LMFDB - Embedded newform 336.4.h.b.239.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(239,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("336.239");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.1
Character	$\chi$	$=$	336.239
Dual form			336.4.h.b.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.19367 - 0.160438i) q^{3} +7.59851i q^{5} +7.00000i q^{7} +(26.9485 + 1.66652i) q^{9} +O(q^{10})$	\(q+(-5.19367 - 0.160438i) q^{3} +7.59851i q^{5} +7.00000i q^{7} +(26.9485 + 1.66652i) q^{9} +0.0298576 q^{11} +40.1686 q^{13} +(1.21909 - 39.4642i) q^{15} +0.439999i q^{17} -0.434338i q^{19} +(1.12306 - 36.3557i) q^{21} -139.817 q^{23} +67.2627 q^{25} +(-139.694 - 12.9789i) q^{27} +273.309i q^{29} -149.527i q^{31} +(-0.155070 - 0.00479027i) q^{33} -53.1896 q^{35} -290.284 q^{37} +(-208.622 - 6.44454i) q^{39} +57.4908i q^{41} +432.992i q^{43} +(-12.6631 + 204.769i) q^{45} -317.042 q^{47} -49.0000 q^{49} +(0.0705924 - 2.28521i) q^{51} +147.100i q^{53} +0.226873i q^{55} +(-0.0696841 + 2.25581i) q^{57} -488.249 q^{59} -441.188 q^{61} +(-11.6656 + 188.640i) q^{63} +305.221i q^{65} +440.204i q^{67} +(726.166 + 22.4319i) q^{69} -202.974 q^{71} -542.719 q^{73} +(-349.341 - 10.7915i) q^{75} +0.209003i q^{77} -325.250i q^{79} +(723.445 + 89.8205i) q^{81} -376.536 q^{83} -3.34334 q^{85} +(43.8491 - 1419.48i) q^{87} -496.611i q^{89} +281.180i q^{91} +(-23.9898 + 776.595i) q^{93} +3.30032 q^{95} +373.344 q^{97} +(0.804617 + 0.0497582i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 136 q^{9}+O(q^{10}) $ 24 * q - 136 * q^9	$ 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) $ 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	−5.19367	−	0.160438i	−0.999523	−	0.0308762i
$3$	−5.19367	−	0.160438i	−0.999523	−	0.0308762i
$4$		0			0
$5$			7.59851i			0.679631i	0.940492	+	0.339816i	$0.110365\pi$
$5$			7.59851i			0.679631i	−0.940492	+	0.339816i	$0.889635\pi$
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		0			0
$9$	26.9485	+	1.66652i	0.998093	+	0.0617230i
$10$		0			0
$11$	0.0298576			0.000818399			0.000409200	−	1.00000i	$-0.499870\pi$
$11$	0.0298576			0.000818399			0.000409200		1.00000i	$0.499870\pi$
$12$		0			0
$13$	40.1686			0.856981			0.428490	−	0.903546i	$-0.359046\pi$
$13$	40.1686			0.856981			0.428490	+	0.903546i	$0.359046\pi$
$14$		0			0
$15$	1.21909	−	39.4642i	0.0209844	−	0.679307i
$16$		0			0
$17$			0.439999i			0.00627738i	0.999995	+	0.00313869i	$0.000999078\pi$
$17$			0.439999i			0.00627738i	−0.999995	+	0.00313869i	$0.999001\pi$
$18$		0			0
$19$		−	0.434338i		−	0.00524441i	−0.999997	−	0.00262221i	$-0.999165\pi$
$19$		−	0.434338i		−	0.00524441i	0.999997	−	0.00262221i	$-0.000834675\pi$
$20$		0			0
$21$	1.12306	−	36.3557i	0.0116701	−	0.377784i
$22$		0			0
$23$	−139.817			−1.26756			−0.633781	−	0.773513i	$-0.718498\pi$
$23$	−139.817			−1.26756			−0.633781	+	0.773513i	$0.718498\pi$
$24$		0			0
$25$	67.2627			0.538101
$26$		0			0
$27$	−139.694	−	12.9789i	−0.995712	−	0.0925109i
$28$		0			0
$29$			273.309i			1.75008i	0.484053	+	0.875039i	$0.339164\pi$
$29$			273.309i			1.75008i	−0.484053	+	0.875039i	$0.660836\pi$
$30$		0			0
$31$		−	149.527i		−	0.866318i	−0.901318	−	0.433159i	$-0.857399\pi$
$31$		−	149.527i		−	0.866318i	0.901318	−	0.433159i	$-0.142601\pi$
$32$		0			0
$33$	−0.155070	−	0.00479027i	−0.000818009	−	2.52691e-5i
$34$		0			0
$35$	−53.1896			−0.256876
$36$		0			0
$37$	−290.284			−1.28979			−0.644897	−	0.764269i	$-0.723100\pi$
$37$	−290.284			−1.28979			−0.644897	+	0.764269i	$0.723100\pi$
$38$		0			0
$39$	−208.622	−	6.44454i	−0.856572	−	0.0264603i
$40$		0			0
$41$			57.4908i			0.218989i	0.993987	+	0.109495i	$0.0349232\pi$
$41$			57.4908i			0.218989i	−0.993987	+	0.109495i	$0.965077\pi$
$42$		0			0
$43$			432.992i			1.53560i	0.640692	+	0.767798i	$0.278648\pi$
$43$			432.992i			1.53560i	−0.640692	+	0.767798i	$0.721352\pi$
$44$		0			0
$45$	−12.6631	+	204.769i	−0.0419489	+	0.678335i
$46$		0			0
$47$	−317.042			−0.983942			−0.491971	−	0.870612i	$-0.663723\pi$
$47$	−317.042			−0.983942			−0.491971	+	0.870612i	$0.663723\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	0.0705924	−	2.28521i	0.000193822	−	0.00627439i
$52$		0			0
$53$			147.100i			0.381239i	0.981664	+	0.190620i	$0.0610497\pi$
$53$			147.100i			0.381239i	−0.981664	+	0.190620i	$0.938950\pi$
$54$		0			0
$55$			0.226873i			0.000556210i
$56$		0			0
$57$	−0.0696841	+	2.25581i	−0.000161928	+	0.00524191i
$58$		0			0
$59$	−488.249			−1.07737			−0.538683	−	0.842508i	$-0.681078\pi$
$59$	−488.249			−1.07737			−0.538683	+	0.842508i	$0.681078\pi$
$60$		0			0
$61$	−441.188			−0.926038			−0.463019	−	0.886348i	$-0.653234\pi$
$61$	−441.188			−0.926038			−0.463019	+	0.886348i	$0.653234\pi$
$62$		0			0
$63$	−11.6656	+	188.640i	−0.0233291	+	0.377244i
$64$		0			0
$65$			305.221i			0.582431i
$66$		0			0
$67$			440.204i			0.802678i	0.915930	+	0.401339i	$0.131455\pi$
$67$			440.204i			0.802678i	−0.915930	+	0.401339i	$0.868545\pi$
$68$		0			0
$69$	726.166	+	22.4319i	1.26696	+	0.0391375i
$70$		0			0
$71$	−202.974			−0.339275			−0.169638	−	0.985507i	$-0.554260\pi$
$71$	−202.974			−0.339275			−0.169638	+	0.985507i	$0.554260\pi$
$72$		0			0
$73$	−542.719			−0.870144			−0.435072	−	0.900396i	$-0.643277\pi$
$73$	−542.719			−0.870144			−0.435072	+	0.900396i	$0.643277\pi$
$74$		0			0
$75$	−349.341	−	10.7915i	−0.537845	−	0.0166145i
$76$		0			0
$77$			0.209003i			0.000309326i
$78$		0			0
$79$		−	325.250i		−	0.463209i	−0.972810	−	0.231605i	$-0.925602\pi$
$79$		−	325.250i		−	0.463209i	0.972810	−	0.231605i	$-0.0743976\pi$
$80$		0			0
$81$	723.445	+	89.8205i	0.992381	+	0.123211i
$82$		0			0
$83$	−376.536			−0.497955			−0.248977	−	0.968509i	$-0.580094\pi$
$83$	−376.536			−0.497955			−0.248977	+	0.968509i	$0.580094\pi$
$84$		0			0
$85$	−3.34334			−0.00426631
$86$		0			0
$87$	43.8491	−	1419.48i	0.0540358	−	1.74924i
$88$		0			0
$89$		−	496.611i		−	0.591468i	−0.955270	−	0.295734i	$-0.904436\pi$
$89$		−	496.611i		−	0.591468i	0.955270	−	0.295734i	$-0.0955641\pi$
$90$		0			0
$91$			281.180i			0.323908i
$92$		0			0
$93$	−23.9898	+	776.595i	−0.0267486	+	0.865905i
$94$		0			0
$95$	3.30032			0.00356427
$96$		0			0
$97$	373.344			0.390797			0.195398	−	0.980724i	$-0.437400\pi$
$97$	373.344			0.390797			0.195398	+	0.980724i	$0.437400\pi$
$98$		0			0
$99$	0.804617	+	0.0497582i	0.000816839	+	5.05141e-5i
$100$		0			0
$101$		−	336.295i		−	0.331313i	−0.986183	−	0.165657i	$-0.947026\pi$
$101$		−	336.295i		−	0.331313i	0.986183	−	0.165657i	$-0.0529743\pi$
$102$		0			0
$103$		−	372.072i		−	0.355936i	−0.984036	−	0.177968i	$-0.943048\pi$
$103$		−	372.072i		−	0.355936i	0.984036	−	0.177968i	$-0.0569523\pi$
$104$		0			0
$105$	276.249	+	8.53360i	0.256754	+	0.00793137i
$106$		0			0
$107$	1350.90			1.22053			0.610263	−	0.792199i	$-0.291064\pi$
$107$	1350.90			1.22053			0.610263	+	0.792199i	$0.291064\pi$
$108$		0			0
$109$	171.760			0.150932			0.0754662	−	0.997148i	$-0.475955\pi$
$109$	171.760			0.150932			0.0754662	+	0.997148i	$0.475955\pi$
$110$		0			0
$111$	1507.64	+	46.5725i	1.28918	+	0.0398240i
$112$		0			0
$113$		−	1132.11i		−	0.942479i	−0.882005	−	0.471240i	$-0.843807\pi$
$113$		−	1132.11i		−	0.942479i	0.882005	−	0.471240i	$-0.156193\pi$
$114$		0			0
$115$		−	1062.40i		−	0.861475i
$116$		0			0
$117$	1082.48	+	66.9417i	0.855347	+	0.0528954i
$118$		0			0
$119$	−3.08000			−0.00237263
$120$		0			0
$121$	−1331.00			−0.999999
$122$		0			0
$123$	9.22368	−	298.588i	0.00676156	−	0.218885i
$124$		0			0
$125$			1460.91i			1.04534i
$126$		0			0
$127$			2285.26i			1.59673i	0.602177	+	0.798363i	$0.294300\pi$
$127$			2285.26i			1.59673i	−0.602177	+	0.798363i	$0.705700\pi$
$128$		0			0
$129$	69.4682	−	2248.82i	0.0474134	−	1.53486i
$130$		0			0
$131$	1583.22			1.05593			0.527963	−	0.849267i	$-0.322956\pi$
$131$	1583.22			1.05593			0.527963	+	0.849267i	$0.322956\pi$
$132$		0			0
$133$	3.04036			0.00198220
$134$		0			0
$135$	98.6204	−	1061.47i	0.0628733	−	0.676717i
$136$		0			0
$137$		−	2999.39i		−	1.87047i	−0.354023	−	0.935237i	$-0.615186\pi$
$137$		−	2999.39i		−	1.87047i	0.354023	−	0.935237i	$-0.384814\pi$
$138$		0			0
$139$			1324.48i			0.808210i	0.914712	+	0.404105i	$0.132417\pi$
$139$			1324.48i			0.808210i	−0.914712	+	0.404105i	$0.867583\pi$
$140$		0			0
$141$	1646.61	+	50.8654i	0.983473	+	0.0303804i
$142$		0			0
$143$	1.19933			0.000701353
$144$		0			0
$145$	−2076.74			−1.18941
$146$		0			0
$147$	254.490	+	7.86144i	0.142789	+	0.00441089i
$148$		0			0
$149$			2415.47i			1.32808i	0.747699	+	0.664038i	$0.231159\pi$
$149$			2415.47i			1.32808i	−0.747699	+	0.664038i	$0.768841\pi$
$150$		0			0
$151$			2734.78i			1.47386i	0.675967	+	0.736932i	$0.263726\pi$
$151$			2734.78i			1.47386i	−0.675967	+	0.736932i	$0.736274\pi$
$152$		0			0
$153$	−0.733268	+	11.8573i	−0.000387459	+	0.00626541i
$154$		0			0
$155$	1136.18			0.588777
$156$		0			0
$157$	−2957.87			−1.50359			−0.751794	−	0.659398i	$-0.770811\pi$
$157$	−2957.87			−1.50359			−0.751794	+	0.659398i	$0.770811\pi$
$158$		0			0
$159$	23.6003	−	763.987i	0.0117712	−	0.381057i
$160$		0			0
$161$		−	978.721i		−	0.479093i
$162$		0			0
$163$		−	691.183i		−	0.332133i	−0.986115	−	0.166066i	$-0.946893\pi$
$163$		−	691.183i		−	0.332133i	0.986115	−	0.166066i	$-0.0531066\pi$
$164$		0			0
$165$	0.0363989	−	1.17830i	1.71737e−5	−	0.000555945i
$166$		0			0
$167$	1944.18			0.900868			0.450434	−	0.892810i	$-0.351269\pi$
$167$	1944.18			0.900868			0.450434	+	0.892810i	$0.351269\pi$
$168$		0			0
$169$	−583.488			−0.265584
$170$		0			0
$171$	0.723833	−	11.7048i	0.000323701	−	0.00523441i
$172$		0			0
$173$		−	1459.20i		−	0.641275i	−0.947202	−	0.320637i	$-0.896103\pi$
$173$		−	1459.20i		−	0.641275i	0.947202	−	0.320637i	$-0.103897\pi$
$174$		0			0
$175$			470.839i			0.203383i
$176$		0			0
$177$	2535.81	+	78.3335i	1.07685	+	0.0332650i
$178$		0			0
$179$	2398.75			1.00163			0.500813	−	0.865555i	$-0.333034\pi$
$179$	2398.75			1.00163			0.500813	+	0.865555i	$0.333034\pi$
$180$		0			0
$181$	−2169.01			−0.890724			−0.445362	−	0.895351i	$-0.646925\pi$
$181$	−2169.01			−0.890724			−0.445362	+	0.895351i	$0.646925\pi$
$182$		0			0
$183$	2291.39	+	70.7831i	0.925596	+	0.0285925i
$184$		0			0
$185$		−	2205.73i		−	0.876585i
$186$		0			0
$187$			0.0131373i			5.13741e-6i
$188$		0			0
$189$	90.8525	−	977.861i	0.0349658	−	0.376344i
$190$		0			0
$191$	3345.58			1.26742			0.633712	−	0.773569i	$-0.281530\pi$
$191$	3345.58			1.26742			0.633712	+	0.773569i	$0.281530\pi$
$192$		0			0
$193$	−1400.64			−0.522383			−0.261192	−	0.965287i	$-0.584115\pi$
$193$	−1400.64			−0.522383			−0.261192	+	0.965287i	$0.584115\pi$
$194$		0			0
$195$	48.9689	−	1585.22i	0.0179833	−	0.582153i
$196$		0			0
$197$			3747.32i			1.35526i	0.735405	+	0.677628i	$0.236992\pi$
$197$			3747.32i			1.35526i	−0.735405	+	0.677628i	$0.763008\pi$
$198$		0			0
$199$		−	1160.47i		−	0.413385i	−0.978406	−	0.206692i	$-0.933730\pi$
$199$		−	1160.47i		−	0.413385i	0.978406	−	0.206692i	$-0.0662699\pi$
$200$		0			0
$201$	70.6252	−	2286.27i	0.0247837	−	0.802295i
$202$		0			0
$203$	−1913.16			−0.661467
$204$		0			0
$205$	−436.844			−0.148832
$206$		0			0
$207$	−3767.87	−	233.008i	−1.26515	−	0.0782377i
$208$		0			0
$209$		−	0.0129683i		−	4.29203e-6i
$210$		0			0
$211$		−	3860.25i		−	1.25948i	−0.776806	−	0.629740i	$-0.783161\pi$
$211$		−	3860.25i		−	1.25948i	0.776806	−	0.629740i	$-0.216839\pi$
$212$		0			0
$213$	1054.18	+	32.5646i	0.339114	+	0.0104755i
$214$		0			0
$215$	−3290.09			−1.04364
$216$		0			0
$217$	1046.69			0.327437
$218$		0			0
$219$	2818.71	+	87.0725i	0.869729	+	0.0268667i
$220$		0			0
$221$			17.6741i			0.00537960i
$222$		0			0
$223$			4974.72i			1.49386i	0.664901	+	0.746932i	$0.268474\pi$
$223$			4974.72i			1.49386i	−0.664901	+	0.746932i	$0.731526\pi$
$224$		0			0
$225$	1812.63	+	112.095i	0.537076	+	0.0332132i
$226$		0			0
$227$	−3501.02			−1.02366			−0.511830	−	0.859087i	$-0.671032\pi$
$227$	−3501.02			−1.02366			−0.511830	+	0.859087i	$0.671032\pi$
$228$		0			0
$229$	4196.81			1.21106			0.605530	−	0.795822i	$-0.292961\pi$
$229$	4196.81			1.21106			0.605530	+	0.795822i	$0.292961\pi$
$230$		0			0
$231$	0.0335319	−	1.08549i	9.55081e−6	−	0.000309178i
$232$		0			0
$233$			4918.36i			1.38288i	0.722432	+	0.691442i	$0.243024\pi$
$233$			4918.36i			1.38288i	−0.722432	+	0.691442i	$0.756976\pi$
$234$		0			0
$235$		−	2409.04i		−	0.668717i
$236$		0			0
$237$	−52.1824	+	1689.24i	−0.0143022	+	0.462988i
$238$		0			0
$239$	−1041.63			−0.281913			−0.140957	−	0.990016i	$-0.545018\pi$
$239$	−1041.63			−0.281913			−0.140957	+	0.990016i	$0.545018\pi$
$240$		0			0
$241$	2047.75			0.547332			0.273666	−	0.961825i	$-0.411764\pi$
$241$	2047.75			0.547332			0.273666	+	0.961825i	$0.411764\pi$
$242$		0			0
$243$	−3742.93	−	582.567i	−0.988103	−	0.153793i
$244$		0			0
$245$		−	372.327i		−	0.0970902i
$246$		0			0
$247$		−	17.4467i		−	0.00449436i
$248$		0			0
$249$	1955.61	+	60.4106i	0.497717	+	0.0153750i
$250$		0			0
$251$	−741.223			−0.186397			−0.0931983	−	0.995648i	$-0.529709\pi$
$251$	−741.223			−0.186397			−0.0931983	+	0.995648i	$0.529709\pi$
$252$		0			0
$253$	−4.17460			−0.00103737
$254$		0			0
$255$	17.3642	+	0.536397i	0.00426427	+	0.000131727i
$256$		0			0
$257$		−	7849.47i		−	1.90520i	−0.304225	−	0.952600i	$-0.598398\pi$
$257$		−	7849.47i		−	1.90520i	0.304225	−	0.952600i	$-0.401602\pi$
$258$		0			0
$259$		−	2031.99i		−	0.487497i
$260$		0			0
$261$	−455.475	+	7365.28i	−0.108020	+	1.74674i
$262$		0			0
$263$	−1518.51			−0.356029			−0.178014	−	0.984028i	$-0.556967\pi$
$263$	−1518.51			−0.356029			−0.178014	+	0.984028i	$0.556967\pi$
$264$		0			0
$265$	−1117.74			−0.259102
$266$		0			0
$267$	−79.6750	+	2579.23i	−0.0182623	+	0.591186i
$268$		0			0
$269$		−	2768.04i		−	0.627398i	−0.949522	−	0.313699i	$-0.898432\pi$
$269$		−	2768.04i		−	0.627398i	0.949522	−	0.313699i	$-0.101568\pi$
$270$		0			0
$271$			2789.15i			0.625199i	0.949885	+	0.312599i	$0.101200\pi$
$271$			2789.15i			0.625199i	−0.949885	+	0.312599i	$0.898800\pi$
$272$		0			0
$273$	45.1118	−	1460.36i	0.0100011	−	0.323754i
$274$		0			0
$275$	2.00830			0.000440382
$276$		0			0
$277$	−2896.05			−0.628184			−0.314092	−	0.949393i	$-0.601700\pi$
$277$	−2896.05			−0.628184			−0.314092	+	0.949393i	$0.601700\pi$
$278$		0			0
$279$	249.190	−	4029.53i	0.0534717	−	0.864666i
$280$		0			0
$281$			4683.57i			0.994300i	0.867665	+	0.497150i	$0.165620\pi$
$281$			4683.57i			0.994300i	−0.867665	+	0.497150i	$0.834380\pi$
$282$		0			0
$283$			8527.52i			1.79120i	0.444864	+	0.895598i	$0.353252\pi$
$283$			8527.52i			1.79120i	−0.444864	+	0.895598i	$0.646748\pi$
$284$		0			0
$285$	−17.1408	−	0.529495i	−0.00356257	−	0.000110051i
$286$		0			0
$287$	−402.436			−0.0827701
$288$		0			0
$289$	4912.81			0.999961
$290$		0			0
$291$	−1939.03	−	59.8983i	−0.390611	−	0.0120663i
$292$		0			0
$293$			8621.57i			1.71904i	0.511106	+	0.859518i	$0.329236\pi$
$293$			8621.57i			1.71904i	−0.511106	+	0.859518i	$0.670764\pi$
$294$		0			0
$295$		−	3709.96i		−	0.732212i
$296$		0			0
$297$	−4.17094	−	0.387519i	−0.000814890	−	7.57109e-5i
$298$		0			0
$299$	−5616.26			−1.08628
$300$		0			0
$301$	−3030.94			−0.580401
$302$		0			0
$303$	−53.9544	+	1746.61i	−0.0102297	+	0.331155i
$304$		0			0
$305$		−	3352.37i		−	0.629364i
$306$		0			0
$307$		−	4270.07i		−	0.793830i	−0.917855	−	0.396915i	$-0.870081\pi$
$307$		−	4270.07i		−	0.793830i	0.917855	−	0.396915i	$-0.129919\pi$
$308$		0			0
$309$	−59.6944	+	1932.42i	−0.0109900	+	0.355766i
$310$		0			0
$311$	−8868.46			−1.61699			−0.808496	−	0.588502i	$-0.799718\pi$
$311$	−8868.46			−1.61699			−0.808496	+	0.588502i	$0.799718\pi$
$312$		0			0
$313$	−6273.41			−1.13289			−0.566444	−	0.824100i	$-0.691681\pi$
$313$	−6273.41			−1.13289			−0.566444	+	0.824100i	$0.691681\pi$
$314$		0			0
$315$	−1433.38	−	88.6415i	−0.256387	−	0.0158552i
$316$		0			0
$317$			11021.1i			1.95271i	0.216178	+	0.976354i	$0.430641\pi$
$317$			11021.1i			1.95271i	−0.216178	+	0.976354i	$0.569359\pi$
$318$		0			0
$319$			8.16034i			0.00143226i
$320$		0			0
$321$	−7016.13	−	216.735i	−1.21994	−	0.0376852i
$322$		0			0
$323$	0.191108			3.29212e−5
$324$		0			0
$325$	2701.84			0.461143
$326$		0			0
$327$	−892.066	−	27.5568i	−0.150860	−	0.00466022i
$328$		0			0
$329$		−	2219.29i		−	0.371895i
$330$		0			0
$331$			1923.51i			0.319414i	0.987164	+	0.159707i	$0.0510549\pi$
$331$			1923.51i			0.319414i	−0.987164	+	0.159707i	$0.948945\pi$
$332$		0			0
$333$	−7822.73	−	483.765i	−1.28734	−	0.0796100i
$334$		0			0
$335$	−3344.89			−0.545525
$336$		0			0
$337$	−7227.06			−1.16820			−0.584100	−	0.811682i	$-0.698552\pi$
$337$	−7227.06			−1.16820			−0.584100	+	0.811682i	$0.698552\pi$
$338$		0			0
$339$	−181.633	+	5879.82i	−0.0291002	+	0.942030i
$340$		0			0
$341$		−	4.46451i		−	0.000708994i
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$		0			0
$345$	−170.449	+	5517.78i	−0.0265991	+	0.861064i
$346$		0			0
$347$	−7023.96			−1.08665			−0.543323	−	0.839524i	$-0.682834\pi$
$347$	−7023.96			−1.08665			−0.543323	+	0.839524i	$0.682834\pi$
$348$		0			0
$349$	10241.7			1.57085			0.785425	−	0.618957i	$-0.212444\pi$
$349$	10241.7			1.57085			0.785425	+	0.618957i	$0.212444\pi$
$350$		0			0
$351$	−5611.32	−	521.345i	−0.853306	−	0.0792801i
$352$		0			0
$353$		−	10999.2i		−	1.65843i	−0.558930	−	0.829215i	$-0.688788\pi$
$353$		−	10999.2i		−	1.65843i	0.558930	−	0.829215i	$-0.311212\pi$
$354$		0			0
$355$		−	1542.30i		−	0.230582i
$356$		0			0
$357$	15.9965	+	0.494147i	0.00237150	+	7.32578e-5i
$358$		0			0
$359$	10462.6			1.53815			0.769076	−	0.639157i	$-0.220717\pi$
$359$	10462.6			1.53815			0.769076	+	0.639157i	$0.220717\pi$
$360$		0			0
$361$	6858.81			0.999972
$362$		0			0
$363$	6912.78	+	213.542i	0.999523	+	0.0308762i
$364$		0			0
$365$		−	4123.86i		−	0.591377i
$366$		0			0
$367$			2957.89i			0.420711i	0.977625	+	0.210355i	$0.0674621\pi$
$367$			2957.89i			0.420711i	−0.977625	+	0.210355i	$0.932538\pi$
$368$		0			0
$369$	−95.8096	+	1549.29i	−0.0135167	+	0.218572i
$370$		0			0
$371$	−1029.70			−0.144095
$372$		0			0
$373$	−4947.61			−0.686803			−0.343402	−	0.939189i	$-0.611579\pi$
$373$	−4947.61			−0.686803			−0.343402	+	0.939189i	$0.611579\pi$
$374$		0			0
$375$	234.385	−	7587.49i	0.0322762	−	1.04484i
$376$		0			0
$377$			10978.4i			1.49978i
$378$		0			0
$379$		−	7756.85i		−	1.05130i	−0.850701	−	0.525650i	$-0.823822\pi$
$379$		−	7756.85i		−	1.05130i	0.850701	−	0.525650i	$-0.176178\pi$
$380$		0			0
$381$	366.642	−	11868.9i	0.0493009	−	1.59596i
$382$		0			0
$383$	7742.65			1.03298			0.516489	−	0.856294i	$-0.327239\pi$
$383$	7742.65			1.03298			0.516489	+	0.856294i	$0.327239\pi$
$384$		0			0
$385$	−1.58811			−0.000210228
$386$		0			0
$387$	−721.590	+	11668.5i	−0.0947816	+	1.53267i
$388$		0			0
$389$		−	7875.98i		−	1.02655i	−0.858224	−	0.513275i	$-0.828432\pi$
$389$		−	7875.98i		−	1.02655i	0.858224	−	0.513275i	$-0.171568\pi$
$390$		0			0
$391$		−	61.5195i		−	0.00795697i
$392$		0			0
$393$	−8222.72	−	254.008i	−1.05542	−	0.0326030i
$394$		0			0
$395$	2471.42			0.314811
$396$		0			0
$397$	10206.3			1.29028			0.645140	−	0.764065i	$-0.276799\pi$
$397$	10206.3			1.29028			0.645140	+	0.764065i	$0.276799\pi$
$398$		0			0
$399$	−15.7907	−	0.487788i	−0.00198126	−	6.12029e-5i
$400$		0			0
$401$			11841.0i			1.47459i	0.675570	+	0.737296i	$0.263897\pi$
$401$			11841.0i			1.47459i	−0.675570	+	0.737296i	$0.736103\pi$
$402$		0			0
$403$		−	6006.28i		−	0.742418i
$404$		0			0
$405$	−682.502	+	5497.11i	−0.0837378	+	0.674453i
$406$		0			0
$407$	−8.66717			−0.00105557
$408$		0			0
$409$	13143.7			1.58903			0.794517	−	0.607241i	$-0.207724\pi$
$409$	13143.7			1.58903			0.794517	+	0.607241i	$0.207724\pi$
$410$		0			0
$411$	−481.214	+	15577.8i	−0.0577532	+	1.86958i
$412$		0			0
$413$		−	3417.74i		−	0.407206i
$414$		0			0
$415$		−	2861.11i		−	0.338426i
$416$		0			0
$417$	212.497	−	6878.94i	0.0249545	−	0.807825i
$418$		0			0
$419$	9151.59			1.06703			0.533514	−	0.845791i	$-0.320871\pi$
$419$	9151.59			1.06703			0.533514	+	0.845791i	$0.320871\pi$
$420$		0			0
$421$	4546.98			0.526380			0.263190	−	0.964744i	$-0.415225\pi$
$421$	4546.98			0.526380			0.263190	+	0.964744i	$0.415225\pi$
$422$		0			0
$423$	−8543.80	−	528.356i	−0.982066	−	0.0607318i
$424$		0			0
$425$			29.5955i			0.00337787i
$426$		0			0
$427$		−	3088.31i		−	0.350009i
$428$		0			0
$429$	−6.22896	−	0.192418i	−0.000701018	−	2.16551e-5i
$430$		0			0
$431$	1673.79			0.187061			0.0935307	−	0.995616i	$-0.470185\pi$
$431$	1673.79			0.187061			0.0935307	+	0.995616i	$0.470185\pi$
$432$		0			0
$433$	7444.60			0.826246			0.413123	−	0.910675i	$-0.364438\pi$
$433$	7444.60			0.826246			0.413123	+	0.910675i	$0.364438\pi$
$434$		0			0
$435$	10785.9	+	333.187i	1.18884	+	0.0367244i
$436$		0			0
$437$			60.7279i			0.00664762i
$438$		0			0
$439$		−	7622.30i		−	0.828685i	−0.910121	−	0.414343i	$-0.864012\pi$
$439$		−	7622.30i		−	0.828685i	0.910121	−	0.414343i	$-0.135988\pi$
$440$		0			0
$441$	−1320.48	−	81.6595i	−0.142585	−	0.00881757i
$442$		0			0
$443$	11283.3			1.21012			0.605062	−	0.796178i	$-0.293148\pi$
$443$	11283.3			1.21012			0.605062	+	0.796178i	$0.293148\pi$
$444$		0			0
$445$	3773.50			0.401980
$446$		0			0
$447$	387.533	−	12545.2i	0.0410060	−	1.32744i
$448$		0			0
$449$			1115.19i			0.117214i	0.998281	+	0.0586072i	$0.0186660\pi$
$449$			1115.19i			0.117214i	−0.998281	+	0.0586072i	$0.981334\pi$
$450$		0			0
$451$			1.71653i			0.000179221i
$452$		0			0
$453$	438.762	−	14203.6i	0.0455074	−	1.47316i
$454$		0			0
$455$	−2136.55			−0.220138
$456$		0			0
$457$	5761.30			0.589720			0.294860	−	0.955540i	$-0.404727\pi$
$457$	5761.30			0.589720			0.294860	+	0.955540i	$0.404727\pi$
$458$		0			0
$459$	5.71072	−	61.4655i	0.000580727	−	0.00625046i
$460$		0			0
$461$		−	13482.7i		−	1.36215i	−0.732212	−	0.681077i	$-0.761512\pi$
$461$		−	13482.7i		−	1.36215i	0.732212	−	0.681077i	$-0.238488\pi$
$462$		0			0
$463$			2543.21i			0.255276i	0.991821	+	0.127638i	$0.0407396\pi$
$463$			2543.21i			0.255276i	−0.991821	+	0.127638i	$0.959260\pi$
$464$		0			0
$465$	−5900.96	−	182.286i	−0.588496	−	0.0181792i
$466$		0			0
$467$	−10843.8			−1.07450			−0.537248	−	0.843424i	$-0.680536\pi$
$467$	−10843.8			−1.07450			−0.537248	+	0.843424i	$0.680536\pi$
$468$		0			0
$469$	−3081.43			−0.303384
$470$		0			0
$471$	15362.2	+	474.553i	1.50287	+	0.0464251i
$472$		0			0
$473$			12.9281i			0.00125673i
$474$		0			0
$475$		−	29.2147i		−	0.00282203i
$476$		0			0
$477$	−245.144	+	3964.11i	−0.0235312	+	0.380512i
$478$		0			0
$479$	3170.33			0.302414			0.151207	−	0.988502i	$-0.451684\pi$
$479$	3170.33			0.302414			0.151207	+	0.988502i	$0.451684\pi$
$480$		0			0
$481$	−11660.3			−1.10533
$482$		0			0
$483$	−157.024	+	5083.16i	−0.0147926	+	0.478865i
$484$		0			0
$485$			2836.85i			0.265598i
$486$		0			0
$487$			5722.95i			0.532509i	0.963903	+	0.266254i	$0.0857861\pi$
$487$			5722.95i			0.532509i	−0.963903	+	0.266254i	$0.914214\pi$
$488$		0			0
$489$	−110.892	+	3589.78i	−0.0102550	+	0.331974i
$490$		0			0
$491$	13139.2			1.20767			0.603834	−	0.797110i	$-0.293639\pi$
$491$	13139.2			1.20767			0.603834	+	0.797110i	$0.293639\pi$
$492$		0			0
$493$	−120.256			−0.0109859
$494$		0			0
$495$	−0.378088	+	6.11389i	−3.43309e−5	+	0.000555149i
$496$		0			0
$497$		−	1420.82i		−	0.128234i
$498$		0			0
$499$		−	13089.1i		−	1.17425i	−0.809498	−	0.587123i	$-0.800261\pi$
$499$		−	13089.1i		−	1.17425i	0.809498	−	0.587123i	$-0.199739\pi$
$500$		0			0
$501$	−10097.4	−	311.919i	−0.900439	−	0.0278154i
$502$		0			0
$503$	17972.9			1.59318			0.796592	−	0.604518i	$-0.206634\pi$
$503$	17972.9			1.59318			0.796592	+	0.604518i	$0.206634\pi$
$504$		0			0
$505$	2555.34			0.225171
$506$		0			0
$507$	3030.44	+	93.6133i	0.265457	+	0.00820022i
$508$		0			0
$509$		−	18517.0i		−	1.61248i	−0.591588	−	0.806240i	$-0.701499\pi$
$509$		−	18517.0i		−	1.61248i	0.591588	−	0.806240i	$-0.298501\pi$
$510$		0			0
$511$		−	3799.03i		−	0.328883i
$512$		0			0
$513$	−5.63723	+	60.6746i	−0.000485166	+	0.00522192i
$514$		0			0
$515$	2827.20			0.241905
$516$		0			0
$517$	−9.46609			−0.000805257
$518$		0			0
$519$	−234.110	+	7578.59i	−0.0198001	+	0.640969i
$520$		0			0
$521$		−	5545.36i		−	0.466308i	−0.972440	−	0.233154i	$-0.925095\pi$
$521$		−	5545.36i		−	0.466308i	0.972440	−	0.233154i	$-0.0749047\pi$
$522$		0			0
$523$		−	10763.5i		−	0.899912i	−0.893051	−	0.449956i	$-0.851440\pi$
$523$		−	10763.5i		−	0.899912i	0.893051	−	0.449956i	$-0.148560\pi$
$524$		0			0
$525$	75.5402	−	2445.38i	0.00627971	−	0.203286i
$526$		0			0
$527$	65.7918			0.00543821
$528$		0			0
$529$	7381.88			0.606714
$530$		0			0
$531$	−13157.6	−	813.677i	−1.07531	−	0.0664983i
$532$		0			0
$533$			2309.32i			0.187669i
$534$		0			0
$535$			10264.8i			0.829508i
$536$		0			0
$537$	−12458.3	−	384.850i	−1.00115	−	0.0309264i
$538$		0			0
$539$	−1.46302			−0.000116914
$540$		0			0
$541$	−9149.30			−0.727097			−0.363548	−	0.931575i	$-0.618435\pi$
$541$	−9149.30			−0.727097			−0.363548	+	0.931575i	$0.618435\pi$
$542$		0			0
$543$	11265.1	+	347.990i	0.890300	+	0.0275022i
$544$		0			0
$545$			1305.12i			0.102578i
$546$		0			0
$547$		−	4297.91i		−	0.335951i	−0.985791	−	0.167976i	$-0.946277\pi$
$547$		−	4297.91i		−	0.335951i	0.985791	−	0.167976i	$-0.0537230\pi$
$548$		0			0
$549$	−11889.4	−	735.249i	−0.924272	−	0.0571578i
$550$		0			0
$551$	118.708			0.00917813
$552$		0			0
$553$	2276.75			0.175077
$554$		0			0
$555$	−353.881	+	11455.8i	−0.0270656	+	0.876167i
$556$		0			0
$557$		−	1753.57i		−	0.133395i	−0.997773	−	0.0666975i	$-0.978754\pi$
$557$		−	1753.57i		−	0.133395i	0.997773	−	0.0666975i	$-0.0212462\pi$
$558$		0			0
$559$			17392.7i			1.31598i
$560$		0			0
$561$	0.00210772	−	0.0682309i	1.58624e−7	−	5.13496e-6i
$562$		0			0
$563$	10323.1			0.772764			0.386382	−	0.922339i	$-0.373725\pi$
$563$	10323.1			0.772764			0.386382	+	0.922339i	$0.373725\pi$
$564$		0			0
$565$	8602.36			0.640538
$566$		0			0
$567$	−628.744	+	5064.12i	−0.0465692	+	0.375085i
$568$		0			0
$569$			15103.5i			1.11278i	0.830920	+	0.556391i	$0.187814\pi$
$569$			15103.5i			1.11278i	−0.830920	+	0.556391i	$0.812186\pi$
$570$		0			0
$571$			9072.75i			0.664944i	0.943113	+	0.332472i	$0.107883\pi$
$571$			9072.75i			0.664944i	−0.943113	+	0.332472i	$0.892117\pi$
$572$		0			0
$573$	−17375.9	−	536.757i	−1.26682	−	0.0391333i
$574$		0			0
$575$	−9404.49			−0.682077
$576$		0			0
$577$	3540.84			0.255472			0.127736	−	0.991808i	$-0.459229\pi$
$577$	3540.84			0.255472			0.127736	+	0.991808i	$0.459229\pi$
$578$		0			0
$579$	7274.45	+	224.715i	0.522134	+	0.0161292i
$580$		0			0
$581$		−	2635.75i		−	0.188209i
$582$		0			0
$583$			4.39203i			0.000312006i
$584$		0			0
$585$	−508.657	+	8225.25i	−0.0359494	+	0.581320i
$586$		0			0
$587$	−11465.3			−0.806171			−0.403085	−	0.915162i	$-0.632062\pi$
$587$	−11465.3			−0.806171			−0.403085	+	0.915162i	$0.632062\pi$
$588$		0			0
$589$	−64.9452			−0.00454333
$590$		0			0
$591$	601.210	−	19462.3i	0.0418452	−	1.35461i
$592$		0			0
$593$		−	4829.33i		−	0.334430i	−0.985920	−	0.167215i	$-0.946523\pi$
$593$		−	4829.33i		−	0.334430i	0.985920	−	0.167215i	$-0.0534774\pi$
$594$		0			0
$595$		−	23.4034i		−	0.00161251i
$596$		0			0
$597$	−186.183	+	6027.10i	−0.0127638	+	0.413188i
$598$		0			0
$599$	−9235.49			−0.629970			−0.314985	−	0.949097i	$-0.601999\pi$
$599$	−9235.49			−0.629970			−0.314985	+	0.949097i	$0.601999\pi$
$600$		0			0
$601$	7969.74			0.540919			0.270460	−	0.962731i	$-0.412824\pi$
$601$	7969.74			0.540919			0.270460	+	0.962731i	$0.412824\pi$
$602$		0			0
$603$	−733.609	+	11862.8i	−0.0495437	+	0.801148i
$604$		0			0
$605$		−	10113.6i		−	0.679631i
$606$		0			0
$607$			12277.3i			0.820953i	0.911871	+	0.410477i	$0.134638\pi$
$607$			12277.3i			0.820953i	−0.911871	+	0.410477i	$0.865362\pi$
$608$		0			0
$609$	9936.35	+	306.943i	0.661152	+	0.0204236i
$610$		0			0
$611$	−12735.1			−0.843219
$612$		0			0
$613$	2671.59			0.176027			0.0880135	−	0.996119i	$-0.471948\pi$
$613$	2671.59			0.176027			0.0880135	+	0.996119i	$0.471948\pi$
$614$		0			0
$615$	2268.83	+	70.0862i	0.148761	+	0.00459536i
$616$		0			0
$617$			16638.0i			1.08561i	0.839860	+	0.542803i	$0.182637\pi$
$617$			16638.0i			1.08561i	−0.839860	+	0.542803i	$0.817363\pi$
$618$		0			0
$619$		−	30122.8i		−	1.95596i	−0.208698	−	0.977980i	$-0.566923\pi$
$619$		−	30122.8i		−	1.95596i	0.208698	−	0.977980i	$-0.433077\pi$
$620$		0			0
$621$	19531.7	+	1814.68i	1.26213	+	0.117263i
$622$		0			0
$623$	3476.27			0.223554
$624$		0			0
$625$	−2692.90			−0.172345
$626$		0			0
$627$	−0.00208060	+	0.0673529i	−1.32522e−7	+	4.28998e-6i
$628$		0			0
$629$		−	127.725i		−	0.00809654i
$630$		0			0
$631$		−	4373.21i		−	0.275903i	−0.990439	−	0.137952i	$-0.955948\pi$
$631$		−	4373.21i		−	0.275903i	0.990439	−	0.137952i	$-0.0440518\pi$
$632$		0			0
$633$	−619.329	+	20048.9i	−0.0388880	+	1.25888i
$634$		0			0
$635$	−17364.6			−1.08518
$636$		0			0
$637$	−1968.26			−0.122426
$638$		0			0
$639$	−5469.84	−	338.260i	−0.338628	−	0.0209411i
$640$		0			0
$641$			9337.02i			0.575335i	0.957730	+	0.287668i	$0.0928799\pi$
$641$			9337.02i			0.575335i	−0.957730	+	0.287668i	$0.907120\pi$
$642$		0			0
$643$			22548.8i			1.38295i	0.722400	+	0.691476i	$0.243039\pi$
$643$			22548.8i			1.38295i	−0.722400	+	0.691476i	$0.756961\pi$
$644$		0			0
$645$	17087.7	+	527.854i	1.04314	+	0.0322236i
$646$		0			0
$647$	−21092.7			−1.28167			−0.640834	−	0.767680i	$-0.721411\pi$
$647$	−21092.7			−1.28167			−0.640834	+	0.767680i	$0.721411\pi$
$648$		0			0
$649$	−14.5779			−0.000881716
$650$		0			0
$651$	−5436.16	−	167.928i	−0.327281	−	0.0101100i
$652$		0			0
$653$		−	23059.3i		−	1.38190i	−0.722902	−	0.690951i	$-0.757192\pi$
$653$		−	23059.3i		−	1.38190i	0.722902	−	0.690951i	$-0.242808\pi$
$654$		0			0
$655$			12030.1i			0.717641i
$656$		0			0
$657$	−14625.5	−	904.453i	−0.868485	−	0.0537079i
$658$		0			0
$659$	22080.3			1.30520			0.652601	−	0.757702i	$-0.273678\pi$
$659$	22080.3			1.30520			0.652601	+	0.757702i	$0.273678\pi$
$660$		0			0
$661$	−22653.9			−1.33303			−0.666516	−	0.745491i	$-0.732215\pi$
$661$	−22653.9			−1.33303			−0.666516	+	0.745491i	$0.732215\pi$
$662$		0			0
$663$	2.83560	−	91.7937i	0.000166102	−	0.00537703i
$664$		0			0
$665$			23.1022i			0.00134717i
$666$		0			0
$667$		−	38213.4i		−	2.21833i
$668$		0			0
$669$	798.131	−	25837.1i	0.0461249	−	1.49315i
$670$		0			0
$671$	−13.1728			−0.000757869
$672$		0			0
$673$	3686.80			0.211168			0.105584	−	0.994410i	$-0.466329\pi$
$673$	3686.80			0.211168			0.105584	+	0.994410i	$0.466329\pi$
$674$		0			0
$675$	−9396.23	−	872.997i	−0.535794	−	0.0497803i
$676$		0			0
$677$			23120.2i			1.31253i	0.754531	+	0.656264i	$0.227864\pi$
$677$			23120.2i			1.31253i	−0.754531	+	0.656264i	$0.772136\pi$
$678$		0			0
$679$			2613.41i			0.147707i
$680$		0			0
$681$	18183.2	+	561.695i	1.02317	+	0.0316068i
$682$		0			0
$683$	−14836.8			−0.831207			−0.415603	−	0.909546i	$-0.636430\pi$
$683$	−14836.8			−0.831207			−0.415603	+	0.909546i	$0.636430\pi$
$684$		0			0
$685$	22790.9			1.27123
$686$		0			0
$687$	−21796.9	−	673.326i	−1.21048	−	0.0373930i
$688$		0			0
$689$			5908.77i			0.326715i
$690$		0			0
$691$		−	25813.4i		−	1.42111i	−0.703642	−	0.710555i	$-0.748444\pi$
$691$		−	25813.4i		−	1.42111i	0.703642	−	0.710555i	$-0.251556\pi$
$692$		0			0
$693$	−0.348308	+	5.63232i	−1.90925e−5	+	0.000308736i
$694$		0			0
$695$	−10064.1			−0.549285
$696$		0			0
$697$	−25.2959			−0.00137468
$698$		0			0
$699$	789.089	−	25544.3i	0.0426983	−	1.38223i
$700$		0			0
$701$		−	16480.0i		−	0.887931i	−0.896044	−	0.443965i	$-0.853571\pi$
$701$		−	16480.0i		−	0.887931i	0.896044	−	0.443965i	$-0.146429\pi$
$702$		0			0
$703$			126.081i			0.00676422i
$704$		0			0
$705$	−386.501	+	12511.8i	−0.0206475	+	0.668399i
$706$		0			0
$707$	2354.07			0.125225
$708$		0			0
$709$	2993.67			0.158575			0.0792875	−	0.996852i	$-0.474735\pi$
$709$	2993.67			0.158575			0.0792875	+	0.996852i	$0.474735\pi$
$710$		0			0
$711$	542.037	−	8765.02i	0.0285907	−	0.462326i
$712$		0			0
$713$			20906.5i			1.09811i
$714$		0			0
$715$			9.11316i			0.000476661i
$716$		0			0
$717$	5409.87	+	167.116i	0.281779	+	0.00870442i
$718$		0			0
$719$	17596.5			0.912711			0.456355	−	0.889798i	$-0.349155\pi$
$719$	17596.5			0.912711			0.456355	+	0.889798i	$0.349155\pi$
$720$		0			0
$721$	2604.51			0.134531
$722$		0			0
$723$	−10635.3	−	328.535i	−0.547071	−	0.0168995i
$724$		0			0
$725$			18383.5i			0.941719i
$726$		0			0
$727$			33368.5i			1.70229i	0.524927	+	0.851147i	$0.324093\pi$
$727$			33368.5i			1.70229i	−0.524927	+	0.851147i	$0.675907\pi$
$728$		0			0
$729$	19346.1	+	3626.17i	0.982883	+	0.184228i
$730$		0			0
$731$	−190.516			−0.00963953
$732$		0			0
$733$	−23542.7			−1.18632			−0.593158	−	0.805086i	$-0.702119\pi$
$733$	−23542.7			−1.18632			−0.593158	+	0.805086i	$0.702119\pi$
$734$		0			0
$735$	−59.7352	+	1933.74i	−0.00299778	+	0.0970439i
$736$		0			0
$737$			13.1434i			0.000656911i
$738$		0			0
$739$			23280.9i			1.15886i	0.815021	+	0.579432i	$0.196726\pi$
$739$			23280.9i			1.15886i	−0.815021	+	0.579432i	$0.803274\pi$
$740$		0			0
$741$	−2.79911	+	90.6125i	−0.000138769	+	0.00449222i
$742$		0			0
$743$	−35371.0			−1.74648			−0.873242	−	0.487287i	$-0.837987\pi$
$743$	−35371.0			−1.74648			−0.873242	+	0.487287i	$0.837987\pi$
$744$		0			0
$745$	−18354.0			−0.902602
$746$		0			0
$747$	−10147.1	−	627.506i	−0.497005	−	0.0307353i
$748$		0			0
$749$			9456.29i			0.461316i
$750$		0			0
$751$		−	8592.26i		−	0.417492i	−0.977970	−	0.208746i	$-0.933062\pi$
$751$		−	8592.26i		−	0.417492i	0.977970	−	0.208746i	$-0.0669381\pi$
$752$		0			0
$753$	3849.67	+	118.920i	0.186308	+	0.00575522i
$754$		0			0
$755$	−20780.3			−1.00168
$756$		0			0
$757$	5401.34			0.259333			0.129666	−	0.991558i	$-0.458609\pi$
$757$	5401.34			0.259333			0.129666	+	0.991558i	$0.458609\pi$
$758$		0			0
$759$	21.6815	+	0.669763i	0.00103688	+	3.20301e-5i
$760$		0			0
$761$			34301.0i			1.63391i	0.576698	+	0.816957i	$0.304341\pi$
$761$			34301.0i			1.63391i	−0.576698	+	0.816957i	$0.695659\pi$
$762$		0			0
$763$			1202.32i			0.0570471i
$764$		0			0
$765$	−90.0980	−	5.57174i	−0.00425817	−	0.000263329i
$766$		0			0
$767$	−19612.3			−0.923282
$768$		0			0
$769$	−23217.6			−1.08875			−0.544375	−	0.838842i	$-0.683233\pi$
$769$	−23217.6			−1.08875			−0.544375	+	0.838842i	$0.683233\pi$
$770$		0			0
$771$	−1259.35	+	40767.6i	−0.0588254	+	1.90429i
$772$		0			0
$773$			9655.19i			0.449254i	0.974445	+	0.224627i	$0.0721163\pi$
$773$			9655.19i			0.449254i	−0.974445	+	0.224627i	$0.927884\pi$
$774$		0			0
$775$		−	10057.6i		−	0.466167i
$776$		0			0
$777$	−326.007	+	10553.5i	−0.0150521	+	0.487264i
$778$		0			0
$779$	24.9704			0.00114847
$780$		0			0
$781$	−6.06030			−0.000277663
$782$		0			0
$783$	3547.26	−	38179.8i	0.161901	−	1.74257i
$784$		0			0
$785$		−	22475.4i		−	1.02189i
$786$		0			0
$787$		−	7935.72i		−	0.359438i	−0.983718	−	0.179719i	$-0.942481\pi$
$787$		−	7935.72i		−	0.359438i	0.983718	−	0.179719i	$-0.0575189\pi$
$788$		0			0
$789$	7886.66	+	243.626i	0.355859	+	0.0109928i
$790$		0			0
$791$	7924.79			0.356224
$792$		0			0
$793$	−17721.9			−0.793597
$794$		0			0
$795$	5805.16	+	179.327i	0.258978	+	0.00800009i
$796$		0			0
$797$		−	13066.5i		−	0.580726i	−0.956917	−	0.290363i	$-0.906224\pi$
$797$		−	13066.5i		−	0.580726i	0.956917	−	0.290363i	$-0.0937760\pi$
$798$		0			0
$799$		−	139.498i		−	0.00617658i
$800$		0			0
$801$	827.612	−	13382.9i	0.0365072	−	0.590340i
$802$		0			0
$803$	−16.2043			−0.000712125
$804$		0			0
$805$	7436.82			0.325607
$806$		0			0
$807$	−444.097	+	14376.3i	−0.0193717	+	0.627099i
$808$		0			0
$809$			5705.93i			0.247973i	0.992284	+	0.123986i	$0.0395679\pi$
$809$			5705.93i			0.247973i	−0.992284	+	0.123986i	$0.960432\pi$
$810$		0			0
$811$			35006.8i			1.51573i	0.652413	+	0.757864i	$0.273757\pi$
$811$			35006.8i			1.51573i	−0.652413	+	0.757864i	$0.726243\pi$
$812$		0			0
$813$	447.484	−	14485.9i	0.0193038	−	0.624900i
$814$		0			0
$815$	5251.96			0.225728
$816$		0			0
$817$	188.065			0.00805330
$818$		0			0
$819$	−468.592	+	7577.38i	−0.0199926	+	0.323291i
$820$		0			0
$821$		−	15875.8i		−	0.674869i	−0.941349	−	0.337435i	$-0.890441\pi$
$821$		−	15875.8i		−	0.674869i	0.941349	−	0.337435i	$-0.109559\pi$
$822$		0			0
$823$			32848.1i			1.39127i	0.718397	+	0.695634i	$0.244876\pi$
$823$			32848.1i			1.39127i	−0.718397	+	0.695634i	$0.755124\pi$
$824$		0			0
$825$	−10.4305	−	0.322207i	−0.000440172	−	1.35973e-5i
$826$		0			0
$827$	10175.0			0.427836			0.213918	−	0.976852i	$-0.431378\pi$
$827$	10175.0			0.427836			0.213918	+	0.976852i	$0.431378\pi$
$828$		0			0
$829$	8733.63			0.365901			0.182950	−	0.983122i	$-0.441435\pi$
$829$	8733.63			0.365901			0.182950	+	0.983122i	$0.441435\pi$
$830$		0			0
$831$	15041.1	+	464.635i	0.627884	+	0.0193959i
$832$		0			0
$833$		−	21.5600i		−	0.000896769i
$834$		0			0
$835$			14772.9i			0.612258i
$836$		0			0
$837$	−1940.70	+	20888.1i	−0.0801439	+	0.862603i
$838$		0			0
$839$	23404.1			0.963050			0.481525	−	0.876432i	$-0.340083\pi$
$839$	23404.1			0.963050			0.481525	+	0.876432i	$0.340083\pi$
$840$		0			0
$841$	−50308.9			−2.06277
$842$		0			0
$843$	751.421	−	24324.9i	0.0307002	−	0.993826i
$844$		0			0
$845$		−	4433.63i		−	0.180499i
$846$		0			0
$847$		−	9316.99i		−	0.377964i
$848$		0			0
$849$	1368.13	−	44289.2i	0.0553054	−	1.79034i
$850$		0			0
$851$	40586.7			1.63489
$852$		0			0
$853$	4019.79			0.161354			0.0806771	−	0.996740i	$-0.474292\pi$
$853$	4019.79			0.161354			0.0806771	+	0.996740i	$0.474292\pi$
$854$		0			0
$855$	88.9387	+	5.50005i	0.00355747	+	0.000219997i
$856$		0			0
$857$			34890.1i			1.39069i	0.718676	+	0.695345i	$0.244749\pi$
$857$			34890.1i			1.39069i	−0.718676	+	0.695345i	$0.755251\pi$
$858$		0			0
$859$			22977.7i			0.912675i	0.889807	+	0.456338i	$0.150839\pi$
$859$			22977.7i			0.912675i	−0.889807	+	0.456338i	$0.849161\pi$
$860$		0			0
$861$	2090.12	+	64.5658i	0.0827306	+	0.00255563i
$862$		0			0
$863$	−2184.37			−0.0861607			−0.0430804	−	0.999072i	$-0.513717\pi$
$863$	−2184.37			−0.0861607			−0.0430804	+	0.999072i	$0.513717\pi$
$864$		0			0
$865$	11087.7			0.435830
$866$		0			0
$867$	−25515.5	−	788.199i	−0.999484	−	0.0308750i
$868$		0			0
$869$		−	9.71118i		−	0.000379090i
$870$		0			0
$871$			17682.3i			0.687880i
$872$		0			0
$873$	10061.1	+	622.185i	0.390052	+	0.0241212i
$874$		0			0
$875$	−10226.4			−0.395102
$876$		0			0
$877$	−48020.9			−1.84898			−0.924488	−	0.381211i	$-0.875507\pi$
$877$	−48020.9			−1.84898			−0.924488	+	0.381211i	$0.875507\pi$
$878$		0			0
$879$	1383.22	−	44777.6i	0.0530773	−	1.71822i
$880$		0			0
$881$		−	48557.5i		−	1.85692i	−0.371438	−	0.928458i	$-0.621135\pi$
$881$		−	48557.5i		−	1.85692i	0.371438	−	0.928458i	$-0.378865\pi$
$882$		0			0
$883$		−	34418.8i		−	1.31176i	−0.754865	−	0.655880i	$-0.772298\pi$
$883$		−	34418.8i		−	1.31176i	0.754865	−	0.655880i	$-0.227702\pi$
$884$		0			0
$885$	−595.217	+	19268.3i	−0.0226079	+	0.731863i
$886$		0			0
$887$	1694.17			0.0641314			0.0320657	−	0.999486i	$-0.489791\pi$
$887$	1694.17			0.0641314			0.0320657	+	0.999486i	$0.489791\pi$
$888$		0			0
$889$	−15996.8			−0.603506
$890$		0			0
$891$	21.6003	+	2.68182i	0.000812164	+	0.000100835i
$892$		0			0
$893$			137.703i			0.00516020i
$894$		0			0
$895$			18226.9i			0.680736i
$896$		0			0
$897$	29169.0	+	901.059i	1.08576	+	0.0335401i
$898$		0			0
$899$	40867.1			1.51612
$900$		0			0
$901$	−64.7237			−0.00239318
$902$		0			0
$903$	15741.7	+	486.277i	0.580124	+	0.0179206i
$904$		0			0
$905$		−	16481.2i		−	0.605364i
$906$		0			0
$907$		−	18211.1i		−	0.666691i	−0.942805	−	0.333346i	$-0.891822\pi$
$907$		−	18211.1i		−	0.666691i	0.942805	−	0.333346i	$-0.108178\pi$
$908$		0			0
$909$	560.443	−	9062.66i	0.0204496	−	0.330681i
$910$		0			0
$911$	−7379.36			−0.268374			−0.134187	−	0.990956i	$-0.542842\pi$
$911$	−7379.36			−0.268374			−0.134187	+	0.990956i	$0.542842\pi$
$912$		0			0
$913$	−11.2425			−0.000407526
$914$		0			0
$915$	−537.846	+	17411.1i	−0.0194324	+	0.629064i
$916$		0			0
$917$			11082.5i			0.399103i
$918$		0			0
$919$		−	4388.74i		−	0.157531i	−0.996893	−	0.0787656i	$-0.974902\pi$
$919$		−	4388.74i		−	0.157531i	0.996893	−	0.0787656i	$-0.0250979\pi$
$920$		0			0
$921$	−685.080	+	22177.4i	−0.0245105	+	0.793451i
$922$		0			0
$923$	−8153.16			−0.290752
$924$		0			0
$925$	−19525.3			−0.694041
$926$		0			0
$927$	620.067	−	10026.8i	0.0219694	−	0.355257i
$928$		0			0
$929$		−	27520.1i		−	0.971910i	−0.873984	−	0.485955i	$-0.838472\pi$
$929$		−	27520.1i		−	0.971910i	0.873984	−	0.485955i	$-0.161528\pi$
$930$		0			0
$931$			21.2825i			0.000749202i
$932$		0			0
$933$	46059.9	+	1422.83i	1.61622	+	0.0499266i
$934$		0			0
$935$	−0.0998239			−3.49154e−6
$936$		0			0
$937$	46438.8			1.61909			0.809546	−	0.587057i	$-0.199714\pi$
$937$	46438.8			1.61909			0.809546	+	0.587057i	$0.199714\pi$
$938$		0			0
$939$	32582.0	+	1006.49i	1.13235	+	0.0349793i
$940$		0			0
$941$			17701.1i			0.613219i	0.951835	+	0.306610i	$0.0991946\pi$
$941$			17701.1i			0.613219i	−0.951835	+	0.306610i	$0.900805\pi$
$942$		0			0
$943$		−	8038.21i		−	0.277582i
$944$		0			0
$945$	7430.29	+	690.343i	0.255775	+	0.0237639i
$946$		0			0
$947$	−55200.9			−1.89418			−0.947090	−	0.320967i	$-0.895992\pi$
$947$	−55200.9			−1.89418			−0.947090	+	0.320967i	$0.895992\pi$
$948$		0			0
$949$	−21800.2			−0.745696
$950$		0			0
$951$	1768.20	−	57240.2i	0.0602922	−	1.95178i
$952$		0			0
$953$		−	3055.84i		−	0.103870i	−0.998650	−	0.0519351i	$-0.983461\pi$
$953$		−	3055.84i		−	0.103870i	0.998650	−	0.0519351i	$-0.0165389\pi$
$954$		0			0
$955$			25421.4i			0.861381i
$956$		0			0
$957$	1.30923	−	42.3822i	4.42228e−5	−	0.00143158i
$958$		0			0
$959$	20995.7			0.706973
$960$		0			0
$961$	7432.66			0.249494
$962$		0			0
$963$	36404.7	+	2251.30i	1.21820	+	0.0753346i
$964$		0			0
$965$		−	10642.7i		−	0.355028i
$966$		0			0
$967$		−	9976.97i		−	0.331787i	−0.986144	−	0.165893i	$-0.946949\pi$
$967$		−	9976.97i		−	0.331787i	0.986144	−	0.165893i	$-0.0530508\pi$
$968$		0			0
$969$	−0.992554	−	0.0306609i	−3.29055e−5	−	1.01648e-6i
$970$		0			0
$971$	25909.3			0.856301			0.428150	−	0.903707i	$-0.359165\pi$
$971$	25909.3			0.856301			0.428150	+	0.903707i	$0.359165\pi$
$972$		0			0
$973$	−9271.39			−0.305475
$974$		0			0
$975$	−14032.5	−	433.477i	−0.460923	−	0.0142383i
$976$		0			0
$977$		−	27062.6i		−	0.886193i	−0.896474	−	0.443096i	$-0.853880\pi$
$977$		−	27062.6i		−	0.886193i	0.896474	−	0.443096i	$-0.146120\pi$
$978$		0			0
$979$		−	14.8276i		−	0.000484057i
$980$		0			0
$981$	4628.68	+	286.242i	0.150645	+	0.00931600i
$982$		0			0
$983$	44352.2			1.43908			0.719540	−	0.694451i	$-0.244353\pi$
$983$	44352.2			1.43908			0.719540	+	0.694451i	$0.244353\pi$
$984$		0			0
$985$	−28474.0			−0.921074
$986$		0			0
$987$	−356.058	+	11526.3i	−0.0114827	+	0.371718i
$988$		0			0
$989$		−	60539.8i		−	1.94646i
$990$		0			0
$991$		−	21159.0i		−	0.678243i	−0.940742	−	0.339122i	$-0.889870\pi$
$991$		−	21159.0i		−	0.678243i	0.940742	−	0.339122i	$-0.110130\pi$
$992$		0			0
$993$	308.604	−	9990.11i	0.00986229	−	0.319261i
$994$		0			0
$995$	8817.84			0.280949
$996$		0			0
$997$	27664.4			0.878775			0.439388	−	0.898298i	$-0.355195\pi$
$997$	27664.4			0.878775			0.439388	+	0.898298i	$0.355195\pi$
$998$		0			0
$999$	40551.1	+	3767.57i	1.28426	+	0.119320i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.h.b.239.1	✓	24
3.2	odd	2	inner	336.4.h.b.239.23	yes	24
4.3	odd	2	inner	336.4.h.b.239.24	yes	24
12.11	even	2	inner	336.4.h.b.239.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.h.b.239.1	✓	24	1.1	even	1	trivial
336.4.h.b.239.2	yes	24	12.11	even	2	inner
336.4.h.b.239.23	yes	24	3.2	odd	2	inner
336.4.h.b.239.24	yes	24	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-5.19367 - 0.160438i) q^{3} +7.59851i q^{5} +7.00000i q^{7} +(26.9485 + 1.66652i) q^{9} +O(q^{10})\)	\(q+(-5.19367 - 0.160438i) q^{3} +7.59851i q^{5} +7.00000i q^{7} +(26.9485 + 1.66652i) q^{9} +0.0298576 q^{11} +40.1686 q^{13} +(1.21909 - 39.4642i) q^{15} +0.439999i q^{17} -0.434338i q^{19} +(1.12306 - 36.3557i) q^{21} -139.817 q^{23} +67.2627 q^{25} +(-139.694 - 12.9789i) q^{27} +273.309i q^{29} -149.527i q^{31} +(-0.155070 - 0.00479027i) q^{33} -53.1896 q^{35} -290.284 q^{37} +(-208.622 - 6.44454i) q^{39} +57.4908i q^{41} +432.992i q^{43} +(-12.6631 + 204.769i) q^{45} -317.042 q^{47} -49.0000 q^{49} +(0.0705924 - 2.28521i) q^{51} +147.100i q^{53} +0.226873i q^{55} +(-0.0696841 + 2.25581i) q^{57} -488.249 q^{59} -441.188 q^{61} +(-11.6656 + 188.640i) q^{63} +305.221i q^{65} +440.204i q^{67} +(726.166 + 22.4319i) q^{69} -202.974 q^{71} -542.719 q^{73} +(-349.341 - 10.7915i) q^{75} +0.209003i q^{77} -325.250i q^{79} +(723.445 + 89.8205i) q^{81} -376.536 q^{83} -3.34334 q^{85} +(43.8491 - 1419.48i) q^{87} -496.611i q^{89} +281.180i q^{91} +(-23.9898 + 776.595i) q^{93} +3.30032 q^{95} +373.344 q^{97} +(0.804617 + 0.0497582i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 136 q^{9}+O(q^{10}) \) 24 * q - 136 * q^9	\( 24 q - 136 q^{9} + 96 q^{13} - 112 q^{21} + 168 q^{25} - 320 q^{33} - 864 q^{37} + 592 q^{45} - 1176 q^{49} - 1120 q^{57} - 2304 q^{61} + 1168 q^{69} + 3312 q^{73} - 968 q^{81} - 5808 q^{85} + 1776 q^{93} + 6480 q^{97}+O(q^{100}) \) 24 * q - 136 * q^9 + 96 * q^13 - 112 * q^21 + 168 * q^25 - 320 * q^33 - 864 * q^37 + 592 * q^45 - 1176 * q^49 - 1120 * q^57 - 2304 * q^61 + 1168 * q^69 + 3312 * q^73 - 968 * q^81 - 5808 * q^85 + 1776 * q^93 + 6480 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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