Properties

 Label 90.6.c.c Level $90$ Weight $6$ Character orbit 90.c Analytic conductor $14.435$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 90.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.4345437832$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{1249})$$ Defining polynomial: $$x^{4} + 625x^{2} + 97344$$ x^4 + 625*x^2 + 97344 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 5$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 16 q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 15 \beta_1 - 1) q^{7} - 16 \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 - 16 * q^4 + (-b2 - 1) * q^5 + (-b3 + 3*b2 - 15*b1 - 1) * q^7 - 16*b1 * q^8 $$q + \beta_1 q^{2} - 16 q^{4} + ( - \beta_{2} - 1) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 15 \beta_1 - 1) q^{7} - 16 \beta_1 q^{8} + ( - 4 \beta_{3} - 3 \beta_1 + 4) q^{10} + (3 \beta_{3} + \beta_{2} + \beta_1 - 89) q^{11} + ( - 3 \beta_{3} + 9 \beta_{2} + 78 \beta_1 - 3) q^{13} + (12 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 220) q^{14} + 256 q^{16} + ( - 3 \beta_{3} + 9 \beta_{2} + 187 \beta_1 - 3) q^{17} + (36 \beta_{3} + 12 \beta_{2} + 12 \beta_1 + 36) q^{19} + (16 \beta_{2} + 16) q^{20} + (4 \beta_{3} - 12 \beta_{2} - 84 \beta_1 + 4) q^{22} + ( - 12 \beta_{3} + 36 \beta_{2} - 553 \beta_1 - 12) q^{23} + (\beta_{3} + 3 \beta_{2} - 468 \beta_1 - 2498) q^{25} + (36 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 1308) q^{26} + (16 \beta_{3} - 48 \beta_{2} + 240 \beta_1 + 16) q^{28} + (81 \beta_{3} + 27 \beta_{2} + 27 \beta_1 - 1269) q^{29} + (18 \beta_{3} + 6 \beta_{2} + 6 \beta_1 + 5830) q^{31} + 256 \beta_1 q^{32} + (36 \beta_{3} + 12 \beta_{2} + 12 \beta_1 - 3052) q^{34} + (57 \beta_{3} - 5 \beta_{2} + 824 \beta_1 + 9313) q^{35} + ( - 19 \beta_{3} + 57 \beta_{2} + 2184 \beta_1 - 19) q^{37} + (48 \beta_{3} - 144 \beta_{2} + 96 \beta_1 + 48) q^{38} + (64 \beta_{3} + 48 \beta_1 - 64) q^{40} + (102 \beta_{3} + 34 \beta_{2} + 34 \beta_1 - 12560) q^{41} + ( - 56 \beta_{3} + 168 \beta_{2} - 3063 \beta_1 - 56) q^{43} + ( - 48 \beta_{3} - 16 \beta_{2} - 16 \beta_1 + 1424) q^{44} + (144 \beta_{3} + 48 \beta_{2} + 48 \beta_1 + 8608) q^{46} + ( - 6 \beta_{3} + 18 \beta_{2} - 1669 \beta_1 - 6) q^{47} + ( - 342 \beta_{3} - 114 \beta_{2} - 114 \beta_1 - 17439) q^{49} + (12 \beta_{3} - 4 \beta_{2} - 2491 \beta_1 + 7484) q^{50} + (48 \beta_{3} - 144 \beta_{2} - 1248 \beta_1 + 48) q^{52} + ( - 111 \beta_{3} + 333 \beta_{2} + 1629 \beta_1 - 111) q^{53} + ( - 5 \beta_{3} + 87 \beta_{2} + 2340 \beta_1 - 3033) q^{55} + ( - 192 \beta_{3} - 64 \beta_{2} - 64 \beta_1 - 3520) q^{56} + (108 \beta_{3} - 324 \beta_{2} - 1134 \beta_1 + 108) q^{58} + ( - 555 \beta_{3} - 185 \beta_{2} - 185 \beta_1 - 11495) q^{59} + ( - 288 \beta_{3} - 96 \beta_{2} - 96 \beta_1 + 28754) q^{61} + (24 \beta_{3} - 72 \beta_{2} + 5860 \beta_1 + 24) q^{62} - 4096 q^{64} + ( - 321 \beta_{3} - 15 \beta_{2} + 2103 \beta_1 + 28431) q^{65} + ( - 86 \beta_{3} + 258 \beta_{2} + 4725 \beta_1 - 86) q^{67} + (48 \beta_{3} - 144 \beta_{2} - 2992 \beta_1 + 48) q^{68} + ( - 20 \beta_{3} - 228 \beta_{2} + 9360 \beta_1 - 12708) q^{70} + ( - 882 \beta_{3} - 294 \beta_{2} - 294 \beta_1 + 6126) q^{71} + ( - 98 \beta_{3} + 294 \beta_{2} - 11142 \beta_1 - 98) q^{73} + (228 \beta_{3} + 76 \beta_{2} + 76 \beta_1 - 35324) q^{74} + ( - 576 \beta_{3} - 192 \beta_{2} - 192 \beta_1 - 576) q^{76} + (30 \beta_{3} - 90 \beta_{2} - 6544 \beta_1 + 30) q^{77} + (594 \beta_{3} + 198 \beta_{2} + 198 \beta_1 - 7506) q^{79} + ( - 256 \beta_{2} - 256) q^{80} + (136 \beta_{3} - 408 \beta_{2} - 12390 \beta_1 + 136) q^{82} + (432 \beta_{3} - 1296 \beta_{2} - 6025 \beta_1 + 432) q^{83} + ( - 757 \beta_{3} - 15 \beta_{2} + 1776 \beta_1 + 28867) q^{85} + (672 \beta_{3} + 224 \beta_{2} + 224 \beta_1 + 47888) q^{86} + ( - 64 \beta_{3} + 192 \beta_{2} + 1344 \beta_1 - 64) q^{88} + (600 \beta_{3} + 200 \beta_{2} + 200 \beta_1 - 30610) q^{89} + (450 \beta_{3} + 150 \beta_{2} + 150 \beta_1 - 75678) q^{91} + (192 \beta_{3} - 576 \beta_{2} + 8848 \beta_1 + 192) q^{92} + (72 \beta_{3} + 24 \beta_{2} + 24 \beta_1 + 26584) q^{94} + ( - 60 \beta_{3} - 60 \beta_{2} + 28080 \beta_1 - 37500) q^{95} + (16 \beta_{3} - 48 \beta_{2} + 21414 \beta_1 + 16) q^{97} + ( - 456 \beta_{3} + 1368 \beta_{2} - 18009 \beta_1 - 456) q^{98}+O(q^{100})$$ q + b1 * q^2 - 16 * q^4 + (-b2 - 1) * q^5 + (-b3 + 3*b2 - 15*b1 - 1) * q^7 - 16*b1 * q^8 + (-4*b3 - 3*b1 + 4) * q^10 + (3*b3 + b2 + b1 - 89) * q^11 + (-3*b3 + 9*b2 + 78*b1 - 3) * q^13 + (12*b3 + 4*b2 + 4*b1 + 220) * q^14 + 256 * q^16 + (-3*b3 + 9*b2 + 187*b1 - 3) * q^17 + (36*b3 + 12*b2 + 12*b1 + 36) * q^19 + (16*b2 + 16) * q^20 + (4*b3 - 12*b2 - 84*b1 + 4) * q^22 + (-12*b3 + 36*b2 - 553*b1 - 12) * q^23 + (b3 + 3*b2 - 468*b1 - 2498) * q^25 + (36*b3 + 12*b2 + 12*b1 - 1308) * q^26 + (16*b3 - 48*b2 + 240*b1 + 16) * q^28 + (81*b3 + 27*b2 + 27*b1 - 1269) * q^29 + (18*b3 + 6*b2 + 6*b1 + 5830) * q^31 + 256*b1 * q^32 + (36*b3 + 12*b2 + 12*b1 - 3052) * q^34 + (57*b3 - 5*b2 + 824*b1 + 9313) * q^35 + (-19*b3 + 57*b2 + 2184*b1 - 19) * q^37 + (48*b3 - 144*b2 + 96*b1 + 48) * q^38 + (64*b3 + 48*b1 - 64) * q^40 + (102*b3 + 34*b2 + 34*b1 - 12560) * q^41 + (-56*b3 + 168*b2 - 3063*b1 - 56) * q^43 + (-48*b3 - 16*b2 - 16*b1 + 1424) * q^44 + (144*b3 + 48*b2 + 48*b1 + 8608) * q^46 + (-6*b3 + 18*b2 - 1669*b1 - 6) * q^47 + (-342*b3 - 114*b2 - 114*b1 - 17439) * q^49 + (12*b3 - 4*b2 - 2491*b1 + 7484) * q^50 + (48*b3 - 144*b2 - 1248*b1 + 48) * q^52 + (-111*b3 + 333*b2 + 1629*b1 - 111) * q^53 + (-5*b3 + 87*b2 + 2340*b1 - 3033) * q^55 + (-192*b3 - 64*b2 - 64*b1 - 3520) * q^56 + (108*b3 - 324*b2 - 1134*b1 + 108) * q^58 + (-555*b3 - 185*b2 - 185*b1 - 11495) * q^59 + (-288*b3 - 96*b2 - 96*b1 + 28754) * q^61 + (24*b3 - 72*b2 + 5860*b1 + 24) * q^62 - 4096 * q^64 + (-321*b3 - 15*b2 + 2103*b1 + 28431) * q^65 + (-86*b3 + 258*b2 + 4725*b1 - 86) * q^67 + (48*b3 - 144*b2 - 2992*b1 + 48) * q^68 + (-20*b3 - 228*b2 + 9360*b1 - 12708) * q^70 + (-882*b3 - 294*b2 - 294*b1 + 6126) * q^71 + (-98*b3 + 294*b2 - 11142*b1 - 98) * q^73 + (228*b3 + 76*b2 + 76*b1 - 35324) * q^74 + (-576*b3 - 192*b2 - 192*b1 - 576) * q^76 + (30*b3 - 90*b2 - 6544*b1 + 30) * q^77 + (594*b3 + 198*b2 + 198*b1 - 7506) * q^79 + (-256*b2 - 256) * q^80 + (136*b3 - 408*b2 - 12390*b1 + 136) * q^82 + (432*b3 - 1296*b2 - 6025*b1 + 432) * q^83 + (-757*b3 - 15*b2 + 1776*b1 + 28867) * q^85 + (672*b3 + 224*b2 + 224*b1 + 47888) * q^86 + (-64*b3 + 192*b2 + 1344*b1 - 64) * q^88 + (600*b3 + 200*b2 + 200*b1 - 30610) * q^89 + (450*b3 + 150*b2 + 150*b1 - 75678) * q^91 + (192*b3 - 576*b2 + 8848*b1 + 192) * q^92 + (72*b3 + 24*b2 + 24*b1 + 26584) * q^94 + (-60*b3 - 60*b2 + 28080*b1 - 37500) * q^95 + (16*b3 - 48*b2 + 21414*b1 + 16) * q^97 + (-456*b3 + 1368*b2 - 18009*b1 - 456) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 64 q^{4} - 6 q^{5}+O(q^{10})$$ 4 * q - 64 * q^4 - 6 * q^5 $$4 q - 64 q^{4} - 6 q^{5} + 8 q^{10} - 348 q^{11} + 912 q^{14} + 1024 q^{16} + 240 q^{19} + 96 q^{20} - 9984 q^{25} - 5136 q^{26} - 4860 q^{29} + 23368 q^{31} - 12112 q^{34} + 37356 q^{35} - 128 q^{40} - 49968 q^{41} + 5568 q^{44} + 34816 q^{46} - 70668 q^{49} + 29952 q^{50} - 11968 q^{55} - 14592 q^{56} - 47460 q^{59} + 114248 q^{61} - 16384 q^{64} + 113052 q^{65} - 51328 q^{70} + 22152 q^{71} - 140688 q^{74} - 3840 q^{76} - 28440 q^{79} - 1536 q^{80} + 113924 q^{85} + 193344 q^{86} - 120840 q^{89} - 301512 q^{91} + 106528 q^{94} - 150240 q^{95}+O(q^{100})$$ 4 * q - 64 * q^4 - 6 * q^5 + 8 * q^10 - 348 * q^11 + 912 * q^14 + 1024 * q^16 + 240 * q^19 + 96 * q^20 - 9984 * q^25 - 5136 * q^26 - 4860 * q^29 + 23368 * q^31 - 12112 * q^34 + 37356 * q^35 - 128 * q^40 - 49968 * q^41 + 5568 * q^44 + 34816 * q^46 - 70668 * q^49 + 29952 * q^50 - 11968 * q^55 - 14592 * q^56 - 47460 * q^59 + 114248 * q^61 - 16384 * q^64 + 113052 * q^65 - 51328 * q^70 + 22152 * q^71 - 140688 * q^74 - 3840 * q^76 - 28440 * q^79 - 1536 * q^80 + 113924 * q^85 + 193344 * q^86 - 120840 * q^89 - 301512 * q^91 + 106528 * q^94 - 150240 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 625x^{2} + 97344$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} - 313\nu ) / 78$$ (-v^3 - 313*v) / 78 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 312\nu^{2} + 1249\nu + 97656 ) / 312$$ (v^3 + 312*v^2 + 1249*v + 97656) / 312 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 936\nu^{2} + \nu + 292656 ) / 312$$ (v^3 + 936*v^2 + v + 292656) / 312
 $$\nu$$ $$=$$ $$( -2\beta_{3} + 6\beta_{2} + \beta _1 - 2 ) / 20$$ (-2*b3 + 6*b2 + b1 - 2) / 20 $$\nu^{2}$$ $$=$$ $$( 3\beta_{3} + \beta_{2} + \beta _1 - 3127 ) / 10$$ (3*b3 + b2 + b1 - 3127) / 10 $$\nu^{3}$$ $$=$$ $$( 626\beta_{3} - 1878\beta_{2} - 1873\beta _1 + 626 ) / 20$$ (626*b3 - 1878*b2 - 1873*b1 + 626) / 20

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/90\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$37$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 17.1706i − 18.1706i − 17.1706i 18.1706i
4.00000i 0 −16.0000 −19.1706 52.5118i 0 233.706i 64.0000i 0 −210.047 + 76.6824i
19.2 4.00000i 0 −16.0000 16.1706 + 53.5118i 0 119.706i 64.0000i 0 214.047 64.6824i
19.3 4.00000i 0 −16.0000 −19.1706 + 52.5118i 0 233.706i 64.0000i 0 −210.047 76.6824i
19.4 4.00000i 0 −16.0000 16.1706 53.5118i 0 119.706i 64.0000i 0 214.047 + 64.6824i
 $$n$$: 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
5.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.6.c.c 4
3.b odd 2 1 30.6.c.b 4
4.b odd 2 1 720.6.f.i 4
5.b even 2 1 inner 90.6.c.c 4
5.c odd 4 1 450.6.a.bb 2
5.c odd 4 1 450.6.a.bc 2
12.b even 2 1 240.6.f.b 4
15.d odd 2 1 30.6.c.b 4
15.e even 4 1 150.6.a.n 2
15.e even 4 1 150.6.a.o 2
20.d odd 2 1 720.6.f.i 4
60.h even 2 1 240.6.f.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.b 4 3.b odd 2 1
30.6.c.b 4 15.d odd 2 1
90.6.c.c 4 1.a even 1 1 trivial
90.6.c.c 4 5.b even 2 1 inner
150.6.a.n 2 15.e even 4 1
150.6.a.o 2 15.e even 4 1
240.6.f.b 4 12.b even 2 1
240.6.f.b 4 60.h even 2 1
450.6.a.bb 2 5.c odd 4 1
450.6.a.bc 2 5.c odd 4 1
720.6.f.i 4 4.b odd 2 1
720.6.f.i 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 68948T_{7}^{2} + 782656576$$ acting on $$S_{6}^{\mathrm{new}}(90, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 16)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6 T^{3} + 5010 T^{2} + \cdots + 9765625$$
$7$ $$T^{4} + 68948 T^{2} + \cdots + 782656576$$
$11$ $$(T^{2} + 174 T - 23656)^{2}$$
$13$ $$T^{4} + 768132 T^{2} + \cdots + 31678304256$$
$17$ $$T^{4} + 1708148 T^{2} + \cdots + 85278016576$$
$19$ $$(T^{2} - 120 T - 4492800)^{2}$$
$23$ $$T^{4} + 18462752 T^{2} + \cdots + 56918507776$$
$29$ $$(T^{2} + 2430 T - 21286800)^{2}$$
$31$ $$(T^{2} - 11684 T + 33004864)^{2}$$
$37$ $$T^{4} + 177178148 T^{2} + \cdots + 43\!\cdots\!76$$
$41$ $$(T^{2} + 24984 T + 119953964)^{2}$$
$43$ $$T^{4} + 487889312 T^{2} + \cdots + 23\!\cdots\!36$$
$47$ $$T^{4} + 90906128 T^{2} + \cdots + 18\!\cdots\!96$$
$53$ $$T^{4} + 863264052 T^{2} + \cdots + 11\!\cdots\!76$$
$59$ $$(T^{2} + 23730 T - 927897400)^{2}$$
$61$ $$(T^{2} - 57124 T + 528018244)^{2}$$
$67$ $$T^{4} + 1195938128 T^{2} + \cdots + 18\!\cdots\!96$$
$71$ $$(T^{2} - 11076 T - 2668294656)^{2}$$
$73$ $$T^{4} + 4520143952 T^{2} + \cdots + 27\!\cdots\!76$$
$79$ $$(T^{2} + 14220 T - 1173592800)^{2}$$
$83$ $$T^{4} + 12944582432 T^{2} + \cdots + 26\!\cdots\!56$$
$89$ $$(T^{2} + 60420 T - 336355900)^{2}$$
$97$ $$T^{4} + 14673446528 T^{2} + \cdots + 53\!\cdots\!96$$