Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [225,12,Mod(1,225)] 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("225.1"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,46,0,4648,0,0,-68372] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(172.877215626\)
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: \( x^{4} - 2x^{3} - 6154x^{2} - 41770x + 5647125 \) 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: no (minimal twist has level 75)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-40.8785\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-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)
 
\(f(q)\) \(=\) \(q-29.8785 q^{2} -1155.27 q^{4} +60161.8 q^{7} +95709.1 q^{8} +770245. q^{11} +293566. q^{13} -1.79754e6 q^{14} -493642. q^{16} -497023. q^{17} -1.40960e7 q^{19} -2.30138e7 q^{22} +5.26889e6 q^{23} -8.77130e6 q^{26} -6.95034e7 q^{28} -1.78246e8 q^{29} -2.16036e8 q^{31} -1.81263e8 q^{32} +1.48503e7 q^{34} +5.23052e8 q^{37} +4.21168e8 q^{38} -7.77482e8 q^{41} +1.14173e9 q^{43} -8.89844e8 q^{44} -1.57427e8 q^{46} -2.31913e8 q^{47} +1.64212e9 q^{49} -3.39149e8 q^{52} -2.44201e9 q^{53} +5.75803e9 q^{56} +5.32572e9 q^{58} -3.13948e9 q^{59} +1.03302e10 q^{61} +6.45484e9 q^{62} +6.42684e9 q^{64} -1.56856e10 q^{67} +5.74198e8 q^{68} -2.49558e8 q^{71} -1.64453e10 q^{73} -1.56280e10 q^{74} +1.62848e10 q^{76} +4.63393e10 q^{77} +3.02025e10 q^{79} +2.32300e10 q^{82} -1.86048e10 q^{83} -3.41133e10 q^{86} +7.37194e10 q^{88} -8.20974e10 q^{89} +1.76614e10 q^{91} -6.08701e9 q^{92} +6.92923e9 q^{94} -7.91004e10 q^{97} -4.90640e10 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 46 q^{2} + 4648 q^{4} - 68372 q^{7} + 386172 q^{8} + 99944 q^{11} - 2306276 q^{13} - 3107022 q^{14} + 7622200 q^{16} - 3443816 q^{17} - 4214548 q^{19} + 34106532 q^{22} + 52691304 q^{23} + 8766722 q^{26}+ \cdots - 139912937716 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −29.8785 −0.660228 −0.330114 − 0.943941i \(-0.607087\pi\)
−0.330114 + 0.943941i \(0.607087\pi\)
\(3\) 0 0
\(4\) −1155.27 −0.564099
\(5\) 0 0
\(6\) 0 0
\(7\) 60161.8 1.35295 0.676475 − 0.736466i \(-0.263507\pi\)
0.676475 + 0.736466i \(0.263507\pi\)
\(8\) 95709.1 1.03266
\(9\) 0 0
\(10\) 0 0
\(11\) 770245. 1.44201 0.721006 − 0.692929i \(-0.243680\pi\)
0.721006 + 0.692929i \(0.243680\pi\)
\(12\) 0 0
\(13\) 293566. 0.219289 0.109644 − 0.993971i \(-0.465029\pi\)
0.109644 + 0.993971i \(0.465029\pi\)
\(14\) −1.79754e6 −0.893255
\(15\) 0 0
\(16\) −493642. −0.117693
\(17\) −497023. −0.0848999 −0.0424500 − 0.999099i \(-0.513516\pi\)
−0.0424500 + 0.999099i \(0.513516\pi\)
\(18\) 0 0
\(19\) −1.40960e7 −1.30602 −0.653012 − 0.757347i \(-0.726495\pi\)
−0.653012 + 0.757347i \(0.726495\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.30138e7 −0.952057
\(23\) 5.26889e6 0.170693 0.0853466 − 0.996351i \(-0.472800\pi\)
0.0853466 + 0.996351i \(0.472800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.77130e6 −0.144781
\(27\) 0 0
\(28\) −6.95034e7 −0.763197
\(29\) −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 \(0.737003\pi\)
\(32\) −1.81263e8 −0.954957
\(33\) 0 0
\(34\) 1.48503e7 0.0560533
\(35\) 0 0
\(36\) 0 0
\(37\) 5.23052e8 1.24004 0.620019 − 0.784587i \(-0.287125\pi\)
0.620019 + 0.784587i \(0.287125\pi\)
\(38\) 4.21168e8 0.862274
\(39\) 0 0
\(40\) 0 0
\(41\) −7.77482e8 −1.04804 −0.524022 − 0.851705i \(-0.675569\pi\)
−0.524022 + 0.851705i \(0.675569\pi\)
\(42\) 0 0
\(43\) 1.14173e9 1.18437 0.592186 − 0.805801i \(-0.298265\pi\)
0.592186 + 0.805801i \(0.298265\pi\)
\(44\) −8.89844e8 −0.813438
\(45\) 0 0
\(46\) −1.57427e8 −0.112696
\(47\) −2.31913e8 −0.147499 −0.0737493 − 0.997277i \(-0.523496\pi\)
−0.0737493 + 0.997277i \(0.523496\pi\)
\(48\) 0 0
\(49\) 1.64212e9 0.830472
\(50\) 0 0
\(51\) 0 0
\(52\) −3.39149e8 −0.123701
\(53\) −2.44201e9 −0.802102 −0.401051 − 0.916056i \(-0.631355\pi\)
−0.401051 + 0.916056i \(0.631355\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 5.75803e9 1.39714
\(57\) 0 0
\(58\) 5.32572e9 1.06543
\(59\) −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 \(0.213686\pi\)
\(62\) 6.45484e9 0.894811
\(63\) 0 0
\(64\) 6.42684e9 0.748183
\(65\) 0 0
\(66\) 0 0
\(67\) −1.56856e10 −1.41935 −0.709674 − 0.704530i \(-0.751158\pi\)
−0.709674 + 0.704530i \(0.751158\pi\)
\(68\) 5.74198e8 0.0478919
\(69\) 0 0
\(70\) 0 0
\(71\) −2.49558e8 −0.0164154 −0.00820768 − 0.999966i \(-0.502613\pi\)
−0.00820768 + 0.999966i \(0.502613\pi\)
\(72\) 0 0
\(73\) −1.64453e10 −0.928469 −0.464234 − 0.885712i \(-0.653670\pi\)
−0.464234 + 0.885712i \(0.653670\pi\)
\(74\) −1.56280e10 −0.818708
\(75\) 0 0
\(76\) 1.62848e10 0.736727
\(77\) 4.63393e10 1.95097
\(78\) 0 0
\(79\) 3.02025e10 1.10432 0.552159 − 0.833739i \(-0.313804\pi\)
0.552159 + 0.833739i \(0.313804\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.32300e10 0.691948
\(83\) −1.86048e10 −0.518437 −0.259218 − 0.965819i \(-0.583465\pi\)
−0.259218 + 0.965819i \(0.583465\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −3.41133e10 −0.781956
\(87\) 0 0
\(88\) 7.37194e10 1.48911
\(89\) −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
\(92\) −6.08701e9 −0.0962878
\(93\) 0 0
\(94\) 6.92923e9 0.0973827
\(95\) 0 0
\(96\) 0 0
\(97\) −7.91004e10 −0.935264 −0.467632 − 0.883923i \(-0.654893\pi\)
−0.467632 + 0.883923i \(0.654893\pi\)
\(98\) −4.90640e10 −0.548301
\(99\) 0 0
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 225.12.a.s.1.2 4
3.2 odd 2 75.12.a.h.1.3 ✓ 4
5.2 odd 4 225.12.b.o.199.4 8
5.3 odd 4 225.12.b.o.199.5 8
5.4 even 2 225.12.a.q.1.3 4
15.2 even 4 75.12.b.g.49.5 8
15.8 even 4 75.12.b.g.49.4 8
15.14 odd 2 75.12.a.i.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.12.a.h.1.3 ✓ 4 3.2 odd 2
75.12.a.i.1.2 yes 4 15.14 odd 2
75.12.b.g.49.4 8 15.8 even 4
75.12.b.g.49.5 8 15.2 even 4
225.12.a.q.1.3 4 5.4 even 2
225.12.a.s.1.2 4 1.1 even 1 trivial
225.12.b.o.199.4 8 5.2 odd 4
225.12.b.o.199.5 8 5.3 odd 4