# Properties

 Label 42.4.e.c Level $42$ Weight $4$ Character orbit 42.e Analytic conductor $2.478$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 42.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.47808022024$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{1345})$$ Defining polynomial: $$x^{4} - x^{3} + 337 x^{2} + 336 x + 112896$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{2} q^{2} + ( -3 + 3 \beta_{2} ) q^{3} + ( -4 + 4 \beta_{2} ) q^{4} + ( \beta_{1} - 3 \beta_{2} ) q^{5} + 6 q^{6} + ( 2 - 3 \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} -9 \beta_{2} q^{9} +O(q^{10})$$ $$q -2 \beta_{2} q^{2} + ( -3 + 3 \beta_{2} ) q^{3} + ( -4 + 4 \beta_{2} ) q^{4} + ( \beta_{1} - 3 \beta_{2} ) q^{5} + 6 q^{6} + ( 2 - 3 \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} -9 \beta_{2} q^{9} + ( -4 - 2 \beta_{1} + 6 \beta_{2} - 2 \beta_{3} ) q^{10} + ( -34 + \beta_{1} + 33 \beta_{2} + \beta_{3} ) q^{11} -12 \beta_{2} q^{12} + ( 21 - \beta_{3} ) q^{13} + ( -6 - 2 \beta_{1} + 4 \beta_{2} ) q^{14} + ( 6 + 3 \beta_{3} ) q^{15} -16 \beta_{2} q^{16} + ( 44 + 4 \beta_{1} - 48 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -18 + 18 \beta_{2} ) q^{18} + ( -3 \beta_{1} - 20 \beta_{2} ) q^{19} + ( 8 + 4 \beta_{3} ) q^{20} + ( 3 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{21} + ( 68 - 2 \beta_{3} ) q^{22} + ( -4 \beta_{1} - 72 \beta_{2} ) q^{23} + ( -24 + 24 \beta_{2} ) q^{24} + ( -215 - 5 \beta_{1} + 220 \beta_{2} - 5 \beta_{3} ) q^{25} + ( -2 \beta_{1} - 40 \beta_{2} ) q^{26} + 27 q^{27} + ( 4 + 4 \beta_{1} + 4 \beta_{2} + 4 \beta_{3} ) q^{28} + ( 44 - 11 \beta_{3} ) q^{29} + ( 6 \beta_{1} - 18 \beta_{2} ) q^{30} + ( 261 - 2 \beta_{1} - 259 \beta_{2} - 2 \beta_{3} ) q^{31} + ( -32 + 32 \beta_{2} ) q^{32} + ( -3 \beta_{1} - 99 \beta_{2} ) q^{33} + ( -88 - 8 \beta_{3} ) q^{34} + ( -6 - 4 \beta_{1} + 342 \beta_{2} - 3 \beta_{3} ) q^{35} + 36 q^{36} + ( 9 \beta_{1} - 8 \beta_{2} ) q^{37} + ( -46 + 6 \beta_{1} + 40 \beta_{2} + 6 \beta_{3} ) q^{38} + ( -63 + 3 \beta_{1} + 60 \beta_{2} + 3 \beta_{3} ) q^{39} + ( 8 \beta_{1} - 24 \beta_{2} ) q^{40} + ( -210 - 6 \beta_{3} ) q^{41} + ( 12 - 18 \beta_{2} - 6 \beta_{3} ) q^{42} + ( -61 + 15 \beta_{3} ) q^{43} + ( -4 \beta_{1} - 132 \beta_{2} ) q^{44} + ( -18 - 9 \beta_{1} + 27 \beta_{2} - 9 \beta_{3} ) q^{45} + ( -152 + 8 \beta_{1} + 144 \beta_{2} + 8 \beta_{3} ) q^{46} + ( 12 \beta_{1} - 294 \beta_{2} ) q^{47} + 48 q^{48} + ( 331 - 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{49} + ( 430 + 10 \beta_{3} ) q^{50} + ( -12 \beta_{1} + 144 \beta_{2} ) q^{51} + ( -84 + 4 \beta_{1} + 80 \beta_{2} + 4 \beta_{3} ) q^{52} + ( 120 + 3 \beta_{1} - 123 \beta_{2} + 3 \beta_{3} ) q^{53} -54 \beta_{2} q^{54} + ( -268 + 31 \beta_{3} ) q^{55} + ( 16 - 24 \beta_{2} - 8 \beta_{3} ) q^{56} + ( 69 - 9 \beta_{3} ) q^{57} + ( -22 \beta_{1} - 66 \beta_{2} ) q^{58} + ( 16 - 25 \beta_{1} + 9 \beta_{2} - 25 \beta_{3} ) q^{59} + ( -24 - 12 \beta_{1} + 36 \beta_{2} - 12 \beta_{3} ) q^{60} + ( -4 \beta_{1} - 110 \beta_{2} ) q^{61} + ( -522 + 4 \beta_{3} ) q^{62} + ( -27 - 9 \beta_{1} + 18 \beta_{2} ) q^{63} + 64 q^{64} + ( 18 \beta_{1} + 276 \beta_{2} ) q^{65} + ( -204 + 6 \beta_{1} + 198 \beta_{2} + 6 \beta_{3} ) q^{66} + ( -349 + 11 \beta_{1} + 338 \beta_{2} + 11 \beta_{3} ) q^{67} + ( -16 \beta_{1} + 192 \beta_{2} ) q^{68} + ( 228 - 12 \beta_{3} ) q^{69} + ( 676 + 2 \beta_{1} - 666 \beta_{2} + 8 \beta_{3} ) q^{70} + ( 226 + 20 \beta_{3} ) q^{71} -72 \beta_{2} q^{72} + ( 443 + 35 \beta_{1} - 478 \beta_{2} + 35 \beta_{3} ) q^{73} + ( 2 - 18 \beta_{1} + 16 \beta_{2} - 18 \beta_{3} ) q^{74} + ( 15 \beta_{1} - 660 \beta_{2} ) q^{75} + ( 92 - 12 \beta_{3} ) q^{76} + ( -302 + 35 \beta_{1} + 369 \beta_{2} + 32 \beta_{3} ) q^{77} + ( 126 - 6 \beta_{3} ) q^{78} + ( 8 \beta_{1} + 259 \beta_{2} ) q^{79} + ( -32 - 16 \beta_{1} + 48 \beta_{2} - 16 \beta_{3} ) q^{80} + ( -81 + 81 \beta_{2} ) q^{81} + ( -12 \beta_{1} + 432 \beta_{2} ) q^{82} + ( -98 - 25 \beta_{3} ) q^{83} + ( -36 - 12 \beta_{1} + 24 \beta_{2} ) q^{84} + ( -1432 - 56 \beta_{3} ) q^{85} + ( 30 \beta_{1} + 92 \beta_{2} ) q^{86} + ( -132 + 33 \beta_{1} + 99 \beta_{2} + 33 \beta_{3} ) q^{87} + ( -272 + 8 \beta_{1} + 264 \beta_{2} + 8 \beta_{3} ) q^{88} + ( -42 \beta_{1} - 366 \beta_{2} ) q^{89} + ( 36 + 18 \beta_{3} ) q^{90} + ( 378 - 3 \beta_{1} - 60 \beta_{2} - 22 \beta_{3} ) q^{91} + ( 304 - 16 \beta_{3} ) q^{92} + ( 6 \beta_{1} + 777 \beta_{2} ) q^{93} + ( -564 - 24 \beta_{1} + 588 \beta_{2} - 24 \beta_{3} ) q^{94} + ( 962 - 14 \beta_{1} - 948 \beta_{2} - 14 \beta_{3} ) q^{95} -96 \beta_{2} q^{96} + ( 994 - 35 \beta_{3} ) q^{97} + ( -6 + 6 \beta_{1} - 662 \beta_{2} + 12 \beta_{3} ) q^{98} + ( 306 - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 6q^{3} - 8q^{4} - 5q^{5} + 24q^{6} + 32q^{8} - 18q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 6q^{3} - 8q^{4} - 5q^{5} + 24q^{6} + 32q^{8} - 18q^{9} - 10q^{10} - 67q^{11} - 24q^{12} + 82q^{13} - 18q^{14} + 30q^{15} - 32q^{16} + 92q^{17} - 36q^{18} - 43q^{19} + 40q^{20} + 27q^{21} + 268q^{22} - 148q^{23} - 48q^{24} - 435q^{25} - 82q^{26} + 108q^{27} + 36q^{28} + 154q^{29} - 30q^{30} + 520q^{31} - 64q^{32} - 201q^{33} - 368q^{34} + 650q^{35} + 144q^{36} - 7q^{37} - 86q^{38} - 123q^{39} - 40q^{40} - 852q^{41} - 214q^{43} - 268q^{44} - 45q^{45} - 296q^{46} - 576q^{47} + 192q^{48} + 1318q^{49} + 1740q^{50} + 276q^{51} - 164q^{52} + 243q^{53} - 108q^{54} - 1010q^{55} + 258q^{57} - 154q^{58} + 7q^{59} - 60q^{60} - 224q^{61} - 2080q^{62} - 81q^{63} + 256q^{64} + 570q^{65} - 402q^{66} - 687q^{67} + 368q^{68} + 888q^{69} + 1390q^{70} + 944q^{71} - 144q^{72} + 921q^{73} - 14q^{74} - 1305q^{75} + 344q^{76} - 371q^{77} + 492q^{78} + 526q^{79} - 80q^{80} - 162q^{81} + 852q^{82} - 442q^{83} - 108q^{84} - 5840q^{85} + 214q^{86} - 231q^{87} - 536q^{88} - 774q^{89} + 180q^{90} + 1345q^{91} + 1184q^{92} + 1560q^{93} - 1152q^{94} + 1910q^{95} - 192q^{96} + 3906q^{97} - 1318q^{98} + 1206q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 337 x^{2} + 336 x + 112896$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 337 \nu^{2} - 337 \nu + 112896$$$$)/113232$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 673$$$$)/337$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 336 \beta_{2} + \beta_{1} - 337$$ $$\nu^{3}$$ $$=$$ $$337 \beta_{3} - 673$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 −8.91856 − 15.4474i 9.41856 + 16.3134i −8.91856 + 15.4474i 9.41856 − 16.3134i
−1.00000 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i −10.4186 18.0455i 6.00000 −18.3371 2.59808i 8.00000 −4.50000 7.79423i −20.8371 + 36.0910i
25.2 −1.00000 1.73205i −1.50000 + 2.59808i −2.00000 + 3.46410i 7.91856 + 13.7153i 6.00000 18.3371 2.59808i 8.00000 −4.50000 7.79423i 15.8371 27.4307i
37.1 −1.00000 + 1.73205i −1.50000 2.59808i −2.00000 3.46410i −10.4186 + 18.0455i 6.00000 −18.3371 + 2.59808i 8.00000 −4.50000 + 7.79423i −20.8371 36.0910i
37.2 −1.00000 + 1.73205i −1.50000 2.59808i −2.00000 3.46410i 7.91856 13.7153i 6.00000 18.3371 + 2.59808i 8.00000 −4.50000 + 7.79423i 15.8371 + 27.4307i
 $$n$$: 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 42.4.e.c 4
3.b odd 2 1 126.4.g.g 4
4.b odd 2 1 336.4.q.j 4
7.b odd 2 1 294.4.e.l 4
7.c even 3 1 inner 42.4.e.c 4
7.c even 3 1 294.4.a.n 2
7.d odd 6 1 294.4.a.m 2
7.d odd 6 1 294.4.e.l 4
21.c even 2 1 882.4.g.bf 4
21.g even 6 1 882.4.a.z 2
21.g even 6 1 882.4.g.bf 4
21.h odd 6 1 126.4.g.g 4
21.h odd 6 1 882.4.a.v 2
28.f even 6 1 2352.4.a.ca 2
28.g odd 6 1 336.4.q.j 4
28.g odd 6 1 2352.4.a.bq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 1.a even 1 1 trivial
42.4.e.c 4 7.c even 3 1 inner
126.4.g.g 4 3.b odd 2 1
126.4.g.g 4 21.h odd 6 1
294.4.a.m 2 7.d odd 6 1
294.4.a.n 2 7.c even 3 1
294.4.e.l 4 7.b odd 2 1
294.4.e.l 4 7.d odd 6 1
336.4.q.j 4 4.b odd 2 1
336.4.q.j 4 28.g odd 6 1
882.4.a.v 2 21.h odd 6 1
882.4.a.z 2 21.g even 6 1
882.4.g.bf 4 21.c even 2 1
882.4.g.bf 4 21.g even 6 1
2352.4.a.bq 2 28.g odd 6 1
2352.4.a.ca 2 28.f even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 5 T_{5}^{3} + 355 T_{5}^{2} - 1650 T_{5} + 108900$$ acting on $$S_{4}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + 2 T + T^{2} )^{2}$$
$3$ $$( 9 + 3 T + T^{2} )^{2}$$
$5$ $$108900 - 1650 T + 355 T^{2} + 5 T^{3} + T^{4}$$
$7$ $$117649 - 659 T^{2} + T^{4}$$
$11$ $$617796 + 52662 T + 3703 T^{2} + 67 T^{3} + T^{4}$$
$13$ $$( 84 - 41 T + T^{2} )^{2}$$
$17$ $$10653696 + 300288 T + 11728 T^{2} - 92 T^{3} + T^{4}$$
$19$ $$6574096 - 110252 T + 4413 T^{2} + 43 T^{3} + T^{4}$$
$23$ $$9216 + 14208 T + 21808 T^{2} + 148 T^{3} + T^{4}$$
$29$ $$( -39204 - 77 T + T^{2} )^{2}$$
$31$ $$4389725025 - 34452600 T + 204145 T^{2} - 520 T^{3} + T^{4}$$
$37$ $$741146176 - 190568 T + 27273 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$( 33264 + 426 T + T^{2} )^{2}$$
$43$ $$( -72794 + 107 T + T^{2} )^{2}$$
$47$ $$1191906576 + 19885824 T + 297252 T^{2} + 576 T^{3} + T^{4}$$
$53$ $$137733696 - 2851848 T + 47313 T^{2} - 243 T^{3} + T^{4}$$
$59$ $$44160500736 + 1471008 T + 210193 T^{2} - 7 T^{3} + T^{4}$$
$61$ $$51322896 + 1604736 T + 43012 T^{2} + 224 T^{3} + T^{4}$$
$67$ $$5976217636 + 53109222 T + 394663 T^{2} + 687 T^{3} + T^{4}$$
$71$ $$( -78804 - 472 T + T^{2} )^{2}$$
$73$ $$39938423716 + 184058166 T + 1048087 T^{2} - 921 T^{3} + T^{4}$$
$79$ $$2270427201 - 25063374 T + 229027 T^{2} - 526 T^{3} + T^{4}$$
$83$ $$( -197946 + 221 T + T^{2} )^{2}$$
$89$ $$196582277376 - 343173024 T + 1042452 T^{2} + 774 T^{3} + T^{4}$$
$97$ $$( 541646 - 1953 T + T^{2} )^{2}$$