# Properties

 Label 880.2.bo.h Level $880$ Weight $2$ Character orbit 880.bo Analytic conductor $7.027$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$880 = 2^{4} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 880.bo (of order $$5$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.02683537787$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.13140625.1 Defining polynomial: $$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} + 4 x^{4} + 3 x^{3} + 5 x^{2} + 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{3} + ( 1 - \beta_{3} + \beta_{4} + \beta_{7} ) q^{5} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{7} + ( -2 + \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{3} + ( 1 - \beta_{3} + \beta_{4} + \beta_{7} ) q^{5} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{7} + ( -2 + \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{9} + ( -1 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{11} + ( -2 - 2 \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{13} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{15} + ( -1 + \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{17} + ( -3 \beta_{1} - 5 \beta_{3} + \beta_{5} + \beta_{6} ) q^{19} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{21} + ( -2 + \beta_{1} + \beta_{2} ) q^{23} -\beta_{3} q^{25} + ( -4 + \beta_{1} + \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{27} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} - 3 \beta_{7} ) q^{29} + 5 \beta_{6} q^{31} + ( 2 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} + 3 \beta_{7} ) q^{33} + ( 1 + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{35} + ( -3 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} + 3 \beta_{6} - 4 \beta_{7} ) q^{37} + ( -5 + 2 \beta_{1} - \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{39} + ( 1 - 4 \beta_{1} + \beta_{3} + \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{41} + ( 6 - 2 \beta_{1} - 2 \beta_{2} + \beta_{5} ) q^{43} + ( -1 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{45} + ( 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} - 3 \beta_{6} ) q^{47} + ( 1 - 4 \beta_{1} - \beta_{2} - \beta_{3} + 4 \beta_{5} + \beta_{6} ) q^{49} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{51} + ( 1 - 5 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} + \beta_{6} + \beta_{7} ) q^{53} + ( \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{55} + ( -8 + 7 \beta_{2} + \beta_{4} - 7 \beta_{5} - \beta_{6} - 8 \beta_{7} ) q^{57} + ( -4 \beta_{1} - \beta_{2} + 5 \beta_{3} - 5 \beta_{4} + 4 \beta_{6} - 7 \beta_{7} ) q^{59} + ( 2 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{61} + ( -6 - \beta_{1} + 3 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{63} + ( 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{7} ) q^{65} + ( 5 + 4 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - \beta_{5} + 6 \beta_{7} ) q^{67} + ( -2 + 3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{69} + ( 3 + 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{71} + ( 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{6} - \beta_{7} ) q^{73} + ( -1 + \beta_{2} - \beta_{5} - \beta_{7} ) q^{75} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} - 6 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{77} + ( -6 - 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} - 6 \beta_{7} ) q^{79} + ( 2 \beta_{1} + 6 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{81} + ( 1 - 4 \beta_{1} - \beta_{2} - \beta_{3} + 7 \beta_{4} + 4 \beta_{5} + \beta_{6} + 7 \beta_{7} ) q^{83} + ( -3 + \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{85} + ( -7 + 3 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - 8 \beta_{5} - 4 \beta_{7} ) q^{87} + ( 1 - 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{5} ) q^{89} + ( -8 - 2 \beta_{1} + \beta_{3} - 8 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{91} + ( -5 - 5 \beta_{2} + 5 \beta_{3} + 5 \beta_{6} ) q^{93} + ( -\beta_{1} + 2 \beta_{2} + \beta_{6} - 5 \beta_{7} ) q^{95} + ( 8 + 2 \beta_{2} + 13 \beta_{4} - 2 \beta_{5} + 8 \beta_{7} ) q^{97} + ( 6 - \beta_{1} - 3 \beta_{2} - 11 \beta_{3} - \beta_{4} + 5 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 5q^{3} + 2q^{5} + q^{7} - 5q^{9} + O(q^{10})$$ $$8q + 5q^{3} + 2q^{5} + q^{7} - 5q^{9} - 3q^{11} - 2q^{13} - 5q^{15} - 13q^{17} - 15q^{19} - 20q^{21} - 10q^{23} - 2q^{25} - 10q^{27} - 9q^{29} + 10q^{31} + 5q^{33} + 4q^{35} + 24q^{37} - 21q^{39} + 8q^{41} + 38q^{43} + q^{49} - q^{51} + 13q^{53} - 7q^{55} - 45q^{57} + 27q^{59} + 6q^{61} - 25q^{63} + 2q^{65} + 38q^{67} - q^{69} + 20q^{71} + 13q^{73} - 5q^{75} + 34q^{77} - 37q^{79} + 8q^{81} - 27q^{83} - 12q^{85} - 38q^{87} - 16q^{89} - 44q^{91} - 35q^{93} + 15q^{95} + 24q^{97} + 20q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 2 \nu^{6} - 3 \nu^{5} - 4 \nu^{3} - 7 \nu^{2} - 12 \nu - 7$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 7 \nu^{5} + 20 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} + 6 \nu + 9$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 9 \nu^{5} + 12 \nu^{4} - 16 \nu^{3} + 13 \nu^{2} - 10 \nu - 1$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 10 \nu^{6} - 17 \nu^{5} + 8 \nu^{4} - 4 \nu^{3} - 13 \nu^{2} - 8 \nu - 5$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - 12 \nu^{6} + 23 \nu^{5} - 20 \nu^{4} + 16 \nu^{3} + \nu^{2} + 6 \nu - 1$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 18 \nu^{6} - 35 \nu^{5} + 32 \nu^{4} - 28 \nu^{3} - 11 \nu^{2} - 12 \nu - 7$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} + 2 \beta_{5} + \beta_{4} - 3 \beta_{2} - 2 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$3 \beta_{7} + 4 \beta_{6} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 4 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{7} - 6 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} - 1$$ $$\nu^{6}$$ $$=$$ $$-16 \beta_{6} - 16 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 7 \beta_{1} - 6$$ $$\nu^{7}$$ $$=$$ $$-16 \beta_{7} - 51 \beta_{6} - 29 \beta_{5} - 23 \beta_{4} + 29 \beta_{2} - 16$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$$.

 $$n$$ $$111$$ $$177$$ $$321$$ $$661$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{7}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.418926 − 1.28932i −0.227943 + 0.701538i −0.386111 − 0.280526i 1.69513 + 1.23158i 0.418926 + 1.28932i −0.227943 − 0.701538i −0.386111 + 0.280526i 1.69513 − 1.23158i
0 −0.177837 + 0.547326i 0 0.809017 0.587785i 0 1.12773 + 3.47080i 0 2.15911 + 1.56869i 0
81.2 0 0.868820 2.67395i 0 0.809017 0.587785i 0 −0.318714 0.980901i 0 −3.96813 2.88301i 0
401.1 0 0.261370 + 0.189896i 0 −0.309017 + 0.951057i 0 2.17239 1.57833i 0 −0.894797 2.75390i 0
401.2 0 1.54765 + 1.12443i 0 −0.309017 + 0.951057i 0 −2.48141 + 1.80285i 0 0.203814 + 0.627276i 0
641.1 0 −0.177837 0.547326i 0 0.809017 + 0.587785i 0 1.12773 3.47080i 0 2.15911 1.56869i 0
641.2 0 0.868820 + 2.67395i 0 0.809017 + 0.587785i 0 −0.318714 + 0.980901i 0 −3.96813 + 2.88301i 0
801.1 0 0.261370 0.189896i 0 −0.309017 0.951057i 0 2.17239 + 1.57833i 0 −0.894797 + 2.75390i 0
801.2 0 1.54765 1.12443i 0 −0.309017 0.951057i 0 −2.48141 1.80285i 0 0.203814 0.627276i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 801.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bo.h 8
4.b odd 2 1 55.2.g.b 8
11.c even 5 1 inner 880.2.bo.h 8
11.c even 5 1 9680.2.a.cn 4
11.d odd 10 1 9680.2.a.cm 4
12.b even 2 1 495.2.n.e 8
20.d odd 2 1 275.2.h.a 8
20.e even 4 2 275.2.z.a 16
44.c even 2 1 605.2.g.k 8
44.g even 10 1 605.2.a.k 4
44.g even 10 2 605.2.g.e 8
44.g even 10 1 605.2.g.k 8
44.h odd 10 1 55.2.g.b 8
44.h odd 10 1 605.2.a.j 4
44.h odd 10 2 605.2.g.m 8
132.n odd 10 1 5445.2.a.bi 4
132.o even 10 1 495.2.n.e 8
132.o even 10 1 5445.2.a.bp 4
220.n odd 10 1 275.2.h.a 8
220.n odd 10 1 3025.2.a.bd 4
220.o even 10 1 3025.2.a.w 4
220.v even 20 2 275.2.z.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b 8 4.b odd 2 1
55.2.g.b 8 44.h odd 10 1
275.2.h.a 8 20.d odd 2 1
275.2.h.a 8 220.n odd 10 1
275.2.z.a 16 20.e even 4 2
275.2.z.a 16 220.v even 20 2
495.2.n.e 8 12.b even 2 1
495.2.n.e 8 132.o even 10 1
605.2.a.j 4 44.h odd 10 1
605.2.a.k 4 44.g even 10 1
605.2.g.e 8 44.g even 10 2
605.2.g.k 8 44.c even 2 1
605.2.g.k 8 44.g even 10 1
605.2.g.m 8 44.h odd 10 2
880.2.bo.h 8 1.a even 1 1 trivial
880.2.bo.h 8 11.c even 5 1 inner
3025.2.a.w 4 220.o even 10 1
3025.2.a.bd 4 220.n odd 10 1
5445.2.a.bi 4 132.n odd 10 1
5445.2.a.bp 4 132.o even 10 1
9680.2.a.cm 4 11.d odd 10 1
9680.2.a.cn 4 11.c even 5 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(880, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$1 - 5 T + 12 T^{2} - 15 T^{3} + 39 T^{4} - 35 T^{5} + 18 T^{6} - 5 T^{7} + T^{8}$$
$5$ $$( 1 - T + T^{2} - T^{3} + T^{4} )^{2}$$
$7$ $$961 + 341 T + 777 T^{2} - 197 T^{3} + 30 T^{4} + 17 T^{5} + 7 T^{6} - T^{7} + T^{8}$$
$11$ $$14641 + 3993 T + 2178 T^{2} - 99 T^{3} + 75 T^{4} - 9 T^{5} + 18 T^{6} + 3 T^{7} + T^{8}$$
$13$ $$19321 - 3336 T + 1439 T^{2} + 128 T^{3} + 129 T^{4} - 4 T^{5} + 21 T^{6} + 2 T^{7} + T^{8}$$
$17$ $$361 + 931 T + 2544 T^{2} + 2537 T^{3} + 1379 T^{4} + 449 T^{5} + 96 T^{6} + 13 T^{7} + T^{8}$$
$19$ $$625 - 3750 T + 67125 T^{2} + 33875 T^{3} + 7950 T^{4} + 1025 T^{5} + 135 T^{6} + 15 T^{7} + T^{8}$$
$23$ $$( -11 - 10 T + 4 T^{2} + 5 T^{3} + T^{4} )^{2}$$
$29$ $$203401 + 18942 T - 2329 T^{2} - 429 T^{3} + 1384 T^{4} + 363 T^{5} + 99 T^{6} + 9 T^{7} + T^{8}$$
$31$ $$390625 + 156250 T + 78125 T^{2} + 6250 T^{3} - 625 T^{4} - 250 T^{5} + 125 T^{6} - 10 T^{7} + T^{8}$$
$37$ $$1324801 + 740093 T + 124041 T^{2} - 46831 T^{3} + 19344 T^{4} - 3133 T^{5} + 359 T^{6} - 24 T^{7} + T^{8}$$
$41$ $$101761 - 120263 T + 69513 T^{2} - 23171 T^{3} + 5430 T^{4} - 831 T^{5} + 93 T^{6} - 8 T^{7} + T^{8}$$
$43$ $$( 211 - 289 T + 121 T^{2} - 19 T^{3} + T^{4} )^{2}$$
$47$ $$28561 + 3887 T^{2} + 1170 T^{3} + 259 T^{4} - 90 T^{5} + 23 T^{6} + T^{8}$$
$53$ $$885481 + 313353 T + 81665 T^{2} + 13753 T^{3} + 5824 T^{4} - 133 T^{5} + 85 T^{6} - 13 T^{7} + T^{8}$$
$59$ $$687241 - 395433 T + 157155 T^{2} - 54243 T^{3} + 18994 T^{4} - 3177 T^{5} + 385 T^{6} - 27 T^{7} + T^{8}$$
$61$ $$28561 + 26364 T + 17745 T^{2} + 6396 T^{3} + 1504 T^{4} + 96 T^{5} + 10 T^{6} - 6 T^{7} + T^{8}$$
$67$ $$( -4079 + 1014 T + 22 T^{2} - 19 T^{3} + T^{4} )^{2}$$
$71$ $$17161 - 9825 T + 11477 T^{2} - 10445 T^{3} + 5634 T^{4} - 1325 T^{5} + 213 T^{6} - 20 T^{7} + T^{8}$$
$73$ $$121 + 319 T + 584 T^{2} + 633 T^{3} + 479 T^{4} + 201 T^{5} + 56 T^{6} - 13 T^{7} + T^{8}$$
$79$ $$45954841 + 17171207 T + 6346418 T^{2} + 1002329 T^{3} + 104405 T^{4} + 8839 T^{5} + 698 T^{6} + 37 T^{7} + T^{8}$$
$83$ $$2886601 - 2101663 T + 461353 T^{2} + 174129 T^{3} + 42080 T^{4} + 5379 T^{5} + 493 T^{6} + 27 T^{7} + T^{8}$$
$89$ $$( 1861 - 472 T - 102 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$97$ $$9066121 + 1951128 T - 356419 T^{2} - 209856 T^{3} + 96589 T^{4} - 7968 T^{5} + 749 T^{6} - 24 T^{7} + T^{8}$$