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 43.2
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.43
Dual form 75.5.f.d.7.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(5.44949 - 5.44949i) q^{2} +(-3.67423 - 3.67423i) q^{3} -43.3939i q^{4} -40.0454 q^{6} +(-19.2247 + 19.2247i) q^{7} +(-149.283 - 149.283i) q^{8} +27.0000i q^{9} +185.060 q^{11} +(-159.439 + 159.439i) q^{12} +(-48.5074 - 48.5074i) q^{13} +209.530i q^{14} -932.727 q^{16} +(147.551 - 147.551i) q^{17} +(147.136 + 147.136i) q^{18} -140.242i q^{19} +141.272 q^{21} +(1008.48 - 1008.48i) q^{22} +(-177.267 - 177.267i) q^{23} +1097.00i q^{24} -528.681 q^{26} +(99.2043 - 99.2043i) q^{27} +(834.236 + 834.236i) q^{28} +588.756i q^{29} +1413.57 q^{31} +(-2694.36 + 2694.36i) q^{32} +(-679.955 - 679.955i) q^{33} -1608.15i q^{34} +1171.63 q^{36} +(763.868 - 763.868i) q^{37} +(-764.246 - 764.246i) q^{38} +356.455i q^{39} +995.850 q^{41} +(769.863 - 769.863i) q^{42} +(-657.277 - 657.277i) q^{43} -8030.48i q^{44} -1932.03 q^{46} +(-102.436 + 102.436i) q^{47} +(3427.06 + 3427.06i) q^{48} +1661.82i q^{49} -1084.27 q^{51} +(-2104.92 + 2104.92i) q^{52} +(-365.928 - 365.928i) q^{53} -1081.23i q^{54} +5739.84 q^{56} +(-515.281 + 515.281i) q^{57} +(3208.42 + 3208.42i) q^{58} +5805.99i q^{59} +6282.87 q^{61} +(7703.26 - 7703.26i) q^{62} +(-519.068 - 519.068i) q^{63} +14442.2i q^{64} -7410.81 q^{66} +(-5494.12 + 5494.12i) q^{67} +(-6402.79 - 6402.79i) q^{68} +1302.64i q^{69} -7666.03 q^{71} +(4030.63 - 4030.63i) q^{72} +(6406.59 + 6406.59i) q^{73} -8325.39i q^{74} -6085.64 q^{76} +(-3557.74 + 3557.74i) q^{77} +(1942.50 + 1942.50i) q^{78} -5699.38i q^{79} -729.000 q^{81} +(5426.87 - 5426.87i) q^{82} +(999.189 + 999.189i) q^{83} -6130.36i q^{84} -7163.65 q^{86} +(2163.23 - 2163.23i) q^{87} +(-27626.3 - 27626.3i) q^{88} -3090.84i q^{89} +1865.08 q^{91} +(-7692.29 + 7692.29i) q^{92} +(-5193.80 - 5193.80i) q^{93} +1116.45i q^{94} +19799.4 q^{96} +(-5031.88 + 5031.88i) q^{97} +(9056.06 + 9056.06i) q^{98} +4996.63i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{2} - 72 q^{6} - 72 q^{7} - 264 q^{8} - 24 q^{11} - 432 q^{12} + 144 q^{13} - 2320 q^{16} + 600 q^{17} + 324 q^{18} + 36 q^{21} + 1800 q^{22} + 888 q^{23} - 792 q^{26} + 1152 q^{28} + 3244 q^{31}+ \cdots + 20136 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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 0.491488 − 0.870884i \(-0.336453\pi\)
0.870884 − 0.491488i \(-0.163547\pi\)
\(3\) −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 \(-0.839482\pi\)
0.483180 + 0.875521i \(0.339482\pi\)
\(8\) −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 \(-0.315056\pi\)
−0.835903 + 0.548877i \(0.815056\pi\)
\(14\) 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 \(-0.632430\pi\)
0.914697 + 0.404141i \(0.132430\pi\)
\(18\) 147.136 + 147.136i 0.454124 + 0.454124i
\(19\) − 140.242i − 0.388482i −0.980954 − 0.194241i \(-0.937776\pi\)
0.980954 − 0.194241i \(-0.0622243\pi\)
\(20\) 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 \(-0.326148\pi\)
−0.854519 + 0.519421i \(0.826148\pi\)
\(24\) 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 \(0.113830\pi\)
−0.936737 + 0.350033i \(0.886170\pi\)
\(30\) 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 \(-0.620901\pi\)
0.928731 + 0.370755i \(0.120901\pi\)
\(38\) −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 \(-0.330878\pi\)
−0.862143 + 0.506666i \(0.830878\pi\)
\(44\) − 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 \(-0.760439\pi\)
0.683540 + 0.729913i \(0.260439\pi\)
\(48\) 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 \(-0.279363\pi\)
−0.769235 + 0.638966i \(0.779363\pi\)
\(54\) − 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 \(0.313929\pi\)
−0.551833 + 0.833955i \(0.686071\pi\)
\(60\) 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 −0.257677 + 0.966231i \(0.582957\pi\)
−0.966231 + 0.257677i \(0.917043\pi\)
\(68\) −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 0.973492 + 0.228721i \(0.0734543\pi\)
0.228721 + 0.973492i \(0.426546\pi\)
\(74\) − 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 \(-0.849064\pi\)
0.889668 − 0.456608i \(-0.150936\pi\)
\(80\) 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 \(-0.217297\pi\)
−0.630857 + 0.775899i \(0.717297\pi\)
\(84\) − 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 \(-0.937496\pi\)
0.980783 − 0.195104i \(-0.0625045\pi\)
\(90\) 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 \(-0.873442\pi\)
0.387201 + 0.921995i \(0.373442\pi\)
\(98\) 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.d.43.2 yes 4
3.2 odd 2 225.5.g.d.118.1 4
5.2 odd 4 inner 75.5.f.d.7.2 yes 4
5.3 odd 4 75.5.f.a.7.1 ✓ 4
5.4 even 2 75.5.f.a.43.1 yes 4
15.2 even 4 225.5.g.d.82.1 4
15.8 even 4 225.5.g.l.82.2 4
15.14 odd 2 225.5.g.l.118.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.5.f.a.7.1 ✓ 4 5.3 odd 4
75.5.f.a.43.1 yes 4 5.4 even 2
75.5.f.d.7.2 yes 4 5.2 odd 4 inner
75.5.f.d.43.2 yes 4 1.1 even 1 trivial
225.5.g.d.82.1 4 15.2 even 4
225.5.g.d.118.1 4 3.2 odd 2
225.5.g.l.82.2 4 15.8 even 4
225.5.g.l.118.2 4 15.14 odd 2