# Properties

 Label 675.4.b.k 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: $$x^{6} + 47x^{4} + 541x^{2} + 36$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{4} - \beta_{2} - 7) q^{4} + (14 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{5} - 9 \beta_{3} - 8 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b4 - b2 - 7) * q^4 + (14*b3 + 2*b1) * q^7 + (b5 - 9*b3 - 8*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{4} - \beta_{2} - 7) q^{4} + (14 \beta_{3} + 2 \beta_1) q^{7} + (\beta_{5} - 9 \beta_{3} - 8 \beta_1) q^{8} + (4 \beta_{4} + 2 \beta_{2} - 12) q^{11} + ( - 4 \beta_{5} - 14 \beta_{3} + 10 \beta_1) q^{13} + (2 \beta_{4} - 16 \beta_{2} - 30) q^{14} + (17 \beta_{2} + 58) q^{16} + ( - 8 \beta_{5} + 3 \beta_{3} + 2 \beta_1) q^{17} + ( - 4 \beta_{4} - 14 \beta_{2} - 59) q^{19} + ( - 2 \beta_{5} + 54 \beta_{3} - 42 \beta_1) q^{22} + ( - 4 \beta_{5} - 39 \beta_{3} + 32 \beta_1) q^{23} + (10 \beta_{4} - 28 \beta_{2} - 126) q^{26} + (16 \beta_{5} - 116 \beta_{3} - 46 \beta_1) q^{28} + (8 \beta_{4} + 42 \beta_{2} + 42) q^{29} + ( - 8 \beta_{4} - 30 \beta_{2} + 83) q^{31} + ( - 9 \beta_{5} + 183 \beta_{3} + 11 \beta_1) q^{32} + (2 \beta_{4} - 69 \beta_{2} + 18) q^{34} + (8 \beta_{5} + 50 \beta_{3} - 64 \beta_1) q^{37} + (14 \beta_{5} - 234 \beta_{3} - 41 \beta_1) q^{38} + (16 \beta_{4} - 42 \beta_{2} + 132) q^{41} + ( - 8 \beta_{5} + 16 \beta_{3} - 78 \beta_1) q^{43} + ( - 10 \beta_{4} - 12 \beta_{2} + 546) q^{44} + (32 \beta_{4} - 25 \beta_{2} - 456) q^{46} + (4 \beta_{5} + 168 \beta_{3} - 28 \beta_1) q^{47} + (4 \beta_{4} - 60 \beta_{2} + 87) q^{49} + ( - 4 \beta_{5} - 472 \beta_{3} - 154 \beta_1) q^{52} + ( - 12 \beta_{5} + 165 \beta_{3} + 14 \beta_1) q^{53} + ( - 30 \beta_{4} + 162 \beta_{2} + 354) q^{56} + ( - 42 \beta_{5} + 678 \beta_{3} + 20 \beta_1) q^{58} + (12 \beta_{4} - 82 \beta_{2} + 78) q^{59} + ( - 16 \beta_{4} + 60 \beta_{2} + 173) q^{61} + (30 \beta_{5} - 498 \beta_{3} + 117 \beta_1) q^{62} + (11 \beta_{4} - 130 \beta_{2} + 353) q^{64} + ( - 24 \beta_{5} + 302 \beta_{3} - 52 \beta_1) q^{67} + (5 \beta_{5} - 999 \beta_{3} - 51 \beta_1) q^{68} + ( - 60 \beta_{4} - 34 \beta_{2} + 192) q^{71} + (60 \beta_{5} - 404 \beta_{3} - 22 \beta_1) q^{73} + ( - 64 \beta_{4} + 78 \beta_{2} + 912) q^{74} + ( - 73 \beta_{4} + 275 \beta_{2} + 59) q^{76} + (52 \beta_{5} - 60 \beta_{3} - 56 \beta_1) q^{77} + ( - 12 \beta_{4} + 22 \beta_{2} - 221) q^{79} + (42 \beta_{5} - 534 \beta_{3} - 38 \beta_1) q^{82} + ( - 60 \beta_{5} - 489 \beta_{3} + 120 \beta_1) q^{83} + ( - 78 \beta_{4} - 2 \beta_{2} + 1218) q^{86} + ( - 4 \beta_{5} + 192 \beta_{3} + 278 \beta_1) q^{88} + (48 \beta_{4} - 66 \beta_{2} + 756) q^{89} + (76 \beta_{4} - 196 \beta_{2} - 56) q^{91} + ( - 7 \beta_{5} - 495 \beta_{3} - 481 \beta_1) q^{92} + ( - 28 \beta_{4} - 108 \beta_{2} + 396) q^{94} + ( - 8 \beta_{5} + 440 \beta_{3} + 64 \beta_1) q^{97} + (60 \beta_{5} - 876 \beta_{3} - 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b4 - b2 - 7) * q^4 + (14*b3 + 2*b1) * q^7 + (b5 - 9*b3 - 8*b1) * q^8 + (4*b4 + 2*b2 - 12) * q^11 + (-4*b5 - 14*b3 + 10*b1) * q^13 + (2*b4 - 16*b2 - 30) * q^14 + (17*b2 + 58) * q^16 + (-8*b5 + 3*b3 + 2*b1) * q^17 + (-4*b4 - 14*b2 - 59) * q^19 + (-2*b5 + 54*b3 - 42*b1) * q^22 + (-4*b5 - 39*b3 + 32*b1) * q^23 + (10*b4 - 28*b2 - 126) * q^26 + (16*b5 - 116*b3 - 46*b1) * q^28 + (8*b4 + 42*b2 + 42) * q^29 + (-8*b4 - 30*b2 + 83) * q^31 + (-9*b5 + 183*b3 + 11*b1) * q^32 + (2*b4 - 69*b2 + 18) * q^34 + (8*b5 + 50*b3 - 64*b1) * q^37 + (14*b5 - 234*b3 - 41*b1) * q^38 + (16*b4 - 42*b2 + 132) * q^41 + (-8*b5 + 16*b3 - 78*b1) * q^43 + (-10*b4 - 12*b2 + 546) * q^44 + (32*b4 - 25*b2 - 456) * q^46 + (4*b5 + 168*b3 - 28*b1) * q^47 + (4*b4 - 60*b2 + 87) * q^49 + (-4*b5 - 472*b3 - 154*b1) * q^52 + (-12*b5 + 165*b3 + 14*b1) * q^53 + (-30*b4 + 162*b2 + 354) * q^56 + (-42*b5 + 678*b3 + 20*b1) * q^58 + (12*b4 - 82*b2 + 78) * q^59 + (-16*b4 + 60*b2 + 173) * q^61 + (30*b5 - 498*b3 + 117*b1) * q^62 + (11*b4 - 130*b2 + 353) * q^64 + (-24*b5 + 302*b3 - 52*b1) * q^67 + (5*b5 - 999*b3 - 51*b1) * q^68 + (-60*b4 - 34*b2 + 192) * q^71 + (60*b5 - 404*b3 - 22*b1) * q^73 + (-64*b4 + 78*b2 + 912) * q^74 + (-73*b4 + 275*b2 + 59) * q^76 + (52*b5 - 60*b3 - 56*b1) * q^77 + (-12*b4 + 22*b2 - 221) * q^79 + (42*b5 - 534*b3 - 38*b1) * q^82 + (-60*b5 - 489*b3 + 120*b1) * q^83 + (-78*b4 - 2*b2 + 1218) * q^86 + (-4*b5 + 192*b3 + 278*b1) * q^88 + (48*b4 - 66*b2 + 756) * q^89 + (76*b4 - 196*b2 - 56) * q^91 + (-7*b5 - 495*b3 - 481*b1) * q^92 + (-28*b4 - 108*b2 + 396) * q^94 + (-8*b5 + 440*b3 + 64*b1) * q^97 + (60*b5 - 876*b3 - 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 46 q^{4}+O(q^{10})$$ 6 * q - 46 * q^4 $$6 q - 46 q^{4} - 76 q^{11} - 216 q^{14} + 382 q^{16} - 374 q^{19} - 832 q^{26} + 320 q^{29} + 454 q^{31} - 34 q^{34} + 676 q^{41} + 3272 q^{44} - 2850 q^{46} + 394 q^{49} + 2508 q^{56} + 280 q^{59} + 1190 q^{61} + 1836 q^{64} + 1204 q^{71} + 5756 q^{74} + 1050 q^{76} - 1258 q^{79} + 7460 q^{86} + 4308 q^{89} - 880 q^{91} + 2216 q^{94}+O(q^{100})$$ 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

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + 24\nu^{2} + 6 ) / 17$$ (v^4 + 24*v^2 + 6) / 17 $$\beta_{3}$$ $$=$$ $$( \nu^{5} + 41\nu^{3} + 397\nu ) / 102$$ (v^5 + 41*v^3 + 397*v) / 102 $$\beta_{4}$$ $$=$$ $$( \nu^{4} + 41\nu^{2} + 261 ) / 17$$ (v^4 + 41*v^2 + 261) / 17 $$\beta_{5}$$ $$=$$ $$( 3\nu^{5} + 157\nu^{3} + 2007\nu ) / 34$$ (3*v^5 + 157*v^3 + 2007*v) / 34
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{2} - 15$$ b4 - b2 - 15 $$\nu^{3}$$ $$=$$ $$\beta_{5} - 9\beta_{3} - 24\beta_1$$ b5 - 9*b3 - 24*b1 $$\nu^{4}$$ $$=$$ $$-24\beta_{4} + 41\beta_{2} + 354$$ -24*b4 + 41*b2 + 354 $$\nu^{5}$$ $$=$$ $$-41\beta_{5} + 471\beta_{3} + 587\beta_1$$ -41*b5 + 471*b3 + 587*b1

## 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.k 6
3.b odd 2 1 675.4.b.l 6
5.b even 2 1 inner 675.4.b.k 6
5.c odd 4 1 135.4.a.g yes 3
5.c odd 4 1 675.4.a.q 3
15.d odd 2 1 675.4.b.l 6
15.e even 4 1 135.4.a.f 3
15.e even 4 1 675.4.a.r 3
20.e even 4 1 2160.4.a.be 3
45.k odd 12 2 405.4.e.r 6
45.l even 12 2 405.4.e.t 6
60.l odd 4 1 2160.4.a.bm 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.a.f 3 15.e even 4 1
135.4.a.g yes 3 5.c odd 4 1
405.4.e.r 6 45.k odd 12 2
405.4.e.t 6 45.l even 12 2
675.4.a.q 3 5.c odd 4 1
675.4.a.r 3 15.e even 4 1
675.4.b.k 6 1.a even 1 1 trivial
675.4.b.k 6 5.b even 2 1 inner
675.4.b.l 6 3.b odd 2 1
675.4.b.l 6 15.d odd 2 1
2160.4.a.be 3 20.e even 4 1
2160.4.a.bm 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 $$S_{4}^{\mathrm{new}}(675, [\chi])$$:

 $$T_{2}^{6} + 47T_{2}^{4} + 541T_{2}^{2} + 36$$ T2^6 + 47*T2^4 + 541*T2^2 + 36 $$T_{7}^{6} + 832T_{7}^{4} + 146304T_{7}^{2} + 3240000$$ T7^6 + 832*T7^4 + 146304*T7^2 + 3240000 $$T_{11}^{3} + 38T_{11}^{2} - 2612T_{11} - 83280$$ T11^3 + 38*T11^2 - 2612*T11 - 83280

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 47 T^{4} + 541 T^{2} + \cdots + 36$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$T^{6} + 832 T^{4} + 146304 T^{2} + \cdots + 3240000$$
$11$ $$(T^{3} + 38 T^{2} - 2612 T - 83280)^{2}$$
$13$ $$T^{6} + 9936 T^{4} + \cdots + 10024014400$$
$17$ $$T^{6} + 23315 T^{4} + \cdots + 306790808769$$
$19$ $$(T^{3} + 187 T^{2} + 3587 T - 525871)^{2}$$
$23$ $$T^{6} + 53163 T^{4} + \cdots + 4177858328361$$
$29$ $$(T^{3} - 160 T^{2} - 47768 T + 7892760)^{2}$$
$31$ $$(T^{3} - 227 T^{2} - 17973 T - 246321)^{2}$$
$37$ $$T^{6} + \cdots + 185969950926400$$
$41$ $$(T^{3} - 338 T^{2} - 42812 T + 12116640)^{2}$$
$43$ $$T^{6} + \cdots + 340939975993600$$
$47$ $$T^{6} + 114368 T^{4} + \cdots + 80202240000$$
$53$ $$T^{6} + 148667 T^{4} + \cdots + 883203364521$$
$59$ $$(T^{3} - 140 T^{2} - 166448 T + 34131480)^{2}$$
$61$ $$(T^{3} - 595 T^{2} - 2749 T + 1782607)^{2}$$
$67$ $$T^{6} + \cdots + 127577025000000$$
$71$ $$(T^{3} - 602 T^{2} - 583652 T + 280550880)^{2}$$
$73$ $$T^{6} + 1866252 T^{4} + \cdots + 16\!\cdots\!00$$
$79$ $$(T^{3} + 629 T^{2} + 97059 T + 2010303)^{2}$$
$83$ $$T^{6} + 2388123 T^{4} + \cdots + 11\!\cdots\!21$$
$89$ $$(T^{3} - 2154 T^{2} + 1057572 T - 74325600)^{2}$$
$97$ $$T^{6} + 849792 T^{4} + \cdots + 40\!\cdots\!00$$