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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,12,Mod(49,75)] 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("75.49"); 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, 1])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-9296] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.6257385420\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 2 x^{6} - 20886 x^{5} + 6668329 x^{4} - 80402480 x^{3} + 365580800 x^{2} + \cdots + 1959440040000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(20.9393 + 20.9393i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.12.b.g.49.5

$q$-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)
 
\(f(q)\) \(=\) \(q-29.8785i q^{2} -243.000i q^{3} +1155.27 q^{4} -7260.48 q^{6} -60161.8i q^{7} -95709.1i q^{8} -59049.0 q^{9} -770245. q^{11} -280732. i q^{12} +293566. i q^{13} -1.79754e6 q^{14} -493642. q^{16} -497023. i q^{17} +1.76430e6i q^{18} +1.40960e7 q^{19} -1.46193e7 q^{21} +2.30138e7i q^{22} -5.26889e6i q^{23} -2.32573e7 q^{24} +8.77130e6 q^{26} +1.43489e7i q^{27} -6.95034e7i q^{28} -1.78246e8 q^{29} -2.16036e8 q^{31} -1.81263e8i q^{32} +1.87169e8i q^{33} -1.48503e7 q^{34} -6.82178e7 q^{36} -5.23052e8i q^{37} -4.21168e8i q^{38} +7.13364e7 q^{39} +7.77482e8 q^{41} +4.36803e8i q^{42} +1.14173e9i q^{43} -8.89844e8 q^{44} -1.57427e8 q^{46} -2.31913e8i q^{47} +1.19955e8i q^{48} -1.64212e9 q^{49} -1.20777e8 q^{51} +3.39149e8i q^{52} +2.44201e9i q^{53} +4.28724e8 q^{54} -5.75803e9 q^{56} -3.42533e9i q^{57} +5.32572e9i q^{58} -3.13948e9 q^{59} +1.03302e10 q^{61} +6.45484e9i q^{62} +3.55249e9i q^{63} -6.42684e9 q^{64} +5.59234e9 q^{66} +1.56856e10i q^{67} -5.74198e8i q^{68} -1.28034e9 q^{69} +2.49558e8 q^{71} +5.65152e9i q^{72} -1.64453e10i q^{73} -1.56280e10 q^{74} +1.62848e10 q^{76} +4.63393e10i q^{77} -2.13143e9i q^{78} -3.02025e10 q^{79} +3.48678e9 q^{81} -2.32300e10i q^{82} +1.86048e10i q^{83} -1.68893e10 q^{84} +3.41133e10 q^{86} +4.33138e10i q^{87} +7.37194e10i q^{88} -8.20974e10 q^{89} +1.76614e10 q^{91} -6.08701e9i q^{92} +5.24968e10i q^{93} -6.92923e9 q^{94} -4.40469e10 q^{96} +7.91004e10i q^{97} +4.90640e10i q^{98} +4.54822e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 9296 q^{4} + 22356 q^{6} - 472392 q^{9} - 199888 q^{11} - 6214044 q^{14} + 15244400 q^{16} + 8429096 q^{19} + 33228792 q^{21} - 187679592 q^{24} - 17533444 q^{26} - 434786608 q^{29} - 652634072 q^{31}+ \cdots + 11803186512 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 29.8785i − 0.660228i −0.943941 − 0.330114i \(-0.892913\pi\)
0.943941 − 0.330114i \(-0.107087\pi\)
\(3\) − 243.000i − 0.577350i
\(4\) 1155.27 0.564099
\(5\) 0 0
\(6\) −7260.48 −0.381183
\(7\) − 60161.8i − 1.35295i −0.736466 − 0.676475i \(-0.763507\pi\)
0.736466 − 0.676475i \(-0.236493\pi\)
\(8\) − 95709.1i − 1.03266i
\(9\) −59049.0 −0.333333
\(10\) 0 0
\(11\) −770245. −1.44201 −0.721006 − 0.692929i \(-0.756320\pi\)
−0.721006 + 0.692929i \(0.756320\pi\)
\(12\) − 280732.i − 0.325683i
\(13\) 293566.i 0.219289i 0.993971 + 0.109644i \(0.0349712\pi\)
−0.993971 + 0.109644i \(0.965029\pi\)
\(14\) −1.79754e6 −0.893255
\(15\) 0 0
\(16\) −493642. −0.117693
\(17\) − 497023.i − 0.0848999i −0.999099 − 0.0424500i \(-0.986484\pi\)
0.999099 − 0.0424500i \(-0.0135163\pi\)
\(18\) 1.76430e6i 0.220076i
\(19\) 1.40960e7 1.30602 0.653012 − 0.757347i \(-0.273505\pi\)
0.653012 + 0.757347i \(0.273505\pi\)
\(20\) 0 0
\(21\) −1.46193e7 −0.781126
\(22\) 2.30138e7i 0.952057i
\(23\) − 5.26889e6i − 0.170693i −0.996351 − 0.0853466i \(-0.972800\pi\)
0.996351 − 0.0853466i \(-0.0271997\pi\)
\(24\) −2.32573e7 −0.596208
\(25\) 0 0
\(26\) 8.77130e6 0.144781
\(27\) 1.43489e7i 0.192450i
\(28\) − 6.95034e7i − 0.763197i
\(29\) −1.78246e8 −1.61373 −0.806865 − 0.590736i \(-0.798837\pi\)
−0.806865 + 0.590736i \(0.798837\pi\)
\(30\) 0 0
\(31\) −2.16036e8 −1.35531 −0.677653 − 0.735382i \(-0.737003\pi\)
−0.677653 + 0.735382i \(0.737003\pi\)
\(32\) − 1.81263e8i − 0.954957i
\(33\) 1.87169e8i 0.832546i
\(34\) −1.48503e7 −0.0560533
\(35\) 0 0
\(36\) −6.82178e7 −0.188033
\(37\) − 5.23052e8i − 1.24004i −0.784587 − 0.620019i \(-0.787125\pi\)
0.784587 − 0.620019i \(-0.212875\pi\)
\(38\) − 4.21168e8i − 0.862274i
\(39\) 7.13364e7 0.126606
\(40\) 0 0
\(41\) 7.77482e8 1.04804 0.524022 − 0.851705i \(-0.324431\pi\)
0.524022 + 0.851705i \(0.324431\pi\)
\(42\) 4.36803e8i 0.515721i
\(43\) 1.14173e9i 1.18437i 0.805801 + 0.592186i \(0.201735\pi\)
−0.805801 + 0.592186i \(0.798265\pi\)
\(44\) −8.89844e8 −0.813438
\(45\) 0 0
\(46\) −1.57427e8 −0.112696
\(47\) − 2.31913e8i − 0.147499i −0.997277 − 0.0737493i \(-0.976504\pi\)
0.997277 − 0.0737493i \(-0.0234965\pi\)
\(48\) 1.19955e8i 0.0679503i
\(49\) −1.64212e9 −0.830472
\(50\) 0 0
\(51\) −1.20777e8 −0.0490170
\(52\) 3.39149e8i 0.123701i
\(53\) 2.44201e9i 0.802102i 0.916056 + 0.401051i \(0.131355\pi\)
−0.916056 + 0.401051i \(0.868645\pi\)
\(54\) 4.28724e8 0.127061
\(55\) 0 0
\(56\) −5.75803e9 −1.39714
\(57\) − 3.42533e9i − 0.754034i
\(58\) 5.32572e9i 1.06543i
\(59\) −3.13948e9 −0.571705 −0.285853 − 0.958274i \(-0.592277\pi\)
−0.285853 + 0.958274i \(0.592277\pi\)
\(60\) 0 0
\(61\) 1.03302e10 1.56601 0.783005 − 0.622016i \(-0.213686\pi\)
0.783005 + 0.622016i \(0.213686\pi\)
\(62\) 6.45484e9i 0.894811i
\(63\) 3.55249e9i 0.450983i
\(64\) −6.42684e9 −0.748183
\(65\) 0 0
\(66\) 5.59234e9 0.549670
\(67\) 1.56856e10i 1.41935i 0.704530 + 0.709674i \(0.251158\pi\)
−0.704530 + 0.709674i \(0.748842\pi\)
\(68\) − 5.74198e8i − 0.0478919i
\(69\) −1.28034e9 −0.0985497
\(70\) 0 0
\(71\) 2.49558e8 0.0164154 0.00820768 − 0.999966i \(-0.497387\pi\)
0.00820768 + 0.999966i \(0.497387\pi\)
\(72\) 5.65152e9i 0.344221i
\(73\) − 1.64453e10i − 0.928469i −0.885712 − 0.464234i \(-0.846330\pi\)
0.885712 − 0.464234i \(-0.153670\pi\)
\(74\) −1.56280e10 −0.818708
\(75\) 0 0
\(76\) 1.62848e10 0.736727
\(77\) 4.63393e10i 1.95097i
\(78\) − 2.13143e9i − 0.0835892i
\(79\) −3.02025e10 −1.10432 −0.552159 − 0.833739i \(-0.686196\pi\)
−0.552159 + 0.833739i \(0.686196\pi\)
\(80\) 0 0
\(81\) 3.48678e9 0.111111
\(82\) − 2.32300e10i − 0.691948i
\(83\) 1.86048e10i 0.518437i 0.965819 + 0.259218i \(0.0834649\pi\)
−0.965819 + 0.259218i \(0.916535\pi\)
\(84\) −1.68893e10 −0.440632
\(85\) 0 0
\(86\) 3.41133e10 0.781956
\(87\) 4.33138e10i 0.931687i
\(88\) 7.37194e10i 1.48911i
\(89\) −8.20974e10 −1.55842 −0.779210 − 0.626763i \(-0.784380\pi\)
−0.779210 + 0.626763i \(0.784380\pi\)
\(90\) 0 0
\(91\) 1.76614e10 0.296687
\(92\) − 6.08701e9i − 0.0962878i
\(93\) 5.24968e10i 0.782486i
\(94\) −6.92923e9 −0.0973827
\(95\) 0 0
\(96\) −4.40469e10 −0.551345
\(97\) 7.91004e10i 0.935264i 0.883923 + 0.467632i \(0.154893\pi\)
−0.883923 + 0.467632i \(0.845107\pi\)
\(98\) 4.90640e10i 0.548301i
\(99\) 4.54822e10 0.480671
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.b.g.49.4 8
3.2 odd 2 225.12.b.o.199.5 8
5.2 odd 4 75.12.a.h.1.3 ✓ 4
5.3 odd 4 75.12.a.i.1.2 yes 4
5.4 even 2 inner 75.12.b.g.49.5 8
15.2 even 4 225.12.a.s.1.2 4
15.8 even 4 225.12.a.q.1.3 4
15.14 odd 2 225.12.b.o.199.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.a.h.1.3 ✓ 4 5.2 odd 4
75.12.a.i.1.2 yes 4 5.3 odd 4
75.12.b.g.49.4 8 1.1 even 1 trivial
75.12.b.g.49.5 8 5.4 even 2 inner
225.12.a.q.1.3 4 15.8 even 4
225.12.a.s.1.2 4 15.2 even 4
225.12.b.o.199.4 8 15.14 odd 2
225.12.b.o.199.5 8 3.2 odd 2