LMFDB - Embedded newform 48.5.e.b.17.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,5,Mod(17,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("48.17");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	48.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$4.96175822802$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 6)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	48.17
Dual form			48.5.e.b.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.00000 + 8.48528i) q^{3} +16.9706i q^{5} -26.0000 q^{7} +(-63.0000 + 50.9117i) q^{9} +O(q^{10})$	\(q+(3.00000 + 8.48528i) q^{3} +16.9706i q^{5} -26.0000 q^{7} +(-63.0000 + 50.9117i) q^{9} +118.794i q^{11} +50.0000 q^{13} +(-144.000 + 50.9117i) q^{15} +203.647i q^{17} +358.000 q^{19} +(-78.0000 - 220.617i) q^{21} -373.352i q^{23} +337.000 q^{25} +(-621.000 - 381.838i) q^{27} -1442.50i q^{29} +742.000 q^{31} +(-1008.00 + 356.382i) q^{33} -441.235i q^{35} +1874.00 q^{37} +(150.000 + 424.264i) q^{39} +2409.82i q^{41} +262.000 q^{43} +(-864.000 - 1069.15i) q^{45} +1697.06i q^{47} -1725.00 q^{49} +(-1728.00 + 610.940i) q^{51} -458.205i q^{53} -2016.00 q^{55} +(1074.00 + 3037.73i) q^{57} +1815.85i q^{59} -1486.00 q^{61} +(1638.00 - 1323.70i) q^{63} +848.528i q^{65} +4486.00 q^{67} +(3168.00 - 1120.06i) q^{69} -3563.82i q^{71} +290.000 q^{73} +(1011.00 + 2859.54i) q^{75} -3088.64i q^{77} -9818.00 q^{79} +(1377.00 - 6414.87i) q^{81} -7110.67i q^{83} -3456.00 q^{85} +(12240.0 - 4327.49i) q^{87} -7840.40i q^{89} -1300.00 q^{91} +(2226.00 + 6296.08i) q^{93} +6075.46i q^{95} -478.000 q^{97} +(-6048.00 - 7484.02i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 52 q^{7} - 126 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 52 * q^7 - 126 * q^9	$ 2 q + 6 q^{3} - 52 q^{7} - 126 q^{9} + 100 q^{13} - 288 q^{15} + 716 q^{19} - 156 q^{21} + 674 q^{25} - 1242 q^{27} + 1484 q^{31} - 2016 q^{33} + 3748 q^{37} + 300 q^{39} + 524 q^{43} - 1728 q^{45} - 3450 q^{49} - 3456 q^{51} - 4032 q^{55} + 2148 q^{57} - 2972 q^{61} + 3276 q^{63} + 8972 q^{67} + 6336 q^{69} + 580 q^{73} + 2022 q^{75} - 19636 q^{79} + 2754 q^{81} - 6912 q^{85} + 24480 q^{87} - 2600 q^{91} + 4452 q^{93} - 956 q^{97} - 12096 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 52 * q^7 - 126 * q^9 + 100 * q^13 - 288 * q^15 + 716 * q^19 - 156 * q^21 + 674 * q^25 - 1242 * q^27 + 1484 * q^31 - 2016 * q^33 + 3748 * q^37 + 300 * q^39 + 524 * q^43 - 1728 * q^45 - 3450 * q^49 - 3456 * q^51 - 4032 * q^55 + 2148 * q^57 - 2972 * q^61 + 3276 * q^63 + 8972 * q^67 + 6336 * q^69 + 580 * q^73 + 2022 * q^75 - 19636 * q^79 + 2754 * q^81 - 6912 * q^85 + 24480 * q^87 - 2600 * q^91 + 4452 * q^93 - 956 * q^97 - 12096 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	3.00000	+	8.48528i	0.333333	+	0.942809i
$3$	3.00000	+	8.48528i	0.333333	+	0.942809i
$4$		0			0
$5$			16.9706i			0.678823i	0.940638	+	0.339411i	$0.110228\pi$
$5$			16.9706i			0.678823i	−0.940638	+	0.339411i	$0.889772\pi$
$6$		0			0
$7$	−26.0000			−0.530612			−0.265306	−	0.964164i	$-0.585473\pi$
$7$	−26.0000			−0.530612			−0.265306	+	0.964164i	$0.585473\pi$
$8$		0			0
$9$	−63.0000	+	50.9117i	−0.777778	+	0.628539i
$10$		0			0
$11$			118.794i			0.981768i	0.871225	+	0.490884i	$0.163326\pi$
$11$			118.794i			0.981768i	−0.871225	+	0.490884i	$0.836674\pi$
$12$		0			0
$13$	50.0000			0.295858			0.147929	−	0.988998i	$-0.452739\pi$
$13$	50.0000			0.295858			0.147929	+	0.988998i	$0.452739\pi$
$14$		0			0
$15$	−144.000	+	50.9117i	−0.640000	+	0.226274i
$16$		0			0
$17$			203.647i			0.704660i	0.935876	+	0.352330i	$0.114611\pi$
$17$			203.647i			0.704660i	−0.935876	+	0.352330i	$0.885389\pi$
$18$		0			0
$19$	358.000			0.991690			0.495845	−	0.868411i	$-0.334858\pi$
$19$	358.000			0.991690			0.495845	+	0.868411i	$0.334858\pi$
$20$		0			0
$21$	−78.0000	−	220.617i	−0.176871	−	0.500266i
$22$		0			0
$23$		−	373.352i		−	0.705770i	−0.935667	−	0.352885i	$-0.885201\pi$
$23$		−	373.352i		−	0.705770i	0.935667	−	0.352885i	$-0.114799\pi$
$24$		0			0
$25$	337.000			0.539200
$26$		0			0
$27$	−621.000	−	381.838i	−0.851852	−	0.523783i
$28$		0			0
$29$		−	1442.50i		−	1.71522i	−0.514303	−	0.857609i	$-0.671949\pi$
$29$		−	1442.50i		−	1.71522i	0.514303	−	0.857609i	$-0.328051\pi$
$30$		0			0
$31$	742.000			0.772112			0.386056	−	0.922475i	$-0.373837\pi$
$31$	742.000			0.772112			0.386056	+	0.922475i	$0.373837\pi$
$32$		0			0
$33$	−1008.00	+	356.382i	−0.925620	+	0.327256i
$34$		0			0
$35$		−	441.235i		−	0.360192i
$36$		0			0
$37$	1874.00			1.36888			0.684441	−	0.729068i	$-0.260046\pi$
$37$	1874.00			1.36888			0.684441	+	0.729068i	$0.260046\pi$
$38$		0			0
$39$	150.000	+	424.264i	0.0986193	+	0.278938i
$40$		0			0
$41$			2409.82i			1.43356i	0.697298	+	0.716782i	$0.254386\pi$
$41$			2409.82i			1.43356i	−0.697298	+	0.716782i	$0.745614\pi$
$42$		0			0
$43$	262.000			0.141698			0.0708491	−	0.997487i	$-0.477429\pi$
$43$	262.000			0.141698			0.0708491	+	0.997487i	$0.477429\pi$
$44$		0			0
$45$	−864.000	−	1069.15i	−0.426667	−	0.527973i
$46$		0			0
$47$			1697.06i			0.768246i	0.923282	+	0.384123i	$0.125496\pi$
$47$			1697.06i			0.768246i	−0.923282	+	0.384123i	$0.874504\pi$
$48$		0			0
$49$	−1725.00			−0.718451
$50$		0			0
$51$	−1728.00	+	610.940i	−0.664360	+	0.234887i
$52$		0			0
$53$		−	458.205i		−	0.163120i	−0.996668	−	0.0815602i	$-0.974010\pi$
$53$		−	458.205i		−	0.163120i	0.996668	−	0.0815602i	$-0.0259903\pi$
$54$		0			0
$55$	−2016.00			−0.666446
$56$		0			0
$57$	1074.00	+	3037.73i	0.330563	+	0.934974i
$58$		0			0
$59$			1815.85i			0.521646i	0.965387	+	0.260823i	$0.0839939\pi$
$59$			1815.85i			0.521646i	−0.965387	+	0.260823i	$0.916006\pi$
$60$		0			0
$61$	−1486.00			−0.399355			−0.199678	−	0.979862i	$-0.563989\pi$
$61$	−1486.00			−0.399355			−0.199678	+	0.979862i	$0.563989\pi$
$62$		0			0
$63$	1638.00	−	1323.70i	0.412698	−	0.333511i
$64$		0			0
$65$			848.528i			0.200835i
$66$		0			0
$67$	4486.00			0.999332			0.499666	−	0.866218i	$-0.333456\pi$
$67$	4486.00			0.999332			0.499666	+	0.866218i	$0.333456\pi$
$68$		0			0
$69$	3168.00	−	1120.06i	0.665406	−	0.235257i
$70$		0			0
$71$		−	3563.82i		−	0.706967i	−0.935441	−	0.353483i	$-0.884997\pi$
$71$		−	3563.82i		−	0.706967i	0.935441	−	0.353483i	$-0.115003\pi$
$72$		0			0
$73$	290.000			0.0544192			0.0272096	−	0.999630i	$-0.491338\pi$
$73$	290.000			0.0544192			0.0272096	+	0.999630i	$0.491338\pi$
$74$		0			0
$75$	1011.00	+	2859.54i	0.179733	+	0.508363i
$76$		0			0
$77$		−	3088.64i		−	0.520938i
$78$		0			0
$79$	−9818.00			−1.57315			−0.786573	−	0.617498i	$-0.788146\pi$
$79$	−9818.00			−1.57315			−0.786573	+	0.617498i	$0.788146\pi$
$80$		0			0
$81$	1377.00	−	6414.87i	0.209877	−	0.977728i
$82$		0			0
$83$		−	7110.67i		−	1.03218i	−0.856535	−	0.516088i	$-0.827388\pi$
$83$		−	7110.67i		−	1.03218i	0.856535	−	0.516088i	$-0.172612\pi$
$84$		0			0
$85$	−3456.00			−0.478339
$86$		0			0
$87$	12240.0	−	4327.49i	1.61712	−	0.571739i
$88$		0			0
$89$		−	7840.40i		−	0.989825i	−0.868943	−	0.494912i	$-0.835200\pi$
$89$		−	7840.40i		−	0.989825i	0.868943	−	0.494912i	$-0.164800\pi$
$90$		0			0
$91$	−1300.00			−0.156986
$92$		0			0
$93$	2226.00	+	6296.08i	0.257371	+	0.727955i
$94$		0			0
$95$			6075.46i			0.673181i
$96$		0			0
$97$	−478.000			−0.0508024			−0.0254012	−	0.999677i	$-0.508086\pi$
$97$	−478.000			−0.0508024			−0.0254012	+	0.999677i	$0.508086\pi$
$98$		0			0
$99$	−6048.00	−	7484.02i	−0.617080	−	0.763597i
$100$		0			0
$101$			13729.2i			1.34587i	0.739703	+	0.672933i	$0.234966\pi$
$101$			13729.2i			1.34587i	−0.739703	+	0.672933i	$0.765034\pi$
$102$		0			0
$103$	−2138.00			−0.201527			−0.100764	−	0.994910i	$-0.532129\pi$
$103$	−2138.00			−0.201527			−0.100764	+	0.994910i	$0.532129\pi$
$104$		0			0
$105$	3744.00	−	1323.70i	0.339592	−	0.120064i
$106$		0			0
$107$			10844.2i			0.947174i	0.880747	+	0.473587i	$0.157041\pi$
$107$			10844.2i			0.947174i	−0.880747	+	0.473587i	$0.842959\pi$
$108$		0			0
$109$	−4750.00			−0.399798			−0.199899	−	0.979817i	$-0.564061\pi$
$109$	−4750.00			−0.399798			−0.199899	+	0.979817i	$0.564061\pi$
$110$		0			0
$111$	5622.00	+	15901.4i	0.456294	+	1.29059i
$112$		0			0
$113$		−	3190.47i		−	0.249860i	−0.992166	−	0.124930i	$-0.960129\pi$
$113$		−	3190.47i		−	0.249860i	0.992166	−	0.124930i	$-0.0398707\pi$
$114$		0			0
$115$	6336.00			0.479093
$116$		0			0
$117$	−3150.00	+	2545.58i	−0.230112	+	0.185958i
$118$		0			0
$119$		−	5294.82i		−	0.373901i
$120$		0			0
$121$	529.000			0.0361314
$122$		0			0
$123$	−20448.0	+	7229.46i	−1.35158	+	0.477854i
$124$		0			0
$125$			16325.7i			1.04484i
$126$		0			0
$127$	−8282.00			−0.513485			−0.256743	−	0.966480i	$-0.582649\pi$
$127$	−8282.00			−0.513485			−0.256743	+	0.966480i	$0.582649\pi$
$128$		0			0
$129$	786.000	+	2223.14i	0.0472327	+	0.133594i
$130$		0			0
$131$		−	5413.61i		−	0.315460i	−0.987482	−	0.157730i	$-0.949582\pi$
$131$		−	5413.61i		−	0.315460i	0.987482	−	0.157730i	$-0.0504176\pi$
$132$		0			0
$133$	−9308.00			−0.526203
$134$		0			0
$135$	6480.00	−	10538.7i	0.355556	−	0.578256i
$136$		0			0
$137$		−	18090.6i		−	0.963856i	−0.876211	−	0.481928i	$-0.839937\pi$
$137$		−	18090.6i		−	0.963856i	0.876211	−	0.481928i	$-0.160063\pi$
$138$		0			0
$139$	23206.0			1.20108			0.600538	−	0.799596i	$-0.294953\pi$
$139$	23206.0			1.20108			0.600538	+	0.799596i	$0.294953\pi$
$140$		0			0
$141$	−14400.0	+	5091.17i	−0.724310	+	0.256082i
$142$		0			0
$143$			5939.70i			0.290464i
$144$		0			0
$145$	24480.0			1.16433
$146$		0			0
$147$	−5175.00	−	14637.1i	−0.239484	−	0.677362i
$148$		0			0
$149$		−	11183.6i		−	0.503743i	−0.967761	−	0.251872i	$-0.918954\pi$
$149$		−	11183.6i		−	0.503743i	0.967761	−	0.251872i	$-0.0810460\pi$
$150$		0			0
$151$	−14426.0			−0.632692			−0.316346	−	0.948644i	$-0.602456\pi$
$151$	−14426.0			−0.632692			−0.316346	+	0.948644i	$0.602456\pi$
$152$		0			0
$153$	−10368.0	−	12829.7i	−0.442907	−	0.548069i
$154$		0			0
$155$			12592.2i			0.524127i
$156$		0			0
$157$	49010.0			1.98832			0.994158	−	0.107935i	$-0.0344238\pi$
$157$	49010.0			1.98832			0.994158	+	0.107935i	$0.0344238\pi$
$158$		0			0
$159$	3888.00	−	1374.62i	0.153791	−	0.0543735i
$160$		0			0
$161$			9707.16i			0.374490i
$162$		0			0
$163$	42982.0			1.61775			0.808875	−	0.587981i	$-0.200077\pi$
$163$	42982.0			1.61775			0.808875	+	0.587981i	$0.200077\pi$
$164$		0			0
$165$	−6048.00	−	17106.3i	−0.222149	−	0.628332i
$166$		0			0
$167$		−	44157.4i		−	1.58333i	−0.610957	−	0.791663i	$-0.709215\pi$
$167$		−	44157.4i		−	1.58333i	0.610957	−	0.791663i	$-0.290785\pi$
$168$		0			0
$169$	−26061.0			−0.912468
$170$		0			0
$171$	−22554.0	+	18226.4i	−0.771314	+	0.623316i
$172$		0			0
$173$		−	22418.1i		−	0.749043i	−0.927218	−	0.374522i	$-0.877807\pi$
$173$		−	22418.1i		−	0.749043i	0.927218	−	0.374522i	$-0.122193\pi$
$174$		0			0
$175$	−8762.00			−0.286106
$176$		0			0
$177$	−15408.0	+	5447.55i	−0.491813	+	0.173882i
$178$		0			0
$179$		−	30597.9i		−	0.954962i	−0.878642	−	0.477481i	$-0.841550\pi$
$179$		−	30597.9i		−	0.954962i	0.878642	−	0.477481i	$-0.158450\pi$
$180$		0			0
$181$	−13102.0			−0.399927			−0.199963	−	0.979803i	$-0.564082\pi$
$181$	−13102.0			−0.399927			−0.199963	+	0.979803i	$0.564082\pi$
$182$		0			0
$183$	−4458.00	−	12609.1i	−0.133118	−	0.376516i
$184$		0			0
$185$			31802.8i			0.929228i
$186$		0			0
$187$	−24192.0			−0.691813
$188$		0			0
$189$	16146.0	+	9927.78i	0.452003	+	0.277926i
$190$		0			0
$191$			70326.0i			1.92774i	0.266367	+	0.963872i	$0.414177\pi$
$191$			70326.0i			1.92774i	−0.266367	+	0.963872i	$0.585823\pi$
$192$		0			0
$193$	18050.0			0.484577			0.242288	−	0.970204i	$-0.422102\pi$
$193$	18050.0			0.484577			0.242288	+	0.970204i	$0.422102\pi$
$194$		0			0
$195$	−7200.00	+	2545.58i	−0.189349	+	0.0669450i
$196$		0			0
$197$		−	26321.3i		−	0.678228i	−0.940745	−	0.339114i	$-0.889873\pi$
$197$		−	26321.3i		−	0.678228i	0.940745	−	0.339114i	$-0.110127\pi$
$198$		0			0
$199$	37222.0			0.939926			0.469963	−	0.882686i	$-0.344267\pi$
$199$	37222.0			0.939926			0.469963	+	0.882686i	$0.344267\pi$
$200$		0			0
$201$	13458.0	+	38065.0i	0.333111	+	0.942179i
$202$		0			0
$203$			37504.9i			0.910115i
$204$		0			0
$205$	−40896.0			−0.973135
$206$		0			0
$207$	19008.0	+	23521.2i	0.443604	+	0.548932i
$208$		0			0
$209$			42528.2i			0.973609i
$210$		0			0
$211$	−15098.0			−0.339121			−0.169560	−	0.985520i	$-0.554235\pi$
$211$	−15098.0			−0.339121			−0.169560	+	0.985520i	$0.554235\pi$
$212$		0			0
$213$	30240.0	−	10691.5i	0.666534	−	0.235656i
$214$		0			0
$215$			4446.29i			0.0961879i
$216$		0			0
$217$	−19292.0			−0.409692
$218$		0			0
$219$	870.000	+	2460.73i	0.0181397	+	0.0513069i
$220$		0			0
$221$			10182.3i			0.208479i
$222$		0			0
$223$	−58778.0			−1.18197			−0.590983	−	0.806684i	$-0.701260\pi$
$223$	−58778.0			−1.18197			−0.590983	+	0.806684i	$0.701260\pi$
$224$		0			0
$225$	−21231.0	+	17157.2i	−0.419378	+	0.338908i
$226$		0			0
$227$		−	68001.0i		−	1.31967i	−0.751412	−	0.659833i	$-0.770627\pi$
$227$		−	68001.0i		−	1.31967i	0.751412	−	0.659833i	$-0.229373\pi$
$228$		0			0
$229$	28562.0			0.544650			0.272325	−	0.962205i	$-0.412207\pi$
$229$	28562.0			0.544650			0.272325	+	0.962205i	$0.412207\pi$
$230$		0			0
$231$	26208.0	−	9265.93i	0.491145	−	0.173646i
$232$		0			0
$233$			22503.0i			0.414503i	0.978288	+	0.207252i	$0.0664519\pi$
$233$			22503.0i			0.414503i	−0.978288	+	0.207252i	$0.933548\pi$
$234$		0			0
$235$	−28800.0			−0.521503
$236$		0			0
$237$	−29454.0	−	83308.5i	−0.524382	−	1.48318i
$238$		0			0
$239$		−	1289.76i		−	0.0225795i	−0.999936	−	0.0112897i	$-0.996406\pi$
$239$		−	1289.76i		−	0.0225795i	0.999936	−	0.0112897i	$-0.00359371\pi$
$240$		0			0
$241$	−61246.0			−1.05449			−0.527246	−	0.849712i	$-0.676776\pi$
$241$	−61246.0			−1.05449			−0.527246	+	0.849712i	$0.676776\pi$
$242$		0			0
$243$	58563.0	−	7560.39i	0.991770	−	0.128036i
$244$		0			0
$245$		−	29274.2i		−	0.487700i
$246$		0			0
$247$	17900.0			0.293399
$248$		0			0
$249$	60336.0	−	21332.0i	0.973146	−	0.344059i
$250$		0			0
$251$			45260.5i			0.718409i	0.933259	+	0.359205i	$0.116952\pi$
$251$			45260.5i			0.718409i	−0.933259	+	0.359205i	$0.883048\pi$
$252$		0			0
$253$	44352.0			0.692903
$254$		0			0
$255$	−10368.0	−	29325.1i	−0.159446	−	0.450982i
$256$		0			0
$257$		−	85260.1i		−	1.29086i	−0.763819	−	0.645431i	$-0.776678\pi$
$257$		−	85260.1i		−	1.29086i	0.763819	−	0.645431i	$-0.223322\pi$
$258$		0			0
$259$	−48724.0			−0.726346
$260$		0			0
$261$	73440.0	+	90877.4i	1.07808	+	1.33406i
$262$		0			0
$263$		−	84751.0i		−	1.22527i	−0.790364	−	0.612637i	$-0.790109\pi$
$263$		−	84751.0i		−	1.22527i	0.790364	−	0.612637i	$-0.209891\pi$
$264$		0			0
$265$	7776.00			0.110730
$266$		0			0
$267$	66528.0	−	23521.2i	0.933216	−	0.329942i
$268$		0			0
$269$		−	42918.6i		−	0.593117i	−0.955015	−	0.296559i	$-0.904161\pi$
$269$		−	42918.6i		−	0.593117i	0.955015	−	0.296559i	$-0.0958390\pi$
$270$		0			0
$271$	27430.0			0.373497			0.186749	−	0.982408i	$-0.440205\pi$
$271$	27430.0			0.373497			0.186749	+	0.982408i	$0.440205\pi$
$272$		0			0
$273$	−3900.00	−	11030.9i	−0.0523286	−	0.148008i
$274$		0			0
$275$			40033.6i			0.529369i
$276$		0			0
$277$	−93934.0			−1.22423			−0.612115	−	0.790768i	$-0.709681\pi$
$277$	−93934.0			−1.22423			−0.612115	+	0.790768i	$0.709681\pi$
$278$		0			0
$279$	−46746.0	+	37776.5i	−0.600532	+	0.485303i
$280$		0			0
$281$			24471.6i			0.309919i	0.987921	+	0.154960i	$0.0495248\pi$
$281$			24471.6i			0.309919i	−0.987921	+	0.154960i	$0.950475\pi$
$282$		0			0
$283$	65830.0			0.821961			0.410980	−	0.911644i	$-0.365187\pi$
$283$	65830.0			0.821961			0.410980	+	0.911644i	$0.365187\pi$
$284$		0			0
$285$	−51552.0	+	18226.4i	−0.634681	+	0.224394i
$286$		0			0
$287$		−	62655.3i		−	0.760666i
$288$		0			0
$289$	42049.0			0.503454
$290$		0			0
$291$	−1434.00	−	4055.96i	−0.0169341	−	0.0478970i
$292$		0			0
$293$			90028.8i			1.04869i	0.851506	+	0.524344i	$0.175689\pi$
$293$			90028.8i			1.04869i	−0.851506	+	0.524344i	$0.824311\pi$
$294$		0			0
$295$	−30816.0			−0.354105
$296$		0			0
$297$	45360.0	−	73771.0i	0.514233	−	0.836321i
$298$		0			0
$299$		−	18667.6i		−	0.208808i
$300$		0			0
$301$	−6812.00			−0.0751868
$302$		0			0
$303$	−116496.	+	41187.6i	−1.26890	+	0.448622i
$304$		0			0
$305$		−	25218.3i		−	0.271091i
$306$		0			0
$307$	−67322.0			−0.714299			−0.357150	−	0.934047i	$-0.616251\pi$
$307$	−67322.0			−0.714299			−0.357150	+	0.934047i	$0.616251\pi$
$308$		0			0
$309$	−6414.00	−	18141.5i	−0.0671757	−	0.190001i
$310$		0			0
$311$			131997.i			1.36472i	0.731017	+	0.682360i	$0.239046\pi$
$311$			131997.i			1.36472i	−0.731017	+	0.682360i	$0.760954\pi$
$312$		0			0
$313$	−22078.0			−0.225357			−0.112679	−	0.993631i	$-0.535943\pi$
$313$	−22078.0			−0.225357			−0.112679	+	0.993631i	$0.535943\pi$
$314$		0			0
$315$	22464.0	+	27797.8i	0.226395	+	0.280149i
$316$		0			0
$317$			117894.i			1.17321i	0.809874	+	0.586604i	$0.199535\pi$
$317$			117894.i			1.17321i	−0.809874	+	0.586604i	$0.800465\pi$
$318$		0			0
$319$	171360.			1.68395
$320$		0			0
$321$	−92016.0	+	32532.6i	−0.893004	+	0.315725i
$322$		0			0
$323$			72905.5i			0.698804i
$324$		0			0
$325$	16850.0			0.159527
$326$		0			0
$327$	−14250.0	−	40305.1i	−0.133266	−	0.376933i
$328$		0			0
$329$		−	44123.5i		−	0.407641i
$330$		0			0
$331$	−167642.			−1.53012			−0.765062	−	0.643956i	$-0.777292\pi$
$331$	−167642.			−1.53012			−0.765062	+	0.643956i	$0.777292\pi$
$332$		0			0
$333$	−118062.	+	95408.5i	−1.06469	+	0.860396i
$334$		0			0
$335$			76129.9i			0.678369i
$336$		0			0
$337$	162914.			1.43449			0.717247	−	0.696819i	$-0.245402\pi$
$337$	162914.			1.43449			0.717247	+	0.696819i	$0.245402\pi$
$338$		0			0
$339$	27072.0	−	9571.40i	0.235571	−	0.0832868i
$340$		0			0
$341$			88145.1i			0.758035i
$342$		0			0
$343$	107276.			0.911831
$344$		0			0
$345$	19008.0	+	53762.7i	0.159698	+	0.451693i
$346$		0			0
$347$		−	132184.i		−	1.09779i	−0.835892	−	0.548895i	$-0.815049\pi$
$347$		−	132184.i		−	1.09779i	0.835892	−	0.548895i	$-0.184951\pi$
$348$		0			0
$349$	53234.0			0.437057			0.218529	−	0.975831i	$-0.429874\pi$
$349$	53234.0			0.437057			0.218529	+	0.975831i	$0.429874\pi$
$350$		0			0
$351$	−31050.0	−	19091.9i	−0.252027	−	0.154965i
$352$		0			0
$353$		−	144861.i		−	1.16252i	−0.813717	−	0.581261i	$-0.802560\pi$
$353$		−	144861.i		−	1.16252i	0.813717	−	0.581261i	$-0.197440\pi$
$354$		0			0
$355$	60480.0			0.479905
$356$		0			0
$357$	44928.0	−	15884.4i	0.352517	−	0.124634i
$358$		0			0
$359$		−	12931.6i		−	0.100337i	−0.998741	−	0.0501686i	$-0.984024\pi$
$359$		−	12931.6i		−	0.100337i	0.998741	−	0.0501686i	$-0.0159759\pi$
$360$		0			0
$361$	−2157.00			−0.0165514
$362$		0			0
$363$	1587.00	+	4488.71i	0.0120438	+	0.0340650i
$364$		0			0
$365$			4921.46i			0.0369410i
$366$		0			0
$367$	44326.0			0.329099			0.164549	−	0.986369i	$-0.447383\pi$
$367$	44326.0			0.329099			0.164549	+	0.986369i	$0.447383\pi$
$368$		0			0
$369$	−122688.	−	151819.i	−0.901051	−	1.11499i
$370$		0			0
$371$			11913.3i			0.0865537i
$372$		0			0
$373$	−60718.0			−0.436415			−0.218208	−	0.975902i	$-0.570021\pi$
$373$	−60718.0			−0.436415			−0.218208	+	0.975902i	$0.570021\pi$
$374$		0			0
$375$	−138528.	+	48977.0i	−0.985088	+	0.348281i
$376$		0			0
$377$		−	72124.9i		−	0.507461i
$378$		0			0
$379$	−30458.0			−0.212043			−0.106021	−	0.994364i	$-0.533811\pi$
$379$	−30458.0			−0.212043			−0.106021	+	0.994364i	$0.533811\pi$
$380$		0			0
$381$	−24846.0	−	70275.1i	−0.171162	−	0.484118i
$382$		0			0
$383$		−	235687.i		−	1.60671i	−0.595498	−	0.803357i	$-0.703045\pi$
$383$		−	235687.i		−	1.60671i	0.595498	−	0.803357i	$-0.296955\pi$
$384$		0			0
$385$	52416.0			0.353625
$386$		0			0
$387$	−16506.0	+	13338.9i	−0.110210	+	0.0890629i
$388$		0			0
$389$		−	150410.i		−	0.993980i	−0.867756	−	0.496990i	$-0.834439\pi$
$389$		−	150410.i		−	0.993980i	0.867756	−	0.496990i	$-0.165561\pi$
$390$		0			0
$391$	76032.0			0.497328
$392$		0			0
$393$	45936.0	−	16240.8i	0.297419	−	0.105153i
$394$		0			0
$395$		−	166617.i		−	1.06789i
$396$		0			0
$397$	172658.			1.09548			0.547742	−	0.836648i	$-0.315488\pi$
$397$	172658.			1.09548			0.547742	+	0.836648i	$0.315488\pi$
$398$		0			0
$399$	−27924.0	−	78981.0i	−0.175401	−	0.496109i
$400$		0			0
$401$			167466.i			1.04145i	0.853726	+	0.520723i	$0.174338\pi$
$401$			167466.i			1.04145i	−0.853726	+	0.520723i	$0.825662\pi$
$402$		0			0
$403$	37100.0			0.228436
$404$		0			0
$405$	108864.	+	23368.5i	0.663704	+	0.142469i
$406$		0			0
$407$			222620.i			1.34393i
$408$		0			0
$409$	−150430.			−0.899265			−0.449633	−	0.893214i	$-0.648445\pi$
$409$	−150430.			−0.899265			−0.449633	+	0.893214i	$0.648445\pi$
$410$		0			0
$411$	153504.	−	54271.9i	0.908732	−	0.321285i
$412$		0			0
$413$		−	47212.1i		−	0.276792i
$414$		0			0
$415$	120672.			0.700665
$416$		0			0
$417$	69618.0	+	196909.i	0.400359	+	1.13239i
$418$		0			0
$419$			178276.i			1.01546i	0.861515	+	0.507732i	$0.169516\pi$
$419$			178276.i			1.01546i	−0.861515	+	0.507732i	$0.830484\pi$
$420$		0			0
$421$	−216046.			−1.21894			−0.609470	−	0.792809i	$-0.708618\pi$
$421$	−216046.			−1.21894			−0.609470	+	0.792809i	$0.708618\pi$
$422$		0			0
$423$	−86400.0	−	106915.i	−0.482873	−	0.597525i
$424$		0			0
$425$			68629.0i			0.379953i
$426$		0			0
$427$	38636.0			0.211903
$428$		0			0
$429$	−50400.0	+	17819.1i	−0.273852	+	0.0968213i
$430$		0			0
$431$			5498.46i			0.0295997i	0.999890	+	0.0147998i	$0.00471110\pi$
$431$			5498.46i			0.0295997i	−0.999890	+	0.0147998i	$0.995289\pi$
$432$		0			0
$433$	108002.			0.576044			0.288022	−	0.957624i	$-0.407002\pi$
$433$	108002.			0.576044			0.288022	+	0.957624i	$0.407002\pi$
$434$		0			0
$435$	73440.0	+	207720.i	0.388109	+	1.09774i
$436$		0			0
$437$		−	133660.i		−	0.699905i
$438$		0			0
$439$	−357722.			−1.85617			−0.928083	−	0.372374i	$-0.878544\pi$
$439$	−357722.			−1.85617			−0.928083	+	0.372374i	$0.878544\pi$
$440$		0			0
$441$	108675.	−	87822.7i	0.558795	−	0.451575i
$442$		0			0
$443$			86261.4i			0.439551i	0.975551	+	0.219775i	$0.0705324\pi$
$443$			86261.4i			0.439551i	−0.975551	+	0.219775i	$0.929468\pi$
$444$		0			0
$445$	133056.			0.671915
$446$		0			0
$447$	94896.0	−	33550.8i	0.474934	−	0.167914i
$448$		0			0
$449$			301397.i			1.49502i	0.664251	+	0.747509i	$0.268750\pi$
$449$			301397.i			1.49502i	−0.664251	+	0.747509i	$0.731250\pi$
$450$		0			0
$451$	−286272.			−1.40743
$452$		0			0
$453$	−43278.0	−	122409.i	−0.210897	−	0.596507i
$454$		0			0
$455$		−	22061.7i		−	0.106566i
$456$		0			0
$457$	−399070.			−1.91081			−0.955403	−	0.295305i	$-0.904579\pi$
$457$	−399070.			−1.91081			−0.955403	+	0.295305i	$0.904579\pi$
$458$		0			0
$459$	77760.0	−	126465.i	0.369089	−	0.600266i
$460$		0			0
$461$			38268.6i			0.180070i	0.995939	+	0.0900349i	$0.0286979\pi$
$461$			38268.6i			0.180070i	−0.995939	+	0.0900349i	$0.971302\pi$
$462$		0			0
$463$	−144410.			−0.673652			−0.336826	−	0.941567i	$-0.609353\pi$
$463$	−144410.			−0.673652			−0.336826	+	0.941567i	$0.609353\pi$
$464$		0			0
$465$	−106848.	+	37776.5i	−0.494152	+	0.174709i
$466$		0			0
$467$			148204.i			0.679557i	0.940505	+	0.339779i	$0.110352\pi$
$467$			148204.i			0.679557i	−0.940505	+	0.339779i	$0.889648\pi$
$468$		0			0
$469$	−116636.			−0.530258
$470$		0			0
$471$	147030.	+	415864.i	0.662772	+	1.87460i
$472$		0			0
$473$			31124.0i			0.139115i
$474$		0			0
$475$	120646.			0.534719
$476$		0			0
$477$	23328.0	+	28866.9i	0.102528	+	0.126871i
$478$		0			0
$479$		−	305606.i		−	1.33196i	−0.745970	−	0.665979i	$-0.768014\pi$
$479$		−	305606.i		−	1.33196i	0.745970	−	0.665979i	$-0.231986\pi$
$480$		0			0
$481$	93700.0			0.404995
$482$		0			0
$483$	−82368.0	+	29121.5i	−0.353073	+	0.124830i
$484$		0			0
$485$		−	8111.93i		−	0.0344858i
$486$		0			0
$487$	196774.			0.829678			0.414839	−	0.909895i	$-0.363838\pi$
$487$	196774.			0.829678			0.414839	+	0.909895i	$0.363838\pi$
$488$		0			0
$489$	128946.	+	364714.i	0.539250	+	1.52523i
$490$		0			0
$491$		−	166193.i		−	0.689365i	−0.938719	−	0.344682i	$-0.887987\pi$
$491$		−	166193.i		−	0.689365i	0.938719	−	0.344682i	$-0.112013\pi$
$492$		0			0
$493$	293760.			1.20865
$494$		0			0
$495$	127008.	−	102638.i	0.518347	−	0.418888i
$496$		0			0
$497$			92659.3i			0.375125i
$498$		0			0
$499$	−189050.			−0.759234			−0.379617	−	0.925144i	$-0.623944\pi$
$499$	−189050.			−0.759234			−0.379617	+	0.925144i	$0.623944\pi$
$500$		0			0
$501$	374688.	−	132472.i	1.49277	−	0.527776i
$502$		0			0
$503$			344061.i			1.35988i	0.733269	+	0.679939i	$0.237994\pi$
$503$			344061.i			1.35988i	−0.733269	+	0.679939i	$0.762006\pi$
$504$		0			0
$505$	−232992.			−0.913605
$506$		0			0
$507$	−78183.0	−	221135.i	−0.304156	−	0.860283i
$508$		0			0
$509$		−	353208.i		−	1.36331i	−0.731673	−	0.681656i	$-0.761260\pi$
$509$		−	353208.i		−	1.36331i	0.731673	−	0.681656i	$-0.238740\pi$
$510$		0			0
$511$	−7540.00			−0.0288755
$512$		0			0
$513$	−222318.	−	136698.i	−0.844773	−	0.519430i
$514$		0			0
$515$		−	36283.1i		−	0.136801i
$516$		0			0
$517$	−201600.			−0.754240
$518$		0			0
$519$	190224.	−	67254.3i	0.706205	−	0.249681i
$520$		0			0
$521$		−	276043.i		−	1.01695i	−0.861075	−	0.508477i	$-0.830209\pi$
$521$		−	276043.i		−	1.01695i	0.861075	−	0.508477i	$-0.169791\pi$
$522$		0			0
$523$	146950.			0.537237			0.268619	−	0.963247i	$-0.413433\pi$
$523$	146950.			0.537237			0.268619	+	0.963247i	$0.413433\pi$
$524$		0			0
$525$	−26286.0	−	74348.0i	−0.0953687	−	0.269743i
$526$		0			0
$527$			151106.i			0.544077i
$528$		0			0
$529$	140449.			0.501889
$530$		0			0
$531$	−92448.0	−	114399.i	−0.327875	−	0.405725i
$532$		0			0
$533$			120491.i			0.424131i
$534$		0			0
$535$	−184032.			−0.642963
$536$		0			0
$537$	259632.	−	91793.8i	0.900346	−	0.318321i
$538$		0			0
$539$		−	204920.i		−	0.705352i
$540$		0			0
$541$	−244942.			−0.836891			−0.418445	−	0.908242i	$-0.637425\pi$
$541$	−244942.			−0.836891			−0.418445	+	0.908242i	$0.637425\pi$
$542$		0			0
$543$	−39306.0	−	111174.i	−0.133309	−	0.377055i
$544$		0			0
$545$		−	80610.2i		−	0.271392i
$546$		0			0
$547$	283366.			0.947050			0.473525	−	0.880780i	$-0.342981\pi$
$547$	283366.			0.947050			0.473525	+	0.880780i	$0.342981\pi$
$548$		0			0
$549$	93618.0	−	75654.8i	0.310609	−	0.251010i
$550$		0			0
$551$		−	516414.i		−	1.70096i
$552$		0			0
$553$	255268.			0.834730
$554$		0			0
$555$	−269856.	+	95408.5i	−0.876085	+	0.309743i
$556$		0			0
$557$			47093.3i			0.151792i	0.997116	+	0.0758960i	$0.0241817\pi$
$557$			47093.3i			0.151792i	−0.997116	+	0.0758960i	$0.975818\pi$
$558$		0			0
$559$	13100.0			0.0419225
$560$		0			0
$561$	−72576.0	−	205276.i	−0.230604	−	0.652247i
$562$		0			0
$563$			84.8528i			0.000267701i	1.00000		0.000133850i	$4.26059e-5\pi$
$563$			84.8528i			0.000267701i	−1.00000		0.000133850i	$0.999957\pi$
$564$		0			0
$565$	54144.0			0.169611
$566$		0			0
$567$	−35802.0	+	166787.i	−0.111363	+	0.518794i
$568$		0			0
$569$			239115.i			0.738555i	0.929319	+	0.369277i	$0.120395\pi$
$569$			239115.i			0.738555i	−0.929319	+	0.369277i	$0.879605\pi$
$570$		0			0
$571$	140710.			0.431571			0.215786	−	0.976441i	$-0.430769\pi$
$571$	140710.			0.431571			0.215786	+	0.976441i	$0.430769\pi$
$572$		0			0
$573$	−596736.	+	210978.i	−1.81749	+	0.642581i
$574$		0			0
$575$		−	125820.i		−	0.380551i
$576$		0			0
$577$	36002.0			0.108137			0.0540686	−	0.998537i	$-0.482781\pi$
$577$	36002.0			0.108137			0.0540686	+	0.998537i	$0.482781\pi$
$578$		0			0
$579$	54150.0	+	153159.i	0.161526	+	0.456863i
$580$		0			0
$581$			184877.i			0.547686i
$582$		0			0
$583$	54432.0			0.160146
$584$		0			0
$585$	−43200.0	−	53457.3i	−0.126233	−	0.156205i
$586$		0			0
$587$			316179.i			0.917606i	0.888538	+	0.458803i	$0.151722\pi$
$587$			316179.i			0.917606i	−0.888538	+	0.458803i	$0.848278\pi$
$588$		0			0
$589$	265636.			0.765696
$590$		0			0
$591$	223344.	−	78964.0i	0.639439	−	0.226076i
$592$		0			0
$593$		−	262093.i		−	0.745327i	−0.927967	−	0.372663i	$-0.878445\pi$
$593$		−	262093.i		−	0.745327i	0.927967	−	0.372663i	$-0.121555\pi$
$594$		0			0
$595$	89856.0			0.253813
$596$		0			0
$597$	111666.	+	315839.i	0.313309	+	0.886171i
$598$		0			0
$599$			606494.i			1.69034i	0.534501	+	0.845168i	$0.320499\pi$
$599$			606494.i			1.69034i	−0.534501	+	0.845168i	$0.679501\pi$
$600$		0			0
$601$	306530.			0.848641			0.424321	−	0.905512i	$-0.360513\pi$
$601$	306530.			0.848641			0.424321	+	0.905512i	$0.360513\pi$
$602$		0			0
$603$	−282618.	+	228390.i	−0.777258	+	0.628119i
$604$		0			0
$605$			8977.43i			0.0245268i
$606$		0			0
$607$	−563162.			−1.52847			−0.764233	−	0.644940i	$-0.776882\pi$
$607$	−563162.			−1.52847			−0.764233	+	0.644940i	$0.776882\pi$
$608$		0			0
$609$	−318240.	+	112515.i	−0.858065	+	0.303372i
$610$		0			0
$611$			84852.8i			0.227292i
$612$		0			0
$613$	111314.			0.296230			0.148115	−	0.988970i	$-0.452679\pi$
$613$	111314.			0.296230			0.148115	+	0.988970i	$0.452679\pi$
$614$		0			0
$615$	−122688.	−	347014.i	−0.324378	−	0.917481i
$616$		0			0
$617$			121340.i			0.318737i	0.987219	+	0.159368i	$0.0509457\pi$
$617$			121340.i			0.318737i	−0.987219	+	0.159368i	$0.949054\pi$
$618$		0			0
$619$	581158.			1.51675			0.758373	−	0.651821i	$-0.225995\pi$
$619$	581158.			1.51675			0.758373	+	0.651821i	$0.225995\pi$
$620$		0			0
$621$	−142560.	+	231852.i	−0.369670	+	0.601212i
$622$		0			0
$623$			203850.i			0.525213i
$624$		0			0
$625$	−66431.0			−0.170063
$626$		0			0
$627$	−360864.	+	127585.i	−0.917928	+	0.324536i
$628$		0			0
$629$			381634.i			0.964597i
$630$		0			0
$631$	−232346.			−0.583548			−0.291774	−	0.956487i	$-0.594245\pi$
$631$	−232346.			−0.583548			−0.291774	+	0.956487i	$0.594245\pi$
$632$		0			0
$633$	−45294.0	−	128111.i	−0.113040	−	0.319726i
$634$		0			0
$635$		−	140550.i		−	0.348565i
$636$		0			0
$637$	−86250.0			−0.212559
$638$		0			0
$639$	181440.	+	224521.i	0.444356	+	0.549863i
$640$		0			0
$641$		−	653570.i		−	1.59066i	−0.606179	−	0.795328i	$-0.707299\pi$
$641$		−	653570.i		−	1.59066i	0.606179	−	0.795328i	$-0.292701\pi$
$642$		0			0
$643$	424678.			1.02716			0.513580	−	0.858042i	$-0.328319\pi$
$643$	424678.			1.02716			0.513580	+	0.858042i	$0.328319\pi$
$644$		0			0
$645$	−37728.0	+	13338.9i	−0.0906869	+	0.0320626i
$646$		0			0
$647$		−	427964.i		−	1.02235i	−0.859478	−	0.511173i	$-0.829211\pi$
$647$		−	427964.i		−	1.02235i	0.859478	−	0.511173i	$-0.170789\pi$
$648$		0			0
$649$	−215712.			−0.512136
$650$		0			0
$651$	−57876.0	−	163698.i	−0.136564	−	0.386262i
$652$		0			0
$653$			720621.i			1.68998i	0.534785	+	0.844988i	$0.320393\pi$
$653$			720621.i			1.68998i	−0.534785	+	0.844988i	$0.679607\pi$
$654$		0			0
$655$	91872.0			0.214141
$656$		0			0
$657$	−18270.0	+	14764.4i	−0.0423261	+	0.0342046i
$658$		0			0
$659$			40135.4i			0.0924180i	0.998932	+	0.0462090i	$0.0147140\pi$
$659$			40135.4i			0.0924180i	−0.998932	+	0.0462090i	$0.985286\pi$
$660$		0			0
$661$	−358510.			−0.820537			−0.410269	−	0.911965i	$-0.634565\pi$
$661$	−358510.			−0.820537			−0.410269	+	0.911965i	$0.634565\pi$
$662$		0			0
$663$	−86400.0	+	30547.0i	−0.196556	+	0.0694931i
$664$		0			0
$665$		−	157962.i		−	0.357198i
$666$		0			0
$667$	−538560.			−1.21055
$668$		0			0
$669$	−176334.	−	498748.i	−0.393989	−	1.11437i
$670$		0			0
$671$		−	176528.i		−	0.392074i
$672$		0			0
$673$	582434.			1.28593			0.642964	−	0.765896i	$-0.277705\pi$
$673$	582434.			1.28593			0.642964	+	0.765896i	$0.277705\pi$
$674$		0			0
$675$	−209277.	−	128679.i	−0.459319	−	0.282424i
$676$		0			0
$677$			37352.2i			0.0814965i	0.999169	+	0.0407482i	$0.0129742\pi$
$677$			37352.2i			0.0814965i	−0.999169	+	0.0407482i	$0.987026\pi$
$678$		0			0
$679$	12428.0			0.0269564
$680$		0			0
$681$	577008.	−	204003.i	1.24419	−	0.439889i
$682$		0			0
$683$		−	161848.i		−	0.346950i	−0.984838	−	0.173475i	$-0.944500\pi$
$683$		−	161848.i		−	0.346950i	0.984838	−	0.173475i	$-0.0554995\pi$
$684$		0			0
$685$	307008.			0.654287
$686$		0			0
$687$	85686.0	+	242357.i	0.181550	+	0.513501i
$688$		0			0
$689$		−	22910.3i		−	0.0482605i
$690$		0			0
$691$	630118.			1.31967			0.659836	−	0.751410i	$-0.270626\pi$
$691$	630118.			1.31967			0.659836	+	0.751410i	$0.270626\pi$
$692$		0			0
$693$	157248.	+	194584.i	0.327430	+	0.405174i
$694$		0			0
$695$			393819.i			0.815318i
$696$		0			0
$697$	−490752.			−1.01017
$698$		0			0
$699$	−190944.	+	67508.9i	−0.390797	+	0.138168i
$700$		0			0
$701$		−	457747.i		−	0.931514i	−0.884913	−	0.465757i	$-0.845782\pi$
$701$		−	457747.i		−	0.931514i	0.884913	−	0.465757i	$-0.154218\pi$
$702$		0			0
$703$	670892.			1.35751
$704$		0			0
$705$	−86400.0	−	244376.i	−0.173834	−	0.491678i
$706$		0			0
$707$		−	356959.i		−	0.714133i
$708$		0			0
$709$	274130.			0.545336			0.272668	−	0.962108i	$-0.412094\pi$
$709$	274130.			0.545336			0.272668	+	0.962108i	$0.412094\pi$
$710$		0			0
$711$	618534.	−	499851.i	1.22356	−	0.988784i
$712$		0			0
$713$		−	277027.i		−	0.544934i
$714$		0			0
$715$	−100800.			−0.197173
$716$		0			0
$717$	10944.0	−	3869.29i	0.0212881	−	0.00752650i
$718$		0			0
$719$			304045.i			0.588138i	0.955784	+	0.294069i	$0.0950096\pi$
$719$			304045.i			0.588138i	−0.955784	+	0.294069i	$0.904990\pi$
$720$		0			0
$721$	55588.0			0.106933
$722$		0			0
$723$	−183738.	−	519690.i	−0.351498	−	0.994185i
$724$		0			0
$725$		−	486122.i		−	0.924845i
$726$		0			0
$727$	−364058.			−0.688814			−0.344407	−	0.938821i	$-0.611920\pi$
$727$	−364058.			−0.688814			−0.344407	+	0.938821i	$0.611920\pi$
$728$		0			0
$729$	239841.	+	474242.i	0.451303	+	0.892371i
$730$		0			0
$731$			53355.4i			0.0998491i
$732$		0			0
$733$	−301198.			−0.560588			−0.280294	−	0.959914i	$-0.590432\pi$
$733$	−301198.			−0.560588			−0.280294	+	0.959914i	$0.590432\pi$
$734$		0			0
$735$	248400.	−	87822.7i	0.459808	−	0.162567i
$736$		0			0
$737$			532910.i			0.981112i
$738$		0			0
$739$	−872570.			−1.59776			−0.798880	−	0.601491i	$-0.794574\pi$
$739$	−872570.			−1.59776			−0.798880	+	0.601491i	$0.794574\pi$
$740$		0			0
$741$	53700.0	+	151887.i	0.0977998	+	0.276620i
$742$		0			0
$743$		−	874222.i		−	1.58359i	−0.610784	−	0.791797i	$-0.709146\pi$
$743$		−	874222.i		−	1.58359i	0.610784	−	0.791797i	$-0.290854\pi$
$744$		0			0
$745$	189792.			0.341952
$746$		0			0
$747$	362016.	+	447972.i	0.648764	+	0.802804i
$748$		0			0
$749$		−	281949.i		−	0.502582i
$750$		0			0
$751$	−916250.			−1.62455			−0.812277	−	0.583272i	$-0.801772\pi$
$751$	−916250.			−1.62455			−0.812277	+	0.583272i	$0.801772\pi$
$752$		0			0
$753$	−384048.	+	135781.i	−0.677323	+	0.239470i
$754$		0			0
$755$		−	244817.i		−	0.429485i
$756$		0			0
$757$	−691630.			−1.20693			−0.603465	−	0.797390i	$-0.706214\pi$
$757$	−691630.			−1.20693			−0.603465	+	0.797390i	$0.706214\pi$
$758$		0			0
$759$	133056.	+	376339.i	0.230968	+	0.653275i
$760$		0			0
$761$		−	90249.5i		−	0.155839i	−0.996960	−	0.0779193i	$-0.975172\pi$
$761$		−	90249.5i		−	0.155839i	0.996960	−	0.0779193i	$-0.0248277\pi$
$762$		0			0
$763$	123500.			0.212138
$764$		0			0
$765$	217728.	−	175951.i	0.372042	−	0.300655i
$766$		0			0
$767$			90792.5i			0.154333i
$768$		0			0
$769$	−515326.			−0.871424			−0.435712	−	0.900086i	$-0.643503\pi$
$769$	−515326.			−0.871424			−0.435712	+	0.900086i	$0.643503\pi$
$770$		0			0
$771$	723456.	−	255780.i	1.21704	−	0.430287i
$772$		0			0
$773$		−	100449.i		−	0.168107i	−0.996461	−	0.0840535i	$-0.973213\pi$
$773$		−	100449.i		−	0.168107i	0.996461	−	0.0840535i	$-0.0267867\pi$
$774$		0			0
$775$	250054.			0.416323
$776$		0			0
$777$	−146172.	−	413437.i	−0.242115	−	0.684805i
$778$		0			0
$779$			862716.i			1.42165i
$780$		0			0
$781$	423360.			0.694077
$782$		0			0
$783$	−550800.	+	895791.i	−0.898401	+	1.46111i
$784$		0			0
$785$			831727.i			1.34971i
$786$		0			0
$787$	107878.			0.174174			0.0870870	−	0.996201i	$-0.472244\pi$
$787$	107878.			0.174174			0.0870870	+	0.996201i	$0.472244\pi$
$788$		0			0
$789$	719136.	−	254253.i	1.15520	−	0.408425i
$790$		0			0
$791$			82952.1i			0.132579i
$792$		0			0
$793$	−74300.0			−0.118152
$794$		0			0
$795$	23328.0	+	65981.5i	0.0369099	+	0.104397i
$796$		0			0
$797$			489482.i			0.770584i	0.922795	+	0.385292i	$0.125899\pi$
$797$			489482.i			0.770584i	−0.922795	+	0.385292i	$0.874101\pi$
$798$		0			0
$799$	−345600.			−0.541353
$800$		0			0
$801$	399168.	+	493945.i	0.622144	+	0.769864i
$802$		0			0
$803$			34450.2i			0.0534270i
$804$		0			0
$805$	−164736.			−0.254212
$806$		0			0
$807$	364176.	−	128756.i	0.559196	−	0.197706i
$808$		0			0
$809$			1.04074e6i			1.59017i	0.606497	+	0.795086i	$0.292574\pi$
$809$			1.04074e6i			1.59017i	−0.606497	+	0.795086i	$0.707426\pi$
$810$		0			0
$811$	−611066.			−0.929066			−0.464533	−	0.885556i	$-0.653778\pi$
$811$	−611066.			−0.929066			−0.464533	+	0.885556i	$0.653778\pi$
$812$		0			0
$813$	82290.0	+	232751.i	0.124499	+	0.352136i
$814$		0			0
$815$			729429.i			1.09817i
$816$		0			0
$817$	93796.0			0.140521
$818$		0			0
$819$	81900.0	−	66185.2i	0.122100	−	0.0986718i
$820$		0			0
$821$		−	345300.i		−	0.512283i	−0.966639	−	0.256142i	$-0.917549\pi$
$821$		−	345300.i		−	0.512283i	0.966639	−	0.256142i	$-0.0824514\pi$
$822$		0			0
$823$	178150.			0.263018			0.131509	−	0.991315i	$-0.458018\pi$
$823$	178150.			0.263018			0.131509	+	0.991315i	$0.458018\pi$
$824$		0			0
$825$	−339696.	+	120101.i	−0.499094	+	0.176456i
$826$		0			0
$827$			1.13844e6i			1.66455i	0.554361	+	0.832277i	$0.312963\pi$
$827$			1.13844e6i			1.66455i	−0.554361	+	0.832277i	$0.687037\pi$
$828$		0			0
$829$	100082.			0.145629			0.0728143	−	0.997346i	$-0.476802\pi$
$829$	100082.			0.145629			0.0728143	+	0.997346i	$0.476802\pi$
$830$		0			0
$831$	−281802.	−	797056.i	−0.408077	−	1.15422i
$832$		0			0
$833$		−	351291.i		−	0.506263i
$834$		0			0
$835$	749376.			1.07480
$836$		0			0
$837$	−460782.	−	283324.i	−0.657725	−	0.404419i
$838$		0			0
$839$		−	1.13964e6i		−	1.61899i	−0.587127	−	0.809495i	$-0.699741\pi$
$839$		−	1.13964e6i		−	1.61899i	0.587127	−	0.809495i	$-0.300259\pi$
$840$		0			0
$841$	−1.37352e6			−1.94197
$842$		0			0
$843$	−207648.	+	73414.7i	−0.292195	+	0.103306i
$844$		0			0
$845$		−	442270.i		−	0.619404i
$846$		0			0
$847$	−13754.0			−0.0191718
$848$		0			0
$849$	197490.	+	558586.i	0.273987	+	0.774952i
$850$		0			0
$851$		−	699662.i		−	0.966116i
$852$		0			0
$853$	172754.			0.237427			0.118713	−	0.992929i	$-0.462123\pi$
$853$	172754.			0.237427			0.118713	+	0.992929i	$0.462123\pi$
$854$		0			0
$855$	−309312.	−	382754.i	−0.423121	−	0.523585i
$856$		0			0
$857$			299802.i			0.408200i	0.978950	+	0.204100i	$0.0654267\pi$
$857$			299802.i			0.408200i	−0.978950	+	0.204100i	$0.934573\pi$
$858$		0			0
$859$	−8186.00			−0.0110939			−0.00554696	−	0.999985i	$-0.501766\pi$
$859$	−8186.00			−0.0110939			−0.00554696	+	0.999985i	$0.501766\pi$
$860$		0			0
$861$	531648.	−	187966.i	0.717163	−	0.253555i
$862$		0			0
$863$			387336.i			0.520076i	0.965598	+	0.260038i	$0.0837350\pi$
$863$			387336.i			0.520076i	−0.965598	+	0.260038i	$0.916265\pi$
$864$		0			0
$865$	380448.			0.508467
$866$		0			0
$867$	126147.	+	356798.i	0.167818	+	0.474661i
$868$		0			0
$869$		−	1.16632e6i		−	1.54446i
$870$		0			0
$871$	224300.			0.295660
$872$		0			0
$873$	30114.0	−	24335.8i	0.0395130	−	0.0319313i
$874$		0			0
$875$		−	424468.i		−	0.554407i
$876$		0			0
$877$	245426.			0.319096			0.159548	−	0.987190i	$-0.448996\pi$
$877$	245426.			0.319096			0.159548	+	0.987190i	$0.448996\pi$
$878$		0			0
$879$	−763920.	+	270087.i	−0.988713	+	0.349563i
$880$		0			0
$881$		−	607071.i		−	0.782146i	−0.920360	−	0.391073i	$-0.872104\pi$
$881$		−	607071.i		−	0.782146i	0.920360	−	0.391073i	$-0.127896\pi$
$882$		0			0
$883$	−759962.			−0.974699			−0.487349	−	0.873207i	$-0.662036\pi$
$883$	−759962.			−0.974699			−0.487349	+	0.873207i	$0.662036\pi$
$884$		0			0
$885$	−92448.0	−	261482.i	−0.118035	−	0.333854i
$886$		0			0
$887$			5736.05i			0.00729064i	0.999993	+	0.00364532i	$0.00116034\pi$
$887$			5736.05i			0.00729064i	−0.999993	+	0.00364532i	$0.998840\pi$
$888$		0			0
$889$	215332.			0.272461
$890$		0			0
$891$	762048.	+	163579.i	0.959902	+	0.206050i
$892$		0			0
$893$			607546.i			0.761862i
$894$		0			0
$895$	519264.			0.648249
$896$		0			0
$897$	158400.	−	56002.9i	0.196866	−	0.0696026i
$898$		0			0
$899$		−	1.07033e6i		−	1.32434i
$900$		0			0
$901$	93312.0			0.114944
$902$		0			0
$903$	−20436.0	−	57801.7i	−0.0250623	−	0.0708868i
$904$		0			0
$905$		−	222348.i		−	0.271479i
$906$		0			0
$907$	138502.			0.168361			0.0841805	−	0.996451i	$-0.473173\pi$
$907$	138502.			0.168361			0.0841805	+	0.996451i	$0.473173\pi$
$908$		0			0
$909$	−698976.	−	864939.i	−0.845930	−	1.04679i
$910$		0			0
$911$			266845.i			0.321531i	0.986993	+	0.160765i	$0.0513962\pi$
$911$			266845.i			0.321531i	−0.986993	+	0.160765i	$0.948604\pi$
$912$		0			0
$913$	844704.			1.01336
$914$		0			0
$915$	213984.	−	75654.8i	0.255587	−	0.0903637i
$916$		0			0
$917$			140754.i			0.167387i
$918$		0			0
$919$	1.44266e6			1.70818			0.854090	−	0.520125i	$-0.174115\pi$
$919$	1.44266e6			1.70818			0.854090	+	0.520125i	$0.174115\pi$
$920$		0			0
$921$	−201966.	−	571246.i	−0.238100	−	0.673448i
$922$		0			0
$923$		−	178191.i		−	0.209162i
$924$		0			0
$925$	631538.			0.738101
$926$		0			0
$927$	134694.	−	108849.i	0.156743	−	0.126668i
$928$		0			0
$929$			511696.i			0.592899i	0.955048	+	0.296450i	$0.0958027\pi$
$929$			511696.i			0.592899i	−0.955048	+	0.296450i	$0.904197\pi$
$930$		0			0
$931$	−617550.			−0.712480
$932$		0			0
$933$	−1.12003e6	+	395991.i	−1.28667	+	0.454907i
$934$		0			0
$935$		−	410552.i		−	0.469618i
$936$		0			0
$937$	868610.			0.989340			0.494670	−	0.869081i	$-0.335289\pi$
$937$	868610.			0.989340			0.494670	+	0.869081i	$0.335289\pi$
$938$		0			0
$939$	−66234.0	−	187338.i	−0.0751190	−	0.212469i
$940$		0			0
$941$			101942.i			0.115126i	0.998342	+	0.0575632i	$0.0183331\pi$
$941$			101942.i			0.115126i	−0.998342	+	0.0575632i	$0.981667\pi$
$942$		0			0
$943$	899712.			1.01177
$944$		0			0
$945$	−168480.	+	274007.i	−0.188662	+	0.306830i
$946$		0			0
$947$		−	233159.i		−	0.259987i	−0.991515	−	0.129993i	$-0.958504\pi$
$947$		−	233159.i		−	0.259987i	0.991515	−	0.129993i	$-0.0414956\pi$
$948$		0			0
$949$	14500.0			0.0161004
$950$		0			0
$951$	−1.00037e6	+	353683.i	−1.10611	+	0.391069i
$952$		0			0
$953$		−	927916.i		−	1.02170i	−0.859670	−	0.510850i	$-0.829331\pi$
$953$		−	927916.i		−	1.02170i	0.859670	−	0.510850i	$-0.170669\pi$
$954$		0			0
$955$	−1.19347e6			−1.30860
$956$		0			0
$957$	514080.	+	1.45404e6i	0.561315	+	1.58764i
$958$		0			0
$959$			470356.i			0.511434i
$960$		0			0
$961$	−372957.			−0.403842
$962$		0			0
$963$	−552096.	−	683184.i	−0.595336	−	0.736691i
$964$		0			0
$965$			306319.i			0.328942i
$966$		0			0
$967$	−150746.			−0.161210			−0.0806052	−	0.996746i	$-0.525685\pi$
$967$	−150746.			−0.161210			−0.0806052	+	0.996746i	$0.525685\pi$
$968$		0			0
$969$	−618624.	+	218717.i	−0.658839	+	0.232935i
$970$		0			0
$971$			508150.i			0.538956i	0.963007	+	0.269478i	$0.0868511\pi$
$971$			508150.i			0.538956i	−0.963007	+	0.269478i	$0.913149\pi$
$972$		0			0
$973$	−603356.			−0.637306
$974$		0			0
$975$	50550.0	+	142977.i	0.0531755	+	0.150403i
$976$		0			0
$977$		−	214576.i		−	0.224798i	−0.993663	−	0.112399i	$-0.964147\pi$
$977$		−	214576.i		−	0.224798i	0.993663	−	0.112399i	$-0.0358534\pi$
$978$		0			0
$979$	931392.			0.971778
$980$		0			0
$981$	299250.	−	241831.i	0.310954	−	0.251289i
$982$		0			0
$983$			1.34831e6i			1.39535i	0.716414	+	0.697675i	$0.245782\pi$
$983$			1.34831e6i			1.39535i	−0.716414	+	0.697675i	$0.754218\pi$
$984$		0			0
$985$	446688.			0.460396
$986$		0			0
$987$	374400.	−	132370.i	0.384328	−	0.135880i
$988$		0			0
$989$		−	97818.3i		−	0.100006i
$990$		0			0
$991$	879910.			0.895965			0.447982	−	0.894042i	$-0.352143\pi$
$991$	879910.			0.895965			0.447982	+	0.894042i	$0.352143\pi$
$992$		0			0
$993$	−502926.	−	1.42249e6i	−0.510042	−	1.44262i
$994$		0			0
$995$			631678.i			0.638043i
$996$		0			0
$997$	935378.			0.941016			0.470508	−	0.882396i	$-0.344071\pi$
$997$	935378.			0.941016			0.470508	+	0.882396i	$0.344071\pi$
$998$		0			0
$999$	−1.16375e6	−	715564.i	−1.16609	−	0.716997i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.5.e.b.17.2		2
3.2	odd	2	inner	48.5.e.b.17.1		2
4.3	odd	2		6.5.b.a.5.2	yes	2
8.3	odd	2		192.5.e.d.65.2		2
8.5	even	2		192.5.e.c.65.1		2
12.11	even	2		6.5.b.a.5.1	✓	2
20.3	even	4		150.5.b.a.149.3		4
20.7	even	4		150.5.b.a.149.2		4
20.19	odd	2		150.5.d.a.101.1		2
24.5	odd	2		192.5.e.c.65.2		2
24.11	even	2		192.5.e.d.65.1		2
28.27	even	2		294.5.b.a.197.2		2
36.7	odd	6		162.5.d.a.53.2		4
36.11	even	6		162.5.d.a.53.1		4
36.23	even	6		162.5.d.a.107.2		4
36.31	odd	6		162.5.d.a.107.1		4
60.23	odd	4		150.5.b.a.149.1		4
60.47	odd	4		150.5.b.a.149.4		4
60.59	even	2		150.5.d.a.101.2		2
84.83	odd	2		294.5.b.a.197.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.5.b.a.5.1	✓	2	12.11	even	2
6.5.b.a.5.2	yes	2	4.3	odd	2
48.5.e.b.17.1		2	3.2	odd	2	inner
48.5.e.b.17.2		2	1.1	even	1	trivial
150.5.b.a.149.1		4	60.23	odd	4
150.5.b.a.149.2		4	20.7	even	4
150.5.b.a.149.3		4	20.3	even	4
150.5.b.a.149.4		4	60.47	odd	4
150.5.d.a.101.1		2	20.19	odd	2
150.5.d.a.101.2		2	60.59	even	2
162.5.d.a.53.1		4	36.11	even	6
162.5.d.a.53.2		4	36.7	odd	6
162.5.d.a.107.1		4	36.31	odd	6
162.5.d.a.107.2		4	36.23	even	6
192.5.e.c.65.1		2	8.5	even	2
192.5.e.c.65.2		2	24.5	odd	2
192.5.e.d.65.1		2	24.11	even	2
192.5.e.d.65.2		2	8.3	odd	2
294.5.b.a.197.1		2	84.83	odd	2
294.5.b.a.197.2		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	48.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(3.00000 + 8.48528i) q^{3} +16.9706i q^{5} -26.0000 q^{7} +(-63.0000 + 50.9117i) q^{9} +O(q^{10})\)	\(q+(3.00000 + 8.48528i) q^{3} +16.9706i q^{5} -26.0000 q^{7} +(-63.0000 + 50.9117i) q^{9} +118.794i q^{11} +50.0000 q^{13} +(-144.000 + 50.9117i) q^{15} +203.647i q^{17} +358.000 q^{19} +(-78.0000 - 220.617i) q^{21} -373.352i q^{23} +337.000 q^{25} +(-621.000 - 381.838i) q^{27} -1442.50i q^{29} +742.000 q^{31} +(-1008.00 + 356.382i) q^{33} -441.235i q^{35} +1874.00 q^{37} +(150.000 + 424.264i) q^{39} +2409.82i q^{41} +262.000 q^{43} +(-864.000 - 1069.15i) q^{45} +1697.06i q^{47} -1725.00 q^{49} +(-1728.00 + 610.940i) q^{51} -458.205i q^{53} -2016.00 q^{55} +(1074.00 + 3037.73i) q^{57} +1815.85i q^{59} -1486.00 q^{61} +(1638.00 - 1323.70i) q^{63} +848.528i q^{65} +4486.00 q^{67} +(3168.00 - 1120.06i) q^{69} -3563.82i q^{71} +290.000 q^{73} +(1011.00 + 2859.54i) q^{75} -3088.64i q^{77} -9818.00 q^{79} +(1377.00 - 6414.87i) q^{81} -7110.67i q^{83} -3456.00 q^{85} +(12240.0 - 4327.49i) q^{87} -7840.40i q^{89} -1300.00 q^{91} +(2226.00 + 6296.08i) q^{93} +6075.46i q^{95} -478.000 q^{97} +(-6048.00 - 7484.02i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 52 q^{7} - 126 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 52 * q^7 - 126 * q^9	\( 2 q + 6 q^{3} - 52 q^{7} - 126 q^{9} + 100 q^{13} - 288 q^{15} + 716 q^{19} - 156 q^{21} + 674 q^{25} - 1242 q^{27} + 1484 q^{31} - 2016 q^{33} + 3748 q^{37} + 300 q^{39} + 524 q^{43} - 1728 q^{45} - 3450 q^{49} - 3456 q^{51} - 4032 q^{55} + 2148 q^{57} - 2972 q^{61} + 3276 q^{63} + 8972 q^{67} + 6336 q^{69} + 580 q^{73} + 2022 q^{75} - 19636 q^{79} + 2754 q^{81} - 6912 q^{85} + 24480 q^{87} - 2600 q^{91} + 4452 q^{93} - 956 q^{97} - 12096 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 52 * q^7 - 126 * q^9 + 100 * q^13 - 288 * q^15 + 716 * q^19 - 156 * q^21 + 674 * q^25 - 1242 * q^27 + 1484 * q^31 - 2016 * q^33 + 3748 * q^37 + 300 * q^39 + 524 * q^43 - 1728 * q^45 - 3450 * q^49 - 3456 * q^51 - 4032 * q^55 + 2148 * q^57 - 2972 * q^61 + 3276 * q^63 + 8972 * q^67 + 6336 * q^69 + 580 * q^73 + 2022 * q^75 - 19636 * q^79 + 2754 * q^81 - 6912 * q^85 + 24480 * q^87 - 2600 * q^91 + 4452 * q^93 - 956 * q^97 - 12096 * q^99

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