LMFDB - Embedded newform 2178.3.d.m.1693.12

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,-8] 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:	16.0.6879707136000000000000.7
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4x^{14} + 15x^{12} - 56x^{10} + 209x^{8} - 56x^{6} + 15x^{4} - 4x^{2} + 1 $ x^16 - 4x^14 + 15x^12 - 56x^10 + 209x^8 - 56x^6 + 15x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 66)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1693.12
Root			$-0.492303 - 0.159959i$ of defining polynomial
Character	$\chi$	$=$	2178.1693
Dual form			2178.3.d.m.1693.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -0.353386 q^{5} -0.790744i q^{7} -2.82843i q^{8} -0.499764i q^{10} -0.209228i q^{13} +1.11828 q^{14} +4.00000 q^{16} +14.9696i q^{17} -2.16953i q^{19} +0.706772 q^{20} +24.1342 q^{23} -24.8751 q^{25} +0.295893 q^{26} +1.58149i q^{28} -17.8240i q^{29} +33.8376 q^{31} +5.65685i q^{32} -21.1702 q^{34} +0.279438i q^{35} -60.0856 q^{37} +3.06818 q^{38} +0.999527i q^{40} -53.5564i q^{41} +15.5727i q^{43} +34.1309i q^{46} +61.1556 q^{47} +48.3747 q^{49} -35.1787i q^{50} +0.418456i q^{52} -58.9673 q^{53} -2.23656 q^{56} +25.2069 q^{58} +51.4857 q^{59} +40.7522i q^{61} +47.8535i q^{62} -8.00000 q^{64} +0.0739383i q^{65} -72.9302 q^{67} -29.9392i q^{68} -0.395185 q^{70} +95.0520 q^{71} +113.523i q^{73} -84.9738i q^{74} +4.33906i q^{76} +147.127i q^{79} -1.41354 q^{80} +75.7401 q^{82} +19.8752i q^{83} -5.29005i q^{85} -22.0232 q^{86} +74.9710 q^{89} -0.165446 q^{91} -48.2684 q^{92} +86.4870i q^{94} +0.766683i q^{95} +101.506 q^{97} +68.4122i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} - 8 q^{5} - 48 q^{14} + 64 q^{16} + 16 q^{20} + 8 q^{23} - 8 q^{25} - 128 q^{26} - 8 q^{31} - 64 q^{34} + 96 q^{37} + 208 q^{38} + 128 q^{47} - 64 q^{49} - 16 q^{53} + 96 q^{56} + 64 q^{58}+ \cdots + 184 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 - 8 * q^5 - 48 * q^14 + 64 * q^16 + 16 * q^20 + 8 * q^23 - 8 * q^25 - 128 * q^26 - 8 * q^31 - 64 * q^34 + 96 * q^37 + 208 * q^38 + 128 * q^47 - 64 * q^49 - 16 * q^53 + 96 * q^56 + 64 * q^58 + 152 * q^59 - 128 * q^64 + 248 * q^67 - 32 * q^70 + 72 * q^71 - 32 * q^80 + 176 * q^82 - 464 * q^86 + 360 * q^89 - 32 * q^91 - 16 * q^92 + 184 * 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
$5$	−0.353386	−0.0706772	−0.0353386	−	0.999375i	$-0.511251\pi$
$5$	−0.353386	−0.0706772	−0.0353386	+	0.999375i	$0.511251\pi$
$6$	0	0
$7$	− 0.790744i	− 0.112963i	−0.998404	−	0.0564817i	$-0.982012\pi$
$7$	− 0.790744i	− 0.112963i	0.998404	−	0.0564817i	$-0.0179883\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	− 0.499764i	− 0.0499764i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 0.209228i	− 0.0160945i	−0.999968	−	0.00804724i	$-0.997438\pi$
$13$	− 0.209228i	− 0.0160945i	0.999968	−	0.00804724i	$-0.00256154\pi$
$14$	1.11828	0.0798772
$15$	0	0
$16$	4.00000	0.250000
$17$	14.9696i	0.880565i	0.897859	+	0.440282i	$0.145122\pi$
$17$	14.9696i	0.880565i	−0.897859	+	0.440282i	$0.854878\pi$
$18$	0	0
$19$	− 2.16953i	− 0.114186i	−0.998369	−	0.0570930i	$-0.981817\pi$
$19$	− 2.16953i	− 0.114186i	0.998369	−	0.0570930i	$-0.0181831\pi$
$20$	0.706772	0.0353386
$21$	0	0
$22$	0	0
$23$	24.1342	1.04931	0.524656	−	0.851314i	$-0.324194\pi$
$23$	24.1342	1.04931	0.524656	+	0.851314i	$0.324194\pi$
$24$	0	0
$25$	−24.8751	−0.995005
$26$	0.295893	0.0113805
$27$	0	0
$28$	1.58149i	0.0564817i
$29$	− 17.8240i	− 0.614621i	−0.951609	−	0.307310i	$-0.900571\pi$
$29$	− 17.8240i	− 0.614621i	0.951609	−	0.307310i	$-0.0994290\pi$
$30$	0	0
$31$	33.8376	1.09153	0.545767	−	0.837937i	$-0.316238\pi$
$31$	33.8376	1.09153	0.545767	+	0.837937i	$0.316238\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−21.1702	−0.622653
$35$	0.279438i	0.00798394i
$36$	0	0
$37$	−60.0856	−1.62393	−0.811967	−	0.583703i	$-0.801603\pi$
$37$	−60.0856	−1.62393	−0.811967	+	0.583703i	$0.801603\pi$
$38$	3.06818	0.0807416
$39$	0	0
$40$	0.999527i	0.0249882i
$41$	− 53.5564i	− 1.30625i	−0.757249	−	0.653126i	$-0.773457\pi$
$41$	− 53.5564i	− 1.30625i	0.757249	−	0.653126i	$-0.226543\pi$
$42$	0	0
$43$	15.5727i	0.362157i	0.983469	+	0.181078i	$0.0579588\pi$
$43$	15.5727i	0.362157i	−0.983469	+	0.181078i	$0.942041\pi$
$44$	0	0
$45$	0	0
$46$	34.1309i	0.741976i
$47$	61.1556	1.30118	0.650591	−	0.759428i	$-0.274521\pi$
$47$	61.1556	1.30118	0.650591	+	0.759428i	$0.274521\pi$
$48$	0	0
$49$	48.3747	0.987239
$50$	− 35.1787i	− 0.703575i
$51$	0	0
$52$	0.418456i	0.00804724i
$53$	−58.9673	−1.11259	−0.556295	−	0.830985i	$-0.687778\pi$
$53$	−58.9673	−1.11259	−0.556295	+	0.830985i	$0.687778\pi$
$54$	0	0
$55$	0	0
$56$	−2.23656	−0.0399386
$57$	0	0
$58$	25.2069	0.434602
$59$	51.4857	0.872639	0.436320	−	0.899792i	$-0.356282\pi$
$59$	51.4857	0.872639	0.436320	+	0.899792i	$0.356282\pi$
$60$	0	0
$61$	40.7522i	0.668069i	0.942561	+	0.334034i	$0.108410\pi$
$61$	40.7522i	0.668069i	−0.942561	+	0.334034i	$0.891590\pi$
$62$	47.8535i	0.771831i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	0.0739383i	0.00113751i
$66$	0	0
$67$	−72.9302	−1.08851	−0.544255	−	0.838920i	$-0.683188\pi$
$67$	−72.9302	−1.08851	−0.544255	+	0.838920i	$0.683188\pi$
$68$	− 29.9392i	− 0.440282i
$69$	0	0
$70$	−0.395185	−0.00564550
$71$	95.0520	1.33876	0.669380	−	0.742920i	$-0.266560\pi$
$71$	95.0520	1.33876	0.669380	+	0.742920i	$0.266560\pi$
$72$	0	0
$73$	113.523i	1.55511i	0.628814	+	0.777555i	$0.283541\pi$
$73$	113.523i	1.55511i	−0.628814	+	0.777555i	$0.716459\pi$
$74$	− 84.9738i	− 1.14829i
$75$	0	0
$76$	4.33906i	0.0570930i
$77$	0	0
$78$	0	0
$79$	147.127i	1.86236i	0.364556	+	0.931182i	$0.381221\pi$
$79$	147.127i	1.86236i	−0.364556	+	0.931182i	$0.618779\pi$
$80$	−1.41354	−0.0176693
$81$	0	0
$82$	75.7401	0.923660
$83$	19.8752i	0.239461i	0.992806	+	0.119730i	$0.0382030\pi$
$83$	19.8752i	0.239461i	−0.992806	+	0.119730i	$0.961797\pi$
$84$	0	0
$85$	− 5.29005i	− 0.0622359i
$86$	−22.0232	−0.256084
$87$	0	0
$88$	0	0
$89$	74.9710	0.842371	0.421185	−	0.906975i	$-0.361614\pi$
$89$	74.9710	0.842371	0.421185	+	0.906975i	$0.361614\pi$
$90$	0	0
$91$	−0.165446	−0.00181809
$92$	−48.2684	−0.524656
$93$	0	0
$94$	86.4870i	0.920075i
$95$	0.766683i	0.00807034i
$96$	0	0
$97$	101.506	1.04645	0.523227	−	0.852194i	$-0.324728\pi$
$97$	101.506	1.04645	0.523227	+	0.852194i	$0.324728\pi$
$98$	68.4122i	0.698084i
$99$	0	0
$100$	49.7502	0.497502
$101$	26.0855i	0.258272i	0.991627	+	0.129136i	$0.0412204\pi$
$101$	26.0855i	0.258272i	−0.991627	+	0.129136i	$0.958780\pi$
$102$	0	0
$103$	−31.5362	−0.306177	−0.153088	−	0.988212i	$-0.548922\pi$
$103$	−31.5362	−0.306177	−0.153088	+	0.988212i	$0.548922\pi$
$104$	−0.591787	−0.00569026
$105$	0	0
$106$	− 83.3923i	− 0.786720i
$107$	− 80.5378i	− 0.752690i	−0.926480	−	0.376345i	$-0.877181\pi$
$107$	− 80.5378i	− 0.752690i	0.926480	−	0.376345i	$-0.122819\pi$
$108$	0	0
$109$	− 32.1811i	− 0.295239i	−0.989044	−	0.147620i	$-0.952839\pi$
$109$	− 32.1811i	− 0.295239i	0.989044	−	0.147620i	$-0.0471612\pi$
$110$	0	0
$111$	0	0
$112$	− 3.16297i	− 0.0282408i
$113$	29.2039	0.258441	0.129221	−	0.991616i	$-0.458752\pi$
$113$	29.2039	0.258441	0.129221	+	0.991616i	$0.458752\pi$
$114$	0	0
$115$	−8.52869	−0.0741625
$116$	35.6480i	0.307310i
$117$	0	0
$118$	72.8118i	0.617049i
$119$	11.8371	0.0994716
$120$	0	0
$121$	0	0
$122$	−57.6323	−0.472396
$123$	0	0
$124$	−67.6751	−0.545767
$125$	17.6252	0.141001
$126$	0	0
$127$	149.052i	1.17364i	0.809719	+	0.586818i	$0.199620\pi$
$127$	149.052i	1.17364i	−0.809719	+	0.586818i	$0.800380\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−0.104565	−0.000804343 0
$131$	165.965i	1.26691i	0.773780	+	0.633454i	$0.218364\pi$
$131$	165.965i	1.26691i	−0.773780	+	0.633454i	$0.781636\pi$
$132$	0	0
$133$	−1.71554	−0.0128988
$134$	− 103.139i	− 0.769693i
$135$	0	0
$136$	42.3404	0.311327
$137$	146.868	1.07203	0.536014	−	0.844209i	$-0.319930\pi$
$137$	146.868	1.07203	0.536014	+	0.844209i	$0.319930\pi$
$138$	0	0
$139$	213.774i	1.53795i	0.639281	+	0.768973i	$0.279232\pi$
$139$	213.774i	1.53795i	−0.639281	+	0.768973i	$0.720768\pi$
$140$	− 0.558876i	− 0.00399197i
$141$	0	0
$142$	134.424i	0.946646i
$143$	0	0
$144$	0	0
$145$	6.29876i	0.0434397i
$146$	−160.546	−1.09963
$147$	0	0
$148$	120.171	0.811967
$149$	231.242i	1.55196i	0.630757	+	0.775981i	$0.282745\pi$
$149$	231.242i	1.55196i	−0.630757	+	0.775981i	$0.717255\pi$
$150$	0	0
$151$	− 266.405i	− 1.76427i	−0.470992	−	0.882137i	$-0.656104\pi$
$151$	− 266.405i	− 1.76427i	0.470992	−	0.882137i	$-0.343896\pi$
$152$	−6.13636	−0.0403708
$153$	0	0
$154$	0	0
$155$	−11.9577	−0.0771466
$156$	0	0
$157$	−104.278	−0.664193	−0.332097	−	0.943245i	$-0.607756\pi$
$157$	−104.278	−0.664193	−0.332097	+	0.943245i	$0.607756\pi$
$158$	−208.069	−1.31689
$159$	0	0
$160$	− 1.99905i	− 0.0124941i
$161$	− 19.0839i	− 0.118534i
$162$	0	0
$163$	−279.668	−1.71575	−0.857877	−	0.513855i	$-0.828217\pi$
$163$	−279.668	−1.71575	−0.857877	+	0.513855i	$0.828217\pi$
$164$	107.113i	0.653126i
$165$	0	0
$166$	−28.1078	−0.169324
$167$	256.946i	1.53860i	0.638889	+	0.769299i	$0.279394\pi$
$167$	256.946i	1.53860i	−0.638889	+	0.769299i	$0.720606\pi$
$168$	0	0
$169$	168.956	0.999741
$170$	7.48126	0.0440074
$171$	0	0
$172$	− 31.1455i	− 0.181078i
$173$	110.189i	0.636929i	0.947935	+	0.318464i	$0.103167\pi$
$173$	110.189i	0.636929i	−0.947935	+	0.318464i	$0.896833\pi$
$174$	0	0
$175$	19.6698i	0.112399i
$176$	0	0
$177$	0	0
$178$	106.025i	0.595646i
$179$	−27.1267	−0.151546	−0.0757729	−	0.997125i	$-0.524142\pi$
$179$	−27.1267	−0.151546	−0.0757729	+	0.997125i	$0.524142\pi$
$180$	0	0
$181$	259.571	1.43410	0.717048	−	0.697024i	$-0.245493\pi$
$181$	259.571	1.43410	0.717048	+	0.697024i	$0.245493\pi$
$182$	− 0.233976i	− 0.00128558i
$183$	0	0
$184$	− 68.2618i	− 0.370988i
$185$	21.2334	0.114775
$186$	0	0
$187$	0	0
$188$	−122.311	−0.650591
$189$	0	0
$190$	−1.08425	−0.00570660
$191$	124.782	0.653310	0.326655	−	0.945144i	$-0.394078\pi$
$191$	124.782	0.653310	0.326655	+	0.945144i	$0.394078\pi$
$192$	0	0
$193$	41.9892i	0.217560i	0.994066	+	0.108780i	$0.0346945\pi$
$193$	41.9892i	0.217560i	−0.994066	+	0.108780i	$0.965306\pi$
$194$	143.551i	0.739954i
$195$	0	0
$196$	−96.7494	−0.493620
$197$	− 274.338i	− 1.39258i	−0.717760	−	0.696290i	$-0.754833\pi$
$197$	− 274.338i	− 1.39258i	0.717760	−	0.696290i	$-0.245167\pi$
$198$	0	0
$199$	−150.979	−0.758690	−0.379345	−	0.925255i	$-0.623851\pi$
$199$	−150.979	−0.758690	−0.379345	+	0.925255i	$0.623851\pi$
$200$	70.3575i	0.351787i
$201$	0	0
$202$	−36.8904	−0.182626
$203$	−14.0942	−0.0694296
$204$	0	0
$205$	18.9261i	0.0923223i
$206$	− 44.5989i	− 0.216500i
$207$	0	0
$208$	− 0.836913i	− 0.00402362i
$209$	0	0
$210$	0	0
$211$	18.6837i	0.0885486i	0.999019	+	0.0442743i	$0.0140976\pi$
$211$	18.6837i	0.0885486i	−0.999019	+	0.0442743i	$0.985902\pi$
$212$	117.935	0.556295
$213$	0	0
$214$	113.898	0.532232
$215$	− 5.50319i	− 0.0255963i
$216$	0	0
$217$	− 26.7568i	− 0.123303i
$218$	45.5109	0.208766
$219$	0	0
$220$	0	0
$221$	3.13206	0.0141722
$222$	0	0
$223$	−94.2808	−0.422784	−0.211392	−	0.977401i	$-0.567800\pi$
$223$	−94.2808	−0.422784	−0.211392	+	0.977401i	$0.567800\pi$
$224$	4.47312	0.0199693
$225$	0	0
$226$	41.3005i	0.182746i
$227$	289.791i	1.27661i	0.769782	+	0.638306i	$0.220365\pi$
$227$	289.791i	1.27661i	−0.769782	+	0.638306i	$0.779635\pi$
$228$	0	0
$229$	−228.228	−0.996631	−0.498315	−	0.866996i	$-0.666048\pi$
$229$	−228.228	−0.996631	−0.498315	+	0.866996i	$0.666048\pi$
$230$	− 12.0614i	− 0.0524408i
$231$	0	0
$232$	−50.4139	−0.217301
$233$	298.827i	1.28252i	0.767324	+	0.641260i	$0.221588\pi$
$233$	298.827i	1.28252i	−0.767324	+	0.641260i	$0.778412\pi$
$234$	0	0
$235$	−21.6115	−0.0919640
$236$	−102.971	−0.436320
$237$	0	0
$238$	16.7402i	0.0703370i
$239$	245.929i	1.02899i	0.857492	+	0.514496i	$0.172021\pi$
$239$	245.929i	1.02899i	−0.857492	+	0.514496i	$0.827979\pi$
$240$	0	0
$241$	373.418i	1.54945i	0.632297	+	0.774726i	$0.282112\pi$
$241$	373.418i	1.54945i	−0.632297	+	0.774726i	$0.717888\pi$
$242$	0	0
$243$	0	0
$244$	− 81.5044i	− 0.334034i
$245$	−17.0950	−0.0697753
$246$	0	0
$247$	−0.453927	−0.00183776
$248$	− 95.7071i	− 0.385916i
$249$	0	0
$250$	24.9258i	0.0997031i
$251$	359.422	1.43196	0.715980	−	0.698121i	$-0.245980\pi$
$251$	359.422	1.43196	0.715980	+	0.698121i	$0.245980\pi$
$252$	0	0
$253$	0	0
$254$	−210.791	−0.829886
$255$	0	0
$256$	16.0000	0.0625000
$257$	35.7842	0.139238	0.0696191	−	0.997574i	$-0.477822\pi$
$257$	35.7842	0.139238	0.0696191	+	0.997574i	$0.477822\pi$
$258$	0	0
$259$	47.5123i	0.183445i
$260$	− 0.147877i	0 0.000568756i
$261$	0	0
$262$	−234.710	−0.895840
$263$	− 364.668i	− 1.38657i	−0.720663	−	0.693286i	$-0.756162\pi$
$263$	− 364.668i	− 1.38657i	0.720663	−	0.693286i	$-0.243838\pi$
$264$	0	0
$265$	20.8382	0.0786348
$266$	− 2.42615i	− 0.00912085i
$267$	0	0
$268$	145.860	0.544255
$269$	13.3769	0.0497283	0.0248641	−	0.999691i	$-0.492085\pi$
$269$	13.3769	0.0497283	0.0248641	+	0.999691i	$0.492085\pi$
$270$	0	0
$271$	− 70.8425i	− 0.261411i	−0.991421	−	0.130706i	$-0.958276\pi$
$271$	− 70.8425i	− 0.261411i	0.991421	−	0.130706i	$-0.0417243\pi$
$272$	59.8784i	0.220141i
$273$	0	0
$274$	207.702i	0.758038i
$275$	0	0
$276$	0	0
$277$	79.3467i	0.286450i	0.989690	+	0.143225i	$0.0457473\pi$
$277$	79.3467i	0.286450i	−0.989690	+	0.143225i	$0.954253\pi$
$278$	−302.323	−1.08749
$279$	0	0
$280$	0.790370	0.00282275
$281$	343.092i	1.22097i	0.792029	+	0.610484i	$0.209025\pi$
$281$	343.092i	1.22097i	−0.792029	+	0.610484i	$0.790975\pi$
$282$	0	0
$283$	− 119.679i	− 0.422895i	−0.977389	−	0.211447i	$-0.932182\pi$
$283$	− 119.679i	− 0.422895i	0.977389	−	0.211447i	$-0.0678177\pi$
$284$	−190.104	−0.669380
$285$	0	0
$286$	0	0
$287$	−42.3493	−0.147559
$288$	0	0
$289$	64.9111	0.224606
$290$	−8.90779	−0.0307165
$291$	0	0
$292$	− 227.046i	− 0.777555i
$293$	11.5161i	0.0393042i	0.999807	+	0.0196521i	$0.00625585\pi$
$293$	11.5161i	0.0393042i	−0.999807	+	0.0196521i	$0.993744\pi$
$294$	0	0
$295$	−18.1943	−0.0616757
$296$	169.948i	0.574147i
$297$	0	0
$298$	−327.026	−1.09740
$299$	− 5.04955i	− 0.0168881i
$300$	0	0
$301$	12.3141	0.0409105
$302$	376.754	1.24753
$303$	0	0
$304$	− 8.67813i	− 0.0285465i
$305$	− 14.4013i	− 0.0472172i
$306$	0	0
$307$	− 196.858i	− 0.641231i	−0.947210	−	0.320615i	$-0.896110\pi$
$307$	− 196.858i	− 0.641231i	0.947210	−	0.320615i	$-0.103890\pi$
$308$	0	0
$309$	0	0
$310$	− 16.9108i	− 0.0545509i
$311$	−270.416	−0.869504	−0.434752	−	0.900550i	$-0.643164\pi$
$311$	−270.416	−0.869504	−0.434752	+	0.900550i	$0.643164\pi$
$312$	0	0
$313$	344.610	1.10099	0.550495	−	0.834839i	$-0.314439\pi$
$313$	344.610	1.10099	0.550495	+	0.834839i	$0.314439\pi$
$314$	− 147.472i	− 0.469656i
$315$	0	0
$316$	− 294.253i	− 0.931182i
$317$	−65.5714	−0.206850	−0.103425	−	0.994637i	$-0.532980\pi$
$317$	−65.5714	−0.206850	−0.103425	+	0.994637i	$0.532980\pi$
$318$	0	0
$319$	0	0
$320$	2.82709	0.00883465
$321$	0	0
$322$	26.9888	0.0838161
$323$	32.4770	0.100548
$324$	0	0
$325$	5.20458i	0.0160141i
$326$	− 395.510i	− 1.21322i
$327$	0	0
$328$	−151.480	−0.461830
$329$	− 48.3584i	− 0.146986i
$330$	0	0
$331$	105.818	0.319693	0.159847	−	0.987142i	$-0.448900\pi$
$331$	105.818	0.319693	0.159847	+	0.987142i	$0.448900\pi$
$332$	− 39.7505i	− 0.119730i
$333$	0	0
$334$	−363.376	−1.08795
$335$	25.7725	0.0769329
$336$	0	0
$337$	− 225.621i	− 0.669498i	−0.942307	−	0.334749i	$-0.891348\pi$
$337$	− 225.621i	− 0.669498i	0.942307	−	0.334749i	$-0.108652\pi$
$338$	238.940i	0.706924i
$339$	0	0
$340$	10.5801i	0.0311179i
$341$	0	0
$342$	0	0
$343$	− 76.9984i	− 0.224485i
$344$	44.0464	0.128042
$345$	0	0
$346$	−155.830	−0.450377
$347$	− 135.569i	− 0.390687i	−0.980735	−	0.195344i	$-0.937418\pi$
$347$	− 135.569i	− 0.390687i	0.980735	−	0.195344i	$-0.0625822\pi$
$348$	0	0
$349$	500.154i	1.43311i	0.697533	+	0.716553i	$0.254281\pi$
$349$	500.154i	1.43311i	−0.697533	+	0.716553i	$0.745719\pi$
$350$	−27.8174	−0.0794782
$351$	0	0
$352$	0	0
$353$	−320.592	−0.908192	−0.454096	−	0.890953i	$-0.650038\pi$
$353$	−320.592	−0.908192	−0.454096	+	0.890953i	$0.650038\pi$
$354$	0	0
$355$	−33.5900	−0.0946199
$356$	−149.942	−0.421185
$357$	0	0
$358$	− 38.3630i	− 0.107159i
$359$	113.433i	0.315971i	0.987441	+	0.157985i	$0.0504999\pi$
$359$	113.433i	0.315971i	−0.987441	+	0.157985i	$0.949500\pi$
$360$	0	0
$361$	356.293	0.986962
$362$	367.089i	1.01406i
$363$	0	0
$364$	0.330892	0.000909043 0
$365$	− 40.1175i	− 0.109911i
$366$	0	0
$367$	89.5681	0.244055	0.122027	−	0.992527i	$-0.461060\pi$
$367$	89.5681	0.244055	0.122027	+	0.992527i	$0.461060\pi$
$368$	96.5367	0.262328
$369$	0	0
$370$	30.0286i	0.0811583i
$371$	46.6280i	0.125682i
$372$	0	0
$373$	− 275.552i	− 0.738744i	−0.929282	−	0.369372i	$-0.879573\pi$
$373$	− 275.552i	− 0.738744i	0.929282	−	0.369372i	$-0.120427\pi$
$374$	0	0
$375$	0	0
$376$	− 172.974i	− 0.460037i
$377$	−3.72928	−0.00989200
$378$	0	0
$379$	−175.545	−0.463179	−0.231590	−	0.972814i	$-0.574393\pi$
$379$	−175.545	−0.463179	−0.231590	+	0.972814i	$0.574393\pi$
$380$	− 1.53337i	− 0.00403517i
$381$	0	0
$382$	176.469i	0.461960i
$383$	51.1788	0.133626	0.0668131	−	0.997766i	$-0.478717\pi$
$383$	51.1788	0.133626	0.0668131	+	0.997766i	$0.478717\pi$
$384$	0	0
$385$	0	0
$386$	−59.3817	−0.153838
$387$	0	0
$388$	−203.012	−0.523227
$389$	22.9064	0.0588854	0.0294427	−	0.999566i	$-0.490627\pi$
$389$	22.9064	0.0588854	0.0294427	+	0.999566i	$0.490627\pi$
$390$	0	0
$391$	361.279i	0.923987i
$392$	− 136.824i	− 0.349042i
$393$	0	0
$394$	387.973	0.984703
$395$	− 51.9925i	− 0.131627i
$396$	0	0
$397$	131.054	0.330112	0.165056	−	0.986284i	$-0.447220\pi$
$397$	131.054	0.330112	0.165056	+	0.986284i	$0.447220\pi$
$398$	− 213.517i	− 0.536475i
$399$	0	0
$400$	−99.5005	−0.248751
$401$	−775.849	−1.93479	−0.967393	−	0.253279i	$-0.918491\pi$
$401$	−775.849	−1.93479	−0.967393	+	0.253279i	$0.918491\pi$
$402$	0	0
$403$	− 7.07977i	− 0.0175677i
$404$	− 52.1710i	− 0.129136i
$405$	0	0
$406$	− 19.9322i	− 0.0490942i
$407$	0	0
$408$	0	0
$409$	747.379i	1.82733i	0.406464	+	0.913667i	$0.366762\pi$
$409$	747.379i	1.82733i	−0.406464	+	0.913667i	$0.633238\pi$
$410$	−26.7655	−0.0652817
$411$	0	0
$412$	63.0724	0.153088
$413$	− 40.7120i	− 0.0985763i
$414$	0	0
$415$	− 7.02364i	− 0.0169244i
$416$	1.18357	0.00284513
$417$	0	0
$418$	0	0
$419$	408.427	0.974765	0.487383	−	0.873189i	$-0.337952\pi$
$419$	408.427	0.974765	0.487383	+	0.873189i	$0.337952\pi$
$420$	0	0
$421$	104.511	0.248245	0.124123	−	0.992267i	$-0.460388\pi$
$421$	104.511	0.248245	0.124123	+	0.992267i	$0.460388\pi$
$422$	−26.4228	−0.0626133
$423$	0	0
$424$	166.785i	0.393360i
$425$	− 372.371i	− 0.876166i
$426$	0	0
$427$	32.2245	0.0754673
$428$	161.076i	0.376345i
$429$	0	0
$430$	7.78269	0.0180993
$431$	− 289.063i	− 0.670680i	−0.942097	−	0.335340i	$-0.891149\pi$
$431$	− 289.063i	− 0.670680i	0.942097	−	0.335340i	$-0.108851\pi$
$432$	0	0
$433$	550.999	1.27252	0.636258	−	0.771476i	$-0.280481\pi$
$433$	550.999	1.27252	0.636258	+	0.771476i	$0.280481\pi$
$434$	37.8399	0.0871886
$435$	0	0
$436$	64.3622i	0.147620i
$437$	− 52.3599i	− 0.119817i
$438$	0	0
$439$	− 576.891i	− 1.31410i	−0.753846	−	0.657051i	$-0.771804\pi$
$439$	− 576.891i	− 1.31410i	0.753846	−	0.657051i	$-0.228196\pi$
$440$	0	0
$441$	0	0
$442$	4.42940i	0.0100213i
$443$	178.241	0.402349	0.201175	−	0.979555i	$-0.435524\pi$
$443$	178.241	0.402349	0.201175	+	0.979555i	$0.435524\pi$
$444$	0	0
$445$	−26.4937	−0.0595364
$446$	− 133.333i	− 0.298953i
$447$	0	0
$448$	6.32595i	0.0141204i
$449$	−622.710	−1.38688	−0.693441	−	0.720513i	$-0.743906\pi$
$449$	−622.710	−1.38688	−0.693441	+	0.720513i	$0.743906\pi$
$450$	0	0
$451$	0	0
$452$	−58.4078	−0.129221
$453$	0	0
$454$	−409.827	−0.902702
$455$	0.0584663	0.000128497 0
$456$	0	0
$457$	− 787.808i	− 1.72387i	−0.507021	−	0.861934i	$-0.669253\pi$
$457$	− 787.808i	− 1.72387i	0.507021	−	0.861934i	$-0.330747\pi$
$458$	− 322.764i	− 0.704725i
$459$	0	0
$460$	17.0574	0.0370812
$461$	− 377.985i	− 0.819924i	−0.912103	−	0.409962i	$-0.865542\pi$
$461$	− 377.985i	− 0.819924i	0.912103	−	0.409962i	$-0.134458\pi$
$462$	0	0
$463$	−49.3159	−0.106514	−0.0532569	−	0.998581i	$-0.516960\pi$
$463$	−49.3159	−0.106514	−0.0532569	+	0.998581i	$0.516960\pi$
$464$	− 71.2960i	− 0.153655i
$465$	0	0
$466$	−422.605	−0.906878
$467$	117.187	0.250936	0.125468	−	0.992098i	$-0.459957\pi$
$467$	117.187	0.250936	0.125468	+	0.992098i	$0.459957\pi$
$468$	0	0
$469$	57.6691i	0.122962i
$470$	− 30.5633i	− 0.0650283i
$471$	0	0
$472$	− 145.624i	− 0.308525i
$473$	0	0
$474$	0	0
$475$	53.9674i	0.113616i
$476$	−23.6742	−0.0497358
$477$	0	0
$478$	−347.797	−0.727608
$479$	− 920.732i	− 1.92220i	−0.276209	−	0.961098i	$-0.589078\pi$
$479$	− 920.732i	− 1.92220i	0.276209	−	0.961098i	$-0.410922\pi$
$480$	0	0
$481$	12.5716i	0.0261364i
$482$	−528.093	−1.09563
$483$	0	0
$484$	0	0
$485$	−35.8708	−0.0739604
$486$	0	0
$487$	388.108	0.796936	0.398468	−	0.917182i	$-0.369542\pi$
$487$	388.108	0.796936	0.398468	+	0.917182i	$0.369542\pi$
$488$	115.265	0.236198
$489$	0	0
$490$	− 24.1759i	− 0.0493386i
$491$	364.974i	0.743328i	0.928367	+	0.371664i	$0.121213\pi$
$491$	364.974i	0.743328i	−0.928367	+	0.371664i	$0.878787\pi$
$492$	0	0
$493$	266.818	0.541213
$494$	− 0.641950i	− 0.00129949i
$495$	0	0
$496$	135.350	0.272884
$497$	− 75.1617i	− 0.151231i
$498$	0	0
$499$	−245.125	−0.491232	−0.245616	−	0.969367i	$-0.578990\pi$
$499$	−245.125	−0.491232	−0.245616	+	0.969367i	$0.578990\pi$
$500$	−35.2504	−0.0705007
$501$	0	0
$502$	508.299i	1.01255i
$503$	− 294.256i	− 0.585002i	−0.956265	−	0.292501i	$-0.905513\pi$
$503$	− 294.256i	− 0.585002i	0.956265	−	0.292501i	$-0.0944875\pi$
$504$	0	0
$505$	− 9.21825i	− 0.0182540i
$506$	0	0
$507$	0	0
$508$	− 298.103i	− 0.586818i
$509$	855.189	1.68013	0.840067	−	0.542482i	$-0.182515\pi$
$509$	855.189	1.68013	0.840067	+	0.542482i	$0.182515\pi$
$510$	0	0
$511$	89.7677	0.175671
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	50.6065i	0.0984563i
$515$	11.1445	0.0216397
$516$	0	0
$517$	0	0
$518$	−67.1925	−0.129715
$519$	0	0
$520$	0.209129	0.000402172 0
$521$	607.086	1.16523	0.582617	−	0.812747i	$-0.302029\pi$
$521$	607.086	1.16523	0.582617	+	0.812747i	$0.302029\pi$
$522$	0	0
$523$	674.935i	1.29051i	0.763969	+	0.645253i	$0.223248\pi$
$523$	674.935i	1.29051i	−0.763969	+	0.645253i	$0.776752\pi$
$524$	− 331.930i	− 0.633454i
$525$	0	0
$526$	515.719	0.980454
$527$	506.535i	0.961166i
$528$	0	0
$529$	53.4586	0.101056
$530$	29.4697i	0.0556032i
$531$	0	0
$532$	3.43109	0.00644941
$533$	−11.2055	−0.0210234
$534$	0	0
$535$	28.4609i	0.0531980i
$536$	206.278i	0.384846i
$537$	0	0
$538$	18.9178i	0.0351632i
$539$	0	0
$540$	0	0
$541$	233.688i	0.431956i	0.976398	+	0.215978i	$0.0692940\pi$
$541$	233.688i	0.431956i	−0.976398	+	0.215978i	$0.930706\pi$
$542$	100.186	0.184846
$543$	0	0
$544$	−84.6808	−0.155663
$545$	11.3724i	0.0208667i
$546$	0	0
$547$	953.254i	1.74269i	0.490667	+	0.871347i	$0.336753\pi$
$547$	953.254i	1.74269i	−0.490667	+	0.871347i	$0.663247\pi$
$548$	−293.735	−0.536014
$549$	0	0
$550$	0	0
$551$	−38.6697	−0.0701810
$552$	0	0
$553$	116.339	0.210379
$554$	−112.213	−0.202551
$555$	0	0
$556$	− 427.549i	− 0.768973i
$557$	450.083i	0.808049i	0.914748	+	0.404025i	$0.132389\pi$
$557$	450.083i	0.808049i	−0.914748	+	0.404025i	$0.867611\pi$
$558$	0	0
$559$	3.25826	0.00582873
$560$	1.11775i	0.00199598i
$561$	0	0
$562$	−485.205	−0.863354
$563$	312.343i	0.554783i	0.960757	+	0.277391i	$0.0894698\pi$
$563$	312.343i	0.554783i	−0.960757	+	0.277391i	$0.910530\pi$
$564$	0	0
$565$	−10.3202	−0.0182659
$566$	169.252	0.299032
$567$	0	0
$568$	− 268.848i	− 0.473323i
$569$	− 564.562i	− 0.992200i	−0.868265	−	0.496100i	$-0.834765\pi$
$569$	− 564.562i	− 0.992200i	0.868265	−	0.496100i	$-0.165235\pi$
$570$	0	0
$571$	− 224.552i	− 0.393260i	−0.980478	−	0.196630i	$-0.937000\pi$
$571$	− 224.552i	− 0.393260i	0.980478	−	0.196630i	$-0.0629998\pi$
$572$	0	0
$573$	0	0
$574$	− 59.8910i	− 0.104340i
$575$	−600.341	−1.04407
$576$	0	0
$577$	−1055.36	−1.82905	−0.914524	−	0.404532i	$-0.867434\pi$
$577$	−1055.36	−1.82905	−0.914524	+	0.404532i	$0.867434\pi$
$578$	91.7981i	0.158820i
$579$	0	0
$580$	− 12.5975i	− 0.0217198i
$581$	15.7162	0.0270503
$582$	0	0
$583$	0	0
$584$	321.092	0.549815
$585$	0	0
$586$	−16.2862	−0.0277922
$587$	79.4137	0.135287	0.0676437	−	0.997710i	$-0.478452\pi$
$587$	79.4137	0.135287	0.0676437	+	0.997710i	$0.478452\pi$
$588$	0	0
$589$	− 73.4117i	− 0.124638i
$590$	− 25.7307i	− 0.0436113i
$591$	0	0
$592$	−240.342	−0.405984
$593$	− 709.519i	− 1.19649i	−0.801313	−	0.598245i	$-0.795865\pi$
$593$	− 709.519i	− 1.19649i	0.801313	−	0.598245i	$-0.204135\pi$
$594$	0	0
$595$	−4.18307	−0.00703037
$596$	− 462.484i	− 0.775981i
$597$	0	0
$598$	7.14114	0.0119417
$599$	−698.251	−1.16569	−0.582847	−	0.812582i	$-0.698061\pi$
$599$	−698.251	−1.16569	−0.582847	+	0.812582i	$0.698061\pi$
$600$	0	0
$601$	− 223.491i	− 0.371865i	−0.982562	−	0.185933i	$-0.940469\pi$
$601$	− 223.491i	− 0.371865i	0.982562	−	0.185933i	$-0.0595306\pi$
$602$	17.4147i	0.0289281i
$603$	0	0
$604$	532.811i	0.882137i
$605$	0	0
$606$	0	0
$607$	− 953.367i	− 1.57062i	−0.619102	−	0.785310i	$-0.712503\pi$
$607$	− 953.367i	− 1.57062i	0.619102	−	0.785310i	$-0.287497\pi$
$608$	12.2727	0.0201854
$609$	0	0
$610$	20.3665	0.0333876
$611$	− 12.7955i	− 0.0209418i
$612$	0	0
$613$	− 27.8874i	− 0.0454934i	−0.999741	−	0.0227467i	$-0.992759\pi$
$613$	− 27.8874i	− 0.0454934i	0.999741	−	0.0227467i	$-0.00724112\pi$
$614$	278.399	0.453418
$615$	0	0
$616$	0	0
$617$	997.165	1.61615	0.808075	−	0.589079i	$-0.200509\pi$
$617$	997.165	1.61615	0.808075	+	0.589079i	$0.200509\pi$
$618$	0	0
$619$	−23.2638	−0.0375829	−0.0187915	−	0.999823i	$-0.505982\pi$
$619$	−23.2638	−0.0375829	−0.0187915	+	0.999823i	$0.505982\pi$
$620$	23.9154	0.0385733
$621$	0	0
$622$	− 382.426i	− 0.614832i
$623$	− 59.2828i	− 0.0951571i
$624$	0	0
$625$	615.649	0.985039
$626$	487.352i	0.778517i
$627$	0	0
$628$	208.557	0.332097
$629$	− 899.457i	− 1.42998i
$630$	0	0
$631$	−523.766	−0.830058	−0.415029	−	0.909808i	$-0.636229\pi$
$631$	−523.766	−0.830058	−0.415029	+	0.909808i	$0.636229\pi$
$632$	416.137	0.658445
$633$	0	0
$634$	− 92.7320i	− 0.146265i
$635$	− 52.6728i	− 0.0829493i
$636$	0	0
$637$	− 10.1214i	− 0.0158891i
$638$	0	0
$639$	0	0
$640$	3.99811i	0.00624704i
$641$	67.4115	0.105166	0.0525831	−	0.998617i	$-0.483255\pi$
$641$	67.4115	0.105166	0.0525831	+	0.998617i	$0.483255\pi$
$642$	0	0
$643$	−806.459	−1.25421	−0.627106	−	0.778934i	$-0.715761\pi$
$643$	−806.459	−1.25421	−0.627106	+	0.778934i	$0.715761\pi$
$644$	38.1679i	0.0592669i
$645$	0	0
$646$	45.9295i	0.0710982i
$647$	970.912	1.50064	0.750319	−	0.661076i	$-0.229900\pi$
$647$	970.912	1.50064	0.750319	+	0.661076i	$0.229900\pi$
$648$	0	0
$649$	0	0
$650$	−7.36038	−0.0113237
$651$	0	0
$652$	559.336	0.857877
$653$	−28.0428	−0.0429445	−0.0214723	−	0.999769i	$-0.506835\pi$
$653$	−28.0428	−0.0429445	−0.0214723	+	0.999769i	$0.506835\pi$
$654$	0	0
$655$	− 58.6497i	− 0.0895416i
$656$	− 214.225i	− 0.326563i
$657$	0	0
$658$	68.3891	0.103935
$659$	− 570.449i	− 0.865628i	−0.901483	−	0.432814i	$-0.857521\pi$
$659$	− 570.449i	− 0.865628i	0.901483	−	0.432814i	$-0.142479\pi$
$660$	0	0
$661$	854.772	1.29315	0.646575	−	0.762851i	$-0.276201\pi$
$661$	854.772	1.29315	0.646575	+	0.762851i	$0.276201\pi$
$662$	149.650i	0.226057i
$663$	0	0
$664$	56.2157	0.0846622
$665$	0.606249	0.000911653 0
$666$	0	0
$667$	− 430.168i	− 0.644929i
$668$	− 513.892i	− 0.769299i
$669$	0	0
$670$	36.4478i	0.0543998i
$671$	0	0
$672$	0	0
$673$	− 353.405i	− 0.525119i	−0.964916	−	0.262560i	$-0.915433\pi$
$673$	− 353.405i	− 0.525119i	0.964916	−	0.262560i	$-0.0845667\pi$
$674$	319.076	0.473407
$675$	0	0
$676$	−337.912	−0.499870
$677$	1242.34i	1.83506i	0.397665	+	0.917531i	$0.369821\pi$
$677$	1242.34i	1.83506i	−0.397665	+	0.917531i	$0.630179\pi$
$678$	0	0
$679$	− 80.2652i	− 0.118211i
$680$	−14.9625	−0.0220037
$681$	0	0
$682$	0	0
$683$	818.617	1.19856	0.599281	−	0.800539i	$-0.295453\pi$
$683$	818.617	1.19856	0.599281	+	0.800539i	$0.295453\pi$
$684$	0	0
$685$	−51.9010	−0.0757679
$686$	108.892	0.158735
$687$	0	0
$688$	62.2910i	0.0905392i
$689$	12.3376i	0.0179066i
$690$	0	0
$691$	273.464	0.395751	0.197875	−	0.980227i	$-0.436596\pi$
$691$	273.464	0.395751	0.197875	+	0.980227i	$0.436596\pi$
$692$	− 220.377i	− 0.318464i
$693$	0	0
$694$	191.723	0.276258
$695$	− 75.5450i	− 0.108698i
$696$	0	0
$697$	801.717	1.15024
$698$	−707.324	−1.01336
$699$	0	0
$700$	− 39.3397i	− 0.0561995i
$701$	507.111i	0.723411i	0.932292	+	0.361706i	$0.117805\pi$
$701$	507.111i	0.723411i	−0.932292	+	0.361706i	$0.882195\pi$
$702$	0	0
$703$	130.358i	0.185430i
$704$	0	0
$705$	0	0
$706$	− 453.385i	− 0.642189i
$707$	20.6269	0.0291753
$708$	0	0
$709$	578.587	0.816060	0.408030	−	0.912968i	$-0.366216\pi$
$709$	578.587	0.816060	0.408030	+	0.912968i	$0.366216\pi$
$710$	− 47.5035i	− 0.0669063i
$711$	0	0
$712$	− 212.050i	− 0.297823i
$713$	816.642	1.14536
$714$	0	0
$715$	0	0
$716$	54.2534	0.0757729
$717$	0	0
$718$	−160.419	−0.223425
$719$	−39.2537	−0.0545948	−0.0272974	−	0.999627i	$-0.508690\pi$
$719$	−39.2537	−0.0545948	−0.0272974	+	0.999627i	$0.508690\pi$
$720$	0	0
$721$	24.9371i	0.0345868i
$722$	503.875i	0.697887i
$723$	0	0
$724$	−519.143	−0.717048
$725$	443.374i	0.611550i
$726$	0	0
$727$	918.828	1.26386	0.631932	−	0.775024i	$-0.282262\pi$
$727$	918.828	1.26386	0.631932	+	0.775024i	$0.282262\pi$
$728$	0.467951i	0 0.000642790i
$729$	0	0
$730$	56.7347	0.0777188
$731$	−233.118	−0.318903
$732$	0	0
$733$	− 738.905i	− 1.00806i	−0.863687	−	0.504028i	$-0.831851\pi$
$733$	− 738.905i	− 1.00806i	0.863687	−	0.504028i	$-0.168149\pi$
$734$	126.668i	0.172573i
$735$	0	0
$736$	136.524i	0.185494i
$737$	0	0
$738$	0	0
$739$	− 1137.91i	− 1.53979i	−0.638168	−	0.769897i	$-0.720308\pi$
$739$	− 1137.91i	− 1.53979i	0.638168	−	0.769897i	$-0.279692\pi$
$740$	−42.4668	−0.0573876
$741$	0	0
$742$	−65.9420	−0.0888706
$743$	− 188.316i	− 0.253454i	−0.991938	−	0.126727i	$-0.959553\pi$
$743$	− 188.316i	− 0.253454i	0.991938	−	0.126727i	$-0.0404472\pi$
$744$	0	0
$745$	− 81.7178i	− 0.109688i
$746$	389.689	0.522371
$747$	0	0
$748$	0	0
$749$	−63.6848	−0.0850264
$750$	0	0
$751$	605.397	0.806122	0.403061	−	0.915173i	$-0.367946\pi$
$751$	605.397	0.806122	0.403061	+	0.915173i	$0.367946\pi$
$752$	244.622	0.325296
$753$	0	0
$754$	− 5.27400i	− 0.00699470i
$755$	94.1440i	0.124694i
$756$	0	0
$757$	−388.614	−0.513361	−0.256680	−	0.966496i	$-0.582629\pi$
$757$	−388.614	−0.513361	−0.256680	+	0.966496i	$0.582629\pi$
$758$	− 248.258i	− 0.327517i
$759$	0	0
$760$	2.16851	0.00285330
$761$	699.156i	0.918734i	0.888247	+	0.459367i	$0.151924\pi$
$761$	699.156i	0.918734i	−0.888247	+	0.459367i	$0.848076\pi$
$762$	0	0
$763$	−25.4470	−0.0333512
$764$	−249.564	−0.326655
$765$	0	0
$766$	72.3778i	0.0944880i
$767$	− 10.7723i	− 0.0140447i
$768$	0	0
$769$	337.074i	0.438328i	0.975688	+	0.219164i	$0.0703330\pi$
$769$	337.074i	0.438328i	−0.975688	+	0.219164i	$0.929667\pi$
$770$	0	0
$771$	0	0
$772$	− 83.9783i	− 0.108780i
$773$	795.193	1.02871	0.514355	−	0.857577i	$-0.328031\pi$
$773$	795.193	1.02871	0.514355	+	0.857577i	$0.328031\pi$
$774$	0	0
$775$	−841.713	−1.08608
$776$	− 287.102i	− 0.369977i
$777$	0	0
$778$	32.3946i	0.0416383i
$779$	−116.192	−0.149156
$780$	0	0
$781$	0	0
$782$	−510.926	−0.653358
$783$	0	0
$784$	193.499	0.246810
$785$	36.8505	0.0469433
$786$	0	0
$787$	− 535.000i	− 0.679796i	−0.940462	−	0.339898i	$-0.889607\pi$
$787$	− 535.000i	− 0.679796i	0.940462	−	0.339898i	$-0.110393\pi$
$788$	548.677i	0.696290i
$789$	0	0
$790$	73.5286	0.0930741
$791$	− 23.0928i	− 0.0291944i
$792$	0	0
$793$	8.52651	0.0107522
$794$	185.339i	0.233424i
$795$	0	0
$796$	301.959	0.379345
$797$	711.043	0.892149	0.446074	−	0.894996i	$-0.352822\pi$
$797$	711.043	0.892149	0.446074	+	0.894996i	$0.352822\pi$
$798$	0	0
$799$	915.474i	1.14578i
$800$	− 140.715i	− 0.175894i
$801$	0	0
$802$	− 1097.22i	− 1.36810i
$803$	0	0
$804$	0	0
$805$	6.74400i	0.00837764i
$806$	10.0123	0.0124222
$807$	0	0
$808$	73.7809	0.0913130
$809$	107.587i	0.132987i	0.997787	+	0.0664937i	$0.0211812\pi$
$809$	107.587i	0.132987i	−0.997787	+	0.0664937i	$0.978819\pi$
$810$	0	0
$811$	− 1256.75i	− 1.54963i	−0.632189	−	0.774814i	$-0.717844\pi$
$811$	− 1256.75i	− 1.54963i	0.632189	−	0.774814i	$-0.282156\pi$
$812$	28.1884	0.0347148
$813$	0	0
$814$	0	0
$815$	98.8308	0.121265
$816$	0	0
$817$	33.7856	0.0413532
$818$	−1056.95	−1.29212
$819$	0	0
$820$	− 37.8522i	− 0.0461612i
$821$	− 1034.51i	− 1.26006i	−0.776572	−	0.630028i	$-0.783043\pi$
$821$	− 1034.51i	− 1.26006i	0.776572	−	0.630028i	$-0.216957\pi$
$822$	0	0
$823$	−535.829	−0.651067	−0.325534	−	0.945530i	$-0.605544\pi$
$823$	−535.829	−0.651067	−0.325534	+	0.945530i	$0.605544\pi$
$824$	89.1979i	0.108250i
$825$	0	0
$826$	57.5755	0.0697039
$827$	137.866i	0.166706i	0.996520	+	0.0833532i	$0.0265630\pi$
$827$	137.866i	0.166706i	−0.996520	+	0.0833532i	$0.973437\pi$
$828$	0	0
$829$	−9.58047	−0.0115567	−0.00577833	−	0.999983i	$-0.501839\pi$
$829$	−9.58047	−0.0115567	−0.00577833	+	0.999983i	$0.501839\pi$
$830$	9.93292	0.0119674
$831$	0	0
$832$	1.67383i	0.00201181i
$833$	724.150i	0.869328i
$834$	0	0
$835$	− 90.8011i	− 0.108744i
$836$	0	0
$837$	0	0
$838$	577.602i	0.689263i
$839$	480.651	0.572886	0.286443	−	0.958097i	$-0.407527\pi$
$839$	480.651	0.572886	0.286443	+	0.958097i	$0.407527\pi$
$840$	0	0
$841$	523.305	0.622241
$842$	147.801i	0.175536i
$843$	0	0
$844$	− 37.3675i	− 0.0442743i
$845$	−59.7068	−0.0706589
$846$	0	0
$847$	0	0
$848$	−235.869	−0.278148
$849$	0	0
$850$	526.612	0.619543
$851$	−1450.12	−1.70401
$852$	0	0
$853$	− 1518.02i	− 1.77962i	−0.456328	−	0.889812i	$-0.650835\pi$
$853$	− 1518.02i	− 1.77962i	0.456328	−	0.889812i	$-0.349165\pi$
$854$	45.5724i	0.0533634i
$855$	0	0
$856$	−227.795	−0.266116
$857$	229.336i	0.267603i	0.991008	+	0.133802i	$0.0427185\pi$
$857$	229.336i	0.267603i	−0.991008	+	0.133802i	$0.957282\pi$
$858$	0	0
$859$	−308.792	−0.359478	−0.179739	−	0.983714i	$-0.557525\pi$
$859$	−308.792	−0.359478	−0.179739	+	0.983714i	$0.557525\pi$
$860$	11.0064i	0.0127981i
$861$	0	0
$862$	408.797	0.474242
$863$	−517.131	−0.599225	−0.299613	−	0.954061i	$-0.596857\pi$
$863$	−517.131	−0.599225	−0.299613	+	0.954061i	$0.596857\pi$
$864$	0	0
$865$	− 38.9391i	− 0.0450164i
$866$	779.231i	0.899805i
$867$	0	0
$868$	53.5137i	0.0616517i
$869$	0	0
$870$	0	0
$871$	15.2590i	0.0175190i
$872$	−91.0219	−0.104383
$873$	0	0
$874$	74.0481	0.0847232
$875$	− 13.9370i	− 0.0159280i
$876$	0	0
$877$	252.227i	0.287602i	0.989607	+	0.143801i	$0.0459325\pi$
$877$	252.227i	0.287602i	−0.989607	+	0.143801i	$0.954067\pi$
$878$	815.847	0.929210
$879$	0	0
$880$	0	0
$881$	−830.794	−0.943012	−0.471506	−	0.881863i	$-0.656289\pi$
$881$	−830.794	−0.943012	−0.471506	+	0.881863i	$0.656289\pi$
$882$	0	0
$883$	537.950	0.609229	0.304615	−	0.952476i	$-0.401472\pi$
$883$	537.950	0.609229	0.304615	+	0.952476i	$0.401472\pi$
$884$	−6.26412	−0.00708611
$885$	0	0
$886$	252.071i	0.284504i
$887$	1430.06i	1.61224i	0.591753	+	0.806120i	$0.298436\pi$
$887$	1430.06i	1.61224i	−0.591753	+	0.806120i	$0.701564\pi$
$888$	0	0
$889$	117.862	0.132578
$890$	− 37.4678i	− 0.0420986i
$891$	0	0
$892$	188.562	0.211392
$893$	− 132.679i	− 0.148577i
$894$	0	0
$895$	9.58620	0.0107108
$896$	−8.94624	−0.00998465
$897$	0	0
$898$	− 880.645i	− 0.980674i
$899$	− 603.121i	− 0.670879i
$900$	0	0
$901$	− 882.717i	− 0.979708i
$902$	0	0
$903$	0	0
$904$	− 82.6010i	− 0.0913728i
$905$	−91.7289	−0.101358
$906$	0	0
$907$	−1352.76	−1.49147	−0.745734	−	0.666244i	$-0.767901\pi$
$907$	−1352.76	−1.49147	−0.745734	+	0.666244i	$0.767901\pi$
$908$	− 579.582i	− 0.638306i
$909$	0	0
$910$	0.0826838i	0 9.08613e-5i
$911$	854.208	0.937660	0.468830	−	0.883288i	$-0.344676\pi$
$911$	854.208	0.937660	0.468830	+	0.883288i	$0.344676\pi$
$912$	0	0
$913$	0	0
$914$	1114.13	1.21896
$915$	0	0
$916$	456.457	0.498315
$917$	131.236	0.143114
$918$	0	0
$919$	− 330.219i	− 0.359324i	−0.983728	−	0.179662i	$-0.942500\pi$
$919$	− 330.219i	− 0.359324i	0.983728	−	0.179662i	$-0.0575004\pi$
$920$	24.1228i	0.0262204i
$921$	0	0
$922$	534.551	0.579774
$923$	− 19.8875i	− 0.0215466i
$924$	0	0
$925$	1494.64	1.61582
$926$	− 69.7432i	− 0.0753166i
$927$	0	0
$928$	100.828	0.108651
$929$	114.412	0.123156	0.0615778	−	0.998102i	$-0.480387\pi$
$929$	114.412	0.123156	0.0615778	+	0.998102i	$0.480387\pi$
$930$	0	0
$931$	− 104.951i	− 0.112729i
$932$	− 597.654i	− 0.641260i
$933$	0	0
$934$	165.728i	0.177439i
$935$	0	0
$936$	0	0
$937$	− 1619.74i	− 1.72864i	−0.502938	−	0.864322i	$-0.667748\pi$
$937$	− 1619.74i	− 1.72864i	0.502938	−	0.864322i	$-0.332252\pi$
$938$	−81.5564	−0.0869471
$939$	0	0
$940$	43.2231	0.0459820
$941$	− 759.245i	− 0.806849i	−0.915013	−	0.403424i	$-0.867820\pi$
$941$	− 759.245i	− 0.806849i	0.915013	−	0.403424i	$-0.132180\pi$
$942$	0	0
$943$	− 1292.54i	− 1.37067i
$944$	205.943	0.218160
$945$	0	0
$946$	0	0
$947$	−1855.32	−1.95915	−0.979575	−	0.201079i	$-0.935555\pi$
$947$	−1855.32	−1.95915	−0.979575	+	0.201079i	$0.935555\pi$
$948$	0	0
$949$	23.7522	0.0250287
$950$	−76.3214	−0.0803383
$951$	0	0
$952$	− 33.4804i	− 0.0351685i
$953$	− 1624.78i	− 1.70492i	−0.522796	−	0.852458i	$-0.675111\pi$
$953$	− 1624.78i	− 1.70492i	0.522796	−	0.852458i	$-0.324889\pi$
$954$	0	0
$955$	−44.0963	−0.0461742
$956$	− 491.859i	− 0.514496i
$957$	0	0
$958$	1302.11	1.35920
$959$	− 116.135i	− 0.121100i
$960$	0	0
$961$	183.980	0.191447
$962$	−17.7789	−0.0184812
$963$	0	0
$964$	− 746.836i	− 0.774726i
$965$	− 14.8384i	− 0.0153766i
$966$	0	0
$967$	− 325.104i	− 0.336199i	−0.985770	−	0.168099i	$-0.946237\pi$
$967$	− 325.104i	− 0.336199i	0.985770	−	0.168099i	$-0.0537629\pi$
$968$	0	0
$969$	0	0
$970$	− 50.7290i	− 0.0522979i
$971$	−280.767	−0.289152	−0.144576	−	0.989494i	$-0.546182\pi$
$971$	−280.767	−0.289152	−0.144576	+	0.989494i	$0.546182\pi$
$972$	0	0
$973$	169.041	0.173732
$974$	548.867i	0.563519i
$975$	0	0
$976$	163.009i	0.167017i
$977$	−1609.79	−1.64769	−0.823843	−	0.566818i	$-0.808174\pi$
$977$	−1609.79	−1.64769	−0.823843	+	0.566818i	$0.808174\pi$
$978$	0	0
$979$	0	0
$980$	34.1899	0.0348877
$981$	0	0
$982$	−516.151	−0.525612
$983$	374.382	0.380856	0.190428	−	0.981701i	$-0.439012\pi$
$983$	374.382	0.380856	0.190428	+	0.981701i	$0.439012\pi$
$984$	0	0
$985$	96.9474i	0.0984238i
$986$	377.338i	0.382696i
$987$	0	0
$988$	0.907854	0.000918881 0
$989$	375.836i	0.380016i
$990$	0	0
$991$	−943.875	−0.952447	−0.476223	−	0.879324i	$-0.657995\pi$
$991$	−943.875	−0.952447	−0.476223	+	0.879324i	$0.657995\pi$
$992$	191.414i	0.192958i
$993$	0	0
$994$	106.295	0.106936
$995$	53.3540	0.0536221
$996$	0	0
$997$	− 681.654i	− 0.683705i	−0.939754	−	0.341853i	$-0.888946\pi$
$997$	− 681.654i	− 0.683705i	0.939754	−	0.341853i	$-0.111054\pi$
$998$	− 346.659i	− 0.347353i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.3.d.m.1693.12		16
3.2	odd	2		726.3.d.e.241.2		16
11.4	even	5		198.3.j.b.127.2		16
11.8	odd	10		198.3.j.b.145.2		16
11.10	odd	2	inner	2178.3.d.m.1693.4		16
33.8	even	10		66.3.f.a.13.4	✓	16
33.26	odd	10		66.3.f.a.61.4	yes	16
33.32	even	2		726.3.d.e.241.10		16
132.59	even	10		528.3.bf.c.193.1		16
132.107	odd	10		528.3.bf.c.145.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.3.f.a.13.4	✓	16	33.8	even	10
66.3.f.a.61.4	yes	16	33.26	odd	10
198.3.j.b.127.2		16	11.4	even	5
198.3.j.b.145.2		16	11.8	odd	10
528.3.bf.c.145.1		16	132.107	odd	10
528.3.bf.c.193.1		16	132.59	even	10
726.3.d.e.241.2		16	3.2	odd	2
726.3.d.e.241.10		16	33.32	even	2
2178.3.d.m.1693.4		16	11.10	odd	2	inner
2178.3.d.m.1693.12		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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -0.353386 q^{5} -0.790744i q^{7} -2.82843i q^{8} -0.499764i q^{10} -0.209228i q^{13} +1.11828 q^{14} +4.00000 q^{16} +14.9696i q^{17} -2.16953i q^{19} +0.706772 q^{20} +24.1342 q^{23} -24.8751 q^{25} +0.295893 q^{26} +1.58149i q^{28} -17.8240i q^{29} +33.8376 q^{31} +5.65685i q^{32} -21.1702 q^{34} +0.279438i q^{35} -60.0856 q^{37} +3.06818 q^{38} +0.999527i q^{40} -53.5564i q^{41} +15.5727i q^{43} +34.1309i q^{46} +61.1556 q^{47} +48.3747 q^{49} -35.1787i q^{50} +0.418456i q^{52} -58.9673 q^{53} -2.23656 q^{56} +25.2069 q^{58} +51.4857 q^{59} +40.7522i q^{61} +47.8535i q^{62} -8.00000 q^{64} +0.0739383i q^{65} -72.9302 q^{67} -29.9392i q^{68} -0.395185 q^{70} +95.0520 q^{71} +113.523i q^{73} -84.9738i q^{74} +4.33906i q^{76} +147.127i q^{79} -1.41354 q^{80} +75.7401 q^{82} +19.8752i q^{83} -5.29005i q^{85} -22.0232 q^{86} +74.9710 q^{89} -0.165446 q^{91} -48.2684 q^{92} +86.4870i q^{94} +0.766683i q^{95} +101.506 q^{97} +68.4122i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 8 q^{5} - 48 q^{14} + 64 q^{16} + 16 q^{20} + 8 q^{23} - 8 q^{25} - 128 q^{26} - 8 q^{31} - 64 q^{34} + 96 q^{37} + 208 q^{38} + 128 q^{47} - 64 q^{49} - 16 q^{53} + 96 q^{56} + 64 q^{58}+ \cdots + 184 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 - 8 * q^5 - 48 * q^14 + 64 * q^16 + 16 * q^20 + 8 * q^23 - 8 * q^25 - 128 * q^26 - 8 * q^31 - 64 * q^34 + 96 * q^37 + 208 * q^38 + 128 * q^47 - 64 * q^49 - 16 * q^53 + 96 * q^56 + 64 * q^58 + 152 * q^59 - 128 * q^64 + 248 * q^67 - 32 * q^70 + 72 * q^71 - 32 * q^80 + 176 * q^82 - 464 * q^86 + 360 * q^89 - 32 * q^91 - 16 * q^92 + 184 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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