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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(32,75)] 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("75.32"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,150,0,0,-4548] 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: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 32.11
Character \(\chi\) \(=\) 75.32
Dual form 75.10.e.f.68.11

$q$-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+(10.0225 - 10.0225i) q^{2} +(-112.058 - 84.4162i) q^{3} +311.097i q^{4} +(-1969.17 + 277.038i) q^{6} +(-2707.78 - 2707.78i) q^{7} +(8249.53 + 8249.53i) q^{8} +(5430.82 + 18919.0i) q^{9} +25188.1i q^{11} +(26261.6 - 34860.8i) q^{12} +(131855. - 131855. i) q^{13} -54277.7 q^{14} +6080.90 q^{16} +(-274335. + 274335. i) q^{17} +(244047. + 135185. i) q^{18} -504083. i q^{19} +(74846.9 + 532007. i) q^{21} +(252449. + 252449. i) q^{22} +(-357367. - 357367. i) q^{23} +(-228029. - 1.62082e6i) q^{24} -2.64304e6i q^{26} +(988501. - 2.57846e6i) q^{27} +(842382. - 842382. i) q^{28} +1.44978e6 q^{29} -1.57256e6 q^{31} +(-4.16281e6 + 4.16281e6i) q^{32} +(2.12629e6 - 2.82252e6i) q^{33} +5.49908e6i q^{34} +(-5.88563e6 + 1.68951e6i) q^{36} +(-1.41364e7 - 1.41364e7i) q^{37} +(-5.05220e6 - 5.05220e6i) q^{38} +(-2.59060e7 + 3.64466e6i) q^{39} -1.52870e7i q^{41} +(6.08223e6 + 4.58191e6i) q^{42} +(-1.45503e7 + 1.45503e7i) q^{43} -7.83596e6 q^{44} -7.16345e6 q^{46} +(-2.49426e7 + 2.49426e7i) q^{47} +(-681411. - 513326. i) q^{48} -2.56895e7i q^{49} +(5.38997e7 - 7.58302e6i) q^{51} +(4.10196e7 + 4.10196e7i) q^{52} +(-2.38697e7 - 2.38697e7i) q^{53} +(-1.59354e7 - 3.57501e7i) q^{54} -4.46758e7i q^{56} +(-4.25528e7 + 5.64864e7i) q^{57} +(1.45305e7 - 1.45305e7i) q^{58} -1.41279e8 q^{59} +2.77276e7 q^{61} +(-1.57611e7 + 1.57611e7i) q^{62} +(3.65229e7 - 6.59338e7i) q^{63} +8.65574e7i q^{64} +(-6.97806e6 - 4.95997e7i) q^{66} +(-5.16735e7 - 5.16735e7i) q^{67} +(-8.53449e7 - 8.53449e7i) q^{68} +(9.87812e6 + 7.02132e7i) q^{69} -2.50202e8i q^{71} +(-1.11271e8 + 2.00874e8i) q^{72} +(8.81264e7 - 8.81264e7i) q^{73} -2.83365e8 q^{74} +1.56819e8 q^{76} +(6.82039e7 - 6.82039e7i) q^{77} +(-2.23116e8 + 2.96173e8i) q^{78} -3.89035e8i q^{79} +(-3.28433e8 + 2.05491e8i) q^{81} +(-1.53214e8 - 1.53214e8i) q^{82} +(-9.47356e6 - 9.47356e6i) q^{83} +(-1.65506e8 + 2.32846e7i) q^{84} +2.91662e8i q^{86} +(-1.62459e8 - 1.22385e8i) q^{87} +(-2.07790e8 + 2.07790e8i) q^{88} +4.88281e8 q^{89} -7.14067e8 q^{91} +(1.11176e8 - 1.11176e8i) q^{92} +(1.76218e8 + 1.32750e8i) q^{93} +4.99976e8i q^{94} +(8.17884e8 - 1.15066e8i) q^{96} +(-7.00398e7 - 7.00398e7i) q^{97} +(-2.57474e8 - 2.57474e8i) q^{98} +(-4.76533e8 + 1.36792e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 150 q^{3} - 4548 q^{6} + 9760 q^{7} - 191940 q^{12} - 114260 q^{13} - 1494724 q^{16} - 994800 q^{18} + 592212 q^{21} - 308260 q^{22} - 4786830 q^{27} + 12953300 q^{28} + 16124104 q^{31} - 19638120 q^{33}+ \cdots - 3987642200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0225 − 10.0225i 0.442938 − 0.442938i −0.450060 − 0.892998i \(-0.648597\pi\)
0.892998 + 0.450060i \(0.148597\pi\)
\(3\) −112.058 − 84.4162i −0.798722 − 0.601700i
\(4\) 311.097i 0.607611i
\(5\) 0 0
\(6\) −1969.17 + 277.038i −0.620300 + 0.0872686i
\(7\) −2707.78 − 2707.78i −0.426257 − 0.426257i 0.461094 − 0.887351i \(-0.347457\pi\)
−0.887351 + 0.461094i \(0.847457\pi\)
\(8\) 8249.53 + 8249.53i 0.712073 + 0.712073i
\(9\) 5430.82 + 18919.0i 0.275914 + 0.961182i
\(10\) 0 0
\(11\) 25188.1i 0.518715i 0.965781 + 0.259358i \(0.0835109\pi\)
−0.965781 + 0.259358i \(0.916489\pi\)
\(12\) 26261.6 − 34860.8i 0.365600 − 0.485313i
\(13\) 131855. − 131855.i 1.28042 − 1.28042i 0.339984 − 0.940431i \(-0.389578\pi\)
0.940431 − 0.339984i \(-0.110422\pi\)
\(14\) −54277.7 −0.377611
\(15\) 0 0
\(16\) 6080.90 0.0231968
\(17\) −274335. + 274335.i −0.796639 + 0.796639i −0.982564 − 0.185925i \(-0.940472\pi\)
0.185925 + 0.982564i \(0.440472\pi\)
\(18\) 244047. + 135185.i 0.547957 + 0.303531i
\(19\) − 504083.i − 0.887383i −0.896180 − 0.443691i \(-0.853669\pi\)
0.896180 − 0.443691i \(-0.146331\pi\)
\(20\) 0 0
\(21\) 74846.9 + 532007.i 0.0839821 + 0.596940i
\(22\) 252449. + 252449.i 0.229759 + 0.229759i
\(23\) −357367. − 357367.i −0.266280 − 0.266280i 0.561319 − 0.827599i \(-0.310294\pi\)
−0.827599 + 0.561319i \(0.810294\pi\)
\(24\) −228029. − 1.62082e6i −0.140294 − 0.997202i
\(25\) 0 0
\(26\) − 2.64304e6i − 1.13429i
\(27\) 988501. − 2.57846e6i 0.357965 − 0.933735i
\(28\) 842382. − 842382.i 0.258999 − 0.258999i
\(29\) 1.44978e6 0.380636 0.190318 − 0.981722i \(-0.439048\pi\)
0.190318 + 0.981722i \(0.439048\pi\)
\(30\) 0 0
\(31\) −1.57256e6 −0.305831 −0.152915 − 0.988239i \(-0.548866\pi\)
−0.152915 + 0.988239i \(0.548866\pi\)
\(32\) −4.16281e6 + 4.16281e6i −0.701798 + 0.701798i
\(33\) 2.12629e6 − 2.82252e6i 0.312111 − 0.414309i
\(34\) 5.49908e6i 0.705723i
\(35\) 0 0
\(36\) −5.88563e6 + 1.68951e6i −0.584025 + 0.167649i
\(37\) −1.41364e7 − 1.41364e7i −1.24002 − 1.24002i −0.959990 − 0.280034i \(-0.909654\pi\)
−0.280034 − 0.959990i \(-0.590346\pi\)
\(38\) −5.05220e6 − 5.05220e6i −0.393056 − 0.393056i
\(39\) −2.59060e7 + 3.64466e6i −1.79312 + 0.252270i
\(40\) 0 0
\(41\) − 1.52870e7i − 0.844878i −0.906391 − 0.422439i \(-0.861174\pi\)
0.906391 − 0.422439i \(-0.138826\pi\)
\(42\) 6.08223e6 + 4.58191e6i 0.301607 + 0.227209i
\(43\) −1.45503e7 + 1.45503e7i −0.649027 + 0.649027i −0.952758 − 0.303731i \(-0.901768\pi\)
0.303731 + 0.952758i \(0.401768\pi\)
\(44\) −7.83596e6 −0.315177
\(45\) 0 0
\(46\) −7.16345e6 −0.235891
\(47\) −2.49426e7 + 2.49426e7i −0.745592 + 0.745592i −0.973648 − 0.228056i \(-0.926763\pi\)
0.228056 + 0.973648i \(0.426763\pi\)
\(48\) −681411. − 513326.i −0.0185278 − 0.0139575i
\(49\) − 2.56895e7i − 0.636609i
\(50\) 0 0
\(51\) 5.38997e7 − 7.58302e6i 1.11563 − 0.156955i
\(52\) 4.10196e7 + 4.10196e7i 0.777995 + 0.777995i
\(53\) −2.38697e7 − 2.38697e7i −0.415534 − 0.415534i 0.468127 − 0.883661i \(-0.344929\pi\)
−0.883661 + 0.468127i \(0.844929\pi\)
\(54\) −1.59354e7 − 3.57501e7i −0.255031 − 0.572143i
\(55\) 0 0
\(56\) − 4.46758e7i − 0.607052i
\(57\) −4.25528e7 + 5.64864e7i −0.533938 + 0.708772i
\(58\) 1.45305e7 − 1.45305e7i 0.168598 − 0.168598i
\(59\) −1.41279e8 −1.51790 −0.758951 − 0.651148i \(-0.774288\pi\)
−0.758951 + 0.651148i \(0.774288\pi\)
\(60\) 0 0
\(61\) 2.77276e7 0.256406 0.128203 − 0.991748i \(-0.459079\pi\)
0.128203 + 0.991748i \(0.459079\pi\)
\(62\) −1.57611e7 + 1.57611e7i −0.135464 + 0.135464i
\(63\) 3.65229e7 − 6.59338e7i 0.292101 − 0.527321i
\(64\) 8.65574e7i 0.644903i
\(65\) 0 0
\(66\) −6.97806e6 − 4.95997e7i −0.0452676 − 0.321759i
\(67\) −5.16735e7 − 5.16735e7i −0.313279 − 0.313279i 0.532900 − 0.846179i \(-0.321102\pi\)
−0.846179 + 0.532900i \(0.821102\pi\)
\(68\) −8.53449e7 − 8.53449e7i −0.484047 − 0.484047i
\(69\) 9.87812e6 + 7.02132e7i 0.0524630 + 0.372905i
\(70\) 0 0
\(71\) − 2.50202e8i − 1.16850i −0.811575 − 0.584249i \(-0.801389\pi\)
0.811575 − 0.584249i \(-0.198611\pi\)
\(72\) −1.11271e8 + 2.00874e8i −0.487961 + 0.880902i
\(73\) 8.81264e7 − 8.81264e7i 0.363206 − 0.363206i −0.501786 − 0.864992i \(-0.667323\pi\)
0.864992 + 0.501786i \(0.167323\pi\)
\(74\) −2.83365e8 −1.09851
\(75\) 0 0
\(76\) 1.56819e8 0.539184
\(77\) 6.82039e7 − 6.82039e7i 0.221106 − 0.221106i
\(78\) −2.23116e8 + 2.96173e8i −0.682502 + 0.905982i
\(79\) − 3.89035e8i − 1.12374i −0.827225 − 0.561871i \(-0.810082\pi\)
0.827225 − 0.561871i \(-0.189918\pi\)
\(80\) 0 0
\(81\) −3.28433e8 + 2.05491e8i −0.847743 + 0.530407i
\(82\) −1.53214e8 − 1.53214e8i −0.374229 − 0.374229i
\(83\) −9.47356e6 − 9.47356e6i −0.0219110 − 0.0219110i 0.696066 − 0.717977i \(-0.254932\pi\)
−0.717977 + 0.696066i \(0.754932\pi\)
\(84\) −1.65506e8 + 2.32846e7i −0.362708 + 0.0510285i
\(85\) 0 0
\(86\) 2.91662e8i 0.574958i
\(87\) −1.62459e8 − 1.22385e8i −0.304023 − 0.229029i
\(88\) −2.07790e8 + 2.07790e8i −0.369363 + 0.369363i
\(89\) 4.88281e8 0.824925 0.412462 − 0.910975i \(-0.364669\pi\)
0.412462 + 0.910975i \(0.364669\pi\)
\(90\) 0 0
\(91\) −7.14067e8 −1.09157
\(92\) 1.11176e8 − 1.11176e8i 0.161795 − 0.161795i
\(93\) 1.76218e8 + 1.32750e8i 0.244274 + 0.184018i
\(94\) 4.99976e8i 0.660502i
\(95\) 0 0
\(96\) 8.17884e8 − 1.15066e8i 0.982813 − 0.138270i
\(97\) −7.00398e7 − 7.00398e7i −0.0803289 − 0.0803289i 0.665801 − 0.746130i \(-0.268090\pi\)
−0.746130 + 0.665801i \(0.768090\pi\)
\(98\) −2.57474e8 − 2.57474e8i −0.281979 − 0.281979i
\(99\) −4.76533e8 + 1.36792e8i −0.498580 + 0.143121i
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 75.10.e.f.32.11 32
3.2 odd 2 inner 75.10.e.f.32.6 32
5.2 odd 4 15.10.e.a.8.11 yes 32
5.3 odd 4 inner 75.10.e.f.68.6 32
5.4 even 2 15.10.e.a.2.6 ✓ 32
15.2 even 4 15.10.e.a.8.6 yes 32
15.8 even 4 inner 75.10.e.f.68.11 32
15.14 odd 2 15.10.e.a.2.11 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.e.a.2.6 ✓ 32 5.4 even 2
15.10.e.a.2.11 yes 32 15.14 odd 2
15.10.e.a.8.6 yes 32 15.2 even 4
15.10.e.a.8.11 yes 32 5.2 odd 4
75.10.e.f.32.6 32 3.2 odd 2 inner
75.10.e.f.32.11 32 1.1 even 1 trivial
75.10.e.f.68.6 32 5.3 odd 4 inner
75.10.e.f.68.11 32 15.8 even 4 inner