Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [490,4,Mod(361,490)] 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("490.361"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 490.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,1,-4,5,4,0,-16,26,-10,65] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(28.9109359028\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 490.361
Dual form 490.4.e.o.471.1

$q$-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+(1.00000 + 1.73205i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} +2.00000 q^{6} -8.00000 q^{8} +(13.0000 + 22.5167i) q^{9} +(-5.00000 + 8.66025i) q^{10} +(32.5000 - 56.2917i) q^{11} +(2.00000 + 3.46410i) q^{12} +13.0000 q^{13} +5.00000 q^{15} +(-8.00000 - 13.8564i) q^{16} +(36.5000 - 63.2199i) q^{17} +(-26.0000 + 45.0333i) q^{18} +(71.0000 + 122.976i) q^{19} -20.0000 q^{20} +130.000 q^{22} +(-65.0000 - 112.583i) q^{23} +(-4.00000 + 6.92820i) q^{24} +(-12.5000 + 21.6506i) q^{25} +(13.0000 + 22.5167i) q^{26} +53.0000 q^{27} +111.000 q^{29} +(5.00000 + 8.66025i) q^{30} +(-128.000 + 221.703i) q^{31} +(16.0000 - 27.7128i) q^{32} +(-32.5000 - 56.2917i) q^{33} +146.000 q^{34} -104.000 q^{36} +(133.000 + 230.363i) q^{37} +(-142.000 + 245.951i) q^{38} +(6.50000 - 11.2583i) q^{39} +(-20.0000 - 34.6410i) q^{40} -424.000 q^{41} +534.000 q^{43} +(130.000 + 225.167i) q^{44} +(-65.0000 + 112.583i) q^{45} +(130.000 - 225.167i) q^{46} +(134.500 + 232.961i) q^{47} -16.0000 q^{48} -50.0000 q^{50} +(-36.5000 - 63.2199i) q^{51} +(-26.0000 + 45.0333i) q^{52} +(66.0000 - 114.315i) q^{53} +(53.0000 + 91.7987i) q^{54} +325.000 q^{55} +142.000 q^{57} +(111.000 + 192.258i) q^{58} +(112.000 - 193.990i) q^{59} +(-10.0000 + 17.3205i) q^{60} +(286.000 + 495.367i) q^{61} -512.000 q^{62} +64.0000 q^{64} +(32.5000 + 56.2917i) q^{65} +(65.0000 - 112.583i) q^{66} +(54.0000 - 93.5307i) q^{67} +(146.000 + 252.879i) q^{68} -130.000 q^{69} +560.000 q^{71} +(-104.000 - 180.133i) q^{72} +(-293.000 + 507.491i) q^{73} +(-266.000 + 460.726i) q^{74} +(12.5000 + 21.6506i) q^{75} -568.000 q^{76} +26.0000 q^{78} +(-28.5000 - 49.3634i) q^{79} +(40.0000 - 69.2820i) q^{80} +(-324.500 + 562.050i) q^{81} +(-424.000 - 734.390i) q^{82} +252.000 q^{83} +365.000 q^{85} +(534.000 + 924.915i) q^{86} +(55.5000 - 96.1288i) q^{87} +(-260.000 + 450.333i) q^{88} +(92.0000 + 159.349i) q^{89} -260.000 q^{90} +520.000 q^{92} +(128.000 + 221.703i) q^{93} +(-269.000 + 465.922i) q^{94} +(-355.000 + 614.878i) q^{95} +(-16.0000 - 27.7128i) q^{96} -605.000 q^{97} +1690.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} - 4 q^{4} + 5 q^{5} + 4 q^{6} - 16 q^{8} + 26 q^{9} - 10 q^{10} + 65 q^{11} + 4 q^{12} + 26 q^{13} + 10 q^{15} - 16 q^{16} + 73 q^{17} - 52 q^{18} + 142 q^{19} - 40 q^{20} + 260 q^{22}+ \cdots + 3380 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/490\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 + 1.73205i 0.353553 + 0.612372i
\(3\) 0.500000 − 0.866025i 0.0962250 − 0.166667i −0.813894 − 0.581013i \(-0.802656\pi\)
0.910119 + 0.414346i \(0.135990\pi\)
\(4\) −2.00000 + 3.46410i −0.250000 + 0.433013i
\(5\) 2.50000 + 4.33013i 0.223607 + 0.387298i
\(6\) 2.00000 0.136083
\(7\) 0 0
\(8\) −8.00000 −0.353553
\(9\) 13.0000 + 22.5167i 0.481481 + 0.833950i
\(10\) −5.00000 + 8.66025i −0.158114 + 0.273861i
\(11\) 32.5000 − 56.2917i 0.890829 − 1.54296i 0.0519455 − 0.998650i \(-0.483458\pi\)
0.838883 − 0.544311i \(-0.183209\pi\)
\(12\) 2.00000 + 3.46410i 0.0481125 + 0.0833333i
\(13\) 13.0000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 5.00000 0.0860663
\(16\) −8.00000 − 13.8564i −0.125000 − 0.216506i
\(17\) 36.5000 − 63.2199i 0.520738 − 0.901945i −0.478971 − 0.877831i \(-0.658990\pi\)
0.999709 − 0.0241144i \(-0.00767658\pi\)
\(18\) −26.0000 + 45.0333i −0.340459 + 0.589692i
\(19\) 71.0000 + 122.976i 0.857290 + 1.48487i 0.874504 + 0.485019i \(0.161187\pi\)
−0.0172134 + 0.999852i \(0.505479\pi\)
\(20\) −20.0000 −0.223607
\(21\) 0 0
\(22\) 130.000 1.25982
\(23\) −65.0000 − 112.583i −0.589280 − 1.02066i −0.994327 − 0.106367i \(-0.966078\pi\)
0.405047 − 0.914296i \(-0.367255\pi\)
\(24\) −4.00000 + 6.92820i −0.0340207 + 0.0589256i
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 13.0000 + 22.5167i 0.0980581 + 0.169842i
\(27\) 53.0000 0.377772
\(28\) 0 0
\(29\) 111.000 0.710765 0.355382 − 0.934721i \(-0.384351\pi\)
0.355382 + 0.934721i \(0.384351\pi\)
\(30\) 5.00000 + 8.66025i 0.0304290 + 0.0527046i
\(31\) −128.000 + 221.703i −0.741596 + 1.28448i 0.210172 + 0.977664i \(0.432598\pi\)
−0.951768 + 0.306818i \(0.900736\pi\)
\(32\) 16.0000 − 27.7128i 0.0883883 − 0.153093i
\(33\) −32.5000 − 56.2917i −0.171440 − 0.296943i
\(34\) 146.000 0.736435
\(35\) 0 0
\(36\) −104.000 −0.481481
\(37\) 133.000 + 230.363i 0.590948 + 1.02355i 0.994105 + 0.108421i \(0.0345794\pi\)
−0.403157 + 0.915131i \(0.632087\pi\)
\(38\) −142.000 + 245.951i −0.606196 + 1.04996i
\(39\) 6.50000 − 11.2583i 0.0266880 − 0.0462250i
\(40\) −20.0000 − 34.6410i −0.0790569 − 0.136931i
\(41\) −424.000 −1.61507 −0.807533 − 0.589823i \(-0.799198\pi\)
−0.807533 + 0.589823i \(0.799198\pi\)
\(42\) 0 0
\(43\) 534.000 1.89382 0.946910 − 0.321500i \(-0.104187\pi\)
0.946910 + 0.321500i \(0.104187\pi\)
\(44\) 130.000 + 225.167i 0.445414 + 0.771481i
\(45\) −65.0000 + 112.583i −0.215325 + 0.372954i
\(46\) 130.000 − 225.167i 0.416684 − 0.721717i
\(47\) 134.500 + 232.961i 0.417422 + 0.722996i 0.995679 − 0.0928582i \(-0.0296003\pi\)
−0.578257 + 0.815855i \(0.696267\pi\)
\(48\) −16.0000 −0.0481125
\(49\) 0 0
\(50\) −50.0000 −0.141421
\(51\) −36.5000 − 63.2199i −0.100216 − 0.173579i
\(52\) −26.0000 + 45.0333i −0.0693375 + 0.120096i
\(53\) 66.0000 − 114.315i 0.171053 − 0.296272i −0.767735 − 0.640767i \(-0.778616\pi\)
0.938788 + 0.344495i \(0.111950\pi\)
\(54\) 53.0000 + 91.7987i 0.133563 + 0.231337i
\(55\) 325.000 0.796782
\(56\) 0 0
\(57\) 142.000 0.329971
\(58\) 111.000 + 192.258i 0.251293 + 0.435253i
\(59\) 112.000 − 193.990i 0.247138 − 0.428056i −0.715592 − 0.698518i \(-0.753843\pi\)
0.962731 + 0.270462i \(0.0871765\pi\)
\(60\) −10.0000 + 17.3205i −0.0215166 + 0.0372678i
\(61\) 286.000 + 495.367i 0.600304 + 1.03976i 0.992775 + 0.119993i \(0.0382871\pi\)
−0.392471 + 0.919764i \(0.628380\pi\)
\(62\) −512.000 −1.04878
\(63\) 0 0
\(64\) 64.0000 0.125000
\(65\) 32.5000 + 56.2917i 0.0620174 + 0.107417i
\(66\) 65.0000 − 112.583i 0.121226 − 0.209970i
\(67\) 54.0000 − 93.5307i 0.0984649 − 0.170546i −0.812585 − 0.582843i \(-0.801940\pi\)
0.911049 + 0.412297i \(0.135273\pi\)
\(68\) 146.000 + 252.879i 0.260369 + 0.450973i
\(69\) −130.000 −0.226814
\(70\) 0 0
\(71\) 560.000 0.936053 0.468027 − 0.883714i \(-0.344965\pi\)
0.468027 + 0.883714i \(0.344965\pi\)
\(72\) −104.000 − 180.133i −0.170229 − 0.294846i
\(73\) −293.000 + 507.491i −0.469768 + 0.813662i −0.999402 − 0.0345641i \(-0.988996\pi\)
0.529635 + 0.848226i \(0.322329\pi\)
\(74\) −266.000 + 460.726i −0.417863 + 0.723760i
\(75\) 12.5000 + 21.6506i 0.0192450 + 0.0333333i
\(76\) −568.000 −0.857290
\(77\) 0 0
\(78\) 26.0000 0.0377426
\(79\) −28.5000 − 49.3634i −0.0405886 − 0.0703015i 0.845017 − 0.534739i \(-0.179590\pi\)
−0.885606 + 0.464437i \(0.846257\pi\)
\(80\) 40.0000 − 69.2820i 0.0559017 − 0.0968246i
\(81\) −324.500 + 562.050i −0.445130 + 0.770988i
\(82\) −424.000 − 734.390i −0.571012 − 0.989021i
\(83\) 252.000 0.333260 0.166630 − 0.986019i \(-0.446711\pi\)
0.166630 + 0.986019i \(0.446711\pi\)
\(84\) 0 0
\(85\) 365.000 0.465762
\(86\) 534.000 + 924.915i 0.669566 + 1.15972i
\(87\) 55.5000 − 96.1288i 0.0683934 − 0.118461i
\(88\) −260.000 + 450.333i −0.314956 + 0.545519i
\(89\) 92.0000 + 159.349i 0.109573 + 0.189786i 0.915597 − 0.402097i \(-0.131718\pi\)
−0.806024 + 0.591882i \(0.798385\pi\)
\(90\) −260.000 −0.304516
\(91\) 0 0
\(92\) 520.000 0.589280
\(93\) 128.000 + 221.703i 0.142720 + 0.247199i
\(94\) −269.000 + 465.922i −0.295162 + 0.511236i
\(95\) −355.000 + 614.878i −0.383392 + 0.664054i
\(96\) −16.0000 − 27.7128i −0.0170103 − 0.0294628i
\(97\) −605.000 −0.633283 −0.316641 − 0.948545i \(-0.602555\pi\)
−0.316641 + 0.948545i \(0.602555\pi\)
\(98\) 0 0
\(99\) 1690.00 1.71567
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 490.4.e.o.361.1 2
7.2 even 3 inner 490.4.e.o.471.1 2
7.3 odd 6 490.4.a.d.1.1 1
7.4 even 3 70.4.a.c.1.1 ✓ 1
7.5 odd 6 490.4.e.n.471.1 2
7.6 odd 2 490.4.e.n.361.1 2
21.11 odd 6 630.4.a.x.1.1 1
28.11 odd 6 560.4.a.i.1.1 1
35.4 even 6 350.4.a.r.1.1 1
35.18 odd 12 350.4.c.h.99.2 2
35.24 odd 6 2450.4.a.bc.1.1 1
35.32 odd 12 350.4.c.h.99.1 2
56.11 odd 6 2240.4.a.r.1.1 1
56.53 even 6 2240.4.a.v.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
70.4.a.c.1.1 ✓ 1 7.4 even 3
350.4.a.r.1.1 1 35.4 even 6
350.4.c.h.99.1 2 35.32 odd 12
350.4.c.h.99.2 2 35.18 odd 12
490.4.a.d.1.1 1 7.3 odd 6
490.4.e.n.361.1 2 7.6 odd 2
490.4.e.n.471.1 2 7.5 odd 6
490.4.e.o.361.1 2 1.1 even 1 trivial
490.4.e.o.471.1 2 7.2 even 3 inner
560.4.a.i.1.1 1 28.11 odd 6
630.4.a.x.1.1 1 21.11 odd 6
2240.4.a.r.1.1 1 56.11 odd 6
2240.4.a.v.1.1 1 56.53 even 6
2450.4.a.bc.1.1 1 35.24 odd 6