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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 24.6
Character \(\chi\) \(=\) 95.24
Dual form 95.4.p.a.4.6

$q$-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+(-1.17190 - 3.21976i) q^{2} +(-1.04177 - 0.183692i) q^{3} +(-2.86517 + 2.40416i) q^{4} +(8.45885 + 7.31080i) q^{5} +(0.629403 + 3.56952i) q^{6} +(-30.3869 + 17.5439i) q^{7} +(-12.6403 - 7.29787i) q^{8} +(-24.3202 - 8.85181i) q^{9} +(13.6261 - 35.8030i) q^{10} +(-22.7759 + 39.4489i) q^{11} +(3.42647 - 1.97827i) q^{12} +(32.5306 - 5.73602i) q^{13} +(92.0975 + 77.2790i) q^{14} +(-7.46924 - 9.17000i) q^{15} +(-13.8801 + 78.7182i) q^{16} +(20.5569 + 56.4797i) q^{17} +88.6785i q^{18} +(-6.63144 - 82.5532i) q^{19} +(-41.8123 - 0.610227i) q^{20} +(34.8789 - 12.6949i) q^{21} +(153.707 + 27.1027i) q^{22} +(-79.8017 - 95.1039i) q^{23} +(11.8277 + 9.92463i) q^{24} +(18.1044 + 123.682i) q^{25} +(-56.5911 - 98.0187i) q^{26} +(48.4452 + 27.9698i) q^{27} +(44.8852 - 123.321i) q^{28} +(-67.2479 - 24.4762i) q^{29} +(-20.7720 + 34.7955i) q^{30} +(-63.1459 - 109.372i) q^{31} +(154.728 - 27.2827i) q^{32} +(30.9737 - 36.9130i) q^{33} +(157.761 - 132.377i) q^{34} +(-385.299 - 73.7515i) q^{35} +(90.9625 - 33.1076i) q^{36} +344.348i q^{37} +(-258.030 + 118.095i) q^{38} -34.9430 q^{39} +(-53.5690 - 154.142i) q^{40} +(-0.865889 + 4.91070i) q^{41} +(-81.7489 - 97.4246i) q^{42} +(-58.5206 + 69.7422i) q^{43} +(-29.5849 - 167.785i) q^{44} +(-141.007 - 252.676i) q^{45} +(-212.693 + 368.394i) q^{46} +(32.6977 - 89.8363i) q^{47} +(28.9198 - 79.4566i) q^{48} +(444.077 - 769.164i) q^{49} +(377.010 - 203.234i) q^{50} +(-11.0407 - 62.6151i) q^{51} +(-79.4152 + 94.6433i) q^{52} +(-200.720 - 239.209i) q^{53} +(33.2834 - 188.760i) q^{54} +(-481.061 + 167.183i) q^{55} +512.133 q^{56} +(-8.25593 + 87.2196i) q^{57} +245.206i q^{58} +(-194.872 + 70.9278i) q^{59} +(43.4468 + 8.31632i) q^{60} +(-209.212 + 175.550i) q^{61} +(-278.151 + 331.488i) q^{62} +(894.310 - 157.691i) q^{63} +(50.5613 + 87.5747i) q^{64} +(317.106 + 189.304i) q^{65} +(-155.149 - 56.4696i) q^{66} +(-150.415 + 413.261i) q^{67} +(-194.685 - 112.402i) q^{68} +(65.6652 + 113.735i) q^{69} +(214.068 + 1327.00i) q^{70} +(-604.959 - 507.621i) q^{71} +(242.814 + 289.375i) q^{72} +(425.522 + 75.0311i) q^{73} +(1108.72 - 403.541i) q^{74} +(3.85882 - 132.174i) q^{75} +(217.471 + 220.585i) q^{76} -1598.31i q^{77} +(40.9497 + 112.508i) q^{78} +(-186.620 + 1058.37i) q^{79} +(-692.903 + 564.390i) q^{80} +(489.970 + 411.134i) q^{81} +(16.8260 - 2.96688i) q^{82} +(-312.556 + 180.455i) q^{83} +(-69.4133 + 120.227i) q^{84} +(-239.024 + 628.041i) q^{85} +(293.133 + 106.692i) q^{86} +(65.5608 + 37.8515i) q^{87} +(575.787 - 332.431i) q^{88} +(-242.212 - 1373.65i) q^{89} +(-648.311 + 750.119i) q^{90} +(-887.872 + 745.013i) q^{91} +(457.290 + 80.6326i) q^{92} +(45.6928 + 125.540i) q^{93} -327.570 q^{94} +(547.435 - 746.786i) q^{95} -166.202 q^{96} +(-340.020 - 934.196i) q^{97} +(-2996.94 - 528.441i) q^{98} +(903.107 - 757.797i) 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 \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.17190 − 3.21976i −0.414328 − 1.13836i −0.954866 − 0.297038i \(-0.904001\pi\)
0.540537 − 0.841320i \(-0.318221\pi\)
\(3\) −1.04177 − 0.183692i −0.200489 − 0.0353516i 0.0725018 − 0.997368i \(-0.476902\pi\)
−0.272991 + 0.962017i \(0.588013\pi\)
\(4\) −2.86517 + 2.40416i −0.358146 + 0.300520i
\(5\) 8.45885 + 7.31080i 0.756583 + 0.653898i
\(6\) 0.629403 + 3.56952i 0.0428254 + 0.242875i
\(7\) −30.3869 + 17.5439i −1.64074 + 0.947282i −0.660169 + 0.751117i \(0.729515\pi\)
−0.980571 + 0.196164i \(0.937151\pi\)
\(8\) −12.6403 − 7.29787i −0.558627 − 0.322523i
\(9\) −24.3202 − 8.85181i −0.900747 − 0.327845i
\(10\) 13.6261 − 35.8030i 0.430896 − 1.13219i
\(11\) −22.7759 + 39.4489i −0.624289 + 1.08130i 0.364389 + 0.931247i \(0.381278\pi\)
−0.988678 + 0.150053i \(0.952055\pi\)
\(12\) 3.42647 − 1.97827i 0.0824280 − 0.0475899i
\(13\) 32.5306 − 5.73602i 0.694028 − 0.122376i 0.184503 − 0.982832i \(-0.440932\pi\)
0.509525 + 0.860456i \(0.329821\pi\)
\(14\) 92.0975 + 77.2790i 1.75815 + 1.47526i
\(15\) −7.46924 − 9.17000i −0.128570 − 0.157846i
\(16\) −13.8801 + 78.7182i −0.216877 + 1.22997i
\(17\) 20.5569 + 56.4797i 0.293282 + 0.805785i 0.995581 + 0.0939036i \(0.0299345\pi\)
−0.702299 + 0.711882i \(0.747843\pi\)
\(18\) 88.6785i 1.16121i
\(19\) −6.63144 − 82.5532i −0.0800714 − 0.996789i
\(20\) −41.8123 − 0.610227i −0.467476 − 0.00682254i
\(21\) 34.8789 − 12.6949i 0.362438 − 0.131917i
\(22\) 153.707 + 27.1027i 1.48957 + 0.262651i
\(23\) −79.8017 − 95.1039i −0.723470 − 0.862197i 0.271493 − 0.962440i \(-0.412482\pi\)
−0.994963 + 0.100243i \(0.968038\pi\)
\(24\) 11.8277 + 9.92463i 0.100597 + 0.0844107i
\(25\) 18.1044 + 123.682i 0.144835 + 0.989456i
\(26\) −56.5911 − 98.0187i −0.426863 − 0.739348i
\(27\) 48.4452 + 27.9698i 0.345307 + 0.199363i
\(28\) 44.8852 − 123.321i 0.302947 − 0.832340i
\(29\) −67.2479 − 24.4762i −0.430608 − 0.156728i 0.117618 − 0.993059i \(-0.462474\pi\)
−0.548226 + 0.836331i \(0.684696\pi\)
\(30\) −20.7720 + 34.7955i −0.126415 + 0.211759i
\(31\) −63.1459 − 109.372i −0.365850 − 0.633670i 0.623062 − 0.782172i \(-0.285888\pi\)
−0.988912 + 0.148502i \(0.952555\pi\)
\(32\) 154.728 − 27.2827i 0.854758 − 0.150717i
\(33\) 30.9737 − 36.9130i 0.163389 − 0.194719i
\(34\) 157.761 − 132.377i 0.795757 − 0.667719i
\(35\) −385.299 − 73.7515i −1.86078 − 0.356179i
\(36\) 90.9625 − 33.1076i 0.421122 − 0.153276i
\(37\) 344.348i 1.53001i 0.644022 + 0.765007i \(0.277265\pi\)
−0.644022 + 0.765007i \(0.722735\pi\)
\(38\) −258.030 + 118.095i −1.10153 + 0.504148i
\(39\) −34.9430 −0.143471
\(40\) −53.5690 − 154.142i −0.211750 − 0.609301i
\(41\) −0.865889 + 4.91070i −0.00329827 + 0.0187054i −0.986412 − 0.164288i \(-0.947467\pi\)
0.983114 + 0.182993i \(0.0585786\pi\)
\(42\) −81.7489 − 97.4246i −0.300336 − 0.357927i
\(43\) −58.5206 + 69.7422i −0.207542 + 0.247339i −0.859767 − 0.510686i \(-0.829391\pi\)
0.652225 + 0.758025i \(0.273836\pi\)
\(44\) −29.5849 − 167.785i −0.101366 − 0.574874i
\(45\) −141.007 − 252.676i −0.467112 − 0.837038i
\(46\) −212.693 + 368.394i −0.681735 + 1.18080i
\(47\) 32.6977 − 89.8363i 0.101478 − 0.278808i −0.878556 − 0.477640i \(-0.841493\pi\)
0.980034 + 0.198832i \(0.0637147\pi\)
\(48\) 28.9198 − 79.4566i 0.0869628 − 0.238928i
\(49\) 444.077 − 769.164i 1.29468 − 2.24246i
\(50\) 377.010 − 203.234i 1.06635 − 0.574834i
\(51\) −11.0407 − 62.6151i −0.0303139 − 0.171919i
\(52\) −79.4152 + 94.6433i −0.211787 + 0.252397i
\(53\) −200.720 − 239.209i −0.520208 − 0.619959i 0.440422 − 0.897791i \(-0.354829\pi\)
−0.960630 + 0.277831i \(0.910384\pi\)
\(54\) 33.2834 − 188.760i 0.0838759 − 0.475684i
\(55\) −481.061 + 167.183i −1.17939 + 0.409872i
\(56\) 512.133 1.22208
\(57\) −8.25593 + 87.2196i −0.0191846 + 0.202676i
\(58\) 245.206i 0.555123i
\(59\) −194.872 + 70.9278i −0.430004 + 0.156509i −0.547949 − 0.836511i \(-0.684591\pi\)
0.117946 + 0.993020i \(0.462369\pi\)
\(60\) 43.4468 + 8.31632i 0.0934825 + 0.0178939i
\(61\) −209.212 + 175.550i −0.439129 + 0.368473i −0.835383 − 0.549668i \(-0.814754\pi\)
0.396254 + 0.918141i \(0.370310\pi\)
\(62\) −278.151 + 331.488i −0.569762 + 0.679016i
\(63\) 894.310 − 157.691i 1.78845 − 0.315352i
\(64\) 50.5613 + 87.5747i 0.0987525 + 0.171044i
\(65\) 317.106 + 189.304i 0.605111 + 0.361236i
\(66\) −155.149 − 56.4696i −0.289356 − 0.105317i
\(67\) −150.415 + 413.261i −0.274270 + 0.753550i 0.723715 + 0.690099i \(0.242433\pi\)
−0.997985 + 0.0634514i \(0.979789\pi\)
\(68\) −194.685 − 112.402i −0.347192 − 0.200451i
\(69\) 65.6652 + 113.735i 0.114567 + 0.198437i
\(70\) 214.068 + 1327.00i 0.365515 + 2.26581i
\(71\) −604.959 − 507.621i −1.01120 − 0.848500i −0.0227057 − 0.999742i \(-0.507228\pi\)
−0.988497 + 0.151243i \(0.951673\pi\)
\(72\) 242.814 + 289.375i 0.397444 + 0.473655i
\(73\) 425.522 + 75.0311i 0.682242 + 0.120298i 0.504019 − 0.863693i \(-0.331854\pi\)
0.178223 + 0.983990i \(0.442965\pi\)
\(74\) 1108.72 − 403.541i 1.74170 − 0.633928i
\(75\) 3.85882 − 132.174i 0.00594104 − 0.203495i
\(76\) 217.471 + 220.585i 0.328232 + 0.332933i
\(77\) − 1598.31i − 2.36551i
\(78\) 40.9497 + 112.508i 0.0594441 + 0.163321i
\(79\) −186.620 + 1058.37i −0.265777 + 1.50729i 0.501038 + 0.865425i \(0.332952\pi\)
−0.766814 + 0.641869i \(0.778159\pi\)
\(80\) −692.903 + 564.390i −0.968361 + 0.788759i
\(81\) 489.970 + 411.134i 0.672113 + 0.563970i
\(82\) 16.8260 − 2.96688i 0.0226600 − 0.00399557i
\(83\) −312.556 + 180.455i −0.413344 + 0.238644i −0.692225 − 0.721681i \(-0.743370\pi\)
0.278882 + 0.960325i \(0.410036\pi\)
\(84\) −69.4133 + 120.227i −0.0901620 + 0.156165i
\(85\) −239.024 + 628.041i −0.305009 + 0.801420i
\(86\) 293.133 + 106.692i 0.367551 + 0.133778i
\(87\) 65.5608 + 37.8515i 0.0807914 + 0.0466449i
\(88\) 575.787 − 332.431i 0.697489 − 0.402696i
\(89\) −242.212 − 1373.65i −0.288477 − 1.63603i −0.692594 − 0.721328i \(-0.743532\pi\)
0.404117 − 0.914707i \(-0.367579\pi\)
\(90\) −648.311 + 750.119i −0.759311 + 0.878549i
\(91\) −887.872 + 745.013i −1.02279 + 0.858226i
\(92\) 457.290 + 80.6326i 0.518215 + 0.0913753i
\(93\) 45.6928 + 125.540i 0.0509475 + 0.139977i
\(94\) −327.570 −0.359428
\(95\) 547.435 − 746.786i 0.591218 − 0.806512i
\(96\) −166.202 −0.176697
\(97\) −340.020 − 934.196i −0.355915 − 0.977868i −0.980432 − 0.196857i \(-0.936926\pi\)
0.624517 − 0.781011i \(-0.285296\pi\)
\(98\) −2996.94 − 528.441i −3.08915 − 0.544700i
\(99\) 903.107 − 757.797i 0.916825 − 0.769308i
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.24.6 yes 168
5.4 even 2 inner 95.4.p.a.24.23 yes 168
19.4 even 9 inner 95.4.p.a.4.23 yes 168
95.4 even 18 inner 95.4.p.a.4.6 ✓ 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.6 ✓ 168 95.4 even 18 inner
95.4.p.a.4.23 yes 168 19.4 even 9 inner
95.4.p.a.24.6 yes 168 1.1 even 1 trivial
95.4.p.a.24.23 yes 168 5.4 even 2 inner