LMFDB - Embedded newform 1380.4.a.h.1.3

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			1.3
Root			$2.09705$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} -2.81836 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} -2.81836 q^{7} +9.00000 q^{9} -4.15252 q^{11} -45.5039 q^{13} +15.0000 q^{15} +9.65930 q^{17} -13.2618 q^{19} -8.45507 q^{21} -23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -44.1841 q^{29} -120.646 q^{31} -12.4576 q^{33} -14.0918 q^{35} +282.248 q^{37} -136.512 q^{39} -463.208 q^{41} -0.571016 q^{43} +45.0000 q^{45} -272.743 q^{47} -335.057 q^{49} +28.9779 q^{51} -489.984 q^{53} -20.7626 q^{55} -39.7853 q^{57} +858.426 q^{59} +275.908 q^{61} -25.3652 q^{63} -227.519 q^{65} -799.495 q^{67} -69.0000 q^{69} -542.094 q^{71} +488.444 q^{73} +75.0000 q^{75} +11.7033 q^{77} -3.73805 q^{79} +81.0000 q^{81} -911.825 q^{83} +48.2965 q^{85} -132.552 q^{87} -545.482 q^{89} +128.246 q^{91} -361.938 q^{93} -66.3088 q^{95} -924.669 q^{97} -37.3727 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−2.81836	−0.152177	−0.0760885	−	0.997101i	$-0.524243\pi$
$7$	−2.81836	−0.152177	−0.0760885	+	0.997101i	$0.524243\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−4.15252	−0.113821	−0.0569106	−	0.998379i	$-0.518125\pi$
$11$	−4.15252	−0.113821	−0.0569106	+	0.998379i	$0.518125\pi$
$12$	0	0
$13$	−45.5039	−0.970808	−0.485404	−	0.874290i	$-0.661328\pi$
$13$	−45.5039	−0.970808	−0.485404	+	0.874290i	$0.661328\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	9.65930	0.137807	0.0689036	−	0.997623i	$-0.478050\pi$
$17$	9.65930	0.137807	0.0689036	+	0.997623i	$0.478050\pi$
$18$	0	0
$19$	−13.2618	−0.160129	−0.0800647	−	0.996790i	$-0.525513\pi$
$19$	−13.2618	−0.160129	−0.0800647	+	0.996790i	$0.525513\pi$
$20$	0	0
$21$	−8.45507	−0.0878594
$22$	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
$29$	−44.1841	−0.282923	−0.141462	−	0.989944i	$-0.545180\pi$
$29$	−44.1841	−0.282923	−0.141462	+	0.989944i	$0.545180\pi$
$30$	0	0
$31$	−120.646	−0.698988	−0.349494	−	0.936939i	$-0.613647\pi$
$31$	−120.646	−0.698988	−0.349494	+	0.936939i	$0.613647\pi$
$32$	0	0
$33$	−12.4576	−0.0657147
$34$	0	0
$35$	−14.0918	−0.0680556
$36$	0	0
$37$	282.248	1.25409	0.627045	−	0.778983i	$-0.284264\pi$
$37$	282.248	1.25409	0.627045	+	0.778983i	$0.284264\pi$
$38$	0	0
$39$	−136.512	−0.560496
$40$	0	0
$41$	−463.208	−1.76441	−0.882207	−	0.470862i	$-0.843943\pi$
$41$	−463.208	−1.76441	−0.882207	+	0.470862i	$0.843943\pi$
$42$	0	0
$43$	−0.571016	−0.00202509	−0.00101255	−	0.999999i	$-0.500322\pi$
$43$	−0.571016	−0.00202509	−0.00101255	+	0.999999i	$0.500322\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−272.743	−0.846461	−0.423230	−	0.906022i	$-0.639104\pi$
$47$	−272.743	−0.846461	−0.423230	+	0.906022i	$0.639104\pi$
$48$	0	0
$49$	−335.057	−0.976842
$50$	0	0
$51$	28.9779	0.0795631
$52$	0	0
$53$	−489.984	−1.26990	−0.634948	−	0.772555i	$-0.718978\pi$
$53$	−489.984	−1.26990	−0.634948	+	0.772555i	$0.718978\pi$
$54$	0	0
$55$	−20.7626	−0.0509024
$56$	0	0
$57$	−39.7853	−0.0924507
$58$	0	0
$59$	858.426	1.89420	0.947098	−	0.320945i	$-0.104001\pi$
$59$	858.426	1.89420	0.947098	+	0.320945i	$0.104001\pi$
$60$	0	0
$61$	275.908	0.579122	0.289561	−	0.957160i	$-0.406491\pi$
$61$	275.908	0.579122	0.289561	+	0.957160i	$0.406491\pi$
$62$	0	0
$63$	−25.3652	−0.0507257
$64$	0	0
$65$	−227.519	−0.434159
$66$	0	0
$67$	−799.495	−1.45782	−0.728909	−	0.684611i	$-0.759972\pi$
$67$	−799.495	−1.45782	−0.728909	+	0.684611i	$0.759972\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	−542.094	−0.906123	−0.453062	−	0.891479i	$-0.649668\pi$
$71$	−542.094	−0.906123	−0.453062	+	0.891479i	$0.649668\pi$
$72$	0	0
$73$	488.444	0.783123	0.391562	−	0.920152i	$-0.371935\pi$
$73$	488.444	0.783123	0.391562	+	0.920152i	$0.371935\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	11.7033	0.0173210
$78$	0	0
$79$	−3.73805	−0.00532359	−0.00266180	−	0.999996i	$-0.500847\pi$
$79$	−3.73805	−0.00532359	−0.00266180	+	0.999996i	$0.500847\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−911.825	−1.20585	−0.602927	−	0.797797i	$-0.705999\pi$
$83$	−911.825	−1.20585	−0.602927	+	0.797797i	$0.705999\pi$
$84$	0	0
$85$	48.2965	0.0616293
$86$	0	0
$87$	−132.552	−0.163346
$88$	0	0
$89$	−545.482	−0.649674	−0.324837	−	0.945770i	$-0.605309\pi$
$89$	−545.482	−0.649674	−0.324837	+	0.945770i	$0.605309\pi$
$90$	0	0
$91$	128.246	0.147735
$92$	0	0
$93$	−361.938	−0.403561
$94$	0	0
$95$	−66.3088	−0.0716120
$96$	0	0
$97$	−924.669	−0.967896	−0.483948	−	0.875097i	$-0.660798\pi$
$97$	−924.669	−0.967896	−0.483948	+	0.875097i	$0.660798\pi$
$98$	0	0
$99$	−37.3727	−0.0379404
$100$	0	0
$101$	1947.47	1.91862	0.959309	−	0.282359i	$-0.0911169\pi$
$101$	1947.47	1.91862	0.959309	+	0.282359i	$0.0911169\pi$
$102$	0	0
$103$	−935.701	−0.895120	−0.447560	−	0.894254i	$-0.647707\pi$
$103$	−935.701	−0.895120	−0.447560	+	0.894254i	$0.647707\pi$
$104$	0	0
$105$	−42.2754	−0.0392919
$106$	0	0
$107$	532.345	0.480969	0.240485	−	0.970653i	$-0.422694\pi$
$107$	532.345	0.480969	0.240485	+	0.970653i	$0.422694\pi$
$108$	0	0
$109$	−377.521	−0.331743	−0.165871	−	0.986147i	$-0.553044\pi$
$109$	−377.521	−0.331743	−0.165871	+	0.986147i	$0.553044\pi$
$110$	0	0
$111$	846.744	0.724049
$112$	0	0
$113$	561.136	0.467144	0.233572	−	0.972340i	$-0.424959\pi$
$113$	561.136	0.467144	0.233572	+	0.972340i	$0.424959\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−409.535	−0.323603
$118$	0	0
$119$	−27.2234	−0.0209711
$120$	0	0
$121$	−1313.76	−0.987045
$122$	0	0
$123$	−1389.62	−1.01868
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1405.43	0.981982	0.490991	−	0.871165i	$-0.336635\pi$
$127$	1405.43	0.981982	0.490991	+	0.871165i	$0.336635\pi$
$128$	0	0
$129$	−1.71305	−0.00116919
$130$	0	0
$131$	−1744.99	−1.16382	−0.581909	−	0.813254i	$-0.697694\pi$
$131$	−1744.99	−1.16382	−0.581909	+	0.813254i	$0.697694\pi$
$132$	0	0
$133$	37.3764	0.0243680
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−1372.41	−0.855863	−0.427931	−	0.903811i	$-0.640757\pi$
$137$	−1372.41	−0.855863	−0.427931	+	0.903811i	$0.640757\pi$
$138$	0	0
$139$	2508.32	1.53060	0.765298	−	0.643676i	$-0.222592\pi$
$139$	2508.32	1.53060	0.765298	+	0.643676i	$0.222592\pi$
$140$	0	0
$141$	−818.229	−0.488704
$142$	0	0
$143$	188.956	0.110499
$144$	0	0
$145$	−220.920	−0.126527
$146$	0	0
$147$	−1005.17	−0.563980
$148$	0	0
$149$	−2629.93	−1.44599	−0.722993	−	0.690855i	$-0.757234\pi$
$149$	−2629.93	−1.44599	−0.722993	+	0.690855i	$0.757234\pi$
$150$	0	0
$151$	−1160.09	−0.625211	−0.312605	−	0.949883i	$-0.601202\pi$
$151$	−1160.09	−0.625211	−0.312605	+	0.949883i	$0.601202\pi$
$152$	0	0
$153$	86.9337	0.0459358
$154$	0	0
$155$	−603.229	−0.312597
$156$	0	0
$157$	−3375.14	−1.71571	−0.857853	−	0.513896i	$-0.828202\pi$
$157$	−3375.14	−1.71571	−0.857853	+	0.513896i	$0.828202\pi$
$158$	0	0
$159$	−1469.95	−0.733174
$160$	0	0
$161$	64.8222	0.0317311
$162$	0	0
$163$	2807.76	1.34921	0.674604	−	0.738179i	$-0.264314\pi$
$163$	2807.76	1.34921	0.674604	+	0.738179i	$0.264314\pi$
$164$	0	0
$165$	−62.2878	−0.0293885
$166$	0	0
$167$	2581.08	1.19599	0.597994	−	0.801501i	$-0.295965\pi$
$167$	2581.08	1.19599	0.597994	+	0.801501i	$0.295965\pi$
$168$	0	0
$169$	−126.396	−0.0575313
$170$	0	0
$171$	−119.356	−0.0533764
$172$	0	0
$173$	492.259	0.216334	0.108167	−	0.994133i	$-0.465502\pi$
$173$	492.259	0.216334	0.108167	+	0.994133i	$0.465502\pi$
$174$	0	0
$175$	−70.4589	−0.0304354
$176$	0	0
$177$	2575.28	1.09361
$178$	0	0
$179$	−3170.11	−1.32371	−0.661857	−	0.749630i	$-0.730232\pi$
$179$	−3170.11	−1.32371	−0.661857	+	0.749630i	$0.730232\pi$
$180$	0	0
$181$	143.676	0.0590018	0.0295009	−	0.999565i	$-0.490608\pi$
$181$	143.676	0.0590018	0.0295009	+	0.999565i	$0.490608\pi$
$182$	0	0
$183$	827.725	0.334356
$184$	0	0
$185$	1411.24	0.560846
$186$	0	0
$187$	−40.1105	−0.0156854
$188$	0	0
$189$	−76.0957	−0.0292865
$190$	0	0
$191$	3664.73	1.38833	0.694164	−	0.719817i	$-0.255774\pi$
$191$	3664.73	1.38833	0.694164	+	0.719817i	$0.255774\pi$
$192$	0	0
$193$	−3646.00	−1.35982	−0.679910	−	0.733296i	$-0.737981\pi$
$193$	−3646.00	−1.35982	−0.679910	+	0.733296i	$0.737981\pi$
$194$	0	0
$195$	−682.558	−0.250662
$196$	0	0
$197$	1062.37	0.384216	0.192108	−	0.981374i	$-0.438468\pi$
$197$	1062.37	0.384216	0.192108	+	0.981374i	$0.438468\pi$
$198$	0	0
$199$	−4804.43	−1.71144	−0.855722	−	0.517436i	$-0.826887\pi$
$199$	−4804.43	−1.71144	−0.855722	+	0.517436i	$0.826887\pi$
$200$	0	0
$201$	−2398.48	−0.841672
$202$	0	0
$203$	124.527	0.0430544
$204$	0	0
$205$	−2316.04	−0.789070
$206$	0	0
$207$	−207.000	−0.0695048
$208$	0	0
$209$	55.0698	0.0182261
$210$	0	0
$211$	−948.300	−0.309401	−0.154701	−	0.987961i	$-0.549441\pi$
$211$	−948.300	−0.309401	−0.154701	+	0.987961i	$0.549441\pi$
$212$	0	0
$213$	−1626.28	−0.523150
$214$	0	0
$215$	−2.85508	−0.000905650 0
$216$	0	0
$217$	340.023	0.106370
$218$	0	0
$219$	1465.33	0.452136
$220$	0	0
$221$	−439.536	−0.133784
$222$	0	0
$223$	−356.402	−0.107025	−0.0535123	−	0.998567i	$-0.517042\pi$
$223$	−356.402	−0.107025	−0.0535123	+	0.998567i	$0.517042\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−4609.76	−1.34784	−0.673922	−	0.738803i	$-0.735392\pi$
$227$	−4609.76	−1.34784	−0.673922	+	0.738803i	$0.735392\pi$
$228$	0	0
$229$	3634.97	1.04893	0.524466	−	0.851431i	$-0.324265\pi$
$229$	3634.97	1.04893	0.524466	+	0.851431i	$0.324265\pi$
$230$	0	0
$231$	35.1099	0.0100003
$232$	0	0
$233$	5598.28	1.57406	0.787029	−	0.616916i	$-0.211618\pi$
$233$	5598.28	1.57406	0.787029	+	0.616916i	$0.211618\pi$
$234$	0	0
$235$	−1363.72	−0.378549
$236$	0	0
$237$	−11.2142	−0.00307358
$238$	0	0
$239$	−2106.42	−0.570095	−0.285047	−	0.958513i	$-0.592009\pi$
$239$	−2106.42	−0.570095	−0.285047	+	0.958513i	$0.592009\pi$
$240$	0	0
$241$	6056.69	1.61886	0.809431	−	0.587215i	$-0.199776\pi$
$241$	6056.69	1.61886	0.809431	+	0.587215i	$0.199776\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1675.28	−0.436857
$246$	0	0
$247$	603.462	0.155455
$248$	0	0
$249$	−2735.48	−0.696200
$250$	0	0
$251$	−1704.53	−0.428642	−0.214321	−	0.976763i	$-0.568754\pi$
$251$	−1704.53	−0.428642	−0.214321	+	0.976763i	$0.568754\pi$
$252$	0	0
$253$	95.5080	0.0237334
$254$	0	0
$255$	144.889	0.0355817
$256$	0	0
$257$	−6973.11	−1.69249	−0.846246	−	0.532792i	$-0.821143\pi$
$257$	−6973.11	−1.69249	−0.846246	+	0.532792i	$0.821143\pi$
$258$	0	0
$259$	−795.476	−0.190844
$260$	0	0
$261$	−397.657	−0.0943077
$262$	0	0
$263$	5074.80	1.18983	0.594916	−	0.803788i	$-0.297185\pi$
$263$	5074.80	1.18983	0.594916	+	0.803788i	$0.297185\pi$
$264$	0	0
$265$	−2449.92	−0.567914
$266$	0	0
$267$	−1636.45	−0.375090
$268$	0	0
$269$	6654.25	1.50824	0.754120	−	0.656736i	$-0.228063\pi$
$269$	6654.25	1.50824	0.754120	+	0.656736i	$0.228063\pi$
$270$	0	0
$271$	5392.17	1.20868	0.604338	−	0.796728i	$-0.293438\pi$
$271$	5392.17	1.20868	0.604338	+	0.796728i	$0.293438\pi$
$272$	0	0
$273$	384.739	0.0852947
$274$	0	0
$275$	−103.813	−0.0227642
$276$	0	0
$277$	−7849.22	−1.70258	−0.851288	−	0.524698i	$-0.824178\pi$
$277$	−7849.22	−1.70258	−0.851288	+	0.524698i	$0.824178\pi$
$278$	0	0
$279$	−1085.81	−0.232996
$280$	0	0
$281$	−1220.06	−0.259014	−0.129507	−	0.991579i	$-0.541339\pi$
$281$	−1220.06	−0.259014	−0.129507	+	0.991579i	$0.541339\pi$
$282$	0	0
$283$	−6649.39	−1.39670	−0.698349	−	0.715758i	$-0.746082\pi$
$283$	−6649.39	−1.39670	−0.698349	+	0.715758i	$0.746082\pi$
$284$	0	0
$285$	−198.926	−0.0413452
$286$	0	0
$287$	1305.49	0.268503
$288$	0	0
$289$	−4819.70	−0.981009
$290$	0	0
$291$	−2774.01	−0.558815
$292$	0	0
$293$	−7321.50	−1.45982	−0.729908	−	0.683545i	$-0.760437\pi$
$293$	−7321.50	−1.45982	−0.729908	+	0.683545i	$0.760437\pi$
$294$	0	0
$295$	4292.13	0.847110
$296$	0	0
$297$	−112.118	−0.0219049
$298$	0	0
$299$	1046.59	0.202428
$300$	0	0
$301$	1.60933	0.000308173 0
$302$	0	0
$303$	5842.41	1.10771
$304$	0	0
$305$	1379.54	0.258991
$306$	0	0
$307$	−4130.67	−0.767915	−0.383957	−	0.923351i	$-0.625439\pi$
$307$	−4130.67	−0.767915	−0.383957	+	0.923351i	$0.625439\pi$
$308$	0	0
$309$	−2807.10	−0.516798
$310$	0	0
$311$	−6812.52	−1.24213	−0.621065	−	0.783759i	$-0.713300\pi$
$311$	−6812.52	−1.24213	−0.621065	+	0.783759i	$0.713300\pi$
$312$	0	0
$313$	4828.57	0.871971	0.435986	−	0.899954i	$-0.356400\pi$
$313$	4828.57	0.871971	0.435986	+	0.899954i	$0.356400\pi$
$314$	0	0
$315$	−126.826	−0.0226852
$316$	0	0
$317$	3583.17	0.634862	0.317431	−	0.948281i	$-0.397180\pi$
$317$	3583.17	0.634862	0.317431	+	0.948281i	$0.397180\pi$
$318$	0	0
$319$	183.475	0.0322027
$320$	0	0
$321$	1597.04	0.277688
$322$	0	0
$323$	−128.099	−0.0220670
$324$	0	0
$325$	−1137.60	−0.194162
$326$	0	0
$327$	−1132.56	−0.191532
$328$	0	0
$329$	768.687	0.128812
$330$	0	0
$331$	−8137.48	−1.35129	−0.675644	−	0.737228i	$-0.736135\pi$
$331$	−8137.48	−1.35129	−0.675644	+	0.737228i	$0.736135\pi$
$332$	0	0
$333$	2540.23	0.418030
$334$	0	0
$335$	−3997.47	−0.651956
$336$	0	0
$337$	−1399.59	−0.226233	−0.113116	−	0.993582i	$-0.536083\pi$
$337$	−1399.59	−0.226233	−0.113116	+	0.993582i	$0.536083\pi$
$338$	0	0
$339$	1683.41	0.269706
$340$	0	0
$341$	500.985	0.0795597
$342$	0	0
$343$	1911.01	0.300830
$344$	0	0
$345$	−345.000	−0.0538382
$346$	0	0
$347$	9122.94	1.41137	0.705685	−	0.708526i	$-0.250640\pi$
$347$	9122.94	1.41137	0.705685	+	0.708526i	$0.250640\pi$
$348$	0	0
$349$	−10894.3	−1.67094	−0.835472	−	0.549533i	$-0.814806\pi$
$349$	−10894.3	−1.67094	−0.835472	+	0.549533i	$0.814806\pi$
$350$	0	0
$351$	−1228.60	−0.186832
$352$	0	0
$353$	−2653.50	−0.400089	−0.200045	−	0.979787i	$-0.564109\pi$
$353$	−2653.50	−0.400089	−0.200045	+	0.979787i	$0.564109\pi$
$354$	0	0
$355$	−2710.47	−0.405231
$356$	0	0
$357$	−81.6701	−0.0121077
$358$	0	0
$359$	−9594.50	−1.41052	−0.705262	−	0.708946i	$-0.749171\pi$
$359$	−9594.50	−1.41052	−0.705262	+	0.708946i	$0.749171\pi$
$360$	0	0
$361$	−6683.13	−0.974359
$362$	0	0
$363$	−3941.27	−0.569871
$364$	0	0
$365$	2442.22	0.350223
$366$	0	0
$367$	−6702.20	−0.953275	−0.476638	−	0.879100i	$-0.658145\pi$
$367$	−6702.20	−0.953275	−0.476638	+	0.879100i	$0.658145\pi$
$368$	0	0
$369$	−4168.87	−0.588138
$370$	0	0
$371$	1380.95	0.193249
$372$	0	0
$373$	9216.73	1.27942	0.639711	−	0.768616i	$-0.279054\pi$
$373$	9216.73	1.27942	0.639711	+	0.768616i	$0.279054\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	2010.55	0.274664
$378$	0	0
$379$	1890.65	0.256244	0.128122	−	0.991758i	$-0.459105\pi$
$379$	1890.65	0.256244	0.128122	+	0.991758i	$0.459105\pi$
$380$	0	0
$381$	4216.29	0.566948
$382$	0	0
$383$	13094.7	1.74702	0.873508	−	0.486810i	$-0.161840\pi$
$383$	13094.7	1.74702	0.873508	+	0.486810i	$0.161840\pi$
$384$	0	0
$385$	58.5165	0.00774617
$386$	0	0
$387$	−5.13914	−0.000675032 0
$388$	0	0
$389$	13421.1	1.74930	0.874651	−	0.484754i	$-0.161091\pi$
$389$	13421.1	1.74930	0.874651	+	0.484754i	$0.161091\pi$
$390$	0	0
$391$	−222.164	−0.0287348
$392$	0	0
$393$	−5234.96	−0.671931
$394$	0	0
$395$	−18.6903	−0.00238078
$396$	0	0
$397$	−1871.72	−0.236622	−0.118311	−	0.992977i	$-0.537748\pi$
$397$	−1871.72	−0.236622	−0.118311	+	0.992977i	$0.537748\pi$
$398$	0	0
$399$	112.129	0.0140689
$400$	0	0
$401$	14183.2	1.76628	0.883138	−	0.469114i	$-0.155427\pi$
$401$	14183.2	1.76628	0.883138	+	0.469114i	$0.155427\pi$
$402$	0	0
$403$	5489.86	0.678584
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−1172.04	−0.142742
$408$	0	0
$409$	1116.13	0.134937	0.0674685	−	0.997721i	$-0.478508\pi$
$409$	1116.13	0.134937	0.0674685	+	0.997721i	$0.478508\pi$
$410$	0	0
$411$	−4117.24	−0.494132
$412$	0	0
$413$	−2419.35	−0.288253
$414$	0	0
$415$	−4559.13	−0.539274
$416$	0	0
$417$	7524.96	0.883690
$418$	0	0
$419$	14731.7	1.71764	0.858820	−	0.512278i	$-0.171198\pi$
$419$	14731.7	1.71764	0.858820	+	0.512278i	$0.171198\pi$
$420$	0	0
$421$	9404.95	1.08876	0.544382	−	0.838838i	$-0.316764\pi$
$421$	9404.95	1.08876	0.544382	+	0.838838i	$0.316764\pi$
$422$	0	0
$423$	−2454.69	−0.282154
$424$	0	0
$425$	241.482	0.0275615
$426$	0	0
$427$	−777.609	−0.0881291
$428$	0	0
$429$	566.868	0.0637964
$430$	0	0
$431$	3936.43	0.439933	0.219967	−	0.975507i	$-0.429405\pi$
$431$	3936.43	0.439933	0.219967	+	0.975507i	$0.429405\pi$
$432$	0	0
$433$	2574.34	0.285716	0.142858	−	0.989743i	$-0.454371\pi$
$433$	2574.34	0.285716	0.142858	+	0.989743i	$0.454371\pi$
$434$	0	0
$435$	−662.761	−0.0730505
$436$	0	0
$437$	305.021	0.0333893
$438$	0	0
$439$	4728.76	0.514104	0.257052	−	0.966398i	$-0.417249\pi$
$439$	4728.76	0.514104	0.257052	+	0.966398i	$0.417249\pi$
$440$	0	0
$441$	−3015.51	−0.325614
$442$	0	0
$443$	5640.42	0.604931	0.302465	−	0.953160i	$-0.402190\pi$
$443$	5640.42	0.604931	0.302465	+	0.953160i	$0.402190\pi$
$444$	0	0
$445$	−2727.41	−0.290543
$446$	0	0
$447$	−7889.78	−0.834841
$448$	0	0
$449$	−2935.03	−0.308491	−0.154246	−	0.988033i	$-0.549295\pi$
$449$	−2935.03	−0.308491	−0.154246	+	0.988033i	$0.549295\pi$
$450$	0	0
$451$	1923.48	0.200828
$452$	0	0
$453$	−3480.27	−0.360966
$454$	0	0
$455$	641.231	0.0660690
$456$	0	0
$457$	−13303.8	−1.36176	−0.680880	−	0.732395i	$-0.738402\pi$
$457$	−13303.8	−1.36176	−0.680880	+	0.732395i	$0.738402\pi$
$458$	0	0
$459$	260.801	0.0265210
$460$	0	0
$461$	−9088.99	−0.918257	−0.459129	−	0.888370i	$-0.651838\pi$
$461$	−9088.99	−0.918257	−0.459129	+	0.888370i	$0.651838\pi$
$462$	0	0
$463$	−17373.9	−1.74391	−0.871957	−	0.489582i	$-0.837149\pi$
$463$	−17373.9	−1.74391	−0.871957	+	0.489582i	$0.837149\pi$
$464$	0	0
$465$	−1809.69	−0.180478
$466$	0	0
$467$	5445.15	0.539554	0.269777	−	0.962923i	$-0.413050\pi$
$467$	5445.15	0.539554	0.269777	+	0.962923i	$0.413050\pi$
$468$	0	0
$469$	2253.26	0.221846
$470$	0	0
$471$	−10125.4	−0.990563
$472$	0	0
$473$	2.37116	0.000230499 0
$474$	0	0
$475$	−331.544	−0.0320259
$476$	0	0
$477$	−4409.85	−0.423298
$478$	0	0
$479$	10900.0	1.03973	0.519866	−	0.854248i	$-0.325982\pi$
$479$	10900.0	1.03973	0.519866	+	0.854248i	$0.325982\pi$
$480$	0	0
$481$	−12843.4	−1.21748
$482$	0	0
$483$	194.467	0.0183200
$484$	0	0
$485$	−4623.35	−0.432856
$486$	0	0
$487$	10831.1	1.00781	0.503905	−	0.863759i	$-0.331896\pi$
$487$	10831.1	1.00781	0.503905	+	0.863759i	$0.331896\pi$
$488$	0	0
$489$	8423.29	0.778966
$490$	0	0
$491$	16025.9	1.47299	0.736495	−	0.676443i	$-0.236479\pi$
$491$	16025.9	1.47299	0.736495	+	0.676443i	$0.236479\pi$
$492$	0	0
$493$	−426.787	−0.0389889
$494$	0	0
$495$	−186.864	−0.0169675
$496$	0	0
$497$	1527.82	0.137891
$498$	0	0
$499$	16878.5	1.51420	0.757101	−	0.653298i	$-0.226615\pi$
$499$	16878.5	1.51420	0.757101	+	0.653298i	$0.226615\pi$
$500$	0	0
$501$	7743.24	0.690504
$502$	0	0
$503$	2222.20	0.196984	0.0984920	−	0.995138i	$-0.468598\pi$
$503$	2222.20	0.196984	0.0984920	+	0.995138i	$0.468598\pi$
$504$	0	0
$505$	9737.34	0.858032
$506$	0	0
$507$	−379.189	−0.0332157
$508$	0	0
$509$	−11334.9	−0.987051	−0.493526	−	0.869731i	$-0.664292\pi$
$509$	−11334.9	−0.987051	−0.493526	+	0.869731i	$0.664292\pi$
$510$	0	0
$511$	−1376.61	−0.119173
$512$	0	0
$513$	−358.068	−0.0308169
$514$	0	0
$515$	−4678.50	−0.400310
$516$	0	0
$517$	1132.57	0.0963452
$518$	0	0
$519$	1476.78	0.124900
$520$	0	0
$521$	−3769.98	−0.317017	−0.158508	−	0.987358i	$-0.550668\pi$
$521$	−3769.98	−0.317017	−0.158508	+	0.987358i	$0.550668\pi$
$522$	0	0
$523$	2407.40	0.201278	0.100639	−	0.994923i	$-0.467911\pi$
$523$	2407.40	0.201278	0.100639	+	0.994923i	$0.467911\pi$
$524$	0	0
$525$	−211.377	−0.0175719
$526$	0	0
$527$	−1165.35	−0.0963257
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	7725.83	0.631399
$532$	0	0
$533$	21077.8	1.71291
$534$	0	0
$535$	2661.73	0.215096
$536$	0	0
$537$	−9510.32	−0.764247
$538$	0	0
$539$	1391.33	0.111185
$540$	0	0
$541$	10114.5	0.803803	0.401901	−	0.915683i	$-0.368349\pi$
$541$	10114.5	0.803803	0.401901	+	0.915683i	$0.368349\pi$
$542$	0	0
$543$	431.027	0.0340647
$544$	0	0
$545$	−1887.61	−0.148360
$546$	0	0
$547$	−5971.89	−0.466800	−0.233400	−	0.972381i	$-0.574985\pi$
$547$	−5971.89	−0.466800	−0.233400	+	0.972381i	$0.574985\pi$
$548$	0	0
$549$	2483.18	0.193041
$550$	0	0
$551$	585.959	0.0453043
$552$	0	0
$553$	10.5352	0.000810128 0
$554$	0	0
$555$	4233.72	0.323804
$556$	0	0
$557$	−11500.3	−0.874833	−0.437417	−	0.899259i	$-0.644107\pi$
$557$	−11500.3	−0.874833	−0.437417	+	0.899259i	$0.644107\pi$
$558$	0	0
$559$	25.9834	0.00196598
$560$	0	0
$561$	−120.331	−0.00905596
$562$	0	0
$563$	12070.1	0.903544	0.451772	−	0.892133i	$-0.350792\pi$
$563$	12070.1	0.903544	0.451772	+	0.892133i	$0.350792\pi$
$564$	0	0
$565$	2805.68	0.208913
$566$	0	0
$567$	−228.287	−0.0169086
$568$	0	0
$569$	2263.26	0.166750	0.0833749	−	0.996518i	$-0.473430\pi$
$569$	2263.26	0.166750	0.0833749	+	0.996518i	$0.473430\pi$
$570$	0	0
$571$	−14157.0	−1.03757	−0.518784	−	0.854905i	$-0.673615\pi$
$571$	−14157.0	−1.03757	−0.518784	+	0.854905i	$0.673615\pi$
$572$	0	0
$573$	10994.2	0.801552
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−25133.0	−1.81334	−0.906671	−	0.421838i	$-0.861385\pi$
$577$	−25133.0	−1.81334	−0.906671	+	0.421838i	$0.861385\pi$
$578$	0	0
$579$	−10938.0	−0.785092
$580$	0	0
$581$	2569.85	0.183503
$582$	0	0
$583$	2034.67	0.144541
$584$	0	0
$585$	−2047.67	−0.144720
$586$	0	0
$587$	15763.6	1.10840	0.554202	−	0.832382i	$-0.313023\pi$
$587$	15763.6	1.10840	0.554202	+	0.832382i	$0.313023\pi$
$588$	0	0
$589$	1599.98	0.111929
$590$	0	0
$591$	3187.10	0.221827
$592$	0	0
$593$	−3241.32	−0.224461	−0.112230	−	0.993682i	$-0.535799\pi$
$593$	−3241.32	−0.224461	−0.112230	+	0.993682i	$0.535799\pi$
$594$	0	0
$595$	−136.117	−0.00937856
$596$	0	0
$597$	−14413.3	−0.988102
$598$	0	0
$599$	23478.0	1.60148	0.800740	−	0.599013i	$-0.204440\pi$
$599$	23478.0	1.60148	0.800740	+	0.599013i	$0.204440\pi$
$600$	0	0
$601$	5379.62	0.365124	0.182562	−	0.983194i	$-0.441561\pi$
$601$	5379.62	0.365124	0.182562	+	0.983194i	$0.441561\pi$
$602$	0	0
$603$	−7195.45	−0.485939
$604$	0	0
$605$	−6568.78	−0.441420
$606$	0	0
$607$	16621.2	1.11142	0.555712	−	0.831375i	$-0.312446\pi$
$607$	16621.2	1.11142	0.555712	+	0.831375i	$0.312446\pi$
$608$	0	0
$609$	373.580	0.0248575
$610$	0	0
$611$	12410.9	0.821751
$612$	0	0
$613$	−7586.63	−0.499872	−0.249936	−	0.968262i	$-0.580410\pi$
$613$	−7586.63	−0.499872	−0.249936	+	0.968262i	$0.580410\pi$
$614$	0	0
$615$	−6948.12	−0.455570
$616$	0	0
$617$	−6805.35	−0.444040	−0.222020	−	0.975042i	$-0.571265\pi$
$617$	−6805.35	−0.444040	−0.222020	+	0.975042i	$0.571265\pi$
$618$	0	0
$619$	−3599.90	−0.233751	−0.116876	−	0.993147i	$-0.537288\pi$
$619$	−3599.90	−0.233751	−0.116876	+	0.993147i	$0.537288\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	1537.36	0.0988655
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	165.209	0.0105228
$628$	0	0
$629$	2726.32	0.172823
$630$	0	0
$631$	29765.6	1.87789	0.938946	−	0.344064i	$-0.111804\pi$
$631$	29765.6	1.87789	0.938946	+	0.344064i	$0.111804\pi$
$632$	0	0
$633$	−2844.90	−0.178633
$634$	0	0
$635$	7027.15	0.439156
$636$	0	0
$637$	15246.4	0.948326
$638$	0	0
$639$	−4878.85	−0.302041
$640$	0	0
$641$	23325.2	1.43727	0.718636	−	0.695387i	$-0.244767\pi$
$641$	23325.2	1.43727	0.718636	+	0.695387i	$0.244767\pi$
$642$	0	0
$643$	22231.9	1.36352	0.681758	−	0.731578i	$-0.261216\pi$
$643$	22231.9	1.36352	0.681758	+	0.731578i	$0.261216\pi$
$644$	0	0
$645$	−8.56524	−0.000522877 0
$646$	0	0
$647$	9684.03	0.588437	0.294218	−	0.955738i	$-0.404941\pi$
$647$	9684.03	0.588437	0.294218	+	0.955738i	$0.404941\pi$
$648$	0	0
$649$	−3564.63	−0.215600
$650$	0	0
$651$	1020.07	0.0614127
$652$	0	0
$653$	2986.42	0.178970	0.0894851	−	0.995988i	$-0.471478\pi$
$653$	2986.42	0.178970	0.0894851	+	0.995988i	$0.471478\pi$
$654$	0	0
$655$	−8724.93	−0.520475
$656$	0	0
$657$	4395.99	0.261041
$658$	0	0
$659$	2479.40	0.146561	0.0732805	−	0.997311i	$-0.476653\pi$
$659$	2479.40	0.146561	0.0732805	+	0.997311i	$0.476653\pi$
$660$	0	0
$661$	−12159.9	−0.715528	−0.357764	−	0.933812i	$-0.616461\pi$
$661$	−12159.9	−0.715528	−0.357764	+	0.933812i	$0.616461\pi$
$662$	0	0
$663$	−1318.61	−0.0772405
$664$	0	0
$665$	186.882	0.0108977
$666$	0	0
$667$	1016.23	0.0589936
$668$	0	0
$669$	−1069.21	−0.0617906
$670$	0	0
$671$	−1145.72	−0.0659164
$672$	0	0
$673$	4838.41	0.277128	0.138564	−	0.990354i	$-0.455751\pi$
$673$	4838.41	0.277128	0.138564	+	0.990354i	$0.455751\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	2422.77	0.137540	0.0687702	−	0.997633i	$-0.478092\pi$
$677$	2422.77	0.137540	0.0687702	+	0.997633i	$0.478092\pi$
$678$	0	0
$679$	2606.05	0.147292
$680$	0	0
$681$	−13829.3	−0.778178
$682$	0	0
$683$	19874.7	1.11345	0.556723	−	0.830698i	$-0.312058\pi$
$683$	19874.7	1.11345	0.556723	+	0.830698i	$0.312058\pi$
$684$	0	0
$685$	−6862.07	−0.382753
$686$	0	0
$687$	10904.9	0.605601
$688$	0	0
$689$	22296.2	1.23282
$690$	0	0
$691$	10554.3	0.581047	0.290523	−	0.956868i	$-0.406171\pi$
$691$	10554.3	0.581047	0.290523	+	0.956868i	$0.406171\pi$
$692$	0	0
$693$	105.330	0.00577366
$694$	0	0
$695$	12541.6	0.684503
$696$	0	0
$697$	−4474.27	−0.243149
$698$	0	0
$699$	16794.8	0.908782
$700$	0	0
$701$	−5986.73	−0.322562	−0.161281	−	0.986909i	$-0.551562\pi$
$701$	−5986.73	−0.322562	−0.161281	+	0.986909i	$0.551562\pi$
$702$	0	0
$703$	−3743.11	−0.200816
$704$	0	0
$705$	−4091.15	−0.218555
$706$	0	0
$707$	−5488.66	−0.291970
$708$	0	0
$709$	14549.9	0.770711	0.385355	−	0.922768i	$-0.374079\pi$
$709$	14549.9	0.770711	0.385355	+	0.922768i	$0.374079\pi$
$710$	0	0
$711$	−33.6425	−0.00177453
$712$	0	0
$713$	2774.85	0.145749
$714$	0	0
$715$	944.780	0.0494164
$716$	0	0
$717$	−6319.25	−0.329144
$718$	0	0
$719$	27729.5	1.43830	0.719149	−	0.694856i	$-0.244532\pi$
$719$	27729.5	1.43830	0.719149	+	0.694856i	$0.244532\pi$
$720$	0	0
$721$	2637.14	0.136217
$722$	0	0
$723$	18170.1	0.934651
$724$	0	0
$725$	−1104.60	−0.0565846
$726$	0	0
$727$	−5284.68	−0.269598	−0.134799	−	0.990873i	$-0.543039\pi$
$727$	−5284.68	−0.269598	−0.134799	+	0.990873i	$0.543039\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−5.51561	−0.000279073 0
$732$	0	0
$733$	−19356.0	−0.975349	−0.487675	−	0.873025i	$-0.662155\pi$
$733$	−19356.0	−0.975349	−0.487675	+	0.873025i	$0.662155\pi$
$734$	0	0
$735$	−5025.85	−0.252220
$736$	0	0
$737$	3319.92	0.165931
$738$	0	0
$739$	17508.2	0.871514	0.435757	−	0.900064i	$-0.356481\pi$
$739$	17508.2	0.871514	0.435757	+	0.900064i	$0.356481\pi$
$740$	0	0
$741$	1810.39	0.0897519
$742$	0	0
$743$	813.123	0.0401488	0.0200744	−	0.999798i	$-0.493610\pi$
$743$	813.123	0.0401488	0.0200744	+	0.999798i	$0.493610\pi$
$744$	0	0
$745$	−13149.6	−0.646665
$746$	0	0
$747$	−8206.43	−0.401951
$748$	0	0
$749$	−1500.34	−0.0731925
$750$	0	0
$751$	23149.6	1.12482	0.562411	−	0.826858i	$-0.309874\pi$
$751$	23149.6	1.12482	0.562411	+	0.826858i	$0.309874\pi$
$752$	0	0
$753$	−5113.59	−0.247476
$754$	0	0
$755$	−5800.45	−0.279603
$756$	0	0
$757$	22091.2	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$757$	22091.2	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$758$	0	0
$759$	286.524	0.0137025
$760$	0	0
$761$	15088.9	0.718756	0.359378	−	0.933192i	$-0.382989\pi$
$761$	15088.9	0.718756	0.359378	+	0.933192i	$0.382989\pi$
$762$	0	0
$763$	1063.99	0.0504837
$764$	0	0
$765$	434.668	0.0205431
$766$	0	0
$767$	−39061.7	−1.83890
$768$	0	0
$769$	−8054.76	−0.377714	−0.188857	−	0.982005i	$-0.560478\pi$
$769$	−8054.76	−0.377714	−0.188857	+	0.982005i	$0.560478\pi$
$770$	0	0
$771$	−20919.3	−0.977161
$772$	0	0
$773$	12958.5	0.602956	0.301478	−	0.953473i	$-0.402520\pi$
$773$	12958.5	0.602956	0.301478	+	0.953473i	$0.402520\pi$
$774$	0	0
$775$	−3016.15	−0.139798
$776$	0	0
$777$	−2386.43	−0.110184
$778$	0	0
$779$	6142.96	0.282534
$780$	0	0
$781$	2251.06	0.103136
$782$	0	0
$783$	−1192.97	−0.0544486
$784$	0	0
$785$	−16875.7	−0.767287
$786$	0	0
$787$	−10184.0	−0.461273	−0.230636	−	0.973040i	$-0.574081\pi$
$787$	−10184.0	−0.461273	−0.230636	+	0.973040i	$0.574081\pi$
$788$	0	0
$789$	15224.4	0.686949
$790$	0	0
$791$	−1581.48	−0.0710886
$792$	0	0
$793$	−12554.9	−0.562217
$794$	0	0
$795$	−7349.76	−0.327886
$796$	0	0
$797$	−7864.72	−0.349539	−0.174770	−	0.984609i	$-0.555918\pi$
$797$	−7864.72	−0.349539	−0.174770	+	0.984609i	$0.555918\pi$
$798$	0	0
$799$	−2634.51	−0.116648
$800$	0	0
$801$	−4909.34	−0.216558
$802$	0	0
$803$	−2028.27	−0.0891360
$804$	0	0
$805$	324.111	0.0141906
$806$	0	0
$807$	19962.7	0.870783
$808$	0	0
$809$	306.058	0.0133009	0.00665046	−	0.999978i	$-0.497883\pi$
$809$	306.058	0.0133009	0.00665046	+	0.999978i	$0.497883\pi$
$810$	0	0
$811$	3958.24	0.171384	0.0856922	−	0.996322i	$-0.472690\pi$
$811$	3958.24	0.171384	0.0856922	+	0.996322i	$0.472690\pi$
$812$	0	0
$813$	16176.5	0.697829
$814$	0	0
$815$	14038.8	0.603385
$816$	0	0
$817$	7.57267	0.000324277 0
$818$	0	0
$819$	1154.22	0.0492449
$820$	0	0
$821$	6757.79	0.287270	0.143635	−	0.989631i	$-0.454121\pi$
$821$	6757.79	0.287270	0.143635	+	0.989631i	$0.454121\pi$
$822$	0	0
$823$	16294.4	0.690143	0.345071	−	0.938576i	$-0.387855\pi$
$823$	16294.4	0.690143	0.345071	+	0.938576i	$0.387855\pi$
$824$	0	0
$825$	−311.439	−0.0131429
$826$	0	0
$827$	−6376.11	−0.268101	−0.134050	−	0.990975i	$-0.542798\pi$
$827$	−6376.11	−0.268101	−0.134050	+	0.990975i	$0.542798\pi$
$828$	0	0
$829$	9832.26	0.411928	0.205964	−	0.978560i	$-0.433967\pi$
$829$	9832.26	0.411928	0.205964	+	0.978560i	$0.433967\pi$
$830$	0	0
$831$	−23547.6	−0.982983
$832$	0	0
$833$	−3236.41	−0.134616
$834$	0	0
$835$	12905.4	0.534862
$836$	0	0
$837$	−3257.44	−0.134520
$838$	0	0
$839$	36428.1	1.49897	0.749486	−	0.662020i	$-0.230301\pi$
$839$	36428.1	1.49897	0.749486	+	0.662020i	$0.230301\pi$
$840$	0	0
$841$	−22436.8	−0.919954
$842$	0	0
$843$	−3660.19	−0.149542
$844$	0	0
$845$	−631.982	−0.0257288
$846$	0	0
$847$	3702.64	0.150206
$848$	0	0
$849$	−19948.2	−0.806384
$850$	0	0
$851$	−6491.71	−0.261496
$852$	0	0
$853$	33936.3	1.36220	0.681100	−	0.732190i	$-0.261502\pi$
$853$	33936.3	1.36220	0.681100	+	0.732190i	$0.261502\pi$
$854$	0	0
$855$	−596.779	−0.0238707
$856$	0	0
$857$	−36424.7	−1.45186	−0.725930	−	0.687769i	$-0.758590\pi$
$857$	−36424.7	−1.45186	−0.725930	+	0.687769i	$0.758590\pi$
$858$	0	0
$859$	11219.9	0.445657	0.222828	−	0.974858i	$-0.428471\pi$
$859$	11219.9	0.445657	0.222828	+	0.974858i	$0.428471\pi$
$860$	0	0
$861$	3916.46	0.155020
$862$	0	0
$863$	38388.6	1.51421	0.757104	−	0.653294i	$-0.226613\pi$
$863$	38388.6	1.51421	0.757104	+	0.653294i	$0.226613\pi$
$864$	0	0
$865$	2461.29	0.0967474
$866$	0	0
$867$	−14459.1	−0.566386
$868$	0	0
$869$	15.5223	0.000605937 0
$870$	0	0
$871$	36380.1	1.41526
$872$	0	0
$873$	−8322.02	−0.322632
$874$	0	0
$875$	−352.295	−0.0136111
$876$	0	0
$877$	−49445.1	−1.90381	−0.951905	−	0.306392i	$-0.900878\pi$
$877$	−49445.1	−1.90381	−0.951905	+	0.306392i	$0.900878\pi$
$878$	0	0
$879$	−21964.5	−0.842826
$880$	0	0
$881$	28208.7	1.07875	0.539374	−	0.842067i	$-0.318661\pi$
$881$	28208.7	1.07875	0.539374	+	0.842067i	$0.318661\pi$
$882$	0	0
$883$	−32335.8	−1.23238	−0.616188	−	0.787599i	$-0.711324\pi$
$883$	−32335.8	−1.23238	−0.616188	+	0.787599i	$0.711324\pi$
$884$	0	0
$885$	12876.4	0.489079
$886$	0	0
$887$	−10776.7	−0.407943	−0.203972	−	0.978977i	$-0.565385\pi$
$887$	−10776.7	−0.407943	−0.203972	+	0.978977i	$0.565385\pi$
$888$	0	0
$889$	−3961.00	−0.149435
$890$	0	0
$891$	−336.354	−0.0126468
$892$	0	0
$893$	3617.05	0.135543
$894$	0	0
$895$	−15850.5	−0.591983
$896$	0	0
$897$	3139.77	0.116872
$898$	0	0
$899$	5330.62	0.197760
$900$	0	0
$901$	−4732.90	−0.175001
$902$	0	0
$903$	4.82798	0.000177924 0
$904$	0	0
$905$	718.378	0.0263864
$906$	0	0
$907$	−16913.9	−0.619201	−0.309601	−	0.950867i	$-0.600195\pi$
$907$	−16913.9	−0.619201	−0.309601	+	0.950867i	$0.600195\pi$
$908$	0	0
$909$	17527.2	0.639539
$910$	0	0
$911$	25925.8	0.942876	0.471438	−	0.881899i	$-0.343735\pi$
$911$	25925.8	0.942876	0.471438	+	0.881899i	$0.343735\pi$
$912$	0	0
$913$	3786.38	0.137252
$914$	0	0
$915$	4138.63	0.149529
$916$	0	0
$917$	4918.00	0.177106
$918$	0	0
$919$	−51261.6	−1.84001	−0.920003	−	0.391912i	$-0.871814\pi$
$919$	−51261.6	−1.84001	−0.920003	+	0.391912i	$0.871814\pi$
$920$	0	0
$921$	−12392.0	−0.443356
$922$	0	0
$923$	24667.4	0.879672
$924$	0	0
$925$	7056.20	0.250818
$926$	0	0
$927$	−8421.31	−0.298373
$928$	0	0
$929$	25172.1	0.888989	0.444495	−	0.895782i	$-0.353383\pi$
$929$	25172.1	0.888989	0.444495	+	0.895782i	$0.353383\pi$
$930$	0	0
$931$	4443.44	0.156421
$932$	0	0
$933$	−20437.6	−0.717144
$934$	0	0
$935$	−200.552	−0.00701472
$936$	0	0
$937$	−45771.6	−1.59583	−0.797914	−	0.602771i	$-0.794063\pi$
$937$	−45771.6	−1.59583	−0.797914	+	0.602771i	$0.794063\pi$
$938$	0	0
$939$	14485.7	0.503433
$940$	0	0
$941$	−21708.3	−0.752040	−0.376020	−	0.926612i	$-0.622708\pi$
$941$	−21708.3	−0.752040	−0.376020	+	0.926612i	$0.622708\pi$
$942$	0	0
$943$	10653.8	0.367906
$944$	0	0
$945$	−380.478	−0.0130973
$946$	0	0
$947$	25052.6	0.859661	0.429830	−	0.902910i	$-0.358573\pi$
$947$	25052.6	0.859661	0.429830	+	0.902910i	$0.358573\pi$
$948$	0	0
$949$	−22226.1	−0.760263
$950$	0	0
$951$	10749.5	0.366538
$952$	0	0
$953$	−57074.0	−1.93999	−0.969994	−	0.243129i	$-0.921826\pi$
$953$	−57074.0	−1.93999	−0.969994	+	0.243129i	$0.921826\pi$
$954$	0	0
$955$	18323.7	0.620879
$956$	0	0
$957$	550.426	0.0185922
$958$	0	0
$959$	3867.95	0.130243
$960$	0	0
$961$	−15235.6	−0.511415
$962$	0	0
$963$	4791.11	0.160323
$964$	0	0
$965$	−18230.0	−0.608130
$966$	0	0
$967$	−54292.7	−1.80552	−0.902760	−	0.430146i	$-0.858462\pi$
$967$	−54292.7	−1.80552	−0.902760	+	0.430146i	$0.858462\pi$
$968$	0	0
$969$	−384.298	−0.0127404
$970$	0	0
$971$	−4057.42	−0.134098	−0.0670489	−	0.997750i	$-0.521358\pi$
$971$	−4057.42	−0.134098	−0.0670489	+	0.997750i	$0.521358\pi$
$972$	0	0
$973$	−7069.34	−0.232922
$974$	0	0
$975$	−3412.79	−0.112099
$976$	0	0
$977$	2655.91	0.0869704	0.0434852	−	0.999054i	$-0.486154\pi$
$977$	2655.91	0.0869704	0.0434852	+	0.999054i	$0.486154\pi$
$978$	0	0
$979$	2265.13	0.0739467
$980$	0	0
$981$	−3397.69	−0.110581
$982$	0	0
$983$	18208.3	0.590800	0.295400	−	0.955374i	$-0.404547\pi$
$983$	18208.3	0.590800	0.295400	+	0.955374i	$0.404547\pi$
$984$	0	0
$985$	5311.83	0.171826
$986$	0	0
$987$	2306.06	0.0743696
$988$	0	0
$989$	13.1334	0.000422261 0
$990$	0	0
$991$	−51491.3	−1.65053	−0.825266	−	0.564745i	$-0.808975\pi$
$991$	−51491.3	−1.65053	−0.825266	+	0.564745i	$0.808975\pi$
$992$	0	0
$993$	−24412.5	−0.780167
$994$	0	0
$995$	−24022.2	−0.765381
$996$	0	0
$997$	7773.75	0.246938	0.123469	−	0.992348i	$-0.460598\pi$
$997$	7773.75	0.246938	0.123469	+	0.992348i	$0.460598\pi$
$998$	0	0
$999$	7620.70	0.241350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.h.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.h.1.3	✓	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)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} -2.81836 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} -2.81836 q^{7} +9.00000 q^{9} -4.15252 q^{11} -45.5039 q^{13} +15.0000 q^{15} +9.65930 q^{17} -13.2618 q^{19} -8.45507 q^{21} -23.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -44.1841 q^{29} -120.646 q^{31} -12.4576 q^{33} -14.0918 q^{35} +282.248 q^{37} -136.512 q^{39} -463.208 q^{41} -0.571016 q^{43} +45.0000 q^{45} -272.743 q^{47} -335.057 q^{49} +28.9779 q^{51} -489.984 q^{53} -20.7626 q^{55} -39.7853 q^{57} +858.426 q^{59} +275.908 q^{61} -25.3652 q^{63} -227.519 q^{65} -799.495 q^{67} -69.0000 q^{69} -542.094 q^{71} +488.444 q^{73} +75.0000 q^{75} +11.7033 q^{77} -3.73805 q^{79} +81.0000 q^{81} -911.825 q^{83} +48.2965 q^{85} -132.552 q^{87} -545.482 q^{89} +128.246 q^{91} -361.938 q^{93} -66.3088 q^{95} -924.669 q^{97} -37.3727 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} + 25 q^{5} - 23 q^{7} + 45 q^{9}+O(q^{10}) \) 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

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