Properties

Label 75.12.b.a
Level $75$
Weight $12$
Character orbit 75.b
Analytic conductor $57.626$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(57.6257385420\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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.

\(f(q)\) \(=\) \( q + 78 i q^{2} + 243 i q^{3} - 4036 q^{4} - 18954 q^{6} - 27760 i q^{7} - 155064 i q^{8} - 59049 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 78 i q^{2} + 243 i q^{3} - 4036 q^{4} - 18954 q^{6} - 27760 i q^{7} - 155064 i q^{8} - 59049 q^{9} + 637836 q^{11} - 980748 i q^{12} - 766214 i q^{13} + 2165280 q^{14} + 3829264 q^{16} + 3084354 i q^{17} - 4605822 i q^{18} + 19511404 q^{19} + 6745680 q^{21} + 49751208 i q^{22} - 15312360 i q^{23} + 37680552 q^{24} + 59764692 q^{26} - 14348907 i q^{27} + 112039360 i q^{28} - 10751262 q^{29} - 50937400 q^{31} - 18888480 i q^{32} + 154994148 i q^{33} - 240579612 q^{34} + 238321764 q^{36} + 664740830 i q^{37} + 1521889512 i q^{38} + 186190002 q^{39} + 898833450 q^{41} + 526163040 i q^{42} + 957947188 i q^{43} - 2574306096 q^{44} + 1194364080 q^{46} - 1555741344 i q^{47} + 930511152 i q^{48} + 1206709143 q^{49} - 749498022 q^{51} + 3092439704 i q^{52} - 3792417030 i q^{53} + 1119214746 q^{54} - 4304576640 q^{56} + 4741271172 i q^{57} - 838598436 i q^{58} - 555306924 q^{59} + 4950420998 q^{61} - 3973117200 i q^{62} + 1639200240 i q^{63} + 9315634112 q^{64} - 12089543544 q^{66} + 5292399284 i q^{67} - 12448452744 i q^{68} + 3720903480 q^{69} - 14831086248 q^{71} + 9156374136 i q^{72} - 13971005210 i q^{73} - 51849784740 q^{74} - 78748026544 q^{76} - 17706327360 i q^{77} + 14522820156 i q^{78} - 3720542360 q^{79} + 3486784401 q^{81} + 70109009100 i q^{82} - 8768454036 i q^{83} - 27225564480 q^{84} - 74719880664 q^{86} - 2612556666 i q^{87} - 98905401504 i q^{88} + 25472769174 q^{89} - 21270100640 q^{91} + 61800684960 i q^{92} - 12377788200 i q^{93} + 121347824832 q^{94} + 4589900640 q^{96} - 39092494846 i q^{97} + 94123313154 i q^{98} - 37663577964 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8072 q^{4} - 37908 q^{6} - 118098 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8072 q^{4} - 37908 q^{6} - 118098 q^{9} + 1275672 q^{11} + 4330560 q^{14} + 7658528 q^{16} + 39022808 q^{19} + 13491360 q^{21} + 75361104 q^{24} + 119529384 q^{26} - 21502524 q^{29} - 101874800 q^{31} - 481159224 q^{34} + 476643528 q^{36} + 372380004 q^{39} + 1797666900 q^{41} - 5148612192 q^{44} + 2388728160 q^{46} + 2413418286 q^{49} - 1498996044 q^{51} + 2238429492 q^{54} - 8609153280 q^{56} - 1110613848 q^{59} + 9900841996 q^{61} + 18631268224 q^{64} - 24179087088 q^{66} + 7441806960 q^{69} - 29662172496 q^{71} - 103699569480 q^{74} - 157496053088 q^{76} - 7441084720 q^{79} + 6973568802 q^{81} - 54451128960 q^{84} - 149439761328 q^{86} + 50945538348 q^{89} - 42540201280 q^{91} + 242695649664 q^{94} + 9179801280 q^{96} - 75327155928 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
78.0000i 243.000i −4036.00 0 −18954.0 27760.0i 155064.i −59049.0 0
49.2 78.0000i 243.000i −4036.00 0 −18954.0 27760.0i 155064.i −59049.0 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 75.12.b.a 2
3.b odd 2 1 225.12.b.a 2
5.b even 2 1 inner 75.12.b.a 2
5.c odd 4 1 3.12.a.a 1
5.c odd 4 1 75.12.a.a 1
15.d odd 2 1 225.12.b.a 2
15.e even 4 1 9.12.a.a 1
15.e even 4 1 225.12.a.f 1
20.e even 4 1 48.12.a.f 1
35.f even 4 1 147.12.a.c 1
40.i odd 4 1 192.12.a.q 1
40.k even 4 1 192.12.a.g 1
45.k odd 12 2 81.12.c.a 2
45.l even 12 2 81.12.c.e 2
60.l odd 4 1 144.12.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 5.c odd 4 1
9.12.a.a 1 15.e even 4 1
48.12.a.f 1 20.e even 4 1
75.12.a.a 1 5.c odd 4 1
75.12.b.a 2 1.a even 1 1 trivial
75.12.b.a 2 5.b even 2 1 inner
81.12.c.a 2 45.k odd 12 2
81.12.c.e 2 45.l even 12 2
144.12.a.l 1 60.l odd 4 1
147.12.a.c 1 35.f even 4 1
192.12.a.g 1 40.k even 4 1
192.12.a.q 1 40.i odd 4 1
225.12.a.f 1 15.e even 4 1
225.12.b.a 2 3.b odd 2 1
225.12.b.a 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 6084 \) acting on \(S_{12}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6084 \) Copy content Toggle raw display
$3$ \( T^{2} + 59049 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 770617600 \) Copy content Toggle raw display
$11$ \( (T - 637836)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 587083893796 \) Copy content Toggle raw display
$17$ \( T^{2} + 9513239597316 \) Copy content Toggle raw display
$19$ \( (T - 19511404)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 234468368769600 \) Copy content Toggle raw display
$29$ \( (T + 10751262)^{2} \) Copy content Toggle raw display
$31$ \( (T + 50937400)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 44\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T - 898833450)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 91\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + 24\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{2} + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T + 555306924)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4950420998)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 28\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T + 14831086248)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 19\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T + 3720542360)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 76\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T - 25472769174)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 15\!\cdots\!16 \) Copy content Toggle raw display
show more
show less