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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,8,Mod(2,75)] 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("75.2"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 2.10
Character \(\chi\) \(=\) 75.2
Dual form 75.8.l.a.38.10

$q$-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+(-2.68688 + 16.9643i) q^{2} +(2.70038 + 46.6873i) q^{3} +(-158.832 - 51.6076i) q^{4} +(277.208 + 35.7878i) q^{5} +(-799.272 - 79.6331i) q^{6} +(577.729 + 577.729i) q^{7} +(304.152 - 596.931i) q^{8} +(-2172.42 + 252.147i) q^{9} +(-1351.94 + 4606.47i) q^{10} +(-2395.00 - 3296.44i) q^{11} +(1980.51 - 7554.80i) q^{12} +(-2856.20 + 452.377i) q^{13} +(-11353.0 + 8248.46i) q^{14} +(-922.271 + 13038.7i) q^{15} +(-7984.86 - 5801.34i) q^{16} +(-31668.7 - 16136.0i) q^{17} +(1559.51 - 37530.9i) q^{18} +(-631.539 + 205.199i) q^{19} +(-42182.5 - 19990.3i) q^{20} +(-25412.5 + 28532.7i) q^{21} +(62356.7 - 31772.4i) q^{22} +(-39841.6 - 6310.29i) q^{23} +(28690.4 + 12588.1i) q^{24} +(75563.5 + 19841.3i) q^{25} -49668.8i q^{26} +(-17638.5 - 100743. i) q^{27} +(-61946.5 - 121577. i) q^{28} +(-59068.5 + 181794. i) q^{29} +(-218715. - 50679.1i) q^{30} +(80579.6 + 247999. i) q^{31} +(180507. - 180507. i) q^{32} +(147435. - 120718. i) q^{33} +(358825. - 493880. i) q^{34} +(139475. + 180827. i) q^{35} +(358062. + 72064.1i) q^{36} +(11041.6 + 69713.9i) q^{37} +(-1784.19 - 11264.9i) q^{38} +(-28833.1 - 132127. i) q^{39} +(105676. - 154589. i) q^{40} +(-278445. + 383247. i) q^{41} +(-415756. - 507769. i) q^{42} +(198183. - 198183. i) q^{43} +(210281. + 647180. i) q^{44} +(-611235. - 7848.79i) q^{45} +(214099. - 658928. i) q^{46} +(-168111. - 329936. i) q^{47} +(249287. - 388458. i) q^{48} -156002. i q^{49} +(-539624. + 1.22857e6i) q^{50} +(667829. - 1.52210e6i) q^{51} +(477001. + 75549.5i) q^{52} +(1.55977e6 - 794740. i) q^{53} +(1.75643e6 - 28538.3i) q^{54} +(-545941. - 999511. i) q^{55} +(520581. - 169147. i) q^{56} +(-11285.6 - 28930.7i) q^{57} +(-2.92529e6 - 1.49051e6i) q^{58} +(779855. + 566597. i) q^{59} +(819384. - 2.02337e6i) q^{60} +(-2.05588e6 + 1.49368e6i) q^{61} +(-4.42362e6 + 700633. i) q^{62} +(-1.40074e6 - 1.10939e6i) q^{63} +(1.83460e6 + 2.52510e6i) q^{64} +(-807950. + 23185.4i) q^{65} +(1.65175e6 + 2.82547e6i) q^{66} +(-653220. + 1.28202e6i) q^{67} +(4.19725e6 + 4.19725e6i) q^{68} +(187023. - 1.87714e6i) q^{69} +(-3.44234e6 + 1.88024e6i) q^{70} +(-2.18798e6 - 710919. i) q^{71} +(-510229. + 1.37347e6i) q^{72} +(-911175. + 5.75293e6i) q^{73} -1.21231e6 q^{74} +(-722289. + 3.58144e6i) q^{75} +110898. q^{76} +(520785. - 3.28811e6i) q^{77} +(2.31890e6 - 134125. i) q^{78} +(4.85392e6 + 1.57714e6i) q^{79} +(-2.00585e6 - 1.89394e6i) q^{80} +(4.65581e6 - 1.09554e6i) q^{81} +(-5.75336e6 - 5.75336e6i) q^{82} +(-2.91328e6 + 5.71763e6i) q^{83} +(5.50882e6 - 3.22042e6i) q^{84} +(-8.20133e6 - 5.60638e6i) q^{85} +(2.82954e6 + 3.89453e6i) q^{86} +(-8.64699e6 - 2.26684e6i) q^{87} +(-2.69619e6 + 427035. i) q^{88} +(1.01174e6 - 735071. i) q^{89} +(1.77546e6 - 1.03481e7i) q^{90} +(-1.91146e6 - 1.38876e6i) q^{91} +(6.00245e6 + 3.05840e6i) q^{92} +(-1.13608e7 + 4.43174e6i) q^{93} +(6.04882e6 - 1.96538e6i) q^{94} +(-182411. + 34281.5i) q^{95} +(8.91482e6 + 7.93995e6i) q^{96} +(7.24399e6 - 3.69100e6i) q^{97} +(2.64646e6 + 419158. i) q^{98} +(6.03413e6 + 6.55734e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 544 q - 34 q^{3} - 20 q^{4} - 6 q^{6} - 1364 q^{7} - 10 q^{9} + 3160 q^{10} + 2018 q^{12} - 16868 q^{13} - 37570 q^{15} + 507892 q^{16} + 106550 q^{18} + 70800 q^{19} - 6 q^{21} - 195160 q^{22} - 444200 q^{25}+ \cdots - 22089384 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.68688 + 16.9643i −0.237489 + 1.49944i 0.524253 + 0.851563i \(0.324345\pi\)
−0.761742 + 0.647881i \(0.775655\pi\)
\(3\) 2.70038 + 46.6873i 0.0577432 + 0.998331i
\(4\) −158.832 − 51.6076i −1.24087 − 0.403184i
\(5\) 277.208 + 35.7878i 0.991769 + 0.128038i
\(6\) −799.272 − 79.6331i −1.51066 − 0.150510i
\(7\) 577.729 + 577.729i 0.636621 + 0.636621i 0.949720 − 0.313100i \(-0.101367\pi\)
−0.313100 + 0.949720i \(0.601367\pi\)
\(8\) 304.152 − 596.931i 0.210027 − 0.412201i
\(9\) −2172.42 + 252.147i −0.993331 + 0.115294i
\(10\) −1351.94 + 4606.47i −0.427520 + 1.45669i
\(11\) −2395.00 − 3296.44i −0.542540 − 0.746742i 0.446437 − 0.894815i \(-0.352693\pi\)
−0.988976 + 0.148073i \(0.952693\pi\)
\(12\) 1980.51 − 7554.80i 0.330860 − 1.26208i
\(13\) −2856.20 + 452.377i −0.360567 + 0.0571082i −0.334090 − 0.942541i \(-0.608429\pi\)
−0.0264772 + 0.999649i \(0.508429\pi\)
\(14\) −11353.0 + 8248.46i −1.10577 + 0.803387i
\(15\) −922.271 + 13038.7i −0.0705569 + 0.997508i
\(16\) −7984.86 − 5801.34i −0.487357 − 0.354086i
\(17\) −31668.7 − 16136.0i −1.56336 − 0.796571i −0.563789 − 0.825919i \(-0.690657\pi\)
−0.999569 + 0.0293477i \(0.990657\pi\)
\(18\) 1559.51 − 37530.9i 0.0630283 − 1.51683i
\(19\) −631.539 + 205.199i −0.0211233 + 0.00686338i −0.319560 − 0.947566i \(-0.603535\pi\)
0.298436 + 0.954430i \(0.403535\pi\)
\(20\) −42182.5 − 19990.3i −1.17904 − 0.558745i
\(21\) −25412.5 + 28532.7i −0.598798 + 0.672319i
\(22\) 62356.7 − 31772.4i 1.24854 − 0.636165i
\(23\) −39841.6 − 6310.29i −0.682793 − 0.108144i −0.194604 − 0.980882i \(-0.562342\pi\)
−0.488189 + 0.872738i \(0.662342\pi\)
\(24\) 28690.4 + 12588.1i 0.423641 + 0.185875i
\(25\) 75563.5 + 19841.3i 0.967212 + 0.253969i
\(26\) − 49668.8i − 0.554213i
\(27\) −17638.5 − 100743.i −0.172460 − 0.985017i
\(28\) −61946.5 − 121577.i −0.533291 − 1.04664i
\(29\) −59068.5 + 181794.i −0.449741 + 1.38416i 0.427458 + 0.904035i \(0.359409\pi\)
−0.877199 + 0.480126i \(0.840591\pi\)
\(30\) −218715. − 50679.1i −1.47895 − 0.342693i
\(31\) 80579.6 + 247999.i 0.485802 + 1.49514i 0.830816 + 0.556547i \(0.187874\pi\)
−0.345014 + 0.938597i \(0.612126\pi\)
\(32\) 180507. − 180507.i 0.973798 − 0.973798i
\(33\) 147435. − 120718.i 0.714168 − 0.584754i
\(34\) 358825. − 493880.i 1.56569 − 2.15499i
\(35\) 139475. + 180827.i 0.549869 + 0.712893i
\(36\) 358062. + 72064.1i 1.27908 + 0.257431i
\(37\) 11041.6 + 69713.9i 0.0358365 + 0.226263i 0.999106 − 0.0422735i \(-0.0134601\pi\)
−0.963270 + 0.268536i \(0.913460\pi\)
\(38\) −1784.19 − 11264.9i −0.00527471 − 0.0333032i
\(39\) −28833.1 − 132127.i −0.0778333 − 0.356668i
\(40\) 105676. − 154589.i 0.261076 − 0.381917i
\(41\) −278445. + 383247.i −0.630952 + 0.868431i −0.998093 − 0.0617346i \(-0.980337\pi\)
0.367141 + 0.930165i \(0.380337\pi\)
\(42\) −415756. − 507769.i −0.865897 − 1.05753i
\(43\) 198183. − 198183.i 0.380126 − 0.380126i −0.491022 − 0.871147i \(-0.663376\pi\)
0.871147 + 0.491022i \(0.163376\pi\)
\(44\) 210281. + 647180.i 0.372149 + 1.14536i
\(45\) −611235. − 7848.79i −0.999918 − 0.0128398i
\(46\) 214099. − 658928.i 0.324311 − 0.998127i
\(47\) −168111. − 329936.i −0.236186 − 0.463541i 0.742241 − 0.670133i \(-0.233763\pi\)
−0.978427 + 0.206592i \(0.933763\pi\)
\(48\) 249287. − 388458.i 0.325353 − 0.506990i
\(49\) − 156002.i − 0.189428i
\(50\) −539624. + 1.22857e6i −0.610514 + 1.38997i
\(51\) 667829. − 1.52210e6i 0.704968 − 1.60675i
\(52\) 477001. + 75549.5i 0.470444 + 0.0745109i
\(53\) 1.55977e6 − 794740.i 1.43911 − 0.733263i 0.451807 − 0.892116i \(-0.350780\pi\)
0.987302 + 0.158853i \(0.0507796\pi\)
\(54\) 1.75643e6 − 28538.3i 1.51793 − 0.0246633i
\(55\) −545941. − 999511.i −0.442463 − 0.810062i
\(56\) 520581. − 169147.i 0.396123 − 0.128708i
\(57\) −11285.6 − 28930.7i −0.00807166 − 0.0206918i
\(58\) −2.92529e6 − 1.49051e6i −1.96866 − 1.00308i
\(59\) 779855. + 566597.i 0.494347 + 0.359164i 0.806853 − 0.590752i \(-0.201169\pi\)
−0.312507 + 0.949916i \(0.601169\pi\)
\(60\) 819384. − 2.02337e6i 0.489732 − 1.20933i
\(61\) −2.05588e6 + 1.49368e6i −1.15969 + 0.842566i −0.989739 − 0.142886i \(-0.954362\pi\)
−0.169954 + 0.985452i \(0.554362\pi\)
\(62\) −4.42362e6 + 700633.i −2.35726 + 0.373353i
\(63\) −1.40074e6 − 1.10939e6i −0.705774 − 0.558977i
\(64\) 1.83460e6 + 2.52510e6i 0.874803 + 1.20406i
\(65\) −807950. + 23185.4i −0.364912 + 0.0104717i
\(66\) 1.65175e6 + 2.82547e6i 0.707199 + 1.20973i
\(67\) −653220. + 1.28202e6i −0.265337 + 0.520753i −0.984782 − 0.173797i \(-0.944396\pi\)
0.719445 + 0.694550i \(0.244396\pi\)
\(68\) 4.19725e6 + 4.19725e6i 1.61877 + 1.61877i
\(69\) 187023. − 1.87714e6i 0.0685367 − 0.687899i
\(70\) −3.44234e6 + 1.88024e6i −1.19953 + 0.655194i
\(71\) −2.18798e6 − 710919.i −0.725504 − 0.235730i −0.0770959 − 0.997024i \(-0.524565\pi\)
−0.648408 + 0.761293i \(0.724565\pi\)
\(72\) −510229. + 1.37347e6i −0.161102 + 0.433667i
\(73\) −911175. + 5.75293e6i −0.274140 + 1.73085i 0.338923 + 0.940814i \(0.389938\pi\)
−0.613062 + 0.790035i \(0.710062\pi\)
\(74\) −1.21231e6 −0.347779
\(75\) −722289. + 3.58144e6i −0.197696 + 0.980263i
\(76\) 110898. 0.0289786
\(77\) 520785. − 3.28811e6i 0.129999 − 0.820784i
\(78\) 2.31890e6 − 134125.i 0.553288 − 0.0320020i
\(79\) 4.85392e6 + 1.57714e6i 1.10764 + 0.359894i 0.805038 − 0.593224i \(-0.202145\pi\)
0.302601 + 0.953117i \(0.402145\pi\)
\(80\) −2.00585e6 − 1.89394e6i −0.438009 − 0.413572i
\(81\) 4.65581e6 − 1.09554e6i 0.973415 − 0.229050i
\(82\) −5.75336e6 − 5.75336e6i −1.15232 − 1.15232i
\(83\) −2.91328e6 + 5.71763e6i −0.559254 + 1.09760i 0.422308 + 0.906452i \(0.361220\pi\)
−0.981562 + 0.191145i \(0.938780\pi\)
\(84\) 5.50882e6 − 3.22042e6i 1.01410 − 0.592837i
\(85\) −8.20133e6 − 5.60638e6i −1.44850 − 0.990185i
\(86\) 2.82954e6 + 3.89453e6i 0.479702 + 0.660253i
\(87\) −8.64699e6 − 2.26684e6i −1.40782 − 0.369065i
\(88\) −2.69619e6 + 427035.i −0.421756 + 0.0667996i
\(89\) 1.01174e6 − 735071.i 0.152126 − 0.110526i −0.509119 − 0.860696i \(-0.670029\pi\)
0.661245 + 0.750170i \(0.270029\pi\)
\(90\) 1.77546e6 − 1.03481e7i 0.256722 − 1.49627i
\(91\) −1.91146e6 − 1.38876e6i −0.265901 − 0.193188i
\(92\) 6.00245e6 + 3.05840e6i 0.803658 + 0.409484i
\(93\) −1.13608e7 + 4.43174e6i −1.46460 + 0.571326i
\(94\) 6.04882e6 − 1.96538e6i 0.751144 − 0.244062i
\(95\) −182411. + 34281.5i −0.0218282 + 0.00410229i
\(96\) 8.91482e6 + 7.93995e6i 1.02840 + 0.915943i
\(97\) 7.24399e6 − 3.69100e6i 0.805892 − 0.410622i −0.00196739 − 0.999998i \(-0.500626\pi\)
0.807859 + 0.589376i \(0.200626\pi\)
\(98\) 2.64646e6 + 419158.i 0.284036 + 0.0449869i
\(99\) 6.03413e6 + 6.55734e6i 0.625017 + 0.679211i
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 75.8.l.a.2.10 ✓ 544
3.2 odd 2 inner 75.8.l.a.2.59 yes 544
25.13 odd 20 inner 75.8.l.a.38.59 yes 544
75.38 even 20 inner 75.8.l.a.38.10 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.8.l.a.2.10 ✓ 544 1.1 even 1 trivial
75.8.l.a.2.59 yes 544 3.2 odd 2 inner
75.8.l.a.38.10 yes 544 75.38 even 20 inner
75.8.l.a.38.59 yes 544 25.13 odd 20 inner