LMFDB - Embedded newform 1344.4.b.i.895.7

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			895.7
Character	$\chi$	$=$	1344.895
Dual form			1344.4.b.i.895.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -12.4907i q^{5} +(-8.98608 - 16.1941i) q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -12.4907i q^{5} +(-8.98608 - 16.1941i) q^{7} +9.00000 q^{9} +66.0589i q^{11} -82.6584i q^{13} +37.4722i q^{15} +30.0199i q^{17} +4.86333 q^{19} +(26.9583 + 48.5824i) q^{21} +113.091i q^{23} -31.0182 q^{25} -27.0000 q^{27} -233.621 q^{29} +100.776 q^{31} -198.177i q^{33} +(-202.277 + 112.243i) q^{35} -157.539 q^{37} +247.975i q^{39} +433.541i q^{41} -217.482i q^{43} -112.417i q^{45} -328.446 q^{47} +(-181.501 + 291.044i) q^{49} -90.0596i q^{51} +117.713 q^{53} +825.124 q^{55} -14.5900 q^{57} +384.529 q^{59} +87.8653i q^{61} +(-80.8748 - 145.747i) q^{63} -1032.46 q^{65} -305.387i q^{67} -339.274i q^{69} +1072.27i q^{71} -239.138i q^{73} +93.0545 q^{75} +(1069.77 - 593.611i) q^{77} +6.66481i q^{79} +81.0000 q^{81} -443.207 q^{83} +374.970 q^{85} +700.864 q^{87} -113.277i q^{89} +(-1338.58 + 742.775i) q^{91} -302.329 q^{93} -60.7465i q^{95} +356.754i q^{97} +594.530i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	12.4907i		−	1.11720i	−0.829436	−	0.558602i	$-0.811338\pi$
$5$		−	12.4907i		−	1.11720i	0.829436	−	0.558602i	$-0.188662\pi$
$6$		0			0
$7$	−8.98608	−	16.1941i	−0.485203	−	0.874402i
$7$	−8.98608	−	16.1941i	−0.485203	−	0.874402i
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			66.0589i			1.81068i	0.424684	+	0.905342i	$0.360385\pi$
$11$			66.0589i			1.81068i	−0.424684	+	0.905342i	$0.639615\pi$
$12$		0			0
$13$		−	82.6584i		−	1.76349i	−0.471731	−	0.881743i	$-0.656371\pi$
$13$		−	82.6584i		−	1.76349i	0.471731	−	0.881743i	$-0.343629\pi$
$14$		0			0
$15$			37.4722i			0.645018i
$16$		0			0
$17$			30.0199i			0.428288i	0.976802	+	0.214144i	$0.0686961\pi$
$17$			30.0199i			0.428288i	−0.976802	+	0.214144i	$0.931304\pi$
$18$		0			0
$19$	4.86333			0.0587223			0.0293612	−	0.999569i	$-0.490653\pi$
$19$	4.86333			0.0587223			0.0293612	+	0.999569i	$0.490653\pi$
$20$		0			0
$21$	26.9583	+	48.5824i	0.280132	+	0.504836i
$22$		0			0
$23$			113.091i			1.02527i	0.858607	+	0.512634i	$0.171330\pi$
$23$			113.091i			1.02527i	−0.858607	+	0.512634i	$0.828670\pi$
$24$		0			0
$25$	−31.0182			−0.248145
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−233.621			−1.49594			−0.747972	−	0.663730i	$-0.768972\pi$
$29$	−233.621			−1.49594			−0.747972	+	0.663730i	$0.768972\pi$
$30$		0			0
$31$	100.776			0.583870			0.291935	−	0.956438i	$-0.405701\pi$
$31$	100.776			0.583870			0.291935	+	0.956438i	$0.405701\pi$
$32$		0			0
$33$		−	198.177i		−	1.04540i
$34$		0			0
$35$	−202.277	+	112.243i	−0.976885	+	0.542071i
$36$		0			0
$37$	−157.539			−0.699980			−0.349990	−	0.936753i	$-0.613815\pi$
$37$	−157.539			−0.699980			−0.349990	+	0.936753i	$0.613815\pi$
$38$		0			0
$39$			247.975i			1.01815i
$40$		0			0
$41$			433.541i			1.65141i	0.564104	+	0.825704i	$0.309222\pi$
$41$			433.541i			1.65141i	−0.564104	+	0.825704i	$0.690778\pi$
$42$		0			0
$43$		−	217.482i		−	0.771297i	−0.922646	−	0.385648i	$-0.873978\pi$
$43$		−	217.482i		−	0.771297i	0.922646	−	0.385648i	$-0.126022\pi$
$44$		0			0
$45$		−	112.417i		−	0.372401i
$46$		0			0
$47$	−328.446			−1.01934			−0.509668	−	0.860371i	$-0.670232\pi$
$47$	−328.446			−1.01934			−0.509668	+	0.860371i	$0.670232\pi$
$48$		0			0
$49$	−181.501	+	291.044i	−0.529156	+	0.848524i
$50$		0			0
$51$		−	90.0596i		−	0.247272i
$52$		0			0
$53$	117.713			0.305077			0.152539	−	0.988298i	$-0.451255\pi$
$53$	117.713			0.305077			0.152539	+	0.988298i	$0.451255\pi$
$54$		0			0
$55$	825.124			2.02290
$56$		0			0
$57$	−14.5900			−0.0339034
$58$		0			0
$59$	384.529			0.848499			0.424250	−	0.905545i	$-0.360538\pi$
$59$	384.529			0.848499			0.424250	+	0.905545i	$0.360538\pi$
$60$		0			0
$61$			87.8653i			0.184426i	0.995739	+	0.0922131i	$0.0293941\pi$
$61$			87.8653i			0.184426i	−0.995739	+	0.0922131i	$0.970606\pi$
$62$		0			0
$63$	−80.8748	−	145.747i	−0.161734	−	0.291467i
$64$		0			0
$65$	−1032.46			−1.97017
$66$		0			0
$67$		−	305.387i		−	0.556850i	−0.960458	−	0.278425i	$-0.910188\pi$
$67$		−	305.387i		−	0.556850i	0.960458	−	0.278425i	$-0.0898124\pi$
$68$		0			0
$69$		−	339.274i		−	0.591939i
$70$		0			0
$71$			1072.27i			1.79232i	0.443728	+	0.896161i	$0.353656\pi$
$71$			1072.27i			1.79232i	−0.443728	+	0.896161i	$0.646344\pi$
$72$		0			0
$73$		−	239.138i		−	0.383411i	−0.981453	−	0.191705i	$-0.938598\pi$
$73$		−	239.138i		−	0.383411i	0.981453	−	0.191705i	$-0.0614018\pi$
$74$		0			0
$75$	93.0545			0.143267
$76$		0			0
$77$	1069.77	−	593.611i	1.58326	−	0.878549i
$78$		0			0
$79$			6.66481i			0.00949177i	0.999989	+	0.00474588i	$0.00151067\pi$
$79$			6.66481i			0.00949177i	−0.999989	+	0.00474588i	$0.998489\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	−443.207			−0.586123			−0.293062	−	0.956094i	$-0.594674\pi$
$83$	−443.207			−0.586123			−0.293062	+	0.956094i	$0.594674\pi$
$84$		0			0
$85$	374.970			0.478485
$86$		0			0
$87$	700.864			0.863684
$88$		0			0
$89$		−	113.277i		−	0.134914i	−0.997722	−	0.0674568i	$-0.978512\pi$
$89$		−	113.277i		−	0.134914i	0.997722	−	0.0674568i	$-0.0214885\pi$
$90$		0			0
$91$	−1338.58	+	742.775i	−1.54199	+	0.855648i
$92$		0			0
$93$	−302.329			−0.337098
$94$		0			0
$95$		−	60.7465i		−	0.0656049i
$96$		0			0
$97$			356.754i			0.373431i	0.982414	+	0.186716i	$0.0597843\pi$
$97$			356.754i			0.373431i	−0.982414	+	0.186716i	$0.940216\pi$
$98$		0			0
$99$			594.530i			0.603561i
$100$		0			0
$101$		−	1209.11i		−	1.19120i	−0.803282	−	0.595599i	$-0.796915\pi$
$101$		−	1209.11i		−	1.19120i	0.803282	−	0.595599i	$-0.203085\pi$
$102$		0			0
$103$	1866.19			1.78525			0.892625	−	0.450800i	$-0.148861\pi$
$103$	1866.19			1.78525			0.892625	+	0.450800i	$0.148861\pi$
$104$		0			0
$105$	606.830	−	336.728i	0.564005	−	0.312965i
$106$		0			0
$107$			358.014i			0.323463i	0.986835	+	0.161731i	$0.0517078\pi$
$107$			358.014i			0.323463i	−0.986835	+	0.161731i	$0.948292\pi$
$108$		0			0
$109$	46.0800			0.0404923			0.0202462	−	0.999795i	$-0.493555\pi$
$109$	46.0800			0.0404923			0.0202462	+	0.999795i	$0.493555\pi$
$110$		0			0
$111$	472.617			0.404133
$112$		0			0
$113$	1893.26			1.57613			0.788067	−	0.615589i	$-0.211082\pi$
$113$	1893.26			1.57613			0.788067	+	0.615589i	$0.211082\pi$
$114$		0			0
$115$	1412.59			1.14543
$116$		0			0
$117$		−	743.925i		−	0.587828i
$118$		0			0
$119$	486.146	−	269.761i	0.374495	−	0.207806i
$120$		0			0
$121$	−3032.78			−2.27857
$122$		0			0
$123$		−	1300.62i		−	0.953441i
$124$		0			0
$125$		−	1173.90i		−	0.839975i
$126$		0			0
$127$		−	1890.19i		−	1.32069i	−0.750964	−	0.660343i	$-0.770411\pi$
$127$		−	1890.19i		−	1.32069i	0.750964	−	0.660343i	$-0.229589\pi$
$128$		0			0
$129$			652.447i			0.445308i
$130$		0			0
$131$	115.824			0.0772485			0.0386243	−	0.999254i	$-0.487702\pi$
$131$	115.824			0.0772485			0.0386243	+	0.999254i	$0.487702\pi$
$132$		0			0
$133$	−43.7023	−	78.7575i	−0.0284923	−	0.0513469i
$134$		0			0
$135$			337.250i			0.215006i
$136$		0			0
$137$	649.045			0.404757			0.202378	−	0.979307i	$-0.435133\pi$
$137$	649.045			0.404757			0.202378	+	0.979307i	$0.435133\pi$
$138$		0			0
$139$	2243.69			1.36912			0.684558	−	0.728959i	$-0.259995\pi$
$139$	2243.69			1.36912			0.684558	+	0.728959i	$0.259995\pi$
$140$		0			0
$141$	985.339			0.588514
$142$		0			0
$143$	5460.32			3.19311
$144$		0			0
$145$			2918.10i			1.67128i
$146$		0			0
$147$	544.502	−	873.132i	0.305508	−	0.489896i
$148$		0			0
$149$	74.6162			0.0410255			0.0205127	−	0.999790i	$-0.493470\pi$
$149$	74.6162			0.0410255			0.0205127	+	0.999790i	$0.493470\pi$
$150$		0			0
$151$			3168.06i			1.70737i	0.520789	+	0.853685i	$0.325638\pi$
$151$			3168.06i			1.70737i	−0.520789	+	0.853685i	$0.674362\pi$
$152$		0			0
$153$			270.179i			0.142763i
$154$		0			0
$155$		−	1258.77i		−	0.652302i
$156$		0			0
$157$			2402.76i			1.22141i	0.791859	+	0.610703i	$0.209113\pi$
$157$			2402.76i			1.22141i	−0.791859	+	0.610703i	$0.790887\pi$
$158$		0			0
$159$	−353.138			−0.176136
$160$		0			0
$161$	1831.42	−	1016.25i	0.896497	−	0.497463i
$162$		0			0
$163$			1717.82i			0.825462i	0.910853	+	0.412731i	$0.135425\pi$
$163$			1717.82i			0.825462i	−0.910853	+	0.412731i	$0.864575\pi$
$164$		0			0
$165$	−2475.37			−1.16792
$166$		0			0
$167$	−637.463			−0.295380			−0.147690	−	0.989034i	$-0.547184\pi$
$167$	−637.463			−0.295380			−0.147690	+	0.989034i	$0.547184\pi$
$168$		0			0
$169$	−4635.41			−2.10988
$170$		0			0
$171$	43.7700			0.0195741
$172$		0			0
$173$			2057.56i			0.904238i	0.891958	+	0.452119i	$0.149332\pi$
$173$			2057.56i			0.904238i	−0.891958	+	0.452119i	$0.850668\pi$
$174$		0			0
$175$	278.732	+	502.313i	0.120401	+	0.216979i
$176$		0			0
$177$	−1153.59			−0.489881
$178$		0			0
$179$		−	2927.84i		−	1.22255i	−0.791418	−	0.611276i	$-0.790657\pi$
$179$		−	2927.84i		−	1.22255i	0.791418	−	0.611276i	$-0.209343\pi$
$180$		0			0
$181$			2192.81i			0.900497i	0.892903	+	0.450248i	$0.148665\pi$
$181$			2192.81i			0.900497i	−0.892903	+	0.450248i	$0.851335\pi$
$182$		0			0
$183$		−	263.596i		−	0.106479i
$184$		0			0
$185$			1967.78i			0.782020i
$186$		0			0
$187$	−1983.08			−0.775493
$188$		0			0
$189$	242.624	+	437.242i	0.0933773	+	0.168279i
$190$		0			0
$191$			3140.58i			1.18976i	0.803814	+	0.594881i	$0.202801\pi$
$191$			3140.58i			1.18976i	−0.803814	+	0.594881i	$0.797199\pi$
$192$		0			0
$193$	4286.25			1.59861			0.799303	−	0.600928i	$-0.205202\pi$
$193$	4286.25			1.59861			0.799303	+	0.600928i	$0.205202\pi$
$194$		0			0
$195$	3097.39			1.13748
$196$		0			0
$197$	−3512.85			−1.27046			−0.635229	−	0.772324i	$-0.719094\pi$
$197$	−3512.85			−1.27046			−0.635229	+	0.772324i	$0.719094\pi$
$198$		0			0
$199$	1419.23			0.505561			0.252780	−	0.967524i	$-0.418655\pi$
$199$	1419.23			0.505561			0.252780	+	0.967524i	$0.418655\pi$
$200$		0			0
$201$			916.161i			0.321497i
$202$		0			0
$203$	2099.34	+	3783.30i	0.725837	+	1.30806i
$204$		0			0
$205$	5415.24			1.84496
$206$		0			0
$207$			1017.82i			0.341756i
$208$		0			0
$209$			321.266i			0.106328i
$210$		0			0
$211$			5471.84i			1.78529i	0.450757	+	0.892647i	$0.351154\pi$
$211$			5471.84i			1.78529i	−0.450757	+	0.892647i	$0.648846\pi$
$212$		0			0
$213$		−	3216.81i		−	1.03480i
$214$		0			0
$215$	−2716.51			−0.861696
$216$		0			0
$217$	−905.585	−	1631.99i	−0.283296	−	0.510537i
$218$		0			0
$219$			717.414i			0.221362i
$220$		0			0
$221$	2481.39			0.755279
$222$		0			0
$223$	−1076.11			−0.323145			−0.161573	−	0.986861i	$-0.551657\pi$
$223$	−1076.11			−0.323145			−0.161573	+	0.986861i	$0.551657\pi$
$224$		0			0
$225$	−279.163			−0.0827151
$226$		0			0
$227$	5371.69			1.57062			0.785312	−	0.619100i	$-0.212502\pi$
$227$	5371.69			1.57062			0.785312	+	0.619100i	$0.212502\pi$
$228$		0			0
$229$		−	4116.75i		−	1.18796i	−0.804480	−	0.593980i	$-0.797556\pi$
$229$		−	4116.75i		−	1.18796i	0.804480	−	0.593980i	$-0.202444\pi$
$230$		0			0
$231$	−3209.30	+	1780.83i	−0.914098	+	0.507230i
$232$		0			0
$233$	3435.20			0.965868			0.482934	−	0.875657i	$-0.339571\pi$
$233$	3435.20			0.965868			0.482934	+	0.875657i	$0.339571\pi$
$234$		0			0
$235$			4102.53i			1.13881i
$236$		0			0
$237$		−	19.9944i		−	0.00548007i
$238$		0			0
$239$			4307.16i			1.16572i	0.812573	+	0.582860i	$0.198066\pi$
$239$			4307.16i			1.16572i	−0.812573	+	0.582860i	$0.801934\pi$
$240$		0			0
$241$		−	6222.50i		−	1.66318i	−0.555389	−	0.831590i	$-0.687431\pi$
$241$		−	6222.50i		−	1.66318i	0.555389	−	0.831590i	$-0.312569\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$	3635.35	+	2267.07i	0.947975	+	0.591176i
$246$		0			0
$247$		−	401.995i		−	0.103556i
$248$		0			0
$249$	1329.62			0.338398
$250$		0			0
$251$	1364.67			0.343177			0.171589	−	0.985169i	$-0.445110\pi$
$251$	1364.67			0.343177			0.171589	+	0.985169i	$0.445110\pi$
$252$		0			0
$253$	−7470.70			−1.85644
$254$		0			0
$255$	−1124.91			−0.276253
$256$		0			0
$257$			757.000i			0.183737i	0.995771	+	0.0918685i	$0.0292839\pi$
$257$			757.000i			0.183737i	−0.995771	+	0.0918685i	$0.970716\pi$
$258$		0			0
$259$	1415.66	+	2551.21i	0.339632	+	0.612063i
$260$		0			0
$261$	−2102.59			−0.498648
$262$		0			0
$263$			286.534i			0.0671803i	0.999436	+	0.0335901i	$0.0106941\pi$
$263$			286.534i			0.0671803i	−0.999436	+	0.0335901i	$0.989306\pi$
$264$		0			0
$265$		−	1470.32i		−	0.340834i
$266$		0			0
$267$			339.830i			0.0778924i
$268$		0			0
$269$			8010.21i			1.81558i	0.419424	+	0.907790i	$0.362232\pi$
$269$			8010.21i			1.81558i	−0.419424	+	0.907790i	$0.637768\pi$
$270$		0			0
$271$	5666.22			1.27010			0.635052	−	0.772469i	$-0.280979\pi$
$271$	5666.22			1.27010			0.635052	+	0.772469i	$0.280979\pi$
$272$		0			0
$273$	4015.74	−	2228.33i	0.890271	−	0.494009i
$274$		0			0
$275$		−	2049.03i		−	0.449313i
$276$		0			0
$277$	7730.93			1.67692			0.838459	−	0.544964i	$-0.183457\pi$
$277$	7730.93			1.67692			0.838459	+	0.544964i	$0.183457\pi$
$278$		0			0
$279$	906.988			0.194623
$280$		0			0
$281$	3444.60			0.731272			0.365636	−	0.930758i	$-0.380852\pi$
$281$	3444.60			0.731272			0.365636	+	0.930758i	$0.380852\pi$
$282$		0			0
$283$	−5639.98			−1.18467			−0.592335	−	0.805691i	$-0.701794\pi$
$283$	−5639.98			−1.18467			−0.592335	+	0.805691i	$0.701794\pi$
$284$		0			0
$285$			182.240i			0.0378770i
$286$		0			0
$287$	7020.82	−	3895.83i	1.44399	−	0.801268i
$288$		0			0
$289$	4011.81			0.816570
$290$		0			0
$291$		−	1070.26i		−	0.215601i
$292$		0			0
$293$		−	3346.54i		−	0.667259i	−0.942704	−	0.333629i	$-0.891727\pi$
$293$		−	3346.54i		−	0.667259i	0.942704	−	0.333629i	$-0.108273\pi$
$294$		0			0
$295$		−	4803.05i		−	0.947947i
$296$		0			0
$297$		−	1783.59i		−	0.348466i
$298$		0			0
$299$	9347.95			1.80805
$300$		0			0
$301$	−3521.94	+	1954.32i	−0.674423	+	0.374235i
$302$		0			0
$303$			3627.33i			0.687738i
$304$		0			0
$305$	1097.50			0.206042
$306$		0			0
$307$	4402.91			0.818526			0.409263	−	0.912417i	$-0.365786\pi$
$307$	4402.91			0.818526			0.409263	+	0.912417i	$0.365786\pi$
$308$		0			0
$309$	−5598.56			−1.03071
$310$		0			0
$311$	−4179.33			−0.762020			−0.381010	−	0.924571i	$-0.624424\pi$
$311$	−4179.33			−0.762020			−0.381010	+	0.924571i	$0.624424\pi$
$312$		0			0
$313$		−	7718.53i		−	1.39386i	−0.717141	−	0.696928i	$-0.754550\pi$
$313$		−	7718.53i		−	1.39386i	0.717141	−	0.696928i	$-0.245450\pi$
$314$		0			0
$315$	−1820.49	+	1010.18i	−0.325628	+	0.180690i
$316$		0			0
$317$	−3231.78			−0.572602			−0.286301	−	0.958140i	$-0.592426\pi$
$317$	−3231.78			−0.572602			−0.286301	+	0.958140i	$0.592426\pi$
$318$		0			0
$319$		−	15432.8i		−	2.70868i
$320$		0			0
$321$		−	1074.04i		−	0.186751i
$322$		0			0
$323$			145.997i			0.0251500i
$324$		0			0
$325$			2563.91i			0.437601i
$326$		0			0
$327$	−138.240			−0.0233783
$328$		0			0
$329$	2951.45	+	5318.91i	0.494585	+	0.891309i
$330$		0			0
$331$			3627.37i			0.602352i	0.953569	+	0.301176i	$0.0973791\pi$
$331$			3627.37i			0.602352i	−0.953569	+	0.301176i	$0.902621\pi$
$332$		0			0
$333$	−1417.85			−0.233327
$334$		0			0
$335$	−3814.50			−0.622115
$336$		0			0
$337$	−10308.1			−1.66623			−0.833113	−	0.553103i	$-0.813444\pi$
$337$	−10308.1			−1.66623			−0.833113	+	0.553103i	$0.813444\pi$
$338$		0			0
$339$	−5679.79			−0.909982
$340$		0			0
$341$			6657.18i			1.05720i
$342$		0			0
$343$	6344.19	+	323.902i	0.998699	+	0.0509885i
$344$		0			0
$345$	−4237.78			−0.661317
$346$		0			0
$347$			7643.84i			1.18254i	0.806472	+	0.591272i	$0.201374\pi$
$347$			7643.84i			1.18254i	−0.806472	+	0.591272i	$0.798626\pi$
$348$		0			0
$349$		−	7329.32i		−	1.12415i	−0.827085	−	0.562077i	$-0.810002\pi$
$349$		−	7329.32i		−	1.12415i	0.827085	−	0.562077i	$-0.189998\pi$
$350$		0			0
$351$			2231.78i			0.339383i
$352$		0			0
$353$		−	8452.93i		−	1.27452i	−0.770651	−	0.637258i	$-0.780069\pi$
$353$		−	8452.93i		−	1.27452i	0.770651	−	0.637258i	$-0.219931\pi$
$354$		0			0
$355$	13393.4			2.00239
$356$		0			0
$357$	−1458.44	+	809.283i	−0.216215	+	0.119977i
$358$		0			0
$359$			3141.03i			0.461775i	0.972980	+	0.230887i	$0.0741629\pi$
$359$			3141.03i			0.461775i	−0.972980	+	0.230887i	$0.925837\pi$
$360$		0			0
$361$	−6835.35			−0.996552
$362$		0			0
$363$	9098.35			1.31554
$364$		0			0
$365$	−2987.01			−0.428348
$366$		0			0
$367$	−5554.76			−0.790072			−0.395036	−	0.918666i	$-0.629268\pi$
$367$	−5554.76			−0.790072			−0.395036	+	0.918666i	$0.629268\pi$
$368$		0			0
$369$			3901.87i			0.550469i
$370$		0			0
$371$	−1057.78	−	1906.26i	−0.148024	−	0.266760i
$372$		0			0
$373$	10891.7			1.51193			0.755967	−	0.654609i	$-0.227167\pi$
$373$	10891.7			1.51193			0.755967	+	0.654609i	$0.227167\pi$
$374$		0			0
$375$			3521.70i			0.484960i
$376$		0			0
$377$			19310.8i			2.63808i
$378$		0			0
$379$		−	3162.06i		−	0.428559i	−0.976772	−	0.214280i	$-0.931260\pi$
$379$		−	3162.06i		−	0.428559i	0.976772	−	0.214280i	$-0.0687404\pi$
$380$		0			0
$381$			5670.56i			0.762498i
$382$		0			0
$383$	−12001.6			−1.60119			−0.800594	−	0.599207i	$-0.795483\pi$
$383$	−12001.6			−1.60119			−0.800594	+	0.599207i	$0.795483\pi$
$384$		0			0
$385$	−7414.63	−	13362.2i	−0.981519	−	1.76883i
$386$		0			0
$387$		−	1957.34i		−	0.257099i
$388$		0			0
$389$	6566.48			0.855870			0.427935	−	0.903809i	$-0.359241\pi$
$389$	6566.48			0.855870			0.427935	+	0.903809i	$0.359241\pi$
$390$		0			0
$391$	−3394.99			−0.439110
$392$		0			0
$393$	−347.471			−0.0445995
$394$		0			0
$395$	83.2483			0.0106042
$396$		0			0
$397$			14986.7i			1.89461i	0.320326	+	0.947307i	$0.396208\pi$
$397$			14986.7i			1.89461i	−0.320326	+	0.947307i	$0.603792\pi$
$398$		0			0
$399$	131.107	+	236.272i	0.0164500	+	0.0296452i
$400$		0			0
$401$	1702.23			0.211983			0.105992	−	0.994367i	$-0.466198\pi$
$401$	1702.23			0.211983			0.105992	+	0.994367i	$0.466198\pi$
$402$		0			0
$403$		−	8330.01i		−	1.02965i
$404$		0			0
$405$		−	1011.75i		−	0.124134i
$406$		0			0
$407$		−	10406.9i		−	1.26744i
$408$		0			0
$409$			2752.77i			0.332801i	0.986058	+	0.166401i	$0.0532145\pi$
$409$			2752.77i			0.332801i	−0.986058	+	0.166401i	$0.946785\pi$
$410$		0			0
$411$	−1947.14			−0.233686
$412$		0			0
$413$	−3455.41	−	6227.12i	−0.411694	−	0.741929i
$414$		0			0
$415$			5535.97i			0.654820i
$416$		0			0
$417$	−6731.06			−0.790459
$418$		0			0
$419$	−4997.44			−0.582675			−0.291337	−	0.956620i	$-0.594100\pi$
$419$	−4997.44			−0.582675			−0.291337	+	0.956620i	$0.594100\pi$
$420$		0			0
$421$	−7648.38			−0.885414			−0.442707	−	0.896666i	$-0.645982\pi$
$421$	−7648.38			−0.885414			−0.442707	+	0.896666i	$0.645982\pi$
$422$		0			0
$423$	−2956.02			−0.339779
$424$		0			0
$425$		−	931.161i		−	0.106278i
$426$		0			0
$427$	1422.90	−	789.565i	0.161263	−	0.0894842i
$428$		0			0
$429$	−16381.0			−1.84354
$430$		0			0
$431$		−	7778.05i		−	0.869271i	−0.900607	−	0.434635i	$-0.856877\pi$
$431$		−	7778.05i		−	0.869271i	0.900607	−	0.434635i	$-0.143123\pi$
$432$		0			0
$433$		−	13339.2i		−	1.48046i	−0.672352	−	0.740232i	$-0.734716\pi$
$433$		−	13339.2i		−	1.48046i	0.672352	−	0.740232i	$-0.265284\pi$
$434$		0			0
$435$		−	8754.30i		−	0.964911i
$436$		0			0
$437$			550.001i			0.0602062i
$438$		0			0
$439$	3699.19			0.402171			0.201085	−	0.979574i	$-0.435553\pi$
$439$	3699.19			0.402171			0.201085	+	0.979574i	$0.435553\pi$
$440$		0			0
$441$	−1633.51	+	2619.39i	−0.176385	+	0.282841i
$442$		0			0
$443$		−	3394.79i		−	0.364088i	−0.983290	−	0.182044i	$-0.941729\pi$
$443$		−	3394.79i		−	0.364088i	0.983290	−	0.182044i	$-0.0582714\pi$
$444$		0			0
$445$	−1414.91			−0.150726
$446$		0			0
$447$	−223.849			−0.0236861
$448$		0			0
$449$	401.735			0.0422251			0.0211125	−	0.999777i	$-0.493279\pi$
$449$	401.735			0.0422251			0.0211125	+	0.999777i	$0.493279\pi$
$450$		0			0
$451$	−28639.2			−2.99018
$452$		0			0
$453$		−	9504.18i		−	0.985751i
$454$		0			0
$455$	9277.80	+	16719.9i	0.955934	+	1.72272i
$456$		0			0
$457$	9692.22			0.992085			0.496043	−	0.868298i	$-0.334786\pi$
$457$	9692.22			0.992085			0.496043	+	0.868298i	$0.334786\pi$
$458$		0			0
$459$		−	810.537i		−	0.0824240i
$460$		0			0
$461$		−	14822.9i		−	1.49756i	−0.662821	−	0.748778i	$-0.730641\pi$
$461$		−	14822.9i		−	1.49756i	0.662821	−	0.748778i	$-0.269359\pi$
$462$		0			0
$463$			4159.46i			0.417508i	0.977968	+	0.208754i	$0.0669408\pi$
$463$			4159.46i			0.417508i	−0.977968	+	0.208754i	$0.933059\pi$
$464$		0			0
$465$			3776.31i			0.376607i
$466$		0			0
$467$	−2027.66			−0.200919			−0.100459	−	0.994941i	$-0.532031\pi$
$467$	−2027.66			−0.200919			−0.100459	+	0.994941i	$0.532031\pi$
$468$		0			0
$469$	−4945.48	+	2744.23i	−0.486911	+	0.270185i
$470$		0			0
$471$		−	7208.27i		−	0.705180i
$472$		0			0
$473$	14366.7			1.39657
$474$		0			0
$475$	−150.852			−0.0145717
$476$		0			0
$477$	1059.42			0.101692
$478$		0			0
$479$	5863.44			0.559306			0.279653	−	0.960101i	$-0.409781\pi$
$479$	5863.44			0.559306			0.279653	+	0.960101i	$0.409781\pi$
$480$		0			0
$481$			13021.9i			1.23440i
$482$		0			0
$483$	−5494.25	+	3048.75i	−0.517593	+	0.287211i
$484$		0			0
$485$	4456.11			0.417199
$486$		0			0
$487$		−	8660.45i		−	0.805836i	−0.915236	−	0.402918i	$-0.867996\pi$
$487$		−	8660.45i		−	0.805836i	0.915236	−	0.402918i	$-0.132004\pi$
$488$		0			0
$489$		−	5153.47i		−	0.476581i
$490$		0			0
$491$			2031.87i			0.186756i	0.995631	+	0.0933779i	$0.0297665\pi$
$491$			2031.87i			0.186756i	−0.995631	+	0.0933779i	$0.970234\pi$
$492$		0			0
$493$		−	7013.28i		−	0.640694i
$494$		0			0
$495$	7426.11			0.674301
$496$		0			0
$497$	17364.5	−	9635.50i	1.56721	−	0.869640i
$498$		0			0
$499$			706.839i			0.0634117i	0.999497	+	0.0317059i	$0.0100940\pi$
$499$			706.839i			0.0634117i	−0.999497	+	0.0317059i	$0.989906\pi$
$500$		0			0
$501$	1912.39			0.170538
$502$		0			0
$503$	−20321.5			−1.80138			−0.900688	−	0.434467i	$-0.856937\pi$
$503$	−20321.5			−1.80138			−0.900688	+	0.434467i	$0.856937\pi$
$504$		0			0
$505$	−15102.7			−1.33081
$506$		0			0
$507$	13906.2			1.21814
$508$		0			0
$509$			11672.0i			1.01641i	0.861236	+	0.508205i	$0.169691\pi$
$509$			11672.0i			1.01641i	−0.861236	+	0.508205i	$0.830309\pi$
$510$		0			0
$511$	−3872.63	+	2148.91i	−0.335255	+	0.186032i
$512$		0			0
$513$	−131.310			−0.0113011
$514$		0			0
$515$		−	23310.0i		−	1.99449i
$516$		0			0
$517$		−	21696.8i		−	1.84570i
$518$		0			0
$519$		−	6172.67i		−	0.522062i
$520$		0			0
$521$			9249.26i			0.777768i	0.921287	+	0.388884i	$0.127139\pi$
$521$			9249.26i			0.777768i	−0.921287	+	0.388884i	$0.872861\pi$
$522$		0			0
$523$	−6464.38			−0.540474			−0.270237	−	0.962794i	$-0.587102\pi$
$523$	−6464.38			−0.540474			−0.270237	+	0.962794i	$0.587102\pi$
$524$		0			0
$525$	−836.196	−	1506.94i	−0.0695135	−	0.125273i
$526$		0			0
$527$			3025.29i			0.250064i
$528$		0			0
$529$	−622.658			−0.0511760
$530$		0			0
$531$	3460.76			0.282833
$532$		0			0
$533$	35835.8			2.91223
$534$		0			0
$535$	4471.85			0.361374
$536$		0			0
$537$			8783.51i			0.705841i
$538$		0			0
$539$	−19226.0	−	11989.7i	−1.53641	−	0.958134i
$540$		0			0
$541$	−3979.54			−0.316254			−0.158127	−	0.987419i	$-0.550546\pi$
$541$	−3979.54			−0.316254			−0.158127	+	0.987419i	$0.550546\pi$
$542$		0			0
$543$		−	6578.42i		−	0.519902i
$544$		0			0
$545$		−	575.572i		−	0.0452382i
$546$		0			0
$547$			17317.7i			1.35366i	0.736139	+	0.676831i	$0.236647\pi$
$547$			17317.7i			1.35366i	−0.736139	+	0.676831i	$0.763353\pi$
$548$		0			0
$549$			790.788i			0.0614754i
$550$		0			0
$551$	−1136.18			−0.0878454
$552$		0			0
$553$	107.931	−	59.8905i	0.00829962	−	0.00460543i
$554$		0			0
$555$		−	5903.33i		−	0.451500i
$556$		0			0
$557$	4948.25			0.376417			0.188208	−	0.982129i	$-0.439732\pi$
$557$	4948.25			0.376417			0.188208	+	0.982129i	$0.439732\pi$
$558$		0			0
$559$	−17976.7			−1.36017
$560$		0			0
$561$	5949.24			0.447731
$562$		0			0
$563$	−8441.80			−0.631935			−0.315967	−	0.948770i	$-0.602329\pi$
$563$	−8441.80			−0.631935			−0.315967	+	0.948770i	$0.602329\pi$
$564$		0			0
$565$		−	23648.2i		−	1.76086i
$566$		0			0
$567$	−727.873	−	1311.73i	−0.0539114	−	0.0971557i
$568$		0			0
$569$	−6165.26			−0.454237			−0.227119	−	0.973867i	$-0.572931\pi$
$569$	−6165.26			−0.454237			−0.227119	+	0.973867i	$0.572931\pi$
$570$		0			0
$571$			15956.2i			1.16943i	0.811238	+	0.584716i	$0.198794\pi$
$571$			15956.2i			1.16943i	−0.811238	+	0.584716i	$0.801206\pi$
$572$		0			0
$573$		−	9421.74i		−	0.686909i
$574$		0			0
$575$		−	3507.89i		−	0.254416i
$576$		0			0
$577$			670.197i			0.0483547i	0.999708	+	0.0241773i	$0.00769664\pi$
$577$			670.197i			0.0483547i	−0.999708	+	0.0241773i	$0.992303\pi$
$578$		0			0
$579$	−12858.8			−0.922956
$580$		0			0
$581$	3982.69	+	7177.35i	0.284389	+	0.512507i
$582$		0			0
$583$			7775.98i			0.552398i
$584$		0			0
$585$	−9292.16			−0.656724
$586$		0			0
$587$	−15030.4			−1.05685			−0.528426	−	0.848979i	$-0.677218\pi$
$587$	−15030.4			−1.05685			−0.528426	+	0.848979i	$0.677218\pi$
$588$		0			0
$589$	490.109			0.0342862
$590$		0			0
$591$	10538.6			0.733500
$592$		0			0
$593$			22800.0i			1.57890i	0.613818	+	0.789448i	$0.289633\pi$
$593$			22800.0i			1.57890i	−0.613818	+	0.789448i	$0.710367\pi$
$594$		0			0
$595$	−3369.51	−	6072.32i	−0.232162	−	0.418388i
$596$		0			0
$597$	−4257.69			−0.291886
$598$		0			0
$599$			16997.4i			1.15942i	0.814822	+	0.579712i	$0.196835\pi$
$599$			16997.4i			1.15942i	−0.814822	+	0.579712i	$0.803165\pi$
$600$		0			0
$601$			28259.3i			1.91801i	0.283393	+	0.959004i	$0.408540\pi$
$601$			28259.3i			1.91801i	−0.283393	+	0.959004i	$0.591460\pi$
$602$		0			0
$603$		−	2748.48i		−	0.185617i
$604$		0			0
$605$			37881.6i			2.54563i
$606$		0			0
$607$	−1593.77			−0.106572			−0.0532859	−	0.998579i	$-0.516969\pi$
$607$	−1593.77			−0.106572			−0.0532859	+	0.998579i	$0.516969\pi$
$608$		0			0
$609$	−6298.02	−	11349.9i	−0.419062	−	0.755207i
$610$		0			0
$611$			27148.8i			1.79758i
$612$		0			0
$613$	−8449.13			−0.556700			−0.278350	−	0.960480i	$-0.589787\pi$
$613$	−8449.13			−0.556700			−0.278350	+	0.960480i	$0.589787\pi$
$614$		0			0
$615$	−16245.7			−1.06519
$616$		0			0
$617$	1037.28			0.0676811			0.0338405	−	0.999427i	$-0.489226\pi$
$617$	1037.28			0.0676811			0.0338405	+	0.999427i	$0.489226\pi$
$618$		0			0
$619$	−29119.2			−1.89079			−0.945396	−	0.325924i	$-0.894325\pi$
$619$	−29119.2			−1.89079			−0.945396	+	0.325924i	$0.894325\pi$
$620$		0			0
$621$		−	3053.47i		−	0.197313i
$622$		0			0
$623$	−1834.42	+	1017.91i	−0.117969	+	0.0654605i
$624$		0			0
$625$	−18540.1			−1.18657
$626$		0			0
$627$		−	963.799i		−	0.0613883i
$628$		0			0
$629$		−	4729.30i		−	0.299793i
$630$		0			0
$631$			4094.27i			0.258305i	0.991625	+	0.129152i	$0.0412256\pi$
$631$			4094.27i			0.258305i	−0.991625	+	0.129152i	$0.958774\pi$
$632$		0			0
$633$		−	16415.5i		−	1.03074i
$634$		0			0
$635$	−23609.8			−1.47548
$636$		0			0
$637$	24057.2	+	15002.5i	1.49636	+	0.933159i
$638$		0			0
$639$			9650.42i			0.597441i
$640$		0			0
$641$	4884.78			0.300994			0.150497	−	0.988610i	$-0.451913\pi$
$641$	4884.78			0.300994			0.150497	+	0.988610i	$0.451913\pi$
$642$		0			0
$643$	15955.3			0.978561			0.489281	−	0.872126i	$-0.337259\pi$
$643$	15955.3			0.978561			0.489281	+	0.872126i	$0.337259\pi$
$644$		0			0
$645$	8149.54			0.497500
$646$		0			0
$647$	19751.5			1.20017			0.600086	−	0.799935i	$-0.295133\pi$
$647$	19751.5			1.20017			0.600086	+	0.799935i	$0.295133\pi$
$648$		0			0
$649$			25401.6i			1.53636i
$650$		0			0
$651$	2716.76	+	4895.96i	0.163561	+	0.294759i
$652$		0			0
$653$	−10834.1			−0.649266			−0.324633	−	0.945840i	$-0.605241\pi$
$653$	−10834.1			−0.649266			−0.324633	+	0.945840i	$0.605241\pi$
$654$		0			0
$655$		−	1446.72i		−	0.0863024i
$656$		0			0
$657$		−	2152.24i		−	0.127804i
$658$		0			0
$659$		−	8108.01i		−	0.479276i	−0.970862	−	0.239638i	$-0.922971\pi$
$659$		−	8108.01i		−	0.479276i	0.970862	−	0.239638i	$-0.0770288\pi$
$660$		0			0
$661$			3494.01i			0.205599i	0.994702	+	0.102800i	$0.0327800\pi$
$661$			3494.01i			0.205599i	−0.994702	+	0.102800i	$0.967220\pi$
$662$		0			0
$663$	−7444.18			−0.436060
$664$		0			0
$665$	−983.738	+	545.873i	−0.0573650	+	0.0318317i
$666$		0			0
$667$		−	26420.6i		−	1.53375i
$668$		0			0
$669$	3228.32			0.186568
$670$		0			0
$671$	−5804.29			−0.333938
$672$		0			0
$673$	27547.3			1.57782			0.788908	−	0.614512i	$-0.210647\pi$
$673$	27547.3			1.57782			0.788908	+	0.614512i	$0.210647\pi$
$674$		0			0
$675$	837.490			0.0477556
$676$		0			0
$677$		−	7562.95i		−	0.429347i	−0.976686	−	0.214673i	$-0.931131\pi$
$677$		−	7562.95i		−	0.429347i	0.976686	−	0.214673i	$-0.0688687\pi$
$678$		0			0
$679$	5777.32	−	3205.82i	0.326529	−	0.181190i
$680$		0			0
$681$	−16115.1			−0.906801
$682$		0			0
$683$		−	18268.3i		−	1.02345i	−0.859149	−	0.511726i	$-0.829006\pi$
$683$		−	18268.3i		−	1.02345i	0.859149	−	0.511726i	$-0.170994\pi$
$684$		0			0
$685$		−	8107.04i		−	0.452196i
$686$		0			0
$687$			12350.3i			0.685869i
$688$		0			0
$689$		−	9729.95i		−	0.537999i
$690$		0			0
$691$	−7813.52			−0.430160			−0.215080	−	0.976596i	$-0.569001\pi$
$691$	−7813.52			−0.430160			−0.215080	+	0.976596i	$0.569001\pi$
$692$		0			0
$693$	9627.91	−	5342.50i	0.527755	−	0.292850i
$694$		0			0
$695$		−	28025.3i		−	1.52958i
$696$		0			0
$697$	−13014.8			−0.707277
$698$		0			0
$699$	−10305.6			−0.557644
$700$		0			0
$701$	27501.0			1.48174			0.740868	−	0.671650i	$-0.234414\pi$
$701$	27501.0			1.48174			0.740868	+	0.671650i	$0.234414\pi$
$702$		0			0
$703$	−766.164			−0.0411045
$704$		0			0
$705$		−	12307.6i		−	0.657490i
$706$		0			0
$707$	−19580.5	+	10865.2i	−1.04158	+	0.577972i
$708$		0			0
$709$	−12702.5			−0.672851			−0.336426	−	0.941710i	$-0.609218\pi$
$709$	−12702.5			−0.672851			−0.336426	+	0.941710i	$0.609218\pi$
$710$		0			0
$711$			59.9833i			0.00316392i
$712$		0			0
$713$			11396.9i			0.598624i
$714$		0			0
$715$		−	68203.4i		−	3.56736i
$716$		0			0
$717$		−	12921.5i		−	0.673028i
$718$		0			0
$719$	−11491.3			−0.596039			−0.298019	−	0.954560i	$-0.596326\pi$
$719$	−11491.3			−0.596039			−0.298019	+	0.954560i	$0.596326\pi$
$720$		0			0
$721$	−16769.7	−	30221.3i	−0.866208	−	1.56103i
$722$		0			0
$723$			18667.5i			0.960238i
$724$		0			0
$725$	7246.50			0.371212
$726$		0			0
$727$	−10841.8			−0.553095			−0.276548	−	0.961000i	$-0.589190\pi$
$727$	−10841.8			−0.553095			−0.276548	+	0.961000i	$0.589190\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	6528.79			0.330337
$732$		0			0
$733$			25482.7i			1.28407i	0.766674	+	0.642037i	$0.221910\pi$
$733$			25482.7i			1.28407i	−0.766674	+	0.642037i	$0.778090\pi$
$734$		0			0
$735$	−10906.0	−	6801.22i	−0.547314	−	0.341315i
$736$		0			0
$737$	20173.5			1.00828
$738$		0			0
$739$		−	8545.18i		−	0.425358i	−0.977122	−	0.212679i	$-0.931781\pi$
$739$		−	8545.18i		−	0.425358i	0.977122	−	0.212679i	$-0.0682189\pi$
$740$		0			0
$741$			1205.98i			0.0597881i
$742$		0			0
$743$			9590.28i			0.473531i	0.971567	+	0.236765i	$0.0760873\pi$
$743$			9590.28i			0.473531i	−0.971567	+	0.236765i	$0.923913\pi$
$744$		0			0
$745$		−	932.010i		−	0.0458338i
$746$		0			0
$747$	−3988.86			−0.195374
$748$		0			0
$749$	5797.73	−	3217.14i	0.282836	−	0.156945i
$750$		0			0
$751$		−	19602.6i		−	0.952475i	−0.879317	−	0.476237i	$-0.842000\pi$
$751$		−	19602.6i		−	0.952475i	0.879317	−	0.476237i	$-0.158000\pi$
$752$		0			0
$753$	−4094.02			−0.198133
$754$		0			0
$755$	39571.3			1.90748
$756$		0			0
$757$	−7083.47			−0.340097			−0.170048	−	0.985436i	$-0.554392\pi$
$757$	−7083.47			−0.340097			−0.170048	+	0.985436i	$0.554392\pi$
$758$		0			0
$759$	22412.1			1.07181
$760$		0			0
$761$			19592.2i			0.933267i	0.884451	+	0.466633i	$0.154533\pi$
$761$			19592.2i			0.933267i	−0.884451	+	0.466633i	$0.845467\pi$
$762$		0			0
$763$	−414.079	−	746.226i	−0.0196470	−	0.0354065i
$764$		0			0
$765$	3374.73			0.159495
$766$		0			0
$767$		−	31784.6i		−	1.49632i
$768$		0			0
$769$			7802.71i			0.365895i	0.983123	+	0.182947i	$0.0585638\pi$
$769$			7802.71i			0.365895i	−0.983123	+	0.182947i	$0.941436\pi$
$770$		0			0
$771$		−	2271.00i		−	0.106081i
$772$		0			0
$773$		−	18903.5i		−	0.879574i	−0.898102	−	0.439787i	$-0.855054\pi$
$773$		−	18903.5i		−	0.879574i	0.898102	−	0.439787i	$-0.144946\pi$
$774$		0			0
$775$	−3125.90			−0.144885
$776$		0			0
$777$	−4246.98	−	7653.63i	−0.196087	−	0.353375i
$778$		0			0
$779$			2108.45i			0.0969745i
$780$		0			0
$781$	−70832.9			−3.24533
$782$		0			0
$783$	6307.78			0.287895
$784$		0			0
$785$	30012.2			1.36456
$786$		0			0
$787$	13763.6			0.623407			0.311703	−	0.950179i	$-0.399101\pi$
$787$	13763.6			0.623407			0.311703	+	0.950179i	$0.399101\pi$
$788$		0			0
$789$		−	859.601i		−	0.0387866i
$790$		0			0
$791$	−17013.0	−	30659.8i	−0.764745	−	1.37817i
$792$		0			0
$793$	7262.81			0.325233
$794$		0			0
$795$			4410.95i			0.196780i
$796$		0			0
$797$		−	28670.1i		−	1.27421i	−0.770777	−	0.637106i	$-0.780132\pi$
$797$		−	28670.1i		−	1.27421i	0.770777	−	0.637106i	$-0.219868\pi$
$798$		0			0
$799$		−	9859.91i		−	0.436569i
$800$		0			0
$801$		−	1019.49i		−	0.0449712i
$802$		0			0
$803$	15797.2			0.694235
$804$		0			0
$805$	−12693.7	−	22875.7i	−0.555768	−	1.00157i
$806$		0			0
$807$		−	24030.6i		−	1.04823i
$808$		0			0
$809$	−21154.7			−0.919356			−0.459678	−	0.888086i	$-0.652035\pi$
$809$	−21154.7			−0.919356			−0.459678	+	0.888086i	$0.652035\pi$
$810$		0			0
$811$	3553.64			0.153866			0.0769328	−	0.997036i	$-0.475487\pi$
$811$	3553.64			0.153866			0.0769328	+	0.997036i	$0.475487\pi$
$812$		0			0
$813$	−16998.7			−0.733295
$814$		0			0
$815$	21456.9			0.922210
$816$		0			0
$817$		−	1057.69i		−	0.0452923i
$818$		0			0
$819$	−12047.2	+	6684.98i	−0.513998	+	0.285216i
$820$		0			0
$821$	34509.7			1.46699			0.733495	−	0.679695i	$-0.237888\pi$
$821$	34509.7			1.46699			0.733495	+	0.679695i	$0.237888\pi$
$822$		0			0
$823$		−	4122.47i		−	0.174605i	−0.996182	−	0.0873026i	$-0.972175\pi$
$823$		−	4122.47i		−	0.174605i	0.996182	−	0.0873026i	$-0.0278247\pi$
$824$		0			0
$825$			6147.08i			0.259411i
$826$		0			0
$827$			28383.1i			1.19344i	0.802448	+	0.596722i	$0.203531\pi$
$827$			28383.1i			1.19344i	−0.802448	+	0.596722i	$0.796469\pi$
$828$		0			0
$829$			36675.1i			1.53652i	0.640135	+	0.768262i	$0.278878\pi$
$829$			36675.1i			1.53652i	−0.640135	+	0.768262i	$0.721122\pi$
$830$		0			0
$831$	−23192.8			−0.968169
$832$		0			0
$833$	−8737.10	−	5448.62i	−0.363412	−	0.226631i
$834$		0			0
$835$			7962.38i			0.329999i
$836$		0			0
$837$	−2720.96			−0.112366
$838$		0			0
$839$	30440.4			1.25259			0.626293	−	0.779588i	$-0.284571\pi$
$839$	30440.4			1.25259			0.626293	+	0.779588i	$0.284571\pi$
$840$		0			0
$841$	30189.9			1.23785
$842$		0			0
$843$	−10333.8			−0.422200
$844$		0			0
$845$			57899.6i			2.35717i
$846$		0			0
$847$	27252.8	+	49113.3i	1.10557	+	1.99239i
$848$		0			0
$849$	16919.9			0.683970
$850$		0			0
$851$		−	17816.3i		−	0.717667i
$852$		0			0
$853$		−	23267.3i		−	0.933948i	−0.884271	−	0.466974i	$-0.845344\pi$
$853$		−	23267.3i		−	0.933948i	0.884271	−	0.466974i	$-0.154656\pi$
$854$		0			0
$855$		−	546.719i		−	0.0218683i
$856$		0			0
$857$			146.070i			0.00582224i	0.999996	+	0.00291112i	$0.000926639\pi$
$857$			146.070i			0.00582224i	−0.999996	+	0.00291112i	$0.999073\pi$
$858$		0			0
$859$	31628.9			1.25630			0.628151	−	0.778092i	$-0.283812\pi$
$859$	31628.9			1.25630			0.628151	+	0.778092i	$0.283812\pi$
$860$		0			0
$861$	−21062.5	+	11687.5i	−0.833690	+	0.462612i
$862$		0			0
$863$			6979.90i			0.275317i	0.990480	+	0.137659i	$0.0439577\pi$
$863$			6979.90i			0.275317i	−0.990480	+	0.137659i	$0.956042\pi$
$864$		0			0
$865$	25700.4			1.01022
$866$		0			0
$867$	−12035.4			−0.471447
$868$		0			0
$869$	−440.270			−0.0171866
$870$		0			0
$871$	−25242.8			−0.981997
$872$		0			0
$873$			3210.78i			0.124477i
$874$		0			0
$875$	−19010.3	+	10548.8i	−0.734476	+	0.407558i
$876$		0			0
$877$	−8550.80			−0.329236			−0.164618	−	0.986357i	$-0.552639\pi$
$877$	−8550.80			−0.329236			−0.164618	+	0.986357i	$0.552639\pi$
$878$		0			0
$879$			10039.6i			0.385242i
$880$		0			0
$881$		−	26867.5i		−	1.02746i	−0.857952	−	0.513729i	$-0.828264\pi$
$881$		−	26867.5i		−	1.02746i	0.857952	−	0.513729i	$-0.171736\pi$
$882$		0			0
$883$			6801.11i			0.259202i	0.991566	+	0.129601i	$0.0413697\pi$
$883$			6801.11i			0.259202i	−0.991566	+	0.129601i	$0.958630\pi$
$884$		0			0
$885$			14409.1i			0.547297i
$886$		0			0
$887$	−43046.1			−1.62948			−0.814739	−	0.579829i	$-0.803120\pi$
$887$	−43046.1			−1.62948			−0.814739	+	0.579829i	$0.803120\pi$
$888$		0			0
$889$	−30610.0	+	16985.4i	−1.15481	+	0.640800i
$890$		0			0
$891$			5350.77i			0.201187i
$892$		0			0
$893$	−1597.34			−0.0598578
$894$		0			0
$895$	−36570.8			−1.36584
$896$		0			0
$897$	−28043.8			−1.04388
$898$		0			0
$899$	−23543.5			−0.873437
$900$		0			0
$901$			3533.72i			0.130661i
$902$		0			0
$903$	10565.8	−	5862.95i	0.389378	−	0.216065i
$904$		0			0
$905$	27389.7			1.00604
$906$		0			0
$907$			1908.15i			0.0698557i	0.999390	+	0.0349278i	$0.0111201\pi$
$907$			1908.15i			0.0698557i	−0.999390	+	0.0349278i	$0.988880\pi$
$908$		0			0
$909$		−	10882.0i		−	0.397066i
$910$		0			0
$911$			4076.86i			0.148268i	0.997248	+	0.0741341i	$0.0236193\pi$
$911$			4076.86i			0.148268i	−0.997248	+	0.0741341i	$0.976381\pi$
$912$		0			0
$913$		−	29277.8i		−	1.06128i
$914$		0			0
$915$	−3292.50			−0.118958
$916$		0			0
$917$	−1040.80	−	1875.66i	−0.0374812	−	0.0675463i
$918$		0			0
$919$			21044.3i			0.755371i	0.925934	+	0.377686i	$0.123280\pi$
$919$			21044.3i			0.755371i	−0.925934	+	0.377686i	$0.876720\pi$
$920$		0			0
$921$	−13208.7			−0.472576
$922$		0			0
$923$	88632.0			3.16073
$924$		0			0
$925$	4886.57			0.173697
$926$		0			0
$927$	16795.7			0.595083
$928$		0			0
$929$			6829.17i			0.241182i	0.992702	+	0.120591i	$0.0384789\pi$
$929$			6829.17i			0.241182i	−0.992702	+	0.120591i	$0.961521\pi$
$930$		0			0
$931$	−882.697	+	1415.44i	−0.0310733	+	0.0498273i
$932$		0			0
$933$	12538.0			0.439952
$934$		0			0
$935$			24770.1i			0.866384i
$936$		0			0
$937$		−	17036.5i		−	0.593980i	−0.954881	−	0.296990i	$-0.904017\pi$
$937$		−	17036.5i		−	0.593980i	0.954881	−	0.296990i	$-0.0959827\pi$
$938$		0			0
$939$			23155.6i			0.804743i
$940$		0			0
$941$			33692.0i			1.16719i	0.812044	+	0.583596i	$0.198355\pi$
$941$			33692.0i			1.16719i	−0.812044	+	0.583596i	$0.801645\pi$
$942$		0			0
$943$	−49029.7			−1.69314
$944$		0			0
$945$	5461.47	−	3030.55i	0.188002	−	0.104322i
$946$		0			0
$947$			7828.40i			0.268626i	0.990939	+	0.134313i	$0.0428827\pi$
$947$			7828.40i			0.268626i	−0.990939	+	0.134313i	$0.957117\pi$
$948$		0			0
$949$	−19766.7			−0.676139
$950$		0			0
$951$	9695.33			0.330592
$952$		0			0
$953$	−17284.0			−0.587495			−0.293748	−	0.955883i	$-0.594903\pi$
$953$	−17284.0			−0.587495			−0.293748	+	0.955883i	$0.594903\pi$
$954$		0			0
$955$	39228.1			1.32921
$956$		0			0
$957$			46298.3i			1.56386i
$958$		0			0
$959$	−5832.37	−	10510.7i	−0.196389	−	0.353920i
$960$		0			0
$961$	−19635.1			−0.659096
$962$		0			0
$963$			3222.13i			0.107821i
$964$		0			0
$965$		−	53538.4i		−	1.78597i
$966$		0			0
$967$			17034.8i			0.566497i	0.959047	+	0.283248i	$0.0914121\pi$
$967$			17034.8i			0.566497i	−0.959047	+	0.283248i	$0.908588\pi$
$968$		0			0
$969$		−	437.990i		−	0.0145204i
$970$		0			0
$971$	16317.6			0.539296			0.269648	−	0.962959i	$-0.413093\pi$
$971$	16317.6			0.539296			0.269648	+	0.962959i	$0.413093\pi$
$972$		0			0
$973$	−20162.0	−	36334.6i	−0.664299	−	1.19716i
$974$		0			0
$975$		−	7691.73i		−	0.252649i
$976$		0			0
$977$	17314.0			0.566963			0.283481	−	0.958978i	$-0.408511\pi$
$977$	17314.0			0.566963			0.283481	+	0.958978i	$0.408511\pi$
$978$		0			0
$979$	7482.94			0.244286
$980$		0			0
$981$	414.720			0.0134974
$982$		0			0
$983$	−7069.16			−0.229371			−0.114685	−	0.993402i	$-0.536586\pi$
$983$	−7069.16			−0.229371			−0.114685	+	0.993402i	$0.536586\pi$
$984$		0			0
$985$			43878.1i			1.41936i
$986$		0			0
$987$	−8854.34	−	15956.7i	−0.285549	−	0.514598i
$988$		0			0
$989$	24595.4			0.790786
$990$		0			0
$991$			39591.8i			1.26910i	0.772883	+	0.634548i	$0.218814\pi$
$991$			39591.8i			1.26910i	−0.772883	+	0.634548i	$0.781186\pi$
$992$		0			0
$993$		−	10882.1i		−	0.347768i
$994$		0			0
$995$		−	17727.2i		−	0.564814i
$996$		0			0
$997$			35779.5i			1.13656i	0.822836	+	0.568278i	$0.192390\pi$
$997$			35779.5i			1.13656i	−0.822836	+	0.568278i	$0.807610\pi$
$998$		0			0
$999$	4253.55			0.134711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.b.i.895.7		24
4.3	odd	2		1344.4.b.j.895.7		24
7.6	odd	2		1344.4.b.j.895.18		24
8.3	odd	2		672.4.b.a.223.18	yes	24
8.5	even	2		672.4.b.b.223.18	yes	24
28.27	even	2	inner	1344.4.b.i.895.18		24
56.13	odd	2		672.4.b.a.223.7	✓	24
56.27	even	2		672.4.b.b.223.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.b.a.223.7	✓	24	56.13	odd	2
672.4.b.a.223.18	yes	24	8.3	odd	2
672.4.b.b.223.7	yes	24	56.27	even	2
672.4.b.b.223.18	yes	24	8.5	even	2
1344.4.b.i.895.7		24	1.1	even	1	trivial
1344.4.b.i.895.18		24	28.27	even	2	inner
1344.4.b.j.895.7		24	4.3	odd	2
1344.4.b.j.895.18		24	7.6	odd	2

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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -12.4907i q^{5} +(-8.98608 - 16.1941i) q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -12.4907i q^{5} +(-8.98608 - 16.1941i) q^{7} +9.00000 q^{9} +66.0589i q^{11} -82.6584i q^{13} +37.4722i q^{15} +30.0199i q^{17} +4.86333 q^{19} +(26.9583 + 48.5824i) q^{21} +113.091i q^{23} -31.0182 q^{25} -27.0000 q^{27} -233.621 q^{29} +100.776 q^{31} -198.177i q^{33} +(-202.277 + 112.243i) q^{35} -157.539 q^{37} +247.975i q^{39} +433.541i q^{41} -217.482i q^{43} -112.417i q^{45} -328.446 q^{47} +(-181.501 + 291.044i) q^{49} -90.0596i q^{51} +117.713 q^{53} +825.124 q^{55} -14.5900 q^{57} +384.529 q^{59} +87.8653i q^{61} +(-80.8748 - 145.747i) q^{63} -1032.46 q^{65} -305.387i q^{67} -339.274i q^{69} +1072.27i q^{71} -239.138i q^{73} +93.0545 q^{75} +(1069.77 - 593.611i) q^{77} +6.66481i q^{79} +81.0000 q^{81} -443.207 q^{83} +374.970 q^{85} +700.864 q^{87} -113.277i q^{89} +(-1338.58 + 742.775i) q^{91} -302.329 q^{93} -60.7465i q^{95} +356.754i q^{97} +594.530i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 72 q^{3} + 20 q^{7} + 216 q^{9}+O(q^{10}) \) 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

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Database

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