LMFDB - Embedded newform 1344.4.b.f.895.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1344,4,Mod(895,1344)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 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("1344.895"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,24,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 158x^{6} + 8461x^{4} + 180672x^{2} + 1232100 $ x^8 + 158x^6 + 8461x^4 + 180672*x^2 + 1232100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 336)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.8
Root			$-6.15149i$ of defining polynomial
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.f.895.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +20.9291i q^{5} +(-17.6950 + 5.46693i) q^{7} +9.00000 q^{9} +23.8516i q^{11} +74.6121i q^{13} +62.7872i q^{15} +68.6323i q^{17} +26.6585 q^{19} +(-53.0850 + 16.4008i) q^{21} +74.6991i q^{23} -313.026 q^{25} +27.0000 q^{27} -128.393 q^{29} +212.201 q^{31} +71.5548i q^{33} +(-114.418 - 370.340i) q^{35} +329.440 q^{37} +223.836i q^{39} +182.595i q^{41} -260.938i q^{43} +188.362i q^{45} -401.555 q^{47} +(283.225 - 193.475i) q^{49} +205.897i q^{51} +76.7726 q^{53} -499.192 q^{55} +79.9755 q^{57} +901.219 q^{59} +271.635i q^{61} +(-159.255 + 49.2024i) q^{63} -1561.56 q^{65} -499.962i q^{67} +224.097i q^{69} +299.147i q^{71} +452.227i q^{73} -939.079 q^{75} +(-130.395 - 422.054i) q^{77} -347.233i q^{79} +81.0000 q^{81} -775.624 q^{83} -1436.41 q^{85} -385.180 q^{87} +48.7979i q^{89} +(-407.899 - 1320.26i) q^{91} +636.604 q^{93} +557.938i q^{95} -1054.44i q^{97} +214.665i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9} + 56 q^{19} - 12 q^{21} - 656 q^{25} + 216 q^{27} - 240 q^{29} + 320 q^{31} - 600 q^{35} - 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 168 q^{57}+ \cdots + 960 q^{93}+O(q^{100}) $ 8 * q + 24 * q^3 - 4 * q^7 + 72 * q^9 + 56 * q^19 - 12 * q^21 - 656 * q^25 + 216 * q^27 - 240 * q^29 + 320 * q^31 - 600 * q^35 - 392 * q^37 - 816 * q^47 - 16 * q^49 - 288 * q^53 - 456 * q^55 + 168 * q^57 + 1824 * q^59 - 36 * q^63 - 816 * q^65 - 1968 * q^75 + 2064 * q^77 + 648 * q^81 - 1680 * q^83 - 2568 * q^85 - 720 * q^87 + 864 * q^91 + 960 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$			20.9291i			1.87195i	0.352062	+	0.935977i	$0.385481\pi$
$5$			20.9291i			1.87195i	−0.352062	+	0.935977i	$0.614519\pi$
$6$		0			0
$7$	−17.6950	+	5.46693i	−0.955440	+	0.295187i
$7$	−17.6950	+	5.46693i	−0.955440	+	0.295187i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			23.8516i			0.653776i	0.945063	+	0.326888i	$0.106000\pi$
$11$			23.8516i			0.653776i	−0.945063	+	0.326888i	$0.894000\pi$
$12$		0			0
$13$			74.6121i			1.59182i	0.605415	+	0.795910i	$0.293007\pi$
$13$			74.6121i			1.59182i	−0.605415	+	0.795910i	$0.706993\pi$
$14$		0			0
$15$			62.7872i			1.08077i
$16$		0			0
$17$			68.6323i			0.979164i	0.871957	+	0.489582i	$0.162851\pi$
$17$			68.6323i			0.979164i	−0.871957	+	0.489582i	$0.837149\pi$
$18$		0			0
$19$	26.6585			0.321888			0.160944	−	0.986964i	$-0.448546\pi$
$19$	26.6585			0.321888			0.160944	+	0.986964i	$0.448546\pi$
$20$		0			0
$21$	−53.0850	+	16.4008i	−0.551623	+	0.170426i
$22$		0			0
$23$			74.6991i			0.677210i	0.940929	+	0.338605i	$0.109955\pi$
$23$			74.6991i			0.677210i	−0.940929	+	0.338605i	$0.890045\pi$
$24$		0			0
$25$	−313.026			−2.50421
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−128.393			−0.822140			−0.411070	−	0.911604i	$-0.634845\pi$
$29$	−128.393			−0.822140			−0.411070	+	0.911604i	$0.634845\pi$
$30$		0			0
$31$	212.201			1.22943			0.614717	−	0.788748i	$-0.289270\pi$
$31$	212.201			1.22943			0.614717	+	0.788748i	$0.289270\pi$
$32$		0			0
$33$			71.5548i			0.377458i
$34$		0			0
$35$	−114.418	−	370.340i	−0.552576	−	1.78854i
$36$		0			0
$37$	329.440			1.46377			0.731887	−	0.681426i	$-0.238640\pi$
$37$	329.440			1.46377			0.731887	+	0.681426i	$0.238640\pi$
$38$		0			0
$39$			223.836i			0.919038i
$40$		0			0
$41$			182.595i			0.695527i	0.937582	+	0.347763i	$0.113059\pi$
$41$			182.595i			0.695527i	−0.937582	+	0.347763i	$0.886941\pi$
$42$		0			0
$43$		−	260.938i		−	0.925410i	−0.886512	−	0.462705i	$-0.846879\pi$
$43$		−	260.938i		−	0.925410i	0.886512	−	0.462705i	$-0.153121\pi$
$44$		0			0
$45$			188.362i			0.623985i
$46$		0			0
$47$	−401.555			−1.24623			−0.623115	−	0.782130i	$-0.714133\pi$
$47$	−401.555			−1.24623			−0.623115	+	0.782130i	$0.714133\pi$
$48$		0			0
$49$	283.225	−	193.475i	0.825730	−	0.564066i
$50$		0			0
$51$			205.897i			0.565320i
$52$		0			0
$53$	76.7726			0.198972			0.0994862	−	0.995039i	$-0.468280\pi$
$53$	76.7726			0.198972			0.0994862	+	0.995039i	$0.468280\pi$
$54$		0			0
$55$	−499.192			−1.22384
$56$		0			0
$57$	79.9755			0.185842
$58$		0			0
$59$	901.219			1.98862			0.994311	−	0.106518i	$-0.0339703\pi$
$59$	901.219			1.98862			0.994311	+	0.106518i	$0.0339703\pi$
$60$		0			0
$61$			271.635i			0.570152i	0.958505	+	0.285076i	$0.0920189\pi$
$61$			271.635i			0.570152i	−0.958505	+	0.285076i	$0.907981\pi$
$62$		0			0
$63$	−159.255	+	49.2024i	−0.318480	+	0.0983956i
$64$		0			0
$65$	−1561.56			−2.97981
$66$		0			0
$67$		−	499.962i		−	0.911644i	−0.890071	−	0.455822i	$-0.849345\pi$
$67$		−	499.962i		−	0.911644i	0.890071	−	0.455822i	$-0.150655\pi$
$68$		0			0
$69$			224.097i			0.390988i
$70$		0			0
$71$			299.147i			0.500032i	0.968242	+	0.250016i	$0.0804358\pi$
$71$			299.147i			0.500032i	−0.968242	+	0.250016i	$0.919564\pi$
$72$		0			0
$73$			452.227i			0.725057i	0.931973	+	0.362528i	$0.118086\pi$
$73$			452.227i			0.725057i	−0.931973	+	0.362528i	$0.881914\pi$
$74$		0			0
$75$	−939.079			−1.44581
$76$		0			0
$77$	−130.395	−	422.054i	−0.192986	−	0.624643i
$78$		0			0
$79$		−	347.233i		−	0.494516i	−0.968950	−	0.247258i	$-0.920470\pi$
$79$		−	347.233i		−	0.494516i	0.968950	−	0.247258i	$-0.0795295\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−775.624			−1.02573			−0.512866	−	0.858468i	$-0.671416\pi$
$83$	−775.624			−1.02573			−0.512866	+	0.858468i	$0.671416\pi$
$84$		0			0
$85$	−1436.41			−1.83295
$86$		0			0
$87$	−385.180			−0.474663
$88$		0			0
$89$			48.7979i			0.0581188i	0.999578	+	0.0290594i	$0.00925119\pi$
$89$			48.7979i			0.0581188i	−0.999578	+	0.0290594i	$0.990749\pi$
$90$		0			0
$91$	−407.899	−	1320.26i	−0.469884	−	1.52089i
$92$		0			0
$93$	636.604			0.709814
$94$		0			0
$95$			557.938i			0.602560i
$96$		0			0
$97$		−	1054.44i		−	1.10373i	−0.833933	−	0.551865i	$-0.813916\pi$
$97$		−	1054.44i		−	1.10373i	0.833933	−	0.551865i	$-0.186084\pi$
$98$		0			0
$99$			214.665i			0.217925i
$100$		0			0
$101$		−	615.627i		−	0.606506i	−0.952910	−	0.303253i	$-0.901927\pi$
$101$		−	615.627i		−	0.606506i	0.952910	−	0.303253i	$-0.0980728\pi$
$102$		0			0
$103$	609.844			0.583396			0.291698	−	0.956511i	$-0.405780\pi$
$103$	609.844			0.583396			0.291698	+	0.956511i	$0.405780\pi$
$104$		0			0
$105$	−343.254	−	1111.02i	−0.319030	−	1.03261i
$106$		0			0
$107$			1905.47i			1.72157i	0.508966	+	0.860787i	$0.330028\pi$
$107$			1905.47i			1.72157i	−0.508966	+	0.860787i	$0.669972\pi$
$108$		0			0
$109$	−152.454			−0.133968			−0.0669838	−	0.997754i	$-0.521338\pi$
$109$	−152.454			−0.133968			−0.0669838	+	0.997754i	$0.521338\pi$
$110$		0			0
$111$	988.321			0.845110
$112$		0			0
$113$	928.079			0.772622			0.386311	−	0.922368i	$-0.373749\pi$
$113$	928.079			0.772622			0.386311	+	0.922368i	$0.373749\pi$
$114$		0			0
$115$	−1563.38			−1.26771
$116$		0			0
$117$			671.509i			0.530607i
$118$		0			0
$119$	−375.208	−	1214.45i	−0.289036	−	0.935532i
$120$		0			0
$121$	762.100			0.572577
$122$		0			0
$123$			547.786i			0.401563i
$124$		0			0
$125$		−	3935.22i		−	2.81581i
$126$		0			0
$127$		−	1115.88i		−	0.779672i	−0.920884	−	0.389836i	$-0.872532\pi$
$127$		−	1115.88i		−	0.779672i	0.920884	−	0.389836i	$-0.127468\pi$
$128$		0			0
$129$		−	782.814i		−	0.534286i
$130$		0			0
$131$	936.794			0.624795			0.312397	−	0.949952i	$-0.398868\pi$
$131$	936.794			0.624795			0.312397	+	0.949952i	$0.398868\pi$
$132$		0			0
$133$	−471.722	+	145.740i	−0.307545	+	0.0950172i
$134$		0			0
$135$			565.085i			0.360258i
$136$		0			0
$137$	−2044.46			−1.27496			−0.637481	−	0.770466i	$-0.720024\pi$
$137$	−2044.46			−1.27496			−0.637481	+	0.770466i	$0.720024\pi$
$138$		0			0
$139$	2914.20			1.77827			0.889133	−	0.457650i	$-0.151309\pi$
$139$	2914.20			1.77827			0.889133	+	0.457650i	$0.151309\pi$
$140$		0			0
$141$	−1204.66			−0.719511
$142$		0			0
$143$	−1779.62			−1.04069
$144$		0			0
$145$		−	2687.16i		−	1.53901i
$146$		0			0
$147$	849.676	−	580.424i	0.476735	−	0.325664i
$148$		0			0
$149$	−3427.89			−1.88472			−0.942360	−	0.334600i	$-0.891399\pi$
$149$	−3427.89			−1.88472			−0.942360	+	0.334600i	$0.891399\pi$
$150$		0			0
$151$		−	2984.28i		−	1.60832i	−0.594410	−	0.804162i	$-0.702614\pi$
$151$		−	2984.28i		−	1.60832i	0.594410	−	0.804162i	$-0.297386\pi$
$152$		0			0
$153$			617.691i			0.326388i
$154$		0			0
$155$			4441.18i			2.30144i
$156$		0			0
$157$		−	3074.64i		−	1.56295i	−0.623938	−	0.781474i	$-0.714468\pi$
$157$		−	3074.64i		−	1.56295i	0.623938	−	0.781474i	$-0.285532\pi$
$158$		0			0
$159$	230.318			0.114877
$160$		0			0
$161$	−408.375	−	1321.80i	−0.199904	−	0.647034i
$162$		0			0
$163$			87.6924i			0.0421386i	0.999778	+	0.0210693i	$0.00670707\pi$
$163$			87.6924i			0.0421386i	−0.999778	+	0.0210693i	$0.993293\pi$
$164$		0			0
$165$	−1497.58			−0.706583
$166$		0			0
$167$	199.541			0.0924608			0.0462304	−	0.998931i	$-0.485279\pi$
$167$	199.541			0.0924608			0.0462304	+	0.998931i	$0.485279\pi$
$168$		0			0
$169$	−3369.96			−1.53389
$170$		0			0
$171$	239.927			0.107296
$172$		0			0
$173$		−	1698.44i		−	0.746418i	−0.927747	−	0.373209i	$-0.878257\pi$
$173$		−	1698.44i		−	0.746418i	0.927747	−	0.373209i	$-0.121743\pi$
$174$		0			0
$175$	5539.00	−	1711.29i	2.39262	−	0.739209i
$176$		0			0
$177$	2703.66			1.14813
$178$		0			0
$179$		−	709.691i		−	0.296340i	−0.988962	−	0.148170i	$-0.952662\pi$
$179$		−	709.691i		−	0.296340i	0.988962	−	0.148170i	$-0.0473382\pi$
$180$		0			0
$181$			4297.66i			1.76488i	0.470427	+	0.882439i	$0.344100\pi$
$181$			4297.66i			1.76488i	−0.470427	+	0.882439i	$0.655900\pi$
$182$		0			0
$183$			814.904i			0.329177i
$184$		0			0
$185$			6894.88i			2.74012i
$186$		0			0
$187$	−1636.99			−0.640153
$188$		0			0
$189$	−477.765	+	147.607i	−0.183874	+	0.0568087i
$190$		0			0
$191$			3487.40i			1.32115i	0.750760	+	0.660575i	$0.229688\pi$
$191$			3487.40i			1.32115i	−0.750760	+	0.660575i	$0.770312\pi$
$192$		0			0
$193$	−210.502			−0.0785090			−0.0392545	−	0.999229i	$-0.512498\pi$
$193$	−210.502			−0.0785090			−0.0392545	+	0.999229i	$0.512498\pi$
$194$		0			0
$195$	−4684.69			−1.72040
$196$		0			0
$197$	2245.04			0.811943			0.405972	−	0.913886i	$-0.366933\pi$
$197$	2245.04			0.811943			0.405972	+	0.913886i	$0.366933\pi$
$198$		0			0
$199$	976.701			0.347922			0.173961	−	0.984753i	$-0.444343\pi$
$199$	976.701			0.347922			0.173961	+	0.984753i	$0.444343\pi$
$200$		0			0
$201$		−	1499.89i		−	0.526338i
$202$		0			0
$203$	2271.92	−	701.918i	0.785505	−	0.242685i
$204$		0			0
$205$	−3821.55			−1.30199
$206$		0			0
$207$			672.292i			0.225737i
$208$		0			0
$209$			635.849i			0.210443i
$210$		0			0
$211$			4470.74i			1.45867i	0.684159	+	0.729333i	$0.260169\pi$
$211$			4470.74i			1.45867i	−0.684159	+	0.729333i	$0.739831\pi$
$212$		0			0
$213$			897.442i			0.288693i
$214$		0			0
$215$	5461.19			1.73233
$216$		0			0
$217$	−3754.90	+	1160.09i	−1.17465	+	0.362913i
$218$		0			0
$219$			1356.68i			0.418612i
$220$		0			0
$221$	−5120.80			−1.55865
$222$		0			0
$223$	−5256.06			−1.57835			−0.789174	−	0.614169i	$-0.789491\pi$
$223$	−5256.06			−1.57835			−0.789174	+	0.614169i	$0.789491\pi$
$224$		0			0
$225$	−2817.24			−0.834737
$226$		0			0
$227$	6251.50			1.82787			0.913935	−	0.405860i	$-0.133028\pi$
$227$	6251.50			1.82787			0.913935	+	0.405860i	$0.133028\pi$
$228$		0			0
$229$			3326.07i			0.959794i	0.877325	+	0.479897i	$0.159326\pi$
$229$			3326.07i			0.959794i	−0.877325	+	0.479897i	$0.840674\pi$
$230$		0			0
$231$	−391.186	−	1266.16i	−0.111420	−	0.360638i
$232$		0			0
$233$	−1823.44			−0.512695			−0.256347	−	0.966585i	$-0.582519\pi$
$233$	−1823.44			−0.512695			−0.256347	+	0.966585i	$0.582519\pi$
$234$		0			0
$235$		−	8404.17i		−	2.33288i
$236$		0			0
$237$		−	1041.70i		−	0.285509i
$238$		0			0
$239$			2536.33i			0.686450i	0.939253	+	0.343225i	$0.111519\pi$
$239$			2536.33i			0.686450i	−0.939253	+	0.343225i	$0.888481\pi$
$240$		0			0
$241$			959.148i			0.256366i	0.991751	+	0.128183i	$0.0409144\pi$
$241$			959.148i			0.256366i	−0.991751	+	0.128183i	$0.959086\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	4049.25	+	5927.64i	1.05591	+	1.54573i
$246$		0			0
$247$			1989.05i			0.512389i
$248$		0			0
$249$	−2326.87			−0.592207
$250$		0			0
$251$	3386.13			0.851516			0.425758	−	0.904837i	$-0.360008\pi$
$251$	3386.13			0.851516			0.425758	+	0.904837i	$0.360008\pi$
$252$		0			0
$253$	−1781.69			−0.442744
$254$		0			0
$255$	−4309.23			−1.05825
$256$		0			0
$257$		−	2333.34i		−	0.566343i	−0.959069	−	0.283171i	$-0.908614\pi$
$257$		−	2333.34i		−	0.566343i	0.959069	−	0.283171i	$-0.0913865\pi$
$258$		0			0
$259$	−5829.44	+	1801.03i	−1.39855	+	0.432087i
$260$		0			0
$261$	−1155.54			−0.274047
$262$		0			0
$263$		−	5769.92i		−	1.35281i	−0.736531	−	0.676404i	$-0.763537\pi$
$263$		−	5769.92i		−	1.35281i	0.736531	−	0.676404i	$-0.236463\pi$
$264$		0			0
$265$			1606.78i			0.372467i
$266$		0			0
$267$			146.394i			0.0335549i
$268$		0			0
$269$		−	64.3624i		−	0.0145883i	−0.999973	−	0.00729413i	$-0.997678\pi$
$269$		−	64.3624i		−	0.0145883i	0.999973	−	0.00729413i	$-0.00232182\pi$
$270$		0			0
$271$	−6269.92			−1.40543			−0.702713	−	0.711473i	$-0.748028\pi$
$271$	−6269.92			−1.40543			−0.702713	+	0.711473i	$0.748028\pi$
$272$		0			0
$273$	−1223.70	−	3960.78i	−0.271288	−	0.878085i
$274$		0			0
$275$		−	7466.18i		−	1.63719i
$276$		0			0
$277$	−5895.37			−1.27877			−0.639383	−	0.768888i	$-0.720810\pi$
$277$	−5895.37			−1.27877			−0.639383	+	0.768888i	$0.720810\pi$
$278$		0			0
$279$	1909.81			0.409812
$280$		0			0
$281$	−4455.82			−0.945949			−0.472974	−	0.881076i	$-0.656820\pi$
$281$	−4455.82			−0.945949			−0.472974	+	0.881076i	$0.656820\pi$
$282$		0			0
$283$	1357.37			0.285115			0.142557	−	0.989787i	$-0.454467\pi$
$283$	1357.37			0.285115			0.142557	+	0.989787i	$0.454467\pi$
$284$		0			0
$285$			1673.81i			0.347888i
$286$		0			0
$287$	−998.236	−	3231.02i	−0.205310	−	0.664534i
$288$		0			0
$289$	202.606			0.0412388
$290$		0			0
$291$		−	3163.31i		−	0.637239i
$292$		0			0
$293$		−	3600.78i		−	0.717952i	−0.933347	−	0.358976i	$-0.883126\pi$
$293$		−	3600.78i		−	0.717952i	0.933347	−	0.358976i	$-0.116874\pi$
$294$		0			0
$295$			18861.7i			3.72261i
$296$		0			0
$297$			643.994i			0.125819i
$298$		0			0
$299$	−5573.46			−1.07800
$300$		0			0
$301$	1426.53	+	4617.29i	0.273169	+	0.884174i
$302$		0			0
$303$		−	1846.88i		−	0.350167i
$304$		0			0
$305$	−5685.07			−1.06730
$306$		0			0
$307$	3344.04			0.621677			0.310838	−	0.950463i	$-0.399390\pi$
$307$	3344.04			0.621677			0.310838	+	0.950463i	$0.399390\pi$
$308$		0			0
$309$	1829.53			0.336824
$310$		0			0
$311$	−597.591			−0.108959			−0.0544795	−	0.998515i	$-0.517350\pi$
$311$	−597.591			−0.108959			−0.0544795	+	0.998515i	$0.517350\pi$
$312$		0			0
$313$		−	6906.72i		−	1.24726i	−0.781721	−	0.623628i	$-0.785658\pi$
$313$		−	6906.72i		−	1.24726i	0.781721	−	0.623628i	$-0.214342\pi$
$314$		0			0
$315$	−1029.76	−	3333.06i	−0.184192	−	0.596180i
$316$		0			0
$317$	6668.30			1.18148			0.590740	−	0.806862i	$-0.298836\pi$
$317$	6668.30			1.18148			0.590740	+	0.806862i	$0.298836\pi$
$318$		0			0
$319$		−	3062.39i		−	0.537495i
$320$		0			0
$321$			5716.40i			0.993951i
$322$		0			0
$323$			1829.64i			0.315181i
$324$		0			0
$325$		−	23355.5i		−	3.98625i
$326$		0			0
$327$	−457.363			−0.0773462
$328$		0			0
$329$	7105.51	−	2195.27i	1.19070	−	0.367870i
$330$		0			0
$331$			1117.68i			0.185599i	0.995685	+	0.0927997i	$0.0295816\pi$
$331$			1117.68i			0.185599i	−0.995685	+	0.0927997i	$0.970418\pi$
$332$		0			0
$333$	2964.96			0.487925
$334$		0			0
$335$	10463.8			1.70655
$336$		0			0
$337$	−987.236			−0.159579			−0.0797896	−	0.996812i	$-0.525425\pi$
$337$	−987.236			−0.159579			−0.0797896	+	0.996812i	$0.525425\pi$
$338$		0			0
$339$	2784.24			0.446074
$340$		0			0
$341$			5061.34i			0.803774i
$342$		0			0
$343$	−3953.95	+	4971.91i	−0.622430	+	0.782675i
$344$		0			0
$345$	−4690.15			−0.731911
$346$		0			0
$347$		−	8322.10i		−	1.28747i	−0.765247	−	0.643737i	$-0.777383\pi$
$347$		−	8322.10i		−	1.28747i	0.765247	−	0.643737i	$-0.222617\pi$
$348$		0			0
$349$		−	878.983i		−	0.134816i	−0.997725	−	0.0674081i	$-0.978527\pi$
$349$		−	878.983i		−	0.134816i	0.997725	−	0.0674081i	$-0.0214730\pi$
$350$		0			0
$351$			2014.53i			0.306346i
$352$		0			0
$353$			4253.69i			0.641362i	0.947187	+	0.320681i	$0.103912\pi$
$353$			4253.69i			0.641362i	−0.947187	+	0.320681i	$0.896088\pi$
$354$		0			0
$355$	−6260.88			−0.936036
$356$		0			0
$357$	−1125.62	−	3643.34i	−0.166875	−	0.540129i
$358$		0			0
$359$			2174.96i			0.319750i	0.987137	+	0.159875i	$0.0511091\pi$
$359$			2174.96i			0.319750i	−0.987137	+	0.159875i	$0.948891\pi$
$360$		0			0
$361$	−6148.32			−0.896388
$362$		0			0
$363$	2286.30			0.330578
$364$		0			0
$365$	−9464.69			−1.35727
$366$		0			0
$367$	−3958.85			−0.563080			−0.281540	−	0.959549i	$-0.590845\pi$
$367$	−3958.85			−0.563080			−0.281540	+	0.959549i	$0.590845\pi$
$368$		0			0
$369$			1643.36i			0.231842i
$370$		0			0
$371$	−1358.49	+	419.711i	−0.190106	+	0.0587340i
$372$		0			0
$373$	11819.9			1.64078			0.820388	−	0.571807i	$-0.193758\pi$
$373$	11819.9			1.64078			0.820388	+	0.571807i	$0.193758\pi$
$374$		0			0
$375$		−	11805.6i		−	1.62571i
$376$		0			0
$377$		−	9579.70i		−	1.30870i
$378$		0			0
$379$			8146.54i			1.10411i	0.833806	+	0.552057i	$0.186157\pi$
$379$			8146.54i			1.10411i	−0.833806	+	0.552057i	$0.813843\pi$
$380$		0			0
$381$		−	3347.64i		−	0.450144i
$382$		0			0
$383$	−781.310			−0.104238			−0.0521189	−	0.998641i	$-0.516597\pi$
$383$	−781.310			−0.104238			−0.0521189	+	0.998641i	$0.516597\pi$
$384$		0			0
$385$	8833.20	−	2729.05i	1.16930	−	0.361261i
$386$		0			0
$387$		−	2348.44i		−	0.308470i
$388$		0			0
$389$	4647.45			0.605746			0.302873	−	0.953031i	$-0.402054\pi$
$389$	4647.45			0.605746			0.302873	+	0.953031i	$0.402054\pi$
$390$		0			0
$391$	−5126.77			−0.663100
$392$		0			0
$393$	2810.38			0.360725
$394$		0			0
$395$	7267.27			0.925711
$396$		0			0
$397$			7571.92i			0.957239i	0.878022	+	0.478620i	$0.158863\pi$
$397$			7571.92i			0.957239i	−0.878022	+	0.478620i	$0.841137\pi$
$398$		0			0
$399$	−1415.17	+	437.221i	−0.177561	+	0.0548582i
$400$		0			0
$401$	4013.10			0.499762			0.249881	−	0.968277i	$-0.419609\pi$
$401$	4013.10			0.499762			0.249881	+	0.968277i	$0.419609\pi$
$402$		0			0
$403$			15832.8i			1.95704i
$404$		0			0
$405$			1695.26i			0.207995i
$406$		0			0
$407$			7857.68i			0.956980i
$408$		0			0
$409$		−	5888.31i		−	0.711879i	−0.934509	−	0.355939i	$-0.884161\pi$
$409$		−	5888.31i		−	0.711879i	0.934509	−	0.355939i	$-0.115839\pi$
$410$		0			0
$411$	−6133.37			−0.736100
$412$		0			0
$413$	−15947.1	+	4926.90i	−1.90001	+	0.587015i
$414$		0			0
$415$		−	16233.1i		−	1.92012i
$416$		0			0
$417$	8742.59			1.02668
$418$		0			0
$419$	8575.63			0.999873			0.499936	−	0.866062i	$-0.333357\pi$
$419$	8575.63			0.999873			0.499936	+	0.866062i	$0.333357\pi$
$420$		0			0
$421$	7655.57			0.886246			0.443123	−	0.896461i	$-0.353871\pi$
$421$	7655.57			0.886246			0.443123	+	0.896461i	$0.353871\pi$
$422$		0			0
$423$	−3613.99			−0.415410
$424$		0			0
$425$		−	21483.7i		−	2.45203i
$426$		0			0
$427$	−1485.01	−	4806.58i	−0.168301	−	0.544746i
$428$		0			0
$429$	−5338.86			−0.600845
$430$		0			0
$431$			5829.43i			0.651493i	0.945457	+	0.325747i	$0.105616\pi$
$431$			5829.43i			0.651493i	−0.945457	+	0.325747i	$0.894384\pi$
$432$		0			0
$433$			10072.7i			1.11793i	0.829191	+	0.558965i	$0.188801\pi$
$433$			10072.7i			1.11793i	−0.829191	+	0.558965i	$0.811199\pi$
$434$		0			0
$435$		−	8061.47i		−	0.888547i
$436$		0			0
$437$			1991.37i			0.217986i
$438$		0			0
$439$	5808.36			0.631476			0.315738	−	0.948846i	$-0.397748\pi$
$439$	5808.36			0.631476			0.315738	+	0.948846i	$0.397748\pi$
$440$		0			0
$441$	2549.03	−	1741.27i	0.275243	−	0.188022i
$442$		0			0
$443$			8923.50i			0.957039i	0.878077	+	0.478520i	$0.158826\pi$
$443$			8923.50i			0.957039i	−0.878077	+	0.478520i	$0.841174\pi$
$444$		0			0
$445$	−1021.30			−0.108796
$446$		0			0
$447$	−10283.7			−1.08814
$448$		0			0
$449$	14002.7			1.47178			0.735888	−	0.677103i	$-0.236765\pi$
$449$	14002.7			1.47178			0.735888	+	0.677103i	$0.236765\pi$
$450$		0			0
$451$	−4355.19			−0.454719
$452$		0			0
$453$		−	8952.83i		−	0.928567i
$454$		0			0
$455$	27631.8	−	8536.96i	2.84703	−	0.879602i
$456$		0			0
$457$	−5652.19			−0.578552			−0.289276	−	0.957246i	$-0.593415\pi$
$457$	−5652.19			−0.578552			−0.289276	+	0.957246i	$0.593415\pi$
$458$		0			0
$459$			1853.07i			0.188440i
$460$		0			0
$461$		−	4689.73i		−	0.473801i	−0.971534	−	0.236901i	$-0.923868\pi$
$461$		−	4689.73i		−	0.473801i	0.971534	−	0.236901i	$-0.0761316\pi$
$462$		0			0
$463$		−	252.859i		−	0.0253809i	−0.999919	−	0.0126904i	$-0.995960\pi$
$463$		−	252.859i		−	0.0253809i	0.999919	−	0.0126904i	$-0.00403960\pi$
$464$		0			0
$465$			13323.5i			1.32874i
$466$		0			0
$467$	−5019.67			−0.497393			−0.248697	−	0.968581i	$-0.580002\pi$
$467$	−5019.67			−0.497393			−0.248697	+	0.968581i	$0.580002\pi$
$468$		0			0
$469$	2733.26	+	8846.83i	0.269105	+	0.871021i
$470$		0			0
$471$		−	9223.91i		−	0.902368i
$472$		0			0
$473$	6223.79			0.605011
$474$		0			0
$475$	−8344.81			−0.806076
$476$		0			0
$477$	690.954			0.0663241
$478$		0			0
$479$	4266.51			0.406977			0.203488	−	0.979077i	$-0.434772\pi$
$479$	4266.51			0.406977			0.203488	+	0.979077i	$0.434772\pi$
$480$		0			0
$481$			24580.2i			2.33007i
$482$		0			0
$483$	−1225.13	−	3965.40i	−0.115414	−	0.373565i
$484$		0			0
$485$	22068.4			2.06613
$486$		0			0
$487$		−	4599.91i		−	0.428012i	−0.976832	−	0.214006i	$-0.931349\pi$
$487$		−	4599.91i		−	0.428012i	0.976832	−	0.214006i	$-0.0686512\pi$
$488$		0			0
$489$			263.077i			0.0243288i
$490$		0			0
$491$			10713.6i			0.984725i	0.870390	+	0.492362i	$0.163867\pi$
$491$			10713.6i			0.984725i	−0.870390	+	0.492362i	$0.836133\pi$
$492$		0			0
$493$		−	8811.94i		−	0.805009i
$494$		0			0
$495$	−4492.73			−0.407946
$496$		0			0
$497$	−1635.42	−	5293.41i	−0.147603	−	0.477750i
$498$		0			0
$499$			5256.23i			0.471545i	0.971808	+	0.235773i	$0.0757621\pi$
$499$			5256.23i			0.471545i	−0.971808	+	0.235773i	$0.924238\pi$
$500$		0			0
$501$	598.623			0.0533823
$502$		0			0
$503$	12369.0			1.09644			0.548219	−	0.836335i	$-0.315306\pi$
$503$	12369.0			1.09644			0.548219	+	0.836335i	$0.315306\pi$
$504$		0			0
$505$	12884.5			1.13535
$506$		0			0
$507$	−10109.9			−0.885594
$508$		0			0
$509$			15337.1i			1.33557i	0.744354	+	0.667785i	$0.232758\pi$
$509$			15337.1i			1.33557i	−0.744354	+	0.667785i	$0.767242\pi$
$510$		0			0
$511$	−2472.29	−	8002.15i	−0.214027	−	0.692748i
$512$		0			0
$513$	719.780			0.0619475
$514$		0			0
$515$			12763.5i			1.09209i
$516$		0			0
$517$		−	9577.73i		−	0.814755i
$518$		0			0
$519$		−	5095.33i		−	0.430945i
$520$		0			0
$521$			17351.3i			1.45906i	0.683947	+	0.729532i	$0.260262\pi$
$521$			17351.3i			1.45906i	−0.683947	+	0.729532i	$0.739738\pi$
$522$		0			0
$523$	−7260.17			−0.607007			−0.303504	−	0.952830i	$-0.598156\pi$
$523$	−7260.17			−0.607007			−0.303504	+	0.952830i	$0.598156\pi$
$524$		0			0
$525$	16617.0	−	5133.88i	1.38138	−	0.426783i
$526$		0			0
$527$			14563.9i			1.20382i
$528$		0			0
$529$	6587.04			0.541386
$530$		0			0
$531$	8110.97			0.662874
$532$		0			0
$533$	−13623.8			−1.10715
$534$		0			0
$535$	−39879.6			−3.22271
$536$		0			0
$537$		−	2129.07i		−	0.171092i
$538$		0			0
$539$	4614.68	+	6755.38i	0.368773	+	0.539842i
$540$		0			0
$541$	−14127.5			−1.12271			−0.561357	−	0.827573i	$-0.689721\pi$
$541$	−14127.5			−1.12271			−0.561357	+	0.827573i	$0.689721\pi$
$542$		0			0
$543$			12893.0i			1.01895i
$544$		0			0
$545$		−	3190.73i		−	0.250781i
$546$		0			0
$547$		−	10564.8i		−	0.825810i	−0.910774	−	0.412905i	$-0.864514\pi$
$547$		−	10564.8i		−	0.825810i	0.910774	−	0.412905i	$-0.135486\pi$
$548$		0			0
$549$			2444.71i			0.190051i
$550$		0			0
$551$	−3422.78			−0.264637
$552$		0			0
$553$	1898.30	+	6144.28i	0.145975	+	0.472480i
$554$		0			0
$555$			20684.6i			1.58201i
$556$		0			0
$557$	−15271.5			−1.16171			−0.580855	−	0.814007i	$-0.697282\pi$
$557$	−15271.5			−1.16171			−0.580855	+	0.814007i	$0.697282\pi$
$558$		0			0
$559$	19469.1			1.47309
$560$		0			0
$561$	−4910.97			−0.369593
$562$		0			0
$563$	−11971.0			−0.896124			−0.448062	−	0.894003i	$-0.647886\pi$
$563$	−11971.0			−0.896124			−0.448062	+	0.894003i	$0.647886\pi$
$564$		0			0
$565$			19423.8i			1.44631i
$566$		0			0
$567$	−1433.29	+	442.822i	−0.106160	+	0.0327985i
$568$		0			0
$569$	9396.66			0.692317			0.346158	−	0.938176i	$-0.387486\pi$
$569$	9396.66			0.692317			0.346158	+	0.938176i	$0.387486\pi$
$570$		0			0
$571$		−	15035.4i		−	1.10194i	−0.834523	−	0.550972i	$-0.814257\pi$
$571$		−	15035.4i		−	1.10194i	0.834523	−	0.550972i	$-0.185743\pi$
$572$		0			0
$573$			10462.2i			0.762767i
$574$		0			0
$575$		−	23382.8i		−	1.69588i
$576$		0			0
$577$			5019.96i			0.362190i	0.983466	+	0.181095i	$0.0579642\pi$
$577$			5019.96i			0.362190i	−0.983466	+	0.181095i	$0.942036\pi$
$578$		0			0
$579$	−631.505			−0.0453272
$580$		0			0
$581$	13724.7	−	4240.29i	0.980026	−	0.302783i
$582$		0			0
$583$			1831.15i			0.130083i
$584$		0			0
$585$	−14054.1			−0.993271
$586$		0			0
$587$	10231.3			0.719404			0.359702	−	0.933067i	$-0.382878\pi$
$587$	10231.3			0.719404			0.359702	+	0.933067i	$0.382878\pi$
$588$		0			0
$589$	5656.97			0.395741
$590$		0			0
$591$	6735.13			0.468776
$592$		0			0
$593$			4278.75i			0.296302i	0.988965	+	0.148151i	$0.0473322\pi$
$593$			4278.75i			0.296302i	−0.988965	+	0.148151i	$0.952668\pi$
$594$		0			0
$595$	25417.3	−	7852.76i	1.75127	−	0.541062i
$596$		0			0
$597$	2930.10			0.200873
$598$		0			0
$599$			114.830i			0.00783279i	0.999992	+	0.00391639i	$0.00124663\pi$
$599$			114.830i			0.00783279i	−0.999992	+	0.00391639i	$0.998753\pi$
$600$		0			0
$601$		−	2048.91i		−	0.139063i	−0.997580	−	0.0695315i	$-0.977850\pi$
$601$		−	2048.91i		−	0.139063i	0.997580	−	0.0695315i	$-0.0221504\pi$
$602$		0			0
$603$		−	4499.66i		−	0.303881i
$604$		0			0
$605$			15950.1i			1.07184i
$606$		0			0
$607$	1447.87			0.0968162			0.0484081	−	0.998828i	$-0.484585\pi$
$607$	1447.87			0.0968162			0.0484081	+	0.998828i	$0.484585\pi$
$608$		0			0
$609$	6815.76	−	2105.75i	0.453512	−	0.140114i
$610$		0			0
$611$		−	29960.8i		−	1.98377i
$612$		0			0
$613$	13450.7			0.886245			0.443123	−	0.896461i	$-0.353871\pi$
$613$	13450.7			0.886245			0.443123	+	0.896461i	$0.353871\pi$
$614$		0			0
$615$	−11464.7			−0.751706
$616$		0			0
$617$	−19623.3			−1.28039			−0.640197	−	0.768211i	$-0.721147\pi$
$617$	−19623.3			−1.28039			−0.640197	+	0.768211i	$0.721147\pi$
$618$		0			0
$619$	15910.6			1.03312			0.516561	−	0.856250i	$-0.327212\pi$
$619$	15910.6			1.03312			0.516561	+	0.856250i	$0.327212\pi$
$620$		0			0
$621$			2016.88i			0.130329i
$622$		0			0
$623$	−266.775	−	863.479i	−0.0171559	−	0.0555290i
$624$		0			0
$625$	43232.2			2.76686
$626$		0			0
$627$			1907.55i			0.121499i
$628$		0			0
$629$			22610.2i			1.43327i
$630$		0			0
$631$		−	20416.1i		−	1.28804i	−0.765008	−	0.644021i	$-0.777265\pi$
$631$		−	20416.1i		−	1.28804i	0.765008	−	0.644021i	$-0.222735\pi$
$632$		0			0
$633$			13412.2i			0.842161i
$634$		0			0
$635$	23354.3			1.45951
$636$		0			0
$637$	14435.5	+	21132.0i	0.897892	+	1.31441i
$638$		0			0
$639$			2692.33i			0.166677i
$640$		0			0
$641$	699.877			0.0431256			0.0215628	−	0.999767i	$-0.493136\pi$
$641$	699.877			0.0431256			0.0215628	+	0.999767i	$0.493136\pi$
$642$		0			0
$643$	−11449.1			−0.702192			−0.351096	−	0.936339i	$-0.614191\pi$
$643$	−11449.1			−0.702192			−0.351096	+	0.936339i	$0.614191\pi$
$644$		0			0
$645$	16383.6			1.00016
$646$		0			0
$647$	20822.8			1.26527			0.632634	−	0.774451i	$-0.281974\pi$
$647$	20822.8			1.26527			0.632634	+	0.774451i	$0.281974\pi$
$648$		0			0
$649$			21495.5i			1.30011i
$650$		0			0
$651$	−11264.7	+	3480.27i	−0.678185	+	0.209528i
$652$		0			0
$653$	13036.2			0.781236			0.390618	−	0.920553i	$-0.372261\pi$
$653$	13036.2			0.781236			0.390618	+	0.920553i	$0.372261\pi$
$654$		0			0
$655$			19606.2i			1.16959i
$656$		0			0
$657$			4070.04i			0.241686i
$658$		0			0
$659$			13602.0i			0.804034i	0.915632	+	0.402017i	$0.131691\pi$
$659$			13602.0i			0.804034i	−0.915632	+	0.402017i	$0.868309\pi$
$660$		0			0
$661$			13817.6i			0.813076i	0.913634	+	0.406538i	$0.133264\pi$
$661$			13817.6i			0.813076i	−0.913634	+	0.406538i	$0.866736\pi$
$662$		0			0
$663$	−15362.4			−0.899889
$664$		0			0
$665$	−3050.21	−	9872.71i	−0.177868	−	0.575710i
$666$		0			0
$667$		−	9590.87i		−	0.556762i
$668$		0			0
$669$	−15768.2			−0.911260
$670$		0			0
$671$	−6478.93			−0.372752
$672$		0			0
$673$	−14766.4			−0.845771			−0.422886	−	0.906183i	$-0.638983\pi$
$673$	−14766.4			−0.845771			−0.422886	+	0.906183i	$0.638983\pi$
$674$		0			0
$675$	−8451.71			−0.481935
$676$		0			0
$677$		−	19480.1i		−	1.10588i	−0.833222	−	0.552939i	$-0.813506\pi$
$677$		−	19480.1i		−	1.10588i	0.833222	−	0.552939i	$-0.186494\pi$
$678$		0			0
$679$	5764.54	+	18658.2i	0.325806	+	1.05455i
$680$		0			0
$681$	18754.5			1.05532
$682$		0			0
$683$		−	17335.1i		−	0.971170i	−0.874189	−	0.485585i	$-0.838607\pi$
$683$		−	17335.1i		−	0.971170i	0.874189	−	0.485585i	$-0.161393\pi$
$684$		0			0
$685$		−	42788.6i		−	2.38667i
$686$		0			0
$687$			9978.20i			0.554137i
$688$		0			0
$689$			5728.17i			0.316728i
$690$		0			0
$691$	−29783.9			−1.63970			−0.819850	−	0.572578i	$-0.805943\pi$
$691$	−29783.9			−1.63970			−0.819850	+	0.572578i	$0.805943\pi$
$692$		0			0
$693$	−1173.56	−	3798.49i	−0.0643286	−	0.208214i
$694$		0			0
$695$			60991.4i			3.32883i
$696$		0			0
$697$	−12531.9			−0.681034
$698$		0			0
$699$	−5470.33			−0.296004
$700$		0			0
$701$	23567.6			1.26981			0.634903	−	0.772592i	$-0.281040\pi$
$701$	23567.6			1.26981			0.634903	+	0.772592i	$0.281040\pi$
$702$		0			0
$703$	8782.39			0.471172
$704$		0			0
$705$		−	25212.5i		−	1.34689i
$706$		0			0
$707$	3365.59	+	10893.5i	0.179033	+	0.579480i
$708$		0			0
$709$	−9637.20			−0.510484			−0.255242	−	0.966877i	$-0.582155\pi$
$709$	−9637.20			−0.510484			−0.255242	+	0.966877i	$0.582155\pi$
$710$		0			0
$711$		−	3125.10i		−	0.164839i
$712$		0			0
$713$			15851.2i			0.832586i
$714$		0			0
$715$		−	37245.8i		−	1.94813i
$716$		0			0
$717$			7609.00i			0.396322i
$718$		0			0
$719$	26050.6			1.35121			0.675606	−	0.737263i	$-0.263882\pi$
$719$	26050.6			1.35121			0.675606	+	0.737263i	$0.263882\pi$
$720$		0			0
$721$	−10791.2	+	3333.98i	−0.557399	+	0.172211i
$722$		0			0
$723$			2877.44i			0.148013i
$724$		0			0
$725$	40190.5			2.05881
$726$		0			0
$727$	35153.8			1.79337			0.896685	−	0.442668i	$-0.145968\pi$
$727$	35153.8			1.79337			0.896685	+	0.442668i	$0.145968\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	17908.8			0.906128
$732$		0			0
$733$			12635.0i			0.636677i	0.947977	+	0.318338i	$0.103125\pi$
$733$			12635.0i			0.636677i	−0.947977	+	0.318338i	$0.896875\pi$
$734$		0			0
$735$	12147.7	+	17782.9i	0.609627	+	0.892426i
$736$		0			0
$737$	11924.9			0.596011
$738$		0			0
$739$			24477.7i			1.21844i	0.793001	+	0.609220i	$0.208517\pi$
$739$			24477.7i			1.21844i	−0.793001	+	0.609220i	$0.791483\pi$
$740$		0			0
$741$			5967.14i			0.295828i
$742$		0			0
$743$		−	38742.2i		−	1.91294i	−0.291833	−	0.956469i	$-0.594265\pi$
$743$		−	38742.2i		−	1.91294i	0.291833	−	0.956469i	$-0.405735\pi$
$744$		0			0
$745$		−	71742.5i		−	3.52811i
$746$		0			0
$747$	−6980.62			−0.341911
$748$		0			0
$749$	−10417.1	−	33717.2i	−0.508186	−	1.64486i
$750$		0			0
$751$		−	17887.8i		−	0.869153i	−0.900635	−	0.434577i	$-0.856898\pi$
$751$		−	17887.8i		−	0.869153i	0.900635	−	0.434577i	$-0.143102\pi$
$752$		0			0
$753$	10158.4			0.491623
$754$		0			0
$755$	62458.2			3.01071
$756$		0			0
$757$	34118.1			1.63810			0.819052	−	0.573720i	$-0.194500\pi$
$757$	34118.1			1.63810			0.819052	+	0.573720i	$0.194500\pi$
$758$		0			0
$759$	−5345.08			−0.255618
$760$		0			0
$761$		−	4807.40i		−	0.228999i	−0.993423	−	0.114499i	$-0.963474\pi$
$761$		−	4807.40i		−	0.228999i	0.993423	−	0.114499i	$-0.0365264\pi$
$762$		0			0
$763$	2697.68	−	833.457i	0.127998	−	0.0395454i
$764$		0			0
$765$	−12927.7			−0.610983
$766$		0			0
$767$			67241.8i			3.16553i
$768$		0			0
$769$		−	25362.7i		−	1.18934i	−0.803970	−	0.594670i	$-0.797283\pi$
$769$		−	25362.7i		−	1.18934i	0.803970	−	0.594670i	$-0.202717\pi$
$770$		0			0
$771$		−	7000.03i		−	0.326978i
$772$		0			0
$773$		−	8758.34i		−	0.407523i	−0.979021	−	0.203762i	$-0.934683\pi$
$773$		−	8758.34i		−	0.407523i	0.979021	−	0.203762i	$-0.0653167\pi$
$774$		0			0
$775$	−66424.6			−3.07876
$776$		0			0
$777$	−17488.3	+	5403.08i	−0.807452	+	0.249465i
$778$		0			0
$779$			4867.72i			0.223882i
$780$		0			0
$781$	−7135.15			−0.326909
$782$		0			0
$783$	−3466.62			−0.158221
$784$		0			0
$785$	64349.3			2.92576
$786$		0			0
$787$	13007.0			0.589137			0.294569	−	0.955630i	$-0.404824\pi$
$787$	13007.0			0.589137			0.294569	+	0.955630i	$0.404824\pi$
$788$		0			0
$789$		−	17309.8i		−	0.781044i
$790$		0			0
$791$	−16422.3	+	5073.75i	−0.738194	+	0.228068i
$792$		0			0
$793$	−20267.2			−0.907580
$794$		0			0
$795$			4820.34i			0.215044i
$796$		0			0
$797$			27449.5i			1.21997i	0.792415	+	0.609983i	$0.208824\pi$
$797$			27449.5i			1.21997i	−0.792415	+	0.609983i	$0.791176\pi$
$798$		0			0
$799$		−	27559.6i		−	1.22026i
$800$		0			0
$801$			439.181i			0.0193729i
$802$		0			0
$803$	−10786.3			−0.474025
$804$		0			0
$805$	27664.1	−	8546.91i	1.21122	−	0.374210i
$806$		0			0
$807$		−	193.087i		−	0.00842254i
$808$		0			0
$809$	−11638.3			−0.505788			−0.252894	−	0.967494i	$-0.581382\pi$
$809$	−11638.3			−0.505788			−0.252894	+	0.967494i	$0.581382\pi$
$810$		0			0
$811$	−732.648			−0.0317222			−0.0158611	−	0.999874i	$-0.505049\pi$
$811$	−732.648			−0.0317222			−0.0158611	+	0.999874i	$0.505049\pi$
$812$		0			0
$813$	−18809.8			−0.811423
$814$		0			0
$815$	−1835.32			−0.0788816
$816$		0			0
$817$		−	6956.21i		−	0.297879i
$818$		0			0
$819$	−3671.09	−	11882.3i	−0.156628	−	0.506963i
$820$		0			0
$821$	−2380.21			−0.101181			−0.0505907	−	0.998719i	$-0.516110\pi$
$821$	−2380.21			−0.101181			−0.0505907	+	0.998719i	$0.516110\pi$
$822$		0			0
$823$		−	19256.2i		−	0.815588i	−0.913074	−	0.407794i	$-0.866298\pi$
$823$		−	19256.2i		−	0.815588i	0.913074	−	0.407794i	$-0.133702\pi$
$824$		0			0
$825$		−	22398.5i		−	0.945233i
$826$		0			0
$827$		−	8930.67i		−	0.375514i	−0.982216	−	0.187757i	$-0.939878\pi$
$827$		−	8930.67i		−	0.375514i	0.982216	−	0.187757i	$-0.0601217\pi$
$828$		0			0
$829$		−	28554.2i		−	1.19630i	−0.801386	−	0.598148i	$-0.795903\pi$
$829$		−	28554.2i		−	1.19630i	0.801386	−	0.598148i	$-0.204097\pi$
$830$		0			0
$831$	−17686.1			−0.738296
$832$		0			0
$833$	13278.6	+	19438.4i	0.552313	+	0.808524i
$834$		0			0
$835$			4176.21i			0.173082i
$836$		0			0
$837$	5729.43			0.236605
$838$		0			0
$839$	5568.93			0.229155			0.114577	−	0.993414i	$-0.463449\pi$
$839$	5568.93			0.229155			0.114577	+	0.993414i	$0.463449\pi$
$840$		0			0
$841$	−7904.13			−0.324086
$842$		0			0
$843$	−13367.4			−0.546144
$844$		0			0
$845$		−	70530.2i		−	2.87138i
$846$		0			0
$847$	−13485.4	+	4166.35i	−0.547063	+	0.169017i
$848$		0			0
$849$	4072.12			0.164611
$850$		0			0
$851$			24608.9i			0.991283i
$852$		0			0
$853$			26913.6i			1.08031i	0.841566	+	0.540155i	$0.181634\pi$
$853$			26913.6i			1.08031i	−0.841566	+	0.540155i	$0.818366\pi$
$854$		0			0
$855$			5021.44i			0.200853i
$856$		0			0
$857$			18275.6i			0.728452i	0.931311	+	0.364226i	$0.118666\pi$
$857$			18275.6i			0.728452i	−0.931311	+	0.364226i	$0.881334\pi$
$858$		0			0
$859$	−25266.9			−1.00360			−0.501802	−	0.864982i	$-0.667330\pi$
$859$	−25266.9			−1.00360			−0.501802	+	0.864982i	$0.667330\pi$
$860$		0			0
$861$	−2994.71	−	9693.07i	−0.118536	−	0.383669i
$862$		0			0
$863$		−	30621.0i		−	1.20782i	−0.797051	−	0.603912i	$-0.793608\pi$
$863$		−	30621.0i		−	1.20782i	0.797051	−	0.603912i	$-0.206392\pi$
$864$		0			0
$865$	35546.9			1.39726
$866$		0			0
$867$	607.818			0.0238092
$868$		0			0
$869$	8282.07			0.323303
$870$		0			0
$871$	37303.2			1.45117
$872$		0			0
$873$		−	9489.93i		−	0.367910i
$874$		0			0
$875$	21513.6	+	69633.6i	0.831190	+	2.69034i
$876$		0			0
$877$	26961.9			1.03813			0.519064	−	0.854735i	$-0.326280\pi$
$877$	26961.9			1.03813			0.519064	+	0.854735i	$0.326280\pi$
$878$		0			0
$879$		−	10802.3i		−	0.414510i
$880$		0			0
$881$			34254.9i			1.30996i	0.755645	+	0.654982i	$0.227324\pi$
$881$			34254.9i			1.30996i	−0.755645	+	0.654982i	$0.772676\pi$
$882$		0			0
$883$		−	35366.4i		−	1.34788i	−0.738788	−	0.673938i	$-0.764602\pi$
$883$		−	35366.4i		−	1.34788i	0.738788	−	0.673938i	$-0.235398\pi$
$884$		0			0
$885$			56585.0i			2.14925i
$886$		0			0
$887$	−38236.0			−1.44739			−0.723697	−	0.690118i	$-0.757559\pi$
$887$	−38236.0			−1.44739			−0.723697	+	0.690118i	$0.757559\pi$
$888$		0			0
$889$	6100.44	+	19745.5i	0.230149	+	0.744929i
$890$		0			0
$891$			1931.98i			0.0726417i
$892$		0			0
$893$	−10704.9			−0.401147
$894$		0			0
$895$	14853.2			0.554734
$896$		0			0
$897$	−16720.4			−0.622382
$898$		0			0
$899$	−27245.2			−1.01077
$900$		0			0
$901$			5269.08i			0.194826i
$902$		0			0
$903$	4279.59	+	13851.9i	0.157714	+	0.510478i
$904$		0			0
$905$	−89946.1			−3.30377
$906$		0			0
$907$		−	36703.5i		−	1.34368i	−0.740695	−	0.671842i	$-0.765503\pi$
$907$		−	36703.5i		−	1.34368i	0.740695	−	0.671842i	$-0.234497\pi$
$908$		0			0
$909$		−	5540.64i		−	0.202169i
$910$		0			0
$911$		−	9583.16i		−	0.348523i	−0.984699	−	0.174261i	$-0.944246\pi$
$911$		−	9583.16i		−	0.348523i	0.984699	−	0.174261i	$-0.0557538\pi$
$912$		0			0
$913$		−	18499.9i		−	0.670599i
$914$		0			0
$915$	−17055.2			−0.616205
$916$		0			0
$917$	−16576.6	+	5121.39i	−0.596954	+	0.184431i
$918$		0			0
$919$		−	32765.0i		−	1.17608i	−0.808832	−	0.588040i	$-0.799900\pi$
$919$		−	32765.0i		−	1.17608i	0.808832	−	0.588040i	$-0.200100\pi$
$920$		0			0
$921$	10032.1			0.358925
$922$		0			0
$923$	−22320.0			−0.795961
$924$		0			0
$925$	−103123.			−3.66560
$926$		0			0
$927$	5488.60			0.194465
$928$		0			0
$929$			25650.6i			0.905885i	0.891540	+	0.452943i	$0.149626\pi$
$929$			25650.6i			0.905885i	−0.891540	+	0.452943i	$0.850374\pi$
$930$		0			0
$931$	7550.36	−	5157.75i	0.265793	−	0.181566i
$932$		0			0
$933$	−1792.77			−0.0629075
$934$		0			0
$935$		−	34260.7i		−	1.19834i
$936$		0			0
$937$			14746.2i			0.514129i	0.966394	+	0.257064i	$0.0827553\pi$
$937$			14746.2i			0.514129i	−0.966394	+	0.257064i	$0.917245\pi$
$938$		0			0
$939$		−	20720.2i		−	0.720103i
$940$		0			0
$941$			22330.7i			0.773604i	0.922163	+	0.386802i	$0.126420\pi$
$941$			22330.7i			0.773604i	−0.922163	+	0.386802i	$0.873580\pi$
$942$		0			0
$943$	−13639.7			−0.471018
$944$		0			0
$945$	−3089.28	−	9999.17i	−0.106343	−	0.344204i
$946$		0			0
$947$		−	34804.7i		−	1.19430i	−0.802130	−	0.597149i	$-0.796300\pi$
$947$		−	34804.7i		−	1.19430i	0.802130	−	0.597149i	$-0.203700\pi$
$948$		0			0
$949$	−33741.6			−1.15416
$950$		0			0
$951$	20004.9			0.682127
$952$		0			0
$953$	10582.9			0.359719			0.179860	−	0.983692i	$-0.442436\pi$
$953$	10582.9			0.359719			0.179860	+	0.983692i	$0.442436\pi$
$954$		0			0
$955$	−72988.2			−2.47313
$956$		0			0
$957$		−	9187.17i		−	0.310323i
$958$		0			0
$959$	36176.7	−	11176.9i	1.21815	−	0.376352i
$960$		0			0
$961$	15238.4			0.511509
$962$		0			0
$963$			17149.2i			0.573858i
$964$		0			0
$965$		−	4405.60i		−	0.146965i
$966$		0			0
$967$		−	4491.20i		−	0.149356i	−0.997208	−	0.0746780i	$-0.976207\pi$
$967$		−	4491.20i		−	0.149356i	0.997208	−	0.0746780i	$-0.0237929\pi$
$968$		0			0
$969$			5488.91i			0.181970i
$970$		0			0
$971$	−28182.0			−0.931413			−0.465706	−	0.884939i	$-0.654200\pi$
$971$	−28182.0			−0.931413			−0.465706	+	0.884939i	$0.654200\pi$
$972$		0			0
$973$	−51566.7	+	15931.7i	−1.69902	+	0.524920i
$974$		0			0
$975$		−	70066.6i		−	2.30146i
$976$		0			0
$977$	−22874.1			−0.749036			−0.374518	−	0.927220i	$-0.622192\pi$
$977$	−22874.1			−0.749036			−0.374518	+	0.927220i	$0.622192\pi$
$978$		0			0
$979$	−1163.91			−0.0379966
$980$		0			0
$981$	−1372.09			−0.0446559
$982$		0			0
$983$	−671.036			−0.0217729			−0.0108864	−	0.999941i	$-0.503465\pi$
$983$	−671.036			−0.0217729			−0.0108864	+	0.999941i	$0.503465\pi$
$984$		0			0
$985$			46986.7i			1.51992i
$986$		0			0
$987$	21316.5	−	6585.82i	0.687449	−	0.212390i
$988$		0			0
$989$	19491.8			0.626698
$990$		0			0
$991$			11573.2i			0.370975i	0.982647	+	0.185487i	$0.0593864\pi$
$991$			11573.2i			0.370975i	−0.982647	+	0.185487i	$0.940614\pi$
$992$		0			0
$993$			3353.05i			0.107156i
$994$		0			0
$995$			20441.5i			0.651294i
$996$		0			0
$997$			33269.8i			1.05684i	0.848984	+	0.528419i	$0.177215\pi$
$997$			33269.8i			1.05684i	−0.848984	+	0.528419i	$0.822785\pi$
$998$		0			0
$999$	8894.89			0.281703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.f.895.8		8
4.3	odd	2		1344.4.b.e.895.8		8
7.6	odd	2		1344.4.b.e.895.1		8
8.3	odd	2		336.4.b.f.223.1	yes	8
8.5	even	2		336.4.b.e.223.1	✓	8
24.5	odd	2		1008.4.b.i.559.8		8
24.11	even	2		1008.4.b.k.559.8		8
28.27	even	2	inner	1344.4.b.f.895.1		8
56.13	odd	2		336.4.b.f.223.8	yes	8
56.27	even	2		336.4.b.e.223.8	yes	8
168.83	odd	2		1008.4.b.i.559.1		8
168.125	even	2		1008.4.b.k.559.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.b.e.223.1	✓	8	8.5	even	2
336.4.b.e.223.8	yes	8	56.27	even	2
336.4.b.f.223.1	yes	8	8.3	odd	2
336.4.b.f.223.8	yes	8	56.13	odd	2
1008.4.b.i.559.1		8	168.83	odd	2
1008.4.b.i.559.8		8	24.5	odd	2
1008.4.b.k.559.1		8	168.125	even	2
1008.4.b.k.559.8		8	24.11	even	2
1344.4.b.e.895.1		8	7.6	odd	2
1344.4.b.e.895.8		8	4.3	odd	2
1344.4.b.f.895.1		8	28.27	even	2	inner
1344.4.b.f.895.8		8	1.1	even	1	trivial

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +20.9291i q^{5} +(-17.6950 + 5.46693i) q^{7} +9.00000 q^{9} +23.8516i q^{11} +74.6121i q^{13} +62.7872i q^{15} +68.6323i q^{17} +26.6585 q^{19} +(-53.0850 + 16.4008i) q^{21} +74.6991i q^{23} -313.026 q^{25} +27.0000 q^{27} -128.393 q^{29} +212.201 q^{31} +71.5548i q^{33} +(-114.418 - 370.340i) q^{35} +329.440 q^{37} +223.836i q^{39} +182.595i q^{41} -260.938i q^{43} +188.362i q^{45} -401.555 q^{47} +(283.225 - 193.475i) q^{49} +205.897i q^{51} +76.7726 q^{53} -499.192 q^{55} +79.9755 q^{57} +901.219 q^{59} +271.635i q^{61} +(-159.255 + 49.2024i) q^{63} -1561.56 q^{65} -499.962i q^{67} +224.097i q^{69} +299.147i q^{71} +452.227i q^{73} -939.079 q^{75} +(-130.395 - 422.054i) q^{77} -347.233i q^{79} +81.0000 q^{81} -775.624 q^{83} -1436.41 q^{85} -385.180 q^{87} +48.7979i q^{89} +(-407.899 - 1320.26i) q^{91} +636.604 q^{93} +557.938i q^{95} -1054.44i q^{97} +214.665i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{3} - 4 q^{7} + 72 q^{9} + 56 q^{19} - 12 q^{21} - 656 q^{25} + 216 q^{27} - 240 q^{29} + 320 q^{31} - 600 q^{35} - 392 q^{37} - 816 q^{47} - 16 q^{49} - 288 q^{53} - 456 q^{55} + 168 q^{57}+ \cdots + 960 q^{93}+O(q^{100}) \) 8 * q + 24 * q^3 - 4 * q^7 + 72 * q^9 + 56 * q^19 - 12 * q^21 - 656 * q^25 + 216 * q^27 - 240 * q^29 + 320 * q^31 - 600 * q^35 - 392 * q^37 - 816 * q^47 - 16 * q^49 - 288 * q^53 - 456 * q^55 + 168 * q^57 + 1824 * q^59 - 36 * q^63 - 816 * q^65 - 1968 * q^75 + 2064 * q^77 + 648 * q^81 - 1680 * q^83 - 2568 * q^85 - 720 * q^87 + 864 * q^91 + 960 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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