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 11.7
Root \(-1.24804 + 2.16167i\) of defining polynomial
Character \(\chi\) \(=\) 95.11
Dual form 95.4.e.b.26.7

$q$-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+(1.74804 - 3.02770i) q^{2} +(4.62018 - 8.00239i) q^{3} +(-2.11129 - 3.65686i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(-16.1525 - 27.9770i) q^{6} -29.1065 q^{7} +13.2062 q^{8} +(-29.1921 - 50.5622i) q^{9} +(8.74020 + 15.1385i) q^{10} +59.8286 q^{11} -39.0182 q^{12} +(7.68761 + 13.3153i) q^{13} +(-50.8794 + 88.1257i) q^{14} +(23.1009 + 40.0119i) q^{15} +(39.9752 - 69.2391i) q^{16} +(-17.2772 + 29.9249i) q^{17} -204.116 q^{18} +(78.1571 - 27.3947i) q^{19} +21.1129 q^{20} +(-134.477 + 232.922i) q^{21} +(104.583 - 181.143i) q^{22} +(-16.7556 - 29.0215i) q^{23} +(61.0148 - 105.681i) q^{24} +(-12.5000 - 21.6506i) q^{25} +53.7530 q^{26} -290.002 q^{27} +(61.4524 + 106.439i) q^{28} +(119.923 + 207.713i) q^{29} +161.525 q^{30} +56.3531 q^{31} +(-86.9320 - 150.571i) q^{32} +(276.419 - 478.772i) q^{33} +(60.4023 + 104.620i) q^{34} +(72.7663 - 126.035i) q^{35} +(-123.266 + 213.503i) q^{36} -336.855 q^{37} +(53.6791 - 284.523i) q^{38} +142.073 q^{39} +(-33.0154 + 57.1843i) q^{40} +(-52.2601 + 90.5171i) q^{41} +(470.144 + 814.313i) q^{42} +(-131.537 + 227.829i) q^{43} +(-126.316 - 218.785i) q^{44} +291.921 q^{45} -117.158 q^{46} +(119.581 + 207.120i) q^{47} +(-369.385 - 639.794i) q^{48} +504.190 q^{49} -87.4020 q^{50} +(159.647 + 276.517i) q^{51} +(32.4616 - 56.2251i) q^{52} +(-13.9570 - 24.1741i) q^{53} +(-506.934 + 878.036i) q^{54} +(-149.572 + 259.066i) q^{55} -384.385 q^{56} +(141.877 - 752.011i) q^{57} +838.522 q^{58} +(319.477 - 553.350i) q^{59} +(97.5455 - 168.954i) q^{60} +(-173.918 - 301.235i) q^{61} +(98.5075 - 170.620i) q^{62} +(849.681 + 1471.69i) q^{63} +31.7608 q^{64} -76.8761 q^{65} +(-966.383 - 1673.83i) q^{66} +(352.011 + 609.702i) q^{67} +145.908 q^{68} -309.655 q^{69} +(-254.397 - 440.628i) q^{70} +(-109.195 + 189.132i) q^{71} +(-385.516 - 667.733i) q^{72} +(-34.6550 + 60.0242i) q^{73} +(-588.837 + 1019.89i) q^{74} -231.009 q^{75} +(-265.191 - 227.972i) q^{76} -1741.40 q^{77} +(248.349 - 430.152i) q^{78} +(-272.464 + 471.921i) q^{79} +(199.876 + 346.196i) q^{80} +(-551.672 + 955.524i) q^{81} +(182.705 + 316.455i) q^{82} -1272.58 q^{83} +1135.68 q^{84} +(-86.3858 - 149.625i) q^{85} +(459.864 + 796.508i) q^{86} +2216.26 q^{87} +790.106 q^{88} +(-604.779 - 1047.51i) q^{89} +(510.290 - 883.848i) q^{90} +(-223.760 - 387.563i) q^{91} +(-70.7518 + 122.546i) q^{92} +(260.362 - 450.959i) q^{93} +836.127 q^{94} +(-76.7704 + 406.917i) q^{95} -1606.57 q^{96} +(-529.365 + 916.887i) q^{97} +(881.345 - 1526.53i) q^{98} +(-1746.52 - 3025.07i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.74804 − 3.02770i 0.618026 − 1.07045i −0.371820 − 0.928305i \(-0.621266\pi\)
0.989846 − 0.142147i \(-0.0454006\pi\)
\(3\) 4.62018 − 8.00239i 0.889154 − 1.54006i 0.0482766 − 0.998834i \(-0.484627\pi\)
0.840877 − 0.541226i \(-0.182040\pi\)
\(4\) −2.11129 − 3.65686i −0.263911 − 0.457108i
\(5\) −2.50000 + 4.33013i −0.223607 + 0.387298i
\(6\) −16.1525 − 27.9770i −1.09904 − 1.90359i
\(7\) −29.1065 −1.57160 −0.785802 − 0.618478i \(-0.787750\pi\)
−0.785802 + 0.618478i \(0.787750\pi\)
\(8\) 13.2062 0.583635
\(9\) −29.1921 − 50.5622i −1.08119 − 1.87268i
\(10\) 8.74020 + 15.1385i 0.276389 + 0.478721i
\(11\) 59.8286 1.63991 0.819955 − 0.572428i \(-0.193998\pi\)
0.819955 + 0.572428i \(0.193998\pi\)
\(12\) −39.0182 −0.938632
\(13\) 7.68761 + 13.3153i 0.164012 + 0.284078i 0.936304 − 0.351190i \(-0.114223\pi\)
−0.772292 + 0.635268i \(0.780890\pi\)
\(14\) −50.8794 + 88.1257i −0.971292 + 1.68233i
\(15\) 23.1009 + 40.0119i 0.397642 + 0.688736i
\(16\) 39.9752 − 69.2391i 0.624613 − 1.08186i
\(17\) −17.2772 + 29.9249i −0.246490 + 0.426933i −0.962549 − 0.271106i \(-0.912611\pi\)
0.716060 + 0.698039i \(0.245944\pi\)
\(18\) −204.116 −2.67281
\(19\) 78.1571 − 27.3947i 0.943709 − 0.330777i
\(20\) 21.1129 0.236050
\(21\) −134.477 + 232.922i −1.39740 + 2.42037i
\(22\) 104.583 − 181.143i 1.01351 − 1.75544i
\(23\) −16.7556 − 29.0215i −0.151903 − 0.263104i 0.780024 − 0.625750i \(-0.215207\pi\)
−0.931927 + 0.362646i \(0.881874\pi\)
\(24\) 61.0148 − 105.681i 0.518941 − 0.898833i
\(25\) −12.5000 − 21.6506i −0.100000 − 0.173205i
\(26\) 53.7530 0.405455
\(27\) −290.002 −2.06707
\(28\) 61.4524 + 106.439i 0.414765 + 0.718393i
\(29\) 119.923 + 207.713i 0.767902 + 1.33005i 0.938699 + 0.344739i \(0.112033\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(30\) 161.525 0.983011
\(31\) 56.3531 0.326494 0.163247 − 0.986585i \(-0.447803\pi\)
0.163247 + 0.986585i \(0.447803\pi\)
\(32\) −86.9320 − 150.571i −0.480236 − 0.831793i
\(33\) 276.419 − 478.772i 1.45813 − 2.52556i
\(34\) 60.4023 + 104.620i 0.304674 + 0.527711i
\(35\) 72.7663 − 126.035i 0.351422 − 0.608680i
\(36\) −123.266 + 213.503i −0.570677 + 0.988441i
\(37\) −336.855 −1.49672 −0.748360 − 0.663293i \(-0.769159\pi\)
−0.748360 + 0.663293i \(0.769159\pi\)
\(38\) 53.6791 − 284.523i 0.229155 − 1.21462i
\(39\) 142.073 0.583328
\(40\) −33.0154 + 57.1843i −0.130505 + 0.226041i
\(41\) −52.2601 + 90.5171i −0.199065 + 0.344790i −0.948225 − 0.317598i \(-0.897124\pi\)
0.749161 + 0.662388i \(0.230457\pi\)
\(42\) 470.144 + 814.313i 1.72726 + 2.99170i
\(43\) −131.537 + 227.829i −0.466493 + 0.807990i −0.999268 − 0.0382676i \(-0.987816\pi\)
0.532774 + 0.846257i \(0.321149\pi\)
\(44\) −126.316 − 218.785i −0.432791 − 0.749616i
\(45\) 291.921 0.967045
\(46\) −117.158 −0.375521
\(47\) 119.581 + 207.120i 0.371119 + 0.642798i 0.989738 − 0.142894i \(-0.0456407\pi\)
−0.618619 + 0.785691i \(0.712307\pi\)
\(48\) −369.385 − 639.794i −1.11075 − 1.92388i
\(49\) 504.190 1.46994
\(50\) −87.4020 −0.247210
\(51\) 159.647 + 276.517i 0.438335 + 0.759218i
\(52\) 32.4616 − 56.2251i 0.0865694 − 0.149943i
\(53\) −13.9570 − 24.1741i −0.0361724 − 0.0626524i 0.847372 − 0.530999i \(-0.178183\pi\)
−0.883545 + 0.468347i \(0.844850\pi\)
\(54\) −506.934 + 878.036i −1.27750 + 2.21270i
\(55\) −149.572 + 259.066i −0.366695 + 0.635134i
\(56\) −384.385 −0.917244
\(57\) 141.877 − 752.011i 0.329686 − 1.74748i
\(58\) 838.522 1.89833
\(59\) 319.477 − 553.350i 0.704955 − 1.22102i −0.261753 − 0.965135i \(-0.584301\pi\)
0.966708 − 0.255883i \(-0.0823661\pi\)
\(60\) 97.5455 − 168.954i 0.209884 − 0.363530i
\(61\) −173.918 − 301.235i −0.365048 − 0.632282i 0.623735 − 0.781635i \(-0.285614\pi\)
−0.988784 + 0.149353i \(0.952281\pi\)
\(62\) 98.5075 − 170.620i 0.201782 − 0.349496i
\(63\) 849.681 + 1471.69i 1.69920 + 2.94311i
\(64\) 31.7608 0.0620329
\(65\) −76.8761 −0.146697
\(66\) −966.383 − 1673.83i −1.80233 − 3.12172i
\(67\) 352.011 + 609.702i 0.641866 + 1.11174i 0.985016 + 0.172464i \(0.0551729\pi\)
−0.343150 + 0.939281i \(0.611494\pi\)
\(68\) 145.908 0.260206
\(69\) −309.655 −0.540262
\(70\) −254.397 − 440.628i −0.434375 − 0.752360i
\(71\) −109.195 + 189.132i −0.182523 + 0.316138i −0.942739 − 0.333532i \(-0.891760\pi\)
0.760216 + 0.649670i \(0.225093\pi\)
\(72\) −385.516 − 667.733i −0.631020 − 1.09296i
\(73\) −34.6550 + 60.0242i −0.0555625 + 0.0962370i −0.892469 − 0.451109i \(-0.851029\pi\)
0.836906 + 0.547346i \(0.184362\pi\)
\(74\) −588.837 + 1019.89i −0.925012 + 1.60217i
\(75\) −231.009 −0.355662
\(76\) −265.191 − 227.972i −0.400256 − 0.344081i
\(77\) −1741.40 −2.57729
\(78\) 248.349 − 430.152i 0.360512 − 0.624425i
\(79\) −272.464 + 471.921i −0.388032 + 0.672092i −0.992185 − 0.124777i \(-0.960178\pi\)
0.604152 + 0.796869i \(0.293512\pi\)
\(80\) 199.876 + 346.196i 0.279335 + 0.483823i
\(81\) −551.672 + 955.524i −0.756752 + 1.31073i
\(82\) 182.705 + 316.455i 0.246054 + 0.426178i
\(83\) −1272.58 −1.68293 −0.841466 − 0.540310i \(-0.818307\pi\)
−0.841466 + 0.540310i \(0.818307\pi\)
\(84\) 1135.68 1.47516
\(85\) −86.3858 − 149.625i −0.110234 − 0.190930i
\(86\) 459.864 + 796.508i 0.576609 + 0.998717i
\(87\) 2216.26 2.73113
\(88\) 790.106 0.957109
\(89\) −604.779 − 1047.51i −0.720298 − 1.24759i −0.960880 − 0.276964i \(-0.910672\pi\)
0.240583 − 0.970629i \(-0.422661\pi\)
\(90\) 510.290 − 883.848i 0.597659 − 1.03518i
\(91\) −223.760 − 387.563i −0.257762 − 0.446458i
\(92\) −70.7518 + 122.546i −0.0801780 + 0.138872i
\(93\) 260.362 − 450.959i 0.290304 − 0.502821i
\(94\) 836.127 0.917445
\(95\) −76.7704 + 406.917i −0.0829103 + 0.439461i
\(96\) −1606.57 −1.70802
\(97\) −529.365 + 916.887i −0.554112 + 0.959751i 0.443860 + 0.896096i \(0.353609\pi\)
−0.997972 + 0.0636544i \(0.979724\pi\)
\(98\) 881.345 − 1526.53i 0.908462 − 1.57350i
\(99\) −1746.52 − 3025.07i −1.77305 − 3.07102i
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.11.7 ✓ 18
19.7 even 3 inner 95.4.e.b.26.7 yes 18
19.8 odd 6 1805.4.a.o.1.7 9
19.11 even 3 1805.4.a.n.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.e.b.11.7 ✓ 18 1.1 even 1 trivial
95.4.e.b.26.7 yes 18 19.7 even 3 inner
1805.4.a.n.1.3 9 19.11 even 3
1805.4.a.o.1.7 9 19.8 odd 6