LMFDB - Embedded newform 1840.4.a.u.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 80x^{4} + 105x^{3} + 1328x^{2} - 816x - 4032 $ x^6 - 2x^5 - 80x^4 + 105x^3 + 1328x^2 - 816*x - 4032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.24528$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.24528 q^{3} +5.00000 q^{5} +19.3962 q^{7} -25.4493 q^{9} +O(q^{10})$	$q-1.24528 q^{3} +5.00000 q^{5} +19.3962 q^{7} -25.4493 q^{9} +2.63073 q^{11} +36.8906 q^{13} -6.22638 q^{15} +6.64946 q^{17} -151.260 q^{19} -24.1537 q^{21} -23.0000 q^{23} +25.0000 q^{25} +65.3138 q^{27} +132.292 q^{29} +120.197 q^{31} -3.27599 q^{33} +96.9812 q^{35} -356.931 q^{37} -45.9390 q^{39} -341.455 q^{41} +233.466 q^{43} -127.246 q^{45} -345.816 q^{47} +33.2144 q^{49} -8.28042 q^{51} -663.089 q^{53} +13.1537 q^{55} +188.360 q^{57} +855.729 q^{59} -449.199 q^{61} -493.621 q^{63} +184.453 q^{65} -225.651 q^{67} +28.6413 q^{69} -70.3957 q^{71} -832.022 q^{73} -31.1319 q^{75} +51.0264 q^{77} +771.546 q^{79} +605.797 q^{81} +304.702 q^{83} +33.2473 q^{85} -164.740 q^{87} +774.090 q^{89} +715.540 q^{91} -149.678 q^{93} -756.299 q^{95} +1077.84 q^{97} -66.9503 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	$ 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * 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
$3$	−1.24528	−0.239653	−0.119827	−	0.992795i	$-0.538234\pi$
$3$	−1.24528	−0.239653	−0.119827	+	0.992795i	$0.538234\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	19.3962	1.04730	0.523649	−	0.851934i	$-0.324570\pi$
$7$	19.3962	1.04730	0.523649	+	0.851934i	$0.324570\pi$
$8$	0	0
$9$	−25.4493	−0.942566
$10$	0	0
$11$	2.63073	0.0721087	0.0360544	−	0.999350i	$-0.488521\pi$
$11$	2.63073	0.0721087	0.0360544	+	0.999350i	$0.488521\pi$
$12$	0	0
$13$	36.8906	0.787048	0.393524	−	0.919314i	$-0.371256\pi$
$13$	36.8906	0.787048	0.393524	+	0.919314i	$0.371256\pi$
$14$	0	0
$15$	−6.22638	−0.107176
$16$	0	0
$17$	6.64946	0.0948666	0.0474333	−	0.998874i	$-0.484896\pi$
$17$	6.64946	0.0948666	0.0474333	+	0.998874i	$0.484896\pi$
$18$	0	0
$19$	−151.260	−1.82639	−0.913195	−	0.407524i	$-0.866392\pi$
$19$	−151.260	−1.82639	−0.913195	+	0.407524i	$0.866392\pi$
$20$	0	0
$21$	−24.1537	−0.250989
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	65.3138	0.465543
$28$	0	0
$29$	132.292	0.847104	0.423552	−	0.905872i	$-0.360783\pi$
$29$	132.292	0.847104	0.423552	+	0.905872i	$0.360783\pi$
$30$	0	0
$31$	120.197	0.696385	0.348193	−	0.937423i	$-0.386795\pi$
$31$	120.197	0.696385	0.348193	+	0.937423i	$0.386795\pi$
$32$	0	0
$33$	−3.27599	−0.0172811
$34$	0	0
$35$	96.9812	0.468366
$36$	0	0
$37$	−356.931	−1.58592	−0.792960	−	0.609274i	$-0.791461\pi$
$37$	−356.931	−1.58592	−0.792960	+	0.609274i	$0.791461\pi$
$38$	0	0
$39$	−45.9390	−0.188619
$40$	0	0
$41$	−341.455	−1.30064	−0.650322	−	0.759659i	$-0.725366\pi$
$41$	−341.455	−1.30064	−0.650322	+	0.759659i	$0.725366\pi$
$42$	0	0
$43$	233.466	0.827983	0.413991	−	0.910281i	$-0.364134\pi$
$43$	233.466	0.827983	0.413991	+	0.910281i	$0.364134\pi$
$44$	0	0
$45$	−127.246	−0.421528
$46$	0	0
$47$	−345.816	−1.07324	−0.536621	−	0.843823i	$-0.680300\pi$
$47$	−345.816	−1.07324	−0.536621	+	0.843823i	$0.680300\pi$
$48$	0	0
$49$	33.2144	0.0968350
$50$	0	0
$51$	−8.28042	−0.0227351
$52$	0	0
$53$	−663.089	−1.71853	−0.859266	−	0.511529i	$-0.829079\pi$
$53$	−663.089	−1.71853	−0.859266	+	0.511529i	$0.829079\pi$
$54$	0	0
$55$	13.1537	0.0322480
$56$	0	0
$57$	188.360	0.437700
$58$	0	0
$59$	855.729	1.88825	0.944123	−	0.329594i	$-0.106912\pi$
$59$	855.729	1.88825	0.944123	+	0.329594i	$0.106912\pi$
$60$	0	0
$61$	−449.199	−0.942854	−0.471427	−	0.881905i	$-0.656261\pi$
$61$	−449.199	−0.942854	−0.471427	+	0.881905i	$0.656261\pi$
$62$	0	0
$63$	−493.621	−0.987149
$64$	0	0
$65$	184.453	0.351978
$66$	0	0
$67$	−225.651	−0.411457	−0.205728	−	0.978609i	$-0.565956\pi$
$67$	−225.651	−0.411457	−0.205728	+	0.978609i	$0.565956\pi$
$68$	0	0
$69$	28.6413	0.0499712
$70$	0	0
$71$	−70.3957	−0.117668	−0.0588340	−	0.998268i	$-0.518738\pi$
$71$	−70.3957	−0.117668	−0.0588340	+	0.998268i	$0.518738\pi$
$72$	0	0
$73$	−832.022	−1.33398	−0.666992	−	0.745065i	$-0.732418\pi$
$73$	−832.022	−1.33398	−0.666992	+	0.745065i	$0.732418\pi$
$74$	0	0
$75$	−31.1319	−0.0479307
$76$	0	0
$77$	51.0264	0.0755194
$78$	0	0
$79$	771.546	1.09881	0.549403	−	0.835557i	$-0.314855\pi$
$79$	771.546	1.09881	0.549403	+	0.835557i	$0.314855\pi$
$80$	0	0
$81$	605.797	0.830997
$82$	0	0
$83$	304.702	0.402957	0.201479	−	0.979493i	$-0.435425\pi$
$83$	304.702	0.402957	0.201479	+	0.979493i	$0.435425\pi$
$84$	0	0
$85$	33.2473	0.0424256
$86$	0	0
$87$	−164.740	−0.203012
$88$	0	0
$89$	774.090	0.921948	0.460974	−	0.887414i	$-0.347500\pi$
$89$	774.090	0.921948	0.460974	+	0.887414i	$0.347500\pi$
$90$	0	0
$91$	715.540	0.824274
$92$	0	0
$93$	−149.678	−0.166891
$94$	0	0
$95$	−756.299	−0.816786
$96$	0	0
$97$	1077.84	1.12823	0.564116	−	0.825695i	$-0.309217\pi$
$97$	1077.84	1.12823	0.564116	+	0.825695i	$0.309217\pi$
$98$	0	0
$99$	−66.9503	−0.0679673
$100$	0	0
$101$	−335.238	−0.330272	−0.165136	−	0.986271i	$-0.552806\pi$
$101$	−335.238	−0.330272	−0.165136	+	0.986271i	$0.552806\pi$
$102$	0	0
$103$	−475.661	−0.455032	−0.227516	−	0.973774i	$-0.573060\pi$
$103$	−475.661	−0.455032	−0.227516	+	0.973774i	$0.573060\pi$
$104$	0	0
$105$	−120.768	−0.112246
$106$	0	0
$107$	739.766	0.668373	0.334186	−	0.942507i	$-0.391539\pi$
$107$	739.766	0.668373	0.334186	+	0.942507i	$0.391539\pi$
$108$	0	0
$109$	328.647	0.288795	0.144397	−	0.989520i	$-0.453876\pi$
$109$	328.647	0.288795	0.144397	+	0.989520i	$0.453876\pi$
$110$	0	0
$111$	444.477	0.380071
$112$	0	0
$113$	−1559.05	−1.29791	−0.648954	−	0.760828i	$-0.724793\pi$
$113$	−1559.05	−1.29791	−0.648954	+	0.760828i	$0.724793\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−938.840	−0.741845
$118$	0	0
$119$	128.975	0.0993537
$120$	0	0
$121$	−1324.08	−0.994800
$122$	0	0
$123$	425.206	0.311704
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−744.833	−0.520419	−0.260209	−	0.965552i	$-0.583792\pi$
$127$	−744.833	−0.520419	−0.260209	+	0.965552i	$0.583792\pi$
$128$	0	0
$129$	−290.730	−0.198429
$130$	0	0
$131$	−1588.99	−1.05977	−0.529887	−	0.848068i	$-0.677766\pi$
$131$	−1588.99	−1.05977	−0.529887	+	0.848068i	$0.677766\pi$
$132$	0	0
$133$	−2933.87	−1.91278
$134$	0	0
$135$	326.569	0.208197
$136$	0	0
$137$	331.167	0.206522	0.103261	−	0.994654i	$-0.467072\pi$
$137$	331.167	0.206522	0.103261	+	0.994654i	$0.467072\pi$
$138$	0	0
$139$	−1452.26	−0.886181	−0.443090	−	0.896477i	$-0.646118\pi$
$139$	−1452.26	−0.886181	−0.443090	+	0.896477i	$0.646118\pi$
$140$	0	0
$141$	430.636	0.257206
$142$	0	0
$143$	97.0495	0.0567530
$144$	0	0
$145$	661.461	0.378837
$146$	0	0
$147$	−41.3611	−0.0232069
$148$	0	0
$149$	−621.069	−0.341476	−0.170738	−	0.985316i	$-0.554615\pi$
$149$	−621.069	−0.341476	−0.170738	+	0.985316i	$0.554615\pi$
$150$	0	0
$151$	2088.97	1.12581	0.562906	−	0.826521i	$-0.309683\pi$
$151$	2088.97	1.12581	0.562906	+	0.826521i	$0.309683\pi$
$152$	0	0
$153$	−169.224	−0.0894180
$154$	0	0
$155$	600.983	0.311433
$156$	0	0
$157$	−1235.97	−0.628288	−0.314144	−	0.949375i	$-0.601717\pi$
$157$	−1235.97	−0.628288	−0.314144	+	0.949375i	$0.601717\pi$
$158$	0	0
$159$	825.728	0.411852
$160$	0	0
$161$	−446.114	−0.218377
$162$	0	0
$163$	−271.806	−0.130610	−0.0653052	−	0.997865i	$-0.520802\pi$
$163$	−271.806	−0.130610	−0.0653052	+	0.997865i	$0.520802\pi$
$164$	0	0
$165$	−16.3799	−0.00772835
$166$	0	0
$167$	1431.46	0.663293	0.331646	−	0.943404i	$-0.392396\pi$
$167$	1431.46	0.663293	0.331646	+	0.943404i	$0.392396\pi$
$168$	0	0
$169$	−836.081	−0.380556
$170$	0	0
$171$	3849.46	1.72149
$172$	0	0
$173$	−1422.63	−0.625206	−0.312603	−	0.949884i	$-0.601201\pi$
$173$	−1422.63	−0.625206	−0.312603	+	0.949884i	$0.601201\pi$
$174$	0	0
$175$	484.906	0.209460
$176$	0	0
$177$	−1065.62	−0.452525
$178$	0	0
$179$	2759.84	1.15240	0.576202	−	0.817307i	$-0.304534\pi$
$179$	2759.84	1.15240	0.576202	+	0.817307i	$0.304534\pi$
$180$	0	0
$181$	1353.93	0.556003	0.278002	−	0.960581i	$-0.410328\pi$
$181$	1353.93	0.556003	0.278002	+	0.960581i	$0.410328\pi$
$182$	0	0
$183$	559.377	0.225958
$184$	0	0
$185$	−1784.65	−0.709245
$186$	0	0
$187$	17.4930	0.00684071
$188$	0	0
$189$	1266.84	0.487562
$190$	0	0
$191$	−3749.60	−1.42048	−0.710239	−	0.703960i	$-0.751413\pi$
$191$	−3749.60	−1.42048	−0.710239	+	0.703960i	$0.751413\pi$
$192$	0	0
$193$	3352.20	1.25024	0.625120	−	0.780528i	$-0.285050\pi$
$193$	3352.20	1.25024	0.625120	+	0.780528i	$0.285050\pi$
$194$	0	0
$195$	−229.695	−0.0843529
$196$	0	0
$197$	−4024.47	−1.45549	−0.727745	−	0.685848i	$-0.759431\pi$
$197$	−4024.47	−1.45549	−0.727745	+	0.685848i	$0.759431\pi$
$198$	0	0
$199$	−1066.38	−0.379868	−0.189934	−	0.981797i	$-0.560827\pi$
$199$	−1066.38	−0.379868	−0.189934	+	0.981797i	$0.560827\pi$
$200$	0	0
$201$	280.997	0.0986071
$202$	0	0
$203$	2565.97	0.887172
$204$	0	0
$205$	−1707.28	−0.581666
$206$	0	0
$207$	585.334	0.196539
$208$	0	0
$209$	−397.924	−0.131699
$210$	0	0
$211$	−5051.85	−1.64826	−0.824131	−	0.566399i	$-0.808336\pi$
$211$	−5051.85	−1.64826	−0.824131	+	0.566399i	$0.808336\pi$
$212$	0	0
$213$	87.6620	0.0281995
$214$	0	0
$215$	1167.33	0.370285
$216$	0	0
$217$	2331.36	0.729324
$218$	0	0
$219$	1036.10	0.319694
$220$	0	0
$221$	245.303	0.0746645
$222$	0	0
$223$	−3300.58	−0.991134	−0.495567	−	0.868570i	$-0.665040\pi$
$223$	−3300.58	−0.991134	−0.495567	+	0.868570i	$0.665040\pi$
$224$	0	0
$225$	−636.232	−0.188513
$226$	0	0
$227$	4833.21	1.41318	0.706588	−	0.707625i	$-0.250233\pi$
$227$	4833.21	1.41318	0.706588	+	0.707625i	$0.250233\pi$
$228$	0	0
$229$	1242.83	0.358640	0.179320	−	0.983791i	$-0.442610\pi$
$229$	1242.83	0.358640	0.179320	+	0.983791i	$0.442610\pi$
$230$	0	0
$231$	−63.5419	−0.0180985
$232$	0	0
$233$	−4085.93	−1.14883	−0.574416	−	0.818563i	$-0.694771\pi$
$233$	−4085.93	−1.14883	−0.574416	+	0.818563i	$0.694771\pi$
$234$	0	0
$235$	−1729.08	−0.479968
$236$	0	0
$237$	−960.788	−0.263333
$238$	0	0
$239$	−3840.39	−1.03939	−0.519695	−	0.854352i	$-0.673954\pi$
$239$	−3840.39	−1.03939	−0.519695	+	0.854352i	$0.673954\pi$
$240$	0	0
$241$	−1645.48	−0.439811	−0.219906	−	0.975521i	$-0.570575\pi$
$241$	−1645.48	−0.439811	−0.219906	+	0.975521i	$0.570575\pi$
$242$	0	0
$243$	−2517.86	−0.664694
$244$	0	0
$245$	166.072	0.0433059
$246$	0	0
$247$	−5580.07	−1.43746
$248$	0	0
$249$	−379.439	−0.0965700
$250$	0	0
$251$	−4470.47	−1.12420	−0.562098	−	0.827070i	$-0.690006\pi$
$251$	−4470.47	−1.12420	−0.562098	+	0.827070i	$0.690006\pi$
$252$	0	0
$253$	−60.5069	−0.0150357
$254$	0	0
$255$	−41.4021	−0.0101674
$256$	0	0
$257$	−683.157	−0.165814	−0.0829069	−	0.996557i	$-0.526420\pi$
$257$	−683.157	−0.165814	−0.0829069	+	0.996557i	$0.526420\pi$
$258$	0	0
$259$	−6923.11	−1.66093
$260$	0	0
$261$	−3366.74	−0.798452
$262$	0	0
$263$	−2153.60	−0.504930	−0.252465	−	0.967606i	$-0.581241\pi$
$263$	−2153.60	−0.504930	−0.252465	+	0.967606i	$0.581241\pi$
$264$	0	0
$265$	−3315.44	−0.768551
$266$	0	0
$267$	−963.956	−0.220948
$268$	0	0
$269$	−5208.56	−1.18056	−0.590282	−	0.807197i	$-0.700983\pi$
$269$	−5208.56	−1.18056	−0.590282	+	0.807197i	$0.700983\pi$
$270$	0	0
$271$	6409.31	1.43667	0.718335	−	0.695697i	$-0.244904\pi$
$271$	6409.31	1.43667	0.718335	+	0.695697i	$0.244904\pi$
$272$	0	0
$273$	−891.045	−0.197540
$274$	0	0
$275$	65.7684	0.0144217
$276$	0	0
$277$	1045.30	0.226737	0.113369	−	0.993553i	$-0.463836\pi$
$277$	1045.30	0.226737	0.113369	+	0.993553i	$0.463836\pi$
$278$	0	0
$279$	−3058.92	−0.656389
$280$	0	0
$281$	−3110.25	−0.660291	−0.330145	−	0.943930i	$-0.607098\pi$
$281$	−3110.25	−0.660291	−0.330145	+	0.943930i	$0.607098\pi$
$282$	0	0
$283$	−1518.31	−0.318919	−0.159460	−	0.987204i	$-0.550975\pi$
$283$	−1518.31	−0.318919	−0.159460	+	0.987204i	$0.550975\pi$
$284$	0	0
$285$	941.801	0.195746
$286$	0	0
$287$	−6622.96	−1.36216
$288$	0	0
$289$	−4868.78	−0.991000
$290$	0	0
$291$	−1342.21	−0.270385
$292$	0	0
$293$	−370.249	−0.0738231	−0.0369116	−	0.999319i	$-0.511752\pi$
$293$	−370.249	−0.0738231	−0.0369116	+	0.999319i	$0.511752\pi$
$294$	0	0
$295$	4278.65	0.844449
$296$	0	0
$297$	171.823	0.0335697
$298$	0	0
$299$	−848.485	−0.164111
$300$	0	0
$301$	4528.37	0.867145
$302$	0	0
$303$	417.464	0.0791508
$304$	0	0
$305$	−2246.00	−0.421657
$306$	0	0
$307$	7158.05	1.33072	0.665360	−	0.746522i	$-0.268278\pi$
$307$	7158.05	1.33072	0.665360	+	0.746522i	$0.268278\pi$
$308$	0	0
$309$	592.329	0.109050
$310$	0	0
$311$	−8087.70	−1.47463	−0.737317	−	0.675547i	$-0.763908\pi$
$311$	−8087.70	−1.47463	−0.737317	+	0.675547i	$0.763908\pi$
$312$	0	0
$313$	−9528.29	−1.72067	−0.860337	−	0.509726i	$-0.829747\pi$
$313$	−9528.29	−1.72067	−0.860337	+	0.509726i	$0.829747\pi$
$314$	0	0
$315$	−2468.10	−0.441466
$316$	0	0
$317$	10529.7	1.86563	0.932817	−	0.360352i	$-0.117343\pi$
$317$	10529.7	1.86563	0.932817	+	0.360352i	$0.117343\pi$
$318$	0	0
$319$	348.025	0.0610836
$320$	0	0
$321$	−921.213	−0.160178
$322$	0	0
$323$	−1005.80	−0.173263
$324$	0	0
$325$	922.266	0.157410
$326$	0	0
$327$	−409.256	−0.0692107
$328$	0	0
$329$	−6707.52	−1.12401
$330$	0	0
$331$	725.130	0.120413	0.0602066	−	0.998186i	$-0.480824\pi$
$331$	725.130	0.120413	0.0602066	+	0.998186i	$0.480824\pi$
$332$	0	0
$333$	9083.63	1.49483
$334$	0	0
$335$	−1128.25	−0.184009
$336$	0	0
$337$	9691.45	1.56655	0.783274	−	0.621676i	$-0.213548\pi$
$337$	9691.45	1.56655	0.783274	+	0.621676i	$0.213548\pi$
$338$	0	0
$339$	1941.45	0.311048
$340$	0	0
$341$	316.205	0.0502155
$342$	0	0
$343$	−6008.68	−0.945884
$344$	0	0
$345$	143.207	0.0223478
$346$	0	0
$347$	−3860.13	−0.597184	−0.298592	−	0.954381i	$-0.596517\pi$
$347$	−3860.13	−0.597184	−0.298592	+	0.954381i	$0.596517\pi$
$348$	0	0
$349$	−7254.79	−1.11272	−0.556361	−	0.830941i	$-0.687803\pi$
$349$	−7254.79	−1.11272	−0.556361	+	0.830941i	$0.687803\pi$
$350$	0	0
$351$	2409.47	0.366404
$352$	0	0
$353$	−12373.8	−1.86569	−0.932847	−	0.360273i	$-0.882683\pi$
$353$	−12373.8	−1.86569	−0.932847	+	0.360273i	$0.882683\pi$
$354$	0	0
$355$	−351.978	−0.0526227
$356$	0	0
$357$	−160.609	−0.0238104
$358$	0	0
$359$	−10364.1	−1.52367	−0.761836	−	0.647770i	$-0.775702\pi$
$359$	−10364.1	−1.52367	−0.761836	+	0.647770i	$0.775702\pi$
$360$	0	0
$361$	16020.5	2.33570
$362$	0	0
$363$	1648.84	0.238407
$364$	0	0
$365$	−4160.11	−0.596575
$366$	0	0
$367$	4484.12	0.637791	0.318895	−	0.947790i	$-0.396688\pi$
$367$	4484.12	0.637791	0.318895	+	0.947790i	$0.396688\pi$
$368$	0	0
$369$	8689.80	1.22594
$370$	0	0
$371$	−12861.4	−1.79982
$372$	0	0
$373$	−7155.47	−0.993287	−0.496644	−	0.867955i	$-0.665434\pi$
$373$	−7155.47	−0.993287	−0.496644	+	0.867955i	$0.665434\pi$
$374$	0	0
$375$	−155.659	−0.0214353
$376$	0	0
$377$	4880.34	0.666712
$378$	0	0
$379$	8472.83	1.14834	0.574169	−	0.818737i	$-0.305325\pi$
$379$	8472.83	1.14834	0.574169	+	0.818737i	$0.305325\pi$
$380$	0	0
$381$	927.522	0.124720
$382$	0	0
$383$	−1624.48	−0.216729	−0.108364	−	0.994111i	$-0.534561\pi$
$383$	−1624.48	−0.216729	−0.108364	+	0.994111i	$0.534561\pi$
$384$	0	0
$385$	255.132	0.0337733
$386$	0	0
$387$	−5941.55	−0.780429
$388$	0	0
$389$	−545.022	−0.0710379	−0.0355189	−	0.999369i	$-0.511308\pi$
$389$	−545.022	−0.0710379	−0.0355189	+	0.999369i	$0.511308\pi$
$390$	0	0
$391$	−152.938	−0.0197810
$392$	0	0
$393$	1978.73	0.253978
$394$	0	0
$395$	3857.73	0.491401
$396$	0	0
$397$	40.7685	0.00515393	0.00257697	−	0.999997i	$-0.499180\pi$
$397$	40.7685	0.00515393	0.00257697	+	0.999997i	$0.499180\pi$
$398$	0	0
$399$	3653.48	0.458403
$400$	0	0
$401$	−11593.5	−1.44377	−0.721885	−	0.692013i	$-0.756724\pi$
$401$	−11593.5	−1.44377	−0.721885	+	0.692013i	$0.756724\pi$
$402$	0	0
$403$	4434.13	0.548089
$404$	0	0
$405$	3028.99	0.371633
$406$	0	0
$407$	−938.989	−0.114359
$408$	0	0
$409$	4602.74	0.556457	0.278228	−	0.960515i	$-0.410253\pi$
$409$	4602.74	0.556457	0.278228	+	0.960515i	$0.410253\pi$
$410$	0	0
$411$	−412.394	−0.0494937
$412$	0	0
$413$	16597.9	1.97756
$414$	0	0
$415$	1523.51	0.180208
$416$	0	0
$417$	1808.47	0.212376
$418$	0	0
$419$	15407.0	1.79637	0.898187	−	0.439613i	$-0.144884\pi$
$419$	15407.0	1.79637	0.898187	+	0.439613i	$0.144884\pi$
$420$	0	0
$421$	3555.48	0.411600	0.205800	−	0.978594i	$-0.434020\pi$
$421$	3555.48	0.411600	0.205800	+	0.978594i	$0.434020\pi$
$422$	0	0
$423$	8800.76	1.01160
$424$	0	0
$425$	166.237	0.0189733
$426$	0	0
$427$	−8712.78	−0.987450
$428$	0	0
$429$	−120.853	−0.0136011
$430$	0	0
$431$	−3688.86	−0.412265	−0.206132	−	0.978524i	$-0.566088\pi$
$431$	−3688.86	−0.412265	−0.206132	+	0.978524i	$0.566088\pi$
$432$	0	0
$433$	−6691.42	−0.742654	−0.371327	−	0.928502i	$-0.621097\pi$
$433$	−6691.42	−0.742654	−0.371327	+	0.928502i	$0.621097\pi$
$434$	0	0
$435$	−823.701	−0.0907895
$436$	0	0
$437$	3478.98	0.380828
$438$	0	0
$439$	−13233.3	−1.43870	−0.719350	−	0.694647i	$-0.755560\pi$
$439$	−13233.3	−1.43870	−0.719350	+	0.694647i	$0.755560\pi$
$440$	0	0
$441$	−845.283	−0.0912734
$442$	0	0
$443$	−5453.47	−0.584881	−0.292440	−	0.956284i	$-0.594467\pi$
$443$	−5453.47	−0.584881	−0.292440	+	0.956284i	$0.594467\pi$
$444$	0	0
$445$	3870.45	0.412308
$446$	0	0
$447$	773.402	0.0818360
$448$	0	0
$449$	14550.4	1.52934	0.764670	−	0.644422i	$-0.222902\pi$
$449$	14550.4	1.52934	0.764670	+	0.644422i	$0.222902\pi$
$450$	0	0
$451$	−898.279	−0.0937878
$452$	0	0
$453$	−2601.34	−0.269805
$454$	0	0
$455$	3577.70	0.368627
$456$	0	0
$457$	−12778.3	−1.30797	−0.653984	−	0.756508i	$-0.726904\pi$
$457$	−12778.3	−1.30797	−0.653984	+	0.756508i	$0.726904\pi$
$458$	0	0
$459$	434.302	0.0441644
$460$	0	0
$461$	5832.03	0.589208	0.294604	−	0.955619i	$-0.404812\pi$
$461$	5832.03	0.589208	0.294604	+	0.955619i	$0.404812\pi$
$462$	0	0
$463$	15500.1	1.55583	0.777915	−	0.628370i	$-0.216277\pi$
$463$	15500.1	1.55583	0.777915	+	0.628370i	$0.216277\pi$
$464$	0	0
$465$	−748.390	−0.0746360
$466$	0	0
$467$	−10882.0	−1.07828	−0.539141	−	0.842215i	$-0.681251\pi$
$467$	−10882.0	−1.07828	−0.539141	+	0.842215i	$0.681251\pi$
$468$	0	0
$469$	−4376.77	−0.430918
$470$	0	0
$471$	1539.12	0.150571
$472$	0	0
$473$	614.187	0.0597048
$474$	0	0
$475$	−3781.50	−0.365278
$476$	0	0
$477$	16875.1	1.61983
$478$	0	0
$479$	−6132.03	−0.584926	−0.292463	−	0.956277i	$-0.594475\pi$
$479$	−6132.03	−0.584926	−0.292463	+	0.956277i	$0.594475\pi$
$480$	0	0
$481$	−13167.4	−1.24819
$482$	0	0
$483$	555.535	0.0523348
$484$	0	0
$485$	5389.22	0.504561
$486$	0	0
$487$	1164.11	0.108318	0.0541589	−	0.998532i	$-0.482752\pi$
$487$	1164.11	0.108318	0.0541589	+	0.998532i	$0.482752\pi$
$488$	0	0
$489$	338.473	0.0313012
$490$	0	0
$491$	−14810.0	−1.36124	−0.680618	−	0.732638i	$-0.738289\pi$
$491$	−14810.0	−1.36124	−0.680618	+	0.732638i	$0.738289\pi$
$492$	0	0
$493$	879.672	0.0803619
$494$	0	0
$495$	−334.752	−0.0303959
$496$	0	0
$497$	−1365.41	−0.123234
$498$	0	0
$499$	−13281.1	−1.19147	−0.595737	−	0.803180i	$-0.703140\pi$
$499$	−13281.1	−1.19147	−0.595737	+	0.803180i	$0.703140\pi$
$500$	0	0
$501$	−1782.57	−0.158960
$502$	0	0
$503$	6902.85	0.611894	0.305947	−	0.952049i	$-0.401027\pi$
$503$	6902.85	0.611894	0.305947	+	0.952049i	$0.401027\pi$
$504$	0	0
$505$	−1676.19	−0.147702
$506$	0	0
$507$	1041.15	0.0912015
$508$	0	0
$509$	11033.5	0.960810	0.480405	−	0.877047i	$-0.340490\pi$
$509$	11033.5	0.960810	0.480405	+	0.877047i	$0.340490\pi$
$510$	0	0
$511$	−16138.1	−1.39708
$512$	0	0
$513$	−9879.36	−0.850262
$514$	0	0
$515$	−2378.30	−0.203496
$516$	0	0
$517$	−909.749	−0.0773901
$518$	0	0
$519$	1771.57	0.149833
$520$	0	0
$521$	20434.8	1.71836	0.859181	−	0.511671i	$-0.170973\pi$
$521$	20434.8	1.71836	0.859181	+	0.511671i	$0.170973\pi$
$522$	0	0
$523$	3722.02	0.311191	0.155595	−	0.987821i	$-0.450270\pi$
$523$	3722.02	0.311191	0.155595	+	0.987821i	$0.450270\pi$
$524$	0	0
$525$	−603.842	−0.0501978
$526$	0	0
$527$	799.243	0.0660637
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−21777.7	−1.77980
$532$	0	0
$533$	−12596.5	−1.02367
$534$	0	0
$535$	3698.83	0.298905
$536$	0	0
$537$	−3436.76	−0.276178
$538$	0	0
$539$	87.3783	0.00698265
$540$	0	0
$541$	10599.6	0.842348	0.421174	−	0.906980i	$-0.361618\pi$
$541$	10599.6	0.842348	0.421174	+	0.906980i	$0.361618\pi$
$542$	0	0
$543$	−1686.01	−0.133248
$544$	0	0
$545$	1643.23	0.129153
$546$	0	0
$547$	−10928.8	−0.854262	−0.427131	−	0.904190i	$-0.640476\pi$
$547$	−10928.8	−0.854262	−0.427131	+	0.904190i	$0.640476\pi$
$548$	0	0
$549$	11431.8	0.888702
$550$	0	0
$551$	−20010.5	−1.54714
$552$	0	0
$553$	14965.1	1.15078
$554$	0	0
$555$	2222.39	0.169973
$556$	0	0
$557$	−7472.30	−0.568423	−0.284211	−	0.958762i	$-0.591732\pi$
$557$	−7472.30	−0.568423	−0.284211	+	0.958762i	$0.591732\pi$
$558$	0	0
$559$	8612.72	0.651662
$560$	0	0
$561$	−21.7836	−0.00163940
$562$	0	0
$563$	24216.0	1.81276	0.906378	−	0.422468i	$-0.138836\pi$
$563$	24216.0	1.81276	0.906378	+	0.422468i	$0.138836\pi$
$564$	0	0
$565$	−7795.27	−0.580442
$566$	0	0
$567$	11750.2	0.870303
$568$	0	0
$569$	2492.45	0.183636	0.0918181	−	0.995776i	$-0.470732\pi$
$569$	2492.45	0.183636	0.0918181	+	0.995776i	$0.470732\pi$
$570$	0	0
$571$	25619.8	1.87768	0.938841	−	0.344352i	$-0.111901\pi$
$571$	25619.8	1.87768	0.938841	+	0.344352i	$0.111901\pi$
$572$	0	0
$573$	4669.29	0.340423
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−10833.3	−0.781623	−0.390811	−	0.920471i	$-0.627806\pi$
$577$	−10833.3	−0.781623	−0.390811	+	0.920471i	$0.627806\pi$
$578$	0	0
$579$	−4174.41	−0.299624
$580$	0	0
$581$	5910.08	0.422016
$582$	0	0
$583$	−1744.41	−0.123921
$584$	0	0
$585$	−4694.20	−0.331763
$586$	0	0
$587$	7427.51	0.522259	0.261130	−	0.965304i	$-0.415905\pi$
$587$	7427.51	0.522259	0.261130	+	0.965304i	$0.415905\pi$
$588$	0	0
$589$	−18180.9	−1.27187
$590$	0	0
$591$	5011.57	0.348813
$592$	0	0
$593$	−1853.93	−0.128384	−0.0641919	−	0.997938i	$-0.520447\pi$
$593$	−1853.93	−0.128384	−0.0641919	+	0.997938i	$0.520447\pi$
$594$	0	0
$595$	644.873	0.0444323
$596$	0	0
$597$	1327.94	0.0910366
$598$	0	0
$599$	−4549.60	−0.310336	−0.155168	−	0.987888i	$-0.549592\pi$
$599$	−4549.60	−0.310336	−0.155168	+	0.987888i	$0.549592\pi$
$600$	0	0
$601$	−12996.4	−0.882086	−0.441043	−	0.897486i	$-0.645391\pi$
$601$	−12996.4	−0.882086	−0.441043	+	0.897486i	$0.645391\pi$
$602$	0	0
$603$	5742.65	0.387825
$604$	0	0
$605$	−6620.40	−0.444888
$606$	0	0
$607$	22097.2	1.47759	0.738795	−	0.673930i	$-0.235395\pi$
$607$	22097.2	1.47759	0.738795	+	0.673930i	$0.235395\pi$
$608$	0	0
$609$	−3195.34	−0.212614
$610$	0	0
$611$	−12757.4	−0.844693
$612$	0	0
$613$	−663.328	−0.0437057	−0.0218528	−	0.999761i	$-0.506957\pi$
$613$	−663.328	−0.0437057	−0.0218528	+	0.999761i	$0.506957\pi$
$614$	0	0
$615$	2126.03	0.139398
$616$	0	0
$617$	24621.2	1.60650	0.803252	−	0.595640i	$-0.203101\pi$
$617$	24621.2	1.60650	0.803252	+	0.595640i	$0.203101\pi$
$618$	0	0
$619$	−21546.3	−1.39906	−0.699530	−	0.714603i	$-0.746607\pi$
$619$	−21546.3	−1.39906	−0.699530	+	0.714603i	$0.746607\pi$
$620$	0	0
$621$	−1502.22	−0.0970724
$622$	0	0
$623$	15014.4	0.965556
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	495.526	0.0315620
$628$	0	0
$629$	−2373.40	−0.150451
$630$	0	0
$631$	−5279.22	−0.333063	−0.166531	−	0.986036i	$-0.553257\pi$
$631$	−5279.22	−0.333063	−0.166531	+	0.986036i	$0.553257\pi$
$632$	0	0
$633$	6290.94	0.395012
$634$	0	0
$635$	−3724.16	−0.232738
$636$	0	0
$637$	1225.30	0.0762138
$638$	0	0
$639$	1791.52	0.110910
$640$	0	0
$641$	−11877.4	−0.731874	−0.365937	−	0.930640i	$-0.619251\pi$
$641$	−11877.4	−0.731874	−0.365937	+	0.930640i	$0.619251\pi$
$642$	0	0
$643$	13942.5	0.855114	0.427557	−	0.903988i	$-0.359374\pi$
$643$	13942.5	0.855114	0.427557	+	0.903988i	$0.359374\pi$
$644$	0	0
$645$	−1453.65	−0.0887401
$646$	0	0
$647$	11344.9	0.689355	0.344677	−	0.938721i	$-0.387988\pi$
$647$	11344.9	0.689355	0.344677	+	0.938721i	$0.387988\pi$
$648$	0	0
$649$	2251.20	0.136159
$650$	0	0
$651$	−2903.19	−0.174785
$652$	0	0
$653$	217.847	0.0130551	0.00652757	−	0.999979i	$-0.497922\pi$
$653$	217.847	0.0130551	0.00652757	+	0.999979i	$0.497922\pi$
$654$	0	0
$655$	−7944.93	−0.473945
$656$	0	0
$657$	21174.4	1.25737
$658$	0	0
$659$	−511.626	−0.0302430	−0.0151215	−	0.999886i	$-0.504814\pi$
$659$	−511.626	−0.0302430	−0.0151215	+	0.999886i	$0.504814\pi$
$660$	0	0
$661$	−4566.41	−0.268703	−0.134352	−	0.990934i	$-0.542895\pi$
$661$	−4566.41	−0.268703	−0.134352	+	0.990934i	$0.542895\pi$
$662$	0	0
$663$	−305.470	−0.0178936
$664$	0	0
$665$	−14669.4	−0.855419
$666$	0	0
$667$	−3042.72	−0.176633
$668$	0	0
$669$	4110.13	0.237529
$670$	0	0
$671$	−1181.72	−0.0679880
$672$	0	0
$673$	−18864.7	−1.08051	−0.540253	−	0.841502i	$-0.681672\pi$
$673$	−18864.7	−1.08051	−0.540253	+	0.841502i	$0.681672\pi$
$674$	0	0
$675$	1632.85	0.0931085
$676$	0	0
$677$	3765.50	0.213766	0.106883	−	0.994272i	$-0.465913\pi$
$677$	3765.50	0.213766	0.106883	+	0.994272i	$0.465913\pi$
$678$	0	0
$679$	20906.1	1.18160
$680$	0	0
$681$	−6018.68	−0.338673
$682$	0	0
$683$	16552.1	0.927303	0.463652	−	0.886018i	$-0.346539\pi$
$683$	16552.1	0.927303	0.463652	+	0.886018i	$0.346539\pi$
$684$	0	0
$685$	1655.83	0.0923594
$686$	0	0
$687$	−1547.67	−0.0859493
$688$	0	0
$689$	−24461.8	−1.35257
$690$	0	0
$691$	12497.1	0.688003	0.344002	−	0.938969i	$-0.388217\pi$
$691$	12497.1	0.688003	0.344002	+	0.938969i	$0.388217\pi$
$692$	0	0
$693$	−1298.58	−0.0711820
$694$	0	0
$695$	−7261.30	−0.396312
$696$	0	0
$697$	−2270.50	−0.123388
$698$	0	0
$699$	5088.11	0.275322
$700$	0	0
$701$	9288.82	0.500476	0.250238	−	0.968184i	$-0.419491\pi$
$701$	9288.82	0.500476	0.250238	+	0.968184i	$0.419491\pi$
$702$	0	0
$703$	53989.3	2.89651
$704$	0	0
$705$	2153.18	0.115026
$706$	0	0
$707$	−6502.37	−0.345894
$708$	0	0
$709$	36368.6	1.92645	0.963224	−	0.268701i	$-0.0865944\pi$
$709$	36368.6	1.92645	0.963224	+	0.268701i	$0.0865944\pi$
$710$	0	0
$711$	−19635.3	−1.03570
$712$	0	0
$713$	−2764.52	−0.145206
$714$	0	0
$715$	485.247	0.0253807
$716$	0	0
$717$	4782.34	0.249093
$718$	0	0
$719$	−16354.0	−0.848266	−0.424133	−	0.905600i	$-0.639421\pi$
$719$	−16354.0	−0.848266	−0.424133	+	0.905600i	$0.639421\pi$
$720$	0	0
$721$	−9226.03	−0.476554
$722$	0	0
$723$	2049.07	0.105402
$724$	0	0
$725$	3307.30	0.169421
$726$	0	0
$727$	9116.20	0.465064	0.232532	−	0.972589i	$-0.425299\pi$
$727$	9116.20	0.465064	0.232532	+	0.972589i	$0.425299\pi$
$728$	0	0
$729$	−13221.1	−0.671701
$730$	0	0
$731$	1552.42	0.0785479
$732$	0	0
$733$	−14838.4	−0.747707	−0.373854	−	0.927488i	$-0.621964\pi$
$733$	−14838.4	−0.747707	−0.373854	+	0.927488i	$0.621964\pi$
$734$	0	0
$735$	−206.806	−0.0103784
$736$	0	0
$737$	−593.627	−0.0296696
$738$	0	0
$739$	19768.3	0.984017	0.492009	−	0.870590i	$-0.336263\pi$
$739$	19768.3	0.984017	0.492009	+	0.870590i	$0.336263\pi$
$740$	0	0
$741$	6948.73	0.344491
$742$	0	0
$743$	5426.35	0.267932	0.133966	−	0.990986i	$-0.457229\pi$
$743$	5426.35	0.267932	0.133966	+	0.990986i	$0.457229\pi$
$744$	0	0
$745$	−3105.35	−0.152713
$746$	0	0
$747$	−7754.46	−0.379814
$748$	0	0
$749$	14348.7	0.699986
$750$	0	0
$751$	6514.42	0.316531	0.158265	−	0.987397i	$-0.449410\pi$
$751$	6514.42	0.316531	0.158265	+	0.987397i	$0.449410\pi$
$752$	0	0
$753$	5566.96	0.269418
$754$	0	0
$755$	10444.8	0.503479
$756$	0	0
$757$	38354.3	1.84149	0.920747	−	0.390161i	$-0.127581\pi$
$757$	38354.3	1.84149	0.920747	+	0.390161i	$0.127581\pi$
$758$	0	0
$759$	75.3478	0.00360336
$760$	0	0
$761$	4008.93	0.190964	0.0954820	−	0.995431i	$-0.469561\pi$
$761$	4008.93	0.190964	0.0954820	+	0.995431i	$0.469561\pi$
$762$	0	0
$763$	6374.51	0.302455
$764$	0	0
$765$	−846.121	−0.0399890
$766$	0	0
$767$	31568.4	1.48614
$768$	0	0
$769$	−11093.1	−0.520190	−0.260095	−	0.965583i	$-0.583754\pi$
$769$	−11093.1	−0.520190	−0.260095	+	0.965583i	$0.583754\pi$
$770$	0	0
$771$	850.719	0.0397379
$772$	0	0
$773$	−160.328	−0.00746001	−0.00373000	−	0.999993i	$-0.501187\pi$
$773$	−160.328	−0.00746001	−0.00373000	+	0.999993i	$0.501187\pi$
$774$	0	0
$775$	3004.91	0.139277
$776$	0	0
$777$	8621.19	0.398048
$778$	0	0
$779$	51648.5	2.37548
$780$	0	0
$781$	−185.192	−0.00848489
$782$	0	0
$783$	8640.51	0.394363
$784$	0	0
$785$	−6179.85	−0.280979
$786$	0	0
$787$	13488.8	0.610956	0.305478	−	0.952199i	$-0.401184\pi$
$787$	13488.8	0.610956	0.305478	+	0.952199i	$0.401184\pi$
$788$	0	0
$789$	2681.83	0.121008
$790$	0	0
$791$	−30239.8	−1.35930
$792$	0	0
$793$	−16571.2	−0.742071
$794$	0	0
$795$	4128.64	0.184186
$796$	0	0
$797$	−35733.5	−1.58814	−0.794068	−	0.607828i	$-0.792041\pi$
$797$	−35733.5	−1.58814	−0.794068	+	0.607828i	$0.792041\pi$
$798$	0	0
$799$	−2299.49	−0.101815
$800$	0	0
$801$	−19700.0	−0.868997
$802$	0	0
$803$	−2188.83	−0.0961918
$804$	0	0
$805$	−2230.57	−0.0976611
$806$	0	0
$807$	6486.10	0.282926
$808$	0	0
$809$	19429.2	0.844367	0.422184	−	0.906510i	$-0.361264\pi$
$809$	19429.2	0.844367	0.422184	+	0.906510i	$0.361264\pi$
$810$	0	0
$811$	12229.7	0.529522	0.264761	−	0.964314i	$-0.414707\pi$
$811$	12229.7	0.529522	0.264761	+	0.964314i	$0.414707\pi$
$812$	0	0
$813$	−7981.36	−0.344303
$814$	0	0
$815$	−1359.03	−0.0584107
$816$	0	0
$817$	−35314.1	−1.51222
$818$	0	0
$819$	−18210.0	−0.776933
$820$	0	0
$821$	41610.7	1.76885	0.884424	−	0.466685i	$-0.154552\pi$
$821$	41610.7	1.76885	0.884424	+	0.466685i	$0.154552\pi$
$822$	0	0
$823$	−8938.00	−0.378565	−0.189283	−	0.981923i	$-0.560616\pi$
$823$	−8938.00	−0.378565	−0.189283	+	0.981923i	$0.560616\pi$
$824$	0	0
$825$	−81.8997	−0.00345622
$826$	0	0
$827$	25288.4	1.06332	0.531659	−	0.846959i	$-0.321569\pi$
$827$	25288.4	1.06332	0.531659	+	0.846959i	$0.321569\pi$
$828$	0	0
$829$	33100.3	1.38675	0.693377	−	0.720575i	$-0.256122\pi$
$829$	33100.3	1.38675	0.693377	+	0.720575i	$0.256122\pi$
$830$	0	0
$831$	−1301.69	−0.0543384
$832$	0	0
$833$	220.858	0.00918641
$834$	0	0
$835$	7157.31	0.296633
$836$	0	0
$837$	7850.50	0.324197
$838$	0	0
$839$	−17652.9	−0.726395	−0.363198	−	0.931712i	$-0.618315\pi$
$839$	−17652.9	−0.726395	−0.363198	+	0.931712i	$0.618315\pi$
$840$	0	0
$841$	−6887.80	−0.282414
$842$	0	0
$843$	3873.11	0.158241
$844$	0	0
$845$	−4180.40	−0.170190
$846$	0	0
$847$	−25682.2	−1.04185
$848$	0	0
$849$	1890.72	0.0764301
$850$	0	0
$851$	8209.40	0.330687
$852$	0	0
$853$	3135.75	0.125869	0.0629344	−	0.998018i	$-0.479954\pi$
$853$	3135.75	0.125869	0.0629344	+	0.998018i	$0.479954\pi$
$854$	0	0
$855$	19247.3	0.769875
$856$	0	0
$857$	45623.0	1.81850	0.909249	−	0.416252i	$-0.136657\pi$
$857$	45623.0	1.81850	0.909249	+	0.416252i	$0.136657\pi$
$858$	0	0
$859$	35976.9	1.42901	0.714503	−	0.699632i	$-0.246653\pi$
$859$	35976.9	1.42901	0.714503	+	0.699632i	$0.246653\pi$
$860$	0	0
$861$	8247.41	0.326447
$862$	0	0
$863$	−43748.2	−1.72561	−0.862807	−	0.505533i	$-0.831296\pi$
$863$	−43748.2	−1.72561	−0.862807	+	0.505533i	$0.831296\pi$
$864$	0	0
$865$	−7113.16	−0.279601
$866$	0	0
$867$	6062.98	0.237497
$868$	0	0
$869$	2029.73	0.0792335
$870$	0	0
$871$	−8324.39	−0.323836
$872$	0	0
$873$	−27430.4	−1.06343
$874$	0	0
$875$	2424.53	0.0936733
$876$	0	0
$877$	33646.3	1.29550	0.647751	−	0.761852i	$-0.275710\pi$
$877$	33646.3	1.29550	0.647751	+	0.761852i	$0.275710\pi$
$878$	0	0
$879$	461.062	0.0176920
$880$	0	0
$881$	26047.7	0.996106	0.498053	−	0.867146i	$-0.334048\pi$
$881$	26047.7	0.996106	0.498053	+	0.867146i	$0.334048\pi$
$882$	0	0
$883$	24428.8	0.931025	0.465512	−	0.885041i	$-0.345870\pi$
$883$	24428.8	0.931025	0.465512	+	0.885041i	$0.345870\pi$
$884$	0	0
$885$	−5328.10	−0.202375
$886$	0	0
$887$	−19248.4	−0.728632	−0.364316	−	0.931275i	$-0.618697\pi$
$887$	−19248.4	−0.728632	−0.364316	+	0.931275i	$0.618697\pi$
$888$	0	0
$889$	−14447.0	−0.545034
$890$	0	0
$891$	1593.69	0.0599222
$892$	0	0
$893$	52308.0	1.96016
$894$	0	0
$895$	13799.2	0.515371
$896$	0	0
$897$	1056.60	0.0393297
$898$	0	0
$899$	15901.1	0.589911
$900$	0	0
$901$	−4409.18	−0.163031
$902$	0	0
$903$	−5639.07	−0.207814
$904$	0	0
$905$	6769.64	0.248652
$906$	0	0
$907$	−46791.9	−1.71301	−0.856505	−	0.516140i	$-0.827369\pi$
$907$	−46791.9	−1.71301	−0.856505	+	0.516140i	$0.827369\pi$
$908$	0	0
$909$	8531.58	0.311303
$910$	0	0
$911$	−19753.3	−0.718394	−0.359197	−	0.933262i	$-0.616949\pi$
$911$	−19753.3	−0.718394	−0.359197	+	0.933262i	$0.616949\pi$
$912$	0	0
$913$	801.591	0.0290567
$914$	0	0
$915$	2796.89	0.101052
$916$	0	0
$917$	−30820.4	−1.10990
$918$	0	0
$919$	46182.6	1.65770	0.828848	−	0.559473i	$-0.188997\pi$
$919$	46182.6	1.65770	0.828848	+	0.559473i	$0.188997\pi$
$920$	0	0
$921$	−8913.74	−0.318912
$922$	0	0
$923$	−2596.94	−0.0926103
$924$	0	0
$925$	−8923.26	−0.317184
$926$	0	0
$927$	12105.2	0.428897
$928$	0	0
$929$	−9725.75	−0.343478	−0.171739	−	0.985142i	$-0.554939\pi$
$929$	−9725.75	−0.343478	−0.171739	+	0.985142i	$0.554939\pi$
$930$	0	0
$931$	−5024.01	−0.176858
$932$	0	0
$933$	10071.4	0.353401
$934$	0	0
$935$	87.4648	0.00305926
$936$	0	0
$937$	−56305.5	−1.96309	−0.981547	−	0.191220i	$-0.938756\pi$
$937$	−56305.5	−1.96309	−0.981547	+	0.191220i	$0.938756\pi$
$938$	0	0
$939$	11865.4	0.412365
$940$	0	0
$941$	−44046.9	−1.52592	−0.762958	−	0.646447i	$-0.776254\pi$
$941$	−44046.9	−1.52592	−0.762958	+	0.646447i	$0.776254\pi$
$942$	0	0
$943$	7853.48	0.271203
$944$	0	0
$945$	6334.22	0.218045
$946$	0	0
$947$	35545.1	1.21970	0.609852	−	0.792515i	$-0.291229\pi$
$947$	35545.1	1.21970	0.609852	+	0.792515i	$0.291229\pi$
$948$	0	0
$949$	−30693.8	−1.04991
$950$	0	0
$951$	−13112.4	−0.447105
$952$	0	0
$953$	−1104.08	−0.0375285	−0.0187643	−	0.999824i	$-0.505973\pi$
$953$	−1104.08	−0.0375285	−0.0187643	+	0.999824i	$0.505973\pi$
$954$	0	0
$955$	−18748.0	−0.635257
$956$	0	0
$957$	−433.388	−0.0146389
$958$	0	0
$959$	6423.39	0.216290
$960$	0	0
$961$	−15343.8	−0.515047
$962$	0	0
$963$	−18826.5	−0.629985
$964$	0	0
$965$	16761.0	0.559125
$966$	0	0
$967$	−2734.09	−0.0909228	−0.0454614	−	0.998966i	$-0.514476\pi$
$967$	−2734.09	−0.0909228	−0.0454614	+	0.998966i	$0.514476\pi$
$968$	0	0
$969$	1252.49	0.0415231
$970$	0	0
$971$	−20452.8	−0.675965	−0.337982	−	0.941152i	$-0.609744\pi$
$971$	−20452.8	−0.675965	−0.337982	+	0.941152i	$0.609744\pi$
$972$	0	0
$973$	−28168.4	−0.928096
$974$	0	0
$975$	−1148.48	−0.0377237
$976$	0	0
$977$	11174.9	0.365932	0.182966	−	0.983119i	$-0.441430\pi$
$977$	11174.9	0.365932	0.182966	+	0.983119i	$0.441430\pi$
$978$	0	0
$979$	2036.43	0.0664805
$980$	0	0
$981$	−8363.83	−0.272208
$982$	0	0
$983$	−2414.48	−0.0783418	−0.0391709	−	0.999233i	$-0.512472\pi$
$983$	−2414.48	−0.0783418	−0.0391709	+	0.999233i	$0.512472\pi$
$984$	0	0
$985$	−20122.3	−0.650915
$986$	0	0
$987$	8352.72	0.269372
$988$	0	0
$989$	−5369.72	−0.172646
$990$	0	0
$991$	−35774.1	−1.14672	−0.573361	−	0.819303i	$-0.694361\pi$
$991$	−35774.1	−1.14672	−0.573361	+	0.819303i	$0.694361\pi$
$992$	0	0
$993$	−902.987	−0.0288574
$994$	0	0
$995$	−5331.90	−0.169882
$996$	0	0
$997$	9808.42	0.311570	0.155785	−	0.987791i	$-0.450209\pi$
$997$	9808.42	0.311570	0.155785	+	0.987791i	$0.450209\pi$
$998$	0	0
$999$	−23312.5	−0.738313

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.u.1.3		6
4.3	odd	2		920.4.a.a.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.a.1.4	✓	6	4.3	odd	2
1840.4.a.u.1.3		6	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.24528 q^{3} +5.00000 q^{5} +19.3962 q^{7} -25.4493 q^{9} +O(q^{10})\)	\(q-1.24528 q^{3} +5.00000 q^{5} +19.3962 q^{7} -25.4493 q^{9} +2.63073 q^{11} +36.8906 q^{13} -6.22638 q^{15} +6.64946 q^{17} -151.260 q^{19} -24.1537 q^{21} -23.0000 q^{23} +25.0000 q^{25} +65.3138 q^{27} +132.292 q^{29} +120.197 q^{31} -3.27599 q^{33} +96.9812 q^{35} -356.931 q^{37} -45.9390 q^{39} -341.455 q^{41} +233.466 q^{43} -127.246 q^{45} -345.816 q^{47} +33.2144 q^{49} -8.28042 q^{51} -663.089 q^{53} +13.1537 q^{55} +188.360 q^{57} +855.729 q^{59} -449.199 q^{61} -493.621 q^{63} +184.453 q^{65} -225.651 q^{67} +28.6413 q^{69} -70.3957 q^{71} -832.022 q^{73} -31.1319 q^{75} +51.0264 q^{77} +771.546 q^{79} +605.797 q^{81} +304.702 q^{83} +33.2473 q^{85} -164.740 q^{87} +774.090 q^{89} +715.540 q^{91} -149.678 q^{93} -756.299 q^{95} +1077.84 q^{97} -66.9503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9	\( 6 q + 4 q^{3} + 30 q^{5} + 14 q^{7} + 4 q^{9} + 23 q^{11} - 42 q^{13} + 20 q^{15} - 124 q^{17} - 53 q^{19} - 114 q^{21} - 138 q^{23} + 150 q^{25} + 103 q^{27} - 320 q^{29} + 229 q^{31} - 583 q^{33} + 70 q^{35} - 377 q^{37} - 37 q^{39} - 683 q^{41} + 168 q^{43} + 20 q^{45} - 211 q^{47} - 374 q^{49} - 777 q^{51} - 613 q^{53} + 115 q^{55} - 316 q^{57} - 1029 q^{59} - 1169 q^{61} - 183 q^{63} - 210 q^{65} + 1227 q^{67} - 92 q^{69} - 237 q^{71} - 1001 q^{73} + 100 q^{75} - 1498 q^{77} + 898 q^{79} - 838 q^{81} + 1281 q^{83} - 620 q^{85} + 695 q^{87} - 2780 q^{89} - 857 q^{91} - 1569 q^{93} - 265 q^{95} + 91 q^{97} + 1015 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 + 30 * q^5 + 14 * q^7 + 4 * q^9 + 23 * q^11 - 42 * q^13 + 20 * q^15 - 124 * q^17 - 53 * q^19 - 114 * q^21 - 138 * q^23 + 150 * q^25 + 103 * q^27 - 320 * q^29 + 229 * q^31 - 583 * q^33 + 70 * q^35 - 377 * q^37 - 37 * q^39 - 683 * q^41 + 168 * q^43 + 20 * q^45 - 211 * q^47 - 374 * q^49 - 777 * q^51 - 613 * q^53 + 115 * q^55 - 316 * q^57 - 1029 * q^59 - 1169 * q^61 - 183 * q^63 - 210 * q^65 + 1227 * q^67 - 92 * q^69 - 237 * q^71 - 1001 * q^73 + 100 * q^75 - 1498 * q^77 + 898 * q^79 - 838 * q^81 + 1281 * q^83 - 620 * q^85 + 695 * q^87 - 2780 * q^89 - 857 * q^91 - 1569 * q^93 - 265 * q^95 + 91 * q^97 + 1015 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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