# Properties

 Label 12.13.c.a Level 12 Weight 13 Character orbit 12.c Analytic conductor 10.968 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$13$$ Character orbit: $$[\chi]$$ = 12.c (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$10.9679258073$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{7}$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 75 + \beta_{1} ) q^{3} + ( -3 \beta_{1} + \beta_{2} ) q^{5} + ( 3950 + 28 \beta_{1} - \beta_{3} ) q^{7} + ( 24201 + 69 \beta_{1} - 27 \beta_{2} - 12 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 75 + \beta_{1} ) q^{3} + ( -3 \beta_{1} + \beta_{2} ) q^{5} + ( 3950 + 28 \beta_{1} - \beta_{3} ) q^{7} + ( 24201 + 69 \beta_{1} - 27 \beta_{2} - 12 \beta_{3} ) q^{9} + ( -1794 \beta_{1} + 52 \beta_{2} - 63 \beta_{3} ) q^{11} + ( 858050 + 6608 \beta_{1} - 236 \beta_{3} ) q^{13} + ( 1653480 - 162 \beta_{1} + 324 \beta_{2} - 585 \beta_{3} ) q^{15} + ( -26016 \beta_{1} - 1000 \beta_{2} - 1116 \beta_{3} ) q^{17} + ( -512506 + 32900 \beta_{1} - 1175 \beta_{3} ) q^{19} + ( 14994834 + 1763 \beta_{1} - 729 \beta_{2} - 324 \beta_{3} ) q^{21} + ( 68988 \beta_{1} + 6800 \beta_{2} + 3438 \beta_{3} ) q^{23} + ( -109362815 - 216720 \beta_{1} + 7740 \beta_{3} ) q^{25} + ( 127089675 + 4419 \beta_{1} - 8100 \beta_{2} + 16083 \beta_{3} ) q^{27} + ( 460743 \beta_{1} - 20669 \beta_{2} + 15336 \beta_{3} ) q^{29} + ( -629752066 - 573300 \beta_{1} + 20475 \beta_{3} ) q^{31} + ( 948796200 - 125793 \beta_{1} + 62775 \beta_{2} - 10008 \beta_{3} ) q^{33} + ( -457008 \beta_{1} + 17396 \beta_{2} - 15570 \beta_{3} ) q^{35} + ( -1866677950 + 2947504 \beta_{1} - 105268 \beta_{3} ) q^{37} + ( 3533219574 + 341918 \beta_{1} - 172044 \beta_{2} - 76464 \beta_{3} ) q^{39} + ( -5597562 \beta_{1} + 30046 \beta_{2} - 211824 \beta_{3} ) q^{41} + ( -6529982650 + 251748 \beta_{1} - 8991 \beta_{3} ) q^{43} + ( 8969181840 + 415017 \beta_{1} + 98901 \beta_{2} - 192240 \beta_{3} ) q^{45} + ( 8463888 \beta_{1} + 121800 \beta_{2} + 339588 \beta_{3} ) q^{47} + ( -13031961165 + 221200 \beta_{1} - 7900 \beta_{3} ) q^{49} + ( 13630675680 - 1917108 \beta_{1} + 489564 \beta_{2} + 946584 \beta_{3} ) q^{51} + ( 4976817 \beta_{1} - 887675 \beta_{2} + 88992 \beta_{3} ) q^{53} + ( -24007317840 - 39528720 \beta_{1} + 1411740 \beta_{3} ) q^{55} + ( 17232398250 - 3082231 \beta_{1} - 856575 \beta_{2} - 380700 \beta_{3} ) q^{57} + ( 31893834 \beta_{1} + 817328 \beta_{2} + 1320993 \beta_{3} ) q^{59} + ( -2326808926 + 48476400 \beta_{1} - 1731300 \beta_{3} ) q^{61} + ( 4505169150 + 14461320 \beta_{1} - 216000 \beta_{2} + 435441 \beta_{3} ) q^{63} + ( -107631438 \beta_{1} + 4031306 \beta_{2} - 3674520 \beta_{3} ) q^{65} + ( 22263896150 + 13071828 \beta_{1} - 466851 \beta_{3} ) q^{67} + ( -35841396240 + 5303394 \beta_{1} - 303102 \beta_{2} - 5091912 \beta_{3} ) q^{69} + ( 19570764 \beta_{1} - 10716712 \beta_{2} - 483822 \beta_{3} ) q^{71} + ( 116035618850 + 122090304 \beta_{1} - 4360368 \beta_{3} ) q^{73} + ( -121969251285 - 92435435 \beta_{1} + 5642460 \beta_{2} + 2507760 \beta_{3} ) q^{75} + ( -15650142 \beta_{1} + 3384650 \beta_{2} - 211392 \beta_{3} ) q^{77} + ( 141755506622 - 178563700 \beta_{1} + 6377275 \beta_{3} ) q^{79} + ( -232262879679 + 160992738 \beta_{1} - 2521854 \beta_{2} + 4784076 \beta_{3} ) q^{81} + ( 206986626 \beta_{1} + 19092900 \beta_{2} + 10164051 \beta_{3} ) q^{83} + ( 253858121280 - 283872960 \beta_{1} + 10138320 \beta_{3} ) q^{85} + ( -244209659400 + 31919346 \beta_{1} - 17876700 \beta_{2} + 7122501 \beta_{3} ) q^{87} + ( 293742462 \beta_{1} - 26619346 \beta_{2} + 8226324 \beta_{3} ) q^{89} + ( 190708051996 + 50127000 \beta_{1} - 1790250 \beta_{3} ) q^{91} + ( -348184912350 - 584973241 \beta_{1} + 14926275 \beta_{2} + 6633900 \beta_{3} ) q^{93} + ( -521523132 \beta_{1} + 15286544 \beta_{2} - 18294750 \beta_{3} ) q^{95} + ( 215739631250 + 143975888 \beta_{1} - 5141996 \beta_{3} ) q^{97} + ( 318016504080 + 918045306 \beta_{1} + 18920952 \beta_{2} - 37353663 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 300q^{3} + 15800q^{7} + 96804q^{9} + O(q^{10})$$ $$4q + 300q^{3} + 15800q^{7} + 96804q^{9} + 3432200q^{13} + 6613920q^{15} - 2050024q^{19} + 59979336q^{21} - 437451260q^{25} + 508358700q^{27} - 2519008264q^{31} + 3795184800q^{33} - 7466711800q^{37} + 14132878296q^{39} - 26119930600q^{43} + 35876727360q^{45} - 52127844660q^{49} + 54522702720q^{51} - 96029271360q^{55} + 68929593000q^{57} - 9307235704q^{61} + 18020676600q^{63} + 89055584600q^{67} - 143365584960q^{69} + 464142475400q^{73} - 487877005140q^{75} + 567022026488q^{79} - 929051518716q^{81} + 1015432485120q^{85} - 976838637600q^{87} + 762832207984q^{91} - 1392739649400q^{93} + 862958525000q^{97} + 1272066016320q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4105 x^{2} + 385000$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 20 \nu^{2} + 4505 \nu - 41050$$$$)/75$$ $$\beta_{2}$$ $$=$$ $$($$$$11 \nu^{3} - 20 \nu^{2} + 35155 \nu - 41050$$$$)/25$$ $$\beta_{3}$$ $$=$$ $$($$$$28 \nu^{3} + 520 \nu^{2} + 126140 \nu + 1067300$$$$)/75$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$5 \beta_{3} - 9 \beta_{2} + 157 \beta_{1}$$$$)/5184$$ $$\nu^{2}$$ $$=$$ $$($$$$5 \beta_{3} - 140 \beta_{1} - 147780$$$$)/72$$ $$\nu^{3}$$ $$=$$ $$($$$$-15325 \beta_{3} + 40545 \beta_{2} - 520085 \beta_{1}$$$$)/5184$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 − 9.79973i 9.79973i − 63.3164i 63.3164i
0 −446.724 576.089i 0 11638.0i 0 −24223.1 0 −132316. + 514706.i 0
5.2 0 −446.724 + 576.089i 0 11638.0i 0 −24223.1 0 −132316. 514706.i 0
5.3 0 596.724 418.762i 0 23907.4i 0 32123.1 0 180718. 499770.i 0
5.4 0 596.724 + 418.762i 0 23907.4i 0 32123.1 0 180718. + 499770.i 0
 $$n$$: 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
3.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{13}^{\mathrm{new}}(12, [\chi])$$.