# Properties

 Label 3332.1.be.c Level $3332$ Weight $1$ Character orbit 3332.be Analytic conductor $1.663$ Analytic rank $0$ Dimension $12$ Projective image $D_{14}$ CM discriminant -68 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.be (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{14})$$ Coefficient field: $$\Q(\zeta_{28})$$ Defining polynomial: $$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{14}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{14} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{28}^{8} q^{2} + (\zeta_{28}^{13} + \zeta_{28}^{7}) q^{3} - \zeta_{28}^{2} q^{4} + (\zeta_{28}^{7} + \zeta_{28}) q^{6} + \zeta_{28}^{3} q^{7} + \zeta_{28}^{10} q^{8} + ( - \zeta_{28}^{12} - \zeta_{28}^{6} - 1) q^{9} +O(q^{10})$$ q - z^8 * q^2 + (z^13 + z^7) * q^3 - z^2 * q^4 + (z^7 + z) * q^6 + z^3 * q^7 + z^10 * q^8 + (-z^12 - z^6 - 1) * q^9 $$q - \zeta_{28}^{8} q^{2} + (\zeta_{28}^{13} + \zeta_{28}^{7}) q^{3} - \zeta_{28}^{2} q^{4} + (\zeta_{28}^{7} + \zeta_{28}) q^{6} + \zeta_{28}^{3} q^{7} + \zeta_{28}^{10} q^{8} + ( - \zeta_{28}^{12} - \zeta_{28}^{6} - 1) q^{9} + (\zeta_{28}^{9} + \zeta_{28}^{7}) q^{11} + ( - \zeta_{28}^{9} + \zeta_{28}) q^{12} + (\zeta_{28}^{2} - 1) q^{13} - \zeta_{28}^{11} q^{14} + \zeta_{28}^{4} q^{16} + \zeta_{28}^{12} q^{17} + (\zeta_{28}^{8} - \zeta_{28}^{6} - 1) q^{18} + (\zeta_{28}^{10} - \zeta_{28}^{2}) q^{21} + (\zeta_{28}^{3} + \zeta_{28}) q^{22} + ( - \zeta_{28}^{3} - \zeta_{28}) q^{23} + ( - \zeta_{28}^{9} - \zeta_{28}^{3}) q^{24} - \zeta_{28}^{6} q^{25} + ( - \zeta_{28}^{10} + \zeta_{28}^{8}) q^{26} + ( - \zeta_{28}^{13} + \zeta_{28}^{11} - \zeta_{28}^{7} - \zeta_{28}^{5}) q^{27} - \zeta_{28}^{5} q^{28} + ( - \zeta_{28}^{11} + \zeta_{28}^{3}) q^{31} - \zeta_{28}^{12} q^{32} + ( - \zeta_{28}^{8} - \zeta_{28}^{6} - \zeta_{28}^{2} - 1) q^{33} + \zeta_{28}^{6} q^{34} + (\zeta_{28}^{8} + \zeta_{28}^{2} - 1) q^{36} + ( - \zeta_{28}^{13} + \zeta_{28}^{9} - \zeta_{28}^{7} - \zeta_{28}) q^{39} + (\zeta_{28}^{10} + \zeta_{28}^{4}) q^{42} + ( - \zeta_{28}^{11} - \zeta_{28}^{9}) q^{44} + (\zeta_{28}^{11} + \zeta_{28}^{9}) q^{46} + (\zeta_{28}^{11} - \zeta_{28}^{3}) q^{48} + \zeta_{28}^{6} q^{49} - q^{50} + ( - \zeta_{28}^{11} - \zeta_{28}^{5}) q^{51} + ( - \zeta_{28}^{4} + \zeta_{28}^{2}) q^{52} + (\zeta_{28}^{12} - \zeta_{28}^{6}) q^{53} + ( - \zeta_{28}^{13} - \zeta_{28}^{7} + \zeta_{28}^{5} - \zeta_{28}) q^{54} + \zeta_{28}^{13} q^{56} + ( - \zeta_{28}^{11} - \zeta_{28}^{5}) q^{62} + ( - \zeta_{28}^{9} - \zeta_{28}^{3} + \zeta_{28}) q^{63} - \zeta_{28}^{6} q^{64} + (\zeta_{28}^{10} + \zeta_{28}^{8} - \zeta_{28}^{2} - 1) q^{66} + q^{68} + ( - \zeta_{28}^{10} - \zeta_{28}^{8} + \zeta_{28}^{2} + 1) q^{69} + (\zeta_{28}^{13} - \zeta_{28}^{5}) q^{71} + ( - \zeta_{28}^{10} + \zeta_{28}^{8} + \zeta_{28}^{2}) q^{72} + ( - \zeta_{28}^{13} + \zeta_{28}^{5}) q^{75} + (\zeta_{28}^{12} + \zeta_{28}^{10}) q^{77} + (\zeta_{28}^{9} - \zeta_{28}^{7} + \zeta_{28}^{3} - \zeta_{28}) q^{78} + (\zeta_{28}^{13} - \zeta_{28}) q^{79} + (\zeta_{28}^{12} - \zeta_{28}^{10} + \zeta_{28}^{6} + \zeta_{28}^{4} + 1) q^{81} + ( - \zeta_{28}^{12} + \zeta_{28}^{4}) q^{84} + ( - \zeta_{28}^{5} - \zeta_{28}^{3}) q^{88} + (\zeta_{28}^{10} + \zeta_{28}^{2}) q^{89} + (\zeta_{28}^{5} - \zeta_{28}^{3}) q^{91} + (\zeta_{28}^{5} + \zeta_{28}^{3}) q^{92} + (\zeta_{28}^{10} + \zeta_{28}^{4} - \zeta_{28}^{2}) q^{93} + (\zeta_{28}^{11} + \zeta_{28}^{5}) q^{96} + q^{98} + ( - \zeta_{28}^{13} - \zeta_{28}^{9} + \zeta_{28}^{7} + \zeta_{28}^{5} + \zeta_{28}) q^{99} +O(q^{100})$$ q - z^8 * q^2 + (z^13 + z^7) * q^3 - z^2 * q^4 + (z^7 + z) * q^6 + z^3 * q^7 + z^10 * q^8 + (-z^12 - z^6 - 1) * q^9 + (z^9 + z^7) * q^11 + (-z^9 + z) * q^12 + (z^2 - 1) * q^13 - z^11 * q^14 + z^4 * q^16 + z^12 * q^17 + (z^8 - z^6 - 1) * q^18 + (z^10 - z^2) * q^21 + (z^3 + z) * q^22 + (-z^3 - z) * q^23 + (-z^9 - z^3) * q^24 - z^6 * q^25 + (-z^10 + z^8) * q^26 + (-z^13 + z^11 - z^7 - z^5) * q^27 - z^5 * q^28 + (-z^11 + z^3) * q^31 - z^12 * q^32 + (-z^8 - z^6 - z^2 - 1) * q^33 + z^6 * q^34 + (z^8 + z^2 - 1) * q^36 + (-z^13 + z^9 - z^7 - z) * q^39 + (z^10 + z^4) * q^42 + (-z^11 - z^9) * q^44 + (z^11 + z^9) * q^46 + (z^11 - z^3) * q^48 + z^6 * q^49 - q^50 + (-z^11 - z^5) * q^51 + (-z^4 + z^2) * q^52 + (z^12 - z^6) * q^53 + (-z^13 - z^7 + z^5 - z) * q^54 + z^13 * q^56 + (-z^11 - z^5) * q^62 + (-z^9 - z^3 + z) * q^63 - z^6 * q^64 + (z^10 + z^8 - z^2 - 1) * q^66 + q^68 + (-z^10 - z^8 + z^2 + 1) * q^69 + (z^13 - z^5) * q^71 + (-z^10 + z^8 + z^2) * q^72 + (-z^13 + z^5) * q^75 + (z^12 + z^10) * q^77 + (z^9 - z^7 + z^3 - z) * q^78 + (z^13 - z) * q^79 + (z^12 - z^10 + z^6 + z^4 + 1) * q^81 + (-z^12 + z^4) * q^84 + (-z^5 - z^3) * q^88 + (z^10 + z^2) * q^89 + (z^5 - z^3) * q^91 + (z^5 + z^3) * q^92 + (z^10 + z^4 - z^2) * q^93 + (z^11 + z^5) * q^96 + q^98 + (-z^13 - z^9 + z^7 + z^5 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 12 q^{9}+O(q^{10})$$ 12 * q + 2 * q^2 - 2 * q^4 + 2 * q^8 - 12 * q^9 $$12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 12 q^{9} - 10 q^{13} - 2 q^{16} - 2 q^{17} - 16 q^{18} - 2 q^{25} - 4 q^{26} + 2 q^{32} - 14 q^{33} + 2 q^{34} - 12 q^{36} + 2 q^{49} - 12 q^{50} + 4 q^{52} - 4 q^{53} - 2 q^{64} - 14 q^{66} + 12 q^{68} + 14 q^{69} - 2 q^{72} + 12 q^{81} + 4 q^{89} + 12 q^{98}+O(q^{100})$$ 12 * q + 2 * q^2 - 2 * q^4 + 2 * q^8 - 12 * q^9 - 10 * q^13 - 2 * q^16 - 2 * q^17 - 16 * q^18 - 2 * q^25 - 4 * q^26 + 2 * q^32 - 14 * q^33 + 2 * q^34 - 12 * q^36 + 2 * q^49 - 12 * q^50 + 4 * q^52 - 4 * q^53 - 2 * q^64 - 14 * q^66 + 12 * q^68 + 14 * q^69 - 2 * q^72 + 12 * q^81 + 4 * q^89 + 12 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{28}^{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
407.1
 0.974928 + 0.222521i −0.974928 − 0.222521i 0.974928 − 0.222521i −0.974928 + 0.222521i 0.433884 + 0.900969i −0.433884 − 0.900969i 0.781831 − 0.623490i −0.781831 + 0.623490i 0.781831 + 0.623490i −0.781831 − 0.623490i 0.433884 − 0.900969i −0.433884 + 0.900969i
0.222521 0.974928i −0.974928 + 1.22252i −0.900969 0.433884i 0 0.974928 + 1.22252i 0.781831 + 0.623490i −0.623490 + 0.781831i −0.321552 1.40881i 0
407.2 0.222521 0.974928i 0.974928 1.22252i −0.900969 0.433884i 0 −0.974928 1.22252i −0.781831 0.623490i −0.623490 + 0.781831i −0.321552 1.40881i 0
1359.1 0.222521 + 0.974928i −0.974928 1.22252i −0.900969 + 0.433884i 0 0.974928 1.22252i 0.781831 0.623490i −0.623490 0.781831i −0.321552 + 1.40881i 0
1359.2 0.222521 + 0.974928i 0.974928 + 1.22252i −0.900969 + 0.433884i 0 −0.974928 + 1.22252i −0.781831 + 0.623490i −0.623490 0.781831i −0.321552 + 1.40881i 0
1835.1 0.900969 0.433884i −0.433884 + 1.90097i 0.623490 0.781831i 0 0.433884 + 1.90097i −0.974928 0.222521i 0.222521 0.974928i −2.52446 1.21572i 0
1835.2 0.900969 0.433884i 0.433884 1.90097i 0.623490 0.781831i 0 −0.433884 1.90097i 0.974928 + 0.222521i 0.222521 0.974928i −2.52446 1.21572i 0
2311.1 −0.623490 0.781831i −0.781831 + 0.376510i −0.222521 + 0.974928i 0 0.781831 + 0.376510i −0.433884 0.900969i 0.900969 0.433884i −0.153989 + 0.193096i 0
2311.2 −0.623490 0.781831i 0.781831 0.376510i −0.222521 + 0.974928i 0 −0.781831 0.376510i 0.433884 + 0.900969i 0.900969 0.433884i −0.153989 + 0.193096i 0
2787.1 −0.623490 + 0.781831i −0.781831 0.376510i −0.222521 0.974928i 0 0.781831 0.376510i −0.433884 + 0.900969i 0.900969 + 0.433884i −0.153989 0.193096i 0
2787.2 −0.623490 + 0.781831i 0.781831 + 0.376510i −0.222521 0.974928i 0 −0.781831 + 0.376510i 0.433884 0.900969i 0.900969 + 0.433884i −0.153989 0.193096i 0
3263.1 0.900969 + 0.433884i −0.433884 1.90097i 0.623490 + 0.781831i 0 0.433884 1.90097i −0.974928 + 0.222521i 0.222521 + 0.974928i −2.52446 + 1.21572i 0
3263.2 0.900969 + 0.433884i 0.433884 + 1.90097i 0.623490 + 0.781831i 0 −0.433884 + 1.90097i 0.974928 0.222521i 0.222521 + 0.974928i −2.52446 + 1.21572i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3263.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
4.b odd 2 1 inner
17.b even 2 1 inner
49.e even 7 1 inner
196.k odd 14 1 inner
833.r even 14 1 inner
3332.be odd 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.be.c 12
4.b odd 2 1 inner 3332.1.be.c 12
17.b even 2 1 inner 3332.1.be.c 12
49.e even 7 1 inner 3332.1.be.c 12
68.d odd 2 1 CM 3332.1.be.c 12
196.k odd 14 1 inner 3332.1.be.c 12
833.r even 14 1 inner 3332.1.be.c 12
3332.be odd 14 1 inner 3332.1.be.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.be.c 12 1.a even 1 1 trivial
3332.1.be.c 12 4.b odd 2 1 inner
3332.1.be.c 12 17.b even 2 1 inner
3332.1.be.c 12 49.e even 7 1 inner
3332.1.be.c 12 68.d odd 2 1 CM
3332.1.be.c 12 196.k odd 14 1 inner
3332.1.be.c 12 833.r even 14 1 inner
3332.1.be.c 12 3332.be odd 14 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 7T_{3}^{10} + 21T_{3}^{8} + 35T_{3}^{6} + 49T_{3}^{4} - 49T_{3}^{2} + 49$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$3$ $$T^{12} + 7 T^{10} + 21 T^{8} + 35 T^{6} + \cdots + 49$$
$5$ $$T^{12}$$
$7$ $$T^{12} - T^{10} + T^{8} - T^{6} + T^{4} - T^{2} + \cdots + 1$$
$11$ $$T^{12} + 7 T^{10} + 21 T^{8} + 35 T^{6} + \cdots + 49$$
$13$ $$(T^{6} + 5 T^{5} + 11 T^{4} + 13 T^{3} + \cdots + 1)^{2}$$
$17$ $$(T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$19$ $$T^{12}$$
$23$ $$T^{12} + 35 T^{6} + 98 T^{4} + 49 T^{2} + \cdots + 49$$
$29$ $$T^{12}$$
$31$ $$(T^{6} - 7 T^{4} + 14 T^{2} - 7)^{2}$$
$37$ $$T^{12}$$
$41$ $$T^{12}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$(T^{6} + 2 T^{5} + 4 T^{4} + 8 T^{3} + 9 T^{2} + \cdots + 1)^{2}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12} + 14 T^{8} - 14 T^{6} + 49 T^{4} + \cdots + 49$$
$73$ $$T^{12}$$
$79$ $$(T^{6} - 7 T^{4} + 14 T^{2} - 7)^{2}$$
$83$ $$T^{12}$$
$89$ $$(T^{6} - 2 T^{5} + 4 T^{4} - T^{3} + 2 T^{2} + \cdots + 1)^{2}$$
$97$ $$T^{12}$$