LMFDB - Embedded newform 2178.3.c.m.485.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,3,Mod(485,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2178.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$59.3462015777$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 102x^{6} + 3599x^{4} + 51708x^{2} + 249001 $ x^8 + 102x^6 + 3599x^4 + 51708*x^2 + 249001
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 198)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.7
Root			$-3.15782i$ of defining polynomial
Character	$\chi$	$=$	2178.485
Dual form			2178.3.c.m.485.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +2.82122i q^{5} +2.22977 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} +2.82122i q^{5} +2.22977 q^{7} -2.82843i q^{8} -3.98981 q^{10} +4.52397 q^{13} +3.15337i q^{14} +4.00000 q^{16} -14.7266i q^{17} -11.0878 q^{19} -5.64244i q^{20} +11.0813i q^{23} +17.0407 q^{25} +6.39786i q^{26} -4.45954 q^{28} +22.6350i q^{29} -23.0257 q^{31} +5.65685i q^{32} +20.8265 q^{34} +6.29068i q^{35} +20.0315 q^{37} -15.6805i q^{38} +7.97962 q^{40} +40.5345i q^{41} +45.6612 q^{43} -15.6713 q^{46} -11.5931i q^{47} -44.0281 q^{49} +24.0992i q^{50} -9.04794 q^{52} +39.1410i q^{53} -6.30674i q^{56} -32.0107 q^{58} +93.1486i q^{59} +74.1575 q^{61} -32.5633i q^{62} -8.00000 q^{64} +12.7631i q^{65} +87.7093 q^{67} +29.4531i q^{68} -8.89636 q^{70} -67.1772i q^{71} -35.6952 q^{73} +28.3288i q^{74} +22.1755 q^{76} -25.4014 q^{79} +11.2849i q^{80} -57.3244 q^{82} -21.9098i q^{83} +41.5469 q^{85} +64.5747i q^{86} +34.9908i q^{89} +10.0874 q^{91} -22.1626i q^{92} +16.3951 q^{94} -31.2810i q^{95} -109.118 q^{97} -62.2652i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 - 8 * q^7	$ 8 q - 16 q^{4} - 8 q^{7} + 12 q^{10} + 36 q^{13} + 32 q^{16} - 72 q^{19} - 64 q^{25} + 16 q^{28} - 88 q^{31} - 56 q^{34} + 108 q^{37} - 24 q^{40} + 220 q^{43} + 68 q^{46} + 56 q^{49} - 72 q^{52} + 112 q^{58} + 160 q^{61} - 64 q^{64} - 276 q^{67} - 92 q^{70} + 488 q^{73} + 144 q^{76} + 368 q^{79} - 388 q^{82} - 248 q^{85} - 356 q^{91} + 120 q^{94} - 832 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 - 8 * q^7 + 12 * q^10 + 36 * q^13 + 32 * q^16 - 72 * q^19 - 64 * q^25 + 16 * q^28 - 88 * q^31 - 56 * q^34 + 108 * q^37 - 24 * q^40 + 220 * q^43 + 68 * q^46 + 56 * q^49 - 72 * q^52 + 112 * q^58 + 160 * q^61 - 64 * q^64 - 276 * q^67 - 92 * q^70 + 488 * q^73 + 144 * q^76 + 368 * q^79 - 388 * q^82 - 248 * q^85 - 356 * q^91 + 120 * q^94 - 832 * q^97

Display 100 coefficients 10 coefficients

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
$5$	2.82122i	0.564244i	0.959378	+	0.282122i	$0.0910384\pi$
$5$	2.82122i	0.564244i	−0.959378	+	0.282122i	$0.908962\pi$
$6$	0	0
$7$	2.22977	0.318539	0.159269	−	0.987235i	$-0.449086\pi$
$7$	2.22977	0.318539	0.159269	+	0.987235i	$0.449086\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−3.98981	−0.398981
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.52397	0.347998	0.173999	−	0.984746i	$-0.444331\pi$
$13$	4.52397	0.347998	0.173999	+	0.984746i	$0.444331\pi$
$14$	3.15337i	0.225241i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 14.7266i	− 0.866269i	−0.901329	−	0.433134i	$-0.857408\pi$
$17$	− 14.7266i	− 0.866269i	0.901329	−	0.433134i	$-0.142592\pi$
$18$	0	0
$19$	−11.0878	−0.583567	−0.291783	−	0.956484i	$-0.594249\pi$
$19$	−11.0878	−0.583567	−0.291783	+	0.956484i	$0.594249\pi$
$20$	− 5.64244i	− 0.282122i
$21$	0	0
$22$	0	0
$23$	11.0813i	0.481796i	0.970550	+	0.240898i	$0.0774419\pi$
$23$	11.0813i	0.481796i	−0.970550	+	0.240898i	$0.922558\pi$
$24$	0	0
$25$	17.0407	0.681628
$26$	6.39786i	0.246072i
$27$	0	0
$28$	−4.45954	−0.159269
$29$	22.6350i	0.780517i	0.920705	+	0.390259i	$0.127614\pi$
$29$	22.6350i	0.780517i	−0.920705	+	0.390259i	$0.872386\pi$
$30$	0	0
$31$	−23.0257	−0.742766	−0.371383	−	0.928480i	$-0.621116\pi$
$31$	−23.0257	−0.742766	−0.371383	+	0.928480i	$0.621116\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	20.8265	0.612545
$35$	6.29068i	0.179734i
$36$	0	0
$37$	20.0315	0.541391	0.270695	−	0.962665i	$-0.412746\pi$
$37$	20.0315	0.541391	0.270695	+	0.962665i	$0.412746\pi$
$38$	− 15.6805i	− 0.412644i
$39$	0	0
$40$	7.97962	0.199491
$41$	40.5345i	0.988646i	0.869278	+	0.494323i	$0.164584\pi$
$41$	40.5345i	0.988646i	−0.869278	+	0.494323i	$0.835416\pi$
$42$	0	0
$43$	45.6612	1.06189	0.530944	−	0.847407i	$-0.321837\pi$
$43$	45.6612	1.06189	0.530944	+	0.847407i	$0.321837\pi$
$44$	0	0
$45$	0	0
$46$	−15.6713	−0.340681
$47$	− 11.5931i	− 0.246662i	−0.992366	−	0.123331i	$-0.960642\pi$
$47$	− 11.5931i	− 0.246662i	0.992366	−	0.123331i	$-0.0393577\pi$
$48$	0	0
$49$	−44.0281	−0.898533
$50$	24.0992i	0.481984i
$51$	0	0
$52$	−9.04794	−0.173999
$53$	39.1410i	0.738510i	0.929328	+	0.369255i	$0.120387\pi$
$53$	39.1410i	0.738510i	−0.929328	+	0.369255i	$0.879613\pi$
$54$	0	0
$55$	0	0
$56$	− 6.30674i	− 0.112620i
$57$	0	0
$58$	−32.0107	−0.551909
$59$	93.1486i	1.57879i	0.613885	+	0.789395i	$0.289606\pi$
$59$	93.1486i	1.57879i	−0.613885	+	0.789395i	$0.710394\pi$
$60$	0	0
$61$	74.1575	1.21570	0.607849	−	0.794053i	$-0.292033\pi$
$61$	74.1575	1.21570	0.607849	+	0.794053i	$0.292033\pi$
$62$	− 32.5633i	− 0.525215i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	12.7631i	0.196356i
$66$	0	0
$67$	87.7093	1.30909	0.654547	−	0.756021i	$-0.272859\pi$
$67$	87.7093	1.30909	0.654547	+	0.756021i	$0.272859\pi$
$68$	29.4531i	0.433134i
$69$	0	0
$70$	−8.89636	−0.127091
$71$	− 67.1772i	− 0.946158i	−0.881020	−	0.473079i	$-0.843143\pi$
$71$	− 67.1772i	− 0.946158i	0.881020	−	0.473079i	$-0.156857\pi$
$72$	0	0
$73$	−35.6952	−0.488976	−0.244488	−	0.969652i	$-0.578620\pi$
$73$	−35.6952	−0.488976	−0.244488	+	0.969652i	$0.578620\pi$
$74$	28.3288i	0.382821i
$75$	0	0
$76$	22.1755	0.291783
$77$	0	0
$78$	0	0
$79$	−25.4014	−0.321537	−0.160768	−	0.986992i	$-0.551397\pi$
$79$	−25.4014	−0.321537	−0.160768	+	0.986992i	$0.551397\pi$
$80$	11.2849i	0.141061i
$81$	0	0
$82$	−57.3244	−0.699078
$83$	− 21.9098i	− 0.263974i	−0.991251	−	0.131987i	$-0.957864\pi$
$83$	− 21.9098i	− 0.263974i	0.991251	−	0.131987i	$-0.0421357\pi$
$84$	0	0
$85$	41.5469	0.488787
$86$	64.5747i	0.750868i
$87$	0	0
$88$	0	0
$89$	34.9908i	0.393155i	0.980488	+	0.196577i	$0.0629827\pi$
$89$	34.9908i	0.393155i	−0.980488	+	0.196577i	$0.937017\pi$
$90$	0	0
$91$	10.0874	0.110851
$92$	− 22.1626i	− 0.240898i
$93$	0	0
$94$	16.3951	0.174416
$95$	− 31.2810i	− 0.329274i
$96$	0	0
$97$	−109.118	−1.12493	−0.562463	−	0.826823i	$-0.690146\pi$
$97$	−109.118	−1.12493	−0.562463	+	0.826823i	$0.690146\pi$
$98$	− 62.2652i	− 0.635359i
$99$	0	0
$100$	−34.0814	−0.340814
$101$	180.838i	1.79047i	0.445592	+	0.895236i	$0.352993\pi$
$101$	180.838i	1.79047i	−0.445592	+	0.895236i	$0.647007\pi$
$102$	0	0
$103$	−201.556	−1.95685	−0.978427	−	0.206594i	$-0.933762\pi$
$103$	−201.556	−1.95685	−0.978427	+	0.206594i	$0.933762\pi$
$104$	− 12.7957i	− 0.123036i
$105$	0	0
$106$	−55.3538	−0.522205
$107$	91.4378i	0.854559i	0.904120	+	0.427279i	$0.140528\pi$
$107$	91.4378i	0.854559i	−0.904120	+	0.427279i	$0.859472\pi$
$108$	0	0
$109$	207.818	1.90659	0.953295	−	0.302040i	$-0.0976678\pi$
$109$	207.818	1.90659	0.953295	+	0.302040i	$0.0976678\pi$
$110$	0	0
$111$	0	0
$112$	8.91908	0.0796347
$113$	− 63.4902i	− 0.561860i	−0.959728	−	0.280930i	$-0.909357\pi$
$113$	− 63.4902i	− 0.561860i	0.959728	−	0.280930i	$-0.0906429\pi$
$114$	0	0
$115$	−31.2628	−0.271851
$116$	− 45.2700i	− 0.390259i
$117$	0	0
$118$	−131.732	−1.11637
$119$	− 32.8369i	− 0.275940i
$120$	0	0
$121$	0	0
$122$	104.875i	0.859628i
$123$	0	0
$124$	46.0515	0.371383
$125$	118.606i	0.948849i
$126$	0	0
$127$	−65.9902	−0.519608	−0.259804	−	0.965661i	$-0.583658\pi$
$127$	−65.9902	−0.519608	−0.259804	+	0.965661i	$0.583658\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−18.0498	−0.138845
$131$	59.5648i	0.454693i	0.973814	+	0.227347i	$0.0730050\pi$
$131$	59.5648i	0.454693i	−0.973814	+	0.227347i	$0.926995\pi$
$132$	0	0
$133$	−24.7232	−0.185888
$134$	124.040i	0.925669i
$135$	0	0
$136$	−41.6530	−0.306272
$137$	190.202i	1.38834i	0.719811	+	0.694170i	$0.244228\pi$
$137$	190.202i	1.38834i	−0.719811	+	0.694170i	$0.755772\pi$
$138$	0	0
$139$	234.628	1.68797	0.843986	−	0.536365i	$-0.180203\pi$
$139$	234.628	1.68797	0.843986	+	0.536365i	$0.180203\pi$
$140$	− 12.5814i	− 0.0898668i
$141$	0	0
$142$	95.0029	0.669035
$143$	0	0
$144$	0	0
$145$	−63.8583	−0.440402
$146$	− 50.4807i	− 0.345758i
$147$	0	0
$148$	−40.0629	−0.270695
$149$	− 150.403i	− 1.00942i	−0.863290	−	0.504708i	$-0.831600\pi$
$149$	− 150.403i	− 1.00942i	0.863290	−	0.504708i	$-0.168400\pi$
$150$	0	0
$151$	50.8256	0.336593	0.168297	−	0.985736i	$-0.446173\pi$
$151$	50.8256	0.336593	0.168297	+	0.985736i	$0.446173\pi$
$152$	31.3609i	0.206322i
$153$	0	0
$154$	0	0
$155$	− 64.9607i	− 0.419101i
$156$	0	0
$157$	−103.192	−0.657277	−0.328638	−	0.944456i	$-0.606590\pi$
$157$	−103.192	−0.657277	−0.328638	+	0.944456i	$0.606590\pi$
$158$	− 35.9230i	− 0.227361i
$159$	0	0
$160$	−15.9592	−0.0997453
$161$	24.7088i	0.153471i
$162$	0	0
$163$	−269.471	−1.65319	−0.826597	−	0.562794i	$-0.809726\pi$
$163$	−269.471	−1.65319	−0.826597	+	0.562794i	$0.809726\pi$
$164$	− 81.0690i	− 0.494323i
$165$	0	0
$166$	30.9852	0.186658
$167$	211.841i	1.26851i	0.773124	+	0.634255i	$0.218693\pi$
$167$	211.841i	1.26851i	−0.773124	+	0.634255i	$0.781307\pi$
$168$	0	0
$169$	−148.534	−0.878898
$170$	58.7562i	0.345625i
$171$	0	0
$172$	−91.3224	−0.530944
$173$	120.944i	0.699099i	0.936918	+	0.349549i	$0.113665\pi$
$173$	120.944i	0.699099i	−0.936918	+	0.349549i	$0.886335\pi$
$174$	0	0
$175$	37.9969	0.217125
$176$	0	0
$177$	0	0
$178$	−49.4844	−0.278002
$179$	− 46.7700i	− 0.261285i	−0.991430	−	0.130642i	$-0.958296\pi$
$179$	− 46.7700i	− 0.261285i	0.991430	−	0.130642i	$-0.0417040\pi$
$180$	0	0
$181$	−282.456	−1.56053	−0.780264	−	0.625450i	$-0.784916\pi$
$181$	−282.456	−1.56053	−0.780264	+	0.625450i	$0.784916\pi$
$182$	14.2658i	0.0783833i
$183$	0	0
$184$	31.3427	0.170341
$185$	56.5132i	0.305477i
$186$	0	0
$187$	0	0
$188$	23.1862i	0.123331i
$189$	0	0
$190$	44.2381	0.232832
$191$	357.599i	1.87224i	0.351676	+	0.936122i	$0.385612\pi$
$191$	357.599i	1.87224i	−0.351676	+	0.936122i	$0.614388\pi$
$192$	0	0
$193$	37.1925	0.192707	0.0963537	−	0.995347i	$-0.469282\pi$
$193$	37.1925	0.192707	0.0963537	+	0.995347i	$0.469282\pi$
$194$	− 154.316i	− 0.795442i
$195$	0	0
$196$	88.0562	0.449267
$197$	− 129.220i	− 0.655937i	−0.944689	−	0.327969i	$-0.893636\pi$
$197$	− 129.220i	− 0.655937i	0.944689	−	0.327969i	$-0.106364\pi$
$198$	0	0
$199$	−93.6837	−0.470772	−0.235386	−	0.971902i	$-0.575635\pi$
$199$	−93.6837	−0.470772	−0.235386	+	0.971902i	$0.575635\pi$
$200$	− 48.1984i	− 0.240992i
$201$	0	0
$202$	−255.743	−1.26606
$203$	50.4708i	0.248625i
$204$	0	0
$205$	−114.357	−0.557838
$206$	− 285.043i	− 1.38370i
$207$	0	0
$208$	18.0959	0.0869995
$209$	0	0
$210$	0	0
$211$	−289.622	−1.37261	−0.686307	−	0.727312i	$-0.740769\pi$
$211$	−289.622	−1.37261	−0.686307	+	0.727312i	$0.740769\pi$
$212$	− 78.2821i	− 0.369255i
$213$	0	0
$214$	−129.313	−0.604264
$215$	128.820i	0.599164i
$216$	0	0
$217$	−51.3421	−0.236600
$218$	293.900i	1.34816i
$219$	0	0
$220$	0	0
$221$	− 66.6226i	− 0.301460i
$222$	0	0
$223$	143.972	0.645612	0.322806	−	0.946465i	$-0.395374\pi$
$223$	143.972	0.645612	0.322806	+	0.946465i	$0.395374\pi$
$224$	12.6135i	0.0563102i
$225$	0	0
$226$	89.7887	0.397295
$227$	41.5246i	0.182928i	0.995808	+	0.0914639i	$0.0291546\pi$
$227$	41.5246i	0.182928i	−0.995808	+	0.0914639i	$0.970845\pi$
$228$	0	0
$229$	−126.536	−0.552561	−0.276280	−	0.961077i	$-0.589102\pi$
$229$	−126.536	−0.552561	−0.276280	+	0.961077i	$0.589102\pi$
$230$	− 44.2123i	− 0.192227i
$231$	0	0
$232$	64.0214	0.275954
$233$	− 199.110i	− 0.854547i	−0.904122	−	0.427274i	$-0.859474\pi$
$233$	− 199.110i	− 0.854547i	0.904122	−	0.427274i	$-0.140526\pi$
$234$	0	0
$235$	32.7068	0.139178
$236$	− 186.297i	− 0.789395i
$237$	0	0
$238$	46.4384	0.195119
$239$	246.440i	1.03113i	0.856850	+	0.515566i	$0.172418\pi$
$239$	246.440i	1.03113i	−0.856850	+	0.515566i	$0.827582\pi$
$240$	0	0
$241$	439.621	1.82415	0.912076	−	0.410021i	$-0.134479\pi$
$241$	439.621	1.82415	0.912076	+	0.410021i	$0.134479\pi$
$242$	0	0
$243$	0	0
$244$	−148.315	−0.607849
$245$	− 124.213i	− 0.506992i
$246$	0	0
$247$	−50.1607	−0.203080
$248$	65.1266i	0.262607i
$249$	0	0
$250$	−167.734	−0.670938
$251$	161.114i	0.641887i	0.947098	+	0.320944i	$0.104000\pi$
$251$	161.114i	0.641887i	−0.947098	+	0.320944i	$0.896000\pi$
$252$	0	0
$253$	0	0
$254$	− 93.3242i	− 0.367418i
$255$	0	0
$256$	16.0000	0.0625000
$257$	108.341i	0.421562i	0.977533	+	0.210781i	$0.0676007\pi$
$257$	108.341i	0.421562i	−0.977533	+	0.210781i	$0.932399\pi$
$258$	0	0
$259$	44.6655	0.172454
$260$	− 25.5263i	− 0.0981779i
$261$	0	0
$262$	−84.2373	−0.321517
$263$	− 263.126i	− 1.00048i	−0.865887	−	0.500240i	$-0.833245\pi$
$263$	− 263.126i	− 1.00048i	0.865887	−	0.500240i	$-0.166755\pi$
$264$	0	0
$265$	−110.426	−0.416700
$266$	− 34.9638i	− 0.131443i
$267$	0	0
$268$	−175.419	−0.654547
$269$	317.693i	1.18101i	0.807033	+	0.590507i	$0.201072\pi$
$269$	317.693i	1.18101i	−0.807033	+	0.590507i	$0.798928\pi$
$270$	0	0
$271$	−43.4187	−0.160216	−0.0801082	−	0.996786i	$-0.525527\pi$
$271$	−43.4187	−0.160216	−0.0801082	+	0.996786i	$0.525527\pi$
$272$	− 58.9063i	− 0.216567i
$273$	0	0
$274$	−268.987	−0.981704
$275$	0	0
$276$	0	0
$277$	−352.762	−1.27351	−0.636754	−	0.771067i	$-0.719723\pi$
$277$	−352.762	−1.27351	−0.636754	+	0.771067i	$0.719723\pi$
$278$	331.814i	1.19358i
$279$	0	0
$280$	17.7927	0.0635454
$281$	179.361i	0.638294i	0.947705	+	0.319147i	$0.103396\pi$
$281$	179.361i	0.638294i	−0.947705	+	0.319147i	$0.896604\pi$
$282$	0	0
$283$	−207.445	−0.733021	−0.366510	−	0.930414i	$-0.619448\pi$
$283$	−207.445	−0.733021	−0.366510	+	0.930414i	$0.619448\pi$
$284$	134.354i	0.473079i
$285$	0	0
$286$	0	0
$287$	90.3826i	0.314922i
$288$	0	0
$289$	72.1281	0.249578
$290$	− 90.3093i	− 0.311412i
$291$	0	0
$292$	71.3905	0.244488
$293$	391.679i	1.33679i	0.743807	+	0.668395i	$0.233018\pi$
$293$	391.679i	1.33679i	−0.743807	+	0.668395i	$0.766982\pi$
$294$	0	0
$295$	−262.793	−0.890823
$296$	− 56.6575i	− 0.191411i
$297$	0	0
$298$	212.702	0.713764
$299$	50.1315i	0.167664i
$300$	0	0
$301$	101.814	0.338252
$302$	71.8782i	0.238007i
$303$	0	0
$304$	−44.3511	−0.145892
$305$	209.215i	0.685950i
$306$	0	0
$307$	−82.4090	−0.268433	−0.134217	−	0.990952i	$-0.542852\pi$
$307$	−82.4090	−0.268433	−0.134217	+	0.990952i	$0.542852\pi$
$308$	0	0
$309$	0	0
$310$	91.8683	0.296349
$311$	358.954i	1.15419i	0.816676	+	0.577097i	$0.195814\pi$
$311$	358.954i	1.15419i	−0.816676	+	0.577097i	$0.804186\pi$
$312$	0	0
$313$	422.642	1.35029	0.675147	−	0.737684i	$-0.264080\pi$
$313$	422.642	1.35029	0.675147	+	0.737684i	$0.264080\pi$
$314$	− 145.936i	− 0.464765i
$315$	0	0
$316$	50.8028	0.160768
$317$	− 374.953i	− 1.18282i	−0.806372	−	0.591409i	$-0.798572\pi$
$317$	− 374.953i	− 1.18282i	0.806372	−	0.591409i	$-0.201428\pi$
$318$	0	0
$319$	0	0
$320$	− 22.5698i	− 0.0705305i
$321$	0	0
$322$	−34.9435	−0.108520
$323$	163.285i	0.505526i
$324$	0	0
$325$	77.0917	0.237205
$326$	− 381.089i	− 1.16898i
$327$	0	0
$328$	114.649	0.349539
$329$	− 25.8500i	− 0.0785714i
$330$	0	0
$331$	−166.263	−0.502305	−0.251152	−	0.967948i	$-0.580810\pi$
$331$	−166.263	−0.502305	−0.251152	+	0.967948i	$0.580810\pi$
$332$	43.8197i	0.131987i
$333$	0	0
$334$	−299.589	−0.896972
$335$	247.447i	0.738649i
$336$	0	0
$337$	628.279	1.86433	0.932165	−	0.362034i	$-0.117918\pi$
$337$	628.279	1.86433	0.932165	+	0.362034i	$0.117918\pi$
$338$	− 210.058i	− 0.621474i
$339$	0	0
$340$	−83.0939	−0.244394
$341$	0	0
$342$	0	0
$343$	−207.431	−0.604756
$344$	− 129.149i	− 0.375434i
$345$	0	0
$346$	−171.041	−0.494337
$347$	− 569.371i	− 1.64084i	−0.571763	−	0.820419i	$-0.693740\pi$
$347$	− 569.371i	− 1.64084i	0.571763	−	0.820419i	$-0.306260\pi$
$348$	0	0
$349$	89.1980	0.255582	0.127791	−	0.991801i	$-0.459211\pi$
$349$	89.1980	0.255582	0.127791	+	0.991801i	$0.459211\pi$
$350$	53.7357i	0.153531i
$351$	0	0
$352$	0	0
$353$	− 490.203i	− 1.38868i	−0.719649	−	0.694338i	$-0.755697\pi$
$353$	− 490.203i	− 1.38868i	0.719649	−	0.694338i	$-0.244303\pi$
$354$	0	0
$355$	189.522	0.533864
$356$	− 69.9815i	− 0.196577i
$357$	0	0
$358$	66.1427	0.184756
$359$	479.086i	1.33450i	0.744833	+	0.667251i	$0.232529\pi$
$359$	479.086i	1.33450i	−0.744833	+	0.667251i	$0.767471\pi$
$360$	0	0
$361$	−238.061	−0.659450
$362$	− 399.453i	− 1.10346i
$363$	0	0
$364$	−20.1748	−0.0554254
$365$	− 100.704i	− 0.275902i
$366$	0	0
$367$	4.59096	0.0125094	0.00625471	−	0.999980i	$-0.498009\pi$
$367$	4.59096	0.0125094	0.00625471	+	0.999980i	$0.498009\pi$
$368$	44.3252i	0.120449i
$369$	0	0
$370$	−79.9217	−0.216005
$371$	87.2755i	0.235244i
$372$	0	0
$373$	127.560	0.341984	0.170992	−	0.985272i	$-0.445303\pi$
$373$	127.560	0.341984	0.170992	+	0.985272i	$0.445303\pi$
$374$	0	0
$375$	0	0
$376$	−32.7903	−0.0872082
$377$	102.400i	0.271618i
$378$	0	0
$379$	315.967	0.833686	0.416843	−	0.908979i	$-0.363137\pi$
$379$	315.967	0.833686	0.416843	+	0.908979i	$0.363137\pi$
$380$	62.5621i	0.164637i
$381$	0	0
$382$	−505.721	−1.32388
$383$	− 283.220i	− 0.739478i	−0.929136	−	0.369739i	$-0.879447\pi$
$383$	− 283.220i	− 0.739478i	0.929136	−	0.369739i	$-0.120553\pi$
$384$	0	0
$385$	0	0
$386$	52.5982i	0.136265i
$387$	0	0
$388$	218.236	0.562463
$389$	− 294.629i	− 0.757401i	−0.925519	−	0.378700i	$-0.876371\pi$
$389$	− 294.629i	− 0.757401i	0.925519	−	0.378700i	$-0.123629\pi$
$390$	0	0
$391$	163.190	0.417365
$392$	124.530i	0.317679i
$393$	0	0
$394$	182.744	0.463818
$395$	− 71.6630i	− 0.181425i
$396$	0	0
$397$	354.576	0.893138	0.446569	−	0.894749i	$-0.352646\pi$
$397$	354.576	0.893138	0.446569	+	0.894749i	$0.352646\pi$
$398$	− 132.489i	− 0.332886i
$399$	0	0
$400$	68.1628	0.170407
$401$	40.8024i	0.101752i	0.998705	+	0.0508758i	$0.0162013\pi$
$401$	40.8024i	0.101752i	−0.998705	+	0.0508758i	$0.983799\pi$
$402$	0	0
$403$	−104.168	−0.258481
$404$	− 361.675i	− 0.895236i
$405$	0	0
$406$	−71.3765	−0.175804
$407$	0	0
$408$	0	0
$409$	477.416	1.16728	0.583638	−	0.812014i	$-0.301629\pi$
$409$	477.416	1.16728	0.583638	+	0.812014i	$0.301629\pi$
$410$	− 161.725i	− 0.394451i
$411$	0	0
$412$	403.112	0.978427
$413$	207.700i	0.502906i
$414$	0	0
$415$	61.8125	0.148946
$416$	25.5914i	0.0615179i
$417$	0	0
$418$	0	0
$419$	− 75.9631i	− 0.181296i	−0.995883	−	0.0906481i	$-0.971106\pi$
$419$	− 75.9631i	− 0.181296i	0.995883	−	0.0906481i	$-0.0288938\pi$
$420$	0	0
$421$	−279.302	−0.663425	−0.331713	−	0.943381i	$-0.607626\pi$
$421$	−279.302	−0.663425	−0.331713	+	0.943381i	$0.607626\pi$
$422$	− 409.587i	− 0.970585i
$423$	0	0
$424$	110.708	0.261103
$425$	− 250.951i	− 0.590473i
$426$	0	0
$427$	165.354	0.387247
$428$	− 182.876i	− 0.427279i
$429$	0	0
$430$	−182.179	−0.423673
$431$	− 594.555i	− 1.37948i	−0.724058	−	0.689739i	$-0.757725\pi$
$431$	− 594.555i	− 1.37948i	0.724058	−	0.689739i	$-0.242275\pi$
$432$	0	0
$433$	−157.674	−0.364144	−0.182072	−	0.983285i	$-0.558280\pi$
$433$	−157.674	−0.364144	−0.182072	+	0.983285i	$0.558280\pi$
$434$	− 72.6087i	− 0.167301i
$435$	0	0
$436$	−415.637	−0.953295
$437$	− 122.867i	− 0.281160i
$438$	0	0
$439$	−265.463	−0.604699	−0.302349	−	0.953197i	$-0.597771\pi$
$439$	−265.463	−0.604699	−0.302349	+	0.953197i	$0.597771\pi$
$440$	0	0
$441$	0	0
$442$	94.2186	0.213164
$443$	− 59.7786i	− 0.134940i	−0.997721	−	0.0674702i	$-0.978507\pi$
$443$	− 59.7786i	− 0.134940i	0.997721	−	0.0674702i	$-0.0214928\pi$
$444$	0	0
$445$	−98.7167	−0.221835
$446$	203.607i	0.456517i
$447$	0	0
$448$	−17.8382	−0.0398173
$449$	278.016i	0.619189i	0.950869	+	0.309595i	$0.100193\pi$
$449$	278.016i	0.619189i	−0.950869	+	0.309595i	$0.899807\pi$
$450$	0	0
$451$	0	0
$452$	126.980i	0.280930i
$453$	0	0
$454$	−58.7247	−0.129350
$455$	28.4588i	0.0625469i
$456$	0	0
$457$	−329.376	−0.720736	−0.360368	−	0.932810i	$-0.617349\pi$
$457$	−329.376	−0.720736	−0.360368	+	0.932810i	$0.617349\pi$
$458$	− 178.949i	− 0.390719i
$459$	0	0
$460$	62.5257	0.135925
$461$	628.919i	1.36425i	0.731236	+	0.682124i	$0.238944\pi$
$461$	628.919i	1.36425i	−0.731236	+	0.682124i	$0.761056\pi$
$462$	0	0
$463$	−509.930	−1.10136	−0.550680	−	0.834716i	$-0.685632\pi$
$463$	−509.930	−1.10136	−0.550680	+	0.834716i	$0.685632\pi$
$464$	90.5400i	0.195129i
$465$	0	0
$466$	281.583	0.604256
$467$	− 221.291i	− 0.473856i	−0.971527	−	0.236928i	$-0.923859\pi$
$467$	− 221.291i	− 0.473856i	0.971527	−	0.236928i	$-0.0761406\pi$
$468$	0	0
$469$	195.572	0.416997
$470$	46.2543i	0.0984135i
$471$	0	0
$472$	263.464	0.558187
$473$	0	0
$474$	0	0
$475$	−188.943	−0.397775
$476$	65.6737i	0.137970i
$477$	0	0
$478$	−348.519	−0.729120
$479$	− 258.547i	− 0.539763i	−0.962894	−	0.269882i	$-0.913015\pi$
$479$	− 258.547i	− 0.539763i	0.962894	−	0.269882i	$-0.0869846\pi$
$480$	0	0
$481$	90.6217	0.188403
$482$	621.718i	1.28987i
$483$	0	0
$484$	0	0
$485$	− 307.845i	− 0.634733i
$486$	0	0
$487$	−446.632	−0.917108	−0.458554	−	0.888667i	$-0.651632\pi$
$487$	−446.632	−0.917108	−0.458554	+	0.888667i	$0.651632\pi$
$488$	− 209.749i	− 0.429814i
$489$	0	0
$490$	175.664	0.358498
$491$	− 444.393i	− 0.905078i	−0.891745	−	0.452539i	$-0.850518\pi$
$491$	− 444.393i	− 0.905078i	0.891745	−	0.452539i	$-0.149482\pi$
$492$	0	0
$493$	333.336	0.676138
$494$	− 70.9380i	− 0.143599i
$495$	0	0
$496$	−92.1030	−0.185691
$497$	− 149.790i	− 0.301388i
$498$	0	0
$499$	752.146	1.50731	0.753654	−	0.657272i	$-0.228290\pi$
$499$	752.146	1.50731	0.753654	+	0.657272i	$0.228290\pi$
$500$	− 237.212i	− 0.474425i
$501$	0	0
$502$	−227.849	−0.453883
$503$	333.651i	0.663321i	0.943399	+	0.331661i	$0.107609\pi$
$503$	333.651i	0.663321i	−0.943399	+	0.331661i	$0.892391\pi$
$504$	0	0
$505$	−510.183	−1.01026
$506$	0	0
$507$	0	0
$508$	131.980	0.259804
$509$	− 353.913i	− 0.695310i	−0.937622	−	0.347655i	$-0.886978\pi$
$509$	− 353.913i	− 0.695310i	0.937622	−	0.347655i	$-0.113022\pi$
$510$	0	0
$511$	−79.5922	−0.155758
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−153.218	−0.298089
$515$	− 568.634i	− 1.10414i
$516$	0	0
$517$	0	0
$518$	63.1666i	0.121943i
$519$	0	0
$520$	36.0996	0.0694223
$521$	783.964i	1.50473i	0.658747	+	0.752365i	$0.271087\pi$
$521$	783.964i	1.50473i	−0.658747	+	0.752365i	$0.728913\pi$
$522$	0	0
$523$	−417.872	−0.798990	−0.399495	−	0.916735i	$-0.630815\pi$
$523$	−417.872	−0.798990	−0.399495	+	0.916735i	$0.630815\pi$
$524$	− 119.130i	− 0.227347i
$525$	0	0
$526$	372.117	0.707446
$527$	339.090i	0.643435i
$528$	0	0
$529$	406.205	0.767873
$530$	− 156.165i	− 0.294652i
$531$	0	0
$532$	49.4463	0.0929442
$533$	183.377i	0.344047i
$534$	0	0
$535$	−257.966	−0.482180
$536$	− 248.079i	− 0.462835i
$537$	0	0
$538$	−449.285	−0.835103
$539$	0	0
$540$	0	0
$541$	38.1922	0.0705956	0.0352978	−	0.999377i	$-0.488762\pi$
$541$	38.1922	0.0705956	0.0352978	+	0.999377i	$0.488762\pi$
$542$	− 61.4033i	− 0.113290i
$543$	0	0
$544$	83.3061	0.153136
$545$	586.302i	1.07578i
$546$	0	0
$547$	916.000	1.67459	0.837294	−	0.546752i	$-0.184136\pi$
$547$	916.000	1.67459	0.837294	+	0.546752i	$0.184136\pi$
$548$	− 380.405i	− 0.694170i
$549$	0	0
$550$	0	0
$551$	− 250.972i	− 0.455484i
$552$	0	0
$553$	−56.6393	−0.102422
$554$	− 498.880i	− 0.900506i
$555$	0	0
$556$	−469.256	−0.843986
$557$	776.071i	1.39330i	0.717409	+	0.696652i	$0.245328\pi$
$557$	776.071i	1.39330i	−0.717409	+	0.696652i	$0.754672\pi$
$558$	0	0
$559$	206.570	0.369535
$560$	25.1627i	0.0449334i
$561$	0	0
$562$	−253.654	−0.451342
$563$	390.137i	0.692961i	0.938057	+	0.346481i	$0.112623\pi$
$563$	390.137i	0.692961i	−0.938057	+	0.346481i	$0.887377\pi$
$564$	0	0
$565$	179.120	0.317026
$566$	− 293.371i	− 0.518324i
$567$	0	0
$568$	−190.006	−0.334517
$569$	− 104.399i	− 0.183479i	−0.995783	−	0.0917393i	$-0.970757\pi$
$569$	− 104.399i	− 0.183479i	0.995783	−	0.0917393i	$-0.0292427\pi$
$570$	0	0
$571$	66.1095	0.115778	0.0578892	−	0.998323i	$-0.481563\pi$
$571$	66.1095	0.115778	0.0578892	+	0.998323i	$0.481563\pi$
$572$	0	0
$573$	0	0
$574$	−127.820	−0.222683
$575$	188.833i	0.328406i
$576$	0	0
$577$	961.516	1.66641	0.833203	−	0.552967i	$-0.186505\pi$
$577$	961.516	1.66641	0.833203	+	0.552967i	$0.186505\pi$
$578$	102.004i	0.176478i
$579$	0	0
$580$	127.717	0.220201
$581$	− 48.8539i	− 0.0840859i
$582$	0	0
$583$	0	0
$584$	100.961i	0.172879i
$585$	0	0
$586$	−553.918	−0.945253
$587$	− 660.093i	− 1.12452i	−0.826961	−	0.562260i	$-0.809932\pi$
$587$	− 660.093i	− 1.12452i	0.826961	−	0.562260i	$-0.190068\pi$
$588$	0	0
$589$	255.304	0.433453
$590$	− 371.645i	− 0.629907i
$591$	0	0
$592$	80.1258	0.135348
$593$	883.670i	1.49017i	0.666970	+	0.745084i	$0.267591\pi$
$593$	883.670i	1.49017i	−0.666970	+	0.745084i	$0.732409\pi$
$594$	0	0
$595$	92.6401	0.155698
$596$	300.806i	0.504708i
$597$	0	0
$598$	−70.8967	−0.118556
$599$	− 234.670i	− 0.391769i	−0.980627	−	0.195885i	$-0.937242\pi$
$599$	− 234.670i	− 0.391769i	0.980627	−	0.195885i	$-0.0627578\pi$
$600$	0	0
$601$	155.942	0.259470	0.129735	−	0.991549i	$-0.458587\pi$
$601$	155.942	0.259470	0.129735	+	0.991549i	$0.458587\pi$
$602$	143.987i	0.239180i
$603$	0	0
$604$	−101.651	−0.168297
$605$	0	0
$606$	0	0
$607$	1043.58	1.71924	0.859618	−	0.510937i	$-0.170702\pi$
$607$	1043.58	1.71924	0.859618	+	0.510937i	$0.170702\pi$
$608$	− 62.7219i	− 0.103161i
$609$	0	0
$610$	−295.874	−0.485040
$611$	− 52.4469i	− 0.0858379i
$612$	0	0
$613$	824.714	1.34537	0.672686	−	0.739928i	$-0.265140\pi$
$613$	824.714	1.34537	0.672686	+	0.739928i	$0.265140\pi$
$614$	− 116.544i	− 0.189811i
$615$	0	0
$616$	0	0
$617$	513.392i	0.832078i	0.909347	+	0.416039i	$0.136582\pi$
$617$	513.392i	0.832078i	−0.909347	+	0.416039i	$0.863418\pi$
$618$	0	0
$619$	812.562	1.31270	0.656350	−	0.754456i	$-0.272099\pi$
$619$	812.562	1.31270	0.656350	+	0.754456i	$0.272099\pi$
$620$	129.921i	0.209551i
$621$	0	0
$622$	−507.638	−0.816138
$623$	78.0214i	0.125235i
$624$	0	0
$625$	91.4034	0.146245
$626$	597.706i	0.954802i
$627$	0	0
$628$	206.385	0.328638
$629$	− 294.995i	− 0.468990i
$630$	0	0
$631$	597.643	0.947136	0.473568	−	0.880757i	$-0.342966\pi$
$631$	597.643	0.947136	0.473568	+	0.880757i	$0.342966\pi$
$632$	71.8460i	0.113680i
$633$	0	0
$634$	530.264	0.836378
$635$	− 186.173i	− 0.293186i
$636$	0	0
$637$	−199.182	−0.312688
$638$	0	0
$639$	0	0
$640$	31.9185	0.0498726
$641$	− 815.883i	− 1.27283i	−0.771347	−	0.636414i	$-0.780417\pi$
$641$	− 815.883i	− 1.27283i	0.771347	−	0.636414i	$-0.219583\pi$
$642$	0	0
$643$	−463.073	−0.720176	−0.360088	−	0.932918i	$-0.617253\pi$
$643$	−463.073	−0.720176	−0.360088	+	0.932918i	$0.617253\pi$
$644$	− 49.4175i	− 0.0767353i
$645$	0	0
$646$	−230.920	−0.357461
$647$	− 993.955i	− 1.53625i	−0.640298	−	0.768126i	$-0.721189\pi$
$647$	− 993.955i	− 1.53625i	0.640298	−	0.768126i	$-0.278811\pi$
$648$	0	0
$649$	0	0
$650$	109.024i	0.167729i
$651$	0	0
$652$	538.941	0.826597
$653$	− 610.848i	− 0.935449i	−0.883874	−	0.467725i	$-0.845074\pi$
$653$	− 610.848i	− 0.935449i	0.883874	−	0.467725i	$-0.154926\pi$
$654$	0	0
$655$	−168.045	−0.256558
$656$	162.138i	0.247161i
$657$	0	0
$658$	36.5574	0.0555584
$659$	− 249.471i	− 0.378560i	−0.981923	−	0.189280i	$-0.939385\pi$
$659$	− 249.471i	− 0.378560i	0.981923	−	0.189280i	$-0.0606154\pi$
$660$	0	0
$661$	−1210.43	−1.83121	−0.915604	−	0.402081i	$-0.868287\pi$
$661$	−1210.43	−1.83121	−0.915604	+	0.402081i	$0.868287\pi$
$662$	− 235.131i	− 0.355183i
$663$	0	0
$664$	−61.9704	−0.0933289
$665$	− 69.7495i	− 0.104887i
$666$	0	0
$667$	−250.825	−0.376050
$668$	− 423.682i	− 0.634255i
$669$	0	0
$670$	−349.943	−0.522304
$671$	0	0
$672$	0	0
$673$	1121.51	1.66643	0.833214	−	0.552950i	$-0.186498\pi$
$673$	1121.51	1.66643	0.833214	+	0.552950i	$0.186498\pi$
$674$	888.521i	1.31828i
$675$	0	0
$676$	297.067	0.439449
$677$	− 238.834i	− 0.352782i	−0.984320	−	0.176391i	$-0.943558\pi$
$677$	− 238.834i	− 0.352782i	0.984320	−	0.176391i	$-0.0564424\pi$
$678$	0	0
$679$	−243.308	−0.358332
$680$	− 117.512i	− 0.172812i
$681$	0	0
$682$	0	0
$683$	1328.40i	1.94494i	0.233023	+	0.972471i	$0.425138\pi$
$683$	1328.40i	1.94494i	−0.233023	+	0.972471i	$0.574862\pi$
$684$	0	0
$685$	−536.603	−0.783363
$686$	− 293.352i	− 0.427627i
$687$	0	0
$688$	182.645	0.265472
$689$	177.073i	0.257000i
$690$	0	0
$691$	−40.9558	−0.0592703	−0.0296352	−	0.999561i	$-0.509435\pi$
$691$	−40.9558	−0.0592703	−0.0296352	+	0.999561i	$0.509435\pi$
$692$	− 241.888i	− 0.349549i
$693$	0	0
$694$	805.212	1.16025
$695$	661.938i	0.952429i
$696$	0	0
$697$	596.934	0.856433
$698$	126.145i	0.180724i
$699$	0	0
$700$	−75.9937	−0.108562
$701$	− 232.085i	− 0.331078i	−0.986203	−	0.165539i	$-0.947064\pi$
$701$	− 232.085i	− 0.331078i	0.986203	−	0.165539i	$-0.0529363\pi$
$702$	0	0
$703$	−222.104	−0.315938
$704$	0	0
$705$	0	0
$706$	693.251	0.981942
$707$	403.227i	0.570335i
$708$	0	0
$709$	215.214	0.303546	0.151773	−	0.988415i	$-0.451502\pi$
$709$	215.214	0.303546	0.151773	+	0.988415i	$0.451502\pi$
$710$	268.024i	0.377499i
$711$	0	0
$712$	98.9688	0.139001
$713$	− 255.155i	− 0.357862i
$714$	0	0
$715$	0	0
$716$	93.5399i	0.130642i
$717$	0	0
$718$	−677.530	−0.943636
$719$	152.742i	0.212437i	0.994343	+	0.106218i	$0.0338743\pi$
$719$	152.742i	0.212437i	−0.994343	+	0.106218i	$0.966126\pi$
$720$	0	0
$721$	−449.423	−0.623333
$722$	− 336.670i	− 0.466302i
$723$	0	0
$724$	564.911	0.780264
$725$	385.716i	0.532023i
$726$	0	0
$727$	610.733	0.840073	0.420037	−	0.907507i	$-0.362017\pi$
$727$	610.733	0.840073	0.420037	+	0.907507i	$0.362017\pi$
$728$	− 28.5315i	− 0.0391917i
$729$	0	0
$730$	142.417	0.195092
$731$	− 672.433i	− 0.919881i
$732$	0	0
$733$	222.773	0.303920	0.151960	−	0.988387i	$-0.451442\pi$
$733$	222.773	0.303920	0.151960	+	0.988387i	$0.451442\pi$
$734$	6.49259i	0.00884550i
$735$	0	0
$736$	−62.6853	−0.0851703
$737$	0	0
$738$	0	0
$739$	320.857	0.434178	0.217089	−	0.976152i	$-0.430344\pi$
$739$	320.857	0.434178	0.217089	+	0.976152i	$0.430344\pi$
$740$	− 113.026i	− 0.152738i
$741$	0	0
$742$	−123.426	−0.166343
$743$	− 276.852i	− 0.372614i	−0.982492	−	0.186307i	$-0.940348\pi$
$743$	− 276.852i	− 0.372614i	0.982492	−	0.186307i	$-0.0596518\pi$
$744$	0	0
$745$	424.320	0.569557
$746$	180.397i	0.241819i
$747$	0	0
$748$	0	0
$749$	203.885i	0.272210i
$750$	0	0
$751$	535.544	0.713107	0.356554	−	0.934275i	$-0.383952\pi$
$751$	535.544	0.713107	0.356554	+	0.934275i	$0.383952\pi$
$752$	− 46.3725i	− 0.0616655i
$753$	0	0
$754$	−144.816	−0.192063
$755$	143.390i	0.189921i
$756$	0	0
$757$	678.968	0.896919	0.448460	−	0.893803i	$-0.351973\pi$
$757$	678.968	0.896919	0.448460	+	0.893803i	$0.351973\pi$
$758$	446.845i	0.589505i
$759$	0	0
$760$	−88.4762	−0.116416
$761$	− 562.037i	− 0.738551i	−0.929320	−	0.369276i	$-0.879606\pi$
$761$	− 562.037i	− 0.738551i	0.929320	−	0.369276i	$-0.120394\pi$
$762$	0	0
$763$	463.387	0.607323
$764$	− 715.197i	− 0.936122i
$765$	0	0
$766$	400.534	0.522890
$767$	421.402i	0.549416i
$768$	0	0
$769$	113.242	0.147258	0.0736291	−	0.997286i	$-0.476542\pi$
$769$	113.242	0.147258	0.0736291	+	0.997286i	$0.476542\pi$
$770$	0	0
$771$	0	0
$772$	−74.3851	−0.0963537
$773$	1070.60i	1.38499i	0.721422	+	0.692495i	$0.243489\pi$
$773$	1070.60i	1.38499i	−0.721422	+	0.692495i	$0.756511\pi$
$774$	0	0
$775$	−392.375	−0.506290
$776$	308.632i	0.397721i
$777$	0	0
$778$	416.668	0.535563
$779$	− 449.437i	− 0.576941i
$780$	0	0
$781$	0	0
$782$	230.785i	0.295122i
$783$	0	0
$784$	−176.112	−0.224633
$785$	− 291.129i	− 0.370865i
$786$	0	0
$787$	−208.201	−0.264550	−0.132275	−	0.991213i	$-0.542228\pi$
$787$	−208.201	−0.264550	−0.132275	+	0.991213i	$0.542228\pi$
$788$	258.439i	0.327969i
$789$	0	0
$790$	101.347	0.128287
$791$	− 141.569i	− 0.178974i
$792$	0	0
$793$	335.487	0.423060
$794$	501.446i	0.631544i
$795$	0	0
$796$	187.367	0.235386
$797$	− 263.612i	− 0.330755i	−0.986230	−	0.165377i	$-0.947116\pi$
$797$	− 263.612i	− 0.330755i	0.986230	−	0.165377i	$-0.0528842\pi$
$798$	0	0
$799$	−170.727	−0.213676
$800$	96.3968i	0.120496i
$801$	0	0
$802$	−57.7033	−0.0719492
$803$	0	0
$804$	0	0
$805$	−69.7089	−0.0865949
$806$	− 147.316i	− 0.182774i
$807$	0	0
$808$	511.486	0.633028
$809$	967.704i	1.19617i	0.801431	+	0.598087i	$0.204072\pi$
$809$	967.704i	1.19617i	−0.801431	+	0.598087i	$0.795928\pi$
$810$	0	0
$811$	438.776	0.541030	0.270515	−	0.962716i	$-0.412806\pi$
$811$	438.776	0.541030	0.270515	+	0.962716i	$0.412806\pi$
$812$	− 100.942i	− 0.124312i
$813$	0	0
$814$	0	0
$815$	− 760.236i	− 0.932805i
$816$	0	0
$817$	−506.280	−0.619682
$818$	675.169i	0.825389i
$819$	0	0
$820$	228.714	0.278919
$821$	− 275.623i	− 0.335716i	−0.985811	−	0.167858i	$-0.946315\pi$
$821$	− 275.623i	− 0.335716i	0.985811	−	0.167858i	$-0.0536850\pi$
$822$	0	0
$823$	1210.40	1.47072	0.735360	−	0.677677i	$-0.237013\pi$
$823$	1210.40	1.47072	0.735360	+	0.677677i	$0.237013\pi$
$824$	570.086i	0.691852i
$825$	0	0
$826$	−293.732	−0.355608
$827$	− 1090.50i	− 1.31862i	−0.751871	−	0.659311i	$-0.770848\pi$
$827$	− 1090.50i	− 1.31862i	0.751871	−	0.659311i	$-0.229152\pi$
$828$	0	0
$829$	1284.18	1.54907	0.774536	−	0.632530i	$-0.217983\pi$
$829$	1284.18	1.54907	0.774536	+	0.632530i	$0.217983\pi$
$830$	87.4161i	0.105321i
$831$	0	0
$832$	−36.1918	−0.0434997
$833$	648.383i	0.778371i
$834$	0	0
$835$	−597.651	−0.715750
$836$	0	0
$837$	0	0
$838$	107.428	0.128196
$839$	− 529.962i	− 0.631659i	−0.948816	−	0.315829i	$-0.897717\pi$
$839$	− 529.962i	− 0.631659i	0.948816	−	0.315829i	$-0.102283\pi$
$840$	0	0
$841$	328.657	0.390793
$842$	− 394.993i	− 0.469112i
$843$	0	0
$844$	579.243	0.686307
$845$	− 419.046i	− 0.495913i
$846$	0	0
$847$	0	0
$848$	156.564i	0.184628i
$849$	0	0
$850$	354.899	0.417528
$851$	221.975i	0.260840i
$852$	0	0
$853$	1222.42	1.43308	0.716542	−	0.697544i	$-0.245724\pi$
$853$	1222.42	1.43308	0.716542	+	0.697544i	$0.245724\pi$
$854$	233.846i	0.273825i
$855$	0	0
$856$	258.625	0.302132
$857$	1479.73i	1.72664i	0.504653	+	0.863322i	$0.331620\pi$
$857$	1479.73i	1.72664i	−0.504653	+	0.863322i	$0.668380\pi$
$858$	0	0
$859$	95.9499	0.111700	0.0558498	−	0.998439i	$-0.482213\pi$
$859$	95.9499	0.111700	0.0558498	+	0.998439i	$0.482213\pi$
$860$	− 257.641i	− 0.299582i
$861$	0	0
$862$	840.828	0.975439
$863$	− 1277.20i	− 1.47996i	−0.672630	−	0.739979i	$-0.734835\pi$
$863$	− 1277.20i	− 1.47996i	0.672630	−	0.739979i	$-0.265165\pi$
$864$	0	0
$865$	−341.210	−0.394462
$866$	− 222.985i	− 0.257488i
$867$	0	0
$868$	102.684	0.118300
$869$	0	0
$870$	0	0
$871$	396.794	0.455562
$872$	− 587.799i	− 0.674081i
$873$	0	0
$874$	173.760	0.198810
$875$	264.464i	0.302245i
$876$	0	0
$877$	−327.910	−0.373900	−0.186950	−	0.982369i	$-0.559860\pi$
$877$	−327.910	−0.373900	−0.186950	+	0.982369i	$0.559860\pi$
$878$	− 375.421i	− 0.427586i
$879$	0	0
$880$	0	0
$881$	768.903i	0.872761i	0.899762	+	0.436381i	$0.143740\pi$
$881$	768.903i	0.872761i	−0.899762	+	0.436381i	$0.856260\pi$
$882$	0	0
$883$	−698.075	−0.790572	−0.395286	−	0.918558i	$-0.629355\pi$
$883$	−698.075	−0.790572	−0.395286	+	0.918558i	$0.629355\pi$
$884$	133.245i	0.150730i
$885$	0	0
$886$	84.5397	0.0954173
$887$	− 836.619i	− 0.943200i	−0.881812	−	0.471600i	$-0.843677\pi$
$887$	− 836.619i	− 0.943200i	0.881812	−	0.471600i	$-0.156323\pi$
$888$	0	0
$889$	−147.143	−0.165515
$890$	− 139.607i	− 0.156861i
$891$	0	0
$892$	−287.943	−0.322806
$893$	128.542i	0.143944i
$894$	0	0
$895$	131.948	0.147428
$896$	− 25.2270i	− 0.0281551i
$897$	0	0
$898$	−393.174	−0.437833
$899$	− 521.187i	− 0.579741i
$900$	0	0
$901$	576.413	0.639748
$902$	0	0
$903$	0	0
$904$	−179.577	−0.198648
$905$	− 796.870i	− 0.880520i
$906$	0	0
$907$	−109.786	−0.121043	−0.0605215	−	0.998167i	$-0.519276\pi$
$907$	−109.786	−0.121043	−0.0605215	+	0.998167i	$0.519276\pi$
$908$	− 83.0493i	− 0.0914639i
$909$	0	0
$910$	−40.2469	−0.0442273
$911$	1011.19i	1.10998i	0.831858	+	0.554988i	$0.187277\pi$
$911$	1011.19i	1.10998i	−0.831858	+	0.554988i	$0.812723\pi$
$912$	0	0
$913$	0	0
$914$	− 465.808i	− 0.509637i
$915$	0	0
$916$	253.073	0.276280
$917$	132.816i	0.144837i
$918$	0	0
$919$	−744.228	−0.809823	−0.404912	−	0.914356i	$-0.632698\pi$
$919$	−744.228	−0.809823	−0.404912	+	0.914356i	$0.632698\pi$
$920$	88.4246i	0.0961137i
$921$	0	0
$922$	−889.425	−0.964670
$923$	− 303.908i	− 0.329261i
$924$	0	0
$925$	341.350	0.369027
$926$	− 721.150i	− 0.778779i
$927$	0	0
$928$	−128.043	−0.137977
$929$	− 181.064i	− 0.194902i	−0.995240	−	0.0974511i	$-0.968931\pi$
$929$	− 181.064i	− 0.194902i	0.995240	−	0.0974511i	$-0.0310690\pi$
$930$	0	0
$931$	488.173	0.524354
$932$	398.219i	0.427274i
$933$	0	0
$934$	312.952	0.335067
$935$	0	0
$936$	0	0
$937$	−808.409	−0.862764	−0.431382	−	0.902169i	$-0.641974\pi$
$937$	−808.409	−0.862764	−0.431382	+	0.902169i	$0.641974\pi$
$938$	276.580i	0.294861i
$939$	0	0
$940$	−65.4135	−0.0695888
$941$	− 131.022i	− 0.139237i	−0.997574	−	0.0696185i	$-0.977822\pi$
$941$	− 131.022i	− 0.139237i	0.997574	−	0.0696185i	$-0.0221782\pi$
$942$	0	0
$943$	−449.175	−0.476326
$944$	372.594i	0.394698i
$945$	0	0
$946$	0	0
$947$	− 816.092i	− 0.861765i	−0.902408	−	0.430883i	$-0.858202\pi$
$947$	− 816.092i	− 0.861765i	0.902408	−	0.430883i	$-0.141798\pi$
$948$	0	0
$949$	−161.484	−0.170163
$950$	− 267.206i	− 0.281270i
$951$	0	0
$952$	−92.8767	−0.0975596
$953$	1000.66i	1.05001i	0.851098	+	0.525007i	$0.175937\pi$
$953$	1000.66i	1.05001i	−0.851098	+	0.525007i	$0.824063\pi$
$954$	0	0
$955$	−1008.86	−1.05640
$956$	− 492.881i	− 0.515566i
$957$	0	0
$958$	365.640	0.381670
$959$	424.108i	0.442240i
$960$	0	0
$961$	−430.815	−0.448299
$962$	128.159i	0.133221i
$963$	0	0
$964$	−879.241	−0.912076
$965$	104.928i	0.108734i
$966$	0	0
$967$	−1675.29	−1.73247	−0.866233	−	0.499640i	$-0.833466\pi$
$967$	−1675.29	−1.73247	−0.866233	+	0.499640i	$0.833466\pi$
$968$	0	0
$969$	0	0
$970$	435.359	0.448824
$971$	1409.87i	1.45197i	0.687709	+	0.725987i	$0.258617\pi$
$971$	1409.87i	1.45197i	−0.687709	+	0.725987i	$0.741383\pi$
$972$	0	0
$973$	523.167	0.537684
$974$	− 631.633i	− 0.648493i
$975$	0	0
$976$	296.630	0.303924
$977$	653.115i	0.668490i	0.942486	+	0.334245i	$0.108481\pi$
$977$	653.115i	0.668490i	−0.942486	+	0.334245i	$0.891519\pi$
$978$	0	0
$979$	0	0
$980$	248.426i	0.253496i
$981$	0	0
$982$	628.467	0.639987
$983$	− 641.147i	− 0.652235i	−0.945329	−	0.326118i	$-0.894259\pi$
$983$	− 641.147i	− 0.652235i	0.945329	−	0.326118i	$-0.105741\pi$
$984$	0	0
$985$	364.557	0.370109
$986$	471.408i	0.478102i
$987$	0	0
$988$	100.321	0.101540
$989$	505.986i	0.511613i
$990$	0	0
$991$	435.813	0.439771	0.219886	−	0.975526i	$-0.429432\pi$
$991$	435.813	0.439771	0.219886	+	0.975526i	$0.429432\pi$
$992$	− 130.253i	− 0.131304i
$993$	0	0
$994$	211.835	0.213113
$995$	− 264.302i	− 0.265631i
$996$	0	0
$997$	−422.219	−0.423490	−0.211745	−	0.977325i	$-0.567915\pi$
$997$	−422.219	−0.423490	−0.211745	+	0.977325i	$0.567915\pi$
$998$	1063.70i	1.06583i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.3.c.m.485.7		8
3.2	odd	2	inner	2178.3.c.m.485.2		8
11.7	odd	10		198.3.k.a.71.4	yes	16
11.8	odd	10		198.3.k.a.53.1	✓	16
11.10	odd	2		2178.3.c.p.485.3		8
33.8	even	10		198.3.k.a.53.4	yes	16
33.29	even	10		198.3.k.a.71.1	yes	16
33.32	even	2		2178.3.c.p.485.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.3.k.a.53.1	✓	16	11.8	odd	10
198.3.k.a.53.4	yes	16	33.8	even	10
198.3.k.a.71.1	yes	16	33.29	even	10
198.3.k.a.71.4	yes	16	11.7	odd	10
2178.3.c.m.485.2		8	3.2	odd	2	inner
2178.3.c.m.485.7		8	1.1	even	1	trivial
2178.3.c.p.485.3		8	11.10	odd	2
2178.3.c.p.485.6		8	33.32	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.82122i q^{5} +2.22977 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +2.82122i q^{5} +2.22977 q^{7} -2.82843i q^{8} -3.98981 q^{10} +4.52397 q^{13} +3.15337i q^{14} +4.00000 q^{16} -14.7266i q^{17} -11.0878 q^{19} -5.64244i q^{20} +11.0813i q^{23} +17.0407 q^{25} +6.39786i q^{26} -4.45954 q^{28} +22.6350i q^{29} -23.0257 q^{31} +5.65685i q^{32} +20.8265 q^{34} +6.29068i q^{35} +20.0315 q^{37} -15.6805i q^{38} +7.97962 q^{40} +40.5345i q^{41} +45.6612 q^{43} -15.6713 q^{46} -11.5931i q^{47} -44.0281 q^{49} +24.0992i q^{50} -9.04794 q^{52} +39.1410i q^{53} -6.30674i q^{56} -32.0107 q^{58} +93.1486i q^{59} +74.1575 q^{61} -32.5633i q^{62} -8.00000 q^{64} +12.7631i q^{65} +87.7093 q^{67} +29.4531i q^{68} -8.89636 q^{70} -67.1772i q^{71} -35.6952 q^{73} +28.3288i q^{74} +22.1755 q^{76} -25.4014 q^{79} +11.2849i q^{80} -57.3244 q^{82} -21.9098i q^{83} +41.5469 q^{85} +64.5747i q^{86} +34.9908i q^{89} +10.0874 q^{91} -22.1626i q^{92} +16.3951 q^{94} -31.2810i q^{95} -109.118 q^{97} -62.2652i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 - 8 * q^7	\( 8 q - 16 q^{4} - 8 q^{7} + 12 q^{10} + 36 q^{13} + 32 q^{16} - 72 q^{19} - 64 q^{25} + 16 q^{28} - 88 q^{31} - 56 q^{34} + 108 q^{37} - 24 q^{40} + 220 q^{43} + 68 q^{46} + 56 q^{49} - 72 q^{52} + 112 q^{58} + 160 q^{61} - 64 q^{64} - 276 q^{67} - 92 q^{70} + 488 q^{73} + 144 q^{76} + 368 q^{79} - 388 q^{82} - 248 q^{85} - 356 q^{91} + 120 q^{94} - 832 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 - 8 * q^7 + 12 * q^10 + 36 * q^13 + 32 * q^16 - 72 * q^19 - 64 * q^25 + 16 * q^28 - 88 * q^31 - 56 * q^34 + 108 * q^37 - 24 * q^40 + 220 * q^43 + 68 * q^46 + 56 * q^49 - 72 * q^52 + 112 * q^58 + 160 * q^61 - 64 * q^64 - 276 * q^67 - 92 * q^70 + 488 * q^73 + 144 * q^76 + 368 * q^79 - 388 * q^82 - 248 * q^85 - 356 * q^91 + 120 * q^94 - 832 * q^97

Introduction

L-functions

Modular forms

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