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Properties

Label 10.3.c.a.7.1
Level $10$
Weight $3$
Character 10.7
Analytic conductor $0.272$
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] = [10,3,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, 3); 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, 3, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.272480264360\)
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: \(\Z[a_1, a_2]\)
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 10.3.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+(-1.00000 - 1.00000i) q^{2} +(-2.00000 + 2.00000i) q^{3} +2.00000i q^{4} -5.00000i q^{5} +4.00000 q^{6} +(2.00000 + 2.00000i) q^{7} +(2.00000 - 2.00000i) q^{8} +1.00000i q^{9} +(-5.00000 + 5.00000i) q^{10} -8.00000 q^{11} +(-4.00000 - 4.00000i) q^{12} +(3.00000 - 3.00000i) q^{13} -4.00000i q^{14} +(10.0000 + 10.0000i) q^{15} -4.00000 q^{16} +(7.00000 + 7.00000i) q^{17} +(1.00000 - 1.00000i) q^{18} -20.0000i q^{19} +10.0000 q^{20} -8.00000 q^{21} +(8.00000 + 8.00000i) q^{22} +(-2.00000 + 2.00000i) q^{23} +8.00000i q^{24} -25.0000 q^{25} -6.00000 q^{26} +(-20.0000 - 20.0000i) q^{27} +(-4.00000 + 4.00000i) q^{28} +40.0000i q^{29} -20.0000i q^{30} +52.0000 q^{31} +(4.00000 + 4.00000i) q^{32} +(16.0000 - 16.0000i) q^{33} -14.0000i q^{34} +(10.0000 - 10.0000i) q^{35} -2.00000 q^{36} +(-3.00000 - 3.00000i) q^{37} +(-20.0000 + 20.0000i) q^{38} +12.0000i q^{39} +(-10.0000 - 10.0000i) q^{40} -8.00000 q^{41} +(8.00000 + 8.00000i) q^{42} +(-42.0000 + 42.0000i) q^{43} -16.0000i q^{44} +5.00000 q^{45} +4.00000 q^{46} +(-18.0000 - 18.0000i) q^{47} +(8.00000 - 8.00000i) q^{48} -41.0000i q^{49} +(25.0000 + 25.0000i) q^{50} -28.0000 q^{51} +(6.00000 + 6.00000i) q^{52} +(53.0000 - 53.0000i) q^{53} +40.0000i q^{54} +40.0000i q^{55} +8.00000 q^{56} +(40.0000 + 40.0000i) q^{57} +(40.0000 - 40.0000i) q^{58} -20.0000i q^{59} +(-20.0000 + 20.0000i) q^{60} -48.0000 q^{61} +(-52.0000 - 52.0000i) q^{62} +(-2.00000 + 2.00000i) q^{63} -8.00000i q^{64} +(-15.0000 - 15.0000i) q^{65} -32.0000 q^{66} +(62.0000 + 62.0000i) q^{67} +(-14.0000 + 14.0000i) q^{68} -8.00000i q^{69} -20.0000 q^{70} -28.0000 q^{71} +(2.00000 + 2.00000i) q^{72} +(-47.0000 + 47.0000i) q^{73} +6.00000i q^{74} +(50.0000 - 50.0000i) q^{75} +40.0000 q^{76} +(-16.0000 - 16.0000i) q^{77} +(12.0000 - 12.0000i) q^{78} +20.0000i q^{80} +71.0000 q^{81} +(8.00000 + 8.00000i) q^{82} +(18.0000 - 18.0000i) q^{83} -16.0000i q^{84} +(35.0000 - 35.0000i) q^{85} +84.0000 q^{86} +(-80.0000 - 80.0000i) q^{87} +(-16.0000 + 16.0000i) q^{88} +80.0000i q^{89} +(-5.00000 - 5.00000i) q^{90} +12.0000 q^{91} +(-4.00000 - 4.00000i) q^{92} +(-104.000 + 104.000i) q^{93} +36.0000i q^{94} -100.000 q^{95} -16.0000 q^{96} +(-63.0000 - 63.0000i) q^{97} +(-41.0000 + 41.0000i) q^{98} -8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 4 q^{3} + 8 q^{6} + 4 q^{7} + 4 q^{8} - 10 q^{10} - 16 q^{11} - 8 q^{12} + 6 q^{13} + 20 q^{15} - 8 q^{16} + 14 q^{17} + 2 q^{18} + 20 q^{20} - 16 q^{21} + 16 q^{22} - 4 q^{23} - 50 q^{25}+ \cdots - 82 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 − 1.00000i −0.500000 − 0.500000i
\(3\) −2.00000 + 2.00000i −0.666667 + 0.666667i −0.956943 − 0.290276i \(-0.906253\pi\)
0.290276 + 0.956943i \(0.406253\pi\)
\(4\) 2.00000i 0.500000i
\(5\) − 5.00000i − 1.00000i
\(6\) 4.00000 0.666667
\(7\) 2.00000 + 2.00000i 0.285714 + 0.285714i 0.835383 − 0.549669i \(-0.185246\pi\)
−0.549669 + 0.835383i \(0.685246\pi\)
\(8\) 2.00000 − 2.00000i 0.250000 − 0.250000i
\(9\) 1.00000i 0.111111i
\(10\) −5.00000 + 5.00000i −0.500000 + 0.500000i
\(11\) −8.00000 −0.727273 −0.363636 − 0.931541i \(-0.618465\pi\)
−0.363636 + 0.931541i \(0.618465\pi\)
\(12\) −4.00000 − 4.00000i −0.333333 − 0.333333i
\(13\) 3.00000 − 3.00000i 0.230769 − 0.230769i −0.582245 − 0.813014i \(-0.697825\pi\)
0.813014 + 0.582245i \(0.197825\pi\)
\(14\) − 4.00000i − 0.285714i
\(15\) 10.0000 + 10.0000i 0.666667 + 0.666667i
\(16\) −4.00000 −0.250000
\(17\) 7.00000 + 7.00000i 0.411765 + 0.411765i 0.882353 − 0.470588i \(-0.155958\pi\)
−0.470588 + 0.882353i \(0.655958\pi\)
\(18\) 1.00000 − 1.00000i 0.0555556 − 0.0555556i
\(19\) − 20.0000i − 1.05263i −0.850289 − 0.526316i \(-0.823573\pi\)
0.850289 − 0.526316i \(-0.176427\pi\)
\(20\) 10.0000 0.500000
\(21\) −8.00000 −0.380952
\(22\) 8.00000 + 8.00000i 0.363636 + 0.363636i
\(23\) −2.00000 + 2.00000i −0.0869565 + 0.0869565i −0.749247 − 0.662291i \(-0.769584\pi\)
0.662291 + 0.749247i \(0.269584\pi\)
\(24\) 8.00000i 0.333333i
\(25\) −25.0000 −1.00000
\(26\) −6.00000 −0.230769
\(27\) −20.0000 − 20.0000i −0.740741 − 0.740741i
\(28\) −4.00000 + 4.00000i −0.142857 + 0.142857i
\(29\) 40.0000i 1.37931i 0.724138 + 0.689655i \(0.242238\pi\)
−0.724138 + 0.689655i \(0.757762\pi\)
\(30\) − 20.0000i − 0.666667i
\(31\) 52.0000 1.67742 0.838710 − 0.544579i \(-0.183310\pi\)
0.838710 + 0.544579i \(0.183310\pi\)
\(32\) 4.00000 + 4.00000i 0.125000 + 0.125000i
\(33\) 16.0000 − 16.0000i 0.484848 − 0.484848i
\(34\) − 14.0000i − 0.411765i
\(35\) 10.0000 − 10.0000i 0.285714 − 0.285714i
\(36\) −2.00000 −0.0555556
\(37\) −3.00000 − 3.00000i −0.0810811 − 0.0810811i 0.665403 − 0.746484i \(-0.268260\pi\)
−0.746484 + 0.665403i \(0.768260\pi\)
\(38\) −20.0000 + 20.0000i −0.526316 + 0.526316i
\(39\) 12.0000i 0.307692i
\(40\) −10.0000 − 10.0000i −0.250000 − 0.250000i
\(41\) −8.00000 −0.195122 −0.0975610 − 0.995230i \(-0.531104\pi\)
−0.0975610 + 0.995230i \(0.531104\pi\)
\(42\) 8.00000 + 8.00000i 0.190476 + 0.190476i
\(43\) −42.0000 + 42.0000i −0.976744 + 0.976744i −0.999736 − 0.0229915i \(-0.992681\pi\)
0.0229915 + 0.999736i \(0.492681\pi\)
\(44\) − 16.0000i − 0.363636i
\(45\) 5.00000 0.111111
\(46\) 4.00000 0.0869565
\(47\) −18.0000 − 18.0000i −0.382979 − 0.382979i 0.489195 − 0.872174i \(-0.337290\pi\)
−0.872174 + 0.489195i \(0.837290\pi\)
\(48\) 8.00000 − 8.00000i 0.166667 − 0.166667i
\(49\) − 41.0000i − 0.836735i
\(50\) 25.0000 + 25.0000i 0.500000 + 0.500000i
\(51\) −28.0000 −0.549020
\(52\) 6.00000 + 6.00000i 0.115385 + 0.115385i
\(53\) 53.0000 − 53.0000i 1.00000 − 1.00000i − 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(54\) 40.0000i 0.740741i
\(55\) 40.0000i 0.727273i
\(56\) 8.00000 0.142857
\(57\) 40.0000 + 40.0000i 0.701754 + 0.701754i
\(58\) 40.0000 − 40.0000i 0.689655 − 0.689655i
\(59\) − 20.0000i − 0.338983i −0.985532 − 0.169492i \(-0.945787\pi\)
0.985532 − 0.169492i \(-0.0542125\pi\)
\(60\) −20.0000 + 20.0000i −0.333333 + 0.333333i
\(61\) −48.0000 −0.786885 −0.393443 − 0.919349i \(-0.628716\pi\)
−0.393443 + 0.919349i \(0.628716\pi\)
\(62\) −52.0000 − 52.0000i −0.838710 − 0.838710i
\(63\) −2.00000 + 2.00000i −0.0317460 + 0.0317460i
\(64\) − 8.00000i − 0.125000i
\(65\) −15.0000 − 15.0000i −0.230769 − 0.230769i
\(66\) −32.0000 −0.484848
\(67\) 62.0000 + 62.0000i 0.925373 + 0.925373i 0.997403 − 0.0720294i \(-0.0229475\pi\)
−0.0720294 + 0.997403i \(0.522948\pi\)
\(68\) −14.0000 + 14.0000i −0.205882 + 0.205882i
\(69\) − 8.00000i − 0.115942i
\(70\) −20.0000 −0.285714
\(71\) −28.0000 −0.394366 −0.197183 − 0.980367i \(-0.563179\pi\)
−0.197183 + 0.980367i \(0.563179\pi\)
\(72\) 2.00000 + 2.00000i 0.0277778 + 0.0277778i
\(73\) −47.0000 + 47.0000i −0.643836 + 0.643836i −0.951496 − 0.307661i \(-0.900454\pi\)
0.307661 + 0.951496i \(0.400454\pi\)
\(74\) 6.00000i 0.0810811i
\(75\) 50.0000 − 50.0000i 0.666667 − 0.666667i
\(76\) 40.0000 0.526316
\(77\) −16.0000 − 16.0000i −0.207792 − 0.207792i
\(78\) 12.0000 − 12.0000i 0.153846 − 0.153846i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 20.0000i 0.250000i
\(81\) 71.0000 0.876543
\(82\) 8.00000 + 8.00000i 0.0975610 + 0.0975610i
\(83\) 18.0000 − 18.0000i 0.216867 − 0.216867i −0.590310 − 0.807177i \(-0.700994\pi\)
0.807177 + 0.590310i \(0.200994\pi\)
\(84\) − 16.0000i − 0.190476i
\(85\) 35.0000 − 35.0000i 0.411765 − 0.411765i
\(86\) 84.0000 0.976744
\(87\) −80.0000 − 80.0000i −0.919540 − 0.919540i
\(88\) −16.0000 + 16.0000i −0.181818 + 0.181818i
\(89\) 80.0000i 0.898876i 0.893311 + 0.449438i \(0.148376\pi\)
−0.893311 + 0.449438i \(0.851624\pi\)
\(90\) −5.00000 − 5.00000i −0.0555556 − 0.0555556i
\(91\) 12.0000 0.131868
\(92\) −4.00000 − 4.00000i −0.0434783 − 0.0434783i
\(93\) −104.000 + 104.000i −1.11828 + 1.11828i
\(94\) 36.0000i 0.382979i
\(95\) −100.000 −1.05263
\(96\) −16.0000 −0.166667
\(97\) −63.0000 − 63.0000i −0.649485 − 0.649485i 0.303384 − 0.952868i \(-0.401884\pi\)
−0.952868 + 0.303384i \(0.901884\pi\)
\(98\) −41.0000 + 41.0000i −0.418367 + 0.418367i
\(99\) − 8.00000i − 0.0808081i
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 10.3.c.a.7.1 yes 2
3.2 odd 2 90.3.g.b.37.1 2
4.3 odd 2 80.3.p.c.17.1 2
5.2 odd 4 50.3.c.c.43.1 2
5.3 odd 4 inner 10.3.c.a.3.1 ✓ 2
5.4 even 2 50.3.c.c.7.1 2
7.6 odd 2 490.3.f.b.197.1 2
8.3 odd 2 320.3.p.a.257.1 2
8.5 even 2 320.3.p.h.257.1 2
12.11 even 2 720.3.bh.c.577.1 2
15.2 even 4 450.3.g.b.343.1 2
15.8 even 4 90.3.g.b.73.1 2
15.14 odd 2 450.3.g.b.307.1 2
20.3 even 4 80.3.p.c.33.1 2
20.7 even 4 400.3.p.b.193.1 2
20.19 odd 2 400.3.p.b.257.1 2
35.13 even 4 490.3.f.b.393.1 2
40.3 even 4 320.3.p.a.193.1 2
40.13 odd 4 320.3.p.h.193.1 2
60.23 odd 4 720.3.bh.c.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.3.c.a.3.1 ✓ 2 5.3 odd 4 inner
10.3.c.a.7.1 yes 2 1.1 even 1 trivial
50.3.c.c.7.1 2 5.4 even 2
50.3.c.c.43.1 2 5.2 odd 4
80.3.p.c.17.1 2 4.3 odd 2
80.3.p.c.33.1 2 20.3 even 4
90.3.g.b.37.1 2 3.2 odd 2
90.3.g.b.73.1 2 15.8 even 4
320.3.p.a.193.1 2 40.3 even 4
320.3.p.a.257.1 2 8.3 odd 2
320.3.p.h.193.1 2 40.13 odd 4
320.3.p.h.257.1 2 8.5 even 2
400.3.p.b.193.1 2 20.7 even 4
400.3.p.b.257.1 2 20.19 odd 2
450.3.g.b.307.1 2 15.14 odd 2
450.3.g.b.343.1 2 15.2 even 4
490.3.f.b.197.1 2 7.6 odd 2
490.3.f.b.393.1 2 35.13 even 4
720.3.bh.c.433.1 2 60.23 odd 4
720.3.bh.c.577.1 2 12.11 even 2