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Properties

Label 371.2.w.a.3.6
Level $371$
Weight $2$
Character 371.3
Analytic conductor $2.962$
Analytic rank $0$
Dimension $1632$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [371,2,Mod(3,371)] 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("371.3"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(371, base_ring=CyclotomicField(156)) chi = DirichletCharacter(H, H._module([26, 51])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 371 = 7 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 371.w (of order \(156\), degree \(48\), 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: \(2.96244991499\)
Analytic rank: \(0\)
Dimension: \(1632\)
Relative dimension: \(34\) over \(\Q(\zeta_{156})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{156}]$

Embedding invariants

Embedding label 3.6
Character \(\chi\) \(=\) 371.3
Dual form 371.2.w.a.124.6

$q$-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+(-1.03883 + 1.88602i) q^{2} +(0.613883 - 0.721703i) q^{3} +(-1.40899 - 2.22814i) q^{4} +(-0.771282 - 0.555636i) q^{5} +(0.723429 + 1.90753i) q^{6} +(-2.47803 + 0.927015i) q^{7} +(1.36748 - 0.0827175i) q^{8} +(0.337231 + 2.07507i) q^{9} +(1.84917 - 0.877444i) q^{10} +(-0.227363 + 0.533641i) q^{11} +(-2.47301 - 0.350945i) q^{12} +(-0.397042 - 0.756500i) q^{13} +(0.825881 - 5.63664i) q^{14} +(-0.874481 + 0.215540i) q^{15} +(0.995711 - 2.09842i) q^{16} +(0.0659229 + 0.322912i) q^{17} +(-4.26395 - 1.51962i) q^{18} +(-8.20472 + 1.84786i) q^{19} +(-0.151307 + 2.50141i) q^{20} +(-0.852193 + 2.35748i) q^{21} +(-0.770268 - 0.983175i) q^{22} +(-7.30665 - 1.95781i) q^{23} +(0.779778 - 1.03770i) q^{24} +(-1.29720 - 3.88558i) q^{25} +(1.83924 + 0.0370443i) q^{26} +(4.13710 + 2.50096i) q^{27} +(5.55703 + 4.21524i) q^{28} +(-0.354321 - 0.0430224i) q^{29} +(0.501923 - 1.87320i) q^{30} +(-5.26972 + 0.747827i) q^{31} +(4.52486 + 6.28098i) q^{32} +(0.245556 + 0.491682i) q^{33} +(-0.677502 - 0.211118i) q^{34} +(2.42634 + 0.661895i) q^{35} +(4.14838 - 3.67514i) q^{36} +(-9.09222 - 4.31431i) q^{37} +(5.03821 - 17.3939i) q^{38} +(-0.789706 - 0.177857i) q^{39} +(-1.10068 - 0.696025i) q^{40} +(7.73921 + 6.06328i) q^{41} +(-3.56098 - 4.05628i) q^{42} +(0.941542 + 0.649900i) q^{43} +(1.50938 - 0.245298i) q^{44} +(0.892883 - 1.78784i) q^{45} +(11.2828 - 11.7467i) q^{46} +(8.88128 + 7.25134i) q^{47} +(-0.903183 - 2.00679i) q^{48} +(5.28129 - 4.59435i) q^{49} +(8.67586 + 1.58991i) q^{50} +(0.273515 + 0.150653i) q^{51} +(-1.12616 + 1.95056i) q^{52} +(-5.45947 + 4.81603i) q^{53} +(-9.01461 + 5.20459i) q^{54} +(0.471871 - 0.285256i) q^{55} +(-3.31199 + 1.47265i) q^{56} +(-3.70314 + 7.05574i) q^{57} +(0.449221 - 0.623565i) q^{58} +(7.22333 + 1.17391i) q^{59} +(1.71239 + 1.64477i) q^{60} +(5.65047 - 3.73451i) q^{61} +(4.06392 - 10.7157i) q^{62} +(-2.75929 - 4.82947i) q^{63} +(-11.9352 + 1.44919i) q^{64} +(-0.114108 + 0.804086i) q^{65} +(-1.18241 - 0.0476497i) q^{66} +(-2.10873 + 9.36302i) q^{67} +(0.626607 - 0.601864i) q^{68} +(-5.89839 + 4.07136i) q^{69} +(-3.76891 + 3.88855i) q^{70} +(-1.17458 - 6.40948i) q^{71} +(0.632803 + 2.80973i) q^{72} +(2.27201 + 1.50162i) q^{73} +(17.5822 - 12.6663i) q^{74} +(-3.60056 - 1.44910i) q^{75} +(15.6776 + 15.6776i) q^{76} +(0.0687201 - 1.53315i) q^{77} +(1.15581 - 1.30464i) q^{78} +(-0.0445141 + 2.21011i) q^{79} +(-1.93393 + 1.06522i) q^{80} +(-1.63766 + 0.546730i) q^{81} +(-19.4752 + 8.29762i) q^{82} +(-4.67460 + 4.67460i) q^{83} +(6.45352 - 1.42286i) q^{84} +(0.128576 - 0.285685i) q^{85} +(-2.20383 + 1.10064i) q^{86} +(-0.248561 + 0.229304i) q^{87} +(-0.266774 + 0.748552i) q^{88} +(-4.51665 + 13.5290i) q^{89} +(2.44436 + 3.54126i) q^{90} +(1.68517 + 1.50657i) q^{91} +(5.93272 + 19.0388i) q^{92} +(-2.69528 + 4.26225i) q^{93} +(-22.9024 + 9.21740i) q^{94} +(7.35489 + 3.13363i) q^{95} +(7.31074 + 0.590184i) q^{96} +(-10.4217 - 3.95242i) q^{97} +(3.17869 + 14.7334i) q^{98} +(-1.18402 - 0.291834i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 1632 q - 26 q^{2} - 72 q^{3} - 26 q^{4} - 72 q^{5} - 52 q^{7} - 92 q^{8} - 26 q^{9} - 66 q^{10} - 26 q^{11} - 90 q^{12} - 62 q^{14} - 104 q^{15} - 98 q^{16} - 78 q^{17} - 22 q^{18} - 84 q^{19} - 56 q^{21}+ \cdots - 408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/371\mathbb{Z}\right)^\times\).

\(n\) \(213\) \(267\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{52}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.03883 + 1.88602i −0.734564 + 1.33362i 0.200694 + 0.979654i \(0.435680\pi\)
−0.935258 + 0.353967i \(0.884833\pi\)
\(3\) 0.613883 − 0.721703i 0.354426 − 0.416675i −0.555454 − 0.831547i \(-0.687456\pi\)
0.909880 + 0.414872i \(0.136174\pi\)
\(4\) −1.40899 − 2.22814i −0.704494 − 1.11407i
\(5\) −0.771282 − 0.555636i −0.344928 − 0.248488i 0.398701 − 0.917081i \(-0.369461\pi\)
−0.743629 + 0.668593i \(0.766897\pi\)
\(6\) 0.723429 + 1.90753i 0.295339 + 0.778744i
\(7\) −2.47803 + 0.927015i −0.936608 + 0.350379i
\(8\) 1.36748 − 0.0827175i 0.483478 − 0.0292451i
\(9\) 0.337231 + 2.07507i 0.112410 + 0.691689i
\(10\) 1.84917 − 0.877444i 0.584760 − 0.277472i
\(11\) −0.227363 + 0.533641i −0.0685526 + 0.160899i −0.950731 − 0.310016i \(-0.899666\pi\)
0.882179 + 0.470914i \(0.156076\pi\)
\(12\) −2.47301 − 0.350945i −0.713896 − 0.101309i
\(13\) −0.397042 − 0.756500i −0.110120 − 0.209815i 0.824160 − 0.566358i \(-0.191648\pi\)
−0.934279 + 0.356542i \(0.883956\pi\)
\(14\) 0.825881 − 5.63664i 0.220726 − 1.50646i
\(15\) −0.874481 + 0.215540i −0.225790 + 0.0556523i
\(16\) 0.995711 − 2.09842i 0.248928 − 0.524604i
\(17\) 0.0659229 + 0.322912i 0.0159887 + 0.0783176i 0.987002 − 0.160707i \(-0.0513774\pi\)
−0.971013 + 0.239025i \(0.923172\pi\)
\(18\) −4.26395 − 1.51962i −1.00502 − 0.358177i
\(19\) −8.20472 + 1.84786i −1.88229 + 0.423927i −0.998992 − 0.0448800i \(-0.985709\pi\)
−0.883300 + 0.468807i \(0.844684\pi\)
\(20\) −0.151307 + 2.50141i −0.0338333 + 0.559331i
\(21\) −0.852193 + 2.35748i −0.185964 + 0.514445i
\(22\) −0.770268 − 0.983175i −0.164222 − 0.209614i
\(23\) −7.30665 − 1.95781i −1.52354 − 0.408232i −0.602636 − 0.798016i \(-0.705883\pi\)
−0.920906 + 0.389784i \(0.872550\pi\)
\(24\) 0.779778 − 1.03770i 0.159171 − 0.211819i
\(25\) −1.29720 − 3.88558i −0.259439 − 0.777116i
\(26\) 1.83924 + 0.0370443i 0.360704 + 0.00726499i
\(27\) 4.13710 + 2.50096i 0.796185 + 0.481310i
\(28\) 5.55703 + 4.21524i 1.05018 + 0.796606i
\(29\) −0.354321 − 0.0430224i −0.0657958 − 0.00798905i 0.0875726 − 0.996158i \(-0.472089\pi\)
−0.153368 + 0.988169i \(0.549012\pi\)
\(30\) 0.501923 − 1.87320i 0.0916382 − 0.341998i
\(31\) −5.26972 + 0.747827i −0.946470 + 0.134314i −0.596648 − 0.802503i \(-0.703501\pi\)
−0.349821 + 0.936816i \(0.613758\pi\)
\(32\) 4.52486 + 6.28098i 0.799890 + 1.11033i
\(33\) 0.245556 + 0.491682i 0.0427458 + 0.0855908i
\(34\) −0.677502 − 0.211118i −0.116191 − 0.0362065i
\(35\) 2.42634 + 0.661895i 0.410127 + 0.111881i
\(36\) 4.14838 − 3.67514i 0.691397 − 0.612524i
\(37\) −9.09222 − 4.31431i −1.49475 − 0.709269i −0.506520 − 0.862228i \(-0.669068\pi\)
−0.988232 + 0.152959i \(0.951120\pi\)
\(38\) 5.03821 − 17.3939i 0.817306 − 2.82167i
\(39\) −0.789706 − 0.177857i −0.126454 − 0.0284798i
\(40\) −1.10068 − 0.696025i −0.174032 − 0.110051i
\(41\) 7.73921 + 6.06328i 1.20866 + 0.946926i 0.999521 − 0.0309598i \(-0.00985638\pi\)
0.209141 + 0.977886i \(0.432933\pi\)
\(42\) −3.56098 − 4.05628i −0.549472 − 0.625898i
\(43\) 0.941542 + 0.649900i 0.143584 + 0.0991088i 0.637691 − 0.770292i \(-0.279890\pi\)
−0.494107 + 0.869401i \(0.664505\pi\)
\(44\) 1.50938 − 0.245298i 0.227547 − 0.0369800i
\(45\) 0.892883 − 1.78784i 0.133103 − 0.266515i
\(46\) 11.2828 − 11.7467i 1.66357 − 1.73196i
\(47\) 8.88128 + 7.25134i 1.29547 + 1.05772i 0.994571 + 0.104061i \(0.0331836\pi\)
0.300897 + 0.953657i \(0.402714\pi\)
\(48\) −0.903183 − 2.00679i −0.130363 − 0.289655i
\(49\) 5.28129 − 4.59435i 0.754469 − 0.656335i
\(50\) 8.67586 + 1.58991i 1.22695 + 0.224847i
\(51\) 0.273515 + 0.150653i 0.0382998 + 0.0210957i
\(52\) −1.12616 + 1.95056i −0.156170 + 0.270495i
\(53\) −5.45947 + 4.81603i −0.749916 + 0.661533i
\(54\) −9.01461 + 5.20459i −1.22673 + 0.708255i
\(55\) 0.471871 − 0.285256i 0.0636271 − 0.0384639i
\(56\) −3.31199 + 1.47265i −0.442583 + 0.196792i
\(57\) −3.70314 + 7.05574i −0.490493 + 0.934556i
\(58\) 0.449221 − 0.623565i 0.0589856 − 0.0818781i
\(59\) 7.22333 + 1.17391i 0.940398 + 0.152830i 0.611240 − 0.791445i \(-0.290671\pi\)
0.329158 + 0.944275i \(0.393235\pi\)
\(60\) 1.71239 + 1.64477i 0.221068 + 0.212339i
\(61\) 5.65047 − 3.73451i 0.723468 − 0.478155i −0.135115 − 0.990830i \(-0.543140\pi\)
0.858583 + 0.512675i \(0.171345\pi\)
\(62\) 4.06392 − 10.7157i 0.516119 − 1.36089i
\(63\) −2.75929 − 4.82947i −0.347638 − 0.608456i
\(64\) −11.9352 + 1.44919i −1.49190 + 0.181149i
\(65\) −0.114108 + 0.804086i −0.0141533 + 0.0997346i
\(66\) −1.18241 − 0.0476497i −0.145545 − 0.00586527i
\(67\) −2.10873 + 9.36302i −0.257622 + 1.14387i 0.661514 + 0.749933i \(0.269914\pi\)
−0.919136 + 0.393941i \(0.871111\pi\)
\(68\) 0.626607 − 0.601864i 0.0759872 − 0.0729867i
\(69\) −5.89839 + 4.07136i −0.710083 + 0.490135i
\(70\) −3.76891 + 3.88855i −0.450471 + 0.464770i
\(71\) −1.17458 − 6.40948i −0.139397 − 0.760665i −0.976812 − 0.214098i \(-0.931319\pi\)
0.837415 − 0.546567i \(-0.184066\pi\)
\(72\) 0.632803 + 2.80973i 0.0745765 + 0.331129i
\(73\) 2.27201 + 1.50162i 0.265918 + 0.175751i 0.677289 − 0.735717i \(-0.263155\pi\)
−0.411370 + 0.911468i \(0.634950\pi\)
\(74\) 17.5822 − 12.6663i 2.04389 − 1.47243i
\(75\) −3.60056 − 1.44910i −0.415757 − 0.167328i
\(76\) 15.6776 + 15.6776i 1.79835 + 1.79835i
\(77\) 0.0687201 − 1.53315i 0.00783138 − 0.174718i
\(78\) 1.15581 − 1.30464i 0.130870 − 0.147722i
\(79\) −0.0445141 + 2.21011i −0.00500823 + 0.248657i 0.989533 + 0.144309i \(0.0460960\pi\)
−0.994541 + 0.104348i \(0.966725\pi\)
\(80\) −1.93393 + 1.06522i −0.216220 + 0.119095i
\(81\) −1.63766 + 0.546730i −0.181962 + 0.0607478i
\(82\) −19.4752 + 8.29762i −2.15068 + 0.916318i
\(83\) −4.67460 + 4.67460i −0.513104 + 0.513104i −0.915476 − 0.402372i \(-0.868186\pi\)
0.402372 + 0.915476i \(0.368186\pi\)
\(84\) 6.45352 − 1.42286i 0.704137 − 0.155247i
\(85\) 0.128576 − 0.285685i 0.0139461 − 0.0309869i
\(86\) −2.20383 + 1.10064i −0.237645 + 0.118685i
\(87\) −0.248561 + 0.229304i −0.0266486 + 0.0245840i
\(88\) −0.266774 + 0.748552i −0.0284382 + 0.0797959i
\(89\) −4.51665 + 13.5290i −0.478764 + 1.43407i 0.381568 + 0.924341i \(0.375384\pi\)
−0.860333 + 0.509733i \(0.829744\pi\)
\(90\) 2.44436 + 3.54126i 0.257658 + 0.373282i
\(91\) 1.68517 + 1.50657i 0.176654 + 0.157931i
\(92\) 5.93272 + 19.0388i 0.618528 + 1.98493i
\(93\) −2.69528 + 4.26225i −0.279488 + 0.441975i
\(94\) −22.9024 + 9.21740i −2.36220 + 0.950703i
\(95\) 7.35489 + 3.13363i 0.754596 + 0.321503i
\(96\) 7.31074 + 0.590184i 0.746149 + 0.0602354i
\(97\) −10.4217 − 3.95242i −1.05816 − 0.401307i −0.236706 − 0.971581i \(-0.576068\pi\)
−0.821454 + 0.570274i \(0.806837\pi\)
\(98\) 3.17869 + 14.7334i 0.321096 + 1.48830i
\(99\) −1.18402 − 0.291834i −0.118998 − 0.0293304i
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 371.2.w.a.3.6 ✓ 1632
7.5 odd 6 inner 371.2.w.a.215.29 yes 1632
53.18 odd 52 inner 371.2.w.a.283.29 yes 1632
371.124 even 156 inner 371.2.w.a.124.6 yes 1632
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
371.2.w.a.3.6 ✓ 1632 1.1 even 1 trivial
371.2.w.a.124.6 yes 1632 371.124 even 156 inner
371.2.w.a.215.29 yes 1632 7.5 odd 6 inner
371.2.w.a.283.29 yes 1632 53.18 odd 52 inner