LMFDB - Embedded newform 1840.4.a.bb.1.5

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 204 x^{8} + 42 x^{7} + 12958 x^{6} + 5872 x^{5} - 259871 x^{4} - 149461 x^{3} + \cdots - 43712 $ x^10 - x^9 - 204x^8 + 42x^7 + 12958x^6 + 5872x^5 - 259871x^4 - 149461x^3 + 1222472x^2 + 627136x - 43712
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.575045$ 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)

$f(q)$	$=$	$q-0.575045 q^{3} -5.00000 q^{5} +24.8747 q^{7} -26.6693 q^{9} +O(q^{10})$	$q-0.575045 q^{3} -5.00000 q^{5} +24.8747 q^{7} -26.6693 q^{9} +5.05844 q^{11} -93.1604 q^{13} +2.87522 q^{15} -95.9572 q^{17} -58.3273 q^{19} -14.3041 q^{21} -23.0000 q^{23} +25.0000 q^{25} +30.8623 q^{27} +85.0540 q^{29} -172.280 q^{31} -2.90883 q^{33} -124.373 q^{35} +222.779 q^{37} +53.5714 q^{39} +171.389 q^{41} +406.942 q^{43} +133.347 q^{45} +166.455 q^{47} +275.750 q^{49} +55.1797 q^{51} +69.4544 q^{53} -25.2922 q^{55} +33.5408 q^{57} -662.361 q^{59} +190.331 q^{61} -663.391 q^{63} +465.802 q^{65} -106.999 q^{67} +13.2260 q^{69} +920.046 q^{71} -649.441 q^{73} -14.3761 q^{75} +125.827 q^{77} +543.783 q^{79} +702.325 q^{81} +131.724 q^{83} +479.786 q^{85} -48.9099 q^{87} +788.269 q^{89} -2317.34 q^{91} +99.0689 q^{93} +291.637 q^{95} +900.688 q^{97} -134.905 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) $ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9	$ 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) $ 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 + 14 * q^11 + 11 * q^13 - 5 * q^15 + 68 * q^17 - 114 * q^19 - 232 * q^21 - 230 * q^23 + 250 * q^25 + 433 * q^27 - 273 * q^29 + 129 * q^31 + 98 * q^33 + 140 * q^35 + 62 * q^37 - 283 * q^39 + 767 * q^41 - 332 * q^43 - 695 * q^45 + 323 * q^47 + 1162 * q^49 - 176 * q^51 + 558 * q^53 - 70 * q^55 + 46 * q^57 - 822 * q^59 + 318 * q^61 - 2698 * q^63 - 55 * q^65 - 1152 * q^67 - 23 * q^69 - 1247 * q^71 + 1941 * q^73 + 25 * q^75 + 528 * q^77 - 3134 * q^79 + 6210 * q^81 - 482 * q^83 - 340 * q^85 - 1797 * q^87 + 4734 * q^89 - 4992 * q^91 + 4647 * q^93 + 570 * q^95 + 2326 * q^97 - 4356 * 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
$3$	−0.575045	−0.110667	−0.0553337	−	0.998468i	$-0.517622\pi$
$3$	−0.575045	−0.110667	−0.0553337	+	0.998468i	$0.517622\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	24.8747	1.34311	0.671554	−	0.740956i	$-0.265627\pi$
$7$	24.8747	1.34311	0.671554	+	0.740956i	$0.265627\pi$
$8$	0	0
$9$	−26.6693	−0.987753
$10$	0	0
$11$	5.05844	0.138653	0.0693263	−	0.997594i	$-0.477915\pi$
$11$	5.05844	0.138653	0.0693263	+	0.997594i	$0.477915\pi$
$12$	0	0
$13$	−93.1604	−1.98754	−0.993771	−	0.111437i	$-0.964455\pi$
$13$	−93.1604	−1.98754	−0.993771	+	0.111437i	$0.964455\pi$
$14$	0	0
$15$	2.87522	0.0494920
$16$	0	0
$17$	−95.9572	−1.36900	−0.684501	−	0.729012i	$-0.739980\pi$
$17$	−95.9572	−1.36900	−0.684501	+	0.729012i	$0.739980\pi$
$18$	0	0
$19$	−58.3273	−0.704274	−0.352137	−	0.935949i	$-0.614545\pi$
$19$	−58.3273	−0.704274	−0.352137	+	0.935949i	$0.614545\pi$
$20$	0	0
$21$	−14.3041	−0.148638
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	30.8623	0.219979
$28$	0	0
$29$	85.0540	0.544625	0.272313	−	0.962209i	$-0.412211\pi$
$29$	85.0540	0.544625	0.272313	+	0.962209i	$0.412211\pi$
$30$	0	0
$31$	−172.280	−0.998143	−0.499072	−	0.866561i	$-0.666326\pi$
$31$	−172.280	−0.998143	−0.499072	+	0.866561i	$0.666326\pi$
$32$	0	0
$33$	−2.90883	−0.0153443
$34$	0	0
$35$	−124.373	−0.600656
$36$	0	0
$37$	222.779	0.989857	0.494929	−	0.868934i	$-0.335194\pi$
$37$	222.779	0.989857	0.494929	+	0.868934i	$0.335194\pi$
$38$	0	0
$39$	53.5714	0.219956
$40$	0	0
$41$	171.389	0.652842	0.326421	−	0.945225i	$-0.394157\pi$
$41$	171.389	0.652842	0.326421	+	0.945225i	$0.394157\pi$
$42$	0	0
$43$	406.942	1.44321	0.721606	−	0.692304i	$-0.243404\pi$
$43$	406.942	1.44321	0.721606	+	0.692304i	$0.243404\pi$
$44$	0	0
$45$	133.347	0.441736
$46$	0	0
$47$	166.455	0.516595	0.258298	−	0.966065i	$-0.416838\pi$
$47$	166.455	0.516595	0.258298	+	0.966065i	$0.416838\pi$
$48$	0	0
$49$	275.750	0.803937
$50$	0	0
$51$	55.1797	0.151504
$52$	0	0
$53$	69.4544	0.180006	0.0900028	−	0.995942i	$-0.471312\pi$
$53$	69.4544	0.180006	0.0900028	+	0.995942i	$0.471312\pi$
$54$	0	0
$55$	−25.2922	−0.0620073
$56$	0	0
$57$	33.5408	0.0779402
$58$	0	0
$59$	−662.361	−1.46156	−0.730780	−	0.682613i	$-0.760843\pi$
$59$	−662.361	−1.46156	−0.730780	+	0.682613i	$0.760843\pi$
$60$	0	0
$61$	190.331	0.399498	0.199749	−	0.979847i	$-0.435987\pi$
$61$	190.331	0.399498	0.199749	+	0.979847i	$0.435987\pi$
$62$	0	0
$63$	−663.391	−1.32666
$64$	0	0
$65$	465.802	0.888856
$66$	0	0
$67$	−106.999	−0.195105	−0.0975527	−	0.995230i	$-0.531101\pi$
$67$	−106.999	−0.195105	−0.0975527	+	0.995230i	$0.531101\pi$
$68$	0	0
$69$	13.2260	0.0230758
$70$	0	0
$71$	920.046	1.53788	0.768939	−	0.639322i	$-0.220785\pi$
$71$	920.046	1.53788	0.768939	+	0.639322i	$0.220785\pi$
$72$	0	0
$73$	−649.441	−1.04125	−0.520625	−	0.853785i	$-0.674301\pi$
$73$	−649.441	−1.04125	−0.520625	+	0.853785i	$0.674301\pi$
$74$	0	0
$75$	−14.3761	−0.0221335
$76$	0	0
$77$	125.827	0.186225
$78$	0	0
$79$	543.783	0.774435	0.387218	−	0.921988i	$-0.373436\pi$
$79$	543.783	0.774435	0.387218	+	0.921988i	$0.373436\pi$
$80$	0	0
$81$	702.325	0.963408
$82$	0	0
$83$	131.724	0.174200	0.0870998	−	0.996200i	$-0.472240\pi$
$83$	131.724	0.174200	0.0870998	+	0.996200i	$0.472240\pi$
$84$	0	0
$85$	479.786	0.612236
$86$	0	0
$87$	−48.9099	−0.0602723
$88$	0	0
$89$	788.269	0.938835	0.469417	−	0.882976i	$-0.344464\pi$
$89$	788.269	0.938835	0.469417	+	0.882976i	$0.344464\pi$
$90$	0	0
$91$	−2317.34	−2.66948
$92$	0	0
$93$	99.0689	0.110462
$94$	0	0
$95$	291.637	0.314961
$96$	0	0
$97$	900.688	0.942794	0.471397	−	0.881921i	$-0.343750\pi$
$97$	900.688	0.942794	0.471397	+	0.881921i	$0.343750\pi$
$98$	0	0
$99$	−134.905	−0.136954
$100$	0	0
$101$	−728.024	−0.717239	−0.358620	−	0.933484i	$-0.616752\pi$
$101$	−728.024	−0.717239	−0.358620	+	0.933484i	$0.616752\pi$
$102$	0	0
$103$	102.184	0.0977528	0.0488764	−	0.998805i	$-0.484436\pi$
$103$	102.184	0.0977528	0.0488764	+	0.998805i	$0.484436\pi$
$104$	0	0
$105$	71.5203	0.0664730
$106$	0	0
$107$	−24.6710	−0.0222900	−0.0111450	−	0.999938i	$-0.503548\pi$
$107$	−24.6710	−0.0222900	−0.0111450	+	0.999938i	$0.503548\pi$
$108$	0	0
$109$	−1947.95	−1.71174	−0.855871	−	0.517190i	$-0.826978\pi$
$109$	−1947.95	−1.71174	−0.855871	+	0.517190i	$0.826978\pi$
$110$	0	0
$111$	−128.108	−0.109545
$112$	0	0
$113$	−922.086	−0.767633	−0.383817	−	0.923409i	$-0.625391\pi$
$113$	−922.086	−0.767633	−0.383817	+	0.923409i	$0.625391\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	2484.53	1.96320
$118$	0	0
$119$	−2386.91	−1.83872
$120$	0	0
$121$	−1305.41	−0.980775
$122$	0	0
$123$	−98.5566	−0.0722483
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	837.709	0.585313	0.292656	−	0.956218i	$-0.405461\pi$
$127$	837.709	0.585313	0.292656	+	0.956218i	$0.405461\pi$
$128$	0	0
$129$	−234.010	−0.159717
$130$	0	0
$131$	−1738.68	−1.15961	−0.579805	−	0.814755i	$-0.696871\pi$
$131$	−1738.68	−1.15961	−0.579805	+	0.814755i	$0.696871\pi$
$132$	0	0
$133$	−1450.87	−0.945915
$134$	0	0
$135$	−154.311	−0.0983778
$136$	0	0
$137$	1705.94	1.06385	0.531927	−	0.846790i	$-0.321468\pi$
$137$	1705.94	1.06385	0.531927	+	0.846790i	$0.321468\pi$
$138$	0	0
$139$	2865.66	1.74865	0.874323	−	0.485344i	$-0.161306\pi$
$139$	2865.66	1.74865	0.874323	+	0.485344i	$0.161306\pi$
$140$	0	0
$141$	−95.7192	−0.0571703
$142$	0	0
$143$	−471.247	−0.275578
$144$	0	0
$145$	−425.270	−0.243564
$146$	0	0
$147$	−158.569	−0.0889696
$148$	0	0
$149$	3164.78	1.74006	0.870030	−	0.492998i	$-0.164099\pi$
$149$	3164.78	1.74006	0.870030	+	0.492998i	$0.164099\pi$
$150$	0	0
$151$	3587.98	1.93368	0.966840	−	0.255384i	$-0.0822020\pi$
$151$	3587.98	1.93368	0.966840	+	0.255384i	$0.0822020\pi$
$152$	0	0
$153$	2559.11	1.35224
$154$	0	0
$155$	861.401	0.446383
$156$	0	0
$157$	547.591	0.278360	0.139180	−	0.990267i	$-0.455553\pi$
$157$	547.591	0.278360	0.139180	+	0.990267i	$0.455553\pi$
$158$	0	0
$159$	−39.9394	−0.0199208
$160$	0	0
$161$	−572.118	−0.280057
$162$	0	0
$163$	−4092.84	−1.96673	−0.983363	−	0.181652i	$-0.941856\pi$
$163$	−4092.84	−1.96673	−0.983363	+	0.181652i	$0.941856\pi$
$164$	0	0
$165$	14.5442	0.00686219
$166$	0	0
$167$	−551.846	−0.255707	−0.127854	−	0.991793i	$-0.540809\pi$
$167$	−551.846	−0.255707	−0.127854	+	0.991793i	$0.540809\pi$
$168$	0	0
$169$	6481.87	2.95033
$170$	0	0
$171$	1555.55	0.695648
$172$	0	0
$173$	−3854.36	−1.69388	−0.846941	−	0.531687i	$-0.821558\pi$
$173$	−3854.36	−1.69388	−0.846941	+	0.531687i	$0.821558\pi$
$174$	0	0
$175$	621.867	0.268621
$176$	0	0
$177$	380.887	0.161747
$178$	0	0
$179$	−1919.21	−0.801388	−0.400694	−	0.916212i	$-0.631231\pi$
$179$	−1919.21	−0.801388	−0.400694	+	0.916212i	$0.631231\pi$
$180$	0	0
$181$	3630.00	1.49069	0.745347	−	0.666676i	$-0.232284\pi$
$181$	3630.00	1.49069	0.745347	+	0.666676i	$0.232284\pi$
$182$	0	0
$183$	−109.449	−0.0442114
$184$	0	0
$185$	−1113.90	−0.442678
$186$	0	0
$187$	−485.394	−0.189816
$188$	0	0
$189$	767.689	0.295456
$190$	0	0
$191$	1430.20	0.541811	0.270905	−	0.962606i	$-0.412677\pi$
$191$	1430.20	0.541811	0.270905	+	0.962606i	$0.412677\pi$
$192$	0	0
$193$	1018.83	0.379984	0.189992	−	0.981786i	$-0.439154\pi$
$193$	1018.83	0.379984	0.189992	+	0.981786i	$0.439154\pi$
$194$	0	0
$195$	−267.857	−0.0983674
$196$	0	0
$197$	3397.75	1.22883	0.614415	−	0.788983i	$-0.289392\pi$
$197$	3397.75	1.22883	0.614415	+	0.788983i	$0.289392\pi$
$198$	0	0
$199$	−473.774	−0.168769	−0.0843843	−	0.996433i	$-0.526892\pi$
$199$	−473.774	−0.168769	−0.0843843	+	0.996433i	$0.526892\pi$
$200$	0	0
$201$	61.5295	0.0215918
$202$	0	0
$203$	2115.69	0.731490
$204$	0	0
$205$	−856.947	−0.291960
$206$	0	0
$207$	613.394	0.205961
$208$	0	0
$209$	−295.045	−0.0976494
$210$	0	0
$211$	−186.758	−0.0609336	−0.0304668	−	0.999536i	$-0.509699\pi$
$211$	−186.758	−0.0609336	−0.0304668	+	0.999536i	$0.509699\pi$
$212$	0	0
$213$	−529.067	−0.170193
$214$	0	0
$215$	−2034.71	−0.645424
$216$	0	0
$217$	−4285.42	−1.34061
$218$	0	0
$219$	373.458	0.115233
$220$	0	0
$221$	8939.42	2.72095
$222$	0	0
$223$	1934.74	0.580987	0.290493	−	0.956877i	$-0.406181\pi$
$223$	1934.74	0.580987	0.290493	+	0.956877i	$0.406181\pi$
$224$	0	0
$225$	−666.733	−0.197551
$226$	0	0
$227$	1225.37	0.358285	0.179142	−	0.983823i	$-0.442668\pi$
$227$	1225.37	0.358285	0.179142	+	0.983823i	$0.442668\pi$
$228$	0	0
$229$	−71.2139	−0.0205500	−0.0102750	−	0.999947i	$-0.503271\pi$
$229$	−71.2139	−0.0205500	−0.0102750	+	0.999947i	$0.503271\pi$
$230$	0	0
$231$	−72.3563	−0.0206091
$232$	0	0
$233$	3716.48	1.04496	0.522478	−	0.852652i	$-0.325008\pi$
$233$	3716.48	1.04496	0.522478	+	0.852652i	$0.325008\pi$
$234$	0	0
$235$	−832.276	−0.231028
$236$	0	0
$237$	−312.700	−0.0857048
$238$	0	0
$239$	−396.290	−0.107255	−0.0536274	−	0.998561i	$-0.517078\pi$
$239$	−396.290	−0.107255	−0.0536274	+	0.998561i	$0.517078\pi$
$240$	0	0
$241$	1578.82	0.421994	0.210997	−	0.977487i	$-0.432329\pi$
$241$	1578.82	0.421994	0.210997	+	0.977487i	$0.432329\pi$
$242$	0	0
$243$	−1237.15	−0.326597
$244$	0	0
$245$	−1378.75	−0.359531
$246$	0	0
$247$	5433.80	1.39977
$248$	0	0
$249$	−75.7471	−0.0192782
$250$	0	0
$251$	6159.70	1.54899	0.774495	−	0.632580i	$-0.218004\pi$
$251$	6159.70	1.54899	0.774495	+	0.632580i	$0.218004\pi$
$252$	0	0
$253$	−116.344	−0.0289111
$254$	0	0
$255$	−275.898	−0.0677546
$256$	0	0
$257$	3986.76	0.967655	0.483828	−	0.875163i	$-0.339246\pi$
$257$	3986.76	0.967655	0.483828	+	0.875163i	$0.339246\pi$
$258$	0	0
$259$	5541.57	1.32948
$260$	0	0
$261$	−2268.33	−0.537955
$262$	0	0
$263$	3290.92	0.771586	0.385793	−	0.922585i	$-0.373928\pi$
$263$	3290.92	0.771586	0.385793	+	0.922585i	$0.373928\pi$
$264$	0	0
$265$	−347.272	−0.0805009
$266$	0	0
$267$	−453.290	−0.103898
$268$	0	0
$269$	−3018.98	−0.684276	−0.342138	−	0.939650i	$-0.611151\pi$
$269$	−3018.98	−0.684276	−0.342138	+	0.939650i	$0.611151\pi$
$270$	0	0
$271$	−4998.53	−1.12044	−0.560220	−	0.828344i	$-0.689283\pi$
$271$	−4998.53	−1.12044	−0.560220	+	0.828344i	$0.689283\pi$
$272$	0	0
$273$	1332.57	0.295425
$274$	0	0
$275$	126.461	0.0277305
$276$	0	0
$277$	2231.80	0.484101	0.242050	−	0.970264i	$-0.422180\pi$
$277$	2231.80	0.484101	0.242050	+	0.970264i	$0.422180\pi$
$278$	0	0
$279$	4594.60	0.985919
$280$	0	0
$281$	2363.90	0.501845	0.250922	−	0.968007i	$-0.419266\pi$
$281$	2363.90	0.501845	0.250922	+	0.968007i	$0.419266\pi$
$282$	0	0
$283$	−5348.26	−1.12340	−0.561698	−	0.827343i	$-0.689852\pi$
$283$	−5348.26	−1.12340	−0.561698	+	0.827343i	$0.689852\pi$
$284$	0	0
$285$	−167.704	−0.0348559
$286$	0	0
$287$	4263.26	0.876837
$288$	0	0
$289$	4294.78	0.874167
$290$	0	0
$291$	−517.936	−0.104337
$292$	0	0
$293$	−6131.67	−1.22258	−0.611290	−	0.791407i	$-0.709349\pi$
$293$	−6131.67	−1.22258	−0.611290	+	0.791407i	$0.709349\pi$
$294$	0	0
$295$	3311.80	0.653629
$296$	0	0
$297$	156.115	0.0305007
$298$	0	0
$299$	2142.69	0.414431
$300$	0	0
$301$	10122.6	1.93839
$302$	0	0
$303$	418.647	0.0793750
$304$	0	0
$305$	−951.655	−0.178661
$306$	0	0
$307$	−3031.94	−0.563654	−0.281827	−	0.959465i	$-0.590940\pi$
$307$	−3031.94	−0.563654	−0.281827	+	0.959465i	$0.590940\pi$
$308$	0	0
$309$	−58.7607	−0.0108181
$310$	0	0
$311$	−869.411	−0.158520	−0.0792601	−	0.996854i	$-0.525256\pi$
$311$	−869.411	−0.158520	−0.0792601	+	0.996854i	$0.525256\pi$
$312$	0	0
$313$	−7040.54	−1.27142	−0.635711	−	0.771928i	$-0.719293\pi$
$313$	−7040.54	−1.27142	−0.635711	+	0.771928i	$0.719293\pi$
$314$	0	0
$315$	3316.96	0.593299
$316$	0	0
$317$	−3710.72	−0.657460	−0.328730	−	0.944424i	$-0.606621\pi$
$317$	−3710.72	−0.657460	−0.328730	+	0.944424i	$0.606621\pi$
$318$	0	0
$319$	430.241	0.0755137
$320$	0	0
$321$	14.1869	0.00246678
$322$	0	0
$323$	5596.93	0.964152
$324$	0	0
$325$	−2329.01	−0.397509
$326$	0	0
$327$	1120.16	0.189434
$328$	0	0
$329$	4140.52	0.693843
$330$	0	0
$331$	−7815.80	−1.29787	−0.648935	−	0.760843i	$-0.724785\pi$
$331$	−7815.80	−1.29787	−0.648935	+	0.760843i	$0.724785\pi$
$332$	0	0
$333$	−5941.38	−0.977734
$334$	0	0
$335$	534.997	0.0872538
$336$	0	0
$337$	−980.947	−0.158563	−0.0792813	−	0.996852i	$-0.525263\pi$
$337$	−980.947	−0.158563	−0.0792813	+	0.996852i	$0.525263\pi$
$338$	0	0
$339$	530.241	0.0849520
$340$	0	0
$341$	−871.470	−0.138395
$342$	0	0
$343$	−1672.82	−0.263334
$344$	0	0
$345$	−66.1302	−0.0103198
$346$	0	0
$347$	8010.73	1.23930	0.619652	−	0.784877i	$-0.287274\pi$
$347$	8010.73	1.23930	0.619652	+	0.784877i	$0.287274\pi$
$348$	0	0
$349$	−8207.15	−1.25879	−0.629396	−	0.777084i	$-0.716698\pi$
$349$	−8207.15	−1.25879	−0.629396	+	0.777084i	$0.716698\pi$
$350$	0	0
$351$	−2875.14	−0.437219
$352$	0	0
$353$	11093.5	1.67265	0.836324	−	0.548235i	$-0.184700\pi$
$353$	11093.5	1.67265	0.836324	+	0.548235i	$0.184700\pi$
$354$	0	0
$355$	−4600.23	−0.687760
$356$	0	0
$357$	1372.58	0.203486
$358$	0	0
$359$	5353.52	0.787041	0.393521	−	0.919316i	$-0.371257\pi$
$359$	5353.52	0.787041	0.393521	+	0.919316i	$0.371257\pi$
$360$	0	0
$361$	−3456.93	−0.503998
$362$	0	0
$363$	750.671	0.108540
$364$	0	0
$365$	3247.20	0.465661
$366$	0	0
$367$	−12377.0	−1.76042	−0.880210	−	0.474584i	$-0.842598\pi$
$367$	−12377.0	−1.76042	−0.880210	+	0.474584i	$0.842598\pi$
$368$	0	0
$369$	−4570.84	−0.644846
$370$	0	0
$371$	1727.66	0.241767
$372$	0	0
$373$	9763.80	1.35536	0.677682	−	0.735355i	$-0.262985\pi$
$373$	9763.80	1.35536	0.677682	+	0.735355i	$0.262985\pi$
$374$	0	0
$375$	71.8806	0.00989840
$376$	0	0
$377$	−7923.67	−1.08247
$378$	0	0
$379$	5663.20	0.767543	0.383772	−	0.923428i	$-0.374625\pi$
$379$	5663.20	0.767543	0.383772	+	0.923428i	$0.374625\pi$
$380$	0	0
$381$	−481.720	−0.0647750
$382$	0	0
$383$	9683.43	1.29191	0.645953	−	0.763377i	$-0.276460\pi$
$383$	9683.43	1.29191	0.645953	+	0.763377i	$0.276460\pi$
$384$	0	0
$385$	−629.136	−0.0832825
$386$	0	0
$387$	−10852.9	−1.42554
$388$	0	0
$389$	−4736.17	−0.617309	−0.308655	−	0.951174i	$-0.599879\pi$
$389$	−4736.17	−0.617309	−0.308655	+	0.951174i	$0.599879\pi$
$390$	0	0
$391$	2207.02	0.285457
$392$	0	0
$393$	999.818	0.128331
$394$	0	0
$395$	−2718.92	−0.346338
$396$	0	0
$397$	−1102.97	−0.139437	−0.0697186	−	0.997567i	$-0.522210\pi$
$397$	−1102.97	−0.139437	−0.0697186	+	0.997567i	$0.522210\pi$
$398$	0	0
$399$	834.317	0.104682
$400$	0	0
$401$	5299.21	0.659924	0.329962	−	0.943994i	$-0.392964\pi$
$401$	5299.21	0.659924	0.329962	+	0.943994i	$0.392964\pi$
$402$	0	0
$403$	16049.7	1.98385
$404$	0	0
$405$	−3511.62	−0.430849
$406$	0	0
$407$	1126.92	0.137246
$408$	0	0
$409$	−4989.73	−0.603242	−0.301621	−	0.953428i	$-0.597528\pi$
$409$	−4989.73	−0.603242	−0.301621	+	0.953428i	$0.597528\pi$
$410$	0	0
$411$	−980.990	−0.117734
$412$	0	0
$413$	−16476.0	−1.96303
$414$	0	0
$415$	−658.619	−0.0779044
$416$	0	0
$417$	−1647.88	−0.193518
$418$	0	0
$419$	4153.22	0.484243	0.242122	−	0.970246i	$-0.422157\pi$
$419$	4153.22	0.484243	0.242122	+	0.970246i	$0.422157\pi$
$420$	0	0
$421$	5131.27	0.594021	0.297011	−	0.954874i	$-0.404010\pi$
$421$	5131.27	0.594021	0.297011	+	0.954874i	$0.404010\pi$
$422$	0	0
$423$	−4439.25	−0.510268
$424$	0	0
$425$	−2398.93	−0.273800
$426$	0	0
$427$	4734.43	0.536569
$428$	0	0
$429$	270.988	0.0304975
$430$	0	0
$431$	898.575	0.100424	0.0502121	−	0.998739i	$-0.484010\pi$
$431$	898.575	0.100424	0.0502121	+	0.998739i	$0.484010\pi$
$432$	0	0
$433$	8286.07	0.919638	0.459819	−	0.888013i	$-0.347914\pi$
$433$	8286.07	0.919638	0.459819	+	0.888013i	$0.347914\pi$
$434$	0	0
$435$	244.549	0.0269546
$436$	0	0
$437$	1341.53	0.146851
$438$	0	0
$439$	4756.46	0.517115	0.258557	−	0.965996i	$-0.416753\pi$
$439$	4756.46	0.517115	0.258557	+	0.965996i	$0.416753\pi$
$440$	0	0
$441$	−7354.07	−0.794091
$442$	0	0
$443$	12491.0	1.33966	0.669828	−	0.742516i	$-0.266368\pi$
$443$	12491.0	1.33966	0.669828	+	0.742516i	$0.266368\pi$
$444$	0	0
$445$	−3941.34	−0.419860
$446$	0	0
$447$	−1819.89	−0.192568
$448$	0	0
$449$	4369.07	0.459218	0.229609	−	0.973283i	$-0.426255\pi$
$449$	4369.07	0.459218	0.229609	+	0.973283i	$0.426255\pi$
$450$	0	0
$451$	866.963	0.0905182
$452$	0	0
$453$	−2063.25	−0.213995
$454$	0	0
$455$	11586.7	1.19383
$456$	0	0
$457$	16759.0	1.71543	0.857717	−	0.514122i	$-0.171882\pi$
$457$	16759.0	1.71543	0.857717	+	0.514122i	$0.171882\pi$
$458$	0	0
$459$	−2961.46	−0.301152
$460$	0	0
$461$	−16347.4	−1.65157	−0.825786	−	0.563983i	$-0.809268\pi$
$461$	−16347.4	−1.65157	−0.825786	+	0.563983i	$0.809268\pi$
$462$	0	0
$463$	6673.38	0.669845	0.334923	−	0.942246i	$-0.391290\pi$
$463$	6673.38	0.669845	0.334923	+	0.942246i	$0.391290\pi$
$464$	0	0
$465$	−495.344	−0.0494001
$466$	0	0
$467$	11780.4	1.16731	0.583655	−	0.812002i	$-0.301622\pi$
$467$	11780.4	1.16731	0.583655	+	0.812002i	$0.301622\pi$
$468$	0	0
$469$	−2661.58	−0.262047
$470$	0	0
$471$	−314.889	−0.0308054
$472$	0	0
$473$	2058.49	0.200105
$474$	0	0
$475$	−1458.18	−0.140855
$476$	0	0
$477$	−1852.30	−0.177801
$478$	0	0
$479$	−8786.73	−0.838154	−0.419077	−	0.907951i	$-0.637646\pi$
$479$	−8786.73	−0.838154	−0.419077	+	0.907951i	$0.637646\pi$
$480$	0	0
$481$	−20754.2	−1.96738
$482$	0	0
$483$	328.993	0.0309932
$484$	0	0
$485$	−4503.44	−0.421630
$486$	0	0
$487$	−10352.0	−0.963227	−0.481614	−	0.876384i	$-0.659949\pi$
$487$	−10352.0	−0.963227	−0.481614	+	0.876384i	$0.659949\pi$
$488$	0	0
$489$	2353.57	0.217652
$490$	0	0
$491$	−20167.2	−1.85363	−0.926814	−	0.375522i	$-0.877464\pi$
$491$	−20167.2	−1.85363	−0.926814	+	0.375522i	$0.877464\pi$
$492$	0	0
$493$	−8161.55	−0.745593
$494$	0	0
$495$	674.526	0.0612479
$496$	0	0
$497$	22885.9	2.06553
$498$	0	0
$499$	4066.31	0.364795	0.182398	−	0.983225i	$-0.441614\pi$
$499$	4066.31	0.364795	0.182398	+	0.983225i	$0.441614\pi$
$500$	0	0
$501$	317.336	0.0282985
$502$	0	0
$503$	10149.6	0.899698	0.449849	−	0.893105i	$-0.351478\pi$
$503$	10149.6	0.899698	0.449849	+	0.893105i	$0.351478\pi$
$504$	0	0
$505$	3640.12	0.320759
$506$	0	0
$507$	−3727.37	−0.326505
$508$	0	0
$509$	6513.81	0.567229	0.283614	−	0.958938i	$-0.408466\pi$
$509$	6513.81	0.567229	0.283614	+	0.958938i	$0.408466\pi$
$510$	0	0
$511$	−16154.6	−1.39851
$512$	0	0
$513$	−1800.11	−0.154926
$514$	0	0
$515$	−510.922	−0.0437164
$516$	0	0
$517$	842.004	0.0716272
$518$	0	0
$519$	2216.43	0.187458
$520$	0	0
$521$	−6341.03	−0.533216	−0.266608	−	0.963805i	$-0.585903\pi$
$521$	−6341.03	−0.533216	−0.266608	+	0.963805i	$0.585903\pi$
$522$	0	0
$523$	19016.9	1.58997	0.794983	−	0.606631i	$-0.207480\pi$
$523$	19016.9	1.58997	0.794983	+	0.606631i	$0.207480\pi$
$524$	0	0
$525$	−357.602	−0.0297276
$526$	0	0
$527$	16531.5	1.36646
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	17664.7	1.44366
$532$	0	0
$533$	−15966.7	−1.29755
$534$	0	0
$535$	123.355	0.00996840
$536$	0	0
$537$	1103.63	0.0886876
$538$	0	0
$539$	1394.87	0.111468
$540$	0	0
$541$	7118.12	0.565678	0.282839	−	0.959167i	$-0.408724\pi$
$541$	7118.12	0.565678	0.282839	+	0.959167i	$0.408724\pi$
$542$	0	0
$543$	−2087.41	−0.164971
$544$	0	0
$545$	9739.75	0.765514
$546$	0	0
$547$	−19062.0	−1.49001	−0.745004	−	0.667060i	$-0.767552\pi$
$547$	−19062.0	−1.49001	−0.745004	+	0.667060i	$0.767552\pi$
$548$	0	0
$549$	−5076.00	−0.394605
$550$	0	0
$551$	−4960.97	−0.383565
$552$	0	0
$553$	13526.4	1.04015
$554$	0	0
$555$	640.541	0.0489900
$556$	0	0
$557$	6622.46	0.503775	0.251888	−	0.967757i	$-0.418949\pi$
$557$	6622.46	0.503775	0.251888	+	0.967757i	$0.418949\pi$
$558$	0	0
$559$	−37910.9	−2.86845
$560$	0	0
$561$	279.123	0.0210064
$562$	0	0
$563$	12912.3	0.966585	0.483292	−	0.875459i	$-0.339441\pi$
$563$	12912.3	0.966585	0.483292	+	0.875459i	$0.339441\pi$
$564$	0	0
$565$	4610.43	0.343296
$566$	0	0
$567$	17470.1	1.29396
$568$	0	0
$569$	23465.2	1.72885	0.864423	−	0.502765i	$-0.167684\pi$
$569$	23465.2	1.72885	0.864423	+	0.502765i	$0.167684\pi$
$570$	0	0
$571$	8463.04	0.620258	0.310129	−	0.950694i	$-0.399628\pi$
$571$	8463.04	0.620258	0.310129	+	0.950694i	$0.399628\pi$
$572$	0	0
$573$	−822.431	−0.0599608
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−3360.91	−0.242490	−0.121245	−	0.992623i	$-0.538689\pi$
$577$	−3360.91	−0.242490	−0.121245	+	0.992623i	$0.538689\pi$
$578$	0	0
$579$	−585.872	−0.0420518
$580$	0	0
$581$	3276.59	0.233969
$582$	0	0
$583$	351.331	0.0249582
$584$	0	0
$585$	−12422.6	−0.877970
$586$	0	0
$587$	3773.05	0.265299	0.132649	−	0.991163i	$-0.457652\pi$
$587$	3773.05	0.265299	0.132649	+	0.991163i	$0.457652\pi$
$588$	0	0
$589$	10048.6	0.702966
$590$	0	0
$591$	−1953.86	−0.135991
$592$	0	0
$593$	−9825.30	−0.680399	−0.340200	−	0.940353i	$-0.610495\pi$
$593$	−9825.30	−0.680399	−0.340200	+	0.940353i	$0.610495\pi$
$594$	0	0
$595$	11934.5	0.822299
$596$	0	0
$597$	272.441	0.0186772
$598$	0	0
$599$	−13167.6	−0.898188	−0.449094	−	0.893485i	$-0.648253\pi$
$599$	−13167.6	−0.898188	−0.449094	+	0.893485i	$0.648253\pi$
$600$	0	0
$601$	−10396.6	−0.705631	−0.352815	−	0.935693i	$-0.614776\pi$
$601$	−10396.6	−0.705631	−0.352815	+	0.935693i	$0.614776\pi$
$602$	0	0
$603$	2853.60	0.192716
$604$	0	0
$605$	6527.06	0.438616
$606$	0	0
$607$	−6354.69	−0.424924	−0.212462	−	0.977169i	$-0.568148\pi$
$607$	−6354.69	−0.424924	−0.212462	+	0.977169i	$0.568148\pi$
$608$	0	0
$609$	−1216.62	−0.0809521
$610$	0	0
$611$	−15507.0	−1.02676
$612$	0	0
$613$	−14376.1	−0.947218	−0.473609	−	0.880735i	$-0.657049\pi$
$613$	−14376.1	−0.947218	−0.473609	+	0.880735i	$0.657049\pi$
$614$	0	0
$615$	492.783	0.0323104
$616$	0	0
$617$	8677.34	0.566185	0.283093	−	0.959093i	$-0.408640\pi$
$617$	8677.34	0.566185	0.283093	+	0.959093i	$0.408640\pi$
$618$	0	0
$619$	−14081.8	−0.914374	−0.457187	−	0.889371i	$-0.651143\pi$
$619$	−14081.8	−0.914374	−0.457187	+	0.889371i	$0.651143\pi$
$620$	0	0
$621$	−709.832	−0.0458689
$622$	0	0
$623$	19607.9	1.26096
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	169.664	0.0108066
$628$	0	0
$629$	−21377.3	−1.35512
$630$	0	0
$631$	−20207.5	−1.27488	−0.637439	−	0.770501i	$-0.720006\pi$
$631$	−20207.5	−1.27488	−0.637439	+	0.770501i	$0.720006\pi$
$632$	0	0
$633$	107.394	0.00674336
$634$	0	0
$635$	−4188.55	−0.261760
$636$	0	0
$637$	−25689.0	−1.59786
$638$	0	0
$639$	−24537.0	−1.51904
$640$	0	0
$641$	30269.8	1.86519	0.932593	−	0.360929i	$-0.117540\pi$
$641$	30269.8	1.86519	0.932593	+	0.360929i	$0.117540\pi$
$642$	0	0
$643$	23781.6	1.45856	0.729279	−	0.684216i	$-0.239856\pi$
$643$	23781.6	1.45856	0.729279	+	0.684216i	$0.239856\pi$
$644$	0	0
$645$	1170.05	0.0714274
$646$	0	0
$647$	25995.2	1.57956	0.789782	−	0.613388i	$-0.210194\pi$
$647$	25995.2	1.57956	0.789782	+	0.613388i	$0.210194\pi$
$648$	0	0
$649$	−3350.51	−0.202649
$650$	0	0
$651$	2464.31	0.148362
$652$	0	0
$653$	−27117.4	−1.62509	−0.812547	−	0.582895i	$-0.801920\pi$
$653$	−27117.4	−1.62509	−0.812547	+	0.582895i	$0.801920\pi$
$654$	0	0
$655$	8693.39	0.518594
$656$	0	0
$657$	17320.1	1.02850
$658$	0	0
$659$	−5413.67	−0.320010	−0.160005	−	0.987116i	$-0.551151\pi$
$659$	−5413.67	−0.320010	−0.160005	+	0.987116i	$0.551151\pi$
$660$	0	0
$661$	15097.2	0.888371	0.444186	−	0.895935i	$-0.353493\pi$
$661$	15097.2	0.888371	0.444186	+	0.895935i	$0.353493\pi$
$662$	0	0
$663$	−5140.56	−0.301121
$664$	0	0
$665$	7254.37	0.423026
$666$	0	0
$667$	−1956.24	−0.113562
$668$	0	0
$669$	−1112.56	−0.0642963
$670$	0	0
$671$	962.779	0.0553914
$672$	0	0
$673$	−954.849	−0.0546905	−0.0273453	−	0.999626i	$-0.508705\pi$
$673$	−954.849	−0.0546905	−0.0273453	+	0.999626i	$0.508705\pi$
$674$	0	0
$675$	771.557	0.0439959
$676$	0	0
$677$	27311.2	1.55045	0.775224	−	0.631686i	$-0.217637\pi$
$677$	27311.2	1.55045	0.775224	+	0.631686i	$0.217637\pi$
$678$	0	0
$679$	22404.3	1.26627
$680$	0	0
$681$	−704.642	−0.0396505
$682$	0	0
$683$	2417.09	0.135413	0.0677066	−	0.997705i	$-0.478432\pi$
$683$	2417.09	0.135413	0.0677066	+	0.997705i	$0.478432\pi$
$684$	0	0
$685$	−8529.69	−0.475770
$686$	0	0
$687$	40.9512	0.00227422
$688$	0	0
$689$	−6470.40	−0.357769
$690$	0	0
$691$	−30614.2	−1.68541	−0.842705	−	0.538375i	$-0.819038\pi$
$691$	−30614.2	−1.68541	−0.842705	+	0.538375i	$0.819038\pi$
$692$	0	0
$693$	−3355.73	−0.183944
$694$	0	0
$695$	−14328.3	−0.782018
$696$	0	0
$697$	−16446.0	−0.893742
$698$	0	0
$699$	−2137.14	−0.115643
$700$	0	0
$701$	−14708.7	−0.792498	−0.396249	−	0.918143i	$-0.629688\pi$
$701$	−14708.7	−0.792498	−0.396249	+	0.918143i	$0.629688\pi$
$702$	0	0
$703$	−12994.1	−0.697130
$704$	0	0
$705$	478.596	0.0255673
$706$	0	0
$707$	−18109.4	−0.963329
$708$	0	0
$709$	14604.1	0.773580	0.386790	−	0.922168i	$-0.373584\pi$
$709$	14604.1	0.773580	0.386790	+	0.922168i	$0.373584\pi$
$710$	0	0
$711$	−14502.3	−0.764951
$712$	0	0
$713$	3962.45	0.208127
$714$	0	0
$715$	2356.23	0.123242
$716$	0	0
$717$	227.885	0.0118696
$718$	0	0
$719$	−12227.6	−0.634231	−0.317116	−	0.948387i	$-0.602714\pi$
$719$	−12227.6	−0.634231	−0.317116	+	0.948387i	$0.602714\pi$
$720$	0	0
$721$	2541.81	0.131292
$722$	0	0
$723$	−907.891	−0.0467010
$724$	0	0
$725$	2126.35	0.108925
$726$	0	0
$727$	−16943.7	−0.864386	−0.432193	−	0.901781i	$-0.642260\pi$
$727$	−16943.7	−0.864386	−0.432193	+	0.901781i	$0.642260\pi$
$728$	0	0
$729$	−18251.3	−0.927264
$730$	0	0
$731$	−39049.0	−1.97576
$732$	0	0
$733$	23122.5	1.16514	0.582571	−	0.812780i	$-0.302047\pi$
$733$	23122.5	1.16514	0.582571	+	0.812780i	$0.302047\pi$
$734$	0	0
$735$	792.844	0.0397884
$736$	0	0
$737$	−541.251	−0.0270519
$738$	0	0
$739$	15011.8	0.747251	0.373625	−	0.927580i	$-0.378115\pi$
$739$	15011.8	0.747251	0.373625	+	0.927580i	$0.378115\pi$
$740$	0	0
$741$	−3124.68	−0.154909
$742$	0	0
$743$	−7090.99	−0.350125	−0.175063	−	0.984557i	$-0.556013\pi$
$743$	−7090.99	−0.350125	−0.175063	+	0.984557i	$0.556013\pi$
$744$	0	0
$745$	−15823.9	−0.778179
$746$	0	0
$747$	−3512.99	−0.172066
$748$	0	0
$749$	−613.683	−0.0299379
$750$	0	0
$751$	−27962.9	−1.35870	−0.679349	−	0.733815i	$-0.737738\pi$
$751$	−27962.9	−1.35870	−0.679349	+	0.733815i	$0.737738\pi$
$752$	0	0
$753$	−3542.10	−0.171423
$754$	0	0
$755$	−17939.9	−0.864768
$756$	0	0
$757$	19108.3	0.917440	0.458720	−	0.888581i	$-0.348308\pi$
$757$	19108.3	0.917440	0.458720	+	0.888581i	$0.348308\pi$
$758$	0	0
$759$	66.9031	0.00319951
$760$	0	0
$761$	8071.02	0.384460	0.192230	−	0.981350i	$-0.438428\pi$
$761$	8071.02	0.384460	0.192230	+	0.981350i	$0.438428\pi$
$762$	0	0
$763$	−48454.7	−2.29905
$764$	0	0
$765$	−12795.6	−0.604738
$766$	0	0
$767$	61705.8	2.90491
$768$	0	0
$769$	21088.3	0.988899	0.494449	−	0.869206i	$-0.335370\pi$
$769$	21088.3	0.988899	0.494449	+	0.869206i	$0.335370\pi$
$770$	0	0
$771$	−2292.57	−0.107088
$772$	0	0
$773$	21990.6	1.02322	0.511609	−	0.859218i	$-0.329050\pi$
$773$	21990.6	1.02322	0.511609	+	0.859218i	$0.329050\pi$
$774$	0	0
$775$	−4307.01	−0.199629
$776$	0	0
$777$	−3186.65	−0.147131
$778$	0	0
$779$	−9996.68	−0.459779
$780$	0	0
$781$	4654.00	0.213231
$782$	0	0
$783$	2624.96	0.119806
$784$	0	0
$785$	−2737.95	−0.124486
$786$	0	0
$787$	−7396.43	−0.335012	−0.167506	−	0.985871i	$-0.553571\pi$
$787$	−7396.43	−0.335012	−0.167506	+	0.985871i	$0.553571\pi$
$788$	0	0
$789$	−1892.43	−0.0853894
$790$	0	0
$791$	−22936.6	−1.03101
$792$	0	0
$793$	−17731.3	−0.794020
$794$	0	0
$795$	199.697	0.00890883
$796$	0	0
$797$	5606.78	0.249187	0.124594	−	0.992208i	$-0.460237\pi$
$797$	5606.78	0.249187	0.124594	+	0.992208i	$0.460237\pi$
$798$	0	0
$799$	−15972.6	−0.707220
$800$	0	0
$801$	−21022.6	−0.927337
$802$	0	0
$803$	−3285.16	−0.144372
$804$	0	0
$805$	2860.59	0.125245
$806$	0	0
$807$	1736.05	0.0757270
$808$	0	0
$809$	−27817.9	−1.20893	−0.604466	−	0.796631i	$-0.706614\pi$
$809$	−27817.9	−1.20893	−0.604466	+	0.796631i	$0.706614\pi$
$810$	0	0
$811$	−26920.6	−1.16561	−0.582806	−	0.812612i	$-0.698045\pi$
$811$	−26920.6	−1.16561	−0.582806	+	0.812612i	$0.698045\pi$
$812$	0	0
$813$	2874.38	0.123996
$814$	0	0
$815$	20464.2	0.879547
$816$	0	0
$817$	−23735.8	−1.01642
$818$	0	0
$819$	61801.8	2.63679
$820$	0	0
$821$	12611.9	0.536123	0.268062	−	0.963402i	$-0.413617\pi$
$821$	12611.9	0.536123	0.268062	+	0.963402i	$0.413617\pi$
$822$	0	0
$823$	−5136.76	−0.217565	−0.108783	−	0.994066i	$-0.534695\pi$
$823$	−5136.76	−0.217565	−0.108783	+	0.994066i	$0.534695\pi$
$824$	0	0
$825$	−72.7208	−0.00306886
$826$	0	0
$827$	−19140.6	−0.804817	−0.402408	−	0.915460i	$-0.631827\pi$
$827$	−19140.6	−0.804817	−0.402408	+	0.915460i	$0.631827\pi$
$828$	0	0
$829$	−31698.5	−1.32803	−0.664015	−	0.747720i	$-0.731149\pi$
$829$	−31698.5	−1.32803	−0.664015	+	0.747720i	$0.731149\pi$
$830$	0	0
$831$	−1283.39	−0.0535742
$832$	0	0
$833$	−26460.2	−1.10059
$834$	0	0
$835$	2759.23	0.114356
$836$	0	0
$837$	−5316.96	−0.219571
$838$	0	0
$839$	5440.08	0.223853	0.111926	−	0.993717i	$-0.464298\pi$
$839$	5440.08	0.223853	0.111926	+	0.993717i	$0.464298\pi$
$840$	0	0
$841$	−17154.8	−0.703383
$842$	0	0
$843$	−1359.35	−0.0555379
$844$	0	0
$845$	−32409.3	−1.31943
$846$	0	0
$847$	−32471.7	−1.31729
$848$	0	0
$849$	3075.49	0.124323
$850$	0	0
$851$	−5123.93	−0.206399
$852$	0	0
$853$	−27051.0	−1.08583	−0.542913	−	0.839789i	$-0.682679\pi$
$853$	−27051.0	−1.08583	−0.542913	+	0.839789i	$0.682679\pi$
$854$	0	0
$855$	−7777.75	−0.311103
$856$	0	0
$857$	29947.2	1.19367	0.596835	−	0.802364i	$-0.296425\pi$
$857$	29947.2	1.19367	0.596835	+	0.802364i	$0.296425\pi$
$858$	0	0
$859$	−35287.3	−1.40162	−0.700808	−	0.713350i	$-0.747177\pi$
$859$	−35287.3	−1.40162	−0.700808	+	0.713350i	$0.747177\pi$
$860$	0	0
$861$	−2451.56	−0.0970373
$862$	0	0
$863$	19405.5	0.765436	0.382718	−	0.923865i	$-0.374988\pi$
$863$	19405.5	0.765436	0.382718	+	0.923865i	$0.374988\pi$
$864$	0	0
$865$	19271.8	0.757527
$866$	0	0
$867$	−2469.69	−0.0967419
$868$	0	0
$869$	2750.70	0.107377
$870$	0	0
$871$	9968.12	0.387780
$872$	0	0
$873$	−24020.7	−0.931247
$874$	0	0
$875$	−3109.34	−0.120131
$876$	0	0
$877$	−15209.3	−0.585611	−0.292806	−	0.956172i	$-0.594589\pi$
$877$	−15209.3	−0.585611	−0.292806	+	0.956172i	$0.594589\pi$
$878$	0	0
$879$	3525.98	0.135300
$880$	0	0
$881$	3311.62	0.126642	0.0633208	−	0.997993i	$-0.479831\pi$
$881$	3311.62	0.126642	0.0633208	+	0.997993i	$0.479831\pi$
$882$	0	0
$883$	6831.60	0.260364	0.130182	−	0.991490i	$-0.458444\pi$
$883$	6831.60	0.260364	0.130182	+	0.991490i	$0.458444\pi$
$884$	0	0
$885$	−1904.44	−0.0723355
$886$	0	0
$887$	13884.3	0.525580	0.262790	−	0.964853i	$-0.415357\pi$
$887$	13884.3	0.525580	0.262790	+	0.964853i	$0.415357\pi$
$888$	0	0
$889$	20837.8	0.786137
$890$	0	0
$891$	3552.67	0.133579
$892$	0	0
$893$	−9708.88	−0.363824
$894$	0	0
$895$	9596.05	0.358392
$896$	0	0
$897$	−1232.14	−0.0458641
$898$	0	0
$899$	−14653.1	−0.543614
$900$	0	0
$901$	−6664.65	−0.246428
$902$	0	0
$903$	−5820.93	−0.214516
$904$	0	0
$905$	−18150.0	−0.666659
$906$	0	0
$907$	−37030.3	−1.35564	−0.677822	−	0.735226i	$-0.737076\pi$
$907$	−37030.3	−1.35564	−0.677822	+	0.735226i	$0.737076\pi$
$908$	0	0
$909$	19415.9	0.708455
$910$	0	0
$911$	−47449.9	−1.72567	−0.862835	−	0.505486i	$-0.831313\pi$
$911$	−47449.9	−1.72567	−0.862835	+	0.505486i	$0.831313\pi$
$912$	0	0
$913$	666.318	0.0241532
$914$	0	0
$915$	547.244	0.0197720
$916$	0	0
$917$	−43249.1	−1.55748
$918$	0	0
$919$	29898.3	1.07318	0.536591	−	0.843842i	$-0.319712\pi$
$919$	29898.3	1.07318	0.536591	+	0.843842i	$0.319712\pi$
$920$	0	0
$921$	1743.50	0.0623781
$922$	0	0
$923$	−85711.9	−3.05660
$924$	0	0
$925$	5569.49	0.197971
$926$	0	0
$927$	−2725.19	−0.0965556
$928$	0	0
$929$	49717.8	1.75585	0.877926	−	0.478796i	$-0.158927\pi$
$929$	49717.8	1.75585	0.877926	+	0.478796i	$0.158927\pi$
$930$	0	0
$931$	−16083.8	−0.566192
$932$	0	0
$933$	499.950	0.0175430
$934$	0	0
$935$	2426.97	0.0848882
$936$	0	0
$937$	52313.2	1.82390	0.911952	−	0.410296i	$-0.134575\pi$
$937$	52313.2	1.82390	0.911952	+	0.410296i	$0.134575\pi$
$938$	0	0
$939$	4048.63	0.140705
$940$	0	0
$941$	16317.5	0.565288	0.282644	−	0.959225i	$-0.408788\pi$
$941$	16317.5	0.565288	0.282644	+	0.959225i	$0.408788\pi$
$942$	0	0
$943$	−3941.95	−0.136127
$944$	0	0
$945$	−3838.45	−0.132132
$946$	0	0
$947$	24217.1	0.830991	0.415496	−	0.909595i	$-0.363608\pi$
$947$	24217.1	0.830991	0.415496	+	0.909595i	$0.363608\pi$
$948$	0	0
$949$	60502.2	2.06953
$950$	0	0
$951$	2133.83	0.0727595
$952$	0	0
$953$	−47824.3	−1.62558	−0.812792	−	0.582554i	$-0.802053\pi$
$953$	−47824.3	−1.62558	−0.812792	+	0.582554i	$0.802053\pi$
$954$	0	0
$955$	−7151.01	−0.242305
$956$	0	0
$957$	−247.408	−0.00835691
$958$	0	0
$959$	42434.7	1.42887
$960$	0	0
$961$	−110.513	−0.00370960
$962$	0	0
$963$	657.958	0.0220170
$964$	0	0
$965$	−5094.14	−0.169934
$966$	0	0
$967$	−10700.9	−0.355861	−0.177930	−	0.984043i	$-0.556940\pi$
$967$	−10700.9	−0.355861	−0.177930	+	0.984043i	$0.556940\pi$
$968$	0	0
$969$	−3218.48	−0.106700
$970$	0	0
$971$	55.2528	0.00182610	0.000913052	−	1.00000i	$-0.499709\pi$
$971$	55.2528	0.00182610	0.000913052		1.00000i	$0.499709\pi$
$972$	0	0
$973$	71282.3	2.34862
$974$	0	0
$975$	1339.29	0.0439913
$976$	0	0
$977$	18487.8	0.605402	0.302701	−	0.953086i	$-0.402112\pi$
$977$	18487.8	0.605402	0.302701	+	0.953086i	$0.402112\pi$
$978$	0	0
$979$	3987.41	0.130172
$980$	0	0
$981$	51950.5	1.69078
$982$	0	0
$983$	−34125.0	−1.10724	−0.553621	−	0.832769i	$-0.686754\pi$
$983$	−34125.0	−1.10724	−0.553621	+	0.832769i	$0.686754\pi$
$984$	0	0
$985$	−16988.7	−0.549549
$986$	0	0
$987$	−2380.98	−0.0767858
$988$	0	0
$989$	−9359.67	−0.300930
$990$	0	0
$991$	−59074.7	−1.89361	−0.946806	−	0.321804i	$-0.895711\pi$
$991$	−59074.7	−1.89361	−0.946806	+	0.321804i	$0.895711\pi$
$992$	0	0
$993$	4494.44	0.143632
$994$	0	0
$995$	2368.87	0.0754756
$996$	0	0
$997$	25881.1	0.822130	0.411065	−	0.911606i	$-0.365157\pi$
$997$	25881.1	0.822130	0.411065	+	0.911606i	$0.365157\pi$
$998$	0	0
$999$	6875.48	0.217748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.bb.1.5		10
4.3	odd	2		920.4.a.g.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.g.1.6	✓	10	4.3	odd	2
1840.4.a.bb.1.5		10	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)

\(f(q)\)	\(=\)	\(q-0.575045 q^{3} -5.00000 q^{5} +24.8747 q^{7} -26.6693 q^{9} +O(q^{10})\)	\(q-0.575045 q^{3} -5.00000 q^{5} +24.8747 q^{7} -26.6693 q^{9} +5.05844 q^{11} -93.1604 q^{13} +2.87522 q^{15} -95.9572 q^{17} -58.3273 q^{19} -14.3041 q^{21} -23.0000 q^{23} +25.0000 q^{25} +30.8623 q^{27} +85.0540 q^{29} -172.280 q^{31} -2.90883 q^{33} -124.373 q^{35} +222.779 q^{37} +53.5714 q^{39} +171.389 q^{41} +406.942 q^{43} +133.347 q^{45} +166.455 q^{47} +275.750 q^{49} +55.1797 q^{51} +69.4544 q^{53} -25.2922 q^{55} +33.5408 q^{57} -662.361 q^{59} +190.331 q^{61} -663.391 q^{63} +465.802 q^{65} -106.999 q^{67} +13.2260 q^{69} +920.046 q^{71} -649.441 q^{73} -14.3761 q^{75} +125.827 q^{77} +543.783 q^{79} +702.325 q^{81} +131.724 q^{83} +479.786 q^{85} -48.9099 q^{87} +788.269 q^{89} -2317.34 q^{91} +99.0689 q^{93} +291.637 q^{95} +900.688 q^{97} -134.905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9}+O(q^{10}) \) 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9	\( 10 q + q^{3} - 50 q^{5} - 28 q^{7} + 139 q^{9} + 14 q^{11} + 11 q^{13} - 5 q^{15} + 68 q^{17} - 114 q^{19} - 232 q^{21} - 230 q^{23} + 250 q^{25} + 433 q^{27} - 273 q^{29} + 129 q^{31} + 98 q^{33} + 140 q^{35} + 62 q^{37} - 283 q^{39} + 767 q^{41} - 332 q^{43} - 695 q^{45} + 323 q^{47} + 1162 q^{49} - 176 q^{51} + 558 q^{53} - 70 q^{55} + 46 q^{57} - 822 q^{59} + 318 q^{61} - 2698 q^{63} - 55 q^{65} - 1152 q^{67} - 23 q^{69} - 1247 q^{71} + 1941 q^{73} + 25 q^{75} + 528 q^{77} - 3134 q^{79} + 6210 q^{81} - 482 q^{83} - 340 q^{85} - 1797 q^{87} + 4734 q^{89} - 4992 q^{91} + 4647 q^{93} + 570 q^{95} + 2326 q^{97} - 4356 q^{99}+O(q^{100}) \) 10 * q + q^3 - 50 * q^5 - 28 * q^7 + 139 * q^9 + 14 * q^11 + 11 * q^13 - 5 * q^15 + 68 * q^17 - 114 * q^19 - 232 * q^21 - 230 * q^23 + 250 * q^25 + 433 * q^27 - 273 * q^29 + 129 * q^31 + 98 * q^33 + 140 * q^35 + 62 * q^37 - 283 * q^39 + 767 * q^41 - 332 * q^43 - 695 * q^45 + 323 * q^47 + 1162 * q^49 - 176 * q^51 + 558 * q^53 - 70 * q^55 + 46 * q^57 - 822 * q^59 + 318 * q^61 - 2698 * q^63 - 55 * q^65 - 1152 * q^67 - 23 * q^69 - 1247 * q^71 + 1941 * q^73 + 25 * q^75 + 528 * q^77 - 3134 * q^79 + 6210 * q^81 - 482 * q^83 - 340 * q^85 - 1797 * q^87 + 4734 * q^89 - 4992 * q^91 + 4647 * q^93 + 570 * q^95 + 2326 * q^97 - 4356 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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