# Properties

 Label 252.2.e.a Level 252 Weight 2 Character orbit 252.e Analytic conductor 2.012 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 252.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.01223013094$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: 12.0.653473922154496.1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{7} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{6} q^{7} + ( \beta_{1} + \beta_{3} + \beta_{8} + \beta_{10} ) q^{8} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} + ( \beta_{7} - \beta_{9} - \beta_{10} ) q^{5} -\beta_{6} q^{7} + ( \beta_{1} + \beta_{3} + \beta_{8} + \beta_{10} ) q^{8} + ( -1 + \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{10} + ( \beta_{1} - \beta_{3} + 2 \beta_{8} ) q^{11} + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{11} ) q^{13} + \beta_{10} q^{14} + ( -1 + \beta_{2} - 2 \beta_{6} + \beta_{11} ) q^{16} + ( -\beta_{7} - \beta_{9} - \beta_{10} ) q^{17} + ( -2 \beta_{2} - \beta_{5} - 2 \beta_{6} + \beta_{11} ) q^{19} + ( 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{10} ) q^{20} + ( 3 + \beta_{2} + \beta_{4} + \beta_{6} + 2 \beta_{11} ) q^{22} + ( -3 \beta_{1} - \beta_{3} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{23} + ( -2 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{11} ) q^{25} + ( 2 \beta_{3} + 2 \beta_{7} - 4 \beta_{8} ) q^{26} + ( \beta_{4} - \beta_{6} ) q^{28} + \beta_{7} q^{29} + ( 2 + 2 \beta_{4} - 3 \beta_{5} + \beta_{11} ) q^{31} + ( -\beta_{1} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 3 \beta_{10} ) q^{32} + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{34} + ( \beta_{1} - \beta_{3} + \beta_{8} ) q^{35} + ( 1 - \beta_{2} + 3 \beta_{4} + \beta_{5} - \beta_{6} ) q^{37} + ( -2 \beta_{3} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{38} + ( -2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{11} ) q^{40} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{7} + \beta_{9} - 3 \beta_{10} ) q^{41} + ( -2 - 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{43} + ( 2 \beta_{1} - \beta_{3} - 3 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{44} + ( -3 - \beta_{2} + 3 \beta_{4} + 3 \beta_{6} - 2 \beta_{11} ) q^{46} + ( 4 \beta_{3} - 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{47} - q^{49} + ( -\beta_{1} + 2 \beta_{3} + 2 \beta_{7} - 4 \beta_{8} ) q^{50} + ( -4 - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{11} ) q^{52} + ( -2 \beta_{1} - 2 \beta_{3} - 3 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} ) q^{53} + ( -2 - 2 \beta_{4} + 3 \beta_{5} + 8 \beta_{6} - \beta_{11} ) q^{55} + ( -\beta_{1} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{56} + \beta_{5} q^{58} + ( -4 \beta_{1} + 4 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{59} + ( 6 + 2 \beta_{2} + \beta_{5} + 2 \beta_{6} + 3 \beta_{11} ) q^{61} + ( 2 \beta_{3} - 6 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{62} + ( -2 \beta_{2} + 3 \beta_{4} - \beta_{5} - 5 \beta_{6} + 2 \beta_{11} ) q^{64} + ( 2 \beta_{1} + 2 \beta_{3} - 4 \beta_{7} + 2 \beta_{9} + 6 \beta_{10} ) q^{65} + ( -1 + 3 \beta_{2} - \beta_{4} + 3 \beta_{5} - \beta_{6} - 2 \beta_{11} ) q^{67} + ( 2 \beta_{3} + 2 \beta_{9} - 2 \beta_{10} ) q^{68} + ( 3 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{11} ) q^{70} + ( 3 \beta_{1} + \beta_{3} - 6 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{71} + ( 1 - \beta_{2} + 5 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{11} ) q^{73} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{7} - 4 \beta_{9} ) q^{74} + ( 6 - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{11} ) q^{76} + ( -2 \beta_{7} + \beta_{9} + \beta_{10} ) q^{77} + ( 3 + 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} ) q^{79} + ( 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{10} ) q^{80} + ( 3 - 3 \beta_{2} - \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{82} + ( 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{9} - 2 \beta_{10} ) q^{83} + ( -5 + \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{85} + ( -4 \beta_{3} + 4 \beta_{7} ) q^{86} + ( 5 + \beta_{2} + \beta_{4} - 3 \beta_{5} - \beta_{6} + 3 \beta_{11} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{3} + 5 \beta_{7} - 3 \beta_{9} + \beta_{10} ) q^{89} + ( -1 + \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{11} ) q^{91} + ( -6 \beta_{1} + \beta_{3} - \beta_{7} - 3 \beta_{8} - 5 \beta_{9} - 4 \beta_{10} ) q^{92} + ( -6 - 2 \beta_{2} - 6 \beta_{4} - 6 \beta_{6} - 4 \beta_{11} ) q^{94} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{95} + ( 1 + 3 \beta_{2} - 3 \beta_{4} + 3 \beta_{6} + 3 \beta_{11} ) q^{97} -\beta_{1} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 8q^{4} + O(q^{10})$$ $$12q + 8q^{4} - 8q^{10} - 20q^{16} + 20q^{22} - 12q^{25} - 4q^{28} - 16q^{34} + 8q^{37} + 8q^{40} - 36q^{46} - 12q^{49} - 16q^{52} + 4q^{58} + 56q^{61} - 16q^{64} + 24q^{70} + 72q^{76} + 56q^{82} - 56q^{85} + 28q^{88} - 24q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{10} + 13 x^{8} - 28 x^{6} + 52 x^{4} - 64 x^{2} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{9} - 2 \nu^{7} + \nu^{5} + 6 \nu^{3} - 8 \nu$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{10} - 2 \nu^{8} + 23 \nu^{6} - 58 \nu^{4} + 96 \nu^{2} - 160$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{8} + 2 \nu^{6} - 5 \nu^{4} + 6 \nu^{2} - 8$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{10} + 10 \nu^{8} - 27 \nu^{6} + 34 \nu^{4} - 64 \nu^{2} + 32$$$$)/64$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{11} - 4 \nu^{9} + 5 \nu^{7} - 12 \nu^{5} + 12 \nu^{3} - 16 \nu$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{11} + 6 \nu^{9} - 19 \nu^{7} + 30 \nu^{5} - 24 \nu^{3}$$$$)/64$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} + 10 \nu^{9} - 33 \nu^{7} + 82 \nu^{5} - 120 \nu^{3} + 128 \nu$$$$)/64$$ $$\beta_{10}$$ $$=$$ $$($$$$3 \nu^{11} - 10 \nu^{9} + 27 \nu^{7} - 34 \nu^{5} + 64 \nu^{3} - 32 \nu$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{10} + 10 \nu^{8} - 27 \nu^{6} + 66 \nu^{4} - 96 \nu^{2} + 96$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} + \beta_{8} + \beta_{3} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} - 2 \beta_{6} + \beta_{2} - 1$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{11} - 5 \beta_{6} - \beta_{5} + 3 \beta_{4} - 2 \beta_{2}$$ $$\nu^{7}$$ $$=$$ $$3 \beta_{10} - \beta_{8} - 6 \beta_{7} - 3 \beta_{3} - 3 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-\beta_{11} - 6 \beta_{5} + 6 \beta_{4} - 3 \beta_{2} + 3$$ $$\nu^{9}$$ $$=$$ $$-3 \beta_{10} - \beta_{9} - 10 \beta_{8} - 11 \beta_{7} + 4 \beta_{3} - 3 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-10 \beta_{11} + \beta_{6} - 11 \beta_{5} - 7 \beta_{4} - 2 \beta_{2} - 12$$ $$\nu^{11}$$ $$=$$ $$-3 \beta_{10} + 8 \beta_{9} - 23 \beta_{8} + 6 \beta_{7} + 19 \beta_{3} - 5 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$73$$ $$127$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
71.1
 −1.35489 − 0.405301i −1.35489 + 0.405301i −1.16947 − 0.795191i −1.16947 + 0.795191i −0.892524 − 1.09700i −0.892524 + 1.09700i 0.892524 − 1.09700i 0.892524 + 1.09700i 1.16947 − 0.795191i 1.16947 + 0.795191i 1.35489 − 0.405301i 1.35489 + 0.405301i
−1.35489 0.405301i 0 1.67146 + 1.09828i 3.31339i 0 1.00000i −1.81951 2.16549i 0 1.34292 4.48929i
71.2 −1.35489 + 0.405301i 0 1.67146 1.09828i 3.31339i 0 1.00000i −1.81951 + 2.16549i 0 1.34292 + 4.48929i
71.3 −1.16947 0.795191i 0 0.735342 + 1.85991i 0.665647i 0 1.00000i 0.619022 2.75986i 0 −0.529317 + 0.778457i
71.4 −1.16947 + 0.795191i 0 0.735342 1.85991i 0.665647i 0 1.00000i 0.619022 + 2.75986i 0 −0.529317 0.778457i
71.5 −0.892524 1.09700i 0 −0.406803 + 1.95819i 2.56483i 0 1.00000i 2.51121 1.30147i 0 −2.81361 + 2.28917i
71.6 −0.892524 + 1.09700i 0 −0.406803 1.95819i 2.56483i 0 1.00000i 2.51121 + 1.30147i 0 −2.81361 2.28917i
71.7 0.892524 1.09700i 0 −0.406803 1.95819i 2.56483i 0 1.00000i −2.51121 1.30147i 0 −2.81361 2.28917i
71.8 0.892524 + 1.09700i 0 −0.406803 + 1.95819i 2.56483i 0 1.00000i −2.51121 + 1.30147i 0 −2.81361 + 2.28917i
71.9 1.16947 0.795191i 0 0.735342 1.85991i 0.665647i 0 1.00000i −0.619022 2.75986i 0 −0.529317 0.778457i
71.10 1.16947 + 0.795191i 0 0.735342 + 1.85991i 0.665647i 0 1.00000i −0.619022 + 2.75986i 0 −0.529317 + 0.778457i
71.11 1.35489 0.405301i 0 1.67146 1.09828i 3.31339i 0 1.00000i 1.81951 2.16549i 0 1.34292 + 4.48929i
71.12 1.35489 + 0.405301i 0 1.67146 + 1.09828i 3.31339i 0 1.00000i 1.81951 + 2.16549i 0 1.34292 4.48929i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.e.a 12
3.b odd 2 1 inner 252.2.e.a 12
4.b odd 2 1 inner 252.2.e.a 12
7.b odd 2 1 1764.2.e.g 12
8.b even 2 1 4032.2.h.h 12
8.d odd 2 1 4032.2.h.h 12
12.b even 2 1 inner 252.2.e.a 12
21.c even 2 1 1764.2.e.g 12
24.f even 2 1 4032.2.h.h 12
24.h odd 2 1 4032.2.h.h 12
28.d even 2 1 1764.2.e.g 12
84.h odd 2 1 1764.2.e.g 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.e.a 12 1.a even 1 1 trivial
252.2.e.a 12 3.b odd 2 1 inner
252.2.e.a 12 4.b odd 2 1 inner
252.2.e.a 12 12.b even 2 1 inner
1764.2.e.g 12 7.b odd 2 1
1764.2.e.g 12 21.c even 2 1
1764.2.e.g 12 28.d even 2 1
1764.2.e.g 12 84.h odd 2 1
4032.2.h.h 12 8.b even 2 1
4032.2.h.h 12 8.d odd 2 1
4032.2.h.h 12 24.f even 2 1
4032.2.h.h 12 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(252, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 4 T^{2} + 13 T^{4} - 28 T^{6} + 52 T^{8} - 64 T^{10} + 64 T^{12}$$
$3$ 
$5$ $$( 1 - 12 T^{2} + 95 T^{4} - 568 T^{6} + 2375 T^{8} - 7500 T^{10} + 15625 T^{12} )^{2}$$
$7$ $$( 1 + T^{2} )^{6}$$
$11$ $$( 1 + 38 T^{2} + 715 T^{4} + 9068 T^{6} + 86515 T^{8} + 556358 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 + 11 T^{2} - 16 T^{3} + 143 T^{4} + 2197 T^{6} )^{4}$$
$17$ $$( 1 - 68 T^{2} + 2167 T^{4} - 44168 T^{6} + 626263 T^{8} - 5679428 T^{10} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 - 50 T^{2} + 1639 T^{4} - 35804 T^{6} + 591679 T^{8} - 6516050 T^{10} + 47045881 T^{12} )^{2}$$
$23$ $$( 1 - 2 T^{2} + 979 T^{4} + 3772 T^{6} + 517891 T^{8} - 559682 T^{10} + 148035889 T^{12} )^{2}$$
$29$ $$( 1 - 56 T^{2} + 841 T^{4} )^{6}$$
$31$ $$( 1 - 58 T^{2} + 3727 T^{4} - 113644 T^{6} + 3581647 T^{8} - 53564218 T^{10} + 887503681 T^{12} )^{2}$$
$37$ $$( 1 - 2 T + 47 T^{2} - 276 T^{3} + 1739 T^{4} - 2738 T^{5} + 50653 T^{6} )^{4}$$
$41$ $$( 1 - 132 T^{2} + 10823 T^{4} - 527752 T^{6} + 18193463 T^{8} - 373000452 T^{10} + 4750104241 T^{12} )^{2}$$
$43$ $$( 1 - 98 T^{2} + 4567 T^{4} - 172988 T^{6} + 8444383 T^{8} - 335042498 T^{10} + 6321363049 T^{12} )^{2}$$
$47$ $$( 1 + 10 T^{2} + 4591 T^{4} + 70732 T^{6} + 10141519 T^{8} + 48796810 T^{10} + 10779215329 T^{12} )^{2}$$
$53$ $$( 1 - 184 T^{2} + 16171 T^{4} - 975856 T^{6} + 45424339 T^{8} - 1451848504 T^{10} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 + 146 T^{2} + 7799 T^{4} + 306396 T^{6} + 27148319 T^{8} + 1769134706 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 - 14 T + 187 T^{2} - 1700 T^{3} + 11407 T^{4} - 52094 T^{5} + 226981 T^{6} )^{4}$$
$67$ $$( 1 - 186 T^{2} + 13175 T^{4} - 680684 T^{6} + 59142575 T^{8} - 3748108506 T^{10} + 90458382169 T^{12} )^{2}$$
$71$ $$( 1 + 158 T^{2} + 5683 T^{4} - 107140 T^{6} + 28648003 T^{8} + 4015045598 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 + 47 T^{2} - 352 T^{3} + 3431 T^{4} + 389017 T^{6} )^{4}$$
$79$ $$( 1 - 162 T^{2} + 19359 T^{4} - 2006332 T^{6} + 120819519 T^{8} - 6309913122 T^{10} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 + 370 T^{2} + 65191 T^{4} + 6834652 T^{6} + 449100799 T^{8} + 17559578770 T^{10} + 326940373369 T^{12} )^{2}$$
$89$ $$( 1 - 308 T^{2} + 49895 T^{4} - 5241384 T^{6} + 395218295 T^{8} - 19324610228 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 + 135 T^{2} + 128 T^{3} + 13095 T^{4} + 912673 T^{6} )^{4}$$