LMFDB - Embedded newform 2178.3.d.n.1693.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2178,3,Mod(1693,2178)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2178.1693"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2178, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2178.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-32,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$59.3462015777$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 889x^{12} - 18240x^{10} + 235606x^{8} - 2565840x^{6} + 30314764x^{4} - 184688100x^{2} + 519885601 $ x^16 + 889x^12 - 18240x^10 + 235606x^8 - 2565840x^6 + 30314764x^4 - 184688100x^2 + 519885601
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 198)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1693.15
Root			$2.29126 - 0.744477i$ of defining polynomial
Character	$\chi$	$=$	2178.1693
Dual form			2178.3.d.n.1693.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +5.68763 q^{5} -10.8351i q^{7} -2.82843i q^{8} +8.04353i q^{10} -4.50286i q^{13} +15.3231 q^{14} +4.00000 q^{16} -17.4537i q^{17} -33.6518i q^{19} -11.3753 q^{20} +7.03543 q^{23} +7.34916 q^{25} +6.36801 q^{26} +21.6701i q^{28} +54.2307i q^{29} -38.4817 q^{31} +5.65685i q^{32} +24.6833 q^{34} -61.6259i q^{35} -20.0715 q^{37} +47.5908 q^{38} -16.0871i q^{40} +15.0259i q^{41} -67.5682i q^{43} +9.94960i q^{46} -29.8783 q^{47} -68.3988 q^{49} +10.3933i q^{50} +9.00572i q^{52} -15.5208 q^{53} -30.6462 q^{56} -76.6938 q^{58} -38.7571 q^{59} +40.9048i q^{61} -54.4213i q^{62} -8.00000 q^{64} -25.6106i q^{65} -33.1092 q^{67} +34.9075i q^{68} +87.1522 q^{70} +86.4062 q^{71} +101.171i q^{73} -28.3854i q^{74} +67.3036i q^{76} -50.0905i q^{79} +22.7505 q^{80} -21.2498 q^{82} +14.6875i q^{83} -99.2704i q^{85} +95.5559 q^{86} -81.0524 q^{89} -48.7888 q^{91} -14.0709 q^{92} -42.2543i q^{94} -191.399i q^{95} +24.7275 q^{97} -96.7305i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 64 q^{16} + 136 q^{25} + 40 q^{31} - 64 q^{34} + 240 q^{37} - 544 q^{49} - 320 q^{58} - 128 q^{64} - 280 q^{67} + 256 q^{70} - 400 q^{82} - 32 q^{91} + 520 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 64 * q^16 + 136 * q^25 + 40 * q^31 - 64 * q^34 + 240 * q^37 - 544 * q^49 - 320 * q^58 - 128 * q^64 - 280 * q^67 + 256 * q^70 - 400 * q^82 - 32 * q^91 + 520 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	5.68763	1.13753	0.568763	−	0.822501i	$-0.307422\pi$
$5$	5.68763	1.13753	0.568763	+	0.822501i	$0.307422\pi$
$6$	0	0
$7$	− 10.8351i	− 1.54787i	−0.633267	−	0.773934i	$-0.718286\pi$
$7$	− 10.8351i	− 1.54787i	0.633267	−	0.773934i	$-0.281714\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	8.04353i	0.804353i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 4.50286i	− 0.346374i	−0.984889	−	0.173187i	$-0.944594\pi$
$13$	− 4.50286i	− 0.346374i	0.984889	−	0.173187i	$-0.0554065\pi$
$14$	15.3231	1.09451
$15$	0	0
$16$	4.00000	0.250000
$17$	− 17.4537i	− 1.02669i	−0.858182	−	0.513345i	$-0.828406\pi$
$17$	− 17.4537i	− 1.02669i	0.858182	−	0.513345i	$-0.171594\pi$
$18$	0	0
$19$	− 33.6518i	− 1.77115i	−0.464499	−	0.885574i	$-0.653765\pi$
$19$	− 33.6518i	− 1.77115i	0.464499	−	0.885574i	$-0.346235\pi$
$20$	−11.3753	−0.568763
$21$	0	0
$22$	0	0
$23$	7.03543	0.305888	0.152944	−	0.988235i	$-0.451125\pi$
$23$	7.03543	0.305888	0.152944	+	0.988235i	$0.451125\pi$
$24$	0	0
$25$	7.34916	0.293966
$26$	6.36801	0.244923
$27$	0	0
$28$	21.6701i	0.773934i
$29$	54.2307i	1.87003i	0.354614	+	0.935013i	$0.384612\pi$
$29$	54.2307i	1.87003i	−0.354614	+	0.935013i	$0.615388\pi$
$30$	0	0
$31$	−38.4817	−1.24134	−0.620672	−	0.784070i	$-0.713140\pi$
$31$	−38.4817	−1.24134	−0.620672	+	0.784070i	$0.713140\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	24.6833	0.725979
$35$	− 61.6259i	− 1.76074i
$36$	0	0
$37$	−20.0715	−0.542474	−0.271237	−	0.962513i	$-0.587433\pi$
$37$	−20.0715	−0.542474	−0.271237	+	0.962513i	$0.587433\pi$
$38$	47.5908	1.25239
$39$	0	0
$40$	− 16.0871i	− 0.402176i
$41$	15.0259i	0.366485i	0.983068	+	0.183243i	$0.0586594\pi$
$41$	15.0259i	0.366485i	−0.983068	+	0.183243i	$0.941341\pi$
$42$	0	0
$43$	− 67.5682i	− 1.57135i	−0.618637	−	0.785677i	$-0.712315\pi$
$43$	− 67.5682i	− 1.57135i	0.618637	−	0.785677i	$-0.287685\pi$
$44$	0	0
$45$	0	0
$46$	9.94960i	0.216296i
$47$	−29.8783	−0.635709	−0.317854	−	0.948140i	$-0.602962\pi$
$47$	−29.8783	−0.635709	−0.317854	+	0.948140i	$0.602962\pi$
$48$	0	0
$49$	−68.3988	−1.39589
$50$	10.3933i	0.207866i
$51$	0	0
$52$	9.00572i	0.173187i
$53$	−15.5208	−0.292845	−0.146422	−	0.989222i	$-0.546776\pi$
$53$	−15.5208	−0.292845	−0.146422	+	0.989222i	$0.546776\pi$
$54$	0	0
$55$	0	0
$56$	−30.6462	−0.547254
$57$	0	0
$58$	−76.6938	−1.32231
$59$	−38.7571	−0.656901	−0.328450	−	0.944521i	$-0.606526\pi$
$59$	−38.7571	−0.656901	−0.328450	+	0.944521i	$0.606526\pi$
$60$	0	0
$61$	40.9048i	0.670571i	0.942117	+	0.335286i	$0.108833\pi$
$61$	40.9048i	0.670571i	−0.942117	+	0.335286i	$0.891167\pi$
$62$	− 54.4213i	− 0.877763i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 25.6106i	− 0.394010i
$66$	0	0
$67$	−33.1092	−0.494167	−0.247083	−	0.968994i	$-0.579472\pi$
$67$	−33.1092	−0.494167	−0.247083	+	0.968994i	$0.579472\pi$
$68$	34.9075i	0.513345i
$69$	0	0
$70$	87.1522	1.24503
$71$	86.4062	1.21699	0.608495	−	0.793558i	$-0.291774\pi$
$71$	86.4062	1.21699	0.608495	+	0.793558i	$0.291774\pi$
$72$	0	0
$73$	101.171i	1.38590i	0.720986	+	0.692950i	$0.243689\pi$
$73$	101.171i	1.38590i	−0.720986	+	0.692950i	$0.756311\pi$
$74$	− 28.3854i	− 0.383587i
$75$	0	0
$76$	67.3036i	0.885574i
$77$	0	0
$78$	0	0
$79$	− 50.0905i	− 0.634057i	−0.948416	−	0.317029i	$-0.897315\pi$
$79$	− 50.0905i	− 0.634057i	0.948416	−	0.317029i	$-0.102685\pi$
$80$	22.7505	0.284382
$81$	0	0
$82$	−21.2498	−0.259144
$83$	14.6875i	0.176958i	0.996078	+	0.0884789i	$0.0282006\pi$
$83$	14.6875i	0.176958i	−0.996078	+	0.0884789i	$0.971799\pi$
$84$	0	0
$85$	− 99.2704i	− 1.16789i
$86$	95.5559	1.11112
$87$	0	0
$88$	0	0
$89$	−81.0524	−0.910701	−0.455350	−	0.890312i	$-0.650486\pi$
$89$	−81.0524	−0.910701	−0.455350	+	0.890312i	$0.650486\pi$
$90$	0	0
$91$	−48.7888	−0.536141
$92$	−14.0709	−0.152944
$93$	0	0
$94$	− 42.2543i	− 0.449514i
$95$	− 191.399i	− 2.01473i
$96$	0	0
$97$	24.7275	0.254923	0.127462	−	0.991844i	$-0.459317\pi$
$97$	24.7275	0.254923	0.127462	+	0.991844i	$0.459317\pi$
$98$	− 96.7305i	− 0.987046i
$99$	0	0
$100$	−14.6983	−0.146983
$101$	− 113.069i	− 1.11950i	−0.828662	−	0.559750i	$-0.810897\pi$
$101$	− 113.069i	− 1.11950i	0.828662	−	0.559750i	$-0.189103\pi$
$102$	0	0
$103$	−36.8203	−0.357478	−0.178739	−	0.983896i	$-0.557202\pi$
$103$	−36.8203	−0.357478	−0.178739	+	0.983896i	$0.557202\pi$
$104$	−12.7360	−0.122462
$105$	0	0
$106$	− 21.9497i	− 0.207073i
$107$	201.032i	1.87880i	0.342819	+	0.939401i	$0.388618\pi$
$107$	201.032i	1.87880i	−0.342819	+	0.939401i	$0.611382\pi$
$108$	0	0
$109$	116.981i	1.07322i	0.843831	+	0.536608i	$0.180295\pi$
$109$	116.981i	1.07322i	−0.843831	+	0.536608i	$0.819705\pi$
$110$	0	0
$111$	0	0
$112$	− 43.3403i	− 0.386967i
$113$	−194.964	−1.72534	−0.862672	−	0.505764i	$-0.831211\pi$
$113$	−194.964	−1.72534	−0.862672	+	0.505764i	$0.831211\pi$
$114$	0	0
$115$	40.0149	0.347956
$116$	− 108.461i	− 0.935013i
$117$	0	0
$118$	− 54.8109i	− 0.464499i
$119$	−189.112	−1.58918
$120$	0	0
$121$	0	0
$122$	−57.8482	−0.474165
$123$	0	0
$124$	76.9633	0.620672
$125$	−100.391	−0.803132
$126$	0	0
$127$	80.7585i	0.635894i	0.948109	+	0.317947i	$0.102993\pi$
$127$	80.7585i	0.635894i	−0.948109	+	0.317947i	$0.897007\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	36.2189	0.278607
$131$	14.5488i	0.111060i	0.998457	+	0.0555298i	$0.0176848\pi$
$131$	14.5488i	0.111060i	−0.998457	+	0.0555298i	$0.982315\pi$
$132$	0	0
$133$	−364.620	−2.74150
$134$	− 46.8235i	− 0.349429i
$135$	0	0
$136$	−49.3666	−0.362990
$137$	112.819	0.823499	0.411750	−	0.911297i	$-0.364918\pi$
$137$	112.819	0.823499	0.411750	+	0.911297i	$0.364918\pi$
$138$	0	0
$139$	36.4088i	0.261934i	0.991387	+	0.130967i	$0.0418082\pi$
$139$	36.4088i	0.261934i	−0.991387	+	0.130967i	$0.958192\pi$
$140$	123.252i	0.880370i
$141$	0	0
$142$	122.197i	0.860541i
$143$	0	0
$144$	0	0
$145$	308.444i	2.12720i
$146$	−143.077	−0.979979
$147$	0	0
$148$	40.1431	0.271237
$149$	− 272.954i	− 1.83190i	−0.401288	−	0.915952i	$-0.631437\pi$
$149$	− 272.954i	− 1.83190i	0.401288	−	0.915952i	$-0.368563\pi$
$150$	0	0
$151$	− 40.6044i	− 0.268904i	−0.990920	−	0.134452i	$-0.957073\pi$
$151$	− 40.6044i	− 0.268904i	0.990920	−	0.134452i	$-0.0429273\pi$
$152$	−95.1817	−0.626195
$153$	0	0
$154$	0	0
$155$	−218.870	−1.41206
$156$	0	0
$157$	−165.215	−1.05232	−0.526161	−	0.850385i	$-0.676369\pi$
$157$	−165.215	−1.05232	−0.526161	+	0.850385i	$0.676369\pi$
$158$	70.8387	0.448346
$159$	0	0
$160$	32.1741i	0.201088i
$161$	− 76.2294i	− 0.473474i
$162$	0	0
$163$	−220.927	−1.35538	−0.677691	−	0.735347i	$-0.737019\pi$
$163$	−220.927	−1.35538	−0.677691	+	0.735347i	$0.737019\pi$
$164$	− 30.0518i	− 0.183243i
$165$	0	0
$166$	−20.7713	−0.125128
$167$	− 203.934i	− 1.22116i	−0.791954	−	0.610581i	$-0.790936\pi$
$167$	− 203.934i	− 1.22116i	0.791954	−	0.610581i	$-0.209064\pi$
$168$	0	0
$169$	148.724	0.880025
$170$	140.390	0.825821
$171$	0	0
$172$	135.136i	0.785677i
$173$	− 281.078i	− 1.62473i	−0.583150	−	0.812365i	$-0.698180\pi$
$173$	− 281.078i	− 1.62473i	0.583150	−	0.812365i	$-0.301820\pi$
$174$	0	0
$175$	− 79.6287i	− 0.455021i
$176$	0	0
$177$	0	0
$178$	− 114.625i	− 0.643963i
$179$	145.463	0.812645	0.406323	−	0.913730i	$-0.366811\pi$
$179$	145.463	0.812645	0.406323	+	0.913730i	$0.366811\pi$
$180$	0	0
$181$	225.091	1.24360	0.621798	−	0.783178i	$-0.286403\pi$
$181$	225.091	1.24360	0.621798	+	0.783178i	$0.286403\pi$
$182$	− 68.9978i	− 0.379109i
$183$	0	0
$184$	− 19.8992i	− 0.108148i
$185$	−114.159	−0.617078
$186$	0	0
$187$	0	0
$188$	59.7566	0.317854
$189$	0	0
$190$	270.679	1.42463
$191$	249.843	1.30808	0.654039	−	0.756460i	$-0.273073\pi$
$191$	249.843	1.30808	0.654039	+	0.756460i	$0.273073\pi$
$192$	0	0
$193$	6.69041i	0.0346653i	0.999850	+	0.0173327i	$0.00551743\pi$
$193$	6.69041i	0.0346653i	−0.999850	+	0.0173327i	$0.994483\pi$
$194$	34.9700i	0.180258i
$195$	0	0
$196$	136.798	0.697947
$197$	− 258.538i	− 1.31237i	−0.754598	−	0.656187i	$-0.772168\pi$
$197$	− 258.538i	− 1.31237i	0.754598	−	0.656187i	$-0.227832\pi$
$198$	0	0
$199$	235.298	1.18240	0.591200	−	0.806525i	$-0.298654\pi$
$199$	235.298	1.18240	0.591200	+	0.806525i	$0.298654\pi$
$200$	− 20.7866i	− 0.103933i
$201$	0	0
$202$	159.904	0.791606
$203$	587.594	2.89455
$204$	0	0
$205$	85.4618i	0.416887i
$206$	− 52.0717i	− 0.252775i
$207$	0	0
$208$	− 18.0114i	− 0.0865935i
$209$	0	0
$210$	0	0
$211$	32.1659i	0.152445i	0.997091	+	0.0762224i	$0.0242859\pi$
$211$	32.1659i	0.152445i	−0.997091	+	0.0762224i	$0.975714\pi$
$212$	31.0416	0.146422
$213$	0	0
$214$	−284.302	−1.32851
$215$	− 384.303i	− 1.78746i
$216$	0	0
$217$	416.952i	1.92144i
$218$	−165.436	−0.758879
$219$	0	0
$220$	0	0
$221$	−78.5917	−0.355619
$222$	0	0
$223$	−173.304	−0.777148	−0.388574	−	0.921418i	$-0.627032\pi$
$223$	−173.304	−0.777148	−0.388574	+	0.921418i	$0.627032\pi$
$224$	61.2924	0.273627
$225$	0	0
$226$	− 275.720i	− 1.22000i
$227$	− 2.22668i	− 0.00980918i	−0.999988	−	0.00490459i	$-0.998439\pi$
$227$	− 2.22668i	− 0.00980918i	0.999988	−	0.00490459i	$-0.00156119\pi$
$228$	0	0
$229$	314.248	1.37226	0.686130	−	0.727479i	$-0.259308\pi$
$229$	314.248	1.37226	0.686130	+	0.727479i	$0.259308\pi$
$230$	56.5896i	0.246042i
$231$	0	0
$232$	153.388	0.661154
$233$	− 127.192i	− 0.545888i	−0.962030	−	0.272944i	$-0.912003\pi$
$233$	− 127.192i	− 0.545888i	0.962030	−	0.272944i	$-0.0879975\pi$
$234$	0	0
$235$	−169.937	−0.723136
$236$	77.5143	0.328450
$237$	0	0
$238$	− 267.445i	− 1.12372i
$239$	159.300i	0.666528i	0.942834	+	0.333264i	$0.108150\pi$
$239$	159.300i	0.666528i	−0.942834	+	0.333264i	$0.891850\pi$
$240$	0	0
$241$	− 265.125i	− 1.10010i	−0.835131	−	0.550051i	$-0.814608\pi$
$241$	− 265.125i	− 1.10010i	0.835131	−	0.550051i	$-0.185392\pi$
$242$	0	0
$243$	0	0
$244$	− 81.8097i	− 0.335286i
$245$	−389.027	−1.58787
$246$	0	0
$247$	−151.529	−0.613479
$248$	108.843i	0.438881i
$249$	0	0
$250$	− 141.975i	− 0.567900i
$251$	188.313	0.750250	0.375125	−	0.926974i	$-0.377600\pi$
$251$	188.313	0.750250	0.375125	+	0.926974i	$0.377600\pi$
$252$	0	0
$253$	0	0
$254$	−114.210	−0.449645
$255$	0	0
$256$	16.0000	0.0625000
$257$	301.419	1.17284	0.586418	−	0.810009i	$-0.300538\pi$
$257$	301.419	1.17284	0.586418	+	0.810009i	$0.300538\pi$
$258$	0	0
$259$	217.476i	0.839677i
$260$	51.2212i	0.197005i
$261$	0	0
$262$	−20.5751	−0.0785310
$263$	− 409.440i	− 1.55681i	−0.627764	−	0.778404i	$-0.716030\pi$
$263$	− 409.440i	− 1.55681i	0.627764	−	0.778404i	$-0.283970\pi$
$264$	0	0
$265$	−88.2765	−0.333119
$266$	− 515.650i	− 1.93853i
$267$	0	0
$268$	66.2184	0.247083
$269$	195.927	0.728352	0.364176	−	0.931330i	$-0.381351\pi$
$269$	195.927	0.728352	0.364176	+	0.931330i	$0.381351\pi$
$270$	0	0
$271$	− 109.464i	− 0.403928i	−0.979393	−	0.201964i	$-0.935268\pi$
$271$	− 109.464i	− 0.403928i	0.979393	−	0.201964i	$-0.0647324\pi$
$272$	− 69.8149i	− 0.256672i
$273$	0	0
$274$	159.551i	0.582302i
$275$	0	0
$276$	0	0
$277$	− 202.397i	− 0.730676i	−0.930875	−	0.365338i	$-0.880953\pi$
$277$	− 202.397i	− 0.730676i	0.930875	−	0.365338i	$-0.119047\pi$
$278$	−51.4898	−0.185215
$279$	0	0
$280$	−174.304	−0.622516
$281$	217.045i	0.772402i	0.922415	+	0.386201i	$0.126213\pi$
$281$	217.045i	0.772402i	−0.922415	+	0.386201i	$0.873787\pi$
$282$	0	0
$283$	− 181.771i	− 0.642302i	−0.947028	−	0.321151i	$-0.895930\pi$
$283$	− 181.771i	− 0.642302i	0.947028	−	0.321151i	$-0.104070\pi$
$284$	−172.812	−0.608495
$285$	0	0
$286$	0	0
$287$	162.807	0.567271
$288$	0	0
$289$	−15.6326	−0.0540920
$290$	−436.206	−1.50416
$291$	0	0
$292$	− 202.341i	− 0.692950i
$293$	66.2418i	0.226081i	0.993590	+	0.113041i	$0.0360590\pi$
$293$	66.2418i	0.226081i	−0.993590	+	0.113041i	$0.963941\pi$
$294$	0	0
$295$	−220.436	−0.747242
$296$	56.7709i	0.191793i
$297$	0	0
$298$	386.015	1.29535
$299$	− 31.6796i	− 0.105952i
$300$	0	0
$301$	−732.107	−2.43225
$302$	57.4234	0.190144
$303$	0	0
$304$	− 134.607i	− 0.442787i
$305$	232.652i	0.762792i
$306$	0	0
$307$	− 596.201i	− 1.94202i	−0.239034	−	0.971011i	$-0.576831\pi$
$307$	− 596.201i	− 1.94202i	0.239034	−	0.971011i	$-0.423169\pi$
$308$	0	0
$309$	0	0
$310$	− 309.528i	− 0.998479i
$311$	165.904	0.533453	0.266726	−	0.963772i	$-0.414058\pi$
$311$	165.904	0.533453	0.266726	+	0.963772i	$0.414058\pi$
$312$	0	0
$313$	350.003	1.11822	0.559110	−	0.829094i	$-0.311143\pi$
$313$	350.003	1.11822	0.559110	+	0.829094i	$0.311143\pi$
$314$	− 233.649i	− 0.744104i
$315$	0	0
$316$	100.181i	0.317029i
$317$	−76.1277	−0.240150	−0.120075	−	0.992765i	$-0.538314\pi$
$317$	−76.1277	−0.240150	−0.120075	+	0.992765i	$0.538314\pi$
$318$	0	0
$319$	0	0
$320$	−45.5011	−0.142191
$321$	0	0
$322$	107.805	0.334797
$323$	−587.349	−1.81842
$324$	0	0
$325$	− 33.0922i	− 0.101822i
$326$	− 312.438i	− 0.958399i
$327$	0	0
$328$	42.4997	0.129572
$329$	323.734i	0.983993i
$330$	0	0
$331$	498.082	1.50478	0.752390	−	0.658718i	$-0.228901\pi$
$331$	498.082	1.50478	0.752390	+	0.658718i	$0.228901\pi$
$332$	− 29.3750i	− 0.0884789i
$333$	0	0
$334$	288.406	0.863491
$335$	−188.313	−0.562128
$336$	0	0
$337$	− 282.275i	− 0.837611i	−0.908076	−	0.418805i	$-0.862449\pi$
$337$	− 282.275i	− 0.837611i	0.908076	−	0.418805i	$-0.137551\pi$
$338$	210.328i	0.622272i
$339$	0	0
$340$	198.541i	0.583943i
$341$	0	0
$342$	0	0
$343$	210.187i	0.612791i
$344$	−191.112	−0.555558
$345$	0	0
$346$	397.505	1.14886
$347$	− 197.453i	− 0.569029i	−0.958672	−	0.284514i	$-0.908168\pi$
$347$	− 197.453i	− 0.569029i	0.958672	−	0.284514i	$-0.0918324\pi$
$348$	0	0
$349$	214.148i	0.613604i	0.951773	+	0.306802i	$0.0992589\pi$
$349$	214.148i	0.613604i	−0.951773	+	0.306802i	$0.900741\pi$
$350$	112.612	0.321748
$351$	0	0
$352$	0	0
$353$	−610.069	−1.72824	−0.864120	−	0.503285i	$-0.832124\pi$
$353$	−610.069	−1.72824	−0.864120	+	0.503285i	$0.832124\pi$
$354$	0	0
$355$	491.447	1.38436
$356$	162.105	0.455350
$357$	0	0
$358$	205.716i	0.574627i
$359$	− 585.274i	− 1.63029i	−0.579258	−	0.815144i	$-0.696658\pi$
$359$	− 585.274i	− 1.63029i	0.579258	−	0.815144i	$-0.303342\pi$
$360$	0	0
$361$	−771.444	−2.13696
$362$	318.327i	0.879355i
$363$	0	0
$364$	97.5776	0.268070
$365$	575.422i	1.57650i
$366$	0	0
$367$	−244.776	−0.666964	−0.333482	−	0.942756i	$-0.608224\pi$
$367$	−244.776	−0.666964	−0.333482	+	0.942756i	$0.608224\pi$
$368$	28.1417	0.0764720
$369$	0	0
$370$	− 161.446i	− 0.436340i
$371$	168.169i	0.453285i
$372$	0	0
$373$	− 141.374i	− 0.379017i	−0.981879	−	0.189509i	$-0.939310\pi$
$373$	− 141.374i	− 0.379017i	0.981879	−	0.189509i	$-0.0606895\pi$
$374$	0	0
$375$	0	0
$376$	84.5086i	0.224757i
$377$	244.193	0.647728
$378$	0	0
$379$	−6.22172	−0.0164162	−0.00820808	−	0.999966i	$-0.502613\pi$
$379$	−6.22172	−0.0164162	−0.00820808	+	0.999966i	$0.502613\pi$
$380$	382.798i	1.00736i
$381$	0	0
$382$	353.331i	0.924951i
$383$	335.463	0.875883	0.437942	−	0.899003i	$-0.355708\pi$
$383$	335.463	0.875883	0.437942	+	0.899003i	$0.355708\pi$
$384$	0	0
$385$	0	0
$386$	−9.46166	−0.0245121
$387$	0	0
$388$	−49.4551	−0.127462
$389$	−128.680	−0.330798	−0.165399	−	0.986227i	$-0.552891\pi$
$389$	−128.680	−0.330798	−0.165399	+	0.986227i	$0.552891\pi$
$390$	0	0
$391$	− 122.794i	− 0.314052i
$392$	193.461i	0.493523i
$393$	0	0
$394$	365.628	0.927989
$395$	− 284.896i	− 0.721257i
$396$	0	0
$397$	−386.412	−0.973331	−0.486666	−	0.873588i	$-0.661787\pi$
$397$	−386.412	−0.973331	−0.486666	+	0.873588i	$0.661787\pi$
$398$	332.761i	0.836083i
$399$	0	0
$400$	29.3966	0.0734916
$401$	−72.7085	−0.181318	−0.0906590	−	0.995882i	$-0.528897\pi$
$401$	−72.7085	−0.181318	−0.0906590	+	0.995882i	$0.528897\pi$
$402$	0	0
$403$	173.278i	0.429969i
$404$	226.139i	0.559750i
$405$	0	0
$406$	830.983i	2.04676i
$407$	0	0
$408$	0	0
$409$	395.368i	0.966670i	0.875435	+	0.483335i	$0.160575\pi$
$409$	395.368i	0.966670i	−0.875435	+	0.483335i	$0.839425\pi$
$410$	−120.861	−0.294783
$411$	0	0
$412$	73.6406	0.178739
$413$	419.936i	1.01680i
$414$	0	0
$415$	83.5371i	0.201294i
$416$	25.4720	0.0612308
$417$	0	0
$418$	0	0
$419$	33.9782	0.0810936	0.0405468	−	0.999178i	$-0.487090\pi$
$419$	33.9782	0.0810936	0.0405468	+	0.999178i	$0.487090\pi$
$420$	0	0
$421$	16.5930	0.0394133	0.0197066	−	0.999806i	$-0.493727\pi$
$421$	16.5930	0.0394133	0.0197066	+	0.999806i	$0.493727\pi$
$422$	−45.4894	−0.107795
$423$	0	0
$424$	43.8994i	0.103536i
$425$	− 128.270i	− 0.301812i
$426$	0	0
$427$	443.207	1.03796
$428$	− 402.064i	− 0.939401i
$429$	0	0
$430$	543.487	1.26392
$431$	− 123.648i	− 0.286885i	−0.989659	−	0.143443i	$-0.954183\pi$
$431$	− 123.648i	− 0.286885i	0.989659	−	0.143443i	$-0.0458173\pi$
$432$	0	0
$433$	341.768	0.789303	0.394651	−	0.918831i	$-0.370865\pi$
$433$	341.768	0.789303	0.394651	+	0.918831i	$0.370865\pi$
$434$	−589.659	−1.35866
$435$	0	0
$436$	− 233.961i	− 0.536608i
$437$	− 236.755i	− 0.541773i
$438$	0	0
$439$	− 552.255i	− 1.25798i	−0.777412	−	0.628992i	$-0.783468\pi$
$439$	− 552.255i	− 1.25798i	0.777412	−	0.628992i	$-0.216532\pi$
$440$	0	0
$441$	0	0
$442$	− 111.145i	− 0.251460i
$443$	5.10301	0.0115192	0.00575960	−	0.999983i	$-0.498167\pi$
$443$	5.10301	0.0115192	0.00575960	+	0.999983i	$0.498167\pi$
$444$	0	0
$445$	−460.996	−1.03595
$446$	− 245.089i	− 0.549527i
$447$	0	0
$448$	86.6806i	0.193483i
$449$	−267.632	−0.596062	−0.298031	−	0.954556i	$-0.596330\pi$
$449$	−267.632	−0.596062	−0.298031	+	0.954556i	$0.596330\pi$
$450$	0	0
$451$	0	0
$452$	389.928	0.862672
$453$	0	0
$454$	3.14901	0.00693614
$455$	−277.493	−0.609874
$456$	0	0
$457$	− 209.812i	− 0.459107i	−0.973296	−	0.229553i	$-0.926273\pi$
$457$	− 209.812i	− 0.459107i	0.973296	−	0.229553i	$-0.0737265\pi$
$458$	444.413i	0.970335i
$459$	0	0
$460$	−80.0298	−0.173978
$461$	710.734i	1.54172i	0.637003	+	0.770862i	$0.280174\pi$
$461$	710.734i	1.54172i	−0.637003	+	0.770862i	$0.719826\pi$
$462$	0	0
$463$	−546.690	−1.18076	−0.590378	−	0.807127i	$-0.701021\pi$
$463$	−546.690	−1.18076	−0.590378	+	0.807127i	$0.701021\pi$
$464$	216.923i	0.467506i
$465$	0	0
$466$	179.877	0.386001
$467$	−317.845	−0.680610	−0.340305	−	0.940315i	$-0.610530\pi$
$467$	−317.845	−0.680610	−0.340305	+	0.940315i	$0.610530\pi$
$468$	0	0
$469$	358.740i	0.764905i
$470$	− 240.327i	− 0.511334i
$471$	0	0
$472$	109.622i	0.232250i
$473$	0	0
$474$	0	0
$475$	− 247.312i	− 0.520658i
$476$	378.225	0.794590
$477$	0	0
$478$	−225.284	−0.471306
$479$	7.87229i	0.0164348i	0.999966	+	0.00821742i	$0.00261572\pi$
$479$	7.87229i	0.0164348i	−0.999966	+	0.00821742i	$0.997384\pi$
$480$	0	0
$481$	90.3793i	0.187899i
$482$	374.943	0.777890
$483$	0	0
$484$	0	0
$485$	140.641	0.289982
$486$	0	0
$487$	281.751	0.578544	0.289272	−	0.957247i	$-0.406587\pi$
$487$	281.751	0.578544	0.289272	+	0.957247i	$0.406587\pi$
$488$	115.696	0.237083
$489$	0	0
$490$	− 550.167i	− 1.12279i
$491$	819.603i	1.66925i	0.550816	+	0.834627i	$0.314317\pi$
$491$	819.603i	1.66925i	−0.550816	+	0.834627i	$0.685683\pi$
$492$	0	0
$493$	946.528	1.91994
$494$	− 214.295i	− 0.433795i
$495$	0	0
$496$	−153.927	−0.310336
$497$	− 936.218i	− 1.88374i
$498$	0	0
$499$	9.99727	0.0200346	0.0100173	−	0.999950i	$-0.496811\pi$
$499$	9.99727	0.0200346	0.0100173	+	0.999950i	$0.496811\pi$
$500$	200.783	0.401566
$501$	0	0
$502$	266.315i	0.530507i
$503$	− 250.288i	− 0.497591i	−0.968556	−	0.248796i	$-0.919965\pi$
$503$	− 250.288i	− 0.497591i	0.968556	−	0.248796i	$-0.0800348\pi$
$504$	0	0
$505$	− 643.097i	− 1.27346i
$506$	0	0
$507$	0	0
$508$	− 161.517i	− 0.317947i
$509$	−23.3774	−0.0459281	−0.0229640	−	0.999736i	$-0.507310\pi$
$509$	−23.3774	−0.0459281	−0.0229640	+	0.999736i	$0.507310\pi$
$510$	0	0
$511$	1096.19	2.14519
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	426.271i	0.829320i
$515$	−209.420	−0.406641
$516$	0	0
$517$	0	0
$518$	−307.558	−0.593742
$519$	0	0
$520$	−72.4378	−0.139303
$521$	525.279	1.00821	0.504106	−	0.863642i	$-0.331822\pi$
$521$	525.279	1.00821	0.504106	+	0.863642i	$0.331822\pi$
$522$	0	0
$523$	− 426.352i	− 0.815204i	−0.913160	−	0.407602i	$-0.866365\pi$
$523$	− 426.352i	− 0.815204i	0.913160	−	0.407602i	$-0.133635\pi$
$524$	− 29.0976i	− 0.0555298i
$525$	0	0
$526$	579.036	1.10083
$527$	671.649i	1.27448i
$528$	0	0
$529$	−479.503	−0.906432
$530$	− 124.842i	− 0.235551i
$531$	0	0
$532$	729.239	1.37075
$533$	67.6595	0.126941
$534$	0	0
$535$	1143.40i	2.13719i
$536$	93.6469i	0.174714i
$537$	0	0
$538$	277.082i	0.515022i
$539$	0	0
$540$	0	0
$541$	572.341i	1.05793i	0.848643	+	0.528966i	$0.177420\pi$
$541$	572.341i	1.05793i	−0.848643	+	0.528966i	$0.822580\pi$
$542$	154.806	0.285620
$543$	0	0
$544$	98.7332	0.181495
$545$	665.343i	1.22081i
$546$	0	0
$547$	− 178.116i	− 0.325623i	−0.986657	−	0.162811i	$-0.947944\pi$
$547$	− 178.116i	− 0.325623i	0.986657	−	0.162811i	$-0.0520562\pi$
$548$	−225.639	−0.411750
$549$	0	0
$550$	0	0
$551$	1824.96	3.31209
$552$	0	0
$553$	−542.734	−0.981436
$554$	286.233	0.516666
$555$	0	0
$556$	− 72.8176i	− 0.130967i
$557$	404.515i	0.726239i	0.931743	+	0.363120i	$0.118288\pi$
$557$	404.515i	0.726239i	−0.931743	+	0.363120i	$0.881712\pi$
$558$	0	0
$559$	−304.250	−0.544276
$560$	− 246.504i	− 0.440185i
$561$	0	0
$562$	−306.948	−0.546171
$563$	1018.64i	1.80930i	0.426151	+	0.904652i	$0.359869\pi$
$563$	1018.64i	1.80930i	−0.426151	+	0.904652i	$0.640131\pi$
$564$	0	0
$565$	−1108.88	−1.96262
$566$	257.064	0.454176
$567$	0	0
$568$	− 244.394i	− 0.430271i
$569$	− 738.074i	− 1.29714i	−0.761154	−	0.648571i	$-0.775367\pi$
$569$	− 738.074i	− 1.29714i	0.761154	−	0.648571i	$-0.224633\pi$
$570$	0	0
$571$	650.748i	1.13966i	0.821762	+	0.569832i	$0.192992\pi$
$571$	650.748i	1.13966i	−0.821762	+	0.569832i	$0.807008\pi$
$572$	0	0
$573$	0	0
$574$	230.243i	0.401121i
$575$	51.7045	0.0899208
$576$	0	0
$577$	−829.069	−1.43686	−0.718431	−	0.695598i	$-0.755139\pi$
$577$	−829.069	−1.43686	−0.718431	+	0.695598i	$0.755139\pi$
$578$	− 22.1078i	− 0.0382488i
$579$	0	0
$580$	− 616.889i	− 1.06360i
$581$	159.140	0.273907
$582$	0	0
$583$	0	0
$584$	286.154	0.489990
$585$	0	0
$586$	−93.6801	−0.159864
$587$	742.508	1.26492	0.632460	−	0.774593i	$-0.282045\pi$
$587$	742.508	1.26492	0.632460	+	0.774593i	$0.282045\pi$
$588$	0	0
$589$	1294.98i	2.19860i
$590$	− 311.744i	− 0.528380i
$591$	0	0
$592$	−80.2861	−0.135618
$593$	− 360.480i	− 0.607891i	−0.952689	−	0.303946i	$-0.901696\pi$
$593$	− 360.480i	− 0.607891i	0.952689	−	0.303946i	$-0.0983041\pi$
$594$	0	0
$595$	−1075.60	−1.80773
$596$	545.907i	0.915952i
$597$	0	0
$598$	44.8017	0.0749192
$599$	−291.934	−0.487369	−0.243684	−	0.969855i	$-0.578356\pi$
$599$	−291.934	−0.487369	−0.243684	+	0.969855i	$0.578356\pi$
$600$	0	0
$601$	− 806.103i	− 1.34127i	−0.741788	−	0.670635i	$-0.766022\pi$
$601$	− 806.103i	− 1.34127i	0.741788	−	0.670635i	$-0.233978\pi$
$602$	− 1035.36i	− 1.71986i
$603$	0	0
$604$	81.2089i	0.134452i
$605$	0	0
$606$	0	0
$607$	− 143.096i	− 0.235742i	−0.993029	−	0.117871i	$-0.962393\pi$
$607$	− 143.096i	− 0.235742i	0.993029	−	0.117871i	$-0.0376070\pi$
$608$	190.363	0.313098
$609$	0	0
$610$	−329.019	−0.539376
$611$	134.538i	0.220193i
$612$	0	0
$613$	399.809i	0.652217i	0.945332	+	0.326108i	$0.105737\pi$
$613$	399.809i	0.652217i	−0.945332	+	0.326108i	$0.894263\pi$
$614$	843.155	1.37322
$615$	0	0
$616$	0	0
$617$	−366.869	−0.594602	−0.297301	−	0.954784i	$-0.596086\pi$
$617$	−366.869	−0.594602	−0.297301	+	0.954784i	$0.596086\pi$
$618$	0	0
$619$	−837.536	−1.35305	−0.676523	−	0.736421i	$-0.736514\pi$
$619$	−837.536	−1.35305	−0.676523	+	0.736421i	$0.736514\pi$
$620$	437.739	0.706031
$621$	0	0
$622$	234.624i	0.377208i
$623$	878.208i	1.40964i
$624$	0	0
$625$	−754.719	−1.20755
$626$	494.979i	0.790701i
$627$	0	0
$628$	330.429	0.526161
$629$	350.323i	0.556952i
$630$	0	0
$631$	444.227	0.704004	0.352002	−	0.935999i	$-0.385501\pi$
$631$	444.227	0.704004	0.352002	+	0.935999i	$0.385501\pi$
$632$	−141.677	−0.224173
$633$	0	0
$634$	− 107.661i	− 0.169812i
$635$	459.325i	0.723346i
$636$	0	0
$637$	307.990i	0.483501i
$638$	0	0
$639$	0	0
$640$	− 64.3482i	− 0.100544i
$641$	1122.83	1.75168	0.875841	−	0.482599i	$-0.160307\pi$
$641$	1122.83	1.75168	0.875841	+	0.482599i	$0.160307\pi$
$642$	0	0
$643$	772.429	1.20129	0.600645	−	0.799516i	$-0.294911\pi$
$643$	772.429	1.20129	0.600645	+	0.799516i	$0.294911\pi$
$644$	152.459i	0.236737i
$645$	0	0
$646$	− 830.637i	− 1.28582i
$647$	373.909	0.577912	0.288956	−	0.957342i	$-0.406692\pi$
$647$	373.909	0.577912	0.288956	+	0.957342i	$0.406692\pi$
$648$	0	0
$649$	0	0
$650$	46.7995	0.0719992
$651$	0	0
$652$	441.854	0.677691
$653$	−246.864	−0.378045	−0.189023	−	0.981973i	$-0.560532\pi$
$653$	−246.864	−0.378045	−0.189023	+	0.981973i	$0.560532\pi$
$654$	0	0
$655$	82.7482i	0.126333i
$656$	60.1036i	0.0916213i
$657$	0	0
$658$	−457.829	−0.695788
$659$	− 609.418i	− 0.924762i	−0.886681	−	0.462381i	$-0.846995\pi$
$659$	− 609.418i	− 0.924762i	0.886681	−	0.462381i	$-0.153005\pi$
$660$	0	0
$661$	1129.61	1.70893	0.854467	−	0.519506i	$-0.173884\pi$
$661$	1129.61	1.70893	0.854467	+	0.519506i	$0.173884\pi$
$662$	704.394i	1.06404i
$663$	0	0
$664$	41.5425	0.0625641
$665$	−2073.82	−3.11853
$666$	0	0
$667$	381.536i	0.572019i
$668$	407.868i	0.610581i
$669$	0	0
$670$	− 266.315i	− 0.397484i
$671$	0	0
$672$	0	0
$673$	436.864i	0.649130i	0.945863	+	0.324565i	$0.105218\pi$
$673$	436.864i	0.649130i	−0.945863	+	0.324565i	$0.894782\pi$
$674$	399.197	0.592280
$675$	0	0
$676$	−297.448	−0.440013
$677$	− 479.008i	− 0.707545i	−0.935331	−	0.353773i	$-0.884899\pi$
$677$	− 479.008i	− 0.707545i	0.935331	−	0.353773i	$-0.115101\pi$
$678$	0	0
$679$	− 267.925i	− 0.394587i
$680$	−280.779	−0.412910
$681$	0	0
$682$	0	0
$683$	−614.419	−0.899588	−0.449794	−	0.893132i	$-0.648503\pi$
$683$	−614.419	−0.899588	−0.449794	+	0.893132i	$0.648503\pi$
$684$	0	0
$685$	641.675	0.936752
$686$	−297.250	−0.433308
$687$	0	0
$688$	− 270.273i	− 0.392839i
$689$	69.8879i	0.101434i
$690$	0	0
$691$	−958.142	−1.38660	−0.693301	−	0.720648i	$-0.743844\pi$
$691$	−958.142	−1.38660	−0.693301	+	0.720648i	$0.743844\pi$
$692$	562.156i	0.812365i
$693$	0	0
$694$	279.241	0.402364
$695$	207.080i	0.297957i
$696$	0	0
$697$	262.258	0.376267
$698$	−302.851	−0.433883
$699$	0	0
$700$	159.257i	0.227510i
$701$	− 193.665i	− 0.276270i	−0.990413	−	0.138135i	$-0.955889\pi$
$701$	− 193.665i	− 0.276270i	0.990413	−	0.138135i	$-0.0441107\pi$
$702$	0	0
$703$	675.443i	0.960801i
$704$	0	0
$705$	0	0
$706$	− 862.768i	− 1.22205i
$707$	−1225.12	−1.73284
$708$	0	0
$709$	29.5819	0.0417235	0.0208617	−	0.999782i	$-0.493359\pi$
$709$	29.5819	0.0417235	0.0208617	+	0.999782i	$0.493359\pi$
$710$	695.011i	0.978889i
$711$	0	0
$712$	229.251i	0.321981i
$713$	−270.735	−0.379713
$714$	0	0
$715$	0	0
$716$	−290.927	−0.406323
$717$	0	0
$718$	827.702	1.15279
$719$	30.7450	0.0427608	0.0213804	−	0.999771i	$-0.493194\pi$
$719$	30.7450	0.0427608	0.0213804	+	0.999771i	$0.493194\pi$
$720$	0	0
$721$	398.950i	0.553329i
$722$	− 1090.99i	− 1.51106i
$723$	0	0
$724$	−450.182	−0.621798
$725$	398.550i	0.549725i
$726$	0	0
$727$	252.046	0.346694	0.173347	−	0.984861i	$-0.444542\pi$
$727$	252.046	0.346694	0.173347	+	0.984861i	$0.444542\pi$
$728$	137.996i	0.189554i
$729$	0	0
$730$	−813.769	−1.11475
$731$	−1179.32	−1.61329
$732$	0	0
$733$	− 905.286i	− 1.23504i	−0.786554	−	0.617521i	$-0.788137\pi$
$733$	− 905.286i	− 1.23504i	0.786554	−	0.617521i	$-0.211863\pi$
$734$	− 346.165i	− 0.471615i
$735$	0	0
$736$	39.7984i	0.0540739i
$737$	0	0
$738$	0	0
$739$	− 251.971i	− 0.340962i	−0.985361	−	0.170481i	$-0.945468\pi$
$739$	− 251.971i	− 0.340962i	0.985361	−	0.170481i	$-0.0545321\pi$
$740$	228.319	0.308539
$741$	0	0
$742$	−237.827	−0.320521
$743$	558.799i	0.752085i	0.926602	+	0.376042i	$0.122715\pi$
$743$	558.799i	0.752085i	−0.926602	+	0.376042i	$0.877285\pi$
$744$	0	0
$745$	− 1552.46i	− 2.08384i
$746$	199.932	0.268006
$747$	0	0
$748$	0	0
$749$	2178.19	2.90814
$750$	0	0
$751$	−914.374	−1.21754	−0.608771	−	0.793346i	$-0.708337\pi$
$751$	−914.374	−1.21754	−0.608771	+	0.793346i	$0.708337\pi$
$752$	−119.513	−0.158927
$753$	0	0
$754$	345.342i	0.458013i
$755$	− 230.943i	− 0.305885i
$756$	0	0
$757$	−251.388	−0.332084	−0.166042	−	0.986119i	$-0.553099\pi$
$757$	−251.388	−0.332084	−0.166042	+	0.986119i	$0.553099\pi$
$758$	− 8.79885i	− 0.0116080i
$759$	0	0
$760$	−541.358	−0.712314
$761$	619.866i	0.814541i	0.913308	+	0.407271i	$0.133519\pi$
$761$	619.866i	0.814541i	−0.913308	+	0.407271i	$0.866481\pi$
$762$	0	0
$763$	1267.49	1.66120
$764$	−499.686	−0.654039
$765$	0	0
$766$	474.417i	0.619343i
$767$	174.518i	0.227533i
$768$	0	0
$769$	− 298.190i	− 0.387763i	−0.981025	−	0.193882i	$-0.937892\pi$
$769$	− 298.190i	− 0.387763i	0.981025	−	0.193882i	$-0.0621078\pi$
$770$	0	0
$771$	0	0
$772$	− 13.3808i	− 0.0173327i
$773$	−606.758	−0.784939	−0.392470	−	0.919765i	$-0.628379\pi$
$773$	−606.758	−0.784939	−0.392470	+	0.919765i	$0.628379\pi$
$774$	0	0
$775$	−282.808	−0.364913
$776$	− 69.9400i	− 0.0901289i
$777$	0	0
$778$	− 181.981i	− 0.233909i
$779$	505.649	0.649100
$780$	0	0
$781$	0	0
$782$	173.658	0.222068
$783$	0	0
$784$	−273.595	−0.348973
$785$	−939.680	−1.19704
$786$	0	0
$787$	− 59.7811i	− 0.0759608i	−0.999278	−	0.0379804i	$-0.987908\pi$
$787$	− 59.7811i	− 0.0759608i	0.999278	−	0.0379804i	$-0.0120924\pi$
$788$	517.075i	0.656187i
$789$	0	0
$790$	402.904	0.510005
$791$	2112.45i	2.67060i
$792$	0	0
$793$	184.189	0.232268
$794$	− 546.470i	− 0.688249i
$795$	0	0
$796$	−470.595	−0.591200
$797$	−912.063	−1.14437	−0.572185	−	0.820125i	$-0.693904\pi$
$797$	−912.063	−1.14437	−0.572185	+	0.820125i	$0.693904\pi$
$798$	0	0
$799$	521.488i	0.652676i
$800$	41.5731i	0.0519664i
$801$	0	0
$802$	− 102.825i	− 0.128211i
$803$	0	0
$804$	0	0
$805$	− 433.565i	− 0.538590i
$806$	−245.052	−0.304034
$807$	0	0
$808$	−319.809	−0.395803
$809$	938.105i	1.15959i	0.814764	+	0.579793i	$0.196867\pi$
$809$	938.105i	1.15959i	−0.814764	+	0.579793i	$0.803133\pi$
$810$	0	0
$811$	− 382.379i	− 0.471491i	−0.971815	−	0.235746i	$-0.924247\pi$
$811$	− 382.379i	− 0.471491i	0.971815	−	0.235746i	$-0.0757532\pi$
$812$	−1175.19	−1.44728
$813$	0	0
$814$	0	0
$815$	−1256.55	−1.54178
$816$	0	0
$817$	−2273.79	−2.78310
$818$	−559.135	−0.683539
$819$	0	0
$820$	− 170.924i	− 0.208443i
$821$	− 626.026i	− 0.762516i	−0.924469	−	0.381258i	$-0.875491\pi$
$821$	− 626.026i	− 0.762516i	0.924469	−	0.381258i	$-0.124509\pi$
$822$	0	0
$823$	42.8763	0.0520976	0.0260488	−	0.999661i	$-0.491707\pi$
$823$	42.8763	0.0520976	0.0260488	+	0.999661i	$0.491707\pi$
$824$	104.143i	0.126388i
$825$	0	0
$826$	−593.880	−0.718983
$827$	1083.94i	1.31069i	0.755331	+	0.655343i	$0.227476\pi$
$827$	1083.94i	1.31069i	−0.755331	+	0.655343i	$0.772524\pi$
$828$	0	0
$829$	−1409.55	−1.70030	−0.850150	−	0.526540i	$-0.823489\pi$
$829$	−1409.55	−1.70030	−0.850150	+	0.526540i	$0.823489\pi$
$830$	−118.139	−0.142337
$831$	0	0
$832$	36.0229i	0.0432967i
$833$	1193.81i	1.43315i
$834$	0	0
$835$	− 1159.90i	− 1.38910i
$836$	0	0
$837$	0	0
$838$	48.0525i	0.0573418i
$839$	889.398	1.06007	0.530035	−	0.847976i	$-0.322179\pi$
$839$	889.398	1.06007	0.530035	+	0.847976i	$0.322179\pi$
$840$	0	0
$841$	−2099.97	−2.49699
$842$	23.4660i	0.0278694i
$843$	0	0
$844$	− 64.3317i	− 0.0762224i
$845$	845.889	1.00105
$846$	0	0
$847$	0	0
$848$	−62.0831	−0.0732112
$849$	0	0
$850$	181.401	0.213414
$851$	−141.212	−0.165936
$852$	0	0
$853$	1457.54i	1.70872i	0.519679	+	0.854362i	$0.326052\pi$
$853$	1457.54i	1.70872i	−0.519679	+	0.854362i	$0.673948\pi$
$854$	626.789i	0.733945i
$855$	0	0
$856$	568.604	0.664257
$857$	− 731.392i	− 0.853433i	−0.904385	−	0.426716i	$-0.859670\pi$
$857$	− 731.392i	− 0.853433i	0.904385	−	0.426716i	$-0.140330\pi$
$858$	0	0
$859$	85.1091	0.0990792	0.0495396	−	0.998772i	$-0.484225\pi$
$859$	85.1091	0.0990792	0.0495396	+	0.998772i	$0.484225\pi$
$860$	768.607i	0.893729i
$861$	0	0
$862$	174.864	0.202859
$863$	−592.606	−0.686682	−0.343341	−	0.939211i	$-0.611559\pi$
$863$	−592.606	−0.686682	−0.343341	+	0.939211i	$0.611559\pi$
$864$	0	0
$865$	− 1598.67i	− 1.84817i
$866$	483.333i	0.558121i
$867$	0	0
$868$	− 833.903i	− 0.960718i
$869$	0	0
$870$	0	0
$871$	149.086i	0.171167i
$872$	330.871	0.379439
$873$	0	0
$874$	334.822	0.383091
$875$	1087.75i	1.24314i
$876$	0	0
$877$	810.847i	0.924569i	0.886732	+	0.462284i	$0.152970\pi$
$877$	810.847i	0.924569i	−0.886732	+	0.462284i	$0.847030\pi$
$878$	781.007	0.889529
$879$	0	0
$880$	0	0
$881$	162.880	0.184881	0.0924404	−	0.995718i	$-0.470533\pi$
$881$	162.880	0.184881	0.0924404	+	0.995718i	$0.470533\pi$
$882$	0	0
$883$	861.628	0.975797	0.487898	−	0.872900i	$-0.337764\pi$
$883$	861.628	0.975797	0.487898	+	0.872900i	$0.337764\pi$
$884$	157.183	0.177809
$885$	0	0
$886$	7.21674i	0.00814531i
$887$	25.1678i	0.0283740i	0.999899	+	0.0141870i	$0.00451602\pi$
$887$	25.1678i	0.0283740i	−0.999899	+	0.0141870i	$0.995484\pi$
$888$	0	0
$889$	875.024	0.984279
$890$	− 651.947i	− 0.732524i
$891$	0	0
$892$	346.608	0.388574
$893$	1005.46i	1.12593i
$894$	0	0
$895$	827.343	0.924405
$896$	−122.585	−0.136813
$897$	0	0
$898$	− 378.488i	− 0.421479i
$899$	− 2086.89i	− 2.32135i
$900$	0	0
$901$	270.895i	0.300661i
$902$	0	0
$903$	0	0
$904$	551.441i	0.610001i
$905$	1280.23	1.41462
$906$	0	0
$907$	725.098	0.799447	0.399723	−	0.916636i	$-0.369106\pi$
$907$	725.098	0.799447	0.399723	+	0.916636i	$0.369106\pi$
$908$	4.45337i	0.00490459i
$909$	0	0
$910$	− 392.434i	− 0.431246i
$911$	855.259	0.938814	0.469407	−	0.882982i	$-0.344468\pi$
$911$	855.259	0.938814	0.469407	+	0.882982i	$0.344468\pi$
$912$	0	0
$913$	0	0
$914$	296.719	0.324637
$915$	0	0
$916$	−628.495	−0.686130
$917$	157.637	0.171905
$918$	0	0
$919$	− 1432.42i	− 1.55868i	−0.626603	−	0.779339i	$-0.715555\pi$
$919$	− 1432.42i	− 1.55868i	0.626603	−	0.779339i	$-0.284445\pi$
$920$	− 113.179i	− 0.123021i
$921$	0	0
$922$	−1005.13	−1.09016
$923$	− 389.075i	− 0.421533i
$924$	0	0
$925$	−147.509	−0.159469
$926$	− 773.136i	− 0.834921i
$927$	0	0
$928$	−306.775	−0.330577
$929$	1517.47	1.63345	0.816724	−	0.577028i	$-0.195788\pi$
$929$	1517.47	1.63345	0.816724	+	0.577028i	$0.195788\pi$
$930$	0	0
$931$	2301.74i	2.47233i
$932$	254.384i	0.272944i
$933$	0	0
$934$	− 449.501i	− 0.481264i
$935$	0	0
$936$	0	0
$937$	− 187.686i	− 0.200306i	−0.994972	−	0.100153i	$-0.968067\pi$
$937$	− 187.686i	− 0.200306i	0.994972	−	0.100153i	$-0.0319332\pi$
$938$	−507.336	−0.540869
$939$	0	0
$940$	339.874	0.361568
$941$	− 440.952i	− 0.468599i	−0.972164	−	0.234300i	$-0.924720\pi$
$941$	− 440.952i	− 0.468599i	0.972164	−	0.234300i	$-0.0752797\pi$
$942$	0	0
$943$	105.714i	0.112104i
$944$	−155.029	−0.164225
$945$	0	0
$946$	0	0
$947$	744.379	0.786039	0.393019	−	0.919530i	$-0.371431\pi$
$947$	744.379	0.786039	0.393019	+	0.919530i	$0.371431\pi$
$948$	0	0
$949$	455.558	0.480040
$950$	349.753	0.368161
$951$	0	0
$952$	534.891i	0.561860i
$953$	399.851i	0.419571i	0.977747	+	0.209786i	$0.0672766\pi$
$953$	399.851i	0.419571i	−0.977747	+	0.209786i	$0.932723\pi$
$954$	0	0
$955$	1421.02	1.48797
$956$	− 318.600i	− 0.333264i
$957$	0	0
$958$	−11.1331	−0.0116212
$959$	− 1222.41i	− 1.27467i
$960$	0	0
$961$	519.839	0.540936
$962$	−127.816	−0.132864
$963$	0	0
$964$	530.250i	0.550051i
$965$	38.0526i	0.0394327i
$966$	0	0
$967$	1646.99i	1.70320i	0.524196	+	0.851598i	$0.324366\pi$
$967$	1646.99i	1.70320i	−0.524196	+	0.851598i	$0.675634\pi$
$968$	0	0
$969$	0	0
$970$	198.897i	0.205048i
$971$	662.526	0.682313	0.341156	−	0.940007i	$-0.389181\pi$
$971$	662.526	0.682313	0.341156	+	0.940007i	$0.389181\pi$
$972$	0	0
$973$	394.492	0.405439
$974$	398.456i	0.409093i
$975$	0	0
$976$	163.619i	0.167643i
$977$	1487.18	1.52219	0.761095	−	0.648641i	$-0.224662\pi$
$977$	1487.18	1.52219	0.761095	+	0.648641i	$0.224662\pi$
$978$	0	0
$979$	0	0
$980$	778.054	0.793933
$981$	0	0
$982$	−1159.09	−1.18034
$983$	374.686	0.381165	0.190583	−	0.981671i	$-0.438962\pi$
$983$	374.686	0.381165	0.190583	+	0.981671i	$0.438962\pi$
$984$	0	0
$985$	− 1470.47i	− 1.49286i
$986$	1338.59i	1.35760i
$987$	0	0
$988$	303.059	0.306740
$989$	− 475.371i	− 0.480659i
$990$	0	0
$991$	−455.590	−0.459727	−0.229864	−	0.973223i	$-0.573828\pi$
$991$	−455.590	−0.459727	−0.229864	+	0.973223i	$0.573828\pi$
$992$	− 217.685i	− 0.219441i
$993$	0	0
$994$	1324.01	1.33200
$995$	1338.29	1.34501
$996$	0	0
$997$	− 451.423i	− 0.452781i	−0.974037	−	0.226390i	$-0.927307\pi$
$997$	− 451.423i	− 0.452781i	0.974037	−	0.226390i	$-0.0726925\pi$
$998$	14.1383i	0.0141666i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.3.d.n.1693.15		16
3.2	odd	2	inner	2178.3.d.n.1693.2		16
11.3	even	5		198.3.j.c.145.4	yes	16
11.7	odd	10		198.3.j.c.127.4	yes	16
11.10	odd	2	inner	2178.3.d.n.1693.7		16
33.14	odd	10		198.3.j.c.145.1	yes	16
33.29	even	10		198.3.j.c.127.1	✓	16
33.32	even	2	inner	2178.3.d.n.1693.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.3.j.c.127.1	✓	16	33.29	even	10
198.3.j.c.127.4	yes	16	11.7	odd	10
198.3.j.c.145.1	yes	16	33.14	odd	10
198.3.j.c.145.4	yes	16	11.3	even	5
2178.3.d.n.1693.2		16	3.2	odd	2	inner
2178.3.d.n.1693.7		16	11.10	odd	2	inner
2178.3.d.n.1693.10		16	33.32	even	2	inner
2178.3.d.n.1693.15		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2178.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +5.68763 q^{5} -10.8351i q^{7} -2.82843i q^{8} +8.04353i q^{10} -4.50286i q^{13} +15.3231 q^{14} +4.00000 q^{16} -17.4537i q^{17} -33.6518i q^{19} -11.3753 q^{20} +7.03543 q^{23} +7.34916 q^{25} +6.36801 q^{26} +21.6701i q^{28} +54.2307i q^{29} -38.4817 q^{31} +5.65685i q^{32} +24.6833 q^{34} -61.6259i q^{35} -20.0715 q^{37} +47.5908 q^{38} -16.0871i q^{40} +15.0259i q^{41} -67.5682i q^{43} +9.94960i q^{46} -29.8783 q^{47} -68.3988 q^{49} +10.3933i q^{50} +9.00572i q^{52} -15.5208 q^{53} -30.6462 q^{56} -76.6938 q^{58} -38.7571 q^{59} +40.9048i q^{61} -54.4213i q^{62} -8.00000 q^{64} -25.6106i q^{65} -33.1092 q^{67} +34.9075i q^{68} +87.1522 q^{70} +86.4062 q^{71} +101.171i q^{73} -28.3854i q^{74} +67.3036i q^{76} -50.0905i q^{79} +22.7505 q^{80} -21.2498 q^{82} +14.6875i q^{83} -99.2704i q^{85} +95.5559 q^{86} -81.0524 q^{89} -48.7888 q^{91} -14.0709 q^{92} -42.2543i q^{94} -191.399i q^{95} +24.7275 q^{97} -96.7305i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 64 q^{16} + 136 q^{25} + 40 q^{31} - 64 q^{34} + 240 q^{37} - 544 q^{49} - 320 q^{58} - 128 q^{64} - 280 q^{67} + 256 q^{70} - 400 q^{82} - 32 q^{91} + 520 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 64 * q^16 + 136 * q^25 + 40 * q^31 - 64 * q^34 + 240 * q^37 - 544 * q^49 - 320 * q^58 - 128 * q^64 - 280 * q^67 + 256 * q^70 - 400 * q^82 - 32 * q^91 + 520 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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