# Properties

 Label 10.11.c.b 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.

 $$f(q)$$ $$=$$ $$q + ( 16 + 16 i ) q^{2} + ( 57 - 57 i ) q^{3} + 512 i q^{4} + ( 2925 + 1100 i ) q^{5} + 1824 q^{6} + ( 6953 + 6953 i ) q^{7} + ( -8192 + 8192 i ) q^{8} + 52551 i q^{9} +O(q^{10})$$ $$q + ( 16 + 16 i ) q^{2} + ( 57 - 57 i ) q^{3} + 512 i q^{4} + ( 2925 + 1100 i ) q^{5} + 1824 q^{6} + ( 6953 + 6953 i ) q^{7} + ( -8192 + 8192 i ) q^{8} + 52551 i q^{9} + ( 29200 + 64400 i ) q^{10} + 75242 q^{11} + ( 29184 + 29184 i ) q^{12} + ( 109857 - 109857 i ) q^{13} + 222496 i q^{14} + ( 229425 - 104025 i ) q^{15} -262144 q^{16} + ( -1528927 - 1528927 i ) q^{17} + ( -840816 + 840816 i ) q^{18} -4038680 i q^{19} + ( -563200 + 1497600 i ) q^{20} + 792642 q^{21} + ( 1203872 + 1203872 i ) q^{22} + ( -712423 + 712423 i ) q^{23} + 933888 i q^{24} + ( 7345625 + 6435000 i ) q^{25} + 3515424 q^{26} + ( 6361200 + 6361200 i ) q^{27} + ( -3559936 + 3559936 i ) q^{28} + 446120 i q^{29} + ( 5335200 + 2006400 i ) q^{30} -29080718 q^{31} + ( -4194304 - 4194304 i ) q^{32} + ( 4288794 - 4288794 i ) q^{33} -48925664 i q^{34} + ( 12689225 + 27985825 i ) q^{35} -26906112 q^{36} + ( -911847 - 911847 i ) q^{37} + ( 64618880 - 64618880 i ) q^{38} -12523698 i q^{39} + ( -32972800 + 14950400 i ) q^{40} -163945678 q^{41} + ( 12682272 + 12682272 i ) q^{42} + ( 118422777 - 118422777 i ) q^{43} + 38523904 i q^{44} + ( -57806100 + 153711675 i ) q^{45} -22797536 q^{46} + ( 276320313 + 276320313 i ) q^{47} + ( -14942208 + 14942208 i ) q^{48} -185786831 i q^{49} + ( 14570000 + 220490000 i ) q^{50} -174297678 q^{51} + ( 56246784 + 56246784 i ) q^{52} + ( 308460097 - 308460097 i ) q^{53} + 203558400 i q^{54} + ( 220082850 + 82766200 i ) q^{55} -113917952 q^{56} + ( -230204760 - 230204760 i ) q^{57} + ( -7137920 + 7137920 i ) q^{58} -940888360 i q^{59} + ( 53260800 + 117465600 i ) q^{60} -1353610038 q^{61} + ( -465291488 - 465291488 i ) q^{62} + ( -365387103 + 365387103 i ) q^{63} -134217728 i q^{64} + ( 442174425 - 200489025 i ) q^{65} + 137241408 q^{66} + ( 853570913 + 853570913 i ) q^{67} + ( 782810624 - 782810624 i ) q^{68} + 81216222 i q^{69} + ( -244745600 + 650800800 i ) q^{70} + 2827014562 q^{71} + ( -430497792 - 430497792 i ) q^{72} + ( -2753297183 + 2753297183 i ) q^{73} -29179104 i q^{74} + ( 785495625 - 51905625 i ) q^{75} + 2067804160 q^{76} + ( 523157626 + 523157626 i ) q^{77} + ( 200379168 - 200379168 i ) q^{78} + 3324500640 i q^{79} + ( -766771200 - 288358400 i ) q^{80} -2377907199 q^{81} + ( -2623130848 - 2623130848 i ) q^{82} + ( 1346339097 - 1346339097 i ) q^{83} + 405832704 i q^{84} + ( -2790291775 - 6153931175 i ) q^{85} + 3789528864 q^{86} + ( 25428840 + 25428840 i ) q^{87} + ( -616382464 + 616382464 i ) q^{88} -2667450320 i q^{89} + ( -3384284400 + 1534489200 i ) q^{90} + 1527671442 q^{91} + ( -364760576 - 364760576 i ) q^{92} + ( -1657600926 + 1657600926 i ) q^{93} + 8842250016 i q^{94} + ( 4442548000 - 11813139000 i ) q^{95} -478150656 q^{96} + ( -526562847 - 526562847 i ) q^{97} + ( 2972589296 - 2972589296 i ) q^{98} + 3954042342 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 32q^{2} + 114q^{3} + 5850q^{5} + 3648q^{6} + 13906q^{7} - 16384q^{8} + O(q^{10})$$ $$2q + 32q^{2} + 114q^{3} + 5850q^{5} + 3648q^{6} + 13906q^{7} - 16384q^{8} + 58400q^{10} + 150484q^{11} + 58368q^{12} + 219714q^{13} + 458850q^{15} - 524288q^{16} - 3057854q^{17} - 1681632q^{18} - 1126400q^{20} + 1585284q^{21} + 2407744q^{22} - 1424846q^{23} + 14691250q^{25} + 7030848q^{26} + 12722400q^{27} - 7119872q^{28} + 10670400q^{30} - 58161436q^{31} - 8388608q^{32} + 8577588q^{33} + 25378450q^{35} - 53812224q^{36} - 1823694q^{37} + 129237760q^{38} - 65945600q^{40} - 327891356q^{41} + 25364544q^{42} + 236845554q^{43} - 115612200q^{45} - 45595072q^{46} + 552640626q^{47} - 29884416q^{48} + 29140000q^{50} - 348595356q^{51} + 112493568q^{52} + 616920194q^{53} + 440165700q^{55} - 227835904q^{56} - 460409520q^{57} - 14275840q^{58} + 106521600q^{60} - 2707220076q^{61} - 930582976q^{62} - 730774206q^{63} + 884348850q^{65} + 274482816q^{66} + 1707141826q^{67} + 1565621248q^{68} - 489491200q^{70} + 5654029124q^{71} - 860995584q^{72} - 5506594366q^{73} + 1570991250q^{75} + 4135608320q^{76} + 1046315252q^{77} + 400758336q^{78} - 1533542400q^{80} - 4755814398q^{81} - 5246261696q^{82} + 2692678194q^{83} - 5580583550q^{85} + 7579057728q^{86} + 50857680q^{87} - 1232764928q^{88} - 6768568800q^{90} + 3055342884q^{91} - 729521152q^{92} - 3315201852q^{93} + 8885096000q^{95} - 956301312q^{96} - 1053125694q^{97} + 5945178592q^{98} + O(q^{100})$$

## 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 57.0000 + 57.0000i 512.000i 2925.00 1100.00i 1824.00 6953.00 6953.00i −8192.00 8192.00i 52551.0i 29200.0 64400.0i
7.1 16.0000 + 16.0000i 57.0000 57.0000i 512.000i 2925.00 + 1100.00i 1824.00 6953.00 + 6953.00i −8192.00 + 8192.00i 52551.0i 29200.0 + 64400.0i
 $$n$$: 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.b 2
3.b odd 2 1 90.11.g.a 2
4.b odd 2 1 80.11.p.a 2
5.b even 2 1 50.11.c.b 2
5.c odd 4 1 inner 10.11.c.b 2
5.c odd 4 1 50.11.c.b 2
15.e even 4 1 90.11.g.a 2
20.e even 4 1 80.11.p.a 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 114 T_{3} + 6498$$ acting on $$S_{11}^{\mathrm{new}}(10, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 32 T + 512 T^{2}$$
$3$ $$1 - 114 T + 6498 T^{2} - 6731586 T^{3} + 3486784401 T^{4}$$
$5$ $$1 - 5850 T + 9765625 T^{2}$$
$7$ $$1 - 13906 T + 96688418 T^{2} - 3928100812594 T^{3} + 79792266297612001 T^{4}$$
$11$ $$( 1 - 75242 T + 25937424601 T^{2} )^{2}$$
$13$ $$1 - 219714 T + 24137120898 T^{2} - 30289440678111186 T^{3} +$$$$19\!\cdots\!01$$$$T^{4}$$
$17$ $$1 + 3057854 T + 4675235542658 T^{2} + 6164615012463576446 T^{3} +$$$$40\!\cdots\!01$$$$T^{4}$$
$19$ $$1 + 4048803626798 T^{2} +$$$$37\!\cdots\!01$$$$T^{4}$$
$23$ $$1 + 1424846 T + 1015093061858 T^{2} + 59026398796722923054 T^{3} +$$$$17\!\cdots\!01$$$$T^{4}$$
$29$ $$1 - 841215443546002 T^{2} +$$$$17\!\cdots\!01$$$$T^{4}$$
$31$ $$( 1 + 29080718 T + 819628286980801 T^{2} )^{2}$$
$37$ $$1 + 1823694 T + 1662929902818 T^{2} +$$$$87\!\cdots\!06$$$$T^{3} +$$$$23\!\cdots\!01$$$$T^{4}$$
$41$ $$( 1 + 163945678 T + 13422659310152401 T^{2} )^{2}$$
$43$ $$1 - 236845554 T + 28047908224783458 T^{2} -$$$$51\!\cdots\!46$$$$T^{3} +$$$$46\!\cdots\!01$$$$T^{4}$$
$47$ $$1 - 552640626 T + 152705830752835938 T^{2} -$$$$29\!\cdots\!74$$$$T^{3} +$$$$27\!\cdots\!01$$$$T^{4}$$
$53$ $$1 - 616920194 T + 190295262882498818 T^{2} -$$$$10\!\cdots\!06$$$$T^{3} +$$$$30\!\cdots\!01$$$$T^{4}$$
$59$ $$1 - 136962600617793202 T^{2} +$$$$26\!\cdots\!01$$$$T^{4}$$
$61$ $$( 1 + 1353610038 T + 713342911662882601 T^{2} )^{2}$$
$67$ $$1 - 1707141826 T + 1457166607039307138 T^{2} -$$$$31\!\cdots\!74$$$$T^{3} +$$$$33\!\cdots\!01$$$$T^{4}$$
$71$ $$( 1 - 2827014562 T + 3255243551009881201 T^{2} )^{2}$$
$73$ $$1 + 5506594366 T + 15161290755831470978 T^{2} +$$$$23\!\cdots\!34$$$$T^{3} +$$$$18\!\cdots\!01$$$$T^{4}$$
$79$ $$1 - 7884247659893284802 T^{2} +$$$$89\!\cdots\!01$$$$T^{4}$$
$83$ $$1 - 2692678194 T + 3625257928221550818 T^{2} -$$$$41\!\cdots\!06$$$$T^{3} +$$$$24\!\cdots\!01$$$$T^{4}$$
$89$ $$1 - 55248148650264264802 T^{2} +$$$$97\!\cdots\!01$$$$T^{4}$$
$97$ $$1 + 1053125694 T + 554536863681490818 T^{2} +$$$$77\!\cdots\!06$$$$T^{3} +$$$$54\!\cdots\!01$$$$T^{4}$$