LMFDB - Embedded newform 425.2.d.a.101.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(101,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("425.101");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.93924352.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 17x^{4} + 73x^{2} + 67 $ x^6 + 17x^4 + 73x^2 + 67
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.1
Root			$2.21547i$ of defining polynomial
Character	$\chi$	$=$	425.101
Dual form			425.2.d.a.101.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.17009 q^{2} -2.21547i q^{3} +2.70928 q^{4} +4.80775i q^{6} +1.02091i q^{7} -1.53919 q^{8} -1.90829 q^{9} +4.98140i q^{11} -6.00231i q^{12} -3.87936 q^{13} -2.21547i q^{14} -2.07838 q^{16} +(-1.63090 + 3.78684i) q^{17} +4.14116 q^{18} -4.04945 q^{19} +2.26180 q^{21} -10.8101i q^{22} +7.02322i q^{23} +3.41002i q^{24} +8.41855 q^{26} -2.41864i q^{27} +2.76593i q^{28} -0.644091i q^{29} -4.25729i q^{31} +7.58864 q^{32} +11.0361 q^{33} +(3.53919 - 8.21777i) q^{34} -5.17009 q^{36} +10.2596i q^{37} +8.78765 q^{38} +8.59460i q^{39} +5.82867i q^{41} -4.90829 q^{42} -5.86603 q^{43} +13.4960i q^{44} -15.2410i q^{46} +10.3402 q^{47} +4.60458i q^{48} +5.95774 q^{49} +(8.38962 + 3.61320i) q^{51} -10.5103 q^{52} -11.3896 q^{53} +5.24867i q^{54} -1.57138i q^{56} +8.97142i q^{57} +1.39773i q^{58} -0.447480 q^{59} -3.14275i q^{61} +9.23869i q^{62} -1.94820i q^{63} -12.3112 q^{64} -23.9493 q^{66} +5.07838 q^{67} +(-4.41855 + 10.2596i) q^{68} +15.5597 q^{69} +3.41002i q^{71} +2.93722 q^{72} -3.78684i q^{73} -22.2642i q^{74} -10.9711 q^{76} -5.08557 q^{77} -18.6510i q^{78} -0.376821i q^{79} -11.0833 q^{81} -12.6487i q^{82} +3.57531 q^{83} +6.12783 q^{84} +12.7298 q^{86} -1.42696 q^{87} -7.66731i q^{88} -2.63090 q^{89} -3.96049i q^{91} +19.0278i q^{92} -9.43188 q^{93} -22.4391 q^{94} -16.8124i q^{96} -10.9037i q^{97} -12.9288 q^{98} -9.50596i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 2 q^{4} - 6 q^{8} - 16 q^{9} + 2 q^{13} - 6 q^{16} - 2 q^{17} - 16 q^{18} + 12 q^{19} - 2 q^{21} + 22 q^{26} + 6 q^{32} + 28 q^{33} + 18 q^{34} - 20 q^{36} + 32 q^{38} - 34 q^{42} - 8 q^{43}+ \cdots - 16 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 2 * q^4 - 6 * q^8 - 16 * q^9 + 2 * q^13 - 6 * q^16 - 2 * q^17 - 16 * q^18 + 12 * q^19 - 2 * q^21 + 22 * q^26 + 6 * q^32 + 28 * q^33 + 18 * q^34 - 20 * q^36 + 32 * q^38 - 34 * q^42 - 8 * q^43 + 40 * q^47 + 4 * q^49 - 8 * q^51 - 30 * q^52 - 10 * q^53 - 4 * q^59 - 22 * q^64 - 16 * q^66 + 24 * q^67 + 2 * q^68 + 24 * q^69 + 52 * q^72 - 36 * q^76 + 44 * q^77 + 34 * q^81 - 20 * q^83 - 6 * q^84 - 4 * q^86 - 76 * q^87 - 8 * q^89 - 30 * q^93 - 40 * q^94 - 16 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$1$	$-1$

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 $

$2$	−2.17009			−1.53448			−0.767241	−	0.641358i	$-0.778371\pi$
$2$	−2.17009			−1.53448			−0.767241	+	0.641358i	$0.778371\pi$
$3$		−	2.21547i		−	1.27910i	−0.768749	−	0.639550i	$-0.779121\pi$
$3$		−	2.21547i		−	1.27910i	0.768749	−	0.639550i	$-0.220879\pi$
$4$	2.70928			1.35464
$5$		0			0
$5$		0			0
$6$			4.80775i			1.96276i
$7$			1.02091i			0.385868i	0.981212	+	0.192934i	$0.0618004\pi$
$7$			1.02091i			0.385868i	−0.981212	+	0.192934i	$0.938200\pi$
$8$	−1.53919			−0.544185
$9$	−1.90829			−0.636097
$10$		0			0
$11$			4.98140i			1.50195i	0.660332	+	0.750974i	$0.270416\pi$
$11$			4.98140i			1.50195i	−0.660332	+	0.750974i	$0.729584\pi$
$12$		−	6.00231i		−	1.73272i
$13$	−3.87936			−1.07594			−0.537971	−	0.842964i	$-0.680809\pi$
$13$	−3.87936			−1.07594			−0.537971	+	0.842964i	$0.680809\pi$
$14$		−	2.21547i		−	0.592108i
$15$		0			0
$16$	−2.07838			−0.519594
$17$	−1.63090	+	3.78684i	−0.395551	+	0.918444i
$17$	−1.63090	+	3.78684i	−0.395551	+	0.918444i
$18$	4.14116			0.976080
$19$	−4.04945			−0.929007			−0.464504	−	0.885571i	$-0.653767\pi$
$19$	−4.04945			−0.929007			−0.464504	+	0.885571i	$0.653767\pi$
$20$		0			0
$21$	2.26180			0.493564
$22$		−	10.8101i		−	2.30471i
$23$			7.02322i			1.46444i	0.681067	+	0.732221i	$0.261516\pi$
$23$			7.02322i			1.46444i	−0.681067	+	0.732221i	$0.738484\pi$
$24$			3.41002i			0.696068i
$25$		0			0
$26$	8.41855			1.65101
$27$		−	2.41864i		−	0.465468i
$28$			2.76593i			0.522712i
$29$		−	0.644091i		−	0.119605i	−0.998210	−	0.0598023i	$-0.980953\pi$
$29$		−	0.644091i		−	0.119605i	0.998210	−	0.0598023i	$-0.0190470\pi$
$30$		0			0
$31$		−	4.25729i		−	0.764632i	−0.924032	−	0.382316i	$-0.875127\pi$
$31$		−	4.25729i		−	0.764632i	0.924032	−	0.382316i	$-0.124873\pi$
$32$	7.58864			1.34149
$33$	11.0361			1.92114
$34$	3.53919	−	8.21777i	0.606966	−	1.40934i
$35$		0			0
$36$	−5.17009			−0.861681
$37$			10.2596i			1.68667i	0.537390	+	0.843334i	$0.319410\pi$
$37$			10.2596i			1.68667i	−0.537390	+	0.843334i	$0.680590\pi$
$38$	8.78765			1.42555
$39$			8.59460i			1.37624i
$40$		0			0
$41$			5.82867i			0.910285i	0.890419	+	0.455142i	$0.150412\pi$
$41$			5.82867i			0.910285i	−0.890419	+	0.455142i	$0.849588\pi$
$42$	−4.90829			−0.757366
$43$	−5.86603			−0.894561			−0.447281	−	0.894394i	$-0.647607\pi$
$43$	−5.86603			−0.894561			−0.447281	+	0.894394i	$0.647607\pi$
$44$			13.4960i			2.03459i
$45$		0			0
$46$		−	15.2410i		−	2.24716i
$47$	10.3402			1.50827			0.754135	−	0.656720i	$-0.228057\pi$
$47$	10.3402			1.50827			0.754135	+	0.656720i	$0.228057\pi$
$48$			4.60458i			0.664613i
$49$	5.95774			0.851106
$50$		0			0
$51$	8.38962	+	3.61320i	1.17478	+	0.505949i
$52$	−10.5103			−1.45751
$53$	−11.3896			−1.56448			−0.782242	−	0.622974i	$-0.785924\pi$
$53$	−11.3896			−1.56448			−0.782242	+	0.622974i	$0.785924\pi$
$54$			5.24867i			0.714253i
$55$		0			0
$56$		−	1.57138i		−	0.209984i
$57$			8.97142i			1.18829i
$58$			1.39773i			0.183531i
$59$	−0.447480			−0.0582570			−0.0291285	−	0.999576i	$-0.509273\pi$
$59$	−0.447480			−0.0582570			−0.0291285	+	0.999576i	$0.509273\pi$
$60$		0			0
$61$		−	3.14275i		−	0.402388i	−0.979551	−	0.201194i	$-0.935518\pi$
$61$		−	3.14275i		−	0.402388i	0.979551	−	0.201194i	$-0.0644822\pi$
$62$			9.23869i			1.17331i
$63$		−	1.94820i		−	0.245450i
$64$	−12.3112			−1.53891
$65$		0			0
$66$	−23.9493			−2.94796
$67$	5.07838			0.620423			0.310211	−	0.950668i	$-0.399600\pi$
$67$	5.07838			0.620423			0.310211	+	0.950668i	$0.399600\pi$
$68$	−4.41855	+	10.2596i	−0.535828	+	1.24416i
$69$	15.5597			1.87317
$70$		0			0
$71$			3.41002i			0.404695i	0.979314	+	0.202348i	$0.0648571\pi$
$71$			3.41002i			0.404695i	−0.979314	+	0.202348i	$0.935143\pi$
$72$	2.93722			0.346155
$73$		−	3.78684i		−	0.443216i	−0.975136	−	0.221608i	$-0.928869\pi$
$73$		−	3.78684i		−	0.443216i	0.975136	−	0.221608i	$-0.0711306\pi$
$74$		−	22.2642i		−	2.58816i
$75$		0			0
$76$	−10.9711			−1.25847
$77$	−5.08557			−0.579554
$78$		−	18.6510i		−	2.11181i
$79$		−	0.376821i		−	0.0423957i	−0.999775	−	0.0211978i	$-0.993252\pi$
$79$		−	0.376821i		−	0.0423957i	0.999775	−	0.0211978i	$-0.00674798\pi$
$80$		0			0
$81$	−11.0833			−1.23148
$82$		−	12.6487i		−	1.39682i
$83$	3.57531			0.392441			0.196220	−	0.980560i	$-0.437133\pi$
$83$	3.57531			0.392441			0.196220	+	0.980560i	$0.437133\pi$
$84$	6.12783			0.668601
$85$		0			0
$86$	12.7298			1.37269
$87$	−1.42696			−0.152986
$88$		−	7.66731i		−	0.817338i
$89$	−2.63090			−0.278875			−0.139437	−	0.990231i	$-0.544529\pi$
$89$	−2.63090			−0.278875			−0.139437	+	0.990231i	$0.544529\pi$
$90$		0			0
$91$		−	3.96049i		−	0.415172i
$92$			19.0278i			1.98379i
$93$	−9.43188			−0.978041
$94$	−22.4391			−2.31441
$95$		0			0
$96$		−	16.8124i		−	1.71591i
$97$		−	10.9037i		−	1.10710i	−0.832815	−	0.553551i	$-0.813272\pi$
$97$		−	10.9037i		−	1.10710i	0.832815	−	0.553551i	$-0.186728\pi$
$98$	−12.9288			−1.30601
$99$		−	9.50596i		−	0.955385i
$100$		0			0
$101$	−6.21953			−0.618867			−0.309433	−	0.950921i	$-0.600139\pi$
$101$	−6.21953			−0.618867			−0.309433	+	0.950921i	$0.600139\pi$
$102$	−18.2062	−	7.84095i	−1.80268	−	0.776370i
$103$	10.4969			1.03429			0.517147	−	0.855897i	$-0.326994\pi$
$103$	10.4969			1.03429			0.517147	+	0.855897i	$0.326994\pi$
$104$	5.97107			0.585512
$105$		0			0
$106$	24.7165			2.40068
$107$			5.15504i			0.498357i	0.968458	+	0.249178i	$0.0801605\pi$
$107$			5.15504i			0.498357i	−0.968458	+	0.249178i	$0.919839\pi$
$108$		−	6.55277i		−	0.630541i
$109$		−	6.92959i		−	0.663735i	−0.943326	−	0.331867i	$-0.892321\pi$
$109$		−	6.92959i		−	0.663735i	0.943326	−	0.331867i	$-0.107679\pi$
$110$		0			0
$111$	22.7298			2.15742
$112$		−	2.12184i		−	0.200495i
$113$		−	19.5783i		−	1.84177i	−0.389832	−	0.920886i	$-0.627467\pi$
$113$		−	19.5783i		−	1.84177i	0.389832	−	0.920886i	$-0.372533\pi$
$114$		−	19.4687i		−	1.82342i
$115$		0			0
$116$		−	1.74502i		−	0.162021i
$117$	7.40295			0.684403
$118$	0.971071			0.0893943
$119$	−3.86603	−	1.66500i	−0.354398	−	0.152631i
$120$		0			0
$121$	−13.8143			−1.25585
$122$			6.82004i			0.617458i
$123$	12.9132			1.16435
$124$		−	11.5342i		−	1.03580i
$125$		0			0
$126$			4.22776i			0.376638i
$127$	2.20620			0.195769			0.0978845	−	0.995198i	$-0.468792\pi$
$127$	2.20620			0.195769			0.0978845	+	0.995198i	$0.468792\pi$
$128$	11.5392			1.01993
$129$			12.9960i			1.14423i
$130$		0			0
$131$			19.2015i			1.67764i	0.544408	+	0.838821i	$0.316754\pi$
$131$			19.2015i			1.67764i	−0.544408	+	0.838821i	$0.683246\pi$
$132$	29.8999			2.60245
$133$		−	4.13413i		−	0.358474i
$134$	−11.0205			−0.952028
$135$		0			0
$136$	2.51026	−	5.82867i	0.215253	−	0.499804i
$137$	−0.304056			−0.0259772			−0.0129886	−	0.999916i	$-0.504135\pi$
$137$	−0.304056			−0.0259772			−0.0129886	+	0.999916i	$0.504135\pi$
$138$	−33.7659			−2.87435
$139$			18.9478i			1.60713i	0.595215	+	0.803567i	$0.297067\pi$
$139$			18.9478i			1.60713i	−0.595215	+	0.803567i	$0.702933\pi$
$140$		0			0
$141$		−	22.9083i		−	1.92923i
$142$		−	7.40004i		−	0.620998i
$143$		−	19.3246i		−	1.61601i
$144$	3.96615			0.330513
$145$		0			0
$146$			8.21777i			0.680108i
$147$		−	13.1992i		−	1.08865i
$148$			27.7961i			2.28482i
$149$	−19.4969			−1.59725			−0.798625	−	0.601829i	$-0.794439\pi$
$149$	−19.4969			−1.59725			−0.798625	+	0.601829i	$0.794439\pi$
$150$		0			0
$151$	−19.1773			−1.56062			−0.780312	−	0.625390i	$-0.784940\pi$
$151$	−19.1773			−1.56062			−0.780312	+	0.625390i	$0.784940\pi$
$152$	6.23287			0.505552
$153$	3.11223	−	7.22640i	0.251609	−	0.584220i
$154$	11.0361			0.889316
$155$		0			0
$156$			23.2851i			1.86430i
$157$	−14.6248			−1.16718			−0.583591	−	0.812048i	$-0.698353\pi$
$157$	−14.6248			−1.16718			−0.583591	+	0.812048i	$0.698353\pi$
$158$			0.817734i			0.0650554i
$159$			25.2333i			2.00113i
$160$		0			0
$161$	−7.17009			−0.565082
$162$	24.0517			1.88968
$163$			13.5760i			1.06335i	0.846947	+	0.531677i	$0.178438\pi$
$163$			13.5760i			1.06335i	−0.846947	+	0.531677i	$0.821562\pi$
$164$			15.7915i			1.23311i
$165$		0			0
$166$	−7.75872			−0.602194
$167$			2.59229i			0.200597i	0.994957	+	0.100299i	$0.0319798\pi$
$167$			2.59229i			0.200597i	−0.994957	+	0.100299i	$0.968020\pi$
$168$	−3.48133			−0.268590
$169$	2.04945			0.157650
$170$		0			0
$171$	7.72753			0.590939
$172$	−15.8927			−1.21181
$173$			11.0132i			0.837321i	0.908143	+	0.418661i	$0.137500\pi$
$173$			11.0132i			0.837321i	−0.908143	+	0.418661i	$0.862500\pi$
$174$	3.09663			0.234755
$175$		0			0
$176$		−	10.3532i		−	0.780404i
$177$			0.991377i			0.0745165i
$178$	5.70928			0.427928
$179$	24.9854			1.86750			0.933750	−	0.357926i	$-0.116516\pi$
$179$	24.9854			1.86750			0.933750	+	0.357926i	$0.116516\pi$
$180$		0			0
$181$			14.3432i			1.06612i	0.846076	+	0.533062i	$0.178959\pi$
$181$			14.3432i			1.06612i	−0.846076	+	0.533062i	$0.821041\pi$
$182$			8.59460i			0.637074i
$183$	−6.96266			−0.514695
$184$		−	10.8101i		−	0.796928i
$185$		0			0
$186$	20.4680			1.50079
$187$	−18.8638	−	8.12415i	−1.37946	−	0.594097i
$188$	28.0144			2.04316
$189$	2.46922			0.179609
$190$		0			0
$191$	5.39576			0.390424			0.195212	−	0.980761i	$-0.437461\pi$
$191$	5.39576			0.390424			0.195212	+	0.980761i	$0.437461\pi$
$192$			27.2751i			1.96841i
$193$			16.1387i			1.16169i	0.814013	+	0.580846i	$0.197278\pi$
$193$			16.1387i			1.16169i	−0.814013	+	0.580846i	$0.802722\pi$
$194$			23.6619i			1.69883i
$195$		0			0
$196$	16.1412			1.15294
$197$			9.26822i			0.660333i	0.943923	+	0.330167i	$0.107105\pi$
$197$			9.26822i			0.660333i	−0.943923	+	0.330167i	$0.892895\pi$
$198$			20.6287i			1.46602i
$199$		−	4.63411i		−	0.328503i	−0.986418	−	0.164252i	$-0.947479\pi$
$199$		−	4.63411i		−	0.328503i	0.986418	−	0.164252i	$-0.0525209\pi$
$200$		0			0
$201$		−	11.2510i		−	0.793583i
$202$	13.4969			0.949641
$203$	0.657560			0.0461516
$204$	22.7298	+	9.78915i	1.59140	+	0.685378i
$205$		0			0
$206$	−22.7792			−1.58711
$207$		−	13.4023i		−	0.931528i
$208$	8.06278			0.559053
$209$		−	20.1719i		−	1.39532i
$210$		0			0
$211$		−	6.09593i		−	0.419661i	−0.977738	−	0.209831i	$-0.932709\pi$
$211$		−	6.09593i		−	0.419661i	0.977738	−	0.209831i	$-0.0672913\pi$
$212$	−30.8576			−2.11931
$213$	7.55479			0.517645
$214$		−	11.1869i		−	0.764720i
$215$		0			0
$216$			3.72275i			0.253301i
$217$	4.34632			0.295047
$218$			15.0378i			1.01849i
$219$	−8.38962			−0.566918
$220$		0			0
$221$	6.32684	−	14.6905i	0.425589	−	0.988192i
$222$	−49.3256			−3.31052
$223$	17.7009			1.18534			0.592669	−	0.805446i	$-0.298074\pi$
$223$	17.7009			1.18534			0.592669	+	0.805446i	$0.298074\pi$
$224$			7.74733i			0.517640i
$225$		0			0
$226$			42.4866i			2.82617i
$227$		−	25.5806i		−	1.69784i	−0.528518	−	0.848922i	$-0.677252\pi$
$227$		−	25.5806i		−	1.69784i	0.528518	−	0.848922i	$-0.322748\pi$
$228$			24.3060i			1.60971i
$229$	11.9867			0.792101			0.396051	−	0.918229i	$-0.370380\pi$
$229$	11.9867			0.792101			0.396051	+	0.918229i	$0.370380\pi$
$230$		0			0
$231$			11.2669i			0.741308i
$232$			0.991377i			0.0650871i
$233$			4.72774i			0.309724i	0.987936	+	0.154862i	$0.0494933\pi$
$233$			4.72774i			0.309724i	−0.987936	+	0.154862i	$0.950507\pi$
$234$	−16.0650			−1.05020
$235$		0			0
$236$	−1.21235			−0.0789171
$237$	−0.834834			−0.0542283
$238$	8.38962	+	3.61320i	0.543818	+	0.234209i
$239$	10.7649			0.696321			0.348161	−	0.937435i	$-0.386806\pi$
$239$	10.7649			0.696321			0.348161	+	0.937435i	$0.386806\pi$
$240$		0			0
$241$		−	21.6201i		−	1.39267i	−0.717715	−	0.696337i	$-0.754812\pi$
$241$		−	21.6201i		−	1.39267i	0.717715	−	0.696337i	$-0.245188\pi$
$242$	29.9783			1.92708
$243$			17.2987i			1.10971i
$244$		−	8.51458i		−	0.545090i
$245$		0			0
$246$	−28.0228			−1.78667
$247$	15.7093			0.999557
$248$			6.55277i			0.416101i
$249$		−	7.92097i		−	0.501971i
$250$		0			0
$251$	−0.837101			−0.0528374			−0.0264187	−	0.999651i	$-0.508410\pi$
$251$	−0.837101			−0.0528374			−0.0264187	+	0.999651i	$0.508410\pi$
$252$		−	5.27820i		−	0.332495i
$253$	−34.9854			−2.19952
$254$	−4.78765			−0.300404
$255$		0			0
$256$	−0.418551			−0.0261594
$257$	−7.67316			−0.478638			−0.239319	−	0.970941i	$-0.576924\pi$
$257$	−7.67316			−0.478638			−0.239319	+	0.970941i	$0.576924\pi$
$258$		−	28.2024i		−	1.75581i
$259$	−10.4741			−0.650832
$260$		0			0
$261$			1.22911i			0.0760802i
$262$		−	41.6689i		−	2.57431i
$263$	−26.8371			−1.65485			−0.827423	−	0.561579i	$-0.810194\pi$
$263$	−26.8371			−1.65485			−0.827423	+	0.561579i	$0.810194\pi$
$264$	−16.9867			−1.04546
$265$		0			0
$266$			8.97142i			0.550073i
$267$			5.82867i			0.356709i
$268$	13.7587			0.840448
$269$			20.5192i			1.25108i	0.780193	+	0.625539i	$0.215121\pi$
$269$			20.5192i			1.25108i	−0.780193	+	0.625539i	$0.784879\pi$
$270$		0			0
$271$	12.2062			0.741474			0.370737	−	0.928738i	$-0.379105\pi$
$271$	12.2062			0.741474			0.370737	+	0.928738i	$0.379105\pi$
$272$	3.38962	−	7.87049i	0.205526	−	0.477218i
$273$	−8.77432			−0.531046
$274$	0.659827			0.0398616
$275$		0			0
$276$	42.1555			2.53746
$277$		−	6.52324i		−	0.391943i	−0.980610	−	0.195972i	$-0.937214\pi$
$277$		−	6.52324i		−	0.391943i	0.980610	−	0.195972i	$-0.0627861\pi$
$278$		−	41.1184i		−	2.46612i
$279$			8.12415i			0.486380i
$280$		0			0
$281$	1.60811			0.0959319			0.0479659	−	0.998849i	$-0.484726\pi$
$281$	1.60811			0.0959319			0.0479659	+	0.998849i	$0.484726\pi$
$282$			49.7130i			2.96037i
$283$			2.18593i			0.129940i	0.997887	+	0.0649701i	$0.0206952\pi$
$283$			2.18593i			0.129940i	−0.997887	+	0.0649701i	$0.979305\pi$
$284$			9.23869i			0.548215i
$285$		0			0
$286$			41.9361i			2.47974i
$287$	−5.95055			−0.351250
$288$	−14.4813			−0.853321
$289$	−11.6803	−	12.3519i	−0.687079	−	0.726583i
$290$		0			0
$291$	−24.1568			−1.41609
$292$		−	10.2596i		−	0.600398i
$293$	1.50307			0.0878104			0.0439052	−	0.999036i	$-0.486020\pi$
$293$	1.50307			0.0878104			0.0439052	+	0.999036i	$0.486020\pi$
$294$			28.6433i			1.67051i
$295$		0			0
$296$		−	15.7915i		−	0.917860i
$297$	12.0482			0.699109
$298$	42.3100			2.45095
$299$		−	27.2456i		−	1.57565i
$300$		0			0
$301$		−	5.98870i		−	0.345183i
$302$	41.6163			2.39475
$303$			13.7792i			0.791593i
$304$	8.41628			0.482707
$305$		0			0
$306$	−6.75380	+	15.6819i	−0.386089	+	0.896475i
$307$	0.340173			0.0194147			0.00970735	−	0.999953i	$-0.496910\pi$
$307$	0.340173			0.0194147			0.00970735	+	0.999953i	$0.496910\pi$
$308$	−13.7782			−0.785086
$309$		−	23.2556i		−	1.32296i
$310$		0			0
$311$			23.1915i			1.31507i	0.753424	+	0.657535i	$0.228401\pi$
$311$			23.1915i			1.31507i	−0.753424	+	0.657535i	$0.771599\pi$
$312$		−	13.2287i		−	0.748928i
$313$		−	25.4070i		−	1.43609i	−0.695998	−	0.718043i	$-0.745038\pi$
$313$		−	25.4070i		−	1.43609i	0.695998	−	0.718043i	$-0.254962\pi$
$314$	31.7370			1.79102
$315$		0			0
$316$		−	1.02091i		−	0.0574308i
$317$		−	6.76956i		−	0.380216i	−0.981763	−	0.190108i	$-0.939116\pi$
$317$		−	6.76956i		−	0.380216i	0.981763	−	0.190108i	$-0.0608839\pi$
$318$		−	54.7585i		−	3.07070i
$319$	3.20847			0.179640
$320$		0			0
$321$	11.4208			0.637448
$322$	15.5597			0.867109
$323$	6.60424	−	15.3346i	0.367469	−	0.853241i
$324$	−30.0277			−1.66821
$325$		0			0
$326$		−	29.4611i		−	1.63170i
$327$	−15.3523			−0.848983
$328$		−	8.97142i		−	0.495364i
$329$			10.5564i			0.581993i
$330$		0			0
$331$	21.8888			1.20312			0.601559	−	0.798828i	$-0.294546\pi$
$331$	21.8888			1.20312			0.601559	+	0.798828i	$0.294546\pi$
$332$	9.68649			0.531615
$333$		−	19.5783i		−	1.07288i
$334$		−	5.62549i		−	0.307813i
$335$		0			0
$336$	−4.70086			−0.256453
$337$			29.7283i			1.61941i	0.586840	+	0.809703i	$0.300372\pi$
$337$			29.7283i			1.61941i	−0.586840	+	0.809703i	$0.699628\pi$
$338$	−4.44748			−0.241911
$339$	−43.3751			−2.35581
$340$		0			0
$341$	21.2072			1.14844
$342$	−16.7694			−0.906785
$343$			13.2287i			0.714283i
$344$	9.02893			0.486807
$345$		0			0
$346$		−	23.8997i		−	1.28485i
$347$		−	22.0474i		−	1.18357i	−0.806097	−	0.591784i	$-0.798424\pi$
$347$		−	22.0474i		−	1.18357i	0.806097	−	0.591784i	$-0.201576\pi$
$348$	−3.86603			−0.207241
$349$	−15.0761			−0.807006			−0.403503	−	0.914978i	$-0.632207\pi$
$349$	−15.0761			−0.807006			−0.403503	+	0.914978i	$0.632207\pi$
$350$		0			0
$351$			9.38280i			0.500817i
$352$			37.8020i			2.01485i
$353$	14.1194			0.751501			0.375750	−	0.926721i	$-0.377385\pi$
$353$	14.1194			0.751501			0.375750	+	0.926721i	$0.377385\pi$
$354$		−	2.15137i		−	0.114344i
$355$		0			0
$356$	−7.12783			−0.377774
$357$	−3.68876	+	8.56506i	−0.195230	+	0.453311i
$358$	−54.2206			−2.86565
$359$	23.8082			1.25655			0.628274	−	0.777992i	$-0.283762\pi$
$359$	23.8082			1.25655			0.628274	+	0.777992i	$0.283762\pi$
$360$		0			0
$361$	−2.60197			−0.136946
$362$		−	31.1261i		−	1.63595i
$363$			30.6052i			1.60635i
$364$		−	10.7300i		−	0.562407i
$365$		0			0
$366$	15.1096			0.789790
$367$			3.77323i			0.196961i	0.995139	+	0.0984806i	$0.0313982\pi$
$367$			3.77323i			0.196961i	−0.995139	+	0.0984806i	$0.968602\pi$
$368$		−	14.5969i		−	0.760916i
$369$		−	11.1228i		−	0.579029i
$370$		0			0
$371$		−	11.6278i		−	0.603685i
$372$	−25.5536			−1.32489
$373$	1.54760			0.0801317			0.0400658	−	0.999197i	$-0.487243\pi$
$373$	1.54760			0.0801317			0.0400658	+	0.999197i	$0.487243\pi$
$374$	40.9360	+	17.6301i	2.11675	+	0.911631i
$375$		0			0
$376$	−15.9155			−0.820778
$377$			2.49866i			0.128688i
$378$	−5.35842			−0.275608
$379$		−	13.4824i		−	0.692543i	−0.938134	−	0.346271i	$-0.887448\pi$
$379$		−	13.4824i		−	0.692543i	0.938134	−	0.346271i	$-0.112552\pi$
$380$		0			0
$381$		−	4.88777i		−	0.250408i
$382$	−11.7093			−0.599099
$383$	−4.44748			−0.227256			−0.113628	−	0.993523i	$-0.536247\pi$
$383$	−4.44748			−0.227256			−0.113628	+	0.993523i	$0.536247\pi$
$384$		−	25.5647i		−	1.30459i
$385$		0			0
$386$		−	35.0225i		−	1.78260i
$387$	11.1941			0.569028
$388$		−	29.5411i		−	1.49972i
$389$	−25.4680			−1.29128			−0.645639	−	0.763642i	$-0.723409\pi$
$389$	−25.4680			−1.29128			−0.645639	+	0.763642i	$0.723409\pi$
$390$		0			0
$391$	−26.5958	−	11.4542i	−1.34501	−	0.579261i
$392$	−9.17009			−0.463159
$393$	42.5402			2.14587
$394$		−	20.1128i		−	1.01327i
$395$		0			0
$396$		−	25.7543i		−	1.29420i
$397$			14.3937i			0.722400i	0.932488	+	0.361200i	$0.117633\pi$
$397$			14.3937i			0.722400i	−0.932488	+	0.361200i	$0.882367\pi$
$398$			10.0564i			0.504083i
$399$	−9.15902			−0.458525
$400$		0			0
$401$		−	0.456838i		−	0.0228134i	−0.999935	−	0.0114067i	$-0.996369\pi$
$401$		−	0.456838i		−	0.0228134i	0.999935	−	0.0114067i	$-0.00363094\pi$
$402$			24.4156i			1.21774i
$403$			16.5156i			0.822699i
$404$	−16.8504			−0.838340
$405$		0			0
$406$	−1.42696			−0.0708189
$407$	−51.1071			−2.53329
$408$	−12.9132	−	5.56140i	−0.639299	−	0.275330i
$409$	26.7770			1.32404			0.662018	−	0.749488i	$-0.269700\pi$
$409$	26.7770			1.32404			0.662018	+	0.749488i	$0.269700\pi$
$410$		0			0
$411$			0.673625i			0.0332275i
$412$	28.4391			1.40109
$413$		−	0.456838i		−	0.0224795i
$414$			29.0843i			1.42941i
$415$		0			0
$416$	−29.4391			−1.44337
$417$	41.9783			2.05568
$418$			43.7748i			2.14109i
$419$		−	16.2914i		−	0.795889i	−0.917409	−	0.397944i	$-0.869724\pi$
$419$		−	16.2914i		−	0.795889i	0.917409	−	0.397944i	$-0.130276\pi$
$420$		0			0
$421$	−6.21953			−0.303122			−0.151561	−	0.988448i	$-0.548430\pi$
$421$	−6.21953			−0.303122			−0.151561	+	0.988448i	$0.548430\pi$
$422$			13.2287i			0.643963i
$423$	−19.7321			−0.959406
$424$	17.5308			0.851370
$425$		0			0
$426$	−16.3945			−0.794318
$427$	3.20847			0.155269
$428$			13.9664i			0.675093i
$429$	−42.8131			−2.06704
$430$		0			0
$431$			24.4292i			1.17671i	0.808602	+	0.588357i	$0.200225\pi$
$431$			24.4292i			1.17671i	−0.808602	+	0.588357i	$0.799775\pi$
$432$			5.02686i			0.241855i
$433$	6.85989			0.329665			0.164833	−	0.986322i	$-0.447292\pi$
$433$	6.85989			0.329665			0.164833	+	0.986322i	$0.447292\pi$
$434$	−9.43188			−0.452745
$435$		0			0
$436$		−	18.7742i		−	0.899120i
$437$		−	28.4402i		−	1.36048i
$438$	18.2062			0.869926
$439$		−	13.0255i		−	0.621675i	−0.950463	−	0.310837i	$-0.899391\pi$
$439$		−	13.0255i		−	0.621675i	0.950463	−	0.310837i	$-0.100609\pi$
$440$		0			0
$441$	−11.3691			−0.541386
$442$	−13.7298	+	31.8797i	−0.653060	+	1.51636i
$443$	2.71542			0.129013			0.0645067	−	0.997917i	$-0.479453\pi$
$443$	2.71542			0.129013			0.0645067	+	0.997917i	$0.479453\pi$
$444$	61.5813			2.92252
$445$		0			0
$446$	−38.4124			−1.81888
$447$			43.1948i			2.04304i
$448$		−	12.5687i		−	0.593815i
$449$		−	20.5697i		−	0.970743i	−0.874308	−	0.485372i	$-0.838684\pi$
$449$		−	20.5697i		−	0.970743i	0.874308	−	0.485372i	$-0.161316\pi$
$450$		0			0
$451$	−29.0349			−1.36720
$452$		−	53.0430i		−	2.49493i
$453$			42.4866i			1.99619i
$454$			55.5121i			2.60531i
$455$		0			0
$456$		−	13.8087i		−	0.646652i
$457$	15.0338			0.703254			0.351627	−	0.936140i	$-0.385629\pi$
$457$	15.0338			0.703254			0.351627	+	0.936140i	$0.385629\pi$
$458$	−26.0121			−1.21547
$459$	9.15902	+	3.94456i	0.427507	+	0.184116i
$460$		0			0
$461$	−12.7382			−0.593277			−0.296639	−	0.954990i	$-0.595866\pi$
$461$	−12.7382			−0.593277			−0.296639	+	0.954990i	$0.595866\pi$
$462$		−	24.4501i		−	1.13752i
$463$	5.68261			0.264093			0.132047	−	0.991243i	$-0.457845\pi$
$463$	5.68261			0.264093			0.132047	+	0.991243i	$0.457845\pi$
$464$			1.33866i			0.0621459i
$465$		0			0
$466$		−	10.2596i		−	0.475267i
$467$	−23.9155			−1.10668			−0.553338	−	0.832957i	$-0.686646\pi$
$467$	−23.9155			−1.10668			−0.553338	+	0.832957i	$0.686646\pi$
$468$	20.0566			0.927118
$469$			5.18457i			0.239401i
$470$		0			0
$471$			32.4007i			1.49294i
$472$	0.688756			0.0317026
$473$		−	29.2210i		−	1.34358i
$474$	1.81166			0.0832124
$475$		0			0
$476$	−10.4741	−	4.51095i	−0.480082	−	0.206759i
$477$	21.7347			0.995164
$478$	−23.3607			−1.06849
$479$			11.6142i			0.530666i	0.964157	+	0.265333i	$0.0854818\pi$
$479$			11.6142i			0.530666i	−0.964157	+	0.265333i	$0.914518\pi$
$480$		0			0
$481$		−	39.8007i		−	1.81476i
$482$			46.9175i			2.13704i
$483$			15.8851i			0.722796i
$484$	−37.4268			−1.70122
$485$		0			0
$486$		−	37.5398i		−	1.70284i
$487$			7.96411i			0.360888i	0.983585	+	0.180444i	$0.0577535\pi$
$487$			7.96411i			0.360888i	−0.983585	+	0.180444i	$0.942246\pi$
$488$			4.83729i			0.218974i
$489$	30.0772			1.36014
$490$		0			0
$491$	36.2557			1.63619			0.818097	−	0.575080i	$-0.195029\pi$
$491$	36.2557			1.63619			0.818097	+	0.575080i	$0.195029\pi$
$492$	34.9854			1.57727
$493$	2.43907	+	1.05045i	0.109850	+	0.0473097i
$494$	−34.0905			−1.53380
$495$		0			0
$496$			8.84826i			0.397298i
$497$	−3.48133			−0.156159
$498$			17.1892i			0.770266i
$499$		−	20.1583i		−	0.902409i	−0.892420	−	0.451205i	$-0.850994\pi$
$499$		−	20.1583i		−	0.902409i	0.892420	−	0.451205i	$-0.149006\pi$
$500$		0			0
$501$	5.74313			0.256584
$502$	1.81658			0.0810780
$503$			19.2151i			0.856759i	0.903599	+	0.428379i	$0.140915\pi$
$503$			19.2151i			0.856759i	−0.903599	+	0.428379i	$0.859085\pi$
$504$			2.99864i			0.133570i
$505$		0			0
$506$	75.9214			3.37512
$507$		−	4.54048i		−	0.201650i
$508$	5.97721			0.265196
$509$	1.52813			0.0677330			0.0338665	−	0.999426i	$-0.489218\pi$
$509$	1.52813			0.0677330			0.0338665	+	0.999426i	$0.489218\pi$
$510$		0			0
$511$	3.86603			0.171023
$512$	−22.1701			−0.979789
$513$			9.79417i			0.432423i
$514$	16.6514			0.734462
$515$		0			0
$516$			35.2097i			1.55002i
$517$			51.5085i			2.26534i
$518$	22.7298			0.998690
$519$	24.3995			1.07102
$520$		0			0
$521$		−	9.96279i		−	0.436478i	−0.975895	−	0.218239i	$-0.929969\pi$
$521$		−	9.96279i		−	0.436478i	0.975895	−	0.218239i	$-0.0700312\pi$
$522$		−	2.66728i		−	0.116744i
$523$	31.2618			1.36698			0.683491	−	0.729959i	$-0.260461\pi$
$523$	31.2618			1.36698			0.683491	+	0.729959i	$0.260461\pi$
$524$			52.0221i			2.27260i
$525$		0			0
$526$	58.2388			2.53933
$527$	16.1217	+	6.94320i	0.702272	+	0.302451i
$528$	−22.9372			−0.998214
$529$	−26.3256			−1.14459
$530$		0			0
$531$	0.853922			0.0370571
$532$		−	11.2005i		−	0.485603i
$533$		−	22.6115i		−	0.979413i
$534$		−	12.6487i		−	0.547363i
$535$		0			0
$536$	−7.81658			−0.337625
$537$		−	55.3544i		−	2.38872i
$538$		−	44.5284i		−	1.91976i
$539$			29.6779i			1.27832i
$540$		0			0
$541$		−	29.1433i		−	1.25297i	−0.779434	−	0.626485i	$-0.784493\pi$
$541$		−	29.1433i		−	1.25297i	0.779434	−	0.626485i	$-0.215507\pi$
$542$	−26.4885			−1.13778
$543$	31.7770			1.36368
$544$	−12.3763	+	28.7370i	−0.530629	+	1.23209i
$545$		0			0
$546$	19.0410			0.814881
$547$			18.5069i			0.791298i	0.918402	+	0.395649i	$0.129480\pi$
$547$			18.5069i			0.791298i	−0.918402	+	0.395649i	$0.870520\pi$
$548$	−0.823770			−0.0351897
$549$			5.99729i			0.255958i
$550$		0			0
$551$			2.60821i			0.111114i
$552$	−23.9493			−1.01935
$553$	0.384701			0.0163591
$554$			14.1560i			0.601430i
$555$		0			0
$556$			51.3349i			2.17708i
$557$	−1.44029			−0.0610271			−0.0305136	−	0.999534i	$-0.509714\pi$
$557$	−1.44029			−0.0610271			−0.0305136	+	0.999534i	$0.509714\pi$
$558$		−	17.6301i		−	0.746342i
$559$	22.7565			0.962496
$560$		0			0
$561$	−17.9988	+	41.7920i	−0.759909	+	1.76446i
$562$	−3.48974			−0.147206
$563$	2.47414			0.104273			0.0521363	−	0.998640i	$-0.483397\pi$
$563$	2.47414			0.104273			0.0521363	+	0.998640i	$0.483397\pi$
$564$		−	62.0649i		−	2.61340i
$565$		0			0
$566$		−	4.74366i		−	0.199391i
$567$		−	11.3151i		−	0.475188i
$568$		−	5.24867i		−	0.220229i
$569$	15.3919			0.645262			0.322631	−	0.946525i	$-0.395433\pi$
$569$	15.3919			0.645262			0.322631	+	0.946525i	$0.395433\pi$
$570$		0			0
$571$			17.8974i			0.748982i	0.927231	+	0.374491i	$0.122183\pi$
$571$			17.8974i			0.748982i	−0.927231	+	0.374491i	$0.877817\pi$
$572$		−	52.3558i		−	2.18910i
$573$		−	11.9541i		−	0.499391i
$574$	12.9132			0.538987
$575$		0			0
$576$	23.4934			0.978893
$577$	44.1894			1.83963			0.919814	−	0.392355i	$-0.128339\pi$
$577$	44.1894			1.83963			0.919814	+	0.392355i	$0.128339\pi$
$578$	25.3474	+	26.8047i	1.05431	+	1.11493i
$579$	35.7548			1.48592
$580$		0			0
$581$			3.65007i			0.151430i
$582$	52.4222			2.17297
$583$		−	56.7362i		−	2.34977i
$584$			5.82867i			0.241192i
$585$		0			0
$586$	−3.26180			−0.134744
$587$	3.04718			0.125771			0.0628853	−	0.998021i	$-0.479970\pi$
$587$	3.04718			0.125771			0.0628853	+	0.998021i	$0.479970\pi$
$588$		−	35.7602i		−	1.47473i
$589$			17.2397i			0.710348i
$590$		0			0
$591$	20.5334			0.844633
$592$		−	21.3233i		−	0.876383i
$593$	−9.68876			−0.397870			−0.198935	−	0.980013i	$-0.563748\pi$
$593$	−9.68876			−0.397870			−0.198935	+	0.980013i	$0.563748\pi$
$594$	−26.1457			−1.07277
$595$		0			0
$596$	−52.8225			−2.16370
$597$	−10.2667			−0.420189
$598$			59.1253i			2.41781i
$599$	−33.2846			−1.35997			−0.679986	−	0.733225i	$-0.738014\pi$
$599$	−33.2846			−1.35997			−0.679986	+	0.733225i	$0.738014\pi$
$600$		0			0
$601$			13.1055i			0.534586i	0.963615	+	0.267293i	$0.0861292\pi$
$601$			13.1055i			0.534586i	−0.963615	+	0.267293i	$0.913871\pi$
$602$			12.9960i			0.529677i
$603$	−9.69102			−0.394649
$604$	−51.9565			−2.11408
$605$		0			0
$606$		−	29.9020i		−	1.21469i
$607$		−	8.34094i		−	0.338548i	−0.985569	−	0.169274i	$-0.945858\pi$
$607$		−	8.34094i		−	0.338548i	0.985569	−	0.169274i	$-0.0541423\pi$
$608$	−30.7298			−1.24626
$609$		−	1.45680i		−	0.0590326i
$610$		0			0
$611$	−40.1133			−1.62281
$612$	8.43188	−	19.5783i	0.340839	−	0.791406i
$613$	−32.6430			−1.31844			−0.659219	−	0.751951i	$-0.729113\pi$
$613$	−32.6430			−1.31844			−0.659219	+	0.751951i	$0.729113\pi$
$614$	−0.738205			−0.0297915
$615$		0			0
$616$	7.82765			0.315385
$617$		−	31.5324i		−	1.26945i	−0.772739	−	0.634724i	$-0.781114\pi$
$617$		−	31.5324i		−	1.26945i	0.772739	−	0.634724i	$-0.218886\pi$
$618$			50.4666i			2.03007i
$619$		−	42.1234i		−	1.69308i	−0.532323	−	0.846541i	$-0.678681\pi$
$619$		−	42.1234i		−	1.69308i	0.532323	−	0.846541i	$-0.321319\pi$
$620$		0			0
$621$	16.9867			0.681652
$622$		−	50.3276i		−	2.01795i
$623$		−	2.68591i		−	0.107609i
$624$		−	17.8628i		−	0.715085i
$625$		0			0
$626$			55.1353i			2.20365i
$627$	−44.6902			−1.78475
$628$	−39.6225			−1.58111
$629$	−38.8515	−	16.7324i	−1.54911	−	0.667163i
$630$		0			0
$631$	16.7031			0.664941			0.332471	−	0.943114i	$-0.392118\pi$
$631$	16.7031			0.664941			0.332471	+	0.943114i	$0.392118\pi$
$632$			0.579999i			0.0230711i
$633$	−13.5053			−0.536789
$634$			14.6905i			0.583436i
$635$		0			0
$636$			68.3640i			2.71081i
$637$	−23.1122			−0.915740
$638$	−6.96266			−0.275654
$639$		−	6.50731i		−	0.257425i
$640$		0			0
$641$			16.3260i			0.644838i	0.946597	+	0.322419i	$0.104496\pi$
$641$			16.3260i			0.644838i	−0.946597	+	0.322419i	$0.895504\pi$
$642$	−24.7842			−0.978153
$643$			28.0633i			1.10671i	0.832945	+	0.553355i	$0.186653\pi$
$643$			28.0633i			1.10671i	−0.832945	+	0.553355i	$0.813347\pi$
$644$	−19.4257			−0.765481
$645$		0			0
$646$	−14.3318	+	33.2775i	−0.563876	+	1.30928i
$647$	39.7815			1.56397			0.781986	−	0.623296i	$-0.214207\pi$
$647$	39.7815			1.56397			0.781986	+	0.623296i	$0.214207\pi$
$648$	17.0593			0.670152
$649$		−	2.22908i		−	0.0874989i
$650$		0			0
$651$		−	9.62912i		−	0.377395i
$652$			36.7811i			1.44046i
$653$			35.7761i			1.40003i	0.714129	+	0.700014i	$0.246823\pi$
$653$			35.7761i			1.40003i	−0.714129	+	0.700014i	$0.753177\pi$
$654$	33.3158			1.30275
$655$		0			0
$656$		−	12.1142i		−	0.472979i
$657$			7.22640i			0.281929i
$658$		−	22.9083i		−	0.893059i
$659$	−5.18568			−0.202006			−0.101003	−	0.994886i	$-0.532205\pi$
$659$	−5.18568			−0.202006			−0.101003	+	0.994886i	$0.532205\pi$
$660$		0			0
$661$	31.4079			1.22162			0.610812	−	0.791775i	$-0.290843\pi$
$661$	31.4079			1.22162			0.610812	+	0.791775i	$0.290843\pi$
$662$	−47.5006			−1.84616
$663$	−32.5464	−	14.0169i	−1.26400	−	0.544372i
$664$	−5.50307			−0.213561
$665$		0			0
$666$			42.4866i			1.64632i
$667$	4.52359			0.175154
$668$			7.02322i			0.271737i
$669$		−	39.2157i		−	1.51617i
$670$		0			0
$671$	15.6553			0.604366
$672$	17.1639			0.662113
$673$		−	17.8838i		−	0.689368i	−0.938719	−	0.344684i	$-0.887986\pi$
$673$		−	17.8838i		−	0.689368i	0.938719	−	0.344684i	$-0.112014\pi$
$674$		−	64.5131i		−	2.48495i
$675$		0			0
$676$	5.55252			0.213558
$677$		−	3.30278i		−	0.126936i	−0.997984	−	0.0634682i	$-0.979784\pi$
$677$		−	3.30278i		−	0.126936i	0.997984	−	0.0634682i	$-0.0202161\pi$
$678$	94.1276			3.61495
$679$	11.1317			0.427196
$680$		0			0
$681$	−56.6730			−2.17171
$682$	−46.0216			−1.76226
$683$		−	10.8742i		−	0.416088i	−0.978119	−	0.208044i	$-0.933290\pi$
$683$		−	10.8742i		−	0.416088i	0.978119	−	0.208044i	$-0.0667097\pi$
$684$	20.9360			0.800508
$685$		0			0
$686$		−	28.7074i		−	1.09606i
$687$		−	26.5561i		−	1.01318i
$688$	12.1918			0.464809
$689$	44.1845			1.68329
$690$		0			0
$691$		−	3.41002i		−	0.129723i	−0.997894	−	0.0648617i	$-0.979339\pi$
$691$		−	3.41002i		−	0.129723i	0.997894	−	0.0648617i	$-0.0206606\pi$
$692$			29.8379i			1.13427i
$693$	9.70474			0.368653
$694$			47.8448i			1.81616i
$695$		0			0
$696$	2.19636			0.0832529
$697$	−22.0722	−	9.50596i	−0.836046	−	0.360064i
$698$	32.7165			1.23834
$699$	10.4741			0.396168
$700$		0			0
$701$	−2.68422			−0.101382			−0.0506908	−	0.998714i	$-0.516142\pi$
$701$	−2.68422			−0.101382			−0.0506908	+	0.998714i	$0.516142\pi$
$702$		−	20.3615i		−	0.768494i
$703$		−	41.5457i		−	1.56693i
$704$		−	61.3272i		−	2.31136i
$705$		0			0
$706$	−30.6404			−1.15316
$707$		−	6.34959i		−	0.238801i
$708$			2.68591i			0.100943i
$709$			46.9766i			1.76424i	0.471021	+	0.882122i	$0.343885\pi$
$709$			46.9766i			1.76424i	−0.471021	+	0.882122i	$0.656115\pi$
$710$		0			0
$711$			0.719084i			0.0269678i
$712$	4.04945			0.151759
$713$	29.8999			1.11976
$714$	8.00492	−	18.5869i	0.299577	−	0.695598i
$715$		0			0
$716$	67.6925			2.52979
$717$		−	23.8492i		−	0.890665i
$718$	−51.6658			−1.92815
$719$			37.4252i			1.39572i	0.716232	+	0.697862i	$0.245865\pi$
$719$			37.4252i			1.39572i	−0.716232	+	0.697862i	$0.754135\pi$
$720$		0			0
$721$			10.7164i			0.399101i
$722$	5.64650			0.210141
$723$	−47.8987			−1.78137
$724$			38.8598i			1.44421i
$725$		0			0
$726$		−	66.4158i		−	2.46492i
$727$	−37.8537			−1.40392			−0.701959	−	0.712218i	$-0.747691\pi$
$727$	−37.8537			−1.40392			−0.701959	+	0.712218i	$0.747691\pi$
$728$			6.09593i			0.225930i
$729$	5.07489			0.187959
$730$		0			0
$731$	9.56690	−	22.2137i	0.353844	−	0.821605i
$732$	−18.8638			−0.697225
$733$	36.0205			1.33045			0.665224	−	0.746644i	$-0.268336\pi$
$733$	36.0205			1.33045			0.665224	+	0.746644i	$0.268336\pi$
$734$		−	8.18824i		−	0.302233i
$735$		0			0
$736$			53.2967i			1.96454i
$737$			25.2974i			0.931842i
$738$			24.1374i			0.888511i
$739$	19.0349			0.700210			0.350105	−	0.936710i	$-0.386146\pi$
$739$	19.0349			0.700210			0.350105	+	0.936710i	$0.386146\pi$
$740$		0			0
$741$		−	34.8034i		−	1.27853i
$742$			25.2333i			0.926344i
$743$			10.8901i			0.399518i	0.979845	+	0.199759i	$0.0640160\pi$
$743$			10.8901i			0.399518i	−0.979845	+	0.199759i	$0.935984\pi$
$744$	14.5174			0.532235
$745$		0			0
$746$	−3.35842			−0.122961
$747$	−6.82273			−0.249630
$748$	−51.1071	−	22.0106i	−1.86866	−	0.804786i
$749$	−5.26284			−0.192300
$750$		0			0
$751$		−	1.59232i		−	0.0581047i	−0.999578	−	0.0290524i	$-0.990751\pi$
$751$		−	1.59232i		−	0.0581047i	0.999578	−	0.0290524i	$-0.00924895\pi$
$752$	−21.4908			−0.783688
$753$			1.85457i			0.0675843i
$754$		−	5.42231i		−	0.197469i
$755$		0			0
$756$	6.68980			0.243306
$757$	7.42696			0.269937			0.134969	−	0.990850i	$-0.456907\pi$
$757$	7.42696			0.269937			0.134969	+	0.990850i	$0.456907\pi$
$758$			29.2579i			1.06269i
$759$			77.5091i			2.81340i
$760$		0			0
$761$	42.3679			1.53583			0.767917	−	0.640549i	$-0.221293\pi$
$761$	42.3679			1.53583			0.767917	+	0.640549i	$0.221293\pi$
$762$			10.6069i			0.384247i
$763$	7.07450			0.256114
$764$	14.6186			0.528883
$765$		0			0
$766$	9.65142			0.348720
$767$	1.73594			0.0626811
$768$			0.927285i			0.0334605i
$769$	−3.40522			−0.122795			−0.0613977	−	0.998113i	$-0.519556\pi$
$769$	−3.40522			−0.122795			−0.0613977	+	0.998113i	$0.519556\pi$
$770$		0			0
$771$			16.9996i			0.612226i
$772$			43.7243i			1.57367i
$773$	−49.7058			−1.78779			−0.893896	−	0.448274i	$-0.852039\pi$
$773$	−49.7058			−1.78779			−0.893896	+	0.448274i	$0.852039\pi$
$774$	−24.2922			−0.873163
$775$		0			0
$776$			16.7828i			0.602469i
$777$			23.2051i			0.832479i
$778$	55.2678			1.98145
$779$		−	23.6029i		−	0.845661i
$780$		0			0
$781$	−16.9867			−0.607831
$782$	57.7152	+	24.8565i	2.06389	+	0.888867i
$783$	−1.55783			−0.0556722
$784$	−12.3824			−0.442230
$785$		0			0
$786$	−92.3160			−3.29280
$787$		−	3.96049i		−	0.141176i	−0.997506	−	0.0705880i	$-0.977512\pi$
$787$		−	3.96049i		−	0.141176i	0.997506	−	0.0705880i	$-0.0224876\pi$
$788$			25.1102i			0.894512i
$789$			59.4567i			2.11671i
$790$		0			0
$791$	19.9877			0.710681

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.17009 q^{2} -2.21547i q^{3} +2.70928 q^{4} +4.80775i q^{6} +1.02091i q^{7} -1.53919 q^{8} -1.90829 q^{9} +4.98140i q^{11} -6.00231i q^{12} -3.87936 q^{13} -2.21547i q^{14} -2.07838 q^{16} +(-1.63090 + 3.78684i) q^{17} +4.14116 q^{18} -4.04945 q^{19} +2.26180 q^{21} -10.8101i q^{22} +7.02322i q^{23} +3.41002i q^{24} +8.41855 q^{26} -2.41864i q^{27} +2.76593i q^{28} -0.644091i q^{29} -4.25729i q^{31} +7.58864 q^{32} +11.0361 q^{33} +(3.53919 - 8.21777i) q^{34} -5.17009 q^{36} +10.2596i q^{37} +8.78765 q^{38} +8.59460i q^{39} +5.82867i q^{41} -4.90829 q^{42} -5.86603 q^{43} +13.4960i q^{44} -15.2410i q^{46} +10.3402 q^{47} +4.60458i q^{48} +5.95774 q^{49} +(8.38962 + 3.61320i) q^{51} -10.5103 q^{52} -11.3896 q^{53} +5.24867i q^{54} -1.57138i q^{56} +8.97142i q^{57} +1.39773i q^{58} -0.447480 q^{59} -3.14275i q^{61} +9.23869i q^{62} -1.94820i q^{63} -12.3112 q^{64} -23.9493 q^{66} +5.07838 q^{67} +(-4.41855 + 10.2596i) q^{68} +15.5597 q^{69} +3.41002i q^{71} +2.93722 q^{72} -3.78684i q^{73} -22.2642i q^{74} -10.9711 q^{76} -5.08557 q^{77} -18.6510i q^{78} -0.376821i q^{79} -11.0833 q^{81} -12.6487i q^{82} +3.57531 q^{83} +6.12783 q^{84} +12.7298 q^{86} -1.42696 q^{87} -7.66731i q^{88} -2.63090 q^{89} -3.96049i q^{91} +19.0278i q^{92} -9.43188 q^{93} -22.4391 q^{94} -16.8124i q^{96} -10.9037i q^{97} -12.9288 q^{98} -9.50596i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 2 q^{4} - 6 q^{8} - 16 q^{9} + 2 q^{13} - 6 q^{16} - 2 q^{17} - 16 q^{18} + 12 q^{19} - 2 q^{21} + 22 q^{26} + 6 q^{32} + 28 q^{33} + 18 q^{34} - 20 q^{36} + 32 q^{38} - 34 q^{42} - 8 q^{43}+ \cdots - 16 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 2 * q^4 - 6 * q^8 - 16 * q^9 + 2 * q^13 - 6 * q^16 - 2 * q^17 - 16 * q^18 + 12 * q^19 - 2 * q^21 + 22 * q^26 + 6 * q^32 + 28 * q^33 + 18 * q^34 - 20 * q^36 + 32 * q^38 - 34 * q^42 - 8 * q^43 + 40 * q^47 + 4 * q^49 - 8 * q^51 - 30 * q^52 - 10 * q^53 - 4 * q^59 - 22 * q^64 - 16 * q^66 + 24 * q^67 + 2 * q^68 + 24 * q^69 + 52 * q^72 - 36 * q^76 + 44 * q^77 + 34 * q^81 - 20 * q^83 - 6 * q^84 - 4 * q^86 - 76 * q^87 - 8 * q^89 - 30 * q^93 - 40 * q^94 - 16 * q^98

Introduction

L-functions

Modular forms

Varieties

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