Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [10,5,Mod(3,10)] 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("10.3"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4] 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: no
Analytic conductor: \(1.03369963084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 7.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 10.7
Dual form 10.5.c.a.3.1

$q$-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+(-2.00000 - 2.00000i) q^{2} +(9.00000 - 9.00000i) q^{3} +8.00000i q^{4} +(-15.0000 + 20.0000i) q^{5} -36.0000 q^{6} +(29.0000 + 29.0000i) q^{7} +(16.0000 - 16.0000i) q^{8} -81.0000i q^{9} +(70.0000 - 10.0000i) q^{10} -118.000 q^{11} +(72.0000 + 72.0000i) q^{12} +(69.0000 - 69.0000i) q^{13} -116.000i q^{14} +(45.0000 + 315.000i) q^{15} -64.0000 q^{16} +(-271.000 - 271.000i) q^{17} +(-162.000 + 162.000i) q^{18} +280.000i q^{19} +(-160.000 - 120.000i) q^{20} +522.000 q^{21} +(236.000 + 236.000i) q^{22} +(269.000 - 269.000i) q^{23} -288.000i q^{24} +(-175.000 - 600.000i) q^{25} -276.000 q^{26} +(-232.000 + 232.000i) q^{28} +680.000i q^{29} +(540.000 - 720.000i) q^{30} +202.000 q^{31} +(128.000 + 128.000i) q^{32} +(-1062.00 + 1062.00i) q^{33} +1084.00i q^{34} +(-1015.00 + 145.000i) q^{35} +648.000 q^{36} +(-651.000 - 651.000i) q^{37} +(560.000 - 560.000i) q^{38} -1242.00i q^{39} +(80.0000 + 560.000i) q^{40} +1682.00 q^{41} +(-1044.00 - 1044.00i) q^{42} +(1089.00 - 1089.00i) q^{43} -944.000i q^{44} +(1620.00 + 1215.00i) q^{45} -1076.00 q^{46} +(1269.00 + 1269.00i) q^{47} +(-576.000 + 576.000i) q^{48} -719.000i q^{49} +(-850.000 + 1550.00i) q^{50} -4878.00 q^{51} +(552.000 + 552.000i) q^{52} +(-611.000 + 611.000i) q^{53} +(1770.00 - 2360.00i) q^{55} +928.000 q^{56} +(2520.00 + 2520.00i) q^{57} +(1360.00 - 1360.00i) q^{58} +1160.00i q^{59} +(-2520.00 + 360.000i) q^{60} -5598.00 q^{61} +(-404.000 - 404.000i) q^{62} +(2349.00 - 2349.00i) q^{63} -512.000i q^{64} +(345.000 + 2415.00i) q^{65} +4248.00 q^{66} +(-751.000 - 751.000i) q^{67} +(2168.00 - 2168.00i) q^{68} -4842.00i q^{69} +(2320.00 + 1740.00i) q^{70} +6442.00 q^{71} +(-1296.00 - 1296.00i) q^{72} +(-2951.00 + 2951.00i) q^{73} +2604.00i q^{74} +(-6975.00 - 3825.00i) q^{75} -2240.00 q^{76} +(-3422.00 - 3422.00i) q^{77} +(-2484.00 + 2484.00i) q^{78} +10560.0i q^{79} +(960.000 - 1280.00i) q^{80} +6561.00 q^{81} +(-3364.00 - 3364.00i) q^{82} +(-6231.00 + 6231.00i) q^{83} +4176.00i q^{84} +(9485.00 - 1355.00i) q^{85} -4356.00 q^{86} +(6120.00 + 6120.00i) q^{87} +(-1888.00 + 1888.00i) q^{88} -14480.0i q^{89} +(-810.000 - 5670.00i) q^{90} +4002.00 q^{91} +(2152.00 + 2152.00i) q^{92} +(1818.00 - 1818.00i) q^{93} -5076.00i q^{94} +(-5600.00 - 4200.00i) q^{95} +2304.00 q^{96} +(-7311.00 - 7311.00i) q^{97} +(-1438.00 + 1438.00i) q^{98} +9558.00i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 18 q^{3} - 30 q^{5} - 72 q^{6} + 58 q^{7} + 32 q^{8} + 140 q^{10} - 236 q^{11} + 144 q^{12} + 138 q^{13} + 90 q^{15} - 128 q^{16} - 542 q^{17} - 324 q^{18} - 320 q^{20} + 1044 q^{21} + 472 q^{22}+ \cdots - 2876 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 − 2.00000i −0.500000 − 0.500000i
\(3\) 9.00000 − 9.00000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(4\) 8.00000i 0.500000i
\(5\) −15.0000 + 20.0000i −0.600000 + 0.800000i
\(6\) −36.0000 −1.00000
\(7\) 29.0000 + 29.0000i 0.591837 + 0.591837i 0.938127 − 0.346291i \(-0.112559\pi\)
−0.346291 + 0.938127i \(0.612559\pi\)
\(8\) 16.0000 − 16.0000i 0.250000 − 0.250000i
\(9\) − 81.0000i − 1.00000i
\(10\) 70.0000 − 10.0000i 0.700000 − 0.100000i
\(11\) −118.000 −0.975207 −0.487603 − 0.873065i \(-0.662129\pi\)
−0.487603 + 0.873065i \(0.662129\pi\)
\(12\) 72.0000 + 72.0000i 0.500000 + 0.500000i
\(13\) 69.0000 − 69.0000i 0.408284 − 0.408284i −0.472856 − 0.881140i \(-0.656777\pi\)
0.881140 + 0.472856i \(0.156777\pi\)
\(14\) − 116.000i − 0.591837i
\(15\) 45.0000 + 315.000i 0.200000 + 1.40000i
\(16\) −64.0000 −0.250000
\(17\) −271.000 − 271.000i −0.937716 − 0.937716i 0.0604547 − 0.998171i \(-0.480745\pi\)
−0.998171 + 0.0604547i \(0.980745\pi\)
\(18\) −162.000 + 162.000i −0.500000 + 0.500000i
\(19\) 280.000i 0.775623i 0.921739 + 0.387812i \(0.126769\pi\)
−0.921739 + 0.387812i \(0.873231\pi\)
\(20\) −160.000 − 120.000i −0.400000 − 0.300000i
\(21\) 522.000 1.18367
\(22\) 236.000 + 236.000i 0.487603 + 0.487603i
\(23\) 269.000 − 269.000i 0.508507 − 0.508507i −0.405561 − 0.914068i \(-0.632924\pi\)
0.914068 + 0.405561i \(0.132924\pi\)
\(24\) − 288.000i − 0.500000i
\(25\) −175.000 − 600.000i −0.280000 − 0.960000i
\(26\) −276.000 −0.408284
\(27\) 0 0
\(28\) −232.000 + 232.000i −0.295918 + 0.295918i
\(29\) 680.000i 0.808561i 0.914635 + 0.404281i \(0.132478\pi\)
−0.914635 + 0.404281i \(0.867522\pi\)
\(30\) 540.000 − 720.000i 0.600000 − 0.800000i
\(31\) 202.000 0.210198 0.105099 − 0.994462i \(-0.466484\pi\)
0.105099 + 0.994462i \(0.466484\pi\)
\(32\) 128.000 + 128.000i 0.125000 + 0.125000i
\(33\) −1062.00 + 1062.00i −0.975207 + 0.975207i
\(34\) 1084.00i 0.937716i
\(35\) −1015.00 + 145.000i −0.828571 + 0.118367i
\(36\) 648.000 0.500000
\(37\) −651.000 − 651.000i −0.475530 − 0.475530i 0.428169 − 0.903699i \(-0.359159\pi\)
−0.903699 + 0.428169i \(0.859159\pi\)
\(38\) 560.000 − 560.000i 0.387812 − 0.387812i
\(39\) − 1242.00i − 0.816568i
\(40\) 80.0000 + 560.000i 0.0500000 + 0.350000i
\(41\) 1682.00 1.00059 0.500297 − 0.865854i \(-0.333224\pi\)
0.500297 + 0.865854i \(0.333224\pi\)
\(42\) −1044.00 − 1044.00i −0.591837 − 0.591837i
\(43\) 1089.00 − 1089.00i 0.588967 − 0.588967i −0.348385 − 0.937352i \(-0.613270\pi\)
0.937352 + 0.348385i \(0.113270\pi\)
\(44\) − 944.000i − 0.487603i
\(45\) 1620.00 + 1215.00i 0.800000 + 0.600000i
\(46\) −1076.00 −0.508507
\(47\) 1269.00 + 1269.00i 0.574468 + 0.574468i 0.933374 − 0.358906i \(-0.116850\pi\)
−0.358906 + 0.933374i \(0.616850\pi\)
\(48\) −576.000 + 576.000i −0.250000 + 0.250000i
\(49\) − 719.000i − 0.299459i
\(50\) −850.000 + 1550.00i −0.340000 + 0.620000i
\(51\) −4878.00 −1.87543
\(52\) 552.000 + 552.000i 0.204142 + 0.204142i
\(53\) −611.000 + 611.000i −0.217515 + 0.217515i −0.807450 − 0.589935i \(-0.799153\pi\)
0.589935 + 0.807450i \(0.299153\pi\)
\(54\) 0 0
\(55\) 1770.00 − 2360.00i 0.585124 − 0.780165i
\(56\) 928.000 0.295918
\(57\) 2520.00 + 2520.00i 0.775623 + 0.775623i
\(58\) 1360.00 − 1360.00i 0.404281 − 0.404281i
\(59\) 1160.00i 0.333238i 0.986021 + 0.166619i \(0.0532849\pi\)
−0.986021 + 0.166619i \(0.946715\pi\)
\(60\) −2520.00 + 360.000i −0.700000 + 0.100000i
\(61\) −5598.00 −1.50443 −0.752217 − 0.658915i \(-0.771016\pi\)
−0.752217 + 0.658915i \(0.771016\pi\)
\(62\) −404.000 − 404.000i −0.105099 − 0.105099i
\(63\) 2349.00 − 2349.00i 0.591837 − 0.591837i
\(64\) − 512.000i − 0.125000i
\(65\) 345.000 + 2415.00i 0.0816568 + 0.571598i
\(66\) 4248.00 0.975207
\(67\) −751.000 − 751.000i −0.167298 − 0.167298i 0.618493 − 0.785791i \(-0.287744\pi\)
−0.785791 + 0.618493i \(0.787744\pi\)
\(68\) 2168.00 − 2168.00i 0.468858 − 0.468858i
\(69\) − 4842.00i − 1.01701i
\(70\) 2320.00 + 1740.00i 0.473469 + 0.355102i
\(71\) 6442.00 1.27792 0.638961 − 0.769240i \(-0.279365\pi\)
0.638961 + 0.769240i \(0.279365\pi\)
\(72\) −1296.00 − 1296.00i −0.250000 − 0.250000i
\(73\) −2951.00 + 2951.00i −0.553762 + 0.553762i −0.927525 − 0.373762i \(-0.878068\pi\)
0.373762 + 0.927525i \(0.378068\pi\)
\(74\) 2604.00i 0.475530i
\(75\) −6975.00 − 3825.00i −1.24000 − 0.680000i
\(76\) −2240.00 −0.387812
\(77\) −3422.00 − 3422.00i −0.577163 − 0.577163i
\(78\) −2484.00 + 2484.00i −0.408284 + 0.408284i
\(79\) 10560.0i 1.69204i 0.533154 + 0.846018i \(0.321007\pi\)
−0.533154 + 0.846018i \(0.678993\pi\)
\(80\) 960.000 − 1280.00i 0.150000 − 0.200000i
\(81\) 6561.00 1.00000
\(82\) −3364.00 − 3364.00i −0.500297 − 0.500297i
\(83\) −6231.00 + 6231.00i −0.904485 + 0.904485i −0.995820 − 0.0913348i \(-0.970887\pi\)
0.0913348 + 0.995820i \(0.470887\pi\)
\(84\) 4176.00i 0.591837i
\(85\) 9485.00 − 1355.00i 1.31280 − 0.187543i
\(86\) −4356.00 −0.588967
\(87\) 6120.00 + 6120.00i 0.808561 + 0.808561i
\(88\) −1888.00 + 1888.00i −0.243802 + 0.243802i
\(89\) − 14480.0i − 1.82805i −0.405656 − 0.914026i \(-0.632957\pi\)
0.405656 − 0.914026i \(-0.367043\pi\)
\(90\) −810.000 − 5670.00i −0.100000 − 0.700000i
\(91\) 4002.00 0.483275
\(92\) 2152.00 + 2152.00i 0.254253 + 0.254253i
\(93\) 1818.00 − 1818.00i 0.210198 − 0.210198i
\(94\) − 5076.00i − 0.574468i
\(95\) −5600.00 − 4200.00i −0.620499 − 0.465374i
\(96\) 2304.00 0.250000
\(97\) −7311.00 − 7311.00i −0.777022 − 0.777022i 0.202301 − 0.979323i \(-0.435158\pi\)
−0.979323 + 0.202301i \(0.935158\pi\)
\(98\) −1438.00 + 1438.00i −0.149729 + 0.149729i
\(99\) 9558.00i 0.975207i
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 10.5.c.a.7.1 yes 2
3.2 odd 2 90.5.g.b.37.1 2
4.3 odd 2 80.5.p.b.17.1 2
5.2 odd 4 50.5.c.b.43.1 2
5.3 odd 4 inner 10.5.c.a.3.1 ✓ 2
5.4 even 2 50.5.c.b.7.1 2
8.3 odd 2 320.5.p.i.257.1 2
8.5 even 2 320.5.p.b.257.1 2
15.2 even 4 450.5.g.a.343.1 2
15.8 even 4 90.5.g.b.73.1 2
15.14 odd 2 450.5.g.a.307.1 2
20.3 even 4 80.5.p.b.33.1 2
20.7 even 4 400.5.p.c.193.1 2
20.19 odd 2 400.5.p.c.257.1 2
40.3 even 4 320.5.p.i.193.1 2
40.13 odd 4 320.5.p.b.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.5.c.a.3.1 ✓ 2 5.3 odd 4 inner
10.5.c.a.7.1 yes 2 1.1 even 1 trivial
50.5.c.b.7.1 2 5.4 even 2
50.5.c.b.43.1 2 5.2 odd 4
80.5.p.b.17.1 2 4.3 odd 2
80.5.p.b.33.1 2 20.3 even 4
90.5.g.b.37.1 2 3.2 odd 2
90.5.g.b.73.1 2 15.8 even 4
320.5.p.b.193.1 2 40.13 odd 4
320.5.p.b.257.1 2 8.5 even 2
320.5.p.i.193.1 2 40.3 even 4
320.5.p.i.257.1 2 8.3 odd 2
400.5.p.c.193.1 2 20.7 even 4
400.5.p.c.257.1 2 20.19 odd 2
450.5.g.a.307.1 2 15.14 odd 2
450.5.g.a.343.1 2 15.2 even 4