# 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$4 = 2^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 4.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.62951444024$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-39})$$ Defining polynomial: $$x^{2} - x + 10$$ 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

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.

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

## 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.

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

$p$ $F_p(T)$
$2$ $$1 + 20 T + 256 T^{2}$$
$3$ $$1 - 3138 T^{2} + 43046721 T^{4}$$
$5$ $$( 1 - 610 T + 390625 T^{2} )^{2}$$
$7$ $$1 - 9572738 T^{2} + 33232930569601 T^{4}$$
$11$ $$1 - 87015362 T^{2} + 45949729863572161 T^{4}$$
$13$ $$( 1 + 5470 T + 815730721 T^{2} )^{2}$$
$17$ $$( 1 - 73090 T + 6975757441 T^{2} )^{2}$$
$19$ $$1 - 33587484482 T^{2} +$$$$28\!\cdots\!81$$$$T^{4}$$
$23$ $$1 - 100353384578 T^{2} +$$$$61\!\cdots\!61$$$$T^{4}$$
$29$ $$( 1 + 128222 T + 500246412961 T^{2} )^{2}$$
$31$ $$1 - 1701165473282 T^{2} +$$$$72\!\cdots\!81$$$$T^{4}$$
$37$ $$( 1 + 3472030 T + 3512479453921 T^{2} )^{2}$$
$41$ $$( 1 - 2146882 T + 7984925229121 T^{2} )^{2}$$
$43$ $$1 + 11766582970942 T^{2} +$$$$13\!\cdots\!01$$$$T^{4}$$
$47$ $$1 + 10537750788862 T^{2} +$$$$56\!\cdots\!21$$$$T^{4}$$
$53$ $$( 1 - 824290 T + 62259690411361 T^{2} )^{2}$$
$59$ $$1 - 279781405698242 T^{2} +$$$$21\!\cdots\!41$$$$T^{4}$$
$61$ $$( 1 + 14746078 T + 191707312997281 T^{2} )^{2}$$
$67$ $$1 - 579369070794818 T^{2} +$$$$16\!\cdots\!81$$$$T^{4}$$
$71$ $$1 - 1290076545985922 T^{2} +$$$$41\!\cdots\!21$$$$T^{4}$$
$73$ $$( 1 + 5725630 T + 806460091894081 T^{2} )^{2}$$
$79$ $$1 - 1744457179595522 T^{2} +$$$$23\!\cdots\!21$$$$T^{4}$$
$83$ $$1 - 1804713576833858 T^{2} +$$$$50\!\cdots\!81$$$$T^{4}$$
$89$ $$( 1 + 83324222 T + 3936588805702081 T^{2} )^{2}$$
$97$ $$( 1 - 120619010 T + 7837433594376961 T^{2} )^{2}$$