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.d.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)
 
\(f(q)\) \(=\) \(q+(5.44949 + 5.44949i) q^{2} +(-3.67423 + 3.67423i) q^{3} +43.3939i q^{4} -40.0454 q^{6} +(-19.2247 - 19.2247i) q^{7} +(-149.283 + 149.283i) q^{8} -27.0000i q^{9} +185.060 q^{11} +(-159.439 - 159.439i) q^{12} +(-48.5074 + 48.5074i) q^{13} -209.530i q^{14} -932.727 q^{16} +(147.551 + 147.551i) q^{17} +(147.136 - 147.136i) q^{18} +140.242i q^{19} +141.272 q^{21} +(1008.48 + 1008.48i) q^{22} +(-177.267 + 177.267i) q^{23} -1097.00i q^{24} -528.681 q^{26} +(99.2043 + 99.2043i) q^{27} +(834.236 - 834.236i) q^{28} -588.756i q^{29} +1413.57 q^{31} +(-2694.36 - 2694.36i) q^{32} +(-679.955 + 679.955i) q^{33} +1608.15i q^{34} +1171.63 q^{36} +(763.868 + 763.868i) q^{37} +(-764.246 + 764.246i) q^{38} -356.455i q^{39} +995.850 q^{41} +(769.863 + 769.863i) q^{42} +(-657.277 + 657.277i) q^{43} +8030.48i q^{44} -1932.03 q^{46} +(-102.436 - 102.436i) q^{47} +(3427.06 - 3427.06i) q^{48} -1661.82i q^{49} -1084.27 q^{51} +(-2104.92 - 2104.92i) q^{52} +(-365.928 + 365.928i) q^{53} +1081.23i q^{54} +5739.84 q^{56} +(-515.281 - 515.281i) q^{57} +(3208.42 - 3208.42i) q^{58} -5805.99i q^{59} +6282.87 q^{61} +(7703.26 + 7703.26i) q^{62} +(-519.068 + 519.068i) q^{63} -14442.2i q^{64} -7410.81 q^{66} +(-5494.12 - 5494.12i) q^{67} +(-6402.79 + 6402.79i) q^{68} -1302.64i q^{69} -7666.03 q^{71} +(4030.63 + 4030.63i) q^{72} +(6406.59 - 6406.59i) q^{73} +8325.39i q^{74} -6085.64 q^{76} +(-3557.74 - 3557.74i) q^{77} +(1942.50 - 1942.50i) q^{78} +5699.38i q^{79} -729.000 q^{81} +(5426.87 + 5426.87i) q^{82} +(999.189 - 999.189i) q^{83} +6130.36i q^{84} -7163.65 q^{86} +(2163.23 + 2163.23i) q^{87} +(-27626.3 + 27626.3i) q^{88} +3090.84i q^{89} +1865.08 q^{91} +(-7692.29 - 7692.29i) q^{92} +(-5193.80 + 5193.80i) q^{93} -1116.45i q^{94} +19799.4 q^{96} +(-5031.88 - 5031.88i) q^{97} +(9056.06 - 9056.06i) q^{98} -4996.63i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{2} - 72 q^{6} - 72 q^{7} - 264 q^{8} - 24 q^{11} - 432 q^{12} + 144 q^{13} - 2320 q^{16} + 600 q^{17} + 324 q^{18} + 36 q^{21} + 1800 q^{22} + 888 q^{23} - 792 q^{26} + 1152 q^{28} + 3244 q^{31}+ \cdots + 20136 q^{98}+O(q^{100}) \) 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 \(0.163547\pi\)
0.491488 + 0.870884i \(0.336453\pi\)
\(3\) −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 0.483180 − 0.875521i \(-0.339482\pi\)
−0.875521 + 0.483180i \(0.839482\pi\)
\(8\) −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 −0.835903 − 0.548877i \(-0.815056\pi\)
0.548877 + 0.835903i \(0.315056\pi\)
\(14\) − 209.530i − 1.06903i
\(15\) 0 0
\(16\) −932.727 −3.64346
\(17\) 147.551 + 147.551i 0.510555 + 0.510555i 0.914697 − 0.404141i \(-0.132430\pi\)
−0.404141 + 0.914697i \(0.632430\pi\)
\(18\) 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 −0.854519 − 0.519421i \(-0.826148\pi\)
0.519421 + 0.854519i \(0.326148\pi\)
\(24\) − 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 0.928731 − 0.370755i \(-0.120901\pi\)
−0.370755 + 0.928731i \(0.620901\pi\)
\(38\) −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 −0.862143 − 0.506666i \(-0.830878\pi\)
0.506666 + 0.862143i \(0.330878\pi\)
\(44\) 8030.48i 4.14797i
\(45\) 0 0
\(46\) −1932.03 −0.913056
\(47\) −102.436 − 102.436i −0.0463722 − 0.0463722i 0.683540 − 0.729913i \(-0.260439\pi\)
−0.729913 + 0.683540i \(0.760439\pi\)
\(48\) 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 −0.769235 − 0.638966i \(-0.779363\pi\)
0.638966 + 0.769235i \(0.279363\pi\)
\(54\) 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 \(-0.917043\pi\)
−0.257677 − 0.966231i \(-0.582957\pi\)
\(68\) −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 \(-0.426546\pi\)
0.973492 − 0.228721i \(-0.0734543\pi\)
\(74\) 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 −0.630857 − 0.775899i \(-0.717297\pi\)
0.775899 + 0.630857i \(0.217297\pi\)
\(84\) 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 0.387201 − 0.921995i \(-0.373442\pi\)
−0.921995 + 0.387201i \(0.873442\pi\)
\(98\) 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.7.2 yes 4
3.2 odd 2 225.5.g.d.82.1 4
5.2 odd 4 75.5.f.a.43.1 yes 4
5.3 odd 4 inner 75.5.f.d.43.2 yes 4
5.4 even 2 75.5.f.a.7.1 ✓ 4
15.2 even 4 225.5.g.l.118.2 4
15.8 even 4 225.5.g.d.118.1 4
15.14 odd 2 225.5.g.l.82.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.5.f.a.7.1 ✓ 4 5.4 even 2
75.5.f.a.43.1 yes 4 5.2 odd 4
75.5.f.d.7.2 yes 4 1.1 even 1 trivial
75.5.f.d.43.2 yes 4 5.3 odd 4 inner
225.5.g.d.82.1 4 3.2 odd 2
225.5.g.d.118.1 4 15.8 even 4
225.5.g.l.82.2 4 15.14 odd 2
225.5.g.l.118.2 4 15.2 even 4