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Properties

Label 75.3.f.a.43.2
Level $75$
Weight $3$
Character 75.43
Analytic conductor $2.044$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,3,Mod(7,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.7"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.43
Dual form 75.3.f.a.7.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 - 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} +1.00000i q^{4} +3.00000 q^{6} +(7.34847 - 7.34847i) q^{7} +(6.12372 + 6.12372i) q^{8} +3.00000i q^{9} -18.0000 q^{11} +(-1.22474 + 1.22474i) q^{12} +(-7.34847 - 7.34847i) q^{13} -18.0000i q^{14} +11.0000 q^{16} +(-4.89898 + 4.89898i) q^{17} +(3.67423 + 3.67423i) q^{18} +10.0000i q^{19} +18.0000 q^{21} +(-22.0454 + 22.0454i) q^{22} +(-19.5959 - 19.5959i) q^{23} +15.0000i q^{24} -18.0000 q^{26} +(-3.67423 + 3.67423i) q^{27} +(7.34847 + 7.34847i) q^{28} +22.0000 q^{31} +(-11.0227 + 11.0227i) q^{32} +(-22.0454 - 22.0454i) q^{33} +12.0000i q^{34} -3.00000 q^{36} +(7.34847 - 7.34847i) q^{37} +(12.2474 + 12.2474i) q^{38} -18.0000i q^{39} -18.0000 q^{41} +(22.0454 - 22.0454i) q^{42} +(29.3939 + 29.3939i) q^{43} -18.0000i q^{44} -48.0000 q^{46} +(44.0908 - 44.0908i) q^{47} +(13.4722 + 13.4722i) q^{48} -59.0000i q^{49} -12.0000 q^{51} +(7.34847 - 7.34847i) q^{52} +(4.89898 + 4.89898i) q^{53} +9.00000i q^{54} +90.0000 q^{56} +(-12.2474 + 12.2474i) q^{57} +90.0000i q^{59} +2.00000 q^{61} +(26.9444 - 26.9444i) q^{62} +(22.0454 + 22.0454i) q^{63} +71.0000i q^{64} -54.0000 q^{66} +(44.0908 - 44.0908i) q^{67} +(-4.89898 - 4.89898i) q^{68} -48.0000i q^{69} +72.0000 q^{71} +(-18.3712 + 18.3712i) q^{72} +(-44.0908 - 44.0908i) q^{73} -18.0000i q^{74} -10.0000 q^{76} +(-132.272 + 132.272i) q^{77} +(-22.0454 - 22.0454i) q^{78} -70.0000i q^{79} -9.00000 q^{81} +(-22.0454 + 22.0454i) q^{82} +(53.8888 + 53.8888i) q^{83} +18.0000i q^{84} +72.0000 q^{86} +(-110.227 - 110.227i) q^{88} +90.0000i q^{89} -108.000 q^{91} +(19.5959 - 19.5959i) q^{92} +(26.9444 + 26.9444i) q^{93} -108.000i q^{94} -27.0000 q^{96} +(-102.879 + 102.879i) q^{97} +(-72.2599 - 72.2599i) q^{98} -54.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{6} - 72 q^{11} + 44 q^{16} + 72 q^{21} - 72 q^{26} + 88 q^{31} - 12 q^{36} - 72 q^{41} - 192 q^{46} - 48 q^{51} + 360 q^{56} + 8 q^{61} - 216 q^{66} + 288 q^{71} - 40 q^{76} - 36 q^{81} + 288 q^{86}+ \cdots - 108 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22474 − 1.22474i 0.612372 − 0.612372i −0.331191 − 0.943564i \(-0.607451\pi\)
0.943564 + 0.331191i \(0.107451\pi\)
\(3\) 1.22474 + 1.22474i 0.408248 + 0.408248i
\(4\) 1.00000i 0.250000i
\(5\) 0 0
\(6\) 3.00000 0.500000
\(7\) 7.34847 − 7.34847i 1.04978 − 1.04978i 0.0510871 − 0.998694i \(-0.483731\pi\)
0.998694 − 0.0510871i \(-0.0162686\pi\)
\(8\) 6.12372 + 6.12372i 0.765466 + 0.765466i
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −18.0000 −1.63636 −0.818182 − 0.574960i \(-0.805018\pi\)
−0.818182 + 0.574960i \(0.805018\pi\)
\(12\) −1.22474 + 1.22474i −0.102062 + 0.102062i
\(13\) −7.34847 − 7.34847i −0.565267 − 0.565267i 0.365532 − 0.930799i \(-0.380887\pi\)
−0.930799 + 0.365532i \(0.880887\pi\)
\(14\) − 18.0000i − 1.28571i
\(15\) 0 0
\(16\) 11.0000 0.687500
\(17\) −4.89898 + 4.89898i −0.288175 + 0.288175i −0.836358 − 0.548183i \(-0.815320\pi\)
0.548183 + 0.836358i \(0.315320\pi\)
\(18\) 3.67423 + 3.67423i 0.204124 + 0.204124i
\(19\) 10.0000i 0.526316i 0.964753 + 0.263158i \(0.0847640\pi\)
−0.964753 + 0.263158i \(0.915236\pi\)
\(20\) 0 0
\(21\) 18.0000 0.857143
\(22\) −22.0454 + 22.0454i −1.00206 + 1.00206i
\(23\) −19.5959 − 19.5959i −0.851996 − 0.851996i 0.138382 − 0.990379i \(-0.455810\pi\)
−0.990379 + 0.138382i \(0.955810\pi\)
\(24\) 15.0000i 0.625000i
\(25\) 0 0
\(26\) −18.0000 −0.692308
\(27\) −3.67423 + 3.67423i −0.136083 + 0.136083i
\(28\) 7.34847 + 7.34847i 0.262445 + 0.262445i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 22.0000 0.709677 0.354839 − 0.934928i \(-0.384536\pi\)
0.354839 + 0.934928i \(0.384536\pi\)
\(32\) −11.0227 + 11.0227i −0.344459 + 0.344459i
\(33\) −22.0454 − 22.0454i −0.668043 − 0.668043i
\(34\) 12.0000i 0.352941i
\(35\) 0 0
\(36\) −3.00000 −0.0833333
\(37\) 7.34847 − 7.34847i 0.198607 − 0.198607i −0.600795 − 0.799403i \(-0.705149\pi\)
0.799403 + 0.600795i \(0.205149\pi\)
\(38\) 12.2474 + 12.2474i 0.322301 + 0.322301i
\(39\) − 18.0000i − 0.461538i
\(40\) 0 0
\(41\) −18.0000 −0.439024 −0.219512 − 0.975610i \(-0.570447\pi\)
−0.219512 + 0.975610i \(0.570447\pi\)
\(42\) 22.0454 − 22.0454i 0.524891 − 0.524891i
\(43\) 29.3939 + 29.3939i 0.683579 + 0.683579i 0.960805 − 0.277226i \(-0.0894151\pi\)
−0.277226 + 0.960805i \(0.589415\pi\)
\(44\) − 18.0000i − 0.409091i
\(45\) 0 0
\(46\) −48.0000 −1.04348
\(47\) 44.0908 − 44.0908i 0.938102 − 0.938102i −0.0600905 − 0.998193i \(-0.519139\pi\)
0.998193 + 0.0600905i \(0.0191389\pi\)
\(48\) 13.4722 + 13.4722i 0.280671 + 0.280671i
\(49\) − 59.0000i − 1.20408i
\(50\) 0 0
\(51\) −12.0000 −0.235294
\(52\) 7.34847 − 7.34847i 0.141317 − 0.141317i
\(53\) 4.89898 + 4.89898i 0.0924336 + 0.0924336i 0.751812 − 0.659378i \(-0.229180\pi\)
−0.659378 + 0.751812i \(0.729180\pi\)
\(54\) 9.00000i 0.166667i
\(55\) 0 0
\(56\) 90.0000 1.60714
\(57\) −12.2474 + 12.2474i −0.214868 + 0.214868i
\(58\) 0 0
\(59\) 90.0000i 1.52542i 0.646738 + 0.762712i \(0.276133\pi\)
−0.646738 + 0.762712i \(0.723867\pi\)
\(60\) 0 0
\(61\) 2.00000 0.0327869 0.0163934 − 0.999866i \(-0.494782\pi\)
0.0163934 + 0.999866i \(0.494782\pi\)
\(62\) 26.9444 − 26.9444i 0.434587 − 0.434587i
\(63\) 22.0454 + 22.0454i 0.349927 + 0.349927i
\(64\) 71.0000i 1.10938i
\(65\) 0 0
\(66\) −54.0000 −0.818182
\(67\) 44.0908 − 44.0908i 0.658072 − 0.658072i −0.296852 − 0.954924i \(-0.595937\pi\)
0.954924 + 0.296852i \(0.0959367\pi\)
\(68\) −4.89898 − 4.89898i −0.0720438 − 0.0720438i
\(69\) − 48.0000i − 0.695652i
\(70\) 0 0
\(71\) 72.0000 1.01408 0.507042 − 0.861921i \(-0.330739\pi\)
0.507042 + 0.861921i \(0.330739\pi\)
\(72\) −18.3712 + 18.3712i −0.255155 + 0.255155i
\(73\) −44.0908 − 44.0908i −0.603984 − 0.603984i 0.337384 − 0.941367i \(-0.390458\pi\)
−0.941367 + 0.337384i \(0.890458\pi\)
\(74\) − 18.0000i − 0.243243i
\(75\) 0 0
\(76\) −10.0000 −0.131579
\(77\) −132.272 + 132.272i −1.71782 + 1.71782i
\(78\) −22.0454 − 22.0454i −0.282633 − 0.282633i
\(79\) − 70.0000i − 0.886076i −0.896503 − 0.443038i \(-0.853901\pi\)
0.896503 − 0.443038i \(-0.146099\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) −22.0454 + 22.0454i −0.268846 + 0.268846i
\(83\) 53.8888 + 53.8888i 0.649262 + 0.649262i 0.952815 − 0.303552i \(-0.0981727\pi\)
−0.303552 + 0.952815i \(0.598173\pi\)
\(84\) 18.0000i 0.214286i
\(85\) 0 0
\(86\) 72.0000 0.837209
\(87\) 0 0
\(88\) −110.227 − 110.227i −1.25258 − 1.25258i
\(89\) 90.0000i 1.01124i 0.862757 + 0.505618i \(0.168735\pi\)
−0.862757 + 0.505618i \(0.831265\pi\)
\(90\) 0 0
\(91\) −108.000 −1.18681
\(92\) 19.5959 − 19.5959i 0.212999 − 0.212999i
\(93\) 26.9444 + 26.9444i 0.289725 + 0.289725i
\(94\) − 108.000i − 1.14894i
\(95\) 0 0
\(96\) −27.0000 −0.281250
\(97\) −102.879 + 102.879i −1.06060 + 1.06060i −0.0625628 + 0.998041i \(0.519927\pi\)
−0.998041 + 0.0625628i \(0.980073\pi\)
\(98\) −72.2599 − 72.2599i −0.737346 − 0.737346i
\(99\) − 54.0000i − 0.545455i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.3.f.a.43.2 yes 4
3.2 odd 2 225.3.g.f.118.1 4
4.3 odd 2 1200.3.bg.j.193.1 4
5.2 odd 4 inner 75.3.f.a.7.2 yes 4
5.3 odd 4 inner 75.3.f.a.7.1 ✓ 4
5.4 even 2 inner 75.3.f.a.43.1 yes 4
15.2 even 4 225.3.g.f.82.1 4
15.8 even 4 225.3.g.f.82.2 4
15.14 odd 2 225.3.g.f.118.2 4
20.3 even 4 1200.3.bg.j.1057.2 4
20.7 even 4 1200.3.bg.j.1057.1 4
20.19 odd 2 1200.3.bg.j.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.3.f.a.7.1 ✓ 4 5.3 odd 4 inner
75.3.f.a.7.2 yes 4 5.2 odd 4 inner
75.3.f.a.43.1 yes 4 5.4 even 2 inner
75.3.f.a.43.2 yes 4 1.1 even 1 trivial
225.3.g.f.82.1 4 15.2 even 4
225.3.g.f.82.2 4 15.8 even 4
225.3.g.f.118.1 4 3.2 odd 2
225.3.g.f.118.2 4 15.14 odd 2
1200.3.bg.j.193.1 4 4.3 odd 2
1200.3.bg.j.193.2 4 20.19 odd 2
1200.3.bg.j.1057.1 4 20.7 even 4
1200.3.bg.j.1057.2 4 20.3 even 4