Properties

Label 75.12.a.h.1.2
Level $75$
Weight $12$
Character 75.1
Self dual yes
Analytic conductor $57.626$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,12,Mod(1,75)] 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("75.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-46,-972] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.6257385420\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6154x^{2} - 41770x + 5647125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(28.7345\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q-39.7345 q^{2} -243.000 q^{3} -469.173 q^{4} +9655.47 q^{6} +8306.83 q^{7} +100019. q^{8} +59049.0 q^{9} +337742. q^{11} +114009. q^{12} +332445. q^{13} -330067. q^{14} -3.01331e6 q^{16} -4.09893e6 q^{17} -2.34628e6 q^{18} +9.11010e6 q^{19} -2.01856e6 q^{21} -1.34200e7 q^{22} -2.88443e7 q^{23} -2.43045e7 q^{24} -1.32095e7 q^{26} -1.43489e7 q^{27} -3.89734e6 q^{28} -4.37488e7 q^{29} +9.03196e7 q^{31} -8.51055e7 q^{32} -8.20712e7 q^{33} +1.62869e8 q^{34} -2.77042e7 q^{36} -7.41610e8 q^{37} -3.61985e8 q^{38} -8.07841e7 q^{39} +6.88364e8 q^{41} +8.02063e7 q^{42} +2.14585e8 q^{43} -1.58459e8 q^{44} +1.14611e9 q^{46} -1.50451e9 q^{47} +7.32235e8 q^{48} -1.90832e9 q^{49} +9.96041e8 q^{51} -1.55974e8 q^{52} +2.39637e9 q^{53} +5.70146e8 q^{54} +8.30836e8 q^{56} -2.21376e9 q^{57} +1.73833e9 q^{58} +6.31089e9 q^{59} -2.31809e9 q^{61} -3.58880e9 q^{62} +4.90510e8 q^{63} +9.55289e9 q^{64} +3.26106e9 q^{66} -2.51799e9 q^{67} +1.92311e9 q^{68} +7.00917e9 q^{69} +2.10654e9 q^{71} +5.90599e9 q^{72} +2.23379e10 q^{73} +2.94675e10 q^{74} -4.27422e9 q^{76} +2.80556e9 q^{77} +3.20991e9 q^{78} +4.01526e10 q^{79} +3.48678e9 q^{81} -2.73518e10 q^{82} +9.34317e9 q^{83} +9.47054e8 q^{84} -8.52641e9 q^{86} +1.06309e10 q^{87} +3.37804e10 q^{88} +5.40157e10 q^{89} +2.76156e9 q^{91} +1.35330e10 q^{92} -2.19477e10 q^{93} +5.97807e10 q^{94} +2.06806e10 q^{96} -1.27556e11 q^{97} +7.58262e10 q^{98} +1.99433e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 46 q^{2} - 972 q^{3} + 4648 q^{4} + 11178 q^{6} - 68372 q^{7} - 386172 q^{8} + 236196 q^{9} - 99944 q^{11} - 1129464 q^{12} - 2306276 q^{13} + 3107022 q^{14} + 7622200 q^{16} + 3443816 q^{17} - 2716254 q^{18}+ \cdots - 5901593256 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −39.7345 −0.878016 −0.439008 − 0.898483i \(-0.644670\pi\)
−0.439008 + 0.898483i \(0.644670\pi\)
\(3\) −243.000 −0.577350
\(4\) −469.173 −0.229089
\(5\) 0 0
\(6\) 9655.47 0.506923
\(7\) 8306.83 0.186808 0.0934041 − 0.995628i \(-0.470225\pi\)
0.0934041 + 0.995628i \(0.470225\pi\)
\(8\) 100019. 1.07916
\(9\) 59049.0 0.333333
\(10\) 0 0
\(11\) 337742. 0.632303 0.316151 − 0.948709i \(-0.397609\pi\)
0.316151 + 0.948709i \(0.397609\pi\)
\(12\) 114009. 0.132264
\(13\) 332445. 0.248331 0.124166 − 0.992262i \(-0.460375\pi\)
0.124166 + 0.992262i \(0.460375\pi\)
\(14\) −330067. −0.164021
\(15\) 0 0
\(16\) −3.01331e6 −0.718430
\(17\) −4.09893e6 −0.700167 −0.350084 − 0.936718i \(-0.613847\pi\)
−0.350084 + 0.936718i \(0.613847\pi\)
\(18\) −2.34628e6 −0.292672
\(19\) 9.11010e6 0.844070 0.422035 − 0.906579i \(-0.361316\pi\)
0.422035 + 0.906579i \(0.361316\pi\)
\(20\) 0 0
\(21\) −2.01856e6 −0.107854
\(22\) −1.34200e7 −0.555172
\(23\) −2.88443e7 −0.934452 −0.467226 − 0.884138i \(-0.654747\pi\)
−0.467226 + 0.884138i \(0.654747\pi\)
\(24\) −2.43045e7 −0.623053
\(25\) 0 0
\(26\) −1.32095e7 −0.218039
\(27\) −1.43489e7 −0.192450
\(28\) −3.89734e6 −0.0427956
\(29\) −4.37488e7 −0.396074 −0.198037 − 0.980195i \(-0.563457\pi\)
−0.198037 + 0.980195i \(0.563457\pi\)
\(30\) 0 0
\(31\) 9.03196e7 0.566621 0.283310 − 0.959028i \(-0.408567\pi\)
0.283310 + 0.959028i \(0.408567\pi\)
\(32\) −8.51055e7 −0.448366
\(33\) −8.20712e7 −0.365060
\(34\) 1.62869e8 0.614758
\(35\) 0 0
\(36\) −2.77042e7 −0.0763628
\(37\) −7.41610e8 −1.75819 −0.879096 − 0.476645i \(-0.841853\pi\)
−0.879096 + 0.476645i \(0.841853\pi\)
\(38\) −3.61985e8 −0.741107
\(39\) −8.07841e7 −0.143374
\(40\) 0 0
\(41\) 6.88364e8 0.927912 0.463956 − 0.885858i \(-0.346430\pi\)
0.463956 + 0.885858i \(0.346430\pi\)
\(42\) 8.02063e7 0.0946973
\(43\) 2.14585e8 0.222599 0.111299 − 0.993787i \(-0.464499\pi\)
0.111299 + 0.993787i \(0.464499\pi\)
\(44\) −1.58459e8 −0.144853
\(45\) 0 0
\(46\) 1.14611e9 0.820464
\(47\) −1.50451e9 −0.956876 −0.478438 − 0.878121i \(-0.658797\pi\)
−0.478438 + 0.878121i \(0.658797\pi\)
\(48\) 7.32235e8 0.414786
\(49\) −1.90832e9 −0.965103
\(50\) 0 0
\(51\) 9.96041e8 0.404242
\(52\) −1.55974e8 −0.0568898
\(53\) 2.39637e9 0.787111 0.393556 − 0.919301i \(-0.371245\pi\)
0.393556 + 0.919301i \(0.371245\pi\)
\(54\) 5.70146e8 0.168974
\(55\) 0 0
\(56\) 8.30836e8 0.201596
\(57\) −2.21376e9 −0.487324
\(58\) 1.73833e9 0.347759
\(59\) 6.31089e9 1.14922 0.574612 − 0.818426i \(-0.305153\pi\)
0.574612 + 0.818426i \(0.305153\pi\)
\(60\) 0 0
\(61\) −2.31809e9 −0.351412 −0.175706 − 0.984443i \(-0.556221\pi\)
−0.175706 + 0.984443i \(0.556221\pi\)
\(62\) −3.58880e9 −0.497502
\(63\) 4.90510e8 0.0622694
\(64\) 9.55289e9 1.11210
\(65\) 0 0
\(66\) 3.26106e9 0.320528
\(67\) −2.51799e9 −0.227847 −0.113923 − 0.993490i \(-0.536342\pi\)
−0.113923 + 0.993490i \(0.536342\pi\)
\(68\) 1.92311e9 0.160400
\(69\) 7.00917e9 0.539506
\(70\) 0 0
\(71\) 2.10654e9 0.138563 0.0692816 − 0.997597i \(-0.477929\pi\)
0.0692816 + 0.997597i \(0.477929\pi\)
\(72\) 5.90599e9 0.359720
\(73\) 2.23379e10 1.26115 0.630575 − 0.776128i \(-0.282819\pi\)
0.630575 + 0.776128i \(0.282819\pi\)
\(74\) 2.94675e10 1.54372
\(75\) 0 0
\(76\) −4.27422e9 −0.193367
\(77\) 2.80556e9 0.118119
\(78\) 3.20991e9 0.125885
\(79\) 4.01526e10 1.46813 0.734065 − 0.679079i \(-0.237621\pi\)
0.734065 + 0.679079i \(0.237621\pi\)
\(80\) 0 0
\(81\) 3.48678e9 0.111111
\(82\) −2.73518e10 −0.814721
\(83\) 9.34317e9 0.260354 0.130177 − 0.991491i \(-0.458445\pi\)
0.130177 + 0.991491i \(0.458445\pi\)
\(84\) 9.47054e8 0.0247081
\(85\) 0 0
\(86\) −8.52641e9 −0.195445
\(87\) 1.06309e10 0.228674
\(88\) 3.37804e10 0.682355
\(89\) 5.40157e10 1.02536 0.512678 − 0.858581i \(-0.328653\pi\)
0.512678 + 0.858581i \(0.328653\pi\)
\(90\) 0 0
\(91\) 2.76156e9 0.0463903
\(92\) 1.35330e10 0.214072
\(93\) −2.19477e10 −0.327139
\(94\) 5.97807e10 0.840152
\(95\) 0 0
\(96\) 2.06806e10 0.258864
\(97\) −1.27556e11 −1.50819 −0.754097 − 0.656763i \(-0.771925\pi\)
−0.754097 + 0.656763i \(0.771925\pi\)
\(98\) 7.58262e10 0.847375
\(99\) 1.99433e10 0.210768
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 75.12.a.h.1.2 ✓ 4
3.2 odd 2 225.12.a.s.1.3 4
5.2 odd 4 75.12.b.g.49.3 8
5.3 odd 4 75.12.b.g.49.6 8
5.4 even 2 75.12.a.i.1.3 yes 4
15.2 even 4 225.12.b.o.199.6 8
15.8 even 4 225.12.b.o.199.3 8
15.14 odd 2 225.12.a.q.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.a.h.1.2 ✓ 4 1.1 even 1 trivial
75.12.a.i.1.3 yes 4 5.4 even 2
75.12.b.g.49.3 8 5.2 odd 4
75.12.b.g.49.6 8 5.3 odd 4
225.12.a.q.1.2 4 15.14 odd 2
225.12.a.s.1.3 4 3.2 odd 2
225.12.b.o.199.3 8 15.8 even 4
225.12.b.o.199.6 8 15.2 even 4