LMFDB - Embedded newform 315.4.d.c.64.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,4,Mod(64,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.64"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$315 = 3^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	315.d (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5856016518$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{2}\cdot 7^{4}$
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.7
Root			$2.67516i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.4.d.c.64.4

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67516i q^{2} +5.19383 q^{4} +(-6.35505 + 9.19855i) q^{5} +7.00000i q^{7} +22.1018i q^{8} +(-15.4091 - 10.6457i) q^{10} +57.5880 q^{11} +45.5159i q^{13} -11.7261 q^{14} +4.52655 q^{16} -92.0051i q^{17} -125.177 q^{19} +(-33.0070 + 47.7757i) q^{20} +96.4692i q^{22} +158.496i q^{23} +(-44.2268 - 116.914i) q^{25} -76.2466 q^{26} +36.3568i q^{28} -40.1708 q^{29} +49.5590 q^{31} +184.397i q^{32} +154.123 q^{34} +(-64.3899 - 44.4853i) q^{35} +231.307i q^{37} -209.692i q^{38} +(-203.305 - 140.458i) q^{40} -169.556 q^{41} +147.428i q^{43} +299.102 q^{44} -265.507 q^{46} +67.0327i q^{47} -49.0000 q^{49} +(195.851 - 74.0870i) q^{50} +236.402i q^{52} +268.647i q^{53} +(-365.974 + 529.726i) q^{55} -154.713 q^{56} -67.2926i q^{58} -240.843 q^{59} +90.4579 q^{61} +83.0194i q^{62} -272.683 q^{64} +(-418.681 - 289.256i) q^{65} -406.498i q^{67} -477.859i q^{68} +(74.5201 - 107.863i) q^{70} -330.782 q^{71} -546.255i q^{73} -387.477 q^{74} -650.149 q^{76} +403.116i q^{77} +25.3087 q^{79} +(-28.7664 + 41.6377i) q^{80} -284.034i q^{82} +376.255i q^{83} +(846.314 + 584.697i) q^{85} -246.965 q^{86} +1272.80i q^{88} +1026.44 q^{89} -318.612 q^{91} +823.203i q^{92} -112.291 q^{94} +(795.506 - 1151.45i) q^{95} -942.660i q^{97} -82.0829i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 36 q^{4} - 6 q^{5} - 16 q^{10} - 84 q^{11} + 56 q^{14} + 148 q^{16} + 72 q^{19} + 68 q^{20} - 362 q^{25} + 620 q^{26} - 88 q^{29} + 120 q^{31} + 964 q^{34} + 28 q^{35} + 1396 q^{40} + 852 q^{41}+ \cdots + 1628 q^{95}+O(q^{100})$ 10 * q - 36 * q^4 - 6 * q^5 - 16 * q^10 - 84 * q^11 + 56 * q^14 + 148 * q^16 + 72 * q^19 + 68 * q^20 - 362 * q^25 + 620 * q^26 - 88 * q^29 + 120 * q^31 + 964 * q^34 + 28 * q^35 + 1396 * q^40 + 852 * q^41 + 1424 * q^44 + 176 * q^46 - 490 * q^49 - 1644 * q^50 - 1136 * q^55 - 588 * q^56 - 864 * q^59 - 884 * q^61 - 2724 * q^64 - 520 * q^65 + 756 * q^70 - 880 * q^71 - 3024 * q^74 - 2672 * q^76 - 3580 * q^79 + 2316 * q^80 + 1572 * q^85 - 2432 * q^86 + 1492 * q^89 + 476 * q^91 + 1652 * q^94 + 1628 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$ .

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

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	$0.904304\pi$
$3$		0			0
$3$		0			0
$4$	5.19383			0.649229
$5$	−6.35505	+	9.19855i	−0.568413	+	0.822744i
$5$	−6.35505	+	9.19855i	−0.568413	+	0.822744i
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$			22.1018i			0.976771i
$9$		0			0
$10$	−15.4091	−	10.6457i	−0.487278	−	0.336648i
$11$	57.5880			1.57849			0.789247	−	0.614076i	$-0.210471\pi$
$11$	57.5880			1.57849			0.789247	+	0.614076i	$0.210471\pi$
$12$		0			0
$13$			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
$15$		0			0
$16$	4.52655			0.0707273
$17$		−	92.0051i		−	1.31262i	−0.754492	−	0.656309i	$-0.772117\pi$
$17$		−	92.0051i		−	1.31262i	0.754492	−	0.656309i	$-0.227883\pi$
$18$		0			0
$19$	−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
$21$		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	$0.744851\pi$
$24$		0			0
$25$	−44.2268	−	116.914i	−0.353814	−	0.935316i
$26$	−76.2466			−0.575123
$27$		0			0
$28$			36.3568i			0.245386i
$29$	−40.1708			−0.257225			−0.128613	−	0.991695i	$-0.541052\pi$
$29$	−40.1708			−0.257225			−0.128613	+	0.991695i	$0.541052\pi$
$30$		0			0
$31$	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
$33$		0			0
$34$	154.123			0.777410
$35$	−64.3899	−	44.4853i	−0.310968	−	0.214840i
$36$		0			0
$37$			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
$39$		0			0
$40$	−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	$0.0841923\pi$
$43$			147.428i			0.522849i	−0.965224	+	0.261425i	$0.915808\pi$
$44$	299.102			1.02480
$45$		0			0
$46$	−265.507			−0.851018
$47$			67.0327i			0.208037i	0.994575	+	0.104018i	$0.0331701\pi$
$47$			67.0327i			0.208037i	−0.994575	+	0.104018i	$0.966830\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	195.851	−	74.0870i	0.553949	−	0.209550i
$51$		0			0
$52$			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	$0.886818\pi$
$54$		0			0
$55$	−365.974	+	529.726i	−0.897236	+	1.29870i
$56$	−154.713			−0.369185
$57$		0			0
$58$		−	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	$0.585610\pi$
$60$		0			0
$61$	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
$63$		0			0
$64$	−272.683			−0.532583
$65$	−418.681	−	289.256i	−0.798938	−	0.551966i
$66$		0			0
$67$		−	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
$69$		0			0
$70$	74.5201	−	107.863i	0.127241	−	0.184174i
$71$	−330.782			−0.552910			−0.276455	−	0.961027i	$-0.589160\pi$
$71$	−330.782			−0.552910			−0.276455	+	0.961027i	$0.589160\pi$
$72$		0			0
$73$		−	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
$75$		0			0
$76$	−650.149			−0.981279
$77$			403.116i			0.596615i
$78$		0			0
$79$	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
$81$		0			0
$82$		−	284.034i		−	0.382516i
$83$			376.255i			0.497582i	0.968557	+	0.248791i	$0.0800333\pi$
$83$			376.255i			0.497582i	−0.968557	+	0.248791i	$0.919967\pi$
$84$		0			0
$85$	846.314	+	584.697i	1.07995	+	0.746109i
$86$	−246.965			−0.309662
$87$		0			0
$88$			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	$0.290668\pi$
$90$		0			0
$91$	−318.612			−0.367028
$92$			823.203i			0.932878i
$93$		0			0
$94$	−112.291			−0.123212
$95$	795.506	−	1151.45i	0.859128	−	1.24354i
$96$		0			0
$97$		−	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
$99$		0			0
$100$	−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	$0.403737\pi$
$102$		0			0
$103$			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	$-0.239276\pi$
$108$		0			0
$109$	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
$111$		0			0
$112$			31.6858i			0.0267324i
$113$			1045.27i			0.870182i	0.900387	+	0.435091i	$0.143284\pi$
$113$			1045.27i			0.870182i	−0.900387	+	0.435091i	$0.856716\pi$
$114$		0			0
$115$	−1457.94	−	1007.25i	−1.18220	−	0.816753i
$116$	−208.640			−0.166998
$117$		0			0
$118$		−	403.451i		−	0.314751i
$119$	644.036			0.496123
$120$		0			0
$121$	1985.38			1.49164
$122$			151.532i			0.112451i
$123$		0			0
$124$	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	$0.0290337\pi$
$127$			260.727i			0.182171i	−0.995843	+	0.0910857i	$0.970966\pi$
$128$			1018.39i			0.703233i
$129$		0			0
$130$	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	$0.422434\pi$
$132$		0			0
$133$		−	876.240i		−	0.571275i
$134$	680.950			0.438993
$135$		0			0
$136$	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	$-0.0775557\pi$
$138$		0			0
$139$	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
$141$		0			0
$142$		−	554.114i		−	0.327466i
$143$			2621.17i			1.53282i
$144$		0			0
$145$	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	$-0.257116\pi$
$149$	2514.00			1.38225			0.691124	+	0.722736i	$0.257116\pi$
$150$		0			0
$151$	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
$153$		0			0
$154$	−675.285			−0.353351
$155$	−314.950	+	455.872i	−0.163209	+	0.236235i
$156$		0			0
$157$			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
$159$		0			0
$160$	−1696.19	−	1171.85i	−0.838096	−	0.579019i
$161$	−1109.47			−0.543098
$162$		0			0
$163$			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
$165$		0			0
$166$	−630.288			−0.294698
$167$		−	2086.20i		−	0.966675i	−0.875434	−	0.483338i	$-0.839424\pi$
$167$		−	2086.20i		−	0.966675i	0.875434	−	0.483338i	$-0.160576\pi$
$168$		0			0
$169$	125.299			0.0570317
$170$	−979.462	+	1417.71i	−0.441890	+	0.639609i
$171$		0			0
$172$			765.715i			0.339449i
$173$			1918.19i			0.842990i	0.906831	+	0.421495i	$0.138495\pi$
$173$			1918.19i			0.842990i	−0.906831	+	0.421495i	$0.861505\pi$
$174$		0			0
$175$	818.401	−	309.587i	0.353516	−	0.133729i
$176$	260.675			0.111643
$177$		0			0
$178$			1719.45i			0.724035i
$179$	629.046			0.262665			0.131333	−	0.991338i	$-0.458074\pi$
$179$	629.046			0.262665			0.131333	+	0.991338i	$0.458074\pi$
$180$		0			0
$181$	−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
$183$		0			0
$184$	−3503.05			−1.40352
$185$	−2127.69	−	1469.97i	−0.845573	−	0.584185i
$186$		0			0
$187$		−	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	$-0.544780\pi$
$191$	−740.255			−0.280434			−0.140217	+	0.990121i	$0.544780\pi$
$192$		0			0
$193$		−	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
$195$		0			0
$196$	−254.498			−0.0927470
$197$		−	3414.89i		−	1.23503i	−0.786559	−	0.617515i	$-0.788140\pi$
$197$		−	3414.89i		−	1.23503i	0.786559	−	0.617515i	$-0.211860\pi$
$198$		0			0
$199$	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
$201$		0			0
$202$			1012.83i			0.352785i
$203$		−	281.196i		−	0.0972220i
$204$		0			0
$205$	1077.54	−	1559.67i	0.367114	−	0.531376i
$206$	−504.169			−0.170520
$207$		0			0
$208$			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
$213$		0			0
$214$	2532.28			0.808892
$215$	−1356.12	−	936.910i	−0.430171	−	0.297194i
$216$		0			0
$217$			346.913i			0.108525i
$218$			2960.16i			0.919667i
$219$		0			0
$220$	−1900.81	+	2751.31i	−0.582512	+	0.843151i
$221$	4187.70			1.27464
$222$		0			0
$223$		−	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
$225$		0			0
$226$	−1750.99			−0.515373
$227$			3152.33i			0.921707i	0.887476	+	0.460854i	$0.152457\pi$
$227$			3152.33i			0.921707i	−0.887476	+	0.460854i	$0.847543\pi$
$228$		0			0
$229$	−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
$231$		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	$-0.0422175\pi$
$234$		0			0
$235$	−616.604	−	425.996i	−0.171161	−	0.118251i
$236$	−1250.90			−0.345027
$237$		0			0
$238$			1078.86i			0.293833i
$239$	5158.82			1.39622			0.698109	−	0.715991i	$-0.254025\pi$
$239$	5158.82			1.39622			0.698109	+	0.715991i	$0.254025\pi$
$240$		0			0
$241$	−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
$243$		0			0
$244$	469.823			0.123268
$245$	311.397	−	450.729i	0.0812018	−	0.117535i
$246$		0			0
$247$		−	5697.55i		−	1.46772i
$248$			1095.34i			0.280461i
$249$		0			0
$250$	−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
$255$		0			0
$256$	−3887.43			−0.949079
$257$			802.202i			0.194708i	0.995250	+	0.0973541i	$0.0310379\pi$
$257$			802.202i			0.194708i	−0.995250	+	0.0973541i	$0.968962\pi$
$258$		0			0
$259$	−1619.15			−0.388452
$260$	−2174.56	−	1502.35i	−0.518694	−	0.358352i
$261$		0			0
$262$			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	$-0.0107107\pi$
$264$		0			0
$265$	−2471.16	−	1707.26i	−0.572839	−	0.395760i
$266$	1467.84			0.338343
$267$		0			0
$268$		−	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	$0.367762\pi$
$270$		0			0
$271$	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
$273$		0			0
$274$	1296.06			0.285759
$275$	−2546.93	−	6732.87i	−0.558494	−	1.47639i
$276$		0			0
$277$			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
$279$		0			0
$280$	983.206	−	1423.13i	0.209849	−	0.303744i
$281$	−815.552			−0.173138			−0.0865689	−	0.996246i	$-0.527590\pi$
$281$	−815.552			−0.173138			−0.0865689	+	0.996246i	$0.527590\pi$
$282$		0			0
$283$			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
$285$		0			0
$286$	−4390.89			−0.907828
$287$		−	1186.89i		−	0.244112i
$288$		0			0
$289$	−3551.94			−0.722967
$290$	618.995	+	427.648i	0.125340	+	0.0865943i
$291$		0			0
$292$		−	2837.16i		−	0.568603i
$293$			435.520i			0.0868373i	0.999057	+	0.0434186i	$0.0138249\pi$
$293$			435.520i			0.0868373i	−0.999057	+	0.0434186i	$0.986175\pi$
$294$		0			0
$295$	1530.57	−	2215.40i	0.302078	−	0.437240i
$296$	−5112.31			−1.00387
$297$		0			0
$298$			4211.36i			0.818650i
$299$	−7214.10			−1.39533
$300$		0			0
$301$	−1031.99			−0.197618
$302$			169.279i			0.0322547i
$303$		0			0
$304$	−566.620			−0.106901
$305$	−574.864	+	832.082i	−0.107923	+	0.156213i
$306$		0			0
$307$		−	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
$309$		0			0
$310$	−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	$0.446613\pi$
$312$		0			0
$313$			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	$-0.0471570\pi$
$318$		0			0
$319$	−2313.36			−0.406029
$320$	1732.91	−	2508.29i	0.302727	−	0.438180i
$321$		0			0
$322$		−	1858.55i		−	0.321655i
$323$			11516.9i			1.98396i
$324$		0			0
$325$	5321.47	−	2013.02i	0.908253	−	0.343577i
$326$	−2219.93			−0.377149
$327$		0			0
$328$		−	3747.50i		−	0.630856i
$329$	−469.229			−0.0786305
$330$		0			0
$331$	−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
$333$		0			0
$334$	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	$0.330027\pi$
$337$			10650.5i			1.72157i	−0.508970	+	0.860784i	$0.669973\pi$
$338$			209.896i			0.0337776i
$339$		0			0
$340$	4395.61	+	3036.82i	0.701134	+	0.484395i
$341$	2854.01			0.453235
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$	−3258.42			−0.510704
$345$		0			0
$346$	−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	$-0.100627\pi$
$348$		0			0
$349$	−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
$351$		0			0
$352$			10619.1i			1.60795i
$353$			2959.98i			0.446300i	0.974784	+	0.223150i	$0.0716340\pi$
$353$			2959.98i			0.446300i	−0.974784	+	0.223150i	$0.928366\pi$
$354$		0			0
$355$	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	$-0.449005\pi$
$359$	2170.17			0.319045			0.159523	+	0.987194i	$0.449005\pi$
$360$		0			0
$361$	8810.30			1.28449
$362$		−	4691.88i		−	0.681215i
$363$		0			0
$364$	−1654.82			−0.238285
$365$	5024.76	+	3471.48i	0.720569	+	0.497823i
$366$		0			0
$367$		−	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
$369$		0			0
$370$	2462.43	−	3564.23i	0.345989	−	0.500798i
$371$	−1880.53			−0.263159
$372$		0			0
$373$		−	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
$375$		0			0
$376$	−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
$381$		0			0
$382$		−	1240.05i		−	0.166090i
$383$		−	13216.7i		−	1.76330i	−0.471905	−	0.881649i	$-0.656434\pi$
$383$		−	13216.7i		−	1.76330i	0.471905	−	0.881649i	$-0.343566\pi$
$384$		0			0
$385$	−3708.08	−	2561.82i	−0.490861	−	0.339123i
$386$	6838.97			0.901799
$387$		0			0
$388$		−	4896.02i		−	0.640612i
$389$	−7755.01			−1.01078			−0.505391	−	0.862890i	$-0.668652\pi$
$389$	−7755.01			−1.01078			−0.505391	+	0.862890i	$0.668652\pi$
$390$		0			0
$391$	14582.5			1.88610
$392$		−	1082.99i		−	0.139539i
$393$		0			0
$394$	5720.49			0.731457
$395$	−160.838	+	232.803i	−0.0204876	+	0.0296547i
$396$		0			0
$397$		−	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	$-0.390201\pi$
$401$	5430.61			0.676288			0.338144	+	0.941094i	$0.390201\pi$
$402$		0			0
$403$			2255.73i			0.278823i
$404$	3140.28			0.386719
$405$		0			0
$406$	471.048			0.0575806
$407$			13320.5i			1.62229i
$408$		0			0
$409$	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
$411$		0			0
$412$			1563.17i			0.186922i
$413$		−	1685.90i		−	0.200866i
$414$		0			0
$415$	−3461.00	−	2391.12i	−0.409383	−	0.282832i
$416$	−8393.01			−0.989186
$417$		0			0
$418$		−	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
$423$		0			0
$424$	−5937.58			−0.680081
$425$	−10756.7	+	4069.09i	−1.22771	+	0.464423i
$426$		0			0
$427$			633.205i			0.0717634i
$428$		−	7851.31i		−	0.886699i
$429$		0			0
$430$	1569.48	−	2271.72i	0.176016	−	0.254773i
$431$	−8174.07			−0.913530			−0.456765	−	0.889588i	$-0.650992\pi$
$431$	−8174.07			−0.913530			−0.456765	+	0.889588i	$0.650992\pi$
$432$		0			0
$433$		−	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
$435$		0			0
$436$	9177.96			1.00813
$437$		−	19840.1i		−	2.17181i
$438$		0			0
$439$	−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
$441$		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	$0.931641\pi$
$444$		0			0
$445$	−6523.06	+	9441.74i	−0.694882	+	1.00580i
$446$	305.903			0.0324774
$447$		0			0
$448$		−	1908.78i		−	0.201298i
$449$	−15243.1			−1.60216			−0.801078	−	0.598559i	$-0.795740\pi$
$449$	−15243.1			−1.60216			−0.801078	+	0.598559i	$0.795740\pi$
$450$		0			0
$451$	−9764.40			−1.01948
$452$			5428.95i			0.564947i
$453$		0			0
$454$	−5280.67			−0.545890
$455$	2024.79	−	2930.77i	0.208623	−	0.301970i
$456$		0			0
$457$		−	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
$459$		0			0
$460$	−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	$0.135103\pi$
$463$			8205.35i			0.823618i	−0.911270	+	0.411809i	$0.864897\pi$
$464$	−181.835			−0.0181929
$465$		0			0
$466$	1575.76			0.156643
$467$			167.628i			0.0166100i	0.999966	+	0.00830501i	$0.00264360\pi$
$467$			167.628i			0.0166100i	−0.999966	+	0.00830501i	$0.997356\pi$
$468$		0			0
$469$	2845.49			0.280154
$470$	713.612	−	1032.91i	0.0700351	−	0.101372i
$471$		0			0
$472$		−	5323.06i		−	0.519097i
$473$			8490.07i			0.825315i
$474$		0			0
$475$	5536.18	+	14635.0i	0.534773	+	1.41368i
$476$	3345.01			0.322098
$477$		0			0
$478$			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	$0.397611\pi$
$480$		0			0
$481$	−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
$486$		0			0
$487$		−	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
$489$		0			0
$490$	755.044	+	521.641i	0.0696111	+	0.0480925i
$491$	16710.8			1.53594			0.767972	−	0.640484i	$-0.221266\pi$
$491$	16710.8			1.53594			0.767972	+	0.640484i	$0.221266\pi$
$492$		0			0
$493$			3695.92i			0.337639i
$494$	9544.32			0.869270
$495$		0			0
$496$	224.331			0.0203080
$497$		−	2315.47i		−	0.208980i
$498$		0			0
$499$	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
$501$		0			0
$502$		−	3836.54i		−	0.341102i
$503$		−	19523.7i		−	1.73065i	−0.501209	−	0.865326i	$-0.667111\pi$
$503$		−	19523.7i		−	1.73065i	0.501209	−	0.865326i	$-0.332889\pi$
$504$		0			0
$505$	−3842.37	+	5561.60i	−0.338580	+	0.490075i
$506$	−15290.0			−1.34333
$507$		0			0
$508$			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	$0.623487\pi$
$510$		0			0
$511$	3823.78			0.331026
$512$			1635.04i			0.141131i
$513$		0			0
$514$	−1343.82			−0.115318
$515$	−2768.46	−	1912.66i	−0.236880	−	0.163654i
$516$		0			0
$517$			3860.28i			0.328385i
$518$		−	2712.34i		−	0.230064i
$519$		0			0
$520$	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	$0.591894\pi$
$522$		0			0
$523$			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
$528$		0			0
$529$	−12954.0			−1.06469
$530$	2859.94	−	4139.60i	0.234392	−	0.339269i
$531$		0			0
$532$		−	4551.04i		−	0.370888i
$533$		−	7717.51i		−	0.627171i
$534$		0			0
$535$	13905.1	+	9606.67i	1.12368	+	0.776322i
$536$	8984.34			0.724001
$537$		0			0
$538$			5965.62i			0.478060i
$539$	−2821.81			−0.225499
$540$		0			0
$541$	−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
$543$		0			0
$544$	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	$-0.850883\pi$
$547$		−	11552.7i		−	0.903033i	0.892263	−	0.451516i	$-0.149117\pi$
$548$		−	4018.43i		−	0.313246i
$549$		0			0
$550$	11278.6	−	4266.52i	0.874406	−	0.330773i
$551$	5028.47			0.388784
$552$		0			0
$553$			177.161i			0.0136232i
$554$	−11037.1			−0.846430
$555$		0			0
$556$	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	$0.785501\pi$
$558$		0			0
$559$	−6710.31			−0.507721
$560$	−291.464	−	201.365i	−0.0219939	−	0.0151950i
$561$		0			0
$562$		−	1366.18i		−	0.102542i
$563$			13631.9i			1.02045i	0.860040	+	0.510227i	$0.170439\pi$
$563$			13631.9i			1.02045i	−0.860040	+	0.510227i	$0.829561\pi$
$564$		0			0
$565$	−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	$-0.536275\pi$
$569$	−3086.83			−0.227428			−0.113714	+	0.993514i	$0.536275\pi$
$570$		0			0
$571$	−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
$573$		0			0
$574$	1988.24			0.144577
$575$	18530.5	−	7009.78i	1.34396	−	0.508396i
$576$		0			0
$577$			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
$579$		0			0
$580$	1325.92	−	1919.19i	0.0949238	−	0.137397i
$581$	−2633.78			−0.188068
$582$		0			0
$583$			15470.8i			1.09903i
$584$	12073.2			0.855468
$585$		0			0
$586$	−729.566			−0.0514302
$587$		−	596.893i		−	0.0419701i	−0.999780	−	0.0209850i	$-0.993320\pi$
$587$		−	596.893i		−	0.0419701i	0.999780	−	0.0209850i	$-0.00668023\pi$
$588$		0			0
$589$	−6203.66			−0.433985
$590$	3711.16	+	2563.95i	0.258960	+	0.178909i
$591$		0			0
$592$			1047.02i			0.0726898i
$593$			19496.3i			1.35012i	0.737765	+	0.675058i	$0.235881\pi$
$593$			19496.3i			1.35012i	−0.737765	+	0.675058i	$0.764119\pi$
$594$		0			0
$595$	−4092.88	+	5924.20i	−0.282003	+	0.408182i
$596$	13057.3			0.897396
$597$		0			0
$598$		−	12084.8i		−	0.826395i
$599$	3797.02			0.259001			0.129501	−	0.991579i	$-0.458663\pi$
$599$	3797.02			0.259001			0.129501	+	0.991579i	$0.458663\pi$
$600$		0			0
$601$	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
$603$		0			0
$604$	524.850			0.0353573
$605$	−12617.2	+	18262.6i	−0.847869	+	1.22724i
$606$		0			0
$607$			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
$612$		0			0
$613$		−	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
$615$		0			0
$616$	−8909.59			−0.582756
$617$		−	22964.9i		−	1.49843i	−0.662327	−	0.749215i	$-0.730431\pi$
$617$		−	22964.9i		−	1.49843i	0.662327	−	0.749215i	$-0.269569\pi$
$618$		0			0
$619$	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
$621$		0			0
$622$			3067.41i			0.197736i
$623$			7185.06i			0.462060i
$624$		0			0
$625$	−11713.0	+	10341.5i	−0.749631	+	0.661856i
$626$	−4092.35			−0.261283
$627$		0			0
$628$			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
$633$		0			0
$634$	2791.14			0.174843
$635$	−2398.31	−	1656.93i	−0.149880	−	0.103548i
$636$		0			0
$637$		−	2230.28i		−	0.138724i
$638$		−	3875.25i		−	0.240474i
$639$		0			0
$640$	−9367.71	−	6471.91i	−0.578580	−	0.399726i
$641$	−30367.1			−1.87118			−0.935592	−	0.353084i	$-0.885133\pi$
$641$	−30367.1			−1.87118			−0.935592	+	0.353084i	$0.885133\pi$
$642$		0			0
$643$		−	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
$645$		0			0
$646$	−19292.7			−1.17502
$647$			14507.9i			0.881555i	0.897616	+	0.440778i	$0.145297\pi$
$647$			14507.9i			0.881555i	−0.897616	+	0.440778i	$0.854703\pi$
$648$		0			0
$649$	−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	$0.932737\pi$
$654$		0			0
$655$	−4598.01	+	6655.35i	−0.274289	+	0.397017i
$656$	−767.504			−0.0456799
$657$		0			0
$658$		−	786.035i		−	0.0465696i
$659$	7308.92			0.432041			0.216020	−	0.976389i	$-0.430692\pi$
$659$	7308.92			0.432041			0.216020	+	0.976389i	$0.430692\pi$
$660$		0			0
$661$	−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
$663$		0			0
$664$	−8315.91			−0.486024
$665$	8060.14	+	5568.54i	0.470013	+	0.324720i
$666$		0			0
$667$		−	6366.92i		−	0.369608i
$668$		−	10835.4i		−	0.627594i
$669$		0			0
$670$	−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	$0.0494264\pi$
$673$			5400.26i			0.309309i	−0.987969	+	0.154654i	$0.950574\pi$
$674$	−17841.3			−1.01962
$675$		0			0
$676$	650.781			0.0370267
$677$		−	6431.09i		−	0.365091i	−0.983197	−	0.182546i	$-0.941566\pi$
$677$		−	6431.09i		−	0.365091i	0.983197	−	0.182546i	$-0.0584338\pi$
$678$		0			0
$679$	6598.62			0.372948
$680$	−12922.9	+	18705.1i	−0.728778	+	1.05486i
$681$		0			0
$682$			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	$-0.198704\pi$
$684$		0			0
$685$	7116.86	+	4916.86i	0.396965	+	0.274253i
$686$	574.581			0.0319790
$687$		0			0
$688$			667.339i			0.0369797i
$689$	−12227.7			−0.676109
$690$		0			0
$691$	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
$693$		0			0
$694$	6732.70			0.368256
$695$	−18766.2	+	27163.0i	−1.02424	+	1.48252i
$696$		0			0
$697$			15600.0i			0.847766i
$698$		−	17664.4i		−	0.957890i
$699$		0			0
$700$	4250.64	−	1607.95i	0.229513	−	0.0868209i
$701$	−12639.3			−0.680996			−0.340498	−	0.940245i	$-0.610596\pi$
$701$	−12639.3			−0.680996			−0.340498	+	0.940245i	$0.610596\pi$
$702$		0			0
$703$		−	28954.4i		−	1.55339i
$704$	−15703.3			−0.840680
$705$		0			0
$706$	−4958.45			−0.264325
$707$			4232.32i			0.225138i
$708$		0			0
$709$	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
$711$		0			0
$712$			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
$717$		0			0
$718$			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	$0.714836\pi$
$720$		0			0
$721$	−2106.77			−0.108821
$722$			14758.7i			0.760750i
$723$		0			0
$724$	−14547.2			−0.746741
$725$	1776.63	+	4696.55i	0.0910100	+	0.240587i
$726$		0			0
$727$			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
$729$		0			0
$730$	−5815.28	+	8417.28i	−0.294840	+	0.426764i
$731$	13564.1			0.686302
$732$		0			0
$733$			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
$738$		0			0
$739$	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
$741$		0			0
$742$		−	3150.19i		−	0.155859i
$743$			7438.65i			0.367292i	0.982992	+	0.183646i	$0.0587900\pi$
$743$			7438.65i			0.367292i	−0.982992	+	0.183646i	$0.941210\pi$
$744$		0			0
$745$	−15976.6	+	23125.2i	−0.785688	+	1.13724i
$746$	7782.84			0.381970
$747$		0			0
$748$		−	27518.9i		−	1.34518i
$749$	10581.6			0.516214
$750$		0			0
$751$	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
$753$		0			0
$754$	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	$-0.921305\pi$
$757$		−	10193.8i		−	0.489432i	0.969595	−	0.244716i	$-0.0786947\pi$
$758$		−	2403.59i		−	0.115174i
$759$		0			0
$760$	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	$0.0648618\pi$
$762$		0			0
$763$			12369.6i			0.586908i
$764$	−3844.76			−0.182066
$765$		0			0
$766$	22140.2			1.04433
$767$		−	10962.2i		−	0.516065i
$768$		0			0
$769$	−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
$771$		0			0
$772$		−	21204.2i		−	0.988544i
$773$		−	21799.2i		−	1.01431i	−0.861854	−	0.507156i	$-0.830697\pi$
$773$		−	21799.2i		−	1.01431i	0.861854	−	0.507156i	$-0.169303\pi$
$774$		0			0
$775$	−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
$780$		0			0
$781$	−19049.1			−0.872765
$782$			24428.0i			1.11706i
$783$		0			0
$784$	−221.801			−0.0101039
$785$	−21509.4	−	14860.3i	−0.977967	−	0.675652i
$786$		0			0
$787$		−	24080.3i		−	1.09068i	−0.838214	−	0.545342i	$-0.816400\pi$
$787$		−	24080.3i		−	1.09068i	0.838214	−	0.545342i	$-0.183600\pi$
$788$		−	17736.4i		−	0.801817i
$789$		0			0
$790$	−389.983	−	269.429i	−0.0175632	−	0.0121340i
$791$	−7316.87			−0.328898
$792$		0			0
$793$			4117.28i			0.184374i
$794$	5964.96			0.266610
$795$		0			0
$796$	17619.8			0.784569
$797$			17194.3i			0.764184i	0.924124	+	0.382092i	$0.124796\pi$
$797$			17194.3i			0.764184i	−0.924124	+	0.382092i	$0.875204\pi$
$798$		0			0
$799$	6167.35			0.273073
$800$	21558.7	−	8155.29i	0.952769	−	0.360416i
$801$		0			0
$802$			9097.15i			0.400538i
$803$		−	31457.7i		−	1.38246i
$804$		0			0
$805$	7050.75	−	10205.6i	0.308704	−	0.446830i
$806$	−3778.71			−0.165136
$807$		0			0
$808$			13363.1i			0.581823i
$809$	33349.1			1.44931			0.724656	−	0.689111i	$-0.241999\pi$
$809$	33349.1			1.44931			0.724656	+	0.689111i	$0.241999\pi$
$810$		0			0
$811$	4577.87			0.198213			0.0991066	−	0.995077i	$-0.468402\pi$
$811$	4577.87			0.198213			0.0991066	+	0.995077i	$0.468402\pi$
$812$		−	1460.48i		−	0.0631194i
$813$		0			0
$814$	−22314.0			−0.960819
$815$	−12190.0	−	8421.74i	−0.523921	−	0.361964i
$816$		0			0
$817$		−	18454.6i		−	0.790262i
$818$			16247.0i			0.694455i
$819$		0			0
$820$	5596.55	−	8100.67i	0.238341	−	0.344985i
$821$	5832.52			0.247937			0.123969	−	0.992286i	$-0.460438\pi$
$821$	5832.52			0.247937			0.123969	+	0.992286i	$0.460438\pi$
$822$		0			0
$823$		−	37974.3i		−	1.60838i	−0.594369	−	0.804192i	$-0.702598\pi$
$823$		−	37974.3i		−	1.60838i	0.594369	−	0.804192i	$-0.297402\pi$
$824$	−6651.92			−0.281226
$825$		0			0
$826$	2824.15			0.118965
$827$		−	15796.2i		−	0.664193i	−0.943245	−	0.332096i	$-0.892244\pi$
$827$		−	15796.2i		−	0.664193i	0.943245	−	0.332096i	$-0.107756\pi$
$828$		0			0
$829$	12714.1			0.532666			0.266333	−	0.963881i	$-0.414188\pi$
$829$	12714.1			0.532666			0.266333	+	0.963881i	$0.414188\pi$
$830$	4005.51	−	5797.74i	0.167510	−	0.242461i
$831$		0			0
$832$		−	12411.4i		−	0.517173i
$833$			4508.25i			0.187517i
$834$		0			0
$835$	19190.0	+	13257.9i	0.795326	+	0.549470i
$836$	−37440.8			−1.54894
$837$		0			0
$838$			23169.0i			0.955085i
$839$	−42924.2			−1.76628			−0.883140	−	0.469109i	$-0.844575\pi$
$839$	−42924.2			−1.76628			−0.883140	+	0.469109i	$0.844575\pi$
$840$		0			0
$841$	−22775.3			−0.933835
$842$			27980.2i			1.14520i
$843$		0			0
$844$	17648.8			0.719784
$845$	−796.279	+	1152.57i	−0.0324176	+	0.0469225i
$846$		0			0
$847$			13897.6i			0.563788i
$848$			1216.04i			0.0492442i
$849$		0			0
$850$	−6816.38	−	18019.3i	−0.275059	−	0.727124i
$851$	−36661.3			−1.47677
$852$		0			0
$853$			36172.5i			1.45196i	0.687716	+	0.725980i	$0.258613\pi$
$853$			36172.5i			1.45196i	−0.687716	+	0.725980i	$0.741387\pi$
$854$	−1060.72			−0.0425025
$855$		0			0
$856$	33410.4			1.33405
$857$			32039.9i			1.27709i	0.769586	+	0.638544i	$0.220463\pi$
$857$			32039.9i			1.27709i	−0.769586	+	0.638544i	$0.779537\pi$
$858$		0			0
$859$	6798.07			0.270020			0.135010	−	0.990844i	$-0.456893\pi$
$859$	6798.07			0.270020			0.135010	+	0.990844i	$0.456893\pi$
$860$	−7043.47	−	4866.15i	−0.279280	−	0.192947i
$861$		0			0
$862$		−	13692.9i		−	0.541046i
$863$		−	30179.3i		−	1.19040i	−0.803577	−	0.595200i	$-0.797073\pi$
$863$		−	30179.3i		−	1.19040i	0.803577	−	0.595200i	$-0.202927\pi$
$864$		0			0
$865$	−17644.6	−	12190.2i	−0.693565	−	0.479166i
$866$	23825.5			0.934901
$867$		0			0
$868$			1801.81i			0.0704578i
$869$	1457.47			0.0568946
$870$		0			0
$871$	18502.1			0.719772
$872$			39055.9i			1.51674i
$873$		0			0
$874$	33235.4			1.28627
$875$	−2353.22	+	9495.55i	−0.0909181	+	0.366867i
$876$		0			0
$877$		−	1700.51i		−	0.0654758i	−0.999464	−	0.0327379i	$-0.989577\pi$
$877$		−	1700.51i		−	0.0654758i	0.999464	−	0.0327379i	$-0.0104227\pi$
$878$		−	9276.02i		−	0.356549i
$879$		0			0
$880$	−1656.60	+	2397.83i	−0.0634591	+	0.0918533i
$881$	−1678.46			−0.0641869			−0.0320935	−	0.999485i	$-0.510217\pi$
$881$	−1678.46			−0.0641869			−0.0320935	+	0.999485i	$0.510217\pi$
$882$		0			0
$883$		−	10285.8i		−	0.392009i	−0.980603	−	0.196005i	$-0.937203\pi$
$883$		−	10285.8i		−	0.392009i	0.980603	−	0.196005i	$-0.0627968\pi$
$884$	21750.2			0.827532
$885$		0			0
$886$	−6657.24			−0.252432
$887$			12167.5i			0.460593i	0.973121	+	0.230296i	$0.0739696\pi$
$887$			12167.5i			0.460593i	−0.973121	+	0.230296i	$0.926030\pi$
$888$		0			0
$889$	−1825.09			−0.0688543
$890$	−15816.4	−	10927.2i	−0.595695	−	0.411550i
$891$		0			0
$892$		−	948.452i		−	0.0356015i
$893$		−	8390.96i		−	0.314438i
$894$		0			0
$895$	−3997.62	+	5786.31i	−0.149302	+	0.216106i
$896$	−7128.73			−0.265797
$897$		0			0
$898$		−	25534.7i		−	0.948892i
$899$	−1990.83			−0.0738574
$900$		0			0
$901$	24716.9			0.913916
$902$		−	16356.9i		−	0.603799i
$903$		0			0
$904$	−23102.3			−0.849968
$905$	17799.5	−	25763.8i	0.653787	−	0.946317i
$906$		0			0
$907$		−	50766.0i		−	1.85850i	−0.369452	−	0.929250i	$-0.620455\pi$
$907$		−	50766.0i		−	1.85850i	0.369452	−	0.929250i	$-0.379545\pi$
$908$			16372.7i			0.598399i
$909$		0			0
$910$	4909.51	+	3391.85i	0.178845	+	0.123559i
$911$	−18451.1			−0.671033			−0.335517	−	0.942034i	$-0.608911\pi$
$911$	−18451.1			−0.671033			−0.335517	+	0.942034i	$0.608911\pi$
$912$		0			0
$913$			21667.8i			0.785431i
$914$	18039.6			0.652841
$915$		0			0
$916$	−31227.1			−1.12639
$917$			5064.65i			0.182388i
$918$		0			0
$919$	−393.861			−0.0141374			−0.00706870	−	0.999975i	$-0.502250\pi$
$919$	−393.861			−0.0141374			−0.00706870	+	0.999975i	$0.502250\pi$
$920$	22262.1	−	32223.0i	0.797781	−	1.15474i
$921$		0			0
$922$			557.167i			0.0199016i
$923$		−	15055.9i		−	0.536912i
$924$		0			0
$925$	27043.2	−	10230.0i	0.961269	−	0.363632i
$926$	−13745.3			−0.487795
$927$		0			0
$928$		−	7407.39i		−	0.262025i
$929$	27452.3			0.969517			0.484759	−	0.874648i	$-0.338907\pi$
$929$	27452.3			0.969517			0.484759	+	0.874648i	$0.338907\pi$
$930$		0			0
$931$	6133.68			0.215922
$932$		−	4885.63i		−	0.171710i
$933$		0			0
$934$	−280.803			−0.00983744
$935$	48737.5	+	33671.5i	1.70469	+	1.17773i
$936$		0			0
$937$			21608.3i			0.753375i	0.926340	+	0.376688i	$0.122937\pi$
$937$			21608.3i			0.753375i	−0.926340	+	0.376688i	$0.877063\pi$
$938$			4766.65i			0.165924i
$939$		0			0
$940$	−3202.54	−	2212.55i	−0.111123	−	0.0767718i
$941$	16708.5			0.578832			0.289416	−	0.957203i	$-0.406539\pi$
$941$	16708.5			0.578832			0.289416	+	0.957203i	$0.406539\pi$
$942$		0			0
$943$		−	26874.0i		−	0.928036i
$944$	−1090.19			−0.0375874
$945$		0			0
$946$	−14222.2			−0.488800
$947$		−	13972.8i		−	0.479466i	−0.970839	−	0.239733i	$-0.922940\pi$
$947$		−	13972.8i		−	0.479466i	0.970839	−	0.239733i	$-0.0770599\pi$
$948$		0			0
$949$	24863.3			0.850471
$950$	−24516.0	+	9274.00i	−0.837268	+	0.316724i
$951$		0			0
$952$			14234.4i			0.484599i
$953$		−	11155.9i		−	0.379197i	−0.981862	−	0.189598i	$-0.939281\pi$
$953$		−	11155.9i		−	0.379197i	0.981862	−	0.189598i	$-0.0607186\pi$
$954$		0			0
$955$	4704.35	−	6809.27i	0.159402	−	0.230726i
$956$	26794.0			0.906466
$957$		0			0
$958$			11104.0i			0.374481i
$959$	5415.85			0.182364
$960$		0			0
$961$	−27334.9			−0.917556
$962$		−	17636.4i		−	0.591081i
$963$		0			0
$964$	−2409.09			−0.0804891
$965$	37553.8	+	25944.9i	1.25274	+	0.865489i
$966$		0			0
$967$		−	1212.55i		−	0.0403238i	−0.999797	−	0.0201619i	$-0.993582\pi$
$967$		−	1212.55i		−	0.0403238i	0.999797	−	0.0201619i	$-0.00641816\pi$
$968$			43880.4i			1.45699i
$969$		0			0
$970$	−10035.3	+	14525.5i	−0.332180	+	0.480810i
$971$	48832.2			1.61390			0.806952	−	0.590617i	$-0.201115\pi$
$971$	48832.2			1.61390			0.806952	+	0.590617i	$0.201115\pi$
$972$		0			0
$973$			20670.8i			0.681063i
$974$	34578.0			1.13753
$975$		0			0
$976$	409.462			0.0134289
$977$		−	5939.19i		−	0.194485i	−0.995261	−	0.0972424i	$-0.968998\pi$
$977$		−	5939.19i		−	0.194485i	0.995261	−	0.0972424i	$-0.0310022\pi$
$978$		0			0
$979$	59110.5			1.92970
$980$	1617.35	−	2341.01i	0.0527186	−	0.0763070i
$981$		0			0
$982$			27993.3i			0.909677i
$983$		−	21863.5i		−	0.709398i	−0.934980	−	0.354699i	$-0.884583\pi$
$983$		−	21863.5i		−	0.709398i	0.934980	−	0.354699i	$-0.115417\pi$
$984$		0			0
$985$	31412.0	+	21701.8i	1.01611	+	0.702006i
$986$	−6191.26			−0.199970
$987$		0			0
$988$		−	29592.1i		−	0.952886i
$989$	−23366.7			−0.751283
$990$		0			0
$991$	−44618.5			−1.43023			−0.715113	−	0.699009i	$-0.753625\pi$
$991$	−44618.5			−1.43023			−0.715113	+	0.699009i	$0.753625\pi$
$992$			9138.55i			0.292489i
$993$		0			0
$994$	3878.80			0.123771
$995$	−21559.1	+	31205.6i	−0.686905	+	0.994255i
$996$		0			0
$997$		−	37923.9i		−	1.20468i	−0.798241	−	0.602338i	$-0.794236\pi$
$997$		−	37923.9i		−	1.20468i	0.798241	−	0.602338i	$-0.205764\pi$
$998$			22997.8i			0.729441i
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.d.c.64.7		10
3.2	odd	2		35.4.b.a.29.4	✓	10
5.2	odd	4		1575.4.a.bq.1.2		5
5.3	odd	4		1575.4.a.bn.1.4		5
5.4	even	2	inner	315.4.d.c.64.4		10
12.11	even	2		560.4.g.f.449.4		10
15.2	even	4		175.4.a.i.1.4		5
15.8	even	4		175.4.a.j.1.2		5
15.14	odd	2		35.4.b.a.29.7	yes	10
21.2	odd	6		245.4.j.e.214.4		20
21.5	even	6		245.4.j.f.214.4		20
21.11	odd	6		245.4.j.e.79.7		20
21.17	even	6		245.4.j.f.79.7		20
21.20	even	2		245.4.b.d.99.4		10
60.59	even	2		560.4.g.f.449.7		10
105.44	odd	6		245.4.j.e.214.7		20
105.59	even	6		245.4.j.f.79.4		20
105.62	odd	4		1225.4.a.be.1.4		5
105.74	odd	6		245.4.j.e.79.4		20
105.83	odd	4		1225.4.a.bh.1.2		5
105.89	even	6		245.4.j.f.214.7		20
105.104	even	2		245.4.b.d.99.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.4.b.a.29.4	✓	10	3.2	odd	2
35.4.b.a.29.7	yes	10	15.14	odd	2
175.4.a.i.1.4		5	15.2	even	4
175.4.a.j.1.2		5	15.8	even	4
245.4.b.d.99.4		10	21.20	even	2
245.4.b.d.99.7		10	105.104	even	2
245.4.j.e.79.4		20	105.74	odd	6
245.4.j.e.79.7		20	21.11	odd	6
245.4.j.e.214.4		20	21.2	odd	6
245.4.j.e.214.7		20	105.44	odd	6
245.4.j.f.79.4		20	105.59	even	6
245.4.j.f.79.7		20	21.17	even	6
245.4.j.f.214.4		20	21.5	even	6
245.4.j.f.214.7		20	105.89	even	6
315.4.d.c.64.4		10	5.4	even	2	inner
315.4.d.c.64.7		10	1.1	even	1	trivial
560.4.g.f.449.4		10	12.11	even	2
560.4.g.f.449.7		10	60.59	even	2
1225.4.a.be.1.4		5	105.62	odd	4
1225.4.a.bh.1.2		5	105.83	odd	4
1575.4.a.bn.1.4		5	5.3	odd	4
1575.4.a.bq.1.2		5	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	315.4.d.c.64.7

Level	$315$
Weight	$4$
Character	315.64
Analytic conductor	$18.586$
Analytic rank	$0$
Dimension	$10$
Inner twists	$2$