# Properties

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

# Related objects

## 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: 6.0.2033649216.1 Defining polynomial: $$x^{6} + 47 x^{4} + 541 x^{2} + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -7 - \beta_{2} + \beta_{4} ) q^{4} + ( -2 \beta_{1} - 14 \beta_{3} ) q^{7} + ( -8 \beta_{1} - 9 \beta_{3} + \beta_{5} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -7 - \beta_{2} + \beta_{4} ) q^{4} + ( -2 \beta_{1} - 14 \beta_{3} ) q^{7} + ( -8 \beta_{1} - 9 \beta_{3} + \beta_{5} ) q^{8} + ( 12 - 2 \beta_{2} - 4 \beta_{4} ) q^{11} + ( -10 \beta_{1} + 14 \beta_{3} + 4 \beta_{5} ) q^{13} + ( 30 + 16 \beta_{2} - 2 \beta_{4} ) q^{14} + ( 58 + 17 \beta_{2} ) q^{16} + ( 2 \beta_{1} + 3 \beta_{3} - 8 \beta_{5} ) q^{17} + ( -59 - 14 \beta_{2} - 4 \beta_{4} ) q^{19} + ( 42 \beta_{1} - 54 \beta_{3} + 2 \beta_{5} ) q^{22} + ( 32 \beta_{1} - 39 \beta_{3} - 4 \beta_{5} ) q^{23} + ( 126 + 28 \beta_{2} - 10 \beta_{4} ) q^{26} + ( 46 \beta_{1} + 116 \beta_{3} - 16 \beta_{5} ) q^{28} + ( -42 - 42 \beta_{2} - 8 \beta_{4} ) q^{29} + ( 83 - 30 \beta_{2} - 8 \beta_{4} ) q^{31} + ( 11 \beta_{1} + 183 \beta_{3} - 9 \beta_{5} ) q^{32} + ( 18 - 69 \beta_{2} + 2 \beta_{4} ) q^{34} + ( 64 \beta_{1} - 50 \beta_{3} - 8 \beta_{5} ) q^{37} + ( -41 \beta_{1} - 234 \beta_{3} + 14 \beta_{5} ) q^{38} + ( -132 + 42 \beta_{2} - 16 \beta_{4} ) q^{41} + ( 78 \beta_{1} - 16 \beta_{3} + 8 \beta_{5} ) q^{43} + ( -546 + 12 \beta_{2} + 10 \beta_{4} ) q^{44} + ( -456 - 25 \beta_{2} + 32 \beta_{4} ) q^{46} + ( -28 \beta_{1} + 168 \beta_{3} + 4 \beta_{5} ) q^{47} + ( 87 - 60 \beta_{2} + 4 \beta_{4} ) q^{49} + ( 154 \beta_{1} + 472 \beta_{3} + 4 \beta_{5} ) q^{52} + ( 14 \beta_{1} + 165 \beta_{3} - 12 \beta_{5} ) q^{53} + ( -354 - 162 \beta_{2} + 30 \beta_{4} ) q^{56} + ( -20 \beta_{1} - 678 \beta_{3} + 42 \beta_{5} ) q^{58} + ( -78 + 82 \beta_{2} - 12 \beta_{4} ) q^{59} + ( 173 + 60 \beta_{2} - 16 \beta_{4} ) q^{61} + ( 117 \beta_{1} - 498 \beta_{3} + 30 \beta_{5} ) q^{62} + ( 353 - 130 \beta_{2} + 11 \beta_{4} ) q^{64} + ( 52 \beta_{1} - 302 \beta_{3} + 24 \beta_{5} ) q^{67} + ( -51 \beta_{1} - 999 \beta_{3} + 5 \beta_{5} ) q^{68} + ( -192 + 34 \beta_{2} + 60 \beta_{4} ) q^{71} + ( 22 \beta_{1} + 404 \beta_{3} - 60 \beta_{5} ) q^{73} + ( -912 - 78 \beta_{2} + 64 \beta_{4} ) q^{74} + ( 59 + 275 \beta_{2} - 73 \beta_{4} ) q^{76} + ( -56 \beta_{1} - 60 \beta_{3} + 52 \beta_{5} ) q^{77} + ( -221 + 22 \beta_{2} - 12 \beta_{4} ) q^{79} + ( 38 \beta_{1} + 534 \beta_{3} - 42 \beta_{5} ) q^{82} + ( 120 \beta_{1} - 489 \beta_{3} - 60 \beta_{5} ) q^{83} + ( -1218 + 2 \beta_{2} + 78 \beta_{4} ) q^{86} + ( -278 \beta_{1} - 192 \beta_{3} + 4 \beta_{5} ) q^{88} + ( -756 + 66 \beta_{2} - 48 \beta_{4} ) q^{89} + ( -56 - 196 \beta_{2} + 76 \beta_{4} ) q^{91} + ( -481 \beta_{1} - 495 \beta_{3} - 7 \beta_{5} ) q^{92} + ( 396 - 108 \beta_{2} - 28 \beta_{4} ) q^{94} + ( -64 \beta_{1} - 440 \beta_{3} + 8 \beta_{5} ) q^{97} + ( -5 \beta_{1} - 876 \beta_{3} + 60 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 46 q^{4} + O(q^{10})$$ $$6 q - 46 q^{4} + 76 q^{11} + 216 q^{14} + 382 q^{16} - 374 q^{19} + 832 q^{26} - 320 q^{29} + 454 q^{31} - 34 q^{34} - 676 q^{41} - 3272 q^{44} - 2850 q^{46} + 394 q^{49} - 2508 q^{56} - 280 q^{59} + 1190 q^{61} + 1836 q^{64} - 1204 q^{71} - 5756 q^{74} + 1050 q^{76} - 1258 q^{79} - 7460 q^{86} - 4308 q^{89} - 880 q^{91} + 2216 q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 47 x^{4} + 541 x^{2} + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 24 \nu^{2} + 6$$$$)/17$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 41 \nu^{3} + 397 \nu$$$$)/102$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 41 \nu^{2} + 261$$$$)/17$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{5} + 157 \nu^{3} + 2007 \nu$$$$)/34$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{2} - 15$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} - 9 \beta_{3} - 24 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-24 \beta_{4} + 41 \beta_{2} + 354$$ $$\nu^{5}$$ $$=$$ $$-41 \beta_{5} + 471 \beta_{3} + 587 \beta_{1}$$

## 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
 − 5.20067i − 4.45938i − 0.258712i 0.258712i 4.45938i 5.20067i
5.20067i 0 −19.0470 0 0 24.4013i 57.4517i 0 0
649.2 4.45938i 0 −11.8861 0 0 5.08123i 17.3296i 0 0
649.3 0.258712i 0 7.93307 0 0 14.5174i 4.12208i 0 0
649.4 0.258712i 0 7.93307 0 0 14.5174i 4.12208i 0 0
649.5 4.45938i 0 −11.8861 0 0 5.08123i 17.3296i 0 0
649.6 5.20067i 0 −19.0470 0 0 24.4013i 57.4517i 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.l 6
3.b odd 2 1 675.4.b.k 6
5.b even 2 1 inner 675.4.b.l 6
5.c odd 4 1 135.4.a.f 3
5.c odd 4 1 675.4.a.r 3
15.d odd 2 1 675.4.b.k 6
15.e even 4 1 135.4.a.g yes 3
15.e even 4 1 675.4.a.q 3
20.e even 4 1 2160.4.a.bm 3
45.k odd 12 2 405.4.e.t 6
45.l even 12 2 405.4.e.r 6
60.l odd 4 1 2160.4.a.be 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 5.c odd 4 1
135.4.a.g yes 3 15.e even 4 1
405.4.e.r 6 45.l even 12 2
405.4.e.t 6 45.k odd 12 2
675.4.a.q 3 15.e even 4 1
675.4.a.r 3 5.c odd 4 1
675.4.b.k 6 3.b odd 2 1
675.4.b.k 6 15.d odd 2 1
675.4.b.l 6 1.a even 1 1 trivial
675.4.b.l 6 5.b even 2 1 inner
2160.4.a.be 3 60.l odd 4 1
2160.4.a.bm 3 20.e even 4 1

## Hecke kernels

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

 $$T_{2}^{6} + 47 T_{2}^{4} + 541 T_{2}^{2} + 36$$ $$T_{7}^{6} + 832 T_{7}^{4} + 146304 T_{7}^{2} + 3240000$$ $$T_{11}^{3} - 38 T_{11}^{2} - 2612 T_{11} + 83280$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$36 + 541 T^{2} + 47 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$3240000 + 146304 T^{2} + 832 T^{4} + T^{6}$$
$11$ $$( 83280 - 2612 T - 38 T^{2} + T^{3} )^{2}$$
$13$ $$10024014400 + 26546496 T^{2} + 9936 T^{4} + T^{6}$$
$17$ $$306790808769 + 152769235 T^{2} + 23315 T^{4} + T^{6}$$
$19$ $$( -525871 + 3587 T + 187 T^{2} + T^{3} )^{2}$$
$23$ $$4177858328361 + 874061523 T^{2} + 53163 T^{4} + T^{6}$$
$29$ $$( -7892760 - 47768 T + 160 T^{2} + T^{3} )^{2}$$
$31$ $$( -246321 - 17973 T - 227 T^{2} + T^{3} )^{2}$$
$37$ $$185969950926400 + 12112190256 T^{2} + 205932 T^{4} + T^{6}$$
$41$ $$( -12116640 - 42812 T + 338 T^{2} + T^{3} )^{2}$$
$43$ $$340939975993600 + 26385567696 T^{2} + 320316 T^{4} + T^{6}$$
$47$ $$80202240000 + 3205848064 T^{2} + 114368 T^{4} + T^{6}$$
$53$ $$883203364521 + 4747623907 T^{2} + 148667 T^{4} + T^{6}$$
$59$ $$( -34131480 - 166448 T + 140 T^{2} + T^{3} )^{2}$$
$61$ $$( 1782607 - 2749 T - 595 T^{2} + T^{3} )^{2}$$
$67$ $$127577025000000 + 25602118704 T^{2} + 618988 T^{4} + T^{6}$$
$71$ $$( -280550880 - 583652 T + 602 T^{2} + T^{3} )^{2}$$
$73$ $$163289762572038400 + 1054986577104 T^{2} + 1866252 T^{4} + T^{6}$$
$79$ $$( 2010303 + 97059 T + 629 T^{2} + T^{3} )^{2}$$
$83$ $$119996015960256921 + 1025510933043 T^{2} + 2388123 T^{4} + T^{6}$$
$89$ $$( 74325600 + 1057572 T + 2154 T^{2} + T^{3} )^{2}$$
$97$ $$4044390164070400 + 118816444416 T^{2} + 849792 T^{4} + T^{6}$$