LMFDB - Embedded newform 1840.4.a.v.1.1

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 31x^{5} + 750x^{4} + 249x^{3} - 2892x^{2} - 620x + 2400 $ x^8 - 2x^7 - 49x^6 + 31x^5 + 750x^4 + 249x^3 - 2892x^2 - 620*x + 2400
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.16661$ 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)

$f(q)$	$=$	$q-8.59146 q^{3} +5.00000 q^{5} -9.86011 q^{7} +46.8131 q^{9} +O(q^{10})$	$q-8.59146 q^{3} +5.00000 q^{5} -9.86011 q^{7} +46.8131 q^{9} -12.1995 q^{11} +20.6010 q^{13} -42.9573 q^{15} -7.09451 q^{17} +155.809 q^{19} +84.7127 q^{21} -23.0000 q^{23} +25.0000 q^{25} -170.224 q^{27} -213.296 q^{29} +95.7095 q^{31} +104.811 q^{33} -49.3006 q^{35} +174.131 q^{37} -176.993 q^{39} +122.549 q^{41} -398.567 q^{43} +234.066 q^{45} +490.851 q^{47} -245.778 q^{49} +60.9522 q^{51} -161.490 q^{53} -60.9974 q^{55} -1338.63 q^{57} +560.053 q^{59} -321.451 q^{61} -461.583 q^{63} +103.005 q^{65} -280.311 q^{67} +197.604 q^{69} -649.370 q^{71} +426.808 q^{73} -214.786 q^{75} +120.288 q^{77} +161.318 q^{79} +198.514 q^{81} -801.673 q^{83} -35.4725 q^{85} +1832.53 q^{87} +1100.38 q^{89} -203.128 q^{91} -822.284 q^{93} +779.046 q^{95} -1179.42 q^{97} -571.096 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9}+O(q^{10}) $ 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9	$ 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9} - 41 q^{11} + 28 q^{13} + 71 q^{17} - 177 q^{19} + 292 q^{21} - 184 q^{23} + 200 q^{25} + 495 q^{27} + 225 q^{29} + 36 q^{31} - 53 q^{33} - 55 q^{35} - 348 q^{37} + 1077 q^{39} + 620 q^{41} + 390 q^{43} + 790 q^{45} - 123 q^{47} + 881 q^{49} - 957 q^{51} + 1406 q^{53} - 205 q^{55} - 1142 q^{57} + 676 q^{59} + 1447 q^{61} + 58 q^{63} + 140 q^{65} + 1582 q^{67} - 1396 q^{71} + 17 q^{73} + 488 q^{77} - 708 q^{79} + 4316 q^{81} - 1486 q^{83} + 355 q^{85} + 803 q^{87} + 1360 q^{89} - 2693 q^{91} + 3833 q^{93} - 885 q^{95} - 855 q^{97} - 1319 q^{99}+O(q^{100}) $ 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9 - 41 * q^11 + 28 * q^13 + 71 * q^17 - 177 * q^19 + 292 * q^21 - 184 * q^23 + 200 * q^25 + 495 * q^27 + 225 * q^29 + 36 * q^31 - 53 * q^33 - 55 * q^35 - 348 * q^37 + 1077 * q^39 + 620 * q^41 + 390 * q^43 + 790 * q^45 - 123 * q^47 + 881 * q^49 - 957 * q^51 + 1406 * q^53 - 205 * q^55 - 1142 * q^57 + 676 * q^59 + 1447 * q^61 + 58 * q^63 + 140 * q^65 + 1582 * q^67 - 1396 * q^71 + 17 * q^73 + 488 * q^77 - 708 * q^79 + 4316 * q^81 - 1486 * q^83 + 355 * q^85 + 803 * q^87 + 1360 * q^89 - 2693 * q^91 + 3833 * q^93 - 885 * q^95 - 855 * q^97 - 1319 * 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
$3$	−8.59146	−1.65343	−0.826713	−	0.562623i	$-0.809792\pi$
$3$	−8.59146	−1.65343	−0.826713	+	0.562623i	$0.809792\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−9.86011	−0.532396	−0.266198	−	0.963918i	$-0.585767\pi$
$7$	−9.86011	−0.532396	−0.266198	+	0.963918i	$0.585767\pi$
$8$	0	0
$9$	46.8131	1.73382
$10$	0	0
$11$	−12.1995	−0.334389	−0.167195	−	0.985924i	$-0.553471\pi$
$11$	−12.1995	−0.334389	−0.167195	+	0.985924i	$0.553471\pi$
$12$	0	0
$13$	20.6010	0.439515	0.219758	−	0.975555i	$-0.429473\pi$
$13$	20.6010	0.439515	0.219758	+	0.975555i	$0.429473\pi$
$14$	0	0
$15$	−42.9573	−0.739435
$16$	0	0
$17$	−7.09451	−0.101216	−0.0506080	−	0.998719i	$-0.516116\pi$
$17$	−7.09451	−0.101216	−0.0506080	+	0.998719i	$0.516116\pi$
$18$	0	0
$19$	155.809	1.88132	0.940660	−	0.339350i	$-0.110207\pi$
$19$	155.809	1.88132	0.940660	+	0.339350i	$0.110207\pi$
$20$	0	0
$21$	84.7127	0.880278
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−170.224	−1.21332
$28$	0	0
$29$	−213.296	−1.36580	−0.682899	−	0.730513i	$-0.739281\pi$
$29$	−213.296	−1.36580	−0.682899	+	0.730513i	$0.739281\pi$
$30$	0	0
$31$	95.7095	0.554514	0.277257	−	0.960796i	$-0.410575\pi$
$31$	95.7095	0.554514	0.277257	+	0.960796i	$0.410575\pi$
$32$	0	0
$33$	104.811	0.552888
$34$	0	0
$35$	−49.3006	−0.238095
$36$	0	0
$37$	174.131	0.773704	0.386852	−	0.922142i	$-0.373563\pi$
$37$	174.131	0.773704	0.386852	+	0.922142i	$0.373563\pi$
$38$	0	0
$39$	−176.993	−0.726706
$40$	0	0
$41$	122.549	0.466803	0.233402	−	0.972380i	$-0.425014\pi$
$41$	122.549	0.466803	0.233402	+	0.972380i	$0.425014\pi$
$42$	0	0
$43$	−398.567	−1.41351	−0.706755	−	0.707458i	$-0.749842\pi$
$43$	−398.567	−1.41351	−0.706755	+	0.707458i	$0.749842\pi$
$44$	0	0
$45$	234.066	0.775388
$46$	0	0
$47$	490.851	1.52336	0.761681	−	0.647952i	$-0.224374\pi$
$47$	490.851	1.52336	0.761681	+	0.647952i	$0.224374\pi$
$48$	0	0
$49$	−245.778	−0.716555
$50$	0	0
$51$	60.9522	0.167353
$52$	0	0
$53$	−161.490	−0.418535	−0.209268	−	0.977858i	$-0.567108\pi$
$53$	−161.490	−0.418535	−0.209268	+	0.977858i	$0.567108\pi$
$54$	0	0
$55$	−60.9974	−0.149543
$56$	0	0
$57$	−1338.63	−3.11063
$58$	0	0
$59$	560.053	1.23581	0.617904	−	0.786253i	$-0.287982\pi$
$59$	560.053	1.23581	0.617904	+	0.786253i	$0.287982\pi$
$60$	0	0
$61$	−321.451	−0.674714	−0.337357	−	0.941377i	$-0.609533\pi$
$61$	−321.451	−0.674714	−0.337357	+	0.941377i	$0.609533\pi$
$62$	0	0
$63$	−461.583	−0.923078
$64$	0	0
$65$	103.005	0.196557
$66$	0	0
$67$	−280.311	−0.511126	−0.255563	−	0.966792i	$-0.582261\pi$
$67$	−280.311	−0.511126	−0.255563	+	0.966792i	$0.582261\pi$
$68$	0	0
$69$	197.604	0.344763
$70$	0	0
$71$	−649.370	−1.08544	−0.542719	−	0.839914i	$-0.682605\pi$
$71$	−649.370	−1.08544	−0.542719	+	0.839914i	$0.682605\pi$
$72$	0	0
$73$	426.808	0.684303	0.342152	−	0.939645i	$-0.388844\pi$
$73$	426.808	0.684303	0.342152	+	0.939645i	$0.388844\pi$
$74$	0	0
$75$	−214.786	−0.330685
$76$	0	0
$77$	120.288	0.178028
$78$	0	0
$79$	161.318	0.229744	0.114872	−	0.993380i	$-0.463354\pi$
$79$	161.318	0.229744	0.114872	+	0.993380i	$0.463354\pi$
$80$	0	0
$81$	198.514	0.272310
$82$	0	0
$83$	−801.673	−1.06018	−0.530091	−	0.847941i	$-0.677842\pi$
$83$	−801.673	−1.06018	−0.530091	+	0.847941i	$0.677842\pi$
$84$	0	0
$85$	−35.4725	−0.0452651
$86$	0	0
$87$	1832.53	2.25825
$88$	0	0
$89$	1100.38	1.31056	0.655280	−	0.755386i	$-0.272551\pi$
$89$	1100.38	1.31056	0.655280	+	0.755386i	$0.272551\pi$
$90$	0	0
$91$	−203.128	−0.233996
$92$	0	0
$93$	−822.284	−0.916848
$94$	0	0
$95$	779.046	0.841352
$96$	0	0
$97$	−1179.42	−1.23455	−0.617276	−	0.786747i	$-0.711764\pi$
$97$	−1179.42	−1.23455	−0.617276	+	0.786747i	$0.711764\pi$
$98$	0	0
$99$	−571.096	−0.579771
$100$	0	0
$101$	1653.36	1.62887	0.814434	−	0.580256i	$-0.197047\pi$
$101$	1653.36	1.62887	0.814434	+	0.580256i	$0.197047\pi$
$102$	0	0
$103$	−1779.09	−1.70193	−0.850964	−	0.525224i	$-0.823982\pi$
$103$	−1779.09	−1.70193	−0.850964	+	0.525224i	$0.823982\pi$
$104$	0	0
$105$	423.564	0.393672
$106$	0	0
$107$	−1970.09	−1.77996	−0.889979	−	0.456001i	$-0.849281\pi$
$107$	−1970.09	−1.77996	−0.889979	+	0.456001i	$0.849281\pi$
$108$	0	0
$109$	564.158	0.495748	0.247874	−	0.968792i	$-0.420268\pi$
$109$	564.158	0.495748	0.247874	+	0.968792i	$0.420268\pi$
$110$	0	0
$111$	−1496.04	−1.27926
$112$	0	0
$113$	−1078.71	−0.898020	−0.449010	−	0.893527i	$-0.648223\pi$
$113$	−1078.71	−0.898020	−0.449010	+	0.893527i	$0.648223\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	964.398	0.762040
$118$	0	0
$119$	69.9526	0.0538870
$120$	0	0
$121$	−1182.17	−0.888184
$122$	0	0
$123$	−1052.87	−0.771825
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−26.7341	−0.0186793	−0.00933965	−	0.999956i	$-0.502973\pi$
$127$	−26.7341	−0.0186793	−0.00933965	+	0.999956i	$0.502973\pi$
$128$	0	0
$129$	3424.27	2.33714
$130$	0	0
$131$	367.682	0.245225	0.122613	−	0.992455i	$-0.460873\pi$
$131$	367.682	0.245225	0.122613	+	0.992455i	$0.460873\pi$
$132$	0	0
$133$	−1536.30	−1.00161
$134$	0	0
$135$	−851.118	−0.542612
$136$	0	0
$137$	1258.57	0.784867	0.392433	−	0.919780i	$-0.371633\pi$
$137$	1258.57	0.784867	0.392433	+	0.919780i	$0.371633\pi$
$138$	0	0
$139$	439.168	0.267984	0.133992	−	0.990982i	$-0.457220\pi$
$139$	439.168	0.267984	0.133992	+	0.990982i	$0.457220\pi$
$140$	0	0
$141$	−4217.13	−2.51877
$142$	0	0
$143$	−251.322	−0.146969
$144$	0	0
$145$	−1066.48	−0.610803
$146$	0	0
$147$	2111.59	1.18477
$148$	0	0
$149$	1929.98	1.06114	0.530571	−	0.847641i	$-0.321978\pi$
$149$	1929.98	1.06114	0.530571	+	0.847641i	$0.321978\pi$
$150$	0	0
$151$	−2151.53	−1.15953	−0.579766	−	0.814783i	$-0.696856\pi$
$151$	−2151.53	−1.15953	−0.579766	+	0.814783i	$0.696856\pi$
$152$	0	0
$153$	−332.116	−0.175490
$154$	0	0
$155$	478.547	0.247986
$156$	0	0
$157$	−414.312	−0.210609	−0.105305	−	0.994440i	$-0.533582\pi$
$157$	−414.312	−0.210609	−0.105305	+	0.994440i	$0.533582\pi$
$158$	0	0
$159$	1387.43	0.692017
$160$	0	0
$161$	226.783	0.111012
$162$	0	0
$163$	2973.76	1.42897	0.714487	−	0.699649i	$-0.246660\pi$
$163$	2973.76	1.42897	0.714487	+	0.699649i	$0.246660\pi$
$164$	0	0
$165$	524.057	0.247259
$166$	0	0
$167$	−926.002	−0.429079	−0.214539	−	0.976715i	$-0.568825\pi$
$167$	−926.002	−0.429079	−0.214539	+	0.976715i	$0.568825\pi$
$168$	0	0
$169$	−1772.60	−0.806826
$170$	0	0
$171$	7293.92	3.26187
$172$	0	0
$173$	1816.26	0.798194	0.399097	−	0.916909i	$-0.369324\pi$
$173$	1816.26	0.798194	0.399097	+	0.916909i	$0.369324\pi$
$174$	0	0
$175$	−246.503	−0.106479
$176$	0	0
$177$	−4811.67	−2.04332
$178$	0	0
$179$	4310.47	1.79989	0.899944	−	0.436006i	$-0.143608\pi$
$179$	4310.47	1.79989	0.899944	+	0.436006i	$0.143608\pi$
$180$	0	0
$181$	2906.21	1.19346	0.596731	−	0.802441i	$-0.296466\pi$
$181$	2906.21	1.19346	0.596731	+	0.802441i	$0.296466\pi$
$182$	0	0
$183$	2761.73	1.11559
$184$	0	0
$185$	870.657	0.346011
$186$	0	0
$187$	86.5493	0.0338455
$188$	0	0
$189$	1678.42	0.645965
$190$	0	0
$191$	317.405	0.120244	0.0601221	−	0.998191i	$-0.480851\pi$
$191$	317.405	0.120244	0.0601221	+	0.998191i	$0.480851\pi$
$192$	0	0
$193$	−1547.68	−0.577223	−0.288612	−	0.957446i	$-0.593194\pi$
$193$	−1547.68	−0.577223	−0.288612	+	0.957446i	$0.593194\pi$
$194$	0	0
$195$	−884.964	−0.324993
$196$	0	0
$197$	1699.95	0.614804	0.307402	−	0.951580i	$-0.400540\pi$
$197$	1699.95	0.614804	0.307402	+	0.951580i	$0.400540\pi$
$198$	0	0
$199$	1899.65	0.676696	0.338348	−	0.941021i	$-0.390132\pi$
$199$	1899.65	0.676696	0.338348	+	0.941021i	$0.390132\pi$
$200$	0	0
$201$	2408.28	0.845109
$202$	0	0
$203$	2103.13	0.727145
$204$	0	0
$205$	612.745	0.208761
$206$	0	0
$207$	−1076.70	−0.361526
$208$	0	0
$209$	−1900.79	−0.629094
$210$	0	0
$211$	3877.88	1.26523	0.632617	−	0.774465i	$-0.281981\pi$
$211$	3877.88	1.26523	0.632617	+	0.774465i	$0.281981\pi$
$212$	0	0
$213$	5579.04	1.79469
$214$	0	0
$215$	−1992.84	−0.632141
$216$	0	0
$217$	−943.706	−0.295221
$218$	0	0
$219$	−3666.91	−1.13145
$220$	0	0
$221$	−146.154	−0.0444859
$222$	0	0
$223$	3490.41	1.04814	0.524070	−	0.851675i	$-0.324413\pi$
$223$	3490.41	1.04814	0.524070	+	0.851675i	$0.324413\pi$
$224$	0	0
$225$	1170.33	0.346764
$226$	0	0
$227$	4860.89	1.42127	0.710635	−	0.703561i	$-0.248408\pi$
$227$	4860.89	1.42127	0.710635	+	0.703561i	$0.248408\pi$
$228$	0	0
$229$	6155.12	1.77616	0.888082	−	0.459684i	$-0.152037\pi$
$229$	6155.12	1.77616	0.888082	+	0.459684i	$0.152037\pi$
$230$	0	0
$231$	−1033.45	−0.294355
$232$	0	0
$233$	2507.33	0.704981	0.352491	−	0.935815i	$-0.385335\pi$
$233$	2507.33	0.704981	0.352491	+	0.935815i	$0.385335\pi$
$234$	0	0
$235$	2454.26	0.681268
$236$	0	0
$237$	−1385.96	−0.379864
$238$	0	0
$239$	−6037.58	−1.63405	−0.817026	−	0.576600i	$-0.804379\pi$
$239$	−6037.58	−1.63405	−0.817026	+	0.576600i	$0.804379\pi$
$240$	0	0
$241$	−2923.22	−0.781332	−0.390666	−	0.920532i	$-0.627755\pi$
$241$	−2923.22	−0.781332	−0.390666	+	0.920532i	$0.627755\pi$
$242$	0	0
$243$	2890.51	0.763071
$244$	0	0
$245$	−1228.89	−0.320453
$246$	0	0
$247$	3209.83	0.826869
$248$	0	0
$249$	6887.54	1.75293
$250$	0	0
$251$	−3053.30	−0.767820	−0.383910	−	0.923371i	$-0.625423\pi$
$251$	−3053.30	−0.767820	−0.383910	+	0.923371i	$0.625423\pi$
$252$	0	0
$253$	280.588	0.0697250
$254$	0	0
$255$	304.761	0.0748426
$256$	0	0
$257$	1538.07	0.373316	0.186658	−	0.982425i	$-0.440234\pi$
$257$	1538.07	0.373316	0.186658	+	0.982425i	$0.440234\pi$
$258$	0	0
$259$	−1716.96	−0.411917
$260$	0	0
$261$	−9985.07	−2.36805
$262$	0	0
$263$	171.263	0.0401542	0.0200771	−	0.999798i	$-0.493609\pi$
$263$	171.263	0.0401542	0.0200771	+	0.999798i	$0.493609\pi$
$264$	0	0
$265$	−807.450	−0.187175
$266$	0	0
$267$	−9453.85	−2.16692
$268$	0	0
$269$	−423.515	−0.0959932	−0.0479966	−	0.998847i	$-0.515284\pi$
$269$	−423.515	−0.0959932	−0.0479966	+	0.998847i	$0.515284\pi$
$270$	0	0
$271$	6100.55	1.36746	0.683731	−	0.729734i	$-0.260356\pi$
$271$	6100.55	1.36746	0.683731	+	0.729734i	$0.260356\pi$
$272$	0	0
$273$	1745.17	0.386895
$274$	0	0
$275$	−304.987	−0.0668779
$276$	0	0
$277$	−3334.85	−0.723365	−0.361682	−	0.932301i	$-0.617797\pi$
$277$	−3334.85	−0.723365	−0.361682	+	0.932301i	$0.617797\pi$
$278$	0	0
$279$	4480.46	0.961427
$280$	0	0
$281$	6431.68	1.36542	0.682708	−	0.730691i	$-0.260802\pi$
$281$	6431.68	1.36542	0.682708	+	0.730691i	$0.260802\pi$
$282$	0	0
$283$	4338.70	0.911339	0.455670	−	0.890149i	$-0.349400\pi$
$283$	4338.70	0.911339	0.455670	+	0.890149i	$0.349400\pi$
$284$	0	0
$285$	−6693.14	−1.39111
$286$	0	0
$287$	−1208.35	−0.248524
$288$	0	0
$289$	−4862.67	−0.989755
$290$	0	0
$291$	10132.9	2.04124
$292$	0	0
$293$	−8845.77	−1.76374	−0.881870	−	0.471493i	$-0.843715\pi$
$293$	−8845.77	−1.76374	−0.881870	+	0.471493i	$0.843715\pi$
$294$	0	0
$295$	2800.26	0.552670
$296$	0	0
$297$	2076.64	0.405720
$298$	0	0
$299$	−473.824	−0.0916452
$300$	0	0
$301$	3929.92	0.752547
$302$	0	0
$303$	−14204.8	−2.69321
$304$	0	0
$305$	−1607.25	−0.301741
$306$	0	0
$307$	7586.73	1.41042	0.705208	−	0.709000i	$-0.250854\pi$
$307$	7586.73	1.41042	0.705208	+	0.709000i	$0.250854\pi$
$308$	0	0
$309$	15284.9	2.81401
$310$	0	0
$311$	1856.72	0.338536	0.169268	−	0.985570i	$-0.445860\pi$
$311$	1856.72	0.338536	0.169268	+	0.985570i	$0.445860\pi$
$312$	0	0
$313$	−519.905	−0.0938875	−0.0469437	−	0.998898i	$-0.514948\pi$
$313$	−519.905	−0.0938875	−0.0469437	+	0.998898i	$0.514948\pi$
$314$	0	0
$315$	−2307.91	−0.412813
$316$	0	0
$317$	8088.54	1.43312	0.716558	−	0.697528i	$-0.245717\pi$
$317$	8088.54	1.43312	0.716558	+	0.697528i	$0.245717\pi$
$318$	0	0
$319$	2602.11	0.456708
$320$	0	0
$321$	16925.9	2.94303
$322$	0	0
$323$	−1105.39	−0.190420
$324$	0	0
$325$	515.026	0.0879030
$326$	0	0
$327$	−4846.94	−0.819684
$328$	0	0
$329$	−4839.85	−0.811032
$330$	0	0
$331$	10581.9	1.75720	0.878598	−	0.477562i	$-0.158479\pi$
$331$	10581.9	1.75720	0.878598	+	0.477562i	$0.158479\pi$
$332$	0	0
$333$	8151.64	1.34146
$334$	0	0
$335$	−1401.55	−0.228582
$336$	0	0
$337$	3054.24	0.493694	0.246847	−	0.969054i	$-0.420606\pi$
$337$	3054.24	0.493694	0.246847	+	0.969054i	$0.420606\pi$
$338$	0	0
$339$	9267.67	1.48481
$340$	0	0
$341$	−1167.61	−0.185424
$342$	0	0
$343$	5805.42	0.913887
$344$	0	0
$345$	988.018	0.154183
$346$	0	0
$347$	−3307.98	−0.511763	−0.255881	−	0.966708i	$-0.582366\pi$
$347$	−3307.98	−0.511763	−0.255881	+	0.966708i	$0.582366\pi$
$348$	0	0
$349$	9721.77	1.49110	0.745551	−	0.666449i	$-0.232187\pi$
$349$	9721.77	1.49110	0.745551	+	0.666449i	$0.232187\pi$
$350$	0	0
$351$	−3506.78	−0.533271
$352$	0	0
$353$	4071.86	0.613947	0.306974	−	0.951718i	$-0.400684\pi$
$353$	4071.86	0.613947	0.306974	+	0.951718i	$0.400684\pi$
$354$	0	0
$355$	−3246.85	−0.485423
$356$	0	0
$357$	−600.995	−0.0890981
$358$	0	0
$359$	−2110.15	−0.310221	−0.155110	−	0.987897i	$-0.549573\pi$
$359$	−2110.15	−0.310221	−0.155110	+	0.987897i	$0.549573\pi$
$360$	0	0
$361$	17417.5	2.53937
$362$	0	0
$363$	10156.6	1.46855
$364$	0	0
$365$	2134.04	0.306030
$366$	0	0
$367$	−7472.02	−1.06277	−0.531385	−	0.847131i	$-0.678328\pi$
$367$	−7472.02	−1.06277	−0.531385	+	0.847131i	$0.678328\pi$
$368$	0	0
$369$	5736.90	0.809352
$370$	0	0
$371$	1592.31	0.222826
$372$	0	0
$373$	−7814.69	−1.08480	−0.542398	−	0.840121i	$-0.682484\pi$
$373$	−7814.69	−1.08480	−0.542398	+	0.840121i	$0.682484\pi$
$374$	0	0
$375$	−1073.93	−0.147887
$376$	0	0
$377$	−4394.12	−0.600289
$378$	0	0
$379$	1340.12	0.181629	0.0908145	−	0.995868i	$-0.471053\pi$
$379$	1340.12	0.181629	0.0908145	+	0.995868i	$0.471053\pi$
$380$	0	0
$381$	229.685	0.0308848
$382$	0	0
$383$	11715.8	1.56306	0.781530	−	0.623868i	$-0.214440\pi$
$383$	11715.8	1.56306	0.781530	+	0.623868i	$0.214440\pi$
$384$	0	0
$385$	601.441	0.0796163
$386$	0	0
$387$	−18658.2	−2.45077
$388$	0	0
$389$	−9375.49	−1.22200	−0.610998	−	0.791632i	$-0.709232\pi$
$389$	−9375.49	−1.22200	−0.610998	+	0.791632i	$0.709232\pi$
$390$	0	0
$391$	163.174	0.0211050
$392$	0	0
$393$	−3158.92	−0.405462
$394$	0	0
$395$	806.592	0.102744
$396$	0	0
$397$	−5651.29	−0.714433	−0.357217	−	0.934022i	$-0.616274\pi$
$397$	−5651.29	−0.714433	−0.357217	+	0.934022i	$0.616274\pi$
$398$	0	0
$399$	13199.0	1.65608
$400$	0	0
$401$	−2429.11	−0.302504	−0.151252	−	0.988495i	$-0.548331\pi$
$401$	−2429.11	−0.302504	−0.151252	+	0.988495i	$0.548331\pi$
$402$	0	0
$403$	1971.71	0.243717
$404$	0	0
$405$	992.572	0.121781
$406$	0	0
$407$	−2124.31	−0.258718
$408$	0	0
$409$	9604.87	1.16120	0.580599	−	0.814189i	$-0.302818\pi$
$409$	9604.87	1.16120	0.580599	+	0.814189i	$0.302818\pi$
$410$	0	0
$411$	−10812.9	−1.29772
$412$	0	0
$413$	−5522.18	−0.657939
$414$	0	0
$415$	−4008.37	−0.474128
$416$	0	0
$417$	−3773.09	−0.443092
$418$	0	0
$419$	131.430	0.0153240	0.00766201	−	0.999971i	$-0.497561\pi$
$419$	131.430	0.0153240	0.00766201	+	0.999971i	$0.497561\pi$
$420$	0	0
$421$	−1188.69	−0.137608	−0.0688041	−	0.997630i	$-0.521918\pi$
$421$	−1188.69	−0.137608	−0.0688041	+	0.997630i	$0.521918\pi$
$422$	0	0
$423$	22978.3	2.64124
$424$	0	0
$425$	−177.363	−0.0202432
$426$	0	0
$427$	3169.54	0.359215
$428$	0	0
$429$	2159.22	0.243003
$430$	0	0
$431$	−6506.68	−0.727183	−0.363591	−	0.931559i	$-0.618450\pi$
$431$	−6506.68	−0.727183	−0.363591	+	0.931559i	$0.618450\pi$
$432$	0	0
$433$	−419.537	−0.0465627	−0.0232814	−	0.999729i	$-0.507411\pi$
$433$	−419.537	−0.0465627	−0.0232814	+	0.999729i	$0.507411\pi$
$434$	0	0
$435$	9162.63	1.00992
$436$	0	0
$437$	−3583.61	−0.392282
$438$	0	0
$439$	−17604.6	−1.91394	−0.956970	−	0.290186i	$-0.906283\pi$
$439$	−17604.6	−1.91394	−0.956970	+	0.290186i	$0.906283\pi$
$440$	0	0
$441$	−11505.6	−1.24238
$442$	0	0
$443$	7269.73	0.779674	0.389837	−	0.920884i	$-0.372531\pi$
$443$	7269.73	0.779674	0.389837	+	0.920884i	$0.372531\pi$
$444$	0	0
$445$	5501.89	0.586101
$446$	0	0
$447$	−16581.3	−1.75452
$448$	0	0
$449$	12466.8	1.31034	0.655171	−	0.755481i	$-0.272597\pi$
$449$	12466.8	1.31034	0.655171	+	0.755481i	$0.272597\pi$
$450$	0	0
$451$	−1495.03	−0.156094
$452$	0	0
$453$	18484.8	1.91720
$454$	0	0
$455$	−1015.64	−0.104646
$456$	0	0
$457$	−12746.8	−1.30475	−0.652374	−	0.757897i	$-0.726227\pi$
$457$	−12746.8	−1.30475	−0.652374	+	0.757897i	$0.726227\pi$
$458$	0	0
$459$	1207.65	0.122807
$460$	0	0
$461$	6143.85	0.620711	0.310355	−	0.950621i	$-0.399552\pi$
$461$	6143.85	0.620711	0.310355	+	0.950621i	$0.399552\pi$
$462$	0	0
$463$	−13870.3	−1.39224	−0.696119	−	0.717926i	$-0.745092\pi$
$463$	−13870.3	−1.39224	−0.696119	+	0.717926i	$0.745092\pi$
$464$	0	0
$465$	−4111.42	−0.410027
$466$	0	0
$467$	−6191.68	−0.613526	−0.306763	−	0.951786i	$-0.599246\pi$
$467$	−6191.68	−0.613526	−0.306763	+	0.951786i	$0.599246\pi$
$468$	0	0
$469$	2763.90	0.272121
$470$	0	0
$471$	3559.54	0.348227
$472$	0	0
$473$	4862.32	0.472663
$474$	0	0
$475$	3895.23	0.376264
$476$	0	0
$477$	−7559.85	−0.725664
$478$	0	0
$479$	16327.9	1.55749	0.778747	−	0.627338i	$-0.215856\pi$
$479$	16327.9	1.55749	0.778747	+	0.627338i	$0.215856\pi$
$480$	0	0
$481$	3587.29	0.340054
$482$	0	0
$483$	−1948.39	−0.183551
$484$	0	0
$485$	−5897.08	−0.552108
$486$	0	0
$487$	−12533.2	−1.16619	−0.583096	−	0.812404i	$-0.698159\pi$
$487$	−12533.2	−1.16619	−0.583096	+	0.812404i	$0.698159\pi$
$488$	0	0
$489$	−25548.9	−2.36270
$490$	0	0
$491$	−5589.44	−0.513743	−0.256871	−	0.966446i	$-0.582692\pi$
$491$	−5589.44	−0.513743	−0.256871	+	0.966446i	$0.582692\pi$
$492$	0	0
$493$	1513.23	0.138241
$494$	0	0
$495$	−2855.48	−0.259281
$496$	0	0
$497$	6402.86	0.577883
$498$	0	0
$499$	−11815.3	−1.05997	−0.529985	−	0.848007i	$-0.677803\pi$
$499$	−11815.3	−1.05997	−0.529985	+	0.848007i	$0.677803\pi$
$500$	0	0
$501$	7955.71	0.709450
$502$	0	0
$503$	17154.1	1.52061	0.760303	−	0.649568i	$-0.225050\pi$
$503$	17154.1	1.52061	0.760303	+	0.649568i	$0.225050\pi$
$504$	0	0
$505$	8266.81	0.728452
$506$	0	0
$507$	15229.2	1.33403
$508$	0	0
$509$	−4104.09	−0.357388	−0.178694	−	0.983905i	$-0.557187\pi$
$509$	−4104.09	−0.357388	−0.178694	+	0.983905i	$0.557187\pi$
$510$	0	0
$511$	−4208.38	−0.364320
$512$	0	0
$513$	−26522.4	−2.28264
$514$	0	0
$515$	−8895.43	−0.761126
$516$	0	0
$517$	−5988.13	−0.509396
$518$	0	0
$519$	−15604.3	−1.31975
$520$	0	0
$521$	−3884.54	−0.326650	−0.163325	−	0.986572i	$-0.552222\pi$
$521$	−3884.54	−0.326650	−0.163325	+	0.986572i	$0.552222\pi$
$522$	0	0
$523$	19826.1	1.65762	0.828808	−	0.559533i	$-0.189019\pi$
$523$	19826.1	1.65762	0.828808	+	0.559533i	$0.189019\pi$
$524$	0	0
$525$	2117.82	0.176056
$526$	0	0
$527$	−679.012	−0.0561256
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	26217.8	2.14267
$532$	0	0
$533$	2524.63	0.205167
$534$	0	0
$535$	−9850.44	−0.796021
$536$	0	0
$537$	−37033.2	−2.97598
$538$	0	0
$539$	2998.37	0.239608
$540$	0	0
$541$	−8868.82	−0.704806	−0.352403	−	0.935848i	$-0.614635\pi$
$541$	−8868.82	−0.704806	−0.352403	+	0.935848i	$0.614635\pi$
$542$	0	0
$543$	−24968.5	−1.97330
$544$	0	0
$545$	2820.79	0.221705
$546$	0	0
$547$	18998.3	1.48503	0.742513	−	0.669832i	$-0.233634\pi$
$547$	18998.3	1.48503	0.742513	+	0.669832i	$0.233634\pi$
$548$	0	0
$549$	−15048.1	−1.16983
$550$	0	0
$551$	−33233.5	−2.56950
$552$	0	0
$553$	−1590.62	−0.122315
$554$	0	0
$555$	−7480.21	−0.572103
$556$	0	0
$557$	14212.2	1.08113	0.540565	−	0.841302i	$-0.318211\pi$
$557$	14212.2	1.08113	0.540565	+	0.841302i	$0.318211\pi$
$558$	0	0
$559$	−8210.90	−0.621259
$560$	0	0
$561$	−743.585	−0.0559611
$562$	0	0
$563$	−15281.3	−1.14392	−0.571961	−	0.820281i	$-0.693817\pi$
$563$	−15281.3	−1.14392	−0.571961	+	0.820281i	$0.693817\pi$
$564$	0	0
$565$	−5393.54	−0.401607
$566$	0	0
$567$	−1957.37	−0.144977
$568$	0	0
$569$	10332.6	0.761275	0.380638	−	0.924724i	$-0.375704\pi$
$569$	10332.6	0.761275	0.380638	+	0.924724i	$0.375704\pi$
$570$	0	0
$571$	−3715.85	−0.272335	−0.136168	−	0.990686i	$-0.543479\pi$
$571$	−3715.85	−0.272335	−0.136168	+	0.990686i	$0.543479\pi$
$572$	0	0
$573$	−2726.97	−0.198815
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−25130.6	−1.81318	−0.906588	−	0.422017i	$-0.861322\pi$
$577$	−25130.6	−1.81318	−0.906588	+	0.422017i	$0.861322\pi$
$578$	0	0
$579$	13296.8	0.954397
$580$	0	0
$581$	7904.59	0.564437
$582$	0	0
$583$	1970.10	0.139954
$584$	0	0
$585$	4821.99	0.340795
$586$	0	0
$587$	−2519.27	−0.177140	−0.0885701	−	0.996070i	$-0.528230\pi$
$587$	−2519.27	−0.177140	−0.0885701	+	0.996070i	$0.528230\pi$
$588$	0	0
$589$	14912.4	1.04322
$590$	0	0
$591$	−14605.0	−1.01653
$592$	0	0
$593$	13182.7	0.912896	0.456448	−	0.889750i	$-0.349122\pi$
$593$	13182.7	0.912896	0.456448	+	0.889750i	$0.349122\pi$
$594$	0	0
$595$	349.763	0.0240990
$596$	0	0
$597$	−16320.8	−1.11887
$598$	0	0
$599$	11988.7	0.817775	0.408887	−	0.912585i	$-0.365917\pi$
$599$	11988.7	0.817775	0.408887	+	0.912585i	$0.365917\pi$
$600$	0	0
$601$	5851.85	0.397174	0.198587	−	0.980083i	$-0.436365\pi$
$601$	5851.85	0.397174	0.198587	+	0.980083i	$0.436365\pi$
$602$	0	0
$603$	−13122.2	−0.886199
$604$	0	0
$605$	−5910.86	−0.397208
$606$	0	0
$607$	−12145.1	−0.812114	−0.406057	−	0.913848i	$-0.633097\pi$
$607$	−12145.1	−0.812114	−0.406057	+	0.913848i	$0.633097\pi$
$608$	0	0
$609$	−18068.9	−1.20228
$610$	0	0
$611$	10112.0	0.669541
$612$	0	0
$613$	23776.8	1.56662	0.783308	−	0.621634i	$-0.213531\pi$
$613$	23776.8	1.56662	0.783308	+	0.621634i	$0.213531\pi$
$614$	0	0
$615$	−5264.37	−0.345170
$616$	0	0
$617$	−26599.7	−1.73559	−0.867797	−	0.496918i	$-0.834465\pi$
$617$	−26599.7	−1.73559	−0.867797	+	0.496918i	$0.834465\pi$
$618$	0	0
$619$	−14977.0	−0.972498	−0.486249	−	0.873820i	$-0.661635\pi$
$619$	−14977.0	−0.972498	−0.486249	+	0.873820i	$0.661635\pi$
$620$	0	0
$621$	3915.14	0.252994
$622$	0	0
$623$	−10849.9	−0.697737
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	16330.6	1.04016
$628$	0	0
$629$	−1235.38	−0.0783111
$630$	0	0
$631$	31325.1	1.97628	0.988140	−	0.153556i	$-0.0490724\pi$
$631$	31325.1	1.97628	0.988140	+	0.153556i	$0.0490724\pi$
$632$	0	0
$633$	−33316.7	−2.09197
$634$	0	0
$635$	−133.671	−0.00835363
$636$	0	0
$637$	−5063.28	−0.314937
$638$	0	0
$639$	−30399.1	−1.88195
$640$	0	0
$641$	−2027.16	−0.124911	−0.0624554	−	0.998048i	$-0.519893\pi$
$641$	−2027.16	−0.124911	−0.0624554	+	0.998048i	$0.519893\pi$
$642$	0	0
$643$	−13089.0	−0.802767	−0.401383	−	0.915910i	$-0.631470\pi$
$643$	−13089.0	−0.802767	−0.401383	+	0.915910i	$0.631470\pi$
$644$	0	0
$645$	17121.4	1.04520
$646$	0	0
$647$	−355.998	−0.0216317	−0.0108159	−	0.999942i	$-0.503443\pi$
$647$	−355.998	−0.0216317	−0.0108159	+	0.999942i	$0.503443\pi$
$648$	0	0
$649$	−6832.36	−0.413241
$650$	0	0
$651$	8107.81	0.488126
$652$	0	0
$653$	23689.6	1.41967	0.709836	−	0.704367i	$-0.248769\pi$
$653$	23689.6	1.41967	0.709836	+	0.704367i	$0.248769\pi$
$654$	0	0
$655$	1838.41	0.109668
$656$	0	0
$657$	19980.2	1.18646
$658$	0	0
$659$	−8056.35	−0.476223	−0.238111	−	0.971238i	$-0.576528\pi$
$659$	−8056.35	−0.476223	−0.238111	+	0.971238i	$0.576528\pi$
$660$	0	0
$661$	−10758.0	−0.633040	−0.316520	−	0.948586i	$-0.602514\pi$
$661$	−10758.0	−0.633040	−0.316520	+	0.948586i	$0.602514\pi$
$662$	0	0
$663$	1255.68	0.0735542
$664$	0	0
$665$	−7681.48	−0.447932
$666$	0	0
$667$	4905.82	0.284789
$668$	0	0
$669$	−29987.7	−1.73302
$670$	0	0
$671$	3921.53	0.225617
$672$	0	0
$673$	8247.58	0.472393	0.236197	−	0.971705i	$-0.424099\pi$
$673$	8247.58	0.472393	0.236197	+	0.971705i	$0.424099\pi$
$674$	0	0
$675$	−4255.59	−0.242663
$676$	0	0
$677$	6788.40	0.385376	0.192688	−	0.981260i	$-0.438279\pi$
$677$	6788.40	0.385376	0.192688	+	0.981260i	$0.438279\pi$
$678$	0	0
$679$	11629.2	0.657270
$680$	0	0
$681$	−41762.1	−2.34997
$682$	0	0
$683$	−28268.2	−1.58368	−0.791838	−	0.610731i	$-0.790876\pi$
$683$	−28268.2	−1.58368	−0.791838	+	0.610731i	$0.790876\pi$
$684$	0	0
$685$	6292.84	0.351003
$686$	0	0
$687$	−52881.4	−2.93676
$688$	0	0
$689$	−3326.86	−0.183953
$690$	0	0
$691$	25464.6	1.40191	0.700953	−	0.713207i	$-0.252758\pi$
$691$	25464.6	1.40191	0.700953	+	0.713207i	$0.252758\pi$
$692$	0	0
$693$	5631.07	0.308668
$694$	0	0
$695$	2195.84	0.119846
$696$	0	0
$697$	−869.424	−0.0472479
$698$	0	0
$699$	−21541.6	−1.16563
$700$	0	0
$701$	−4502.11	−0.242571	−0.121285	−	0.992618i	$-0.538702\pi$
$701$	−4502.11	−0.242571	−0.121285	+	0.992618i	$0.538702\pi$
$702$	0	0
$703$	27131.3	1.45558
$704$	0	0
$705$	−21085.6	−1.12643
$706$	0	0
$707$	−16302.3	−0.867203
$708$	0	0
$709$	−470.632	−0.0249294	−0.0124647	−	0.999922i	$-0.503968\pi$
$709$	−470.632	−0.0249294	−0.0124647	+	0.999922i	$0.503968\pi$
$710$	0	0
$711$	7551.82	0.398334
$712$	0	0
$713$	−2201.32	−0.115624
$714$	0	0
$715$	−1256.61	−0.0657266
$716$	0	0
$717$	51871.6	2.70179
$718$	0	0
$719$	27091.4	1.40520	0.702600	−	0.711585i	$-0.252022\pi$
$719$	27091.4	1.40520	0.702600	+	0.711585i	$0.252022\pi$
$720$	0	0
$721$	17542.0	0.906100
$722$	0	0
$723$	25114.7	1.29188
$724$	0	0
$725$	−5332.41	−0.273160
$726$	0	0
$727$	−7654.83	−0.390512	−0.195256	−	0.980752i	$-0.562554\pi$
$727$	−7654.83	−0.390512	−0.195256	+	0.980752i	$0.562554\pi$
$728$	0	0
$729$	−30193.6	−1.53399
$730$	0	0
$731$	2827.64	0.143070
$732$	0	0
$733$	−9163.05	−0.461726	−0.230863	−	0.972986i	$-0.574155\pi$
$733$	−9163.05	−0.461726	−0.230863	+	0.972986i	$0.574155\pi$
$734$	0	0
$735$	10558.0	0.529845
$736$	0	0
$737$	3419.65	0.170915
$738$	0	0
$739$	24532.2	1.22115	0.610576	−	0.791958i	$-0.290938\pi$
$739$	24532.2	1.22115	0.610576	+	0.791958i	$0.290938\pi$
$740$	0	0
$741$	−27577.1	−1.36717
$742$	0	0
$743$	2482.64	0.122583	0.0612915	−	0.998120i	$-0.480478\pi$
$743$	2482.64	0.122583	0.0612915	+	0.998120i	$0.480478\pi$
$744$	0	0
$745$	9649.90	0.474557
$746$	0	0
$747$	−37528.8	−1.83816
$748$	0	0
$749$	19425.3	0.947643
$750$	0	0
$751$	22441.2	1.09040	0.545200	−	0.838306i	$-0.316454\pi$
$751$	22441.2	1.09040	0.545200	+	0.838306i	$0.316454\pi$
$752$	0	0
$753$	26232.3	1.26953
$754$	0	0
$755$	−10757.7	−0.518558
$756$	0	0
$757$	−1552.93	−0.0745603	−0.0372802	−	0.999305i	$-0.511869\pi$
$757$	−1552.93	−0.0745603	−0.0372802	+	0.999305i	$0.511869\pi$
$758$	0	0
$759$	−2410.66	−0.115285
$760$	0	0
$761$	−37793.9	−1.80030	−0.900150	−	0.435579i	$-0.856544\pi$
$761$	−37793.9	−1.80030	−0.900150	+	0.435579i	$0.856544\pi$
$762$	0	0
$763$	−5562.66	−0.263934
$764$	0	0
$765$	−1660.58	−0.0784816
$766$	0	0
$767$	11537.7	0.543156
$768$	0	0
$769$	3268.52	0.153272	0.0766358	−	0.997059i	$-0.475582\pi$
$769$	3268.52	0.153272	0.0766358	+	0.997059i	$0.475582\pi$
$770$	0	0
$771$	−13214.3	−0.617251
$772$	0	0
$773$	26314.1	1.22439	0.612194	−	0.790708i	$-0.290287\pi$
$773$	26314.1	1.22439	0.612194	+	0.790708i	$0.290287\pi$
$774$	0	0
$775$	2392.74	0.110903
$776$	0	0
$777$	14751.1	0.681074
$778$	0	0
$779$	19094.3	0.878206
$780$	0	0
$781$	7921.98	0.362959
$782$	0	0
$783$	36308.1	1.65715
$784$	0	0
$785$	−2071.56	−0.0941874
$786$	0	0
$787$	−20651.1	−0.935366	−0.467683	−	0.883896i	$-0.654911\pi$
$787$	−20651.1	−0.935366	−0.467683	+	0.883896i	$0.654911\pi$
$788$	0	0
$789$	−1471.40	−0.0663920
$790$	0	0
$791$	10636.2	0.478102
$792$	0	0
$793$	−6622.21	−0.296547
$794$	0	0
$795$	6937.17	0.309479
$796$	0	0
$797$	23478.7	1.04349	0.521743	−	0.853103i	$-0.325282\pi$
$797$	23478.7	1.04349	0.521743	+	0.853103i	$0.325282\pi$
$798$	0	0
$799$	−3482.35	−0.154189
$800$	0	0
$801$	51512.2	2.27228
$802$	0	0
$803$	−5206.84	−0.228824
$804$	0	0
$805$	1133.91	0.0496462
$806$	0	0
$807$	3638.61	0.158718
$808$	0	0
$809$	30473.5	1.32434	0.662170	−	0.749354i	$-0.269636\pi$
$809$	30473.5	1.32434	0.662170	+	0.749354i	$0.269636\pi$
$810$	0	0
$811$	−4113.26	−0.178096	−0.0890482	−	0.996027i	$-0.528383\pi$
$811$	−4113.26	−0.178096	−0.0890482	+	0.996027i	$0.528383\pi$
$812$	0	0
$813$	−52412.6	−2.26100
$814$	0	0
$815$	14868.8	0.639056
$816$	0	0
$817$	−62100.5	−2.65927
$818$	0	0
$819$	−9509.07	−0.405707
$820$	0	0
$821$	41965.9	1.78395	0.891973	−	0.452089i	$-0.149321\pi$
$821$	41965.9	1.78395	0.891973	+	0.452089i	$0.149321\pi$
$822$	0	0
$823$	−10196.4	−0.431863	−0.215931	−	0.976409i	$-0.569279\pi$
$823$	−10196.4	−0.431863	−0.215931	+	0.976409i	$0.569279\pi$
$824$	0	0
$825$	2620.28	0.110578
$826$	0	0
$827$	6996.77	0.294198	0.147099	−	0.989122i	$-0.453006\pi$
$827$	6996.77	0.294198	0.147099	+	0.989122i	$0.453006\pi$
$828$	0	0
$829$	18040.5	0.755819	0.377909	−	0.925843i	$-0.376643\pi$
$829$	18040.5	0.755819	0.377909	+	0.925843i	$0.376643\pi$
$830$	0	0
$831$	28651.3	1.19603
$832$	0	0
$833$	1743.68	0.0725267
$834$	0	0
$835$	−4630.01	−0.191890
$836$	0	0
$837$	−16292.0	−0.672801
$838$	0	0
$839$	−2983.92	−0.122785	−0.0613924	−	0.998114i	$-0.519554\pi$
$839$	−2983.92	−0.122785	−0.0613924	+	0.998114i	$0.519554\pi$
$840$	0	0
$841$	21106.3	0.865404
$842$	0	0
$843$	−55257.5	−2.25761
$844$	0	0
$845$	−8862.99	−0.360824
$846$	0	0
$847$	11656.4	0.472865
$848$	0	0
$849$	−37275.8	−1.50683
$850$	0	0
$851$	−4005.02	−0.161328
$852$	0	0
$853$	18524.2	0.743561	0.371780	−	0.928321i	$-0.378747\pi$
$853$	18524.2	0.743561	0.371780	+	0.928321i	$0.378747\pi$
$854$	0	0
$855$	36469.6	1.45875
$856$	0	0
$857$	−25430.0	−1.01362	−0.506811	−	0.862057i	$-0.669176\pi$
$857$	−25430.0	−1.01362	−0.506811	+	0.862057i	$0.669176\pi$
$858$	0	0
$859$	25755.3	1.02300	0.511502	−	0.859282i	$-0.329089\pi$
$859$	25755.3	1.02300	0.511502	+	0.859282i	$0.329089\pi$
$860$	0	0
$861$	10381.5	0.410916
$862$	0	0
$863$	38221.5	1.50762	0.753809	−	0.657094i	$-0.228214\pi$
$863$	38221.5	1.50762	0.753809	+	0.657094i	$0.228214\pi$
$864$	0	0
$865$	9081.29	0.356963
$866$	0	0
$867$	41777.4	1.63649
$868$	0	0
$869$	−1968.00	−0.0768238
$870$	0	0
$871$	−5774.69	−0.224647
$872$	0	0
$873$	−55212.1	−2.14049
$874$	0	0
$875$	−1232.51	−0.0476189
$876$	0	0
$877$	−6231.05	−0.239917	−0.119959	−	0.992779i	$-0.538276\pi$
$877$	−6231.05	−0.239917	−0.119959	+	0.992779i	$0.538276\pi$
$878$	0	0
$879$	75998.1	2.91621
$880$	0	0
$881$	32067.0	1.22629	0.613147	−	0.789969i	$-0.289903\pi$
$881$	32067.0	1.22629	0.613147	+	0.789969i	$0.289903\pi$
$882$	0	0
$883$	5522.62	0.210477	0.105238	−	0.994447i	$-0.466439\pi$
$883$	5522.62	0.210477	0.105238	+	0.994447i	$0.466439\pi$
$884$	0	0
$885$	−24058.4	−0.913800
$886$	0	0
$887$	24136.9	0.913685	0.456843	−	0.889548i	$-0.348980\pi$
$887$	24136.9	0.913685	0.456843	+	0.889548i	$0.348980\pi$
$888$	0	0
$889$	263.601	0.00994478
$890$	0	0
$891$	−2421.77	−0.0910577
$892$	0	0
$893$	76479.2	2.86593
$894$	0	0
$895$	21552.4	0.804934
$896$	0	0
$897$	4070.83	0.151529
$898$	0	0
$899$	−20414.5	−0.757354
$900$	0	0
$901$	1145.69	0.0423624
$902$	0	0
$903$	−33763.7	−1.24428
$904$	0	0
$905$	14531.0	0.533732
$906$	0	0
$907$	−47038.5	−1.72204	−0.861020	−	0.508572i	$-0.830174\pi$
$907$	−47038.5	−1.72204	−0.861020	+	0.508572i	$0.830174\pi$
$908$	0	0
$909$	77399.1	2.82416
$910$	0	0
$911$	−7552.66	−0.274677	−0.137339	−	0.990524i	$-0.543855\pi$
$911$	−7552.66	−0.274677	−0.137339	+	0.990524i	$0.543855\pi$
$912$	0	0
$913$	9780.00	0.354514
$914$	0	0
$915$	13808.6	0.498907
$916$	0	0
$917$	−3625.38	−0.130557
$918$	0	0
$919$	−11248.9	−0.403774	−0.201887	−	0.979409i	$-0.564707\pi$
$919$	−11248.9	−0.403774	−0.201887	+	0.979409i	$0.564707\pi$
$920$	0	0
$921$	−65181.1	−2.33202
$922$	0	0
$923$	−13377.7	−0.477066
$924$	0	0
$925$	4353.29	0.154741
$926$	0	0
$927$	−83284.6	−2.95084
$928$	0	0
$929$	16685.8	0.589281	0.294641	−	0.955608i	$-0.404800\pi$
$929$	16685.8	0.589281	0.294641	+	0.955608i	$0.404800\pi$
$930$	0	0
$931$	−38294.5	−1.34807
$932$	0	0
$933$	−15951.9	−0.559745
$934$	0	0
$935$	432.747	0.0151362
$936$	0	0
$937$	−20880.0	−0.727984	−0.363992	−	0.931402i	$-0.618586\pi$
$937$	−20880.0	−0.727984	−0.363992	+	0.931402i	$0.618586\pi$
$938$	0	0
$939$	4466.74	0.155236
$940$	0	0
$941$	14017.4	0.485606	0.242803	−	0.970076i	$-0.421933\pi$
$941$	14017.4	0.485606	0.242803	+	0.970076i	$0.421933\pi$
$942$	0	0
$943$	−2818.63	−0.0973352
$944$	0	0
$945$	8392.12	0.288884
$946$	0	0
$947$	16036.4	0.550278	0.275139	−	0.961405i	$-0.411276\pi$
$947$	16036.4	0.550278	0.275139	+	0.961405i	$0.411276\pi$
$948$	0	0
$949$	8792.69	0.300762
$950$	0	0
$951$	−69492.3	−2.36955
$952$	0	0
$953$	−4657.29	−0.158305	−0.0791523	−	0.996863i	$-0.525221\pi$
$953$	−4657.29	−0.158305	−0.0791523	+	0.996863i	$0.525221\pi$
$954$	0	0
$955$	1587.03	0.0537748
$956$	0	0
$957$	−22355.9	−0.755134
$958$	0	0
$959$	−12409.6	−0.417860
$960$	0	0
$961$	−20630.7	−0.692514
$962$	0	0
$963$	−92225.9	−3.08613
$964$	0	0
$965$	−7738.38	−0.258142
$966$	0	0
$967$	20869.1	0.694007	0.347004	−	0.937864i	$-0.387199\pi$
$967$	20869.1	0.694007	0.347004	+	0.937864i	$0.387199\pi$
$968$	0	0
$969$	9496.91	0.314845
$970$	0	0
$971$	56406.4	1.86423	0.932114	−	0.362164i	$-0.117962\pi$
$971$	56406.4	1.86423	0.932114	+	0.362164i	$0.117962\pi$
$972$	0	0
$973$	−4330.25	−0.142674
$974$	0	0
$975$	−4424.82	−0.145341
$976$	0	0
$977$	24506.4	0.802485	0.401243	−	0.915972i	$-0.368578\pi$
$977$	24506.4	0.802485	0.401243	+	0.915972i	$0.368578\pi$
$978$	0	0
$979$	−13424.1	−0.438238
$980$	0	0
$981$	26410.0	0.859538
$982$	0	0
$983$	−27401.1	−0.889075	−0.444537	−	0.895760i	$-0.646632\pi$
$983$	−27401.1	−0.889075	−0.444537	+	0.895760i	$0.646632\pi$
$984$	0	0
$985$	8499.74	0.274949
$986$	0	0
$987$	41581.4	1.34098
$988$	0	0
$989$	9167.05	0.294737
$990$	0	0
$991$	5803.32	0.186023	0.0930113	−	0.995665i	$-0.470351\pi$
$991$	5803.32	0.186023	0.0930113	+	0.995665i	$0.470351\pi$
$992$	0	0
$993$	−90913.6	−2.90539
$994$	0	0
$995$	9498.25	0.302628
$996$	0	0
$997$	17677.7	0.561542	0.280771	−	0.959775i	$-0.409410\pi$
$997$	17677.7	0.561542	0.280771	+	0.959775i	$0.409410\pi$
$998$	0	0
$999$	−29641.3	−0.938747

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.v.1.1		8
4.3	odd	2		115.4.a.f.1.5	✓	8
12.11	even	2		1035.4.a.r.1.4		8
20.3	even	4		575.4.b.k.24.6		16
20.7	even	4		575.4.b.k.24.11		16
20.19	odd	2		575.4.a.n.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.f.1.5	✓	8	4.3	odd	2
575.4.a.n.1.4		8	20.19	odd	2
575.4.b.k.24.6		16	20.3	even	4
575.4.b.k.24.11		16	20.7	even	4
1035.4.a.r.1.4		8	12.11	even	2
1840.4.a.v.1.1		8	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)

\(f(q)\)	\(=\)	\(q-8.59146 q^{3} +5.00000 q^{5} -9.86011 q^{7} +46.8131 q^{9} +O(q^{10})\)	\(q-8.59146 q^{3} +5.00000 q^{5} -9.86011 q^{7} +46.8131 q^{9} -12.1995 q^{11} +20.6010 q^{13} -42.9573 q^{15} -7.09451 q^{17} +155.809 q^{19} +84.7127 q^{21} -23.0000 q^{23} +25.0000 q^{25} -170.224 q^{27} -213.296 q^{29} +95.7095 q^{31} +104.811 q^{33} -49.3006 q^{35} +174.131 q^{37} -176.993 q^{39} +122.549 q^{41} -398.567 q^{43} +234.066 q^{45} +490.851 q^{47} -245.778 q^{49} +60.9522 q^{51} -161.490 q^{53} -60.9974 q^{55} -1338.63 q^{57} +560.053 q^{59} -321.451 q^{61} -461.583 q^{63} +103.005 q^{65} -280.311 q^{67} +197.604 q^{69} -649.370 q^{71} +426.808 q^{73} -214.786 q^{75} +120.288 q^{77} +161.318 q^{79} +198.514 q^{81} -801.673 q^{83} -35.4725 q^{85} +1832.53 q^{87} +1100.38 q^{89} -203.128 q^{91} -822.284 q^{93} +779.046 q^{95} -1179.42 q^{97} -571.096 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9}+O(q^{10}) \) 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9	\( 8 q + 40 q^{5} - 11 q^{7} + 158 q^{9} - 41 q^{11} + 28 q^{13} + 71 q^{17} - 177 q^{19} + 292 q^{21} - 184 q^{23} + 200 q^{25} + 495 q^{27} + 225 q^{29} + 36 q^{31} - 53 q^{33} - 55 q^{35} - 348 q^{37} + 1077 q^{39} + 620 q^{41} + 390 q^{43} + 790 q^{45} - 123 q^{47} + 881 q^{49} - 957 q^{51} + 1406 q^{53} - 205 q^{55} - 1142 q^{57} + 676 q^{59} + 1447 q^{61} + 58 q^{63} + 140 q^{65} + 1582 q^{67} - 1396 q^{71} + 17 q^{73} + 488 q^{77} - 708 q^{79} + 4316 q^{81} - 1486 q^{83} + 355 q^{85} + 803 q^{87} + 1360 q^{89} - 2693 q^{91} + 3833 q^{93} - 885 q^{95} - 855 q^{97} - 1319 q^{99}+O(q^{100}) \) 8 * q + 40 * q^5 - 11 * q^7 + 158 * q^9 - 41 * q^11 + 28 * q^13 + 71 * q^17 - 177 * q^19 + 292 * q^21 - 184 * q^23 + 200 * q^25 + 495 * q^27 + 225 * q^29 + 36 * q^31 - 53 * q^33 - 55 * q^35 - 348 * q^37 + 1077 * q^39 + 620 * q^41 + 390 * q^43 + 790 * q^45 - 123 * q^47 + 881 * q^49 - 957 * q^51 + 1406 * q^53 - 205 * q^55 - 1142 * q^57 + 676 * q^59 + 1447 * q^61 + 58 * q^63 + 140 * q^65 + 1582 * q^67 - 1396 * q^71 + 17 * q^73 + 488 * q^77 - 708 * q^79 + 4316 * q^81 - 1486 * q^83 + 355 * q^85 + 803 * q^87 + 1360 * q^89 - 2693 * q^91 + 3833 * q^93 - 885 * q^95 - 855 * q^97 - 1319 * q^99

Introduction

L-functions

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