LMFDB - Embedded newform 1089.4.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1089.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:	$64.2530799963$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.88301$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.50105 q^{2} -1.74477 q^{4} -10.4342 q^{5} -32.7556 q^{7} +24.3721 q^{8} +O(q^{10})$	$q-2.50105 q^{2} -1.74477 q^{4} -10.4342 q^{5} -32.7556 q^{7} +24.3721 q^{8} +26.0964 q^{10} -73.2583 q^{13} +81.9233 q^{14} -46.9976 q^{16} +55.2769 q^{17} +90.5561 q^{19} +18.2053 q^{20} +91.1987 q^{23} -16.1272 q^{25} +183.222 q^{26} +57.1511 q^{28} +76.7184 q^{29} +23.0275 q^{31} -77.4339 q^{32} -138.250 q^{34} +341.779 q^{35} +125.831 q^{37} -226.485 q^{38} -254.304 q^{40} -140.059 q^{41} +146.015 q^{43} -228.092 q^{46} +221.506 q^{47} +729.931 q^{49} +40.3348 q^{50} +127.819 q^{52} +43.9542 q^{53} -798.324 q^{56} -191.876 q^{58} +401.785 q^{59} -406.495 q^{61} -57.5928 q^{62} +569.646 q^{64} +764.393 q^{65} +221.234 q^{67} -96.4457 q^{68} -854.805 q^{70} +749.522 q^{71} -222.002 q^{73} -314.710 q^{74} -158.000 q^{76} -1180.26 q^{79} +490.383 q^{80} +350.295 q^{82} -676.655 q^{83} -576.771 q^{85} -365.189 q^{86} -1054.06 q^{89} +2399.62 q^{91} -159.121 q^{92} -553.995 q^{94} -944.882 q^{95} -245.450 q^{97} -1825.59 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 11 q^{7} + 42 q^{8}+O(q^{10}) $ 4 * q + 3 * q^2 - 3 * q^4 - 12 * q^5 - 11 * q^7 + 42 * q^8	$ 4 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 11 q^{7} + 42 q^{8} + 4 q^{10} - 182 q^{13} + 144 q^{14} + 97 q^{16} + 57 q^{17} - 173 q^{19} - 99 q^{20} + 42 q^{23} - 186 q^{25} - 273 q^{26} + 116 q^{28} + 651 q^{29} - 170 q^{31} - 51 q^{32} - 356 q^{34} + 669 q^{35} + 244 q^{37} - 927 q^{38} - 532 q^{40} - 102 q^{41} - 322 q^{43} + 9 q^{46} + 633 q^{47} + 689 q^{49} - 390 q^{50} - 1037 q^{52} + 468 q^{53} - 735 q^{56} + 675 q^{58} + 996 q^{59} - 413 q^{61} - 87 q^{62} - 1044 q^{64} + 477 q^{65} - 259 q^{67} - 1119 q^{68} - 937 q^{70} - 237 q^{71} - 2309 q^{73} - 399 q^{74} - 1138 q^{76} - 2045 q^{79} + 510 q^{80} - 1383 q^{82} + 639 q^{83} - 1036 q^{85} + 669 q^{86} - 894 q^{89} - 400 q^{91} + 1761 q^{92} + 792 q^{94} - 537 q^{95} + 1432 q^{97} - 2775 q^{98}+O(q^{100}) $ 4 * q + 3 * q^2 - 3 * q^4 - 12 * q^5 - 11 * q^7 + 42 * q^8 + 4 * q^10 - 182 * q^13 + 144 * q^14 + 97 * q^16 + 57 * q^17 - 173 * q^19 - 99 * q^20 + 42 * q^23 - 186 * q^25 - 273 * q^26 + 116 * q^28 + 651 * q^29 - 170 * q^31 - 51 * q^32 - 356 * q^34 + 669 * q^35 + 244 * q^37 - 927 * q^38 - 532 * q^40 - 102 * q^41 - 322 * q^43 + 9 * q^46 + 633 * q^47 + 689 * q^49 - 390 * q^50 - 1037 * q^52 + 468 * q^53 - 735 * q^56 + 675 * q^58 + 996 * q^59 - 413 * q^61 - 87 * q^62 - 1044 * q^64 + 477 * q^65 - 259 * q^67 - 1119 * q^68 - 937 * q^70 - 237 * q^71 - 2309 * q^73 - 399 * q^74 - 1138 * q^76 - 2045 * q^79 + 510 * q^80 - 1383 * q^82 + 639 * q^83 - 1036 * q^85 + 669 * q^86 - 894 * q^89 - 400 * q^91 + 1761 * q^92 + 792 * q^94 - 537 * q^95 + 1432 * q^97 - 2775 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.50105	−0.884253	−0.442126	−	0.896953i	$-0.645776\pi$
$2$	−2.50105	−0.884253	−0.442126	+	0.896953i	$0.645776\pi$
$3$	0	0
$3$	0	0
$4$	−1.74477	−0.218097
$5$	−10.4342	−0.933264	−0.466632	−	0.884451i	$-0.654533\pi$
$5$	−10.4342	−0.933264	−0.466632	+	0.884451i	$0.654533\pi$
$6$	0	0
$7$	−32.7556	−1.76864	−0.884319	−	0.466884i	$-0.845377\pi$
$7$	−32.7556	−1.76864	−0.884319	+	0.466884i	$0.845377\pi$
$8$	24.3721	1.07711
$9$	0	0
$10$	26.0964	0.825242
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−73.2583	−1.56294	−0.781469	−	0.623944i	$-0.785529\pi$
$13$	−73.2583	−1.56294	−0.781469	+	0.623944i	$0.785529\pi$
$14$	81.9233	1.56392
$15$	0	0
$16$	−46.9976	−0.734337
$17$	55.2769	0.788625	0.394312	−	0.918976i	$-0.370983\pi$
$17$	55.2769	0.788625	0.394312	+	0.918976i	$0.370983\pi$
$18$	0	0
$19$	90.5561	1.09342	0.546710	−	0.837322i	$-0.315880\pi$
$19$	90.5561	1.09342	0.546710	+	0.837322i	$0.315880\pi$
$20$	18.2053	0.203542
$21$	0	0
$22$	0	0
$23$	91.1987	0.826793	0.413397	−	0.910551i	$-0.364342\pi$
$23$	91.1987	0.826793	0.413397	+	0.910551i	$0.364342\pi$
$24$	0	0
$25$	−16.1272	−0.129017
$26$	183.222	1.38203
$27$	0	0
$28$	57.1511	0.385734
$29$	76.7184	0.491250	0.245625	−	0.969365i	$-0.421007\pi$
$29$	76.7184	0.491250	0.245625	+	0.969365i	$0.421007\pi$
$30$	0	0
$31$	23.0275	0.133415	0.0667074	−	0.997773i	$-0.478751\pi$
$31$	23.0275	0.133415	0.0667074	+	0.997773i	$0.478751\pi$
$32$	−77.4339	−0.427766
$33$	0	0
$34$	−138.250	−0.697344
$35$	341.779	1.65061
$36$	0	0
$37$	125.831	0.559096	0.279548	−	0.960132i	$-0.409815\pi$
$37$	125.831	0.559096	0.279548	+	0.960132i	$0.409815\pi$
$38$	−226.485	−0.966861
$39$	0	0
$40$	−254.304	−1.00522
$41$	−140.059	−0.533502	−0.266751	−	0.963765i	$-0.585950\pi$
$41$	−140.059	−0.533502	−0.266751	+	0.963765i	$0.585950\pi$
$42$	0	0
$43$	146.015	0.517838	0.258919	−	0.965899i	$-0.416634\pi$
$43$	146.015	0.517838	0.258919	+	0.965899i	$0.416634\pi$
$44$	0	0
$45$	0	0
$46$	−228.092	−0.731095
$47$	221.506	0.687445	0.343722	−	0.939071i	$-0.388312\pi$
$47$	221.506	0.687445	0.343722	+	0.939071i	$0.388312\pi$
$48$	0	0
$49$	729.931	2.12808
$50$	40.3348	0.114084
$51$	0	0
$52$	127.819	0.340872
$53$	43.9542	0.113917	0.0569583	−	0.998377i	$-0.481860\pi$
$53$	43.9542	0.113917	0.0569583	+	0.998377i	$0.481860\pi$
$54$	0	0
$55$	0	0
$56$	−798.324	−1.90501
$57$	0	0
$58$	−191.876	−0.434389
$59$	401.785	0.886576	0.443288	−	0.896379i	$-0.353812\pi$
$59$	401.785	0.886576	0.443288	+	0.896379i	$0.353812\pi$
$60$	0	0
$61$	−406.495	−0.853219	−0.426609	−	0.904436i	$-0.640292\pi$
$61$	−406.495	−0.853219	−0.426609	+	0.904436i	$0.640292\pi$
$62$	−57.5928	−0.117972
$63$	0	0
$64$	569.646	1.11259
$65$	764.393	1.45863
$66$	0	0
$67$	221.234	0.403404	0.201702	−	0.979447i	$-0.435353\pi$
$67$	221.234	0.403404	0.201702	+	0.979447i	$0.435353\pi$
$68$	−96.4457	−0.171996
$69$	0	0
$70$	−854.805	−1.45955
$71$	749.522	1.25284	0.626422	−	0.779484i	$-0.284519\pi$
$71$	749.522	1.25284	0.626422	+	0.779484i	$0.284519\pi$
$72$	0	0
$73$	−222.002	−0.355936	−0.177968	−	0.984036i	$-0.556952\pi$
$73$	−222.002	−0.355936	−0.177968	+	0.984036i	$0.556952\pi$
$74$	−314.710	−0.494383
$75$	0	0
$76$	−158.000	−0.238471
$77$	0	0
$78$	0	0
$79$	−1180.26	−1.68088	−0.840440	−	0.541904i	$-0.817704\pi$
$79$	−1180.26	−1.68088	−0.840440	+	0.541904i	$0.817704\pi$
$80$	490.383	0.685331
$81$	0	0
$82$	350.295	0.471751
$83$	−676.655	−0.894849	−0.447425	−	0.894322i	$-0.647659\pi$
$83$	−676.655	−0.894849	−0.447425	+	0.894322i	$0.647659\pi$
$84$	0	0
$85$	−576.771	−0.735995
$86$	−365.189	−0.457900
$87$	0	0
$88$	0	0
$89$	−1054.06	−1.25539	−0.627695	−	0.778460i	$-0.716001\pi$
$89$	−1054.06	−1.25539	−0.627695	+	0.778460i	$0.716001\pi$
$90$	0	0
$91$	2399.62	2.76427
$92$	−159.121	−0.180321
$93$	0	0
$94$	−553.995	−0.607875
$95$	−944.882	−1.02045
$96$	0	0
$97$	−245.450	−0.256925	−0.128462	−	0.991714i	$-0.541004\pi$
$97$	−245.450	−0.256925	−0.128462	+	0.991714i	$0.541004\pi$
$98$	−1825.59	−1.88176
$99$	0	0
$100$	28.1383	0.0281383
$101$	−22.6917	−0.0223555	−0.0111778	−	0.999938i	$-0.503558\pi$
$101$	−22.6917	−0.0223555	−0.0111778	+	0.999938i	$0.503558\pi$
$102$	0	0
$103$	981.399	0.938836	0.469418	−	0.882976i	$-0.344464\pi$
$103$	981.399	0.938836	0.469418	+	0.882976i	$0.344464\pi$
$104$	−1785.46	−1.68345
$105$	0	0
$106$	−109.931	−0.100731
$107$	665.970	0.601698	0.300849	−	0.953672i	$-0.402730\pi$
$107$	665.970	0.601698	0.300849	+	0.953672i	$0.402730\pi$
$108$	0	0
$109$	85.6516	0.0752654	0.0376327	−	0.999292i	$-0.488018\pi$
$109$	85.6516	0.0752654	0.0376327	+	0.999292i	$0.488018\pi$
$110$	0	0
$111$	0	0
$112$	1539.43	1.29878
$113$	440.286	0.366536	0.183268	−	0.983063i	$-0.441332\pi$
$113$	440.286	0.366536	0.183268	+	0.983063i	$0.441332\pi$
$114$	0	0
$115$	−951.587	−0.771617
$116$	−133.856	−0.107140
$117$	0	0
$118$	−1004.88	−0.783958
$119$	−1810.63	−1.39479
$120$	0	0
$121$	0	0
$122$	1016.66	0.754461
$123$	0	0
$124$	−40.1778	−0.0290973
$125$	1472.55	1.05367
$126$	0	0
$127$	1625.39	1.13567	0.567836	−	0.823142i	$-0.307781\pi$
$127$	1625.39	1.13567	0.567836	+	0.823142i	$0.307781\pi$
$128$	−805.240	−0.556046
$129$	0	0
$130$	−1911.78	−1.28980
$131$	−1049.60	−0.700030	−0.350015	−	0.936744i	$-0.613823\pi$
$131$	−1049.60	−0.700030	−0.350015	+	0.936744i	$0.613823\pi$
$132$	0	0
$133$	−2966.22	−1.93386
$134$	−553.316	−0.356711
$135$	0	0
$136$	1347.22	0.849432
$137$	3073.32	1.91658	0.958290	−	0.285796i	$-0.0922580\pi$
$137$	3073.32	1.91658	0.958290	+	0.285796i	$0.0922580\pi$
$138$	0	0
$139$	−1066.87	−0.651012	−0.325506	−	0.945540i	$-0.605535\pi$
$139$	−1066.87	−0.651012	−0.325506	+	0.945540i	$0.605535\pi$
$140$	−596.327	−0.359992
$141$	0	0
$142$	−1874.59	−1.10783
$143$	0	0
$144$	0	0
$145$	−800.496	−0.458466
$146$	555.237	0.314738
$147$	0	0
$148$	−219.547	−0.121937
$149$	−2109.87	−1.16005	−0.580025	−	0.814598i	$-0.696957\pi$
$149$	−2109.87	−1.16005	−0.580025	+	0.814598i	$0.696957\pi$
$150$	0	0
$151$	1187.10	0.639768	0.319884	−	0.947457i	$-0.396356\pi$
$151$	1187.10	0.639768	0.319884	+	0.947457i	$0.396356\pi$
$152$	2207.04	1.17773
$153$	0	0
$154$	0	0
$155$	−240.274	−0.124511
$156$	0	0
$157$	−1848.94	−0.939883	−0.469942	−	0.882697i	$-0.655725\pi$
$157$	−1848.94	−0.939883	−0.469942	+	0.882697i	$0.655725\pi$
$158$	2951.88	1.48632
$159$	0	0
$160$	807.962	0.399219
$161$	−2987.27	−1.46230
$162$	0	0
$163$	1812.96	0.871175	0.435588	−	0.900146i	$-0.356541\pi$
$163$	1812.96	0.871175	0.435588	+	0.900146i	$0.356541\pi$
$164$	244.372	0.116355
$165$	0	0
$166$	1692.34	0.791273
$167$	−264.384	−0.122507	−0.0612534	−	0.998122i	$-0.519510\pi$
$167$	−264.384	−0.122507	−0.0612534	+	0.998122i	$0.519510\pi$
$168$	0	0
$169$	3169.78	1.44278
$170$	1442.53	0.650806
$171$	0	0
$172$	−254.763	−0.112939
$173$	665.157	0.292318	0.146159	−	0.989261i	$-0.453309\pi$
$173$	665.157	0.292318	0.146159	+	0.989261i	$0.453309\pi$
$174$	0	0
$175$	528.256	0.228185
$176$	0	0
$177$	0	0
$178$	2636.24	1.11008
$179$	65.0751	0.0271729	0.0135864	−	0.999908i	$-0.495675\pi$
$179$	65.0751	0.0271729	0.0135864	+	0.999908i	$0.495675\pi$
$180$	0	0
$181$	−40.9576	−0.0168196	−0.00840982	−	0.999965i	$-0.502677\pi$
$181$	−40.9576	−0.0168196	−0.00840982	+	0.999965i	$0.502677\pi$
$182$	−6001.56	−2.44431
$183$	0	0
$184$	2222.71	0.890544
$185$	−1312.95	−0.521785
$186$	0	0
$187$	0	0
$188$	−386.477	−0.149929
$189$	0	0
$190$	2363.19	0.902337
$191$	−4579.92	−1.73503	−0.867517	−	0.497407i	$-0.834285\pi$
$191$	−4579.92	−1.73503	−0.867517	+	0.497407i	$0.834285\pi$
$192$	0	0
$193$	−5044.62	−1.88145	−0.940725	−	0.339171i	$-0.889854\pi$
$193$	−5044.62	−1.88145	−0.940725	+	0.339171i	$0.889854\pi$
$194$	613.882	0.227187
$195$	0	0
$196$	−1273.56	−0.464127
$197$	−1703.26	−0.616001	−0.308000	−	0.951386i	$-0.599660\pi$
$197$	−1703.26	−0.616001	−0.308000	+	0.951386i	$0.599660\pi$
$198$	0	0
$199$	−3326.11	−1.18483	−0.592416	−	0.805632i	$-0.701826\pi$
$199$	−3326.11	−1.18483	−0.592416	+	0.805632i	$0.701826\pi$
$200$	−393.053	−0.138965
$201$	0	0
$202$	56.7530	0.0197680
$203$	−2512.96	−0.868843
$204$	0	0
$205$	1461.41	0.497899
$206$	−2454.52	−0.830169
$207$	0	0
$208$	3442.96	1.14772
$209$	0	0
$210$	0	0
$211$	4960.93	1.61860	0.809300	−	0.587396i	$-0.199847\pi$
$211$	4960.93	1.61860	0.809300	+	0.587396i	$0.199847\pi$
$212$	−76.6901	−0.0248448
$213$	0	0
$214$	−1665.62	−0.532054
$215$	−1523.55	−0.483280
$216$	0	0
$217$	−754.280	−0.235962
$218$	−214.218	−0.0665537
$219$	0	0
$220$	0	0
$221$	−4049.49	−1.23257
$222$	0	0
$223$	−3679.14	−1.10481	−0.552406	−	0.833575i	$-0.686290\pi$
$223$	−3679.14	−1.10481	−0.552406	+	0.833575i	$0.686290\pi$
$224$	2536.39	0.756562
$225$	0	0
$226$	−1101.17	−0.324111
$227$	−401.723	−0.117460	−0.0587298	−	0.998274i	$-0.518705\pi$
$227$	−401.723	−0.117460	−0.0587298	+	0.998274i	$0.518705\pi$
$228$	0	0
$229$	−2951.86	−0.851810	−0.425905	−	0.904768i	$-0.640044\pi$
$229$	−2951.86	−0.851810	−0.425905	+	0.904768i	$0.640044\pi$
$230$	2379.96	0.682305
$231$	0	0
$232$	1869.79	0.529128
$233$	2465.80	0.693305	0.346653	−	0.937994i	$-0.387318\pi$
$233$	2465.80	0.693305	0.346653	+	0.937994i	$0.387318\pi$
$234$	0	0
$235$	−2311.24	−0.641568
$236$	−701.025	−0.193359
$237$	0	0
$238$	4528.47	1.23335
$239$	4245.43	1.14901	0.574506	−	0.818500i	$-0.305194\pi$
$239$	4245.43	1.14901	0.574506	+	0.818500i	$0.305194\pi$
$240$	0	0
$241$	−2686.25	−0.717994	−0.358997	−	0.933339i	$-0.616881\pi$
$241$	−2686.25	−0.717994	−0.358997	+	0.933339i	$0.616881\pi$
$242$	0	0
$243$	0	0
$244$	709.242	0.186084
$245$	−7616.25	−1.98606
$246$	0	0
$247$	−6633.98	−1.70895
$248$	561.229	0.143702
$249$	0	0
$250$	−3682.92	−0.931712
$251$	−7355.98	−1.84982	−0.924911	−	0.380184i	$-0.875861\pi$
$251$	−7355.98	−1.84982	−0.924911	+	0.380184i	$0.875861\pi$
$252$	0	0
$253$	0	0
$254$	−4065.18	−1.00422
$255$	0	0
$256$	−2543.23	−0.620905
$257$	478.264	0.116083	0.0580414	−	0.998314i	$-0.481514\pi$
$257$	478.264	0.116083	0.0580414	+	0.998314i	$0.481514\pi$
$258$	0	0
$259$	−4121.69	−0.988839
$260$	−1333.69	−0.318123
$261$	0	0
$262$	2625.09	0.619003
$263$	−3955.38	−0.927373	−0.463686	−	0.885999i	$-0.653474\pi$
$263$	−3955.38	−0.927373	−0.463686	+	0.885999i	$0.653474\pi$
$264$	0	0
$265$	−458.628	−0.106314
$266$	7418.65	1.71003
$267$	0	0
$268$	−386.003	−0.0879810
$269$	1310.11	0.296948	0.148474	−	0.988916i	$-0.452564\pi$
$269$	1310.11	0.296948	0.148474	+	0.988916i	$0.452564\pi$
$270$	0	0
$271$	−5454.75	−1.22270	−0.611352	−	0.791359i	$-0.709374\pi$
$271$	−5454.75	−1.22270	−0.611352	+	0.791359i	$0.709374\pi$
$272$	−2597.88	−0.579116
$273$	0	0
$274$	−7686.52	−1.69474
$275$	0	0
$276$	0	0
$277$	−4762.21	−1.03297	−0.516487	−	0.856295i	$-0.672760\pi$
$277$	−4762.21	−1.03297	−0.516487	+	0.856295i	$0.672760\pi$
$278$	2668.29	0.575659
$279$	0	0
$280$	8329.88	1.77788
$281$	558.653	0.118599	0.0592997	−	0.998240i	$-0.481113\pi$
$281$	558.653	0.118599	0.0592997	+	0.998240i	$0.481113\pi$
$282$	0	0
$283$	−2065.08	−0.433768	−0.216884	−	0.976197i	$-0.569589\pi$
$283$	−2065.08	−0.433768	−0.216884	+	0.976197i	$0.569589\pi$
$284$	−1307.75	−0.273241
$285$	0	0
$286$	0	0
$287$	4587.73	0.943572
$288$	0	0
$289$	−1857.46	−0.378071
$290$	2002.08	0.405400
$291$	0	0
$292$	387.343	0.0776285
$293$	7496.15	1.49464	0.747320	−	0.664464i	$-0.231340\pi$
$293$	7496.15	1.49464	0.747320	+	0.664464i	$0.231340\pi$
$294$	0	0
$295$	−4192.32	−0.827410
$296$	3066.78	0.602206
$297$	0	0
$298$	5276.89	1.02578
$299$	−6681.06	−1.29223
$300$	0	0
$301$	−4782.80	−0.915868
$302$	−2969.00	−0.565717
$303$	0	0
$304$	−4255.92	−0.802939
$305$	4241.46	0.796279
$306$	0	0
$307$	1036.23	0.192640	0.0963201	−	0.995350i	$-0.469293\pi$
$307$	1036.23	0.192640	0.0963201	+	0.995350i	$0.469293\pi$
$308$	0	0
$309$	0	0
$310$	600.936	0.110100
$311$	10334.7	1.88432	0.942162	−	0.335157i	$-0.108789\pi$
$311$	10334.7	1.88432	0.942162	+	0.335157i	$0.108789\pi$
$312$	0	0
$313$	7130.94	1.28775	0.643873	−	0.765132i	$-0.277326\pi$
$313$	7130.94	1.28775	0.643873	+	0.765132i	$0.277326\pi$
$314$	4624.29	0.831095
$315$	0	0
$316$	2059.29	0.366594
$317$	6446.29	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$317$	6446.29	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$318$	0	0
$319$	0	0
$320$	−5943.81	−1.03834
$321$	0	0
$322$	7471.30	1.29304
$323$	5005.66	0.862298
$324$	0	0
$325$	1181.45	0.201646
$326$	−4534.28	−0.770340
$327$	0	0
$328$	−3413.54	−0.574638
$329$	−7255.55	−1.21584
$330$	0	0
$331$	5634.51	0.935652	0.467826	−	0.883821i	$-0.345037\pi$
$331$	5634.51	0.935652	0.467826	+	0.883821i	$0.345037\pi$
$332$	1180.61	0.195164
$333$	0	0
$334$	661.235	0.108327
$335$	−2308.40	−0.376482
$336$	0	0
$337$	3058.45	0.494375	0.247187	−	0.968968i	$-0.420494\pi$
$337$	3058.45	0.494375	0.247187	+	0.968968i	$0.420494\pi$
$338$	−7927.76	−1.27578
$339$	0	0
$340$	1006.33	0.160518
$341$	0	0
$342$	0	0
$343$	−12674.2	−1.99516
$344$	3558.69	0.557766
$345$	0	0
$346$	−1663.59	−0.258483
$347$	10516.3	1.62693	0.813464	−	0.581616i	$-0.197579\pi$
$347$	10516.3	1.62693	0.813464	+	0.581616i	$0.197579\pi$
$348$	0	0
$349$	1504.94	0.230823	0.115412	−	0.993318i	$-0.463181\pi$
$349$	1504.94	0.230823	0.115412	+	0.993318i	$0.463181\pi$
$350$	−1321.19	−0.201773
$351$	0	0
$352$	0	0
$353$	11810.2	1.78071	0.890356	−	0.455264i	$-0.150455\pi$
$353$	11810.2	1.78071	0.890356	+	0.455264i	$0.150455\pi$
$354$	0	0
$355$	−7820.67	−1.16923
$356$	1839.09	0.273796
$357$	0	0
$358$	−162.756	−0.0240277
$359$	−3306.66	−0.486125	−0.243063	−	0.970011i	$-0.578152\pi$
$359$	−3306.66	−0.486125	−0.243063	+	0.970011i	$0.578152\pi$
$360$	0	0
$361$	1341.41	0.195569
$362$	102.437	0.0148728
$363$	0	0
$364$	−4186.79	−0.602878
$365$	2316.41	0.332183
$366$	0	0
$367$	−3587.62	−0.510279	−0.255139	−	0.966904i	$-0.582121\pi$
$367$	−3587.62	−0.510279	−0.255139	+	0.966904i	$0.582121\pi$
$368$	−4286.12	−0.607145
$369$	0	0
$370$	3283.75	0.461390
$371$	−1439.75	−0.201477
$372$	0	0
$373$	−2022.36	−0.280734	−0.140367	−	0.990100i	$-0.544828\pi$
$373$	−2022.36	−0.280734	−0.140367	+	0.990100i	$0.544828\pi$
$374$	0	0
$375$	0	0
$376$	5398.56	0.740451
$377$	−5620.26	−0.767793
$378$	0	0
$379$	−5804.80	−0.786734	−0.393367	−	0.919381i	$-0.628690\pi$
$379$	−5804.80	−0.786734	−0.393367	+	0.919381i	$0.628690\pi$
$380$	1648.60	0.222557
$381$	0	0
$382$	11454.6	1.53421
$383$	5238.25	0.698856	0.349428	−	0.936963i	$-0.386376\pi$
$383$	5238.25	0.698856	0.349428	+	0.936963i	$0.386376\pi$
$384$	0	0
$385$	0	0
$386$	12616.8	1.66368
$387$	0	0
$388$	428.255	0.0560344
$389$	84.9846	0.0110768	0.00553842	−	0.999985i	$-0.498237\pi$
$389$	84.9846	0.0110768	0.00553842	+	0.999985i	$0.498237\pi$
$390$	0	0
$391$	5041.18	0.652030
$392$	17790.0	2.29216
$393$	0	0
$394$	4259.93	0.544700
$395$	12315.1	1.56871
$396$	0	0
$397$	−7187.71	−0.908667	−0.454334	−	0.890832i	$-0.650123\pi$
$397$	−7187.71	−0.908667	−0.454334	+	0.890832i	$0.650123\pi$
$398$	8318.75	1.04769
$399$	0	0
$400$	757.938	0.0947423
$401$	−2274.07	−0.283196	−0.141598	−	0.989924i	$-0.545224\pi$
$401$	−2274.07	−0.283196	−0.141598	+	0.989924i	$0.545224\pi$
$402$	0	0
$403$	−1686.96	−0.208519
$404$	39.5919	0.00487567
$405$	0	0
$406$	6285.02	0.768277
$407$	0	0
$408$	0	0
$409$	−9607.22	−1.16148	−0.580741	−	0.814088i	$-0.697237\pi$
$409$	−9607.22	−1.16148	−0.580741	+	0.814088i	$0.697237\pi$
$410$	−3655.05	−0.440269
$411$	0	0
$412$	−1712.32	−0.204757
$413$	−13160.7	−1.56803
$414$	0	0
$415$	7060.36	0.835131
$416$	5672.67	0.668571
$417$	0	0
$418$	0	0
$419$	−10777.3	−1.25658	−0.628291	−	0.777979i	$-0.716245\pi$
$419$	−10777.3	−1.25658	−0.628291	+	0.777979i	$0.716245\pi$
$420$	0	0
$421$	1231.36	0.142549	0.0712744	−	0.997457i	$-0.477293\pi$
$421$	1231.36	0.142549	0.0712744	+	0.997457i	$0.477293\pi$
$422$	−12407.5	−1.43125
$423$	0	0
$424$	1071.26	0.122700
$425$	−891.460	−0.101746
$426$	0	0
$427$	13315.0	1.50903
$428$	−1161.97	−0.131228
$429$	0	0
$430$	3810.46	0.427342
$431$	−8711.04	−0.973540	−0.486770	−	0.873530i	$-0.661825\pi$
$431$	−8711.04	−0.973540	−0.486770	+	0.873530i	$0.661825\pi$
$432$	0	0
$433$	7487.16	0.830969	0.415485	−	0.909600i	$-0.363612\pi$
$433$	7487.16	0.830969	0.415485	+	0.909600i	$0.363612\pi$
$434$	1886.49	0.208651
$435$	0	0
$436$	−149.443	−0.0164151
$437$	8258.60	0.904033
$438$	0	0
$439$	9595.95	1.04326	0.521628	−	0.853173i	$-0.325325\pi$
$439$	9595.95	1.04326	0.521628	+	0.853173i	$0.325325\pi$
$440$	0	0
$441$	0	0
$442$	10128.0	1.08990
$443$	−9290.91	−0.996443	−0.498221	−	0.867050i	$-0.666013\pi$
$443$	−9290.91	−0.996443	−0.498221	+	0.867050i	$0.666013\pi$
$444$	0	0
$445$	10998.2	1.17161
$446$	9201.69	0.976934
$447$	0	0
$448$	−18659.1	−1.96777
$449$	6860.56	0.721091	0.360546	−	0.932742i	$-0.382591\pi$
$449$	6860.56	0.721091	0.360546	+	0.932742i	$0.382591\pi$
$450$	0	0
$451$	0	0
$452$	−768.199	−0.0799403
$453$	0	0
$454$	1004.73	0.103864
$455$	−25038.2	−2.57980
$456$	0	0
$457$	1156.29	0.118357	0.0591783	−	0.998247i	$-0.481152\pi$
$457$	1156.29	0.118357	0.0591783	+	0.998247i	$0.481152\pi$
$458$	7382.74	0.753216
$459$	0	0
$460$	1660.30	0.168287
$461$	8394.87	0.848131	0.424065	−	0.905632i	$-0.360603\pi$
$461$	8394.87	0.848131	0.424065	+	0.905632i	$0.360603\pi$
$462$	0	0
$463$	1801.45	0.180822	0.0904108	−	0.995905i	$-0.471182\pi$
$463$	1801.45	0.180822	0.0904108	+	0.995905i	$0.471182\pi$
$464$	−3605.58	−0.360743
$465$	0	0
$466$	−6167.08	−0.613057
$467$	19022.3	1.88490	0.942450	−	0.334347i	$-0.108516\pi$
$467$	19022.3	1.88490	0.942450	+	0.334347i	$0.108516\pi$
$468$	0	0
$469$	−7246.66	−0.713475
$470$	5780.51	0.567308
$471$	0	0
$472$	9792.36	0.954936
$473$	0	0
$474$	0	0
$475$	−1460.41	−0.141070
$476$	3159.14	0.304199
$477$	0	0
$478$	−10618.0	−1.01602
$479$	16918.0	1.61378	0.806891	−	0.590701i	$-0.201149\pi$
$479$	16918.0	1.61378	0.806891	+	0.590701i	$0.201149\pi$
$480$	0	0
$481$	−9218.20	−0.873833
$482$	6718.44	0.634889
$483$	0	0
$484$	0	0
$485$	2561.08	0.239779
$486$	0	0
$487$	−9205.88	−0.856588	−0.428294	−	0.903639i	$-0.640885\pi$
$487$	−9205.88	−0.856588	−0.428294	+	0.903639i	$0.640885\pi$
$488$	−9907.14	−0.919007
$489$	0	0
$490$	19048.6	1.75618
$491$	−16404.5	−1.50779	−0.753895	−	0.656995i	$-0.771827\pi$
$491$	−16404.5	−1.50779	−0.753895	+	0.656995i	$0.771827\pi$
$492$	0	0
$493$	4240.75	0.387412
$494$	16591.9	1.51114
$495$	0	0
$496$	−1082.24	−0.0979715
$497$	−24551.1	−2.21583
$498$	0	0
$499$	−4631.36	−0.415487	−0.207744	−	0.978183i	$-0.566612\pi$
$499$	−4631.36	−0.415487	−0.207744	+	0.978183i	$0.566612\pi$
$500$	−2569.27	−0.229802
$501$	0	0
$502$	18397.6	1.63571
$503$	−19601.9	−1.73759	−0.868794	−	0.495173i	$-0.835105\pi$
$503$	−19601.9	−1.73759	−0.868794	+	0.495173i	$0.835105\pi$
$504$	0	0
$505$	236.770	0.0208636
$506$	0	0
$507$	0	0
$508$	−2835.94	−0.247686
$509$	−16283.4	−1.41797	−0.708986	−	0.705222i	$-0.750847\pi$
$509$	−16283.4	−1.41797	−0.708986	+	0.705222i	$0.750847\pi$
$510$	0	0
$511$	7271.81	0.629522
$512$	12802.6	1.10508
$513$	0	0
$514$	−1196.16	−0.102647
$515$	−10240.1	−0.876182
$516$	0	0
$517$	0	0
$518$	10308.5	0.874384
$519$	0	0
$520$	18629.9	1.57110
$521$	10259.0	0.862679	0.431340	−	0.902190i	$-0.358041\pi$
$521$	10259.0	0.862679	0.431340	+	0.902190i	$0.358041\pi$
$522$	0	0
$523$	21372.0	1.78687	0.893435	−	0.449192i	$-0.148288\pi$
$523$	21372.0	1.78687	0.893435	+	0.449192i	$0.148288\pi$
$524$	1831.31	0.152674
$525$	0	0
$526$	9892.58	0.820032
$527$	1272.89	0.105214
$528$	0	0
$529$	−3849.79	−0.316413
$530$	1147.05	0.0940087
$531$	0	0
$532$	5175.38	0.421769
$533$	10260.5	0.833831
$534$	0	0
$535$	−6948.87	−0.561544
$536$	5391.94	0.434508
$537$	0	0
$538$	−3276.65	−0.262577
$539$	0	0
$540$	0	0
$541$	21133.1	1.67945	0.839726	−	0.543011i	$-0.182716\pi$
$541$	21133.1	1.67945	0.839726	+	0.543011i	$0.182716\pi$
$542$	13642.6	1.08118
$543$	0	0
$544$	−4280.30	−0.337347
$545$	−893.707	−0.0702426
$546$	0	0
$547$	694.732	0.0543045	0.0271523	−	0.999631i	$-0.491356\pi$
$547$	694.732	0.0543045	0.0271523	+	0.999631i	$0.491356\pi$
$548$	−5362.25	−0.418000
$549$	0	0
$550$	0	0
$551$	6947.32	0.537143
$552$	0	0
$553$	38660.1	2.97287
$554$	11910.5	0.913410
$555$	0	0
$556$	1861.44	0.141984
$557$	−17901.0	−1.36174	−0.680871	−	0.732403i	$-0.738399\pi$
$557$	−17901.0	−1.36174	−0.680871	+	0.732403i	$0.738399\pi$
$558$	0	0
$559$	−10696.8	−0.809349
$560$	−16062.8	−1.21210
$561$	0	0
$562$	−1397.22	−0.104872
$563$	−15238.5	−1.14072	−0.570361	−	0.821394i	$-0.693197\pi$
$563$	−15238.5	−1.14072	−0.570361	+	0.821394i	$0.693197\pi$
$564$	0	0
$565$	−4594.03	−0.342075
$566$	5164.86	0.383560
$567$	0	0
$568$	18267.4	1.34944
$569$	24653.6	1.81640	0.908201	−	0.418535i	$-0.137456\pi$
$569$	24653.6	1.81640	0.908201	+	0.418535i	$0.137456\pi$
$570$	0	0
$571$	20893.8	1.53131	0.765656	−	0.643250i	$-0.222415\pi$
$571$	20893.8	1.53131	0.765656	+	0.643250i	$0.222415\pi$
$572$	0	0
$573$	0	0
$574$	−11474.1	−0.834357
$575$	−1470.78	−0.106671
$576$	0	0
$577$	−14487.2	−1.04525	−0.522625	−	0.852563i	$-0.675047\pi$
$577$	−14487.2	−1.04525	−0.522625	+	0.852563i	$0.675047\pi$
$578$	4645.60	0.334311
$579$	0	0
$580$	1396.68	0.0999899
$581$	22164.2	1.58266
$582$	0	0
$583$	0	0
$584$	−5410.65	−0.383381
$585$	0	0
$586$	−18748.2	−1.32164
$587$	−9546.40	−0.671247	−0.335623	−	0.941996i	$-0.608947\pi$
$587$	−9546.40	−0.671247	−0.335623	+	0.941996i	$0.608947\pi$
$588$	0	0
$589$	2085.28	0.145879
$590$	10485.2	0.731640
$591$	0	0
$592$	−5913.78	−0.410565
$593$	15771.0	1.09213	0.546067	−	0.837741i	$-0.316124\pi$
$593$	15771.0	1.09213	0.546067	+	0.837741i	$0.316124\pi$
$594$	0	0
$595$	18892.5	1.30171
$596$	3681.25	0.253003
$597$	0	0
$598$	16709.6	1.14266
$599$	−19740.4	−1.34653	−0.673264	−	0.739402i	$-0.735108\pi$
$599$	−19740.4	−1.34653	−0.673264	+	0.739402i	$0.735108\pi$
$600$	0	0
$601$	2159.01	0.146535	0.0732677	−	0.997312i	$-0.476657\pi$
$601$	2159.01	0.146535	0.0732677	+	0.997312i	$0.476657\pi$
$602$	11962.0	0.809859
$603$	0	0
$604$	−2071.22	−0.139531
$605$	0	0
$606$	0	0
$607$	5063.85	0.338609	0.169304	−	0.985564i	$-0.445848\pi$
$607$	5063.85	0.338609	0.169304	+	0.985564i	$0.445848\pi$
$608$	−7012.11	−0.467728
$609$	0	0
$610$	−10608.1	−0.704112
$611$	−16227.1	−1.07443
$612$	0	0
$613$	24425.7	1.60937	0.804685	−	0.593702i	$-0.202334\pi$
$613$	24425.7	1.60937	0.804685	+	0.593702i	$0.202334\pi$
$614$	−2591.65	−0.170343
$615$	0	0
$616$	0	0
$617$	13814.4	0.901369	0.450685	−	0.892683i	$-0.351180\pi$
$617$	13814.4	0.901369	0.450685	+	0.892683i	$0.351180\pi$
$618$	0	0
$619$	−4371.99	−0.283886	−0.141943	−	0.989875i	$-0.545335\pi$
$619$	−4371.99	−0.283886	−0.141943	+	0.989875i	$0.545335\pi$
$620$	419.223	0.0271555
$621$	0	0
$622$	−25847.5	−1.66622
$623$	34526.2	2.22033
$624$	0	0
$625$	−13349.0	−0.854337
$626$	−17834.8	−1.13869
$627$	0	0
$628$	3225.99	0.204985
$629$	6955.57	0.440917
$630$	0	0
$631$	−2311.85	−0.145853	−0.0729266	−	0.997337i	$-0.523234\pi$
$631$	−2311.85	−0.145853	−0.0729266	+	0.997337i	$0.523234\pi$
$632$	−28765.4	−1.81049
$633$	0	0
$634$	−16122.5	−1.00994
$635$	−16959.7	−1.05988
$636$	0	0
$637$	−53473.5	−3.32605
$638$	0	0
$639$	0	0
$640$	8402.05	0.518938
$641$	18458.6	1.13740	0.568699	−	0.822546i	$-0.307447\pi$
$641$	18458.6	1.13740	0.568699	+	0.822546i	$0.307447\pi$
$642$	0	0
$643$	29409.5	1.80373	0.901863	−	0.432022i	$-0.142200\pi$
$643$	29409.5	1.80373	0.901863	+	0.432022i	$0.142200\pi$
$644$	5212.11	0.318922
$645$	0	0
$646$	−12519.4	−0.762490
$647$	2217.21	0.134726	0.0673628	−	0.997729i	$-0.478542\pi$
$647$	2217.21	0.134726	0.0673628	+	0.997729i	$0.478542\pi$
$648$	0	0
$649$	0	0
$650$	−2954.86	−0.178306
$651$	0	0
$652$	−3163.20	−0.190000
$653$	−22501.3	−1.34846	−0.674230	−	0.738521i	$-0.735524\pi$
$653$	−22501.3	−1.34846	−0.674230	+	0.738521i	$0.735524\pi$
$654$	0	0
$655$	10951.7	0.653313
$656$	6582.45	0.391771
$657$	0	0
$658$	18146.5	1.07511
$659$	−5908.12	−0.349238	−0.174619	−	0.984636i	$-0.555869\pi$
$659$	−5908.12	−0.349238	−0.174619	+	0.984636i	$0.555869\pi$
$660$	0	0
$661$	22387.2	1.31734	0.658670	−	0.752432i	$-0.271120\pi$
$661$	22387.2	1.31734	0.658670	+	0.752432i	$0.271120\pi$
$662$	−14092.2	−0.827353
$663$	0	0
$664$	−16491.5	−0.963847
$665$	30950.2	1.80481
$666$	0	0
$667$	6996.62	0.406162
$668$	461.290	0.0267183
$669$	0	0
$670$	5773.42	0.332906
$671$	0	0
$672$	0	0
$673$	−31757.5	−1.81896	−0.909481	−	0.415746i	$-0.863520\pi$
$673$	−31757.5	−1.81896	−0.909481	+	0.415746i	$0.863520\pi$
$674$	−7649.32	−0.437152
$675$	0	0
$676$	−5530.54	−0.314664
$677$	3671.89	0.208452	0.104226	−	0.994554i	$-0.466763\pi$
$677$	3671.89	0.208452	0.104226	+	0.994554i	$0.466763\pi$
$678$	0	0
$679$	8039.88	0.454407
$680$	−14057.1	−0.792745
$681$	0	0
$682$	0	0
$683$	11719.2	0.656549	0.328274	−	0.944582i	$-0.393533\pi$
$683$	11719.2	0.656549	0.328274	+	0.944582i	$0.393533\pi$
$684$	0	0
$685$	−32067.7	−1.78868
$686$	31698.6	1.76423
$687$	0	0
$688$	−6862.34	−0.380268
$689$	−3220.01	−0.178044
$690$	0	0
$691$	612.909	0.0337426	0.0168713	−	0.999858i	$-0.494629\pi$
$691$	612.909	0.0337426	0.0168713	+	0.999858i	$0.494629\pi$
$692$	−1160.55	−0.0637535
$693$	0	0
$694$	−26301.7	−1.43862
$695$	11131.9	0.607566
$696$	0	0
$697$	−7742.05	−0.420733
$698$	−3763.91	−0.204106
$699$	0	0
$700$	−921.686	−0.0497664
$701$	−28016.0	−1.50949	−0.754743	−	0.656021i	$-0.772238\pi$
$701$	−28016.0	−1.50949	−0.754743	+	0.656021i	$0.772238\pi$
$702$	0	0
$703$	11394.8	0.611328
$704$	0	0
$705$	0	0
$706$	−29537.8	−1.57460
$707$	743.281	0.0395388
$708$	0	0
$709$	11305.4	0.598849	0.299425	−	0.954120i	$-0.403205\pi$
$709$	11305.4	0.598849	0.299425	+	0.954120i	$0.403205\pi$
$710$	19559.9	1.03390
$711$	0	0
$712$	−25689.6	−1.35219
$713$	2100.08	0.110307
$714$	0	0
$715$	0	0
$716$	−113.541	−0.00592631
$717$	0	0
$718$	8270.11	0.429857
$719$	−11180.8	−0.579933	−0.289966	−	0.957037i	$-0.593644\pi$
$719$	−11180.8	−0.579933	−0.289966	+	0.957037i	$0.593644\pi$
$720$	0	0
$721$	−32146.3	−1.66046
$722$	−3354.92	−0.172932
$723$	0	0
$724$	71.4618	0.00366831
$725$	−1237.25	−0.0633798
$726$	0	0
$727$	−31428.7	−1.60334	−0.801669	−	0.597769i	$-0.796054\pi$
$727$	−31428.7	−1.60334	−0.801669	+	0.597769i	$0.796054\pi$
$728$	58483.8	2.97741
$729$	0	0
$730$	−5793.46	−0.293734
$731$	8071.24	0.408380
$732$	0	0
$733$	1270.58	0.0640246	0.0320123	−	0.999487i	$-0.489808\pi$
$733$	1270.58	0.0640246	0.0320123	+	0.999487i	$0.489808\pi$
$734$	8972.80	0.451215
$735$	0	0
$736$	−7061.87	−0.353674
$737$	0	0
$738$	0	0
$739$	13509.9	0.672492	0.336246	−	0.941774i	$-0.390843\pi$
$739$	13509.9	0.672492	0.336246	+	0.941774i	$0.390843\pi$
$740$	2290.80	0.113800
$741$	0	0
$742$	3600.87	0.178157
$743$	−32103.2	−1.58513	−0.792565	−	0.609787i	$-0.791255\pi$
$743$	−32103.2	−1.58513	−0.792565	+	0.609787i	$0.791255\pi$
$744$	0	0
$745$	22014.9	1.08263
$746$	5058.01	0.248240
$747$	0	0
$748$	0	0
$749$	−21814.3	−1.06419
$750$	0	0
$751$	−23717.2	−1.15240	−0.576200	−	0.817308i	$-0.695465\pi$
$751$	−23717.2	−1.15240	−0.576200	+	0.817308i	$0.695465\pi$
$752$	−10410.2	−0.504816
$753$	0	0
$754$	14056.5	0.678923
$755$	−12386.5	−0.597073
$756$	0	0
$757$	−23133.8	−1.11072	−0.555359	−	0.831611i	$-0.687419\pi$
$757$	−23133.8	−1.11072	−0.555359	+	0.831611i	$0.687419\pi$
$758$	14518.1	0.695672
$759$	0	0
$760$	−23028.8	−1.09913
$761$	−825.536	−0.0393241	−0.0196621	−	0.999807i	$-0.506259\pi$
$761$	−825.536	−0.0393241	−0.0196621	+	0.999807i	$0.506259\pi$
$762$	0	0
$763$	−2805.57	−0.133117
$764$	7990.93	0.378405
$765$	0	0
$766$	−13101.1	−0.617966
$767$	−29434.1	−1.38566
$768$	0	0
$769$	9591.21	0.449763	0.224882	−	0.974386i	$-0.427800\pi$
$769$	9591.21	0.449763	0.224882	+	0.974386i	$0.427800\pi$
$770$	0	0
$771$	0	0
$772$	8801.72	0.410338
$773$	−2343.87	−0.109060	−0.0545300	−	0.998512i	$-0.517366\pi$
$773$	−2343.87	−0.109060	−0.0545300	+	0.998512i	$0.517366\pi$
$774$	0	0
$775$	−371.368	−0.0172128
$776$	−5982.14	−0.276735
$777$	0	0
$778$	−212.550	−0.00979472
$779$	−12683.2	−0.583343
$780$	0	0
$781$	0	0
$782$	−12608.2	−0.576559
$783$	0	0
$784$	−34305.0	−1.56273
$785$	19292.3	0.877160
$786$	0	0
$787$	−26054.8	−1.18012	−0.590059	−	0.807360i	$-0.700896\pi$
$787$	−26054.8	−1.18012	−0.590059	+	0.807360i	$0.700896\pi$
$788$	2971.80	0.134348
$789$	0	0
$790$	−30800.6	−1.38713
$791$	−14421.8	−0.648269
$792$	0	0
$793$	29779.1	1.33353
$794$	17976.8	0.803492
$795$	0	0
$796$	5803.31	0.258408
$797$	−43397.7	−1.92877	−0.964383	−	0.264511i	$-0.914790\pi$
$797$	−43397.7	−1.92877	−0.964383	+	0.264511i	$0.914790\pi$
$798$	0	0
$799$	12244.1	0.542136
$800$	1248.79	0.0551892
$801$	0	0
$802$	5687.54	0.250417
$803$	0	0
$804$	0	0
$805$	31169.8	1.36471
$806$	4219.15	0.184384
$807$	0	0
$808$	−553.045	−0.0240793
$809$	8483.58	0.368686	0.184343	−	0.982862i	$-0.440984\pi$
$809$	8483.58	0.368686	0.184343	+	0.982862i	$0.440984\pi$
$810$	0	0
$811$	5551.78	0.240381	0.120191	−	0.992751i	$-0.461649\pi$
$811$	5551.78	0.240381	0.120191	+	0.992751i	$0.461649\pi$
$812$	4384.54	0.189492
$813$	0	0
$814$	0	0
$815$	−18916.8	−0.813037
$816$	0	0
$817$	13222.5	0.566215
$818$	24028.1	1.02704
$819$	0	0
$820$	−2549.83	−0.108590
$821$	105.932	0.00450312	0.00225156	−	0.999997i	$-0.499283\pi$
$821$	105.932	0.00450312	0.00225156	+	0.999997i	$0.499283\pi$
$822$	0	0
$823$	37329.8	1.58109	0.790545	−	0.612404i	$-0.209798\pi$
$823$	37329.8	1.58109	0.790545	+	0.612404i	$0.209798\pi$
$824$	23918.8	1.01123
$825$	0	0
$826$	32915.6	1.38654
$827$	1741.41	0.0732222	0.0366111	−	0.999330i	$-0.488344\pi$
$827$	1741.41	0.0732222	0.0366111	+	0.999330i	$0.488344\pi$
$828$	0	0
$829$	−5271.84	−0.220867	−0.110433	−	0.993884i	$-0.535224\pi$
$829$	−5271.84	−0.220867	−0.110433	+	0.993884i	$0.535224\pi$
$830$	−17658.3	−0.738467
$831$	0	0
$832$	−41731.3	−1.73891
$833$	40348.3	1.67825
$834$	0	0
$835$	2758.64	0.114331
$836$	0	0
$837$	0	0
$838$	26954.6	1.11114
$839$	−24098.4	−0.991619	−0.495810	−	0.868431i	$-0.665129\pi$
$839$	−24098.4	−0.991619	−0.495810	+	0.868431i	$0.665129\pi$
$840$	0	0
$841$	−18503.3	−0.758674
$842$	−3079.70	−0.126049
$843$	0	0
$844$	−8655.70	−0.353011
$845$	−33074.1	−1.34649
$846$	0	0
$847$	0	0
$848$	−2065.74	−0.0836531
$849$	0	0
$850$	2229.58	0.0899695
$851$	11475.7	0.462257
$852$	0	0
$853$	−28409.1	−1.14034	−0.570170	−	0.821527i	$-0.693123\pi$
$853$	−28409.1	−1.14034	−0.570170	+	0.821527i	$0.693123\pi$
$854$	−33301.4	−1.33437
$855$	0	0
$856$	16231.1	0.648093
$857$	6941.16	0.276669	0.138335	−	0.990386i	$-0.455825\pi$
$857$	6941.16	0.276669	0.138335	+	0.990386i	$0.455825\pi$
$858$	0	0
$859$	21637.2	0.859430	0.429715	−	0.902965i	$-0.358614\pi$
$859$	21637.2	0.859430	0.429715	+	0.902965i	$0.358614\pi$
$860$	2658.25	0.105402
$861$	0	0
$862$	21786.7	0.860856
$863$	−7440.92	−0.293502	−0.146751	−	0.989173i	$-0.546882\pi$
$863$	−7440.92	−0.293502	−0.146751	+	0.989173i	$0.546882\pi$
$864$	0	0
$865$	−6940.39	−0.272810
$866$	−18725.7	−0.734787
$867$	0	0
$868$	1316.05	0.0514626
$869$	0	0
$870$	0	0
$871$	−16207.2	−0.630495
$872$	2087.51	0.0810688
$873$	0	0
$874$	−20655.1	−0.799394
$875$	−48234.3	−1.86356
$876$	0	0
$877$	−32311.6	−1.24411	−0.622055	−	0.782973i	$-0.713702\pi$
$877$	−32311.6	−1.24411	−0.622055	+	0.782973i	$0.713702\pi$
$878$	−23999.9	−0.922503
$879$	0	0
$880$	0	0
$881$	−21907.4	−0.837774	−0.418887	−	0.908038i	$-0.637580\pi$
$881$	−21907.4	−0.837774	−0.418887	+	0.908038i	$0.637580\pi$
$882$	0	0
$883$	−1200.30	−0.0457456	−0.0228728	−	0.999738i	$-0.507281\pi$
$883$	−1200.30	−0.0457456	−0.0228728	+	0.999738i	$0.507281\pi$
$884$	7065.44	0.268820
$885$	0	0
$886$	23237.0	0.881108
$887$	23760.5	0.899437	0.449718	−	0.893170i	$-0.351524\pi$
$887$	23760.5	0.899437	0.449718	+	0.893170i	$0.351524\pi$
$888$	0	0
$889$	−53240.8	−2.00859
$890$	−27507.1	−1.03600
$891$	0	0
$892$	6419.26	0.240956
$893$	20058.7	0.751666
$894$	0	0
$895$	−679.008	−0.0253595
$896$	26376.1	0.983443
$897$	0	0
$898$	−17158.6	−0.637627
$899$	1766.63	0.0655400
$900$	0	0
$901$	2429.65	0.0898374
$902$	0	0
$903$	0	0
$904$	10730.7	0.394798
$905$	427.361	0.0156972
$906$	0	0
$907$	−12647.9	−0.463028	−0.231514	−	0.972832i	$-0.574368\pi$
$907$	−12647.9	−0.463028	−0.231514	+	0.972832i	$0.574368\pi$
$908$	700.916	0.0256175
$909$	0	0
$910$	62621.6	2.28119
$911$	13367.4	0.486150	0.243075	−	0.970007i	$-0.421844\pi$
$911$	13367.4	0.486150	0.243075	+	0.970007i	$0.421844\pi$
$912$	0	0
$913$	0	0
$914$	−2891.93	−0.104657
$915$	0	0
$916$	5150.33	0.185777
$917$	34380.3	1.23810
$918$	0	0
$919$	−18534.6	−0.665289	−0.332645	−	0.943052i	$-0.607941\pi$
$919$	−18534.6	−0.665289	−0.332645	+	0.943052i	$0.607941\pi$
$920$	−23192.2	−0.831113
$921$	0	0
$922$	−20996.0	−0.749962
$923$	−54908.7	−1.95812
$924$	0	0
$925$	−2029.31	−0.0721332
$926$	−4505.50	−0.159892
$927$	0	0
$928$	−5940.60	−0.210140
$929$	−26233.0	−0.926454	−0.463227	−	0.886240i	$-0.653309\pi$
$929$	−26233.0	−0.926454	−0.463227	+	0.886240i	$0.653309\pi$
$930$	0	0
$931$	66099.7	2.32688
$932$	−4302.27	−0.151208
$933$	0	0
$934$	−47575.7	−1.66673
$935$	0	0
$936$	0	0
$937$	−13905.0	−0.484800	−0.242400	−	0.970176i	$-0.577935\pi$
$937$	−13905.0	−0.484800	−0.242400	+	0.970176i	$0.577935\pi$
$938$	18124.2	0.630892
$939$	0	0
$940$	4032.58	0.139924
$941$	−36656.9	−1.26990	−0.634952	−	0.772552i	$-0.718980\pi$
$941$	−36656.9	−1.26990	−0.634952	+	0.772552i	$0.718980\pi$
$942$	0	0
$943$	−12773.2	−0.441096
$944$	−18882.9	−0.651046
$945$	0	0
$946$	0	0
$947$	9454.30	0.324417	0.162209	−	0.986756i	$-0.448138\pi$
$947$	9454.30	0.324417	0.162209	+	0.986756i	$0.448138\pi$
$948$	0	0
$949$	16263.5	0.556306
$950$	3652.56	0.124742
$951$	0	0
$952$	−44128.9	−1.50234
$953$	−30562.7	−1.03885	−0.519424	−	0.854517i	$-0.673853\pi$
$953$	−30562.7	−1.03885	−0.519424	+	0.854517i	$0.673853\pi$
$954$	0	0
$955$	47787.9	1.61925
$956$	−7407.31	−0.250596
$957$	0	0
$958$	−42312.6	−1.42699
$959$	−100669.	−3.38974
$960$	0	0
$961$	−29260.7	−0.982200
$962$	23055.1	0.772689
$963$	0	0
$964$	4686.90	0.156592
$965$	52636.7	1.75589
$966$	0	0
$967$	−30119.6	−1.00164	−0.500818	−	0.865553i	$-0.666967\pi$
$967$	−30119.6	−1.00164	−0.500818	+	0.865553i	$0.666967\pi$
$968$	0	0
$969$	0	0
$970$	−6405.38	−0.212025
$971$	32210.2	1.06455	0.532274	−	0.846572i	$-0.321338\pi$
$971$	32210.2	1.06455	0.532274	+	0.846572i	$0.321338\pi$
$972$	0	0
$973$	34946.0	1.15140
$974$	23024.3	0.757441
$975$	0	0
$976$	19104.3	0.626550
$977$	8490.33	0.278024	0.139012	−	0.990291i	$-0.455607\pi$
$977$	8490.33	0.278024	0.139012	+	0.990291i	$0.455607\pi$
$978$	0	0
$979$	0	0
$980$	13288.6	0.433153
$981$	0	0
$982$	41028.4	1.33327
$983$	3376.83	0.109567	0.0547833	−	0.998498i	$-0.482553\pi$
$983$	3376.83	0.109567	0.0547833	+	0.998498i	$0.482553\pi$
$984$	0	0
$985$	17772.2	0.574892
$986$	−10606.3	−0.342570
$987$	0	0
$988$	11574.8	0.372716
$989$	13316.4	0.428145
$990$	0	0
$991$	−20081.5	−0.643703	−0.321851	−	0.946790i	$-0.604305\pi$
$991$	−20081.5	−0.643703	−0.321851	+	0.946790i	$0.604305\pi$
$992$	−1783.11	−0.0570703
$993$	0	0
$994$	61403.3	1.95935
$995$	34705.3	1.10576
$996$	0	0
$997$	−34383.3	−1.09221	−0.546103	−	0.837718i	$-0.683889\pi$
$997$	−34383.3	−1.09221	−0.546103	+	0.837718i	$0.683889\pi$
$998$	11583.2	0.367396
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.bg.1.1		4
3.2	odd	2		363.4.a.p.1.4		4
11.5	even	5		99.4.f.b.91.2		8
11.9	even	5		99.4.f.b.37.2		8
11.10	odd	2		1089.4.a.z.1.4		4
33.5	odd	10		33.4.e.b.25.1	yes	8
33.20	odd	10		33.4.e.b.4.1	✓	8
33.32	even	2		363.4.a.t.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.e.b.4.1	✓	8	33.20	odd	10
33.4.e.b.25.1	yes	8	33.5	odd	10
99.4.f.b.37.2		8	11.9	even	5
99.4.f.b.91.2		8	11.5	even	5
363.4.a.p.1.4		4	3.2	odd	2
363.4.a.t.1.1		4	33.32	even	2
1089.4.a.z.1.4		4	11.10	odd	2
1089.4.a.bg.1.1		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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q-2.50105 q^{2} -1.74477 q^{4} -10.4342 q^{5} -32.7556 q^{7} +24.3721 q^{8} +O(q^{10})\)	\(q-2.50105 q^{2} -1.74477 q^{4} -10.4342 q^{5} -32.7556 q^{7} +24.3721 q^{8} +26.0964 q^{10} -73.2583 q^{13} +81.9233 q^{14} -46.9976 q^{16} +55.2769 q^{17} +90.5561 q^{19} +18.2053 q^{20} +91.1987 q^{23} -16.1272 q^{25} +183.222 q^{26} +57.1511 q^{28} +76.7184 q^{29} +23.0275 q^{31} -77.4339 q^{32} -138.250 q^{34} +341.779 q^{35} +125.831 q^{37} -226.485 q^{38} -254.304 q^{40} -140.059 q^{41} +146.015 q^{43} -228.092 q^{46} +221.506 q^{47} +729.931 q^{49} +40.3348 q^{50} +127.819 q^{52} +43.9542 q^{53} -798.324 q^{56} -191.876 q^{58} +401.785 q^{59} -406.495 q^{61} -57.5928 q^{62} +569.646 q^{64} +764.393 q^{65} +221.234 q^{67} -96.4457 q^{68} -854.805 q^{70} +749.522 q^{71} -222.002 q^{73} -314.710 q^{74} -158.000 q^{76} -1180.26 q^{79} +490.383 q^{80} +350.295 q^{82} -676.655 q^{83} -576.771 q^{85} -365.189 q^{86} -1054.06 q^{89} +2399.62 q^{91} -159.121 q^{92} -553.995 q^{94} -944.882 q^{95} -245.450 q^{97} -1825.59 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 11 q^{7} + 42 q^{8}+O(q^{10}) \) 4 * q + 3 * q^2 - 3 * q^4 - 12 * q^5 - 11 * q^7 + 42 * q^8	\( 4 q + 3 q^{2} - 3 q^{4} - 12 q^{5} - 11 q^{7} + 42 q^{8} + 4 q^{10} - 182 q^{13} + 144 q^{14} + 97 q^{16} + 57 q^{17} - 173 q^{19} - 99 q^{20} + 42 q^{23} - 186 q^{25} - 273 q^{26} + 116 q^{28} + 651 q^{29} - 170 q^{31} - 51 q^{32} - 356 q^{34} + 669 q^{35} + 244 q^{37} - 927 q^{38} - 532 q^{40} - 102 q^{41} - 322 q^{43} + 9 q^{46} + 633 q^{47} + 689 q^{49} - 390 q^{50} - 1037 q^{52} + 468 q^{53} - 735 q^{56} + 675 q^{58} + 996 q^{59} - 413 q^{61} - 87 q^{62} - 1044 q^{64} + 477 q^{65} - 259 q^{67} - 1119 q^{68} - 937 q^{70} - 237 q^{71} - 2309 q^{73} - 399 q^{74} - 1138 q^{76} - 2045 q^{79} + 510 q^{80} - 1383 q^{82} + 639 q^{83} - 1036 q^{85} + 669 q^{86} - 894 q^{89} - 400 q^{91} + 1761 q^{92} + 792 q^{94} - 537 q^{95} + 1432 q^{97} - 2775 q^{98}+O(q^{100}) \) 4 * q + 3 * q^2 - 3 * q^4 - 12 * q^5 - 11 * q^7 + 42 * q^8 + 4 * q^10 - 182 * q^13 + 144 * q^14 + 97 * q^16 + 57 * q^17 - 173 * q^19 - 99 * q^20 + 42 * q^23 - 186 * q^25 - 273 * q^26 + 116 * q^28 + 651 * q^29 - 170 * q^31 - 51 * q^32 - 356 * q^34 + 669 * q^35 + 244 * q^37 - 927 * q^38 - 532 * q^40 - 102 * q^41 - 322 * q^43 + 9 * q^46 + 633 * q^47 + 689 * q^49 - 390 * q^50 - 1037 * q^52 + 468 * q^53 - 735 * q^56 + 675 * q^58 + 996 * q^59 - 413 * q^61 - 87 * q^62 - 1044 * q^64 + 477 * q^65 - 259 * q^67 - 1119 * q^68 - 937 * q^70 - 237 * q^71 - 2309 * q^73 - 399 * q^74 - 1138 * q^76 - 2045 * q^79 + 510 * q^80 - 1383 * q^82 + 639 * q^83 - 1036 * q^85 + 669 * q^86 - 894 * q^89 - 400 * q^91 + 1761 * q^92 + 792 * q^94 - 537 * q^95 + 1432 * q^97 - 2775 * q^98

Introduction

L-functions

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