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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 4.16
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.16

$q$-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} +(-3.01843 + 10.7652i) 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} +(4.35200 + 3.12295i) 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} +(2.81458 - 28.3846i) 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} +(-106.778 - 64.9878i) 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} +(-12.3179 - 5.91836i) 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} +(223.295 - 218.188i) q^{35} +(-149.623 - 54.4583i) q^{36} +389.973i q^{37} +(38.4264 + 9.89120i) q^{38} +78.1505 q^{39} +(20.9184 + 81.8444i) 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} +(-17.3256 - 228.442i) 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} +(-46.7553 + 37.4236i) 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} +(-106.112 + 27.1208i) q^{55} -210.983 q^{56} +(-56.6966 - 203.543i) q^{57} -42.4754i q^{58} +(571.777 + 208.110i) q^{59} +(158.528 - 154.902i) 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} +(148.319 - 308.696i) 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} +(-61.6406 - 136.283i) 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} +(-651.344 - 64.5863i) 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} +(-591.931 - 424.764i) 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} +(-105.686 - 29.6331i) q^{90} +(655.248 + 549.819i) q^{91} +(182.367 - 32.1562i) q^{92} +(-54.4864 + 149.700i) q^{93} -92.8516 q^{94} +(-911.603 - 162.345i) 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 −0.907349 − 0.420378i \(-0.861897\pi\)
0.965284 + 0.261204i \(0.0841196\pi\)
\(3\) −2.51249 + 0.443020i −0.483529 + 0.0852592i −0.410099 − 0.912041i \(-0.634506\pi\)
−0.0734303 + 0.997300i \(0.523395\pi\)
\(4\) 5.95252 + 4.99475i 0.744065 + 0.624344i
\(5\) −3.01843 + 10.7652i −0.269976 + 0.962867i
\(6\) −0.212253 + 1.20375i −0.0144420 + 0.0819046i
\(7\) −24.1826 − 13.9618i −1.30574 − 0.753869i −0.324357 − 0.945935i \(-0.605148\pi\)
−0.981382 + 0.192066i \(0.938481\pi\)
\(8\) 6.54343 − 3.77785i 0.289182 − 0.166959i
\(9\) −19.2554 + 7.00838i −0.713161 + 0.259570i
\(10\) 4.35200 + 3.12295i 0.137622 + 0.0987564i
\(11\) 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 −0.157684 − 0.987490i \(-0.550403\pi\)
−0.485915 + 0.874006i \(0.661514\pi\)
\(14\) −10.2484 + 8.59946i −0.195644 + 0.164164i
\(15\) 2.81458 − 28.3846i 0.0484480 − 0.488592i
\(16\) 10.1660 + 57.6543i 0.158844 + 0.900848i
\(17\) −22.2877 + 61.2349i −0.317974 + 0.873627i 0.673008 + 0.739635i \(0.265002\pi\)
−0.990982 + 0.133992i \(0.957220\pi\)
\(18\) 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 −0.692124 − 0.721778i \(-0.743325\pi\)
0.830999 + 0.556274i \(0.187769\pi\)
\(24\) −14.7667 + 12.3907i −0.125593 + 0.105385i
\(25\) −106.778 − 64.9878i −0.854226 − 0.519902i
\(26\) −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 \(0.297279\pi\)
\(30\) −12.3179 − 5.91836i −0.0749642 − 0.0360180i
\(31\) 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
\(35\) 223.295 − 218.188i 1.07839 − 1.05373i
\(36\) −149.623 − 54.4583i −0.692699 − 0.252122i
\(37\) 389.973i 1.73273i 0.499408 + 0.866367i \(0.333551\pi\)
−0.499408 + 0.866367i \(0.666449\pi\)
\(38\) 38.4264 + 9.89120i 0.164042 + 0.0422254i
\(39\) 78.1505 0.320874
\(40\) 20.9184 + 81.8444i 0.0826873 + 0.323518i
\(41\) −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 \(0.0688224\pi\)
0.0416671 + 0.999132i \(0.486733\pi\)
\(44\) −13.2180 + 74.9630i −0.0452884 + 0.256843i
\(45\) −17.3256 − 228.442i −0.0573943 − 0.756757i
\(46\) −5.70886 − 9.88803i −0.0182984 − 0.0316937i
\(47\) −66.2842 − 182.114i −0.205714 − 0.565194i 0.793336 − 0.608784i \(-0.208343\pi\)
−0.999049 + 0.0435905i \(0.986120\pi\)
\(48\) −51.0840 − 140.352i −0.153611 − 0.422043i
\(49\) 218.367 + 378.222i 0.636637 + 1.10269i
\(50\) −46.7553 + 37.4236i −0.132244 + 0.105850i
\(51\) 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 −0.912917 − 0.408146i \(-0.866175\pi\)
0.560472 + 0.828174i \(0.310620\pi\)
\(54\) −10.0801 − 57.1673i −0.0254025 − 0.144065i
\(55\) −106.112 + 27.1208i −0.260147 + 0.0664903i
\(56\) −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 \(0.376837\pi\)
\(60\) 158.528 − 154.902i 0.341098 − 0.333296i
\(61\) −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
\(65\) 148.319 − 308.696i 0.283026 − 0.589062i
\(66\) −11.2517 + 4.09529i −0.0209847 + 0.00763780i
\(67\) 11.3382 + 31.1515i 0.0206744 + 0.0568025i 0.949601 − 0.313463i \(-0.101489\pi\)
−0.928926 + 0.370265i \(0.879267\pi\)
\(68\) −438.521 + 253.180i −0.782037 + 0.451509i
\(69\) −30.3998 + 52.6541i −0.0530393 + 0.0918667i
\(70\) −61.6406 − 136.283i −0.105249 − 0.232699i
\(71\) −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 0.0479266 − 0.998851i \(-0.484739\pi\)
0.386663 + 0.922221i \(0.373628\pi\)
\(74\) 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 \(0.819783\pi\)
\(80\) −651.344 − 64.5863i −0.910281 − 0.0902621i
\(81\) 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 −0.215000 − 0.976614i \(-0.568975\pi\)
−0.953273 + 0.302111i \(0.902309\pi\)
\(84\) 276.785 + 479.406i 0.359520 + 0.622708i
\(85\) −591.931 − 424.764i −0.755341 − 0.542025i
\(86\) 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 \(0.814026\pi\)
\(90\) −105.686 − 29.6331i −0.123781 − 0.0347067i
\(91\) 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
\(95\) −911.603 − 162.345i −0.984510 − 0.175328i
\(96\) −225.771 −0.240028
\(97\) −246.073 + 676.079i −0.257576 + 0.707685i 0.741739 + 0.670688i \(0.234001\pi\)
−0.999315 + 0.0369963i \(0.988221\pi\)
\(98\) 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.4.16 yes 168
5.4 even 2 inner 95.4.p.a.4.13 ✓ 168
19.5 even 9 inner 95.4.p.a.24.13 yes 168
95.24 even 18 inner 95.4.p.a.24.16 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.13 ✓ 168 5.4 even 2 inner
95.4.p.a.4.16 yes 168 1.1 even 1 trivial
95.4.p.a.24.13 yes 168 19.5 even 9 inner
95.4.p.a.24.16 yes 168 95.24 even 18 inner