LMFDB - Embedded newform 672.4.j.a.239.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,4,Mod(239,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("672.239");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	672.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$39.6492835239$
Analytic rank:	$0$
Dimension:	$72$
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.9
Character	$\chi$	$=$	672.239
Dual form			672.4.j.a.239.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.65040 - 2.31814i) q^{3} -0.943399 q^{5} +7.00000i q^{7} +(16.2524 + 21.5606i) q^{9} +O(q^{10})$	\(q+(-4.65040 - 2.31814i) q^{3} -0.943399 q^{5} +7.00000i q^{7} +(16.2524 + 21.5606i) q^{9} -17.0428i q^{11} -27.1431i q^{13} +(4.38718 + 2.18693i) q^{15} -6.69346i q^{17} -24.6789 q^{19} +(16.2270 - 32.5528i) q^{21} +67.7672 q^{23} -124.110 q^{25} +(-25.5999 - 137.941i) q^{27} -112.072 q^{29} +223.003i q^{31} +(-39.5075 + 79.2557i) q^{33} -6.60379i q^{35} -260.864i q^{37} +(-62.9215 + 126.226i) q^{39} +164.002i q^{41} -96.6519 q^{43} +(-15.3325 - 20.3402i) q^{45} -8.05254 q^{47} -49.0000 q^{49} +(-15.5164 + 31.1273i) q^{51} +122.944 q^{53} +16.0781i q^{55} +(114.767 + 57.2093i) q^{57} +750.290i q^{59} +145.035i q^{61} +(-150.924 + 113.767i) q^{63} +25.6068i q^{65} +843.774 q^{67} +(-315.145 - 157.094i) q^{69} +816.642 q^{71} +657.420 q^{73} +(577.161 + 287.705i) q^{75} +119.299 q^{77} -624.022i q^{79} +(-200.716 + 700.824i) q^{81} +1156.43i q^{83} +6.31460i q^{85} +(521.181 + 259.800i) q^{87} +358.915i q^{89} +190.001 q^{91} +(516.953 - 1037.05i) q^{93} +23.2821 q^{95} +721.607 q^{97} +(367.452 - 276.986i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q+O(q^{10}) $ 72 * q	$ 72 q - 48 q^{19} + 1800 q^{25} - 264 q^{27} + 232 q^{33} + 864 q^{43} - 3528 q^{49} - 344 q^{57} - 1632 q^{67} - 3304 q^{75} + 920 q^{81} + 5312 q^{99}+O(q^{100}) $ 72 * q - 48 * q^19 + 1800 * q^25 - 264 * q^27 + 232 * q^33 + 864 * q^43 - 3528 * q^49 - 344 * q^57 - 1632 * q^67 - 3304 * q^75 + 920 * q^81 + 5312 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−4.65040	−	2.31814i	−0.894970	−	0.446127i
$3$	−4.65040	−	2.31814i	−0.894970	−	0.446127i
$4$		0			0
$5$	−0.943399			−0.0843802			−0.0421901	−	0.999110i	$-0.513434\pi$
$5$	−0.943399			−0.0843802			−0.0421901	+	0.999110i	$0.513434\pi$
$6$		0			0
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		0			0
$9$	16.2524	+	21.5606i	0.601942	+	0.798540i
$10$		0			0
$11$		−	17.0428i		−	0.467144i	−0.972339	−	0.233572i	$-0.924959\pi$
$11$		−	17.0428i		−	0.467144i	0.972339	−	0.233572i	$-0.0750415\pi$
$12$		0			0
$13$		−	27.1431i		−	0.579087i	−0.957165	−	0.289544i	$-0.906496\pi$
$13$		−	27.1431i		−	0.579087i	0.957165	−	0.289544i	$-0.0935035\pi$
$14$		0			0
$15$	4.38718	+	2.18693i	0.0755177	+	0.0376442i
$16$		0			0
$17$		−	6.69346i		−	0.0954942i	−0.998859	−	0.0477471i	$-0.984796\pi$
$17$		−	6.69346i		−	0.0954942i	0.998859	−	0.0477471i	$-0.0152042\pi$
$18$		0			0
$19$	−24.6789			−0.297986			−0.148993	−	0.988838i	$-0.547603\pi$
$19$	−24.6789			−0.297986			−0.148993	+	0.988838i	$0.547603\pi$
$20$		0			0
$21$	16.2270	−	32.5528i	0.168620	−	0.338267i
$22$		0			0
$23$	67.7672			0.614367			0.307183	−	0.951650i	$-0.400614\pi$
$23$	67.7672			0.614367			0.307183	+	0.951650i	$0.400614\pi$
$24$		0			0
$25$	−124.110			−0.992880
$26$		0			0
$27$	−25.5999	−	137.941i	−0.182470	−	0.983211i
$28$		0			0
$29$	−112.072			−0.717632			−0.358816	−	0.933408i	$-0.616819\pi$
$29$	−112.072			−0.717632			−0.358816	+	0.933408i	$0.616819\pi$
$30$		0			0
$31$			223.003i			1.29202i	0.763330	+	0.646009i	$0.223563\pi$
$31$			223.003i			1.29202i	−0.763330	+	0.646009i	$0.776437\pi$
$32$		0			0
$33$	−39.5075	+	79.2557i	−0.208405	+	0.418080i
$34$		0			0
$35$		−	6.60379i		−	0.0318927i
$36$		0			0
$37$		−	260.864i		−	1.15907i	−0.814946	−	0.579537i	$-0.803233\pi$
$37$		−	260.864i		−	1.15907i	0.814946	−	0.579537i	$-0.196767\pi$
$38$		0			0
$39$	−62.9215	+	126.226i	−0.258346	+	0.518266i
$40$		0			0
$41$			164.002i			0.624702i	0.949967	+	0.312351i	$0.101116\pi$
$41$			164.002i			0.624702i	−0.949967	+	0.312351i	$0.898884\pi$
$42$		0			0
$43$	−96.6519			−0.342774			−0.171387	−	0.985204i	$-0.554825\pi$
$43$	−96.6519			−0.342774			−0.171387	+	0.985204i	$0.554825\pi$
$44$		0			0
$45$	−15.3325	−	20.3402i	−0.0507920	−	0.0673809i
$46$		0			0
$47$	−8.05254			−0.0249911			−0.0124956	−	0.999922i	$-0.503978\pi$
$47$	−8.05254			−0.0249911			−0.0124956	+	0.999922i	$0.503978\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$		0			0
$51$	−15.5164	+	31.1273i	−0.0426025	+	0.0854645i
$52$		0			0
$53$	122.944			0.318636			0.159318	−	0.987227i	$-0.449070\pi$
$53$	122.944			0.318636			0.159318	+	0.987227i	$0.449070\pi$
$54$		0			0
$55$			16.0781i			0.0394177i
$56$		0			0
$57$	114.767	+	57.2093i	0.266689	+	0.132940i
$58$		0			0
$59$			750.290i			1.65558i	0.561035	+	0.827792i	$0.310403\pi$
$59$			750.290i			1.65558i	−0.561035	+	0.827792i	$0.689597\pi$
$60$		0			0
$61$			145.035i			0.304423i	0.988348	+	0.152212i	$0.0486395\pi$
$61$			145.035i			0.304423i	−0.988348	+	0.152212i	$0.951360\pi$
$62$		0			0
$63$	−150.924	+	113.767i	−0.301820	+	0.227513i
$64$		0			0
$65$			25.6068i			0.0488635i
$66$		0			0
$67$	843.774			1.53856			0.769279	−	0.638913i	$-0.220616\pi$
$67$	843.774			1.53856			0.769279	+	0.638913i	$0.220616\pi$
$68$		0			0
$69$	−315.145	−	157.094i	−0.549840	−	0.274085i
$70$		0			0
$71$	816.642			1.36504			0.682518	−	0.730868i	$-0.260885\pi$
$71$	816.642			1.36504			0.682518	+	0.730868i	$0.260885\pi$
$72$		0			0
$73$	657.420			1.05404			0.527022	−	0.849852i	$-0.323309\pi$
$73$	657.420			1.05404			0.527022	+	0.849852i	$0.323309\pi$
$74$		0			0
$75$	577.161	+	287.705i	0.888598	+	0.442950i
$76$		0			0
$77$	119.299			0.176564
$78$		0			0
$79$		−	624.022i		−	0.888708i	−0.895851	−	0.444354i	$-0.853433\pi$
$79$		−	624.022i		−	0.888708i	0.895851	−	0.444354i	$-0.146567\pi$
$80$		0			0
$81$	−200.716	+	700.824i	−0.275331	+	0.961349i
$82$		0			0
$83$			1156.43i			1.52934i	0.644424	+	0.764668i	$0.277097\pi$
$83$			1156.43i			1.52934i	−0.644424	+	0.764668i	$0.722903\pi$
$84$		0			0
$85$			6.31460i			0.00805782i
$86$		0			0
$87$	521.181	+	259.800i	0.642259	+	0.320155i
$88$		0			0
$89$			358.915i			0.427470i	0.976892	+	0.213735i	$0.0685630\pi$
$89$			358.915i			0.427470i	−0.976892	+	0.213735i	$0.931437\pi$
$90$		0			0
$91$	190.001			0.218874
$92$		0			0
$93$	516.953	−	1037.05i	0.576404	−	1.15632i
$94$		0			0
$95$	23.2821			0.0251441
$96$		0			0
$97$	721.607			0.755341			0.377671	−	0.925940i	$-0.376725\pi$
$97$	721.607			0.755341			0.377671	+	0.925940i	$0.376725\pi$
$98$		0			0
$99$	367.452	−	276.986i	0.373033	−	0.281194i
$100$		0			0
$101$	−954.039			−0.939905			−0.469953	−	0.882692i	$-0.655729\pi$
$101$	−954.039			−0.939905			−0.469953	+	0.882692i	$0.655729\pi$
$102$		0			0
$103$		−	224.441i		−	0.214707i	−0.994221	−	0.107353i	$-0.965762\pi$
$103$		−	224.441i		−	0.214707i	0.994221	−	0.107353i	$-0.0342376\pi$
$104$		0			0
$105$	−15.3085	+	30.7103i	−0.0142282	+	0.0285430i
$106$		0			0
$107$			1820.43i			1.64474i	0.568953	+	0.822370i	$0.307349\pi$
$107$			1820.43i			1.64474i	−0.568953	+	0.822370i	$0.692651\pi$
$108$		0			0
$109$			1382.49i			1.21485i	0.794376	+	0.607427i	$0.207798\pi$
$109$			1382.49i			1.21485i	−0.794376	+	0.607427i	$0.792202\pi$
$110$		0			0
$111$	−604.719	+	1213.12i	−0.517094	+	1.03734i
$112$		0			0
$113$			1996.99i			1.66249i	0.555909	+	0.831243i	$0.312370\pi$
$113$			1996.99i			1.66249i	−0.555909	+	0.831243i	$0.687630\pi$
$114$		0			0
$115$	−63.9315			−0.0518404
$116$		0			0
$117$	585.220	−	441.141i	0.462424	−	0.348577i
$118$		0			0
$119$	46.8542			0.0360934
$120$		0			0
$121$	1040.54			0.781776
$122$		0			0
$123$	380.179	−	762.673i	0.278696	−	0.559089i
$124$		0			0
$125$	235.010			0.168160
$126$		0			0
$127$		−	1683.94i		−	1.17658i	−0.808651	−	0.588288i	$-0.799802\pi$
$127$		−	1683.94i		−	1.17658i	0.808651	−	0.588288i	$-0.200198\pi$
$128$		0			0
$129$	449.470	+	224.053i	0.306772	+	0.152921i
$130$		0			0
$131$			794.391i			0.529819i	0.964273	+	0.264909i	$0.0853420\pi$
$131$			794.391i			0.529819i	−0.964273	+	0.264909i	$0.914658\pi$
$132$		0			0
$133$		−	172.753i		−	0.112628i
$134$		0			0
$135$	24.1509	+	130.133i	0.0153969	+	0.0829636i
$136$		0			0
$137$			1732.52i			1.08043i	0.841526	+	0.540217i	$0.181658\pi$
$137$			1732.52i			1.08043i	−0.841526	+	0.540217i	$0.818342\pi$
$138$		0			0
$139$	205.405			0.125340			0.0626699	−	0.998034i	$-0.480038\pi$
$139$	205.405			0.125340			0.0626699	+	0.998034i	$0.480038\pi$
$140$		0			0
$141$	37.4475	+	18.6669i	0.0223663	+	0.0111492i
$142$		0			0
$143$	−462.593			−0.270517
$144$		0			0
$145$	105.729			0.0605539
$146$		0			0
$147$	227.870	+	113.589i	0.127853	+	0.0637324i
$148$		0			0
$149$	1375.72			0.756396			0.378198	−	0.925725i	$-0.376544\pi$
$149$	1375.72			0.756396			0.378198	+	0.925725i	$0.376544\pi$
$150$		0			0
$151$		−	1982.95i		−	1.06867i	−0.845271	−	0.534337i	$-0.820561\pi$
$151$		−	1982.95i		−	1.06867i	0.845271	−	0.534337i	$-0.179439\pi$
$152$		0			0
$153$	144.315	−	108.785i	0.0762559	−	0.0574820i
$154$		0			0
$155$		−	210.381i		−	0.109021i
$156$		0			0
$157$			2914.24i			1.48141i	0.671830	+	0.740705i	$0.265508\pi$
$157$			2914.24i			1.48141i	−0.671830	+	0.740705i	$0.734492\pi$
$158$		0			0
$159$	−571.741	−	285.003i	−0.285170	−	0.142152i
$160$		0			0
$161$			474.370i			0.232209i
$162$		0			0
$163$	−1902.61			−0.914260			−0.457130	−	0.889400i	$-0.651123\pi$
$163$	−1902.61			−0.914260			−0.457130	+	0.889400i	$0.651123\pi$
$164$		0			0
$165$	37.2714	−	74.7697i	0.0175853	−	0.0352777i
$166$		0			0
$167$	−3785.09			−1.75389			−0.876944	−	0.480593i	$-0.840422\pi$
$167$	−3785.09			−1.75389			−0.876944	+	0.480593i	$0.840422\pi$
$168$		0			0
$169$	1460.25			0.664658
$170$		0			0
$171$	−401.093	−	532.092i	−0.179370	−	0.237954i
$172$		0			0
$173$	3733.70			1.64085			0.820427	−	0.571751i	$-0.193736\pi$
$173$	3733.70			1.64085			0.820427	+	0.571751i	$0.193736\pi$
$174$		0			0
$175$		−	868.770i		−	0.375273i
$176$		0			0
$177$	1739.28	−	3489.15i	0.738600	−	1.48170i
$178$		0			0
$179$			129.596i			0.0541144i	0.999634	+	0.0270572i	$0.00861363\pi$
$179$			129.596i			0.0541144i	−0.999634	+	0.0270572i	$0.991386\pi$
$180$		0			0
$181$			3010.96i			1.23648i	0.785990	+	0.618240i	$0.212154\pi$
$181$			3010.96i			1.23648i	−0.785990	+	0.618240i	$0.787846\pi$
$182$		0			0
$183$	336.212	−	674.470i	0.135811	−	0.272450i
$184$		0			0
$185$			246.099i			0.0978029i
$186$		0			0
$187$	−114.075			−0.0446096
$188$		0			0
$189$	965.585	−	179.199i	0.371619	−	0.0689673i
$190$		0			0
$191$	1629.16			0.617181			0.308590	−	0.951195i	$-0.400143\pi$
$191$	1629.16			0.617181			0.308590	+	0.951195i	$0.400143\pi$
$192$		0			0
$193$	−1558.63			−0.581310			−0.290655	−	0.956828i	$-0.593873\pi$
$193$	−1558.63			−0.581310			−0.290655	+	0.956828i	$0.593873\pi$
$194$		0			0
$195$	59.3601	−	119.082i	0.0217993	−	0.0437313i
$196$		0			0
$197$	−3002.76			−1.08598			−0.542990	−	0.839739i	$-0.682708\pi$
$197$	−3002.76			−1.08598			−0.542990	+	0.839739i	$0.682708\pi$
$198$		0			0
$199$		−	2083.44i		−	0.742167i	−0.928600	−	0.371083i	$-0.878986\pi$
$199$		−	2083.44i		−	0.742167i	0.928600	−	0.371083i	$-0.121014\pi$
$200$		0			0
$201$	−3923.89	−	1955.99i	−1.37696	−	0.686392i
$202$		0			0
$203$		−	784.507i		−	0.271239i
$204$		0			0
$205$		−	154.719i		−	0.0527124i
$206$		0			0
$207$	1101.38	+	1461.10i	0.369813	+	0.490596i
$208$		0			0
$209$			420.597i			0.139202i
$210$		0			0
$211$	−2971.67			−0.969566			−0.484783	−	0.874634i	$-0.661101\pi$
$211$	−2971.67			−0.969566			−0.484783	+	0.874634i	$0.661101\pi$
$212$		0			0
$213$	−3797.71	−	1893.09i	−1.22167	−	0.608979i
$214$		0			0
$215$	91.1813			0.0289233
$216$		0			0
$217$	−1561.02			−0.488337
$218$		0			0
$219$	−3057.27	−	1523.99i	−0.943337	−	0.470237i
$220$		0			0
$221$	−181.681			−0.0552995
$222$		0			0
$223$			6292.13i			1.88947i	0.327833	+	0.944736i	$0.393682\pi$
$223$			6292.13i			1.88947i	−0.327833	+	0.944736i	$0.606318\pi$
$224$		0			0
$225$	−2017.09	−	2675.88i	−0.597656	−	0.792854i
$226$		0			0
$227$		−	1673.87i		−	0.489421i	−0.969596	−	0.244711i	$-0.921307\pi$
$227$		−	1673.87i		−	0.489421i	0.969596	−	0.244711i	$-0.0786930\pi$
$228$		0			0
$229$		−	4746.28i		−	1.36962i	−0.728722	−	0.684810i	$-0.759885\pi$
$229$		−	4746.28i		−	1.36962i	0.728722	−	0.684810i	$-0.240115\pi$
$230$		0			0
$231$	−554.790	−	276.553i	−0.158019	−	0.0787698i
$232$		0			0
$233$			223.319i			0.0627901i	0.999507	+	0.0313951i	$0.00999500\pi$
$233$			223.319i			0.0627901i	−0.999507	+	0.0313951i	$0.990005\pi$
$234$		0			0
$235$	7.59676			0.00210876
$236$		0			0
$237$	−1446.57	+	2901.95i	−0.396476	+	0.795367i
$238$		0			0
$239$	4344.72			1.17588			0.587942	−	0.808903i	$-0.299938\pi$
$239$	4344.72			1.17588			0.587942	+	0.808903i	$0.299938\pi$
$240$		0			0
$241$	−5974.00			−1.59676			−0.798379	−	0.602155i	$-0.794309\pi$
$241$	−5974.00			−1.59676			−0.798379	+	0.602155i	$0.794309\pi$
$242$		0			0
$243$	2558.02	−	2793.82i	0.675297	−	0.737546i
$244$		0			0
$245$	46.2266			0.0120543
$246$		0			0
$247$			669.862i			0.172560i
$248$		0			0
$249$	2680.77	−	5377.87i	0.682278	−	1.36871i
$250$		0			0
$251$			1308.95i			0.329164i	0.986363	+	0.164582i	$0.0526275\pi$
$251$			1308.95i			0.329164i	−0.986363	+	0.164582i	$0.947372\pi$
$252$		0			0
$253$		−	1154.94i		−	0.286998i
$254$		0			0
$255$	14.6381	−	29.3654i	0.00359481	−	0.00721151i
$256$		0			0
$257$			631.339i			0.153237i	0.997060	+	0.0766183i	$0.0244123\pi$
$257$			631.339i			0.153237i	−0.997060	+	0.0766183i	$0.975588\pi$
$258$		0			0
$259$	1826.05			0.438089
$260$		0			0
$261$	−1821.45	−	2416.34i	−0.431973	−	0.573057i
$262$		0			0
$263$	3898.73			0.914092			0.457046	−	0.889443i	$-0.348908\pi$
$263$	3898.73			0.914092			0.457046	+	0.889443i	$0.348908\pi$
$264$		0			0
$265$	−115.986			−0.0268866
$266$		0			0
$267$	832.015	−	1669.10i	0.190706	−	0.382573i
$268$		0			0
$269$	2413.41			0.547019			0.273509	−	0.961869i	$-0.411816\pi$
$269$	2413.41			0.547019			0.273509	+	0.961869i	$0.411816\pi$
$270$		0			0
$271$		−	251.809i		−	0.0564440i	−0.999602	−	0.0282220i	$-0.991015\pi$
$271$		−	251.809i		−	0.0564440i	0.999602	−	0.0282220i	$-0.00898453\pi$
$272$		0			0
$273$	−883.583	−	440.450i	−0.195886	−	0.0976457i
$274$		0			0
$275$			2115.18i			0.463818i
$276$		0			0
$277$		−	2104.47i		−	0.456480i	−0.973605	−	0.228240i	$-0.926703\pi$
$277$		−	2104.47i		−	0.456480i	0.973605	−	0.228240i	$-0.0732972\pi$
$278$		0			0
$279$	−4808.08	+	3624.35i	−1.03173	+	0.777720i
$280$		0			0
$281$		−	258.883i		−	0.0549596i	−0.999622	−	0.0274798i	$-0.991252\pi$
$281$		−	258.883i		−	0.0549596i	0.999622	−	0.0274798i	$-0.00874819\pi$
$282$		0			0
$283$	7910.71			1.66164			0.830818	−	0.556544i	$-0.187873\pi$
$283$	7910.71			1.66164			0.830818	+	0.556544i	$0.187873\pi$
$284$		0			0
$285$	−108.271	−	53.9712i	−0.0225032	−	0.0112175i
$286$		0			0
$287$	−1148.01			−0.236115
$288$		0			0
$289$	4868.20			0.990881
$290$		0			0
$291$	−3355.76	−	1672.79i	−0.676008	−	0.336978i
$292$		0			0
$293$	−9510.79			−1.89634			−0.948168	−	0.317771i	$-0.897066\pi$
$293$	−9510.79			−1.89634			−0.948168	+	0.317771i	$0.897066\pi$
$294$		0			0
$295$		−	707.823i		−	0.139699i
$296$		0			0
$297$	−2350.89	+	436.293i	−0.459301	+	0.0852400i
$298$		0			0
$299$		−	1839.41i		−	0.355772i
$300$		0			0
$301$		−	676.563i		−	0.129556i
$302$		0			0
$303$	4436.66	+	2211.60i	0.841187	+	0.419317i
$304$		0			0
$305$		−	136.826i		−	0.0256873i
$306$		0			0
$307$	−2430.42			−0.451829			−0.225914	−	0.974147i	$-0.572537\pi$
$307$	−2430.42			−0.451829			−0.225914	+	0.974147i	$0.572537\pi$
$308$		0			0
$309$	−520.285	+	1043.74i	−0.0957864	+	0.192156i
$310$		0			0
$311$	−7697.22			−1.40344			−0.701719	−	0.712453i	$-0.747584\pi$
$311$	−7697.22			−1.40344			−0.701719	+	0.712453i	$0.747584\pi$
$312$		0			0
$313$	−76.6858			−0.0138484			−0.00692418	−	0.999976i	$-0.502204\pi$
$313$	−76.6858			−0.0138484			−0.00692418	+	0.999976i	$0.502204\pi$
$314$		0			0
$315$	142.382	−	107.328i	0.0254676	−	0.0191976i
$316$		0			0
$317$	4721.34			0.836520			0.418260	−	0.908327i	$-0.362640\pi$
$317$	4721.34			0.836520			0.418260	+	0.908327i	$0.362640\pi$
$318$		0			0
$319$			1910.02i			0.335237i
$320$		0			0
$321$	4220.00	−	8465.71i	0.733762	−	1.47199i
$322$		0			0
$323$			165.187i			0.0284560i
$324$		0			0
$325$			3368.73i			0.574964i
$326$		0			0
$327$	3204.82	−	6429.15i	0.541978	−	1.08726i
$328$		0			0
$329$		−	56.3678i		−	0.00944577i
$330$		0			0
$331$	−7112.34			−1.18106			−0.590528	−	0.807017i	$-0.701080\pi$
$331$	−7112.34			−1.18106			−0.590528	+	0.807017i	$0.701080\pi$
$332$		0			0
$333$	5624.37	−	4239.67i	0.925566	−	0.697695i
$334$		0			0
$335$	−796.016			−0.129824
$336$		0			0
$337$	3971.89			0.642026			0.321013	−	0.947075i	$-0.395977\pi$
$337$	3971.89			0.642026			0.321013	+	0.947075i	$0.395977\pi$
$338$		0			0
$339$	4629.30	−	9286.80i	0.741679	−	1.48787i
$340$		0			0
$341$	3800.59			0.603559
$342$		0			0
$343$		−	343.000i		−	0.0539949i
$344$		0			0
$345$	297.307	+	148.202i	0.0463956	+	0.0231274i
$346$		0			0
$347$		−	1927.26i		−	0.298157i	−0.988825	−	0.149079i	$-0.952369\pi$
$347$		−	1927.26i		−	0.298157i	0.988825	−	0.149079i	$-0.0476307\pi$
$348$		0			0
$349$			2406.79i			0.369148i	0.982819	+	0.184574i	$0.0590906\pi$
$349$			2406.79i			0.369148i	−0.982819	+	0.184574i	$0.940909\pi$
$350$		0			0
$351$	−3744.13	+	694.860i	−0.569365	+	0.105666i
$352$		0			0
$353$			10899.0i			1.64332i	0.569976	+	0.821662i	$0.306953\pi$
$353$			10899.0i			1.64332i	−0.569976	+	0.821662i	$0.693047\pi$
$354$		0			0
$355$	−770.420			−0.115182
$356$		0			0
$357$	−217.891	−	108.615i	−0.0323025	−	0.0161022i
$358$		0			0
$359$	−1741.52			−0.256028			−0.128014	−	0.991772i	$-0.540860\pi$
$359$	−1741.52			−0.256028			−0.128014	+	0.991772i	$0.540860\pi$
$360$		0			0
$361$	−6249.95			−0.911204
$362$		0			0
$363$	−4838.95	−	2412.13i	−0.699666	−	0.348771i
$364$		0			0
$365$	−620.209			−0.0889404
$366$		0			0
$367$			7142.09i			1.01584i	0.861404	+	0.507921i	$0.169586\pi$
$367$			7142.09i			1.01584i	−0.861404	+	0.507921i	$0.830414\pi$
$368$		0			0
$369$	−3535.97	+	2665.43i	−0.498849	+	0.376034i
$370$		0			0
$371$			860.611i			0.120433i
$372$		0			0
$373$		−	11655.7i		−	1.61799i	−0.587816	−	0.808994i	$-0.700012\pi$
$373$		−	11655.7i		−	1.61799i	0.587816	−	0.808994i	$-0.299988\pi$
$374$		0			0
$375$	−1092.89	−	544.787i	−0.150498	−	0.0750205i
$376$		0			0
$377$			3041.99i			0.415571i
$378$		0			0
$379$	−6201.44			−0.840492			−0.420246	−	0.907410i	$-0.638056\pi$
$379$	−6201.44			−0.840492			−0.420246	+	0.907410i	$0.638056\pi$
$380$		0			0
$381$	−3903.60	+	7830.98i	−0.524902	+	1.05300i
$382$		0			0
$383$	8251.22			1.10083			0.550415	−	0.834891i	$-0.314470\pi$
$383$	8251.22			1.10083			0.550415	+	0.834891i	$0.314470\pi$
$384$		0			0
$385$	−112.547			−0.0148985
$386$		0			0
$387$	−1570.83	−	2083.87i	−0.206330	−	0.273719i
$388$		0			0
$389$	4402.39			0.573804			0.286902	−	0.957960i	$-0.407375\pi$
$389$	4402.39			0.573804			0.286902	+	0.957960i	$0.407375\pi$
$390$		0			0
$391$		−	453.597i		−	0.0586685i
$392$		0			0
$393$	1841.51	−	3694.23i	0.236366	−	0.474172i
$394$		0			0
$395$			588.702i			0.0749893i
$396$		0			0
$397$			1788.95i			0.226158i	0.993586	+	0.113079i	$0.0360713\pi$
$397$			1788.95i			0.226158i	−0.993586	+	0.113079i	$0.963929\pi$
$398$		0			0
$399$	−400.465	+	803.369i	−0.0502464	+	0.100799i
$400$		0			0
$401$		−	11736.2i		−	1.46154i	−0.682625	−	0.730769i	$-0.739162\pi$
$401$		−	11736.2i		−	1.46154i	0.682625	−	0.730769i	$-0.260838\pi$
$402$		0			0
$403$	6052.99			0.748191
$404$		0			0
$405$	189.356	−	661.157i	0.0232325	−	0.0811188i
$406$		0			0
$407$	−4445.84			−0.541455
$408$		0			0
$409$	−7905.07			−0.955698			−0.477849	−	0.878442i	$-0.658583\pi$
$409$	−7905.07			−0.955698			−0.477849	+	0.878442i	$0.658583\pi$
$410$		0			0
$411$	4016.23	−	8056.92i	0.482010	−	0.966956i
$412$		0			0
$413$	−5252.03			−0.625752
$414$		0			0
$415$		−	1090.98i		−	0.129046i
$416$		0			0
$417$	−955.216	−	476.158i	−0.112175	−	0.0559174i
$418$		0			0
$419$			5886.94i			0.686386i	0.939265	+	0.343193i	$0.111508\pi$
$419$			5886.94i			0.686386i	−0.939265	+	0.343193i	$0.888492\pi$
$420$		0			0
$421$		−	3040.33i		−	0.351964i	−0.984393	−	0.175982i	$-0.943690\pi$
$421$		−	3040.33i		−	0.351964i	0.984393	−	0.175982i	$-0.0563101\pi$
$422$		0			0
$423$	−130.873	−	173.617i	−0.0150432	−	0.0199564i
$424$		0			0
$425$			830.725i			0.0948143i
$426$		0			0
$427$	−1015.24			−0.115061
$428$		0			0
$429$	2151.24	+	1072.36i	0.242105	+	0.120685i
$430$		0			0
$431$	13242.3			1.47995			0.739973	−	0.672636i	$-0.234838\pi$
$431$	13242.3			1.47995			0.739973	+	0.672636i	$0.234838\pi$
$432$		0			0
$433$	2796.60			0.310384			0.155192	−	0.987884i	$-0.450400\pi$
$433$	2796.60			0.310384			0.155192	+	0.987884i	$0.450400\pi$
$434$		0			0
$435$	−491.682	−	245.095i	−0.0541939	−	0.0270147i
$436$		0			0
$437$	−1672.42			−0.183073
$438$		0			0
$439$		−	6183.96i		−	0.672311i	−0.941807	−	0.336155i	$-0.890873\pi$
$439$		−	6183.96i		−	0.672311i	0.941807	−	0.336155i	$-0.109127\pi$
$440$		0			0
$441$	−796.370	−	1056.47i	−0.0859917	−	0.114077i
$442$		0			0
$443$			6941.75i			0.744497i	0.928133	+	0.372249i	$0.121413\pi$
$443$			6941.75i			0.744497i	−0.928133	+	0.372249i	$0.878587\pi$
$444$		0			0
$445$		−	338.600i		−	0.0360700i
$446$		0			0
$447$	−6397.63	−	3189.10i	−0.676951	−	0.337448i
$448$		0			0
$449$		−	6553.46i		−	0.688813i	−0.938821	−	0.344406i	$-0.888080\pi$
$449$		−	6553.46i		−	0.688813i	0.938821	−	0.344406i	$-0.111920\pi$
$450$		0			0
$451$	2795.04			0.291826
$452$		0			0
$453$	−4596.75	+	9221.49i	−0.476764	+	0.956432i
$454$		0			0
$455$	−179.247			−0.0184687
$456$		0			0
$457$	6911.03			0.707406			0.353703	−	0.935358i	$-0.384922\pi$
$457$	6911.03			0.707406			0.353703	+	0.935358i	$0.384922\pi$
$458$		0			0
$459$	−923.301	+	171.352i	−0.0938910	+	0.0174249i
$460$		0			0
$461$	−10678.6			−1.07886			−0.539428	−	0.842032i	$-0.681359\pi$
$461$	−10678.6			−1.07886			−0.539428	+	0.842032i	$0.681359\pi$
$462$		0			0
$463$			17439.1i			1.75046i	0.483706	+	0.875231i	$0.339290\pi$
$463$			17439.1i			1.75046i	−0.483706	+	0.875231i	$0.660710\pi$
$464$		0			0
$465$	−487.693	+	978.356i	−0.0486370	+	0.0975703i
$466$		0			0
$467$			154.522i			0.0153114i	0.999971	+	0.00765569i	$0.00243690\pi$
$467$			154.522i			0.0153114i	−0.999971	+	0.00765569i	$0.997563\pi$
$468$		0			0
$469$			5906.42i			0.581520i
$470$		0			0
$471$	6755.61	−	13552.4i	0.660897	−	1.32582i
$472$		0			0
$473$			1647.22i			0.160125i
$474$		0			0
$475$	3062.90			0.295864
$476$		0			0
$477$	1998.15	+	2650.75i	0.191801	+	0.254444i
$478$		0			0
$479$	−18220.9			−1.73807			−0.869036	−	0.494749i	$-0.835260\pi$
$479$	−18220.9			−1.73807			−0.869036	+	0.494749i	$0.835260\pi$
$480$		0			0
$481$	−7080.64			−0.671205
$482$		0			0
$483$	1099.66	−	2206.01i	0.103595	−	0.207820i
$484$		0			0
$485$	−680.764			−0.0637358
$486$		0			0
$487$		−	16412.8i		−	1.52718i	−0.645702	−	0.763590i	$-0.723435\pi$
$487$		−	16412.8i		−	1.52718i	0.645702	−	0.763590i	$-0.276565\pi$
$488$		0			0
$489$	8847.92	+	4410.53i	0.818235	+	0.407875i
$490$		0			0
$491$			17707.5i			1.62756i	0.581175	+	0.813778i	$0.302593\pi$
$491$			17707.5i			1.62756i	−0.581175	+	0.813778i	$0.697407\pi$
$492$		0			0
$493$			750.152i			0.0685297i
$494$		0			0
$495$	−346.654	+	261.309i	−0.0314766	+	0.0237272i
$496$		0			0
$497$			5716.50i			0.515935i
$498$		0			0
$499$	−9056.68			−0.812490			−0.406245	−	0.913764i	$-0.633162\pi$
$499$	−9056.68			−0.812490			−0.406245	+	0.913764i	$0.633162\pi$
$500$		0			0
$501$	17602.2	+	8774.38i	1.56968	+	0.782456i
$502$		0			0
$503$	5887.79			0.521915			0.260958	−	0.965350i	$-0.415962\pi$
$503$	5887.79			0.521915			0.260958	+	0.965350i	$0.415962\pi$
$504$		0			0
$505$	900.040			0.0793094
$506$		0			0
$507$	−6790.76	−	3385.07i	−0.594849	−	0.296522i
$508$		0			0
$509$	9171.58			0.798670			0.399335	−	0.916805i	$-0.369241\pi$
$509$	9171.58			0.798670			0.399335	+	0.916805i	$0.369241\pi$
$510$		0			0
$511$			4601.94i			0.398391i
$512$		0			0
$513$	631.778	+	3404.23i	0.0543737	+	0.292983i
$514$		0			0
$515$			211.737i			0.0181170i
$516$		0			0
$517$			137.238i			0.0116745i
$518$		0			0
$519$	−17363.2	−	8655.24i	−1.46852	−	0.732029i
$520$		0			0
$521$		−	12082.2i		−	1.01599i	−0.861361	−	0.507993i	$-0.830387\pi$
$521$		−	12082.2i		−	1.01599i	0.861361	−	0.507993i	$-0.169613\pi$
$522$		0			0
$523$	5494.18			0.459357			0.229678	−	0.973267i	$-0.426233\pi$
$523$	5494.18			0.459357			0.229678	+	0.973267i	$0.426233\pi$
$524$		0			0
$525$	−2013.93	+	4040.13i	−0.167419	+	0.335858i
$526$		0			0
$527$	1492.66			0.123380
$528$		0			0
$529$	−7574.61			−0.622554
$530$		0			0
$531$	−16176.7	+	12194.1i	−1.32205	+	0.996566i
$532$		0			0
$533$	4451.51			0.361757
$534$		0			0
$535$		−	1717.39i		−	0.138783i
$536$		0			0
$537$	300.422	−	602.674i	0.0241419	−	0.0484308i
$538$		0			0
$539$			835.095i			0.0667349i
$540$		0			0
$541$			4495.37i			0.357248i	0.983917	+	0.178624i	$0.0571646\pi$
$541$			4495.37i			0.357248i	−0.983917	+	0.178624i	$0.942835\pi$
$542$		0			0
$543$	6979.83	−	14002.2i	0.551626	−	1.10661i
$544$		0			0
$545$		−	1304.24i		−	0.102510i
$546$		0			0
$547$	3606.54			0.281910			0.140955	−	0.990016i	$-0.454983\pi$
$547$	3606.54			0.281910			0.140955	+	0.990016i	$0.454983\pi$
$548$		0			0
$549$	−3127.04	+	2357.17i	−0.243094	+	0.183245i
$550$		0			0
$551$	2765.83			0.213844
$552$		0			0
$553$	4368.15			0.335900
$554$		0			0
$555$	570.491	−	1144.46i	0.0436325	−	0.0875306i
$556$		0			0
$557$	20947.3			1.59347			0.796737	−	0.604326i	$-0.206557\pi$
$557$	20947.3			1.59347			0.796737	+	0.604326i	$0.206557\pi$
$558$		0			0
$559$			2623.43i			0.198496i
$560$		0			0
$561$	530.494	+	264.442i	0.0399242	+	0.0199015i
$562$		0			0
$563$			21809.7i			1.63263i	0.577606	+	0.816315i	$0.303987\pi$
$563$			21809.7i			1.63263i	−0.577606	+	0.816315i	$0.696013\pi$
$564$		0			0
$565$		−	1883.96i		−	0.140281i
$566$		0			0
$567$	−4905.77	−	1405.01i	−0.363356	−	0.104065i
$568$		0			0
$569$			14204.3i			1.04653i	0.852171	+	0.523263i	$0.175286\pi$
$569$			14204.3i			1.04653i	−0.852171	+	0.523263i	$0.824714\pi$
$570$		0			0
$571$	13202.4			0.967605			0.483803	−	0.875177i	$-0.339255\pi$
$571$	13202.4			0.967605			0.483803	+	0.875177i	$0.339255\pi$
$572$		0			0
$573$	−7576.22	−	3776.61i	−0.552358	−	0.275341i
$574$		0			0
$575$	−8410.58			−0.609992
$576$		0			0
$577$	8318.78			0.600200			0.300100	−	0.953908i	$-0.402980\pi$
$577$	8318.78			0.600200			0.300100	+	0.953908i	$0.402980\pi$
$578$		0			0
$579$	7248.27	+	3613.13i	0.520255	+	0.259338i
$580$		0			0
$581$	−8095.03			−0.578035
$582$		0			0
$583$		−	2095.31i		−	0.148849i
$584$		0			0
$585$	−552.096	+	416.172i	−0.0390194	+	0.0294130i
$586$		0			0
$587$		−	10925.7i		−	0.768234i	−0.923284	−	0.384117i	$-0.874506\pi$
$587$		−	10925.7i		−	0.768234i	0.923284	−	0.384117i	$-0.125494\pi$
$588$		0			0
$589$		−	5503.48i		−	0.385004i
$590$		0			0
$591$	13964.0	+	6960.83i	0.971919	+	0.484484i
$592$		0			0
$593$			18048.3i			1.24984i	0.780689	+	0.624919i	$0.214868\pi$
$593$			18048.3i			1.24984i	−0.780689	+	0.624919i	$0.785132\pi$
$594$		0			0
$595$	−44.2022			−0.00304557
$596$		0			0
$597$	−4829.71	+	9688.83i	−0.331100	+	0.664217i
$598$		0			0
$599$	19992.5			1.36372			0.681861	−	0.731482i	$-0.261171\pi$
$599$	19992.5			1.36372			0.681861	+	0.731482i	$0.261171\pi$
$600$		0			0
$601$	−2502.34			−0.169838			−0.0849191	−	0.996388i	$-0.527063\pi$
$601$	−2502.34			−0.169838			−0.0849191	+	0.996388i	$0.527063\pi$
$602$		0			0
$603$	13713.4	+	18192.2i	0.926123	+	1.22860i
$604$		0			0
$605$	−981.649			−0.0659664
$606$		0			0
$607$			8274.67i			0.553309i	0.960969	+	0.276655i	$0.0892258\pi$
$607$			8274.67i			0.553309i	−0.960969	+	0.276655i	$0.910774\pi$
$608$		0			0
$609$	−1818.60	+	3648.27i	−0.121007	+	0.242751i
$610$		0			0
$611$			218.571i			0.0144721i
$612$		0			0
$613$		−	20454.3i		−	1.34770i	−0.738866	−	0.673852i	$-0.764639\pi$
$613$		−	20454.3i		−	1.34770i	0.738866	−	0.673852i	$-0.235361\pi$
$614$		0			0
$615$	−358.661	+	719.506i	−0.0235164	+	0.0471760i
$616$		0			0
$617$			18686.3i			1.21925i	0.792688	+	0.609627i	$0.208681\pi$
$617$			18686.3i			1.21925i	−0.792688	+	0.609627i	$0.791319\pi$
$618$		0			0
$619$	6393.01			0.415116			0.207558	−	0.978223i	$-0.433448\pi$
$619$	6393.01			0.415116			0.207558	+	0.978223i	$0.433448\pi$
$620$		0			0
$621$	−1734.83	−	9347.85i	−0.112104	−	0.604052i
$622$		0			0
$623$	−2512.40			−0.161569
$624$		0			0
$625$	15292.0			0.978691
$626$		0			0
$627$	975.004	−	1955.95i	0.0621019	−	0.124582i
$628$		0			0
$629$	−1746.08			−0.110685
$630$		0			0
$631$			7369.39i			0.464930i	0.972605	+	0.232465i	$0.0746791\pi$
$631$			7369.39i			0.464930i	−0.972605	+	0.232465i	$0.925321\pi$
$632$		0			0
$633$	13819.5	+	6888.76i	0.867733	+	0.432549i
$634$		0			0
$635$			1588.62i			0.0992797i
$636$		0			0
$637$			1330.01i			0.0827267i
$638$		0			0
$639$	13272.4	+	17607.3i	0.821673	+	1.09004i
$640$		0			0
$641$			19161.7i			1.18072i	0.807141	+	0.590359i	$0.201014\pi$
$641$			19161.7i			1.18072i	−0.807141	+	0.590359i	$0.798986\pi$
$642$		0			0
$643$	18736.5			1.14914			0.574569	−	0.818456i	$-0.305170\pi$
$643$	18736.5			1.14914			0.574569	+	0.818456i	$0.305170\pi$
$644$		0			0
$645$	−424.030	−	211.371i	−0.0258855	−	0.0129035i
$646$		0			0
$647$	−22636.9			−1.37550			−0.687751	−	0.725947i	$-0.741402\pi$
$647$	−22636.9			−1.37550			−0.687751	+	0.725947i	$0.741402\pi$
$648$		0			0
$649$	12787.0			0.773397
$650$		0			0
$651$	7259.38	+	3618.67i	0.437047	+	0.217860i
$652$		0			0
$653$	−10248.5			−0.614172			−0.307086	−	0.951682i	$-0.599354\pi$
$653$	−10248.5			−0.614172			−0.307086	+	0.951682i	$0.599354\pi$
$654$		0			0
$655$		−	749.428i		−	0.0447062i
$656$		0			0
$657$	10684.7	+	14174.3i	0.634473	+	0.841696i
$658$		0			0
$659$		−	21468.5i		−	1.26904i	−0.772908	−	0.634518i	$-0.781198\pi$
$659$		−	21468.5i		−	1.26904i	0.772908	−	0.634518i	$-0.218802\pi$
$660$		0			0
$661$		−	12401.7i		−	0.729760i	−0.931055	−	0.364880i	$-0.881110\pi$
$661$		−	12401.7i		−	0.729760i	0.931055	−	0.364880i	$-0.118890\pi$
$662$		0			0
$663$	844.889	+	421.162i	0.0494914	+	0.0246706i
$664$		0			0
$665$			162.975i			0.00950359i
$666$		0			0
$667$	−7594.83			−0.440889
$668$		0			0
$669$	14586.0	−	29260.9i	0.842943	−	1.69102i
$670$		0			0
$671$	2471.80			0.142210
$672$		0			0
$673$	9037.77			0.517653			0.258827	−	0.965924i	$-0.416664\pi$
$673$	9037.77			0.517653			0.258827	+	0.965924i	$0.416664\pi$
$674$		0			0
$675$	3177.20	+	17119.8i	0.181171	+	0.976211i
$676$		0			0
$677$	19176.7			1.08866			0.544328	−	0.838872i	$-0.316785\pi$
$677$	19176.7			1.08866			0.544328	+	0.838872i	$0.316785\pi$
$678$		0			0
$679$			5051.25i			0.285492i
$680$		0			0
$681$	−3880.27	+	7784.16i	−0.218344	+	0.438017i
$682$		0			0
$683$		−	5498.48i		−	0.308043i	−0.988067	−	0.154022i	$-0.950777\pi$
$683$		−	5498.48i		−	0.308043i	0.988067	−	0.154022i	$-0.0492225\pi$
$684$		0			0
$685$		−	1634.46i		−	0.0911672i
$686$		0			0
$687$	−11002.5	+	22072.1i	−0.611024	+	1.22577i
$688$		0			0
$689$		−	3337.09i		−	0.184518i
$690$		0			0
$691$	3584.97			0.197364			0.0986820	−	0.995119i	$-0.468537\pi$
$691$	3584.97			0.197364			0.0986820	+	0.995119i	$0.468537\pi$
$692$		0			0
$693$	1938.91	+	2572.16i	0.106281	+	0.140993i
$694$		0			0
$695$	−193.779			−0.0105762
$696$		0			0
$697$	1097.74			0.0596554
$698$		0			0
$699$	517.685	−	1038.52i	0.0280123	−	0.0561953i
$700$		0			0
$701$	−28915.0			−1.55792			−0.778962	−	0.627071i	$-0.784254\pi$
$701$	−28915.0			−1.55792			−0.778962	+	0.627071i	$0.784254\pi$
$702$		0			0
$703$			6437.84i			0.345388i
$704$		0			0
$705$	−35.3280	−	17.6104i	−0.00188727	−	0.000940773i
$706$		0			0
$707$		−	6678.27i		−	0.355251i
$708$		0			0
$709$			9248.53i			0.489895i	0.969536	+	0.244948i	$0.0787708\pi$
$709$			9248.53i			0.489895i	−0.969536	+	0.244948i	$0.921229\pi$
$710$		0			0
$711$	13454.3	−	10141.9i	0.709669	−	0.534951i
$712$		0			0
$713$			15112.3i			0.793773i
$714$		0			0
$715$	436.410			0.0228263
$716$		0			0
$717$	−20204.7	−	10071.7i	−1.05238	−	0.524593i
$718$		0			0
$719$	36559.1			1.89628			0.948139	−	0.317856i	$-0.102963\pi$
$719$	36559.1			1.89628			0.948139	+	0.317856i	$0.102963\pi$
$720$		0			0
$721$	1571.09			0.0811516
$722$		0			0
$723$	27781.5	+	13848.6i	1.42905	+	0.712357i
$724$		0			0
$725$	13909.3			0.712522
$726$		0			0
$727$		−	23985.0i		−	1.22360i	−0.791014	−	0.611798i	$-0.790446\pi$
$727$		−	23985.0i		−	1.22360i	0.791014	−	0.611798i	$-0.209554\pi$
$728$		0			0
$729$	−18372.3	+	7062.54i	−0.933409	+	0.358814i
$730$		0			0
$731$			646.936i			0.0327329i
$732$		0			0
$733$			57.3809i			0.00289142i	0.999999	+	0.00144571i	$0.000460184\pi$
$733$			57.3809i			0.00289142i	−0.999999	+	0.00144571i	$0.999540\pi$
$734$		0			0
$735$	−214.972	−	107.160i	−0.0107882	−	0.00537775i
$736$		0			0
$737$		−	14380.2i		−	0.718728i
$738$		0			0
$739$	−15466.2			−0.769868			−0.384934	−	0.922944i	$-0.625776\pi$
$739$	−15466.2			−0.769868			−0.384934	+	0.922944i	$0.625776\pi$
$740$		0			0
$741$	1552.84	−	3115.13i	0.0769836	−	0.154436i
$742$		0			0
$743$	−16398.3			−0.809686			−0.404843	−	0.914386i	$-0.632674\pi$
$743$	−16398.3			−0.809686			−0.404843	+	0.914386i	$0.632674\pi$
$744$		0			0
$745$	−1297.85			−0.0638248
$746$		0			0
$747$	−24933.3	+	18794.9i	−1.22124	+	0.920572i
$748$		0			0
$749$	−12743.0			−0.621653
$750$		0			0
$751$			31215.7i			1.51675i	0.651821	+	0.758373i	$0.274005\pi$
$751$			31215.7i			1.51675i	−0.651821	+	0.758373i	$0.725995\pi$
$752$		0			0
$753$	3034.33	−	6087.14i	0.146849	−	0.294592i
$754$		0			0
$755$			1870.71i			0.0901750i
$756$		0			0
$757$			35561.7i			1.70741i	0.520753	+	0.853707i	$0.325651\pi$
$757$			35561.7i			1.70741i	−0.520753	+	0.853707i	$0.674349\pi$
$758$		0			0
$759$	−2677.31	+	5370.93i	−0.128037	+	0.256854i
$760$		0			0
$761$		−	34821.0i		−	1.65869i	−0.558740	−	0.829343i	$-0.688715\pi$
$761$		−	34821.0i		−	1.65869i	0.558740	−	0.829343i	$-0.311285\pi$
$762$		0			0
$763$	−9677.46			−0.459171
$764$		0			0
$765$	−136.146	+	102.628i	−0.00643449	+	0.00485034i
$766$		0			0
$767$	20365.2			0.958728
$768$		0			0
$769$	−38667.0			−1.81322			−0.906610	−	0.421969i	$-0.861339\pi$
$769$	−38667.0			−1.81322			−0.906610	+	0.421969i	$0.861339\pi$
$770$		0			0
$771$	1463.53	−	2935.98i	0.0683630	−	0.137142i
$772$		0			0
$773$	−422.626			−0.0196647			−0.00983234	−	0.999952i	$-0.503130\pi$
$773$	−422.626			−0.0196647			−0.00983234	+	0.999952i	$0.503130\pi$
$774$		0			0
$775$		−	27676.9i		−	1.28282i
$776$		0			0
$777$	−8491.84	−	4233.03i	−0.392076	−	0.195443i
$778$		0			0
$779$		−	4047.39i		−	0.186152i
$780$		0			0
$781$		−	13917.8i		−	0.637669i
$782$		0			0
$783$	2869.04	+	15459.3i	0.130947	+	0.705584i
$784$		0			0
$785$		−	2749.29i		−	0.125002i
$786$		0			0
$787$	−25891.9			−1.17274			−0.586371	−	0.810043i	$-0.699444\pi$
$787$	−25891.9			−1.17274			−0.586371	+	0.810043i	$0.699444\pi$
$788$		0			0
$789$	−18130.7	−	9037.81i	−0.818085	−	0.407801i
$790$		0			0
$791$	−13978.9			−0.628361
$792$		0			0
$793$	3936.69			0.176288
$794$		0			0
$795$	539.380	+	268.871i	0.0240627	+	0.0119948i
$796$		0			0
$797$	4440.34			0.197346			0.0986731	−	0.995120i	$-0.468540\pi$
$797$	4440.34			0.197346			0.0986731	+	0.995120i	$0.468540\pi$
$798$		0			0
$799$			53.8994i			0.00238651i
$800$		0			0
$801$	−7738.40	+	5833.24i	−0.341352	+	0.257313i
$802$		0			0
$803$		−	11204.3i		−	0.492390i
$804$		0			0
$805$		−	447.521i		−	0.0195938i
$806$		0			0
$807$	−11223.3	−	5594.62i	−0.489565	−	0.244040i
$808$		0			0
$809$		−	38309.0i		−	1.66486i	−0.554128	−	0.832431i	$-0.686948\pi$
$809$		−	38309.0i		−	1.66486i	0.554128	−	0.832431i	$-0.313052\pi$
$810$		0			0
$811$	27549.7			1.19285			0.596426	−	0.802668i	$-0.296587\pi$
$811$	27549.7			1.19285			0.596426	+	0.802668i	$0.296587\pi$
$812$		0			0
$813$	−583.729	+	1171.01i	−0.0251812	+	0.0505157i
$814$		0			0
$815$	1794.93			0.0771454
$816$		0			0
$817$	2385.27			0.102142
$818$		0			0
$819$	3087.99	+	4096.54i	0.131750	+	0.174780i
$820$		0			0
$821$	10855.6			0.461467			0.230734	−	0.973017i	$-0.425887\pi$
$821$	10855.6			0.461467			0.230734	+	0.973017i	$0.425887\pi$
$822$		0			0
$823$		−	27797.2i		−	1.17734i	−0.808375	−	0.588668i	$-0.799652\pi$
$823$		−	27797.2i		−	1.17734i	0.808375	−	0.588668i	$-0.200348\pi$
$824$		0			0
$825$	4903.28	−	9836.42i	0.206922	−	0.415103i
$826$		0			0
$827$		−	26237.2i		−	1.10321i	−0.834105	−	0.551606i	$-0.814015\pi$
$827$		−	26237.2i		−	1.10321i	0.834105	−	0.551606i	$-0.185985\pi$
$828$		0			0
$829$			10692.2i			0.447956i	0.974594	+	0.223978i	$0.0719043\pi$
$829$			10692.2i			0.447956i	−0.974594	+	0.223978i	$0.928096\pi$
$830$		0			0
$831$	−4878.45	+	9786.61i	−0.203648	+	0.408536i
$832$		0			0
$833$			327.979i			0.0136420i
$834$		0			0
$835$	3570.85			0.147993
$836$		0			0
$837$	30761.2	−	5708.86i	1.27033	−	0.235755i
$838$		0			0
$839$	−36877.9			−1.51748			−0.758741	−	0.651392i	$-0.774185\pi$
$839$	−36877.9			−1.51748			−0.758741	+	0.651392i	$0.774185\pi$
$840$		0			0
$841$	−11828.8			−0.485005
$842$		0			0
$843$	−600.127	+	1203.91i	−0.0245189	+	0.0491872i
$844$		0			0
$845$	−1377.60			−0.0560840
$846$		0			0
$847$			7283.81i			0.295484i
$848$		0			0
$849$	−36788.0	−	18338.2i	−1.48711	−	0.741300i
$850$		0			0
$851$		−	17678.0i		−	0.712096i
$852$		0			0
$853$			23040.3i			0.924836i	0.886662	+	0.462418i	$0.153018\pi$
$853$			23040.3i			0.924836i	−0.886662	+	0.462418i	$0.846982\pi$
$854$		0			0
$855$	378.391	+	501.975i	0.0151353	+	0.0200786i
$856$		0			0
$857$		−	16033.1i		−	0.639069i	−0.947575	−	0.319534i	$-0.896474\pi$
$857$		−	16033.1i		−	0.639069i	0.947575	−	0.319534i	$-0.103526\pi$
$858$		0			0
$859$	−30817.2			−1.22406			−0.612030	−	0.790834i	$-0.709647\pi$
$859$	−30817.2			−1.22406			−0.612030	+	0.790834i	$0.709647\pi$
$860$		0			0
$861$	5338.71	+	2661.25i	0.211316	+	0.105337i
$862$		0			0
$863$	−23768.6			−0.937534			−0.468767	−	0.883322i	$-0.655302\pi$
$863$	−23768.6			−0.937534			−0.468767	+	0.883322i	$0.655302\pi$
$864$		0			0
$865$	−3522.37			−0.138456
$866$		0			0
$867$	−22639.1	−	11285.2i	−0.886809	−	0.442058i
$868$		0			0
$869$	−10635.1			−0.415155
$870$		0			0
$871$		−	22902.6i		−	0.890959i
$872$		0			0
$873$	11727.9	+	15558.3i	0.454672	+	0.603170i
$874$		0			0
$875$			1645.07i			0.0635584i
$876$		0			0
$877$		−	8398.79i		−	0.323383i	−0.986841	−	0.161692i	$-0.948305\pi$
$877$		−	8398.79i		−	0.323383i	0.986841	−	0.161692i	$-0.0516950\pi$
$878$		0			0
$879$	44229.0	+	22047.4i	1.69716	+	0.846006i
$880$		0			0
$881$		−	31711.6i		−	1.21270i	−0.795196	−	0.606352i	$-0.792632\pi$
$881$		−	31711.6i		−	1.21270i	0.795196	−	0.606352i	$-0.207368\pi$
$882$		0			0
$883$	8074.26			0.307724			0.153862	−	0.988092i	$-0.450829\pi$
$883$	8074.26			0.307724			0.153862	+	0.988092i	$0.450829\pi$
$884$		0			0
$885$	−1640.83	+	3291.66i	−0.0623232	+	0.125026i
$886$		0			0
$887$	3865.92			0.146341			0.0731707	−	0.997319i	$-0.476688\pi$
$887$	3865.92			0.146341			0.0731707	+	0.997319i	$0.476688\pi$
$888$		0			0
$889$	11787.6			0.444704
$890$		0			0
$891$	11944.0	+	3420.76i	0.449089	+	0.128619i
$892$		0			0
$893$	198.728			0.00744702
$894$		0			0
$895$		−	122.261i		−	0.00456618i
$896$		0			0
$897$	−4264.01	+	8553.99i	−0.158719	+	0.318405i
$898$		0			0
$899$		−	24992.5i		−	0.927193i
$900$		0			0
$901$		−	822.923i		−	0.0304279i
$902$		0			0
$903$	−1568.37	+	3146.29i	−0.0577985	+	0.115949i
$904$		0			0
$905$		−	2840.54i		−	0.104334i
$906$		0			0
$907$	−24928.5			−0.912610			−0.456305	−	0.889823i	$-0.650827\pi$
$907$	−24928.5			−0.912610			−0.456305	+	0.889823i	$0.650827\pi$
$908$		0			0
$909$	−15505.5	−	20569.6i	−0.565769	−	0.750552i
$910$		0			0
$911$	26550.2			0.965583			0.482791	−	0.875735i	$-0.339623\pi$
$911$	26550.2			0.965583			0.482791	+	0.875735i	$0.339623\pi$
$912$		0			0
$913$	19708.8			0.714421
$914$		0			0
$915$	−317.182	+	636.295i	−0.0114598	+	0.0229894i
$916$		0			0
$917$	−5560.73			−0.200253
$918$		0			0
$919$		−	10168.7i		−	0.365001i	−0.983206	−	0.182501i	$-0.941581\pi$
$919$		−	10168.7i		−	0.365001i	0.983206	−	0.182501i	$-0.0584191\pi$
$920$		0			0
$921$	11302.4	+	5634.06i	0.404373	+	0.201573i
$922$		0			0
$923$		−	22166.2i		−	0.790475i
$924$		0			0
$925$			32375.8i			1.15082i
$926$		0			0
$927$	4839.07	−	3647.71i	0.171452	−	0.129241i
$928$		0			0
$929$		−	25735.5i		−	0.908885i	−0.890776	−	0.454443i	$-0.849838\pi$
$929$		−	25735.5i		−	0.908885i	0.890776	−	0.454443i	$-0.150162\pi$
$930$		0			0
$931$	1209.27			0.0425694
$932$		0			0
$933$	35795.2	+	17843.3i	1.25604	+	0.626111i
$934$		0			0
$935$	107.618			0.00376416
$936$		0			0
$937$	−47302.6			−1.64921			−0.824605	−	0.565709i	$-0.808602\pi$
$937$	−47302.6			−1.64921			−0.824605	+	0.565709i	$0.808602\pi$
$938$		0			0
$939$	356.619	+	177.768i	0.0123939	+	0.00617812i
$940$		0			0
$941$	−30283.2			−1.04910			−0.524550	−	0.851379i	$-0.675767\pi$
$941$	−30283.2			−1.04910			−0.524550	+	0.851379i	$0.675767\pi$
$942$		0			0
$943$			11113.9i			0.383796i
$944$		0			0
$945$	−910.932	+	169.056i	−0.0313573	+	0.00581948i
$946$		0			0
$947$		−	38657.7i		−	1.32651i	−0.748393	−	0.663256i	$-0.769174\pi$
$947$		−	38657.7i		−	1.32651i	0.748393	−	0.663256i	$-0.230826\pi$
$948$		0			0
$949$		−	17844.4i		−	0.610383i
$950$		0			0
$951$	−21956.1	−	10944.7i	−0.748660	−	0.373194i
$952$		0			0
$953$		−	13831.0i		−	0.470125i	−0.971980	−	0.235062i	$-0.924471\pi$
$953$		−	13831.0i		−	0.470125i	0.971980	−	0.235062i	$-0.0755295\pi$
$954$		0			0
$955$	−1536.94			−0.0520778
$956$		0			0
$957$	4427.70	−	8882.37i	0.149558	−	0.300027i
$958$		0			0
$959$	−12127.7			−0.408366
$960$		0			0
$961$	−19939.4			−0.669311
$962$		0			0
$963$	−39249.4	+	29586.4i	−1.31339	+	0.990038i
$964$		0			0
$965$	1470.41			0.0490511
$966$		0			0
$967$		−	21472.2i		−	0.714062i	−0.934092	−	0.357031i	$-0.883789\pi$
$967$		−	21472.2i		−	0.714062i	0.934092	−	0.357031i	$-0.116211\pi$
$968$		0			0
$969$	382.928	−	768.188i	0.0126950	−	0.0254672i
$970$		0			0
$971$		−	15485.7i		−	0.511803i	−0.966703	−	0.255901i	$-0.917628\pi$
$971$		−	15485.7i		−	0.511803i	0.966703	−	0.255901i	$-0.0823722\pi$
$972$		0			0
$973$			1437.84i			0.0473740i
$974$		0			0
$975$	7809.18	−	15665.9i	0.256507	−	0.514576i
$976$		0			0
$977$		−	34470.1i		−	1.12876i	−0.825516	−	0.564379i	$-0.809116\pi$
$977$		−	34470.1i		−	1.12876i	0.825516	−	0.564379i	$-0.190884\pi$
$978$		0			0
$979$	6116.90			0.199690
$980$		0			0
$981$	−29807.4	+	22468.9i	−0.970108	+	0.731271i
$982$		0			0
$983$	−17916.9			−0.581343			−0.290671	−	0.956823i	$-0.593879\pi$
$983$	−17916.9			−0.581343			−0.290671	+	0.956823i	$0.593879\pi$
$984$		0			0
$985$	2832.80			0.0916352
$986$		0			0
$987$	−130.669	+	262.133i	−0.00421401	+	0.00845368i
$988$		0			0
$989$	−6549.83			−0.210589
$990$		0			0
$991$		−	8661.39i		−	0.277637i	−0.990318	−	0.138818i	$-0.955670\pi$
$991$		−	8661.39i		−	0.277637i	0.990318	−	0.138818i	$-0.0443305\pi$
$992$		0			0
$993$	33075.2	+	16487.4i	1.05701	+	0.526900i
$994$		0			0
$995$			1965.52i			0.0626242i
$996$		0			0
$997$			37796.8i			1.20064i	0.799760	+	0.600320i	$0.204960\pi$
$997$			37796.8i			1.20064i	−0.799760	+	0.600320i	$0.795040\pi$
$998$		0			0
$999$	−35983.7	+	6678.08i	−1.13961	+	0.211497i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.4.j.a.239.9		72
3.2	odd	2	inner	672.4.j.a.239.12		72
4.3	odd	2		168.4.j.a.155.72	yes	72
8.3	odd	2	inner	672.4.j.a.239.10		72
8.5	even	2		168.4.j.a.155.2	yes	72
12.11	even	2		168.4.j.a.155.1	✓	72
24.5	odd	2		168.4.j.a.155.71	yes	72
24.11	even	2	inner	672.4.j.a.239.11		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.j.a.155.1	✓	72	12.11	even	2
168.4.j.a.155.2	yes	72	8.5	even	2
168.4.j.a.155.71	yes	72	24.5	odd	2
168.4.j.a.155.72	yes	72	4.3	odd	2
672.4.j.a.239.9		72	1.1	even	1	trivial
672.4.j.a.239.10		72	8.3	odd	2	inner
672.4.j.a.239.11		72	24.11	even	2	inner
672.4.j.a.239.12		72	3.2	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	672.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.65040 - 2.31814i) q^{3} -0.943399 q^{5} +7.00000i q^{7} +(16.2524 + 21.5606i) q^{9} +O(q^{10})\)	\(q+(-4.65040 - 2.31814i) q^{3} -0.943399 q^{5} +7.00000i q^{7} +(16.2524 + 21.5606i) q^{9} -17.0428i q^{11} -27.1431i q^{13} +(4.38718 + 2.18693i) q^{15} -6.69346i q^{17} -24.6789 q^{19} +(16.2270 - 32.5528i) q^{21} +67.7672 q^{23} -124.110 q^{25} +(-25.5999 - 137.941i) q^{27} -112.072 q^{29} +223.003i q^{31} +(-39.5075 + 79.2557i) q^{33} -6.60379i q^{35} -260.864i q^{37} +(-62.9215 + 126.226i) q^{39} +164.002i q^{41} -96.6519 q^{43} +(-15.3325 - 20.3402i) q^{45} -8.05254 q^{47} -49.0000 q^{49} +(-15.5164 + 31.1273i) q^{51} +122.944 q^{53} +16.0781i q^{55} +(114.767 + 57.2093i) q^{57} +750.290i q^{59} +145.035i q^{61} +(-150.924 + 113.767i) q^{63} +25.6068i q^{65} +843.774 q^{67} +(-315.145 - 157.094i) q^{69} +816.642 q^{71} +657.420 q^{73} +(577.161 + 287.705i) q^{75} +119.299 q^{77} -624.022i q^{79} +(-200.716 + 700.824i) q^{81} +1156.43i q^{83} +6.31460i q^{85} +(521.181 + 259.800i) q^{87} +358.915i q^{89} +190.001 q^{91} +(516.953 - 1037.05i) q^{93} +23.2821 q^{95} +721.607 q^{97} +(367.452 - 276.986i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q+O(q^{10}) \) 72 * q	\( 72 q - 48 q^{19} + 1800 q^{25} - 264 q^{27} + 232 q^{33} + 864 q^{43} - 3528 q^{49} - 344 q^{57} - 1632 q^{67} - 3304 q^{75} + 920 q^{81} + 5312 q^{99}+O(q^{100}) \) 72 * q - 48 * q^19 + 1800 * q^25 - 264 * q^27 + 232 * q^33 + 864 * q^43 - 3528 * q^49 - 344 * q^57 - 1632 * q^67 - 3304 * q^75 + 920 * q^81 + 5312 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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