# Properties

 Label 3332.1.cc.c Level $3332$ Weight $1$ Character orbit 3332.cc Analytic conductor $1.663$ Analytic rank $0$ Dimension $24$ Projective image $D_{42}$ CM discriminant -68 Inner twists $8$

# 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: $$24$$ Relative dimension: $$2$$ over $$\Q(\zeta_{42})$$ Coefficient field: $$\Q(\zeta_{84})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1$$ 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: $$D_{42}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{42} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{84}^{40} q^{2} + ( - \zeta_{84}^{37} + \zeta_{84}^{21}) q^{3} - \zeta_{84}^{38} q^{4} + ( - \zeta_{84}^{35} + \zeta_{84}^{19}) q^{6} - \zeta_{84}^{15} q^{7} - \zeta_{84}^{36} q^{8} + ( - \zeta_{84}^{32} + \zeta_{84}^{16} - 1) q^{9} +O(q^{10})$$ q - z^40 * q^2 + (-z^37 + z^21) * q^3 - z^38 * q^4 + (-z^35 + z^19) * q^6 - z^15 * q^7 - z^36 * q^8 + (-z^32 + z^16 - 1) * q^9 $$q - \zeta_{84}^{40} q^{2} + ( - \zeta_{84}^{37} + \zeta_{84}^{21}) q^{3} - \zeta_{84}^{38} q^{4} + ( - \zeta_{84}^{35} + \zeta_{84}^{19}) q^{6} - \zeta_{84}^{15} q^{7} - \zeta_{84}^{36} q^{8} + ( - \zeta_{84}^{32} + \zeta_{84}^{16} - 1) q^{9} + ( - \zeta_{84}^{7} + \zeta_{84}^{3}) q^{11} + ( - \zeta_{84}^{33} + \zeta_{84}^{17}) q^{12} + (\zeta_{84}^{14} + \zeta_{84}^{10}) q^{13} - \zeta_{84}^{13} q^{14} - \zeta_{84}^{34} q^{16} + \zeta_{84}^{32} q^{17} + (\zeta_{84}^{40} - \zeta_{84}^{30} + \zeta_{84}^{14}) q^{18} + ( - \zeta_{84}^{36} - \zeta_{84}^{10}) q^{21} + ( - \zeta_{84}^{5} + \zeta_{84}) q^{22} + ( - \zeta_{84}^{19} - \zeta_{84}) q^{23} + ( - \zeta_{84}^{31} + \zeta_{84}^{15}) q^{24} - \zeta_{84}^{2} q^{25} + (\zeta_{84}^{12} + \zeta_{84}^{8}) q^{26} + (\zeta_{84}^{37} - \zeta_{84}^{27} - \zeta_{84}^{21} - \zeta_{84}^{11}) q^{27} - \zeta_{84}^{11} q^{28} + ( - \zeta_{84}^{41} + \zeta_{84}^{29}) q^{31} - \zeta_{84}^{32} q^{32} + ( - \zeta_{84}^{40} - \zeta_{84}^{28} + \zeta_{84}^{24} - \zeta_{84}^{2}) q^{33} + \zeta_{84}^{30} q^{34} + (\zeta_{84}^{38} - \zeta_{84}^{28} + \zeta_{84}^{12}) q^{36} + (\zeta_{84}^{35} + \zeta_{84}^{31} + \zeta_{84}^{9} + \zeta_{84}^{5}) q^{39} + ( - \zeta_{84}^{34} - \zeta_{84}^{8}) q^{42} + ( - \zeta_{84}^{41} - \zeta_{84}^{3}) q^{44} + (\zeta_{84}^{41} - \zeta_{84}^{17}) q^{46} + ( - \zeta_{84}^{29} + \zeta_{84}^{13}) q^{48} + \zeta_{84}^{30} q^{49} - q^{50} + (\zeta_{84}^{27} - \zeta_{84}^{11}) q^{51} + (\zeta_{84}^{10} + \zeta_{84}^{6}) q^{52} + ( - \zeta_{84}^{18} + \zeta_{84}^{16}) q^{53} + (\zeta_{84}^{35} - \zeta_{84}^{25} - \zeta_{84}^{19} + \zeta_{84}^{9}) q^{54} - \zeta_{84}^{9} q^{56} + ( - \zeta_{84}^{39} + \zeta_{84}^{27}) q^{62} + ( - \zeta_{84}^{31} + \zeta_{84}^{15} - \zeta_{84}^{5}) q^{63} - \zeta_{84}^{30} q^{64} + ( - \zeta_{84}^{38} - \zeta_{84}^{26} + \zeta_{84}^{22} - 1) q^{66} + \zeta_{84}^{28} q^{68} + ( - \zeta_{84}^{40} + \zeta_{84}^{38} - \zeta_{84}^{22} - \zeta_{84}^{14}) q^{69} + (\zeta_{84}^{25} + \zeta_{84}^{23}) q^{71} + (\zeta_{84}^{36} - \zeta_{84}^{26} + \zeta_{84}^{10}) q^{72} + (\zeta_{84}^{39} - \zeta_{84}^{23}) q^{75} + (\zeta_{84}^{22} - \zeta_{84}^{18}) q^{77} + (\zeta_{84}^{33} + \zeta_{84}^{29} + \zeta_{84}^{7} + \zeta_{84}^{3}) q^{78} + (\zeta_{84}^{33} + \zeta_{84}^{23}) q^{79} + (\zeta_{84}^{32} - \zeta_{84}^{22} - \zeta_{84}^{16} - \zeta_{84}^{6} + 1) q^{81} + ( - \zeta_{84}^{32} - \zeta_{84}^{6}) q^{84} + ( - \zeta_{84}^{39} - \zeta_{84}) q^{88} + ( - \zeta_{84}^{24} - \zeta_{84}^{8}) q^{89} + ( - \zeta_{84}^{29} - \zeta_{84}^{25}) q^{91} + (\zeta_{84}^{39} - \zeta_{84}^{15}) q^{92} + ( - \zeta_{84}^{36} + \zeta_{84}^{24} + \zeta_{84}^{20} - \zeta_{84}^{8}) q^{93} + ( - \zeta_{84}^{27} + \zeta_{84}^{11}) q^{96} + \zeta_{84}^{28} q^{98} + (\zeta_{84}^{39} - \zeta_{84}^{35} - \zeta_{84}^{23} + \zeta_{84}^{19} + \zeta_{84}^{7} - \zeta_{84}^{3}) q^{99} +O(q^{100})$$ q - z^40 * q^2 + (-z^37 + z^21) * q^3 - z^38 * q^4 + (-z^35 + z^19) * q^6 - z^15 * q^7 - z^36 * q^8 + (-z^32 + z^16 - 1) * q^9 + (-z^7 + z^3) * q^11 + (-z^33 + z^17) * q^12 + (z^14 + z^10) * q^13 - z^13 * q^14 - z^34 * q^16 + z^32 * q^17 + (z^40 - z^30 + z^14) * q^18 + (-z^36 - z^10) * q^21 + (-z^5 + z) * q^22 + (-z^19 - z) * q^23 + (-z^31 + z^15) * q^24 - z^2 * q^25 + (z^12 + z^8) * q^26 + (z^37 - z^27 - z^21 - z^11) * q^27 - z^11 * q^28 + (-z^41 + z^29) * q^31 - z^32 * q^32 + (-z^40 - z^28 + z^24 - z^2) * q^33 + z^30 * q^34 + (z^38 - z^28 + z^12) * q^36 + (z^35 + z^31 + z^9 + z^5) * q^39 + (-z^34 - z^8) * q^42 + (-z^41 - z^3) * q^44 + (z^41 - z^17) * q^46 + (-z^29 + z^13) * q^48 + z^30 * q^49 - q^50 + (z^27 - z^11) * q^51 + (z^10 + z^6) * q^52 + (-z^18 + z^16) * q^53 + (z^35 - z^25 - z^19 + z^9) * q^54 - z^9 * q^56 + (-z^39 + z^27) * q^62 + (-z^31 + z^15 - z^5) * q^63 - z^30 * q^64 + (-z^38 - z^26 + z^22 - 1) * q^66 + z^28 * q^68 + (-z^40 + z^38 - z^22 - z^14) * q^69 + (z^25 + z^23) * q^71 + (z^36 - z^26 + z^10) * q^72 + (z^39 - z^23) * q^75 + (z^22 - z^18) * q^77 + (z^33 + z^29 + z^7 + z^3) * q^78 + (z^33 + z^23) * q^79 + (z^32 - z^22 - z^16 - z^6 + 1) * q^81 + (-z^32 - z^6) * q^84 + (-z^39 - z) * q^88 + (-z^24 - z^8) * q^89 + (-z^29 - z^25) * q^91 + (z^39 - z^15) * q^92 + (-z^36 + z^24 + z^20 - z^8) * q^93 + (-z^27 + z^11) * q^96 + z^28 * q^98 + (z^39 - z^35 - z^23 + z^19 + z^7 - z^3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9}+O(q^{10})$$ 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9 $$24 q - 2 q^{2} + 2 q^{4} + 4 q^{8} - 24 q^{9} + 10 q^{13} + 2 q^{16} + 2 q^{17} + 10 q^{18} + 6 q^{21} + 2 q^{25} - 2 q^{26} - 2 q^{32} + 8 q^{33} + 4 q^{34} + 6 q^{36} + 4 q^{49} - 24 q^{50} + 2 q^{52} - 2 q^{53} - 4 q^{64} - 22 q^{66} - 12 q^{68} - 14 q^{69} - 4 q^{72} - 6 q^{77} + 30 q^{81} - 6 q^{84} + 2 q^{89} - 12 q^{98}+O(q^{100})$$ 24 * q - 2 * q^2 + 2 * q^4 + 4 * q^8 - 24 * q^9 + 10 * q^13 + 2 * q^16 + 2 * q^17 + 10 * q^18 + 6 * q^21 + 2 * q^25 - 2 * q^26 - 2 * q^32 + 8 * q^33 + 4 * q^34 + 6 * q^36 + 4 * q^49 - 24 * q^50 + 2 * q^52 - 2 * q^53 - 4 * q^64 - 22 * q^66 - 12 * q^68 - 14 * q^69 - 4 * q^72 - 6 * q^77 + 30 * q^81 - 6 * q^84 + 2 * q^89 - 12 * q^98

## Character values

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

 $$n$$ $$785$$ $$885$$ $$1667$$ $$\chi(n)$$ $$-1$$ $$\zeta_{84}^{16}$$ $$-1$$

## 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
 −0.294755 + 0.955573i 0.294755 − 0.955573i −0.294755 − 0.955573i 0.294755 + 0.955573i 0.680173 − 0.733052i −0.680173 + 0.733052i −0.997204 + 0.0747301i 0.997204 − 0.0747301i 0.930874 − 0.365341i −0.930874 + 0.365341i −0.149042 + 0.988831i 0.149042 − 0.988831i −0.997204 − 0.0747301i 0.997204 + 0.0747301i 0.930874 + 0.365341i −0.930874 − 0.365341i −0.563320 + 0.826239i 0.563320 − 0.826239i −0.563320 − 0.826239i 0.563320 + 0.826239i
−0.826239 + 0.563320i −0.997204 + 0.925270i 0.365341 0.930874i 0 0.302705 1.32624i 0.974928 0.222521i 0.222521 + 0.974928i 0.0635609 0.848162i 0
135.2 −0.826239 + 0.563320i 0.997204 0.925270i 0.365341 0.930874i 0 −0.302705 + 1.32624i −0.974928 + 0.222521i 0.222521 + 0.974928i 0.0635609 0.848162i 0
543.1 −0.826239 0.563320i −0.997204 0.925270i 0.365341 + 0.930874i 0 0.302705 + 1.32624i 0.974928 + 0.222521i 0.222521 0.974928i 0.0635609 + 0.848162i 0
543.2 −0.826239 0.563320i 0.997204 + 0.925270i 0.365341 + 0.930874i 0 −0.302705 1.32624i −0.974928 0.222521i 0.222521 0.974928i 0.0635609 + 0.848162i 0
611.1 −0.0747301 + 0.997204i −0.563320 + 0.173761i −0.988831 0.149042i 0 −0.131178 0.574730i −0.974928 0.222521i 0.222521 0.974928i −0.539102 + 0.367554i 0
611.2 −0.0747301 + 0.997204i 0.563320 0.173761i −0.988831 0.149042i 0 0.131178 + 0.574730i 0.974928 + 0.222521i 0.222521 0.974928i −0.539102 + 0.367554i 0
1019.1 0.988831 + 0.149042i −0.930874 + 0.634659i 0.955573 + 0.294755i 0 −1.01507 + 0.488831i 0.433884 0.900969i 0.900969 + 0.433884i 0.0983929 0.250701i 0
1019.2 0.988831 + 0.149042i 0.930874 0.634659i 0.955573 + 0.294755i 0 1.01507 0.488831i −0.433884 + 0.900969i 0.900969 + 0.433884i 0.0983929 0.250701i 0
1087.1 0.733052 + 0.680173i −0.294755 0.0444272i 0.0747301 + 0.997204i 0 −0.185853 0.233052i −0.781831 0.623490i −0.623490 + 0.781831i −0.870666 0.268565i 0
1087.2 0.733052 + 0.680173i 0.294755 + 0.0444272i 0.0747301 + 0.997204i 0 0.185853 + 0.233052i 0.781831 + 0.623490i −0.623490 + 0.781831i −0.870666 0.268565i 0
1495.1 −0.955573 + 0.294755i −0.680173 1.73305i 0.826239 0.563320i 0 1.16078 + 1.45557i −0.781831 0.623490i −0.623490 + 0.781831i −1.80778 + 1.67738i 0
1495.2 −0.955573 + 0.294755i 0.680173 + 1.73305i 0.826239 0.563320i 0 −1.16078 1.45557i 0.781831 + 0.623490i −0.623490 + 0.781831i −1.80778 + 1.67738i 0
1563.1 0.988831 0.149042i −0.930874 0.634659i 0.955573 0.294755i 0 −1.01507 0.488831i 0.433884 + 0.900969i 0.900969 0.433884i 0.0983929 + 0.250701i 0
1563.2 0.988831 0.149042i 0.930874 + 0.634659i 0.955573 0.294755i 0 1.01507 + 0.488831i −0.433884 0.900969i 0.900969 0.433884i 0.0983929 + 0.250701i 0
1971.1 0.733052 0.680173i −0.294755 + 0.0444272i 0.0747301 0.997204i 0 −0.185853 + 0.233052i −0.781831 + 0.623490i −0.623490 0.781831i −0.870666 + 0.268565i 0
1971.2 0.733052 0.680173i 0.294755 0.0444272i 0.0747301 0.997204i 0 0.185853 0.233052i 0.781831 0.623490i −0.623490 0.781831i −0.870666 + 0.268565i 0
2447.1 −0.365341 + 0.930874i −0.149042 + 1.98883i −0.733052 0.680173i 0 −1.79690 0.865341i −0.433884 0.900969i 0.900969 0.433884i −2.94440 0.443797i 0
2447.2 −0.365341 + 0.930874i 0.149042 1.98883i −0.733052 0.680173i 0 1.79690 + 0.865341i 0.433884 + 0.900969i 0.900969 0.433884i −2.94440 0.443797i 0
2515.1 −0.365341 0.930874i −0.149042 1.98883i −0.733052 + 0.680173i 0 −1.79690 + 0.865341i −0.433884 + 0.900969i 0.900969 + 0.433884i −2.94440 + 0.443797i 0
2515.2 −0.365341 0.930874i 0.149042 + 1.98883i −0.733052 + 0.680173i 0 1.79690 0.865341i 0.433884 0.900969i 0.900969 + 0.433884i −2.94440 + 0.443797i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 135.2 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.g even 21 1 inner
196.o odd 42 1 inner
833.z even 42 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.c 24
4.b odd 2 1 inner 3332.1.cc.c 24
17.b even 2 1 inner 3332.1.cc.c 24
49.g even 21 1 inner 3332.1.cc.c 24
68.d odd 2 1 CM 3332.1.cc.c 24
196.o odd 42 1 inner 3332.1.cc.c 24
833.z even 42 1 inner 3332.1.cc.c 24
3332.cc odd 42 1 inner 3332.1.cc.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.cc.c 24 1.a even 1 1 trivial
3332.1.cc.c 24 4.b odd 2 1 inner
3332.1.cc.c 24 17.b even 2 1 inner
3332.1.cc.c 24 49.g even 21 1 inner
3332.1.cc.c 24 68.d odd 2 1 CM
3332.1.cc.c 24 196.o odd 42 1 inner
3332.1.cc.c 24 833.z even 42 1 inner
3332.1.cc.c 24 3332.cc odd 42 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} + 11 T_{3}^{22} + 49 T_{3}^{20} + 118 T_{3}^{18} + 192 T_{3}^{16} + 336 T_{3}^{14} + 421 T_{3}^{12} - 168 T_{3}^{10} + 999 T_{3}^{8} - 736 T_{3}^{6} + 231 T_{3}^{4} - 26 T_{3}^{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3332, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{12} + T^{11} - T^{9} - T^{8} + T^{6} - T^{4} - T^{3} + \cdots + 1)^{2}$$
$3$ $$T^{24} + 11 T^{22} + 49 T^{20} + 118 T^{18} + \cdots + 1$$
$5$ $$T^{24}$$
$7$ $$(T^{12} - T^{10} + T^{8} - T^{6} + T^{4} - T^{2} + 1)^{2}$$
$11$ $$T^{24} - 10 T^{22} + 49 T^{20} - 134 T^{18} + \cdots + 1$$
$13$ $$(T^{12} - 5 T^{11} + 17 T^{10} - 38 T^{9} + \cdots + 1)^{2}$$
$17$ $$(T^{12} - T^{11} + T^{9} - T^{8} + T^{6} - T^{4} + T^{3} + \cdots + 1)^{2}$$
$19$ $$T^{24}$$
$23$ $$T^{24} + 70 T^{18} - 98 T^{16} + \cdots + 2401$$
$29$ $$T^{24}$$
$31$ $$(T^{12} + 7 T^{10} + 35 T^{8} + 84 T^{6} + \cdots + 49)^{2}$$
$37$ $$T^{24}$$
$41$ $$T^{24}$$
$43$ $$T^{24}$$
$47$ $$T^{24}$$
$53$ $$(T^{12} + T^{11} - T^{9} + 6 T^{8} - 21 T^{7} + \cdots + 1)^{2}$$
$59$ $$T^{24}$$
$61$ $$T^{24}$$
$67$ $$T^{24}$$
$71$ $$T^{24} + 6 T^{22} + 13 T^{20} + 38 T^{18} + \cdots + 1$$
$73$ $$T^{24}$$
$79$ $$T^{24} + 11 T^{22} + 77 T^{20} + 328 T^{18} + \cdots + 1$$
$83$ $$T^{24}$$
$89$ $$(T^{12} - T^{11} - 6 T^{9} + 6 T^{8} - 7 T^{7} + \cdots + 1)^{2}$$
$97$ $$T^{24}$$