# Properties

 Label 31.2.d.a Level 31 Weight 2 Character orbit 31.d Analytic conductor 0.248 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$31$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 31.d (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.247536246266$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{10}^{2} ) q^{2} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{3} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{4} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + 3 \zeta_{10}^{2} q^{7} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{8} + 2 \zeta_{10}^{3} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{10}^{2} ) q^{2} + ( 1 - \zeta_{10} + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{3} + ( \zeta_{10} - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{4} + ( -1 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( -1 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{6} + 3 \zeta_{10}^{2} q^{7} + ( -2 + 2 \zeta_{10} + \zeta_{10}^{3} ) q^{8} + 2 \zeta_{10}^{3} q^{9} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{10} + ( -2 \zeta_{10} - 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{11} + ( 1 - \zeta_{10} + \zeta_{10}^{2} ) q^{12} + ( 3 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{13} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{14} + ( -1 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{15} + ( 3 - 3 \zeta_{10}^{3} ) q^{16} + ( -2 + 2 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{17} + ( 2 - 2 \zeta_{10}^{3} ) q^{18} -5 \zeta_{10} q^{19} + ( -2 \zeta_{10} + 3 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{20} + 3 \zeta_{10} q^{21} + ( -4 + 4 \zeta_{10} + 6 \zeta_{10}^{3} ) q^{22} + ( 4 - 4 \zeta_{10} - \zeta_{10}^{3} ) q^{23} + ( 2 \zeta_{10} - \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{24} + ( -3 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{25} -3 q^{26} + 5 \zeta_{10}^{2} q^{27} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{28} + ( -1 + 8 \zeta_{10} - \zeta_{10}^{2} ) q^{29} + ( \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{30} + ( -3 \zeta_{10} + 2 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{31} + ( -4 + \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{32} + ( -2 - 2 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{33} + ( 5 - 2 \zeta_{10} + 2 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{34} + ( 3 \zeta_{10} - 6 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{35} + ( -2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{36} + ( -3 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{37} + ( 5 \zeta_{10} + 5 \zeta_{10}^{3} ) q^{38} + ( 3 - 3 \zeta_{10} ) q^{39} + ( 3 - 3 \zeta_{10} + \zeta_{10}^{3} ) q^{40} + ( 4 - 4 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{41} + ( -3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{42} + ( -1 + 4 \zeta_{10} - \zeta_{10}^{2} ) q^{43} + ( 2 - 2 \zeta_{10}^{3} ) q^{44} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{3} ) q^{45} + ( -5 + 4 \zeta_{10} - 4 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{46} + ( 5 - 4 \zeta_{10} + 4 \zeta_{10}^{2} - 5 \zeta_{10}^{3} ) q^{47} + ( 3 - 3 \zeta_{10} - 3 \zeta_{10}^{3} ) q^{48} + ( -2 + 2 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{49} + ( 3 + 3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{50} + ( 2 \zeta_{10} + \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{51} + ( -3 + 6 \zeta_{10} - 3 \zeta_{10}^{2} ) q^{52} + ( 6 - 6 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{53} + ( 5 - 5 \zeta_{10} - 5 \zeta_{10}^{3} ) q^{54} + 2 \zeta_{10}^{2} q^{55} + ( -3 - 6 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{56} -5 q^{57} + ( -7 \zeta_{10} + \zeta_{10}^{2} - 7 \zeta_{10}^{3} ) q^{58} + ( -3 + 7 \zeta_{10} - 7 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{59} + ( -2 + 3 \zeta_{10} - 2 \zeta_{10}^{2} ) q^{60} + ( 6 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{61} + ( -4 + \zeta_{10} + 7 \zeta_{10}^{3} ) q^{62} -6 q^{63} + ( -2 - \zeta_{10} - 2 \zeta_{10}^{2} ) q^{64} + ( 3 - 9 \zeta_{10} + 9 \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{65} + ( 4 \zeta_{10} + 2 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{66} + ( -1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{67} + ( -2 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{68} + ( -4 \zeta_{10} + 3 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{69} + ( -3 + 3 \zeta_{10} ) q^{70} + ( -5 + 5 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{71} + ( -4 + 2 \zeta_{10} - 4 \zeta_{10}^{2} ) q^{72} + ( 9 \zeta_{10} - 3 \zeta_{10}^{2} + 9 \zeta_{10}^{3} ) q^{73} + ( 3 + 2 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{74} + ( -3 + 3 \zeta_{10}^{3} ) q^{75} + ( 5 - 5 \zeta_{10} ) q^{76} + ( 12 - 6 \zeta_{10} + 6 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{77} + ( -3 + 3 \zeta_{10} - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{78} + ( -3 \zeta_{10} + 3 \zeta_{10}^{2} ) q^{80} -\zeta_{10} q^{81} -4 \zeta_{10}^{2} q^{82} + ( -5 + \zeta_{10} - 5 \zeta_{10}^{2} ) q^{83} + ( -3 + 3 \zeta_{10} ) q^{84} + ( 1 - \zeta_{10} - \zeta_{10}^{3} ) q^{85} + ( -3 \zeta_{10} + \zeta_{10}^{2} - 3 \zeta_{10}^{3} ) q^{86} + ( 7 - \zeta_{10}^{2} + \zeta_{10}^{3} ) q^{87} + ( 8 + 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{88} + ( -\zeta_{10} - 7 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{89} + ( -2 \zeta_{10} + 2 \zeta_{10}^{2} ) q^{90} + ( 9 - 9 \zeta_{10} + 9 \zeta_{10}^{2} ) q^{91} + ( 4 - 3 \zeta_{10}^{2} + 3 \zeta_{10}^{3} ) q^{92} + ( -3 + 2 \zeta_{10} - 6 \zeta_{10}^{2} ) q^{93} + ( -6 - 5 \zeta_{10}^{2} + 5 \zeta_{10}^{3} ) q^{94} + ( -5 + 10 \zeta_{10} - 5 \zeta_{10}^{2} ) q^{95} + ( -4 + 5 \zeta_{10} - 5 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{96} + ( -6 \zeta_{10} - 9 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{97} + ( 2 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{98} + ( 8 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3q^{2} + q^{3} + 3q^{4} - 6q^{5} - 2q^{6} - 3q^{7} - 5q^{8} + 2q^{9} + O(q^{10})$$ $$4q - 3q^{2} + q^{3} + 3q^{4} - 6q^{5} - 2q^{6} - 3q^{7} - 5q^{8} + 2q^{9} + 2q^{10} - 2q^{11} + 2q^{12} + 6q^{13} + 6q^{14} + q^{15} + 9q^{16} - 3q^{17} + 6q^{18} - 5q^{19} - 7q^{20} + 3q^{21} - 6q^{22} + 11q^{23} + 5q^{24} - 6q^{25} - 12q^{26} - 5q^{27} - 6q^{28} + 5q^{29} - 2q^{30} - 11q^{31} - 18q^{32} - 8q^{33} + 11q^{34} + 12q^{35} + 4q^{36} - 8q^{37} + 10q^{38} + 9q^{39} + 10q^{40} + 8q^{41} - 6q^{42} + q^{43} + 6q^{44} - 8q^{45} - 7q^{46} + 7q^{47} + 6q^{48} - 2q^{49} + 12q^{50} + 3q^{51} - 3q^{52} + 21q^{53} + 10q^{54} - 2q^{55} - 20q^{57} - 15q^{58} + 5q^{59} - 3q^{60} + 8q^{61} - 8q^{62} - 24q^{63} - 7q^{64} - 9q^{65} + 6q^{66} - 8q^{67} - 6q^{68} - 11q^{69} - 9q^{70} - 7q^{71} - 10q^{72} + 21q^{73} + 11q^{74} - 9q^{75} + 15q^{76} + 24q^{77} - 3q^{78} - 6q^{80} - q^{81} + 4q^{82} - 14q^{83} - 9q^{84} + 2q^{85} - 7q^{86} + 30q^{87} + 20q^{88} + 5q^{89} - 4q^{90} + 18q^{91} + 22q^{92} - 4q^{93} - 14q^{94} - 5q^{95} - 2q^{96} - 3q^{97} + 4q^{98} + 24q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\zeta_{10}^{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.809017 + 0.587785i −0.309017 − 0.951057i −0.309017 + 0.951057i 0.809017 − 0.587785i
−1.30902 0.951057i 0.809017 0.587785i 0.190983 + 0.587785i −0.381966 −1.61803 0.927051 + 2.85317i −0.690983 + 2.12663i −0.618034 + 1.90211i 0.500000 + 0.363271i
4.1 −0.190983 0.587785i −0.309017 + 0.951057i 1.30902 0.951057i −2.61803 0.618034 −2.42705 + 1.76336i −1.80902 1.31433i 1.61803 + 1.17557i 0.500000 + 1.53884i
8.1 −0.190983 + 0.587785i −0.309017 0.951057i 1.30902 + 0.951057i −2.61803 0.618034 −2.42705 1.76336i −1.80902 + 1.31433i 1.61803 1.17557i 0.500000 1.53884i
16.1 −1.30902 + 0.951057i 0.809017 + 0.587785i 0.190983 0.587785i −0.381966 −1.61803 0.927051 2.85317i −0.690983 2.12663i −0.618034 1.90211i 0.500000 0.363271i
 $$n$$: 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
31.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.2.d.a 4
3.b odd 2 1 279.2.i.a 4
4.b odd 2 1 496.2.n.b 4
5.b even 2 1 775.2.k.c 4
5.c odd 4 2 775.2.bf.a 8
31.b odd 2 1 961.2.d.b 4
31.c even 3 2 961.2.g.b 8
31.d even 5 1 inner 31.2.d.a 4
31.d even 5 1 961.2.a.d 2
31.d even 5 2 961.2.d.f 4
31.e odd 6 2 961.2.g.c 8
31.f odd 10 1 961.2.a.e 2
31.f odd 10 1 961.2.d.b 4
31.f odd 10 2 961.2.d.e 4
31.g even 15 2 961.2.c.f 4
31.g even 15 2 961.2.g.b 8
31.g even 15 4 961.2.g.f 8
31.h odd 30 2 961.2.c.d 4
31.h odd 30 2 961.2.g.c 8
31.h odd 30 4 961.2.g.g 8
93.k even 10 1 8649.2.a.f 2
93.l odd 10 1 279.2.i.a 4
93.l odd 10 1 8649.2.a.g 2
124.l odd 10 1 496.2.n.b 4
155.n even 10 1 775.2.k.c 4
155.s odd 20 2 775.2.bf.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.d.a 4 1.a even 1 1 trivial
31.2.d.a 4 31.d even 5 1 inner
279.2.i.a 4 3.b odd 2 1
279.2.i.a 4 93.l odd 10 1
496.2.n.b 4 4.b odd 2 1
496.2.n.b 4 124.l odd 10 1
775.2.k.c 4 5.b even 2 1
775.2.k.c 4 155.n even 10 1
775.2.bf.a 8 5.c odd 4 2
775.2.bf.a 8 155.s odd 20 2
961.2.a.d 2 31.d even 5 1
961.2.a.e 2 31.f odd 10 1
961.2.c.d 4 31.h odd 30 2
961.2.c.f 4 31.g even 15 2
961.2.d.b 4 31.b odd 2 1
961.2.d.b 4 31.f odd 10 1
961.2.d.e 4 31.f odd 10 2
961.2.d.f 4 31.d even 5 2
961.2.g.b 8 31.c even 3 2
961.2.g.b 8 31.g even 15 2
961.2.g.c 8 31.e odd 6 2
961.2.g.c 8 31.h odd 30 2
961.2.g.f 8 31.g even 15 4
961.2.g.g 8 31.h odd 30 4
8649.2.a.f 2 93.k even 10 1
8649.2.a.g 2 93.l odd 10 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(31, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 2 T^{2} + T^{4} + 8 T^{6} + 24 T^{7} + 16 T^{8}$$
$3$ $$1 - T - 2 T^{2} + 5 T^{3} + T^{4} + 15 T^{5} - 18 T^{6} - 27 T^{7} + 81 T^{8}$$
$5$ $$( 1 + 3 T + 11 T^{2} + 15 T^{3} + 25 T^{4} )^{2}$$
$7$ $$1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 105 T^{5} + 98 T^{6} + 1029 T^{7} + 2401 T^{8}$$
$11$ $$1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8}$$
$13$ $$1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 1560 T^{5} + 3887 T^{6} - 13182 T^{7} + 28561 T^{8}$$
$17$ $$1 + 3 T + 2 T^{2} + 75 T^{3} + 511 T^{4} + 1275 T^{5} + 578 T^{6} + 14739 T^{7} + 83521 T^{8}$$
$19$ $$1 + 5 T + 6 T^{2} - 65 T^{3} - 439 T^{4} - 1235 T^{5} + 2166 T^{6} + 34295 T^{7} + 130321 T^{8}$$
$23$ $$1 - 11 T + 38 T^{2} - 125 T^{3} + 821 T^{4} - 2875 T^{5} + 20102 T^{6} - 133837 T^{7} + 279841 T^{8}$$
$29$ $$1 - 5 T + 31 T^{2} - 115 T^{3} + 96 T^{4} - 3335 T^{5} + 26071 T^{6} - 121945 T^{7} + 707281 T^{8}$$
$31$ $$1 + 11 T + 61 T^{2} + 341 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 4 T + 73 T^{2} + 148 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$1 - 8 T + 23 T^{2} - 356 T^{3} + 3905 T^{4} - 14596 T^{5} + 38663 T^{6} - 551368 T^{7} + 2825761 T^{8}$$
$43$ $$1 - T - 27 T^{2} + 235 T^{3} + 1196 T^{4} + 10105 T^{5} - 49923 T^{6} - 79507 T^{7} + 3418801 T^{8}$$
$47$ $$1 - 7 T - 23 T^{2} + 385 T^{3} - 1284 T^{4} + 18095 T^{5} - 50807 T^{6} - 726761 T^{7} + 4879681 T^{8}$$
$53$ $$1 - 21 T + 118 T^{2} + 555 T^{3} - 9989 T^{4} + 29415 T^{5} + 331462 T^{6} - 3126417 T^{7} + 7890481 T^{8}$$
$59$ $$1 - 5 T + 26 T^{2} - 515 T^{3} + 6161 T^{4} - 30385 T^{5} + 90506 T^{6} - 1026895 T^{7} + 12117361 T^{8}$$
$61$ $$( 1 - 4 T + 46 T^{2} - 244 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$1 + 7 T + 53 T^{2} + 799 T^{3} + 10580 T^{4} + 56729 T^{5} + 267173 T^{6} + 2505377 T^{7} + 25411681 T^{8}$$
$73$ $$1 - 21 T + 233 T^{2} - 2595 T^{3} + 26956 T^{4} - 189435 T^{5} + 1241657 T^{6} - 8169357 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8}$$
$83$ $$1 + 14 T + 13 T^{2} + 70 T^{3} + 6651 T^{4} + 5810 T^{5} + 89557 T^{6} + 8005018 T^{7} + 47458321 T^{8}$$
$89$ $$1 - 5 T - 29 T^{2} + 785 T^{3} - 624 T^{4} + 69865 T^{5} - 229709 T^{6} - 3524845 T^{7} + 62742241 T^{8}$$
$97$ $$1 + 3 T + 182 T^{2} - 345 T^{3} + 16591 T^{4} - 33465 T^{5} + 1712438 T^{6} + 2738019 T^{7} + 88529281 T^{8}$$