LMFDB - Embedded newform 35.4.b.a.29.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [35,4,Mod(29,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, 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("35.29");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 35 = 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	35.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:	$2.06506685020$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			29.4
Root			$-2.67516i$ of defining polynomial
Character	$\chi$	$=$	35.29
Dual form			35.4.b.a.29.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.67516i q^{2} +2.49396i q^{3} +5.19383 q^{4} +(6.35505 - 9.19855i) q^{5} +4.17779 q^{6} +7.00000i q^{7} -22.1018i q^{8} +20.7802 q^{9} +(-15.4091 - 10.6457i) q^{10} -57.5880 q^{11} +12.9532i q^{12} +45.5159i q^{13} +11.7261 q^{14} +(22.9408 + 15.8492i) q^{15} +4.52655 q^{16} +92.0051i q^{17} -34.8101i q^{18} -125.177 q^{19} +(33.0070 - 47.7757i) q^{20} -17.4577 q^{21} +96.4692i q^{22} -158.496i q^{23} +55.1211 q^{24} +(-44.2268 - 116.914i) q^{25} +76.2466 q^{26} +119.162i q^{27} +36.3568i q^{28} +40.1708 q^{29} +(26.5500 - 38.4296i) q^{30} +49.5590 q^{31} -184.397i q^{32} -143.622i q^{33} +154.123 q^{34} +(64.3899 + 44.4853i) q^{35} +107.929 q^{36} +231.307i q^{37} +209.692i q^{38} -113.515 q^{39} +(-203.305 - 140.458i) q^{40} +169.556 q^{41} +29.2445i q^{42} +147.428i q^{43} -299.102 q^{44} +(132.059 - 191.147i) q^{45} -265.507 q^{46} -67.0327i q^{47} +11.2890i q^{48} -49.0000 q^{49} +(-195.851 + 74.0870i) q^{50} -229.457 q^{51} +236.402i q^{52} -268.647i q^{53} +199.615 q^{54} +(-365.974 + 529.726i) q^{55} +154.713 q^{56} -312.187i q^{57} -67.2926i q^{58} +240.843 q^{59} +(119.151 + 82.3183i) q^{60} +90.4579 q^{61} -83.0194i q^{62} +145.461i q^{63} -272.683 q^{64} +(418.681 + 289.256i) q^{65} -240.591 q^{66} -406.498i q^{67} +477.859i q^{68} +395.283 q^{69} +(74.5201 - 107.863i) q^{70} +330.782 q^{71} -459.279i q^{72} -546.255i q^{73} +387.477 q^{74} +(291.580 - 110.300i) q^{75} -650.149 q^{76} -403.116i q^{77} +190.156i q^{78} +25.3087 q^{79} +(28.7664 - 41.6377i) q^{80} +263.879 q^{81} -284.034i q^{82} -376.255i q^{83} -90.6725 q^{84} +(846.314 + 584.697i) q^{85} +246.965 q^{86} +100.184i q^{87} +1272.80i q^{88} -1026.44 q^{89} +(-320.203 - 221.220i) q^{90} -318.612 q^{91} -823.203i q^{92} +123.598i q^{93} -112.291 q^{94} +(-795.506 + 1151.45i) q^{95} +459.879 q^{96} -942.660i q^{97} +82.0829i q^{98} -1196.69 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9} - 16 q^{10} + 84 q^{11} - 56 q^{14} + 8 q^{15} + 148 q^{16} + 72 q^{19} - 68 q^{20} + 140 q^{21} + 72 q^{24} - 362 q^{25} - 620 q^{26} + 88 q^{29} + 52 q^{30}+ \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 - 56 * q^14 + 8 * q^15 + 148 * q^16 + 72 * q^19 - 68 * q^20 + 140 * q^21 + 72 * q^24 - 362 * q^25 - 620 * q^26 + 88 * q^29 + 52 * q^30 + 120 * q^31 + 964 * q^34 - 28 * q^35 - 420 * q^36 + 212 * q^39 + 1396 * q^40 - 852 * q^41 - 1424 * q^44 - 510 * q^45 + 176 * q^46 - 490 * q^49 + 1644 * q^50 + 1276 * q^51 - 996 * q^54 - 1136 * q^55 + 588 * q^56 + 864 * q^59 - 1692 * q^60 - 884 * q^61 - 2724 * q^64 + 520 * q^65 + 1148 * q^66 + 4808 * q^69 + 756 * q^70 + 880 * q^71 + 3024 * q^74 + 720 * q^75 - 2672 * q^76 - 3580 * q^79 - 2316 * q^80 - 302 * q^81 - 1904 * q^84 + 1572 * q^85 + 2432 * q^86 - 1492 * q^89 - 7748 * q^90 + 476 * q^91 + 1652 * q^94 - 1628 * q^95 + 4080 * q^96 - 5304 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$.

$n$	$22$	$31$
$\chi(n)$	$-1$	$1$

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	$-0.904304\pi$
$2$		−	1.67516i		−	0.592259i	0.955148	−	0.296130i	$-0.0956960\pi$
$3$			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
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		−	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	$0.838626\pi$
$14$	11.7261			0.223853
$15$	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	$-0.772716\pi$
$19$	−125.177			−1.51145			−0.755726	+	0.654888i	$0.772716\pi$
$20$	33.0070	−	47.7757i	0.369030	−	0.534149i
$21$	−17.4577			−0.181409
$22$			96.4692i			0.934878i
$23$		−	158.496i		−	1.43690i	−0.695578	−	0.718451i	$-0.744851\pi$
$23$		−	158.496i		−	1.43690i	0.695578	−	0.718451i	$-0.255149\pi$
$24$	55.1211			0.468814
$25$	−44.2268	−	116.914i	−0.353814	−	0.935316i
$26$	76.2466			0.575123
$27$			119.162i			0.849360i
$28$			36.3568i			0.245386i
$29$	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	$-0.454143\pi$
$31$	49.5590			0.287131			0.143566	+	0.989641i	$0.454143\pi$
$32$		−	184.397i		−	1.01866i
$33$		−	143.622i		−	0.757619i
$34$	154.123			0.777410
$35$	64.3899	+	44.4853i	0.310968	+	0.214840i
$36$	107.929			0.499670
$37$			231.307i			1.02775i	0.857866	+	0.513874i	$0.171790\pi$
$37$			231.307i			1.02775i	−0.857866	+	0.513874i	$0.828210\pi$
$38$			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	$-0.395332\pi$
$41$	169.556			0.645859			0.322929	+	0.946423i	$0.395332\pi$
$42$			29.2445i			0.107441i
$43$			147.428i			0.522849i	0.965224	+	0.261425i	$0.0841923\pi$
$43$			147.428i			0.522849i	−0.965224	+	0.261425i	$0.915808\pi$
$44$	−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
$49$	−49.0000			−0.142857
$50$	−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	$-0.886818\pi$
$53$		−	268.647i		−	0.696254i	0.937447	−	0.348127i	$-0.113182\pi$
$54$	199.615			0.503041
$55$	−365.974	+	529.726i	−0.897236	+	1.29870i
$56$	154.713			0.369185
$57$		−	312.187i		−	0.725441i
$58$		−	67.2926i		−	0.152344i
$59$	240.843			0.531442			0.265721	−	0.964050i	$-0.414390\pi$
$59$	240.843			0.531442			0.265721	+	0.964050i	$0.414390\pi$
$60$	119.151	+	82.3183i	0.256372	+	0.177121i
$61$	90.4579			0.189868			0.0949340	−	0.995484i	$-0.469736\pi$
$61$	90.4579			0.189868			0.0949340	+	0.995484i	$0.469736\pi$
$62$		−	83.0194i		−	0.170056i
$63$			145.461i			0.290895i
$64$	−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	$-0.879149\pi$
$67$		−	406.498i		−	0.741218i	0.928789	−	0.370609i	$-0.120851\pi$
$68$			477.859i			0.852190i
$69$	395.283			0.689660
$70$	74.5201	−	107.863i	0.127241	−	0.184174i
$71$	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
$77$		−	403.116i		−	0.596615i
$78$			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	$-0.0800333\pi$
$84$	−90.6725			−0.117776
$85$	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	$-0.709332\pi$
$89$	−1026.44			−1.22250			−0.611248	+	0.791439i	$0.709332\pi$
$90$	−320.203	−	221.220i	−0.375026	−	0.259096i
$91$	−318.612			−0.367028
$92$		−	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	$-0.164233\pi$
$98$			82.0829i			0.0846085i
$99$	−1196.69			−1.21487
$100$	−229.706	−	607.234i	−0.229706	−	0.607234i
$101$	−604.617			−0.595659			−0.297830	−	0.954619i	$-0.596263\pi$
$101$	−604.617			−0.595659			−0.297830	+	0.954619i	$0.596263\pi$
$102$			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
$105$	−110.945	+	160.586i	−0.103115	+	0.149253i
$106$	−450.027			−0.412363
$107$			1511.66i			1.36577i	0.730525	+	0.682886i	$0.239276\pi$
$107$			1511.66i			1.36577i	−0.730525	+	0.682886i	$0.760724\pi$
$108$			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
$112$			31.6858i			0.0267324i
$113$		−	1045.27i		−	0.870182i	−0.900387	−	0.435091i	$-0.856716\pi$
$113$		−	1045.27i		−	0.870182i	0.900387	−	0.435091i	$-0.143284\pi$
$114$	−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
$119$	−644.036			−0.496123
$120$	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
$126$	243.671			0.172285
$127$			260.727i			0.182171i	0.995843	+	0.0910857i	$0.0290337\pi$
$127$			260.727i			0.182171i	−0.995843	+	0.0910857i	$0.970966\pi$
$128$		−	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	$-0.577566\pi$
$131$	−723.522			−0.482553			−0.241276	+	0.970456i	$0.577566\pi$
$132$		−	745.950i		−	0.491868i
$133$		−	876.240i		−	0.571275i
$134$	−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	$0.0775557\pi$
$137$			773.693i			0.482490i	−0.970464	+	0.241245i	$0.922444\pi$
$138$		−	662.164i		−	0.408457i
$139$	2952.97			1.80192			0.900961	−	0.433899i	$-0.142863\pi$
$139$	2952.97			1.80192			0.900961	+	0.433899i	$0.142863\pi$
$140$	334.430	+	231.049i	0.201889	+	0.139480i
$141$	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
$147$		−	122.204i		−	0.0685662i
$148$			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
$154$	−675.285			−0.353351
$155$	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
$161$	1109.47			0.543098
$162$		−	442.040i		−	0.214383i
$163$			1325.20i			0.636798i	0.947957	+	0.318399i	$0.103145\pi$
$163$			1325.20i			0.636798i	−0.947957	+	0.318399i	$0.896855\pi$
$164$	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	$0.839424\pi$
$168$			385.847i			0.177195i
$169$	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
$175$	818.401	−	309.587i	0.353516	−	0.133729i
$176$	−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	$-0.695036\pi$
$181$	−2800.85			−1.15020			−0.575099	+	0.818084i	$0.695036\pi$
$182$			533.726i			0.217376i
$183$			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
$189$	−834.133			−0.321028
$190$	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	$-0.724551\pi$
$193$		−	4082.57i		−	1.52264i	0.648375	−	0.761321i	$-0.275449\pi$
$194$	−1579.11			−0.584399
$195$	−721.393	+	1044.17i	−0.264923	+	0.383461i
$196$	−254.498			−0.0927470
$197$			3414.89i			1.23503i	0.786559	+	0.617515i	$0.211860\pi$
$197$			3414.89i			1.23503i	−0.786559	+	0.617515i	$0.788140\pi$
$198$			2004.65i			0.719515i
$199$	3392.44			1.20846			0.604231	−	0.796809i	$-0.293480\pi$
$199$	3392.44			1.20846			0.604231	+	0.796809i	$0.293480\pi$
$200$	−2584.02	+	977.492i	−0.913589	+	0.345596i
$201$	1013.79			0.355757
$202$			1012.83i			0.352785i
$203$			281.196i			0.0972220i
$204$	−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
$210$	269.007	+	185.850i	0.0883965	+	0.0610709i
$211$	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
$217$			346.913i			0.108525i
$218$		−	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	$-0.00872860\pi$
$224$	1290.78			0.385017
$225$	−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	$-0.834264\pi$
$229$	−6012.35			−1.73497			−0.867483	+	0.497466i	$0.834264\pi$
$230$	−1687.31	+	2442.28i	−0.483730	+	0.700170i
$231$	1005.36			0.286353
$232$		−	887.848i		−	0.251250i
$233$			940.660i			0.264484i	0.991217	+	0.132242i	$0.0422175\pi$
$233$			940.660i			0.264484i	−0.991217	+	0.132242i	$0.957782\pi$
$234$	1584.42			0.442635
$235$	−616.604	−	425.996i	−0.171161	−	0.118251i
$236$	1250.90			0.345027
$237$			63.1188i			0.0172996i
$238$			1078.86i			0.293833i
$239$	−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	$-0.519744\pi$
$241$	−463.836			−0.123976			−0.0619882	+	0.998077i	$0.519744\pi$
$242$		−	3325.83i		−	0.883440i
$243$			3875.47i			1.02309i
$244$	469.823			0.123268
$245$	−311.397	+	450.729i	−0.0812018	+	0.117535i
$246$	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	$-0.407021\pi$
$251$	2290.25			0.575934			0.287967	+	0.957640i	$0.407021\pi$
$252$			755.501i			0.188857i
$253$			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
$259$	−1619.15			−0.388452
$260$	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	$0.0107107\pi$
$263$			286.978i			0.0672845i	−0.999434	+	0.0336423i	$0.989289\pi$
$264$	−3174.31			−0.740020
$265$	−2471.16	−	1707.26i	−0.572839	−	0.395760i
$266$	−1467.84			−0.338343
$267$		−	2559.90i		−	0.586753i
$268$		−	2111.28i		−	0.481221i
$269$	−3561.22			−0.807180			−0.403590	−	0.914940i	$-0.632238\pi$
$269$	−3561.22			−0.807180			−0.403590	+	0.914940i	$0.632238\pi$
$270$	1268.57	−	1836.17i	0.285935	−	0.413874i
$271$	1928.81			0.432349			0.216175	−	0.976355i	$-0.430642\pi$
$271$	1928.81			0.432349			0.216175	+	0.976355i	$0.430642\pi$
$272$			416.466i			0.0928380i
$273$		−	794.605i		−	0.176160i
$274$	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	$0.253381\pi$
$277$			6588.69i			1.42916i	−0.699556	+	0.714578i	$0.746619\pi$
$278$		−	4946.70i		−	1.06721i
$279$	1029.84			0.220986
$280$	983.206	−	1423.13i	0.209849	−	0.303744i
$281$	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
$287$			1186.89i			0.244112i
$288$		−	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	$-0.0138249\pi$
$294$	−204.712			−0.0406089
$295$	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
$301$	−1031.99			−0.197618
$302$		−	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	$-0.151060\pi$
$308$		−	2093.72i		−	0.387340i
$309$	−750.601			−0.138188
$310$	−763.659	−	527.592i	−0.139913	−	0.0966620i
$311$	−1831.11			−0.333868			−0.166934	−	0.985968i	$-0.553387\pi$
$311$	−1831.11			−0.333868			−0.166934	+	0.985968i	$0.553387\pi$
$312$			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
$315$	1338.03	+	924.412i	0.239332	+	0.165348i
$316$	131.449			0.0234006
$317$			1666.19i			0.295214i	0.989046	+	0.147607i	$0.0471570\pi$
$317$			1666.19i			0.295214i	−0.989046	+	0.147607i	$0.952843\pi$
$318$		−	1122.35i		−	0.197919i
$319$	−2313.36			−0.406029
$320$	−1732.91	+	2508.29i	−0.302727	+	0.438180i
$321$	−3770.02			−0.655521
$322$		−	1858.55i		−	0.321655i
$323$		−	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
$329$	469.229			0.0786305
$330$	−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
$336$	−79.0233			−0.0128306
$337$			10650.5i			1.72157i	0.508970	+	0.860784i	$0.330027\pi$
$337$			10650.5i			1.72157i	−0.508970	+	0.860784i	$0.669973\pi$
$338$		−	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
$343$		−	343.000i		−	0.0539949i
$344$	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	$0.100627\pi$
$347$			4019.13i			0.621782i	−0.950446	+	0.310891i	$0.899373\pi$
$348$			520.341i			0.0801529i
$349$	−10544.9			−1.61735			−0.808674	−	0.588256i	$-0.799815\pi$
$349$	−10544.9			−1.61735			−0.808674	+	0.588256i	$0.799815\pi$
$350$	−518.609	−	1370.95i	−0.0792024	−	0.209373i
$351$	−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
$357$		−	1606.20i		−	0.238121i
$358$			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
$364$	−1654.82			−0.238285
$365$	−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
$371$	1880.53			0.263159
$372$			641.949i			0.0894718i
$373$		−	4646.02i		−	0.644938i	−0.946580	−	0.322469i	$-0.895487\pi$
$373$		−	4646.02i		−	0.644938i	0.946580	−	0.322469i	$-0.104513\pi$
$374$	−8875.66			−1.22714
$375$	838.406	−	3383.08i	0.115454	−	0.465870i
$376$	−1481.54			−0.203204
$377$			1828.41i			0.249783i
$378$			1397.31i			0.190132i
$379$	−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
$385$	−3708.08	−	2561.82i	−0.490861	−	0.339123i
$386$	−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
$392$			1082.99i			0.139539i
$393$		−	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
$399$	2185.31			0.274191
$400$	−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
$406$	471.048			0.0575806
$407$		−	13320.5i		−	1.62229i
$408$			5071.42i			0.615374i
$409$	9698.79			1.17255			0.586277	−	0.810111i	$-0.300593\pi$
$409$	9698.79			1.17255			0.586277	+	0.810111i	$0.300593\pi$
$410$	−2612.70	−	1805.05i	−0.314713	−	0.217427i
$411$	−1929.56			−0.231577
$412$			1563.17i			0.186922i
$413$			1685.90i			0.200866i
$414$	−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	$-0.798537\pi$
$419$	−13830.9			−1.61261			−0.806307	+	0.591498i	$0.798537\pi$
$420$	−576.228	+	834.056i	−0.0669453	+	0.0968994i
$421$	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
$427$			633.205i			0.0717634i
$428$			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	$-0.289539\pi$
$434$	581.136			0.0642752
$435$	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	$-0.597323\pi$
$439$	−5537.38			−0.602016			−0.301008	+	0.953622i	$0.597323\pi$
$440$	11707.9	+	8088.70i	1.26853	+	0.876394i
$441$	−1018.23			−0.109948
$442$			7015.07i			0.754916i
$443$		−	3974.09i		−	0.426218i	−0.977028	−	0.213109i	$-0.931641\pi$
$443$		−	3974.09i		−	0.426218i	0.977028	−	0.213109i	$-0.0683589\pi$
$444$	−2996.17			−0.320252
$445$	−6523.06	+	9441.74i	−0.694882	+	1.00580i
$446$	−305.903			−0.0324774
$447$		−	6269.82i		−	0.663428i
$448$		−	1908.78i		−	0.201298i
$449$	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
$455$	−2024.79	+	2930.77i	−0.208623	+	0.301970i
$456$	−6899.89			−0.708590
$457$		−	10768.9i		−	1.10229i	−0.834410	−	0.551145i	$-0.814191\pi$
$457$		−	10768.9i		−	1.10229i	0.834410	−	0.551145i	$-0.185809\pi$
$458$			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	$-0.505348\pi$
$461$	−332.605			−0.0336029			−0.0168015	+	0.999859i	$0.505348\pi$
$462$		−	1684.13i		−	0.169595i
$463$			8205.35i			0.823618i	0.911270	+	0.411809i	$0.135103\pi$
$463$			8205.35i			0.823618i	−0.911270	+	0.411809i	$0.864897\pi$
$464$	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
$469$	2845.49			0.280154
$470$	−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
$476$	−3345.01			−0.322098
$477$		−	5582.52i		−	0.535862i
$478$			8641.86i			0.826924i
$479$	−6628.58			−0.632292			−0.316146	−	0.948711i	$-0.602389\pi$
$479$	−6628.58			−0.632292			−0.316146	+	0.948711i	$0.602389\pi$
$480$	2922.55	−	4230.23i	0.277908	−	0.402255i
$481$	−10528.2			−0.998010
$482$			777.001i			0.0734262i
$483$			2766.98i			0.260667i
$484$	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	$-0.589961\pi$
$487$		−	20641.6i		−	1.92065i	0.278875	−	0.960327i	$-0.410039\pi$
$488$		−	1999.28i		−	0.185458i
$489$	−3305.01			−0.305640
$490$	755.044	+	521.641i	0.0696111	+	0.0480925i
$491$	−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
$497$			2315.47i			0.208980i
$498$		−	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	$0.667111\pi$
$504$	3214.95			0.284138
$505$	−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	$-0.376513\pi$
$509$	8688.17			0.756574			0.378287	+	0.925688i	$0.376513\pi$
$510$	3535.72	+	2442.74i	0.306989	+	0.212091i
$511$	3823.78			0.331026
$512$		−	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
$518$			2712.34i			0.230064i
$519$	4783.89			0.404604
$520$	6393.08	−	9253.60i	0.539144	−	0.780380i
$521$	6771.36			0.569402			0.284701	−	0.958616i	$-0.408106\pi$
$521$	6771.36			0.569402			0.284701	+	0.958616i	$0.408106\pi$
$522$		−	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
$525$	772.099	+	2041.06i	0.0641851	+	0.169675i
$526$	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
$532$		−	4551.04i		−	0.370888i
$533$			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
$539$	2821.81			0.225499
$540$	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
$546$	−1331.09			−0.104332
$547$		−	11552.7i		−	0.903033i	−0.892263	−	0.451516i	$-0.850883\pi$
$547$		−	11552.7i		−	0.903033i	0.892263	−	0.451516i	$-0.149117\pi$
$548$			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
$553$			177.161i			0.0136232i
$554$	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	$-0.785501\pi$
$557$		−	16406.2i		−	1.24803i	0.781413	−	0.624014i	$-0.214499\pi$
$558$		−	1725.16i		−	0.130881i
$559$	−6710.31			−0.507721
$560$	291.464	+	201.365i	0.0219939	+	0.0151950i
$561$	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
$567$			1847.15i			0.136813i
$568$		−	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
$574$	1988.24			0.144577
$575$	−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
$581$	2633.78			0.188068
$582$		−	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	$0.993320\pi$
$588$		−	634.708i		−	0.0445151i
$589$	−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
$595$	−4092.88	+	5924.20i	−0.282003	+	0.408182i
$596$	−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	$-0.437054\pi$
$601$	5789.33			0.392931			0.196466	+	0.980511i	$0.437054\pi$
$602$			1728.76i			0.117041i
$603$		−	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
$609$	−701.291			−0.0466630
$610$	−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	$-0.977294\pi$
$613$		−	2163.47i		−	0.142548i	0.997457	−	0.0712738i	$-0.0227064\pi$
$614$	−8235.08			−0.541272
$615$	3889.76	+	2687.34i	0.255041	+	0.176201i
$616$	−8909.59			−0.582756
$617$			22964.9i			1.49843i	0.662327	+	0.749215i	$0.269569\pi$
$617$			22964.9i			1.49843i	−0.662327	+	0.749215i	$0.730431\pi$
$618$			1257.38i			0.0818433i
$619$	1386.67			0.0900401			0.0450200	−	0.998986i	$-0.485665\pi$
$619$	1386.67			0.0900401			0.0450200	+	0.998986i	$0.485665\pi$
$620$	1635.80	−	2367.72i	0.105960	−	0.153371i
$621$	18886.7			1.22045
$622$			3067.41i			0.197736i
$623$		−	7185.06i		−	0.462060i
$624$	−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
$630$	1548.54	−	2241.42i	0.0979291	−	0.141747i
$631$	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
$637$		−	2230.28i		−	0.138724i
$638$			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	$-0.340328\pi$
$644$	5762.42			0.352595
$645$	−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
$651$	−865.188			−0.0520882
$652$			6882.89i			0.413428i
$653$		−	6999.85i		−	0.419488i	−0.977756	−	0.209744i	$-0.932737\pi$
$653$		−	6999.85i		−	0.419488i	0.977756	−	0.209744i	$-0.0672630\pi$
$654$	7382.53			0.441406
$655$	−4598.01	+	6655.35i	−0.274289	+	0.397017i
$656$	767.504			0.0456799
$657$		−	11351.3i		−	0.674056i
$658$		−	786.035i		−	0.0465696i
$659$	−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	$-0.846192\pi$
$661$	−30097.2			−1.77102			−0.885512	+	0.464617i	$0.846192\pi$
$662$			9157.07i			0.537613i
$663$		−	10444.0i		−	0.611779i
$664$	−8315.91			−0.486024
$665$	−8060.14	−	5568.54i	−0.470013	−	0.324720i
$666$	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
$672$			3219.16i			0.184794i
$673$			5400.26i			0.309309i	0.987969	+	0.154654i	$0.0494264\pi$
$673$			5400.26i			0.309309i	−0.987969	+	0.154654i	$0.950574\pi$
$674$	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
$679$	6598.62			0.372948
$680$	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	$0.198704\pi$
$683$			20865.8i			1.16897i	−0.811404	+	0.584486i	$0.801296\pi$
$684$	−13510.2			−0.755227
$685$	7116.86	+	4916.86i	0.396965	+	0.274253i
$686$	−574.581			−0.0319790
$687$		−	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	$-0.330432\pi$
$691$	18450.3			1.01575			0.507873	+	0.861432i	$0.330432\pi$
$692$		−	9962.76i		−	0.547294i
$693$		−	8376.81i		−	0.459176i
$694$	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
$700$	4250.64	−	1607.95i	0.229513	−	0.0868209i
$701$	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
$707$		−	4232.32i		−	0.225138i
$708$			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
$714$	−2690.65			−0.141029
$715$	−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	$-0.285164\pi$
$719$	24093.1			1.24968			0.624841	+	0.780752i	$0.285164\pi$
$720$	597.771	−	865.238i	0.0309411	−	0.0447854i
$721$	−2106.77			−0.108821
$722$		−	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	$0.629925\pi$
$728$			7041.89i			0.358503i
$729$	−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
$735$	−1124.10	−	776.613i	−0.0564124	−	0.0389739i
$736$	−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
$742$		−	3150.19i		−	0.155859i
$743$		−	7438.65i		−	0.367292i	−0.982992	−	0.183646i	$-0.941210\pi$
$743$		−	7438.65i		−	0.367292i	0.982992	−	0.183646i	$-0.0587900\pi$
$744$	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
$749$	−10581.6			−0.516214
$750$	−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
$756$	−4332.35			−0.208421
$757$		−	10193.8i		−	0.489432i	−0.969595	−	0.244716i	$-0.921305\pi$
$757$		−	10193.8i		−	0.489432i	0.969595	−	0.244716i	$-0.0786947\pi$
$758$			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	$-0.935138\pi$
$761$	−41117.6			−1.95862			−0.979311	+	0.202362i	$0.935138\pi$
$762$			1089.26i			0.0517845i
$763$			12369.6i			0.586908i
$764$	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	$-0.586799\pi$
$769$	−11486.6			−0.538642			−0.269321	+	0.963050i	$0.586799\pi$
$770$	−4291.47	+	6211.64i	−0.200849	+	0.290717i
$771$	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
$777$		−	4038.10i		−	0.186443i
$778$		−	12990.9i		−	0.598645i
$779$	−21224.5			−0.976185

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:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	35.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.67516i q^{2} +2.49396i q^{3} +5.19383 q^{4} +(6.35505 - 9.19855i) q^{5} +4.17779 q^{6} +7.00000i q^{7} -22.1018i q^{8} +20.7802 q^{9} +(-15.4091 - 10.6457i) q^{10} -57.5880 q^{11} +12.9532i q^{12} +45.5159i q^{13} +11.7261 q^{14} +(22.9408 + 15.8492i) q^{15} +4.52655 q^{16} +92.0051i q^{17} -34.8101i q^{18} -125.177 q^{19} +(33.0070 - 47.7757i) q^{20} -17.4577 q^{21} +96.4692i q^{22} -158.496i q^{23} +55.1211 q^{24} +(-44.2268 - 116.914i) q^{25} +76.2466 q^{26} +119.162i q^{27} +36.3568i q^{28} +40.1708 q^{29} +(26.5500 - 38.4296i) q^{30} +49.5590 q^{31} -184.397i q^{32} -143.622i q^{33} +154.123 q^{34} +(64.3899 + 44.4853i) q^{35} +107.929 q^{36} +231.307i q^{37} +209.692i q^{38} -113.515 q^{39} +(-203.305 - 140.458i) q^{40} +169.556 q^{41} +29.2445i q^{42} +147.428i q^{43} -299.102 q^{44} +(132.059 - 191.147i) q^{45} -265.507 q^{46} -67.0327i q^{47} +11.2890i q^{48} -49.0000 q^{49} +(-195.851 + 74.0870i) q^{50} -229.457 q^{51} +236.402i q^{52} -268.647i q^{53} +199.615 q^{54} +(-365.974 + 529.726i) q^{55} +154.713 q^{56} -312.187i q^{57} -67.2926i q^{58} +240.843 q^{59} +(119.151 + 82.3183i) q^{60} +90.4579 q^{61} -83.0194i q^{62} +145.461i q^{63} -272.683 q^{64} +(418.681 + 289.256i) q^{65} -240.591 q^{66} -406.498i q^{67} +477.859i q^{68} +395.283 q^{69} +(74.5201 - 107.863i) q^{70} +330.782 q^{71} -459.279i q^{72} -546.255i q^{73} +387.477 q^{74} +(291.580 - 110.300i) q^{75} -650.149 q^{76} -403.116i q^{77} +190.156i q^{78} +25.3087 q^{79} +(28.7664 - 41.6377i) q^{80} +263.879 q^{81} -284.034i q^{82} -376.255i q^{83} -90.6725 q^{84} +(846.314 + 584.697i) q^{85} +246.965 q^{86} +100.184i q^{87} +1272.80i q^{88} -1026.44 q^{89} +(-320.203 - 221.220i) q^{90} -318.612 q^{91} -823.203i q^{92} +123.598i q^{93} -112.291 q^{94} +(-795.506 + 1151.45i) q^{95} +459.879 q^{96} -942.660i q^{97} +82.0829i q^{98} -1196.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9} - 16 q^{10} + 84 q^{11} - 56 q^{14} + 8 q^{15} + 148 q^{16} + 72 q^{19} - 68 q^{20} + 140 q^{21} + 72 q^{24} - 362 q^{25} - 620 q^{26} + 88 q^{29} + 52 q^{30}+ \cdots - 5304 q^{99}+O(q^{100}) \) 10 * q - 36 * q^4 + 6 * q^5 + 12 * q^6 - 46 * q^9 - 16 * q^10 + 84 * q^11 - 56 * q^14 + 8 * q^15 + 148 * q^16 + 72 * q^19 - 68 * q^20 + 140 * q^21 + 72 * q^24 - 362 * q^25 - 620 * q^26 + 88 * q^29 + 52 * q^30 + 120 * q^31 + 964 * q^34 - 28 * q^35 - 420 * q^36 + 212 * q^39 + 1396 * q^40 - 852 * q^41 - 1424 * q^44 - 510 * q^45 + 176 * q^46 - 490 * q^49 + 1644 * q^50 + 1276 * q^51 - 996 * q^54 - 1136 * q^55 + 588 * q^56 + 864 * q^59 - 1692 * q^60 - 884 * q^61 - 2724 * q^64 + 520 * q^65 + 1148 * q^66 + 4808 * q^69 + 756 * q^70 + 880 * q^71 + 3024 * q^74 + 720 * q^75 - 2672 * q^76 - 3580 * q^79 - 2316 * q^80 - 302 * q^81 - 1904 * q^84 + 1572 * q^85 + 2432 * q^86 - 1492 * q^89 - 7748 * q^90 + 476 * q^91 + 1652 * q^94 - 1628 * q^95 + 4080 * q^96 - 5304 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more