Properties

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

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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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 22 i q^{2} + 81 i q^{3} + 28 q^{4} -1782 q^{6} -5988 i q^{7} + 11880 i q^{8} -6561 q^{9} +O(q^{10})\) \( q + 22 i q^{2} + 81 i q^{3} + 28 q^{4} -1782 q^{6} -5988 i q^{7} + 11880 i q^{8} -6561 q^{9} -14648 q^{11} + 2268 i q^{12} -37906 i q^{13} + 131736 q^{14} -247024 q^{16} -441098 i q^{17} -144342 i q^{18} -441820 q^{19} + 485028 q^{21} -322256 i q^{22} -2264136 i q^{23} -962280 q^{24} + 833932 q^{26} -531441 i q^{27} -167664 i q^{28} + 1049350 q^{29} -7910568 q^{31} + 648032 i q^{32} -1186488 i q^{33} + 9704156 q^{34} -183708 q^{36} -20992558 i q^{37} -9720040 i q^{38} + 3070386 q^{39} + 13285562 q^{41} + 10670616 i q^{42} + 23130764 i q^{43} -410144 q^{44} + 49810992 q^{46} -13873688 i q^{47} -20008944 i q^{48} + 4497463 q^{49} + 35728938 q^{51} -1061368 i q^{52} + 57635174 i q^{53} + 11691702 q^{54} + 71137440 q^{56} -35787420 i q^{57} + 23085700 i q^{58} + 32042120 q^{59} + 110664022 q^{61} -174032496 i q^{62} + 39287268 i q^{63} -140732992 q^{64} + 26102736 q^{66} -118568268 i q^{67} -12350744 i q^{68} + 183395016 q^{69} + 276679712 q^{71} -77944680 i q^{72} + 264023294 i q^{73} + 461836276 q^{74} -12370960 q^{76} + 87712224 i q^{77} + 67548492 i q^{78} -448202760 q^{79} + 43046721 q^{81} + 292282364 i q^{82} -851015796 i q^{83} + 13580784 q^{84} -508876808 q^{86} + 84997350 i q^{87} -174018240 i q^{88} -189894930 q^{89} -226981128 q^{91} -63395808 i q^{92} -640756008 i q^{93} + 305221136 q^{94} -52490592 q^{96} -1014149278 i q^{97} + 98944186 i q^{98} + 96105528 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 56q^{4} - 3564q^{6} - 13122q^{9} + O(q^{10}) \) \( 2q + 56q^{4} - 3564q^{6} - 13122q^{9} - 29296q^{11} + 263472q^{14} - 494048q^{16} - 883640q^{19} + 970056q^{21} - 1924560q^{24} + 1667864q^{26} + 2098700q^{29} - 15821136q^{31} + 19408312q^{34} - 367416q^{36} + 6140772q^{39} + 26571124q^{41} - 820288q^{44} + 99621984q^{46} + 8994926q^{49} + 71457876q^{51} + 23383404q^{54} + 142274880q^{56} + 64084240q^{59} + 221328044q^{61} - 281465984q^{64} + 52205472q^{66} + 366790032q^{69} + 553359424q^{71} + 923672552q^{74} - 24741920q^{76} - 896405520q^{79} + 86093442q^{81} + 27161568q^{84} - 1017753616q^{86} - 379789860q^{89} - 453962256q^{91} + 610442272q^{94} - 104981184q^{96} + 192211056q^{99} + O(q^{100}) \)

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
1.00000i
1.00000i
22.0000i 81.0000i 28.0000 0 −1782.00 5988.00i 11880.0i −6561.00 0
49.2 22.0000i 81.0000i 28.0000 0 −1782.00 5988.00i 11880.0i −6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 484 \) acting on \(S_{10}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 540 T^{2} + 262144 T^{4} \)
$3$ \( 1 + 6561 T^{2} \)
$5$ 1
$7$ \( 1 - 44851070 T^{2} + 1628413597910449 T^{4} \)
$11$ \( ( 1 + 14648 T + 2357947691 T^{2} )^{2} \)
$13$ \( 1 - 19772133910 T^{2} + \)\(11\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 - 42608307390 T^{2} + \)\(14\!\cdots\!09\)\( T^{4} \)
$19$ \( ( 1 + 441820 T + 322687697779 T^{2} )^{2} \)
$23$ \( 1 + 1524006503570 T^{2} + \)\(32\!\cdots\!69\)\( T^{4} \)
$29$ \( ( 1 - 1049350 T + 14507145975869 T^{2} )^{2} \)
$31$ \( ( 1 + 7910568 T + 26439622160671 T^{2} )^{2} \)
$37$ \( 1 + 180764011793210 T^{2} + \)\(16\!\cdots\!29\)\( T^{4} \)
$41$ \( ( 1 - 13285562 T + 327381934393961 T^{2} )^{2} \)
$43$ \( 1 - 470152980649990 T^{2} + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 2045781727484190 T^{2} + \)\(12\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 - 3277713901593990 T^{2} + \)\(10\!\cdots\!89\)\( T^{4} \)
$59$ \( ( 1 - 32042120 T + 8662995818654939 T^{2} )^{2} \)
$61$ \( ( 1 - 110664022 T + 11694146092834141 T^{2} )^{2} \)
$67$ \( 1 - 40354634616070070 T^{2} + \)\(74\!\cdots\!09\)\( T^{4} \)
$71$ \( ( 1 - 276679712 T + 45848500718449031 T^{2} )^{2} \)
$73$ \( 1 - 48034873641925390 T^{2} + \)\(34\!\cdots\!69\)\( T^{4} \)
$79$ \( ( 1 + 448202760 T + 119851595982618319 T^{2} )^{2} \)
$83$ \( 1 + 350347374506432810 T^{2} + \)\(34\!\cdots\!09\)\( T^{4} \)
$89$ \( ( 1 + 189894930 T + 350356403707485209 T^{2} )^{2} \)
$97$ \( 1 - 491963359241209150 T^{2} + \)\(57\!\cdots\!89\)\( T^{4} \)
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