# Properties

 Label 112.4.p.g Level $112$ Weight $4$ Character orbit 112.p Analytic conductor $6.608$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 112.p (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.60821392064$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.12258833328.1 Defining polynomial: $$x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 \beta_{3} - \beta_{5} ) q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -4 + 9 \beta_{3} - \beta_{4} ) q^{7} + ( -25 - \beta_{1} + \beta_{2} - 25 \beta_{3} - 2 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -2 \beta_{3} - \beta_{5} ) q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + ( -4 + 9 \beta_{3} - \beta_{4} ) q^{7} + ( -25 - \beta_{1} + \beta_{2} - 25 \beta_{3} - 2 \beta_{4} ) q^{9} + ( -22 - \beta_{1} - 11 \beta_{3} + \beta_{4} - \beta_{5} ) q^{11} + ( -23 - \beta_{2} - 46 \beta_{3} + \beta_{4} ) q^{13} + ( 19 + \beta_{1} - 7 \beta_{2} + 38 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} ) q^{15} + ( 11 \beta_{1} - 5 \beta_{4} + 8 \beta_{5} ) q^{17} + ( -49 + 2 \beta_{1} - \beta_{2} - 49 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{19} + ( -1 + 2 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - \beta_{4} + 16 \beta_{5} ) q^{21} + ( 5 \beta_{1} + 6 \beta_{2} + \beta_{5} ) q^{23} + ( -94 \beta_{3} - 24 \beta_{5} ) q^{25} + ( -53 - 15 \beta_{1} - 7 \beta_{2} - 7 \beta_{4} ) q^{27} + ( -51 - 22 \beta_{1} + \beta_{2} + \beta_{4} ) q^{29} + ( \beta_{1} + 2 \beta_{2} + 77 \beta_{3} - \beta_{4} - 23 \beta_{5} ) q^{31} + ( -51 - 4 \beta_{1} + 4 \beta_{2} + 51 \beta_{3} + 8 \beta_{5} ) q^{33} + ( 200 + 22 \beta_{1} + 8 \beta_{2} + 19 \beta_{3} - 9 \beta_{4} + \beta_{5} ) q^{35} + ( -73 - 26 \beta_{1} + 2 \beta_{2} - 73 \beta_{3} - 4 \beta_{4} - 24 \beta_{5} ) q^{37} + ( -54 - 37 \beta_{1} - 27 \beta_{3} + 7 \beta_{4} - 22 \beta_{5} ) q^{39} + ( 141 - 8 \beta_{1} - 9 \beta_{2} + 282 \beta_{3} + 9 \beta_{4} - 16 \beta_{5} ) q^{41} + ( -20 + 22 \beta_{1} + 16 \beta_{2} - 40 \beta_{3} - 16 \beta_{4} + 44 \beta_{5} ) q^{43} + ( 438 + 73 \beta_{1} + 219 \beta_{3} + 25 \beta_{4} + 24 \beta_{5} ) q^{45} + ( -33 + 22 \beta_{1} - \beta_{2} - 33 \beta_{3} + 2 \beta_{4} + 21 \beta_{5} ) q^{47} + ( 126 + 26 \beta_{1} + 19 \beta_{2} + 28 \beta_{3} - \beta_{4} + 40 \beta_{5} ) q^{49} + ( 289 - 32 \beta_{1} - 11 \beta_{2} - 289 \beta_{3} + 21 \beta_{5} ) q^{51} + ( 6 \beta_{1} + 12 \beta_{2} + 237 \beta_{3} - 6 \beta_{4} - 48 \beta_{5} ) q^{53} + ( -162 + 11 \beta_{1} + 16 \beta_{2} + 16 \beta_{4} ) q^{55} + ( 7 - 32 \beta_{1} + 8 \beta_{2} + 8 \beta_{4} ) q^{57} + ( 2 \beta_{1} + 4 \beta_{2} + 108 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{59} + ( -189 - 2 \beta_{1} + 6 \beta_{2} + 189 \beta_{3} + 8 \beta_{5} ) q^{61} + ( 487 + 13 \beta_{1} - 15 \beta_{2} + 481 \beta_{3} + 32 \beta_{4} + 20 \beta_{5} ) q^{63} + ( -219 - \beta_{1} - 23 \beta_{2} - 219 \beta_{3} + 46 \beta_{4} - 24 \beta_{5} ) q^{65} + ( -88 + 8 \beta_{1} - 44 \beta_{3} - 42 \beta_{4} + 25 \beta_{5} ) q^{67} + ( -66 - 8 \beta_{1} + 39 \beta_{2} - 132 \beta_{3} - 39 \beta_{4} - 16 \beta_{5} ) q^{69} + ( 322 - 26 \beta_{1} + 14 \beta_{2} + 644 \beta_{3} - 14 \beta_{4} - 52 \beta_{5} ) q^{71} + ( 190 - 58 \beta_{1} + 95 \beta_{3} - 42 \beta_{4} - 8 \beta_{5} ) q^{73} + ( -1340 - 70 \beta_{1} + 24 \beta_{2} - 1340 \beta_{3} - 48 \beta_{4} - 46 \beta_{5} ) q^{75} + ( -33 - 11 \beta_{1} - 25 \beta_{2} - 246 \beta_{3} + 30 \beta_{4} - 32 \beta_{5} ) q^{77} + ( 372 + 33 \beta_{1} - 14 \beta_{2} - 372 \beta_{3} - 47 \beta_{5} ) q^{79} + ( -23 \beta_{1} - 46 \beta_{2} - 122 \beta_{3} + 23 \beta_{4} + 72 \beta_{5} ) q^{81} + ( -42 + 92 \beta_{1} + 22 \beta_{2} + 22 \beta_{4} ) q^{83} + ( 639 + 16 \beta_{1} - 40 \beta_{2} - 40 \beta_{4} ) q^{85} + ( -17 \beta_{1} - 34 \beta_{2} + 1197 \beta_{3} + 17 \beta_{4} ) q^{87} + ( -291 + 4 \beta_{1} - 28 \beta_{2} + 291 \beta_{3} - 32 \beta_{5} ) q^{89} + ( 525 - 54 \beta_{1} - 47 \beta_{2} + 210 \beta_{3} + 15 \beta_{4} - 68 \beta_{5} ) q^{91} + ( -1007 + 90 \beta_{1} + 30 \beta_{2} - 1007 \beta_{3} - 60 \beta_{4} + 120 \beta_{5} ) q^{93} + ( -476 - 6 \beta_{1} - 238 \beta_{3} + 44 \beta_{4} - 25 \beta_{5} ) q^{95} + ( 45 + 48 \beta_{1} - \beta_{2} + 90 \beta_{3} + \beta_{4} + 96 \beta_{5} ) q^{97} + ( 113 + 4 \beta_{1} - 23 \beta_{2} + 226 \beta_{3} + 23 \beta_{4} + 8 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 7q^{3} - 3q^{5} - 52q^{7} - 78q^{9} + O(q^{10})$$ $$6q + 7q^{3} - 3q^{5} - 52q^{7} - 78q^{9} - 99q^{11} + 9q^{17} - 143q^{19} - 15q^{21} + 15q^{23} + 306q^{25} - 362q^{27} - 348q^{29} - 205q^{31} - 471q^{33} + 1185q^{35} - 249q^{37} - 288q^{39} + 2118q^{45} - 75q^{47} + 702q^{49} + 2505q^{51} - 645q^{53} - 918q^{55} - 6q^{57} - 321q^{59} - 1707q^{61} + 1502q^{63} - 612q^{65} - 447q^{67} + 705q^{73} - 4138q^{75} + 555q^{77} + 3447q^{79} + 225q^{81} - 24q^{83} + 3786q^{85} - 3642q^{87} - 2607q^{89} + 2448q^{91} - 2991q^{93} - 2085q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 29 x^{4} - 20 x^{3} + 808 x^{2} - 672 x + 576$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$29 \nu^{5} - 841 \nu^{4} + 1629 \nu^{3} - 23432 \nu^{2} + 19488 \nu - 428592$$$$)/45520$$ $$\beta_{2}$$ $$=$$ $$($$$$64 \nu^{5} + 989 \nu^{4} + 5459 \nu^{3} + 30793 \nu^{2} - 82172 \nu + 596328$$$$)/68280$$ $$\beta_{3}$$ $$=$$ $$($$$$-203 \nu^{5} + 197 \nu^{4} - 5713 \nu^{3} - 986 \nu^{2} - 159176 \nu - 4176$$$$)/136560$$ $$\beta_{4}$$ $$=$$ $$($$$$-133 \nu^{5} + 1012 \nu^{4} + 4792 \nu^{3} + 24959 \nu^{2} + 35804 \nu + 543504$$$$)/68280$$ $$\beta_{5}$$ $$=$$ $$($$$$581 \nu^{5} + 221 \nu^{4} + 16351 \nu^{3} + 2822 \nu^{2} + 481472 \nu + 11952$$$$)/45520$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - 3 \beta_{3} - 2 \beta_{2} - \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-12 \beta_{5} + 2 \beta_{4} - 111 \beta_{3} - \beta_{2} - 11 \beta_{1} - 111$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$29 \beta_{4} + 29 \beta_{2} + 46 \beta_{1} - 51$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$116 \beta_{5} - 11 \beta_{4} + 1029 \beta_{3} + 22 \beta_{2} + 11 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$396 \beta_{5} - 1642 \beta_{4} + 1851 \beta_{3} + 821 \beta_{2} - 425 \beta_{1} + 1851$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$$.

 $$n$$ $$15$$ $$17$$ $$85$$ $$\chi(n)$$ $$-1$$ $$1 + \beta_{3}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 0.426664 − 0.739004i 2.68858 − 4.65676i −2.61524 + 4.52973i 0.426664 + 0.739004i 2.68858 + 4.65676i −2.61524 − 4.52973i
0 −3.53129 + 6.11637i 0 1.05999 0.611983i 0 −16.5026 + 8.40621i 0 −11.4400 19.8147i 0
31.2 0 2.38417 4.12950i 0 14.6315 8.44749i 0 8.89981 + 16.2417i 0 2.13148 + 3.69182i 0
31.3 0 4.64712 8.04905i 0 −17.1915 + 9.92549i 0 −18.3972 2.13127i 0 −29.6915 51.4271i 0
47.1 0 −3.53129 6.11637i 0 1.05999 + 0.611983i 0 −16.5026 8.40621i 0 −11.4400 + 19.8147i 0
47.2 0 2.38417 + 4.12950i 0 14.6315 + 8.44749i 0 8.89981 16.2417i 0 2.13148 3.69182i 0
47.3 0 4.64712 + 8.04905i 0 −17.1915 9.92549i 0 −18.3972 + 2.13127i 0 −29.6915 + 51.4271i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 47.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.4.p.g yes 6
4.b odd 2 1 112.4.p.f 6
7.c even 3 1 784.4.f.g 6
7.d odd 6 1 112.4.p.f 6
7.d odd 6 1 784.4.f.h 6
8.b even 2 1 448.4.p.f 6
8.d odd 2 1 448.4.p.g 6
28.f even 6 1 inner 112.4.p.g yes 6
28.f even 6 1 784.4.f.g 6
28.g odd 6 1 784.4.f.h 6
56.j odd 6 1 448.4.p.g 6
56.m even 6 1 448.4.p.f 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.f 6 4.b odd 2 1
112.4.p.f 6 7.d odd 6 1
112.4.p.g yes 6 1.a even 1 1 trivial
112.4.p.g yes 6 28.f even 6 1 inner
448.4.p.f 6 8.b even 2 1
448.4.p.f 6 56.m even 6 1
448.4.p.g 6 8.d odd 2 1
448.4.p.g 6 56.j odd 6 1
784.4.f.g 6 7.c even 3 1
784.4.f.g 6 28.f even 6 1
784.4.f.h 6 7.d odd 6 1
784.4.f.h 6 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - 7 T_{3}^{5} + 104 T_{3}^{4} - 241 T_{3}^{3} + 5216 T_{3}^{2} - 17215 T_{3} + 97969$$ acting on $$S_{4}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$97969 - 17215 T + 5216 T^{2} - 241 T^{3} + 104 T^{4} - 7 T^{5} + T^{6}$$
$5$ $$168507 - 241029 T + 115632 T^{2} - 1017 T^{3} - 336 T^{4} + 3 T^{5} + T^{6}$$
$7$ $$40353607 + 6117748 T + 343343 T^{2} + 14056 T^{3} + 1001 T^{4} + 52 T^{5} + T^{6}$$
$11$ $$4876875 + 2765475 T + 648954 T^{2} + 71577 T^{3} + 3990 T^{4} + 99 T^{5} + T^{6}$$
$13$ $$2167603200 + 7385616 T^{2} + 5304 T^{4} + T^{6}$$
$17$ $$710833347 - 677307393 T + 214982352 T^{2} + 132003 T^{3} - 14640 T^{4} - 9 T^{5} + T^{6}$$
$19$ $$1106959441 + 184221527 T + 25900616 T^{2} + 725249 T^{3} + 14912 T^{4} + 143 T^{5} + T^{6}$$
$23$ $$8915001507 - 1848154239 T + 126894906 T^{2} + 169515 T^{3} - 11226 T^{4} - 15 T^{5} + T^{6}$$
$29$ $$( -4622400 - 27504 T + 174 T^{2} + T^{3} )^{2}$$
$31$ $$2349411462841 + 44559418309 T + 1159342736 T^{2} - 2893997 T^{3} + 71096 T^{4} + 205 T^{5} + T^{6}$$
$37$ $$14539922265625 + 127835015625 T + 2073393750 T^{2} - 721475 T^{3} + 95526 T^{4} + 249 T^{5} + T^{6}$$
$41$ $$38591270836992 + 14002779024 T^{2} + 235176 T^{4} + T^{6}$$
$43$ $$83474849412288 + 16477161840 T^{2} + 251796 T^{4} + T^{6}$$
$47$ $$20686781241 + 4969435779 T + 1204558776 T^{2} - 2303667 T^{3} + 40176 T^{4} + 75 T^{5} + T^{6}$$
$53$ $$3742269164979921 + 7129289506149 T + 53039092086 T^{2} + 47179233 T^{3} + 532566 T^{4} + 645 T^{5} + T^{6}$$
$59$ $$825904716849 + 27948111129 T + 654024456 T^{2} + 8054127 T^{3} + 72288 T^{4} + 321 T^{5} + T^{6}$$
$61$ $$912934776095523 + 16058508837363 T + 123934119804 T^{2} + 523791243 T^{3} + 1278132 T^{4} + 1707 T^{5} + T^{6}$$
$67$ $$34861164967167075 + 166194940681935 T + 215916796746 T^{2} - 229717323 T^{3} - 447306 T^{4} + 447 T^{5} + T^{6}$$
$71$ $$1026472074810048 + 436555595376 T^{2} + 1416948 T^{4} + T^{6}$$
$73$ $$48340453761263403 - 236639118427497 T + 296644791996 T^{2} + 438086295 T^{3} - 455724 T^{4} - 705 T^{5} + T^{6}$$
$79$ $$6567218609777523 + 126963434082393 T + 656914190058 T^{2} - 3117945933 T^{3} + 4865142 T^{4} - 3447 T^{5} + T^{6}$$
$83$ $$( -308212992 - 856512 T + 12 T^{2} + T^{3} )^{2}$$
$89$ $$6090737949537003 - 59194781610489 T + 74300935212 T^{2} + 1141639191 T^{3} + 2703396 T^{4} + 2607 T^{5} + T^{6}$$
$97$ $$19259496900926208 + 308008959120 T^{2} + 1038696 T^{4} + T^{6}$$