Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,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, 10); 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, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6276877123\)
Analytic rank: \(1\)
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: \( x^{4} - x^{3} - 469x^{2} + 4449x - 5580 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.48693\) 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)
 
\(f(q)\) \(=\) \(q+29.7516 q^{2} +81.0000 q^{3} +373.156 q^{4} +2409.88 q^{6} -10707.3 q^{7} -4130.82 q^{8} +6561.00 q^{9} +54425.4 q^{11} +30225.7 q^{12} -53507.4 q^{13} -318559. q^{14} -313954. q^{16} -644691. q^{17} +195200. q^{18} -210320. q^{19} -867290. q^{21} +1.61924e6 q^{22} -95214.7 q^{23} -334596. q^{24} -1.59193e6 q^{26} +531441. q^{27} -3.99549e6 q^{28} -2.25330e6 q^{29} -548893. q^{31} -7.22566e6 q^{32} +4.40846e6 q^{33} -1.91806e7 q^{34} +2.44828e6 q^{36} +1.19273e7 q^{37} -6.25734e6 q^{38} -4.33410e6 q^{39} +1.48678e7 q^{41} -2.58033e7 q^{42} -2.80141e7 q^{43} +2.03092e7 q^{44} -2.83279e6 q^{46} -5.85976e6 q^{47} -2.54303e7 q^{48} +7.42924e7 q^{49} -5.22200e7 q^{51} -1.99666e7 q^{52} -5.05528e7 q^{53} +1.58112e7 q^{54} +4.42298e7 q^{56} -1.70359e7 q^{57} -6.70392e7 q^{58} -5.84637e6 q^{59} -1.07474e7 q^{61} -1.63304e7 q^{62} -7.02505e7 q^{63} -5.42302e7 q^{64} +1.31159e8 q^{66} +7.46365e7 q^{67} -2.40571e8 q^{68} -7.71239e6 q^{69} +4.06259e7 q^{71} -2.71023e7 q^{72} +3.31290e8 q^{73} +3.54857e8 q^{74} -7.84821e7 q^{76} -5.82749e8 q^{77} -1.28946e8 q^{78} -8.40451e7 q^{79} +4.30467e7 q^{81} +4.42342e8 q^{82} -6.88232e8 q^{83} -3.23635e8 q^{84} -8.33465e8 q^{86} -1.82517e8 q^{87} -2.24821e8 q^{88} -1.04545e9 q^{89} +5.72919e8 q^{91} -3.55300e7 q^{92} -4.44603e7 q^{93} -1.74337e8 q^{94} -5.85279e8 q^{96} +1.28245e9 q^{97} +2.21032e9 q^{98} +3.57085e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 324 q^{3} + 597 q^{4} + 243 q^{6} - 9834 q^{7} - 7671 q^{8} + 26244 q^{9} - 35994 q^{11} + 48357 q^{12} - 79998 q^{13} - 208182 q^{14} - 752815 q^{16} - 667878 q^{17} + 19683 q^{18} - 425792 q^{19}+ \cdots - 236156634 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 29.7516 1.31485 0.657423 − 0.753522i \(-0.271646\pi\)
0.657423 + 0.753522i \(0.271646\pi\)
\(3\) 81.0000 0.577350
\(4\) 373.156 0.728821
\(5\) 0 0
\(6\) 2409.88 0.759127
\(7\) −10707.3 −1.68554 −0.842768 − 0.538276i \(-0.819076\pi\)
−0.842768 + 0.538276i \(0.819076\pi\)
\(8\) −4130.82 −0.356559
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) 54425.4 1.12082 0.560409 − 0.828216i \(-0.310644\pi\)
0.560409 + 0.828216i \(0.310644\pi\)
\(12\) 30225.7 0.420785
\(13\) −53507.4 −0.519599 −0.259800 − 0.965663i \(-0.583657\pi\)
−0.259800 + 0.965663i \(0.583657\pi\)
\(14\) −318559. −2.21622
\(15\) 0 0
\(16\) −313954. −1.19764
\(17\) −644691. −1.87211 −0.936056 − 0.351852i \(-0.885552\pi\)
−0.936056 + 0.351852i \(0.885552\pi\)
\(18\) 195200. 0.438282
\(19\) −210320. −0.370244 −0.185122 − 0.982716i \(-0.559268\pi\)
−0.185122 + 0.982716i \(0.559268\pi\)
\(20\) 0 0
\(21\) −867290. −0.973145
\(22\) 1.61924e6 1.47370
\(23\) −95214.7 −0.0709461 −0.0354731 − 0.999371i \(-0.511294\pi\)
−0.0354731 + 0.999371i \(0.511294\pi\)
\(24\) −334596. −0.205859
\(25\) 0 0
\(26\) −1.59193e6 −0.683193
\(27\) 531441. 0.192450
\(28\) −3.99549e6 −1.22845
\(29\) −2.25330e6 −0.591600 −0.295800 − 0.955250i \(-0.595586\pi\)
−0.295800 + 0.955250i \(0.595586\pi\)
\(30\) 0 0
\(31\) −548893. −0.106748 −0.0533741 − 0.998575i \(-0.516998\pi\)
−0.0533741 + 0.998575i \(0.516998\pi\)
\(32\) −7.22566e6 −1.21816
\(33\) 4.40846e6 0.647104
\(34\) −1.91806e7 −2.46154
\(35\) 0 0
\(36\) 2.44828e6 0.242940
\(37\) 1.19273e7 1.04625 0.523125 − 0.852256i \(-0.324766\pi\)
0.523125 + 0.852256i \(0.324766\pi\)
\(38\) −6.25734e6 −0.486815
\(39\) −4.33410e6 −0.299991
\(40\) 0 0
\(41\) 1.48678e7 0.821714 0.410857 − 0.911700i \(-0.365230\pi\)
0.410857 + 0.911700i \(0.365230\pi\)
\(42\) −2.58033e7 −1.27954
\(43\) −2.80141e7 −1.24960 −0.624798 − 0.780787i \(-0.714819\pi\)
−0.624798 + 0.780787i \(0.714819\pi\)
\(44\) 2.03092e7 0.816875
\(45\) 0 0
\(46\) −2.83279e6 −0.0932832
\(47\) −5.85976e6 −0.175162 −0.0875810 − 0.996157i \(-0.527914\pi\)
−0.0875810 + 0.996157i \(0.527914\pi\)
\(48\) −2.54303e7 −0.691458
\(49\) 7.42924e7 1.84103
\(50\) 0 0
\(51\) −5.22200e7 −1.08086
\(52\) −1.99666e7 −0.378695
\(53\) −5.05528e7 −0.880042 −0.440021 − 0.897987i \(-0.645029\pi\)
−0.440021 + 0.897987i \(0.645029\pi\)
\(54\) 1.58112e7 0.253042
\(55\) 0 0
\(56\) 4.42298e7 0.600993
\(57\) −1.70359e7 −0.213761
\(58\) −6.70392e7 −0.777863
\(59\) −5.84637e6 −0.0628134 −0.0314067 − 0.999507i \(-0.509999\pi\)
−0.0314067 + 0.999507i \(0.509999\pi\)
\(60\) 0 0
\(61\) −1.07474e7 −0.0993849 −0.0496925 − 0.998765i \(-0.515824\pi\)
−0.0496925 + 0.998765i \(0.515824\pi\)
\(62\) −1.63304e7 −0.140357
\(63\) −7.02505e7 −0.561846
\(64\) −5.42302e7 −0.404046
\(65\) 0 0
\(66\) 1.31159e8 0.850843
\(67\) 7.46365e7 0.452496 0.226248 − 0.974070i \(-0.427354\pi\)
0.226248 + 0.974070i \(0.427354\pi\)
\(68\) −2.40571e8 −1.36443
\(69\) −7.71239e6 −0.0409608
\(70\) 0 0
\(71\) 4.06259e7 0.189732 0.0948660 − 0.995490i \(-0.469758\pi\)
0.0948660 + 0.995490i \(0.469758\pi\)
\(72\) −2.71023e7 −0.118853
\(73\) 3.31290e8 1.36538 0.682692 − 0.730706i \(-0.260809\pi\)
0.682692 + 0.730706i \(0.260809\pi\)
\(74\) 3.54857e8 1.37566
\(75\) 0 0
\(76\) −7.84821e7 −0.269842
\(77\) −5.82749e8 −1.88918
\(78\) −1.28946e8 −0.394442
\(79\) −8.40451e7 −0.242767 −0.121384 − 0.992606i \(-0.538733\pi\)
−0.121384 + 0.992606i \(0.538733\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 4.42342e8 1.08043
\(83\) −6.88232e8 −1.59178 −0.795891 − 0.605440i \(-0.792997\pi\)
−0.795891 + 0.605440i \(0.792997\pi\)
\(84\) −3.23635e8 −0.709249
\(85\) 0 0
\(86\) −8.33465e8 −1.64303
\(87\) −1.82517e8 −0.341560
\(88\) −2.24821e8 −0.399637
\(89\) −1.04545e9 −1.76624 −0.883119 − 0.469149i \(-0.844561\pi\)
−0.883119 + 0.469149i \(0.844561\pi\)
\(90\) 0 0
\(91\) 5.72919e8 0.875804
\(92\) −3.55300e7 −0.0517070
\(93\) −4.44603e7 −0.0616310
\(94\) −1.74337e8 −0.230311
\(95\) 0 0
\(96\) −5.85279e8 −0.703302
\(97\) 1.28245e9 1.47085 0.735423 − 0.677608i \(-0.236983\pi\)
0.735423 + 0.677608i \(0.236983\pi\)
\(98\) 2.21032e9 2.42068
\(99\) 3.57085e8 0.373606
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.a.l.1.4 4
3.2 odd 2 225.10.a.q.1.1 4
5.2 odd 4 15.10.b.a.4.7 yes 8
5.3 odd 4 15.10.b.a.4.2 ✓ 8
5.4 even 2 75.10.a.i.1.1 4
15.2 even 4 45.10.b.c.19.2 8
15.8 even 4 45.10.b.c.19.7 8
15.14 odd 2 225.10.a.u.1.4 4
20.3 even 4 240.10.f.c.49.3 8
20.7 even 4 240.10.f.c.49.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.b.a.4.2 ✓ 8 5.3 odd 4
15.10.b.a.4.7 yes 8 5.2 odd 4
45.10.b.c.19.2 8 15.2 even 4
45.10.b.c.19.7 8 15.8 even 4
75.10.a.i.1.1 4 5.4 even 2
75.10.a.l.1.4 4 1.1 even 1 trivial
225.10.a.q.1.1 4 3.2 odd 2
225.10.a.u.1.4 4 15.14 odd 2
240.10.f.c.49.3 8 20.3 even 4
240.10.f.c.49.7 8 20.7 even 4