LMFDB - Embedded newform 1344.4.b.j.895.1

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			895.1
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.j.895.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -19.8256i q^{5} +(13.1775 + 13.0136i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -19.8256i q^{5} +(13.1775 + 13.0136i) q^{7} +9.00000 q^{9} +11.9649i q^{11} +82.5422i q^{13} -59.4769i q^{15} -100.713i q^{17} -104.043 q^{19} +(39.5325 + 39.0407i) q^{21} -144.803i q^{23} -268.056 q^{25} +27.0000 q^{27} -92.0491 q^{29} +195.901 q^{31} +35.8947i q^{33} +(258.002 - 261.252i) q^{35} -281.692 q^{37} +247.627i q^{39} -435.067i q^{41} -362.219i q^{43} -178.431i q^{45} -35.6802 q^{47} +(4.29353 + 342.973i) q^{49} -302.139i q^{51} +68.5367 q^{53} +237.212 q^{55} -312.128 q^{57} +460.066 q^{59} +443.029i q^{61} +(118.598 + 117.122i) q^{63} +1636.45 q^{65} -867.589i q^{67} -434.408i q^{69} -577.682i q^{71} -590.851i q^{73} -804.167 q^{75} +(-155.706 + 157.668i) q^{77} -282.598i q^{79} +81.0000 q^{81} -816.242 q^{83} -1996.70 q^{85} -276.147 q^{87} -849.591i q^{89} +(-1074.17 + 1087.70i) q^{91} +587.704 q^{93} +2062.71i q^{95} +711.190i q^{97} +107.684i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	19.8256i		−	1.77326i	−0.462481	−	0.886629i	$-0.653041\pi$
$5$		−	19.8256i		−	1.77326i	0.462481	−	0.886629i	$-0.346959\pi$
$6$		0			0
$7$	13.1775	+	13.0136i	0.711519	+	0.702667i
$7$	13.1775	+	13.0136i	0.711519	+	0.702667i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			11.9649i			0.327960i	0.986464	+	0.163980i	$0.0524332\pi$
$11$			11.9649i			0.327960i	−0.986464	+	0.163980i	$0.947567\pi$
$12$		0			0
$13$			82.5422i			1.76101i	0.474040	+	0.880503i	$0.342795\pi$
$13$			82.5422i			1.76101i	−0.474040	+	0.880503i	$0.657205\pi$
$14$		0			0
$15$		−	59.4769i		−	1.02379i
$16$		0			0
$17$		−	100.713i		−	1.43685i	−0.695602	−	0.718427i	$-0.744862\pi$
$17$		−	100.713i		−	1.43685i	0.695602	−	0.718427i	$-0.255138\pi$
$18$		0			0
$19$	−104.043			−1.25627			−0.628133	−	0.778106i	$-0.716181\pi$
$19$	−104.043			−1.25627			−0.628133	+	0.778106i	$0.716181\pi$
$20$		0			0
$21$	39.5325	+	39.0407i	0.410795	+	0.405685i
$22$		0			0
$23$		−	144.803i		−	1.31276i	−0.754432	−	0.656379i	$-0.772087\pi$
$23$		−	144.803i		−	1.31276i	0.754432	−	0.656379i	$-0.227913\pi$
$24$		0			0
$25$	−268.056			−2.14445
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	−92.0491			−0.589417			−0.294709	−	0.955587i	$-0.595223\pi$
$29$	−92.0491			−0.589417			−0.294709	+	0.955587i	$0.595223\pi$
$30$		0			0
$31$	195.901			1.13500			0.567499	−	0.823374i	$-0.307911\pi$
$31$	195.901			1.13500			0.567499	+	0.823374i	$0.307911\pi$
$32$		0			0
$33$			35.8947i			0.189348i
$34$		0			0
$35$	258.002	−	261.252i	1.24601	−	1.26171i
$36$		0			0
$37$	−281.692			−1.25162			−0.625808	−	0.779977i	$-0.715231\pi$
$37$	−281.692			−1.25162			−0.625808	+	0.779977i	$0.715231\pi$
$38$		0			0
$39$			247.627i			1.01672i
$40$		0			0
$41$		−	435.067i		−	1.65722i	−0.559826	−	0.828610i	$-0.689132\pi$
$41$		−	435.067i		−	1.65722i	0.559826	−	0.828610i	$-0.310868\pi$
$42$		0			0
$43$		−	362.219i		−	1.28460i	−0.766452	−	0.642301i	$-0.777980\pi$
$43$		−	362.219i		−	1.28460i	0.766452	−	0.642301i	$-0.222020\pi$
$44$		0			0
$45$		−	178.431i		−	0.591086i
$46$		0			0
$47$	−35.6802			−0.110734			−0.0553670	−	0.998466i	$-0.517633\pi$
$47$	−35.6802			−0.110734			−0.0553670	+	0.998466i	$0.517633\pi$
$48$		0			0
$49$	4.29353	+	342.973i	0.0125176	+	0.999922i
$50$		0			0
$51$		−	302.139i		−	0.829568i
$52$		0			0
$53$	68.5367			0.177627			0.0888135	−	0.996048i	$-0.471692\pi$
$53$	68.5367			0.177627			0.0888135	+	0.996048i	$0.471692\pi$
$54$		0			0
$55$	237.212			0.581557
$56$		0			0
$57$	−312.128			−0.725305
$58$		0			0
$59$	460.066			1.01518			0.507589	−	0.861599i	$-0.330537\pi$
$59$	460.066			1.01518			0.507589	+	0.861599i	$0.330537\pi$
$60$		0			0
$61$			443.029i			0.929903i	0.885336	+	0.464952i	$0.153928\pi$
$61$			443.029i			0.929903i	−0.885336	+	0.464952i	$0.846072\pi$
$62$		0			0
$63$	118.598	+	117.122i	0.237173	+	0.234222i
$64$		0			0
$65$	1636.45			3.12272
$66$		0			0
$67$		−	867.589i		−	1.58198i	−0.611827	−	0.790992i	$-0.709565\pi$
$67$		−	867.589i		−	1.58198i	0.611827	−	0.790992i	$-0.290435\pi$
$68$		0			0
$69$		−	434.408i		−	0.757921i
$70$		0			0
$71$		−	577.682i		−	0.965609i	−0.875728	−	0.482805i	$-0.839618\pi$
$71$		−	577.682i		−	0.965609i	0.875728	−	0.482805i	$-0.160382\pi$
$72$		0			0
$73$		−	590.851i		−	0.947313i	−0.880710	−	0.473657i	$-0.842934\pi$
$73$		−	590.851i		−	0.947313i	0.880710	−	0.473657i	$-0.157066\pi$
$74$		0			0
$75$	−804.167			−1.23810
$76$		0			0
$77$	−155.706	+	157.668i	−0.230446	+	0.233349i
$78$		0			0
$79$		−	282.598i		−	0.402466i	−0.979543	−	0.201233i	$-0.935505\pi$
$79$		−	282.598i		−	0.402466i	0.979543	−	0.201233i	$-0.0644948\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−816.242			−1.07945			−0.539724	−	0.841842i	$-0.681471\pi$
$83$	−816.242			−1.07945			−0.539724	+	0.841842i	$0.681471\pi$
$84$		0			0
$85$	−1996.70			−2.54791
$86$		0			0
$87$	−276.147			−0.340300
$88$		0			0
$89$		−	849.591i		−	1.01187i	−0.862572	−	0.505935i	$-0.831148\pi$
$89$		−	849.591i		−	1.01187i	0.862572	−	0.505935i	$-0.168852\pi$
$90$		0			0
$91$	−1074.17	+	1087.70i	−1.23740	+	1.25299i
$92$		0			0
$93$	587.704			0.655291
$94$		0			0
$95$			2062.71i			2.22768i
$96$		0			0
$97$			711.190i			0.744437i	0.928145	+	0.372218i	$0.121403\pi$
$97$			711.190i			0.744437i	−0.928145	+	0.372218i	$0.878597\pi$
$98$		0			0
$99$			107.684i			0.109320i
$100$		0			0
$101$		−	319.737i		−	0.315000i	−0.987519	−	0.157500i	$-0.949657\pi$
$101$		−	319.737i		−	0.315000i	0.987519	−	0.157500i	$-0.0503434\pi$
$102$		0			0
$103$	−732.971			−0.701183			−0.350591	−	0.936529i	$-0.614019\pi$
$103$	−732.971			−0.701183			−0.350591	+	0.936529i	$0.614019\pi$
$104$		0			0
$105$	774.007	−	783.757i	0.719385	−	0.728447i
$106$		0			0
$107$			425.131i			0.384103i	0.981385	+	0.192051i	$0.0615140\pi$
$107$			425.131i			0.384103i	−0.981385	+	0.192051i	$0.938486\pi$
$108$		0			0
$109$	1081.99			0.950787			0.475394	−	0.879773i	$-0.342306\pi$
$109$	1081.99			0.950787			0.475394	+	0.879773i	$0.342306\pi$
$110$		0			0
$111$	−845.075			−0.722621
$112$		0			0
$113$	−1582.56			−1.31747			−0.658737	−	0.752373i	$-0.728909\pi$
$113$	−1582.56			−1.31747			−0.658737	+	0.752373i	$0.728909\pi$
$114$		0			0
$115$	−2870.80			−2.32786
$116$		0			0
$117$			742.880i			0.587002i
$118$		0			0
$119$	1310.64	−	1327.15i	1.00963	−	1.02235i
$120$		0			0
$121$	1187.84			0.892443
$122$		0			0
$123$		−	1305.20i		−	0.956797i
$124$		0			0
$125$			2836.17i			2.02940i
$126$		0			0
$127$			664.704i			0.464433i	0.972664	+	0.232216i	$0.0745978\pi$
$127$			664.704i			0.464433i	−0.972664	+	0.232216i	$0.925402\pi$
$128$		0			0
$129$		−	1086.66i		−	0.741666i
$130$		0			0
$131$	−2109.94			−1.40722			−0.703612	−	0.710584i	$-0.748431\pi$
$131$	−2109.94			−1.40722			−0.703612	+	0.710584i	$0.748431\pi$
$132$		0			0
$133$	−1371.02	−	1353.97i	−0.893856	−	0.882737i
$134$		0			0
$135$		−	535.292i		−	0.341264i
$136$		0			0
$137$	−596.623			−0.372065			−0.186033	−	0.982544i	$-0.559563\pi$
$137$	−596.623			−0.372065			−0.186033	+	0.982544i	$0.559563\pi$
$138$		0			0
$139$	−310.445			−0.189436			−0.0947180	−	0.995504i	$-0.530195\pi$
$139$	−310.445			−0.189436			−0.0947180	+	0.995504i	$0.530195\pi$
$140$		0			0
$141$	−107.041			−0.0639323
$142$		0			0
$143$	−987.610			−0.577539
$144$		0			0
$145$			1824.93i			1.04519i
$146$		0			0
$147$	12.8806	+	1028.92i	0.00722703	+	0.577305i
$148$		0			0
$149$	162.635			0.0894198			0.0447099	−	0.999000i	$-0.485764\pi$
$149$	162.635			0.0894198			0.0447099	+	0.999000i	$0.485764\pi$
$150$		0			0
$151$		−	514.083i		−	0.277056i	−0.990359	−	0.138528i	$-0.955763\pi$
$151$		−	514.083i		−	0.277056i	0.990359	−	0.138528i	$-0.0442371\pi$
$152$		0			0
$153$		−	906.418i		−	0.478951i
$154$		0			0
$155$		−	3883.87i		−	2.01264i
$156$		0			0
$157$			2690.41i			1.36763i	0.729655	+	0.683815i	$0.239681\pi$
$157$			2690.41i			1.36763i	−0.729655	+	0.683815i	$0.760319\pi$
$158$		0			0
$159$	205.610			0.102553
$160$		0			0
$161$	1884.40	−	1908.14i	0.922431	−	0.934051i
$162$		0			0
$163$		−	2139.11i		−	1.02790i	−0.857820	−	0.513950i	$-0.828182\pi$
$163$		−	2139.11i		−	1.02790i	0.857820	−	0.513950i	$-0.171818\pi$
$164$		0			0
$165$	711.636			0.335762
$166$		0			0
$167$	−760.756			−0.352509			−0.176255	−	0.984345i	$-0.556398\pi$
$167$	−760.756			−0.352509			−0.176255	+	0.984345i	$0.556398\pi$
$168$		0			0
$169$	−4616.22			−2.10115
$170$		0			0
$171$	−936.385			−0.418755
$172$		0			0
$173$		−	2118.18i		−	0.930879i	−0.885080	−	0.465440i	$-0.845896\pi$
$173$		−	2118.18i		−	0.930879i	0.885080	−	0.465440i	$-0.154104\pi$
$174$		0			0
$175$	−3532.31	−	3488.36i	−1.52581	−	1.50683i
$176$		0			0
$177$	1380.20			0.586114
$178$		0			0
$179$		−	3182.09i		−	1.32872i	−0.747414	−	0.664359i	$-0.768705\pi$
$179$		−	3182.09i		−	1.32872i	0.747414	−	0.664359i	$-0.231295\pi$
$180$		0			0
$181$		−	1314.83i		−	0.539947i	−0.962868	−	0.269974i	$-0.912985\pi$
$181$		−	1314.83i		−	0.539947i	0.962868	−	0.269974i	$-0.0870150\pi$
$182$		0			0
$183$			1329.09i			0.536880i
$184$		0			0
$185$			5584.71i			2.21944i
$186$		0			0
$187$	1205.02			0.471230
$188$		0			0
$189$	355.793	+	351.367i	0.136932	+	0.135228i
$190$		0			0
$191$			1197.77i			0.453758i	0.973923	+	0.226879i	$0.0728522\pi$
$191$			1197.77i			0.453758i	−0.973923	+	0.226879i	$0.927148\pi$
$192$		0			0
$193$	4675.95			1.74395			0.871975	−	0.489551i	$-0.162839\pi$
$193$	4675.95			1.74395			0.871975	+	0.489551i	$0.162839\pi$
$194$		0			0
$195$	4909.35			1.80290
$196$		0			0
$197$	−192.288			−0.0695428			−0.0347714	−	0.999395i	$-0.511070\pi$
$197$	−192.288			−0.0695428			−0.0347714	+	0.999395i	$0.511070\pi$
$198$		0			0
$199$	3862.98			1.37608			0.688038	−	0.725675i	$-0.258472\pi$
$199$	3862.98			1.37608			0.688038	+	0.725675i	$0.258472\pi$
$200$		0			0
$201$		−	2602.77i		−	0.913358i
$202$		0			0
$203$	−1212.98	−	1197.89i	−0.419381	−	0.414164i
$204$		0			0
$205$	−8625.48			−2.93868
$206$		0			0
$207$		−	1303.22i		−	0.437586i
$208$		0			0
$209$		−	1244.86i		−	0.412004i
$210$		0			0
$211$			4553.94i			1.48581i	0.669397	+	0.742905i	$0.266553\pi$
$211$			4553.94i			1.48581i	−0.669397	+	0.742905i	$0.733447\pi$
$212$		0			0
$213$		−	1733.05i		−	0.557495i
$214$		0			0
$215$	−7181.23			−2.27793
$216$		0			0
$217$	2581.49	+	2549.38i	0.807572	+	0.797526i
$218$		0			0
$219$		−	1772.55i		−	0.546932i
$220$		0			0
$221$	8313.08			2.53031
$222$		0			0
$223$	−1317.35			−0.395588			−0.197794	−	0.980244i	$-0.563378\pi$
$223$	−1317.35			−0.395588			−0.197794	+	0.980244i	$0.563378\pi$
$224$		0			0
$225$	−2412.50			−0.714815
$226$		0			0
$227$	1135.03			0.331871			0.165935	−	0.986137i	$-0.446936\pi$
$227$	1135.03			0.331871			0.165935	+	0.986137i	$0.446936\pi$
$228$		0			0
$229$			854.930i			0.246705i	0.992363	+	0.123352i	$0.0393645\pi$
$229$			854.930i			0.246705i	−0.992363	+	0.123352i	$0.960635\pi$
$230$		0			0
$231$	−467.119	+	473.003i	−0.133048	+	0.134724i
$232$		0			0
$233$	−1250.24			−0.351526			−0.175763	−	0.984432i	$-0.556239\pi$
$233$	−1250.24			−0.351526			−0.175763	+	0.984432i	$0.556239\pi$
$234$		0			0
$235$			707.383i			0.196360i
$236$		0			0
$237$		−	847.795i		−	0.232364i
$238$		0			0
$239$		−	2208.27i		−	0.597662i	−0.954306	−	0.298831i	$-0.903403\pi$
$239$		−	2208.27i		−	0.597662i	0.954306	−	0.298831i	$-0.0965967\pi$
$240$		0			0
$241$		−	121.331i		−	0.0324301i	−0.999869	−	0.0162150i	$-0.994838\pi$
$241$		−	121.331i		−	0.0324301i	0.999869	−	0.0162150i	$-0.00516163\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$	6799.66	−	85.1220i	1.77312	−	0.0221969i
$246$		0			0
$247$		−	8587.92i		−	2.21229i
$248$		0			0
$249$	−2448.73			−0.623220
$250$		0			0
$251$	−5317.50			−1.33720			−0.668601	−	0.743621i	$-0.733107\pi$
$251$	−5317.50			−1.33720			−0.668601	+	0.743621i	$0.733107\pi$
$252$		0			0
$253$	1732.55			0.430531
$254$		0			0
$255$	−5990.10			−1.47104
$256$		0			0
$257$			3458.39i			0.839409i	0.907661	+	0.419705i	$0.137866\pi$
$257$			3458.39i			0.839409i	−0.907661	+	0.419705i	$0.862134\pi$
$258$		0			0
$259$	−3711.99	−	3665.81i	−0.890548	−	0.879470i
$260$		0			0
$261$	−828.442			−0.196472
$262$		0			0
$263$		−	1302.82i		−	0.305457i	−0.988268	−	0.152728i	$-0.951194\pi$
$263$		−	1302.82i		−	0.305457i	0.988268	−	0.152728i	$-0.0488059\pi$
$264$		0			0
$265$		−	1358.78i		−	0.314979i
$266$		0			0
$267$		−	2548.77i		−	0.584203i
$268$		0			0
$269$			872.315i			0.197717i	0.995101	+	0.0988587i	$0.0315192\pi$
$269$			872.315i			0.197717i	−0.995101	+	0.0988587i	$0.968481\pi$
$270$		0			0
$271$	7358.10			1.64934			0.824672	−	0.565611i	$-0.191359\pi$
$271$	7358.10			1.64934			0.824672	+	0.565611i	$0.191359\pi$
$272$		0			0
$273$	−3222.51	+	3263.10i	−0.714414	+	0.723414i
$274$		0			0
$275$		−	3207.26i		−	0.703292i
$276$		0			0
$277$	4496.10			0.975250			0.487625	−	0.873053i	$-0.337863\pi$
$277$	4496.10			0.975250			0.487625	+	0.873053i	$0.337863\pi$
$278$		0			0
$279$	1763.11			0.378333
$280$		0			0
$281$	−3403.74			−0.722599			−0.361299	−	0.932450i	$-0.617667\pi$
$281$	−3403.74			−0.722599			−0.361299	+	0.932450i	$0.617667\pi$
$282$		0			0
$283$	637.261			0.133856			0.0669280	−	0.997758i	$-0.478680\pi$
$283$	637.261			0.133856			0.0669280	+	0.997758i	$0.478680\pi$
$284$		0			0
$285$			6188.14i			1.28615i
$286$		0			0
$287$	5661.78	−	5733.10i	1.16447	−	1.17914i
$288$		0			0
$289$	−5230.13			−1.06455
$290$		0			0
$291$			2133.57i			0.429801i
$292$		0			0
$293$			6781.18i			1.35209i	0.736863	+	0.676043i	$0.236306\pi$
$293$			6781.18i			1.35209i	−0.736863	+	0.676043i	$0.763694\pi$
$294$		0			0
$295$		−	9121.11i		−	1.80017i
$296$		0			0
$297$			323.052i			0.0631158i
$298$		0			0
$299$	11952.3			2.31177
$300$		0			0
$301$	4713.77	−	4773.15i	0.902648	−	0.914019i
$302$		0			0
$303$		−	959.210i		−	0.181865i
$304$		0			0
$305$	8783.34			1.64896
$306$		0			0
$307$	−424.769			−0.0789669			−0.0394835	−	0.999220i	$-0.512571\pi$
$307$	−424.769			−0.0789669			−0.0394835	+	0.999220i	$0.512571\pi$
$308$		0			0
$309$	−2198.91			−0.404828
$310$		0			0
$311$	−2089.42			−0.380966			−0.190483	−	0.981691i	$-0.561005\pi$
$311$	−2089.42			−0.380966			−0.190483	+	0.981691i	$0.561005\pi$
$312$		0			0
$313$			1053.05i			0.190166i	0.995469	+	0.0950828i	$0.0303116\pi$
$313$			1053.05i			0.190166i	−0.995469	+	0.0950828i	$0.969688\pi$
$314$		0			0
$315$	2322.02	−	2351.27i	0.415337	−	0.420569i
$316$		0			0
$317$	−779.541			−0.138118			−0.0690589	−	0.997613i	$-0.522000\pi$
$317$	−779.541			−0.138118			−0.0690589	+	0.997613i	$0.522000\pi$
$318$		0			0
$319$		−	1101.36i		−	0.193305i
$320$		0			0
$321$			1275.39i			0.221762i
$322$		0			0
$323$			10478.5i			1.80507i
$324$		0			0
$325$		−	22125.9i		−	3.77638i
$326$		0			0
$327$	3245.97			0.548937
$328$		0			0
$329$	−470.176	−	464.327i	−0.0787892	−	0.0778091i
$330$		0			0
$331$			2649.53i			0.439973i	0.975503	+	0.219987i	$0.0706014\pi$
$331$			2649.53i			0.439973i	−0.975503	+	0.219987i	$0.929399\pi$
$332$		0			0
$333$	−2535.22			−0.417205
$334$		0			0
$335$	−17200.5			−2.80527
$336$		0			0
$337$	4912.46			0.794061			0.397031	−	0.917805i	$-0.370041\pi$
$337$	4912.46			0.794061			0.397031	+	0.917805i	$0.370041\pi$
$338$		0			0
$339$	−4747.68			−0.760644
$340$		0			0
$341$			2343.94i			0.372233i
$342$		0			0
$343$	−4406.73	+	4575.41i	−0.693706	+	0.720259i
$344$		0			0
$345$	−8612.41			−1.34399
$346$		0			0
$347$			5626.04i			0.870379i	0.900339	+	0.435189i	$0.143319\pi$
$347$			5626.04i			0.870379i	−0.900339	+	0.435189i	$0.856681\pi$
$348$		0			0
$349$		−	5812.55i		−	0.891515i	−0.895154	−	0.445757i	$-0.852934\pi$
$349$		−	5812.55i		−	0.891515i	0.895154	−	0.445757i	$-0.147066\pi$
$350$		0			0
$351$			2228.64i			0.338906i
$352$		0			0
$353$			432.142i			0.0651575i	0.999469	+	0.0325788i	$0.0103720\pi$
$353$			432.142i			0.0651575i	−0.999469	+	0.0325788i	$0.989628\pi$
$354$		0			0
$355$	−11452.9			−1.71227
$356$		0			0
$357$	3931.91	−	3981.44i	0.582910	−	0.590253i
$358$		0			0
$359$			2440.33i			0.358763i	0.983780	+	0.179381i	$0.0574096\pi$
$359$			2440.33i			0.358763i	−0.983780	+	0.179381i	$0.942590\pi$
$360$		0			0
$361$	3965.90			0.578203
$362$		0			0
$363$	3563.52			0.515252
$364$		0			0
$365$	−11714.0			−1.67983
$366$		0			0
$367$	9515.33			1.35340			0.676698	−	0.736261i	$-0.263410\pi$
$367$	9515.33			1.35340			0.676698	+	0.736261i	$0.263410\pi$
$368$		0			0
$369$		−	3915.60i		−	0.552407i
$370$		0			0
$371$	903.142	+	891.907i	0.126385	+	0.124813i
$372$		0			0
$373$	2815.31			0.390808			0.195404	−	0.980723i	$-0.437398\pi$
$373$	2815.31			0.390808			0.195404	+	0.980723i	$0.437398\pi$
$374$		0			0
$375$			8508.51i			1.17167i
$376$		0			0
$377$		−	7597.94i		−	1.03797i
$378$		0			0
$379$			78.5889i			0.0106513i	0.999986	+	0.00532565i	$0.00169522\pi$
$379$			78.5889i			0.0106513i	−0.999986	+	0.00532565i	$0.998305\pi$
$380$		0			0
$381$			1994.11i			0.268140i
$382$		0			0
$383$	−12838.1			−1.71278			−0.856390	−	0.516329i	$-0.827298\pi$
$383$	−12838.1			−1.71278			−0.856390	+	0.516329i	$0.827298\pi$
$384$		0			0
$385$	3125.86	+	3086.98i	0.413789	+	0.408641i
$386$		0			0
$387$		−	3259.97i		−	0.428201i
$388$		0			0
$389$	−4082.44			−0.532103			−0.266051	−	0.963959i	$-0.585719\pi$
$389$	−4082.44			−0.532103			−0.266051	+	0.963959i	$0.585719\pi$
$390$		0			0
$391$	−14583.5			−1.88624
$392$		0			0
$393$	−6329.82			−0.812461
$394$		0			0
$395$	−5602.69			−0.713676
$396$		0			0
$397$		−	5972.68i		−	0.755064i	−0.925997	−	0.377532i	$-0.876773\pi$
$397$		−	5972.68i		−	0.755064i	0.925997	−	0.377532i	$-0.123227\pi$
$398$		0			0
$399$	−4113.07	−	4061.91i	−0.516068	−	0.509648i
$400$		0			0
$401$	−9694.72			−1.20731			−0.603655	−	0.797246i	$-0.706289\pi$
$401$	−9694.72			−1.20731			−0.603655	+	0.797246i	$0.706289\pi$
$402$		0			0
$403$			16170.1i			1.99874i
$404$		0			0
$405$		−	1605.88i		−	0.197029i
$406$		0			0
$407$		−	3370.41i		−	0.410480i
$408$		0			0
$409$			7717.51i			0.933023i	0.884515	+	0.466511i	$0.154489\pi$
$409$			7717.51i			0.933023i	−0.884515	+	0.466511i	$0.845511\pi$
$410$		0			0
$411$	−1789.87			−0.214812
$412$		0			0
$413$	6062.53	+	5987.11i	0.722319	+	0.713333i
$414$		0			0
$415$			16182.5i			1.91414i
$416$		0			0
$417$	−931.335			−0.109371
$418$		0			0
$419$	1095.47			0.127726			0.0638630	−	0.997959i	$-0.479658\pi$
$419$	1095.47			0.127726			0.0638630	+	0.997959i	$0.479658\pi$
$420$		0			0
$421$	−10221.9			−1.18334			−0.591671	−	0.806180i	$-0.701531\pi$
$421$	−10221.9			−1.18334			−0.591671	+	0.806180i	$0.701531\pi$
$422$		0			0
$423$	−321.122			−0.0369113
$424$		0			0
$425$			26996.7i			3.08126i
$426$		0			0
$427$	−5765.40	+	5838.03i	−0.653413	+	0.661644i
$428$		0			0
$429$	−2962.83			−0.333442
$430$		0			0
$431$			1388.32i			0.155158i	0.996986	+	0.0775789i	$0.0247190\pi$
$431$			1388.32i			0.155158i	−0.996986	+	0.0775789i	$0.975281\pi$
$432$		0			0
$433$		−	13096.9i		−	1.45357i	−0.686864	−	0.726786i	$-0.741013\pi$
$433$		−	13096.9i		−	1.45357i	0.686864	−	0.726786i	$-0.258987\pi$
$434$		0			0
$435$			5474.80i			0.603440i
$436$		0			0
$437$			15065.7i			1.64917i
$438$		0			0
$439$	1292.73			0.140543			0.0702717	−	0.997528i	$-0.477613\pi$
$439$	1292.73			0.140543			0.0702717	+	0.997528i	$0.477613\pi$
$440$		0			0
$441$	38.6418	+	3086.76i	0.00417253	+	0.333307i
$442$		0			0
$443$		−	10995.9i		−	1.17930i	−0.807659	−	0.589650i	$-0.799266\pi$
$443$		−	10995.9i		−	1.17930i	0.807659	−	0.589650i	$-0.200734\pi$
$444$		0			0
$445$	−16843.7			−1.79431
$446$		0			0
$447$	487.904			0.0516265
$448$		0			0
$449$	−4456.62			−0.468421			−0.234210	−	0.972186i	$-0.575250\pi$
$449$	−4456.62			−0.468421			−0.234210	+	0.972186i	$0.575250\pi$
$450$		0			0
$451$	5205.54			0.543501
$452$		0			0
$453$		−	1542.25i		−	0.159958i
$454$		0			0
$455$	21564.4	+	21296.1i	2.22187	+	2.19423i
$456$		0			0
$457$	821.728			0.0841112			0.0420556	−	0.999115i	$-0.486609\pi$
$457$	821.728			0.0841112			0.0420556	+	0.999115i	$0.486609\pi$
$458$		0			0
$459$		−	2719.25i		−	0.276523i
$460$		0			0
$461$			5744.64i			0.580379i	0.956969	+	0.290189i	$0.0937183\pi$
$461$			5744.64i			0.580379i	−0.956969	+	0.290189i	$0.906282\pi$
$462$		0			0
$463$			4594.86i			0.461212i	0.973047	+	0.230606i	$0.0740709\pi$
$463$			4594.86i			0.461212i	−0.973047	+	0.230606i	$0.925929\pi$
$464$		0			0
$465$		−	11651.6i		−	1.16200i
$466$		0			0
$467$	14583.8			1.44509			0.722546	−	0.691323i	$-0.242972\pi$
$467$	14583.8			1.44509			0.722546	+	0.691323i	$0.242972\pi$
$468$		0			0
$469$	11290.4	−	11432.7i	1.11161	−	1.12561i
$470$		0			0
$471$			8071.23i			0.789602i
$472$		0			0
$473$	4333.92			0.421298
$474$		0			0
$475$	27889.3			2.69399
$476$		0			0
$477$	616.830			0.0592090
$478$		0			0
$479$	9838.97			0.938526			0.469263	−	0.883059i	$-0.344520\pi$
$479$	9838.97			0.938526			0.469263	+	0.883059i	$0.344520\pi$
$480$		0			0
$481$		−	23251.4i		−	2.20410i
$482$		0			0
$483$	5653.20	−	5724.41i	0.532566	−	0.539275i
$484$		0			0
$485$	14099.8			1.32008
$486$		0			0
$487$		−	13678.2i		−	1.27273i	−0.771389	−	0.636364i	$-0.780438\pi$
$487$		−	13678.2i		−	1.27273i	0.771389	−	0.636364i	$-0.219562\pi$
$488$		0			0
$489$		−	6417.32i		−	0.593458i
$490$		0			0
$491$		−	14860.5i		−	1.36587i	−0.730478	−	0.682937i	$-0.760703\pi$
$491$		−	14860.5i		−	1.36587i	0.730478	−	0.682937i	$-0.239297\pi$
$492$		0			0
$493$			9270.55i			0.846906i
$494$		0			0
$495$	2134.91			0.193852
$496$		0			0
$497$	7517.71	−	7612.41i	0.678502	−	0.687049i
$498$		0			0
$499$		−	12929.8i		−	1.15995i	−0.814633	−	0.579977i	$-0.803062\pi$
$499$		−	12929.8i		−	1.15995i	0.814633	−	0.579977i	$-0.196938\pi$
$500$		0			0
$501$	−2282.27			−0.203521
$502$		0			0
$503$	17268.2			1.53072			0.765360	−	0.643603i	$-0.222561\pi$
$503$	17268.2			1.53072			0.765360	+	0.643603i	$0.222561\pi$
$504$		0			0
$505$	−6338.98			−0.558576
$506$		0			0
$507$	−13848.6			−1.21310
$508$		0			0
$509$			12924.4i			1.12547i	0.826636	+	0.562737i	$0.190251\pi$
$509$			12924.4i			1.12547i	−0.826636	+	0.562737i	$0.809749\pi$
$510$		0			0
$511$	7689.09	−	7785.94i	0.665646	−	0.674031i
$512$		0			0
$513$	−2809.15			−0.241768
$514$		0			0
$515$			14531.6i			1.24338i
$516$		0			0
$517$		−	426.910i		−	0.0363162i
$518$		0			0
$519$		−	6354.53i		−	0.537443i
$520$		0			0
$521$			20808.5i			1.74978i	0.484321	+	0.874890i	$0.339067\pi$
$521$			20808.5i			1.74978i	−0.484321	+	0.874890i	$0.660933\pi$
$522$		0			0
$523$	−15728.2			−1.31500			−0.657500	−	0.753455i	$-0.728386\pi$
$523$	−15728.2			−1.31500			−0.657500	+	0.753455i	$0.728386\pi$
$524$		0			0
$525$	−10596.9	−	10465.1i	−0.880929	−	0.869970i
$526$		0			0
$527$		−	19729.8i		−	1.63083i
$528$		0			0
$529$	−8800.78			−0.723332
$530$		0			0
$531$	4140.60			0.338393
$532$		0			0
$533$	35911.4			2.91838
$534$		0			0
$535$	8428.50			0.681114
$536$		0			0
$537$		−	9546.26i		−	0.767135i
$538$		0			0
$539$	−4103.64	+	51.3717i	−0.327934	+	0.00410526i
$540$		0			0
$541$	309.414			0.0245892			0.0122946	−	0.999924i	$-0.496086\pi$
$541$	309.414			0.0245892			0.0122946	+	0.999924i	$0.496086\pi$
$542$		0			0
$543$		−	3944.49i		−	0.311739i
$544$		0			0
$545$		−	21451.1i		−	1.68599i
$546$		0			0
$547$		−	6179.69i		−	0.483043i	−0.970395	−	0.241521i	$-0.922354\pi$
$547$		−	6179.69i		−	0.483043i	0.970395	−	0.241521i	$-0.0776464\pi$
$548$		0			0
$549$			3987.27i			0.309968i
$550$		0			0
$551$	9577.05			0.740464
$552$		0			0
$553$	3677.62	−	3723.94i	0.282800	−	0.286362i
$554$		0			0
$555$			16754.1i			1.28139i
$556$		0			0
$557$	−6841.12			−0.520408			−0.260204	−	0.965554i	$-0.583790\pi$
$557$	−6841.12			−0.520408			−0.260204	+	0.965554i	$0.583790\pi$
$558$		0			0
$559$	29898.4			2.26219
$560$		0			0
$561$	3615.07			0.272065
$562$		0			0
$563$	−436.446			−0.0326714			−0.0163357	−	0.999867i	$-0.505200\pi$
$563$	−436.446			−0.0326714			−0.0163357	+	0.999867i	$0.505200\pi$
$564$		0			0
$565$			31375.2i			2.33622i
$566$		0			0
$567$	1067.38	+	1054.10i	0.0790576	+	0.0780741i
$568$		0			0
$569$	26786.0			1.97351			0.986754	−	0.162226i	$-0.0518674\pi$
$569$	26786.0			1.97351			0.986754	+	0.162226i	$0.0518674\pi$
$570$		0			0
$571$			22710.8i			1.66448i	0.554416	+	0.832240i	$0.312942\pi$
$571$			22710.8i			1.66448i	−0.554416	+	0.832240i	$0.687058\pi$
$572$		0			0
$573$			3593.32i			0.261977i
$574$		0			0
$575$			38815.1i			2.81514i
$576$		0			0
$577$			15075.7i			1.08771i	0.839179	+	0.543856i	$0.183036\pi$
$577$			15075.7i			1.08771i	−0.839179	+	0.543856i	$0.816964\pi$
$578$		0			0
$579$	14027.8			1.00687
$580$		0			0
$581$	−10756.0	−	10622.2i	−0.768047	−	0.758493i
$582$		0			0
$583$			820.035i			0.0582545i
$584$		0			0
$585$	14728.1			1.04091
$586$		0			0
$587$	6844.12			0.481238			0.240619	−	0.970620i	$-0.422650\pi$
$587$	6844.12			0.481238			0.240619	+	0.970620i	$0.422650\pi$
$588$		0			0
$589$	−20382.1			−1.42586
$590$		0			0
$591$	−576.863			−0.0401506
$592$		0			0
$593$			7305.99i			0.505938i	0.967474	+	0.252969i	$0.0814071\pi$
$593$			7305.99i			0.505938i	−0.967474	+	0.252969i	$0.918593\pi$
$594$		0			0
$595$	−26311.5	−	25984.2i	−1.81289	−	1.79034i
$596$		0			0
$597$	11588.9			0.794478
$598$		0			0
$599$			10829.6i			0.738706i	0.929289	+	0.369353i	$0.120421\pi$
$599$			10829.6i			0.738706i	−0.929289	+	0.369353i	$0.879579\pi$
$600$		0			0
$601$		−	2352.33i		−	0.159656i	−0.996809	−	0.0798282i	$-0.974563\pi$
$601$		−	2352.33i		−	0.159656i	0.996809	−	0.0798282i	$-0.0254372\pi$
$602$		0			0
$603$		−	7808.30i		−	0.527328i
$604$		0			0
$605$		−	23549.7i		−	1.58253i
$606$		0			0
$607$	−12029.9			−0.804416			−0.402208	−	0.915548i	$-0.631757\pi$
$607$	−12029.9			−0.804416			−0.402208	+	0.915548i	$0.631757\pi$
$608$		0			0
$609$	−3638.94	−	3593.67i	−0.242130	−	0.239118i
$610$		0			0
$611$		−	2945.12i		−	0.195003i
$612$		0			0
$613$	3523.87			0.232182			0.116091	−	0.993239i	$-0.462964\pi$
$613$	3523.87			0.232182			0.116091	+	0.993239i	$0.462964\pi$
$614$		0			0
$615$	−25876.4			−1.69665
$616$		0			0
$617$	23175.6			1.51218			0.756090	−	0.654467i	$-0.227107\pi$
$617$	23175.6			1.51218			0.756090	+	0.654467i	$0.227107\pi$
$618$		0			0
$619$	10803.4			0.701497			0.350749	−	0.936470i	$-0.385927\pi$
$619$	10803.4			0.701497			0.350749	+	0.936470i	$0.385927\pi$
$620$		0			0
$621$		−	3909.67i		−	0.252640i
$622$		0			0
$623$	11056.2	−	11195.5i	0.711008	−	0.719964i
$624$		0			0
$625$	22721.9			1.45420
$626$		0			0
$627$		−	3734.59i		−	0.237871i
$628$		0			0
$629$			28370.0i			1.79839i
$630$		0			0
$631$			14142.2i			0.892219i	0.894978	+	0.446110i	$0.147191\pi$
$631$			14142.2i			0.892219i	−0.894978	+	0.446110i	$0.852809\pi$
$632$		0			0
$633$			13661.8i			0.857833i
$634$		0			0
$635$	13178.2			0.823560
$636$		0			0
$637$	−28309.8	+	354.398i	−1.76087	+	0.0220436i
$638$		0			0
$639$		−	5199.14i		−	0.321870i
$640$		0			0
$641$	26163.1			1.61214			0.806070	−	0.591820i	$-0.201590\pi$
$641$	26163.1			1.61214			0.806070	+	0.591820i	$0.201590\pi$
$642$		0			0
$643$	25045.3			1.53607			0.768033	−	0.640410i	$-0.221236\pi$
$643$	25045.3			1.53607			0.768033	+	0.640410i	$0.221236\pi$
$644$		0			0
$645$	−21543.7			−1.31516
$646$		0			0
$647$	16332.9			0.992446			0.496223	−	0.868195i	$-0.334720\pi$
$647$	16332.9			0.992446			0.496223	+	0.868195i	$0.334720\pi$
$648$		0			0
$649$			5504.65i			0.332938i
$650$		0			0
$651$	7744.48	+	7648.14i	0.466252	+	0.460452i
$652$		0			0
$653$	16066.0			0.962801			0.481401	−	0.876501i	$-0.340128\pi$
$653$	16066.0			0.962801			0.481401	+	0.876501i	$0.340128\pi$
$654$		0			0
$655$			41830.9i			2.49537i
$656$		0			0
$657$		−	5317.66i		−	0.315771i
$658$		0			0
$659$			20211.8i			1.19475i	0.801962	+	0.597375i	$0.203790\pi$
$659$			20211.8i			1.19475i	−0.801962	+	0.597375i	$0.796210\pi$
$660$		0			0
$661$			2892.79i			0.170222i	0.996371	+	0.0851108i	$0.0271244\pi$
$661$			2892.79i			0.170222i	−0.996371	+	0.0851108i	$0.972876\pi$
$662$		0			0
$663$	24939.2			1.46087
$664$		0			0
$665$	−26843.3	+	27181.4i	−1.56532	+	1.58504i
$666$		0			0
$667$			13328.9i			0.773762i
$668$		0			0
$669$	−3952.04			−0.228393
$670$		0			0
$671$	−5300.81			−0.304971
$672$		0			0
$673$	2015.37			0.115434			0.0577168	−	0.998333i	$-0.481618\pi$
$673$	2015.37			0.115434			0.0577168	+	0.998333i	$0.481618\pi$
$674$		0			0
$675$	−7237.50			−0.412699
$676$		0			0
$677$		−	16959.2i		−	0.962769i	−0.876509	−	0.481385i	$-0.840134\pi$
$677$		−	16959.2i		−	0.962769i	0.876509	−	0.481385i	$-0.159866\pi$
$678$		0			0
$679$	−9255.12	+	9371.71i	−0.523091	+	0.529681i
$680$		0			0
$681$	3405.09			0.191606
$682$		0			0
$683$		−	24114.8i		−	1.35099i	−0.737363	−	0.675497i	$-0.763929\pi$
$683$		−	24114.8i		−	1.35099i	0.737363	−	0.675497i	$-0.236071\pi$
$684$		0			0
$685$			11828.4i			0.659768i
$686$		0			0
$687$			2564.79i			0.142435i
$688$		0			0
$689$			5657.17i			0.312802i
$690$		0			0
$691$	−4280.39			−0.235649			−0.117825	−	0.993034i	$-0.537592\pi$
$691$	−4280.39			−0.235649			−0.117825	+	0.993034i	$0.537592\pi$
$692$		0			0
$693$	−1401.36	+	1419.01i	−0.0768155	+	0.0777831i
$694$		0			0
$695$			6154.77i			0.335919i
$696$		0			0
$697$	−43816.9			−2.38118
$698$		0			0
$699$	−3750.71			−0.202954
$700$		0			0
$701$	−11078.2			−0.596890			−0.298445	−	0.954427i	$-0.596468\pi$
$701$	−11078.2			−0.596890			−0.298445	+	0.954427i	$0.596468\pi$
$702$		0			0
$703$	29308.0			1.57236
$704$		0			0
$705$			2122.15i			0.113368i
$706$		0			0
$707$	4160.92	−	4213.33i	0.221340	−	0.224128i
$708$		0			0
$709$	−4759.61			−0.252117			−0.126059	−	0.992023i	$-0.540233\pi$
$709$	−4759.61			−0.252117			−0.126059	+	0.992023i	$0.540233\pi$
$710$		0			0
$711$		−	2543.39i		−	0.134155i
$712$		0			0
$713$		−	28367.0i		−	1.48998i
$714$		0			0
$715$			19580.0i			1.02413i
$716$		0			0
$717$		−	6624.82i		−	0.345060i
$718$		0			0
$719$	14788.5			0.767061			0.383530	−	0.923528i	$-0.374708\pi$
$719$	14788.5			0.767061			0.383530	+	0.923528i	$0.374708\pi$
$720$		0			0
$721$	−9658.74	−	9538.58i	−0.498905	−	0.492698i
$722$		0			0
$723$		−	363.994i		−	0.0187235i
$724$		0			0
$725$	24674.3			1.26397
$726$		0			0
$727$	−10230.5			−0.521912			−0.260956	−	0.965351i	$-0.584038\pi$
$727$	−10230.5			−0.521912			−0.260956	+	0.965351i	$0.584038\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−36480.2			−1.84579
$732$		0			0
$733$			31599.8i			1.59232i	0.605089	+	0.796158i	$0.293137\pi$
$733$			31599.8i			1.59232i	−0.605089	+	0.796158i	$0.706863\pi$
$734$		0			0
$735$	20399.0	−	255.366i	1.02371	−	0.0128154i
$736$		0			0
$737$	10380.6			0.518827
$738$		0			0
$739$		−	28960.7i		−	1.44159i	−0.693146	−	0.720797i	$-0.743776\pi$
$739$		−	28960.7i		−	1.44159i	0.693146	−	0.720797i	$-0.256224\pi$
$740$		0			0
$741$		−	25763.8i		−	1.27727i
$742$		0			0
$743$		−	11860.9i		−	0.585645i	−0.956167	−	0.292823i	$-0.905405\pi$
$743$		−	11860.9i		−	0.585645i	0.956167	−	0.292823i	$-0.0945946\pi$
$744$		0			0
$745$		−	3224.33i		−	0.158564i
$746$		0			0
$747$	−7346.18			−0.359816
$748$		0			0
$749$	−5532.48	+	5602.17i	−0.269897	+	0.273296i
$750$		0			0
$751$		−	10317.3i		−	0.501307i	−0.968077	−	0.250654i	$-0.919354\pi$
$751$		−	10317.3i		−	0.501307i	0.968077	−	0.250654i	$-0.0806455\pi$
$752$		0			0
$753$	−15952.5			−0.772034
$754$		0			0
$755$	−10192.0			−0.491292
$756$		0			0
$757$	28862.3			1.38576			0.692878	−	0.721055i	$-0.256343\pi$
$757$	28862.3			1.38576			0.692878	+	0.721055i	$0.256343\pi$
$758$		0			0
$759$	5197.65			0.248567
$760$		0			0
$761$			31946.4i			1.52176i	0.648895	+	0.760878i	$0.275232\pi$
$761$			31946.4i			1.52176i	−0.648895	+	0.760878i	$0.724768\pi$
$762$		0			0
$763$	14257.9	+	14080.6i	0.676503	+	0.668087i
$764$		0			0
$765$	−17970.3			−0.849304
$766$		0			0
$767$			37974.9i			1.78774i
$768$		0			0
$769$		−	36107.9i		−	1.69322i	−0.532216	−	0.846609i	$-0.678640\pi$
$769$		−	36107.9i		−	1.69322i	0.532216	−	0.846609i	$-0.321360\pi$
$770$		0			0
$771$			10375.2i			0.484633i
$772$		0			0
$773$		−	25498.3i		−	1.18643i	−0.805044	−	0.593216i	$-0.797858\pi$
$773$		−	25498.3i		−	1.18643i	0.805044	−	0.593216i	$-0.202142\pi$
$774$		0			0
$775$	−52512.5			−2.43394
$776$		0			0
$777$	−11136.0	−	10997.4i	−0.514158	−	0.507762i
$778$		0			0
$779$			45265.6i			2.08191i
$780$		0			0
$781$	6911.91			0.316681
$782$		0			0
$783$	−2485.33			−0.113433
$784$		0			0
$785$	53339.1			2.42516
$786$		0			0
$787$	25949.5			1.17535			0.587675	−	0.809097i	$-0.300043\pi$
$787$	25949.5			1.17535			0.587675	+	0.809097i	$0.300043\pi$
$788$		0			0
$789$		−	3908.45i		−	0.176355i
$790$		0			0
$791$	−20854.2	−	20594.8i	−0.937408	−	0.925746i
$792$		0			0
$793$	−36568.6			−1.63757
$794$		0			0
$795$		−	4076.35i		−	0.181853i
$796$		0			0
$797$			30584.1i			1.35928i	0.733548	+	0.679638i	$0.237863\pi$
$797$			30584.1i			1.35928i	−0.733548	+	0.679638i	$0.762137\pi$
$798$		0			0
$799$			3593.46i			0.159108i
$800$		0			0
$801$		−	7646.32i		−	0.337290i
$802$		0			0
$803$	7069.48			0.310681
$804$		0			0
$805$	−37830.0	−	37359.4i	−1.65631	−	1.63571i
$806$		0			0
$807$			2616.95i			0.114152i
$808$		0			0
$809$	−27120.4			−1.17862			−0.589308	−	0.807908i	$-0.700600\pi$
$809$	−27120.4			−1.17862			−0.589308	+	0.807908i	$0.700600\pi$
$810$		0			0
$811$	18778.1			0.813057			0.406529	−	0.913638i	$-0.366739\pi$
$811$	18778.1			0.813057			0.406529	+	0.913638i	$0.366739\pi$
$812$		0			0
$813$	22074.3			0.952250
$814$		0			0
$815$	−42409.1			−1.82273
$816$		0			0
$817$			37686.3i			1.61380i
$818$		0			0
$819$	−9667.53	+	9789.31i	−0.412467	+	0.417663i
$820$		0			0
$821$	13990.5			0.594729			0.297364	−	0.954764i	$-0.403892\pi$
$821$	13990.5			0.594729			0.297364	+	0.954764i	$0.403892\pi$
$822$		0			0
$823$		−	17997.8i		−	0.762291i	−0.924515	−	0.381145i	$-0.875530\pi$
$823$		−	17997.8i		−	0.762291i	0.924515	−	0.381145i	$-0.124470\pi$
$824$		0			0
$825$		−	9621.79i		−	0.406046i
$826$		0			0
$827$		−	3321.88i		−	0.139677i	−0.997558	−	0.0698386i	$-0.977752\pi$
$827$		−	3321.88i		−	0.139677i	0.997558	−	0.0698386i	$-0.0222484\pi$
$828$		0			0
$829$			5335.05i			0.223515i	0.993736	+	0.111758i	$0.0356480\pi$
$829$			5335.05i			0.223515i	−0.993736	+	0.111758i	$0.964352\pi$
$830$		0			0
$831$	13488.3			0.563061
$832$		0			0
$833$	34541.9	−	432.415i	1.43674	−	0.0179859i
$834$		0			0
$835$			15082.5i			0.625090i
$836$		0			0
$837$	5289.34			0.218430
$838$		0			0
$839$	−26147.0			−1.07592			−0.537959	−	0.842971i	$-0.680805\pi$
$839$	−26147.0			−1.07592			−0.537959	+	0.842971i	$0.680805\pi$
$840$		0			0
$841$	−15916.0			−0.652588
$842$		0			0
$843$	−10211.2			−0.417193
$844$		0			0
$845$			91519.4i			3.72587i
$846$		0			0
$847$	15652.8	+	15458.1i	0.634989	+	0.627090i
$848$		0			0
$849$	1911.78			0.0772818
$850$		0			0
$851$			40789.6i			1.64307i
$852$		0			0
$853$		−	12749.4i		−	0.511759i	−0.966709	−	0.255880i	$-0.917635\pi$
$853$		−	12749.4i		−	0.511759i	0.966709	−	0.255880i	$-0.0823650\pi$
$854$		0			0
$855$			18564.4i			0.742561i
$856$		0			0
$857$		−	16685.5i		−	0.665071i	−0.943091	−	0.332536i	$-0.892096\pi$
$857$		−	16685.5i		−	0.665071i	0.943091	−	0.332536i	$-0.107904\pi$
$858$		0			0
$859$	−10700.8			−0.425037			−0.212518	−	0.977157i	$-0.568167\pi$
$859$	−10700.8			−0.425037			−0.212518	+	0.977157i	$0.568167\pi$
$860$		0			0
$861$	16985.3	−	17199.3i	0.672310	−	0.680779i
$862$		0			0
$863$		−	31393.5i		−	1.23829i	−0.785275	−	0.619147i	$-0.787478\pi$
$863$		−	31393.5i		−	1.23829i	0.785275	−	0.619147i	$-0.212522\pi$
$864$		0			0
$865$	−41994.2			−1.65069
$866$		0			0
$867$	−15690.4			−0.614618
$868$		0			0
$869$	3381.26			0.131993
$870$		0			0
$871$	71612.7			2.78588
$872$		0			0
$873$			6400.71i			0.248146i
$874$		0			0
$875$	−36908.7	+	37373.7i	−1.42599	+	1.44395i
$876$		0			0
$877$	−1539.12			−0.0592616			−0.0296308	−	0.999561i	$-0.509433\pi$
$877$	−1539.12			−0.0592616			−0.0296308	+	0.999561i	$0.509433\pi$
$878$		0			0
$879$			20343.6i			0.780627i
$880$		0			0
$881$			18655.1i			0.713402i	0.934219	+	0.356701i	$0.116099\pi$
$881$			18655.1i			0.713402i	−0.934219	+	0.356701i	$0.883901\pi$
$882$		0			0
$883$			1920.47i			0.0731923i	0.999330	+	0.0365962i	$0.0116515\pi$
$883$			1920.47i			0.0731923i	−0.999330	+	0.0365962i	$0.988348\pi$
$884$		0			0
$885$		−	27363.3i		−	1.03933i
$886$		0			0
$887$	6877.01			0.260324			0.130162	−	0.991493i	$-0.458450\pi$
$887$	6877.01			0.260324			0.130162	+	0.991493i	$0.458450\pi$
$888$		0			0
$889$	−8650.18	+	8759.15i	−0.326342	+	0.330453i
$890$		0			0
$891$			969.157i			0.0364400i
$892$		0			0
$893$	3712.27			0.139111
$894$		0			0
$895$	−63086.9			−2.35616
$896$		0			0
$897$	35857.0			1.33470
$898$		0			0
$899$	−18032.6			−0.668987
$900$		0			0
$901$		−	6902.54i		−	0.255224i
$902$		0			0
$903$	14141.3	−	14319.4i	0.521144	−	0.527709i
$904$		0			0
$905$	−26067.3			−0.957466
$906$		0			0
$907$		−	50605.5i		−	1.85262i	−0.376761	−	0.926310i	$-0.622962\pi$
$907$		−	50605.5i		−	1.85262i	0.376761	−	0.926310i	$-0.377038\pi$
$908$		0			0
$909$		−	2877.63i		−	0.105000i
$910$		0			0
$911$		−	46531.9i		−	1.69228i	−0.532958	−	0.846142i	$-0.678920\pi$
$911$		−	46531.9i		−	1.69228i	0.532958	−	0.846142i	$-0.321080\pi$
$912$		0			0
$913$		−	9766.26i		−	0.354015i
$914$		0			0
$915$	26350.0			0.952027
$916$		0			0
$917$	−27803.8	−	27457.9i	−1.00127	−	0.988810i
$918$		0			0
$919$			35288.2i			1.26665i	0.773887	+	0.633324i	$0.218310\pi$
$919$			35288.2i			1.26665i	−0.773887	+	0.633324i	$0.781690\pi$
$920$		0			0
$921$	−1274.31			−0.0455916
$922$		0			0
$923$	47683.1			1.70044
$924$		0			0
$925$	75509.0			2.68402
$926$		0			0
$927$	−6596.74			−0.233728
$928$		0			0
$929$		−	26951.7i		−	0.951837i	−0.879489	−	0.475918i	$-0.842116\pi$
$929$		−	26951.7i		−	0.951837i	0.879489	−	0.475918i	$-0.157884\pi$
$930$		0			0
$931$	−446.711	−	35683.9i	−0.0157254	−	1.25617i
$932$		0			0
$933$	−6268.27			−0.219951
$934$		0			0
$935$		−	23890.3i		−	0.835613i
$936$		0			0
$937$		−	6716.04i		−	0.234155i	−0.993123	−	0.117078i	$-0.962647\pi$
$937$		−	6716.04i		−	0.234155i	0.993123	−	0.117078i	$-0.0373526\pi$
$938$		0			0
$939$			3159.15i			0.109792i
$940$		0			0
$941$			1401.32i			0.0485460i	0.999705	+	0.0242730i	$0.00772709\pi$
$941$			1401.32i			0.0485460i	−0.999705	+	0.0242730i	$0.992273\pi$
$942$		0			0
$943$	−62998.8			−2.17553
$944$		0			0
$945$	6966.07	−	7053.82i	0.239795	−	0.242816i
$946$		0			0
$947$			2337.55i			0.0802115i	0.999195	+	0.0401057i	$0.0127695\pi$
$947$			2337.55i			0.0802115i	−0.999195	+	0.0401057i	$0.987231\pi$
$948$		0			0
$949$	48770.1			1.66823
$950$		0			0
$951$	−2338.62			−0.0797424
$952$		0			0
$953$	361.004			0.0122708			0.00613539	−	0.999981i	$-0.498047\pi$
$953$	361.004			0.0122708			0.00613539	+	0.999981i	$0.498047\pi$
$954$		0			0
$955$	23746.6			0.804631
$956$		0			0
$957$		−	3304.08i		−	0.111605i
$958$		0			0
$959$	−7862.01	−	7764.20i	−0.264731	−	0.261438i
$960$		0			0
$961$	8586.37			0.288220
$962$		0			0
$963$			3826.18i			0.128034i
$964$		0			0
$965$		−	92703.7i		−	3.09247i
$966$		0			0
$967$			22036.2i			0.732821i	0.930453	+	0.366410i	$0.119413\pi$
$967$			22036.2i			0.732821i	−0.930453	+	0.366410i	$0.880587\pi$
$968$		0			0
$969$			31435.4i			1.04216i
$970$		0			0
$971$	28352.3			0.937042			0.468521	−	0.883452i	$-0.344787\pi$
$971$	28352.3			0.937042			0.468521	+	0.883452i	$0.344787\pi$
$972$		0			0
$973$	−4090.89	−	4040.00i	−0.134787	−	0.133110i
$974$		0			0
$975$		−	66377.7i		−	2.18030i
$976$		0			0
$977$	−27467.7			−0.899456			−0.449728	−	0.893166i	$-0.648479\pi$
$977$	−27467.7			−0.899456			−0.449728	+	0.893166i	$0.648479\pi$
$978$		0			0
$979$	10165.3			0.331852
$980$		0			0
$981$	9737.91			0.316929
$982$		0			0
$983$	−31642.8			−1.02670			−0.513351	−	0.858179i	$-0.671596\pi$
$983$	−31642.8			−1.02670			−0.513351	+	0.858179i	$0.671596\pi$
$984$		0			0
$985$			3812.23i			0.123317i
$986$		0			0
$987$	−1410.53	−	1392.98i	−0.0454890	−	0.0449231i
$988$		0			0
$989$	−52450.3			−1.68637
$990$		0			0
$991$		−	57299.5i		−	1.83671i	−0.395759	−	0.918354i	$-0.629518\pi$
$991$		−	57299.5i		−	1.83671i	0.395759	−	0.918354i	$-0.370482\pi$
$992$		0			0
$993$			7948.58i			0.254019i
$994$		0			0
$995$		−	76586.0i		−	2.44014i
$996$		0			0
$997$		−	7311.21i		−	0.232245i	−0.993235	−	0.116122i	$-0.962953\pi$
$997$		−	7311.21i		−	0.232245i	0.993235	−	0.116122i	$-0.0370465\pi$
$998$		0			0
$999$	−7605.67			−0.240874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.j.895.1		24
4.3	odd	2		1344.4.b.i.895.1		24
7.6	odd	2		1344.4.b.i.895.24		24
8.3	odd	2		672.4.b.b.223.24	yes	24
8.5	even	2		672.4.b.a.223.24	yes	24
28.27	even	2	inner	1344.4.b.j.895.24		24
56.13	odd	2		672.4.b.b.223.1	yes	24
56.27	even	2		672.4.b.a.223.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.1	✓	24	56.27	even	2
672.4.b.a.223.24	yes	24	8.5	even	2
672.4.b.b.223.1	yes	24	56.13	odd	2
672.4.b.b.223.24	yes	24	8.3	odd	2
1344.4.b.i.895.1		24	4.3	odd	2
1344.4.b.i.895.24		24	7.6	odd	2
1344.4.b.j.895.1		24	1.1	even	1	trivial
1344.4.b.j.895.24		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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -19.8256i q^{5} +(13.1775 + 13.0136i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -19.8256i q^{5} +(13.1775 + 13.0136i) q^{7} +9.00000 q^{9} +11.9649i q^{11} +82.5422i q^{13} -59.4769i q^{15} -100.713i q^{17} -104.043 q^{19} +(39.5325 + 39.0407i) q^{21} -144.803i q^{23} -268.056 q^{25} +27.0000 q^{27} -92.0491 q^{29} +195.901 q^{31} +35.8947i q^{33} +(258.002 - 261.252i) q^{35} -281.692 q^{37} +247.627i q^{39} -435.067i q^{41} -362.219i q^{43} -178.431i q^{45} -35.6802 q^{47} +(4.29353 + 342.973i) q^{49} -302.139i q^{51} +68.5367 q^{53} +237.212 q^{55} -312.128 q^{57} +460.066 q^{59} +443.029i q^{61} +(118.598 + 117.122i) q^{63} +1636.45 q^{65} -867.589i q^{67} -434.408i q^{69} -577.682i q^{71} -590.851i q^{73} -804.167 q^{75} +(-155.706 + 157.668i) q^{77} -282.598i q^{79} +81.0000 q^{81} -816.242 q^{83} -1996.70 q^{85} -276.147 q^{87} -849.591i q^{89} +(-1074.17 + 1087.70i) q^{91} +587.704 q^{93} +2062.71i q^{95} +711.190i q^{97} +107.684i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more