LMFDB - Embedded newform 1815.4.a.bb.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1815,4,Mod(1,1815)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1815, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1815.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1815 = 3 \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1815.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:	$107.088466660$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 44x^{4} + 495x^{2} - 1200 $ x^6 - 44x^4 + 495x^2 - 1200
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.60164$ of defining polynomial
Character	$\chi$	$=$	1815.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.60164 q^{2} +3.00000 q^{3} +4.97180 q^{4} +5.00000 q^{5} +10.8049 q^{6} +31.6129 q^{7} -10.9065 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.60164 q^{2} +3.00000 q^{3} +4.97180 q^{4} +5.00000 q^{5} +10.8049 q^{6} +31.6129 q^{7} -10.9065 q^{8} +9.00000 q^{9} +18.0082 q^{10} +14.9154 q^{12} +38.8162 q^{13} +113.858 q^{14} +15.0000 q^{15} -79.0556 q^{16} +138.261 q^{17} +32.4148 q^{18} +12.5997 q^{19} +24.8590 q^{20} +94.8387 q^{21} -179.716 q^{23} -32.7194 q^{24} +25.0000 q^{25} +139.802 q^{26} +27.0000 q^{27} +157.173 q^{28} +119.045 q^{29} +54.0246 q^{30} +75.6616 q^{31} -197.478 q^{32} +497.968 q^{34} +158.064 q^{35} +44.7462 q^{36} -133.716 q^{37} +45.3797 q^{38} +116.449 q^{39} -54.5324 q^{40} +47.0124 q^{41} +341.575 q^{42} -74.0172 q^{43} +45.0000 q^{45} -647.274 q^{46} -375.658 q^{47} -237.167 q^{48} +656.375 q^{49} +90.0410 q^{50} +414.784 q^{51} +192.986 q^{52} -76.7013 q^{53} +97.2443 q^{54} -344.785 q^{56} +37.7992 q^{57} +428.758 q^{58} +866.360 q^{59} +74.5770 q^{60} +442.580 q^{61} +272.506 q^{62} +284.516 q^{63} -78.7994 q^{64} +194.081 q^{65} -633.006 q^{67} +687.409 q^{68} -539.149 q^{69} +569.291 q^{70} -484.728 q^{71} -98.1583 q^{72} -436.143 q^{73} -481.598 q^{74} +75.0000 q^{75} +62.6434 q^{76} +419.405 q^{78} -472.439 q^{79} -395.278 q^{80} +81.0000 q^{81} +169.322 q^{82} +249.160 q^{83} +471.519 q^{84} +691.307 q^{85} -266.583 q^{86} +357.136 q^{87} +236.982 q^{89} +162.074 q^{90} +1227.09 q^{91} -893.515 q^{92} +226.985 q^{93} -1352.99 q^{94} +62.9986 q^{95} -592.434 q^{96} +1522.39 q^{97} +2364.03 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) $ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9	$ 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) $ 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * q^97

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 $

$2$	3.60164	1.27337	0.636686	−	0.771123i	$-0.280305\pi$
$2$	3.60164	1.27337	0.636686	+	0.771123i	$0.280305\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.97180	0.621475
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	10.8049	0.735181
$7$	31.6129	1.70694	0.853468	−	0.521146i	$-0.174495\pi$
$7$	31.6129	1.70694	0.853468	+	0.521146i	$0.174495\pi$
$8$	−10.9065	−0.482003
$9$	9.00000	0.333333
$10$	18.0082	0.569469
$11$	0	0
$11$	0	0
$12$	14.9154	0.358809
$13$	38.8162	0.828128	0.414064	−	0.910248i	$-0.364109\pi$
$13$	38.8162	0.828128	0.414064	+	0.910248i	$0.364109\pi$
$14$	113.858	2.17356
$15$	15.0000	0.258199
$16$	−79.0556	−1.23524
$17$	138.261	1.97255	0.986274	−	0.165115i	$-0.0527995\pi$
$17$	138.261	1.97255	0.986274	+	0.165115i	$0.0527995\pi$
$18$	32.4148	0.424457
$19$	12.5997	0.152136	0.0760678	−	0.997103i	$-0.475763\pi$
$19$	12.5997	0.152136	0.0760678	+	0.997103i	$0.475763\pi$
$20$	24.8590	0.277932
$21$	94.8387	0.985500
$22$	0	0
$23$	−179.716	−1.62928	−0.814641	−	0.579966i	$-0.803066\pi$
$23$	−179.716	−1.62928	−0.814641	+	0.579966i	$0.803066\pi$
$24$	−32.7194	−0.278284
$25$	25.0000	0.200000
$26$	139.802	1.05452
$27$	27.0000	0.192450
$28$	157.173	1.06082
$29$	119.045	0.762280	0.381140	−	0.924517i	$-0.375532\pi$
$29$	119.045	0.762280	0.381140	+	0.924517i	$0.375532\pi$
$30$	54.0246	0.328783
$31$	75.6616	0.438362	0.219181	−	0.975684i	$-0.429661\pi$
$31$	75.6616	0.438362	0.219181	+	0.975684i	$0.429661\pi$
$32$	−197.478	−1.09092
$33$	0	0
$34$	497.968	2.51179
$35$	158.064	0.763365
$36$	44.7462	0.207158
$37$	−133.716	−0.594131	−0.297066	−	0.954857i	$-0.596008\pi$
$37$	−133.716	−0.594131	−0.297066	+	0.954857i	$0.596008\pi$
$38$	45.3797	0.193725
$39$	116.449	0.478120
$40$	−54.5324	−0.215558
$41$	47.0124	0.179076	0.0895378	−	0.995983i	$-0.471461\pi$
$41$	47.0124	0.179076	0.0895378	+	0.995983i	$0.471461\pi$
$42$	341.575	1.25491
$43$	−74.0172	−0.262501	−0.131250	−	0.991349i	$-0.541899\pi$
$43$	−74.0172	−0.262501	−0.131250	+	0.991349i	$0.541899\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	−647.274	−2.07468
$47$	−375.658	−1.16586	−0.582930	−	0.812522i	$-0.698094\pi$
$47$	−375.658	−1.16586	−0.582930	+	0.812522i	$0.698094\pi$
$48$	−237.167	−0.713168
$49$	656.375	1.91363
$50$	90.0410	0.254674
$51$	414.784	1.13885
$52$	192.986	0.514661
$53$	−76.7013	−0.198788	−0.0993938	−	0.995048i	$-0.531690\pi$
$53$	−76.7013	−0.198788	−0.0993938	+	0.995048i	$0.531690\pi$
$54$	97.2443	0.245060
$55$	0	0
$56$	−344.785	−0.822747
$57$	37.7992	0.0878355
$58$	428.758	0.970666
$59$	866.360	1.91170	0.955851	−	0.293851i	$-0.0949370\pi$
$59$	866.360	1.91170	0.955851	+	0.293851i	$0.0949370\pi$
$60$	74.5770	0.160464
$61$	442.580	0.928961	0.464481	−	0.885583i	$-0.346241\pi$
$61$	442.580	0.928961	0.464481	+	0.885583i	$0.346241\pi$
$62$	272.506	0.558198
$63$	284.516	0.568979
$64$	−78.7994	−0.153905
$65$	194.081	0.370350
$66$	0	0
$67$	−633.006	−1.15424	−0.577120	−	0.816660i	$-0.695823\pi$
$67$	−633.006	−1.15424	−0.577120	+	0.816660i	$0.695823\pi$
$68$	687.409	1.22589
$69$	−539.149	−0.940666
$70$	569.291	0.972047
$71$	−484.728	−0.810235	−0.405117	−	0.914265i	$-0.632769\pi$
$71$	−484.728	−0.810235	−0.405117	+	0.914265i	$0.632769\pi$
$72$	−98.1583	−0.160668
$73$	−436.143	−0.699270	−0.349635	−	0.936886i	$-0.613694\pi$
$73$	−436.143	−0.699270	−0.349635	+	0.936886i	$0.613694\pi$
$74$	−481.598	−0.756550
$75$	75.0000	0.115470
$76$	62.6434	0.0945485
$77$	0	0
$78$	419.405	0.608825
$79$	−472.439	−0.672830	−0.336415	−	0.941714i	$-0.609214\pi$
$79$	−472.439	−0.672830	−0.336415	+	0.941714i	$0.609214\pi$
$80$	−395.278	−0.552418
$81$	81.0000	0.111111
$82$	169.322	0.228030
$83$	249.160	0.329504	0.164752	−	0.986335i	$-0.447318\pi$
$83$	249.160	0.329504	0.164752	+	0.986335i	$0.447318\pi$
$84$	471.519	0.612464
$85$	691.307	0.882151
$86$	−266.583	−0.334261
$87$	357.136	0.440103
$88$	0	0
$89$	236.982	0.282247	0.141124	−	0.989992i	$-0.454928\pi$
$89$	236.982	0.282247	0.141124	+	0.989992i	$0.454928\pi$
$90$	162.074	0.189823
$91$	1227.09	1.41356
$92$	−893.515	−1.01256
$93$	226.985	0.253089
$94$	−1352.99	−1.48457
$95$	62.9986	0.0680371
$96$	−592.434	−0.629844
$97$	1522.39	1.59356	0.796778	−	0.604272i	$-0.206536\pi$
$97$	1522.39	1.59356	0.796778	+	0.604272i	$0.206536\pi$
$98$	2364.03	2.43676
$99$	0	0
$100$	124.295	0.124295
$101$	−1551.66	−1.52867	−0.764335	−	0.644819i	$-0.776933\pi$
$101$	−1551.66	−1.52867	−0.764335	+	0.644819i	$0.776933\pi$
$102$	1493.90	1.45018
$103$	−1454.92	−1.39182	−0.695911	−	0.718128i	$-0.744999\pi$
$103$	−1454.92	−1.39182	−0.695911	+	0.718128i	$0.744999\pi$
$104$	−423.347	−0.399160
$105$	474.193	0.440729
$106$	−276.251	−0.253130
$107$	179.206	0.161912	0.0809558	−	0.996718i	$-0.474203\pi$
$107$	179.206	0.161912	0.0809558	+	0.996718i	$0.474203\pi$
$108$	134.239	0.119603
$109$	2011.72	1.76777	0.883887	−	0.467700i	$-0.154917\pi$
$109$	2011.72	1.76777	0.883887	+	0.467700i	$0.154917\pi$
$110$	0	0
$111$	−401.149	−0.343022
$112$	−2499.18	−2.10848
$113$	2041.59	1.69962	0.849808	−	0.527092i	$-0.176718\pi$
$113$	2041.59	1.69962	0.849808	+	0.527092i	$0.176718\pi$
$114$	136.139	0.111847
$115$	−898.582	−0.728637
$116$	591.869	0.473739
$117$	349.346	0.276043
$118$	3120.32	2.43431
$119$	4370.84	3.36701
$120$	−163.597	−0.124453
$121$	0	0
$122$	1594.02	1.18291
$123$	141.037	0.103389
$124$	376.175	0.272431
$125$	125.000	0.0894427
$126$	1024.72	0.724521
$127$	−804.825	−0.562336	−0.281168	−	0.959659i	$-0.590722\pi$
$127$	−804.825	−0.562336	−0.281168	+	0.959659i	$0.590722\pi$
$128$	1296.02	0.894943
$129$	−222.052	−0.151555
$130$	699.009	0.471594
$131$	1699.61	1.13356	0.566778	−	0.823871i	$-0.308190\pi$
$131$	1699.61	1.13356	0.566778	+	0.823871i	$0.308190\pi$
$132$	0	0
$133$	398.314	0.259686
$134$	−2279.86	−1.46978
$135$	135.000	0.0860663
$136$	−1507.94	−0.950774
$137$	1187.12	0.740311	0.370156	−	0.928970i	$-0.379304\pi$
$137$	1187.12	0.740311	0.370156	+	0.928970i	$0.379304\pi$
$138$	−1941.82	−1.19782
$139$	−2687.64	−1.64002	−0.820010	−	0.572350i	$-0.806032\pi$
$139$	−2687.64	−1.64002	−0.820010	+	0.572350i	$0.806032\pi$
$140$	785.865	0.474412
$141$	−1126.98	−0.673110
$142$	−1745.82	−1.03173
$143$	0	0
$144$	−711.500	−0.411748
$145$	595.226	0.340902
$146$	−1570.83	−0.890430
$147$	1969.12	1.10483
$148$	−664.812	−0.369238
$149$	2893.54	1.59092	0.795462	−	0.606004i	$-0.207228\pi$
$149$	2893.54	1.59092	0.795462	+	0.606004i	$0.207228\pi$
$150$	270.123	0.147036
$151$	25.1251	0.0135408	0.00677038	−	0.999977i	$-0.497845\pi$
$151$	25.1251	0.0135408	0.00677038	+	0.999977i	$0.497845\pi$
$152$	−137.419	−0.0733297
$153$	1244.35	0.657516
$154$	0	0
$155$	378.308	0.196042
$156$	578.959	0.297140
$157$	−1109.99	−0.564246	−0.282123	−	0.959378i	$-0.591039\pi$
$157$	−1109.99	−0.564246	−0.282123	+	0.959378i	$0.591039\pi$
$158$	−1701.55	−0.856762
$159$	−230.104	−0.114770
$160$	−987.390	−0.487875
$161$	−5681.36	−2.78108
$162$	291.733	0.141486
$163$	−3513.43	−1.68830	−0.844151	−	0.536105i	$-0.819895\pi$
$163$	−3513.43	−1.68830	−0.844151	+	0.536105i	$0.819895\pi$
$164$	233.736	0.111291
$165$	0	0
$166$	897.384	0.419581
$167$	−3213.84	−1.48919	−0.744594	−	0.667517i	$-0.767357\pi$
$167$	−3213.84	−1.48919	−0.744594	+	0.667517i	$0.767357\pi$
$168$	−1034.36	−0.475013
$169$	−690.305	−0.314203
$170$	2489.84	1.12331
$171$	113.398	0.0507118
$172$	−367.999	−0.163138
$173$	2783.27	1.22317	0.611583	−	0.791180i	$-0.290533\pi$
$173$	2783.27	1.22317	0.611583	+	0.791180i	$0.290533\pi$
$174$	1286.27	0.560414
$175$	790.322	0.341387
$176$	0	0
$177$	2599.08	1.10372
$178$	853.523	0.359406
$179$	1532.11	0.639751	0.319876	−	0.947460i	$-0.396359\pi$
$179$	1532.11	0.639751	0.319876	+	0.947460i	$0.396359\pi$
$180$	223.731	0.0926441
$181$	−23.1715	−0.00951561	−0.00475780	−	0.999989i	$-0.501514\pi$
$181$	−23.1715	−0.00951561	−0.00475780	+	0.999989i	$0.501514\pi$
$182$	4419.54	1.79999
$183$	1327.74	0.536336
$184$	1960.07	0.785318
$185$	−668.582	−0.265703
$186$	817.518	0.322276
$187$	0	0
$188$	−1867.70	−0.724553
$189$	853.548	0.328500
$190$	226.898	0.0866365
$191$	−4258.34	−1.61321	−0.806604	−	0.591092i	$-0.798697\pi$
$191$	−4258.34	−1.61321	−0.806604	+	0.591092i	$0.798697\pi$
$192$	−236.398	−0.0888572
$193$	−122.310	−0.0456169	−0.0228085	−	0.999740i	$-0.507261\pi$
$193$	−122.310	−0.0456169	−0.0228085	+	0.999740i	$0.507261\pi$
$194$	5483.09	2.02919
$195$	582.243	0.213822
$196$	3263.37	1.18927
$197$	1512.84	0.547134	0.273567	−	0.961853i	$-0.411797\pi$
$197$	1512.84	0.547134	0.273567	+	0.961853i	$0.411797\pi$
$198$	0	0
$199$	1398.64	0.498227	0.249114	−	0.968474i	$-0.419861\pi$
$199$	1398.64	0.498227	0.249114	+	0.968474i	$0.419861\pi$
$200$	−272.662	−0.0964005
$201$	−1899.02	−0.666400
$202$	−5588.51	−1.94657
$203$	3763.36	1.30116
$204$	2062.23	0.707768
$205$	235.062	0.0800851
$206$	−5240.10	−1.77231
$207$	−1617.45	−0.543094
$208$	−3068.64	−1.02294
$209$	0	0
$210$	1707.87	0.561212
$211$	609.235	0.198775	0.0993873	−	0.995049i	$-0.468312\pi$
$211$	609.235	0.198775	0.0993873	+	0.995049i	$0.468312\pi$
$212$	−381.344	−0.123542
$213$	−1454.19	−0.467789
$214$	645.437	0.206174
$215$	−370.086	−0.117394
$216$	−294.475	−0.0927614
$217$	2391.88	0.748256
$218$	7245.47	2.25103
$219$	−1308.43	−0.403723
$220$	0	0
$221$	5366.78	1.63352
$222$	−1444.79	−0.436794
$223$	−2983.97	−0.896060	−0.448030	−	0.894019i	$-0.647874\pi$
$223$	−2983.97	−0.896060	−0.448030	+	0.894019i	$0.647874\pi$
$224$	−6242.85	−1.86213
$225$	225.000	0.0666667
$226$	7353.07	2.16424
$227$	−762.600	−0.222976	−0.111488	−	0.993766i	$-0.535562\pi$
$227$	−762.600	−0.222976	−0.111488	+	0.993766i	$0.535562\pi$
$228$	187.930	0.0545876
$229$	−4537.22	−1.30929	−0.654646	−	0.755935i	$-0.727182\pi$
$229$	−4537.22	−1.30929	−0.654646	+	0.755935i	$0.727182\pi$
$230$	−3236.37	−0.927825
$231$	0	0
$232$	−1298.36	−0.367421
$233$	399.725	0.112390	0.0561950	−	0.998420i	$-0.482103\pi$
$233$	399.725	0.112390	0.0561950	+	0.998420i	$0.482103\pi$
$234$	1258.22	0.351505
$235$	−1878.29	−0.521388
$236$	4307.37	1.18808
$237$	−1417.32	−0.388458
$238$	15742.2	4.28746
$239$	−3303.30	−0.894029	−0.447015	−	0.894527i	$-0.647513\pi$
$239$	−3303.30	−0.894029	−0.447015	+	0.894527i	$0.647513\pi$
$240$	−1185.83	−0.318939
$241$	2398.59	0.641107	0.320553	−	0.947230i	$-0.396131\pi$
$241$	2398.59	0.641107	0.320553	+	0.947230i	$0.396131\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	2200.42	0.577326
$245$	3281.87	0.855801
$246$	507.965	0.131653
$247$	489.073	0.125988
$248$	−825.202	−0.211292
$249$	747.480	0.190239
$250$	450.205	0.113894
$251$	2613.23	0.657153	0.328577	−	0.944477i	$-0.393431\pi$
$251$	2613.23	0.657153	0.328577	+	0.944477i	$0.393431\pi$
$252$	1414.56	0.353606
$253$	0	0
$254$	−2898.69	−0.716063
$255$	2073.92	0.509310
$256$	5298.18	1.29350
$257$	5774.18	1.40149	0.700746	−	0.713411i	$-0.252851\pi$
$257$	5774.18	1.40149	0.700746	+	0.713411i	$0.252851\pi$
$258$	−799.750	−0.192986
$259$	−4227.16	−1.01414
$260$	964.932	0.230164
$261$	1071.41	0.254093
$262$	6121.39	1.44344
$263$	−6772.15	−1.58779	−0.793895	−	0.608055i	$-0.791950\pi$
$263$	−6772.15	−1.58779	−0.793895	+	0.608055i	$0.791950\pi$
$264$	0	0
$265$	−383.507	−0.0889005
$266$	1434.58	0.330676
$267$	710.945	0.162956
$268$	−3147.18	−0.717331
$269$	−3059.42	−0.693442	−0.346721	−	0.937968i	$-0.612705\pi$
$269$	−3059.42	−0.693442	−0.346721	+	0.937968i	$0.612705\pi$
$270$	486.221	0.109594
$271$	−1420.57	−0.318427	−0.159213	−	0.987244i	$-0.550896\pi$
$271$	−1420.57	−0.318427	−0.159213	+	0.987244i	$0.550896\pi$
$272$	−10930.3	−2.43658
$273$	3681.27	0.816120
$274$	4275.58	0.942691
$275$	0	0
$276$	−2680.54	−0.584601
$277$	614.227	0.133232	0.0666161	−	0.997779i	$-0.478780\pi$
$277$	614.227	0.133232	0.0666161	+	0.997779i	$0.478780\pi$
$278$	−9679.91	−2.08835
$279$	680.955	0.146121
$280$	−1723.93	−0.367944
$281$	5797.52	1.23079	0.615393	−	0.788220i	$-0.288997\pi$
$281$	5797.52	1.23079	0.615393	+	0.788220i	$0.288997\pi$
$282$	−4058.96	−0.857119
$283$	4077.91	0.856560	0.428280	−	0.903646i	$-0.359120\pi$
$283$	4077.91	0.856560	0.428280	+	0.903646i	$0.359120\pi$
$284$	−2409.97	−0.503541
$285$	188.996	0.0392812
$286$	0	0
$287$	1486.20	0.305671
$288$	−1777.30	−0.363641
$289$	14203.2	2.89095
$290$	2143.79	0.434095
$291$	4567.16	0.920040
$292$	−2168.42	−0.434579
$293$	−5029.67	−1.00286	−0.501428	−	0.865200i	$-0.667192\pi$
$293$	−5029.67	−1.00286	−0.501428	+	0.865200i	$0.667192\pi$
$294$	7092.08	1.40686
$295$	4331.80	0.854939
$296$	1458.37	0.286373
$297$	0	0
$298$	10421.5	2.02584
$299$	−6975.90	−1.34925
$300$	372.885	0.0717618
$301$	−2339.90	−0.448072
$302$	90.4917	0.0172424
$303$	−4654.97	−0.882579
$304$	−996.079	−0.187924
$305$	2212.90	0.415444
$306$	4481.71	0.837263
$307$	3667.16	0.681746	0.340873	−	0.940109i	$-0.389277\pi$
$307$	3667.16	0.681746	0.340873	+	0.940109i	$0.389277\pi$
$308$	0	0
$309$	−4364.76	−0.803568
$310$	1362.53	0.249634
$311$	−2328.46	−0.424549	−0.212274	−	0.977210i	$-0.568087\pi$
$311$	−2328.46	−0.424549	−0.212274	+	0.977210i	$0.568087\pi$
$312$	−1270.04	−0.230455
$313$	5698.19	1.02901	0.514506	−	0.857487i	$-0.327975\pi$
$313$	5698.19	1.02901	0.514506	+	0.857487i	$0.327975\pi$
$314$	−3997.77	−0.718495
$315$	1422.58	0.254455
$316$	−2348.87	−0.418147
$317$	3389.61	0.600567	0.300283	−	0.953850i	$-0.402919\pi$
$317$	3389.61	0.600567	0.300283	+	0.953850i	$0.402919\pi$
$318$	−828.752	−0.146145
$319$	0	0
$320$	−393.997	−0.0688285
$321$	537.619	0.0934797
$322$	−20462.2	−3.54135
$323$	1742.06	0.300095
$324$	402.716	0.0690528
$325$	970.404	0.165626
$326$	−12654.1	−2.14984
$327$	6035.15	1.02063
$328$	−512.739	−0.0863149
$329$	−11875.6	−1.99005
$330$	0	0
$331$	10574.4	1.75595	0.877977	−	0.478702i	$-0.158893\pi$
$331$	10574.4	1.75595	0.877977	+	0.478702i	$0.158893\pi$
$332$	1238.77	0.204779
$333$	−1203.45	−0.198044
$334$	−11575.1	−1.89629
$335$	−3165.03	−0.516191
$336$	−7497.53	−1.21733
$337$	−1929.14	−0.311831	−0.155916	−	0.987770i	$-0.549833\pi$
$337$	−1929.14	−0.311831	−0.155916	+	0.987770i	$0.549833\pi$
$338$	−2486.23	−0.400098
$339$	6124.77	0.981274
$340$	3437.04	0.548235
$341$	0	0
$342$	408.417	0.0645750
$343$	9906.69	1.55951
$344$	807.267	0.126526
$345$	−2695.75	−0.420679
$346$	10024.3	1.55755
$347$	−10407.4	−1.61008	−0.805039	−	0.593221i	$-0.797856\pi$
$347$	−10407.4	−1.61008	−0.805039	+	0.593221i	$0.797856\pi$
$348$	1775.61	0.273513
$349$	−11338.1	−1.73901	−0.869503	−	0.493928i	$-0.835561\pi$
$349$	−11338.1	−1.73901	−0.869503	+	0.493928i	$0.835561\pi$
$350$	2846.46	0.434713
$351$	1048.04	0.159373
$352$	0	0
$353$	−3084.18	−0.465027	−0.232513	−	0.972593i	$-0.574695\pi$
$353$	−3084.18	−0.465027	−0.232513	+	0.972593i	$0.574695\pi$
$354$	9360.95	1.40545
$355$	−2423.64	−0.362348
$356$	1178.23	0.175410
$357$	13112.5	1.94395
$358$	5518.12	0.814641
$359$	2292.87	0.337083	0.168542	−	0.985695i	$-0.446094\pi$
$359$	2292.87	0.337083	0.168542	+	0.985695i	$0.446094\pi$
$360$	−490.791	−0.0718527
$361$	−6700.25	−0.976855
$362$	−83.4554	−0.0121169
$363$	0	0
$364$	6100.86	0.878494
$365$	−2180.71	−0.312723
$366$	4782.05	0.682955
$367$	−1121.90	−0.159572	−0.0797858	−	0.996812i	$-0.525424\pi$
$367$	−1121.90	−0.159572	−0.0797858	+	0.996812i	$0.525424\pi$
$368$	14207.6	2.01256
$369$	423.112	0.0596919
$370$	−2407.99	−0.338339
$371$	−2424.75	−0.339318
$372$	1128.52	0.157288
$373$	−10327.8	−1.43365	−0.716826	−	0.697253i	$-0.754406\pi$
$373$	−10327.8	−1.43365	−0.716826	+	0.697253i	$0.754406\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	4097.11	0.561947
$377$	4620.88	0.631266
$378$	3074.17	0.418302
$379$	−3836.25	−0.519934	−0.259967	−	0.965617i	$-0.583712\pi$
$379$	−3836.25	−0.519934	−0.259967	+	0.965617i	$0.583712\pi$
$380$	313.217	0.0422834
$381$	−2414.48	−0.324665
$382$	−15337.0	−2.05421
$383$	−9206.79	−1.22832	−0.614158	−	0.789183i	$-0.710504\pi$
$383$	−9206.79	−1.22832	−0.614158	+	0.789183i	$0.710504\pi$
$384$	3888.05	0.516696
$385$	0	0
$386$	−440.517	−0.0580873
$387$	−666.155	−0.0875002
$388$	7569.01	0.990356
$389$	−3057.27	−0.398483	−0.199241	−	0.979950i	$-0.563848\pi$
$389$	−3057.27	−0.398483	−0.199241	+	0.979950i	$0.563848\pi$
$390$	2097.03	0.272275
$391$	−24847.9	−3.21384
$392$	−7158.73	−0.922374
$393$	5098.84	0.654459
$394$	5448.70	0.696704
$395$	−2362.19	−0.300899
$396$	0	0
$397$	−9682.72	−1.22409	−0.612043	−	0.790825i	$-0.709652\pi$
$397$	−9682.72	−1.22409	−0.612043	+	0.790825i	$0.709652\pi$
$398$	5037.41	0.634428
$399$	1194.94	0.149930
$400$	−1976.39	−0.247049
$401$	−924.364	−0.115114	−0.0575568	−	0.998342i	$-0.518331\pi$
$401$	−924.364	−0.115114	−0.0575568	+	0.998342i	$0.518331\pi$
$402$	−6839.58	−0.848575
$403$	2936.89	0.363020
$404$	−7714.54	−0.950031
$405$	405.000	0.0496904
$406$	13554.3	1.65687
$407$	0	0
$408$	−4523.83	−0.548929
$409$	4949.78	0.598413	0.299207	−	0.954188i	$-0.403278\pi$
$409$	4949.78	0.598413	0.299207	+	0.954188i	$0.403278\pi$
$410$	846.608	0.101978
$411$	3561.36	0.427419
$412$	−7233.58	−0.864983
$413$	27388.1	3.26315
$414$	−5825.46	−0.691560
$415$	1245.80	0.147359
$416$	−7665.34	−0.903423
$417$	−8062.92	−0.946865
$418$	0	0
$419$	−8647.87	−1.00830	−0.504148	−	0.863617i	$-0.668193\pi$
$419$	−8647.87	−1.00830	−0.504148	+	0.863617i	$0.668193\pi$
$420$	2357.60	0.273902
$421$	−9322.71	−1.07924	−0.539621	−	0.841908i	$-0.681432\pi$
$421$	−9322.71	−1.07924	−0.539621	+	0.841908i	$0.681432\pi$
$422$	2194.24	0.253114
$423$	−3380.93	−0.388620
$424$	836.541	0.0958161
$425$	3456.54	0.394510
$426$	−5237.45	−0.595670
$427$	13991.2	1.58568
$428$	890.979	0.100624
$429$	0	0
$430$	−1332.92	−0.149486
$431$	−8160.16	−0.911975	−0.455987	−	0.889986i	$-0.650714\pi$
$431$	−8160.16	−0.911975	−0.455987	+	0.889986i	$0.650714\pi$
$432$	−2134.50	−0.237723
$433$	−7199.03	−0.798991	−0.399496	−	0.916735i	$-0.630815\pi$
$433$	−7199.03	−0.798991	−0.399496	+	0.916735i	$0.630815\pi$
$434$	8614.70	0.952808
$435$	1785.68	0.196820
$436$	10001.9	1.09863
$437$	−2264.38	−0.247872
$438$	−4712.49	−0.514090
$439$	−10297.5	−1.11953	−0.559766	−	0.828651i	$-0.689109\pi$
$439$	−10297.5	−1.11953	−0.559766	+	0.828651i	$0.689109\pi$
$440$	0	0
$441$	5907.37	0.637876
$442$	19329.2	2.08008
$443$	12847.2	1.37786	0.688928	−	0.724830i	$-0.258082\pi$
$443$	12847.2	1.37786	0.688928	+	0.724830i	$0.258082\pi$
$444$	−1994.44	−0.213180
$445$	1184.91	0.126225
$446$	−10747.2	−1.14102
$447$	8680.61	0.918520
$448$	−2491.08	−0.262706
$449$	12493.1	1.31311	0.656555	−	0.754278i	$-0.272013\pi$
$449$	12493.1	1.31311	0.656555	+	0.754278i	$0.272013\pi$
$450$	810.369	0.0848914
$451$	0	0
$452$	10150.4	1.05627
$453$	75.3754	0.00781776
$454$	−2746.61	−0.283931
$455$	6135.46	0.632164
$456$	−412.256	−0.0423369
$457$	−14968.1	−1.53212	−0.766061	−	0.642767i	$-0.777786\pi$
$457$	−14968.1	−1.53212	−0.766061	+	0.642767i	$0.777786\pi$
$458$	−16341.4	−1.66722
$459$	3733.06	0.379617
$460$	−4467.57	−0.452830
$461$	−13649.5	−1.37900	−0.689501	−	0.724285i	$-0.742170\pi$
$461$	−13649.5	−1.37900	−0.689501	+	0.724285i	$0.742170\pi$
$462$	0	0
$463$	−4655.26	−0.467275	−0.233638	−	0.972324i	$-0.575063\pi$
$463$	−4655.26	−0.467275	−0.233638	+	0.972324i	$0.575063\pi$
$464$	−9411.19	−0.941602
$465$	1134.92	0.113185
$466$	1439.67	0.143114
$467$	−1053.94	−0.104434	−0.0522170	−	0.998636i	$-0.516629\pi$
$467$	−1053.94	−0.104434	−0.0522170	+	0.998636i	$0.516629\pi$
$468$	1736.88	0.171554
$469$	−20011.2	−1.97021
$470$	−6764.93	−0.663921
$471$	−3329.96	−0.325768
$472$	−9448.93	−0.921445
$473$	0	0
$474$	−5104.66	−0.494652
$475$	314.993	0.0304271
$476$	21731.0	2.09252
$477$	−690.312	−0.0662625
$478$	−11897.3	−1.13843
$479$	−2045.84	−0.195150	−0.0975749	−	0.995228i	$-0.531109\pi$
$479$	−2045.84	−0.195150	−0.0975749	+	0.995228i	$0.531109\pi$
$480$	−2962.17	−0.281675
$481$	−5190.36	−0.492017
$482$	8638.85	0.816367
$483$	−17044.1	−1.60566
$484$	0	0
$485$	7611.93	0.712660
$486$	875.198	0.0816868
$487$	11771.6	1.09532	0.547661	−	0.836700i	$-0.315518\pi$
$487$	11771.6	1.09532	0.547661	+	0.836700i	$0.315518\pi$
$488$	−4826.99	−0.447762
$489$	−10540.3	−0.974742
$490$	11820.1	1.08975
$491$	−7886.33	−0.724858	−0.362429	−	0.932011i	$-0.618052\pi$
$491$	−7886.33	−0.724858	−0.362429	+	0.932011i	$0.618052\pi$
$492$	701.209	0.0642540
$493$	16459.4	1.50364
$494$	1761.46	0.160429
$495$	0	0
$496$	−5981.48	−0.541484
$497$	−15323.7	−1.38302
$498$	2692.15	0.242245
$499$	13390.6	1.20129	0.600645	−	0.799516i	$-0.294910\pi$
$499$	13390.6	1.20129	0.600645	+	0.799516i	$0.294910\pi$
$500$	621.475	0.0555864
$501$	−9641.52	−0.859784
$502$	9411.90	0.836800
$503$	−28.9098	−0.00256267	−0.00128134	−	0.999999i	$-0.500408\pi$
$503$	−28.9098	−0.00256267	−0.00128134	+	0.999999i	$0.500408\pi$
$504$	−3103.07	−0.274249
$505$	−7758.29	−0.683642
$506$	0	0
$507$	−2070.92	−0.181405
$508$	−4001.43	−0.349478
$509$	8883.68	0.773599	0.386800	−	0.922164i	$-0.373580\pi$
$509$	8883.68	0.773599	0.386800	+	0.922164i	$0.373580\pi$
$510$	7469.52	0.648541
$511$	−13787.7	−1.19361
$512$	8714.00	0.752164
$513$	340.193	0.0292785
$514$	20796.5	1.78462
$515$	−7274.60	−0.622441
$516$	−1104.00	−0.0941876
$517$	0	0
$518$	−15224.7	−1.29138
$519$	8349.80	0.706196
$520$	−2116.74	−0.178510
$521$	−1371.72	−0.115348	−0.0576738	−	0.998335i	$-0.518368\pi$
$521$	−1371.72	−0.115348	−0.0576738	+	0.998335i	$0.518368\pi$
$522$	3858.82	0.323555
$523$	6234.89	0.521286	0.260643	−	0.965435i	$-0.416065\pi$
$523$	6234.89	0.521286	0.260643	+	0.965435i	$0.416065\pi$
$524$	8450.14	0.704477
$525$	2370.97	0.197100
$526$	−24390.8	−2.02185
$527$	10461.1	0.864691
$528$	0	0
$529$	20131.0	1.65456
$530$	−1381.25	−0.113203
$531$	7797.24	0.637234
$532$	1980.34	0.161388
$533$	1824.84	0.148298
$534$	2560.57	0.207503
$535$	896.032	0.0724090
$536$	6903.87	0.556346
$537$	4596.34	0.369361
$538$	−11018.9	−0.883009
$539$	0	0
$540$	671.193	0.0534881
$541$	−11629.7	−0.924212	−0.462106	−	0.886825i	$-0.652906\pi$
$541$	−11629.7	−0.924212	−0.462106	+	0.886825i	$0.652906\pi$
$542$	−5116.39	−0.405475
$543$	−69.5145	−0.00549384
$544$	−27303.6	−2.15190
$545$	10058.6	0.790573
$546$	13258.6	1.03922
$547$	22397.1	1.75070	0.875349	−	0.483492i	$-0.160632\pi$
$547$	22397.1	1.75070	0.875349	+	0.483492i	$0.160632\pi$
$548$	5902.13	0.460085
$549$	3983.22	0.309654
$550$	0	0
$551$	1499.94	0.115970
$552$	5880.22	0.453403
$553$	−14935.2	−1.14848
$554$	2212.22	0.169654
$555$	−2005.75	−0.153404
$556$	−13362.4	−1.01923
$557$	3202.74	0.243635	0.121817	−	0.992553i	$-0.461128\pi$
$557$	3202.74	0.243635	0.121817	+	0.992553i	$0.461128\pi$
$558$	2452.55	0.186066
$559$	−2873.07	−0.217384
$560$	−12495.9	−0.942942
$561$	0	0
$562$	20880.6	1.56725
$563$	19681.1	1.47329	0.736644	−	0.676281i	$-0.236410\pi$
$563$	19681.1	1.47329	0.736644	+	0.676281i	$0.236410\pi$
$564$	−5603.10	−0.418321
$565$	10208.0	0.760092
$566$	14687.2	1.09072
$567$	2560.64	0.189660
$568$	5286.68	0.390535
$569$	−21352.3	−1.57317	−0.786585	−	0.617482i	$-0.788153\pi$
$569$	−21352.3	−1.57317	−0.786585	+	0.617482i	$0.788153\pi$
$570$	680.695	0.0500196
$571$	8197.73	0.600813	0.300407	−	0.953811i	$-0.402878\pi$
$571$	8197.73	0.600813	0.300407	+	0.953811i	$0.402878\pi$
$572$	0	0
$573$	−12775.0	−0.931387
$574$	5352.75	0.389232
$575$	−4492.91	−0.325856
$576$	−709.195	−0.0513017
$577$	−17324.5	−1.24996	−0.624982	−	0.780640i	$-0.714894\pi$
$577$	−17324.5	−1.24996	−0.624982	+	0.780640i	$0.714894\pi$
$578$	51154.9	3.68125
$579$	−366.930	−0.0263370
$580$	2959.35	0.211862
$581$	7876.67	0.562443
$582$	16449.3	1.17155
$583$	0	0
$584$	4756.78	0.337050
$585$	1746.73	0.123450
$586$	−18115.1	−1.27701
$587$	−829.014	−0.0582914	−0.0291457	−	0.999575i	$-0.509279\pi$
$587$	−829.014	−0.0582914	−0.0291457	+	0.999575i	$0.509279\pi$
$588$	9790.10	0.686627
$589$	953.316	0.0666905
$590$	15601.6	1.08866
$591$	4538.52	0.315888
$592$	10571.0	0.733897
$593$	−20619.8	−1.42791	−0.713957	−	0.700189i	$-0.753099\pi$
$593$	−20619.8	−1.42791	−0.713957	+	0.700189i	$0.753099\pi$
$594$	0	0
$595$	21854.2	1.50577
$596$	14386.1	0.988720
$597$	4195.93	0.287652
$598$	−25124.7	−1.71810
$599$	6047.56	0.412515	0.206258	−	0.978498i	$-0.433871\pi$
$599$	6047.56	0.412515	0.206258	+	0.978498i	$0.433871\pi$
$600$	−817.985	−0.0556569
$601$	3645.81	0.247447	0.123724	−	0.992317i	$-0.460516\pi$
$601$	3645.81	0.247447	0.123724	+	0.992317i	$0.460516\pi$
$602$	−8427.47	−0.570562
$603$	−5697.06	−0.384746
$604$	124.917	0.00841525
$605$	0	0
$606$	−16765.5	−1.12385
$607$	−7474.16	−0.499780	−0.249890	−	0.968274i	$-0.580395\pi$
$607$	−7474.16	−0.499780	−0.249890	+	0.968274i	$0.580395\pi$
$608$	−2488.17	−0.165968
$609$	11290.1	0.751227
$610$	7970.08	0.529015
$611$	−14581.6	−0.965482
$612$	6186.68	0.408630
$613$	−10709.8	−0.705652	−0.352826	−	0.935689i	$-0.614779\pi$
$613$	−10709.8	−0.705652	−0.352826	+	0.935689i	$0.614779\pi$
$614$	13207.8	0.868116
$615$	705.186	0.0462371
$616$	0	0
$617$	−26312.2	−1.71684	−0.858418	−	0.512950i	$-0.828552\pi$
$617$	−26312.2	−1.71684	−0.858418	+	0.512950i	$0.828552\pi$
$618$	−15720.3	−1.02324
$619$	−20408.5	−1.32518	−0.662590	−	0.748983i	$-0.730543\pi$
$619$	−20408.5	−1.32518	−0.662590	+	0.748983i	$0.730543\pi$
$620$	1880.87	0.121835
$621$	−4852.34	−0.313555
$622$	−8386.26	−0.540608
$623$	7491.68	0.481778
$624$	−9205.91	−0.590595
$625$	625.000	0.0400000
$626$	20522.8	1.31032
$627$	0	0
$628$	−5518.64	−0.350665
$629$	−18487.8	−1.17195
$630$	5123.62	0.324016
$631$	−13017.6	−0.821273	−0.410636	−	0.911799i	$-0.634693\pi$
$631$	−13017.6	−0.821273	−0.410636	+	0.911799i	$0.634693\pi$
$632$	5152.64	0.324306
$633$	1827.70	0.114763
$634$	12208.2	0.764745
$635$	−4024.13	−0.251484
$636$	−1144.03	−0.0713267
$637$	25478.0	1.58473
$638$	0	0
$639$	−4362.56	−0.270078
$640$	6480.08	0.400231
$641$	22044.5	1.35836	0.679178	−	0.733974i	$-0.262337\pi$
$641$	22044.5	1.35836	0.679178	+	0.733974i	$0.262337\pi$
$642$	1936.31	0.119034
$643$	−22105.6	−1.35577	−0.677884	−	0.735169i	$-0.737103\pi$
$643$	−22105.6	−1.35577	−0.677884	+	0.735169i	$0.737103\pi$
$644$	−28246.6	−1.72837
$645$	−1110.26	−0.0677774
$646$	6274.26	0.382132
$647$	−25733.7	−1.56367	−0.781836	−	0.623484i	$-0.785717\pi$
$647$	−25733.7	−1.56367	−0.781836	+	0.623484i	$0.785717\pi$
$648$	−883.424	−0.0535558
$649$	0	0
$650$	3495.05	0.210903
$651$	7175.65	0.432006
$652$	−17468.1	−1.04924
$653$	−13778.1	−0.825697	−0.412849	−	0.910800i	$-0.635466\pi$
$653$	−13778.1	−0.825697	−0.412849	+	0.910800i	$0.635466\pi$
$654$	21736.4	1.29964
$655$	8498.06	0.506942
$656$	−3716.59	−0.221202
$657$	−3925.29	−0.233090
$658$	−42771.8	−2.53407
$659$	9441.03	0.558074	0.279037	−	0.960280i	$-0.409985\pi$
$659$	9441.03	0.558074	0.279037	+	0.960280i	$0.409985\pi$
$660$	0	0
$661$	12160.3	0.715553	0.357776	−	0.933807i	$-0.383535\pi$
$661$	12160.3	0.715553	0.357776	+	0.933807i	$0.383535\pi$
$662$	38085.1	2.23598
$663$	16100.3	0.943115
$664$	−2717.46	−0.158822
$665$	1991.57	0.116135
$666$	−4334.38	−0.252183
$667$	−21394.4	−1.24197
$668$	−15978.6	−0.925494
$669$	−8951.91	−0.517341
$670$	−11399.3	−0.657304
$671$	0	0
$672$	−18728.5	−1.07510
$673$	−26325.7	−1.50784	−0.753922	−	0.656964i	$-0.771841\pi$
$673$	−26325.7	−1.50784	−0.753922	+	0.656964i	$0.771841\pi$
$674$	−6948.08	−0.397077
$675$	675.000	0.0384900
$676$	−3432.06	−0.195270
$677$	21595.7	1.22598	0.612991	−	0.790090i	$-0.289966\pi$
$677$	21595.7	1.22598	0.612991	+	0.790090i	$0.289966\pi$
$678$	22059.2	1.24953
$679$	48127.0	2.72010
$680$	−7539.72	−0.425199
$681$	−2287.80	−0.128735
$682$	0	0
$683$	15507.2	0.868766	0.434383	−	0.900728i	$-0.356966\pi$
$683$	15507.2	0.868766	0.434383	+	0.900728i	$0.356966\pi$
$684$	563.790	0.0315162
$685$	5935.61	0.331077
$686$	35680.3	1.98583
$687$	−13611.7	−0.755921
$688$	5851.48	0.324252
$689$	−2977.25	−0.164622
$690$	−9709.11	−0.535680
$691$	−16616.7	−0.914804	−0.457402	−	0.889260i	$-0.651220\pi$
$691$	−16616.7	−0.914804	−0.457402	+	0.889260i	$0.651220\pi$
$692$	13837.9	0.760168
$693$	0	0
$694$	−37483.6	−2.05023
$695$	−13438.2	−0.733439
$696$	−3895.09	−0.212131
$697$	6500.00	0.353235
$698$	−40835.6	−2.21440
$699$	1199.18	0.0648884
$700$	3929.33	0.212164
$701$	−15303.7	−0.824552	−0.412276	−	0.911059i	$-0.635266\pi$
$701$	−15303.7	−0.824552	−0.412276	+	0.911059i	$0.635266\pi$
$702$	3774.65	0.202942
$703$	−1684.79	−0.0903884
$704$	0	0
$705$	−5634.88	−0.301024
$706$	−11108.1	−0.592152
$707$	−49052.4	−2.60934
$708$	12922.1	0.685936
$709$	−15341.9	−0.812664	−0.406332	−	0.913726i	$-0.633192\pi$
$709$	−15341.9	−0.812664	−0.406332	+	0.913726i	$0.633192\pi$
$710$	−8729.08	−0.461404
$711$	−4251.95	−0.224277
$712$	−2584.63	−0.136044
$713$	−13597.6	−0.714215
$714$	47226.6	2.47537
$715$	0	0
$716$	7617.36	0.397590
$717$	−9909.91	−0.516168
$718$	8258.08	0.429232
$719$	−4365.48	−0.226432	−0.113216	−	0.993570i	$-0.536115\pi$
$719$	−4365.48	−0.226432	−0.113216	+	0.993570i	$0.536115\pi$
$720$	−3557.50	−0.184139
$721$	−45994.2	−2.37575
$722$	−24131.9	−1.24390
$723$	7195.77	0.370143
$724$	−115.204	−0.00591372
$725$	2976.13	0.152456
$726$	0	0
$727$	33165.7	1.69195	0.845975	−	0.533222i	$-0.179019\pi$
$727$	33165.7	1.69195	0.845975	+	0.533222i	$0.179019\pi$
$728$	−13383.2	−0.681340
$729$	729.000	0.0370370
$730$	−7854.15	−0.398212
$731$	−10233.7	−0.517795
$732$	6601.27	0.333320
$733$	14858.5	0.748717	0.374359	−	0.927284i	$-0.377863\pi$
$733$	14858.5	0.748717	0.374359	+	0.927284i	$0.377863\pi$
$734$	−4040.68	−0.203194
$735$	9845.62	0.494097
$736$	35490.0	1.77742
$737$	0	0
$738$	1523.90	0.0760100
$739$	19969.8	0.994048	0.497024	−	0.867737i	$-0.334426\pi$
$739$	19969.8	0.994048	0.497024	+	0.867737i	$0.334426\pi$
$740$	−3324.06	−0.165128
$741$	1467.22	0.0727391
$742$	−8733.08	−0.432077
$743$	11701.0	0.577750	0.288875	−	0.957367i	$-0.406719\pi$
$743$	11701.0	0.577750	0.288875	+	0.957367i	$0.406719\pi$
$744$	−2475.60	−0.121989
$745$	14467.7	0.711483
$746$	−37196.9	−1.82557
$747$	2242.44	0.109835
$748$	0	0
$749$	5665.23	0.276373
$750$	1350.61	0.0657566
$751$	29554.0	1.43600	0.718002	−	0.696041i	$-0.245057\pi$
$751$	29554.0	1.43600	0.718002	+	0.696041i	$0.245057\pi$
$752$	29697.9	1.44012
$753$	7839.68	0.379408
$754$	16642.7	0.803836
$755$	125.626	0.00605561
$756$	4243.67	0.204155
$757$	−10649.1	−0.511292	−0.255646	−	0.966770i	$-0.582288\pi$
$757$	−10649.1	−0.511292	−0.255646	+	0.966770i	$0.582288\pi$
$758$	−13816.8	−0.662069
$759$	0	0
$760$	−687.093	−0.0327940
$761$	9031.14	0.430195	0.215098	−	0.976593i	$-0.430993\pi$
$761$	9031.14	0.430195	0.215098	+	0.976593i	$0.430993\pi$
$762$	−8696.07	−0.413419
$763$	63596.2	3.01748
$764$	−21171.6	−1.00257
$765$	6221.77	0.294050
$766$	−33159.5	−1.56410
$767$	33628.8	1.58313
$768$	15894.5	0.746803
$769$	−15844.9	−0.743017	−0.371509	−	0.928429i	$-0.621159\pi$
$769$	−15844.9	−0.743017	−0.371509	+	0.928429i	$0.621159\pi$
$770$	0	0
$771$	17322.5	0.809151
$772$	−608.101	−0.0283498
$773$	18651.1	0.867832	0.433916	−	0.900953i	$-0.357131\pi$
$773$	18651.1	0.867832	0.433916	+	0.900953i	$0.357131\pi$
$774$	−2399.25	−0.111420
$775$	1891.54	0.0876725
$776$	−16603.9	−0.768098
$777$	−12681.5	−0.585516
$778$	−11011.2	−0.507417
$779$	592.343	0.0272438
$780$	2894.80	0.132885
$781$	0	0
$782$	−89493.0	−4.09241
$783$	3214.22	0.146701
$784$	−51890.1	−2.36380
$785$	−5549.94	−0.252339
$786$	18364.2	0.833369
$787$	25474.1	1.15382	0.576908	−	0.816809i	$-0.304259\pi$
$787$	25474.1	1.15382	0.576908	+	0.816809i	$0.304259\pi$
$788$	7521.54	0.340030
$789$	−20316.5	−0.916711
$790$	−8507.77	−0.383156
$791$	64540.6	2.90114
$792$	0	0
$793$	17179.3	0.769299
$794$	−34873.7	−1.55872
$795$	−1150.52	−0.0513267
$796$	6953.78	0.309636
$797$	−22591.3	−1.00405	−0.502024	−	0.864854i	$-0.667411\pi$
$797$	−22591.3	−1.00405	−0.502024	+	0.864854i	$0.667411\pi$
$798$	4303.75	0.190916
$799$	−51939.1	−2.29972
$800$	−4936.95	−0.218184
$801$	2132.84	0.0940824
$802$	−3329.22	−0.146582
$803$	0	0
$804$	−9441.55	−0.414151
$805$	−28406.8	−1.24374
$806$	10577.6	0.462260
$807$	−9178.25	−0.400359
$808$	16923.1	0.736823
$809$	−10434.9	−0.453489	−0.226745	−	0.973954i	$-0.572808\pi$
$809$	−10434.9	−0.453489	−0.226745	+	0.973954i	$0.572808\pi$
$810$	1458.66	0.0632743
$811$	12216.2	0.528937	0.264469	−	0.964394i	$-0.414803\pi$
$811$	12216.2	0.528937	0.264469	+	0.964394i	$0.414803\pi$
$812$	18710.7	0.808641
$813$	−4261.72	−0.183844
$814$	0	0
$815$	−17567.2	−0.755032
$816$	−32791.0	−1.40676
$817$	−932.597	−0.0399357
$818$	17827.3	0.762002
$819$	11043.8	0.471187
$820$	1168.68	0.0497709
$821$	−21314.2	−0.906054	−0.453027	−	0.891497i	$-0.649656\pi$
$821$	−21314.2	−0.906054	−0.453027	+	0.891497i	$0.649656\pi$
$822$	12826.7	0.544263
$823$	30062.2	1.27327	0.636635	−	0.771165i	$-0.280326\pi$
$823$	30062.2	1.27327	0.636635	+	0.771165i	$0.280326\pi$
$824$	15868.1	0.670861
$825$	0	0
$826$	98642.2	4.15521
$827$	−30802.9	−1.29519	−0.647595	−	0.761985i	$-0.724225\pi$
$827$	−30802.9	−1.29519	−0.647595	+	0.761985i	$0.724225\pi$
$828$	−8041.63	−0.337519
$829$	−15852.0	−0.664130	−0.332065	−	0.943257i	$-0.607745\pi$
$829$	−15852.0	−0.664130	−0.332065	+	0.943257i	$0.607745\pi$
$830$	4486.92	0.187642
$831$	1842.68	0.0769217
$832$	−3058.69	−0.127453
$833$	90751.3	3.77473
$834$	−29039.7	−1.20571
$835$	−16069.2	−0.665985
$836$	0	0
$837$	2042.86	0.0843629
$838$	−31146.5	−1.28394
$839$	28627.5	1.17799	0.588994	−	0.808137i	$-0.299524\pi$
$839$	28627.5	1.17799	0.588994	+	0.808137i	$0.299524\pi$
$840$	−5171.78	−0.212432
$841$	−10217.2	−0.418928
$842$	−33577.0	−1.37428
$843$	17392.6	0.710595
$844$	3029.00	0.123534
$845$	−3451.53	−0.140516
$846$	−12176.9	−0.494858
$847$	0	0
$848$	6063.67	0.245551
$849$	12233.7	0.494535
$850$	12449.2	0.502358
$851$	24031.0	0.968006
$852$	−7229.92	−0.290720
$853$	22333.6	0.896468	0.448234	−	0.893916i	$-0.352053\pi$
$853$	22333.6	0.896468	0.448234	+	0.893916i	$0.352053\pi$
$854$	50391.4	2.01916
$855$	566.988	0.0226790
$856$	−1954.51	−0.0780418
$857$	20331.4	0.810392	0.405196	−	0.914230i	$-0.367203\pi$
$857$	20331.4	0.810392	0.405196	+	0.914230i	$0.367203\pi$
$858$	0	0
$859$	−41004.7	−1.62871	−0.814355	−	0.580367i	$-0.802909\pi$
$859$	−41004.7	−1.62871	−0.814355	+	0.580367i	$0.802909\pi$
$860$	−1840.00	−0.0729574
$861$	4458.59	0.176479
$862$	−29390.0	−1.16128
$863$	27576.4	1.08773	0.543865	−	0.839173i	$-0.316960\pi$
$863$	27576.4	1.08773	0.543865	+	0.839173i	$0.316960\pi$
$864$	−5331.90	−0.209948
$865$	13916.3	0.547017
$866$	−25928.3	−1.01741
$867$	42609.7	1.66909
$868$	11892.0	0.465023
$869$	0	0
$870$	6431.37	0.250625
$871$	−24570.9	−0.955858
$872$	−21940.7	−0.852072
$873$	13701.5	0.531185
$874$	−8155.47	−0.315633
$875$	3951.61	0.152673
$876$	−6505.25	−0.250904
$877$	−35904.7	−1.38246	−0.691229	−	0.722636i	$-0.742931\pi$
$877$	−35904.7	−1.38246	−0.691229	+	0.722636i	$0.742931\pi$
$878$	−37088.0	−1.42558
$879$	−15089.0	−0.578999
$880$	0	0
$881$	−8200.35	−0.313595	−0.156797	−	0.987631i	$-0.550117\pi$
$881$	−8200.35	−0.313595	−0.156797	+	0.987631i	$0.550117\pi$
$882$	21276.2	0.812254
$883$	−37817.9	−1.44131	−0.720654	−	0.693295i	$-0.756158\pi$
$883$	−37817.9	−1.44131	−0.720654	+	0.693295i	$0.756158\pi$
$884$	26682.6	1.01519
$885$	12995.4	0.493599
$886$	46271.1	1.75452
$887$	7045.46	0.266701	0.133350	−	0.991069i	$-0.457426\pi$
$887$	7045.46	0.266701	0.133350	+	0.991069i	$0.457426\pi$
$888$	4375.12	0.165337
$889$	−25442.8	−0.959871
$890$	4267.61	0.160731
$891$	0	0
$892$	−14835.7	−0.556879
$893$	−4733.19	−0.177369
$894$	31264.4	1.16962
$895$	7660.56	0.286105
$896$	40970.8	1.52761
$897$	−20927.7	−0.778992
$898$	44995.7	1.67208
$899$	9007.15	0.334155
$900$	1118.66	0.0414317
$901$	−10604.8	−0.392118
$902$	0	0
$903$	−7019.70	−0.258694
$904$	−22266.6	−0.819220
$905$	−115.858	−0.00425551
$906$	271.475	0.00995492
$907$	−31139.3	−1.13998	−0.569992	−	0.821651i	$-0.693054\pi$
$907$	−31139.3	−1.13998	−0.569992	+	0.821651i	$0.693054\pi$
$908$	−3791.50	−0.138574
$909$	−13964.9	−0.509557
$910$	22097.7	0.804980
$911$	5053.79	0.183798	0.0918988	−	0.995768i	$-0.470706\pi$
$911$	5053.79	0.183798	0.0918988	+	0.995768i	$0.470706\pi$
$912$	−2988.24	−0.108498
$913$	0	0
$914$	−53909.8	−1.95096
$915$	6638.71	0.239857
$916$	−22558.2	−0.813693
$917$	53729.7	1.93491
$918$	13445.1	0.483394
$919$	32201.2	1.15584	0.577922	−	0.816092i	$-0.303864\pi$
$919$	32201.2	1.15584	0.577922	+	0.816092i	$0.303864\pi$
$920$	9800.36	0.351205
$921$	11001.5	0.393606
$922$	−49160.5	−1.75598
$923$	−18815.3	−0.670978
$924$	0	0
$925$	−3342.91	−0.118826
$926$	−16766.6	−0.595015
$927$	−13094.3	−0.463940
$928$	−23508.8	−0.831588
$929$	18704.0	0.660557	0.330279	−	0.943883i	$-0.392857\pi$
$929$	18704.0	0.660557	0.330279	+	0.943883i	$0.392857\pi$
$930$	4087.59	0.144126
$931$	8270.14	0.291131
$932$	1987.36	0.0698477
$933$	−6985.37	−0.245113
$934$	−3795.92	−0.132983
$935$	0	0
$936$	−3810.13	−0.133053
$937$	−27848.6	−0.970944	−0.485472	−	0.874252i	$-0.661352\pi$
$937$	−27848.6	−0.970944	−0.485472	+	0.874252i	$0.661352\pi$
$938$	−72073.0	−2.50881
$939$	17094.6	0.594101
$940$	−9338.50	−0.324030
$941$	24235.0	0.839573	0.419786	−	0.907623i	$-0.362105\pi$
$941$	24235.0	0.839573	0.419786	+	0.907623i	$0.362105\pi$
$942$	−11993.3	−0.414823
$943$	−8448.90	−0.291765
$944$	−68490.6	−2.36142
$945$	4267.74	0.146910
$946$	0	0
$947$	−35474.1	−1.21727	−0.608635	−	0.793451i	$-0.708282\pi$
$947$	−35474.1	−1.21727	−0.608635	+	0.793451i	$0.708282\pi$
$948$	−7046.62	−0.241417
$949$	−16929.4	−0.579085
$950$	1134.49	0.0387450
$951$	10168.8	0.346737
$952$	−47670.5	−1.62291
$953$	6194.48	0.210555	0.105277	−	0.994443i	$-0.466427\pi$
$953$	6194.48	0.210555	0.105277	+	0.994443i	$0.466427\pi$
$954$	−2486.25	−0.0843768
$955$	−21291.7	−0.721449
$956$	−16423.4	−0.555617
$957$	0	0
$958$	−7368.37	−0.248498
$959$	37528.3	1.26366
$960$	−1181.99	−0.0397381
$961$	−24066.3	−0.807839
$962$	−18693.8	−0.626520
$963$	1612.86	0.0539705
$964$	11925.3	0.398432
$965$	−611.550	−0.0204005
$966$	−61386.6	−2.04460
$967$	14858.3	0.494115	0.247058	−	0.969001i	$-0.420536\pi$
$967$	14858.3	0.494115	0.247058	+	0.969001i	$0.420536\pi$
$968$	0	0
$969$	5226.17	0.173260
$970$	27415.4	0.907481
$971$	3584.21	0.118458	0.0592290	−	0.998244i	$-0.481136\pi$
$971$	3584.21	0.118458	0.0592290	+	0.998244i	$0.481136\pi$
$972$	1208.15	0.0398677
$973$	−84964.1	−2.79941
$974$	42397.0	1.39475
$975$	2911.21	0.0956240
$976$	−34988.5	−1.14749
$977$	39576.7	1.29598	0.647989	−	0.761650i	$-0.275610\pi$
$977$	39576.7	1.29598	0.647989	+	0.761650i	$0.275610\pi$
$978$	−37962.4	−1.24121
$979$	0	0
$980$	16316.8	0.531859
$981$	18105.4	0.589258
$982$	−28403.7	−0.923014
$983$	−74.1406	−0.00240561	−0.00120281	−	0.999999i	$-0.500383\pi$
$983$	−74.1406	−0.00240561	−0.00120281	+	0.999999i	$0.500383\pi$
$984$	−1538.22	−0.0498340
$985$	7564.20	0.244686
$986$	59280.7	1.91469
$987$	−35626.9	−1.14895
$988$	2431.57	0.0782983
$989$	13302.1	0.427687
$990$	0	0
$991$	−44698.7	−1.43280	−0.716399	−	0.697691i	$-0.754211\pi$
$991$	−44698.7	−1.43280	−0.716399	+	0.697691i	$0.754211\pi$
$992$	−14941.5	−0.478219
$993$	31723.2	1.01380
$994$	−55190.3	−1.76110
$995$	6993.22	0.222814
$996$	3716.32	0.118229
$997$	−7086.74	−0.225115	−0.112557	−	0.993645i	$-0.535904\pi$
$997$	−7086.74	−0.225115	−0.112557	+	0.993645i	$0.535904\pi$
$998$	48228.0	1.52969
$999$	−3610.34	−0.114341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1815.4.a.bb.1.5	yes	6
11.10	odd	2	inner	1815.4.a.bb.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.bb.1.2	✓	6	11.10	odd	2	inner
1815.4.a.bb.1.5	yes	6	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:	\( N \)	\(=\)	\( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1815.a (trivial)

\(f(q)\)	\(=\)	\(q+3.60164 q^{2} +3.00000 q^{3} +4.97180 q^{4} +5.00000 q^{5} +10.8049 q^{6} +31.6129 q^{7} -10.9065 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.60164 q^{2} +3.00000 q^{3} +4.97180 q^{4} +5.00000 q^{5} +10.8049 q^{6} +31.6129 q^{7} -10.9065 q^{8} +9.00000 q^{9} +18.0082 q^{10} +14.9154 q^{12} +38.8162 q^{13} +113.858 q^{14} +15.0000 q^{15} -79.0556 q^{16} +138.261 q^{17} +32.4148 q^{18} +12.5997 q^{19} +24.8590 q^{20} +94.8387 q^{21} -179.716 q^{23} -32.7194 q^{24} +25.0000 q^{25} +139.802 q^{26} +27.0000 q^{27} +157.173 q^{28} +119.045 q^{29} +54.0246 q^{30} +75.6616 q^{31} -197.478 q^{32} +497.968 q^{34} +158.064 q^{35} +44.7462 q^{36} -133.716 q^{37} +45.3797 q^{38} +116.449 q^{39} -54.5324 q^{40} +47.0124 q^{41} +341.575 q^{42} -74.0172 q^{43} +45.0000 q^{45} -647.274 q^{46} -375.658 q^{47} -237.167 q^{48} +656.375 q^{49} +90.0410 q^{50} +414.784 q^{51} +192.986 q^{52} -76.7013 q^{53} +97.2443 q^{54} -344.785 q^{56} +37.7992 q^{57} +428.758 q^{58} +866.360 q^{59} +74.5770 q^{60} +442.580 q^{61} +272.506 q^{62} +284.516 q^{63} -78.7994 q^{64} +194.081 q^{65} -633.006 q^{67} +687.409 q^{68} -539.149 q^{69} +569.291 q^{70} -484.728 q^{71} -98.1583 q^{72} -436.143 q^{73} -481.598 q^{74} +75.0000 q^{75} +62.6434 q^{76} +419.405 q^{78} -472.439 q^{79} -395.278 q^{80} +81.0000 q^{81} +169.322 q^{82} +249.160 q^{83} +471.519 q^{84} +691.307 q^{85} -266.583 q^{86} +357.136 q^{87} +236.982 q^{89} +162.074 q^{90} +1227.09 q^{91} -893.515 q^{92} +226.985 q^{93} -1352.99 q^{94} +62.9986 q^{95} -592.434 q^{96} +1522.39 q^{97} +2364.03 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) \) 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9	\( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) \) 6 * q + 18 * q^3 + 40 * q^4 + 30 * q^5 + 54 * q^9 + 120 * q^12 + 116 * q^14 + 90 * q^15 + 164 * q^16 + 200 * q^20 + 56 * q^23 + 150 * q^25 + 292 * q^26 + 162 * q^27 + 576 * q^31 - 92 * q^34 + 360 * q^36 + 332 * q^37 + 496 * q^38 + 348 * q^42 + 270 * q^45 + 96 * q^47 + 492 * q^48 + 454 * q^49 + 308 * q^53 - 652 * q^56 + 1784 * q^58 + 2080 * q^59 + 600 * q^60 + 1928 * q^64 - 1168 * q^67 + 168 * q^69 + 580 * q^70 + 1064 * q^71 + 450 * q^75 + 876 * q^78 + 820 * q^80 + 486 * q^81 + 24 * q^82 + 5412 * q^86 - 684 * q^89 + 2744 * q^91 + 1368 * q^92 + 1728 * q^93 + 2812 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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