LMFDB - Embedded newform 425.2.d.a.101.5

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			101.5
Root			$3.29112i$ of defining polynomial
Character	$\chi$	$=$	425.101
Dual form			425.2.d.a.101.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.48119 q^{2} -3.29112i q^{3} +0.193937 q^{4} -4.87478i q^{6} -2.22194i q^{7} -2.67513 q^{8} -7.83146 q^{9} +2.86020i q^{11} -0.638268i q^{12} +2.28726 q^{13} -3.29112i q^{14} -4.35026 q^{16} +(3.15633 - 2.65285i) q^{17} -11.5999 q^{18} +5.76845 q^{19} -7.31265 q^{21} +4.23652i q^{22} -1.58367i q^{23} +8.80417i q^{24} +3.38787 q^{26} +15.9009i q^{27} -0.430914i q^{28} -9.23509i q^{29} +1.15275i q^{31} -1.09332 q^{32} +9.41327 q^{33} +(4.67513 - 3.92939i) q^{34} -1.51881 q^{36} -0.514485i q^{37} +8.54420 q^{38} -7.52763i q^{39} -7.09672i q^{41} -10.8315 q^{42} -7.89446 q^{43} +0.554698i q^{44} -2.34572i q^{46} +3.03761 q^{47} +14.3172i q^{48} +2.06300 q^{49} +(-8.73084 - 10.3878i) q^{51} +0.443583 q^{52} +5.73084 q^{53} +23.5523i q^{54} +5.94397i q^{56} -18.9847i q^{57} -13.6790i q^{58} -7.50659 q^{59} +11.8879i q^{61} +1.70745i q^{62} +17.4010i q^{63} +7.08110 q^{64} +13.9429 q^{66} +7.35026 q^{67} +(0.612127 - 0.514485i) q^{68} -5.21203 q^{69} +8.80417i q^{71} +20.9502 q^{72} +2.65285i q^{73} -0.762052i q^{74} +1.11871 q^{76} +6.35519 q^{77} -11.1499i q^{78} +11.4570i q^{79} +28.8373 q^{81} -10.5116i q^{82} +3.08840 q^{83} -1.41819 q^{84} -11.6932 q^{86} -30.3938 q^{87} -7.65142i q^{88} +2.15633 q^{89} -5.08214i q^{91} -0.307131i q^{92} +3.79384 q^{93} +4.49929 q^{94} +3.59825i q^{96} -8.72060i q^{97} +3.05571 q^{98} -22.3996i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.48119			1.04736			0.523681	−	0.851914i	$-0.324558\pi$
$2$	1.48119			1.04736			0.523681	+	0.851914i	$0.324558\pi$
$3$		−	3.29112i		−	1.90013i	−0.312056	−	0.950064i	$-0.601018\pi$
$3$		−	3.29112i		−	1.90013i	0.312056	−	0.950064i	$-0.398982\pi$
$4$	0.193937			0.0969683
$5$		0			0
$5$		0			0
$6$		−	4.87478i		−	1.99012i
$7$		−	2.22194i		−	0.839813i	−0.907567	−	0.419906i	$-0.862063\pi$
$7$		−	2.22194i		−	0.839813i	0.907567	−	0.419906i	$-0.137937\pi$
$8$	−2.67513			−0.945802
$9$	−7.83146			−2.61049
$10$		0			0
$11$			2.86020i			0.862384i	0.902260	+	0.431192i	$0.141907\pi$
$11$			2.86020i			0.862384i	−0.902260	+	0.431192i	$0.858093\pi$
$12$		−	0.638268i		−	0.184252i
$13$	2.28726			0.634371			0.317186	−	0.948363i	$-0.397262\pi$
$13$	2.28726			0.634371			0.317186	+	0.948363i	$0.397262\pi$
$14$		−	3.29112i		−	0.879588i
$15$		0			0
$16$	−4.35026			−1.08757
$17$	3.15633	−	2.65285i	0.765521	−	0.643411i
$17$	3.15633	−	2.65285i	0.765521	−	0.643411i
$18$	−11.5999			−2.73412
$19$	5.76845			1.32337			0.661687	−	0.749780i	$-0.269841\pi$
$19$	5.76845			1.32337			0.661687	+	0.749780i	$0.269841\pi$
$20$		0			0
$21$	−7.31265			−1.59575
$22$			4.23652i			0.903228i
$23$		−	1.58367i		−	0.330217i	−0.986275	−	0.165109i	$-0.947203\pi$
$23$		−	1.58367i		−	0.330217i	0.986275	−	0.165109i	$-0.0527975\pi$
$24$			8.80417i			1.79714i
$25$		0			0
$26$	3.38787			0.664417
$27$			15.9009i			3.06013i
$28$		−	0.430914i		−	0.0814352i
$29$		−	9.23509i		−	1.71491i	−0.514557	−	0.857456i	$-0.672044\pi$
$29$		−	9.23509i		−	1.71491i	0.514557	−	0.857456i	$-0.327956\pi$
$30$		0			0
$31$			1.15275i			0.207040i	0.994627	+	0.103520i	$0.0330107\pi$
$31$			1.15275i			0.207040i	−0.994627	+	0.103520i	$0.966989\pi$
$32$	−1.09332			−0.193274
$33$	9.41327			1.63864
$34$	4.67513	−	3.92939i	0.801778	−	0.673884i
$35$		0			0
$36$	−1.51881			−0.253134
$37$		−	0.514485i		−	0.0845807i	−0.999105	−	0.0422904i	$-0.986535\pi$
$37$		−	0.514485i		−	0.0845807i	0.999105	−	0.0422904i	$-0.0134655\pi$
$38$	8.54420			1.38605
$39$		−	7.52763i		−	1.20539i
$40$		0			0
$41$		−	7.09672i		−	1.10832i	−0.832410	−	0.554161i	$-0.813039\pi$
$41$		−	7.09672i		−	1.10832i	0.832410	−	0.554161i	$-0.186961\pi$
$42$	−10.8315			−1.67133
$43$	−7.89446			−1.20389			−0.601947	−	0.798536i	$-0.705608\pi$
$43$	−7.89446			−1.20389			−0.601947	+	0.798536i	$0.705608\pi$
$44$			0.554698i			0.0836239i
$45$		0			0
$46$		−	2.34572i		−	0.345857i
$47$	3.03761			0.443081			0.221541	−	0.975151i	$-0.428891\pi$
$47$	3.03761			0.443081			0.221541	+	0.975151i	$0.428891\pi$
$48$			14.3172i			2.06651i
$49$	2.06300			0.294715
$50$		0			0
$51$	−8.73084	−	10.3878i	−1.22256	−	1.45459i
$52$	0.443583			0.0615139
$53$	5.73084			0.787192			0.393596	−	0.919284i	$-0.371231\pi$
$53$	5.73084			0.787192			0.393596	+	0.919284i	$0.371231\pi$
$54$			23.5523i			3.20506i
$55$		0			0
$56$			5.94397i			0.794296i
$57$		−	18.9847i		−	2.51458i
$58$		−	13.6790i		−	1.79613i
$59$	−7.50659			−0.977274			−0.488637	−	0.872487i	$-0.662506\pi$
$59$	−7.50659			−0.977274			−0.488637	+	0.872487i	$0.662506\pi$
$60$		0			0
$61$			11.8879i			1.52209i	0.648697	+	0.761047i	$0.275314\pi$
$61$			11.8879i			1.52209i	−0.648697	+	0.761047i	$0.724686\pi$
$62$			1.70745i			0.216846i
$63$			17.4010i			2.19232i
$64$	7.08110			0.885138
$65$		0			0
$66$	13.9429			1.71625
$67$	7.35026			0.897977			0.448989	−	0.893537i	$-0.351784\pi$
$67$	7.35026			0.897977			0.448989	+	0.893537i	$0.351784\pi$
$68$	0.612127	−	0.514485i	0.0742313	−	0.0623904i
$69$	−5.21203			−0.627455
$70$		0			0
$71$			8.80417i			1.04486i	0.852681	+	0.522431i	$0.174975\pi$
$71$			8.80417i			1.04486i	−0.852681	+	0.522431i	$0.825025\pi$
$72$	20.9502			2.46900
$73$			2.65285i			0.310493i	0.987876	+	0.155246i	$0.0496171\pi$
$73$			2.65285i			0.310493i	−0.987876	+	0.155246i	$0.950383\pi$
$74$		−	0.762052i		−	0.0885867i
$75$		0			0
$76$	1.11871			0.128325
$77$	6.35519			0.724241
$78$		−	11.1499i		−	1.26248i
$79$			11.4570i			1.28902i	0.764598	+	0.644508i	$0.222938\pi$
$79$			11.4570i			1.28902i	−0.764598	+	0.644508i	$0.777062\pi$
$80$		0			0
$81$	28.8373			3.20415
$82$		−	10.5116i		−	1.16081i
$83$	3.08840			0.338996			0.169498	−	0.985531i	$-0.445785\pi$
$83$	3.08840			0.338996			0.169498	+	0.985531i	$0.445785\pi$
$84$	−1.41819			−0.154737
$85$		0			0
$86$	−11.6932			−1.26091
$87$	−30.3938			−3.25855
$88$		−	7.65142i		−	0.815644i
$89$	2.15633			0.228570			0.114285	−	0.993448i	$-0.463542\pi$
$89$	2.15633			0.228570			0.114285	+	0.993448i	$0.463542\pi$
$90$		0			0
$91$		−	5.08214i		−	0.532753i
$92$		−	0.307131i		−	0.0320206i
$93$	3.79384			0.393403
$94$	4.49929			0.464067
$95$		0			0
$96$			3.59825i			0.367245i
$97$		−	8.72060i		−	0.885443i	−0.896659	−	0.442721i	$-0.854013\pi$
$97$		−	8.72060i		−	0.885443i	0.896659	−	0.442721i	$-0.145987\pi$
$98$	3.05571			0.308673
$99$		−	22.3996i		−	2.25124i
$100$		0			0
$101$	7.24965			0.721367			0.360683	−	0.932688i	$-0.382543\pi$
$101$	7.24965			0.721367			0.360683	+	0.932688i	$0.382543\pi$
$102$	−12.9321	−	15.3864i	−1.28047	−	1.52348i
$103$	7.73813			0.762461			0.381231	−	0.924480i	$-0.375500\pi$
$103$	7.73813			0.762461			0.381231	+	0.924480i	$0.375500\pi$
$104$	−6.11871			−0.599989
$105$		0			0
$106$	8.48849			0.824475
$107$			10.5952i			1.02428i	0.858903	+	0.512138i	$0.171146\pi$
$107$			10.5952i			1.02428i	−0.858903	+	0.512138i	$0.828854\pi$
$108$			3.08376i			0.296735i
$109$			14.5408i			1.39275i	0.717676	+	0.696377i	$0.245206\pi$
$109$			14.5408i			1.39275i	−0.717676	+	0.696377i	$0.754794\pi$
$110$		0			0
$111$	−1.69323			−0.160714
$112$			9.66600i			0.913351i
$113$			4.02916i			0.379032i	0.981878	+	0.189516i	$0.0606919\pi$
$113$			4.02916i			0.379032i	−0.981878	+	0.189516i	$0.939308\pi$
$114$		−	28.1200i		−	2.63368i
$115$		0			0
$116$		−	1.79102i		−	0.166292i
$117$	−17.9126			−1.65602
$118$	−11.1187			−1.02356
$119$	−5.89446	−	7.01315i	−0.540344	−	0.642894i
$120$		0			0
$121$	2.81924			0.256294
$122$			17.6083i			1.59418i
$123$	−23.3561			−2.10595
$124$			0.223561i			0.0200764i
$125$		0			0
$126$			25.7742i			2.29615i
$127$	−3.06793			−0.272235			−0.136117	−	0.990693i	$-0.543462\pi$
$127$	−3.06793			−0.272235			−0.136117	+	0.990693i	$0.543462\pi$
$128$	12.6751			1.12033
$129$			25.9816i			2.28755i
$130$		0			0
$131$			7.42786i			0.648975i	0.945890	+	0.324487i	$0.105192\pi$
$131$			7.42786i			0.648975i	−0.945890	+	0.324487i	$0.894808\pi$
$132$	1.82558			0.158896
$133$		−	12.8171i		−	1.11139i
$134$	10.8872			0.940508
$135$		0			0
$136$	−8.44358	+	7.09672i	−0.724031	+	0.608539i
$137$	5.37565			0.459273			0.229637	−	0.973276i	$-0.426246\pi$
$137$	5.37565			0.459273			0.229637	+	0.973276i	$0.426246\pi$
$138$	−7.72004			−0.657173
$139$			4.91500i			0.416885i	0.978035	+	0.208442i	$0.0668394\pi$
$139$			4.91500i			0.416885i	−0.978035	+	0.208442i	$0.933161\pi$
$140$		0			0
$141$		−	9.99714i		−	0.841911i
$142$			13.0407i			1.09435i
$143$			6.54202i			0.547071i
$144$	34.0689			2.83907
$145$		0			0
$146$			3.92939i			0.325198i
$147$		−	6.78959i		−	0.559996i
$148$		−	0.0997774i		−	0.00820165i
$149$	−16.7381			−1.37124			−0.685621	−	0.727959i	$-0.740469\pi$
$149$	−16.7381			−1.37124			−0.685621	+	0.727959i	$0.740469\pi$
$150$		0			0
$151$	−1.81336			−0.147569			−0.0737845	−	0.997274i	$-0.523508\pi$
$151$	−1.81336			−0.147569			−0.0737845	+	0.997274i	$0.523508\pi$
$152$	−15.4314			−1.25165
$153$	−24.7186	+	20.7757i	−1.99838	+	1.67961i
$154$	9.41327			0.758543
$155$		0			0
$156$		−	1.45988i		−	0.116884i
$157$	−4.31994			−0.344769			−0.172385	−	0.985030i	$-0.555147\pi$
$157$	−4.31994			−0.344769			−0.172385	+	0.985030i	$0.555147\pi$
$158$			16.9701i			1.35007i
$159$		−	18.8609i		−	1.49576i
$160$		0			0
$161$	−3.51881			−0.277321
$162$	42.7137			3.35591
$163$		−	4.66743i		−	0.365581i	−0.983152	−	0.182791i	$-0.941487\pi$
$163$		−	4.66743i		−	0.365581i	0.983152	−	0.182791i	$-0.0585131\pi$
$164$		−	1.37631i		−	0.107472i
$165$		0			0
$166$	4.57452			0.355051
$167$		−	8.16590i		−	0.631897i	−0.948776	−	0.315948i	$-0.897677\pi$
$167$		−	8.16590i		−	0.631897i	0.948776	−	0.315948i	$-0.102323\pi$
$168$	19.5623			1.50926
$169$	−7.76845			−0.597573
$170$		0			0
$171$	−45.1754			−3.45465
$172$	−1.53102			−0.116740
$173$		−	23.4285i		−	1.78124i	−0.454750	−	0.890619i	$-0.650272\pi$
$173$		−	23.4285i		−	1.78124i	0.454750	−	0.890619i	$-0.349728\pi$
$174$	−45.0191			−3.41289
$175$		0			0
$176$		−	12.4426i		−	0.937899i
$177$			24.7051i			1.85695i
$178$	3.19394			0.239396
$179$	−14.5296			−1.08599			−0.542997	−	0.839735i	$-0.682711\pi$
$179$	−14.5296			−1.08599			−0.542997	+	0.839735i	$0.682711\pi$
$180$		0			0
$181$		−	9.40223i		−	0.698862i	−0.936962	−	0.349431i	$-0.886375\pi$
$181$		−	9.40223i		−	0.698862i	0.936962	−	0.349431i	$-0.113625\pi$
$182$		−	7.52763i		−	0.557985i
$183$	39.1246			2.89217
$184$			4.23652i			0.312320i
$185$		0			0
$186$	5.61942			0.412036
$187$	7.58769	+	9.02773i	0.554867	+	0.660173i
$188$	0.589104			0.0429648
$189$	35.3307			2.56993
$190$		0			0
$191$	−6.20711			−0.449131			−0.224565	−	0.974459i	$-0.572096\pi$
$191$	−6.20711			−0.449131			−0.224565	+	0.974459i	$0.572096\pi$
$192$		−	23.3047i		−	1.68187i
$193$			14.0937i			1.01448i	0.861804	+	0.507242i	$0.169335\pi$
$193$			14.0937i			1.01448i	−0.861804	+	0.507242i	$0.830665\pi$
$194$		−	12.9169i		−	0.927380i
$195$		0			0
$196$	0.400092			0.0285780
$197$		−	25.2195i		−	1.79682i	−0.439159	−	0.898409i	$-0.644724\pi$
$197$		−	25.2195i		−	1.79682i	0.439159	−	0.898409i	$-0.355276\pi$
$198$		−	33.1781i		−	2.35786i
$199$			12.6098i			0.893883i	0.894563	+	0.446942i	$0.147487\pi$
$199$			12.6098i			0.893883i	−0.894563	+	0.446942i	$0.852513\pi$
$200$		0			0
$201$		−	24.1906i		−	1.70627i
$202$	10.7381			0.755533
$203$	−20.5198			−1.44020
$204$	−1.69323	−	2.01458i	−0.118550	−	0.141049i
$205$		0			0
$206$	11.4617			0.798573
$207$			12.4024i			0.862028i
$208$	−9.95017			−0.689920
$209$			16.4989i			1.14126i
$210$		0			0
$211$		−	13.5954i		−	0.935945i	−0.883743	−	0.467972i	$-0.844985\pi$
$211$		−	13.5954i		−	0.935945i	0.883743	−	0.467972i	$-0.155015\pi$
$212$	1.11142			0.0763326
$213$	28.9756			1.98537
$214$			15.6935i			1.07279i
$215$		0			0
$216$		−	42.5370i		−	2.89427i
$217$	2.56134			0.173875
$218$			21.5377i			1.45872i
$219$	8.73084			0.589976
$220$		0			0
$221$	7.21933	−	6.06775i	0.485625	−	0.408161i
$222$	−2.50800			−0.168326
$223$	−18.8119			−1.25974			−0.629870	−	0.776700i	$-0.716892\pi$
$223$	−18.8119			−1.25974			−0.629870	+	0.776700i	$0.716892\pi$
$224$			2.42929i			0.162314i
$225$		0			0
$226$			5.96797i			0.396984i
$227$			3.39090i			0.225062i	0.993648	+	0.112531i	$0.0358957\pi$
$227$			3.39090i			0.225062i	−0.993648	+	0.112531i	$0.964104\pi$
$228$		−	3.68182i		−	0.243834i
$229$	20.1817			1.33365			0.666823	−	0.745216i	$-0.267654\pi$
$229$	20.1817			1.33365			0.666823	+	0.745216i	$0.267654\pi$
$230$		0			0
$231$		−	20.9157i		−	1.37615i
$232$			24.7051i			1.62197i
$233$			0.347344i			0.0227553i	0.999935	+	0.0113776i	$0.00362169\pi$
$233$			0.347344i			0.0227553i	−0.999935	+	0.0113776i	$0.996378\pi$
$234$	−26.5320			−1.73445
$235$		0			0
$236$	−1.45580			−0.0947646
$237$	37.7064			2.44929
$238$	−8.73084	−	10.3878i	−0.565936	−	0.673344i
$239$	3.94921			0.255453			0.127727	−	0.991809i	$-0.459232\pi$
$239$	3.94921			0.255453			0.127727	+	0.991809i	$0.459232\pi$
$240$		0			0
$241$			8.47303i			0.545796i	0.962043	+	0.272898i	$0.0879822\pi$
$241$			8.47303i			0.545796i	−0.962043	+	0.272898i	$0.912018\pi$
$242$	4.17584			0.268433
$243$		−	47.2044i		−	3.02816i
$244$			2.30551i			0.147595i
$245$		0			0
$246$	−34.5950			−2.20570
$247$	13.1939			0.839510
$248$		−	3.08376i		−	0.195819i
$249$		−	10.1643i		−	0.644135i
$250$		0			0
$251$	9.22425			0.582230			0.291115	−	0.956688i	$-0.405974\pi$
$251$	9.22425			0.582230			0.291115	+	0.956688i	$0.405974\pi$
$252$			3.37469i			0.212585i
$253$	4.52961			0.284774
$254$	−4.54420			−0.285128
$255$		0			0
$256$	4.61213			0.288258
$257$	−6.78067			−0.422967			−0.211483	−	0.977382i	$-0.567829\pi$
$257$	−6.78067			−0.422967			−0.211483	+	0.977382i	$0.567829\pi$
$258$			38.4838i			2.39590i
$259$	−1.14315			−0.0710320
$260$		0			0
$261$			72.3242i			4.47675i
$262$			11.0021i			0.679712i
$263$	−16.7757			−1.03444			−0.517218	−	0.855853i	$-0.673033\pi$
$263$	−16.7757			−1.03444			−0.517218	+	0.855853i	$0.673033\pi$
$264$	−25.1817			−1.54983
$265$		0			0
$266$		−	18.9847i		−	1.16402i
$267$		−	7.09672i		−	0.434312i
$268$	1.42548			0.0870753
$269$		−	1.02897i		−	0.0627374i	−0.999508	−	0.0313687i	$-0.990013\pi$
$269$		−	1.02897i		−	0.0627374i	0.999508	−	0.0313687i	$-0.00998660\pi$
$270$		0			0
$271$	6.93207			0.421093			0.210547	−	0.977584i	$-0.432476\pi$
$271$	6.93207			0.421093			0.210547	+	0.977584i	$0.432476\pi$
$272$	−13.7308	+	11.5406i	−0.832555	+	0.699751i
$273$	−16.7259			−1.01230
$274$	7.96239			0.481025
$275$		0			0
$276$	−1.01080			−0.0608433
$277$		−	23.8432i		−	1.43260i	−0.697792	−	0.716300i	$-0.745834\pi$
$277$		−	23.8432i		−	1.43260i	0.697792	−	0.716300i	$-0.254166\pi$
$278$			7.28007i			0.436629i
$279$		−	9.02773i		−	0.540476i
$280$		0			0
$281$	−9.75131			−0.581714			−0.290857	−	0.956766i	$-0.593940\pi$
$281$	−9.75131			−0.581714			−0.290857	+	0.956766i	$0.593940\pi$
$282$		−	14.8077i		−	0.881786i
$283$			30.2181i			1.79628i	0.439709	+	0.898140i	$0.355082\pi$
$283$			30.2181i			1.79628i	−0.439709	+	0.898140i	$0.644918\pi$
$284$			1.70745i			0.101319i
$285$		0			0
$286$			9.69001i			0.572982i
$287$	−15.7685			−0.930782
$288$	8.56230			0.504538
$289$	2.92478	−	16.7465i	0.172046	−	0.985089i
$290$		0			0
$291$	−28.7005			−1.68245
$292$			0.514485i			0.0301079i
$293$	4.26187			0.248981			0.124490	−	0.992221i	$-0.460270\pi$
$293$	4.26187			0.248981			0.124490	+	0.992221i	$0.460270\pi$
$294$		−	10.0567i		−	0.586519i
$295$		0			0
$296$			1.37631i			0.0799966i
$297$	−45.4798			−2.63900
$298$	−24.7924			−1.43619
$299$		−	3.62225i		−	0.209480i
$300$		0			0
$301$			17.5410i			1.01105i
$302$	−2.68594			−0.154558
$303$		−	23.8594i		−	1.37069i
$304$	−25.0943			−1.43926
$305$		0			0
$306$	−36.6131	+	30.7728i	−2.09303	+	1.75916i
$307$	−6.96239			−0.397365			−0.198682	−	0.980064i	$-0.563666\pi$
$307$	−6.96239			−0.397365			−0.198682	+	0.980064i	$0.563666\pi$
$308$	1.23250			0.0702284
$309$		−	25.4671i		−	1.44877i
$310$		0			0
$311$		−	14.4170i		−	0.817513i	−0.912643	−	0.408757i	$-0.865963\pi$
$311$		−	14.4170i		−	0.817513i	0.912643	−	0.408757i	$-0.134037\pi$
$312$			20.1374i			1.14006i
$313$			11.1259i			0.628872i	0.949279	+	0.314436i	$0.101815\pi$
$313$			11.1259i			0.628872i	−0.949279	+	0.314436i	$0.898185\pi$
$314$	−6.39868			−0.361098
$315$		0			0
$316$			2.22194i			0.124994i
$317$			4.09653i			0.230084i	0.993361	+	0.115042i	$0.0367002\pi$
$317$			4.09653i			0.230084i	−0.993361	+	0.115042i	$0.963300\pi$
$318$		−	27.9366i		−	1.56661i
$319$	26.4142			1.47891
$320$		0			0
$321$	34.8700			1.94625
$322$	−5.21203			−0.290455
$323$	18.2071	−	15.3028i	1.01307	−	0.851473i
$324$	5.59261			0.310701
$325$		0			0
$326$		−	6.91337i		−	0.382896i
$327$	47.8554			2.64641
$328$			18.9847i			1.04825i
$329$		−	6.74938i		−	0.372105i
$330$		0			0
$331$	30.4894			1.67585			0.837926	−	0.545784i	$-0.183768\pi$
$331$	30.4894			1.67585			0.837926	+	0.545784i	$0.183768\pi$
$332$	0.598953			0.0328718
$333$			4.02916i			0.220797i
$334$		−	12.0953i		−	0.661825i
$335$		0			0
$336$	31.8119			1.73548
$337$			27.6055i			1.50377i	0.659296	+	0.751883i	$0.270854\pi$
$337$			27.6055i			1.50377i	−0.659296	+	0.751883i	$0.729146\pi$
$338$	−11.5066			−0.625876
$339$	13.2605			0.720209
$340$		0			0
$341$	−3.29711			−0.178548
$342$	−66.9135			−3.61827
$343$		−	20.1374i		−	1.08732i
$344$	21.1187			1.13864
$345$		0			0
$346$		−	34.7022i		−	1.86560i
$347$		−	1.77481i		−	0.0952770i	−0.998865	−	0.0476385i	$-0.984830\pi$
$347$		−	1.77481i		−	0.0952770i	0.998865	−	0.0476385i	$-0.0151695\pi$
$348$	−5.89446			−0.315976
$349$	11.1319			0.595876			0.297938	−	0.954585i	$-0.403701\pi$
$349$	11.1319			0.595876			0.297938	+	0.954585i	$0.403701\pi$
$350$		0			0
$351$			36.3694i			1.94126i
$352$		−	3.12712i		−	0.166676i
$353$	−27.4241			−1.45964			−0.729818	−	0.683642i	$-0.760395\pi$
$353$	−27.4241			−1.45964			−0.729818	+	0.683642i	$0.760395\pi$
$354$			36.5930i			1.94490i
$355$		0			0
$356$	0.418190			0.0221640
$357$	−23.0811	+	19.3994i	−1.22158	+	1.02672i
$358$	−21.5212			−1.13743
$359$	1.65703			0.0874548			0.0437274	−	0.999043i	$-0.486077\pi$
$359$	1.65703			0.0874548			0.0437274	+	0.999043i	$0.486077\pi$
$360$		0			0
$361$	14.2750			0.751318
$362$		−	13.9265i		−	0.731962i
$363$		−	9.27844i		−	0.486992i
$364$		−	0.985612i		−	0.0516601i
$365$		0			0
$366$	57.9511			3.02915
$367$		−	20.8321i		−	1.08743i	−0.839271	−	0.543713i	$-0.817018\pi$
$367$		−	20.8321i		−	1.08743i	0.839271	−	0.543713i	$-0.182982\pi$
$368$			6.88937i			0.359133i
$369$			55.5777i			2.89326i
$370$		0			0
$371$		−	12.7336i		−	0.661093i
$372$	0.735765			0.0381476
$373$	36.6810			1.89927			0.949635	−	0.313357i	$-0.101454\pi$
$373$	36.6810			1.89927			0.949635	+	0.313357i	$0.101454\pi$
$374$	11.2388	+	13.3718i	0.581147	+	0.691441i
$375$		0			0
$376$	−8.12601			−0.419067
$377$		−	21.1230i		−	1.08789i
$378$	52.3317			2.69165
$379$			17.6245i			0.905312i	0.891685	+	0.452656i	$0.149523\pi$
$379$			17.6245i			0.905312i	−0.891685	+	0.452656i	$0.850477\pi$
$380$		0			0
$381$			10.0969i			0.517281i
$382$	−9.19394			−0.470403
$383$	−11.5066			−0.587959			−0.293980	−	0.955812i	$-0.594980\pi$
$383$	−11.5066			−0.587959			−0.293980	+	0.955812i	$0.594980\pi$
$384$		−	41.7153i		−	2.12878i
$385$		0			0
$386$			20.8755i			1.06253i
$387$	61.8251			3.14275
$388$		−	1.69124i		−	0.0858599i
$389$	−10.6194			−0.538426			−0.269213	−	0.963081i	$-0.586764\pi$
$389$	−10.6194			−0.538426			−0.269213	+	0.963081i	$0.586764\pi$
$390$		0			0
$391$	−4.20123	−	4.99857i	−0.212465	−	0.252788i
$392$	−5.51881			−0.278742
$393$	24.4460			1.23314
$394$		−	37.3550i		−	1.88192i
$395$		0			0
$396$		−	4.34409i		−	0.218299i
$397$			12.3026i			0.617452i	0.951151	+	0.308726i	$0.0999026\pi$
$397$			12.3026i			0.617452i	−0.951151	+	0.308726i	$0.900097\pi$
$398$			18.6775i			0.936220i
$399$	−42.1827			−2.11178
$400$		0			0
$401$			16.6791i			0.832917i	0.909155	+	0.416458i	$0.136729\pi$
$401$			16.6791i			0.832917i	−0.909155	+	0.416458i	$0.863271\pi$
$402$		−	35.8309i		−	1.78709i
$403$			2.63664i			0.131341i
$404$	1.40597			0.0699497
$405$		0			0
$406$	−30.3938			−1.50842
$407$	1.47153			0.0729411
$408$	23.3561	+	27.7888i	1.15630	+	1.37575i
$409$	−35.9438			−1.77731			−0.888654	−	0.458578i	$-0.848359\pi$
$409$	−35.9438			−1.77731			−0.888654	+	0.458578i	$0.848359\pi$
$410$		0			0
$411$		−	17.6919i		−	0.872678i
$412$	1.50071			0.0739345
$413$			16.6791i			0.820727i
$414$			18.3704i			0.902856i
$415$		0			0
$416$	−2.50071			−0.122607
$417$	16.1758			0.792134
$418$			24.4381i			1.19531i
$419$			26.8032i			1.30942i	0.755879	+	0.654711i	$0.227210\pi$
$419$			26.8032i			1.30942i	−0.755879	+	0.654711i	$0.772790\pi$
$420$		0			0
$421$	7.24965			0.353326			0.176663	−	0.984271i	$-0.443470\pi$
$421$	7.24965			0.353326			0.176663	+	0.984271i	$0.443470\pi$
$422$		−	20.1374i		−	0.980274i
$423$	−23.7889			−1.15666
$424$	−15.3307			−0.744527
$425$		0			0
$426$	42.9184			2.07941
$427$	26.4142			1.27827
$428$			2.05480i			0.0993223i
$429$	21.5306			1.03951
$430$		0			0
$431$		−	17.6517i		−	0.850252i	−0.905134	−	0.425126i	$-0.860230\pi$
$431$		−	17.6517i		−	0.850252i	0.905134	−	0.425126i	$-0.139770\pi$
$432$		−	69.1730i		−	3.32809i
$433$	3.37073			0.161987			0.0809935	−	0.996715i	$-0.474191\pi$
$433$	3.37073			0.161987			0.0809935	+	0.996715i	$0.474191\pi$
$434$	3.79384			0.182110
$435$		0			0
$436$			2.81999i			0.135053i
$437$		−	9.13531i		−	0.437001i
$438$	12.9321			0.617918
$439$			0.945399i			0.0451214i	0.999745	+	0.0225607i	$0.00718191\pi$
$439$			0.945399i			0.0451214i	−0.999745	+	0.0225607i	$0.992818\pi$
$440$		0			0
$441$	−16.1563			−0.769349
$442$	10.6932	−	8.98752i	0.508625	−	0.427493i
$443$	5.71767			0.271655			0.135827	−	0.990733i	$-0.456631\pi$
$443$	5.71767			0.271655			0.135827	+	0.990733i	$0.456631\pi$
$444$	−0.328379			−0.0155842
$445$		0			0
$446$	−27.8641			−1.31941
$447$			55.0872i			2.60553i
$448$		−	15.7338i		−	0.743350i
$449$		−	20.6759i		−	0.975756i	−0.872912	−	0.487878i	$-0.837771\pi$
$449$		−	20.6759i		−	0.975756i	0.872912	−	0.487878i	$-0.162229\pi$
$450$		0			0
$451$	20.2981			0.955798
$452$			0.781402i			0.0367541i
$453$			5.96797i			0.280400i
$454$			5.02257i			0.235721i
$455$		0			0
$456$			50.7864i			2.37829i
$457$	−15.0689			−0.704893			−0.352446	−	0.935832i	$-0.614650\pi$
$457$	−15.0689			−0.704893			−0.352446	+	0.935832i	$0.614650\pi$
$458$	29.8930			1.39681
$459$	42.1827	+	50.1884i	1.96892	+	2.34259i
$460$		0			0
$461$	−22.3127			−1.03920			−0.519602	−	0.854409i	$-0.673920\pi$
$461$	−22.3127			−1.03920			−0.519602	+	0.854409i	$0.673920\pi$
$462$		−	30.9802i		−	1.44133i
$463$	19.5574			0.908908			0.454454	−	0.890770i	$-0.349834\pi$
$463$	19.5574			0.908908			0.454454	+	0.890770i	$0.349834\pi$
$464$			40.1750i			1.86508i
$465$		0			0
$466$			0.514485i			0.0238330i
$467$	−16.1260			−0.746223			−0.373111	−	0.927787i	$-0.621709\pi$
$467$	−16.1260			−0.746223			−0.373111	+	0.927787i	$0.621709\pi$
$468$	−3.47390			−0.160581
$469$		−	16.3318i		−	0.754133i
$470$		0			0
$471$			14.2174i			0.655105i
$472$	20.0811			0.924308
$473$		−	22.5798i		−	1.03822i
$474$	55.8505			2.56530
$475$		0			0
$476$	−1.14315	−	1.36011i	−0.0523963	−	0.0623404i
$477$	−44.8808			−2.05495
$478$	5.84955			0.267552
$479$		−	5.44569i		−	0.248820i	−0.992231	−	0.124410i	$-0.960296\pi$
$479$		−	5.44569i		−	0.248820i	0.992231	−	0.124410i	$-0.0397038\pi$
$480$		0			0
$481$		−	1.17676i		−	0.0536556i
$482$			12.5502i			0.571646i
$483$			11.5808i			0.526945i
$484$	0.546753			0.0248524
$485$		0			0
$486$		−	69.9189i		−	3.17158i
$487$			1.41653i			0.0641890i	0.999485	+	0.0320945i	$0.0102177\pi$
$487$			1.41653i			0.0641890i	−0.999485	+	0.0320945i	$0.989782\pi$
$488$		−	31.8018i		−	1.43960i
$489$	−15.3611			−0.694651
$490$		0			0
$491$	21.1636			0.955101			0.477550	−	0.878604i	$-0.341525\pi$
$491$	21.1636			0.955101			0.477550	+	0.878604i	$0.341525\pi$
$492$	−4.52961			−0.204211
$493$	−24.4993	−	29.1489i	−1.10339	−	1.31280i
$494$	19.5428			0.879271
$495$		0			0
$496$		−	5.01478i		−	0.225170i
$497$	19.5623			0.877489
$498$		−	15.0553i		−	0.674643i
$499$			34.6782i			1.55241i	0.630481	+	0.776205i	$0.282858\pi$
$499$			34.6782i			1.55241i	−0.630481	+	0.776205i	$0.717142\pi$
$500$		0			0
$501$	−26.8749			−1.20068
$502$	13.6629			0.609806
$503$			25.6071i			1.14176i	0.821032	+	0.570882i	$0.193399\pi$
$503$			25.6071i			1.14176i	−0.821032	+	0.570882i	$0.806601\pi$
$504$		−	46.5499i		−	2.07350i
$505$		0			0
$506$	6.70923			0.298262
$507$			25.5669i			1.13547i
$508$	−0.594984			−0.0263981
$509$	39.3390			1.74367			0.871835	−	0.489799i	$-0.162930\pi$
$509$	39.3390			1.74367			0.871835	+	0.489799i	$0.162930\pi$
$510$		0			0
$511$	5.89446			0.260756
$512$	−18.5188			−0.818423
$513$			91.7235i			4.04969i
$514$	−10.0435			−0.442999
$515$		0			0
$516$			5.03878i			0.221820i
$517$			8.68819i			0.382106i
$518$	−1.69323			−0.0743962
$519$	−77.1060			−3.38458
$520$		0			0
$521$		−	5.72041i		−	0.250616i	−0.992118	−	0.125308i	$-0.960008\pi$
$521$		−	5.72041i		−	0.250616i	0.992118	−	0.125308i	$-0.0399918\pi$
$522$			107.126i			4.68878i
$523$	21.6873			0.948322			0.474161	−	0.880438i	$-0.342752\pi$
$523$	21.6873			0.948322			0.474161	+	0.880438i	$0.342752\pi$
$524$			1.44053i			0.0629300i
$525$		0			0
$526$	−24.8481			−1.08343
$527$	3.05808	+	3.63846i	0.133212	+	0.158494i
$528$	−40.9502			−1.78213
$529$	20.4920			0.890956
$530$		0			0
$531$	58.7875			2.55116
$532$		−	2.48571i		−	0.107769i
$533$		−	16.2320i		−	0.703087i
$534$		−	10.5116i		−	0.454882i
$535$		0			0
$536$	−19.6629			−0.849308
$537$			47.8187i			2.06353i
$538$		−	1.52410i		−	0.0657088i
$539$			5.90061i			0.254157i
$540$		0			0
$541$			35.4836i			1.52556i	0.646659	+	0.762780i	$0.276166\pi$
$541$			35.4836i			1.52556i	−0.646659	+	0.762780i	$0.723834\pi$
$542$	10.2677			0.441037
$543$	−30.9438			−1.32793
$544$	−3.45088	+	2.90042i	−0.147955	+	0.124354i
$545$		0			0
$546$	−24.7743			−1.06024
$547$		−	23.5121i		−	1.00530i	−0.864489	−	0.502652i	$-0.832358\pi$
$547$		−	23.5121i		−	1.00530i	0.864489	−	0.502652i	$-0.167642\pi$
$548$	1.04254			0.0445349
$549$		−	93.0998i		−	3.97340i
$550$		0			0
$551$		−	53.2721i		−	2.26947i
$552$	13.9429			0.593448
$553$	25.4568			1.08253
$554$		−	35.3165i		−	1.50045i
$555$		0			0
$556$			0.953198i			0.0404246i
$557$	−22.2120			−0.941154			−0.470577	−	0.882359i	$-0.655954\pi$
$557$	−22.2120			−0.941154			−0.470577	+	0.882359i	$0.655954\pi$
$558$		−	13.3718i		−	0.566074i
$559$	−18.0567			−0.763716
$560$		0			0
$561$	29.7113	−	24.9720i	1.25441	−	1.05432i
$562$	−14.4436			−0.609266
$563$	−6.85685			−0.288982			−0.144491	−	0.989506i	$-0.546154\pi$
$563$	−6.85685			−0.288982			−0.144491	+	0.989506i	$0.546154\pi$
$564$		−	1.93881i		−	0.0816386i
$565$		0			0
$566$			44.7589i			1.88136i
$567$		−	64.0747i		−	2.69088i
$568$		−	23.5523i		−	0.988233i
$569$	26.7513			1.12147			0.560737	−	0.827994i	$-0.310518\pi$
$569$	26.7513			1.12147			0.560737	+	0.827994i	$0.310518\pi$
$570$		0			0
$571$			34.0639i			1.42553i	0.701402	+	0.712766i	$0.252558\pi$
$571$			34.0639i			1.42553i	−0.701402	+	0.712766i	$0.747442\pi$
$572$			1.26874i			0.0530486i
$573$			20.4283i			0.853406i
$574$	−23.3561			−0.974867
$575$		0			0
$576$	−55.4553			−2.31064
$577$	−29.0797			−1.21060			−0.605302	−	0.795996i	$-0.706947\pi$
$577$	−29.0797			−1.21060			−0.605302	+	0.795996i	$0.706947\pi$
$578$	4.33216	−	24.8048i	0.180194	−	1.03175i
$579$	46.3839			1.92765
$580$		0			0
$581$		−	6.86222i		−	0.284693i
$582$	−42.5111			−1.76214
$583$			16.3914i			0.678861i
$584$		−	7.09672i		−	0.293664i
$585$		0			0
$586$	6.31265			0.260773
$587$	−35.2506			−1.45495			−0.727474	−	0.686135i	$-0.759306\pi$
$587$	−35.2506			−1.45495			−0.727474	+	0.686135i	$0.759306\pi$
$588$		−	1.31675i		−	0.0543018i
$589$			6.64960i			0.273992i
$590$		0			0
$591$	−83.0005			−3.41418
$592$			2.23814i			0.0919871i
$593$	−29.0811			−1.19422			−0.597109	−	0.802160i	$-0.703684\pi$
$593$	−29.0811			−1.19422			−0.597109	+	0.802160i	$0.703684\pi$
$594$	−67.3644			−2.76399
$595$		0			0
$596$	−3.24614			−0.132967
$597$	41.5002			1.69849
$598$		−	5.36526i		−	0.219402i
$599$	−30.2823			−1.23730			−0.618651	−	0.785666i	$-0.712321\pi$
$599$	−30.2823			−1.23730			−0.618651	+	0.785666i	$0.712321\pi$
$600$		0			0
$601$		−	6.16753i		−	0.251579i	−0.992057	−	0.125789i	$-0.959854\pi$
$601$		−	6.16753i		−	0.251579i	0.992057	−	0.125789i	$-0.0401463\pi$
$602$			25.9816i			1.05893i
$603$	−57.5633			−2.34416
$604$	−0.351676			−0.0143095
$605$		0			0
$606$		−	35.3405i		−	1.43561i
$607$			10.0405i			0.407531i	0.979020	+	0.203766i	$0.0653180\pi$
$607$			10.0405i			0.407531i	−0.979020	+	0.203766i	$0.934682\pi$
$608$	−6.30677			−0.255773
$609$			67.5329i			2.73657i
$610$		0			0
$611$	6.94780			0.281078
$612$	−4.79384	+	4.02916i	−0.193780	+	0.162869i
$613$	28.0494			1.13290			0.566452	−	0.824095i	$-0.308316\pi$
$613$	28.0494			1.13290			0.566452	+	0.824095i	$0.308316\pi$
$614$	−10.3127			−0.416185
$615$		0			0
$616$	−17.0010			−0.684988
$617$			24.4575i			0.984622i	0.870419	+	0.492311i	$0.163848\pi$
$617$			24.4575i			0.984622i	−0.870419	+	0.492311i	$0.836152\pi$
$618$		−	37.7217i		−	1.51739i
$619$		−	35.6042i		−	1.43106i	−0.698584	−	0.715528i	$-0.746186\pi$
$619$		−	35.6042i		−	1.43106i	0.698584	−	0.715528i	$-0.253814\pi$
$620$		0			0
$621$	25.1817			1.01051
$622$		−	21.3544i		−	0.856233i
$623$		−	4.79121i		−	0.191956i
$624$			32.7472i			1.31094i
$625$		0			0
$626$			16.4796i			0.658657i
$627$	54.3000			2.16853
$628$	−0.837795			−0.0334317
$629$	−1.36485	−	1.62388i	−0.0544201	−	0.0647484i
$630$		0			0
$631$	8.67021			0.345155			0.172578	−	0.984996i	$-0.444790\pi$
$631$	8.67021			0.345155			0.172578	+	0.984996i	$0.444790\pi$
$632$		−	30.6490i		−	1.21915i
$633$	−44.7440			−1.77841
$634$			6.06775i			0.240981i
$635$		0			0
$636$		−	3.65781i		−	0.145042i
$637$	4.71862			0.186959
$638$	39.1246			1.54896
$639$		−	68.9495i		−	2.72760i
$640$		0			0
$641$			40.0079i			1.58022i	0.612967	+	0.790108i	$0.289976\pi$
$641$			40.0079i			1.58022i	−0.612967	+	0.790108i	$0.710024\pi$
$642$	51.6493			2.03843
$643$			20.5923i			0.812082i	0.913855	+	0.406041i	$0.133091\pi$
$643$			20.5923i			0.812082i	−0.913855	+	0.406041i	$0.866909\pi$
$644$	−0.682425			−0.0268913
$645$		0			0
$646$	26.9683	−	22.6665i	1.06105	−	0.891800i
$647$	34.0205			1.33748			0.668741	−	0.743495i	$-0.266833\pi$
$647$	34.0205			1.33748			0.668741	+	0.743495i	$0.266833\pi$
$648$	−77.1436			−3.03049
$649$		−	21.4704i		−	0.842786i
$650$		0			0
$651$		−	8.42968i		−	0.330385i
$652$		−	0.905186i		−	0.0354498i
$653$		−	43.7895i		−	1.71362i	−0.515636	−	0.856808i	$-0.672444\pi$
$653$		−	43.7895i		−	1.71362i	0.515636	−	0.856808i	$-0.327556\pi$
$654$	70.8832			2.77175
$655$		0			0
$656$			30.8726i			1.20537i
$657$		−	20.7757i		−	0.810536i
$658$		−	9.99714i		−	0.389729i
$659$	−21.8192			−0.849957			−0.424978	−	0.905203i	$-0.639718\pi$
$659$	−21.8192			−0.849957			−0.424978	+	0.905203i	$0.639718\pi$
$660$		0			0
$661$	−36.1002			−1.40413			−0.702067	−	0.712111i	$-0.747739\pi$
$661$	−36.1002			−1.40413			−0.702067	+	0.712111i	$0.747739\pi$
$662$	45.1608			1.75522
$663$	−19.9697	−	23.7597i	−0.775558	−	0.922749i
$664$	−8.26187			−0.320623
$665$		0			0
$666$			5.96797i			0.231254i
$667$	−14.6253			−0.566294
$668$		−	1.58367i		−	0.0612739i
$669$			61.9123i			2.39367i
$670$		0			0
$671$	−34.0019			−1.31263
$672$	7.99508			0.308417
$673$		−	15.8847i		−	0.612310i	−0.951982	−	0.306155i	$-0.900957\pi$
$673$		−	15.8847i		−	0.612310i	0.951982	−	0.306155i	$-0.0990425\pi$
$674$			40.8891i			1.57499i
$675$		0			0
$676$	−1.50659			−0.0579457
$677$			22.3322i			0.858296i	0.903234	+	0.429148i	$0.141186\pi$
$677$			22.3322i			0.858296i	−0.903234	+	0.429148i	$0.858814\pi$
$678$	19.6413			0.754320
$679$	−19.3766			−0.743606
$680$		0			0
$681$	11.1598			0.427646
$682$	−4.88366			−0.187005
$683$		−	35.6476i		−	1.36402i	−0.731344	−	0.682009i	$-0.761107\pi$
$683$		−	35.6476i		−	1.36402i	0.731344	−	0.682009i	$-0.238893\pi$
$684$	−8.76116			−0.334991
$685$		0			0
$686$		−	29.8274i		−	1.13882i
$687$		−	66.4204i		−	2.53410i
$688$	34.3430			1.30931
$689$	13.1079			0.499372
$690$		0			0
$691$		−	8.80417i		−	0.334926i	−0.985878	−	0.167463i	$-0.946442\pi$
$691$		−	8.80417i		−	0.334926i	0.985878	−	0.167463i	$-0.0535575\pi$
$692$		−	4.54365i		−	0.172724i
$693$	−49.7704			−1.89062
$694$		−	2.62884i		−	0.0997895i
$695$		0			0
$696$	81.3073			3.08194
$697$	−18.8265	−	22.3996i	−0.713106	−	0.848444i
$698$	16.4885			0.624098
$699$	1.14315			0.0432380
$700$		0			0
$701$	34.8832			1.31752			0.658760	−	0.752353i	$-0.271081\pi$
$701$	34.8832			1.31752			0.658760	+	0.752353i	$0.271081\pi$
$702$			53.8702i			2.03320i
$703$		−	2.96778i		−	0.111932i
$704$			20.2534i			0.763328i
$705$		0			0
$706$	−40.6204			−1.52877
$707$		−	16.1082i		−	0.605813i
$708$			4.79121i			0.180065i
$709$		−	41.3038i		−	1.55120i	−0.631227	−	0.775598i	$-0.717448\pi$
$709$		−	41.3038i		−	1.55120i	0.631227	−	0.775598i	$-0.282552\pi$
$710$		0			0
$711$		−	89.7252i		−	3.36496i
$712$	−5.76845			−0.216182
$713$	1.82558			0.0683684
$714$	−34.1876	+	28.7342i	−1.27944	+	1.07535i
$715$		0			0
$716$	−2.81782			−0.105307
$717$		−	12.9973i		−	0.485394i
$718$	2.45439			0.0915969
$719$			8.32990i			0.310653i	0.987863	+	0.155326i	$0.0496429\pi$
$719$			8.32990i			0.310653i	−0.987863	+	0.155326i	$0.950357\pi$
$720$		0			0
$721$		−	17.1936i		−	0.640324i
$722$	21.1441			0.786902
$723$	27.8858			1.03708
$724$		−	1.82344i		−	0.0677674i
$725$		0			0
$726$		−	13.7432i		−	0.510057i
$727$	−28.8470			−1.06988			−0.534938	−	0.844891i	$-0.679665\pi$
$727$	−28.8470			−1.06988			−0.534938	+	0.844891i	$0.679665\pi$
$728$			13.5954i			0.503879i
$729$	−68.8432			−2.54975
$730$		0			0
$731$	−24.9175	+	20.9428i	−0.921606	+	0.774598i
$732$	7.58769			0.280449
$733$	14.1128			0.521269			0.260635	−	0.965437i	$-0.416068\pi$
$733$	14.1128			0.521269			0.260635	+	0.965437i	$0.416068\pi$
$734$		−	30.8564i		−	1.13893i
$735$		0			0
$736$			1.73146i			0.0638223i
$737$			21.0232i			0.774401i
$738$			82.3213i			3.03029i
$739$	−30.2981			−1.11453			−0.557266	−	0.830334i	$-0.688150\pi$
$739$	−30.2981			−1.11453			−0.557266	+	0.830334i	$0.688150\pi$
$740$		0			0
$741$		−	43.4228i		−	1.59518i
$742$		−	18.8609i		−	0.692404i
$743$		−	9.45865i		−	0.347004i	−0.984834	−	0.173502i	$-0.944492\pi$
$743$		−	9.45865i		−	0.347004i	0.984834	−	0.173502i	$-0.0555083\pi$
$744$	−10.1490			−0.372082
$745$		0			0
$746$	54.3317			1.98923
$747$	−24.1866			−0.884943
$748$	1.47153	+	1.75081i	0.0538045	+	0.0640159i
$749$	23.5418			0.860200
$750$		0			0
$751$		−	42.6879i		−	1.55770i	−0.627208	−	0.778852i	$-0.715802\pi$
$751$		−	42.6879i		−	1.55770i	0.627208	−	0.778852i	$-0.284198\pi$
$752$	−13.2144			−0.481880
$753$		−	30.3581i		−	1.10631i
$754$		−	31.2873i		−	1.13942i
$755$		0			0
$756$	6.85192			0.249202
$757$	36.3938			1.32275			0.661377	−	0.750054i	$-0.269972\pi$
$757$	36.3938			1.32275			0.661377	+	0.750054i	$0.269972\pi$
$758$			26.1054i			0.948190i
$759$		−	14.9075i		−	0.541107i
$760$		0			0
$761$	−0.555002			−0.0201188			−0.0100594	−	0.999949i	$-0.503202\pi$
$761$	−0.555002			−0.0201188			−0.0100594	+	0.999949i	$0.503202\pi$
$762$			14.9555i			0.541780i
$763$	32.3087			1.16965
$764$	−1.20379			−0.0435514
$765$		0			0
$766$	−17.0435			−0.615806
$767$	−17.1695			−0.619955
$768$		−	15.1791i		−	0.547727i
$769$	−6.56959			−0.236906			−0.118453	−	0.992960i	$-0.537793\pi$
$769$	−6.56959			−0.236906			−0.118453	+	0.992960i	$0.537793\pi$
$770$		0			0
$771$			22.3160i			0.803691i
$772$			2.73328i			0.0983728i
$773$	28.9995			1.04304			0.521520	−	0.853239i	$-0.325365\pi$
$773$	28.9995			1.04304			0.521520	+	0.853239i	$0.325365\pi$
$774$	91.5750			3.29160
$775$		0			0
$776$			23.3287i			0.837453i
$777$			3.76225i			0.134970i
$778$	−15.7294			−0.563927
$779$		−	40.9371i		−	1.46672i
$780$		0			0
$781$	−25.1817			−0.901073
$782$	−6.22284	−	7.40385i	−0.222528	−	0.264761i
$783$	146.846			5.24785
$784$	−8.97461			−0.320522
$785$		0			0
$786$	36.2092			1.29154
$787$		−	5.08214i		−	0.181159i	−0.995889	−	0.0905793i	$-0.971128\pi$
$787$		−	5.08214i		−	0.181159i	0.995889	−	0.0905793i	$-0.0288719\pi$
$788$		−	4.89099i		−	0.174234i
$789$			55.2110i			1.96556i
$790$		0			0
$791$	8.95254			0.318316

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)

\(f(q)\)	\(=\)	\(q+1.48119 q^{2} -3.29112i q^{3} +0.193937 q^{4} -4.87478i q^{6} -2.22194i q^{7} -2.67513 q^{8} -7.83146 q^{9} +2.86020i q^{11} -0.638268i q^{12} +2.28726 q^{13} -3.29112i q^{14} -4.35026 q^{16} +(3.15633 - 2.65285i) q^{17} -11.5999 q^{18} +5.76845 q^{19} -7.31265 q^{21} +4.23652i q^{22} -1.58367i q^{23} +8.80417i q^{24} +3.38787 q^{26} +15.9009i q^{27} -0.430914i q^{28} -9.23509i q^{29} +1.15275i q^{31} -1.09332 q^{32} +9.41327 q^{33} +(4.67513 - 3.92939i) q^{34} -1.51881 q^{36} -0.514485i q^{37} +8.54420 q^{38} -7.52763i q^{39} -7.09672i q^{41} -10.8315 q^{42} -7.89446 q^{43} +0.554698i q^{44} -2.34572i q^{46} +3.03761 q^{47} +14.3172i q^{48} +2.06300 q^{49} +(-8.73084 - 10.3878i) q^{51} +0.443583 q^{52} +5.73084 q^{53} +23.5523i q^{54} +5.94397i q^{56} -18.9847i q^{57} -13.6790i q^{58} -7.50659 q^{59} +11.8879i q^{61} +1.70745i q^{62} +17.4010i q^{63} +7.08110 q^{64} +13.9429 q^{66} +7.35026 q^{67} +(0.612127 - 0.514485i) q^{68} -5.21203 q^{69} +8.80417i q^{71} +20.9502 q^{72} +2.65285i q^{73} -0.762052i q^{74} +1.11871 q^{76} +6.35519 q^{77} -11.1499i q^{78} +11.4570i q^{79} +28.8373 q^{81} -10.5116i q^{82} +3.08840 q^{83} -1.41819 q^{84} -11.6932 q^{86} -30.3938 q^{87} -7.65142i q^{88} +2.15633 q^{89} -5.08214i q^{91} -0.307131i q^{92} +3.79384 q^{93} +4.49929 q^{94} +3.59825i q^{96} -8.72060i q^{97} +3.05571 q^{98} -22.3996i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 2 q^{4} - 6 q^{8} - 16 q^{9} + 2 q^{13} - 6 q^{16} - 2 q^{17} - 16 q^{18} + 12 q^{19} - 2 q^{21} + 22 q^{26} + 6 q^{32} + 28 q^{33} + 18 q^{34} - 20 q^{36} + 32 q^{38} - 34 q^{42} - 8 q^{43}+ \cdots - 16 q^{98}+O(q^{100}) \) 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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