LMFDB - Embedded newform 960.4.f.n.769.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [960,4,Mod(769,960)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(960, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("960.769"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	960.f (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-22,0,0,0,-36,0,-148] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$56.6418336055$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{129})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 65x^{2} + 1024$ x^4 + 65*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.1
Root			$5.17891i$ of defining polynomial
Character	$\chi$	$=$	960.769
Dual form			960.4.f.n.769.3

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +(-11.1789 - 0.178908i) q^{5} -33.0735i q^{7} -9.00000 q^{9} -48.3578 q^{11} -60.3578i q^{13} +(-0.536725 + 33.5367i) q^{15} -17.7891i q^{17} -130.863 q^{19} -99.2204 q^{21} -70.8625i q^{23} +(124.936 + 4.00000i) q^{25} +27.0000i q^{27} +104.505 q^{29} -210.441 q^{31} +145.073i q^{33} +(-5.91712 + 369.725i) q^{35} -300.945i q^{37} -181.073 q^{39} +240.147 q^{41} -108.000i q^{43} +(100.610 + 1.61018i) q^{45} -278.991i q^{47} -750.853 q^{49} -53.3673 q^{51} +328.358i q^{53} +(540.588 + 8.65162i) q^{55} +392.588i q^{57} +889.533 q^{59} +241.450 q^{61} +297.661i q^{63} +(-10.7985 + 674.735i) q^{65} -103.834i q^{67} -212.588 q^{69} -277.597 q^{71} -274.403i q^{73} +(12.0000 - 374.808i) q^{75} +1599.36i q^{77} -366.991 q^{79} +81.0000 q^{81} +57.7251i q^{83} +(-3.18262 + 198.863i) q^{85} -313.514i q^{87} +203.175 q^{89} -1996.24 q^{91} +631.322i q^{93} +(1462.90 + 23.4124i) q^{95} +1283.45i q^{97} +435.220 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 22 q^{5} - 36 q^{9} - 148 q^{11} + 66 q^{15} - 160 q^{19} + 12 q^{21} + 100 q^{29} - 24 q^{31} - 796 q^{35} - 588 q^{39} + 688 q^{41} + 198 q^{45} - 3276 q^{49} + 468 q^{51} + 1072 q^{55} + 1332 q^{59}+ \cdots + 1332 q^{99}+O(q^{100})$ 4 * q - 22 * q^5 - 36 * q^9 - 148 * q^11 + 66 * q^15 - 160 * q^19 + 12 * q^21 + 100 * q^29 - 24 * q^31 - 796 * q^35 - 588 * q^39 + 688 * q^41 + 198 * q^45 - 3276 * q^49 + 468 * q^51 + 1072 * q^55 + 1332 * q^59 - 488 * q^61 + 820 * q^65 + 240 * q^69 + 616 * q^71 + 48 * q^75 - 2104 * q^79 + 324 * q^81 - 2148 * q^85 - 1368 * q^89 - 1352 * q^91 + 2944 * q^95 + 1332 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$ .

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$		0			0
$2$		0			0
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−11.1789	−	0.178908i	−0.999872	−	0.0160020i
$5$	−11.1789	−	0.178908i	−0.999872	−	0.0160020i
$6$		0			0
$7$		−	33.0735i		−	1.78580i	−0.450257	−	0.892899i	$-0.648667\pi$
$7$		−	33.0735i		−	1.78580i	0.450257	−	0.892899i	$-0.351333\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−48.3578			−1.32549			−0.662747	−	0.748844i	$-0.730609\pi$
$11$	−48.3578			−1.32549			−0.662747	+	0.748844i	$0.730609\pi$
$12$		0			0
$13$		−	60.3578i		−	1.28771i	−0.765147	−	0.643856i	$-0.777334\pi$
$13$		−	60.3578i		−	1.28771i	0.765147	−	0.643856i	$-0.222666\pi$
$14$		0			0
$15$	−0.536725	+	33.5367i	−0.00923879	+	0.577276i
$16$		0			0
$17$		−	17.7891i		−	0.253793i	−0.991916	−	0.126897i	$-0.959498\pi$
$17$		−	17.7891i		−	0.253793i	0.991916	−	0.126897i	$-0.0405017\pi$
$18$		0			0
$19$	−130.863			−1.58010			−0.790051	−	0.613042i	$-0.789946\pi$
$19$	−130.863			−1.58010			−0.790051	+	0.613042i	$0.789946\pi$
$20$		0			0
$21$	−99.2204			−1.03103
$22$		0			0
$23$		−	70.8625i		−	0.642429i	−0.947007	−	0.321214i	$-0.895909\pi$
$23$		−	70.8625i		−	0.642429i	0.947007	−	0.321214i	$-0.104091\pi$
$24$		0			0
$25$	124.936	+	4.00000i	0.999488	+	0.0320000i
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$	104.505			0.669174			0.334587	−	0.942365i	$-0.391403\pi$
$29$	104.505			0.669174			0.334587	+	0.942365i	$0.391403\pi$
$30$		0			0
$31$	−210.441			−1.21923			−0.609617	−	0.792696i	$-0.708677\pi$
$31$	−210.441			−1.21923			−0.609617	+	0.792696i	$0.708677\pi$
$32$		0			0
$33$			145.073i			0.765274i
$34$		0			0
$35$	−5.91712	+	369.725i	−0.0285764	+	1.78557i
$36$		0			0
$37$		−	300.945i		−	1.33717i	−0.743638	−	0.668583i	$-0.766901\pi$
$37$		−	300.945i		−	1.33717i	0.743638	−	0.668583i	$-0.233099\pi$
$38$		0			0
$39$	−181.073			−0.743460
$40$		0			0
$41$	240.147			0.914747			0.457374	−	0.889275i	$-0.348790\pi$
$41$	240.147			0.914747			0.457374	+	0.889275i	$0.348790\pi$
$42$		0			0
$43$		−	108.000i		−	0.383020i	−0.981491	−	0.191510i	$-0.938662\pi$
$43$		−	108.000i		−	0.383020i	0.981491	−	0.191510i	$-0.0613384\pi$
$44$		0			0
$45$	100.610	+	1.61018i	0.333291	+	0.00533402i
$46$		0			0
$47$		−	278.991i		−	0.865850i	−0.901430	−	0.432925i	$-0.857481\pi$
$47$		−	278.991i		−	0.865850i	0.901430	−	0.432925i	$-0.142519\pi$
$48$		0			0
$49$	−750.853			−2.18908
$50$		0			0
$51$	−53.3673			−0.146528
$52$		0			0
$53$			328.358i			0.851008i	0.904957	+	0.425504i	$0.139903\pi$
$53$			328.358i			0.851008i	−0.904957	+	0.425504i	$0.860097\pi$
$54$		0			0
$55$	540.588	+	8.65162i	1.32532	+	0.0212106i
$56$		0			0
$57$			392.588i			0.912272i
$58$		0			0
$59$	889.533			1.96284			0.981418	−	0.191882i	$-0.0614589\pi$
$59$	889.533			1.96284			0.981418	+	0.191882i	$0.0614589\pi$
$60$		0			0
$61$	241.450			0.506795			0.253398	−	0.967362i	$-0.418452\pi$
$61$	241.450			0.506795			0.253398	+	0.967362i	$0.418452\pi$
$62$		0			0
$63$			297.661i			0.595266i
$64$		0			0
$65$	−10.7985	+	674.735i	−0.0206060	+	1.28755i
$66$		0			0
$67$		−	103.834i		−	0.189334i	−0.995509	−	0.0946669i	$-0.969821\pi$
$67$		−	103.834i		−	0.189334i	0.995509	−	0.0946669i	$-0.0301786\pi$
$68$		0			0
$69$	−212.588			−0.370906
$70$		0			0
$71$	−277.597			−0.464010			−0.232005	−	0.972715i	$-0.574529\pi$
$71$	−277.597			−0.464010			−0.232005	+	0.972715i	$0.574529\pi$
$72$		0			0
$73$		−	274.403i		−	0.439951i	−0.975505	−	0.219976i	$-0.929402\pi$
$73$		−	274.403i		−	0.439951i	0.975505	−	0.219976i	$-0.0705978\pi$
$74$		0			0
$75$	12.0000	−	374.808i	0.0184752	−	0.577055i
$76$		0			0
$77$			1599.36i			2.36706i
$78$		0			0
$79$	−366.991			−0.522654			−0.261327	−	0.965250i	$-0.584160\pi$
$79$	−366.991			−0.522654			−0.261327	+	0.965250i	$0.584160\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			57.7251i			0.0763391i	0.999271	+	0.0381696i	$0.0121527\pi$
$83$			57.7251i			0.0763391i	−0.999271	+	0.0381696i	$0.987847\pi$
$84$		0			0
$85$	−3.18262	+	198.863i	−0.00406121	+	0.253761i
$86$		0			0
$87$		−	313.514i		−	0.386348i
$88$		0			0
$89$	203.175			0.241983			0.120992	−	0.992654i	$-0.461393\pi$
$89$	203.175			0.241983			0.120992	+	0.992654i	$0.461393\pi$
$90$		0			0
$91$	−1996.24			−2.29959
$92$		0			0
$93$			631.322i			0.703925i
$94$		0			0
$95$	1462.90	+	23.4124i	1.57990	+	0.0252849i
$96$		0			0
$97$			1283.45i			1.34345i	0.740801	+	0.671725i	$0.234446\pi$
$97$			1283.45i			1.34345i	−0.740801	+	0.671725i	$0.765554\pi$
$98$		0			0
$99$	435.220			0.441831
$100$		0			0
$101$	−886.908			−0.873768			−0.436884	−	0.899518i	$-0.643918\pi$
$101$	−886.908			−0.873768			−0.436884	+	0.899518i	$0.643918\pi$
$102$		0			0
$103$			783.055i			0.749094i	0.927208	+	0.374547i	$0.122202\pi$
$103$			783.055i			0.749094i	−0.927208	+	0.374547i	$0.877798\pi$
$104$		0			0
$105$	1109.18	+	17.7513i	1.03090	+	0.0164986i
$106$		0			0
$107$		−	1301.51i		−	1.17590i	−0.808897	−	0.587950i	$-0.799935\pi$
$107$		−	1301.51i		−	1.17590i	0.808897	−	0.587950i	$-0.200065\pi$
$108$		0			0
$109$	1161.91			1.02102			0.510508	−	0.859873i	$-0.329457\pi$
$109$	1161.91			1.02102			0.510508	+	0.859873i	$0.329457\pi$
$110$		0			0
$111$	−902.836			−0.772013
$112$		0			0
$113$		−	507.808i		−	0.422748i	−0.977405	−	0.211374i	$-0.932206\pi$
$113$		−	507.808i		−	0.422748i	0.977405	−	0.211374i	$-0.0677938\pi$
$114$		0			0
$115$	−12.6779	+	792.166i	−0.0102802	+	0.642346i
$116$		0			0
$117$			543.220i			0.429237i
$118$		0			0
$119$	−588.346			−0.453224
$120$		0			0
$121$	1007.48			0.756933
$122$		0			0
$123$		−	720.441i		−	0.528130i
$124$		0			0
$125$	−1395.93	−	67.0677i	−0.998848	−	0.0479898i
$126$		0			0
$127$			796.064i			0.556215i	0.960550	+	0.278107i	$0.0897071\pi$
$127$			796.064i			0.556215i	−0.960550	+	0.278107i	$0.910293\pi$
$128$		0			0
$129$	−324.000			−0.221137
$130$		0			0
$131$	91.4764			0.0610102			0.0305051	−	0.999535i	$-0.490288\pi$
$131$	91.4764			0.0610102			0.0305051	+	0.999535i	$0.490288\pi$
$132$		0			0
$133$			4328.08i			2.82174i
$134$		0			0
$135$	4.83053	−	301.831i	0.00307960	−	0.192425i
$136$		0			0
$137$			2273.28i			1.41766i	0.705380	+	0.708829i	$0.250776\pi$
$137$			2273.28i			1.41766i	−0.705380	+	0.708829i	$0.749224\pi$
$138$		0			0
$139$	−738.735			−0.450782			−0.225391	−	0.974268i	$-0.572366\pi$
$139$	−738.735			−0.450782			−0.225391	+	0.974268i	$0.572366\pi$
$140$		0			0
$141$	−836.972			−0.499899
$142$		0			0
$143$			2918.77i			1.70685i
$144$		0			0
$145$	−1168.25	−	18.6968i	−0.669088	−	0.0107082i
$146$		0			0
$147$			2252.56i			1.26386i
$148$		0			0
$149$	1507.15			0.828661			0.414330	−	0.910127i	$-0.364016\pi$
$149$	1507.15			0.828661			0.414330	+	0.910127i	$0.364016\pi$
$150$		0			0
$151$	154.365			0.0831925			0.0415962	−	0.999135i	$-0.486756\pi$
$151$	154.365			0.0831925			0.0415962	+	0.999135i	$0.486756\pi$
$152$		0			0
$153$			160.102i			0.0845978i
$154$		0			0
$155$	2352.50	+	37.6496i	1.21908	+	0.0195102i
$156$		0			0
$157$			315.514i			0.160387i	0.996779	+	0.0801935i	$0.0255538\pi$
$157$			315.514i			0.160387i	−0.996779	+	0.0801935i	$0.974446\pi$
$158$		0			0
$159$	985.073			0.491330
$160$		0			0
$161$	−2343.67			−1.14725
$162$		0			0
$163$		−	1457.85i		−	0.700539i	−0.936649	−	0.350270i	$-0.886090\pi$
$163$		−	1457.85i		−	0.700539i	0.936649	−	0.350270i	$-0.113910\pi$
$164$		0			0
$165$	25.9549	−	1621.76i	0.0122460	−	0.765176i
$166$		0			0
$167$		−	2198.61i		−	1.01876i	−0.860541	−	0.509381i	$-0.829874\pi$
$167$		−	2198.61i		−	1.01876i	0.860541	−	0.509381i	$-0.170126\pi$
$168$		0			0
$169$	−1446.07			−0.658200
$170$		0			0
$171$	1177.76			0.526700
$172$		0			0
$173$		−	2030.49i		−	0.892343i	−0.894948	−	0.446171i	$-0.852787\pi$
$173$		−	2030.49i		−	0.892343i	0.894948	−	0.446171i	$-0.147213\pi$
$174$		0			0
$175$	132.294	−	4132.06i	0.0571456	−	1.78488i
$176$		0			0
$177$		−	2668.60i		−	1.13324i
$178$		0			0
$179$	427.220			0.178391			0.0891954	−	0.996014i	$-0.471570\pi$
$179$	427.220			0.178391			0.0891954	+	0.996014i	$0.471570\pi$
$180$		0			0
$181$	3779.97			1.55228			0.776140	−	0.630561i	$-0.217175\pi$
$181$	3779.97			1.55228			0.776140	+	0.630561i	$0.217175\pi$
$182$		0			0
$183$		−	724.350i		−	0.292598i
$184$		0			0
$185$	−53.8416	+	3364.24i	−0.0213974	+	1.33699i
$186$		0			0
$187$			860.241i			0.336401i
$188$		0			0
$189$	892.983			0.343677
$190$		0			0
$191$	1565.60			0.593103			0.296551	−	0.955017i	$-0.404163\pi$
$191$	1565.60			0.593103			0.296551	+	0.955017i	$0.404163\pi$
$192$		0			0
$193$			2642.63i			0.985599i	0.870143	+	0.492800i	$0.164026\pi$
$193$			2642.63i			0.985599i	−0.870143	+	0.492800i	$0.835974\pi$
$194$		0			0
$195$	2024.20	+	32.3956i	0.743365	+	0.0118969i
$196$		0			0
$197$		−	98.4522i		−	0.0356062i	−0.999842	−	0.0178031i	$-0.994333\pi$
$197$		−	98.4522i		−	0.0356062i	0.999842	−	0.0178031i	$-0.00566721\pi$
$198$		0			0
$199$	−1131.62			−0.403106			−0.201553	−	0.979478i	$-0.564599\pi$
$199$	−1131.62			−0.403106			−0.201553	+	0.979478i	$0.564599\pi$
$200$		0			0
$201$	−311.503			−0.109312
$202$		0			0
$203$		−	3456.33i		−	1.19501i
$204$		0			0
$205$	−2684.58	−	42.9643i	−0.914630	−	0.0146378i
$206$		0			0
$207$			637.763i			0.214143i
$208$		0			0
$209$	6328.23			2.09441
$210$		0			0
$211$	1902.85			0.620841			0.310420	−	0.950599i	$-0.399530\pi$
$211$	1902.85			0.620841			0.310420	+	0.950599i	$0.399530\pi$
$212$		0			0
$213$			832.791i			0.267896i
$214$		0			0
$215$	−19.3221	+	1207.32i	−0.00612910	+	0.382971i
$216$		0			0
$217$			6960.00i			2.17731i
$218$		0			0
$219$	−823.209			−0.254006
$220$		0			0
$221$	−1073.71			−0.326813
$222$		0			0
$223$		−	4855.59i		−	1.45809i	−0.684465	−	0.729046i	$-0.739964\pi$
$223$		−	4855.59i		−	1.45809i	0.684465	−	0.729046i	$-0.260036\pi$
$224$		0			0
$225$	−1124.42	−	36.0000i	−0.333163	−	0.0106667i
$226$		0			0
$227$			6536.19i			1.91111i	0.294812	+	0.955555i	$0.404743\pi$
$227$			6536.19i			1.91111i	−0.294812	+	0.955555i	$0.595257\pi$
$228$		0			0
$229$	−5510.35			−1.59011			−0.795053	−	0.606539i	$-0.792557\pi$
$229$	−5510.35			−1.59011			−0.795053	+	0.606539i	$0.792557\pi$
$230$		0			0
$231$	4798.08			1.36663
$232$		0			0
$233$		−	5915.65i		−	1.66329i	−0.555306	−	0.831646i	$-0.687399\pi$
$233$		−	5915.65i		−	1.66329i	0.555306	−	0.831646i	$-0.312601\pi$
$234$		0			0
$235$	−49.9137	+	3118.81i	−0.0138554	+	0.865739i
$236$		0			0
$237$			1100.97i			0.301754i
$238$		0			0
$239$	−2263.25			−0.612543			−0.306272	−	0.951944i	$-0.599082\pi$
$239$	−2263.25			−0.612543			−0.306272	+	0.951944i	$0.599082\pi$
$240$		0			0
$241$	772.493			0.206476			0.103238	−	0.994657i	$-0.467080\pi$
$241$	772.493			0.206476			0.103238	+	0.994657i	$0.467080\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	8393.72	+	134.334i	2.18880	+	0.0350297i
$246$		0			0
$247$			7898.58i			2.03471i
$248$		0			0
$249$	173.175			0.0440744
$250$		0			0
$251$	−596.192			−0.149926			−0.0749628	−	0.997186i	$-0.523884\pi$
$251$	−596.192			−0.149926			−0.0749628	+	0.997186i	$0.523884\pi$
$252$		0			0
$253$			3426.76i			0.851535i
$254$		0			0
$255$	596.588	+	9.54785i	0.146509	+	0.00234474i
$256$		0			0
$257$			4139.19i			1.00465i	0.864678	+	0.502326i	$0.167522\pi$
$257$			4139.19i			1.00465i	−0.864678	+	0.502326i	$0.832478\pi$
$258$		0			0
$259$	−9953.30			−2.38791
$260$		0			0
$261$	−940.542			−0.223058
$262$		0			0
$263$			1611.34i			0.377792i	0.981997	+	0.188896i	$0.0604909\pi$
$263$			1611.34i			0.377792i	−0.981997	+	0.188896i	$0.939509\pi$
$264$		0			0
$265$	58.7460	−	3670.68i	0.0136179	−	0.850899i
$266$		0			0
$267$		−	609.526i		−	0.139709i
$268$		0			0
$269$	−7031.33			−1.59371			−0.796855	−	0.604171i	$-0.793504\pi$
$269$	−7031.33			−1.59371			−0.796855	+	0.604171i	$0.793504\pi$
$270$		0			0
$271$	−5441.27			−1.21968			−0.609840	−	0.792524i	$-0.708766\pi$
$271$	−5441.27			−1.21968			−0.609840	+	0.792524i	$0.708766\pi$
$272$		0			0
$273$			5988.72i			1.32767i
$274$		0			0
$275$	−6041.63	−	193.431i	−1.32481	−	0.0424158i
$276$		0			0
$277$		−	1080.43i		−	0.234355i	−0.993111	−	0.117178i	$-0.962615\pi$
$277$		−	1080.43i		−	0.234355i	0.993111	−	0.117178i	$-0.0373847\pi$
$278$		0			0
$279$	1893.97			0.406411
$280$		0			0
$281$	−1602.99			−0.340308			−0.170154	−	0.985417i	$-0.554427\pi$
$281$	−1602.99			−0.340308			−0.170154	+	0.985417i	$0.554427\pi$
$282$		0			0
$283$			334.810i			0.0703265i	0.999382	+	0.0351632i	$0.0111951\pi$
$283$			334.810i			0.0703265i	−0.999382	+	0.0351632i	$0.988805\pi$
$284$		0			0
$285$	70.2372	−	4388.70i	0.0145982	−	0.912155i
$286$		0			0
$287$		−	7942.49i		−	1.63355i
$288$		0			0
$289$	4596.55			0.935589
$290$		0			0
$291$	3850.35			0.775641
$292$		0			0
$293$			539.250i			0.107520i	0.998554	+	0.0537599i	$0.0171206\pi$
$293$			539.250i			0.107520i	−0.998554	+	0.0537599i	$0.982879\pi$
$294$		0			0
$295$	−9944.01	−	159.145i	−1.96258	−	0.0314094i
$296$		0			0
$297$		−	1305.66i		−	0.255091i
$298$		0			0
$299$	−4277.11			−0.827263
$300$		0			0
$301$	−3571.93			−0.683996
$302$		0			0
$303$			2660.72i			0.504470i
$304$		0			0
$305$	−2699.15	−	43.1974i	−0.506731	−	0.00810977i
$306$		0			0
$307$		−	8477.58i		−	1.57603i	−0.615656	−	0.788015i	$-0.711109\pi$
$307$		−	8477.58i		−	1.57603i	0.615656	−	0.788015i	$-0.288891\pi$
$308$		0			0
$309$	2349.16			0.432490
$310$		0			0
$311$	3646.92			0.664945			0.332473	−	0.943113i	$-0.392117\pi$
$311$	3646.92			0.664945			0.332473	+	0.943113i	$0.392117\pi$
$312$		0			0
$313$		−	7537.05i		−	1.36108i	−0.732709	−	0.680542i	$-0.761745\pi$
$313$		−	7537.05i		−	1.36108i	0.732709	−	0.680542i	$-0.238255\pi$
$314$		0			0
$315$	53.2540	−	3327.53i	0.00952548	−	0.595190i
$316$		0			0
$317$			10721.2i			1.89956i	0.312916	+	0.949781i	$0.398694\pi$
$317$			10721.2i			1.89956i	−0.312916	+	0.949781i	$0.601306\pi$
$318$		0			0
$319$	−5053.62			−0.886986
$320$		0			0
$321$	−3904.52			−0.678907
$322$		0			0
$323$			2327.92i			0.401019i
$324$		0			0
$325$	241.431	−	7540.86i	0.0412068	−	1.28705i
$326$		0			0
$327$		−	3485.73i		−	0.589484i
$328$		0			0
$329$	−9227.18			−1.54623
$330$		0			0
$331$	4405.14			0.731505			0.365753	−	0.930712i	$-0.380812\pi$
$331$	4405.14			0.731505			0.365753	+	0.930712i	$0.380812\pi$
$332$		0			0
$333$			2708.51i			0.445722i
$334$		0			0
$335$	−18.5768	+	1160.75i	−0.00302973	+	0.189310i
$336$		0			0
$337$		−	186.117i		−	0.0300844i	−0.999887	−	0.0150422i	$-0.995212\pi$
$337$		−	186.117i		−	0.0300844i	0.999887	−	0.0150422i	$-0.00478827\pi$
$338$		0			0
$339$	−1523.42			−0.244074
$340$		0			0
$341$	10176.5			1.61609
$342$		0			0
$343$			13489.1i			2.12345i
$344$		0			0
$345$	2376.50	+	38.0337i	0.370859	+	0.00593526i
$346$		0			0
$347$		−	5547.38i		−	0.858211i	−0.903254	−	0.429105i	$-0.858829\pi$
$347$		−	5547.38i		−	0.858211i	0.903254	−	0.429105i	$-0.141171\pi$
$348$		0			0
$349$	9078.82			1.39249			0.696244	−	0.717805i	$-0.254853\pi$
$349$	9078.82			1.39249			0.696244	+	0.717805i	$0.254853\pi$
$350$		0			0
$351$	1629.66			0.247820
$352$		0			0
$353$		−	10678.1i		−	1.61002i	−0.593262	−	0.805009i	$-0.702160\pi$
$353$		−	10678.1i		−	1.61002i	0.593262	−	0.805009i	$-0.297840\pi$
$354$		0			0
$355$	3103.23	+	49.6644i	0.463951	+	0.00742511i
$356$		0			0
$357$			1765.04i			0.261669i
$358$		0			0
$359$	265.733			0.0390665			0.0195332	−	0.999809i	$-0.493782\pi$
$359$	265.733			0.0390665			0.0195332	+	0.999809i	$0.493782\pi$
$360$		0			0
$361$	10266.0			1.49672
$362$		0			0
$363$		−	3022.44i		−	0.437016i
$364$		0			0
$365$	−49.0930	+	3067.53i	−0.00704012	+	0.439895i
$366$		0			0
$367$		−	5854.21i		−	0.832663i	−0.909213	−	0.416332i	$-0.863316\pi$
$367$		−	5854.21i		−	0.832663i	0.909213	−	0.416332i	$-0.136684\pi$
$368$		0			0
$369$	−2161.32			−0.304916
$370$		0			0
$371$	10859.9			1.51973
$372$		0			0
$373$		−	10134.5i		−	1.40682i	−0.710787	−	0.703408i	$-0.751661\pi$
$373$		−	10134.5i		−	1.40682i	0.710787	−	0.703408i	$-0.248339\pi$
$374$		0			0
$375$	−201.203	+	4187.80i	−0.0277069	+	0.576685i
$376$		0			0
$377$		−	6307.68i		−	0.861703i
$378$		0			0
$379$	−4235.70			−0.574072			−0.287036	−	0.957920i	$-0.592670\pi$
$379$	−4235.70			−0.574072			−0.287036	+	0.957920i	$0.592670\pi$
$380$		0			0
$381$	2388.19			0.321131
$382$		0			0
$383$		−	8100.84i		−	1.08077i	−0.841419	−	0.540383i	$-0.818279\pi$
$383$		−	8100.84i		−	1.08077i	0.841419	−	0.540383i	$-0.181721\pi$
$384$		0			0
$385$	286.139	−	17879.1i	0.0378779	−	2.36676i
$386$		0			0
$387$			972.000i			0.127673i
$388$		0			0
$389$	−13820.2			−1.80131			−0.900656	−	0.434532i	$-0.856914\pi$
$389$	−13820.2			−1.80131			−0.900656	+	0.434532i	$0.856914\pi$
$390$		0			0
$391$	−1260.58			−0.163044
$392$		0			0
$393$		−	274.429i		−	0.0352242i
$394$		0			0
$395$	4102.55	+	65.6577i	0.522587	+	0.00836353i
$396$		0			0
$397$		−	11523.9i		−	1.45685i	−0.685125	−	0.728425i	$-0.740252\pi$
$397$		−	11523.9i		−	1.45685i	0.685125	−	0.728425i	$-0.259748\pi$
$398$		0			0
$399$	12984.2			1.62913
$400$		0			0
$401$	700.915			0.0872869			0.0436434	−	0.999047i	$-0.486103\pi$
$401$	700.915			0.0872869			0.0436434	+	0.999047i	$0.486103\pi$
$402$		0			0
$403$			12701.7i			1.57002i
$404$		0			0
$405$	−905.492	−	14.4916i	−0.111097	−	0.00177801i
$406$		0			0
$407$			14553.1i			1.77240i
$408$		0			0
$409$	−6650.69			−0.804047			−0.402024	−	0.915629i	$-0.631693\pi$
$409$	−6650.69			−0.804047			−0.402024	+	0.915629i	$0.631693\pi$
$410$		0			0
$411$	6819.83			0.818485
$412$		0			0
$413$		−	29419.9i		−	3.50523i
$414$		0			0
$415$	10.3275	−	645.303i	0.00122158	−	0.0763294i
$416$		0			0
$417$			2216.20i			0.260259i
$418$		0			0
$419$	13844.4			1.61418			0.807091	−	0.590426i	$-0.201040\pi$
$419$	13844.4			1.61418			0.807091	+	0.590426i	$0.201040\pi$
$420$		0			0
$421$	−12576.3			−1.45590			−0.727949	−	0.685631i	$-0.759526\pi$
$421$	−12576.3			−1.45590			−0.727949	+	0.685631i	$0.759526\pi$
$422$		0			0
$423$			2510.92i			0.288617i
$424$		0			0
$425$	71.1563	−	2222.50i	0.00812139	−	0.253663i
$426$		0			0
$427$		−	7985.59i		−	0.905035i
$428$		0			0
$429$	8756.32			0.985452
$430$		0			0
$431$	13440.6			1.50212			0.751059	−	0.660236i	$-0.229544\pi$
$431$	13440.6			1.50212			0.751059	+	0.660236i	$0.229544\pi$
$432$		0			0
$433$		−	3012.87i		−	0.334387i	−0.985924	−	0.167193i	$-0.946530\pi$
$433$		−	3012.87i		−	0.334387i	0.985924	−	0.167193i	$-0.0534704\pi$
$434$		0			0
$435$	−56.0903	+	3504.75i	−0.00618235	+	0.386298i
$436$		0			0
$437$			9273.25i			1.01510i
$438$		0			0
$439$	−10675.0			−1.16057			−0.580283	−	0.814415i	$-0.697058\pi$
$439$	−10675.0			−1.16057			−0.580283	+	0.814415i	$0.697058\pi$
$440$		0			0
$441$	6757.68			0.729692
$442$		0			0
$443$			125.868i			0.0134992i	0.999977	+	0.00674962i	$0.00214849\pi$
$443$			125.868i			0.0134992i	−0.999977	+	0.00674962i	$0.997852\pi$
$444$		0			0
$445$	−2271.28	−	36.3497i	−0.241952	−	0.00387223i
$446$		0			0
$447$		−	4521.45i		−	0.478428i
$448$		0			0
$449$	9707.21			1.02029			0.510146	−	0.860088i	$-0.329591\pi$
$449$	9707.21			1.02029			0.510146	+	0.860088i	$0.329591\pi$
$450$		0			0
$451$	−11613.0			−1.21249
$452$		0			0
$453$		−	463.096i		−	0.0480312i
$454$		0			0
$455$	22315.8	+	357.144i	2.29930	+	0.0367982i
$456$		0			0
$457$		−	1279.97i		−	0.131016i	−0.997852	−	0.0655082i	$-0.979133\pi$
$457$		−	1279.97i		−	0.131016i	0.997852	−	0.0655082i	$-0.0208668\pi$
$458$		0			0
$459$	480.305			0.0488425
$460$		0			0
$461$	−3080.48			−0.311220			−0.155610	−	0.987819i	$-0.549734\pi$
$461$	−3080.48			−0.311220			−0.155610	+	0.987819i	$0.549734\pi$
$462$		0			0
$463$		−	18017.3i		−	1.80850i	−0.427008	−	0.904248i	$-0.640432\pi$
$463$		−	18017.3i		−	1.80850i	0.427008	−	0.904248i	$-0.359568\pi$
$464$		0			0
$465$	112.949	−	7057.49i	0.0112642	−	0.703835i
$466$		0			0
$467$			7236.77i			0.717083i	0.933514	+	0.358541i	$0.116726\pi$
$467$			7236.77i			0.717083i	−0.933514	+	0.358541i	$0.883274\pi$
$468$		0			0
$469$	−3434.16			−0.338112
$470$		0			0
$471$	946.542			0.0925995
$472$		0			0
$473$			5222.64i			0.507690i
$474$		0			0
$475$	−16349.4	−	523.450i	−1.57929	−	0.0505632i
$476$		0			0
$477$		−	2955.22i		−	0.283669i
$478$		0			0
$479$	−4932.78			−0.470532			−0.235266	−	0.971931i	$-0.575596\pi$
$479$	−4932.78			−0.470532			−0.235266	+	0.971931i	$0.575596\pi$
$480$		0			0
$481$	−18164.4			−1.72188
$482$		0			0
$483$			7031.01i			0.662364i
$484$		0			0
$485$	229.620	−	14347.6i	0.0214979	−	1.34328i
$486$		0			0
$487$		−	5937.91i		−	0.552510i	−0.961084	−	0.276255i	$-0.910907\pi$
$487$		−	5937.91i		−	0.552510i	0.961084	−	0.276255i	$-0.0890934\pi$
$488$		0			0
$489$	−4373.56			−0.404456
$490$		0			0
$491$	−15703.5			−1.44336			−0.721678	−	0.692229i	$-0.756629\pi$
$491$	−15703.5			−1.44336			−0.721678	+	0.692229i	$0.756629\pi$
$492$		0			0
$493$		−	1859.04i		−	0.169832i
$494$		0			0
$495$	−4865.29	−	77.8646i	−0.441775	−	0.00707020i
$496$		0			0
$497$			9181.09i			0.828628i
$498$		0			0
$499$	−5656.77			−0.507478			−0.253739	−	0.967273i	$-0.581660\pi$
$499$	−5656.77			−0.507478			−0.253739	+	0.967273i	$0.581660\pi$
$500$		0			0
$501$	−6595.82			−0.588183
$502$		0			0
$503$		−	290.441i		−	0.0257457i	−0.999917	−	0.0128729i	$-0.995902\pi$
$503$		−	290.441i		−	0.0257457i	0.999917	−	0.0128729i	$-0.00409768\pi$
$504$		0			0
$505$	9914.66	+	158.675i	0.873657	+	0.0139821i
$506$		0			0
$507$			4338.20i			0.380012i
$508$		0			0
$509$	17330.6			1.50916			0.754582	−	0.656206i	$-0.227840\pi$
$509$	17330.6			1.50916			0.754582	+	0.656206i	$0.227840\pi$
$510$		0			0
$511$	−9075.45			−0.785664
$512$		0			0
$513$		−	3533.29i		−	0.304091i
$514$		0			0
$515$	140.095	−	8753.70i	0.0119870	−	0.748998i
$516$		0			0
$517$			13491.4i			1.14768i
$518$		0			0
$519$	−6091.47			−0.515194
$520$		0			0
$521$	6174.36			0.519201			0.259601	−	0.965716i	$-0.416409\pi$
$521$	6174.36			0.519201			0.259601	+	0.965716i	$0.416409\pi$
$522$		0			0
$523$		−	13389.4i		−	1.11946i	−0.828675	−	0.559730i	$-0.810905\pi$
$523$		−	13389.4i		−	1.11946i	0.828675	−	0.559730i	$-0.189095\pi$
$524$		0			0
$525$	−12396.2	−	396.881i	−1.03050	−	0.0329930i
$526$		0			0
$527$			3743.55i			0.309434i
$528$		0			0
$529$	7145.50			0.587285
$530$		0			0
$531$	−8005.80			−0.654279
$532$		0			0
$533$		−	14494.7i		−	1.17793i
$534$		0			0
$535$	−232.850	+	14549.4i	−0.0188168	+	1.17575i
$536$		0			0
$537$		−	1281.66i		−	0.102994i
$538$		0			0
$539$	36309.6			2.90161
$540$		0			0
$541$	14355.5			1.14084			0.570418	−	0.821354i	$-0.306781\pi$
$541$	14355.5			1.14084			0.570418	+	0.821354i	$0.306781\pi$
$542$		0			0
$543$		−	11339.9i		−	0.896209i
$544$		0			0
$545$	−12988.9	−	207.875i	−1.02089	−	0.0163384i
$546$		0			0
$547$			21133.9i			1.65195i	0.563704	+	0.825977i	$0.309376\pi$
$547$			21133.9i			1.65195i	−0.563704	+	0.825977i	$0.690624\pi$
$548$		0			0
$549$	−2173.05			−0.168932
$550$		0			0
$551$	−13675.8			−1.05736
$552$		0			0
$553$			12137.6i			0.933355i
$554$		0			0
$555$	10092.7	+	161.525i	0.771914	+	0.0123538i
$556$		0			0
$557$		−	3098.60i		−	0.235713i	−0.993031	−	0.117856i	$-0.962398\pi$
$557$		−	3098.60i		−	0.235713i	0.993031	−	0.117856i	$-0.0376023\pi$
$558$		0			0
$559$	−6518.64			−0.493219
$560$		0			0
$561$	2580.72			0.194221
$562$		0			0
$563$			7908.77i			0.592033i	0.955183	+	0.296017i	$0.0956584\pi$
$563$			7908.77i			0.592033i	−0.955183	+	0.296017i	$0.904342\pi$
$564$		0			0
$565$	−90.8511	+	5676.74i	−0.00676484	+	0.422694i
$566$		0			0
$567$		−	2678.95i		−	0.198422i
$568$		0			0
$569$	10740.7			0.791345			0.395673	−	0.918392i	$-0.370512\pi$
$569$	10740.7			0.791345			0.395673	+	0.918392i	$0.370512\pi$
$570$		0			0
$571$	−14701.2			−1.07745			−0.538725	−	0.842482i	$-0.681094\pi$
$571$	−14701.2			−1.07745			−0.538725	+	0.842482i	$0.681094\pi$
$572$		0			0
$573$		−	4696.79i		−	0.342428i
$574$		0			0
$575$	283.450	−	8853.28i	0.0205577	−	0.642100i
$576$		0			0
$577$			15788.1i			1.13911i	0.821954	+	0.569554i	$0.192884\pi$
$577$			15788.1i			1.13911i	−0.821954	+	0.569554i	$0.807116\pi$
$578$		0			0
$579$	7927.89			0.569036
$580$		0			0
$581$	1909.17			0.136326
$582$		0			0
$583$		−	15878.7i		−	1.12801i
$584$		0			0
$585$	97.1867	−	6072.61i	0.00686867	−	0.429182i
$586$		0			0
$587$		−	14778.5i		−	1.03913i	−0.854430	−	0.519567i	$-0.826093\pi$
$587$		−	14778.5i		−	1.03913i	0.854430	−	0.519567i	$-0.173907\pi$
$588$		0			0
$589$	27538.8			1.92651
$590$		0			0
$591$	−295.356			−0.0205573
$592$		0			0
$593$		−	11143.9i		−	0.771713i	−0.922559	−	0.385857i	$-0.873906\pi$
$593$		−	11143.9i		−	0.771713i	0.922559	−	0.385857i	$-0.126094\pi$
$594$		0			0
$595$	6577.07	+	105.260i	0.453166	+	0.00725251i
$596$		0			0
$597$			3394.85i			0.232733i
$598$		0			0
$599$	7707.87			0.525768			0.262884	−	0.964827i	$-0.415326\pi$
$599$	7707.87			0.525768			0.262884	+	0.964827i	$0.415326\pi$
$600$		0			0
$601$	−13681.4			−0.928580			−0.464290	−	0.885683i	$-0.653691\pi$
$601$	−13681.4			−0.928580			−0.464290	+	0.885683i	$0.653691\pi$
$602$		0			0
$603$			934.508i			0.0631113i
$604$		0			0
$605$	−11262.5	−	180.246i	−0.756837	−	0.0121125i
$606$		0			0
$607$			11552.4i			0.772481i	0.922398	+	0.386241i	$0.126227\pi$
$607$			11552.4i			0.772481i	−0.922398	+	0.386241i	$0.873773\pi$
$608$		0			0
$609$	−10369.0			−0.689939
$610$		0			0
$611$	−16839.3			−1.11496
$612$		0			0
$613$		−	9904.66i		−	0.652603i	−0.945266	−	0.326301i	$-0.894198\pi$
$613$		−	9904.66i		−	0.652603i	0.945266	−	0.326301i	$-0.105802\pi$
$614$		0			0
$615$	−128.893	+	8053.74i	−0.00845116	+	0.528062i
$616$		0			0
$617$			20323.5i			1.32608i	0.748582	+	0.663042i	$0.230735\pi$
$617$			20323.5i			1.32608i	−0.748582	+	0.663042i	$0.769265\pi$
$618$		0			0
$619$	−9223.99			−0.598940			−0.299470	−	0.954106i	$-0.596810\pi$
$619$	−9223.99			−0.598940			−0.299470	+	0.954106i	$0.596810\pi$
$620$		0			0
$621$	1913.29			0.123635
$622$		0			0
$623$		−	6719.70i		−	0.432134i
$624$		0			0
$625$	15593.0	+	999.488i	0.997952	+	0.0639672i
$626$		0			0
$627$		−	18984.7i		−	1.20921i
$628$		0			0
$629$	−5353.54			−0.339364
$630$		0			0
$631$	−16916.0			−1.06722			−0.533610	−	0.845730i	$-0.679165\pi$
$631$	−16916.0			−1.06722			−0.533610	+	0.845730i	$0.679165\pi$
$632$		0			0
$633$		−	5708.54i		−	0.358443i
$634$		0			0
$635$	142.422	−	8899.13i	0.00890057	−	0.556143i
$636$		0			0
$637$			45319.9i			2.81890i
$638$		0			0
$639$	2498.37			0.154670
$640$		0			0
$641$	−5811.35			−0.358088			−0.179044	−	0.983841i	$-0.557300\pi$
$641$	−5811.35			−0.358088			−0.179044	+	0.983841i	$0.557300\pi$
$642$		0			0
$643$			27931.7i			1.71309i	0.516069	+	0.856547i	$0.327395\pi$
$643$			27931.7i			1.71309i	−0.516069	+	0.856547i	$0.672605\pi$
$644$		0			0
$645$	3621.97	+	57.9663i	0.221108	+	0.00353864i
$646$		0			0
$647$			25437.8i			1.54569i	0.634595	+	0.772845i	$0.281167\pi$
$647$			25437.8i			1.54569i	−0.634595	+	0.772845i	$0.718833\pi$
$648$		0			0
$649$	−43015.9			−2.60173
$650$		0			0
$651$	20880.0			1.25707
$652$		0			0
$653$			22647.7i			1.35723i	0.734493	+	0.678617i	$0.237420\pi$
$653$			22647.7i			1.35723i	−0.734493	+	0.678617i	$0.762580\pi$
$654$		0			0
$655$	−1022.61	−	16.3659i	−0.0610024	−	0.000976288i
$656$		0			0
$657$			2469.63i			0.146650i
$658$		0			0
$659$	25158.5			1.48716			0.743579	−	0.668649i	$-0.233127\pi$
$659$	25158.5			1.48716			0.743579	+	0.668649i	$0.233127\pi$
$660$		0			0
$661$	−23441.0			−1.37935			−0.689675	−	0.724119i	$-0.742247\pi$
$661$	−23441.0			−1.37935			−0.689675	+	0.724119i	$0.742247\pi$
$662$		0			0
$663$			3221.13i			0.188685i
$664$		0			0
$665$	774.329	−	48383.2i	0.0451537	−	2.82138i
$666$		0			0
$667$		−	7405.47i		−	0.429896i
$668$		0			0
$669$	−14566.8			−0.841829
$670$		0			0
$671$	−11676.0			−0.671754
$672$		0			0
$673$			3145.33i			0.180154i	0.995935	+	0.0900770i	$0.0287113\pi$
$673$			3145.33i			0.180154i	−0.995935	+	0.0900770i	$0.971289\pi$
$674$		0			0
$675$	−108.000	+	3373.27i	−0.00615840	+	0.192352i
$676$		0			0
$677$			10606.2i			0.602109i	0.953607	+	0.301054i	$0.0973386\pi$
$677$			10606.2i			0.602109i	−0.953607	+	0.301054i	$0.902661\pi$
$678$		0			0
$679$	42448.1			2.39913
$680$		0			0
$681$	19608.6			1.10338
$682$		0			0
$683$		−	7825.06i		−	0.438386i	−0.975682	−	0.219193i	$-0.929658\pi$
$683$		−	7825.06i		−	0.438386i	0.975682	−	0.219193i	$-0.0703424\pi$
$684$		0			0
$685$	406.708	−	25412.8i	0.0226854	−	1.41748i
$686$		0			0
$687$			16531.1i			0.918049i
$688$		0			0
$689$	19819.0			1.09585
$690$		0			0
$691$	−22750.9			−1.25251			−0.626256	−	0.779618i	$-0.715413\pi$
$691$	−22750.9			−1.25251			−0.626256	+	0.779618i	$0.715413\pi$
$692$		0			0
$693$		−	14394.2i		−	0.789022i
$694$		0			0
$695$	8258.25	+	132.166i	0.450724	+	0.00721343i
$696$		0			0
$697$		−	4271.99i		−	0.232157i
$698$		0			0
$699$	−17747.0			−0.960302
$700$		0			0
$701$	133.598			0.00719817			0.00359908	−	0.999994i	$-0.498854\pi$
$701$	133.598			0.00719817			0.00359908	+	0.999994i	$0.498854\pi$
$702$		0			0
$703$			39382.5i			2.11286i
$704$		0			0
$705$	9356.43	+	149.741i	0.499835	+	0.00799941i
$706$		0			0
$707$			29333.1i			1.56037i
$708$		0			0
$709$	6886.26			0.364766			0.182383	−	0.983228i	$-0.441619\pi$
$709$	6886.26			0.364766			0.182383	+	0.983228i	$0.441619\pi$
$710$		0			0
$711$	3302.92			0.174218
$712$		0			0
$713$			14912.4i			0.783271i
$714$		0			0
$715$	522.193	−	32628.7i	0.0273131	−	1.70663i
$716$		0			0
$717$			6789.76i			0.353652i
$718$		0			0
$719$	1570.18			0.0814436			0.0407218	−	0.999171i	$-0.487034\pi$
$719$	1570.18			0.0814436			0.0407218	+	0.999171i	$0.487034\pi$
$720$		0			0
$721$	25898.3			1.33773
$722$		0			0
$723$		−	2317.48i		−	0.119209i
$724$		0			0
$725$	13056.4	+	418.019i	0.668831	+	0.0214136i
$726$		0			0
$727$		−	3399.18i		−	0.173409i	−0.996234	−	0.0867047i	$-0.972366\pi$
$727$		−	3399.18i		−	0.173409i	0.996234	−	0.0867047i	$-0.0276337\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	−1921.22			−0.0972078
$732$		0			0
$733$		−	14152.1i		−	0.713125i	−0.934272	−	0.356562i	$-0.883949\pi$
$733$		−	14152.1i		−	0.713125i	0.934272	−	0.356562i	$-0.116051\pi$
$734$		0			0
$735$	403.002	−	25181.2i	0.0202244	−	1.26370i
$736$		0			0
$737$			5021.20i			0.250961i
$738$		0			0
$739$	−14919.5			−0.742655			−0.371328	−	0.928502i	$-0.621097\pi$
$739$	−14919.5			−0.742655			−0.371328	+	0.928502i	$0.621097\pi$
$740$		0			0
$741$	23695.7			1.17474
$742$		0			0
$743$		−	7287.29i		−	0.359818i	−0.983683	−	0.179909i	$-0.942420\pi$
$743$		−	7287.29i		−	0.359818i	0.983683	−	0.179909i	$-0.0575803\pi$
$744$		0			0
$745$	−16848.3	−	269.642i	−0.828555	−	0.0132603i
$746$		0			0
$747$		−	519.526i		−	0.0254464i
$748$		0			0
$749$	−43045.3			−2.09992
$750$		0			0
$751$	18783.4			0.912670			0.456335	−	0.889808i	$-0.349162\pi$
$751$	18783.4			0.912670			0.456335	+	0.889808i	$0.349162\pi$
$752$		0			0
$753$			1788.58i			0.0865595i
$754$		0			0
$755$	−1725.63	−	27.6172i	−0.0831818	−	0.00133125i
$756$		0			0
$757$			30614.3i			1.46988i	0.678134	+	0.734939i	$0.262789\pi$
$757$			30614.3i			1.46988i	−0.678134	+	0.734939i	$0.737211\pi$
$758$		0			0
$759$	10280.3			0.491634
$760$		0			0
$761$	−15277.2			−0.727723			−0.363861	−	0.931453i	$-0.618542\pi$
$761$	−15277.2			−0.727723			−0.363861	+	0.931453i	$0.618542\pi$
$762$		0			0
$763$		−	38428.4i		−	1.82333i
$764$		0			0
$765$	28.6435	−	1789.76i	0.00135374	−	0.0845869i
$766$		0			0
$767$		−	53690.3i		−	2.52757i
$768$		0			0
$769$	17700.1			0.830016			0.415008	−	0.909818i	$-0.363779\pi$
$769$	17700.1			0.830016			0.415008	+	0.909818i	$0.363779\pi$
$770$		0			0
$771$	12417.6			0.580036
$772$		0			0
$773$			29362.5i			1.36623i	0.730310	+	0.683116i	$0.239375\pi$
$773$			29362.5i			1.36623i	−0.730310	+	0.683116i	$0.760625\pi$
$774$		0			0
$775$	−26291.6	−	841.763i	−1.21861	−	0.0390155i
$776$		0			0
$777$			29859.9i			1.37866i
$778$		0			0
$779$	−31426.2			−1.44539
$780$		0			0
$781$	13424.0			0.615042
$782$		0			0
$783$			2821.63i			0.128783i
$784$		0			0
$785$	56.4481	−	3527.10i	0.00256652	−	0.160367i
$786$		0			0
$787$			2816.92i			0.127589i	0.997963	+	0.0637943i	$0.0203202\pi$
$787$			2816.92i			0.127589i	−0.997963	+	0.0637943i	$0.979680\pi$
$788$		0			0
$789$	4834.01			0.218118
$790$		0			0
$791$	−16795.0			−0.754943
$792$		0			0
$793$		−	14573.4i		−	0.652606i
$794$		0			0
$795$	−11012.0	−	176.238i	−0.491267	−	0.00786228i
$796$		0			0
$797$			18433.9i			0.819276i	0.912248	+	0.409638i	$0.134345\pi$
$797$			18433.9i			0.819276i	−0.912248	+	0.409638i	$0.865655\pi$
$798$		0			0
$799$	−4962.99			−0.219747
$800$		0			0
$801$	−1828.58			−0.0806612
$802$		0			0
$803$			13269.5i			0.583153i
$804$		0			0
$805$	26199.7	+	419.302i	1.14710	+	0.0183583i
$806$		0			0
$807$			21094.0i			0.920128i
$808$		0			0
$809$	13276.6			0.576983			0.288492	−	0.957482i	$-0.406846\pi$
$809$	13276.6			0.576983			0.288492	+	0.957482i	$0.406846\pi$
$810$		0			0
$811$	18842.5			0.815845			0.407923	−	0.913016i	$-0.366253\pi$
$811$	18842.5			0.815845			0.407923	+	0.913016i	$0.366253\pi$
$812$		0			0
$813$			16323.8i			0.704183i
$814$		0			0
$815$	−260.822	+	16297.2i	−0.0112101	+	0.700449i
$816$		0			0
$817$			14133.2i			0.605210i
$818$		0			0
$819$	17966.2			0.766531
$820$		0			0
$821$	−5863.62			−0.249259			−0.124629	−	0.992203i	$-0.539774\pi$
$821$	−5863.62			−0.249259			−0.124629	+	0.992203i	$0.539774\pi$
$822$		0			0
$823$		−	5018.31i		−	0.212548i	−0.994337	−	0.106274i	$-0.966108\pi$
$823$		−	5018.31i		−	0.212548i	0.994337	−	0.106274i	$-0.0338921\pi$
$824$		0			0
$825$	−580.294	+	18124.9i	−0.0244888	+	0.764882i
$826$		0			0
$827$		−	20303.9i		−	0.853730i	−0.904315	−	0.426865i	$-0.859618\pi$
$827$		−	20303.9i		−	0.853730i	0.904315	−	0.426865i	$-0.140382\pi$
$828$		0			0
$829$	−38156.2			−1.59857			−0.799287	−	0.600949i	$-0.794789\pi$
$829$	−38156.2			−1.59857			−0.799287	+	0.600949i	$0.794789\pi$
$830$		0			0
$831$	−3241.28			−0.135305
$832$		0			0
$833$			13357.0i			0.555573i
$834$		0			0
$835$	−393.349	+	24578.0i	−0.0163023	+	1.01863i
$836$		0			0
$837$		−	5681.90i		−	0.234642i
$838$		0			0
$839$	−41004.6			−1.68729			−0.843645	−	0.536901i	$-0.819595\pi$
$839$	−41004.6			−1.68729			−0.843645	+	0.536901i	$0.819595\pi$
$840$		0			0
$841$	−13467.8			−0.552206
$842$		0			0
$843$			4808.98i			0.196477i
$844$		0			0
$845$	16165.4	+	258.713i	0.658116	+	0.0105326i
$846$		0			0
$847$		−	33320.8i		−	1.35173i
$848$		0			0
$849$	1004.43			0.0406030
$850$		0			0
$851$	−21325.8			−0.859033
$852$		0			0
$853$		−	22451.8i		−	0.901214i	−0.892722	−	0.450607i	$-0.851208\pi$
$853$		−	22451.8i		−	0.901214i	0.892722	−	0.450607i	$-0.148792\pi$
$854$		0			0
$855$	−13166.1	−	210.712i	−0.526633	−	0.00842829i
$856$		0			0
$857$			39858.0i			1.58871i	0.607455	+	0.794354i	$0.292191\pi$
$857$			39858.0i			1.58871i	−0.607455	+	0.794354i	$0.707809\pi$
$858$		0			0
$859$	32585.0			1.29428			0.647139	−	0.762372i	$-0.275965\pi$
$859$	32585.0			1.29428			0.647139	+	0.762372i	$0.275965\pi$
$860$		0			0
$861$	−23827.5			−0.943133
$862$		0			0
$863$			15593.6i			0.615078i	0.951535	+	0.307539i	$0.0995055\pi$
$863$			15593.6i			0.615078i	−0.951535	+	0.307539i	$0.900495\pi$
$864$		0			0
$865$	−363.272	+	22698.7i	−0.0142793	+	0.892228i
$866$		0			0
$867$		−	13789.6i		−	0.540163i
$868$		0			0
$869$	17746.9			0.692775
$870$		0			0
$871$	−6267.21			−0.243807
$872$		0			0
$873$		−	11551.1i		−	0.447817i
$874$		0			0
$875$	−2218.16	+	46168.3i	−0.0857000	+	1.78374i
$876$		0			0
$877$			1804.75i			0.0694892i	0.999396	+	0.0347446i	$0.0110618\pi$
$877$			1804.75i			0.0694892i	−0.999396	+	0.0347446i	$0.988938\pi$
$878$		0			0
$879$	1617.75			0.0620766
$880$		0			0
$881$	−14788.0			−0.565515			−0.282758	−	0.959191i	$-0.591249\pi$
$881$	−14788.0			−0.565515			−0.282758	+	0.959191i	$0.591249\pi$
$882$		0			0
$883$			1999.24i			0.0761947i	0.999274	+	0.0380973i	$0.0121297\pi$
$883$			1999.24i			0.0761947i	−0.999274	+	0.0380973i	$0.987870\pi$
$884$		0			0
$885$	−477.435	+	29832.0i	−0.0181342	+	1.13310i
$886$		0			0
$887$			32474.4i			1.22929i	0.788802	+	0.614647i	$0.210701\pi$
$887$			32474.4i			1.22929i	−0.788802	+	0.614647i	$0.789299\pi$
$888$		0			0
$889$	26328.6			0.993287
$890$		0			0
$891$	−3916.98			−0.147277
$892$		0			0
$893$			36509.4i			1.36813i
$894$		0			0
$895$	−4775.86	−	76.4333i	−0.178368	−	0.00285462i
$896$		0			0
$897$			12831.3i			0.477620i
$898$		0			0
$899$	−21992.0			−0.815880
$900$		0			0
$901$	5841.18			0.215980
$902$		0			0
$903$			10715.8i			0.394905i
$904$		0			0
$905$	−42255.9	−	676.268i	−1.55208	−	0.0248397i
$906$		0			0
$907$			27597.9i			1.01033i	0.863022	+	0.505167i	$0.168569\pi$
$907$			27597.9i			1.01033i	−0.863022	+	0.505167i	$0.831431\pi$
$908$		0			0
$909$	7982.17			0.291256
$910$		0			0
$911$	−19861.4			−0.722325			−0.361163	−	0.932503i	$-0.617620\pi$
$911$	−19861.4			−0.722325			−0.361163	+	0.932503i	$0.617620\pi$
$912$		0			0
$913$		−	2791.46i		−	0.101187i
$914$		0			0
$915$	−129.592	+	8097.45i	−0.00468218	+	0.292561i
$916$		0			0
$917$		−	3025.44i		−	0.108952i
$918$		0			0
$919$	20886.0			0.749691			0.374845	−	0.927087i	$-0.377696\pi$
$919$	20886.0			0.749691			0.374845	+	0.927087i	$0.377696\pi$
$920$		0			0
$921$	−25432.7			−0.909921
$922$		0			0
$923$			16755.2i			0.597511i
$924$		0			0
$925$	1203.78	−	37598.9i	0.0427893	−	1.33648i
$926$		0			0
$927$		−	7047.49i		−	0.249698i
$928$		0			0
$929$	−37461.2			−1.32299			−0.661497	−	0.749948i	$-0.730078\pi$
$929$	−37461.2			−1.32299			−0.661497	+	0.749948i	$0.730078\pi$
$930$		0			0
$931$	98258.5			3.45896
$932$		0			0
$933$		−	10940.8i		−	0.383906i
$934$		0			0
$935$	153.904	−	9616.56i	0.00538311	−	0.336358i
$936$		0			0
$937$			27149.0i			0.946552i	0.880914	+	0.473276i	$0.156929\pi$
$937$			27149.0i			0.946552i	−0.880914	+	0.473276i	$0.843071\pi$
$938$		0			0
$939$	−22611.1			−0.785822
$940$		0			0
$941$	43068.3			1.49202			0.746008	−	0.665937i	$-0.231968\pi$
$941$	43068.3			1.49202			0.746008	+	0.665937i	$0.231968\pi$
$942$		0			0
$943$		−	17017.4i		−	0.587660i
$944$		0			0
$945$	−9982.58	−	159.762i	−0.343633	−	0.00549954i
$946$		0			0
$947$		−	30021.1i		−	1.03015i	−0.857145	−	0.515076i	$-0.827764\pi$
$947$		−	30021.1i		−	1.03015i	0.857145	−	0.515076i	$-0.172236\pi$
$948$		0			0
$949$	−16562.4			−0.566530
$950$		0			0
$951$	32163.5			1.09671
$952$		0			0
$953$			53820.4i			1.82939i	0.404140	+	0.914697i	$0.367571\pi$
$953$			53820.4i			1.82939i	−0.404140	+	0.914697i	$0.632429\pi$
$954$		0			0
$955$	−17501.7	−	280.098i	−0.593027	−	0.00949086i
$956$		0			0
$957$			15160.9i			0.512101i
$958$		0			0
$959$	75185.1			2.53165
$960$		0			0
$961$	14494.3			0.486532
$962$		0			0
$963$			11713.6i			0.391967i
$964$		0			0
$965$	472.788	−	29541.7i	0.0157716	−	0.985473i
$966$		0			0
$967$		−	32210.2i		−	1.07116i	−0.844485	−	0.535580i	$-0.820093\pi$
$967$		−	32210.2i		−	1.07116i	0.844485	−	0.535580i	$-0.179907\pi$
$968$		0			0
$969$	6983.77			0.231529
$970$		0			0
$971$	21616.1			0.714411			0.357206	−	0.934026i	$-0.383730\pi$
$971$	21616.1			0.714411			0.357206	+	0.934026i	$0.383730\pi$
$972$		0			0
$973$			24432.5i			0.805005i
$974$		0			0
$975$	−22622.6	−	724.294i	−0.743080	−	0.0237907i
$976$		0			0
$977$			46002.8i			1.50641i	0.657787	+	0.753204i	$0.271493\pi$
$977$			46002.8i			1.50641i	−0.657787	+	0.753204i	$0.728507\pi$
$978$		0			0
$979$	−9825.11			−0.320748
$980$		0			0
$981$	−10457.2			−0.340339
$982$		0			0
$983$		−	3926.43i		−	0.127400i	−0.997969	−	0.0636998i	$-0.979710\pi$
$983$		−	3926.43i		−	0.127400i	0.997969	−	0.0636998i	$-0.0202900\pi$
$984$		0			0
$985$	−17.6139	+	1100.59i	−0.000569772	+	0.0356017i
$986$		0			0
$987$			27681.5i			0.892719i
$988$		0			0
$989$	−7653.15			−0.246063
$990$		0			0
$991$	55802.3			1.78872			0.894359	−	0.447351i	$-0.147632\pi$
$991$	55802.3			1.78872			0.894359	+	0.447351i	$0.147632\pi$
$992$		0			0
$993$		−	13215.4i		−	0.422335i
$994$		0			0
$995$	12650.2	+	202.456i	0.403055	+	0.00645052i
$996$		0			0
$997$			43880.8i			1.39390i	0.717119	+	0.696951i	$0.245460\pi$
$997$			43880.8i			1.39390i	−0.717119	+	0.696951i	$0.754540\pi$
$998$		0			0
$999$	8125.53			0.257338

Display

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p

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a_n

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a_n

instead) (See

a_n

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a_n

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up to: 50 250 1000 (See only

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.4.f.n.769.1		4
4.3	odd	2		960.4.f.o.769.3		4
5.4	even	2	inner	960.4.f.n.769.3		4
8.3	odd	2		240.4.f.g.49.2		4
8.5	even	2		120.4.f.d.49.4	yes	4
20.19	odd	2		960.4.f.o.769.1		4
24.5	odd	2		360.4.f.d.289.1		4
24.11	even	2		720.4.f.i.289.1		4
40.3	even	4		1200.4.a.bq.1.2		2
40.13	odd	4		600.4.a.t.1.1		2
40.19	odd	2		240.4.f.g.49.4		4
40.27	even	4		1200.4.a.bo.1.1		2
40.29	even	2		120.4.f.d.49.2	✓	4
40.37	odd	4		600.4.a.v.1.2		2
120.29	odd	2		360.4.f.d.289.2		4
120.53	even	4		1800.4.a.bn.1.1		2
120.59	even	2		720.4.f.i.289.2		4
120.77	even	4		1800.4.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.f.d.49.2	✓	4	40.29	even	2
120.4.f.d.49.4	yes	4	8.5	even	2
240.4.f.g.49.2		4	8.3	odd	2
240.4.f.g.49.4		4	40.19	odd	2
360.4.f.d.289.1		4	24.5	odd	2
360.4.f.d.289.2		4	120.29	odd	2
600.4.a.t.1.1		2	40.13	odd	4
600.4.a.v.1.2		2	40.37	odd	4
720.4.f.i.289.1		4	24.11	even	2
720.4.f.i.289.2		4	120.59	even	2
960.4.f.n.769.1		4	1.1	even	1	trivial
960.4.f.n.769.3		4	5.4	even	2	inner
960.4.f.o.769.1		4	20.19	odd	2
960.4.f.o.769.3		4	4.3	odd	2
1200.4.a.bo.1.1		2	40.27	even	4
1200.4.a.bq.1.2		2	40.3	even	4
1800.4.a.bl.1.2		2	120.77	even	4
1800.4.a.bn.1.1		2	120.53	even	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

Representations

Groups

Database

Properties

Related objects

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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	960.4.f.n.769.1

Level	$960$
Weight	$4$
Character	960.769
Analytic conductor	$56.642$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$