# Properties

 Label 12.5.d.a Level 12 Weight 5 Character orbit 12.d Analytic conductor 1.240 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 12.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.24043955701$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{13})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + ( 2 - \beta_{1} ) q^{2} -\beta_{2} q^{3} + ( -4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} + ( 6 + 4 \beta_{1} - 4 \beta_{3} ) q^{5} + ( -6 - 2 \beta_{2} + 3 \beta_{3} ) q^{6} + ( -8 + 12 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{7} + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{8} -27 q^{9} +O(q^{10})$$ $$q + ( 2 - \beta_{1} ) q^{2} -\beta_{2} q^{3} + ( -4 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{4} + ( 6 + 4 \beta_{1} - 4 \beta_{3} ) q^{5} + ( -6 - 2 \beta_{2} + 3 \beta_{3} ) q^{6} + ( -8 + 12 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{7} + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{8} -27 q^{9} + ( -36 - 6 \beta_{1} - 16 \beta_{2} - 8 \beta_{3} ) q^{10} + ( 16 - 24 \beta_{1} + 12 \beta_{2} - 8 \beta_{3} ) q^{11} + ( 36 + 9 \beta_{1} + 2 \beta_{2} + 9 \beta_{3} ) q^{12} + ( 74 - 24 \beta_{1} + 24 \beta_{3} ) q^{13} + ( 136 + 24 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} ) q^{14} + ( 24 - 36 \beta_{1} + 2 \beta_{2} - 12 \beta_{3} ) q^{15} + ( 48 + 16 \beta_{2} - 40 \beta_{3} ) q^{16} + ( -150 - 8 \beta_{1} + 8 \beta_{3} ) q^{17} + ( -54 + 27 \beta_{1} ) q^{18} + ( -48 + 72 \beta_{1} + 36 \beta_{2} + 24 \beta_{3} ) q^{19} + ( -376 + 22 \beta_{1} - 36 \beta_{2} + 46 \beta_{3} ) q^{20} + ( -36 + 36 \beta_{1} - 36 \beta_{3} ) q^{21} + ( -248 - 48 \beta_{1} + 56 \beta_{2} - 20 \beta_{3} ) q^{22} + ( -16 + 24 \beta_{1} - 96 \beta_{2} + 8 \beta_{3} ) q^{23} + ( 336 - 18 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{24} + ( 243 + 48 \beta_{1} - 48 \beta_{3} ) q^{25} + ( 436 - 74 \beta_{1} + 96 \beta_{2} + 48 \beta_{3} ) q^{26} + 27 \beta_{2} q^{27} + ( 400 - 108 \beta_{1} - 88 \beta_{2} + 52 \beta_{3} ) q^{28} + ( 222 + 28 \beta_{1} - 28 \beta_{3} ) q^{29} + ( -468 - 72 \beta_{1} + 52 \beta_{2} + 18 \beta_{3} ) q^{30} + ( 56 - 84 \beta_{1} - 140 \beta_{2} - 28 \beta_{3} ) q^{31} + ( -608 - 88 \beta_{1} - 48 \beta_{2} - 88 \beta_{3} ) q^{32} + ( 180 - 72 \beta_{1} + 72 \beta_{3} ) q^{33} + ( -204 + 150 \beta_{1} + 32 \beta_{2} + 16 \beta_{3} ) q^{34} + ( -16 + 24 \beta_{1} + 264 \beta_{2} + 8 \beta_{3} ) q^{35} + ( 108 + 81 \beta_{1} - 54 \beta_{2} - 27 \beta_{3} ) q^{36} + ( -1102 - 48 \beta_{1} + 48 \beta_{3} ) q^{37} + ( 1176 + 144 \beta_{1} - 24 \beta_{2} - 156 \beta_{3} ) q^{38} + ( -144 + 216 \beta_{1} - 122 \beta_{2} + 72 \beta_{3} ) q^{39} + ( 128 + 444 \beta_{1} - 24 \beta_{2} + 132 \beta_{3} ) q^{40} + ( 138 - 56 \beta_{1} + 56 \beta_{3} ) q^{41} + ( -504 + 36 \beta_{1} - 144 \beta_{2} - 72 \beta_{3} ) q^{42} + ( 304 - 456 \beta_{1} + 332 \beta_{2} - 152 \beta_{3} ) q^{43} + ( -944 + 180 \beta_{1} + 168 \beta_{2} - 140 \beta_{3} ) q^{44} + ( -162 - 108 \beta_{1} + 108 \beta_{3} ) q^{45} + ( -256 + 48 \beta_{1} - 224 \beta_{2} + 272 \beta_{3} ) q^{46} + ( 16 - 24 \beta_{1} - 216 \beta_{2} - 8 \beta_{3} ) q^{47} + ( 432 - 360 \beta_{1} + 32 \beta_{2} ) q^{48} + ( -143 + 96 \beta_{1} - 96 \beta_{3} ) q^{49} + ( -90 - 243 \beta_{1} - 192 \beta_{2} - 96 \beta_{3} ) q^{50} + ( -48 + 72 \beta_{1} + 134 \beta_{2} + 24 \beta_{3} ) q^{51} + ( 1816 - 462 \beta_{1} + 436 \beta_{2} - 166 \beta_{3} ) q^{52} + ( 1278 - 68 \beta_{1} + 68 \beta_{3} ) q^{53} + ( 162 + 54 \beta_{2} - 81 \beta_{3} ) q^{54} + ( -64 + 96 \beta_{1} - 536 \beta_{2} + 32 \beta_{3} ) q^{55} + ( 448 - 456 \beta_{1} + 144 \beta_{2} + 424 \beta_{3} ) q^{56} + ( 1404 + 216 \beta_{1} - 216 \beta_{3} ) q^{57} + ( 108 - 222 \beta_{1} - 112 \beta_{2} - 56 \beta_{3} ) q^{58} + ( -448 + 672 \beta_{1} - 108 \beta_{2} + 224 \beta_{3} ) q^{59} + ( -840 + 414 \beta_{1} + 284 \beta_{2} - 66 \beta_{3} ) q^{60} + ( 1058 + 480 \beta_{1} - 480 \beta_{3} ) q^{61} + ( -1960 - 168 \beta_{1} - 168 \beta_{2} + 476 \beta_{3} ) q^{62} + ( 216 - 324 \beta_{1} + 108 \beta_{2} - 108 \beta_{3} ) q^{63} + ( -3968 + 432 \beta_{1} - 96 \beta_{2} + 144 \beta_{3} ) q^{64} + ( -4548 + 152 \beta_{1} - 152 \beta_{3} ) q^{65} + ( 1224 - 180 \beta_{1} + 288 \beta_{2} + 144 \beta_{3} ) q^{66} + ( -448 + 672 \beta_{1} + 316 \beta_{2} + 224 \beta_{3} ) q^{67} + ( 1304 + 370 \beta_{1} - 204 \beta_{2} - 230 \beta_{3} ) q^{68} + ( -2448 + 72 \beta_{1} - 72 \beta_{3} ) q^{69} + ( 1904 + 48 \beta_{1} + 496 \beta_{2} - 808 \beta_{3} ) q^{70} + ( 432 - 648 \beta_{1} - 240 \beta_{2} - 216 \beta_{3} ) q^{71} + ( -54 \beta_{1} - 324 \beta_{2} + 54 \beta_{3} ) q^{72} + ( 2210 - 1056 \beta_{1} + 1056 \beta_{3} ) q^{73} + ( -1628 + 1102 \beta_{1} + 192 \beta_{2} + 96 \beta_{3} ) q^{74} + ( 288 - 432 \beta_{1} - 147 \beta_{2} - 144 \beta_{3} ) q^{75} + ( 240 - 1188 \beta_{1} - 648 \beta_{2} - 228 \beta_{3} ) q^{76} + ( 5232 - 336 \beta_{1} + 336 \beta_{3} ) q^{77} + ( 2148 + 432 \beta_{1} - 532 \beta_{2} + 222 \beta_{3} ) q^{78} + ( -296 + 444 \beta_{1} + 980 \beta_{2} + 148 \beta_{3} ) q^{79} + ( 6304 + 448 \beta_{1} - 672 \beta_{2} - 240 \beta_{3} ) q^{80} + 729 q^{81} + ( 948 - 138 \beta_{1} + 224 \beta_{2} + 112 \beta_{3} ) q^{82} + ( 1552 - 2328 \beta_{1} + 660 \beta_{2} - 776 \beta_{3} ) q^{83} + ( -3024 + 468 \beta_{1} - 504 \beta_{2} + 324 \beta_{3} ) q^{84} + ( -2564 - 648 \beta_{1} + 648 \beta_{3} ) q^{85} + ( -4088 - 912 \beta_{1} + 1272 \beta_{2} - 692 \beta_{3} ) q^{86} + ( 168 - 252 \beta_{1} - 166 \beta_{2} - 84 \beta_{3} ) q^{87} + ( -2240 + 984 \beta_{1} - 304 \beta_{2} - 824 \beta_{3} ) q^{88} + ( -6270 + 784 \beta_{1} - 784 \beta_{3} ) q^{89} + ( 972 + 162 \beta_{1} + 432 \beta_{2} + 216 \beta_{3} ) q^{90} + ( -784 + 1176 \beta_{1} - 2024 \beta_{2} + 392 \beta_{3} ) q^{91} + ( 3968 + 576 \beta_{1} + 896 \beta_{3} ) q^{92} + ( -4284 - 252 \beta_{1} + 252 \beta_{3} ) q^{93} + ( -1616 - 48 \beta_{1} - 400 \beta_{2} + 664 \beta_{3} ) q^{94} + ( -1536 + 2304 \beta_{1} + 1464 \beta_{2} + 768 \beta_{3} ) q^{95} + ( -1824 - 792 \beta_{1} + 784 \beta_{2} + 264 \beta_{3} ) q^{96} + ( 5762 + 1104 \beta_{1} - 1104 \beta_{3} ) q^{97} + ( -1438 + 143 \beta_{1} - 384 \beta_{2} - 192 \beta_{3} ) q^{98} + ( -432 + 648 \beta_{1} - 324 \beta_{2} + 216 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{2} - 20q^{4} + 24q^{5} - 18q^{6} - 108q^{9} + O(q^{10})$$ $$4q + 6q^{2} - 20q^{4} + 24q^{5} - 18q^{6} - 108q^{9} - 172q^{10} + 180q^{12} + 296q^{13} + 600q^{14} + 112q^{16} - 600q^{17} - 162q^{18} - 1368q^{20} - 144q^{21} - 1128q^{22} + 1296q^{24} + 972q^{25} + 1692q^{26} + 1488q^{28} + 888q^{29} - 1980q^{30} - 2784q^{32} + 720q^{33} - 484q^{34} + 540q^{36} - 4408q^{37} + 4680q^{38} + 1664q^{40} + 552q^{41} - 2088q^{42} - 3696q^{44} - 648q^{45} - 384q^{46} + 1008q^{48} - 572q^{49} - 1038q^{50} + 6008q^{52} + 5112q^{53} + 486q^{54} + 1728q^{56} + 5616q^{57} - 124q^{58} - 2664q^{60} + 4232q^{61} - 7224q^{62} - 14720q^{64} - 18192q^{65} + 4824q^{66} + 5496q^{68} - 9792q^{69} + 6096q^{70} + 8840q^{73} - 4116q^{74} - 1872q^{76} + 20928q^{77} + 9900q^{78} + 25632q^{80} + 2916q^{81} + 3740q^{82} - 10512q^{84} - 10256q^{85} - 19560q^{86} - 8640q^{88} - 25080q^{89} + 4644q^{90} + 18816q^{92} - 17136q^{93} - 5232q^{94} - 8352q^{96} + 23048q^{97} - 5850q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 4 x^{2} + 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 4 \nu^{2} - 4 \nu + 3$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1} - 8$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{1} - 5$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 1.15139 + 1.99426i 1.15139 − 1.99426i −0.651388 + 1.12824i −0.651388 − 1.12824i
−0.302776 3.98852i 5.19615i −15.8167 + 2.41526i 34.8444 −20.7250 + 1.57327i 43.0318i 14.4222 + 62.3538i −27.0000 −10.5500 138.978i
7.2 −0.302776 + 3.98852i 5.19615i −15.8167 2.41526i 34.8444 −20.7250 1.57327i 43.0318i 14.4222 62.3538i −27.0000 −10.5500 + 138.978i
7.3 3.30278 2.25647i 5.19615i 5.81665 14.9053i −22.8444 11.7250 + 17.1617i 56.8882i −14.4222 62.3538i −27.0000 −75.4500 + 51.5478i
7.4 3.30278 + 2.25647i 5.19615i 5.81665 + 14.9053i −22.8444 11.7250 17.1617i 56.8882i −14.4222 + 62.3538i −27.0000 −75.4500 51.5478i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 12.5.d.a 4
3.b odd 2 1 36.5.d.b 4
4.b odd 2 1 inner 12.5.d.a 4
5.b even 2 1 300.5.c.a 4
5.c odd 4 2 300.5.f.a 8
8.b even 2 1 192.5.g.d 4
8.d odd 2 1 192.5.g.d 4
12.b even 2 1 36.5.d.b 4
16.e even 4 2 768.5.b.g 8
16.f odd 4 2 768.5.b.g 8
20.d odd 2 1 300.5.c.a 4
20.e even 4 2 300.5.f.a 8
24.f even 2 1 576.5.g.m 4
24.h odd 2 1 576.5.g.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.5.d.a 4 1.a even 1 1 trivial
12.5.d.a 4 4.b odd 2 1 inner
36.5.d.b 4 3.b odd 2 1
36.5.d.b 4 12.b even 2 1
192.5.g.d 4 8.b even 2 1
192.5.g.d 4 8.d odd 2 1
300.5.c.a 4 5.b even 2 1
300.5.c.a 4 20.d odd 2 1
300.5.f.a 8 5.c odd 4 2
300.5.f.a 8 20.e even 4 2
576.5.g.m 4 24.f even 2 1
576.5.g.m 4 24.h odd 2 1
768.5.b.g 8 16.e even 4 2
768.5.b.g 8 16.f odd 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(12, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 6 T + 28 T^{2} - 96 T^{3} + 256 T^{4}$$
$3$ $$( 1 + 27 T^{2} )^{2}$$
$5$ $$( 1 - 12 T + 454 T^{2} - 7500 T^{3} + 390625 T^{4} )^{2}$$
$7$ $$1 - 4516 T^{2} + 16148934 T^{4} - 26033841316 T^{6} + 33232930569601 T^{8}$$
$11$ $$1 - 36196 T^{2} + 708332166 T^{4} - 7758934056676 T^{6} + 45949729863572161 T^{8}$$
$13$ $$( 1 - 148 T + 32646 T^{2} - 4227028 T^{3} + 815730721 T^{4} )^{2}$$
$17$ $$( 1 + 300 T + 186214 T^{2} + 25056300 T^{3} + 6975757441 T^{4} )^{2}$$
$19$ $$1 - 195556 T^{2} + 17286835974 T^{4} - 3321237654045796 T^{6} +$$$$28\!\cdots\!81$$$$T^{8}$$
$23$ $$1 - 655492 T^{2} + 255175536006 T^{4} - 51332224363813252 T^{6} +$$$$61\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 444 T + 1423078 T^{2} - 314032764 T^{3} + 500246412961 T^{4} )^{2}$$
$31$ $$1 - 2090020 T^{2} + 2465294161734 T^{4} - 1782559326072438820 T^{6} +$$$$72\!\cdots\!81$$$$T^{8}$$
$37$ $$( 1 + 2204 T + 4842918 T^{2} + 4130650844 T^{3} + 3512479453921 T^{4} )^{2}$$
$41$ $$( 1 - 276 T + 5507494 T^{2} - 779910036 T^{3} + 7984925229121 T^{4} )^{2}$$
$43$ $$1 - 3593572 T^{2} + 5893643026566 T^{4} - 42002389247979180772 T^{6} +$$$$13\!\cdots\!01$$$$T^{8}$$
$47$ $$1 - 16853380 T^{2} + 118578854811654 T^{4} -$$$$40\!\cdots\!80$$$$T^{6} +$$$$56\!\cdots\!21$$$$T^{8}$$
$53$ $$( 1 - 2556 T + 17173798 T^{2} - 20168069436 T^{3} + 62259690411361 T^{4} )^{2}$$
$59$ $$1 - 32722276 T^{2} + 559903452302214 T^{4} -$$$$48\!\cdots\!96$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8}$$
$61$ $$( 1 - 2116 T + 16830246 T^{2} - 29297799556 T^{3} + 191707312997281 T^{4} )^{2}$$
$67$ $$1 - 53256676 T^{2} + 1338152481850374 T^{4} -$$$$21\!\cdots\!16$$$$T^{6} +$$$$16\!\cdots\!81$$$$T^{8}$$
$71$ $$1 - 79127428 T^{2} + 2740885222137990 T^{4} -$$$$51\!\cdots\!08$$$$T^{6} +$$$$41\!\cdots\!21$$$$T^{8}$$
$73$ $$( 1 - 4420 T + 3693510 T^{2} - 125520225220 T^{3} + 806460091894081 T^{4} )^{2}$$
$79$ $$1 - 86136100 T^{2} + 4459690111983174 T^{4} -$$$$13\!\cdots\!00$$$$T^{6} +$$$$23\!\cdots\!21$$$$T^{8}$$
$83$ $$1 - 855268 T^{2} + 4298268871421190 T^{4} -$$$$19\!\cdots\!88$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8}$$
$89$ $$( 1 + 12540 T + 132835270 T^{2} + 786787702140 T^{3} + 3936588805702081 T^{4} )^{2}$$
$97$ $$( 1 - 11524 T + 146880774 T^{2} - 1020211434244 T^{3} + 7837433594376961 T^{4} )^{2}$$