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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 4.12
Character \(\chi\) \(=\) 95.4
Dual form 95.4.p.a.24.12

$q$-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+(-0.358626 + 0.985318i) q^{2} +(4.97216 - 0.876727i) q^{3} +(5.28612 + 4.43558i) q^{4} +(8.56332 + 7.18815i) q^{5} +(-0.919294 + 5.21358i) q^{6} +(1.32457 + 0.764739i) q^{7} +(-13.5308 + 7.81200i) q^{8} +(-1.41794 + 0.516087i) q^{9} +(-10.1536 + 5.85973i) q^{10} +(-30.3961 - 52.6475i) q^{11} +(30.1722 + 17.4199i) q^{12} +(53.7837 + 9.48352i) q^{13} +(-1.22853 + 1.03086i) q^{14} +(48.8803 + 28.2330i) q^{15} +(6.74132 + 38.2319i) q^{16} +(-24.4784 + 67.2538i) q^{17} -1.58220i q^{18} +(-48.9868 - 66.7779i) q^{19} +(13.3831 + 75.9807i) q^{20} +(7.25643 + 2.64112i) q^{21} +(62.7754 - 11.0690i) q^{22} +(101.076 - 120.458i) q^{23} +(-60.4283 + 50.7054i) q^{24} +(21.6610 + 123.109i) q^{25} +(-28.6325 + 49.5930i) q^{26} +(-124.654 + 71.9689i) q^{27} +(3.60975 + 9.91772i) q^{28} +(180.307 - 65.6265i) q^{29} +(-45.3482 + 38.0375i) q^{30} +(4.52453 - 7.83671i) q^{31} +(-163.181 - 28.7733i) q^{32} +(-197.292 - 235.123i) q^{33} +(-57.4878 - 48.2380i) q^{34} +(5.84563 + 16.0699i) q^{35} +(-9.78453 - 3.56128i) q^{36} -56.2291i q^{37} +(83.3654 - 24.3193i) q^{38} +275.736 q^{39} +(-172.022 - 30.3646i) q^{40} +(-22.3721 - 126.878i) q^{41} +(-5.20469 + 6.20271i) q^{42} +(-67.6797 - 80.6575i) q^{43} +(72.8451 - 413.125i) q^{44} +(-15.8520 - 5.77293i) q^{45} +(82.4406 + 142.791i) q^{46} +(-148.737 - 408.650i) q^{47} +(67.0379 + 184.185i) q^{48} +(-170.330 - 295.021i) q^{49} +(-129.070 - 22.8071i) q^{50} +(-62.7473 + 355.858i) q^{51} +(242.242 + 288.693i) q^{52} +(-184.419 + 219.782i) q^{53} +(-26.2081 - 148.634i) q^{54} +(118.147 - 669.329i) q^{55} -23.8966 q^{56} +(-302.117 - 289.083i) q^{57} +201.195i q^{58} +(-647.459 - 235.656i) q^{59} +(133.157 + 366.055i) q^{60} +(619.373 + 519.716i) q^{61} +(6.09903 + 7.26854i) q^{62} +(-2.27282 - 0.400760i) q^{63} +(-68.4149 + 118.498i) q^{64} +(392.399 + 467.816i) q^{65} +(302.425 - 110.074i) q^{66} +(183.562 + 504.332i) q^{67} +(-427.705 + 246.936i) q^{68} +(396.958 - 687.551i) q^{69} +(-17.9303 - 0.00327656i) q^{70} +(-611.916 + 513.458i) q^{71} +(15.1541 - 18.0600i) q^{72} +(498.678 - 87.9304i) q^{73} +(55.4035 + 20.1652i) q^{74} +(215.635 + 593.127i) q^{75} +(37.2485 - 570.281i) q^{76} -92.9802i q^{77} +(-98.8862 + 271.688i) q^{78} +(21.6320 + 122.681i) q^{79} +(-217.089 + 375.850i) q^{80} +(-525.493 + 440.941i) q^{81} +(133.039 + 23.4583i) q^{82} +(191.888 + 110.787i) q^{83} +(26.6434 + 46.1477i) q^{84} +(-693.047 + 399.962i) q^{85} +(103.745 - 37.7601i) q^{86} +(838.981 - 484.386i) q^{87} +(822.565 + 474.908i) q^{88} +(178.187 - 1010.55i) q^{89} +(11.3731 - 13.5489i) q^{90} +(63.9877 + 53.6921i) q^{91} +(1068.60 - 188.423i) q^{92} +(15.6260 - 42.9322i) q^{93} +455.991 q^{94} +(60.5193 - 923.966i) q^{95} -836.590 q^{96} +(-425.726 + 1169.67i) q^{97} +(351.774 - 62.0273i) q^{98} +(70.2705 + 58.9639i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.358626 + 0.985318i −0.126794 + 0.348362i −0.986805 − 0.161912i \(-0.948234\pi\)
0.860012 + 0.510274i \(0.170456\pi\)
\(3\) 4.97216 − 0.876727i 0.956893 − 0.168726i 0.326669 − 0.945139i \(-0.394074\pi\)
0.630225 + 0.776413i \(0.282963\pi\)
\(4\) 5.28612 + 4.43558i 0.660765 + 0.554447i
\(5\) 8.56332 + 7.18815i 0.765927 + 0.642928i
\(6\) −0.919294 + 5.21358i −0.0625501 + 0.354739i
\(7\) 1.32457 + 0.764739i 0.0715199 + 0.0412920i 0.535333 − 0.844641i \(-0.320186\pi\)
−0.463814 + 0.885933i \(0.653519\pi\)
\(8\) −13.5308 + 7.81200i −0.597982 + 0.345245i
\(9\) −1.41794 + 0.516087i −0.0525162 + 0.0191143i
\(10\) −10.1536 + 5.85973i −0.321086 + 0.185301i
\(11\) −30.3961 − 52.6475i −0.833160 − 1.44308i −0.895520 − 0.445022i \(-0.853196\pi\)
0.0623600 − 0.998054i \(-0.480137\pi\)
\(12\) 30.1722 + 17.4199i 0.725831 + 0.419059i
\(13\) 53.7837 + 9.48352i 1.14746 + 0.202327i 0.714865 − 0.699263i \(-0.246488\pi\)
0.432591 + 0.901590i \(0.357599\pi\)
\(14\) −1.22853 + 1.03086i −0.0234528 + 0.0196793i
\(15\) 48.8803 + 28.2330i 0.841389 + 0.485981i
\(16\) 6.74132 + 38.2319i 0.105333 + 0.597374i
\(17\) −24.4784 + 67.2538i −0.349228 + 0.959497i 0.633386 + 0.773836i \(0.281665\pi\)
−0.982614 + 0.185660i \(0.940558\pi\)
\(18\) − 1.58220i − 0.0207183i
\(19\) −48.9868 − 66.7779i −0.591492 − 0.806311i
\(20\) 13.3831 + 75.9807i 0.149628 + 0.849490i
\(21\) 7.25643 + 2.64112i 0.0754039 + 0.0274448i
\(22\) 62.7754 − 11.0690i 0.608352 − 0.107269i
\(23\) 101.076 − 120.458i 0.916339 − 1.09205i −0.0791203 − 0.996865i \(-0.525211\pi\)
0.995460 − 0.0951855i \(-0.0303444\pi\)
\(24\) −60.4283 + 50.7054i −0.513953 + 0.431258i
\(25\) 21.6610 + 123.109i 0.173288 + 0.984871i
\(26\) −28.6325 + 49.5930i −0.215973 + 0.374077i
\(27\) −124.654 + 71.9689i −0.888505 + 0.512979i
\(28\) 3.60975 + 9.91772i 0.0243635 + 0.0669383i
\(29\) 180.307 − 65.6265i 1.15456 − 0.420225i 0.307409 − 0.951578i \(-0.400538\pi\)
0.847151 + 0.531352i \(0.178316\pi\)
\(30\) −45.3482 + 38.0375i −0.275980 + 0.231489i
\(31\) 4.52453 − 7.83671i 0.0262138 − 0.0454037i −0.852621 − 0.522530i \(-0.824988\pi\)
0.878835 + 0.477126i \(0.158322\pi\)
\(32\) −163.181 − 28.7733i −0.901458 − 0.158951i
\(33\) −197.292 − 235.123i −1.04073 − 1.24029i
\(34\) −57.4878 − 48.2380i −0.289973 − 0.243316i
\(35\) 5.84563 + 16.0699i 0.0282312 + 0.0776087i
\(36\) −9.78453 − 3.56128i −0.0452988 − 0.0164874i
\(37\) − 56.2291i − 0.249838i −0.992167 − 0.124919i \(-0.960133\pi\)
0.992167 − 0.124919i \(-0.0398671\pi\)
\(38\) 83.3654 − 24.3193i 0.355886 − 0.103819i
\(39\) 275.736 1.13213
\(40\) −172.022 − 30.3646i −0.679978 − 0.120027i
\(41\) −22.3721 − 126.878i −0.0852177 − 0.483294i −0.997309 − 0.0733074i \(-0.976645\pi\)
0.912092 − 0.409986i \(-0.134467\pi\)
\(42\) −5.20469 + 6.20271i −0.0191215 + 0.0227881i
\(43\) −67.6797 − 80.6575i −0.240024 − 0.286050i 0.632562 − 0.774510i \(-0.282003\pi\)
−0.872586 + 0.488460i \(0.837559\pi\)
\(44\) 72.8451 − 413.125i 0.249587 − 1.41548i
\(45\) −15.8520 − 5.77293i −0.0525127 − 0.0191239i
\(46\) 82.4406 + 142.791i 0.264243 + 0.457683i
\(47\) −148.737 − 408.650i −0.461605 − 1.26825i −0.924278 − 0.381721i \(-0.875332\pi\)
0.462672 − 0.886529i \(-0.346891\pi\)
\(48\) 67.0379 + 184.185i 0.201585 + 0.553850i
\(49\) −170.330 − 295.021i −0.496590 − 0.860119i
\(50\) −129.070 − 22.8071i −0.365064 − 0.0645082i
\(51\) −62.7473 + 355.858i −0.172282 + 0.977060i
\(52\) 242.242 + 288.693i 0.646018 + 0.769895i
\(53\) −184.419 + 219.782i −0.477961 + 0.569612i −0.950113 − 0.311905i \(-0.899033\pi\)
0.472152 + 0.881517i \(0.343477\pi\)
\(54\) −26.2081 − 148.634i −0.0660458 − 0.374564i
\(55\) 118.147 − 669.329i 0.289653 − 1.64095i
\(56\) −23.8966 −0.0570234
\(57\) −302.117 − 289.083i −0.702041 − 0.671753i
\(58\) 201.195i 0.455487i
\(59\) −647.459 − 235.656i −1.42868 − 0.519996i −0.492126 − 0.870524i \(-0.663780\pi\)
−0.936552 + 0.350528i \(0.886002\pi\)
\(60\) 133.157 + 366.055i 0.286509 + 0.787625i
\(61\) 619.373 + 519.716i 1.30004 + 1.09087i 0.990138 + 0.140098i \(0.0447419\pi\)
0.309906 + 0.950767i \(0.399703\pi\)
\(62\) 6.09903 + 7.26854i 0.0124932 + 0.0148888i
\(63\) −2.27282 − 0.400760i −0.00454522 − 0.000801445i
\(64\) −68.4149 + 118.498i −0.133623 + 0.231441i
\(65\) 392.399 + 467.816i 0.748785 + 0.892699i
\(66\) 302.425 − 110.074i 0.564029 − 0.205290i
\(67\) 183.562 + 504.332i 0.334711 + 0.919612i 0.986868 + 0.161528i \(0.0516421\pi\)
−0.652157 + 0.758084i \(0.726136\pi\)
\(68\) −427.705 + 246.936i −0.762748 + 0.440373i
\(69\) 396.958 − 687.551i 0.692581 − 1.19959i
\(70\) −17.9303 − 0.00327656i −0.0306155 − 5.59463e-6i
\(71\) −611.916 + 513.458i −1.02283 + 0.858258i −0.989981 − 0.141203i \(-0.954903\pi\)
−0.0328510 + 0.999460i \(0.510459\pi\)
\(72\) 15.1541 − 18.0600i 0.0248046 − 0.0295610i
\(73\) 498.678 − 87.9304i 0.799532 − 0.140979i 0.241069 − 0.970508i \(-0.422502\pi\)
0.558464 + 0.829529i \(0.311391\pi\)
\(74\) 55.4035 + 20.1652i 0.0870342 + 0.0316778i
\(75\) 215.635 + 593.127i 0.331992 + 0.913178i
\(76\) 37.2485 − 570.281i 0.0562197 − 0.860733i
\(77\) − 92.9802i − 0.137611i
\(78\) −98.8862 + 271.688i −0.143547 + 0.394392i
\(79\) 21.6320 + 122.681i 0.0308075 + 0.174718i 0.996329 − 0.0856046i \(-0.0272822\pi\)
−0.965522 + 0.260323i \(0.916171\pi\)
\(80\) −217.089 + 375.850i −0.303390 + 0.525266i
\(81\) −525.493 + 440.941i −0.720840 + 0.604857i
\(82\) 133.039 + 23.4583i 0.179166 + 0.0315919i
\(83\) 191.888 + 110.787i 0.253765 + 0.146511i 0.621487 − 0.783425i \(-0.286529\pi\)
−0.367722 + 0.929936i \(0.619862\pi\)
\(84\) 26.6434 + 46.1477i 0.0346076 + 0.0599420i
\(85\) −693.047 + 399.962i −0.884370 + 0.510376i
\(86\) 103.745 − 37.7601i 0.130083 − 0.0473462i
\(87\) 838.981 − 484.386i 1.03389 − 0.596915i
\(88\) 822.565 + 474.908i 0.996429 + 0.575289i
\(89\) 178.187 − 1010.55i 0.212222 − 1.20357i −0.673441 − 0.739241i \(-0.735184\pi\)
0.885663 − 0.464329i \(-0.153705\pi\)
\(90\) 11.3731 − 13.5489i 0.0133203 − 0.0158687i
\(91\) 63.9877 + 53.6921i 0.0737114 + 0.0618512i
\(92\) 1068.60 − 188.423i 1.21097 − 0.213527i
\(93\) 15.6260 − 42.9322i 0.0174231 − 0.0478694i
\(94\) 455.991 0.500339
\(95\) 60.5193 − 923.966i 0.0653595 − 0.997862i
\(96\) −836.590 −0.889418
\(97\) −425.726 + 1169.67i −0.445628 + 1.22435i 0.490111 + 0.871660i \(0.336956\pi\)
−0.935739 + 0.352693i \(0.885266\pi\)
\(98\) 351.774 − 62.0273i 0.362598 − 0.0639357i
\(99\) 70.2705 + 58.9639i 0.0713378 + 0.0598596i
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.4.12 ✓ 168
5.4 even 2 inner 95.4.p.a.4.17 yes 168
19.5 even 9 inner 95.4.p.a.24.17 yes 168
95.24 even 18 inner 95.4.p.a.24.12 yes 168
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.4.p.a.4.12 ✓ 168 1.1 even 1 trivial
95.4.p.a.4.17 yes 168 5.4 even 2 inner
95.4.p.a.24.12 yes 168 95.24 even 18 inner
95.4.p.a.24.17 yes 168 19.5 even 9 inner