LMFDB - Embedded newform 2240.4.a.cj.1.4

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:	4.4.505876.1
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:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.15507$ 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)

$f(q)$	$=$	$q+8.94308 q^{3} -5.00000 q^{5} -7.00000 q^{7} +52.9787 q^{9} +35.8927 q^{11} -7.01392 q^{13} -44.7154 q^{15} -31.3558 q^{17} -90.2089 q^{19} -62.6016 q^{21} -99.9189 q^{23} +25.0000 q^{25} +232.330 q^{27} -192.069 q^{29} -164.095 q^{31} +320.991 q^{33} +35.0000 q^{35} -95.6928 q^{37} -62.7261 q^{39} -146.753 q^{41} -323.890 q^{43} -264.894 q^{45} +101.260 q^{47} +49.0000 q^{49} -280.418 q^{51} +120.176 q^{53} -179.464 q^{55} -806.745 q^{57} -608.600 q^{59} -832.769 q^{61} -370.851 q^{63} +35.0696 q^{65} +652.930 q^{67} -893.583 q^{69} +217.953 q^{71} +607.890 q^{73} +223.577 q^{75} -251.249 q^{77} +1039.99 q^{79} +647.318 q^{81} +736.325 q^{83} +156.779 q^{85} -1717.69 q^{87} -602.294 q^{89} +49.0975 q^{91} -1467.52 q^{93} +451.044 q^{95} +547.366 q^{97} +1901.55 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.170121\pi$
$3$	8.94308	1.72110	0.860548	+	0.509369i	$0.170121\pi$
$4$	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	$-0.336298\pi$
$11$	35.8927	0.983824	0.491912	+	0.870645i	$0.336298\pi$
$12$	0	0
$13$	−7.01392	−0.149639	−0.0748197	−	0.997197i	$-0.523838\pi$
$13$	−7.01392	−0.149639	−0.0748197	+	0.997197i	$0.523838\pi$
$14$	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	$-0.683324\pi$
$19$	−90.2089	−1.08923	−0.544614	+	0.838687i	$0.683324\pi$
$20$	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	$-0.710819\pi$
$29$	−192.069	−1.22987	−0.614937	+	0.788576i	$0.710819\pi$
$30$	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	$-0.568191\pi$
$37$	−95.6928	−0.425184	−0.212592	+	0.977141i	$0.568191\pi$
$38$	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	$-0.694739\pi$
$43$	−323.890	−1.14867	−0.574335	+	0.818621i	$0.694739\pi$
$44$	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	$-0.450227\pi$
$53$	120.176	0.311461	0.155731	+	0.987800i	$0.450227\pi$
$54$	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	$-0.734335\pi$
$59$	−608.600	−1.34293	−0.671466	+	0.741035i	$0.734335\pi$
$60$	0	0
$61$	−832.769	−1.74795	−0.873977	−	0.485968i	$-0.838467\pi$
$61$	−832.769	−1.74795	−0.873977	+	0.485968i	$0.838467\pi$
$62$	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	$-0.297040\pi$
$67$	652.930	1.19057	0.595284	+	0.803515i	$0.297040\pi$
$68$	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	$-0.338135\pi$
$83$	736.325	0.973761	0.486881	+	0.873469i	$0.338135\pi$
$84$	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	$-0.395144\pi$
$101$	656.708	0.646979	0.323490	+	0.946232i	$0.395144\pi$
$102$	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	$-0.415153\pi$
$107$	583.092	0.526819	0.263410	+	0.964684i	$0.415153\pi$
$108$	0	0
$109$	1119.34	0.983608	0.491804	−	0.870706i	$-0.336338\pi$
$109$	1119.34	0.983608	0.491804	+	0.870706i	$0.336338\pi$
$110$	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	$-0.236439\pi$
$131$	2208.80	1.47316	0.736580	+	0.676350i	$0.236439\pi$
$132$	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	$-0.433444\pi$
$139$	680.329	0.415142	0.207571	+	0.978220i	$0.433444\pi$
$140$	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	$-0.759173\pi$
$149$	−2645.18	−1.45437	−0.727187	+	0.686439i	$0.759173\pi$
$150$	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	$-0.320491\pi$
$157$	2103.03	1.06905	0.534523	+	0.845154i	$0.320491\pi$
$158$	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	$-0.737111\pi$
$163$	−2821.49	−1.35581	−0.677903	+	0.735151i	$0.737111\pi$
$164$	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	$-0.515573\pi$
$173$	−222.556	−0.0978071	−0.0489036	+	0.998804i	$0.515573\pi$
$174$	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	$-0.462265\pi$
$179$	566.489	0.236544	0.118272	+	0.992981i	$0.462265\pi$
$180$	0	0
$181$	−3087.49	−1.26791	−0.633954	−	0.773371i	$-0.718569\pi$
$181$	−3087.49	−1.26791	−0.633954	+	0.773371i	$0.718569\pi$
$182$	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	$-0.356008\pi$
$197$	2417.15	0.874185	0.437093	+	0.899417i	$0.356008\pi$
$198$	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	$-0.452131\pi$
$211$	918.379	0.299639	0.149819	+	0.988713i	$0.452131\pi$
$212$	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	$-0.831792\pi$
$227$	−5907.16	−1.72719	−0.863594	+	0.504187i	$0.831792\pi$
$228$	0	0
$229$	5044.70	1.45573	0.727867	−	0.685719i	$-0.240512\pi$
$229$	5044.70	1.45573	0.727867	+	0.685719i	$0.240512\pi$
$230$	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	$-0.819028\pi$
$251$	−6702.05	−1.68538	−0.842688	+	0.538402i	$0.819028\pi$
$252$	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	$-0.443275\pi$
$269$	1564.17	0.354532	0.177266	+	0.984163i	$0.443275\pi$
$270$	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	$-0.332290\pi$
$277$	4636.34	1.00567	0.502835	+	0.864382i	$0.332290\pi$
$278$	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	$-0.379452\pi$
$283$	3520.36	0.739448	0.369724	+	0.929142i	$0.379452\pi$
$284$	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	$-0.598985\pi$
$293$	−3069.22	−0.611965	−0.305983	+	0.952037i	$0.598985\pi$
$294$	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	$-0.545687\pi$
$307$	−1538.81	−0.286073	−0.143037	+	0.989717i	$0.545687\pi$
$308$	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	$-0.354116\pi$
$317$	4994.18	0.884862	0.442431	+	0.896802i	$0.354116\pi$
$318$	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	$-0.511998\pi$
$331$	−453.883	−0.0753706	−0.0376853	+	0.999290i	$0.511998\pi$
$332$	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	$-0.429548\pi$
$347$	2838.04	0.439060	0.219530	+	0.975606i	$0.429548\pi$
$348$	0	0
$349$	2421.05	0.371334	0.185667	−	0.982613i	$-0.440555\pi$
$349$	2421.05	0.371334	0.185667	+	0.982613i	$0.440555\pi$
$350$	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	$-0.205784\pi$
$373$	11500.2	1.59641	0.798204	+	0.602387i	$0.205784\pi$
$374$	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	$-0.442060\pi$
$379$	2671.25	0.362039	0.181020	+	0.983480i	$0.442060\pi$
$380$	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	$-0.630749\pi$
$389$	−6127.19	−0.798614	−0.399307	+	0.916817i	$0.630749\pi$
$390$	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	$-0.238214\pi$
$397$	11593.1	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$398$	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	$-0.535004\pi$
$419$	−1882.54	−0.219494	−0.109747	+	0.993960i	$0.535004\pi$
$420$	0	0
$421$	−7920.80	−0.916950	−0.458475	−	0.888707i	$-0.651604\pi$
$421$	−7920.80	−0.916950	−0.458475	+	0.888707i	$0.651604\pi$
$422$	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	$-0.503227\pi$
$443$	−189.022	−0.0202725	−0.0101363	+	0.999949i	$0.503227\pi$
$444$	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	$-0.206623\pi$
$461$	15769.9	1.59322	0.796612	+	0.604491i	$0.206623\pi$
$462$	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	$-0.319577\pi$
$467$	10837.7	1.07390	0.536949	+	0.843614i	$0.319577\pi$
$468$	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	$-0.154763\pi$
$491$	19238.0	1.76823	0.884114	+	0.467271i	$0.154763\pi$
$492$	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	$-0.699416\pi$
$499$	−13070.7	−1.17260	−0.586299	+	0.810094i	$0.699416\pi$
$500$	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	$-0.183609\pi$
$509$	19251.0	1.67640	0.838198	+	0.545366i	$0.183609\pi$
$510$	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	$-0.534295\pi$
$523$	−2572.34	−0.215068	−0.107534	+	0.994201i	$0.534295\pi$
$524$	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	$-0.246680\pi$
$541$	17980.2	1.42889	0.714444	+	0.699692i	$0.246680\pi$
$542$	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	$-0.671077\pi$
$547$	−13099.0	−1.02390	−0.511951	+	0.859014i	$0.671077\pi$
$548$	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	$-0.346481\pi$
$557$	12194.3	0.927627	0.463814	+	0.885933i	$0.346481\pi$
$558$	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	$-0.627266\pi$
$563$	−10399.7	−0.778499	−0.389249	+	0.921132i	$0.627266\pi$
$564$	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	$-0.719016\pi$
$571$	−17329.5	−1.27008	−0.635040	+	0.772479i	$0.719016\pi$
$572$	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	$-0.391678\pi$
$587$	9493.79	0.667548	0.333774	+	0.942653i	$0.391678\pi$
$588$	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	$-0.747056\pi$
$613$	−21264.3	−1.40107	−0.700537	+	0.713616i	$0.747056\pi$
$614$	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	$-0.605978\pi$
$619$	−10066.5	−0.653648	−0.326824	+	0.945085i	$0.605978\pi$
$620$	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	$-0.555806\pi$
$643$	−5687.92	−0.348848	−0.174424	+	0.984671i	$0.555806\pi$
$644$	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	$-0.279803\pi$
$653$	21288.9	1.27580	0.637901	+	0.770119i	$0.279803\pi$
$654$	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	$-0.390172\pi$
$659$	11443.8	0.676460	0.338230	+	0.941063i	$0.390172\pi$
$660$	0	0
$661$	9948.40	0.585398	0.292699	−	0.956205i	$-0.405447\pi$
$661$	9948.40	0.585398	0.292699	+	0.956205i	$0.405447\pi$
$662$	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	$-0.661276\pi$
$677$	−17095.8	−0.970524	−0.485262	+	0.874369i	$0.661276\pi$
$678$	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	$-0.494582\pi$
$683$	607.606	0.0340401	0.0170201	+	0.999855i	$0.494582\pi$
$684$	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	$-0.323064\pi$
$691$	19169.7	1.05535	0.527676	+	0.849446i	$0.323064\pi$
$692$	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	$-0.369747\pi$
$701$	14769.1	0.795753	0.397876	+	0.917439i	$0.369747\pi$
$702$	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	$-0.640174\pi$
$709$	−16094.9	−0.852549	−0.426274	+	0.904594i	$0.640174\pi$
$710$	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	$-0.788538\pi$
$733$	−31249.5	−1.57466	−0.787331	+	0.616530i	$0.788538\pi$
$734$	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	$-0.369876\pi$
$739$	15971.2	0.795009	0.397504	+	0.917600i	$0.369876\pi$
$740$	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	$-0.311138\pi$
$757$	23290.6	1.11824	0.559122	+	0.829085i	$0.311138\pi$
$758$	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	$-0.213710\pi$
$773$	33654.0	1.56591	0.782957	+	0.622075i	$0.213710\pi$
$774$	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	$-0.669449\pi$
$787$	−22411.6	−1.01510	−0.507552	+	0.861621i	$0.669449\pi$
$788$	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	$-0.528958\pi$
$797$	−4088.30	−0.181700	−0.0908501	+	0.995865i	$0.528958\pi$
$798$	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	$-0.705836\pi$
$811$	−27831.2	−1.20504	−0.602519	+	0.798104i	$0.705836\pi$
$812$	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	$-0.559214\pi$
$821$	−8701.78	−0.369908	−0.184954	+	0.982747i	$0.559214\pi$
$822$	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	$-0.625432\pi$
$827$	−18262.0	−0.767874	−0.383937	+	0.923359i	$0.625432\pi$
$828$	0	0
$829$	−15237.0	−0.638364	−0.319182	−	0.947693i	$-0.603408\pi$
$829$	−15237.0	−0.638364	−0.319182	+	0.947693i	$0.603408\pi$
$830$	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	$-0.555994\pi$
$853$	−8719.68	−0.350007	−0.175004	+	0.984568i	$0.555994\pi$
$854$	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	$-0.370782\pi$
$859$	19883.7	0.789782	0.394891	+	0.918728i	$0.370782\pi$
$860$	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	$-0.361054\pi$
$877$	21960.6	0.845560	0.422780	+	0.906232i	$0.361054\pi$
$878$	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	$-0.547531\pi$
$883$	−7807.00	−0.297538	−0.148769	+	0.988872i	$0.547531\pi$
$884$	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	$-0.910275\pi$
$907$	−52475.2	−1.92107	−0.960534	+	0.278162i	$0.910275\pi$
$908$	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	$-0.659461\pi$
$941$	−27726.8	−0.960540	−0.480270	+	0.877121i	$0.659461\pi$
$942$	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	$-0.550817\pi$
$947$	−9265.39	−0.317935	−0.158968	+	0.987284i	$0.550817\pi$
$948$	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	$-0.403104\pi$
$971$	18137.9	0.599457	0.299729	+	0.954024i	$0.403104\pi$
$972$	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	$-0.660168\pi$
$997$	−30360.9	−0.964434	−0.482217	+	0.876052i	$0.660168\pi$
$998$	0	0
$999$	−22232.3	−0.704102

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.4.a.g.1.1	✓	4	8.5	even	2
1120.4.a.o.1.4	yes	4	8.3	odd	2
2240.4.a.cb.1.1		4	4.3	odd	2
2240.4.a.cj.1.4		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)

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