# Properties

 Label 3332.1.cc.b Level $3332$ Weight $1$ Character orbit 3332.cc Analytic conductor $1.663$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -68 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(135,3332)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([21, 32, 21]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3332.135");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3332 = 2^{2} \cdot 7^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3332.cc (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.66288462209$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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

 $$f(q)$$ $$=$$ $$q - \zeta_{42}^{17} q^{2} + ( - \zeta_{42}^{11} + 1) q^{3} - \zeta_{42}^{13} q^{4} + ( - \zeta_{42}^{17} - \zeta_{42}^{7}) q^{6} - \zeta_{42}^{9} q^{7} - \zeta_{42}^{9} q^{8} + ( - \zeta_{42}^{11} - \zeta_{42} + 1) q^{9} +O(q^{10})$$ q - z^17 * q^2 + (-z^11 + 1) * q^3 - z^13 * q^4 + (-z^17 - z^7) * q^6 - z^9 * q^7 - z^9 * q^8 + (-z^11 - z + 1) * q^9 $$q - \zeta_{42}^{17} q^{2} + ( - \zeta_{42}^{11} + 1) q^{3} - \zeta_{42}^{13} q^{4} + ( - \zeta_{42}^{17} - \zeta_{42}^{7}) q^{6} - \zeta_{42}^{9} q^{7} - \zeta_{42}^{9} q^{8} + ( - \zeta_{42}^{11} - \zeta_{42} + 1) q^{9} + (\zeta_{42}^{14} + \zeta_{42}^{6}) q^{11} + ( - \zeta_{42}^{13} - \zeta_{42}^{3}) q^{12} + (\zeta_{42}^{20} - \zeta_{42}^{7}) q^{13} - \zeta_{42}^{5} q^{14} - \zeta_{42}^{5} q^{16} - \zeta_{42} q^{17} + (\zeta_{42}^{18} - \zeta_{42}^{17} - \zeta_{42}^{7}) q^{18} + (\zeta_{42}^{20} - \zeta_{42}^{9}) q^{21} + (\zeta_{42}^{10} + \zeta_{42}^{2}) q^{22} + ( - \zeta_{42}^{17} + \zeta_{42}^{2}) q^{23} + (\zeta_{42}^{20} - \zeta_{42}^{9}) q^{24} + \zeta_{42}^{4} q^{25} + (\zeta_{42}^{16} - \zeta_{42}^{3}) q^{26} + (\zeta_{42}^{12} - \zeta_{42}^{11} + \zeta_{42} + 1) q^{27} - \zeta_{42} q^{28} + ( - \zeta_{42}^{19} + \zeta_{42}^{16}) q^{31} - \zeta_{42} q^{32} + ( - \zeta_{42}^{17} + \zeta_{42}^{14} + \zeta_{42}^{6} + \zeta_{42}^{4}) q^{33} + \zeta_{42}^{18} q^{34} + (\zeta_{42}^{14} - \zeta_{42}^{13} - \zeta_{42}^{3}) q^{36} + (\zeta_{42}^{20} + \zeta_{42}^{18} + \zeta_{42}^{10} - \zeta_{42}^{7}) q^{39} + (\zeta_{42}^{16} - \zeta_{42}^{5}) q^{42} + ( - \zeta_{42}^{19} + \zeta_{42}^{6}) q^{44} + ( - \zeta_{42}^{19} - \zeta_{42}^{13}) q^{46} + (\zeta_{42}^{16} - \zeta_{42}^{5}) q^{48} + \zeta_{42}^{18} q^{49} + q^{50} + (\zeta_{42}^{12} - \zeta_{42}) q^{51} + (\zeta_{42}^{20} + \zeta_{42}^{12}) q^{52} + ( - \zeta_{42}^{15} - \zeta_{42}^{11}) q^{53} + (\zeta_{42}^{18} - \zeta_{42}^{17} + \zeta_{42}^{8} - \zeta_{42}^{7}) q^{54} + \zeta_{42}^{18} q^{56} + ( - \zeta_{42}^{15} + \zeta_{42}^{12}) q^{62} + (\zeta_{42}^{20} + \zeta_{42}^{10} - \zeta_{42}^{9}) q^{63} + \zeta_{42}^{18} q^{64} + ( - \zeta_{42}^{13} + \zeta_{42}^{10} + \zeta_{42}^{2} + 1) q^{66} + \zeta_{42}^{14} q^{68} + ( - \zeta_{42}^{17} - \zeta_{42}^{13} - \zeta_{42}^{7} + \zeta_{42}^{2}) q^{69} + (\zeta_{42}^{8} + \zeta_{42}^{4}) q^{71} + (\zeta_{42}^{20} + \zeta_{42}^{10} - \zeta_{42}^{9}) q^{72} + ( - \zeta_{42}^{15} + \zeta_{42}^{4}) q^{75} + ( - \zeta_{42}^{15} + \zeta_{42}^{2}) q^{77} + (\zeta_{42}^{16} + \zeta_{42}^{14} + \zeta_{42}^{6} - \zeta_{42}^{3}) q^{78} + (\zeta_{42}^{4} - \zeta_{42}^{3}) q^{79} + (\zeta_{42}^{12} - \zeta_{42}^{11} + \zeta_{42}^{2} + \zeta_{42} + 1) q^{81} + (\zeta_{42}^{12} - \zeta_{42}) q^{84} + ( - \zeta_{42}^{15} + \zeta_{42}^{2}) q^{88} + (\zeta_{42}^{16} + \zeta_{42}^{6}) q^{89} + (\zeta_{42}^{16} + \zeta_{42}^{8}) q^{91} + ( - \zeta_{42}^{15} - \zeta_{42}^{9}) q^{92} + ( - \zeta_{42}^{19} + \zeta_{42}^{16} - \zeta_{42}^{9} + \zeta_{42}^{6}) q^{93} + (\zeta_{42}^{12} - \zeta_{42}) q^{96} + \zeta_{42}^{14} q^{98} + ( - \zeta_{42}^{17} - \zeta_{42}^{15} + \zeta_{42}^{14} - \zeta_{42}^{7} + \zeta_{42}^{6} + \zeta_{42}^{4}) q^{99} +O(q^{100})$$ q - z^17 * q^2 + (-z^11 + 1) * q^3 - z^13 * q^4 + (-z^17 - z^7) * q^6 - z^9 * q^7 - z^9 * q^8 + (-z^11 - z + 1) * q^9 + (z^14 + z^6) * q^11 + (-z^13 - z^3) * q^12 + (z^20 - z^7) * q^13 - z^5 * q^14 - z^5 * q^16 - z * q^17 + (z^18 - z^17 - z^7) * q^18 + (z^20 - z^9) * q^21 + (z^10 + z^2) * q^22 + (-z^17 + z^2) * q^23 + (z^20 - z^9) * q^24 + z^4 * q^25 + (z^16 - z^3) * q^26 + (z^12 - z^11 + z + 1) * q^27 - z * q^28 + (-z^19 + z^16) * q^31 - z * q^32 + (-z^17 + z^14 + z^6 + z^4) * q^33 + z^18 * q^34 + (z^14 - z^13 - z^3) * q^36 + (z^20 + z^18 + z^10 - z^7) * q^39 + (z^16 - z^5) * q^42 + (-z^19 + z^6) * q^44 + (-z^19 - z^13) * q^46 + (z^16 - z^5) * q^48 + z^18 * q^49 + q^50 + (z^12 - z) * q^51 + (z^20 + z^12) * q^52 + (-z^15 - z^11) * q^53 + (z^18 - z^17 + z^8 - z^7) * q^54 + z^18 * q^56 + (-z^15 + z^12) * q^62 + (z^20 + z^10 - z^9) * q^63 + z^18 * q^64 + (-z^13 + z^10 + z^2 + 1) * q^66 + z^14 * q^68 + (-z^17 - z^13 - z^7 + z^2) * q^69 + (z^8 + z^4) * q^71 + (z^20 + z^10 - z^9) * q^72 + (-z^15 + z^4) * q^75 + (-z^15 + z^2) * q^77 + (z^16 + z^14 + z^6 - z^3) * q^78 + (z^4 - z^3) * q^79 + (z^12 - z^11 + z^2 + z + 1) * q^81 + (z^12 - z) * q^84 + (-z^15 + z^2) * q^88 + (z^16 + z^6) * q^89 + (z^16 + z^8) * q^91 + (-z^15 - z^9) * q^92 + (-z^19 + z^16 - z^9 + z^6) * q^93 + (z^12 - z) * q^96 + z^14 * q^98 + (-z^17 - z^15 + z^14 - z^7 + z^6 + z^4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9}+O(q^{10})$$ 12 * q + q^2 + 13 * q^3 + q^4 - 5 * q^6 - 2 * q^7 - 2 * q^8 + 14 * q^9 $$12 q + q^{2} + 13 q^{3} + q^{4} - 5 q^{6} - 2 q^{7} - 2 q^{8} + 14 q^{9} - 8 q^{11} - q^{12} - 5 q^{13} + q^{14} + q^{16} + q^{17} - 7 q^{18} - q^{21} + 2 q^{22} + 2 q^{23} - q^{24} + q^{25} - q^{26} + 12 q^{27} + q^{28} + 2 q^{31} + q^{32} - 6 q^{33} - 2 q^{34} - 7 q^{36} - 6 q^{39} + 2 q^{42} - q^{44} + 2 q^{46} + 2 q^{48} - 2 q^{49} + 12 q^{50} - q^{51} - q^{52} - q^{53} - 6 q^{54} - 2 q^{56} - 4 q^{62} - 2 q^{64} + 15 q^{66} - 6 q^{68} - 3 q^{69} + 2 q^{71} - q^{75} - q^{77} - 9 q^{78} - q^{79} + 13 q^{81} - q^{84} - q^{88} - q^{89} + 2 q^{91} - 4 q^{92} - 2 q^{93} - q^{96} - 6 q^{98} - 14 q^{99}+O(q^{100})$$ 12 * q + q^2 + 13 * q^3 + q^4 - 5 * q^6 - 2 * q^7 - 2 * q^8 + 14 * q^9 - 8 * q^11 - q^12 - 5 * q^13 + q^14 + q^16 + q^17 - 7 * q^18 - q^21 + 2 * q^22 + 2 * q^23 - q^24 + q^25 - q^26 + 12 * q^27 + q^28 + 2 * q^31 + q^32 - 6 * q^33 - 2 * q^34 - 7 * q^36 - 6 * q^39 + 2 * q^42 - q^44 + 2 * q^46 + 2 * q^48 - 2 * q^49 + 12 * q^50 - q^51 - q^52 - q^53 - 6 * q^54 - 2 * q^56 - 4 * q^62 - 2 * q^64 + 15 * q^66 - 6 * q^68 - 3 * q^69 + 2 * q^71 - q^75 - q^77 - 9 * q^78 - q^79 + 13 * q^81 - q^84 - q^88 - q^89 + 2 * q^91 - 4 * q^92 - 2 * q^93 - q^96 - 6 * q^98 - 14 * q^99

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{42}^{11}$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
135.1
 −0.988831 − 0.149042i −0.988831 + 0.149042i 0.365341 − 0.930874i −0.733052 − 0.680173i 0.826239 + 0.563320i 0.0747301 − 0.997204i −0.733052 + 0.680173i 0.826239 − 0.563320i 0.955573 + 0.294755i 0.955573 − 0.294755i 0.365341 + 0.930874i 0.0747301 + 0.997204i
0.826239 0.563320i 1.07473 0.997204i 0.365341 0.930874i 0 0.326239 1.42935i −0.222521 0.974928i −0.222521 0.974928i 0.0858993 1.14625i 0
543.1 0.826239 + 0.563320i 1.07473 + 0.997204i 0.365341 + 0.930874i 0 0.326239 + 1.42935i −0.222521 + 0.974928i −0.222521 + 0.974928i 0.0858993 + 1.14625i 0
611.1 0.0747301 0.997204i 1.82624 0.563320i −0.988831 0.149042i 0 −0.425270 1.86323i −0.222521 + 0.974928i −0.222521 + 0.974928i 2.19158 1.49419i 0
1019.1 −0.988831 0.149042i 1.36534 0.930874i 0.955573 + 0.294755i 0 −1.48883 + 0.716983i −0.900969 0.433884i −0.900969 0.433884i 0.632289 1.61105i 0
1087.1 −0.733052 0.680173i 1.95557 + 0.294755i 0.0747301 + 0.997204i 0 −1.23305 1.54620i 0.623490 0.781831i 0.623490 0.781831i 2.78181 + 0.858075i 0
1495.1 0.955573 0.294755i 0.266948 + 0.680173i 0.826239 0.563320i 0 0.455573 + 0.571270i 0.623490 0.781831i 0.623490 0.781831i 0.341678 0.317031i 0
1563.1 −0.988831 + 0.149042i 1.36534 + 0.930874i 0.955573 0.294755i 0 −1.48883 0.716983i −0.900969 + 0.433884i −0.900969 + 0.433884i 0.632289 + 1.61105i 0
1971.1 −0.733052 + 0.680173i 1.95557 0.294755i 0.0747301 0.997204i 0 −1.23305 + 1.54620i 0.623490 + 0.781831i 0.623490 + 0.781831i 2.78181 0.858075i 0
2447.1 0.365341 0.930874i 0.0111692 0.149042i −0.733052 0.680173i 0 −0.134659 0.0648483i −0.900969 + 0.433884i −0.900969 + 0.433884i 0.966742 + 0.145713i 0
2515.1 0.365341 + 0.930874i 0.0111692 + 0.149042i −0.733052 + 0.680173i 0 −0.134659 + 0.0648483i −0.900969 0.433884i −0.900969 0.433884i 0.966742 0.145713i 0
2923.1 0.0747301 + 0.997204i 1.82624 + 0.563320i −0.988831 + 0.149042i 0 −0.425270 + 1.86323i −0.222521 0.974928i −0.222521 0.974928i 2.19158 + 1.49419i 0
2991.1 0.955573 + 0.294755i 0.266948 0.680173i 0.826239 + 0.563320i 0 0.455573 0.571270i 0.623490 + 0.781831i 0.623490 + 0.781831i 0.341678 + 0.317031i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2991.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by $$\Q(\sqrt{-17})$$
49.g even 21 1 inner
3332.cc odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.cc.b yes 12
4.b odd 2 1 3332.1.cc.a 12
17.b even 2 1 3332.1.cc.a 12
49.g even 21 1 inner 3332.1.cc.b yes 12
68.d odd 2 1 CM 3332.1.cc.b yes 12
196.o odd 42 1 3332.1.cc.a 12
833.z even 42 1 3332.1.cc.a 12
3332.cc odd 42 1 inner 3332.1.cc.b yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.cc.a 12 4.b odd 2 1
3332.1.cc.a 12 17.b even 2 1
3332.1.cc.a 12 196.o odd 42 1
3332.1.cc.a 12 833.z even 42 1
3332.1.cc.b yes 12 1.a even 1 1 trivial
3332.1.cc.b yes 12 49.g even 21 1 inner
3332.1.cc.b yes 12 68.d odd 2 1 CM
3332.1.cc.b yes 12 3332.cc odd 42 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 13 T_{3}^{11} + 77 T_{3}^{10} - 274 T_{3}^{9} + 650 T_{3}^{8} - 1078 T_{3}^{7} + 1275 T_{3}^{6} - 1078 T_{3}^{5} + 643 T_{3}^{4} - 260 T_{3}^{3} + 63 T_{3}^{2} - 6 T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1$$
$3$ $$T^{12} - 13 T^{11} + 77 T^{10} - 274 T^{9} + \cdots + 1$$
$5$ $$T^{12}$$
$7$ $$(T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$11$ $$T^{12} + 8 T^{11} + 35 T^{10} + 104 T^{9} + \cdots + 1$$
$13$ $$T^{12} + 5 T^{11} + 17 T^{10} + 38 T^{9} + \cdots + 1$$
$17$ $$T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + \cdots + 1$$
$19$ $$T^{12}$$
$23$ $$T^{12} - 2 T^{11} - 6 T^{9} + 12 T^{8} + \cdots + 1$$
$29$ $$T^{12}$$
$31$ $$(T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2}$$
$37$ $$T^{12}$$
$41$ $$T^{12}$$
$43$ $$T^{12}$$
$47$ $$T^{12}$$
$53$ $$T^{12} + T^{11} - T^{9} + 6 T^{8} - 21 T^{7} + \cdots + 1$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$T^{12}$$
$71$ $$T^{12} - 2 T^{11} + 3 T^{10} - 4 T^{9} + \cdots + 1$$
$73$ $$T^{12}$$
$79$ $$T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1$$
$83$ $$T^{12}$$
$89$ $$T^{12} + T^{11} + 6 T^{9} + 6 T^{8} + 7 T^{7} + \cdots + 1$$
$97$ $$T^{12}$$