LMFDB - Embedded newform 616.4.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [616,4,Mod(1,616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(616, 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("616.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 616 = 2^{3} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	616.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:	$36.3451765635$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 94x^{4} + 161x^{3} + 533x^{2} - 384x - 468 $ x^6 - x^5 - 94x^4 + 161x^3 + 533x^2 - 384x - 468
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.85526$ of defining polynomial
Character	$\chi$	$=$	616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.18857 q^{3} +2.34292 q^{5} +7.00000 q^{7} -22.2102 q^{9} -11.0000 q^{11} -21.0600 q^{13} +5.12765 q^{15} +86.0692 q^{17} -129.304 q^{19} +15.3200 q^{21} +109.478 q^{23} -119.511 q^{25} -107.700 q^{27} -149.527 q^{29} -284.780 q^{31} -24.0743 q^{33} +16.4004 q^{35} -107.219 q^{37} -46.0913 q^{39} +505.363 q^{41} +26.8319 q^{43} -52.0366 q^{45} -12.1730 q^{47} +49.0000 q^{49} +188.368 q^{51} -571.361 q^{53} -25.7721 q^{55} -282.990 q^{57} -53.0539 q^{59} -469.509 q^{61} -155.471 q^{63} -49.3419 q^{65} +732.490 q^{67} +239.601 q^{69} +338.314 q^{71} -1142.10 q^{73} -261.558 q^{75} -77.0000 q^{77} -156.015 q^{79} +363.966 q^{81} -960.073 q^{83} +201.653 q^{85} -327.250 q^{87} -466.178 q^{89} -147.420 q^{91} -623.260 q^{93} -302.948 q^{95} -126.097 q^{97} +244.312 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{5} + 42 q^{7} + 28 q^{9} - 66 q^{11} - 70 q^{13} - 50 q^{15} - 102 q^{17} - 136 q^{19} - 146 q^{23} + 44 q^{25} - 30 q^{27} - 148 q^{29} + 308 q^{31} - 98 q^{35} - 6 q^{37} - 540 q^{39} - 90 q^{41}+ \cdots - 308 q^{99}+O(q^{100}) $ 6 * q - 14 * q^5 + 42 * q^7 + 28 * q^9 - 66 * q^11 - 70 * q^13 - 50 * q^15 - 102 * q^17 - 136 * q^19 - 146 * q^23 + 44 * q^25 - 30 * q^27 - 148 * q^29 + 308 * q^31 - 98 * q^35 - 6 * q^37 - 540 * q^39 - 90 * q^41 - 396 * q^43 - 812 * q^45 - 346 * q^47 + 294 * q^49 - 600 * q^51 - 984 * q^53 + 154 * q^55 - 1580 * q^57 - 44 * q^59 - 1414 * q^61 + 196 * q^63 - 1208 * q^65 - 926 * q^67 - 2550 * q^69 - 1478 * q^71 - 678 * q^73 - 1350 * q^75 - 462 * q^77 - 700 * q^79 - 966 * q^81 - 1192 * q^83 - 1596 * q^85 + 1900 * q^87 - 3314 * q^89 - 490 * q^91 - 2330 * q^93 + 668 * q^95 - 346 * q^97 - 308 * q^99

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
$3$	2.18857	0.421190	0.210595	−	0.977573i	$-0.432460\pi$
$3$	2.18857	0.421190	0.210595	+	0.977573i	$0.432460\pi$
$4$	0	0
$5$	2.34292	0.209557	0.104779	−	0.994496i	$-0.466587\pi$
$5$	2.34292	0.209557	0.104779	+	0.994496i	$0.466587\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−22.2102	−0.822599
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−21.0600	−0.449307	−0.224654	−	0.974439i	$-0.572125\pi$
$13$	−21.0600	−0.449307	−0.224654	+	0.974439i	$0.572125\pi$
$14$	0	0
$15$	5.12765	0.0882635
$16$	0	0
$17$	86.0692	1.22793	0.613966	−	0.789332i	$-0.289573\pi$
$17$	86.0692	1.22793	0.613966	+	0.789332i	$0.289573\pi$
$18$	0	0
$19$	−129.304	−1.56128	−0.780639	−	0.624983i	$-0.785106\pi$
$19$	−129.304	−1.56128	−0.780639	+	0.624983i	$0.785106\pi$
$20$	0	0
$21$	15.3200	0.159195
$22$	0	0
$23$	109.478	0.992514	0.496257	−	0.868176i	$-0.334707\pi$
$23$	109.478	0.992514	0.496257	+	0.868176i	$0.334707\pi$
$24$	0	0
$25$	−119.511	−0.956086
$26$	0	0
$27$	−107.700	−0.767661
$28$	0	0
$29$	−149.527	−0.957463	−0.478731	−	0.877961i	$-0.658903\pi$
$29$	−149.527	−0.957463	−0.478731	+	0.877961i	$0.658903\pi$
$30$	0	0
$31$	−284.780	−1.64993	−0.824967	−	0.565181i	$-0.808806\pi$
$31$	−284.780	−1.64993	−0.824967	+	0.565181i	$0.808806\pi$
$32$	0	0
$33$	−24.0743	−0.126994
$34$	0	0
$35$	16.4004	0.0792052
$36$	0	0
$37$	−107.219	−0.476399	−0.238199	−	0.971216i	$-0.576557\pi$
$37$	−107.219	−0.476399	−0.238199	+	0.971216i	$0.576557\pi$
$38$	0	0
$39$	−46.0913	−0.189244
$40$	0	0
$41$	505.363	1.92499	0.962493	−	0.271308i	$-0.0874562\pi$
$41$	505.363	1.92499	0.962493	+	0.271308i	$0.0874562\pi$
$42$	0	0
$43$	26.8319	0.0951586	0.0475793	−	0.998867i	$-0.484849\pi$
$43$	26.8319	0.0951586	0.0475793	+	0.998867i	$0.484849\pi$
$44$	0	0
$45$	−52.0366	−0.172381
$46$	0	0
$47$	−12.1730	−0.0377791	−0.0188896	−	0.999822i	$-0.506013\pi$
$47$	−12.1730	−0.0377791	−0.0188896	+	0.999822i	$0.506013\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	188.368	0.517193
$52$	0	0
$53$	−571.361	−1.48080	−0.740401	−	0.672166i	$-0.765364\pi$
$53$	−571.361	−1.48080	−0.740401	+	0.672166i	$0.765364\pi$
$54$	0	0
$55$	−25.7721	−0.0631839
$56$	0	0
$57$	−282.990	−0.657595
$58$	0	0
$59$	−53.0539	−0.117068	−0.0585341	−	0.998285i	$-0.518643\pi$
$59$	−53.0539	−0.117068	−0.0585341	+	0.998285i	$0.518643\pi$
$60$	0	0
$61$	−469.509	−0.985484	−0.492742	−	0.870176i	$-0.664005\pi$
$61$	−469.509	−0.985484	−0.492742	+	0.870176i	$0.664005\pi$
$62$	0	0
$63$	−155.471	−0.310913
$64$	0	0
$65$	−49.3419	−0.0941556
$66$	0	0
$67$	732.490	1.33564	0.667820	−	0.744323i	$-0.267228\pi$
$67$	732.490	1.33564	0.667820	+	0.744323i	$0.267228\pi$
$68$	0	0
$69$	239.601	0.418037
$70$	0	0
$71$	338.314	0.565500	0.282750	−	0.959194i	$-0.408753\pi$
$71$	338.314	0.565500	0.282750	+	0.959194i	$0.408753\pi$
$72$	0	0
$73$	−1142.10	−1.83114	−0.915569	−	0.402160i	$-0.868259\pi$
$73$	−1142.10	−1.83114	−0.915569	+	0.402160i	$0.868259\pi$
$74$	0	0
$75$	−261.558	−0.402694
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−156.015	−0.222191	−0.111095	−	0.993810i	$-0.535436\pi$
$79$	−156.015	−0.222191	−0.111095	+	0.993810i	$0.535436\pi$
$80$	0	0
$81$	363.966	0.499267
$82$	0	0
$83$	−960.073	−1.26966	−0.634830	−	0.772652i	$-0.718930\pi$
$83$	−960.073	−1.26966	−0.634830	+	0.772652i	$0.718930\pi$
$84$	0	0
$85$	201.653	0.257322
$86$	0	0
$87$	−327.250	−0.403274
$88$	0	0
$89$	−466.178	−0.555222	−0.277611	−	0.960694i	$-0.589543\pi$
$89$	−466.178	−0.555222	−0.277611	+	0.960694i	$0.589543\pi$
$90$	0	0
$91$	−147.420	−0.169822
$92$	0	0
$93$	−623.260	−0.694936
$94$	0	0
$95$	−302.948	−0.327177
$96$	0	0
$97$	−126.097	−0.131992	−0.0659960	−	0.997820i	$-0.521022\pi$
$97$	−126.097	−0.131992	−0.0659960	+	0.997820i	$0.521022\pi$
$98$	0	0
$99$	244.312	0.248023
$100$	0	0
$101$	−575.121	−0.566600	−0.283300	−	0.959031i	$-0.591429\pi$
$101$	−575.121	−0.566600	−0.283300	+	0.959031i	$0.591429\pi$
$102$	0	0
$103$	21.2830	0.0203600	0.0101800	−	0.999948i	$-0.496760\pi$
$103$	21.2830	0.0203600	0.0101800	+	0.999948i	$0.496760\pi$
$104$	0	0
$105$	35.8935	0.0333605
$106$	0	0
$107$	−1738.63	−1.57083	−0.785417	−	0.618967i	$-0.787551\pi$
$107$	−1738.63	−1.57083	−0.785417	+	0.618967i	$0.787551\pi$
$108$	0	0
$109$	1730.40	1.52057	0.760287	−	0.649587i	$-0.225058\pi$
$109$	1730.40	1.52057	0.760287	+	0.649587i	$0.225058\pi$
$110$	0	0
$111$	−234.657	−0.200655
$112$	0	0
$113$	−1525.71	−1.27015	−0.635075	−	0.772451i	$-0.719031\pi$
$113$	−1525.71	−1.27015	−0.635075	+	0.772451i	$0.719031\pi$
$114$	0	0
$115$	256.499	0.207988
$116$	0	0
$117$	467.746	0.369600
$118$	0	0
$119$	602.484	0.464115
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1106.02	0.810785
$124$	0	0
$125$	−572.869	−0.409912
$126$	0	0
$127$	74.4626	0.0520274	0.0260137	−	0.999662i	$-0.491719\pi$
$127$	74.4626	0.0520274	0.0260137	+	0.999662i	$0.491719\pi$
$128$	0	0
$129$	58.7234	0.0400799
$130$	0	0
$131$	2364.83	1.57722	0.788610	−	0.614893i	$-0.210801\pi$
$131$	2364.83	1.57722	0.788610	+	0.614893i	$0.210801\pi$
$132$	0	0
$133$	−905.125	−0.590107
$134$	0	0
$135$	−252.332	−0.160869
$136$	0	0
$137$	−2120.20	−1.32220	−0.661098	−	0.750300i	$-0.729909\pi$
$137$	−2120.20	−1.32220	−0.661098	+	0.750300i	$0.729909\pi$
$138$	0	0
$139$	67.9829	0.0414837	0.0207418	−	0.999785i	$-0.493397\pi$
$139$	67.9829	0.0414837	0.0207418	+	0.999785i	$0.493397\pi$
$140$	0	0
$141$	−26.6415	−0.0159122
$142$	0	0
$143$	231.660	0.135471
$144$	0	0
$145$	−350.329	−0.200643
$146$	0	0
$147$	107.240	0.0601701
$148$	0	0
$149$	69.6488	0.0382943	0.0191472	−	0.999817i	$-0.493905\pi$
$149$	69.6488	0.0382943	0.0191472	+	0.999817i	$0.493905\pi$
$150$	0	0
$151$	−1558.63	−0.839998	−0.419999	−	0.907525i	$-0.637970\pi$
$151$	−1558.63	−0.839998	−0.419999	+	0.907525i	$0.637970\pi$
$152$	0	0
$153$	−1911.61	−1.01010
$154$	0	0
$155$	−667.216	−0.345756
$156$	0	0
$157$	−883.194	−0.448959	−0.224479	−	0.974479i	$-0.572068\pi$
$157$	−883.194	−0.448959	−0.224479	+	0.974479i	$0.572068\pi$
$158$	0	0
$159$	−1250.46	−0.623699
$160$	0	0
$161$	766.348	0.375135
$162$	0	0
$163$	2924.36	1.40524	0.702618	−	0.711567i	$-0.252014\pi$
$163$	2924.36	1.40524	0.702618	+	0.711567i	$0.252014\pi$
$164$	0	0
$165$	−56.4041	−0.0266124
$166$	0	0
$167$	2628.35	1.21789	0.608945	−	0.793212i	$-0.291593\pi$
$167$	2628.35	1.21789	0.608945	+	0.793212i	$0.291593\pi$
$168$	0	0
$169$	−1753.48	−0.798123
$170$	0	0
$171$	2871.85	1.28430
$172$	0	0
$173$	−4201.35	−1.84638	−0.923188	−	0.384349i	$-0.874426\pi$
$173$	−4201.35	−1.84638	−0.923188	+	0.384349i	$0.874426\pi$
$174$	0	0
$175$	−836.575	−0.361366
$176$	0	0
$177$	−116.112	−0.0493080
$178$	0	0
$179$	−1671.06	−0.697770	−0.348885	−	0.937166i	$-0.613440\pi$
$179$	−1671.06	−0.697770	−0.348885	+	0.937166i	$0.613440\pi$
$180$	0	0
$181$	2368.14	0.972500	0.486250	−	0.873820i	$-0.338365\pi$
$181$	2368.14	0.972500	0.486250	+	0.873820i	$0.338365\pi$
$182$	0	0
$183$	−1027.55	−0.415076
$184$	0	0
$185$	−251.206	−0.0998327
$186$	0	0
$187$	−946.761	−0.370235
$188$	0	0
$189$	−753.899	−0.290149
$190$	0	0
$191$	4229.06	1.60212	0.801059	−	0.598586i	$-0.204271\pi$
$191$	4229.06	1.60212	0.801059	+	0.598586i	$0.204271\pi$
$192$	0	0
$193$	2734.96	1.02004	0.510018	−	0.860164i	$-0.329639\pi$
$193$	2734.96	1.02004	0.510018	+	0.860164i	$0.329639\pi$
$194$	0	0
$195$	−107.988	−0.0396574
$196$	0	0
$197$	280.720	0.101525	0.0507626	−	0.998711i	$-0.483835\pi$
$197$	280.720	0.101525	0.0507626	+	0.998711i	$0.483835\pi$
$198$	0	0
$199$	1484.60	0.528847	0.264423	−	0.964407i	$-0.414818\pi$
$199$	1484.60	0.528847	0.264423	+	0.964407i	$0.414818\pi$
$200$	0	0
$201$	1603.11	0.562559
$202$	0	0
$203$	−1046.69	−0.361887
$204$	0	0
$205$	1184.02	0.403394
$206$	0	0
$207$	−2431.53	−0.816440
$208$	0	0
$209$	1422.34	0.470743
$210$	0	0
$211$	977.779	0.319019	0.159510	−	0.987196i	$-0.449009\pi$
$211$	977.779	0.319019	0.159510	+	0.987196i	$0.449009\pi$
$212$	0	0
$213$	740.424	0.238183
$214$	0	0
$215$	62.8649	0.0199412
$216$	0	0
$217$	−1993.46	−0.623616
$218$	0	0
$219$	−2499.57	−0.771258
$220$	0	0
$221$	−1812.62	−0.551719
$222$	0	0
$223$	5596.83	1.68068	0.840340	−	0.542060i	$-0.182356\pi$
$223$	5596.83	1.68068	0.840340	+	0.542060i	$0.182356\pi$
$224$	0	0
$225$	2654.35	0.786475
$226$	0	0
$227$	−1441.89	−0.421592	−0.210796	−	0.977530i	$-0.567606\pi$
$227$	−1441.89	−0.421592	−0.210796	+	0.977530i	$0.567606\pi$
$228$	0	0
$229$	−6321.63	−1.82422	−0.912108	−	0.409950i	$-0.865546\pi$
$229$	−6321.63	−1.82422	−0.912108	+	0.409950i	$0.865546\pi$
$230$	0	0
$231$	−168.520	−0.0479991
$232$	0	0
$233$	−6821.76	−1.91806	−0.959030	−	0.283303i	$-0.908570\pi$
$233$	−6821.76	−1.91806	−0.959030	+	0.283303i	$0.908570\pi$
$234$	0	0
$235$	−28.5204	−0.00791689
$236$	0	0
$237$	−341.450	−0.0935847
$238$	0	0
$239$	6951.18	1.88131	0.940657	−	0.339358i	$-0.110210\pi$
$239$	6951.18	1.88131	0.940657	+	0.339358i	$0.110210\pi$
$240$	0	0
$241$	−161.993	−0.0432982	−0.0216491	−	0.999766i	$-0.506892\pi$
$241$	−161.993	−0.0432982	−0.0216491	+	0.999766i	$0.506892\pi$
$242$	0	0
$243$	3704.46	0.977948
$244$	0	0
$245$	114.803	0.0299367
$246$	0	0
$247$	2723.13	0.701493
$248$	0	0
$249$	−2101.19	−0.534768
$250$	0	0
$251$	4527.58	1.13856	0.569279	−	0.822144i	$-0.307222\pi$
$251$	4527.58	1.13856	0.569279	+	0.822144i	$0.307222\pi$
$252$	0	0
$253$	−1204.26	−0.299254
$254$	0	0
$255$	441.332	0.108382
$256$	0	0
$257$	7111.31	1.72604	0.863018	−	0.505173i	$-0.168571\pi$
$257$	7111.31	1.72604	0.863018	+	0.505173i	$0.168571\pi$
$258$	0	0
$259$	−750.535	−0.180062
$260$	0	0
$261$	3321.01	0.787608
$262$	0	0
$263$	6203.83	1.45454	0.727271	−	0.686350i	$-0.240788\pi$
$263$	6203.83	1.45454	0.727271	+	0.686350i	$0.240788\pi$
$264$	0	0
$265$	−1338.65	−0.310313
$266$	0	0
$267$	−1020.26	−0.233854
$268$	0	0
$269$	658.858	0.149336	0.0746678	−	0.997208i	$-0.476210\pi$
$269$	658.858	0.149336	0.0746678	+	0.997208i	$0.476210\pi$
$270$	0	0
$271$	−1810.66	−0.405867	−0.202933	−	0.979193i	$-0.565048\pi$
$271$	−1810.66	−0.405867	−0.202933	+	0.979193i	$0.565048\pi$
$272$	0	0
$273$	−322.639	−0.0715275
$274$	0	0
$275$	1314.62	0.288271
$276$	0	0
$277$	7278.51	1.57878	0.789392	−	0.613889i	$-0.210396\pi$
$277$	7278.51	1.57878	0.789392	+	0.613889i	$0.210396\pi$
$278$	0	0
$279$	6325.00	1.35723
$280$	0	0
$281$	−3301.95	−0.700988	−0.350494	−	0.936565i	$-0.613986\pi$
$281$	−3301.95	−0.700988	−0.350494	+	0.936565i	$0.613986\pi$
$282$	0	0
$283$	2303.26	0.483797	0.241898	−	0.970302i	$-0.422230\pi$
$283$	2303.26	0.483797	0.241898	+	0.970302i	$0.422230\pi$
$284$	0	0
$285$	−663.023	−0.137804
$286$	0	0
$287$	3537.54	0.727576
$288$	0	0
$289$	2494.90	0.507817
$290$	0	0
$291$	−275.972	−0.0555938
$292$	0	0
$293$	7617.46	1.51883	0.759414	−	0.650608i	$-0.225486\pi$
$293$	7617.46	1.51883	0.759414	+	0.650608i	$0.225486\pi$
$294$	0	0
$295$	−124.301	−0.0245325
$296$	0	0
$297$	1184.70	0.231459
$298$	0	0
$299$	−2305.61	−0.445944
$300$	0	0
$301$	187.823	0.0359666
$302$	0	0
$303$	−1258.69	−0.238647
$304$	0	0
$305$	−1100.02	−0.206515
$306$	0	0
$307$	−7069.50	−1.31426	−0.657130	−	0.753778i	$-0.728230\pi$
$307$	−7069.50	−1.31426	−0.657130	+	0.753778i	$0.728230\pi$
$308$	0	0
$309$	46.5793	0.00857542
$310$	0	0
$311$	−3185.90	−0.580887	−0.290444	−	0.956892i	$-0.593803\pi$
$311$	−3185.90	−0.580887	−0.290444	+	0.956892i	$0.593803\pi$
$312$	0	0
$313$	4496.13	0.811937	0.405968	−	0.913887i	$-0.366934\pi$
$313$	4496.13	0.811937	0.405968	+	0.913887i	$0.366934\pi$
$314$	0	0
$315$	−364.257	−0.0651541
$316$	0	0
$317$	6385.29	1.13134	0.565668	−	0.824633i	$-0.308618\pi$
$317$	6385.29	1.13134	0.565668	+	0.824633i	$0.308618\pi$
$318$	0	0
$319$	1644.79	0.288686
$320$	0	0
$321$	−3805.10	−0.661620
$322$	0	0
$323$	−11129.1	−1.91714
$324$	0	0
$325$	2516.90	0.429576
$326$	0	0
$327$	3787.11	0.640451
$328$	0	0
$329$	−85.2112	−0.0142792
$330$	0	0
$331$	−4218.42	−0.700500	−0.350250	−	0.936656i	$-0.613903\pi$
$331$	−4218.42	−0.700500	−0.350250	+	0.936656i	$0.613903\pi$
$332$	0	0
$333$	2381.36	0.391885
$334$	0	0
$335$	1716.17	0.279893
$336$	0	0
$337$	−10964.8	−1.77237	−0.886184	−	0.463333i	$-0.846653\pi$
$337$	−10964.8	−1.77237	−0.886184	+	0.463333i	$0.846653\pi$
$338$	0	0
$339$	−3339.13	−0.534975
$340$	0	0
$341$	3132.58	0.497474
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	561.366	0.0876027
$346$	0	0
$347$	3382.36	0.523269	0.261635	−	0.965167i	$-0.415738\pi$
$347$	3382.36	0.523269	0.261635	+	0.965167i	$0.415738\pi$
$348$	0	0
$349$	4210.80	0.645843	0.322921	−	0.946426i	$-0.395335\pi$
$349$	4210.80	0.645843	0.322921	+	0.946426i	$0.395335\pi$
$350$	0	0
$351$	2268.16	0.344916
$352$	0	0
$353$	2407.73	0.363032	0.181516	−	0.983388i	$-0.441900\pi$
$353$	2407.73	0.363032	0.181516	+	0.983388i	$0.441900\pi$
$354$	0	0
$355$	792.643	0.118505
$356$	0	0
$357$	1318.58	0.195481
$358$	0	0
$359$	−8561.82	−1.25871	−0.629353	−	0.777119i	$-0.716680\pi$
$359$	−8561.82	−1.25871	−0.629353	+	0.777119i	$0.716680\pi$
$360$	0	0
$361$	9860.41	1.43759
$362$	0	0
$363$	264.817	0.0382900
$364$	0	0
$365$	−2675.86	−0.383728
$366$	0	0
$367$	6176.88	0.878557	0.439279	−	0.898351i	$-0.355234\pi$
$367$	6176.88	0.878557	0.439279	+	0.898351i	$0.355234\pi$
$368$	0	0
$369$	−11224.2	−1.58349
$370$	0	0
$371$	−3999.53	−0.559690
$372$	0	0
$373$	−7251.55	−1.00662	−0.503312	−	0.864105i	$-0.667885\pi$
$373$	−7251.55	−1.00662	−0.503312	+	0.864105i	$0.667885\pi$
$374$	0	0
$375$	−1253.76	−0.172651
$376$	0	0
$377$	3149.03	0.430195
$378$	0	0
$379$	13777.1	1.86724	0.933618	−	0.358271i	$-0.116634\pi$
$379$	13777.1	1.86724	0.933618	+	0.358271i	$0.116634\pi$
$380$	0	0
$381$	162.966	0.0219135
$382$	0	0
$383$	−7034.63	−0.938519	−0.469260	−	0.883060i	$-0.655479\pi$
$383$	−7034.63	−0.938519	−0.469260	+	0.883060i	$0.655479\pi$
$384$	0	0
$385$	−180.405	−0.0238813
$386$	0	0
$387$	−595.940	−0.0782774
$388$	0	0
$389$	3261.11	0.425051	0.212525	−	0.977156i	$-0.431831\pi$
$389$	3261.11	0.425051	0.212525	+	0.977156i	$0.431831\pi$
$390$	0	0
$391$	9422.71	1.21874
$392$	0	0
$393$	5175.59	0.664310
$394$	0	0
$395$	−365.531	−0.0465617
$396$	0	0
$397$	5777.15	0.730345	0.365172	−	0.930940i	$-0.381010\pi$
$397$	5777.15	0.730345	0.365172	+	0.930940i	$0.381010\pi$
$398$	0	0
$399$	−1980.93	−0.248548
$400$	0	0
$401$	2182.14	0.271748	0.135874	−	0.990726i	$-0.456616\pi$
$401$	2182.14	0.271748	0.135874	+	0.990726i	$0.456616\pi$
$402$	0	0
$403$	5997.46	0.741327
$404$	0	0
$405$	852.743	0.104625
$406$	0	0
$407$	1179.41	0.143640
$408$	0	0
$409$	10596.4	1.28107	0.640537	−	0.767927i	$-0.278712\pi$
$409$	10596.4	1.28107	0.640537	+	0.767927i	$0.278712\pi$
$410$	0	0
$411$	−4640.20	−0.556896
$412$	0	0
$413$	−371.377	−0.0442476
$414$	0	0
$415$	−2249.38	−0.266066
$416$	0	0
$417$	148.785	0.0174725
$418$	0	0
$419$	5499.18	0.641176	0.320588	−	0.947219i	$-0.396120\pi$
$419$	5499.18	0.641176	0.320588	+	0.947219i	$0.396120\pi$
$420$	0	0
$421$	−9544.04	−1.10486	−0.552432	−	0.833558i	$-0.686300\pi$
$421$	−9544.04	−1.10486	−0.552432	+	0.833558i	$0.686300\pi$
$422$	0	0
$423$	270.365	0.0310771
$424$	0	0
$425$	−10286.2	−1.17401
$426$	0	0
$427$	−3286.56	−0.372478
$428$	0	0
$429$	507.004	0.0570592
$430$	0	0
$431$	−16427.9	−1.83597	−0.917985	−	0.396616i	$-0.870184\pi$
$431$	−16427.9	−1.83597	−0.917985	+	0.396616i	$0.870184\pi$
$432$	0	0
$433$	3266.92	0.362583	0.181291	−	0.983429i	$-0.441972\pi$
$433$	3266.92	0.362583	0.181291	+	0.983429i	$0.441972\pi$
$434$	0	0
$435$	−766.720	−0.0845090
$436$	0	0
$437$	−14155.9	−1.54959
$438$	0	0
$439$	−3974.88	−0.432142	−0.216071	−	0.976378i	$-0.569324\pi$
$439$	−3974.88	−0.432142	−0.216071	+	0.976378i	$0.569324\pi$
$440$	0	0
$441$	−1088.30	−0.117514
$442$	0	0
$443$	−12617.3	−1.35320	−0.676598	−	0.736353i	$-0.736546\pi$
$443$	−12617.3	−1.35320	−0.676598	+	0.736353i	$0.736546\pi$
$444$	0	0
$445$	−1092.22	−0.116351
$446$	0	0
$447$	152.431	0.0161292
$448$	0	0
$449$	−1589.76	−0.167094	−0.0835470	−	0.996504i	$-0.526625\pi$
$449$	−1589.76	−0.167094	−0.0835470	+	0.996504i	$0.526625\pi$
$450$	0	0
$451$	−5558.99	−0.580405
$452$	0	0
$453$	−3411.18	−0.353799
$454$	0	0
$455$	−345.393	−0.0355875
$456$	0	0
$457$	−8883.91	−0.909347	−0.454673	−	0.890658i	$-0.650244\pi$
$457$	−8883.91	−0.909347	−0.454673	+	0.890658i	$0.650244\pi$
$458$	0	0
$459$	−9269.64	−0.942636
$460$	0	0
$461$	1614.34	0.163096	0.0815480	−	0.996669i	$-0.474014\pi$
$461$	1614.34	0.163096	0.0815480	+	0.996669i	$0.474014\pi$
$462$	0	0
$463$	−969.325	−0.0972967	−0.0486483	−	0.998816i	$-0.515491\pi$
$463$	−969.325	−0.0972967	−0.0486483	+	0.998816i	$0.515491\pi$
$464$	0	0
$465$	−1460.25	−0.145629
$466$	0	0
$467$	−11104.3	−1.10031	−0.550156	−	0.835062i	$-0.685432\pi$
$467$	−11104.3	−1.10031	−0.550156	+	0.835062i	$0.685432\pi$
$468$	0	0
$469$	5127.43	0.504824
$470$	0	0
$471$	−1932.93	−0.189097
$472$	0	0
$473$	−295.151	−0.0286914
$474$	0	0
$475$	15453.2	1.49272
$476$	0	0
$477$	12690.0	1.21811
$478$	0	0
$479$	16429.7	1.56721	0.783605	−	0.621260i	$-0.213379\pi$
$479$	16429.7	1.56721	0.783605	+	0.621260i	$0.213379\pi$
$480$	0	0
$481$	2258.04	0.214049
$482$	0	0
$483$	1677.21	0.158003
$484$	0	0
$485$	−295.436	−0.0276599
$486$	0	0
$487$	8110.29	0.754645	0.377323	−	0.926082i	$-0.376845\pi$
$487$	8110.29	0.754645	0.377323	+	0.926082i	$0.376845\pi$
$488$	0	0
$489$	6400.17	0.591872
$490$	0	0
$491$	−10499.5	−0.965041	−0.482520	−	0.875885i	$-0.660279\pi$
$491$	−10499.5	−0.965041	−0.482520	+	0.875885i	$0.660279\pi$
$492$	0	0
$493$	−12869.6	−1.17570
$494$	0	0
$495$	572.403	0.0519750
$496$	0	0
$497$	2368.20	0.213739
$498$	0	0
$499$	−14154.9	−1.26986	−0.634931	−	0.772569i	$-0.718971\pi$
$499$	−14154.9	−1.26986	−0.634931	+	0.772569i	$0.718971\pi$
$500$	0	0
$501$	5752.32	0.512964
$502$	0	0
$503$	−6130.51	−0.543431	−0.271716	−	0.962378i	$-0.587591\pi$
$503$	−6130.51	−0.543431	−0.271716	+	0.962378i	$0.587591\pi$
$504$	0	0
$505$	−1347.46	−0.118735
$506$	0	0
$507$	−3837.60	−0.336162
$508$	0	0
$509$	−7607.48	−0.662467	−0.331233	−	0.943549i	$-0.607465\pi$
$509$	−7607.48	−0.662467	−0.331233	+	0.943549i	$0.607465\pi$
$510$	0	0
$511$	−7994.72	−0.692105
$512$	0	0
$513$	13926.0	1.19853
$514$	0	0
$515$	49.8644	0.00426658
$516$	0	0
$517$	133.903	0.0113908
$518$	0	0
$519$	−9194.96	−0.777676
$520$	0	0
$521$	10821.4	0.909970	0.454985	−	0.890499i	$-0.349645\pi$
$521$	10821.4	0.909970	0.454985	+	0.890499i	$0.349645\pi$
$522$	0	0
$523$	−2274.36	−0.190155	−0.0950773	−	0.995470i	$-0.530310\pi$
$523$	−2274.36	−0.190155	−0.0950773	+	0.995470i	$0.530310\pi$
$524$	0	0
$525$	−1830.90	−0.152204
$526$	0	0
$527$	−24510.8	−2.02601
$528$	0	0
$529$	−181.491	−0.0149167
$530$	0	0
$531$	1178.33	0.0963001
$532$	0	0
$533$	−10642.9	−0.864910
$534$	0	0
$535$	−4073.46	−0.329180
$536$	0	0
$537$	−3657.23	−0.293894
$538$	0	0
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−20620.6	−1.63872	−0.819362	−	0.573276i	$-0.805672\pi$
$541$	−20620.6	−1.63872	−0.819362	+	0.573276i	$0.805672\pi$
$542$	0	0
$543$	5182.84	0.409608
$544$	0	0
$545$	4054.20	0.318647
$546$	0	0
$547$	−8404.71	−0.656964	−0.328482	−	0.944510i	$-0.606537\pi$
$547$	−8404.71	−0.656964	−0.328482	+	0.944510i	$0.606537\pi$
$548$	0	0
$549$	10427.9	0.810657
$550$	0	0
$551$	19334.3	1.49487
$552$	0	0
$553$	−1092.11	−0.0839803
$554$	0	0
$555$	−549.783	−0.0420486
$556$	0	0
$557$	−8712.75	−0.662785	−0.331393	−	0.943493i	$-0.607518\pi$
$557$	−8712.75	−0.662785	−0.331393	+	0.943493i	$0.607518\pi$
$558$	0	0
$559$	−565.079	−0.0427555
$560$	0	0
$561$	−2072.05	−0.155940
$562$	0	0
$563$	−20867.8	−1.56212	−0.781058	−	0.624459i	$-0.785320\pi$
$563$	−20867.8	−1.56212	−0.781058	+	0.624459i	$0.785320\pi$
$564$	0	0
$565$	−3574.62	−0.266169
$566$	0	0
$567$	2547.76	0.188705
$568$	0	0
$569$	−9186.30	−0.676818	−0.338409	−	0.940999i	$-0.609889\pi$
$569$	−9186.30	−0.676818	−0.338409	+	0.940999i	$0.609889\pi$
$570$	0	0
$571$	−18199.1	−1.33382	−0.666908	−	0.745140i	$-0.732383\pi$
$571$	−18199.1	−1.33382	−0.666908	+	0.745140i	$0.732383\pi$
$572$	0	0
$573$	9255.60	0.674796
$574$	0	0
$575$	−13083.8	−0.948928
$576$	0	0
$577$	6516.00	0.470130	0.235065	−	0.971980i	$-0.424470\pi$
$577$	6516.00	0.470130	0.235065	+	0.971980i	$0.424470\pi$
$578$	0	0
$579$	5985.65	0.429629
$580$	0	0
$581$	−6720.51	−0.479886
$582$	0	0
$583$	6284.97	0.446478
$584$	0	0
$585$	1095.89	0.0774522
$586$	0	0
$587$	13888.2	0.976539	0.488269	−	0.872693i	$-0.337628\pi$
$587$	13888.2	0.976539	0.488269	+	0.872693i	$0.337628\pi$
$588$	0	0
$589$	36823.0	2.57600
$590$	0	0
$591$	614.375	0.0427614
$592$	0	0
$593$	−8551.96	−0.592221	−0.296110	−	0.955154i	$-0.595690\pi$
$593$	−8551.96	−0.592221	−0.296110	+	0.955154i	$0.595690\pi$
$594$	0	0
$595$	1411.57	0.0972586
$596$	0	0
$597$	3249.15	0.222745
$598$	0	0
$599$	20061.7	1.36845	0.684224	−	0.729272i	$-0.260141\pi$
$599$	20061.7	1.36845	0.684224	+	0.729272i	$0.260141\pi$
$600$	0	0
$601$	−3216.14	−0.218285	−0.109142	−	0.994026i	$-0.534810\pi$
$601$	−3216.14	−0.218285	−0.109142	+	0.994026i	$0.534810\pi$
$602$	0	0
$603$	−16268.7	−1.09870
$604$	0	0
$605$	283.493	0.0190507
$606$	0	0
$607$	8911.67	0.595903	0.297952	−	0.954581i	$-0.403697\pi$
$607$	8911.67	0.595903	0.297952	+	0.954581i	$0.403697\pi$
$608$	0	0
$609$	−2290.75	−0.152423
$610$	0	0
$611$	256.364	0.0169744
$612$	0	0
$613$	8512.29	0.560862	0.280431	−	0.959874i	$-0.409523\pi$
$613$	8512.29	0.560862	0.280431	+	0.959874i	$0.409523\pi$
$614$	0	0
$615$	2591.32	0.169906
$616$	0	0
$617$	−22502.8	−1.46828	−0.734140	−	0.678998i	$-0.762414\pi$
$617$	−22502.8	−1.46828	−0.734140	+	0.678998i	$0.762414\pi$
$618$	0	0
$619$	−2927.69	−0.190103	−0.0950515	−	0.995472i	$-0.530302\pi$
$619$	−2927.69	−0.190103	−0.0950515	+	0.995472i	$0.530302\pi$
$620$	0	0
$621$	−11790.8	−0.761914
$622$	0	0
$623$	−3263.25	−0.209854
$624$	0	0
$625$	13596.7	0.870186
$626$	0	0
$627$	3112.89	0.198272
$628$	0	0
$629$	−9228.28	−0.584985
$630$	0	0
$631$	21948.4	1.38471	0.692354	−	0.721558i	$-0.256574\pi$
$631$	21948.4	1.38471	0.692354	+	0.721558i	$0.256574\pi$
$632$	0	0
$633$	2139.94	0.134368
$634$	0	0
$635$	174.460	0.0109027
$636$	0	0
$637$	−1031.94	−0.0641868
$638$	0	0
$639$	−7514.01	−0.465179
$640$	0	0
$641$	6200.42	0.382062	0.191031	−	0.981584i	$-0.438817\pi$
$641$	6200.42	0.382062	0.191031	+	0.981584i	$0.438817\pi$
$642$	0	0
$643$	−1727.66	−0.105960	−0.0529799	−	0.998596i	$-0.516872\pi$
$643$	−1727.66	−0.105960	−0.0529799	+	0.998596i	$0.516872\pi$
$644$	0	0
$645$	137.584	0.00839903
$646$	0	0
$647$	7608.11	0.462296	0.231148	−	0.972919i	$-0.425752\pi$
$647$	7608.11	0.462296	0.231148	+	0.972919i	$0.425752\pi$
$648$	0	0
$649$	583.592	0.0352974
$650$	0	0
$651$	−4362.82	−0.262661
$652$	0	0
$653$	12218.9	0.732254	0.366127	−	0.930565i	$-0.380684\pi$
$653$	12218.9	0.732254	0.366127	+	0.930565i	$0.380684\pi$
$654$	0	0
$655$	5540.60	0.330518
$656$	0	0
$657$	25366.3	1.50629
$658$	0	0
$659$	−20384.1	−1.20493	−0.602466	−	0.798145i	$-0.705815\pi$
$659$	−20384.1	−1.20493	−0.602466	+	0.798145i	$0.705815\pi$
$660$	0	0
$661$	−4458.36	−0.262345	−0.131173	−	0.991360i	$-0.541874\pi$
$661$	−4458.36	−0.262345	−0.131173	+	0.991360i	$0.541874\pi$
$662$	0	0
$663$	−3967.04	−0.232379
$664$	0	0
$665$	−2120.64	−0.123661
$666$	0	0
$667$	−16369.9	−0.950295
$668$	0	0
$669$	12249.1	0.707886
$670$	0	0
$671$	5164.60	0.297134
$672$	0	0
$673$	−30812.3	−1.76483	−0.882413	−	0.470476i	$-0.844082\pi$
$673$	−30812.3	−1.76483	−0.882413	+	0.470476i	$0.844082\pi$
$674$	0	0
$675$	12871.3	0.733950
$676$	0	0
$677$	18805.9	1.06761	0.533804	−	0.845608i	$-0.320762\pi$
$677$	18805.9	1.06761	0.533804	+	0.845608i	$0.320762\pi$
$678$	0	0
$679$	−882.680	−0.0498883
$680$	0	0
$681$	−3155.67	−0.177571
$682$	0	0
$683$	−9735.21	−0.545399	−0.272699	−	0.962099i	$-0.587916\pi$
$683$	−9735.21	−0.545399	−0.272699	+	0.962099i	$0.587916\pi$
$684$	0	0
$685$	−4967.46	−0.277076
$686$	0	0
$687$	−13835.3	−0.768342
$688$	0	0
$689$	12032.9	0.665335
$690$	0	0
$691$	−10894.4	−0.599774	−0.299887	−	0.953975i	$-0.596949\pi$
$691$	−10894.4	−0.599774	−0.299887	+	0.953975i	$0.596949\pi$
$692$	0	0
$693$	1710.18	0.0937438
$694$	0	0
$695$	159.278	0.00869320
$696$	0	0
$697$	43496.2	2.36375
$698$	0	0
$699$	−14929.9	−0.807869
$700$	0	0
$701$	−22249.4	−1.19878	−0.599392	−	0.800456i	$-0.704591\pi$
$701$	−22249.4	−1.19878	−0.599392	+	0.800456i	$0.704591\pi$
$702$	0	0
$703$	13863.8	0.743790
$704$	0	0
$705$	−62.4190	−0.00333452
$706$	0	0
$707$	−4025.84	−0.214155
$708$	0	0
$709$	−18986.7	−1.00573	−0.502863	−	0.864366i	$-0.667720\pi$
$709$	−18986.7	−1.00573	−0.502863	+	0.864366i	$0.667720\pi$
$710$	0	0
$711$	3465.12	0.182774
$712$	0	0
$713$	−31177.2	−1.63758
$714$	0	0
$715$	542.761	0.0283890
$716$	0	0
$717$	15213.1	0.792392
$718$	0	0
$719$	−23210.6	−1.20391	−0.601953	−	0.798532i	$-0.705611\pi$
$719$	−23210.6	−1.20391	−0.601953	+	0.798532i	$0.705611\pi$
$720$	0	0
$721$	148.981	0.00769534
$722$	0	0
$723$	−354.532	−0.0182368
$724$	0	0
$725$	17870.1	0.915417
$726$	0	0
$727$	36607.6	1.86754	0.933769	−	0.357875i	$-0.116499\pi$
$727$	36607.6	1.86754	0.933769	+	0.357875i	$0.116499\pi$
$728$	0	0
$729$	−1719.60	−0.0873649
$730$	0	0
$731$	2309.40	0.116848
$732$	0	0
$733$	−36919.3	−1.86036	−0.930182	−	0.367100i	$-0.880351\pi$
$733$	−36919.3	−1.86036	−0.930182	+	0.367100i	$0.880351\pi$
$734$	0	0
$735$	251.255	0.0126091
$736$	0	0
$737$	−8057.39	−0.402711
$738$	0	0
$739$	−33782.6	−1.68161	−0.840806	−	0.541336i	$-0.817919\pi$
$739$	−33782.6	−1.68161	−0.840806	+	0.541336i	$0.817919\pi$
$740$	0	0
$741$	5959.77	0.295462
$742$	0	0
$743$	−12495.2	−0.616964	−0.308482	−	0.951230i	$-0.599821\pi$
$743$	−12495.2	−0.616964	−0.308482	+	0.951230i	$0.599821\pi$
$744$	0	0
$745$	163.182	0.00802485
$746$	0	0
$747$	21323.4	1.04442
$748$	0	0
$749$	−12170.4	−0.593719
$750$	0	0
$751$	29369.7	1.42705	0.713526	−	0.700629i	$-0.247097\pi$
$751$	29369.7	1.42705	0.713526	+	0.700629i	$0.247097\pi$
$752$	0	0
$753$	9908.92	0.479550
$754$	0	0
$755$	−3651.75	−0.176028
$756$	0	0
$757$	−21027.1	−1.00957	−0.504784	−	0.863246i	$-0.668428\pi$
$757$	−21027.1	−1.00957	−0.504784	+	0.863246i	$0.668428\pi$
$758$	0	0
$759$	−2635.61	−0.126043
$760$	0	0
$761$	9511.24	0.453065	0.226532	−	0.974004i	$-0.427261\pi$
$761$	9511.24	0.453065	0.226532	+	0.974004i	$0.427261\pi$
$762$	0	0
$763$	12112.8	0.574723
$764$	0	0
$765$	−4478.75	−0.211673
$766$	0	0
$767$	1117.31	0.0525996
$768$	0	0
$769$	17789.8	0.834223	0.417111	−	0.908855i	$-0.363043\pi$
$769$	17789.8	0.834223	0.417111	+	0.908855i	$0.363043\pi$
$770$	0	0
$771$	15563.6	0.726990
$772$	0	0
$773$	21162.6	0.984690	0.492345	−	0.870400i	$-0.336140\pi$
$773$	21162.6	0.984690	0.492345	+	0.870400i	$0.336140\pi$
$774$	0	0
$775$	34034.2	1.57748
$776$	0	0
$777$	−1642.60	−0.0758403
$778$	0	0
$779$	−65345.2	−3.00544
$780$	0	0
$781$	−3721.45	−0.170505
$782$	0	0
$783$	16104.0	0.735007
$784$	0	0
$785$	−2069.25	−0.0940826
$786$	0	0
$787$	−13182.0	−0.597062	−0.298531	−	0.954400i	$-0.596497\pi$
$787$	−13182.0	−0.597062	−0.298531	+	0.954400i	$0.596497\pi$
$788$	0	0
$789$	13577.5	0.612639
$790$	0	0
$791$	−10680.0	−0.480071
$792$	0	0
$793$	9887.87	0.442785
$794$	0	0
$795$	−2929.74	−0.130701
$796$	0	0
$797$	5657.68	0.251450	0.125725	−	0.992065i	$-0.459874\pi$
$797$	5657.68	0.251450	0.125725	+	0.992065i	$0.459874\pi$
$798$	0	0
$799$	−1047.72	−0.0463902
$800$	0	0
$801$	10353.9	0.456725
$802$	0	0
$803$	12563.1	0.552109
$804$	0	0
$805$	1795.49	0.0786122
$806$	0	0
$807$	1441.96	0.0628987
$808$	0	0
$809$	20138.9	0.875212	0.437606	−	0.899167i	$-0.355826\pi$
$809$	20138.9	0.875212	0.437606	+	0.899167i	$0.355826\pi$
$810$	0	0
$811$	32209.2	1.39460	0.697298	−	0.716782i	$-0.254386\pi$
$811$	32209.2	1.39460	0.697298	+	0.716782i	$0.254386\pi$
$812$	0	0
$813$	−3962.76	−0.170947
$814$	0	0
$815$	6851.55	0.294478
$816$	0	0
$817$	−3469.46	−0.148569
$818$	0	0
$819$	3274.22	0.139695
$820$	0	0
$821$	−17618.4	−0.748947	−0.374473	−	0.927238i	$-0.622177\pi$
$821$	−17618.4	−0.748947	−0.374473	+	0.927238i	$0.622177\pi$
$822$	0	0
$823$	−29098.2	−1.23244	−0.616222	−	0.787573i	$-0.711337\pi$
$823$	−29098.2	−1.23244	−0.616222	+	0.787573i	$0.711337\pi$
$824$	0	0
$825$	2877.13	0.121417
$826$	0	0
$827$	24917.8	1.04773	0.523866	−	0.851800i	$-0.324489\pi$
$827$	24917.8	1.04773	0.523866	+	0.851800i	$0.324489\pi$
$828$	0	0
$829$	31331.9	1.31267	0.656335	−	0.754470i	$-0.272106\pi$
$829$	31331.9	1.31267	0.656335	+	0.754470i	$0.272106\pi$
$830$	0	0
$831$	15929.5	0.664969
$832$	0	0
$833$	4217.39	0.175419
$834$	0	0
$835$	6158.01	0.255218
$836$	0	0
$837$	30670.7	1.26659
$838$	0	0
$839$	−27675.8	−1.13883	−0.569413	−	0.822051i	$-0.692830\pi$
$839$	−27675.8	−1.13883	−0.569413	+	0.822051i	$0.692830\pi$
$840$	0	0
$841$	−2030.74	−0.0832647
$842$	0	0
$843$	−7226.55	−0.295250
$844$	0	0
$845$	−4108.26	−0.167252
$846$	0	0
$847$	847.000	0.0343604
$848$	0	0
$849$	5040.84	0.203771
$850$	0	0
$851$	−11738.2	−0.472832
$852$	0	0
$853$	31985.7	1.28390	0.641952	−	0.766745i	$-0.278125\pi$
$853$	31985.7	1.28390	0.641952	+	0.766745i	$0.278125\pi$
$854$	0	0
$855$	6728.52	0.269135
$856$	0	0
$857$	−10590.3	−0.422119	−0.211060	−	0.977473i	$-0.567691\pi$
$857$	−10590.3	−0.422119	−0.211060	+	0.977473i	$0.567691\pi$
$858$	0	0
$859$	5241.49	0.208192	0.104096	−	0.994567i	$-0.466805\pi$
$859$	5241.49	0.208192	0.104096	+	0.994567i	$0.466805\pi$
$860$	0	0
$861$	7742.15	0.306448
$862$	0	0
$863$	−2295.89	−0.0905596	−0.0452798	−	0.998974i	$-0.514418\pi$
$863$	−2295.89	−0.0905596	−0.0452798	+	0.998974i	$0.514418\pi$
$864$	0	0
$865$	−9843.44	−0.386921
$866$	0	0
$867$	5460.27	0.213888
$868$	0	0
$869$	1716.17	0.0669931
$870$	0	0
$871$	−15426.2	−0.600113
$872$	0	0
$873$	2800.64	0.108576
$874$	0	0
$875$	−4010.08	−0.154932
$876$	0	0
$877$	−34939.0	−1.34528	−0.672638	−	0.739972i	$-0.734839\pi$
$877$	−34939.0	−1.34528	−0.672638	+	0.739972i	$0.734839\pi$
$878$	0	0
$879$	16671.3	0.639716
$880$	0	0
$881$	31320.9	1.19776	0.598880	−	0.800839i	$-0.295613\pi$
$881$	31320.9	1.19776	0.598880	+	0.800839i	$0.295613\pi$
$882$	0	0
$883$	12551.8	0.478372	0.239186	−	0.970974i	$-0.423119\pi$
$883$	12551.8	0.478372	0.239186	+	0.970974i	$0.423119\pi$
$884$	0	0
$885$	−272.041	−0.0103328
$886$	0	0
$887$	−9061.43	−0.343014	−0.171507	−	0.985183i	$-0.554864\pi$
$887$	−9061.43	−0.343014	−0.171507	+	0.985183i	$0.554864\pi$
$888$	0	0
$889$	521.238	0.0196645
$890$	0	0
$891$	−4003.62	−0.150535
$892$	0	0
$893$	1574.02	0.0589837
$894$	0	0
$895$	−3915.16	−0.146223
$896$	0	0
$897$	−5046.00	−0.187827
$898$	0	0
$899$	42582.2	1.57975
$900$	0	0
$901$	−49176.6	−1.81832
$902$	0	0
$903$	411.064	0.0151488
$904$	0	0
$905$	5548.36	0.203794
$906$	0	0
$907$	7541.25	0.276079	0.138039	−	0.990427i	$-0.455920\pi$
$907$	7541.25	0.276079	0.138039	+	0.990427i	$0.455920\pi$
$908$	0	0
$909$	12773.5	0.466085
$910$	0	0
$911$	49191.8	1.78902	0.894511	−	0.447047i	$-0.147524\pi$
$911$	49191.8	1.78902	0.894511	+	0.447047i	$0.147524\pi$
$912$	0	0
$913$	10560.8	0.382817
$914$	0	0
$915$	−2407.48	−0.0869822
$916$	0	0
$917$	16553.8	0.596133
$918$	0	0
$919$	42169.9	1.51366	0.756831	−	0.653610i	$-0.226746\pi$
$919$	42169.9	1.51366	0.756831	+	0.653610i	$0.226746\pi$
$920$	0	0
$921$	−15472.1	−0.553553
$922$	0	0
$923$	−7124.89	−0.254083
$924$	0	0
$925$	12813.9	0.455478
$926$	0	0
$927$	−472.699	−0.0167481
$928$	0	0
$929$	−56429.2	−1.99288	−0.996438	−	0.0843275i	$-0.973126\pi$
$929$	−56429.2	−1.99288	−0.996438	+	0.0843275i	$0.973126\pi$
$930$	0	0
$931$	−6335.87	−0.223040
$932$	0	0
$933$	−6972.57	−0.244664
$934$	0	0
$935$	−2218.19	−0.0775855
$936$	0	0
$937$	−31947.1	−1.11384	−0.556919	−	0.830567i	$-0.688017\pi$
$937$	−31947.1	−1.11384	−0.556919	+	0.830567i	$0.688017\pi$
$938$	0	0
$939$	9840.09	0.341980
$940$	0	0
$941$	−27012.6	−0.935799	−0.467900	−	0.883782i	$-0.654989\pi$
$941$	−27012.6	−0.935799	−0.467900	+	0.883782i	$0.654989\pi$
$942$	0	0
$943$	55326.3	1.91057
$944$	0	0
$945$	−1766.33	−0.0608027
$946$	0	0
$947$	−25388.4	−0.871184	−0.435592	−	0.900144i	$-0.643461\pi$
$947$	−25388.4	−0.871184	−0.435592	+	0.900144i	$0.643461\pi$
$948$	0	0
$949$	24052.7	0.822744
$950$	0	0
$951$	13974.7	0.476508
$952$	0	0
$953$	39733.5	1.35057	0.675285	−	0.737557i	$-0.264021\pi$
$953$	39733.5	1.35057	0.675285	+	0.737557i	$0.264021\pi$
$954$	0	0
$955$	9908.36	0.335735
$956$	0	0
$957$	3599.75	0.121592
$958$	0	0
$959$	−14841.4	−0.499743
$960$	0	0
$961$	51308.5	1.72228
$962$	0	0
$963$	38615.1	1.29217
$964$	0	0
$965$	6407.80	0.213756
$966$	0	0
$967$	−4200.60	−0.139692	−0.0698460	−	0.997558i	$-0.522251\pi$
$967$	−4200.60	−0.139692	−0.0698460	+	0.997558i	$0.522251\pi$
$968$	0	0
$969$	−24356.7	−0.807482
$970$	0	0
$971$	−36383.1	−1.20246	−0.601231	−	0.799075i	$-0.705323\pi$
$971$	−36383.1	−1.20246	−0.601231	+	0.799075i	$0.705323\pi$
$972$	0	0
$973$	475.880	0.0156794
$974$	0	0
$975$	5508.40	0.180933
$976$	0	0
$977$	5474.38	0.179264	0.0896319	−	0.995975i	$-0.471431\pi$
$977$	5474.38	0.179264	0.0896319	+	0.995975i	$0.471431\pi$
$978$	0	0
$979$	5127.96	0.167406
$980$	0	0
$981$	−38432.5	−1.25082
$982$	0	0
$983$	−20660.9	−0.670376	−0.335188	−	0.942151i	$-0.608800\pi$
$983$	−20660.9	−0.670376	−0.335188	+	0.942151i	$0.608800\pi$
$984$	0	0
$985$	657.704	0.0212753
$986$	0	0
$987$	−186.491	−0.00601425
$988$	0	0
$989$	2937.51	0.0944462
$990$	0	0
$991$	36458.2	1.16865	0.584325	−	0.811520i	$-0.301359\pi$
$991$	36458.2	1.16865	0.584325	+	0.811520i	$0.301359\pi$
$992$	0	0
$993$	−9232.31	−0.295044
$994$	0	0
$995$	3478.30	0.110824
$996$	0	0
$997$	34040.5	1.08132	0.540658	−	0.841242i	$-0.318175\pi$
$997$	34040.5	1.08132	0.540658	+	0.841242i	$0.318175\pi$
$998$	0	0
$999$	11547.5	0.365713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	616.4.a.h.1.4	✓	6
4.3	odd	2		1232.4.a.ba.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.h.1.4	✓	6	1.1	even	1	trivial
1232.4.a.ba.1.3		6	4.3	odd	2

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:	\( N \)	\(=\)	\( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	616.a (trivial)

\(f(q)\)	\(=\)	\(q+2.18857 q^{3} +2.34292 q^{5} +7.00000 q^{7} -22.2102 q^{9} -11.0000 q^{11} -21.0600 q^{13} +5.12765 q^{15} +86.0692 q^{17} -129.304 q^{19} +15.3200 q^{21} +109.478 q^{23} -119.511 q^{25} -107.700 q^{27} -149.527 q^{29} -284.780 q^{31} -24.0743 q^{33} +16.4004 q^{35} -107.219 q^{37} -46.0913 q^{39} +505.363 q^{41} +26.8319 q^{43} -52.0366 q^{45} -12.1730 q^{47} +49.0000 q^{49} +188.368 q^{51} -571.361 q^{53} -25.7721 q^{55} -282.990 q^{57} -53.0539 q^{59} -469.509 q^{61} -155.471 q^{63} -49.3419 q^{65} +732.490 q^{67} +239.601 q^{69} +338.314 q^{71} -1142.10 q^{73} -261.558 q^{75} -77.0000 q^{77} -156.015 q^{79} +363.966 q^{81} -960.073 q^{83} +201.653 q^{85} -327.250 q^{87} -466.178 q^{89} -147.420 q^{91} -623.260 q^{93} -302.948 q^{95} -126.097 q^{97} +244.312 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{5} + 42 q^{7} + 28 q^{9} - 66 q^{11} - 70 q^{13} - 50 q^{15} - 102 q^{17} - 136 q^{19} - 146 q^{23} + 44 q^{25} - 30 q^{27} - 148 q^{29} + 308 q^{31} - 98 q^{35} - 6 q^{37} - 540 q^{39} - 90 q^{41}+ \cdots - 308 q^{99}+O(q^{100}) \) 6 * q - 14 * q^5 + 42 * q^7 + 28 * q^9 - 66 * q^11 - 70 * q^13 - 50 * q^15 - 102 * q^17 - 136 * q^19 - 146 * q^23 + 44 * q^25 - 30 * q^27 - 148 * q^29 + 308 * q^31 - 98 * q^35 - 6 * q^37 - 540 * q^39 - 90 * q^41 - 396 * q^43 - 812 * q^45 - 346 * q^47 + 294 * q^49 - 600 * q^51 - 984 * q^53 + 154 * q^55 - 1580 * q^57 - 44 * q^59 - 1414 * q^61 + 196 * q^63 - 1208 * q^65 - 926 * q^67 - 2550 * q^69 - 1478 * q^71 - 678 * q^73 - 1350 * q^75 - 462 * q^77 - 700 * q^79 - 966 * q^81 - 1192 * q^83 - 1596 * q^85 + 1900 * q^87 - 3314 * q^89 - 490 * q^91 - 2330 * q^93 + 668 * q^95 - 346 * q^97 - 308 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more