# Properties

 Label 75.9.f.c Level $75$ Weight $9$ Character orbit 75.f Analytic conductor $30.553$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.151613669376.2 Defining polynomial: $$x^{8} - 7x^{4} + 256$$ x^8 - 7*x^4 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}\cdot 3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} + 2 \beta_{3}) q^{2} + 27 \beta_{2} q^{3} + ( - 4 \beta_{4} + 224 \beta_1) q^{4} + ( - 27 \beta_{6} + 162) q^{6} + (96 \beta_{5} - 1471 \beta_{3}) q^{7} + (8 \beta_{7} - 1808 \beta_{2}) q^{8} - 2187 \beta_1 q^{9}+O(q^{10})$$ q + (-b5 + 2*b3) * q^2 + 27*b2 * q^3 + (-4*b4 + 224*b1) * q^4 + (-27*b6 + 162) * q^6 + (96*b5 - 1471*b3) * q^7 + (8*b7 - 1808*b2) * q^8 - 2187*b1 * q^9 $$q + ( - \beta_{5} + 2 \beta_{3}) q^{2} + 27 \beta_{2} q^{3} + ( - 4 \beta_{4} + 224 \beta_1) q^{4} + ( - 27 \beta_{6} + 162) q^{6} + (96 \beta_{5} - 1471 \beta_{3}) q^{7} + (8 \beta_{7} - 1808 \beta_{2}) q^{8} - 2187 \beta_1 q^{9} + ( - 116 \beta_{6} + 13494) q^{11} + ( - 324 \beta_{5} + 6048 \beta_{3}) q^{12} + (1092 \beta_{7} + 3531 \beta_{2}) q^{13} + (1663 \beta_{4} - 53754 \beta_1) q^{14} + (768 \beta_{6} + 50240) q^{16} + ( - 620 \beta_{5} + 1318 \beta_{3}) q^{17} + (2187 \beta_{7} + 4374 \beta_{2}) q^{18} + (2332 \beta_{4} + 91571 \beta_1) q^{19} + (2592 \beta_{6} - 119151) q^{21} + ( - 14190 \beta_{5} + 81276 \beta_{3}) q^{22} + (13724 \beta_{7} - 12986 \beta_{2}) q^{23} + (216 \beta_{4} + 146448 \beta_1) q^{24} + ( - 5715 \beta_{6} + 532242) q^{26} - 59049 \beta_{3} q^{27} + (39156 \beta_{7} + 509216 \beta_{2}) q^{28} + (23404 \beta_{4} + 148062 \beta_1) q^{29} + (20168 \beta_{6} - 363539) q^{31} + ( - 43584 \beta_{5} + 203904 \beta_{3}) q^{32} + (9396 \beta_{7} + 364338 \beta_{2}) q^{33} + ( - 2558 \beta_{4} + 298068 \beta_1) q^{34} + ( - 8748 \beta_{6} + 489888) q^{36} + (32016 \beta_{5} - 1456796 \beta_{3}) q^{37} + ( - 77579 \beta_{7} + 908234 \beta_{2}) q^{38} + (29484 \beta_{4} - 286011 \beta_1) q^{39} + ( - 27652 \beta_{6} + 4239276) q^{41} + (134703 \beta_{5} - 1451358 \beta_{3}) q^{42} + ( - 121332 \beta_{7} + 444685 \beta_{2}) q^{43} + ( - 79960 \beta_{4} + 3674112 \beta_1) q^{44} + ( - 14462 \beta_{6} + 6344916) q^{46} + (161712 \beta_{5} + 1614270 \beta_{3}) q^{47} + ( - 62208 \beta_{7} + 1356480 \beta_{2}) q^{48} + ( - 282432 \beta_{4} + 5039810 \beta_1) q^{49} + ( - 16740 \beta_{6} + 106758) q^{51} + ( - 286980 \beta_{5} + 2835168 \beta_{3}) q^{52} + ( - 168356 \beta_{7} + 1243604 \beta_{2}) q^{53} + (59049 \beta_{4} - 354294 \beta_1) q^{54} + ( - 161800 \beta_{6} + 7619280) q^{56} + (188892 \beta_{5} + 2472417 \beta_{3}) q^{57} + ( - 7638 \beta_{7} + 10656948 \beta_{2}) q^{58} + (148848 \beta_{4} + 7828614 \beta_1) q^{59} + (538636 \beta_{6} + 3705875) q^{61} + (484547 \beta_{5} - 10165702 \beta_{3}) q^{62} + ( - 209952 \beta_{7} - 3217077 \beta_{2}) q^{63} + ( - 487680 \beta_{4} + 8759296 \beta_1) q^{64} + ( - 383130 \beta_{6} + 6583356) q^{66} + ( - 1122900 \beta_{5} + 379023 \beta_{3}) q^{67} + ( - 154696 \beta_{7} - 1455872 \beta_{2}) q^{68} + (370548 \beta_{4} + 1051866 \beta_1) q^{69} + (101260 \beta_{6} - 3530868) q^{71} + (17496 \beta_{5} + 3954096 \beta_{3}) q^{72} + (1523544 \beta_{7} - 5871476 \beta_{2}) q^{73} + (1520828 \beta_{4} - 23724264 \beta_1) q^{74} + ( - 156084 \beta_{6} - 7415392) q^{76} + (1807332 \beta_{5} - 25061322 \beta_{3}) q^{77} + (462915 \beta_{7} + 14370534 \beta_{2}) q^{78} + ( - 232600 \beta_{4} - 24785990 \beta_1) q^{79} - 4782969 q^{81} + ( - 4405188 \beta_{5} + 21419688 \beta_{3}) q^{82} + (3756056 \beta_{7} + 6252442 \beta_{2}) q^{83} + (1057212 \beta_{4} - 41246496 \beta_1) q^{84} + ( - 202021 \beta_{6} - 54115266) q^{86} + (1895724 \beta_{5} + 3997674 \beta_{3}) q^{87} + ( - 521232 \beta_{7} - 23962848 \beta_{2}) q^{88} + ( - 535288 \beta_{4} + 85933356 \beta_1) q^{89} + (1945308 \beta_{6} - 64643679) q^{91} + ( - 2918344 \beta_{5} + 22782464 \beta_{3}) q^{92} + ( - 1633608 \beta_{7} - 9815553 \beta_{2}) q^{93} + ( - 1290846 \beta_{4} - 65995596 \beta_1) q^{94} + ( - 1176768 \beta_{6} + 16516224) q^{96} + (3498360 \beta_{5} + 23773483 \beta_{3}) q^{97} + ( - 6734402 \beta_{7} - 142257796 \beta_{2}) q^{98} + (253692 \beta_{4} - 29511378 \beta_1) q^{99}+O(q^{100})$$ q + (-b5 + 2*b3) * q^2 + 27*b2 * q^3 + (-4*b4 + 224*b1) * q^4 + (-27*b6 + 162) * q^6 + (96*b5 - 1471*b3) * q^7 + (8*b7 - 1808*b2) * q^8 - 2187*b1 * q^9 + (-116*b6 + 13494) * q^11 + (-324*b5 + 6048*b3) * q^12 + (1092*b7 + 3531*b2) * q^13 + (1663*b4 - 53754*b1) * q^14 + (768*b6 + 50240) * q^16 + (-620*b5 + 1318*b3) * q^17 + (2187*b7 + 4374*b2) * q^18 + (2332*b4 + 91571*b1) * q^19 + (2592*b6 - 119151) * q^21 + (-14190*b5 + 81276*b3) * q^22 + (13724*b7 - 12986*b2) * q^23 + (216*b4 + 146448*b1) * q^24 + (-5715*b6 + 532242) * q^26 - 59049*b3 * q^27 + (39156*b7 + 509216*b2) * q^28 + (23404*b4 + 148062*b1) * q^29 + (20168*b6 - 363539) * q^31 + (-43584*b5 + 203904*b3) * q^32 + (9396*b7 + 364338*b2) * q^33 + (-2558*b4 + 298068*b1) * q^34 + (-8748*b6 + 489888) * q^36 + (32016*b5 - 1456796*b3) * q^37 + (-77579*b7 + 908234*b2) * q^38 + (29484*b4 - 286011*b1) * q^39 + (-27652*b6 + 4239276) * q^41 + (134703*b5 - 1451358*b3) * q^42 + (-121332*b7 + 444685*b2) * q^43 + (-79960*b4 + 3674112*b1) * q^44 + (-14462*b6 + 6344916) * q^46 + (161712*b5 + 1614270*b3) * q^47 + (-62208*b7 + 1356480*b2) * q^48 + (-282432*b4 + 5039810*b1) * q^49 + (-16740*b6 + 106758) * q^51 + (-286980*b5 + 2835168*b3) * q^52 + (-168356*b7 + 1243604*b2) * q^53 + (59049*b4 - 354294*b1) * q^54 + (-161800*b6 + 7619280) * q^56 + (188892*b5 + 2472417*b3) * q^57 + (-7638*b7 + 10656948*b2) * q^58 + (148848*b4 + 7828614*b1) * q^59 + (538636*b6 + 3705875) * q^61 + (484547*b5 - 10165702*b3) * q^62 + (-209952*b7 - 3217077*b2) * q^63 + (-487680*b4 + 8759296*b1) * q^64 + (-383130*b6 + 6583356) * q^66 + (-1122900*b5 + 379023*b3) * q^67 + (-154696*b7 - 1455872*b2) * q^68 + (370548*b4 + 1051866*b1) * q^69 + (101260*b6 - 3530868) * q^71 + (17496*b5 + 3954096*b3) * q^72 + (1523544*b7 - 5871476*b2) * q^73 + (1520828*b4 - 23724264*b1) * q^74 + (-156084*b6 - 7415392) * q^76 + (1807332*b5 - 25061322*b3) * q^77 + (462915*b7 + 14370534*b2) * q^78 + (-232600*b4 - 24785990*b1) * q^79 - 4782969 * q^81 + (-4405188*b5 + 21419688*b3) * q^82 + (3756056*b7 + 6252442*b2) * q^83 + (1057212*b4 - 41246496*b1) * q^84 + (-202021*b6 - 54115266) * q^86 + (1895724*b5 + 3997674*b3) * q^87 + (-521232*b7 - 23962848*b2) * q^88 + (-535288*b4 + 85933356*b1) * q^89 + (1945308*b6 - 64643679) * q^91 + (-2918344*b5 + 22782464*b3) * q^92 + (-1633608*b7 - 9815553*b2) * q^93 + (-1290846*b4 - 65995596*b1) * q^94 + (-1176768*b6 + 16516224) * q^96 + (3498360*b5 + 23773483*b3) * q^97 + (-6734402*b7 - 142257796*b2) * q^98 + (253692*b4 - 29511378*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 1296 q^{6}+O(q^{10})$$ 8 * q + 1296 * q^6 $$8 q + 1296 q^{6} + 107952 q^{11} + 401920 q^{16} - 953208 q^{21} + 4257936 q^{26} - 2908312 q^{31} + 3919104 q^{36} + 33914208 q^{41} + 50759328 q^{46} + 854064 q^{51} + 60954240 q^{56} + 29647000 q^{61} + 52666848 q^{66} - 28246944 q^{71} - 59323136 q^{76} - 38263752 q^{81} - 432922128 q^{86} - 517149432 q^{91} + 132129792 q^{96}+O(q^{100})$$ 8 * q + 1296 * q^6 + 107952 * q^11 + 401920 * q^16 - 953208 * q^21 + 4257936 * q^26 - 2908312 * q^31 + 3919104 * q^36 + 33914208 * q^41 + 50759328 * q^46 + 854064 * q^51 + 60954240 * q^56 + 29647000 * q^61 + 52666848 * q^66 - 28246944 * q^71 - 59323136 * q^76 - 38263752 * q^81 - 432922128 * q^86 - 517149432 * q^91 + 132129792 * q^96

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7x^{4} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{6} + 9\nu^{2} ) / 80$$ (v^6 + 9*v^2) / 80 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + 11\nu ) / 20$$ (-v^5 + 11*v) / 20 $$\beta_{3}$$ $$=$$ $$( -\nu^{7} + 71\nu^{3} ) / 320$$ (-v^7 + 71*v^3) / 320 $$\beta_{4}$$ $$=$$ $$( 12\nu^{4} - 42 ) / 5$$ (12*v^4 - 42) / 5 $$\beta_{5}$$ $$=$$ $$( 3\nu^{5} + 87\nu ) / 10$$ (3*v^5 + 87*v) / 10 $$\beta_{6}$$ $$=$$ $$( -3\nu^{6} + 69\nu^{2} ) / 8$$ (-3*v^6 + 69*v^2) / 8 $$\beta_{7}$$ $$=$$ $$( 27\nu^{7} + 3\nu^{3} ) / 160$$ (27*v^7 + 3*v^3) / 160
 $$\nu$$ $$=$$ $$( \beta_{5} + 6\beta_{2} ) / 12$$ (b5 + 6*b2) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{6} + 30\beta_1 ) / 12$$ (b6 + 30*b1) / 12 $$\nu^{3}$$ $$=$$ $$( \beta_{7} + 54\beta_{3} ) / 12$$ (b7 + 54*b3) / 12 $$\nu^{4}$$ $$=$$ $$( 5\beta_{4} + 42 ) / 12$$ (5*b4 + 42) / 12 $$\nu^{5}$$ $$=$$ $$( 11\beta_{5} - 174\beta_{2} ) / 12$$ (11*b5 - 174*b2) / 12 $$\nu^{6}$$ $$=$$ $$( -3\beta_{6} + 230\beta_1 ) / 4$$ (-3*b6 + 230*b1) / 4 $$\nu^{7}$$ $$=$$ $$( 71\beta_{7} - 6\beta_{3} ) / 12$$ (71*b7 - 6*b3) / 12

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.662382 + 1.88713i 1.88713 + 0.662382i −1.88713 − 0.662382i −0.662382 − 1.88713i 0.662382 − 1.88713i 1.88713 − 0.662382i −1.88713 + 0.662382i −0.662382 + 1.88713i
−17.7465 17.7465i −33.0681 + 33.0681i 373.880i 0 1173.69 3270.12 + 3270.12i 2091.96 2091.96i 2187.00i 0
7.2 −12.8476 12.8476i 33.0681 33.0681i 74.1200i 0 −849.690 −333.082 333.082i −2336.72 + 2336.72i 2187.00i 0
7.3 12.8476 + 12.8476i −33.0681 + 33.0681i 74.1200i 0 −849.690 333.082 + 333.082i 2336.72 2336.72i 2187.00i 0
7.4 17.7465 + 17.7465i 33.0681 33.0681i 373.880i 0 1173.69 −3270.12 3270.12i −2091.96 + 2091.96i 2187.00i 0
43.1 −17.7465 + 17.7465i −33.0681 33.0681i 373.880i 0 1173.69 3270.12 3270.12i 2091.96 + 2091.96i 2187.00i 0
43.2 −12.8476 + 12.8476i 33.0681 + 33.0681i 74.1200i 0 −849.690 −333.082 + 333.082i −2336.72 2336.72i 2187.00i 0
43.3 12.8476 12.8476i −33.0681 33.0681i 74.1200i 0 −849.690 333.082 333.082i 2336.72 + 2336.72i 2187.00i 0
43.4 17.7465 17.7465i 33.0681 + 33.0681i 373.880i 0 1173.69 −3270.12 + 3270.12i −2091.96 2091.96i 2187.00i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 43.4 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.c 8
5.b even 2 1 inner 75.9.f.c 8
5.c odd 4 2 inner 75.9.f.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.f.c 8 1.a even 1 1 trivial
75.9.f.c 8 5.b even 2 1 inner
75.9.f.c 8 5.c odd 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 505728T_{2}^{4} + 43237380096$$ acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + 505728 T^{4} + \cdots + 43237380096$$
$3$ $$(T^{4} + 4782969)^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8} + 457467317346834 T^{4} + \cdots + 22\!\cdots\!25$$
$11$ $$(T^{2} - 26988 T + 163195812)^{4}$$
$13$ $$T^{8} + \cdots + 73\!\cdots\!21$$
$17$ $$T^{8} + \cdots + 93\!\cdots\!56$$
$19$ $$(T^{4} + 32041029074 T^{2} + \cdots + 56\!\cdots\!25)^{2}$$
$23$ $$T^{8} + \cdots + 58\!\cdots\!00$$
$29$ $$(T^{4} + 1581918894216 T^{2} + \cdots + 55\!\cdots\!00)^{2}$$
$31$ $$(T^{2} + 727078 T - 438913901975)^{4}$$
$37$ $$T^{8} + \cdots + 12\!\cdots\!00$$
$41$ $$(T^{2} - 8478552 T + 16897916126160)^{4}$$
$43$ $$T^{8} + \cdots + 15\!\cdots\!01$$
$47$ $$T^{8} + \cdots + 38\!\cdots\!96$$
$53$ $$T^{8} + \cdots + 55\!\cdots\!00$$
$59$ $$(T^{4} + 184787676030024 T^{2} + \cdots + 91\!\cdots\!00)^{2}$$
$61$ $$(T^{2} - 7411750 T - 393607242140759)^{4}$$
$67$ $$T^{8} + \cdots + 12\!\cdots\!61$$
$71$ $$(T^{2} + 7061736 T - 1929008156976)^{4}$$
$73$ $$T^{8} + \cdots + 93\!\cdots\!00$$
$79$ $$(T^{4} + \cdots + 28\!\cdots\!00)^{2}$$
$83$ $$T^{8} + \cdots + 17\!\cdots\!96$$
$89$ $$(T^{4} + \cdots + 48\!\cdots\!00)^{2}$$
$97$ $$T^{8} + \cdots + 26\!\cdots\!21$$