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.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)
 
\(f(q)\) \(=\) \(q+(-5.44949 - 5.44949i) q^{2} +(3.67423 - 3.67423i) q^{3} +43.3939i q^{4} -40.0454 q^{6} +(19.2247 + 19.2247i) q^{7} +(149.283 - 149.283i) q^{8} -27.0000i q^{9} +185.060 q^{11} +(159.439 + 159.439i) q^{12} +(48.5074 - 48.5074i) q^{13} -209.530i q^{14} -932.727 q^{16} +(-147.551 - 147.551i) q^{17} +(-147.136 + 147.136i) q^{18} +140.242i q^{19} +141.272 q^{21} +(-1008.48 - 1008.48i) q^{22} +(177.267 - 177.267i) q^{23} -1097.00i q^{24} -528.681 q^{26} +(-99.2043 - 99.2043i) q^{27} +(-834.236 + 834.236i) q^{28} -588.756i q^{29} +1413.57 q^{31} +(2694.36 + 2694.36i) q^{32} +(679.955 - 679.955i) q^{33} +1608.15i q^{34} +1171.63 q^{36} +(-763.868 - 763.868i) q^{37} +(764.246 - 764.246i) q^{38} -356.455i q^{39} +995.850 q^{41} +(-769.863 - 769.863i) q^{42} +(657.277 - 657.277i) q^{43} +8030.48i q^{44} -1932.03 q^{46} +(102.436 + 102.436i) q^{47} +(-3427.06 + 3427.06i) q^{48} -1661.82i q^{49} -1084.27 q^{51} +(2104.92 + 2104.92i) q^{52} +(365.928 - 365.928i) q^{53} +1081.23i q^{54} +5739.84 q^{56} +(515.281 + 515.281i) q^{57} +(-3208.42 + 3208.42i) q^{58} -5805.99i q^{59} +6282.87 q^{61} +(-7703.26 - 7703.26i) q^{62} +(519.068 - 519.068i) q^{63} -14442.2i q^{64} -7410.81 q^{66} +(5494.12 + 5494.12i) q^{67} +(6402.79 - 6402.79i) q^{68} -1302.64i q^{69} -7666.03 q^{71} +(-4030.63 - 4030.63i) q^{72} +(-6406.59 + 6406.59i) q^{73} +8325.39i q^{74} -6085.64 q^{76} +(3557.74 + 3557.74i) q^{77} +(-1942.50 + 1942.50i) q^{78} +5699.38i q^{79} -729.000 q^{81} +(-5426.87 - 5426.87i) q^{82} +(-999.189 + 999.189i) q^{83} +6130.36i q^{84} -7163.65 q^{86} +(-2163.23 - 2163.23i) q^{87} +(27626.3 - 27626.3i) q^{88} +3090.84i q^{89} +1865.08 q^{91} +(7692.29 + 7692.29i) q^{92} +(5193.80 - 5193.80i) q^{93} -1116.45i q^{94} +19799.4 q^{96} +(5031.88 + 5031.88i) q^{97} +(-9056.06 + 9056.06i) q^{98} -4996.63i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{2} - 72 q^{6} + 72 q^{7} + 264 q^{8} - 24 q^{11} + 432 q^{12} - 144 q^{13} - 2320 q^{16} - 600 q^{17} - 324 q^{18} + 36 q^{21} - 1800 q^{22} - 888 q^{23} - 792 q^{26} - 1152 q^{28} + 3244 q^{31}+ \cdots - 20136 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.44949 − 5.44949i −1.36237 − 1.36237i −0.870884 − 0.491488i \(-0.836453\pi\)
−0.491488 − 0.870884i \(-0.663547\pi\)
\(3\) 3.67423 − 3.67423i 0.408248 − 0.408248i
\(4\) 43.3939i 2.71212i
\(5\) 0 0
\(6\) −40.0454 −1.11237
\(7\) 19.2247 + 19.2247i 0.392342 + 0.392342i 0.875521 − 0.483180i \(-0.160518\pi\)
−0.483180 + 0.875521i \(0.660518\pi\)
\(8\) 149.283 − 149.283i 2.33254 − 2.33254i
\(9\) − 27.0000i − 0.333333i
\(10\) 0 0
\(11\) 185.060 1.52942 0.764712 − 0.644373i \(-0.222881\pi\)
0.764712 + 0.644373i \(0.222881\pi\)
\(12\) 159.439 + 159.439i 1.10722 + 1.10722i
\(13\) 48.5074 − 48.5074i 0.287026 − 0.287026i −0.548877 − 0.835903i \(-0.684944\pi\)
0.835903 + 0.548877i \(0.184944\pi\)
\(14\) − 209.530i − 1.06903i
\(15\) 0 0
\(16\) −932.727 −3.64346
\(17\) −147.551 − 147.551i −0.510555 − 0.510555i 0.404141 − 0.914697i \(-0.367570\pi\)
−0.914697 + 0.404141i \(0.867570\pi\)
\(18\) −147.136 + 147.136i −0.454124 + 0.454124i
\(19\) 140.242i 0.388482i 0.980954 + 0.194241i \(0.0622243\pi\)
−0.980954 + 0.194241i \(0.937776\pi\)
\(20\) 0 0
\(21\) 141.272 0.320346
\(22\) −1008.48 − 1008.48i −2.08364 − 2.08364i
\(23\) 177.267 − 177.267i 0.335098 − 0.335098i −0.519421 − 0.854519i \(-0.673852\pi\)
0.854519 + 0.519421i \(0.173852\pi\)
\(24\) − 1097.00i − 1.90451i
\(25\) 0 0
\(26\) −528.681 −0.782073
\(27\) −99.2043 − 99.2043i −0.136083 − 0.136083i
\(28\) −834.236 + 834.236i −1.06408 + 1.06408i
\(29\) − 588.756i − 0.700067i −0.936737 − 0.350033i \(-0.886170\pi\)
0.936737 − 0.350033i \(-0.113830\pi\)
\(30\) 0 0
\(31\) 1413.57 1.47094 0.735471 − 0.677557i \(-0.236961\pi\)
0.735471 + 0.677557i \(0.236961\pi\)
\(32\) 2694.36 + 2694.36i 2.63121 + 2.63121i
\(33\) 679.955 − 679.955i 0.624384 − 0.624384i
\(34\) 1608.15i 1.39113i
\(35\) 0 0
\(36\) 1171.63 0.904039
\(37\) −763.868 − 763.868i −0.557975 − 0.557975i 0.370755 − 0.928731i \(-0.379099\pi\)
−0.928731 + 0.370755i \(0.879099\pi\)
\(38\) 764.246 − 764.246i 0.529257 − 0.529257i
\(39\) − 356.455i − 0.234356i
\(40\) 0 0
\(41\) 995.850 0.592415 0.296208 − 0.955124i \(-0.404278\pi\)
0.296208 + 0.955124i \(0.404278\pi\)
\(42\) −769.863 − 769.863i −0.436430 − 0.436430i
\(43\) 657.277 − 657.277i 0.355477 − 0.355477i −0.506666 − 0.862143i \(-0.669122\pi\)
0.862143 + 0.506666i \(0.169122\pi\)
\(44\) 8030.48i 4.14797i
\(45\) 0 0
\(46\) −1932.03 −0.913056
\(47\) 102.436 + 102.436i 0.0463722 + 0.0463722i 0.729913 − 0.683540i \(-0.239561\pi\)
−0.683540 + 0.729913i \(0.739561\pi\)
\(48\) −3427.06 + 3427.06i −1.48744 + 1.48744i
\(49\) − 1661.82i − 0.692136i
\(50\) 0 0
\(51\) −1084.27 −0.416867
\(52\) 2104.92 + 2104.92i 0.778448 + 0.778448i
\(53\) 365.928 − 365.928i 0.130270 − 0.130270i −0.638966 − 0.769235i \(-0.720637\pi\)
0.769235 + 0.638966i \(0.220637\pi\)
\(54\) 1081.23i 0.370791i
\(55\) 0 0
\(56\) 5739.84 1.83031
\(57\) 515.281 + 515.281i 0.158597 + 0.158597i
\(58\) −3208.42 + 3208.42i −0.953752 + 0.953752i
\(59\) − 5805.99i − 1.66791i −0.551833 − 0.833955i \(-0.686071\pi\)
0.551833 − 0.833955i \(-0.313929\pi\)
\(60\) 0 0
\(61\) 6282.87 1.68849 0.844245 − 0.535957i \(-0.180049\pi\)
0.844245 + 0.535957i \(0.180049\pi\)
\(62\) −7703.26 − 7703.26i −2.00397 − 2.00397i
\(63\) 519.068 − 519.068i 0.130781 − 0.130781i
\(64\) − 14442.2i − 3.52592i
\(65\) 0 0
\(66\) −7410.81 −1.70129
\(67\) 5494.12 + 5494.12i 1.22391 + 1.22391i 0.966231 + 0.257677i \(0.0829571\pi\)
0.257677 + 0.966231i \(0.417043\pi\)
\(68\) 6402.79 − 6402.79i 1.38469 − 1.38469i
\(69\) − 1302.64i − 0.273606i
\(70\) 0 0
\(71\) −7666.03 −1.52074 −0.760368 − 0.649493i \(-0.774981\pi\)
−0.760368 + 0.649493i \(0.774981\pi\)
\(72\) −4030.63 − 4030.63i −0.777514 − 0.777514i
\(73\) −6406.59 + 6406.59i −1.20221 + 1.20221i −0.228721 + 0.973492i \(0.573454\pi\)
−0.973492 + 0.228721i \(0.926546\pi\)
\(74\) 8325.39i 1.52034i
\(75\) 0 0
\(76\) −6085.64 −1.05361
\(77\) 3557.74 + 3557.74i 0.600057 + 0.600057i
\(78\) −1942.50 + 1942.50i −0.319280 + 0.319280i
\(79\) 5699.38i 0.913215i 0.889668 + 0.456608i \(0.150936\pi\)
−0.889668 + 0.456608i \(0.849064\pi\)
\(80\) 0 0
\(81\) −729.000 −0.111111
\(82\) −5426.87 − 5426.87i −0.807090 − 0.807090i
\(83\) −999.189 + 999.189i −0.145041 + 0.145041i −0.775899 − 0.630857i \(-0.782703\pi\)
0.630857 + 0.775899i \(0.282703\pi\)
\(84\) 6130.36i 0.868815i
\(85\) 0 0
\(86\) −7163.65 −0.968584
\(87\) −2163.23 − 2163.23i −0.285801 − 0.285801i
\(88\) 27626.3 − 27626.3i 3.56744 − 3.56744i
\(89\) 3090.84i 0.390208i 0.980783 + 0.195104i \(0.0625045\pi\)
−0.980783 + 0.195104i \(0.937496\pi\)
\(90\) 0 0
\(91\) 1865.08 0.225225
\(92\) 7692.29 + 7692.29i 0.908825 + 0.908825i
\(93\) 5193.80 − 5193.80i 0.600509 − 0.600509i
\(94\) − 1116.45i − 0.126352i
\(95\) 0 0
\(96\) 19799.4 2.14838
\(97\) 5031.88 + 5031.88i 0.534794 + 0.534794i 0.921995 − 0.387201i \(-0.126558\pi\)
−0.387201 + 0.921995i \(0.626558\pi\)
\(98\) −9056.06 + 9056.06i −0.942947 + 0.942947i
\(99\) − 4996.63i − 0.509808i
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.7.1 ✓ 4
3.2 odd 2 225.5.g.l.82.2 4
5.2 odd 4 75.5.f.d.43.2 yes 4
5.3 odd 4 inner 75.5.f.a.43.1 yes 4
5.4 even 2 75.5.f.d.7.2 yes 4
15.2 even 4 225.5.g.d.118.1 4
15.8 even 4 225.5.g.l.118.2 4
15.14 odd 2 225.5.g.d.82.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.5.f.a.7.1 ✓ 4 1.1 even 1 trivial
75.5.f.a.43.1 yes 4 5.3 odd 4 inner
75.5.f.d.7.2 yes 4 5.4 even 2
75.5.f.d.43.2 yes 4 5.2 odd 4
225.5.g.d.82.1 4 15.14 odd 2
225.5.g.d.118.1 4 15.2 even 4
225.5.g.l.82.2 4 3.2 odd 2
225.5.g.l.118.2 4 15.8 even 4