# Properties

 Label 10.11.c.a Level 10 Weight 11 Character orbit 10.c Analytic conductor 6.354 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.35357252674$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 16 + 16 i ) q^{2} + ( -183 + 183 i ) q^{3} + 512 i q^{4} + ( -1875 - 2500 i ) q^{5} -5856 q^{6} + ( -8407 - 8407 i ) q^{7} + ( -8192 + 8192 i ) q^{8} -7929 i q^{9} +O(q^{10})$$ $$q + ( 16 + 16 i ) q^{2} + ( -183 + 183 i ) q^{3} + 512 i q^{4} + ( -1875 - 2500 i ) q^{5} -5856 q^{6} + ( -8407 - 8407 i ) q^{7} + ( -8192 + 8192 i ) q^{8} -7929 i q^{9} + ( 10000 - 70000 i ) q^{10} -173398 q^{11} + ( -93696 - 93696 i ) q^{12} + ( -232623 + 232623 i ) q^{13} -269024 i q^{14} + ( 800625 + 114375 i ) q^{15} -262144 q^{16} + ( 1880033 + 1880033 i ) q^{17} + ( 126864 - 126864 i ) q^{18} -1101560 i q^{19} + ( 1280000 - 960000 i ) q^{20} + 3076962 q^{21} + ( -2774368 - 2774368 i ) q^{22} + ( -5228263 + 5228263 i ) q^{23} -2998272 i q^{24} + ( -2734375 + 9375000 i ) q^{25} -7443936 q^{26} + ( -9354960 - 9354960 i ) q^{27} + ( 4304384 - 4304384 i ) q^{28} -24790840 i q^{29} + ( 10980000 + 14640000 i ) q^{30} -10065998 q^{31} + ( -4194304 - 4194304 i ) q^{32} + ( 31731834 - 31731834 i ) q^{33} + 60161056 i q^{34} + ( -5254375 + 36780625 i ) q^{35} + 4059648 q^{36} + ( 56387913 + 56387913 i ) q^{37} + ( 17624960 - 17624960 i ) q^{38} -85140018 i q^{39} + ( 35840000 + 5120000 i ) q^{40} -153003598 q^{41} + ( 49231392 + 49231392 i ) q^{42} + ( 59372457 - 59372457 i ) q^{43} -88779776 i q^{44} + ( -19822500 + 14866875 i ) q^{45} -167304416 q^{46} + ( 172339353 + 172339353 i ) q^{47} + ( 47972352 - 47972352 i ) q^{48} -141119951 i q^{49} + ( -193750000 + 106250000 i ) q^{50} -688092078 q^{51} + ( -119102976 - 119102976 i ) q^{52} + ( 196385617 - 196385617 i ) q^{53} -299358720 i q^{54} + ( 325121250 + 433495000 i ) q^{55} + 137740288 q^{56} + ( 201585480 + 201585480 i ) q^{57} + ( 396653440 - 396653440 i ) q^{58} + 694068920 i q^{59} + ( -58560000 + 409920000 i ) q^{60} + 906185802 q^{61} + ( -161055968 - 161055968 i ) q^{62} + ( -66659103 + 66659103 i ) q^{63} -134217728 i q^{64} + ( 1017725625 + 145389375 i ) q^{65} + 1015418688 q^{66} + ( -962073967 - 962073967 i ) q^{67} + ( -962576896 + 962576896 i ) q^{68} -1913544258 i q^{69} + ( -672560000 + 504420000 i ) q^{70} -3120877598 q^{71} + ( 64954368 + 64954368 i ) q^{72} + ( -636339263 + 636339263 i ) q^{73} + 1804413216 i q^{74} + ( -1215234375 - 2216015625 i ) q^{75} + 563998720 q^{76} + ( 1457756986 + 1457756986 i ) q^{77} + ( 1362240288 - 1362240288 i ) q^{78} -1967996640 i q^{79} + ( 491520000 + 655360000 i ) q^{80} + 3892114881 q^{81} + ( -2448057568 - 2448057568 i ) q^{82} + ( -5183822103 + 5183822103 i ) q^{83} + 1575404544 i q^{84} + ( 1175020625 - 8225144375 i ) q^{85} + 1899918624 q^{86} + ( 4536723720 + 4536723720 i ) q^{87} + ( 1420476416 - 1420476416 i ) q^{88} + 7771383280 i q^{89} + ( -555030000 - 79290000 i ) q^{90} + 3911323122 q^{91} + ( -2676870656 - 2676870656 i ) q^{92} + ( 1842077634 - 1842077634 i ) q^{93} + 5514859296 i q^{94} + ( -2753900000 + 2065425000 i ) q^{95} + 1535115264 q^{96} + ( -640361247 - 640361247 i ) q^{97} + ( 2257919216 - 2257919216 i ) q^{98} + 1374872742 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 32q^{2} - 366q^{3} - 3750q^{5} - 11712q^{6} - 16814q^{7} - 16384q^{8} + O(q^{10})$$ $$2q + 32q^{2} - 366q^{3} - 3750q^{5} - 11712q^{6} - 16814q^{7} - 16384q^{8} + 20000q^{10} - 346796q^{11} - 187392q^{12} - 465246q^{13} + 1601250q^{15} - 524288q^{16} + 3760066q^{17} + 253728q^{18} + 2560000q^{20} + 6153924q^{21} - 5548736q^{22} - 10456526q^{23} - 5468750q^{25} - 14887872q^{26} - 18709920q^{27} + 8608768q^{28} + 21960000q^{30} - 20131996q^{31} - 8388608q^{32} + 63463668q^{33} - 10508750q^{35} + 8119296q^{36} + 112775826q^{37} + 35249920q^{38} + 71680000q^{40} - 306007196q^{41} + 98462784q^{42} + 118744914q^{43} - 39645000q^{45} - 334608832q^{46} + 344678706q^{47} + 95944704q^{48} - 387500000q^{50} - 1376184156q^{51} - 238205952q^{52} + 392771234q^{53} + 650242500q^{55} + 275480576q^{56} + 403170960q^{57} + 793306880q^{58} - 117120000q^{60} + 1812371604q^{61} - 322111936q^{62} - 133318206q^{63} + 2035451250q^{65} + 2030837376q^{66} - 1924147934q^{67} - 1925153792q^{68} - 1345120000q^{70} - 6241755196q^{71} + 129908736q^{72} - 1272678526q^{73} - 2430468750q^{75} + 1127997440q^{76} + 2915513972q^{77} + 2724480576q^{78} + 983040000q^{80} + 7784229762q^{81} - 4896115136q^{82} - 10367644206q^{83} + 2350041250q^{85} + 3799837248q^{86} + 9073447440q^{87} + 2840952832q^{88} - 1110060000q^{90} + 7822646244q^{91} - 5353741312q^{92} + 3684155268q^{93} - 5507800000q^{95} + 3070230528q^{96} - 1280722494q^{97} + 4515838432q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$i$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 1.00000i 1.00000i
16.0000 16.0000i −183.000 183.000i 512.000i −1875.00 + 2500.00i −5856.00 −8407.00 + 8407.00i −8192.00 8192.00i 7929.00i 10000.0 + 70000.0i
7.1 16.0000 + 16.0000i −183.000 + 183.000i 512.000i −1875.00 2500.00i −5856.00 −8407.00 8407.00i −8192.00 + 8192.00i 7929.00i 10000.0 70000.0i
 $$n$$: 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
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.11.c.a 2
3.b odd 2 1 90.11.g.b 2
4.b odd 2 1 80.11.p.b 2
5.b even 2 1 50.11.c.c 2
5.c odd 4 1 inner 10.11.c.a 2
5.c odd 4 1 50.11.c.c 2
15.e even 4 1 90.11.g.b 2
20.e even 4 1 80.11.p.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.a 2 1.a even 1 1 trivial
10.11.c.a 2 5.c odd 4 1 inner
50.11.c.c 2 5.b even 2 1
50.11.c.c 2 5.c odd 4 1
80.11.p.b 2 4.b odd 2 1
80.11.p.b 2 20.e even 4 1
90.11.g.b 2 3.b odd 2 1
90.11.g.b 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 366 T_{3} + 66978$$ acting on $$S_{11}^{\mathrm{new}}(10, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 32 T + 512 T^{2}$$
$3$ $$1 + 366 T + 66978 T^{2} + 21611934 T^{3} + 3486784401 T^{4}$$
$5$ $$1 + 3750 T + 9765625 T^{2}$$
$7$ $$1 + 16814 T + 141355298 T^{2} + 4749538836686 T^{3} + 79792266297612001 T^{4}$$
$11$ $$( 1 + 173398 T + 25937424601 T^{2} )^{2}$$
$13$ $$1 + 465246 T + 108226920258 T^{2} + 64138111898779854 T^{3} +$$$$19\!\cdots\!01$$$$T^{4}$$
$17$ $$1 - 3760066 T + 7069048162178 T^{2} - 7580270121285669634 T^{3} +$$$$40\!\cdots\!01$$$$T^{4}$$
$19$ $$1 - 11048698082002 T^{2} +$$$$37\!\cdots\!01$$$$T^{4}$$
$23$ $$1 + 10456526 T + 54669467994338 T^{2} +$$$$43\!\cdots\!74$$$$T^{3} +$$$$17\!\cdots\!01$$$$T^{4}$$
$29$ $$1 - 226828718694802 T^{2} +$$$$17\!\cdots\!01$$$$T^{4}$$
$31$ $$( 1 + 10065998 T + 819628286980801 T^{2} )^{2}$$
$37$ $$1 - 112775826 T + 6359193464991138 T^{2} -$$$$54\!\cdots\!74$$$$T^{3} +$$$$23\!\cdots\!01$$$$T^{4}$$
$41$ $$( 1 + 153003598 T + 13422659310152401 T^{2} )^{2}$$
$43$ $$1 - 118744914 T + 7050177300433698 T^{2} -$$$$25\!\cdots\!86$$$$T^{3} +$$$$46\!\cdots\!01$$$$T^{4}$$
$47$ $$1 - 344678706 T + 59401705184917218 T^{2} -$$$$18\!\cdots\!94$$$$T^{3} +$$$$27\!\cdots\!01$$$$T^{4}$$
$53$ $$1 - 392771234 T + 77134621128941378 T^{2} -$$$$68\!\cdots\!66$$$$T^{3} +$$$$30\!\cdots\!01$$$$T^{4}$$
$59$ $$1 - 540501840891316402 T^{2} +$$$$26\!\cdots\!01$$$$T^{4}$$
$61$ $$( 1 - 906185802 T + 713342911662882601 T^{2} )^{2}$$
$67$ $$1 + 1924147934 T + 1851172635958234178 T^{2} +$$$$35\!\cdots\!66$$$$T^{3} +$$$$33\!\cdots\!01$$$$T^{4}$$
$71$ $$( 1 + 3120877598 T + 3255243551009881201 T^{2} )^{2}$$
$73$ $$1 + 1272678526 T + 809855315270766338 T^{2} +$$$$54\!\cdots\!74$$$$T^{3} +$$$$18\!\cdots\!01$$$$T^{4}$$
$79$ $$1 - 15063541390202404802 T^{2} +$$$$89\!\cdots\!01$$$$T^{4}$$
$83$ $$1 + 10367644206 T + 53744023191102685218 T^{2} +$$$$16\!\cdots\!94$$$$T^{3} +$$$$24\!\cdots\!01$$$$T^{4}$$
$89$ $$1 - 1969041775268808802 T^{2} +$$$$97\!\cdots\!01$$$$T^{4}$$
$97$ $$1 + 1280722494 T + 820125053318790018 T^{2} +$$$$94\!\cdots\!06$$$$T^{3} +$$$$54\!\cdots\!01$$$$T^{4}$$