LMFDB - Embedded newform 2240.4.a.cn.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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 99x^{3} - 98x^{2} + 924x + 168 $ x^5 - 99x^3 - 98x^2 + 924*x + 168
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.179033$ 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)

$f(q)$	$=$	$q-1.17903 q^{3} +5.00000 q^{5} -7.00000 q^{7} -25.6099 q^{9} -52.0666 q^{11} +63.8274 q^{13} -5.89517 q^{15} +51.2353 q^{17} -56.1226 q^{19} +8.25323 q^{21} -97.1253 q^{23} +25.0000 q^{25} +62.0288 q^{27} +300.236 q^{29} +181.133 q^{31} +61.3882 q^{33} -35.0000 q^{35} -95.8142 q^{37} -75.2546 q^{39} +247.130 q^{41} +414.927 q^{43} -128.049 q^{45} -24.1145 q^{47} +49.0000 q^{49} -60.4081 q^{51} -399.051 q^{53} -260.333 q^{55} +66.1704 q^{57} -353.455 q^{59} -586.387 q^{61} +179.269 q^{63} +319.137 q^{65} -413.604 q^{67} +114.514 q^{69} -36.8541 q^{71} -472.474 q^{73} -29.4758 q^{75} +364.466 q^{77} +1101.84 q^{79} +618.333 q^{81} -267.578 q^{83} +256.176 q^{85} -353.988 q^{87} -336.727 q^{89} -446.792 q^{91} -213.562 q^{93} -280.613 q^{95} +375.268 q^{97} +1333.42 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + 25 q^{5} - 35 q^{7} + 68 q^{9} - 35 q^{11} + 39 q^{13} - 25 q^{15} - 29 q^{17} - 84 q^{19} + 35 q^{21} - 128 q^{23} + 125 q^{25} - 35 q^{27} - 73 q^{29} - 318 q^{31} + 205 q^{33} - 175 q^{35}+ \cdots - 3734 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + 25 * q^5 - 35 * q^7 + 68 * q^9 - 35 * q^11 + 39 * q^13 - 25 * q^15 - 29 * q^17 - 84 * q^19 + 35 * q^21 - 128 * q^23 + 125 * q^25 - 35 * q^27 - 73 * q^29 - 318 * q^31 + 205 * q^33 - 175 * q^35 + 236 * q^37 - q^39 + 262 * q^41 - 372 * q^43 + 340 * q^45 - 603 * q^47 + 245 * q^49 - 777 * q^51 - 20 * q^53 - 175 * q^55 - 164 * q^57 - 960 * q^59 - 246 * q^61 - 476 * q^63 + 195 * q^65 - 1360 * q^67 - 1576 * q^69 + 696 * q^71 + 1122 * q^73 - 125 * q^75 + 245 * q^77 + 93 * q^79 + 3125 * q^81 + 1412 * q^83 - 145 * q^85 + 231 * q^87 + 2854 * q^89 - 273 * q^91 - 3262 * q^93 - 420 * q^95 + 1739 * q^97 - 3734 * 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
$3$	−1.17903	−0.226905	−0.113453	−	0.993543i	$-0.536191\pi$
$3$	−1.17903	−0.226905	−0.113453	+	0.993543i	$0.536191\pi$
$4$	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
$9$	−25.6099	−0.948514
$10$	0	0
$11$	−52.0666	−1.42715	−0.713576	−	0.700578i	$-0.752926\pi$
$11$	−52.0666	−1.42715	−0.713576	+	0.700578i	$0.752926\pi$
$12$	0	0
$13$	63.8274	1.36173	0.680867	−	0.732407i	$-0.261603\pi$
$13$	63.8274	1.36173	0.680867	+	0.732407i	$0.261603\pi$
$14$	0	0
$15$	−5.89517	−0.101475
$16$	0	0
$17$	51.2353	0.730963	0.365482	−	0.930819i	$-0.380904\pi$
$17$	51.2353	0.730963	0.365482	+	0.930819i	$0.380904\pi$
$18$	0	0
$19$	−56.1226	−0.677653	−0.338827	−	0.940849i	$-0.610030\pi$
$19$	−56.1226	−0.677653	−0.338827	+	0.940849i	$0.610030\pi$
$20$	0	0
$21$	8.25323	0.0857621
$22$	0	0
$23$	−97.1253	−0.880523	−0.440262	−	0.897870i	$-0.645114\pi$
$23$	−97.1253	−0.880523	−0.440262	+	0.897870i	$0.645114\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	62.0288	0.442128
$28$	0	0
$29$	300.236	1.92250	0.961249	−	0.275681i	$-0.0889034\pi$
$29$	300.236	1.92250	0.961249	+	0.275681i	$0.0889034\pi$
$30$	0	0
$31$	181.133	1.04943	0.524717	−	0.851277i	$-0.324171\pi$
$31$	181.133	1.04943	0.524717	+	0.851277i	$0.324171\pi$
$32$	0	0
$33$	61.3882	0.323828
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−95.8142	−0.425723	−0.212862	−	0.977082i	$-0.568278\pi$
$37$	−95.8142	−0.425723	−0.212862	+	0.977082i	$0.568278\pi$
$38$	0	0
$39$	−75.2546	−0.308984
$40$	0	0
$41$	247.130	0.941346	0.470673	−	0.882308i	$-0.344011\pi$
$41$	247.130	0.941346	0.470673	+	0.882308i	$0.344011\pi$
$42$	0	0
$43$	414.927	1.47153	0.735765	−	0.677237i	$-0.236823\pi$
$43$	414.927	1.47153	0.735765	+	0.677237i	$0.236823\pi$
$44$	0	0
$45$	−128.049	−0.424188
$46$	0	0
$47$	−24.1145	−0.0748396	−0.0374198	−	0.999300i	$-0.511914\pi$
$47$	−24.1145	−0.0748396	−0.0374198	+	0.999300i	$0.511914\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−60.4081	−0.165859
$52$	0	0
$53$	−399.051	−1.03422	−0.517112	−	0.855918i	$-0.672993\pi$
$53$	−399.051	−1.03422	−0.517112	+	0.855918i	$0.672993\pi$
$54$	0	0
$55$	−260.333	−0.638242
$56$	0	0
$57$	66.1704	0.153763
$58$	0	0
$59$	−353.455	−0.779930	−0.389965	−	0.920830i	$-0.627513\pi$
$59$	−353.455	−0.779930	−0.389965	+	0.920830i	$0.627513\pi$
$60$	0	0
$61$	−586.387	−1.23081	−0.615403	−	0.788213i	$-0.711007\pi$
$61$	−586.387	−1.23081	−0.615403	+	0.788213i	$0.711007\pi$
$62$	0	0
$63$	179.269	0.358505
$64$	0	0
$65$	319.137	0.608986
$66$	0	0
$67$	−413.604	−0.754175	−0.377087	−	0.926178i	$-0.623074\pi$
$67$	−413.604	−0.754175	−0.377087	+	0.926178i	$0.623074\pi$
$68$	0	0
$69$	114.514	0.199795
$70$	0	0
$71$	−36.8541	−0.0616024	−0.0308012	−	0.999526i	$-0.509806\pi$
$71$	−36.8541	−0.0616024	−0.0308012	+	0.999526i	$0.509806\pi$
$72$	0	0
$73$	−472.474	−0.757520	−0.378760	−	0.925495i	$-0.623649\pi$
$73$	−472.474	−0.757520	−0.378760	+	0.925495i	$0.623649\pi$
$74$	0	0
$75$	−29.4758	−0.0453810
$76$	0	0
$77$	364.466	0.539413
$78$	0	0
$79$	1101.84	1.56919	0.784597	−	0.620005i	$-0.212870\pi$
$79$	1101.84	1.56919	0.784597	+	0.620005i	$0.212870\pi$
$80$	0	0
$81$	618.333	0.848193
$82$	0	0
$83$	−267.578	−0.353861	−0.176931	−	0.984223i	$-0.556617\pi$
$83$	−267.578	−0.353861	−0.176931	+	0.984223i	$0.556617\pi$
$84$	0	0
$85$	256.176	0.326897
$86$	0	0
$87$	−353.988	−0.436225
$88$	0	0
$89$	−336.727	−0.401044	−0.200522	−	0.979689i	$-0.564264\pi$
$89$	−336.727	−0.401044	−0.200522	+	0.979689i	$0.564264\pi$
$90$	0	0
$91$	−446.792	−0.514687
$92$	0	0
$93$	−213.562	−0.238122
$94$	0	0
$95$	−280.613	−0.303056
$96$	0	0
$97$	375.268	0.392812	0.196406	−	0.980523i	$-0.437073\pi$
$97$	375.268	0.392812	0.196406	+	0.980523i	$0.437073\pi$
$98$	0	0
$99$	1333.42	1.35367
$100$	0	0
$101$	138.134	0.136088	0.0680438	−	0.997682i	$-0.478324\pi$
$101$	138.134	0.136088	0.0680438	+	0.997682i	$0.478324\pi$
$102$	0	0
$103$	−1341.72	−1.28353	−0.641766	−	0.766900i	$-0.721798\pi$
$103$	−1341.72	−1.28353	−0.641766	+	0.766900i	$0.721798\pi$
$104$	0	0
$105$	41.2662	0.0383540
$106$	0	0
$107$	−1696.25	−1.53254	−0.766272	−	0.642516i	$-0.777890\pi$
$107$	−1696.25	−1.53254	−0.766272	+	0.642516i	$0.777890\pi$
$108$	0	0
$109$	−2204.86	−1.93750	−0.968749	−	0.248042i	$-0.920213\pi$
$109$	−2204.86	−1.93750	−0.968749	+	0.248042i	$0.920213\pi$
$110$	0	0
$111$	112.968	0.0965987
$112$	0	0
$113$	1100.76	0.916381	0.458190	−	0.888854i	$-0.348498\pi$
$113$	1100.76	0.916381	0.458190	+	0.888854i	$0.348498\pi$
$114$	0	0
$115$	−485.627	−0.393782
$116$	0	0
$117$	−1634.61	−1.29162
$118$	0	0
$119$	−358.647	−0.276278
$120$	0	0
$121$	1379.93	1.03676
$122$	0	0
$123$	−291.374	−0.213596
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−127.572	−0.0891354	−0.0445677	−	0.999006i	$-0.514191\pi$
$127$	−127.572	−0.0891354	−0.0445677	+	0.999006i	$0.514191\pi$
$128$	0	0
$129$	−489.213	−0.333898
$130$	0	0
$131$	160.270	0.106892	0.0534459	−	0.998571i	$-0.482980\pi$
$131$	160.270	0.106892	0.0534459	+	0.998571i	$0.482980\pi$
$132$	0	0
$133$	392.858	0.256129
$134$	0	0
$135$	310.144	0.197726
$136$	0	0
$137$	980.400	0.611396	0.305698	−	0.952129i	$-0.401110\pi$
$137$	980.400	0.611396	0.305698	+	0.952129i	$0.401110\pi$
$138$	0	0
$139$	−1263.68	−0.771105	−0.385552	−	0.922686i	$-0.625989\pi$
$139$	−1263.68	−0.771105	−0.385552	+	0.922686i	$0.625989\pi$
$140$	0	0
$141$	28.4318	0.0169815
$142$	0	0
$143$	−3323.27	−1.94340
$144$	0	0
$145$	1501.18	0.859767
$146$	0	0
$147$	−57.7726	−0.0324150
$148$	0	0
$149$	426.329	0.234405	0.117202	−	0.993108i	$-0.462607\pi$
$149$	426.329	0.234405	0.117202	+	0.993108i	$0.462607\pi$
$150$	0	0
$151$	−1511.76	−0.814739	−0.407369	−	0.913263i	$-0.633554\pi$
$151$	−1511.76	−0.814739	−0.407369	+	0.913263i	$0.633554\pi$
$152$	0	0
$153$	−1312.13	−0.693329
$154$	0	0
$155$	905.666	0.469321
$156$	0	0
$157$	−3871.11	−1.96782	−0.983910	−	0.178663i	$-0.942823\pi$
$157$	−3871.11	−1.96782	−0.983910	+	0.178663i	$0.942823\pi$
$158$	0	0
$159$	470.495	0.234671
$160$	0	0
$161$	679.877	0.332806
$162$	0	0
$163$	−1702.44	−0.818070	−0.409035	−	0.912519i	$-0.634135\pi$
$163$	−1702.44	−0.818070	−0.409035	+	0.912519i	$0.634135\pi$
$164$	0	0
$165$	306.941	0.144820
$166$	0	0
$167$	−2853.48	−1.32221	−0.661104	−	0.750295i	$-0.729912\pi$
$167$	−2853.48	−1.32221	−0.661104	+	0.750295i	$0.729912\pi$
$168$	0	0
$169$	1876.93	0.854317
$170$	0	0
$171$	1437.29	0.642763
$172$	0	0
$173$	−25.2070	−0.0110778	−0.00553889	−	0.999985i	$-0.501763\pi$
$173$	−25.2070	−0.0110778	−0.00553889	+	0.999985i	$0.501763\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	416.735	0.176970
$178$	0	0
$179$	3107.71	1.29766	0.648830	−	0.760933i	$-0.275259\pi$
$179$	3107.71	1.29766	0.648830	+	0.760933i	$0.275259\pi$
$180$	0	0
$181$	1165.44	0.478598	0.239299	−	0.970946i	$-0.423082\pi$
$181$	1165.44	0.478598	0.239299	+	0.970946i	$0.423082\pi$
$182$	0	0
$183$	691.370	0.279276
$184$	0	0
$185$	−479.071	−0.190389
$186$	0	0
$187$	−2667.65	−1.04320
$188$	0	0
$189$	−434.202	−0.167109
$190$	0	0
$191$	−2472.20	−0.936555	−0.468277	−	0.883581i	$-0.655125\pi$
$191$	−2472.20	−0.936555	−0.468277	+	0.883581i	$0.655125\pi$
$192$	0	0
$193$	1179.58	0.439939	0.219970	−	0.975507i	$-0.429404\pi$
$193$	1179.58	0.439939	0.219970	+	0.975507i	$0.429404\pi$
$194$	0	0
$195$	−376.273	−0.138182
$196$	0	0
$197$	−2437.76	−0.881642	−0.440821	−	0.897595i	$-0.645313\pi$
$197$	−2437.76	−0.881642	−0.440821	+	0.897595i	$0.645313\pi$
$198$	0	0
$199$	3655.05	1.30201	0.651004	−	0.759075i	$-0.274348\pi$
$199$	3655.05	1.30201	0.651004	+	0.759075i	$0.274348\pi$
$200$	0	0
$201$	487.652	0.171126
$202$	0	0
$203$	−2101.65	−0.726636
$204$	0	0
$205$	1235.65	0.420983
$206$	0	0
$207$	2487.37	0.835189
$208$	0	0
$209$	2922.11	0.967113
$210$	0	0
$211$	−3001.88	−0.979420	−0.489710	−	0.871885i	$-0.662897\pi$
$211$	−3001.88	−0.979420	−0.489710	+	0.871885i	$0.662897\pi$
$212$	0	0
$213$	43.4522	0.0139779
$214$	0	0
$215$	2074.63	0.658088
$216$	0	0
$217$	−1267.93	−0.396649
$218$	0	0
$219$	557.063	0.171885
$220$	0	0
$221$	3270.21	0.995377
$222$	0	0
$223$	2898.33	0.870343	0.435171	−	0.900348i	$-0.356688\pi$
$223$	2898.33	0.870343	0.435171	+	0.900348i	$0.356688\pi$
$224$	0	0
$225$	−640.247	−0.189703
$226$	0	0
$227$	4222.64	1.23465	0.617326	−	0.786707i	$-0.288216\pi$
$227$	4222.64	1.23465	0.617326	+	0.786707i	$0.288216\pi$
$228$	0	0
$229$	−902.201	−0.260345	−0.130173	−	0.991491i	$-0.541553\pi$
$229$	−902.201	−0.260345	−0.130173	+	0.991491i	$0.541553\pi$
$230$	0	0
$231$	−429.718	−0.122395
$232$	0	0
$233$	4943.38	1.38992	0.694961	−	0.719048i	$-0.255422\pi$
$233$	4943.38	1.38992	0.694961	+	0.719048i	$0.255422\pi$
$234$	0	0
$235$	−120.572	−0.0334693
$236$	0	0
$237$	−1299.10	−0.356058
$238$	0	0
$239$	2973.64	0.804806	0.402403	−	0.915463i	$-0.368175\pi$
$239$	2973.64	0.804806	0.402403	+	0.915463i	$0.368175\pi$
$240$	0	0
$241$	1041.58	0.278399	0.139199	−	0.990264i	$-0.455547\pi$
$241$	1041.58	0.278399	0.139199	+	0.990264i	$0.455547\pi$
$242$	0	0
$243$	−2403.81	−0.634587
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−3582.16	−0.922783
$248$	0	0
$249$	315.483	0.0802929
$250$	0	0
$251$	−444.000	−0.111654	−0.0558268	−	0.998440i	$-0.517779\pi$
$251$	−444.000	−0.111654	−0.0558268	+	0.998440i	$0.517779\pi$
$252$	0	0
$253$	5056.98	1.25664
$254$	0	0
$255$	−302.040	−0.0741745
$256$	0	0
$257$	−923.976	−0.224265	−0.112132	−	0.993693i	$-0.535768\pi$
$257$	−923.976	−0.224265	−0.112132	+	0.993693i	$0.535768\pi$
$258$	0	0
$259$	670.699	0.160908
$260$	0	0
$261$	−7689.01	−1.82352
$262$	0	0
$263$	1911.63	0.448199	0.224099	−	0.974566i	$-0.428056\pi$
$263$	1911.63	0.448199	0.224099	+	0.974566i	$0.428056\pi$
$264$	0	0
$265$	−1995.26	−0.462519
$266$	0	0
$267$	397.012	0.0909990
$268$	0	0
$269$	−8315.36	−1.88475	−0.942373	−	0.334564i	$-0.891411\pi$
$269$	−8315.36	−1.88475	−0.942373	+	0.334564i	$0.891411\pi$
$270$	0	0
$271$	−6774.68	−1.51857	−0.759285	−	0.650758i	$-0.774451\pi$
$271$	−6774.68	−1.51857	−0.759285	+	0.650758i	$0.774451\pi$
$272$	0	0
$273$	526.782	0.116785
$274$	0	0
$275$	−1301.66	−0.285430
$276$	0	0
$277$	3301.83	0.716202	0.358101	−	0.933683i	$-0.383424\pi$
$277$	3301.83	0.716202	0.358101	+	0.933683i	$0.383424\pi$
$278$	0	0
$279$	−4638.80	−0.995403
$280$	0	0
$281$	2726.25	0.578771	0.289385	−	0.957213i	$-0.406549\pi$
$281$	2726.25	0.578771	0.289385	+	0.957213i	$0.406549\pi$
$282$	0	0
$283$	−799.847	−0.168007	−0.0840035	−	0.996465i	$-0.526771\pi$
$283$	−799.847	−0.168007	−0.0840035	+	0.996465i	$0.526771\pi$
$284$	0	0
$285$	330.852	0.0687649
$286$	0	0
$287$	−1729.91	−0.355795
$288$	0	0
$289$	−2287.95	−0.465693
$290$	0	0
$291$	−442.454	−0.0891309
$292$	0	0
$293$	−6817.14	−1.35925	−0.679627	−	0.733557i	$-0.737859\pi$
$293$	−6817.14	−1.35925	−0.679627	+	0.733557i	$0.737859\pi$
$294$	0	0
$295$	−1767.27	−0.348795
$296$	0	0
$297$	−3229.63	−0.630983
$298$	0	0
$299$	−6199.26	−1.19904
$300$	0	0
$301$	−2904.49	−0.556186
$302$	0	0
$303$	−162.865	−0.0308790
$304$	0	0
$305$	−2931.93	−0.550433
$306$	0	0
$307$	85.4466	0.0158850	0.00794250	−	0.999968i	$-0.497472\pi$
$307$	85.4466	0.0158850	0.00794250	+	0.999968i	$0.497472\pi$
$308$	0	0
$309$	1581.94	0.291240
$310$	0	0
$311$	−5561.82	−1.01409	−0.507045	−	0.861920i	$-0.669262\pi$
$311$	−5561.82	−1.01409	−0.507045	+	0.861920i	$0.669262\pi$
$312$	0	0
$313$	−8203.36	−1.48141	−0.740705	−	0.671830i	$-0.765508\pi$
$313$	−8203.36	−1.48141	−0.740705	+	0.671830i	$0.765508\pi$
$314$	0	0
$315$	896.346	0.160328
$316$	0	0
$317$	7442.96	1.31873	0.659366	−	0.751822i	$-0.270825\pi$
$317$	7442.96	1.31873	0.659366	+	0.751822i	$0.270825\pi$
$318$	0	0
$319$	−15632.3	−2.74370
$320$	0	0
$321$	1999.93	0.347742
$322$	0	0
$323$	−2875.46	−0.495340
$324$	0	0
$325$	1595.68	0.272347
$326$	0	0
$327$	2599.60	0.439628
$328$	0	0
$329$	168.801	0.0282867
$330$	0	0
$331$	−7622.62	−1.26579	−0.632896	−	0.774237i	$-0.718134\pi$
$331$	−7622.62	−1.26579	−0.632896	+	0.774237i	$0.718134\pi$
$332$	0	0
$333$	2453.79	0.403804
$334$	0	0
$335$	−2068.02	−0.337277
$336$	0	0
$337$	−2162.74	−0.349591	−0.174795	−	0.984605i	$-0.555926\pi$
$337$	−2162.74	−0.349591	−0.174795	+	0.984605i	$0.555926\pi$
$338$	0	0
$339$	−1297.84	−0.207931
$340$	0	0
$341$	−9430.98	−1.49770
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	572.570	0.0893511
$346$	0	0
$347$	−8061.11	−1.24710	−0.623549	−	0.781784i	$-0.714310\pi$
$347$	−8061.11	−1.24710	−0.623549	+	0.781784i	$0.714310\pi$
$348$	0	0
$349$	5166.88	0.792483	0.396242	−	0.918146i	$-0.370314\pi$
$349$	5166.88	0.792483	0.396242	+	0.918146i	$0.370314\pi$
$350$	0	0
$351$	3959.14	0.602060
$352$	0	0
$353$	729.077	0.109929	0.0549644	−	0.998488i	$-0.482495\pi$
$353$	729.077	0.109929	0.0549644	+	0.998488i	$0.482495\pi$
$354$	0	0
$355$	−184.270	−0.0275494
$356$	0	0
$357$	422.857	0.0626889
$358$	0	0
$359$	4185.19	0.615281	0.307641	−	0.951503i	$-0.400461\pi$
$359$	4185.19	0.615281	0.307641	+	0.951503i	$0.400461\pi$
$360$	0	0
$361$	−3709.25	−0.540786
$362$	0	0
$363$	−1626.98	−0.235246
$364$	0	0
$365$	−2362.37	−0.338773
$366$	0	0
$367$	2576.21	0.366422	0.183211	−	0.983074i	$-0.441351\pi$
$367$	2576.21	0.366422	0.183211	+	0.983074i	$0.441351\pi$
$368$	0	0
$369$	−6328.97	−0.892880
$370$	0	0
$371$	2793.36	0.390900
$372$	0	0
$373$	10692.8	1.48432	0.742162	−	0.670220i	$-0.233800\pi$
$373$	10692.8	1.48432	0.742162	+	0.670220i	$0.233800\pi$
$374$	0	0
$375$	−147.379	−0.0202950
$376$	0	0
$377$	19163.3	2.61793
$378$	0	0
$379$	−11212.9	−1.51971	−0.759853	−	0.650095i	$-0.774729\pi$
$379$	−11212.9	−1.51971	−0.759853	+	0.650095i	$0.774729\pi$
$380$	0	0
$381$	150.412	0.0202253
$382$	0	0
$383$	−9737.40	−1.29911	−0.649553	−	0.760316i	$-0.725044\pi$
$383$	−9737.40	−1.29911	−0.649553	+	0.760316i	$0.725044\pi$
$384$	0	0
$385$	1822.33	0.241233
$386$	0	0
$387$	−10626.2	−1.39577
$388$	0	0
$389$	−594.346	−0.0774667	−0.0387333	−	0.999250i	$-0.512332\pi$
$389$	−594.346	−0.0774667	−0.0387333	+	0.999250i	$0.512332\pi$
$390$	0	0
$391$	−4976.24	−0.643630
$392$	0	0
$393$	−188.963	−0.0242543
$394$	0	0
$395$	5509.19	0.701765
$396$	0	0
$397$	2273.63	0.287431	0.143716	−	0.989619i	$-0.454095\pi$
$397$	2273.63	0.287431	0.143716	+	0.989619i	$0.454095\pi$
$398$	0	0
$399$	−463.193	−0.0581169
$400$	0	0
$401$	695.727	0.0866407	0.0433204	−	0.999061i	$-0.486206\pi$
$401$	695.727	0.0866407	0.0433204	+	0.999061i	$0.486206\pi$
$402$	0	0
$403$	11561.3	1.42905
$404$	0	0
$405$	3091.66	0.379323
$406$	0	0
$407$	4988.72	0.607571
$408$	0	0
$409$	9528.76	1.15200	0.575999	−	0.817450i	$-0.304613\pi$
$409$	9528.76	1.15200	0.575999	+	0.817450i	$0.304613\pi$
$410$	0	0
$411$	−1155.92	−0.138729
$412$	0	0
$413$	2474.18	0.294786
$414$	0	0
$415$	−1337.89	−0.158251
$416$	0	0
$417$	1489.92	0.174968
$418$	0	0
$419$	−11894.2	−1.38680	−0.693398	−	0.720555i	$-0.743887\pi$
$419$	−11894.2	−1.38680	−0.693398	+	0.720555i	$0.743887\pi$
$420$	0	0
$421$	−14056.1	−1.62720	−0.813599	−	0.581426i	$-0.802495\pi$
$421$	−14056.1	−1.62720	−0.813599	+	0.581426i	$0.802495\pi$
$422$	0	0
$423$	617.569	0.0709864
$424$	0	0
$425$	1280.88	0.146193
$426$	0	0
$427$	4104.71	0.465201
$428$	0	0
$429$	3918.25	0.440967
$430$	0	0
$431$	3906.42	0.436579	0.218290	−	0.975884i	$-0.429952\pi$
$431$	3906.42	0.436579	0.218290	+	0.975884i	$0.429952\pi$
$432$	0	0
$433$	−16442.4	−1.82488	−0.912438	−	0.409216i	$-0.865802\pi$
$433$	−16442.4	−1.82488	−0.912438	+	0.409216i	$0.865802\pi$
$434$	0	0
$435$	−1769.94	−0.195086
$436$	0	0
$437$	5450.93	0.596689
$438$	0	0
$439$	−13106.6	−1.42493	−0.712464	−	0.701709i	$-0.752421\pi$
$439$	−13106.6	−1.42493	−0.712464	+	0.701709i	$0.752421\pi$
$440$	0	0
$441$	−1254.88	−0.135502
$442$	0	0
$443$	−10627.2	−1.13976	−0.569879	−	0.821729i	$-0.693010\pi$
$443$	−10627.2	−1.13976	−0.569879	+	0.821729i	$0.693010\pi$
$444$	0	0
$445$	−1683.63	−0.179352
$446$	0	0
$447$	−502.657	−0.0531876
$448$	0	0
$449$	−17887.5	−1.88010	−0.940051	−	0.341035i	$-0.889223\pi$
$449$	−17887.5	−1.88010	−0.940051	+	0.341035i	$0.889223\pi$
$450$	0	0
$451$	−12867.2	−1.34344
$452$	0	0
$453$	1782.42	0.184868
$454$	0	0
$455$	−2233.96	−0.230175
$456$	0	0
$457$	−4945.24	−0.506190	−0.253095	−	0.967441i	$-0.581448\pi$
$457$	−4945.24	−0.506190	−0.253095	+	0.967441i	$0.581448\pi$
$458$	0	0
$459$	3178.06	0.323179
$460$	0	0
$461$	15226.3	1.53831	0.769156	−	0.639061i	$-0.220677\pi$
$461$	15226.3	1.53831	0.769156	+	0.639061i	$0.220677\pi$
$462$	0	0
$463$	−12609.0	−1.26564	−0.632819	−	0.774300i	$-0.718102\pi$
$463$	−12609.0	−1.26564	−0.632819	+	0.774300i	$0.718102\pi$
$464$	0	0
$465$	−1067.81	−0.106491
$466$	0	0
$467$	17980.8	1.78170	0.890850	−	0.454298i	$-0.150110\pi$
$467$	17980.8	1.78170	0.890850	+	0.454298i	$0.150110\pi$
$468$	0	0
$469$	2895.22	0.285051
$470$	0	0
$471$	4564.16	0.446508
$472$	0	0
$473$	−21603.8	−2.10010
$474$	0	0
$475$	−1403.07	−0.135531
$476$	0	0
$477$	10219.7	0.980976
$478$	0	0
$479$	−7115.27	−0.678716	−0.339358	−	0.940657i	$-0.610210\pi$
$479$	−7115.27	−0.678716	−0.339358	+	0.940657i	$0.610210\pi$
$480$	0	0
$481$	−6115.57	−0.579721
$482$	0	0
$483$	−801.598	−0.0755155
$484$	0	0
$485$	1876.34	0.175671
$486$	0	0
$487$	5895.10	0.548527	0.274263	−	0.961655i	$-0.411566\pi$
$487$	5895.10	0.548527	0.274263	+	0.961655i	$0.411566\pi$
$488$	0	0
$489$	2007.23	0.185624
$490$	0	0
$491$	−6475.05	−0.595143	−0.297571	−	0.954700i	$-0.596177\pi$
$491$	−6475.05	−0.595143	−0.297571	+	0.954700i	$0.596177\pi$
$492$	0	0
$493$	15382.7	1.40528
$494$	0	0
$495$	6667.09	0.605381
$496$	0	0
$497$	257.978	0.0232835
$498$	0	0
$499$	17255.0	1.54798	0.773990	−	0.633198i	$-0.218258\pi$
$499$	17255.0	1.54798	0.773990	+	0.633198i	$0.218258\pi$
$500$	0	0
$501$	3364.34	0.300016
$502$	0	0
$503$	−2573.05	−0.228085	−0.114042	−	0.993476i	$-0.536380\pi$
$503$	−2573.05	−0.228085	−0.114042	+	0.993476i	$0.536380\pi$
$504$	0	0
$505$	690.670	0.0608602
$506$	0	0
$507$	−2212.97	−0.193849
$508$	0	0
$509$	1053.20	0.0917139	0.0458569	−	0.998948i	$-0.485398\pi$
$509$	1053.20	0.0917139	0.0458569	+	0.998948i	$0.485398\pi$
$510$	0	0
$511$	3307.32	0.286316
$512$	0	0
$513$	−3481.22	−0.299609
$514$	0	0
$515$	−6708.61	−0.574013
$516$	0	0
$517$	1255.56	0.106807
$518$	0	0
$519$	29.7199	0.00251360
$520$	0	0
$521$	8924.40	0.750452	0.375226	−	0.926933i	$-0.377565\pi$
$521$	8924.40	0.750452	0.375226	+	0.926933i	$0.377565\pi$
$522$	0	0
$523$	13200.4	1.10366	0.551828	−	0.833958i	$-0.313930\pi$
$523$	13200.4	1.10366	0.551828	+	0.833958i	$0.313930\pi$
$524$	0	0
$525$	206.331	0.0171524
$526$	0	0
$527$	9280.40	0.767098
$528$	0	0
$529$	−2733.67	−0.224679
$530$	0	0
$531$	9051.93	0.739775
$532$	0	0
$533$	15773.7	1.28186
$534$	0	0
$535$	−8481.23	−0.685375
$536$	0	0
$537$	−3664.09	−0.294446
$538$	0	0
$539$	−2551.26	−0.203879
$540$	0	0
$541$	−18907.7	−1.50260	−0.751300	−	0.659961i	$-0.770573\pi$
$541$	−18907.7	−1.50260	−0.751300	+	0.659961i	$0.770573\pi$
$542$	0	0
$543$	−1374.09	−0.108596
$544$	0	0
$545$	−11024.3	−0.866476
$546$	0	0
$547$	9216.11	0.720388	0.360194	−	0.932877i	$-0.382710\pi$
$547$	9216.11	0.720388	0.360194	+	0.932877i	$0.382710\pi$
$548$	0	0
$549$	15017.3	1.16744
$550$	0	0
$551$	−16850.0	−1.30279
$552$	0	0
$553$	−7712.86	−0.593100
$554$	0	0
$555$	564.841	0.0432003
$556$	0	0
$557$	−23700.4	−1.80291	−0.901453	−	0.432876i	$-0.857499\pi$
$557$	−23700.4	−1.80291	−0.901453	+	0.432876i	$0.857499\pi$
$558$	0	0
$559$	26483.7	2.00383
$560$	0	0
$561$	3145.24	0.236706
$562$	0	0
$563$	−12363.7	−0.925522	−0.462761	−	0.886483i	$-0.653141\pi$
$563$	−12363.7	−0.925522	−0.462761	+	0.886483i	$0.653141\pi$
$564$	0	0
$565$	5503.81	0.409818
$566$	0	0
$567$	−4328.33	−0.320587
$568$	0	0
$569$	7244.03	0.533718	0.266859	−	0.963736i	$-0.414014\pi$
$569$	7244.03	0.533718	0.266859	+	0.963736i	$0.414014\pi$
$570$	0	0
$571$	−12777.9	−0.936495	−0.468247	−	0.883597i	$-0.655114\pi$
$571$	−12777.9	−0.936495	−0.468247	+	0.883597i	$0.655114\pi$
$572$	0	0
$573$	2914.80	0.212509
$574$	0	0
$575$	−2428.13	−0.176105
$576$	0	0
$577$	16191.9	1.16825	0.584123	−	0.811666i	$-0.301439\pi$
$577$	16191.9	1.16825	0.584123	+	0.811666i	$0.301439\pi$
$578$	0	0
$579$	−1390.77	−0.0998244
$580$	0	0
$581$	1873.04	0.133747
$582$	0	0
$583$	20777.2	1.47599
$584$	0	0
$585$	−8173.06	−0.577631
$586$	0	0
$587$	18119.5	1.27406	0.637030	−	0.770839i	$-0.280163\pi$
$587$	18119.5	1.27406	0.637030	+	0.770839i	$0.280163\pi$
$588$	0	0
$589$	−10165.7	−0.711152
$590$	0	0
$591$	2874.20	0.200049
$592$	0	0
$593$	−4445.27	−0.307834	−0.153917	−	0.988084i	$-0.549189\pi$
$593$	−4445.27	−0.307834	−0.153917	+	0.988084i	$0.549189\pi$
$594$	0	0
$595$	−1793.23	−0.123555
$596$	0	0
$597$	−4309.42	−0.295432
$598$	0	0
$599$	3944.36	0.269052	0.134526	−	0.990910i	$-0.457049\pi$
$599$	3944.36	0.269052	0.134526	+	0.990910i	$0.457049\pi$
$600$	0	0
$601$	−26153.8	−1.77510	−0.887550	−	0.460712i	$-0.847594\pi$
$601$	−26153.8	−1.77510	−0.887550	+	0.460712i	$0.847594\pi$
$602$	0	0
$603$	10592.3	0.715345
$604$	0	0
$605$	6899.65	0.463654
$606$	0	0
$607$	9392.23	0.628038	0.314019	−	0.949417i	$-0.398325\pi$
$607$	9392.23	0.628038	0.314019	+	0.949417i	$0.398325\pi$
$608$	0	0
$609$	2477.92	0.164877
$610$	0	0
$611$	−1539.16	−0.101912
$612$	0	0
$613$	−2559.16	−0.168619	−0.0843097	−	0.996440i	$-0.526868\pi$
$613$	−2559.16	−0.168619	−0.0843097	+	0.996440i	$0.526868\pi$
$614$	0	0
$615$	−1456.87	−0.0955231
$616$	0	0
$617$	−4395.13	−0.286777	−0.143388	−	0.989666i	$-0.545800\pi$
$617$	−4395.13	−0.286777	−0.143388	+	0.989666i	$0.545800\pi$
$618$	0	0
$619$	21458.0	1.39333	0.696664	−	0.717398i	$-0.254667\pi$
$619$	21458.0	1.39333	0.696664	+	0.717398i	$0.254667\pi$
$620$	0	0
$621$	−6024.57	−0.389304
$622$	0	0
$623$	2357.09	0.151580
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−3445.27	−0.219443
$628$	0	0
$629$	−4909.07	−0.311188
$630$	0	0
$631$	−26755.9	−1.68801	−0.844007	−	0.536333i	$-0.819809\pi$
$631$	−26755.9	−1.68801	−0.844007	+	0.536333i	$0.819809\pi$
$632$	0	0
$633$	3539.31	0.222235
$634$	0	0
$635$	−637.861	−0.0398626
$636$	0	0
$637$	3127.54	0.194533
$638$	0	0
$639$	943.828	0.0584308
$640$	0	0
$641$	23583.6	1.45319	0.726596	−	0.687065i	$-0.241101\pi$
$641$	23583.6	1.45319	0.726596	+	0.687065i	$0.241101\pi$
$642$	0	0
$643$	−8540.49	−0.523801	−0.261900	−	0.965095i	$-0.584349\pi$
$643$	−8540.49	−0.523801	−0.261900	+	0.965095i	$0.584349\pi$
$644$	0	0
$645$	−2446.06	−0.149324
$646$	0	0
$647$	−31576.4	−1.91869	−0.959347	−	0.282231i	$-0.908926\pi$
$647$	−31576.4	−1.91869	−0.959347	+	0.282231i	$0.908926\pi$
$648$	0	0
$649$	18403.2	1.11308
$650$	0	0
$651$	1494.93	0.0900017
$652$	0	0
$653$	5913.61	0.354391	0.177196	−	0.984176i	$-0.443297\pi$
$653$	5913.61	0.354391	0.177196	+	0.984176i	$0.443297\pi$
$654$	0	0
$655$	801.348	0.0478035
$656$	0	0
$657$	12100.0	0.718518
$658$	0	0
$659$	−28032.9	−1.65707	−0.828534	−	0.559939i	$-0.810825\pi$
$659$	−28032.9	−1.65707	−0.828534	+	0.559939i	$0.810825\pi$
$660$	0	0
$661$	11830.1	0.696121	0.348061	−	0.937472i	$-0.386840\pi$
$661$	11830.1	0.696121	0.348061	+	0.937472i	$0.386840\pi$
$662$	0	0
$663$	−3855.69	−0.225856
$664$	0	0
$665$	1964.29	0.114544
$666$	0	0
$667$	−29160.5	−1.69280
$668$	0	0
$669$	−3417.22	−0.197485
$670$	0	0
$671$	30531.2	1.75655
$672$	0	0
$673$	−6267.23	−0.358966	−0.179483	−	0.983761i	$-0.557442\pi$
$673$	−6267.23	−0.358966	−0.179483	+	0.983761i	$0.557442\pi$
$674$	0	0
$675$	1550.72	0.0884255
$676$	0	0
$677$	−22641.1	−1.28533	−0.642663	−	0.766149i	$-0.722171\pi$
$677$	−22641.1	−1.28533	−0.642663	+	0.766149i	$0.722171\pi$
$678$	0	0
$679$	−2626.88	−0.148469
$680$	0	0
$681$	−4978.63	−0.280149
$682$	0	0
$683$	484.764	0.0271581	0.0135790	−	0.999908i	$-0.495678\pi$
$683$	484.764	0.0271581	0.0135790	+	0.999908i	$0.495678\pi$
$684$	0	0
$685$	4902.00	0.273424
$686$	0	0
$687$	1063.72	0.0590737
$688$	0	0
$689$	−25470.4	−1.40834
$690$	0	0
$691$	12456.4	0.685764	0.342882	−	0.939378i	$-0.388597\pi$
$691$	12456.4	0.685764	0.342882	+	0.939378i	$0.388597\pi$
$692$	0	0
$693$	−9333.93	−0.511640
$694$	0	0
$695$	−6318.38	−0.344848
$696$	0	0
$697$	12661.8	0.688090
$698$	0	0
$699$	−5828.41	−0.315380
$700$	0	0
$701$	−10798.1	−0.581793	−0.290897	−	0.956755i	$-0.593954\pi$
$701$	−10798.1	−0.581793	−0.290897	+	0.956755i	$0.593954\pi$
$702$	0	0
$703$	5377.34	0.288493
$704$	0	0
$705$	142.159	0.00759435
$706$	0	0
$707$	−966.938	−0.0514363
$708$	0	0
$709$	22997.2	1.21816	0.609082	−	0.793107i	$-0.291538\pi$
$709$	22997.2	1.21816	0.609082	+	0.793107i	$0.291538\pi$
$710$	0	0
$711$	−28217.9	−1.48840
$712$	0	0
$713$	−17592.6	−0.924051
$714$	0	0
$715$	−16616.4	−0.869115
$716$	0	0
$717$	−3506.02	−0.182615
$718$	0	0
$719$	9243.86	0.479469	0.239734	−	0.970839i	$-0.422940\pi$
$719$	9243.86	0.479469	0.239734	+	0.970839i	$0.422940\pi$
$720$	0	0
$721$	9392.05	0.485130
$722$	0	0
$723$	−1228.06	−0.0631701
$724$	0	0
$725$	7505.90	0.384500
$726$	0	0
$727$	24919.8	1.27128	0.635642	−	0.771984i	$-0.280735\pi$
$727$	24919.8	1.27128	0.635642	+	0.771984i	$0.280735\pi$
$728$	0	0
$729$	−13860.8	−0.704202
$730$	0	0
$731$	21258.9	1.07563
$732$	0	0
$733$	25726.5	1.29636	0.648180	−	0.761487i	$-0.275531\pi$
$733$	25726.5	1.29636	0.648180	+	0.761487i	$0.275531\pi$
$734$	0	0
$735$	−288.863	−0.0144964
$736$	0	0
$737$	21534.9	1.07632
$738$	0	0
$739$	32124.9	1.59910	0.799548	−	0.600602i	$-0.205072\pi$
$739$	32124.9	1.59910	0.799548	+	0.600602i	$0.205072\pi$
$740$	0	0
$741$	4223.48	0.209384
$742$	0	0
$743$	4511.62	0.222766	0.111383	−	0.993778i	$-0.464472\pi$
$743$	4511.62	0.222766	0.111383	+	0.993778i	$0.464472\pi$
$744$	0	0
$745$	2131.65	0.104829
$746$	0	0
$747$	6852.63	0.335642
$748$	0	0
$749$	11873.7	0.579247
$750$	0	0
$751$	20080.5	0.975697	0.487849	−	0.872928i	$-0.337782\pi$
$751$	20080.5	0.975697	0.487849	+	0.872928i	$0.337782\pi$
$752$	0	0
$753$	523.491	0.0253348
$754$	0	0
$755$	−7558.82	−0.364362
$756$	0	0
$757$	34752.0	1.66854	0.834268	−	0.551359i	$-0.185890\pi$
$757$	34752.0	1.66854	0.834268	+	0.551359i	$0.185890\pi$
$758$	0	0
$759$	−5962.35	−0.285138
$760$	0	0
$761$	32963.4	1.57020	0.785099	−	0.619370i	$-0.212612\pi$
$761$	32963.4	1.57020	0.785099	+	0.619370i	$0.212612\pi$
$762$	0	0
$763$	15434.0	0.732306
$764$	0	0
$765$	−6560.65	−0.310066
$766$	0	0
$767$	−22560.1	−1.06206
$768$	0	0
$769$	24879.6	1.16669	0.583343	−	0.812226i	$-0.301744\pi$
$769$	24879.6	1.16669	0.583343	+	0.812226i	$0.301744\pi$
$770$	0	0
$771$	1089.40	0.0508868
$772$	0	0
$773$	23772.1	1.10611	0.553054	−	0.833145i	$-0.313462\pi$
$773$	23772.1	1.10611	0.553054	+	0.833145i	$0.313462\pi$
$774$	0	0
$775$	4528.33	0.209887
$776$	0	0
$777$	−790.777	−0.0365109
$778$	0	0
$779$	−13869.6	−0.637906
$780$	0	0
$781$	1918.86	0.0879160
$782$	0	0
$783$	18623.3	0.849990
$784$	0	0
$785$	−19355.5	−0.880036
$786$	0	0
$787$	22996.4	1.04159	0.520797	−	0.853681i	$-0.325635\pi$
$787$	22996.4	1.04159	0.520797	+	0.853681i	$0.325635\pi$
$788$	0	0
$789$	−2253.88	−0.101699
$790$	0	0
$791$	−7705.34	−0.346359
$792$	0	0
$793$	−37427.5	−1.67603
$794$	0	0
$795$	2352.47	0.104948
$796$	0	0
$797$	−25470.0	−1.13199	−0.565994	−	0.824410i	$-0.691507\pi$
$797$	−25470.0	−1.13199	−0.565994	+	0.824410i	$0.691507\pi$
$798$	0	0
$799$	−1235.51	−0.0547050
$800$	0	0
$801$	8623.53	0.380396
$802$	0	0
$803$	24600.1	1.08110
$804$	0	0
$805$	3399.39	0.148836
$806$	0	0
$807$	9804.09	0.427658
$808$	0	0
$809$	127.081	0.00552278	0.00276139	−	0.999996i	$-0.499121\pi$
$809$	127.081	0.00552278	0.00276139	+	0.999996i	$0.499121\pi$
$810$	0	0
$811$	−27366.1	−1.18490	−0.592450	−	0.805607i	$-0.701839\pi$
$811$	−27366.1	−1.18490	−0.592450	+	0.805607i	$0.701839\pi$
$812$	0	0
$813$	7987.57	0.344571
$814$	0	0
$815$	−8512.20	−0.365852
$816$	0	0
$817$	−23286.8	−0.997187
$818$	0	0
$819$	11442.3	0.488188
$820$	0	0
$821$	−37015.9	−1.57352	−0.786762	−	0.617256i	$-0.788244\pi$
$821$	−37015.9	−1.57352	−0.786762	+	0.617256i	$0.788244\pi$
$822$	0	0
$823$	21835.3	0.924824	0.462412	−	0.886665i	$-0.346984\pi$
$823$	21835.3	0.924824	0.462412	+	0.886665i	$0.346984\pi$
$824$	0	0
$825$	1534.71	0.0647656
$826$	0	0
$827$	23311.6	0.980197	0.490099	−	0.871667i	$-0.336961\pi$
$827$	23311.6	0.980197	0.490099	+	0.871667i	$0.336961\pi$
$828$	0	0
$829$	−29151.7	−1.22133	−0.610663	−	0.791890i	$-0.709097\pi$
$829$	−29151.7	−1.22133	−0.610663	+	0.791890i	$0.709097\pi$
$830$	0	0
$831$	−3892.97	−0.162510
$832$	0	0
$833$	2510.53	0.104423
$834$	0	0
$835$	−14267.4	−0.591309
$836$	0	0
$837$	11235.5	0.463984
$838$	0	0
$839$	21059.0	0.866552	0.433276	−	0.901261i	$-0.357358\pi$
$839$	21059.0	0.866552	0.433276	+	0.901261i	$0.357358\pi$
$840$	0	0
$841$	65752.7	2.69600
$842$	0	0
$843$	−3214.34	−0.131326
$844$	0	0
$845$	9384.67	0.382062
$846$	0	0
$847$	−9659.50	−0.391859
$848$	0	0
$849$	943.046	0.0381216
$850$	0	0
$851$	9305.99	0.374859
$852$	0	0
$853$	−14309.1	−0.574364	−0.287182	−	0.957876i	$-0.592719\pi$
$853$	−14309.1	−0.574364	−0.287182	+	0.957876i	$0.592719\pi$
$854$	0	0
$855$	7186.47	0.287453
$856$	0	0
$857$	3887.04	0.154934	0.0774672	−	0.996995i	$-0.475317\pi$
$857$	3887.04	0.154934	0.0774672	+	0.996995i	$0.475317\pi$
$858$	0	0
$859$	−20261.9	−0.804804	−0.402402	−	0.915463i	$-0.631825\pi$
$859$	−20261.9	−0.804804	−0.402402	+	0.915463i	$0.631825\pi$
$860$	0	0
$861$	2039.62	0.0807318
$862$	0	0
$863$	32304.6	1.27423	0.637116	−	0.770768i	$-0.280127\pi$
$863$	32304.6	1.27423	0.637116	+	0.770768i	$0.280127\pi$
$864$	0	0
$865$	−126.035	−0.00495413
$866$	0	0
$867$	2697.57	0.105668
$868$	0	0
$869$	−57368.9	−2.23948
$870$	0	0
$871$	−26399.2	−1.02698
$872$	0	0
$873$	−9610.57	−0.372587
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−7220.02	−0.277996	−0.138998	−	0.990293i	$-0.544388\pi$
$877$	−7220.02	−0.277996	−0.138998	+	0.990293i	$0.544388\pi$
$878$	0	0
$879$	8037.64	0.308422
$880$	0	0
$881$	38531.6	1.47351	0.736756	−	0.676159i	$-0.236357\pi$
$881$	38531.6	1.47351	0.736756	+	0.676159i	$0.236357\pi$
$882$	0	0
$883$	−331.921	−0.0126501	−0.00632505	−	0.999980i	$-0.502013\pi$
$883$	−331.921	−0.0126501	−0.00632505	+	0.999980i	$0.502013\pi$
$884$	0	0
$885$	2083.67	0.0791434
$886$	0	0
$887$	36815.6	1.39363	0.696814	−	0.717252i	$-0.254600\pi$
$887$	36815.6	1.39363	0.696814	+	0.717252i	$0.254600\pi$
$888$	0	0
$889$	893.005	0.0336900
$890$	0	0
$891$	−32194.5	−1.21050
$892$	0	0
$893$	1353.37	0.0507153
$894$	0	0
$895$	15538.5	0.580331
$896$	0	0
$897$	7309.13	0.272068
$898$	0	0
$899$	54382.7	2.01754
$900$	0	0
$901$	−20445.5	−0.755980
$902$	0	0
$903$	3424.49	0.126201
$904$	0	0
$905$	5827.18	0.214035
$906$	0	0
$907$	−8709.19	−0.318836	−0.159418	−	0.987211i	$-0.550962\pi$
$907$	−8709.19	−0.318836	−0.159418	+	0.987211i	$0.550962\pi$
$908$	0	0
$909$	−3537.60	−0.129081
$910$	0	0
$911$	30540.1	1.11069	0.555346	−	0.831620i	$-0.312586\pi$
$911$	30540.1	1.11069	0.555346	+	0.831620i	$0.312586\pi$
$912$	0	0
$913$	13931.9	0.505013
$914$	0	0
$915$	3456.85	0.124896
$916$	0	0
$917$	−1121.89	−0.0404013
$918$	0	0
$919$	18480.0	0.663329	0.331665	−	0.943397i	$-0.392390\pi$
$919$	18480.0	0.663329	0.331665	+	0.943397i	$0.392390\pi$
$920$	0	0
$921$	−100.744	−0.00360439
$922$	0	0
$923$	−2352.30	−0.0838861
$924$	0	0
$925$	−2395.35	−0.0851446
$926$	0	0
$927$	34361.3	1.21745
$928$	0	0
$929$	−7677.61	−0.271146	−0.135573	−	0.990767i	$-0.543287\pi$
$929$	−7677.61	−0.271146	−0.135573	+	0.990767i	$0.543287\pi$
$930$	0	0
$931$	−2750.01	−0.0968076
$932$	0	0
$933$	6557.57	0.230102
$934$	0	0
$935$	−13338.2	−0.466531
$936$	0	0
$937$	−25506.0	−0.889270	−0.444635	−	0.895712i	$-0.646667\pi$
$937$	−25506.0	−0.889270	−0.444635	+	0.895712i	$0.646667\pi$
$938$	0	0
$939$	9672.03	0.336139
$940$	0	0
$941$	−40321.6	−1.39686	−0.698431	−	0.715677i	$-0.746118\pi$
$941$	−40321.6	−1.39686	−0.698431	+	0.715677i	$0.746118\pi$
$942$	0	0
$943$	−24002.6	−0.828877
$944$	0	0
$945$	−2171.01	−0.0747332
$946$	0	0
$947$	20348.5	0.698245	0.349122	−	0.937077i	$-0.386480\pi$
$947$	20348.5	0.698245	0.349122	+	0.937077i	$0.386480\pi$
$948$	0	0
$949$	−30156.8	−1.03154
$950$	0	0
$951$	−8775.49	−0.299227
$952$	0	0
$953$	8822.60	0.299887	0.149943	−	0.988695i	$-0.452091\pi$
$953$	8822.60	0.299887	0.149943	+	0.988695i	$0.452091\pi$
$954$	0	0
$955$	−12361.0	−0.418840
$956$	0	0
$957$	18431.0	0.622559
$958$	0	0
$959$	−6862.80	−0.231086
$960$	0	0
$961$	3018.21	0.101313
$962$	0	0
$963$	43440.6	1.45364
$964$	0	0
$965$	5897.91	0.196747
$966$	0	0
$967$	−699.183	−0.0232515	−0.0116258	−	0.999932i	$-0.503701\pi$
$967$	−699.183	−0.0232515	−0.0116258	+	0.999932i	$0.503701\pi$
$968$	0	0
$969$	3390.26	0.112395
$970$	0	0
$971$	−15428.0	−0.509895	−0.254948	−	0.966955i	$-0.582058\pi$
$971$	−15428.0	−0.509895	−0.254948	+	0.966955i	$0.582058\pi$
$972$	0	0
$973$	8845.73	0.291450
$974$	0	0
$975$	−1881.37	−0.0617968
$976$	0	0
$977$	−17840.7	−0.584210	−0.292105	−	0.956386i	$-0.594356\pi$
$977$	−17840.7	−0.584210	−0.292105	+	0.956386i	$0.594356\pi$
$978$	0	0
$979$	17532.2	0.572351
$980$	0	0
$981$	56466.2	1.83774
$982$	0	0
$983$	−27934.9	−0.906393	−0.453197	−	0.891411i	$-0.649716\pi$
$983$	−27934.9	−0.906393	−0.453197	+	0.891411i	$0.649716\pi$
$984$	0	0
$985$	−12188.8	−0.394282
$986$	0	0
$987$	−199.023	−0.00641840
$988$	0	0
$989$	−40299.9	−1.29572
$990$	0	0
$991$	−8360.69	−0.267998	−0.133999	−	0.990981i	$-0.542782\pi$
$991$	−8360.69	−0.267998	−0.133999	+	0.990981i	$0.542782\pi$
$992$	0	0
$993$	8987.32	0.287215
$994$	0	0
$995$	18275.2	0.582275
$996$	0	0
$997$	−40993.9	−1.30220	−0.651099	−	0.758993i	$-0.725692\pi$
$997$	−40993.9	−1.30220	−0.651099	+	0.758993i	$0.725692\pi$
$998$	0	0
$999$	−5943.24	−0.188224

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.cn.1.3		5
4.3	odd	2		2240.4.a.co.1.3		5
8.3	odd	2		1120.4.a.r.1.3	✓	5
8.5	even	2		1120.4.a.s.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.4.a.r.1.3	✓	5	8.3	odd	2
1120.4.a.s.1.3	yes	5	8.5	even	2
2240.4.a.cn.1.3		5	1.1	even	1	trivial
2240.4.a.co.1.3		5	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:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q-1.17903 q^{3} +5.00000 q^{5} -7.00000 q^{7} -25.6099 q^{9} -52.0666 q^{11} +63.8274 q^{13} -5.89517 q^{15} +51.2353 q^{17} -56.1226 q^{19} +8.25323 q^{21} -97.1253 q^{23} +25.0000 q^{25} +62.0288 q^{27} +300.236 q^{29} +181.133 q^{31} +61.3882 q^{33} -35.0000 q^{35} -95.8142 q^{37} -75.2546 q^{39} +247.130 q^{41} +414.927 q^{43} -128.049 q^{45} -24.1145 q^{47} +49.0000 q^{49} -60.4081 q^{51} -399.051 q^{53} -260.333 q^{55} +66.1704 q^{57} -353.455 q^{59} -586.387 q^{61} +179.269 q^{63} +319.137 q^{65} -413.604 q^{67} +114.514 q^{69} -36.8541 q^{71} -472.474 q^{73} -29.4758 q^{75} +364.466 q^{77} +1101.84 q^{79} +618.333 q^{81} -267.578 q^{83} +256.176 q^{85} -353.988 q^{87} -336.727 q^{89} -446.792 q^{91} -213.562 q^{93} -280.613 q^{95} +375.268 q^{97} +1333.42 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + 25 q^{5} - 35 q^{7} + 68 q^{9} - 35 q^{11} + 39 q^{13} - 25 q^{15} - 29 q^{17} - 84 q^{19} + 35 q^{21} - 128 q^{23} + 125 q^{25} - 35 q^{27} - 73 q^{29} - 318 q^{31} + 205 q^{33} - 175 q^{35}+ \cdots - 3734 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + 25 * q^5 - 35 * q^7 + 68 * q^9 - 35 * q^11 + 39 * q^13 - 25 * q^15 - 29 * q^17 - 84 * q^19 + 35 * q^21 - 128 * q^23 + 125 * q^25 - 35 * q^27 - 73 * q^29 - 318 * q^31 + 205 * q^33 - 175 * q^35 + 236 * q^37 - q^39 + 262 * q^41 - 372 * q^43 + 340 * q^45 - 603 * q^47 + 245 * q^49 - 777 * q^51 - 20 * q^53 - 175 * q^55 - 164 * q^57 - 960 * q^59 - 246 * q^61 - 476 * q^63 + 195 * q^65 - 1360 * q^67 - 1576 * q^69 + 696 * q^71 + 1122 * q^73 - 125 * q^75 + 245 * q^77 + 93 * q^79 + 3125 * q^81 + 1412 * q^83 - 145 * q^85 + 231 * q^87 + 2854 * q^89 - 273 * q^91 - 3262 * q^93 - 420 * q^95 + 1739 * q^97 - 3734 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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