} + ( -\zeta_{42}^{8} - \zeta_{42}^{10} - \zeta_{42}^{12} - \zeta_{42}^{14} ) q^{87} + ( -\zeta_{42}^{9} - \zeta_{42}^{11} ) q^{89} + ( -\zeta_{42}^{8} - \zeta_{42}^{12} - \zeta_{42}^{16} ) q^{90} + ( -\zeta_{42}^{4} - \zeta_{42}^{14} ) q^{92} + ( -\zeta_{42}^{17} + \zeta_{42}^{20} ) q^{94} + ( \zeta_{42}^{2} + \zeta_{42}^{6} ) q^{96} -\zeta_{42}^{20} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - q^{2} + 8q^{3} + q^{4} + q^{5} - 5q^{6} - q^{7} + 2q^{8} - 7q^{9} + O(q^{10})$$ $$12q - q^{2} + 8q^{3} + q^{4} + q^{5} - 5q^{6} - q^{7} + 2q^{8} - 7q^{9} - q^{10} + q^{12} - 2q^{14} - 2q^{15} + q^{16} + 7q^{18} - 2q^{20} - q^{21} + q^{23} - q^{24} + q^{25} - 12q^{27} - q^{28} - 5q^{29} - q^{30} - q^{32} + 2q^{35} - 7q^{36} - q^{40} + 2q^{41} + q^{42} - 2q^{43} + 13q^{46} - 2q^{47} - 2q^{48} + q^{49} - 12q^{50} + 15q^{54} + q^{56} + q^{58} + q^{60} - q^{61} + 7q^{63} - 2q^{64} - 6q^{67} - 2q^{69} + q^{70} + q^{75} - 6q^{80} - 8q^{81} + q^{82} - 2q^{83} + 2q^{84} - q^{86} + 6q^{87} - q^{89} + 5q^{92} + 2q^{94} - q^{96} - q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$\zeta_{42}^{10}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
39.1
 −0.733052 + 0.680173i 0.365341 − 0.930874i 0.365341 + 0.930874i 0.0747301 + 0.997204i 0.955573 + 0.294755i 0.826239 − 0.563320i 0.826239 + 0.563320i 0.0747301 − 0.997204i −0.988831 − 0.149042i −0.733052 − 0.680173i 0.955573 − 0.294755i −0.988831 + 0.149042i
0.988831 + 0.149042i −0.123490 + 0.0841939i 0.955573 + 0.294755i 0.0747301 + 0.997204i −0.134659 + 0.0648483i −0.955573 0.294755i 0.900969 + 0.433884i −0.357180 + 0.910080i −0.0747301 + 0.997204i
179.1 −0.0747301 0.997204i 1.40097 + 0.432142i −0.988831 + 0.149042i −0.733052 + 0.680173i 0.326239 1.42935i 0.988831 0.149042i 0.222521 + 0.974928i 0.949729 + 0.647514i 0.733052 + 0.680173i
219.1 −0.0747301 + 0.997204i 1.40097 0.432142i −0.988831 0.149042i −0.733052 0.680173i 0.326239 + 1.42935i 0.988831 + 0.149042i 0.222521 0.974928i 0.949729 0.647514i 0.733052 0.680173i
319.1 −0.955573 + 0.294755i 0.722521 + 1.84095i 0.826239 0.563320i −0.988831 0.149042i −1.23305 1.54620i −0.826239 + 0.563320i −0.623490 + 0.781831i −2.13402 + 1.98008i 0.988831 0.149042i
359.1 −0.365341 0.930874i −0.123490 1.64786i −0.733052 + 0.680173i 0.826239 0.563320i −1.48883 + 0.716983i 0.733052 0.680173i 0.900969 + 0.433884i −1.71135 + 0.257945i −0.826239 0.563320i
499.1 0.733052 + 0.680173i 0.722521 + 0.108903i 0.0747301 + 0.997204i 0.365341 + 0.930874i 0.455573 + 0.571270i −0.0747301 0.997204i −0.623490 + 0.781831i −0.445396 0.137386i −0.365341 + 0.930874i
599.1 0.733052 0.680173i 0.722521 0.108903i 0.0747301 0.997204i 0.365341 0.930874i 0.455573 0.571270i −0.0747301 + 0.997204i −0.623490 0.781831i −0.445396 + 0.137386i −0.365341 0.930874i
639.1 −0.955573 0.294755i 0.722521 1.84095i 0.826239 + 0.563320i −0.988831 + 0.149042i −1.23305 + 1.54620i −0.826239 0.563320i −0.623490 0.781831i −2.13402 1.98008i 0.988831 + 0.149042i
739.1 −0.826239 0.563320i 1.40097 + 1.29991i 0.365341 + 0.930874i 0.955573 0.294755i −0.425270 1.86323i −0.365341 0.930874i 0.222521 0.974928i 0.198220 + 2.64506i −0.955573 0.294755i
779.1 0.988831 0.149042i −0.123490 0.0841939i 0.955573 0.294755i 0.0747301 0.997204i −0.134659 0.0648483i −0.955573 + 0.294755i 0.900969 0.433884i −0.357180 0.910080i −0.0747301 0.997204i
879.1 −0.365341 + 0.930874i −0.123490 + 1.64786i −0.733052 0.680173i 0.826239 + 0.563320i −1.48883 0.716983i 0.733052 + 0.680173i 0.900969 0.433884i −1.71135 0.257945i −0.826239 + 0.563320i
919.1 −0.826239 + 0.563320i 1.40097 1.29991i 0.365341 0.930874i 0.955573 + 0.294755i −0.425270 + 1.86323i −0.365341 + 0.930874i 0.222521 + 0.974928i 0.198220 2.64506i −0.955573 + 0.294755i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 919.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
49.g even 21 1 inner
980.bq odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.bq.a 12
4.b odd 2 1 980.1.bq.b yes 12
5.b even 2 1 980.1.bq.b yes 12
20.d odd 2 1 CM 980.1.bq.a 12
49.g even 21 1 inner 980.1.bq.a 12
196.o odd 42 1 980.1.bq.b yes 12
245.t even 42 1 980.1.bq.b yes 12
980.bq odd 42 1 inner 980.1.bq.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.bq.a 12 1.a even 1 1 trivial
980.1.bq.a 12 20.d odd 2 1 CM
980.1.bq.a 12 49.g even 21 1 inner
980.1.bq.a 12 980.bq odd 42 1 inner
980.1.bq.b yes 12 4.b odd 2 1
980.1.bq.b yes 12 5.b even 2 1
980.1.bq.b yes 12 196.o odd 42 1
980.1.bq.b yes 12 245.t even 42 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - \cdots$$ acting on $$S_{1}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12}$$
$3$ $$1 + 6 T - 118 T^{3} + 349 T^{4} - 518 T^{5} + 519 T^{6} - 392 T^{7} + 230 T^{8} - 104 T^{9} + 35 T^{10} - 8 T^{11} + T^{12}$$
$5$ $$1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12}$$
$7$ $$1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12}$$
$11$ $$T^{12}$$
$13$ $$T^{12}$$
$17$ $$T^{12}$$
$19$ $$T^{12}$$
$23$ $$1 + 13 T + 49 T^{2} + 29 T^{3} + 69 T^{4} - 20 T^{6} + 21 T^{7} + 6 T^{8} + T^{9} - T^{11} + T^{12}$$
$29$ $$1 + 12 T + 45 T^{2} + 10 T^{3} + 61 T^{4} + 92 T^{5} + 105 T^{6} + 92 T^{7} + 68 T^{8} + 38 T^{9} + 17 T^{10} + 5 T^{11} + T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$1 + 5 T + 52 T^{2} + 94 T^{3} + 54 T^{4} - 6 T^{5} + 7 T^{6} - 6 T^{7} + 12 T^{8} - 4 T^{9} + 3 T^{10} - 2 T^{11} + T^{12}$$
$43$ $$1 - 5 T + 52 T^{2} - 94 T^{3} + 54 T^{4} + 6 T^{5} + 7 T^{6} + 6 T^{7} + 12 T^{8} + 4 T^{9} + 3 T^{10} + 2 T^{11} + T^{12}$$
$47$ $$1 - 3 T + 7 T^{2} - 8 T^{3} + 3 T^{4} + 28 T^{5} + T^{6} - 7 T^{7} + 12 T^{8} + 6 T^{9} + 2 T^{11} + T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$1 + 8 T + 28 T^{2} - 71 T^{3} + 48 T^{4} + 21 T^{5} + 22 T^{6} - 15 T^{8} - T^{9} + T^{11} + T^{12}$$
$67$ $$( 1 + T + T^{2} )^{6}$$
$71$ $$T^{12}$$
$73$ $$T^{12}$$
$79$ $$T^{12}$$
$83$ $$1 - 5 T + 52 T^{2} - 94 T^{3} + 54 T^{4} + 6 T^{5} + 7 T^{6} + 6 T^{7} + 12 T^{8} + 4 T^{9} + 3 T^{10} + 2 T^{11} + T^{12}$$
$89$ $$1 + 8 T + 28 T^{2} - 71 T^{3} + 48 T^{4} + 21 T^{5} + 22 T^{6} - 15 T^{8} - T^{9} + T^{11} + T^{12}$$
$97$ $$T^{12}$$