LMFDB - Embedded newform 544.2.c.b.273.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [544,2,Mod(273,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("544.273"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 544 = 2^{5} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	544.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.34386186996$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.4469724736.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 2x^{5} - 4x^{4} + 4x^{3} - 8x + 16 $ x^8 - x^7 + 2x^5 - 4x^4 + 4x^3 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			273.6
Root			$0.733159 + 1.20933i$ of defining polynomial
Character	$\chi$	$=$	544.273
Dual form			544.2.c.b.273.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.826905i q^{3} -1.12786i q^{5} -1.74755 q^{7} +2.31623 q^{9} +5.05364i q^{11} +3.09887i q^{13} +0.932637 q^{15} +1.00000 q^{17} -1.04322i q^{19} -1.44506i q^{21} +7.54284 q^{23} +3.72792 q^{25} +4.39601i q^{27} +1.12786i q^{29} +2.68019 q^{31} -4.17888 q^{33} +1.97100i q^{35} +9.14869i q^{37} -2.56247 q^{39} -4.72792 q^{41} -1.52970i q^{43} -2.61239i q^{45} -7.66056 q^{47} -3.94606 q^{49} +0.826905i q^{51} +3.90954i q^{53} +5.69982 q^{55} +0.862647 q^{57} -11.0351i q^{59} +1.12786i q^{61} -4.04773 q^{63} +3.49510 q^{65} -2.86631i q^{67} +6.23721i q^{69} +3.01963 q^{71} +12.2230 q^{73} +3.08263i q^{75} -8.83150i q^{77} -3.95227 q^{79} +3.31360 q^{81} -9.34878i q^{83} -1.12786i q^{85} -0.932637 q^{87} -18.2672 q^{89} -5.41543i q^{91} +2.21626i q^{93} -1.17662 q^{95} -1.13735 q^{97} +11.7054i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{7} - 8 q^{9} - 12 q^{15} + 8 q^{17} + 16 q^{23} - 8 q^{25} - 24 q^{31} - 8 q^{33} + 12 q^{39} - 4 q^{47} + 8 q^{49} + 12 q^{55} + 8 q^{57} - 40 q^{63} - 24 q^{65} + 36 q^{71} + 8 q^{73} - 24 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) $ 8 * q + 12 * q^7 - 8 * q^9 - 12 * q^15 + 8 * q^17 + 16 * q^23 - 8 * q^25 - 24 * q^31 - 8 * q^33 + 12 * q^39 - 4 * q^47 + 8 * q^49 + 12 * q^55 + 8 * q^57 - 40 * q^63 - 24 * q^65 + 36 * q^71 + 8 * q^73 - 24 * q^79 + 40 * q^81 + 12 * q^87 + 8 * q^89 - 32 * q^95 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/544\mathbb{Z}\right)^\times$.

$n$	$69$	$511$	$513$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	0.826905i	0.477414i	0.971092	+	0.238707i	$0.0767235\pi$
$3$	0.826905i	0.477414i	−0.971092	+	0.238707i	$0.923277\pi$
$4$	0	0
$5$	− 1.12786i	− 0.504397i	−0.967676	−	0.252198i	$-0.918846\pi$
$5$	− 1.12786i	− 0.504397i	0.967676	−	0.252198i	$-0.0811535\pi$
$6$	0	0
$7$	−1.74755	−0.660513	−0.330256	−	0.943891i	$-0.607135\pi$
$7$	−1.74755	−0.660513	−0.330256	+	0.943891i	$0.607135\pi$
$8$	0	0
$9$	2.31623	0.772076
$10$	0	0
$11$	5.05364i	1.52373i	0.647736	+	0.761865i	$0.275716\pi$
$11$	5.05364i	1.52373i	−0.647736	+	0.761865i	$0.724284\pi$
$12$	0	0
$13$	3.09887i	0.859471i	0.902955	+	0.429736i	$0.141393\pi$
$13$	3.09887i	0.859471i	−0.902955	+	0.429736i	$0.858607\pi$
$14$	0	0
$15$	0.932637	0.240806
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	− 1.04322i	− 0.239332i	−0.992814	−	0.119666i	$-0.961818\pi$
$19$	− 1.04322i	− 0.239332i	0.992814	−	0.119666i	$-0.0381824\pi$
$20$	0	0
$21$	− 1.44506i	− 0.315338i
$22$	0	0
$23$	7.54284	1.57279	0.786395	−	0.617724i	$-0.211945\pi$
$23$	7.54284	1.57279	0.786395	+	0.617724i	$0.211945\pi$
$24$	0	0
$25$	3.72792	0.745584
$26$	0	0
$27$	4.39601i	0.846013i
$28$	0	0
$29$	1.12786i	0.209439i	0.994502	+	0.104720i	$0.0333945\pi$
$29$	1.12786i	0.209439i	−0.994502	+	0.104720i	$0.966605\pi$
$30$	0	0
$31$	2.68019	0.481376	0.240688	−	0.970603i	$-0.422627\pi$
$31$	2.68019	0.481376	0.240688	+	0.970603i	$0.422627\pi$
$32$	0	0
$33$	−4.17888	−0.727449
$34$	0	0
$35$	1.97100i	0.333160i
$36$	0	0
$37$	9.14869i	1.50404i	0.659143	+	0.752018i	$0.270919\pi$
$37$	9.14869i	1.50404i	−0.659143	+	0.752018i	$0.729081\pi$
$38$	0	0
$39$	−2.56247	−0.410323
$40$	0	0
$41$	−4.72792	−0.738377	−0.369189	−	0.929355i	$-0.620364\pi$
$41$	−4.72792	−0.738377	−0.369189	+	0.929355i	$0.620364\pi$
$42$	0	0
$43$	− 1.52970i	− 0.233277i	−0.993174	−	0.116638i	$-0.962788\pi$
$43$	− 1.52970i	− 0.233277i	0.993174	−	0.116638i	$-0.0372119\pi$
$44$	0	0
$45$	− 2.61239i	− 0.389433i
$46$	0	0
$47$	−7.66056	−1.11741	−0.558704	−	0.829367i	$-0.688701\pi$
$47$	−7.66056	−1.11741	−0.558704	+	0.829367i	$0.688701\pi$
$48$	0	0
$49$	−3.94606	−0.563723
$50$	0	0
$51$	0.826905i	0.115790i
$52$	0	0
$53$	3.90954i	0.537016i	0.963277	+	0.268508i	$0.0865307\pi$
$53$	3.90954i	0.537016i	−0.963277	+	0.268508i	$0.913469\pi$
$54$	0	0
$55$	5.69982	0.768564
$56$	0	0
$57$	0.862647	0.114260
$58$	0	0
$59$	− 11.0351i	− 1.43664i	−0.695712	−	0.718321i	$-0.744911\pi$
$59$	− 11.0351i	− 1.43664i	0.695712	−	0.718321i	$-0.255089\pi$
$60$	0	0
$61$	1.12786i	0.144408i	0.997390	+	0.0722042i	$0.0230033\pi$
$61$	1.12786i	0.144408i	−0.997390	+	0.0722042i	$0.976997\pi$
$62$	0	0
$63$	−4.04773	−0.509966
$64$	0	0
$65$	3.49510	0.433514
$66$	0	0
$67$	− 2.86631i	− 0.350176i	−0.984553	−	0.175088i	$-0.943979\pi$
$67$	− 2.86631i	− 0.350176i	0.984553	−	0.175088i	$-0.0560210\pi$
$68$	0	0
$69$	6.23721i	0.750871i
$70$	0	0
$71$	3.01963	0.358364	0.179182	−	0.983816i	$-0.442655\pi$
$71$	3.01963	0.358364	0.179182	+	0.983816i	$0.442655\pi$
$72$	0	0
$73$	12.2230	1.43060	0.715298	−	0.698819i	$-0.246291\pi$
$73$	12.2230	1.43060	0.715298	+	0.698819i	$0.246291\pi$
$74$	0	0
$75$	3.08263i	0.355952i
$76$	0	0
$77$	− 8.83150i	− 1.00644i
$78$	0	0
$79$	−3.95227	−0.444665	−0.222332	−	0.974971i	$-0.571367\pi$
$79$	−3.95227	−0.444665	−0.222332	+	0.974971i	$0.571367\pi$
$80$	0	0
$81$	3.31360	0.368178
$82$	0	0
$83$	− 9.34878i	− 1.02616i	−0.858340	−	0.513081i	$-0.828504\pi$
$83$	− 9.34878i	− 1.02616i	0.858340	−	0.513081i	$-0.171496\pi$
$84$	0	0
$85$	− 1.12786i	− 0.122334i
$86$	0	0
$87$	−0.932637	−0.0999891
$88$	0	0
$89$	−18.2672	−1.93632	−0.968158	−	0.250339i	$-0.919458\pi$
$89$	−18.2672	−1.93632	−0.968158	+	0.250339i	$0.919458\pi$
$90$	0	0
$91$	− 5.41543i	− 0.567692i
$92$	0	0
$93$	2.21626i	0.229816i
$94$	0	0
$95$	−1.17662	−0.120718
$96$	0	0
$97$	−1.13735	−0.115481	−0.0577403	−	0.998332i	$-0.518390\pi$
$97$	−1.13735	−0.115481	−0.0577403	+	0.998332i	$0.518390\pi$
$98$	0	0
$99$	11.7054i	1.17644i
$100$	0	0
$101$	− 15.1985i	− 1.51231i	−0.654393	−	0.756154i	$-0.727076\pi$
$101$	− 15.1985i	− 1.51231i	0.654393	−	0.756154i	$-0.272924\pi$
$102$	0	0
$103$	14.6325	1.44178	0.720889	−	0.693050i	$-0.243734\pi$
$103$	14.6325	1.44178	0.720889	+	0.693050i	$0.243734\pi$
$104$	0	0
$105$	−1.62983	−0.159055
$106$	0	0
$107$	3.08263i	0.298010i	0.988836	+	0.149005i	$0.0476070\pi$
$107$	3.08263i	0.298010i	−0.988836	+	0.149005i	$0.952393\pi$
$108$	0	0
$109$	− 17.6022i	− 1.68598i	−0.537929	−	0.842990i	$-0.680793\pi$
$109$	− 17.6022i	− 1.68598i	0.537929	−	0.842990i	$-0.319207\pi$
$110$	0	0
$111$	−7.56509	−0.718047
$112$	0	0
$113$	5.49510	0.516936	0.258468	−	0.966020i	$-0.416782\pi$
$113$	5.49510	0.516936	0.258468	+	0.966020i	$0.416782\pi$
$114$	0	0
$115$	− 8.50730i	− 0.793310i
$116$	0	0
$117$	7.17769i	0.663577i
$118$	0	0
$119$	−1.74755	−0.160198
$120$	0	0
$121$	−14.5393	−1.32175
$122$	0	0
$123$	− 3.90954i	− 0.352511i
$124$	0	0
$125$	− 9.84392i	− 0.880467i
$126$	0	0
$127$	−14.7279	−1.30689	−0.653446	−	0.756973i	$-0.726677\pi$
$127$	−14.7279	−1.30689	−0.653446	+	0.756973i	$0.726677\pi$
$128$	0	0
$129$	1.26492	0.111370
$130$	0	0
$131$	− 10.5015i	− 0.917524i	−0.888559	−	0.458762i	$-0.848293\pi$
$131$	− 10.5015i	− 0.917524i	0.888559	−	0.458762i	$-0.151707\pi$
$132$	0	0
$133$	1.82309i	0.158082i
$134$	0	0
$135$	4.95811	0.426726
$136$	0	0
$137$	5.31623	0.454196	0.227098	−	0.973872i	$-0.427076\pi$
$137$	5.31623	0.454196	0.227098	+	0.973872i	$0.427076\pi$
$138$	0	0
$139$	2.36527i	0.200619i	0.994956	+	0.100310i	$0.0319834\pi$
$139$	2.36527i	0.200619i	−0.994956	+	0.100310i	$0.968017\pi$
$140$	0	0
$141$	− 6.33455i	− 0.533465i
$142$	0	0
$143$	−15.6606	−1.30960
$144$	0	0
$145$	1.27208	0.105640
$146$	0	0
$147$	− 3.26302i	− 0.269129i
$148$	0	0
$149$	− 21.4357i	− 1.75608i	−0.478585	−	0.878041i	$-0.658850\pi$
$149$	− 21.4357i	− 1.75608i	0.478585	−	0.878041i	$-0.341150\pi$
$150$	0	0
$151$	18.1276	1.47520	0.737600	−	0.675238i	$-0.235959\pi$
$151$	18.1276	1.47520	0.737600	+	0.675238i	$0.235959\pi$
$152$	0	0
$153$	2.31623	0.187256
$154$	0	0
$155$	− 3.02289i	− 0.242804i
$156$	0	0
$157$	14.6187i	1.16670i	0.812220	+	0.583351i	$0.198259\pi$
$157$	14.6187i	1.16670i	−0.812220	+	0.583351i	$0.801741\pi$
$158$	0	0
$159$	−3.23282	−0.256379
$160$	0	0
$161$	−13.1815	−1.03885
$162$	0	0
$163$	19.4090i	1.52023i	0.649788	+	0.760116i	$0.274858\pi$
$163$	19.4090i	1.52023i	−0.649788	+	0.760116i	$0.725142\pi$
$164$	0	0
$165$	4.71321i	0.366923i
$166$	0	0
$167$	−11.8751	−0.918924	−0.459462	−	0.888197i	$-0.651958\pi$
$167$	−11.8751	−0.918924	−0.459462	+	0.888197i	$0.651958\pi$
$168$	0	0
$169$	3.39702	0.261309
$170$	0	0
$171$	− 2.41635i	− 0.184783i
$172$	0	0
$173$	18.0197i	1.37001i	0.728539	+	0.685005i	$0.240200\pi$
$173$	18.0197i	1.37001i	−0.728539	+	0.685005i	$0.759800\pi$
$174$	0	0
$175$	−6.51474	−0.492468
$176$	0	0
$177$	9.12494	0.685872
$178$	0	0
$179$	15.8637i	1.18571i	0.805310	+	0.592855i	$0.201999\pi$
$179$	15.8637i	1.18571i	−0.805310	+	0.592855i	$0.798001\pi$
$180$	0	0
$181$	11.0659i	0.822519i	0.911518	+	0.411259i	$0.134911\pi$
$181$	11.0659i	0.822519i	−0.911518	+	0.411259i	$0.865089\pi$
$182$	0	0
$183$	−0.932637	−0.0689425
$184$	0	0
$185$	10.3185	0.758630
$186$	0	0
$187$	5.05364i	0.369559i
$188$	0	0
$189$	− 7.68226i	− 0.558803i
$190$	0	0
$191$	−14.2930	−1.03421	−0.517103	−	0.855923i	$-0.672990\pi$
$191$	−14.2930	−1.03421	−0.517103	+	0.855923i	$0.672990\pi$
$192$	0	0
$193$	−21.3480	−1.53666	−0.768330	−	0.640054i	$-0.778912\pi$
$193$	−21.3480	−1.53666	−0.768330	+	0.640054i	$0.778912\pi$
$194$	0	0
$195$	2.89012i	0.206966i
$196$	0	0
$197$	− 22.8269i	− 1.62635i	−0.582018	−	0.813176i	$-0.697737\pi$
$197$	− 22.8269i	− 1.62635i	0.582018	−	0.813176i	$-0.302263\pi$
$198$	0	0
$199$	8.81492	0.624873	0.312436	−	0.949939i	$-0.398855\pi$
$199$	8.81492	0.624873	0.312436	+	0.949939i	$0.398855\pi$
$200$	0	0
$201$	2.37017	0.167179
$202$	0	0
$203$	− 1.97100i	− 0.138337i
$204$	0	0
$205$	5.33246i	0.372435i
$206$	0	0
$207$	17.4709	1.21431
$208$	0	0
$209$	5.27208	0.364677
$210$	0	0
$211$	− 8.67845i	− 0.597449i	−0.954339	−	0.298725i	$-0.903439\pi$
$211$	− 8.67845i	− 0.597449i	0.954339	−	0.298725i	$-0.0965612\pi$
$212$	0	0
$213$	2.49695i	0.171088i
$214$	0	0
$215$	−1.72529	−0.117664
$216$	0	0
$217$	−4.68377	−0.317955
$218$	0	0
$219$	10.1073i	0.682986i
$220$	0	0
$221$	3.09887i	0.208452i
$222$	0	0
$223$	4.33228	0.290111	0.145055	−	0.989424i	$-0.453664\pi$
$223$	4.33228	0.290111	0.145055	+	0.989424i	$0.453664\pi$
$224$	0	0
$225$	8.63472	0.575648
$226$	0	0
$227$	− 25.6392i	− 1.70173i	−0.525381	−	0.850867i	$-0.676077\pi$
$227$	− 25.6392i	− 1.70173i	0.525381	−	0.850867i	$-0.323923\pi$
$228$	0	0
$229$	2.46448i	0.162857i	0.996679	+	0.0814287i	$0.0259483\pi$
$229$	2.46448i	0.162857i	−0.996679	+	0.0814287i	$0.974052\pi$
$230$	0	0
$231$	7.30281	0.480489
$232$	0	0
$233$	−2.67172	−0.175030	−0.0875151	−	0.996163i	$-0.527893\pi$
$233$	−2.67172	−0.175030	−0.0875151	+	0.996163i	$0.527893\pi$
$234$	0	0
$235$	8.64007i	0.563616i
$236$	0	0
$237$	− 3.26815i	− 0.212289i
$238$	0	0
$239$	2.89337	0.187157	0.0935784	−	0.995612i	$-0.470169\pi$
$239$	2.89337	0.187157	0.0935784	+	0.995612i	$0.470169\pi$
$240$	0	0
$241$	13.8260	0.890612	0.445306	−	0.895379i	$-0.353095\pi$
$241$	13.8260	0.890612	0.445306	+	0.895379i	$0.353095\pi$
$242$	0	0
$243$	15.9281i	1.02179i
$244$	0	0
$245$	4.45062i	0.284340i
$246$	0	0
$247$	3.23282	0.205699
$248$	0	0
$249$	7.73055	0.489903
$250$	0	0
$251$	− 26.7270i	− 1.68700i	−0.537132	−	0.843498i	$-0.680492\pi$
$251$	− 26.7270i	− 1.68700i	0.537132	−	0.843498i	$-0.319508\pi$
$252$	0	0
$253$	38.1188i	2.39651i
$254$	0	0
$255$	0.932637	0.0584040
$256$	0	0
$257$	−0.460746	−0.0287405	−0.0143703	−	0.999897i	$-0.504574\pi$
$257$	−0.460746	−0.0287405	−0.0143703	+	0.999897i	$0.504574\pi$
$258$	0	0
$259$	− 15.9878i	− 0.993435i
$260$	0	0
$261$	2.61239i	0.161703i
$262$	0	0
$263$	16.2623	1.00278	0.501388	−	0.865223i	$-0.332823\pi$
$263$	16.2623	1.00278	0.501388	+	0.865223i	$0.332823\pi$
$264$	0	0
$265$	4.40943	0.270869
$266$	0	0
$267$	− 15.1052i	− 0.924424i
$268$	0	0
$269$	20.0596i	1.22306i	0.791222	+	0.611529i	$0.209445\pi$
$269$	20.0596i	1.22306i	−0.791222	+	0.611529i	$0.790555\pi$
$270$	0	0
$271$	−0.670347	−0.0407207	−0.0203604	−	0.999793i	$-0.506481\pi$
$271$	−0.670347	−0.0407207	−0.0203604	+	0.999793i	$0.506481\pi$
$272$	0	0
$273$	4.47805	0.271024
$274$	0	0
$275$	18.8396i	1.13607i
$276$	0	0
$277$	3.58534i	0.215422i	0.994182	+	0.107711i	$0.0343522\pi$
$277$	3.58534i	0.215422i	−0.994182	+	0.107711i	$0.965648\pi$
$278$	0	0
$279$	6.20793	0.371659
$280$	0	0
$281$	0.397016	0.0236840	0.0118420	−	0.999930i	$-0.496230\pi$
$281$	0.397016	0.0236840	0.0118420	+	0.999930i	$0.496230\pi$
$282$	0	0
$283$	9.96859i	0.592571i	0.955099	+	0.296286i	$0.0957481\pi$
$283$	9.96859i	0.592571i	−0.955099	+	0.296286i	$0.904252\pi$
$284$	0	0
$285$	− 0.972950i	− 0.0576326i
$286$	0	0
$287$	8.26229	0.487708
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	− 0.940482i	− 0.0551320i
$292$	0	0
$293$	− 5.39407i	− 0.315125i	−0.987509	−	0.157562i	$-0.949636\pi$
$293$	− 5.39407i	− 0.315125i	0.987509	−	0.157562i	$-0.0503635\pi$
$294$	0	0
$295$	−12.4461	−0.724637
$296$	0	0
$297$	−22.2159	−1.28910
$298$	0	0
$299$	23.3743i	1.35177i
$300$	0	0
$301$	2.67323i	0.154082i
$302$	0	0
$303$	12.5677	0.721997
$304$	0	0
$305$	1.27208	0.0728391
$306$	0	0
$307$	28.9614i	1.65291i	0.562999	+	0.826457i	$0.309647\pi$
$307$	28.9614i	1.65291i	−0.562999	+	0.826457i	$0.690353\pi$
$308$	0	0
$309$	12.0996i	0.688325i
$310$	0	0
$311$	13.5049	0.765795	0.382898	−	0.923791i	$-0.374926\pi$
$311$	13.5049	0.765795	0.382898	+	0.923791i	$0.374926\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	4.56529i	0.257225i
$316$	0	0
$317$	− 14.7272i	− 0.827161i	−0.910468	−	0.413580i	$-0.864278\pi$
$317$	− 14.7272i	− 0.827161i	0.910468	−	0.413580i	$-0.135722\pi$
$318$	0	0
$319$	−5.69982	−0.319129
$320$	0	0
$321$	−2.54904	−0.142274
$322$	0	0
$323$	− 1.04322i	− 0.0580466i
$324$	0	0
$325$	11.5523i	0.640808i
$326$	0	0
$327$	14.5553	0.804910
$328$	0	0
$329$	13.3872	0.738062
$330$	0	0
$331$	− 2.58159i	− 0.141897i	−0.997480	−	0.0709484i	$-0.977397\pi$
$331$	− 2.58159i	− 0.141897i	0.997480	−	0.0709484i	$-0.0226026\pi$
$332$	0	0
$333$	21.1905i	1.16123i
$334$	0	0
$335$	−3.23282	−0.176628
$336$	0	0
$337$	7.00263	0.381457	0.190729	−	0.981643i	$-0.438915\pi$
$337$	7.00263	0.381457	0.190729	+	0.981643i	$0.438915\pi$
$338$	0	0
$339$	4.54393i	0.246792i
$340$	0	0
$341$	13.5447i	0.733487i
$342$	0	0
$343$	19.1288	1.03286
$344$	0	0
$345$	7.03473	0.378737
$346$	0	0
$347$	− 1.76335i	− 0.0946614i	−0.998879	−	0.0473307i	$-0.984929\pi$
$347$	− 1.76335i	− 0.0946614i	0.998879	−	0.0473307i	$-0.0150715\pi$
$348$	0	0
$349$	− 36.8581i	− 1.97297i	−0.163851	−	0.986485i	$-0.552392\pi$
$349$	− 36.8581i	− 1.97297i	0.163851	−	0.986485i	$-0.447608\pi$
$350$	0	0
$351$	−13.6227	−0.727124
$352$	0	0
$353$	−32.9830	−1.75551	−0.877755	−	0.479109i	$-0.840960\pi$
$353$	−32.9830	−1.75551	−0.877755	+	0.479109i	$0.840960\pi$
$354$	0	0
$355$	− 3.40574i	− 0.180758i
$356$	0	0
$357$	− 1.44506i	− 0.0764806i
$358$	0	0
$359$	−10.1093	−0.533546	−0.266773	−	0.963759i	$-0.585957\pi$
$359$	−10.1093	−0.533546	−0.266773	+	0.963759i	$0.585957\pi$
$360$	0	0
$361$	17.9117	0.942720
$362$	0	0
$363$	− 12.0226i	− 0.631022i
$364$	0	0
$365$	− 13.7859i	− 0.721588i
$366$	0	0
$367$	−36.8261	−1.92230	−0.961152	−	0.276018i	$-0.910985\pi$
$367$	−36.8261	−1.92230	−0.961152	+	0.276018i	$0.910985\pi$
$368$	0	0
$369$	−10.9509	−0.570083
$370$	0	0
$371$	− 6.83212i	− 0.354706i
$372$	0	0
$373$	− 15.2775i	− 0.791037i	−0.918458	−	0.395518i	$-0.870565\pi$
$373$	− 15.2775i	− 0.791037i	0.918458	−	0.395518i	$-0.129435\pi$
$374$	0	0
$375$	8.13998	0.420347
$376$	0	0
$377$	−3.49510	−0.180007
$378$	0	0
$379$	14.9805i	0.769498i	0.923021	+	0.384749i	$0.125712\pi$
$379$	14.9805i	0.769498i	−0.923021	+	0.384749i	$0.874288\pi$
$380$	0	0
$381$	− 12.1786i	− 0.623928i
$382$	0	0
$383$	5.19355	0.265378	0.132689	−	0.991158i	$-0.457639\pi$
$383$	5.19355	0.265378	0.132689	+	0.991158i	$0.457639\pi$
$384$	0	0
$385$	−9.96074	−0.507646
$386$	0	0
$387$	− 3.54313i	− 0.180108i
$388$	0	0
$389$	9.29660i	0.471357i	0.971831	+	0.235678i	$0.0757312\pi$
$389$	9.29660i	0.471357i	−0.971831	+	0.235678i	$0.924269\pi$
$390$	0	0
$391$	7.54284	0.381458
$392$	0	0
$393$	8.68377	0.438038
$394$	0	0
$395$	4.45763i	0.224287i
$396$	0	0
$397$	− 12.1177i	− 0.608172i	−0.952645	−	0.304086i	$-0.901649\pi$
$397$	− 12.1177i	− 0.608172i	0.952645	−	0.304086i	$-0.0983511\pi$
$398$	0	0
$399$	−1.50752	−0.0754705
$400$	0	0
$401$	30.3506	1.51564	0.757818	−	0.652466i	$-0.226265\pi$
$401$	30.3506	1.51564	0.757818	+	0.652466i	$0.226265\pi$
$402$	0	0
$403$	8.30555i	0.413729i
$404$	0	0
$405$	− 3.73730i	− 0.185708i
$406$	0	0
$407$	−46.2342	−2.29174
$408$	0	0
$409$	23.3015	1.15219	0.576093	−	0.817384i	$-0.304576\pi$
$409$	23.3015	1.15219	0.576093	+	0.817384i	$0.304576\pi$
$410$	0	0
$411$	4.39601i	0.216839i
$412$	0	0
$413$	19.2843i	0.948920i
$414$	0	0
$415$	−10.5442	−0.517592
$416$	0	0
$417$	−1.95585	−0.0957784
$418$	0	0
$419$	8.50917i	0.415700i	0.978161	+	0.207850i	$0.0666466\pi$
$419$	8.50917i	0.415700i	−0.978161	+	0.207850i	$0.933353\pi$
$420$	0	0
$421$	− 31.3191i	− 1.52640i	−0.646162	−	0.763200i	$-0.723627\pi$
$421$	− 31.3191i	− 1.52640i	0.646162	−	0.763200i	$-0.276373\pi$
$422$	0	0
$423$	−17.7436	−0.862724
$424$	0	0
$425$	3.72792	0.180831
$426$	0	0
$427$	− 1.97100i	− 0.0953835i
$428$	0	0
$429$	− 12.9498i	− 0.625222i
$430$	0	0
$431$	−19.2734	−0.928366	−0.464183	−	0.885739i	$-0.653652\pi$
$431$	−19.2734	−0.928366	−0.464183	+	0.885739i	$0.653652\pi$
$432$	0	0
$433$	21.9363	1.05419	0.527095	−	0.849806i	$-0.323281\pi$
$433$	21.9363	1.05419	0.527095	+	0.849806i	$0.323281\pi$
$434$	0	0
$435$	1.05189i	0.0504342i
$436$	0	0
$437$	− 7.86887i	− 0.376419i
$438$	0	0
$439$	5.43358	0.259331	0.129665	−	0.991558i	$-0.458610\pi$
$439$	5.43358	0.259331	0.129665	+	0.991558i	$0.458610\pi$
$440$	0	0
$441$	−9.13998	−0.435237
$442$	0	0
$443$	5.06043i	0.240428i	0.992748	+	0.120214i	$0.0383581\pi$
$443$	5.06043i	0.240428i	−0.992748	+	0.120214i	$0.961642\pi$
$444$	0	0
$445$	20.6029i	0.976671i
$446$	0	0
$447$	17.7253	0.838378
$448$	0	0
$449$	−15.8653	−0.748729	−0.374364	−	0.927282i	$-0.622139\pi$
$449$	−15.8653	−0.748729	−0.374364	+	0.927282i	$0.622139\pi$
$450$	0	0
$451$	− 23.8932i	− 1.12509i
$452$	0	0
$453$	14.9898i	0.704281i
$454$	0	0
$455$	−6.10788	−0.286342
$456$	0	0
$457$	−10.1272	−0.473730	−0.236865	−	0.971543i	$-0.576120\pi$
$457$	−10.1272	−0.473730	−0.236865	+	0.971543i	$0.576120\pi$
$458$	0	0
$459$	4.39601i	0.205188i
$460$	0	0
$461$	− 22.6244i	− 1.05372i	−0.849951	−	0.526862i	$-0.823368\pi$
$461$	− 22.6244i	− 1.05372i	0.849951	−	0.526862i	$-0.176632\pi$
$462$	0	0
$463$	7.18114	0.333736	0.166868	−	0.985979i	$-0.446635\pi$
$463$	7.18114	0.333736	0.166868	+	0.985979i	$0.446635\pi$
$464$	0	0
$465$	2.49964	0.115918
$466$	0	0
$467$	− 38.2074i	− 1.76803i	−0.467459	−	0.884015i	$-0.654830\pi$
$467$	− 38.2074i	− 1.76803i	0.467459	−	0.884015i	$-0.345170\pi$
$468$	0	0
$469$	5.00903i	0.231296i
$470$	0	0
$471$	−12.0883	−0.556999
$472$	0	0
$473$	7.73055	0.355451
$474$	0	0
$475$	− 3.88906i	− 0.178442i
$476$	0	0
$477$	9.05539i	0.414618i
$478$	0	0
$479$	−0.136030	−0.00621538	−0.00310769	−	0.999995i	$-0.500989\pi$
$479$	−0.136030	−0.00621538	−0.00310769	+	0.999995i	$0.500989\pi$
$480$	0	0
$481$	−28.3506	−1.29268
$482$	0	0
$483$	− 10.8998i	− 0.495960i
$484$	0	0
$485$	1.28278i	0.0582481i
$486$	0	0
$487$	38.5514	1.74693	0.873464	−	0.486888i	$-0.161868\pi$
$487$	38.5514	1.74693	0.873464	+	0.486888i	$0.161868\pi$
$488$	0	0
$489$	−16.0494	−0.725779
$490$	0	0
$491$	18.6889i	0.843418i	0.906731	+	0.421709i	$0.138570\pi$
$491$	18.6889i	0.843418i	−0.906731	+	0.421709i	$0.861430\pi$
$492$	0	0
$493$	1.12786i	0.0507965i
$494$	0	0
$495$	13.2021	0.593390
$496$	0	0
$497$	−5.27697	−0.236704
$498$	0	0
$499$	18.3033i	0.819368i	0.912228	+	0.409684i	$0.134361\pi$
$499$	18.3033i	0.819368i	−0.912228	+	0.409684i	$0.865639\pi$
$500$	0	0
$501$	− 9.81959i	− 0.438707i
$502$	0	0
$503$	26.6030	1.18617	0.593085	−	0.805140i	$-0.297910\pi$
$503$	26.6030	1.18617	0.593085	+	0.805140i	$0.297910\pi$
$504$	0	0
$505$	−17.1419	−0.762803
$506$	0	0
$507$	2.80901i	0.124752i
$508$	0	0
$509$	− 7.57086i	− 0.335572i	−0.985823	−	0.167786i	$-0.946338\pi$
$509$	− 7.57086i	− 0.335572i	0.985823	−	0.167786i	$-0.0536618\pi$
$510$	0	0
$511$	−21.3604	−0.944928
$512$	0	0
$513$	4.58603	0.202478
$514$	0	0
$515$	− 16.5034i	− 0.727228i
$516$	0	0
$517$	− 38.7137i	− 1.70263i
$518$	0	0
$519$	−14.9005	−0.654061
$520$	0	0
$521$	−22.4389	−0.983065	−0.491533	−	0.870859i	$-0.663563\pi$
$521$	−22.4389	−0.983065	−0.491533	+	0.870859i	$0.663563\pi$
$522$	0	0
$523$	− 13.5755i	− 0.593616i	−0.954937	−	0.296808i	$-0.904078\pi$
$523$	− 13.5755i	− 0.593616i	0.954937	−	0.296808i	$-0.0959221\pi$
$524$	0	0
$525$	− 5.38707i	− 0.235111i
$526$	0	0
$527$	2.68019	0.116751
$528$	0	0
$529$	33.8944	1.47367
$530$	0	0
$531$	− 25.5597i	− 1.10920i
$532$	0	0
$533$	− 14.6512i	− 0.634614i
$534$	0	0
$535$	3.47680	0.150315
$536$	0	0
$537$	−13.1178	−0.566074
$538$	0	0
$539$	− 19.9420i	− 0.858961i
$540$	0	0
$541$	− 14.1426i	− 0.608037i	−0.952666	−	0.304019i	$-0.901671\pi$
$541$	− 14.1426i	− 0.608037i	0.952666	−	0.304019i	$-0.0983285\pi$
$542$	0	0
$543$	−9.15041	−0.392682
$544$	0	0
$545$	−19.8529	−0.850403
$546$	0	0
$547$	29.0623i	1.24261i	0.783567	+	0.621307i	$0.213398\pi$
$547$	29.0623i	1.24261i	−0.783567	+	0.621307i	$0.786602\pi$
$548$	0	0
$549$	2.61239i	0.111494i
$550$	0	0
$551$	1.17662	0.0501255
$552$	0	0
$553$	6.90680	0.293707
$554$	0	0
$555$	8.53241i	0.362180i
$556$	0	0
$557$	26.1386i	1.10753i	0.832674	+	0.553764i	$0.186809\pi$
$557$	26.1386i	1.10753i	−0.832674	+	0.553764i	$0.813191\pi$
$558$	0	0
$559$	4.74034	0.200495
$560$	0	0
$561$	−4.17888	−0.176432
$562$	0	0
$563$	− 17.8812i	− 0.753602i	−0.926294	−	0.376801i	$-0.877024\pi$
$563$	− 17.8812i	− 0.753602i	0.926294	−	0.376801i	$-0.122976\pi$
$564$	0	0
$565$	− 6.19774i	− 0.260741i
$566$	0	0
$567$	−5.79069	−0.243186
$568$	0	0
$569$	−21.4434	−0.898955	−0.449478	−	0.893292i	$-0.648390\pi$
$569$	−21.4434	−0.898955	−0.449478	+	0.893292i	$0.648390\pi$
$570$	0	0
$571$	− 34.9469i	− 1.46248i	−0.682120	−	0.731240i	$-0.738942\pi$
$571$	− 34.9469i	− 1.46248i	0.682120	−	0.731240i	$-0.261058\pi$
$572$	0	0
$573$	− 11.8190i	− 0.493744i
$574$	0	0
$575$	28.1191	1.17265
$576$	0	0
$577$	−8.95848	−0.372946	−0.186473	−	0.982460i	$-0.559706\pi$
$577$	−8.95848	−0.372946	−0.186473	+	0.982460i	$0.559706\pi$
$578$	0	0
$579$	− 17.6527i	− 0.733622i
$580$	0	0
$581$	16.3375i	0.677793i
$582$	0	0
$583$	−19.7574	−0.818268
$584$	0	0
$585$	8.09546	0.334706
$586$	0	0
$587$	− 6.94513i	− 0.286656i	−0.989675	−	0.143328i	$-0.954220\pi$
$587$	− 6.94513i	− 0.286656i	0.989675	−	0.143328i	$-0.0457804\pi$
$588$	0	0
$589$	− 2.79604i	− 0.115209i
$590$	0	0
$591$	18.8757	0.776443
$592$	0	0
$593$	18.5932	0.763531	0.381765	−	0.924259i	$-0.375316\pi$
$593$	18.5932	0.763531	0.381765	+	0.924259i	$0.375316\pi$
$594$	0	0
$595$	1.97100i	0.0808033i
$596$	0	0
$597$	7.28909i	0.298323i
$598$	0	0
$599$	−31.7057	−1.29546	−0.647730	−	0.761870i	$-0.724282\pi$
$599$	−31.7057	−1.29546	−0.647730	+	0.761870i	$0.724282\pi$
$600$	0	0
$601$	−29.2257	−1.19214	−0.596070	−	0.802933i	$-0.703272\pi$
$601$	−29.2257	−1.19214	−0.596070	+	0.802933i	$0.703272\pi$
$602$	0	0
$603$	− 6.63904i	− 0.270363i
$604$	0	0
$605$	16.3983i	0.666686i
$606$	0	0
$607$	4.90321	0.199015	0.0995077	−	0.995037i	$-0.468273\pi$
$607$	4.90321	0.199015	0.0995077	+	0.995037i	$0.468273\pi$
$608$	0	0
$609$	1.62983	0.0660441
$610$	0	0
$611$	− 23.7391i	− 0.960379i
$612$	0	0
$613$	33.8739i	1.36816i	0.729409	+	0.684078i	$0.239795\pi$
$613$	33.8739i	1.36816i	−0.729409	+	0.684078i	$0.760205\pi$
$614$	0	0
$615$	−4.40943	−0.177805
$616$	0	0
$617$	−34.5174	−1.38962	−0.694809	−	0.719194i	$-0.744511\pi$
$617$	−34.5174	−1.38962	−0.694809	+	0.719194i	$0.744511\pi$
$618$	0	0
$619$	− 21.3435i	− 0.857868i	−0.903336	−	0.428934i	$-0.858889\pi$
$619$	− 21.3435i	− 0.857868i	0.903336	−	0.428934i	$-0.141111\pi$
$620$	0	0
$621$	33.1584i	1.33060i
$622$	0	0
$623$	31.9228	1.27896
$624$	0	0
$625$	7.53699	0.301480
$626$	0	0
$627$	4.35951i	0.174102i
$628$	0	0
$629$	9.14869i	0.364782i
$630$	0	0
$631$	−44.3689	−1.76630	−0.883149	−	0.469093i	$-0.844581\pi$
$631$	−44.3689	−1.76630	−0.883149	+	0.469093i	$0.844581\pi$
$632$	0	0
$633$	7.17625	0.285230
$634$	0	0
$635$	16.6111i	0.659192i
$636$	0	0
$637$	− 12.2283i	− 0.484504i
$638$	0	0
$639$	6.99416	0.276685
$640$	0	0
$641$	−3.68151	−0.145411	−0.0727055	−	0.997353i	$-0.523163\pi$
$641$	−3.68151	−0.145411	−0.0727055	+	0.997353i	$0.523163\pi$
$642$	0	0
$643$	30.6988i	1.21064i	0.795982	+	0.605321i	$0.206955\pi$
$643$	30.6988i	1.21064i	−0.795982	+	0.605321i	$0.793045\pi$
$644$	0	0
$645$	− 1.42665i	− 0.0561744i
$646$	0	0
$647$	20.1066	0.790472	0.395236	−	0.918580i	$-0.370663\pi$
$647$	20.1066	0.790472	0.395236	+	0.918580i	$0.370663\pi$
$648$	0	0
$649$	55.7672	2.18905
$650$	0	0
$651$	− 3.87303i	− 0.151796i
$652$	0	0
$653$	4.85298i	0.189912i	0.995481	+	0.0949560i	$0.0302710\pi$
$653$	4.85298i	0.189912i	−0.995481	+	0.0949560i	$0.969729\pi$
$654$	0	0
$655$	−11.8443	−0.462796
$656$	0	0
$657$	28.3113	1.10453
$658$	0	0
$659$	0.820109i	0.0319469i	0.999872	+	0.0159735i	$0.00508473\pi$
$659$	0.820109i	0.0319469i	−0.999872	+	0.0159735i	$0.994915\pi$
$660$	0	0
$661$	− 33.6446i	− 1.30862i	−0.756225	−	0.654311i	$-0.772959\pi$
$661$	− 33.6446i	− 1.30862i	0.756225	−	0.654311i	$-0.227041\pi$
$662$	0	0
$663$	−2.56247	−0.0995180
$664$	0	0
$665$	2.05620	0.0797360
$666$	0	0
$667$	8.50730i	0.329404i
$668$	0	0
$669$	3.58238i	0.138503i
$670$	0	0
$671$	−5.69982	−0.220039
$672$	0	0
$673$	14.3061	0.551459	0.275729	−	0.961235i	$-0.411081\pi$
$673$	14.3061	0.551459	0.275729	+	0.961235i	$0.411081\pi$
$674$	0	0
$675$	16.3880i	0.630774i
$676$	0	0
$677$	25.2693i	0.971177i	0.874187	+	0.485589i	$0.161395\pi$
$677$	25.2693i	0.971177i	−0.874187	+	0.485589i	$0.838605\pi$
$678$	0	0
$679$	1.98758	0.0762765
$680$	0	0
$681$	21.2012	0.812431
$682$	0	0
$683$	16.5149i	0.631923i	0.948772	+	0.315962i	$0.102327\pi$
$683$	16.5149i	0.631923i	−0.948772	+	0.315962i	$0.897673\pi$
$684$	0	0
$685$	− 5.99599i	− 0.229095i
$686$	0	0
$687$	−2.03789	−0.0777503
$688$	0	0
$689$	−12.1151	−0.461550
$690$	0	0
$691$	− 3.61558i	− 0.137543i	−0.997632	−	0.0687716i	$-0.978092\pi$
$691$	− 3.61558i	− 0.137543i	0.997632	−	0.0687716i	$-0.0219080\pi$
$692$	0	0
$693$	− 20.4558i	− 0.777050i
$694$	0	0
$695$	2.66770	0.101192
$696$	0	0
$697$	−4.72792	−0.179083
$698$	0	0
$699$	− 2.20926i	− 0.0835618i
$700$	0	0
$701$	− 20.4233i	− 0.771377i	−0.922629	−	0.385689i	$-0.873964\pi$
$701$	− 20.4233i	− 0.771377i	0.922629	−	0.385689i	$-0.126036\pi$
$702$	0	0
$703$	9.54414	0.359964
$704$	0	0
$705$	−7.14452	−0.269078
$706$	0	0
$707$	26.5602i	0.998899i
$708$	0	0
$709$	− 1.88393i	− 0.0707523i	−0.999374	−	0.0353762i	$-0.988737\pi$
$709$	− 1.88393i	− 0.0707523i	0.999374	−	0.0353762i	$-0.0112629\pi$
$710$	0	0
$711$	−9.15436	−0.343315
$712$	0	0
$713$	20.2162	0.757104
$714$	0	0
$715$	17.6630i	0.660559i
$716$	0	0
$717$	2.39254i	0.0893512i
$718$	0	0
$719$	29.2872	1.09223	0.546114	−	0.837711i	$-0.316107\pi$
$719$	29.2872	1.09223	0.546114	+	0.837711i	$0.316107\pi$
$720$	0	0
$721$	−25.5710	−0.952313
$722$	0	0
$723$	11.4328i	0.425190i
$724$	0	0
$725$	4.20459i	0.156155i
$726$	0	0
$727$	−31.1485	−1.15523	−0.577617	−	0.816308i	$-0.696017\pi$
$727$	−31.1485	−1.15523	−0.577617	+	0.816308i	$0.696017\pi$
$728$	0	0
$729$	−3.23019	−0.119637
$730$	0	0
$731$	− 1.52970i	− 0.0565780i
$732$	0	0
$733$	6.44596i	0.238087i	0.992889	+	0.119043i	$0.0379828\pi$
$733$	6.44596i	0.238087i	−0.992889	+	0.119043i	$0.962017\pi$
$734$	0	0
$735$	−3.68024	−0.135748
$736$	0	0
$737$	14.4853	0.533573
$738$	0	0
$739$	19.5888i	0.720587i	0.932839	+	0.360293i	$0.117323\pi$
$739$	19.5888i	0.720587i	−0.932839	+	0.360293i	$0.882677\pi$
$740$	0	0
$741$	2.67323i	0.0982036i
$742$	0	0
$743$	16.2315	0.595476	0.297738	−	0.954648i	$-0.403768\pi$
$743$	16.2315	0.595476	0.297738	+	0.954648i	$0.403768\pi$
$744$	0	0
$745$	−24.1766	−0.885762
$746$	0	0
$747$	− 21.6539i	− 0.792275i
$748$	0	0
$749$	− 5.38707i	− 0.196839i
$750$	0	0
$751$	−2.87965	−0.105080	−0.0525400	−	0.998619i	$-0.516732\pi$
$751$	−2.87965	−0.105080	−0.0525400	+	0.998619i	$0.516732\pi$
$752$	0	0
$753$	22.1007	0.805395
$754$	0	0
$755$	− 20.4454i	− 0.744086i
$756$	0	0
$757$	19.8538i	0.721600i	0.932643	+	0.360800i	$0.117496\pi$
$757$	19.8538i	0.721600i	−0.932643	+	0.360800i	$0.882504\pi$
$758$	0	0
$759$	−31.5206	−1.14412
$760$	0	0
$761$	21.1619	0.767119	0.383560	−	0.923516i	$-0.374698\pi$
$761$	21.1619	0.767119	0.383560	+	0.923516i	$0.374698\pi$
$762$	0	0
$763$	30.7607i	1.11361i
$764$	0	0
$765$	− 2.61239i	− 0.0944513i
$766$	0	0
$767$	34.1962	1.23475
$768$	0	0
$769$	−28.0879	−1.01288	−0.506438	−	0.862276i	$-0.669038\pi$
$769$	−28.0879	−1.01288	−0.506438	+	0.862276i	$0.669038\pi$
$770$	0	0
$771$	− 0.380993i	− 0.0137211i
$772$	0	0
$773$	21.4752i	0.772409i	0.922413	+	0.386204i	$0.126214\pi$
$773$	21.4752i	0.772409i	−0.922413	+	0.386204i	$0.873786\pi$
$774$	0	0
$775$	9.99153	0.358906
$776$	0	0
$777$	13.2204	0.474279
$778$	0	0
$779$	4.93228i	0.176717i
$780$	0	0
$781$	15.2601i	0.546050i
$782$	0	0
$783$	−4.95811	−0.177188
$784$	0	0
$785$	16.4880	0.588481
$786$	0	0
$787$	− 1.22304i	− 0.0435966i	−0.999762	−	0.0217983i	$-0.993061\pi$
$787$	− 1.22304i	− 0.0435966i	0.999762	−	0.0217983i	$-0.00693916\pi$
$788$	0	0
$789$	13.4474i	0.478739i
$790$	0	0
$791$	−9.60298	−0.341443
$792$	0	0
$793$	−3.49510	−0.124115
$794$	0	0
$795$	3.64618i	0.129317i
$796$	0	0
$797$	− 27.2300i	− 0.964535i	−0.876024	−	0.482267i	$-0.839813\pi$
$797$	− 27.2300i	− 0.964535i	0.876024	−	0.482267i	$-0.160187\pi$
$798$	0	0
$799$	−7.66056	−0.271011
$800$	0	0
$801$	−42.3110	−1.49498
$802$	0	0
$803$	61.7707i	2.17984i
$804$	0	0
$805$	14.8670i	0.523991i
$806$	0	0
$807$	−16.5874	−0.583904
$808$	0	0
$809$	12.6939	0.446294	0.223147	−	0.974785i	$-0.428367\pi$
$809$	12.6939	0.446294	0.223147	+	0.974785i	$0.428367\pi$
$810$	0	0
$811$	30.6006i	1.07453i	0.843412	+	0.537267i	$0.180543\pi$
$811$	30.6006i	1.07453i	−0.843412	+	0.537267i	$0.819457\pi$
$812$	0	0
$813$	− 0.554313i	− 0.0194406i
$814$	0	0
$815$	21.8907	0.766799
$816$	0	0
$817$	−1.59582	−0.0558307
$818$	0	0
$819$	− 12.5434i	− 0.438301i
$820$	0	0
$821$	34.8617i	1.21668i	0.793676	+	0.608340i	$0.208164\pi$
$821$	34.8617i	1.21668i	−0.793676	+	0.608340i	$0.791836\pi$
$822$	0	0
$823$	25.7292	0.896865	0.448433	−	0.893817i	$-0.351982\pi$
$823$	25.7292	0.896865	0.448433	+	0.893817i	$0.351982\pi$
$824$	0	0
$825$	−15.5785	−0.542374
$826$	0	0
$827$	− 31.5024i	− 1.09545i	−0.836659	−	0.547723i	$-0.815495\pi$
$827$	− 31.5024i	− 1.09545i	0.836659	−	0.547723i	$-0.184505\pi$
$828$	0	0
$829$	− 24.2760i	− 0.843142i	−0.906795	−	0.421571i	$-0.861479\pi$
$829$	− 24.2760i	− 0.843142i	0.906795	−	0.421571i	$-0.138521\pi$
$830$	0	0
$831$	−2.96474	−0.102846
$832$	0	0
$833$	−3.94606	−0.136723
$834$	0	0
$835$	13.3935i	0.463502i
$836$	0	0
$837$	11.7821i	0.407251i
$838$	0	0
$839$	−34.6299	−1.19556	−0.597778	−	0.801662i	$-0.703950\pi$
$839$	−34.6299	−1.19556	−0.597778	+	0.801662i	$0.703950\pi$
$840$	0	0
$841$	27.7279	0.956135
$842$	0	0
$843$	0.328294i	0.0113071i
$844$	0	0
$845$	− 3.83138i	− 0.131803i
$846$	0	0
$847$	25.4081	0.873033
$848$	0	0
$849$	−8.24308	−0.282902
$850$	0	0
$851$	69.0071i	2.36553i
$852$	0	0
$853$	− 24.3542i	− 0.833872i	−0.908936	−	0.416936i	$-0.863104\pi$
$853$	− 24.3542i	− 0.833872i	0.908936	−	0.416936i	$-0.136896\pi$
$854$	0	0
$855$	−2.72531	−0.0932037
$856$	0	0
$857$	1.86002	0.0635371	0.0317686	−	0.999495i	$-0.489886\pi$
$857$	1.86002	0.0635371	0.0317686	+	0.999495i	$0.489886\pi$
$858$	0	0
$859$	− 20.8140i	− 0.710166i	−0.934835	−	0.355083i	$-0.884453\pi$
$859$	− 20.8140i	− 0.710166i	0.934835	−	0.355083i	$-0.115547\pi$
$860$	0	0
$861$	6.83212i	0.232838i
$862$	0	0
$863$	31.1302	1.05968	0.529842	−	0.848096i	$-0.322251\pi$
$863$	31.1302	1.05968	0.529842	+	0.848096i	$0.322251\pi$
$864$	0	0
$865$	20.3237	0.691028
$866$	0	0
$867$	0.826905i	0.0280832i
$868$	0	0
$869$	− 19.9733i	− 0.677549i
$870$	0	0
$871$	8.88233	0.300966
$872$	0	0
$873$	−2.63437	−0.0891599
$874$	0	0
$875$	17.2028i	0.581559i
$876$	0	0
$877$	− 0.991052i	− 0.0334654i	−0.999860	−	0.0167327i	$-0.994674\pi$
$877$	− 0.991052i	− 0.0334654i	0.999860	−	0.0167327i	$-0.00532644\pi$
$878$	0	0
$879$	4.46038	0.150445
$880$	0	0
$881$	19.3087	0.650527	0.325263	−	0.945623i	$-0.394547\pi$
$881$	19.3087	0.650527	0.325263	+	0.945623i	$0.394547\pi$
$882$	0	0
$883$	− 23.9889i	− 0.807290i	−0.914916	−	0.403645i	$-0.867743\pi$
$883$	− 23.9889i	− 0.807290i	0.914916	−	0.403645i	$-0.132257\pi$
$884$	0	0
$885$	− 10.2917i	− 0.345952i
$886$	0	0
$887$	−2.58473	−0.0867866	−0.0433933	−	0.999058i	$-0.513817\pi$
$887$	−2.58473	−0.0867866	−0.0433933	+	0.999058i	$0.513817\pi$
$888$	0	0
$889$	25.7378	0.863219
$890$	0	0
$891$	16.7457i	0.561004i
$892$	0	0
$893$	7.99168i	0.267431i
$894$	0	0
$895$	17.8921	0.598068
$896$	0	0
$897$	−19.3283	−0.645352
$898$	0	0
$899$	3.02289i	0.100819i
$900$	0	0
$901$	3.90954i	0.130246i
$902$	0	0
$903$	−2.21051	−0.0735610
$904$	0	0
$905$	12.4808	0.414876
$906$	0	0
$907$	− 42.7770i	− 1.42039i	−0.704006	−	0.710194i	$-0.748607\pi$
$907$	− 42.7770i	− 1.42039i	0.704006	−	0.710194i	$-0.251393\pi$
$908$	0	0
$909$	− 35.2032i	− 1.16762i
$910$	0	0
$911$	42.6128	1.41183	0.705913	−	0.708299i	$-0.250537\pi$
$911$	42.6128	1.41183	0.705913	+	0.708299i	$0.250537\pi$
$912$	0	0
$913$	47.2453	1.56359
$914$	0	0
$915$	1.05189i	0.0347744i
$916$	0	0
$917$	18.3520i	0.606036i
$918$	0	0
$919$	−4.78549	−0.157859	−0.0789294	−	0.996880i	$-0.525150\pi$
$919$	−4.78549	−0.157859	−0.0789294	+	0.996880i	$0.525150\pi$
$920$	0	0
$921$	−23.9483	−0.789124
$922$	0	0
$923$	9.35744i	0.308004i
$924$	0	0
$925$	34.1056i	1.12139i
$926$	0	0
$927$	33.8921	1.11316
$928$	0	0
$929$	27.2819	0.895088	0.447544	−	0.894262i	$-0.352299\pi$
$929$	27.2819	0.895088	0.447544	+	0.894262i	$0.352299\pi$
$930$	0	0
$931$	4.11663i	0.134917i
$932$	0	0
$933$	11.1673i	0.365601i
$934$	0	0
$935$	5.69982	0.186404
$936$	0	0
$937$	0.0837798	0.00273697	0.00136848	−	0.999999i	$-0.499564\pi$
$937$	0.0837798	0.00273697	0.00136848	+	0.999999i	$0.499564\pi$
$938$	0	0
$939$	− 4.96143i	− 0.161910i
$940$	0	0
$941$	− 22.3002i	− 0.726967i	−0.931601	−	0.363483i	$-0.881587\pi$
$941$	− 22.3002i	− 0.726967i	0.931601	−	0.363483i	$-0.118413\pi$
$942$	0	0
$943$	−35.6619	−1.16131
$944$	0	0
$945$	−8.66456	−0.281858
$946$	0	0
$947$	− 18.5335i	− 0.602257i	−0.953584	−	0.301128i	$-0.902637\pi$
$947$	− 18.5335i	− 0.602257i	0.953584	−	0.301128i	$-0.0973633\pi$
$948$	0	0
$949$	37.8775i	1.22956i
$950$	0	0
$951$	12.1780	0.394898
$952$	0	0
$953$	−20.9782	−0.679549	−0.339775	−	0.940507i	$-0.610351\pi$
$953$	−20.9782	−0.679549	−0.339775	+	0.940507i	$0.610351\pi$
$954$	0	0
$955$	16.1206i	0.521650i
$956$	0	0
$957$	− 4.71321i	− 0.152356i
$958$	0	0
$959$	−9.29039	−0.300002
$960$	0	0
$961$	−23.8166	−0.768277
$962$	0	0
$963$	7.14009i	0.230086i
$964$	0	0
$965$	24.0776i	0.775086i
$966$	0	0
$967$	−7.73192	−0.248642	−0.124321	−	0.992242i	$-0.539675\pi$
$967$	−7.73192	−0.248642	−0.124321	+	0.992242i	$0.539675\pi$
$968$	0	0
$969$	0.862647	0.0277122
$970$	0	0
$971$	17.2155i	0.552471i	0.961090	+	0.276235i	$0.0890869\pi$
$971$	17.2155i	0.552471i	−0.961090	+	0.276235i	$0.910913\pi$
$972$	0	0
$973$	− 4.13343i	− 0.132512i
$974$	0	0
$975$	−9.55268	−0.305931
$976$	0	0
$977$	25.7377	0.823422	0.411711	−	0.911314i	$-0.364931\pi$
$977$	25.7377	0.823422	0.411711	+	0.911314i	$0.364931\pi$
$978$	0	0
$979$	− 92.3157i	− 2.95042i
$980$	0	0
$981$	− 40.7706i	− 1.30171i
$982$	0	0
$983$	−33.9385	−1.08247	−0.541235	−	0.840872i	$-0.682043\pi$
$983$	−33.9385	−1.08247	−0.541235	+	0.840872i	$0.682043\pi$
$984$	0	0
$985$	−25.7457	−0.820327
$986$	0	0
$987$	11.0700i	0.352361i
$988$	0	0
$989$	− 11.5383i	− 0.366896i
$990$	0	0
$991$	−3.70829	−0.117798	−0.0588988	−	0.998264i	$-0.518759\pi$
$991$	−3.70829	−0.117798	−0.0588988	+	0.998264i	$0.518759\pi$
$992$	0	0
$993$	2.13473	0.0677435
$994$	0	0
$995$	− 9.94203i	− 0.315184i
$996$	0	0
$997$	− 26.8766i	− 0.851191i	−0.904913	−	0.425596i	$-0.860065\pi$
$997$	− 26.8766i	− 0.851191i	0.904913	−	0.425596i	$-0.139935\pi$
$998$	0	0
$999$	−40.2178	−1.27243

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	544.2.c.b.273.6		8
3.2	odd	2		4896.2.f.d.2449.5		8
4.3	odd	2		136.2.c.b.69.4	yes	8
8.3	odd	2		136.2.c.b.69.3	✓	8
8.5	even	2	inner	544.2.c.b.273.3		8
12.11	even	2		1224.2.f.c.613.5		8
16.3	odd	4		4352.2.a.bf.1.6		8
16.5	even	4		4352.2.a.bb.1.6		8
16.11	odd	4		4352.2.a.bf.1.3		8
16.13	even	4		4352.2.a.bb.1.3		8
24.5	odd	2		4896.2.f.d.2449.4		8
24.11	even	2		1224.2.f.c.613.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.c.b.69.3	✓	8	8.3	odd	2
136.2.c.b.69.4	yes	8	4.3	odd	2
544.2.c.b.273.3		8	8.5	even	2	inner
544.2.c.b.273.6		8	1.1	even	1	trivial
1224.2.f.c.613.5		8	12.11	even	2
1224.2.f.c.613.6		8	24.11	even	2
4352.2.a.bb.1.3		8	16.13	even	4
4352.2.a.bb.1.6		8	16.5	even	4
4352.2.a.bf.1.3		8	16.11	odd	4
4352.2.a.bf.1.6		8	16.3	odd	4
4896.2.f.d.2449.4		8	24.5	odd	2
4896.2.f.d.2449.5		8	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 544 = 2^{5} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	544.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.826905i q^{3} -1.12786i q^{5} -1.74755 q^{7} +2.31623 q^{9} +5.05364i q^{11} +3.09887i q^{13} +0.932637 q^{15} +1.00000 q^{17} -1.04322i q^{19} -1.44506i q^{21} +7.54284 q^{23} +3.72792 q^{25} +4.39601i q^{27} +1.12786i q^{29} +2.68019 q^{31} -4.17888 q^{33} +1.97100i q^{35} +9.14869i q^{37} -2.56247 q^{39} -4.72792 q^{41} -1.52970i q^{43} -2.61239i q^{45} -7.66056 q^{47} -3.94606 q^{49} +0.826905i q^{51} +3.90954i q^{53} +5.69982 q^{55} +0.862647 q^{57} -11.0351i q^{59} +1.12786i q^{61} -4.04773 q^{63} +3.49510 q^{65} -2.86631i q^{67} +6.23721i q^{69} +3.01963 q^{71} +12.2230 q^{73} +3.08263i q^{75} -8.83150i q^{77} -3.95227 q^{79} +3.31360 q^{81} -9.34878i q^{83} -1.12786i q^{85} -0.932637 q^{87} -18.2672 q^{89} -5.41543i q^{91} +2.21626i q^{93} -1.17662 q^{95} -1.13735 q^{97} +11.7054i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{7} - 8 q^{9} - 12 q^{15} + 8 q^{17} + 16 q^{23} - 8 q^{25} - 24 q^{31} - 8 q^{33} + 12 q^{39} - 4 q^{47} + 8 q^{49} + 12 q^{55} + 8 q^{57} - 40 q^{63} - 24 q^{65} + 36 q^{71} + 8 q^{73} - 24 q^{79}+ \cdots - 8 q^{97}+O(q^{100}) \) 8 * q + 12 * q^7 - 8 * q^9 - 12 * q^15 + 8 * q^17 + 16 * q^23 - 8 * q^25 - 24 * q^31 - 8 * q^33 + 12 * q^39 - 4 * q^47 + 8 * q^49 + 12 * q^55 + 8 * q^57 - 40 * q^63 - 24 * q^65 + 36 * q^71 + 8 * q^73 - 24 * q^79 + 40 * q^81 + 12 * q^87 + 8 * q^89 - 32 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more