LMFDB - Embedded newform 1344.4.a.bj.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} +0.876894 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +0.876894 q^{5} +7.00000 q^{7} +9.00000 q^{9} +56.3542 q^{11} +3.75379 q^{13} -2.63068 q^{15} -29.6458 q^{17} -135.939 q^{19} -21.0000 q^{21} -177.062 q^{23} -124.231 q^{25} -27.0000 q^{27} +149.939 q^{29} -38.4621 q^{31} -169.062 q^{33} +6.13826 q^{35} -86.2765 q^{37} -11.2614 q^{39} +58.5094 q^{41} +485.356 q^{43} +7.89205 q^{45} +315.542 q^{47} +49.0000 q^{49} +88.9375 q^{51} -154.216 q^{53} +49.4166 q^{55} +407.818 q^{57} +727.663 q^{59} -688.220 q^{61} +63.0000 q^{63} +3.29168 q^{65} +710.739 q^{67} +531.187 q^{69} -777.771 q^{71} -853.913 q^{73} +372.693 q^{75} +394.479 q^{77} +231.049 q^{79} +81.0000 q^{81} -444.189 q^{83} -25.9963 q^{85} -449.818 q^{87} -1083.28 q^{89} +26.2765 q^{91} +115.386 q^{93} -119.204 q^{95} +634.155 q^{97} +507.187 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} + 24 q^{13} - 30 q^{15} - 150 q^{17} - 8 q^{19} - 42 q^{21} - 82 q^{23} - 166 q^{25} - 54 q^{27} + 36 q^{29} + 88 q^{31} - 66 q^{33} + 70 q^{35} - 288 q^{37} - 72 q^{39} - 386 q^{41} + 344 q^{43} + 90 q^{45} - 276 q^{47} + 98 q^{49} + 450 q^{51} - 160 q^{53} - 264 q^{55} + 24 q^{57} + 1076 q^{59} - 156 q^{61} + 126 q^{63} + 188 q^{65} + 1372 q^{67} + 246 q^{69} - 1102 q^{71} - 240 q^{73} + 498 q^{75} + 154 q^{77} - 412 q^{79} + 162 q^{81} + 464 q^{83} - 1124 q^{85} - 108 q^{87} - 1746 q^{89} + 168 q^{91} - 264 q^{93} + 1048 q^{95} + 856 q^{97} + 198 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 + 24 * q^13 - 30 * q^15 - 150 * q^17 - 8 * q^19 - 42 * q^21 - 82 * q^23 - 166 * q^25 - 54 * q^27 + 36 * q^29 + 88 * q^31 - 66 * q^33 + 70 * q^35 - 288 * q^37 - 72 * q^39 - 386 * q^41 + 344 * q^43 + 90 * q^45 - 276 * q^47 + 98 * q^49 + 450 * q^51 - 160 * q^53 - 264 * q^55 + 24 * q^57 + 1076 * q^59 - 156 * q^61 + 126 * q^63 + 188 * q^65 + 1372 * q^67 + 246 * q^69 - 1102 * q^71 - 240 * q^73 + 498 * q^75 + 154 * q^77 - 412 * q^79 + 162 * q^81 + 464 * q^83 - 1124 * q^85 - 108 * q^87 - 1746 * q^89 + 168 * q^91 - 264 * q^93 + 1048 * q^95 + 856 * q^97 + 198 * 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
$5$	0.876894	0.0784318	0.0392159	−	0.999231i	$-0.487514\pi$
$5$	0.876894	0.0784318	0.0392159	+	0.999231i	$0.487514\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	56.3542	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	56.3542	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	0	0
$13$	3.75379	0.0800857	0.0400428	−	0.999198i	$-0.487251\pi$
$13$	3.75379	0.0800857	0.0400428	+	0.999198i	$0.487251\pi$
$14$	0	0
$15$	−2.63068	−0.0452826
$16$	0	0
$17$	−29.6458	−0.422951	−0.211476	−	0.977383i	$-0.567827\pi$
$17$	−29.6458	−0.422951	−0.211476	+	0.977383i	$0.567827\pi$
$18$	0	0
$19$	−135.939	−1.64140	−0.820701	−	0.571358i	$-0.806417\pi$
$19$	−135.939	−1.64140	−0.820701	+	0.571358i	$0.806417\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−177.062	−1.60522	−0.802610	−	0.596504i	$-0.796556\pi$
$23$	−177.062	−1.60522	−0.802610	+	0.596504i	$0.796556\pi$
$24$	0	0
$25$	−124.231	−0.993848
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	149.939	0.960105	0.480052	−	0.877240i	$-0.340618\pi$
$29$	149.939	0.960105	0.480052	+	0.877240i	$0.340618\pi$
$30$	0	0
$31$	−38.4621	−0.222839	−0.111419	−	0.993773i	$-0.535540\pi$
$31$	−38.4621	−0.222839	−0.111419	+	0.993773i	$0.535540\pi$
$32$	0	0
$33$	−169.062	−0.891818
$34$	0	0
$35$	6.13826	0.0296444
$36$	0	0
$37$	−86.2765	−0.383345	−0.191673	−	0.981459i	$-0.561391\pi$
$37$	−86.2765	−0.383345	−0.191673	+	0.981459i	$0.561391\pi$
$38$	0	0
$39$	−11.2614	−0.0462375
$40$	0	0
$41$	58.5094	0.222869	0.111435	−	0.993772i	$-0.464455\pi$
$41$	58.5094	0.222869	0.111435	+	0.993772i	$0.464455\pi$
$42$	0	0
$43$	485.356	1.72130	0.860652	−	0.509193i	$-0.170056\pi$
$43$	485.356	1.72130	0.860652	+	0.509193i	$0.170056\pi$
$44$	0	0
$45$	7.89205	0.0261439
$46$	0	0
$47$	315.542	0.979287	0.489643	−	0.871923i	$-0.337127\pi$
$47$	315.542	0.979287	0.489643	+	0.871923i	$0.337127\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	88.9375	0.244191
$52$	0	0
$53$	−154.216	−0.399683	−0.199841	−	0.979828i	$-0.564043\pi$
$53$	−154.216	−0.399683	−0.199841	+	0.979828i	$0.564043\pi$
$54$	0	0
$55$	49.4166	0.121152
$56$	0	0
$57$	407.818	0.947664
$58$	0	0
$59$	727.663	1.60565	0.802827	−	0.596211i	$-0.203328\pi$
$59$	727.663	1.60565	0.802827	+	0.596211i	$0.203328\pi$
$60$	0	0
$61$	−688.220	−1.44455	−0.722275	−	0.691606i	$-0.756903\pi$
$61$	−688.220	−1.44455	−0.722275	+	0.691606i	$0.756903\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	3.29168	0.00628126
$66$	0	0
$67$	710.739	1.29598	0.647989	−	0.761650i	$-0.275610\pi$
$67$	710.739	1.29598	0.647989	+	0.761650i	$0.275610\pi$
$68$	0	0
$69$	531.187	0.926775
$70$	0	0
$71$	−777.771	−1.30006	−0.650031	−	0.759908i	$-0.725244\pi$
$71$	−777.771	−1.30006	−0.650031	+	0.759908i	$0.725244\pi$
$72$	0	0
$73$	−853.913	−1.36908	−0.684541	−	0.728975i	$-0.739997\pi$
$73$	−853.913	−1.36908	−0.684541	+	0.728975i	$0.739997\pi$
$74$	0	0
$75$	372.693	0.573799
$76$	0	0
$77$	394.479	0.583832
$78$	0	0
$79$	231.049	0.329051	0.164526	−	0.986373i	$-0.447391\pi$
$79$	231.049	0.329051	0.164526	+	0.986373i	$0.447391\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−444.189	−0.587423	−0.293712	−	0.955894i	$-0.594891\pi$
$83$	−444.189	−0.587423	−0.293712	+	0.955894i	$0.594891\pi$
$84$	0	0
$85$	−25.9963	−0.0331728
$86$	0	0
$87$	−449.818	−0.554317
$88$	0	0
$89$	−1083.28	−1.29019	−0.645097	−	0.764101i	$-0.723183\pi$
$89$	−1083.28	−1.29019	−0.645097	+	0.764101i	$0.723183\pi$
$90$	0	0
$91$	26.2765	0.0302695
$92$	0	0
$93$	115.386	0.128656
$94$	0	0
$95$	−119.204	−0.128738
$96$	0	0
$97$	634.155	0.663801	0.331901	−	0.943314i	$-0.392310\pi$
$97$	634.155	0.663801	0.331901	+	0.943314i	$0.392310\pi$
$98$	0	0
$99$	507.187	0.514891
$100$	0	0
$101$	−1960.64	−1.93160	−0.965798	−	0.259296i	$-0.916509\pi$
$101$	−1960.64	−1.93160	−0.965798	+	0.259296i	$0.916509\pi$
$102$	0	0
$103$	−1806.65	−1.72830	−0.864149	−	0.503236i	$-0.832143\pi$
$103$	−1806.65	−1.72830	−0.864149	+	0.503236i	$0.832143\pi$
$104$	0	0
$105$	−18.4148	−0.0171152
$106$	0	0
$107$	120.718	0.109068	0.0545338	−	0.998512i	$-0.482633\pi$
$107$	120.718	0.109068	0.0545338	+	0.998512i	$0.482633\pi$
$108$	0	0
$109$	−1219.17	−1.07133	−0.535665	−	0.844431i	$-0.679939\pi$
$109$	−1219.17	−1.07133	−0.535665	+	0.844431i	$0.679939\pi$
$110$	0	0
$111$	258.830	0.221324
$112$	0	0
$113$	−608.212	−0.506334	−0.253167	−	0.967423i	$-0.581472\pi$
$113$	−608.212	−0.506334	−0.253167	+	0.967423i	$0.581472\pi$
$114$	0	0
$115$	−155.265	−0.125900
$116$	0	0
$117$	33.7841	0.0266952
$118$	0	0
$119$	−207.521	−0.159861
$120$	0	0
$121$	1844.79	1.38602
$122$	0	0
$123$	−175.528	−0.128674
$124$	0	0
$125$	−218.549	−0.156381
$126$	0	0
$127$	−1356.77	−0.947982	−0.473991	−	0.880530i	$-0.657187\pi$
$127$	−1356.77	−0.947982	−0.473991	+	0.880530i	$0.657187\pi$
$128$	0	0
$129$	−1456.07	−0.993796
$130$	0	0
$131$	1782.23	1.18866	0.594328	−	0.804223i	$-0.297418\pi$
$131$	1782.23	1.18866	0.594328	+	0.804223i	$0.297418\pi$
$132$	0	0
$133$	−951.576	−0.620392
$134$	0	0
$135$	−23.6761	−0.0150942
$136$	0	0
$137$	−979.701	−0.610960	−0.305480	−	0.952199i	$-0.598817\pi$
$137$	−979.701	−0.610960	−0.305480	+	0.952199i	$0.598817\pi$
$138$	0	0
$139$	636.257	0.388249	0.194125	−	0.980977i	$-0.437813\pi$
$139$	636.257	0.388249	0.194125	+	0.980977i	$0.437813\pi$
$140$	0	0
$141$	−946.625	−0.565391
$142$	0	0
$143$	211.542	0.123706
$144$	0	0
$145$	131.481	0.0753028
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	903.269	0.496635	0.248318	−	0.968679i	$-0.420122\pi$
$149$	903.269	0.496635	0.248318	+	0.968679i	$0.420122\pi$
$150$	0	0
$151$	1373.30	0.740118	0.370059	−	0.929008i	$-0.379337\pi$
$151$	1373.30	0.740118	0.370059	+	0.929008i	$0.379337\pi$
$152$	0	0
$153$	−266.813	−0.140984
$154$	0	0
$155$	−33.7272	−0.0174776
$156$	0	0
$157$	347.060	0.176423	0.0882116	−	0.996102i	$-0.471885\pi$
$157$	347.060	0.176423	0.0882116	+	0.996102i	$0.471885\pi$
$158$	0	0
$159$	462.648	0.230757
$160$	0	0
$161$	−1239.44	−0.606716
$162$	0	0
$163$	2667.49	1.28180	0.640901	−	0.767623i	$-0.278561\pi$
$163$	2667.49	1.28180	0.640901	+	0.767623i	$0.278561\pi$
$164$	0	0
$165$	−148.250	−0.0699469
$166$	0	0
$167$	−2333.51	−1.08127	−0.540636	−	0.841256i	$-0.681816\pi$
$167$	−2333.51	−1.08127	−0.540636	+	0.841256i	$0.681816\pi$
$168$	0	0
$169$	−2182.91	−0.993586
$170$	0	0
$171$	−1223.45	−0.547134
$172$	0	0
$173$	−2511.35	−1.10367	−0.551833	−	0.833955i	$-0.686071\pi$
$173$	−2511.35	−1.10367	−0.551833	+	0.833955i	$0.686071\pi$
$174$	0	0
$175$	−869.617	−0.375639
$176$	0	0
$177$	−2182.99	−0.927025
$178$	0	0
$179$	−1799.61	−0.751449	−0.375724	−	0.926731i	$-0.622606\pi$
$179$	−1799.61	−0.751449	−0.375724	+	0.926731i	$0.622606\pi$
$180$	0	0
$181$	−1567.89	−0.643869	−0.321935	−	0.946762i	$-0.604333\pi$
$181$	−1567.89	−0.643869	−0.321935	+	0.946762i	$0.604333\pi$
$182$	0	0
$183$	2064.66	0.834011
$184$	0	0
$185$	−75.6554	−0.0300665
$186$	0	0
$187$	−1670.67	−0.653322
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−3768.49	−1.42764	−0.713818	−	0.700331i	$-0.753036\pi$
$191$	−3768.49	−1.42764	−0.713818	+	0.700331i	$0.753036\pi$
$192$	0	0
$193$	2440.06	0.910050	0.455025	−	0.890479i	$-0.349630\pi$
$193$	2440.06	0.910050	0.455025	+	0.890479i	$0.349630\pi$
$194$	0	0
$195$	−9.87503	−0.00362649
$196$	0	0
$197$	2543.70	0.919955	0.459978	−	0.887931i	$-0.347857\pi$
$197$	2543.70	0.919955	0.459978	+	0.887931i	$0.347857\pi$
$198$	0	0
$199$	1080.43	0.384873	0.192437	−	0.981309i	$-0.438361\pi$
$199$	1080.43	0.384873	0.192437	+	0.981309i	$0.438361\pi$
$200$	0	0
$201$	−2132.22	−0.748233
$202$	0	0
$203$	1049.58	0.362886
$204$	0	0
$205$	51.3066	0.0174800
$206$	0	0
$207$	−1593.56	−0.535074
$208$	0	0
$209$	−7660.75	−2.53543
$210$	0	0
$211$	−3766.55	−1.22891	−0.614454	−	0.788953i	$-0.710624\pi$
$211$	−3766.55	−1.22891	−0.614454	+	0.788953i	$0.710624\pi$
$212$	0	0
$213$	2333.31	0.750591
$214$	0	0
$215$	425.606	0.135005
$216$	0	0
$217$	−269.235	−0.0842251
$218$	0	0
$219$	2561.74	0.790439
$220$	0	0
$221$	−111.284	−0.0338723
$222$	0	0
$223$	5461.86	1.64015	0.820074	−	0.572258i	$-0.193932\pi$
$223$	5461.86	1.64015	0.820074	+	0.572258i	$0.193932\pi$
$224$	0	0
$225$	−1118.08	−0.331283
$226$	0	0
$227$	−3480.41	−1.01763	−0.508816	−	0.860875i	$-0.669917\pi$
$227$	−3480.41	−1.01763	−0.508816	+	0.860875i	$0.669917\pi$
$228$	0	0
$229$	−3465.22	−0.999949	−0.499975	−	0.866040i	$-0.666657\pi$
$229$	−3465.22	−0.999949	−0.499975	+	0.866040i	$0.666657\pi$
$230$	0	0
$231$	−1183.44	−0.337076
$232$	0	0
$233$	4469.14	1.25658	0.628290	−	0.777979i	$-0.283755\pi$
$233$	4469.14	1.25658	0.628290	+	0.777979i	$0.283755\pi$
$234$	0	0
$235$	276.697	0.0768072
$236$	0	0
$237$	−693.148	−0.189978
$238$	0	0
$239$	2879.32	0.779278	0.389639	−	0.920968i	$-0.372600\pi$
$239$	2879.32	0.779278	0.389639	+	0.920968i	$0.372600\pi$
$240$	0	0
$241$	1014.11	0.271056	0.135528	−	0.990774i	$-0.456727\pi$
$241$	1014.11	0.271056	0.135528	+	0.990774i	$0.456727\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	42.9678	0.0112045
$246$	0	0
$247$	−510.288	−0.131453
$248$	0	0
$249$	1332.57	0.339149
$250$	0	0
$251$	−7235.24	−1.81946	−0.909729	−	0.415201i	$-0.863711\pi$
$251$	−7235.24	−1.81946	−0.909729	+	0.415201i	$0.863711\pi$
$252$	0	0
$253$	−9978.21	−2.47954
$254$	0	0
$255$	77.9888	0.0191523
$256$	0	0
$257$	719.449	0.174622	0.0873112	−	0.996181i	$-0.472173\pi$
$257$	719.449	0.174622	0.0873112	+	0.996181i	$0.472173\pi$
$258$	0	0
$259$	−603.936	−0.144891
$260$	0	0
$261$	1349.45	0.320035
$262$	0	0
$263$	1340.74	0.314348	0.157174	−	0.987571i	$-0.449762\pi$
$263$	1340.74	0.314348	0.157174	+	0.987571i	$0.449762\pi$
$264$	0	0
$265$	−135.231	−0.0313478
$266$	0	0
$267$	3249.84	0.744894
$268$	0	0
$269$	1189.06	0.269510	0.134755	−	0.990879i	$-0.456975\pi$
$269$	1189.06	0.269510	0.134755	+	0.990879i	$0.456975\pi$
$270$	0	0
$271$	−2661.82	−0.596656	−0.298328	−	0.954463i	$-0.596429\pi$
$271$	−2661.82	−0.596656	−0.298328	+	0.954463i	$0.596429\pi$
$272$	0	0
$273$	−78.8296	−0.0174761
$274$	0	0
$275$	−7000.94	−1.53517
$276$	0	0
$277$	−8638.72	−1.87383	−0.936914	−	0.349559i	$-0.886331\pi$
$277$	−8638.72	−1.87383	−0.936914	+	0.349559i	$0.886331\pi$
$278$	0	0
$279$	−346.159	−0.0742796
$280$	0	0
$281$	−6951.67	−1.47581	−0.737904	−	0.674906i	$-0.764184\pi$
$281$	−6951.67	−1.47581	−0.737904	+	0.674906i	$0.764184\pi$
$282$	0	0
$283$	3231.21	0.678712	0.339356	−	0.940658i	$-0.389791\pi$
$283$	3231.21	0.678712	0.339356	+	0.940658i	$0.389791\pi$
$284$	0	0
$285$	357.613	0.0743270
$286$	0	0
$287$	409.566	0.0842367
$288$	0	0
$289$	−4034.12	−0.821112
$290$	0	0
$291$	−1902.47	−0.383246
$292$	0	0
$293$	4270.89	0.851563	0.425781	−	0.904826i	$-0.359999\pi$
$293$	4270.89	0.851563	0.425781	+	0.904826i	$0.359999\pi$
$294$	0	0
$295$	638.083	0.125934
$296$	0	0
$297$	−1521.56	−0.297273
$298$	0	0
$299$	−664.655	−0.128555
$300$	0	0
$301$	3397.49	0.650592
$302$	0	0
$303$	5881.93	1.11521
$304$	0	0
$305$	−603.496	−0.113299
$306$	0	0
$307$	−285.402	−0.0530578	−0.0265289	−	0.999648i	$-0.508445\pi$
$307$	−285.402	−0.0530578	−0.0265289	+	0.999648i	$0.508445\pi$
$308$	0	0
$309$	5419.95	0.997833
$310$	0	0
$311$	1366.10	0.249082	0.124541	−	0.992214i	$-0.460254\pi$
$311$	1366.10	0.249082	0.124541	+	0.992214i	$0.460254\pi$
$312$	0	0
$313$	753.970	0.136156	0.0680781	−	0.997680i	$-0.478313\pi$
$313$	753.970	0.136156	0.0680781	+	0.997680i	$0.478313\pi$
$314$	0	0
$315$	55.2443	0.00988148
$316$	0	0
$317$	−5379.11	−0.953063	−0.476531	−	0.879157i	$-0.658106\pi$
$317$	−5379.11	−0.953063	−0.476531	+	0.879157i	$0.658106\pi$
$318$	0	0
$319$	8449.71	1.48305
$320$	0	0
$321$	−362.154	−0.0629702
$322$	0	0
$323$	4030.04	0.694233
$324$	0	0
$325$	−466.337	−0.0795930
$326$	0	0
$327$	3657.50	0.618533
$328$	0	0
$329$	2208.79	0.370136
$330$	0	0
$331$	−8010.26	−1.33016	−0.665081	−	0.746771i	$-0.731603\pi$
$331$	−8010.26	−1.33016	−0.665081	+	0.746771i	$0.731603\pi$
$332$	0	0
$333$	−776.489	−0.127782
$334$	0	0
$335$	623.243	0.101646
$336$	0	0
$337$	5653.31	0.913814	0.456907	−	0.889514i	$-0.348957\pi$
$337$	5653.31	0.913814	0.456907	+	0.889514i	$0.348957\pi$
$338$	0	0
$339$	1824.64	0.292332
$340$	0	0
$341$	−2167.50	−0.344213
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	465.795	0.0726886
$346$	0	0
$347$	−770.445	−0.119192	−0.0595961	−	0.998223i	$-0.518981\pi$
$347$	−770.445	−0.119192	−0.0595961	+	0.998223i	$0.518981\pi$
$348$	0	0
$349$	−904.811	−0.138778	−0.0693889	−	0.997590i	$-0.522105\pi$
$349$	−904.811	−0.138778	−0.0693889	+	0.997590i	$0.522105\pi$
$350$	0	0
$351$	−101.352	−0.0154125
$352$	0	0
$353$	−409.934	−0.0618090	−0.0309045	−	0.999522i	$-0.509839\pi$
$353$	−409.934	−0.0618090	−0.0309045	+	0.999522i	$0.509839\pi$
$354$	0	0
$355$	−682.023	−0.101966
$356$	0	0
$357$	622.563	0.0922955
$358$	0	0
$359$	−2446.32	−0.359642	−0.179821	−	0.983699i	$-0.557552\pi$
$359$	−2446.32	−0.359642	−0.179821	+	0.983699i	$0.557552\pi$
$360$	0	0
$361$	11620.5	1.69420
$362$	0	0
$363$	−5534.37	−0.800219
$364$	0	0
$365$	−748.791	−0.107380
$366$	0	0
$367$	8076.32	1.14872	0.574360	−	0.818603i	$-0.305251\pi$
$367$	8076.32	1.14872	0.574360	+	0.818603i	$0.305251\pi$
$368$	0	0
$369$	526.585	0.0742898
$370$	0	0
$371$	−1079.51	−0.151066
$372$	0	0
$373$	−8458.38	−1.17415	−0.587075	−	0.809532i	$-0.699721\pi$
$373$	−8458.38	−1.17415	−0.587075	+	0.809532i	$0.699721\pi$
$374$	0	0
$375$	655.648	0.0902867
$376$	0	0
$377$	562.841	0.0768906
$378$	0	0
$379$	11138.6	1.50963	0.754817	−	0.655936i	$-0.227726\pi$
$379$	11138.6	1.50963	0.754817	+	0.655936i	$0.227726\pi$
$380$	0	0
$381$	4070.31	0.547318
$382$	0	0
$383$	499.363	0.0666221	0.0333110	−	0.999445i	$-0.489395\pi$
$383$	499.363	0.0666221	0.0333110	+	0.999445i	$0.489395\pi$
$384$	0	0
$385$	345.917	0.0457910
$386$	0	0
$387$	4368.20	0.573768
$388$	0	0
$389$	−227.788	−0.0296897	−0.0148449	−	0.999890i	$-0.504725\pi$
$389$	−227.788	−0.0296897	−0.0148449	+	0.999890i	$0.504725\pi$
$390$	0	0
$391$	5249.17	0.678930
$392$	0	0
$393$	−5346.68	−0.686271
$394$	0	0
$395$	202.606	0.0258081
$396$	0	0
$397$	−7979.71	−1.00879	−0.504396	−	0.863473i	$-0.668285\pi$
$397$	−7979.71	−1.00879	−0.504396	+	0.863473i	$0.668285\pi$
$398$	0	0
$399$	2854.73	0.358183
$400$	0	0
$401$	−12358.0	−1.53898	−0.769489	−	0.638660i	$-0.779489\pi$
$401$	−12358.0	−1.53898	−0.769489	+	0.638660i	$0.779489\pi$
$402$	0	0
$403$	−144.379	−0.0178462
$404$	0	0
$405$	71.0284	0.00871465
$406$	0	0
$407$	−4862.04	−0.592144
$408$	0	0
$409$	372.125	0.0449887	0.0224944	−	0.999747i	$-0.492839\pi$
$409$	372.125	0.0449887	0.0224944	+	0.999747i	$0.492839\pi$
$410$	0	0
$411$	2939.10	0.352738
$412$	0	0
$413$	5093.64	0.606880
$414$	0	0
$415$	−389.507	−0.0460727
$416$	0	0
$417$	−1908.77	−0.224156
$418$	0	0
$419$	−8281.97	−0.965635	−0.482817	−	0.875721i	$-0.660386\pi$
$419$	−8281.97	−0.965635	−0.482817	+	0.875721i	$0.660386\pi$
$420$	0	0
$421$	12771.4	1.47848	0.739242	−	0.673440i	$-0.235184\pi$
$421$	12771.4	1.47848	0.739242	+	0.673440i	$0.235184\pi$
$422$	0	0
$423$	2839.87	0.326429
$424$	0	0
$425$	3682.93	0.420349
$426$	0	0
$427$	−4817.54	−0.545988
$428$	0	0
$429$	−634.625	−0.0714219
$430$	0	0
$431$	−9372.55	−1.04747	−0.523735	−	0.851881i	$-0.675462\pi$
$431$	−9372.55	−1.04747	−0.523735	+	0.851881i	$0.675462\pi$
$432$	0	0
$433$	−7381.75	−0.819271	−0.409635	−	0.912249i	$-0.634344\pi$
$433$	−7381.75	−0.819271	−0.409635	+	0.912249i	$0.634344\pi$
$434$	0	0
$435$	−394.443	−0.0434761
$436$	0	0
$437$	24069.8	2.63481
$438$	0	0
$439$	1508.09	0.163957	0.0819786	−	0.996634i	$-0.473876\pi$
$439$	1508.09	0.163957	0.0819786	+	0.996634i	$0.473876\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	9229.84	0.989894	0.494947	−	0.868923i	$-0.335187\pi$
$443$	9229.84	0.989894	0.494947	+	0.868923i	$0.335187\pi$
$444$	0	0
$445$	−949.921	−0.101192
$446$	0	0
$447$	−2709.81	−0.286733
$448$	0	0
$449$	−1747.05	−0.183627	−0.0918136	−	0.995776i	$-0.529266\pi$
$449$	−1747.05	−0.183627	−0.0918136	+	0.995776i	$0.529266\pi$
$450$	0	0
$451$	3297.25	0.344260
$452$	0	0
$453$	−4119.91	−0.427307
$454$	0	0
$455$	23.0417	0.00237409
$456$	0	0
$457$	14462.8	1.48040	0.740198	−	0.672388i	$-0.234732\pi$
$457$	14462.8	1.48040	0.740198	+	0.672388i	$0.234732\pi$
$458$	0	0
$459$	800.438	0.0813970
$460$	0	0
$461$	712.086	0.0719417	0.0359709	−	0.999353i	$-0.488548\pi$
$461$	712.086	0.0719417	0.0359709	+	0.999353i	$0.488548\pi$
$462$	0	0
$463$	−3749.73	−0.376382	−0.188191	−	0.982132i	$-0.560262\pi$
$463$	−3749.73	−0.376382	−0.188191	+	0.982132i	$0.560262\pi$
$464$	0	0
$465$	101.182	0.0100907
$466$	0	0
$467$	−17963.3	−1.77997	−0.889983	−	0.455994i	$-0.849284\pi$
$467$	−17963.3	−1.77997	−0.889983	+	0.455994i	$0.849284\pi$
$468$	0	0
$469$	4975.17	0.489834
$470$	0	0
$471$	−1041.18	−0.101858
$472$	0	0
$473$	27351.8	2.65886
$474$	0	0
$475$	16887.9	1.63130
$476$	0	0
$477$	−1387.94	−0.133228
$478$	0	0
$479$	−7260.06	−0.692527	−0.346264	−	0.938137i	$-0.612550\pi$
$479$	−7260.06	−0.692527	−0.346264	+	0.938137i	$0.612550\pi$
$480$	0	0
$481$	−323.864	−0.0307005
$482$	0	0
$483$	3718.31	0.350288
$484$	0	0
$485$	556.087	0.0520631
$486$	0	0
$487$	−3182.75	−0.296148	−0.148074	−	0.988976i	$-0.547307\pi$
$487$	−3182.75	−0.296148	−0.148074	+	0.988976i	$0.547307\pi$
$488$	0	0
$489$	−8002.47	−0.740049
$490$	0	0
$491$	−12187.9	−1.12022	−0.560112	−	0.828417i	$-0.689242\pi$
$491$	−12187.9	−1.12022	−0.560112	+	0.828417i	$0.689242\pi$
$492$	0	0
$493$	−4445.08	−0.406078
$494$	0	0
$495$	444.750	0.0403839
$496$	0	0
$497$	−5444.40	−0.491377
$498$	0	0
$499$	368.606	0.0330683	0.0165341	−	0.999863i	$-0.494737\pi$
$499$	368.606	0.0330683	0.0165341	+	0.999863i	$0.494737\pi$
$500$	0	0
$501$	7000.53	0.624273
$502$	0	0
$503$	−12794.1	−1.13412	−0.567060	−	0.823677i	$-0.691919\pi$
$503$	−12794.1	−1.13412	−0.567060	+	0.823677i	$0.691919\pi$
$504$	0	0
$505$	−1719.28	−0.151499
$506$	0	0
$507$	6548.73	0.573647
$508$	0	0
$509$	−603.881	−0.0525866	−0.0262933	−	0.999654i	$-0.508370\pi$
$509$	−603.881	−0.0525866	−0.0262933	+	0.999654i	$0.508370\pi$
$510$	0	0
$511$	−5977.39	−0.517464
$512$	0	0
$513$	3670.36	0.315888
$514$	0	0
$515$	−1584.24	−0.135554
$516$	0	0
$517$	17782.1	1.51268
$518$	0	0
$519$	7534.04	0.637202
$520$	0	0
$521$	−11964.9	−1.00613	−0.503064	−	0.864249i	$-0.667794\pi$
$521$	−11964.9	−1.00613	−0.503064	+	0.864249i	$0.667794\pi$
$522$	0	0
$523$	4926.73	0.411913	0.205957	−	0.978561i	$-0.433969\pi$
$523$	4926.73	0.411913	0.205957	+	0.978561i	$0.433969\pi$
$524$	0	0
$525$	2608.85	0.216876
$526$	0	0
$527$	1140.24	0.0942499
$528$	0	0
$529$	19184.1	1.57673
$530$	0	0
$531$	6548.97	0.535218
$532$	0	0
$533$	219.632	0.0178486
$534$	0	0
$535$	105.857	0.00855437
$536$	0	0
$537$	5398.84	0.433849
$538$	0	0
$539$	2761.35	0.220668
$540$	0	0
$541$	2094.84	0.166477	0.0832385	−	0.996530i	$-0.473474\pi$
$541$	2094.84	0.166477	0.0832385	+	0.996530i	$0.473474\pi$
$542$	0	0
$543$	4703.67	0.371738
$544$	0	0
$545$	−1069.08	−0.0840264
$546$	0	0
$547$	8445.47	0.660150	0.330075	−	0.943955i	$-0.392926\pi$
$547$	8445.47	0.660150	0.330075	+	0.943955i	$0.392926\pi$
$548$	0	0
$549$	−6193.98	−0.481516
$550$	0	0
$551$	−20382.7	−1.57592
$552$	0	0
$553$	1617.34	0.124370
$554$	0	0
$555$	226.966	0.0173589
$556$	0	0
$557$	−18621.4	−1.41654	−0.708271	−	0.705940i	$-0.750525\pi$
$557$	−18621.4	−1.41654	−0.708271	+	0.705940i	$0.750525\pi$
$558$	0	0
$559$	1821.92	0.137852
$560$	0	0
$561$	5012.00	0.377196
$562$	0	0
$563$	8094.87	0.605964	0.302982	−	0.952996i	$-0.402018\pi$
$563$	8094.87	0.605964	0.302982	+	0.952996i	$0.402018\pi$
$564$	0	0
$565$	−533.338	−0.0397127
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	8187.82	0.603254	0.301627	−	0.953426i	$-0.402470\pi$
$569$	8187.82	0.603254	0.301627	+	0.953426i	$0.402470\pi$
$570$	0	0
$571$	11758.0	0.861744	0.430872	−	0.902413i	$-0.358206\pi$
$571$	11758.0	0.861744	0.430872	+	0.902413i	$0.358206\pi$
$572$	0	0
$573$	11305.5	0.824246
$574$	0	0
$575$	21996.7	1.59535
$576$	0	0
$577$	−9198.93	−0.663703	−0.331851	−	0.943332i	$-0.607673\pi$
$577$	−9198.93	−0.663703	−0.331851	+	0.943332i	$0.607673\pi$
$578$	0	0
$579$	−7320.19	−0.525418
$580$	0	0
$581$	−3109.33	−0.222025
$582$	0	0
$583$	−8690.71	−0.617380
$584$	0	0
$585$	29.6251	0.00209375
$586$	0	0
$587$	106.239	0.00747009	0.00373504	−	0.999993i	$-0.498811\pi$
$587$	106.239	0.00747009	0.00373504	+	0.999993i	$0.498811\pi$
$588$	0	0
$589$	5228.52	0.365768
$590$	0	0
$591$	−7631.10	−0.531136
$592$	0	0
$593$	−11280.0	−0.781136	−0.390568	−	0.920574i	$-0.627721\pi$
$593$	−11280.0	−0.781136	−0.390568	+	0.920574i	$0.627721\pi$
$594$	0	0
$595$	−181.974	−0.0125382
$596$	0	0
$597$	−3241.30	−0.222207
$598$	0	0
$599$	−12945.7	−0.883053	−0.441527	−	0.897248i	$-0.645563\pi$
$599$	−12945.7	−0.883053	−0.441527	+	0.897248i	$0.645563\pi$
$600$	0	0
$601$	−3077.87	−0.208900	−0.104450	−	0.994530i	$-0.533308\pi$
$601$	−3077.87	−0.208900	−0.104450	+	0.994530i	$0.533308\pi$
$602$	0	0
$603$	6396.65	0.431993
$604$	0	0
$605$	1617.69	0.108708
$606$	0	0
$607$	−16004.8	−1.07020	−0.535102	−	0.844787i	$-0.679727\pi$
$607$	−16004.8	−1.07020	−0.535102	+	0.844787i	$0.679727\pi$
$608$	0	0
$609$	−3148.73	−0.209512
$610$	0	0
$611$	1184.48	0.0784268
$612$	0	0
$613$	11740.7	0.773578	0.386789	−	0.922168i	$-0.373584\pi$
$613$	11740.7	0.773578	0.386789	+	0.922168i	$0.373584\pi$
$614$	0	0
$615$	−153.920	−0.0100921
$616$	0	0
$617$	−3.63255	−0.000237019 0	−0.000118510	−	1.00000i	$-0.500038\pi$
$617$	−3.63255	−0.000237019 0	−0.000118510		1.00000i	$0.500038\pi$
$618$	0	0
$619$	−6041.73	−0.392307	−0.196153	−	0.980573i	$-0.562845\pi$
$619$	−6041.73	−0.392307	−0.196153	+	0.980573i	$0.562845\pi$
$620$	0	0
$621$	4780.69	0.308925
$622$	0	0
$623$	−7582.95	−0.487648
$624$	0	0
$625$	15337.2	0.981583
$626$	0	0
$627$	22982.2	1.46383
$628$	0	0
$629$	2557.74	0.162136
$630$	0	0
$631$	26746.9	1.68745	0.843723	−	0.536779i	$-0.180359\pi$
$631$	26746.9	1.68745	0.843723	+	0.536779i	$0.180359\pi$
$632$	0	0
$633$	11299.6	0.709511
$634$	0	0
$635$	−1189.74	−0.0743520
$636$	0	0
$637$	183.936	0.0114408
$638$	0	0
$639$	−6999.94	−0.433354
$640$	0	0
$641$	23798.5	1.46643	0.733217	−	0.679994i	$-0.238018\pi$
$641$	23798.5	1.46643	0.733217	+	0.679994i	$0.238018\pi$
$642$	0	0
$643$	12067.9	0.740140	0.370070	−	0.929004i	$-0.379334\pi$
$643$	12067.9	0.740140	0.370070	+	0.929004i	$0.379334\pi$
$644$	0	0
$645$	−1276.82	−0.0779452
$646$	0	0
$647$	−6510.14	−0.395580	−0.197790	−	0.980244i	$-0.563376\pi$
$647$	−6510.14	−0.395580	−0.197790	+	0.980244i	$0.563376\pi$
$648$	0	0
$649$	41006.8	2.48021
$650$	0	0
$651$	807.704	0.0486274
$652$	0	0
$653$	29875.2	1.79036	0.895181	−	0.445703i	$-0.147046\pi$
$653$	29875.2	1.79036	0.895181	+	0.445703i	$0.147046\pi$
$654$	0	0
$655$	1562.82	0.0932284
$656$	0	0
$657$	−7685.22	−0.456360
$658$	0	0
$659$	16941.0	1.00141	0.500705	−	0.865618i	$-0.333074\pi$
$659$	16941.0	1.00141	0.500705	+	0.865618i	$0.333074\pi$
$660$	0	0
$661$	31618.1	1.86052	0.930259	−	0.366905i	$-0.119583\pi$
$661$	31618.1	1.86052	0.930259	+	0.366905i	$0.119583\pi$
$662$	0	0
$663$	333.853	0.0195562
$664$	0	0
$665$	−834.431	−0.0486584
$666$	0	0
$667$	−26548.6	−1.54118
$668$	0	0
$669$	−16385.6	−0.946940
$670$	0	0
$671$	−38784.0	−2.23136
$672$	0	0
$673$	−27915.4	−1.59890	−0.799449	−	0.600734i	$-0.794875\pi$
$673$	−27915.4	−1.59890	−0.799449	+	0.600734i	$0.794875\pi$
$674$	0	0
$675$	3354.24	0.191266
$676$	0	0
$677$	−14052.2	−0.797741	−0.398871	−	0.917007i	$-0.630598\pi$
$677$	−14052.2	−0.797741	−0.398871	+	0.917007i	$0.630598\pi$
$678$	0	0
$679$	4439.09	0.250893
$680$	0	0
$681$	10441.2	0.587530
$682$	0	0
$683$	−9826.72	−0.550526	−0.275263	−	0.961369i	$-0.588765\pi$
$683$	−9826.72	−0.550526	−0.275263	+	0.961369i	$0.588765\pi$
$684$	0	0
$685$	−859.094	−0.0479187
$686$	0	0
$687$	10395.7	0.577321
$688$	0	0
$689$	−578.894	−0.0320089
$690$	0	0
$691$	28241.2	1.55477	0.777386	−	0.629024i	$-0.216545\pi$
$691$	28241.2	1.55477	0.777386	+	0.629024i	$0.216545\pi$
$692$	0	0
$693$	3550.31	0.194611
$694$	0	0
$695$	557.931	0.0304511
$696$	0	0
$697$	−1734.56	−0.0942628
$698$	0	0
$699$	−13407.4	−0.725487
$700$	0	0
$701$	−13561.0	−0.730658	−0.365329	−	0.930879i	$-0.619043\pi$
$701$	−13561.0	−0.730658	−0.365329	+	0.930879i	$0.619043\pi$
$702$	0	0
$703$	11728.4	0.629223
$704$	0	0
$705$	−830.090	−0.0443447
$706$	0	0
$707$	−13724.5	−0.730075
$708$	0	0
$709$	12419.2	0.657848	0.328924	−	0.944356i	$-0.393314\pi$
$709$	12419.2	0.657848	0.328924	+	0.944356i	$0.393314\pi$
$710$	0	0
$711$	2079.44	0.109684
$712$	0	0
$713$	6810.20	0.357705
$714$	0	0
$715$	185.500	0.00970251
$716$	0	0
$717$	−8637.95	−0.449916
$718$	0	0
$719$	−23656.1	−1.22702	−0.613508	−	0.789688i	$-0.710242\pi$
$719$	−23656.1	−1.22702	−0.613508	+	0.789688i	$0.710242\pi$
$720$	0	0
$721$	−12646.6	−0.653235
$722$	0	0
$723$	−3042.33	−0.156494
$724$	0	0
$725$	−18627.1	−0.954199
$726$	0	0
$727$	−32584.6	−1.66231	−0.831153	−	0.556044i	$-0.812319\pi$
$727$	−32584.6	−1.66231	−0.831153	+	0.556044i	$0.812319\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−14388.8	−0.728028
$732$	0	0
$733$	−27826.8	−1.40219	−0.701095	−	0.713068i	$-0.747305\pi$
$733$	−27826.8	−1.40219	−0.701095	+	0.713068i	$0.747305\pi$
$734$	0	0
$735$	−128.903	−0.00646895
$736$	0	0
$737$	40053.1	2.00186
$738$	0	0
$739$	−17634.8	−0.877818	−0.438909	−	0.898531i	$-0.644635\pi$
$739$	−17634.8	−0.877818	−0.438909	+	0.898531i	$0.644635\pi$
$740$	0	0
$741$	1530.86	0.0758943
$742$	0	0
$743$	−17572.6	−0.867669	−0.433834	−	0.900993i	$-0.642840\pi$
$743$	−17572.6	−0.867669	−0.433834	+	0.900993i	$0.642840\pi$
$744$	0	0
$745$	792.071	0.0389520
$746$	0	0
$747$	−3997.70	−0.195808
$748$	0	0
$749$	845.025	0.0412237
$750$	0	0
$751$	−8719.41	−0.423669	−0.211835	−	0.977306i	$-0.567944\pi$
$751$	−8719.41	−0.423669	−0.211835	+	0.977306i	$0.567944\pi$
$752$	0	0
$753$	21705.7	1.05047
$754$	0	0
$755$	1204.24	0.0580488
$756$	0	0
$757$	37363.8	1.79394	0.896969	−	0.442093i	$-0.145764\pi$
$757$	37363.8	1.79394	0.896969	+	0.442093i	$0.145764\pi$
$758$	0	0
$759$	29934.6	1.43157
$760$	0	0
$761$	19533.4	0.930468	0.465234	−	0.885188i	$-0.345970\pi$
$761$	19533.4	0.930468	0.465234	+	0.885188i	$0.345970\pi$
$762$	0	0
$763$	−8534.17	−0.404925
$764$	0	0
$765$	−233.966	−0.0110576
$766$	0	0
$767$	2731.49	0.128590
$768$	0	0
$769$	−22424.6	−1.05156	−0.525781	−	0.850620i	$-0.676227\pi$
$769$	−22424.6	−1.05156	−0.525781	+	0.850620i	$0.676227\pi$
$770$	0	0
$771$	−2158.35	−0.100818
$772$	0	0
$773$	34994.7	1.62830	0.814148	−	0.580658i	$-0.197205\pi$
$773$	34994.7	1.62830	0.814148	+	0.580658i	$0.197205\pi$
$774$	0	0
$775$	4778.19	0.221468
$776$	0	0
$777$	1811.81	0.0836528
$778$	0	0
$779$	−7953.74	−0.365818
$780$	0	0
$781$	−43830.6	−2.00817
$782$	0	0
$783$	−4048.36	−0.184772
$784$	0	0
$785$	304.335	0.0138372
$786$	0	0
$787$	30981.7	1.40328	0.701639	−	0.712532i	$-0.252452\pi$
$787$	30981.7	1.40328	0.701639	+	0.712532i	$0.252452\pi$
$788$	0	0
$789$	−4022.22	−0.181489
$790$	0	0
$791$	−4257.49	−0.191376
$792$	0	0
$793$	−2583.43	−0.115688
$794$	0	0
$795$	405.693	0.0180987
$796$	0	0
$797$	38915.7	1.72957	0.864784	−	0.502144i	$-0.167455\pi$
$797$	38915.7	1.72957	0.864784	+	0.502144i	$0.167455\pi$
$798$	0	0
$799$	−9354.50	−0.414191
$800$	0	0
$801$	−9749.51	−0.430065
$802$	0	0
$803$	−48121.5	−2.11478
$804$	0	0
$805$	−1086.86	−0.0475859
$806$	0	0
$807$	−3567.18	−0.155602
$808$	0	0
$809$	3956.52	0.171946	0.0859728	−	0.996297i	$-0.472600\pi$
$809$	3956.52	0.171946	0.0859728	+	0.996297i	$0.472600\pi$
$810$	0	0
$811$	−21542.7	−0.932759	−0.466379	−	0.884585i	$-0.654442\pi$
$811$	−21542.7	−0.932759	−0.466379	+	0.884585i	$0.654442\pi$
$812$	0	0
$813$	7985.45	0.344480
$814$	0	0
$815$	2339.11	0.100534
$816$	0	0
$817$	−65979.0	−2.82535
$818$	0	0
$819$	236.489	0.0100898
$820$	0	0
$821$	3362.75	0.142948	0.0714742	−	0.997442i	$-0.477230\pi$
$821$	3362.75	0.142948	0.0714742	+	0.997442i	$0.477230\pi$
$822$	0	0
$823$	35734.9	1.51354	0.756768	−	0.653683i	$-0.226777\pi$
$823$	35734.9	1.51354	0.756768	+	0.653683i	$0.226777\pi$
$824$	0	0
$825$	21002.8	0.886332
$826$	0	0
$827$	3687.52	0.155052	0.0775258	−	0.996990i	$-0.475298\pi$
$827$	3687.52	0.155052	0.0775258	+	0.996990i	$0.475298\pi$
$828$	0	0
$829$	8365.15	0.350463	0.175231	−	0.984527i	$-0.443933\pi$
$829$	8365.15	0.350463	0.175231	+	0.984527i	$0.443933\pi$
$830$	0	0
$831$	25916.2	1.08186
$832$	0	0
$833$	−1452.65	−0.0604216
$834$	0	0
$835$	−2046.24	−0.0848062
$836$	0	0
$837$	1038.48	0.0428853
$838$	0	0
$839$	33414.8	1.37498	0.687490	−	0.726194i	$-0.258713\pi$
$839$	33414.8	1.37498	0.687490	+	0.726194i	$0.258713\pi$
$840$	0	0
$841$	−1907.18	−0.0781985
$842$	0	0
$843$	20855.0	0.852058
$844$	0	0
$845$	−1914.18	−0.0779288
$846$	0	0
$847$	12913.5	0.523866
$848$	0	0
$849$	−9693.64	−0.391855
$850$	0	0
$851$	15276.3	0.615354
$852$	0	0
$853$	22959.6	0.921597	0.460798	−	0.887505i	$-0.347563\pi$
$853$	22959.6	0.921597	0.460798	+	0.887505i	$0.347563\pi$
$854$	0	0
$855$	−1072.84	−0.0429127
$856$	0	0
$857$	1835.34	0.0731553	0.0365777	−	0.999331i	$-0.488354\pi$
$857$	1835.34	0.0731553	0.0365777	+	0.999331i	$0.488354\pi$
$858$	0	0
$859$	−15520.2	−0.616463	−0.308232	−	0.951311i	$-0.599737\pi$
$859$	−15520.2	−0.616463	−0.308232	+	0.951311i	$0.599737\pi$
$860$	0	0
$861$	−1228.70	−0.0486341
$862$	0	0
$863$	−3343.74	−0.131891	−0.0659456	−	0.997823i	$-0.521006\pi$
$863$	−3343.74	−0.131891	−0.0659456	+	0.997823i	$0.521006\pi$
$864$	0	0
$865$	−2202.19	−0.0865625
$866$	0	0
$867$	12102.4	0.474069
$868$	0	0
$869$	13020.6	0.508277
$870$	0	0
$871$	2667.96	0.103789
$872$	0	0
$873$	5707.40	0.221267
$874$	0	0
$875$	−1529.85	−0.0591065
$876$	0	0
$877$	12103.7	0.466035	0.233017	−	0.972473i	$-0.425140\pi$
$877$	12103.7	0.466035	0.233017	+	0.972473i	$0.425140\pi$
$878$	0	0
$879$	−12812.7	−0.491650
$880$	0	0
$881$	−30396.5	−1.16241	−0.581206	−	0.813757i	$-0.697419\pi$
$881$	−30396.5	−1.16241	−0.581206	+	0.813757i	$0.697419\pi$
$882$	0	0
$883$	48003.3	1.82949	0.914746	−	0.404030i	$-0.132391\pi$
$883$	48003.3	1.82949	0.914746	+	0.404030i	$0.132391\pi$
$884$	0	0
$885$	−1914.25	−0.0727083
$886$	0	0
$887$	46162.9	1.74746	0.873730	−	0.486411i	$-0.161694\pi$
$887$	46162.9	1.74746	0.873730	+	0.486411i	$0.161694\pi$
$888$	0	0
$889$	−9497.38	−0.358304
$890$	0	0
$891$	4564.69	0.171630
$892$	0	0
$893$	−42894.5	−1.60740
$894$	0	0
$895$	−1578.07	−0.0589375
$896$	0	0
$897$	1993.97	0.0742214
$898$	0	0
$899$	−5766.99	−0.213949
$900$	0	0
$901$	4571.86	0.169046
$902$	0	0
$903$	−10192.5	−0.375619
$904$	0	0
$905$	−1374.87	−0.0504998
$906$	0	0
$907$	33924.6	1.24195	0.620975	−	0.783830i	$-0.286737\pi$
$907$	33924.6	1.24195	0.620975	+	0.783830i	$0.286737\pi$
$908$	0	0
$909$	−17645.8	−0.643865
$910$	0	0
$911$	53863.4	1.95892	0.979459	−	0.201644i	$-0.0646284\pi$
$911$	53863.4	1.95892	0.979459	+	0.201644i	$0.0646284\pi$
$912$	0	0
$913$	−25031.9	−0.907377
$914$	0	0
$915$	1810.49	0.0654130
$916$	0	0
$917$	12475.6	0.449270
$918$	0	0
$919$	42709.2	1.53302	0.766511	−	0.642231i	$-0.221991\pi$
$919$	42709.2	1.53302	0.766511	+	0.642231i	$0.221991\pi$
$920$	0	0
$921$	856.205	0.0306329
$922$	0	0
$923$	−2919.59	−0.104116
$924$	0	0
$925$	10718.2	0.380987
$926$	0	0
$927$	−16259.9	−0.576099
$928$	0	0
$929$	27676.5	0.977434	0.488717	−	0.872442i	$-0.337465\pi$
$929$	27676.5	0.977434	0.488717	+	0.872442i	$0.337465\pi$
$930$	0	0
$931$	−6661.03	−0.234486
$932$	0	0
$933$	−4098.31	−0.143808
$934$	0	0
$935$	−1465.00	−0.0512412
$936$	0	0
$937$	1942.63	0.0677298	0.0338649	−	0.999426i	$-0.489218\pi$
$937$	1942.63	0.0677298	0.0338649	+	0.999426i	$0.489218\pi$
$938$	0	0
$939$	−2261.91	−0.0786099
$940$	0	0
$941$	−38950.3	−1.34936	−0.674679	−	0.738112i	$-0.735718\pi$
$941$	−38950.3	−1.34936	−0.674679	+	0.738112i	$0.735718\pi$
$942$	0	0
$943$	−10359.8	−0.357754
$944$	0	0
$945$	−165.733	−0.00570508
$946$	0	0
$947$	−36704.5	−1.25949	−0.629744	−	0.776803i	$-0.716840\pi$
$947$	−36704.5	−1.25949	−0.629744	+	0.776803i	$0.716840\pi$
$948$	0	0
$949$	−3205.41	−0.109644
$950$	0	0
$951$	16137.3	0.550251
$952$	0	0
$953$	21115.4	0.717729	0.358865	−	0.933390i	$-0.383164\pi$
$953$	21115.4	0.717729	0.358865	+	0.933390i	$0.383164\pi$
$954$	0	0
$955$	−3304.57	−0.111972
$956$	0	0
$957$	−25349.1	−0.856239
$958$	0	0
$959$	−6857.90	−0.230921
$960$	0	0
$961$	−28311.7	−0.950343
$962$	0	0
$963$	1086.46	0.0363559
$964$	0	0
$965$	2139.68	0.0713769
$966$	0	0
$967$	48141.5	1.60096	0.800479	−	0.599360i	$-0.204578\pi$
$967$	48141.5	1.60096	0.800479	+	0.599360i	$0.204578\pi$
$968$	0	0
$969$	−12090.1	−0.400816
$970$	0	0
$971$	−23997.8	−0.793128	−0.396564	−	0.918007i	$-0.629798\pi$
$971$	−23997.8	−0.793128	−0.396564	+	0.918007i	$0.629798\pi$
$972$	0	0
$973$	4453.80	0.146744
$974$	0	0
$975$	1399.01	0.0459530
$976$	0	0
$977$	−16720.0	−0.547513	−0.273756	−	0.961799i	$-0.588266\pi$
$977$	−16720.0	−0.547513	−0.273756	+	0.961799i	$0.588266\pi$
$978$	0	0
$979$	−61047.2	−1.99293
$980$	0	0
$981$	−10972.5	−0.357110
$982$	0	0
$983$	−29992.3	−0.973150	−0.486575	−	0.873639i	$-0.661754\pi$
$983$	−29992.3	−0.973150	−0.486575	+	0.873639i	$0.661754\pi$
$984$	0	0
$985$	2230.56	0.0721538
$986$	0	0
$987$	−6626.37	−0.213698
$988$	0	0
$989$	−85938.3	−2.76307
$990$	0	0
$991$	−10445.9	−0.334838	−0.167419	−	0.985886i	$-0.553543\pi$
$991$	−10445.9	−0.334838	−0.167419	+	0.985886i	$0.553543\pi$
$992$	0	0
$993$	24030.8	0.767969
$994$	0	0
$995$	947.425	0.0301863
$996$	0	0
$997$	−1464.38	−0.0465168	−0.0232584	−	0.999729i	$-0.507404\pi$
$997$	−1464.38	−0.0465168	−0.0232584	+	0.999729i	$0.507404\pi$
$998$	0	0
$999$	2329.47	0.0737748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bj.1.1		2
4.3	odd	2		1344.4.a.br.1.1		2
8.3	odd	2		672.4.a.e.1.2	✓	2
8.5	even	2		672.4.a.j.1.2	yes	2
24.5	odd	2		2016.4.a.r.1.1		2
24.11	even	2		2016.4.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.e.1.2	✓	2	8.3	odd	2
672.4.a.j.1.2	yes	2	8.5	even	2
1344.4.a.bj.1.1		2	1.1	even	1	trivial
1344.4.a.br.1.1		2	4.3	odd	2
2016.4.a.q.1.1		2	24.11	even	2
2016.4.a.r.1.1		2	24.5	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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +0.876894 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +0.876894 q^{5} +7.00000 q^{7} +9.00000 q^{9} +56.3542 q^{11} +3.75379 q^{13} -2.63068 q^{15} -29.6458 q^{17} -135.939 q^{19} -21.0000 q^{21} -177.062 q^{23} -124.231 q^{25} -27.0000 q^{27} +149.939 q^{29} -38.4621 q^{31} -169.062 q^{33} +6.13826 q^{35} -86.2765 q^{37} -11.2614 q^{39} +58.5094 q^{41} +485.356 q^{43} +7.89205 q^{45} +315.542 q^{47} +49.0000 q^{49} +88.9375 q^{51} -154.216 q^{53} +49.4166 q^{55} +407.818 q^{57} +727.663 q^{59} -688.220 q^{61} +63.0000 q^{63} +3.29168 q^{65} +710.739 q^{67} +531.187 q^{69} -777.771 q^{71} -853.913 q^{73} +372.693 q^{75} +394.479 q^{77} +231.049 q^{79} +81.0000 q^{81} -444.189 q^{83} -25.9963 q^{85} -449.818 q^{87} -1083.28 q^{89} +26.2765 q^{91} +115.386 q^{93} -119.204 q^{95} +634.155 q^{97} +507.187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} + 10 q^{5} + 14 q^{7} + 18 q^{9} + 22 q^{11} + 24 q^{13} - 30 q^{15} - 150 q^{17} - 8 q^{19} - 42 q^{21} - 82 q^{23} - 166 q^{25} - 54 q^{27} + 36 q^{29} + 88 q^{31} - 66 q^{33} + 70 q^{35} - 288 q^{37} - 72 q^{39} - 386 q^{41} + 344 q^{43} + 90 q^{45} - 276 q^{47} + 98 q^{49} + 450 q^{51} - 160 q^{53} - 264 q^{55} + 24 q^{57} + 1076 q^{59} - 156 q^{61} + 126 q^{63} + 188 q^{65} + 1372 q^{67} + 246 q^{69} - 1102 q^{71} - 240 q^{73} + 498 q^{75} + 154 q^{77} - 412 q^{79} + 162 q^{81} + 464 q^{83} - 1124 q^{85} - 108 q^{87} - 1746 q^{89} + 168 q^{91} - 264 q^{93} + 1048 q^{95} + 856 q^{97} + 198 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 10 * q^5 + 14 * q^7 + 18 * q^9 + 22 * q^11 + 24 * q^13 - 30 * q^15 - 150 * q^17 - 8 * q^19 - 42 * q^21 - 82 * q^23 - 166 * q^25 - 54 * q^27 + 36 * q^29 + 88 * q^31 - 66 * q^33 + 70 * q^35 - 288 * q^37 - 72 * q^39 - 386 * q^41 + 344 * q^43 + 90 * q^45 - 276 * q^47 + 98 * q^49 + 450 * q^51 - 160 * q^53 - 264 * q^55 + 24 * q^57 + 1076 * q^59 - 156 * q^61 + 126 * q^63 + 188 * q^65 + 1372 * q^67 + 246 * q^69 - 1102 * q^71 - 240 * q^73 + 498 * q^75 + 154 * q^77 - 412 * q^79 + 162 * q^81 + 464 * q^83 - 1124 * q^85 - 108 * q^87 - 1746 * q^89 + 168 * q^91 - 264 * q^93 + 1048 * q^95 + 856 * q^97 + 198 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more