# Properties

 Label 5.12.b.a Level 5 Weight 12 Character orbit 5.b Analytic conductor 3.842 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$5$$ Weight: $$k$$ = $$12$$ Character orbit: $$[\chi]$$ = 5.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$3.8417159028$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}\cdot 3^{2}\cdot 5^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( -18 - \beta_{2} ) q^{4}$$ $$+ ( -75 - 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{5}$$ $$+ ( -438 - 11 \beta_{2} ) q^{6}$$ $$+ ( -156 \beta_{1} - 119 \beta_{3} ) q^{7}$$ $$+ ( 1196 \beta_{1} + 112 \beta_{3} ) q^{8}$$ $$+ ( 6363 + 78 \beta_{2} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$-\beta_{3} q^{3}$$ $$+ ( -18 - \beta_{2} ) q^{4}$$ $$+ ( -75 - 10 \beta_{1} - 5 \beta_{2} + 5 \beta_{3} ) q^{5}$$ $$+ ( -438 - 11 \beta_{2} ) q^{6}$$ $$+ ( -156 \beta_{1} - 119 \beta_{3} ) q^{7}$$ $$+ ( 1196 \beta_{1} + 112 \beta_{3} ) q^{8}$$ $$+ ( 6363 + 78 \beta_{2} ) q^{9}$$ $$+ ( 22850 - 4245 \beta_{1} + 65 \beta_{2} + 560 \beta_{3} ) q^{10}$$ $$+ ( -81588 + 220 \beta_{2} ) q^{11}$$ $$+ ( -9612 \beta_{1} - 816 \beta_{3} ) q^{12}$$ $$+ ( 39468 \beta_{1} + 110 \beta_{3} ) q^{13}$$ $$+ ( 270174 - 1153 \beta_{2} ) q^{14}$$ $$+ ( 858300 - 48060 \beta_{1} - 280 \beta_{2} - 4095 \beta_{3} ) q^{15}$$ $$+ ( -2458744 - 2012 \beta_{2} ) q^{16}$$ $$+ ( -29480 \beta_{1} + 12648 \beta_{3} ) q^{17}$$ $$+ ( 71415 \beta_{1} - 8736 \beta_{3} ) q^{18}$$ $$+ ( 3865220 + 7436 \beta_{2} ) q^{19}$$ $$+ ( 8861850 + 56580 \beta_{1} + 165 \beta_{2} + 2960 \beta_{3} ) q^{20}$$ $$+ ( -20254968 + 10998 \beta_{2} ) q^{21}$$ $$+ ( 101892 \beta_{1} - 24640 \beta_{3} ) q^{22}$$ $$+ ( -624404 \beta_{1} + 46641 \beta_{3} ) q^{23}$$ $$+ ( 18603960 - 21892 \beta_{2} ) q^{24}$$ $$+ ( 39788125 + 565500 \beta_{1} + 1500 \beta_{2} + 29750 \beta_{3} ) q^{25}$$ $$+ ( -81492708 - 38258 \beta_{2} ) q^{26}$$ $$+ ( 749736 \beta_{1} - 118458 \beta_{3} ) q^{27}$$ $$+ ( -1010916 \beta_{1} - 114576 \beta_{3} ) q^{28}$$ $$+ ( 54060630 + 4024 \beta_{2} ) q^{29}$$ $$+ ( 97498350 + 624780 \beta_{1} + 3015 \beta_{2} + 31360 \beta_{3} ) q^{30}$$ $$+ ( -171010768 + 75560 \beta_{2} ) q^{31}$$ $$+ ( -1687344 \beta_{1} + 454720 \beta_{3} ) q^{32}$$ $$+ ( 2114640 \beta_{1} + 265068 \beta_{3} ) q^{33}$$ $$+ ( 66445504 + 168608 \beta_{2} ) q^{34}$$ $$+ ( 98573100 - 5056920 \beta_{1} - 43460 \beta_{2} - 574665 \beta_{3} ) q^{35}$$ $$+ ( -138338334 - 7767 \beta_{2} ) q^{36}$$ $$+ ( 661212 \beta_{1} - 35238 \beta_{3} ) q^{37}$$ $$+ ( 10066844 \beta_{1} - 832832 \beta_{3} ) q^{38}$$ $$+ ( 1499256 - 442728 \beta_{2} ) q^{39}$$ $$+ ( -68801000 + 305700 \beta_{1} + 109100 \beta_{2} + 1128400 \beta_{3} ) q^{40}$$ $$+ ( 253218342 - 426490 \beta_{2} ) q^{41}$$ $$+ ( -11082636 \beta_{1} - 1231776 \beta_{3} ) q^{42}$$ $$+ ( -4432008 \beta_{1} + 1380883 \beta_{3} ) q^{43}$$ $$+ ( -388393416 + 77628 \beta_{2} ) q^{44}$$ $$+ ( -691596225 - 4462830 \beta_{1} - 37665 \beta_{2} - 206085 \beta_{3} ) q^{45}$$ $$+ ( 1310447422 + 1137455 \beta_{2} ) q^{46}$$ $$+ ( 32038228 \beta_{1} - 761891 \beta_{3} ) q^{47}$$ $$+ ( -19339344 \beta_{1} + 780736 \beta_{3} ) q^{48}$$ $$+ ( -475161593 + 1488630 \beta_{2} ) q^{49}$$ $$+ ( -1155292500 + 41039125 \beta_{1} - 238250 \beta_{2} - 168000 \beta_{3} ) q^{50}$$ $$+ ( 2172988272 - 662264 \beta_{2} ) q^{51}$$ $$+ ( -32569416 \beta_{1} + 4510176 \beta_{3} ) q^{52}$$ $$+ ( -56138540 \beta_{1} - 4395006 \beta_{3} ) q^{53}$$ $$+ ( -1600839180 - 2052774 \beta_{2} ) q^{54}$$ $$+ ( -1943190900 - 11592120 \beta_{1} + 391440 \beta_{2} - 1078940 \beta_{3} ) q^{55}$$ $$+ ( 2591684520 - 2610764 \beta_{2} ) q^{56}$$ $$+ ( 71474832 \beta_{1} + 2336404 \beta_{3} ) q^{57}$$ $$+ ( 57416646 \beta_{1} - 450688 \beta_{3} ) q^{58}$$ $$+ ( 800242860 - 1049212 \beta_{2} ) q^{59}$$ $$+ ( 480738600 + 1585980 \beta_{1} - 853260 \beta_{2} - 8724240 \beta_{3} ) q^{60}$$ $$+ ( -577617838 + 5036150 \beta_{2} ) q^{61}$$ $$+ ( -107993728 \beta_{1} - 8462720 \beta_{3} ) q^{62}$$ $$+ ( 78077844 \beta_{1} + 8346807 \beta_{3} ) q^{63}$$ $$+ ( -1350287648 + 2568688 \beta_{2} ) q^{64}$$ $$+ ( 807430800 - 162255060 \beta_{1} + 2596220 \beta_{2} + 22552530 \beta_{3} ) q^{65}$$ $$+ ( -4252746456 + 801108 \beta_{2} ) q^{66}$$ $$+ ( -36958200 \beta_{1} - 2121217 \beta_{3} ) q^{67}$$ $$+ ( 146689536 \beta_{1} + 7019008 \beta_{3} ) q^{68}$$ $$+ ( 8239025496 + 3230446 \beta_{2} ) q^{69}$$ $$+ ( 10195893450 + 62327460 \beta_{1} - 1264395 \beta_{2} + 4867520 \beta_{3} ) q^{70}$$ $$+ ( -15083866728 - 7672800 \beta_{2} ) q^{71}$$ $$+ ( 1441908 \beta_{1} - 17021424 \beta_{3} ) q^{72}$$ $$+ ( -16271424 \beta_{1} - 35900124 \beta_{3} ) q^{73}$$ $$+ ( -1381498236 - 1048830 \beta_{2} ) q^{74}$$ $$+ ( 4833135000 + 14418000 \beta_{1} - 8541000 \beta_{2} - 38537125 \beta_{3} ) q^{75}$$ $$+ ( -13246909560 - 3999068 \beta_{2} ) q^{76}$$ $$+ ( 235747008 \beta_{1} + 35386932 \beta_{3} ) q^{77}$$ $$+ ( -367735896 \beta_{1} + 49585536 \beta_{3} ) q^{78}$$ $$+ ( 18659442080 - 20151976 \beta_{2} ) q^{79}$$ $$+ ( 18011731800 + 138064240 \beta_{1} + 12444620 \beta_{2} - 6157120 \beta_{3} ) q^{80}$$ $$+ ( -19431929079 + 14810094 \beta_{2} ) q^{81}$$ $$+ ( -102474318 \beta_{1} + 47766880 \beta_{3} ) q^{82}$$ $$+ ( -478282616 \beta_{1} - 44237433 \beta_{3} ) q^{83}$$ $$+ ( -19124966376 + 20057004 \beta_{2} ) q^{84}$$ $$+ ( -11529396400 + 733005480 \beta_{1} + 1625240 \beta_{2} + 35284760 \beta_{3} ) q^{85}$$ $$+ ( 9761355282 + 19621721 \beta_{2} ) q^{86}$$ $$+ ( 38678688 \beta_{1} - 50704614 \beta_{3} ) q^{87}$$ $$+ ( -114976848 \beta_{1} - 59157056 \beta_{3} ) q^{88}$$ $$+ ( 29568124890 + 24794472 \beta_{2} ) q^{89}$$ $$+ ( 9129941550 - 723008835 \beta_{1} + 2195895 \beta_{2} + 4218480 \beta_{3} ) q^{90}$$ $$+ ( 12891273912 - 46716384 \beta_{2} ) q^{91}$$ $$+ ( 980305500 \beta_{1} - 31874192 \beta_{3} ) q^{92}$$ $$+ ( 726282720 \beta_{1} + 234027808 \beta_{3} ) q^{93}$$ $$+ ( -66524687306 - 40419029 \beta_{2} ) q^{94}$$ $$+ ( -66176569500 - 458042600 \beta_{1} - 19883800 \beta_{2} - 3353700 \beta_{3} ) q^{95}$$ $$+ ( 78397957152 - 16907376 \beta_{2} ) q^{96}$$ $$+ ( -2749674072 \beta_{1} - 158351216 \beta_{3} ) q^{97}$$ $$+ ( 766355827 \beta_{1} - 166726560 \beta_{3} ) q^{98}$$ $$+ ( 29890091556 - 4964004 \beta_{2} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut -\mathstrut 72q^{4}$$ $$\mathstrut -\mathstrut 300q^{5}$$ $$\mathstrut -\mathstrut 1752q^{6}$$ $$\mathstrut +\mathstrut 25452q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut -\mathstrut 72q^{4}$$ $$\mathstrut -\mathstrut 300q^{5}$$ $$\mathstrut -\mathstrut 1752q^{6}$$ $$\mathstrut +\mathstrut 25452q^{9}$$ $$\mathstrut +\mathstrut 91400q^{10}$$ $$\mathstrut -\mathstrut 326352q^{11}$$ $$\mathstrut +\mathstrut 1080696q^{14}$$ $$\mathstrut +\mathstrut 3433200q^{15}$$ $$\mathstrut -\mathstrut 9834976q^{16}$$ $$\mathstrut +\mathstrut 15460880q^{19}$$ $$\mathstrut +\mathstrut 35447400q^{20}$$ $$\mathstrut -\mathstrut 81019872q^{21}$$ $$\mathstrut +\mathstrut 74415840q^{24}$$ $$\mathstrut +\mathstrut 159152500q^{25}$$ $$\mathstrut -\mathstrut 325970832q^{26}$$ $$\mathstrut +\mathstrut 216242520q^{29}$$ $$\mathstrut +\mathstrut 389993400q^{30}$$ $$\mathstrut -\mathstrut 684043072q^{31}$$ $$\mathstrut +\mathstrut 265782016q^{34}$$ $$\mathstrut +\mathstrut 394292400q^{35}$$ $$\mathstrut -\mathstrut 553353336q^{36}$$ $$\mathstrut +\mathstrut 5997024q^{39}$$ $$\mathstrut -\mathstrut 275204000q^{40}$$ $$\mathstrut +\mathstrut 1012873368q^{41}$$ $$\mathstrut -\mathstrut 1553573664q^{44}$$ $$\mathstrut -\mathstrut 2766384900q^{45}$$ $$\mathstrut +\mathstrut 5241789688q^{46}$$ $$\mathstrut -\mathstrut 1900646372q^{49}$$ $$\mathstrut -\mathstrut 4621170000q^{50}$$ $$\mathstrut +\mathstrut 8691953088q^{51}$$ $$\mathstrut -\mathstrut 6403356720q^{54}$$ $$\mathstrut -\mathstrut 7772763600q^{55}$$ $$\mathstrut +\mathstrut 10366738080q^{56}$$ $$\mathstrut +\mathstrut 3200971440q^{59}$$ $$\mathstrut +\mathstrut 1922954400q^{60}$$ $$\mathstrut -\mathstrut 2310471352q^{61}$$ $$\mathstrut -\mathstrut 5401150592q^{64}$$ $$\mathstrut +\mathstrut 3229723200q^{65}$$ $$\mathstrut -\mathstrut 17010985824q^{66}$$ $$\mathstrut +\mathstrut 32956101984q^{69}$$ $$\mathstrut +\mathstrut 40783573800q^{70}$$ $$\mathstrut -\mathstrut 60335466912q^{71}$$ $$\mathstrut -\mathstrut 5525992944q^{74}$$ $$\mathstrut +\mathstrut 19332540000q^{75}$$ $$\mathstrut -\mathstrut 52987638240q^{76}$$ $$\mathstrut +\mathstrut 74637768320q^{79}$$ $$\mathstrut +\mathstrut 72046927200q^{80}$$ $$\mathstrut -\mathstrut 77727716316q^{81}$$ $$\mathstrut -\mathstrut 76499865504q^{84}$$ $$\mathstrut -\mathstrut 46117585600q^{85}$$ $$\mathstrut +\mathstrut 39045421128q^{86}$$ $$\mathstrut +\mathstrut 118272499560q^{89}$$ $$\mathstrut +\mathstrut 36519766200q^{90}$$ $$\mathstrut +\mathstrut 51565095648q^{91}$$ $$\mathstrut -\mathstrut 266098749224q^{94}$$ $$\mathstrut -\mathstrut 264706278000q^{95}$$ $$\mathstrut +\mathstrut 313591828608q^{96}$$ $$\mathstrut +\mathstrut 119560366224q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$x^{3}\mathstrut -\mathstrut$$ $$142$$ $$x^{2}\mathstrut -\mathstrut$$ $$2144$$ $$x\mathstrut +\mathstrut$$ $$28656$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} - 12 \nu^{2} + 31 \nu + 2562$$$$)/15$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 3 \nu^{2} + 316 \nu + 1422$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{3} + 57 \nu^{2} + 34 \nu - 22932$$$$)/30$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$6$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$30$$$$)/120$$ $$\nu^{2}$$ $$=$$ $$($$$$-$$$$38$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$5$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$234$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$8550$$$$)/120$$ $$\nu^{3}$$ $$=$$ $$($$$$518$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$29$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$1194$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$205770$$$$)/120$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 11.3434 + 1.39818i −10.8434 − 10.0894i −10.8434 + 10.0894i 11.3434 − 1.39818i
58.2855i 258.747i −1349.20 −6731.01 + 1876.59i −15081.2 21698.4i 40729.8i 110197. 109378. + 392321.i
4.2 27.1071i 524.040i 1313.20 6581.01 2349.13i 14205.2 66589.5i 91112.6i −97470.8 −63678.2 178392.i
4.3 27.1071i 524.040i 1313.20 6581.01 + 2349.13i 14205.2 66589.5i 91112.6i −97470.8 −63678.2 + 178392.i
4.4 58.2855i 258.747i −1349.20 −6731.01 1876.59i −15081.2 21698.4i 40729.8i 110197. 109378. 392321.i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

## Hecke kernels

There are no other newforms in $$S_{12}^{\mathrm{new}}(5, [\chi])$$.