LMFDB - Embedded newform 702.2.b.g.649.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [702,2,Mod(649,702)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("702.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 702 = 2 \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	702.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,-4,0,0,-8,4,0,4,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$5.60549822189$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	702.649
Dual form			702.2.b.g.649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{4} +4.16228i q^{5} -4.16228i q^{7} -1.00000i q^{8} -4.16228 q^{10} +3.00000i q^{11} +(-0.418861 + 3.58114i) q^{13} +4.16228 q^{14} +1.00000 q^{16} -1.16228 q^{17} +6.00000i q^{19} -4.16228i q^{20} -3.00000 q^{22} -1.16228 q^{23} -12.3246 q^{25} +(-3.58114 - 0.418861i) q^{26} +4.16228i q^{28} -8.32456 q^{29} +1.83772i q^{31} +1.00000i q^{32} -1.16228i q^{34} +17.3246 q^{35} -8.32456i q^{37} -6.00000 q^{38} +4.16228 q^{40} -7.16228i q^{41} +3.16228 q^{43} -3.00000i q^{44} -1.16228i q^{46} +6.00000i q^{47} -10.3246 q^{49} -12.3246i q^{50} +(0.418861 - 3.58114i) q^{52} +6.48683 q^{53} -12.4868 q^{55} -4.16228 q^{56} -8.32456i q^{58} +8.32456i q^{59} -3.16228 q^{61} -1.83772 q^{62} -1.00000 q^{64} +(-14.9057 - 1.74342i) q^{65} -7.16228i q^{67} +1.16228 q^{68} +17.3246i q^{70} +9.48683i q^{71} +13.6491i q^{73} +8.32456 q^{74} -6.00000i q^{76} +12.4868 q^{77} +8.64911 q^{79} +4.16228i q^{80} +7.16228 q^{82} -5.32456i q^{83} -4.83772i q^{85} +3.16228i q^{86} +3.00000 q^{88} +9.48683i q^{89} +(14.9057 + 1.74342i) q^{91} +1.16228 q^{92} -6.00000 q^{94} -24.9737 q^{95} +3.00000i q^{97} -10.3246i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{10} - 8 q^{13} + 4 q^{14} + 4 q^{16} + 8 q^{17} - 12 q^{22} + 8 q^{23} - 24 q^{25} - 8 q^{26} - 8 q^{29} + 44 q^{35} - 24 q^{38} + 4 q^{40} - 16 q^{49} + 8 q^{52} - 12 q^{53} - 12 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^10 - 8 * q^13 + 4 * q^14 + 4 * q^16 + 8 * q^17 - 12 * q^22 + 8 * q^23 - 24 * q^25 - 8 * q^26 - 8 * q^29 + 44 * q^35 - 24 * q^38 + 4 * q^40 - 16 * q^49 + 8 * q^52 - 12 * q^53 - 12 * q^55 - 4 * q^56 - 20 * q^62 - 4 * q^64 - 28 * q^65 - 8 * q^68 + 8 * q^74 + 12 * q^77 - 16 * q^79 + 16 * q^82 + 12 * q^88 + 28 * q^91 - 8 * q^92 - 24 * q^94 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/702\mathbb{Z}\right)^\times$.

$n$	$379$	$677$
$\chi(n)$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$			4.16228i			1.86143i	0.365749	+	0.930714i	$0.380813\pi$
$5$			4.16228i			1.86143i	−0.365749	+	0.930714i	$0.619187\pi$
$6$		0			0
$7$		−	4.16228i		−	1.57319i	−0.617467	−	0.786597i	$-0.711841\pi$
$7$		−	4.16228i		−	1.57319i	0.617467	−	0.786597i	$-0.288159\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	−4.16228			−1.31623
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$	−0.418861	+	3.58114i	−0.116171	+	0.993229i
$13$	−0.418861	+	3.58114i	−0.116171	+	0.993229i
$14$	4.16228			1.11242
$15$		0			0
$16$	1.00000			0.250000
$17$	−1.16228			−0.281894			−0.140947	−	0.990017i	$-0.545015\pi$
$17$	−1.16228			−0.281894			−0.140947	+	0.990017i	$0.545015\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		−	4.16228i		−	0.930714i
$21$		0			0
$22$	−3.00000			−0.639602
$23$	−1.16228			−0.242352			−0.121176	−	0.992631i	$-0.538666\pi$
$23$	−1.16228			−0.242352			−0.121176	+	0.992631i	$0.538666\pi$
$24$		0			0
$25$	−12.3246			−2.46491
$26$	−3.58114	−	0.418861i	−0.702319	−	0.0821454i
$27$		0			0
$28$			4.16228i			0.786597i
$29$	−8.32456			−1.54583			−0.772916	−	0.634509i	$-0.781202\pi$
$29$	−8.32456			−1.54583			−0.772916	+	0.634509i	$0.781202\pi$
$30$		0			0
$31$			1.83772i			0.330065i	0.986288	+	0.165032i	$0.0527729\pi$
$31$			1.83772i			0.330065i	−0.986288	+	0.165032i	$0.947227\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	1.16228i		−	0.199329i
$35$	17.3246			2.92838
$36$		0			0
$37$		−	8.32456i		−	1.36855i	−0.729225	−	0.684274i	$-0.760119\pi$
$37$		−	8.32456i		−	1.36855i	0.729225	−	0.684274i	$-0.239881\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$	4.16228			0.658114
$41$		−	7.16228i		−	1.11856i	−0.828979	−	0.559280i	$-0.811078\pi$
$41$		−	7.16228i		−	1.11856i	0.828979	−	0.559280i	$-0.188922\pi$
$42$		0			0
$43$	3.16228			0.482243			0.241121	−	0.970495i	$-0.422485\pi$
$43$	3.16228			0.482243			0.241121	+	0.970495i	$0.422485\pi$
$44$		−	3.00000i		−	0.452267i
$45$		0			0
$46$		−	1.16228i		−	0.171368i
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	−10.3246			−1.47494
$50$		−	12.3246i		−	1.74296i
$51$		0			0
$52$	0.418861	−	3.58114i	0.0580856	−	0.496615i
$53$	6.48683			0.891035			0.445518	−	0.895273i	$-0.353020\pi$
$53$	6.48683			0.891035			0.445518	+	0.895273i	$0.353020\pi$
$54$		0			0
$55$	−12.4868			−1.68372
$56$	−4.16228			−0.556208
$57$		0			0
$58$		−	8.32456i		−	1.09307i
$59$			8.32456i			1.08376i	0.840454	+	0.541882i	$0.182288\pi$
$59$			8.32456i			1.08376i	−0.840454	+	0.541882i	$0.817712\pi$
$60$		0			0
$61$	−3.16228			−0.404888			−0.202444	−	0.979294i	$-0.564888\pi$
$61$	−3.16228			−0.404888			−0.202444	+	0.979294i	$0.564888\pi$
$62$	−1.83772			−0.233391
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−14.9057	−	1.74342i	−1.84882	−	0.216244i
$66$		0			0
$67$		−	7.16228i		−	0.875011i	−0.899216	−	0.437506i	$-0.855862\pi$
$67$		−	7.16228i		−	0.875011i	0.899216	−	0.437506i	$-0.144138\pi$
$68$	1.16228			0.140947
$69$		0			0
$70$			17.3246i			2.07068i
$71$			9.48683i			1.12588i	0.826498	+	0.562940i	$0.190330\pi$
$71$			9.48683i			1.12588i	−0.826498	+	0.562940i	$0.809670\pi$
$72$		0			0
$73$			13.6491i			1.59751i	0.601658	+	0.798754i	$0.294507\pi$
$73$			13.6491i			1.59751i	−0.601658	+	0.798754i	$0.705493\pi$
$74$	8.32456			0.967710
$75$		0			0
$76$		−	6.00000i		−	0.688247i
$77$	12.4868			1.42301
$78$		0			0
$79$	8.64911			0.973101			0.486550	−	0.873652i	$-0.338255\pi$
$79$	8.64911			0.973101			0.486550	+	0.873652i	$0.338255\pi$
$80$			4.16228i			0.465357i
$81$		0			0
$82$	7.16228			0.790941
$83$		−	5.32456i		−	0.584446i	−0.956350	−	0.292223i	$-0.905605\pi$
$83$		−	5.32456i		−	0.584446i	0.956350	−	0.292223i	$-0.0943949\pi$
$84$		0			0
$85$		−	4.83772i		−	0.524725i
$86$			3.16228i			0.340997i
$87$		0			0
$88$	3.00000			0.319801
$89$			9.48683i			1.00560i	0.864402	+	0.502801i	$0.167697\pi$
$89$			9.48683i			1.00560i	−0.864402	+	0.502801i	$0.832303\pi$
$90$		0			0
$91$	14.9057	+	1.74342i	1.56254	+	0.182760i
$92$	1.16228			0.121176
$93$		0			0
$94$	−6.00000			−0.618853
$95$	−24.9737			−2.56224
$96$		0			0
$97$			3.00000i			0.304604i	0.988334	+	0.152302i	$0.0486686\pi$
$97$			3.00000i			0.304604i	−0.988334	+	0.152302i	$0.951331\pi$
$98$		−	10.3246i		−	1.04294i
$99$		0			0
$100$	12.3246			1.23246
$101$	−1.83772			−0.182860			−0.0914301	−	0.995811i	$-0.529144\pi$
$101$	−1.83772			−0.182860			−0.0914301	+	0.995811i	$0.529144\pi$
$102$		0			0
$103$	14.0000			1.37946			0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000			1.37946			0.689730	+	0.724066i	$0.257729\pi$
$104$	3.58114	+	0.418861i	0.351160	+	0.0410727i
$105$		0			0
$106$			6.48683i			0.630057i
$107$	6.67544			0.645340			0.322670	−	0.946512i	$-0.395420\pi$
$107$	6.67544			0.645340			0.322670	+	0.946512i	$0.395420\pi$
$108$		0			0
$109$			10.8377i			1.03807i	0.854754	+	0.519033i	$0.173708\pi$
$109$			10.8377i			1.03807i	−0.854754	+	0.519033i	$0.826292\pi$
$110$		−	12.4868i		−	1.19057i
$111$		0			0
$112$		−	4.16228i		−	0.393298i
$113$	7.16228			0.673770			0.336885	−	0.941546i	$-0.390627\pi$
$113$	7.16228			0.673770			0.336885	+	0.941546i	$0.390627\pi$
$114$		0			0
$115$		−	4.83772i		−	0.451120i
$116$	8.32456			0.772916
$117$		0			0
$118$	−8.32456			−0.766337
$119$			4.83772i			0.443473i
$120$		0			0
$121$	2.00000			0.181818
$122$		−	3.16228i		−	0.286299i
$123$		0			0
$124$		−	1.83772i		−	0.165032i
$125$		−	30.4868i		−	2.72683i
$126$		0			0
$127$	−6.16228			−0.546814			−0.273407	−	0.961898i	$-0.588151\pi$
$127$	−6.16228			−0.546814			−0.273407	+	0.961898i	$0.588151\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$	1.74342	−	14.9057i	0.152908	−	1.30732i
$131$	5.32456			0.465209			0.232604	−	0.972571i	$-0.425275\pi$
$131$	5.32456			0.465209			0.232604	+	0.972571i	$0.425275\pi$
$132$		0			0
$133$	24.9737			2.16549
$134$	7.16228			0.618727
$135$		0			0
$136$			1.16228i			0.0996645i
$137$			8.32456i			0.711215i	0.934635	+	0.355607i	$0.115726\pi$
$137$			8.32456i			0.711215i	−0.934635	+	0.355607i	$0.884274\pi$
$138$		0			0
$139$	9.16228			0.777134			0.388567	−	0.921420i	$-0.372970\pi$
$139$	9.16228			0.777134			0.388567	+	0.921420i	$0.372970\pi$
$140$	−17.3246			−1.46419
$141$		0			0
$142$	−9.48683			−0.796117
$143$	−10.7434	−	1.25658i	−0.898410	−	0.105081i
$144$		0			0
$145$		−	34.6491i		−	2.87745i
$146$	−13.6491			−1.12961
$147$		0			0
$148$			8.32456i			0.684274i
$149$		−	14.8114i		−	1.21340i	−0.794932	−	0.606698i	$-0.792494\pi$
$149$		−	14.8114i		−	1.21340i	0.794932	−	0.606698i	$-0.207506\pi$
$150$		0			0
$151$		−	10.1623i		−	0.826994i	−0.910505	−	0.413497i	$-0.864307\pi$
$151$		−	10.1623i		−	0.826994i	0.910505	−	0.413497i	$-0.135693\pi$
$152$	6.00000			0.486664
$153$		0			0
$154$			12.4868i			1.00622i
$155$	−7.64911			−0.614391
$156$		0			0
$157$	5.67544			0.452950			0.226475	−	0.974017i	$-0.427280\pi$
$157$	5.67544			0.452950			0.226475	+	0.974017i	$0.427280\pi$
$158$			8.64911i			0.688086i
$159$		0			0
$160$	−4.16228			−0.329057
$161$			4.83772i			0.381266i
$162$		0			0
$163$		−	9.67544i		−	0.757839i	−0.925430	−	0.378920i	$-0.876296\pi$
$163$		−	9.67544i		−	0.757839i	0.925430	−	0.378920i	$-0.123704\pi$
$164$			7.16228i			0.559280i
$165$		0			0
$166$	5.32456			0.413266
$167$			15.4868i			1.19841i	0.800597	+	0.599204i	$0.204516\pi$
$167$			15.4868i			1.19841i	−0.800597	+	0.599204i	$0.795484\pi$
$168$		0			0
$169$	−12.6491	−	3.00000i	−0.973009	−	0.230769i
$170$	4.83772			0.371036
$171$		0			0
$172$	−3.16228			−0.241121
$173$	6.48683			0.493185			0.246592	−	0.969119i	$-0.420689\pi$
$173$	6.48683			0.493185			0.246592	+	0.969119i	$0.420689\pi$
$174$		0			0
$175$			51.2982i			3.87778i
$176$			3.00000i			0.226134i
$177$		0			0
$178$	−9.48683			−0.711068
$179$	−3.00000			−0.224231			−0.112115	−	0.993695i	$-0.535763\pi$
$179$	−3.00000			−0.224231			−0.112115	+	0.993695i	$0.535763\pi$
$180$		0			0
$181$	22.1359			1.64535			0.822676	−	0.568511i	$-0.192480\pi$
$181$	22.1359			1.64535			0.822676	+	0.568511i	$0.192480\pi$
$182$	−1.74342	+	14.9057i	−0.129231	+	1.10488i
$183$		0			0
$184$			1.16228i			0.0856842i
$185$	34.6491			2.54745
$186$		0			0
$187$		−	3.48683i		−	0.254982i
$188$		−	6.00000i		−	0.437595i
$189$		0			0
$190$		−	24.9737i		−	1.81178i
$191$	12.0000			0.868290			0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000			0.868290			0.434145	+	0.900843i	$0.357051\pi$
$192$		0			0
$193$			5.32456i			0.383270i	0.981466	+	0.191635i	$0.0613790\pi$
$193$			5.32456i			0.383270i	−0.981466	+	0.191635i	$0.938621\pi$
$194$	−3.00000			−0.215387
$195$		0			0
$196$	10.3246			0.737468
$197$			4.16228i			0.296550i	0.988946	+	0.148275i	$0.0473721\pi$
$197$			4.16228i			0.296550i	−0.988946	+	0.148275i	$0.952628\pi$
$198$		0			0
$199$	2.16228			0.153280			0.0766399	−	0.997059i	$-0.475581\pi$
$199$	2.16228			0.153280			0.0766399	+	0.997059i	$0.475581\pi$
$200$			12.3246i			0.871478i
$201$		0			0
$202$		−	1.83772i		−	0.129302i
$203$			34.6491i			2.43189i
$204$		0			0
$205$	29.8114			2.08212
$206$			14.0000i			0.975426i
$207$		0			0
$208$	−0.418861	+	3.58114i	−0.0290428	+	0.248307i
$209$	−18.0000			−1.24509
$210$		0			0
$211$	−4.32456			−0.297715			−0.148857	−	0.988859i	$-0.547560\pi$
$211$	−4.32456			−0.297715			−0.148857	+	0.988859i	$0.547560\pi$
$212$	−6.48683			−0.445518
$213$		0			0
$214$			6.67544i			0.456324i
$215$			13.1623i			0.897660i
$216$		0			0
$217$	7.64911			0.519255
$218$	−10.8377			−0.734023
$219$		0			0
$220$	12.4868			0.841862
$221$	0.486833	−	4.16228i	0.0327479	−	0.279985i
$222$		0			0
$223$		−	28.6491i		−	1.91849i	−0.282582	−	0.959243i	$-0.591191\pi$
$223$		−	28.6491i		−	1.91849i	0.282582	−	0.959243i	$-0.408809\pi$
$224$	4.16228			0.278104
$225$		0			0
$226$			7.16228i			0.476428i
$227$			22.6491i			1.50327i	0.659577	+	0.751637i	$0.270736\pi$
$227$			22.6491i			1.50327i	−0.659577	+	0.751637i	$0.729264\pi$
$228$		0			0
$229$			20.3246i			1.34308i	0.740966	+	0.671542i	$0.234368\pi$
$229$			20.3246i			1.34308i	−0.740966	+	0.671542i	$0.765632\pi$
$230$	4.83772			0.318990
$231$		0			0
$232$			8.32456i			0.546534i
$233$	6.00000			0.393073			0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000			0.393073			0.196537	+	0.980497i	$0.437031\pi$
$234$		0			0
$235$	−24.9737			−1.62910
$236$		−	8.32456i		−	0.541882i
$237$		0			0
$238$	−4.83772			−0.313583
$239$		−	22.8377i		−	1.47725i	−0.674117	−	0.738625i	$-0.735476\pi$
$239$		−	22.8377i		−	1.47725i	0.674117	−	0.738625i	$-0.264524\pi$
$240$		0			0
$241$		−	1.35089i		−	0.0870184i	−0.999053	−	0.0435092i	$-0.986146\pi$
$241$		−	1.35089i		−	0.0870184i	0.999053	−	0.0435092i	$-0.0138538\pi$
$242$			2.00000i			0.128565i
$243$		0			0
$244$	3.16228			0.202444
$245$		−	42.9737i		−	2.74549i
$246$		0			0
$247$	−21.4868	−	2.51317i	−1.36717	−	0.159909i
$248$	1.83772			0.116695
$249$		0			0
$250$	30.4868			1.92816
$251$	10.6491			0.672166			0.336083	−	0.941832i	$-0.390898\pi$
$251$	10.6491			0.672166			0.336083	+	0.941832i	$0.390898\pi$
$252$		0			0
$253$		−	3.48683i		−	0.219215i
$254$		−	6.16228i		−	0.386656i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−27.4868			−1.71458			−0.857291	−	0.514833i	$-0.827854\pi$
$257$	−27.4868			−1.71458			−0.857291	+	0.514833i	$0.827854\pi$
$258$		0			0
$259$	−34.6491			−2.15299
$260$	14.9057	+	1.74342i	0.924412	+	0.108122i
$261$		0			0
$262$			5.32456i			0.328952i
$263$	6.00000			0.369976			0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000			0.369976			0.184988	+	0.982741i	$0.440775\pi$
$264$		0			0
$265$			27.0000i			1.65860i
$266$			24.9737i			1.53123i
$267$		0			0
$268$			7.16228i			0.437506i
$269$	−26.3246			−1.60504			−0.802518	−	0.596628i	$-0.796507\pi$
$269$	−26.3246			−1.60504			−0.802518	+	0.596628i	$0.796507\pi$
$270$		0			0
$271$		−	7.83772i		−	0.476108i	−0.971252	−	0.238054i	$-0.923491\pi$
$271$		−	7.83772i		−	0.476108i	0.971252	−	0.238054i	$-0.0765095\pi$
$272$	−1.16228			−0.0704734
$273$		0			0
$274$	−8.32456			−0.502905
$275$		−	36.9737i		−	2.22960i
$276$		0			0
$277$	−3.81139			−0.229004			−0.114502	−	0.993423i	$-0.536527\pi$
$277$	−3.81139			−0.229004			−0.114502	+	0.993423i	$0.536527\pi$
$278$			9.16228i			0.549517i
$279$		0			0
$280$		−	17.3246i		−	1.03534i
$281$		−	17.8114i		−	1.06254i	−0.847203	−	0.531269i	$-0.821715\pi$
$281$		−	17.8114i		−	1.06254i	0.847203	−	0.531269i	$-0.178285\pi$
$282$		0			0
$283$	25.4868			1.51503			0.757517	−	0.652815i	$-0.226412\pi$
$283$	25.4868			1.51503			0.757517	+	0.652815i	$0.226412\pi$
$284$		−	9.48683i		−	0.562940i
$285$		0			0
$286$	1.25658	−	10.7434i	0.0743033	−	0.635272i
$287$	−29.8114			−1.75971
$288$		0			0
$289$	−15.6491			−0.920536
$290$	34.6491			2.03467
$291$		0			0
$292$		−	13.6491i		−	0.798754i
$293$		−	22.6491i		−	1.32318i	−0.749868	−	0.661588i	$-0.769883\pi$
$293$		−	22.6491i		−	1.32318i	0.749868	−	0.661588i	$-0.230117\pi$
$294$		0			0
$295$	−34.6491			−2.01735
$296$	−8.32456			−0.483855
$297$		0			0
$298$	14.8114			0.858001
$299$	0.486833	−	4.16228i	0.0281543	−	0.240711i
$300$		0			0
$301$		−	13.1623i		−	0.758661i
$302$	10.1623			0.584773
$303$		0			0
$304$			6.00000i			0.344124i
$305$		−	13.1623i		−	0.753670i
$306$		0			0
$307$			22.8377i			1.30342i	0.758469	+	0.651709i	$0.225948\pi$
$307$			22.8377i			1.30342i	−0.758469	+	0.651709i	$0.774052\pi$
$308$	−12.4868			−0.711503
$309$		0			0
$310$		−	7.64911i		−	0.434440i
$311$	13.1623			0.746364			0.373182	−	0.927758i	$-0.378267\pi$
$311$	13.1623			0.746364			0.373182	+	0.927758i	$0.378267\pi$
$312$		0			0
$313$	−17.0000			−0.960897			−0.480448	−	0.877023i	$-0.659526\pi$
$313$	−17.0000			−0.960897			−0.480448	+	0.877023i	$0.659526\pi$
$314$			5.67544i			0.320284i
$315$		0			0
$316$	−8.64911			−0.486550
$317$			0.486833i			0.0273433i	0.999907	+	0.0136716i	$0.00435195\pi$
$317$			0.486833i			0.0273433i	−0.999907	+	0.0136716i	$0.995648\pi$
$318$		0			0
$319$		−	24.9737i		−	1.39826i
$320$		−	4.16228i		−	0.232678i
$321$		0			0
$322$	−4.83772			−0.269596
$323$		−	6.97367i		−	0.388025i
$324$		0			0
$325$	5.16228	−	44.1359i	0.286352	−	2.44822i
$326$	9.67544			0.535873
$327$		0			0
$328$	−7.16228			−0.395471
$329$	24.9737			1.37684
$330$		0			0
$331$			19.1623i			1.05325i	0.850096	+	0.526627i	$0.176544\pi$
$331$			19.1623i			1.05325i	−0.850096	+	0.526627i	$0.823456\pi$
$332$			5.32456i			0.292223i
$333$		0			0
$334$	−15.4868			−0.847402
$335$	29.8114			1.62877
$336$		0			0
$337$	4.00000			0.217894			0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000			0.217894			0.108947	+	0.994048i	$0.465252\pi$
$338$	3.00000	−	12.6491i	0.163178	−	0.688021i
$339$		0			0
$340$			4.83772i			0.262362i
$341$	−5.51317			−0.298555
$342$		0			0
$343$			13.8377i			0.747167i
$344$		−	3.16228i		−	0.170499i
$345$		0			0
$346$			6.48683i			0.348734i
$347$	−9.97367			−0.535414			−0.267707	−	0.963500i	$-0.586266\pi$
$347$	−9.97367			−0.535414			−0.267707	+	0.963500i	$0.586266\pi$
$348$		0			0
$349$		−	15.2982i		−	0.818895i	−0.912334	−	0.409448i	$-0.865721\pi$
$349$		−	15.2982i		−	0.818895i	0.912334	−	0.409448i	$-0.134279\pi$
$350$	−51.2982			−2.74201
$351$		0			0
$352$	−3.00000			−0.159901
$353$			26.1359i			1.39108i	0.718489	+	0.695538i	$0.244834\pi$
$353$			26.1359i			1.39108i	−0.718489	+	0.695538i	$0.755166\pi$
$354$		0			0
$355$	−39.4868			−2.09574
$356$		−	9.48683i		−	0.502801i
$357$		0			0
$358$		−	3.00000i		−	0.158555i
$359$		−	15.6754i		−	0.827318i	−0.910432	−	0.413659i	$-0.864251\pi$
$359$		−	15.6754i		−	0.827318i	0.910432	−	0.413659i	$-0.135749\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$			22.1359i			1.16344i
$363$		0			0
$364$	−14.9057	−	1.74342i	−0.781271	−	0.0913799i
$365$	−56.8114			−2.97364
$366$		0			0
$367$	27.4605			1.43343			0.716713	−	0.697368i	$-0.245646\pi$
$367$	27.4605			1.43343			0.716713	+	0.697368i	$0.245646\pi$
$368$	−1.16228			−0.0605879
$369$		0			0
$370$			34.6491i			1.80132i
$371$		−	27.0000i		−	1.40177i
$372$		0			0
$373$	13.6754			0.708088			0.354044	−	0.935229i	$-0.384806\pi$
$373$	13.6754			0.708088			0.354044	+	0.935229i	$0.384806\pi$
$374$	3.48683			0.180300
$375$		0			0
$376$	6.00000			0.309426
$377$	3.48683	−	29.8114i	0.179581	−	1.53536i
$378$		0			0
$379$			26.3246i			1.35220i	0.736809	+	0.676101i	$0.236332\pi$
$379$			26.3246i			1.35220i	−0.736809	+	0.676101i	$0.763668\pi$
$380$	24.9737			1.28112
$381$		0			0
$382$			12.0000i			0.613973i
$383$		−	2.32456i		−	0.118779i	−0.998235	−	0.0593896i	$-0.981085\pi$
$383$		−	2.32456i		−	0.118779i	0.998235	−	0.0593896i	$-0.0189154\pi$
$384$		0			0
$385$			51.9737i			2.64882i
$386$	−5.32456			−0.271013
$387$		0			0
$388$		−	3.00000i		−	0.152302i
$389$	0.486833			0.0246834			0.0123417	−	0.999924i	$-0.496071\pi$
$389$	0.486833			0.0246834			0.0123417	+	0.999924i	$0.496071\pi$
$390$		0			0
$391$	1.35089			0.0683174
$392$			10.3246i			0.521469i
$393$		0			0
$394$	−4.16228			−0.209693
$395$			36.0000i			1.81136i
$396$		0			0
$397$			4.64911i			0.233332i	0.993171	+	0.116666i	$0.0372207\pi$
$397$			4.64911i			0.233332i	−0.993171	+	0.116666i	$0.962779\pi$
$398$			2.16228i			0.108385i
$399$		0			0
$400$	−12.3246			−0.616228
$401$			18.9737i			0.947500i	0.880659	+	0.473750i	$0.157100\pi$
$401$			18.9737i			0.947500i	−0.880659	+	0.473750i	$0.842900\pi$
$402$		0			0
$403$	−6.58114	−	0.769751i	−0.327830	−	0.0383440i
$404$	1.83772			0.0914301
$405$		0			0
$406$	−34.6491			−1.71961
$407$	24.9737			1.23790
$408$		0			0
$409$		−	32.6228i		−	1.61309i	−0.591171	−	0.806546i	$-0.701334\pi$
$409$		−	32.6228i		−	1.61309i	0.591171	−	0.806546i	$-0.298666\pi$
$410$			29.8114i			1.47228i
$411$		0			0
$412$	−14.0000			−0.689730
$413$	34.6491			1.70497
$414$		0			0
$415$	22.1623			1.08790
$416$	−3.58114	−	0.418861i	−0.175580	−	0.0205364i
$417$		0			0
$418$		−	18.0000i		−	0.880409i
$419$	−3.67544			−0.179557			−0.0897786	−	0.995962i	$-0.528616\pi$
$419$	−3.67544			−0.179557			−0.0897786	+	0.995962i	$0.528616\pi$
$420$		0			0
$421$			31.1623i			1.51876i	0.650649	+	0.759378i	$0.274497\pi$
$421$			31.1623i			1.51876i	−0.650649	+	0.759378i	$0.725503\pi$
$422$		−	4.32456i		−	0.210516i
$423$		0			0
$424$		−	6.48683i		−	0.315028i
$425$	14.3246			0.694843
$426$		0			0
$427$			13.1623i			0.636967i
$428$	−6.67544			−0.322670
$429$		0			0
$430$	−13.1623			−0.634741
$431$			13.3509i			0.643090i	0.946894	+	0.321545i	$0.104202\pi$
$431$			13.3509i			0.643090i	−0.946894	+	0.321545i	$0.895798\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$			7.64911i			0.367169i
$435$		0			0
$436$		−	10.8377i		−	0.519033i
$437$		−	6.97367i		−	0.333596i
$438$		0			0
$439$	−14.1623			−0.675929			−0.337964	−	0.941159i	$-0.609738\pi$
$439$	−14.1623			−0.675929			−0.337964	+	0.941159i	$0.609738\pi$
$440$			12.4868i			0.595286i
$441$		0			0
$442$	4.16228	+	0.486833i	0.197979	+	0.0231563i
$443$	−27.2982			−1.29698			−0.648489	−	0.761224i	$-0.724599\pi$
$443$	−27.2982			−1.29698			−0.648489	+	0.761224i	$0.724599\pi$
$444$		0			0
$445$	−39.4868			−1.87186
$446$	28.6491			1.35657
$447$		0			0
$448$			4.16228i			0.196649i
$449$		−	12.9737i		−	0.612265i	−0.951989	−	0.306133i	$-0.900965\pi$
$449$		−	12.9737i		−	0.612265i	0.951989	−	0.306133i	$-0.0990351\pi$
$450$		0			0
$451$	21.4868			1.01178
$452$	−7.16228			−0.336885
$453$		0			0
$454$	−22.6491			−1.06298
$455$	−7.25658	+	62.0416i	−0.340194	+	2.90856i
$456$		0			0
$457$		−	6.67544i		−	0.312264i	−0.987736	−	0.156132i	$-0.950097\pi$
$457$		−	6.67544i		−	0.312264i	0.987736	−	0.156132i	$-0.0499025\pi$
$458$	−20.3246			−0.949704
$459$		0			0
$460$			4.83772i			0.225560i
$461$		−	11.5132i		−	0.536222i	−0.963388	−	0.268111i	$-0.913601\pi$
$461$		−	11.5132i		−	0.536222i	0.963388	−	0.268111i	$-0.0863993\pi$
$462$		0			0
$463$		−	31.4605i		−	1.46209i	−0.682327	−	0.731047i	$-0.739032\pi$
$463$		−	31.4605i		−	1.46209i	0.682327	−	0.731047i	$-0.260968\pi$
$464$	−8.32456			−0.386458
$465$		0			0
$466$			6.00000i			0.277945i
$467$	15.0000			0.694117			0.347059	−	0.937843i	$-0.387180\pi$
$467$	15.0000			0.694117			0.347059	+	0.937843i	$0.387180\pi$
$468$		0			0
$469$	−29.8114			−1.37656
$470$		−	24.9737i		−	1.15195i
$471$		0			0
$472$	8.32456			0.383169
$473$			9.48683i			0.436205i
$474$		0			0
$475$		−	73.9473i		−	3.39294i
$476$		−	4.83772i		−	0.221737i
$477$		0			0
$478$	22.8377			1.04457
$479$			3.48683i			0.159317i	0.996822	+	0.0796587i	$0.0253831\pi$
$479$			3.48683i			0.159317i	−0.996822	+	0.0796587i	$0.974617\pi$
$480$		0			0
$481$	29.8114	+	3.48683i	1.35928	+	0.158986i
$482$	1.35089			0.0615313
$483$		0			0
$484$	−2.00000			−0.0909091
$485$	−12.4868			−0.566998
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$			3.16228i			0.143150i
$489$		0			0
$490$	42.9737			1.94135
$491$	18.6754			0.842811			0.421406	−	0.906872i	$-0.361537\pi$
$491$	18.6754			0.842811			0.421406	+	0.906872i	$0.361537\pi$
$492$		0			0
$493$	9.67544			0.435760
$494$	2.51317	−	21.4868i	0.113073	−	0.966738i
$495$		0			0
$496$			1.83772i			0.0825162i
$497$	39.4868			1.77123
$498$		0			0
$499$		−	8.32456i		−	0.372658i	−0.982487	−	0.186329i	$-0.940341\pi$
$499$		−	8.32456i		−	0.372658i	0.982487	−	0.186329i	$-0.0596591\pi$
$500$			30.4868i			1.36341i
$501$		0			0
$502$			10.6491i			0.475293i
$503$	−12.9737			−0.578467			−0.289234	−	0.957259i	$-0.593400\pi$
$503$	−12.9737			−0.578467			−0.289234	+	0.957259i	$0.593400\pi$
$504$		0			0
$505$		−	7.64911i		−	0.340381i
$506$	3.48683			0.155009
$507$		0			0
$508$	6.16228			0.273407
$509$		−	23.5132i		−	1.04220i	−0.853495	−	0.521101i	$-0.825521\pi$
$509$		−	23.5132i		−	1.04220i	0.853495	−	0.521101i	$-0.174479\pi$
$510$		0			0
$511$	56.8114			2.51319
$512$			1.00000i			0.0441942i
$513$		0			0
$514$		−	27.4868i		−	1.21239i
$515$			58.2719i			2.56777i
$516$		0			0
$517$	−18.0000			−0.791639
$518$		−	34.6491i		−	1.52239i
$519$		0			0
$520$	−1.74342	+	14.9057i	−0.0764539	+	0.653658i
$521$	−42.9737			−1.88271			−0.941355	−	0.337417i	$-0.890447\pi$
$521$	−42.9737			−1.88271			−0.941355	+	0.337417i	$0.890447\pi$
$522$		0			0
$523$	33.1623			1.45009			0.725043	−	0.688704i	$-0.241820\pi$
$523$	33.1623			1.45009			0.725043	+	0.688704i	$0.241820\pi$
$524$	−5.32456			−0.232604
$525$		0			0
$526$			6.00000i			0.261612i
$527$		−	2.13594i		−	0.0930432i
$528$		0			0
$529$	−21.6491			−0.941266
$530$	−27.0000			−1.17281
$531$		0			0
$532$	−24.9737			−1.08275
$533$	25.6491	+	3.00000i	1.11099	+	0.129944i
$534$		0			0
$535$			27.7851i			1.20125i
$536$	−7.16228			−0.309363
$537$		0			0
$538$		−	26.3246i		−	1.13493i
$539$		−	30.9737i		−	1.33413i
$540$		0			0
$541$			11.8114i			0.507811i	0.967229	+	0.253906i	$0.0817153\pi$
$541$			11.8114i			0.507811i	−0.967229	+	0.253906i	$0.918285\pi$
$542$	7.83772			0.336659
$543$		0			0
$544$		−	1.16228i		−	0.0498322i
$545$	−45.1096			−1.93228
$546$		0			0
$547$	31.6228			1.35209			0.676046	−	0.736859i	$-0.263692\pi$
$547$	31.6228			1.35209			0.676046	+	0.736859i	$0.263692\pi$
$548$		−	8.32456i		−	0.355607i
$549$		0			0
$550$	36.9737			1.57656
$551$		−	49.9473i		−	2.12783i
$552$		0			0
$553$		−	36.0000i		−	1.53088i
$554$		−	3.81139i		−	0.161930i
$555$		0			0
$556$	−9.16228			−0.388567
$557$			11.5132i			0.487829i	0.969797	+	0.243914i	$0.0784316\pi$
$557$			11.5132i			0.487829i	−0.969797	+	0.243914i	$0.921568\pi$
$558$		0			0
$559$	−1.32456	+	11.3246i	−0.0560227	+	0.478978i
$560$	17.3246			0.732096
$561$		0			0
$562$	17.8114			0.751328
$563$	1.64911			0.0695017			0.0347509	−	0.999396i	$-0.488936\pi$
$563$	1.64911			0.0695017			0.0347509	+	0.999396i	$0.488936\pi$
$564$		0			0
$565$			29.8114i			1.25417i
$566$			25.4868i			1.07129i
$567$		0			0
$568$	9.48683			0.398059
$569$	19.3509			0.811231			0.405616	−	0.914044i	$-0.367057\pi$
$569$	19.3509			0.811231			0.405616	+	0.914044i	$0.367057\pi$
$570$		0			0
$571$	−2.18861			−0.0915905			−0.0457953	−	0.998951i	$-0.514582\pi$
$571$	−2.18861			−0.0915905			−0.0457953	+	0.998951i	$0.514582\pi$
$572$	10.7434	+	1.25658i	0.449205	+	0.0525404i
$573$		0			0
$574$		−	29.8114i		−	1.24430i
$575$	14.3246			0.597375
$576$		0			0
$577$			35.6228i			1.48300i	0.670955	+	0.741498i	$0.265884\pi$
$577$			35.6228i			1.48300i	−0.670955	+	0.741498i	$0.734116\pi$
$578$		−	15.6491i		−	0.650917i
$579$		0			0
$580$			34.6491i			1.43873i
$581$	−22.1623			−0.919446
$582$		0			0
$583$			19.4605i			0.805972i
$584$	13.6491			0.564804
$585$		0			0
$586$	22.6491			0.935626
$587$			6.29822i			0.259955i	0.991517	+	0.129978i	$0.0414906\pi$
$587$			6.29822i			0.259955i	−0.991517	+	0.129978i	$0.958509\pi$
$588$		0			0
$589$	−11.0263			−0.454332
$590$		−	34.6491i		−	1.42648i
$591$		0			0
$592$		−	8.32456i		−	0.342137i
$593$			14.1359i			0.580494i	0.956952	+	0.290247i	$0.0937374\pi$
$593$			14.1359i			0.580494i	−0.956952	+	0.290247i	$0.906263\pi$
$594$		0			0
$595$	−20.1359			−0.825493
$596$			14.8114i			0.606698i
$597$		0			0
$598$	4.16228	+	0.486833i	0.170208	+	0.0199081i
$599$	27.4868			1.12308			0.561541	−	0.827449i	$-0.310209\pi$
$599$	27.4868			1.12308			0.561541	+	0.827449i	$0.310209\pi$
$600$		0			0
$601$	−30.9473			−1.26237			−0.631184	−	0.775633i	$-0.717431\pi$
$601$	−30.9473			−1.26237			−0.631184	+	0.775633i	$0.717431\pi$
$602$	13.1623			0.536454
$603$		0			0
$604$			10.1623i			0.413497i
$605$			8.32456i			0.338441i
$606$		0			0
$607$	5.67544			0.230359			0.115180	−	0.993345i	$-0.463256\pi$
$607$	5.67544			0.230359			0.115180	+	0.993345i	$0.463256\pi$
$608$	−6.00000			−0.243332
$609$		0			0
$610$	13.1623			0.532925
$611$	−21.4868	−	2.51317i	−0.869264	−	0.101672i
$612$		0			0
$613$			3.29822i			0.133214i	0.997779	+	0.0666070i	$0.0212174\pi$
$613$			3.29822i			0.133214i	−0.997779	+	0.0666070i	$0.978783\pi$
$614$	−22.8377			−0.921655
$615$		0			0
$616$		−	12.4868i		−	0.503109i
$617$			15.6754i			0.631070i	0.948914	+	0.315535i	$0.102184\pi$
$617$			15.6754i			0.631070i	−0.948914	+	0.315535i	$0.897816\pi$
$618$		0			0
$619$		−	35.6228i		−	1.43180i	−0.698203	−	0.715900i	$-0.746017\pi$
$619$		−	35.6228i		−	1.43180i	0.698203	−	0.715900i	$-0.253983\pi$
$620$	7.64911			0.307196
$621$		0			0
$622$			13.1623i			0.527759i
$623$	39.4868			1.58201
$624$		0			0
$625$	65.2719			2.61088
$626$		−	17.0000i		−	0.679457i
$627$		0			0
$628$	−5.67544			−0.226475
$629$			9.67544i			0.385785i
$630$		0			0
$631$		−	22.1623i		−	0.882266i	−0.897442	−	0.441133i	$-0.854577\pi$
$631$		−	22.1623i		−	0.882266i	0.897442	−	0.441133i	$-0.145423\pi$
$632$		−	8.64911i		−	0.344043i
$633$		0			0
$634$	−0.486833			−0.0193346
$635$		−	25.6491i		−	1.01785i
$636$		0			0
$637$	4.32456	−	36.9737i	0.171345	−	1.46495i
$638$	24.9737			0.988717
$639$		0			0
$640$	4.16228			0.164528
$641$	−24.9737			−0.986400			−0.493200	−	0.869916i	$-0.664173\pi$
$641$	−24.9737			−0.986400			−0.493200	+	0.869916i	$0.664173\pi$
$642$		0			0
$643$			46.4605i			1.83222i	0.400923	+	0.916112i	$0.368689\pi$
$643$			46.4605i			1.83222i	−0.400923	+	0.916112i	$0.631311\pi$
$644$		−	4.83772i		−	0.190633i
$645$		0			0
$646$	6.97367			0.274375
$647$	−26.1359			−1.02751			−0.513755	−	0.857937i	$-0.671746\pi$
$647$	−26.1359			−1.02751			−0.513755	+	0.857937i	$0.671746\pi$
$648$		0			0
$649$	−24.9737			−0.980302
$650$	44.1359	+	5.16228i	1.73115	+	0.202481i
$651$		0			0
$652$			9.67544i			0.378920i
$653$	−29.1359			−1.14018			−0.570089	−	0.821583i	$-0.693091\pi$
$653$	−29.1359			−1.14018			−0.570089	+	0.821583i	$0.693091\pi$
$654$		0			0
$655$			22.1623i			0.865952i
$656$		−	7.16228i		−	0.279640i
$657$		0			0
$658$			24.9737i			0.973575i
$659$	−35.3246			−1.37605			−0.688025	−	0.725687i	$-0.741522\pi$
$659$	−35.3246			−1.37605			−0.688025	+	0.725687i	$0.741522\pi$
$660$		0			0
$661$		−	10.8377i		−	0.421539i	−0.977536	−	0.210769i	$-0.932403\pi$
$661$		−	10.8377i		−	0.421539i	0.977536	−	0.210769i	$-0.0675969\pi$
$662$	−19.1623			−0.744763
$663$		0			0
$664$	−5.32456			−0.206633
$665$			103.947i			4.03090i
$666$		0			0
$667$	9.67544			0.374635
$668$		−	15.4868i		−	0.599204i
$669$		0			0
$670$			29.8114i			1.15171i
$671$		−	9.48683i		−	0.366235i
$672$		0			0
$673$	18.3509			0.707375			0.353687	−	0.935364i	$-0.384928\pi$
$673$	18.3509			0.707375			0.353687	+	0.935364i	$0.384928\pi$
$674$			4.00000i			0.154074i
$675$		0			0
$676$	12.6491	+	3.00000i	0.486504	+	0.115385i
$677$	31.9473			1.22784			0.613918	−	0.789370i	$-0.289593\pi$
$677$	31.9473			1.22784			0.613918	+	0.789370i	$0.289593\pi$
$678$		0			0
$679$	12.4868			0.479201
$680$	−4.83772			−0.185518
$681$		0			0
$682$		−	5.51317i		−	0.211110i
$683$			31.9473i			1.22243i	0.791464	+	0.611215i	$0.209319\pi$
$683$			31.9473i			1.22243i	−0.791464	+	0.611215i	$0.790681\pi$
$684$		0			0
$685$	−34.6491			−1.32387
$686$	−13.8377			−0.528327
$687$		0			0
$688$	3.16228			0.120561
$689$	−2.71708	+	23.2302i	−0.103513	+	0.885002i
$690$		0			0
$691$		−	10.8377i		−	0.412286i	−0.978522	−	0.206143i	$-0.933909\pi$
$691$		−	10.8377i		−	0.412286i	0.978522	−	0.206143i	$-0.0660913\pi$
$692$	−6.48683			−0.246592
$693$		0			0
$694$		−	9.97367i		−	0.378595i
$695$			38.1359i			1.44658i
$696$		0			0
$697$			8.32456i			0.315315i
$698$	15.2982			0.579046
$699$		0			0
$700$		−	51.2982i		−	1.93889i
$701$	−16.1623			−0.610441			−0.305220	−	0.952282i	$-0.598730\pi$
$701$	−16.1623			−0.610441			−0.305220	+	0.952282i	$0.598730\pi$
$702$		0			0
$703$	49.9473			1.88380
$704$		−	3.00000i		−	0.113067i
$705$		0			0
$706$	−26.1359			−0.983639
$707$			7.64911i			0.287674i
$708$		0			0
$709$			37.9473i			1.42514i	0.701600	+	0.712571i	$0.252469\pi$
$709$			37.9473i			1.42514i	−0.701600	+	0.712571i	$0.747531\pi$
$710$		−	39.4868i		−	1.48191i
$711$		0			0
$712$	9.48683			0.355534
$713$		−	2.13594i		−	0.0799917i
$714$		0			0
$715$	5.23025	−	44.7171i	0.195600	−	1.67232i
$716$	3.00000			0.112115
$717$		0			0
$718$	15.6754			0.585002
$719$	−40.8377			−1.52299			−0.761495	−	0.648171i	$-0.775534\pi$
$719$	−40.8377			−1.52299			−0.761495	+	0.648171i	$0.775534\pi$
$720$		0			0
$721$		−	58.2719i		−	2.17016i
$722$		−	17.0000i		−	0.632674i
$723$		0			0
$724$	−22.1359			−0.822676
$725$	102.596			3.81034
$726$		0			0
$727$	−30.1623			−1.11866			−0.559328	−	0.828946i	$-0.688941\pi$
$727$	−30.1623			−1.11866			−0.559328	+	0.828946i	$0.688941\pi$
$728$	1.74342	−	14.9057i	0.0646153	−	0.552442i
$729$		0			0
$730$		−	56.8114i		−	2.10268i
$731$	−3.67544			−0.135941
$732$		0			0
$733$			5.81139i			0.214649i	0.994224	+	0.107324i	$0.0342283\pi$
$733$			5.81139i			0.214649i	−0.994224	+	0.107324i	$0.965772\pi$
$734$			27.4605i			1.01359i
$735$		0			0
$736$		−	1.16228i		−	0.0428421i
$737$	21.4868			0.791478
$738$		0			0
$739$			41.8114i			1.53806i	0.639215	+	0.769028i	$0.279260\pi$
$739$			41.8114i			1.53806i	−0.639215	+	0.769028i	$0.720740\pi$
$740$	−34.6491			−1.27373
$741$		0			0
$742$	27.0000			0.991201
$743$		−	28.6491i		−	1.05103i	−0.850783	−	0.525517i	$-0.823872\pi$
$743$		−	28.6491i		−	1.05103i	0.850783	−	0.525517i	$-0.176128\pi$
$744$		0			0
$745$	61.6491			2.25865
$746$			13.6754i			0.500694i
$747$		0			0
$748$			3.48683i			0.127491i
$749$		−	27.7851i		−	1.01524i
$750$		0			0
$751$	−16.4868			−0.601613			−0.300807	−	0.953685i	$-0.597256\pi$
$751$	−16.4868			−0.601613			−0.300807	+	0.953685i	$0.597256\pi$
$752$			6.00000i			0.218797i
$753$		0			0
$754$	29.8114	+	3.48683i	1.08567	+	0.126983i
$755$	42.2982			1.53939
$756$		0			0
$757$	45.8114			1.66504			0.832522	−	0.553993i	$-0.186896\pi$
$757$	45.8114			1.66504			0.832522	+	0.553993i	$0.186896\pi$
$758$	−26.3246			−0.956151
$759$		0			0
$760$			24.9737i			0.905890i
$761$		−	12.0000i		−	0.435000i	−0.976060	−	0.217500i	$-0.930210\pi$
$761$		−	12.0000i		−	0.435000i	0.976060	−	0.217500i	$-0.0697902\pi$
$762$		0			0
$763$	45.1096			1.63308
$764$	−12.0000			−0.434145
$765$		0			0
$766$	2.32456			0.0839896
$767$	−29.8114	−	3.48683i	−1.07643	−	0.125902i
$768$		0			0
$769$		−	28.9473i		−	1.04387i	−0.852986	−	0.521934i	$-0.825211\pi$
$769$		−	28.9473i		−	1.04387i	0.852986	−	0.521934i	$-0.174789\pi$
$770$	−51.9737			−1.87300
$771$		0			0
$772$		−	5.32456i		−	0.191635i
$773$		−	48.9737i		−	1.76146i	−0.473618	−	0.880730i	$-0.657052\pi$
$773$		−	48.9737i		−	1.76146i	0.473618	−	0.880730i	$-0.342948\pi$
$774$		0			0
$775$		−	22.6491i		−	0.813580i
$776$	3.00000			0.107694
$777$		0			0
$778$			0.486833i			0.0174538i
$779$	42.9737			1.53969
$780$		0			0
$781$	−28.4605			−1.01840
$782$			1.35089i			0.0483077i
$783$		0			0
$784$	−10.3246			−0.368734
$785$			23.6228i			0.843133i
$786$		0			0
$787$			4.64911i			0.165723i	0.996561	+	0.0828614i	$0.0264059\pi$
$787$			4.64911i			0.165723i	−0.996561	+	0.0828614i	$0.973594\pi$
$788$		−	4.16228i		−	0.148275i
$789$		0			0
$790$	−36.0000			−1.28082
$791$		−	29.8114i		−	1.05997i
$792$		0			0
$793$	1.32456	−	11.3246i	0.0470363	−	0.402147i
$794$	−4.64911			−0.164991
$795$		0			0
$796$	−2.16228			−0.0766399
$797$	−22.1623			−0.785028			−0.392514	−	0.919746i	$-0.628395\pi$
$797$	−22.1623			−0.785028			−0.392514	+	0.919746i	$0.628395\pi$
$798$		0			0
$799$		−	6.97367i		−	0.246711i
$800$		−	12.3246i		−	0.435739i
$801$		0			0
$802$	−18.9737			−0.669983
$803$	−40.9473			−1.44500
$804$		0			0
$805$	−20.1359			−0.709699
$806$	0.769751	−	6.58114i	0.0271133	−	0.231811i
$807$		0			0
$808$			1.83772i			0.0646508i
$809$	25.3509			0.891290			0.445645	−	0.895210i	$-0.352974\pi$
$809$	25.3509			0.891290			0.445645	+	0.895210i	$0.352974\pi$
$810$		0			0
$811$			23.8114i			0.836131i	0.908417	+	0.418065i	$0.137292\pi$
$811$			23.8114i			0.836131i	−0.908417	+	0.418065i	$0.862708\pi$
$812$		−	34.6491i		−	1.21595i
$813$		0			0
$814$			24.9737i			0.875327i
$815$	40.2719			1.41066
$816$		0			0
$817$			18.9737i			0.663805i
$818$	32.6228			1.14063
$819$		0			0
$820$	−29.8114			−1.04106
$821$		−	0.973666i		−	0.0339812i	−0.999856	−	0.0169906i	$-0.994591\pi$
$821$		−	0.973666i		−	0.0339812i	0.999856	−	0.0169906i	$-0.00540853\pi$
$822$		0			0
$823$	−8.16228			−0.284519			−0.142260	−	0.989829i	$-0.545437\pi$
$823$	−8.16228			−0.284519			−0.142260	+	0.989829i	$0.545437\pi$
$824$		−	14.0000i		−	0.487713i
$825$		0			0
$826$			34.6491i			1.20560i
$827$		−	30.0000i		−	1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$		−	30.0000i		−	1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$		0			0
$829$	14.6491			0.508785			0.254392	−	0.967101i	$-0.418125\pi$
$829$	14.6491			0.508785			0.254392	+	0.967101i	$0.418125\pi$
$830$			22.1623i			0.769264i
$831$		0			0
$832$	0.418861	−	3.58114i	0.0145214	−	0.124154i
$833$	12.0000			0.415775
$834$		0			0
$835$	−64.4605			−2.23075
$836$	18.0000			0.622543
$837$		0			0
$838$		−	3.67544i		−	0.126966i
$839$		−	9.29822i		−	0.321010i	−0.987035	−	0.160505i	$-0.948688\pi$
$839$		−	9.29822i		−	0.321010i	0.987035	−	0.160505i	$-0.0513123\pi$
$840$		0			0
$841$	40.2982			1.38959
$842$	−31.1623			−1.07392
$843$		0			0
$844$	4.32456			0.148857
$845$	12.4868	−	52.6491i	0.429560	−	1.81118i
$846$		0			0
$847$		−	8.32456i		−	0.286035i
$848$	6.48683			0.222759
$849$		0			0
$850$			14.3246i			0.491328i
$851$			9.67544i			0.331670i
$852$		0			0
$853$			1.16228i			0.0397956i	0.999802	+	0.0198978i	$0.00633409\pi$
$853$			1.16228i			0.0397956i	−0.999802	+	0.0198978i	$0.993666\pi$
$854$	−13.1623			−0.450404
$855$		0			0
$856$		−	6.67544i		−	0.228162i
$857$	52.6491			1.79846			0.899230	−	0.437477i	$-0.144128\pi$
$857$	52.6491			1.79846			0.899230	+	0.437477i	$0.144128\pi$
$858$		0			0
$859$	−2.97367			−0.101460			−0.0507301	−	0.998712i	$-0.516155\pi$
$859$	−2.97367			−0.101460			−0.0507301	+	0.998712i	$0.516155\pi$
$860$		−	13.1623i		−	0.448830i
$861$		0			0
$862$	−13.3509			−0.454733
$863$			33.2982i			1.13348i	0.823895	+	0.566742i	$0.191796\pi$
$863$			33.2982i			1.13348i	−0.823895	+	0.566742i	$0.808204\pi$
$864$		0			0
$865$			27.0000i			0.918028i
$866$		−	11.0000i		−	0.373795i
$867$		0			0
$868$	−7.64911			−0.259628
$869$			25.9473i			0.880203i
$870$		0			0
$871$	25.6491	+	3.00000i	0.869087	+	0.101651i
$872$	10.8377			0.367012
$873$		0			0
$874$	6.97367			0.235888
$875$	−126.895			−4.28982
$876$		0			0
$877$		−	29.8114i		−	1.00666i	−0.864095	−	0.503330i	$-0.832108\pi$
$877$		−	29.8114i		−	1.00666i	0.864095	−	0.503330i	$-0.167892\pi$
$878$		−	14.1623i		−	0.477954i
$879$		0			0
$880$	−12.4868			−0.420931
$881$	5.02633			0.169341			0.0846707	−	0.996409i	$-0.473016\pi$
$881$	5.02633			0.169341			0.0846707	+	0.996409i	$0.473016\pi$
$882$		0			0
$883$	4.18861			0.140958			0.0704790	−	0.997513i	$-0.477547\pi$
$883$	4.18861			0.140958			0.0704790	+	0.997513i	$0.477547\pi$
$884$	−0.486833	+	4.16228i	−0.0163740	+	0.139993i
$885$		0			0
$886$		−	27.2982i		−	0.917102i
$887$	−40.4605			−1.35853			−0.679265	−	0.733893i	$-0.737701\pi$
$887$	−40.4605			−1.35853			−0.679265	+	0.733893i	$0.737701\pi$
$888$		0			0
$889$			25.6491i			0.860244i
$890$		−	39.4868i		−	1.32360i
$891$		0			0
$892$			28.6491i			0.959243i
$893$	−36.0000			−1.20469
$894$		0			0
$895$		−	12.4868i		−	0.417389i
$896$	−4.16228			−0.139052
$897$		0			0
$898$	12.9737			0.432937
$899$		−	15.2982i		−	0.510224i
$900$		0			0
$901$	−7.53950			−0.251177
$902$			21.4868i			0.715433i
$903$		0			0
$904$		−	7.16228i		−	0.238214i
$905$			92.1359i			3.06270i
$906$		0			0
$907$	10.0000			0.332045			0.166022	−	0.986122i	$-0.446908\pi$
$907$	10.0000			0.332045			0.166022	+	0.986122i	$0.446908\pi$
$908$		−	22.6491i		−	0.751637i
$909$		0			0
$910$	−62.0416	−	7.25658i	−2.05666	−	0.240553i
$911$	43.9473			1.45604			0.728020	−	0.685556i	$-0.240441\pi$
$911$	43.9473			1.45604			0.728020	+	0.685556i	$0.240441\pi$
$912$		0			0
$913$	15.9737			0.528651
$914$	6.67544			0.220804
$915$		0			0
$916$		−	20.3246i		−	0.671542i
$917$		−	22.1623i		−	0.731863i
$918$		0			0
$919$	−22.8114			−0.752478			−0.376239	−	0.926523i	$-0.622783\pi$
$919$	−22.8114			−0.752478			−0.376239	+	0.926523i	$0.622783\pi$
$920$	−4.83772			−0.159495
$921$		0			0
$922$	11.5132			0.379166
$923$	−33.9737	−	3.97367i	−1.11826	−	0.130795i
$924$		0			0
$925$			102.596i			3.37335i
$926$	31.4605			1.03386
$927$		0			0
$928$		−	8.32456i		−	0.273267i
$929$			36.0000i			1.18112i	0.806993	+	0.590561i	$0.201093\pi$
$929$			36.0000i			1.18112i	−0.806993	+	0.590561i	$0.798907\pi$
$930$		0			0
$931$		−	61.9473i		−	2.03024i
$932$	−6.00000			−0.196537
$933$		0			0
$934$			15.0000i			0.490815i
$935$	14.5132			0.474631
$936$		0			0
$937$	38.9473			1.27235			0.636177	−	0.771543i	$-0.280515\pi$
$937$	38.9473			1.27235			0.636177	+	0.771543i	$0.280515\pi$
$938$		−	29.8114i		−	0.973376i
$939$		0			0
$940$	24.9737			0.814551
$941$		−	31.8377i		−	1.03788i	−0.854811	−	0.518940i	$-0.826327\pi$
$941$		−	31.8377i		−	1.03788i	0.854811	−	0.518940i	$-0.173673\pi$
$942$		0			0
$943$			8.32456i			0.271085i
$944$			8.32456i			0.270941i
$945$		0			0
$946$	−9.48683			−0.308444
$947$		−	35.3246i		−	1.14789i	−0.818893	−	0.573947i	$-0.805412\pi$
$947$		−	35.3246i		−	1.14789i	0.818893	−	0.573947i	$-0.194588\pi$
$948$		0			0
$949$	−48.8794	−	5.71708i	−1.58669	−	0.185584i
$950$	73.9473			2.39917
$951$		0			0
$952$	4.83772			0.156791
$953$	−10.4605			−0.338849			−0.169424	−	0.985543i	$-0.554191\pi$
$953$	−10.4605			−0.338849			−0.169424	+	0.985543i	$0.554191\pi$
$954$		0			0
$955$			49.9473i			1.61626i
$956$			22.8377i			0.738625i
$957$		0			0
$958$	−3.48683			−0.112654
$959$	34.6491			1.11888
$960$		0			0
$961$	27.6228			0.891057
$962$	−3.48683	+	29.8114i	−0.112420	+	0.961158i
$963$		0			0
$964$			1.35089i			0.0435092i
$965$	−22.1623			−0.713429
$966$		0			0
$967$			9.18861i			0.295486i	0.989026	+	0.147743i	$0.0472008\pi$
$967$			9.18861i			0.295486i	−0.989026	+	0.147743i	$0.952799\pi$
$968$		−	2.00000i		−	0.0642824i
$969$		0			0
$970$		−	12.4868i		−	0.400928i
$971$	15.9737			0.512619			0.256310	−	0.966595i	$-0.417493\pi$
$971$	15.9737			0.512619			0.256310	+	0.966595i	$0.417493\pi$
$972$		0			0
$973$		−	38.1359i		−	1.22258i
$974$	−12.0000			−0.384505
$975$		0			0
$976$	−3.16228			−0.101222
$977$		−	42.0000i		−	1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$		−	42.0000i		−	1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$		0			0
$979$	−28.4605			−0.909601
$980$			42.9737i			1.37274i
$981$		0			0
$982$			18.6754i			0.595957i
$983$		−	3.48683i		−	0.111213i	−0.998453	−	0.0556064i	$-0.982291\pi$
$983$		−	3.48683i		−	0.111213i	0.998453	−	0.0556064i	$-0.0177092\pi$
$984$		0			0
$985$	−17.3246			−0.552006
$986$			9.67544i			0.308129i
$987$		0			0
$988$	21.4868	+	2.51317i	0.683587	+	0.0799545i
$989$	−3.67544			−0.116872
$990$		0			0
$991$	−29.8377			−0.947826			−0.473913	−	0.880572i	$-0.657159\pi$
$991$	−29.8377			−0.947826			−0.473913	+	0.880572i	$0.657159\pi$
$992$	−1.83772			−0.0583477
$993$		0			0
$994$			39.4868i			1.25245i
$995$			9.00000i			0.285319i
$996$		0			0
$997$	12.8377			0.406575			0.203287	−	0.979119i	$-0.434837\pi$
$997$	12.8377			0.406575			0.203287	+	0.979119i	$0.434837\pi$
$998$	8.32456			0.263509
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	702.2.b.g.649.4	yes	4
3.2	odd	2		702.2.b.f.649.1	✓	4
13.5	odd	4		9126.2.a.cc.1.2		2
13.8	odd	4		9126.2.a.bn.1.1		2
13.12	even	2	inner	702.2.b.g.649.1	yes	4
39.5	even	4		9126.2.a.bo.1.1		2
39.8	even	4		9126.2.a.cb.1.2		2
39.38	odd	2		702.2.b.f.649.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
702.2.b.f.649.1	✓	4	3.2	odd	2
702.2.b.f.649.4	yes	4	39.38	odd	2
702.2.b.g.649.1	yes	4	13.12	even	2	inner
702.2.b.g.649.4	yes	4	1.1	even	1	trivial
9126.2.a.bn.1.1		2	13.8	odd	4
9126.2.a.bo.1.1		2	39.5	even	4
9126.2.a.cb.1.2		2	39.8	even	4
9126.2.a.cc.1.2		2	13.5	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	702.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +4.16228i q^{5} -4.16228i q^{7} -1.00000i q^{8} -4.16228 q^{10} +3.00000i q^{11} +(-0.418861 + 3.58114i) q^{13} +4.16228 q^{14} +1.00000 q^{16} -1.16228 q^{17} +6.00000i q^{19} -4.16228i q^{20} -3.00000 q^{22} -1.16228 q^{23} -12.3246 q^{25} +(-3.58114 - 0.418861i) q^{26} +4.16228i q^{28} -8.32456 q^{29} +1.83772i q^{31} +1.00000i q^{32} -1.16228i q^{34} +17.3246 q^{35} -8.32456i q^{37} -6.00000 q^{38} +4.16228 q^{40} -7.16228i q^{41} +3.16228 q^{43} -3.00000i q^{44} -1.16228i q^{46} +6.00000i q^{47} -10.3246 q^{49} -12.3246i q^{50} +(0.418861 - 3.58114i) q^{52} +6.48683 q^{53} -12.4868 q^{55} -4.16228 q^{56} -8.32456i q^{58} +8.32456i q^{59} -3.16228 q^{61} -1.83772 q^{62} -1.00000 q^{64} +(-14.9057 - 1.74342i) q^{65} -7.16228i q^{67} +1.16228 q^{68} +17.3246i q^{70} +9.48683i q^{71} +13.6491i q^{73} +8.32456 q^{74} -6.00000i q^{76} +12.4868 q^{77} +8.64911 q^{79} +4.16228i q^{80} +7.16228 q^{82} -5.32456i q^{83} -4.83772i q^{85} +3.16228i q^{86} +3.00000 q^{88} +9.48683i q^{89} +(14.9057 + 1.74342i) q^{91} +1.16228 q^{92} -6.00000 q^{94} -24.9737 q^{95} +3.00000i q^{97} -10.3246i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{10} - 8 q^{13} + 4 q^{14} + 4 q^{16} + 8 q^{17} - 12 q^{22} + 8 q^{23} - 24 q^{25} - 8 q^{26} - 8 q^{29} + 44 q^{35} - 24 q^{38} + 4 q^{40} - 16 q^{49} + 8 q^{52} - 12 q^{53} - 12 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^10 - 8 * q^13 + 4 * q^14 + 4 * q^16 + 8 * q^17 - 12 * q^22 + 8 * q^23 - 24 * q^25 - 8 * q^26 - 8 * q^29 + 44 * q^35 - 24 * q^38 + 4 * q^40 - 16 * q^49 + 8 * q^52 - 12 * q^53 - 12 * q^55 - 4 * q^56 - 20 * q^62 - 4 * q^64 - 28 * q^65 - 8 * q^68 + 8 * q^74 + 12 * q^77 - 16 * q^79 + 16 * q^82 + 12 * q^88 + 28 * q^91 - 8 * q^92 - 24 * q^94 - 24 * q^95

Introduction

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Varieties

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Groups

Database

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