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Properties

Label 10.5.c.b.3.1
Level $10$
Weight $5$
Character 10.3
Analytic conductor $1.034$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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 3.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 10.3
Dual form 10.5.c.b.7.1

$q$-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} +(1.00000 + 1.00000i) q^{3} -8.00000i q^{4} +(-15.0000 + 20.0000i) q^{5} +4.00000 q^{6} +(-19.0000 + 19.0000i) q^{7} +(-16.0000 - 16.0000i) q^{8} -79.0000i q^{9} +(10.0000 + 70.0000i) q^{10} +202.000 q^{11} +(8.00000 - 8.00000i) q^{12} +(-99.0000 - 99.0000i) q^{13} +76.0000i q^{14} +(-35.0000 + 5.00000i) q^{15} -64.0000 q^{16} +(-239.000 + 239.000i) q^{17} +(-158.000 - 158.000i) q^{18} -40.0000i q^{19} +(160.000 + 120.000i) q^{20} -38.0000 q^{21} +(404.000 - 404.000i) q^{22} +(541.000 + 541.000i) q^{23} -32.0000i q^{24} +(-175.000 - 600.000i) q^{25} -396.000 q^{26} +(160.000 - 160.000i) q^{27} +(152.000 + 152.000i) q^{28} +200.000i q^{29} +(-60.0000 + 80.0000i) q^{30} -758.000 q^{31} +(-128.000 + 128.000i) q^{32} +(202.000 + 202.000i) q^{33} +956.000i q^{34} +(-95.0000 - 665.000i) q^{35} -632.000 q^{36} +(141.000 - 141.000i) q^{37} +(-80.0000 - 80.0000i) q^{38} -198.000i q^{39} +(560.000 - 80.0000i) q^{40} +1042.00 q^{41} +(-76.0000 + 76.0000i) q^{42} +(-759.000 - 759.000i) q^{43} -1616.00i q^{44} +(1580.00 + 1185.00i) q^{45} +2164.00 q^{46} +(-459.000 + 459.000i) q^{47} +(-64.0000 - 64.0000i) q^{48} +1679.00i q^{49} +(-1550.00 - 850.000i) q^{50} -478.000 q^{51} +(-792.000 + 792.000i) q^{52} +(-1819.00 - 1819.00i) q^{53} -640.000i q^{54} +(-3030.00 + 4040.00i) q^{55} +608.000 q^{56} +(40.0000 - 40.0000i) q^{57} +(400.000 + 400.000i) q^{58} -4600.00i q^{59} +(40.0000 + 280.000i) q^{60} +2082.00 q^{61} +(-1516.00 + 1516.00i) q^{62} +(1501.00 + 1501.00i) q^{63} +512.000i q^{64} +(3465.00 - 495.000i) q^{65} +808.000 q^{66} +(5081.00 - 5081.00i) q^{67} +(1912.00 + 1912.00i) q^{68} +1082.00i q^{69} +(-1520.00 - 1140.00i) q^{70} -3478.00 q^{71} +(-1264.00 + 1264.00i) q^{72} +(-3479.00 - 3479.00i) q^{73} -564.000i q^{74} +(425.000 - 775.000i) q^{75} -320.000 q^{76} +(-3838.00 + 3838.00i) q^{77} +(-396.000 - 396.000i) q^{78} +7680.00i q^{79} +(960.000 - 1280.00i) q^{80} -6079.00 q^{81} +(2084.00 - 2084.00i) q^{82} +(6081.00 + 6081.00i) q^{83} +304.000i q^{84} +(-1195.00 - 8365.00i) q^{85} -3036.00 q^{86} +(-200.000 + 200.000i) q^{87} +(-3232.00 - 3232.00i) q^{88} +5680.00i q^{89} +(5530.00 - 790.000i) q^{90} +3762.00 q^{91} +(4328.00 - 4328.00i) q^{92} +(-758.000 - 758.000i) q^{93} +1836.00i q^{94} +(800.000 + 600.000i) q^{95} -256.000 q^{96} +(561.000 - 561.000i) q^{97} +(3358.00 + 3358.00i) q^{98} -15958.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} + 2 q^{3} - 30 q^{5} + 8 q^{6} - 38 q^{7} - 32 q^{8} + 20 q^{10} + 404 q^{11} + 16 q^{12} - 198 q^{13} - 70 q^{15} - 128 q^{16} - 478 q^{17} - 316 q^{18} + 320 q^{20} - 76 q^{21} + 808 q^{22}+ \cdots + 6716 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(7\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 1.00000 + 1.00000i 0.111111 + 0.111111i 0.760477 − 0.649365i \(-0.224965\pi\)
−0.649365 + 0.760477i \(0.724965\pi\)
\(4\) − 8.00000i − 0.500000i
\(5\) −15.0000 + 20.0000i −0.600000 + 0.800000i
\(6\) 4.00000 0.111111
\(7\) −19.0000 + 19.0000i −0.387755 + 0.387755i −0.873886 − 0.486131i \(-0.838408\pi\)
0.486131 + 0.873886i \(0.338408\pi\)
\(8\) −16.0000 − 16.0000i −0.250000 − 0.250000i
\(9\) − 79.0000i − 0.975309i
\(10\) 10.0000 + 70.0000i 0.100000 + 0.700000i
\(11\) 202.000 1.66942 0.834711 − 0.550689i \(-0.185635\pi\)
0.834711 + 0.550689i \(0.185635\pi\)
\(12\) 8.00000 − 8.00000i 0.0555556 − 0.0555556i
\(13\) −99.0000 − 99.0000i −0.585799 − 0.585799i 0.350692 − 0.936491i \(-0.385946\pi\)
−0.936491 + 0.350692i \(0.885946\pi\)
\(14\) 76.0000i 0.387755i
\(15\) −35.0000 + 5.00000i −0.155556 + 0.0222222i
\(16\) −64.0000 −0.250000
\(17\) −239.000 + 239.000i −0.826990 + 0.826990i −0.987099 − 0.160110i \(-0.948815\pi\)
0.160110 + 0.987099i \(0.448815\pi\)
\(18\) −158.000 − 158.000i −0.487654 − 0.487654i
\(19\) − 40.0000i − 0.110803i −0.998464 − 0.0554017i \(-0.982356\pi\)
0.998464 − 0.0554017i \(-0.0176439\pi\)
\(20\) 160.000 + 120.000i 0.400000 + 0.300000i
\(21\) −38.0000 −0.0861678
\(22\) 404.000 − 404.000i 0.834711 − 0.834711i
\(23\) 541.000 + 541.000i 1.02268 + 1.02268i 0.999737 + 0.0229476i \(0.00730510\pi\)
0.0229476 + 0.999737i \(0.492695\pi\)
\(24\) − 32.0000i − 0.0555556i
\(25\) −175.000 − 600.000i −0.280000 − 0.960000i
\(26\) −396.000 −0.585799
\(27\) 160.000 − 160.000i 0.219479 − 0.219479i
\(28\) 152.000 + 152.000i 0.193878 + 0.193878i
\(29\) 200.000i 0.237812i 0.992906 + 0.118906i \(0.0379387\pi\)
−0.992906 + 0.118906i \(0.962061\pi\)
\(30\) −60.0000 + 80.0000i −0.0666667 + 0.0888889i
\(31\) −758.000 −0.788762 −0.394381 − 0.918947i \(-0.629041\pi\)
−0.394381 + 0.918947i \(0.629041\pi\)
\(32\) −128.000 + 128.000i −0.125000 + 0.125000i
\(33\) 202.000 + 202.000i 0.185491 + 0.185491i
\(34\) 956.000i 0.826990i
\(35\) −95.0000 − 665.000i −0.0775510 − 0.542857i
\(36\) −632.000 −0.487654
\(37\) 141.000 − 141.000i 0.102995 − 0.102995i −0.653732 − 0.756726i \(-0.726797\pi\)
0.756726 + 0.653732i \(0.226797\pi\)
\(38\) −80.0000 − 80.0000i −0.0554017 − 0.0554017i
\(39\) − 198.000i − 0.130178i
\(40\) 560.000 − 80.0000i 0.350000 − 0.0500000i
\(41\) 1042.00 0.619869 0.309935 − 0.950758i \(-0.399693\pi\)
0.309935 + 0.950758i \(0.399693\pi\)
\(42\) −76.0000 + 76.0000i −0.0430839 + 0.0430839i
\(43\) −759.000 − 759.000i −0.410492 − 0.410492i 0.471418 − 0.881910i \(-0.343742\pi\)
−0.881910 + 0.471418i \(0.843742\pi\)
\(44\) − 1616.00i − 0.834711i
\(45\) 1580.00 + 1185.00i 0.780247 + 0.585185i
\(46\) 2164.00 1.02268
\(47\) −459.000 + 459.000i −0.207786 + 0.207786i −0.803326 − 0.595540i \(-0.796938\pi\)
0.595540 + 0.803326i \(0.296938\pi\)
\(48\) −64.0000 − 64.0000i −0.0277778 − 0.0277778i
\(49\) 1679.00i 0.699292i
\(50\) −1550.00 − 850.000i −0.620000 − 0.340000i
\(51\) −478.000 −0.183775
\(52\) −792.000 + 792.000i −0.292899 + 0.292899i
\(53\) −1819.00 − 1819.00i −0.647561 − 0.647561i 0.304842 − 0.952403i \(-0.401396\pi\)
−0.952403 + 0.304842i \(0.901396\pi\)
\(54\) − 640.000i − 0.219479i
\(55\) −3030.00 + 4040.00i −1.00165 + 1.33554i
\(56\) 608.000 0.193878
\(57\) 40.0000 − 40.0000i 0.0123115 − 0.0123115i
\(58\) 400.000 + 400.000i 0.118906 + 0.118906i
\(59\) − 4600.00i − 1.32146i −0.750624 − 0.660730i \(-0.770247\pi\)
0.750624 − 0.660730i \(-0.229753\pi\)
\(60\) 40.0000 + 280.000i 0.0111111 + 0.0777778i
\(61\) 2082.00 0.559527 0.279764 − 0.960069i \(-0.409744\pi\)
0.279764 + 0.960069i \(0.409744\pi\)
\(62\) −1516.00 + 1516.00i −0.394381 + 0.394381i
\(63\) 1501.00 + 1501.00i 0.378181 + 0.378181i
\(64\) 512.000i 0.125000i
\(65\) 3465.00 − 495.000i 0.820118 − 0.117160i
\(66\) 808.000 0.185491
\(67\) 5081.00 − 5081.00i 1.13188 − 1.13188i 0.142013 − 0.989865i \(-0.454642\pi\)
0.989865 − 0.142013i \(-0.0453575\pi\)
\(68\) 1912.00 + 1912.00i 0.413495 + 0.413495i
\(69\) 1082.00i 0.227263i
\(70\) −1520.00 − 1140.00i −0.310204 − 0.232653i
\(71\) −3478.00 −0.689942 −0.344971 − 0.938613i \(-0.612111\pi\)
−0.344971 + 0.938613i \(0.612111\pi\)
\(72\) −1264.00 + 1264.00i −0.243827 + 0.243827i
\(73\) −3479.00 − 3479.00i −0.652843 − 0.652843i 0.300834 − 0.953677i \(-0.402735\pi\)
−0.953677 + 0.300834i \(0.902735\pi\)
\(74\) − 564.000i − 0.102995i
\(75\) 425.000 − 775.000i 0.0755556 − 0.137778i
\(76\) −320.000 −0.0554017
\(77\) −3838.00 + 3838.00i −0.647327 + 0.647327i
\(78\) −396.000 − 396.000i −0.0650888 − 0.0650888i
\(79\) 7680.00i 1.23057i 0.788304 + 0.615286i \(0.210959\pi\)
−0.788304 + 0.615286i \(0.789041\pi\)
\(80\) 960.000 − 1280.00i 0.150000 − 0.200000i
\(81\) −6079.00 −0.926536
\(82\) 2084.00 − 2084.00i 0.309935 − 0.309935i
\(83\) 6081.00 + 6081.00i 0.882712 + 0.882712i 0.993809 − 0.111098i \(-0.0354367\pi\)
−0.111098 + 0.993809i \(0.535437\pi\)
\(84\) 304.000i 0.0430839i
\(85\) −1195.00 − 8365.00i −0.165398 − 1.15779i
\(86\) −3036.00 −0.410492
\(87\) −200.000 + 200.000i −0.0264236 + 0.0264236i
\(88\) −3232.00 − 3232.00i −0.417355 − 0.417355i
\(89\) 5680.00i 0.717081i 0.933514 + 0.358541i \(0.116726\pi\)
−0.933514 + 0.358541i \(0.883274\pi\)
\(90\) 5530.00 − 790.000i 0.682716 − 0.0975309i
\(91\) 3762.00 0.454293
\(92\) 4328.00 − 4328.00i 0.511342 − 0.511342i
\(93\) −758.000 − 758.000i −0.0876402 − 0.0876402i
\(94\) 1836.00i 0.207786i
\(95\) 800.000 + 600.000i 0.0886427 + 0.0664820i
\(96\) −256.000 −0.0277778
\(97\) 561.000 − 561.000i 0.0596238 − 0.0596238i −0.676666 − 0.736290i \(-0.736576\pi\)
0.736290 + 0.676666i \(0.236576\pi\)
\(98\) 3358.00 + 3358.00i 0.349646 + 0.349646i
\(99\) − 15958.0i − 1.62820i
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.b.3.1 ✓ 2
3.2 odd 2 90.5.g.a.73.1 2
4.3 odd 2 80.5.p.c.33.1 2
5.2 odd 4 inner 10.5.c.b.7.1 yes 2
5.3 odd 4 50.5.c.a.7.1 2
5.4 even 2 50.5.c.a.43.1 2
8.3 odd 2 320.5.p.g.193.1 2
8.5 even 2 320.5.p.d.193.1 2
15.2 even 4 90.5.g.a.37.1 2
15.8 even 4 450.5.g.b.307.1 2
15.14 odd 2 450.5.g.b.343.1 2
20.3 even 4 400.5.p.b.257.1 2
20.7 even 4 80.5.p.c.17.1 2
20.19 odd 2 400.5.p.b.193.1 2
40.27 even 4 320.5.p.g.257.1 2
40.37 odd 4 320.5.p.d.257.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.5.c.b.3.1 ✓ 2 1.1 even 1 trivial
10.5.c.b.7.1 yes 2 5.2 odd 4 inner
50.5.c.a.7.1 2 5.3 odd 4
50.5.c.a.43.1 2 5.4 even 2
80.5.p.c.17.1 2 20.7 even 4
80.5.p.c.33.1 2 4.3 odd 2
90.5.g.a.37.1 2 15.2 even 4
90.5.g.a.73.1 2 3.2 odd 2
320.5.p.d.193.1 2 8.5 even 2
320.5.p.d.257.1 2 40.37 odd 4
320.5.p.g.193.1 2 8.3 odd 2
320.5.p.g.257.1 2 40.27 even 4
400.5.p.b.193.1 2 20.19 odd 2
400.5.p.b.257.1 2 20.3 even 4
450.5.g.b.307.1 2 15.8 even 4
450.5.g.b.343.1 2 15.14 odd 2