Properties

 Label 4056.2.c.p Level $4056$ Weight $2$ Character orbit 4056.c Analytic conductor $32.387$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

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Newspace parameters

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.c (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$32.3873230598$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.649638144.4 Defining polynomial: $$x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169$$ x^8 - 14*x^6 + 75*x^4 - 170*x^2 + 169 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: no (minimal twist has level 312) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{3} - \beta_1 q^{5} + ( - \beta_{4} - \beta_1) q^{7} + q^{9}+O(q^{10})$$ q + q^3 - b1 * q^5 + (-b4 - b1) * q^7 + q^9 $$q + q^{3} - \beta_1 q^{5} + ( - \beta_{4} - \beta_1) q^{7} + q^{9} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{11} - \beta_1 q^{15} + ( - \beta_{2} - 3) q^{17} + (\beta_{7} - \beta_{4} + \beta_1) q^{19} + ( - \beta_{4} - \beta_1) q^{21} + ( - \beta_{3} - \beta_{2} + 1) q^{23} + (2 \beta_{6} - \beta_{3} - 1) q^{25} + q^{27} + (\beta_{6} - 2) q^{29} + (\beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{31} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{33} + (2 \beta_{6} - \beta_{3} - \beta_{2} - 3) q^{35} + (\beta_{5} - \beta_{4} + \beta_1) q^{37} + ( - 3 \beta_{5} - \beta_{4} - 3 \beta_1) q^{41} + (\beta_{6} - \beta_{3} + \beta_{2} + 2) q^{43} - \beta_1 q^{45} + ( - \beta_{7} + 2 \beta_{5} - \beta_{4} - \beta_1) q^{47} + ( - \beta_{3} - 2 \beta_{2} - 4) q^{49} + ( - \beta_{2} - 3) q^{51} + (\beta_{3} - 2 \beta_{2}) q^{53} + ( - 3 \beta_{6} + 2 \beta_{3} + \beta_{2} + 3) q^{55} + (\beta_{7} - \beta_{4} + \beta_1) q^{57} + ( - \beta_{7} + 3 \beta_{5} - \beta_1) q^{59} + (\beta_{6} - 2 \beta_{3} - 3 \beta_{2}) q^{61} + ( - \beta_{4} - \beta_1) q^{63} + (4 \beta_{5} - \beta_{4} + \beta_1) q^{67} + ( - \beta_{3} - \beta_{2} + 1) q^{69} + ( - 2 \beta_{7} - \beta_{5} + 3 \beta_{4}) q^{71} + (3 \beta_{5} + 2 \beta_1) q^{73} + (2 \beta_{6} - \beta_{3} - 1) q^{75} + ( - 2 \beta_{6} - 2 \beta_{2} + 8) q^{77} + ( - \beta_{3} - 2 \beta_{2} - 1) q^{79} + q^{81} + (2 \beta_{7} - \beta_{5} + \beta_{4}) q^{83} + ( - \beta_{5} + \beta_{4} + 3 \beta_1) q^{85} + (\beta_{6} - 2) q^{87} + ( - 3 \beta_{7} + \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{89} + (\beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{93} + ( - \beta_{6} - \beta_{2} + 9) q^{95} + ( - \beta_{7} + 6 \beta_{5}) q^{97} + ( - \beta_{7} + \beta_{4} + \beta_1) q^{99}+O(q^{100})$$ q + q^3 - b1 * q^5 + (-b4 - b1) * q^7 + q^9 + (-b7 + b4 + b1) * q^11 - b1 * q^15 + (-b2 - 3) * q^17 + (b7 - b4 + b1) * q^19 + (-b4 - b1) * q^21 + (-b3 - b2 + 1) * q^23 + (2*b6 - b3 - 1) * q^25 + q^27 + (b6 - 2) * q^29 + (b7 - 2*b5 - 2*b1) * q^31 + (-b7 + b4 + b1) * q^33 + (2*b6 - b3 - b2 - 3) * q^35 + (b5 - b4 + b1) * q^37 + (-3*b5 - b4 - 3*b1) * q^41 + (b6 - b3 + b2 + 2) * q^43 - b1 * q^45 + (-b7 + 2*b5 - b4 - b1) * q^47 + (-b3 - 2*b2 - 4) * q^49 + (-b2 - 3) * q^51 + (b3 - 2*b2) * q^53 + (-3*b6 + 2*b3 + b2 + 3) * q^55 + (b7 - b4 + b1) * q^57 + (-b7 + 3*b5 - b1) * q^59 + (b6 - 2*b3 - 3*b2) * q^61 + (-b4 - b1) * q^63 + (4*b5 - b4 + b1) * q^67 + (-b3 - b2 + 1) * q^69 + (-2*b7 - b5 + 3*b4) * q^71 + (3*b5 + 2*b1) * q^73 + (2*b6 - b3 - 1) * q^75 + (-2*b6 - 2*b2 + 8) * q^77 + (-b3 - 2*b2 - 1) * q^79 + q^81 + (2*b7 - b5 + b4) * q^83 + (-b5 + b4 + 3*b1) * q^85 + (b6 - 2) * q^87 + (-3*b7 + b5 + 2*b4 + 3*b1) * q^89 + (b7 - 2*b5 - 2*b1) * q^93 + (-b6 - b2 + 9) * q^95 + (-b7 + 6*b5) * q^97 + (-b7 + b4 + b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{3} + 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^3 + 8 * q^9 $$8 q + 8 q^{3} + 8 q^{9} - 24 q^{17} + 4 q^{23} - 4 q^{25} + 8 q^{27} - 12 q^{29} - 20 q^{35} + 16 q^{43} - 36 q^{49} - 24 q^{51} + 4 q^{53} + 20 q^{55} - 4 q^{61} + 4 q^{69} - 4 q^{75} + 56 q^{77} - 12 q^{79} + 8 q^{81} - 12 q^{87} + 68 q^{95}+O(q^{100})$$ 8 * q + 8 * q^3 + 8 * q^9 - 24 * q^17 + 4 * q^23 - 4 * q^25 + 8 * q^27 - 12 * q^29 - 20 * q^35 + 16 * q^43 - 36 * q^49 - 24 * q^51 + 4 * q^53 + 20 * q^55 - 4 * q^61 + 4 * q^69 - 4 * q^75 + 56 * q^77 - 12 * q^79 + 8 * q^81 - 12 * q^87 + 68 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 14x^{6} + 75x^{4} - 170x^{2} + 169$$ :

 $$\beta_{1}$$ $$=$$ $$( -3\nu^{7} + 26\nu^{6} + 16\nu^{5} - 260\nu^{4} - 56\nu^{3} + 1092\nu^{2} + 55\nu - 1508 ) / 364$$ (-3*v^7 + 26*v^6 + 16*v^5 - 260*v^4 - 56*v^3 + 1092*v^2 + 55*v - 1508) / 364 $$\beta_{2}$$ $$=$$ $$( -5\nu^{7} - 39\nu^{6} + 57\nu^{5} + 390\nu^{4} - 245\nu^{3} - 1092\nu^{2} + 668\nu + 351 ) / 364$$ (-5*v^7 - 39*v^6 + 57*v^5 + 390*v^4 - 245*v^3 - 1092*v^2 + 668*v + 351) / 364 $$\beta_{3}$$ $$=$$ $$( -9\nu^{7} + 48\nu^{5} + 196\nu^{3} - 1655\nu + 182 ) / 364$$ (-9*v^7 + 48*v^5 + 196*v^3 - 1655*v + 182) / 364 $$\beta_{4}$$ $$=$$ $$( 8\nu^{7} - 39\nu^{6} - 73\nu^{5} + 572\nu^{4} + 301\nu^{3} - 2730\nu^{2} - 359\nu + 3991 ) / 364$$ (8*v^7 - 39*v^6 - 73*v^5 + 572*v^4 + 301*v^3 - 2730*v^2 - 359*v + 3991) / 364 $$\beta_{5}$$ $$=$$ $$( -\nu^{7} + 14\nu^{5} - 62\nu^{3} + 79\nu ) / 26$$ (-v^7 + 14*v^5 - 62*v^3 + 79*v) / 26 $$\beta_{6}$$ $$=$$ $$( -19\nu^{7} + 26\nu^{6} + 162\nu^{5} - 260\nu^{4} - 294\nu^{3} + 728\nu^{2} - 319\nu - 52 ) / 364$$ (-19*v^7 + 26*v^6 + 162*v^5 - 260*v^4 - 294*v^3 + 728*v^2 - 319*v - 52) / 364 $$\beta_{7}$$ $$=$$ $$( 23\nu^{7} + 26\nu^{6} - 244\nu^{5} - 260\nu^{4} + 1036\nu^{3} + 1092\nu^{2} - 1271\nu - 1508 ) / 364$$ (23*v^7 + 26*v^6 - 244*v^5 - 260*v^4 + 1036*v^3 + 1092*v^2 - 1271*v - 1508) / 364
 $$\nu$$ $$=$$ $$( \beta_{7} + 3\beta_{6} - \beta_{5} - 3\beta_{3} + 2\beta_{2} - \beta_1 ) / 8$$ (b7 + 3*b6 - b5 - 3*b3 + 2*b2 - b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_{2} + 3\beta _1 + 14 ) / 4$$ (b7 - b6 + b5 + b3 + 2*b2 + 3*b1 + 14) / 4 $$\nu^{3}$$ $$=$$ $$( 11\beta_{7} + 15\beta_{6} + \beta_{5} - 7\beta_{3} + 10\beta_{2} - 11\beta _1 - 4 ) / 8$$ (11*b7 + 15*b6 + b5 - 7*b3 + 10*b2 - 11*b1 - 4) / 8 $$\nu^{4}$$ $$=$$ $$( 3\beta_{7} - 3\beta_{6} + 4\beta_{5} + 4\beta_{4} + 3\beta_{3} + 6\beta_{2} + 15\beta _1 + 23 ) / 2$$ (3*b7 - 3*b6 + 4*b5 + 4*b4 + 3*b3 + 6*b2 + 15*b1 + 23) / 2 $$\nu^{5}$$ $$=$$ $$( 38\beta_{7} + 27\beta_{6} + 32\beta_{5} - 7\beta_{3} + 18\beta_{2} - 38\beta _1 - 10 ) / 4$$ (38*b7 + 27*b6 + 32*b5 - 7*b3 + 18*b2 - 38*b1 - 10) / 4 $$\nu^{6}$$ $$=$$ $$( 16\beta_{7} - 9\beta_{6} + 26\beta_{5} + 40\beta_{4} + 9\beta_{3} + 18\beta_{2} + 108\beta _1 + 52 ) / 2$$ (16*b7 - 9*b6 + 26*b5 + 40*b4 + 9*b3 + 18*b2 + 108*b1 + 52) / 2 $$\nu^{7}$$ $$=$$ $$( 461\beta_{7} + 63\beta_{6} + 547\beta_{5} + \beta_{3} + 42\beta_{2} - 461\beta _1 - 32 ) / 8$$ (461*b7 + 63*b6 + 547*b5 + b3 + 42*b2 - 461*b1 - 32) / 8

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4056\mathbb{Z}\right)^\times$$.

 $$n$$ $$1015$$ $$2029$$ $$2705$$ $$3889$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 −2.34138 − 0.500000i −1.42055 − 0.500000i 1.42055 + 0.500000i 2.34138 + 0.500000i 2.34138 − 0.500000i 1.42055 − 0.500000i −1.42055 + 0.500000i −2.34138 + 0.500000i
0 1.00000 0 4.20740i 0 3.55539i 0 1.00000 0
337.2 0 1.00000 0 1.55452i 0 2.96046i 0 1.00000 0
337.3 0 1.00000 0 1.28657i 0 1.96046i 0 1.00000 0
337.4 0 1.00000 0 0.475353i 0 4.55539i 0 1.00000 0
337.5 0 1.00000 0 0.475353i 0 4.55539i 0 1.00000 0
337.6 0 1.00000 0 1.28657i 0 1.96046i 0 1.00000 0
337.7 0 1.00000 0 1.55452i 0 2.96046i 0 1.00000 0
337.8 0 1.00000 0 4.20740i 0 3.55539i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 337.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4056.2.c.p 8
13.b even 2 1 inner 4056.2.c.p 8
13.c even 3 1 312.2.bf.b 8
13.d odd 4 1 4056.2.a.bd 4
13.d odd 4 1 4056.2.a.be 4
13.e even 6 1 312.2.bf.b 8
39.h odd 6 1 936.2.bi.c 8
39.i odd 6 1 936.2.bi.c 8
52.f even 4 1 8112.2.a.cq 4
52.f even 4 1 8112.2.a.cs 4
52.i odd 6 1 624.2.bv.g 8
52.j odd 6 1 624.2.bv.g 8
156.p even 6 1 1872.2.by.m 8
156.r even 6 1 1872.2.by.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.bf.b 8 13.c even 3 1
312.2.bf.b 8 13.e even 6 1
624.2.bv.g 8 52.i odd 6 1
624.2.bv.g 8 52.j odd 6 1
936.2.bi.c 8 39.h odd 6 1
936.2.bi.c 8 39.i odd 6 1
1872.2.by.m 8 156.p even 6 1
1872.2.by.m 8 156.r even 6 1
4056.2.a.bd 4 13.d odd 4 1
4056.2.a.be 4 13.d odd 4 1
4056.2.c.p 8 1.a even 1 1 trivial
4056.2.c.p 8 13.b even 2 1 inner
8112.2.a.cq 4 52.f even 4 1
8112.2.a.cs 4 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4056, [\chi])$$:

 $$T_{5}^{8} + 22T_{5}^{6} + 81T_{5}^{4} + 88T_{5}^{2} + 16$$ T5^8 + 22*T5^6 + 81*T5^4 + 88*T5^2 + 16 $$T_{7}^{8} + 46T_{7}^{6} + 717T_{7}^{4} + 4432T_{7}^{2} + 8836$$ T7^8 + 46*T7^6 + 717*T7^4 + 4432*T7^2 + 8836 $$T_{11}^{8} + 64T_{11}^{6} + 1236T_{11}^{4} + 6880T_{11}^{2} + 10816$$ T11^8 + 64*T11^6 + 1236*T11^4 + 6880*T11^2 + 10816

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T - 1)^{8}$$
$5$ $$T^{8} + 22 T^{6} + 81 T^{4} + 88 T^{2} + \cdots + 16$$
$7$ $$T^{8} + 46 T^{6} + 717 T^{4} + \cdots + 8836$$
$11$ $$T^{8} + 64 T^{6} + 1236 T^{4} + \cdots + 10816$$
$13$ $$T^{8}$$
$17$ $$(T^{4} + 12 T^{3} + 33 T^{2} - 48)^{2}$$
$19$ $$T^{8} + 112 T^{6} + 3636 T^{4} + \cdots + 141376$$
$23$ $$(T^{4} - 2 T^{3} - 30 T^{2} - 32 T - 8)^{2}$$
$29$ $$(T^{4} + 6 T^{3} - 15 T^{2} - 36 T + 36)^{2}$$
$31$ $$T^{8} + 118 T^{6} + 4473 T^{4} + \cdots + 141376$$
$37$ $$T^{8} + 82 T^{6} + 1425 T^{4} + \cdots + 16384$$
$41$ $$T^{8} + 210 T^{6} + 13809 T^{4} + \cdots + 3154176$$
$43$ $$(T^{4} - 8 T^{3} - 45 T^{2} + 190 T + 694)^{2}$$
$47$ $$T^{8} + 160 T^{6} + 9300 T^{4} + \cdots + 2096704$$
$53$ $$(T^{4} - 2 T^{3} - 147 T^{2} - 248 T + 208)^{2}$$
$59$ $$T^{8} + 184 T^{6} + 9744 T^{4} + \cdots + 262144$$
$61$ $$(T^{4} + 2 T^{3} - 198 T^{2} - 430 T + 6481)^{2}$$
$67$ $$T^{8} + 190 T^{6} + 3693 T^{4} + \cdots + 45796$$
$71$ $$T^{8} + 432 T^{6} + 48756 T^{4} + \cdots + 1272384$$
$73$ $$T^{8} + 124 T^{6} + 4518 T^{4} + \cdots + 185761$$
$79$ $$(T^{4} + 6 T^{3} - 63 T^{2} - 108 T + 216)^{2}$$
$83$ $$T^{8} + 312 T^{6} + 22692 T^{4} + \cdots + 1557504$$
$89$ $$T^{8} + 480 T^{6} + 68160 T^{4} + \cdots + 589824$$
$97$ $$T^{8} + 414 T^{6} + \cdots + 33039504$$
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