Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(11,95)] 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("95.11"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 3 x^{17} + 64 x^{16} - 83 x^{15} + 2369 x^{14} - 2209 x^{13} + 52787 x^{12} - 15807 x^{11} + \cdots + 156250000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 11.3
Root \(1.85387 - 3.21100i\) of defining polynomial
Character \(\chi\) \(=\) 95.11
Dual form 95.4.e.b.26.3

$q$-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+(-1.35387 + 2.34498i) q^{2} +(2.16510 - 3.75006i) q^{3} +(0.334056 + 0.578603i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(5.86254 + 10.1542i) q^{6} +9.08880 q^{7} -23.4710 q^{8} +(4.12469 + 7.14417i) q^{9} +(-6.76936 - 11.7249i) q^{10} +29.7669 q^{11} +2.89306 q^{12} +(33.3072 + 57.6897i) q^{13} +(-12.3051 + 21.3130i) q^{14} +(10.8255 + 18.7503i) q^{15} +(29.1044 - 50.4102i) q^{16} +(-43.3610 + 75.1034i) q^{17} -22.3372 q^{18} +(-11.3672 + 82.0353i) q^{19} -3.34056 q^{20} +(19.6781 - 34.0836i) q^{21} +(-40.3006 + 69.8028i) q^{22} +(-109.366 - 189.427i) q^{23} +(-50.8171 + 88.0179i) q^{24} +(-12.5000 - 21.6506i) q^{25} -180.375 q^{26} +152.637 q^{27} +(3.03617 + 5.25880i) q^{28} +(127.375 + 220.620i) q^{29} -58.6254 q^{30} +11.1060 q^{31} +(-15.0770 - 26.1141i) q^{32} +(64.4484 - 111.628i) q^{33} +(-117.410 - 203.361i) q^{34} +(-22.7220 + 39.3556i) q^{35} +(-2.75576 + 4.77311i) q^{36} +150.747 q^{37} +(-176.981 - 137.721i) q^{38} +288.453 q^{39} +(58.6776 - 101.633i) q^{40} +(167.237 - 289.663i) q^{41} +(53.2834 + 92.2896i) q^{42} +(190.281 - 329.576i) q^{43} +(9.94383 + 17.2232i) q^{44} -41.2469 q^{45} +592.270 q^{46} +(-282.758 - 489.751i) q^{47} +(-126.028 - 218.286i) q^{48} -260.394 q^{49} +67.6936 q^{50} +(187.762 + 325.213i) q^{51} +(-22.2529 + 38.5432i) q^{52} +(-81.4232 - 141.029i) q^{53} +(-206.651 + 357.930i) q^{54} +(-74.4173 + 128.895i) q^{55} -213.324 q^{56} +(283.026 + 220.242i) q^{57} -689.799 q^{58} +(192.766 - 333.881i) q^{59} +(-7.23265 + 12.5273i) q^{60} +(183.182 + 317.281i) q^{61} +(-15.0361 + 26.0432i) q^{62} +(37.4885 + 64.9319i) q^{63} +547.319 q^{64} -333.072 q^{65} +(174.510 + 302.260i) q^{66} +(-67.2137 - 116.418i) q^{67} -57.9401 q^{68} -947.152 q^{69} +(-61.5254 - 106.565i) q^{70} +(-398.844 + 690.817i) q^{71} +(-96.8108 - 167.681i) q^{72} +(13.3128 - 23.0585i) q^{73} +(-204.093 + 353.499i) q^{74} -108.255 q^{75} +(-51.2631 + 20.8273i) q^{76} +270.546 q^{77} +(-390.529 + 676.416i) q^{78} +(411.728 - 713.133i) q^{79} +(145.522 + 252.051i) q^{80} +(219.107 - 379.505i) q^{81} +(452.835 + 784.333i) q^{82} -946.123 q^{83} +26.2945 q^{84} +(-216.805 - 375.517i) q^{85} +(515.232 + 892.408i) q^{86} +1103.12 q^{87} -698.661 q^{88} +(241.295 + 417.936i) q^{89} +(55.8431 - 96.7230i) q^{90} +(302.722 + 524.330i) q^{91} +(73.0688 - 126.559i) q^{92} +(24.0455 - 41.6481i) q^{93} +1531.27 q^{94} +(-326.805 - 254.310i) q^{95} -130.573 q^{96} +(-257.473 + 445.956i) q^{97} +(352.540 - 610.617i) q^{98} +(122.779 + 212.660i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 6 q^{2} + 2 q^{3} - 50 q^{4} - 45 q^{5} + 22 q^{6} - 90 q^{7} - 222 q^{8} - 115 q^{9} + 30 q^{10} - 54 q^{11} - 208 q^{12} + 88 q^{13} - 6 q^{14} + 10 q^{15} - 270 q^{16} - 174 q^{17} - 382 q^{18}+ \cdots + 557 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.35387 + 2.34498i −0.478666 + 0.829074i −0.999701 − 0.0244612i \(-0.992213\pi\)
0.521034 + 0.853536i \(0.325546\pi\)
\(3\) 2.16510 − 3.75006i 0.416674 − 0.721700i −0.578929 − 0.815378i \(-0.696529\pi\)
0.995603 + 0.0936782i \(0.0298625\pi\)
\(4\) 0.334056 + 0.578603i 0.0417571 + 0.0723253i
\(5\) −2.50000 + 4.33013i −0.223607 + 0.387298i
\(6\) 5.86254 + 10.1542i 0.398895 + 0.690907i
\(7\) 9.08880 0.490749 0.245374 − 0.969428i \(-0.421089\pi\)
0.245374 + 0.969428i \(0.421089\pi\)
\(8\) −23.4710 −1.03728
\(9\) 4.12469 + 7.14417i 0.152766 + 0.264599i
\(10\) −6.76936 − 11.7249i −0.214066 − 0.370773i
\(11\) 29.7669 0.815915 0.407958 − 0.913001i \(-0.366241\pi\)
0.407958 + 0.913001i \(0.366241\pi\)
\(12\) 2.89306 0.0695962
\(13\) 33.3072 + 57.6897i 0.710596 + 1.23079i 0.964634 + 0.263594i \(0.0849078\pi\)
−0.254038 + 0.967194i \(0.581759\pi\)
\(14\) −12.3051 + 21.3130i −0.234905 + 0.406867i
\(15\) 10.8255 + 18.7503i 0.186342 + 0.322754i
\(16\) 29.1044 − 50.4102i 0.454756 − 0.787660i
\(17\) −43.3610 + 75.1034i −0.618622 + 1.07149i 0.371115 + 0.928587i \(0.378976\pi\)
−0.989737 + 0.142899i \(0.954358\pi\)
\(18\) −22.3372 −0.292496
\(19\) −11.3672 + 82.0353i −0.137254 + 0.990536i
\(20\) −3.34056 −0.0373486
\(21\) 19.6781 − 34.0836i 0.204482 − 0.354173i
\(22\) −40.3006 + 69.8028i −0.390551 + 0.676454i
\(23\) −109.366 − 189.427i −0.991494 − 1.71732i −0.608462 − 0.793583i \(-0.708213\pi\)
−0.383032 − 0.923735i \(-0.625120\pi\)
\(24\) −50.8171 + 88.0179i −0.432209 + 0.748607i
\(25\) −12.5000 − 21.6506i −0.100000 − 0.173205i
\(26\) −180.375 −1.36055
\(27\) 152.637 1.08796
\(28\) 3.03617 + 5.25880i 0.0204922 + 0.0354936i
\(29\) 127.375 + 220.620i 0.815620 + 1.41270i 0.908882 + 0.417054i \(0.136937\pi\)
−0.0932619 + 0.995642i \(0.529729\pi\)
\(30\) −58.6254 −0.356783
\(31\) 11.1060 0.0643449 0.0321724 − 0.999482i \(-0.489757\pi\)
0.0321724 + 0.999482i \(0.489757\pi\)
\(32\) −15.0770 − 26.1141i −0.0832893 − 0.144261i
\(33\) 64.4484 − 111.628i 0.339970 − 0.588846i
\(34\) −117.410 − 203.361i −0.592227 − 1.02577i
\(35\) −22.7220 + 39.3556i −0.109735 + 0.190066i
\(36\) −2.75576 + 4.77311i −0.0127581 + 0.0220977i
\(37\) 150.747 0.669803 0.334901 − 0.942253i \(-0.391297\pi\)
0.334901 + 0.942253i \(0.391297\pi\)
\(38\) −176.981 − 137.721i −0.755529 − 0.587930i
\(39\) 288.453 1.18435
\(40\) 58.6776 − 101.633i 0.231944 − 0.401738i
\(41\) 167.237 − 289.663i 0.637025 − 1.10336i −0.349058 − 0.937101i \(-0.613498\pi\)
0.986082 − 0.166258i \(-0.0531684\pi\)
\(42\) 53.2834 + 92.2896i 0.195757 + 0.339062i
\(43\) 190.281 − 329.576i 0.674827 − 1.16883i −0.301693 − 0.953405i \(-0.597552\pi\)
0.976519 − 0.215429i \(-0.0691150\pi\)
\(44\) 9.94383 + 17.2232i 0.0340702 + 0.0590113i
\(45\) −41.2469 −0.136638
\(46\) 592.270 1.89838
\(47\) −282.758 − 489.751i −0.877541 − 1.51995i −0.854031 − 0.520223i \(-0.825849\pi\)
−0.0235109 − 0.999724i \(-0.507484\pi\)
\(48\) −126.028 − 218.286i −0.378969 − 0.656394i
\(49\) −260.394 −0.759165
\(50\) 67.6936 0.191467
\(51\) 187.762 + 325.213i 0.515527 + 0.892919i
\(52\) −22.2529 + 38.5432i −0.0593448 + 0.102788i
\(53\) −81.4232 − 141.029i −0.211025 − 0.365507i 0.741010 − 0.671494i \(-0.234347\pi\)
−0.952036 + 0.305987i \(0.901014\pi\)
\(54\) −206.651 + 357.930i −0.520771 + 0.902001i
\(55\) −74.4173 + 128.895i −0.182444 + 0.316003i
\(56\) −213.324 −0.509046
\(57\) 283.026 + 220.242i 0.657680 + 0.511786i
\(58\) −689.799 −1.56164
\(59\) 192.766 − 333.881i 0.425357 − 0.736739i −0.571097 − 0.820883i \(-0.693482\pi\)
0.996454 + 0.0841431i \(0.0268153\pi\)
\(60\) −7.23265 + 12.5273i −0.0155622 + 0.0269545i
\(61\) 183.182 + 317.281i 0.384493 + 0.665962i 0.991699 − 0.128583i \(-0.0410428\pi\)
−0.607205 + 0.794545i \(0.707710\pi\)
\(62\) −15.0361 + 26.0432i −0.0307997 + 0.0533467i
\(63\) 37.4885 + 64.9319i 0.0749699 + 0.129852i
\(64\) 547.319 1.06898
\(65\) −333.072 −0.635576
\(66\) 174.510 + 302.260i 0.325465 + 0.563721i
\(67\) −67.2137 − 116.418i −0.122559 − 0.212279i 0.798217 − 0.602370i \(-0.205777\pi\)
−0.920776 + 0.390091i \(0.872443\pi\)
\(68\) −57.9401 −0.103327
\(69\) −947.152 −1.65252
\(70\) −61.5254 − 106.565i −0.105053 − 0.181957i
\(71\) −398.844 + 690.817i −0.666677 + 1.15472i 0.312151 + 0.950032i \(0.398950\pi\)
−0.978828 + 0.204685i \(0.934383\pi\)
\(72\) −96.8108 − 167.681i −0.158462 − 0.274464i
\(73\) 13.3128 − 23.0585i 0.0213445 − 0.0369697i −0.855156 − 0.518371i \(-0.826539\pi\)
0.876500 + 0.481401i \(0.159872\pi\)
\(74\) −204.093 + 353.499i −0.320612 + 0.555316i
\(75\) −108.255 −0.166669
\(76\) −51.2631 + 20.8273i −0.0773722 + 0.0314350i
\(77\) 270.546 0.400409
\(78\) −390.529 + 676.416i −0.566907 + 0.981911i
\(79\) 411.728 − 713.133i 0.586367 − 1.01562i −0.408336 − 0.912831i \(-0.633891\pi\)
0.994703 − 0.102786i \(-0.0327757\pi\)
\(80\) 145.522 + 252.051i 0.203373 + 0.352252i
\(81\) 219.107 − 379.505i 0.300559 − 0.520583i
\(82\) 452.835 + 784.333i 0.609845 + 1.05628i
\(83\) −946.123 −1.25121 −0.625605 − 0.780140i \(-0.715148\pi\)
−0.625605 + 0.780140i \(0.715148\pi\)
\(84\) 26.2945 0.0341543
\(85\) −216.805 − 375.517i −0.276656 − 0.479183i
\(86\) 515.232 + 892.408i 0.646034 + 1.11896i
\(87\) 1103.12 1.35939
\(88\) −698.661 −0.846335
\(89\) 241.295 + 417.936i 0.287385 + 0.497765i 0.973185 − 0.230025i \(-0.0738808\pi\)
−0.685800 + 0.727790i \(0.740548\pi\)
\(90\) 55.8431 − 96.7230i 0.0654042 − 0.113283i
\(91\) 302.722 + 524.330i 0.348724 + 0.604008i
\(92\) 73.0688 − 126.559i 0.0828038 − 0.143420i
\(93\) 24.0455 − 41.6481i 0.0268108 − 0.0464377i
\(94\) 1531.27 1.68020
\(95\) −326.805 − 254.310i −0.352942 − 0.274649i
\(96\) −130.573 −0.138818
\(97\) −257.473 + 445.956i −0.269509 + 0.466803i −0.968735 − 0.248097i \(-0.920195\pi\)
0.699226 + 0.714901i \(0.253528\pi\)
\(98\) 352.540 − 610.617i 0.363387 − 0.629405i
\(99\) 122.779 + 212.660i 0.124644 + 0.215890i
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 95.4.e.b.11.3 ✓ 18
19.7 even 3 inner 95.4.e.b.26.3 yes 18
19.8 odd 6 1805.4.a.o.1.3 9
19.11 even 3 1805.4.a.n.1.7 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.e.b.11.3 ✓ 18 1.1 even 1 trivial
95.4.e.b.26.3 yes 18 19.7 even 3 inner
1805.4.a.n.1.7 9 19.11 even 3
1805.4.a.o.1.3 9 19.8 odd 6