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Properties

Label 100.4.g.a.21.2
Level $100$
Weight $4$
Character 100.21
Analytic conductor $5.900$
Analytic rank $0$
Dimension $28$
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] = [100,4,Mod(21,100)] 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("100.21"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.g (of order \(5\), degree \(4\), 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: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 21.2
Character \(\chi\) \(=\) 100.21
Dual form 100.4.g.a.81.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.80168 - 3.48863i) q^{3} +(-10.9094 + 2.44643i) q^{5} +21.1171 q^{7} +(2.54218 + 7.82402i) q^{9} +(-19.7734 + 60.8562i) q^{11} +(21.5451 + 66.3089i) q^{13} +(60.9181 + 26.3118i) q^{15} +(22.4469 - 16.3086i) q^{17} +(51.8191 - 37.6487i) q^{19} +(-101.398 - 73.6698i) q^{21} +(-59.5819 + 183.374i) q^{23} +(113.030 - 53.3782i) q^{25} +(-34.4318 + 105.970i) q^{27} +(-152.193 - 110.575i) q^{29} +(-123.304 + 89.5852i) q^{31} +(307.250 - 223.230i) q^{33} +(-230.375 + 51.6616i) q^{35} +(-17.8034 - 54.7932i) q^{37} +(127.874 - 393.557i) q^{39} +(-73.7479 - 226.973i) q^{41} -113.814 q^{43} +(-46.8746 - 79.1361i) q^{45} +(271.048 + 196.928i) q^{47} +102.933 q^{49} -164.677 q^{51} +(344.395 + 250.217i) q^{53} +(66.8351 - 712.278i) q^{55} -380.161 q^{57} +(140.427 + 432.189i) q^{59} +(-205.577 + 632.702i) q^{61} +(53.6835 + 165.221i) q^{63} +(-397.264 - 670.681i) q^{65} +(-302.002 + 219.418i) q^{67} +(925.818 - 672.646i) q^{69} +(-589.123 - 428.023i) q^{71} +(103.889 - 319.736i) q^{73} +(-728.950 - 138.014i) q^{75} +(-417.557 + 1285.11i) q^{77} +(635.856 + 461.977i) q^{79} +(714.720 - 519.274i) q^{81} +(230.207 - 167.256i) q^{83} +(-204.984 + 232.832i) q^{85} +(345.029 + 1061.89i) q^{87} +(-177.591 + 546.570i) q^{89} +(454.970 + 1400.25i) q^{91} +904.594 q^{93} +(-473.210 + 537.497i) q^{95} +(100.433 + 72.9685i) q^{97} -526.407 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q - 4 q^{3} - 25 q^{5} + 16 q^{7} - 13 q^{9} - 20 q^{11} - 86 q^{13} - 240 q^{15} - 178 q^{17} + 2 q^{19} + 108 q^{21} + 102 q^{23} + 95 q^{25} + 92 q^{27} + 192 q^{29} + 378 q^{31} + 560 q^{33} - 350 q^{35}+ \cdots - 2100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(51\) \(77\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{5}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.80168 − 3.48863i −0.924084 − 0.671386i 0.0204531 − 0.999791i \(-0.493489\pi\)
−0.944537 + 0.328404i \(0.893489\pi\)
\(4\) 0 0
\(5\) −10.9094 + 2.44643i −0.975766 + 0.218816i
\(6\) 0 0
\(7\) 21.1171 1.14022 0.570109 − 0.821569i \(-0.306901\pi\)
0.570109 + 0.821569i \(0.306901\pi\)
\(8\) 0 0
\(9\) 2.54218 + 7.82402i 0.0941548 + 0.289779i
\(10\) 0 0
\(11\) −19.7734 + 60.8562i −0.541990 + 1.66807i 0.186050 + 0.982540i \(0.440431\pi\)
−0.728040 + 0.685535i \(0.759569\pi\)
\(12\) 0 0
\(13\) 21.5451 + 66.3089i 0.459656 + 1.41468i 0.865581 + 0.500769i \(0.166949\pi\)
−0.405925 + 0.913906i \(0.633051\pi\)
\(14\) 0 0
\(15\) 60.9181 + 26.3118i 1.04860 + 0.452912i
\(16\) 0 0
\(17\) 22.4469 − 16.3086i 0.320245 − 0.232672i −0.416035 − 0.909349i \(-0.636581\pi\)
0.736280 + 0.676677i \(0.236581\pi\)
\(18\) 0 0
\(19\) 51.8191 − 37.6487i 0.625690 − 0.454590i −0.229214 − 0.973376i \(-0.573616\pi\)
0.854904 + 0.518786i \(0.173616\pi\)
\(20\) 0 0
\(21\) −101.398 − 73.6698i −1.05366 − 0.765527i
\(22\) 0 0
\(23\) −59.5819 + 183.374i −0.540161 + 1.66244i 0.192066 + 0.981382i \(0.438481\pi\)
−0.732226 + 0.681061i \(0.761519\pi\)
\(24\) 0 0
\(25\) 113.030 − 53.3782i 0.904240 − 0.427026i
\(26\) 0 0
\(27\) −34.4318 + 105.970i −0.245422 + 0.755332i
\(28\) 0 0
\(29\) −152.193 − 110.575i −0.974537 − 0.708042i −0.0180557 − 0.999837i \(-0.505748\pi\)
−0.956481 + 0.291795i \(0.905748\pi\)
\(30\) 0 0
\(31\) −123.304 + 89.5852i −0.714386 + 0.519032i −0.884586 − 0.466378i \(-0.845559\pi\)
0.170200 + 0.985410i \(0.445559\pi\)
\(32\) 0 0
\(33\) 307.250 − 223.230i 1.62077 − 1.17756i
\(34\) 0 0
\(35\) −230.375 + 51.6616i −1.11259 + 0.249497i
\(36\) 0 0
\(37\) −17.8034 − 54.7932i −0.0791043 − 0.243458i 0.903682 − 0.428204i \(-0.140854\pi\)
−0.982786 + 0.184746i \(0.940854\pi\)
\(38\) 0 0
\(39\) 127.874 − 393.557i 0.525033 − 1.61589i
\(40\) 0 0
\(41\) −73.7479 − 226.973i −0.280914 − 0.864566i −0.987594 − 0.157032i \(-0.949808\pi\)
0.706679 − 0.707534i \(-0.250192\pi\)
\(42\) 0 0
\(43\) −113.814 −0.403639 −0.201819 − 0.979423i \(-0.564685\pi\)
−0.201819 + 0.979423i \(0.564685\pi\)
\(44\) 0 0
\(45\) −46.8746 − 79.1361i −0.155281 − 0.262154i
\(46\) 0 0
\(47\) 271.048 + 196.928i 0.841201 + 0.611168i 0.922706 − 0.385505i \(-0.125973\pi\)
−0.0815052 + 0.996673i \(0.525973\pi\)
\(48\) 0 0
\(49\) 102.933 0.300097
\(50\) 0 0
\(51\) −164.677 −0.452146
\(52\) 0 0
\(53\) 344.395 + 250.217i 0.892571 + 0.648491i 0.936547 − 0.350542i \(-0.114003\pi\)
−0.0439762 + 0.999033i \(0.514003\pi\)
\(54\) 0 0
\(55\) 66.8351 − 712.278i 0.163855 − 1.74625i
\(56\) 0 0
\(57\) −380.161 −0.883396
\(58\) 0 0
\(59\) 140.427 + 432.189i 0.309865 + 0.953665i 0.977817 + 0.209461i \(0.0671710\pi\)
−0.667952 + 0.744204i \(0.732829\pi\)
\(60\) 0 0
\(61\) −205.577 + 632.702i −0.431500 + 1.32802i 0.465132 + 0.885241i \(0.346007\pi\)
−0.896631 + 0.442778i \(0.853993\pi\)
\(62\) 0 0
\(63\) 53.6835 + 165.221i 0.107357 + 0.330411i
\(64\) 0 0
\(65\) −397.264 − 670.681i −0.758069 − 1.27981i
\(66\) 0 0
\(67\) −302.002 + 219.418i −0.550679 + 0.400091i −0.828036 − 0.560675i \(-0.810542\pi\)
0.277357 + 0.960767i \(0.410542\pi\)
\(68\) 0 0
\(69\) 925.818 − 672.646i 1.61530 − 1.17358i
\(70\) 0 0
\(71\) −589.123 − 428.023i −0.984734 − 0.715451i −0.0259722 − 0.999663i \(-0.508268\pi\)
−0.958762 + 0.284212i \(0.908268\pi\)
\(72\) 0 0
\(73\) 103.889 − 319.736i 0.166565 − 0.512634i −0.832583 − 0.553900i \(-0.813139\pi\)
0.999148 + 0.0412656i \(0.0131390\pi\)
\(74\) 0 0
\(75\) −728.950 − 138.014i −1.12229 − 0.212487i
\(76\) 0 0
\(77\) −417.557 + 1285.11i −0.617987 + 1.90197i
\(78\) 0 0
\(79\) 635.856 + 461.977i 0.905562 + 0.657930i 0.939889 − 0.341481i \(-0.110929\pi\)
−0.0343263 + 0.999411i \(0.510929\pi\)
\(80\) 0 0
\(81\) 714.720 − 519.274i 0.980411 − 0.712310i
\(82\) 0 0
\(83\) 230.207 − 167.256i 0.304440 − 0.221189i −0.425067 − 0.905162i \(-0.639749\pi\)
0.729507 + 0.683973i \(0.239749\pi\)
\(84\) 0 0
\(85\) −204.984 + 232.832i −0.261572 + 0.297108i
\(86\) 0 0
\(87\) 345.029 + 1061.89i 0.425184 + 1.30858i
\(88\) 0 0
\(89\) −177.591 + 546.570i −0.211513 + 0.650970i 0.787870 + 0.615842i \(0.211184\pi\)
−0.999383 + 0.0351281i \(0.988816\pi\)
\(90\) 0 0
\(91\) 454.970 + 1400.25i 0.524108 + 1.61304i
\(92\) 0 0
\(93\) 904.594 1.00862
\(94\) 0 0
\(95\) −473.210 + 537.497i −0.511056 + 0.580485i
\(96\) 0 0
\(97\) 100.433 + 72.9685i 0.105128 + 0.0763797i 0.639107 − 0.769118i \(-0.279304\pi\)
−0.533979 + 0.845497i \(0.679304\pi\)
\(98\) 0 0
\(99\) −526.407 −0.534403
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 100.4.g.a.21.2 ✓ 28
5.2 odd 4 500.4.i.b.149.3 56
5.3 odd 4 500.4.i.b.149.12 56
5.4 even 2 500.4.g.a.101.6 28
25.6 even 5 inner 100.4.g.a.81.2 yes 28
25.8 odd 20 500.4.i.b.349.3 56
25.9 even 10 2500.4.a.d.1.3 14
25.16 even 5 2500.4.a.c.1.12 14
25.17 odd 20 500.4.i.b.349.12 56
25.19 even 10 500.4.g.a.401.6 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
100.4.g.a.21.2 ✓ 28 1.1 even 1 trivial
100.4.g.a.81.2 yes 28 25.6 even 5 inner
500.4.g.a.101.6 28 5.4 even 2
500.4.g.a.401.6 28 25.19 even 10
500.4.i.b.149.3 56 5.2 odd 4
500.4.i.b.149.12 56 5.3 odd 4
500.4.i.b.349.3 56 25.8 odd 20
500.4.i.b.349.12 56 25.17 odd 20
2500.4.a.c.1.12 14 25.16 even 5
2500.4.a.d.1.3 14 25.9 even 10