LMFDB - Embedded newform 1344.4.b.j.895.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(895,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1344.895");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			895.3
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.j.895.22

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -15.0520i q^{5} +(-17.2151 + 6.82941i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -15.0520i q^{5} +(-17.2151 + 6.82941i) q^{7} +9.00000 q^{9} +27.2041i q^{11} +25.6016i q^{13} -45.1561i q^{15} -13.7552i q^{17} +147.077 q^{19} +(-51.6453 + 20.4882i) q^{21} -87.4067i q^{23} -101.563 q^{25} +27.0000 q^{27} +19.9059 q^{29} +24.8121 q^{31} +81.6123i q^{33} +(102.796 + 259.122i) q^{35} -338.738 q^{37} +76.8048i q^{39} +195.330i q^{41} -216.529i q^{43} -135.468i q^{45} +527.838 q^{47} +(249.718 - 235.138i) q^{49} -41.2657i q^{51} +371.290 q^{53} +409.476 q^{55} +441.232 q^{57} -77.9067 q^{59} -671.685i q^{61} +(-154.936 + 61.4647i) q^{63} +385.356 q^{65} +706.188i q^{67} -262.220i q^{69} -1063.56i q^{71} +254.746i q^{73} -304.690 q^{75} +(-185.788 - 468.321i) q^{77} +38.8019i q^{79} +81.0000 q^{81} -33.1828 q^{83} -207.044 q^{85} +59.7178 q^{87} -188.045i q^{89} +(-174.844 - 440.734i) q^{91} +74.4363 q^{93} -2213.81i q^{95} -1340.43i q^{97} +244.837i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	$ 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) $ 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$		−	15.0520i		−	1.34629i	−0.739509	−	0.673147i	$-0.764942\pi$
$5$		−	15.0520i		−	1.34629i	0.739509	−	0.673147i	$-0.235058\pi$
$6$		0			0
$7$	−17.2151	+	6.82941i	−0.929527	+	0.368753i
$7$	−17.2151	+	6.82941i	−0.929527	+	0.368753i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			27.2041i			0.745667i	0.927898	+	0.372834i	$0.121614\pi$
$11$			27.2041i			0.745667i	−0.927898	+	0.372834i	$0.878386\pi$
$12$		0			0
$13$			25.6016i			0.546200i	0.961986	+	0.273100i	$0.0880490\pi$
$13$			25.6016i			0.546200i	−0.961986	+	0.273100i	$0.911951\pi$
$14$		0			0
$15$		−	45.1561i		−	0.777283i
$16$		0			0
$17$		−	13.7552i		−	0.196243i	−0.995174	−	0.0981217i	$-0.968717\pi$
$17$		−	13.7552i		−	0.196243i	0.995174	−	0.0981217i	$-0.0312834\pi$
$18$		0			0
$19$	147.077			1.77589			0.887943	−	0.459953i	$-0.152134\pi$
$19$	147.077			1.77589			0.887943	+	0.459953i	$0.152134\pi$
$20$		0			0
$21$	−51.6453	+	20.4882i	−0.536663	+	0.212900i
$22$		0			0
$23$		−	87.4067i		−	0.792416i	−0.918161	−	0.396208i	$-0.870326\pi$
$23$		−	87.4067i		−	0.792416i	0.918161	−	0.396208i	$-0.129674\pi$
$24$		0			0
$25$	−101.563			−0.812506
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	19.9059			0.127463			0.0637317	−	0.997967i	$-0.479700\pi$
$29$	19.9059			0.127463			0.0637317	+	0.997967i	$0.479700\pi$
$30$		0			0
$31$	24.8121			0.143754			0.0718771	−	0.997413i	$-0.477101\pi$
$31$	24.8121			0.143754			0.0718771	+	0.997413i	$0.477101\pi$
$32$		0			0
$33$			81.6123i			0.430511i
$34$		0			0
$35$	102.796	+	259.122i	0.496450	+	1.25142i
$36$		0			0
$37$	−338.738			−1.50509			−0.752543	−	0.658544i	$-0.771173\pi$
$37$	−338.738			−1.50509			−0.752543	+	0.658544i	$0.771173\pi$
$38$		0			0
$39$			76.8048i			0.315349i
$40$		0			0
$41$			195.330i			0.744035i	0.928226	+	0.372018i	$0.121334\pi$
$41$			195.330i			0.744035i	−0.928226	+	0.372018i	$0.878666\pi$
$42$		0			0
$43$		−	216.529i		−	0.767916i	−0.923351	−	0.383958i	$-0.874561\pi$
$43$		−	216.529i		−	0.767916i	0.923351	−	0.383958i	$-0.125439\pi$
$44$		0			0
$45$		−	135.468i		−	0.448764i
$46$		0			0
$47$	527.838			1.63815			0.819076	−	0.573685i	$-0.194487\pi$
$47$	527.838			1.63815			0.819076	+	0.573685i	$0.194487\pi$
$48$		0			0
$49$	249.718	−	235.138i	0.728042	−	0.685532i
$50$		0			0
$51$		−	41.2657i		−	0.113301i
$52$		0			0
$53$	371.290			0.962276			0.481138	−	0.876645i	$-0.340224\pi$
$53$	371.290			0.962276			0.481138	+	0.876645i	$0.340224\pi$
$54$		0			0
$55$	409.476			1.00389
$56$		0			0
$57$	441.232			1.02531
$58$		0			0
$59$	−77.9067			−0.171908			−0.0859541	−	0.996299i	$-0.527394\pi$
$59$	−77.9067			−0.171908			−0.0859541	+	0.996299i	$0.527394\pi$
$60$		0			0
$61$		−	671.685i		−	1.40984i	−0.709285	−	0.704922i	$-0.750982\pi$
$61$		−	671.685i		−	1.40984i	0.709285	−	0.704922i	$-0.249018\pi$
$62$		0			0
$63$	−154.936	+	61.4647i	−0.309842	+	0.122918i
$64$		0			0
$65$	385.356			0.735346
$66$		0			0
$67$			706.188i			1.28768i	0.765160	+	0.643841i	$0.222660\pi$
$67$			706.188i			1.28768i	−0.765160	+	0.643841i	$0.777340\pi$
$68$		0			0
$69$		−	262.220i		−	0.457501i
$70$		0			0
$71$		−	1063.56i		−	1.77777i	−0.458131	−	0.888885i	$-0.651481\pi$
$71$		−	1063.56i		−	1.77777i	0.458131	−	0.888885i	$-0.348519\pi$
$72$		0			0
$73$			254.746i			0.408435i	0.978926	+	0.204218i	$0.0654650\pi$
$73$			254.746i			0.408435i	−0.978926	+	0.204218i	$0.934535\pi$
$74$		0			0
$75$	−304.690			−0.469100
$76$		0			0
$77$	−185.788	−	468.321i	−0.274967	−	0.693118i
$78$		0			0
$79$			38.8019i			0.0552602i	0.999618	+	0.0276301i	$0.00879606\pi$
$79$			38.8019i			0.0552602i	−0.999618	+	0.0276301i	$0.991204\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−33.1828			−0.0438829			−0.0219415	−	0.999759i	$-0.506985\pi$
$83$	−33.1828			−0.0438829			−0.0219415	+	0.999759i	$0.506985\pi$
$84$		0			0
$85$	−207.044			−0.264201
$86$		0			0
$87$	59.7178			0.0735910
$88$		0			0
$89$		−	188.045i		−	0.223963i	−0.993710	−	0.111982i	$-0.964280\pi$
$89$		−	188.045i		−	0.223963i	0.993710	−	0.111982i	$-0.0357198\pi$
$90$		0			0
$91$	−174.844	−	440.734i	−0.201413	−	0.507708i
$92$		0			0
$93$	74.4363			0.0829966
$94$		0			0
$95$		−	2213.81i		−	2.39086i
$96$		0			0
$97$		−	1340.43i		−	1.40310i	−0.712622	−	0.701548i	$-0.752493\pi$
$97$		−	1340.43i		−	1.40310i	0.712622	−	0.701548i	$-0.247507\pi$
$98$		0			0
$99$			244.837i			0.248556i
$100$		0			0
$101$			694.627i			0.684336i	0.939639	+	0.342168i	$0.111161\pi$
$101$			694.627i			0.684336i	−0.939639	+	0.342168i	$0.888839\pi$
$102$		0			0
$103$	511.735			0.489541			0.244771	−	0.969581i	$-0.421287\pi$
$103$	511.735			0.489541			0.244771	+	0.969581i	$0.421287\pi$
$104$		0			0
$105$	308.389	+	777.365i	0.286626	+	0.722506i
$106$		0			0
$107$		−	1320.82i		−	1.19335i	−0.802482	−	0.596676i	$-0.796488\pi$
$107$		−	1320.82i		−	1.19335i	0.802482	−	0.596676i	$-0.203512\pi$
$108$		0			0
$109$	−1487.98			−1.30754			−0.653772	−	0.756692i	$-0.726814\pi$
$109$	−1487.98			−1.30754			−0.653772	+	0.756692i	$0.726814\pi$
$110$		0			0
$111$	−1016.21			−0.868961
$112$		0			0
$113$	1685.79			1.40342			0.701708	−	0.712465i	$-0.252421\pi$
$113$	1685.79			1.40342			0.701708	+	0.712465i	$0.252421\pi$
$114$		0			0
$115$	−1315.65			−1.06682
$116$		0			0
$117$			230.414i			0.182067i
$118$		0			0
$119$	93.9402	+	236.798i	0.0723654	+	0.182414i
$120$		0			0
$121$	590.937			0.443980
$122$		0			0
$123$			585.990i			0.429569i
$124$		0			0
$125$		−	352.771i		−	0.252422i
$126$		0			0
$127$		−	2132.01i		−	1.48965i	−0.667262	−	0.744823i	$-0.732534\pi$
$127$		−	2132.01i		−	1.48965i	0.667262	−	0.744823i	$-0.267466\pi$
$128$		0			0
$129$		−	649.587i		−	0.443356i
$130$		0			0
$131$	1557.84			1.03900			0.519499	−	0.854471i	$-0.326119\pi$
$131$	1557.84			1.03900			0.519499	+	0.854471i	$0.326119\pi$
$132$		0			0
$133$	−2531.95	+	1004.45i	−1.65073	+	0.654864i
$134$		0			0
$135$		−	406.404i		−	0.259094i
$136$		0			0
$137$	−1366.82			−0.852375			−0.426188	−	0.904635i	$-0.640144\pi$
$137$	−1366.82			−0.852375			−0.426188	+	0.904635i	$0.640144\pi$
$138$		0			0
$139$	−1834.60			−1.11949			−0.559744	−	0.828665i	$-0.689101\pi$
$139$	−1834.60			−1.11949			−0.559744	+	0.828665i	$0.689101\pi$
$140$		0			0
$141$	1583.52			0.945788
$142$		0			0
$143$	−696.468			−0.407284
$144$		0			0
$145$		−	299.624i		−	0.171603i
$146$		0			0
$147$	749.155	−	705.413i	0.420335	−	0.395792i
$148$		0			0
$149$	−1968.21			−1.08216			−0.541081	−	0.840970i	$-0.681985\pi$
$149$	−1968.21			−1.08216			−0.541081	+	0.840970i	$0.681985\pi$
$150$		0			0
$151$		−	2344.49i		−	1.26352i	−0.775164	−	0.631761i	$-0.782333\pi$
$151$		−	2344.49i		−	1.26352i	0.775164	−	0.631761i	$-0.217667\pi$
$152$		0			0
$153$		−	123.797i		−	0.0654145i
$154$		0			0
$155$		−	373.472i		−	0.193535i
$156$		0			0
$157$		−	1573.89i		−	0.800063i	−0.916501	−	0.400031i	$-0.868999\pi$
$157$		−	1573.89i		−	0.800063i	0.916501	−	0.400031i	$-0.131001\pi$
$158$		0			0
$159$	1113.87			0.555570
$160$		0			0
$161$	596.936	+	1504.71i	0.292206	+	0.736572i
$162$		0			0
$163$		−	2400.33i		−	1.15342i	−0.816947	−	0.576712i	$-0.804335\pi$
$163$		−	2400.33i		−	1.15342i	0.816947	−	0.576712i	$-0.195665\pi$
$164$		0			0
$165$	1228.43			0.579595
$166$		0			0
$167$	1555.74			0.720879			0.360439	−	0.932783i	$-0.382627\pi$
$167$	1555.74			0.720879			0.360439	+	0.932783i	$0.382627\pi$
$168$		0			0
$169$	1541.56			0.701665
$170$		0			0
$171$	1323.70			0.591962
$172$		0			0
$173$			418.477i			0.183909i	0.995763	+	0.0919543i	$0.0293114\pi$
$173$			418.477i			0.183909i	−0.995763	+	0.0919543i	$0.970689\pi$
$174$		0			0
$175$	1748.42	−	693.616i	0.755246	−	0.299614i
$176$		0			0
$177$	−233.720			−0.0992512
$178$		0			0
$179$		−	1397.09i		−	0.583371i	−0.956514	−	0.291686i	$-0.905784\pi$
$179$		−	1397.09i		−	0.583371i	0.956514	−	0.291686i	$-0.0942161\pi$
$180$		0			0
$181$			1954.09i			0.802466i	0.915976	+	0.401233i	$0.131418\pi$
$181$			1954.09i			0.802466i	−0.915976	+	0.401233i	$0.868582\pi$
$182$		0			0
$183$		−	2015.06i		−	0.813974i
$184$		0			0
$185$			5098.69i			2.02629i
$186$		0			0
$187$	374.199			0.146332
$188$		0			0
$189$	−464.807	+	184.394i	−0.178888	+	0.0709666i
$190$		0			0
$191$			602.233i			0.228147i	0.993472	+	0.114073i	$0.0363899\pi$
$191$			602.233i			0.228147i	−0.993472	+	0.114073i	$0.963610\pi$
$192$		0			0
$193$	−4869.54			−1.81615			−0.908076	−	0.418806i	$-0.862449\pi$
$193$	−4869.54			−1.81615			−0.908076	+	0.418806i	$0.862449\pi$
$194$		0			0
$195$	1156.07			0.424552
$196$		0			0
$197$	1735.63			0.627710			0.313855	−	0.949471i	$-0.398379\pi$
$197$	1735.63			0.627710			0.313855	+	0.949471i	$0.398379\pi$
$198$		0			0
$199$	1026.68			0.365724			0.182862	−	0.983139i	$-0.441464\pi$
$199$	1026.68			0.365724			0.182862	+	0.983139i	$0.441464\pi$
$200$		0			0
$201$			2118.57i			0.743443i
$202$		0			0
$203$	−342.682	+	135.946i	−0.118481	+	0.0470025i
$204$		0			0
$205$	2940.11			1.00169
$206$		0			0
$207$		−	786.661i		−	0.264139i
$208$		0			0
$209$			4001.10i			1.32422i
$210$		0			0
$211$			1008.49i			0.329040i	0.986374	+	0.164520i	$0.0526075\pi$
$211$			1008.49i			0.329040i	−0.986374	+	0.164520i	$0.947392\pi$
$212$		0			0
$213$		−	3190.69i		−	1.02640i
$214$		0			0
$215$	−3259.20			−1.03384
$216$		0			0
$217$	−427.142	+	169.452i	−0.133624	+	0.0530099i
$218$		0			0
$219$			764.238i			0.235810i
$220$		0			0
$221$	352.156			0.107188
$222$		0			0
$223$	2969.77			0.891796			0.445898	−	0.895084i	$-0.352884\pi$
$223$	2969.77			0.891796			0.445898	+	0.895084i	$0.352884\pi$
$224$		0			0
$225$	−914.069			−0.270835
$226$		0			0
$227$	5537.65			1.61915			0.809575	−	0.587017i	$-0.199698\pi$
$227$	5537.65			1.61915			0.809575	+	0.587017i	$0.199698\pi$
$228$		0			0
$229$			4407.78i			1.27194i	0.771714	+	0.635969i	$0.219400\pi$
$229$			4407.78i			1.27194i	−0.771714	+	0.635969i	$0.780600\pi$
$230$		0			0
$231$	−557.363	−	1404.96i	−0.158752	−	0.400172i
$232$		0			0
$233$	−2332.33			−0.655777			−0.327888	−	0.944716i	$-0.606337\pi$
$233$	−2332.33			−0.655777			−0.327888	+	0.944716i	$0.606337\pi$
$234$		0			0
$235$		−	7945.03i		−	2.20543i
$236$		0			0
$237$			116.406i			0.0319045i
$238$		0			0
$239$			2968.57i			0.803435i	0.915764	+	0.401717i	$0.131587\pi$
$239$			2968.57i			0.803435i	−0.915764	+	0.401717i	$0.868413\pi$
$240$		0			0
$241$		−	5652.00i		−	1.51069i	−0.655324	−	0.755347i	$-0.727468\pi$
$241$		−	5652.00i		−	1.51069i	0.655324	−	0.755347i	$-0.272532\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	−3539.30	−	3758.77i	−0.922928	−	0.980158i
$246$		0			0
$247$			3765.41i			0.969989i
$248$		0			0
$249$	−99.5483			−0.0253358
$250$		0			0
$251$	−88.2042			−0.0221809			−0.0110904	−	0.999938i	$-0.503530\pi$
$251$	−88.2042			−0.0221809			−0.0110904	+	0.999938i	$0.503530\pi$
$252$		0			0
$253$	2377.82			0.590879
$254$		0			0
$255$	−621.133			−0.152537
$256$		0			0
$257$		−	5659.62i		−	1.37369i	−0.726805	−	0.686844i	$-0.758996\pi$
$257$		−	5659.62i		−	1.37369i	0.726805	−	0.686844i	$-0.241004\pi$
$258$		0			0
$259$	5831.40	−	2313.38i	1.39902	−	0.555005i
$260$		0			0
$261$	179.153			0.0424878
$262$		0			0
$263$		−	3236.02i		−	0.758713i	−0.925251	−	0.379356i	$-0.876145\pi$
$263$		−	3236.02i		−	0.758713i	0.925251	−	0.379356i	$-0.123855\pi$
$264$		0			0
$265$		−	5588.66i		−	1.29551i
$266$		0			0
$267$		−	564.135i		−	0.129305i
$268$		0			0
$269$			6864.53i			1.55590i	0.628325	+	0.777951i	$0.283741\pi$
$269$			6864.53i			1.55590i	−0.628325	+	0.777951i	$0.716259\pi$
$270$		0			0
$271$	7664.56			1.71804			0.859020	−	0.511942i	$-0.171074\pi$
$271$	7664.56			1.71804			0.859020	+	0.511942i	$0.171074\pi$
$272$		0			0
$273$	−524.531	−	1322.20i	−0.116286	−	0.293125i
$274$		0			0
$275$		−	2762.94i		−	0.605859i
$276$		0			0
$277$	−733.283			−0.159057			−0.0795283	−	0.996833i	$-0.525341\pi$
$277$	−733.283			−0.159057			−0.0795283	+	0.996833i	$0.525341\pi$
$278$		0			0
$279$	223.309			0.0479181
$280$		0			0
$281$	4574.58			0.971163			0.485581	−	0.874191i	$-0.338608\pi$
$281$	4574.58			0.971163			0.485581	+	0.874191i	$0.338608\pi$
$282$		0			0
$283$	−520.292			−0.109287			−0.0546434	−	0.998506i	$-0.517402\pi$
$283$	−520.292			−0.109287			−0.0546434	+	0.998506i	$0.517402\pi$
$284$		0			0
$285$		−	6641.43i		−	1.38037i
$286$		0			0
$287$	−1333.99	−	3362.63i	−0.274365	−	0.691601i
$288$		0			0
$289$	4723.79			0.961489
$290$		0			0
$291$		−	4021.30i		−	0.810078i
$292$		0			0
$293$			3855.51i			0.768741i	0.923179	+	0.384371i	$0.125582\pi$
$293$			3855.51i			0.768741i	−0.923179	+	0.384371i	$0.874418\pi$
$294$		0			0
$295$			1172.65i			0.231439i
$296$		0			0
$297$			734.510i			0.143504i
$298$		0			0
$299$	2237.75			0.432818
$300$		0			0
$301$	1478.77	+	3727.57i	0.283172	+	0.713799i
$302$		0			0
$303$			2083.88i			0.395102i
$304$		0			0
$305$	−10110.2			−1.89806
$306$		0			0
$307$	−209.678			−0.0389804			−0.0194902	−	0.999810i	$-0.506204\pi$
$307$	−209.678			−0.0389804			−0.0194902	+	0.999810i	$0.506204\pi$
$308$		0			0
$309$	1535.20			0.282637
$310$		0			0
$311$	−6281.97			−1.14539			−0.572697	−	0.819767i	$-0.694103\pi$
$311$	−6281.97			−1.14539			−0.572697	+	0.819767i	$0.694103\pi$
$312$		0			0
$313$		−	5531.66i		−	0.998940i	−0.866331	−	0.499470i	$-0.833528\pi$
$313$		−	5531.66i		−	0.998940i	0.866331	−	0.499470i	$-0.166472\pi$
$314$		0			0
$315$	925.167	+	2332.10i	0.165483	+	0.417139i
$316$		0			0
$317$	10746.8			1.90410			0.952049	−	0.305945i	$-0.0989725\pi$
$317$	10746.8			1.90410			0.952049	+	0.305945i	$0.0989725\pi$
$318$		0			0
$319$			541.523i			0.0950453i
$320$		0			0
$321$		−	3962.47i		−	0.688982i
$322$		0			0
$323$		−	2023.08i		−	0.348506i
$324$		0			0
$325$		−	2600.18i		−	0.443791i
$326$		0			0
$327$	−4463.93			−0.754910
$328$		0			0
$329$	−9086.78	+	3604.82i	−1.52271	+	0.604074i
$330$		0			0
$331$		−	4144.42i		−	0.688212i	−0.938931	−	0.344106i	$-0.888182\pi$
$331$		−	4144.42i		−	0.688212i	0.938931	−	0.344106i	$-0.111818\pi$
$332$		0			0
$333$	−3048.64			−0.501695
$334$		0			0
$335$	10629.6			1.73360
$336$		0			0
$337$	4349.21			0.703017			0.351509	−	0.936185i	$-0.385669\pi$
$337$	4349.21			0.703017			0.351509	+	0.936185i	$0.385669\pi$
$338$		0			0
$339$	5057.38			0.810263
$340$		0			0
$341$			674.990i			0.107193i
$342$		0			0
$343$	−2693.07	+	5753.34i	−0.423943	+	0.905689i
$344$		0			0
$345$	−3946.94			−0.615931
$346$		0			0
$347$		−	3023.08i		−	0.467687i	−0.972274	−	0.233843i	$-0.924870\pi$
$347$		−	3023.08i		−	0.467687i	0.972274	−	0.233843i	$-0.0751303\pi$
$348$		0			0
$349$			7824.55i			1.20011i	0.799958	+	0.600056i	$0.204855\pi$
$349$			7824.55i			1.20011i	−0.799958	+	0.600056i	$0.795145\pi$
$350$		0			0
$351$			691.243i			0.105116i
$352$		0			0
$353$			7085.27i			1.06830i	0.845389	+	0.534152i	$0.179369\pi$
$353$			7085.27i			1.06830i	−0.845389	+	0.534152i	$0.820631\pi$
$354$		0			0
$355$	−16008.8			−2.39340
$356$		0			0
$357$	281.821	+	710.393i	0.0417802	+	0.105317i
$358$		0			0
$359$		−	1102.42i		−	0.162072i	−0.996711	−	0.0810358i	$-0.974177\pi$
$359$		−	1102.42i		−	0.162072i	0.996711	−	0.0810358i	$-0.0258228\pi$
$360$		0			0
$361$	14772.7			2.15377
$362$		0			0
$363$	1772.81			0.256332
$364$		0			0
$365$	3834.44			0.549873
$366$		0			0
$367$	−913.703			−0.129959			−0.0649794	−	0.997887i	$-0.520698\pi$
$367$	−913.703			−0.129959			−0.0649794	+	0.997887i	$0.520698\pi$
$368$		0			0
$369$			1757.97i			0.248012i
$370$		0			0
$371$	−6391.79	+	2535.69i	−0.894461	+	0.354842i
$372$		0			0
$373$	11782.9			1.63565			0.817823	−	0.575470i	$-0.195181\pi$
$373$	11782.9			1.63565			0.817823	+	0.575470i	$0.195181\pi$
$374$		0			0
$375$		−	1058.31i		−	0.145736i
$376$		0			0
$377$			509.623i			0.0696205i
$378$		0			0
$379$		−	12418.1i		−	1.68305i	−0.540216	−	0.841527i	$-0.681657\pi$
$379$		−	12418.1i		−	1.68305i	0.540216	−	0.841527i	$-0.318343\pi$
$380$		0			0
$381$		−	6396.03i		−	0.860048i
$382$		0			0
$383$	−12254.5			−1.63492			−0.817461	−	0.575984i	$-0.804619\pi$
$383$	−12254.5			−1.63492			−0.817461	+	0.575984i	$0.804619\pi$
$384$		0			0
$385$	−7049.17	+	2796.48i	−0.933141	+	0.370187i
$386$		0			0
$387$		−	1948.76i		−	0.255972i
$388$		0			0
$389$	−10242.7			−1.33503			−0.667515	−	0.744597i	$-0.732642\pi$
$389$	−10242.7			−1.33503			−0.667515	+	0.744597i	$0.732642\pi$
$390$		0			0
$391$	−1202.30			−0.155506
$392$		0			0
$393$	4673.51			0.599866
$394$		0			0
$395$	584.047			0.0743965
$396$		0			0
$397$		−	5209.88i		−	0.658631i	−0.944220	−	0.329316i	$-0.893182\pi$
$397$		−	5209.88i		−	0.658631i	0.944220	−	0.329316i	$-0.106818\pi$
$398$		0			0
$399$	−7595.84	+	3013.35i	−0.953052	+	0.378086i
$400$		0			0
$401$	−11924.3			−1.48496			−0.742480	−	0.669868i	$-0.766350\pi$
$401$	−11924.3			−1.48496			−0.742480	+	0.669868i	$0.766350\pi$
$402$		0			0
$403$			635.229i			0.0785186i
$404$		0			0
$405$		−	1219.21i		−	0.149588i
$406$		0			0
$407$		−	9215.05i		−	1.12229i
$408$		0			0
$409$			2042.83i			0.246972i	0.992346	+	0.123486i	$0.0394074\pi$
$409$			2042.83i			0.246972i	−0.992346	+	0.123486i	$0.960593\pi$
$410$		0			0
$411$	−4100.46			−0.492119
$412$		0			0
$413$	1341.17	−	532.056i	0.159793	−	0.0633917i
$414$		0			0
$415$			499.468i			0.0590793i
$416$		0			0
$417$	−5503.80			−0.646337
$418$		0			0
$419$	−7143.53			−0.832899			−0.416449	−	0.909159i	$-0.636726\pi$
$419$	−7143.53			−0.832899			−0.416449	+	0.909159i	$0.636726\pi$
$420$		0			0
$421$	−1207.16			−0.139746			−0.0698731	−	0.997556i	$-0.522259\pi$
$421$	−1207.16			−0.139746			−0.0698731	+	0.997556i	$0.522259\pi$
$422$		0			0
$423$	4750.55			0.546051
$424$		0			0
$425$			1397.03i			0.159449i
$426$		0			0
$427$	4587.21	+	11563.1i	0.519885	+	1.31049i
$428$		0			0
$429$	−2089.40			−0.235145
$430$		0			0
$431$			15173.9i			1.69582i	0.530139	+	0.847911i	$0.322140\pi$
$431$			15173.9i			1.69582i	−0.530139	+	0.847911i	$0.677860\pi$
$432$		0			0
$433$		−	9092.52i		−	1.00914i	−0.863370	−	0.504571i	$-0.831651\pi$
$433$		−	9092.52i		−	1.00914i	0.863370	−	0.504571i	$-0.168349\pi$
$434$		0			0
$435$		−	898.873i		−	0.0990751i
$436$		0			0
$437$		−	12855.5i		−	1.40724i
$438$		0			0
$439$	10351.1			1.12536			0.562678	−	0.826676i	$-0.309771\pi$
$439$	10351.1			1.12536			0.562678	+	0.826676i	$0.309771\pi$
$440$		0			0
$441$	2247.47	−	2116.24i	0.242681	−	0.228511i
$442$		0			0
$443$			14724.0i			1.57914i	0.613659	+	0.789571i	$0.289697\pi$
$443$			14724.0i			1.57914i	−0.613659	+	0.789571i	$0.710303\pi$
$444$		0			0
$445$	−2830.46			−0.301520
$446$		0			0
$447$	−5904.64			−0.624787
$448$		0			0
$449$	−7509.51			−0.789300			−0.394650	−	0.918832i	$-0.629134\pi$
$449$	−7509.51			−0.789300			−0.394650	+	0.918832i	$0.629134\pi$
$450$		0			0
$451$	−5313.78			−0.554803
$452$		0			0
$453$		−	7033.46i		−	0.729494i
$454$		0			0
$455$	−6633.93	+	2631.75i	−0.683524	+	0.271161i
$456$		0			0
$457$	3404.50			0.348481			0.174240	−	0.984703i	$-0.444253\pi$
$457$	3404.50			0.348481			0.174240	+	0.984703i	$0.444253\pi$
$458$		0			0
$459$		−	371.392i		−	0.0377671i
$460$		0			0
$461$			16179.2i			1.63457i	0.576231	+	0.817287i	$0.304523\pi$
$461$			16179.2i			1.63457i	−0.576231	+	0.817287i	$0.695477\pi$
$462$		0			0
$463$		−	2257.76i		−	0.226625i	−0.993559	−	0.113312i	$-0.963854\pi$
$463$		−	2257.76i		−	0.226625i	0.993559	−	0.113312i	$-0.0361461\pi$
$464$		0			0
$465$		−	1120.42i		−	0.111738i
$466$		0			0
$467$	14871.9			1.47364			0.736821	−	0.676088i	$-0.236326\pi$
$467$	14871.9			1.47364			0.736821	+	0.676088i	$0.236326\pi$
$468$		0			0
$469$	−4822.85	−	12157.1i	−0.474837	−	1.19694i
$470$		0			0
$471$		−	4721.66i		−	0.461917i
$472$		0			0
$473$	5890.48			0.572610
$474$		0			0
$475$	−14937.6			−1.44292
$476$		0			0
$477$	3341.61			0.320759
$478$		0			0
$479$	1190.52			0.113562			0.0567811	−	0.998387i	$-0.481916\pi$
$479$	1190.52			0.113562			0.0567811	+	0.998387i	$0.481916\pi$
$480$		0			0
$481$		−	8672.23i		−	0.822078i
$482$		0			0
$483$	1790.81	+	4514.14i	0.168705	+	0.425260i
$484$		0			0
$485$	−20176.2			−1.88898
$486$		0			0
$487$			5110.86i			0.475555i	0.971320	+	0.237777i	$0.0764189\pi$
$487$			5110.86i			0.475555i	−0.971320	+	0.237777i	$0.923581\pi$
$488$		0			0
$489$		−	7200.98i		−	0.665930i
$490$		0			0
$491$			2491.35i			0.228988i	0.993424	+	0.114494i	$0.0365247\pi$
$491$			2491.35i			0.228988i	−0.993424	+	0.114494i	$0.963475\pi$
$492$		0			0
$493$		−	273.811i		−	0.0250139i
$494$		0			0
$495$	3685.29			0.334629
$496$		0			0
$497$	7263.50	+	18309.3i	0.655558	+	1.65249i
$498$		0			0
$499$			19390.1i			1.73952i	0.493477	+	0.869759i	$0.335726\pi$
$499$			19390.1i			1.73952i	−0.493477	+	0.869759i	$0.664274\pi$
$500$		0			0
$501$	4667.22			0.416200
$502$		0			0
$503$	9029.44			0.800403			0.400202	−	0.916427i	$-0.368940\pi$
$503$	9029.44			0.800403			0.400202	+	0.916427i	$0.368940\pi$
$504$		0			0
$505$	10455.5			0.921318
$506$		0			0
$507$	4624.68			0.405107
$508$		0			0
$509$			20322.2i			1.76967i	0.465901	+	0.884837i	$0.345730\pi$
$509$			20322.2i			1.76967i	−0.465901	+	0.884837i	$0.654270\pi$
$510$		0			0
$511$	−1739.76	−	4385.47i	−0.150612	−	0.379652i
$512$		0			0
$513$	3971.09			0.341769
$514$		0			0
$515$		−	7702.64i		−	0.659066i
$516$		0			0
$517$			14359.4i			1.22152i
$518$		0			0
$519$			1255.43i			0.106180i
$520$		0			0
$521$		−	2042.29i		−	0.171736i	−0.996307	−	0.0858679i	$-0.972634\pi$
$521$		−	2042.29i		−	0.171736i	0.996307	−	0.0858679i	$-0.0273663\pi$
$522$		0			0
$523$	12849.8			1.07434			0.537171	−	0.843473i	$-0.319493\pi$
$523$	12849.8			1.07434			0.537171	+	0.843473i	$0.319493\pi$
$524$		0			0
$525$	5245.26	−	2080.85i	0.436042	−	0.172982i
$526$		0			0
$527$		−	341.296i		−	0.0282108i
$528$		0			0
$529$	4527.06			0.372077
$530$		0			0
$531$	−701.160			−0.0573027
$532$		0			0
$533$	−5000.76			−0.406392
$534$		0			0
$535$	−19881.0			−1.60660
$536$		0			0
$537$		−	4191.27i		−	0.336810i
$538$		0			0
$539$	6396.71	+	6793.36i	0.511179	+	0.542877i
$540$		0			0
$541$	1688.96			0.134222			0.0671111	−	0.997746i	$-0.478622\pi$
$541$	1688.96			0.134222			0.0671111	+	0.997746i	$0.478622\pi$
$542$		0			0
$543$			5862.27i			0.463304i
$544$		0			0
$545$			22397.0i			1.76034i
$546$		0			0
$547$			16668.3i			1.30290i	0.758692	+	0.651450i	$0.225839\pi$
$547$			16668.3i			1.30290i	−0.758692	+	0.651450i	$0.774161\pi$
$548$		0			0
$549$		−	6045.17i		−	0.469948i
$550$		0			0
$551$	2927.71			0.226361
$552$		0			0
$553$	−264.994	−	667.979i	−0.0203774	−	0.0513659i
$554$		0			0
$555$			15296.1i			1.16988i
$556$		0			0
$557$	−3901.51			−0.296790			−0.148395	−	0.988928i	$-0.547411\pi$
$557$	−3901.51			−0.296790			−0.148395	+	0.988928i	$0.547411\pi$
$558$		0			0
$559$	5543.49			0.419436
$560$		0			0
$561$	1122.60			0.0844850
$562$		0			0
$563$	25800.8			1.93139			0.965696	−	0.259674i	$-0.0836150\pi$
$563$	25800.8			1.93139			0.965696	+	0.259674i	$0.0836150\pi$
$564$		0			0
$565$		−	25374.6i		−	1.88941i
$566$		0			0
$567$	−1394.42	+	553.182i	−0.103281	+	0.0409726i
$568$		0			0
$569$	22155.0			1.63231			0.816156	−	0.577831i	$-0.196101\pi$
$569$	22155.0			1.63231			0.816156	+	0.577831i	$0.196101\pi$
$570$		0			0
$571$			9303.79i			0.681877i	0.940086	+	0.340938i	$0.110745\pi$
$571$			9303.79i			0.681877i	−0.940086	+	0.340938i	$0.889255\pi$
$572$		0			0
$573$			1806.70i			0.131721i
$574$		0			0
$575$			8877.31i			0.643842i
$576$		0			0
$577$			1625.68i			0.117293i	0.998279	+	0.0586464i	$0.0186784\pi$
$577$			1625.68i			0.117293i	−0.998279	+	0.0586464i	$0.981322\pi$
$578$		0			0
$579$	−14608.6			−1.04856
$580$		0			0
$581$	571.244	−	226.619i	0.0407904	−	0.0161820i
$582$		0			0
$583$			10100.6i			0.717538i
$584$		0			0
$585$	3468.20			0.245115
$586$		0			0
$587$	−15931.2			−1.12019			−0.560096	−	0.828428i	$-0.689236\pi$
$587$	−15931.2			−1.12019			−0.560096	+	0.828428i	$0.689236\pi$
$588$		0			0
$589$	3649.29			0.255291
$590$		0			0
$591$	5206.90			0.362408
$592$		0			0
$593$		−	7893.90i		−	0.546650i	−0.961922	−	0.273325i	$-0.911876\pi$
$593$		−	7893.90i		−	0.546650i	0.961922	−	0.273325i	$-0.0881235\pi$
$594$		0			0
$595$	3564.28	−	1413.99i	0.245582	−	0.0974250i
$596$		0			0
$597$	3080.03			0.211151
$598$		0			0
$599$		−	17823.9i		−	1.21580i	−0.794014	−	0.607899i	$-0.792012\pi$
$599$		−	17823.9i		−	1.21580i	0.794014	−	0.607899i	$-0.207988\pi$
$600$		0			0
$601$		−	7390.73i		−	0.501621i	−0.968036	−	0.250810i	$-0.919303\pi$
$601$		−	7390.73i		−	0.501621i	0.968036	−	0.250810i	$-0.0806970\pi$
$602$		0			0
$603$			6355.70i			0.429227i
$604$		0			0
$605$		−	8894.80i		−	0.597727i
$606$		0			0
$607$	10493.1			0.701652			0.350826	−	0.936441i	$-0.385901\pi$
$607$	10493.1			0.701652			0.350826	+	0.936441i	$0.385901\pi$
$608$		0			0
$609$	−1028.05	+	407.837i	−0.0684049	+	0.0271369i
$610$		0			0
$611$			13513.5i			0.894759i
$612$		0			0
$613$	−10049.7			−0.662159			−0.331079	−	0.943603i	$-0.607413\pi$
$613$	−10049.7			−0.662159			−0.331079	+	0.943603i	$0.607413\pi$
$614$		0			0
$615$	8820.34			0.578326
$616$		0			0
$617$	−4531.65			−0.295685			−0.147842	−	0.989011i	$-0.547233\pi$
$617$	−4531.65			−0.295685			−0.147842	+	0.989011i	$0.547233\pi$
$618$		0			0
$619$	−6471.90			−0.420239			−0.210119	−	0.977676i	$-0.567385\pi$
$619$	−6471.90			−0.420239			−0.210119	+	0.977676i	$0.567385\pi$
$620$		0			0
$621$		−	2359.98i		−	0.152500i
$622$		0			0
$623$	1284.24	+	3237.21i	0.0825872	+	0.208180i
$624$		0			0
$625$	−18005.3			−1.15234
$626$		0			0
$627$			12003.3i			0.764539i
$628$		0			0
$629$			4659.42i			0.295363i
$630$		0			0
$631$			15116.5i			0.953691i	0.878987	+	0.476846i	$0.158220\pi$
$631$			15116.5i			0.953691i	−0.878987	+	0.476846i	$0.841780\pi$
$632$		0			0
$633$			3025.48i			0.189971i
$634$		0			0
$635$	−32091.0			−2.00550
$636$		0			0
$637$	6019.90	+	6393.19i	0.374438	+	0.397657i
$638$		0			0
$639$		−	9572.06i		−	0.592590i
$640$		0			0
$641$	12466.9			0.768198			0.384099	−	0.923292i	$-0.374512\pi$
$641$	12466.9			0.768198			0.384099	+	0.923292i	$0.374512\pi$
$642$		0			0
$643$	−24488.3			−1.50190			−0.750951	−	0.660358i	$-0.770405\pi$
$643$	−24488.3			−1.50190			−0.750951	+	0.660358i	$0.770405\pi$
$644$		0			0
$645$	−9777.60			−0.596888
$646$		0			0
$647$	9980.07			0.606425			0.303212	−	0.952923i	$-0.401941\pi$
$647$	9980.07			0.606425			0.303212	+	0.952923i	$0.401941\pi$
$648$		0			0
$649$		−	2119.38i		−	0.128186i
$650$		0			0
$651$	−1281.43	+	508.355i	−0.0771476	+	0.0306053i
$652$		0			0
$653$	−16110.1			−0.965450			−0.482725	−	0.875772i	$-0.660353\pi$
$653$	−16110.1			−0.965450			−0.482725	+	0.875772i	$0.660353\pi$
$654$		0			0
$655$		−	23448.6i		−	1.39880i
$656$		0			0
$657$			2292.71i			0.136145i
$658$		0			0
$659$			18120.5i			1.07113i	0.844494	+	0.535565i	$0.179901\pi$
$659$			18120.5i			1.07113i	−0.844494	+	0.535565i	$0.820099\pi$
$660$		0			0
$661$		−	21636.0i		−	1.27314i	−0.771221	−	0.636568i	$-0.780354\pi$
$661$		−	21636.0i		−	1.27314i	0.771221	−	0.636568i	$-0.219646\pi$
$662$		0			0
$663$	1056.47			0.0618851
$664$		0			0
$665$	15119.0	+	38110.9i	0.881639	+	2.22237i
$666$		0			0
$667$		−	1739.91i		−	0.101004i
$668$		0			0
$669$	8909.31			0.514879
$670$		0			0
$671$	18272.6			1.05127
$672$		0			0
$673$	−19991.0			−1.14501			−0.572507	−	0.819900i	$-0.694029\pi$
$673$	−19991.0			−1.14501			−0.572507	+	0.819900i	$0.694029\pi$
$674$		0			0
$675$	−2742.21			−0.156367
$676$		0			0
$677$			24729.4i			1.40388i	0.712234	+	0.701942i	$0.247683\pi$
$677$			24729.4i			1.40388i	−0.712234	+	0.701942i	$0.752317\pi$
$678$		0			0
$679$	9154.36	+	23075.7i	0.517396	+	1.30422i
$680$		0			0
$681$	16613.0			0.934816
$682$		0			0
$683$		−	10700.8i		−	0.599494i	−0.954019	−	0.299747i	$-0.903098\pi$
$683$		−	10700.8i		−	0.599494i	0.954019	−	0.299747i	$-0.0969023\pi$
$684$		0			0
$685$			20573.4i			1.14755i
$686$		0			0
$687$			13223.3i			0.734354i
$688$		0			0
$689$			9505.61i			0.525595i
$690$		0			0
$691$	18990.5			1.04549			0.522745	−	0.852489i	$-0.324908\pi$
$691$	18990.5			1.04549			0.522745	+	0.852489i	$0.324908\pi$
$692$		0			0
$693$	−1672.09	−	4214.89i	−0.0916558	−	0.231039i
$694$		0			0
$695$			27614.5i			1.50716i
$696$		0			0
$697$	2686.81			0.146012
$698$		0			0
$699$	−6996.99			−0.378613
$700$		0			0
$701$	−7572.89			−0.408023			−0.204011	−	0.978968i	$-0.565398\pi$
$701$	−7572.89			−0.408023			−0.204011	+	0.978968i	$0.565398\pi$
$702$		0			0
$703$	−49820.6			−2.67286
$704$		0			0
$705$		−	23835.1i		−	1.27331i
$706$		0			0
$707$	−4743.89	−	11958.1i	−0.252351	−	0.636109i
$708$		0			0
$709$	−32690.5			−1.73162			−0.865809	−	0.500375i	$-0.833196\pi$
$709$	−32690.5			−1.73162			−0.865809	+	0.500375i	$0.833196\pi$
$710$		0			0
$711$			349.217i			0.0184201i
$712$		0			0
$713$		−	2168.74i		−	0.113913i
$714$		0			0
$715$			10483.2i			0.548323i
$716$		0			0
$717$			8905.71i			0.463863i
$718$		0			0
$719$	−10789.1			−0.559617			−0.279808	−	0.960056i	$-0.590271\pi$
$719$	−10789.1			−0.559617			−0.279808	+	0.960056i	$0.590271\pi$
$720$		0			0
$721$	−8809.56	+	3494.84i	−0.455042	+	0.180520i
$722$		0			0
$723$		−	16956.0i		−	0.872200i
$724$		0			0
$725$	−2021.71			−0.103565
$726$		0			0
$727$	12723.6			0.649094			0.324547	−	0.945870i	$-0.394788\pi$
$727$	12723.6			0.649094			0.324547	+	0.945870i	$0.394788\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−2978.41			−0.150698
$732$		0			0
$733$		−	31732.2i		−	1.59898i	−0.600678	−	0.799491i	$-0.705103\pi$
$733$		−	31732.2i		−	1.59898i	0.600678	−	0.799491i	$-0.294897\pi$
$734$		0			0
$735$	−10617.9	−	11276.3i	−0.532853	−	0.565895i
$736$		0			0
$737$	−19211.2			−0.960182
$738$		0			0
$739$			37386.9i			1.86103i	0.366254	+	0.930515i	$0.380640\pi$
$739$			37386.9i			1.86103i	−0.366254	+	0.930515i	$0.619360\pi$
$740$		0			0
$741$			11296.2i			0.560024i
$742$		0			0
$743$			7652.31i			0.377841i	0.981992	+	0.188921i	$0.0604989\pi$
$743$			7652.31i			0.377841i	−0.981992	+	0.188921i	$0.939501\pi$
$744$		0			0
$745$			29625.6i			1.45691i
$746$		0			0
$747$	−298.645			−0.0146276
$748$		0			0
$749$	9020.43	+	22738.1i	0.440053	+	1.10925i
$750$		0			0
$751$		−	6610.99i		−	0.321223i	−0.987018	−	0.160612i	$-0.948653\pi$
$751$		−	6610.99i		−	0.321223i	0.987018	−	0.160612i	$-0.0513466\pi$
$752$		0			0
$753$	−264.613			−0.0128061
$754$		0			0
$755$	−35289.3			−1.70107
$756$		0			0
$757$	−40156.8			−1.92804			−0.964018	−	0.265838i	$-0.914351\pi$
$757$	−40156.8			−1.92804			−0.964018	+	0.265838i	$0.914351\pi$
$758$		0			0
$759$	7133.46			0.341144
$760$		0			0
$761$			3028.46i			0.144259i	0.997395	+	0.0721297i	$0.0229796\pi$
$761$			3028.46i			0.144259i	−0.997395	+	0.0721297i	$0.977020\pi$
$762$		0			0
$763$	25615.6	−	10162.0i	1.21540	−	0.482161i
$764$		0			0
$765$	−1863.40			−0.0880671
$766$		0			0
$767$		−	1994.53i		−	0.0938963i
$768$		0			0
$769$		−	8219.82i		−	0.385454i	−0.981252	−	0.192727i	$-0.938267\pi$
$769$		−	8219.82i		−	0.385454i	0.981252	−	0.192727i	$-0.0617332\pi$
$770$		0			0
$771$		−	16978.9i		−	0.793099i
$772$		0			0
$773$			25171.1i			1.17120i	0.810599	+	0.585602i	$0.199142\pi$
$773$			25171.1i			1.17120i	−0.810599	+	0.585602i	$0.800858\pi$
$774$		0			0
$775$	−2520.00			−0.116801
$776$		0			0
$777$	17494.2	−	6940.13i	0.807723	−	0.320432i
$778$		0			0
$779$			28728.6i			1.32132i
$780$		0			0
$781$	28933.3			1.32563
$782$		0			0
$783$	537.460			0.0245303
$784$		0			0
$785$	−23690.2			−1.07712
$786$		0			0
$787$	−10829.9			−0.490525			−0.245262	−	0.969457i	$-0.578874\pi$
$787$	−10829.9			−0.490525			−0.245262	+	0.969457i	$0.578874\pi$
$788$		0			0
$789$		−	9708.06i		−	0.438043i
$790$		0			0
$791$	−29021.1	+	11513.0i	−1.30451	+	0.517514i
$792$		0			0
$793$	17196.2			0.770057
$794$		0			0
$795$		−	16766.0i		−	0.747960i
$796$		0			0
$797$			39662.5i			1.76276i	0.472412	+	0.881378i	$0.343383\pi$
$797$			39662.5i			1.76276i	−0.472412	+	0.881378i	$0.656617\pi$
$798$		0			0
$799$		−	7260.55i		−	0.321477i
$800$		0			0
$801$		−	1692.41i		−	0.0746544i
$802$		0			0
$803$	−6930.13			−0.304557
$804$		0			0
$805$	22649.0	−	8985.09i	0.991642	−	0.393395i
$806$		0			0
$807$			20593.6i			0.898301i
$808$		0			0
$809$	−24004.9			−1.04322			−0.521611	−	0.853183i	$-0.674669\pi$
$809$	−24004.9			−1.04322			−0.521611	+	0.853183i	$0.674669\pi$
$810$		0			0
$811$	13219.3			0.572372			0.286186	−	0.958174i	$-0.407612\pi$
$811$	13219.3			0.572372			0.286186	+	0.958174i	$0.407612\pi$
$812$		0			0
$813$	22993.7			0.991911
$814$		0			0
$815$	−36129.8			−1.55285
$816$		0			0
$817$		−	31846.5i		−	1.36373i
$818$		0			0
$819$	−1573.59	−	3966.60i	−0.0671377	−	0.169236i
$820$		0			0
$821$	11274.9			0.479288			0.239644	−	0.970861i	$-0.422969\pi$
$821$	11274.9			0.479288			0.239644	+	0.970861i	$0.422969\pi$
$822$		0			0
$823$		−	33943.0i		−	1.43764i	−0.695195	−	0.718821i	$-0.744682\pi$
$823$		−	33943.0i		−	1.43764i	0.695195	−	0.718821i	$-0.255318\pi$
$824$		0			0
$825$		−	8288.81i		−	0.349793i
$826$		0			0
$827$		−	4372.89i		−	0.183870i	−0.995765	−	0.0919349i	$-0.970695\pi$
$827$		−	4372.89i		−	0.183870i	0.995765	−	0.0919349i	$-0.0293052\pi$
$828$		0			0
$829$		−	38161.3i		−	1.59879i	−0.600807	−	0.799394i	$-0.705154\pi$
$829$		−	38161.3i		−	1.59879i	0.600807	−	0.799394i	$-0.294846\pi$
$830$		0			0
$831$	−2199.85			−0.0918314
$832$		0			0
$833$	−3234.38	−	3434.94i	−0.134531	−	0.142873i
$834$		0			0
$835$		−	23417.0i		−	0.970515i
$836$		0			0
$837$	669.926			0.0276655
$838$		0			0
$839$	−12748.2			−0.524575			−0.262287	−	0.964990i	$-0.584477\pi$
$839$	−12748.2			−0.524575			−0.262287	+	0.964990i	$0.584477\pi$
$840$		0			0
$841$	−23992.8			−0.983753
$842$		0			0
$843$	13723.8			0.560701
$844$		0			0
$845$		−	23203.6i		−	0.944647i
$846$		0			0
$847$	−10173.0	+	4035.75i	−0.412692	+	0.163719i
$848$		0			0
$849$	−1560.88			−0.0630967
$850$		0			0
$851$			29608.0i			1.19265i
$852$		0			0
$853$		−	19709.8i		−	0.791150i	−0.918434	−	0.395575i	$-0.870545\pi$
$853$		−	19709.8i		−	0.791150i	0.918434	−	0.395575i	$-0.129455\pi$
$854$		0			0
$855$		−	19924.3i		−	0.796955i
$856$		0			0
$857$			10857.7i			0.432779i	0.976307	+	0.216389i	$0.0694281\pi$
$857$			10857.7i			0.432779i	−0.976307	+	0.216389i	$0.930572\pi$
$858$		0			0
$859$	26714.9			1.06112			0.530560	−	0.847647i	$-0.321982\pi$
$859$	26714.9			1.06112			0.530560	+	0.847647i	$0.321982\pi$
$860$		0			0
$861$	−4001.97	−	10087.9i	−0.158405	−	0.399296i
$862$		0			0
$863$		−	18640.7i		−	0.735267i	−0.929971	−	0.367634i	$-0.880168\pi$
$863$		−	18640.7i		−	0.735267i	0.929971	−	0.367634i	$-0.119832\pi$
$864$		0			0
$865$	6298.92			0.247595
$866$		0			0
$867$	14171.4			0.555116
$868$		0			0
$869$	−1055.57			−0.0412058
$870$		0			0
$871$	−18079.5			−0.703332
$872$		0			0
$873$		−	12063.9i		−	0.467699i
$874$		0			0
$875$	2409.22	+	6072.98i	0.0930815	+	0.234633i
$876$		0			0
$877$	8149.11			0.313770			0.156885	−	0.987617i	$-0.449855\pi$
$877$	8149.11			0.313770			0.156885	+	0.987617i	$0.449855\pi$
$878$		0			0
$879$			11566.5i			0.443833i
$880$		0			0
$881$		−	42756.6i		−	1.63508i	−0.575872	−	0.817540i	$-0.695337\pi$
$881$		−	42756.6i		−	1.63508i	0.575872	−	0.817540i	$-0.304663\pi$
$882$		0			0
$883$		−	14109.0i		−	0.537718i	−0.963180	−	0.268859i	$-0.913353\pi$
$883$		−	14109.0i		−	0.537718i	0.963180	−	0.268859i	$-0.0866465\pi$
$884$		0			0
$885$			3517.96i			0.133621i
$886$		0			0
$887$	15410.5			0.583352			0.291676	−	0.956517i	$-0.405787\pi$
$887$	15410.5			0.583352			0.291676	+	0.956517i	$0.405787\pi$
$888$		0			0
$889$	14560.4	+	36702.7i	0.549312	+	1.38467i
$890$		0			0
$891$			2203.53i			0.0828519i
$892$		0			0
$893$	77633.0			2.90917
$894$		0			0
$895$	−21029.0			−0.785389
$896$		0			0
$897$	6713.25			0.249887
$898$		0			0
$899$	493.908			0.0183234
$900$		0			0
$901$		−	5107.19i		−	0.188840i
$902$		0			0
$903$	4436.30	+	11182.7i	0.163489	+	0.412112i
$904$		0			0
$905$	29413.0			1.08035
$906$		0			0
$907$		−	11061.2i		−	0.404940i	−0.979288	−	0.202470i	$-0.935103\pi$
$907$		−	11061.2i		−	0.404940i	0.979288	−	0.202470i	$-0.0648969\pi$
$908$		0			0
$909$			6251.64i			0.228112i
$910$		0			0
$911$			34878.4i			1.26847i	0.773142	+	0.634233i	$0.218684\pi$
$911$			34878.4i			1.26847i	−0.773142	+	0.634233i	$0.781316\pi$
$912$		0			0
$913$		−	902.707i		−	0.0327221i
$914$		0			0
$915$	−30330.7			−1.09585
$916$		0			0
$917$	−26818.3	+	10639.1i	−0.965777	+	0.383134i
$918$		0			0
$919$			15646.3i			0.561614i	0.959764	+	0.280807i	$0.0906021\pi$
$919$			15646.3i			0.561614i	−0.959764	+	0.280807i	$0.909398\pi$
$920$		0			0
$921$	−629.035			−0.0225053
$922$		0			0
$923$	27228.9			0.971018
$924$		0			0
$925$	34403.3			1.22289
$926$		0			0
$927$	4605.61			0.163180
$928$		0			0
$929$		−	32537.6i		−	1.14911i	−0.818465	−	0.574556i	$-0.805175\pi$
$929$		−	32537.6i		−	1.14911i	0.818465	−	0.574556i	$-0.194825\pi$
$930$		0			0
$931$	36727.9	−	34583.4i	1.29292	−	1.21743i
$932$		0			0
$933$	−18845.9			−0.661294
$934$		0			0
$935$		−	5632.45i		−	0.197006i
$936$		0			0
$937$			31430.6i			1.09583i	0.836534	+	0.547916i	$0.184579\pi$
$937$			31430.6i			1.09583i	−0.836534	+	0.547916i	$0.815421\pi$
$938$		0			0
$939$		−	16595.0i		−	0.576738i
$940$		0			0
$941$		−	39964.0i		−	1.38447i	−0.721671	−	0.692236i	$-0.756626\pi$
$941$		−	39964.0i		−	1.38447i	0.721671	−	0.692236i	$-0.243374\pi$
$942$		0			0
$943$	17073.2			0.589585
$944$		0			0
$945$	2775.50	+	6996.29i	0.0955419	+	0.240835i
$946$		0			0
$947$		−	52953.8i		−	1.81707i	−0.417805	−	0.908537i	$-0.637200\pi$
$947$		−	52953.8i		−	1.81707i	0.417805	−	0.908537i	$-0.362800\pi$
$948$		0			0
$949$	−6521.90			−0.223087
$950$		0			0
$951$	32240.3			1.09933
$952$		0			0
$953$	−23246.8			−0.790177			−0.395088	−	0.918643i	$-0.629286\pi$
$953$	−23246.8			−0.790177			−0.395088	+	0.918643i	$0.629286\pi$
$954$		0			0
$955$	9064.82			0.307152
$956$		0			0
$957$			1624.57i			0.0548744i
$958$		0			0
$959$	23529.9	−	9334.58i	0.792306	−	0.314316i
$960$		0			0
$961$	−29175.4			−0.979335
$962$		0			0
$963$		−	11887.4i		−	0.397784i
$964$		0			0
$965$			73296.4i			2.44507i
$966$		0			0
$967$		−	19719.0i		−	0.655760i	−0.944719	−	0.327880i	$-0.893666\pi$
$967$		−	19719.0i		−	0.655760i	0.944719	−	0.327880i	$-0.106334\pi$
$968$		0			0
$969$		−	6069.25i		−	0.201210i
$970$		0			0
$971$	−31665.7			−1.04655			−0.523275	−	0.852164i	$-0.675290\pi$
$971$	−31665.7			−1.04655			−0.523275	+	0.852164i	$0.675290\pi$
$972$		0			0
$973$	31582.8	−	12529.2i	1.04059	−	0.412815i
$974$		0			0
$975$		−	7800.54i		−	0.256223i
$976$		0			0
$977$	−36738.2			−1.20303			−0.601514	−	0.798862i	$-0.705436\pi$
$977$	−36738.2			−1.20303			−0.601514	+	0.798862i	$0.705436\pi$
$978$		0			0
$979$	5115.59			0.167002
$980$		0			0
$981$	−13391.8			−0.435848
$982$		0			0
$983$	33040.6			1.07206			0.536029	−	0.844200i	$-0.319924\pi$
$983$	33040.6			1.07206			0.536029	+	0.844200i	$0.319924\pi$
$984$		0			0
$985$		−	26124.8i		−	0.845081i
$986$		0			0
$987$	−27260.4	+	10814.5i	−0.879135	+	0.348762i
$988$		0			0
$989$	−18926.1			−0.608509
$990$		0			0
$991$			57138.0i			1.83153i	0.401713	+	0.915766i	$0.368415\pi$
$991$			57138.0i			1.83153i	−0.401713	+	0.915766i	$0.631585\pi$
$992$		0			0
$993$		−	12433.3i		−	0.397339i
$994$		0			0
$995$		−	15453.5i		−	0.492372i
$996$		0			0
$997$		−	26619.7i		−	0.845590i	−0.906225	−	0.422795i	$-0.861049\pi$
$997$		−	26619.7i		−	0.845590i	0.906225	−	0.422795i	$-0.138951\pi$
$998$		0			0
$999$	−9145.92			−0.289654

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.j.895.3		24
4.3	odd	2		1344.4.b.i.895.3		24
7.6	odd	2		1344.4.b.i.895.22		24
8.3	odd	2		672.4.b.b.223.22	yes	24
8.5	even	2		672.4.b.a.223.22	yes	24
28.27	even	2	inner	1344.4.b.j.895.22		24
56.13	odd	2		672.4.b.b.223.3	yes	24
56.27	even	2		672.4.b.a.223.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.3	✓	24	56.27	even	2
672.4.b.a.223.22	yes	24	8.5	even	2
672.4.b.b.223.3	yes	24	56.13	odd	2
672.4.b.b.223.22	yes	24	8.3	odd	2
1344.4.b.i.895.3		24	4.3	odd	2
1344.4.b.i.895.22		24	7.6	odd	2
1344.4.b.j.895.3		24	1.1	even	1	trivial
1344.4.b.j.895.22		24	28.27	even	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -15.0520i q^{5} +(-17.2151 + 6.82941i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -15.0520i q^{5} +(-17.2151 + 6.82941i) q^{7} +9.00000 q^{9} +27.2041i q^{11} +25.6016i q^{13} -45.1561i q^{15} -13.7552i q^{17} +147.077 q^{19} +(-51.6453 + 20.4882i) q^{21} -87.4067i q^{23} -101.563 q^{25} +27.0000 q^{27} +19.9059 q^{29} +24.8121 q^{31} +81.6123i q^{33} +(102.796 + 259.122i) q^{35} -338.738 q^{37} +76.8048i q^{39} +195.330i q^{41} -216.529i q^{43} -135.468i q^{45} +527.838 q^{47} +(249.718 - 235.138i) q^{49} -41.2657i q^{51} +371.290 q^{53} +409.476 q^{55} +441.232 q^{57} -77.9067 q^{59} -671.685i q^{61} +(-154.936 + 61.4647i) q^{63} +385.356 q^{65} +706.188i q^{67} -262.220i q^{69} -1063.56i q^{71} +254.746i q^{73} -304.690 q^{75} +(-185.788 - 468.321i) q^{77} +38.8019i q^{79} +81.0000 q^{81} -33.1828 q^{83} -207.044 q^{85} +59.7178 q^{87} -188.045i q^{89} +(-174.844 - 440.734i) q^{91} +74.4363 q^{93} -2213.81i q^{95} -1340.43i q^{97} +244.837i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) \) 24 * q + 72 * q^3 - 20 * q^7 + 216 * q^9 - 56 * q^19 - 60 * q^21 - 432 * q^25 + 648 * q^27 + 464 * q^31 + 568 * q^35 - 504 * q^37 + 560 * q^47 - 128 * q^49 + 784 * q^53 + 424 * q^55 - 168 * q^57 + 800 * q^59 - 180 * q^63 + 560 * q^65 - 1296 * q^75 + 1568 * q^77 + 1944 * q^81 + 1936 * q^83 - 3000 * q^85 - 496 * q^91 + 1392 * q^93

Introduction

L-functions

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