Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,5,Mod(7,75)] 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("75.7"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-12,0,0,0,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.75274723129\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 7.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.7
Dual form 75.5.f.a.43.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.550510 - 0.550510i) q^{2} +(-3.67423 + 3.67423i) q^{3} -15.3939i q^{4} +4.04541 q^{6} +(16.7753 + 16.7753i) q^{7} +(-17.2827 + 17.2827i) q^{8} -27.0000i q^{9} -197.060 q^{11} +(56.5607 + 56.5607i) q^{12} +(-120.507 + 120.507i) q^{13} -18.4699i q^{14} -227.273 q^{16} +(-152.449 - 152.449i) q^{17} +(-14.8638 + 14.8638i) q^{18} -418.242i q^{19} -123.272 q^{21} +(108.484 + 108.484i) q^{22} +(-621.267 + 621.267i) q^{23} -127.001i q^{24} +132.681 q^{26} +(99.2043 + 99.2043i) q^{27} +(258.236 - 258.236i) q^{28} +792.756i q^{29} +208.426 q^{31} +(401.639 + 401.639i) q^{32} +(724.045 - 724.045i) q^{33} +167.850i q^{34} -415.635 q^{36} +(-460.132 - 460.132i) q^{37} +(-230.246 + 230.246i) q^{38} -885.545i q^{39} +2436.15 q^{41} +(67.8627 + 67.8627i) q^{42} +(2114.72 - 2114.72i) q^{43} +3033.52i q^{44} +684.028 q^{46} +(-2910.44 - 2910.44i) q^{47} +(835.056 - 835.056i) q^{48} -1838.18i q^{49} +1120.27 q^{51} +(1855.08 + 1855.08i) q^{52} +(-1289.93 + 1289.93i) q^{53} -109.226i q^{54} -579.842 q^{56} +(1536.72 + 1536.72i) q^{57} +(436.420 - 436.420i) q^{58} +1953.99i q^{59} +1227.13 q^{61} +(-114.740 - 114.740i) q^{62} +(452.932 - 452.932i) q^{63} +3194.16i q^{64} -797.189 q^{66} +(1165.88 + 1165.88i) q^{67} +(-2346.79 + 2346.79i) q^{68} -4565.36i q^{69} -3109.97 q^{71} +(466.632 + 466.632i) q^{72} +(-3457.41 + 3457.41i) q^{73} +506.614i q^{74} -6438.36 q^{76} +(-3305.74 - 3305.74i) q^{77} +(-487.502 + 487.502i) q^{78} -9879.38i q^{79} -729.000 q^{81} +(-1341.13 - 1341.13i) q^{82} +(-3296.81 + 3296.81i) q^{83} +1897.64i q^{84} -2328.35 q^{86} +(-2912.77 - 2912.77i) q^{87} +(3405.72 - 3405.72i) q^{88} -6138.84i q^{89} -4043.08 q^{91} +(9563.71 + 9563.71i) q^{92} +(-765.804 + 765.804i) q^{93} +3204.45i q^{94} -2951.43 q^{96} +(728.123 + 728.123i) q^{97} +(-1011.94 + 1011.94i) q^{98} +5320.63i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{2} - 72 q^{6} + 72 q^{7} + 264 q^{8} - 24 q^{11} + 432 q^{12} - 144 q^{13} - 2320 q^{16} - 600 q^{17} - 324 q^{18} + 36 q^{21} - 1800 q^{22} - 888 q^{23} - 792 q^{26} - 1152 q^{28} + 3244 q^{31}+ \cdots - 20136 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.550510 − 0.550510i −0.137628 − 0.137628i 0.634937 − 0.772564i \(-0.281026\pi\)
−0.772564 + 0.634937i \(0.781026\pi\)
\(3\) −3.67423 + 3.67423i −0.408248 + 0.408248i
\(4\) − 15.3939i − 0.962117i
\(5\) 0 0
\(6\) 4.04541 0.112372
\(7\) 16.7753 + 16.7753i 0.342352 + 0.342352i 0.857251 − 0.514899i \(-0.172171\pi\)
−0.514899 + 0.857251i \(0.672171\pi\)
\(8\) −17.2827 + 17.2827i −0.270041 + 0.270041i
\(9\) − 27.0000i − 0.333333i
\(10\) 0 0
\(11\) −197.060 −1.62860 −0.814298 − 0.580447i \(-0.802878\pi\)
−0.814298 + 0.580447i \(0.802878\pi\)
\(12\) 56.5607 + 56.5607i 0.392783 + 0.392783i
\(13\) −120.507 + 120.507i −0.713062 + 0.713062i −0.967175 − 0.254113i \(-0.918216\pi\)
0.254113 + 0.967175i \(0.418216\pi\)
\(14\) − 18.4699i − 0.0942342i
\(15\) 0 0
\(16\) −227.273 −0.887787
\(17\) −152.449 − 152.449i −0.527507 − 0.527507i 0.392321 − 0.919828i \(-0.371672\pi\)
−0.919828 + 0.392321i \(0.871672\pi\)
\(18\) −14.8638 + 14.8638i −0.0458759 + 0.0458759i
\(19\) − 418.242i − 1.15856i −0.815127 − 0.579282i \(-0.803333\pi\)
0.815127 − 0.579282i \(-0.196667\pi\)
\(20\) 0 0
\(21\) −123.272 −0.279529
\(22\) 108.484 + 108.484i 0.224140 + 0.224140i
\(23\) −621.267 + 621.267i −1.17442 + 1.17442i −0.193272 + 0.981145i \(0.561910\pi\)
−0.981145 + 0.193272i \(0.938090\pi\)
\(24\) − 127.001i − 0.220488i
\(25\) 0 0
\(26\) 132.681 0.196274
\(27\) 99.2043 + 99.2043i 0.136083 + 0.136083i
\(28\) 258.236 − 258.236i 0.329383 − 0.329383i
\(29\) 792.756i 0.942635i 0.881964 + 0.471318i \(0.156221\pi\)
−0.881964 + 0.471318i \(0.843779\pi\)
\(30\) 0 0
\(31\) 208.426 0.216884 0.108442 − 0.994103i \(-0.465414\pi\)
0.108442 + 0.994103i \(0.465414\pi\)
\(32\) 401.639 + 401.639i 0.392225 + 0.392225i
\(33\) 724.045 − 724.045i 0.664872 − 0.664872i
\(34\) 167.850i 0.145199i
\(35\) 0 0
\(36\) −415.635 −0.320706
\(37\) −460.132 − 460.132i −0.336108 − 0.336108i 0.518792 − 0.854900i \(-0.326382\pi\)
−0.854900 + 0.518792i \(0.826382\pi\)
\(38\) −230.246 + 230.246i −0.159450 + 0.159450i
\(39\) − 885.545i − 0.582212i
\(40\) 0 0
\(41\) 2436.15 1.44923 0.724613 − 0.689156i \(-0.242018\pi\)
0.724613 + 0.689156i \(0.242018\pi\)
\(42\) 67.8627 + 67.8627i 0.0384709 + 0.0384709i
\(43\) 2114.72 − 2114.72i 1.14371 − 1.14371i 0.155946 − 0.987766i \(-0.450157\pi\)
0.987766 − 0.155946i \(-0.0498427\pi\)
\(44\) 3033.52i 1.56690i
\(45\) 0 0
\(46\) 684.028 0.323264
\(47\) −2910.44 − 2910.44i −1.31754 − 1.31754i −0.915721 − 0.401815i \(-0.868380\pi\)
−0.401815 − 0.915721i \(-0.631620\pi\)
\(48\) 835.056 − 835.056i 0.362438 − 0.362438i
\(49\) − 1838.18i − 0.765590i
\(50\) 0 0
\(51\) 1120.27 0.430708
\(52\) 1855.08 + 1855.08i 0.686049 + 0.686049i
\(53\) −1289.93 + 1289.93i −0.459212 + 0.459212i −0.898397 − 0.439185i \(-0.855267\pi\)
0.439185 + 0.898397i \(0.355267\pi\)
\(54\) − 109.226i − 0.0374575i
\(55\) 0 0
\(56\) −579.842 −0.184899
\(57\) 1536.72 + 1536.72i 0.472982 + 0.472982i
\(58\) 436.420 − 436.420i 0.129733 − 0.129733i
\(59\) 1953.99i 0.561331i 0.959806 + 0.280665i \(0.0905551\pi\)
−0.959806 + 0.280665i \(0.909445\pi\)
\(60\) 0 0
\(61\) 1227.13 0.329784 0.164892 − 0.986312i \(-0.447272\pi\)
0.164892 + 0.986312i \(0.447272\pi\)
\(62\) −114.740 − 114.740i −0.0298492 − 0.0298492i
\(63\) 452.932 − 452.932i 0.114117 − 0.114117i
\(64\) 3194.16i 0.779825i
\(65\) 0 0
\(66\) −797.189 −0.183009
\(67\) 1165.88 + 1165.88i 0.259718 + 0.259718i 0.824939 − 0.565221i \(-0.191209\pi\)
−0.565221 + 0.824939i \(0.691209\pi\)
\(68\) −2346.79 + 2346.79i −0.507524 + 0.507524i
\(69\) − 4565.36i − 0.958908i
\(70\) 0 0
\(71\) −3109.97 −0.616936 −0.308468 − 0.951235i \(-0.599816\pi\)
−0.308468 + 0.951235i \(0.599816\pi\)
\(72\) 466.632 + 466.632i 0.0900138 + 0.0900138i
\(73\) −3457.41 + 3457.41i −0.648791 + 0.648791i −0.952701 − 0.303910i \(-0.901708\pi\)
0.303910 + 0.952701i \(0.401708\pi\)
\(74\) 506.614i 0.0925154i
\(75\) 0 0
\(76\) −6438.36 −1.11468
\(77\) −3305.74 − 3305.74i −0.557554 − 0.557554i
\(78\) −487.502 + 487.502i −0.0801285 + 0.0801285i
\(79\) − 9879.38i − 1.58298i −0.611182 − 0.791490i \(-0.709306\pi\)
0.611182 − 0.791490i \(-0.290694\pi\)
\(80\) 0 0
\(81\) −729.000 −0.111111
\(82\) −1341.13 − 1341.13i −0.199454 − 0.199454i
\(83\) −3296.81 + 3296.81i −0.478562 + 0.478562i −0.904671 − 0.426110i \(-0.859884\pi\)
0.426110 + 0.904671i \(0.359884\pi\)
\(84\) 1897.64i 0.268940i
\(85\) 0 0
\(86\) −2328.35 −0.314813
\(87\) −2912.77 − 2912.77i −0.384829 − 0.384829i
\(88\) 3405.72 − 3405.72i 0.439789 − 0.439789i
\(89\) − 6138.84i − 0.775008i −0.921868 − 0.387504i \(-0.873337\pi\)
0.921868 − 0.387504i \(-0.126663\pi\)
\(90\) 0 0
\(91\) −4043.08 −0.488236
\(92\) 9563.71 + 9563.71i 1.12993 + 1.12993i
\(93\) −765.804 + 765.804i −0.0885425 + 0.0885425i
\(94\) 3204.45i 0.362658i
\(95\) 0 0
\(96\) −2951.43 −0.320251
\(97\) 728.123 + 728.123i 0.0773858 + 0.0773858i 0.744740 − 0.667354i \(-0.232573\pi\)
−0.667354 + 0.744740i \(0.732573\pi\)
\(98\) −1011.94 + 1011.94i −0.105366 + 0.105366i
\(99\) 5320.63i 0.542866i
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.f.a.7.2 ✓ 4
3.2 odd 2 225.5.g.l.82.1 4
5.2 odd 4 75.5.f.d.43.1 yes 4
5.3 odd 4 inner 75.5.f.a.43.2 yes 4
5.4 even 2 75.5.f.d.7.1 yes 4
15.2 even 4 225.5.g.d.118.2 4
15.8 even 4 225.5.g.l.118.1 4
15.14 odd 2 225.5.g.d.82.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.5.f.a.7.2 ✓ 4 1.1 even 1 trivial
75.5.f.a.43.2 yes 4 5.3 odd 4 inner
75.5.f.d.7.1 yes 4 5.4 even 2
75.5.f.d.43.1 yes 4 5.2 odd 4
225.5.g.d.82.2 4 15.14 odd 2
225.5.g.d.118.2 4 15.2 even 4
225.5.g.l.82.1 4 3.2 odd 2
225.5.g.l.118.1 4 15.8 even 4