LMFDB - Embedded newform 4600.2.e.v.4049.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 18x^{8} + 103x^{6} + 239x^{4} + 197x^{2} + 4 $ x^10 + 18x^8 + 103x^6 + 239x^4 + 197x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$0.144312i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.v.4049.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.08344i q^{3} -0.555022i q^{7} -6.50759 q^{9} +O(q^{10})$	$q-3.08344i q^{3} -0.555022i q^{7} -6.50759 q^{9} +4.65190 q^{11} -5.02256i q^{13} +1.32867i q^{17} +0.196402 q^{19} -1.71138 q^{21} -1.00000i q^{23} +10.8154i q^{27} +0.812298 q^{29} -2.11145 q^{31} -14.3439i q^{33} -5.64564i q^{37} -15.4868 q^{39} -4.89714 q^{41} -1.66507i q^{43} -9.89310i q^{47} +6.69195 q^{49} +4.09688 q^{51} -2.23261i q^{53} -0.605593i q^{57} -2.43488 q^{59} -5.71138 q^{61} +3.61185i q^{63} +6.91830i q^{67} -3.08344 q^{69} +0.0120411 q^{71} -15.2989i q^{73} -2.58191i q^{77} -10.6351 q^{79} +13.8260 q^{81} +5.64020i q^{83} -2.50467i q^{87} -9.06693 q^{89} -2.78763 q^{91} +6.51051i q^{93} -14.0720i q^{97} -30.2727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{9}+O(q^{10}) $ 10 * q - 8 * q^9	$ 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) $ 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.08344i	− 1.78022i	−0.455742	−	0.890112i	$-0.650626\pi$
$3$	− 3.08344i	− 1.78022i	0.455742	−	0.890112i	$-0.349374\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.555022i	− 0.209779i	−0.994484	−	0.104889i	$-0.966551\pi$
$7$	− 0.555022i	− 0.209779i	0.994484	−	0.104889i	$-0.0334488\pi$
$8$	0	0
$9$	−6.50759	−2.16920
$10$	0	0
$11$	4.65190	1.40260	0.701301	−	0.712866i	$-0.252603\pi$
$11$	4.65190	1.40260	0.701301	+	0.712866i	$0.252603\pi$
$12$	0	0
$13$	− 5.02256i	− 1.39301i	−0.717553	−	0.696504i	$-0.754738\pi$
$13$	− 5.02256i	− 1.39301i	0.717553	−	0.696504i	$-0.245262\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.32867i	0.322251i	0.986934	+	0.161125i	$0.0515123\pi$
$17$	1.32867i	0.322251i	−0.986934	+	0.161125i	$0.948488\pi$
$18$	0	0
$19$	0.196402	0.0450577	0.0225288	−	0.999746i	$-0.492828\pi$
$19$	0.196402	0.0450577	0.0225288	+	0.999746i	$0.492828\pi$
$20$	0	0
$21$	−1.71138	−0.373453
$22$	0	0
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	10.8154i	2.08143i
$28$	0	0
$29$	0.812298	0.150840	0.0754200	−	0.997152i	$-0.475970\pi$
$29$	0.812298	0.150840	0.0754200	+	0.997152i	$0.475970\pi$
$30$	0	0
$31$	−2.11145	−0.379227	−0.189613	−	0.981859i	$-0.560723\pi$
$31$	−2.11145	−0.379227	−0.189613	+	0.981859i	$0.560723\pi$
$32$	0	0
$33$	− 14.3439i	− 2.49694i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.64564i	− 0.928138i	−0.885799	−	0.464069i	$-0.846389\pi$
$37$	− 5.64564i	− 0.928138i	0.885799	−	0.464069i	$-0.153611\pi$
$38$	0	0
$39$	−15.4868	−2.47987
$40$	0	0
$41$	−4.89714	−0.764804	−0.382402	−	0.923996i	$-0.624903\pi$
$41$	−4.89714	−0.764804	−0.382402	+	0.923996i	$0.624903\pi$
$42$	0	0
$43$	− 1.66507i	− 0.253920i	−0.991908	−	0.126960i	$-0.959478\pi$
$43$	− 1.66507i	− 0.253920i	0.991908	−	0.126960i	$-0.0405220\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.89310i	− 1.44306i	−0.692385	−	0.721528i	$-0.743440\pi$
$47$	− 9.89310i	− 1.44306i	0.692385	−	0.721528i	$-0.256560\pi$
$48$	0	0
$49$	6.69195	0.955993
$50$	0	0
$51$	4.09688	0.573678
$52$	0	0
$53$	− 2.23261i	− 0.306673i	−0.988174	−	0.153336i	$-0.950998\pi$
$53$	− 2.23261i	− 0.306673i	0.988174	−	0.153336i	$-0.0490018\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.605593i	− 0.0802127i
$58$	0	0
$59$	−2.43488	−0.316994	−0.158497	−	0.987359i	$-0.550665\pi$
$59$	−2.43488	−0.316994	−0.158497	+	0.987359i	$0.550665\pi$
$60$	0	0
$61$	−5.71138	−0.731267	−0.365633	−	0.930759i	$-0.619148\pi$
$61$	−5.71138	−0.731267	−0.365633	+	0.930759i	$0.619148\pi$
$62$	0	0
$63$	3.61185i	0.455051i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.91830i	0.845205i	0.906315	+	0.422602i	$0.138883\pi$
$67$	6.91830i	0.845205i	−0.906315	+	0.422602i	$0.861117\pi$
$68$	0	0
$69$	−3.08344	−0.371202
$70$	0	0
$71$	0.0120411	0.00142902	0.000714510	−	1.00000i	$-0.499773\pi$
$71$	0.0120411	0.00142902	0.000714510		1.00000i	$0.499773\pi$
$72$	0	0
$73$	− 15.2989i	− 1.79061i	−0.445458	−	0.895303i	$-0.646959\pi$
$73$	− 15.2989i	− 1.79061i	0.445458	−	0.895303i	$-0.353041\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.58191i	− 0.294236i
$78$	0	0
$79$	−10.6351	−1.19654	−0.598272	−	0.801293i	$-0.704146\pi$
$79$	−10.6351	−1.19654	−0.598272	+	0.801293i	$0.704146\pi$
$80$	0	0
$81$	13.8260	1.53622
$82$	0	0
$83$	5.64020i	0.619092i	0.950884	+	0.309546i	$0.100177\pi$
$83$	5.64020i	0.619092i	−0.950884	+	0.309546i	$0.899823\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 2.50467i	− 0.268529i
$88$	0	0
$89$	−9.06693	−0.961093	−0.480547	−	0.876969i	$-0.659562\pi$
$89$	−9.06693	−0.961093	−0.480547	+	0.876969i	$0.659562\pi$
$90$	0	0
$91$	−2.78763	−0.292223
$92$	0	0
$93$	6.51051i	0.675108i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 14.0720i	− 1.42880i	−0.699739	−	0.714398i	$-0.746701\pi$
$97$	− 14.0720i	− 1.42880i	0.699739	−	0.714398i	$-0.253299\pi$
$98$	0	0
$99$	−30.2727	−3.04252
$100$	0	0
$101$	7.53015	0.749278	0.374639	−	0.927171i	$-0.377767\pi$
$101$	7.53015	0.749278	0.374639	+	0.927171i	$0.377767\pi$
$102$	0	0
$103$	14.4476i	1.42357i	0.702399	+	0.711784i	$0.252112\pi$
$103$	14.4476i	1.42357i	−0.702399	+	0.711784i	$0.747888\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.35862i	0.228016i	0.993480	+	0.114008i	$0.0363690\pi$
$107$	2.35862i	0.228016i	−0.993480	+	0.114008i	$0.963631\pi$
$108$	0	0
$109$	10.3578	0.992101	0.496050	−	0.868294i	$-0.334783\pi$
$109$	10.3578	0.992101	0.496050	+	0.868294i	$0.334783\pi$
$110$	0	0
$111$	−17.4080	−1.65229
$112$	0	0
$113$	19.2723i	1.81298i	0.422226	+	0.906491i	$0.361249\pi$
$113$	19.2723i	1.81298i	−0.422226	+	0.906491i	$0.638751\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	32.6848i	3.02171i
$118$	0	0
$119$	0.737442	0.0676012
$120$	0	0
$121$	10.6402	0.967291
$122$	0	0
$123$	15.1000i	1.36152i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 18.4934i	− 1.64102i	−0.571632	−	0.820510i	$-0.693689\pi$
$127$	− 18.4934i	− 1.64102i	0.571632	−	0.820510i	$-0.306311\pi$
$128$	0	0
$129$	−5.13413	−0.452035
$130$	0	0
$131$	−14.7727	−1.29070	−0.645351	−	0.763887i	$-0.723289\pi$
$131$	−14.7727	−1.29070	−0.645351	+	0.763887i	$0.723289\pi$
$132$	0	0
$133$	− 0.109007i	− 0.00945213i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.6173i	0.992533i	0.868170	+	0.496266i	$0.165296\pi$
$137$	11.6173i	0.992533i	−0.868170	+	0.496266i	$0.834704\pi$
$138$	0	0
$139$	−11.1389	−0.944787	−0.472393	−	0.881388i	$-0.656610\pi$
$139$	−11.1389	−0.944787	−0.472393	+	0.881388i	$0.656610\pi$
$140$	0	0
$141$	−30.5047	−2.56896
$142$	0	0
$143$	− 23.3645i	− 1.95384i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 20.6342i	− 1.70188i
$148$	0	0
$149$	12.4796	1.02237	0.511184	−	0.859472i	$-0.329207\pi$
$149$	12.4796	1.02237	0.511184	+	0.859472i	$0.329207\pi$
$150$	0	0
$151$	1.32673	0.107968	0.0539840	−	0.998542i	$-0.482808\pi$
$151$	1.32673	0.107968	0.0539840	+	0.998542i	$0.482808\pi$
$152$	0	0
$153$	− 8.64646i	− 0.699025i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	23.1754i	1.84960i	0.380458	+	0.924798i	$0.375766\pi$
$157$	23.1754i	1.84960i	−0.380458	+	0.924798i	$0.624234\pi$
$158$	0	0
$159$	−6.88412	−0.545946
$160$	0	0
$161$	−0.555022	−0.0437418
$162$	0	0
$163$	− 25.0682i	− 1.96349i	−0.190196	−	0.981746i	$-0.560912\pi$
$163$	− 25.0682i	− 1.96349i	0.190196	−	0.981746i	$-0.439088\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.39603i	− 0.340175i	−0.985429	−	0.170087i	$-0.945595\pi$
$167$	− 4.39603i	− 0.340175i	0.985429	−	0.170087i	$-0.0544050\pi$
$168$	0	0
$169$	−12.2261	−0.940473
$170$	0	0
$171$	−1.27810	−0.0977389
$172$	0	0
$173$	− 19.4211i	− 1.47656i	−0.674493	−	0.738281i	$-0.735638\pi$
$173$	− 19.4211i	− 1.47656i	0.674493	−	0.738281i	$-0.264362\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.50779i	0.564320i
$178$	0	0
$179$	−10.4169	−0.778592	−0.389296	−	0.921113i	$-0.627282\pi$
$179$	−10.4169	−0.778592	−0.389296	+	0.921113i	$0.627282\pi$
$180$	0	0
$181$	−9.13693	−0.679143	−0.339571	−	0.940580i	$-0.610282\pi$
$181$	−9.13693	−0.679143	−0.339571	+	0.940580i	$0.610282\pi$
$182$	0	0
$183$	17.6107i	1.30182i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.18086i	0.451989i
$188$	0	0
$189$	6.00280	0.436640
$190$	0	0
$191$	−1.49819	−0.108405	−0.0542026	−	0.998530i	$-0.517262\pi$
$191$	−1.49819	−0.108405	−0.0542026	+	0.998530i	$0.517262\pi$
$192$	0	0
$193$	− 14.4113i	− 1.03735i	−0.854972	−	0.518675i	$-0.826425\pi$
$193$	− 14.4113i	− 1.03735i	0.854972	−	0.518675i	$-0.173575\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.4686i	1.74331i	0.490116	+	0.871657i	$0.336954\pi$
$197$	24.4686i	1.74331i	−0.490116	+	0.871657i	$0.663046\pi$
$198$	0	0
$199$	1.61185	0.114261	0.0571307	−	0.998367i	$-0.481805\pi$
$199$	1.61185	0.114261	0.0571307	+	0.998367i	$0.481805\pi$
$200$	0	0
$201$	21.3321	1.50465
$202$	0	0
$203$	− 0.450843i	− 0.0316430i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.50759i	0.452309i
$208$	0	0
$209$	0.913642	0.0631979
$210$	0	0
$211$	6.25081	0.430324	0.215162	−	0.976578i	$-0.430972\pi$
$211$	6.25081	0.430324	0.215162	+	0.976578i	$0.430972\pi$
$212$	0	0
$213$	− 0.0371281i	− 0.00254398i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.17190i	0.0795536i
$218$	0	0
$219$	−47.1733	−3.18768
$220$	0	0
$221$	6.67334	0.448898
$222$	0	0
$223$	− 21.2953i	− 1.42604i	−0.701144	−	0.713019i	$-0.747327\pi$
$223$	− 21.2953i	− 1.42604i	0.701144	−	0.713019i	$-0.252673\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.5174i	0.698061i	0.937111	+	0.349031i	$0.113489\pi$
$227$	10.5174i	0.698061i	−0.937111	+	0.349031i	$0.886511\pi$
$228$	0	0
$229$	24.0063	1.58638	0.793190	−	0.608975i	$-0.208419\pi$
$229$	24.0063	1.58638	0.793190	+	0.608975i	$0.208419\pi$
$230$	0	0
$231$	−7.96115	−0.523805
$232$	0	0
$233$	11.3112i	0.741021i	0.928828	+	0.370510i	$0.120817\pi$
$233$	11.3112i	0.741021i	−0.928828	+	0.370510i	$0.879183\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	32.7927i	2.13012i
$238$	0	0
$239$	25.5592	1.65329	0.826644	−	0.562726i	$-0.190247\pi$
$239$	25.5592	1.65329	0.826644	+	0.562726i	$0.190247\pi$
$240$	0	0
$241$	−16.1617	−1.04106	−0.520532	−	0.853842i	$-0.674266\pi$
$241$	−16.1617	−1.04106	−0.520532	+	0.853842i	$0.674266\pi$
$242$	0	0
$243$	− 10.1852i	− 0.653380i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 0.986440i	− 0.0627657i
$248$	0	0
$249$	17.3912	1.10212
$250$	0	0
$251$	−13.3497	−0.842627	−0.421313	−	0.906915i	$-0.638431\pi$
$251$	−13.3497	−0.842627	−0.421313	+	0.906915i	$0.638431\pi$
$252$	0	0
$253$	− 4.65190i	− 0.292463i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.1834i	0.635222i	0.948221	+	0.317611i	$0.102881\pi$
$257$	10.1834i	0.635222i	−0.948221	+	0.317611i	$0.897119\pi$
$258$	0	0
$259$	−3.13345	−0.194703
$260$	0	0
$261$	−5.28610	−0.327201
$262$	0	0
$263$	23.3073i	1.43719i	0.695430	+	0.718594i	$0.255214\pi$
$263$	23.3073i	1.43719i	−0.695430	+	0.718594i	$0.744786\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	27.9573i	1.71096i
$268$	0	0
$269$	−9.80817	−0.598015	−0.299007	−	0.954251i	$-0.596656\pi$
$269$	−9.80817	−0.598015	−0.299007	+	0.954251i	$0.596656\pi$
$270$	0	0
$271$	−6.08332	−0.369535	−0.184768	−	0.982782i	$-0.559153\pi$
$271$	−6.08332	−0.369535	−0.184768	+	0.982782i	$0.559153\pi$
$272$	0	0
$273$	8.59549i	0.520223i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0.0445722i	0.00267809i	0.999999	+	0.00133904i	$0.000426231\pi$
$277$	0.0445722i	0.00267809i	−0.999999	+	0.00133904i	$0.999574\pi$
$278$	0	0
$279$	13.7404	0.822617
$280$	0	0
$281$	27.9996	1.67032	0.835158	−	0.550010i	$-0.185376\pi$
$281$	27.9996	1.67032	0.835158	+	0.550010i	$0.185376\pi$
$282$	0	0
$283$	12.1798i	0.724013i	0.932176	+	0.362006i	$0.117908\pi$
$283$	12.1798i	0.724013i	−0.932176	+	0.362006i	$0.882092\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.71802i	0.160440i
$288$	0	0
$289$	15.2346	0.896155
$290$	0	0
$291$	−43.3902	−2.54358
$292$	0	0
$293$	− 2.71056i	− 0.158352i	−0.996861	−	0.0791762i	$-0.974771\pi$
$293$	− 2.71056i	− 0.158352i	0.996861	−	0.0791762i	$-0.0252290\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	50.3124i	2.91942i
$298$	0	0
$299$	−5.02256	−0.290462
$300$	0	0
$301$	−0.924148	−0.0532670
$302$	0	0
$303$	− 23.2188i	− 1.33388i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.1631i	1.83564i	0.396994	+	0.917821i	$0.370053\pi$
$307$	32.1631i	1.83564i	−0.396994	+	0.917821i	$0.629947\pi$
$308$	0	0
$309$	44.5484	2.53427
$310$	0	0
$311$	−4.07951	−0.231328	−0.115664	−	0.993288i	$-0.536900\pi$
$311$	−4.07951	−0.231328	−0.115664	+	0.993288i	$0.536900\pi$
$312$	0	0
$313$	13.7806i	0.778925i	0.921042	+	0.389462i	$0.127339\pi$
$313$	13.7806i	0.778925i	−0.921042	+	0.389462i	$0.872661\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.4630i	− 0.587658i	−0.955858	−	0.293829i	$-0.905070\pi$
$317$	− 10.4630i	− 0.587658i	0.955858	−	0.293829i	$-0.0949297\pi$
$318$	0	0
$319$	3.77873	0.211568
$320$	0	0
$321$	7.27266	0.405920
$322$	0	0
$323$	0.260954i	0.0145199i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 31.9377i	− 1.76616i
$328$	0	0
$329$	−5.49088	−0.302722
$330$	0	0
$331$	−12.6451	−0.695038	−0.347519	−	0.937673i	$-0.612976\pi$
$331$	−12.6451	−0.695038	−0.347519	+	0.937673i	$0.612976\pi$
$332$	0	0
$333$	36.7395i	2.01331i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 24.5733i	− 1.33859i	−0.742997	−	0.669295i	$-0.766596\pi$
$337$	− 24.5733i	− 1.33859i	0.742997	−	0.669295i	$-0.233404\pi$
$338$	0	0
$339$	59.4248	3.22751
$340$	0	0
$341$	−9.82224	−0.531904
$342$	0	0
$343$	− 7.59933i	− 0.410325i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3.50569i	0.188195i	0.995563	+	0.0940977i	$0.0299966\pi$
$347$	3.50569i	0.188195i	−0.995563	+	0.0940977i	$0.970003\pi$
$348$	0	0
$349$	18.1085	0.969327	0.484664	−	0.874701i	$-0.338942\pi$
$349$	18.1085	0.969327	0.484664	+	0.874701i	$0.338942\pi$
$350$	0	0
$351$	54.3212	2.89945
$352$	0	0
$353$	− 20.1915i	− 1.07469i	−0.843364	−	0.537343i	$-0.819428\pi$
$353$	− 20.1915i	− 1.07469i	0.843364	−	0.537343i	$-0.180572\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 2.27386i	− 0.120345i
$358$	0	0
$359$	9.43409	0.497912	0.248956	−	0.968515i	$-0.419912\pi$
$359$	9.43409	0.497912	0.248956	+	0.968515i	$0.419912\pi$
$360$	0	0
$361$	−18.9614	−0.997970
$362$	0	0
$363$	− 32.8084i	− 1.72199i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.3412i	1.58380i	0.610652	+	0.791899i	$0.290908\pi$
$367$	30.3412i	1.58380i	−0.610652	+	0.791899i	$0.709092\pi$
$368$	0	0
$369$	31.8686	1.65901
$370$	0	0
$371$	−1.23915	−0.0643333
$372$	0	0
$373$	14.7893i	0.765758i	0.923798	+	0.382879i	$0.125067\pi$
$373$	14.7893i	0.765758i	−0.923798	+	0.382879i	$0.874933\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.07982i	− 0.210121i
$378$	0	0
$379$	11.2405	0.577385	0.288693	−	0.957422i	$-0.406779\pi$
$379$	11.2405	0.577385	0.288693	+	0.957422i	$0.406779\pi$
$380$	0	0
$381$	−57.0231	−2.92138
$382$	0	0
$383$	− 30.4602i	− 1.55644i	−0.627991	−	0.778221i	$-0.716122\pi$
$383$	− 30.4602i	− 1.55644i	0.627991	−	0.778221i	$-0.283878\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.8356i	0.550803i
$388$	0	0
$389$	−1.94011	−0.0983673	−0.0491836	−	0.998790i	$-0.515662\pi$
$389$	−1.94011	−0.0983673	−0.0491836	+	0.998790i	$0.515662\pi$
$390$	0	0
$391$	1.32867	0.0671939
$392$	0	0
$393$	45.5509i	2.29774i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 18.3206i	− 0.919487i	−0.888052	−	0.459743i	$-0.847941\pi$
$397$	− 18.3206i	− 0.919487i	0.888052	−	0.459743i	$-0.152059\pi$
$398$	0	0
$399$	−0.336117	−0.0168269
$400$	0	0
$401$	−1.32199	−0.0660170	−0.0330085	−	0.999455i	$-0.510509\pi$
$401$	−1.32199	−0.0660170	−0.0330085	+	0.999455i	$0.510509\pi$
$402$	0	0
$403$	10.6049i	0.528266i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 26.2630i	− 1.30181i
$408$	0	0
$409$	−21.9963	−1.08765	−0.543823	−	0.839200i	$-0.683024\pi$
$409$	−21.9963	−1.08765	−0.543823	+	0.839200i	$0.683024\pi$
$410$	0	0
$411$	35.8212	1.76693
$412$	0	0
$413$	1.35141i	0.0664985i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	34.3460i	1.68193i
$418$	0	0
$419$	−28.3194	−1.38349	−0.691746	−	0.722141i	$-0.743158\pi$
$419$	−28.3194	−1.38349	−0.691746	+	0.722141i	$0.743158\pi$
$420$	0	0
$421$	14.7650	0.719602	0.359801	−	0.933029i	$-0.382845\pi$
$421$	14.7650	0.719602	0.359801	+	0.933029i	$0.382845\pi$
$422$	0	0
$423$	64.3802i	3.13027i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.16994i	0.153404i
$428$	0	0
$429$	−72.0429	−3.47826
$430$	0	0
$431$	−27.0514	−1.30302	−0.651510	−	0.758640i	$-0.725864\pi$
$431$	−27.0514	−1.30302	−0.651510	+	0.758640i	$0.725864\pi$
$432$	0	0
$433$	38.2700i	1.83914i	0.392926	+	0.919570i	$0.371463\pi$
$433$	38.2700i	1.83914i	−0.392926	+	0.919570i	$0.628537\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.196402i	− 0.00939517i
$438$	0	0
$439$	2.39959	0.114526	0.0572630	−	0.998359i	$-0.481763\pi$
$439$	2.39959	0.114526	0.0572630	+	0.998359i	$0.481763\pi$
$440$	0	0
$441$	−43.5485	−2.07374
$442$	0	0
$443$	− 10.6542i	− 0.506198i	−0.967440	−	0.253099i	$-0.918550\pi$
$443$	− 10.6542i	− 0.506198i	0.967440	−	0.253099i	$-0.0814499\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 38.4800i	− 1.82004i
$448$	0	0
$449$	28.9533	1.36639	0.683195	−	0.730236i	$-0.260590\pi$
$449$	28.9533	1.36639	0.683195	+	0.730236i	$0.260590\pi$
$450$	0	0
$451$	−22.7810	−1.07272
$452$	0	0
$453$	− 4.09090i	− 0.192207i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 38.5855i	− 1.80496i	−0.430736	−	0.902478i	$-0.641746\pi$
$457$	− 38.5855i	− 1.80496i	0.430736	−	0.902478i	$-0.358254\pi$
$458$	0	0
$459$	−14.3702	−0.670743
$460$	0	0
$461$	−8.07659	−0.376164	−0.188082	−	0.982153i	$-0.560227\pi$
$461$	−8.07659	−0.376164	−0.188082	+	0.982153i	$0.560227\pi$
$462$	0	0
$463$	− 6.07771i	− 0.282455i	−0.989977	−	0.141228i	$-0.954895\pi$
$463$	− 6.07771i	− 0.282455i	0.989977	−	0.141228i	$-0.0451050\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 10.2961i	− 0.476446i	−0.971210	−	0.238223i	$-0.923435\pi$
$467$	− 10.2961i	− 0.476446i	0.971210	−	0.238223i	$-0.0765649\pi$
$468$	0	0
$469$	3.83981	0.177306
$470$	0	0
$471$	71.4598	3.29270
$472$	0	0
$473$	− 7.74572i	− 0.356149i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	14.5289i	0.665233i
$478$	0	0
$479$	33.9760	1.55240	0.776201	−	0.630486i	$-0.217144\pi$
$479$	33.9760	1.55240	0.776201	+	0.630486i	$0.217144\pi$
$480$	0	0
$481$	−28.3556	−1.29290
$482$	0	0
$483$	1.71138i	0.0778703i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.4951i	0.747464i	0.927537	+	0.373732i	$0.121922\pi$
$487$	16.4951i	0.747464i	−0.927537	+	0.373732i	$0.878078\pi$
$488$	0	0
$489$	−77.2962	−3.49546
$490$	0	0
$491$	27.4876	1.24050	0.620249	−	0.784405i	$-0.287032\pi$
$491$	27.4876	1.24050	0.620249	+	0.784405i	$0.287032\pi$
$492$	0	0
$493$	1.07928i	0.0486082i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.00668310i	0 0.000299778i
$498$	0	0
$499$	−18.2961	−0.819045	−0.409523	−	0.912300i	$-0.634305\pi$
$499$	−18.2961	−0.819045	−0.409523	+	0.912300i	$0.634305\pi$
$500$	0	0
$501$	−13.5549	−0.605587
$502$	0	0
$503$	1.97389i	0.0880115i	0.999031	+	0.0440058i	$0.0140120\pi$
$503$	1.97389i	0.0880115i	−0.999031	+	0.0440058i	$0.985988\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	37.6986i	1.67425i
$508$	0	0
$509$	38.3441	1.69957	0.849787	−	0.527126i	$-0.176731\pi$
$509$	38.3441	1.69957	0.849787	+	0.527126i	$0.176731\pi$
$510$	0	0
$511$	−8.49125	−0.375631
$512$	0	0
$513$	2.12417i	0.0937844i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 46.0217i	− 2.02403i
$518$	0	0
$519$	−59.8839	−2.62861
$520$	0	0
$521$	−23.1814	−1.01560	−0.507799	−	0.861476i	$-0.669541\pi$
$521$	−23.1814	−1.01560	−0.507799	+	0.861476i	$0.669541\pi$
$522$	0	0
$523$	43.5123i	1.90266i	0.308172	+	0.951331i	$0.400283\pi$
$523$	43.5123i	1.90266i	−0.308172	+	0.951331i	$0.599717\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.80542i	− 0.122206i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	15.8452	0.687622
$532$	0	0
$533$	24.5962i	1.06538i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	32.1197i	1.38607i
$538$	0	0
$539$	31.1303	1.34088
$540$	0	0
$541$	18.6112	0.800158	0.400079	−	0.916481i	$-0.368983\pi$
$541$	18.6112	0.800158	0.400079	+	0.916481i	$0.368983\pi$
$542$	0	0
$543$	28.1732i	1.20903i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.78956i	0.290300i	0.989410	+	0.145150i	$0.0463665\pi$
$547$	6.78956i	0.290300i	−0.989410	+	0.145150i	$0.953633\pi$
$548$	0	0
$549$	37.1673	1.58626
$550$	0	0
$551$	0.159537	0.00679649
$552$	0	0
$553$	5.90272i	0.251009i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.9869i	− 1.05873i	−0.848395	−	0.529364i	$-0.822430\pi$
$557$	− 24.9869i	− 1.05873i	0.848395	−	0.529364i	$-0.177570\pi$
$558$	0	0
$559$	−8.36290	−0.353713
$560$	0	0
$561$	19.0583	0.804642
$562$	0	0
$563$	− 44.7551i	− 1.88620i	−0.332508	−	0.943100i	$-0.607895\pi$
$563$	− 44.7551i	− 1.88620i	0.332508	−	0.943100i	$-0.392105\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 7.67371i	− 0.322265i
$568$	0	0
$569$	27.4895	1.15242	0.576211	−	0.817301i	$-0.304531\pi$
$569$	27.4895	1.15242	0.576211	+	0.817301i	$0.304531\pi$
$570$	0	0
$571$	−1.40495	−0.0587952	−0.0293976	−	0.999568i	$-0.509359\pi$
$571$	−1.40495	−0.0587952	−0.0293976	+	0.999568i	$0.509359\pi$
$572$	0	0
$573$	4.61957i	0.192985i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	19.1987i	0.799253i	0.916678	+	0.399627i	$0.130860\pi$
$577$	19.1987i	0.799253i	−0.916678	+	0.399627i	$0.869140\pi$
$578$	0	0
$579$	−44.4364	−1.84671
$580$	0	0
$581$	3.13043	0.129872
$582$	0	0
$583$	− 10.3859i	− 0.430139i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.8565i	0.943390i	0.881762	+	0.471695i	$0.156358\pi$
$587$	22.8565i	0.943390i	−0.881762	+	0.471695i	$0.843642\pi$
$588$	0	0
$589$	−0.414692	−0.0170871
$590$	0	0
$591$	75.4473	3.10349
$592$	0	0
$593$	− 26.5387i	− 1.08981i	−0.838497	−	0.544906i	$-0.816565\pi$
$593$	− 26.5387i	− 1.08981i	0.838497	−	0.544906i	$-0.183435\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 4.97005i	− 0.203411i
$598$	0	0
$599$	8.82768	0.360689	0.180345	−	0.983603i	$-0.442279\pi$
$599$	8.82768	0.360689	0.180345	+	0.983603i	$0.442279\pi$
$600$	0	0
$601$	−30.3072	−1.23626	−0.618129	−	0.786077i	$-0.712109\pi$
$601$	−30.3072	−1.23626	−0.618129	+	0.786077i	$0.712109\pi$
$602$	0	0
$603$	− 45.0215i	− 1.83342i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 38.0102i	− 1.54278i	−0.636360	−	0.771392i	$-0.719561\pi$
$607$	− 38.0102i	− 1.54278i	0.636360	−	0.771392i	$-0.280439\pi$
$608$	0	0
$609$	−1.39015	−0.0563316
$610$	0	0
$611$	−49.6887	−2.01019
$612$	0	0
$613$	− 31.7417i	− 1.28204i	−0.767526	−	0.641018i	$-0.778512\pi$
$613$	− 31.7417i	− 1.28204i	0.767526	−	0.641018i	$-0.221488\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 46.6699i	− 1.87886i	−0.342741	−	0.939430i	$-0.611355\pi$
$617$	− 46.6699i	− 1.87886i	0.342741	−	0.939430i	$-0.388645\pi$
$618$	0	0
$619$	−23.2223	−0.933383	−0.466692	−	0.884420i	$-0.654554\pi$
$619$	−23.2223	−0.933383	−0.466692	+	0.884420i	$0.654554\pi$
$620$	0	0
$621$	10.8154	0.434009
$622$	0	0
$623$	5.03235i	0.201617i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.81716i	− 0.112506i
$628$	0	0
$629$	7.50121	0.299093
$630$	0	0
$631$	−13.0906	−0.521129	−0.260565	−	0.965456i	$-0.583909\pi$
$631$	−13.0906	−0.521129	−0.260565	+	0.965456i	$0.583909\pi$
$632$	0	0
$633$	− 19.2740i	− 0.766072i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 33.6107i	− 1.33171i
$638$	0	0
$639$	−0.0783588	−0.00309983
$640$	0	0
$641$	22.8746	0.903494	0.451747	−	0.892146i	$-0.350801\pi$
$641$	22.8746	0.903494	0.451747	+	0.892146i	$0.350801\pi$
$642$	0	0
$643$	− 30.6333i	− 1.20806i	−0.796962	−	0.604030i	$-0.793561\pi$
$643$	− 30.6333i	− 1.20806i	0.796962	−	0.604030i	$-0.206439\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.74454i	0.343783i	0.985116	+	0.171892i	$0.0549879\pi$
$647$	8.74454i	0.343783i	−0.985116	+	0.171892i	$0.945012\pi$
$648$	0	0
$649$	−11.3268	−0.444616
$650$	0	0
$651$	3.61348	0.141623
$652$	0	0
$653$	− 24.7443i	− 0.968320i	−0.874979	−	0.484160i	$-0.839125\pi$
$653$	− 24.7443i	− 0.968320i	0.874979	−	0.484160i	$-0.160875\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	99.5593i	3.88418i
$658$	0	0
$659$	−8.30726	−0.323605	−0.161803	−	0.986823i	$-0.551731\pi$
$659$	−8.30726	−0.323605	−0.161803	+	0.986823i	$0.551731\pi$
$660$	0	0
$661$	−14.1755	−0.551364	−0.275682	−	0.961249i	$-0.588904\pi$
$661$	−14.1755	−0.551364	−0.275682	+	0.961249i	$0.588904\pi$
$662$	0	0
$663$	− 20.5768i	− 0.799138i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 0.812298i	− 0.0314523i
$668$	0	0
$669$	−65.6627	−2.53867
$670$	0	0
$671$	−26.5688	−1.02568
$672$	0	0
$673$	7.30229i	0.281482i	0.990046	+	0.140741i	$0.0449486\pi$
$673$	7.30229i	0.281482i	−0.990046	+	0.140741i	$0.955051\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 39.7090i	− 1.52614i	−0.646315	−	0.763071i	$-0.723691\pi$
$677$	− 39.7090i	− 1.52614i	0.646315	−	0.763071i	$-0.276309\pi$
$678$	0	0
$679$	−7.81027	−0.299731
$680$	0	0
$681$	32.4296	1.24271
$682$	0	0
$683$	− 38.5993i	− 1.47696i	−0.674274	−	0.738481i	$-0.735543\pi$
$683$	− 38.5993i	− 1.47696i	0.674274	−	0.738481i	$-0.264457\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 74.0219i	− 2.82411i
$688$	0	0
$689$	−11.2134	−0.427198
$690$	0	0
$691$	−21.4163	−0.814713	−0.407357	−	0.913269i	$-0.633549\pi$
$691$	−21.4163	−0.814713	−0.407357	+	0.913269i	$0.633549\pi$
$692$	0	0
$693$	16.8020i	0.638255i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 6.50669i	− 0.246459i
$698$	0	0
$699$	34.8773	1.31918
$700$	0	0
$701$	−5.40147	−0.204011	−0.102005	−	0.994784i	$-0.532526\pi$
$701$	−5.40147	−0.204011	−0.102005	+	0.994784i	$0.532526\pi$
$702$	0	0
$703$	− 1.10881i	− 0.0418197i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.17940i	− 0.157182i
$708$	0	0
$709$	15.3161	0.575208	0.287604	−	0.957749i	$-0.407141\pi$
$709$	15.3161	0.575208	0.287604	+	0.957749i	$0.407141\pi$
$710$	0	0
$711$	69.2090	2.59554
$712$	0	0
$713$	2.11145i	0.0790742i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 78.8102i	− 2.94322i
$718$	0	0
$719$	−11.6505	−0.434491	−0.217245	−	0.976117i	$-0.569707\pi$
$719$	−11.6505	−0.434491	−0.217245	+	0.976117i	$0.569707\pi$
$720$	0	0
$721$	8.01875	0.298634
$722$	0	0
$723$	49.8334i	1.85333i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	28.8188i	1.06883i	0.845223	+	0.534415i	$0.179468\pi$
$727$	28.8188i	1.06883i	−0.845223	+	0.534415i	$0.820532\pi$
$728$	0	0
$729$	10.0725	0.373056
$730$	0	0
$731$	2.21233	0.0818259
$732$	0	0
$733$	25.2304i	0.931906i	0.884810	+	0.465953i	$0.154288\pi$
$733$	25.2304i	0.931906i	−0.884810	+	0.465953i	$0.845712\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32.1833i	1.18549i
$738$	0	0
$739$	1.22800	0.0451728	0.0225864	−	0.999745i	$-0.492810\pi$
$739$	1.22800	0.0451728	0.0225864	+	0.999745i	$0.492810\pi$
$740$	0	0
$741$	−3.04163	−0.111737
$742$	0	0
$743$	− 35.1117i	− 1.28812i	−0.764974	−	0.644061i	$-0.777248\pi$
$743$	− 35.1117i	− 1.28812i	0.764974	−	0.644061i	$-0.222752\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 36.7041i	− 1.34293i
$748$	0	0
$749$	1.30909	0.0478329
$750$	0	0
$751$	31.7367	1.15809	0.579044	−	0.815296i	$-0.303426\pi$
$751$	31.7367	1.15809	0.579044	+	0.815296i	$0.303426\pi$
$752$	0	0
$753$	41.1630i	1.50006i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.22287i	− 0.117137i	−0.998283	−	0.0585685i	$-0.981346\pi$
$757$	− 3.22287i	− 0.117137i	0.998283	−	0.0585685i	$-0.0186536\pi$
$758$	0	0
$759$	−14.3439	−0.520649
$760$	0	0
$761$	−9.60077	−0.348027	−0.174014	−	0.984743i	$-0.555674\pi$
$761$	−9.60077	−0.348027	−0.174014	+	0.984743i	$0.555674\pi$
$762$	0	0
$763$	− 5.74882i	− 0.208121i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.2293i	0.441575i
$768$	0	0
$769$	−14.9778	−0.540113	−0.270056	−	0.962844i	$-0.587042\pi$
$769$	−14.9778	−0.540113	−0.270056	+	0.962844i	$0.587042\pi$
$770$	0	0
$771$	31.3998	1.13084
$772$	0	0
$773$	− 28.7430i	− 1.03381i	−0.856042	−	0.516906i	$-0.827084\pi$
$773$	− 28.7430i	− 1.03381i	0.856042	−	0.516906i	$-0.172916\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.66181i	0.346616i
$778$	0	0
$779$	−0.961806	−0.0344603
$780$	0	0
$781$	0.0560142	0.00200435
$782$	0	0
$783$	8.78536i	0.313963i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 34.1965i	− 1.21897i	−0.792797	−	0.609486i	$-0.791376\pi$
$787$	− 34.1965i	− 1.21897i	0.792797	−	0.609486i	$-0.208624\pi$
$788$	0	0
$789$	71.8666	2.55852
$790$	0	0
$791$	10.6965	0.380324
$792$	0	0
$793$	28.6857i	1.01866i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 0.303391i	− 0.0107467i	−0.999986	−	0.00537334i	$-0.998290\pi$
$797$	− 0.303391i	− 0.0107467i	0.999986	−	0.00537334i	$-0.00171039\pi$
$798$	0	0
$799$	13.1447	0.465026
$800$	0	0
$801$	59.0039	2.08480
$802$	0	0
$803$	− 71.1692i	− 2.51151i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.2429i	1.06460i
$808$	0	0
$809$	13.1865	0.463611	0.231806	−	0.972762i	$-0.425537\pi$
$809$	13.1865	0.463611	0.231806	+	0.972762i	$0.425537\pi$
$810$	0	0
$811$	14.7284	0.517185	0.258593	−	0.965986i	$-0.416741\pi$
$811$	14.7284	0.517185	0.258593	+	0.965986i	$0.416741\pi$
$812$	0	0
$813$	18.7575i	0.657856i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 0.327022i	− 0.0114410i
$818$	0	0
$819$	18.1408	0.633890
$820$	0	0
$821$	36.6177	1.27797	0.638984	−	0.769220i	$-0.279355\pi$
$821$	36.6177	1.27797	0.638984	+	0.769220i	$0.279355\pi$
$822$	0	0
$823$	− 0.420867i	− 0.0146705i	−0.999973	−	0.00733525i	$-0.997665\pi$
$823$	− 0.420867i	− 0.0146705i	0.999973	−	0.00733525i	$-0.00233490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.1118i	1.22096i	0.792032	+	0.610479i	$0.209023\pi$
$827$	35.1118i	1.22096i	−0.792032	+	0.610479i	$0.790977\pi$
$828$	0	0
$829$	35.6234	1.23725	0.618626	−	0.785685i	$-0.287689\pi$
$829$	35.6234	1.23725	0.618626	+	0.785685i	$0.287689\pi$
$830$	0	0
$831$	0.137436	0.00476759
$832$	0	0
$833$	8.89141i	0.308069i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 22.8362i	− 0.789335i
$838$	0	0
$839$	19.9998	0.690469	0.345235	−	0.938516i	$-0.387799\pi$
$839$	19.9998	0.690469	0.345235	+	0.938516i	$0.387799\pi$
$840$	0	0
$841$	−28.3402	−0.977247
$842$	0	0
$843$	− 86.3350i	− 2.97354i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 5.90554i	− 0.202917i
$848$	0	0
$849$	37.5556	1.28890
$850$	0	0
$851$	−5.64564	−0.193530
$852$	0	0
$853$	− 42.4994i	− 1.45515i	−0.686026	−	0.727577i	$-0.740647\pi$
$853$	− 42.4994i	− 1.45515i	0.686026	−	0.727577i	$-0.259353\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 18.0100i	− 0.615208i	−0.951514	−	0.307604i	$-0.900473\pi$
$857$	− 18.0100i	− 0.615208i	0.951514	−	0.307604i	$-0.0995273\pi$
$858$	0	0
$859$	−22.3869	−0.763832	−0.381916	−	0.924197i	$-0.624736\pi$
$859$	−22.3869	−0.763832	−0.381916	+	0.924197i	$0.624736\pi$
$860$	0	0
$861$	8.38084	0.285618
$862$	0	0
$863$	− 18.7613i	− 0.638642i	−0.947647	−	0.319321i	$-0.896545\pi$
$863$	− 18.7613i	− 0.638642i	0.947647	−	0.319321i	$-0.103455\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 46.9750i	− 1.59536i
$868$	0	0
$869$	−49.4735	−1.67827
$870$	0	0
$871$	34.7476	1.17738
$872$	0	0
$873$	91.5749i	3.09934i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 31.6345i	− 1.06822i	−0.845415	−	0.534111i	$-0.820647\pi$
$877$	− 31.6345i	− 1.06822i	0.845415	−	0.534111i	$-0.179353\pi$
$878$	0	0
$879$	−8.35783	−0.281903
$880$	0	0
$881$	39.6850	1.33702	0.668512	−	0.743702i	$-0.266932\pi$
$881$	39.6850	1.33702	0.668512	+	0.743702i	$0.266932\pi$
$882$	0	0
$883$	10.3778i	0.349242i	0.984636	+	0.174621i	$0.0558701\pi$
$883$	10.3778i	0.349242i	−0.984636	+	0.174621i	$0.944130\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.0199i	0.806508i	0.915088	+	0.403254i	$0.132121\pi$
$887$	24.0199i	0.806508i	−0.915088	+	0.403254i	$0.867879\pi$
$888$	0	0
$889$	−10.2642	−0.344251
$890$	0	0
$891$	64.3170	2.15470
$892$	0	0
$893$	− 1.94302i	− 0.0650207i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	15.4868i	0.517088i
$898$	0	0
$899$	−1.71512	−0.0572025
$900$	0	0
$901$	2.96641	0.0988254
$902$	0	0
$903$	2.84955i	0.0948271i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 18.5181i	− 0.614885i	−0.951567	−	0.307442i	$-0.900527\pi$
$907$	− 18.5181i	− 0.614885i	0.951567	−	0.307442i	$-0.0994731\pi$
$908$	0	0
$909$	−49.0032	−1.62533
$910$	0	0
$911$	−3.34956	−0.110976	−0.0554879	−	0.998459i	$-0.517671\pi$
$911$	−3.34956	−0.110976	−0.0554879	+	0.998459i	$0.517671\pi$
$912$	0	0
$913$	26.2376i	0.868339i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.19920i	0.270761i
$918$	0	0
$919$	32.9754	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$919$	32.9754	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$920$	0	0
$921$	99.1728	3.26785
$922$	0	0
$923$	− 0.0604774i	− 0.00199064i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 94.0193i	− 3.08800i
$928$	0	0
$929$	36.5008	1.19755	0.598777	−	0.800916i	$-0.295654\pi$
$929$	36.5008	1.19755	0.598777	+	0.800916i	$0.295654\pi$
$930$	0	0
$931$	1.31431	0.0430748
$932$	0	0
$933$	12.5789i	0.411816i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 12.4598i	− 0.407043i	−0.979071	−	0.203521i	$-0.934761\pi$
$937$	− 12.4598i	− 0.407043i	0.979071	−	0.203521i	$-0.0652386\pi$
$938$	0	0
$939$	42.4916	1.38666
$940$	0	0
$941$	−45.2938	−1.47653	−0.738267	−	0.674508i	$-0.764356\pi$
$941$	−45.2938	−1.47653	−0.738267	+	0.674508i	$0.764356\pi$
$942$	0	0
$943$	4.89714i	0.159473i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 41.4016i	− 1.34537i	−0.739929	−	0.672685i	$-0.765141\pi$
$947$	− 41.4016i	− 1.34537i	0.739929	−	0.672685i	$-0.234859\pi$
$948$	0	0
$949$	−76.8399	−2.49433
$950$	0	0
$951$	−32.2619	−1.04616
$952$	0	0
$953$	− 15.3738i	− 0.498006i	−0.968503	−	0.249003i	$-0.919897\pi$
$953$	− 15.3738i	− 0.498006i	0.968503	−	0.249003i	$-0.0801029\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 11.6515i	− 0.376639i
$958$	0	0
$959$	6.44785	0.208212
$960$	0	0
$961$	−26.5418	−0.856187
$962$	0	0
$963$	− 15.3489i	− 0.494612i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 53.4925i	− 1.72020i	−0.510124	−	0.860101i	$-0.670400\pi$
$967$	− 53.4925i	− 1.72020i	0.510124	−	0.860101i	$-0.329600\pi$
$968$	0	0
$969$	0.804635	0.0258486
$970$	0	0
$971$	19.8881	0.638241	0.319120	−	0.947714i	$-0.396613\pi$
$971$	19.8881	0.638241	0.319120	+	0.947714i	$0.396613\pi$
$972$	0	0
$973$	6.18231i	0.198196i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.8758i	0.475919i	0.971275	+	0.237960i	$0.0764786\pi$
$977$	14.8758i	0.475919i	−0.971275	+	0.237960i	$0.923521\pi$
$978$	0	0
$979$	−42.1785	−1.34803
$980$	0	0
$981$	−67.4045	−2.15206
$982$	0	0
$983$	− 1.16403i	− 0.0371269i	−0.999828	−	0.0185634i	$-0.994091\pi$
$983$	− 1.16403i	− 0.0371269i	0.999828	−	0.0185634i	$-0.00590926\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	16.9308i	0.538913i
$988$	0	0
$989$	−1.66507	−0.0529460
$990$	0	0
$991$	−47.8693	−1.52062	−0.760310	−	0.649561i	$-0.774953\pi$
$991$	−47.8693	−1.52062	−0.760310	+	0.649561i	$0.774953\pi$
$992$	0	0
$993$	38.9904i	1.23732i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	11.4353i	0.362159i	0.983468	+	0.181079i	$0.0579591\pi$
$997$	11.4353i	0.362159i	−0.983468	+	0.181079i	$0.942041\pi$
$998$	0	0
$999$	61.0601	1.93186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.v.4049.1		10
5.2	odd	4		4600.2.a.bc.1.1	✓	5
5.3	odd	4		4600.2.a.bg.1.5	yes	5
5.4	even	2	inner	4600.2.e.v.4049.10		10
20.3	even	4		9200.2.a.cs.1.1		5
20.7	even	4		9200.2.a.cw.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.bc.1.1	✓	5	5.2	odd	4
4600.2.a.bg.1.5	yes	5	5.3	odd	4
4600.2.e.v.4049.1		10	1.1	even	1	trivial
4600.2.e.v.4049.10		10	5.4	even	2	inner
9200.2.a.cs.1.1		5	20.3	even	4
9200.2.a.cw.1.5		5	20.7	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.08344i q^{3} -0.555022i q^{7} -6.50759 q^{9} +O(q^{10})\)	\(q-3.08344i q^{3} -0.555022i q^{7} -6.50759 q^{9} +4.65190 q^{11} -5.02256i q^{13} +1.32867i q^{17} +0.196402 q^{19} -1.71138 q^{21} -1.00000i q^{23} +10.8154i q^{27} +0.812298 q^{29} -2.11145 q^{31} -14.3439i q^{33} -5.64564i q^{37} -15.4868 q^{39} -4.89714 q^{41} -1.66507i q^{43} -9.89310i q^{47} +6.69195 q^{49} +4.09688 q^{51} -2.23261i q^{53} -0.605593i q^{57} -2.43488 q^{59} -5.71138 q^{61} +3.61185i q^{63} +6.91830i q^{67} -3.08344 q^{69} +0.0120411 q^{71} -15.2989i q^{73} -2.58191i q^{77} -10.6351 q^{79} +13.8260 q^{81} +5.64020i q^{83} -2.50467i q^{87} -9.06693 q^{89} -2.78763 q^{91} +6.51051i q^{93} -14.0720i q^{97} -30.2727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{9}+O(q^{10}) \) 10 * q - 8 * q^9	\( 10 q - 8 q^{9} - 8 q^{11} - 8 q^{19} - 12 q^{21} + 22 q^{29} + 8 q^{31} - 62 q^{39} - 16 q^{41} + 4 q^{49} - 10 q^{51} - 46 q^{59} - 52 q^{61} - 6 q^{69} - 4 q^{71} - 86 q^{79} - 6 q^{81} - 30 q^{89} - 38 q^{91} - 74 q^{99}+O(q^{100}) \) 10 * q - 8 * q^9 - 8 * q^11 - 8 * q^19 - 12 * q^21 + 22 * q^29 + 8 * q^31 - 62 * q^39 - 16 * q^41 + 4 * q^49 - 10 * q^51 - 46 * q^59 - 52 * q^61 - 6 * q^69 - 4 * q^71 - 86 * q^79 - 6 * q^81 - 30 * q^89 - 38 * q^91 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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