# Properties

 Label 4100.2.d.c Level $4100$ Weight $2$ Character orbit 4100.d Analytic conductor $32.739$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4100 = 2^{2} \cdot 5^{2} \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4100.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.7386648287$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 14 x^{4} - 14 x^{3} + 8 x^{2} + 4 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 164) 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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{4} + \beta_{6} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{7} ) q^{7} + ( -2 + \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{4} + \beta_{6} ) q^{3} + ( \beta_{2} - \beta_{4} - \beta_{7} ) q^{7} + ( -2 + \beta_{3} + \beta_{5} ) q^{9} + ( -\beta_{1} - \beta_{3} - \beta_{5} ) q^{11} + ( 2 \beta_{4} - \beta_{6} ) q^{13} + ( \beta_{6} + 2 \beta_{7} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} ) q^{19} + ( -1 - 2 \beta_{1} + \beta_{3} - 3 \beta_{5} ) q^{21} + ( 2 \beta_{2} - 2 \beta_{6} ) q^{23} + ( 4 \beta_{2} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{27} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{29} + ( -2 - 2 \beta_{3} + 2 \beta_{5} ) q^{31} + ( -5 \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{33} + ( -\beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{37} + ( 6 + 2 \beta_{1} ) q^{39} - q^{41} + ( 2 \beta_{4} + 2 \beta_{7} ) q^{43} + ( \beta_{2} + 3 \beta_{4} - \beta_{6} ) q^{47} + ( -6 - 4 \beta_{1} - \beta_{3} - 3 \beta_{5} ) q^{49} + ( -2 - 2 \beta_{1} - 6 \beta_{3} + 4 \beta_{5} ) q^{51} + ( 2 \beta_{2} + 2 \beta_{4} - 3 \beta_{6} + 2 \beta_{7} ) q^{53} + ( 3 \beta_{2} + 4 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{57} + ( -4 - 2 \beta_{3} ) q^{59} + ( 6 - 2 \beta_{1} ) q^{61} + ( -3 \beta_{2} - \beta_{4} + \beta_{6} + 4 \beta_{7} ) q^{63} + ( \beta_{2} + \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{67} + ( 6 - 2 \beta_{5} ) q^{69} + ( -1 - \beta_{1} + 3 \beta_{3} - 4 \beta_{5} ) q^{71} + ( \beta_{2} - 3 \beta_{7} ) q^{73} + ( \beta_{2} + 4 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{77} + ( 3 - \beta_{1} - 3 \beta_{3} ) q^{79} + ( 4 - 6 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} ) q^{81} + ( -4 \beta_{4} - 2 \beta_{7} ) q^{83} + ( -4 \beta_{2} - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{7} ) q^{87} + ( -2 - 4 \beta_{1} - 2 \beta_{3} ) q^{89} + ( 12 + 2 \beta_{1} + 6 \beta_{5} ) q^{91} + ( 4 \beta_{2} - 6 \beta_{6} - 6 \beta_{7} ) q^{93} + ( -3 \beta_{6} - 4 \beta_{7} ) q^{97} + ( -9 + 7 \beta_{1} + 7 \beta_{3} + 4 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 24q^{9} + O(q^{10})$$ $$8q - 24q^{9} + 8q^{11} - 12q^{19} + 8q^{29} - 16q^{31} + 48q^{39} - 8q^{41} - 32q^{49} - 8q^{51} - 24q^{59} + 48q^{61} + 56q^{69} - 4q^{71} + 36q^{79} + 56q^{81} - 8q^{89} + 72q^{91} - 116q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 2 x^{6} + 4 x^{5} + 14 x^{4} - 14 x^{3} + 8 x^{2} + 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-19 \nu^{7} + 33 \nu^{6} - 46 \nu^{5} + 12 \nu^{4} - 413 \nu^{3} + 174 \nu^{2} + 94 \nu + 516$$$$)/317$$ $$\beta_{2}$$ $$=$$ $$($$$$100 \nu^{7} - 157 \nu^{6} + 142 \nu^{5} + 404 \nu^{4} + 1840 \nu^{3} - 799 \nu^{2} + 1474 \nu + 254$$$$)/951$$ $$\beta_{3}$$ $$=$$ $$($$$$100 \nu^{7} - 157 \nu^{6} + 142 \nu^{5} + 404 \nu^{4} + 1840 \nu^{3} - 799 \nu^{2} - 428 \nu + 254$$$$)/951$$ $$\beta_{4}$$ $$=$$ $$($$$$154 \nu^{7} - 451 \nu^{6} + 523 \nu^{5} + 470 \nu^{4} + 1312 \nu^{3} - 3646 \nu^{2} + 1357 \nu + 239$$$$)/951$$ $$\beta_{5}$$ $$=$$ $$($$$$-358 \nu^{7} + 505 \nu^{6} - 166 \nu^{5} - 2093 \nu^{4} - 5446 \nu^{3} + 2461 \nu^{2} + 1304 \nu - 3458$$$$)/951$$ $$\beta_{6}$$ $$=$$ $$($$$$-508 \nu^{7} + 1216 \nu^{6} - 1330 \nu^{5} - 1748 \nu^{4} - 6304 \nu^{3} + 10792 \nu^{2} - 5662 \nu - 986$$$$)/951$$ $$\beta_{7}$$ $$=$$ $$($$$$-836 \nu^{7} + 1769 \nu^{6} - 2024 \nu^{5} - 2959 \nu^{4} - 11198 \nu^{3} + 11777 \nu^{2} - 9812 \nu - 1705$$$$)/951$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + 2 \beta_{4} + 2 \beta_{2}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} + \beta_{5} + 2 \beta_{4} + 5 \beta_{3} + 5 \beta_{2} + 2 \beta_{1} - 1$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$\beta_{5} + 7 \beta_{3} + 6 \beta_{1} - 8$$ $$\nu^{5}$$ $$=$$ $$($$$$-6 \beta_{7} - \beta_{6} + 6 \beta_{5} - 16 \beta_{4} + 29 \beta_{3} - 29 \beta_{2} + 16 \beta_{1} - 12$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-16 \beta_{7} - 13 \beta_{6} - 70 \beta_{4} - 92 \beta_{2}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-35 \beta_{7} - 11 \beta_{6} - 35 \beta_{5} - 108 \beta_{4} - 175 \beta_{3} - 175 \beta_{2} - 108 \beta_{1} + 95$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4100\mathbb{Z}\right)^\times$$.

 $$n$$ $$1477$$ $$2051$$ $$3901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1149.1
 1.76639 + 1.76639i −0.187509 + 0.187509i 0.626295 − 0.626295i −1.20518 + 1.20518i −1.20518 − 1.20518i 0.626295 + 0.626295i −0.187509 − 0.187509i 1.76639 − 1.76639i
0 3.24028i 0 0 0 0.858626i 0 −7.49944 0
1149.2 0 2.92968i 0 0 0 2.77840i 0 −5.58303 0
1149.3 0 2.21551i 0 0 0 5.06479i 0 −1.90849 0
1149.4 0 0.0950939i 0 0 0 3.14501i 0 2.99096 0
1149.5 0 0.0950939i 0 0 0 3.14501i 0 2.99096 0
1149.6 0 2.21551i 0 0 0 5.06479i 0 −1.90849 0
1149.7 0 2.92968i 0 0 0 2.77840i 0 −5.58303 0
1149.8 0 3.24028i 0 0 0 0.858626i 0 −7.49944 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1149.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 4100.2.d.c 8
5.b even 2 1 inner 4100.2.d.c 8
5.c odd 4 1 164.2.a.a 4
5.c odd 4 1 4100.2.a.c 4
15.e even 4 1 1476.2.a.g 4
20.e even 4 1 656.2.a.i 4
35.f even 4 1 8036.2.a.i 4
40.i odd 4 1 2624.2.a.v 4
40.k even 4 1 2624.2.a.y 4
60.l odd 4 1 5904.2.a.bp 4
205.g odd 4 1 6724.2.a.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.2.a.a 4 5.c odd 4 1
656.2.a.i 4 20.e even 4 1
1476.2.a.g 4 15.e even 4 1
2624.2.a.v 4 40.i odd 4 1
2624.2.a.y 4 40.k even 4 1
4100.2.a.c 4 5.c odd 4 1
4100.2.d.c 8 1.a even 1 1 trivial
4100.2.d.c 8 5.b even 2 1 inner
5904.2.a.bp 4 60.l odd 4 1
6724.2.a.c 4 205.g odd 4 1
8036.2.a.i 4 35.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 $$S_{2}^{\mathrm{new}}(4100, [\chi])$$:

 $$T_{3}^{8} + 24 T_{3}^{6} + 184 T_{3}^{4} + 444 T_{3}^{2} + 4$$ $$T_{7}^{8} + 44 T_{7}^{6} + 560 T_{7}^{4} + 2348 T_{7}^{2} + 1444$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$4 + 444 T^{2} + 184 T^{4} + 24 T^{6} + T^{8}$$
$5$ $$T^{8}$$
$7$ $$1444 + 2348 T^{2} + 560 T^{4} + 44 T^{6} + T^{8}$$
$11$ $$( 54 + 18 T - 18 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$13$ $$20736 + 13824 T^{2} + 1888 T^{4} + 80 T^{6} + T^{8}$$
$17$ $$186624 + 47872 T^{2} + 3808 T^{4} + 112 T^{6} + T^{8}$$
$19$ $$( -186 - 134 T - 14 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$23$ $$36864 + 22528 T^{2} + 2944 T^{4} + 112 T^{6} + T^{8}$$
$29$ $$( 144 - 40 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$31$ $$( 64 - 32 T - 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$37$ $$104976 + 42768 T^{2} + 4600 T^{4} + 128 T^{6} + T^{8}$$
$41$ $$( 1 + T )^{8}$$
$43$ $$82944 + 46336 T^{2} + 3904 T^{4} + 112 T^{6} + T^{8}$$
$47$ $$1397124 + 189004 T^{2} + 8680 T^{4} + 160 T^{6} + T^{8}$$
$53$ $$1679616 + 518400 T^{2} + 20448 T^{4} + 256 T^{6} + T^{8}$$
$59$ $$( -192 - 128 T + 16 T^{2} + 12 T^{3} + T^{4} )^{2}$$
$61$ $$( 288 - 432 T + 176 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$67$ $$1196836 + 429340 T^{2} + 18528 T^{4} + 244 T^{6} + T^{8}$$
$71$ $$( -426 - 694 T - 186 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$73$ $$163216 + 414224 T^{2} + 16664 T^{4} + 224 T^{6} + T^{8}$$
$79$ $$( -18 + 42 T + 50 T^{2} - 18 T^{3} + T^{4} )^{2}$$
$83$ $$11943936 + 1253376 T^{2} + 31744 T^{4} + 304 T^{6} + T^{8}$$
$89$ $$( 4272 - 272 T - 128 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$97$ $$24443136 + 2824960 T^{2} + 65248 T^{4} + 496 T^{6} + T^{8}$$