LMFDB - Embedded newform 1120.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,4,Mod(1,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, 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("1120.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1120.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:	$66.0821392064$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.505876.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 14x + 14 $ x^4 - x^3 - 12x^2 + 14x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.15507$ of defining polynomial
Character	$\chi$	$=$	1120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.94308 q^{3} +5.00000 q^{5} -7.00000 q^{7} +52.9787 q^{9} -35.8927 q^{11} +7.01392 q^{13} -44.7154 q^{15} -31.3558 q^{17} +90.2089 q^{19} +62.6016 q^{21} -99.9189 q^{23} +25.0000 q^{25} -232.330 q^{27} +192.069 q^{29} -164.095 q^{31} +320.991 q^{33} -35.0000 q^{35} +95.6928 q^{37} -62.7261 q^{39} -146.753 q^{41} +323.890 q^{43} +264.894 q^{45} +101.260 q^{47} +49.0000 q^{49} +280.418 q^{51} -120.176 q^{53} -179.464 q^{55} -806.745 q^{57} +608.600 q^{59} +832.769 q^{61} -370.851 q^{63} +35.0696 q^{65} -652.930 q^{67} +893.583 q^{69} +217.953 q^{71} +607.890 q^{73} -223.577 q^{75} +251.249 q^{77} +1039.99 q^{79} +647.318 q^{81} -736.325 q^{83} -156.779 q^{85} -1717.69 q^{87} -602.294 q^{89} -49.0975 q^{91} +1467.52 q^{93} +451.044 q^{95} +547.366 q^{97} -1901.55 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 7 q^{3} + 20 q^{5} - 28 q^{7} + 47 q^{9} - 91 q^{11} + 113 q^{13} - 35 q^{15} + 3 q^{17} - 112 q^{19} + 49 q^{21} - 168 q^{23} + 100 q^{25} - 133 q^{27} + 31 q^{29} - 126 q^{31} - 33 q^{33} - 140 q^{35}+ \cdots - 1050 q^{99}+O(q^{100}) $ 4 * q - 7 * q^3 + 20 * q^5 - 28 * q^7 + 47 * q^9 - 91 * q^11 + 113 * q^13 - 35 * q^15 + 3 * q^17 - 112 * q^19 + 49 * q^21 - 168 * q^23 + 100 * q^25 - 133 * q^27 + 31 * q^29 - 126 * q^31 - 33 * q^33 - 140 * q^35 + 186 * q^37 + 161 * q^39 - 8 * q^41 - 28 * q^43 + 235 * q^45 - 231 * q^47 + 196 * q^49 - 525 * q^51 + 110 * q^53 - 455 * q^55 - 728 * q^57 - 196 * q^59 - 432 * q^61 - 329 * q^63 + 565 * q^65 + 336 * q^67 - 480 * q^69 - 1708 * q^71 + 712 * q^73 - 175 * q^75 + 637 * q^77 + 455 * q^79 - 180 * q^81 - 1512 * q^83 + 15 * q^85 - 2541 * q^87 + 732 * q^89 - 791 * q^91 + 1430 * q^93 - 560 * q^95 - 589 * q^97 - 1050 * 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
$3$	−8.94308	−1.72110	−0.860548	−	0.509369i	$-0.829879\pi$
$3$	−8.94308	−1.72110	−0.860548	+	0.509369i	$0.829879\pi$
$4$	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
$9$	52.9787	1.96217
$10$	0	0
$11$	−35.8927	−0.983824	−0.491912	−	0.870645i	$-0.663702\pi$
$11$	−35.8927	−0.983824	−0.491912	+	0.870645i	$0.663702\pi$
$12$	0	0
$13$	7.01392	0.149639	0.0748197	−	0.997197i	$-0.476162\pi$
$13$	7.01392	0.149639	0.0748197	+	0.997197i	$0.476162\pi$
$14$	0	0
$15$	−44.7154	−0.769698
$16$	0	0
$17$	−31.3558	−0.447348	−0.223674	−	0.974664i	$-0.571805\pi$
$17$	−31.3558	−0.447348	−0.223674	+	0.974664i	$0.571805\pi$
$18$	0	0
$19$	90.2089	1.08923	0.544614	−	0.838687i	$-0.316676\pi$
$19$	90.2089	1.08923	0.544614	+	0.838687i	$0.316676\pi$
$20$	0	0
$21$	62.6016	0.650513
$22$	0	0
$23$	−99.9189	−0.905849	−0.452924	−	0.891549i	$-0.649619\pi$
$23$	−99.9189	−0.905849	−0.452924	+	0.891549i	$0.649619\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−232.330	−1.65599
$28$	0	0
$29$	192.069	1.22987	0.614937	−	0.788576i	$-0.289181\pi$
$29$	192.069	1.22987	0.614937	+	0.788576i	$0.289181\pi$
$30$	0	0
$31$	−164.095	−0.950721	−0.475360	−	0.879791i	$-0.657682\pi$
$31$	−164.095	−0.950721	−0.475360	+	0.879791i	$0.657682\pi$
$32$	0	0
$33$	320.991	1.69326
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	95.6928	0.425184	0.212592	−	0.977141i	$-0.431809\pi$
$37$	95.6928	0.425184	0.212592	+	0.977141i	$0.431809\pi$
$38$	0	0
$39$	−62.7261	−0.257544
$40$	0	0
$41$	−146.753	−0.558998	−0.279499	−	0.960146i	$-0.590168\pi$
$41$	−146.753	−0.558998	−0.279499	+	0.960146i	$0.590168\pi$
$42$	0	0
$43$	323.890	1.14867	0.574335	−	0.818621i	$-0.305261\pi$
$43$	323.890	1.14867	0.574335	+	0.818621i	$0.305261\pi$
$44$	0	0
$45$	264.894	0.877511
$46$	0	0
$47$	101.260	0.314260	0.157130	−	0.987578i	$-0.449776\pi$
$47$	101.260	0.314260	0.157130	+	0.987578i	$0.449776\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	280.418	0.769928
$52$	0	0
$53$	−120.176	−0.311461	−0.155731	−	0.987800i	$-0.549773\pi$
$53$	−120.176	−0.311461	−0.155731	+	0.987800i	$0.549773\pi$
$54$	0	0
$55$	−179.464	−0.439979
$56$	0	0
$57$	−806.745	−1.87467
$58$	0	0
$59$	608.600	1.34293	0.671466	−	0.741035i	$-0.265665\pi$
$59$	608.600	1.34293	0.671466	+	0.741035i	$0.265665\pi$
$60$	0	0
$61$	832.769	1.74795	0.873977	−	0.485968i	$-0.161533\pi$
$61$	832.769	1.74795	0.873977	+	0.485968i	$0.161533\pi$
$62$	0	0
$63$	−370.851	−0.741632
$64$	0	0
$65$	35.0696	0.0669208
$66$	0	0
$67$	−652.930	−1.19057	−0.595284	−	0.803515i	$-0.702960\pi$
$67$	−652.930	−1.19057	−0.595284	+	0.803515i	$0.702960\pi$
$68$	0	0
$69$	893.583	1.55905
$70$	0	0
$71$	217.953	0.364313	0.182156	−	0.983270i	$-0.441692\pi$
$71$	217.953	0.364313	0.182156	+	0.983270i	$0.441692\pi$
$72$	0	0
$73$	607.890	0.974632	0.487316	−	0.873226i	$-0.337976\pi$
$73$	607.890	0.974632	0.487316	+	0.873226i	$0.337976\pi$
$74$	0	0
$75$	−223.577	−0.344219
$76$	0	0
$77$	251.249	0.371850
$78$	0	0
$79$	1039.99	1.48111	0.740557	−	0.671994i	$-0.234562\pi$
$79$	1039.99	1.48111	0.740557	+	0.671994i	$0.234562\pi$
$80$	0	0
$81$	647.318	0.887953
$82$	0	0
$83$	−736.325	−0.973761	−0.486881	−	0.873469i	$-0.661865\pi$
$83$	−736.325	−0.973761	−0.486881	+	0.873469i	$0.661865\pi$
$84$	0	0
$85$	−156.779	−0.200060
$86$	0	0
$87$	−1717.69	−2.11673
$88$	0	0
$89$	−602.294	−0.717337	−0.358668	−	0.933465i	$-0.616769\pi$
$89$	−602.294	−0.717337	−0.358668	+	0.933465i	$0.616769\pi$
$90$	0	0
$91$	−49.0975	−0.0565584
$92$	0	0
$93$	1467.52	1.63628
$94$	0	0
$95$	451.044	0.487118
$96$	0	0
$97$	547.366	0.572955	0.286477	−	0.958087i	$-0.407516\pi$
$97$	547.366	0.572955	0.286477	+	0.958087i	$0.407516\pi$
$98$	0	0
$99$	−1901.55	−1.93043
$100$	0	0
$101$	−656.708	−0.646979	−0.323490	−	0.946232i	$-0.604856\pi$
$101$	−656.708	−0.646979	−0.323490	+	0.946232i	$0.604856\pi$
$102$	0	0
$103$	−93.1931	−0.0891514	−0.0445757	−	0.999006i	$-0.514194\pi$
$103$	−93.1931	−0.0891514	−0.0445757	+	0.999006i	$0.514194\pi$
$104$	0	0
$105$	313.008	0.290918
$106$	0	0
$107$	−583.092	−0.526819	−0.263410	−	0.964684i	$-0.584847\pi$
$107$	−583.092	−0.526819	−0.263410	+	0.964684i	$0.584847\pi$
$108$	0	0
$109$	−1119.34	−0.983608	−0.491804	−	0.870706i	$-0.663662\pi$
$109$	−1119.34	−0.983608	−0.491804	+	0.870706i	$0.663662\pi$
$110$	0	0
$111$	−855.789	−0.731783
$112$	0	0
$113$	−1959.55	−1.63132	−0.815659	−	0.578532i	$-0.803626\pi$
$113$	−1959.55	−1.63132	−0.815659	+	0.578532i	$0.803626\pi$
$114$	0	0
$115$	−499.594	−0.405108
$116$	0	0
$117$	371.589	0.293619
$118$	0	0
$119$	219.491	0.169081
$120$	0	0
$121$	−42.7131	−0.0320910
$122$	0	0
$123$	1312.42	0.962090
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1112.45	−0.777275	−0.388638	−	0.921391i	$-0.627054\pi$
$127$	−1112.45	−0.777275	−0.388638	+	0.921391i	$0.627054\pi$
$128$	0	0
$129$	−2896.58	−1.97697
$130$	0	0
$131$	−2208.80	−1.47316	−0.736580	−	0.676350i	$-0.763561\pi$
$131$	−2208.80	−1.47316	−0.736580	+	0.676350i	$0.763561\pi$
$132$	0	0
$133$	−631.462	−0.411690
$134$	0	0
$135$	−1161.65	−0.740583
$136$	0	0
$137$	−1560.31	−0.973037	−0.486519	−	0.873670i	$-0.661733\pi$
$137$	−1560.31	−0.973037	−0.486519	+	0.873670i	$0.661733\pi$
$138$	0	0
$139$	−680.329	−0.415142	−0.207571	−	0.978220i	$-0.566556\pi$
$139$	−680.329	−0.415142	−0.207571	+	0.978220i	$0.566556\pi$
$140$	0	0
$141$	−905.574	−0.540873
$142$	0	0
$143$	−251.749	−0.147219
$144$	0	0
$145$	960.346	0.550017
$146$	0	0
$147$	−438.211	−0.245871
$148$	0	0
$149$	2645.18	1.45437	0.727187	−	0.686439i	$-0.240827\pi$
$149$	2645.18	1.45437	0.727187	+	0.686439i	$0.240827\pi$
$150$	0	0
$151$	−1824.25	−0.983148	−0.491574	−	0.870836i	$-0.663578\pi$
$151$	−1824.25	−0.983148	−0.491574	+	0.870836i	$0.663578\pi$
$152$	0	0
$153$	−1661.19	−0.877774
$154$	0	0
$155$	−820.475	−0.425175
$156$	0	0
$157$	−2103.03	−1.06905	−0.534523	−	0.845154i	$-0.679509\pi$
$157$	−2103.03	−1.06905	−0.534523	+	0.845154i	$0.679509\pi$
$158$	0	0
$159$	1074.74	0.536055
$160$	0	0
$161$	699.432	0.342379
$162$	0	0
$163$	2821.49	1.35581	0.677903	−	0.735151i	$-0.262889\pi$
$163$	2821.49	1.35581	0.677903	+	0.735151i	$0.262889\pi$
$164$	0	0
$165$	1604.96	0.757247
$166$	0	0
$167$	−321.021	−0.148751	−0.0743753	−	0.997230i	$-0.523696\pi$
$167$	−321.021	−0.148751	−0.0743753	+	0.997230i	$0.523696\pi$
$168$	0	0
$169$	−2147.80	−0.977608
$170$	0	0
$171$	4779.15	2.13726
$172$	0	0
$173$	222.556	0.0978071	0.0489036	−	0.998804i	$-0.484427\pi$
$173$	222.556	0.0978071	0.0489036	+	0.998804i	$0.484427\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−5442.76	−2.31132
$178$	0	0
$179$	−566.489	−0.236544	−0.118272	−	0.992981i	$-0.537735\pi$
$179$	−566.489	−0.236544	−0.118272	+	0.992981i	$0.537735\pi$
$180$	0	0
$181$	3087.49	1.26791	0.633954	−	0.773371i	$-0.281431\pi$
$181$	3087.49	1.26791	0.633954	+	0.773371i	$0.281431\pi$
$182$	0	0
$183$	−7447.52	−3.00840
$184$	0	0
$185$	478.464	0.190148
$186$	0	0
$187$	1125.45	0.440111
$188$	0	0
$189$	1626.31	0.625907
$190$	0	0
$191$	−696.772	−0.263961	−0.131981	−	0.991252i	$-0.542134\pi$
$191$	−696.772	−0.263961	−0.131981	+	0.991252i	$0.542134\pi$
$192$	0	0
$193$	−5071.55	−1.89149	−0.945747	−	0.324905i	$-0.894668\pi$
$193$	−5071.55	−1.89149	−0.945747	+	0.324905i	$0.894668\pi$
$194$	0	0
$195$	−313.630	−0.115177
$196$	0	0
$197$	−2417.15	−0.874185	−0.437093	−	0.899417i	$-0.643992\pi$
$197$	−2417.15	−0.874185	−0.437093	+	0.899417i	$0.643992\pi$
$198$	0	0
$199$	1653.91	0.589157	0.294579	−	0.955627i	$-0.404821\pi$
$199$	1653.91	0.589157	0.294579	+	0.955627i	$0.404821\pi$
$200$	0	0
$201$	5839.20	2.04908
$202$	0	0
$203$	−1344.48	−0.464849
$204$	0	0
$205$	−733.764	−0.249992
$206$	0	0
$207$	−5293.57	−1.77743
$208$	0	0
$209$	−3237.84	−1.07161
$210$	0	0
$211$	−918.379	−0.299639	−0.149819	−	0.988713i	$-0.547869\pi$
$211$	−918.379	−0.299639	−0.149819	+	0.988713i	$0.547869\pi$
$212$	0	0
$213$	−1949.17	−0.627018
$214$	0	0
$215$	1619.45	0.513701
$216$	0	0
$217$	1148.67	0.359339
$218$	0	0
$219$	−5436.41	−1.67744
$220$	0	0
$221$	−219.927	−0.0669408
$222$	0	0
$223$	−6310.29	−1.89493	−0.947463	−	0.319865i	$-0.896362\pi$
$223$	−6310.29	−1.89493	−0.947463	+	0.319865i	$0.896362\pi$
$224$	0	0
$225$	1324.47	0.392435
$226$	0	0
$227$	5907.16	1.72719	0.863594	−	0.504187i	$-0.168208\pi$
$227$	5907.16	1.72719	0.863594	+	0.504187i	$0.168208\pi$
$228$	0	0
$229$	−5044.70	−1.45573	−0.727867	−	0.685719i	$-0.759488\pi$
$229$	−5044.70	−1.45573	−0.727867	+	0.685719i	$0.759488\pi$
$230$	0	0
$231$	−2246.94	−0.639991
$232$	0	0
$233$	−2882.29	−0.810408	−0.405204	−	0.914226i	$-0.632800\pi$
$233$	−2882.29	−0.810408	−0.405204	+	0.914226i	$0.632800\pi$
$234$	0	0
$235$	506.298	0.140542
$236$	0	0
$237$	−9300.71	−2.54914
$238$	0	0
$239$	−4746.48	−1.28462	−0.642310	−	0.766445i	$-0.722024\pi$
$239$	−4746.48	−1.28462	−0.642310	+	0.766445i	$0.722024\pi$
$240$	0	0
$241$	4622.72	1.23558	0.617792	−	0.786342i	$-0.288027\pi$
$241$	4622.72	1.23558	0.617792	+	0.786342i	$0.288027\pi$
$242$	0	0
$243$	483.883	0.127741
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	632.718	0.162991
$248$	0	0
$249$	6585.02	1.67594
$250$	0	0
$251$	6702.05	1.68538	0.842688	−	0.538402i	$-0.180972\pi$
$251$	6702.05	1.68538	0.842688	+	0.538402i	$0.180972\pi$
$252$	0	0
$253$	3586.36	0.891196
$254$	0	0
$255$	1402.09	0.344322
$256$	0	0
$257$	3266.63	0.792866	0.396433	−	0.918064i	$-0.370248\pi$
$257$	3266.63	0.792866	0.396433	+	0.918064i	$0.370248\pi$
$258$	0	0
$259$	−669.850	−0.160704
$260$	0	0
$261$	10175.6	2.41323
$262$	0	0
$263$	−6638.02	−1.55634	−0.778170	−	0.628053i	$-0.783852\pi$
$263$	−6638.02	−1.55634	−0.778170	+	0.628053i	$0.783852\pi$
$264$	0	0
$265$	−600.880	−0.139290
$266$	0	0
$267$	5386.36	1.23461
$268$	0	0
$269$	−1564.17	−0.354532	−0.177266	−	0.984163i	$-0.556725\pi$
$269$	−1564.17	−0.354532	−0.177266	+	0.984163i	$0.556725\pi$
$270$	0	0
$271$	−3322.97	−0.744856	−0.372428	−	0.928061i	$-0.621475\pi$
$271$	−3322.97	−0.744856	−0.372428	+	0.928061i	$0.621475\pi$
$272$	0	0
$273$	439.083	0.0973425
$274$	0	0
$275$	−897.318	−0.196765
$276$	0	0
$277$	−4636.34	−1.00567	−0.502835	−	0.864382i	$-0.667710\pi$
$277$	−4636.34	−1.00567	−0.502835	+	0.864382i	$0.667710\pi$
$278$	0	0
$279$	−8693.54	−1.86548
$280$	0	0
$281$	7272.19	1.54385	0.771926	−	0.635712i	$-0.219293\pi$
$281$	7272.19	1.54385	0.771926	+	0.635712i	$0.219293\pi$
$282$	0	0
$283$	−3520.36	−0.739448	−0.369724	−	0.929142i	$-0.620548\pi$
$283$	−3520.36	−0.739448	−0.369724	+	0.929142i	$0.620548\pi$
$284$	0	0
$285$	−4033.73	−0.838377
$286$	0	0
$287$	1027.27	0.211281
$288$	0	0
$289$	−3929.81	−0.799880
$290$	0	0
$291$	−4895.14	−0.986111
$292$	0	0
$293$	3069.22	0.611965	0.305983	−	0.952037i	$-0.401015\pi$
$293$	3069.22	0.611965	0.305983	+	0.952037i	$0.401015\pi$
$294$	0	0
$295$	3043.00	0.600578
$296$	0	0
$297$	8338.94	1.62921
$298$	0	0
$299$	−700.823	−0.135551
$300$	0	0
$301$	−2267.23	−0.434156
$302$	0	0
$303$	5873.00	1.11351
$304$	0	0
$305$	4163.85	0.781709
$306$	0	0
$307$	1538.81	0.286073	0.143037	−	0.989717i	$-0.454313\pi$
$307$	1538.81	0.286073	0.143037	+	0.989717i	$0.454313\pi$
$308$	0	0
$309$	833.434	0.153438
$310$	0	0
$311$	8510.17	1.55166	0.775832	−	0.630939i	$-0.217330\pi$
$311$	8510.17	1.55166	0.775832	+	0.630939i	$0.217330\pi$
$312$	0	0
$313$	5199.51	0.938958	0.469479	−	0.882944i	$-0.344442\pi$
$313$	5199.51	0.938958	0.469479	+	0.882944i	$0.344442\pi$
$314$	0	0
$315$	−1854.25	−0.331668
$316$	0	0
$317$	−4994.18	−0.884862	−0.442431	−	0.896802i	$-0.645884\pi$
$317$	−4994.18	−0.884862	−0.442431	+	0.896802i	$0.645884\pi$
$318$	0	0
$319$	−6893.89	−1.20998
$320$	0	0
$321$	5214.64	0.906707
$322$	0	0
$323$	−2828.57	−0.487263
$324$	0	0
$325$	175.348	0.0299279
$326$	0	0
$327$	10010.3	1.69288
$328$	0	0
$329$	−708.818	−0.118779
$330$	0	0
$331$	453.883	0.0753706	0.0376853	−	0.999290i	$-0.488002\pi$
$331$	453.883	0.0753706	0.0376853	+	0.999290i	$0.488002\pi$
$332$	0	0
$333$	5069.68	0.834285
$334$	0	0
$335$	−3264.65	−0.532438
$336$	0	0
$337$	231.967	0.0374957	0.0187479	−	0.999824i	$-0.494032\pi$
$337$	231.967	0.0374957	0.0187479	+	0.999824i	$0.494032\pi$
$338$	0	0
$339$	17524.4	2.80766
$340$	0	0
$341$	5889.82	0.935341
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4467.91	0.697230
$346$	0	0
$347$	−2838.04	−0.439060	−0.219530	−	0.975606i	$-0.570452\pi$
$347$	−2838.04	−0.439060	−0.219530	+	0.975606i	$0.570452\pi$
$348$	0	0
$349$	−2421.05	−0.371334	−0.185667	−	0.982613i	$-0.559445\pi$
$349$	−2421.05	−0.371334	−0.185667	+	0.982613i	$0.559445\pi$
$350$	0	0
$351$	−1629.54	−0.247802
$352$	0	0
$353$	3893.76	0.587093	0.293546	−	0.955945i	$-0.405164\pi$
$353$	3893.76	0.587093	0.293546	+	0.955945i	$0.405164\pi$
$354$	0	0
$355$	1089.76	0.162926
$356$	0	0
$357$	−1962.92	−0.291006
$358$	0	0
$359$	8210.64	1.20708	0.603539	−	0.797333i	$-0.293757\pi$
$359$	8210.64	1.20708	0.603539	+	0.797333i	$0.293757\pi$
$360$	0	0
$361$	1278.64	0.186418
$362$	0	0
$363$	381.987	0.0552317
$364$	0	0
$365$	3039.45	0.435869
$366$	0	0
$367$	11323.4	1.61056	0.805280	−	0.592894i	$-0.202015\pi$
$367$	11323.4	1.61056	0.805280	+	0.592894i	$0.202015\pi$
$368$	0	0
$369$	−7774.77	−1.09685
$370$	0	0
$371$	841.232	0.117721
$372$	0	0
$373$	−11500.2	−1.59641	−0.798204	−	0.602387i	$-0.794216\pi$
$373$	−11500.2	−1.59641	−0.798204	+	0.602387i	$0.794216\pi$
$374$	0	0
$375$	−1117.89	−0.153940
$376$	0	0
$377$	1347.16	0.184038
$378$	0	0
$379$	−2671.25	−0.362039	−0.181020	−	0.983480i	$-0.557940\pi$
$379$	−2671.25	−0.362039	−0.181020	+	0.983480i	$0.557940\pi$
$380$	0	0
$381$	9948.73	1.33777
$382$	0	0
$383$	−7751.15	−1.03411	−0.517057	−	0.855951i	$-0.672972\pi$
$383$	−7751.15	−1.03411	−0.517057	+	0.855951i	$0.672972\pi$
$384$	0	0
$385$	1256.24	0.166297
$386$	0	0
$387$	17159.3	2.25389
$388$	0	0
$389$	6127.19	0.798614	0.399307	−	0.916817i	$-0.369251\pi$
$389$	6127.19	0.798614	0.399307	+	0.916817i	$0.369251\pi$
$390$	0	0
$391$	3133.04	0.405229
$392$	0	0
$393$	19753.5	2.53545
$394$	0	0
$395$	5199.95	0.662374
$396$	0	0
$397$	−11593.1	−1.46560	−0.732798	−	0.680446i	$-0.761786\pi$
$397$	−11593.1	−1.46560	−0.732798	+	0.680446i	$0.761786\pi$
$398$	0	0
$399$	5647.22	0.708558
$400$	0	0
$401$	−12315.8	−1.53371	−0.766857	−	0.641818i	$-0.778181\pi$
$401$	−12315.8	−1.53371	−0.766857	+	0.641818i	$0.778181\pi$
$402$	0	0
$403$	−1150.95	−0.142265
$404$	0	0
$405$	3236.59	0.397105
$406$	0	0
$407$	−3434.68	−0.418306
$408$	0	0
$409$	9058.63	1.09516	0.547580	−	0.836753i	$-0.315549\pi$
$409$	9058.63	1.09516	0.547580	+	0.836753i	$0.315549\pi$
$410$	0	0
$411$	13954.0	1.67469
$412$	0	0
$413$	−4260.20	−0.507581
$414$	0	0
$415$	−3681.63	−0.435479
$416$	0	0
$417$	6084.24	0.714500
$418$	0	0
$419$	1882.54	0.219494	0.109747	−	0.993960i	$-0.464996\pi$
$419$	1882.54	0.219494	0.109747	+	0.993960i	$0.464996\pi$
$420$	0	0
$421$	7920.80	0.916950	0.458475	−	0.888707i	$-0.348396\pi$
$421$	7920.80	0.916950	0.458475	+	0.888707i	$0.348396\pi$
$422$	0	0
$423$	5364.61	0.616634
$424$	0	0
$425$	−783.896	−0.0894695
$426$	0	0
$427$	−5829.38	−0.660664
$428$	0	0
$429$	2251.41	0.253378
$430$	0	0
$431$	3734.03	0.417313	0.208657	−	0.977989i	$-0.433091\pi$
$431$	3734.03	0.417313	0.208657	+	0.977989i	$0.433091\pi$
$432$	0	0
$433$	−12255.6	−1.36020	−0.680102	−	0.733118i	$-0.738064\pi$
$433$	−12255.6	−1.36020	−0.680102	+	0.733118i	$0.738064\pi$
$434$	0	0
$435$	−8588.46	−0.946632
$436$	0	0
$437$	−9013.57	−0.986676
$438$	0	0
$439$	158.293	0.0172094	0.00860468	−	0.999963i	$-0.497261\pi$
$439$	158.293	0.0172094	0.00860468	+	0.999963i	$0.497261\pi$
$440$	0	0
$441$	2595.96	0.280311
$442$	0	0
$443$	189.022	0.0202725	0.0101363	−	0.999949i	$-0.496773\pi$
$443$	189.022	0.0202725	0.0101363	+	0.999949i	$0.496773\pi$
$444$	0	0
$445$	−3011.47	−0.320803
$446$	0	0
$447$	−23656.1	−2.50312
$448$	0	0
$449$	−13984.7	−1.46989	−0.734943	−	0.678129i	$-0.762791\pi$
$449$	−13984.7	−1.46989	−0.734943	+	0.678129i	$0.762791\pi$
$450$	0	0
$451$	5267.35	0.549956
$452$	0	0
$453$	16314.4	1.69209
$454$	0	0
$455$	−245.487	−0.0252937
$456$	0	0
$457$	7730.41	0.791277	0.395638	−	0.918406i	$-0.370523\pi$
$457$	7730.41	0.791277	0.395638	+	0.918406i	$0.370523\pi$
$458$	0	0
$459$	7284.89	0.740805
$460$	0	0
$461$	−15769.9	−1.59322	−0.796612	−	0.604491i	$-0.793377\pi$
$461$	−15769.9	−1.59322	−0.796612	+	0.604491i	$0.793377\pi$
$462$	0	0
$463$	5236.42	0.525609	0.262805	−	0.964849i	$-0.415353\pi$
$463$	5236.42	0.525609	0.262805	+	0.964849i	$0.415353\pi$
$464$	0	0
$465$	7337.58	0.731768
$466$	0	0
$467$	−10837.7	−1.07390	−0.536949	−	0.843614i	$-0.680423\pi$
$467$	−10837.7	−1.07390	−0.536949	+	0.843614i	$0.680423\pi$
$468$	0	0
$469$	4570.51	0.449992
$470$	0	0
$471$	18807.6	1.83993
$472$	0	0
$473$	−11625.3	−1.13009
$474$	0	0
$475$	2255.22	0.217846
$476$	0	0
$477$	−6366.77	−0.611141
$478$	0	0
$479$	−8122.11	−0.774757	−0.387379	−	0.921921i	$-0.626619\pi$
$479$	−8122.11	−0.774757	−0.387379	+	0.921921i	$0.626619\pi$
$480$	0	0
$481$	671.182	0.0636243
$482$	0	0
$483$	−6255.08	−0.589267
$484$	0	0
$485$	2736.83	0.256233
$486$	0	0
$487$	−8318.71	−0.774039	−0.387019	−	0.922072i	$-0.626495\pi$
$487$	−8318.71	−0.774039	−0.387019	+	0.922072i	$0.626495\pi$
$488$	0	0
$489$	−25232.8	−2.33347
$490$	0	0
$491$	−19238.0	−1.76823	−0.884114	−	0.467271i	$-0.845237\pi$
$491$	−19238.0	−1.76823	−0.884114	+	0.467271i	$0.845237\pi$
$492$	0	0
$493$	−6022.49	−0.550181
$494$	0	0
$495$	−9507.75	−0.863316
$496$	0	0
$497$	−1525.67	−0.137697
$498$	0	0
$499$	13070.7	1.17260	0.586299	−	0.810094i	$-0.300584\pi$
$499$	13070.7	1.17260	0.586299	+	0.810094i	$0.300584\pi$
$500$	0	0
$501$	2870.91	0.256014
$502$	0	0
$503$	3905.10	0.346162	0.173081	−	0.984908i	$-0.444628\pi$
$503$	3905.10	0.346162	0.173081	+	0.984908i	$0.444628\pi$
$504$	0	0
$505$	−3283.54	−0.289338
$506$	0	0
$507$	19208.0	1.68256
$508$	0	0
$509$	−19251.0	−1.67640	−0.838198	−	0.545366i	$-0.816391\pi$
$509$	−19251.0	−1.67640	−0.838198	+	0.545366i	$0.816391\pi$
$510$	0	0
$511$	−4255.23	−0.368376
$512$	0	0
$513$	−20958.2	−1.80376
$514$	0	0
$515$	−465.966	−0.0398697
$516$	0	0
$517$	−3634.48	−0.309177
$518$	0	0
$519$	−1990.34	−0.168335
$520$	0	0
$521$	−13343.1	−1.12202	−0.561009	−	0.827809i	$-0.689587\pi$
$521$	−13343.1	−1.12202	−0.561009	+	0.827809i	$0.689587\pi$
$522$	0	0
$523$	2572.34	0.215068	0.107534	−	0.994201i	$-0.465705\pi$
$523$	2572.34	0.215068	0.107534	+	0.994201i	$0.465705\pi$
$524$	0	0
$525$	1565.04	0.130103
$526$	0	0
$527$	5145.34	0.425302
$528$	0	0
$529$	−2183.22	−0.179438
$530$	0	0
$531$	32242.9	2.63507
$532$	0	0
$533$	−1029.31	−0.0836482
$534$	0	0
$535$	−2915.46	−0.235601
$536$	0	0
$537$	5066.16	0.407115
$538$	0	0
$539$	−1758.74	−0.140546
$540$	0	0
$541$	−17980.2	−1.42889	−0.714444	−	0.699692i	$-0.753320\pi$
$541$	−17980.2	−1.42889	−0.714444	+	0.699692i	$0.753320\pi$
$542$	0	0
$543$	−27611.7	−2.18219
$544$	0	0
$545$	−5596.69	−0.439883
$546$	0	0
$547$	13099.0	1.02390	0.511951	−	0.859014i	$-0.328923\pi$
$547$	13099.0	1.02390	0.511951	+	0.859014i	$0.328923\pi$
$548$	0	0
$549$	44119.0	3.42979
$550$	0	0
$551$	17326.4	1.33961
$552$	0	0
$553$	−7279.93	−0.559808
$554$	0	0
$555$	−4278.94	−0.327263
$556$	0	0
$557$	−12194.3	−0.927627	−0.463814	−	0.885933i	$-0.653519\pi$
$557$	−12194.3	−0.927627	−0.463814	+	0.885933i	$0.653519\pi$
$558$	0	0
$559$	2271.74	0.171886
$560$	0	0
$561$	−10065.0	−0.757474
$562$	0	0
$563$	10399.7	0.778499	0.389249	−	0.921132i	$-0.372734\pi$
$563$	10399.7	0.778499	0.389249	+	0.921132i	$0.372734\pi$
$564$	0	0
$565$	−9797.75	−0.729548
$566$	0	0
$567$	−4531.23	−0.335615
$568$	0	0
$569$	−5456.69	−0.402033	−0.201016	−	0.979588i	$-0.564424\pi$
$569$	−5456.69	−0.402033	−0.201016	+	0.979588i	$0.564424\pi$
$570$	0	0
$571$	17329.5	1.27008	0.635040	−	0.772479i	$-0.280984\pi$
$571$	17329.5	1.27008	0.635040	+	0.772479i	$0.280984\pi$
$572$	0	0
$573$	6231.28	0.454303
$574$	0	0
$575$	−2497.97	−0.181170
$576$	0	0
$577$	−20784.3	−1.49959	−0.749793	−	0.661673i	$-0.769847\pi$
$577$	−20784.3	−1.49959	−0.749793	+	0.661673i	$0.769847\pi$
$578$	0	0
$579$	45355.3	3.25544
$580$	0	0
$581$	5154.28	0.368047
$582$	0	0
$583$	4313.44	0.306423
$584$	0	0
$585$	1857.94	0.131310
$586$	0	0
$587$	−9493.79	−0.667548	−0.333774	−	0.942653i	$-0.608322\pi$
$587$	−9493.79	−0.667548	−0.333774	+	0.942653i	$0.608322\pi$
$588$	0	0
$589$	−14802.8	−1.03555
$590$	0	0
$591$	21616.7	1.50456
$592$	0	0
$593$	22130.7	1.53254	0.766272	−	0.642516i	$-0.222109\pi$
$593$	22130.7	1.53254	0.766272	+	0.642516i	$0.222109\pi$
$594$	0	0
$595$	1097.45	0.0756155
$596$	0	0
$597$	−14791.0	−1.01400
$598$	0	0
$599$	176.484	0.0120383	0.00601914	−	0.999982i	$-0.498084\pi$
$599$	176.484	0.0120383	0.00601914	+	0.999982i	$0.498084\pi$
$600$	0	0
$601$	17561.7	1.19194	0.595970	−	0.803007i	$-0.296768\pi$
$601$	17561.7	1.19194	0.595970	+	0.803007i	$0.296768\pi$
$602$	0	0
$603$	−34591.4	−2.33610
$604$	0	0
$605$	−213.566	−0.0143515
$606$	0	0
$607$	15946.0	1.06627	0.533137	−	0.846029i	$-0.321013\pi$
$607$	15946.0	1.06627	0.533137	+	0.846029i	$0.321013\pi$
$608$	0	0
$609$	12023.8	0.800050
$610$	0	0
$611$	710.228	0.0470258
$612$	0	0
$613$	21264.3	1.40107	0.700537	−	0.713616i	$-0.252944\pi$
$613$	21264.3	1.40107	0.700537	+	0.713616i	$0.252944\pi$
$614$	0	0
$615$	6562.11	0.430260
$616$	0	0
$617$	23865.8	1.55722	0.778608	−	0.627510i	$-0.215926\pi$
$617$	23865.8	1.55722	0.778608	+	0.627510i	$0.215926\pi$
$618$	0	0
$619$	10066.5	0.653648	0.326824	−	0.945085i	$-0.394022\pi$
$619$	10066.5	0.653648	0.326824	+	0.945085i	$0.394022\pi$
$620$	0	0
$621$	23214.1	1.50008
$622$	0	0
$623$	4216.06	0.271128
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	28956.3	1.84434
$628$	0	0
$629$	−3000.53	−0.190205
$630$	0	0
$631$	21450.8	1.35332	0.676658	−	0.736297i	$-0.263427\pi$
$631$	21450.8	1.35332	0.676658	+	0.736297i	$0.263427\pi$
$632$	0	0
$633$	8213.13	0.515707
$634$	0	0
$635$	−5562.25	−0.347608
$636$	0	0
$637$	343.682	0.0213771
$638$	0	0
$639$	11546.8	0.714845
$640$	0	0
$641$	763.570	0.0470502	0.0235251	−	0.999723i	$-0.492511\pi$
$641$	763.570	0.0470502	0.0235251	+	0.999723i	$0.492511\pi$
$642$	0	0
$643$	5687.92	0.348848	0.174424	−	0.984671i	$-0.444194\pi$
$643$	5687.92	0.348848	0.174424	+	0.984671i	$0.444194\pi$
$644$	0	0
$645$	−14482.9	−0.884129
$646$	0	0
$647$	−11490.6	−0.698209	−0.349104	−	0.937084i	$-0.613514\pi$
$647$	−11490.6	−0.698209	−0.349104	+	0.937084i	$0.613514\pi$
$648$	0	0
$649$	−21844.3	−1.32121
$650$	0	0
$651$	−10272.6	−0.618457
$652$	0	0
$653$	−21288.9	−1.27580	−0.637901	−	0.770119i	$-0.720197\pi$
$653$	−21288.9	−1.27580	−0.637901	+	0.770119i	$0.720197\pi$
$654$	0	0
$655$	−11044.0	−0.658817
$656$	0	0
$657$	32205.2	1.91240
$658$	0	0
$659$	−11443.8	−0.676460	−0.338230	−	0.941063i	$-0.609828\pi$
$659$	−11443.8	−0.676460	−0.338230	+	0.941063i	$0.609828\pi$
$660$	0	0
$661$	−9948.40	−0.585398	−0.292699	−	0.956205i	$-0.594553\pi$
$661$	−9948.40	−0.585398	−0.292699	+	0.956205i	$0.594553\pi$
$662$	0	0
$663$	1966.83	0.115212
$664$	0	0
$665$	−3157.31	−0.184113
$666$	0	0
$667$	−19191.3	−1.11408
$668$	0	0
$669$	56433.5	3.26135
$670$	0	0
$671$	−29890.3	−1.71968
$672$	0	0
$673$	−28899.0	−1.65524	−0.827618	−	0.561292i	$-0.810304\pi$
$673$	−28899.0	−1.65524	−0.827618	+	0.561292i	$0.810304\pi$
$674$	0	0
$675$	−5808.24	−0.331199
$676$	0	0
$677$	17095.8	0.970524	0.485262	−	0.874369i	$-0.338724\pi$
$677$	17095.8	0.970524	0.485262	+	0.874369i	$0.338724\pi$
$678$	0	0
$679$	−3831.56	−0.216557
$680$	0	0
$681$	−52828.2	−2.97266
$682$	0	0
$683$	−607.606	−0.0340401	−0.0170201	−	0.999855i	$-0.505418\pi$
$683$	−607.606	−0.0340401	−0.0170201	+	0.999855i	$0.505418\pi$
$684$	0	0
$685$	−7801.54	−0.435156
$686$	0	0
$687$	45115.1	2.50546
$688$	0	0
$689$	−842.905	−0.0466069
$690$	0	0
$691$	−19169.7	−1.05535	−0.527676	−	0.849446i	$-0.676936\pi$
$691$	−19169.7	−1.05535	−0.527676	+	0.849446i	$0.676936\pi$
$692$	0	0
$693$	13310.8	0.729635
$694$	0	0
$695$	−3401.65	−0.185657
$696$	0	0
$697$	4601.56	0.250066
$698$	0	0
$699$	25776.6	1.39479
$700$	0	0
$701$	−14769.1	−0.795753	−0.397876	−	0.917439i	$-0.630253\pi$
$701$	−14769.1	−0.795753	−0.397876	+	0.917439i	$0.630253\pi$
$702$	0	0
$703$	8632.34	0.463122
$704$	0	0
$705$	−4527.87	−0.241886
$706$	0	0
$707$	4596.96	0.244535
$708$	0	0
$709$	16094.9	0.852549	0.426274	−	0.904594i	$-0.359826\pi$
$709$	16094.9	0.852549	0.426274	+	0.904594i	$0.359826\pi$
$710$	0	0
$711$	55097.3	2.90620
$712$	0	0
$713$	16396.2	0.861209
$714$	0	0
$715$	−1258.74	−0.0658382
$716$	0	0
$717$	42448.1	2.21095
$718$	0	0
$719$	10165.8	0.527286	0.263643	−	0.964620i	$-0.415076\pi$
$719$	10165.8	0.527286	0.263643	+	0.964620i	$0.415076\pi$
$720$	0	0
$721$	652.352	0.0336961
$722$	0	0
$723$	−41341.4	−2.12656
$724$	0	0
$725$	4801.73	0.245975
$726$	0	0
$727$	−33418.7	−1.70486	−0.852429	−	0.522842i	$-0.824872\pi$
$727$	−33418.7	−1.70486	−0.852429	+	0.522842i	$0.824872\pi$
$728$	0	0
$729$	−21805.0	−1.10781
$730$	0	0
$731$	−10155.8	−0.513854
$732$	0	0
$733$	31249.5	1.57466	0.787331	−	0.616530i	$-0.211462\pi$
$733$	31249.5	1.57466	0.787331	+	0.616530i	$0.211462\pi$
$734$	0	0
$735$	−2191.05	−0.109957
$736$	0	0
$737$	23435.4	1.17131
$738$	0	0
$739$	−15971.2	−0.795009	−0.397504	−	0.917600i	$-0.630124\pi$
$739$	−15971.2	−0.795009	−0.397504	+	0.917600i	$0.630124\pi$
$740$	0	0
$741$	−5658.45	−0.280524
$742$	0	0
$743$	−4034.27	−0.199197	−0.0995983	−	0.995028i	$-0.531756\pi$
$743$	−4034.27	−0.199197	−0.0995983	+	0.995028i	$0.531756\pi$
$744$	0	0
$745$	13225.9	0.650416
$746$	0	0
$747$	−39009.6	−1.91069
$748$	0	0
$749$	4081.65	0.199119
$750$	0	0
$751$	8130.93	0.395076	0.197538	−	0.980295i	$-0.436705\pi$
$751$	8130.93	0.395076	0.197538	+	0.980295i	$0.436705\pi$
$752$	0	0
$753$	−59937.0	−2.90070
$754$	0	0
$755$	−9121.25	−0.439677
$756$	0	0
$757$	−23290.6	−1.11824	−0.559122	−	0.829085i	$-0.688862\pi$
$757$	−23290.6	−1.11824	−0.559122	+	0.829085i	$0.688862\pi$
$758$	0	0
$759$	−32073.1	−1.53383
$760$	0	0
$761$	−7215.05	−0.343686	−0.171843	−	0.985124i	$-0.554972\pi$
$761$	−7215.05	−0.343686	−0.171843	+	0.985124i	$0.554972\pi$
$762$	0	0
$763$	7835.37	0.371769
$764$	0	0
$765$	−8305.96	−0.392552
$766$	0	0
$767$	4268.68	0.200956
$768$	0	0
$769$	30628.6	1.43627	0.718137	−	0.695902i	$-0.244995\pi$
$769$	30628.6	1.43627	0.718137	+	0.695902i	$0.244995\pi$
$770$	0	0
$771$	−29213.7	−1.36460
$772$	0	0
$773$	−33654.0	−1.56591	−0.782957	−	0.622075i	$-0.786290\pi$
$773$	−33654.0	−1.56591	−0.782957	+	0.622075i	$0.786290\pi$
$774$	0	0
$775$	−4102.38	−0.190144
$776$	0	0
$777$	5990.52	0.276588
$778$	0	0
$779$	−13238.4	−0.608877
$780$	0	0
$781$	−7822.91	−0.358420
$782$	0	0
$783$	−44623.4	−2.03667
$784$	0	0
$785$	−10515.2	−0.478092
$786$	0	0
$787$	22411.6	1.01510	0.507552	−	0.861621i	$-0.330551\pi$
$787$	22411.6	1.01510	0.507552	+	0.861621i	$0.330551\pi$
$788$	0	0
$789$	59364.3	2.67861
$790$	0	0
$791$	13716.9	0.616581
$792$	0	0
$793$	5840.98	0.261563
$794$	0	0
$795$	5373.72	0.239731
$796$	0	0
$797$	4088.30	0.181700	0.0908501	−	0.995865i	$-0.471042\pi$
$797$	4088.30	0.181700	0.0908501	+	0.995865i	$0.471042\pi$
$798$	0	0
$799$	−3175.08	−0.140584
$800$	0	0
$801$	−31908.7	−1.40754
$802$	0	0
$803$	−21818.8	−0.958866
$804$	0	0
$805$	3497.16	0.153116
$806$	0	0
$807$	13988.5	0.610184
$808$	0	0
$809$	3262.90	0.141802	0.0709009	−	0.997483i	$-0.477413\pi$
$809$	3262.90	0.141802	0.0709009	+	0.997483i	$0.477413\pi$
$810$	0	0
$811$	27831.2	1.20504	0.602519	−	0.798104i	$-0.294164\pi$
$811$	27831.2	1.20504	0.602519	+	0.798104i	$0.294164\pi$
$812$	0	0
$813$	29717.6	1.28197
$814$	0	0
$815$	14107.5	0.606335
$816$	0	0
$817$	29217.8	1.25116
$818$	0	0
$819$	−2601.12	−0.110977
$820$	0	0
$821$	8701.78	0.369908	0.184954	−	0.982747i	$-0.440786\pi$
$821$	8701.78	0.369908	0.184954	+	0.982747i	$0.440786\pi$
$822$	0	0
$823$	−31027.9	−1.31418	−0.657088	−	0.753814i	$-0.728212\pi$
$823$	−31027.9	−1.31418	−0.657088	+	0.753814i	$0.728212\pi$
$824$	0	0
$825$	8024.79	0.338651
$826$	0	0
$827$	18262.0	0.767874	0.383937	−	0.923359i	$-0.374568\pi$
$827$	18262.0	0.767874	0.383937	+	0.923359i	$0.374568\pi$
$828$	0	0
$829$	15237.0	0.638364	0.319182	−	0.947693i	$-0.396592\pi$
$829$	15237.0	0.638364	0.319182	+	0.947693i	$0.396592\pi$
$830$	0	0
$831$	41463.2	1.73086
$832$	0	0
$833$	−1536.44	−0.0639068
$834$	0	0
$835$	−1605.10	−0.0665232
$836$	0	0
$837$	38124.1	1.57439
$838$	0	0
$839$	−26690.2	−1.09827	−0.549134	−	0.835734i	$-0.685042\pi$
$839$	−26690.2	−1.09827	−0.549134	+	0.835734i	$0.685042\pi$
$840$	0	0
$841$	12501.6	0.512592
$842$	0	0
$843$	−65035.8	−2.65712
$844$	0	0
$845$	−10739.0	−0.437200
$846$	0	0
$847$	298.992	0.0121293
$848$	0	0
$849$	31482.9	1.27266
$850$	0	0
$851$	−9561.52	−0.385152
$852$	0	0
$853$	8719.68	0.350007	0.175004	−	0.984568i	$-0.444006\pi$
$853$	8719.68	0.350007	0.175004	+	0.984568i	$0.444006\pi$
$854$	0	0
$855$	23895.7	0.955810
$856$	0	0
$857$	49470.6	1.97186	0.985930	−	0.167160i	$-0.0534597\pi$
$857$	49470.6	1.97186	0.985930	+	0.167160i	$0.0534597\pi$
$858$	0	0
$859$	−19883.7	−0.789782	−0.394891	−	0.918728i	$-0.629218\pi$
$859$	−19883.7	−0.789782	−0.394891	+	0.918728i	$0.629218\pi$
$860$	0	0
$861$	−9186.95	−0.363636
$862$	0	0
$863$	−19430.2	−0.766409	−0.383205	−	0.923664i	$-0.625180\pi$
$863$	−19430.2	−0.766409	−0.383205	+	0.923664i	$0.625180\pi$
$864$	0	0
$865$	1112.78	0.0437407
$866$	0	0
$867$	35144.6	1.37667
$868$	0	0
$869$	−37328.0	−1.45715
$870$	0	0
$871$	−4579.60	−0.178156
$872$	0	0
$873$	28998.8	1.12424
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−21960.6	−0.845560	−0.422780	−	0.906232i	$-0.638946\pi$
$877$	−21960.6	−0.845560	−0.422780	+	0.906232i	$0.638946\pi$
$878$	0	0
$879$	−27448.3	−1.05325
$880$	0	0
$881$	−14525.4	−0.555473	−0.277737	−	0.960657i	$-0.589584\pi$
$881$	−14525.4	−0.555473	−0.277737	+	0.960657i	$0.589584\pi$
$882$	0	0
$883$	7807.00	0.297538	0.148769	−	0.988872i	$-0.452469\pi$
$883$	7807.00	0.297538	0.148769	+	0.988872i	$0.452469\pi$
$884$	0	0
$885$	−27213.8	−1.03365
$886$	0	0
$887$	−20837.4	−0.788785	−0.394393	−	0.918942i	$-0.629045\pi$
$887$	−20837.4	−0.788785	−0.394393	+	0.918942i	$0.629045\pi$
$888$	0	0
$889$	7787.15	0.293782
$890$	0	0
$891$	−23234.0	−0.873590
$892$	0	0
$893$	9134.52	0.342301
$894$	0	0
$895$	−2832.44	−0.105786
$896$	0	0
$897$	6267.52	0.233296
$898$	0	0
$899$	−31517.6	−1.16927
$900$	0	0
$901$	3768.22	0.139331
$902$	0	0
$903$	20276.0	0.747225
$904$	0	0
$905$	15437.4	0.567025
$906$	0	0
$907$	52475.2	1.92107	0.960534	−	0.278162i	$-0.0897251\pi$
$907$	52475.2	1.92107	0.960534	+	0.278162i	$0.0897251\pi$
$908$	0	0
$909$	−34791.6	−1.26949
$910$	0	0
$911$	−8902.53	−0.323769	−0.161885	−	0.986810i	$-0.551757\pi$
$911$	−8902.53	−0.323769	−0.161885	+	0.986810i	$0.551757\pi$
$912$	0	0
$913$	26428.7	0.958009
$914$	0	0
$915$	−37237.6	−1.34540
$916$	0	0
$917$	15461.6	0.556802
$918$	0	0
$919$	46228.4	1.65934	0.829670	−	0.558255i	$-0.188529\pi$
$919$	46228.4	1.65934	0.829670	+	0.558255i	$0.188529\pi$
$920$	0	0
$921$	−13761.7	−0.492360
$922$	0	0
$923$	1528.70	0.0545156
$924$	0	0
$925$	2392.32	0.0850368
$926$	0	0
$927$	−4937.25	−0.174931
$928$	0	0
$929$	−37088.2	−1.30982	−0.654911	−	0.755706i	$-0.727294\pi$
$929$	−37088.2	−1.30982	−0.654911	+	0.755706i	$0.727294\pi$
$930$	0	0
$931$	4420.23	0.155604
$932$	0	0
$933$	−76107.2	−2.67057
$934$	0	0
$935$	5627.23	0.196824
$936$	0	0
$937$	−22304.9	−0.777662	−0.388831	−	0.921309i	$-0.627121\pi$
$937$	−22304.9	−0.777662	−0.388831	+	0.921309i	$0.627121\pi$
$938$	0	0
$939$	−46499.7	−1.61604
$940$	0	0
$941$	27726.8	0.960540	0.480270	−	0.877121i	$-0.340539\pi$
$941$	27726.8	0.960540	0.480270	+	0.877121i	$0.340539\pi$
$942$	0	0
$943$	14663.4	0.506368
$944$	0	0
$945$	8131.54	0.279914
$946$	0	0
$947$	9265.39	0.317935	0.158968	−	0.987284i	$-0.449183\pi$
$947$	9265.39	0.317935	0.158968	+	0.987284i	$0.449183\pi$
$948$	0	0
$949$	4263.70	0.145843
$950$	0	0
$951$	44663.4	1.52293
$952$	0	0
$953$	−7247.40	−0.246344	−0.123172	−	0.992385i	$-0.539307\pi$
$953$	−7247.40	−0.246344	−0.123172	+	0.992385i	$0.539307\pi$
$954$	0	0
$955$	−3483.86	−0.118047
$956$	0	0
$957$	61652.6	2.08249
$958$	0	0
$959$	10922.2	0.367774
$960$	0	0
$961$	−2863.82	−0.0961304
$962$	0	0
$963$	−30891.5	−1.03371
$964$	0	0
$965$	−25357.8	−0.845902
$966$	0	0
$967$	46817.8	1.55694	0.778469	−	0.627683i	$-0.215996\pi$
$967$	46817.8	1.55694	0.778469	+	0.627683i	$0.215996\pi$
$968$	0	0
$969$	25296.2	0.838628
$970$	0	0
$971$	−18137.9	−0.599457	−0.299729	−	0.954024i	$-0.596896\pi$
$971$	−18137.9	−0.599457	−0.299729	+	0.954024i	$0.596896\pi$
$972$	0	0
$973$	4762.30	0.156909
$974$	0	0
$975$	−1568.15	−0.0515088
$976$	0	0
$977$	12235.7	0.400671	0.200336	−	0.979727i	$-0.435797\pi$
$977$	12235.7	0.400671	0.200336	+	0.979727i	$0.435797\pi$
$978$	0	0
$979$	21618.0	0.705733
$980$	0	0
$981$	−59301.1	−1.93001
$982$	0	0
$983$	−37510.9	−1.21710	−0.608551	−	0.793515i	$-0.708249\pi$
$983$	−37510.9	−1.21710	−0.608551	+	0.793515i	$0.708249\pi$
$984$	0	0
$985$	−12085.7	−0.390947
$986$	0	0
$987$	6339.01	0.204431
$988$	0	0
$989$	−32362.7	−1.04052
$990$	0	0
$991$	−28734.1	−0.921060	−0.460530	−	0.887644i	$-0.652341\pi$
$991$	−28734.1	−0.921060	−0.460530	+	0.887644i	$0.652341\pi$
$992$	0	0
$993$	−4059.11	−0.129720
$994$	0	0
$995$	8269.53	0.263479
$996$	0	0
$997$	30360.9	0.964434	0.482217	−	0.876052i	$-0.339832\pi$
$997$	30360.9	0.964434	0.482217	+	0.876052i	$0.339832\pi$
$998$	0	0
$999$	−22232.3	−0.704102

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.4.a.g.1.1	✓	4
4.3	odd	2		1120.4.a.o.1.4	yes	4
8.3	odd	2		2240.4.a.cb.1.1		4
8.5	even	2		2240.4.a.cj.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.4.a.g.1.1	✓	4	1.1	even	1	trivial
1120.4.a.o.1.4	yes	4	4.3	odd	2
2240.4.a.cb.1.1		4	8.3	odd	2
2240.4.a.cj.1.4		4	8.5	even	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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1120.a (trivial)

\(f(q)\)	\(=\)	\(q-8.94308 q^{3} +5.00000 q^{5} -7.00000 q^{7} +52.9787 q^{9} -35.8927 q^{11} +7.01392 q^{13} -44.7154 q^{15} -31.3558 q^{17} +90.2089 q^{19} +62.6016 q^{21} -99.9189 q^{23} +25.0000 q^{25} -232.330 q^{27} +192.069 q^{29} -164.095 q^{31} +320.991 q^{33} -35.0000 q^{35} +95.6928 q^{37} -62.7261 q^{39} -146.753 q^{41} +323.890 q^{43} +264.894 q^{45} +101.260 q^{47} +49.0000 q^{49} +280.418 q^{51} -120.176 q^{53} -179.464 q^{55} -806.745 q^{57} +608.600 q^{59} +832.769 q^{61} -370.851 q^{63} +35.0696 q^{65} -652.930 q^{67} +893.583 q^{69} +217.953 q^{71} +607.890 q^{73} -223.577 q^{75} +251.249 q^{77} +1039.99 q^{79} +647.318 q^{81} -736.325 q^{83} -156.779 q^{85} -1717.69 q^{87} -602.294 q^{89} -49.0975 q^{91} +1467.52 q^{93} +451.044 q^{95} +547.366 q^{97} -1901.55 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 7 q^{3} + 20 q^{5} - 28 q^{7} + 47 q^{9} - 91 q^{11} + 113 q^{13} - 35 q^{15} + 3 q^{17} - 112 q^{19} + 49 q^{21} - 168 q^{23} + 100 q^{25} - 133 q^{27} + 31 q^{29} - 126 q^{31} - 33 q^{33} - 140 q^{35}+ \cdots - 1050 q^{99}+O(q^{100}) \) 4 * q - 7 * q^3 + 20 * q^5 - 28 * q^7 + 47 * q^9 - 91 * q^11 + 113 * q^13 - 35 * q^15 + 3 * q^17 - 112 * q^19 + 49 * q^21 - 168 * q^23 + 100 * q^25 - 133 * q^27 + 31 * q^29 - 126 * q^31 - 33 * q^33 - 140 * q^35 + 186 * q^37 + 161 * q^39 - 8 * q^41 - 28 * q^43 + 235 * q^45 - 231 * q^47 + 196 * q^49 - 525 * q^51 + 110 * q^53 - 455 * q^55 - 728 * q^57 - 196 * q^59 - 432 * q^61 - 329 * q^63 + 565 * q^65 + 336 * q^67 - 480 * q^69 - 1708 * q^71 + 712 * q^73 - 175 * q^75 + 637 * q^77 + 455 * q^79 - 180 * q^81 - 1512 * q^83 + 15 * q^85 - 2541 * q^87 + 732 * q^89 - 791 * q^91 + 1430 * q^93 - 560 * q^95 - 589 * q^97 - 1050 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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