# Properties

 Label 10.7.c.a Level 10 Weight 7 Character orbit 10.c Analytic conductor 2.301 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$7$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30054083620$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -4 - 4 i ) q^{2} + ( -23 + 23 i ) q^{3} + 32 i q^{4} + ( -75 + 100 i ) q^{5} + 184 q^{6} + ( -247 - 247 i ) q^{7} + ( 128 - 128 i ) q^{8} -329 i q^{9} +O(q^{10})$$ $$q + ( -4 - 4 i ) q^{2} + ( -23 + 23 i ) q^{3} + 32 i q^{4} + ( -75 + 100 i ) q^{5} + 184 q^{6} + ( -247 - 247 i ) q^{7} + ( 128 - 128 i ) q^{8} -329 i q^{9} + ( 700 - 100 i ) q^{10} + 1402 q^{11} + ( -736 - 736 i ) q^{12} + ( -2703 + 2703 i ) q^{13} + 1976 i q^{14} + ( -575 - 4025 i ) q^{15} -1024 q^{16} + ( 2593 + 2593 i ) q^{17} + ( -1316 + 1316 i ) q^{18} + 1720 i q^{19} + ( -3200 - 2400 i ) q^{20} + 11362 q^{21} + ( -5608 - 5608 i ) q^{22} + ( 2137 - 2137 i ) q^{23} + 5888 i q^{24} + ( -4375 - 15000 i ) q^{25} + 21624 q^{26} + ( -9200 - 9200 i ) q^{27} + ( 7904 - 7904 i ) q^{28} + 30520 i q^{29} + ( -13800 + 18400 i ) q^{30} -37838 q^{31} + ( 4096 + 4096 i ) q^{32} + ( -32246 + 32246 i ) q^{33} -20744 i q^{34} + ( 43225 - 6175 i ) q^{35} + 10528 q^{36} + ( 37113 + 37113 i ) q^{37} + ( 6880 - 6880 i ) q^{38} -124338 i q^{39} + ( 3200 + 22400 i ) q^{40} -35438 q^{41} + ( -45448 - 45448 i ) q^{42} + ( 39177 - 39177 i ) q^{43} + 44864 i q^{44} + ( 32900 + 24675 i ) q^{45} -17096 q^{46} + ( 95193 + 95193 i ) q^{47} + ( 23552 - 23552 i ) q^{48} + 4369 i q^{49} + ( -42500 + 77500 i ) q^{50} -119278 q^{51} + ( -86496 - 86496 i ) q^{52} + ( 36017 - 36017 i ) q^{53} + 73600 i q^{54} + ( -105150 + 140200 i ) q^{55} -63232 q^{56} + ( -39560 - 39560 i ) q^{57} + ( 122080 - 122080 i ) q^{58} -35960 i q^{59} + ( 128800 - 18400 i ) q^{60} + 83322 q^{61} + ( 151352 + 151352 i ) q^{62} + ( -81263 + 81263 i ) q^{63} -32768 i q^{64} + ( -67575 - 473025 i ) q^{65} + 257968 q^{66} + ( 60833 + 60833 i ) q^{67} + ( -82976 + 82976 i ) q^{68} + 98302 i q^{69} + ( -197600 - 148200 i ) q^{70} -40318 q^{71} + ( -42112 - 42112 i ) q^{72} + ( -129023 + 129023 i ) q^{73} -296904 i q^{74} + ( 445625 + 244375 i ) q^{75} -55040 q^{76} + ( -346294 - 346294 i ) q^{77} + ( -497352 + 497352 i ) q^{78} + 524640 i q^{79} + ( 76800 - 102400 i ) q^{80} + 663041 q^{81} + ( 141752 + 141752 i ) q^{82} + ( -114423 + 114423 i ) q^{83} + 363584 i q^{84} + ( -453775 + 64825 i ) q^{85} -313416 q^{86} + ( -701960 - 701960 i ) q^{87} + ( 179456 - 179456 i ) q^{88} + 187280 i q^{89} + ( -32900 - 230300 i ) q^{90} + 1335282 q^{91} + ( 68384 + 68384 i ) q^{92} + ( 870274 - 870274 i ) q^{93} -761544 i q^{94} + ( -172000 - 129000 i ) q^{95} -188416 q^{96} + ( 532833 + 532833 i ) q^{97} + ( 17476 - 17476 i ) q^{98} -461258 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{2} - 46q^{3} - 150q^{5} + 368q^{6} - 494q^{7} + 256q^{8} + O(q^{10})$$ $$2q - 8q^{2} - 46q^{3} - 150q^{5} + 368q^{6} - 494q^{7} + 256q^{8} + 1400q^{10} + 2804q^{11} - 1472q^{12} - 5406q^{13} - 1150q^{15} - 2048q^{16} + 5186q^{17} - 2632q^{18} - 6400q^{20} + 22724q^{21} - 11216q^{22} + 4274q^{23} - 8750q^{25} + 43248q^{26} - 18400q^{27} + 15808q^{28} - 27600q^{30} - 75676q^{31} + 8192q^{32} - 64492q^{33} + 86450q^{35} + 21056q^{36} + 74226q^{37} + 13760q^{38} + 6400q^{40} - 70876q^{41} - 90896q^{42} + 78354q^{43} + 65800q^{45} - 34192q^{46} + 190386q^{47} + 47104q^{48} - 85000q^{50} - 238556q^{51} - 172992q^{52} + 72034q^{53} - 210300q^{55} - 126464q^{56} - 79120q^{57} + 244160q^{58} + 257600q^{60} + 166644q^{61} + 302704q^{62} - 162526q^{63} - 135150q^{65} + 515936q^{66} + 121666q^{67} - 165952q^{68} - 395200q^{70} - 80636q^{71} - 84224q^{72} - 258046q^{73} + 891250q^{75} - 110080q^{76} - 692588q^{77} - 994704q^{78} + 153600q^{80} + 1326082q^{81} + 283504q^{82} - 228846q^{83} - 907550q^{85} - 626832q^{86} - 1403920q^{87} + 358912q^{88} - 65800q^{90} + 2670564q^{91} + 136768q^{92} + 1740548q^{93} - 344000q^{95} - 376832q^{96} + 1065666q^{97} + 34952q^{98} + O(q^{100})$$

## 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
−4.00000 + 4.00000i −23.0000 23.0000i 32.0000i −75.0000 100.000i 184.000 −247.000 + 247.000i 128.000 + 128.000i 329.000i 700.000 + 100.000i
7.1 −4.00000 4.00000i −23.0000 + 23.0000i 32.0000i −75.0000 + 100.000i 184.000 −247.000 247.000i 128.000 128.000i 329.000i 700.000 100.000i
 $$n$$: 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 10.7.c.a 2
3.b odd 2 1 90.7.g.a 2
4.b odd 2 1 80.7.p.a 2
5.b even 2 1 50.7.c.c 2
5.c odd 4 1 inner 10.7.c.a 2
5.c odd 4 1 50.7.c.c 2
15.d odd 2 1 450.7.g.b 2
15.e even 4 1 90.7.g.a 2
15.e even 4 1 450.7.g.b 2
20.e even 4 1 80.7.p.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.7.c.a 2 1.a even 1 1 trivial
10.7.c.a 2 5.c odd 4 1 inner
50.7.c.c 2 5.b even 2 1
50.7.c.c 2 5.c odd 4 1
80.7.p.a 2 4.b odd 2 1
80.7.p.a 2 20.e even 4 1
90.7.g.a 2 3.b odd 2 1
90.7.g.a 2 15.e even 4 1
450.7.g.b 2 15.d odd 2 1
450.7.g.b 2 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 46 T_{3} + 1058$$ acting on $$S_{7}^{\mathrm{new}}(10, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T + 32 T^{2}$$
$3$ $$1 + 46 T + 1058 T^{2} + 33534 T^{3} + 531441 T^{4}$$
$5$ $$1 + 150 T + 15625 T^{2}$$
$7$ $$1 + 494 T + 122018 T^{2} + 58118606 T^{3} + 13841287201 T^{4}$$
$11$ $$( 1 - 1402 T + 1771561 T^{2} )^{2}$$
$13$ $$1 + 5406 T + 14612418 T^{2} + 26093729454 T^{3} + 23298085122481 T^{4}$$
$17$ $$1 - 5186 T + 13447298 T^{2} - 125177432834 T^{3} + 582622237229761 T^{4}$$
$19$ $$1 - 91133362 T^{2} + 2213314919066161 T^{4}$$
$23$ $$1 - 4274 T + 9133538 T^{2} - 632705389586 T^{3} + 21914624432020321 T^{4}$$
$29$ $$1 - 258176242 T^{2} + 353814783205469041 T^{4}$$
$31$ $$( 1 + 37838 T + 887503681 T^{2} )^{2}$$
$37$ $$1 - 74226 T + 2754749538 T^{2} - 190443608434434 T^{3} + 6582952005840035281 T^{4}$$
$41$ $$( 1 + 35438 T + 4750104241 T^{2} )^{2}$$
$43$ $$1 - 78354 T + 3069674658 T^{2} - 495304080341346 T^{3} + 39959630797262576401 T^{4}$$
$47$ $$1 - 190386 T + 18123414498 T^{2} - 2052211689626994 T^{3} +$$$$11\!\cdots\!41$$$$T^{4}$$
$53$ $$1 - 72034 T + 2594448578 T^{2} - 1596587589566386 T^{3} +$$$$49\!\cdots\!41$$$$T^{4}$$
$59$ $$1 - 83067945682 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$
$61$ $$( 1 - 83322 T + 51520374361 T^{2} )^{2}$$
$67$ $$1 - 121666 T + 7401307778 T^{2} - 11005709524973554 T^{3} +$$$$81\!\cdots\!61$$$$T^{4}$$
$71$ $$( 1 + 40318 T + 128100283921 T^{2} )^{2}$$
$73$ $$1 + 258046 T + 33293869058 T^{2} + 39051191756971294 T^{3} +$$$$22\!\cdots\!21$$$$T^{4}$$
$79$ $$1 - 210927781442 T^{2} +$$$$59\!\cdots\!41$$$$T^{4}$$
$83$ $$1 + 228846 T + 26185245858 T^{2} + 74818996684002174 T^{3} +$$$$10\!\cdots\!61$$$$T^{4}$$
$89$ $$1 - 958888783522 T^{2} +$$$$24\!\cdots\!21$$$$T^{4}$$
$97$ $$1 - 1065666 T + 567822011778 T^{2} - 887669944604667714 T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$