LMFDB - Embedded newform 245.4.b.d.99.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,4,Mod(99,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("245.99");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	245.b (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:	$14.4554679514$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 $ x^10 + 55x^8 + 983x^6 + 6409x^4 + 13560x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.7
Root			$2.67516i$ of defining polynomial
Character	$\chi$	$=$	245.99
Dual form			245.4.b.d.99.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.67516i q^{2} +2.49396i q^{3} +5.19383 q^{4} +(-6.35505 - 9.19855i) q^{5} -4.17779 q^{6} +22.1018i q^{8} +20.7802 q^{9} +(15.4091 - 10.6457i) q^{10} -57.5880 q^{11} +12.9532i q^{12} +45.5159i q^{13} +(22.9408 - 15.8492i) q^{15} +4.52655 q^{16} +92.0051i q^{17} +34.8101i q^{18} +125.177 q^{19} +(-33.0070 - 47.7757i) q^{20} -96.4692i q^{22} +158.496i q^{23} -55.1211 q^{24} +(-44.2268 + 116.914i) q^{25} -76.2466 q^{26} +119.162i q^{27} +40.1708 q^{29} +(26.5500 + 38.4296i) q^{30} -49.5590 q^{31} +184.397i q^{32} -143.622i q^{33} -154.123 q^{34} +107.929 q^{36} -231.307i q^{37} +209.692i q^{38} -113.515 q^{39} +(203.305 - 140.458i) q^{40} -169.556 q^{41} -147.428i q^{43} -299.102 q^{44} +(-132.059 - 191.147i) q^{45} -265.507 q^{46} -67.0327i q^{47} +11.2890i q^{48} +(-195.851 - 74.0870i) q^{50} -229.457 q^{51} +236.402i q^{52} +268.647i q^{53} -199.615 q^{54} +(365.974 + 529.726i) q^{55} +312.187i q^{57} +67.2926i q^{58} -240.843 q^{59} +(119.151 - 82.3183i) q^{60} -90.4579 q^{61} -83.0194i q^{62} -272.683 q^{64} +(418.681 - 289.256i) q^{65} +240.591 q^{66} +406.498i q^{67} +477.859i q^{68} -395.283 q^{69} +330.782 q^{71} +459.279i q^{72} -546.255i q^{73} +387.477 q^{74} +(-291.580 - 110.300i) q^{75} +650.149 q^{76} -190.156i q^{78} +25.3087 q^{79} +(-28.7664 - 41.6377i) q^{80} +263.879 q^{81} -284.034i q^{82} -376.255i q^{83} +(846.314 - 584.697i) q^{85} +246.965 q^{86} +100.184i q^{87} -1272.80i q^{88} +1026.44 q^{89} +(320.203 - 221.220i) q^{90} +823.203i q^{92} -123.598i q^{93} +112.291 q^{94} +(-795.506 - 1151.45i) q^{95} -459.879 q^{96} -942.660i q^{97} -1196.69 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 36 q^{4} - 6 q^{5} - 12 q^{6} - 46 q^{9} + 16 q^{10} + 84 q^{11} + 8 q^{15} + 148 q^{16} - 72 q^{19} + 68 q^{20} - 72 q^{24} - 362 q^{25} + 620 q^{26} + 88 q^{29} + 52 q^{30} - 120 q^{31} - 964 q^{34}+ \cdots - 5304 q^{99}+O(q^{100}) $ 10 * q - 36 * q^4 - 6 * q^5 - 12 * q^6 - 46 * q^9 + 16 * q^10 + 84 * q^11 + 8 * q^15 + 148 * q^16 - 72 * q^19 + 68 * q^20 - 72 * q^24 - 362 * q^25 + 620 * q^26 + 88 * q^29 + 52 * q^30 - 120 * q^31 - 964 * q^34 - 420 * q^36 + 212 * q^39 - 1396 * q^40 + 852 * q^41 - 1424 * q^44 + 510 * q^45 + 176 * q^46 + 1644 * q^50 + 1276 * q^51 + 996 * q^54 + 1136 * q^55 - 864 * q^59 - 1692 * q^60 + 884 * q^61 - 2724 * q^64 + 520 * q^65 - 1148 * q^66 - 4808 * q^69 + 880 * q^71 + 3024 * q^74 - 720 * q^75 + 2672 * q^76 - 3580 * q^79 + 2316 * q^80 - 302 * q^81 + 1572 * q^85 + 2432 * q^86 + 1492 * q^89 + 7748 * q^90 - 1652 * q^94 - 1628 * q^95 - 4080 * q^96 - 5304 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.67516i			0.592259i	0.955148	+	0.296130i	$0.0956960\pi$
$2$			1.67516i			0.592259i	−0.955148	+	0.296130i	$0.904304\pi$
$3$			2.49396i			0.479963i	0.970777	+	0.239982i	$0.0771414\pi$
$3$			2.49396i			0.479963i	−0.970777	+	0.239982i	$0.922859\pi$
$4$	5.19383			0.649229
$5$	−6.35505	−	9.19855i	−0.568413	−	0.822744i
$5$	−6.35505	−	9.19855i	−0.568413	−	0.822744i
$6$	−4.17779			−0.284263
$7$		0			0
$7$		0			0
$8$			22.1018i			0.976771i
$9$	20.7802			0.769635
$10$	15.4091	−	10.6457i	0.487278	−	0.336648i
$11$	−57.5880			−1.57849			−0.789247	−	0.614076i	$-0.789529\pi$
$11$	−57.5880			−1.57849			−0.789247	+	0.614076i	$0.789529\pi$
$12$			12.9532i			0.311606i
$13$			45.5159i			0.971066i	0.874218	+	0.485533i	$0.161374\pi$
$13$			45.5159i			0.971066i	−0.874218	+	0.485533i	$0.838626\pi$
$14$		0			0
$15$	22.9408	−	15.8492i	0.394887	−	0.272817i
$16$	4.52655			0.0707273
$17$			92.0051i			1.31262i	0.754492	+	0.656309i	$0.227883\pi$
$17$			92.0051i			1.31262i	−0.754492	+	0.656309i	$0.772117\pi$
$18$			34.8101i			0.455824i
$19$	125.177			1.51145			0.755726	−	0.654888i	$-0.227284\pi$
$19$	125.177			1.51145			0.755726	+	0.654888i	$0.227284\pi$
$20$	−33.0070	−	47.7757i	−0.369030	−	0.534149i
$21$		0			0
$22$		−	96.4692i		−	0.934878i
$23$			158.496i			1.43690i	0.695578	+	0.718451i	$0.255149\pi$
$23$			158.496i			1.43690i	−0.695578	+	0.718451i	$0.744851\pi$
$24$	−55.1211			−0.468814
$25$	−44.2268	+	116.914i	−0.353814	+	0.935316i
$26$	−76.2466			−0.575123
$27$			119.162i			0.849360i
$28$		0			0
$29$	40.1708			0.257225			0.128613	−	0.991695i	$-0.458948\pi$
$29$	40.1708			0.257225			0.128613	+	0.991695i	$0.458948\pi$
$30$	26.5500	+	38.4296i	0.161578	+	0.233875i
$31$	−49.5590			−0.287131			−0.143566	−	0.989641i	$-0.545857\pi$
$31$	−49.5590			−0.287131			−0.143566	+	0.989641i	$0.545857\pi$
$32$			184.397i			1.01866i
$33$		−	143.622i		−	0.757619i
$34$	−154.123			−0.777410
$35$		0			0
$36$	107.929			0.499670
$37$		−	231.307i		−	1.02775i	−0.857866	−	0.513874i	$-0.828210\pi$
$37$		−	231.307i		−	1.02775i	0.857866	−	0.513874i	$-0.171790\pi$
$38$			209.692i			0.895171i
$39$	−113.515			−0.466076
$40$	203.305	−	140.458i	0.803632	−	0.555209i
$41$	−169.556			−0.645859			−0.322929	−	0.946423i	$-0.604668\pi$
$41$	−169.556			−0.645859			−0.322929	+	0.946423i	$0.604668\pi$
$42$		0			0
$43$		−	147.428i		−	0.522849i	−0.965224	−	0.261425i	$-0.915808\pi$
$43$		−	147.428i		−	0.522849i	0.965224	−	0.261425i	$-0.0841923\pi$
$44$	−299.102			−1.02480
$45$	−132.059	−	191.147i	−0.437470	−	0.633213i
$46$	−265.507			−0.851018
$47$		−	67.0327i		−	0.208037i	−0.994575	−	0.104018i	$-0.966830\pi$
$47$		−	67.0327i		−	0.208037i	0.994575	−	0.104018i	$-0.0331701\pi$
$48$			11.2890i			0.0339465i
$49$		0			0
$50$	−195.851	−	74.0870i	−0.553949	−	0.209550i
$51$	−229.457			−0.630008
$52$			236.402i			0.630444i
$53$			268.647i			0.696254i	0.937447	+	0.348127i	$0.113182\pi$
$53$			268.647i			0.696254i	−0.937447	+	0.348127i	$0.886818\pi$
$54$	−199.615			−0.503041
$55$	365.974	+	529.726i	0.897236	+	1.29870i
$56$		0			0
$57$			312.187i			0.725441i
$58$			67.2926i			0.152344i
$59$	−240.843			−0.531442			−0.265721	−	0.964050i	$-0.585610\pi$
$59$	−240.843			−0.531442			−0.265721	+	0.964050i	$0.585610\pi$
$60$	119.151	−	82.3183i	0.256372	−	0.177121i
$61$	−90.4579			−0.189868			−0.0949340	−	0.995484i	$-0.530264\pi$
$61$	−90.4579			−0.189868			−0.0949340	+	0.995484i	$0.530264\pi$
$62$		−	83.0194i		−	0.170056i
$63$		0			0
$64$	−272.683			−0.532583
$65$	418.681	−	289.256i	0.798938	−	0.551966i
$66$	240.591			0.448707
$67$			406.498i			0.741218i	0.928789	+	0.370609i	$0.120851\pi$
$67$			406.498i			0.741218i	−0.928789	+	0.370609i	$0.879149\pi$
$68$			477.859i			0.852190i
$69$	−395.283			−0.689660
$70$		0			0
$71$	330.782			0.552910			0.276455	−	0.961027i	$-0.410840\pi$
$71$	330.782			0.552910			0.276455	+	0.961027i	$0.410840\pi$
$72$			459.279i			0.751758i
$73$		−	546.255i		−	0.875812i	−0.899021	−	0.437906i	$-0.855720\pi$
$73$		−	546.255i		−	0.875812i	0.899021	−	0.437906i	$-0.144280\pi$
$74$	387.477			0.608693
$75$	−291.580	−	110.300i	−0.448917	−	0.169818i
$76$	650.149			0.981279
$77$		0			0
$78$		−	190.156i		−	0.276038i
$79$	25.3087			0.0360436			0.0180218	−	0.999838i	$-0.494263\pi$
$79$	25.3087			0.0360436			0.0180218	+	0.999838i	$0.494263\pi$
$80$	−28.7664	−	41.6377i	−0.0402023	−	0.0581905i
$81$	263.879			0.361974
$82$		−	284.034i		−	0.382516i
$83$		−	376.255i		−	0.497582i	−0.968557	−	0.248791i	$-0.919967\pi$
$83$		−	376.255i		−	0.497582i	0.968557	−	0.248791i	$-0.0800333\pi$
$84$		0			0
$85$	846.314	−	584.697i	1.07995	−	0.746109i
$86$	246.965			0.309662
$87$			100.184i			0.123459i
$88$		−	1272.80i		−	1.54183i
$89$	1026.44			1.22250			0.611248	−	0.791439i	$-0.290668\pi$
$89$	1026.44			1.22250			0.611248	+	0.791439i	$0.290668\pi$
$90$	320.203	−	221.220i	0.375026	−	0.259096i
$91$		0			0
$92$			823.203i			0.932878i
$93$		−	123.598i		−	0.137812i
$94$	112.291			0.123212
$95$	−795.506	−	1151.45i	−0.859128	−	1.24354i
$96$	−459.879			−0.488919
$97$		−	942.660i		−	0.986728i	−0.869823	−	0.493364i	$-0.835767\pi$
$97$		−	942.660i		−	0.986728i	0.869823	−	0.493364i	$-0.164233\pi$
$98$		0			0
$99$	−1196.69			−1.21487
$100$	−229.706	+	607.234i	−0.229706	+	0.607234i
$101$	604.617			0.595659			0.297830	−	0.954619i	$-0.403737\pi$
$101$	604.617			0.595659			0.297830	+	0.954619i	$0.403737\pi$
$102$		−	384.378i		−	0.373128i
$103$			300.967i			0.287914i	0.989584	+	0.143957i	$0.0459828\pi$
$103$			300.967i			0.287914i	−0.989584	+	0.143957i	$0.954017\pi$
$104$	−1005.98			−0.948509
$105$		0			0
$106$	−450.027			−0.412363
$107$		−	1511.66i		−	1.36577i	−0.730525	−	0.682886i	$-0.760724\pi$
$107$		−	1511.66i		−	1.36577i	0.730525	−	0.682886i	$-0.239276\pi$
$108$			618.907i			0.551429i
$109$	1767.09			1.55281			0.776406	−	0.630233i	$-0.217041\pi$
$109$	1767.09			1.55281			0.776406	+	0.630233i	$0.217041\pi$
$110$	−887.377	+	613.066i	−0.769165	+	0.531396i
$111$	576.871			0.493281
$112$		0			0
$113$			1045.27i			0.870182i	0.900387	+	0.435091i	$0.143284\pi$
$113$			1045.27i			0.870182i	−0.900387	+	0.435091i	$0.856716\pi$
$114$	−522.963			−0.429649
$115$	1457.94	−	1007.25i	1.18220	−	0.816753i
$116$	208.640			0.166998
$117$			945.828i			0.747366i
$118$		−	403.451i		−	0.314751i
$119$		0			0
$120$	350.297	+	507.034i	0.266480	+	0.385714i
$121$	1985.38			1.49164
$122$		−	151.532i		−	0.112451i
$123$		−	422.866i		−	0.309988i
$124$	−257.401			−0.186414
$125$	1356.51	−	336.174i	0.970638	−	0.240547i
$126$		0			0
$127$		−	260.727i		−	0.182171i	−0.995843	−	0.0910857i	$-0.970966\pi$
$127$		−	260.727i		−	0.182171i	0.995843	−	0.0910857i	$-0.0290337\pi$
$128$			1018.39i			0.703233i
$129$	367.679			0.250948
$130$	484.551	+	701.358i	0.326907	+	0.473178i
$131$	723.522			0.482553			0.241276	−	0.970456i	$-0.422434\pi$
$131$	723.522			0.482553			0.241276	+	0.970456i	$0.422434\pi$
$132$		−	745.950i		−	0.491868i
$133$		0			0
$134$	−680.950			−0.438993
$135$	1096.12	−	757.279i	0.698805	−	0.482787i
$136$	−2033.48			−1.28213
$137$		−	773.693i		−	0.482490i	−0.970464	−	0.241245i	$-0.922444\pi$
$137$		−	773.693i		−	0.482490i	0.970464	−	0.241245i	$-0.0775557\pi$
$138$		−	662.164i		−	0.408457i
$139$	−2952.97			−1.80192			−0.900961	−	0.433899i	$-0.857137\pi$
$139$	−2952.97			−1.80192			−0.900961	+	0.433899i	$0.857137\pi$
$140$		0			0
$141$	167.177			0.0998499
$142$			554.114i			0.327466i
$143$		−	2621.17i		−	1.53282i
$144$	94.0624			0.0544343
$145$	−255.287	−	369.513i	−0.146210	−	0.211630i
$146$	915.066			0.518708
$147$		0			0
$148$		−	1201.37i		−	0.667244i
$149$	−2514.00			−1.38225			−0.691124	−	0.722736i	$-0.742884\pi$
$149$	−2514.00			−1.38225			−0.691124	+	0.722736i	$0.742884\pi$
$150$	184.770	−	488.444i	0.100576	−	0.265875i
$151$	101.052			0.0544605			0.0272302	−	0.999629i	$-0.491331\pi$
$151$	101.052			0.0544605			0.0272302	+	0.999629i	$0.491331\pi$
$152$			2766.64i			1.47634i
$153$			1911.88i			1.01024i
$154$		0			0
$155$	314.950	+	455.872i	0.163209	+	0.236235i
$156$	−589.578			−0.302590
$157$			2338.35i			1.18867i	0.804219	+	0.594333i	$0.202584\pi$
$157$			2338.35i			1.18867i	−0.804219	+	0.594333i	$0.797416\pi$
$158$			42.3961i			0.0213472i
$159$	−669.995			−0.334176
$160$	1696.19	−	1171.85i	0.838096	−	0.579019i
$161$		0			0
$162$			442.040i			0.214383i
$163$		−	1325.20i		−	0.636798i	−0.947957	−	0.318399i	$-0.896855\pi$
$163$		−	1325.20i		−	0.636798i	0.947957	−	0.318399i	$-0.103145\pi$
$164$	−880.646			−0.419310
$165$	−1321.12	+	912.726i	−0.623326	+	0.430640i
$166$	630.288			0.294698
$167$			2086.20i			0.966675i	0.875434	+	0.483338i	$0.160576\pi$
$167$			2086.20i			0.966675i	−0.875434	+	0.483338i	$0.839424\pi$
$168$		0			0
$169$	125.299			0.0570317
$170$	979.462	+	1417.71i	0.441890	+	0.639609i
$171$	2601.20			1.16327
$172$		−	765.715i		−	0.339449i
$173$		−	1918.19i		−	0.842990i	−0.906831	−	0.421495i	$-0.861505\pi$
$173$		−	1918.19i		−	0.842990i	0.906831	−	0.421495i	$-0.138495\pi$
$174$	−167.825			−0.0731195
$175$		0			0
$176$	−260.675			−0.111643
$177$		−	600.652i		−	0.255072i
$178$			1719.45i			0.724035i
$179$	−629.046			−0.262665			−0.131333	−	0.991338i	$-0.541926\pi$
$179$	−629.046			−0.262665			−0.131333	+	0.991338i	$0.541926\pi$
$180$	−685.891	−	992.787i	−0.284019	−	0.411100i
$181$	2800.85			1.15020			0.575099	−	0.818084i	$-0.304964\pi$
$181$	2800.85			1.15020			0.575099	+	0.818084i	$0.304964\pi$
$182$		0			0
$183$		−	225.599i		−	0.0911296i
$184$	−3503.05			−1.40352
$185$	−2127.69	+	1469.97i	−0.845573	+	0.584185i
$186$	207.047			0.0816207
$187$		−	5298.39i		−	2.07196i
$188$		−	348.157i		−	0.135063i
$189$		0			0
$190$	1928.86	−	1332.60i	0.736497	−	0.508827i
$191$	740.255			0.280434			0.140217	−	0.990121i	$-0.455220\pi$
$191$	740.255			0.280434			0.140217	+	0.990121i	$0.455220\pi$
$192$		−	680.060i		−	0.255620i
$193$			4082.57i			1.52264i	0.648375	+	0.761321i	$0.275449\pi$
$193$			4082.57i			1.52264i	−0.648375	+	0.761321i	$0.724551\pi$
$194$	1579.11			0.584399
$195$	721.393	+	1044.17i	0.264923	+	0.383461i
$196$		0			0
$197$		−	3414.89i		−	1.23503i	−0.786559	−	0.617515i	$-0.788140\pi$
$197$		−	3414.89i		−	1.23503i	0.786559	−	0.617515i	$-0.211860\pi$
$198$		−	2004.65i		−	0.719515i
$199$	−3392.44			−1.20846			−0.604231	−	0.796809i	$-0.706520\pi$
$199$	−3392.44			−1.20846			−0.604231	+	0.796809i	$0.706520\pi$
$200$	−2584.02	−	977.492i	−0.913589	−	0.345596i
$201$	−1013.79			−0.355757
$202$			1012.83i			0.352785i
$203$		0			0
$204$	−1191.76			−0.409020
$205$	1077.54	+	1559.67i	0.367114	+	0.531376i
$206$	−504.169			−0.170520
$207$			3293.58i			1.10589i
$208$			206.030i			0.0686809i
$209$	−7208.70			−2.38582
$210$		0			0
$211$	3398.04			1.10867			0.554337	−	0.832292i	$-0.312972\pi$
$211$	3398.04			1.10867			0.554337	+	0.832292i	$0.312972\pi$
$212$			1395.31i			0.452029i
$213$			824.958i			0.265376i
$214$	2532.28			0.808892
$215$	−1356.12	+	936.910i	−0.430171	+	0.297194i
$216$	−2633.69			−0.829630
$217$		0			0
$218$			2960.16i			0.919667i
$219$	1362.34			0.420358
$220$	1900.81	+	2751.31i	0.582512	+	0.843151i
$221$	−4187.70			−1.27464
$222$			966.353i			0.292150i
$223$		−	182.611i		−	0.0548365i	−0.999624	−	0.0274183i	$-0.991271\pi$
$223$		−	182.611i		−	0.0548365i	0.999624	−	0.0274183i	$-0.00872860\pi$
$224$		0			0
$225$	−919.040	+	2429.50i	−0.272308	+	0.719852i
$226$	−1750.99			−0.515373
$227$		−	3152.33i		−	0.921707i	−0.887476	−	0.460854i	$-0.847543\pi$
$227$		−	3152.33i		−	0.921707i	0.887476	−	0.460854i	$-0.152457\pi$
$228$			1621.45i			0.470977i
$229$	6012.35			1.73497			0.867483	−	0.497466i	$-0.165736\pi$
$229$	6012.35			1.73497			0.867483	+	0.497466i	$0.165736\pi$
$230$	1687.31	+	2442.28i	0.483730	+	0.700170i
$231$		0			0
$232$			887.848i			0.251250i
$233$		−	940.660i		−	0.264484i	−0.991217	−	0.132242i	$-0.957782\pi$
$233$		−	940.660i		−	0.264484i	0.991217	−	0.132242i	$-0.0422175\pi$
$234$	−1584.42			−0.442635
$235$	−616.604	+	425.996i	−0.171161	+	0.118251i
$236$	−1250.90			−0.345027
$237$			63.1188i			0.0172996i
$238$		0			0
$239$	−5158.82			−1.39622			−0.698109	−	0.715991i	$-0.745975\pi$
$239$	−5158.82			−1.39622			−0.698109	+	0.715991i	$0.745975\pi$
$240$	103.843	−	71.7423i	0.0279293	−	0.0192956i
$241$	463.836			0.123976			0.0619882	−	0.998077i	$-0.480256\pi$
$241$	463.836			0.123976			0.0619882	+	0.998077i	$0.480256\pi$
$242$			3325.83i			0.883440i
$243$			3875.47i			1.02309i
$244$	−469.823			−0.123268
$245$		0			0
$246$	708.370			0.183594
$247$			5697.55i			1.46772i
$248$		−	1095.34i		−	0.280461i
$249$	938.365			0.238821
$250$	563.146	+	2272.37i	0.142466	+	0.574869i
$251$	−2290.25			−0.575934			−0.287967	−	0.957640i	$-0.592979\pi$
$251$	−2290.25			−0.575934			−0.287967	+	0.957640i	$0.592979\pi$
$252$		0			0
$253$		−	9127.48i		−	2.26814i
$254$	436.760			0.107893
$255$	1458.21	+	2110.67i	0.358105	+	0.518335i
$256$	−3887.43			−0.949079
$257$		−	802.202i		−	0.194708i	−0.995250	−	0.0973541i	$-0.968962\pi$
$257$		−	802.202i		−	0.194708i	0.995250	−	0.0973541i	$-0.0310379\pi$
$258$			615.922i			0.148627i
$259$		0			0
$260$	2174.56	−	1502.35i	0.518694	−	0.358352i
$261$	834.756			0.197970
$262$			1212.02i			0.285796i
$263$		−	286.978i		−	0.0672845i	−0.999434	−	0.0336423i	$-0.989289\pi$
$263$		−	286.978i		−	0.0672845i	0.999434	−	0.0336423i	$-0.0107107\pi$
$264$	3174.31			0.740020
$265$	2471.16	−	1707.26i	0.572839	−	0.395760i
$266$		0			0
$267$			2559.90i			0.586753i
$268$			2111.28i			0.481221i
$269$	3561.22			0.807180			0.403590	−	0.914940i	$-0.367762\pi$
$269$	3561.22			0.807180			0.403590	+	0.914940i	$0.367762\pi$
$270$	1268.57	+	1836.17i	0.285935	+	0.413874i
$271$	−1928.81			−0.432349			−0.216175	−	0.976355i	$-0.569358\pi$
$271$	−1928.81			−0.432349			−0.216175	+	0.976355i	$0.569358\pi$
$272$			416.466i			0.0928380i
$273$		0			0
$274$	1296.06			0.285759
$275$	2546.93	−	6732.87i	0.558494	−	1.47639i
$276$	−2053.04			−0.447747
$277$		−	6588.69i		−	1.42916i	−0.699556	−	0.714578i	$-0.746619\pi$
$277$		−	6588.69i		−	1.42916i	0.699556	−	0.714578i	$-0.253381\pi$
$278$		−	4946.70i		−	1.06721i
$279$	−1029.84			−0.220986
$280$		0			0
$281$	815.552			0.173138			0.0865689	−	0.996246i	$-0.472410\pi$
$281$	815.552			0.173138			0.0865689	+	0.996246i	$0.472410\pi$
$282$			280.049i			0.0591371i
$283$			6513.49i			1.36815i	0.729411	+	0.684076i	$0.239794\pi$
$283$			6513.49i			1.36815i	−0.729411	+	0.684076i	$0.760206\pi$
$284$	1718.03			0.358965
$285$	2871.67	−	1983.96i	0.596852	−	0.412350i
$286$	4390.89			0.907828
$287$		0			0
$288$			3831.80i			0.783997i
$289$	−3551.94			−0.722967
$290$	618.995	−	427.648i	0.125340	−	0.0865943i
$291$	2350.96			0.473593
$292$		−	2837.16i		−	0.568603i
$293$		−	435.520i		−	0.0868373i	−0.999057	−	0.0434186i	$-0.986175\pi$
$293$		−	435.520i		−	0.0868373i	0.999057	−	0.0434186i	$-0.0138249\pi$
$294$		0			0
$295$	1530.57	+	2215.40i	0.302078	+	0.437240i
$296$	5112.31			1.00387
$297$		−	6862.29i		−	1.34071i
$298$		−	4211.36i		−	0.818650i
$299$	−7214.10			−1.39533
$300$	−1514.42	−	572.879i	−0.291450	−	0.110251i
$301$		0			0
$302$			169.279i			0.0322547i
$303$			1507.89i			0.285895i
$304$	566.620			0.106901
$305$	574.864	+	832.082i	0.107923	+	0.156213i
$306$	−3202.71			−0.598323
$307$		−	4915.99i		−	0.913910i	−0.889490	−	0.456955i	$-0.848940\pi$
$307$		−	4915.99i		−	0.913910i	0.889490	−	0.456955i	$-0.151060\pi$
$308$		0			0
$309$	−750.601			−0.138188
$310$	−763.659	+	527.592i	−0.139913	+	0.0966620i
$311$	1831.11			0.333868			0.166934	−	0.985968i	$-0.446613\pi$
$311$	1831.11			0.333868			0.166934	+	0.985968i	$0.446613\pi$
$312$		−	2508.89i		−	0.455249i
$313$			2442.96i			0.441163i	0.975369	+	0.220582i	$0.0707955\pi$
$313$			2442.96i			0.441163i	−0.975369	+	0.220582i	$0.929204\pi$
$314$	−3917.11			−0.703998
$315$		0			0
$316$	131.449			0.0234006
$317$		−	1666.19i		−	0.295214i	−0.989046	−	0.147607i	$-0.952843\pi$
$317$		−	1666.19i		−	0.295214i	0.989046	−	0.147607i	$-0.0471570\pi$
$318$		−	1122.35i		−	0.197919i
$319$	−2313.36			−0.406029
$320$	1732.91	+	2508.29i	0.302727	+	0.438180i
$321$	3770.02			0.655521
$322$		0			0
$323$			11516.9i			1.98396i
$324$	1370.54			0.235004
$325$	−5321.47	−	2013.02i	−0.908253	−	0.343577i
$326$	2219.93			0.377149
$327$			4407.05i			0.745292i
$328$		−	3747.50i		−	0.630856i
$329$		0			0
$330$	−1528.96	−	2213.09i	−0.255051	−	0.369171i
$331$	−5466.38			−0.907732			−0.453866	−	0.891070i	$-0.649956\pi$
$331$	−5466.38			−0.907732			−0.453866	+	0.891070i	$0.649956\pi$
$332$		−	1954.20i		−	0.323045i
$333$		−	4806.60i		−	0.790991i
$334$	−3494.72			−0.572522
$335$	3739.19	−	2583.31i	0.609833	−	0.421318i
$336$		0			0
$337$		−	10650.5i		−	1.72157i	−0.508970	−	0.860784i	$-0.669973\pi$
$337$		−	10650.5i		−	1.72157i	0.508970	−	0.860784i	$-0.330027\pi$
$338$			209.896i			0.0337776i
$339$	−2606.86			−0.417655
$340$	4395.61	−	3036.82i	0.701134	−	0.484395i
$341$	2854.01			0.453235
$342$			4357.43i			0.688956i
$343$		0			0
$344$	3258.42			0.510704
$345$	2512.04	+	3636.04i	0.392011	+	0.567413i
$346$	3213.28			0.499269
$347$		−	4019.13i		−	0.621782i	−0.950446	−	0.310891i	$-0.899373\pi$
$347$		−	4019.13i		−	0.621782i	0.950446	−	0.310891i	$-0.100627\pi$
$348$			520.341i			0.0801529i
$349$	10544.9			1.61735			0.808674	−	0.588256i	$-0.200185\pi$
$349$	10544.9			1.61735			0.808674	+	0.588256i	$0.200185\pi$
$350$		0			0
$351$	−5423.76			−0.824784
$352$		−	10619.1i		−	1.60795i
$353$		−	2959.98i		−	0.446300i	−0.974784	−	0.223150i	$-0.928366\pi$
$353$		−	2959.98i		−	0.446300i	0.974784	−	0.223150i	$-0.0716340\pi$
$354$	1006.19			0.151069
$355$	−2102.14	−	3042.72i	−0.314281	−	0.454903i
$356$	5331.14			0.793680
$357$		0			0
$358$		−	1053.75i		−	0.155566i
$359$	−2170.17			−0.319045			−0.159523	−	0.987194i	$-0.550995\pi$
$359$	−2170.17			−0.319045			−0.159523	+	0.987194i	$0.550995\pi$
$360$	4224.70	−	2918.74i	0.618504	−	0.427309i
$361$	8810.30			1.28449
$362$			4691.88i			0.681215i
$363$			4951.46i			0.715934i
$364$		0			0
$365$	−5024.76	+	3471.48i	−0.720569	+	0.497823i
$366$	377.914			0.0539724
$367$		−	1252.20i		−	0.178105i	−0.996027	−	0.0890523i	$-0.971616\pi$
$367$		−	1252.20i		−	0.178105i	0.996027	−	0.0890523i	$-0.0283838\pi$
$368$			717.441i			0.101628i
$369$	−3523.40			−0.497076
$370$	−2462.43	−	3564.23i	−0.345989	−	0.500798i
$371$		0			0
$372$		−	641.949i		−	0.0894718i
$373$			4646.02i			0.644938i	0.946580	+	0.322469i	$0.104513\pi$
$373$			4646.02i			0.644938i	−0.946580	+	0.322469i	$0.895487\pi$
$374$	8875.66			1.22714
$375$	838.406	+	3383.08i	0.115454	+	0.465870i
$376$	1481.54			0.203204
$377$			1828.41i			0.249783i
$378$		0			0
$379$	−1434.84			−0.194466			−0.0972331	−	0.995262i	$-0.530999\pi$
$379$	−1434.84			−0.194466			−0.0972331	+	0.995262i	$0.530999\pi$
$380$	−4131.72	−	5980.43i	−0.557771	−	0.807341i
$381$	650.242			0.0874355
$382$			1240.05i			0.166090i
$383$			13216.7i			1.76330i	0.471905	+	0.881649i	$0.343566\pi$
$383$			13216.7i			1.76330i	−0.471905	+	0.881649i	$0.656434\pi$
$384$	−2539.82			−0.337526
$385$		0			0
$386$	−6838.97			−0.901799
$387$		−	3063.57i		−	0.402403i
$388$		−	4896.02i		−	0.640612i
$389$	7755.01			1.01078			0.505391	−	0.862890i	$-0.331348\pi$
$389$	7755.01			1.01078			0.505391	+	0.862890i	$0.331348\pi$
$390$	−1749.16	+	1208.45i	−0.227108	+	0.156903i
$391$	−14582.5			−1.88610
$392$		0			0
$393$			1804.44i			0.231607i
$394$	5720.49			0.731457
$395$	−160.838	−	232.803i	−0.0204876	−	0.0296547i
$396$	−6215.40			−0.788726
$397$		−	3560.83i		−	0.450158i	−0.974341	−	0.225079i	$-0.927736\pi$
$397$		−	3560.83i		−	0.450158i	0.974341	−	0.225079i	$-0.0722640\pi$
$398$		−	5682.89i		−	0.715723i
$399$		0			0
$400$	−200.195	+	529.219i	−0.0250243	+	0.0661524i
$401$	−5430.61			−0.676288			−0.338144	−	0.941094i	$-0.609799\pi$
$401$	−5430.61			−0.676288			−0.338144	+	0.941094i	$0.609799\pi$
$402$		−	1698.26i		−	0.210701i
$403$		−	2255.73i		−	0.278823i
$404$	3140.28			0.386719
$405$	−1676.96	−	2427.31i	−0.205751	−	0.297812i
$406$		0			0
$407$			13320.5i			1.62229i
$408$		−	5071.42i		−	0.615374i
$409$	−9698.79			−1.17255			−0.586277	−	0.810111i	$-0.699407\pi$
$409$	−9698.79			−1.17255			−0.586277	+	0.810111i	$0.699407\pi$
$410$	−2612.70	+	1805.05i	−0.314713	+	0.217427i
$411$	1929.56			0.231577
$412$			1563.17i			0.186922i
$413$		0			0
$414$	−5517.27			−0.654974
$415$	−3461.00	+	2391.12i	−0.409383	+	0.282832i
$416$	−8393.01			−0.989186
$417$		−	7364.58i		−	0.864856i
$418$		−	12075.7i		−	1.41302i
$419$	13830.9			1.61261			0.806307	−	0.591498i	$-0.201463\pi$
$419$	13830.9			1.61261			0.806307	+	0.591498i	$0.201463\pi$
$420$		0			0
$421$	16703.0			1.93362			0.966810	−	0.255498i	$-0.0822393\pi$
$421$	16703.0			1.93362			0.966810	+	0.255498i	$0.0822393\pi$
$422$			5692.26i			0.656623i
$423$		−	1392.95i		−	0.160112i
$424$	−5937.58			−0.680081
$425$	−10756.7	−	4069.09i	−1.22771	−	0.464423i
$426$	−1381.94			−0.157172
$427$		0			0
$428$		−	7851.31i		−	0.886699i
$429$	6537.10			0.735698
$430$	−1569.48	−	2271.72i	−0.176016	−	0.254773i
$431$	8174.07			0.913530			0.456765	−	0.889588i	$-0.349008\pi$
$431$	8174.07			0.913530			0.456765	+	0.889588i	$0.349008\pi$
$432$			539.392i			0.0600729i
$433$		−	14222.8i		−	1.57853i	−0.614051	−	0.789267i	$-0.710461\pi$
$433$		−	14222.8i		−	1.57853i	0.614051	−	0.789267i	$-0.289539\pi$
$434$		0			0
$435$	921.552	−	636.677i	0.101575	−	0.0701755i
$436$	9177.96			1.00813
$437$			19840.1i			2.17181i
$438$			2282.14i			0.248961i
$439$	5537.38			0.602016			0.301008	−	0.953622i	$-0.402677\pi$
$439$	5537.38			0.602016			0.301008	+	0.953622i	$0.402677\pi$
$440$	−11707.9	+	8088.70i	−1.26853	+	0.876394i
$441$		0			0
$442$		−	7015.07i		−	0.754916i
$443$			3974.09i			0.426218i	0.977028	+	0.213109i	$0.0683589\pi$
$443$			3974.09i			0.426218i	−0.977028	+	0.213109i	$0.931641\pi$
$444$	2996.17			0.320252
$445$	−6523.06	−	9441.74i	−0.694882	−	1.00580i
$446$	305.903			0.0324774
$447$		−	6269.82i		−	0.663428i
$448$		0			0
$449$	15243.1			1.60216			0.801078	−	0.598559i	$-0.204260\pi$
$449$	15243.1			1.60216			0.801078	+	0.598559i	$0.204260\pi$
$450$	−4069.81	−	1539.54i	−0.426339	−	0.161277i
$451$	9764.40			1.01948
$452$			5428.95i			0.564947i
$453$			252.021i			0.0261390i
$454$	5280.67			0.545890
$455$		0			0
$456$	−6899.89			−0.708590
$457$			10768.9i			1.10229i	0.834410	+	0.551145i	$0.185809\pi$
$457$			10768.9i			1.10229i	−0.834410	+	0.551145i	$0.814191\pi$
$458$			10071.7i			1.02755i
$459$	−10963.5			−1.11489
$460$	7572.27	−	5231.49i	0.767520	−	0.530260i
$461$	332.605			0.0336029			0.0168015	−	0.999859i	$-0.494652\pi$
$461$	332.605			0.0336029			0.0168015	+	0.999859i	$0.494652\pi$
$462$		0			0
$463$		−	8205.35i		−	0.823618i	−0.911270	−	0.411809i	$-0.864897\pi$
$463$		−	8205.35i		−	0.823618i	0.911270	−	0.411809i	$-0.135103\pi$
$464$	181.835			0.0181929
$465$	−1136.93	+	785.473i	−0.113384	+	0.0783343i
$466$	1575.76			0.156643
$467$		−	167.628i		−	0.0166100i	−0.999966	−	0.00830501i	$-0.997356\pi$
$467$		−	167.628i		−	0.0166100i	0.999966	−	0.00830501i	$-0.00264360\pi$
$468$			4912.47i			0.485212i
$469$		0			0
$470$	−713.612	−	1032.91i	−0.0700351	−	0.101372i
$471$	−5831.75			−0.570515
$472$		−	5323.06i		−	0.519097i
$473$			8490.07i			0.825315i
$474$	−105.734			−0.0102459
$475$	−5536.18	+	14635.0i	−0.534773	+	1.41368i
$476$		0			0
$477$			5582.52i			0.535862i
$478$		−	8641.86i		−	0.826924i
$479$	6628.58			0.632292			0.316146	−	0.948711i	$-0.397611\pi$
$479$	6628.58			0.632292			0.316146	+	0.948711i	$0.397611\pi$
$480$	2922.55	+	4230.23i	0.277908	+	0.402255i
$481$	10528.2			0.998010
$482$			777.001i			0.0734262i
$483$		0			0
$484$	10311.7			0.968419
$485$	−8671.11	+	5990.65i	−0.811824	+	0.560868i
$486$	−6492.05			−0.605937
$487$			20641.6i			1.92065i	0.278875	+	0.960327i	$0.410039\pi$
$487$			20641.6i			1.92065i	−0.278875	+	0.960327i	$0.589961\pi$
$488$		−	1999.28i		−	0.185458i
$489$	3305.01			0.305640
$490$		0			0
$491$	−16710.8			−1.53594			−0.767972	−	0.640484i	$-0.778734\pi$
$491$	−16710.8			−1.53594			−0.767972	+	0.640484i	$0.778734\pi$
$492$		−	2196.30i		−	0.201253i
$493$			3695.92i			0.337639i
$494$	−9544.32			−0.869270
$495$	7605.01	+	11007.8i	0.690545	+	0.999523i
$496$	−224.331			−0.0203080
$497$		0			0
$498$			1571.91i			0.141444i
$499$	13728.7			1.23162			0.615812	−	0.787893i	$-0.288828\pi$
$499$	13728.7			1.23162			0.615812	+	0.787893i	$0.288828\pi$
$500$	7045.47	−	1746.03i	0.630166	−	0.156170i
$501$	−5202.90			−0.463969
$502$		−	3836.54i		−	0.341102i
$503$			19523.7i			1.73065i	0.501209	+	0.865326i	$0.332889\pi$
$503$			19523.7i			1.73065i	−0.501209	+	0.865326i	$0.667111\pi$
$504$		0			0
$505$	−3842.37	−	5561.60i	−0.338580	−	0.490075i
$506$	15290.0			1.34333
$507$			312.490i			0.0273731i
$508$		−	1354.17i		−	0.118271i
$509$	−8688.17			−0.756574			−0.378287	−	0.925688i	$-0.623487\pi$
$509$	−8688.17			−0.756574			−0.378287	+	0.925688i	$0.623487\pi$
$510$	−3535.72	+	2442.74i	−0.306989	+	0.212091i
$511$		0			0
$512$			1635.04i			0.141131i
$513$			14916.3i			1.28377i
$514$	1343.82			0.115318
$515$	2768.46	−	1912.66i	0.236880	−	0.163654i
$516$	1909.66			0.162923
$517$			3860.28i			0.328385i
$518$		0			0
$519$	4783.89			0.404604
$520$	6393.08	+	9253.60i	0.539144	+	0.780380i
$521$	−6771.36			−0.569402			−0.284701	−	0.958616i	$-0.591894\pi$
$521$	−6771.36			−0.569402			−0.284701	+	0.958616i	$0.591894\pi$
$522$			1398.35i			0.117249i
$523$			1365.89i			0.114200i	0.998368	+	0.0570998i	$0.0181853\pi$
$523$			1365.89i			0.114200i	−0.998368	+	0.0570998i	$0.981815\pi$
$524$	3757.85			0.313287
$525$		0			0
$526$	480.735			0.0398499
$527$		−	4559.68i		−	0.376894i
$528$		−	650.113i		−	0.0535844i
$529$	−12954.0			−1.06469
$530$	2859.94	+	4139.60i	0.234392	+	0.339269i
$531$	−5004.75			−0.409016
$532$		0			0
$533$		−	7717.51i		−	0.627171i
$534$	−4288.24			−0.347510
$535$	−13905.1	+	9606.67i	−1.12368	+	0.776322i
$536$	−8984.34			−0.724001
$537$		−	1568.82i		−	0.126070i
$538$			5965.62i			0.478060i
$539$		0			0
$540$	5693.05	−	3933.18i	0.453685	−	0.313439i
$541$	−23250.1			−1.84769			−0.923844	−	0.382770i	$-0.874970\pi$
$541$	−23250.1			−1.84769			−0.923844	+	0.382770i	$0.874970\pi$
$542$		−	3231.06i		−	0.256063i
$543$			6985.22i			0.552052i
$544$	−16965.5			−1.33711
$545$	−11229.9	−	16254.7i	−0.882638	−	1.27757i
$546$		0			0
$547$			11552.7i			0.903033i	0.892263	+	0.451516i	$0.149117\pi$
$547$			11552.7i			0.903033i	−0.892263	+	0.451516i	$0.850883\pi$
$548$		−	4018.43i		−	0.313246i
$549$	−1879.73			−0.146129
$550$	11278.6	+	4266.52i	0.874406	+	0.330773i
$551$	5028.47			0.388784
$552$		−	8736.48i		−	0.673640i
$553$		0			0
$554$	11037.1			0.846430
$555$	−3666.04	−	5306.38i	−0.280387	−	0.405844i
$556$	−15337.2			−1.16986
$557$			16406.2i			1.24803i	0.781413	+	0.624014i	$0.214499\pi$
$557$			16406.2i			1.24803i	−0.781413	+	0.624014i	$0.785501\pi$
$558$		−	1725.16i		−	0.130881i
$559$	6710.31			0.507721
$560$		0			0
$561$	13214.0			0.994465
$562$			1366.18i			0.102542i
$563$		−	13631.9i		−	1.02045i	−0.860040	−	0.510227i	$-0.829561\pi$
$563$		−	13631.9i		−	1.02045i	0.860040	−	0.510227i	$-0.170439\pi$
$564$	868.289			0.0648255
$565$	9614.95	−	6642.72i	0.715936	−	0.494622i
$566$	−10911.2			−0.810300
$567$		0			0
$568$			7310.88i			0.540067i
$569$	3086.83			0.227428			0.113714	−	0.993514i	$-0.463725\pi$
$569$	3086.83			0.227428			0.113714	+	0.993514i	$0.463725\pi$
$570$	3323.46	+	4810.51i	0.244218	+	0.353491i
$571$	−3258.06			−0.238784			−0.119392	−	0.992847i	$-0.538095\pi$
$571$	−3258.06			−0.238784			−0.119392	+	0.992847i	$0.538095\pi$
$572$		−	13613.9i		−	0.995152i
$573$			1846.17i			0.134598i
$574$		0			0
$575$	−18530.5	−	7009.78i	−1.34396	−	0.508396i
$576$	−5666.39			−0.409895
$577$			23758.4i			1.71417i	0.515178	+	0.857083i	$0.327726\pi$
$577$			23758.4i			1.71417i	−0.515178	+	0.857083i	$0.672274\pi$
$578$		−	5950.07i		−	0.428184i
$579$	−10181.8			−0.730812
$580$	−1325.92	−	1919.19i	−0.0949238	−	0.137397i
$581$		0			0
$582$			3938.23i			0.280490i
$583$		−	15470.8i		−	1.09903i
$584$	12073.2			0.855468
$585$	8700.25	−	6010.78i	0.614891	−	0.424812i
$586$	729.566			0.0514302
$587$			596.893i			0.0419701i	0.999780	+	0.0209850i	$0.00668023\pi$
$587$			596.893i			0.0419701i	−0.999780	+	0.0209850i	$0.993320\pi$
$588$		0			0
$589$	−6203.66			−0.433985
$590$	−3711.16	+	2563.95i	−0.258960	+	0.178909i
$591$	8516.60			0.592768
$592$		−	1047.02i		−	0.0726898i
$593$		−	19496.3i		−	1.35012i	−0.737765	−	0.675058i	$-0.764119\pi$
$593$		−	19496.3i		−	1.35012i	0.737765	−	0.675058i	$-0.235881\pi$
$594$	11495.5			0.794047
$595$		0			0
$596$	−13057.3			−0.897396
$597$		−	8460.62i		−	0.580017i
$598$		−	12084.8i		−	0.826395i
$599$	−3797.02			−0.259001			−0.129501	−	0.991579i	$-0.541337\pi$
$599$	−3797.02			−0.259001			−0.129501	+	0.991579i	$0.541337\pi$
$600$	2437.83	−	6444.45i	0.165873	−	0.438489i
$601$	−5789.33			−0.392931			−0.196466	−	0.980511i	$-0.562946\pi$
$601$	−5789.33			−0.392931			−0.196466	+	0.980511i	$0.562946\pi$
$602$		0			0
$603$			8447.09i			0.570468i
$604$	524.850			0.0353573
$605$	−12617.2	−	18262.6i	−0.847869	−	1.22724i
$606$	−2525.96			−0.169324
$607$			18536.4i			1.23949i	0.784803	+	0.619745i	$0.212764\pi$
$607$			18536.4i			1.23949i	−0.784803	+	0.619745i	$0.787236\pi$
$608$			23082.3i			1.53966i
$609$		0			0
$610$	−1393.87	+	962.991i	−0.0925184	+	0.0639186i
$611$	3051.06			0.202017
$612$			9929.98i			0.655876i
$613$			2163.47i			0.142548i	0.997457	+	0.0712738i	$0.0227064\pi$
$613$			2163.47i			0.142548i	−0.997457	+	0.0712738i	$0.977294\pi$
$614$	8235.08			0.541272
$615$	−3889.76	+	2687.34i	−0.255041	+	0.176201i
$616$		0			0
$617$		−	22964.9i		−	1.49843i	−0.662327	−	0.749215i	$-0.730431\pi$
$617$		−	22964.9i		−	1.49843i	0.662327	−	0.749215i	$-0.269569\pi$
$618$		−	1257.38i		−	0.0818433i
$619$	−1386.67			−0.0900401			−0.0450200	−	0.998986i	$-0.514335\pi$
$619$	−1386.67			−0.0900401			−0.0450200	+	0.998986i	$0.514335\pi$
$620$	1635.80	+	2367.72i	0.105960	+	0.153371i
$621$	−18886.7			−1.22045
$622$			3067.41i			0.197736i
$623$		0			0
$624$	−513.831			−0.0329643
$625$	−11713.0	−	10341.5i	−0.749631	−	0.661856i
$626$	−4092.35			−0.261283
$627$		−	17978.2i		−	1.14510i
$628$			12145.0i			0.771716i
$629$	21281.4			1.34904
$630$		0			0
$631$	5969.39			0.376605			0.188303	−	0.982111i	$-0.439701\pi$
$631$	5969.39			0.376605			0.188303	+	0.982111i	$0.439701\pi$
$632$			559.367i			0.0352064i
$633$			8474.57i			0.532123i
$634$	2791.14			0.174843
$635$	−2398.31	+	1656.93i	−0.149880	+	0.103548i
$636$	−3479.84			−0.216957
$637$		0			0
$638$		−	3875.25i		−	0.240474i
$639$	6873.70			0.425539
$640$	9367.71	−	6471.91i	0.578580	−	0.399726i
$641$	30367.1			1.87118			0.935592	−	0.353084i	$-0.114867\pi$
$641$	30367.1			1.87118			0.935592	+	0.353084i	$0.114867\pi$
$642$			6315.40i			0.388238i
$643$		−	28592.2i		−	1.75360i	−0.480851	−	0.876802i	$-0.659672\pi$
$643$		−	28592.2i		−	1.75360i	0.480851	−	0.876802i	$-0.340328\pi$
$644$		0			0
$645$	−2336.62	−	3382.12i	−0.142642	−	0.206466i
$646$	−19292.7			−1.17502
$647$		−	14507.9i		−	0.881555i	−0.897616	−	0.440778i	$-0.854703\pi$
$647$		−	14507.9i		−	0.881555i	0.897616	−	0.440778i	$-0.145297\pi$
$648$			5832.21i			0.353566i
$649$	13869.7			0.838877
$650$	3372.14	−	8914.33i	0.203487	−	0.537921i
$651$		0			0
$652$		−	6882.89i		−	0.413428i
$653$			6999.85i			0.419488i	0.977756	+	0.209744i	$0.0672630\pi$
$653$			6999.85i			0.419488i	−0.977756	+	0.209744i	$0.932737\pi$
$654$	−7382.53			−0.441406
$655$	−4598.01	−	6655.35i	−0.274289	−	0.397017i
$656$	−767.504			−0.0456799
$657$		−	11351.3i		−	0.674056i
$658$		0			0
$659$	−7308.92			−0.432041			−0.216020	−	0.976389i	$-0.569308\pi$
$659$	−7308.92			−0.432041			−0.216020	+	0.976389i	$0.569308\pi$
$660$	−6861.66	+	4740.55i	−0.404681	+	0.279584i
$661$	30097.2			1.77102			0.885512	−	0.464617i	$-0.153808\pi$
$661$	30097.2			1.77102			0.885512	+	0.464617i	$0.153808\pi$
$662$		−	9157.07i		−	0.537613i
$663$		−	10444.0i		−	0.611779i
$664$	8315.91			0.486024
$665$		0			0
$666$	8051.83			0.468472
$667$			6366.92i			0.369608i
$668$			10835.4i			0.627594i
$669$	455.425			0.0263195
$670$	4327.47	+	6263.76i	0.249529	+	0.361179i
$671$	5209.29			0.299706
$672$		0			0
$673$		−	5400.26i		−	0.309309i	−0.987969	−	0.154654i	$-0.950574\pi$
$673$		−	5400.26i		−	0.309309i	0.987969	−	0.154654i	$-0.0494264\pi$
$674$	17841.3			1.01962
$675$	−13931.7	−	5270.15i	−0.794419	−	0.300516i
$676$	650.781			0.0370267
$677$			6431.09i			0.365091i	0.983197	+	0.182546i	$0.0584338\pi$
$677$			6431.09i			0.365091i	−0.983197	+	0.182546i	$0.941566\pi$
$678$		−	4366.91i		−	0.247360i
$679$		0			0
$680$	12922.9	+	18705.1i	0.728778	+	1.05486i
$681$	7861.79			0.442385
$682$			4780.92i			0.268433i
$683$		−	20865.8i		−	1.16897i	−0.811404	−	0.584486i	$-0.801296\pi$
$683$		−	20865.8i		−	1.16897i	0.811404	−	0.584486i	$-0.198704\pi$
$684$	13510.2			0.755227
$685$	−7116.86	+	4916.86i	−0.396965	+	0.274253i
$686$		0			0
$687$			14994.6i			0.832720i
$688$		−	667.339i		−	0.0369797i
$689$	−12227.7			−0.676109
$690$	−6090.95	+	4208.08i	−0.336056	+	0.232172i
$691$	−18450.3			−1.01575			−0.507873	−	0.861432i	$-0.669568\pi$
$691$	−18450.3			−1.01575			−0.507873	+	0.861432i	$0.669568\pi$
$692$		−	9962.76i		−	0.547294i
$693$		0			0
$694$	6732.70			0.368256
$695$	18766.2	+	27163.0i	1.02424	+	1.48252i
$696$	−2214.26			−0.120591
$697$		−	15600.0i		−	0.847766i
$698$			17664.4i			0.957890i
$699$	2345.97			0.126942
$700$		0			0
$701$	12639.3			0.680996			0.340498	−	0.940245i	$-0.389404\pi$
$701$	12639.3			0.680996			0.340498	+	0.940245i	$0.389404\pi$
$702$		−	9085.69i		−	0.488486i
$703$		−	28954.4i		−	1.55339i
$704$	15703.3			0.840680
$705$	−1062.42	−	1537.79i	−0.0567560	−	0.0821509i
$706$	4958.45			0.264325
$707$		0			0
$708$		−	3119.69i		−	0.165600i
$709$	23126.8			1.22503			0.612514	−	0.790460i	$-0.290158\pi$
$709$	23126.8			1.22503			0.612514	+	0.790460i	$0.290158\pi$
$710$	5097.04	−	3521.42i	0.269421	−	0.186136i
$711$	525.918			0.0277404
$712$			22686.1i			1.19410i
$713$		−	7854.92i		−	0.412579i
$714$		0			0
$715$	−24111.0	+	16657.7i	−1.26112	+	0.871275i
$716$	−3267.16			−0.170530
$717$		−	12865.9i		−	0.670134i
$718$		−	3635.39i		−	0.188957i
$719$	−24093.1			−1.24968			−0.624841	−	0.780752i	$-0.714836\pi$
$719$	−24093.1			−1.24968			−0.624841	+	0.780752i	$0.714836\pi$
$720$	−597.771	−	865.238i	−0.0309411	−	0.0447854i
$721$		0			0
$722$			14758.7i			0.760750i
$723$			1156.79i			0.0595041i
$724$	14547.2			0.746741
$725$	−1776.63	+	4696.55i	−0.0910100	+	0.240587i
$726$	−8294.49			−0.424019
$727$			35983.4i			1.83570i	0.396931	+	0.917849i	$0.370075\pi$
$727$			35983.4i			1.83570i	−0.396931	+	0.917849i	$0.629925\pi$
$728$		0			0
$729$	−2540.55			−0.129073
$730$	−5815.28	−	8417.28i	−0.294840	−	0.426764i
$731$	13564.1			0.686302
$732$		−	1171.72i		−	0.0591640i
$733$			1451.50i			0.0731413i	0.999331	+	0.0365706i	$0.0116434\pi$
$733$			1451.50i			0.0731413i	−0.999331	+	0.0365706i	$0.988357\pi$
$734$	2097.64			0.105484
$735$		0			0
$736$	−29226.3			−1.46371
$737$		−	23409.4i		−	1.17001i
$738$		−	5902.27i		−	0.294398i
$739$	5891.67			0.293273			0.146636	−	0.989190i	$-0.453155\pi$
$739$	5891.67			0.293273			0.146636	+	0.989190i	$0.453155\pi$
$740$	−11050.9	+	7634.77i	−0.548970	+	0.379270i
$741$	−14209.5			−0.704451
$742$		0			0
$743$			7438.65i			0.367292i	0.982992	+	0.183646i	$0.0587900\pi$
$743$			7438.65i			0.367292i	−0.982992	+	0.183646i	$0.941210\pi$
$744$	2731.75			0.134611
$745$	15976.6	+	23125.2i	0.785688	+	1.13724i
$746$	−7782.84			−0.381970
$747$		−	7818.64i		−	0.382957i
$748$		−	27518.9i		−	1.34518i
$749$		0			0
$750$	−5667.20	+	1404.47i	−0.275916	+	0.0683784i
$751$	20272.4			0.985018			0.492509	−	0.870307i	$-0.336080\pi$
$751$	20272.4			0.985018			0.492509	+	0.870307i	$0.336080\pi$
$752$		−	303.427i		−	0.0147139i
$753$		−	5711.80i		−	0.276427i
$754$	−3062.89			−0.147936
$755$	−642.193	−	929.537i	−0.0309560	−	0.0448070i
$756$		0			0
$757$			10193.8i			0.489432i	0.969595	+	0.244716i	$0.0786947\pi$
$757$			10193.8i			0.489432i	−0.969595	+	0.244716i	$0.921305\pi$
$758$		−	2403.59i		−	0.115174i
$759$	22763.6			1.08862
$760$	25449.1	−	17582.1i	1.21465	−	0.839172i
$761$	41117.6			1.95862			0.979311	−	0.202362i	$-0.0648618\pi$
$761$	41117.6			1.95862			0.979311	+	0.202362i	$0.0648618\pi$
$762$			1089.26i			0.0517845i
$763$		0			0
$764$	3844.76			0.182066
$765$	17586.5	−	12150.1i	0.831167	−	0.574232i
$766$	−22140.2			−1.04433
$767$		−	10962.2i		−	0.516065i
$768$		−	9695.10i		−	0.455523i
$769$	11486.6			0.538642			0.269321	−	0.963050i	$-0.413201\pi$
$769$	11486.6			0.538642			0.269321	+	0.963050i	$0.413201\pi$
$770$		0			0
$771$	2000.66			0.0934527
$772$			21204.2i			0.988544i
$773$			21799.2i			1.01431i	0.861854	+	0.507156i	$0.169303\pi$
$773$			21799.2i			1.01431i	−0.861854	+	0.507156i	$0.830697\pi$
$774$	5131.98			0.238327
$775$	2191.84	−	5794.17i	0.101591	−	0.268558i
$776$	20834.5			0.963807
$777$		0			0
$778$			12990.9i			0.598645i
$779$	−21224.5			−0.976185
$780$	3746.79	+	5423.26i	0.171996	+	0.248954i
$781$	−19049.1			−0.872765
$782$		−	24428.0i		−	1.11706i
$783$			4786.83i

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:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	245.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.67516i q^{2} +2.49396i q^{3} +5.19383 q^{4} +(-6.35505 - 9.19855i) q^{5} -4.17779 q^{6} +22.1018i q^{8} +20.7802 q^{9} +(15.4091 - 10.6457i) q^{10} -57.5880 q^{11} +12.9532i q^{12} +45.5159i q^{13} +(22.9408 - 15.8492i) q^{15} +4.52655 q^{16} +92.0051i q^{17} +34.8101i q^{18} +125.177 q^{19} +(-33.0070 - 47.7757i) q^{20} -96.4692i q^{22} +158.496i q^{23} -55.1211 q^{24} +(-44.2268 + 116.914i) q^{25} -76.2466 q^{26} +119.162i q^{27} +40.1708 q^{29} +(26.5500 + 38.4296i) q^{30} -49.5590 q^{31} +184.397i q^{32} -143.622i q^{33} -154.123 q^{34} +107.929 q^{36} -231.307i q^{37} +209.692i q^{38} -113.515 q^{39} +(203.305 - 140.458i) q^{40} -169.556 q^{41} -147.428i q^{43} -299.102 q^{44} +(-132.059 - 191.147i) q^{45} -265.507 q^{46} -67.0327i q^{47} +11.2890i q^{48} +(-195.851 - 74.0870i) q^{50} -229.457 q^{51} +236.402i q^{52} +268.647i q^{53} -199.615 q^{54} +(365.974 + 529.726i) q^{55} +312.187i q^{57} +67.2926i q^{58} -240.843 q^{59} +(119.151 - 82.3183i) q^{60} -90.4579 q^{61} -83.0194i q^{62} -272.683 q^{64} +(418.681 - 289.256i) q^{65} +240.591 q^{66} +406.498i q^{67} +477.859i q^{68} -395.283 q^{69} +330.782 q^{71} +459.279i q^{72} -546.255i q^{73} +387.477 q^{74} +(-291.580 - 110.300i) q^{75} +650.149 q^{76} -190.156i q^{78} +25.3087 q^{79} +(-28.7664 - 41.6377i) q^{80} +263.879 q^{81} -284.034i q^{82} -376.255i q^{83} +(846.314 - 584.697i) q^{85} +246.965 q^{86} +100.184i q^{87} -1272.80i q^{88} +1026.44 q^{89} +(320.203 - 221.220i) q^{90} +823.203i q^{92} -123.598i q^{93} +112.291 q^{94} +(-795.506 - 1151.45i) q^{95} -459.879 q^{96} -942.660i q^{97} -1196.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 36 q^{4} - 6 q^{5} - 12 q^{6} - 46 q^{9} + 16 q^{10} + 84 q^{11} + 8 q^{15} + 148 q^{16} - 72 q^{19} + 68 q^{20} - 72 q^{24} - 362 q^{25} + 620 q^{26} + 88 q^{29} + 52 q^{30} - 120 q^{31} - 964 q^{34}+ \cdots - 5304 q^{99}+O(q^{100}) \) 10 * q - 36 * q^4 - 6 * q^5 - 12 * q^6 - 46 * q^9 + 16 * q^10 + 84 * q^11 + 8 * q^15 + 148 * q^16 - 72 * q^19 + 68 * q^20 - 72 * q^24 - 362 * q^25 + 620 * q^26 + 88 * q^29 + 52 * q^30 - 120 * q^31 - 964 * q^34 - 420 * q^36 + 212 * q^39 - 1396 * q^40 + 852 * q^41 - 1424 * q^44 + 510 * q^45 + 176 * q^46 + 1644 * q^50 + 1276 * q^51 + 996 * q^54 + 1136 * q^55 - 864 * q^59 - 1692 * q^60 + 884 * q^61 - 2724 * q^64 + 520 * q^65 - 1148 * q^66 - 4808 * q^69 + 880 * q^71 + 3024 * q^74 - 720 * q^75 + 2672 * q^76 - 3580 * q^79 + 2316 * q^80 - 302 * q^81 + 1572 * q^85 + 2432 * q^86 + 1492 * q^89 + 7748 * q^90 - 1652 * q^94 - 1628 * q^95 - 4080 * q^96 - 5304 * q^99

Introduction

L-functions

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