LMFDB - Embedded newform 3344.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

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:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} -1.00000 q^{5} +3.41421 q^{7} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} -1.00000 q^{5} +3.41421 q^{7} +2.82843 q^{9} +1.00000 q^{11} +2.24264 q^{13} -2.41421 q^{15} +3.41421 q^{17} +1.00000 q^{19} +8.24264 q^{21} +3.00000 q^{23} -4.00000 q^{25} -0.414214 q^{27} -6.24264 q^{29} +6.41421 q^{31} +2.41421 q^{33} -3.41421 q^{35} +10.0711 q^{37} +5.41421 q^{39} -1.65685 q^{41} -0.343146 q^{43} -2.82843 q^{45} -8.82843 q^{47} +4.65685 q^{49} +8.24264 q^{51} -4.48528 q^{53} -1.00000 q^{55} +2.41421 q^{57} +1.58579 q^{59} -11.0711 q^{61} +9.65685 q^{63} -2.24264 q^{65} +10.4142 q^{67} +7.24264 q^{69} +12.4142 q^{71} -4.48528 q^{73} -9.65685 q^{75} +3.41421 q^{77} +14.5858 q^{79} -9.48528 q^{81} -3.41421 q^{83} -3.41421 q^{85} -15.0711 q^{87} +4.89949 q^{89} +7.65685 q^{91} +15.4853 q^{93} -1.00000 q^{95} +2.41421 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7	$ 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} + 8 q^{21} + 6 q^{23} - 8 q^{25} + 2 q^{27} - 4 q^{29} + 10 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} + 8 q^{39} + 8 q^{41} - 12 q^{43} - 12 q^{47} - 2 q^{49} + 8 q^{51} + 8 q^{53} - 2 q^{55} + 2 q^{57} + 6 q^{59} - 8 q^{61} + 8 q^{63} + 4 q^{65} + 18 q^{67} + 6 q^{69} + 22 q^{71} + 8 q^{73} - 8 q^{75} + 4 q^{77} + 32 q^{79} - 2 q^{81} - 4 q^{83} - 4 q^{85} - 16 q^{87} - 10 q^{89} + 4 q^{91} + 14 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 + 8 * q^21 + 6 * q^23 - 8 * q^25 + 2 * q^27 - 4 * q^29 + 10 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 + 8 * q^39 + 8 * q^41 - 12 * q^43 - 12 * q^47 - 2 * q^49 + 8 * q^51 + 8 * q^53 - 2 * q^55 + 2 * q^57 + 6 * q^59 - 8 * q^61 + 8 * q^63 + 4 * q^65 + 18 * q^67 + 6 * q^69 + 22 * q^71 + 8 * q^73 - 8 * q^75 + 4 * q^77 + 32 * q^79 - 2 * q^81 - 4 * q^83 - 4 * q^85 - 16 * q^87 - 10 * q^89 + 4 * q^91 + 14 * q^93 - 2 * q^95 + 2 * q^97

Display 100 coefficients 10 coefficients

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
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	0	0
$15$	−2.41421	−0.623347
$16$	0	0
$17$	3.41421	0.828068	0.414034	−	0.910261i	$-0.364119\pi$
$17$	3.41421	0.828068	0.414034	+	0.910261i	$0.364119\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.24264	1.79869
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	−6.24264	−1.15923	−0.579615	−	0.814891i	$-0.696797\pi$
$29$	−6.24264	−1.15923	−0.579615	+	0.814891i	$0.696797\pi$
$30$	0	0
$31$	6.41421	1.15203	0.576013	−	0.817440i	$-0.304608\pi$
$31$	6.41421	1.15203	0.576013	+	0.817440i	$0.304608\pi$
$32$	0	0
$33$	2.41421	0.420261
$34$	0	0
$35$	−3.41421	−0.577107
$36$	0	0
$37$	10.0711	1.65567	0.827837	−	0.560969i	$-0.189571\pi$
$37$	10.0711	1.65567	0.827837	+	0.560969i	$0.189571\pi$
$38$	0	0
$39$	5.41421	0.866968
$40$	0	0
$41$	−1.65685	−0.258757	−0.129379	−	0.991595i	$-0.541298\pi$
$41$	−1.65685	−0.258757	−0.129379	+	0.991595i	$0.541298\pi$
$42$	0	0
$43$	−0.343146	−0.0523292	−0.0261646	−	0.999658i	$-0.508329\pi$
$43$	−0.343146	−0.0523292	−0.0261646	+	0.999658i	$0.508329\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	−8.82843	−1.28776	−0.643879	−	0.765127i	$-0.722676\pi$
$47$	−8.82843	−1.28776	−0.643879	+	0.765127i	$0.722676\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	8.24264	1.15420
$52$	0	0
$53$	−4.48528	−0.616101	−0.308050	−	0.951370i	$-0.599677\pi$
$53$	−4.48528	−0.616101	−0.308050	+	0.951370i	$0.599677\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	2.41421	0.319770
$58$	0	0
$59$	1.58579	0.206452	0.103226	−	0.994658i	$-0.467084\pi$
$59$	1.58579	0.206452	0.103226	+	0.994658i	$0.467084\pi$
$60$	0	0
$61$	−11.0711	−1.41750	−0.708752	−	0.705457i	$-0.750742\pi$
$61$	−11.0711	−1.41750	−0.708752	+	0.705457i	$0.750742\pi$
$62$	0	0
$63$	9.65685	1.21665
$64$	0	0
$65$	−2.24264	−0.278165
$66$	0	0
$67$	10.4142	1.27230	0.636149	−	0.771566i	$-0.280526\pi$
$67$	10.4142	1.27230	0.636149	+	0.771566i	$0.280526\pi$
$68$	0	0
$69$	7.24264	0.871911
$70$	0	0
$71$	12.4142	1.47330	0.736648	−	0.676276i	$-0.236407\pi$
$71$	12.4142	1.47330	0.736648	+	0.676276i	$0.236407\pi$
$72$	0	0
$73$	−4.48528	−0.524962	−0.262481	−	0.964937i	$-0.584541\pi$
$73$	−4.48528	−0.524962	−0.262481	+	0.964937i	$0.584541\pi$
$74$	0	0
$75$	−9.65685	−1.11508
$76$	0	0
$77$	3.41421	0.389086
$78$	0	0
$79$	14.5858	1.64103	0.820515	−	0.571626i	$-0.193687\pi$
$79$	14.5858	1.64103	0.820515	+	0.571626i	$0.193687\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	−3.41421	−0.374759	−0.187379	−	0.982288i	$-0.559999\pi$
$83$	−3.41421	−0.374759	−0.187379	+	0.982288i	$0.559999\pi$
$84$	0	0
$85$	−3.41421	−0.370323
$86$	0	0
$87$	−15.0711	−1.61579
$88$	0	0
$89$	4.89949	0.519345	0.259673	−	0.965697i	$-0.416385\pi$
$89$	4.89949	0.519345	0.259673	+	0.965697i	$0.416385\pi$
$90$	0	0
$91$	7.65685	0.802656
$92$	0	0
$93$	15.4853	1.60575
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.41421	0.245126	0.122563	−	0.992461i	$-0.460889\pi$
$97$	2.41421	0.245126	0.122563	+	0.992461i	$0.460889\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	6.24264	0.621166	0.310583	−	0.950546i	$-0.399476\pi$
$101$	6.24264	0.621166	0.310583	+	0.950546i	$0.399476\pi$
$102$	0	0
$103$	−13.6569	−1.34565	−0.672825	−	0.739802i	$-0.734919\pi$
$103$	−13.6569	−1.34565	−0.672825	+	0.739802i	$0.734919\pi$
$104$	0	0
$105$	−8.24264	−0.804399
$106$	0	0
$107$	−15.6569	−1.51361	−0.756803	−	0.653643i	$-0.773240\pi$
$107$	−15.6569	−1.51361	−0.756803	+	0.653643i	$0.773240\pi$
$108$	0	0
$109$	−0.343146	−0.0328674	−0.0164337	−	0.999865i	$-0.505231\pi$
$109$	−0.343146	−0.0328674	−0.0164337	+	0.999865i	$0.505231\pi$
$110$	0	0
$111$	24.3137	2.30776
$112$	0	0
$113$	−3.58579	−0.337322	−0.168661	−	0.985674i	$-0.553944\pi$
$113$	−3.58579	−0.337322	−0.168661	+	0.985674i	$0.553944\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	0	0
$117$	6.34315	0.586424
$118$	0	0
$119$	11.6569	1.06858
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	19.4142	1.72273	0.861366	−	0.507984i	$-0.169609\pi$
$127$	19.4142	1.72273	0.861366	+	0.507984i	$0.169609\pi$
$128$	0	0
$129$	−0.828427	−0.0729389
$130$	0	0
$131$	−14.4853	−1.26558	−0.632792	−	0.774321i	$-0.718091\pi$
$131$	−14.4853	−1.26558	−0.632792	+	0.774321i	$0.718091\pi$
$132$	0	0
$133$	3.41421	0.296050
$134$	0	0
$135$	0.414214	0.0356498
$136$	0	0
$137$	−2.65685	−0.226990	−0.113495	−	0.993539i	$-0.536205\pi$
$137$	−2.65685	−0.226990	−0.113495	+	0.993539i	$0.536205\pi$
$138$	0	0
$139$	−20.3848	−1.72901	−0.864507	−	0.502621i	$-0.832369\pi$
$139$	−20.3848	−1.72901	−0.864507	+	0.502621i	$0.832369\pi$
$140$	0	0
$141$	−21.3137	−1.79494
$142$	0	0
$143$	2.24264	0.187539
$144$	0	0
$145$	6.24264	0.518423
$146$	0	0
$147$	11.2426	0.927277
$148$	0	0
$149$	21.3137	1.74609	0.873044	−	0.487642i	$-0.162143\pi$
$149$	21.3137	1.74609	0.873044	+	0.487642i	$0.162143\pi$
$150$	0	0
$151$	10.4853	0.853280	0.426640	−	0.904422i	$-0.359697\pi$
$151$	10.4853	0.853280	0.426640	+	0.904422i	$0.359697\pi$
$152$	0	0
$153$	9.65685	0.780710
$154$	0	0
$155$	−6.41421	−0.515202
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	0	0
$159$	−10.8284	−0.858750
$160$	0	0
$161$	10.2426	0.807233
$162$	0	0
$163$	20.1421	1.57765	0.788827	−	0.614615i	$-0.210689\pi$
$163$	20.1421	1.57765	0.788827	+	0.614615i	$0.210689\pi$
$164$	0	0
$165$	−2.41421	−0.187946
$166$	0	0
$167$	18.7279	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	1.89949	0.144416	0.0722080	−	0.997390i	$-0.476995\pi$
$173$	1.89949	0.144416	0.0722080	+	0.997390i	$0.476995\pi$
$174$	0	0
$175$	−13.6569	−1.03236
$176$	0	0
$177$	3.82843	0.287762
$178$	0	0
$179$	−16.0711	−1.20121	−0.600604	−	0.799547i	$-0.705073\pi$
$179$	−16.0711	−1.20121	−0.600604	+	0.799547i	$0.705073\pi$
$180$	0	0
$181$	−19.3848	−1.44086	−0.720430	−	0.693528i	$-0.756055\pi$
$181$	−19.3848	−1.44086	−0.720430	+	0.693528i	$0.756055\pi$
$182$	0	0
$183$	−26.7279	−1.97578
$184$	0	0
$185$	−10.0711	−0.740440
$186$	0	0
$187$	3.41421	0.249672
$188$	0	0
$189$	−1.41421	−0.102869
$190$	0	0
$191$	−8.31371	−0.601559	−0.300779	−	0.953694i	$-0.597247\pi$
$191$	−8.31371	−0.601559	−0.300779	+	0.953694i	$0.597247\pi$
$192$	0	0
$193$	1.17157	0.0843317	0.0421658	−	0.999111i	$-0.486574\pi$
$193$	1.17157	0.0843317	0.0421658	+	0.999111i	$0.486574\pi$
$194$	0	0
$195$	−5.41421	−0.387720
$196$	0	0
$197$	15.8995	1.13279	0.566396	−	0.824133i	$-0.308337\pi$
$197$	15.8995	1.13279	0.566396	+	0.824133i	$0.308337\pi$
$198$	0	0
$199$	−12.1421	−0.860733	−0.430367	−	0.902654i	$-0.641616\pi$
$199$	−12.1421	−0.860733	−0.430367	+	0.902654i	$0.641616\pi$
$200$	0	0
$201$	25.1421	1.77339
$202$	0	0
$203$	−21.3137	−1.49593
$204$	0	0
$205$	1.65685	0.115720
$206$	0	0
$207$	8.48528	0.589768
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	12.5858	0.866441	0.433221	−	0.901288i	$-0.357377\pi$
$211$	12.5858	0.866441	0.433221	+	0.901288i	$0.357377\pi$
$212$	0	0
$213$	29.9706	2.05355
$214$	0	0
$215$	0.343146	0.0234023
$216$	0	0
$217$	21.8995	1.48663
$218$	0	0
$219$	−10.8284	−0.731717
$220$	0	0
$221$	7.65685	0.515056
$222$	0	0
$223$	16.4142	1.09918	0.549589	−	0.835435i	$-0.314785\pi$
$223$	16.4142	1.09918	0.549589	+	0.835435i	$0.314785\pi$
$224$	0	0
$225$	−11.3137	−0.754247
$226$	0	0
$227$	−4.92893	−0.327145	−0.163572	−	0.986531i	$-0.552302\pi$
$227$	−4.92893	−0.327145	−0.163572	+	0.986531i	$0.552302\pi$
$228$	0	0
$229$	−14.3137	−0.945876	−0.472938	−	0.881096i	$-0.656807\pi$
$229$	−14.3137	−0.945876	−0.472938	+	0.881096i	$0.656807\pi$
$230$	0	0
$231$	8.24264	0.542326
$232$	0	0
$233$	−0.242641	−0.0158959	−0.00794796	−	0.999968i	$-0.502530\pi$
$233$	−0.242641	−0.0158959	−0.00794796	+	0.999968i	$0.502530\pi$
$234$	0	0
$235$	8.82843	0.575903
$236$	0	0
$237$	35.2132	2.28734
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−16.9706	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	−4.65685	−0.297516
$246$	0	0
$247$	2.24264	0.142696
$248$	0	0
$249$	−8.24264	−0.522356
$250$	0	0
$251$	11.3431	0.715973	0.357987	−	0.933727i	$-0.383463\pi$
$251$	11.3431	0.715973	0.357987	+	0.933727i	$0.383463\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	0	0
$255$	−8.24264	−0.516174
$256$	0	0
$257$	−19.1716	−1.19589	−0.597945	−	0.801537i	$-0.704016\pi$
$257$	−19.1716	−1.19589	−0.597945	+	0.801537i	$0.704016\pi$
$258$	0	0
$259$	34.3848	2.13657
$260$	0	0
$261$	−17.6569	−1.09293
$262$	0	0
$263$	9.51472	0.586703	0.293351	−	0.956005i	$-0.405229\pi$
$263$	9.51472	0.586703	0.293351	+	0.956005i	$0.405229\pi$
$264$	0	0
$265$	4.48528	0.275529
$266$	0	0
$267$	11.8284	0.723888
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−6.14214	−0.373108	−0.186554	−	0.982445i	$-0.559732\pi$
$271$	−6.14214	−0.373108	−0.186554	+	0.982445i	$0.559732\pi$
$272$	0	0
$273$	18.4853	1.11878
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−2.48528	−0.149326	−0.0746630	−	0.997209i	$-0.523788\pi$
$277$	−2.48528	−0.149326	−0.0746630	+	0.997209i	$0.523788\pi$
$278$	0	0
$279$	18.1421	1.08614
$280$	0	0
$281$	23.6569	1.41125	0.705625	−	0.708586i	$-0.250666\pi$
$281$	23.6569	1.41125	0.705625	+	0.708586i	$0.250666\pi$
$282$	0	0
$283$	16.7279	0.994372	0.497186	−	0.867644i	$-0.334367\pi$
$283$	16.7279	0.994372	0.497186	+	0.867644i	$0.334367\pi$
$284$	0	0
$285$	−2.41421	−0.143006
$286$	0	0
$287$	−5.65685	−0.333914
$288$	0	0
$289$	−5.34315	−0.314303
$290$	0	0
$291$	5.82843	0.341668
$292$	0	0
$293$	−31.6985	−1.85185	−0.925923	−	0.377713i	$-0.876711\pi$
$293$	−31.6985	−1.85185	−0.925923	+	0.377713i	$0.876711\pi$
$294$	0	0
$295$	−1.58579	−0.0923281
$296$	0	0
$297$	−0.414214	−0.0240351
$298$	0	0
$299$	6.72792	0.389086
$300$	0	0
$301$	−1.17157	−0.0675283
$302$	0	0
$303$	15.0711	0.865810
$304$	0	0
$305$	11.0711	0.633927
$306$	0	0
$307$	−7.41421	−0.423152	−0.211576	−	0.977362i	$-0.567860\pi$
$307$	−7.41421	−0.423152	−0.211576	+	0.977362i	$0.567860\pi$
$308$	0	0
$309$	−32.9706	−1.87563
$310$	0	0
$311$	11.6569	0.661000	0.330500	−	0.943806i	$-0.392783\pi$
$311$	11.6569	0.661000	0.330500	+	0.943806i	$0.392783\pi$
$312$	0	0
$313$	−13.9706	−0.789663	−0.394831	−	0.918754i	$-0.629197\pi$
$313$	−13.9706	−0.789663	−0.394831	+	0.918754i	$0.629197\pi$
$314$	0	0
$315$	−9.65685	−0.544102
$316$	0	0
$317$	−28.4142	−1.59590	−0.797951	−	0.602723i	$-0.794082\pi$
$317$	−28.4142	−1.59590	−0.797951	+	0.602723i	$0.794082\pi$
$318$	0	0
$319$	−6.24264	−0.349521
$320$	0	0
$321$	−37.7990	−2.10973
$322$	0	0
$323$	3.41421	0.189972
$324$	0	0
$325$	−8.97056	−0.497597
$326$	0	0
$327$	−0.828427	−0.0458121
$328$	0	0
$329$	−30.1421	−1.66179
$330$	0	0
$331$	−34.2132	−1.88053	−0.940264	−	0.340447i	$-0.889422\pi$
$331$	−34.2132	−1.88053	−0.940264	+	0.340447i	$0.889422\pi$
$332$	0	0
$333$	28.4853	1.56098
$334$	0	0
$335$	−10.4142	−0.568989
$336$	0	0
$337$	16.7279	0.911228	0.455614	−	0.890177i	$-0.349420\pi$
$337$	16.7279	0.911228	0.455614	+	0.890177i	$0.349420\pi$
$338$	0	0
$339$	−8.65685	−0.470176
$340$	0	0
$341$	6.41421	0.347349
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	−7.24264	−0.389931
$346$	0	0
$347$	−33.6985	−1.80903	−0.904515	−	0.426442i	$-0.859767\pi$
$347$	−33.6985	−1.80903	−0.904515	+	0.426442i	$0.859767\pi$
$348$	0	0
$349$	23.2132	1.24257	0.621287	−	0.783583i	$-0.286610\pi$
$349$	23.2132	1.24257	0.621287	+	0.783583i	$0.286610\pi$
$350$	0	0
$351$	−0.928932	−0.0495827
$352$	0	0
$353$	−24.3137	−1.29409	−0.647044	−	0.762453i	$-0.723995\pi$
$353$	−24.3137	−1.29409	−0.647044	+	0.762453i	$0.723995\pi$
$354$	0	0
$355$	−12.4142	−0.658878
$356$	0	0
$357$	28.1421	1.48944
$358$	0	0
$359$	−16.4853	−0.870060	−0.435030	−	0.900416i	$-0.643262\pi$
$359$	−16.4853	−0.870060	−0.435030	+	0.900416i	$0.643262\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	2.41421	0.126713
$364$	0	0
$365$	4.48528	0.234770
$366$	0	0
$367$	9.82843	0.513040	0.256520	−	0.966539i	$-0.417424\pi$
$367$	9.82843	0.513040	0.256520	+	0.966539i	$0.417424\pi$
$368$	0	0
$369$	−4.68629	−0.243959
$370$	0	0
$371$	−15.3137	−0.795048
$372$	0	0
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	0	0
$375$	21.7279	1.12203
$376$	0	0
$377$	−14.0000	−0.721037
$378$	0	0
$379$	30.6985	1.57688	0.788438	−	0.615115i	$-0.210890\pi$
$379$	30.6985	1.57688	0.788438	+	0.615115i	$0.210890\pi$
$380$	0	0
$381$	46.8701	2.40123
$382$	0	0
$383$	−33.5269	−1.71315	−0.856573	−	0.516027i	$-0.827411\pi$
$383$	−33.5269	−1.71315	−0.856573	+	0.516027i	$0.827411\pi$
$384$	0	0
$385$	−3.41421	−0.174004
$386$	0	0
$387$	−0.970563	−0.0493365
$388$	0	0
$389$	−28.3137	−1.43556	−0.717781	−	0.696269i	$-0.754842\pi$
$389$	−28.3137	−1.43556	−0.717781	+	0.696269i	$0.754842\pi$
$390$	0	0
$391$	10.2426	0.517993
$392$	0	0
$393$	−34.9706	−1.76403
$394$	0	0
$395$	−14.5858	−0.733891
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	0	0
$399$	8.24264	0.412648
$400$	0	0
$401$	−34.9706	−1.74635	−0.873173	−	0.487410i	$-0.837942\pi$
$401$	−34.9706	−1.74635	−0.873173	+	0.487410i	$0.837942\pi$
$402$	0	0
$403$	14.3848	0.716557
$404$	0	0
$405$	9.48528	0.471327
$406$	0	0
$407$	10.0711	0.499204
$408$	0	0
$409$	0.727922	0.0359934	0.0179967	−	0.999838i	$-0.494271\pi$
$409$	0.727922	0.0359934	0.0179967	+	0.999838i	$0.494271\pi$
$410$	0	0
$411$	−6.41421	−0.316390
$412$	0	0
$413$	5.41421	0.266416
$414$	0	0
$415$	3.41421	0.167597
$416$	0	0
$417$	−49.2132	−2.40998
$418$	0	0
$419$	27.4558	1.34131	0.670653	−	0.741771i	$-0.266014\pi$
$419$	27.4558	1.34131	0.670653	+	0.741771i	$0.266014\pi$
$420$	0	0
$421$	−28.1421	−1.37156	−0.685782	−	0.727807i	$-0.740540\pi$
$421$	−28.1421	−1.37156	−0.685782	+	0.727807i	$0.740540\pi$
$422$	0	0
$423$	−24.9706	−1.21411
$424$	0	0
$425$	−13.6569	−0.662455
$426$	0	0
$427$	−37.7990	−1.82922
$428$	0	0
$429$	5.41421	0.261401
$430$	0	0
$431$	−30.4853	−1.46842	−0.734212	−	0.678920i	$-0.762448\pi$
$431$	−30.4853	−1.46842	−0.734212	+	0.678920i	$0.762448\pi$
$432$	0	0
$433$	11.3848	0.547117	0.273559	−	0.961855i	$-0.411799\pi$
$433$	11.3848	0.547117	0.273559	+	0.961855i	$0.411799\pi$
$434$	0	0
$435$	15.0711	0.722602
$436$	0	0
$437$	3.00000	0.143509
$438$	0	0
$439$	14.2426	0.679764	0.339882	−	0.940468i	$-0.389613\pi$
$439$	14.2426	0.679764	0.339882	+	0.940468i	$0.389613\pi$
$440$	0	0
$441$	13.1716	0.627218
$442$	0	0
$443$	11.9706	0.568739	0.284369	−	0.958715i	$-0.408216\pi$
$443$	11.9706	0.568739	0.284369	+	0.958715i	$0.408216\pi$
$444$	0	0
$445$	−4.89949	−0.232258
$446$	0	0
$447$	51.4558	2.43378
$448$	0	0
$449$	−13.3848	−0.631667	−0.315833	−	0.948815i	$-0.602284\pi$
$449$	−13.3848	−0.631667	−0.315833	+	0.948815i	$0.602284\pi$
$450$	0	0
$451$	−1.65685	−0.0780182
$452$	0	0
$453$	25.3137	1.18934
$454$	0	0
$455$	−7.65685	−0.358959
$456$	0	0
$457$	16.9289	0.791902	0.395951	−	0.918272i	$-0.370415\pi$
$457$	16.9289	0.791902	0.395951	+	0.918272i	$0.370415\pi$
$458$	0	0
$459$	−1.41421	−0.0660098
$460$	0	0
$461$	12.9706	0.604099	0.302050	−	0.953292i	$-0.402329\pi$
$461$	12.9706	0.604099	0.302050	+	0.953292i	$0.402329\pi$
$462$	0	0
$463$	−12.4558	−0.578872	−0.289436	−	0.957197i	$-0.593468\pi$
$463$	−12.4558	−0.578872	−0.289436	+	0.957197i	$0.593468\pi$
$464$	0	0
$465$	−15.4853	−0.718113
$466$	0	0
$467$	−9.68629	−0.448228	−0.224114	−	0.974563i	$-0.571949\pi$
$467$	−9.68629	−0.448228	−0.224114	+	0.974563i	$0.571949\pi$
$468$	0	0
$469$	35.5563	1.64184
$470$	0	0
$471$	12.0711	0.556205
$472$	0	0
$473$	−0.343146	−0.0157779
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−12.6863	−0.580865
$478$	0	0
$479$	40.5269	1.85172	0.925861	−	0.377864i	$-0.123341\pi$
$479$	40.5269	1.85172	0.925861	+	0.377864i	$0.123341\pi$
$480$	0	0
$481$	22.5858	1.02982
$482$	0	0
$483$	24.7279	1.12516
$484$	0	0
$485$	−2.41421	−0.109624
$486$	0	0
$487$	−12.5563	−0.568982	−0.284491	−	0.958679i	$-0.591825\pi$
$487$	−12.5563	−0.568982	−0.284491	+	0.958679i	$0.591825\pi$
$488$	0	0
$489$	48.6274	2.19901
$490$	0	0
$491$	−27.7574	−1.25267	−0.626336	−	0.779553i	$-0.715446\pi$
$491$	−27.7574	−1.25267	−0.626336	+	0.779553i	$0.715446\pi$
$492$	0	0
$493$	−21.3137	−0.959921
$494$	0	0
$495$	−2.82843	−0.127128
$496$	0	0
$497$	42.3848	1.90122
$498$	0	0
$499$	32.6274	1.46060	0.730302	−	0.683125i	$-0.239379\pi$
$499$	32.6274	1.46060	0.730302	+	0.683125i	$0.239379\pi$
$500$	0	0
$501$	45.2132	2.01998
$502$	0	0
$503$	28.1421	1.25480	0.627398	−	0.778699i	$-0.284120\pi$
$503$	28.1421	1.25480	0.627398	+	0.778699i	$0.284120\pi$
$504$	0	0
$505$	−6.24264	−0.277794
$506$	0	0
$507$	−19.2426	−0.854596
$508$	0	0
$509$	−8.21320	−0.364044	−0.182022	−	0.983294i	$-0.558264\pi$
$509$	−8.21320	−0.364044	−0.182022	+	0.983294i	$0.558264\pi$
$510$	0	0
$511$	−15.3137	−0.677439
$512$	0	0
$513$	−0.414214	−0.0182880
$514$	0	0
$515$	13.6569	0.601793
$516$	0	0
$517$	−8.82843	−0.388274
$518$	0	0
$519$	4.58579	0.201294
$520$	0	0
$521$	0.556349	0.0243741	0.0121871	−	0.999926i	$-0.496121\pi$
$521$	0.556349	0.0243741	0.0121871	+	0.999926i	$0.496121\pi$
$522$	0	0
$523$	−2.34315	−0.102459	−0.0512293	−	0.998687i	$-0.516314\pi$
$523$	−2.34315	−0.102459	−0.0512293	+	0.998687i	$0.516314\pi$
$524$	0	0
$525$	−32.9706	−1.43895
$526$	0	0
$527$	21.8995	0.953957
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	4.48528	0.194645
$532$	0	0
$533$	−3.71573	−0.160946
$534$	0	0
$535$	15.6569	0.676905
$536$	0	0
$537$	−38.7990	−1.67430
$538$	0	0
$539$	4.65685	0.200585
$540$	0	0
$541$	9.21320	0.396107	0.198053	−	0.980191i	$-0.436538\pi$
$541$	9.21320	0.396107	0.198053	+	0.980191i	$0.436538\pi$
$542$	0	0
$543$	−46.7990	−2.00834
$544$	0	0
$545$	0.343146	0.0146987
$546$	0	0
$547$	14.7279	0.629720	0.314860	−	0.949138i	$-0.398042\pi$
$547$	14.7279	0.629720	0.314860	+	0.949138i	$0.398042\pi$
$548$	0	0
$549$	−31.3137	−1.33644
$550$	0	0
$551$	−6.24264	−0.265945
$552$	0	0
$553$	49.7990	2.11767
$554$	0	0
$555$	−24.3137	−1.03206
$556$	0	0
$557$	−27.9411	−1.18390	−0.591952	−	0.805973i	$-0.701642\pi$
$557$	−27.9411	−1.18390	−0.591952	+	0.805973i	$0.701642\pi$
$558$	0	0
$559$	−0.769553	−0.0325486
$560$	0	0
$561$	8.24264	0.348005
$562$	0	0
$563$	23.2132	0.978320	0.489160	−	0.872194i	$-0.337303\pi$
$563$	23.2132	0.978320	0.489160	+	0.872194i	$0.337303\pi$
$564$	0	0
$565$	3.58579	0.150855
$566$	0	0
$567$	−32.3848	−1.36003
$568$	0	0
$569$	−26.2426	−1.10015	−0.550074	−	0.835116i	$-0.685401\pi$
$569$	−26.2426	−1.10015	−0.550074	+	0.835116i	$0.685401\pi$
$570$	0	0
$571$	−33.6985	−1.41024	−0.705119	−	0.709089i	$-0.749106\pi$
$571$	−33.6985	−1.41024	−0.705119	+	0.709089i	$0.749106\pi$
$572$	0	0
$573$	−20.0711	−0.838481
$574$	0	0
$575$	−12.0000	−0.500435
$576$	0	0
$577$	29.9706	1.24769	0.623845	−	0.781548i	$-0.285569\pi$
$577$	29.9706	1.24769	0.623845	+	0.781548i	$0.285569\pi$
$578$	0	0
$579$	2.82843	0.117545
$580$	0	0
$581$	−11.6569	−0.483608
$582$	0	0
$583$	−4.48528	−0.185761
$584$	0	0
$585$	−6.34315	−0.262257
$586$	0	0
$587$	−7.65685	−0.316032	−0.158016	−	0.987437i	$-0.550510\pi$
$587$	−7.65685	−0.316032	−0.158016	+	0.987437i	$0.550510\pi$
$588$	0	0
$589$	6.41421	0.264293
$590$	0	0
$591$	38.3848	1.57894
$592$	0	0
$593$	36.9706	1.51820	0.759100	−	0.650975i	$-0.225640\pi$
$593$	36.9706	1.51820	0.759100	+	0.650975i	$0.225640\pi$
$594$	0	0
$595$	−11.6569	−0.477884
$596$	0	0
$597$	−29.3137	−1.19973
$598$	0	0
$599$	−13.3137	−0.543983	−0.271992	−	0.962300i	$-0.587682\pi$
$599$	−13.3137	−0.543983	−0.271992	+	0.962300i	$0.587682\pi$
$600$	0	0
$601$	11.5563	0.471393	0.235697	−	0.971827i	$-0.424263\pi$
$601$	11.5563	0.471393	0.235697	+	0.971827i	$0.424263\pi$
$602$	0	0
$603$	29.4558	1.19953
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	2.10051	0.0852569	0.0426284	−	0.999091i	$-0.486427\pi$
$607$	2.10051	0.0852569	0.0426284	+	0.999091i	$0.486427\pi$
$608$	0	0
$609$	−51.4558	−2.08510
$610$	0	0
$611$	−19.7990	−0.800981
$612$	0	0
$613$	−13.4142	−0.541795	−0.270897	−	0.962608i	$-0.587320\pi$
$613$	−13.4142	−0.541795	−0.270897	+	0.962608i	$0.587320\pi$
$614$	0	0
$615$	4.00000	0.161296
$616$	0	0
$617$	22.8284	0.919038	0.459519	−	0.888168i	$-0.348022\pi$
$617$	22.8284	0.919038	0.459519	+	0.888168i	$0.348022\pi$
$618$	0	0
$619$	−23.4853	−0.943953	−0.471977	−	0.881611i	$-0.656459\pi$
$619$	−23.4853	−0.943953	−0.471977	+	0.881611i	$0.656459\pi$
$620$	0	0
$621$	−1.24264	−0.0498655
$622$	0	0
$623$	16.7279	0.670190
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	2.41421	0.0964144
$628$	0	0
$629$	34.3848	1.37101
$630$	0	0
$631$	8.02944	0.319647	0.159823	−	0.987146i	$-0.448908\pi$
$631$	8.02944	0.319647	0.159823	+	0.987146i	$0.448908\pi$
$632$	0	0
$633$	30.3848	1.20769
$634$	0	0
$635$	−19.4142	−0.770430
$636$	0	0
$637$	10.4437	0.413793
$638$	0	0
$639$	35.1127	1.38904
$640$	0	0
$641$	0.899495	0.0355279	0.0177640	−	0.999842i	$-0.494345\pi$
$641$	0.899495	0.0355279	0.0177640	+	0.999842i	$0.494345\pi$
$642$	0	0
$643$	−24.6569	−0.972371	−0.486186	−	0.873856i	$-0.661612\pi$
$643$	−24.6569	−0.972371	−0.486186	+	0.873856i	$0.661612\pi$
$644$	0	0
$645$	0.828427	0.0326193
$646$	0	0
$647$	30.9411	1.21642	0.608211	−	0.793776i	$-0.291888\pi$
$647$	30.9411	1.21642	0.608211	+	0.793776i	$0.291888\pi$
$648$	0	0
$649$	1.58579	0.0622476
$650$	0	0
$651$	52.8701	2.07214
$652$	0	0
$653$	−8.51472	−0.333207	−0.166603	−	0.986024i	$-0.553280\pi$
$653$	−8.51472	−0.333207	−0.166603	+	0.986024i	$0.553280\pi$
$654$	0	0
$655$	14.4853	0.565987
$656$	0	0
$657$	−12.6863	−0.494939
$658$	0	0
$659$	−21.7990	−0.849168	−0.424584	−	0.905389i	$-0.639580\pi$
$659$	−21.7990	−0.849168	−0.424584	+	0.905389i	$0.639580\pi$
$660$	0	0
$661$	−49.8701	−1.93972	−0.969860	−	0.243662i	$-0.921651\pi$
$661$	−49.8701	−1.93972	−0.969860	+	0.243662i	$0.921651\pi$
$662$	0	0
$663$	18.4853	0.717909
$664$	0	0
$665$	−3.41421	−0.132398
$666$	0	0
$667$	−18.7279	−0.725148
$668$	0	0
$669$	39.6274	1.53208
$670$	0	0
$671$	−11.0711	−0.427394
$672$	0	0
$673$	−3.85786	−0.148710	−0.0743549	−	0.997232i	$-0.523690\pi$
$673$	−3.85786	−0.148710	−0.0743549	+	0.997232i	$0.523690\pi$
$674$	0	0
$675$	1.65685	0.0637723
$676$	0	0
$677$	21.3137	0.819152	0.409576	−	0.912276i	$-0.365677\pi$
$677$	21.3137	0.819152	0.409576	+	0.912276i	$0.365677\pi$
$678$	0	0
$679$	8.24264	0.316324
$680$	0	0
$681$	−11.8995	−0.455990
$682$	0	0
$683$	−23.1127	−0.884383	−0.442191	−	0.896921i	$-0.645799\pi$
$683$	−23.1127	−0.884383	−0.442191	+	0.896921i	$0.645799\pi$
$684$	0	0
$685$	2.65685	0.101513
$686$	0	0
$687$	−34.5563	−1.31841
$688$	0	0
$689$	−10.0589	−0.383213
$690$	0	0
$691$	−25.9706	−0.987967	−0.493983	−	0.869471i	$-0.664460\pi$
$691$	−25.9706	−0.987967	−0.493983	+	0.869471i	$0.664460\pi$
$692$	0	0
$693$	9.65685	0.366834
$694$	0	0
$695$	20.3848	0.773239
$696$	0	0
$697$	−5.65685	−0.214269
$698$	0	0
$699$	−0.585786	−0.0221565
$700$	0	0
$701$	−1.31371	−0.0496181	−0.0248090	−	0.999692i	$-0.507898\pi$
$701$	−1.31371	−0.0496181	−0.0248090	+	0.999692i	$0.507898\pi$
$702$	0	0
$703$	10.0711	0.379838
$704$	0	0
$705$	21.3137	0.802721
$706$	0	0
$707$	21.3137	0.801585
$708$	0	0
$709$	−25.2843	−0.949571	−0.474785	−	0.880102i	$-0.657474\pi$
$709$	−25.2843	−0.949571	−0.474785	+	0.880102i	$0.657474\pi$
$710$	0	0
$711$	41.2548	1.54718
$712$	0	0
$713$	19.2426	0.720643
$714$	0	0
$715$	−2.24264	−0.0838700
$716$	0	0
$717$	−14.4853	−0.540963
$718$	0	0
$719$	−10.0294	−0.374035	−0.187017	−	0.982357i	$-0.559882\pi$
$719$	−10.0294	−0.374035	−0.187017	+	0.982357i	$0.559882\pi$
$720$	0	0
$721$	−46.6274	−1.73650
$722$	0	0
$723$	−40.9706	−1.52371
$724$	0	0
$725$	24.9706	0.927383
$726$	0	0
$727$	18.1127	0.671763	0.335881	−	0.941904i	$-0.390966\pi$
$727$	18.1127	0.671763	0.335881	+	0.941904i	$0.390966\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	−1.17157	−0.0433322
$732$	0	0
$733$	13.4142	0.495465	0.247733	−	0.968828i	$-0.420315\pi$
$733$	13.4142	0.495465	0.247733	+	0.968828i	$0.420315\pi$
$734$	0	0
$735$	−11.2426	−0.414691
$736$	0	0
$737$	10.4142	0.383612
$738$	0	0
$739$	27.4142	1.00845	0.504224	−	0.863573i	$-0.331779\pi$
$739$	27.4142	1.00845	0.504224	+	0.863573i	$0.331779\pi$
$740$	0	0
$741$	5.41421	0.198896
$742$	0	0
$743$	6.92893	0.254198	0.127099	−	0.991890i	$-0.459433\pi$
$743$	6.92893	0.254198	0.127099	+	0.991890i	$0.459433\pi$
$744$	0	0
$745$	−21.3137	−0.780874
$746$	0	0
$747$	−9.65685	−0.353326
$748$	0	0
$749$	−53.4558	−1.95323
$750$	0	0
$751$	−3.44365	−0.125661	−0.0628303	−	0.998024i	$-0.520013\pi$
$751$	−3.44365	−0.125661	−0.0628303	+	0.998024i	$0.520013\pi$
$752$	0	0
$753$	27.3848	0.997957
$754$	0	0
$755$	−10.4853	−0.381598
$756$	0	0
$757$	41.9411	1.52438	0.762188	−	0.647356i	$-0.224125\pi$
$757$	41.9411	1.52438	0.762188	+	0.647356i	$0.224125\pi$
$758$	0	0
$759$	7.24264	0.262891
$760$	0	0
$761$	5.85786	0.212347	0.106174	−	0.994348i	$-0.466140\pi$
$761$	5.85786	0.212347	0.106174	+	0.994348i	$0.466140\pi$
$762$	0	0
$763$	−1.17157	−0.0424138
$764$	0	0
$765$	−9.65685	−0.349144
$766$	0	0
$767$	3.55635	0.128412
$768$	0	0
$769$	1.85786	0.0669963	0.0334982	−	0.999439i	$-0.489335\pi$
$769$	1.85786	0.0669963	0.0334982	+	0.999439i	$0.489335\pi$
$770$	0	0
$771$	−46.2843	−1.66689
$772$	0	0
$773$	19.6569	0.707008	0.353504	−	0.935433i	$-0.384990\pi$
$773$	19.6569	0.707008	0.353504	+	0.935433i	$0.384990\pi$
$774$	0	0
$775$	−25.6569	−0.921621
$776$	0	0
$777$	83.0122	2.97805
$778$	0	0
$779$	−1.65685	−0.0593630
$780$	0	0
$781$	12.4142	0.444215
$782$	0	0
$783$	2.58579	0.0924085
$784$	0	0
$785$	−5.00000	−0.178458
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	0	0
$789$	22.9706	0.817774
$790$	0	0
$791$	−12.2426	−0.435298
$792$	0	0
$793$	−24.8284	−0.881683
$794$	0	0
$795$	10.8284	0.384045
$796$	0	0
$797$	−13.2426	−0.469078	−0.234539	−	0.972107i	$-0.575358\pi$
$797$	−13.2426	−0.469078	−0.234539	+	0.972107i	$0.575358\pi$
$798$	0	0
$799$	−30.1421	−1.06635
$800$	0	0
$801$	13.8579	0.489644
$802$	0	0
$803$	−4.48528	−0.158282
$804$	0	0
$805$	−10.2426	−0.361006
$806$	0	0
$807$	−4.82843	−0.169969
$808$	0	0
$809$	35.5147	1.24863	0.624316	−	0.781172i	$-0.285378\pi$
$809$	35.5147	1.24863	0.624316	+	0.781172i	$0.285378\pi$
$810$	0	0
$811$	−43.7574	−1.53653	−0.768264	−	0.640133i	$-0.778879\pi$
$811$	−43.7574	−1.53653	−0.768264	+	0.640133i	$0.778879\pi$
$812$	0	0
$813$	−14.8284	−0.520056
$814$	0	0
$815$	−20.1421	−0.705548
$816$	0	0
$817$	−0.343146	−0.0120052
$818$	0	0
$819$	21.6569	0.756752
$820$	0	0
$821$	−38.5269	−1.34460	−0.672299	−	0.740279i	$-0.734693\pi$
$821$	−38.5269	−1.34460	−0.672299	+	0.740279i	$0.734693\pi$
$822$	0	0
$823$	23.0000	0.801730	0.400865	−	0.916137i	$-0.368710\pi$
$823$	23.0000	0.801730	0.400865	+	0.916137i	$0.368710\pi$
$824$	0	0
$825$	−9.65685	−0.336209
$826$	0	0
$827$	−23.8995	−0.831067	−0.415533	−	0.909578i	$-0.636405\pi$
$827$	−23.8995	−0.831067	−0.415533	+	0.909578i	$0.636405\pi$
$828$	0	0
$829$	−37.0416	−1.28651	−0.643255	−	0.765652i	$-0.722416\pi$
$829$	−37.0416	−1.28651	−0.643255	+	0.765652i	$0.722416\pi$
$830$	0	0
$831$	−6.00000	−0.208138
$832$	0	0
$833$	15.8995	0.550885
$834$	0	0
$835$	−18.7279	−0.648106
$836$	0	0
$837$	−2.65685	−0.0918343
$838$	0	0
$839$	−9.44365	−0.326031	−0.163016	−	0.986624i	$-0.552122\pi$
$839$	−9.44365	−0.326031	−0.163016	+	0.986624i	$0.552122\pi$
$840$	0	0
$841$	9.97056	0.343813
$842$	0	0
$843$	57.1127	1.96707
$844$	0	0
$845$	7.97056	0.274196
$846$	0	0
$847$	3.41421	0.117314
$848$	0	0
$849$	40.3848	1.38600
$850$	0	0
$851$	30.2132	1.03570
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	−11.6985	−0.399613	−0.199806	−	0.979835i	$-0.564031\pi$
$857$	−11.6985	−0.399613	−0.199806	+	0.979835i	$0.564031\pi$
$858$	0	0
$859$	−27.2843	−0.930927	−0.465464	−	0.885067i	$-0.654112\pi$
$859$	−27.2843	−0.930927	−0.465464	+	0.885067i	$0.654112\pi$
$860$	0	0
$861$	−13.6569	−0.465424
$862$	0	0
$863$	31.5980	1.07561	0.537804	−	0.843070i	$-0.319254\pi$
$863$	31.5980	1.07561	0.537804	+	0.843070i	$0.319254\pi$
$864$	0	0
$865$	−1.89949	−0.0645848
$866$	0	0
$867$	−12.8995	−0.438090
$868$	0	0
$869$	14.5858	0.494789
$870$	0	0
$871$	23.3553	0.791365
$872$	0	0
$873$	6.82843	0.231107
$874$	0	0
$875$	30.7279	1.03879
$876$	0	0
$877$	−38.1421	−1.28797	−0.643984	−	0.765039i	$-0.722720\pi$
$877$	−38.1421	−1.28797	−0.643984	+	0.765039i	$0.722720\pi$
$878$	0	0
$879$	−76.5269	−2.58119
$880$	0	0
$881$	−51.7696	−1.74416	−0.872080	−	0.489363i	$-0.837229\pi$
$881$	−51.7696	−1.74416	−0.872080	+	0.489363i	$0.837229\pi$
$882$	0	0
$883$	−50.4853	−1.69896	−0.849482	−	0.527617i	$-0.823086\pi$
$883$	−50.4853	−1.69896	−0.849482	+	0.527617i	$0.823086\pi$
$884$	0	0
$885$	−3.82843	−0.128691
$886$	0	0
$887$	−39.2548	−1.31805	−0.659024	−	0.752122i	$-0.729031\pi$
$887$	−39.2548	−1.31805	−0.659024	+	0.752122i	$0.729031\pi$
$888$	0	0
$889$	66.2843	2.22310
$890$	0	0
$891$	−9.48528	−0.317769
$892$	0	0
$893$	−8.82843	−0.295432
$894$	0	0
$895$	16.0711	0.537197
$896$	0	0
$897$	16.2426	0.542326
$898$	0	0
$899$	−40.0416	−1.33546
$900$	0	0
$901$	−15.3137	−0.510174
$902$	0	0
$903$	−2.82843	−0.0941242
$904$	0	0
$905$	19.3848	0.644372
$906$	0	0
$907$	21.8579	0.725778	0.362889	−	0.931832i	$-0.381790\pi$
$907$	21.8579	0.725778	0.362889	+	0.931832i	$0.381790\pi$
$908$	0	0
$909$	17.6569	0.585641
$910$	0	0
$911$	−38.4264	−1.27312	−0.636562	−	0.771226i	$-0.719644\pi$
$911$	−38.4264	−1.27312	−0.636562	+	0.771226i	$0.719644\pi$
$912$	0	0
$913$	−3.41421	−0.112994
$914$	0	0
$915$	26.7279	0.883598
$916$	0	0
$917$	−49.4558	−1.63318
$918$	0	0
$919$	56.3848	1.85996	0.929981	−	0.367607i	$-0.119823\pi$
$919$	56.3848	1.85996	0.929981	+	0.367607i	$0.119823\pi$
$920$	0	0
$921$	−17.8995	−0.589808
$922$	0	0
$923$	27.8406	0.916385
$924$	0	0
$925$	−40.2843	−1.32454
$926$	0	0
$927$	−38.6274	−1.26869
$928$	0	0
$929$	0.284271	0.00932664	0.00466332	−	0.999989i	$-0.498516\pi$
$929$	0.284271	0.00932664	0.00466332	+	0.999989i	$0.498516\pi$
$930$	0	0
$931$	4.65685	0.152622
$932$	0	0
$933$	28.1421	0.921332
$934$	0	0
$935$	−3.41421	−0.111657
$936$	0	0
$937$	−7.55635	−0.246855	−0.123428	−	0.992354i	$-0.539389\pi$
$937$	−7.55635	−0.246855	−0.123428	+	0.992354i	$0.539389\pi$
$938$	0	0
$939$	−33.7279	−1.10067
$940$	0	0
$941$	28.2426	0.920684	0.460342	−	0.887742i	$-0.347727\pi$
$941$	28.2426	0.920684	0.460342	+	0.887742i	$0.347727\pi$
$942$	0	0
$943$	−4.97056	−0.161864
$944$	0	0
$945$	1.41421	0.0460044
$946$	0	0
$947$	27.9706	0.908921	0.454461	−	0.890767i	$-0.349832\pi$
$947$	27.9706	0.908921	0.454461	+	0.890767i	$0.349832\pi$
$948$	0	0
$949$	−10.0589	−0.326525
$950$	0	0
$951$	−68.5980	−2.22444
$952$	0	0
$953$	6.10051	0.197615	0.0988074	−	0.995107i	$-0.468497\pi$
$953$	6.10051	0.197615	0.0988074	+	0.995107i	$0.468497\pi$
$954$	0	0
$955$	8.31371	0.269025
$956$	0	0
$957$	−15.0711	−0.487178
$958$	0	0
$959$	−9.07107	−0.292920
$960$	0	0
$961$	10.1421	0.327166
$962$	0	0
$963$	−44.2843	−1.42704
$964$	0	0
$965$	−1.17157	−0.0377143
$966$	0	0
$967$	−3.55635	−0.114364	−0.0571822	−	0.998364i	$-0.518212\pi$
$967$	−3.55635	−0.114364	−0.0571822	+	0.998364i	$0.518212\pi$
$968$	0	0
$969$	8.24264	0.264792
$970$	0	0
$971$	−0.272078	−0.00873140	−0.00436570	−	0.999990i	$-0.501390\pi$
$971$	−0.272078	−0.00873140	−0.00436570	+	0.999990i	$0.501390\pi$
$972$	0	0
$973$	−69.5980	−2.23121
$974$	0	0
$975$	−21.6569	−0.693574
$976$	0	0
$977$	42.8406	1.37059	0.685296	−	0.728264i	$-0.259673\pi$
$977$	42.8406	1.37059	0.685296	+	0.728264i	$0.259673\pi$
$978$	0	0
$979$	4.89949	0.156589
$980$	0	0
$981$	−0.970563	−0.0309877
$982$	0	0
$983$	−32.8995	−1.04933	−0.524665	−	0.851308i	$-0.675810\pi$
$983$	−32.8995	−1.04933	−0.524665	+	0.851308i	$0.675810\pi$
$984$	0	0
$985$	−15.8995	−0.506600
$986$	0	0
$987$	−72.7696	−2.31628
$988$	0	0
$989$	−1.02944	−0.0327342
$990$	0	0
$991$	31.3137	0.994713	0.497356	−	0.867546i	$-0.334304\pi$
$991$	31.3137	0.994713	0.497356	+	0.867546i	$0.334304\pi$
$992$	0	0
$993$	−82.5980	−2.62117
$994$	0	0
$995$	12.1421	0.384932
$996$	0	0
$997$	−6.44365	−0.204072	−0.102036	−	0.994781i	$-0.532536\pi$
$997$	−6.44365	−0.204072	−0.102036	+	0.994781i	$0.532536\pi$
$998$	0	0
$999$	−4.17157	−0.131983

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.n.1.2		2
4.3	odd	2		209.2.a.b.1.2	✓	2
12.11	even	2		1881.2.a.d.1.1		2
20.19	odd	2		5225.2.a.f.1.1		2
44.43	even	2		2299.2.a.f.1.1		2
76.75	even	2		3971.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.b.1.2	✓	2	4.3	odd	2
1881.2.a.d.1.1		2	12.11	even	2
2299.2.a.f.1.1		2	44.43	even	2
3344.2.a.n.1.2		2	1.1	even	1	trivial
3971.2.a.d.1.1		2	76.75	even	2
5225.2.a.f.1.1		2	20.19	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} -1.00000 q^{5} +3.41421 q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} -1.00000 q^{5} +3.41421 q^{7} +2.82843 q^{9} +1.00000 q^{11} +2.24264 q^{13} -2.41421 q^{15} +3.41421 q^{17} +1.00000 q^{19} +8.24264 q^{21} +3.00000 q^{23} -4.00000 q^{25} -0.414214 q^{27} -6.24264 q^{29} +6.41421 q^{31} +2.41421 q^{33} -3.41421 q^{35} +10.0711 q^{37} +5.41421 q^{39} -1.65685 q^{41} -0.343146 q^{43} -2.82843 q^{45} -8.82843 q^{47} +4.65685 q^{49} +8.24264 q^{51} -4.48528 q^{53} -1.00000 q^{55} +2.41421 q^{57} +1.58579 q^{59} -11.0711 q^{61} +9.65685 q^{63} -2.24264 q^{65} +10.4142 q^{67} +7.24264 q^{69} +12.4142 q^{71} -4.48528 q^{73} -9.65685 q^{75} +3.41421 q^{77} +14.5858 q^{79} -9.48528 q^{81} -3.41421 q^{83} -3.41421 q^{85} -15.0711 q^{87} +4.89949 q^{89} +7.65685 q^{91} +15.4853 q^{93} -1.00000 q^{95} +2.41421 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7	\( 2 q + 2 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{11} - 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{19} + 8 q^{21} + 6 q^{23} - 8 q^{25} + 2 q^{27} - 4 q^{29} + 10 q^{31} + 2 q^{33} - 4 q^{35} + 6 q^{37} + 8 q^{39} + 8 q^{41} - 12 q^{43} - 12 q^{47} - 2 q^{49} + 8 q^{51} + 8 q^{53} - 2 q^{55} + 2 q^{57} + 6 q^{59} - 8 q^{61} + 8 q^{63} + 4 q^{65} + 18 q^{67} + 6 q^{69} + 22 q^{71} + 8 q^{73} - 8 q^{75} + 4 q^{77} + 32 q^{79} - 2 q^{81} - 4 q^{83} - 4 q^{85} - 16 q^{87} - 10 q^{89} + 4 q^{91} + 14 q^{93} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^11 - 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^19 + 8 * q^21 + 6 * q^23 - 8 * q^25 + 2 * q^27 - 4 * q^29 + 10 * q^31 + 2 * q^33 - 4 * q^35 + 6 * q^37 + 8 * q^39 + 8 * q^41 - 12 * q^43 - 12 * q^47 - 2 * q^49 + 8 * q^51 + 8 * q^53 - 2 * q^55 + 2 * q^57 + 6 * q^59 - 8 * q^61 + 8 * q^63 + 4 * q^65 + 18 * q^67 + 6 * q^69 + 22 * q^71 + 8 * q^73 - 8 * q^75 + 4 * q^77 + 32 * q^79 - 2 * q^81 - 4 * q^83 - 4 * q^85 - 16 * q^87 - 10 * q^89 + 4 * q^91 + 14 * q^93 - 2 * q^95 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more