LMFDB - Embedded newform 336.4.h.a.239.5

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 152x^{10} + 8222x^{8} + 194132x^{6} + 1882697x^{4} + 5152508x^{2} + 4008004 $ x^12 + 152x^10 + 8222x^8 + 194132x^6 + 1882697x^4 + 5152508*x^2 + 4008004
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.5
Root			$-1.52642i$ of defining polynomial
Character	$\chi$	$=$	336.239
Dual form			336.4.h.a.239.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.18705 - 4.71347i) q^{3} +14.5852i q^{5} +7.00000i q^{7} +(-17.4336 + 20.6172i) q^{9} +O(q^{10})$	\(q+(-2.18705 - 4.71347i) q^{3} +14.5852i q^{5} +7.00000i q^{7} +(-17.4336 + 20.6172i) q^{9} +31.0233 q^{11} -55.7458 q^{13} +(68.7469 - 31.8986i) q^{15} -86.4384i q^{17} -96.8558i q^{19} +(32.9943 - 15.3094i) q^{21} -115.219 q^{23} -87.7283 q^{25} +(135.307 + 37.0818i) q^{27} -49.1735i q^{29} +12.5750i q^{31} +(-67.8496 - 146.228i) q^{33} -102.096 q^{35} -296.822 q^{37} +(121.919 + 262.756i) q^{39} -213.956i q^{41} -165.680i q^{43} +(-300.706 - 254.273i) q^{45} -426.135 q^{47} -49.0000 q^{49} +(-407.425 + 189.045i) q^{51} +460.767i q^{53} +452.482i q^{55} +(-456.527 + 211.829i) q^{57} +686.382 q^{59} -583.424 q^{61} +(-144.320 - 122.035i) q^{63} -813.064i q^{65} -589.155i q^{67} +(251.989 + 543.080i) q^{69} -766.330 q^{71} -904.778 q^{73} +(191.866 + 413.505i) q^{75} +217.163i q^{77} -459.440i q^{79} +(-121.139 - 718.865i) q^{81} +1115.25 q^{83} +1260.72 q^{85} +(-231.778 + 107.545i) q^{87} +119.900i q^{89} -390.221i q^{91} +(59.2717 - 27.5021i) q^{93} +1412.66 q^{95} -331.017 q^{97} +(-540.848 + 639.614i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 76 q^{9}+O(q^{10}) $ 12 * q + 76 * q^9	$ 12 q + 76 q^{9} - 96 q^{13} + 112 q^{21} - 1068 q^{25} - 832 q^{33} - 720 q^{37} + 392 q^{45} - 588 q^{49} - 2336 q^{57} + 432 q^{61} - 424 q^{69} + 1656 q^{73} - 868 q^{81} - 1464 q^{85} + 696 q^{93} - 6264 q^{97}+O(q^{100}) $ 12 * q + 76 * q^9 - 96 * q^13 + 112 * q^21 - 1068 * q^25 - 832 * q^33 - 720 * q^37 + 392 * q^45 - 588 * q^49 - 2336 * q^57 + 432 * q^61 - 424 * q^69 + 1656 * q^73 - 868 * q^81 - 1464 * q^85 + 696 * q^93 - 6264 * 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
$3$	−2.18705	−	4.71347i	−0.420898	−	0.907108i
$3$	−2.18705	−	4.71347i	−0.420898	−	0.907108i
$4$		0			0
$5$			14.5852i			1.30454i	0.757986	+	0.652270i	$0.226183\pi$
$5$			14.5852i			1.30454i	−0.757986	+	0.652270i	$0.773817\pi$
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		0			0
$9$	−17.4336	+	20.6172i	−0.645689	+	0.763600i
$10$		0			0
$11$	31.0233			0.850353			0.425177	−	0.905110i	$-0.360212\pi$
$11$	31.0233			0.850353			0.425177	+	0.905110i	$0.360212\pi$
$12$		0			0
$13$	−55.7458			−1.18932			−0.594658	−	0.803979i	$-0.702713\pi$
$13$	−55.7458			−1.18932			−0.594658	+	0.803979i	$0.702713\pi$
$14$		0			0
$15$	68.7469	−	31.8986i	1.18336	−	0.549079i
$16$		0			0
$17$		−	86.4384i		−	1.23320i	−0.787277	−	0.616600i	$-0.788510\pi$
$17$		−	86.4384i		−	1.23320i	0.787277	−	0.616600i	$-0.211490\pi$
$18$		0			0
$19$		−	96.8558i		−	1.16949i	−0.811218	−	0.584743i	$-0.801195\pi$
$19$		−	96.8558i		−	1.16949i	0.811218	−	0.584743i	$-0.198805\pi$
$20$		0			0
$21$	32.9943	−	15.3094i	0.342855	−	0.159085i
$22$		0			0
$23$	−115.219			−1.04456			−0.522278	−	0.852775i	$-0.674918\pi$
$23$	−115.219			−1.04456			−0.522278	+	0.852775i	$0.674918\pi$
$24$		0			0
$25$	−87.7283			−0.701826
$26$		0			0
$27$	135.307	+	37.0818i	0.964437	+	0.264311i
$28$		0			0
$29$		−	49.1735i		−	0.314872i	−0.987529	−	0.157436i	$-0.949677\pi$
$29$		−	49.1735i		−	0.314872i	0.987529	−	0.157436i	$-0.0503228\pi$
$30$		0			0
$31$			12.5750i			0.0728558i	0.999336	+	0.0364279i	$0.0115979\pi$
$31$			12.5750i			0.0728558i	−0.999336	+	0.0364279i	$0.988402\pi$
$32$		0			0
$33$	−67.8496	−	146.228i	−0.357912	−	0.771362i
$34$		0			0
$35$	−102.096			−0.493070
$36$		0			0
$37$	−296.822			−1.31884			−0.659421	−	0.751774i	$-0.729199\pi$
$37$	−296.822			−1.31884			−0.659421	+	0.751774i	$0.729199\pi$
$38$		0			0
$39$	121.919	+	262.756i	0.500581	+	1.07884i
$40$		0			0
$41$		−	213.956i		−	0.814982i	−0.913209	−	0.407491i	$-0.866404\pi$
$41$		−	213.956i		−	0.814982i	0.913209	−	0.407491i	$-0.133596\pi$
$42$		0			0
$43$		−	165.680i		−	0.587581i	−0.955870	−	0.293790i	$-0.905083\pi$
$43$		−	165.680i		−	0.587581i	0.955870	−	0.293790i	$-0.0949168\pi$
$44$		0			0
$45$	−300.706	−	254.273i	−0.996148	−	0.842328i
$46$		0			0
$47$	−426.135			−1.32251			−0.661257	−	0.750159i	$-0.729977\pi$
$47$	−426.135			−1.32251			−0.661257	+	0.750159i	$0.729977\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	−407.425	+	189.045i	−1.11864	+	0.519052i
$52$		0			0
$53$			460.767i			1.19417i	0.802177	+	0.597087i	$0.203675\pi$
$53$			460.767i			1.19417i	−0.802177	+	0.597087i	$0.796325\pi$
$54$		0			0
$55$			452.482i			1.10932i
$56$		0			0
$57$	−456.527	+	211.829i	−1.06085	+	0.492235i
$58$		0			0
$59$	686.382			1.51456			0.757282	−	0.653088i	$-0.226527\pi$
$59$	686.382			1.51456			0.757282	+	0.653088i	$0.226527\pi$
$60$		0			0
$61$	−583.424			−1.22459			−0.612293	−	0.790631i	$-0.709753\pi$
$61$	−583.424			−1.22459			−0.612293	+	0.790631i	$0.709753\pi$
$62$		0			0
$63$	−144.320	−	122.035i	−0.288614	−	0.244048i
$64$		0			0
$65$		−	813.064i		−	1.55151i
$66$		0			0
$67$		−	589.155i		−	1.07428i	−0.843493	−	0.537140i	$-0.819505\pi$
$67$		−	589.155i		−	1.07428i	0.843493	−	0.537140i	$-0.180495\pi$
$68$		0			0
$69$	251.989	+	543.080i	0.439652	+	0.947524i
$70$		0			0
$71$	−766.330			−1.28094			−0.640469	−	0.767984i	$-0.721260\pi$
$71$	−766.330			−1.28094			−0.640469	+	0.767984i	$0.721260\pi$
$72$		0			0
$73$	−904.778			−1.45063			−0.725317	−	0.688415i	$-0.758307\pi$
$73$	−904.778			−1.45063			−0.725317	+	0.688415i	$0.758307\pi$
$74$		0			0
$75$	191.866	+	413.505i	0.295397	+	0.636632i
$76$		0			0
$77$			217.163i			0.321403i
$78$		0			0
$79$		−	459.440i		−	0.654317i	−0.944969	−	0.327159i	$-0.893909\pi$
$79$		−	459.440i		−	0.654317i	0.944969	−	0.327159i	$-0.106091\pi$
$80$		0			0
$81$	−121.139	−	718.865i	−0.166171	−	0.986097i
$82$		0			0
$83$	1115.25			1.47487			0.737436	−	0.675417i	$-0.236036\pi$
$83$	1115.25			1.47487			0.737436	+	0.675417i	$0.236036\pi$
$84$		0			0
$85$	1260.72			1.60876
$86$		0			0
$87$	−231.778	+	107.545i	−0.285623	+	0.132529i
$88$		0			0
$89$			119.900i			0.142802i	0.997448	+	0.0714011i	$0.0227470\pi$
$89$			119.900i			0.142802i	−0.997448	+	0.0714011i	$0.977253\pi$
$90$		0			0
$91$		−	390.221i		−	0.449519i
$92$		0			0
$93$	59.2717	−	27.5021i	0.0660881	−	0.0306649i
$94$		0			0
$95$	1412.66			1.52564
$96$		0			0
$97$	−331.017			−0.346491			−0.173246	−	0.984879i	$-0.555425\pi$
$97$	−331.017			−0.346491			−0.173246	+	0.984879i	$0.555425\pi$
$98$		0			0
$99$	−540.848	+	639.614i	−0.549064	+	0.649330i
$100$		0			0
$101$		−	1609.16i		−	1.58532i	−0.609662	−	0.792661i	$-0.708695\pi$
$101$		−	1609.16i		−	1.58532i	0.609662	−	0.792661i	$-0.291305\pi$
$102$		0			0
$103$		−	1388.96i		−	1.32872i	−0.747411	−	0.664362i	$-0.768703\pi$
$103$		−	1388.96i		−	1.32872i	0.747411	−	0.664362i	$-0.231297\pi$
$104$		0			0
$105$	223.290	+	481.229i	0.207532	+	0.447268i
$106$		0			0
$107$	−502.365			−0.453883			−0.226941	−	0.973908i	$-0.572873\pi$
$107$	−502.365			−0.453883			−0.226941	+	0.973908i	$0.572873\pi$
$108$		0			0
$109$	1599.88			1.40588			0.702938	−	0.711251i	$-0.251871\pi$
$109$	1599.88			1.40588			0.702938	+	0.711251i	$0.251871\pi$
$110$		0			0
$111$	649.164	+	1399.06i	0.555099	+	1.19633i
$112$		0			0
$113$			934.016i			0.777565i	0.921330	+	0.388782i	$0.127104\pi$
$113$			934.016i			0.777565i	−0.921330	+	0.388782i	$0.872896\pi$
$114$		0			0
$115$		−	1680.49i		−	1.36266i
$116$		0			0
$117$	971.851	−	1149.32i	0.767928	−	0.908162i
$118$		0			0
$119$	605.069			0.466105
$120$		0			0
$121$	−368.553			−0.276900
$122$		0			0
$123$	−1008.47	+	467.932i	−0.739277	+	0.343025i
$124$		0			0
$125$			543.616i			0.388980i
$126$		0			0
$127$		−	301.355i		−	0.210558i	−0.994443	−	0.105279i	$-0.966426\pi$
$127$		−	301.355i		−	0.210558i	0.994443	−	0.105279i	$-0.0335736\pi$
$128$		0			0
$129$	−780.928	+	362.351i	−0.532999	+	0.247312i
$130$		0			0
$131$	389.347			0.259675			0.129837	−	0.991535i	$-0.458555\pi$
$131$	389.347			0.259675			0.129837	+	0.991535i	$0.458555\pi$
$132$		0			0
$133$	677.991			0.442024
$134$		0			0
$135$	−540.846	+	1973.48i	−0.344805	+	1.25815i
$136$		0			0
$137$			3037.04i			1.89395i	0.321302	+	0.946977i	$0.395880\pi$
$137$			3037.04i			1.89395i	−0.321302	+	0.946977i	$0.604120\pi$
$138$		0			0
$139$		−	1287.18i		−	0.785445i	−0.919657	−	0.392723i	$-0.871533\pi$
$139$		−	1287.18i		−	0.785445i	0.919657	−	0.392723i	$-0.128467\pi$
$140$		0			0
$141$	931.979	+	2008.57i	0.556644	+	1.19966i
$142$		0			0
$143$	−1729.42			−1.01134
$144$		0			0
$145$	717.205			0.410763
$146$		0			0
$147$	107.166	+	230.960i	0.0601283	+	0.129587i
$148$		0			0
$149$			2051.44i			1.12792i	0.825801	+	0.563961i	$0.190723\pi$
$149$			2051.44i			1.12792i	−0.825801	+	0.563961i	$0.809277\pi$
$150$		0			0
$151$		−	473.535i		−	0.255204i	−0.991825	−	0.127602i	$-0.959272\pi$
$151$		−	473.535i		−	0.255204i	0.991825	−	0.127602i	$-0.0407280\pi$
$152$		0			0
$153$	1782.12	+	1506.93i	0.941671	+	0.796263i
$154$		0			0
$155$	−183.409			−0.0950434
$156$		0			0
$157$	1184.81			0.602280			0.301140	−	0.953580i	$-0.402633\pi$
$157$	1184.81			0.602280			0.301140	+	0.953580i	$0.402633\pi$
$158$		0			0
$159$	2171.81	−	1007.72i	1.08324	−	0.502626i
$160$		0			0
$161$		−	806.531i		−	0.394805i
$162$		0			0
$163$			4055.24i			1.94866i	0.225132	+	0.974328i	$0.427719\pi$
$163$			4055.24i			1.94866i	−0.225132	+	0.974328i	$0.572281\pi$
$164$		0			0
$165$	2132.76	−	989.601i	1.00627	−	0.466911i
$166$		0			0
$167$	−3681.23			−1.70576			−0.852881	−	0.522105i	$-0.825147\pi$
$167$	−3681.23			−1.70576			−0.852881	+	0.522105i	$0.825147\pi$
$168$		0			0
$169$	910.598			0.414473
$170$		0			0
$171$	1996.90	+	1688.55i	0.893020	+	0.755125i
$172$		0			0
$173$			1069.27i			0.469914i	0.972006	+	0.234957i	$0.0754949\pi$
$173$			1069.27i			0.469914i	−0.972006	+	0.234957i	$0.924505\pi$
$174$		0			0
$175$		−	614.098i		−	0.265265i
$176$		0			0
$177$	−1501.15	−	3235.24i	−0.637478	−	1.37387i
$178$		0			0
$179$	−2666.57			−1.11346			−0.556728	−	0.830695i	$-0.687944\pi$
$179$	−2666.57			−1.11346			−0.556728	+	0.830695i	$0.687944\pi$
$180$		0			0
$181$	2648.47			1.08762			0.543810	−	0.839208i	$-0.316981\pi$
$181$	2648.47			1.08762			0.543810	+	0.839208i	$0.316981\pi$
$182$		0			0
$183$	1275.98	+	2749.95i	0.515427	+	1.11083i
$184$		0			0
$185$		−	4329.20i		−	1.72048i
$186$		0			0
$187$		−	2681.61i		−	1.04865i
$188$		0			0
$189$	−259.573	+	947.148i	−0.0999003	+	0.364523i
$190$		0			0
$191$	−864.174			−0.327379			−0.163690	−	0.986512i	$-0.552340\pi$
$191$	−864.174			−0.327379			−0.163690	+	0.986512i	$0.552340\pi$
$192$		0			0
$193$	−1133.69			−0.422822			−0.211411	−	0.977397i	$-0.567806\pi$
$193$	−1133.69			−0.422822			−0.211411	+	0.977397i	$0.567806\pi$
$194$		0			0
$195$	−3832.36	+	1778.21i	−1.40739	+	0.653029i
$196$		0			0
$197$		−	1642.58i		−	0.594056i	−0.954869	−	0.297028i	$-0.904004\pi$
$197$		−	1642.58i		−	0.594056i	0.954869	−	0.297028i	$-0.0959955\pi$
$198$		0			0
$199$		−	4912.39i		−	1.74990i	−0.484213	−	0.874950i	$-0.660894\pi$
$199$		−	4912.39i		−	1.74990i	0.484213	−	0.874950i	$-0.339106\pi$
$200$		0			0
$201$	−2776.96	+	1288.51i	−0.974487	+	0.452162i
$202$		0			0
$203$	344.214			0.119010
$204$		0			0
$205$	3120.59			1.06318
$206$		0			0
$207$	2008.68	−	2375.49i	0.674458	−	0.797623i
$208$		0			0
$209$		−	3004.79i		−	0.994476i
$210$		0			0
$211$		−	928.566i		−	0.302963i	−0.988460	−	0.151481i	$-0.951596\pi$
$211$		−	928.566i		−	0.302963i	0.988460	−	0.151481i	$-0.0484043\pi$
$212$		0			0
$213$	1676.00	+	3612.07i	0.539145	+	1.16195i
$214$		0			0
$215$	2416.48			0.766523
$216$		0			0
$217$	−88.0248			−0.0275369
$218$		0			0
$219$	1978.80	+	4264.64i	0.610569	+	1.31588i
$220$		0			0
$221$			4818.58i			1.46666i
$222$		0			0
$223$			208.318i			0.0625561i	0.999511	+	0.0312781i	$0.00995774\pi$
$223$			208.318i			0.0625561i	−0.999511	+	0.0312781i	$0.990042\pi$
$224$		0			0
$225$	1529.42	−	1808.71i	0.453161	−	0.535915i
$226$		0			0
$227$	6151.95			1.79876			0.899382	−	0.437165i	$-0.144017\pi$
$227$	6151.95			1.79876			0.899382	+	0.437165i	$0.144017\pi$
$228$		0			0
$229$	1493.71			0.431035			0.215518	−	0.976500i	$-0.430856\pi$
$229$	1493.71			0.431035			0.215518	+	0.976500i	$0.430856\pi$
$230$		0			0
$231$	1023.59	−	474.947i	0.291547	−	0.135278i
$232$		0			0
$233$			1003.48i			0.282146i	0.989999	+	0.141073i	$0.0450552\pi$
$233$			1003.48i			0.282146i	−0.989999	+	0.141073i	$0.954945\pi$
$234$		0			0
$235$		−	6215.27i		−	1.72527i
$236$		0			0
$237$	−2165.56	+	1004.82i	−0.593536	+	0.275401i
$238$		0			0
$239$	6048.49			1.63701			0.818503	−	0.574502i	$-0.194804\pi$
$239$	6048.49			1.63701			0.818503	+	0.574502i	$0.194804\pi$
$240$		0			0
$241$	−5619.61			−1.50204			−0.751018	−	0.660282i	$-0.770437\pi$
$241$	−5619.61			−1.50204			−0.751018	+	0.660282i	$0.770437\pi$
$242$		0			0
$243$	−3123.41	+	2143.18i	−0.824555	+	0.565782i
$244$		0			0
$245$		−	714.675i		−	0.186363i
$246$		0			0
$247$			5399.31i			1.39089i
$248$		0			0
$249$	−2439.11	−	5256.69i	−0.620771	−	1.33787i
$250$		0			0
$251$	−6402.41			−1.61003			−0.805013	−	0.593257i	$-0.797842\pi$
$251$	−6402.41			−1.61003			−0.805013	+	0.593257i	$0.797842\pi$
$252$		0			0
$253$	−3574.47			−0.888241
$254$		0			0
$255$	−2757.26	−	5942.37i	−0.677124	−	1.45932i
$256$		0			0
$257$			1609.20i			0.390580i	0.980746	+	0.195290i	$0.0625648\pi$
$257$			1609.20i			0.390580i	−0.980746	+	0.195290i	$0.937435\pi$
$258$		0			0
$259$		−	2077.75i		−	0.498475i
$260$		0			0
$261$	1013.82	+	857.271i	0.240436	+	0.203309i
$262$		0			0
$263$	1897.73			0.444940			0.222470	−	0.974940i	$-0.428588\pi$
$263$	1897.73			0.444940			0.222470	+	0.974940i	$0.428588\pi$
$264$		0			0
$265$	−6720.38			−1.55785
$266$		0			0
$267$	565.146	−	262.228i	0.129537	−	0.0601053i
$268$		0			0
$269$			3874.72i			0.878238i	0.898429	+	0.439119i	$0.144709\pi$
$269$			3874.72i			0.878238i	−0.898429	+	0.439119i	$0.855291\pi$
$270$		0			0
$271$		−	1292.92i		−	0.289813i	−0.989445	−	0.144906i	$-0.953712\pi$
$271$		−	1292.92i		−	0.289813i	0.989445	−	0.144906i	$-0.0462881\pi$
$272$		0			0
$273$	−1839.29	+	853.433i	−0.407762	+	0.189202i
$274$		0			0
$275$	−2721.62			−0.596800
$276$		0			0
$277$	−5644.74			−1.22440			−0.612202	−	0.790702i	$-0.709716\pi$
$277$	−5644.74			−1.22440			−0.612202	+	0.790702i	$0.709716\pi$
$278$		0			0
$279$	−259.261	−	219.227i	−0.0556328	−	0.0470422i
$280$		0			0
$281$		−	1146.55i		−	0.243407i	−0.992566	−	0.121704i	$-0.961164\pi$
$281$		−	1146.55i		−	0.243407i	0.992566	−	0.121704i	$-0.0388357\pi$
$282$		0			0
$283$		−	5222.40i		−	1.09696i	−0.836164	−	0.548479i	$-0.815207\pi$
$283$		−	5222.40i		−	1.09696i	0.836164	−	0.548479i	$-0.184793\pi$
$284$		0			0
$285$	−3089.57	−	6658.54i	−0.642141	−	1.38392i
$286$		0			0
$287$	1497.69			0.308034
$288$		0			0
$289$	−2558.59			−0.520780
$290$		0			0
$291$	723.951	+	1560.24i	0.145838	+	0.314305i
$292$		0			0
$293$		−	4783.47i		−	0.953765i	−0.878967	−	0.476883i	$-0.841767\pi$
$293$		−	4783.47i		−	0.953765i	0.878967	−	0.476883i	$-0.158233\pi$
$294$		0			0
$295$			10011.0i			1.97581i
$296$		0			0
$297$	4197.67	+	1150.40i	0.820112	+	0.224758i
$298$		0			0
$299$	6422.97			1.24231
$300$		0			0
$301$	1159.76			0.222085
$302$		0			0
$303$	−7584.74	+	3519.32i	−1.43806	+	0.667260i
$304$		0			0
$305$		−	8509.36i		−	1.59752i
$306$		0			0
$307$			9710.23i			1.80519i	0.430494	+	0.902593i	$0.358339\pi$
$307$			9710.23i			1.80519i	−0.430494	+	0.902593i	$0.641661\pi$
$308$		0			0
$309$	−6546.84	+	3037.74i	−1.20530	+	0.559258i
$310$		0			0
$311$	−3463.09			−0.631427			−0.315713	−	0.948855i	$-0.602244\pi$
$311$	−3463.09			−0.631427			−0.315713	+	0.948855i	$0.602244\pi$
$312$		0			0
$313$	2972.44			0.536780			0.268390	−	0.963310i	$-0.413508\pi$
$313$	2972.44			0.536780			0.268390	+	0.963310i	$0.413508\pi$
$314$		0			0
$315$	1779.91	−	2104.94i	0.318370	−	0.376508i
$316$		0			0
$317$		−	1056.65i		−	0.187217i	−0.995609	−	0.0936083i	$-0.970160\pi$
$317$		−	1056.65i		−	0.187217i	0.995609	−	0.0936083i	$-0.0298401\pi$
$318$		0			0
$319$		−	1525.52i		−	0.267752i
$320$		0			0
$321$	1098.70	+	2367.88i	0.191038	+	0.411720i
$322$		0			0
$323$	−8372.06			−1.44221
$324$		0			0
$325$	4890.48			0.834693
$326$		0			0
$327$	−3499.02	−	7540.97i	−0.591731	−	1.27528i
$328$		0			0
$329$		−	2982.94i		−	0.499863i
$330$		0			0
$331$			11409.6i			1.89465i	0.320279	+	0.947323i	$0.396223\pi$
$331$			11409.6i			1.89465i	−0.320279	+	0.947323i	$0.603777\pi$
$332$		0			0
$333$	5174.67	−	6119.63i	0.851562	−	1.00707i
$334$		0			0
$335$	8592.94			1.40144
$336$		0			0
$337$	−6052.69			−0.978371			−0.489185	−	0.872180i	$-0.662706\pi$
$337$	−6052.69			−0.978371			−0.489185	+	0.872180i	$0.662706\pi$
$338$		0			0
$339$	4402.45	−	2042.74i	0.705335	−	0.327276i
$340$		0			0
$341$			390.117i			0.0619532i
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$		0			0
$345$	−7920.94	+	3675.32i	−1.23608	+	0.573543i
$346$		0			0
$347$	1661.68			0.257071			0.128535	−	0.991705i	$-0.458972\pi$
$347$	1661.68			0.257071			0.128535	+	0.991705i	$0.458972\pi$
$348$		0			0
$349$	−5890.92			−0.903535			−0.451768	−	0.892136i	$-0.649206\pi$
$349$	−5890.92			−0.903535			−0.451768	+	0.892136i	$0.649206\pi$
$350$		0			0
$351$	−7542.79	−	2067.16i	−1.14702	−	0.314350i
$352$		0			0
$353$			4847.63i			0.730916i	0.930828	+	0.365458i	$0.119088\pi$
$353$			4847.63i			0.730916i	−0.930828	+	0.365458i	$0.880912\pi$
$354$		0			0
$355$		−	11177.1i		−	1.67104i
$356$		0			0
$357$	−1323.32	−	2851.97i	−0.196183	−	0.422808i
$358$		0			0
$359$	−9508.69			−1.39791			−0.698954	−	0.715166i	$-0.746351\pi$
$359$	−9508.69			−1.39791			−0.698954	+	0.715166i	$0.746351\pi$
$360$		0			0
$361$	−2522.04			−0.367699
$362$		0			0
$363$	806.046	+	1737.17i	0.116547	+	0.251178i
$364$		0			0
$365$		−	13196.4i		−	1.89241i
$366$		0			0
$367$			3663.26i			0.521037i	0.965469	+	0.260518i	$0.0838935\pi$
$367$			3663.26i			0.521037i	−0.965469	+	0.260518i	$0.916106\pi$
$368$		0			0
$369$	4411.17	+	3730.02i	0.622321	+	0.526225i
$370$		0			0
$371$	−3225.37			−0.451355
$372$		0			0
$373$	6227.72			0.864502			0.432251	−	0.901753i	$-0.357720\pi$
$373$	6227.72			0.864502			0.432251	+	0.901753i	$0.357720\pi$
$374$		0			0
$375$	2562.32	−	1188.92i	0.352847	−	0.163721i
$376$		0			0
$377$			2741.22i			0.374482i
$378$		0			0
$379$		−	5881.91i		−	0.797186i	−0.917128	−	0.398593i	$-0.869499\pi$
$379$		−	5881.91i		−	0.797186i	0.917128	−	0.398593i	$-0.130501\pi$
$380$		0			0
$381$	−1420.43	+	659.079i	−0.190999	+	0.0886237i
$382$		0			0
$383$	3238.56			0.432069			0.216035	−	0.976386i	$-0.430688\pi$
$383$	3238.56			0.432069			0.216035	+	0.976386i	$0.430688\pi$
$384$		0			0
$385$	−3167.37			−0.419284
$386$		0			0
$387$	3415.86	+	2888.40i	0.448677	+	0.379395i
$388$		0			0
$389$			7962.71i			1.03785i	0.854818	+	0.518927i	$0.173668\pi$
$389$			7962.71i			1.03785i	−0.854818	+	0.518927i	$0.826332\pi$
$390$		0			0
$391$			9959.32i			1.28814i
$392$		0			0
$393$	−851.521	−	1835.17i	−0.109297	−	0.235553i
$394$		0			0
$395$	6701.03			0.853583
$396$		0			0
$397$	−5784.53			−0.731277			−0.365639	−	0.930757i	$-0.619149\pi$
$397$	−5784.53			−0.731277			−0.365639	+	0.930757i	$0.619149\pi$
$398$		0			0
$399$	−1482.80	−	3195.69i	−0.186047	−	0.400964i
$400$		0			0
$401$			2460.74i			0.306443i	0.988192	+	0.153221i	$0.0489647\pi$
$401$			2460.74i			0.306443i	−0.988192	+	0.153221i	$0.951035\pi$
$402$		0			0
$403$		−	701.002i		−	0.0866486i
$404$		0			0
$405$	10484.8	−	1766.84i	1.28640	−	0.216777i
$406$		0			0
$407$	−9208.39			−1.12148
$408$		0			0
$409$	12763.1			1.54301			0.771507	−	0.636221i	$-0.219503\pi$
$409$	12763.1			1.54301			0.771507	+	0.636221i	$0.219503\pi$
$410$		0			0
$411$	14315.0	−	6642.16i	1.71802	−	0.797162i
$412$		0			0
$413$			4804.67i			0.572452i
$414$		0			0
$415$			16266.1i			1.92403i
$416$		0			0
$417$	−6067.07	+	2815.12i	−0.712483	+	0.330593i
$418$		0			0
$419$	7332.36			0.854915			0.427457	−	0.904035i	$-0.359409\pi$
$419$	7332.36			0.854915			0.427457	+	0.904035i	$0.359409\pi$
$420$		0			0
$421$	11584.3			1.34106			0.670528	−	0.741885i	$-0.266068\pi$
$421$	11584.3			1.34106			0.670528	+	0.741885i	$0.266068\pi$
$422$		0			0
$423$	7429.07	−	8785.71i	0.853933	−	1.00987i
$424$		0			0
$425$			7583.09i			0.865491i
$426$		0			0
$427$		−	4083.97i		−	0.462850i
$428$		0			0
$429$	3782.33	+	8151.57i	0.425671	+	0.917393i
$430$		0			0
$431$	973.251			0.108770			0.0543850	−	0.998520i	$-0.482680\pi$
$431$	973.251			0.108770			0.0543850	+	0.998520i	$0.482680\pi$
$432$		0			0
$433$	−6021.08			−0.668256			−0.334128	−	0.942528i	$-0.608442\pi$
$433$	−6021.08			−0.668256			−0.334128	+	0.942528i	$0.608442\pi$
$434$		0			0
$435$	−1568.56	−	3380.52i	−0.172890	−	0.372606i
$436$		0			0
$437$			11159.6i			1.22159i
$438$		0			0
$439$		−	4643.72i		−	0.504858i	−0.967615	−	0.252429i	$-0.918771\pi$
$439$		−	4643.72i		−	0.504858i	0.967615	−	0.252429i	$-0.0812295\pi$
$440$		0			0
$441$	854.247	−	1010.24i	0.0922413	−	0.109086i
$442$		0			0
$443$	−14381.2			−1.54238			−0.771189	−	0.636606i	$-0.780338\pi$
$443$	−14381.2			−1.54238			−0.771189	+	0.636606i	$0.780338\pi$
$444$		0			0
$445$	−1748.77			−0.186291
$446$		0			0
$447$	9669.40	−	4486.61i	1.02315	−	0.474741i
$448$		0			0
$449$			10241.0i			1.07639i	0.842819	+	0.538197i	$0.180894\pi$
$449$			10241.0i			1.07639i	−0.842819	+	0.538197i	$0.819106\pi$
$450$		0			0
$451$		−	6637.62i		−	0.693023i
$452$		0			0
$453$	−2231.99	+	1035.65i	−0.231497	+	0.107415i
$454$		0			0
$455$	5691.45			0.586416
$456$		0			0
$457$	18144.7			1.85728			0.928638	−	0.370988i	$-0.120981\pi$
$457$	18144.7			1.85728			0.928638	+	0.370988i	$0.120981\pi$
$458$		0			0
$459$	3205.29	−	11695.7i	0.325948	−	1.18934i
$460$		0			0
$461$		−	10005.2i		−	1.01083i	−0.862877	−	0.505413i	$-0.831340\pi$
$461$		−	10005.2i		−	1.01083i	0.862877	−	0.505413i	$-0.168660\pi$
$462$		0			0
$463$			13659.6i			1.37109i	0.728031	+	0.685544i	$0.240436\pi$
$463$			13659.6i			1.37109i	−0.728031	+	0.685544i	$0.759564\pi$
$464$		0			0
$465$	401.124	+	864.491i	0.0400036	+	0.0862146i
$466$		0			0
$467$	−11500.9			−1.13961			−0.569804	−	0.821781i	$-0.692981\pi$
$467$	−11500.9			−1.13961			−0.569804	+	0.821781i	$0.692981\pi$
$468$		0			0
$469$	4124.08			0.406039
$470$		0			0
$471$	−2591.24	−	5584.56i	−0.253499	−	0.546333i
$472$		0			0
$473$		−	5139.95i		−	0.499651i
$474$		0			0
$475$			8496.99i			0.820776i
$476$		0			0
$477$	−9499.73	−	8032.83i	−0.911871	−	0.771065i
$478$		0			0
$479$	3223.54			0.307489			0.153745	−	0.988111i	$-0.450867\pi$
$479$	3223.54			0.307489			0.153745	+	0.988111i	$0.450867\pi$
$480$		0			0
$481$	16546.6			1.56852
$482$		0			0
$483$	−3801.56	+	1763.93i	−0.358131	+	0.166173i
$484$		0			0
$485$		−	4827.95i		−	0.452012i
$486$		0			0
$487$			13316.6i			1.23908i	0.784964	+	0.619542i	$0.212682\pi$
$487$			13316.6i			1.23908i	−0.784964	+	0.619542i	$0.787318\pi$
$488$		0			0
$489$	19114.3	−	8869.03i	1.76764	−	0.820187i
$490$		0			0
$491$	4229.41			0.388739			0.194369	−	0.980928i	$-0.437734\pi$
$491$	4229.41			0.388739			0.194369	+	0.980928i	$0.437734\pi$
$492$		0			0
$493$	−4250.47			−0.388300
$494$		0			0
$495$	−9328.91	−	7888.38i	−0.847077	−	0.716276i
$496$		0			0
$497$		−	5364.31i		−	0.484149i
$498$		0			0
$499$			2412.58i			0.216437i	0.994127	+	0.108219i	$0.0345146\pi$
$499$			2412.58i			0.216437i	−0.994127	+	0.108219i	$0.965485\pi$
$500$		0			0
$501$	8051.05	+	17351.4i	0.717953	+	1.54731i
$502$		0			0
$503$	−6942.70			−0.615427			−0.307714	−	0.951479i	$-0.599564\pi$
$503$	−6942.70			−0.615427			−0.307714	+	0.951479i	$0.599564\pi$
$504$		0			0
$505$	23470.0			2.06812
$506$		0			0
$507$	−1991.52	−	4292.07i	−0.174451	−	0.375972i
$508$		0			0
$509$		−	5335.45i		−	0.464616i	−0.972642	−	0.232308i	$-0.925372\pi$
$509$		−	5335.45i		−	0.464616i	0.972642	−	0.232308i	$-0.0746278\pi$
$510$		0			0
$511$		−	6333.44i		−	0.548288i
$512$		0			0
$513$	3591.59	−	13105.2i	0.309108	−	1.12790i
$514$		0			0
$515$	20258.3			1.73337
$516$		0			0
$517$	−13220.1			−1.12460
$518$		0			0
$519$	5039.97	−	2338.55i	0.426262	−	0.197786i
$520$		0			0
$521$		−	4804.38i		−	0.404000i	−0.979386	−	0.202000i	$-0.935256\pi$
$521$		−	4804.38i		−	0.404000i	0.979386	−	0.202000i	$-0.0647440\pi$
$522$		0			0
$523$			20692.5i			1.73006i	0.501722	+	0.865029i	$0.332700\pi$
$523$			20692.5i			1.73006i	−0.501722	+	0.865029i	$0.667300\pi$
$524$		0			0
$525$	−2894.53	+	1343.06i	−0.240624	+	0.111650i
$526$		0			0
$527$	1086.96			0.0898458
$528$		0			0
$529$	1108.36			0.0910959
$530$		0			0
$531$	−11966.1	+	14151.3i	−0.977938	+	1.15652i
$532$		0			0
$533$			11927.1i			0.969271i
$534$		0			0
$535$		−	7327.09i		−	0.592108i
$536$		0			0
$537$	5831.93	+	12568.8i	0.468652	+	1.01003i
$538$		0			0
$539$	−1520.14			−0.121479
$540$		0			0
$541$	−5176.25			−0.411357			−0.205679	−	0.978620i	$-0.565940\pi$
$541$	−5176.25			−0.411357			−0.205679	+	0.978620i	$0.565940\pi$
$542$		0			0
$543$	−5792.34	−	12483.5i	−0.457777	−	0.986588i
$544$		0			0
$545$			23334.5i			1.83402i
$546$		0			0
$547$		−	439.585i		−	0.0343607i	−0.999852	−	0.0171804i	$-0.994531\pi$
$547$		−	439.585i		−	0.0343607i	0.999852	−	0.0171804i	$-0.00546895\pi$
$548$		0			0
$549$	10171.2	−	12028.6i	0.790702	−	0.935095i
$550$		0			0
$551$	−4762.73			−0.368238
$552$		0			0
$553$	3216.08			0.247309
$554$		0			0
$555$	−20405.6	+	9468.20i	−1.56066	+	0.724149i
$556$		0			0
$557$		−	19663.7i		−	1.49583i	−0.663795	−	0.747914i	$-0.731055\pi$
$557$		−	19663.7i		−	1.49583i	0.663795	−	0.747914i	$-0.268945\pi$
$558$		0			0
$559$			9235.97i			0.698819i
$560$		0			0
$561$	−12639.7	+	5864.81i	−0.951243	+	0.441377i
$562$		0			0
$563$	−3268.52			−0.244675			−0.122337	−	0.992489i	$-0.539039\pi$
$563$	−3268.52			−0.244675			−0.122337	+	0.992489i	$0.539039\pi$
$564$		0			0
$565$	−13622.8			−1.01436
$566$		0			0
$567$	5032.05	−	847.973i	0.372710	−	0.0628069i
$568$		0			0
$569$			12364.5i			0.910980i	0.890241	+	0.455490i	$0.150536\pi$
$569$			12364.5i			0.910980i	−0.890241	+	0.455490i	$0.849464\pi$
$570$		0			0
$571$		−	12593.6i		−	0.922990i	−0.887143	−	0.461495i	$-0.847313\pi$
$571$		−	12593.6i		−	0.922990i	0.887143	−	0.461495i	$-0.152687\pi$
$572$		0			0
$573$	1889.99	+	4073.26i	0.137793	+	0.296968i
$574$		0			0
$575$	10107.9			0.733096
$576$		0			0
$577$	−13016.7			−0.939156			−0.469578	−	0.882891i	$-0.655594\pi$
$577$	−13016.7			−0.939156			−0.469578	+	0.882891i	$0.655594\pi$
$578$		0			0
$579$	2479.43	+	5343.60i	0.177965	+	0.383545i
$580$		0			0
$581$			7806.74i			0.557449i
$582$		0			0
$583$			14294.5i			1.01547i
$584$		0			0
$585$	16763.1	+	14174.6i	1.18473	+	1.00179i
$586$		0			0
$587$	−16247.0			−1.14240			−0.571199	−	0.820812i	$-0.693521\pi$
$587$	−16247.0			−1.14240			−0.571199	+	0.820812i	$0.693521\pi$
$588$		0			0
$589$	1217.96			0.0852039
$590$		0			0
$591$	−7742.26	+	3592.41i	−0.538873	+	0.250037i
$592$		0			0
$593$		−	17711.4i		−	1.22651i	−0.789885	−	0.613256i	$-0.789860\pi$
$593$		−	17711.4i		−	1.22651i	0.789885	−	0.613256i	$-0.210140\pi$
$594$		0			0
$595$			8825.05i			0.608053i
$596$		0			0
$597$	−23154.4	+	10743.7i	−1.58735	+	0.736530i
$598$		0			0
$599$	−8240.46			−0.562097			−0.281048	−	0.959694i	$-0.590682\pi$
$599$	−8240.46			−0.562097			−0.281048	+	0.959694i	$0.590682\pi$
$600$		0			0
$601$	5366.47			0.364231			0.182116	−	0.983277i	$-0.441706\pi$
$601$	5366.47			0.364231			0.182116	+	0.983277i	$0.441706\pi$
$602$		0			0
$603$	12146.7	+	10271.1i	0.820320	+	0.693650i
$604$		0			0
$605$		−	5375.43i		−	0.361227i
$606$		0			0
$607$		−	7742.72i		−	0.517739i	−0.965912	−	0.258869i	$-0.916650\pi$
$607$		−	7742.72i		−	0.517739i	0.965912	−	0.258869i	$-0.0833499\pi$
$608$		0			0
$609$	−752.814	−	1622.44i	−0.0500913	−	0.107955i
$610$		0			0
$611$	23755.2			1.57289
$612$		0			0
$613$	2581.09			0.170064			0.0850320	−	0.996378i	$-0.472901\pi$
$613$	2581.09			0.170064			0.0850320	+	0.996378i	$0.472901\pi$
$614$		0			0
$615$	−6824.89	−	14708.8i	−0.447490	−	0.964416i
$616$		0			0
$617$			5974.41i			0.389823i	0.980821	+	0.194911i	$0.0624419\pi$
$617$			5974.41i			0.389823i	−0.980821	+	0.194911i	$0.937558\pi$
$618$		0			0
$619$		−	6142.33i		−	0.398839i	−0.979914	−	0.199419i	$-0.936094\pi$
$619$		−	6142.33i		−	0.398839i	0.979914	−	0.199419i	$-0.0639056\pi$
$620$		0			0
$621$	−15589.9	−	4272.52i	−1.00741	−	0.276088i
$622$		0			0
$623$	−839.302			−0.0539742
$624$		0			0
$625$	−18894.8			−1.20927
$626$		0			0
$627$	−14163.0	+	6571.63i	−0.902097	+	0.418574i
$628$		0			0
$629$			25656.8i			1.62639i
$630$		0			0
$631$			4185.65i			0.264070i	0.991245	+	0.132035i	$0.0421511\pi$
$631$			4185.65i			0.264070i	−0.991245	+	0.132035i	$0.957849\pi$
$632$		0			0
$633$	−4376.77	+	2030.82i	−0.274820	+	0.127517i
$634$		0			0
$635$	4395.32			0.274682
$636$		0			0
$637$	2731.55			0.169902
$638$		0			0
$639$	13359.9	−	15799.6i	0.827088	−	0.978125i
$640$		0			0
$641$		−	11627.2i		−	0.716452i	−0.933635	−	0.358226i	$-0.883382\pi$
$641$		−	11627.2i		−	0.716452i	0.933635	−	0.358226i	$-0.116618\pi$
$642$		0			0
$643$		−	30093.1i		−	1.84565i	−0.385215	−	0.922827i	$-0.625873\pi$
$643$		−	30093.1i		−	1.84565i	0.385215	−	0.922827i	$-0.374127\pi$
$644$		0			0
$645$	−5284.96	−	11390.0i	−0.322628	−	0.695319i
$646$		0			0
$647$	23676.7			1.43868			0.719340	−	0.694659i	$-0.244445\pi$
$647$	23676.7			1.43868			0.719340	+	0.694659i	$0.244445\pi$
$648$		0			0
$649$	21293.8			1.28791
$650$		0			0
$651$	192.515	+	414.902i	0.0115902	+	0.0249790i
$652$		0			0
$653$		−	13535.0i		−	0.811126i	−0.914067	−	0.405563i	$-0.867076\pi$
$653$		−	13535.0i		−	0.811126i	0.914067	−	0.405563i	$-0.132924\pi$
$654$		0			0
$655$			5678.70i			0.338756i
$656$		0			0
$657$	15773.5	−	18654.0i	0.936658	−	1.10770i
$658$		0			0
$659$	4587.12			0.271152			0.135576	−	0.990767i	$-0.456712\pi$
$659$	4587.12			0.271152			0.135576	+	0.990767i	$0.456712\pi$
$660$		0			0
$661$	−22014.3			−1.29540			−0.647698	−	0.761897i	$-0.724268\pi$
$661$	−22014.3			−1.29540			−0.647698	+	0.761897i	$0.724268\pi$
$662$		0			0
$663$	22712.2	−	10538.5i	1.33042	−	0.617316i
$664$		0			0
$665$			9888.63i			0.576639i
$666$		0			0
$667$			5665.70i			0.328901i
$668$		0			0
$669$	981.902	−	455.603i	0.0567452	−	0.0263298i
$670$		0			0
$671$	−18099.8			−1.04133
$672$		0			0
$673$	5312.83			0.304301			0.152150	−	0.988357i	$-0.451380\pi$
$673$	5312.83			0.304301			0.152150	+	0.988357i	$0.451380\pi$
$674$		0			0
$675$	−11870.2	−	3253.13i	−0.676867	−	0.185501i
$676$		0			0
$677$		−	31630.1i		−	1.79563i	−0.440372	−	0.897815i	$-0.645154\pi$
$677$		−	31630.1i		−	1.79563i	0.440372	−	0.897815i	$-0.354846\pi$
$678$		0			0
$679$		−	2317.12i		−	0.130961i
$680$		0			0
$681$	−13454.6	−	28997.0i	−0.757097	−	1.63167i
$682$		0			0
$683$	−16304.5			−0.913435			−0.456718	−	0.889612i	$-0.650975\pi$
$683$	−16304.5			−0.913435			−0.456718	+	0.889612i	$0.650975\pi$
$684$		0			0
$685$	−44295.8			−2.47074
$686$		0			0
$687$	−3266.82	−	7040.55i	−0.181422	−	0.390995i
$688$		0			0
$689$		−	25685.8i		−	1.42025i
$690$		0			0
$691$		−	9225.62i		−	0.507900i	−0.967217	−	0.253950i	$-0.918270\pi$
$691$		−	9225.62i		−	0.507900i	0.967217	−	0.253950i	$-0.0817299\pi$
$692$		0			0
$693$	−4477.30	−	3785.94i	−0.245424	−	0.207527i
$694$		0			0
$695$	18773.7			1.02464
$696$		0			0
$697$	−18494.0			−1.00504
$698$		0			0
$699$	4729.86	−	2194.66i	0.255937	−	0.118755i
$700$		0			0
$701$			4759.97i			0.256464i	0.991744	+	0.128232i	$0.0409303\pi$
$701$			4759.97i			0.256464i	−0.991744	+	0.128232i	$0.959070\pi$
$702$		0			0
$703$			28748.9i			1.54237i
$704$		0			0
$705$	−29295.5	+	13593.1i	−1.56501	+	0.726165i
$706$		0			0
$707$	11264.1			0.599196
$708$		0			0
$709$	29605.9			1.56823			0.784114	−	0.620617i	$-0.213118\pi$
$709$	29605.9			1.56823			0.784114	+	0.620617i	$0.213118\pi$
$710$		0			0
$711$	9472.38	+	8009.70i	0.499637	+	0.422485i
$712$		0			0
$713$		−	1448.87i		−	0.0761020i
$714$		0			0
$715$		−	25224.0i		−	1.31933i
$716$		0			0
$717$	−13228.4	−	28509.4i	−0.689013	−	1.48494i
$718$		0			0
$719$	34086.2			1.76801			0.884007	−	0.467474i	$-0.154836\pi$
$719$	34086.2			1.76801			0.884007	+	0.467474i	$0.154836\pi$
$720$		0			0
$721$	9722.74			0.502211
$722$		0			0
$723$	12290.4	+	26487.8i	0.632204	+	1.36251i
$724$		0			0
$725$			4313.90i			0.220985i
$726$		0			0
$727$		−	15552.2i		−	0.793396i	−0.917949	−	0.396698i	$-0.870156\pi$
$727$		−	15552.2i		−	0.793396i	0.917949	−	0.396698i	$-0.129844\pi$
$728$		0			0
$729$	16932.9	+	10034.9i	0.860279	+	0.509823i
$730$		0			0
$731$	−14321.1			−0.724604
$732$		0			0
$733$	−1511.48			−0.0761636			−0.0380818	−	0.999275i	$-0.512125\pi$
$733$	−1511.48			−0.0761636			−0.0380818	+	0.999275i	$0.512125\pi$
$734$		0			0
$735$	−3368.60	+	1563.03i	−0.169051	+	0.0784399i
$736$		0			0
$737$		−	18277.5i		−	0.913517i
$738$		0			0
$739$		−	6749.94i		−	0.335995i	−0.985787	−	0.167998i	$-0.946270\pi$
$739$		−	6749.94i		−	0.335995i	0.985787	−	0.167998i	$-0.0537301\pi$
$740$		0			0
$741$	25449.5	−	11808.6i	1.26169	−	0.585423i
$742$		0			0
$743$	−24008.4			−1.18544			−0.592719	−	0.805409i	$-0.701946\pi$
$743$	−24008.4			−1.18544			−0.592719	+	0.805409i	$0.701946\pi$
$744$		0			0
$745$	−29920.7			−1.47142
$746$		0			0
$747$	−19442.8	+	22993.3i	−0.952309	+	1.12621i
$748$		0			0
$749$		−	3516.55i		−	0.171552i
$750$		0			0
$751$		−	35758.5i		−	1.73748i	−0.495272	−	0.868738i	$-0.664931\pi$
$751$		−	35758.5i		−	1.73748i	0.495272	−	0.868738i	$-0.335069\pi$
$752$		0			0
$753$	14002.4	+	30177.6i	0.677658	+	1.46047i
$754$		0			0
$755$	6906.61			0.332924
$756$		0			0
$757$	−12640.7			−0.606916			−0.303458	−	0.952845i	$-0.598141\pi$
$757$	−12640.7			−0.606916			−0.303458	+	0.952845i	$0.598141\pi$
$758$		0			0
$759$	7817.55	+	16848.2i	0.373859	+	0.805730i
$760$		0			0
$761$			28346.1i			1.35026i	0.737699	+	0.675129i	$0.235912\pi$
$761$			28346.1i			1.35026i	−0.737699	+	0.675129i	$0.764088\pi$
$762$		0			0
$763$			11199.1i			0.531371i
$764$		0			0
$765$	−21978.9	+	25992.6i	−1.03876	+	1.22845i
$766$		0			0
$767$	−38262.9			−1.80130
$768$		0			0
$769$	−15226.0			−0.713999			−0.356999	−	0.934105i	$-0.616200\pi$
$769$	−15226.0			−0.713999			−0.356999	+	0.934105i	$0.616200\pi$
$770$		0			0
$771$	7584.91	−	3519.40i	0.354298	−	0.164395i
$772$		0			0
$773$			20084.4i			0.934522i	0.884119	+	0.467261i	$0.154759\pi$
$773$			20084.4i			0.934522i	−0.884119	+	0.467261i	$0.845241\pi$
$774$		0			0
$775$		−	1103.18i		−	0.0511321i
$776$		0			0
$777$	−9793.42	+	4544.15i	−0.452171	+	0.209808i
$778$		0			0
$779$	−20722.8			−0.953110
$780$		0			0
$781$	−23774.1			−1.08925
$782$		0			0
$783$	1823.44	−	6653.50i	0.0832242	−	0.303674i
$784$		0			0
$785$			17280.7i			0.785699i
$786$		0			0
$787$		−	18800.0i		−	0.851522i	−0.904836	−	0.425761i	$-0.860006\pi$
$787$		−	18800.0i		−	0.851522i	0.904836	−	0.425761i	$-0.139994\pi$
$788$		0			0
$789$	−4150.44	−	8944.91i	−0.187275	−	0.403609i
$790$		0			0
$791$	−6538.11			−0.293892
$792$		0			0
$793$	32523.5			1.45642
$794$		0			0
$795$	14697.8	+	31676.3i	0.655696	+	1.41314i
$796$		0			0
$797$		−	28103.9i		−	1.24905i	−0.781005	−	0.624524i	$-0.785293\pi$
$797$		−	28103.9i		−	1.24905i	0.781005	−	0.624524i	$-0.214707\pi$
$798$		0			0
$799$			36834.4i			1.63092i
$800$		0			0
$801$	−2472.01	−	2090.29i	−0.109044	−	0.0922059i
$802$		0			0
$803$	−28069.2			−1.23355
$804$		0			0
$805$	11763.4			0.515039
$806$		0			0
$807$	18263.4	−	8474.22i	0.796656	−	0.369649i
$808$		0			0
$809$		−	7958.95i		−	0.345886i	−0.984932	−	0.172943i	$-0.944672\pi$
$809$		−	7958.95i		−	0.345886i	0.984932	−	0.172943i	$-0.0553276\pi$
$810$		0			0
$811$			18626.6i			0.806494i	0.915091	+	0.403247i	$0.132118\pi$
$811$			18626.6i			0.806494i	−0.915091	+	0.403247i	$0.867882\pi$
$812$		0			0
$813$	−6094.14	+	2827.68i	−0.262892	+	0.121982i
$814$		0			0
$815$	−59146.5			−2.54210
$816$		0			0
$817$	−16047.1			−0.687168
$818$		0			0
$819$	8045.27	+	6802.95i	0.343253	+	0.290250i
$820$		0			0
$821$		−	34766.7i		−	1.47791i	−0.673753	−	0.738957i	$-0.735319\pi$
$821$		−	34766.7i		−	1.47791i	0.673753	−	0.738957i	$-0.264681\pi$
$822$		0			0
$823$		−	22756.3i		−	0.963832i	−0.876217	−	0.481916i	$-0.839941\pi$
$823$		−	22756.3i		−	0.963832i	0.876217	−	0.481916i	$-0.160059\pi$
$824$		0			0
$825$	5952.33	+	12828.3i	0.251192	+	0.541362i
$826$		0			0
$827$	25960.0			1.09156			0.545778	−	0.837930i	$-0.316234\pi$
$827$	25960.0			1.09156			0.545778	+	0.837930i	$0.316234\pi$
$828$		0			0
$829$	−9264.02			−0.388121			−0.194061	−	0.980990i	$-0.562166\pi$
$829$	−9264.02			−0.388121			−0.194061	+	0.980990i	$0.562166\pi$
$830$		0			0
$831$	12345.3	+	26606.3i	0.515349	+	1.11067i
$832$		0			0
$833$			4235.48i			0.176171i
$834$		0			0
$835$		−	53691.6i		−	2.22524i
$836$		0			0
$837$	−466.303	+	1701.48i	−0.0192566	+	0.0702649i
$838$		0			0
$839$	16620.6			0.683917			0.341959	−	0.939715i	$-0.388910\pi$
$839$	16620.6			0.683917			0.341959	+	0.939715i	$0.388910\pi$
$840$		0			0
$841$	21971.0			0.900856
$842$		0			0
$843$	−5404.23	+	2507.56i	−0.220797	+	0.102450i
$844$		0			0
$845$			13281.3i			0.540697i
$846$		0			0
$847$		−	2579.87i		−	0.104658i
$848$		0			0
$849$	−24615.6	+	11421.7i	−0.995060	+	0.461708i
$850$		0			0
$851$	34199.4			1.37760
$852$		0			0
$853$	18494.8			0.742379			0.371190	−	0.928557i	$-0.378950\pi$
$853$	18494.8			0.742379			0.371190	+	0.928557i	$0.378950\pi$
$854$		0			0
$855$	−24627.8	+	29125.1i	−0.985091	+	1.16498i
$856$		0			0
$857$			293.990i			0.0117182i	0.999983	+	0.00585910i	$0.00186502\pi$
$857$			293.990i			0.0117182i	−0.999983	+	0.00585910i	$0.998135\pi$
$858$		0			0
$859$			11212.2i			0.445351i	0.974893	+	0.222675i	$0.0714790\pi$
$859$			11212.2i			0.445351i	−0.974893	+	0.222675i	$0.928521\pi$
$860$		0			0
$861$	−3275.53	−	7059.32i	−0.129651	−	0.279420i
$862$		0			0
$863$	17496.3			0.690129			0.345064	−	0.938579i	$-0.387857\pi$
$863$	17496.3			0.690129			0.345064	+	0.938579i	$0.387857\pi$
$864$		0			0
$865$	−15595.5			−0.613021
$866$		0			0
$867$	5595.77	+	12059.8i	0.219195	+	0.472403i
$868$		0			0
$869$		−	14253.4i		−	0.556401i
$870$		0			0
$871$			32842.9i			1.27766i
$872$		0			0
$873$	5770.81	−	6824.64i	0.223726	−	0.264581i
$874$		0			0
$875$	−3805.31			−0.147021
$876$		0			0
$877$	−15209.2			−0.585607			−0.292804	−	0.956173i	$-0.594588\pi$
$877$	−15209.2			−0.585607			−0.292804	+	0.956173i	$0.594588\pi$
$878$		0			0
$879$	−22546.7	+	10461.7i	−0.865168	+	0.401438i
$880$		0			0
$881$			23485.7i			0.898132i	0.893499	+	0.449066i	$0.148243\pi$
$881$			23485.7i			0.898132i	−0.893499	+	0.449066i	$0.851757\pi$
$882$		0			0
$883$		−	38770.5i		−	1.47761i	−0.673919	−	0.738805i	$-0.735390\pi$
$883$		−	38770.5i		−	1.47761i	0.673919	−	0.738805i	$-0.264610\pi$
$884$		0			0
$885$	47186.7	−	21894.6i	1.79227	−	0.831616i
$886$		0			0
$887$	−24491.0			−0.927089			−0.463544	−	0.886074i	$-0.653422\pi$
$887$	−24491.0			−0.927089			−0.463544	+	0.886074i	$0.653422\pi$
$888$		0			0
$889$	2109.48			0.0795836
$890$		0			0
$891$	−3758.13	−	22301.6i	−0.141304	−	0.838531i
$892$		0			0
$893$			41273.6i			1.54666i
$894$		0			0
$895$		−	38892.5i		−	1.45255i
$896$		0			0
$897$	−14047.4	−	30274.5i	−0.522885	−	1.12691i
$898$		0			0
$899$	618.355			0.0229402
$900$		0			0
$901$	39827.9			1.47265
$902$		0			0
$903$	−2536.46	−	5466.50i	−0.0934751	−	0.201455i
$904$		0			0
$905$			38628.5i			1.41884i
$906$		0			0
$907$			16043.5i			0.587339i	0.955907	+	0.293670i	$0.0948765\pi$
$907$			16043.5i			0.587339i	−0.955907	+	0.293670i	$0.905123\pi$
$908$		0			0
$909$	33176.4	+	28053.5i	1.21055	+	1.02363i
$910$		0			0
$911$	−20367.3			−0.740723			−0.370362	−	0.928888i	$-0.620766\pi$
$911$	−20367.3			−0.740723			−0.370362	+	0.928888i	$0.620766\pi$
$912$		0			0
$913$	34598.7			1.25416
$914$		0			0
$915$	−40108.6	+	18610.4i	−1.44913	+	0.672395i
$916$		0			0
$917$			2725.43i			0.0981478i
$918$		0			0
$919$		−	27011.0i		−	0.969542i	−0.874641	−	0.484771i	$-0.838903\pi$
$919$		−	27011.0i		−	0.969542i	0.874641	−	0.484771i	$-0.161097\pi$
$920$		0			0
$921$	45768.9	−	21236.8i	1.63750	−	0.759800i
$922$		0			0
$923$	42719.7			1.52344
$924$		0			0
$925$	26039.6			0.925598
$926$		0			0
$927$	28636.6	+	24214.6i	1.01461	+	0.857943i
$928$		0			0
$929$		−	29191.8i		−	1.03095i	−0.856905	−	0.515474i	$-0.827616\pi$
$929$		−	29191.8i		−	1.03095i	0.856905	−	0.515474i	$-0.172384\pi$
$930$		0			0
$931$			4745.93i			0.167069i
$932$		0			0
$933$	7573.96	+	16323.2i	0.265767	+	0.572772i
$934$		0			0
$935$	39111.8			1.36801
$936$		0			0
$937$	−44629.5			−1.55601			−0.778006	−	0.628257i	$-0.783769\pi$
$937$	−44629.5			−1.55601			−0.778006	+	0.628257i	$0.783769\pi$
$938$		0			0
$939$	−6500.88	−	14010.5i	−0.225930	−	0.486918i
$940$		0			0
$941$			5317.21i			0.184204i	0.995750	+	0.0921021i	$0.0293586\pi$
$941$			5317.21i			0.184204i	−0.995750	+	0.0921021i	$0.970641\pi$
$942$		0			0
$943$			24651.7i			0.851294i
$944$		0			0
$945$	−13814.3	−	3785.92i	−0.475535	−	0.130324i
$946$		0			0
$947$	−7403.69			−0.254052			−0.127026	−	0.991899i	$-0.540543\pi$
$947$	−7403.69			−0.254052			−0.127026	+	0.991899i	$0.540543\pi$
$948$		0			0
$949$	50437.6			1.72526
$950$		0			0
$951$	−4980.51	+	2310.96i	−0.169826	+	0.0787992i
$952$		0			0
$953$		−	30007.8i		−	1.01999i	−0.860178	−	0.509993i	$-0.829648\pi$
$953$		−	30007.8i		−	1.01999i	0.860178	−	0.509993i	$-0.170352\pi$
$954$		0			0
$955$		−	12604.2i		−	0.427080i
$956$		0			0
$957$	−7190.51	+	3336.40i	−0.242880	+	0.112696i
$958$		0			0
$959$	−21259.3			−0.715847
$960$		0			0
$961$	29632.9			0.994692
$962$		0			0
$963$	8758.03	−	10357.4i	0.293067	−	0.346585i
$964$		0			0
$965$		−	16535.1i		−	0.551588i
$966$		0			0
$967$		−	44735.5i		−	1.48769i	−0.668352	−	0.743846i	$-0.733000\pi$
$967$		−	44735.5i		−	1.48769i	0.668352	−	0.743846i	$-0.267000\pi$
$968$		0			0
$969$	18310.1	+	39461.4i	0.607024	+	1.30824i
$970$		0			0
$971$	9285.42			0.306883			0.153441	−	0.988158i	$-0.450964\pi$
$971$	9285.42			0.306883			0.153441	+	0.988158i	$0.450964\pi$
$972$		0			0
$973$	9010.23			0.296870
$974$		0			0
$975$	−10695.7	−	23051.2i	−0.351321	−	0.757157i
$976$		0			0
$977$		−	17486.1i		−	0.572600i	−0.958140	−	0.286300i	$-0.907575\pi$
$977$		−	17486.1i		−	0.572600i	0.958140	−	0.286300i	$-0.0924254\pi$
$978$		0			0
$979$			3719.71i			0.121432i
$980$		0			0
$981$	−27891.6	+	32985.0i	−0.907759	+	1.07353i
$982$		0			0
$983$	51426.1			1.66860			0.834302	−	0.551307i	$-0.185871\pi$
$983$	51426.1			1.66860			0.834302	+	0.551307i	$0.185871\pi$
$984$		0			0
$985$	23957.4			0.774971
$986$		0			0
$987$	−14060.0	+	6523.85i	−0.453430	+	0.210392i
$988$		0			0
$989$			19089.5i			0.613761i
$990$		0			0
$991$			29365.3i			0.941290i	0.882323	+	0.470645i	$0.155979\pi$
$991$			29365.3i			0.941290i	−0.882323	+	0.470645i	$0.844021\pi$
$992$		0			0
$993$	53778.8	−	24953.4i	1.71865	−	0.797454i
$994$		0			0
$995$	71648.3			2.28282
$996$		0			0
$997$	−8452.97			−0.268514			−0.134257	−	0.990947i	$-0.542865\pi$
$997$	−8452.97			−0.268514			−0.134257	+	0.990947i	$0.542865\pi$
$998$		0			0
$999$	−40162.0	−	11006.7i	−1.27194	−	0.348585i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.h.a.239.5	✓	12
3.2	odd	2	inner	336.4.h.a.239.7	yes	12
4.3	odd	2	inner	336.4.h.a.239.8	yes	12
12.11	even	2	inner	336.4.h.a.239.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.h.a.239.5	✓	12	1.1	even	1	trivial
336.4.h.a.239.6	yes	12	12.11	even	2	inner
336.4.h.a.239.7	yes	12	3.2	odd	2	inner
336.4.h.a.239.8	yes	12	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)

\(f(q)\)	\(=\)	\(q+(-2.18705 - 4.71347i) q^{3} +14.5852i q^{5} +7.00000i q^{7} +(-17.4336 + 20.6172i) q^{9} +O(q^{10})\)	\(q+(-2.18705 - 4.71347i) q^{3} +14.5852i q^{5} +7.00000i q^{7} +(-17.4336 + 20.6172i) q^{9} +31.0233 q^{11} -55.7458 q^{13} +(68.7469 - 31.8986i) q^{15} -86.4384i q^{17} -96.8558i q^{19} +(32.9943 - 15.3094i) q^{21} -115.219 q^{23} -87.7283 q^{25} +(135.307 + 37.0818i) q^{27} -49.1735i q^{29} +12.5750i q^{31} +(-67.8496 - 146.228i) q^{33} -102.096 q^{35} -296.822 q^{37} +(121.919 + 262.756i) q^{39} -213.956i q^{41} -165.680i q^{43} +(-300.706 - 254.273i) q^{45} -426.135 q^{47} -49.0000 q^{49} +(-407.425 + 189.045i) q^{51} +460.767i q^{53} +452.482i q^{55} +(-456.527 + 211.829i) q^{57} +686.382 q^{59} -583.424 q^{61} +(-144.320 - 122.035i) q^{63} -813.064i q^{65} -589.155i q^{67} +(251.989 + 543.080i) q^{69} -766.330 q^{71} -904.778 q^{73} +(191.866 + 413.505i) q^{75} +217.163i q^{77} -459.440i q^{79} +(-121.139 - 718.865i) q^{81} +1115.25 q^{83} +1260.72 q^{85} +(-231.778 + 107.545i) q^{87} +119.900i q^{89} -390.221i q^{91} +(59.2717 - 27.5021i) q^{93} +1412.66 q^{95} -331.017 q^{97} +(-540.848 + 639.614i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 76 q^{9}+O(q^{10}) \) 12 * q + 76 * q^9	\( 12 q + 76 q^{9} - 96 q^{13} + 112 q^{21} - 1068 q^{25} - 832 q^{33} - 720 q^{37} + 392 q^{45} - 588 q^{49} - 2336 q^{57} + 432 q^{61} - 424 q^{69} + 1656 q^{73} - 868 q^{81} - 1464 q^{85} + 696 q^{93} - 6264 q^{97}+O(q^{100}) \) 12 * q + 76 * q^9 - 96 * q^13 + 112 * q^21 - 1068 * q^25 - 832 * q^33 - 720 * q^37 + 392 * q^45 - 588 * q^49 - 2336 * q^57 + 432 * q^61 - 424 * q^69 + 1656 * q^73 - 868 * q^81 - 1464 * q^85 + 696 * q^93 - 6264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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