# Properties

 Label 75.9.f.a Level $75$ Weight $9$ Character orbit 75.f Analytic conductor $30.553$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 6 \beta_1 q^{2} - 27 \beta_{3} q^{3} - 148 \beta_{2} q^{4} + 486 q^{6} + 2345 \beta_1 q^{7} - 2424 \beta_{3} q^{8} - 2187 \beta_{2} q^{9}+O(q^{10})$$ q + 6*b1 * q^2 - 27*b3 * q^3 - 148*b2 * q^4 + 486 * q^6 + 2345*b1 * q^7 - 2424*b3 * q^8 - 2187*b2 * q^9 $$q + 6 \beta_1 q^{2} - 27 \beta_{3} q^{3} - 148 \beta_{2} q^{4} + 486 q^{6} + 2345 \beta_1 q^{7} - 2424 \beta_{3} q^{8} - 2187 \beta_{2} q^{9} - 234 q^{11} - 3996 \beta_1 q^{12} + 9741 \beta_{3} q^{13} + 42210 \beta_{2} q^{14} + 5744 q^{16} + 72678 \beta_1 q^{17} - 13122 \beta_{3} q^{18} - 181693 \beta_{2} q^{19} + 189945 q^{21} - 1404 \beta_1 q^{22} - 219990 \beta_{3} q^{23} - 196344 \beta_{2} q^{24} - 175338 q^{26} - 59049 \beta_1 q^{27} - 347060 \beta_{3} q^{28} + 240174 \beta_{2} q^{29} + 836725 q^{31} - 586080 \beta_1 q^{32} + 6318 \beta_{3} q^{33} + 1308204 \beta_{2} q^{34} - 323676 q^{36} + 496660 \beta_1 q^{37} - 1090158 \beta_{3} q^{38} + 789021 \beta_{2} q^{39} + 2822220 q^{41} + 1139670 \beta_1 q^{42} - 2287837 \beta_{3} q^{43} + 34632 \beta_{2} q^{44} + 3959820 q^{46} - 4403538 \beta_1 q^{47} - 155088 \beta_{3} q^{48} + 10732274 \beta_{2} q^{49} + 5886918 q^{51} + 1441668 \beta_1 q^{52} + 973740 \beta_{3} q^{53} - 1062882 \beta_{2} q^{54} + 17052840 q^{56} - 4905711 \beta_1 q^{57} + 1441044 \beta_{3} q^{58} + 12785958 \beta_{2} q^{59} + 517403 q^{61} + 5020350 \beta_1 q^{62} - 5128515 \beta_{3} q^{63} - 12019904 \beta_{2} q^{64} - 113724 q^{66} + 1687023 \beta_1 q^{67} - 10756344 \beta_{3} q^{68} - 17819190 \beta_{2} q^{69} - 20828628 q^{71} - 5301288 \beta_1 q^{72} - 23574700 \beta_{3} q^{73} + 8939880 \beta_{2} q^{74} - 26890564 q^{76} - 548730 \beta_1 q^{77} + 4734126 \beta_{3} q^{78} + 42187930 \beta_{2} q^{79} - 4782969 q^{81} + 16933320 \beta_1 q^{82} + 54440934 \beta_{3} q^{83} - 28111860 \beta_{2} q^{84} + 41181066 q^{86} + 6484698 \beta_1 q^{87} + 567216 \beta_{3} q^{88} - 95161428 \beta_{2} q^{89} - 68527935 q^{91} - 32558520 \beta_1 q^{92} - 22591575 \beta_{3} q^{93} - 79263684 \beta_{2} q^{94} - 47472480 q^{96} + 57863323 \beta_1 q^{97} + 64393644 \beta_{3} q^{98} + 511758 \beta_{2} q^{99}+O(q^{100})$$ q + 6*b1 * q^2 - 27*b3 * q^3 - 148*b2 * q^4 + 486 * q^6 + 2345*b1 * q^7 - 2424*b3 * q^8 - 2187*b2 * q^9 - 234 * q^11 - 3996*b1 * q^12 + 9741*b3 * q^13 + 42210*b2 * q^14 + 5744 * q^16 + 72678*b1 * q^17 - 13122*b3 * q^18 - 181693*b2 * q^19 + 189945 * q^21 - 1404*b1 * q^22 - 219990*b3 * q^23 - 196344*b2 * q^24 - 175338 * q^26 - 59049*b1 * q^27 - 347060*b3 * q^28 + 240174*b2 * q^29 + 836725 * q^31 - 586080*b1 * q^32 + 6318*b3 * q^33 + 1308204*b2 * q^34 - 323676 * q^36 + 496660*b1 * q^37 - 1090158*b3 * q^38 + 789021*b2 * q^39 + 2822220 * q^41 + 1139670*b1 * q^42 - 2287837*b3 * q^43 + 34632*b2 * q^44 + 3959820 * q^46 - 4403538*b1 * q^47 - 155088*b3 * q^48 + 10732274*b2 * q^49 + 5886918 * q^51 + 1441668*b1 * q^52 + 973740*b3 * q^53 - 1062882*b2 * q^54 + 17052840 * q^56 - 4905711*b1 * q^57 + 1441044*b3 * q^58 + 12785958*b2 * q^59 + 517403 * q^61 + 5020350*b1 * q^62 - 5128515*b3 * q^63 - 12019904*b2 * q^64 - 113724 * q^66 + 1687023*b1 * q^67 - 10756344*b3 * q^68 - 17819190*b2 * q^69 - 20828628 * q^71 - 5301288*b1 * q^72 - 23574700*b3 * q^73 + 8939880*b2 * q^74 - 26890564 * q^76 - 548730*b1 * q^77 + 4734126*b3 * q^78 + 42187930*b2 * q^79 - 4782969 * q^81 + 16933320*b1 * q^82 + 54440934*b3 * q^83 - 28111860*b2 * q^84 + 41181066 * q^86 + 6484698*b1 * q^87 + 567216*b3 * q^88 - 95161428*b2 * q^89 - 68527935 * q^91 - 32558520*b1 * q^92 - 22591575*b3 * q^93 - 79263684*b2 * q^94 - 47472480 * q^96 + 57863323*b1 * q^97 + 64393644*b3 * q^98 + 511758*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 1944 q^{6}+O(q^{10})$$ 4 * q + 1944 * q^6 $$4 q + 1944 q^{6} - 936 q^{11} + 22976 q^{16} + 759780 q^{21} - 701352 q^{26} + 3346900 q^{31} - 1294704 q^{36} + 11288880 q^{41} + 15839280 q^{46} + 23547672 q^{51} + 68211360 q^{56} + 2069612 q^{61} - 454896 q^{66} - 83314512 q^{71} - 107562256 q^{76} - 19131876 q^{81} + 164724264 q^{86} - 274111740 q^{91} - 189889920 q^{96}+O(q^{100})$$ 4 * q + 1944 * q^6 - 936 * q^11 + 22976 * q^16 + 759780 * q^21 - 701352 * q^26 + 3346900 * q^31 - 1294704 * q^36 + 11288880 * q^41 + 15839280 * q^46 + 23547672 * q^51 + 68211360 * q^56 + 2069612 * q^61 - 454896 * q^66 - 83314512 * q^71 - 107562256 * q^76 - 19131876 * q^81 + 164724264 * q^86 - 274111740 * q^91 - 189889920 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 − 1.22474i 1.22474 + 1.22474i −1.22474 + 1.22474i 1.22474 − 1.22474i
−7.34847 7.34847i −33.0681 + 33.0681i 148.000i 0 486.000 −2872.03 2872.03i −2968.78 + 2968.78i 2187.00i 0
7.2 7.34847 + 7.34847i 33.0681 33.0681i 148.000i 0 486.000 2872.03 + 2872.03i 2968.78 2968.78i 2187.00i 0
43.1 −7.34847 + 7.34847i −33.0681 33.0681i 148.000i 0 486.000 −2872.03 + 2872.03i −2968.78 2968.78i 2187.00i 0
43.2 7.34847 7.34847i 33.0681 + 33.0681i 148.000i 0 486.000 2872.03 2872.03i 2968.78 + 2968.78i 2187.00i 0
 $$n$$: 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
5.c odd 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.f.a 4
5.b even 2 1 inner 75.9.f.a 4
5.c odd 4 2 inner 75.9.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.f.a 4 1.a even 1 1 trivial
75.9.f.a 4 5.b even 2 1 inner
75.9.f.a 4 5.c odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 11664$$ acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 11664$$
$3$ $$T^{4} + 4782969$$
$5$ $$T^{4}$$
$7$ $$T^{4} + \cdots + 272153483555625$$
$11$ $$(T + 234)^{4}$$
$13$ $$T^{4} + 81\!\cdots\!49$$
$17$ $$T^{4} + 25\!\cdots\!04$$
$19$ $$(T^{2} + 33012346249)^{2}$$
$23$ $$T^{4} + 21\!\cdots\!00$$
$29$ $$(T^{2} + 57683550276)^{2}$$
$31$ $$(T - 836725)^{4}$$
$37$ $$T^{4} + 54\!\cdots\!00$$
$41$ $$(T - 2822220)^{4}$$
$43$ $$T^{4} + 24\!\cdots\!49$$
$47$ $$T^{4} + 33\!\cdots\!24$$
$53$ $$T^{4} + 80\!\cdots\!00$$
$59$ $$(T^{2} + 163480721977764)^{2}$$
$61$ $$(T - 517403)^{4}$$
$67$ $$T^{4} + 72\!\cdots\!69$$
$71$ $$(T + 20828628)^{4}$$
$73$ $$T^{4} + 27\!\cdots\!00$$
$79$ $$(T^{2} + 17\!\cdots\!00)^{2}$$
$83$ $$T^{4} + 79\!\cdots\!24$$
$89$ $$(T^{2} + 90\!\cdots\!84)^{2}$$
$97$ $$T^{4} + 10\!\cdots\!69$$