# Properties

 Label 624.4.q.i Level $624$ Weight $4$ Character orbit 624.q Analytic conductor $36.817$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 624.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.8171918436$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084$$ x^8 - 2*x^7 + 29*x^6 + 2*x^5 + 595*x^4 - 288*x^3 + 2526*x^2 + 1872*x + 6084 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{8}\cdot 3$$ Twist minimal: no (minimal twist has level 39) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 \beta_{2} q^{3} + (\beta_{4} - 2) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{7} + (9 \beta_{2} - 9) q^{9}+O(q^{10})$$ q - 3*b2 * q^3 + (b4 - 2) * q^5 + (-b3 + 3*b2 - 3) * q^7 + (9*b2 - 9) * q^9 $$q - 3 \beta_{2} q^{3} + (\beta_{4} - 2) q^{5} + ( - \beta_{3} + 3 \beta_{2} - 3) q^{7} + (9 \beta_{2} - 9) q^{9} + (\beta_{7} + 10 \beta_{2} - \beta_1) q^{11} + ( - \beta_{7} + \beta_{5} + 2 \beta_{4} - 23 \beta_{2} - 2 \beta_1 + 4) q^{13} + ( - 3 \beta_{6} - 3 \beta_{4} + 6 \beta_{2}) q^{15} + ( - 5 \beta_{6} - 5 \beta_{5} - \beta_{3} + 29 \beta_{2} - 29) q^{17} + ( - 4 \beta_{6} + 3 \beta_{5} - \beta_{3} - 31 \beta_{2} + 31) q^{19} + ( - 3 \beta_{7} + 3 \beta_{3} + 3 \beta_{2} + 9) q^{21} + (3 \beta_{7} + \beta_{6} + \beta_{4} + 23 \beta_{2} + 2 \beta_1) q^{23} + (2 \beta_{7} + 3 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 8) q^{25} + 27 q^{27} + (6 \beta_{7} + 5 \beta_{6} + 5 \beta_{4} - 56 \beta_{2} + 4 \beta_1) q^{29} + ( - 15 \beta_{5} + 8 \beta_{4} + 15 \beta_1 - 18) q^{31} + (3 \beta_{5} - 3 \beta_{3} - 33 \beta_{2} + 33) q^{33} + ( - 13 \beta_{6} - 12 \beta_{5} - 5 \beta_{3} - 12 \beta_{2} + 12) q^{35} + (\beta_{7} + 4 \beta_{6} + 4 \beta_{4} - 31 \beta_{2} + 6 \beta_1) q^{37} + ( - 6 \beta_{6} + 3 \beta_{5} - 6 \beta_{4} + 3 \beta_{3} + 60 \beta_{2} + \cdots - 72) q^{39}+ \cdots + ( - 9 \beta_{7} - 9 \beta_{5} + 9 \beta_{3} + 9 \beta_{2} + 9 \beta_1 - 99) q^{99}+O(q^{100})$$ q - 3*b2 * q^3 + (b4 - 2) * q^5 + (-b3 + 3*b2 - 3) * q^7 + (9*b2 - 9) * q^9 + (b7 + 10*b2 - b1) * q^11 + (-b7 + b5 + 2*b4 - 23*b2 - 2*b1 + 4) * q^13 + (-3*b6 - 3*b4 + 6*b2) * q^15 + (-5*b6 - 5*b5 - b3 + 29*b2 - 29) * q^17 + (-4*b6 + 3*b5 - b3 - 31*b2 + 31) * q^19 + (-3*b7 + 3*b3 + 3*b2 + 9) * q^21 + (3*b7 + b6 + b4 + 23*b2 + 2*b1) * q^23 + (2*b7 + 3*b5 - 8*b4 - 2*b3 - 2*b2 - 3*b1 - 8) * q^25 + 27 * q^27 + (6*b7 + 5*b6 + 5*b4 - 56*b2 + 4*b1) * q^29 + (-15*b5 + 8*b4 + 15*b1 - 18) * q^31 + (3*b5 - 3*b3 - 33*b2 + 33) * q^33 + (-13*b6 - 12*b5 - 5*b3 - 12*b2 + 12) * q^35 + (b7 + 4*b6 + 4*b4 - 31*b2 + 6*b1) * q^37 + (-6*b6 + 3*b5 - 6*b4 + 3*b3 + 60*b2 + 3*b1 - 72) * q^39 + (-3*b7 + 10*b6 + 10*b4 + 255*b2 + 10*b1) * q^41 + (4*b6 + 18*b5 + b3 - 123*b2 + 123) * q^43 + (9*b6 - 18*b2 + 18) * q^45 + (-3*b7 - 9*b5 - 38*b4 + 3*b3 + 3*b2 + 9*b1 + 37) * q^47 + (2*b7 - 32*b6 - 32*b4 - 251*b2 - 3*b1) * q^49 + (-3*b7 + 15*b5 - 15*b4 + 3*b3 + 3*b2 - 15*b1 + 87) * q^51 + (2*b7 + 3*b5 - 26*b4 - 2*b3 - 2*b2 - 3*b1 + 81) * q^53 + (3*b7 + 32*b6 + 32*b4 + 26*b2 + 15*b1) * q^55 + (-3*b7 - 9*b5 - 12*b4 + 3*b3 + 3*b2 + 9*b1 - 93) * q^57 + (17*b6 - 7*b5 - 14*b3 - 89*b2 + 89) * q^59 + (16*b6 - 15*b5 - 3*b3 - 234*b2 + 234) * q^61 + (9*b7 - 36*b2) * q^63 + (-5*b7 - 35*b6 + 3*b5 - 38*b4 - 2*b3 + 92*b2 - 12*b1 + 263) * q^65 + (-b7 - 22*b6 - 22*b4 - 254*b2 - 36*b1) * q^67 + (-3*b6 - 6*b5 - 9*b3 - 78*b2 + 78) * q^69 + (15*b6 - 34*b5 - 11*b3 - 262*b2 + 262) * q^71 + (-2*b7 - 42*b5 + 66*b4 + 2*b3 + 2*b2 + 42*b1 + 183) * q^73 + (-6*b7 + 24*b6 + 24*b4 + 30*b2 + 9*b1) * q^75 + (16*b7 - 28*b5 + 30*b4 - 16*b3 - 16*b2 + 28*b1 + 606) * q^77 + (2*b7 + 3*b5 - 10*b4 - 2*b3 - 2*b2 - 3*b1 + 194) * q^79 - 81*b2 * q^81 + (-5*b7 + 36*b5 - 35*b4 + 5*b3 + 5*b2 - 36*b1 + 134) * q^83 + (68*b6 + 18*b5 - 25*b3 - 404*b2 + 404) * q^85 + (-15*b6 - 12*b5 - 18*b3 + 150*b2 - 150) * q^87 + (28*b7 - 67*b6 - 67*b4 - 377*b2 - 17*b1) * q^89 + (-21*b7 - 24*b6 - 71*b5 - 34*b4 - b3 - 35*b2 + 22*b1 - 399) * q^91 + (-24*b6 - 24*b4 + 54*b2 - 45*b1) * q^93 + (-38*b6 - 9*b5 - 7*b3 - 531*b2 + 531) * q^95 + (24*b6 - 39*b5 - 22*b3 + 538*b2 - 538) * q^97 + (-9*b7 - 9*b5 + 9*b3 + 9*b2 + 9*b1 - 99) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9}+O(q^{10})$$ 8 * q - 12 * q^3 - 12 * q^5 - 14 * q^7 - 36 * q^9 $$8 q - 12 q^{3} - 12 q^{5} - 14 q^{7} - 36 q^{9} + 40 q^{11} - 60 q^{13} + 18 q^{15} - 98 q^{17} + 124 q^{19} + 84 q^{21} + 104 q^{23} - 116 q^{25} + 216 q^{27} - 194 q^{29} - 52 q^{31} + 120 q^{33} + 88 q^{35} - 102 q^{37} - 342 q^{39} + 1054 q^{41} + 450 q^{43} + 54 q^{45} + 192 q^{47} - 1070 q^{49} + 588 q^{51} + 524 q^{53} + 204 q^{55} - 744 q^{57} + 308 q^{59} + 928 q^{61} - 126 q^{63} + 2346 q^{65} - 1134 q^{67} + 312 q^{69} + 1064 q^{71} + 1904 q^{73} + 174 q^{75} + 5016 q^{77} + 1492 q^{79} - 324 q^{81} + 808 q^{83} + 1394 q^{85} - 582 q^{87} - 1620 q^{89} - 3278 q^{91} + 78 q^{93} + 2204 q^{95} - 2166 q^{97} - 720 q^{99}+O(q^{100})$$ 8 * q - 12 * q^3 - 12 * q^5 - 14 * q^7 - 36 * q^9 + 40 * q^11 - 60 * q^13 + 18 * q^15 - 98 * q^17 + 124 * q^19 + 84 * q^21 + 104 * q^23 - 116 * q^25 + 216 * q^27 - 194 * q^29 - 52 * q^31 + 120 * q^33 + 88 * q^35 - 102 * q^37 - 342 * q^39 + 1054 * q^41 + 450 * q^43 + 54 * q^45 + 192 * q^47 - 1070 * q^49 + 588 * q^51 + 524 * q^53 + 204 * q^55 - 744 * q^57 + 308 * q^59 + 928 * q^61 - 126 * q^63 + 2346 * q^65 - 1134 * q^67 + 312 * q^69 + 1064 * q^71 + 1904 * q^73 + 174 * q^75 + 5016 * q^77 + 1492 * q^79 - 324 * q^81 + 808 * q^83 + 1394 * q^85 - 582 * q^87 - 1620 * q^89 - 3278 * q^91 + 78 * q^93 + 2204 * q^95 - 2166 * q^97 - 720 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2x^{7} + 29x^{6} + 2x^{5} + 595x^{4} - 288x^{3} + 2526x^{2} + 1872x + 6084$$ :

 $$\beta_{1}$$ $$=$$ $$( - 5924 \nu^{7} + 261773 \nu^{6} - 568842 \nu^{5} + 5976355 \nu^{4} + 5492514 \nu^{3} + 104173009 \nu^{2} + 54255204 \nu + 87160788 ) / 37839126$$ (-5924*v^7 + 261773*v^6 - 568842*v^5 + 5976355*v^4 + 5492514*v^3 + 104173009*v^2 + 54255204*v + 87160788) / 37839126 $$\beta_{2}$$ $$=$$ $$( 5984 \nu^{7} - 26943 \nu^{6} + 182186 \nu^{5} - 346833 \nu^{4} + 3020182 \nu^{3} - 11402021 \nu^{2} + 10968108 \nu + 5315076 ) / 37839126$$ (5984*v^7 - 26943*v^6 + 182186*v^5 - 346833*v^4 + 3020182*v^3 - 11402021*v^2 + 10968108*v + 5315076) / 37839126 $$\beta_{3}$$ $$=$$ $$( 4151 \nu^{7} - 166803 \nu^{6} + 1127906 \nu^{5} - 5302947 \nu^{4} + 18697822 \nu^{3} - 70589441 \nu^{2} + 343464684 \nu - 201355050 ) / 18919563$$ (4151*v^7 - 166803*v^6 + 1127906*v^5 - 5302947*v^4 + 18697822*v^3 - 70589441*v^2 + 343464684*v - 201355050) / 18919563 $$\beta_{4}$$ $$=$$ $$( 3550 \nu^{7} - 12579 \nu^{6} + 85058 \nu^{5} + 128084 \nu^{4} + 1410046 \nu^{3} + 983208 \nu^{2} + 1395576 \nu + 73106644 ) / 6306521$$ (3550*v^7 - 12579*v^6 + 85058*v^5 + 128084*v^4 + 1410046*v^3 + 983208*v^2 + 1395576*v + 73106644) / 6306521 $$\beta_{5}$$ $$=$$ $$( - 44524 \nu^{7} + 220899 \nu^{6} - 1493698 \nu^{5} + 4583667 \nu^{4} - 24761726 \nu^{3} + 93482353 \nu^{2} + 39080772 \nu + 266656650 ) / 37839126$$ (-44524*v^7 + 220899*v^6 - 1493698*v^5 + 4583667*v^4 - 24761726*v^3 + 93482353*v^2 + 39080772*v + 266656650) / 37839126 $$\beta_{6}$$ $$=$$ $$( 31238 \nu^{7} - 150864 \nu^{6} + 1020128 \nu^{5} - 2812083 \nu^{4} + 16911136 \nu^{3} - 63844208 \nu^{2} + 72590028 \nu - 182114400 ) / 18919563$$ (31238*v^7 - 150864*v^6 + 1020128*v^5 - 2812083*v^4 + 16911136*v^3 - 63844208*v^2 + 72590028*v - 182114400) / 18919563 $$\beta_{7}$$ $$=$$ $$( 108812 \nu^{7} - 222399 \nu^{6} + 3305704 \nu^{5} + 103215 \nu^{4} + 64570858 \nu^{3} - 22042613 \nu^{2} + 273536628 \nu + 201818916 ) / 12613042$$ (108812*v^7 - 222399*v^6 + 3305704*v^5 + 103215*v^4 + 64570858*v^3 - 22042613*v^2 + 273536628*v + 201818916) / 12613042
 $$\nu$$ $$=$$ $$( \beta_{6} + \beta_{5} - 3\beta_{2} + 3 ) / 4$$ (b6 + b5 - 3*b2 + 3) / 4 $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{4} - 14\beta_{2}$$ b6 + b4 - 14*b2 $$\nu^{3}$$ $$=$$ $$( -2\beta_{7} - 9\beta_{5} + 13\beta_{4} + 2\beta_{3} + 2\beta_{2} + 9\beta _1 - 55 ) / 2$$ (-2*b7 - 9*b5 + 13*b4 + 2*b3 + 2*b2 + 9*b1 - 55) / 2 $$\nu^{4}$$ $$=$$ $$-32\beta_{6} - 3\beta_{5} + 2\beta_{3} + 309\beta_{2} - 309$$ -32*b6 - 3*b5 + 2*b3 + 309*b2 - 309 $$\nu^{5}$$ $$=$$ $$29\beta_{7} - 183\beta_{6} - 183\beta_{4} + 882\beta_{2} - 99\beta_1$$ 29*b7 - 183*b6 - 183*b4 + 882*b2 - 99*b1 $$\nu^{6}$$ $$=$$ $$84\beta_{7} + 165\beta_{5} - 932\beta_{4} - 84\beta_{3} - 84\beta_{2} - 165\beta _1 + 7795$$ 84*b7 + 165*b5 - 932*b4 - 84*b3 - 84*b2 - 165*b1 + 7795 $$\nu^{7}$$ $$=$$ $$5164\beta_{6} + 2382\beta_{5} - 767\beta_{3} - 28804\beta_{2} + 28804$$ 5164*b6 + 2382*b5 - 767*b3 - 28804*b2 + 28804

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$$.

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 2.66520 − 4.61626i −2.11303 + 3.65987i 1.18088 − 2.04535i −0.733051 + 1.26968i 2.66520 + 4.61626i −2.11303 − 3.65987i 1.18088 + 2.04535i −0.733051 − 1.26968i
0 −1.50000 2.59808i 0 −16.4131 0 4.83984 8.38285i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 2.59808i 0 −5.85953 0 −12.0627 + 20.8932i 0 −4.50000 + 7.79423i 0
289.3 0 −1.50000 2.59808i 0 6.42208 0 −14.7469 + 25.5424i 0 −4.50000 + 7.79423i 0
289.4 0 −1.50000 2.59808i 0 9.85055 0 14.9698 25.9285i 0 −4.50000 + 7.79423i 0
529.1 0 −1.50000 + 2.59808i 0 −16.4131 0 4.83984 + 8.38285i 0 −4.50000 7.79423i 0
529.2 0 −1.50000 + 2.59808i 0 −5.85953 0 −12.0627 20.8932i 0 −4.50000 7.79423i 0
529.3 0 −1.50000 + 2.59808i 0 6.42208 0 −14.7469 25.5424i 0 −4.50000 7.79423i 0
529.4 0 −1.50000 + 2.59808i 0 9.85055 0 14.9698 + 25.9285i 0 −4.50000 7.79423i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 529.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.q.i 8
4.b odd 2 1 39.4.e.c 8
12.b even 2 1 117.4.g.e 8
13.c even 3 1 inner 624.4.q.i 8
52.i odd 6 1 507.4.a.i 4
52.j odd 6 1 39.4.e.c 8
52.j odd 6 1 507.4.a.m 4
52.l even 12 2 507.4.b.h 8
156.p even 6 1 117.4.g.e 8
156.p even 6 1 1521.4.a.v 4
156.r even 6 1 1521.4.a.bb 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.c 8 4.b odd 2 1
39.4.e.c 8 52.j odd 6 1
117.4.g.e 8 12.b even 2 1
117.4.g.e 8 156.p even 6 1
507.4.a.i 4 52.i odd 6 1
507.4.a.m 4 52.j odd 6 1
507.4.b.h 8 52.l even 12 2
624.4.q.i 8 1.a even 1 1 trivial
624.4.q.i 8 13.c even 3 1 inner
1521.4.a.v 4 156.p even 6 1
1521.4.a.bb 4 156.r even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(624, [\chi])$$:

 $$T_{5}^{4} + 6T_{5}^{3} - 203T_{5}^{2} - 156T_{5} + 6084$$ T5^4 + 6*T5^3 - 203*T5^2 - 156*T5 + 6084 $$T_{7}^{8} + 14 T_{7}^{7} + 1319 T_{7}^{6} + 9582 T_{7}^{5} + 1232045 T_{7}^{4} + 8434260 T_{7}^{3} + 391649180 T_{7}^{2} - 2608994224 T_{7} + 42523388944$$ T7^8 + 14*T7^7 + 1319*T7^6 + 9582*T7^5 + 1232045*T7^4 + 8434260*T7^3 + 391649180*T7^2 - 2608994224*T7 + 42523388944

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{2} + 3 T + 9)^{4}$$
$5$ $$(T^{4} + 6 T^{3} - 203 T^{2} - 156 T + 6084)^{2}$$
$7$ $$T^{8} + 14 T^{7} + \cdots + 42523388944$$
$11$ $$T^{8} - 40 T^{7} + \cdots + 751198464$$
$13$ $$T^{8} + 60 T^{7} + \cdots + 23298085122481$$
$17$ $$T^{8} + \cdots + 509493017090304$$
$19$ $$T^{8} - 124 T^{7} + \cdots + 2959721107456$$
$23$ $$T^{8} - 104 T^{7} + \cdots + 6612632822016$$
$29$ $$T^{8} + 194 T^{7} + \cdots + 75\!\cdots\!24$$
$31$ $$(T^{4} + 26 T^{3} - 80975 T^{2} + \cdots + 328187792)^{2}$$
$37$ $$T^{8} + \cdots + 738573457717264$$
$41$ $$T^{8} - 1054 T^{7} + \cdots + 10\!\cdots\!04$$
$43$ $$T^{8} - 450 T^{7} + \cdots + 55\!\cdots\!84$$
$47$ $$(T^{4} - 96 T^{3} - 434600 T^{2} + \cdots + 42871452048)^{2}$$
$53$ $$(T^{4} - 262 T^{3} - 111719 T^{2} + \cdots + 744728256)^{2}$$
$59$ $$T^{8} - 308 T^{7} + \cdots + 44\!\cdots\!56$$
$61$ $$T^{8} - 928 T^{7} + \cdots + 27\!\cdots\!21$$
$67$ $$T^{8} + 1134 T^{7} + \cdots + 10\!\cdots\!44$$
$71$ $$T^{8} - 1064 T^{7} + \cdots + 98\!\cdots\!16$$
$73$ $$(T^{4} - 952 T^{3} + \cdots - 120133390247)^{2}$$
$79$ $$(T^{4} - 746 T^{3} + 184337 T^{2} + \cdots + 680937616)^{2}$$
$83$ $$(T^{4} - 404 T^{3} + \cdots + 58964273856)^{2}$$
$89$ $$T^{8} + 1620 T^{7} + \cdots + 72\!\cdots\!96$$
$97$ $$T^{8} + 2166 T^{7} + \cdots + 19\!\cdots\!96$$