# Properties

 Label 75.4.e.c Level $75$ Weight $4$ Character orbit 75.e Analytic conductor $4.425$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.28356903014400.8 Defining polynomial: $$x^{8} + 209 x^{4} + 1600$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 15) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + ( 1 + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( -1 - 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{6} + ( 1 - \beta_{2} - 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{7} + ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( 3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{3} q^{2} + ( 1 + \beta_{2} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{4} + ( -1 - 3 \beta_{1} - 3 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{6} + ( 1 - \beta_{2} - 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{7} + ( 3 \beta_{1} - 2 \beta_{6} + 2 \beta_{7} ) q^{8} + ( 3 \beta_{1} + 9 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} ) q^{9} + ( 7 \beta_{1} + 7 \beta_{3} - 2 \beta_{6} ) q^{11} + ( -17 + 17 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{12} + ( -8 - 8 \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{13} + ( -7 \beta_{1} + 7 \beta_{3} + 8 \beta_{7} ) q^{14} + ( 39 + 7 \beta_{4} - 7 \beta_{5} ) q^{16} + ( 20 \beta_{3} + 7 \beta_{6} + 7 \beta_{7} ) q^{17} + ( 30 - 21 \beta_{1} + 30 \beta_{2} - 6 \beta_{6} + 6 \beta_{7} ) q^{18} + ( -12 \beta_{2} - 6 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{19} + ( -62 - 6 \beta_{1} - 6 \beta_{3} - \beta_{4} + \beta_{5} + 11 \beta_{6} ) q^{21} + ( 65 - 65 \beta_{2} + 10 \beta_{4} - 5 \beta_{6} + 5 \beta_{7} ) q^{22} + ( 14 \beta_{1} + 9 \beta_{6} - 9 \beta_{7} ) q^{23} + ( 3 \beta_{1} + 39 \beta_{2} - 3 \beta_{3} + 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 3 \beta_{7} ) q^{24} + ( 4 \beta_{1} + 4 \beta_{3} - 4 \beta_{6} ) q^{26} + ( -87 + 87 \beta_{2} - 18 \beta_{3} + 3 \beta_{4} - 9 \beta_{6} - 6 \beta_{7} ) q^{27} + ( -63 - 63 \beta_{2} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{28} + ( 3 \beta_{1} - 3 \beta_{3} - 22 \beta_{7} ) q^{29} + ( 62 - 30 \beta_{4} + 30 \beta_{5} ) q^{31} + ( -43 \beta_{3} - 30 \beta_{6} - 30 \beta_{7} ) q^{32} + ( 25 + 36 \beta_{1} + 25 \beta_{2} + 20 \beta_{5} + 7 \beta_{6} - 7 \beta_{7} ) q^{33} + ( -166 \beta_{2} + 34 \beta_{4} + 34 \beta_{5} - 34 \beta_{6} + 34 \beta_{7} ) q^{34} + ( -105 - 6 \beta_{1} - 6 \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 24 \beta_{6} ) q^{36} + ( 146 - 146 \beta_{2} + 6 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{37} + ( -12 \beta_{1} - 12 \beta_{6} + 12 \beta_{7} ) q^{38} + ( 3 \beta_{1} + 16 \beta_{2} - 3 \beta_{3} + 7 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} + 20 \beta_{7} ) q^{39} + ( -32 \beta_{1} - 32 \beta_{3} + 52 \beta_{6} ) q^{41} + ( -65 + 65 \beta_{2} + 66 \beta_{3} + 10 \beta_{4} - 3 \beta_{6} + 7 \beta_{7} ) q^{42} + ( -78 - 78 \beta_{2} + 38 \beta_{5} - 19 \beta_{6} + 19 \beta_{7} ) q^{43} + ( 29 \beta_{1} - 29 \beta_{3} - 36 \beta_{7} ) q^{44} + ( 108 + 32 \beta_{4} - 32 \beta_{5} ) q^{46} + ( 2 \beta_{3} + 21 \beta_{6} + 21 \beta_{7} ) q^{47} + ( 151 + 21 \beta_{1} + 151 \beta_{2} - 46 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{48} + ( 85 \beta_{2} - 16 \beta_{4} - 16 \beta_{5} + 16 \beta_{6} - 16 \beta_{7} ) q^{49} + ( -106 + 81 \beta_{1} + 81 \beta_{3} + \beta_{4} - \beta_{5} + 19 \beta_{6} ) q^{51} + ( -24 + 24 \beta_{2} + 16 \beta_{4} - 8 \beta_{6} + 8 \beta_{7} ) q^{52} + ( -130 \beta_{1} + \beta_{6} - \beta_{7} ) q^{53} + ( -81 \beta_{1} + 147 \beta_{2} + 81 \beta_{3} - 33 \beta_{4} - 33 \beta_{5} + 33 \beta_{6} - 39 \beta_{7} ) q^{54} + ( 3 \beta_{1} + 3 \beta_{3} - 68 \beta_{6} ) q^{56} + ( -84 + 84 \beta_{2} - 18 \beta_{3} + 6 \beta_{4} + 18 \beta_{6} + 24 \beta_{7} ) q^{57} + ( 5 + 5 \beta_{2} - 50 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{58} + ( -39 \beta_{1} + 39 \beta_{3} + 136 \beta_{7} ) q^{59} + 2 q^{61} + ( 58 \beta_{3} + 60 \beta_{6} + 60 \beta_{7} ) q^{62} + ( -183 - 6 \beta_{1} - 183 \beta_{2} + 18 \beta_{5} - 66 \beta_{6} + 66 \beta_{7} ) q^{63} + ( 15 \beta_{2} - 47 \beta_{4} - 47 \beta_{5} + 47 \beta_{6} - 47 \beta_{7} ) q^{64} + ( 290 + 15 \beta_{1} + 15 \beta_{3} + 70 \beta_{4} - 70 \beta_{5} + 40 \beta_{6} ) q^{66} + ( -84 + 84 \beta_{2} - 134 \beta_{4} + 67 \beta_{6} - 67 \beta_{7} ) q^{67} + ( 142 \beta_{1} + 12 \beta_{6} - 12 \beta_{7} ) q^{68} + ( 69 \beta_{1} - 148 \beta_{2} - 69 \beta_{3} - 13 \beta_{4} - 13 \beta_{5} + 13 \beta_{6} - 14 \beta_{7} ) q^{69} + ( -116 \beta_{1} - 116 \beta_{3} - 74 \beta_{6} ) q^{71} + ( 210 - 210 \beta_{2} - 27 \beta_{3} - 60 \beta_{4} - 60 \beta_{7} ) q^{72} + ( 317 + 317 \beta_{2} - 12 \beta_{5} + 6 \beta_{6} - 6 \beta_{7} ) q^{73} + ( 158 \beta_{1} - 158 \beta_{3} - 12 \beta_{7} ) q^{74} + ( -180 + 12 \beta_{4} - 12 \beta_{5} ) q^{76} + ( -74 \beta_{3} - 78 \beta_{6} - 78 \beta_{7} ) q^{77} + ( 40 + 12 \beta_{1} + 40 \beta_{2} + 20 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{78} + ( 298 \beta_{2} - 56 \beta_{4} - 56 \beta_{5} + 56 \beta_{6} - 56 \beta_{7} ) q^{79} + ( -9 - 72 \beta_{1} - 72 \beta_{3} - 90 \beta_{4} + 90 \beta_{5} - 108 \beta_{6} ) q^{81} + ( -340 + 340 \beta_{2} + 40 \beta_{4} - 20 \beta_{6} + 20 \beta_{7} ) q^{82} + ( 86 \beta_{1} - 7 \beta_{6} + 7 \beta_{7} ) q^{83} + ( 3 \beta_{1} - 94 \beta_{2} - 3 \beta_{3} + 62 \beta_{4} + 62 \beta_{5} - 62 \beta_{6} + 130 \beta_{7} ) q^{84} + ( 154 \beta_{1} + 154 \beta_{3} + 76 \beta_{6} ) q^{86} + ( 195 - 195 \beta_{2} - 84 \beta_{3} - 60 \beta_{4} + 57 \beta_{6} - 3 \beta_{7} ) q^{87} + ( -295 - 295 \beta_{2} - 50 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} ) q^{88} + ( -276 \beta_{1} + 276 \beta_{3} - 126 \beta_{7} ) q^{89} + ( -144 + 38 \beta_{4} - 38 \beta_{5} ) q^{91} + ( -124 \beta_{3} + 8 \beta_{6} + 8 \beta_{7} ) q^{92} + ( -418 - 90 \beta_{1} - 418 \beta_{2} - 32 \beta_{5} + 122 \beta_{6} - 122 \beta_{7} ) q^{93} + ( 24 \beta_{2} + 44 \beta_{4} + 44 \beta_{5} - 44 \beta_{6} + 44 \beta_{7} ) q^{94} + ( 497 - 219 \beta_{1} - 219 \beta_{3} - 47 \beta_{4} + 47 \beta_{5} + 4 \beta_{6} ) q^{96} + ( -179 + 179 \beta_{2} + 236 \beta_{4} - 118 \beta_{6} + 118 \beta_{7} ) q^{97} + ( -149 \beta_{1} - 32 \beta_{6} + 32 \beta_{7} ) q^{98} + ( 129 \beta_{1} - 540 \beta_{2} - 129 \beta_{3} - 30 \beta_{4} - 30 \beta_{5} + 30 \beta_{6} - 126 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 6q^{3} - 12q^{6} + 16q^{7} + O(q^{10})$$ $$8q + 6q^{3} - 12q^{6} + 16q^{7} - 132q^{12} - 68q^{13} + 284q^{16} + 240q^{18} - 492q^{21} + 500q^{22} - 702q^{27} - 508q^{28} + 616q^{31} + 240q^{33} - 804q^{36} + 1156q^{37} - 540q^{42} - 548q^{43} + 736q^{46} + 1116q^{48} - 852q^{51} - 224q^{52} - 684q^{57} - 60q^{58} + 16q^{61} - 1428q^{63} + 2040q^{66} - 404q^{67} + 1800q^{72} + 2512q^{73} - 1488q^{76} + 360q^{78} + 288q^{81} - 2800q^{82} + 1680q^{87} - 2460q^{88} - 1304q^{91} - 3408q^{93} + 4164q^{96} - 1904q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 209 x^{4} + 1600$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 249 \nu^{2}$$$$)/680$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 249 \nu^{3}$$$$)/680$$ $$\beta_{4}$$ $$=$$ $$($$$$-9 \nu^{6} - 20 \nu^{5} - 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu - 4520$$$$)/1360$$ $$\beta_{5}$$ $$=$$ $$($$$$-9 \nu^{6} - 20 \nu^{5} + 40 \nu^{4} - 1561 \nu^{2} - 3620 \nu + 4520$$$$)/1360$$ $$\beta_{6}$$ $$=$$ $$($$$$-13 \nu^{7} - 40 \nu^{5} - 2557 \nu^{3} - 7240 \nu$$$$)/2720$$ $$\beta_{7}$$ $$=$$ $$($$$$-13 \nu^{7} + 40 \nu^{5} - 2557 \nu^{3} + 7240 \nu$$$$)/2720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - 9 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{6} + 13 \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$17 \beta_{5} - 17 \beta_{4} - 113$$ $$\nu^{5}$$ $$=$$ $$34 \beta_{7} - 34 \beta_{6} - 181 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-249 \beta_{7} + 249 \beta_{6} - 249 \beta_{5} - 249 \beta_{4} + 1561 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$-498 \beta_{7} - 498 \beta_{6} - 2557 \beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −2.66260 − 2.66260i −1.18766 − 1.18766i 1.18766 + 1.18766i 2.66260 + 2.66260i −2.66260 + 2.66260i −1.18766 + 1.18766i 1.18766 − 1.18766i 2.66260 − 2.66260i
−2.66260 + 2.66260i 2.80471 4.37420i 6.17891i 0 4.17891 + 19.1146i −9.35782 9.35782i −4.84884 4.84884i −11.2672 24.5367i 0
32.2 −1.18766 + 1.18766i 0.932827 + 5.11173i 5.17891i 0 −7.17891 4.96314i 13.3578 + 13.3578i −15.6521 15.6521i −25.2597 + 9.53673i 0
32.3 1.18766 1.18766i −5.11173 0.932827i 5.17891i 0 −7.17891 + 4.96314i 13.3578 + 13.3578i 15.6521 + 15.6521i 25.2597 + 9.53673i 0
32.4 2.66260 2.66260i 4.37420 2.80471i 6.17891i 0 4.17891 19.1146i −9.35782 9.35782i 4.84884 + 4.84884i 11.2672 24.5367i 0
68.1 −2.66260 2.66260i 2.80471 + 4.37420i 6.17891i 0 4.17891 19.1146i −9.35782 + 9.35782i −4.84884 + 4.84884i −11.2672 + 24.5367i 0
68.2 −1.18766 1.18766i 0.932827 5.11173i 5.17891i 0 −7.17891 + 4.96314i 13.3578 13.3578i −15.6521 + 15.6521i −25.2597 9.53673i 0
68.3 1.18766 + 1.18766i −5.11173 + 0.932827i 5.17891i 0 −7.17891 4.96314i 13.3578 13.3578i 15.6521 15.6521i 25.2597 9.53673i 0
68.4 2.66260 + 2.66260i 4.37420 + 2.80471i 6.17891i 0 4.17891 + 19.1146i −9.35782 + 9.35782i 4.84884 4.84884i 11.2672 + 24.5367i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 68.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.e.c 8
3.b odd 2 1 inner 75.4.e.c 8
5.b even 2 1 15.4.e.a 8
5.c odd 4 1 15.4.e.a 8
5.c odd 4 1 inner 75.4.e.c 8
15.d odd 2 1 15.4.e.a 8
15.e even 4 1 15.4.e.a 8
15.e even 4 1 inner 75.4.e.c 8
20.d odd 2 1 240.4.v.c 8
20.e even 4 1 240.4.v.c 8
60.h even 2 1 240.4.v.c 8
60.l odd 4 1 240.4.v.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.e.a 8 5.b even 2 1
15.4.e.a 8 5.c odd 4 1
15.4.e.a 8 15.d odd 2 1
15.4.e.a 8 15.e even 4 1
75.4.e.c 8 1.a even 1 1 trivial
75.4.e.c 8 3.b odd 2 1 inner
75.4.e.c 8 5.c odd 4 1 inner
75.4.e.c 8 15.e even 4 1 inner
240.4.v.c 8 20.d odd 2 1
240.4.v.c 8 20.e even 4 1
240.4.v.c 8 60.h even 2 1
240.4.v.c 8 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 209 T_{2}^{4} + 1600$$ acting on $$S_{4}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1600 + 209 T^{4} + T^{8}$$
$3$ $$531441 - 118098 T + 13122 T^{2} + 5346 T^{3} - 1422 T^{4} + 198 T^{5} + 18 T^{6} - 6 T^{7} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$( 62500 + 2000 T + 32 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$11$ $$( 961000 + 1990 T^{2} + T^{4} )^{2}$$
$13$ $$( 6400 + 2720 T + 578 T^{2} + 34 T^{3} + T^{4} )^{2}$$
$17$ $$32321044225600 + 68589044 T^{4} + T^{8}$$
$19$ $$( 876096 + 2772 T^{2} + T^{4} )^{2}$$
$23$ $$227401574425600 + 36067364 T^{4} + T^{8}$$
$29$ $$( 40401000 - 18390 T^{2} + T^{4} )^{2}$$
$31$ $$( -23096 - 154 T + T^{2} )^{4}$$
$37$ $$( 1695792400 - 23802040 T + 167042 T^{2} - 578 T^{3} + T^{4} )^{2}$$
$41$ $$( 1201216000 + 160240 T^{2} + T^{4} )^{2}$$
$43$ $$( 193210000 - 3808600 T + 37538 T^{2} + 274 T^{3} + T^{4} )^{2}$$
$47$ $$8274187321753600 + 971492324 T^{4} + T^{8}$$
$53$ $$14\!\cdots\!00$$$$+ 59035541924 T^{4} + T^{8}$$
$59$ $$( 94361796000 - 709410 T^{2} + T^{4} )^{2}$$
$61$ $$( -2 + T )^{8}$$
$67$ $$( 80906113600 - 57456880 T + 20402 T^{2} + 202 T^{3} + T^{4} )^{2}$$
$71$ $$( 15888196000 + 568060 T^{2} + T^{4} )^{2}$$
$73$ $$( 37974316900 - 244756720 T + 788768 T^{2} - 1256 T^{3} + T^{4} )^{2}$$
$79$ $$( 797271696 + 348072 T^{2} + T^{4} )^{2}$$
$83$ $$15732888703346713600 + 10775280164 T^{4} + T^{8}$$
$89$ $$( 3249684036000 - 3626460 T^{2} + T^{4} )^{2}$$
$97$ $$( 615926736100 - 747139120 T + 453152 T^{2} + 952 T^{3} + T^{4} )^{2}$$