LMFDB - Embedded newform 2178.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, 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("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 726)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +11.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

$q+2.00000 q^{2} +4.00000 q^{4} +11.0000 q^{7} +8.00000 q^{8} -34.0000 q^{13} +22.0000 q^{14} +16.0000 q^{16} -36.0000 q^{17} -37.0000 q^{19} -6.00000 q^{23} -125.000 q^{25} -68.0000 q^{26} +44.0000 q^{28} +42.0000 q^{29} +113.000 q^{31} +32.0000 q^{32} -72.0000 q^{34} +311.000 q^{37} -74.0000 q^{38} -18.0000 q^{41} -412.000 q^{43} -12.0000 q^{46} -18.0000 q^{47} -222.000 q^{49} -250.000 q^{50} -136.000 q^{52} -750.000 q^{53} +88.0000 q^{56} +84.0000 q^{58} -546.000 q^{59} -25.0000 q^{61} +226.000 q^{62} +64.0000 q^{64} -535.000 q^{67} -144.000 q^{68} -300.000 q^{71} -499.000 q^{73} +622.000 q^{74} -148.000 q^{76} -343.000 q^{79} -36.0000 q^{82} +1386.00 q^{83} -824.000 q^{86} +1392.00 q^{89} -374.000 q^{91} -24.0000 q^{92} -36.0000 q^{94} +53.0000 q^{97} -444.000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	11.0000	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$7$	11.0000	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	22.0000	0.419982
$15$	0	0
$16$	16.0000	0.250000
$17$	−36.0000	−0.513605	−0.256802	−	0.966464i	$-0.582669\pi$
$17$	−36.0000	−0.513605	−0.256802	+	0.966464i	$0.582669\pi$
$18$	0	0
$19$	−37.0000	−0.446757	−0.223378	−	0.974732i	$-0.571709\pi$
$19$	−37.0000	−0.446757	−0.223378	+	0.974732i	$0.571709\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.00000	−0.0543951	−0.0271975	−	0.999630i	$-0.508658\pi$
$23$	−6.00000	−0.0543951	−0.0271975	+	0.999630i	$0.508658\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	−68.0000	−0.512919
$27$	0	0
$28$	44.0000	0.296972
$29$	42.0000	0.268938	0.134469	−	0.990918i	$-0.457067\pi$
$29$	42.0000	0.268938	0.134469	+	0.990918i	$0.457067\pi$
$30$	0	0
$31$	113.000	0.654690	0.327345	−	0.944905i	$-0.393846\pi$
$31$	113.000	0.654690	0.327345	+	0.944905i	$0.393846\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−72.0000	−0.363173
$35$	0	0
$36$	0	0
$37$	311.000	1.38184	0.690920	−	0.722931i	$-0.257206\pi$
$37$	311.000	1.38184	0.690920	+	0.722931i	$0.257206\pi$
$38$	−74.0000	−0.315905
$39$	0	0
$40$	0	0
$41$	−18.0000	−0.0685641	−0.0342820	−	0.999412i	$-0.510914\pi$
$41$	−18.0000	−0.0685641	−0.0342820	+	0.999412i	$0.510914\pi$
$42$	0	0
$43$	−412.000	−1.46115	−0.730575	−	0.682833i	$-0.760748\pi$
$43$	−412.000	−1.46115	−0.730575	+	0.682833i	$0.760748\pi$
$44$	0	0
$45$	0	0
$46$	−12.0000	−0.0384631
$47$	−18.0000	−0.0558632	−0.0279316	−	0.999610i	$-0.508892\pi$
$47$	−18.0000	−0.0558632	−0.0279316	+	0.999610i	$0.508892\pi$
$48$	0	0
$49$	−222.000	−0.647230
$50$	−250.000	−0.707107
$51$	0	0
$52$	−136.000	−0.362689
$53$	−750.000	−1.94378	−0.971891	−	0.235432i	$-0.924349\pi$
$53$	−750.000	−1.94378	−0.971891	+	0.235432i	$0.924349\pi$
$54$	0	0
$55$	0	0
$56$	88.0000	0.209991
$57$	0	0
$58$	84.0000	0.190168
$59$	−546.000	−1.20480	−0.602400	−	0.798195i	$-0.705789\pi$
$59$	−546.000	−1.20480	−0.602400	+	0.798195i	$0.705789\pi$
$60$	0	0
$61$	−25.0000	−0.0524741	−0.0262371	−	0.999656i	$-0.508352\pi$
$61$	−25.0000	−0.0524741	−0.0262371	+	0.999656i	$0.508352\pi$
$62$	226.000	0.462936
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−535.000	−0.975532	−0.487766	−	0.872974i	$-0.662188\pi$
$67$	−535.000	−0.975532	−0.487766	+	0.872974i	$0.662188\pi$
$68$	−144.000	−0.256802
$69$	0	0
$70$	0	0
$71$	−300.000	−0.501457	−0.250729	−	0.968057i	$-0.580670\pi$
$71$	−300.000	−0.501457	−0.250729	+	0.968057i	$0.580670\pi$
$72$	0	0
$73$	−499.000	−0.800048	−0.400024	−	0.916505i	$-0.630998\pi$
$73$	−499.000	−0.800048	−0.400024	+	0.916505i	$0.630998\pi$
$74$	622.000	0.977109
$75$	0	0
$76$	−148.000	−0.223378
$77$	0	0
$78$	0	0
$79$	−343.000	−0.488488	−0.244244	−	0.969714i	$-0.578540\pi$
$79$	−343.000	−0.488488	−0.244244	+	0.969714i	$0.578540\pi$
$80$	0	0
$81$	0	0
$82$	−36.0000	−0.0484821
$83$	1386.00	1.83293	0.916465	−	0.400114i	$-0.131029\pi$
$83$	1386.00	1.83293	0.916465	+	0.400114i	$0.131029\pi$
$84$	0	0
$85$	0	0
$86$	−824.000	−1.03319
$87$	0	0
$88$	0	0
$89$	1392.00	1.65788	0.828942	−	0.559334i	$-0.188943\pi$
$89$	1392.00	1.65788	0.828942	+	0.559334i	$0.188943\pi$
$90$	0	0
$91$	−374.000	−0.430834
$92$	−24.0000	−0.0271975
$93$	0	0
$94$	−36.0000	−0.0395012
$95$	0	0
$96$	0	0
$97$	53.0000	0.0554777	0.0277388	−	0.999615i	$-0.491169\pi$
$97$	53.0000	0.0554777	0.0277388	+	0.999615i	$0.491169\pi$
$98$	−444.000	−0.457661
$99$	0	0
$100$	−500.000	−0.500000
$101$	−210.000	−0.206889	−0.103444	−	0.994635i	$-0.532986\pi$
$101$	−210.000	−0.206889	−0.103444	+	0.994635i	$0.532986\pi$
$102$	0	0
$103$	1139.00	1.08960	0.544801	−	0.838565i	$-0.316605\pi$
$103$	1139.00	1.08960	0.544801	+	0.838565i	$0.316605\pi$
$104$	−272.000	−0.256460
$105$	0	0
$106$	−1500.00	−1.37446
$107$	−30.0000	−0.0271048	−0.0135524	−	0.999908i	$-0.504314\pi$
$107$	−30.0000	−0.0271048	−0.0135524	+	0.999908i	$0.504314\pi$
$108$	0	0
$109$	−1435.00	−1.26099	−0.630496	−	0.776193i	$-0.717148\pi$
$109$	−1435.00	−1.26099	−0.630496	+	0.776193i	$0.717148\pi$
$110$	0	0
$111$	0	0
$112$	176.000	0.148486
$113$	1434.00	1.19380	0.596900	−	0.802316i	$-0.296399\pi$
$113$	1434.00	1.19380	0.596900	+	0.802316i	$0.296399\pi$
$114$	0	0
$115$	0	0
$116$	168.000	0.134469
$117$	0	0
$118$	−1092.00	−0.851922
$119$	−396.000	−0.305053
$120$	0	0
$121$	0	0
$122$	−50.0000	−0.0371048
$123$	0	0
$124$	452.000	0.327345
$125$	0	0
$126$	0	0
$127$	−1183.00	−0.826569	−0.413285	−	0.910602i	$-0.635619\pi$
$127$	−1183.00	−0.826569	−0.413285	+	0.910602i	$0.635619\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	−318.000	−0.212090	−0.106045	−	0.994361i	$-0.533819\pi$
$131$	−318.000	−0.212090	−0.106045	+	0.994361i	$0.533819\pi$
$132$	0	0
$133$	−407.000	−0.265349
$134$	−1070.00	−0.689805
$135$	0	0
$136$	−288.000	−0.181587
$137$	−168.000	−0.104768	−0.0523840	−	0.998627i	$-0.516682\pi$
$137$	−168.000	−0.104768	−0.0523840	+	0.998627i	$0.516682\pi$
$138$	0	0
$139$	−2536.00	−1.54749	−0.773744	−	0.633499i	$-0.781618\pi$
$139$	−2536.00	−1.54749	−0.773744	+	0.633499i	$0.781618\pi$
$140$	0	0
$141$	0	0
$142$	−600.000	−0.354584
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−998.000	−0.565720
$147$	0	0
$148$	1244.00	0.690920
$149$	3336.00	1.83420	0.917100	−	0.398657i	$-0.130524\pi$
$149$	3336.00	1.83420	0.917100	+	0.398657i	$0.130524\pi$
$150$	0	0
$151$	−520.000	−0.280245	−0.140123	−	0.990134i	$-0.544750\pi$
$151$	−520.000	−0.280245	−0.140123	+	0.990134i	$0.544750\pi$
$152$	−296.000	−0.157952
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3661.00	−1.86102	−0.930508	−	0.366271i	$-0.880634\pi$
$157$	−3661.00	−1.86102	−0.930508	+	0.366271i	$0.880634\pi$
$158$	−686.000	−0.345413
$159$	0	0
$160$	0	0
$161$	−66.0000	−0.0323076
$162$	0	0
$163$	−1393.00	−0.669375	−0.334688	−	0.942329i	$-0.608631\pi$
$163$	−1393.00	−0.669375	−0.334688	+	0.942329i	$0.608631\pi$
$164$	−72.0000	−0.0342820
$165$	0	0
$166$	2772.00	1.29608
$167$	−246.000	−0.113988	−0.0569942	−	0.998375i	$-0.518152\pi$
$167$	−246.000	−0.113988	−0.0569942	+	0.998375i	$0.518152\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	0	0
$172$	−1648.00	−0.730575
$173$	1968.00	0.864880	0.432440	−	0.901663i	$-0.357653\pi$
$173$	1968.00	0.864880	0.432440	+	0.901663i	$0.357653\pi$
$174$	0	0
$175$	−1375.00	−0.593944
$176$	0	0
$177$	0	0
$178$	2784.00	1.17230
$179$	2442.00	1.01969	0.509843	−	0.860268i	$-0.329704\pi$
$179$	2442.00	1.01969	0.509843	+	0.860268i	$0.329704\pi$
$180$	0	0
$181$	461.000	0.189314	0.0946571	−	0.995510i	$-0.469825\pi$
$181$	461.000	0.189314	0.0946571	+	0.995510i	$0.469825\pi$
$182$	−748.000	−0.304645
$183$	0	0
$184$	−48.0000	−0.0192316
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−72.0000	−0.0279316
$189$	0	0
$190$	0	0
$191$	−2172.00	−0.822829	−0.411415	−	0.911448i	$-0.634965\pi$
$191$	−2172.00	−0.822829	−0.411415	+	0.911448i	$0.634965\pi$
$192$	0	0
$193$	2315.00	0.863406	0.431703	−	0.902016i	$-0.357913\pi$
$193$	2315.00	0.863406	0.431703	+	0.902016i	$0.357913\pi$
$194$	106.000	0.0392286
$195$	0	0
$196$	−888.000	−0.323615
$197$	−4152.00	−1.50161	−0.750806	−	0.660522i	$-0.770335\pi$
$197$	−4152.00	−1.50161	−0.750806	+	0.660522i	$0.770335\pi$
$198$	0	0
$199$	1457.00	0.519015	0.259508	−	0.965741i	$-0.416440\pi$
$199$	1457.00	0.519015	0.259508	+	0.965741i	$0.416440\pi$
$200$	−1000.00	−0.353553
$201$	0	0
$202$	−420.000	−0.146293
$203$	462.000	0.159734
$204$	0	0
$205$	0	0
$206$	2278.00	0.770465
$207$	0	0
$208$	−544.000	−0.181344
$209$	0	0
$210$	0	0
$211$	−3475.00	−1.13379	−0.566893	−	0.823791i	$-0.691855\pi$
$211$	−3475.00	−1.13379	−0.566893	+	0.823791i	$0.691855\pi$
$212$	−3000.00	−0.971891
$213$	0	0
$214$	−60.0000	−0.0191660
$215$	0	0
$216$	0	0
$217$	1243.00	0.388849
$218$	−2870.00	−0.891656
$219$	0	0
$220$	0	0
$221$	1224.00	0.372557
$222$	0	0
$223$	−4387.00	−1.31738	−0.658689	−	0.752415i	$-0.728889\pi$
$223$	−4387.00	−1.31738	−0.658689	+	0.752415i	$0.728889\pi$
$224$	352.000	0.104995
$225$	0	0
$226$	2868.00	0.844144
$227$	−5724.00	−1.67364	−0.836818	−	0.547482i	$-0.815587\pi$
$227$	−5724.00	−1.67364	−0.836818	+	0.547482i	$0.815587\pi$
$228$	0	0
$229$	−2986.00	−0.861661	−0.430831	−	0.902433i	$-0.641779\pi$
$229$	−2986.00	−0.861661	−0.430831	+	0.902433i	$0.641779\pi$
$230$	0	0
$231$	0	0
$232$	336.000	0.0950840
$233$	474.000	0.133274	0.0666369	−	0.997777i	$-0.478773\pi$
$233$	474.000	0.133274	0.0666369	+	0.997777i	$0.478773\pi$
$234$	0	0
$235$	0	0
$236$	−2184.00	−0.602400
$237$	0	0
$238$	−792.000	−0.215705
$239$	−3174.00	−0.859033	−0.429517	−	0.903059i	$-0.641316\pi$
$239$	−3174.00	−0.859033	−0.429517	+	0.903059i	$0.641316\pi$
$240$	0	0
$241$	−3454.00	−0.923202	−0.461601	−	0.887088i	$-0.652725\pi$
$241$	−3454.00	−0.923202	−0.461601	+	0.887088i	$0.652725\pi$
$242$	0	0
$243$	0	0
$244$	−100.000	−0.0262371
$245$	0	0
$246$	0	0
$247$	1258.00	0.324067
$248$	904.000	0.231468
$249$	0	0
$250$	0	0
$251$	3546.00	0.891719	0.445860	−	0.895103i	$-0.352898\pi$
$251$	3546.00	0.891719	0.445860	+	0.895103i	$0.352898\pi$
$252$	0	0
$253$	0	0
$254$	−2366.00	−0.584473
$255$	0	0
$256$	256.000	0.0625000
$257$	918.000	0.222814	0.111407	−	0.993775i	$-0.464464\pi$
$257$	918.000	0.222814	0.111407	+	0.993775i	$0.464464\pi$
$258$	0	0
$259$	3421.00	0.820736
$260$	0	0
$261$	0	0
$262$	−636.000	−0.149970
$263$	−6162.00	−1.44473	−0.722367	−	0.691510i	$-0.756946\pi$
$263$	−6162.00	−1.44473	−0.722367	+	0.691510i	$0.756946\pi$
$264$	0	0
$265$	0	0
$266$	−814.000	−0.187630
$267$	0	0
$268$	−2140.00	−0.487766
$269$	4890.00	1.10836	0.554179	−	0.832397i	$-0.313032\pi$
$269$	4890.00	1.10836	0.554179	+	0.832397i	$0.313032\pi$
$270$	0	0
$271$	2072.00	0.464447	0.232223	−	0.972662i	$-0.425400\pi$
$271$	2072.00	0.464447	0.232223	+	0.972662i	$0.425400\pi$
$272$	−576.000	−0.128401
$273$	0	0
$274$	−336.000	−0.0740821
$275$	0	0
$276$	0	0
$277$	−6511.00	−1.41230	−0.706152	−	0.708061i	$-0.749570\pi$
$277$	−6511.00	−1.41230	−0.706152	+	0.708061i	$0.749570\pi$
$278$	−5072.00	−1.09424
$279$	0	0
$280$	0	0
$281$	432.000	0.0917116	0.0458558	−	0.998948i	$-0.485399\pi$
$281$	432.000	0.0917116	0.0458558	+	0.998948i	$0.485399\pi$
$282$	0	0
$283$	6827.00	1.43400	0.717002	−	0.697071i	$-0.245514\pi$
$283$	6827.00	1.43400	0.717002	+	0.697071i	$0.245514\pi$
$284$	−1200.00	−0.250729
$285$	0	0
$286$	0	0
$287$	−198.000	−0.0407232
$288$	0	0
$289$	−3617.00	−0.736210
$290$	0	0
$291$	0	0
$292$	−1996.00	−0.400024
$293$	−6216.00	−1.23939	−0.619697	−	0.784841i	$-0.712745\pi$
$293$	−6216.00	−1.23939	−0.619697	+	0.784841i	$0.712745\pi$
$294$	0	0
$295$	0	0
$296$	2488.00	0.488554
$297$	0	0
$298$	6672.00	1.29698
$299$	204.000	0.0394569
$300$	0	0
$301$	−4532.00	−0.867841
$302$	−1040.00	−0.198163
$303$	0	0
$304$	−592.000	−0.111689
$305$	0	0
$306$	0	0
$307$	−8389.00	−1.55956	−0.779781	−	0.626052i	$-0.784670\pi$
$307$	−8389.00	−1.55956	−0.779781	+	0.626052i	$0.784670\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8268.00	−1.50751	−0.753754	−	0.657156i	$-0.771759\pi$
$311$	−8268.00	−1.50751	−0.753754	+	0.657156i	$0.771759\pi$
$312$	0	0
$313$	8318.00	1.50211	0.751056	−	0.660238i	$-0.229545\pi$
$313$	8318.00	1.50211	0.751056	+	0.660238i	$0.229545\pi$
$314$	−7322.00	−1.31594
$315$	0	0
$316$	−1372.00	−0.244244
$317$	372.000	0.0659104	0.0329552	−	0.999457i	$-0.489508\pi$
$317$	372.000	0.0659104	0.0329552	+	0.999457i	$0.489508\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−132.000	−0.0228449
$323$	1332.00	0.229457
$324$	0	0
$325$	4250.00	0.725377
$326$	−2786.00	−0.473320
$327$	0	0
$328$	−144.000	−0.0242411
$329$	−198.000	−0.0331796
$330$	0	0
$331$	10763.0	1.78727	0.893637	−	0.448790i	$-0.148145\pi$
$331$	10763.0	1.78727	0.893637	+	0.448790i	$0.148145\pi$
$332$	5544.00	0.916465
$333$	0	0
$334$	−492.000	−0.0806019
$335$	0	0
$336$	0	0
$337$	8141.00	1.31593	0.657965	−	0.753048i	$-0.271417\pi$
$337$	8141.00	1.31593	0.657965	+	0.753048i	$0.271417\pi$
$338$	−2082.00	−0.335047
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−6215.00	−0.978363
$344$	−3296.00	−0.516594
$345$	0	0
$346$	3936.00	0.611563
$347$	1110.00	0.171723	0.0858616	−	0.996307i	$-0.472636\pi$
$347$	1110.00	0.171723	0.0858616	+	0.996307i	$0.472636\pi$
$348$	0	0
$349$	−6205.00	−0.951708	−0.475854	−	0.879524i	$-0.657861\pi$
$349$	−6205.00	−0.951708	−0.475854	+	0.879524i	$0.657861\pi$
$350$	−2750.00	−0.419982
$351$	0	0
$352$	0	0
$353$	−1752.00	−0.264163	−0.132082	−	0.991239i	$-0.542166\pi$
$353$	−1752.00	−0.264163	−0.132082	+	0.991239i	$0.542166\pi$
$354$	0	0
$355$	0	0
$356$	5568.00	0.828942
$357$	0	0
$358$	4884.00	0.721026
$359$	−1716.00	−0.252276	−0.126138	−	0.992013i	$-0.540258\pi$
$359$	−1716.00	−0.252276	−0.126138	+	0.992013i	$0.540258\pi$
$360$	0	0
$361$	−5490.00	−0.800408
$362$	922.000	0.133865
$363$	0	0
$364$	−1496.00	−0.215417
$365$	0	0
$366$	0	0
$367$	−1024.00	−0.145647	−0.0728234	−	0.997345i	$-0.523201\pi$
$367$	−1024.00	−0.145647	−0.0728234	+	0.997345i	$0.523201\pi$
$368$	−96.0000	−0.0135988
$369$	0	0
$370$	0	0
$371$	−8250.00	−1.15450
$372$	0	0
$373$	−6661.00	−0.924647	−0.462324	−	0.886711i	$-0.652984\pi$
$373$	−6661.00	−0.924647	−0.462324	+	0.886711i	$0.652984\pi$
$374$	0	0
$375$	0	0
$376$	−144.000	−0.0197506
$377$	−1428.00	−0.195082
$378$	0	0
$379$	9560.00	1.29568	0.647842	−	0.761775i	$-0.275672\pi$
$379$	9560.00	1.29568	0.647842	+	0.761775i	$0.275672\pi$
$380$	0	0
$381$	0	0
$382$	−4344.00	−0.581828
$383$	6102.00	0.814093	0.407047	−	0.913407i	$-0.366559\pi$
$383$	6102.00	0.814093	0.407047	+	0.913407i	$0.366559\pi$
$384$	0	0
$385$	0	0
$386$	4630.00	0.610520
$387$	0	0
$388$	212.000	0.0277388
$389$	10002.0	1.30365	0.651827	−	0.758368i	$-0.274003\pi$
$389$	10002.0	1.30365	0.651827	+	0.758368i	$0.274003\pi$
$390$	0	0
$391$	216.000	0.0279376
$392$	−1776.00	−0.228830
$393$	0	0
$394$	−8304.00	−1.06180
$395$	0	0
$396$	0	0
$397$	4259.00	0.538421	0.269210	−	0.963081i	$-0.413237\pi$
$397$	4259.00	0.538421	0.269210	+	0.963081i	$0.413237\pi$
$398$	2914.00	0.366999
$399$	0	0
$400$	−2000.00	−0.250000
$401$	5772.00	0.718803	0.359401	−	0.933183i	$-0.382981\pi$
$401$	5772.00	0.718803	0.359401	+	0.933183i	$0.382981\pi$
$402$	0	0
$403$	−3842.00	−0.474897
$404$	−840.000	−0.103444
$405$	0	0
$406$	924.000	0.112949
$407$	0	0
$408$	0	0
$409$	−3091.00	−0.373692	−0.186846	−	0.982389i	$-0.559827\pi$
$409$	−3091.00	−0.373692	−0.186846	+	0.982389i	$0.559827\pi$
$410$	0	0
$411$	0	0
$412$	4556.00	0.544801
$413$	−6006.00	−0.715583
$414$	0	0
$415$	0	0
$416$	−1088.00	−0.128230
$417$	0	0
$418$	0	0
$419$	−1542.00	−0.179789	−0.0898945	−	0.995951i	$-0.528653\pi$
$419$	−1542.00	−0.179789	−0.0898945	+	0.995951i	$0.528653\pi$
$420$	0	0
$421$	−34.0000	−0.00393601	−0.00196800	−	0.999998i	$-0.500626\pi$
$421$	−34.0000	−0.00393601	−0.00196800	+	0.999998i	$0.500626\pi$
$422$	−6950.00	−0.801708
$423$	0	0
$424$	−6000.00	−0.687231
$425$	4500.00	0.513605
$426$	0	0
$427$	−275.000	−0.0311667
$428$	−120.000	−0.0135524
$429$	0	0
$430$	0	0
$431$	−8334.00	−0.931403	−0.465701	−	0.884942i	$-0.654198\pi$
$431$	−8334.00	−0.931403	−0.465701	+	0.884942i	$0.654198\pi$
$432$	0	0
$433$	1559.00	0.173027	0.0865136	−	0.996251i	$-0.472427\pi$
$433$	1559.00	0.173027	0.0865136	+	0.996251i	$0.472427\pi$
$434$	2486.00	0.274958
$435$	0	0
$436$	−5740.00	−0.630496
$437$	222.000	0.0243014
$438$	0	0
$439$	−3025.00	−0.328873	−0.164437	−	0.986388i	$-0.552581\pi$
$439$	−3025.00	−0.328873	−0.164437	+	0.986388i	$0.552581\pi$
$440$	0	0
$441$	0	0
$442$	2448.00	0.263438
$443$	5448.00	0.584294	0.292147	−	0.956373i	$-0.405630\pi$
$443$	5448.00	0.584294	0.292147	+	0.956373i	$0.405630\pi$
$444$	0	0
$445$	0	0
$446$	−8774.00	−0.931527
$447$	0	0
$448$	704.000	0.0742430
$449$	7272.00	0.764336	0.382168	−	0.924093i	$-0.375178\pi$
$449$	7272.00	0.764336	0.382168	+	0.924093i	$0.375178\pi$
$450$	0	0
$451$	0	0
$452$	5736.00	0.596900
$453$	0	0
$454$	−11448.0	−1.18344
$455$	0	0
$456$	0	0
$457$	5402.00	0.552943	0.276471	−	0.961022i	$-0.410835\pi$
$457$	5402.00	0.552943	0.276471	+	0.961022i	$0.410835\pi$
$458$	−5972.00	−0.609287
$459$	0	0
$460$	0	0
$461$	−9258.00	−0.935332	−0.467666	−	0.883905i	$-0.654905\pi$
$461$	−9258.00	−0.935332	−0.467666	+	0.883905i	$0.654905\pi$
$462$	0	0
$463$	13592.0	1.36431	0.682153	−	0.731209i	$-0.261044\pi$
$463$	13592.0	1.36431	0.682153	+	0.731209i	$0.261044\pi$
$464$	672.000	0.0672345
$465$	0	0
$466$	948.000	0.0942387
$467$	15444.0	1.53033	0.765164	−	0.643836i	$-0.222658\pi$
$467$	15444.0	1.53033	0.765164	+	0.643836i	$0.222658\pi$
$468$	0	0
$469$	−5885.00	−0.579412
$470$	0	0
$471$	0	0
$472$	−4368.00	−0.425961
$473$	0	0
$474$	0	0
$475$	4625.00	0.446757
$476$	−1584.00	−0.152526
$477$	0	0
$478$	−6348.00	−0.607428
$479$	18822.0	1.79541	0.897703	−	0.440602i	$-0.145235\pi$
$479$	18822.0	1.79541	0.897703	+	0.440602i	$0.145235\pi$
$480$	0	0
$481$	−10574.0	−1.00236
$482$	−6908.00	−0.652802
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3656.00	0.340183	0.170092	−	0.985428i	$-0.445594\pi$
$487$	3656.00	0.340183	0.170092	+	0.985428i	$0.445594\pi$
$488$	−200.000	−0.0185524
$489$	0	0
$490$	0	0
$491$	15126.0	1.39028	0.695139	−	0.718875i	$-0.255343\pi$
$491$	15126.0	1.39028	0.695139	+	0.718875i	$0.255343\pi$
$492$	0	0
$493$	−1512.00	−0.138128
$494$	2516.00	0.229150
$495$	0	0
$496$	1808.00	0.163673
$497$	−3300.00	−0.297837
$498$	0	0
$499$	10961.0	0.983330	0.491665	−	0.870784i	$-0.336388\pi$
$499$	10961.0	0.983330	0.491665	+	0.870784i	$0.336388\pi$
$500$	0	0
$501$	0	0
$502$	7092.00	0.630541
$503$	−13068.0	−1.15840	−0.579198	−	0.815187i	$-0.696634\pi$
$503$	−13068.0	−1.15840	−0.579198	+	0.815187i	$0.696634\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−4732.00	−0.413285
$509$	2082.00	0.181303	0.0906513	−	0.995883i	$-0.471105\pi$
$509$	2082.00	0.181303	0.0906513	+	0.995883i	$0.471105\pi$
$510$	0	0
$511$	−5489.00	−0.475184
$512$	512.000	0.0441942
$513$	0	0
$514$	1836.00	0.157553
$515$	0	0
$516$	0	0
$517$	0	0
$518$	6842.00	0.580348
$519$	0	0
$520$	0	0
$521$	2538.00	0.213420	0.106710	−	0.994290i	$-0.465968\pi$
$521$	2538.00	0.213420	0.106710	+	0.994290i	$0.465968\pi$
$522$	0	0
$523$	−3589.00	−0.300069	−0.150034	−	0.988681i	$-0.547938\pi$
$523$	−3589.00	−0.300069	−0.150034	+	0.988681i	$0.547938\pi$
$524$	−1272.00	−0.106045
$525$	0	0
$526$	−12324.0	−1.02158
$527$	−4068.00	−0.336252
$528$	0	0
$529$	−12131.0	−0.997041
$530$	0	0
$531$	0	0
$532$	−1628.00	−0.132674
$533$	612.000	0.0497348
$534$	0	0
$535$	0	0
$536$	−4280.00	−0.344903
$537$	0	0
$538$	9780.00	0.783728
$539$	0	0
$540$	0	0
$541$	10838.0	0.861298	0.430649	−	0.902520i	$-0.358285\pi$
$541$	10838.0	0.861298	0.430649	+	0.902520i	$0.358285\pi$
$542$	4144.00	0.328413
$543$	0	0
$544$	−1152.00	−0.0907934
$545$	0	0
$546$	0	0
$547$	−13372.0	−1.04524	−0.522619	−	0.852566i	$-0.675045\pi$
$547$	−13372.0	−1.04524	−0.522619	+	0.852566i	$0.675045\pi$
$548$	−672.000	−0.0523840
$549$	0	0
$550$	0	0
$551$	−1554.00	−0.120150
$552$	0	0
$553$	−3773.00	−0.290134
$554$	−13022.0	−0.998649
$555$	0	0
$556$	−10144.0	−0.773744
$557$	19332.0	1.47060	0.735299	−	0.677743i	$-0.237042\pi$
$557$	19332.0	1.47060	0.735299	+	0.677743i	$0.237042\pi$
$558$	0	0
$559$	14008.0	1.05988
$560$	0	0
$561$	0	0
$562$	864.000	0.0648499
$563$	−19584.0	−1.46602	−0.733008	−	0.680220i	$-0.761884\pi$
$563$	−19584.0	−1.46602	−0.733008	+	0.680220i	$0.761884\pi$
$564$	0	0
$565$	0	0
$566$	13654.0	1.01399
$567$	0	0
$568$	−2400.00	−0.177292
$569$	20352.0	1.49947	0.749737	−	0.661736i	$-0.230180\pi$
$569$	20352.0	1.49947	0.749737	+	0.661736i	$0.230180\pi$
$570$	0	0
$571$	−6703.00	−0.491264	−0.245632	−	0.969363i	$-0.578995\pi$
$571$	−6703.00	−0.491264	−0.245632	+	0.969363i	$0.578995\pi$
$572$	0	0
$573$	0	0
$574$	−396.000	−0.0287957
$575$	750.000	0.0543951
$576$	0	0
$577$	7103.00	0.512481	0.256241	−	0.966613i	$-0.417516\pi$
$577$	7103.00	0.512481	0.256241	+	0.966613i	$0.417516\pi$
$578$	−7234.00	−0.520579
$579$	0	0
$580$	0	0
$581$	15246.0	1.08866
$582$	0	0
$583$	0	0
$584$	−3992.00	−0.282860
$585$	0	0
$586$	−12432.0	−0.876384
$587$	−7344.00	−0.516387	−0.258194	−	0.966093i	$-0.583127\pi$
$587$	−7344.00	−0.516387	−0.258194	+	0.966093i	$0.583127\pi$
$588$	0	0
$589$	−4181.00	−0.292487
$590$	0	0
$591$	0	0
$592$	4976.00	0.345460
$593$	−13470.0	−0.932794	−0.466397	−	0.884576i	$-0.654448\pi$
$593$	−13470.0	−0.932794	−0.466397	+	0.884576i	$0.654448\pi$
$594$	0	0
$595$	0	0
$596$	13344.0	0.917100
$597$	0	0
$598$	408.000	0.0279003
$599$	−9348.00	−0.637644	−0.318822	−	0.947815i	$-0.603287\pi$
$599$	−9348.00	−0.637644	−0.318822	+	0.947815i	$0.603287\pi$
$600$	0	0
$601$	−17377.0	−1.17941	−0.589703	−	0.807620i	$-0.700755\pi$
$601$	−17377.0	−1.17941	−0.589703	+	0.807620i	$0.700755\pi$
$602$	−9064.00	−0.613656
$603$	0	0
$604$	−2080.00	−0.140123
$605$	0	0
$606$	0	0
$607$	15824.0	1.05812	0.529058	−	0.848586i	$-0.322545\pi$
$607$	15824.0	1.05812	0.529058	+	0.848586i	$0.322545\pi$
$608$	−1184.00	−0.0789762
$609$	0	0
$610$	0	0
$611$	612.000	0.0405219
$612$	0	0
$613$	−3217.00	−0.211963	−0.105982	−	0.994368i	$-0.533798\pi$
$613$	−3217.00	−0.211963	−0.105982	+	0.994368i	$0.533798\pi$
$614$	−16778.0	−1.10278
$615$	0	0
$616$	0	0
$617$	22212.0	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	22212.0	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	−3076.00	−0.199733	−0.0998666	−	0.995001i	$-0.531842\pi$
$619$	−3076.00	−0.199733	−0.0998666	+	0.995001i	$0.531842\pi$
$620$	0	0
$621$	0	0
$622$	−16536.0	−1.06597
$623$	15312.0	0.984691
$624$	0	0
$625$	15625.0	1.00000
$626$	16636.0	1.06215
$627$	0	0
$628$	−14644.0	−0.930508
$629$	−11196.0	−0.709720
$630$	0	0
$631$	10568.0	0.666728	0.333364	−	0.942798i	$-0.391816\pi$
$631$	10568.0	0.666728	0.333364	+	0.942798i	$0.391816\pi$
$632$	−2744.00	−0.172706
$633$	0	0
$634$	744.000	0.0466057
$635$	0	0
$636$	0	0
$637$	7548.00	0.469486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11904.0	−0.733510	−0.366755	−	0.930318i	$-0.619531\pi$
$641$	−11904.0	−0.733510	−0.366755	+	0.930318i	$0.619531\pi$
$642$	0	0
$643$	3005.00	0.184301	0.0921506	−	0.995745i	$-0.470626\pi$
$643$	3005.00	0.184301	0.0921506	+	0.995745i	$0.470626\pi$
$644$	−264.000	−0.0161538
$645$	0	0
$646$	2664.00	0.162250
$647$	−9324.00	−0.566560	−0.283280	−	0.959037i	$-0.591423\pi$
$647$	−9324.00	−0.566560	−0.283280	+	0.959037i	$0.591423\pi$
$648$	0	0
$649$	0	0
$650$	8500.00	0.512919
$651$	0	0
$652$	−5572.00	−0.334688
$653$	21018.0	1.25957	0.629784	−	0.776770i	$-0.283143\pi$
$653$	21018.0	1.25957	0.629784	+	0.776770i	$0.283143\pi$
$654$	0	0
$655$	0	0
$656$	−288.000	−0.0171410
$657$	0	0
$658$	−396.000	−0.0234615
$659$	22482.0	1.32894	0.664472	−	0.747313i	$-0.268656\pi$
$659$	22482.0	1.32894	0.664472	+	0.747313i	$0.268656\pi$
$660$	0	0
$661$	−28273.0	−1.66368	−0.831840	−	0.555016i	$-0.812712\pi$
$661$	−28273.0	−1.66368	−0.831840	+	0.555016i	$0.812712\pi$
$662$	21526.0	1.26379
$663$	0	0
$664$	11088.0	0.648039
$665$	0	0
$666$	0	0
$667$	−252.000	−0.0146289
$668$	−984.000	−0.0569942
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−26101.0	−1.49498	−0.747489	−	0.664275i	$-0.768741\pi$
$673$	−26101.0	−1.49498	−0.747489	+	0.664275i	$0.768741\pi$
$674$	16282.0	0.930503
$675$	0	0
$676$	−4164.00	−0.236914
$677$	−3468.00	−0.196877	−0.0984387	−	0.995143i	$-0.531385\pi$
$677$	−3468.00	−0.196877	−0.0984387	+	0.995143i	$0.531385\pi$
$678$	0	0
$679$	583.000	0.0329506
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14478.0	−0.811106	−0.405553	−	0.914072i	$-0.632921\pi$
$683$	−14478.0	−0.811106	−0.405553	+	0.914072i	$0.632921\pi$
$684$	0	0
$685$	0	0
$686$	−12430.0	−0.691807
$687$	0	0
$688$	−6592.00	−0.365287
$689$	25500.0	1.40997
$690$	0	0
$691$	7349.00	0.404586	0.202293	−	0.979325i	$-0.435161\pi$
$691$	7349.00	0.404586	0.202293	+	0.979325i	$0.435161\pi$
$692$	7872.00	0.432440
$693$	0	0
$694$	2220.00	0.121427
$695$	0	0
$696$	0	0
$697$	648.000	0.0352148
$698$	−12410.0	−0.672959
$699$	0	0
$700$	−5500.00	−0.296972
$701$	16782.0	0.904205	0.452102	−	0.891966i	$-0.350674\pi$
$701$	16782.0	0.904205	0.452102	+	0.891966i	$0.350674\pi$
$702$	0	0
$703$	−11507.0	−0.617347
$704$	0	0
$705$	0	0
$706$	−3504.00	−0.186791
$707$	−2310.00	−0.122880
$708$	0	0
$709$	16418.0	0.869663	0.434831	−	0.900512i	$-0.356808\pi$
$709$	16418.0	0.869663	0.434831	+	0.900512i	$0.356808\pi$
$710$	0	0
$711$	0	0
$712$	11136.0	0.586151
$713$	−678.000	−0.0356119
$714$	0	0
$715$	0	0
$716$	9768.00	0.509843
$717$	0	0
$718$	−3432.00	−0.178386
$719$	−16374.0	−0.849301	−0.424650	−	0.905357i	$-0.639603\pi$
$719$	−16374.0	−0.849301	−0.424650	+	0.905357i	$0.639603\pi$
$720$	0	0
$721$	12529.0	0.647163
$722$	−10980.0	−0.565974
$723$	0	0
$724$	1844.00	0.0946571
$725$	−5250.00	−0.268938
$726$	0	0
$727$	−5596.00	−0.285480	−0.142740	−	0.989760i	$-0.545591\pi$
$727$	−5596.00	−0.285480	−0.142740	+	0.989760i	$0.545591\pi$
$728$	−2992.00	−0.152323
$729$	0	0
$730$	0	0
$731$	14832.0	0.750453
$732$	0	0
$733$	3962.00	0.199645	0.0998225	−	0.995005i	$-0.468172\pi$
$733$	3962.00	0.199645	0.0998225	+	0.995005i	$0.468172\pi$
$734$	−2048.00	−0.102988
$735$	0	0
$736$	−192.000	−0.00961578
$737$	0	0
$738$	0	0
$739$	6815.00	0.339234	0.169617	−	0.985510i	$-0.445747\pi$
$739$	6815.00	0.339234	0.169617	+	0.985510i	$0.445747\pi$
$740$	0	0
$741$	0	0
$742$	−16500.0	−0.816353
$743$	30360.0	1.49906	0.749529	−	0.661971i	$-0.230280\pi$
$743$	30360.0	1.49906	0.749529	+	0.661971i	$0.230280\pi$
$744$	0	0
$745$	0	0
$746$	−13322.0	−0.653824
$747$	0	0
$748$	0	0
$749$	−330.000	−0.0160987
$750$	0	0
$751$	32825.0	1.59494	0.797471	−	0.603357i	$-0.206171\pi$
$751$	32825.0	1.59494	0.797471	+	0.603357i	$0.206171\pi$
$752$	−288.000	−0.0139658
$753$	0	0
$754$	−2856.00	−0.137943
$755$	0	0
$756$	0	0
$757$	21641.0	1.03904	0.519521	−	0.854457i	$-0.326110\pi$
$757$	21641.0	1.03904	0.519521	+	0.854457i	$0.326110\pi$
$758$	19120.0	0.916187
$759$	0	0
$760$	0	0
$761$	1176.00	0.0560184	0.0280092	−	0.999608i	$-0.491083\pi$
$761$	1176.00	0.0560184	0.0280092	+	0.999608i	$0.491083\pi$
$762$	0	0
$763$	−15785.0	−0.748959
$764$	−8688.00	−0.411415
$765$	0	0
$766$	12204.0	0.575651
$767$	18564.0	0.873934
$768$	0	0
$769$	21779.0	1.02129	0.510644	−	0.859792i	$-0.329407\pi$
$769$	21779.0	1.02129	0.510644	+	0.859792i	$0.329407\pi$
$770$	0	0
$771$	0	0
$772$	9260.00	0.431703
$773$	16212.0	0.754340	0.377170	−	0.926144i	$-0.376897\pi$
$773$	16212.0	0.754340	0.377170	+	0.926144i	$0.376897\pi$
$774$	0	0
$775$	−14125.0	−0.654690
$776$	424.000	0.0196143
$777$	0	0
$778$	20004.0	0.921823
$779$	666.000	0.0306315
$780$	0	0
$781$	0	0
$782$	432.000	0.0197548
$783$	0	0
$784$	−3552.00	−0.161808
$785$	0	0
$786$	0	0
$787$	−4084.00	−0.184980	−0.0924898	−	0.995714i	$-0.529483\pi$
$787$	−4084.00	−0.184980	−0.0924898	+	0.995714i	$0.529483\pi$
$788$	−16608.0	−0.750806
$789$	0	0
$790$	0	0
$791$	15774.0	0.709050
$792$	0	0
$793$	850.000	0.0380635
$794$	8518.00	0.380721
$795$	0	0
$796$	5828.00	0.259508
$797$	−38586.0	−1.71491	−0.857457	−	0.514556i	$-0.827957\pi$
$797$	−38586.0	−1.71491	−0.857457	+	0.514556i	$0.827957\pi$
$798$	0	0
$799$	648.000	0.0286916
$800$	−4000.00	−0.176777
$801$	0	0
$802$	11544.0	0.508270
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−7684.00	−0.335803
$807$	0	0
$808$	−1680.00	−0.0731463
$809$	−9132.00	−0.396865	−0.198433	−	0.980115i	$-0.563585\pi$
$809$	−9132.00	−0.396865	−0.198433	+	0.980115i	$0.563585\pi$
$810$	0	0
$811$	37217.0	1.61142	0.805712	−	0.592307i	$-0.201783\pi$
$811$	37217.0	1.61142	0.805712	+	0.592307i	$0.201783\pi$
$812$	1848.00	0.0798671
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15244.0	0.652779
$818$	−6182.00	−0.264240
$819$	0	0
$820$	0	0
$821$	558.000	0.0237203	0.0118601	−	0.999930i	$-0.496225\pi$
$821$	558.000	0.0237203	0.0118601	+	0.999930i	$0.496225\pi$
$822$	0	0
$823$	33839.0	1.43324	0.716618	−	0.697466i	$-0.245689\pi$
$823$	33839.0	1.43324	0.716618	+	0.697466i	$0.245689\pi$
$824$	9112.00	0.385232
$825$	0	0
$826$	−12012.0	−0.505994
$827$	−9402.00	−0.395332	−0.197666	−	0.980269i	$-0.563336\pi$
$827$	−9402.00	−0.395332	−0.197666	+	0.980269i	$0.563336\pi$
$828$	0	0
$829$	5705.00	0.239014	0.119507	−	0.992833i	$-0.461869\pi$
$829$	5705.00	0.239014	0.119507	+	0.992833i	$0.461869\pi$
$830$	0	0
$831$	0	0
$832$	−2176.00	−0.0906721
$833$	7992.00	0.332421
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−3084.00	−0.127130
$839$	24972.0	1.02757	0.513784	−	0.857920i	$-0.328243\pi$
$839$	24972.0	1.02757	0.513784	+	0.857920i	$0.328243\pi$
$840$	0	0
$841$	−22625.0	−0.927672
$842$	−68.0000	−0.00278318
$843$	0	0
$844$	−13900.0	−0.566893
$845$	0	0
$846$	0	0
$847$	0	0
$848$	−12000.0	−0.485945
$849$	0	0
$850$	9000.00	0.363173
$851$	−1866.00	−0.0751653
$852$	0	0
$853$	40487.0	1.62514	0.812572	−	0.582860i	$-0.198067\pi$
$853$	40487.0	1.62514	0.812572	+	0.582860i	$0.198067\pi$
$854$	−550.000	−0.0220382
$855$	0	0
$856$	−240.000	−0.00958298
$857$	−40080.0	−1.59756	−0.798779	−	0.601625i	$-0.794520\pi$
$857$	−40080.0	−1.59756	−0.798779	+	0.601625i	$0.794520\pi$
$858$	0	0
$859$	−2875.00	−0.114195	−0.0570976	−	0.998369i	$-0.518185\pi$
$859$	−2875.00	−0.114195	−0.0570976	+	0.998369i	$0.518185\pi$
$860$	0	0
$861$	0	0
$862$	−16668.0	−0.658601
$863$	2646.00	0.104370	0.0521848	−	0.998637i	$-0.483382\pi$
$863$	2646.00	0.104370	0.0521848	+	0.998637i	$0.483382\pi$
$864$	0	0
$865$	0	0
$866$	3118.00	0.122349
$867$	0	0
$868$	4972.00	0.194425
$869$	0	0
$870$	0	0
$871$	18190.0	0.707629
$872$	−11480.0	−0.445828
$873$	0	0
$874$	444.000	0.0171837
$875$	0	0
$876$	0	0
$877$	−12145.0	−0.467625	−0.233813	−	0.972282i	$-0.575120\pi$
$877$	−12145.0	−0.467625	−0.233813	+	0.972282i	$0.575120\pi$
$878$	−6050.00	−0.232549
$879$	0	0
$880$	0	0
$881$	−27834.0	−1.06442	−0.532208	−	0.846613i	$-0.678638\pi$
$881$	−27834.0	−1.06442	−0.532208	+	0.846613i	$0.678638\pi$
$882$	0	0
$883$	47951.0	1.82750	0.913748	−	0.406281i	$-0.133174\pi$
$883$	47951.0	1.82750	0.913748	+	0.406281i	$0.133174\pi$
$884$	4896.00	0.186279
$885$	0	0
$886$	10896.0	0.413158
$887$	−48246.0	−1.82632	−0.913158	−	0.407606i	$-0.866364\pi$
$887$	−48246.0	−1.82632	−0.913158	+	0.407606i	$0.866364\pi$
$888$	0	0
$889$	−13013.0	−0.490936
$890$	0	0
$891$	0	0
$892$	−17548.0	−0.658689
$893$	666.000	0.0249573
$894$	0	0
$895$	0	0
$896$	1408.00	0.0524977
$897$	0	0
$898$	14544.0	0.540467
$899$	4746.00	0.176071
$900$	0	0
$901$	27000.0	0.998336
$902$	0	0
$903$	0	0
$904$	11472.0	0.422072
$905$	0	0
$906$	0	0
$907$	−27937.0	−1.02275	−0.511374	−	0.859358i	$-0.670863\pi$
$907$	−27937.0	−1.02275	−0.511374	+	0.859358i	$0.670863\pi$
$908$	−22896.0	−0.836818
$909$	0	0
$910$	0	0
$911$	18204.0	0.662048	0.331024	−	0.943622i	$-0.392606\pi$
$911$	18204.0	0.662048	0.331024	+	0.943622i	$0.392606\pi$
$912$	0	0
$913$	0	0
$914$	10804.0	0.390990
$915$	0	0
$916$	−11944.0	−0.430831
$917$	−3498.00	−0.125970
$918$	0	0
$919$	−19279.0	−0.692008	−0.346004	−	0.938233i	$-0.612462\pi$
$919$	−19279.0	−0.692008	−0.346004	+	0.938233i	$0.612462\pi$
$920$	0	0
$921$	0	0
$922$	−18516.0	−0.661380
$923$	10200.0	0.363745
$924$	0	0
$925$	−38875.0	−1.38184
$926$	27184.0	0.964710
$927$	0	0
$928$	1344.00	0.0475420
$929$	−16074.0	−0.567676	−0.283838	−	0.958872i	$-0.591608\pi$
$929$	−16074.0	−0.567676	−0.283838	+	0.958872i	$0.591608\pi$
$930$	0	0
$931$	8214.00	0.289155
$932$	1896.00	0.0666369
$933$	0	0
$934$	30888.0	1.08211
$935$	0	0
$936$	0	0
$937$	12287.0	0.428387	0.214194	−	0.976791i	$-0.431288\pi$
$937$	12287.0	0.428387	0.214194	+	0.976791i	$0.431288\pi$
$938$	−11770.0	−0.409706
$939$	0	0
$940$	0	0
$941$	−26514.0	−0.918525	−0.459262	−	0.888301i	$-0.651886\pi$
$941$	−26514.0	−0.918525	−0.459262	+	0.888301i	$0.651886\pi$
$942$	0	0
$943$	108.000	0.00372955
$944$	−8736.00	−0.301200
$945$	0	0
$946$	0	0
$947$	−55278.0	−1.89683	−0.948413	−	0.317038i	$-0.897312\pi$
$947$	−55278.0	−1.89683	−0.948413	+	0.317038i	$0.897312\pi$
$948$	0	0
$949$	16966.0	0.580337
$950$	9250.00	0.315905
$951$	0	0
$952$	−3168.00	−0.107852
$953$	−17298.0	−0.587972	−0.293986	−	0.955810i	$-0.594982\pi$
$953$	−17298.0	−0.587972	−0.293986	+	0.955810i	$0.594982\pi$
$954$	0	0
$955$	0	0
$956$	−12696.0	−0.429517
$957$	0	0
$958$	37644.0	1.26954
$959$	−1848.00	−0.0622263
$960$	0	0
$961$	−17022.0	−0.571381
$962$	−21148.0	−0.708772
$963$	0	0
$964$	−13816.0	−0.461601
$965$	0	0
$966$	0	0
$967$	29567.0	0.983258	0.491629	−	0.870805i	$-0.336402\pi$
$967$	29567.0	0.983258	0.491629	+	0.870805i	$0.336402\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27294.0	0.902066	0.451033	−	0.892507i	$-0.351056\pi$
$971$	27294.0	0.902066	0.451033	+	0.892507i	$0.351056\pi$
$972$	0	0
$973$	−27896.0	−0.919121
$974$	7312.00	0.240546
$975$	0	0
$976$	−400.000	−0.0131185
$977$	41334.0	1.35352	0.676761	−	0.736202i	$-0.263383\pi$
$977$	41334.0	1.35352	0.676761	+	0.736202i	$0.263383\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	30252.0	0.983075
$983$	41922.0	1.36023	0.680114	−	0.733106i	$-0.261930\pi$
$983$	41922.0	1.36023	0.680114	+	0.733106i	$0.261930\pi$
$984$	0	0
$985$	0	0
$986$	−3024.00	−0.0976712
$987$	0	0
$988$	5032.00	0.162034
$989$	2472.00	0.0794793
$990$	0	0
$991$	44948.0	1.44079	0.720394	−	0.693565i	$-0.243961\pi$
$991$	44948.0	1.44079	0.720394	+	0.693565i	$0.243961\pi$
$992$	3616.00	0.115734
$993$	0	0
$994$	−6600.00	−0.210603
$995$	0	0
$996$	0	0
$997$	60011.0	1.90629	0.953143	−	0.302520i	$-0.0978278\pi$
$997$	60011.0	1.90629	0.953143	+	0.302520i	$0.0978278\pi$
$998$	21922.0	0.695319
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.q.1.1		1
3.2	odd	2		726.4.a.a.1.1	✓	1
11.10	odd	2		2178.4.a.h.1.1		1
33.32	even	2		726.4.a.e.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
726.4.a.a.1.1	✓	1	3.2	odd	2
726.4.a.e.1.1	yes	1	33.32	even	2
2178.4.a.h.1.1		1	11.10	odd	2
2178.4.a.q.1.1		1	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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