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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,4,Mod(2,95)] 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("95.2"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(36)) 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.r (of order \(36\), degree \(12\), 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: \(336\)
Relative dimension: \(28\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 2.6
Character \(\chi\) \(=\) 95.2
Dual form 95.4.r.a.48.6

$q$-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+(-2.17048 - 3.09977i) q^{2} +(-0.0734001 - 0.838967i) q^{3} +(-2.16143 + 5.93847i) q^{4} +(10.3606 - 4.20218i) q^{5} +(-2.44129 + 2.04849i) q^{6} +(4.69640 - 17.5272i) q^{7} +(-6.14222 + 1.64580i) q^{8} +(25.8913 - 4.56534i) q^{9} +(-35.5133 - 22.9947i) q^{10} +(13.7373 - 23.7936i) q^{11} +(5.14083 + 1.37748i) q^{12} +(-59.0628 - 5.16733i) q^{13} +(-64.5238 + 23.4847i) q^{14} +(-4.28595 - 8.38374i) q^{15} +(57.1619 + 47.9645i) q^{16} +(0.588831 - 0.412304i) q^{17} +(-70.3482 - 70.3482i) q^{18} +(-79.6739 + 22.6069i) q^{19} +(2.56086 + 70.6087i) q^{20} +(-15.0495 - 2.65362i) q^{21} +(-103.571 + 9.06132i) q^{22} +(-111.804 + 52.1352i) q^{23} +(1.83161 + 5.03231i) q^{24} +(89.6834 - 87.0740i) q^{25} +(112.177 + 194.297i) q^{26} +(-11.6158 - 43.3507i) q^{27} +(93.9338 + 65.7732i) q^{28} +(-1.45337 - 8.24249i) q^{29} +(-16.6851 + 31.4823i) q^{30} +(100.581 - 58.0704i) q^{31} +(20.1764 - 230.617i) q^{32} +(-20.9704 - 9.77865i) q^{33} +(-2.55609 - 0.930342i) q^{34} +(-24.9950 - 201.327i) q^{35} +(-28.8511 + 163.623i) q^{36} +(296.824 - 296.824i) q^{37} +(243.007 + 197.903i) q^{38} +49.9310i q^{39} +(-56.7210 + 42.8622i) q^{40} +(-176.135 + 209.909i) q^{41} +(24.4390 + 52.4095i) q^{42} +(163.008 - 349.573i) q^{43} +(111.606 + 133.006i) q^{44} +(249.065 - 156.100i) q^{45} +(404.276 + 233.409i) q^{46} +(-28.7351 + 41.0380i) q^{47} +(36.0449 - 51.4775i) q^{48} +(11.9001 + 6.87051i) q^{49} +(-464.566 - 89.0054i) q^{50} +(-0.389129 - 0.463746i) q^{51} +(158.346 - 339.574i) q^{52} +(281.900 + 604.535i) q^{53} +(-109.165 + 130.098i) q^{54} +(42.3410 - 304.242i) q^{55} +115.385i q^{56} +(24.8145 + 65.1844i) q^{57} +(-22.3953 + 22.3953i) q^{58} +(-64.7934 + 367.462i) q^{59} +(59.0504 - 7.33116i) q^{60} +(653.170 + 237.734i) q^{61} +(-398.314 - 185.737i) q^{62} +(41.5784 - 475.243i) q^{63} +(-241.675 + 139.531i) q^{64} +(-633.640 + 194.656i) q^{65} +(15.2043 + 86.2278i) q^{66} +(-157.285 - 110.132i) q^{67} +(1.17574 + 4.38792i) q^{68} +(51.9461 + 89.9733i) q^{69} +(-569.817 + 514.456i) q^{70} +(-211.298 - 580.536i) q^{71} +(-151.517 + 70.6533i) q^{72} +(-788.271 + 68.9648i) q^{73} +(-1564.34 - 275.835i) q^{74} +(-79.6350 - 68.8501i) q^{75} +(37.9587 - 522.004i) q^{76} +(-352.520 - 352.520i) q^{77} +(154.775 - 108.375i) q^{78} +(127.656 + 107.116i) q^{79} +(793.786 + 256.736i) q^{80} +(631.524 - 229.856i) q^{81} +(1032.97 + 90.3731i) q^{82} +(819.194 + 219.502i) q^{83} +(48.2867 - 83.6351i) q^{84} +(4.36806 - 6.74608i) q^{85} +(-1437.40 + 253.453i) q^{86} +(-6.80849 + 1.82433i) q^{87} +(-45.2176 + 168.754i) q^{88} +(593.738 - 498.206i) q^{89} +(-1024.46 - 433.233i) q^{90} +(-367.952 + 1010.94i) q^{91} +(-67.9465 - 776.632i) q^{92} +(-56.1017 - 80.1216i) q^{93} +189.578 q^{94} +(-730.470 + 569.025i) q^{95} -194.961 q^{96} +(397.158 + 567.200i) q^{97} +(-4.53190 - 51.7998i) q^{98} +(247.050 - 678.764i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 336 q - 12 q^{2} - 12 q^{3} - 12 q^{5} + 12 q^{6} + 6 q^{7} - 18 q^{8} - 12 q^{10} - 12 q^{11} - 18 q^{12} - 12 q^{13} - 264 q^{15} - 996 q^{16} + 132 q^{17} + 1116 q^{20} - 600 q^{21} + 468 q^{22} + 564 q^{23}+ \cdots + 1290 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.17048 − 3.09977i −0.767382 − 1.09593i −0.992577 − 0.121621i \(-0.961191\pi\)
0.225195 − 0.974314i \(-0.427698\pi\)
\(3\) −0.0734001 − 0.838967i −0.0141258 − 0.161459i 0.985855 − 0.167599i \(-0.0536013\pi\)
−0.999981 + 0.00613949i \(0.998046\pi\)
\(4\) −2.16143 + 5.93847i −0.270178 + 0.742309i
\(5\) 10.3606 − 4.20218i 0.926679 − 0.375854i
\(6\) −2.44129 + 2.04849i −0.166109 + 0.139382i
\(7\) 4.69640 − 17.5272i 0.253582 − 0.946380i −0.715292 − 0.698825i \(-0.753706\pi\)
0.968874 − 0.247554i \(-0.0796269\pi\)
\(8\) −6.14222 + 1.64580i −0.271450 + 0.0727349i
\(9\) 25.8913 − 4.56534i 0.958938 − 0.169087i
\(10\) −35.5133 − 22.9947i −1.12303 − 0.727156i
\(11\) 13.7373 − 23.7936i 0.376540 − 0.652186i −0.614016 − 0.789293i \(-0.710447\pi\)
0.990556 + 0.137107i \(0.0437804\pi\)
\(12\) 5.14083 + 1.37748i 0.123669 + 0.0331370i
\(13\) −59.0628 − 5.16733i −1.26008 − 0.110243i −0.562531 − 0.826776i \(-0.690172\pi\)
−0.697553 + 0.716533i \(0.745728\pi\)
\(14\) −64.5238 + 23.4847i −1.23176 + 0.448326i
\(15\) −4.28595 − 8.38374i −0.0737752 − 0.144312i
\(16\) 57.1619 + 47.9645i 0.893154 + 0.749445i
\(17\) 0.588831 − 0.412304i 0.00840073 − 0.00588225i −0.569369 − 0.822082i \(-0.692812\pi\)
0.577769 + 0.816200i \(0.303923\pi\)
\(18\) −70.3482 − 70.3482i −0.921180 − 0.921180i
\(19\) −79.6739 + 22.6069i −0.962023 + 0.272968i
\(20\) 2.56086 + 70.6087i 0.0286313 + 0.789429i
\(21\) −15.0495 − 2.65362i −0.156384 − 0.0275747i
\(22\) −103.571 + 9.06132i −1.00370 + 0.0878127i
\(23\) −111.804 + 52.1352i −1.01360 + 0.472649i −0.857167 − 0.515039i \(-0.827777\pi\)
−0.156433 + 0.987689i \(0.550000\pi\)
\(24\) 1.83161 + 5.03231i 0.0155782 + 0.0428007i
\(25\) 89.6834 − 87.0740i 0.717467 − 0.696592i
\(26\) 112.177 + 194.297i 0.846146 + 1.46557i
\(27\) −11.6158 − 43.3507i −0.0827948 − 0.308994i
\(28\) 93.9338 + 65.7732i 0.633994 + 0.443927i
\(29\) −1.45337 − 8.24249i −0.00930636 − 0.0527790i 0.979801 − 0.199975i \(-0.0640862\pi\)
−0.989107 + 0.147196i \(0.952975\pi\)
\(30\) −16.6851 + 31.4823i −0.101542 + 0.191595i
\(31\) 100.581 − 58.0704i 0.582737 − 0.336443i −0.179483 − 0.983761i \(-0.557443\pi\)
0.762220 + 0.647318i \(0.224109\pi\)
\(32\) 20.1764 − 230.617i 0.111460 − 1.27399i
\(33\) −20.9704 − 9.77865i −0.110620 − 0.0515832i
\(34\) −2.55609 − 0.930342i −0.0128931 − 0.00469272i
\(35\) −24.9950 − 201.327i −0.120712 − 0.972300i
\(36\) −28.8511 + 163.623i −0.133570 + 0.757512i
\(37\) 296.824 − 296.824i 1.31885 − 1.31885i 0.404169 − 0.914684i \(-0.367561\pi\)
0.914684 − 0.404169i \(-0.132439\pi\)
\(38\) 243.007 + 197.903i 1.03739 + 0.844844i
\(39\) 49.9310i 0.205009i
\(40\) −56.7210 + 42.8622i −0.224210 + 0.169428i
\(41\) −176.135 + 209.909i −0.670918 + 0.799569i −0.988909 − 0.148525i \(-0.952548\pi\)
0.317990 + 0.948094i \(0.396992\pi\)
\(42\) 24.4390 + 52.4095i 0.0897860 + 0.192547i
\(43\) 163.008 − 349.573i 0.578106 − 1.23975i −0.372119 − 0.928185i \(-0.621369\pi\)
0.950224 − 0.311567i \(-0.100854\pi\)
\(44\) 111.606 + 133.006i 0.382391 + 0.455715i
\(45\) 249.065 − 156.100i 0.825076 − 0.517110i
\(46\) 404.276 + 233.409i 1.29581 + 0.748137i
\(47\) −28.7351 + 41.0380i −0.0891797 + 0.127362i −0.861263 − 0.508160i \(-0.830326\pi\)
0.772083 + 0.635522i \(0.219215\pi\)
\(48\) 36.0449 − 51.4775i 0.108388 − 0.154795i
\(49\) 11.9001 + 6.87051i 0.0346941 + 0.0200306i
\(50\) −464.566 − 89.0054i −1.31399 − 0.251745i
\(51\) −0.389129 − 0.463746i −0.00106841 − 0.00127328i
\(52\) 158.346 − 339.574i 0.422281 − 0.905586i
\(53\) 281.900 + 604.535i 0.730601 + 1.56678i 0.821035 + 0.570877i \(0.193397\pi\)
−0.0904339 + 0.995902i \(0.528825\pi\)
\(54\) −109.165 + 130.098i −0.275102 + 0.327854i
\(55\) 42.3410 − 304.242i 0.103805 − 0.745891i
\(56\) 115.385i 0.275339i
\(57\) 24.8145 + 65.1844i 0.0576625 + 0.151472i
\(58\) −22.3953 + 22.3953i −0.0507008 + 0.0507008i
\(59\) −64.7934 + 367.462i −0.142973 + 0.810838i 0.826000 + 0.563670i \(0.190611\pi\)
−0.968973 + 0.247168i \(0.920500\pi\)
\(60\) 59.0504 − 7.33116i 0.127056 − 0.0157741i
\(61\) 653.170 + 237.734i 1.37098 + 0.498996i 0.919432 − 0.393250i \(-0.128649\pi\)
0.451549 + 0.892246i \(0.350872\pi\)
\(62\) −398.314 − 185.737i −0.815902 − 0.380461i
\(63\) 41.5784 − 475.243i 0.0831490 − 0.950397i
\(64\) −241.675 + 139.531i −0.472021 + 0.272521i
\(65\) −633.640 + 194.656i −1.20913 + 0.371448i
\(66\) 15.2043 + 86.2278i 0.0283563 + 0.160817i
\(67\) −157.285 − 110.132i −0.286797 − 0.200817i 0.421321 − 0.906912i \(-0.361567\pi\)
−0.708118 + 0.706094i \(0.750455\pi\)
\(68\) 1.17574 + 4.38792i 0.00209675 + 0.00782519i
\(69\) 51.9461 + 89.9733i 0.0906315 + 0.156978i
\(70\) −569.817 + 514.456i −0.972945 + 0.878418i
\(71\) −211.298 − 580.536i −0.353189 − 0.970380i −0.981339 − 0.192287i \(-0.938410\pi\)
0.628149 − 0.778093i \(-0.283813\pi\)
\(72\) −151.517 + 70.6533i −0.248006 + 0.115647i
\(73\) −788.271 + 68.9648i −1.26384 + 0.110571i −0.699295 − 0.714833i \(-0.746503\pi\)
−0.564542 + 0.825404i \(0.690947\pi\)
\(74\) −1564.34 − 275.835i −2.45744 − 0.433313i
\(75\) −79.6350 − 68.8501i −0.122606 − 0.106002i
\(76\) 37.9587 − 522.004i 0.0572916 − 0.787868i
\(77\) −352.520 − 352.520i −0.521732 − 0.521732i
\(78\) 154.775 − 108.375i 0.224677 − 0.157320i
\(79\) 127.656 + 107.116i 0.181803 + 0.152551i 0.729147 − 0.684357i \(-0.239917\pi\)
−0.547345 + 0.836907i \(0.684361\pi\)
\(80\) 793.786 + 256.736i 1.10935 + 0.358799i
\(81\) 631.524 − 229.856i 0.866288 − 0.315303i
\(82\) 1032.97 + 90.3731i 1.39113 + 0.121708i
\(83\) 819.194 + 219.502i 1.08335 + 0.290283i 0.755968 − 0.654608i \(-0.227166\pi\)
0.327383 + 0.944892i \(0.393833\pi\)
\(84\) 48.2867 − 83.6351i 0.0627204 − 0.108635i
\(85\) 4.36806 − 6.74608i 0.00557391 − 0.00860841i
\(86\) −1437.40 + 253.453i −1.80232 + 0.317797i
\(87\) −6.80849 + 1.82433i −0.00839020 + 0.00224815i
\(88\) −45.2176 + 168.754i −0.0547752 + 0.204424i
\(89\) 593.738 − 498.206i 0.707147 − 0.593367i −0.216650 − 0.976249i \(-0.569513\pi\)
0.923797 + 0.382882i \(0.125068\pi\)
\(90\) −1024.46 − 433.233i −1.19987 − 0.507409i
\(91\) −367.952 + 1010.94i −0.423866 + 1.16456i
\(92\) −67.9465 − 776.632i −0.0769990 − 0.880103i
\(93\) −56.1017 − 80.1216i −0.0625535 − 0.0893357i
\(94\) 189.578 0.208015
\(95\) −730.470 + 569.025i −0.788891 + 0.614534i
\(96\) −194.961 −0.207272
\(97\) 397.158 + 567.200i 0.415724 + 0.593716i 0.971331 − 0.237729i \(-0.0764031\pi\)
−0.555607 + 0.831445i \(0.687514\pi\)
\(98\) −4.53190 − 51.7998i −0.00467133 − 0.0533936i
\(99\) 247.050 − 678.764i 0.250803 − 0.689074i
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.r.a.2.6 ✓ 336
5.3 odd 4 inner 95.4.r.a.78.6 yes 336
19.10 odd 18 inner 95.4.r.a.67.6 yes 336
95.48 even 36 inner 95.4.r.a.48.6 yes 336
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.r.a.2.6 ✓ 336 1.1 even 1 trivial
95.4.r.a.48.6 yes 336 95.48 even 36 inner
95.4.r.a.67.6 yes 336 19.10 odd 18 inner
95.4.r.a.78.6 yes 336 5.3 odd 4 inner