# Properties

 Label 4.9.b.b Level $4$ Weight $9$ Character orbit 4.b Analytic conductor $1.630$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4,9,Mod(3,4)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4, 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("4.3");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4 = 2^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 4.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: $$1.62951444024$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-39})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 10$$ x^2 - x + 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ 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 $$\beta = 2\sqrt{-39}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 10) q^{2} - 8 \beta q^{3} + (20 \beta - 56) q^{4} + 610 q^{5} + (80 \beta - 1248) q^{6} - 112 \beta q^{7} + ( - 144 \beta + 3680) q^{8} - 3423 q^{9} +O(q^{10})$$ q + (-b - 10) * q^2 - 8*b * q^3 + (20*b - 56) * q^4 + 610 * q^5 + (80*b - 1248) * q^6 - 112*b * q^7 + (-144*b + 3680) * q^8 - 3423 * q^9 $$q + ( - \beta - 10) q^{2} - 8 \beta q^{3} + (20 \beta - 56) q^{4} + 610 q^{5} + (80 \beta - 1248) q^{6} - 112 \beta q^{7} + ( - 144 \beta + 3680) q^{8} - 3423 q^{9} + ( - 610 \beta - 6100) q^{10} + 1480 \beta q^{11} + (448 \beta + 24960) q^{12} - 5470 q^{13} + (1120 \beta - 17472) q^{14} - 4880 \beta q^{15} + ( - 2240 \beta - 59264) q^{16} + 73090 q^{17} + (3423 \beta + 34230) q^{18} + 1560 \beta q^{19} + (12200 \beta - 34160) q^{20} - 139776 q^{21} + ( - 14800 \beta + 230880) q^{22} + 18992 \beta q^{23} + ( - 29440 \beta - 179712) q^{24} - 18525 q^{25} + (5470 \beta + 54700) q^{26} - 25104 \beta q^{27} + (6272 \beta + 349440) q^{28} - 128222 q^{29} + (48800 \beta - 761280) q^{30} + 5440 \beta q^{31} + (81664 \beta + 243200) q^{32} + 1847040 q^{33} + ( - 73090 \beta - 730900) q^{34} - 68320 \beta q^{35} + ( - 68460 \beta + 191688) q^{36} - 3472030 q^{37} + ( - 15600 \beta + 243360) q^{38} + 43760 \beta q^{39} + ( - 87840 \beta + 2244800) q^{40} + 2146882 q^{41} + (139776 \beta + 1397760) q^{42} + 474632 \beta q^{43} + ( - 82880 \beta - 4617600) q^{44} - 2088030 q^{45} + ( - 189920 \beta + 2962752) q^{46} - 610592 \beta q^{47} + (474112 \beta - 2795520) q^{48} + 3807937 q^{49} + (18525 \beta + 185250) q^{50} - 584720 \beta q^{51} + ( - 109400 \beta + 306320) q^{52} + 824290 q^{53} + (251040 \beta - 3916224) q^{54} + 902800 \beta q^{55} + ( - 412160 \beta - 2515968) q^{56} + 1946880 q^{57} + (128222 \beta + 1282220) q^{58} + 298280 \beta q^{59} + (273280 \beta + 15225600) q^{60} - 14746078 q^{61} + ( - 54400 \beta + 848640) q^{62} + 383376 \beta q^{63} + ( - 1059840 \beta + 10307584) q^{64} - 3336700 q^{65} + ( - 1847040 \beta - 18470400) q^{66} - 1221512 \beta q^{67} + (1461800 \beta - 4093040) q^{68} + 23702016 q^{69} + (683200 \beta - 10657920) q^{70} + 95760 \beta q^{71} + (492912 \beta - 12596640) q^{72} - 5725630 q^{73} + (3472030 \beta + 34720300) q^{74} + 148200 \beta q^{75} + ( - 87360 \beta - 4867200) q^{76} + 25858560 q^{77} + ( - 437600 \beta + 6826560) q^{78} - 2875360 \beta q^{79} + ( - 1366400 \beta - 36151040) q^{80} - 53788095 q^{81} + ( - 2146882 \beta - 21468820) q^{82} + 4160152 \beta q^{83} + ( - 2795520 \beta + 7827456) q^{84} + 44584900 q^{85} + ( - 4746320 \beta + 74042592) q^{86} + 1025776 \beta q^{87} + (5446400 \beta + 33246720) q^{88} - 83324222 q^{89} + (2088030 \beta + 20880300) q^{90} + 612640 \beta q^{91} + ( - 1063552 \beta - 59255040) q^{92} + 6789120 q^{93} + (6105920 \beta - 95252352) q^{94} + 951600 \beta q^{95} + ( - 1945600 \beta + 101916672) q^{96} + 120619010 q^{97} + ( - 3807937 \beta - 38079370) q^{98} - 5066040 \beta q^{99} +O(q^{100})$$ q + (-b - 10) * q^2 - 8*b * q^3 + (20*b - 56) * q^4 + 610 * q^5 + (80*b - 1248) * q^6 - 112*b * q^7 + (-144*b + 3680) * q^8 - 3423 * q^9 + (-610*b - 6100) * q^10 + 1480*b * q^11 + (448*b + 24960) * q^12 - 5470 * q^13 + (1120*b - 17472) * q^14 - 4880*b * q^15 + (-2240*b - 59264) * q^16 + 73090 * q^17 + (3423*b + 34230) * q^18 + 1560*b * q^19 + (12200*b - 34160) * q^20 - 139776 * q^21 + (-14800*b + 230880) * q^22 + 18992*b * q^23 + (-29440*b - 179712) * q^24 - 18525 * q^25 + (5470*b + 54700) * q^26 - 25104*b * q^27 + (6272*b + 349440) * q^28 - 128222 * q^29 + (48800*b - 761280) * q^30 + 5440*b * q^31 + (81664*b + 243200) * q^32 + 1847040 * q^33 + (-73090*b - 730900) * q^34 - 68320*b * q^35 + (-68460*b + 191688) * q^36 - 3472030 * q^37 + (-15600*b + 243360) * q^38 + 43760*b * q^39 + (-87840*b + 2244800) * q^40 + 2146882 * q^41 + (139776*b + 1397760) * q^42 + 474632*b * q^43 + (-82880*b - 4617600) * q^44 - 2088030 * q^45 + (-189920*b + 2962752) * q^46 - 610592*b * q^47 + (474112*b - 2795520) * q^48 + 3807937 * q^49 + (18525*b + 185250) * q^50 - 584720*b * q^51 + (-109400*b + 306320) * q^52 + 824290 * q^53 + (251040*b - 3916224) * q^54 + 902800*b * q^55 + (-412160*b - 2515968) * q^56 + 1946880 * q^57 + (128222*b + 1282220) * q^58 + 298280*b * q^59 + (273280*b + 15225600) * q^60 - 14746078 * q^61 + (-54400*b + 848640) * q^62 + 383376*b * q^63 + (-1059840*b + 10307584) * q^64 - 3336700 * q^65 + (-1847040*b - 18470400) * q^66 - 1221512*b * q^67 + (1461800*b - 4093040) * q^68 + 23702016 * q^69 + (683200*b - 10657920) * q^70 + 95760*b * q^71 + (492912*b - 12596640) * q^72 - 5725630 * q^73 + (3472030*b + 34720300) * q^74 + 148200*b * q^75 + (-87360*b - 4867200) * q^76 + 25858560 * q^77 + (-437600*b + 6826560) * q^78 - 2875360*b * q^79 + (-1366400*b - 36151040) * q^80 - 53788095 * q^81 + (-2146882*b - 21468820) * q^82 + 4160152*b * q^83 + (-2795520*b + 7827456) * q^84 + 44584900 * q^85 + (-4746320*b + 74042592) * q^86 + 1025776*b * q^87 + (5446400*b + 33246720) * q^88 - 83324222 * q^89 + (2088030*b + 20880300) * q^90 + 612640*b * q^91 + (-1063552*b - 59255040) * q^92 + 6789120 * q^93 + (6105920*b - 95252352) * q^94 + 951600*b * q^95 + (-1945600*b + 101916672) * q^96 + 120619010 * q^97 + (-3807937*b - 38079370) * q^98 - 5066040*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 20 q^{2} - 112 q^{4} + 1220 q^{5} - 2496 q^{6} + 7360 q^{8} - 6846 q^{9}+O(q^{10})$$ 2 * q - 20 * q^2 - 112 * q^4 + 1220 * q^5 - 2496 * q^6 + 7360 * q^8 - 6846 * q^9 $$2 q - 20 q^{2} - 112 q^{4} + 1220 q^{5} - 2496 q^{6} + 7360 q^{8} - 6846 q^{9} - 12200 q^{10} + 49920 q^{12} - 10940 q^{13} - 34944 q^{14} - 118528 q^{16} + 146180 q^{17} + 68460 q^{18} - 68320 q^{20} - 279552 q^{21} + 461760 q^{22} - 359424 q^{24} - 37050 q^{25} + 109400 q^{26} + 698880 q^{28} - 256444 q^{29} - 1522560 q^{30} + 486400 q^{32} + 3694080 q^{33} - 1461800 q^{34} + 383376 q^{36} - 6944060 q^{37} + 486720 q^{38} + 4489600 q^{40} + 4293764 q^{41} + 2795520 q^{42} - 9235200 q^{44} - 4176060 q^{45} + 5925504 q^{46} - 5591040 q^{48} + 7615874 q^{49} + 370500 q^{50} + 612640 q^{52} + 1648580 q^{53} - 7832448 q^{54} - 5031936 q^{56} + 3893760 q^{57} + 2564440 q^{58} + 30451200 q^{60} - 29492156 q^{61} + 1697280 q^{62} + 20615168 q^{64} - 6673400 q^{65} - 36940800 q^{66} - 8186080 q^{68} + 47404032 q^{69} - 21315840 q^{70} - 25193280 q^{72} - 11451260 q^{73} + 69440600 q^{74} - 9734400 q^{76} + 51717120 q^{77} + 13653120 q^{78} - 72302080 q^{80} - 107576190 q^{81} - 42937640 q^{82} + 15654912 q^{84} + 89169800 q^{85} + 148085184 q^{86} + 66493440 q^{88} - 166648444 q^{89} + 41760600 q^{90} - 118510080 q^{92} + 13578240 q^{93} - 190504704 q^{94} + 203833344 q^{96} + 241238020 q^{97} - 76158740 q^{98}+O(q^{100})$$ 2 * q - 20 * q^2 - 112 * q^4 + 1220 * q^5 - 2496 * q^6 + 7360 * q^8 - 6846 * q^9 - 12200 * q^10 + 49920 * q^12 - 10940 * q^13 - 34944 * q^14 - 118528 * q^16 + 146180 * q^17 + 68460 * q^18 - 68320 * q^20 - 279552 * q^21 + 461760 * q^22 - 359424 * q^24 - 37050 * q^25 + 109400 * q^26 + 698880 * q^28 - 256444 * q^29 - 1522560 * q^30 + 486400 * q^32 + 3694080 * q^33 - 1461800 * q^34 + 383376 * q^36 - 6944060 * q^37 + 486720 * q^38 + 4489600 * q^40 + 4293764 * q^41 + 2795520 * q^42 - 9235200 * q^44 - 4176060 * q^45 + 5925504 * q^46 - 5591040 * q^48 + 7615874 * q^49 + 370500 * q^50 + 612640 * q^52 + 1648580 * q^53 - 7832448 * q^54 - 5031936 * q^56 + 3893760 * q^57 + 2564440 * q^58 + 30451200 * q^60 - 29492156 * q^61 + 1697280 * q^62 + 20615168 * q^64 - 6673400 * q^65 - 36940800 * q^66 - 8186080 * q^68 + 47404032 * q^69 - 21315840 * q^70 - 25193280 * q^72 - 11451260 * q^73 + 69440600 * q^74 - 9734400 * q^76 + 51717120 * q^77 + 13653120 * q^78 - 72302080 * q^80 - 107576190 * q^81 - 42937640 * q^82 + 15654912 * q^84 + 89169800 * q^85 + 148085184 * q^86 + 66493440 * q^88 - 166648444 * q^89 + 41760600 * q^90 - 118510080 * q^92 + 13578240 * q^93 - 190504704 * q^94 + 203833344 * q^96 + 241238020 * q^97 - 76158740 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.5 + 3.12250i 0.5 − 3.12250i
−10.0000 12.4900i 99.9200i −56.0000 + 249.800i 610.000 −1248.00 + 999.200i 1398.88i 3680.00 1798.56i −3423.00 −6100.00 7618.90i
3.2 −10.0000 + 12.4900i 99.9200i −56.0000 249.800i 610.000 −1248.00 999.200i 1398.88i 3680.00 + 1798.56i −3423.00 −6100.00 + 7618.90i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.9.b.b 2
3.b odd 2 1 36.9.d.b 2
4.b odd 2 1 inner 4.9.b.b 2
5.b even 2 1 100.9.b.c 2
5.c odd 4 2 100.9.d.b 4
8.b even 2 1 64.9.c.b 2
8.d odd 2 1 64.9.c.b 2
12.b even 2 1 36.9.d.b 2
16.e even 4 2 256.9.d.e 4
16.f odd 4 2 256.9.d.e 4
20.d odd 2 1 100.9.b.c 2
20.e even 4 2 100.9.d.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.b 2 1.a even 1 1 trivial
4.9.b.b 2 4.b odd 2 1 inner
36.9.d.b 2 3.b odd 2 1
36.9.d.b 2 12.b even 2 1
64.9.c.b 2 8.b even 2 1
64.9.c.b 2 8.d odd 2 1
100.9.b.c 2 5.b even 2 1
100.9.b.c 2 20.d odd 2 1
100.9.d.b 4 5.c odd 4 2
100.9.d.b 4 20.e even 4 2
256.9.d.e 4 16.e even 4 2
256.9.d.e 4 16.f odd 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 9984$$ acting on $$S_{9}^{\mathrm{new}}(4, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 20T + 256$$
$3$ $$T^{2} + 9984$$
$5$ $$(T - 610)^{2}$$
$7$ $$T^{2} + 1956864$$
$11$ $$T^{2} + 341702400$$
$13$ $$(T + 5470)^{2}$$
$17$ $$(T - 73090)^{2}$$
$19$ $$T^{2} + 379641600$$
$23$ $$T^{2} + 56268585984$$
$29$ $$(T + 128222)^{2}$$
$31$ $$T^{2} + 4616601600$$
$37$ $$(T + 3472030)^{2}$$
$41$ $$(T - 2146882)^{2}$$
$43$ $$T^{2} + 35142983526144$$
$47$ $$T^{2} + 58160324112384$$
$53$ $$(T - 824290)^{2}$$
$59$ $$T^{2} + 13879469510400$$
$61$ $$(T + 14746078)^{2}$$
$67$ $$T^{2} + \cdots + 232766284318464$$
$71$ $$T^{2} + 1430516505600$$
$73$ $$(T + 5725630)^{2}$$
$79$ $$T^{2} + 12\!\cdots\!00$$
$83$ $$T^{2} + 26\!\cdots\!24$$
$89$ $$(T + 83324222)^{2}$$
$97$ $$(T - 120619010)^{2}$$