# Properties

 Label 675.4.b.m Level $675$ Weight $4$ Character orbit 675.b Analytic conductor $39.826$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 675.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$39.8262892539$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.12559936.1 Defining polynomial: $$x^{6} - 6 x^{3} + 49 x^{2} - 42 x + 18$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 135) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \beta_{2} - \beta_{4} ) q^{2} + ( -7 + 3 \beta_{1} - \beta_{3} ) q^{4} + ( 2 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{7} + ( -29 \beta_{2} + 7 \beta_{4} + 5 \beta_{5} ) q^{8} +O(q^{10})$$ $$q + ( 2 \beta_{2} - \beta_{4} ) q^{2} + ( -7 + 3 \beta_{1} - \beta_{3} ) q^{4} + ( 2 \beta_{2} + \beta_{4} - 3 \beta_{5} ) q^{7} + ( -29 \beta_{2} + 7 \beta_{4} + 5 \beta_{5} ) q^{8} + ( 3 - 9 \beta_{1} + 5 \beta_{3} ) q^{11} + ( 5 \beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{13} + ( 13 + 16 \beta_{1} - 5 \beta_{3} ) q^{14} + ( 69 - 37 \beta_{1} + 9 \beta_{3} ) q^{16} + ( 51 \beta_{2} - 3 \beta_{4} + 5 \beta_{5} ) q^{17} + ( 28 - 33 \beta_{1} + \beta_{3} ) q^{19} + ( 95 \beta_{2} - 37 \beta_{4} - 19 \beta_{5} ) q^{22} + ( -85 \beta_{2} - 25 \beta_{4} - 5 \beta_{5} ) q^{23} + ( -72 + 21 \beta_{1} - 10 \beta_{3} ) q^{26} + ( -124 \beta_{2} + 36 \beta_{4} + 2 \beta_{5} ) q^{28} + ( 47 - \beta_{1} + 25 \beta_{3} ) q^{29} + ( -39 + 19 \beta_{1} + 17 \beta_{3} ) q^{31} + ( 295 \beta_{2} - 95 \beta_{4} - 15 \beta_{5} ) q^{32} + ( -145 + 29 \beta_{1} + 7 \beta_{3} ) q^{34} + ( 138 \beta_{2} - 55 \beta_{4} + 25 \beta_{5} ) q^{37} + ( 417 \beta_{2} - 66 \beta_{4} - 35 \beta_{5} ) q^{38} + ( -188 - 26 \beta_{1} - 10 \beta_{3} ) q^{41} + ( -281 \beta_{2} + 17 \beta_{4} + 29 \beta_{5} ) q^{43} + ( -535 + 155 \beta_{1} - 35 \beta_{3} ) q^{44} + ( -95 - 35 \beta_{1} - 35 \beta_{3} ) q^{46} + ( -9 \beta_{2} - 123 \beta_{4} + 5 \beta_{5} ) q^{47} + ( -161 - 53 \beta_{1} + 41 \beta_{3} ) q^{49} + ( -315 \beta_{2} + 95 \beta_{4} + 25 \beta_{5} ) q^{52} + ( 136 \beta_{2} - 38 \beta_{4} + 30 \beta_{5} ) q^{53} + ( 744 - 42 \beta_{1} ) q^{56} + ( 55 \beta_{2} - 173 \beta_{4} - 51 \beta_{5} ) q^{58} + ( 104 + 68 \beta_{1} ) q^{59} + ( -26 - 43 \beta_{1} + 31 \beta_{3} ) q^{61} + ( -321 \beta_{2} - 27 \beta_{4} - 15 \beta_{5} ) q^{62} + ( -1053 + 169 \beta_{1} - 53 \beta_{3} ) q^{64} + ( -40 \beta_{2} + 121 \beta_{4} - 3 \beta_{5} ) q^{67} + ( -215 \beta_{2} + 115 \beta_{4} + 55 \beta_{5} ) q^{68} + ( -64 + 182 \beta_{1} + 30 \beta_{3} ) q^{71} + ( -306 \beta_{2} - 59 \beta_{4} - 3 \beta_{5} ) q^{73} + ( -931 + 68 \beta_{1} - 5 \beta_{3} ) q^{74} + ( -1266 + 394 \beta_{1} - 128 \beta_{3} ) q^{76} + ( 658 \beta_{2} - 104 \beta_{4} + 80 \beta_{5} ) q^{77} + ( -343 - 38 \beta_{1} - 54 \beta_{3} ) q^{79} + ( -70 \beta_{2} + 212 \beta_{4} - 6 \beta_{5} ) q^{82} + ( -102 \beta_{2} - 24 \beta_{4} + 60 \beta_{5} ) q^{83} + ( 691 - 443 \beta_{1} + 75 \beta_{3} ) q^{86} + ( -1945 \beta_{2} + 569 \beta_{4} + 73 \beta_{5} ) q^{88} + ( 360 + 60 \beta_{3} ) q^{89} + ( -230 + 81 \beta_{1} + 23 \beta_{3} ) q^{91} + ( -415 \beta_{2} + 35 \beta_{4} - 5 \beta_{5} ) q^{92} + ( -1345 + 89 \beta_{1} - 113 \beta_{3} ) q^{94} + ( -202 \beta_{2} - 113 \beta_{4} - \beta_{5} ) q^{97} + ( 179 \beta_{2} - 97 \beta_{4} - 135 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 34 q^{4} + O(q^{10})$$ $$6 q - 34 q^{4} - 10 q^{11} + 120 q^{14} + 322 q^{16} + 100 q^{19} - 370 q^{26} + 230 q^{29} - 230 q^{31} - 826 q^{34} - 1160 q^{41} - 2830 q^{44} - 570 q^{46} - 1154 q^{49} + 4380 q^{56} + 760 q^{59} - 304 q^{61} - 5874 q^{64} - 80 q^{71} - 5440 q^{74} - 6552 q^{76} - 2026 q^{79} + 3110 q^{86} + 2040 q^{89} - 1264 q^{91} - 7666 q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 6 x^{3} + 49 x^{2} - 42 x + 18$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{5} + 49 \nu^{4} + 21 \nu^{3} - 9 \nu^{2} + 1544$$$$)/334$$ $$\beta_{2}$$ $$=$$ $$($$$$-49 \nu^{5} - 21 \nu^{4} - 9 \nu^{3} + 147 \nu^{2} - 2338 \nu + 1056$$$$)/1002$$ $$\beta_{3}$$ $$=$$ $$($$$$27 \nu^{5} + 107 \nu^{4} + 189 \nu^{3} - 81 \nu^{2} + 1872$$$$)/334$$ $$\beta_{4}$$ $$=$$ $$($$$$-245 \nu^{5} - 105 \nu^{4} - 45 \nu^{3} + 1737 \nu^{2} - 11690 \nu + 5280$$$$)/1002$$ $$\beta_{5}$$ $$=$$ $$($$$$63 \nu^{5} + 27 \nu^{4} - 60 \nu^{3} - 523 \nu^{2} + 2505 \nu - 1143$$$$)/167$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_{1} - 1$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{5} - 14 \beta_{4} + 7 \beta_{3} + 25 \beta_{2} - 14 \beta_{1} + 25$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$-\beta_{3} + 9 \beta_{1} - 36$$ $$\nu^{5}$$ $$=$$ $$($$$$49 \beta_{5} + 116 \beta_{4} + 49 \beta_{3} - 265 \beta_{2} - 116 \beta_{1} + 265$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
649.1
 −2.05655 − 2.05655i 0.455589 − 0.455589i 1.60096 + 1.60096i 1.60096 − 1.60096i 0.455589 + 0.455589i −2.05655 + 2.05655i
5.45876i 0 −21.7980 0 0 11.8065i 75.3201i 0 0
649.2 2.58488i 0 1.31841 0 0 22.8935i 24.0869i 0 0
649.3 2.12612i 0 3.47962 0 0 30.7000i 24.4070i 0 0
649.4 2.12612i 0 3.47962 0 0 30.7000i 24.4070i 0 0
649.5 2.58488i 0 1.31841 0 0 22.8935i 24.0869i 0 0
649.6 5.45876i 0 −21.7980 0 0 11.8065i 75.3201i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 649.6 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 675.4.b.m 6
3.b odd 2 1 675.4.b.n 6
5.b even 2 1 inner 675.4.b.m 6
5.c odd 4 1 135.4.a.e 3
5.c odd 4 1 675.4.a.s 3
15.d odd 2 1 675.4.b.n 6
15.e even 4 1 135.4.a.h yes 3
15.e even 4 1 675.4.a.p 3
20.e even 4 1 2160.4.a.bi 3
45.k odd 12 2 405.4.e.v 6
45.l even 12 2 405.4.e.q 6
60.l odd 4 1 2160.4.a.bq 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.e 3 5.c odd 4 1
135.4.a.h yes 3 15.e even 4 1
405.4.e.q 6 45.l even 12 2
405.4.e.v 6 45.k odd 12 2
675.4.a.p 3 15.e even 4 1
675.4.a.s 3 5.c odd 4 1
675.4.b.m 6 1.a even 1 1 trivial
675.4.b.m 6 5.b even 2 1 inner
675.4.b.n 6 3.b odd 2 1
675.4.b.n 6 15.d odd 2 1
2160.4.a.bi 3 20.e even 4 1
2160.4.a.bq 3 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(675, [\chi])$$:

 $$T_{2}^{6} + 41 T_{2}^{4} + 364 T_{2}^{2} + 900$$ $$T_{7}^{6} + 1606 T_{7}^{4} + 698409 T_{7}^{2} + 68856804$$ $$T_{11}^{3} + 5 T_{11}^{2} - 2888 T_{11} - 31260$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$900 + 364 T^{2} + 41 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$68856804 + 698409 T^{2} + 1606 T^{4} + T^{6}$$
$11$ $$( -31260 - 2888 T + 5 T^{2} + T^{3} )^{2}$$
$13$ $$41280625 + 501411 T^{2} + 1587 T^{4} + T^{6}$$
$17$ $$1743897600 + 18504064 T^{2} + 12809 T^{4} + T^{6}$$
$19$ $$( 368012 - 16663 T - 50 T^{2} + T^{3} )^{2}$$
$23$ $$306362250000 + 577935000 T^{2} + 48825 T^{4} + T^{6}$$
$29$ $$( 6440340 - 47408 T - 115 T^{2} + T^{3} )^{2}$$
$31$ $$( -938304 - 29232 T + 115 T^{2} + T^{3} )^{2}$$
$37$ $$513801865171204 + 22021941993 T^{2} + 283302 T^{4} + T^{6}$$
$41$ $$( 3917280 + 89812 T + 580 T^{2} + T^{3} )^{2}$$
$43$ $$28707478180096 + 28852500384 T^{2} + 350169 T^{4} + T^{6}$$
$47$ $$207021450297600 + 57904129504 T^{2} + 519329 T^{4} + T^{6}$$
$53$ $$160233065222400 + 13510637584 T^{2} + 276344 T^{4} + T^{6}$$
$59$ $$( 5205120 - 27392 T - 380 T^{2} + T^{3} )^{2}$$
$61$ $$( -5069066 - 87475 T + 152 T^{2} + T^{3} )^{2}$$
$67$ $$431363822490000 + 59618589201 T^{2} + 488002 T^{4} + T^{6}$$
$71$ $$( -216071280 - 677372 T + 40 T^{2} + T^{3} )^{2}$$
$73$ $$270535190786116 + 37739718249 T^{2} + 431334 T^{4} + T^{6}$$
$79$ $$( -90596925 + 52215 T + 1013 T^{2} + T^{3} )^{2}$$
$83$ $$7146869573505600 + 136509985584 T^{2} + 675756 T^{4} + T^{6}$$
$89$ $$( 125064000 + 46800 T - 1020 T^{2} + T^{3} )^{2}$$
$97$ $$752435512191364 + 40694604489 T^{2} + 587526 T^{4} + T^{6}$$
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