Properties

Label 75.3.j.a
Level $75$
Weight $3$
Character orbit 75.j
Analytic conductor $2.044$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.j (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72q - q^{3} + 26q^{4} - 11q^{6} - 8q^{7} - 13q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 72q - q^{3} + 26q^{4} - 11q^{6} - 8q^{7} - 13q^{9} - 20q^{10} + 31q^{12} - 42q^{13} + 45q^{15} - 130q^{16} + 30q^{18} - 36q^{19} - 60q^{21} - 70q^{22} - 72q^{24} + 100q^{25} - 154q^{27} - 62q^{28} + 15q^{30} + 114q^{31} - 10q^{33} + 178q^{34} + 271q^{36} - 98q^{37} - 155q^{39} - 120q^{40} - 475q^{42} - 52q^{43} + 35q^{45} + 198q^{46} - 326q^{48} + 112q^{49} + 44q^{51} + 412q^{52} + 304q^{54} + 10q^{55} + 622q^{57} + 190q^{58} + 360q^{60} - 306q^{61} + 293q^{63} + 474q^{64} + 320q^{66} + 472q^{67} + 269q^{69} - 840q^{70} + 175q^{72} + 318q^{73} - 310q^{75} + 112q^{76} + 815q^{78} - 346q^{79} - 373q^{81} - 1620q^{82} - 730q^{84} - 530q^{85} - 370q^{87} - 810q^{88} - 230q^{90} - 550q^{91} - 272q^{93} - 612q^{94} - 698q^{96} + 182q^{97} - 160q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.26850 3.12232i 0.549348 2.94927i −3.36674 + 10.3618i −3.89421 + 3.13610i −10.4548 + 4.97519i −5.08243 25.3081 8.22311i −8.39643 3.24035i 18.6259 + 5.04472i
11.2 −2.06309 2.83960i 2.44039 + 1.74485i −2.57091 + 7.91245i 4.77936 1.46892i −0.0800814 10.5295i 6.80412 14.4196 4.68521i 2.91102 + 8.51622i −14.0314 10.5409i
11.3 −1.71799 2.36461i −2.67836 + 1.35143i −1.40382 + 4.32053i 2.09404 + 4.54038i 7.79701 + 4.01154i 3.73689 1.50905 0.490320i 5.34726 7.23926i 7.13868 12.7519i
11.4 −1.49360 2.05577i −0.154094 + 2.99604i −0.759266 + 2.33678i −4.43527 2.30832i 6.38932 4.15811i −8.13443 −3.72888 + 1.21159i −8.95251 0.923341i 1.87916 + 12.5656i
11.5 −1.37909 1.89816i −1.85450 2.35814i −0.465043 + 1.43126i 3.05671 3.95684i −1.91859 + 6.77224i −3.37306 −5.56759 + 1.80902i −2.12163 + 8.74635i −11.7262 0.345267i
11.6 −1.06164 1.46122i 2.88836 0.810770i 0.227982 0.701657i −4.39665 2.38107i −4.25111 3.35979i 6.57297 −8.13837 + 2.64432i 7.68530 4.68360i 1.18838 + 8.95230i
11.7 −0.733747 1.00992i 1.70510 2.46833i 0.754522 2.32218i 3.80815 + 3.24006i −3.74392 + 0.0891202i −0.783313 −7.64775 + 2.48490i −3.18527 8.41748i 0.477973 6.22330i
11.8 −0.146095 0.201083i −2.97010 0.422474i 1.21698 3.74547i −3.70242 + 3.36037i 0.348966 + 0.658958i −11.1634 −1.87649 + 0.609710i 8.64303 + 2.50958i 1.21662 + 0.253561i
11.9 −0.0749426 0.103150i 1.19040 + 2.75372i 1.23104 3.78877i −0.601131 + 4.96373i 0.194834 0.329160i 8.18661 −0.968106 + 0.314557i −6.16592 + 6.55603i 0.557058 0.309989i
11.10 0.0749426 + 0.103150i −2.58164 + 1.52811i 1.23104 3.78877i 0.601131 4.96373i −0.351099 0.151775i 8.18661 0.968106 0.314557i 4.32978 7.89006i 0.557058 0.309989i
11.11 0.146095 + 0.201083i 2.65119 + 1.40399i 1.21698 3.74547i 3.70242 3.36037i 0.105007 + 0.738225i −11.1634 1.87649 0.609710i 5.05760 + 7.44451i 1.21662 + 0.253561i
11.12 0.733747 + 1.00992i 0.0713918 2.99915i 0.754522 2.32218i −3.80815 3.24006i 3.08127 2.12852i −0.783313 7.64775 2.48490i −8.98981 0.428229i 0.477973 6.22330i
11.13 1.06164 + 1.46122i −1.86018 2.35367i 0.227982 0.701657i 4.39665 + 2.38107i 1.46439 5.21687i 6.57297 8.13837 2.64432i −2.07948 + 8.75647i 1.18838 + 8.95230i
11.14 1.37909 + 1.89816i 2.88640 0.817724i −0.465043 + 1.43126i −3.05671 + 3.95684i 5.53279 + 4.35114i −3.37306 5.56759 1.80902i 7.66265 4.72056i −11.7262 0.345267i
11.15 1.49360 + 2.05577i −1.63636 + 2.51442i −0.759266 + 2.33678i 4.43527 + 2.30832i −7.61315 + 0.391563i −8.13443 3.72888 1.21159i −3.64463 8.22902i 1.87916 + 12.5656i
11.16 1.71799 + 2.36461i 1.37249 + 2.66763i −1.40382 + 4.32053i −2.09404 4.54038i −3.94999 + 7.82837i 3.73689 −1.50905 + 0.490320i −5.23255 + 7.32260i 7.13868 12.7519i
11.17 2.06309 + 2.83960i −2.99991 0.0228156i −2.57091 + 7.91245i −4.77936 + 1.46892i −6.12430 8.56561i 6.80412 −14.4196 + 4.68521i 8.99896 + 0.136890i −14.0314 10.5409i
11.18 2.26850 + 3.12232i 1.28911 2.70891i −3.36674 + 10.3618i 3.89421 3.13610i 11.3824 2.12015i −5.08243 −25.3081 + 8.22311i −5.67640 6.98416i 18.6259 + 5.04472i
41.1 −2.26850 + 3.12232i 0.549348 + 2.94927i −3.36674 10.3618i −3.89421 3.13610i −10.4548 4.97519i −5.08243 25.3081 + 8.22311i −8.39643 + 3.24035i 18.6259 5.04472i
41.2 −2.06309 + 2.83960i 2.44039 1.74485i −2.57091 7.91245i 4.77936 + 1.46892i −0.0800814 + 10.5295i 6.80412 14.4196 + 4.68521i 2.91102 8.51622i −14.0314 + 10.5409i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.j.a 72
3.b odd 2 1 inner 75.3.j.a 72
5.b even 2 1 375.3.j.a 72
5.c odd 4 2 375.3.h.b 144
15.d odd 2 1 375.3.j.a 72
15.e even 4 2 375.3.h.b 144
25.d even 5 1 inner 75.3.j.a 72
25.e even 10 1 375.3.j.a 72
25.f odd 20 2 375.3.h.b 144
75.h odd 10 1 375.3.j.a 72
75.j odd 10 1 inner 75.3.j.a 72
75.l even 20 2 375.3.h.b 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.j.a 72 1.a even 1 1 trivial
75.3.j.a 72 3.b odd 2 1 inner
75.3.j.a 72 25.d even 5 1 inner
75.3.j.a 72 75.j odd 10 1 inner
375.3.h.b 144 5.c odd 4 2
375.3.h.b 144 15.e even 4 2
375.3.h.b 144 25.f odd 20 2
375.3.h.b 144 75.l even 20 2
375.3.j.a 72 5.b even 2 1
375.3.j.a 72 15.d odd 2 1
375.3.j.a 72 25.e even 10 1
375.3.j.a 72 75.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(75, [\chi])\).