# Properties

 Label 3.9.b.a Level $3$ Weight $9$ Character orbit 3.b Analytic conductor $1.222$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3,9,Mod(2,3)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("3.2");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$, degree $$1$$, 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: $$1.22213583018$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 14$$ x^2 + 14 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 6\sqrt{-14}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( - 3 \beta + 45) q^{3} - 248 q^{4} - 10 \beta q^{5} + (45 \beta + 1512) q^{6} - 1750 q^{7} + 8 \beta q^{8} + ( - 270 \beta - 2511) q^{9} +O(q^{10})$$ q + b * q^2 + (-3*b + 45) * q^3 - 248 * q^4 - 10*b * q^5 + (45*b + 1512) * q^6 - 1750 * q^7 + 8*b * q^8 + (-270*b - 2511) * q^9 $$q + \beta q^{2} + ( - 3 \beta + 45) q^{3} - 248 q^{4} - 10 \beta q^{5} + (45 \beta + 1512) q^{6} - 1750 q^{7} + 8 \beta q^{8} + ( - 270 \beta - 2511) q^{9} + 5040 q^{10} + 310 \beta q^{11} + (744 \beta - 11160) q^{12} + 25730 q^{13} - 1750 \beta q^{14} + ( - 450 \beta - 15120) q^{15} - 67520 q^{16} + 3336 \beta q^{17} + ( - 2511 \beta + 136080) q^{18} + 18938 q^{19} + 2480 \beta q^{20} + (5250 \beta - 78750) q^{21} - 156240 q^{22} - 20956 \beta q^{23} + (360 \beta + 12096) q^{24} + 340225 q^{25} + 25730 \beta q^{26} + ( - 4617 \beta - 521235) q^{27} + 434000 q^{28} + 20530 \beta q^{29} + ( - 15120 \beta + 226800) q^{30} - 351478 q^{31} - 65472 \beta q^{32} + (13950 \beta + 468720) q^{33} - 1681344 q^{34} + 17500 \beta q^{35} + (66960 \beta + 622728) q^{36} + 1335170 q^{37} + 18938 \beta q^{38} + ( - 77190 \beta + 1157850) q^{39} + 40320 q^{40} + 83540 \beta q^{41} + ( - 78750 \beta - 2646000) q^{42} - 3526150 q^{43} - 76880 \beta q^{44} + (25110 \beta - 1360800) q^{45} + 10561824 q^{46} - 181784 \beta q^{47} + (202560 \beta - 3038400) q^{48} - 2702301 q^{49} + 340225 \beta q^{50} + (150120 \beta + 5044032) q^{51} - 6381040 q^{52} - 294066 \beta q^{53} + ( - 521235 \beta + 2326968) q^{54} + 1562400 q^{55} - 14000 \beta q^{56} + ( - 56814 \beta + 852210) q^{57} - 10347120 q^{58} + 610910 \beta q^{59} + (111600 \beta + 3749760) q^{60} + 753602 q^{61} - 351478 \beta q^{62} + (472500 \beta + 4394250) q^{63} + 15712768 q^{64} - 257300 \beta q^{65} + (468720 \beta - 7030800) q^{66} + 2268890 q^{67} - 827328 \beta q^{68} + ( - 943020 \beta - 31685472) q^{69} - 8820000 q^{70} + 758220 \beta q^{71} + ( - 20088 \beta + 1088640) q^{72} + 27672770 q^{73} + 1335170 \beta q^{74} + ( - 1020675 \beta + 15310125) q^{75} - 4696624 q^{76} - 542500 \beta q^{77} + (1157850 \beta + 38903760) q^{78} - 22980982 q^{79} + 675200 \beta q^{80} + (1355940 \beta - 30436479) q^{81} - 42104160 q^{82} - 2066606 \beta q^{83} + ( - 1302000 \beta + 19530000) q^{84} + 16813440 q^{85} - 3526150 \beta q^{86} + (923850 \beta + 31041360) q^{87} - 1249920 q^{88} + 3234540 \beta q^{89} + ( - 1360800 \beta - 12655440) q^{90} - 45027500 q^{91} + 5197088 \beta q^{92} + (1054434 \beta - 15816510) q^{93} + 91619136 q^{94} - 189380 \beta q^{95} + ( - 2946240 \beta - 98993664) q^{96} + 147271010 q^{97} - 2702301 \beta q^{98} + ( - 778410 \beta + 42184800) q^{99} +O(q^{100})$$ q + b * q^2 + (-3*b + 45) * q^3 - 248 * q^4 - 10*b * q^5 + (45*b + 1512) * q^6 - 1750 * q^7 + 8*b * q^8 + (-270*b - 2511) * q^9 + 5040 * q^10 + 310*b * q^11 + (744*b - 11160) * q^12 + 25730 * q^13 - 1750*b * q^14 + (-450*b - 15120) * q^15 - 67520 * q^16 + 3336*b * q^17 + (-2511*b + 136080) * q^18 + 18938 * q^19 + 2480*b * q^20 + (5250*b - 78750) * q^21 - 156240 * q^22 - 20956*b * q^23 + (360*b + 12096) * q^24 + 340225 * q^25 + 25730*b * q^26 + (-4617*b - 521235) * q^27 + 434000 * q^28 + 20530*b * q^29 + (-15120*b + 226800) * q^30 - 351478 * q^31 - 65472*b * q^32 + (13950*b + 468720) * q^33 - 1681344 * q^34 + 17500*b * q^35 + (66960*b + 622728) * q^36 + 1335170 * q^37 + 18938*b * q^38 + (-77190*b + 1157850) * q^39 + 40320 * q^40 + 83540*b * q^41 + (-78750*b - 2646000) * q^42 - 3526150 * q^43 - 76880*b * q^44 + (25110*b - 1360800) * q^45 + 10561824 * q^46 - 181784*b * q^47 + (202560*b - 3038400) * q^48 - 2702301 * q^49 + 340225*b * q^50 + (150120*b + 5044032) * q^51 - 6381040 * q^52 - 294066*b * q^53 + (-521235*b + 2326968) * q^54 + 1562400 * q^55 - 14000*b * q^56 + (-56814*b + 852210) * q^57 - 10347120 * q^58 + 610910*b * q^59 + (111600*b + 3749760) * q^60 + 753602 * q^61 - 351478*b * q^62 + (472500*b + 4394250) * q^63 + 15712768 * q^64 - 257300*b * q^65 + (468720*b - 7030800) * q^66 + 2268890 * q^67 - 827328*b * q^68 + (-943020*b - 31685472) * q^69 - 8820000 * q^70 + 758220*b * q^71 + (-20088*b + 1088640) * q^72 + 27672770 * q^73 + 1335170*b * q^74 + (-1020675*b + 15310125) * q^75 - 4696624 * q^76 - 542500*b * q^77 + (1157850*b + 38903760) * q^78 - 22980982 * q^79 + 675200*b * q^80 + (1355940*b - 30436479) * q^81 - 42104160 * q^82 - 2066606*b * q^83 + (-1302000*b + 19530000) * q^84 + 16813440 * q^85 - 3526150*b * q^86 + (923850*b + 31041360) * q^87 - 1249920 * q^88 + 3234540*b * q^89 + (-1360800*b - 12655440) * q^90 - 45027500 * q^91 + 5197088*b * q^92 + (1054434*b - 15816510) * q^93 + 91619136 * q^94 - 189380*b * q^95 + (-2946240*b - 98993664) * q^96 + 147271010 * q^97 - 2702301*b * q^98 + (-778410*b + 42184800) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 90 q^{3} - 496 q^{4} + 3024 q^{6} - 3500 q^{7} - 5022 q^{9}+O(q^{10})$$ 2 * q + 90 * q^3 - 496 * q^4 + 3024 * q^6 - 3500 * q^7 - 5022 * q^9 $$2 q + 90 q^{3} - 496 q^{4} + 3024 q^{6} - 3500 q^{7} - 5022 q^{9} + 10080 q^{10} - 22320 q^{12} + 51460 q^{13} - 30240 q^{15} - 135040 q^{16} + 272160 q^{18} + 37876 q^{19} - 157500 q^{21} - 312480 q^{22} + 24192 q^{24} + 680450 q^{25} - 1042470 q^{27} + 868000 q^{28} + 453600 q^{30} - 702956 q^{31} + 937440 q^{33} - 3362688 q^{34} + 1245456 q^{36} + 2670340 q^{37} + 2315700 q^{39} + 80640 q^{40} - 5292000 q^{42} - 7052300 q^{43} - 2721600 q^{45} + 21123648 q^{46} - 6076800 q^{48} - 5404602 q^{49} + 10088064 q^{51} - 12762080 q^{52} + 4653936 q^{54} + 3124800 q^{55} + 1704420 q^{57} - 20694240 q^{58} + 7499520 q^{60} + 1507204 q^{61} + 8788500 q^{63} + 31425536 q^{64} - 14061600 q^{66} + 4537780 q^{67} - 63370944 q^{69} - 17640000 q^{70} + 2177280 q^{72} + 55345540 q^{73} + 30620250 q^{75} - 9393248 q^{76} + 77807520 q^{78} - 45961964 q^{79} - 60872958 q^{81} - 84208320 q^{82} + 39060000 q^{84} + 33626880 q^{85} + 62082720 q^{87} - 2499840 q^{88} - 25310880 q^{90} - 90055000 q^{91} - 31633020 q^{93} + 183238272 q^{94} - 197987328 q^{96} + 294542020 q^{97} + 84369600 q^{99}+O(q^{100})$$ 2 * q + 90 * q^3 - 496 * q^4 + 3024 * q^6 - 3500 * q^7 - 5022 * q^9 + 10080 * q^10 - 22320 * q^12 + 51460 * q^13 - 30240 * q^15 - 135040 * q^16 + 272160 * q^18 + 37876 * q^19 - 157500 * q^21 - 312480 * q^22 + 24192 * q^24 + 680450 * q^25 - 1042470 * q^27 + 868000 * q^28 + 453600 * q^30 - 702956 * q^31 + 937440 * q^33 - 3362688 * q^34 + 1245456 * q^36 + 2670340 * q^37 + 2315700 * q^39 + 80640 * q^40 - 5292000 * q^42 - 7052300 * q^43 - 2721600 * q^45 + 21123648 * q^46 - 6076800 * q^48 - 5404602 * q^49 + 10088064 * q^51 - 12762080 * q^52 + 4653936 * q^54 + 3124800 * q^55 + 1704420 * q^57 - 20694240 * q^58 + 7499520 * q^60 + 1507204 * q^61 + 8788500 * q^63 + 31425536 * q^64 - 14061600 * q^66 + 4537780 * q^67 - 63370944 * q^69 - 17640000 * q^70 + 2177280 * q^72 + 55345540 * q^73 + 30620250 * q^75 - 9393248 * q^76 + 77807520 * q^78 - 45961964 * q^79 - 60872958 * q^81 - 84208320 * q^82 + 39060000 * q^84 + 33626880 * q^85 + 62082720 * q^87 - 2499840 * q^88 - 25310880 * q^90 - 90055000 * q^91 - 31633020 * q^93 + 183238272 * q^94 - 197987328 * q^96 + 294542020 * q^97 + 84369600 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 − 3.74166i 3.74166i
22.4499i 45.0000 + 67.3498i −248.000 224.499i 1512.00 1010.25i −1750.00 179.600i −2511.00 + 6061.48i 5040.00
2.2 22.4499i 45.0000 67.3498i −248.000 224.499i 1512.00 + 1010.25i −1750.00 179.600i −2511.00 6061.48i 5040.00
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.9.b.a 2
3.b odd 2 1 inner 3.9.b.a 2
4.b odd 2 1 48.9.e.b 2
5.b even 2 1 75.9.c.c 2
5.c odd 4 2 75.9.d.b 4
8.b even 2 1 192.9.e.e 2
8.d odd 2 1 192.9.e.f 2
9.c even 3 2 81.9.d.d 4
9.d odd 6 2 81.9.d.d 4
12.b even 2 1 48.9.e.b 2
15.d odd 2 1 75.9.c.c 2
15.e even 4 2 75.9.d.b 4
24.f even 2 1 192.9.e.f 2
24.h odd 2 1 192.9.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.9.b.a 2 1.a even 1 1 trivial
3.9.b.a 2 3.b odd 2 1 inner
48.9.e.b 2 4.b odd 2 1
48.9.e.b 2 12.b even 2 1
75.9.c.c 2 5.b even 2 1
75.9.c.c 2 15.d odd 2 1
75.9.d.b 4 5.c odd 4 2
75.9.d.b 4 15.e even 4 2
81.9.d.d 4 9.c even 3 2
81.9.d.d 4 9.d odd 6 2
192.9.e.e 2 8.b even 2 1
192.9.e.e 2 24.h odd 2 1
192.9.e.f 2 8.d odd 2 1
192.9.e.f 2 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 504$$
$3$ $$T^{2} - 90T + 6561$$
$5$ $$T^{2} + 50400$$
$7$ $$(T + 1750)^{2}$$
$11$ $$T^{2} + 48434400$$
$13$ $$(T - 25730)^{2}$$
$17$ $$T^{2} + 5608963584$$
$19$ $$(T - 18938)^{2}$$
$23$ $$T^{2} + 221333583744$$
$29$ $$T^{2} + 212426373600$$
$31$ $$(T + 351478)^{2}$$
$37$ $$(T - 1335170)^{2}$$
$41$ $$T^{2} + 3517381526400$$
$43$ $$(T + 3526150)^{2}$$
$47$ $$T^{2} + 16654893018624$$
$53$ $$T^{2} + 43583305427424$$
$59$ $$T^{2} + 188098358162400$$
$61$ $$(T - 753602)^{2}$$
$67$ $$(T - 2268890)^{2}$$
$71$ $$T^{2} + 289748374473600$$
$73$ $$(T - 27672770)^{2}$$
$79$ $$(T + 22980982)^{2}$$
$83$ $$T^{2} + 21\!\cdots\!44$$
$89$ $$T^{2} + 52\!\cdots\!00$$
$97$ $$(T - 147271010)^{2}$$