LMFDB - Embedded newform 1344.4.a.bl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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([0, 0, 0, 0]))

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

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

chi := DirichletCharacter("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.15207$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -20.3041 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -20.3041 q^{5} -7.00000 q^{7} +9.00000 q^{9} -30.9124 q^{11} -50.6083 q^{13} -60.9124 q^{15} -102.737 q^{17} -61.2165 q^{19} -21.0000 q^{21} -148.129 q^{23} +287.258 q^{25} +27.0000 q^{27} -159.217 q^{29} +121.217 q^{31} -92.7372 q^{33} +142.129 q^{35} +357.908 q^{37} -151.825 q^{39} +466.737 q^{41} -185.567 q^{43} -182.737 q^{45} +131.392 q^{47} +49.0000 q^{49} -308.212 q^{51} -200.175 q^{53} +627.650 q^{55} -183.650 q^{57} -591.908 q^{59} -70.5158 q^{61} -63.0000 q^{63} +1027.56 q^{65} -643.041 q^{67} -444.387 q^{69} +522.397 q^{71} -576.175 q^{73} +861.774 q^{75} +216.387 q^{77} -280.774 q^{79} +81.0000 q^{81} -557.732 q^{83} +2085.99 q^{85} -477.650 q^{87} -1228.89 q^{89} +354.258 q^{91} +363.650 q^{93} +1242.95 q^{95} +65.0413 q^{97} -278.212 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 14 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 14 * q^5 - 14 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - 14 q^{5} - 14 q^{7} + 18 q^{9} + 18 q^{11} - 48 q^{13} - 42 q^{15} + 34 q^{17} - 16 q^{19} - 42 q^{21} - 110 q^{23} + 202 q^{25} + 54 q^{27} - 212 q^{29} + 136 q^{31} + 54 q^{33} + 98 q^{35} + 24 q^{37} - 144 q^{39} + 694 q^{41} - 584 q^{43} - 126 q^{45} + 316 q^{47} + 98 q^{49} + 102 q^{51} - 560 q^{53} + 936 q^{55} - 48 q^{57} - 492 q^{59} + 604 q^{61} - 126 q^{63} + 1044 q^{65} - 1020 q^{67} - 330 q^{69} + 1710 q^{71} - 1312 q^{73} + 606 q^{75} - 126 q^{77} + 556 q^{79} + 162 q^{81} - 264 q^{83} + 2948 q^{85} - 636 q^{87} + 70 q^{89} + 336 q^{91} + 408 q^{93} + 1528 q^{95} - 136 q^{97} + 162 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 14 * q^5 - 14 * q^7 + 18 * q^9 + 18 * q^11 - 48 * q^13 - 42 * q^15 + 34 * q^17 - 16 * q^19 - 42 * q^21 - 110 * q^23 + 202 * q^25 + 54 * q^27 - 212 * q^29 + 136 * q^31 + 54 * q^33 + 98 * q^35 + 24 * q^37 - 144 * q^39 + 694 * q^41 - 584 * q^43 - 126 * q^45 + 316 * q^47 + 98 * q^49 + 102 * q^51 - 560 * q^53 + 936 * q^55 - 48 * q^57 - 492 * q^59 + 604 * q^61 - 126 * q^63 + 1044 * q^65 - 1020 * q^67 - 330 * q^69 + 1710 * q^71 - 1312 * q^73 + 606 * q^75 - 126 * q^77 + 556 * q^79 + 162 * q^81 - 264 * q^83 + 2948 * q^85 - 636 * q^87 + 70 * q^89 + 336 * q^91 + 408 * q^93 + 1528 * q^95 - 136 * q^97 + 162 * q^99

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−20.3041	−1.81606	−0.908029	−	0.418908i	$-0.862413\pi$
$5$	−20.3041	−1.81606	−0.908029	+	0.418908i	$0.862413\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−30.9124	−0.847313	−0.423656	−	0.905823i	$-0.639254\pi$
$11$	−30.9124	−0.847313	−0.423656	+	0.905823i	$0.639254\pi$
$12$	0	0
$13$	−50.6083	−1.07971	−0.539854	−	0.841759i	$-0.681521\pi$
$13$	−50.6083	−1.07971	−0.539854	+	0.841759i	$0.681521\pi$
$14$	0	0
$15$	−60.9124	−1.04850
$16$	0	0
$17$	−102.737	−1.46573	−0.732866	−	0.680373i	$-0.761818\pi$
$17$	−102.737	−1.46573	−0.732866	+	0.680373i	$0.761818\pi$
$18$	0	0
$19$	−61.2165	−0.739160	−0.369580	−	0.929199i	$-0.620498\pi$
$19$	−61.2165	−0.739160	−0.369580	+	0.929199i	$0.620498\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−148.129	−1.34291	−0.671457	−	0.741044i	$-0.734331\pi$
$23$	−148.129	−1.34291	−0.671457	+	0.741044i	$0.734331\pi$
$24$	0	0
$25$	287.258	2.29806
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−159.217	−1.01951	−0.509755	−	0.860320i	$-0.670264\pi$
$29$	−159.217	−1.01951	−0.509755	+	0.860320i	$0.670264\pi$
$30$	0	0
$31$	121.217	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$31$	121.217	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$32$	0	0
$33$	−92.7372	−0.489196
$34$	0	0
$35$	142.129	0.686405
$36$	0	0
$37$	357.908	1.59026	0.795130	−	0.606439i	$-0.207402\pi$
$37$	357.908	1.59026	0.795130	+	0.606439i	$0.207402\pi$
$38$	0	0
$39$	−151.825	−0.623370
$40$	0	0
$41$	466.737	1.77786	0.888928	−	0.458047i	$-0.151451\pi$
$41$	466.737	1.77786	0.888928	+	0.458047i	$0.151451\pi$
$42$	0	0
$43$	−185.567	−0.658109	−0.329055	−	0.944311i	$-0.606730\pi$
$43$	−185.567	−0.658109	−0.329055	+	0.944311i	$0.606730\pi$
$44$	0	0
$45$	−182.737	−0.605352
$46$	0	0
$47$	131.392	0.407776	0.203888	−	0.978994i	$-0.434642\pi$
$47$	131.392	0.407776	0.203888	+	0.978994i	$0.434642\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−308.212	−0.846240
$52$	0	0
$53$	−200.175	−0.518796	−0.259398	−	0.965771i	$-0.583524\pi$
$53$	−200.175	−0.518796	−0.259398	+	0.965771i	$0.583524\pi$
$54$	0	0
$55$	627.650	1.53877
$56$	0	0
$57$	−183.650	−0.426754
$58$	0	0
$59$	−591.908	−1.30610	−0.653049	−	0.757316i	$-0.726510\pi$
$59$	−591.908	−1.30610	−0.653049	+	0.757316i	$0.726510\pi$
$60$	0	0
$61$	−70.5158	−0.148010	−0.0740051	−	0.997258i	$-0.523578\pi$
$61$	−70.5158	−0.148010	−0.0740051	+	0.997258i	$0.523578\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	1027.56	1.96081
$66$	0	0
$67$	−643.041	−1.17254	−0.586269	−	0.810117i	$-0.699404\pi$
$67$	−643.041	−1.17254	−0.586269	+	0.810117i	$0.699404\pi$
$68$	0	0
$69$	−444.387	−0.775332
$70$	0	0
$71$	522.397	0.873198	0.436599	−	0.899656i	$-0.356183\pi$
$71$	522.397	0.873198	0.436599	+	0.899656i	$0.356183\pi$
$72$	0	0
$73$	−576.175	−0.923784	−0.461892	−	0.886936i	$-0.652829\pi$
$73$	−576.175	−0.923784	−0.461892	+	0.886936i	$0.652829\pi$
$74$	0	0
$75$	861.774	1.32679
$76$	0	0
$77$	216.387	0.320254
$78$	0	0
$79$	−280.774	−0.399867	−0.199934	−	0.979809i	$-0.564073\pi$
$79$	−280.774	−0.399867	−0.199934	+	0.979809i	$0.564073\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−557.732	−0.737579	−0.368790	−	0.929513i	$-0.620228\pi$
$83$	−557.732	−0.737579	−0.368790	+	0.929513i	$0.620228\pi$
$84$	0	0
$85$	2085.99	2.66185
$86$	0	0
$87$	−477.650	−0.588614
$88$	0	0
$89$	−1228.89	−1.46362	−0.731811	−	0.681508i	$-0.761325\pi$
$89$	−1228.89	−1.46362	−0.731811	+	0.681508i	$0.761325\pi$
$90$	0	0
$91$	354.258	0.408091
$92$	0	0
$93$	363.650	0.405470
$94$	0	0
$95$	1242.95	1.34236
$96$	0	0
$97$	65.0413	0.0680819	0.0340410	−	0.999420i	$-0.489162\pi$
$97$	65.0413	0.0680819	0.0340410	+	0.999420i	$0.489162\pi$
$98$	0	0
$99$	−278.212	−0.282438
$100$	0	0
$101$	−303.511	−0.299014	−0.149507	−	0.988761i	$-0.547769\pi$
$101$	−303.511	−0.299014	−0.149507	+	0.988761i	$0.547769\pi$
$102$	0	0
$103$	−174.268	−0.166710	−0.0833549	−	0.996520i	$-0.526564\pi$
$103$	−174.268	−0.166710	−0.0833549	+	0.996520i	$0.526564\pi$
$104$	0	0
$105$	426.387	0.396296
$106$	0	0
$107$	1945.42	1.75767	0.878835	−	0.477126i	$-0.158321\pi$
$107$	1945.42	1.75767	0.878835	+	0.477126i	$0.158321\pi$
$108$	0	0
$109$	1102.87	0.969132	0.484566	−	0.874755i	$-0.338977\pi$
$109$	1102.87	0.969132	0.484566	+	0.874755i	$0.338977\pi$
$110$	0	0
$111$	1073.72	0.918137
$112$	0	0
$113$	80.0827	0.0666685	0.0333343	−	0.999444i	$-0.489387\pi$
$113$	80.0827	0.0666685	0.0333343	+	0.999444i	$0.489387\pi$
$114$	0	0
$115$	3007.63	2.43881
$116$	0	0
$117$	−455.474	−0.359903
$118$	0	0
$119$	719.160	0.553994
$120$	0	0
$121$	−375.423	−0.282061
$122$	0	0
$123$	1400.21	1.02645
$124$	0	0
$125$	−3294.51	−2.35736
$126$	0	0
$127$	1981.12	1.38422	0.692112	−	0.721791i	$-0.256681\pi$
$127$	1981.12	1.38422	0.692112	+	0.721791i	$0.256681\pi$
$128$	0	0
$129$	−556.701	−0.379959
$130$	0	0
$131$	2854.25	1.90364	0.951820	−	0.306658i	$-0.0992107\pi$
$131$	2854.25	1.90364	0.951820	+	0.306658i	$0.0992107\pi$
$132$	0	0
$133$	428.516	0.279376
$134$	0	0
$135$	−548.212	−0.349500
$136$	0	0
$137$	−535.474	−0.333932	−0.166966	−	0.985963i	$-0.553397\pi$
$137$	−535.474	−0.333932	−0.166966	+	0.985963i	$0.553397\pi$
$138$	0	0
$139$	2326.43	1.41961	0.709804	−	0.704399i	$-0.248784\pi$
$139$	2326.43	1.41961	0.709804	+	0.704399i	$0.248784\pi$
$140$	0	0
$141$	394.175	0.235429
$142$	0	0
$143$	1564.42	0.914851
$144$	0	0
$145$	3232.75	1.85149
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	−2061.04	−1.13320	−0.566601	−	0.823992i	$-0.691742\pi$
$149$	−2061.04	−1.13320	−0.566601	+	0.823992i	$0.691742\pi$
$150$	0	0
$151$	1113.03	0.599849	0.299925	−	0.953963i	$-0.403038\pi$
$151$	1113.03	0.599849	0.299925	+	0.953963i	$0.403038\pi$
$152$	0	0
$153$	−924.635	−0.488577
$154$	0	0
$155$	−2461.20	−1.27541
$156$	0	0
$157$	1537.30	0.781464	0.390732	−	0.920505i	$-0.372222\pi$
$157$	1537.30	0.781464	0.390732	+	0.920505i	$0.372222\pi$
$158$	0	0
$159$	−600.526	−0.299527
$160$	0	0
$161$	1036.90	0.507574
$162$	0	0
$163$	−2043.52	−0.981967	−0.490984	−	0.871169i	$-0.663363\pi$
$163$	−2043.52	−0.981967	−0.490984	+	0.871169i	$0.663363\pi$
$164$	0	0
$165$	1882.95	0.888408
$166$	0	0
$167$	−2513.44	−1.16464	−0.582322	−	0.812958i	$-0.697856\pi$
$167$	−2513.44	−1.16464	−0.582322	+	0.812958i	$0.697856\pi$
$168$	0	0
$169$	364.197	0.165770
$170$	0	0
$171$	−550.949	−0.246387
$172$	0	0
$173$	−1591.00	−0.699197	−0.349599	−	0.936900i	$-0.613682\pi$
$173$	−1591.00	−0.699197	−0.349599	+	0.936900i	$0.613682\pi$
$174$	0	0
$175$	−2010.81	−0.868586
$176$	0	0
$177$	−1775.72	−0.754076
$178$	0	0
$179$	2552.27	1.06573	0.532866	−	0.846200i	$-0.321115\pi$
$179$	2552.27	1.06573	0.532866	+	0.846200i	$0.321115\pi$
$180$	0	0
$181$	3361.33	1.38036	0.690182	−	0.723636i	$-0.257530\pi$
$181$	3361.33	1.38036	0.690182	+	0.723636i	$0.257530\pi$
$182$	0	0
$183$	−211.547	−0.0854537
$184$	0	0
$185$	−7267.00	−2.88800
$186$	0	0
$187$	3175.85	1.24193
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	3868.64	1.46558	0.732789	−	0.680456i	$-0.238218\pi$
$191$	3868.64	1.46558	0.732789	+	0.680456i	$0.238218\pi$
$192$	0	0
$193$	2693.17	1.00445	0.502224	−	0.864738i	$-0.332515\pi$
$193$	2693.17	1.00445	0.502224	+	0.864738i	$0.332515\pi$
$194$	0	0
$195$	3082.67	1.13208
$196$	0	0
$197$	1064.86	0.385116	0.192558	−	0.981286i	$-0.438322\pi$
$197$	1064.86	0.385116	0.192558	+	0.981286i	$0.438322\pi$
$198$	0	0
$199$	−1625.92	−0.579187	−0.289594	−	0.957150i	$-0.593520\pi$
$199$	−1625.92	−0.579187	−0.289594	+	0.957150i	$0.593520\pi$
$200$	0	0
$201$	−1929.12	−0.676965
$202$	0	0
$203$	1114.52	0.385338
$204$	0	0
$205$	−9476.70	−3.22869
$206$	0	0
$207$	−1333.16	−0.447638
$208$	0	0
$209$	1892.35	0.626300
$210$	0	0
$211$	1138.04	0.371309	0.185654	−	0.982615i	$-0.440560\pi$
$211$	1138.04	0.371309	0.185654	+	0.982615i	$0.440560\pi$
$212$	0	0
$213$	1567.19	0.504141
$214$	0	0
$215$	3767.78	1.19516
$216$	0	0
$217$	−848.516	−0.265442
$218$	0	0
$219$	−1728.53	−0.533347
$220$	0	0
$221$	5199.35	1.58256
$222$	0	0
$223$	−3535.78	−1.06176	−0.530881	−	0.847446i	$-0.678139\pi$
$223$	−3535.78	−1.06176	−0.530881	+	0.847446i	$0.678139\pi$
$224$	0	0
$225$	2585.32	0.766021
$226$	0	0
$227$	−3055.21	−0.893309	−0.446655	−	0.894706i	$-0.647385\pi$
$227$	−3055.21	−0.893309	−0.446655	+	0.894706i	$0.647385\pi$
$228$	0	0
$229$	321.703	0.0928329	0.0464164	−	0.998922i	$-0.485220\pi$
$229$	321.703	0.0928329	0.0464164	+	0.998922i	$0.485220\pi$
$230$	0	0
$231$	649.160	0.184899
$232$	0	0
$233$	−906.939	−0.255002	−0.127501	−	0.991838i	$-0.540696\pi$
$233$	−906.939	−0.255002	−0.127501	+	0.991838i	$0.540696\pi$
$234$	0	0
$235$	−2667.80	−0.740544
$236$	0	0
$237$	−842.321	−0.230863
$238$	0	0
$239$	4290.84	1.16130	0.580651	−	0.814152i	$-0.302798\pi$
$239$	4290.84	1.16130	0.580651	+	0.814152i	$0.302798\pi$
$240$	0	0
$241$	−6985.64	−1.86716	−0.933579	−	0.358373i	$-0.883332\pi$
$241$	−6985.64	−1.86716	−0.933579	+	0.358373i	$0.883332\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−994.903	−0.259437
$246$	0	0
$247$	3098.06	0.798077
$248$	0	0
$249$	−1673.20	−0.425842
$250$	0	0
$251$	−5666.63	−1.42500	−0.712499	−	0.701673i	$-0.752437\pi$
$251$	−5666.63	−1.42500	−0.712499	+	0.701673i	$0.752437\pi$
$252$	0	0
$253$	4579.02	1.13787
$254$	0	0
$255$	6257.97	1.53682
$256$	0	0
$257$	840.596	0.204027	0.102013	−	0.994783i	$-0.467472\pi$
$257$	840.596	0.204027	0.102013	+	0.994783i	$0.467472\pi$
$258$	0	0
$259$	−2505.35	−0.601062
$260$	0	0
$261$	−1432.95	−0.339836
$262$	0	0
$263$	−3386.69	−0.794039	−0.397019	−	0.917810i	$-0.629955\pi$
$263$	−3386.69	−0.794039	−0.397019	+	0.917810i	$0.629955\pi$
$264$	0	0
$265$	4064.38	0.942163
$266$	0	0
$267$	−3686.68	−0.845023
$268$	0	0
$269$	−7929.42	−1.79727	−0.898635	−	0.438698i	$-0.855440\pi$
$269$	−7929.42	−1.79727	−0.898635	+	0.438698i	$0.855440\pi$
$270$	0	0
$271$	−5080.25	−1.13876	−0.569379	−	0.822075i	$-0.692816\pi$
$271$	−5080.25	−1.13876	−0.569379	+	0.822075i	$0.692816\pi$
$272$	0	0
$273$	1062.77	0.235612
$274$	0	0
$275$	−8879.83	−1.94718
$276$	0	0
$277$	195.329	0.0423688	0.0211844	−	0.999776i	$-0.493256\pi$
$277$	195.329	0.0423688	0.0211844	+	0.999776i	$0.493256\pi$
$278$	0	0
$279$	1090.95	0.234098
$280$	0	0
$281$	4029.83	0.855514	0.427757	−	0.903894i	$-0.359304\pi$
$281$	4029.83	0.855514	0.427757	+	0.903894i	$0.359304\pi$
$282$	0	0
$283$	6927.28	1.45507	0.727534	−	0.686072i	$-0.240667\pi$
$283$	6927.28	1.45507	0.727534	+	0.686072i	$0.240667\pi$
$284$	0	0
$285$	3728.85	0.775010
$286$	0	0
$287$	−3267.16	−0.671967
$288$	0	0
$289$	5641.93	1.14837
$290$	0	0
$291$	195.124	0.0393071
$292$	0	0
$293$	911.009	0.181644	0.0908221	−	0.995867i	$-0.471051\pi$
$293$	911.009	0.181644	0.0908221	+	0.995867i	$0.471051\pi$
$294$	0	0
$295$	12018.2	2.37195
$296$	0	0
$297$	−834.635	−0.163065
$298$	0	0
$299$	7496.55	1.44996
$300$	0	0
$301$	1298.97	0.248742
$302$	0	0
$303$	−910.533	−0.172636
$304$	0	0
$305$	1431.76	0.268795
$306$	0	0
$307$	3310.72	0.615482	0.307741	−	0.951470i	$-0.400427\pi$
$307$	3310.72	0.615482	0.307741	+	0.951470i	$0.400427\pi$
$308$	0	0
$309$	−522.803	−0.0962499
$310$	0	0
$311$	−4979.13	−0.907847	−0.453924	−	0.891041i	$-0.649976\pi$
$311$	−4979.13	−0.907847	−0.453924	+	0.891041i	$0.649976\pi$
$312$	0	0
$313$	−5116.19	−0.923911	−0.461955	−	0.886903i	$-0.652852\pi$
$313$	−5116.19	−0.923911	−0.461955	+	0.886903i	$0.652852\pi$
$314$	0	0
$315$	1279.16	0.228802
$316$	0	0
$317$	−978.317	−0.173337	−0.0866684	−	0.996237i	$-0.527622\pi$
$317$	−978.317	−0.173337	−0.0866684	+	0.996237i	$0.527622\pi$
$318$	0	0
$319$	4921.77	0.863843
$320$	0	0
$321$	5836.26	1.01479
$322$	0	0
$323$	6289.22	1.08341
$324$	0	0
$325$	−14537.6	−2.48124
$326$	0	0
$327$	3308.60	0.559529
$328$	0	0
$329$	−919.742	−0.154125
$330$	0	0
$331$	9092.36	1.50985	0.754926	−	0.655810i	$-0.227673\pi$
$331$	9092.36	1.50985	0.754926	+	0.655810i	$0.227673\pi$
$332$	0	0
$333$	3221.17	0.530087
$334$	0	0
$335$	13056.4	2.12939
$336$	0	0
$337$	−621.406	−0.100445	−0.0502227	−	0.998738i	$-0.515993\pi$
$337$	−621.406	−0.100445	−0.0502227	+	0.998738i	$0.515993\pi$
$338$	0	0
$339$	240.248	0.0384911
$340$	0	0
$341$	−3747.09	−0.595063
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	9022.89	1.40805
$346$	0	0
$347$	4276.85	0.661652	0.330826	−	0.943692i	$-0.392673\pi$
$347$	4276.85	0.661652	0.330826	+	0.943692i	$0.392673\pi$
$348$	0	0
$349$	7497.41	1.14993	0.574967	−	0.818177i	$-0.305015\pi$
$349$	7497.41	1.14993	0.574967	+	0.818177i	$0.305015\pi$
$350$	0	0
$351$	−1366.42	−0.207790
$352$	0	0
$353$	−2295.77	−0.346151	−0.173076	−	0.984909i	$-0.555371\pi$
$353$	−2295.77	−0.346151	−0.173076	+	0.984909i	$0.555371\pi$
$354$	0	0
$355$	−10606.8	−1.58578
$356$	0	0
$357$	2157.48	0.319849
$358$	0	0
$359$	−681.808	−0.100235	−0.0501176	−	0.998743i	$-0.515960\pi$
$359$	−681.808	−0.100235	−0.0501176	+	0.998743i	$0.515960\pi$
$360$	0	0
$361$	−3111.54	−0.453643
$362$	0	0
$363$	−1126.27	−0.162848
$364$	0	0
$365$	11698.7	1.67764
$366$	0	0
$367$	−8497.77	−1.20867	−0.604333	−	0.796732i	$-0.706560\pi$
$367$	−8497.77	−1.20867	−0.604333	+	0.796732i	$0.706560\pi$
$368$	0	0
$369$	4200.63	0.592619
$370$	0	0
$371$	1401.23	0.196086
$372$	0	0
$373$	−12306.1	−1.70828	−0.854138	−	0.520046i	$-0.825915\pi$
$373$	−12306.1	−1.70828	−0.854138	+	0.520046i	$0.825915\pi$
$374$	0	0
$375$	−9883.52	−1.36102
$376$	0	0
$377$	8057.67	1.10077
$378$	0	0
$379$	7064.19	0.957423	0.478711	−	0.877972i	$-0.341104\pi$
$379$	7064.19	0.957423	0.478711	+	0.877972i	$0.341104\pi$
$380$	0	0
$381$	5943.37	0.799182
$382$	0	0
$383$	−3082.91	−0.411304	−0.205652	−	0.978625i	$-0.565931\pi$
$383$	−3082.91	−0.411304	−0.205652	+	0.978625i	$0.565931\pi$
$384$	0	0
$385$	−4393.55	−0.581600
$386$	0	0
$387$	−1670.10	−0.219370
$388$	0	0
$389$	−9990.11	−1.30210	−0.651052	−	0.759033i	$-0.725672\pi$
$389$	−9990.11	−1.30210	−0.651052	+	0.759033i	$0.725672\pi$
$390$	0	0
$391$	15218.4	1.96835
$392$	0	0
$393$	8562.74	1.09907
$394$	0	0
$395$	5700.87	0.726182
$396$	0	0
$397$	−8327.91	−1.05281	−0.526405	−	0.850234i	$-0.676460\pi$
$397$	−8327.91	−1.05281	−0.526405	+	0.850234i	$0.676460\pi$
$398$	0	0
$399$	1285.55	0.161298
$400$	0	0
$401$	−9715.82	−1.20994	−0.604969	−	0.796249i	$-0.706815\pi$
$401$	−9715.82	−1.20994	−0.604969	+	0.796249i	$0.706815\pi$
$402$	0	0
$403$	−6134.56	−0.758273
$404$	0	0
$405$	−1644.63	−0.201784
$406$	0	0
$407$	−11063.8	−1.34745
$408$	0	0
$409$	13341.8	1.61299	0.806493	−	0.591244i	$-0.201363\pi$
$409$	13341.8	1.61299	0.806493	+	0.591244i	$0.201363\pi$
$410$	0	0
$411$	−1606.42	−0.192796
$412$	0	0
$413$	4143.35	0.493659
$414$	0	0
$415$	11324.3	1.33949
$416$	0	0
$417$	6979.30	0.819611
$418$	0	0
$419$	4281.34	0.499181	0.249591	−	0.968351i	$-0.419704\pi$
$419$	4281.34	0.499181	0.249591	+	0.968351i	$0.419704\pi$
$420$	0	0
$421$	3296.60	0.381631	0.190815	−	0.981626i	$-0.438887\pi$
$421$	3296.60	0.381631	0.190815	+	0.981626i	$0.438887\pi$
$422$	0	0
$423$	1182.53	0.135925
$424$	0	0
$425$	−29512.1	−3.36834
$426$	0	0
$427$	493.610	0.0559426
$428$	0	0
$429$	4693.27	0.528189
$430$	0	0
$431$	−4122.73	−0.460754	−0.230377	−	0.973101i	$-0.573996\pi$
$431$	−4122.73	−0.460754	−0.230377	+	0.973101i	$0.573996\pi$
$432$	0	0
$433$	1070.60	0.118822	0.0594110	−	0.998234i	$-0.481078\pi$
$433$	1070.60	0.118822	0.0594110	+	0.998234i	$0.481078\pi$
$434$	0	0
$435$	9698.26	1.06896
$436$	0	0
$437$	9067.94	0.992628
$438$	0	0
$439$	−6.22847	−0.000677150 0	−0.000338575	−	1.00000i	$-0.500108\pi$
$439$	−6.22847	−0.000677150 0	−0.000338575		1.00000i	$0.500108\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−6835.85	−0.733140	−0.366570	−	0.930390i	$-0.619468\pi$
$443$	−6835.85	−0.733140	−0.366570	+	0.930390i	$0.619468\pi$
$444$	0	0
$445$	24951.6	2.65802
$446$	0	0
$447$	−6183.12	−0.654254
$448$	0	0
$449$	5590.94	0.587646	0.293823	−	0.955860i	$-0.405072\pi$
$449$	5590.94	0.587646	0.293823	+	0.955860i	$0.405072\pi$
$450$	0	0
$451$	−14428.0	−1.50640
$452$	0	0
$453$	3339.09	0.346323
$454$	0	0
$455$	−7192.90	−0.741117
$456$	0	0
$457$	−7615.65	−0.779530	−0.389765	−	0.920914i	$-0.627444\pi$
$457$	−7615.65	−0.779530	−0.389765	+	0.920914i	$0.627444\pi$
$458$	0	0
$459$	−2773.90	−0.282080
$460$	0	0
$461$	−12191.7	−1.23172	−0.615860	−	0.787855i	$-0.711192\pi$
$461$	−12191.7	−1.23172	−0.615860	+	0.787855i	$0.711192\pi$
$462$	0	0
$463$	−4210.28	−0.422610	−0.211305	−	0.977420i	$-0.567771\pi$
$463$	−4210.28	−0.422610	−0.211305	+	0.977420i	$0.567771\pi$
$464$	0	0
$465$	−7383.59	−0.736357
$466$	0	0
$467$	16128.2	1.59812	0.799062	−	0.601248i	$-0.205330\pi$
$467$	16128.2	1.59812	0.799062	+	0.601248i	$0.205330\pi$
$468$	0	0
$469$	4501.29	0.443177
$470$	0	0
$471$	4611.90	0.451178
$472$	0	0
$473$	5736.32	0.557624
$474$	0	0
$475$	−17584.9	−1.69864
$476$	0	0
$477$	−1801.58	−0.172932
$478$	0	0
$479$	3736.70	0.356438	0.178219	−	0.983991i	$-0.442966\pi$
$479$	3736.70	0.356438	0.178219	+	0.983991i	$0.442966\pi$
$480$	0	0
$481$	−18113.1	−1.71702
$482$	0	0
$483$	3110.71	0.293048
$484$	0	0
$485$	−1320.61	−0.123641
$486$	0	0
$487$	−15812.5	−1.47132	−0.735661	−	0.677350i	$-0.763128\pi$
$487$	−15812.5	−1.47132	−0.735661	+	0.677350i	$0.763128\pi$
$488$	0	0
$489$	−6130.55	−0.566939
$490$	0	0
$491$	2564.56	0.235717	0.117858	−	0.993030i	$-0.462397\pi$
$491$	2564.56	0.235717	0.117858	+	0.993030i	$0.462397\pi$
$492$	0	0
$493$	16357.5	1.49433
$494$	0	0
$495$	5648.85	0.512923
$496$	0	0
$497$	−3656.78	−0.330038
$498$	0	0
$499$	−8511.36	−0.763569	−0.381784	−	0.924251i	$-0.624690\pi$
$499$	−8511.36	−0.763569	−0.381784	+	0.924251i	$0.624690\pi$
$500$	0	0
$501$	−7540.31	−0.672407
$502$	0	0
$503$	−6486.89	−0.575022	−0.287511	−	0.957777i	$-0.592828\pi$
$503$	−6486.89	−0.575022	−0.287511	+	0.957777i	$0.592828\pi$
$504$	0	0
$505$	6162.53	0.543027
$506$	0	0
$507$	1092.59	0.0957074
$508$	0	0
$509$	13980.3	1.21742	0.608710	−	0.793393i	$-0.291687\pi$
$509$	13980.3	1.21742	0.608710	+	0.793393i	$0.291687\pi$
$510$	0	0
$511$	4033.23	0.349157
$512$	0	0
$513$	−1652.85	−0.142251
$514$	0	0
$515$	3538.35	0.302754
$516$	0	0
$517$	−4061.63	−0.345513
$518$	0	0
$519$	−4772.99	−0.403682
$520$	0	0
$521$	−10684.8	−0.898485	−0.449242	−	0.893410i	$-0.648306\pi$
$521$	−10684.8	−0.898485	−0.449242	+	0.893410i	$0.648306\pi$
$522$	0	0
$523$	2500.29	0.209044	0.104522	−	0.994523i	$-0.466669\pi$
$523$	2500.29	0.209044	0.104522	+	0.994523i	$0.466669\pi$
$524$	0	0
$525$	−6032.42	−0.501478
$526$	0	0
$527$	−12453.4	−1.02938
$528$	0	0
$529$	9775.18	0.803418
$530$	0	0
$531$	−5327.17	−0.435366
$532$	0	0
$533$	−23620.8	−1.91957
$534$	0	0
$535$	−39500.0	−3.19203
$536$	0	0
$537$	7656.82	0.615300
$538$	0	0
$539$	−1514.71	−0.121045
$540$	0	0
$541$	8780.94	0.697823	0.348911	−	0.937156i	$-0.386551\pi$
$541$	8780.94	0.697823	0.348911	+	0.937156i	$0.386551\pi$
$542$	0	0
$543$	10084.0	0.796954
$544$	0	0
$545$	−22392.7	−1.76000
$546$	0	0
$547$	−10965.6	−0.857140	−0.428570	−	0.903509i	$-0.640982\pi$
$547$	−10965.6	−0.857140	−0.428570	+	0.903509i	$0.640982\pi$
$548$	0	0
$549$	−634.642	−0.0493367
$550$	0	0
$551$	9746.69	0.753580
$552$	0	0
$553$	1965.42	0.151136
$554$	0	0
$555$	−21801.0	−1.66739
$556$	0	0
$557$	−3156.31	−0.240103	−0.120051	−	0.992768i	$-0.538306\pi$
$557$	−3156.31	−0.240103	−0.120051	+	0.992768i	$0.538306\pi$
$558$	0	0
$559$	9391.22	0.710566
$560$	0	0
$561$	9527.56	0.717030
$562$	0	0
$563$	−16805.3	−1.25801	−0.629003	−	0.777403i	$-0.716536\pi$
$563$	−16805.3	−1.25801	−0.629003	+	0.777403i	$0.716536\pi$
$564$	0	0
$565$	−1626.01	−0.121074
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−16899.6	−1.24511	−0.622557	−	0.782575i	$-0.713906\pi$
$569$	−16899.6	−1.24511	−0.622557	+	0.782575i	$0.713906\pi$
$570$	0	0
$571$	−7998.44	−0.586207	−0.293104	−	0.956081i	$-0.594688\pi$
$571$	−7998.44	−0.586207	−0.293104	+	0.956081i	$0.594688\pi$
$572$	0	0
$573$	11605.9	0.846152
$574$	0	0
$575$	−42551.2	−3.08610
$576$	0	0
$577$	23385.2	1.68724	0.843619	−	0.536942i	$-0.180421\pi$
$577$	23385.2	1.68724	0.843619	+	0.536942i	$0.180421\pi$
$578$	0	0
$579$	8079.50	0.579918
$580$	0	0
$581$	3904.13	0.278779
$582$	0	0
$583$	6187.90	0.439582
$584$	0	0
$585$	9248.01	0.653604
$586$	0	0
$587$	−7583.02	−0.533194	−0.266597	−	0.963808i	$-0.585899\pi$
$587$	−7583.02	−0.533194	−0.266597	+	0.963808i	$0.585899\pi$
$588$	0	0
$589$	−7420.46	−0.519108
$590$	0	0
$591$	3194.57	0.222347
$592$	0	0
$593$	10593.1	0.733567	0.366783	−	0.930306i	$-0.380459\pi$
$593$	10593.1	0.733567	0.366783	+	0.930306i	$0.380459\pi$
$594$	0	0
$595$	−14601.9	−1.00609
$596$	0	0
$597$	−4877.75	−0.334394
$598$	0	0
$599$	22125.1	1.50919	0.754597	−	0.656188i	$-0.227832\pi$
$599$	22125.1	1.50919	0.754597	+	0.656188i	$0.227832\pi$
$600$	0	0
$601$	−19841.6	−1.34669	−0.673343	−	0.739331i	$-0.735142\pi$
$601$	−19841.6	−1.34669	−0.673343	+	0.739331i	$0.735142\pi$
$602$	0	0
$603$	−5787.37	−0.390846
$604$	0	0
$605$	7622.64	0.512239
$606$	0	0
$607$	−15507.5	−1.03695	−0.518475	−	0.855093i	$-0.673500\pi$
$607$	−15507.5	−1.03695	−0.518475	+	0.855093i	$0.673500\pi$
$608$	0	0
$609$	3343.55	0.222475
$610$	0	0
$611$	−6649.51	−0.440279
$612$	0	0
$613$	14042.8	0.925255	0.462628	−	0.886553i	$-0.346907\pi$
$613$	14042.8	0.925255	0.462628	+	0.886553i	$0.346907\pi$
$614$	0	0
$615$	−28430.1	−1.86408
$616$	0	0
$617$	13175.9	0.859708	0.429854	−	0.902898i	$-0.358565\pi$
$617$	13175.9	0.859708	0.429854	+	0.902898i	$0.358565\pi$
$618$	0	0
$619$	10828.3	0.703109	0.351555	−	0.936167i	$-0.385653\pi$
$619$	10828.3	0.703109	0.351555	+	0.936167i	$0.385653\pi$
$620$	0	0
$621$	−3999.48	−0.258444
$622$	0	0
$623$	8602.25	0.553197
$624$	0	0
$625$	30984.9	1.98303
$626$	0	0
$627$	5677.05	0.361594
$628$	0	0
$629$	−36770.4	−2.33089
$630$	0	0
$631$	23922.2	1.50923	0.754617	−	0.656166i	$-0.227823\pi$
$631$	23922.2	1.50923	0.754617	+	0.656166i	$0.227823\pi$
$632$	0	0
$633$	3414.13	0.214375
$634$	0	0
$635$	−40225.0	−2.51383
$636$	0	0
$637$	−2479.81	−0.154244
$638$	0	0
$639$	4701.57	0.291066
$640$	0	0
$641$	−21962.6	−1.35331	−0.676654	−	0.736301i	$-0.736571\pi$
$641$	−21962.6	−1.35331	−0.676654	+	0.736301i	$0.736571\pi$
$642$	0	0
$643$	17927.1	1.09950	0.549749	−	0.835330i	$-0.314724\pi$
$643$	17927.1	1.09950	0.549749	+	0.835330i	$0.314724\pi$
$644$	0	0
$645$	11303.3	0.690028
$646$	0	0
$647$	−850.926	−0.0517054	−0.0258527	−	0.999666i	$-0.508230\pi$
$647$	−850.926	−0.0517054	−0.0258527	+	0.999666i	$0.508230\pi$
$648$	0	0
$649$	18297.3	1.10667
$650$	0	0
$651$	−2545.55	−0.153253
$652$	0	0
$653$	−18674.9	−1.11915	−0.559575	−	0.828780i	$-0.689036\pi$
$653$	−18674.9	−1.11915	−0.559575	+	0.828780i	$0.689036\pi$
$654$	0	0
$655$	−57953.0	−3.45712
$656$	0	0
$657$	−5185.58	−0.307928
$658$	0	0
$659$	−777.121	−0.0459368	−0.0229684	−	0.999736i	$-0.507312\pi$
$659$	−777.121	−0.0459368	−0.0229684	+	0.999736i	$0.507312\pi$
$660$	0	0
$661$	−21963.4	−1.29240	−0.646201	−	0.763167i	$-0.723643\pi$
$661$	−21963.4	−1.29240	−0.646201	+	0.763167i	$0.723643\pi$
$662$	0	0
$663$	15598.1	0.913693
$664$	0	0
$665$	−8700.64	−0.507363
$666$	0	0
$667$	23584.6	1.36911
$668$	0	0
$669$	−10607.3	−0.613009
$670$	0	0
$671$	2179.81	0.125411
$672$	0	0
$673$	29380.0	1.68279	0.841393	−	0.540423i	$-0.181736\pi$
$673$	29380.0	1.68279	0.841393	+	0.540423i	$0.181736\pi$
$674$	0	0
$675$	7755.96	0.442262
$676$	0	0
$677$	−7898.58	−0.448400	−0.224200	−	0.974543i	$-0.571977\pi$
$677$	−7898.58	−0.448400	−0.224200	+	0.974543i	$0.571977\pi$
$678$	0	0
$679$	−455.289	−0.0257326
$680$	0	0
$681$	−9165.62	−0.515752
$682$	0	0
$683$	3097.04	0.173507	0.0867533	−	0.996230i	$-0.472351\pi$
$683$	3097.04	0.173507	0.0867533	+	0.996230i	$0.472351\pi$
$684$	0	0
$685$	10872.3	0.606439
$686$	0	0
$687$	965.109	0.0535971
$688$	0	0
$689$	10130.5	0.560148
$690$	0	0
$691$	26926.3	1.48238	0.741189	−	0.671296i	$-0.234262\pi$
$691$	26926.3	1.48238	0.741189	+	0.671296i	$0.234262\pi$
$692$	0	0
$693$	1947.48	0.106751
$694$	0	0
$695$	−47236.2	−2.57809
$696$	0	0
$697$	−47951.3	−2.60586
$698$	0	0
$699$	−2720.82	−0.147226
$700$	0	0
$701$	16209.3	0.873347	0.436674	−	0.899620i	$-0.356156\pi$
$701$	16209.3	0.873347	0.436674	+	0.899620i	$0.356156\pi$
$702$	0	0
$703$	−21909.9	−1.17546
$704$	0	0
$705$	−8003.39	−0.427553
$706$	0	0
$707$	2124.58	0.113017
$708$	0	0
$709$	8277.60	0.438465	0.219233	−	0.975673i	$-0.429645\pi$
$709$	8277.60	0.438465	0.219233	+	0.975673i	$0.429645\pi$
$710$	0	0
$711$	−2526.96	−0.133289
$712$	0	0
$713$	−17955.7	−0.943121
$714$	0	0
$715$	−31764.3	−1.66142
$716$	0	0
$717$	12872.5	0.670478
$718$	0	0
$719$	2334.31	0.121078	0.0605389	−	0.998166i	$-0.480718\pi$
$719$	2334.31	0.121078	0.0605389	+	0.998166i	$0.480718\pi$
$720$	0	0
$721$	1219.87	0.0630104
$722$	0	0
$723$	−20956.9	−1.07800
$724$	0	0
$725$	−45736.2	−2.34290
$726$	0	0
$727$	18885.8	0.963462	0.481731	−	0.876319i	$-0.340008\pi$
$727$	18885.8	0.963462	0.481731	+	0.876319i	$0.340008\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	19064.6	0.964611
$732$	0	0
$733$	−3071.96	−0.154796	−0.0773980	−	0.997000i	$-0.524661\pi$
$733$	−3071.96	−0.154796	−0.0773980	+	0.997000i	$0.524661\pi$
$734$	0	0
$735$	−2984.71	−0.149786
$736$	0	0
$737$	19878.0	0.993506
$738$	0	0
$739$	−27388.4	−1.36333	−0.681663	−	0.731666i	$-0.738743\pi$
$739$	−27388.4	−1.36333	−0.681663	+	0.731666i	$0.738743\pi$
$740$	0	0
$741$	9294.19	0.460770
$742$	0	0
$743$	30541.6	1.50803	0.754013	−	0.656859i	$-0.228115\pi$
$743$	30541.6	1.50803	0.754013	+	0.656859i	$0.228115\pi$
$744$	0	0
$745$	41847.7	2.05796
$746$	0	0
$747$	−5019.59	−0.245860
$748$	0	0
$749$	−13617.9	−0.664337
$750$	0	0
$751$	36868.0	1.79139	0.895695	−	0.444670i	$-0.146679\pi$
$751$	36868.0	1.79139	0.895695	+	0.444670i	$0.146679\pi$
$752$	0	0
$753$	−16999.9	−0.822723
$754$	0	0
$755$	−22599.1	−1.08936
$756$	0	0
$757$	−1804.07	−0.0866184	−0.0433092	−	0.999062i	$-0.513790\pi$
$757$	−1804.07	−0.0866184	−0.0433092	+	0.999062i	$0.513790\pi$
$758$	0	0
$759$	13737.1	0.656948
$760$	0	0
$761$	−17838.8	−0.849746	−0.424873	−	0.905253i	$-0.639681\pi$
$761$	−17838.8	−0.849746	−0.424873	+	0.905253i	$0.639681\pi$
$762$	0	0
$763$	−7720.06	−0.366298
$764$	0	0
$765$	18773.9	0.887284
$766$	0	0
$767$	29955.4	1.41021
$768$	0	0
$769$	6804.58	0.319089	0.159544	−	0.987191i	$-0.448997\pi$
$769$	6804.58	0.319089	0.159544	+	0.987191i	$0.448997\pi$
$770$	0	0
$771$	2521.79	0.117795
$772$	0	0
$773$	23600.9	1.09814	0.549072	−	0.835775i	$-0.314981\pi$
$773$	23600.9	1.09814	0.549072	+	0.835775i	$0.314981\pi$
$774$	0	0
$775$	34820.4	1.61392
$776$	0	0
$777$	−7516.06	−0.347023
$778$	0	0
$779$	−28572.0	−1.31412
$780$	0	0
$781$	−16148.5	−0.739872
$782$	0	0
$783$	−4298.85	−0.196205
$784$	0	0
$785$	−31213.5	−1.41918
$786$	0	0
$787$	9183.98	0.415977	0.207988	−	0.978131i	$-0.433308\pi$
$787$	9183.98	0.415977	0.207988	+	0.978131i	$0.433308\pi$
$788$	0	0
$789$	−10160.1	−0.458438
$790$	0	0
$791$	−560.579	−0.0251983
$792$	0	0
$793$	3568.68	0.159808
$794$	0	0
$795$	12193.2	0.543958
$796$	0	0
$797$	−11076.9	−0.492302	−0.246151	−	0.969231i	$-0.579166\pi$
$797$	−11076.9	−0.492302	−0.246151	+	0.969231i	$0.579166\pi$
$798$	0	0
$799$	−13498.8	−0.597690
$800$	0	0
$801$	−11060.0	−0.487874
$802$	0	0
$803$	17811.0	0.782734
$804$	0	0
$805$	−21053.4	−0.921783
$806$	0	0
$807$	−23788.3	−1.03765
$808$	0	0
$809$	32258.3	1.40190	0.700952	−	0.713209i	$-0.252759\pi$
$809$	32258.3	1.40190	0.700952	+	0.713209i	$0.252759\pi$
$810$	0	0
$811$	17693.9	0.766114	0.383057	−	0.923725i	$-0.374871\pi$
$811$	17693.9	0.766114	0.383057	+	0.923725i	$0.374871\pi$
$812$	0	0
$813$	−15240.8	−0.657462
$814$	0	0
$815$	41491.9	1.78331
$816$	0	0
$817$	11359.8	0.486448
$818$	0	0
$819$	3188.32	0.136030
$820$	0	0
$821$	18430.7	0.783480	0.391740	−	0.920076i	$-0.371873\pi$
$821$	18430.7	0.783480	0.391740	+	0.920076i	$0.371873\pi$
$822$	0	0
$823$	32285.9	1.36746	0.683729	−	0.729736i	$-0.260357\pi$
$823$	32285.9	1.36746	0.683729	+	0.729736i	$0.260357\pi$
$824$	0	0
$825$	−26639.5	−1.12420
$826$	0	0
$827$	36266.1	1.52491	0.762453	−	0.647044i	$-0.223995\pi$
$827$	36266.1	1.52491	0.762453	+	0.647044i	$0.223995\pi$
$828$	0	0
$829$	31156.2	1.30531	0.652654	−	0.757656i	$-0.273655\pi$
$829$	31156.2	1.30531	0.652654	+	0.757656i	$0.273655\pi$
$830$	0	0
$831$	585.986	0.0244616
$832$	0	0
$833$	−5034.12	−0.209390
$834$	0	0
$835$	51033.1	2.11506
$836$	0	0
$837$	3272.85	0.135157
$838$	0	0
$839$	85.5051	0.00351843	0.00175922	−	0.999998i	$-0.499440\pi$
$839$	85.5051	0.00351843	0.00175922	+	0.999998i	$0.499440\pi$
$840$	0	0
$841$	960.906	0.0393992
$842$	0	0
$843$	12089.5	0.493931
$844$	0	0
$845$	−7394.70	−0.301048
$846$	0	0
$847$	2627.96	0.106609
$848$	0	0
$849$	20781.8	0.840083
$850$	0	0
$851$	−53016.5	−2.13558
$852$	0	0
$853$	−1148.37	−0.0460956	−0.0230478	−	0.999734i	$-0.507337\pi$
$853$	−1148.37	−0.0460956	−0.0230478	+	0.999734i	$0.507337\pi$
$854$	0	0
$855$	11186.5	0.447452
$856$	0	0
$857$	−11674.6	−0.465340	−0.232670	−	0.972556i	$-0.574746\pi$
$857$	−11674.6	−0.465340	−0.232670	+	0.972556i	$0.574746\pi$
$858$	0	0
$859$	−47503.1	−1.88683	−0.943413	−	0.331619i	$-0.892405\pi$
$859$	−47503.1	−1.88683	−0.943413	+	0.331619i	$0.892405\pi$
$860$	0	0
$861$	−9801.48	−0.387960
$862$	0	0
$863$	10361.9	0.408717	0.204359	−	0.978896i	$-0.434489\pi$
$863$	10361.9	0.408717	0.204359	+	0.978896i	$0.434489\pi$
$864$	0	0
$865$	32303.8	1.26978
$866$	0	0
$867$	16925.8	0.663011
$868$	0	0
$869$	8679.39	0.338813
$870$	0	0
$871$	32543.2	1.26600
$872$	0	0
$873$	585.372	0.0226940
$874$	0	0
$875$	23061.5	0.890997
$876$	0	0
$877$	26153.1	1.00699	0.503493	−	0.863999i	$-0.332048\pi$
$877$	26153.1	1.00699	0.503493	+	0.863999i	$0.332048\pi$
$878$	0	0
$879$	2733.03	0.104872
$880$	0	0
$881$	35711.7	1.36567	0.682837	−	0.730571i	$-0.260746\pi$
$881$	35711.7	1.36567	0.682837	+	0.730571i	$0.260746\pi$
$882$	0	0
$883$	5654.75	0.215513	0.107756	−	0.994177i	$-0.465633\pi$
$883$	5654.75	0.215513	0.107756	+	0.994177i	$0.465633\pi$
$884$	0	0
$885$	36054.5	1.36945
$886$	0	0
$887$	−8344.44	−0.315873	−0.157936	−	0.987449i	$-0.550484\pi$
$887$	−8344.44	−0.315873	−0.157936	+	0.987449i	$0.550484\pi$
$888$	0	0
$889$	−13867.9	−0.523187
$890$	0	0
$891$	−2503.90	−0.0941459
$892$	0	0
$893$	−8043.35	−0.301411
$894$	0	0
$895$	−51821.7	−1.93543
$896$	0	0
$897$	22489.6	0.837132
$898$	0	0
$899$	−19299.7	−0.715996
$900$	0	0
$901$	20565.4	0.760415
$902$	0	0
$903$	3896.91	0.143611
$904$	0	0
$905$	−68249.0	−2.50682
$906$	0	0
$907$	52612.1	1.92608	0.963040	−	0.269358i	$-0.0868116\pi$
$907$	52612.1	1.92608	0.963040	+	0.269358i	$0.0868116\pi$
$908$	0	0
$909$	−2731.60	−0.0996715
$910$	0	0
$911$	24706.6	0.898536	0.449268	−	0.893397i	$-0.351685\pi$
$911$	24706.6	0.898536	0.449268	+	0.893397i	$0.351685\pi$
$912$	0	0
$913$	17240.8	0.624960
$914$	0	0
$915$	4295.29	0.155189
$916$	0	0
$917$	−19979.7	−0.719508
$918$	0	0
$919$	−42479.2	−1.52477	−0.762383	−	0.647127i	$-0.775970\pi$
$919$	−42479.2	−1.52477	−0.762383	+	0.647127i	$0.775970\pi$
$920$	0	0
$921$	9932.17	0.355349
$922$	0	0
$923$	−26437.6	−0.942799
$924$	0	0
$925$	102812.	3.65452
$926$	0	0
$927$	−1568.41	−0.0555699
$928$	0	0
$929$	9423.54	0.332805	0.166403	−	0.986058i	$-0.446785\pi$
$929$	9423.54	0.332805	0.166403	+	0.986058i	$0.446785\pi$
$930$	0	0
$931$	−2999.61	−0.105594
$932$	0	0
$933$	−14937.4	−0.524146
$934$	0	0
$935$	−64483.0	−2.25542
$936$	0	0
$937$	−33158.7	−1.15608	−0.578041	−	0.816008i	$-0.696183\pi$
$937$	−33158.7	−1.15608	−0.578041	+	0.816008i	$0.696183\pi$
$938$	0	0
$939$	−15348.6	−0.533420
$940$	0	0
$941$	−15005.1	−0.519822	−0.259911	−	0.965633i	$-0.583693\pi$
$941$	−15005.1	−0.519822	−0.259911	+	0.965633i	$0.583693\pi$
$942$	0	0
$943$	−69137.3	−2.38751
$944$	0	0
$945$	3837.48	0.132099
$946$	0	0
$947$	7631.38	0.261865	0.130933	−	0.991391i	$-0.458203\pi$
$947$	7631.38	0.261865	0.130933	+	0.991391i	$0.458203\pi$
$948$	0	0
$949$	29159.2	0.997417
$950$	0	0
$951$	−2934.95	−0.100076
$952$	0	0
$953$	−42615.3	−1.44853	−0.724263	−	0.689524i	$-0.757820\pi$
$953$	−42615.3	−1.44853	−0.724263	+	0.689524i	$0.757820\pi$
$954$	0	0
$955$	−78549.5	−2.66157
$956$	0	0
$957$	14765.3	0.498740
$958$	0	0
$959$	3748.32	0.126214
$960$	0	0
$961$	−15097.6	−0.506782
$962$	0	0
$963$	17508.8	0.585890
$964$	0	0
$965$	−54682.4	−1.82413
$966$	0	0
$967$	−34795.1	−1.15712	−0.578559	−	0.815640i	$-0.696385\pi$
$967$	−34795.1	−1.15712	−0.578559	+	0.815640i	$0.696385\pi$
$968$	0	0
$969$	18867.6	0.625507
$970$	0	0
$971$	7714.99	0.254980	0.127490	−	0.991840i	$-0.459308\pi$
$971$	7714.99	0.254980	0.127490	+	0.991840i	$0.459308\pi$
$972$	0	0
$973$	−16285.0	−0.536561
$974$	0	0
$975$	−43612.9	−1.43254
$976$	0	0
$977$	22817.7	0.747188	0.373594	−	0.927592i	$-0.378125\pi$
$977$	22817.7	0.747188	0.373594	+	0.927592i	$0.378125\pi$
$978$	0	0
$979$	37988.0	1.24015
$980$	0	0
$981$	9925.80	0.323044
$982$	0	0
$983$	10924.8	0.354473	0.177237	−	0.984168i	$-0.443284\pi$
$983$	10924.8	0.354473	0.177237	+	0.984168i	$0.443284\pi$
$984$	0	0
$985$	−21621.0	−0.699393
$986$	0	0
$987$	−2759.23	−0.0889839
$988$	0	0
$989$	27487.8	0.883784
$990$	0	0
$991$	49279.5	1.57963	0.789816	−	0.613343i	$-0.210176\pi$
$991$	49279.5	1.57963	0.789816	+	0.613343i	$0.210176\pi$
$992$	0	0
$993$	27277.1	0.871714
$994$	0	0
$995$	33012.8	1.05184
$996$	0	0
$997$	49601.6	1.57563	0.787813	−	0.615914i	$-0.211213\pi$
$997$	49601.6	1.57563	0.787813	+	0.615914i	$0.211213\pi$
$998$	0	0
$999$	9663.50	0.306046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bl.1.1		2
4.3	odd	2		1344.4.a.bd.1.1		2
8.3	odd	2		168.4.a.h.1.2	✓	2
8.5	even	2		336.4.a.n.1.2		2
24.5	odd	2		1008.4.a.y.1.1		2
24.11	even	2		504.4.a.j.1.1		2
56.13	odd	2		2352.4.a.bv.1.1		2
56.27	even	2		1176.4.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.h.1.2	✓	2	8.3	odd	2
336.4.a.n.1.2		2	8.5	even	2
504.4.a.j.1.1		2	24.11	even	2
1008.4.a.y.1.1		2	24.5	odd	2
1176.4.a.p.1.1		2	56.27	even	2
1344.4.a.bd.1.1		2	4.3	odd	2
1344.4.a.bl.1.1		2	1.1	even	1	trivial
2352.4.a.bv.1.1		2	56.13	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.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -20.3041 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -20.3041 q^{5} -7.00000 q^{7} +9.00000 q^{9} -30.9124 q^{11} -50.6083 q^{13} -60.9124 q^{15} -102.737 q^{17} -61.2165 q^{19} -21.0000 q^{21} -148.129 q^{23} +287.258 q^{25} +27.0000 q^{27} -159.217 q^{29} +121.217 q^{31} -92.7372 q^{33} +142.129 q^{35} +357.908 q^{37} -151.825 q^{39} +466.737 q^{41} -185.567 q^{43} -182.737 q^{45} +131.392 q^{47} +49.0000 q^{49} -308.212 q^{51} -200.175 q^{53} +627.650 q^{55} -183.650 q^{57} -591.908 q^{59} -70.5158 q^{61} -63.0000 q^{63} +1027.56 q^{65} -643.041 q^{67} -444.387 q^{69} +522.397 q^{71} -576.175 q^{73} +861.774 q^{75} +216.387 q^{77} -280.774 q^{79} +81.0000 q^{81} -557.732 q^{83} +2085.99 q^{85} -477.650 q^{87} -1228.89 q^{89} +354.258 q^{91} +363.650 q^{93} +1242.95 q^{95} +65.0413 q^{97} -278.212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 14 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 14 * q^5 - 14 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - 14 q^{5} - 14 q^{7} + 18 q^{9} + 18 q^{11} - 48 q^{13} - 42 q^{15} + 34 q^{17} - 16 q^{19} - 42 q^{21} - 110 q^{23} + 202 q^{25} + 54 q^{27} - 212 q^{29} + 136 q^{31} + 54 q^{33} + 98 q^{35} + 24 q^{37} - 144 q^{39} + 694 q^{41} - 584 q^{43} - 126 q^{45} + 316 q^{47} + 98 q^{49} + 102 q^{51} - 560 q^{53} + 936 q^{55} - 48 q^{57} - 492 q^{59} + 604 q^{61} - 126 q^{63} + 1044 q^{65} - 1020 q^{67} - 330 q^{69} + 1710 q^{71} - 1312 q^{73} + 606 q^{75} - 126 q^{77} + 556 q^{79} + 162 q^{81} - 264 q^{83} + 2948 q^{85} - 636 q^{87} + 70 q^{89} + 336 q^{91} + 408 q^{93} + 1528 q^{95} - 136 q^{97} + 162 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 14 * q^5 - 14 * q^7 + 18 * q^9 + 18 * q^11 - 48 * q^13 - 42 * q^15 + 34 * q^17 - 16 * q^19 - 42 * q^21 - 110 * q^23 + 202 * q^25 + 54 * q^27 - 212 * q^29 + 136 * q^31 + 54 * q^33 + 98 * q^35 + 24 * q^37 - 144 * q^39 + 694 * q^41 - 584 * q^43 - 126 * q^45 + 316 * q^47 + 98 * q^49 + 102 * q^51 - 560 * q^53 + 936 * q^55 - 48 * q^57 - 492 * q^59 + 604 * q^61 - 126 * q^63 + 1044 * q^65 - 1020 * q^67 - 330 * q^69 + 1710 * q^71 - 1312 * q^73 + 606 * q^75 - 126 * q^77 + 556 * q^79 + 162 * q^81 - 264 * q^83 + 2948 * q^85 - 636 * q^87 + 70 * q^89 + 336 * q^91 + 408 * q^93 + 1528 * q^95 - 136 * q^97 + 162 * q^99

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