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Properties

Label 100.2.e.a.7.1
Level $100$
Weight $2$
Character 100.7
Analytic conductor $0.799$
Analytic rank $0$
Dimension $2$
CM discriminant -20
Inner twists $4$

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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] = [100,2,Mod(7,100)] 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("100.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 100.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.798504020213\)
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: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 7.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 100.7
Dual form 100.2.e.a.43.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.00000i) q^{2} +(-1.00000 - 1.00000i) q^{3} -2.00000i q^{4} +2.00000 q^{6} +(3.00000 - 3.00000i) q^{7} +(2.00000 + 2.00000i) q^{8} -1.00000i q^{9} +(-2.00000 + 2.00000i) q^{12} +6.00000i q^{14} -4.00000 q^{16} +(1.00000 + 1.00000i) q^{18} -6.00000 q^{21} +(-1.00000 - 1.00000i) q^{23} -4.00000i q^{24} +(-4.00000 + 4.00000i) q^{27} +(-6.00000 - 6.00000i) q^{28} +6.00000i q^{29} +(4.00000 - 4.00000i) q^{32} -2.00000 q^{36} +12.0000 q^{41} +(6.00000 - 6.00000i) q^{42} +(9.00000 + 9.00000i) q^{43} +2.00000 q^{46} +(-7.00000 + 7.00000i) q^{47} +(4.00000 + 4.00000i) q^{48} -11.0000i q^{49} -8.00000i q^{54} +12.0000 q^{56} +(-6.00000 - 6.00000i) q^{58} -8.00000 q^{61} +(-3.00000 - 3.00000i) q^{63} +8.00000i q^{64} +(3.00000 - 3.00000i) q^{67} +2.00000i q^{69} +(2.00000 - 2.00000i) q^{72} +5.00000 q^{81} +(-12.0000 + 12.0000i) q^{82} +(-11.0000 - 11.0000i) q^{83} +12.0000i q^{84} -18.0000 q^{86} +(6.00000 - 6.00000i) q^{87} +6.00000i q^{89} +(-2.00000 + 2.00000i) q^{92} -14.0000i q^{94} -8.00000 q^{96} +(11.0000 + 11.0000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 4 q^{6} + 6 q^{7} + 4 q^{8} - 4 q^{12} - 8 q^{16} + 2 q^{18} - 12 q^{21} - 2 q^{23} - 8 q^{27} - 12 q^{28} + 8 q^{32} - 4 q^{36} + 24 q^{41} + 12 q^{42} + 18 q^{43} + 4 q^{46}+ \cdots + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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 −0.707107 + 0.707107i
\(3\) −1.00000 − 1.00000i −0.577350 − 0.577350i 0.356822 − 0.934172i \(-0.383860\pi\)
−0.934172 + 0.356822i \(0.883860\pi\)
\(4\) − 2.00000i − 1.00000i
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 3.00000 − 3.00000i 1.13389 − 1.13389i 0.144370 − 0.989524i \(-0.453885\pi\)
0.989524 − 0.144370i \(-0.0461154\pi\)
\(8\) 2.00000 + 2.00000i 0.707107 + 0.707107i
\(9\) − 1.00000i − 0.333333i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −2.00000 + 2.00000i −0.577350 + 0.577350i
\(13\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) 6.00000i 1.60357i
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 1.00000 + 1.00000i 0.235702 + 0.235702i
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −6.00000 −1.30931
\(22\) 0 0
\(23\) −1.00000 − 1.00000i −0.208514 − 0.208514i 0.595121 − 0.803636i \(-0.297104\pi\)
−0.803636 + 0.595121i \(0.797104\pi\)
\(24\) − 4.00000i − 0.816497i
\(25\) 0 0
\(26\) 0 0
\(27\) −4.00000 + 4.00000i −0.769800 + 0.769800i
\(28\) −6.00000 − 6.00000i −1.13389 − 1.13389i
\(29\) 6.00000i 1.11417i 0.830455 + 0.557086i \(0.188081\pi\)
−0.830455 + 0.557086i \(0.811919\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 4.00000 − 4.00000i 0.707107 − 0.707107i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −2.00000 −0.333333
\(37\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 12.0000 1.87409 0.937043 − 0.349215i \(-0.113552\pi\)
0.937043 + 0.349215i \(0.113552\pi\)
\(42\) 6.00000 − 6.00000i 0.925820 − 0.925820i
\(43\) 9.00000 + 9.00000i 1.37249 + 1.37249i 0.856742 + 0.515745i \(0.172485\pi\)
0.515745 + 0.856742i \(0.327515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −7.00000 + 7.00000i −1.02105 + 1.02105i −0.0212814 + 0.999774i \(0.506775\pi\)
−0.999774 + 0.0212814i \(0.993225\pi\)
\(48\) 4.00000 + 4.00000i 0.577350 + 0.577350i
\(49\) − 11.0000i − 1.57143i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) − 8.00000i − 1.08866i
\(55\) 0 0
\(56\) 12.0000 1.60357
\(57\) 0 0
\(58\) −6.00000 − 6.00000i −0.787839 − 0.787839i
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) 0 0
\(63\) −3.00000 − 3.00000i −0.377964 − 0.377964i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 3.00000 − 3.00000i 0.366508 − 0.366508i −0.499694 − 0.866202i \(-0.666554\pi\)
0.866202 + 0.499694i \(0.166554\pi\)
\(68\) 0 0
\(69\) 2.00000i 0.240772i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 2.00000 − 2.00000i 0.235702 − 0.235702i
\(73\) 0 0 −0.707107 − 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 5.00000 0.555556
\(82\) −12.0000 + 12.0000i −1.32518 + 1.32518i
\(83\) −11.0000 − 11.0000i −1.20741 − 1.20741i −0.971864 − 0.235543i \(-0.924313\pi\)
−0.235543 − 0.971864i \(-0.575687\pi\)
\(84\) 12.0000i 1.30931i
\(85\) 0 0
\(86\) −18.0000 −1.94099
\(87\) 6.00000 − 6.00000i 0.643268 − 0.643268i
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.00000 + 2.00000i −0.208514 + 0.208514i
\(93\) 0 0
\(94\) − 14.0000i − 1.44399i
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) 0 0 0.707107 − 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(98\) 11.0000 + 11.0000i 1.11117 + 1.11117i
\(99\) 0 0
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 100.2.e.a.7.1 ✓ 2
3.2 odd 2 900.2.k.e.307.1 2
4.3 odd 2 100.2.e.c.7.1 yes 2
5.2 odd 4 inner 100.2.e.a.43.1 yes 2
5.3 odd 4 100.2.e.c.43.1 yes 2
5.4 even 2 100.2.e.c.7.1 yes 2
8.3 odd 2 1600.2.n.b.1407.1 2
8.5 even 2 1600.2.n.l.1407.1 2
12.11 even 2 900.2.k.a.307.1 2
15.2 even 4 900.2.k.e.343.1 2
15.8 even 4 900.2.k.a.343.1 2
15.14 odd 2 900.2.k.a.307.1 2
20.3 even 4 inner 100.2.e.a.43.1 yes 2
20.7 even 4 100.2.e.c.43.1 yes 2
20.19 odd 2 CM 100.2.e.a.7.1 ✓ 2
40.3 even 4 1600.2.n.l.1343.1 2
40.13 odd 4 1600.2.n.b.1343.1 2
40.19 odd 2 1600.2.n.l.1407.1 2
40.27 even 4 1600.2.n.b.1343.1 2
40.29 even 2 1600.2.n.b.1407.1 2
40.37 odd 4 1600.2.n.l.1343.1 2
60.23 odd 4 900.2.k.e.343.1 2
60.47 odd 4 900.2.k.a.343.1 2
60.59 even 2 900.2.k.e.307.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.2.e.a.7.1 ✓ 2 1.1 even 1 trivial
100.2.e.a.7.1 ✓ 2 20.19 odd 2 CM
100.2.e.a.43.1 yes 2 5.2 odd 4 inner
100.2.e.a.43.1 yes 2 20.3 even 4 inner
100.2.e.c.7.1 yes 2 4.3 odd 2
100.2.e.c.7.1 yes 2 5.4 even 2
100.2.e.c.43.1 yes 2 5.3 odd 4
100.2.e.c.43.1 yes 2 20.7 even 4
900.2.k.a.307.1 2 12.11 even 2
900.2.k.a.307.1 2 15.14 odd 2
900.2.k.a.343.1 2 15.8 even 4
900.2.k.a.343.1 2 60.47 odd 4
900.2.k.e.307.1 2 3.2 odd 2
900.2.k.e.307.1 2 60.59 even 2
900.2.k.e.343.1 2 15.2 even 4
900.2.k.e.343.1 2 60.23 odd 4
1600.2.n.b.1343.1 2 40.13 odd 4
1600.2.n.b.1343.1 2 40.27 even 4
1600.2.n.b.1407.1 2 8.3 odd 2
1600.2.n.b.1407.1 2 40.29 even 2
1600.2.n.l.1343.1 2 40.3 even 4
1600.2.n.l.1343.1 2 40.37 odd 4
1600.2.n.l.1407.1 2 8.5 even 2
1600.2.n.l.1407.1 2 40.19 odd 2