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 7.1
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.7
Dual form 75.5.f.d.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)
 
\(f(q)\) \(=\) \(q+(0.550510 + 0.550510i) q^{2} +(3.67423 - 3.67423i) q^{3} -15.3939i q^{4} +4.04541 q^{6} +(-16.7753 - 16.7753i) q^{7} +(17.2827 - 17.2827i) q^{8} -27.0000i q^{9} -197.060 q^{11} +(-56.5607 - 56.5607i) q^{12} +(120.507 - 120.507i) q^{13} -18.4699i q^{14} -227.273 q^{16} +(152.449 + 152.449i) q^{17} +(14.8638 - 14.8638i) q^{18} -418.242i q^{19} -123.272 q^{21} +(-108.484 - 108.484i) q^{22} +(621.267 - 621.267i) q^{23} -127.001i q^{24} +132.681 q^{26} +(-99.2043 - 99.2043i) q^{27} +(-258.236 + 258.236i) q^{28} +792.756i q^{29} +208.426 q^{31} +(-401.639 - 401.639i) q^{32} +(-724.045 + 724.045i) q^{33} +167.850i q^{34} -415.635 q^{36} +(460.132 + 460.132i) q^{37} +(230.246 - 230.246i) q^{38} -885.545i q^{39} +2436.15 q^{41} +(-67.8627 - 67.8627i) q^{42} +(-2114.72 + 2114.72i) q^{43} +3033.52i q^{44} +684.028 q^{46} +(2910.44 + 2910.44i) q^{47} +(-835.056 + 835.056i) q^{48} -1838.18i q^{49} +1120.27 q^{51} +(-1855.08 - 1855.08i) q^{52} +(1289.93 - 1289.93i) q^{53} -109.226i q^{54} -579.842 q^{56} +(-1536.72 - 1536.72i) q^{57} +(-436.420 + 436.420i) q^{58} +1953.99i q^{59} +1227.13 q^{61} +(114.740 + 114.740i) q^{62} +(-452.932 + 452.932i) q^{63} +3194.16i q^{64} -797.189 q^{66} +(-1165.88 - 1165.88i) q^{67} +(2346.79 - 2346.79i) q^{68} -4565.36i q^{69} -3109.97 q^{71} +(-466.632 - 466.632i) q^{72} +(3457.41 - 3457.41i) q^{73} +506.614i q^{74} -6438.36 q^{76} +(3305.74 + 3305.74i) q^{77} +(487.502 - 487.502i) q^{78} -9879.38i q^{79} -729.000 q^{81} +(1341.13 + 1341.13i) q^{82} +(3296.81 - 3296.81i) q^{83} +1897.64i q^{84} -2328.35 q^{86} +(2912.77 + 2912.77i) q^{87} +(-3405.72 + 3405.72i) q^{88} -6138.84i q^{89} -4043.08 q^{91} +(-9563.71 - 9563.71i) q^{92} +(765.804 - 765.804i) q^{93} +3204.45i q^{94} -2951.43 q^{96} +(-728.123 - 728.123i) q^{97} +(1011.94 - 1011.94i) q^{98} +5320.63i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{2} - 72 q^{6} - 72 q^{7} - 264 q^{8} - 24 q^{11} - 432 q^{12} + 144 q^{13} - 2320 q^{16} + 600 q^{17} + 324 q^{18} + 36 q^{21} + 1800 q^{22} + 888 q^{23} - 792 q^{26} + 1152 q^{28} + 3244 q^{31}+ \cdots + 20136 q^{98}+O(q^{100}) \) 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 0.772564 − 0.634937i \(-0.218974\pi\)
−0.634937 + 0.772564i \(0.718974\pi\)
\(3\) 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 0.514899 − 0.857251i \(-0.327829\pi\)
−0.857251 + 0.514899i \(0.827829\pi\)
\(8\) 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 −0.254113 − 0.967175i \(-0.581784\pi\)
0.967175 + 0.254113i \(0.0817835\pi\)
\(14\) − 18.4699i − 0.0942342i
\(15\) 0 0
\(16\) −227.273 −0.887787
\(17\) 152.449 + 152.449i 0.527507 + 0.527507i 0.919828 − 0.392321i \(-0.128328\pi\)
−0.392321 + 0.919828i \(0.628328\pi\)
\(18\) 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 \(-0.438090\pi\)
0.981145 − 0.193272i \(-0.0619101\pi\)
\(24\) − 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 0.854900 − 0.518792i \(-0.173618\pi\)
−0.518792 + 0.854900i \(0.673618\pi\)
\(38\) 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 \(0.549843\pi\)
−0.987766 + 0.155946i \(0.950157\pi\)
\(44\) 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 \(0.131620\pi\)
0.401815 + 0.915721i \(0.368380\pi\)
\(48\) −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 −0.439185 − 0.898397i \(-0.644733\pi\)
0.898397 + 0.439185i \(0.144733\pi\)
\(54\) − 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 0.565221 − 0.824939i \(-0.308791\pi\)
−0.824939 + 0.565221i \(0.808791\pi\)
\(68\) 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 −0.303910 − 0.952701i \(-0.598292\pi\)
0.952701 + 0.303910i \(0.0982922\pi\)
\(74\) 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 −0.426110 − 0.904671i \(-0.640116\pi\)
0.904671 + 0.426110i \(0.140116\pi\)
\(84\) 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 0.667354 − 0.744740i \(-0.267427\pi\)
−0.744740 + 0.667354i \(0.767427\pi\)
\(98\) 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.d.7.1 yes 4
3.2 odd 2 225.5.g.d.82.2 4
5.2 odd 4 75.5.f.a.43.2 yes 4
5.3 odd 4 inner 75.5.f.d.43.1 yes 4
5.4 even 2 75.5.f.a.7.2 ✓ 4
15.2 even 4 225.5.g.l.118.1 4
15.8 even 4 225.5.g.d.118.2 4
15.14 odd 2 225.5.g.l.82.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.5.f.a.7.2 ✓ 4 5.4 even 2
75.5.f.a.43.2 yes 4 5.2 odd 4
75.5.f.d.7.1 yes 4 1.1 even 1 trivial
75.5.f.d.43.1 yes 4 5.3 odd 4 inner
225.5.g.d.82.2 4 3.2 odd 2
225.5.g.d.118.2 4 15.8 even 4
225.5.g.l.82.1 4 15.14 odd 2
225.5.g.l.118.1 4 15.2 even 4