LMFDB - Embedded newform 1815.4.a.m.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{15}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 15 $ x^2 - 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.87298$ 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)

$f(q)$	$=$	$q+3.87298 q^{2} +3.00000 q^{3} +7.00000 q^{4} -5.00000 q^{5} +11.6190 q^{6} -15.4919 q^{7} -3.87298 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.87298 q^{2} +3.00000 q^{3} +7.00000 q^{4} -5.00000 q^{5} +11.6190 q^{6} -15.4919 q^{7} -3.87298 q^{8} +9.00000 q^{9} -19.3649 q^{10} +21.0000 q^{12} +69.7137 q^{13} -60.0000 q^{14} -15.0000 q^{15} -71.0000 q^{16} -15.4919 q^{17} +34.8569 q^{18} +7.74597 q^{19} -35.0000 q^{20} -46.4758 q^{21} +48.0000 q^{23} -11.6190 q^{24} +25.0000 q^{25} +270.000 q^{26} +27.0000 q^{27} -108.444 q^{28} -209.141 q^{29} -58.0948 q^{30} +160.000 q^{31} -243.998 q^{32} -60.0000 q^{34} +77.4597 q^{35} +63.0000 q^{36} -266.000 q^{37} +30.0000 q^{38} +209.141 q^{39} +19.3649 q^{40} +178.157 q^{41} -180.000 q^{42} -309.839 q^{43} -45.0000 q^{45} +185.903 q^{46} -504.000 q^{47} -213.000 q^{48} -103.000 q^{49} +96.8246 q^{50} -46.4758 q^{51} +487.996 q^{52} -342.000 q^{53} +104.571 q^{54} +60.0000 q^{56} +23.2379 q^{57} -810.000 q^{58} -660.000 q^{59} -105.000 q^{60} -216.887 q^{61} +619.677 q^{62} -139.427 q^{63} -377.000 q^{64} -348.569 q^{65} +496.000 q^{67} -108.444 q^{68} +144.000 q^{69} +300.000 q^{70} -708.000 q^{71} -34.8569 q^{72} -642.915 q^{73} -1030.21 q^{74} +75.0000 q^{75} +54.2218 q^{76} +810.000 q^{78} -178.157 q^{79} +355.000 q^{80} +81.0000 q^{81} +690.000 q^{82} -85.2056 q^{83} -325.331 q^{84} +77.4597 q^{85} -1200.00 q^{86} -627.423 q^{87} -606.000 q^{89} -174.284 q^{90} -1080.00 q^{91} +336.000 q^{92} +480.000 q^{93} -1951.98 q^{94} -38.7298 q^{95} -731.994 q^{96} +254.000 q^{97} -398.917 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 14 q^{4} - 10 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 14 * q^4 - 10 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} + 14 q^{4} - 10 q^{5} + 18 q^{9} + 42 q^{12} - 120 q^{14} - 30 q^{15} - 142 q^{16} - 70 q^{20} + 96 q^{23} + 50 q^{25} + 540 q^{26} + 54 q^{27} + 320 q^{31} - 120 q^{34} + 126 q^{36} - 532 q^{37} + 60 q^{38} - 360 q^{42} - 90 q^{45} - 1008 q^{47} - 426 q^{48} - 206 q^{49} - 684 q^{53} + 120 q^{56} - 1620 q^{58} - 1320 q^{59} - 210 q^{60} - 754 q^{64} + 992 q^{67} + 288 q^{69} + 600 q^{70} - 1416 q^{71} + 150 q^{75} + 1620 q^{78} + 710 q^{80} + 162 q^{81} + 1380 q^{82} - 2400 q^{86} - 1212 q^{89} - 2160 q^{91} + 672 q^{92} + 960 q^{93} + 508 q^{97}+O(q^{100}) $ 2 * q + 6 * q^3 + 14 * q^4 - 10 * q^5 + 18 * q^9 + 42 * q^12 - 120 * q^14 - 30 * q^15 - 142 * q^16 - 70 * q^20 + 96 * q^23 + 50 * q^25 + 540 * q^26 + 54 * q^27 + 320 * q^31 - 120 * q^34 + 126 * q^36 - 532 * q^37 + 60 * q^38 - 360 * q^42 - 90 * q^45 - 1008 * q^47 - 426 * q^48 - 206 * q^49 - 684 * q^53 + 120 * q^56 - 1620 * q^58 - 1320 * q^59 - 210 * q^60 - 754 * q^64 + 992 * q^67 + 288 * q^69 + 600 * q^70 - 1416 * q^71 + 150 * q^75 + 1620 * q^78 + 710 * q^80 + 162 * q^81 + 1380 * q^82 - 2400 * q^86 - 1212 * q^89 - 2160 * q^91 + 672 * q^92 + 960 * q^93 + 508 * 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 $

$2$	3.87298	1.36931	0.684653	−	0.728869i	$-0.259954\pi$
$2$	3.87298	1.36931	0.684653	+	0.728869i	$0.259954\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	7.00000	0.875000
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	11.6190	0.790569
$7$	−15.4919	−0.836486	−0.418243	−	0.908335i	$-0.637354\pi$
$7$	−15.4919	−0.836486	−0.418243	+	0.908335i	$0.637354\pi$
$8$	−3.87298	−0.171163
$9$	9.00000	0.333333
$10$	−19.3649	−0.612372
$11$	0	0
$11$	0	0
$12$	21.0000	0.505181
$13$	69.7137	1.48732	0.743658	−	0.668561i	$-0.233089\pi$
$13$	69.7137	1.48732	0.743658	+	0.668561i	$0.233089\pi$
$14$	−60.0000	−1.14541
$15$	−15.0000	−0.258199
$16$	−71.0000	−1.10938
$17$	−15.4919	−0.221020	−0.110510	−	0.993875i	$-0.535248\pi$
$17$	−15.4919	−0.221020	−0.110510	+	0.993875i	$0.535248\pi$
$18$	34.8569	0.456435
$19$	7.74597	0.0935288	0.0467644	−	0.998906i	$-0.485109\pi$
$19$	7.74597	0.0935288	0.0467644	+	0.998906i	$0.485109\pi$
$20$	−35.0000	−0.391312
$21$	−46.4758	−0.482945
$22$	0	0
$23$	48.0000	0.435161	0.217580	−	0.976042i	$-0.430184\pi$
$23$	48.0000	0.435161	0.217580	+	0.976042i	$0.430184\pi$
$24$	−11.6190	−0.0988212
$25$	25.0000	0.200000
$26$	270.000	2.03659
$27$	27.0000	0.192450
$28$	−108.444	−0.731925
$29$	−209.141	−1.33919	−0.669595	−	0.742726i	$-0.733532\pi$
$29$	−209.141	−1.33919	−0.669595	+	0.742726i	$0.733532\pi$
$30$	−58.0948	−0.353553
$31$	160.000	0.926995	0.463498	−	0.886098i	$-0.346594\pi$
$31$	160.000	0.926995	0.463498	+	0.886098i	$0.346594\pi$
$32$	−243.998	−1.34791
$33$	0	0
$34$	−60.0000	−0.302645
$35$	77.4597	0.374088
$36$	63.0000	0.291667
$37$	−266.000	−1.18190	−0.590948	−	0.806710i	$-0.701246\pi$
$37$	−266.000	−1.18190	−0.590948	+	0.806710i	$0.701246\pi$
$38$	30.0000	0.128070
$39$	209.141	0.858702
$40$	19.3649	0.0765466
$41$	178.157	0.678622	0.339311	−	0.940674i	$-0.389806\pi$
$41$	178.157	0.678622	0.339311	+	0.940674i	$0.389806\pi$
$42$	−180.000	−0.661300
$43$	−309.839	−1.09884	−0.549418	−	0.835548i	$-0.685151\pi$
$43$	−309.839	−1.09884	−0.549418	+	0.835548i	$0.685151\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	185.903	0.595868
$47$	−504.000	−1.56417	−0.782085	−	0.623172i	$-0.785844\pi$
$47$	−504.000	−1.56417	−0.782085	+	0.623172i	$0.785844\pi$
$48$	−213.000	−0.640498
$49$	−103.000	−0.300292
$50$	96.8246	0.273861
$51$	−46.4758	−0.127606
$52$	487.996	1.30140
$53$	−342.000	−0.886364	−0.443182	−	0.896432i	$-0.646151\pi$
$53$	−342.000	−0.886364	−0.443182	+	0.896432i	$0.646151\pi$
$54$	104.571	0.263523
$55$	0	0
$56$	60.0000	0.143176
$57$	23.2379	0.0539989
$58$	−810.000	−1.83376
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	−105.000	−0.225924
$61$	−216.887	−0.455238	−0.227619	−	0.973750i	$-0.573094\pi$
$61$	−216.887	−0.455238	−0.227619	+	0.973750i	$0.573094\pi$
$62$	619.677	1.26934
$63$	−139.427	−0.278829
$64$	−377.000	−0.736328
$65$	−348.569	−0.665148
$66$	0	0
$67$	496.000	0.904419	0.452209	−	0.891912i	$-0.350636\pi$
$67$	496.000	0.904419	0.452209	+	0.891912i	$0.350636\pi$
$68$	−108.444	−0.193393
$69$	144.000	0.251240
$70$	300.000	0.512241
$71$	−708.000	−1.18344	−0.591719	−	0.806144i	$-0.701551\pi$
$71$	−708.000	−1.18344	−0.591719	+	0.806144i	$0.701551\pi$
$72$	−34.8569	−0.0570544
$73$	−642.915	−1.03079	−0.515394	−	0.856953i	$-0.672354\pi$
$73$	−642.915	−1.03079	−0.515394	+	0.856953i	$0.672354\pi$
$74$	−1030.21	−1.61838
$75$	75.0000	0.115470
$76$	54.2218	0.0818377
$77$	0	0
$78$	810.000	1.17583
$79$	−178.157	−0.253725	−0.126862	−	0.991920i	$-0.540491\pi$
$79$	−178.157	−0.253725	−0.126862	+	0.991920i	$0.540491\pi$
$80$	355.000	0.496128
$81$	81.0000	0.111111
$82$	690.000	0.929241
$83$	−85.2056	−0.112681	−0.0563406	−	0.998412i	$-0.517943\pi$
$83$	−85.2056	−0.112681	−0.0563406	+	0.998412i	$0.517943\pi$
$84$	−325.331	−0.422577
$85$	77.4597	0.0988433
$86$	−1200.00	−1.50464
$87$	−627.423	−0.773182
$88$	0	0
$89$	−606.000	−0.721751	−0.360876	−	0.932614i	$-0.617522\pi$
$89$	−606.000	−0.721751	−0.360876	+	0.932614i	$0.617522\pi$
$90$	−174.284	−0.204124
$91$	−1080.00	−1.24412
$92$	336.000	0.380765
$93$	480.000	0.535201
$94$	−1951.98	−2.14183
$95$	−38.7298	−0.0418273
$96$	−731.994	−0.778217
$97$	254.000	0.265874	0.132937	−	0.991124i	$-0.457559\pi$
$97$	254.000	0.265874	0.132937	+	0.991124i	$0.457559\pi$
$98$	−398.917	−0.411191
$99$	0	0
$100$	175.000	0.175000
$101$	1820.30	1.79334	0.896668	−	0.442705i	$-0.145981\pi$
$101$	1820.30	1.79334	0.896668	+	0.442705i	$0.145981\pi$
$102$	−180.000	−0.174732
$103$	1508.00	1.44260	0.721299	−	0.692624i	$-0.243545\pi$
$103$	1508.00	1.44260	0.721299	+	0.692624i	$0.243545\pi$
$104$	−270.000	−0.254574
$105$	232.379	0.215980
$106$	−1324.56	−1.21370
$107$	−890.786	−0.804818	−0.402409	−	0.915460i	$-0.631827\pi$
$107$	−890.786	−0.804818	−0.402409	+	0.915460i	$0.631827\pi$
$108$	189.000	0.168394
$109$	−681.645	−0.598989	−0.299494	−	0.954098i	$-0.596818\pi$
$109$	−681.645	−0.598989	−0.299494	+	0.954098i	$0.596818\pi$
$110$	0	0
$111$	−798.000	−0.682368
$112$	1099.93	0.927976
$113$	18.0000	0.0149849	0.00749247	−	0.999972i	$-0.497615\pi$
$113$	18.0000	0.0149849	0.00749247	+	0.999972i	$0.497615\pi$
$114$	90.0000	0.0739410
$115$	−240.000	−0.194610
$116$	−1463.99	−1.17179
$117$	627.423	0.495772
$118$	−2556.17	−1.99419
$119$	240.000	0.184880
$120$	58.0948	0.0441942
$121$	0	0
$122$	−840.000	−0.623361
$123$	534.472	0.391802
$124$	1120.00	0.811121
$125$	−125.000	−0.0894427
$126$	−540.000	−0.381802
$127$	278.855	0.194837	0.0974187	−	0.995243i	$-0.468941\pi$
$127$	278.855	0.194837	0.0974187	+	0.995243i	$0.468941\pi$
$128$	491.869	0.339652
$129$	−929.516	−0.634413
$130$	−1350.00	−0.910791
$131$	−821.072	−0.547614	−0.273807	−	0.961785i	$-0.588283\pi$
$131$	−821.072	−0.547614	−0.273807	+	0.961785i	$0.588283\pi$
$132$	0	0
$133$	−120.000	−0.0782355
$134$	1921.00	1.23843
$135$	−135.000	−0.0860663
$136$	60.0000	0.0378306
$137$	−1866.00	−1.16367	−0.581836	−	0.813306i	$-0.697666\pi$
$137$	−1866.00	−1.16367	−0.581836	+	0.813306i	$0.697666\pi$
$138$	557.710	0.344025
$139$	426.028	0.259966	0.129983	−	0.991516i	$-0.458508\pi$
$139$	426.028	0.259966	0.129983	+	0.991516i	$0.458508\pi$
$140$	542.218	0.327327
$141$	−1512.00	−0.903074
$142$	−2742.07	−1.62049
$143$	0	0
$144$	−639.000	−0.369792
$145$	1045.71	0.598904
$146$	−2490.00	−1.41146
$147$	−309.000	−0.173373
$148$	−1862.00	−1.03416
$149$	−1634.40	−0.898625	−0.449313	−	0.893375i	$-0.648331\pi$
$149$	−1634.40	−0.898625	−0.449313	+	0.893375i	$0.648331\pi$
$150$	290.474	0.158114
$151$	−1820.30	−0.981020	−0.490510	−	0.871435i	$-0.663190\pi$
$151$	−1820.30	−0.981020	−0.490510	+	0.871435i	$0.663190\pi$
$152$	−30.0000	−0.0160087
$153$	−139.427	−0.0736734
$154$	0	0
$155$	−800.000	−0.414565
$156$	1463.99	0.751364
$157$	274.000	0.139284	0.0696420	−	0.997572i	$-0.477814\pi$
$157$	274.000	0.139284	0.0696420	+	0.997572i	$0.477814\pi$
$158$	−690.000	−0.347427
$159$	−1026.00	−0.511743
$160$	1219.99	0.602804
$161$	−743.613	−0.364006
$162$	313.712	0.152145
$163$	−3688.00	−1.77219	−0.886093	−	0.463507i	$-0.846591\pi$
$163$	−3688.00	−1.77219	−0.886093	+	0.463507i	$0.846591\pi$
$164$	1247.10	0.593794
$165$	0	0
$166$	−330.000	−0.154295
$167$	1789.32	0.829111	0.414556	−	0.910024i	$-0.363937\pi$
$167$	1789.32	0.829111	0.414556	+	0.910024i	$0.363937\pi$
$168$	180.000	0.0826625
$169$	2663.00	1.21211
$170$	300.000	0.135347
$171$	69.7137	0.0311763
$172$	−2168.87	−0.961482
$173$	−790.089	−0.347222	−0.173611	−	0.984814i	$-0.555543\pi$
$173$	−790.089	−0.347222	−0.173611	+	0.984814i	$0.555543\pi$
$174$	−2430.00	−1.05872
$175$	−387.298	−0.167297
$176$	0	0
$177$	−1980.00	−0.840824
$178$	−2347.03	−0.988299
$179$	3204.00	1.33787	0.668934	−	0.743322i	$-0.266751\pi$
$179$	3204.00	1.33787	0.668934	+	0.743322i	$0.266751\pi$
$180$	−315.000	−0.130437
$181$	4490.00	1.84386	0.921931	−	0.387354i	$-0.126611\pi$
$181$	4490.00	1.84386	0.921931	+	0.387354i	$0.126611\pi$
$182$	−4182.82	−1.70358
$183$	−650.661	−0.262832
$184$	−185.903	−0.0744835
$185$	1330.00	0.528560
$186$	1859.03	0.732854
$187$	0	0
$188$	−3528.00	−1.36865
$189$	−418.282	−0.160982
$190$	−150.000	−0.0572744
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1131.00	−0.425119
$193$	3772.29	1.40692	0.703459	−	0.710736i	$-0.251638\pi$
$193$	3772.29	1.40692	0.703459	+	0.710736i	$0.251638\pi$
$194$	983.738	0.364063
$195$	−1045.71	−0.384023
$196$	−721.000	−0.262755
$197$	418.282	0.151276	0.0756380	−	0.997135i	$-0.475901\pi$
$197$	418.282	0.151276	0.0756380	+	0.997135i	$0.475901\pi$
$198$	0	0
$199$	−3680.00	−1.31090	−0.655448	−	0.755240i	$-0.727520\pi$
$199$	−3680.00	−1.31090	−0.655448	+	0.755240i	$0.727520\pi$
$200$	−96.8246	−0.0342327
$201$	1488.00	0.522166
$202$	7050.00	2.45563
$203$	3240.00	1.12021
$204$	−325.331	−0.111655
$205$	−890.786	−0.303489
$206$	5840.46	1.97536
$207$	432.000	0.145054
$208$	−4949.67	−1.64999
$209$	0	0
$210$	900.000	0.295742
$211$	2300.55	0.750600	0.375300	−	0.926903i	$-0.377540\pi$
$211$	2300.55	0.750600	0.375300	+	0.926903i	$0.377540\pi$
$212$	−2394.00	−0.775569
$213$	−2124.00	−0.683259
$214$	−3450.00	−1.10204
$215$	1549.19	0.491414
$216$	−104.571	−0.0329404
$217$	−2478.71	−0.775418
$218$	−2640.00	−0.820199
$219$	−1928.75	−0.595126
$220$	0	0
$221$	−1080.00	−0.328727
$222$	−3090.64	−0.934370
$223$	−3568.00	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−3568.00	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	3780.00	1.12751
$225$	225.000	0.0666667
$226$	69.7137	0.0205190
$227$	4980.66	1.45629	0.728145	−	0.685423i	$-0.240383\pi$
$227$	4980.66	1.45629	0.728145	+	0.685423i	$0.240383\pi$
$228$	162.665	0.0472490
$229$	5170.00	1.49189	0.745946	−	0.666007i	$-0.231998\pi$
$229$	5170.00	1.49189	0.745946	+	0.666007i	$0.231998\pi$
$230$	−929.516	−0.266480
$231$	0	0
$232$	810.000	0.229220
$233$	4663.07	1.31111	0.655554	−	0.755149i	$-0.272435\pi$
$233$	4663.07	1.31111	0.655554	+	0.755149i	$0.272435\pi$
$234$	2430.00	0.678864
$235$	2520.00	0.699518
$236$	−4620.00	−1.27431
$237$	−534.472	−0.146488
$238$	929.516	0.253158
$239$	2912.48	0.788255	0.394127	−	0.919056i	$-0.371047\pi$
$239$	2912.48	0.788255	0.394127	+	0.919056i	$0.371047\pi$
$240$	1065.00	0.286439
$241$	5577.10	1.49067	0.745337	−	0.666688i	$-0.232289\pi$
$241$	5577.10	1.49067	0.745337	+	0.666688i	$0.232289\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	−1518.21	−0.398334
$245$	515.000	0.134294
$246$	2070.00	0.536497
$247$	540.000	0.139107
$248$	−619.677	−0.158668
$249$	−255.617	−0.0650565
$250$	−484.123	−0.122474
$251$	−1272.00	−0.319872	−0.159936	−	0.987127i	$-0.551129\pi$
$251$	−1272.00	−0.319872	−0.159936	+	0.987127i	$0.551129\pi$
$252$	−975.992	−0.243975
$253$	0	0
$254$	1080.00	0.266792
$255$	232.379	0.0570672
$256$	4921.00	1.20142
$257$	1986.00	0.482036	0.241018	−	0.970521i	$-0.422519\pi$
$257$	1986.00	0.482036	0.241018	+	0.970521i	$0.422519\pi$
$258$	−3600.00	−0.868706
$259$	4120.85	0.988639
$260$	−2439.98	−0.582004
$261$	−1882.27	−0.446397
$262$	−3180.00	−0.749851
$263$	−4732.79	−1.10964	−0.554821	−	0.831969i	$-0.687213\pi$
$263$	−4732.79	−1.10964	−0.554821	+	0.831969i	$0.687213\pi$
$264$	0	0
$265$	1710.00	0.396394
$266$	−464.758	−0.107128
$267$	−1818.00	−0.416703
$268$	3472.00	0.791366
$269$	−6570.00	−1.48914	−0.744572	−	0.667542i	$-0.767347\pi$
$269$	−6570.00	−1.48914	−0.744572	+	0.667542i	$0.767347\pi$
$270$	−522.853	−0.117851
$271$	1386.53	0.310795	0.155398	−	0.987852i	$-0.450334\pi$
$271$	1386.53	0.310795	0.155398	+	0.987852i	$0.450334\pi$
$272$	1099.93	0.245194
$273$	−3240.00	−0.718292
$274$	−7226.99	−1.59342
$275$	0	0
$276$	1008.00	0.219835
$277$	−7769.20	−1.68522	−0.842611	−	0.538523i	$-0.818982\pi$
$277$	−7769.20	−1.68522	−0.842611	+	0.538523i	$0.818982\pi$
$278$	1650.00	0.355973
$279$	1440.00	0.308998
$280$	−300.000	−0.0640301
$281$	−6731.25	−1.42901	−0.714506	−	0.699629i	$-0.753349\pi$
$281$	−6731.25	−1.42901	−0.714506	+	0.699629i	$0.753349\pi$
$282$	−5855.95	−1.23658
$283$	3284.29	0.689861	0.344931	−	0.938628i	$-0.387902\pi$
$283$	3284.29	0.689861	0.344931	+	0.938628i	$0.387902\pi$
$284$	−4956.00	−1.03551
$285$	−116.190	−0.0241490
$286$	0	0
$287$	−2760.00	−0.567657
$288$	−2195.98	−0.449304
$289$	−4673.00	−0.951150
$290$	4050.00	0.820083
$291$	762.000	0.153503
$292$	−4500.41	−0.901940
$293$	7699.49	1.53518	0.767592	−	0.640938i	$-0.221455\pi$
$293$	7699.49	1.53518	0.767592	+	0.640938i	$0.221455\pi$
$294$	−1196.75	−0.237401
$295$	3300.00	0.651300
$296$	1030.21	0.202297
$297$	0	0
$298$	−6330.00	−1.23049
$299$	3346.26	0.647221
$300$	525.000	0.101036
$301$	4800.00	0.919161
$302$	−7050.00	−1.34332
$303$	5460.91	1.03538
$304$	−549.964	−0.103758
$305$	1084.44	0.203589
$306$	−540.000	−0.100882
$307$	5778.49	1.07425	0.537127	−	0.843501i	$-0.319510\pi$
$307$	5778.49	1.07425	0.537127	+	0.843501i	$0.319510\pi$
$308$	0	0
$309$	4524.00	0.832885
$310$	−3098.39	−0.567666
$311$	5328.00	0.971457	0.485729	−	0.874110i	$-0.338554\pi$
$311$	5328.00	0.971457	0.485729	+	0.874110i	$0.338554\pi$
$312$	−810.000	−0.146978
$313$	442.000	0.0798189	0.0399095	−	0.999203i	$-0.487293\pi$
$313$	442.000	0.0798189	0.0399095	+	0.999203i	$0.487293\pi$
$314$	1061.20	0.190722
$315$	697.137	0.124696
$316$	−1247.10	−0.222009
$317$	−7794.00	−1.38093	−0.690465	−	0.723366i	$-0.742594\pi$
$317$	−7794.00	−1.38093	−0.690465	+	0.723366i	$0.742594\pi$
$318$	−3973.68	−0.700733
$319$	0	0
$320$	1885.00	0.329296
$321$	−2672.36	−0.464662
$322$	−2880.00	−0.498435
$323$	−120.000	−0.0206718
$324$	567.000	0.0972222
$325$	1742.84	0.297463
$326$	−14283.6	−2.42667
$327$	−2044.94	−0.345826
$328$	−690.000	−0.116155
$329$	7807.93	1.30841
$330$	0	0
$331$	−2140.00	−0.355363	−0.177681	−	0.984088i	$-0.556860\pi$
$331$	−2140.00	−0.355363	−0.177681	+	0.984088i	$0.556860\pi$
$332$	−596.439	−0.0985960
$333$	−2394.00	−0.393965
$334$	6930.00	1.13531
$335$	−2480.00	−0.404468
$336$	3299.78	0.535767
$337$	−487.996	−0.0788808	−0.0394404	−	0.999222i	$-0.512558\pi$
$337$	−487.996	−0.0788808	−0.0394404	+	0.999222i	$0.512558\pi$
$338$	10313.8	1.65975
$339$	54.0000	0.00865156
$340$	542.218	0.0864879
$341$	0	0
$342$	270.000	0.0426898
$343$	6909.40	1.08768
$344$	1200.00	0.188080
$345$	−720.000	−0.112358
$346$	−3060.00	−0.475453
$347$	−9194.46	−1.42243	−0.711217	−	0.702973i	$-0.751856\pi$
$347$	−9194.46	−1.42243	−0.711217	+	0.702973i	$0.751856\pi$
$348$	−4391.96	−0.676534
$349$	−3284.29	−0.503736	−0.251868	−	0.967762i	$-0.581045\pi$
$349$	−3284.29	−0.503736	−0.251868	+	0.967762i	$0.581045\pi$
$350$	−1500.00	−0.229081
$351$	1882.27	0.286234
$352$	0	0
$353$	−4602.00	−0.693880	−0.346940	−	0.937887i	$-0.612779\pi$
$353$	−4602.00	−0.693880	−0.346940	+	0.937887i	$0.612779\pi$
$354$	−7668.51	−1.15135
$355$	3540.00	0.529250
$356$	−4242.00	−0.631532
$357$	720.000	0.106741
$358$	12409.0	1.83195
$359$	7730.47	1.13649	0.568244	−	0.822860i	$-0.307623\pi$
$359$	7730.47	1.13649	0.568244	+	0.822860i	$0.307623\pi$
$360$	174.284	0.0255155
$361$	−6799.00	−0.991252
$362$	17389.7	2.52481
$363$	0	0
$364$	−7560.00	−1.08860
$365$	3214.58	0.460982
$366$	−2520.00	−0.359898
$367$	3176.00	0.451733	0.225866	−	0.974158i	$-0.427479\pi$
$367$	3176.00	0.451733	0.225866	+	0.974158i	$0.427479\pi$
$368$	−3408.00	−0.482756
$369$	1603.42	0.226207
$370$	5151.07	0.723760
$371$	5298.24	0.741431
$372$	3360.00	0.468301
$373$	10356.4	1.43762	0.718809	−	0.695207i	$-0.244687\pi$
$373$	10356.4	1.43762	0.718809	+	0.695207i	$0.244687\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	1951.98	0.267728
$377$	−14580.0	−1.99180
$378$	−1620.00	−0.220433
$379$	−8020.00	−1.08696	−0.543482	−	0.839421i	$-0.682895\pi$
$379$	−8020.00	−1.08696	−0.543482	+	0.839421i	$0.682895\pi$
$380$	−271.109	−0.0365989
$381$	836.564	0.112489
$382$	0	0
$383$	−1128.00	−0.150491	−0.0752456	−	0.997165i	$-0.523974\pi$
$383$	−1128.00	−0.150491	−0.0752456	+	0.997165i	$0.523974\pi$
$384$	1475.61	0.196098
$385$	0	0
$386$	14610.0	1.92650
$387$	−2788.55	−0.366279
$388$	1778.00	0.232640
$389$	−2070.00	−0.269802	−0.134901	−	0.990859i	$-0.543072\pi$
$389$	−2070.00	−0.269802	−0.134901	+	0.990859i	$0.543072\pi$
$390$	−4050.00	−0.525845
$391$	−743.613	−0.0961793
$392$	398.917	0.0513989
$393$	−2463.22	−0.316165
$394$	1620.00	0.207143
$395$	890.786	0.113469
$396$	0	0
$397$	3314.00	0.418954	0.209477	−	0.977814i	$-0.432824\pi$
$397$	3314.00	0.418954	0.209477	+	0.977814i	$0.432824\pi$
$398$	−14252.6	−1.79502
$399$	−360.000	−0.0451693
$400$	−1775.00	−0.221875
$401$	3030.00	0.377334	0.188667	−	0.982041i	$-0.439583\pi$
$401$	3030.00	0.377334	0.188667	+	0.982041i	$0.439583\pi$
$402$	5763.00	0.715006
$403$	11154.2	1.37873
$404$	12742.1	1.56917
$405$	−405.000	−0.0496904
$406$	12548.5	1.53392
$407$	0	0
$408$	180.000	0.0218415
$409$	−12594.9	−1.52269	−0.761344	−	0.648348i	$-0.775460\pi$
$409$	−12594.9	−1.52269	−0.761344	+	0.648348i	$0.775460\pi$
$410$	−3450.00	−0.415569
$411$	−5598.00	−0.671847
$412$	10556.0	1.26227
$413$	10224.7	1.21822
$414$	1673.13	0.198623
$415$	426.028	0.0503925
$416$	−17010.0	−2.00477
$417$	1278.08	0.150091
$418$	0	0
$419$	2436.00	0.284025	0.142012	−	0.989865i	$-0.454643\pi$
$419$	2436.00	0.284025	0.142012	+	0.989865i	$0.454643\pi$
$420$	1626.65	0.188982
$421$	−11810.0	−1.36718	−0.683592	−	0.729865i	$-0.739583\pi$
$421$	−11810.0	−1.36718	−0.683592	+	0.729865i	$0.739583\pi$
$422$	8910.00	1.02780
$423$	−4536.00	−0.521390
$424$	1324.56	0.151713
$425$	−387.298	−0.0442041
$426$	−8226.22	−0.935590
$427$	3360.00	0.380800
$428$	−6235.50	−0.704216
$429$	0	0
$430$	6000.00	0.672897
$431$	−9837.38	−1.09942	−0.549710	−	0.835356i	$-0.685262\pi$
$431$	−9837.38	−1.09942	−0.549710	+	0.835356i	$0.685262\pi$
$432$	−1917.00	−0.213499
$433$	−15658.0	−1.73782	−0.868909	−	0.494971i	$-0.835179\pi$
$433$	−15658.0	−1.73782	−0.868909	+	0.494971i	$0.835179\pi$
$434$	−9600.00	−1.06179
$435$	3137.12	0.345778
$436$	−4771.52	−0.524115
$437$	371.806	0.0407000
$438$	−7470.00	−0.814910
$439$	12091.5	1.31456	0.657282	−	0.753645i	$-0.271706\pi$
$439$	12091.5	1.31456	0.657282	+	0.753645i	$0.271706\pi$
$440$	0	0
$441$	−927.000	−0.100097
$442$	−4182.82	−0.450128
$443$	−13812.0	−1.48133	−0.740664	−	0.671876i	$-0.765489\pi$
$443$	−13812.0	−1.48133	−0.740664	+	0.671876i	$0.765489\pi$
$444$	−5586.00	−0.597072
$445$	3030.00	0.322777
$446$	−13818.8	−1.46713
$447$	−4903.20	−0.518822
$448$	5840.46	0.615928
$449$	8706.00	0.915059	0.457530	−	0.889194i	$-0.348734\pi$
$449$	8706.00	0.915059	0.457530	+	0.889194i	$0.348734\pi$
$450$	871.421	0.0912871
$451$	0	0
$452$	126.000	0.0131118
$453$	−5460.91	−0.566392
$454$	19290.0	1.99411
$455$	5400.00	0.556387
$456$	−90.0000	−0.00924262
$457$	−5460.91	−0.558972	−0.279486	−	0.960150i	$-0.590164\pi$
$457$	−5460.91	−0.558972	−0.279486	+	0.960150i	$0.590164\pi$
$458$	20023.3	2.04286
$459$	−418.282	−0.0425354
$460$	−1680.00	−0.170283
$461$	13206.9	1.33429	0.667143	−	0.744930i	$-0.267517\pi$
$461$	13206.9	1.33429	0.667143	+	0.744930i	$0.267517\pi$
$462$	0	0
$463$	−12328.0	−1.23743	−0.618716	−	0.785615i	$-0.712347\pi$
$463$	−12328.0	−1.23743	−0.618716	+	0.785615i	$0.712347\pi$
$464$	14849.0	1.48566
$465$	−2400.00	−0.239349
$466$	18060.0	1.79531
$467$	−6036.00	−0.598100	−0.299050	−	0.954237i	$-0.596670\pi$
$467$	−6036.00	−0.598100	−0.299050	+	0.954237i	$0.596670\pi$
$468$	4391.96	0.433800
$469$	−7684.00	−0.756533
$470$	9759.92	0.957854
$471$	822.000	0.0804156
$472$	2556.17	0.249274
$473$	0	0
$474$	−2070.00	−0.200587
$475$	193.649	0.0187058
$476$	1680.00	0.161770
$477$	−3078.00	−0.295455
$478$	11280.0	1.07936
$479$	−1859.03	−0.177331	−0.0886653	−	0.996061i	$-0.528260\pi$
$479$	−1859.03	−0.177331	−0.0886653	+	0.996061i	$0.528260\pi$
$480$	3659.97	0.348029
$481$	−18543.8	−1.75785
$482$	21600.0	2.04119
$483$	−2230.84	−0.210159
$484$	0	0
$485$	−1270.00	−0.118903
$486$	941.135	0.0878410
$487$	5716.00	0.531862	0.265931	−	0.963992i	$-0.414321\pi$
$487$	5716.00	0.531862	0.265931	+	0.963992i	$0.414321\pi$
$488$	840.000	0.0779201
$489$	−11064.0	−1.02317
$490$	1994.59	0.183890
$491$	11138.7	1.02379	0.511897	−	0.859047i	$-0.328943\pi$
$491$	11138.7	1.02379	0.511897	+	0.859047i	$0.328943\pi$
$492$	3741.30	0.342827
$493$	3240.00	0.295988
$494$	2091.41	0.190480
$495$	0	0
$496$	−11360.0	−1.02839
$497$	10968.3	0.989930
$498$	−990.000	−0.0890823
$499$	−8420.00	−0.755373	−0.377686	−	0.925934i	$-0.623280\pi$
$499$	−8420.00	−0.755373	−0.377686	+	0.925934i	$0.623280\pi$
$500$	−875.000	−0.0782624
$501$	5367.95	0.478688
$502$	−4926.43	−0.438003
$503$	20782.4	1.84223	0.921116	−	0.389288i	$-0.127279\pi$
$503$	20782.4	1.84223	0.921116	+	0.389288i	$0.127279\pi$
$504$	540.000	0.0477252
$505$	−9101.51	−0.802004
$506$	0	0
$507$	7989.00	0.699811
$508$	1951.98	0.170483
$509$	10470.0	0.911738	0.455869	−	0.890047i	$-0.349329\pi$
$509$	10470.0	0.911738	0.455869	+	0.890047i	$0.349329\pi$
$510$	900.000	0.0781425
$511$	9960.00	0.862240
$512$	15124.0	1.30545
$513$	209.141	0.0179996
$514$	7691.74	0.660055
$515$	−7540.00	−0.645150
$516$	−6506.61	−0.555112
$517$	0	0
$518$	15960.0	1.35375
$519$	−2370.27	−0.200468
$520$	1350.00	0.113849
$521$	−6090.00	−0.512107	−0.256053	−	0.966663i	$-0.582422\pi$
$521$	−6090.00	−0.512107	−0.256053	+	0.966663i	$0.582422\pi$
$522$	−7290.00	−0.611254
$523$	9961.31	0.832845	0.416422	−	0.909171i	$-0.363284\pi$
$523$	9961.31	0.832845	0.416422	+	0.909171i	$0.363284\pi$
$524$	−5747.51	−0.479162
$525$	−1161.90	−0.0965891
$526$	−18330.0	−1.51944
$527$	−2478.71	−0.204885
$528$	0	0
$529$	−9863.00	−0.810635
$530$	6622.80	0.542785
$531$	−5940.00	−0.485450
$532$	−840.000	−0.0684561
$533$	12420.0	1.00932
$534$	−7041.08	−0.570595
$535$	4453.93	0.359926
$536$	−1921.00	−0.154803
$537$	9612.00	0.772418
$538$	−25445.5	−2.03910
$539$	0	0
$540$	−945.000	−0.0753080
$541$	−23640.7	−1.87873	−0.939365	−	0.342920i	$-0.888584\pi$
$541$	−23640.7	−1.87873	−0.939365	+	0.342920i	$0.888584\pi$
$542$	5370.00	0.425574
$543$	13470.0	1.06455
$544$	3780.00	0.297916
$545$	3408.23	0.267876
$546$	−12548.5	−0.983562
$547$	−13648.4	−1.06684	−0.533422	−	0.845850i	$-0.679094\pi$
$547$	−13648.4	−1.06684	−0.533422	+	0.845850i	$0.679094\pi$
$548$	−13062.0	−1.01821
$549$	−1951.98	−0.151746
$550$	0	0
$551$	−1620.00	−0.125253
$552$	−557.710	−0.0430031
$553$	2760.00	0.212237
$554$	−30090.0	−2.30758
$555$	3990.00	0.305164
$556$	2982.20	0.227470
$557$	8675.48	0.659950	0.329975	−	0.943990i	$-0.392960\pi$
$557$	8675.48	0.659950	0.329975	+	0.943990i	$0.392960\pi$
$558$	5577.10	0.423113
$559$	−21600.0	−1.63432
$560$	−5499.64	−0.415004
$561$	0	0
$562$	−26070.0	−1.95676
$563$	21169.7	1.58472	0.792360	−	0.610053i	$-0.208852\pi$
$563$	21169.7	1.58472	0.792360	+	0.610053i	$0.208852\pi$
$564$	−10584.0	−0.790189
$565$	−90.0000	−0.00670147
$566$	12720.0	0.944632
$567$	−1254.85	−0.0929429
$568$	2742.07	0.202561
$569$	−17699.5	−1.30405	−0.652024	−	0.758199i	$-0.726080\pi$
$569$	−17699.5	−1.30405	−0.652024	+	0.758199i	$0.726080\pi$
$570$	−450.000	−0.0330674
$571$	13795.6	1.01108	0.505540	−	0.862803i	$-0.331293\pi$
$571$	13795.6	1.01108	0.505540	+	0.862803i	$0.331293\pi$
$572$	0	0
$573$	0	0
$574$	−10689.4	−0.777297
$575$	1200.00	0.0870321
$576$	−3393.00	−0.245443
$577$	−1514.00	−0.109235	−0.0546175	−	0.998507i	$-0.517394\pi$
$577$	−1514.00	−0.109235	−0.0546175	+	0.998507i	$0.517394\pi$
$578$	−18098.5	−1.30242
$579$	11316.9	0.812284
$580$	7319.94	0.524041
$581$	1320.00	0.0942562
$582$	2951.21	0.210192
$583$	0	0
$584$	2490.00	0.176433
$585$	−3137.12	−0.221716
$586$	29820.0	2.10214
$587$	−10956.0	−0.770362	−0.385181	−	0.922841i	$-0.625861\pi$
$587$	−10956.0	−0.770362	−0.385181	+	0.922841i	$0.625861\pi$
$588$	−2163.00	−0.151702
$589$	1239.35	0.0867007
$590$	12780.8	0.891829
$591$	1254.85	0.0873392
$592$	18886.0	1.31117
$593$	−12982.2	−0.899016	−0.449508	−	0.893276i	$-0.648401\pi$
$593$	−12982.2	−0.899016	−0.449508	+	0.893276i	$0.648401\pi$
$594$	0	0
$595$	−1200.00	−0.0826810
$596$	−11440.8	−0.786297
$597$	−11040.0	−0.756846
$598$	12960.0	0.886244
$599$	15996.0	1.09112	0.545558	−	0.838073i	$-0.316318\pi$
$599$	15996.0	1.09112	0.545558	+	0.838073i	$0.316318\pi$
$600$	−290.474	−0.0197642
$601$	−16003.2	−1.08616	−0.543081	−	0.839681i	$-0.682742\pi$
$601$	−16003.2	−1.08616	−0.543081	+	0.839681i	$0.682742\pi$
$602$	18590.3	1.25861
$603$	4464.00	0.301473
$604$	−12742.1	−0.858393
$605$	0	0
$606$	21150.0	1.41776
$607$	1270.34	0.0849447	0.0424724	−	0.999098i	$-0.486477\pi$
$607$	1270.34	0.0849447	0.0424724	+	0.999098i	$0.486477\pi$
$608$	−1890.00	−0.126068
$609$	9720.00	0.646756
$610$	4200.00	0.278775
$611$	−35135.7	−2.32641
$612$	−975.992	−0.0644643
$613$	131.681	0.00867629	0.00433814	−	0.999991i	$-0.498619\pi$
$613$	131.681	0.00867629	0.00433814	+	0.999991i	$0.498619\pi$
$614$	22380.0	1.47098
$615$	−2672.36	−0.175219
$616$	0	0
$617$	1746.00	0.113924	0.0569622	−	0.998376i	$-0.481859\pi$
$617$	1746.00	0.113924	0.0569622	+	0.998376i	$0.481859\pi$
$618$	17521.4	1.14047
$619$	2036.00	0.132203	0.0661016	−	0.997813i	$-0.478944\pi$
$619$	2036.00	0.132203	0.0661016	+	0.997813i	$0.478944\pi$
$620$	−5600.00	−0.362744
$621$	1296.00	0.0837467
$622$	20635.3	1.33022
$623$	9388.11	0.603735
$624$	−14849.0	−0.952623
$625$	625.000	0.0400000
$626$	1711.86	0.109297
$627$	0	0
$628$	1918.00	0.121873
$629$	4120.85	0.261223
$630$	2700.00	0.170747
$631$	−7328.00	−0.462319	−0.231159	−	0.972916i	$-0.574252\pi$
$631$	−7328.00	−0.462319	−0.231159	+	0.972916i	$0.574252\pi$
$632$	690.000	0.0434284
$633$	6901.66	0.433359
$634$	−30186.0	−1.89092
$635$	−1394.27	−0.0871340
$636$	−7182.00	−0.447775
$637$	−7180.51	−0.446628
$638$	0	0
$639$	−6372.00	−0.394480
$640$	−2459.34	−0.151897
$641$	−5790.00	−0.356773	−0.178386	−	0.983961i	$-0.557088\pi$
$641$	−5790.00	−0.356773	−0.178386	+	0.983961i	$0.557088\pi$
$642$	−10350.0	−0.636265
$643$	24128.0	1.47981	0.739903	−	0.672713i	$-0.234871\pi$
$643$	24128.0	1.47981	0.739903	+	0.672713i	$0.234871\pi$
$644$	−5205.29	−0.318505
$645$	4647.58	0.283718
$646$	−464.758	−0.0283060
$647$	−20904.0	−1.27020	−0.635101	−	0.772429i	$-0.719042\pi$
$647$	−20904.0	−1.27020	−0.635101	+	0.772429i	$0.719042\pi$
$648$	−313.712	−0.0190181
$649$	0	0
$650$	6750.00	0.407318
$651$	−7436.13	−0.447688
$652$	−25816.0	−1.55066
$653$	−13482.0	−0.807950	−0.403975	−	0.914770i	$-0.632372\pi$
$653$	−13482.0	−0.807950	−0.403975	+	0.914770i	$0.632372\pi$
$654$	−7920.00	−0.473542
$655$	4105.36	0.244900
$656$	−12649.2	−0.752846
$657$	−5786.24	−0.343596
$658$	30240.0	1.79161
$659$	25127.9	1.48535	0.742674	−	0.669653i	$-0.233557\pi$
$659$	25127.9	1.48535	0.742674	+	0.669653i	$0.233557\pi$
$660$	0	0
$661$	4498.00	0.264678	0.132339	−	0.991205i	$-0.457751\pi$
$661$	4498.00	0.264678	0.132339	+	0.991205i	$0.457751\pi$
$662$	−8288.18	−0.486600
$663$	−3240.00	−0.189791
$664$	330.000	0.0192869
$665$	600.000	0.0349880
$666$	−9271.92	−0.539459
$667$	−10038.8	−0.582763
$668$	12525.2	0.725472
$669$	−10704.0	−0.618596
$670$	−9605.00	−0.553841
$671$	0	0
$672$	11340.0	0.650967
$673$	25461.0	1.45832	0.729160	−	0.684343i	$-0.239911\pi$
$673$	25461.0	1.45832	0.729160	+	0.684343i	$0.239911\pi$
$674$	−1890.00	−0.108012
$675$	675.000	0.0384900
$676$	18641.0	1.06059
$677$	−12207.6	−0.693025	−0.346512	−	0.938045i	$-0.612634\pi$
$677$	−12207.6	−0.693025	−0.346512	+	0.938045i	$0.612634\pi$
$678$	209.141	0.0118466
$679$	−3934.95	−0.222400
$680$	−300.000	−0.0169183
$681$	14942.0	0.840789
$682$	0	0
$683$	2052.00	0.114960	0.0574799	−	0.998347i	$-0.481693\pi$
$683$	2052.00	0.114960	0.0574799	+	0.998347i	$0.481693\pi$
$684$	487.996	0.0272792
$685$	9330.00	0.520410
$686$	26760.0	1.48936
$687$	15510.0	0.861344
$688$	21998.5	1.21902
$689$	−23842.1	−1.31830
$690$	−2788.55	−0.153852
$691$	12572.0	0.692129	0.346065	−	0.938211i	$-0.387518\pi$
$691$	12572.0	0.692129	0.346065	+	0.938211i	$0.387518\pi$
$692$	−5530.62	−0.303819
$693$	0	0
$694$	−35610.0	−1.94775
$695$	−2130.14	−0.116260
$696$	2430.00	0.132340
$697$	−2760.00	−0.149989
$698$	−12720.0	−0.689769
$699$	13989.2	0.756968
$700$	−2711.09	−0.146385
$701$	−20147.3	−1.08552	−0.542761	−	0.839887i	$-0.682621\pi$
$701$	−20147.3	−1.08552	−0.542761	+	0.839887i	$0.682621\pi$
$702$	7290.00	0.391942
$703$	−2060.43	−0.110541
$704$	0	0
$705$	7560.00	0.403867
$706$	−17823.5	−0.950135
$707$	−28200.0	−1.50010
$708$	−13860.0	−0.735721
$709$	27646.0	1.46441	0.732205	−	0.681084i	$-0.238491\pi$
$709$	27646.0	1.46441	0.732205	+	0.681084i	$0.238491\pi$
$710$	13710.4	0.724705
$711$	−1603.42	−0.0845749
$712$	2347.03	0.123537
$713$	7680.00	0.403392
$714$	2788.55	0.146161
$715$	0	0
$716$	22428.0	1.17063
$717$	8737.45	0.455099
$718$	29940.0	1.55620
$719$	−10896.0	−0.565163	−0.282582	−	0.959243i	$-0.591191\pi$
$719$	−10896.0	−0.565163	−0.282582	+	0.959243i	$0.591191\pi$
$720$	3195.00	0.165376
$721$	−23361.8	−1.20671
$722$	−26332.4	−1.35733
$723$	16731.3	0.860641
$724$	31430.0	1.61338
$725$	−5228.53	−0.267838
$726$	0	0
$727$	−21656.0	−1.10478	−0.552391	−	0.833585i	$-0.686284\pi$
$727$	−21656.0	−1.10478	−0.552391	+	0.833585i	$0.686284\pi$
$728$	4182.82	0.212947
$729$	729.000	0.0370370
$730$	12450.0	0.631226
$731$	4800.00	0.242865
$732$	−4554.63	−0.229978
$733$	27351.0	1.37822	0.689108	−	0.724659i	$-0.258002\pi$
$733$	27351.0	1.37822	0.689108	+	0.724659i	$0.258002\pi$
$734$	12300.6	0.618560
$735$	1545.00	0.0775349
$736$	−11711.9	−0.586558
$737$	0	0
$738$	6210.00	0.309747
$739$	14105.4	0.702132	0.351066	−	0.936351i	$-0.385819\pi$
$739$	14105.4	0.702132	0.351066	+	0.936351i	$0.385819\pi$
$740$	9310.00	0.462490
$741$	1620.00	0.0803133
$742$	20520.0	1.01525
$743$	8667.74	0.427979	0.213990	−	0.976836i	$-0.431354\pi$
$743$	8667.74	0.427979	0.213990	+	0.976836i	$0.431354\pi$
$744$	−1859.03	−0.0916067
$745$	8171.99	0.401877
$746$	40110.0	1.96854
$747$	−766.851	−0.0375604
$748$	0	0
$749$	13800.0	0.673219
$750$	−1452.37	−0.0707107
$751$	26752.0	1.29986	0.649930	−	0.759994i	$-0.274798\pi$
$751$	26752.0	1.29986	0.649930	+	0.759994i	$0.274798\pi$
$752$	35784.0	1.73525
$753$	−3816.00	−0.184678
$754$	−56468.1	−2.72738
$755$	9101.51	0.438726
$756$	−2927.98	−0.140859
$757$	38194.0	1.83380	0.916899	−	0.399120i	$-0.130684\pi$
$757$	38194.0	1.83380	0.916899	+	0.399120i	$0.130684\pi$
$758$	−31061.3	−1.48839
$759$	0	0
$760$	150.000	0.00715931
$761$	29334.0	1.39731	0.698657	−	0.715457i	$-0.253781\pi$
$761$	29334.0	1.39731	0.698657	+	0.715457i	$0.253781\pi$
$762$	3240.00	0.154033
$763$	10560.0	0.501045
$764$	0	0
$765$	697.137	0.0329478
$766$	−4368.73	−0.206068
$767$	−46011.0	−2.16605
$768$	14763.0	0.693638
$769$	−24120.9	−1.13111	−0.565555	−	0.824711i	$-0.691338\pi$
$769$	−24120.9	−1.13111	−0.565555	+	0.824711i	$0.691338\pi$
$770$	0	0
$771$	5958.00	0.278304
$772$	26406.0	1.23105
$773$	7422.00	0.345344	0.172672	−	0.984979i	$-0.444760\pi$
$773$	7422.00	0.345344	0.172672	+	0.984979i	$0.444760\pi$
$774$	−10800.0	−0.501548
$775$	4000.00	0.185399
$776$	−983.738	−0.0455079
$777$	12362.6	0.570791
$778$	−8017.08	−0.369442
$779$	1380.00	0.0634706
$780$	−7319.94	−0.336020
$781$	0	0
$782$	−2880.00	−0.131699
$783$	−5646.81	−0.257727
$784$	7313.00	0.333136
$785$	−1370.00	−0.0622897
$786$	−9540.00	−0.432927
$787$	−10906.3	−0.493988	−0.246994	−	0.969017i	$-0.579443\pi$
$787$	−10906.3	−0.493988	−0.246994	+	0.969017i	$0.579443\pi$
$788$	2927.98	0.132367
$789$	−14198.4	−0.640653
$790$	3450.00	0.155374
$791$	−278.855	−0.0125347
$792$	0	0
$793$	−15120.0	−0.677083
$794$	12835.1	0.573677
$795$	5130.00	0.228858
$796$	−25760.0	−1.14703
$797$	−19626.0	−0.872257	−0.436128	−	0.899884i	$-0.643651\pi$
$797$	−19626.0	−0.872257	−0.436128	+	0.899884i	$0.643651\pi$
$798$	−1394.27	−0.0618506
$799$	7807.93	0.345713
$800$	−6099.95	−0.269582
$801$	−5454.00	−0.240584
$802$	11735.1	0.516686
$803$	0	0
$804$	10416.0	0.456896
$805$	3718.06	0.162788
$806$	43200.0	1.88791
$807$	−19710.0	−0.859758
$808$	−7050.00	−0.306953
$809$	24175.2	1.05062	0.525311	−	0.850910i	$-0.323949\pi$
$809$	24175.2	1.05062	0.525311	+	0.850910i	$0.323949\pi$
$810$	−1568.56	−0.0680414
$811$	−35794.1	−1.54982	−0.774908	−	0.632074i	$-0.782204\pi$
$811$	−35794.1	−1.54982	−0.774908	+	0.632074i	$0.782204\pi$
$812$	22680.0	0.980187
$813$	4159.58	0.179438
$814$	0	0
$815$	18440.0	0.792546
$816$	3299.78	0.141563
$817$	−2400.00	−0.102773
$818$	−48780.0	−2.08503
$819$	−9720.00	−0.414706
$820$	−6235.50	−0.265553
$821$	8838.15	0.375705	0.187852	−	0.982197i	$-0.439847\pi$
$821$	8838.15	0.375705	0.187852	+	0.982197i	$0.439847\pi$
$822$	−21681.0	−0.919964
$823$	−34732.0	−1.47106	−0.735529	−	0.677493i	$-0.763066\pi$
$823$	−34732.0	−1.47106	−0.735529	+	0.677493i	$0.763066\pi$
$824$	−5840.46	−0.246920
$825$	0	0
$826$	39600.0	1.66811
$827$	3632.86	0.152753	0.0763766	−	0.997079i	$-0.475665\pi$
$827$	3632.86	0.152753	0.0763766	+	0.997079i	$0.475665\pi$
$828$	3024.00	0.126922
$829$	4106.00	0.172023	0.0860116	−	0.996294i	$-0.472588\pi$
$829$	4106.00	0.172023	0.0860116	+	0.996294i	$0.472588\pi$
$830$	1650.00	0.0690028
$831$	−23307.6	−0.972963
$832$	−26282.1	−1.09515
$833$	1595.67	0.0663705
$834$	4950.00	0.205521
$835$	−8946.59	−0.370790
$836$	0	0
$837$	4320.00	0.178400
$838$	9434.59	0.388917
$839$	−41280.0	−1.69862	−0.849311	−	0.527893i	$-0.822982\pi$
$839$	−41280.0	−1.69862	−0.849311	+	0.527893i	$0.822982\pi$
$840$	−900.000	−0.0369678
$841$	19351.0	0.793431
$842$	−45739.9	−1.87209
$843$	−20193.7	−0.825041
$844$	16103.9	0.656775
$845$	−13315.0	−0.542071
$846$	−17567.9	−0.713942
$847$	0	0
$848$	24282.0	0.983310
$849$	9852.87	0.398292
$850$	−1500.00	−0.0605289
$851$	−12768.0	−0.514314
$852$	−14868.0	−0.597851
$853$	24516.0	0.984070	0.492035	−	0.870576i	$-0.336253\pi$
$853$	24516.0	0.984070	0.492035	+	0.870576i	$0.336253\pi$
$854$	13013.2	0.521433
$855$	−348.569	−0.0139424
$856$	3450.00	0.137755
$857$	−12780.8	−0.509434	−0.254717	−	0.967016i	$-0.581982\pi$
$857$	−12780.8	−0.509434	−0.254717	+	0.967016i	$0.581982\pi$
$858$	0	0
$859$	11756.0	0.466949	0.233475	−	0.972363i	$-0.424990\pi$
$859$	11756.0	0.466949	0.233475	+	0.972363i	$0.424990\pi$
$860$	10844.4	0.429988
$861$	−8280.00	−0.327737
$862$	−38100.0	−1.50544
$863$	−33048.0	−1.30355	−0.651777	−	0.758411i	$-0.725976\pi$
$863$	−33048.0	−1.30355	−0.651777	+	0.758411i	$0.725976\pi$
$864$	−6587.94	−0.259406
$865$	3950.44	0.155282
$866$	−60643.2	−2.37961
$867$	−14019.0	−0.549147
$868$	−17351.0	−0.678491
$869$	0	0
$870$	12150.0	0.473475
$871$	34578.0	1.34516
$872$	2640.00	0.102525
$873$	2286.00	0.0886247
$874$	1440.00	0.0557308
$875$	1936.49	0.0748176
$876$	−13501.2	−0.520735
$877$	−9814.14	−0.377879	−0.188940	−	0.981989i	$-0.560505\pi$
$877$	−9814.14	−0.377879	−0.188940	+	0.981989i	$0.560505\pi$
$878$	46830.0	1.80004
$879$	23098.5	0.886339
$880$	0	0
$881$	38502.0	1.47238	0.736189	−	0.676776i	$-0.236623\pi$
$881$	38502.0	1.47238	0.736189	+	0.676776i	$0.236623\pi$
$882$	−3590.26	−0.137064
$883$	26332.0	1.00356	0.501779	−	0.864996i	$-0.332679\pi$
$883$	26332.0	1.00356	0.501779	+	0.864996i	$0.332679\pi$
$884$	−7560.00	−0.287636
$885$	9900.00	0.376028
$886$	−53493.6	−2.02839
$887$	−35778.6	−1.35437	−0.677186	−	0.735812i	$-0.736801\pi$
$887$	−35778.6	−1.35437	−0.677186	+	0.735812i	$0.736801\pi$
$888$	3090.64	0.116796
$889$	−4320.00	−0.162979
$890$	11735.1	0.441981
$891$	0	0
$892$	−24976.0	−0.937509
$893$	−3903.97	−0.146295
$894$	−18990.0	−0.710426
$895$	−16020.0	−0.598312
$896$	−7620.00	−0.284114
$897$	10038.8	0.373673
$898$	33718.2	1.25300
$899$	−33462.6	−1.24142
$900$	1575.00	0.0583333
$901$	5298.24	0.195905
$902$	0	0
$903$	14400.0	0.530678
$904$	−69.7137	−0.00256487
$905$	−22450.0	−0.824600
$906$	−21150.0	−0.775565
$907$	9956.00	0.364480	0.182240	−	0.983254i	$-0.441665\pi$
$907$	9956.00	0.364480	0.182240	+	0.983254i	$0.441665\pi$
$908$	34864.6	1.27425
$909$	16382.7	0.597778
$910$	20914.1	0.761864
$911$	−41160.0	−1.49692	−0.748459	−	0.663181i	$-0.769206\pi$
$911$	−41160.0	−1.49692	−0.748459	+	0.663181i	$0.769206\pi$
$912$	−1649.89	−0.0599050
$913$	0	0
$914$	−21150.0	−0.765405
$915$	3253.31	0.117542
$916$	36190.0	1.30541
$917$	12720.0	0.458071
$918$	−1620.00	−0.0582440
$919$	2641.37	0.0948106	0.0474053	−	0.998876i	$-0.484905\pi$
$919$	2641.37	0.0948106	0.0474053	+	0.998876i	$0.484905\pi$
$920$	929.516	0.0333100
$921$	17335.5	0.620221
$922$	51150.0	1.82705
$923$	−49357.3	−1.76015
$924$	0	0
$925$	−6650.00	−0.236379
$926$	−47746.1	−1.69442
$927$	13572.0	0.480866
$928$	51030.0	1.80511
$929$	41430.0	1.46316	0.731579	−	0.681756i	$-0.238784\pi$
$929$	41430.0	1.46316	0.731579	+	0.681756i	$0.238784\pi$
$930$	−9295.16	−0.327742
$931$	−797.835	−0.0280859
$932$	32641.5	1.14722
$933$	15984.0	0.560871
$934$	−23377.3	−0.818982
$935$	0	0
$936$	−2430.00	−0.0848579
$937$	28110.1	0.980061	0.490031	−	0.871705i	$-0.336986\pi$
$937$	28110.1	0.980061	0.490031	+	0.871705i	$0.336986\pi$
$938$	−29760.0	−1.03593
$939$	1326.00	0.0460835
$940$	17640.0	0.612078
$941$	11146.4	0.386146	0.193073	−	0.981184i	$-0.438155\pi$
$941$	11146.4	0.386146	0.193073	+	0.981184i	$0.438155\pi$
$942$	3183.59	0.110114
$943$	8551.55	0.295309
$944$	46860.0	1.61564
$945$	2091.41	0.0719932
$946$	0	0
$947$	−51036.0	−1.75126	−0.875632	−	0.482979i	$-0.839555\pi$
$947$	−51036.0	−1.75126	−0.875632	+	0.482979i	$0.839555\pi$
$948$	−3741.30	−0.128177
$949$	−44820.0	−1.53311
$950$	750.000	0.0256139
$951$	−23382.0	−0.797280
$952$	−929.516	−0.0316447
$953$	23640.7	0.803565	0.401782	−	0.915735i	$-0.368391\pi$
$953$	23640.7	0.803565	0.401782	+	0.915735i	$0.368391\pi$
$954$	−11921.0	−0.404568
$955$	0	0
$956$	20387.4	0.689723
$957$	0	0
$958$	−7200.00	−0.242820
$959$	28907.9	0.973396
$960$	5655.00	0.190119
$961$	−4191.00	−0.140680
$962$	−71820.0	−2.40704
$963$	−8017.08	−0.268273
$964$	39039.7	1.30434
$965$	−18861.4	−0.629193
$966$	−8640.00	−0.287772
$967$	38048.2	1.26530	0.632651	−	0.774437i	$-0.281967\pi$
$967$	38048.2	1.26530	0.632651	+	0.774437i	$0.281967\pi$
$968$	0	0
$969$	−360.000	−0.0119348
$970$	−4918.69	−0.162814
$971$	27720.0	0.916145	0.458073	−	0.888915i	$-0.348540\pi$
$971$	27720.0	0.916145	0.458073	+	0.888915i	$0.348540\pi$
$972$	1701.00	0.0561313
$973$	−6600.00	−0.217458
$974$	22138.0	0.728282
$975$	5228.53	0.171740
$976$	15399.0	0.505030
$977$	−19794.0	−0.648174	−0.324087	−	0.946027i	$-0.605057\pi$
$977$	−19794.0	−0.648174	−0.324087	+	0.946027i	$0.605057\pi$
$978$	−42850.7	−1.40104
$979$	0	0
$980$	3605.00	0.117508
$981$	−6134.81	−0.199663
$982$	43140.0	1.40189
$983$	−51672.0	−1.67658	−0.838291	−	0.545222i	$-0.816445\pi$
$983$	−51672.0	−1.67658	−0.838291	+	0.545222i	$0.816445\pi$
$984$	−2070.00	−0.0670622
$985$	−2091.41	−0.0676527
$986$	12548.5	0.405299
$987$	23423.8	0.755408
$988$	3780.00	0.121718
$989$	−14872.3	−0.478170
$990$	0	0
$991$	48280.0	1.54759	0.773797	−	0.633434i	$-0.218355\pi$
$991$	48280.0	1.54759	0.773797	+	0.633434i	$0.218355\pi$
$992$	−39039.7	−1.24951
$993$	−6420.00	−0.205169
$994$	42480.0	1.35552
$995$	18400.0	0.586250
$996$	−1789.32	−0.0569244
$997$	38272.8	1.21576	0.607880	−	0.794029i	$-0.292020\pi$
$997$	38272.8	1.21576	0.607880	+	0.794029i	$0.292020\pi$
$998$	−32610.5	−1.03434
$999$	−7182.00	−0.227456

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1815.4.a.m.1.1	✓	2	11.10	odd	2	inner
1815.4.a.m.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q+3.87298 q^{2} +3.00000 q^{3} +7.00000 q^{4} -5.00000 q^{5} +11.6190 q^{6} -15.4919 q^{7} -3.87298 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.87298 q^{2} +3.00000 q^{3} +7.00000 q^{4} -5.00000 q^{5} +11.6190 q^{6} -15.4919 q^{7} -3.87298 q^{8} +9.00000 q^{9} -19.3649 q^{10} +21.0000 q^{12} +69.7137 q^{13} -60.0000 q^{14} -15.0000 q^{15} -71.0000 q^{16} -15.4919 q^{17} +34.8569 q^{18} +7.74597 q^{19} -35.0000 q^{20} -46.4758 q^{21} +48.0000 q^{23} -11.6190 q^{24} +25.0000 q^{25} +270.000 q^{26} +27.0000 q^{27} -108.444 q^{28} -209.141 q^{29} -58.0948 q^{30} +160.000 q^{31} -243.998 q^{32} -60.0000 q^{34} +77.4597 q^{35} +63.0000 q^{36} -266.000 q^{37} +30.0000 q^{38} +209.141 q^{39} +19.3649 q^{40} +178.157 q^{41} -180.000 q^{42} -309.839 q^{43} -45.0000 q^{45} +185.903 q^{46} -504.000 q^{47} -213.000 q^{48} -103.000 q^{49} +96.8246 q^{50} -46.4758 q^{51} +487.996 q^{52} -342.000 q^{53} +104.571 q^{54} +60.0000 q^{56} +23.2379 q^{57} -810.000 q^{58} -660.000 q^{59} -105.000 q^{60} -216.887 q^{61} +619.677 q^{62} -139.427 q^{63} -377.000 q^{64} -348.569 q^{65} +496.000 q^{67} -108.444 q^{68} +144.000 q^{69} +300.000 q^{70} -708.000 q^{71} -34.8569 q^{72} -642.915 q^{73} -1030.21 q^{74} +75.0000 q^{75} +54.2218 q^{76} +810.000 q^{78} -178.157 q^{79} +355.000 q^{80} +81.0000 q^{81} +690.000 q^{82} -85.2056 q^{83} -325.331 q^{84} +77.4597 q^{85} -1200.00 q^{86} -627.423 q^{87} -606.000 q^{89} -174.284 q^{90} -1080.00 q^{91} +336.000 q^{92} +480.000 q^{93} -1951.98 q^{94} -38.7298 q^{95} -731.994 q^{96} +254.000 q^{97} -398.917 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 14 q^{4} - 10 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 14 * q^4 - 10 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} + 14 q^{4} - 10 q^{5} + 18 q^{9} + 42 q^{12} - 120 q^{14} - 30 q^{15} - 142 q^{16} - 70 q^{20} + 96 q^{23} + 50 q^{25} + 540 q^{26} + 54 q^{27} + 320 q^{31} - 120 q^{34} + 126 q^{36} - 532 q^{37} + 60 q^{38} - 360 q^{42} - 90 q^{45} - 1008 q^{47} - 426 q^{48} - 206 q^{49} - 684 q^{53} + 120 q^{56} - 1620 q^{58} - 1320 q^{59} - 210 q^{60} - 754 q^{64} + 992 q^{67} + 288 q^{69} + 600 q^{70} - 1416 q^{71} + 150 q^{75} + 1620 q^{78} + 710 q^{80} + 162 q^{81} + 1380 q^{82} - 2400 q^{86} - 1212 q^{89} - 2160 q^{91} + 672 q^{92} + 960 q^{93} + 508 q^{97}+O(q^{100}) \) 2 * q + 6 * q^3 + 14 * q^4 - 10 * q^5 + 18 * q^9 + 42 * q^12 - 120 * q^14 - 30 * q^15 - 142 * q^16 - 70 * q^20 + 96 * q^23 + 50 * q^25 + 540 * q^26 + 54 * q^27 + 320 * q^31 - 120 * q^34 + 126 * q^36 - 532 * q^37 + 60 * q^38 - 360 * q^42 - 90 * q^45 - 1008 * q^47 - 426 * q^48 - 206 * q^49 - 684 * q^53 + 120 * q^56 - 1620 * q^58 - 1320 * q^59 - 210 * q^60 - 754 * q^64 + 992 * q^67 + 288 * q^69 + 600 * q^70 - 1416 * q^71 + 150 * q^75 + 1620 * q^78 + 710 * q^80 + 162 * q^81 + 1380 * q^82 - 2400 * q^86 - 1212 * q^89 - 2160 * q^91 + 672 * q^92 + 960 * q^93 + 508 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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