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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(4,95)] 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("95.4"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([9, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 95.p (of order \(18\), degree \(6\), 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: \(5.60518145055\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(28\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 4.13
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.13

$q$-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+(-0.163864 + 0.450212i) q^{2} +(2.51249 - 0.443020i) q^{3} +(5.95252 + 4.99475i) q^{4} +(-11.1258 + 1.10321i) q^{5} +(-0.212253 + 1.20375i) q^{6} +(24.1826 + 13.9618i) q^{7} +(-6.54343 + 3.77785i) q^{8} +(-19.2554 + 7.00838i) q^{9} +(1.32643 - 5.18973i) q^{10} +(4.89800 + 8.48359i) q^{11} +(17.1684 + 9.91219i) q^{12} +(30.1669 + 5.31924i) q^{13} +(-10.2484 + 8.59946i) q^{14} +(-27.4647 + 7.70076i) q^{15} +(10.1660 + 57.6543i) q^{16} +(22.2877 - 61.2349i) q^{17} -9.81740i q^{18} +(8.03149 + 82.4287i) q^{19} +(-71.7367 - 49.0036i) q^{20} +(66.9440 + 24.3656i) q^{21} +(-4.62201 + 0.814986i) q^{22} +(-15.3185 + 18.2559i) q^{23} +(-14.7667 + 12.3907i) q^{24} +(122.566 - 24.5482i) q^{25} +(-7.33803 + 12.7098i) q^{26} +(-104.929 + 60.5809i) q^{27} +(74.2115 + 203.894i) q^{28} +(83.3092 - 30.3221i) q^{29} +(1.03349 - 13.6268i) q^{30} +(31.2215 - 54.0772i) q^{31} +(-87.1498 - 15.3669i) q^{32} +(16.0646 + 19.1450i) q^{33} +(23.9165 + 20.0683i) q^{34} +(-284.454 - 128.658i) q^{35} +(-149.623 - 54.4583i) q^{36} -389.973i q^{37} +(-38.4264 - 9.89120i) q^{38} +78.1505 q^{39} +(68.6330 - 49.2504i) q^{40} +(-82.0866 - 465.536i) q^{41} +(-21.9394 + 26.1463i) q^{42} +(-287.154 - 342.216i) q^{43} +(-13.2180 + 74.9630i) q^{44} +(206.499 - 99.2165i) q^{45} +(-5.70886 - 9.88803i) q^{46} +(66.2842 + 182.114i) q^{47} +(51.0840 + 140.352i) q^{48} +(218.367 + 378.222i) q^{49} +(-9.03217 + 59.2031i) q^{50} +(28.8693 - 163.726i) q^{51} +(153.001 + 182.339i) q^{52} +(135.989 - 162.066i) q^{53} +(-10.0801 - 57.1673i) q^{54} +(-63.8533 - 88.9829i) q^{55} -210.983 q^{56} +(56.6966 + 203.543i) q^{57} +42.4754i q^{58} +(571.777 + 208.110i) q^{59} +(-201.947 - 91.3404i) q^{60} +(-201.815 - 169.343i) q^{61} +(19.2301 + 22.9176i) q^{62} +(-563.495 - 99.3594i) q^{63} +(-212.976 + 368.885i) q^{64} +(-341.498 - 25.9001i) q^{65} +(-11.2517 + 4.09529i) q^{66} +(-11.3382 - 31.1515i) q^{67} +(438.521 - 253.180i) q^{68} +(-30.3998 + 52.6541i) q^{69} +(104.535 - 106.982i) q^{70} +(-125.395 + 105.219i) q^{71} +(99.5196 - 118.603i) q^{72} +(-271.059 + 47.7950i) q^{73} +(175.570 + 63.9024i) q^{74} +(297.070 - 115.976i) q^{75} +(-363.904 + 530.774i) q^{76} +273.541i q^{77} +(-12.8060 + 35.1843i) q^{78} +(93.0813 + 527.890i) q^{79} +(-176.710 - 630.233i) q^{80} +(187.027 - 156.934i) q^{81} +(223.041 + 39.3281i) q^{82} +(883.408 + 510.036i) q^{83} +(276.785 + 479.406i) q^{84} +(-180.413 + 705.874i) q^{85} +(201.124 - 73.2031i) q^{86} +(195.880 - 113.091i) q^{87} +(-64.0995 - 37.0079i) q^{88} +(-200.632 + 1137.84i) q^{89} +(10.8307 + 109.226i) q^{90} +(655.248 + 549.819i) q^{91} +(-182.367 + 32.1562i) q^{92} +(54.4864 - 149.700i) q^{93} -92.8516 q^{94} +(-180.293 - 908.223i) q^{95} -225.771 q^{96} +(246.073 - 676.079i) q^{97} +(-206.062 + 36.3343i) q^{98} +(-153.769 - 129.027i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 168 q + 6 q^{4} - 6 q^{5} + 6 q^{6} - 12 q^{9} + 3 q^{10} - 90 q^{11} + 162 q^{14} - 72 q^{15} + 426 q^{16} - 444 q^{19} - 966 q^{20} + 114 q^{21} + 108 q^{24} + 270 q^{25} - 642 q^{26} - 60 q^{29} + 396 q^{30}+ \cdots + 22692 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.163864 + 0.450212i −0.0579345 + 0.159174i −0.965284 − 0.261204i \(-0.915880\pi\)
0.907349 + 0.420378i \(0.138103\pi\)
\(3\) 2.51249 − 0.443020i 0.483529 − 0.0852592i 0.0734303 − 0.997300i \(-0.476605\pi\)
0.410099 + 0.912041i \(0.365494\pi\)
\(4\) 5.95252 + 4.99475i 0.744065 + 0.624344i
\(5\) −11.1258 + 1.10321i −0.995120 + 0.0986745i
\(6\) −0.212253 + 1.20375i −0.0144420 + 0.0819046i
\(7\) 24.1826 + 13.9618i 1.30574 + 0.753869i 0.981382 − 0.192066i \(-0.0615186\pi\)
0.324357 + 0.945935i \(0.394852\pi\)
\(8\) −6.54343 + 3.77785i −0.289182 + 0.166959i
\(9\) −19.2554 + 7.00838i −0.713161 + 0.259570i
\(10\) 1.32643 − 5.18973i 0.0419454 − 0.164114i
\(11\) 4.89800 + 8.48359i 0.134255 + 0.232536i 0.925313 − 0.379205i \(-0.123803\pi\)
−0.791058 + 0.611741i \(0.790469\pi\)
\(12\) 17.1684 + 9.91219i 0.413008 + 0.238450i
\(13\) 30.1669 + 5.31924i 0.643599 + 0.113484i 0.485915 − 0.874006i \(-0.338486\pi\)
0.157684 + 0.987490i \(0.449597\pi\)
\(14\) −10.2484 + 8.59946i −0.195644 + 0.164164i
\(15\) −27.4647 + 7.70076i −0.472756 + 0.132555i
\(16\) 10.1660 + 57.6543i 0.158844 + 0.900848i
\(17\) 22.2877 − 61.2349i 0.317974 − 0.873627i −0.673008 − 0.739635i \(-0.734998\pi\)
0.990982 − 0.133992i \(-0.0427795\pi\)
\(18\) − 9.81740i − 0.128555i
\(19\) 8.03149 + 82.4287i 0.0969763 + 0.995287i
\(20\) −71.7367 − 49.0036i −0.802040 − 0.547877i
\(21\) 66.9440 + 24.3656i 0.695637 + 0.253191i
\(22\) −4.62201 + 0.814986i −0.0447917 + 0.00789798i
\(23\) −15.3185 + 18.2559i −0.138875 + 0.165505i −0.830999 − 0.556274i \(-0.812231\pi\)
0.692124 + 0.721778i \(0.256675\pi\)
\(24\) −14.7667 + 12.3907i −0.125593 + 0.105385i
\(25\) 122.566 − 24.5482i 0.980527 − 0.196386i
\(26\) −7.33803 + 12.7098i −0.0553503 + 0.0958695i
\(27\) −104.929 + 60.5809i −0.747912 + 0.431807i
\(28\) 74.2115 + 203.894i 0.500881 + 1.37616i
\(29\) 83.3092 − 30.3221i 0.533453 − 0.194161i −0.0612267 − 0.998124i \(-0.519501\pi\)
0.594679 + 0.803963i \(0.297279\pi\)
\(30\) 1.03349 − 13.6268i 0.00628961 − 0.0829299i
\(31\) 31.2215 − 54.0772i 0.180889 − 0.313308i −0.761295 − 0.648406i \(-0.775436\pi\)
0.942183 + 0.335098i \(0.108769\pi\)
\(32\) −87.1498 − 15.3669i −0.481439 − 0.0848907i
\(33\) 16.0646 + 19.1450i 0.0847419 + 0.100992i
\(34\) 23.9165 + 20.0683i 0.120637 + 0.101226i
\(35\) −284.454 − 128.658i −1.37375 − 0.621347i
\(36\) −149.623 − 54.4583i −0.692699 − 0.252122i
\(37\) − 389.973i − 1.73273i −0.499408 − 0.866367i \(-0.666449\pi\)
0.499408 − 0.866367i \(-0.333551\pi\)
\(38\) −38.4264 − 9.89120i −0.164042 − 0.0422254i
\(39\) 78.1505 0.320874
\(40\) 68.6330 − 49.2504i 0.271296 − 0.194679i
\(41\) −82.0866 − 465.536i −0.312677 − 1.77328i −0.584958 − 0.811063i \(-0.698889\pi\)
0.272281 − 0.962218i \(-0.412222\pi\)
\(42\) −21.9394 + 26.1463i −0.0806028 + 0.0960587i
\(43\) −287.154 − 342.216i −1.01838 − 1.21366i −0.976717 − 0.214531i \(-0.931178\pi\)
−0.0416671 − 0.999132i \(-0.513267\pi\)
\(44\) −13.2180 + 74.9630i −0.0452884 + 0.256843i
\(45\) 206.499 − 99.2165i 0.684068 − 0.328674i
\(46\) −5.70886 − 9.88803i −0.0182984 − 0.0316937i
\(47\) 66.2842 + 182.114i 0.205714 + 0.565194i 0.999049 − 0.0435905i \(-0.0138797\pi\)
−0.793336 + 0.608784i \(0.791657\pi\)
\(48\) 51.0840 + 140.352i 0.153611 + 0.422043i
\(49\) 218.367 + 378.222i 0.636637 + 1.10269i
\(50\) −9.03217 + 59.2031i −0.0255468 + 0.167452i
\(51\) 28.8693 − 163.726i 0.0792650 − 0.449534i
\(52\) 153.001 + 182.339i 0.408026 + 0.486267i
\(53\) 135.989 − 162.066i 0.352445 − 0.420028i −0.560472 − 0.828174i \(-0.689380\pi\)
0.912917 + 0.408146i \(0.133825\pi\)
\(54\) −10.0801 − 57.1673i −0.0254025 − 0.144065i
\(55\) −63.8533 − 88.9829i −0.156545 − 0.218154i
\(56\) −210.983 −0.503461
\(57\) 56.6966 + 203.543i 0.131748 + 0.472982i
\(58\) 42.4754i 0.0961603i
\(59\) 571.777 + 208.110i 1.26168 + 0.459213i 0.884332 − 0.466859i \(-0.154614\pi\)
0.377347 + 0.926072i \(0.376837\pi\)
\(60\) −201.947 − 91.3404i −0.434521 − 0.196533i
\(61\) −201.815 − 169.343i −0.423602 − 0.355445i 0.405929 − 0.913905i \(-0.366948\pi\)
−0.829531 + 0.558460i \(0.811392\pi\)
\(62\) 19.2301 + 22.9176i 0.0393908 + 0.0469441i
\(63\) −563.495 − 99.3594i −1.12688 − 0.198700i
\(64\) −212.976 + 368.885i −0.415968 + 0.720478i
\(65\) −341.498 − 25.9001i −0.651656 − 0.0494232i
\(66\) −11.2517 + 4.09529i −0.0209847 + 0.00763780i
\(67\) −11.3382 − 31.1515i −0.0206744 − 0.0568025i 0.928926 − 0.370265i \(-0.120733\pi\)
−0.949601 + 0.313463i \(0.898511\pi\)
\(68\) 438.521 − 253.180i 0.782037 − 0.451509i
\(69\) −30.3998 + 52.6541i −0.0530393 + 0.0918667i
\(70\) 104.535 − 106.982i 0.178490 − 0.182668i
\(71\) −125.395 + 105.219i −0.209601 + 0.175876i −0.741544 − 0.670904i \(-0.765906\pi\)
0.531943 + 0.846780i \(0.321462\pi\)
\(72\) 99.5196 − 118.603i 0.162896 − 0.194132i
\(73\) −271.059 + 47.7950i −0.434590 + 0.0766299i −0.386663 − 0.922221i \(-0.626372\pi\)
−0.0479266 + 0.998851i \(0.515261\pi\)
\(74\) 175.570 + 63.9024i 0.275806 + 0.100385i
\(75\) 297.070 − 115.976i 0.457369 − 0.178557i
\(76\) −363.904 + 530.774i −0.549245 + 0.801104i
\(77\) 273.541i 0.404842i
\(78\) −12.8060 + 35.1843i −0.0185897 + 0.0510748i
\(79\) 93.0813 + 527.890i 0.132563 + 0.751801i 0.976526 + 0.215400i \(0.0691055\pi\)
−0.843963 + 0.536401i \(0.819783\pi\)
\(80\) −176.710 − 630.233i −0.246959 − 0.880778i
\(81\) 187.027 − 156.934i 0.256553 − 0.215274i
\(82\) 223.041 + 39.3281i 0.300375 + 0.0529642i
\(83\) 883.408 + 510.036i 1.16827 + 0.674503i 0.953273 − 0.302111i \(-0.0976915\pi\)
0.215000 + 0.976614i \(0.431025\pi\)
\(84\) 276.785 + 479.406i 0.359520 + 0.622708i
\(85\) −180.413 + 705.874i −0.230218 + 0.900739i
\(86\) 201.124 − 73.2031i 0.252183 − 0.0917871i
\(87\) 195.880 − 113.091i 0.241386 − 0.139364i
\(88\) −64.0995 − 37.0079i −0.0776481 − 0.0448301i
\(89\) −200.632 + 1137.84i −0.238954 + 1.35518i 0.595169 + 0.803601i \(0.297085\pi\)
−0.834124 + 0.551578i \(0.814026\pi\)
\(90\) 10.8307 + 109.226i 0.0126851 + 0.127927i
\(91\) 655.248 + 549.819i 0.754821 + 0.633370i
\(92\) −182.367 + 32.1562i −0.206664 + 0.0364404i
\(93\) 54.4864 − 149.700i 0.0607525 − 0.166916i
\(94\) −92.8516 −0.101882
\(95\) −180.293 − 908.223i −0.194712 − 0.980860i
\(96\) −225.771 −0.240028
\(97\) 246.073 − 676.079i 0.257576 − 0.707685i −0.741739 − 0.670688i \(-0.765999\pi\)
0.999315 − 0.0369963i \(-0.0117790\pi\)
\(98\) −206.062 + 36.3343i −0.212402 + 0.0374523i
\(99\) −153.769 − 129.027i −0.156105 − 0.130987i
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.p.a.4.13 ✓ 168
5.4 even 2 inner 95.4.p.a.4.16 yes 168
19.5 even 9 inner 95.4.p.a.24.16 yes 168
95.24 even 18 inner 95.4.p.a.24.13 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.13 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.16 yes 168 5.4 even 2 inner
95.4.p.a.24.13 yes 168 95.24 even 18 inner
95.4.p.a.24.16 yes 168 19.5 even 9 inner