LMFDB - Embedded newform 1840.3.k.e.321.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(321,1840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			321.17
Character	$\chi$	$=$	1840.321
Dual form			1840.3.k.e.321.18

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.35475 q^{3} -2.23607i q^{5} +5.28829i q^{7} -3.45517 q^{9} +O(q^{10})$	$q-2.35475 q^{3} -2.23607i q^{5} +5.28829i q^{7} -3.45517 q^{9} +4.16134i q^{11} -19.7006 q^{13} +5.26537i q^{15} -3.83408i q^{17} -8.80476i q^{19} -12.4526i q^{21} +(1.35641 - 22.9600i) q^{23} -5.00000 q^{25} +29.3288 q^{27} -43.8502 q^{29} +3.73337 q^{31} -9.79891i q^{33} +11.8250 q^{35} +39.6217i q^{37} +46.3900 q^{39} -5.34636 q^{41} -0.906562i q^{43} +7.72599i q^{45} -12.8994 q^{47} +21.0340 q^{49} +9.02829i q^{51} -17.6343i q^{53} +9.30505 q^{55} +20.7330i q^{57} +17.6113 q^{59} +14.0501i q^{61} -18.2719i q^{63} +44.0519i q^{65} +110.284i q^{67} +(-3.19400 + 54.0649i) q^{69} +73.0000 q^{71} -3.88326 q^{73} +11.7737 q^{75} -22.0064 q^{77} -12.3124i q^{79} -37.9653 q^{81} -60.6843i q^{83} -8.57327 q^{85} +103.256 q^{87} -128.169i q^{89} -104.182i q^{91} -8.79113 q^{93} -19.6880 q^{95} -100.118i q^{97} -14.3781i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 128 q^{9}+O(q^{10}) $ 48 * q + 128 * q^9	$ 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) $ 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.35475			−0.784916			−0.392458	−	0.919770i	$-0.628375\pi$
$3$	−2.35475			−0.784916			−0.392458	+	0.919770i	$0.628375\pi$
$4$		0			0
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$			5.28829i			0.755469i	0.925914	+	0.377735i	$0.123297\pi$
$7$			5.28829i			0.755469i	−0.925914	+	0.377735i	$0.876703\pi$
$8$		0			0
$9$	−3.45517			−0.383908
$10$		0			0
$11$			4.16134i			0.378304i	0.981948	+	0.189152i	$0.0605739\pi$
$11$			4.16134i			0.378304i	−0.981948	+	0.189152i	$0.939426\pi$
$12$		0			0
$13$	−19.7006			−1.51543			−0.757716	−	0.652585i	$-0.773685\pi$
$13$	−19.7006			−1.51543			−0.757716	+	0.652585i	$0.773685\pi$
$14$		0			0
$15$			5.26537i			0.351025i
$16$		0			0
$17$		−	3.83408i		−	0.225534i	−0.993621	−	0.112767i	$-0.964029\pi$
$17$		−	3.83408i		−	0.225534i	0.993621	−	0.112767i	$-0.0359714\pi$
$18$		0			0
$19$		−	8.80476i		−	0.463408i	−0.972786	−	0.231704i	$-0.925570\pi$
$19$		−	8.80476i		−	0.463408i	0.972786	−	0.231704i	$-0.0744301\pi$
$20$		0			0
$21$		−	12.4526i		−	0.592980i
$22$		0			0
$23$	1.35641	−	22.9600i	0.0589743	−	0.998260i
$23$	1.35641	−	22.9600i	0.0589743	−	0.998260i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	29.3288			1.08625
$28$		0			0
$29$	−43.8502			−1.51208			−0.756039	−	0.654527i	$-0.772868\pi$
$29$	−43.8502			−1.51208			−0.756039	+	0.654527i	$0.772868\pi$
$30$		0			0
$31$	3.73337			0.120431			0.0602156	−	0.998185i	$-0.480821\pi$
$31$	3.73337			0.120431			0.0602156	+	0.998185i	$0.480821\pi$
$32$		0			0
$33$		−	9.79891i		−	0.296937i
$34$		0			0
$35$	11.8250			0.337856
$36$		0			0
$37$			39.6217i			1.07086i	0.844580	+	0.535429i	$0.179850\pi$
$37$			39.6217i			1.07086i	−0.844580	+	0.535429i	$0.820150\pi$
$38$		0			0
$39$	46.3900			1.18949
$40$		0			0
$41$	−5.34636			−0.130399			−0.0651995	−	0.997872i	$-0.520768\pi$
$41$	−5.34636			−0.130399			−0.0651995	+	0.997872i	$0.520768\pi$
$42$		0			0
$43$		−	0.906562i		−	0.0210828i	−0.999944	−	0.0105414i	$-0.996644\pi$
$43$		−	0.906562i		−	0.0210828i	0.999944	−	0.0105414i	$-0.00335550\pi$
$44$		0			0
$45$			7.72599i			0.171689i
$46$		0			0
$47$	−12.8994			−0.274456			−0.137228	−	0.990539i	$-0.543819\pi$
$47$	−12.8994			−0.274456			−0.137228	+	0.990539i	$0.543819\pi$
$48$		0			0
$49$	21.0340			0.429266
$50$		0			0
$51$			9.02829i			0.177025i
$52$		0			0
$53$		−	17.6343i		−	0.332723i	−0.986065	−	0.166362i	$-0.946798\pi$
$53$		−	17.6343i		−	0.332723i	0.986065	−	0.166362i	$-0.0532019\pi$
$54$		0			0
$55$	9.30505			0.169183
$56$		0			0
$57$			20.7330i			0.363736i
$58$		0			0
$59$	17.6113			0.298497			0.149248	−	0.988800i	$-0.452315\pi$
$59$	17.6113			0.298497			0.149248	+	0.988800i	$0.452315\pi$
$60$		0			0
$61$			14.0501i			0.230330i	0.993346	+	0.115165i	$0.0367396\pi$
$61$			14.0501i			0.230330i	−0.993346	+	0.115165i	$0.963260\pi$
$62$		0			0
$63$		−	18.2719i		−	0.290030i
$64$		0			0
$65$			44.0519i			0.677722i
$66$		0			0
$67$			110.284i			1.64603i	0.568023	+	0.823013i	$0.307709\pi$
$67$			110.284i			1.64603i	−0.568023	+	0.823013i	$0.692291\pi$
$68$		0			0
$69$	−3.19400	+	54.0649i	−0.0462898	+	0.783549i
$70$		0			0
$71$	73.0000			1.02817			0.514084	−	0.857740i	$-0.328132\pi$
$71$	73.0000			1.02817			0.514084	+	0.857740i	$0.328132\pi$
$72$		0			0
$73$	−3.88326			−0.0531954			−0.0265977	−	0.999646i	$-0.508467\pi$
$73$	−3.88326			−0.0531954			−0.0265977	+	0.999646i	$0.508467\pi$
$74$		0			0
$75$	11.7737			0.156983
$76$		0			0
$77$	−22.0064			−0.285797
$78$		0			0
$79$		−	12.3124i		−	0.155853i	−0.996959	−	0.0779264i	$-0.975170\pi$
$79$		−	12.3124i		−	0.155853i	0.996959	−	0.0779264i	$-0.0248299\pi$
$80$		0			0
$81$	−37.9653			−0.468708
$82$		0			0
$83$		−	60.6843i		−	0.731137i	−0.930785	−	0.365568i	$-0.880875\pi$
$83$		−	60.6843i		−	0.731137i	0.930785	−	0.365568i	$-0.119125\pi$
$84$		0			0
$85$	−8.57327			−0.100862
$86$		0			0
$87$	103.256			1.18685
$88$		0			0
$89$		−	128.169i		−	1.44010i	−0.693923	−	0.720050i	$-0.744119\pi$
$89$		−	128.169i		−	1.44010i	0.693923	−	0.720050i	$-0.255881\pi$
$90$		0			0
$91$		−	104.182i		−	1.14486i
$92$		0			0
$93$	−8.79113			−0.0945283
$94$		0			0
$95$	−19.6880			−0.207243
$96$		0			0
$97$		−	100.118i		−	1.03214i	−0.856546	−	0.516071i	$-0.827394\pi$
$97$		−	100.118i		−	1.03214i	0.856546	−	0.516071i	$-0.172606\pi$
$98$		0			0
$99$		−	14.3781i		−	0.145234i
$100$		0			0
$101$	−18.5168			−0.183334			−0.0916671	−	0.995790i	$-0.529220\pi$
$101$	−18.5168			−0.183334			−0.0916671	+	0.995790i	$0.529220\pi$
$102$		0			0
$103$			59.8951i			0.581506i	0.956798	+	0.290753i	$0.0939058\pi$
$103$			59.8951i			0.581506i	−0.956798	+	0.290753i	$0.906094\pi$
$104$		0			0
$105$	−27.8448			−0.265189
$106$		0			0
$107$			181.153i			1.69302i	0.532374	+	0.846509i	$0.321300\pi$
$107$			181.153i			1.69302i	−0.532374	+	0.846509i	$0.678700\pi$
$108$		0			0
$109$		−	63.3800i		−	0.581467i	−0.956804	−	0.290734i	$-0.906101\pi$
$109$		−	63.3800i		−	0.581467i	0.956804	−	0.290734i	$-0.0938994\pi$
$110$		0			0
$111$		−	93.2992i		−	0.840533i
$112$		0			0
$113$			125.568i			1.11122i	0.831442	+	0.555612i	$0.187516\pi$
$113$			125.568i			1.11122i	−0.831442	+	0.555612i	$0.812484\pi$
$114$		0			0
$115$	−51.3401	−	3.03302i	−0.446435	−	0.0263741i
$116$		0			0
$117$	68.0689			0.581786
$118$		0			0
$119$	20.2757			0.170384
$120$		0			0
$121$	103.683			0.856886
$122$		0			0
$123$	12.5893			0.102352
$124$		0			0
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	59.5530			0.468921			0.234461	−	0.972126i	$-0.424668\pi$
$127$	59.5530			0.468921			0.234461	+	0.972126i	$0.424668\pi$
$128$		0			0
$129$			2.13472i			0.0165483i
$130$		0			0
$131$	−84.8909			−0.648022			−0.324011	−	0.946053i	$-0.605032\pi$
$131$	−84.8909			−0.648022			−0.324011	+	0.946053i	$0.605032\pi$
$132$		0			0
$133$	46.5621			0.350091
$134$		0			0
$135$		−	65.5811i		−	0.485786i
$136$		0			0
$137$		−	63.6767i		−	0.464794i	−0.972621	−	0.232397i	$-0.925343\pi$
$137$		−	63.6767i		−	0.464794i	0.972621	−	0.232397i	$-0.0746568\pi$
$138$		0			0
$139$	206.504			1.48564			0.742819	−	0.669492i	$-0.233488\pi$
$139$	206.504			1.48564			0.742819	+	0.669492i	$0.233488\pi$
$140$		0			0
$141$	30.3749			0.215425
$142$		0			0
$143$		−	81.9810i		−	0.573294i
$144$		0			0
$145$			98.0521i			0.676221i
$146$		0			0
$147$	−49.5298			−0.336938
$148$		0			0
$149$		−	174.280i		−	1.16966i	−0.811155	−	0.584831i	$-0.801161\pi$
$149$		−	174.280i		−	1.16966i	0.811155	−	0.584831i	$-0.198839\pi$
$150$		0			0
$151$	274.863			1.82028			0.910142	−	0.414297i	$-0.135973\pi$
$151$	274.863			1.82028			0.910142	+	0.414297i	$0.135973\pi$
$152$		0			0
$153$			13.2474i			0.0865843i
$154$		0			0
$155$		−	8.34806i		−	0.0538585i
$156$		0			0
$157$			92.0443i			0.586269i	0.956071	+	0.293135i	$0.0946984\pi$
$157$			92.0443i			0.586269i	−0.956071	+	0.293135i	$0.905302\pi$
$158$		0			0
$159$			41.5244i			0.261160i
$160$		0			0
$161$	121.419	+	7.17308i	0.754154	+	0.0445533i
$162$		0			0
$163$	166.504			1.02150			0.510748	−	0.859730i	$-0.329368\pi$
$163$	166.504			1.02150			0.510748	+	0.859730i	$0.329368\pi$
$164$		0			0
$165$	−21.9110			−0.132794
$166$		0			0
$167$	191.663			1.14768			0.573840	−	0.818967i	$-0.305453\pi$
$167$	191.663			1.14768			0.573840	+	0.818967i	$0.305453\pi$
$168$		0			0
$169$	219.114			1.29653
$170$		0			0
$171$			30.4219i			0.177906i
$172$		0			0
$173$	81.5528			0.471403			0.235702	−	0.971825i	$-0.424261\pi$
$173$	81.5528			0.471403			0.235702	+	0.971825i	$0.424261\pi$
$174$		0			0
$175$		−	26.4414i		−	0.151094i
$176$		0			0
$177$	−41.4702			−0.234295
$178$		0			0
$179$	−7.72138			−0.0431362			−0.0215681	−	0.999767i	$-0.506866\pi$
$179$	−7.72138			−0.0431362			−0.0215681	+	0.999767i	$0.506866\pi$
$180$		0			0
$181$		−	16.0608i		−	0.0887338i	−0.999015	−	0.0443669i	$-0.985873\pi$
$181$		−	16.0608i		−	0.0887338i	0.999015	−	0.0443669i	$-0.0141271\pi$
$182$		0			0
$183$		−	33.0844i		−	0.180789i
$184$		0			0
$185$	88.5969			0.478902
$186$		0			0
$187$	15.9549			0.0853205
$188$		0			0
$189$			155.099i			0.820629i
$190$		0			0
$191$			15.6916i			0.0821551i	0.999156	+	0.0410775i	$0.0130791\pi$
$191$			15.6916i			0.0821551i	−0.999156	+	0.0410775i	$0.986921\pi$
$192$		0			0
$193$	−117.850			−0.610622			−0.305311	−	0.952253i	$-0.598760\pi$
$193$	−117.850			−0.610622			−0.305311	+	0.952253i	$0.598760\pi$
$194$		0			0
$195$		−	103.731i		−	0.531954i
$196$		0			0
$197$	255.020			1.29452			0.647260	−	0.762269i	$-0.275915\pi$
$197$	255.020			1.29452			0.647260	+	0.762269i	$0.275915\pi$
$198$		0			0
$199$			111.646i			0.561034i	0.959849	+	0.280517i	$0.0905060\pi$
$199$			111.646i			0.561034i	−0.959849	+	0.280517i	$0.909494\pi$
$200$		0			0
$201$		−	259.690i		−	1.29199i
$202$		0			0
$203$		−	231.893i		−	1.14233i
$204$		0			0
$205$			11.9548i			0.0583162i
$206$		0			0
$207$	−4.68662	+	79.3305i	−0.0226407	+	0.383239i
$208$		0			0
$209$	36.6396			0.175309
$210$		0			0
$211$	154.426			0.731876			0.365938	−	0.930639i	$-0.380748\pi$
$211$	154.426			0.731876			0.365938	+	0.930639i	$0.380748\pi$
$212$		0			0
$213$	−171.896			−0.807026
$214$		0			0
$215$	−2.02713			−0.00942853
$216$		0			0
$217$			19.7431i			0.0909821i
$218$		0			0
$219$	9.14410			0.0417539
$220$		0			0
$221$			75.5337i			0.341782i
$222$		0			0
$223$	110.049			0.493493			0.246747	−	0.969080i	$-0.420638\pi$
$223$	110.049			0.493493			0.246747	+	0.969080i	$0.420638\pi$
$224$		0			0
$225$	17.2758			0.0767815
$226$		0			0
$227$		−	99.6979i		−	0.439198i	−0.975590	−	0.219599i	$-0.929525\pi$
$227$		−	99.6979i		−	0.439198i	0.975590	−	0.219599i	$-0.0704749\pi$
$228$		0			0
$229$			104.584i			0.456698i	0.973579	+	0.228349i	$0.0733328\pi$
$229$			104.584i			0.456698i	−0.973579	+	0.228349i	$0.926667\pi$
$230$		0			0
$231$	51.8194			0.224327
$232$		0			0
$233$	298.798			1.28240			0.641198	−	0.767376i	$-0.278438\pi$
$233$	298.798			1.28240			0.641198	+	0.767376i	$0.278438\pi$
$234$		0			0
$235$			28.8440i			0.122740i
$236$		0			0
$237$			28.9925i			0.122331i
$238$		0			0
$239$	−159.516			−0.667430			−0.333715	−	0.942674i	$-0.608302\pi$
$239$	−159.516			−0.667430			−0.333715	+	0.942674i	$0.608302\pi$
$240$		0			0
$241$		−	398.701i		−	1.65436i	−0.561938	−	0.827180i	$-0.689944\pi$
$241$		−	398.701i		−	1.65436i	0.561938	−	0.827180i	$-0.310056\pi$
$242$		0			0
$243$	−174.560			−0.718355
$244$		0			0
$245$		−	47.0335i		−	0.191974i
$246$		0			0
$247$			173.459i			0.702264i
$248$		0			0
$249$			142.896i			0.573881i
$250$		0			0
$251$			151.283i			0.602719i	0.953511	+	0.301360i	$0.0974405\pi$
$251$			151.283i			0.602719i	−0.953511	+	0.301360i	$0.902560\pi$
$252$		0			0
$253$	95.5443	+	5.64448i	0.377646	+	0.0223102i
$254$		0			0
$255$	20.1879			0.0791681
$256$		0			0
$257$	349.495			1.35990			0.679951	−	0.733258i	$-0.262001\pi$
$257$	349.495			1.35990			0.679951	+	0.733258i	$0.262001\pi$
$258$		0			0
$259$	−209.531			−0.809000
$260$		0			0
$261$	151.510			0.580498
$262$		0			0
$263$		−	180.231i		−	0.685290i	−0.939465	−	0.342645i	$-0.888677\pi$
$263$		−	180.231i		−	0.685290i	0.939465	−	0.342645i	$-0.111323\pi$
$264$		0			0
$265$	−39.4316			−0.148798
$266$		0			0
$267$			301.805i			1.13036i
$268$		0			0
$269$	89.4756			0.332623			0.166311	−	0.986073i	$-0.446814\pi$
$269$	89.4756			0.332623			0.166311	+	0.986073i	$0.446814\pi$
$270$		0			0
$271$	47.0511			0.173620			0.0868102	−	0.996225i	$-0.472333\pi$
$271$	47.0511			0.173620			0.0868102	+	0.996225i	$0.472333\pi$
$272$		0			0
$273$			245.323i			0.898620i
$274$		0			0
$275$		−	20.8067i		−	0.0756608i
$276$		0			0
$277$	−211.024			−0.761818			−0.380909	−	0.924612i	$-0.624389\pi$
$277$	−211.024			−0.761818			−0.380909	+	0.924612i	$0.624389\pi$
$278$		0			0
$279$	−12.8994			−0.0462344
$280$		0			0
$281$		−	48.9193i		−	0.174090i	−0.996204	−	0.0870450i	$-0.972258\pi$
$281$		−	48.9193i		−	0.174090i	0.996204	−	0.0870450i	$-0.0277424\pi$
$282$		0			0
$283$		−	332.591i		−	1.17523i	−0.809140	−	0.587617i	$-0.800066\pi$
$283$		−	332.591i		−	1.17523i	0.809140	−	0.587617i	$-0.199934\pi$
$284$		0			0
$285$	46.3604			0.162668
$286$		0			0
$287$		−	28.2731i		−	0.0985125i
$288$		0			0
$289$	274.300			0.949134
$290$		0			0
$291$			235.752i			0.810145i
$292$		0			0
$293$			50.9833i			0.174004i	0.996208	+	0.0870022i	$0.0277287\pi$
$293$			50.9833i			0.174004i	−0.996208	+	0.0870022i	$0.972271\pi$
$294$		0			0
$295$		−	39.3801i		−	0.133492i
$296$		0			0
$297$			122.047i			0.410933i
$298$		0			0
$299$	−26.7221	+	452.325i	−0.0893715	+	1.51279i
$300$		0			0
$301$	4.79416			0.0159274
$302$		0			0
$303$	43.6023			0.143902
$304$		0			0
$305$	31.4170			0.103007
$306$		0			0
$307$	−131.275			−0.427604			−0.213802	−	0.976877i	$-0.568585\pi$
$307$	−131.275			−0.427604			−0.213802	+	0.976877i	$0.568585\pi$
$308$		0			0
$309$		−	141.038i		−	0.456433i
$310$		0			0
$311$	66.6581			0.214335			0.107167	−	0.994241i	$-0.465822\pi$
$311$	66.6581			0.214335			0.107167	+	0.994241i	$0.465822\pi$
$312$		0			0
$313$			560.462i			1.79061i	0.445449	+	0.895307i	$0.353044\pi$
$313$			560.462i			1.79061i	−0.445449	+	0.895307i	$0.646956\pi$
$314$		0			0
$315$	−40.8572			−0.129706
$316$		0			0
$317$	−225.933			−0.712722			−0.356361	−	0.934348i	$-0.615983\pi$
$317$	−225.933			−0.712722			−0.356361	+	0.934348i	$0.615983\pi$
$318$		0			0
$319$		−	182.476i		−	0.572025i
$320$		0			0
$321$		−	426.569i		−	1.32888i
$322$		0			0
$323$	−33.7582			−0.104514
$324$		0			0
$325$	98.5031			0.303086
$326$		0			0
$327$			149.244i			0.456403i
$328$		0			0
$329$		−	68.2159i		−	0.207343i
$330$		0			0
$331$	−560.300			−1.69275			−0.846375	−	0.532587i	$-0.821220\pi$
$331$	−560.300			−1.69275			−0.846375	+	0.532587i	$0.821220\pi$
$332$		0			0
$333$		−	136.900i		−	0.411110i
$334$		0			0
$335$	246.602			0.736125
$336$		0			0
$337$			216.133i			0.641343i	0.947191	+	0.320671i	$0.103909\pi$
$337$			216.133i			0.641343i	−0.947191	+	0.320671i	$0.896091\pi$
$338$		0			0
$339$		−	295.682i		−	0.872217i
$340$		0			0
$341$			15.5358i			0.0455596i
$342$		0			0
$343$			370.360i			1.07977i
$344$		0			0
$345$	120.893	+	7.14200i	0.350414	+	0.0207014i
$346$		0			0
$347$	−398.199			−1.14755			−0.573774	−	0.819014i	$-0.694521\pi$
$347$	−398.199			−1.14755			−0.573774	+	0.819014i	$0.694521\pi$
$348$		0			0
$349$	−211.211			−0.605189			−0.302595	−	0.953119i	$-0.597853\pi$
$349$	−211.211			−0.605189			−0.302595	+	0.953119i	$0.597853\pi$
$350$		0			0
$351$	−577.795			−1.64614
$352$		0			0
$353$	416.864			1.18092			0.590458	−	0.807068i	$-0.298947\pi$
$353$	416.864			1.18092			0.590458	+	0.807068i	$0.298947\pi$
$354$		0			0
$355$		−	163.233i		−	0.459811i
$356$		0			0
$357$	−47.7442			−0.133737
$358$		0			0
$359$		−	106.974i		−	0.297976i	−0.988839	−	0.148988i	$-0.952398\pi$
$359$		−	106.974i		−	0.297976i	0.988839	−	0.148988i	$-0.0476016\pi$
$360$		0			0
$361$	283.476			0.785253
$362$		0			0
$363$	−244.148			−0.672583
$364$		0			0
$365$			8.68324i			0.0237897i
$366$		0			0
$367$			602.527i			1.64176i	0.571098	+	0.820882i	$0.306518\pi$
$367$			602.527i			1.64176i	−0.571098	+	0.820882i	$0.693482\pi$
$368$		0			0
$369$	18.4726			0.0500612
$370$		0			0
$371$	93.2554			0.251362
$372$		0			0
$373$			358.988i			0.962436i	0.876601	+	0.481218i	$0.159805\pi$
$373$			358.988i			0.962436i	−0.876601	+	0.481218i	$0.840195\pi$
$374$		0			0
$375$		−	26.3269i		−	0.0702050i
$376$		0			0
$377$	863.876			2.29145
$378$		0			0
$379$		−	350.164i		−	0.923916i	−0.886902	−	0.461958i	$-0.847147\pi$
$379$		−	350.164i		−	0.923916i	0.886902	−	0.461958i	$-0.152853\pi$
$380$		0			0
$381$	−140.232			−0.368063
$382$		0			0
$383$		−	560.121i		−	1.46246i	−0.682132	−	0.731229i	$-0.738947\pi$
$383$		−	560.121i		−	1.46246i	0.682132	−	0.731229i	$-0.261053\pi$
$384$		0			0
$385$			49.2078i			0.127812i
$386$		0			0
$387$			3.13232i			0.00809386i
$388$		0			0
$389$		−	688.226i		−	1.76922i	−0.466333	−	0.884609i	$-0.654425\pi$
$389$		−	688.226i		−	1.76922i	0.466333	−	0.884609i	$-0.345575\pi$
$390$		0			0
$391$	−88.0304	−	5.20058i	−0.225142	−	0.0133007i
$392$		0			0
$393$	199.897			0.508643
$394$		0			0
$395$	−27.5313			−0.0696995
$396$		0			0
$397$	−409.796			−1.03223			−0.516116	−	0.856519i	$-0.672622\pi$
$397$	−409.796			−1.03223			−0.516116	+	0.856519i	$0.672622\pi$
$398$		0			0
$399$	−109.642			−0.274792
$400$		0			0
$401$		−	338.132i		−	0.843221i	−0.906777	−	0.421611i	$-0.861465\pi$
$401$		−	338.132i		−	0.843221i	0.906777	−	0.421611i	$-0.138535\pi$
$402$		0			0
$403$	−73.5496			−0.182505
$404$		0			0
$405$			84.8930i			0.209612i
$406$		0			0
$407$	−164.880			−0.405110
$408$		0			0
$409$	−194.503			−0.475556			−0.237778	−	0.971319i	$-0.576419\pi$
$409$	−194.503			−0.475556			−0.237778	+	0.971319i	$0.576419\pi$
$410$		0			0
$411$			149.943i			0.364824i
$412$		0			0
$413$			93.1336i			0.225505i
$414$		0			0
$415$	−135.694			−0.326974
$416$		0			0
$417$	−486.264			−1.16610
$418$		0			0
$419$		−	61.8558i		−	0.147627i	−0.997272	−	0.0738136i	$-0.976483\pi$
$419$		−	61.8558i		−	0.147627i	0.997272	−	0.0738136i	$-0.0235170\pi$
$420$		0			0
$421$		−	613.421i		−	1.45706i	−0.685016	−	0.728528i	$-0.740205\pi$
$421$		−	613.421i		−	1.45706i	0.685016	−	0.728528i	$-0.259795\pi$
$422$		0			0
$423$	44.5697			0.105366
$424$		0			0
$425$			19.1704i			0.0451068i
$426$		0			0
$427$	−74.3010			−0.174007
$428$		0			0
$429$			193.045i			0.449987i
$430$		0			0
$431$		−	202.052i		−	0.468799i	−0.972140	−	0.234400i	$-0.924688\pi$
$431$		−	202.052i		−	0.468799i	0.972140	−	0.234400i	$-0.0753124\pi$
$432$		0			0
$433$			263.987i			0.609669i	0.952405	+	0.304834i	$0.0986011\pi$
$433$			263.987i			0.609669i	−0.952405	+	0.304834i	$0.901399\pi$
$434$		0			0
$435$		−	230.888i		−	0.530777i
$436$		0			0
$437$	−202.157	−	11.9429i	−0.462602	−	0.0273292i
$438$		0			0
$439$	−220.227			−0.501656			−0.250828	−	0.968032i	$-0.580703\pi$
$439$	−220.227			−0.501656			−0.250828	+	0.968032i	$0.580703\pi$
$440$		0			0
$441$	−72.6761			−0.164798
$442$		0			0
$443$	231.412			0.522376			0.261188	−	0.965288i	$-0.415886\pi$
$443$	231.412			0.522376			0.261188	+	0.965288i	$0.415886\pi$
$444$		0			0
$445$	−286.594			−0.644032
$446$		0			0
$447$			410.384i			0.918086i
$448$		0			0
$449$	392.154			0.873394			0.436697	−	0.899609i	$-0.356148\pi$
$449$	392.154			0.873394			0.436697	+	0.899609i	$0.356148\pi$
$450$		0			0
$451$		−	22.2480i		−	0.0493305i
$452$		0			0
$453$	−647.232			−1.42877
$454$		0			0
$455$	−232.959			−0.511998
$456$		0			0
$457$			505.038i			1.10512i	0.833474	+	0.552558i	$0.186348\pi$
$457$			505.038i			1.10512i	−0.833474	+	0.552558i	$0.813652\pi$
$458$		0			0
$459$		−	112.449i		−	0.244987i
$460$		0			0
$461$	−643.395			−1.39565			−0.697825	−	0.716268i	$-0.745849\pi$
$461$	−643.395			−1.39565			−0.697825	+	0.716268i	$0.745849\pi$
$462$		0			0
$463$	−374.824			−0.809555			−0.404778	−	0.914415i	$-0.632651\pi$
$463$	−374.824			−0.809555			−0.404778	+	0.914415i	$0.632651\pi$
$464$		0			0
$465$			19.6576i			0.0422744i
$466$		0			0
$467$			560.197i			1.19957i	0.800163	+	0.599783i	$0.204746\pi$
$467$			560.197i			1.19957i	−0.800163	+	0.599783i	$0.795254\pi$
$468$		0			0
$469$	−583.212			−1.24352
$470$		0			0
$471$		−	216.741i		−	0.460172i
$472$		0			0
$473$	3.77252			0.00797572
$474$		0			0
$475$			44.0238i			0.0926817i
$476$		0			0
$477$			60.9295i			0.127735i
$478$		0			0
$479$			111.401i			0.232570i	0.993216	+	0.116285i	$0.0370986\pi$
$479$			111.401i			0.232570i	−0.993216	+	0.116285i	$0.962901\pi$
$480$		0			0
$481$		−	780.573i		−	1.62281i
$482$		0			0
$483$	−285.911	−	16.8908i	−0.591948	−	0.0349706i
$484$		0			0
$485$	−223.870			−0.461588
$486$		0			0
$487$	102.519			0.210511			0.105256	−	0.994445i	$-0.466434\pi$
$487$	102.519			0.210511			0.105256	+	0.994445i	$0.466434\pi$
$488$		0			0
$489$	−392.075			−0.801788
$490$		0			0
$491$	194.203			0.395525			0.197762	−	0.980250i	$-0.436633\pi$
$491$	194.203			0.395525			0.197762	+	0.980250i	$0.436633\pi$
$492$		0			0
$493$			168.125i			0.341025i
$494$		0			0
$495$	−32.1505			−0.0649505
$496$		0			0
$497$			386.045i			0.776750i
$498$		0			0
$499$	743.076			1.48913			0.744565	−	0.667550i	$-0.232657\pi$
$499$	743.076			1.48913			0.744565	+	0.667550i	$0.232657\pi$
$500$		0			0
$501$	−451.317			−0.900832
$502$		0			0
$503$		−	24.7740i		−	0.0492524i	−0.999697	−	0.0246262i	$-0.992160\pi$
$503$		−	24.7740i		−	0.0492524i	0.999697	−	0.0246262i	$-0.00783956\pi$
$504$		0			0
$505$			41.4047i			0.0819895i
$506$		0			0
$507$	−515.958			−1.01767
$508$		0			0
$509$	−68.3473			−0.134278			−0.0671388	−	0.997744i	$-0.521387\pi$
$509$	−68.3473			−0.134278			−0.0671388	+	0.997744i	$0.521387\pi$
$510$		0			0
$511$		−	20.5358i		−	0.0401875i
$512$		0			0
$513$		−	258.233i		−	0.503378i
$514$		0			0
$515$	133.930			0.260057
$516$		0			0
$517$		−	53.6790i		−	0.103828i
$518$		0			0
$519$	−192.036			−0.370012
$520$		0			0
$521$		−	675.810i		−	1.29714i	−0.761155	−	0.648570i	$-0.775367\pi$
$521$		−	675.810i		−	1.29714i	0.761155	−	0.648570i	$-0.224633\pi$
$522$		0			0
$523$		−	832.938i		−	1.59262i	−0.604892	−	0.796308i	$-0.706784\pi$
$523$		−	832.938i		−	1.59262i	0.604892	−	0.796308i	$-0.293216\pi$
$524$		0			0
$525$			62.2629i			0.118596i
$526$		0			0
$527$		−	14.3140i		−	0.0271614i
$528$		0			0
$529$	−525.320	−	62.2862i	−0.993044	−	0.117743i
$530$		0			0
$531$	−60.8500			−0.114595
$532$		0			0
$533$	105.327			0.197611
$534$		0			0
$535$	405.070			0.757141
$536$		0			0
$537$	18.1819			0.0338583
$538$		0			0
$539$			87.5299i			0.162393i
$540$		0			0
$541$	−716.526			−1.32445			−0.662224	−	0.749306i	$-0.730387\pi$
$541$	−716.526			−1.32445			−0.662224	+	0.749306i	$0.730387\pi$
$542$		0			0
$543$			37.8192i			0.0696486i
$544$		0			0
$545$	−141.722			−0.260040
$546$		0			0
$547$	252.767			0.462096			0.231048	−	0.972942i	$-0.425784\pi$
$547$	252.767			0.462096			0.231048	+	0.972942i	$0.425784\pi$
$548$		0			0
$549$		−	48.5455i		−	0.0884253i
$550$		0			0
$551$			386.091i			0.700709i
$552$		0			0
$553$	65.1113			0.117742
$554$		0			0
$555$	−208.623			−0.375898
$556$		0			0
$557$			917.398i			1.64703i	0.567292	+	0.823517i	$0.307991\pi$
$557$			917.398i			1.64703i	−0.567292	+	0.823517i	$0.692009\pi$
$558$		0			0
$559$			17.8598i			0.0319496i
$560$		0			0
$561$	−37.5698			−0.0669694
$562$		0			0
$563$		−	910.746i		−	1.61767i	−0.588038	−	0.808833i	$-0.700100\pi$
$563$		−	910.746i		−	1.61767i	0.588038	−	0.808833i	$-0.299900\pi$
$564$		0			0
$565$	280.779			0.496955
$566$		0			0
$567$		−	200.771i		−	0.354094i
$568$		0			0
$569$			781.650i			1.37373i	0.726787	+	0.686863i	$0.241013\pi$
$569$			781.650i			1.37373i	−0.726787	+	0.686863i	$0.758987\pi$
$570$		0			0
$571$			151.657i			0.265598i	0.991143	+	0.132799i	$0.0423965\pi$
$571$			151.657i			0.265598i	−0.991143	+	0.132799i	$0.957603\pi$
$572$		0			0
$573$		−	36.9498i		−	0.0644848i
$574$		0			0
$575$	−6.78204	+	114.800i	−0.0117949	+	0.199652i
$576$		0			0
$577$	−581.544			−1.00787			−0.503937	−	0.863740i	$-0.668116\pi$
$577$	−581.544			−1.00787			−0.503937	+	0.863740i	$0.668116\pi$
$578$		0			0
$579$	277.507			0.479287
$580$		0			0
$581$	320.916			0.552351
$582$		0			0
$583$	73.3825			0.125870
$584$		0			0
$585$		−	152.207i		−	0.260182i
$586$		0			0
$587$	531.175			0.904899			0.452449	−	0.891790i	$-0.350550\pi$
$587$	531.175			0.904899			0.452449	+	0.891790i	$0.350550\pi$
$588$		0			0
$589$		−	32.8714i		−	0.0558088i
$590$		0			0
$591$	−600.508			−1.01609
$592$		0			0
$593$	804.908			1.35735			0.678675	−	0.734439i	$-0.262555\pi$
$593$	804.908			1.35735			0.678675	+	0.734439i	$0.262555\pi$
$594$		0			0
$595$		−	45.3379i		−	0.0761981i
$596$		0			0
$597$		−	262.898i		−	0.440365i
$598$		0			0
$599$	−94.3895			−0.157578			−0.0787892	−	0.996891i	$-0.525105\pi$
$599$	−94.3895			−0.157578			−0.0787892	+	0.996891i	$0.525105\pi$
$600$		0			0
$601$	−681.262			−1.13355			−0.566774	−	0.823874i	$-0.691809\pi$
$601$	−681.262			−1.13355			−0.566774	+	0.823874i	$0.691809\pi$
$602$		0			0
$603$		−	381.049i		−	0.631922i
$604$		0			0
$605$		−	231.843i		−	0.383211i
$606$		0			0
$607$	745.339			1.22791			0.613953	−	0.789342i	$-0.289578\pi$
$607$	745.339			1.22791			0.613953	+	0.789342i	$0.289578\pi$
$608$		0			0
$609$			546.048i			0.896631i
$610$		0			0
$611$	254.127			0.415919
$612$		0			0
$613$			523.208i			0.853521i	0.904365	+	0.426761i	$0.140345\pi$
$613$			523.208i			0.853521i	−0.904365	+	0.426761i	$0.859655\pi$
$614$		0			0
$615$		−	28.1506i		−	0.0457733i
$616$		0			0
$617$			311.646i			0.505098i	0.967584	+	0.252549i	$0.0812689\pi$
$617$			311.646i			0.505098i	−0.967584	+	0.252549i	$0.918731\pi$
$618$		0			0
$619$			713.343i			1.15241i	0.817305	+	0.576206i	$0.195467\pi$
$619$			713.343i			1.15241i	−0.817305	+	0.576206i	$0.804533\pi$
$620$		0			0
$621$	39.7818	−	673.388i	0.0640609	−	1.08436i
$622$		0			0
$623$	677.793			1.08795
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−86.2771			−0.137603
$628$		0			0
$629$	151.913			0.241515
$630$		0			0
$631$			452.931i			0.717799i	0.933376	+	0.358899i	$0.116848\pi$
$631$			452.931i			0.717799i	−0.933376	+	0.358899i	$0.883152\pi$
$632$		0			0
$633$	−363.634			−0.574461
$634$		0			0
$635$		−	133.165i		−	0.209708i
$636$		0			0
$637$	−414.383			−0.650523
$638$		0			0
$639$	−252.227			−0.394722
$640$		0			0
$641$		−	384.308i		−	0.599545i	−0.954011	−	0.299772i	$-0.903089\pi$
$641$		−	384.308i		−	0.599545i	0.954011	−	0.299772i	$-0.0969108\pi$
$642$		0			0
$643$			390.029i			0.606577i	0.952899	+	0.303288i	$0.0980846\pi$
$643$			390.029i			0.606577i	−0.952899	+	0.303288i	$0.901915\pi$
$644$		0			0
$645$	4.77339			0.00740060
$646$		0			0
$647$	−13.3648			−0.0206566			−0.0103283	−	0.999947i	$-0.503288\pi$
$647$	−13.3648			−0.0206566			−0.0103283	+	0.999947i	$0.503288\pi$
$648$		0			0
$649$			73.2867i			0.112922i
$650$		0			0
$651$		−	46.4900i		−	0.0714133i
$652$		0			0
$653$	−1041.26			−1.59458			−0.797292	−	0.603593i	$-0.793735\pi$
$653$	−1041.26			−1.59458			−0.797292	+	0.603593i	$0.793735\pi$
$654$		0			0
$655$			189.822i			0.289804i
$656$		0			0
$657$	13.4173			0.0204221
$658$		0			0
$659$			949.533i			1.44087i	0.693523	+	0.720435i	$0.256058\pi$
$659$			949.533i			1.44087i	−0.693523	+	0.720435i	$0.743942\pi$
$660$		0			0
$661$		−	899.192i		−	1.36035i	−0.733050	−	0.680175i	$-0.761904\pi$
$661$		−	899.192i		−	1.36035i	0.733050	−	0.680175i	$-0.238096\pi$
$662$		0			0
$663$		−	177.863i		−	0.268270i
$664$		0			0
$665$		−	104.116i		−	0.156565i
$666$		0			0
$667$	−59.4789	+	1006.80i	−0.0891737	+	1.50945i
$668$		0			0
$669$	−259.137			−0.387350
$670$		0			0
$671$	−58.4673			−0.0871346
$672$		0			0
$673$	−1068.44			−1.58757			−0.793786	−	0.608197i	$-0.791893\pi$
$673$	−1068.44			−1.58757			−0.793786	+	0.608197i	$0.791893\pi$
$674$		0			0
$675$	−146.644			−0.217250
$676$		0			0
$677$			965.209i			1.42571i	0.701309	+	0.712857i	$0.252599\pi$
$677$			965.209i			1.42571i	−0.701309	+	0.712857i	$0.747401\pi$
$678$		0			0
$679$	529.452			0.779752
$680$		0			0
$681$			234.763i			0.344733i
$682$		0			0
$683$	276.423			0.404719			0.202360	−	0.979311i	$-0.435139\pi$
$683$	276.423			0.404719			0.202360	+	0.979311i	$0.435139\pi$
$684$		0			0
$685$	−142.386			−0.207862
$686$		0			0
$687$		−	246.269i		−	0.358470i
$688$		0			0
$689$			347.407i			0.504219i
$690$		0			0
$691$	1014.50			1.46817			0.734083	−	0.679059i	$-0.237612\pi$
$691$	1014.50			1.46817			0.734083	+	0.679059i	$0.237612\pi$
$692$		0			0
$693$	76.0357			0.109720
$694$		0			0
$695$		−	461.756i		−	0.664398i
$696$		0			0
$697$			20.4984i			0.0294094i
$698$		0			0
$699$	−703.594			−1.00657
$700$		0			0
$701$		−	382.835i		−	0.546126i	−0.961996	−	0.273063i	$-0.911963\pi$
$701$		−	382.835i		−	0.546126i	0.961996	−	0.273063i	$-0.0880368\pi$
$702$		0			0
$703$	348.860			0.496244
$704$		0			0
$705$		−	67.9203i		−	0.0963409i
$706$		0			0
$707$		−	97.9219i		−	0.138503i
$708$		0			0
$709$			487.474i			0.687552i	0.939052	+	0.343776i	$0.111706\pi$
$709$			487.474i			0.687552i	−0.939052	+	0.343776i	$0.888294\pi$
$710$		0			0
$711$			42.5413i			0.0598331i
$712$		0			0
$713$	5.06397	−	85.7180i	0.00710235	−	0.120222i
$714$		0			0
$715$	−183.315			−0.256385
$716$		0			0
$717$	375.619			0.523876
$718$		0			0
$719$	−632.258			−0.879357			−0.439678	−	0.898155i	$-0.644908\pi$
$719$	−632.258			−0.879357			−0.439678	+	0.898155i	$0.644908\pi$
$720$		0			0
$721$	−316.742			−0.439310
$722$		0			0
$723$			938.839i			1.29853i
$724$		0			0
$725$	219.251			0.302415
$726$		0			0
$727$		−	937.310i		−	1.28929i	−0.764484	−	0.644643i	$-0.777006\pi$
$727$		−	937.310i		−	1.28929i	0.764484	−	0.644643i	$-0.222994\pi$
$728$		0			0
$729$	752.733			1.03256
$730$		0			0
$731$	−3.47583			−0.00475490
$732$		0			0
$733$			608.487i			0.830132i	0.909791	+	0.415066i	$0.136242\pi$
$733$			608.487i			0.830132i	−0.909791	+	0.415066i	$0.863758\pi$
$734$		0			0
$735$			110.752i			0.150683i
$736$		0			0
$737$	−458.929			−0.622698
$738$		0			0
$739$	304.462			0.411992			0.205996	−	0.978553i	$-0.433957\pi$
$739$	304.462			0.411992			0.205996	+	0.978553i	$0.433957\pi$
$740$		0			0
$741$		−	408.452i		−	0.551218i
$742$		0			0
$743$			789.160i			1.06213i	0.847332	+	0.531063i	$0.178207\pi$
$743$			789.160i			1.06213i	−0.847332	+	0.531063i	$0.821793\pi$
$744$		0			0
$745$	−389.701			−0.523089
$746$		0			0
$747$			209.675i			0.280689i
$748$		0			0
$749$	−957.988			−1.27902
$750$		0			0
$751$		−	970.882i		−	1.29279i	−0.763005	−	0.646393i	$-0.776277\pi$
$751$		−	970.882i		−	1.29279i	0.763005	−	0.646393i	$-0.223723\pi$
$752$		0			0
$753$		−	356.232i		−	0.473084i
$754$		0			0
$755$		−	614.612i		−	0.814056i
$756$		0			0
$757$		−	534.555i		−	0.706150i	−0.935595	−	0.353075i	$-0.885136\pi$
$757$		−	534.555i		−	0.706150i	0.935595	−	0.353075i	$-0.114864\pi$
$758$		0			0
$759$	−224.983	−	13.2913i	−0.296420	−	0.0175116i
$760$		0			0
$761$	1243.09			1.63349			0.816745	−	0.576999i	$-0.195776\pi$
$761$	1243.09			1.63349			0.816745	+	0.576999i	$0.195776\pi$
$762$		0			0
$763$	335.171			0.439281
$764$		0			0
$765$	29.6221			0.0387217
$766$		0			0
$767$	−346.953			−0.452351
$768$		0			0
$769$		−	646.815i		−	0.841112i	−0.907267	−	0.420556i	$-0.861835\pi$
$769$		−	646.815i		−	0.841112i	0.907267	−	0.420556i	$-0.138165\pi$
$770$		0			0
$771$	−822.971			−1.06741
$772$		0			0
$773$			1349.10i			1.74528i	0.488360	+	0.872642i	$0.337595\pi$
$773$			1349.10i			1.74528i	−0.488360	+	0.872642i	$0.662405\pi$
$774$		0			0
$775$	−18.6668			−0.0240862
$776$		0			0
$777$	493.393			0.634997
$778$		0			0
$779$			47.0734i			0.0604280i
$780$		0			0
$781$			303.778i			0.388960i
$782$		0			0
$783$	−1286.07			−1.64249
$784$		0			0
$785$	205.817			0.262188
$786$		0			0
$787$		−	742.381i		−	0.943305i	−0.881784	−	0.471653i	$-0.843658\pi$
$787$		−	742.381i		−	0.943305i	0.881784	−	0.471653i	$-0.156342\pi$
$788$		0			0
$789$			424.399i			0.537895i
$790$		0			0
$791$	−664.041			−0.839496
$792$		0			0
$793$		−	276.796i		−	0.349049i
$794$		0			0
$795$	92.8513			0.116794
$796$		0			0
$797$		−	974.692i		−	1.22295i	−0.791263	−	0.611476i	$-0.790576\pi$
$797$		−	974.692i		−	1.22295i	0.791263	−	0.611476i	$-0.209424\pi$
$798$		0			0
$799$			49.4575i			0.0618992i
$800$		0			0
$801$			442.845i			0.552865i
$802$		0			0
$803$		−	16.1596i		−	0.0201240i
$804$		0			0
$805$	16.0395	−	271.501i	0.0199248	−	0.337268i
$806$		0			0
$807$	−210.692			−0.261081
$808$		0			0
$809$	660.157			0.816016			0.408008	−	0.912978i	$-0.366224\pi$
$809$	660.157			0.816016			0.408008	+	0.912978i	$0.366224\pi$
$810$		0			0
$811$	400.770			0.494167			0.247084	−	0.968994i	$-0.420528\pi$
$811$	400.770			0.494167			0.247084	+	0.968994i	$0.420528\pi$
$812$		0			0
$813$	−110.793			−0.136277
$814$		0			0
$815$		−	372.314i		−	0.456827i
$816$		0			0
$817$	−7.98206			−0.00976997
$818$		0			0
$819$			359.968i			0.439521i
$820$		0			0
$821$	−649.137			−0.790666			−0.395333	−	0.918538i	$-0.629371\pi$
$821$	−649.137			−0.790666			−0.395333	+	0.918538i	$0.629371\pi$
$822$		0			0
$823$	−602.232			−0.731752			−0.365876	−	0.930664i	$-0.619231\pi$
$823$	−602.232			−0.731752			−0.365876	+	0.930664i	$0.619231\pi$
$824$		0			0
$825$			48.9946i			0.0593873i
$826$		0			0
$827$			1076.03i			1.30112i	0.759454	+	0.650561i	$0.225466\pi$
$827$			1076.03i			1.30112i	−0.759454	+	0.650561i	$0.774534\pi$
$828$		0			0
$829$	183.571			0.221437			0.110718	−	0.993852i	$-0.464685\pi$
$829$	183.571			0.221437			0.110718	+	0.993852i	$0.464685\pi$
$830$		0			0
$831$	496.907			0.597963
$832$		0			0
$833$		−	80.6462i		−	0.0968142i
$834$		0			0
$835$		−	428.571i		−	0.513258i
$836$		0			0
$837$	109.495			0.130818
$838$		0			0
$839$			156.570i			0.186615i	0.995637	+	0.0933075i	$0.0297440\pi$
$839$			156.570i			0.186615i	−0.995637	+	0.0933075i	$0.970256\pi$
$840$		0			0
$841$	1081.84			1.28638
$842$		0			0
$843$			115.193i			0.136646i
$844$		0			0
$845$		−	489.954i		−	0.579827i
$846$		0			0
$847$			548.306i			0.647351i
$848$		0			0
$849$			783.167i			0.922459i
$850$		0			0
$851$	909.714	+	53.7433i	1.06899	+	0.0631531i
$852$		0			0
$853$	−749.390			−0.878535			−0.439267	−	0.898356i	$-0.644762\pi$
$853$	−749.390			−0.878535			−0.439267	+	0.898356i	$0.644762\pi$
$854$		0			0
$855$	68.0255			0.0795620
$856$		0			0
$857$	1596.34			1.86270			0.931351	−	0.364123i	$-0.118631\pi$
$857$	1596.34			1.86270			0.931351	+	0.364123i	$0.118631\pi$
$858$		0			0
$859$	207.405			0.241450			0.120725	−	0.992686i	$-0.461478\pi$
$859$	207.405			0.241450			0.120725	+	0.992686i	$0.461478\pi$
$860$		0			0
$861$			66.5760i			0.0773240i
$862$		0			0
$863$	−471.177			−0.545976			−0.272988	−	0.962017i	$-0.588012\pi$
$863$	−471.177			−0.545976			−0.272988	+	0.962017i	$0.588012\pi$
$864$		0			0
$865$		−	182.358i		−	0.210818i
$866$		0			0
$867$	−645.907			−0.744990
$868$		0			0
$869$	51.2360			0.0589598
$870$		0			0
$871$		−	2172.66i		−	2.49444i
$872$		0			0
$873$			345.924i			0.396247i
$874$		0			0
$875$	−59.1248			−0.0675712
$876$		0			0
$877$	−1305.15			−1.48820			−0.744098	−	0.668070i	$-0.767121\pi$
$877$	−1305.15			−1.48820			−0.744098	+	0.668070i	$0.767121\pi$
$878$		0			0
$879$		−	120.053i		−	0.136579i
$880$		0			0
$881$			733.756i			0.832867i	0.909166	+	0.416434i	$0.136720\pi$
$881$			733.756i			0.832867i	−0.909166	+	0.416434i	$0.863280\pi$
$882$		0			0
$883$	1422.70			1.61121			0.805605	−	0.592452i	$-0.201840\pi$
$883$	1422.70			1.61121			0.805605	+	0.592452i	$0.201840\pi$
$884$		0			0
$885$			92.7301i			0.104780i
$886$		0			0
$887$	267.354			0.301414			0.150707	−	0.988578i	$-0.451845\pi$
$887$	267.354			0.301414			0.150707	+	0.988578i	$0.451845\pi$
$888$		0			0
$889$			314.933i			0.354256i
$890$		0			0
$891$		−	157.987i		−	0.177314i
$892$		0			0
$893$			113.576i			0.127185i
$894$		0			0
$895$			17.2655i			0.0192911i
$896$		0			0
$897$	62.9237	−	1065.11i	0.0701491	−	1.18742i
$898$		0			0
$899$	−163.709			−0.182101
$900$		0			0
$901$	−67.6114			−0.0750404
$902$		0			0
$903$	−11.2890			−0.0125017
$904$		0			0
$905$	−35.9131			−0.0396830
$906$		0			0
$907$			287.288i			0.316745i	0.987379	+	0.158373i	$0.0506247\pi$
$907$			287.288i			0.316745i	−0.987379	+	0.158373i	$0.949375\pi$
$908$		0			0
$909$	63.9785			0.0703834
$910$		0			0
$911$		−	1398.07i		−	1.53465i	−0.641256	−	0.767327i	$-0.721586\pi$
$911$		−	1398.07i		−	1.53465i	0.641256	−	0.767327i	$-0.278414\pi$
$912$		0			0
$913$	252.528			0.276592
$914$		0			0
$915$	−73.9791			−0.0808514
$916$		0			0
$917$		−	448.927i		−	0.489561i
$918$		0			0
$919$			1350.74i			1.46979i	0.678181	+	0.734895i	$0.262769\pi$
$919$			1350.74i			1.46979i	−0.678181	+	0.734895i	$0.737231\pi$
$920$		0			0
$921$	309.118			0.335633
$922$		0			0
$923$	−1438.14			−1.55812
$924$		0			0
$925$		−	198.109i		−	0.214172i
$926$		0			0
$927$		−	206.948i		−	0.223244i
$928$		0			0
$929$	913.546			0.983364			0.491682	−	0.870775i	$-0.336382\pi$
$929$	913.546			0.983364			0.491682	+	0.870775i	$0.336382\pi$
$930$		0			0
$931$		−	185.200i		−	0.198925i
$932$		0			0
$933$	−156.963			−0.168235
$934$		0			0
$935$		−	35.6763i		−	0.0381565i
$936$		0			0
$937$		−	1718.09i		−	1.83361i	−0.399335	−	0.916805i	$-0.630759\pi$
$937$		−	1718.09i		−	1.83361i	0.399335	−	0.916805i	$-0.369241\pi$
$938$		0			0
$939$		−	1319.75i		−	1.40548i
$940$		0			0
$941$		−	556.050i		−	0.590914i	−0.955356	−	0.295457i	$-0.904528\pi$
$941$		−	556.050i		−	0.590914i	0.955356	−	0.295457i	$-0.0954720\pi$
$942$		0			0
$943$	−7.25185	+	122.752i	−0.00769019	+	0.130172i
$944$		0			0
$945$	346.812			0.366996
$946$		0			0
$947$	884.819			0.934339			0.467170	−	0.884168i	$-0.345274\pi$
$947$	884.819			0.934339			0.467170	+	0.884168i	$0.345274\pi$
$948$		0			0
$949$	76.5027			0.0806140
$950$		0			0
$951$	532.015			0.559427
$952$		0			0
$953$		−	715.596i		−	0.750888i	−0.926845	−	0.375444i	$-0.877490\pi$
$953$		−	715.596i		−	0.750888i	0.926845	−	0.375444i	$-0.122510\pi$
$954$		0			0
$955$	35.0875			0.0367409
$956$		0			0
$957$			429.685i			0.448991i
$958$		0			0
$959$	336.741			0.351137
$960$		0			0
$961$	−947.062			−0.985496
$962$		0			0
$963$		−	625.914i		−	0.649962i
$964$		0			0
$965$			263.521i			0.273079i
$966$		0			0
$967$	−485.848			−0.502428			−0.251214	−	0.967932i	$-0.580830\pi$
$967$	−485.848			−0.502428			−0.251214	+	0.967932i	$0.580830\pi$
$968$		0			0
$969$	79.4919			0.0820350
$970$		0			0
$971$		−	828.478i		−	0.853221i	−0.904435	−	0.426611i	$-0.859708\pi$
$971$		−	828.478i		−	0.853221i	0.904435	−	0.426611i	$-0.140292\pi$
$972$		0			0
$973$			1092.05i			1.12235i
$974$		0			0
$975$	−231.950			−0.237897
$976$		0			0
$977$			418.692i			0.428549i	0.976774	+	0.214274i	$0.0687387\pi$
$977$			418.692i			0.428549i	−0.976774	+	0.214274i	$0.931261\pi$
$978$		0			0
$979$	533.355			0.544795
$980$		0			0
$981$			218.988i			0.223230i
$982$		0			0
$983$			1438.96i			1.46384i	0.681389	+	0.731922i	$0.261376\pi$
$983$			1438.96i			1.46384i	−0.681389	+	0.731922i	$0.738624\pi$
$984$		0			0
$985$		−	570.243i		−	0.578927i
$986$		0			0
$987$			160.631i			0.162747i
$988$		0			0
$989$	−20.8146	−	1.22967i	−0.0210461	−	0.00124335i
$990$		0			0
$991$	−476.520			−0.480848			−0.240424	−	0.970668i	$-0.577286\pi$
$991$	−476.520			−0.480848			−0.240424	+	0.970668i	$0.577286\pi$
$992$		0			0
$993$	1319.37			1.32867
$994$		0			0
$995$	249.648			0.250902
$996$		0			0
$997$	−215.632			−0.216281			−0.108140	−	0.994136i	$-0.534490\pi$
$997$	−215.632			−0.216281			−0.108140	+	0.994136i	$0.534490\pi$
$998$		0			0
$999$			1162.06i			1.16322i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.k.e.321.17		48
4.3	odd	2		920.3.k.a.321.31	✓	48
23.22	odd	2	inner	1840.3.k.e.321.18		48
92.91	even	2		920.3.k.a.321.32	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.3.k.a.321.31	✓	48	4.3	odd	2
920.3.k.a.321.32	yes	48	92.91	even	2
1840.3.k.e.321.17		48	1.1	even	1	trivial
1840.3.k.e.321.18		48	23.22	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.35475 q^{3} -2.23607i q^{5} +5.28829i q^{7} -3.45517 q^{9} +O(q^{10})\)	\(q-2.35475 q^{3} -2.23607i q^{5} +5.28829i q^{7} -3.45517 q^{9} +4.16134i q^{11} -19.7006 q^{13} +5.26537i q^{15} -3.83408i q^{17} -8.80476i q^{19} -12.4526i q^{21} +(1.35641 - 22.9600i) q^{23} -5.00000 q^{25} +29.3288 q^{27} -43.8502 q^{29} +3.73337 q^{31} -9.79891i q^{33} +11.8250 q^{35} +39.6217i q^{37} +46.3900 q^{39} -5.34636 q^{41} -0.906562i q^{43} +7.72599i q^{45} -12.8994 q^{47} +21.0340 q^{49} +9.02829i q^{51} -17.6343i q^{53} +9.30505 q^{55} +20.7330i q^{57} +17.6113 q^{59} +14.0501i q^{61} -18.2719i q^{63} +44.0519i q^{65} +110.284i q^{67} +(-3.19400 + 54.0649i) q^{69} +73.0000 q^{71} -3.88326 q^{73} +11.7737 q^{75} -22.0064 q^{77} -12.3124i q^{79} -37.9653 q^{81} -60.6843i q^{83} -8.57327 q^{85} +103.256 q^{87} -128.169i q^{89} -104.182i q^{91} -8.79113 q^{93} -19.6880 q^{95} -100.118i q^{97} -14.3781i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 128 q^{9}+O(q^{10}) \) 48 * q + 128 * q^9	\( 48 q + 128 q^{9} - 8 q^{23} - 240 q^{25} + 72 q^{29} - 32 q^{31} + 40 q^{35} + 96 q^{39} - 104 q^{41} - 128 q^{47} - 344 q^{49} - 80 q^{55} - 248 q^{59} + 292 q^{69} - 208 q^{71} + 224 q^{73} - 288 q^{77} + 184 q^{81} + 48 q^{87} - 672 q^{93}+O(q^{100}) \) 48 * q + 128 * q^9 - 8 * q^23 - 240 * q^25 + 72 * q^29 - 32 * q^31 + 40 * q^35 + 96 * q^39 - 104 * q^41 - 128 * q^47 - 344 * q^49 - 80 * q^55 - 248 * q^59 + 292 * q^69 - 208 * q^71 + 224 * q^73 - 288 * q^77 + 184 * q^81 + 48 * q^87 - 672 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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