Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [370,2,Mod(149,370)] 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("370.149"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,-10,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: 10.0.12837029094400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 2x^{8} - 4x^{7} + 51x^{6} - 124x^{5} + 154x^{4} - 46x^{3} + x^{2} + 4x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 149.2
Root \(-1.95884 + 1.95884i\) of defining polynomial
Character \(\chi\) \(=\) 370.149
Dual form 370.2.b.d.149.9

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -1.09441i q^{3} -1.00000 q^{4} +(1.74265 + 1.40113i) q^{5} -1.09441 q^{6} +3.20984i q^{7} +1.00000i q^{8} +1.80226 q^{9} +(1.40113 - 1.74265i) q^{10} +3.82327 q^{11} +1.09441i q^{12} +0.147332i q^{13} +3.20984 q^{14} +(1.53341 - 1.90718i) q^{15} +1.00000 q^{16} -0.978989i q^{17} -1.80226i q^{18} -2.67594 q^{19} +(-1.74265 - 1.40113i) q^{20} +3.51289 q^{21} -3.82327i q^{22} -2.33616i q^{23} +1.09441 q^{24} +(1.07367 + 4.88336i) q^{25} +0.147332 q^{26} -5.25565i q^{27} -3.20984i q^{28} -6.30425 q^{29} +(-1.90718 - 1.53341i) q^{30} -3.62372 q^{31} -1.00000i q^{32} -4.18424i q^{33} -0.978989 q^{34} +(-4.49740 + 5.59362i) q^{35} -1.80226 q^{36} -1.00000i q^{37} +2.67594i q^{38} +0.161242 q^{39} +(-1.40113 + 1.74265i) q^{40} +11.8265 q^{41} -3.51289i q^{42} -4.53390i q^{43} -3.82327 q^{44} +(3.14071 + 2.52520i) q^{45} -2.33616 q^{46} -6.23085i q^{47} -1.09441i q^{48} -3.30305 q^{49} +(4.88336 - 1.07367i) q^{50} -1.07142 q^{51} -0.147332i q^{52} +11.2978i q^{53} -5.25565 q^{54} +(6.66263 + 5.35690i) q^{55} -3.20984 q^{56} +2.92858i q^{57} +6.30425i q^{58} -6.92858 q^{59} +(-1.53341 + 1.90718i) q^{60} +10.4885 q^{61} +3.62372i q^{62} +5.78496i q^{63} -1.00000 q^{64} +(-0.206432 + 0.256749i) q^{65} -4.18424 q^{66} -2.80936i q^{67} +0.978989i q^{68} -2.55672 q^{69} +(5.59362 + 4.49740i) q^{70} -12.3189 q^{71} +1.80226i q^{72} -13.9966i q^{73} -1.00000 q^{74} +(5.34441 - 1.17503i) q^{75} +2.67594 q^{76} +12.2721i q^{77} -0.161242i q^{78} -15.6057 q^{79} +(1.74265 + 1.40113i) q^{80} -0.345071 q^{81} -11.8265i q^{82} +13.5371i q^{83} -3.51289 q^{84} +(1.37169 - 1.70604i) q^{85} -4.53390 q^{86} +6.89945i q^{87} +3.82327i q^{88} +6.46929 q^{89} +(2.52520 - 3.14071i) q^{90} -0.472913 q^{91} +2.33616i q^{92} +3.96585i q^{93} -6.23085 q^{94} +(-4.66323 - 3.74934i) q^{95} -1.09441 q^{96} +3.07063i q^{97} +3.30305i q^{98} +6.89054 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{4} + 6 q^{5} - 6 q^{9} + 2 q^{10} + 6 q^{11} + 2 q^{14} + 10 q^{16} - 8 q^{19} - 6 q^{20} + 32 q^{21} + 4 q^{25} - 12 q^{26} - 22 q^{29} + 20 q^{30} + 46 q^{31} - 18 q^{34} + 32 q^{35} + 6 q^{36}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 1.09441i − 0.631859i −0.948783 − 0.315930i \(-0.897684\pi\)
0.948783 − 0.315930i \(-0.102316\pi\)
\(4\) −1.00000 −0.500000
\(5\) 1.74265 + 1.40113i 0.779337 + 0.626605i
\(6\) −1.09441 −0.446792
\(7\) 3.20984i 1.21320i 0.795006 + 0.606602i \(0.207468\pi\)
−0.795006 + 0.606602i \(0.792532\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.80226 0.600754
\(10\) 1.40113 − 1.74265i 0.443076 − 0.551075i
\(11\) 3.82327 1.15276 0.576380 − 0.817182i \(-0.304465\pi\)
0.576380 + 0.817182i \(0.304465\pi\)
\(12\) 1.09441i 0.315930i
\(13\) 0.147332i 0.0408626i 0.999791 + 0.0204313i \(0.00650394\pi\)
−0.999791 + 0.0204313i \(0.993496\pi\)
\(14\) 3.20984 0.857865
\(15\) 1.53341 − 1.90718i 0.395926 − 0.492432i
\(16\) 1.00000 0.250000
\(17\) − 0.978989i − 0.237440i −0.992928 − 0.118720i \(-0.962121\pi\)
0.992928 − 0.118720i \(-0.0378790\pi\)
\(18\) − 1.80226i − 0.424797i
\(19\) −2.67594 −0.613903 −0.306951 − 0.951725i \(-0.599309\pi\)
−0.306951 + 0.951725i \(0.599309\pi\)
\(20\) −1.74265 − 1.40113i −0.389669 − 0.313302i
\(21\) 3.51289 0.766574
\(22\) − 3.82327i − 0.815124i
\(23\) − 2.33616i − 0.487122i −0.969886 − 0.243561i \(-0.921684\pi\)
0.969886 − 0.243561i \(-0.0783157\pi\)
\(24\) 1.09441 0.223396
\(25\) 1.07367 + 4.88336i 0.214733 + 0.976673i
\(26\) 0.147332 0.0288942
\(27\) − 5.25565i − 1.01145i
\(28\) − 3.20984i − 0.606602i
\(29\) −6.30425 −1.17067 −0.585335 − 0.810792i \(-0.699037\pi\)
−0.585335 + 0.810792i \(0.699037\pi\)
\(30\) −1.90718 − 1.53341i −0.348202 − 0.279962i
\(31\) −3.62372 −0.650839 −0.325420 − 0.945570i \(-0.605506\pi\)
−0.325420 + 0.945570i \(0.605506\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.18424i − 0.728382i
\(34\) −0.978989 −0.167895
\(35\) −4.49740 + 5.59362i −0.760199 + 0.945495i
\(36\) −1.80226 −0.300377
\(37\) − 1.00000i − 0.164399i
\(38\) 2.67594i 0.434095i
\(39\) 0.161242 0.0258194
\(40\) −1.40113 + 1.74265i −0.221538 + 0.275537i
\(41\) 11.8265 1.84698 0.923491 − 0.383620i \(-0.125323\pi\)
0.923491 + 0.383620i \(0.125323\pi\)
\(42\) − 3.51289i − 0.542050i
\(43\) − 4.53390i − 0.691413i −0.938343 − 0.345706i \(-0.887639\pi\)
0.938343 − 0.345706i \(-0.112361\pi\)
\(44\) −3.82327 −0.576380
\(45\) 3.14071 + 2.52520i 0.468190 + 0.376435i
\(46\) −2.33616 −0.344448
\(47\) − 6.23085i − 0.908863i −0.890782 − 0.454431i \(-0.849843\pi\)
0.890782 − 0.454431i \(-0.150157\pi\)
\(48\) − 1.09441i − 0.157965i
\(49\) −3.30305 −0.471864
\(50\) 4.88336 − 1.07367i 0.690612 − 0.151839i
\(51\) −1.07142 −0.150028
\(52\) − 0.147332i − 0.0204313i
\(53\) 11.2978i 1.55188i 0.630807 + 0.775939i \(0.282724\pi\)
−0.630807 + 0.775939i \(0.717276\pi\)
\(54\) −5.25565 −0.715204
\(55\) 6.66263 + 5.35690i 0.898389 + 0.722325i
\(56\) −3.20984 −0.428932
\(57\) 2.92858i 0.387900i
\(58\) 6.30425i 0.827788i
\(59\) −6.92858 −0.902025 −0.451012 − 0.892518i \(-0.648937\pi\)
−0.451012 + 0.892518i \(0.648937\pi\)
\(60\) −1.53341 + 1.90718i −0.197963 + 0.246216i
\(61\) 10.4885 1.34291 0.671457 − 0.741044i \(-0.265669\pi\)
0.671457 + 0.741044i \(0.265669\pi\)
\(62\) 3.62372i 0.460213i
\(63\) 5.78496i 0.728837i
\(64\) −1.00000 −0.125000
\(65\) −0.206432 + 0.256749i −0.0256047 + 0.0318458i
\(66\) −4.18424 −0.515044
\(67\) − 2.80936i − 0.343218i −0.985165 − 0.171609i \(-0.945103\pi\)
0.985165 − 0.171609i \(-0.0548966\pi\)
\(68\) 0.978989i 0.118720i
\(69\) −2.55672 −0.307793
\(70\) 5.59362 + 4.49740i 0.668566 + 0.537542i
\(71\) −12.3189 −1.46198 −0.730990 − 0.682388i \(-0.760941\pi\)
−0.730990 + 0.682388i \(0.760941\pi\)
\(72\) 1.80226i 0.212399i
\(73\) − 13.9966i − 1.63818i −0.573665 − 0.819090i \(-0.694479\pi\)
0.573665 − 0.819090i \(-0.305521\pi\)
\(74\) −1.00000 −0.116248
\(75\) 5.34441 − 1.17503i 0.617120 − 0.135681i
\(76\) 2.67594 0.306951
\(77\) 12.2721i 1.39853i
\(78\) − 0.161242i − 0.0182571i
\(79\) −15.6057 −1.75578 −0.877890 − 0.478861i \(-0.841050\pi\)
−0.877890 + 0.478861i \(0.841050\pi\)
\(80\) 1.74265 + 1.40113i 0.194834 + 0.156651i
\(81\) −0.345071 −0.0383412
\(82\) − 11.8265i − 1.30601i
\(83\) 13.5371i 1.48589i 0.669354 + 0.742944i \(0.266571\pi\)
−0.669354 + 0.742944i \(0.733429\pi\)
\(84\) −3.51289 −0.383287
\(85\) 1.37169 − 1.70604i 0.148781 − 0.185046i
\(86\) −4.53390 −0.488903
\(87\) 6.89945i 0.739699i
\(88\) 3.82327i 0.407562i
\(89\) 6.46929 0.685743 0.342872 − 0.939382i \(-0.388600\pi\)
0.342872 + 0.939382i \(0.388600\pi\)
\(90\) 2.52520 − 3.14071i 0.266180 − 0.331060i
\(91\) −0.472913 −0.0495747
\(92\) 2.33616i 0.243561i
\(93\) 3.96585i 0.411239i
\(94\) −6.23085 −0.642663
\(95\) −4.66323 − 3.74934i −0.478437 − 0.384674i
\(96\) −1.09441 −0.111698
\(97\) 3.07063i 0.311775i 0.987775 + 0.155887i \(0.0498237\pi\)
−0.987775 + 0.155887i \(0.950176\pi\)
\(98\) 3.30305i 0.333658i
\(99\) 6.89054 0.692525
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 370.2.b.d.149.2 ✓ 10
3.2 odd 2 3330.2.d.p.1999.7 10
5.2 odd 4 1850.2.a.be.1.2 5
5.3 odd 4 1850.2.a.bd.1.4 5
5.4 even 2 inner 370.2.b.d.149.9 yes 10
15.14 odd 2 3330.2.d.p.1999.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.b.d.149.2 ✓ 10 1.1 even 1 trivial
370.2.b.d.149.9 yes 10 5.4 even 2 inner
1850.2.a.bd.1.4 5 5.3 odd 4
1850.2.a.be.1.2 5 5.2 odd 4
3330.2.d.p.1999.2 10 15.14 odd 2
3330.2.d.p.1999.7 10 3.2 odd 2