LMFDB - Embedded newform 1344.4.p.c.223.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(223,1344)]

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, 1, 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("1344.223");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.p (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			223.9
Character	$\chi$	$=$	1344.223
Dual form			1344.4.p.c.223.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +20.7003 q^{5} +(11.8166 - 14.2607i) q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} +20.7003 q^{5} +(11.8166 - 14.2607i) q^{7} -9.00000 q^{9} +39.2814 q^{11} -36.2449 q^{13} -62.1010i q^{15} +92.2048i q^{17} -87.5184i q^{19} +(-42.7820 - 35.4499i) q^{21} +100.990i q^{23} +303.504 q^{25} +27.0000i q^{27} +36.2034i q^{29} +264.306 q^{31} -117.844i q^{33} +(244.609 - 295.201i) q^{35} -229.534i q^{37} +108.735i q^{39} -337.096i q^{41} +368.528 q^{43} -186.303 q^{45} -314.672 q^{47} +(-63.7338 - 337.027i) q^{49} +276.614 q^{51} -108.563i q^{53} +813.139 q^{55} -262.555 q^{57} +185.373i q^{59} -163.504 q^{61} +(-106.350 + 128.346i) q^{63} -750.282 q^{65} +458.260 q^{67} +302.969 q^{69} +584.527i q^{71} +1099.24i q^{73} -910.512i q^{75} +(464.175 - 560.180i) q^{77} -438.627i q^{79} +81.0000 q^{81} +722.114i q^{83} +1908.67i q^{85} +108.610 q^{87} -452.334i q^{89} +(-428.293 + 516.877i) q^{91} -792.917i q^{93} -1811.66i q^{95} -598.000i q^{97} -353.533 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 288 q^{9}+O(q^{10}) $ 32 * q - 288 * q^9	$ 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) $ 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

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
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	20.7003			1.85149			0.925747	−	0.378143i	$-0.123437\pi$
$5$	20.7003			1.85149			0.925747	+	0.378143i	$0.123437\pi$
$6$		0			0
$7$	11.8166	−	14.2607i	0.638039	−	0.770004i
$7$	11.8166	−	14.2607i	0.638039	−	0.770004i
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	39.2814			1.07671			0.538354	−	0.842719i	$-0.319046\pi$
$11$	39.2814			1.07671			0.538354	+	0.842719i	$0.319046\pi$
$12$		0			0
$13$	−36.2449			−0.773271			−0.386636	−	0.922233i	$-0.626363\pi$
$13$	−36.2449			−0.773271			−0.386636	+	0.922233i	$0.626363\pi$
$14$		0			0
$15$		−	62.1010i		−	1.06896i
$16$		0			0
$17$			92.2048i			1.31547i	0.753251	+	0.657733i	$0.228485\pi$
$17$			92.2048i			1.31547i	−0.753251	+	0.657733i	$0.771515\pi$
$18$		0			0
$19$		−	87.5184i		−	1.05674i	−0.849013	−	0.528371i	$-0.822803\pi$
$19$		−	87.5184i		−	1.05674i	0.849013	−	0.528371i	$-0.177197\pi$
$20$		0			0
$21$	−42.7820	−	35.4499i	−0.444562	−	0.368372i
$22$		0			0
$23$			100.990i			0.915558i	0.889066	+	0.457779i	$0.151355\pi$
$23$			100.990i			0.915558i	−0.889066	+	0.457779i	$0.848645\pi$
$24$		0			0
$25$	303.504			2.42803
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$			36.2034i			0.231821i	0.993260	+	0.115911i	$0.0369786\pi$
$29$			36.2034i			0.231821i	−0.993260	+	0.115911i	$0.963021\pi$
$30$		0			0
$31$	264.306			1.53131			0.765656	−	0.643250i	$-0.222414\pi$
$31$	264.306			1.53131			0.765656	+	0.643250i	$0.222414\pi$
$32$		0			0
$33$		−	117.844i		−	0.621638i
$34$		0			0
$35$	244.609	−	295.201i	1.18133	−	1.42566i
$36$		0			0
$37$		−	229.534i		−	1.01987i	−0.860213	−	0.509936i	$-0.829669\pi$
$37$		−	229.534i		−	1.01987i	0.860213	−	0.509936i	$-0.170331\pi$
$38$		0			0
$39$			108.735i			0.446448i
$40$		0			0
$41$		−	337.096i		−	1.28404i	−0.766688	−	0.642019i	$-0.778097\pi$
$41$		−	337.096i		−	1.28404i	0.766688	−	0.642019i	$-0.221903\pi$
$42$		0			0
$43$	368.528			1.30697			0.653487	−	0.756937i	$-0.273305\pi$
$43$	368.528			1.30697			0.653487	+	0.756937i	$0.273305\pi$
$44$		0			0
$45$	−186.303			−0.617165
$46$		0			0
$47$	−314.672			−0.976588			−0.488294	−	0.872679i	$-0.662381\pi$
$47$	−314.672			−0.976588			−0.488294	+	0.872679i	$0.662381\pi$
$48$		0			0
$49$	−63.7338	−	337.027i	−0.185813	−	0.982585i
$50$		0			0
$51$	276.614			0.759485
$52$		0			0
$53$		−	108.563i		−	0.281362i	−0.990055	−	0.140681i	$-0.955071\pi$
$53$		−	108.563i		−	0.281362i	0.990055	−	0.140681i	$-0.0449293\pi$
$54$		0			0
$55$	813.139			1.99352
$56$		0			0
$57$	−262.555			−0.610110
$58$		0			0
$59$			185.373i			0.409043i	0.978862	+	0.204521i	$0.0655638\pi$
$59$			185.373i			0.409043i	−0.978862	+	0.204521i	$0.934436\pi$
$60$		0			0
$61$	−163.504			−0.343189			−0.171594	−	0.985168i	$-0.554892\pi$
$61$	−163.504			−0.343189			−0.171594	+	0.985168i	$0.554892\pi$
$62$		0			0
$63$	−106.350	+	128.346i	−0.212680	+	0.256668i
$64$		0			0
$65$	−750.282			−1.43171
$66$		0			0
$67$	458.260			0.835603			0.417802	−	0.908538i	$-0.362801\pi$
$67$	458.260			0.835603			0.417802	+	0.908538i	$0.362801\pi$
$68$		0			0
$69$	302.969			0.528597
$70$		0			0
$71$			584.527i			0.977051i	0.872550	+	0.488525i	$0.162465\pi$
$71$			584.527i			0.977051i	−0.872550	+	0.488525i	$0.837535\pi$
$72$		0			0
$73$			1099.24i			1.76241i	0.472736	+	0.881204i	$0.343266\pi$
$73$			1099.24i			1.76241i	−0.472736	+	0.881204i	$0.656734\pi$
$74$		0			0
$75$		−	910.512i		−	1.40183i
$76$		0			0
$77$	464.175	−	560.180i	0.686982	−	0.829070i
$78$		0			0
$79$		−	438.627i		−	0.624676i	−0.949971	−	0.312338i	$-0.898888\pi$
$79$		−	438.627i		−	0.624676i	0.949971	−	0.312338i	$-0.101112\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			722.114i			0.954967i	0.878641	+	0.477484i	$0.158451\pi$
$83$			722.114i			0.954967i	−0.878641	+	0.477484i	$0.841549\pi$
$84$		0			0
$85$			1908.67i			2.43558i
$86$		0			0
$87$	108.610			0.133842
$88$		0			0
$89$		−	452.334i		−	0.538734i	−0.963038	−	0.269367i	$-0.913186\pi$
$89$		−	452.334i		−	0.538734i	0.963038	−	0.269367i	$-0.0868144\pi$
$90$		0			0
$91$	−428.293	+	516.877i	−0.493377	+	0.595422i
$92$		0			0
$93$		−	792.917i		−	0.884103i
$94$		0			0
$95$		−	1811.66i		−	1.95655i
$96$		0			0
$97$		−	598.000i		−	0.625956i	−0.949760	−	0.312978i	$-0.898673\pi$
$97$		−	598.000i		−	0.625956i	0.949760	−	0.312978i	$-0.101327\pi$
$98$		0			0
$99$	−353.533			−0.358903
$100$		0			0
$101$	1415.04			1.39407			0.697037	−	0.717035i	$-0.254501\pi$
$101$	1415.04			1.39407			0.697037	+	0.717035i	$0.254501\pi$
$102$		0			0
$103$	1138.86			1.08947			0.544733	−	0.838609i	$-0.316631\pi$
$103$	1138.86			1.08947			0.544733	+	0.838609i	$0.316631\pi$
$104$		0			0
$105$	−885.602	−	733.826i	−0.823104	−	0.682039i
$106$		0			0
$107$	−1503.38			−1.35829			−0.679147	−	0.734002i	$-0.737650\pi$
$107$	−1503.38			−1.35829			−0.679147	+	0.734002i	$0.737650\pi$
$108$		0			0
$109$		−	359.283i		−	0.315716i	−0.987462	−	0.157858i	$-0.949541\pi$
$109$		−	359.283i		−	0.315716i	0.987462	−	0.157858i	$-0.0504589\pi$
$110$		0			0
$111$	−688.603			−0.588823
$112$		0			0
$113$	−2363.20			−1.96736			−0.983679	−	0.179932i	$-0.942412\pi$
$113$	−2363.20			−1.96736			−0.983679	+	0.179932i	$0.942412\pi$
$114$		0			0
$115$			2090.52i			1.69515i
$116$		0			0
$117$	326.204			0.257757
$118$		0			0
$119$	1314.90	+	1089.55i	1.01291	+	0.839319i
$120$		0			0
$121$	212.030			0.159301
$122$		0			0
$123$	−1011.29			−0.741340
$124$		0			0
$125$	3695.09			2.64399
$126$		0			0
$127$		−	1620.69i		−	1.13238i	−0.824274	−	0.566192i	$-0.808416\pi$
$127$		−	1620.69i		−	1.13238i	0.824274	−	0.566192i	$-0.191584\pi$
$128$		0			0
$129$		−	1105.58i		−	0.754582i
$130$		0			0
$131$			503.229i			0.335629i	0.985819	+	0.167814i	$0.0536709\pi$
$131$			503.229i			0.335629i	−0.985819	+	0.167814i	$0.946329\pi$
$132$		0			0
$133$	−1248.07	−	1034.17i	−0.813696	−	0.674243i
$134$		0			0
$135$			558.909i			0.356320i
$136$		0			0
$137$	−577.769			−0.360308			−0.180154	−	0.983638i	$-0.557660\pi$
$137$	−577.769			−0.360308			−0.180154	+	0.983638i	$0.557660\pi$
$138$		0			0
$139$		−	412.449i		−	0.251680i	−0.992051	−	0.125840i	$-0.959837\pi$
$139$		−	412.449i		−	0.251680i	0.992051	−	0.125840i	$-0.0401625\pi$
$140$		0			0
$141$			944.017i			0.563834i
$142$		0			0
$143$	−1423.75			−0.832588
$144$		0			0
$145$			749.423i			0.429215i
$146$		0			0
$147$	−1011.08	+	191.201i	−0.567296	+	0.107279i
$148$		0			0
$149$			2985.68i			1.64158i	0.571227	+	0.820792i	$0.306468\pi$
$149$			2985.68i			1.64158i	−0.571227	+	0.820792i	$0.693532\pi$
$150$		0			0
$151$		−	2788.11i		−	1.50261i	−0.659957	−	0.751303i	$-0.729426\pi$
$151$		−	2788.11i		−	1.50261i	0.659957	−	0.751303i	$-0.270574\pi$
$152$		0			0
$153$		−	829.843i		−	0.438489i
$154$		0			0
$155$	5471.21			2.83522
$156$		0			0
$157$	1616.65			0.821801			0.410900	−	0.911680i	$-0.365214\pi$
$157$	1616.65			0.821801			0.410900	+	0.911680i	$0.365214\pi$
$158$		0			0
$159$	−325.688			−0.162445
$160$		0			0
$161$	1440.18	+	1193.36i	0.704983	+	0.584161i
$162$		0			0
$163$	1934.76			0.929708			0.464854	−	0.885387i	$-0.346107\pi$
$163$	1934.76			0.929708			0.464854	+	0.885387i	$0.346107\pi$
$164$		0			0
$165$		−	2439.42i		−	1.15096i
$166$		0			0
$167$	816.453			0.378318			0.189159	−	0.981947i	$-0.439424\pi$
$167$	816.453			0.378318			0.189159	+	0.981947i	$0.439424\pi$
$168$		0			0
$169$	−883.307			−0.402051
$170$		0			0
$171$			787.666i			0.352247i
$172$		0			0
$173$	−3644.18			−1.60151			−0.800757	−	0.598989i	$-0.795569\pi$
$173$	−3644.18			−1.60151			−0.800757	+	0.598989i	$0.795569\pi$
$174$		0			0
$175$	3586.40	−	4328.17i	1.54918	−	1.86959i
$176$		0			0
$177$	556.119			0.236161
$178$		0			0
$179$	−3415.80			−1.42631			−0.713154	−	0.701008i	$-0.752734\pi$
$179$	−3415.80			−1.42631			−0.713154	+	0.701008i	$0.752734\pi$
$180$		0			0
$181$	−3513.95			−1.44304			−0.721519	−	0.692395i	$-0.756556\pi$
$181$	−3513.95			−1.44304			−0.721519	+	0.692395i	$0.756556\pi$
$182$		0			0
$183$			490.512i			0.198140i
$184$		0			0
$185$		−	4751.44i		−	1.88829i
$186$		0			0
$187$			3621.93i			1.41637i
$188$		0			0
$189$	385.038	+	319.049i	0.148187	+	0.122791i
$190$		0			0
$191$		−	3058.15i		−	1.15854i	−0.815137	−	0.579268i	$-0.803339\pi$
$191$		−	3058.15i		−	1.15854i	0.815137	−	0.579268i	$-0.196661\pi$
$192$		0			0
$193$	−1894.89			−0.706721			−0.353361	−	0.935487i	$-0.614961\pi$
$193$	−1894.89			−0.706721			−0.353361	+	0.935487i	$0.614961\pi$
$194$		0			0
$195$			2250.85i			0.826597i
$196$		0			0
$197$		−	109.557i		−	0.0396224i	−0.999804	−	0.0198112i	$-0.993693\pi$
$197$		−	109.557i		−	0.0396224i	0.999804	−	0.0198112i	$-0.00630651\pi$
$198$		0			0
$199$	1525.71			0.543492			0.271746	−	0.962369i	$-0.412399\pi$
$199$	1525.71			0.543492			0.271746	+	0.962369i	$0.412399\pi$
$200$		0			0
$201$		−	1374.78i		−	0.482436i
$202$		0			0
$203$	516.286	+	427.803i	0.178503	+	0.147911i
$204$		0			0
$205$		−	6978.01i		−	2.37739i
$206$		0			0
$207$		−	908.908i		−	0.305186i
$208$		0			0
$209$		−	3437.85i		−	1.13780i
$210$		0			0
$211$	5626.27			1.83568			0.917841	−	0.396949i	$-0.129931\pi$
$211$	5626.27			1.83568			0.917841	+	0.396949i	$0.129931\pi$
$212$		0			0
$213$	1753.58			0.564101
$214$		0			0
$215$	7628.64			2.41986
$216$		0			0
$217$	3123.21	−	3769.18i	0.977037	−	1.17912i
$218$		0			0
$219$	3297.71			1.01753
$220$		0			0
$221$		−	3341.95i		−	1.01721i
$222$		0			0
$223$	−1409.96			−0.423398			−0.211699	−	0.977335i	$-0.567900\pi$
$223$	−1409.96			−0.423398			−0.211699	+	0.977335i	$0.567900\pi$
$224$		0			0
$225$	−2731.54			−0.809344
$226$		0			0
$227$		−	848.622i		−	0.248128i	−0.992274	−	0.124064i	$-0.960407\pi$
$227$		−	848.622i		−	0.248128i	0.992274	−	0.124064i	$-0.0395928\pi$
$228$		0			0
$229$	5880.63			1.69696			0.848479	−	0.529230i	$-0.177519\pi$
$229$	5880.63			1.69696			0.848479	+	0.529230i	$0.177519\pi$
$230$		0			0
$231$	−1680.54	−	1392.52i	−0.478664	−	0.396629i
$232$		0			0
$233$	−3101.62			−0.872077			−0.436038	−	0.899928i	$-0.643619\pi$
$233$	−3101.62			−0.872077			−0.436038	+	0.899928i	$0.643619\pi$
$234$		0			0
$235$	−6513.82			−1.80815
$236$		0			0
$237$	−1315.88			−0.360657
$238$		0			0
$239$			6225.18i			1.68483i	0.538831	+	0.842414i	$0.318866\pi$
$239$			6225.18i			1.68483i	−0.538831	+	0.842414i	$0.681134\pi$
$240$		0			0
$241$		−	201.092i		−	0.0537489i	−0.999639	−	0.0268744i	$-0.991445\pi$
$241$		−	201.092i		−	0.0537489i	0.999639	−	0.0268744i	$-0.00855543\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−1319.31	−	6976.57i	−0.344031	−	1.81925i
$246$		0			0
$247$			3172.10i			0.817149i
$248$		0			0
$249$	2166.34			0.551351
$250$		0			0
$251$		−	4561.95i		−	1.14720i	−0.819135	−	0.573601i	$-0.805546\pi$
$251$		−	4561.95i		−	1.14720i	0.819135	−	0.573601i	$-0.194454\pi$
$252$		0			0
$253$			3967.02i			0.985789i
$254$		0			0
$255$	5726.01			1.40618
$256$		0			0
$257$		−	1939.24i		−	0.470687i	−0.971912	−	0.235343i	$-0.924379\pi$
$257$		−	1939.24i		−	0.470687i	0.971912	−	0.235343i	$-0.0756215\pi$
$258$		0			0
$259$	−3273.32	−	2712.33i	−0.785305	−	0.650718i
$260$		0			0
$261$		−	325.831i		−	0.0772737i
$262$		0			0
$263$		−	1244.55i		−	0.291796i	−0.989300	−	0.145898i	$-0.953393\pi$
$263$		−	1244.55i		−	0.291796i	0.989300	−	0.145898i	$-0.0466072\pi$
$264$		0			0
$265$		−	2247.28i		−	0.520941i
$266$		0			0
$267$	−1357.00			−0.311038
$268$		0			0
$269$	188.083			0.0426305			0.0213153	−	0.999773i	$-0.493215\pi$
$269$	188.083			0.0426305			0.0213153	+	0.999773i	$0.493215\pi$
$270$		0			0
$271$	504.590			0.113106			0.0565529	−	0.998400i	$-0.481989\pi$
$271$	504.590			0.113106			0.0565529	+	0.998400i	$0.481989\pi$
$272$		0			0
$273$	1550.63	+	1284.88i	0.343767	+	0.284851i
$274$		0			0
$275$	11922.1			2.61428
$276$		0			0
$277$		−	4147.02i		−	0.899532i	−0.893146	−	0.449766i	$-0.851507\pi$
$277$		−	4147.02i		−	0.899532i	0.893146	−	0.449766i	$-0.148493\pi$
$278$		0			0
$279$	−2378.75			−0.510437
$280$		0			0
$281$	−2018.75			−0.428572			−0.214286	−	0.976771i	$-0.568742\pi$
$281$	−2018.75			−0.428572			−0.214286	+	0.976771i	$0.568742\pi$
$282$		0			0
$283$		−	50.2649i		−	0.0105581i	−0.999986	−	0.00527905i	$-0.998320\pi$
$283$		−	50.2649i		−	0.0105581i	0.999986	−	0.00527905i	$-0.00168038\pi$
$284$		0			0
$285$	−5434.98			−1.12962
$286$		0			0
$287$	−4807.22	−	3983.35i	−0.988715	−	0.819267i
$288$		0			0
$289$	−3588.72			−0.730453
$290$		0			0
$291$	−1794.00			−0.361396
$292$		0			0
$293$	2247.98			0.448221			0.224110	−	0.974564i	$-0.428052\pi$
$293$	2247.98			0.448221			0.224110	+	0.974564i	$0.428052\pi$
$294$		0			0
$295$			3837.29i			0.757340i
$296$		0			0
$297$			1060.60i			0.207213i
$298$		0			0
$299$		−	3660.36i		−	0.707974i
$300$		0			0
$301$	4354.76	−	5255.45i	0.833901	−	1.00638i
$302$		0			0
$303$		−	4245.11i		−	0.804869i
$304$		0			0
$305$	−3384.59			−0.635413
$306$		0			0
$307$			1759.91i			0.327177i	0.986529	+	0.163588i	$0.0523069\pi$
$307$			1759.91i			0.327177i	−0.986529	+	0.163588i	$0.947693\pi$
$308$		0			0
$309$		−	3416.57i		−	0.629004i
$310$		0			0
$311$	8517.10			1.55293			0.776464	−	0.630162i	$-0.217012\pi$
$311$	8517.10			1.55293			0.776464	+	0.630162i	$0.217012\pi$
$312$		0			0
$313$		−	10113.5i		−	1.82636i	−0.407556	−	0.913180i	$-0.633619\pi$
$313$		−	10113.5i		−	1.82636i	0.407556	−	0.913180i	$-0.366381\pi$
$314$		0			0
$315$	−2201.48	+	2656.81i	−0.393775	+	0.475220i
$316$		0			0
$317$			6511.01i			1.15361i	0.816882	+	0.576805i	$0.195701\pi$
$317$			6511.01i			1.15361i	−0.816882	+	0.576805i	$0.804299\pi$
$318$		0			0
$319$			1422.12i			0.249604i
$320$		0			0
$321$			4510.15i			0.784212i
$322$		0			0
$323$	8069.62			1.39011
$324$		0			0
$325$	−11000.5			−1.87753
$326$		0			0
$327$	−1077.85			−0.182279
$328$		0			0
$329$	−3718.37	+	4487.44i	−0.623101	+	0.751977i
$330$		0			0
$331$	−2031.11			−0.337280			−0.168640	−	0.985678i	$-0.553938\pi$
$331$	−2031.11			−0.337280			−0.168640	+	0.985678i	$0.553938\pi$
$332$		0			0
$333$			2065.81i			0.339957i
$334$		0			0
$335$	9486.15			1.54712
$336$		0			0
$337$	−9702.01			−1.56826			−0.784128	−	0.620599i	$-0.786889\pi$
$337$	−9702.01			−1.56826			−0.784128	+	0.620599i	$0.786889\pi$
$338$		0			0
$339$			7089.61i			1.13585i
$340$		0			0
$341$	10382.3			1.64878
$342$		0			0
$343$	−5559.35	−	3073.64i	−0.875150	−	0.483851i
$344$		0			0
$345$	6271.57			0.978695
$346$		0			0
$347$	−4652.17			−0.719716			−0.359858	−	0.933007i	$-0.617175\pi$
$347$	−4652.17			−0.719716			−0.359858	+	0.933007i	$0.617175\pi$
$348$		0			0
$349$	5507.67			0.844753			0.422376	−	0.906421i	$-0.361196\pi$
$349$	5507.67			0.844753			0.422376	+	0.906421i	$0.361196\pi$
$350$		0			0
$351$		−	978.612i		−	0.148816i
$352$		0			0
$353$			8066.25i			1.21621i	0.793856	+	0.608106i	$0.208071\pi$
$353$			8066.25i			1.21621i	−0.793856	+	0.608106i	$0.791929\pi$
$354$		0			0
$355$			12099.9i			1.80900i
$356$		0			0
$357$	3268.65	−	3944.71i	0.484581	−	0.584807i
$358$		0			0
$359$			2334.87i			0.343258i	0.985162	+	0.171629i	$0.0549030\pi$
$359$			2334.87i			0.343258i	−0.985162	+	0.171629i	$0.945097\pi$
$360$		0			0
$361$	−800.476			−0.116704
$362$		0			0
$363$		−	636.090i		−	0.0919726i
$364$		0			0
$365$			22754.6i			3.26309i
$366$		0			0
$367$	−8870.84			−1.26173			−0.630864	−	0.775894i	$-0.717299\pi$
$367$	−8870.84			−1.26173			−0.630864	+	0.775894i	$0.717299\pi$
$368$		0			0
$369$			3033.87i			0.428013i
$370$		0			0
$371$	−1548.17	−	1282.84i	−0.216650	−	0.179520i
$372$		0			0
$373$			8302.86i			1.15256i	0.817251	+	0.576281i	$0.195497\pi$
$373$			8302.86i			1.15256i	−0.817251	+	0.576281i	$0.804503\pi$
$374$		0			0
$375$		−	11085.3i		−	1.52651i
$376$		0			0
$377$		−	1312.19i		−	0.179261i
$378$		0			0
$379$	−530.766			−0.0719357			−0.0359679	−	0.999353i	$-0.511451\pi$
$379$	−530.766			−0.0719357			−0.0359679	+	0.999353i	$0.511451\pi$
$380$		0			0
$381$	−4862.06			−0.653782
$382$		0			0
$383$	534.478			0.0713069			0.0356535	−	0.999364i	$-0.488649\pi$
$383$	534.478			0.0713069			0.0356535	+	0.999364i	$0.488649\pi$
$384$		0			0
$385$	9608.57	−	11595.9i	1.27194	−	1.53502i
$386$		0			0
$387$	−3316.75			−0.435658
$388$		0			0
$389$			11253.1i			1.46672i	0.679842	+	0.733358i	$0.262048\pi$
$389$			11253.1i			1.46672i	−0.679842	+	0.733358i	$0.737952\pi$
$390$		0			0
$391$	−9311.74			−1.20439
$392$		0			0
$393$	1509.69			0.193775
$394$		0			0
$395$		−	9079.73i		−	1.15658i
$396$		0			0
$397$	8896.66			1.12471			0.562356	−	0.826895i	$-0.309895\pi$
$397$	8896.66			1.12471			0.562356	+	0.826895i	$0.309895\pi$
$398$		0			0
$399$	−3102.52	+	3744.22i	−0.389274	+	0.469788i
$400$		0			0
$401$	6210.10			0.773360			0.386680	−	0.922214i	$-0.373622\pi$
$401$	6210.10			0.773360			0.386680	+	0.922214i	$0.373622\pi$
$402$		0			0
$403$	−9579.73			−1.18412
$404$		0			0
$405$	1676.73			0.205722
$406$		0			0
$407$		−	9016.44i		−	1.09810i
$408$		0			0
$409$			3899.47i			0.471434i	0.971822	+	0.235717i	$0.0757439\pi$
$409$			3899.47i			0.471434i	−0.971822	+	0.235717i	$0.924256\pi$
$410$		0			0
$411$			1733.31i			0.208024i
$412$		0			0
$413$	2643.54	+	2190.49i	0.314965	+	0.260985i
$414$		0			0
$415$			14948.0i			1.76812i
$416$		0			0
$417$	−1237.35			−0.145307
$418$		0			0
$419$			12136.9i			1.41510i	0.706663	+	0.707550i	$0.250200\pi$
$419$			12136.9i			1.41510i	−0.706663	+	0.707550i	$0.749800\pi$
$420$		0			0
$421$			5388.65i			0.623817i	0.950112	+	0.311908i	$0.100968\pi$
$421$			5388.65i			0.623817i	−0.950112	+	0.311908i	$0.899032\pi$
$422$		0			0
$423$	2832.05			0.325529
$424$		0			0
$425$			27984.5i			3.19400i
$426$		0			0
$427$	−1932.07	+	2331.68i	−0.218968	+	0.264257i
$428$		0			0
$429$			4271.25i			0.480695i
$430$		0			0
$431$		−	2284.44i		−	0.255308i	−0.991819	−	0.127654i	$-0.959255\pi$
$431$		−	2284.44i		−	0.255308i	0.991819	−	0.127654i	$-0.0407447\pi$
$432$		0			0
$433$			1711.79i			0.189985i	0.995478	+	0.0949925i	$0.0302827\pi$
$433$			1711.79i			0.189985i	−0.995478	+	0.0949925i	$0.969717\pi$
$434$		0			0
$435$	2248.27			0.247808
$436$		0			0
$437$	8838.47			0.967509
$438$		0			0
$439$	2820.52			0.306643			0.153322	−	0.988176i	$-0.451003\pi$
$439$	2820.52			0.306643			0.153322	+	0.988176i	$0.451003\pi$
$440$		0			0
$441$	573.604	+	3033.24i	0.0619376	+	0.327528i
$442$		0			0
$443$	13392.2			1.43630			0.718150	−	0.695888i	$-0.244989\pi$
$443$	13392.2			1.43630			0.718150	+	0.695888i	$0.244989\pi$
$444$		0			0
$445$		−	9363.47i		−	0.997463i
$446$		0			0
$447$	8957.03			0.947769
$448$		0			0
$449$	−15186.9			−1.59624			−0.798120	−	0.602498i	$-0.794172\pi$
$449$	−15186.9			−1.59624			−0.798120	+	0.602498i	$0.794172\pi$
$450$		0			0
$451$		−	13241.6i		−	1.38254i
$452$		0			0
$453$	−8364.34			−0.867530
$454$		0			0
$455$	−8865.81	+	10699.5i	−0.913485	+	1.10242i
$456$		0			0
$457$	−19132.7			−1.95840			−0.979202	−	0.202888i	$-0.934967\pi$
$457$	−19132.7			−1.95840			−0.979202	+	0.202888i	$0.934967\pi$
$458$		0			0
$459$	−2489.53			−0.253162
$460$		0			0
$461$	237.892			0.0240341			0.0120170	−	0.999928i	$-0.496175\pi$
$461$	237.892			0.0240341			0.0120170	+	0.999928i	$0.496175\pi$
$462$		0			0
$463$		−	6527.48i		−	0.655200i	−0.944816	−	0.327600i	$-0.893760\pi$
$463$		−	6527.48i		−	0.655200i	0.944816	−	0.327600i	$-0.106240\pi$
$464$		0			0
$465$		−	16413.6i		−	1.63691i
$466$		0			0
$467$		−	488.599i		−	0.0484147i	−0.999707	−	0.0242073i	$-0.992294\pi$
$467$		−	488.599i		−	0.0484147i	0.999707	−	0.0242073i	$-0.00770619\pi$
$468$		0			0
$469$	5415.10	−	6535.10i	0.533147	−	0.643418i
$470$		0			0
$471$		−	4849.95i		−	0.474467i
$472$		0			0
$473$	14476.3			1.40723
$474$		0			0
$475$		−	26562.2i		−	2.56580i
$476$		0			0
$477$			977.063i			0.0937875i
$478$		0			0
$479$	5872.67			0.560186			0.280093	−	0.959973i	$-0.409635\pi$
$479$	5872.67			0.560186			0.280093	+	0.959973i	$0.409635\pi$
$480$		0			0
$481$			8319.45i			0.788637i
$482$		0			0
$483$	3580.08	−	4320.55i	0.337266	−	0.407022i
$484$		0			0
$485$		−	12378.8i		−	1.15895i
$486$		0			0
$487$		−	14307.7i		−	1.33130i	−0.746263	−	0.665652i	$-0.768154\pi$
$487$		−	14307.7i		−	1.33130i	0.746263	−	0.665652i	$-0.231846\pi$
$488$		0			0
$489$		−	5804.29i		−	0.536767i
$490$		0			0
$491$	−10665.1			−0.980264			−0.490132	−	0.871648i	$-0.663052\pi$
$491$	−10665.1			−0.980264			−0.490132	+	0.871648i	$0.663052\pi$
$492$		0			0
$493$	−3338.13			−0.304953
$494$		0			0
$495$	−7318.25			−0.664507
$496$		0			0
$497$	8335.75	+	6907.15i	0.752333	+	0.623397i
$498$		0			0
$499$	−11905.8			−1.06809			−0.534046	−	0.845455i	$-0.679329\pi$
$499$	−11905.8			−1.06809			−0.534046	+	0.845455i	$0.679329\pi$
$500$		0			0
$501$		−	2449.36i		−	0.218422i
$502$		0			0
$503$	−11100.0			−0.983941			−0.491971	−	0.870612i	$-0.663723\pi$
$503$	−11100.0			−0.983941			−0.491971	+	0.870612i	$0.663723\pi$
$504$		0			0
$505$	29291.8			2.58112
$506$		0			0
$507$			2649.92i			0.232125i
$508$		0			0
$509$	−8580.71			−0.747216			−0.373608	−	0.927587i	$-0.621880\pi$
$509$	−8580.71			−0.747216			−0.373608	+	0.927587i	$0.621880\pi$
$510$		0			0
$511$	15675.8	+	12989.3i	1.35706	+	1.12448i
$512$		0			0
$513$	2363.00			0.203370
$514$		0			0
$515$	23574.7			2.01714
$516$		0			0
$517$	−12360.8			−1.05150
$518$		0			0
$519$			10932.5i			0.924635i
$520$		0			0
$521$		−	4351.09i		−	0.365882i	−0.983124	−	0.182941i	$-0.941438\pi$
$521$		−	4351.09i		−	0.365882i	0.983124	−	0.182941i	$-0.0585618\pi$
$522$		0			0
$523$			3685.85i			0.308166i	0.988058	+	0.154083i	$0.0492424\pi$
$523$			3685.85i			0.308166i	−0.988058	+	0.154083i	$0.950758\pi$
$524$		0			0
$525$	−12984.5	−	10759.2i	−1.07941	−	0.894419i
$526$		0			0
$527$			24370.2i			2.01439i
$528$		0			0
$529$	1968.06			0.161754
$530$		0			0
$531$		−	1668.36i		−	0.136348i
$532$		0			0
$533$			12218.0i			0.992910i
$534$		0			0
$535$	−31120.5			−2.51487
$536$		0			0
$537$			10247.4i			0.823479i
$538$		0			0
$539$	−2503.55	−	13238.9i	−0.200066	−	1.05796i
$540$		0			0
$541$		−	21536.0i		−	1.71147i	−0.517413	−	0.855736i	$-0.673105\pi$
$541$		−	21536.0i		−	1.71147i	0.517413	−	0.855736i	$-0.326895\pi$
$542$		0			0
$543$			10541.8i			0.833138i
$544$		0			0
$545$		−	7437.28i		−	0.584547i
$546$		0			0
$547$	5550.29			0.433845			0.216923	−	0.976189i	$-0.430398\pi$
$547$	5550.29			0.433845			0.216923	+	0.976189i	$0.430398\pi$
$548$		0			0
$549$	1471.54			0.114396
$550$		0			0
$551$	3168.47			0.244975
$552$		0			0
$553$	−6255.12	−	5183.10i	−0.481003	−	0.398568i
$554$		0			0
$555$	−14254.3			−1.09020
$556$		0			0
$557$			14848.3i			1.12952i	0.825256	+	0.564758i	$0.191031\pi$
$557$			14848.3i			1.12952i	−0.825256	+	0.564758i	$0.808969\pi$
$558$		0			0
$559$	−13357.2			−1.01065
$560$		0			0
$561$	10865.8			0.817744
$562$		0			0
$563$		−	1189.55i		−	0.0890469i	−0.999008	−	0.0445235i	$-0.985823\pi$
$563$		−	1189.55i		−	0.0890469i	0.999008	−	0.0445235i	$-0.0141769\pi$
$564$		0			0
$565$	−48919.1			−3.64255
$566$		0			0
$567$	957.148	−	1155.11i	0.0708932	−	0.0855560i
$568$		0			0
$569$	9572.43			0.705267			0.352634	−	0.935761i	$-0.385286\pi$
$569$	9572.43			0.705267			0.352634	+	0.935761i	$0.385286\pi$
$570$		0			0
$571$	−19487.5			−1.42825			−0.714123	−	0.700021i	$-0.753174\pi$
$571$	−19487.5			−1.42825			−0.714123	+	0.700021i	$0.753174\pi$
$572$		0			0
$573$	−9174.46			−0.668881
$574$		0			0
$575$			30650.8i			2.22300i
$576$		0			0
$577$			24751.1i			1.78579i	0.450263	+	0.892896i	$0.351330\pi$
$577$			24751.1i			1.78579i	−0.450263	+	0.892896i	$0.648670\pi$
$578$		0			0
$579$			5684.67i			0.408026i
$580$		0			0
$581$	10297.8	+	8532.96i	0.735329	+	0.609306i
$582$		0			0
$583$		−	4264.49i		−	0.302945i
$584$		0			0
$585$	6752.54			0.477236
$586$		0			0
$587$		−	15015.1i		−	1.05577i	−0.849315	−	0.527886i	$-0.822985\pi$
$587$		−	15015.1i		−	1.05577i	0.849315	−	0.527886i	$-0.177015\pi$
$588$		0			0
$589$		−	23131.6i		−	1.61820i
$590$		0			0
$591$	−328.671			−0.0228760
$592$		0			0
$593$			18080.4i			1.25206i	0.779799	+	0.626030i	$0.215321\pi$
$593$			18080.4i			1.25206i	−0.779799	+	0.626030i	$0.784679\pi$
$594$		0			0
$595$	27218.9	+	22554.1i	1.87541	+	1.55399i
$596$		0			0
$597$		−	4577.13i		−	0.313785i
$598$		0			0
$599$			18896.2i			1.28894i	0.764628	+	0.644472i	$0.222923\pi$
$599$			18896.2i			1.28894i	−0.764628	+	0.644472i	$0.777077\pi$
$600$		0			0
$601$		−	316.737i		−	0.0214974i	−0.999942	−	0.0107487i	$-0.996579\pi$
$601$		−	316.737i		−	0.0214974i	0.999942	−	0.0107487i	$-0.00342149\pi$
$602$		0			0
$603$	−4124.34			−0.278534
$604$		0			0
$605$	4389.09			0.294945
$606$		0			0
$607$	24428.2			1.63346			0.816730	−	0.577020i	$-0.195784\pi$
$607$	24428.2			1.63346			0.816730	+	0.577020i	$0.195784\pi$
$608$		0			0
$609$	1283.41	−	1548.86i	0.0853964	−	0.103059i
$610$		0			0
$611$	11405.3			0.755168
$612$		0			0
$613$			24938.5i			1.64316i	0.570092	+	0.821581i	$0.306908\pi$
$613$			24938.5i			1.64316i	−0.570092	+	0.821581i	$0.693092\pi$
$614$		0			0
$615$	−20934.0			−1.37259
$616$		0			0
$617$	−10603.6			−0.691873			−0.345936	−	0.938258i	$-0.612439\pi$
$617$	−10603.6			−0.691873			−0.345936	+	0.938258i	$0.612439\pi$
$618$		0			0
$619$		−	17013.3i		−	1.10472i	−0.833604	−	0.552362i	$-0.813727\pi$
$619$		−	17013.3i		−	1.10472i	0.833604	−	0.552362i	$-0.186273\pi$
$620$		0			0
$621$	−2726.72			−0.176199
$622$		0			0
$623$	−6450.59	−	5345.07i	−0.414827	−	0.343733i
$624$		0			0
$625$	38551.7			2.46731
$626$		0			0
$627$	−10313.5			−0.656911
$628$		0			0
$629$	21164.2			1.34161
$630$		0			0
$631$		−	2157.13i		−	0.136092i	−0.997682	−	0.0680460i	$-0.978324\pi$
$631$		−	2157.13i		−	0.136092i	0.997682	−	0.0680460i	$-0.0216765\pi$
$632$		0			0
$633$		−	16878.8i		−	1.05983i
$634$		0			0
$635$		−	33548.8i		−	2.09660i
$636$		0			0
$637$	2310.02	+	12215.5i	0.143684	+	0.759805i
$638$		0			0
$639$		−	5260.75i		−	0.325684i
$640$		0			0
$641$	−22030.9			−1.35752			−0.678758	−	0.734362i	$-0.737481\pi$
$641$	−22030.9			−1.35752			−0.678758	+	0.734362i	$0.737481\pi$
$642$		0			0
$643$			30067.6i			1.84409i	0.387081	+	0.922046i	$0.373483\pi$
$643$			30067.6i			1.84409i	−0.387081	+	0.922046i	$0.626517\pi$
$644$		0			0
$645$		−	22885.9i		−	1.39711i
$646$		0			0
$647$	6954.93			0.422607			0.211303	−	0.977421i	$-0.432229\pi$
$647$	6954.93			0.422607			0.211303	+	0.977421i	$0.432229\pi$
$648$		0			0
$649$			7281.72i			0.440420i
$650$		0			0
$651$	−11307.5	−	9369.62i	−0.680763	−	0.564092i
$652$		0			0
$653$			1199.25i			0.0718686i	0.999354	+	0.0359343i	$0.0114407\pi$
$653$			1199.25i			0.0718686i	−0.999354	+	0.0359343i	$0.988559\pi$
$654$		0			0
$655$			10417.0i			0.621415i
$656$		0			0
$657$		−	9893.12i		−	0.587469i
$658$		0			0
$659$	−13611.9			−0.804619			−0.402310	−	0.915504i	$-0.631792\pi$
$659$	−13611.9			−0.804619			−0.402310	+	0.915504i	$0.631792\pi$
$660$		0			0
$661$	−30065.6			−1.76916			−0.884582	−	0.466385i	$-0.845556\pi$
$661$	−30065.6			−1.76916			−0.884582	+	0.466385i	$0.845556\pi$
$662$		0			0
$663$	−10025.9			−0.587288
$664$		0			0
$665$	−25835.5	−	21407.8i	−1.50655	−	1.24836i
$666$		0			0
$667$	−3656.18			−0.212246
$668$		0			0
$669$			4229.88i			0.244449i
$670$		0			0
$671$	−6422.67			−0.369515
$672$		0			0
$673$	10262.1			0.587777			0.293888	−	0.955840i	$-0.405051\pi$
$673$	10262.1			0.587777			0.293888	+	0.955840i	$0.405051\pi$
$674$		0			0
$675$			8194.61i			0.467275i
$676$		0			0
$677$	−426.182			−0.0241943			−0.0120971	−	0.999927i	$-0.503851\pi$
$677$	−426.182			−0.0241943			−0.0120971	+	0.999927i	$0.503851\pi$
$678$		0			0
$679$	−8527.88	−	7066.35i	−0.481988	−	0.399384i
$680$		0			0
$681$	−2545.87			−0.143257
$682$		0			0
$683$	−2385.48			−0.133643			−0.0668213	−	0.997765i	$-0.521286\pi$
$683$	−2385.48			−0.133643			−0.0668213	+	0.997765i	$0.521286\pi$
$684$		0			0
$685$	−11960.0			−0.667108
$686$		0			0
$687$		−	17641.9i		−	0.979739i
$688$		0			0
$689$			3934.84i			0.217570i
$690$		0			0
$691$			9485.21i			0.522192i	0.965313	+	0.261096i	$0.0840839\pi$
$691$			9485.21i			0.522192i	−0.965313	+	0.261096i	$0.915916\pi$
$692$		0			0
$693$	−4177.57	+	5041.62i	−0.228994	+	0.276357i
$694$		0			0
$695$		−	8537.84i		−	0.465984i
$696$		0			0
$697$	31081.9			1.68911
$698$		0			0
$699$			9304.86i			0.503494i
$700$		0			0
$701$		−	13463.5i		−	0.725404i	−0.931905	−	0.362702i	$-0.881854\pi$
$701$		−	13463.5i		−	0.725404i	0.931905	−	0.362702i	$-0.118146\pi$
$702$		0			0
$703$	−20088.5			−1.07774
$704$		0			0
$705$			19541.5i			1.04393i
$706$		0			0
$707$	16721.0	−	20179.4i	0.889473	−	1.07344i
$708$		0			0
$709$			1482.11i			0.0785075i	0.999229	+	0.0392538i	$0.0124981\pi$
$709$			1482.11i			0.0785075i	−0.999229	+	0.0392538i	$0.987502\pi$
$710$		0			0
$711$			3947.65i			0.208225i
$712$		0			0
$713$			26692.2i			1.40200i
$714$		0			0
$715$	−29472.1			−1.54153
$716$		0			0
$717$	18675.6			0.972735
$718$		0			0
$719$	−33887.8			−1.75772			−0.878862	−	0.477077i	$-0.841696\pi$
$719$	−33887.8			−1.75772			−0.878862	+	0.477077i	$0.841696\pi$
$720$		0			0
$721$	13457.5	−	16240.9i	0.695122	−	0.838893i
$722$		0			0
$723$	−603.276			−0.0310319
$724$		0			0
$725$			10987.9i			0.562869i
$726$		0			0
$727$	−9840.41			−0.502009			−0.251004	−	0.967986i	$-0.580761\pi$
$727$	−9840.41			−0.502009			−0.251004	+	0.967986i	$0.580761\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			33980.0i			1.71928i
$732$		0			0
$733$	13241.0			0.667212			0.333606	−	0.942713i	$-0.391735\pi$
$733$	13241.0			0.667212			0.333606	+	0.942713i	$0.391735\pi$
$734$		0			0
$735$	−20929.7	+	3957.93i	−1.05035	+	0.198627i
$736$		0			0
$737$	18001.1			0.899701
$738$		0			0
$739$	−6185.80			−0.307914			−0.153957	−	0.988078i	$-0.549202\pi$
$739$	−6185.80			−0.307914			−0.153957	+	0.988078i	$0.549202\pi$
$740$		0			0
$741$	9516.29			0.471781
$742$		0			0
$743$		−	11911.6i		−	0.588148i	−0.955783	−	0.294074i	$-0.904989\pi$
$743$		−	11911.6i		−	0.588148i	0.955783	−	0.294074i	$-0.0950112\pi$
$744$		0			0
$745$			61804.5i			3.03939i
$746$		0			0
$747$		−	6499.02i		−	0.318322i
$748$		0			0
$749$	−17764.9	+	21439.3i	−0.866645	+	1.04589i
$750$		0			0
$751$		−	28902.1i		−	1.40433i	−0.712015	−	0.702165i	$-0.752217\pi$
$751$		−	28902.1i		−	1.40433i	0.712015	−	0.702165i	$-0.247783\pi$
$752$		0			0
$753$	−13685.8			−0.662337
$754$		0			0
$755$		−	57714.9i		−	2.78207i
$756$		0			0
$757$			7827.05i			0.375798i	0.982188	+	0.187899i	$0.0601678\pi$
$757$			7827.05i			0.375798i	−0.982188	+	0.187899i	$0.939832\pi$
$758$		0			0
$759$	11901.1			0.569145
$760$		0			0
$761$			4882.67i			0.232584i	0.993215	+	0.116292i	$0.0371008\pi$
$761$			4882.67i			0.232584i	−0.993215	+	0.116292i	$0.962899\pi$
$762$		0			0
$763$	−5123.62	−	4245.52i	−0.243103	−	0.201439i
$764$		0			0
$765$		−	17178.0i		−	0.811860i
$766$		0			0
$767$		−	6718.83i		−	0.316301i
$768$		0			0
$769$			20071.4i			0.941215i	0.882343	+	0.470607i	$0.155965\pi$
$769$			20071.4i			0.941215i	−0.882343	+	0.470607i	$0.844035\pi$
$770$		0			0
$771$	−5817.72			−0.271751
$772$		0			0
$773$	4965.70			0.231053			0.115526	−	0.993304i	$-0.463145\pi$
$773$	4965.70			0.231053			0.115526	+	0.993304i	$0.463145\pi$
$774$		0			0
$775$	80217.8			3.71808
$776$		0			0
$777$	−8136.98	+	9819.95i	−0.375692	+	0.453396i
$778$		0			0
$779$	−29502.1			−1.35690
$780$		0			0
$781$			22961.1i			1.05200i
$782$		0			0
$783$	−977.493			−0.0446140
$784$		0			0
$785$	33465.2			1.52156
$786$		0			0
$787$			40092.6i			1.81594i	0.419035	+	0.907970i	$0.362369\pi$
$787$			40092.6i			1.81594i	−0.419035	+	0.907970i	$0.637631\pi$
$788$		0			0
$789$	−3733.66			−0.168469
$790$		0			0
$791$	−27925.1	+	33700.9i	−1.25525	+	1.51487i
$792$		0			0
$793$	5926.18			0.265378
$794$		0			0
$795$	−6741.84			−0.300765
$796$		0			0
$797$	26919.4			1.19640			0.598202	−	0.801345i	$-0.295882\pi$
$797$	26919.4			1.19640			0.598202	+	0.801345i	$0.295882\pi$
$798$		0			0
$799$		−	29014.3i		−	1.28467i
$800$		0			0
$801$			4071.01i			0.179578i
$802$		0			0
$803$			43179.5i			1.89760i
$804$		0			0
$805$	29812.3	+	24703.0i	1.30527	+	1.08157i
$806$		0			0
$807$		−	564.248i		−	0.0246127i
$808$		0			0
$809$	−34737.1			−1.50963			−0.754815	−	0.655937i	$-0.772274\pi$
$809$	−34737.1			−1.50963			−0.754815	+	0.655937i	$0.772274\pi$
$810$		0			0
$811$			11900.4i			0.515265i	0.966243	+	0.257632i	$0.0829423\pi$
$811$			11900.4i			0.515265i	−0.966243	+	0.257632i	$0.917058\pi$
$812$		0			0
$813$		−	1513.77i		−	0.0653017i
$814$		0			0
$815$	40050.3			1.72135
$816$		0			0
$817$		−	32252.9i		−	1.38114i
$818$		0			0
$819$	3854.64	−	4651.89i	0.164459	−	0.198474i
$820$		0			0
$821$			20274.1i			0.861841i	0.902390	+	0.430921i	$0.141811\pi$
$821$			20274.1i			0.861841i	−0.902390	+	0.430921i	$0.858189\pi$
$822$		0			0
$823$			11589.9i			0.490887i	0.969411	+	0.245443i	$0.0789335\pi$
$823$			11589.9i			0.490887i	−0.969411	+	0.245443i	$0.921066\pi$
$824$		0			0
$825$		−	35766.2i		−	1.50936i
$826$		0			0
$827$	−12796.5			−0.538064			−0.269032	−	0.963131i	$-0.586704\pi$
$827$	−12796.5			−0.538064			−0.269032	+	0.963131i	$0.586704\pi$
$828$		0			0
$829$	−14976.8			−0.627461			−0.313730	−	0.949512i	$-0.601579\pi$
$829$	−14976.8			−0.627461			−0.313730	+	0.949512i	$0.601579\pi$
$830$		0			0
$831$	−12441.1			−0.519345
$832$		0			0
$833$	31075.5	−	5876.56i	1.29256	−	0.244430i
$834$		0			0
$835$	16900.9			0.700453
$836$		0			0
$837$			7136.25i			0.294701i
$838$		0			0
$839$	2505.42			0.103095			0.0515476	−	0.998671i	$-0.483585\pi$
$839$	2505.42			0.103095			0.0515476	+	0.998671i	$0.483585\pi$
$840$		0			0
$841$	23078.3			0.946259
$842$		0			0
$843$			6056.25i			0.247436i
$844$		0			0
$845$	−18284.8			−0.744396
$846$		0			0
$847$	2505.48	−	3023.69i	0.101640	−	0.122663i
$848$		0			0
$849$	−150.795			−0.00609572
$850$		0			0
$851$	23180.6			0.933751
$852$		0			0
$853$	28769.0			1.15479			0.577393	−	0.816467i	$-0.304070\pi$
$853$	28769.0			1.15479			0.577393	+	0.816467i	$0.304070\pi$
$854$		0			0
$855$			16305.0i			0.652184i
$856$		0			0
$857$			32617.6i			1.30011i	0.759887	+	0.650056i	$0.225254\pi$
$857$			32617.6i			1.30011i	−0.759887	+	0.650056i	$0.774746\pi$
$858$		0			0
$859$			12072.8i			0.479532i	0.970831	+	0.239766i	$0.0770708\pi$
$859$			12072.8i			0.479532i	−0.970831	+	0.239766i	$0.922929\pi$
$860$		0			0
$861$	−11950.0	+	14421.7i	−0.473004	+	0.570835i
$862$		0			0
$863$			15773.6i			0.622179i	0.950381	+	0.311090i	$0.100694\pi$
$863$			15773.6i			0.622179i	−0.950381	+	0.311090i	$0.899306\pi$
$864$		0			0
$865$	−75435.8			−2.96520
$866$		0			0
$867$			10766.2i			0.421727i
$868$		0			0
$869$		−	17229.9i		−	0.672594i
$870$		0			0
$871$	−16609.6			−0.646148
$872$		0			0
$873$			5382.00i			0.208652i
$874$		0			0
$875$	43663.6	−	52694.5i	1.68697	−	2.03589i
$876$		0			0
$877$			23431.2i			0.902182i	0.892478	+	0.451091i	$0.148965\pi$
$877$			23431.2i			0.902182i	−0.892478	+	0.451091i	$0.851035\pi$
$878$		0			0
$879$		−	6743.95i		−	0.258780i
$880$		0			0
$881$		−	47521.7i		−	1.81730i	−0.417554	−	0.908652i	$-0.637113\pi$
$881$		−	47521.7i		−	1.81730i	0.417554	−	0.908652i	$-0.362887\pi$
$882$		0			0
$883$	−33236.8			−1.26671			−0.633357	−	0.773860i	$-0.718324\pi$
$883$	−33236.8			−1.26671			−0.633357	+	0.773860i	$0.718324\pi$
$884$		0			0
$885$	11511.9			0.437251
$886$		0			0
$887$	16319.1			0.617746			0.308873	−	0.951103i	$-0.400048\pi$
$887$	16319.1			0.617746			0.308873	+	0.951103i	$0.400048\pi$
$888$		0			0
$889$	−23112.1	−	19151.1i	−0.871940	−	0.722504i
$890$		0			0
$891$	3181.80			0.119634
$892$		0			0
$893$			27539.6i			1.03200i
$894$		0			0
$895$	−70708.3			−2.64080
$896$		0			0
$897$	−10981.1			−0.408749
$898$		0			0
$899$			9568.77i			0.354990i
$900$		0			0
$901$	10010.0			0.370123
$902$		0			0
$903$	−15766.4	−	13064.3i	−0.581031	−	0.481453i
$904$		0			0
$905$	−72739.9			−2.67178
$906$		0			0
$907$	40088.2			1.46759			0.733797	−	0.679369i	$-0.237746\pi$
$907$	40088.2			1.46759			0.733797	+	0.679369i	$0.237746\pi$
$908$		0			0
$909$	−12735.3			−0.464691
$910$		0			0
$911$		−	47537.5i		−	1.72885i	−0.502758	−	0.864427i	$-0.667681\pi$
$911$		−	47537.5i		−	1.72885i	0.502758	−	0.864427i	$-0.332319\pi$
$912$		0			0
$913$			28365.7i			1.02822i
$914$		0			0
$915$			10153.8i			0.366856i
$916$		0			0
$917$	7176.39	+	5946.48i	0.258435	+	0.214144i
$918$		0			0
$919$			15740.2i			0.564983i	0.959270	+	0.282492i	$0.0911610\pi$
$919$			15740.2i			0.564983i	−0.959270	+	0.282492i	$0.908839\pi$
$920$		0			0
$921$	5279.72			0.188896
$922$		0			0
$923$		−	21186.1i		−	0.755525i
$924$		0			0
$925$		−	69664.6i		−	2.47628i
$926$		0			0
$927$	−10249.7			−0.363155
$928$		0			0
$929$			36693.6i			1.29588i	0.761689	+	0.647942i	$0.224370\pi$
$929$			36693.6i			1.29588i	−0.761689	+	0.647942i	$0.775630\pi$
$930$		0			0
$931$	−29496.0	+	5577.88i	−1.03834	+	0.196356i
$932$		0			0
$933$		−	25551.3i		−	0.896583i
$934$		0			0
$935$			74975.3i			2.62241i
$936$		0			0
$937$		−	31736.7i		−	1.10650i	−0.833015	−	0.553251i	$-0.813387\pi$
$937$		−	31736.7i		−	1.10650i	0.833015	−	0.553251i	$-0.186613\pi$
$938$		0			0
$939$	−30340.6			−1.05445
$940$		0			0
$941$	24588.9			0.851835			0.425917	−	0.904762i	$-0.359951\pi$
$941$	24588.9			0.851835			0.425917	+	0.904762i	$0.359951\pi$
$942$		0			0
$943$	34043.3			1.17561
$944$		0			0
$945$	7970.42	+	6604.43i	0.274368	+	0.227346i
$946$		0			0
$947$	6910.91			0.237143			0.118572	−	0.992946i	$-0.462169\pi$
$947$	6910.91			0.237143			0.118572	+	0.992946i	$0.462169\pi$
$948$		0			0
$949$		−	39841.7i		−	1.36282i
$950$		0			0
$951$	19533.0			0.666037
$952$		0			0
$953$	4158.99			0.141367			0.0706835	−	0.997499i	$-0.477482\pi$
$953$	4158.99			0.141367			0.0706835	+	0.997499i	$0.477482\pi$
$954$		0			0
$955$		−	63304.8i		−	2.14502i
$956$		0			0
$957$	4266.37			0.144109
$958$		0			0
$959$	−6827.29	+	8239.38i	−0.229890	+	0.277438i
$960$		0			0
$961$	40066.4			1.34492
$962$		0			0
$963$	13530.4			0.452765
$964$		0			0
$965$	−39224.9			−1.30849
$966$		0			0
$967$			38144.8i			1.26852i	0.773121	+	0.634258i	$0.218694\pi$
$967$			38144.8i			1.26852i	−0.773121	+	0.634258i	$0.781306\pi$
$968$		0			0
$969$		−	24208.8i		−	0.802580i
$970$		0			0
$971$		−	11358.8i		−	0.375409i	−0.982226	−	0.187705i	$-0.939895\pi$
$971$		−	11358.8i		−	0.375409i	0.982226	−	0.187705i	$-0.0601048\pi$
$972$		0			0
$973$	−5881.80	−	4873.76i	−0.193794	−	0.160581i
$974$		0			0
$975$			33001.4i			1.08399i
$976$		0			0
$977$	−18568.5			−0.608043			−0.304022	−	0.952665i	$-0.598329\pi$
$977$	−18568.5			−0.608043			−0.304022	+	0.952665i	$0.598329\pi$
$978$		0			0
$979$		−	17768.3i		−	0.580059i
$980$		0			0
$981$			3233.55i			0.105239i
$982$		0			0
$983$	61176.0			1.98496			0.992479	−	0.122419i	$-0.0390652\pi$
$983$	61176.0			1.98496			0.992479	+	0.122419i	$0.0390652\pi$
$984$		0			0
$985$		−	2267.86i		−	0.0733606i
$986$		0			0
$987$	13462.3	+	11155.1i	0.434154	+	0.359748i
$988$		0			0
$989$			37217.5i			1.19661i
$990$		0			0
$991$		−	17776.3i		−	0.569810i	−0.958556	−	0.284905i	$-0.908038\pi$
$991$		−	17776.3i		−	0.569810i	0.958556	−	0.284905i	$-0.0919621\pi$
$992$		0			0
$993$			6093.32i			0.194729i
$994$		0			0
$995$	31582.7			1.00627
$996$		0			0
$997$	−28772.7			−0.913982			−0.456991	−	0.889471i	$-0.651073\pi$
$997$	−28772.7			−0.913982			−0.456991	+	0.889471i	$0.651073\pi$
$998$		0			0
$999$	6197.43			0.196274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.p.c.223.9	yes	32
4.3	odd	2	inner	1344.4.p.c.223.6	yes	32
7.6	odd	2		1344.4.p.d.223.14	yes	32
8.3	odd	2		1344.4.p.d.223.13	yes	32
8.5	even	2		1344.4.p.d.223.8	yes	32
28.27	even	2		1344.4.p.d.223.7	yes	32
56.13	odd	2	inner	1344.4.p.c.223.5	✓	32
56.27	even	2	inner	1344.4.p.c.223.10	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.p.c.223.5	✓	32	56.13	odd	2	inner
1344.4.p.c.223.6	yes	32	4.3	odd	2	inner
1344.4.p.c.223.9	yes	32	1.1	even	1	trivial
1344.4.p.c.223.10	yes	32	56.27	even	2	inner
1344.4.p.d.223.7	yes	32	28.27	even	2
1344.4.p.d.223.8	yes	32	8.5	even	2
1344.4.p.d.223.13	yes	32	8.3	odd	2
1344.4.p.d.223.14	yes	32	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.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +20.7003 q^{5} +(11.8166 - 14.2607i) q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} +20.7003 q^{5} +(11.8166 - 14.2607i) q^{7} -9.00000 q^{9} +39.2814 q^{11} -36.2449 q^{13} -62.1010i q^{15} +92.2048i q^{17} -87.5184i q^{19} +(-42.7820 - 35.4499i) q^{21} +100.990i q^{23} +303.504 q^{25} +27.0000i q^{27} +36.2034i q^{29} +264.306 q^{31} -117.844i q^{33} +(244.609 - 295.201i) q^{35} -229.534i q^{37} +108.735i q^{39} -337.096i q^{41} +368.528 q^{43} -186.303 q^{45} -314.672 q^{47} +(-63.7338 - 337.027i) q^{49} +276.614 q^{51} -108.563i q^{53} +813.139 q^{55} -262.555 q^{57} +185.373i q^{59} -163.504 q^{61} +(-106.350 + 128.346i) q^{63} -750.282 q^{65} +458.260 q^{67} +302.969 q^{69} +584.527i q^{71} +1099.24i q^{73} -910.512i q^{75} +(464.175 - 560.180i) q^{77} -438.627i q^{79} +81.0000 q^{81} +722.114i q^{83} +1908.67i q^{85} +108.610 q^{87} -452.334i q^{89} +(-428.293 + 516.877i) q^{91} -792.917i q^{93} -1811.66i q^{95} -598.000i q^{97} -353.533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 288 q^{9}+O(q^{10}) \) 32 * q - 288 * q^9	\( 32 q - 288 q^{9} - 224 q^{13} - 72 q^{21} + 1120 q^{25} - 752 q^{49} - 672 q^{57} - 544 q^{61} + 1536 q^{65} - 144 q^{69} - 1632 q^{77} + 2592 q^{81}+O(q^{100}) \) 32 * q - 288 * q^9 - 224 * q^13 - 72 * q^21 + 1120 * q^25 - 752 * q^49 - 672 * q^57 - 544 * q^61 + 1536 * q^65 - 144 * q^69 - 1632 * q^77 + 2592 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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