LMFDB - Embedded newform 1380.4.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, 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("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 176x^{3} + 306x^{2} + 3519x - 7104 $ x^5 - 2x^4 - 176x^3 + 306x^2 + 3519x - 7104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$12.5752$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} -32.6481 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} -32.6481 q^{7} +9.00000 q^{9} +55.6441 q^{11} +0.905137 q^{13} +15.0000 q^{15} -16.2739 q^{17} -59.2584 q^{19} -97.9443 q^{21} -23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -195.305 q^{29} +70.6041 q^{31} +166.932 q^{33} -163.240 q^{35} +92.4940 q^{37} +2.71541 q^{39} -266.861 q^{41} +64.0830 q^{43} +45.0000 q^{45} -124.884 q^{47} +722.898 q^{49} -48.8216 q^{51} +185.300 q^{53} +278.220 q^{55} -177.775 q^{57} -898.566 q^{59} +77.1427 q^{61} -293.833 q^{63} +4.52569 q^{65} -378.278 q^{67} -69.0000 q^{69} +918.643 q^{71} -142.030 q^{73} +75.0000 q^{75} -1816.67 q^{77} +573.448 q^{79} +81.0000 q^{81} -279.062 q^{83} -81.3694 q^{85} -585.916 q^{87} -797.567 q^{89} -29.5510 q^{91} +211.812 q^{93} -296.292 q^{95} -1822.33 q^{97} +500.797 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 15 q^{3} + 25 q^{5} - 23 q^{7} + 45 q^{9}+O(q^{10}) $ 5 * q + 15 * q^3 + 25 * q^5 - 23 * q^7 + 45 * q^9	$ 5 q + 15 q^{3} + 25 q^{5} - 23 q^{7} + 45 q^{9} - 42 q^{11} - 80 q^{13} + 75 q^{15} - 81 q^{17} - 128 q^{19} - 69 q^{21} - 115 q^{23} + 125 q^{25} + 135 q^{27} - 261 q^{29} - 233 q^{31} - 126 q^{33} - 115 q^{35} - 371 q^{37} - 240 q^{39} - 819 q^{41} - 596 q^{43} + 225 q^{45} - 186 q^{47} - 132 q^{49} - 243 q^{51} - 831 q^{53} - 210 q^{55} - 384 q^{57} - 1275 q^{59} - 152 q^{61} - 207 q^{63} - 400 q^{65} - 485 q^{67} - 345 q^{69} + 531 q^{71} - 788 q^{73} + 375 q^{75} - 1896 q^{77} - 134 q^{79} + 405 q^{81} - 585 q^{83} - 405 q^{85} - 783 q^{87} - 846 q^{89} + 500 q^{91} - 699 q^{93} - 640 q^{95} - 2078 q^{97} - 378 q^{99}+O(q^{100}) $ 5 * q + 15 * q^3 + 25 * q^5 - 23 * q^7 + 45 * q^9 - 42 * q^11 - 80 * q^13 + 75 * q^15 - 81 * q^17 - 128 * q^19 - 69 * q^21 - 115 * q^23 + 125 * q^25 + 135 * q^27 - 261 * q^29 - 233 * q^31 - 126 * q^33 - 115 * q^35 - 371 * q^37 - 240 * q^39 - 819 * q^41 - 596 * q^43 + 225 * q^45 - 186 * q^47 - 132 * q^49 - 243 * q^51 - 831 * q^53 - 210 * q^55 - 384 * q^57 - 1275 * q^59 - 152 * q^61 - 207 * q^63 - 400 * q^65 - 485 * q^67 - 345 * q^69 + 531 * q^71 - 788 * q^73 + 375 * q^75 - 1896 * q^77 - 134 * q^79 + 405 * q^81 - 585 * q^83 - 405 * q^85 - 783 * q^87 - 846 * q^89 + 500 * q^91 - 699 * q^93 - 640 * q^95 - 2078 * q^97 - 378 * 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−32.6481	−1.76283	−0.881416	−	0.472342i	$-0.843409\pi$
$7$	−32.6481	−1.76283	−0.881416	+	0.472342i	$0.843409\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	55.6441	1.52521	0.762605	−	0.646864i	$-0.223920\pi$
$11$	55.6441	1.52521	0.762605	+	0.646864i	$0.223920\pi$
$12$	0	0
$13$	0.905137	0.0193108	0.00965538	−	0.999953i	$-0.496927\pi$
$13$	0.905137	0.0193108	0.00965538	+	0.999953i	$0.496927\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−16.2739	−0.232176	−0.116088	−	0.993239i	$-0.537035\pi$
$17$	−16.2739	−0.232176	−0.116088	+	0.993239i	$0.537035\pi$
$18$	0	0
$19$	−59.2584	−0.715516	−0.357758	−	0.933814i	$-0.616459\pi$
$19$	−59.2584	−0.715516	−0.357758	+	0.933814i	$0.616459\pi$
$20$	0	0
$21$	−97.9443	−1.01777
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−195.305	−1.25060	−0.625299	−	0.780386i	$-0.715023\pi$
$29$	−195.305	−1.25060	−0.625299	+	0.780386i	$0.715023\pi$
$30$	0	0
$31$	70.6041	0.409060	0.204530	−	0.978860i	$-0.434433\pi$
$31$	70.6041	0.409060	0.204530	+	0.978860i	$0.434433\pi$
$32$	0	0
$33$	166.932	0.880581
$34$	0	0
$35$	−163.240	−0.788362
$36$	0	0
$37$	92.4940	0.410971	0.205485	−	0.978660i	$-0.434123\pi$
$37$	92.4940	0.410971	0.205485	+	0.978660i	$0.434123\pi$
$38$	0	0
$39$	2.71541	0.0111491
$40$	0	0
$41$	−266.861	−1.01651	−0.508253	−	0.861208i	$-0.669708\pi$
$41$	−266.861	−1.01651	−0.508253	+	0.861208i	$0.669708\pi$
$42$	0	0
$43$	64.0830	0.227269	0.113634	−	0.993523i	$-0.463751\pi$
$43$	64.0830	0.227269	0.113634	+	0.993523i	$0.463751\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−124.884	−0.387580	−0.193790	−	0.981043i	$-0.562078\pi$
$47$	−124.884	−0.387580	−0.193790	+	0.981043i	$0.562078\pi$
$48$	0	0
$49$	722.898	2.10757
$50$	0	0
$51$	−48.8216	−0.134047
$52$	0	0
$53$	185.300	0.480243	0.240122	−	0.970743i	$-0.422813\pi$
$53$	185.300	0.480243	0.240122	+	0.970743i	$0.422813\pi$
$54$	0	0
$55$	278.220	0.682095
$56$	0	0
$57$	−177.775	−0.413103
$58$	0	0
$59$	−898.566	−1.98277	−0.991385	−	0.130983i	$-0.958187\pi$
$59$	−898.566	−1.98277	−0.991385	+	0.130983i	$0.958187\pi$
$60$	0	0
$61$	77.1427	0.161920	0.0809599	−	0.996717i	$-0.474201\pi$
$61$	77.1427	0.161920	0.0809599	+	0.996717i	$0.474201\pi$
$62$	0	0
$63$	−293.833	−0.587610
$64$	0	0
$65$	4.52569	0.00863604
$66$	0	0
$67$	−378.278	−0.689762	−0.344881	−	0.938646i	$-0.612081\pi$
$67$	−378.278	−0.689762	−0.344881	+	0.938646i	$0.612081\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	918.643	1.53553	0.767766	−	0.640730i	$-0.221368\pi$
$71$	918.643	1.53553	0.767766	+	0.640730i	$0.221368\pi$
$72$	0	0
$73$	−142.030	−0.227716	−0.113858	−	0.993497i	$-0.536321\pi$
$73$	−142.030	−0.227716	−0.113858	+	0.993497i	$0.536321\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	−1816.67	−2.68869
$78$	0	0
$79$	573.448	0.816683	0.408342	−	0.912829i	$-0.366107\pi$
$79$	573.448	0.816683	0.408342	+	0.912829i	$0.366107\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−279.062	−0.369049	−0.184525	−	0.982828i	$-0.559075\pi$
$83$	−279.062	−0.369049	−0.184525	+	0.982828i	$0.559075\pi$
$84$	0	0
$85$	−81.3694	−0.103832
$86$	0	0
$87$	−585.916	−0.722033
$88$	0	0
$89$	−797.567	−0.949909	−0.474955	−	0.880010i	$-0.657536\pi$
$89$	−797.567	−0.949909	−0.474955	+	0.880010i	$0.657536\pi$
$90$	0	0
$91$	−29.5510	−0.0340416
$92$	0	0
$93$	211.812	0.236171
$94$	0	0
$95$	−296.292	−0.319988
$96$	0	0
$97$	−1822.33	−1.90752	−0.953760	−	0.300570i	$-0.902823\pi$
$97$	−1822.33	−1.90752	−0.953760	+	0.300570i	$0.902823\pi$
$98$	0	0
$99$	500.797	0.508404
$100$	0	0
$101$	−1128.81	−1.11209	−0.556046	−	0.831152i	$-0.687682\pi$
$101$	−1128.81	−1.11209	−0.556046	+	0.831152i	$0.687682\pi$
$102$	0	0
$103$	−1232.77	−1.17930	−0.589650	−	0.807659i	$-0.700734\pi$
$103$	−1232.77	−1.17930	−0.589650	+	0.807659i	$0.700734\pi$
$104$	0	0
$105$	−489.721	−0.455161
$106$	0	0
$107$	−547.714	−0.494856	−0.247428	−	0.968906i	$-0.579585\pi$
$107$	−547.714	−0.494856	−0.247428	+	0.968906i	$0.579585\pi$
$108$	0	0
$109$	−1439.15	−1.26464	−0.632321	−	0.774706i	$-0.717898\pi$
$109$	−1439.15	−1.26464	−0.632321	+	0.774706i	$0.717898\pi$
$110$	0	0
$111$	277.482	0.237274
$112$	0	0
$113$	−1446.78	−1.20444	−0.602218	−	0.798331i	$-0.705716\pi$
$113$	−1446.78	−1.20444	−0.602218	+	0.798331i	$0.705716\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	8.14624	0.00643692
$118$	0	0
$119$	531.311	0.409287
$120$	0	0
$121$	1765.26	1.32627
$122$	0	0
$123$	−800.584	−0.586880
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−469.540	−0.328070	−0.164035	−	0.986455i	$-0.552451\pi$
$127$	−469.540	−0.328070	−0.164035	+	0.986455i	$0.552451\pi$
$128$	0	0
$129$	192.249	0.131214
$130$	0	0
$131$	1542.13	1.02852	0.514262	−	0.857633i	$-0.328066\pi$
$131$	1542.13	1.02852	0.514262	+	0.857633i	$0.328066\pi$
$132$	0	0
$133$	1934.67	1.26133
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1599.54	−0.997500	−0.498750	−	0.866746i	$-0.666207\pi$
$137$	−1599.54	−0.997500	−0.498750	+	0.866746i	$0.666207\pi$
$138$	0	0
$139$	−2125.97	−1.29728	−0.648642	−	0.761093i	$-0.724663\pi$
$139$	−2125.97	−1.29728	−0.648642	+	0.761093i	$0.724663\pi$
$140$	0	0
$141$	−374.653	−0.223770
$142$	0	0
$143$	50.3655	0.0294530
$144$	0	0
$145$	−976.527	−0.559284
$146$	0	0
$147$	2168.69	1.21681
$148$	0	0
$149$	1071.73	0.589259	0.294629	−	0.955612i	$-0.404804\pi$
$149$	1071.73	0.589259	0.294629	+	0.955612i	$0.404804\pi$
$150$	0	0
$151$	3321.04	1.78981	0.894907	−	0.446252i	$-0.147241\pi$
$151$	3321.04	1.78981	0.894907	+	0.446252i	$0.147241\pi$
$152$	0	0
$153$	−146.465	−0.0773921
$154$	0	0
$155$	353.021	0.182937
$156$	0	0
$157$	1416.77	0.720197	0.360099	−	0.932914i	$-0.382743\pi$
$157$	1416.77	0.720197	0.360099	+	0.932914i	$0.382743\pi$
$158$	0	0
$159$	555.899	0.277268
$160$	0	0
$161$	750.906	0.367576
$162$	0	0
$163$	−2562.25	−1.23123	−0.615616	−	0.788046i	$-0.711093\pi$
$163$	−2562.25	−1.23123	−0.615616	+	0.788046i	$0.711093\pi$
$164$	0	0
$165$	834.661	0.393808
$166$	0	0
$167$	−1435.47	−0.665148	−0.332574	−	0.943077i	$-0.607917\pi$
$167$	−1435.47	−0.665148	−0.332574	+	0.943077i	$0.607917\pi$
$168$	0	0
$169$	−2196.18	−0.999627
$170$	0	0
$171$	−533.325	−0.238505
$172$	0	0
$173$	968.389	0.425579	0.212790	−	0.977098i	$-0.431745\pi$
$173$	968.389	0.425579	0.212790	+	0.977098i	$0.431745\pi$
$174$	0	0
$175$	−816.202	−0.352566
$176$	0	0
$177$	−2695.70	−1.14475
$178$	0	0
$179$	1549.23	0.646898	0.323449	−	0.946246i	$-0.395158\pi$
$179$	1549.23	0.646898	0.323449	+	0.946246i	$0.395158\pi$
$180$	0	0
$181$	110.840	0.0455176	0.0227588	−	0.999741i	$-0.492755\pi$
$181$	110.840	0.0455176	0.0227588	+	0.999741i	$0.492755\pi$
$182$	0	0
$183$	231.428	0.0934845
$184$	0	0
$185$	462.470	0.183792
$186$	0	0
$187$	−905.545	−0.354118
$188$	0	0
$189$	−881.498	−0.339257
$190$	0	0
$191$	4435.56	1.68035	0.840173	−	0.542318i	$-0.182453\pi$
$191$	4435.56	1.68035	0.840173	+	0.542318i	$0.182453\pi$
$192$	0	0
$193$	2360.94	0.880540	0.440270	−	0.897866i	$-0.354883\pi$
$193$	2360.94	0.880540	0.440270	+	0.897866i	$0.354883\pi$
$194$	0	0
$195$	13.5771	0.00498602
$196$	0	0
$197$	−607.861	−0.219839	−0.109920	−	0.993940i	$-0.535059\pi$
$197$	−607.861	−0.219839	−0.109920	+	0.993940i	$0.535059\pi$
$198$	0	0
$199$	1226.28	0.436826	0.218413	−	0.975856i	$-0.429912\pi$
$199$	1226.28	0.436826	0.218413	+	0.975856i	$0.429912\pi$
$200$	0	0
$201$	−1134.83	−0.398234
$202$	0	0
$203$	6376.35	2.20459
$204$	0	0
$205$	−1334.31	−0.454595
$206$	0	0
$207$	−207.000	−0.0695048
$208$	0	0
$209$	−3297.38	−1.09131
$210$	0	0
$211$	−365.227	−0.119162	−0.0595812	−	0.998223i	$-0.518977\pi$
$211$	−365.227	−0.119162	−0.0595812	+	0.998223i	$0.518977\pi$
$212$	0	0
$213$	2755.93	0.886540
$214$	0	0
$215$	320.415	0.101638
$216$	0	0
$217$	−2305.09	−0.721104
$218$	0	0
$219$	−426.089	−0.131472
$220$	0	0
$221$	−14.7301	−0.00448350
$222$	0	0
$223$	−2840.48	−0.852972	−0.426486	−	0.904494i	$-0.640249\pi$
$223$	−2840.48	−0.852972	−0.426486	+	0.904494i	$0.640249\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−5510.59	−1.61124	−0.805618	−	0.592436i	$-0.798166\pi$
$227$	−5510.59	−1.61124	−0.805618	+	0.592436i	$0.798166\pi$
$228$	0	0
$229$	−735.352	−0.212198	−0.106099	−	0.994356i	$-0.533836\pi$
$229$	−735.352	−0.212198	−0.106099	+	0.994356i	$0.533836\pi$
$230$	0	0
$231$	−5450.02	−1.55232
$232$	0	0
$233$	−5693.04	−1.60070	−0.800351	−	0.599532i	$-0.795353\pi$
$233$	−5693.04	−1.60070	−0.800351	+	0.599532i	$0.795353\pi$
$234$	0	0
$235$	−624.422	−0.173331
$236$	0	0
$237$	1720.34	0.471512
$238$	0	0
$239$	793.063	0.214640	0.107320	−	0.994225i	$-0.465773\pi$
$239$	793.063	0.214640	0.107320	+	0.994225i	$0.465773\pi$
$240$	0	0
$241$	−6630.59	−1.77226	−0.886129	−	0.463440i	$-0.846615\pi$
$241$	−6630.59	−1.77226	−0.886129	+	0.463440i	$0.846615\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	3614.49	0.942535
$246$	0	0
$247$	−53.6370	−0.0138172
$248$	0	0
$249$	−837.187	−0.213071
$250$	0	0
$251$	754.055	0.189624	0.0948119	−	0.995495i	$-0.469775\pi$
$251$	754.055	0.189624	0.0948119	+	0.995495i	$0.469775\pi$
$252$	0	0
$253$	−1279.81	−0.318028
$254$	0	0
$255$	−244.108	−0.0599476
$256$	0	0
$257$	449.676	0.109144	0.0545720	−	0.998510i	$-0.482621\pi$
$257$	449.676	0.109144	0.0545720	+	0.998510i	$0.482621\pi$
$258$	0	0
$259$	−3019.75	−0.724472
$260$	0	0
$261$	−1757.75	−0.416866
$262$	0	0
$263$	−1923.70	−0.451027	−0.225514	−	0.974240i	$-0.572406\pi$
$263$	−1923.70	−0.451027	−0.225514	+	0.974240i	$0.572406\pi$
$264$	0	0
$265$	926.499	0.214771
$266$	0	0
$267$	−2392.70	−0.548430
$268$	0	0
$269$	−8569.95	−1.94245	−0.971224	−	0.238166i	$-0.923454\pi$
$269$	−8569.95	−1.94245	−0.971224	+	0.238166i	$0.923454\pi$
$270$	0	0
$271$	−1347.20	−0.301981	−0.150990	−	0.988535i	$-0.548246\pi$
$271$	−1347.20	−0.301981	−0.150990	+	0.988535i	$0.548246\pi$
$272$	0	0
$273$	−88.6530	−0.0196539
$274$	0	0
$275$	1391.10	0.305042
$276$	0	0
$277$	4134.26	0.896764	0.448382	−	0.893842i	$-0.352000\pi$
$277$	4134.26	0.896764	0.448382	+	0.893842i	$0.352000\pi$
$278$	0	0
$279$	635.437	0.136353
$280$	0	0
$281$	5477.71	1.16289	0.581446	−	0.813585i	$-0.302487\pi$
$281$	5477.71	1.16289	0.581446	+	0.813585i	$0.302487\pi$
$282$	0	0
$283$	−5466.56	−1.14825	−0.574123	−	0.818769i	$-0.694657\pi$
$283$	−5466.56	−1.14825	−0.574123	+	0.818769i	$0.694657\pi$
$284$	0	0
$285$	−888.875	−0.184745
$286$	0	0
$287$	8712.51	1.79193
$288$	0	0
$289$	−4648.16	−0.946094
$290$	0	0
$291$	−5466.98	−1.10131
$292$	0	0
$293$	7620.51	1.51944	0.759719	−	0.650252i	$-0.225337\pi$
$293$	7620.51	1.51944	0.759719	+	0.650252i	$0.225337\pi$
$294$	0	0
$295$	−4492.83	−0.886721
$296$	0	0
$297$	1502.39	0.293527
$298$	0	0
$299$	−20.8182	−0.00402657
$300$	0	0
$301$	−2092.19	−0.400637
$302$	0	0
$303$	−3386.44	−0.642066
$304$	0	0
$305$	385.713	0.0724128
$306$	0	0
$307$	−487.074	−0.0905499	−0.0452749	−	0.998975i	$-0.514416\pi$
$307$	−487.074	−0.0905499	−0.0452749	+	0.998975i	$0.514416\pi$
$308$	0	0
$309$	−3698.30	−0.680869
$310$	0	0
$311$	2250.91	0.410409	0.205205	−	0.978719i	$-0.434214\pi$
$311$	2250.91	0.410409	0.205205	+	0.978719i	$0.434214\pi$
$312$	0	0
$313$	−3074.28	−0.555172	−0.277586	−	0.960701i	$-0.589534\pi$
$313$	−3074.28	−0.555172	−0.277586	+	0.960701i	$0.589534\pi$
$314$	0	0
$315$	−1469.16	−0.262787
$316$	0	0
$317$	−7675.54	−1.35994	−0.679970	−	0.733240i	$-0.738007\pi$
$317$	−7675.54	−1.35994	−0.679970	+	0.733240i	$0.738007\pi$
$318$	0	0
$319$	−10867.6	−1.90742
$320$	0	0
$321$	−1643.14	−0.285705
$322$	0	0
$323$	964.363	0.166126
$324$	0	0
$325$	22.6284	0.00386215
$326$	0	0
$327$	−4317.46	−0.730142
$328$	0	0
$329$	4077.24	0.683239
$330$	0	0
$331$	6520.56	1.08279	0.541393	−	0.840770i	$-0.317897\pi$
$331$	6520.56	1.08279	0.541393	+	0.840770i	$0.317897\pi$
$332$	0	0
$333$	832.446	0.136990
$334$	0	0
$335$	−1891.39	−0.308471
$336$	0	0
$337$	2343.94	0.378880	0.189440	−	0.981892i	$-0.439333\pi$
$337$	2343.94	0.378880	0.189440	+	0.981892i	$0.439333\pi$
$338$	0	0
$339$	−4340.33	−0.695382
$340$	0	0
$341$	3928.70	0.623903
$342$	0	0
$343$	−12402.9	−1.95246
$344$	0	0
$345$	−345.000	−0.0538382
$346$	0	0
$347$	3246.85	0.502305	0.251153	−	0.967947i	$-0.419190\pi$
$347$	3246.85	0.502305	0.251153	+	0.967947i	$0.419190\pi$
$348$	0	0
$349$	11575.7	1.77545	0.887725	−	0.460373i	$-0.152284\pi$
$349$	11575.7	1.77545	0.887725	+	0.460373i	$0.152284\pi$
$350$	0	0
$351$	24.4387	0.00371636
$352$	0	0
$353$	−3160.07	−0.476468	−0.238234	−	0.971208i	$-0.576569\pi$
$353$	−3160.07	−0.476468	−0.238234	+	0.971208i	$0.576569\pi$
$354$	0	0
$355$	4593.21	0.686711
$356$	0	0
$357$	1593.93	0.236302
$358$	0	0
$359$	−1420.00	−0.208760	−0.104380	−	0.994538i	$-0.533286\pi$
$359$	−1420.00	−0.208760	−0.104380	+	0.994538i	$0.533286\pi$
$360$	0	0
$361$	−3347.45	−0.488037
$362$	0	0
$363$	5295.79	0.765722
$364$	0	0
$365$	−710.148	−0.101838
$366$	0	0
$367$	9185.64	1.30650	0.653251	−	0.757141i	$-0.273404\pi$
$367$	9185.64	1.30650	0.653251	+	0.757141i	$0.273404\pi$
$368$	0	0
$369$	−2401.75	−0.338835
$370$	0	0
$371$	−6049.68	−0.846587
$372$	0	0
$373$	4301.37	0.597095	0.298548	−	0.954395i	$-0.403498\pi$
$373$	4301.37	0.597095	0.298548	+	0.954395i	$0.403498\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−176.778	−0.0241500
$378$	0	0
$379$	12368.6	1.67634	0.838169	−	0.545411i	$-0.183626\pi$
$379$	12368.6	1.67634	0.838169	+	0.545411i	$0.183626\pi$
$380$	0	0
$381$	−1408.62	−0.189411
$382$	0	0
$383$	5959.00	0.795015	0.397508	−	0.917599i	$-0.369875\pi$
$383$	5959.00	0.795015	0.397508	+	0.917599i	$0.369875\pi$
$384$	0	0
$385$	−9083.36	−1.20242
$386$	0	0
$387$	576.747	0.0757563
$388$	0	0
$389$	7199.86	0.938425	0.469213	−	0.883085i	$-0.344538\pi$
$389$	7199.86	0.938425	0.469213	+	0.883085i	$0.344538\pi$
$390$	0	0
$391$	374.299	0.0484121
$392$	0	0
$393$	4626.39	0.593818
$394$	0	0
$395$	2867.24	0.365232
$396$	0	0
$397$	−996.121	−0.125929	−0.0629646	−	0.998016i	$-0.520056\pi$
$397$	−996.121	−0.125929	−0.0629646	+	0.998016i	$0.520056\pi$
$398$	0	0
$399$	5804.02	0.728231
$400$	0	0
$401$	−11516.1	−1.43413	−0.717063	−	0.697008i	$-0.754514\pi$
$401$	−11516.1	−1.43413	−0.717063	+	0.697008i	$0.754514\pi$
$402$	0	0
$403$	63.9064	0.00789927
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	5146.75	0.626817
$408$	0	0
$409$	1066.79	0.128972	0.0644860	−	0.997919i	$-0.479459\pi$
$409$	1066.79	0.128972	0.0644860	+	0.997919i	$0.479459\pi$
$410$	0	0
$411$	−4798.61	−0.575907
$412$	0	0
$413$	29336.5	3.49529
$414$	0	0
$415$	−1395.31	−0.165044
$416$	0	0
$417$	−6377.91	−0.748988
$418$	0	0
$419$	136.470	0.0159117	0.00795586	−	0.999968i	$-0.497468\pi$
$419$	136.470	0.0159117	0.00795586	+	0.999968i	$0.497468\pi$
$420$	0	0
$421$	16364.3	1.89441	0.947207	−	0.320623i	$-0.103892\pi$
$421$	16364.3	1.89441	0.947207	+	0.320623i	$0.103892\pi$
$422$	0	0
$423$	−1123.96	−0.129193
$424$	0	0
$425$	−406.847	−0.0464352
$426$	0	0
$427$	−2518.56	−0.285437
$428$	0	0
$429$	151.097	0.0170047
$430$	0	0
$431$	−10075.3	−1.12601	−0.563006	−	0.826453i	$-0.690355\pi$
$431$	−10075.3	−1.12601	−0.563006	+	0.826453i	$0.690355\pi$
$432$	0	0
$433$	12238.7	1.35832	0.679159	−	0.733991i	$-0.262344\pi$
$433$	12238.7	1.35832	0.679159	+	0.733991i	$0.262344\pi$
$434$	0	0
$435$	−2929.58	−0.322903
$436$	0	0
$437$	1362.94	0.149195
$438$	0	0
$439$	−7615.56	−0.827952	−0.413976	−	0.910288i	$-0.635860\pi$
$439$	−7615.56	−0.827952	−0.413976	+	0.910288i	$0.635860\pi$
$440$	0	0
$441$	6506.08	0.702524
$442$	0	0
$443$	7014.24	0.752272	0.376136	−	0.926564i	$-0.377252\pi$
$443$	7014.24	0.752272	0.376136	+	0.926564i	$0.377252\pi$
$444$	0	0
$445$	−3987.84	−0.424812
$446$	0	0
$447$	3215.19	0.340209
$448$	0	0
$449$	5553.24	0.583682	0.291841	−	0.956467i	$-0.405732\pi$
$449$	5553.24	0.583682	0.291841	+	0.956467i	$0.405732\pi$
$450$	0	0
$451$	−14849.3	−1.55039
$452$	0	0
$453$	9963.11	1.03335
$454$	0	0
$455$	−147.755	−0.0152239
$456$	0	0
$457$	−10370.0	−1.06146	−0.530731	−	0.847540i	$-0.678082\pi$
$457$	−10370.0	−1.06146	−0.530731	+	0.847540i	$0.678082\pi$
$458$	0	0
$459$	−439.395	−0.0446823
$460$	0	0
$461$	1177.68	0.118980	0.0594901	−	0.998229i	$-0.481053\pi$
$461$	1177.68	0.118980	0.0594901	+	0.998229i	$0.481053\pi$
$462$	0	0
$463$	−11525.0	−1.15683	−0.578416	−	0.815742i	$-0.696329\pi$
$463$	−11525.0	−1.15683	−0.578416	+	0.815742i	$0.696329\pi$
$464$	0	0
$465$	1059.06	0.105619
$466$	0	0
$467$	11754.0	1.16469	0.582343	−	0.812943i	$-0.302136\pi$
$467$	11754.0	1.16469	0.582343	+	0.812943i	$0.302136\pi$
$468$	0	0
$469$	12350.1	1.21593
$470$	0	0
$471$	4250.32	0.415806
$472$	0	0
$473$	3565.84	0.346633
$474$	0	0
$475$	−1481.46	−0.143103
$476$	0	0
$477$	1667.70	0.160081
$478$	0	0
$479$	4471.02	0.426485	0.213243	−	0.976999i	$-0.431598\pi$
$479$	4471.02	0.426485	0.213243	+	0.976999i	$0.431598\pi$
$480$	0	0
$481$	83.7198	0.00793616
$482$	0	0
$483$	2252.72	0.212220
$484$	0	0
$485$	−9111.64	−0.853069
$486$	0	0
$487$	−7254.71	−0.675036	−0.337518	−	0.941319i	$-0.609587\pi$
$487$	−7254.71	−0.675036	−0.337518	+	0.941319i	$0.609587\pi$
$488$	0	0
$489$	−7686.75	−0.710853
$490$	0	0
$491$	−8504.16	−0.781644	−0.390822	−	0.920466i	$-0.627809\pi$
$491$	−8504.16	−0.781644	−0.390822	+	0.920466i	$0.627809\pi$
$492$	0	0
$493$	3178.38	0.290359
$494$	0	0
$495$	2503.98	0.227365
$496$	0	0
$497$	−29991.9	−2.70688
$498$	0	0
$499$	12535.3	1.12456	0.562281	−	0.826946i	$-0.309924\pi$
$499$	12535.3	1.12456	0.562281	+	0.826946i	$0.309924\pi$
$500$	0	0
$501$	−4306.40	−0.384024
$502$	0	0
$503$	−771.563	−0.0683942	−0.0341971	−	0.999415i	$-0.510887\pi$
$503$	−771.563	−0.0683942	−0.0341971	+	0.999415i	$0.510887\pi$
$504$	0	0
$505$	−5644.07	−0.497343
$506$	0	0
$507$	−6588.54	−0.577135
$508$	0	0
$509$	247.300	0.0215352	0.0107676	−	0.999942i	$-0.496573\pi$
$509$	247.300	0.0215352	0.0107676	+	0.999942i	$0.496573\pi$
$510$	0	0
$511$	4636.99	0.401426
$512$	0	0
$513$	−1599.98	−0.137701
$514$	0	0
$515$	−6163.83	−0.527399
$516$	0	0
$517$	−6949.08	−0.591142
$518$	0	0
$519$	2905.17	0.245708
$520$	0	0
$521$	8570.02	0.720651	0.360326	−	0.932827i	$-0.382666\pi$
$521$	8570.02	0.720651	0.360326	+	0.932827i	$0.382666\pi$
$522$	0	0
$523$	711.197	0.0594617	0.0297309	−	0.999558i	$-0.490535\pi$
$523$	711.197	0.0594617	0.0297309	+	0.999558i	$0.490535\pi$
$524$	0	0
$525$	−2448.61	−0.203554
$526$	0	0
$527$	−1149.00	−0.0949741
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−8087.10	−0.660923
$532$	0	0
$533$	−241.546	−0.0196295
$534$	0	0
$535$	−2738.57	−0.221306
$536$	0	0
$537$	4647.68	0.373487
$538$	0	0
$539$	40225.0	3.21449
$540$	0	0
$541$	−2299.28	−0.182724	−0.0913619	−	0.995818i	$-0.529122\pi$
$541$	−2299.28	−0.182724	−0.0913619	+	0.995818i	$0.529122\pi$
$542$	0	0
$543$	332.521	0.0262796
$544$	0	0
$545$	−7195.77	−0.565565
$546$	0	0
$547$	3507.85	0.274195	0.137098	−	0.990558i	$-0.456223\pi$
$547$	3507.85	0.274195	0.137098	+	0.990558i	$0.456223\pi$
$548$	0	0
$549$	694.284	0.0539733
$550$	0	0
$551$	11573.5	0.894822
$552$	0	0
$553$	−18722.0	−1.43967
$554$	0	0
$555$	1387.41	0.106112
$556$	0	0
$557$	10266.9	0.781011	0.390505	−	0.920601i	$-0.372300\pi$
$557$	10266.9	0.781011	0.390505	+	0.920601i	$0.372300\pi$
$558$	0	0
$559$	58.0039	0.00438873
$560$	0	0
$561$	−2716.64	−0.204450
$562$	0	0
$563$	−4166.24	−0.311876	−0.155938	−	0.987767i	$-0.549840\pi$
$563$	−4166.24	−0.311876	−0.155938	+	0.987767i	$0.549840\pi$
$564$	0	0
$565$	−7233.89	−0.538640
$566$	0	0
$567$	−2644.50	−0.195870
$568$	0	0
$569$	−13713.2	−1.01034	−0.505172	−	0.863019i	$-0.668571\pi$
$569$	−13713.2	−1.01034	−0.505172	+	0.863019i	$0.668571\pi$
$570$	0	0
$571$	20615.4	1.51091	0.755455	−	0.655200i	$-0.227416\pi$
$571$	20615.4	1.51091	0.755455	+	0.655200i	$0.227416\pi$
$572$	0	0
$573$	13306.7	0.970149
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−10217.2	−0.737168	−0.368584	−	0.929594i	$-0.620157\pi$
$577$	−10217.2	−0.737168	−0.368584	+	0.929594i	$0.620157\pi$
$578$	0	0
$579$	7082.82	0.508380
$580$	0	0
$581$	9110.85	0.650571
$582$	0	0
$583$	10310.8	0.732472
$584$	0	0
$585$	40.7312	0.00287868
$586$	0	0
$587$	−3558.28	−0.250197	−0.125099	−	0.992144i	$-0.539925\pi$
$587$	−3558.28	−0.250197	−0.125099	+	0.992144i	$0.539925\pi$
$588$	0	0
$589$	−4183.88	−0.292689
$590$	0	0
$591$	−1823.58	−0.126924
$592$	0	0
$593$	−11544.2	−0.799435	−0.399717	−	0.916638i	$-0.630892\pi$
$593$	−11544.2	−0.799435	−0.399717	+	0.916638i	$0.630892\pi$
$594$	0	0
$595$	2656.56	0.183039
$596$	0	0
$597$	3678.83	0.252202
$598$	0	0
$599$	16089.7	1.09751	0.548754	−	0.835984i	$-0.315102\pi$
$599$	16089.7	1.09751	0.548754	+	0.835984i	$0.315102\pi$
$600$	0	0
$601$	−6821.53	−0.462989	−0.231494	−	0.972836i	$-0.574361\pi$
$601$	−6821.53	−0.462989	−0.231494	+	0.972836i	$0.574361\pi$
$602$	0	0
$603$	−3404.50	−0.229921
$604$	0	0
$605$	8826.32	0.593125
$606$	0	0
$607$	−765.671	−0.0511987	−0.0255994	−	0.999672i	$-0.508149\pi$
$607$	−765.671	−0.0511987	−0.0255994	+	0.999672i	$0.508149\pi$
$608$	0	0
$609$	19129.0	1.27282
$610$	0	0
$611$	−113.038	−0.00748447
$612$	0	0
$613$	−21213.5	−1.39773	−0.698864	−	0.715255i	$-0.746311\pi$
$613$	−21213.5	−1.39773	−0.698864	+	0.715255i	$0.746311\pi$
$614$	0	0
$615$	−4002.92	−0.262461
$616$	0	0
$617$	13540.0	0.883470	0.441735	−	0.897146i	$-0.354363\pi$
$617$	13540.0	0.883470	0.441735	+	0.897146i	$0.354363\pi$
$618$	0	0
$619$	22495.5	1.46070	0.730349	−	0.683074i	$-0.239357\pi$
$619$	22495.5	1.46070	0.730349	+	0.683074i	$0.239357\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	26039.0	1.67453
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−9892.13	−0.630070
$628$	0	0
$629$	−1505.24	−0.0954177
$630$	0	0
$631$	24394.9	1.53906	0.769529	−	0.638612i	$-0.220491\pi$
$631$	24394.9	1.53906	0.769529	+	0.638612i	$0.220491\pi$
$632$	0	0
$633$	−1095.68	−0.0687985
$634$	0	0
$635$	−2347.70	−0.146717
$636$	0	0
$637$	654.322	0.0406989
$638$	0	0
$639$	8267.78	0.511844
$640$	0	0
$641$	22073.7	1.36015	0.680076	−	0.733142i	$-0.261947\pi$
$641$	22073.7	1.36015	0.680076	+	0.733142i	$0.261947\pi$
$642$	0	0
$643$	−2951.48	−0.181019	−0.0905093	−	0.995896i	$-0.528849\pi$
$643$	−2951.48	−0.181019	−0.0905093	+	0.995896i	$0.528849\pi$
$644$	0	0
$645$	961.244	0.0586806
$646$	0	0
$647$	−8077.79	−0.490836	−0.245418	−	0.969417i	$-0.578925\pi$
$647$	−8077.79	−0.490836	−0.245418	+	0.969417i	$0.578925\pi$
$648$	0	0
$649$	−49999.9	−3.02414
$650$	0	0
$651$	−6915.27	−0.416330
$652$	0	0
$653$	21888.8	1.31175	0.655877	−	0.754868i	$-0.272299\pi$
$653$	21888.8	1.31175	0.655877	+	0.754868i	$0.272299\pi$
$654$	0	0
$655$	7710.65	0.459970
$656$	0	0
$657$	−1278.27	−0.0759055
$658$	0	0
$659$	1961.56	0.115951	0.0579753	−	0.998318i	$-0.481536\pi$
$659$	1961.56	0.115951	0.0579753	+	0.998318i	$0.481536\pi$
$660$	0	0
$661$	−16468.2	−0.969042	−0.484521	−	0.874780i	$-0.661006\pi$
$661$	−16468.2	−0.969042	−0.484521	+	0.874780i	$0.661006\pi$
$662$	0	0
$663$	−44.1903	−0.00258855
$664$	0	0
$665$	9673.36	0.564086
$666$	0	0
$667$	4492.03	0.260768
$668$	0	0
$669$	−8521.45	−0.492464
$670$	0	0
$671$	4292.53	0.246962
$672$	0	0
$673$	27933.2	1.59992	0.799958	−	0.600055i	$-0.204855\pi$
$673$	27933.2	1.59992	0.799958	+	0.600055i	$0.204855\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−16233.0	−0.921544	−0.460772	−	0.887519i	$-0.652427\pi$
$677$	−16233.0	−0.921544	−0.460772	+	0.887519i	$0.652427\pi$
$678$	0	0
$679$	59495.5	3.36263
$680$	0	0
$681$	−16531.8	−0.930247
$682$	0	0
$683$	25174.2	1.41034	0.705171	−	0.709038i	$-0.250870\pi$
$683$	25174.2	1.41034	0.705171	+	0.709038i	$0.250870\pi$
$684$	0	0
$685$	−7997.68	−0.446096
$686$	0	0
$687$	−2206.06	−0.122513
$688$	0	0
$689$	167.722	0.00927386
$690$	0	0
$691$	117.621	0.00647541	0.00323770	−	0.999995i	$-0.498969\pi$
$691$	117.621	0.00647541	0.00323770	+	0.999995i	$0.498969\pi$
$692$	0	0
$693$	−16350.1	−0.896230
$694$	0	0
$695$	−10629.9	−0.580163
$696$	0	0
$697$	4342.87	0.236008
$698$	0	0
$699$	−17079.1	−0.924165
$700$	0	0
$701$	−28576.3	−1.53968	−0.769838	−	0.638240i	$-0.779663\pi$
$701$	−28576.3	−1.53968	−0.769838	+	0.638240i	$0.779663\pi$
$702$	0	0
$703$	−5481.04	−0.294056
$704$	0	0
$705$	−1873.27	−0.100073
$706$	0	0
$707$	36853.6	1.96043
$708$	0	0
$709$	26017.6	1.37815	0.689077	−	0.724688i	$-0.258016\pi$
$709$	26017.6	1.37815	0.689077	+	0.724688i	$0.258016\pi$
$710$	0	0
$711$	5161.03	0.272228
$712$	0	0
$713$	−1623.89	−0.0852950
$714$	0	0
$715$	251.828	0.0131718
$716$	0	0
$717$	2379.19	0.123922
$718$	0	0
$719$	28146.5	1.45993	0.729963	−	0.683486i	$-0.239537\pi$
$719$	28146.5	1.45993	0.729963	+	0.683486i	$0.239537\pi$
$720$	0	0
$721$	40247.4	2.07891
$722$	0	0
$723$	−19891.8	−1.02321
$724$	0	0
$725$	−4882.64	−0.250119
$726$	0	0
$727$	17451.9	0.890312	0.445156	−	0.895453i	$-0.353148\pi$
$727$	17451.9	0.890312	0.445156	+	0.895453i	$0.353148\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−1042.88	−0.0527664
$732$	0	0
$733$	−18021.3	−0.908091	−0.454045	−	0.890979i	$-0.650020\pi$
$733$	−18021.3	−0.908091	−0.454045	+	0.890979i	$0.650020\pi$
$734$	0	0
$735$	10843.5	0.544173
$736$	0	0
$737$	−21048.9	−1.05203
$738$	0	0
$739$	−10346.4	−0.515016	−0.257508	−	0.966276i	$-0.582901\pi$
$739$	−10346.4	−0.515016	−0.257508	+	0.966276i	$0.582901\pi$
$740$	0	0
$741$	−160.911	−0.00797734
$742$	0	0
$743$	31440.6	1.55241	0.776206	−	0.630480i	$-0.217142\pi$
$743$	31440.6	1.55241	0.776206	+	0.630480i	$0.217142\pi$
$744$	0	0
$745$	5358.65	0.263524
$746$	0	0
$747$	−2511.56	−0.123016
$748$	0	0
$749$	17881.8	0.872347
$750$	0	0
$751$	11560.7	0.561725	0.280862	−	0.959748i	$-0.409380\pi$
$751$	11560.7	0.561725	0.280862	+	0.959748i	$0.409380\pi$
$752$	0	0
$753$	2262.17	0.109479
$754$	0	0
$755$	16605.2	0.800429
$756$	0	0
$757$	−32400.0	−1.55561	−0.777806	−	0.628504i	$-0.783668\pi$
$757$	−32400.0	−1.55561	−0.777806	+	0.628504i	$0.783668\pi$
$758$	0	0
$759$	−3839.44	−0.183614
$760$	0	0
$761$	−7686.68	−0.366152	−0.183076	−	0.983099i	$-0.558605\pi$
$761$	−7686.68	−0.366152	−0.183076	+	0.983099i	$0.558605\pi$
$762$	0	0
$763$	46985.6	2.22935
$764$	0	0
$765$	−732.325	−0.0346108
$766$	0	0
$767$	−813.326	−0.0382888
$768$	0	0
$769$	−15751.3	−0.738628	−0.369314	−	0.929305i	$-0.620407\pi$
$769$	−15751.3	−0.738628	−0.369314	+	0.929305i	$0.620407\pi$
$770$	0	0
$771$	1349.03	0.0630143
$772$	0	0
$773$	11687.4	0.543811	0.271905	−	0.962324i	$-0.412346\pi$
$773$	11687.4	0.543811	0.271905	+	0.962324i	$0.412346\pi$
$774$	0	0
$775$	1765.10	0.0818121
$776$	0	0
$777$	−9059.26	−0.418274
$778$	0	0
$779$	15813.8	0.727326
$780$	0	0
$781$	51117.0	2.34201
$782$	0	0
$783$	−5273.25	−0.240678
$784$	0	0
$785$	7083.87	0.322082
$786$	0	0
$787$	−32966.6	−1.49318	−0.746590	−	0.665284i	$-0.768310\pi$
$787$	−32966.6	−1.49318	−0.746590	+	0.665284i	$0.768310\pi$
$788$	0	0
$789$	−5771.09	−0.260401
$790$	0	0
$791$	47234.5	2.12322
$792$	0	0
$793$	69.8247	0.00312680
$794$	0	0
$795$	2779.50	0.123998
$796$	0	0
$797$	31489.7	1.39953	0.699763	−	0.714375i	$-0.253289\pi$
$797$	31489.7	1.39953	0.699763	+	0.714375i	$0.253289\pi$
$798$	0	0
$799$	2032.35	0.0899869
$800$	0	0
$801$	−7178.10	−0.316636
$802$	0	0
$803$	−7903.10	−0.347316
$804$	0	0
$805$	3754.53	0.164385
$806$	0	0
$807$	−25709.8	−1.12147
$808$	0	0
$809$	−27731.1	−1.20516	−0.602579	−	0.798059i	$-0.705860\pi$
$809$	−27731.1	−1.20516	−0.602579	+	0.798059i	$0.705860\pi$
$810$	0	0
$811$	15626.4	0.676592	0.338296	−	0.941040i	$-0.390149\pi$
$811$	15626.4	0.676592	0.338296	+	0.941040i	$0.390149\pi$
$812$	0	0
$813$	−4041.61	−0.174349
$814$	0	0
$815$	−12811.3	−0.550624
$816$	0	0
$817$	−3797.45	−0.162614
$818$	0	0
$819$	−265.959	−0.0113472
$820$	0	0
$821$	−12576.0	−0.534598	−0.267299	−	0.963614i	$-0.586131\pi$
$821$	−12576.0	−0.534598	−0.267299	+	0.963614i	$0.586131\pi$
$822$	0	0
$823$	−25810.7	−1.09320	−0.546601	−	0.837393i	$-0.684078\pi$
$823$	−25810.7	−1.09320	−0.546601	+	0.837393i	$0.684078\pi$
$824$	0	0
$825$	4173.31	0.176116
$826$	0	0
$827$	17844.2	0.750307	0.375153	−	0.926963i	$-0.377590\pi$
$827$	17844.2	0.750307	0.375153	+	0.926963i	$0.377590\pi$
$828$	0	0
$829$	−30724.9	−1.28724	−0.643619	−	0.765346i	$-0.722568\pi$
$829$	−30724.9	−1.28724	−0.643619	+	0.765346i	$0.722568\pi$
$830$	0	0
$831$	12402.8	0.517747
$832$	0	0
$833$	−11764.3	−0.489328
$834$	0	0
$835$	−7177.34	−0.297463
$836$	0	0
$837$	1906.31	0.0787237
$838$	0	0
$839$	1226.76	0.0504798	0.0252399	−	0.999681i	$-0.491965\pi$
$839$	1226.76	0.0504798	0.0252399	+	0.999681i	$0.491965\pi$
$840$	0	0
$841$	13755.2	0.563993
$842$	0	0
$843$	16433.1	0.671396
$844$	0	0
$845$	−10980.9	−0.447047
$846$	0	0
$847$	−57632.5	−2.33799
$848$	0	0
$849$	−16399.7	−0.662940
$850$	0	0
$851$	−2127.36	−0.0856934
$852$	0	0
$853$	−30880.2	−1.23953	−0.619764	−	0.784789i	$-0.712772\pi$
$853$	−30880.2	−1.23953	−0.619764	+	0.784789i	$0.712772\pi$
$854$	0	0
$855$	−2666.63	−0.106663
$856$	0	0
$857$	19958.1	0.795513	0.397756	−	0.917491i	$-0.369789\pi$
$857$	19958.1	0.795513	0.397756	+	0.917491i	$0.369789\pi$
$858$	0	0
$859$	42844.6	1.70179	0.850896	−	0.525334i	$-0.176060\pi$
$859$	42844.6	1.70179	0.850896	+	0.525334i	$0.176060\pi$
$860$	0	0
$861$	26137.5	1.03457
$862$	0	0
$863$	32729.7	1.29100	0.645500	−	0.763760i	$-0.276649\pi$
$863$	32729.7	1.29100	0.645500	+	0.763760i	$0.276649\pi$
$864$	0	0
$865$	4841.94	0.190325
$866$	0	0
$867$	−13944.5	−0.546228
$868$	0	0
$869$	31909.0	1.24561
$870$	0	0
$871$	−342.394	−0.0133198
$872$	0	0
$873$	−16401.0	−0.635840
$874$	0	0
$875$	−4081.01	−0.157672
$876$	0	0
$877$	−31265.4	−1.20383	−0.601914	−	0.798561i	$-0.705595\pi$
$877$	−31265.4	−1.20383	−0.601914	+	0.798561i	$0.705595\pi$
$878$	0	0
$879$	22861.5	0.877248
$880$	0	0
$881$	20354.0	0.778371	0.389185	−	0.921159i	$-0.372757\pi$
$881$	20354.0	0.778371	0.389185	+	0.921159i	$0.372757\pi$
$882$	0	0
$883$	−13779.2	−0.525149	−0.262574	−	0.964912i	$-0.584572\pi$
$883$	−13779.2	−0.525149	−0.262574	+	0.964912i	$0.584572\pi$
$884$	0	0
$885$	−13478.5	−0.511949
$886$	0	0
$887$	−34063.8	−1.28946	−0.644729	−	0.764411i	$-0.723030\pi$
$887$	−34063.8	−1.28946	−0.644729	+	0.764411i	$0.723030\pi$
$888$	0	0
$889$	15329.6	0.578332
$890$	0	0
$891$	4507.17	0.169468
$892$	0	0
$893$	7400.45	0.277320
$894$	0	0
$895$	7746.14	0.289302
$896$	0	0
$897$	−62.4545	−0.00232474
$898$	0	0
$899$	−13789.4	−0.511570
$900$	0	0
$901$	−3015.55	−0.111501
$902$	0	0
$903$	−6276.56	−0.231308
$904$	0	0
$905$	554.201	0.0203561
$906$	0	0
$907$	33592.2	1.22978	0.614890	−	0.788613i	$-0.289200\pi$
$907$	33592.2	1.22978	0.614890	+	0.788613i	$0.289200\pi$
$908$	0	0
$909$	−10159.3	−0.370697
$910$	0	0
$911$	−21759.3	−0.791347	−0.395673	−	0.918391i	$-0.629489\pi$
$911$	−21759.3	−0.791347	−0.395673	+	0.918391i	$0.629489\pi$
$912$	0	0
$913$	−15528.2	−0.562878
$914$	0	0
$915$	1157.14	0.0418075
$916$	0	0
$917$	−50347.6	−1.81311
$918$	0	0
$919$	5286.15	0.189743	0.0948717	−	0.995490i	$-0.469756\pi$
$919$	5286.15	0.189743	0.0948717	+	0.995490i	$0.469756\pi$
$920$	0	0
$921$	−1461.22	−0.0522790
$922$	0	0
$923$	831.498	0.0296523
$924$	0	0
$925$	2312.35	0.0821942
$926$	0	0
$927$	−11094.9	−0.393100
$928$	0	0
$929$	3169.17	0.111924	0.0559619	−	0.998433i	$-0.482177\pi$
$929$	3169.17	0.111924	0.0559619	+	0.998433i	$0.482177\pi$
$930$	0	0
$931$	−42837.7	−1.50800
$932$	0	0
$933$	6752.73	0.236950
$934$	0	0
$935$	−4527.73	−0.158366
$936$	0	0
$937$	−34689.5	−1.20945	−0.604726	−	0.796433i	$-0.706717\pi$
$937$	−34689.5	−1.20945	−0.604726	+	0.796433i	$0.706717\pi$
$938$	0	0
$939$	−9222.85	−0.320529
$940$	0	0
$941$	7586.07	0.262804	0.131402	−	0.991329i	$-0.458052\pi$
$941$	7586.07	0.262804	0.131402	+	0.991329i	$0.458052\pi$
$942$	0	0
$943$	6137.81	0.211956
$944$	0	0
$945$	−4407.49	−0.151720
$946$	0	0
$947$	−12471.2	−0.427941	−0.213970	−	0.976840i	$-0.568640\pi$
$947$	−12471.2	−0.427941	−0.213970	+	0.976840i	$0.568640\pi$
$948$	0	0
$949$	−128.556	−0.00439738
$950$	0	0
$951$	−23026.6	−0.785162
$952$	0	0
$953$	−50645.2	−1.72147	−0.860735	−	0.509054i	$-0.829995\pi$
$953$	−50645.2	−1.72147	−0.860735	+	0.509054i	$0.829995\pi$
$954$	0	0
$955$	22177.8	0.751474
$956$	0	0
$957$	−32602.8	−1.10125
$958$	0	0
$959$	52221.8	1.75842
$960$	0	0
$961$	−24806.1	−0.832670
$962$	0	0
$963$	−4929.43	−0.164952
$964$	0	0
$965$	11804.7	0.393789
$966$	0	0
$967$	−7862.31	−0.261463	−0.130732	−	0.991418i	$-0.541733\pi$
$967$	−7862.31	−0.261463	−0.130732	+	0.991418i	$0.541733\pi$
$968$	0	0
$969$	2893.09	0.0959127
$970$	0	0
$971$	−3655.76	−0.120823	−0.0604114	−	0.998174i	$-0.519241\pi$
$971$	−3655.76	−0.120823	−0.0604114	+	0.998174i	$0.519241\pi$
$972$	0	0
$973$	69408.9	2.28689
$974$	0	0
$975$	67.8853	0.00222981
$976$	0	0
$977$	49750.4	1.62913	0.814563	−	0.580075i	$-0.196977\pi$
$977$	49750.4	1.62913	0.814563	+	0.580075i	$0.196977\pi$
$978$	0	0
$979$	−44379.9	−1.44881
$980$	0	0
$981$	−12952.4	−0.421547
$982$	0	0
$983$	11252.8	0.365114	0.182557	−	0.983195i	$-0.441563\pi$
$983$	11252.8	0.365114	0.182557	+	0.983195i	$0.441563\pi$
$984$	0	0
$985$	−3039.31	−0.0983151
$986$	0	0
$987$	12231.7	0.394468
$988$	0	0
$989$	−1473.91	−0.0473888
$990$	0	0
$991$	11165.6	0.357907	0.178954	−	0.983858i	$-0.442729\pi$
$991$	11165.6	0.357907	0.178954	+	0.983858i	$0.442729\pi$
$992$	0	0
$993$	19561.7	0.625147
$994$	0	0
$995$	6131.38	0.195355
$996$	0	0
$997$	−22374.8	−0.710749	−0.355375	−	0.934724i	$-0.615647\pi$
$997$	−22374.8	−0.710749	−0.355375	+	0.934724i	$0.615647\pi$
$998$	0	0
$999$	2497.34	0.0790914

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.h.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.h.1.1	✓	5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} -32.6481 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} -32.6481 q^{7} +9.00000 q^{9} +55.6441 q^{11} +0.905137 q^{13} +15.0000 q^{15} -16.2739 q^{17} -59.2584 q^{19} -97.9443 q^{21} -23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -195.305 q^{29} +70.6041 q^{31} +166.932 q^{33} -163.240 q^{35} +92.4940 q^{37} +2.71541 q^{39} -266.861 q^{41} +64.0830 q^{43} +45.0000 q^{45} -124.884 q^{47} +722.898 q^{49} -48.8216 q^{51} +185.300 q^{53} +278.220 q^{55} -177.775 q^{57} -898.566 q^{59} +77.1427 q^{61} -293.833 q^{63} +4.52569 q^{65} -378.278 q^{67} -69.0000 q^{69} +918.643 q^{71} -142.030 q^{73} +75.0000 q^{75} -1816.67 q^{77} +573.448 q^{79} +81.0000 q^{81} -279.062 q^{83} -81.3694 q^{85} -585.916 q^{87} -797.567 q^{89} -29.5510 q^{91} +211.812 q^{93} -296.292 q^{95} -1822.33 q^{97} +500.797 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} + 25 q^{5} - 23 q^{7} + 45 q^{9}+O(q^{10}) \) 5 * q + 15 * q^3 + 25 * q^5 - 23 * q^7 + 45 * q^9	\( 5 q + 15 q^{3} + 25 q^{5} - 23 q^{7} + 45 q^{9} - 42 q^{11} - 80 q^{13} + 75 q^{15} - 81 q^{17} - 128 q^{19} - 69 q^{21} - 115 q^{23} + 125 q^{25} + 135 q^{27} - 261 q^{29} - 233 q^{31} - 126 q^{33} - 115 q^{35} - 371 q^{37} - 240 q^{39} - 819 q^{41} - 596 q^{43} + 225 q^{45} - 186 q^{47} - 132 q^{49} - 243 q^{51} - 831 q^{53} - 210 q^{55} - 384 q^{57} - 1275 q^{59} - 152 q^{61} - 207 q^{63} - 400 q^{65} - 485 q^{67} - 345 q^{69} + 531 q^{71} - 788 q^{73} + 375 q^{75} - 1896 q^{77} - 134 q^{79} + 405 q^{81} - 585 q^{83} - 405 q^{85} - 783 q^{87} - 846 q^{89} + 500 q^{91} - 699 q^{93} - 640 q^{95} - 2078 q^{97} - 378 q^{99}+O(q^{100}) \) 5 * q + 15 * q^3 + 25 * q^5 - 23 * q^7 + 45 * q^9 - 42 * q^11 - 80 * q^13 + 75 * q^15 - 81 * q^17 - 128 * q^19 - 69 * q^21 - 115 * q^23 + 125 * q^25 + 135 * q^27 - 261 * q^29 - 233 * q^31 - 126 * q^33 - 115 * q^35 - 371 * q^37 - 240 * q^39 - 819 * q^41 - 596 * q^43 + 225 * q^45 - 186 * q^47 - 132 * q^49 - 243 * q^51 - 831 * q^53 - 210 * q^55 - 384 * q^57 - 1275 * q^59 - 152 * q^61 - 207 * q^63 - 400 * q^65 - 485 * q^67 - 345 * q^69 + 531 * q^71 - 788 * q^73 + 375 * q^75 - 1896 * q^77 - 134 * q^79 + 405 * q^81 - 585 * q^83 - 405 * q^85 - 783 * q^87 - 846 * q^89 + 500 * q^91 - 699 * q^93 - 640 * q^95 - 2078 * q^97 - 378 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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