LMFDB - Embedded newform 702.2.b.g.649.1

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

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -4.16228i q^{5} +4.16228i q^{7} +1.00000i q^{8} -4.16228 q^{10} -3.00000i q^{11} +(-0.418861 - 3.58114i) q^{13} +4.16228 q^{14} +1.00000 q^{16} -1.16228 q^{17} -6.00000i q^{19} +4.16228i q^{20} -3.00000 q^{22} -1.16228 q^{23} -12.3246 q^{25} +(-3.58114 + 0.418861i) q^{26} -4.16228i q^{28} -8.32456 q^{29} -1.83772i q^{31} -1.00000i q^{32} +1.16228i q^{34} +17.3246 q^{35} +8.32456i q^{37} -6.00000 q^{38} +4.16228 q^{40} +7.16228i q^{41} +3.16228 q^{43} +3.00000i q^{44} +1.16228i q^{46} -6.00000i q^{47} -10.3246 q^{49} +12.3246i q^{50} +(0.418861 + 3.58114i) q^{52} +6.48683 q^{53} -12.4868 q^{55} -4.16228 q^{56} +8.32456i q^{58} -8.32456i q^{59} -3.16228 q^{61} -1.83772 q^{62} -1.00000 q^{64} +(-14.9057 + 1.74342i) q^{65} +7.16228i q^{67} +1.16228 q^{68} -17.3246i q^{70} -9.48683i q^{71} -13.6491i q^{73} +8.32456 q^{74} +6.00000i q^{76} +12.4868 q^{77} +8.64911 q^{79} -4.16228i q^{80} +7.16228 q^{82} +5.32456i q^{83} +4.83772i q^{85} -3.16228i q^{86} +3.00000 q^{88} -9.48683i q^{89} +(14.9057 - 1.74342i) q^{91} +1.16228 q^{92} -6.00000 q^{94} -24.9737 q^{95} -3.00000i q^{97} +10.3246i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.619187\pi$
$5$		−	4.16228i		−	1.86143i	0.365749	−	0.930714i	$-0.380813\pi$
$6$		0			0
$7$			4.16228i			1.57319i	0.617467	+	0.786597i	$0.288159\pi$
$7$			4.16228i			1.57319i	−0.617467	+	0.786597i	$0.711841\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−4.16228			−1.31623
$11$		−	3.00000i		−	0.904534i	−0.891883	−	0.452267i	$-0.850615\pi$
$11$		−	3.00000i		−	0.904534i	0.891883	−	0.452267i	$-0.149385\pi$
$12$		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	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$			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	$-0.947227\pi$
$31$		−	1.83772i		−	0.330065i	0.986288	−	0.165032i	$-0.0527729\pi$
$32$		−	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	$0.239881\pi$
$37$			8.32456i			1.36855i	−0.729225	+	0.684274i	$0.760119\pi$
$38$	−6.00000			−0.973329
$39$		0			0
$40$	4.16228			0.658114
$41$			7.16228i			1.11856i	0.828979	+	0.559280i	$0.188922\pi$
$41$			7.16228i			1.11856i	−0.828979	+	0.559280i	$0.811078\pi$
$42$		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	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$		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	$-0.817712\pi$
$59$		−	8.32456i		−	1.08376i	0.840454	−	0.541882i	$-0.182288\pi$
$60$		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	$0.144138\pi$
$67$			7.16228i			0.875011i	−0.899216	+	0.437506i	$0.855862\pi$
$68$	1.16228			0.140947
$69$		0			0
$70$		−	17.3246i		−	2.07068i
$71$		−	9.48683i		−	1.12588i	−0.826498	−	0.562940i	$-0.809670\pi$
$71$		−	9.48683i		−	1.12588i	0.826498	−	0.562940i	$-0.190330\pi$
$72$		0			0
$73$		−	13.6491i		−	1.59751i	−0.601658	−	0.798754i	$-0.705493\pi$
$73$		−	13.6491i		−	1.59751i	0.601658	−	0.798754i	$-0.294507\pi$
$74$	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	$0.0943949\pi$
$83$			5.32456i			0.584446i	−0.956350	+	0.292223i	$0.905605\pi$
$84$		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	$-0.832303\pi$
$89$		−	9.48683i		−	1.00560i	0.864402	−	0.502801i	$-0.167697\pi$
$90$		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	$-0.951331\pi$
$97$		−	3.00000i		−	0.304604i	0.988334	−	0.152302i	$-0.0486686\pi$
$98$			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	$-0.826292\pi$
$109$		−	10.8377i		−	1.03807i	0.854754	−	0.519033i	$-0.173708\pi$
$110$			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	$-0.884274\pi$
$137$		−	8.32456i		−	0.711215i	0.934635	−	0.355607i	$-0.115726\pi$
$138$		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	$0.207506\pi$
$149$			14.8114i			1.21340i	−0.794932	+	0.606698i	$0.792494\pi$
$150$		0			0
$151$			10.1623i			0.826994i	0.910505	+	0.413497i	$0.135693\pi$
$151$			10.1623i			0.826994i	−0.910505	+	0.413497i	$0.864307\pi$
$152$	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	$0.123704\pi$
$163$			9.67544i			0.757839i	−0.925430	+	0.378920i	$0.876296\pi$
$164$		−	7.16228i		−	0.559280i
$165$		0			0
$166$	5.32456			0.413266
$167$		−	15.4868i		−	1.19841i	−0.800597	−	0.599204i	$-0.795484\pi$
$167$		−	15.4868i		−	1.19841i	0.800597	−	0.599204i	$-0.204516\pi$
$168$		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	$-0.938621\pi$
$193$		−	5.32456i		−	0.383270i	0.981466	−	0.191635i	$-0.0613790\pi$
$194$	−3.00000			−0.215387
$195$		0			0
$196$	10.3246			0.737468
$197$		−	4.16228i		−	0.296550i	−0.988946	−	0.148275i	$-0.952628\pi$
$197$		−	4.16228i		−	0.296550i	0.988946	−	0.148275i	$-0.0473721\pi$
$198$		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	$0.408809\pi$
$223$			28.6491i			1.91849i	−0.282582	+	0.959243i	$0.591191\pi$
$224$	4.16228			0.278104
$225$		0			0
$226$		−	7.16228i		−	0.476428i
$227$		−	22.6491i		−	1.50327i	−0.659577	−	0.751637i	$-0.729264\pi$
$227$		−	22.6491i		−	1.50327i	0.659577	−	0.751637i	$-0.270736\pi$
$228$		0			0
$229$		−	20.3246i		−	1.34308i	−0.740966	−	0.671542i	$-0.765632\pi$
$229$		−	20.3246i		−	1.34308i	0.740966	−	0.671542i	$-0.234368\pi$
$230$	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	$0.264524\pi$
$239$			22.8377i			1.47725i	−0.674117	+	0.738625i	$0.735476\pi$
$240$		0			0
$241$			1.35089i			0.0870184i	0.999053	+	0.0435092i	$0.0138538\pi$
$241$			1.35089i			0.0870184i	−0.999053	+	0.0435092i	$0.986146\pi$
$242$		−	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	$0.0765095\pi$
$271$			7.83772i			0.476108i	−0.971252	+	0.238054i	$0.923491\pi$
$272$	−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	$0.178285\pi$
$281$			17.8114i			1.06254i	−0.847203	+	0.531269i	$0.821715\pi$
$282$		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	$0.230117\pi$
$293$			22.6491i			1.32318i	−0.749868	+	0.661588i	$0.769883\pi$
$294$		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	$-0.774052\pi$
$307$		−	22.8377i		−	1.30342i	0.758469	−	0.651709i	$-0.225948\pi$
$308$	−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	$-0.995648\pi$
$317$		−	0.486833i		−	0.0273433i	0.999907	−	0.0136716i	$-0.00435195\pi$
$318$		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	$-0.823456\pi$
$331$		−	19.1623i		−	1.05325i	0.850096	−	0.526627i	$-0.176544\pi$
$332$		−	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	$0.134279\pi$
$349$			15.2982i			0.818895i	−0.912334	+	0.409448i	$0.865721\pi$
$350$	−51.2982			−2.74201
$351$		0			0
$352$	−3.00000			−0.159901
$353$		−	26.1359i		−	1.39108i	−0.718489	−	0.695538i	$-0.755166\pi$
$353$		−	26.1359i		−	1.39108i	0.718489	−	0.695538i	$-0.244834\pi$
$354$		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	$0.135749\pi$
$359$			15.6754i			0.827318i	−0.910432	+	0.413659i	$0.864251\pi$
$360$		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	$-0.763668\pi$
$379$		−	26.3246i		−	1.35220i	0.736809	−	0.676101i	$-0.236332\pi$
$380$	24.9737			1.28112
$381$		0			0
$382$		−	12.0000i		−	0.613973i
$383$			2.32456i			0.118779i	0.998235	+	0.0593896i	$0.0189154\pi$
$383$			2.32456i			0.118779i	−0.998235	+	0.0593896i	$0.981085\pi$
$384$		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	$-0.962779\pi$
$397$		−	4.64911i		−	0.233332i	0.993171	−	0.116666i	$-0.0372207\pi$
$398$		−	2.16228i		−	0.108385i
$399$		0			0
$400$	−12.3246			−0.616228
$401$		−	18.9737i		−	0.947500i	−0.880659	−	0.473750i	$-0.842900\pi$
$401$		−	18.9737i		−	0.947500i	0.880659	−	0.473750i	$-0.157100\pi$
$402$		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	$0.298666\pi$
$409$			32.6228i			1.61309i	−0.591171	+	0.806546i	$0.701334\pi$
$410$		−	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	$-0.725503\pi$
$421$		−	31.1623i		−	1.51876i	0.650649	−	0.759378i	$-0.274497\pi$
$422$			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	$-0.895798\pi$
$431$		−	13.3509i		−	0.643090i	0.946894	−	0.321545i	$-0.104202\pi$
$432$		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	$0.0990351\pi$
$449$			12.9737i			0.612265i	−0.951989	+	0.306133i	$0.900965\pi$
$450$		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	$0.0499025\pi$
$457$			6.67544i			0.312264i	−0.987736	+	0.156132i	$0.950097\pi$
$458$	−20.3246			−0.949704
$459$		0			0
$460$		−	4.83772i		−	0.225560i
$461$			11.5132i			0.536222i	0.963388	+	0.268111i	$0.0863993\pi$
$461$			11.5132i			0.536222i	−0.963388	+	0.268111i	$0.913601\pi$
$462$		0			0
$463$			31.4605i			1.46209i	0.682327	+	0.731047i	$0.260968\pi$
$463$			31.4605i			1.46209i	−0.682327	+	0.731047i	$0.739032\pi$
$464$	−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	$-0.974617\pi$
$479$		−	3.48683i		−	0.159317i	0.996822	−	0.0796587i	$-0.0253831\pi$
$480$		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	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		−	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	$0.0596591\pi$
$499$			8.32456i			0.372658i	−0.982487	+	0.186329i	$0.940341\pi$
$500$		−	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	$0.174479\pi$
$509$			23.5132i			1.04220i	−0.853495	+	0.521101i	$0.825521\pi$
$510$		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	$-0.918285\pi$
$541$		−	11.8114i		−	0.507811i	0.967229	−	0.253906i	$-0.0817153\pi$
$542$	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	$-0.921568\pi$
$557$		−	11.5132i		−	0.487829i	0.969797	−	0.243914i	$-0.0784316\pi$
$558$		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	$-0.734116\pi$
$577$		−	35.6228i		−	1.48300i	0.670955	−	0.741498i	$-0.265884\pi$
$578$			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	$-0.958509\pi$
$587$		−	6.29822i		−	0.259955i	0.991517	−	0.129978i	$-0.0414906\pi$
$588$		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	$-0.906263\pi$
$593$		−	14.1359i		−	0.580494i	0.956952	−	0.290247i	$-0.0937374\pi$
$594$		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	$-0.978783\pi$
$613$		−	3.29822i		−	0.133214i	0.997779	−	0.0666070i	$-0.0212174\pi$
$614$	−22.8377			−0.921655
$615$		0			0
$616$			12.4868i			0.503109i
$617$		−	15.6754i		−	0.631070i	−0.948914	−	0.315535i	$-0.897816\pi$
$617$		−	15.6754i		−	0.631070i	0.948914	−	0.315535i	$-0.102184\pi$
$618$		0			0
$619$			35.6228i			1.43180i	0.698203	+	0.715900i	$0.253983\pi$
$619$			35.6228i			1.43180i	−0.698203	+	0.715900i	$0.746017\pi$
$620$	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	$0.145423\pi$
$631$			22.1623i			0.882266i	−0.897442	+	0.441133i	$0.854577\pi$
$632$			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	$-0.631311\pi$
$643$		−	46.4605i		−	1.83222i	0.400923	−	0.916112i	$-0.368689\pi$
$644$			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	$0.0675969\pi$
$661$			10.8377i			0.421539i	−0.977536	+	0.210769i	$0.932403\pi$
$662$	−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	$-0.790681\pi$
$683$		−	31.9473i		−	1.22243i	0.791464	−	0.611215i	$-0.209319\pi$
$684$		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	$0.0660913\pi$
$691$			10.8377i			0.412286i	−0.978522	+	0.206143i	$0.933909\pi$
$692$	−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	$-0.747531\pi$
$709$		−	37.9473i		−	1.42514i	0.701600	−	0.712571i	$-0.252469\pi$
$710$			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	$-0.965772\pi$
$733$		−	5.81139i		−	0.214649i	0.994224	−	0.107324i	$-0.0342283\pi$
$734$		−	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	$-0.720740\pi$
$739$		−	41.8114i		−	1.53806i	0.639215	−	0.769028i	$-0.279260\pi$
$740$	−34.6491			−1.27373
$741$		0			0
$742$	27.0000			0.991201
$743$			28.6491i			1.05103i	0.850783	+	0.525517i	$0.176128\pi$
$743$			28.6491i			1.05103i	−0.850783	+	0.525517i	$0.823872\pi$
$744$		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	$0.0697902\pi$
$761$			12.0000i			0.435000i	−0.976060	+	0.217500i	$0.930210\pi$
$762$		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	$0.174789\pi$
$769$			28.9473i			1.04387i	−0.852986	+	0.521934i	$0.825211\pi$
$770$	−51.9737			−1.87300
$771$		0			0
$772$			5.32456i			0.191635i
$773$			48.9737i			1.76146i	0.473618	+	0.880730i	$0.342948\pi$
$773$			48.9737i			1.76146i	−0.473618	+	0.880730i	$0.657052\pi$
$774$		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	$-0.973594\pi$
$787$		−	4.64911i		−	0.165723i	0.996561	−	0.0828614i	$-0.0264059\pi$
$788$			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	$-0.862708\pi$
$811$		−	23.8114i		−	0.836131i	0.908417	−	0.418065i	$-0.137292\pi$
$812$			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	$0.00540853\pi$
$821$			0.973666i			0.0339812i	−0.999856	+	0.0169906i	$0.994591\pi$
$822$		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	$0.174665\pi$
$827$			30.0000i			1.04320i	−0.853189	+	0.521601i	$0.825335\pi$
$828$		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	$0.0513123\pi$
$839$			9.29822i			0.321010i	−0.987035	+	0.160505i	$0.948688\pi$
$840$		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	$-0.993666\pi$
$853$		−	1.16228i		−	0.0397956i	0.999802	−	0.0198978i	$-0.00633409\pi$
$854$	−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	$-0.808204\pi$
$863$		−	33.2982i		−	1.13348i	0.823895	−	0.566742i	$-0.191796\pi$
$864$		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	$0.167892\pi$
$877$			29.8114i			1.00666i	−0.864095	+	0.503330i	$0.832108\pi$
$878$			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	$-0.798907\pi$
$929$		−	36.0000i		−	1.18112i	0.806993	−	0.590561i	$-0.201093\pi$
$930$		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	$0.173673\pi$
$941$			31.8377i			1.03788i	−0.854811	+	0.518940i	$0.826327\pi$
$942$		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	$0.194588\pi$
$947$			35.3246i			1.14789i	−0.818893	+	0.573947i	$0.805412\pi$
$948$		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	$-0.952799\pi$
$967$		−	9.18861i		−	0.295486i	0.989026	−	0.147743i	$-0.0472008\pi$
$968$			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	$0.234500\pi$
$977$			42.0000i			1.34370i	−0.740688	+	0.671850i	$0.765500\pi$
$978$		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	$0.0177092\pi$
$983$			3.48683i			0.111213i	−0.998453	+	0.0556064i	$0.982291\pi$
$984$		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.649.1	yes	4
3.2	odd	2		702.2.b.f.649.4	yes	4
13.5	odd	4		9126.2.a.bn.1.1		2
13.8	odd	4		9126.2.a.cc.1.2		2
13.12	even	2	inner	702.2.b.g.649.4	yes	4
39.5	even	4		9126.2.a.cb.1.2		2
39.8	even	4		9126.2.a.bo.1.1		2
39.38	odd	2		702.2.b.f.649.1	✓	4

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

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