LMFDB - Embedded newform 1840.4.a.z.1.7

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 148x^{7} + 278x^{6} + 6502x^{5} - 4928x^{4} - 87343x^{3} + 42737x^{2} + 286800x + 53104 $ x^9 - 3x^8 - 148x^7 + 278x^6 + 6502x^5 - 4928x^4 - 87343x^3 + 42737x^2 + 286800x + 53104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.15779$ 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)

$f(q)$	$=$	$q+3.15779 q^{3} -5.00000 q^{5} -11.8460 q^{7} -17.0284 q^{9} +O(q^{10})$	$q+3.15779 q^{3} -5.00000 q^{5} -11.8460 q^{7} -17.0284 q^{9} +63.8452 q^{11} +11.1240 q^{13} -15.7889 q^{15} -3.42310 q^{17} +20.8600 q^{19} -37.4072 q^{21} +23.0000 q^{23} +25.0000 q^{25} -139.032 q^{27} -264.532 q^{29} -129.604 q^{31} +201.610 q^{33} +59.2301 q^{35} +248.679 q^{37} +35.1273 q^{39} -285.414 q^{41} -213.641 q^{43} +85.1419 q^{45} +627.043 q^{47} -202.672 q^{49} -10.8094 q^{51} +402.287 q^{53} -319.226 q^{55} +65.8713 q^{57} -525.629 q^{59} +632.936 q^{61} +201.718 q^{63} -55.6201 q^{65} +21.9976 q^{67} +72.6291 q^{69} +878.981 q^{71} -453.654 q^{73} +78.9447 q^{75} -756.311 q^{77} +631.616 q^{79} +20.7322 q^{81} -1116.40 q^{83} +17.1155 q^{85} -835.334 q^{87} +653.332 q^{89} -131.775 q^{91} -409.261 q^{93} -104.300 q^{95} +333.205 q^{97} -1087.18 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9	$ 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9} - 22 q^{11} + 23 q^{13} - 15 q^{15} - 135 q^{17} - 102 q^{19} + 54 q^{21} + 207 q^{23} + 225 q^{25} + 363 q^{27} + 280 q^{29} - 168 q^{31} + 28 q^{33} + 125 q^{35} + 153 q^{37} + 5 q^{39} - 502 q^{41} - 110 q^{43} - 310 q^{45} + 153 q^{47} + 764 q^{49} - 924 q^{51} - 273 q^{53} + 110 q^{55} + 748 q^{57} - 827 q^{59} + 1976 q^{61} - 2237 q^{63} - 115 q^{65} - 1613 q^{67} + 69 q^{69} - 1370 q^{71} + 425 q^{73} + 75 q^{75} + 2006 q^{77} - 2624 q^{79} + 1729 q^{81} - 2505 q^{83} + 675 q^{85} - 1591 q^{87} + 1120 q^{89} - 2392 q^{91} + 4401 q^{93} + 510 q^{95} + 2026 q^{97} - 3206 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9 - 22 * q^11 + 23 * q^13 - 15 * q^15 - 135 * q^17 - 102 * q^19 + 54 * q^21 + 207 * q^23 + 225 * q^25 + 363 * q^27 + 280 * q^29 - 168 * q^31 + 28 * q^33 + 125 * q^35 + 153 * q^37 + 5 * q^39 - 502 * q^41 - 110 * q^43 - 310 * q^45 + 153 * q^47 + 764 * q^49 - 924 * q^51 - 273 * q^53 + 110 * q^55 + 748 * q^57 - 827 * q^59 + 1976 * q^61 - 2237 * q^63 - 115 * q^65 - 1613 * q^67 + 69 * q^69 - 1370 * q^71 + 425 * q^73 + 75 * q^75 + 2006 * q^77 - 2624 * q^79 + 1729 * q^81 - 2505 * q^83 + 675 * q^85 - 1591 * q^87 + 1120 * q^89 - 2392 * q^91 + 4401 * q^93 + 510 * q^95 + 2026 * q^97 - 3206 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.15779	0.607716	0.303858	−	0.952717i	$-0.401725\pi$
$3$	3.15779	0.607716	0.303858	+	0.952717i	$0.401725\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−11.8460	−0.639624	−0.319812	−	0.947481i	$-0.603620\pi$
$7$	−11.8460	−0.639624	−0.319812	+	0.947481i	$0.603620\pi$
$8$	0	0
$9$	−17.0284	−0.630681
$10$	0	0
$11$	63.8452	1.75001	0.875003	−	0.484118i	$-0.160859\pi$
$11$	63.8452	1.75001	0.875003	+	0.484118i	$0.160859\pi$
$12$	0	0
$13$	11.1240	0.237327	0.118663	−	0.992935i	$-0.462139\pi$
$13$	11.1240	0.237327	0.118663	+	0.992935i	$0.462139\pi$
$14$	0	0
$15$	−15.7889	−0.271779
$16$	0	0
$17$	−3.42310	−0.0488367	−0.0244183	−	0.999702i	$-0.507773\pi$
$17$	−3.42310	−0.0488367	−0.0244183	+	0.999702i	$0.507773\pi$
$18$	0	0
$19$	20.8600	0.251874	0.125937	−	0.992038i	$-0.459806\pi$
$19$	20.8600	0.251874	0.125937	+	0.992038i	$0.459806\pi$
$20$	0	0
$21$	−37.4072	−0.388710
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−139.032	−0.990991
$28$	0	0
$29$	−264.532	−1.69387	−0.846936	−	0.531695i	$-0.821555\pi$
$29$	−264.532	−1.69387	−0.846936	+	0.531695i	$0.821555\pi$
$30$	0	0
$31$	−129.604	−0.750887	−0.375443	−	0.926845i	$-0.622510\pi$
$31$	−129.604	−0.750887	−0.375443	+	0.926845i	$0.622510\pi$
$32$	0	0
$33$	201.610	1.06351
$34$	0	0
$35$	59.2301	0.286049
$36$	0	0
$37$	248.679	1.10494	0.552468	−	0.833534i	$-0.313686\pi$
$37$	248.679	1.10494	0.552468	+	0.833534i	$0.313686\pi$
$38$	0	0
$39$	35.1273	0.144227
$40$	0	0
$41$	−285.414	−1.08717	−0.543587	−	0.839353i	$-0.682934\pi$
$41$	−285.414	−1.08717	−0.543587	+	0.839353i	$0.682934\pi$
$42$	0	0
$43$	−213.641	−0.757673	−0.378837	−	0.925464i	$-0.623676\pi$
$43$	−213.641	−0.757673	−0.378837	+	0.925464i	$0.623676\pi$
$44$	0	0
$45$	85.1419	0.282049
$46$	0	0
$47$	627.043	1.94603	0.973017	−	0.230732i	$-0.0741121\pi$
$47$	627.043	1.94603	0.973017	+	0.230732i	$0.0741121\pi$
$48$	0	0
$49$	−202.672	−0.590881
$50$	0	0
$51$	−10.8094	−0.0296788
$52$	0	0
$53$	402.287	1.04261	0.521306	−	0.853370i	$-0.325445\pi$
$53$	402.287	1.04261	0.521306	+	0.853370i	$0.325445\pi$
$54$	0	0
$55$	−319.226	−0.782626
$56$	0	0
$57$	65.8713	0.153068
$58$	0	0
$59$	−525.629	−1.15985	−0.579924	−	0.814670i	$-0.696918\pi$
$59$	−525.629	−1.15985	−0.579924	+	0.814670i	$0.696918\pi$
$60$	0	0
$61$	632.936	1.32851	0.664255	−	0.747506i	$-0.268749\pi$
$61$	632.936	1.32851	0.664255	+	0.747506i	$0.268749\pi$
$62$	0	0
$63$	201.718	0.403399
$64$	0	0
$65$	−55.6201	−0.106136
$66$	0	0
$67$	21.9976	0.0401110	0.0200555	−	0.999799i	$-0.493616\pi$
$67$	21.9976	0.0401110	0.0200555	+	0.999799i	$0.493616\pi$
$68$	0	0
$69$	72.6291	0.126718
$70$	0	0
$71$	878.981	1.46924	0.734619	−	0.678480i	$-0.237361\pi$
$71$	878.981	1.46924	0.734619	+	0.678480i	$0.237361\pi$
$72$	0	0
$73$	−453.654	−0.727345	−0.363672	−	0.931527i	$-0.618477\pi$
$73$	−453.654	−0.727345	−0.363672	+	0.931527i	$0.618477\pi$
$74$	0	0
$75$	78.9447	0.121543
$76$	0	0
$77$	−756.311	−1.11935
$78$	0	0
$79$	631.616	0.899523	0.449762	−	0.893149i	$-0.351509\pi$
$79$	631.616	0.899523	0.449762	+	0.893149i	$0.351509\pi$
$80$	0	0
$81$	20.7322	0.0284393
$82$	0	0
$83$	−1116.40	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$83$	−1116.40	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$84$	0	0
$85$	17.1155	0.0218404
$86$	0	0
$87$	−835.334	−1.02939
$88$	0	0
$89$	653.332	0.778125	0.389062	−	0.921211i	$-0.372799\pi$
$89$	653.332	0.778125	0.389062	+	0.921211i	$0.372799\pi$
$90$	0	0
$91$	−131.775	−0.151800
$92$	0	0
$93$	−409.261	−0.456326
$94$	0	0
$95$	−104.300	−0.112641
$96$	0	0
$97$	333.205	0.348782	0.174391	−	0.984677i	$-0.444204\pi$
$97$	333.205	0.348782	0.174391	+	0.984677i	$0.444204\pi$
$98$	0	0
$99$	−1087.18	−1.10370
$100$	0	0
$101$	371.547	0.366043	0.183022	−	0.983109i	$-0.441412\pi$
$101$	371.547	0.366043	0.183022	+	0.983109i	$0.441412\pi$
$102$	0	0
$103$	−2079.94	−1.98973	−0.994867	−	0.101188i	$-0.967736\pi$
$103$	−2079.94	−1.98973	−0.994867	+	0.101188i	$0.967736\pi$
$104$	0	0
$105$	187.036	0.173836
$106$	0	0
$107$	−1513.50	−1.36744	−0.683718	−	0.729746i	$-0.739638\pi$
$107$	−1513.50	−1.36744	−0.683718	+	0.729746i	$0.739638\pi$
$108$	0	0
$109$	−1503.45	−1.32114	−0.660569	−	0.750765i	$-0.729685\pi$
$109$	−1503.45	−1.32114	−0.660569	+	0.750765i	$0.729685\pi$
$110$	0	0
$111$	785.276	0.671487
$112$	0	0
$113$	−23.4051	−0.0194847	−0.00974234	−	0.999953i	$-0.503101\pi$
$113$	−23.4051	−0.0194847	−0.00974234	+	0.999953i	$0.503101\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−189.424	−0.149678
$118$	0	0
$119$	40.5501	0.0312371
$120$	0	0
$121$	2745.21	2.06252
$122$	0	0
$123$	−901.276	−0.660694
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2061.58	−1.44044	−0.720218	−	0.693748i	$-0.755958\pi$
$127$	−2061.58	−1.44044	−0.720218	+	0.693748i	$0.755958\pi$
$128$	0	0
$129$	−674.633	−0.460450
$130$	0	0
$131$	−2366.12	−1.57808	−0.789040	−	0.614342i	$-0.789422\pi$
$131$	−2366.12	−1.57808	−0.789040	+	0.614342i	$0.789422\pi$
$132$	0	0
$133$	−247.107	−0.161105
$134$	0	0
$135$	695.161	0.443185
$136$	0	0
$137$	129.817	0.0809563	0.0404781	−	0.999180i	$-0.487112\pi$
$137$	129.817	0.0809563	0.0404781	+	0.999180i	$0.487112\pi$
$138$	0	0
$139$	−2560.37	−1.56236	−0.781178	−	0.624309i	$-0.785381\pi$
$139$	−2560.37	−1.56236	−0.781178	+	0.624309i	$0.785381\pi$
$140$	0	0
$141$	1980.07	1.18264
$142$	0	0
$143$	710.216	0.415324
$144$	0	0
$145$	1322.66	0.757522
$146$	0	0
$147$	−639.995	−0.359088
$148$	0	0
$149$	−1755.48	−0.965198	−0.482599	−	0.875841i	$-0.660307\pi$
$149$	−1755.48	−0.965198	−0.482599	+	0.875841i	$0.660307\pi$
$150$	0	0
$151$	−2963.74	−1.59726	−0.798629	−	0.601824i	$-0.794441\pi$
$151$	−2963.74	−1.59726	−0.798629	+	0.601824i	$0.794441\pi$
$152$	0	0
$153$	58.2898	0.0308004
$154$	0	0
$155$	648.018	0.335807
$156$	0	0
$157$	−338.854	−0.172252	−0.0861259	−	0.996284i	$-0.527449\pi$
$157$	−338.854	−0.172252	−0.0861259	+	0.996284i	$0.527449\pi$
$158$	0	0
$159$	1270.34	0.633612
$160$	0	0
$161$	−272.458	−0.133371
$162$	0	0
$163$	586.416	0.281789	0.140895	−	0.990025i	$-0.455002\pi$
$163$	586.416	0.281789	0.140895	+	0.990025i	$0.455002\pi$
$164$	0	0
$165$	−1008.05	−0.475615
$166$	0	0
$167$	−177.382	−0.0821928	−0.0410964	−	0.999155i	$-0.513085\pi$
$167$	−177.382	−0.0821928	−0.0410964	+	0.999155i	$0.513085\pi$
$168$	0	0
$169$	−2073.26	−0.943676
$170$	0	0
$171$	−355.212	−0.158852
$172$	0	0
$173$	−772.790	−0.339619	−0.169810	−	0.985477i	$-0.554315\pi$
$173$	−772.790	−0.339619	−0.169810	+	0.985477i	$0.554315\pi$
$174$	0	0
$175$	−296.150	−0.127925
$176$	0	0
$177$	−1659.82	−0.704859
$178$	0	0
$179$	−3860.47	−1.61199	−0.805993	−	0.591926i	$-0.798368\pi$
$179$	−3860.47	−1.61199	−0.805993	+	0.591926i	$0.798368\pi$
$180$	0	0
$181$	1839.83	0.755544	0.377772	−	0.925899i	$-0.376690\pi$
$181$	1839.83	0.755544	0.377772	+	0.925899i	$0.376690\pi$
$182$	0	0
$183$	1998.68	0.807357
$184$	0	0
$185$	−1243.40	−0.494142
$186$	0	0
$187$	−218.549	−0.0854645
$188$	0	0
$189$	1646.98	0.633862
$190$	0	0
$191$	227.205	0.0860731	0.0430365	−	0.999073i	$-0.486297\pi$
$191$	227.205	0.0860731	0.0430365	+	0.999073i	$0.486297\pi$
$192$	0	0
$193$	−4160.08	−1.55155	−0.775775	−	0.631010i	$-0.782641\pi$
$193$	−4160.08	−1.55155	−0.775775	+	0.631010i	$0.782641\pi$
$194$	0	0
$195$	−175.637	−0.0645005
$196$	0	0
$197$	−3206.25	−1.15957	−0.579787	−	0.814768i	$-0.696864\pi$
$197$	−3206.25	−1.15957	−0.579787	+	0.814768i	$0.696864\pi$
$198$	0	0
$199$	760.108	0.270767	0.135383	−	0.990793i	$-0.456773\pi$
$199$	760.108	0.270767	0.135383	+	0.990793i	$0.456773\pi$
$200$	0	0
$201$	69.4638	0.0243761
$202$	0	0
$203$	3133.64	1.08344
$204$	0	0
$205$	1427.07	0.486199
$206$	0	0
$207$	−391.653	−0.131506
$208$	0	0
$209$	1331.81	0.440781
$210$	0	0
$211$	−2486.34	−0.811217	−0.405609	−	0.914047i	$-0.632940\pi$
$211$	−2486.34	−0.811217	−0.405609	+	0.914047i	$0.632940\pi$
$212$	0	0
$213$	2775.64	0.892880
$214$	0	0
$215$	1068.21	0.338842
$216$	0	0
$217$	1535.29	0.480286
$218$	0	0
$219$	−1432.54	−0.442019
$220$	0	0
$221$	−38.0787	−0.0115903
$222$	0	0
$223$	−3552.58	−1.06681	−0.533405	−	0.845860i	$-0.679088\pi$
$223$	−3552.58	−1.06681	−0.533405	+	0.845860i	$0.679088\pi$
$224$	0	0
$225$	−425.710	−0.126136
$226$	0	0
$227$	5222.12	1.52689	0.763446	−	0.645872i	$-0.223506\pi$
$227$	5222.12	1.52689	0.763446	+	0.645872i	$0.223506\pi$
$228$	0	0
$229$	4959.47	1.43114	0.715570	−	0.698541i	$-0.246167\pi$
$229$	4959.47	1.43114	0.715570	+	0.698541i	$0.246167\pi$
$230$	0	0
$231$	−2388.27	−0.680245
$232$	0	0
$233$	−2742.66	−0.771149	−0.385574	−	0.922677i	$-0.625997\pi$
$233$	−2742.66	−0.771149	−0.385574	+	0.922677i	$0.625997\pi$
$234$	0	0
$235$	−3135.22	−0.870293
$236$	0	0
$237$	1994.51	0.546655
$238$	0	0
$239$	718.602	0.194487	0.0972437	−	0.995261i	$-0.468997\pi$
$239$	718.602	0.194487	0.0972437	+	0.995261i	$0.468997\pi$
$240$	0	0
$241$	−2192.41	−0.585997	−0.292998	−	0.956113i	$-0.594653\pi$
$241$	−2192.41	−0.585997	−0.292998	+	0.956113i	$0.594653\pi$
$242$	0	0
$243$	3819.34	1.00827
$244$	0	0
$245$	1013.36	0.264250
$246$	0	0
$247$	232.047	0.0597765
$248$	0	0
$249$	−3525.37	−0.897234
$250$	0	0
$251$	140.983	0.0354533	0.0177266	−	0.999843i	$-0.494357\pi$
$251$	140.983	0.0354533	0.0177266	+	0.999843i	$0.494357\pi$
$252$	0	0
$253$	1468.44	0.364901
$254$	0	0
$255$	54.0471	0.0132728
$256$	0	0
$257$	6760.27	1.64083	0.820417	−	0.571766i	$-0.193741\pi$
$257$	6760.27	1.64083	0.820417	+	0.571766i	$0.193741\pi$
$258$	0	0
$259$	−2945.86	−0.706744
$260$	0	0
$261$	4504.54	1.06829
$262$	0	0
$263$	−2710.40	−0.635476	−0.317738	−	0.948179i	$-0.602923\pi$
$263$	−2710.40	−0.635476	−0.317738	+	0.948179i	$0.602923\pi$
$264$	0	0
$265$	−2011.44	−0.466270
$266$	0	0
$267$	2063.08	0.472879
$268$	0	0
$269$	3966.97	0.899146	0.449573	−	0.893244i	$-0.351576\pi$
$269$	3966.97	0.899146	0.449573	+	0.893244i	$0.351576\pi$
$270$	0	0
$271$	5355.51	1.20046	0.600229	−	0.799828i	$-0.295076\pi$
$271$	5355.51	1.20046	0.600229	+	0.799828i	$0.295076\pi$
$272$	0	0
$273$	−416.119	−0.0922514
$274$	0	0
$275$	1596.13	0.350001
$276$	0	0
$277$	−4871.21	−1.05662	−0.528308	−	0.849053i	$-0.677174\pi$
$277$	−4871.21	−1.05662	−0.528308	+	0.849053i	$0.677174\pi$
$278$	0	0
$279$	2206.94	0.473570
$280$	0	0
$281$	8637.61	1.83372	0.916862	−	0.399205i	$-0.130714\pi$
$281$	8637.61	1.83372	0.916862	+	0.399205i	$0.130714\pi$
$282$	0	0
$283$	−3862.43	−0.811299	−0.405649	−	0.914029i	$-0.632955\pi$
$283$	−3862.43	−0.811299	−0.405649	+	0.914029i	$0.632955\pi$
$284$	0	0
$285$	−329.357	−0.0684541
$286$	0	0
$287$	3381.01	0.695383
$288$	0	0
$289$	−4901.28	−0.997615
$290$	0	0
$291$	1052.19	0.211960
$292$	0	0
$293$	−9257.97	−1.84593	−0.922963	−	0.384889i	$-0.874240\pi$
$293$	−9257.97	−1.84593	−0.922963	+	0.384889i	$0.874240\pi$
$294$	0	0
$295$	2628.14	0.518700
$296$	0	0
$297$	−8876.55	−1.73424
$298$	0	0
$299$	255.853	0.0494861
$300$	0	0
$301$	2530.79	0.484626
$302$	0	0
$303$	1173.27	0.222450
$304$	0	0
$305$	−3164.68	−0.594128
$306$	0	0
$307$	−2103.21	−0.390998	−0.195499	−	0.980704i	$-0.562633\pi$
$307$	−2103.21	−0.390998	−0.195499	+	0.980704i	$0.562633\pi$
$308$	0	0
$309$	−6568.01	−1.20919
$310$	0	0
$311$	−9697.14	−1.76808	−0.884042	−	0.467407i	$-0.845188\pi$
$311$	−9697.14	−1.76808	−0.884042	+	0.467407i	$0.845188\pi$
$312$	0	0
$313$	−7079.61	−1.27848	−0.639238	−	0.769009i	$-0.720750\pi$
$313$	−7079.61	−1.27848	−0.639238	+	0.769009i	$0.720750\pi$
$314$	0	0
$315$	−1008.59	−0.180405
$316$	0	0
$317$	8795.99	1.55846	0.779231	−	0.626737i	$-0.215610\pi$
$317$	8795.99	1.55846	0.779231	+	0.626737i	$0.215610\pi$
$318$	0	0
$319$	−16889.1	−2.96429
$320$	0	0
$321$	−4779.31	−0.831013
$322$	0	0
$323$	−71.4057	−0.0123007
$324$	0	0
$325$	278.101	0.0474654
$326$	0	0
$327$	−4747.56	−0.802877
$328$	0	0
$329$	−7427.96	−1.24473
$330$	0	0
$331$	8753.55	1.45359	0.726796	−	0.686854i	$-0.241009\pi$
$331$	8753.55	1.45359	0.726796	+	0.686854i	$0.241009\pi$
$332$	0	0
$333$	−4234.61	−0.696862
$334$	0	0
$335$	−109.988	−0.0179382
$336$	0	0
$337$	2723.80	0.440281	0.220140	−	0.975468i	$-0.429348\pi$
$337$	2723.80	0.440281	0.220140	+	0.975468i	$0.429348\pi$
$338$	0	0
$339$	−73.9084	−0.0118412
$340$	0	0
$341$	−8274.57	−1.31406
$342$	0	0
$343$	6464.04	1.01757
$344$	0	0
$345$	−363.145	−0.0566698
$346$	0	0
$347$	529.354	0.0818939	0.0409470	−	0.999161i	$-0.486963\pi$
$347$	529.354	0.0818939	0.0409470	+	0.999161i	$0.486963\pi$
$348$	0	0
$349$	−8397.03	−1.28792	−0.643958	−	0.765061i	$-0.722709\pi$
$349$	−8397.03	−1.28792	−0.643958	+	0.765061i	$0.722709\pi$
$350$	0	0
$351$	−1546.60	−0.235189
$352$	0	0
$353$	6986.31	1.05338	0.526691	−	0.850057i	$-0.323433\pi$
$353$	6986.31	1.05338	0.526691	+	0.850057i	$0.323433\pi$
$354$	0	0
$355$	−4394.91	−0.657063
$356$	0	0
$357$	128.048	0.0189833
$358$	0	0
$359$	9638.32	1.41697	0.708483	−	0.705728i	$-0.249380\pi$
$359$	9638.32	1.41697	0.708483	+	0.705728i	$0.249380\pi$
$360$	0	0
$361$	−6423.86	−0.936560
$362$	0	0
$363$	8668.80	1.25343
$364$	0	0
$365$	2268.27	0.325278
$366$	0	0
$367$	2460.84	0.350013	0.175006	−	0.984567i	$-0.444005\pi$
$367$	2460.84	0.350013	0.175006	+	0.984567i	$0.444005\pi$
$368$	0	0
$369$	4860.14	0.685660
$370$	0	0
$371$	−4765.50	−0.666880
$372$	0	0
$373$	−8721.68	−1.21070	−0.605350	−	0.795959i	$-0.706967\pi$
$373$	−8721.68	−1.21070	−0.605350	+	0.795959i	$0.706967\pi$
$374$	0	0
$375$	−394.723	−0.0543558
$376$	0	0
$377$	−2942.66	−0.402001
$378$	0	0
$379$	10772.0	1.45995	0.729975	−	0.683474i	$-0.239532\pi$
$379$	10772.0	1.45995	0.729975	+	0.683474i	$0.239532\pi$
$380$	0	0
$381$	−6510.02	−0.875376
$382$	0	0
$383$	−5078.01	−0.677478	−0.338739	−	0.940880i	$-0.610000\pi$
$383$	−5078.01	−0.677478	−0.338739	+	0.940880i	$0.610000\pi$
$384$	0	0
$385$	3781.56	0.500587
$386$	0	0
$387$	3637.96	0.477850
$388$	0	0
$389$	8960.07	1.16785	0.583925	−	0.811808i	$-0.301516\pi$
$389$	8960.07	1.16785	0.583925	+	0.811808i	$0.301516\pi$
$390$	0	0
$391$	−78.7313	−0.0101831
$392$	0	0
$393$	−7471.69	−0.959025
$394$	0	0
$395$	−3158.08	−0.402279
$396$	0	0
$397$	3022.24	0.382070	0.191035	−	0.981583i	$-0.438816\pi$
$397$	3022.24	0.382070	0.191035	+	0.981583i	$0.438816\pi$
$398$	0	0
$399$	−780.313	−0.0979060
$400$	0	0
$401$	−6861.95	−0.854537	−0.427268	−	0.904125i	$-0.640524\pi$
$401$	−6861.95	−0.854537	−0.427268	+	0.904125i	$0.640524\pi$
$402$	0	0
$403$	−1441.71	−0.178206
$404$	0	0
$405$	−103.661	−0.0127184
$406$	0	0
$407$	15877.0	1.93364
$408$	0	0
$409$	2337.14	0.282553	0.141276	−	0.989970i	$-0.454879\pi$
$409$	2337.14	0.282553	0.141276	+	0.989970i	$0.454879\pi$
$410$	0	0
$411$	409.934	0.0491985
$412$	0	0
$413$	6226.61	0.741867
$414$	0	0
$415$	5582.02	0.660267
$416$	0	0
$417$	−8085.09	−0.949469
$418$	0	0
$419$	−7822.12	−0.912018	−0.456009	−	0.889975i	$-0.650722\pi$
$419$	−7822.12	−0.912018	−0.456009	+	0.889975i	$0.650722\pi$
$420$	0	0
$421$	10793.9	1.24956	0.624780	−	0.780801i	$-0.285189\pi$
$421$	10793.9	1.24956	0.624780	+	0.780801i	$0.285189\pi$
$422$	0	0
$423$	−10677.5	−1.22733
$424$	0	0
$425$	−85.5775	−0.00976733
$426$	0	0
$427$	−7497.77	−0.849748
$428$	0	0
$429$	2242.71	0.252399
$430$	0	0
$431$	−7455.77	−0.833253	−0.416626	−	0.909078i	$-0.636788\pi$
$431$	−7455.77	−0.833253	−0.416626	+	0.909078i	$0.636788\pi$
$432$	0	0
$433$	9031.24	1.00234	0.501170	−	0.865349i	$-0.332903\pi$
$433$	9031.24	1.00234	0.501170	+	0.865349i	$0.332903\pi$
$434$	0	0
$435$	4176.67	0.460359
$436$	0	0
$437$	479.779	0.0525194
$438$	0	0
$439$	14691.8	1.59727	0.798635	−	0.601816i	$-0.205556\pi$
$439$	14691.8	1.59727	0.798635	+	0.601816i	$0.205556\pi$
$440$	0	0
$441$	3451.18	0.372657
$442$	0	0
$443$	4081.88	0.437779	0.218890	−	0.975750i	$-0.429757\pi$
$443$	4081.88	0.437779	0.218890	+	0.975750i	$0.429757\pi$
$444$	0	0
$445$	−3266.66	−0.347988
$446$	0	0
$447$	−5543.43	−0.586567
$448$	0	0
$449$	−4719.30	−0.496030	−0.248015	−	0.968756i	$-0.579778\pi$
$449$	−4719.30	−0.496030	−0.248015	+	0.968756i	$0.579778\pi$
$450$	0	0
$451$	−18222.3	−1.90256
$452$	0	0
$453$	−9358.86	−0.970679
$454$	0	0
$455$	658.877	0.0678871
$456$	0	0
$457$	−2758.98	−0.282406	−0.141203	−	0.989981i	$-0.545097\pi$
$457$	−2758.98	−0.282406	−0.141203	+	0.989981i	$0.545097\pi$
$458$	0	0
$459$	475.921	0.0483967
$460$	0	0
$461$	−12349.6	−1.24767	−0.623836	−	0.781555i	$-0.714427\pi$
$461$	−12349.6	−1.24767	−0.623836	+	0.781555i	$0.714427\pi$
$462$	0	0
$463$	4873.41	0.489171	0.244586	−	0.969628i	$-0.421348\pi$
$463$	4873.41	0.489171	0.244586	+	0.969628i	$0.421348\pi$
$464$	0	0
$465$	2046.30	0.204075
$466$	0	0
$467$	−10914.7	−1.08152	−0.540762	−	0.841175i	$-0.681864\pi$
$467$	−10914.7	−1.08152	−0.540762	+	0.841175i	$0.681864\pi$
$468$	0	0
$469$	−260.584	−0.0256560
$470$	0	0
$471$	−1070.03	−0.104680
$472$	0	0
$473$	−13640.0	−1.32593
$474$	0	0
$475$	521.499	0.0503748
$476$	0	0
$477$	−6850.30	−0.657555
$478$	0	0
$479$	6397.59	0.610257	0.305129	−	0.952311i	$-0.401301\pi$
$479$	6397.59	0.610257	0.305129	+	0.952311i	$0.401301\pi$
$480$	0	0
$481$	2766.32	0.262231
$482$	0	0
$483$	−860.365	−0.0810517
$484$	0	0
$485$	−1666.02	−0.155980
$486$	0	0
$487$	4271.21	0.397427	0.198714	−	0.980058i	$-0.436324\pi$
$487$	4271.21	0.397427	0.198714	+	0.980058i	$0.436324\pi$
$488$	0	0
$489$	1851.78	0.171248
$490$	0	0
$491$	−882.327	−0.0810974	−0.0405487	−	0.999178i	$-0.512911\pi$
$491$	−882.327	−0.0810974	−0.0405487	+	0.999178i	$0.512911\pi$
$492$	0	0
$493$	905.518	0.0827230
$494$	0	0
$495$	5435.91	0.493588
$496$	0	0
$497$	−10412.4	−0.939760
$498$	0	0
$499$	6069.02	0.544462	0.272231	−	0.962232i	$-0.412239\pi$
$499$	6069.02	0.544462	0.272231	+	0.962232i	$0.412239\pi$
$500$	0	0
$501$	−560.133	−0.0499499
$502$	0	0
$503$	772.754	0.0684998	0.0342499	−	0.999413i	$-0.489096\pi$
$503$	772.754	0.0684998	0.0342499	+	0.999413i	$0.489096\pi$
$504$	0	0
$505$	−1857.74	−0.163699
$506$	0	0
$507$	−6546.90	−0.573487
$508$	0	0
$509$	10261.3	0.893567	0.446784	−	0.894642i	$-0.352569\pi$
$509$	10261.3	0.893567	0.446784	+	0.894642i	$0.352569\pi$
$510$	0	0
$511$	5373.99	0.465227
$512$	0	0
$513$	−2900.21	−0.249605
$514$	0	0
$515$	10399.7	0.889836
$516$	0	0
$517$	40033.7	3.40557
$518$	0	0
$519$	−2440.31	−0.206392
$520$	0	0
$521$	8569.34	0.720594	0.360297	−	0.932838i	$-0.382675\pi$
$521$	8569.34	0.720594	0.360297	+	0.932838i	$0.382675\pi$
$522$	0	0
$523$	14396.2	1.20364	0.601819	−	0.798633i	$-0.294443\pi$
$523$	14396.2	1.20364	0.601819	+	0.798633i	$0.294443\pi$
$524$	0	0
$525$	−935.179	−0.0777420
$526$	0	0
$527$	443.646	0.0366708
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	8950.61	0.731494
$532$	0	0
$533$	−3174.95	−0.258016
$534$	0	0
$535$	7567.50	0.611536
$536$	0	0
$537$	−12190.6	−0.979630
$538$	0	0
$539$	−12939.6	−1.03404
$540$	0	0
$541$	9590.56	0.762164	0.381082	−	0.924541i	$-0.375552\pi$
$541$	9590.56	0.762164	0.381082	+	0.924541i	$0.375552\pi$
$542$	0	0
$543$	5809.79	0.459156
$544$	0	0
$545$	7517.24	0.590831
$546$	0	0
$547$	−19967.9	−1.56081	−0.780407	−	0.625272i	$-0.784988\pi$
$547$	−19967.9	−1.56081	−0.780407	+	0.625272i	$0.784988\pi$
$548$	0	0
$549$	−10777.9	−0.837866
$550$	0	0
$551$	−5518.12	−0.426642
$552$	0	0
$553$	−7482.13	−0.575357
$554$	0	0
$555$	−3926.38	−0.300298
$556$	0	0
$557$	13556.9	1.03128	0.515640	−	0.856805i	$-0.327554\pi$
$557$	13556.9	1.03128	0.515640	+	0.856805i	$0.327554\pi$
$558$	0	0
$559$	−2376.55	−0.179816
$560$	0	0
$561$	−690.130	−0.0519381
$562$	0	0
$563$	12284.2	0.919570	0.459785	−	0.888030i	$-0.347926\pi$
$563$	12284.2	0.919570	0.459785	+	0.888030i	$0.347926\pi$
$564$	0	0
$565$	117.026	0.00871381
$566$	0	0
$567$	−245.594	−0.0181905
$568$	0	0
$569$	−24780.3	−1.82573	−0.912867	−	0.408257i	$-0.866137\pi$
$569$	−24780.3	−1.82573	−0.912867	+	0.408257i	$0.866137\pi$
$570$	0	0
$571$	−2095.75	−0.153598	−0.0767989	−	0.997047i	$-0.524470\pi$
$571$	−2095.75	−0.153598	−0.0767989	+	0.997047i	$0.524470\pi$
$572$	0	0
$573$	717.464	0.0523080
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	6550.60	0.472625	0.236313	−	0.971677i	$-0.424061\pi$
$577$	6550.60	0.472625	0.236313	+	0.971677i	$0.424061\pi$
$578$	0	0
$579$	−13136.6	−0.942902
$580$	0	0
$581$	13224.9	0.944343
$582$	0	0
$583$	25684.1	1.82458
$584$	0	0
$585$	947.121	0.0669379
$586$	0	0
$587$	4786.38	0.336551	0.168275	−	0.985740i	$-0.446180\pi$
$587$	4786.38	0.336551	0.168275	+	0.985740i	$0.446180\pi$
$588$	0	0
$589$	−2703.53	−0.189129
$590$	0	0
$591$	−10124.7	−0.704692
$592$	0	0
$593$	−11886.7	−0.823148	−0.411574	−	0.911376i	$-0.635021\pi$
$593$	−11886.7	−0.823148	−0.411574	+	0.911376i	$0.635021\pi$
$594$	0	0
$595$	−202.750	−0.0139697
$596$	0	0
$597$	2400.26	0.164549
$598$	0	0
$599$	−17023.7	−1.16122	−0.580609	−	0.814182i	$-0.697186\pi$
$599$	−17023.7	−1.16122	−0.580609	+	0.814182i	$0.697186\pi$
$600$	0	0
$601$	13560.3	0.920358	0.460179	−	0.887826i	$-0.347785\pi$
$601$	13560.3	0.920358	0.460179	+	0.887826i	$0.347785\pi$
$602$	0	0
$603$	−374.584	−0.0252972
$604$	0	0
$605$	−13726.1	−0.922387
$606$	0	0
$607$	−11690.5	−0.781719	−0.390859	−	0.920450i	$-0.627822\pi$
$607$	−11690.5	−0.781719	−0.390859	+	0.920450i	$0.627822\pi$
$608$	0	0
$609$	9895.38	0.658425
$610$	0	0
$611$	6975.25	0.461847
$612$	0	0
$613$	−18922.5	−1.24677	−0.623387	−	0.781913i	$-0.714244\pi$
$613$	−18922.5	−1.24677	−0.623387	+	0.781913i	$0.714244\pi$
$614$	0	0
$615$	4506.38	0.295471
$616$	0	0
$617$	−5618.11	−0.366575	−0.183287	−	0.983059i	$-0.558674\pi$
$617$	−5618.11	−0.366575	−0.183287	+	0.983059i	$0.558674\pi$
$618$	0	0
$619$	2536.69	0.164714	0.0823570	−	0.996603i	$-0.473755\pi$
$619$	2536.69	0.164714	0.0823570	+	0.996603i	$0.473755\pi$
$620$	0	0
$621$	−3197.74	−0.206636
$622$	0	0
$623$	−7739.38	−0.497708
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4205.57	0.267870
$628$	0	0
$629$	−851.254	−0.0539614
$630$	0	0
$631$	−25695.6	−1.62112	−0.810559	−	0.585657i	$-0.800836\pi$
$631$	−25695.6	−1.62112	−0.810559	+	0.585657i	$0.800836\pi$
$632$	0	0
$633$	−7851.34	−0.492990
$634$	0	0
$635$	10307.9	0.644182
$636$	0	0
$637$	−2254.53	−0.140232
$638$	0	0
$639$	−14967.6	−0.926620
$640$	0	0
$641$	10417.8	0.641933	0.320966	−	0.947091i	$-0.395992\pi$
$641$	10417.8	0.641933	0.320966	+	0.947091i	$0.395992\pi$
$642$	0	0
$643$	−4330.25	−0.265580	−0.132790	−	0.991144i	$-0.542394\pi$
$643$	−4330.25	−0.265580	−0.132790	+	0.991144i	$0.542394\pi$
$644$	0	0
$645$	3373.16	0.205920
$646$	0	0
$647$	−21516.3	−1.30741	−0.653703	−	0.756751i	$-0.726785\pi$
$647$	−21516.3	−1.30741	−0.653703	+	0.756751i	$0.726785\pi$
$648$	0	0
$649$	−33558.9	−2.02974
$650$	0	0
$651$	4848.10	0.291877
$652$	0	0
$653$	−26311.9	−1.57682	−0.788411	−	0.615149i	$-0.789096\pi$
$653$	−26311.9	−1.57682	−0.788411	+	0.615149i	$0.789096\pi$
$654$	0	0
$655$	11830.6	0.705739
$656$	0	0
$657$	7724.99	0.458722
$658$	0	0
$659$	4814.49	0.284592	0.142296	−	0.989824i	$-0.454552\pi$
$659$	4814.49	0.284592	0.142296	+	0.989824i	$0.454552\pi$
$660$	0	0
$661$	13320.7	0.783833	0.391917	−	0.920001i	$-0.371812\pi$
$661$	13320.7	0.783833	0.391917	+	0.920001i	$0.371812\pi$
$662$	0	0
$663$	−120.244	−0.00704359
$664$	0	0
$665$	1235.54	0.0720482
$666$	0	0
$667$	−6084.23	−0.353197
$668$	0	0
$669$	−11218.3	−0.648317
$670$	0	0
$671$	40409.9	2.32490
$672$	0	0
$673$	19357.2	1.10871	0.554357	−	0.832279i	$-0.312964\pi$
$673$	19357.2	1.10871	0.554357	+	0.832279i	$0.312964\pi$
$674$	0	0
$675$	−3475.81	−0.198198
$676$	0	0
$677$	−6379.31	−0.362152	−0.181076	−	0.983469i	$-0.557958\pi$
$677$	−6379.31	−0.362152	−0.181076	+	0.983469i	$0.557958\pi$
$678$	0	0
$679$	−3947.15	−0.223089
$680$	0	0
$681$	16490.3	0.927917
$682$	0	0
$683$	10922.8	0.611932	0.305966	−	0.952043i	$-0.401021\pi$
$683$	10922.8	0.611932	0.305966	+	0.952043i	$0.401021\pi$
$684$	0	0
$685$	−649.085	−0.0362048
$686$	0	0
$687$	15660.9	0.869727
$688$	0	0
$689$	4475.06	0.247440
$690$	0	0
$691$	5677.75	0.312579	0.156289	−	0.987711i	$-0.450047\pi$
$691$	5677.75	0.312579	0.156289	+	0.987711i	$0.450047\pi$
$692$	0	0
$693$	12878.8	0.705950
$694$	0	0
$695$	12801.8	0.698707
$696$	0	0
$697$	977.000	0.0530940
$698$	0	0
$699$	−8660.74	−0.468640
$700$	0	0
$701$	6569.06	0.353937	0.176969	−	0.984217i	$-0.443371\pi$
$701$	6569.06	0.353937	0.176969	+	0.984217i	$0.443371\pi$
$702$	0	0
$703$	5187.44	0.278305
$704$	0	0
$705$	−9900.34	−0.528891
$706$	0	0
$707$	−4401.35	−0.234130
$708$	0	0
$709$	15904.0	0.842437	0.421218	−	0.906959i	$-0.361603\pi$
$709$	15904.0	0.842437	0.421218	+	0.906959i	$0.361603\pi$
$710$	0	0
$711$	−10755.4	−0.567312
$712$	0	0
$713$	−2980.88	−0.156571
$714$	0	0
$715$	−3551.08	−0.185738
$716$	0	0
$717$	2269.19	0.118193
$718$	0	0
$719$	26424.9	1.37063	0.685316	−	0.728246i	$-0.259664\pi$
$719$	26424.9	1.37063	0.685316	+	0.728246i	$0.259664\pi$
$720$	0	0
$721$	24639.0	1.27268
$722$	0	0
$723$	−6923.15	−0.356120
$724$	0	0
$725$	−6613.29	−0.338774
$726$	0	0
$727$	−15839.0	−0.808029	−0.404014	−	0.914753i	$-0.632385\pi$
$727$	−15839.0	−0.808029	−0.404014	+	0.914753i	$0.632385\pi$
$728$	0	0
$729$	11500.9	0.584305
$730$	0	0
$731$	731.314	0.0370022
$732$	0	0
$733$	18096.8	0.911899	0.455949	−	0.890006i	$-0.349300\pi$
$733$	18096.8	0.911899	0.455949	+	0.890006i	$0.349300\pi$
$734$	0	0
$735$	3199.98	0.160589
$736$	0	0
$737$	1404.44	0.0701945
$738$	0	0
$739$	−33757.2	−1.68035	−0.840176	−	0.542314i	$-0.817548\pi$
$739$	−33757.2	−1.68035	−0.840176	+	0.542314i	$0.817548\pi$
$740$	0	0
$741$	732.755	0.0363271
$742$	0	0
$743$	−15357.5	−0.758294	−0.379147	−	0.925337i	$-0.623782\pi$
$743$	−15357.5	−0.758294	−0.379147	+	0.925337i	$0.623782\pi$
$744$	0	0
$745$	8777.40	0.431650
$746$	0	0
$747$	19010.6	0.931138
$748$	0	0
$749$	17928.9	0.874645
$750$	0	0
$751$	26534.6	1.28930	0.644648	−	0.764479i	$-0.277004\pi$
$751$	26534.6	1.28930	0.644648	+	0.764479i	$0.277004\pi$
$752$	0	0
$753$	445.194	0.0215455
$754$	0	0
$755$	14818.7	0.714315
$756$	0	0
$757$	10098.8	0.484873	0.242437	−	0.970167i	$-0.422053\pi$
$757$	10098.8	0.484873	0.242437	+	0.970167i	$0.422053\pi$
$758$	0	0
$759$	4637.02	0.221757
$760$	0	0
$761$	15297.4	0.728688	0.364344	−	0.931265i	$-0.381293\pi$
$761$	15297.4	0.728688	0.364344	+	0.931265i	$0.381293\pi$
$762$	0	0
$763$	17809.8	0.845033
$764$	0	0
$765$	−291.449	−0.0137743
$766$	0	0
$767$	−5847.11	−0.275263
$768$	0	0
$769$	−6661.82	−0.312395	−0.156197	−	0.987726i	$-0.549924\pi$
$769$	−6661.82	−0.312395	−0.156197	+	0.987726i	$0.549924\pi$
$770$	0	0
$771$	21347.5	0.997161
$772$	0	0
$773$	26876.6	1.25056	0.625280	−	0.780401i	$-0.284985\pi$
$773$	26876.6	1.25056	0.625280	+	0.780401i	$0.284985\pi$
$774$	0	0
$775$	−3240.09	−0.150177
$776$	0	0
$777$	−9302.39	−0.429500
$778$	0	0
$779$	−5953.72	−0.273831
$780$	0	0
$781$	56118.8	2.57117
$782$	0	0
$783$	36778.4	1.67861
$784$	0	0
$785$	1694.27	0.0770333
$786$	0	0
$787$	−7029.87	−0.318409	−0.159204	−	0.987246i	$-0.550893\pi$
$787$	−7029.87	−0.318409	−0.159204	+	0.987246i	$0.550893\pi$
$788$	0	0
$789$	−8558.85	−0.386189
$790$	0	0
$791$	277.257	0.0124629
$792$	0	0
$793$	7040.80	0.315291
$794$	0	0
$795$	−6351.69	−0.283360
$796$	0	0
$797$	−40917.6	−1.81854	−0.909269	−	0.416210i	$-0.863358\pi$
$797$	−40917.6	−1.81854	−0.909269	+	0.416210i	$0.863358\pi$
$798$	0	0
$799$	−2146.43	−0.0950379
$800$	0	0
$801$	−11125.2	−0.490748
$802$	0	0
$803$	−28963.6	−1.27286
$804$	0	0
$805$	1362.29	0.0596453
$806$	0	0
$807$	12526.8	0.546426
$808$	0	0
$809$	35938.8	1.56186	0.780928	−	0.624621i	$-0.214746\pi$
$809$	35938.8	1.56186	0.780928	+	0.624621i	$0.214746\pi$
$810$	0	0
$811$	30145.8	1.30526	0.652628	−	0.757678i	$-0.273666\pi$
$811$	30145.8	1.30526	0.652628	+	0.757678i	$0.273666\pi$
$812$	0	0
$813$	16911.6	0.729538
$814$	0	0
$815$	−2932.08	−0.126020
$816$	0	0
$817$	−4456.55	−0.190838
$818$	0	0
$819$	2243.92	0.0957374
$820$	0	0
$821$	−20634.1	−0.877142	−0.438571	−	0.898696i	$-0.644515\pi$
$821$	−20634.1	−0.877142	−0.438571	+	0.898696i	$0.644515\pi$
$822$	0	0
$823$	16368.5	0.693280	0.346640	−	0.937998i	$-0.387323\pi$
$823$	16368.5	0.693280	0.346640	+	0.937998i	$0.387323\pi$
$824$	0	0
$825$	5040.24	0.212701
$826$	0	0
$827$	−32696.6	−1.37481	−0.687407	−	0.726273i	$-0.741251\pi$
$827$	−32696.6	−1.37481	−0.687407	+	0.726273i	$0.741251\pi$
$828$	0	0
$829$	30865.6	1.29313	0.646567	−	0.762857i	$-0.276204\pi$
$829$	30865.6	1.29313	0.646567	+	0.762857i	$0.276204\pi$
$830$	0	0
$831$	−15382.3	−0.642123
$832$	0	0
$833$	693.766	0.0288566
$834$	0	0
$835$	886.908	0.0367578
$836$	0	0
$837$	18019.1	0.744122
$838$	0	0
$839$	−4377.37	−0.180124	−0.0900618	−	0.995936i	$-0.528706\pi$
$839$	−4377.37	−0.180124	−0.0900618	+	0.995936i	$0.528706\pi$
$840$	0	0
$841$	45587.9	1.86920
$842$	0	0
$843$	27275.7	1.11438
$844$	0	0
$845$	10366.3	0.422025
$846$	0	0
$847$	−32519.8	−1.31924
$848$	0	0
$849$	−12196.7	−0.493039
$850$	0	0
$851$	5719.62	0.230395
$852$	0	0
$853$	27780.8	1.11512	0.557560	−	0.830137i	$-0.311738\pi$
$853$	27780.8	1.11512	0.557560	+	0.830137i	$0.311738\pi$
$854$	0	0
$855$	1776.06	0.0710408
$856$	0	0
$857$	26654.8	1.06244	0.531220	−	0.847234i	$-0.321734\pi$
$857$	26654.8	1.06244	0.531220	+	0.847234i	$0.321734\pi$
$858$	0	0
$859$	−35222.5	−1.39904	−0.699521	−	0.714612i	$-0.746603\pi$
$859$	−35222.5	−1.39904	−0.699521	+	0.714612i	$0.746603\pi$
$860$	0	0
$861$	10676.5	0.422596
$862$	0	0
$863$	−27958.9	−1.10282	−0.551409	−	0.834235i	$-0.685910\pi$
$863$	−27958.9	−1.10282	−0.551409	+	0.834235i	$0.685910\pi$
$864$	0	0
$865$	3863.95	0.151882
$866$	0	0
$867$	−15477.2	−0.606267
$868$	0	0
$869$	40325.7	1.57417
$870$	0	0
$871$	244.702	0.00951943
$872$	0	0
$873$	−5673.94	−0.219970
$874$	0	0
$875$	1480.75	0.0572098
$876$	0	0
$877$	33115.8	1.27508	0.637539	−	0.770418i	$-0.279953\pi$
$877$	33115.8	1.27508	0.637539	+	0.770418i	$0.279953\pi$
$878$	0	0
$879$	−29234.7	−1.12180
$880$	0	0
$881$	−10198.6	−0.390013	−0.195006	−	0.980802i	$-0.562473\pi$
$881$	−10198.6	−0.390013	−0.195006	+	0.980802i	$0.562473\pi$
$882$	0	0
$883$	8668.83	0.330384	0.165192	−	0.986261i	$-0.447176\pi$
$883$	8668.83	0.330384	0.165192	+	0.986261i	$0.447176\pi$
$884$	0	0
$885$	8299.12	0.315222
$886$	0	0
$887$	−33102.0	−1.25305	−0.626525	−	0.779401i	$-0.715523\pi$
$887$	−33102.0	−1.25305	−0.626525	+	0.779401i	$0.715523\pi$
$888$	0	0
$889$	24421.4	0.921337
$890$	0	0
$891$	1323.65	0.0497689
$892$	0	0
$893$	13080.1	0.490155
$894$	0	0
$895$	19302.4	0.720902
$896$	0	0
$897$	807.928	0.0300735
$898$	0	0
$899$	34284.2	1.27191
$900$	0	0
$901$	−1377.07	−0.0509177
$902$	0	0
$903$	7991.71	0.294515
$904$	0	0
$905$	−9199.15	−0.337890
$906$	0	0
$907$	5631.57	0.206167	0.103083	−	0.994673i	$-0.467129\pi$
$907$	5631.57	0.206167	0.103083	+	0.994673i	$0.467129\pi$
$908$	0	0
$909$	−6326.85	−0.230856
$910$	0	0
$911$	410.486	0.0149287	0.00746433	−	0.999972i	$-0.497624\pi$
$911$	410.486	0.0149287	0.00746433	+	0.999972i	$0.497624\pi$
$912$	0	0
$913$	−71277.1	−2.58371
$914$	0	0
$915$	−9993.38	−0.361061
$916$	0	0
$917$	28029.0	1.00938
$918$	0	0
$919$	21938.1	0.787455	0.393728	−	0.919227i	$-0.371185\pi$
$919$	21938.1	0.787455	0.393728	+	0.919227i	$0.371185\pi$
$920$	0	0
$921$	−6641.48	−0.237616
$922$	0	0
$923$	9777.81	0.348690
$924$	0	0
$925$	6216.98	0.220987
$926$	0	0
$927$	35418.0	1.25489
$928$	0	0
$929$	6555.80	0.231527	0.115764	−	0.993277i	$-0.463068\pi$
$929$	6555.80	0.231527	0.115764	+	0.993277i	$0.463068\pi$
$930$	0	0
$931$	−4227.73	−0.148827
$932$	0	0
$933$	−30621.5	−1.07449
$934$	0	0
$935$	1092.74	0.0382209
$936$	0	0
$937$	−349.446	−0.0121834	−0.00609172	−	0.999981i	$-0.501939\pi$
$937$	−349.446	−0.0121834	−0.00609172	+	0.999981i	$0.501939\pi$
$938$	0	0
$939$	−22355.9	−0.776951
$940$	0	0
$941$	31295.6	1.08417	0.542087	−	0.840322i	$-0.317634\pi$
$941$	31295.6	1.08417	0.542087	+	0.840322i	$0.317634\pi$
$942$	0	0
$943$	−6564.52	−0.226692
$944$	0	0
$945$	−8234.89	−0.283472
$946$	0	0
$947$	−19810.5	−0.679785	−0.339892	−	0.940464i	$-0.610391\pi$
$947$	−19810.5	−0.679785	−0.339892	+	0.940464i	$0.610391\pi$
$948$	0	0
$949$	−5046.46	−0.172619
$950$	0	0
$951$	27775.9	0.947102
$952$	0	0
$953$	−37167.2	−1.26334	−0.631670	−	0.775237i	$-0.717630\pi$
$953$	−37167.2	−1.26334	−0.631670	+	0.775237i	$0.717630\pi$
$954$	0	0
$955$	−1136.02	−0.0384930
$956$	0	0
$957$	−53332.1	−1.80144
$958$	0	0
$959$	−1537.81	−0.0517816
$960$	0	0
$961$	−12993.9	−0.436169
$962$	0	0
$963$	25772.5	0.862416
$964$	0	0
$965$	20800.4	0.693874
$966$	0	0
$967$	19333.9	0.642955	0.321478	−	0.946917i	$-0.395821\pi$
$967$	19333.9	0.642955	0.321478	+	0.946917i	$0.395821\pi$
$968$	0	0
$969$	−225.484	−0.00747533
$970$	0	0
$971$	24183.2	0.799256	0.399628	−	0.916677i	$-0.369139\pi$
$971$	24183.2	0.799256	0.399628	+	0.916677i	$0.369139\pi$
$972$	0	0
$973$	30330.1	0.999321
$974$	0	0
$975$	878.183	0.0288455
$976$	0	0
$977$	−36983.2	−1.21105	−0.605525	−	0.795826i	$-0.707037\pi$
$977$	−36983.2	−1.21105	−0.605525	+	0.795826i	$0.707037\pi$
$978$	0	0
$979$	41712.2	1.36172
$980$	0	0
$981$	25601.3	0.833217
$982$	0	0
$983$	−35614.8	−1.15558	−0.577789	−	0.816186i	$-0.696085\pi$
$983$	−35614.8	−1.15558	−0.577789	+	0.816186i	$0.696085\pi$
$984$	0	0
$985$	16031.3	0.518577
$986$	0	0
$987$	−23455.9	−0.756444
$988$	0	0
$989$	−4913.74	−0.157986
$990$	0	0
$991$	−37477.1	−1.20131	−0.600656	−	0.799507i	$-0.705094\pi$
$991$	−37477.1	−1.20131	−0.600656	+	0.799507i	$0.705094\pi$
$992$	0	0
$993$	27641.8	0.883371
$994$	0	0
$995$	−3800.54	−0.121091
$996$	0	0
$997$	13732.5	0.436222	0.218111	−	0.975924i	$-0.430010\pi$
$997$	13732.5	0.436222	0.218111	+	0.975924i	$0.430010\pi$
$998$	0	0
$999$	−34574.4	−1.09498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.z.1.7		9
4.3	odd	2		920.4.a.e.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.e.1.3	✓	9	4.3	odd	2
1840.4.a.z.1.7		9	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)

\(f(q)\)	\(=\)	\(q+3.15779 q^{3} -5.00000 q^{5} -11.8460 q^{7} -17.0284 q^{9} +O(q^{10})\)	\(q+3.15779 q^{3} -5.00000 q^{5} -11.8460 q^{7} -17.0284 q^{9} +63.8452 q^{11} +11.1240 q^{13} -15.7889 q^{15} -3.42310 q^{17} +20.8600 q^{19} -37.4072 q^{21} +23.0000 q^{23} +25.0000 q^{25} -139.032 q^{27} -264.532 q^{29} -129.604 q^{31} +201.610 q^{33} +59.2301 q^{35} +248.679 q^{37} +35.1273 q^{39} -285.414 q^{41} -213.641 q^{43} +85.1419 q^{45} +627.043 q^{47} -202.672 q^{49} -10.8094 q^{51} +402.287 q^{53} -319.226 q^{55} +65.8713 q^{57} -525.629 q^{59} +632.936 q^{61} +201.718 q^{63} -55.6201 q^{65} +21.9976 q^{67} +72.6291 q^{69} +878.981 q^{71} -453.654 q^{73} +78.9447 q^{75} -756.311 q^{77} +631.616 q^{79} +20.7322 q^{81} -1116.40 q^{83} +17.1155 q^{85} -835.334 q^{87} +653.332 q^{89} -131.775 q^{91} -409.261 q^{93} -104.300 q^{95} +333.205 q^{97} -1087.18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9	\( 9 q + 3 q^{3} - 45 q^{5} - 25 q^{7} + 62 q^{9} - 22 q^{11} + 23 q^{13} - 15 q^{15} - 135 q^{17} - 102 q^{19} + 54 q^{21} + 207 q^{23} + 225 q^{25} + 363 q^{27} + 280 q^{29} - 168 q^{31} + 28 q^{33} + 125 q^{35} + 153 q^{37} + 5 q^{39} - 502 q^{41} - 110 q^{43} - 310 q^{45} + 153 q^{47} + 764 q^{49} - 924 q^{51} - 273 q^{53} + 110 q^{55} + 748 q^{57} - 827 q^{59} + 1976 q^{61} - 2237 q^{63} - 115 q^{65} - 1613 q^{67} + 69 q^{69} - 1370 q^{71} + 425 q^{73} + 75 q^{75} + 2006 q^{77} - 2624 q^{79} + 1729 q^{81} - 2505 q^{83} + 675 q^{85} - 1591 q^{87} + 1120 q^{89} - 2392 q^{91} + 4401 q^{93} + 510 q^{95} + 2026 q^{97} - 3206 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 - 45 * q^5 - 25 * q^7 + 62 * q^9 - 22 * q^11 + 23 * q^13 - 15 * q^15 - 135 * q^17 - 102 * q^19 + 54 * q^21 + 207 * q^23 + 225 * q^25 + 363 * q^27 + 280 * q^29 - 168 * q^31 + 28 * q^33 + 125 * q^35 + 153 * q^37 + 5 * q^39 - 502 * q^41 - 110 * q^43 - 310 * q^45 + 153 * q^47 + 764 * q^49 - 924 * q^51 - 273 * q^53 + 110 * q^55 + 748 * q^57 - 827 * q^59 + 1976 * q^61 - 2237 * q^63 - 115 * q^65 - 1613 * q^67 + 69 * q^69 - 1370 * q^71 + 425 * q^73 + 75 * q^75 + 2006 * q^77 - 2624 * q^79 + 1729 * q^81 - 2505 * q^83 + 675 * q^85 - 1591 * q^87 + 1120 * q^89 - 2392 * q^91 + 4401 * q^93 + 510 * q^95 + 2026 * q^97 - 3206 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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