# Properties

 Label 22.7.b.a Level $22$ Weight $7$ Character orbit 22.b Analytic conductor $5.061$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$22 = 2 \cdot 11$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 22.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.06118983964$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2x^{5} - 1781x^{4} + 14500x^{3} + 786532x^{2} - 11444432x + 42080676$$ x^6 - 2*x^5 - 1781*x^4 + 14500*x^3 + 786532*x^2 - 11444432*x + 42080676 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_1 - 9) q^{3} - 32 q^{4} + (3 \beta_{3} - 2 \beta_1 + 61) q^{5} + ( - \beta_{5} - \beta_{4} - 9 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - 12 \beta_{2}) q^{7} - 32 \beta_{2} q^{8} + (7 \beta_{3} - 30 \beta_1 - 50) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b1 - 9) * q^3 - 32 * q^4 + (3*b3 - 2*b1 + 61) * q^5 + (-b5 - b4 - 9*b2) * q^6 + (-2*b5 - 3*b4 - 12*b2) * q^7 - 32*b2 * q^8 + (7*b3 - 30*b1 - 50) * q^9 $$q + \beta_{2} q^{2} + (\beta_1 - 9) q^{3} - 32 q^{4} + (3 \beta_{3} - 2 \beta_1 + 61) q^{5} + ( - \beta_{5} - \beta_{4} - 9 \beta_{2}) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - 12 \beta_{2}) q^{7} - 32 \beta_{2} q^{8} + (7 \beta_{3} - 30 \beta_1 - 50) q^{9} + (11 \beta_{5} - \beta_{4} + 67 \beta_{2}) q^{10} + (11 \beta_{5} + 11 \beta_{3} + 33 \beta_1 - 209) q^{11} + ( - 32 \beta_1 + 288) q^{12} + ( - 24 \beta_{5} - 15 \beta_{4}) q^{13} + ( - 8 \beta_{3} - 88 \beta_1 + 400) q^{14} + ( - 2 \beta_{3} + 163 \beta_1 - 1139) q^{15} + 1024 q^{16} + ( - 26 \beta_{5} + 33 \beta_{4} - 168 \beta_{2}) q^{17} + (51 \beta_{5} + 23 \beta_{4} - 36 \beta_{2}) q^{18} + ( - 116 \beta_{5} + 9 \beta_{4} - 588 \beta_{2}) q^{19} + ( - 96 \beta_{3} + 64 \beta_1 - 1952) q^{20} + (126 \beta_{5} + 45 \beta_{4} + 1848 \beta_{2}) q^{21} + ( - 44 \beta_{4} - 88 \beta_{3} - 187 \beta_{2} + 88 \beta_1 + 176) q^{22} + ( - 68 \beta_{3} + 141 \beta_1 + 335) q^{23} + (32 \beta_{5} + 32 \beta_{4} + 288 \beta_{2}) q^{24} + (67 \beta_{3} + 62 \beta_1 + 9934) q^{25} + (72 \beta_{3} - 552 \beta_1 - 144) q^{26} + ( - 182 \beta_{3} - 9 \beta_1 - 9515) q^{27} + (64 \beta_{5} + 96 \beta_{4} + 384 \beta_{2}) q^{28} + (62 \beta_{5} - 30 \beta_{4} - 2616 \beta_{2}) q^{29} + ( - 169 \beta_{5} - 161 \beta_{4} - 1143 \beta_{2}) q^{30} + (416 \beta_{3} - 143 \beta_1 - 13169) q^{31} + 1024 \beta_{2} q^{32} + ( - 132 \beta_{5} - 99 \beta_{4} + 275 \beta_{3} - 1056 \beta_{2} + \cdots + 23837) q^{33}+ \cdots + ( - 2409 \beta_{5} + 2739 \beta_{4} - 11693 \beta_{3} + \cdots - 414458) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b1 - 9) * q^3 - 32 * q^4 + (3*b3 - 2*b1 + 61) * q^5 + (-b5 - b4 - 9*b2) * q^6 + (-2*b5 - 3*b4 - 12*b2) * q^7 - 32*b2 * q^8 + (7*b3 - 30*b1 - 50) * q^9 + (11*b5 - b4 + 67*b2) * q^10 + (11*b5 + 11*b3 + 33*b1 - 209) * q^11 + (-32*b1 + 288) * q^12 + (-24*b5 - 15*b4) * q^13 + (-8*b3 - 88*b1 + 400) * q^14 + (-2*b3 + 163*b1 - 1139) * q^15 + 1024 * q^16 + (-26*b5 + 33*b4 - 168*b2) * q^17 + (51*b5 + 23*b4 - 36*b2) * q^18 + (-116*b5 + 9*b4 - 588*b2) * q^19 + (-96*b3 + 64*b1 - 1952) * q^20 + (126*b5 + 45*b4 + 1848*b2) * q^21 + (-44*b4 - 88*b3 - 187*b2 + 88*b1 + 176) * q^22 + (-68*b3 + 141*b1 + 335) * q^23 + (32*b5 + 32*b4 + 288*b2) * q^24 + (67*b3 + 62*b1 + 9934) * q^25 + (72*b3 - 552*b1 - 144) * q^26 + (-182*b3 - 9*b1 - 9515) * q^27 + (64*b5 + 96*b4 + 384*b2) * q^28 + (62*b5 - 30*b4 - 2616*b2) * q^29 + (-169*b5 - 161*b4 - 1143*b2) * q^30 + (416*b3 - 143*b1 - 13169) * q^31 + 1024*b2 * q^32 + (-132*b5 - 99*b4 + 275*b3 - 1056*b2 - 682*b1 + 23837) * q^33 + (472*b3 + 584*b1 + 4432) * q^34 + (-386*b5 - 489*b4 - 612*b2) * q^35 + (-224*b3 + 960*b1 + 1600) * q^36 + (-313*b3 - 302*b1 - 34115) * q^37 + (1000*b3 - 712*b1 + 16816) * q^38 + (738*b5 + 291*b4 + 10044*b2) * q^39 + (-352*b5 + 32*b4 - 2144*b2) * q^40 + (950*b5 - 33*b4 - 1752*b2) * q^41 + (-648*b3 + 2088*b1 - 57840) * q^42 + (-266*b5 + 480*b4 - 5712*b2) * q^43 + (-352*b5 - 352*b3 - 1056*b1 + 6688) * q^44 + (-1054*b3 - 3144*b1 + 62852) * q^45 + (-345*b5 - 73*b4 + 199*b2) * q^46 + (430*b3 - 1206*b1 + 82502) * q^47 + (1024*b1 - 9216) * q^48 + (-2028*b3 + 4092*b1 - 45815) * q^49 + (139*b5 - 129*b4 + 10068*b2) * q^50 + (-510*b5 + 237*b4 - 13020*b2) * q^51 + (768*b5 + 480*b4) * q^52 + (76*b3 - 5724*b1 - 86734) * q^53 + (-537*b5 + 191*b4 - 9879*b2) * q^54 + (-484*b5 + 99*b4 - 44*b3 + 19932*b2 + 6941*b1 + 43527) * q^55 + (256*b3 + 2816*b1 - 12800) * q^56 + (1710*b5 + 1587*b4 + 11784*b2) * q^57 + (-736*b3 - 224*b1 + 85184) * q^58 + (1796*b3 + 6557*b1 - 141681) * q^59 + (64*b3 - 5216*b1 + 36448) * q^60 + (-1070*b5 - 624*b4 + 34296*b2) * q^61 + (1391*b5 - 273*b4 - 12337*b2) * q^62 + (-3252*b5 - 1020*b4 - 43200*b2) * q^63 - 32768 * q^64 + (-654*b5 - 2631*b4 - 24408*b2) * q^65 + (1507*b5 + 407*b4 + 264*b3 + 24387*b2 - 3432*b1 + 33264) * q^66 + (2272*b3 - 1303*b1 + 89247) * q^67 + (832*b5 - 1056*b4 + 5376*b2) * q^68 + (715*b3 - 3986*b1 + 67567) * q^69 + (-824*b3 - 14824*b1 + 21232) * q^70 + (6236*b3 - 4123*b1 - 77433) * q^71 + (-1632*b5 - 736*b4 + 1152*b2) * q^72 + (1714*b5 - 327*b4 - 81096*b2) * q^73 + (-637*b5 + 615*b4 - 34741*b2) * q^74 + (702*b3 + 9972*b1 - 38796) * q^75 + (3712*b5 - 288*b4 + 18816*b2) * q^76 + (3806*b5 + 231*b4 + 396*b3 + 73128*b2 - 9372*b1 + 44616) * q^77 + (-3576*b3 + 12888*b1 - 314256) * q^78 + (-1792*b5 + 1596*b4 + 81504*b2) * q^79 + (3072*b3 - 2048*b1 + 62464) * q^80 + (-5894*b3 + 8904*b1 + 79939) * q^81 + (-7864*b3 + 6808*b1 + 71792) * q^82 + (-8186*b5 - 3510*b4 + 60552*b2) * q^83 + (-4032*b5 - 1440*b4 - 59136*b2) * q^84 + (3058*b5 + 5247*b4 - 100344*b2) * q^85 + (5968*b3 + 9392*b1 + 170848) * q^86 + (2772*b5 + 2208*b4 + 33072*b2) * q^87 + (1408*b4 + 2816*b3 + 5984*b2 - 2816*b1 - 5632) * q^88 + (3379*b3 - 11490*b1 + 399029) * q^89 + (-18*b5 + 4198*b4 + 60744*b2) * q^90 + (-11124*b3 + 27108*b1 - 850104) * q^91 + (2176*b3 - 4512*b1 - 10720) * q^92 + (663*b3 - 1846*b1 + 117039) * q^93 + (2496*b5 + 776*b4 + 83362*b2) * q^94 + (-338*b5 + 993*b4 - 261036*b2) * q^95 + (-1024*b5 - 1024*b4 - 9216*b2) * q^96 + (-17521*b3 - 28718*b1 + 240685) * q^97 + (-10176*b5 - 2064*b4 - 49871*b2) * q^98 + (-2409*b5 + 2739*b4 - 11693*b3 + 73260*b2 + 19602*b1 - 414458) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 52 q^{3} - 192 q^{4} + 368 q^{5} - 346 q^{9}+O(q^{10})$$ 6 * q - 52 * q^3 - 192 * q^4 + 368 * q^5 - 346 * q^9 $$6 q - 52 q^{3} - 192 q^{4} + 368 q^{5} - 346 q^{9} - 1166 q^{11} + 1664 q^{12} + 2208 q^{14} - 6512 q^{15} + 6144 q^{16} - 11776 q^{20} + 1056 q^{22} + 2156 q^{23} + 59862 q^{25} - 1824 q^{26} - 57472 q^{27} - 78468 q^{31} + 142208 q^{33} + 28704 q^{34} + 11072 q^{36} - 205920 q^{37} + 101472 q^{38} - 344160 q^{42} + 37312 q^{44} + 368716 q^{45} + 493460 q^{47} - 53248 q^{48} - 270762 q^{49} - 531700 q^{53} + 274956 q^{55} - 70656 q^{56} + 509184 q^{58} - 833380 q^{59} + 208384 q^{60} - 196608 q^{64} + 193248 q^{66} + 537420 q^{67} + 398860 q^{69} + 96096 q^{70} - 460372 q^{71} - 211428 q^{75} + 249744 q^{77} - 1866912 q^{78} + 376832 q^{80} + 485654 q^{81} + 428640 q^{82} + 1055808 q^{86} - 33792 q^{88} + 2377952 q^{89} - 5068656 q^{91} - 68992 q^{92} + 699868 q^{93} + 1351632 q^{97} - 2470930 q^{99}+O(q^{100})$$ 6 * q - 52 * q^3 - 192 * q^4 + 368 * q^5 - 346 * q^9 - 1166 * q^11 + 1664 * q^12 + 2208 * q^14 - 6512 * q^15 + 6144 * q^16 - 11776 * q^20 + 1056 * q^22 + 2156 * q^23 + 59862 * q^25 - 1824 * q^26 - 57472 * q^27 - 78468 * q^31 + 142208 * q^33 + 28704 * q^34 + 11072 * q^36 - 205920 * q^37 + 101472 * q^38 - 344160 * q^42 + 37312 * q^44 + 368716 * q^45 + 493460 * q^47 - 53248 * q^48 - 270762 * q^49 - 531700 * q^53 + 274956 * q^55 - 70656 * q^56 + 509184 * q^58 - 833380 * q^59 + 208384 * q^60 - 196608 * q^64 + 193248 * q^66 + 537420 * q^67 + 398860 * q^69 + 96096 * q^70 - 460372 * q^71 - 211428 * q^75 + 249744 * q^77 - 1866912 * q^78 + 376832 * q^80 + 485654 * q^81 + 428640 * q^82 + 1055808 * q^86 - 33792 * q^88 + 2377952 * q^89 - 5068656 * q^91 - 68992 * q^92 + 699868 * q^93 + 1351632 * q^97 - 2470930 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 1781x^{4} + 14500x^{3} + 786532x^{2} - 11444432x + 42080676$$ :

 $$\beta_{1}$$ $$=$$ $$( -2707\nu^{5} - 22767\nu^{4} + 4050251\nu^{3} + 3625173\nu^{2} - 657163750\nu + 10557684654 ) / 963515190$$ (-2707*v^5 - 22767*v^4 + 4050251*v^3 + 3625173*v^2 - 657163750*v + 10557684654) / 963515190 $$\beta_{2}$$ $$=$$ $$( 5414\nu^{5} + 45534\nu^{4} - 8100502\nu^{3} - 7250346\nu^{2} + 3241357880\nu - 21115369308 ) / 481757595$$ (5414*v^5 + 45534*v^4 - 8100502*v^3 - 7250346*v^2 + 3241357880*v - 21115369308) / 481757595 $$\beta_{3}$$ $$=$$ $$( - 44423 \nu^{5} - 907518 \nu^{4} + 67178179 \nu^{3} + 1011971817 \nu^{2} - 24365375270 \nu - 111796059144 ) / 3372303165$$ (-44423*v^5 - 907518*v^4 + 67178179*v^3 + 1011971817*v^2 - 24365375270*v - 111796059144) / 3372303165 $$\beta_{4}$$ $$=$$ $$( - 95153 \nu^{5} - 18774978 \nu^{4} - 154836221 \nu^{3} + 16296294372 \nu^{2} + 108700698730 \nu - 1601025981984 ) / 6744606330$$ (-95153*v^5 - 18774978*v^4 - 154836221*v^3 + 16296294372*v^2 + 108700698730*v - 1601025981984) / 6744606330 $$\beta_{5}$$ $$=$$ $$( - 46261 \nu^{5} - 187378 \nu^{4} + 90358871 \nu^{3} - 137288348 \nu^{2} - 45368513086 \nu + 315774403296 ) / 449640422$$ (-46261*v^5 - 187378*v^4 + 90358871*v^3 - 137288348*v^2 - 45368513086*v + 315774403296) / 449640422
 $$\nu$$ $$=$$ $$( \beta_{2} + 4\beta_1 ) / 4$$ (b2 + 4*b1) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{5} - \beta_{4} + 14\beta_{3} - 24\beta _1 + 1192 ) / 2$$ (-b5 - b4 + 14*b3 - 24*b1 + 1192) / 2 $$\nu^{3}$$ $$=$$ $$( 99\beta_{5} + 15\beta_{4} + 28\beta_{3} + 1834\beta_{2} + 3504\beta _1 - 23048 ) / 4$$ (99*b5 + 15*b4 + 28*b3 + 1834*b2 + 3504*b1 - 23048) / 4 $$\nu^{4}$$ $$=$$ $$-859\beta_{5} - 887\beta_{4} + 6181\beta_{3} - 5748\beta_{2} - 16062\beta _1 + 521678$$ -859*b5 - 887*b4 + 6181*b3 - 5748*b2 - 16062*b1 + 521678 $$\nu^{5}$$ $$=$$ $$( 174345\beta_{5} + 49605\beta_{4} - 128548\beta_{3} + 2694664\beta_{2} + 3324008\beta _1 - 33241672 ) / 4$$ (174345*b5 + 49605*b4 - 128548*b3 + 2694664*b2 + 3324008*b1 - 33241672) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/22\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
21.1
 −32.5131 − 1.41421i 7.52788 − 1.41421i 25.9852 − 1.41421i −32.5131 + 1.41421i 7.52788 + 1.41421i 25.9852 + 1.41421i
5.65685i −41.5131 −32.0000 155.573 234.833i 562.567i 181.019i 994.336 880.056i
21.2 5.65685i −1.47212 −32.0000 −147.340 8.32754i 44.7193i 181.019i −726.833 833.480i
21.3 5.65685i 16.9852 −32.0000 175.767 96.0828i 412.125i 181.019i −440.503 994.286i
21.4 5.65685i −41.5131 −32.0000 155.573 234.833i 562.567i 181.019i 994.336 880.056i
21.5 5.65685i −1.47212 −32.0000 −147.340 8.32754i 44.7193i 181.019i −726.833 833.480i
21.6 5.65685i 16.9852 −32.0000 175.767 96.0828i 412.125i 181.019i −440.503 994.286i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 21.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 22.7.b.a 6
3.b odd 2 1 198.7.d.a 6
4.b odd 2 1 176.7.h.e 6
11.b odd 2 1 inner 22.7.b.a 6
33.d even 2 1 198.7.d.a 6
44.c even 2 1 176.7.h.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.7.b.a 6 1.a even 1 1 trivial
22.7.b.a 6 11.b odd 2 1 inner
176.7.h.e 6 4.b odd 2 1
176.7.h.e 6 44.c even 2 1
198.7.d.a 6 3.b odd 2 1
198.7.d.a 6 33.d even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(22, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 32)^{3}$$
$3$ $$(T^{3} + 26 T^{2} - 669 T - 1038)^{2}$$
$5$ $$(T^{3} - 184 T^{2} - 21475 T + 4028950)^{2}$$
$7$ $$T^{6} + \cdots + 107496740487168$$
$11$ $$T^{6} + 1166 T^{5} + \cdots + 55\!\cdots\!81$$
$13$ $$T^{6} + 16642440 T^{4} + \cdots + 91\!\cdots\!32$$
$17$ $$T^{6} + 83290056 T^{4} + \cdots + 29\!\cdots\!72$$
$19$ $$T^{6} + 256000200 T^{4} + \cdots + 61\!\cdots\!88$$
$23$ $$(T^{3} - 1078 T^{2} + \cdots + 6239437498)^{2}$$
$29$ $$T^{6} + 808758528 T^{4} + \cdots + 56\!\cdots\!92$$
$31$ $$(T^{3} + 39234 T^{2} + \cdots - 1132011557222)^{2}$$
$37$ $$(T^{3} + 102960 T^{2} + \cdots + 26364256495174)^{2}$$
$41$ $$T^{6} + 15110795400 T^{4} + \cdots + 22\!\cdots\!08$$
$43$ $$T^{6} + 18236828832 T^{4} + \cdots + 19\!\cdots\!08$$
$47$ $$(T^{3} - 246730 T^{2} + \cdots - 432230445439256)^{2}$$
$53$ $$(T^{3} + 265850 T^{2} + \cdots - 31\!\cdots\!92)^{2}$$
$59$ $$(T^{3} + 416690 T^{2} + \cdots - 90\!\cdots\!90)^{2}$$
$61$ $$T^{6} + 146302417824 T^{4} + \cdots + 14\!\cdots\!00$$
$67$ $$(T^{3} - 268710 T^{2} + \cdots + 18\!\cdots\!74)^{2}$$
$71$ $$(T^{3} + 230186 T^{2} + \cdots + 97\!\cdots\!82)^{2}$$
$73$ $$T^{6} + 704846656392 T^{4} + \cdots + 76\!\cdots\!00$$
$79$ $$T^{6} + 882501800064 T^{4} + \cdots + 15\!\cdots\!68$$
$83$ $$T^{6} + 1744310743296 T^{4} + \cdots + 14\!\cdots\!72$$
$89$ $$(T^{3} - 1188976 T^{2} + \cdots - 22\!\cdots\!14)^{2}$$
$97$ $$(T^{3} - 675816 T^{2} + \cdots + 17\!\cdots\!14)^{2}$$