# Properties

 Label 169.4.e.b Level $169$ Weight $4$ Character orbit 169.e Analytic conductor $9.971$ Analytic rank $0$ Dimension $2$ Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(23,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("169.23");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$169 = 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 169.e (of order $$6$$, degree $$2$$, not 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: $$9.97132279097$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} + 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 14 \zeta_{6} + 28) q^{6} + (13 \zeta_{6} - 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8} - 22 \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z + 2) * q^2 + (-7*z + 7) * q^3 + 4*z * q^4 + (-16*z + 8) * q^5 + (-14*z + 28) * q^6 + (13*z - 26) * q^7 + (-16*z + 8) * q^8 - 22*z * q^9 $$q + (2 \zeta_{6} + 2) q^{2} + ( - 7 \zeta_{6} + 7) q^{3} + 4 \zeta_{6} q^{4} + ( - 16 \zeta_{6} + 8) q^{5} + ( - 14 \zeta_{6} + 28) q^{6} + (13 \zeta_{6} - 26) q^{7} + ( - 16 \zeta_{6} + 8) q^{8} - 22 \zeta_{6} q^{9} + ( - 48 \zeta_{6} + 48) q^{10} + (13 \zeta_{6} + 13) q^{11} + 28 q^{12} - 78 q^{14} + ( - 56 \zeta_{6} - 56) q^{15} + ( - 80 \zeta_{6} + 80) q^{16} - 27 \zeta_{6} q^{17} + ( - 88 \zeta_{6} + 44) q^{18} + ( - 51 \zeta_{6} + 102) q^{19} + ( - 32 \zeta_{6} + 64) q^{20} + (182 \zeta_{6} - 91) q^{21} + 78 \zeta_{6} q^{22} + (57 \zeta_{6} - 57) q^{23} + ( - 56 \zeta_{6} - 56) q^{24} - 67 q^{25} + 35 q^{27} + ( - 52 \zeta_{6} - 52) q^{28} + ( - 69 \zeta_{6} + 69) q^{29} - 336 \zeta_{6} q^{30} + (84 \zeta_{6} - 42) q^{31} + ( - 96 \zeta_{6} + 192) q^{32} + ( - 91 \zeta_{6} + 182) q^{33} + ( - 108 \zeta_{6} + 54) q^{34} + 312 \zeta_{6} q^{35} + ( - 88 \zeta_{6} + 88) q^{36} + (23 \zeta_{6} + 23) q^{37} + 306 q^{38} - 192 q^{40} + (227 \zeta_{6} + 227) q^{41} + (546 \zeta_{6} - 546) q^{42} + 85 \zeta_{6} q^{43} + (104 \zeta_{6} - 52) q^{44} + (176 \zeta_{6} - 352) q^{45} + (114 \zeta_{6} - 228) q^{46} + (396 \zeta_{6} - 198) q^{47} - 560 \zeta_{6} q^{48} + ( - 164 \zeta_{6} + 164) q^{49} + ( - 134 \zeta_{6} - 134) q^{50} - 189 q^{51} + 426 q^{53} + (70 \zeta_{6} + 70) q^{54} + ( - 312 \zeta_{6} + 312) q^{55} + 312 \zeta_{6} q^{56} + ( - 714 \zeta_{6} + 357) q^{57} + ( - 138 \zeta_{6} + 276) q^{58} + ( - 11 \zeta_{6} + 22) q^{59} + ( - 448 \zeta_{6} + 224) q^{60} + 17 \zeta_{6} q^{61} + (252 \zeta_{6} - 252) q^{62} + (286 \zeta_{6} + 286) q^{63} - 64 q^{64} + 546 q^{66} + ( - 95 \zeta_{6} - 95) q^{67} + ( - 108 \zeta_{6} + 108) q^{68} + 399 \zeta_{6} q^{69} + (1248 \zeta_{6} - 624) q^{70} + (337 \zeta_{6} - 674) q^{71} + (176 \zeta_{6} - 352) q^{72} + ( - 1160 \zeta_{6} + 580) q^{73} + 138 \zeta_{6} q^{74} + (469 \zeta_{6} - 469) q^{75} + (204 \zeta_{6} + 204) q^{76} - 507 q^{77} - 1244 q^{79} + ( - 640 \zeta_{6} - 640) q^{80} + ( - 839 \zeta_{6} + 839) q^{81} + 1362 \zeta_{6} q^{82} + (492 \zeta_{6} - 246) q^{83} + (364 \zeta_{6} - 728) q^{84} + (216 \zeta_{6} - 432) q^{85} + (340 \zeta_{6} - 170) q^{86} - 483 \zeta_{6} q^{87} + ( - 312 \zeta_{6} + 312) q^{88} + ( - 177 \zeta_{6} - 177) q^{89} - 1056 q^{90} - 228 q^{92} + (294 \zeta_{6} + 294) q^{93} + (1188 \zeta_{6} - 1188) q^{94} - 1224 \zeta_{6} q^{95} + ( - 1344 \zeta_{6} + 672) q^{96} + (713 \zeta_{6} - 1426) q^{97} + ( - 328 \zeta_{6} + 656) q^{98} + ( - 572 \zeta_{6} + 286) q^{99} +O(q^{100})$$ q + (2*z + 2) * q^2 + (-7*z + 7) * q^3 + 4*z * q^4 + (-16*z + 8) * q^5 + (-14*z + 28) * q^6 + (13*z - 26) * q^7 + (-16*z + 8) * q^8 - 22*z * q^9 + (-48*z + 48) * q^10 + (13*z + 13) * q^11 + 28 * q^12 - 78 * q^14 + (-56*z - 56) * q^15 + (-80*z + 80) * q^16 - 27*z * q^17 + (-88*z + 44) * q^18 + (-51*z + 102) * q^19 + (-32*z + 64) * q^20 + (182*z - 91) * q^21 + 78*z * q^22 + (57*z - 57) * q^23 + (-56*z - 56) * q^24 - 67 * q^25 + 35 * q^27 + (-52*z - 52) * q^28 + (-69*z + 69) * q^29 - 336*z * q^30 + (84*z - 42) * q^31 + (-96*z + 192) * q^32 + (-91*z + 182) * q^33 + (-108*z + 54) * q^34 + 312*z * q^35 + (-88*z + 88) * q^36 + (23*z + 23) * q^37 + 306 * q^38 - 192 * q^40 + (227*z + 227) * q^41 + (546*z - 546) * q^42 + 85*z * q^43 + (104*z - 52) * q^44 + (176*z - 352) * q^45 + (114*z - 228) * q^46 + (396*z - 198) * q^47 - 560*z * q^48 + (-164*z + 164) * q^49 + (-134*z - 134) * q^50 - 189 * q^51 + 426 * q^53 + (70*z + 70) * q^54 + (-312*z + 312) * q^55 + 312*z * q^56 + (-714*z + 357) * q^57 + (-138*z + 276) * q^58 + (-11*z + 22) * q^59 + (-448*z + 224) * q^60 + 17*z * q^61 + (252*z - 252) * q^62 + (286*z + 286) * q^63 - 64 * q^64 + 546 * q^66 + (-95*z - 95) * q^67 + (-108*z + 108) * q^68 + 399*z * q^69 + (1248*z - 624) * q^70 + (337*z - 674) * q^71 + (176*z - 352) * q^72 + (-1160*z + 580) * q^73 + 138*z * q^74 + (469*z - 469) * q^75 + (204*z + 204) * q^76 - 507 * q^77 - 1244 * q^79 + (-640*z - 640) * q^80 + (-839*z + 839) * q^81 + 1362*z * q^82 + (492*z - 246) * q^83 + (364*z - 728) * q^84 + (216*z - 432) * q^85 + (340*z - 170) * q^86 - 483*z * q^87 + (-312*z + 312) * q^88 + (-177*z - 177) * q^89 - 1056 * q^90 - 228 * q^92 + (294*z + 294) * q^93 + (1188*z - 1188) * q^94 - 1224*z * q^95 + (-1344*z + 672) * q^96 + (713*z - 1426) * q^97 + (-328*z + 656) * q^98 + (-572*z + 286) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{2} + 7 q^{3} + 4 q^{4} + 42 q^{6} - 39 q^{7} - 22 q^{9}+O(q^{10})$$ 2 * q + 6 * q^2 + 7 * q^3 + 4 * q^4 + 42 * q^6 - 39 * q^7 - 22 * q^9 $$2 q + 6 q^{2} + 7 q^{3} + 4 q^{4} + 42 q^{6} - 39 q^{7} - 22 q^{9} + 48 q^{10} + 39 q^{11} + 56 q^{12} - 156 q^{14} - 168 q^{15} + 80 q^{16} - 27 q^{17} + 153 q^{19} + 96 q^{20} + 78 q^{22} - 57 q^{23} - 168 q^{24} - 134 q^{25} + 70 q^{27} - 156 q^{28} + 69 q^{29} - 336 q^{30} + 288 q^{32} + 273 q^{33} + 312 q^{35} + 88 q^{36} + 69 q^{37} + 612 q^{38} - 384 q^{40} + 681 q^{41} - 546 q^{42} + 85 q^{43} - 528 q^{45} - 342 q^{46} - 560 q^{48} + 164 q^{49} - 402 q^{50} - 378 q^{51} + 852 q^{53} + 210 q^{54} + 312 q^{55} + 312 q^{56} + 414 q^{58} + 33 q^{59} + 17 q^{61} - 252 q^{62} + 858 q^{63} - 128 q^{64} + 1092 q^{66} - 285 q^{67} + 108 q^{68} + 399 q^{69} - 1011 q^{71} - 528 q^{72} + 138 q^{74} - 469 q^{75} + 612 q^{76} - 1014 q^{77} - 2488 q^{79} - 1920 q^{80} + 839 q^{81} + 1362 q^{82} - 1092 q^{84} - 648 q^{85} - 483 q^{87} + 312 q^{88} - 531 q^{89} - 2112 q^{90} - 456 q^{92} + 882 q^{93} - 1188 q^{94} - 1224 q^{95} - 2139 q^{97} + 984 q^{98}+O(q^{100})$$ 2 * q + 6 * q^2 + 7 * q^3 + 4 * q^4 + 42 * q^6 - 39 * q^7 - 22 * q^9 + 48 * q^10 + 39 * q^11 + 56 * q^12 - 156 * q^14 - 168 * q^15 + 80 * q^16 - 27 * q^17 + 153 * q^19 + 96 * q^20 + 78 * q^22 - 57 * q^23 - 168 * q^24 - 134 * q^25 + 70 * q^27 - 156 * q^28 + 69 * q^29 - 336 * q^30 + 288 * q^32 + 273 * q^33 + 312 * q^35 + 88 * q^36 + 69 * q^37 + 612 * q^38 - 384 * q^40 + 681 * q^41 - 546 * q^42 + 85 * q^43 - 528 * q^45 - 342 * q^46 - 560 * q^48 + 164 * q^49 - 402 * q^50 - 378 * q^51 + 852 * q^53 + 210 * q^54 + 312 * q^55 + 312 * q^56 + 414 * q^58 + 33 * q^59 + 17 * q^61 - 252 * q^62 + 858 * q^63 - 128 * q^64 + 1092 * q^66 - 285 * q^67 + 108 * q^68 + 399 * q^69 - 1011 * q^71 - 528 * q^72 + 138 * q^74 - 469 * q^75 + 612 * q^76 - 1014 * q^77 - 2488 * q^79 - 1920 * q^80 + 839 * q^81 + 1362 * q^82 - 1092 * q^84 - 648 * q^85 - 483 * q^87 + 312 * q^88 - 531 * q^89 - 2112 * q^90 - 456 * q^92 + 882 * q^93 - 1188 * q^94 - 1224 * q^95 - 2139 * q^97 + 984 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
23.1
 0.5 − 0.866025i 0.5 + 0.866025i
3.00000 1.73205i 3.50000 + 6.06218i 2.00000 3.46410i 13.8564i 21.0000 + 12.1244i −19.5000 11.2583i 13.8564i −11.0000 + 19.0526i 24.0000 + 41.5692i
147.1 3.00000 + 1.73205i 3.50000 6.06218i 2.00000 + 3.46410i 13.8564i 21.0000 12.1244i −19.5000 + 11.2583i 13.8564i −11.0000 19.0526i 24.0000 41.5692i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 169.4.e.b 2
13.b even 2 1 13.4.e.a 2
13.c even 3 1 13.4.e.a 2
13.c even 3 1 169.4.b.b 2
13.d odd 4 2 169.4.c.i 4
13.e even 6 1 169.4.b.b 2
13.e even 6 1 inner 169.4.e.b 2
13.f odd 12 2 169.4.a.h 2
13.f odd 12 2 169.4.c.i 4
39.d odd 2 1 117.4.q.c 2
39.i odd 6 1 117.4.q.c 2
39.k even 12 2 1521.4.a.q 2
52.b odd 2 1 208.4.w.a 2
52.j odd 6 1 208.4.w.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 13.b even 2 1
13.4.e.a 2 13.c even 3 1
117.4.q.c 2 39.d odd 2 1
117.4.q.c 2 39.i odd 6 1
169.4.a.h 2 13.f odd 12 2
169.4.b.b 2 13.c even 3 1
169.4.b.b 2 13.e even 6 1
169.4.c.i 4 13.d odd 4 2
169.4.c.i 4 13.f odd 12 2
169.4.e.b 2 1.a even 1 1 trivial
169.4.e.b 2 13.e even 6 1 inner
208.4.w.a 2 52.b odd 2 1
208.4.w.a 2 52.j odd 6 1
1521.4.a.q 2 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 6T_{2} + 12$$ acting on $$S_{4}^{\mathrm{new}}(169, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 6T + 12$$
$3$ $$T^{2} - 7T + 49$$
$5$ $$T^{2} + 192$$
$7$ $$T^{2} + 39T + 507$$
$11$ $$T^{2} - 39T + 507$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 27T + 729$$
$19$ $$T^{2} - 153T + 7803$$
$23$ $$T^{2} + 57T + 3249$$
$29$ $$T^{2} - 69T + 4761$$
$31$ $$T^{2} + 5292$$
$37$ $$T^{2} - 69T + 1587$$
$41$ $$T^{2} - 681T + 154587$$
$43$ $$T^{2} - 85T + 7225$$
$47$ $$T^{2} + 117612$$
$53$ $$(T - 426)^{2}$$
$59$ $$T^{2} - 33T + 363$$
$61$ $$T^{2} - 17T + 289$$
$67$ $$T^{2} + 285T + 27075$$
$71$ $$T^{2} + 1011 T + 340707$$
$73$ $$T^{2} + 1009200$$
$79$ $$(T + 1244)^{2}$$
$83$ $$T^{2} + 181548$$
$89$ $$T^{2} + 531T + 93987$$
$97$ $$T^{2} + 2139 T + 1525107$$
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