LMFDB - Embedded newform 672.4.i.c.209.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,4,Mod(209,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("672.209");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.i (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$80$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.2
Character	$\chi$	$=$	672.209
Dual form			672.4.i.c.209.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-5.19433 - 0.137767i) q^{3} -6.83655i q^{5} +(-13.6972 - 12.4654i) q^{7} +(26.9620 + 1.43121i) q^{9} +4.03973 q^{11} +11.6537 q^{13} +(-0.941848 + 35.5113i) q^{15} +104.163 q^{17} +29.0310 q^{19} +(69.4304 + 66.6365i) q^{21} +104.560i q^{23} +78.2615 q^{25} +(-139.852 - 11.1486i) q^{27} -131.471 q^{29} -113.891i q^{31} +(-20.9837 - 0.556540i) q^{33} +(-85.2206 + 93.6416i) q^{35} -320.606i q^{37} +(-60.5334 - 1.60550i) q^{39} +253.628 q^{41} +356.247i q^{43} +(9.78453 - 184.327i) q^{45} +316.669 q^{47} +(32.2261 + 341.483i) q^{49} +(-541.057 - 14.3502i) q^{51} -470.890 q^{53} -27.6178i q^{55} +(-150.797 - 3.99951i) q^{57} -722.454i q^{59} +141.417 q^{61} +(-351.464 - 355.697i) q^{63} -79.6715i q^{65} -97.1111i q^{67} +(14.4048 - 543.117i) q^{69} -900.777i q^{71} +622.304i q^{73} +(-406.516 - 10.7818i) q^{75} +(-55.3330 - 50.3570i) q^{77} -211.405 q^{79} +(724.903 + 77.1766i) q^{81} -837.420i q^{83} -712.116i q^{85} +(682.901 + 18.1123i) q^{87} -857.227 q^{89} +(-159.624 - 145.269i) q^{91} +(-15.6904 + 591.588i) q^{93} -198.472i q^{95} -1135.63i q^{97} +(108.919 + 5.78170i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) $ 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−5.19433	−	0.137767i	−0.999648	−	0.0265132i
$3$	−5.19433	−	0.137767i	−0.999648	−	0.0265132i
$4$		0			0
$5$		−	6.83655i		−	0.611480i	−0.952115	−	0.305740i	$-0.901096\pi$
$5$		−	6.83655i		−	0.611480i	0.952115	−	0.305740i	$-0.0989038\pi$
$6$		0			0
$7$	−13.6972	−	12.4654i	−0.739579	−	0.673070i
$7$	−13.6972	−	12.4654i	−0.739579	−	0.673070i
$8$		0			0
$9$	26.9620	+	1.43121i	0.998594	+	0.0530077i
$10$		0			0
$11$	4.03973			0.110729			0.0553647	−	0.998466i	$-0.482368\pi$
$11$	4.03973			0.110729			0.0553647	+	0.998466i	$0.482368\pi$
$12$		0			0
$13$	11.6537			0.248628			0.124314	−	0.992243i	$-0.460327\pi$
$13$	11.6537			0.248628			0.124314	+	0.992243i	$0.460327\pi$
$14$		0			0
$15$	−0.941848	+	35.5113i	−0.0162123	+	0.611265i
$16$		0			0
$17$	104.163			1.48607			0.743037	−	0.669250i	$-0.233385\pi$
$17$	104.163			1.48607			0.743037	+	0.669250i	$0.233385\pi$
$18$		0			0
$19$	29.0310			0.350536			0.175268	−	0.984521i	$-0.443921\pi$
$19$	29.0310			0.350536			0.175268	+	0.984521i	$0.443921\pi$
$20$		0			0
$21$	69.4304	+	66.6365i	0.721474	+	0.692442i
$22$		0			0
$23$			104.560i			0.947921i	0.880546	+	0.473960i	$0.157176\pi$
$23$			104.560i			0.947921i	−0.880546	+	0.473960i	$0.842824\pi$
$24$		0			0
$25$	78.2615			0.626092
$26$		0			0
$27$	−139.852	−	11.1486i	−0.996838	−	0.0794650i
$28$		0			0
$29$	−131.471			−0.841844			−0.420922	−	0.907097i	$-0.638293\pi$
$29$	−131.471			−0.841844			−0.420922	+	0.907097i	$0.638293\pi$
$30$		0			0
$31$		−	113.891i		−	0.659854i	−0.944006	−	0.329927i	$-0.892976\pi$
$31$		−	113.891i		−	0.659854i	0.944006	−	0.329927i	$-0.107024\pi$
$32$		0			0
$33$	−20.9837	−	0.556540i	−0.110691	−	0.00293579i
$34$		0			0
$35$	−85.2206	+	93.6416i	−0.411569	+	0.452238i
$36$		0			0
$37$		−	320.606i		−	1.42452i	−0.701915	−	0.712261i	$-0.747671\pi$
$37$		−	320.606i		−	1.42452i	0.701915	−	0.712261i	$-0.252329\pi$
$38$		0			0
$39$	−60.5334	−	1.60550i	−0.248541	−	0.00659193i
$40$		0			0
$41$	253.628			0.966099			0.483050	−	0.875593i	$-0.339529\pi$
$41$	253.628			0.966099			0.483050	+	0.875593i	$0.339529\pi$
$42$		0			0
$43$			356.247i			1.26342i	0.775204	+	0.631711i	$0.217647\pi$
$43$			356.247i			1.26342i	−0.775204	+	0.631711i	$0.782353\pi$
$44$		0			0
$45$	9.78453	−	184.327i	0.0324132	−	0.610620i
$46$		0			0
$47$	316.669			0.982784			0.491392	−	0.870938i	$-0.336488\pi$
$47$	316.669			0.982784			0.491392	+	0.870938i	$0.336488\pi$
$48$		0			0
$49$	32.2261	+	341.483i	0.0939537	+	0.995577i
$50$		0			0
$51$	−541.057	−	14.3502i	−1.48555	−	0.0394005i
$52$		0			0
$53$	−470.890			−1.22041			−0.610205	−	0.792244i	$-0.708913\pi$
$53$	−470.890			−1.22041			−0.610205	+	0.792244i	$0.708913\pi$
$54$		0			0
$55$		−	27.6178i		−	0.0677089i
$56$		0			0
$57$	−150.797	−	3.99951i	−0.350412	−	0.00929382i
$58$		0			0
$59$		−	722.454i		−	1.59416i	−0.603874	−	0.797080i	$-0.706377\pi$
$59$		−	722.454i		−	1.59416i	0.603874	−	0.797080i	$-0.293623\pi$
$60$		0			0
$61$	141.417			0.296828			0.148414	−	0.988925i	$-0.452583\pi$
$61$	141.417			0.296828			0.148414	+	0.988925i	$0.452583\pi$
$62$		0			0
$63$	−351.464	−	355.697i	−0.702861	−	0.711327i
$64$		0			0
$65$		−	79.6715i		−	0.152031i
$66$		0			0
$67$		−	97.1111i		−	0.177075i	−0.996073	−	0.0885374i	$-0.971781\pi$
$67$		−	97.1111i		−	0.177075i	0.996073	−	0.0885374i	$-0.0282193\pi$
$68$		0			0
$69$	14.4048	−	543.117i	0.0251324	−	0.947588i
$70$		0			0
$71$		−	900.777i		−	1.50567i	−0.658209	−	0.752835i	$-0.728686\pi$
$71$		−	900.777i		−	1.50567i	0.658209	−	0.752835i	$-0.271314\pi$
$72$		0			0
$73$			622.304i			0.997742i	0.866676	+	0.498871i	$0.166252\pi$
$73$			622.304i			0.997742i	−0.866676	+	0.498871i	$0.833748\pi$
$74$		0			0
$75$	−406.516	−	10.7818i	−0.625872	−	0.0165997i
$76$		0			0
$77$	−55.3330	−	50.3570i	−0.0818932	−	0.0745287i
$78$		0			0
$79$	−211.405			−0.301074			−0.150537	−	0.988604i	$-0.548100\pi$
$79$	−211.405			−0.301074			−0.150537	+	0.988604i	$0.548100\pi$
$80$		0			0
$81$	724.903	+	77.1766i	0.994380	+	0.105866i
$82$		0			0
$83$		−	837.420i		−	1.10745i	−0.832698	−	0.553727i	$-0.813205\pi$
$83$		−	837.420i		−	1.10745i	0.832698	−	0.553727i	$-0.186795\pi$
$84$		0			0
$85$		−	712.116i		−	0.908704i
$86$		0			0
$87$	682.901	+	18.1123i	0.841548	+	0.0223200i
$88$		0			0
$89$	−857.227			−1.02096			−0.510482	−	0.859888i	$-0.670533\pi$
$89$	−857.227			−1.02096			−0.510482	+	0.859888i	$0.670533\pi$
$90$		0			0
$91$	−159.624	−	145.269i	−0.183880	−	0.167344i
$92$		0			0
$93$	−15.6904	+	591.588i	−0.0174948	+	0.659622i
$94$		0			0
$95$		−	198.472i		−	0.214345i
$96$		0			0
$97$		−	1135.63i		−	1.18872i	−0.804199	−	0.594360i	$-0.797405\pi$
$97$		−	1135.63i		−	1.18872i	0.804199	−	0.594360i	$-0.202595\pi$
$98$		0			0
$99$	108.919	+	5.78170i	0.110574	+	0.00586952i
$100$		0			0
$101$		−	956.403i		−	0.942234i	−0.882071	−	0.471117i	$-0.843851\pi$
$101$		−	956.403i		−	0.942234i	0.882071	−	0.471117i	$-0.156149\pi$
$102$		0			0
$103$		−	474.636i		−	0.454051i	−0.973889	−	0.227025i	$-0.927100\pi$
$103$		−	474.636i		−	0.454051i	0.973889	−	0.227025i	$-0.0729000\pi$
$104$		0			0
$105$	455.564	−	474.664i	0.423414	−	0.441167i
$106$		0			0
$107$	−479.771			−0.433470			−0.216735	−	0.976231i	$-0.569541\pi$
$107$	−479.771			−0.433470			−0.216735	+	0.976231i	$0.569541\pi$
$108$		0			0
$109$			1248.35i			1.09697i	0.836159	+	0.548486i	$0.184796\pi$
$109$			1248.35i			1.09697i	−0.836159	+	0.548486i	$0.815204\pi$
$110$		0			0
$111$	−44.1688	+	1665.33i	−0.0377686	+	1.42402i
$112$		0			0
$113$		−	1349.96i		−	1.12384i	−0.827193	−	0.561918i	$-0.810063\pi$
$113$		−	1349.96i		−	1.12384i	0.827193	−	0.561918i	$-0.189937\pi$
$114$		0			0
$115$	714.827			0.579634
$116$		0			0
$117$	314.209	+	16.6789i	0.248279	+	0.0131792i
$118$		0			0
$119$	−1426.74	−	1298.44i	−1.09907	−	1.00023i
$120$		0			0
$121$	−1314.68			−0.987739
$122$		0			0
$123$	−1317.43	−	34.9415i	−0.965760	−	0.0256144i
$124$		0			0
$125$		−	1389.61i		−	0.994323i
$126$		0			0
$127$	1963.80			1.37212			0.686058	−	0.727547i	$-0.259340\pi$
$127$	1963.80			1.37212			0.686058	+	0.727547i	$0.259340\pi$
$128$		0			0
$129$	49.0789	−	1850.46i	0.0334973	−	1.26298i
$130$		0			0
$131$		−	1775.12i		−	1.18391i	−0.805969	−	0.591957i	$-0.798355\pi$
$131$		−	1775.12i		−	1.18391i	0.805969	−	0.591957i	$-0.201645\pi$
$132$		0			0
$133$	−397.644	−	361.884i	−0.259249	−	0.235935i
$134$		0			0
$135$	−76.2182	+	956.109i	−0.0485913	+	0.609546i
$136$		0			0
$137$		−	573.341i		−	0.357546i	−0.983890	−	0.178773i	$-0.942787\pi$
$137$		−	573.341i		−	0.357546i	0.983890	−	0.178773i	$-0.0572128\pi$
$138$		0			0
$139$	1094.52			0.667883			0.333942	−	0.942594i	$-0.391621\pi$
$139$	1094.52			0.667883			0.333942	+	0.942594i	$0.391621\pi$
$140$		0			0
$141$	−1644.88	−	43.6263i	−0.982439	−	0.0260567i
$142$		0			0
$143$	47.0780			0.0275305
$144$		0			0
$145$			898.806i			0.514771i
$146$		0			0
$147$	−120.348	−	1778.21i	−0.0675248	−	0.997718i
$148$		0			0
$149$	−2573.83			−1.41514			−0.707571	−	0.706643i	$-0.750209\pi$
$149$	−2573.83			−1.41514			−0.707571	+	0.706643i	$0.750209\pi$
$150$		0			0
$151$	1119.27			0.603214			0.301607	−	0.953432i	$-0.402477\pi$
$151$	1119.27			0.603214			0.301607	+	0.953432i	$0.402477\pi$
$152$		0			0
$153$	2808.45	+	149.079i	1.48398	+	0.0787734i
$154$		0			0
$155$	−778.624			−0.403487
$156$		0			0
$157$	−2769.72			−1.40795			−0.703973	−	0.710227i	$-0.748592\pi$
$157$	−2769.72			−1.40795			−0.703973	+	0.710227i	$0.748592\pi$
$158$		0			0
$159$	2445.96	+	64.8729i	1.21998	+	0.0323570i
$160$		0			0
$161$	1303.38	−	1432.17i	0.638017	−	0.701062i
$162$		0			0
$163$			1305.78i			0.627464i	0.949511	+	0.313732i	$0.101579\pi$
$163$			1305.78i			0.627464i	−0.949511	+	0.313732i	$0.898421\pi$
$164$		0			0
$165$	−3.80481	+	143.456i	−0.00179518	+	0.0676851i
$166$		0			0
$167$	−1241.39			−0.575218			−0.287609	−	0.957748i	$-0.592860\pi$
$167$	−1241.39			−0.575218			−0.287609	+	0.957748i	$0.592860\pi$
$168$		0			0
$169$	−2061.19			−0.938184
$170$		0			0
$171$	782.736	+	41.5495i	0.350043	+	0.0185811i
$172$		0			0
$173$		−	415.397i		−	0.182555i	−0.995825	−	0.0912775i	$-0.970905\pi$
$173$		−	415.397i		−	0.182555i	0.995825	−	0.0912775i	$-0.0290950\pi$
$174$		0			0
$175$	−1071.96	−	975.564i	−0.463045	−	0.421404i
$176$		0			0
$177$	−99.5300	+	3752.66i	−0.0422663	+	1.59360i
$178$		0			0
$179$	117.160			0.0489215			0.0244608	−	0.999701i	$-0.492213\pi$
$179$	117.160			0.0489215			0.0244608	+	0.999701i	$0.492213\pi$
$180$		0			0
$181$	−1379.06			−0.566325			−0.283162	−	0.959072i	$-0.591384\pi$
$181$	−1379.06			−0.566325			−0.283162	+	0.959072i	$0.591384\pi$
$182$		0			0
$183$	−734.564	−	19.4825i	−0.296724	−	0.00786987i
$184$		0			0
$185$	−2191.84			−0.871066
$186$		0			0
$187$	420.791			0.164552
$188$		0			0
$189$	1776.61	+	1896.03i	0.683755	+	0.729712i
$190$		0			0
$191$			2323.47i			0.880211i	0.897946	+	0.440106i	$0.145059\pi$
$191$			2323.47i			0.880211i	−0.897946	+	0.440106i	$0.854941\pi$
$192$		0			0
$193$	1284.20			0.478958			0.239479	−	0.970902i	$-0.423023\pi$
$193$	1284.20			0.478958			0.239479	+	0.970902i	$0.423023\pi$
$194$		0			0
$195$	−10.9761	+	413.840i	−0.00403083	+	0.151978i
$196$		0			0
$197$	4935.49			1.78497			0.892485	−	0.451076i	$-0.148960\pi$
$197$	4935.49			1.78497			0.892485	+	0.451076i	$0.148960\pi$
$198$		0			0
$199$			3824.82i			1.36249i	0.732058	+	0.681243i	$0.238560\pi$
$199$			3824.82i			1.36249i	−0.732058	+	0.681243i	$0.761440\pi$
$200$		0			0
$201$	−13.3787	+	504.427i	−0.00469482	+	0.177013i
$202$		0			0
$203$	1800.78	+	1638.84i	0.622610	+	0.566620i
$204$		0			0
$205$		−	1733.94i		−	0.590750i
$206$		0			0
$207$	−149.647	+	2819.14i	−0.0502471	+	0.946588i
$208$		0			0
$209$	117.278			0.0388146
$210$		0			0
$211$			1236.70i			0.403496i	0.979437	+	0.201748i	$0.0646623\pi$
$211$			1236.70i			0.403496i	−0.979437	+	0.201748i	$0.935338\pi$
$212$		0			0
$213$	−124.097	+	4678.93i	−0.0399201	+	1.50514i
$214$		0			0
$215$	2435.50			0.772557
$216$		0			0
$217$	−1419.70	+	1559.99i	−0.444128	+	0.488014i
$218$		0			0
$219$	85.7326	−	3232.45i	0.0264533	−	0.997391i
$220$		0			0
$221$	1213.89			0.369480
$222$		0			0
$223$		−	2567.59i		−	0.771026i	−0.922702	−	0.385513i	$-0.874025\pi$
$223$		−	2567.59i		−	0.771026i	0.922702	−	0.385513i	$-0.125975\pi$
$224$		0			0
$225$	2110.09	+	112.009i	0.625212	+	0.0331877i
$226$		0			0
$227$			5160.79i			1.50896i	0.656323	+	0.754480i	$0.272111\pi$
$227$			5160.79i			1.50896i	−0.656323	+	0.754480i	$0.727889\pi$
$228$		0			0
$229$	2221.09			0.640934			0.320467	−	0.947260i	$-0.396160\pi$
$229$	2221.09			0.640934			0.320467	+	0.947260i	$0.396160\pi$
$230$		0			0
$231$	280.480	+	269.194i	0.0798884	+	0.0766737i
$232$		0			0
$233$			3711.51i			1.04356i	0.853080	+	0.521779i	$0.174732\pi$
$233$			3711.51i			1.04356i	−0.853080	+	0.521779i	$0.825268\pi$
$234$		0			0
$235$		−	2164.92i		−	0.600953i
$236$		0			0
$237$	1098.10	+	29.1245i	0.300968	+	0.00798244i
$238$		0			0
$239$		−	3198.82i		−	0.865751i	−0.901454	−	0.432876i	$-0.857499\pi$
$239$		−	3198.82i		−	0.865751i	0.901454	−	0.432876i	$-0.142501\pi$
$240$		0			0
$241$		−	5785.50i		−	1.54638i	−0.634177	−	0.773188i	$-0.718661\pi$
$241$		−	5785.50i		−	1.54638i	0.634177	−	0.773188i	$-0.281339\pi$
$242$		0			0
$243$	−3754.75	−	500.748i	−0.991224	−	0.132193i
$244$		0			0
$245$	2334.56	−	220.316i	0.608775	−	0.0574508i
$246$		0			0
$247$	338.320			0.0871531
$248$		0			0
$249$	−115.368	+	4349.83i	−0.0293622	+	1.10707i
$250$		0			0
$251$		−	2096.38i		−	0.527180i	−0.964635	−	0.263590i	$-0.915093\pi$
$251$		−	2096.38i		−	0.527180i	0.964635	−	0.263590i	$-0.0849066\pi$
$252$		0			0
$253$			422.392i			0.104963i
$254$		0			0
$255$	−98.1058	+	3698.96i	−0.0240926	+	0.908385i
$256$		0			0
$257$	−5905.29			−1.43331			−0.716657	−	0.697426i	$-0.754329\pi$
$257$	−5905.29			−1.43331			−0.716657	+	0.697426i	$0.754329\pi$
$258$		0			0
$259$	−3996.49	+	4391.40i	−0.958803	+	1.05355i
$260$		0			0
$261$	−3544.72	−	188.162i	−0.840661	−	0.0446243i
$262$		0			0
$263$		−	2143.88i		−	0.502652i	−0.967902	−	0.251326i	$-0.919133\pi$
$263$		−	2143.88i		−	0.502652i	0.967902	−	0.251326i	$-0.0808666\pi$
$264$		0			0
$265$			3219.26i			0.746256i
$266$		0			0
$267$	4452.72	+	118.097i	1.02061	+	0.0270690i
$268$		0			0
$269$			6249.96i			1.41660i	0.705909	+	0.708302i	$0.250539\pi$
$269$			6249.96i			1.41660i	−0.705909	+	0.708302i	$0.749461\pi$
$270$		0			0
$271$		−	4623.10i		−	1.03629i	−0.855294	−	0.518143i	$-0.826624\pi$
$271$		−	4623.10i		−	1.03629i	0.855294	−	0.518143i	$-0.173376\pi$
$272$		0			0
$273$	809.124	+	776.565i	0.179379	+	0.172161i
$274$		0			0
$275$	316.156			0.0693269
$276$		0			0
$277$			282.839i			0.0613507i	0.999529	+	0.0306754i	$0.00976580\pi$
$277$			282.839i			0.0613507i	−0.999529	+	0.0306754i	$0.990234\pi$
$278$		0			0
$279$	163.002	−	3070.74i	0.0349774	−	0.658926i
$280$		0			0
$281$		−	6722.69i		−	1.42720i	−0.700555	−	0.713598i	$-0.747064\pi$
$281$		−	6722.69i		−	1.42720i	0.700555	−	0.713598i	$-0.252936\pi$
$282$		0			0
$283$	7208.18			1.51407			0.757035	−	0.653375i	$-0.226647\pi$
$283$	7208.18			1.51407			0.757035	+	0.653375i	$0.226647\pi$
$284$		0			0
$285$	−27.3428	+	1030.93i	−0.00568298	+	0.214270i
$286$		0			0
$287$	−3473.99	−	3161.58i	−0.714507	−	0.650252i
$288$		0			0
$289$	5936.94			1.20842
$290$		0			0
$291$	−156.452	+	5898.84i	−0.0315168	+	1.18830i
$292$		0			0
$293$		−	3175.54i		−	0.633163i	−0.948565	−	0.316582i	$-0.897465\pi$
$293$		−	3175.54i		−	0.633163i	0.948565	−	0.316582i	$-0.102535\pi$
$294$		0			0
$295$	−4939.09			−0.974797
$296$		0			0
$297$	−564.966	−	45.0375i	−0.110379	−	0.00879912i
$298$		0			0
$299$			1218.51i			0.235680i
$300$		0			0
$301$	4440.77	−	4879.58i	0.850371	−	0.934400i
$302$		0			0
$303$	−131.760	+	4967.87i	−0.0249816	+	0.941903i
$304$		0			0
$305$		−	966.802i		−	0.181505i
$306$		0			0
$307$	−3739.33			−0.695163			−0.347581	−	0.937650i	$-0.612997\pi$
$307$	−3739.33			−0.695163			−0.347581	+	0.937650i	$0.612997\pi$
$308$		0			0
$309$	−65.3889	+	2465.41i	−0.0120383	+	0.453891i
$310$		0			0
$311$	−9117.85			−1.66246			−0.831231	−	0.555927i	$-0.812363\pi$
$311$	−9117.85			−1.66246			−0.831231	+	0.555927i	$0.812363\pi$
$312$		0			0
$313$			9104.29i			1.64410i	0.569412	+	0.822052i	$0.307171\pi$
$313$			9104.29i			1.64410i	−0.569412	+	0.822052i	$0.692829\pi$
$314$		0			0
$315$	−2431.74	+	2402.80i	−0.434962	+	0.429785i
$316$		0			0
$317$	9883.17			1.75109			0.875543	−	0.483141i	$-0.160504\pi$
$317$	9883.17			1.75109			0.875543	+	0.483141i	$0.160504\pi$
$318$		0			0
$319$	−531.106			−0.0932170
$320$		0			0
$321$	2492.09	+	66.0965i	0.433317	+	0.0114927i
$322$		0			0
$323$	3023.96			0.520922
$324$		0			0
$325$	912.040			0.155664
$326$		0			0
$327$	171.981	−	6484.33i	0.0290842	−	1.09659i
$328$		0			0
$329$	−4337.47	−	3947.41i	−0.726846	−	0.661482i
$330$		0			0
$331$		−	452.334i		−	0.0751134i	−0.999294	−	0.0375567i	$-0.988043\pi$
$331$		−	452.334i		−	0.0751134i	0.999294	−	0.0375567i	$-0.0119575\pi$
$332$		0			0
$333$	458.854	−	8644.19i	0.0755107	−	1.42252i
$334$		0			0
$335$	−663.905			−0.108278
$336$		0			0
$337$	4969.87			0.803341			0.401670	−	0.915784i	$-0.368430\pi$
$337$	4969.87			0.803341			0.401670	+	0.915784i	$0.368430\pi$
$338$		0			0
$339$	−185.979	+	7012.13i	−0.0297965	+	1.12344i
$340$		0			0
$341$		−	460.090i		−	0.0730653i
$342$		0			0
$343$	3815.32	−	5079.07i	0.600606	−	0.799545i
$344$		0			0
$345$	−3713.04	−	98.4793i	−0.579431	−	0.0153680i
$346$		0			0
$347$	11013.5			1.70386			0.851928	−	0.523659i	$-0.175433\pi$
$347$	11013.5			1.70386			0.851928	+	0.523659i	$0.175433\pi$
$348$		0			0
$349$	−4530.91			−0.694940			−0.347470	−	0.937691i	$-0.612959\pi$
$349$	−4530.91			−0.694940			−0.347470	+	0.937691i	$0.612959\pi$
$350$		0			0
$351$	−1629.81	−	129.923i	−0.247842	−	0.0197573i
$352$		0			0
$353$	−6152.85			−0.927714			−0.463857	−	0.885910i	$-0.653535\pi$
$353$	−6152.85			−0.927714			−0.463857	+	0.885910i	$0.653535\pi$
$354$		0			0
$355$	−6158.21			−0.920687
$356$		0			0
$357$	7232.08	+	6941.06i	1.07216	+	1.02902i
$358$		0			0
$359$			2822.93i			0.415010i	0.978234	+	0.207505i	$0.0665344\pi$
$359$			2822.93i			0.415010i	−0.978234	+	0.207505i	$0.933466\pi$
$360$		0			0
$361$	−6016.20			−0.877125
$362$		0			0
$363$	6828.88	+	181.119i	0.987392	+	0.0261881i
$364$		0			0
$365$	4254.41			0.610099
$366$		0			0
$367$			6388.51i			0.908658i	0.890834	+	0.454329i	$0.150121\pi$
$367$			6388.51i			0.908658i	−0.890834	+	0.454329i	$0.849879\pi$
$368$		0			0
$369$	6838.33	+	362.995i	0.964741	+	0.0512107i
$370$		0			0
$371$	6449.87	+	5869.85i	0.902589	+	0.821421i
$372$		0			0
$373$			6415.30i			0.890541i	0.895396	+	0.445270i	$0.146892\pi$
$373$			6415.30i			0.890541i	−0.895396	+	0.445270i	$0.853108\pi$
$374$		0			0
$375$	−191.442	+	7218.08i	−0.0263627	+	0.993973i
$376$		0			0
$377$	−1532.13			−0.209306
$378$		0			0
$379$		−	10997.5i		−	1.49051i	−0.666782	−	0.745253i	$-0.732329\pi$
$379$		−	10997.5i		−	1.49051i	0.666782	−	0.745253i	$-0.267671\pi$
$380$		0			0
$381$	−10200.6	−	270.545i	−1.37163	−	0.0363792i
$382$		0			0
$383$	−9990.07			−1.33282			−0.666408	−	0.745587i	$-0.732169\pi$
$383$	−9990.07			−1.33282			−0.666408	+	0.745587i	$0.732169\pi$
$384$		0			0
$385$	−344.268	+	378.287i	−0.0455728	+	0.0500760i
$386$		0			0
$387$	−509.864	+	9605.14i	−0.0669711	+	1.26165i
$388$		0			0
$389$	−3943.66			−0.514014			−0.257007	−	0.966410i	$-0.582736\pi$
$389$	−3943.66			−0.514014			−0.257007	+	0.966410i	$0.582736\pi$
$390$		0			0
$391$			10891.2i			1.40868i
$392$		0			0
$393$	−244.552	+	9220.55i	−0.0313894	+	1.18350i
$394$		0			0
$395$			1445.28i			0.184101i
$396$		0			0
$397$	14238.3			1.80000			0.900002	−	0.435885i	$-0.143565\pi$
$397$	14238.3			1.80000			0.900002	+	0.435885i	$0.143565\pi$
$398$		0			0
$399$	2015.64	+	1934.53i	0.252902	+	0.242726i
$400$		0			0
$401$		−	876.488i		−	0.109151i	−0.998510	−	0.0545757i	$-0.982619\pi$
$401$		−	876.488i		−	0.109151i	0.998510	−	0.0545757i	$-0.0173806\pi$
$402$		0			0
$403$		−	1327.26i		−	0.164058i
$404$		0			0
$405$	527.622	−	4955.84i	0.0647352	−	0.608044i
$406$		0			0
$407$		−	1295.16i		−	0.157737i
$408$		0			0
$409$		−	11741.1i		−	1.41947i	−0.704471	−	0.709733i	$-0.748816\pi$
$409$		−	11741.1i		−	1.41947i	0.704471	−	0.709733i	$-0.251184\pi$
$410$		0			0
$411$	−78.9873	+	2978.12i	−0.00947969	+	0.357421i
$412$		0			0
$413$	−9005.70	+	9895.59i	−1.07298	+	1.17901i
$414$		0			0
$415$	−5725.06			−0.677186
$416$		0			0
$417$	−5685.28	−	150.788i	−0.667648	−	0.0177077i
$418$		0			0
$419$		−	16673.4i		−	1.94403i	−0.234911	−	0.972017i	$-0.575480\pi$
$419$		−	16673.4i		−	1.94403i	0.234911	−	0.972017i	$-0.424520\pi$
$420$		0			0
$421$			2350.61i			0.272118i	0.990701	+	0.136059i	$0.0434436\pi$
$421$			2350.61i			0.272118i	−0.990701	+	0.136059i	$0.956556\pi$
$422$		0			0
$423$	8538.03	+	453.219i	0.981402	+	0.0520952i
$424$		0			0
$425$	8151.96			0.930419
$426$		0			0
$427$	−1937.01	−	1762.82i	−0.219528	−	0.199786i
$428$		0			0
$429$	−244.538	−	6.48577i	−0.0275208	−	0.000729921i
$430$		0			0
$431$		−	6140.09i		−	0.686213i	−0.939297	−	0.343106i	$-0.888521\pi$
$431$		−	6140.09i		−	0.686213i	0.939297	−	0.343106i	$-0.111479\pi$
$432$		0			0
$433$		−	8753.16i		−	0.971478i	−0.874104	−	0.485739i	$-0.838551\pi$
$433$		−	8753.16i		−	0.971478i	0.874104	−	0.485739i	$-0.161449\pi$
$434$		0			0
$435$	123.825	−	4668.69i	0.0136482	−	0.514590i
$436$		0			0
$437$			3035.47i			0.332280i
$438$		0			0
$439$			4097.50i			0.445474i	0.974879	+	0.222737i	$0.0714992\pi$
$439$			4097.50i			0.445474i	−0.974879	+	0.222737i	$0.928501\pi$
$440$		0			0
$441$	380.149	+	9253.19i	0.0410484	+	0.999157i
$442$		0			0
$443$	−7908.50			−0.848181			−0.424091	−	0.905620i	$-0.639406\pi$
$443$	−7908.50			−0.848181			−0.424091	+	0.905620i	$0.639406\pi$
$444$		0			0
$445$			5860.48i			0.624299i
$446$		0			0
$447$	13369.3	+	354.587i	1.41464	+	0.0375199i
$448$		0			0
$449$		−	2363.47i		−	0.248417i	−0.992256	−	0.124208i	$-0.960361\pi$
$449$		−	2363.47i		−	0.248417i	0.992256	−	0.124208i	$-0.0396392\pi$
$450$		0			0
$451$	1024.59			0.106976
$452$		0			0
$453$	−5813.88	−	154.199i	−0.603002	−	0.0159931i
$454$		0			0
$455$	−993.139	+	1091.28i	−0.102328	+	0.112439i
$456$		0			0
$457$	5371.11			0.549781			0.274891	−	0.961476i	$-0.411358\pi$
$457$	5371.11			0.549781			0.274891	+	0.961476i	$0.411358\pi$
$458$		0			0
$459$	−14567.5	−	1161.28i	−1.48137	−	0.118091i
$460$		0			0
$461$		−	14246.7i		−	1.43934i	−0.694319	−	0.719668i	$-0.744294\pi$
$461$		−	14246.7i		−	1.43934i	0.694319	−	0.719668i	$-0.255706\pi$
$462$		0			0
$463$	3194.81			0.320681			0.160341	−	0.987062i	$-0.448741\pi$
$463$	3194.81			0.320681			0.160341	+	0.987062i	$0.448741\pi$
$464$		0			0
$465$	4044.43	+	107.268i	0.403346	+	0.0106977i
$466$		0			0
$467$		−	3317.47i		−	0.328724i	−0.986400	−	0.164362i	$-0.947443\pi$
$467$		−	3317.47i		−	0.328724i	0.986400	−	0.164362i	$-0.0525565\pi$
$468$		0			0
$469$	−1210.53	+	1330.15i	−0.119184	+	0.130961i
$470$		0			0
$471$	14386.8	+	381.574i	1.40745	+	0.0373291i
$472$		0			0
$473$			1439.14i			0.139898i
$474$		0			0
$475$	2272.01			0.219468
$476$		0			0
$477$	−12696.2	−	673.942i	−1.21869	−	0.0646912i
$478$		0			0
$479$	1661.85			0.158522			0.0792608	−	0.996854i	$-0.474744\pi$
$479$	1661.85			0.158522			0.0792608	+	0.996854i	$0.474744\pi$
$480$		0			0
$481$		−	3736.26i		−	0.354176i
$482$		0			0
$483$	−6967.49	+	7259.61i	−0.656380	+	0.683900i
$484$		0			0
$485$	−7763.80			−0.726879
$486$		0			0
$487$	−6158.69			−0.573053			−0.286527	−	0.958072i	$-0.592501\pi$
$487$	−6158.69			−0.573053			−0.286527	+	0.958072i	$0.592501\pi$
$488$		0			0
$489$	179.893	−	6782.66i	0.0166361	−	0.627244i
$490$		0			0
$491$	−7846.32			−0.721180			−0.360590	−	0.932724i	$-0.617425\pi$
$491$	−7846.32			−0.721180			−0.360590	+	0.932724i	$0.617425\pi$
$492$		0			0
$493$	−13694.4			−1.25104
$494$		0			0
$495$	39.5269	−	744.633i	0.00358909	−	0.0676137i
$496$		0			0
$497$	−11228.6	+	12338.1i	−1.01342	+	1.11356i
$498$		0			0
$499$		−	11321.6i		−	1.01568i	−0.861452	−	0.507839i	$-0.830444\pi$
$499$		−	11321.6i		−	1.01568i	0.861452	−	0.507839i	$-0.169556\pi$
$500$		0			0
$501$	6448.17	+	171.022i	0.575016	+	0.0152509i
$502$		0			0
$503$	21262.4			1.88478			0.942391	−	0.334513i	$-0.108572\pi$
$503$	21262.4			1.88478			0.942391	+	0.334513i	$0.108572\pi$
$504$		0			0
$505$	−6538.50			−0.576157
$506$		0			0
$507$	10706.5	+	283.963i	0.937854	+	0.0248742i
$508$		0			0
$509$		−	6748.42i		−	0.587659i	−0.955858	−	0.293829i	$-0.905070\pi$
$509$		−	6748.42i		−	0.587659i	0.955858	−	0.293829i	$-0.0949298\pi$
$510$		0			0
$511$	7757.28	−	8523.81i	0.671550	−	0.737909i
$512$		0			0
$513$	−4060.06	−	323.656i	−0.349427	−	0.0278553i
$514$		0			0
$515$	−3244.87			−0.277643
$516$		0			0
$517$	1279.26			0.108823
$518$		0			0
$519$	−57.2278	+	2157.71i	−0.00484011	+	0.182491i
$520$		0			0
$521$	1669.33			0.140374			0.0701868	−	0.997534i	$-0.477640\pi$
$521$	1669.33			0.140374			0.0701868	+	0.997534i	$0.477640\pi$
$522$		0			0
$523$	8699.40			0.727339			0.363669	−	0.931528i	$-0.381524\pi$
$523$	8699.40			0.727339			0.363669	+	0.931528i	$0.381524\pi$
$524$		0			0
$525$	5433.73	+	5215.08i	0.451709	+	0.433533i
$526$		0			0
$527$		−	11863.3i		−	0.980592i
$528$		0			0
$529$	1234.29			0.101446
$530$		0			0
$531$	1033.98	−	19478.8i	0.0845028	−	1.59192i
$532$		0			0
$533$	2955.72			0.240200
$534$		0			0
$535$			3279.98i			0.265058i
$536$		0			0
$537$	−608.568	−	16.1407i	−0.0489043	−	0.00129707i
$538$		0			0
$539$	130.185	+	1379.50i	0.0104035	+	0.110240i
$540$		0			0
$541$			6335.93i			0.503517i	0.967790	+	0.251759i	$0.0810089\pi$
$541$			6335.93i			0.503517i	−0.967790	+	0.251759i	$0.918991\pi$
$542$		0			0
$543$	7163.29	+	189.988i	0.566126	+	0.0150151i
$544$		0			0
$545$	8534.40			0.670777
$546$		0			0
$547$			25466.5i			1.99062i	0.0967351	+	0.995310i	$0.469160\pi$
$547$			25466.5i			1.99062i	−0.0967351	+	0.995310i	$0.530840\pi$
$548$		0			0
$549$	3812.88	+	202.397i	0.296411	+	0.0157342i
$550$		0			0
$551$	−3816.73			−0.295096
$552$		0			0
$553$	2895.65	+	2635.25i	0.222668	+	0.202644i
$554$		0			0
$555$	11385.1	+	301.962i	0.870760	+	0.0230947i
$556$		0			0
$557$	−2282.80			−0.173654			−0.0868269	−	0.996223i	$-0.527673\pi$
$557$	−2282.80			−0.173654			−0.0868269	+	0.996223i	$0.527673\pi$
$558$		0			0
$559$			4151.61i			0.314122i
$560$		0			0
$561$	−2185.72	−	57.9709i	−0.164494	−	0.00436280i
$562$		0			0
$563$			5471.40i			0.409577i	0.978806	+	0.204789i	$0.0656507\pi$
$563$			5471.40i			0.409577i	−0.978806	+	0.204789i	$0.934349\pi$
$564$		0			0
$565$	−9229.07			−0.687204
$566$		0			0
$567$	−8967.10	−	10093.3i	−0.664167	−	0.747584i
$568$		0			0
$569$		−	10659.3i		−	0.785342i	−0.919679	−	0.392671i	$-0.871551\pi$
$569$		−	10659.3i		−	0.785342i	0.919679	−	0.392671i	$-0.128449\pi$
$570$		0			0
$571$		−	12002.7i		−	0.879680i	−0.898076	−	0.439840i	$-0.855035\pi$
$571$		−	12002.7i		−	0.879680i	0.898076	−	0.439840i	$-0.144965\pi$
$572$		0			0
$573$	320.096	−	12068.9i	0.0233372	−	0.879902i
$574$		0			0
$575$			8182.99i			0.593486i
$576$		0			0
$577$			10078.4i			0.727159i	0.931563	+	0.363579i	$0.118445\pi$
$577$			10078.4i			0.727159i	−0.931563	+	0.363579i	$0.881555\pi$
$578$		0			0
$579$	−6670.56	−	176.920i	−0.478789	−	0.0126987i
$580$		0			0
$581$	−10438.8	+	11470.3i	−0.745395	+	0.819050i
$582$		0			0
$583$	−1902.27			−0.135135
$584$		0			0
$585$	114.026	−	2148.11i	0.00805883	−	0.151817i
$586$		0			0
$587$		−	22343.5i		−	1.57106i	−0.618820	−	0.785532i	$-0.712389\pi$
$587$		−	22343.5i		−	1.57106i	0.618820	−	0.785532i	$-0.287611\pi$
$588$		0			0
$589$		−	3306.38i		−	0.231302i
$590$		0			0
$591$	−25636.6	−	679.946i	−1.78434	−	0.0473253i
$592$		0			0
$593$	21338.0			1.47765			0.738824	−	0.673898i	$-0.235381\pi$
$593$	21338.0			1.47765			0.738824	+	0.673898i	$0.235381\pi$
$594$		0			0
$595$	−8876.84	+	9753.99i	−0.611621	+	0.672058i
$596$		0			0
$597$	526.933	−	19867.4i	0.0361238	−	1.36201i
$598$		0			0
$599$		−	8738.27i		−	0.596054i	−0.954557	−	0.298027i	$-0.903671\pi$
$599$		−	8738.27i		−	0.596054i	0.954557	−	0.298027i	$-0.0963285\pi$
$600$		0			0
$601$		−	7481.20i		−	0.507761i	−0.967236	−	0.253881i	$-0.918293\pi$
$601$		−	7481.20i		−	0.507761i	0.967236	−	0.253881i	$-0.0817070\pi$
$602$		0			0
$603$	138.986	−	2618.31i	0.00938633	−	0.176826i
$604$		0			0
$605$			8987.88i			0.603982i
$606$		0			0
$607$			473.402i			0.0316554i	0.999875	+	0.0158277i	$0.00503832\pi$
$607$			473.402i			0.0316554i	−0.999875	+	0.0158277i	$0.994962\pi$
$608$		0			0
$609$	−9128.05	−	8760.75i	−0.607369	−	0.582928i
$610$		0			0
$611$	3690.38			0.244348
$612$		0			0
$613$		−	7125.38i		−	0.469480i	−0.972058	−	0.234740i	$-0.924576\pi$
$613$		−	7125.38i		−	0.469480i	0.972058	−	0.234740i	$-0.0754239\pi$
$614$		0			0
$615$	−238.879	+	9006.66i	−0.0156627	+	0.590543i
$616$		0			0
$617$			3345.74i			0.218305i	0.994025	+	0.109153i	$0.0348137\pi$
$617$			3345.74i			0.218305i	−0.994025	+	0.109153i	$0.965186\pi$
$618$		0			0
$619$	7950.72			0.516263			0.258131	−	0.966110i	$-0.416893\pi$
$619$	7950.72			0.516263			0.258131	+	0.966110i	$0.416893\pi$
$620$		0			0
$621$	1165.70	−	14622.9i	0.0753265	−	0.944923i
$622$		0			0
$623$	11741.6	+	10685.7i	0.755084	+	0.687181i
$624$		0			0
$625$	282.564			0.0180841
$626$		0			0
$627$	−609.178	−	16.1569i	−0.0388010	−	0.00102910i
$628$		0			0
$629$		−	33395.3i		−	2.11694i
$630$		0			0
$631$	9702.61			0.612131			0.306066	−	0.952010i	$-0.400987\pi$
$631$	9702.61			0.612131			0.306066	+	0.952010i	$0.400987\pi$
$632$		0			0
$633$	170.376	−	6423.81i	0.0106980	−	0.403355i
$634$		0			0
$635$		−	13425.6i		−	0.839021i
$636$		0			0
$637$	375.555	+	3979.55i	0.0233596	+	0.247529i
$638$		0			0
$639$	1289.20	−	24286.8i	0.0798121	−	1.50355i
$640$		0			0
$641$		−	27464.9i		−	1.69235i	−0.532904	−	0.846176i	$-0.678899\pi$
$641$		−	27464.9i		−	1.69235i	0.532904	−	0.846176i	$-0.321101\pi$
$642$		0			0
$643$	−18925.7			−1.16074			−0.580372	−	0.814352i	$-0.697093\pi$
$643$	−18925.7			−1.16074			−0.580372	+	0.814352i	$0.697093\pi$
$644$		0			0
$645$	−12650.8	−	335.530i	−0.772285	−	0.0204829i
$646$		0			0
$647$	−6973.22			−0.423718			−0.211859	−	0.977300i	$-0.567952\pi$
$647$	−6973.22			−0.423718			−0.211859	+	0.977300i	$0.567952\pi$
$648$		0			0
$649$		−	2918.52i		−	0.176521i
$650$		0			0
$651$	7589.32	−	7907.51i	0.456911	−	0.476067i
$652$		0			0
$653$	−5754.57			−0.344860			−0.172430	−	0.985022i	$-0.555162\pi$
$653$	−5754.57			−0.344860			−0.172430	+	0.985022i	$0.555162\pi$
$654$		0			0
$655$	−12135.7			−0.723940
$656$		0			0
$657$	−890.647	+	16778.6i	−0.0528880	+	0.996339i
$658$		0			0
$659$	−30735.0			−1.81679			−0.908394	−	0.418115i	$-0.862691\pi$
$659$	−30735.0			−1.81679			−0.908394	+	0.418115i	$0.862691\pi$
$660$		0			0
$661$	22592.9			1.32944			0.664721	−	0.747092i	$-0.268550\pi$
$661$	22592.9			1.32944			0.664721	+	0.747092i	$0.268550\pi$
$662$		0			0
$663$	−6305.34	−	167.233i	−0.369350	−	0.00979609i
$664$		0			0
$665$	−2474.04	+	2718.51i	−0.144270	+	0.158525i
$666$		0			0
$667$		−	13746.5i		−	0.798002i
$668$		0			0
$669$	−353.728	+	13336.9i	−0.0204423	+	0.770755i
$670$		0			0
$671$	571.285			0.0328677
$672$		0			0
$673$	−12930.6			−0.740621			−0.370311	−	0.928908i	$-0.620749\pi$
$673$	−12930.6			−0.740621			−0.370311	+	0.928908i	$0.620749\pi$
$674$		0			0
$675$	−10945.1	−	872.509i	−0.624112	−	0.0497524i
$676$		0			0
$677$		−	27874.6i		−	1.58244i	−0.611534	−	0.791218i	$-0.709447\pi$
$677$		−	27874.6i		−	1.58244i	0.611534	−	0.791218i	$-0.290553\pi$
$678$		0			0
$679$	−14156.1	+	15555.0i	−0.800092	+	0.879153i
$680$		0			0
$681$	710.985	−	26806.8i	0.0400073	−	1.50843i
$682$		0			0
$683$	374.516			0.0209816			0.0104908	−	0.999945i	$-0.496661\pi$
$683$	374.516			0.0209816			0.0104908	+	0.999945i	$0.496661\pi$
$684$		0			0
$685$	−3919.68			−0.218632
$686$		0			0
$687$	−11537.1	−	305.992i	−0.640709	−	0.0169932i
$688$		0			0
$689$	−5487.63			−0.303428
$690$		0			0
$691$	−15693.7			−0.863988			−0.431994	−	0.901877i	$-0.642190\pi$
$691$	−15693.7			−0.863988			−0.431994	+	0.901877i	$0.642190\pi$
$692$		0			0
$693$	−1419.82	−	1436.92i	−0.0778275	−	0.0787649i
$694$		0			0
$695$		−	7482.73i		−	0.408397i
$696$		0			0
$697$	26418.7			1.43569
$698$		0			0
$699$	511.322	−	19278.8i	0.0276681	−	1.04319i
$700$		0			0
$701$	4484.20			0.241606			0.120803	−	0.992676i	$-0.461453\pi$
$701$	4484.20			0.241606			0.120803	+	0.992676i	$0.461453\pi$
$702$		0			0
$703$		−	9307.52i		−	0.499346i
$704$		0			0
$705$	−298.254	+	11245.3i	−0.0159332	+	0.600741i
$706$		0			0
$707$	−11922.0	+	13100.0i	−0.634190	+	0.696856i
$708$		0			0
$709$			11896.5i			0.630156i	0.949066	+	0.315078i	$0.102031\pi$
$709$			11896.5i			0.630156i	−0.949066	+	0.315078i	$0.897969\pi$
$710$		0			0
$711$	−5699.90	−	302.564i	−0.300651	−	0.0159593i
$712$		0			0
$713$	11908.4			0.625489
$714$		0			0
$715$		−	321.851i		−	0.0168343i
$716$		0			0
$717$	−440.691	+	16615.7i	−0.0229538	+	0.865447i
$718$		0			0
$719$	13003.8			0.674494			0.337247	−	0.941416i	$-0.390504\pi$
$719$	13003.8			0.674494			0.337247	+	0.941416i	$0.390504\pi$
$720$		0			0
$721$	−5916.54	+	6501.17i	−0.305608	+	0.335806i
$722$		0			0
$723$	−797.048	+	30051.8i	−0.0409994	+	1.54583i
$724$		0			0
$725$	−10289.1			−0.527072
$726$		0			0
$727$		−	38446.5i		−	1.96135i	−0.195639	−	0.980676i	$-0.562678\pi$
$727$		−	38446.5i		−	1.96135i	0.195639	−	0.980676i	$-0.437322\pi$
$728$		0			0
$729$	19434.4	+	3118.33i	0.987371	+	0.158427i
$730$		0			0
$731$			37107.8i			1.87754i
$732$		0			0
$733$	17763.3			0.895092			0.447546	−	0.894261i	$-0.352298\pi$
$733$	17763.3			0.895092			0.447546	+	0.894261i	$0.352298\pi$
$734$		0			0
$735$	−12156.8	+	822.766i	−0.610084	+	0.0412901i
$736$		0			0
$737$		−	392.303i		−	0.0196074i
$738$		0			0
$739$			28211.2i			1.40428i	0.712037	+	0.702142i	$0.247773\pi$
$739$			28211.2i			1.40428i	−0.712037	+	0.702142i	$0.752227\pi$
$740$		0			0
$741$	−1757.35	−	46.6092i	−0.0871225	−	0.00231071i
$742$		0			0
$743$			17970.1i			0.887294i	0.896202	+	0.443647i	$0.146316\pi$
$743$			17970.1i			0.887294i	−0.896202	+	0.443647i	$0.853684\pi$
$744$		0			0
$745$			17596.1i			0.865330i
$746$		0			0
$747$	1198.52	−	22578.5i	0.0587037	−	1.10590i
$748$		0			0
$749$	6571.52	+	5980.56i	0.320585	+	0.291755i
$750$		0			0
$751$	−28710.6			−1.39503			−0.697513	−	0.716572i	$-0.745710\pi$
$751$	−28710.6			−1.39503			−0.697513	+	0.716572i	$0.745710\pi$
$752$		0			0
$753$	−288.811	+	10889.3i	−0.0139772	+	0.526994i
$754$		0			0
$755$		−	7651.98i		−	0.368853i
$756$		0			0
$757$		−	34560.4i		−	1.65934i	−0.558256	−	0.829669i	$-0.688529\pi$
$757$		−	34560.4i		−	1.65934i	0.558256	−	0.829669i	$-0.311471\pi$
$758$		0			0
$759$	58.1916	−	2194.04i	0.00278290	−	0.104926i
$760$		0			0
$761$	−2013.29			−0.0959023			−0.0479511	−	0.998850i	$-0.515269\pi$
$761$	−2013.29			−0.0959023			−0.0479511	+	0.998850i	$0.515269\pi$
$762$		0			0
$763$	15561.2	−	17098.9i	0.738339	−	0.811298i
$764$		0			0
$765$	1019.19	−	19200.1i	0.0481683	−	0.907427i
$766$		0			0
$767$		−	8419.29i		−	0.396353i
$768$		0			0
$769$			7245.88i			0.339783i	0.985463	+	0.169891i	$0.0543417\pi$
$769$			7245.88i			0.339783i	−0.985463	+	0.169891i	$0.945658\pi$
$770$		0			0
$771$	30674.0	+	813.551i	1.43281	+	0.0380017i
$772$		0			0
$773$			35166.4i			1.63629i	0.575015	+	0.818143i	$0.304996\pi$
$773$			35166.4i			1.63629i	−0.575015	+	0.818143i	$0.695004\pi$
$774$		0			0
$775$		−	8913.31i		−	0.413130i
$776$		0			0
$777$	21364.1	−	22259.8i	0.986398	−	1.02775i
$778$		0			0
$779$	7363.09			0.338652
$780$		0			0
$781$		−	3638.90i		−	0.166722i
$782$		0			0
$783$	18386.5	+	1465.72i	0.839182	+	0.0668972i
$784$		0			0
$785$			18935.3i			0.860931i
$786$		0			0
$787$	−30974.5			−1.40295			−0.701475	−	0.712694i	$-0.747475\pi$
$787$	−30974.5			−1.40295			−0.701475	+	0.712694i	$0.747475\pi$
$788$		0			0
$789$	−295.355	+	11136.0i	−0.0133269	+	0.502475i
$790$		0			0
$791$	−16827.8	+	18490.7i	−0.756421	+	0.831166i
$792$		0			0
$793$	1648.03			0.0737999
$794$		0			0
$795$	443.507	−	16721.9i	0.0197856	−	0.745994i
$796$		0			0
$797$		−	8981.62i		−	0.399178i	−0.979880	−	0.199589i	$-0.936039\pi$
$797$		−	8981.62i		−	0.399178i	0.979880	−	0.199589i	$-0.0639608\pi$
$798$		0			0
$799$	32985.2			1.46049
$800$		0			0
$801$	−23112.6	−	1226.87i	−1.01953	−	0.0541190i
$802$		0			0
$803$			2513.94i			0.110479i
$804$		0			0
$805$	−9791.12	−	8910.63i	−0.428685	−	0.390135i
$806$		0			0
$807$	861.035	−	32464.3i	0.0375587	−	1.41611i
$808$		0			0
$809$			29145.4i			1.26662i	0.773896	+	0.633312i	$0.218305\pi$
$809$			29145.4i			1.26662i	−0.773896	+	0.633312i	$0.781695\pi$
$810$		0			0
$811$	35758.2			1.54826			0.774132	−	0.633025i	$-0.218187\pi$
$811$	35758.2			1.54826			0.774132	+	0.633025i	$0.218187\pi$
$812$		0			0
$813$	−636.909	+	24013.9i	−0.0274752	+	1.03592i
$814$		0			0
$815$	8927.04			0.383682
$816$		0			0
$817$			10342.2i			0.442874i
$818$		0			0
$819$	−4095.87	−	4145.20i	−0.174751	−	0.176856i
$820$		0			0
$821$	29573.6			1.25716			0.628578	−	0.777746i	$-0.283637\pi$
$821$	29573.6			1.25716			0.628578	+	0.777746i	$0.283637\pi$
$822$		0			0
$823$	26488.9			1.12192			0.560962	−	0.827841i	$-0.310431\pi$
$823$	26488.9			1.12192			0.560962	+	0.827841i	$0.310431\pi$
$824$		0			0
$825$	−1642.21	−	43.5557i	−0.0693025	−	0.00183808i
$826$		0			0
$827$	−6964.53			−0.292842			−0.146421	−	0.989222i	$-0.546775\pi$
$827$	−6964.53			−0.292842			−0.146421	+	0.989222i	$0.546775\pi$
$828$		0			0
$829$	−11031.4			−0.462166			−0.231083	−	0.972934i	$-0.574227\pi$
$829$	−11031.4			−0.462166			−0.231083	+	0.972934i	$0.574227\pi$
$830$		0			0
$831$	38.9658	−	1469.16i	0.00162660	−	0.0613292i
$832$		0			0
$833$	3356.77	+	35569.9i	0.139622	+	1.47950i
$834$		0			0
$835$			8486.81i			0.351734i
$836$		0			0
$837$	−1269.73	+	15928.0i	−0.0524353	+	0.657767i
$838$		0			0
$839$	20561.9			0.846099			0.423049	−	0.906107i	$-0.360960\pi$
$839$	20561.9			0.846099			0.423049	+	0.906107i	$0.360960\pi$
$840$		0			0
$841$	−7104.47			−0.291298
$842$		0			0
$843$	−926.162	+	34919.9i	−0.0378395	+	1.42669i
$844$		0			0
$845$			14091.4i			0.573681i
$846$		0			0
$847$	18007.4	+	16388.1i	0.730511	+	0.664817i
$848$		0			0
$849$	−37441.6	−	993.046i	−1.51354	−	0.0401428i
$850$		0			0
$851$	33522.4			1.35033
$852$		0			0
$853$	9441.72			0.378990			0.189495	−	0.981882i	$-0.439315\pi$
$853$	9441.72			0.378990			0.189495	+	0.981882i	$0.439315\pi$
$854$		0			0
$855$	284.055	−	5351.22i	0.0113620	−	0.214044i
$856$		0			0
$857$	−17877.9			−0.712598			−0.356299	−	0.934372i	$-0.615961\pi$
$857$	−17877.9			−0.712598			−0.356299	+	0.934372i	$0.615961\pi$
$858$		0			0
$859$	43955.9			1.74593			0.872966	−	0.487781i	$-0.162193\pi$
$859$	43955.9			1.74593			0.872966	+	0.487781i	$0.162193\pi$
$860$		0			0
$861$	17609.5	+	16900.9i	0.697015	+	0.668968i
$862$		0			0
$863$			42634.4i			1.68168i	0.541282	+	0.840841i	$0.317939\pi$
$863$			42634.4i			1.68168i	−0.541282	+	0.840841i	$0.682061\pi$
$864$		0			0
$865$	−2839.88			−0.111629
$866$		0			0
$867$	−30838.4	−	817.912i	−1.20799	−	0.0320389i
$868$		0			0
$869$	−854.017			−0.0333378
$870$		0			0
$871$		−	1131.71i		−	0.0440258i
$872$		0			0
$873$	1625.33	−	30618.9i	0.0630114	−	1.18705i
$874$		0			0
$875$	−17322.1	+	19033.7i	−0.669249	+	0.735380i
$876$		0			0
$877$			22335.1i			0.859980i	0.902834	+	0.429990i	$0.141483\pi$
$877$			22335.1i			0.859980i	−0.902834	+	0.429990i	$0.858517\pi$
$878$		0			0
$879$	−437.483	+	16494.8i	−0.0167872	+	0.632940i
$880$		0			0
$881$	31160.2			1.19162			0.595809	−	0.803126i	$-0.296832\pi$
$881$	31160.2			1.19162			0.595809	+	0.803126i	$0.296832\pi$
$882$		0			0
$883$			29300.5i			1.11669i	0.829607	+	0.558347i	$0.188564\pi$
$883$			29300.5i			1.11669i	−0.829607	+	0.558347i	$0.811436\pi$
$884$		0			0
$885$	25655.3	+	680.442i	0.974454	+	0.0258450i
$886$		0			0
$887$	28062.3			1.06228			0.531139	−	0.847285i	$-0.321764\pi$
$887$	28062.3			1.06228			0.531139	+	0.847285i	$0.321764\pi$
$888$		0			0
$889$	−26898.5	−	24479.6i	−1.01479	−	0.923530i
$890$		0			0
$891$	2928.41	+	311.773i	0.110107	+	0.0117225i
$892$		0			0
$893$	9193.22			0.344501
$894$		0			0
$895$		−	800.971i		−	0.0299145i
$896$		0			0
$897$	167.870	−	6329.34i	0.00624863	−	0.235597i
$898$		0			0
$899$			14973.4i			0.555494i
$900$		0			0
$901$	−49049.3			−1.81362
$902$		0			0
$903$	−23739.0	+	24734.3i	−0.874846	+	0.911525i
$904$		0			0
$905$			9428.02i			0.346296i
$906$		0			0
$907$		−	10433.3i		−	0.381953i	−0.981595	−	0.190976i	$-0.938835\pi$
$907$		−	10433.3i		−	0.381953i	0.981595	−	0.190976i	$-0.0611654\pi$
$908$		0			0
$909$	1368.81	−	25786.6i	0.0499457	−	0.940910i
$910$		0			0
$911$			12356.5i			0.449384i	0.974430	+	0.224692i	$0.0721376\pi$
$911$			12356.5i			0.449384i	−0.974430	+	0.224692i	$0.927862\pi$
$912$		0			0
$913$		−	3382.95i		−	0.122628i
$914$		0			0
$915$	−133.193	+	5021.88i	−0.00481226	+	0.181441i
$916$		0			0
$917$	−22127.6	+	24314.1i	−0.796858	+	0.875598i
$918$		0			0
$919$	24591.0			0.882680			0.441340	−	0.897340i	$-0.354503\pi$
$919$	24591.0			0.882680			0.441340	+	0.897340i	$0.354503\pi$
$920$		0			0
$921$	19423.3	+	515.155i	0.694919	+	0.0184310i
$922$		0			0
$923$		−	10497.4i		−	0.374352i
$924$		0			0
$925$		−	25091.1i		−	0.891882i
$926$		0			0
$927$	679.303	−	12797.1i	0.0240682	−	0.453412i
$928$		0			0
$929$	−15512.5			−0.547844			−0.273922	−	0.961752i	$-0.588321\pi$
$929$	−15512.5			−0.547844			−0.273922	+	0.961752i	$0.588321\pi$
$930$		0			0
$931$	935.558	+	9913.60i	0.0329341	+	0.348985i
$932$		0			0
$933$	47361.1	+	1256.13i	1.66188	+	0.0440772i
$934$		0			0
$935$		−	2876.76i		−	0.100620i
$936$		0			0
$937$			31883.8i			1.11163i	0.831306	+	0.555816i	$0.187594\pi$
$937$			31883.8i			1.11163i	−0.831306	+	0.555816i	$0.812406\pi$
$938$		0			0
$939$	1254.27	−	47290.6i	0.0435904	−	1.64353i
$940$		0			0
$941$			26463.4i			0.916773i	0.888753	+	0.458386i	$0.151572\pi$
$941$			26463.4i			0.916773i	−0.888753	+	0.458386i	$0.848428\pi$
$942$		0			0
$943$			26519.3i			0.915786i
$944$		0			0
$945$	12962.3	−	12145.9i	0.446204	−	0.418102i
$946$		0			0
$947$	−6190.46			−0.212421			−0.106211	−	0.994344i	$-0.533872\pi$
$947$	−6190.46			−0.212421			−0.106211	+	0.994344i	$0.533872\pi$
$948$		0			0
$949$			7252.17i			0.248067i
$950$		0			0
$951$	−51336.4	−	1361.57i	−1.75047	−	0.0464269i
$952$		0			0
$953$			24095.6i			0.819027i	0.912304	+	0.409513i	$0.134301\pi$
$953$			24095.6i			0.819027i	−0.912304	+	0.409513i	$0.865699\pi$
$954$		0			0
$955$	15884.5			0.538232
$956$		0			0
$957$	2758.74	+	73.1686i	0.0931842	+	0.00247148i
$958$		0			0
$959$	−7146.95	+	7853.17i	−0.240654	+	0.264434i
$960$		0			0
$961$	16819.8			0.564593
$962$		0			0
$963$	−12935.6	−	686.653i	−0.432860	−	0.0229772i
$964$		0			0
$965$		−	8779.51i		−	0.292873i
$966$		0			0
$967$	−33985.7			−1.13020			−0.565102	−	0.825021i	$-0.691163\pi$
$967$	−33985.7			−1.13020			−0.565102	+	0.825021i	$0.691163\pi$
$968$		0			0
$969$	−15707.4	−	416.601i	−0.520739	−	0.0138113i
$970$		0			0
$971$			17226.4i			0.569331i	0.958627	+	0.284666i	$0.0918826\pi$
$971$			17226.4i			0.569331i	−0.958627	+	0.284666i	$0.908117\pi$
$972$		0			0
$973$	−14991.8	−	13643.6i	−0.493952	−	0.449532i
$974$		0			0
$975$	−4737.44	−	125.649i	−0.155610	−	0.00412716i
$976$		0			0
$977$		−	30772.6i		−	1.00768i	−0.863797	−	0.503840i	$-0.831920\pi$
$977$		−	30772.6i		−	1.00768i	0.863797	−	0.503840i	$-0.168080\pi$
$978$		0			0
$979$	−3462.96			−0.113051
$980$		0			0
$981$	−1786.65	+	33658.0i	−0.0581480	+	1.09543i
$982$		0			0
$983$	−22275.6			−0.722769			−0.361384	−	0.932417i	$-0.617696\pi$
$983$	−22275.6			−0.722769			−0.361384	+	0.932417i	$0.617696\pi$
$984$		0			0
$985$		−	33741.7i		−	1.09147i
$986$		0			0
$987$	21986.4	+	21101.7i	0.709053	+	0.680521i
$988$		0			0
$989$	−37249.0			−1.19762
$990$		0			0
$991$	−107.013			−0.00343025			−0.00171513	−	0.999999i	$-0.500546\pi$
$991$	−107.013			−0.00343025			−0.00171513	+	0.999999i	$0.500546\pi$
$992$		0			0
$993$	−62.3165	+	2349.57i	−0.00199150	+	0.0750870i
$994$		0			0
$995$	26148.6			0.833132
$996$		0			0
$997$	16525.7			0.524950			0.262475	−	0.964939i	$-0.415461\pi$
$997$	16525.7			0.524950			0.262475	+	0.964939i	$0.415461\pi$
$998$		0			0
$999$	−3574.32	+	44837.5i	−0.113200	+	1.42002i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.i.c.209.2		80
3.2	odd	2	inner	672.4.i.c.209.3		80
4.3	odd	2		168.4.i.c.125.34	yes	80
7.6	odd	2	inner	672.4.i.c.209.80		80
8.3	odd	2		168.4.i.c.125.45	yes	80
8.5	even	2	inner	672.4.i.c.209.79		80
12.11	even	2		168.4.i.c.125.48	yes	80
21.20	even	2	inner	672.4.i.c.209.77		80
24.5	odd	2	inner	672.4.i.c.209.78		80
24.11	even	2		168.4.i.c.125.35	yes	80
28.27	even	2		168.4.i.c.125.33	✓	80
56.13	odd	2	inner	672.4.i.c.209.1		80
56.27	even	2		168.4.i.c.125.46	yes	80
84.83	odd	2		168.4.i.c.125.47	yes	80
168.83	odd	2		168.4.i.c.125.36	yes	80
168.125	even	2	inner	672.4.i.c.209.4		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.i.c.125.33	✓	80	28.27	even	2
168.4.i.c.125.34	yes	80	4.3	odd	2
168.4.i.c.125.35	yes	80	24.11	even	2
168.4.i.c.125.36	yes	80	168.83	odd	2
168.4.i.c.125.45	yes	80	8.3	odd	2
168.4.i.c.125.46	yes	80	56.27	even	2
168.4.i.c.125.47	yes	80	84.83	odd	2
168.4.i.c.125.48	yes	80	12.11	even	2
672.4.i.c.209.1		80	56.13	odd	2	inner
672.4.i.c.209.2		80	1.1	even	1	trivial
672.4.i.c.209.3		80	3.2	odd	2	inner
672.4.i.c.209.4		80	168.125	even	2	inner
672.4.i.c.209.77		80	21.20	even	2	inner
672.4.i.c.209.78		80	24.5	odd	2	inner
672.4.i.c.209.79		80	8.5	even	2	inner
672.4.i.c.209.80		80	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.i (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.19433 - 0.137767i) q^{3} -6.83655i q^{5} +(-13.6972 - 12.4654i) q^{7} +(26.9620 + 1.43121i) q^{9} +4.03973 q^{11} +11.6537 q^{13} +(-0.941848 + 35.5113i) q^{15} +104.163 q^{17} +29.0310 q^{19} +(69.4304 + 66.6365i) q^{21} +104.560i q^{23} +78.2615 q^{25} +(-139.852 - 11.1486i) q^{27} -131.471 q^{29} -113.891i q^{31} +(-20.9837 - 0.556540i) q^{33} +(-85.2206 + 93.6416i) q^{35} -320.606i q^{37} +(-60.5334 - 1.60550i) q^{39} +253.628 q^{41} +356.247i q^{43} +(9.78453 - 184.327i) q^{45} +316.669 q^{47} +(32.2261 + 341.483i) q^{49} +(-541.057 - 14.3502i) q^{51} -470.890 q^{53} -27.6178i q^{55} +(-150.797 - 3.99951i) q^{57} -722.454i q^{59} +141.417 q^{61} +(-351.464 - 355.697i) q^{63} -79.6715i q^{65} -97.1111i q^{67} +(14.4048 - 543.117i) q^{69} -900.777i q^{71} +622.304i q^{73} +(-406.516 - 10.7818i) q^{75} +(-55.3330 - 50.3570i) q^{77} -211.405 q^{79} +(724.903 + 77.1766i) q^{81} -837.420i q^{83} -712.116i q^{85} +(682.901 + 18.1123i) q^{87} -857.227 q^{89} +(-159.624 - 145.269i) q^{91} +(-15.6904 + 591.588i) q^{93} -198.472i q^{95} -1135.63i q^{97} +(108.919 + 5.78170i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 64 q^{7} + 104 q^{9} + 8 q^{15} - 976 q^{25} - 568 q^{39} - 4048 q^{49} - 1448 q^{57} + 2152 q^{63} - 4992 q^{79} + 1568 q^{81}+O(q^{100}) \) 80 * q - 64 * q^7 + 104 * q^9 + 8 * q^15 - 976 * q^25 - 568 * q^39 - 4048 * q^49 - 1448 * q^57 + 2152 * q^63 - 4992 * q^79 + 1568 * q^81

Introduction

L-functions

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