# Properties

 Label 75.10.b.e Level $75$ Weight $10$ Character orbit 75.b Analytic conductor $38.628$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 75.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$38.6276877123$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{4729})$$ Defining polynomial: $$x^{4} + 2365 x^{2} + 1397124$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - 9 \beta_{2} ) q^{2} + 81 \beta_{2} q^{3} + ( -770 + 19 \beta_{3} ) q^{4} + ( 810 - 81 \beta_{3} ) q^{6} + ( -56 \beta_{1} + 5908 \beta_{2} ) q^{7} + ( -429 \beta_{1} + 24609 \beta_{2} ) q^{8} -6561 q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - 9 \beta_{2} ) q^{2} + 81 \beta_{2} q^{3} + ( -770 + 19 \beta_{3} ) q^{4} + ( 810 - 81 \beta_{3} ) q^{6} + ( -56 \beta_{1} + 5908 \beta_{2} ) q^{7} + ( -429 \beta_{1} + 24609 \beta_{2} ) q^{8} -6561 q^{9} + ( 16768 + 1952 \beta_{3} ) q^{11} + ( 1539 \beta_{1} - 60831 \beta_{2} ) q^{12} + ( 1384 \beta_{1} + 72530 \beta_{2} ) q^{13} + ( 125832 - 6468 \beta_{3} ) q^{14} + ( 363218 - 19171 \beta_{3} ) q^{16} + ( 2200 \beta_{1} - 191478 \beta_{2} ) q^{17} + ( -6561 \beta_{1} + 59049 \beta_{2} ) q^{18} + ( 201164 + 968 \beta_{3} ) q^{19} + ( -483084 + 4536 \beta_{3} ) q^{21} + ( -800 \beta_{1} + 2138784 \beta_{2} ) q^{22} + ( 64968 \beta_{1} + 144336 \beta_{2} ) q^{23} + ( -2028078 + 34749 \beta_{3} ) q^{24} + ( -924428 - 58690 \beta_{3} ) q^{26} -531441 \beta_{2} q^{27} + ( 155372 \beta_{1} - 5694556 \beta_{2} ) q^{28} + ( 31078 + 12416 \beta_{3} ) q^{29} + ( -2475696 - 75736 \beta_{3} ) q^{31} + ( 316109 \beta_{1} - 13156737 \beta_{2} ) q^{32} + ( 158112 \beta_{1} + 1516320 \beta_{2} ) q^{33} + ( -4537180 + 213478 \beta_{3} ) q^{34} + ( 5051970 - 124659 \beta_{3} ) q^{36} + ( -174696 \beta_{1} - 2774162 \beta_{2} ) q^{37} + ( 192452 \beta_{1} - 675012 \beta_{2} ) q^{38} + ( -5762826 - 112104 \beta_{3} ) q^{39} + ( 7340714 - 470096 \beta_{3} ) q^{41} + ( -523908 \beta_{1} + 9668484 \beta_{2} ) q^{42} + ( -152384 \beta_{1} + 13798268 \beta_{2} ) q^{43} + ( 30926656 - 1147360 \beta_{3} ) q^{44} + ( -75998496 + 505344 \beta_{3} ) q^{46} + ( 431368 \beta_{1} - 47767536 \beta_{2} ) q^{47} + ( -1552851 \beta_{1} + 27867807 \beta_{2} ) q^{48} + ( 1077559 + 664832 \beta_{3} ) q^{49} + ( 15687918 - 178200 \beta_{3} ) q^{51} + ( 312390 \beta_{1} - 23388158 \beta_{2} ) q^{52} + ( -929872 \beta_{1} - 32617734 \beta_{2} ) q^{53} + ( -5314410 + 531441 \beta_{3} ) q^{54} + ( -177723000 + 3936660 \beta_{3} ) q^{56} + ( 78408 \beta_{1} + 16372692 \beta_{2} ) q^{57} + ( -80666 \beta_{1} + 14284266 \beta_{2} ) q^{58} + ( -93124864 - 1613408 \beta_{3} ) q^{59} + ( 75911686 + 2256688 \beta_{3} ) q^{61} + ( -1794072 \beta_{1} - 66557064 \beta_{2} ) q^{62} + ( 367416 \beta_{1} - 38762388 \beta_{2} ) q^{63} + ( -322401682 + 6502275 \beta_{3} ) q^{64} + ( -173306304 + 64800 \beta_{3} ) q^{66} + ( -7444160 \beta_{1} - 20518268 \beta_{2} ) q^{67} + ( -5332082 \beta_{1} + 193207578 \beta_{2} ) q^{68} + ( -6428808 - 5262408 \beta_{3} ) q^{69} + ( -110604928 - 7061120 \beta_{3} ) q^{71} + ( 2814669 \beta_{1} - 161459649 \beta_{2} ) q^{72} + ( 6480208 \beta_{1} - 13321054 \beta_{2} ) q^{73} + ( 180496012 + 1027202 \beta_{3} ) q^{74} + ( -133156936 + 3095148 \beta_{3} ) q^{76} + ( 10593408 \beta_{1} - 18609024 \beta_{2} ) q^{77} + ( -4753890 \beta_{1} - 79632558 \beta_{2} ) q^{78} + ( 467102400 - 1798040 \beta_{3} ) q^{79} + 43046721 q^{81} + ( 11571578 \beta_{1} - 617489034 \beta_{2} ) q^{82} + ( -3161088 \beta_{1} + 101939532 \beta_{2} ) q^{83} + ( 473844168 - 12585132 \beta_{3} ) q^{84} + ( 319624408 - 15322108 \beta_{3} ) q^{86} + ( 1005696 \beta_{1} + 3523014 \beta_{2} ) q^{87} + ( 40843296 \beta_{1} - 529135776 \beta_{2} ) q^{88} + ( -107605986 - 9306192 \beta_{3} ) q^{89} + ( -332705016 - 4192496 \beta_{3} ) q^{91} + ( -47282976 \beta_{1} + 1350655008 \beta_{2} ) q^{92} + ( -6134616 \beta_{1} - 206665992 \beta_{2} ) q^{93} + ( -991866016 + 52081216 \beta_{3} ) q^{94} + ( 1091300526 - 25604829 \beta_{3} ) q^{96} + ( 44039040 \beta_{1} - 171547778 \beta_{2} ) q^{97} + ( -4905929 \beta_{1} + 770149905 \beta_{2} ) q^{98} + ( -110014848 - 12807072 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 3042q^{4} + 3078q^{6} - 26244q^{9} + O(q^{10})$$ $$4q - 3042q^{4} + 3078q^{6} - 26244q^{9} + 70976q^{11} + 490392q^{14} + 1414530q^{16} + 806592q^{19} - 1923264q^{21} - 8042814q^{24} - 3815092q^{26} + 149144q^{29} - 10054256q^{31} - 17721764q^{34} + 19958562q^{36} - 23275512q^{39} + 28422664q^{41} + 121411904q^{44} - 302983296q^{46} + 5639900q^{49} + 62395272q^{51} - 20194758q^{54} - 703018680q^{56} - 375726272q^{59} + 308160120q^{61} - 1276602178q^{64} - 693095616q^{66} - 36240048q^{69} - 456541952q^{71} + 724038452q^{74} - 526437448q^{76} + 1864813520q^{79} + 172186884q^{81} + 1870206408q^{84} + 1247853416q^{86} - 449036328q^{89} - 1339205056q^{91} - 3863301632q^{94} + 4313992446q^{96} - 465673536q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2365 x^{2} + 1397124$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1183 \nu$$$$)/1182$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 1183$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 1183$$ $$\nu^{3}$$ $$=$$ $$1182 \beta_{2} - 1183 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 34.8839i − 33.8839i 33.8839i 34.8839i
43.8839i 81.0000i −1413.79 0 3554.59 7861.50i 39574.2i −6561.00 0
49.2 24.8839i 81.0000i −107.207 0 −2015.59 4010.50i 10072.8i −6561.00 0
49.3 24.8839i 81.0000i −107.207 0 −2015.59 4010.50i 10072.8i −6561.00 0
49.4 43.8839i 81.0000i −1413.79 0 3554.59 7861.50i 39574.2i −6561.00 0
 $$n$$: 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.10.b.e 4
3.b odd 2 1 225.10.b.g 4
5.b even 2 1 inner 75.10.b.e 4
5.c odd 4 1 15.10.a.c 2
5.c odd 4 1 75.10.a.g 2
15.d odd 2 1 225.10.b.g 4
15.e even 4 1 45.10.a.e 2
15.e even 4 1 225.10.a.j 2
20.e even 4 1 240.10.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.10.a.c 2 5.c odd 4 1
45.10.a.e 2 15.e even 4 1
75.10.a.g 2 5.c odd 4 1
75.10.b.e 4 1.a even 1 1 trivial
75.10.b.e 4 5.b even 2 1 inner
225.10.a.j 2 15.e even 4 1
225.10.b.g 4 3.b odd 2 1
225.10.b.g 4 15.d odd 2 1
240.10.a.m 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 2545 T_{2}^{2} + 1192464$$ acting on $$S_{10}^{\mathrm{new}}(75, [\chi])$$.