LMFDB - Embedded newform 2240.4.a.ci.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, 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("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.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:	$132.164278413$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.817716.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 20x^{2} + 12x + 32 $ x^4 - 2x^3 - 20x^2 + 12*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.56756$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.96834 q^{3} -5.00000 q^{5} +7.00000 q^{7} -2.31560 q^{9} -22.8554 q^{11} -0.450727 q^{13} -24.8417 q^{15} -46.9410 q^{17} +113.126 q^{19} +34.7784 q^{21} +175.267 q^{23} +25.0000 q^{25} -145.650 q^{27} -172.586 q^{29} -213.349 q^{31} -113.553 q^{33} -35.0000 q^{35} +400.325 q^{37} -2.23936 q^{39} -375.755 q^{41} +131.262 q^{43} +11.5780 q^{45} +516.159 q^{47} +49.0000 q^{49} -233.219 q^{51} -48.1978 q^{53} +114.277 q^{55} +562.048 q^{57} -452.504 q^{59} -605.168 q^{61} -16.2092 q^{63} +2.25363 q^{65} -503.210 q^{67} +870.787 q^{69} +1149.94 q^{71} -1076.40 q^{73} +124.208 q^{75} -159.988 q^{77} -491.100 q^{79} -661.117 q^{81} +286.987 q^{83} +234.705 q^{85} -857.466 q^{87} -1042.74 q^{89} -3.15509 q^{91} -1059.99 q^{93} -565.630 q^{95} -674.847 q^{97} +52.9240 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 5 q^{3} - 20 q^{5} + 28 q^{7} + 23 q^{9} + 13 q^{11} + 39 q^{13} - 25 q^{15} - 233 q^{17} + 8 q^{19} + 35 q^{21} - 32 q^{23} + 100 q^{25} + 191 q^{27} + 181 q^{29} - 90 q^{31} - 645 q^{33} - 140 q^{35}+ \cdots - 138 q^{99}+O(q^{100}) $ 4 * q + 5 * q^3 - 20 * q^5 + 28 * q^7 + 23 * q^9 + 13 * q^11 + 39 * q^13 - 25 * q^15 - 233 * q^17 + 8 * q^19 + 35 * q^21 - 32 * q^23 + 100 * q^25 + 191 * q^27 + 181 * q^29 - 90 * q^31 - 645 * q^33 - 140 * q^35 + 286 * q^37 + 451 * q^39 - 632 * q^41 - 356 * q^43 - 115 * q^45 - 201 * q^47 + 196 * q^49 - 341 * q^51 + 506 * q^53 - 65 * q^55 - 552 * q^57 + 1116 * q^59 - 480 * q^61 + 161 * q^63 - 195 * q^65 - 1504 * q^67 - 856 * q^69 + 1204 * q^71 - 312 * q^73 + 125 * q^75 + 91 * q^77 - 819 * q^79 - 1420 * q^81 - 2280 * q^83 + 1165 * q^85 + 589 * q^87 - 1404 * q^89 + 273 * q^91 + 2922 * q^93 - 40 * q^95 - 2833 * q^97 - 138 * 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
$3$	4.96834	0.956157	0.478079	−	0.878317i	$-0.341333\pi$
$3$	4.96834	0.956157	0.478079	+	0.878317i	$0.341333\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−2.31560	−0.0857630
$10$	0	0
$11$	−22.8554	−0.626469	−0.313235	−	0.949676i	$-0.601413\pi$
$11$	−22.8554	−0.626469	−0.313235	+	0.949676i	$0.601413\pi$
$12$	0	0
$13$	−0.450727	−0.00961609	−0.00480804	−	0.999988i	$-0.501530\pi$
$13$	−0.450727	−0.00961609	−0.00480804	+	0.999988i	$0.501530\pi$
$14$	0	0
$15$	−24.8417	−0.427607
$16$	0	0
$17$	−46.9410	−0.669697	−0.334849	−	0.942272i	$-0.608685\pi$
$17$	−46.9410	−0.669697	−0.334849	+	0.942272i	$0.608685\pi$
$18$	0	0
$19$	113.126	1.36594	0.682971	−	0.730446i	$-0.260688\pi$
$19$	113.126	1.36594	0.682971	+	0.730446i	$0.260688\pi$
$20$	0	0
$21$	34.7784	0.361394
$22$	0	0
$23$	175.267	1.58894	0.794472	−	0.607300i	$-0.207748\pi$
$23$	175.267	1.58894	0.794472	+	0.607300i	$0.207748\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−145.650	−1.03816
$28$	0	0
$29$	−172.586	−1.10512	−0.552559	−	0.833474i	$-0.686349\pi$
$29$	−172.586	−1.10512	−0.552559	+	0.833474i	$0.686349\pi$
$30$	0	0
$31$	−213.349	−1.23608	−0.618042	−	0.786145i	$-0.712074\pi$
$31$	−213.349	−1.23608	−0.618042	+	0.786145i	$0.712074\pi$
$32$	0	0
$33$	−113.553	−0.599003
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	400.325	1.77873	0.889366	−	0.457196i	$-0.151146\pi$
$37$	400.325	1.77873	0.889366	+	0.457196i	$0.151146\pi$
$38$	0	0
$39$	−2.23936	−0.00919449
$40$	0	0
$41$	−375.755	−1.43130	−0.715648	−	0.698462i	$-0.753868\pi$
$41$	−375.755	−1.43130	−0.715648	+	0.698462i	$0.753868\pi$
$42$	0	0
$43$	131.262	0.465517	0.232759	−	0.972535i	$-0.425225\pi$
$43$	131.262	0.465517	0.232759	+	0.972535i	$0.425225\pi$
$44$	0	0
$45$	11.5780	0.0383544
$46$	0	0
$47$	516.159	1.60190	0.800952	−	0.598728i	$-0.204327\pi$
$47$	516.159	1.60190	0.800952	+	0.598728i	$0.204327\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−233.219	−0.640336
$52$	0	0
$53$	−48.1978	−0.124915	−0.0624573	−	0.998048i	$-0.519894\pi$
$53$	−48.1978	−0.124915	−0.0624573	+	0.998048i	$0.519894\pi$
$54$	0	0
$55$	114.277	0.280165
$56$	0	0
$57$	562.048	1.30605
$58$	0	0
$59$	−452.504	−0.998492	−0.499246	−	0.866460i	$-0.666390\pi$
$59$	−452.504	−0.998492	−0.499246	+	0.866460i	$0.666390\pi$
$60$	0	0
$61$	−605.168	−1.27023	−0.635113	−	0.772419i	$-0.719046\pi$
$61$	−605.168	−1.27023	−0.635113	+	0.772419i	$0.719046\pi$
$62$	0	0
$63$	−16.2092	−0.0324154
$64$	0	0
$65$	2.25363	0.00430044
$66$	0	0
$67$	−503.210	−0.917566	−0.458783	−	0.888548i	$-0.651714\pi$
$67$	−503.210	−0.917566	−0.458783	+	0.888548i	$0.651714\pi$
$68$	0	0
$69$	870.787	1.51928
$70$	0	0
$71$	1149.94	1.92215	0.961077	−	0.276280i	$-0.0891016\pi$
$71$	1149.94	1.92215	0.961077	+	0.276280i	$0.0891016\pi$
$72$	0	0
$73$	−1076.40	−1.72580	−0.862900	−	0.505374i	$-0.831355\pi$
$73$	−1076.40	−1.72580	−0.862900	+	0.505374i	$0.831355\pi$
$74$	0	0
$75$	124.208	0.191231
$76$	0	0
$77$	−159.988	−0.236783
$78$	0	0
$79$	−491.100	−0.699406	−0.349703	−	0.936861i	$-0.613717\pi$
$79$	−491.100	−0.699406	−0.349703	+	0.936861i	$0.613717\pi$
$80$	0	0
$81$	−661.117	−0.906882
$82$	0	0
$83$	286.987	0.379529	0.189764	−	0.981830i	$-0.439228\pi$
$83$	286.987	0.379529	0.189764	+	0.981830i	$0.439228\pi$
$84$	0	0
$85$	234.705	0.299498
$86$	0	0
$87$	−857.466	−1.05667
$88$	0	0
$89$	−1042.74	−1.24191	−0.620956	−	0.783845i	$-0.713256\pi$
$89$	−1042.74	−1.24191	−0.620956	+	0.783845i	$0.713256\pi$
$90$	0	0
$91$	−3.15509	−0.00363454
$92$	0	0
$93$	−1059.99	−1.18189
$94$	0	0
$95$	−565.630	−0.610867
$96$	0	0
$97$	−674.847	−0.706395	−0.353197	−	0.935549i	$-0.614906\pi$
$97$	−674.847	−0.706395	−0.353197	+	0.935549i	$0.614906\pi$
$98$	0	0
$99$	52.9240	0.0537279
$100$	0	0
$101$	−218.795	−0.215554	−0.107777	−	0.994175i	$-0.534373\pi$
$101$	−218.795	−0.215554	−0.107777	+	0.994175i	$0.534373\pi$
$102$	0	0
$103$	−1219.78	−1.16688	−0.583438	−	0.812158i	$-0.698293\pi$
$103$	−1219.78	−1.16688	−0.583438	+	0.812158i	$0.698293\pi$
$104$	0	0
$105$	−173.892	−0.161620
$106$	0	0
$107$	1582.05	1.42937	0.714685	−	0.699446i	$-0.246570\pi$
$107$	1582.05	1.42937	0.714685	+	0.699446i	$0.246570\pi$
$108$	0	0
$109$	−1252.11	−1.10028	−0.550139	−	0.835073i	$-0.685425\pi$
$109$	−1252.11	−1.10028	−0.550139	+	0.835073i	$0.685425\pi$
$110$	0	0
$111$	1988.95	1.70075
$112$	0	0
$113$	−1377.44	−1.14671	−0.573355	−	0.819307i	$-0.694358\pi$
$113$	−1377.44	−1.14671	−0.573355	+	0.819307i	$0.694358\pi$
$114$	0	0
$115$	−876.336	−0.710598
$116$	0	0
$117$	1.04370	0.000824705 0
$118$	0	0
$119$	−328.587	−0.253122
$120$	0	0
$121$	−808.631	−0.607537
$122$	0	0
$123$	−1866.88	−1.36854
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1531.99	−1.07041	−0.535206	−	0.844722i	$-0.679766\pi$
$127$	−1531.99	−1.07041	−0.535206	+	0.844722i	$0.679766\pi$
$128$	0	0
$129$	652.154	0.445108
$130$	0	0
$131$	−684.715	−0.456671	−0.228335	−	0.973583i	$-0.573328\pi$
$131$	−684.715	−0.456671	−0.228335	+	0.973583i	$0.573328\pi$
$132$	0	0
$133$	791.882	0.516277
$134$	0	0
$135$	728.249	0.464279
$136$	0	0
$137$	−47.8521	−0.0298415	−0.0149207	−	0.999889i	$-0.504750\pi$
$137$	−47.8521	−0.0298415	−0.0149207	+	0.999889i	$0.504750\pi$
$138$	0	0
$139$	547.426	0.334043	0.167022	−	0.985953i	$-0.446585\pi$
$139$	547.426	0.334043	0.167022	+	0.985953i	$0.446585\pi$
$140$	0	0
$141$	2564.45	1.53167
$142$	0	0
$143$	10.3015	0.00602418
$144$	0	0
$145$	862.930	0.494224
$146$	0	0
$147$	243.449	0.136594
$148$	0	0
$149$	2459.90	1.35250	0.676250	−	0.736672i	$-0.263604\pi$
$149$	2459.90	1.35250	0.676250	+	0.736672i	$0.263604\pi$
$150$	0	0
$151$	−1525.50	−0.822142	−0.411071	−	0.911603i	$-0.634845\pi$
$151$	−1525.50	−0.822142	−0.411071	+	0.911603i	$0.634845\pi$
$152$	0	0
$153$	108.697	0.0574353
$154$	0	0
$155$	1066.74	0.552793
$156$	0	0
$157$	3098.22	1.57493	0.787467	−	0.616357i	$-0.211392\pi$
$157$	3098.22	1.57493	0.787467	+	0.616357i	$0.211392\pi$
$158$	0	0
$159$	−239.463	−0.119438
$160$	0	0
$161$	1226.87	0.600565
$162$	0	0
$163$	−1282.69	−0.616367	−0.308184	−	0.951327i	$-0.599721\pi$
$163$	−1282.69	−0.616367	−0.308184	+	0.951327i	$0.599721\pi$
$164$	0	0
$165$	567.767	0.267882
$166$	0	0
$167$	−2182.07	−1.01110	−0.505550	−	0.862797i	$-0.668710\pi$
$167$	−2182.07	−1.01110	−0.505550	+	0.862797i	$0.668710\pi$
$168$	0	0
$169$	−2196.80	−0.999908
$170$	0	0
$171$	−261.955	−0.117147
$172$	0	0
$173$	3324.16	1.46088	0.730438	−	0.682979i	$-0.239316\pi$
$173$	3324.16	1.46088	0.730438	+	0.682979i	$0.239316\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	−2248.19	−0.954715
$178$	0	0
$179$	2501.60	1.04457	0.522285	−	0.852771i	$-0.325080\pi$
$179$	2501.60	1.04457	0.522285	+	0.852771i	$0.325080\pi$
$180$	0	0
$181$	−4697.30	−1.92899	−0.964497	−	0.264094i	$-0.914927\pi$
$181$	−4697.30	−1.92899	−0.964497	+	0.264094i	$0.914927\pi$
$182$	0	0
$183$	−3006.68	−1.21454
$184$	0	0
$185$	−2001.63	−0.795473
$186$	0	0
$187$	1072.85	0.419545
$188$	0	0
$189$	−1019.55	−0.392388
$190$	0	0
$191$	−2751.84	−1.04249	−0.521246	−	0.853406i	$-0.674533\pi$
$191$	−2751.84	−1.04249	−0.521246	+	0.853406i	$0.674533\pi$
$192$	0	0
$193$	1138.94	0.424781	0.212390	−	0.977185i	$-0.431875\pi$
$193$	1138.94	0.424781	0.212390	+	0.977185i	$0.431875\pi$
$194$	0	0
$195$	11.1968	0.00411190
$196$	0	0
$197$	−2358.84	−0.853099	−0.426550	−	0.904464i	$-0.640271\pi$
$197$	−2358.84	−0.853099	−0.426550	+	0.904464i	$0.640271\pi$
$198$	0	0
$199$	−2330.05	−0.830014	−0.415007	−	0.909818i	$-0.636221\pi$
$199$	−2330.05	−0.830014	−0.415007	+	0.909818i	$0.636221\pi$
$200$	0	0
$201$	−2500.12	−0.877338
$202$	0	0
$203$	−1208.10	−0.417695
$204$	0	0
$205$	1878.78	0.640095
$206$	0	0
$207$	−405.849	−0.136273
$208$	0	0
$209$	−2585.54	−0.855720
$210$	0	0
$211$	−3133.89	−1.02249	−0.511246	−	0.859435i	$-0.670816\pi$
$211$	−3133.89	−1.02249	−0.511246	+	0.859435i	$0.670816\pi$
$212$	0	0
$213$	5713.30	1.83788
$214$	0	0
$215$	−656.309	−0.208186
$216$	0	0
$217$	−1493.44	−0.467196
$218$	0	0
$219$	−5347.94	−1.65014
$220$	0	0
$221$	21.1575	0.00643987
$222$	0	0
$223$	4203.14	1.26217	0.631083	−	0.775715i	$-0.282611\pi$
$223$	4203.14	1.26217	0.631083	+	0.775715i	$0.282611\pi$
$224$	0	0
$225$	−57.8900	−0.0171526
$226$	0	0
$227$	−1116.15	−0.326351	−0.163175	−	0.986597i	$-0.552174\pi$
$227$	−1116.15	−0.326351	−0.163175	+	0.986597i	$0.552174\pi$
$228$	0	0
$229$	2227.72	0.642847	0.321423	−	0.946936i	$-0.395839\pi$
$229$	2227.72	0.642847	0.321423	+	0.946936i	$0.395839\pi$
$230$	0	0
$231$	−794.873	−0.226402
$232$	0	0
$233$	1360.52	0.382535	0.191268	−	0.981538i	$-0.438740\pi$
$233$	1360.52	0.382535	0.191268	+	0.981538i	$0.438740\pi$
$234$	0	0
$235$	−2580.79	−0.716393
$236$	0	0
$237$	−2439.95	−0.668742
$238$	0	0
$239$	−5361.84	−1.45117	−0.725583	−	0.688135i	$-0.758430\pi$
$239$	−5361.84	−1.45117	−0.725583	+	0.688135i	$0.758430\pi$
$240$	0	0
$241$	−6792.05	−1.81541	−0.907706	−	0.419608i	$-0.862168\pi$
$241$	−6792.05	−1.81541	−0.907706	+	0.419608i	$0.862168\pi$
$242$	0	0
$243$	647.894	0.171039
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−50.9889	−0.0131350
$248$	0	0
$249$	1425.85	0.362889
$250$	0	0
$251$	−3237.44	−0.814124	−0.407062	−	0.913401i	$-0.633447\pi$
$251$	−3237.44	−0.814124	−0.407062	+	0.913401i	$0.633447\pi$
$252$	0	0
$253$	−4005.80	−0.995425
$254$	0	0
$255$	1166.09	0.286367
$256$	0	0
$257$	3371.52	0.818325	0.409162	−	0.912462i	$-0.365821\pi$
$257$	3371.52	0.818325	0.409162	+	0.912462i	$0.365821\pi$
$258$	0	0
$259$	2802.28	0.672297
$260$	0	0
$261$	399.641	0.0947783
$262$	0	0
$263$	−2591.22	−0.607534	−0.303767	−	0.952746i	$-0.598244\pi$
$263$	−2591.22	−0.607534	−0.303767	+	0.952746i	$0.598244\pi$
$264$	0	0
$265$	240.989	0.0558635
$266$	0	0
$267$	−5180.69	−1.18746
$268$	0	0
$269$	−2574.98	−0.583641	−0.291821	−	0.956473i	$-0.594261\pi$
$269$	−2574.98	−0.583641	−0.291821	+	0.956473i	$0.594261\pi$
$270$	0	0
$271$	458.239	0.102716	0.0513580	−	0.998680i	$-0.483645\pi$
$271$	458.239	0.102716	0.0513580	+	0.998680i	$0.483645\pi$
$272$	0	0
$273$	−15.6755	−0.00347519
$274$	0	0
$275$	−571.385	−0.125294
$276$	0	0
$277$	7899.72	1.71353	0.856766	−	0.515706i	$-0.172470\pi$
$277$	7899.72	1.71353	0.856766	+	0.515706i	$0.172470\pi$
$278$	0	0
$279$	494.031	0.106010
$280$	0	0
$281$	5443.55	1.15564	0.577820	−	0.816164i	$-0.303903\pi$
$281$	5443.55	1.15564	0.577820	+	0.816164i	$0.303903\pi$
$282$	0	0
$283$	7375.80	1.54928	0.774639	−	0.632404i	$-0.217932\pi$
$283$	7375.80	1.54928	0.774639	+	0.632404i	$0.217932\pi$
$284$	0	0
$285$	−2810.24	−0.584085
$286$	0	0
$287$	−2630.29	−0.540979
$288$	0	0
$289$	−2709.55	−0.551506
$290$	0	0
$291$	−3352.87	−0.675425
$292$	0	0
$293$	7931.69	1.58148	0.790741	−	0.612150i	$-0.209695\pi$
$293$	7931.69	1.58148	0.790741	+	0.612150i	$0.209695\pi$
$294$	0	0
$295$	2262.52	0.446539
$296$	0	0
$297$	3328.88	0.650375
$298$	0	0
$299$	−78.9976	−0.0152794
$300$	0	0
$301$	918.833	0.175949
$302$	0	0
$303$	−1087.05	−0.206104
$304$	0	0
$305$	3025.84	0.568062
$306$	0	0
$307$	−350.327	−0.0651278	−0.0325639	−	0.999470i	$-0.510367\pi$
$307$	−350.327	−0.0651278	−0.0325639	+	0.999470i	$0.510367\pi$
$308$	0	0
$309$	−6060.27	−1.11572
$310$	0	0
$311$	8242.06	1.50278	0.751390	−	0.659859i	$-0.229384\pi$
$311$	8242.06	1.50278	0.751390	+	0.659859i	$0.229384\pi$
$312$	0	0
$313$	−8490.82	−1.53332	−0.766661	−	0.642052i	$-0.778083\pi$
$313$	−8490.82	−1.53332	−0.766661	+	0.642052i	$0.778083\pi$
$314$	0	0
$315$	81.0461	0.0144966
$316$	0	0
$317$	2789.84	0.494299	0.247150	−	0.968977i	$-0.420506\pi$
$317$	2789.84	0.494299	0.247150	+	0.968977i	$0.420506\pi$
$318$	0	0
$319$	3944.52	0.692322
$320$	0	0
$321$	7860.17	1.36670
$322$	0	0
$323$	−5310.24	−0.914767
$324$	0	0
$325$	−11.2682	−0.00192322
$326$	0	0
$327$	−6220.91	−1.05204
$328$	0	0
$329$	3613.11	0.605463
$330$	0	0
$331$	−6921.94	−1.14944	−0.574719	−	0.818350i	$-0.694889\pi$
$331$	−6921.94	−1.14944	−0.574719	+	0.818350i	$0.694889\pi$
$332$	0	0
$333$	−926.994	−0.152549
$334$	0	0
$335$	2516.05	0.410348
$336$	0	0
$337$	10153.5	1.64124	0.820619	−	0.571475i	$-0.193629\pi$
$337$	10153.5	1.64124	0.820619	+	0.571475i	$0.193629\pi$
$338$	0	0
$339$	−6843.57	−1.09644
$340$	0	0
$341$	4876.17	0.774368
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−4353.93	−0.679443
$346$	0	0
$347$	4729.44	0.731670	0.365835	−	0.930680i	$-0.380783\pi$
$347$	4729.44	0.731670	0.365835	+	0.930680i	$0.380783\pi$
$348$	0	0
$349$	−756.020	−0.115957	−0.0579783	−	0.998318i	$-0.518465\pi$
$349$	−756.020	−0.115957	−0.0579783	+	0.998318i	$0.518465\pi$
$350$	0	0
$351$	65.6483	0.00998304
$352$	0	0
$353$	−3113.40	−0.469432	−0.234716	−	0.972064i	$-0.575416\pi$
$353$	−3113.40	−0.469432	−0.234716	+	0.972064i	$0.575416\pi$
$354$	0	0
$355$	−5749.71	−0.859614
$356$	0	0
$357$	−1632.53	−0.242024
$358$	0	0
$359$	−5835.64	−0.857920	−0.428960	−	0.903323i	$-0.641120\pi$
$359$	−5835.64	−0.857920	−0.428960	+	0.903323i	$0.641120\pi$
$360$	0	0
$361$	5938.49	0.865795
$362$	0	0
$363$	−4017.55	−0.580901
$364$	0	0
$365$	5382.02	0.771802
$366$	0	0
$367$	−9837.20	−1.39918	−0.699588	−	0.714546i	$-0.746633\pi$
$367$	−9837.20	−1.39918	−0.699588	+	0.714546i	$0.746633\pi$
$368$	0	0
$369$	870.099	0.122752
$370$	0	0
$371$	−337.384	−0.0472133
$372$	0	0
$373$	1271.19	0.176461	0.0882303	−	0.996100i	$-0.471879\pi$
$373$	1271.19	0.176461	0.0882303	+	0.996100i	$0.471879\pi$
$374$	0	0
$375$	−621.042	−0.0855213
$376$	0	0
$377$	77.7891	0.0106269
$378$	0	0
$379$	−4604.99	−0.624123	−0.312062	−	0.950062i	$-0.601020\pi$
$379$	−4604.99	−0.624123	−0.312062	+	0.950062i	$0.601020\pi$
$380$	0	0
$381$	−7611.45	−1.02348
$382$	0	0
$383$	−1634.45	−0.218058	−0.109029	−	0.994039i	$-0.534774\pi$
$383$	−1634.45	−0.218058	−0.109029	+	0.994039i	$0.534774\pi$
$384$	0	0
$385$	799.939	0.105893
$386$	0	0
$387$	−303.950	−0.0399242
$388$	0	0
$389$	−6462.01	−0.842254	−0.421127	−	0.907002i	$-0.638365\pi$
$389$	−6462.01	−0.842254	−0.421127	+	0.907002i	$0.638365\pi$
$390$	0	0
$391$	−8227.21	−1.06411
$392$	0	0
$393$	−3401.90	−0.436649
$394$	0	0
$395$	2455.50	0.312784
$396$	0	0
$397$	−7817.65	−0.988304	−0.494152	−	0.869376i	$-0.664521\pi$
$397$	−7817.65	−0.988304	−0.494152	+	0.869376i	$0.664521\pi$
$398$	0	0
$399$	3934.34	0.493642
$400$	0	0
$401$	11331.7	1.41117	0.705585	−	0.708625i	$-0.250684\pi$
$401$	11331.7	1.41117	0.705585	+	0.708625i	$0.250684\pi$
$402$	0	0
$403$	96.1620	0.0118863
$404$	0	0
$405$	3305.58	0.405570
$406$	0	0
$407$	−9149.59	−1.11432
$408$	0	0
$409$	−3964.56	−0.479302	−0.239651	−	0.970859i	$-0.577033\pi$
$409$	−3964.56	−0.479302	−0.239651	+	0.970859i	$0.577033\pi$
$410$	0	0
$411$	−237.746	−0.0285332
$412$	0	0
$413$	−3167.53	−0.377394
$414$	0	0
$415$	−1434.93	−0.169730
$416$	0	0
$417$	2719.80	0.319398
$418$	0	0
$419$	−5153.89	−0.600916	−0.300458	−	0.953795i	$-0.597140\pi$
$419$	−5153.89	−0.600916	−0.300458	+	0.953795i	$0.597140\pi$
$420$	0	0
$421$	−9154.46	−1.05976	−0.529882	−	0.848071i	$-0.677764\pi$
$421$	−9154.46	−1.05976	−0.529882	+	0.848071i	$0.677764\pi$
$422$	0	0
$423$	−1195.22	−0.137384
$424$	0	0
$425$	−1173.52	−0.133939
$426$	0	0
$427$	−4236.17	−0.480100
$428$	0	0
$429$	51.1815	0.00576006
$430$	0	0
$431$	16866.0	1.88494	0.942469	−	0.334294i	$-0.108498\pi$
$431$	16866.0	1.88494	0.942469	+	0.334294i	$0.108498\pi$
$432$	0	0
$433$	7415.34	0.822999	0.411500	−	0.911410i	$-0.365005\pi$
$433$	7415.34	0.822999	0.411500	+	0.911410i	$0.365005\pi$
$434$	0	0
$435$	4287.33	0.472556
$436$	0	0
$437$	19827.3	2.17041
$438$	0	0
$439$	−11509.2	−1.25126	−0.625629	−	0.780121i	$-0.715158\pi$
$439$	−11509.2	−1.25126	−0.625629	+	0.780121i	$0.715158\pi$
$440$	0	0
$441$	−113.464	−0.0122519
$442$	0	0
$443$	−3262.78	−0.349931	−0.174966	−	0.984575i	$-0.555981\pi$
$443$	−3262.78	−0.349931	−0.174966	+	0.984575i	$0.555981\pi$
$444$	0	0
$445$	5213.70	0.555400
$446$	0	0
$447$	12221.6	1.29320
$448$	0	0
$449$	−2342.43	−0.246205	−0.123103	−	0.992394i	$-0.539284\pi$
$449$	−2342.43	−0.246205	−0.123103	+	0.992394i	$0.539284\pi$
$450$	0	0
$451$	8588.03	0.896662
$452$	0	0
$453$	−7579.21	−0.786098
$454$	0	0
$455$	15.7754	0.00162542
$456$	0	0
$457$	16429.3	1.68169	0.840844	−	0.541277i	$-0.182059\pi$
$457$	16429.3	1.68169	0.840844	+	0.541277i	$0.182059\pi$
$458$	0	0
$459$	6836.94	0.695253
$460$	0	0
$461$	−6240.94	−0.630519	−0.315260	−	0.949005i	$-0.602092\pi$
$461$	−6240.94	−0.630519	−0.315260	+	0.949005i	$0.602092\pi$
$462$	0	0
$463$	−7559.25	−0.758764	−0.379382	−	0.925240i	$-0.623863\pi$
$463$	−7559.25	−0.758764	−0.379382	+	0.925240i	$0.623863\pi$
$464$	0	0
$465$	5299.95	0.528557
$466$	0	0
$467$	6127.85	0.607201	0.303601	−	0.952799i	$-0.401811\pi$
$467$	6127.85	0.607201	0.303601	+	0.952799i	$0.401811\pi$
$468$	0	0
$469$	−3522.47	−0.346807
$470$	0	0
$471$	15393.0	1.50588
$472$	0	0
$473$	−3000.04	−0.291632
$474$	0	0
$475$	2828.15	0.273188
$476$	0	0
$477$	111.607	0.0107131
$478$	0	0
$479$	−10823.4	−1.03243	−0.516214	−	0.856459i	$-0.672659\pi$
$479$	−10823.4	−1.03243	−0.516214	+	0.856459i	$0.672659\pi$
$480$	0	0
$481$	−180.437	−0.0171044
$482$	0	0
$483$	6095.51	0.574234
$484$	0	0
$485$	3374.23	0.315909
$486$	0	0
$487$	−7892.91	−0.734419	−0.367209	−	0.930138i	$-0.619687\pi$
$487$	−7892.91	−0.734419	−0.367209	+	0.930138i	$0.619687\pi$
$488$	0	0
$489$	−6372.83	−0.589344
$490$	0	0
$491$	−3104.54	−0.285349	−0.142674	−	0.989770i	$-0.545570\pi$
$491$	−3104.54	−0.285349	−0.142674	+	0.989770i	$0.545570\pi$
$492$	0	0
$493$	8101.35	0.740095
$494$	0	0
$495$	−264.620	−0.0240278
$496$	0	0
$497$	8049.59	0.726506
$498$	0	0
$499$	−5677.84	−0.509369	−0.254684	−	0.967024i	$-0.581972\pi$
$499$	−5677.84	−0.509369	−0.254684	+	0.967024i	$0.581972\pi$
$500$	0	0
$501$	−10841.3	−0.966771
$502$	0	0
$503$	−36.3259	−0.00322006	−0.00161003	−	0.999999i	$-0.500512\pi$
$503$	−36.3259	−0.00322006	−0.00161003	+	0.999999i	$0.500512\pi$
$504$	0	0
$505$	1093.98	0.0963987
$506$	0	0
$507$	−10914.4	−0.956069
$508$	0	0
$509$	−2613.79	−0.227611	−0.113805	−	0.993503i	$-0.536304\pi$
$509$	−2613.79	−0.227611	−0.113805	+	0.993503i	$0.536304\pi$
$510$	0	0
$511$	−7534.82	−0.652291
$512$	0	0
$513$	−16476.8	−1.41807
$514$	0	0
$515$	6098.89	0.521843
$516$	0	0
$517$	−11797.0	−1.00354
$518$	0	0
$519$	16515.6	1.39683
$520$	0	0
$521$	2865.84	0.240988	0.120494	−	0.992714i	$-0.461552\pi$
$521$	2865.84	0.240988	0.120494	+	0.992714i	$0.461552\pi$
$522$	0	0
$523$	−20595.8	−1.72197	−0.860987	−	0.508627i	$-0.830153\pi$
$523$	−20595.8	−1.72197	−0.860987	+	0.508627i	$0.830153\pi$
$524$	0	0
$525$	869.459	0.0722787
$526$	0	0
$527$	10014.8	0.827802
$528$	0	0
$529$	18551.6	1.52475
$530$	0	0
$531$	1047.82	0.0856337
$532$	0	0
$533$	169.363	0.0137635
$534$	0	0
$535$	−7910.26	−0.639234
$536$	0	0
$537$	12428.8	0.998773
$538$	0	0
$539$	−1119.91	−0.0894956
$540$	0	0
$541$	4228.02	0.336001	0.168001	−	0.985787i	$-0.446269\pi$
$541$	4228.02	0.336001	0.168001	+	0.985787i	$0.446269\pi$
$542$	0	0
$543$	−23337.8	−1.84442
$544$	0	0
$545$	6260.55	0.492059
$546$	0	0
$547$	−8575.91	−0.670346	−0.335173	−	0.942157i	$-0.608795\pi$
$547$	−8575.91	−0.670346	−0.335173	+	0.942157i	$0.608795\pi$
$548$	0	0
$549$	1401.33	0.108938
$550$	0	0
$551$	−19524.0	−1.50953
$552$	0	0
$553$	−3437.70	−0.264351
$554$	0	0
$555$	−9944.76	−0.760597
$556$	0	0
$557$	−19783.7	−1.50496	−0.752481	−	0.658614i	$-0.771143\pi$
$557$	−19783.7	−1.50496	−0.752481	+	0.658614i	$0.771143\pi$
$558$	0	0
$559$	−59.1632	−0.00447646
$560$	0	0
$561$	5330.30	0.401151
$562$	0	0
$563$	12145.1	0.909158	0.454579	−	0.890706i	$-0.349790\pi$
$563$	12145.1	0.909158	0.454579	+	0.890706i	$0.349790\pi$
$564$	0	0
$565$	6887.18	0.512824
$566$	0	0
$567$	−4627.82	−0.342769
$568$	0	0
$569$	−14348.4	−1.05715	−0.528575	−	0.848887i	$-0.677273\pi$
$569$	−14348.4	−1.05715	−0.528575	+	0.848887i	$0.677273\pi$
$570$	0	0
$571$	23201.2	1.70042	0.850209	−	0.526445i	$-0.176475\pi$
$571$	23201.2	1.70042	0.850209	+	0.526445i	$0.176475\pi$
$572$	0	0
$573$	−13672.1	−0.996787
$574$	0	0
$575$	4381.68	0.317789
$576$	0	0
$577$	−936.733	−0.0675853	−0.0337926	−	0.999429i	$-0.510759\pi$
$577$	−936.733	−0.0675853	−0.0337926	+	0.999429i	$0.510759\pi$
$578$	0	0
$579$	5658.64	0.406157
$580$	0	0
$581$	2008.91	0.143448
$582$	0	0
$583$	1101.58	0.0782551
$584$	0	0
$585$	−5.21852	−0.000368819 0
$586$	0	0
$587$	12738.1	0.895672	0.447836	−	0.894116i	$-0.352195\pi$
$587$	12738.1	0.895672	0.447836	+	0.894116i	$0.352195\pi$
$588$	0	0
$589$	−24135.3	−1.68842
$590$	0	0
$591$	−11719.5	−0.815697
$592$	0	0
$593$	−11634.9	−0.805714	−0.402857	−	0.915263i	$-0.631983\pi$
$593$	−11634.9	−0.805714	−0.402857	+	0.915263i	$0.631983\pi$
$594$	0	0
$595$	1642.93	0.113200
$596$	0	0
$597$	−11576.5	−0.793624
$598$	0	0
$599$	517.243	0.0352821	0.0176410	−	0.999844i	$-0.494384\pi$
$599$	517.243	0.0352821	0.0176410	+	0.999844i	$0.494384\pi$
$600$	0	0
$601$	9170.49	0.622416	0.311208	−	0.950342i	$-0.399266\pi$
$601$	9170.49	0.622416	0.311208	+	0.950342i	$0.399266\pi$
$602$	0	0
$603$	1165.23	0.0786932
$604$	0	0
$605$	4043.16	0.271699
$606$	0	0
$607$	−4824.78	−0.322622	−0.161311	−	0.986904i	$-0.551572\pi$
$607$	−4824.78	−0.322622	−0.161311	+	0.986904i	$0.551572\pi$
$608$	0	0
$609$	−6002.26	−0.399383
$610$	0	0
$611$	−232.647	−0.0154041
$612$	0	0
$613$	13329.2	0.878243	0.439121	−	0.898428i	$-0.355290\pi$
$613$	13329.2	0.878243	0.439121	+	0.898428i	$0.355290\pi$
$614$	0	0
$615$	9334.40	0.612031
$616$	0	0
$617$	−21964.7	−1.43317	−0.716584	−	0.697501i	$-0.754295\pi$
$617$	−21964.7	−1.43317	−0.716584	+	0.697501i	$0.754295\pi$
$618$	0	0
$619$	8756.27	0.568569	0.284284	−	0.958740i	$-0.408244\pi$
$619$	8756.27	0.568569	0.284284	+	0.958740i	$0.408244\pi$
$620$	0	0
$621$	−25527.6	−1.64958
$622$	0	0
$623$	−7299.18	−0.469399
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−12845.8	−0.818203
$628$	0	0
$629$	−18791.7	−1.19121
$630$	0	0
$631$	−1613.61	−0.101802	−0.0509008	−	0.998704i	$-0.516209\pi$
$631$	−1613.61	−0.101802	−0.0509008	+	0.998704i	$0.516209\pi$
$632$	0	0
$633$	−15570.2	−0.977663
$634$	0	0
$635$	7659.96	0.478702
$636$	0	0
$637$	−22.0856	−0.00137373
$638$	0	0
$639$	−2662.81	−0.164850
$640$	0	0
$641$	26646.3	1.64191	0.820956	−	0.570991i	$-0.193441\pi$
$641$	26646.3	1.64191	0.820956	+	0.570991i	$0.193441\pi$
$642$	0	0
$643$	14540.5	0.891790	0.445895	−	0.895085i	$-0.352885\pi$
$643$	14540.5	0.891790	0.445895	+	0.895085i	$0.352885\pi$
$644$	0	0
$645$	−3260.77	−0.199058
$646$	0	0
$647$	−12367.3	−0.751483	−0.375742	−	0.926724i	$-0.622612\pi$
$647$	−12367.3	−0.751483	−0.375742	+	0.926724i	$0.622612\pi$
$648$	0	0
$649$	10342.2	0.625524
$650$	0	0
$651$	−7419.93	−0.446713
$652$	0	0
$653$	4725.94	0.283217	0.141608	−	0.989923i	$-0.454773\pi$
$653$	4725.94	0.283217	0.141608	+	0.989923i	$0.454773\pi$
$654$	0	0
$655$	3423.58	0.204229
$656$	0	0
$657$	2492.52	0.148010
$658$	0	0
$659$	1139.07	0.0673323	0.0336662	−	0.999433i	$-0.489282\pi$
$659$	1139.07	0.0673323	0.0336662	+	0.999433i	$0.489282\pi$
$660$	0	0
$661$	−13905.2	−0.818228	−0.409114	−	0.912483i	$-0.634162\pi$
$661$	−13905.2	−0.818228	−0.409114	+	0.912483i	$0.634162\pi$
$662$	0	0
$663$	105.118	0.00615753
$664$	0	0
$665$	−3959.41	−0.230886
$666$	0	0
$667$	−30248.7	−1.75597
$668$	0	0
$669$	20882.6	1.20683
$670$	0	0
$671$	13831.3	0.795757
$672$	0	0
$673$	4861.17	0.278431	0.139216	−	0.990262i	$-0.455542\pi$
$673$	4861.17	0.278431	0.139216	+	0.990262i	$0.455542\pi$
$674$	0	0
$675$	−3641.25	−0.207632
$676$	0	0
$677$	13983.8	0.793859	0.396930	−	0.917849i	$-0.370076\pi$
$677$	13983.8	0.793859	0.396930	+	0.917849i	$0.370076\pi$
$678$	0	0
$679$	−4723.93	−0.266992
$680$	0	0
$681$	−5545.42	−0.312043
$682$	0	0
$683$	−11906.5	−0.667039	−0.333520	−	0.942743i	$-0.608236\pi$
$683$	−11906.5	−0.667039	−0.333520	+	0.942743i	$0.608236\pi$
$684$	0	0
$685$	239.261	0.0133455
$686$	0	0
$687$	11068.1	0.614663
$688$	0	0
$689$	21.7240	0.00120119
$690$	0	0
$691$	14847.6	0.817409	0.408705	−	0.912667i	$-0.365981\pi$
$691$	14847.6	0.817409	0.408705	+	0.912667i	$0.365981\pi$
$692$	0	0
$693$	370.468	0.0203072
$694$	0	0
$695$	−2737.13	−0.149389
$696$	0	0
$697$	17638.3	0.958535
$698$	0	0
$699$	6759.54	0.365764
$700$	0	0
$701$	−10193.6	−0.549227	−0.274613	−	0.961555i	$-0.588550\pi$
$701$	−10193.6	−0.549227	−0.274613	+	0.961555i	$0.588550\pi$
$702$	0	0
$703$	45287.2	2.42964
$704$	0	0
$705$	−12822.3	−0.684985
$706$	0	0
$707$	−1531.57	−0.0814718
$708$	0	0
$709$	−15601.0	−0.826388	−0.413194	−	0.910643i	$-0.635587\pi$
$709$	−15601.0	−0.826388	−0.413194	+	0.910643i	$0.635587\pi$
$710$	0	0
$711$	1137.19	0.0599832
$712$	0	0
$713$	−37393.1	−1.96407
$714$	0	0
$715$	−51.5077	−0.00269409
$716$	0	0
$717$	−26639.4	−1.38754
$718$	0	0
$719$	2514.26	0.130412	0.0652058	−	0.997872i	$-0.479230\pi$
$719$	2514.26	0.130412	0.0652058	+	0.997872i	$0.479230\pi$
$720$	0	0
$721$	−8538.44	−0.441038
$722$	0	0
$723$	−33745.2	−1.73582
$724$	0	0
$725$	−4314.65	−0.221024
$726$	0	0
$727$	14012.5	0.714850	0.357425	−	0.933942i	$-0.383655\pi$
$727$	14012.5	0.714850	0.357425	+	0.933942i	$0.383655\pi$
$728$	0	0
$729$	21069.1	1.07042
$730$	0	0
$731$	−6161.56	−0.311756
$732$	0	0
$733$	31576.6	1.59114	0.795572	−	0.605859i	$-0.207170\pi$
$733$	31576.6	1.59114	0.795572	+	0.605859i	$0.207170\pi$
$734$	0	0
$735$	−1217.24	−0.0610867
$736$	0	0
$737$	11501.1	0.574827
$738$	0	0
$739$	4425.41	0.220286	0.110143	−	0.993916i	$-0.464869\pi$
$739$	4425.41	0.220286	0.110143	+	0.993916i	$0.464869\pi$
$740$	0	0
$741$	−253.330	−0.0125591
$742$	0	0
$743$	10718.2	0.529225	0.264612	−	0.964355i	$-0.414756\pi$
$743$	10718.2	0.529225	0.264612	+	0.964355i	$0.414756\pi$
$744$	0	0
$745$	−12299.5	−0.604857
$746$	0	0
$747$	−664.547	−0.0325495
$748$	0	0
$749$	11074.4	0.540251
$750$	0	0
$751$	17956.6	0.872500	0.436250	−	0.899826i	$-0.356306\pi$
$751$	17956.6	0.872500	0.436250	+	0.899826i	$0.356306\pi$
$752$	0	0
$753$	−16084.7	−0.778431
$754$	0	0
$755$	7627.50	0.367673
$756$	0	0
$757$	21607.2	1.03742	0.518711	−	0.854950i	$-0.326412\pi$
$757$	21607.2	1.03742	0.518711	+	0.854950i	$0.326412\pi$
$758$	0	0
$759$	−19902.2	−0.951783
$760$	0	0
$761$	8681.22	0.413527	0.206764	−	0.978391i	$-0.433707\pi$
$761$	8681.22	0.413527	0.206764	+	0.978391i	$0.433707\pi$
$762$	0	0
$763$	−8764.77	−0.415866
$764$	0	0
$765$	−543.483	−0.0256858
$766$	0	0
$767$	203.956	0.00960158
$768$	0	0
$769$	−25645.8	−1.20262	−0.601308	−	0.799017i	$-0.705354\pi$
$769$	−25645.8	−1.20262	−0.601308	+	0.799017i	$0.705354\pi$
$770$	0	0
$771$	16750.8	0.782447
$772$	0	0
$773$	−14965.6	−0.696347	−0.348173	−	0.937430i	$-0.613198\pi$
$773$	−14965.6	−0.696347	−0.348173	+	0.937430i	$0.613198\pi$
$774$	0	0
$775$	−5333.72	−0.247217
$776$	0	0
$777$	13922.7	0.642822
$778$	0	0
$779$	−42507.7	−1.95506
$780$	0	0
$781$	−26282.4	−1.20417
$782$	0	0
$783$	25137.1	1.14729
$784$	0	0
$785$	−15491.1	−0.704332
$786$	0	0
$787$	27057.1	1.22551	0.612757	−	0.790271i	$-0.290060\pi$
$787$	27057.1	1.22551	0.612757	+	0.790271i	$0.290060\pi$
$788$	0	0
$789$	−12874.1	−0.580898
$790$	0	0
$791$	−9642.05	−0.433416
$792$	0	0
$793$	272.765	0.0122146
$794$	0	0
$795$	1197.31	0.0534143
$796$	0	0
$797$	25376.5	1.12783	0.563916	−	0.825832i	$-0.309294\pi$
$797$	25376.5	1.12783	0.563916	+	0.825832i	$0.309294\pi$
$798$	0	0
$799$	−24229.0	−1.07279
$800$	0	0
$801$	2414.57	0.106510
$802$	0	0
$803$	24601.6	1.08116
$804$	0	0
$805$	−6134.35	−0.268581
$806$	0	0
$807$	−12793.4	−0.558053
$808$	0	0
$809$	−31482.0	−1.36817	−0.684083	−	0.729404i	$-0.739798\pi$
$809$	−31482.0	−1.36817	−0.684083	+	0.729404i	$0.739798\pi$
$810$	0	0
$811$	7203.51	0.311898	0.155949	−	0.987765i	$-0.450156\pi$
$811$	7203.51	0.311898	0.155949	+	0.987765i	$0.450156\pi$
$812$	0	0
$813$	2276.69	0.0982126
$814$	0	0
$815$	6413.44	0.275648
$816$	0	0
$817$	14849.1	0.635869
$818$	0	0
$819$	7.30593	0.000311709 0
$820$	0	0
$821$	26185.2	1.11312	0.556558	−	0.830809i	$-0.312122\pi$
$821$	26185.2	1.11312	0.556558	+	0.830809i	$0.312122\pi$
$822$	0	0
$823$	−12956.4	−0.548761	−0.274381	−	0.961621i	$-0.588473\pi$
$823$	−12956.4	−0.548761	−0.274381	+	0.961621i	$0.588473\pi$
$824$	0	0
$825$	−2838.83	−0.119801
$826$	0	0
$827$	−22038.3	−0.926660	−0.463330	−	0.886186i	$-0.653346\pi$
$827$	−22038.3	−0.926660	−0.463330	+	0.886186i	$0.653346\pi$
$828$	0	0
$829$	−5453.92	−0.228495	−0.114248	−	0.993452i	$-0.536446\pi$
$829$	−5453.92	−0.228495	−0.114248	+	0.993452i	$0.536446\pi$
$830$	0	0
$831$	39248.5	1.63841
$832$	0	0
$833$	−2300.11	−0.0956710
$834$	0	0
$835$	10910.4	0.452178
$836$	0	0
$837$	31074.2	1.28325
$838$	0	0
$839$	41299.4	1.69942	0.849711	−	0.527249i	$-0.176777\pi$
$839$	41299.4	1.69942	0.849711	+	0.527249i	$0.176777\pi$
$840$	0	0
$841$	5396.94	0.221286
$842$	0	0
$843$	27045.4	1.10497
$844$	0	0
$845$	10984.0	0.447172
$846$	0	0
$847$	−5660.42	−0.229627
$848$	0	0
$849$	36645.5	1.48135
$850$	0	0
$851$	70163.9	2.82631
$852$	0	0
$853$	20935.0	0.840331	0.420165	−	0.907448i	$-0.361972\pi$
$853$	20935.0	0.840331	0.420165	+	0.907448i	$0.361972\pi$
$854$	0	0
$855$	1309.77	0.0523898
$856$	0	0
$857$	−7754.92	−0.309105	−0.154552	−	0.987985i	$-0.549394\pi$
$857$	−7754.92	−0.309105	−0.154552	+	0.987985i	$0.549394\pi$
$858$	0	0
$859$	−27629.6	−1.09745	−0.548725	−	0.836003i	$-0.684887\pi$
$859$	−27629.6	−1.09745	−0.548725	+	0.836003i	$0.684887\pi$
$860$	0	0
$861$	−13068.2	−0.517261
$862$	0	0
$863$	44014.7	1.73613	0.868064	−	0.496453i	$-0.165364\pi$
$863$	44014.7	1.73613	0.868064	+	0.496453i	$0.165364\pi$
$864$	0	0
$865$	−16620.8	−0.653323
$866$	0	0
$867$	−13461.9	−0.527326
$868$	0	0
$869$	11224.3	0.438156
$870$	0	0
$871$	226.810	0.00882339
$872$	0	0
$873$	1562.68	0.0605825
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−16804.5	−0.647033	−0.323516	−	0.946223i	$-0.604865\pi$
$877$	−16804.5	−0.647033	−0.323516	+	0.946223i	$0.604865\pi$
$878$	0	0
$879$	39407.3	1.51215
$880$	0	0
$881$	−6832.51	−0.261286	−0.130643	−	0.991429i	$-0.541704\pi$
$881$	−6832.51	−0.261286	−0.130643	+	0.991429i	$0.541704\pi$
$882$	0	0
$883$	−15038.8	−0.573154	−0.286577	−	0.958057i	$-0.592517\pi$
$883$	−15038.8	−0.573154	−0.286577	+	0.958057i	$0.592517\pi$
$884$	0	0
$885$	11241.0	0.426962
$886$	0	0
$887$	24951.9	0.944535	0.472268	−	0.881455i	$-0.343436\pi$
$887$	24951.9	0.944535	0.472268	+	0.881455i	$0.343436\pi$
$888$	0	0
$889$	−10723.9	−0.404577
$890$	0	0
$891$	15110.1	0.568133
$892$	0	0
$893$	58391.0	2.18811
$894$	0	0
$895$	−12508.0	−0.467146
$896$	0	0
$897$	−392.487	−0.0146095
$898$	0	0
$899$	36821.0	1.36602
$900$	0	0
$901$	2262.45	0.0836550
$902$	0	0
$903$	4565.08	0.168235
$904$	0	0
$905$	23486.5	0.862672
$906$	0	0
$907$	−1411.04	−0.0516568	−0.0258284	−	0.999666i	$-0.508222\pi$
$907$	−1411.04	−0.0516568	−0.0258284	+	0.999666i	$0.508222\pi$
$908$	0	0
$909$	506.643	0.0184866
$910$	0	0
$911$	8018.14	0.291606	0.145803	−	0.989314i	$-0.453423\pi$
$911$	8018.14	0.291606	0.145803	+	0.989314i	$0.453423\pi$
$912$	0	0
$913$	−6559.20	−0.237763
$914$	0	0
$915$	15033.4	0.543157
$916$	0	0
$917$	−4793.01	−0.172605
$918$	0	0
$919$	26941.6	0.967053	0.483526	−	0.875330i	$-0.339356\pi$
$919$	26941.6	0.967053	0.483526	+	0.875330i	$0.339356\pi$
$920$	0	0
$921$	−1740.54	−0.0622724
$922$	0	0
$923$	−518.309	−0.0184836
$924$	0	0
$925$	10008.1	0.355746
$926$	0	0
$927$	2824.52	0.100075
$928$	0	0
$929$	−46912.7	−1.65679	−0.828395	−	0.560145i	$-0.810745\pi$
$929$	−46912.7	−1.65679	−0.828395	+	0.560145i	$0.810745\pi$
$930$	0	0
$931$	5543.17	0.195134
$932$	0	0
$933$	40949.4	1.43689
$934$	0	0
$935$	−5364.27	−0.187626
$936$	0	0
$937$	3893.83	0.135759	0.0678794	−	0.997694i	$-0.478377\pi$
$937$	3893.83	0.135759	0.0678794	+	0.997694i	$0.478377\pi$
$938$	0	0
$939$	−42185.3	−1.46610
$940$	0	0
$941$	9391.41	0.325347	0.162673	−	0.986680i	$-0.447988\pi$
$941$	9391.41	0.325347	0.162673	+	0.986680i	$0.447988\pi$
$942$	0	0
$943$	−65857.5	−2.27425
$944$	0	0
$945$	5097.75	0.175481
$946$	0	0
$947$	−52526.7	−1.80242	−0.901208	−	0.433386i	$-0.857319\pi$
$947$	−52526.7	−1.80242	−0.901208	+	0.433386i	$0.857319\pi$
$948$	0	0
$949$	485.164	0.0165954
$950$	0	0
$951$	13860.9	0.472628
$952$	0	0
$953$	−44616.6	−1.51655	−0.758276	−	0.651934i	$-0.773958\pi$
$953$	−44616.6	−1.51655	−0.758276	+	0.651934i	$0.773958\pi$
$954$	0	0
$955$	13759.2	0.466217
$956$	0	0
$957$	19597.7	0.661969
$958$	0	0
$959$	−334.965	−0.0112790
$960$	0	0
$961$	15726.7	0.527902
$962$	0	0
$963$	−3663.40	−0.122587
$964$	0	0
$965$	−5694.70	−0.189968
$966$	0	0
$967$	20663.4	0.687166	0.343583	−	0.939122i	$-0.388359\pi$
$967$	20663.4	0.687166	0.343583	+	0.939122i	$0.388359\pi$
$968$	0	0
$969$	−26383.1	−0.874661
$970$	0	0
$971$	34537.9	1.14148	0.570738	−	0.821133i	$-0.306657\pi$
$971$	34537.9	1.14148	0.570738	+	0.821133i	$0.306657\pi$
$972$	0	0
$973$	3831.98	0.126257
$974$	0	0
$975$	−55.9841	−0.00183890
$976$	0	0
$977$	29955.6	0.980925	0.490462	−	0.871462i	$-0.336828\pi$
$977$	29955.6	0.980925	0.490462	+	0.871462i	$0.336828\pi$
$978$	0	0
$979$	23832.2	0.778020
$980$	0	0
$981$	2899.39	0.0943632
$982$	0	0
$983$	−59845.2	−1.94177	−0.970887	−	0.239536i	$-0.923005\pi$
$983$	−59845.2	−1.94177	−0.970887	+	0.239536i	$0.923005\pi$
$984$	0	0
$985$	11794.2	0.381517
$986$	0	0
$987$	17951.2	0.578918
$988$	0	0
$989$	23005.9	0.739682
$990$	0	0
$991$	−29394.8	−0.942236	−0.471118	−	0.882070i	$-0.656149\pi$
$991$	−29394.8	−0.942236	−0.471118	+	0.882070i	$0.656149\pi$
$992$	0	0
$993$	−34390.6	−1.09904
$994$	0	0
$995$	11650.2	0.371194
$996$	0	0
$997$	−13546.2	−0.430303	−0.215151	−	0.976581i	$-0.569024\pi$
$997$	−13546.2	−0.430303	−0.215151	+	0.976581i	$0.569024\pi$
$998$	0	0
$999$	−58307.3	−1.84661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.ci.1.3		4
4.3	odd	2		2240.4.a.cd.1.2		4
8.3	odd	2		1120.4.a.m.1.3	yes	4
8.5	even	2		1120.4.a.h.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.4.a.h.1.2	✓	4	8.5	even	2
1120.4.a.m.1.3	yes	4	8.3	odd	2
2240.4.a.cd.1.2		4	4.3	odd	2
2240.4.a.ci.1.3		4	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q+4.96834 q^{3} -5.00000 q^{5} +7.00000 q^{7} -2.31560 q^{9} -22.8554 q^{11} -0.450727 q^{13} -24.8417 q^{15} -46.9410 q^{17} +113.126 q^{19} +34.7784 q^{21} +175.267 q^{23} +25.0000 q^{25} -145.650 q^{27} -172.586 q^{29} -213.349 q^{31} -113.553 q^{33} -35.0000 q^{35} +400.325 q^{37} -2.23936 q^{39} -375.755 q^{41} +131.262 q^{43} +11.5780 q^{45} +516.159 q^{47} +49.0000 q^{49} -233.219 q^{51} -48.1978 q^{53} +114.277 q^{55} +562.048 q^{57} -452.504 q^{59} -605.168 q^{61} -16.2092 q^{63} +2.25363 q^{65} -503.210 q^{67} +870.787 q^{69} +1149.94 q^{71} -1076.40 q^{73} +124.208 q^{75} -159.988 q^{77} -491.100 q^{79} -661.117 q^{81} +286.987 q^{83} +234.705 q^{85} -857.466 q^{87} -1042.74 q^{89} -3.15509 q^{91} -1059.99 q^{93} -565.630 q^{95} -674.847 q^{97} +52.9240 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 5 q^{3} - 20 q^{5} + 28 q^{7} + 23 q^{9} + 13 q^{11} + 39 q^{13} - 25 q^{15} - 233 q^{17} + 8 q^{19} + 35 q^{21} - 32 q^{23} + 100 q^{25} + 191 q^{27} + 181 q^{29} - 90 q^{31} - 645 q^{33} - 140 q^{35}+ \cdots - 138 q^{99}+O(q^{100}) \) 4 * q + 5 * q^3 - 20 * q^5 + 28 * q^7 + 23 * q^9 + 13 * q^11 + 39 * q^13 - 25 * q^15 - 233 * q^17 + 8 * q^19 + 35 * q^21 - 32 * q^23 + 100 * q^25 + 191 * q^27 + 181 * q^29 - 90 * q^31 - 645 * q^33 - 140 * q^35 + 286 * q^37 + 451 * q^39 - 632 * q^41 - 356 * q^43 - 115 * q^45 - 201 * q^47 + 196 * q^49 - 341 * q^51 + 506 * q^53 - 65 * q^55 - 552 * q^57 + 1116 * q^59 - 480 * q^61 + 161 * q^63 - 195 * q^65 - 1504 * q^67 - 856 * q^69 + 1204 * q^71 - 312 * q^73 + 125 * q^75 + 91 * q^77 - 819 * q^79 - 1420 * q^81 - 2280 * q^83 + 1165 * q^85 + 589 * q^87 - 1404 * q^89 + 273 * q^91 + 2922 * q^93 - 40 * q^95 - 2833 * q^97 - 138 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more