# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 112.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.60821392064$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 14) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} - 7 \zeta_{6} q^{5} + (18 \zeta_{6} + 1) q^{7} + 26 \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 - 7*z * q^5 + (18*z + 1) * q^7 + 26*z * q^9 $$q + (\zeta_{6} - 1) q^{3} - 7 \zeta_{6} q^{5} + (18 \zeta_{6} + 1) q^{7} + 26 \zeta_{6} q^{9} + ( - 35 \zeta_{6} + 35) q^{11} + 66 q^{13} + 7 q^{15} + (59 \zeta_{6} - 59) q^{17} + 137 \zeta_{6} q^{19} + (\zeta_{6} - 19) q^{21} - 7 \zeta_{6} q^{23} + ( - 76 \zeta_{6} + 76) q^{25} - 53 q^{27} + 106 q^{29} + ( - 75 \zeta_{6} + 75) q^{31} + 35 \zeta_{6} q^{33} + ( - 133 \zeta_{6} + 126) q^{35} - 11 \zeta_{6} q^{37} + (66 \zeta_{6} - 66) q^{39} - 498 q^{41} - 260 q^{43} + ( - 182 \zeta_{6} + 182) q^{45} - 171 \zeta_{6} q^{47} + (360 \zeta_{6} - 323) q^{49} - 59 \zeta_{6} q^{51} + ( - 417 \zeta_{6} + 417) q^{53} - 245 q^{55} - 137 q^{57} + (17 \zeta_{6} - 17) q^{59} - 51 \zeta_{6} q^{61} + (494 \zeta_{6} - 468) q^{63} - 462 \zeta_{6} q^{65} + ( - 439 \zeta_{6} + 439) q^{67} + 7 q^{69} + 784 q^{71} + (295 \zeta_{6} - 295) q^{73} + 76 \zeta_{6} q^{75} + ( - 35 \zeta_{6} + 665) q^{77} - 495 \zeta_{6} q^{79} + (649 \zeta_{6} - 649) q^{81} - 932 q^{83} + 413 q^{85} + (106 \zeta_{6} - 106) q^{87} + 873 \zeta_{6} q^{89} + (1188 \zeta_{6} + 66) q^{91} + 75 \zeta_{6} q^{93} + ( - 959 \zeta_{6} + 959) q^{95} - 290 q^{97} + 910 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - 7*z * q^5 + (18*z + 1) * q^7 + 26*z * q^9 + (-35*z + 35) * q^11 + 66 * q^13 + 7 * q^15 + (59*z - 59) * q^17 + 137*z * q^19 + (z - 19) * q^21 - 7*z * q^23 + (-76*z + 76) * q^25 - 53 * q^27 + 106 * q^29 + (-75*z + 75) * q^31 + 35*z * q^33 + (-133*z + 126) * q^35 - 11*z * q^37 + (66*z - 66) * q^39 - 498 * q^41 - 260 * q^43 + (-182*z + 182) * q^45 - 171*z * q^47 + (360*z - 323) * q^49 - 59*z * q^51 + (-417*z + 417) * q^53 - 245 * q^55 - 137 * q^57 + (17*z - 17) * q^59 - 51*z * q^61 + (494*z - 468) * q^63 - 462*z * q^65 + (-439*z + 439) * q^67 + 7 * q^69 + 784 * q^71 + (295*z - 295) * q^73 + 76*z * q^75 + (-35*z + 665) * q^77 - 495*z * q^79 + (649*z - 649) * q^81 - 932 * q^83 + 413 * q^85 + (106*z - 106) * q^87 + 873*z * q^89 + (1188*z + 66) * q^91 + 75*z * q^93 + (-959*z + 959) * q^95 - 290 * q^97 + 910 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9}+O(q^{10})$$ 2 * q - q^3 - 7 * q^5 + 20 * q^7 + 26 * q^9 $$2 q - q^{3} - 7 q^{5} + 20 q^{7} + 26 q^{9} + 35 q^{11} + 132 q^{13} + 14 q^{15} - 59 q^{17} + 137 q^{19} - 37 q^{21} - 7 q^{23} + 76 q^{25} - 106 q^{27} + 212 q^{29} + 75 q^{31} + 35 q^{33} + 119 q^{35} - 11 q^{37} - 66 q^{39} - 996 q^{41} - 520 q^{43} + 182 q^{45} - 171 q^{47} - 286 q^{49} - 59 q^{51} + 417 q^{53} - 490 q^{55} - 274 q^{57} - 17 q^{59} - 51 q^{61} - 442 q^{63} - 462 q^{65} + 439 q^{67} + 14 q^{69} + 1568 q^{71} - 295 q^{73} + 76 q^{75} + 1295 q^{77} - 495 q^{79} - 649 q^{81} - 1864 q^{83} + 826 q^{85} - 106 q^{87} + 873 q^{89} + 1320 q^{91} + 75 q^{93} + 959 q^{95} - 580 q^{97} + 1820 q^{99}+O(q^{100})$$ 2 * q - q^3 - 7 * q^5 + 20 * q^7 + 26 * q^9 + 35 * q^11 + 132 * q^13 + 14 * q^15 - 59 * q^17 + 137 * q^19 - 37 * q^21 - 7 * q^23 + 76 * q^25 - 106 * q^27 + 212 * q^29 + 75 * q^31 + 35 * q^33 + 119 * q^35 - 11 * q^37 - 66 * q^39 - 996 * q^41 - 520 * q^43 + 182 * q^45 - 171 * q^47 - 286 * q^49 - 59 * q^51 + 417 * q^53 - 490 * q^55 - 274 * q^57 - 17 * q^59 - 51 * q^61 - 442 * q^63 - 462 * q^65 + 439 * q^67 + 14 * q^69 + 1568 * q^71 - 295 * q^73 + 76 * q^75 + 1295 * q^77 - 495 * q^79 - 649 * q^81 - 1864 * q^83 + 826 * q^85 - 106 * q^87 + 873 * q^89 + 1320 * q^91 + 75 * q^93 + 959 * q^95 - 580 * q^97 + 1820 * q^99

## Character values

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

 $$n$$ $$15$$ $$17$$ $$85$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
65.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i 0 −3.50000 + 6.06218i 0 10.0000 15.5885i 0 13.0000 22.5167i 0
81.1 0 −0.500000 + 0.866025i 0 −3.50000 6.06218i 0 10.0000 + 15.5885i 0 13.0000 + 22.5167i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.4.i.b 2
4.b odd 2 1 14.4.c.b 2
7.c even 3 1 inner 112.4.i.b 2
7.c even 3 1 784.4.a.l 1
7.d odd 6 1 784.4.a.j 1
8.b even 2 1 448.4.i.d 2
8.d odd 2 1 448.4.i.c 2
12.b even 2 1 126.4.g.c 2
20.d odd 2 1 350.4.e.b 2
20.e even 4 2 350.4.j.d 4
28.d even 2 1 98.4.c.e 2
28.f even 6 1 98.4.a.c 1
28.f even 6 1 98.4.c.e 2
28.g odd 6 1 14.4.c.b 2
28.g odd 6 1 98.4.a.b 1
56.k odd 6 1 448.4.i.c 2
56.p even 6 1 448.4.i.d 2
84.h odd 2 1 882.4.g.d 2
84.j odd 6 1 882.4.a.p 1
84.j odd 6 1 882.4.g.d 2
84.n even 6 1 126.4.g.c 2
84.n even 6 1 882.4.a.k 1
140.p odd 6 1 350.4.e.b 2
140.p odd 6 1 2450.4.a.bh 1
140.s even 6 1 2450.4.a.bf 1
140.w even 12 2 350.4.j.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 4.b odd 2 1
14.4.c.b 2 28.g odd 6 1
98.4.a.b 1 28.g odd 6 1
98.4.a.c 1 28.f even 6 1
98.4.c.e 2 28.d even 2 1
98.4.c.e 2 28.f even 6 1
112.4.i.b 2 1.a even 1 1 trivial
112.4.i.b 2 7.c even 3 1 inner
126.4.g.c 2 12.b even 2 1
126.4.g.c 2 84.n even 6 1
350.4.e.b 2 20.d odd 2 1
350.4.e.b 2 140.p odd 6 1
350.4.j.d 4 20.e even 4 2
350.4.j.d 4 140.w even 12 2
448.4.i.c 2 8.d odd 2 1
448.4.i.c 2 56.k odd 6 1
448.4.i.d 2 8.b even 2 1
448.4.i.d 2 56.p even 6 1
784.4.a.j 1 7.d odd 6 1
784.4.a.l 1 7.c even 3 1
882.4.a.k 1 84.n even 6 1
882.4.a.p 1 84.j odd 6 1
882.4.g.d 2 84.h odd 2 1
882.4.g.d 2 84.j odd 6 1
2450.4.a.bf 1 140.s even 6 1
2450.4.a.bh 1 140.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{4}^{\mathrm{new}}(112, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} + 7T + 49$$
$7$ $$T^{2} - 20T + 343$$
$11$ $$T^{2} - 35T + 1225$$
$13$ $$(T - 66)^{2}$$
$17$ $$T^{2} + 59T + 3481$$
$19$ $$T^{2} - 137T + 18769$$
$23$ $$T^{2} + 7T + 49$$
$29$ $$(T - 106)^{2}$$
$31$ $$T^{2} - 75T + 5625$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$(T + 498)^{2}$$
$43$ $$(T + 260)^{2}$$
$47$ $$T^{2} + 171T + 29241$$
$53$ $$T^{2} - 417T + 173889$$
$59$ $$T^{2} + 17T + 289$$
$61$ $$T^{2} + 51T + 2601$$
$67$ $$T^{2} - 439T + 192721$$
$71$ $$(T - 784)^{2}$$
$73$ $$T^{2} + 295T + 87025$$
$79$ $$T^{2} + 495T + 245025$$
$83$ $$(T + 932)^{2}$$
$89$ $$T^{2} - 873T + 762129$$
$97$ $$(T + 290)^{2}$$