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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,5,Mod(74,75)] 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("75.74"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.75274723129\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{14})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 74.1
Root \(1.87083 + 1.87083i\) of defining polynomial
Character \(\chi\) \(=\) 75.74
Dual form 75.5.d.b.74.2

$q$-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)
 
\(f(q)\) \(=\) \(q-3.74166 q^{2} +(7.48331 - 5.00000i) q^{3} -2.00000 q^{4} +(-28.0000 + 18.7083i) q^{6} +75.0000i q^{7} +67.3498 q^{8} +(31.0000 - 74.8331i) q^{9} -37.4166i q^{11} +(-14.9666 + 10.0000i) q^{12} +55.0000i q^{13} -280.624i q^{14} -220.000 q^{16} +501.382 q^{17} +(-115.991 + 280.000i) q^{18} +347.000 q^{19} +(375.000 + 561.249i) q^{21} +140.000i q^{22} +651.048 q^{23} +(504.000 - 336.749i) q^{24} -205.791i q^{26} +(-142.183 - 715.000i) q^{27} -150.000i q^{28} +860.581i q^{29} -3.00000 q^{31} -254.433 q^{32} +(-187.083 - 280.000i) q^{33} -1876.00 q^{34} +(-62.0000 + 149.666i) q^{36} +2230.00i q^{37} -1298.36 q^{38} +(275.000 + 411.582i) q^{39} -2207.58i q^{41} +(-1403.12 - 2100.00i) q^{42} -1475.00i q^{43} +74.8331i q^{44} -2436.00 q^{46} -1855.86 q^{47} +(-1646.33 + 1100.00i) q^{48} -3224.00 q^{49} +(3752.00 - 2506.91i) q^{51} -110.000i q^{52} -546.282 q^{53} +(532.000 + 2675.29i) q^{54} +5051.24i q^{56} +(2596.71 - 1735.00i) q^{57} -3220.00i q^{58} +2843.66i q^{59} +367.000 q^{61} +11.2250 q^{62} +(5612.49 + 2325.00i) q^{63} +4472.00 q^{64} +(700.000 + 1047.66i) q^{66} +2235.00i q^{67} -1002.76 q^{68} +(4872.00 - 3255.24i) q^{69} +486.415i q^{71} +(2087.84 - 5040.00i) q^{72} +6970.00i q^{73} -8343.90i q^{74} -694.000 q^{76} +2806.24 q^{77} +(-1028.96 - 1540.00i) q^{78} -4518.00 q^{79} +(-4639.00 - 4639.66i) q^{81} +8260.00i q^{82} +314.299 q^{83} +(-750.000 - 1122.50i) q^{84} +5518.94i q^{86} +(4302.91 + 6440.00i) q^{87} -2520.00i q^{88} -8081.98i q^{89} -4125.00 q^{91} -1302.10 q^{92} +(-22.4499 + 15.0000i) q^{93} +6944.00 q^{94} +(-1904.00 + 1272.16i) q^{96} -4535.00i q^{97} +12063.1 q^{98} +(-2800.00 - 1159.91i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} - 112 q^{6} + 124 q^{9} - 880 q^{16} + 1388 q^{19} + 1500 q^{21} + 2016 q^{24} - 12 q^{31} - 7504 q^{34} - 248 q^{36} + 1100 q^{39} - 9744 q^{46} - 12896 q^{49} + 15008 q^{51} + 2128 q^{54}+ \cdots - 11200 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.74166 −0.935414 −0.467707 − 0.883883i \(-0.654920\pi\)
−0.467707 + 0.883883i \(0.654920\pi\)
\(3\) 7.48331 − 5.00000i 0.831479 − 0.555556i
\(4\) −2.00000 −0.125000
\(5\) 0 0
\(6\) −28.0000 + 18.7083i −0.777778 + 0.519675i
\(7\) 75.0000i 1.53061i 0.643666 + 0.765306i \(0.277412\pi\)
−0.643666 + 0.765306i \(0.722588\pi\)
\(8\) 67.3498 1.05234
\(9\) 31.0000 − 74.8331i 0.382716 − 0.923866i
\(10\) 0 0
\(11\) − 37.4166i − 0.309228i −0.987975 − 0.154614i \(-0.950587\pi\)
0.987975 − 0.154614i \(-0.0494134\pi\)
\(12\) −14.9666 + 10.0000i −0.103935 + 0.0694444i
\(13\) 55.0000i 0.325444i 0.986672 + 0.162722i \(0.0520273\pi\)
−0.986672 + 0.162722i \(0.947973\pi\)
\(14\) − 280.624i − 1.43176i
\(15\) 0 0
\(16\) −220.000 −0.859375
\(17\) 501.382 1.73489 0.867443 − 0.497536i \(-0.165762\pi\)
0.867443 + 0.497536i \(0.165762\pi\)
\(18\) −115.991 + 280.000i −0.357998 + 0.864198i
\(19\) 347.000 0.961219 0.480609 − 0.876935i \(-0.340415\pi\)
0.480609 + 0.876935i \(0.340415\pi\)
\(20\) 0 0
\(21\) 375.000 + 561.249i 0.850340 + 1.27267i
\(22\) 140.000i 0.289256i
\(23\) 651.048 1.23072 0.615358 − 0.788248i \(-0.289012\pi\)
0.615358 + 0.788248i \(0.289012\pi\)
\(24\) 504.000 − 336.749i 0.875000 − 0.584634i
\(25\) 0 0
\(26\) − 205.791i − 0.304425i
\(27\) −142.183 − 715.000i −0.195038 − 0.980796i
\(28\) − 150.000i − 0.191327i
\(29\) 860.581i 1.02328i 0.859199 + 0.511642i \(0.170962\pi\)
−0.859199 + 0.511642i \(0.829038\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.00312175 −0.00156087 − 0.999999i \(-0.500497\pi\)
−0.00156087 + 0.999999i \(0.500497\pi\)
\(32\) −254.433 −0.248469
\(33\) −187.083 − 280.000i −0.171793 − 0.257117i
\(34\) −1876.00 −1.62284
\(35\) 0 0
\(36\) −62.0000 + 149.666i −0.0478395 + 0.115483i
\(37\) 2230.00i 1.62893i 0.580215 + 0.814463i \(0.302968\pi\)
−0.580215 + 0.814463i \(0.697032\pi\)
\(38\) −1298.36 −0.899138
\(39\) 275.000 + 411.582i 0.180802 + 0.270600i
\(40\) 0 0
\(41\) − 2207.58i − 1.31325i −0.754216 − 0.656626i \(-0.771983\pi\)
0.754216 − 0.656626i \(-0.228017\pi\)
\(42\) −1403.12 − 2100.00i −0.795420 − 1.19048i
\(43\) − 1475.00i − 0.797729i −0.917010 − 0.398864i \(-0.869404\pi\)
0.917010 − 0.398864i \(-0.130596\pi\)
\(44\) 74.8331i 0.0386535i
\(45\) 0 0
\(46\) −2436.00 −1.15123
\(47\) −1855.86 −0.840137 −0.420068 − 0.907492i \(-0.637994\pi\)
−0.420068 + 0.907492i \(0.637994\pi\)
\(48\) −1646.33 + 1100.00i −0.714553 + 0.477431i
\(49\) −3224.00 −1.34277
\(50\) 0 0
\(51\) 3752.00 − 2506.91i 1.44252 − 0.963826i
\(52\) − 110.000i − 0.0406805i
\(53\) −546.282 −0.194476 −0.0972378 − 0.995261i \(-0.531001\pi\)
−0.0972378 + 0.995261i \(0.531001\pi\)
\(54\) 532.000 + 2675.29i 0.182442 + 0.917450i
\(55\) 0 0
\(56\) 5051.24i 1.61073i
\(57\) 2596.71 − 1735.00i 0.799234 − 0.534010i
\(58\) − 3220.00i − 0.957194i
\(59\) 2843.66i 0.816909i 0.912779 + 0.408454i \(0.133932\pi\)
−0.912779 + 0.408454i \(0.866068\pi\)
\(60\) 0 0
\(61\) 367.000 0.0986294 0.0493147 − 0.998783i \(-0.484296\pi\)
0.0493147 + 0.998783i \(0.484296\pi\)
\(62\) 11.2250 0.00292013
\(63\) 5612.49 + 2325.00i 1.41408 + 0.585790i
\(64\) 4472.00 1.09180
\(65\) 0 0
\(66\) 700.000 + 1047.66i 0.160698 + 0.240511i
\(67\) 2235.00i 0.497884i 0.968518 + 0.248942i \(0.0800828\pi\)
−0.968518 + 0.248942i \(0.919917\pi\)
\(68\) −1002.76 −0.216861
\(69\) 4872.00 − 3255.24i 1.02331 − 0.683731i
\(70\) 0 0
\(71\) 486.415i 0.0964919i 0.998835 + 0.0482459i \(0.0153631\pi\)
−0.998835 + 0.0482459i \(0.984637\pi\)
\(72\) 2087.84 − 5040.00i 0.402748 − 0.972222i
\(73\) 6970.00i 1.30794i 0.756521 + 0.653969i \(0.226897\pi\)
−0.756521 + 0.653969i \(0.773103\pi\)
\(74\) − 8343.90i − 1.52372i
\(75\) 0 0
\(76\) −694.000 −0.120152
\(77\) 2806.24 0.473308
\(78\) −1028.96 − 1540.00i −0.169125 − 0.253123i
\(79\) −4518.00 −0.723922 −0.361961 − 0.932193i \(-0.617893\pi\)
−0.361961 + 0.932193i \(0.617893\pi\)
\(80\) 0 0
\(81\) −4639.00 − 4639.66i −0.707057 − 0.707157i
\(82\) 8260.00i 1.22844i
\(83\) 314.299 0.0456233 0.0228117 − 0.999740i \(-0.492738\pi\)
0.0228117 + 0.999740i \(0.492738\pi\)
\(84\) −750.000 − 1122.50i −0.106293 − 0.159084i
\(85\) 0 0
\(86\) 5518.94i 0.746207i
\(87\) 4302.91 + 6440.00i 0.568491 + 0.850839i
\(88\) − 2520.00i − 0.325413i
\(89\) − 8081.98i − 1.02032i −0.860079 − 0.510162i \(-0.829586\pi\)
0.860079 − 0.510162i \(-0.170414\pi\)
\(90\) 0 0
\(91\) −4125.00 −0.498128
\(92\) −1302.10 −0.153839
\(93\) −22.4499 + 15.0000i −0.00259567 + 0.00173430i
\(94\) 6944.00 0.785876
\(95\) 0 0
\(96\) −1904.00 + 1272.16i −0.206597 + 0.138039i
\(97\) − 4535.00i − 0.481985i −0.970527 − 0.240993i \(-0.922527\pi\)
0.970527 − 0.240993i \(-0.0774730\pi\)
\(98\) 12063.1 1.25605
\(99\) −2800.00 − 1159.91i −0.285685 − 0.118346i
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 75.5.d.b.74.1 4
3.2 odd 2 inner 75.5.d.b.74.3 4
5.2 odd 4 75.5.c.c.26.1 ✓ 2
5.3 odd 4 75.5.c.g.26.2 yes 2
5.4 even 2 inner 75.5.d.b.74.4 4
15.2 even 4 75.5.c.c.26.2 yes 2
15.8 even 4 75.5.c.g.26.1 yes 2
15.14 odd 2 inner 75.5.d.b.74.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.5.c.c.26.1 ✓ 2 5.2 odd 4
75.5.c.c.26.2 yes 2 15.2 even 4
75.5.c.g.26.1 yes 2 15.8 even 4
75.5.c.g.26.2 yes 2 5.3 odd 4
75.5.d.b.74.1 4 1.1 even 1 trivial
75.5.d.b.74.2 4 15.14 odd 2 inner
75.5.d.b.74.3 4 3.2 odd 2 inner
75.5.d.b.74.4 4 5.4 even 2 inner