# Properties

 Label 75.10.a.g Level $75$ Weight $10$ Character orbit 75.a Self dual yes Analytic conductor $38.628$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 75.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$38.6276877123$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{4729})$$ Defining polynomial: $$x^{2} - x - 1182$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{4729})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -9 - \beta ) q^{2} -81 q^{3} + ( 751 + 19 \beta ) q^{4} + ( 729 + 81 \beta ) q^{6} + ( 5908 + 56 \beta ) q^{7} + ( -24609 - 429 \beta ) q^{8} + 6561 q^{9} +O(q^{10})$$ $$q + ( -9 - \beta ) q^{2} -81 q^{3} + ( 751 + 19 \beta ) q^{4} + ( 729 + 81 \beta ) q^{6} + ( 5908 + 56 \beta ) q^{7} + ( -24609 - 429 \beta ) q^{8} + 6561 q^{9} + ( 18720 - 1952 \beta ) q^{11} + ( -60831 - 1539 \beta ) q^{12} + ( -72530 + 1384 \beta ) q^{13} + ( -119364 - 6468 \beta ) q^{14} + ( 344047 + 19171 \beta ) q^{16} + ( -191478 - 2200 \beta ) q^{17} + ( -59049 - 6561 \beta ) q^{18} + ( -202132 + 968 \beta ) q^{19} + ( -478548 - 4536 \beta ) q^{21} + ( 2138784 + 800 \beta ) q^{22} + ( -144336 + 64968 \beta ) q^{23} + ( 1993329 + 34749 \beta ) q^{24} + ( -983118 + 58690 \beta ) q^{26} -531441 q^{27} + ( 5694556 + 155372 \beta ) q^{28} + ( -43494 + 12416 \beta ) q^{29} + ( -2551432 + 75736 \beta ) q^{31} + ( -13156737 - 316109 \beta ) q^{32} + ( -1516320 + 158112 \beta ) q^{33} + ( 4323702 + 213478 \beta ) q^{34} + ( 4927311 + 124659 \beta ) q^{36} + ( -2774162 + 174696 \beta ) q^{37} + ( 675012 + 192452 \beta ) q^{38} + ( 5874930 - 112104 \beta ) q^{39} + ( 6870618 + 470096 \beta ) q^{41} + ( 9668484 + 523908 \beta ) q^{42} + ( -13798268 - 152384 \beta ) q^{43} + ( -29779296 - 1147360 \beta ) q^{44} + ( -75493152 - 505344 \beta ) q^{46} + ( -47767536 - 431368 \beta ) q^{47} + ( -27867807 - 1552851 \beta ) q^{48} + ( -1742391 + 664832 \beta ) q^{49} + ( 15509718 + 178200 \beta ) q^{51} + ( -23388158 - 312390 \beta ) q^{52} + ( 32617734 - 929872 \beta ) q^{53} + ( 4782969 + 531441 \beta ) q^{54} + ( -173786340 - 3936660 \beta ) q^{56} + ( 16372692 - 78408 \beta ) q^{57} + ( -14284266 - 80666 \beta ) q^{58} + ( 94738272 - 1613408 \beta ) q^{59} + ( 78168374 - 2256688 \beta ) q^{61} + ( -66557064 + 1794072 \beta ) q^{62} + ( 38762388 + 367416 \beta ) q^{63} + ( 315899407 + 6502275 \beta ) q^{64} + ( -173241504 - 64800 \beta ) q^{66} + ( -20518268 + 7444160 \beta ) q^{67} + ( -193207578 - 5332082 \beta ) q^{68} + ( 11691216 - 5262408 \beta ) q^{69} + ( -117666048 + 7061120 \beta ) q^{71} + ( -161459649 - 2814669 \beta ) q^{72} + ( 13321054 + 6480208 \beta ) q^{73} + ( -181523214 + 1027202 \beta ) q^{74} + ( -130061788 - 3095148 \beta ) q^{76} + ( -18609024 - 10593408 \beta ) q^{77} + ( 79632558 - 4753890 \beta ) q^{78} + ( -465304360 - 1798040 \beta ) q^{79} + 43046721 q^{81} + ( -617489034 - 11571578 \beta ) q^{82} + ( -101939532 - 3161088 \beta ) q^{83} + ( -461259036 - 12585132 \beta ) q^{84} + ( 304302300 + 15322108 \beta ) q^{86} + ( 3523014 - 1005696 \beta ) q^{87} + ( 529135776 + 40843296 \beta ) q^{88} + ( 116912178 - 9306192 \beta ) q^{89} + ( -336897512 + 4192496 \beta ) q^{91} + ( 1350655008 + 47282976 \beta ) q^{92} + ( 206665992 - 6134616 \beta ) q^{93} + ( 939784800 + 52081216 \beta ) q^{94} + ( 1065695697 + 25604829 \beta ) q^{96} + ( -171547778 - 44039040 \beta ) q^{97} + ( -770149905 - 4905929 \beta ) q^{98} + ( 122821920 - 12807072 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 19q^{2} - 162q^{3} + 1521q^{4} + 1539q^{6} + 11872q^{7} - 49647q^{8} + 13122q^{9} + O(q^{10})$$ $$2q - 19q^{2} - 162q^{3} + 1521q^{4} + 1539q^{6} + 11872q^{7} - 49647q^{8} + 13122q^{9} + 35488q^{11} - 123201q^{12} - 143676q^{13} - 245196q^{14} + 707265q^{16} - 385156q^{17} - 124659q^{18} - 403296q^{19} - 961632q^{21} + 4278368q^{22} - 223704q^{23} + 4021407q^{24} - 1907546q^{26} - 1062882q^{27} + 11544484q^{28} - 74572q^{29} - 5027128q^{31} - 26629583q^{32} - 2874528q^{33} + 8860882q^{34} + 9979281q^{36} - 5373628q^{37} + 1542476q^{38} + 11637756q^{39} + 14211332q^{41} + 19860876q^{42} - 27748920q^{43} - 60705952q^{44} - 151491648q^{46} - 95966440q^{47} - 57288465q^{48} - 2819950q^{49} + 31197636q^{51} - 47088706q^{52} + 64305596q^{53} + 10097379q^{54} - 351509340q^{56} + 32666976q^{57} - 28649198q^{58} + 187863136q^{59} + 154080060q^{61} - 131320056q^{62} + 77892192q^{63} + 638301089q^{64} - 346547808q^{66} - 33592376q^{67} - 391747238q^{68} + 18120024q^{69} - 228270976q^{71} - 325733967q^{72} + 33122316q^{73} - 362019226q^{74} - 263218724q^{76} - 47811456q^{77} + 154511226q^{78} - 932406760q^{79} + 86093442q^{81} - 1246549646q^{82} - 207040152q^{83} - 935103204q^{84} + 623926708q^{86} + 6040332q^{87} + 1099114848q^{88} + 224518164q^{89} - 669602528q^{91} + 2748592992q^{92} + 407197368q^{93} + 1931650816q^{94} + 2156996223q^{96} - 387134596q^{97} - 1545205739q^{98} + 232836768q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 34.8839 −33.8839
−43.8839 −81.0000 1413.79 0 3554.59 7861.50 −39574.2 6561.00 0
1.2 24.8839 −81.0000 107.207 0 −2015.59 4010.50 −10072.8 6561.00 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 19 T_{2} - 1092$$ acting on $$S_{10}^{\mathrm{new}}(\Gamma_0(75))$$.