LMFDB - Embedded newform 1936.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1936,4,Mod(1,1936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1936, 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("1936.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1936 = 2^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1936.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:	$114.227697771$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 121)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1936.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.535898 q^{3} -1.53590 q^{5} -28.2487 q^{7} -26.7128 q^{9} +O(q^{10})$	$q+0.535898 q^{3} -1.53590 q^{5} -28.2487 q^{7} -26.7128 q^{9} -68.4641 q^{13} -0.823085 q^{15} -55.3538 q^{17} -55.1769 q^{19} -15.1384 q^{21} +178.315 q^{23} -122.641 q^{25} -28.7846 q^{27} -113.172 q^{29} -70.8128 q^{31} +43.3872 q^{35} -210.664 q^{37} -36.6898 q^{39} -191.928 q^{41} -208.210 q^{43} +41.0282 q^{45} -512.515 q^{47} +454.990 q^{49} -29.6640 q^{51} -375.449 q^{53} -29.5692 q^{57} +506.508 q^{59} +468.697 q^{61} +754.603 q^{63} +105.154 q^{65} +289.895 q^{67} +95.5589 q^{69} +394.010 q^{71} -289.538 q^{73} -65.7231 q^{75} -169.587 q^{79} +705.820 q^{81} -303.331 q^{83} +85.0179 q^{85} -60.6486 q^{87} -1146.68 q^{89} +1934.02 q^{91} -37.9485 q^{93} +84.7461 q^{95} +641.600 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{3} - 10 q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 8 * q^3 - 10 * q^5 - 8 * q^7 + 2 * q^9	$ 2 q + 8 q^{3} - 10 q^{5} - 8 q^{7} + 2 q^{9} - 130 q^{13} - 64 q^{15} + 14 q^{17} - 48 q^{19} + 136 q^{21} + 128 q^{23} - 176 q^{25} - 16 q^{27} + 30 q^{29} + 184 q^{31} - 128 q^{35} + 126 q^{37} - 496 q^{39} - 370 q^{41} - 264 q^{43} - 202 q^{45} - 256 q^{47} + 522 q^{49} + 488 q^{51} - 162 q^{53} + 24 q^{57} + 1304 q^{59} + 300 q^{61} + 1336 q^{63} + 626 q^{65} + 656 q^{67} - 280 q^{69} + 1176 q^{71} + 668 q^{73} - 464 q^{75} + 416 q^{79} + 26 q^{81} - 960 q^{83} - 502 q^{85} + 1008 q^{87} - 1074 q^{89} + 688 q^{91} + 1864 q^{93} + 24 q^{95} - 338 q^{97}+O(q^{100}) $ 2 * q + 8 * q^3 - 10 * q^5 - 8 * q^7 + 2 * q^9 - 130 * q^13 - 64 * q^15 + 14 * q^17 - 48 * q^19 + 136 * q^21 + 128 * q^23 - 176 * q^25 - 16 * q^27 + 30 * q^29 + 184 * q^31 - 128 * q^35 + 126 * q^37 - 496 * q^39 - 370 * q^41 - 264 * q^43 - 202 * q^45 - 256 * q^47 + 522 * q^49 + 488 * q^51 - 162 * q^53 + 24 * q^57 + 1304 * q^59 + 300 * q^61 + 1336 * q^63 + 626 * q^65 + 656 * q^67 - 280 * q^69 + 1176 * q^71 + 668 * q^73 - 464 * q^75 + 416 * q^79 + 26 * q^81 - 960 * q^83 - 502 * q^85 + 1008 * q^87 - 1074 * q^89 + 688 * q^91 + 1864 * q^93 + 24 * q^95 - 338 * 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 $

$2$	0	0
$2$	0	0
$3$	0.535898	0.103134	0.0515668	−	0.998670i	$-0.483578\pi$
$3$	0.535898	0.103134	0.0515668	+	0.998670i	$0.483578\pi$
$4$	0	0
$5$	−1.53590	−0.137375	−0.0686875	−	0.997638i	$-0.521881\pi$
$5$	−1.53590	−0.137375	−0.0686875	+	0.997638i	$0.521881\pi$
$6$	0	0
$7$	−28.2487	−1.52529	−0.762644	−	0.646819i	$-0.776099\pi$
$7$	−28.2487	−1.52529	−0.762644	+	0.646819i	$0.776099\pi$
$8$	0	0
$9$	−26.7128	−0.989363
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−68.4641	−1.46066	−0.730328	−	0.683097i	$-0.760633\pi$
$13$	−68.4641	−1.46066	−0.730328	+	0.683097i	$0.760633\pi$
$14$	0	0
$15$	−0.823085	−0.0141680
$16$	0	0
$17$	−55.3538	−0.789722	−0.394861	−	0.918741i	$-0.629207\pi$
$17$	−55.3538	−0.789722	−0.394861	+	0.918741i	$0.629207\pi$
$18$	0	0
$19$	−55.1769	−0.666234	−0.333117	−	0.942885i	$-0.608100\pi$
$19$	−55.1769	−0.666234	−0.333117	+	0.942885i	$0.608100\pi$
$20$	0	0
$21$	−15.1384	−0.157308
$22$	0	0
$23$	178.315	1.61658	0.808290	−	0.588785i	$-0.200394\pi$
$23$	178.315	1.61658	0.808290	+	0.588785i	$0.200394\pi$
$24$	0	0
$25$	−122.641	−0.981128
$26$	0	0
$27$	−28.7846	−0.205170
$28$	0	0
$29$	−113.172	−0.724671	−0.362336	−	0.932048i	$-0.618021\pi$
$29$	−113.172	−0.724671	−0.362336	+	0.932048i	$0.618021\pi$
$30$	0	0
$31$	−70.8128	−0.410269	−0.205135	−	0.978734i	$-0.565763\pi$
$31$	−70.8128	−0.410269	−0.205135	+	0.978734i	$0.565763\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	43.3872	0.209536
$36$	0	0
$37$	−210.664	−0.936026	−0.468013	−	0.883722i	$-0.655030\pi$
$37$	−210.664	−0.936026	−0.468013	+	0.883722i	$0.655030\pi$
$38$	0	0
$39$	−36.6898	−0.150643
$40$	0	0
$41$	−191.928	−0.731077	−0.365538	−	0.930796i	$-0.619115\pi$
$41$	−191.928	−0.731077	−0.365538	+	0.930796i	$0.619115\pi$
$42$	0	0
$43$	−208.210	−0.738413	−0.369207	−	0.929347i	$-0.620371\pi$
$43$	−208.210	−0.738413	−0.369207	+	0.929347i	$0.620371\pi$
$44$	0	0
$45$	41.0282	0.135914
$46$	0	0
$47$	−512.515	−1.59060	−0.795298	−	0.606218i	$-0.792686\pi$
$47$	−512.515	−1.59060	−0.795298	+	0.606218i	$0.792686\pi$
$48$	0	0
$49$	454.990	1.32650
$50$	0	0
$51$	−29.6640	−0.0814470
$52$	0	0
$53$	−375.449	−0.973054	−0.486527	−	0.873666i	$-0.661736\pi$
$53$	−375.449	−0.973054	−0.486527	+	0.873666i	$0.661736\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−29.5692	−0.0687112
$58$	0	0
$59$	506.508	1.11766	0.558828	−	0.829284i	$-0.311251\pi$
$59$	506.508	1.11766	0.558828	+	0.829284i	$0.311251\pi$
$60$	0	0
$61$	468.697	0.983779	0.491890	−	0.870658i	$-0.336306\pi$
$61$	468.697	0.983779	0.491890	+	0.870658i	$0.336306\pi$
$62$	0	0
$63$	754.603	1.50906
$64$	0	0
$65$	105.154	0.200657
$66$	0	0
$67$	289.895	0.528601	0.264301	−	0.964440i	$-0.414859\pi$
$67$	289.895	0.528601	0.264301	+	0.964440i	$0.414859\pi$
$68$	0	0
$69$	95.5589	0.166724
$70$	0	0
$71$	394.010	0.658597	0.329299	−	0.944226i	$-0.393188\pi$
$71$	394.010	0.658597	0.329299	+	0.944226i	$0.393188\pi$
$72$	0	0
$73$	−289.538	−0.464218	−0.232109	−	0.972690i	$-0.574563\pi$
$73$	−289.538	−0.464218	−0.232109	+	0.972690i	$0.574563\pi$
$74$	0	0
$75$	−65.7231	−0.101187
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−169.587	−0.241519	−0.120760	−	0.992682i	$-0.538533\pi$
$79$	−169.587	−0.241519	−0.120760	+	0.992682i	$0.538533\pi$
$80$	0	0
$81$	705.820	0.968203
$82$	0	0
$83$	−303.331	−0.401143	−0.200572	−	0.979679i	$-0.564280\pi$
$83$	−303.331	−0.401143	−0.200572	+	0.979679i	$0.564280\pi$
$84$	0	0
$85$	85.0179	0.108488
$86$	0	0
$87$	−60.6486	−0.0747380
$88$	0	0
$89$	−1146.68	−1.36571	−0.682854	−	0.730555i	$-0.739262\pi$
$89$	−1146.68	−1.36571	−0.682854	+	0.730555i	$0.739262\pi$
$90$	0	0
$91$	1934.02	2.22792
$92$	0	0
$93$	−37.9485	−0.0423126
$94$	0	0
$95$	84.7461	0.0915239
$96$	0	0
$97$	641.600	0.671594	0.335797	−	0.941934i	$-0.390994\pi$
$97$	641.600	0.671594	0.335797	+	0.941934i	$0.390994\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1107.57	−1.09116	−0.545580	−	0.838058i	$-0.683691\pi$
$101$	−1107.57	−1.09116	−0.545580	+	0.838058i	$0.683691\pi$
$102$	0	0
$103$	298.297	0.285360	0.142680	−	0.989769i	$-0.454428\pi$
$103$	298.297	0.285360	0.142680	+	0.989769i	$0.454428\pi$
$104$	0	0
$105$	23.2511	0.0216102
$106$	0	0
$107$	−598.126	−0.540402	−0.270201	−	0.962804i	$-0.587090\pi$
$107$	−598.126	−0.540402	−0.270201	+	0.962804i	$0.587090\pi$
$108$	0	0
$109$	1530.16	1.34461	0.672305	−	0.740275i	$-0.265305\pi$
$109$	1530.16	1.34461	0.672305	+	0.740275i	$0.265305\pi$
$110$	0	0
$111$	−112.895	−0.0965358
$112$	0	0
$113$	−151.010	−0.125716	−0.0628578	−	0.998022i	$-0.520021\pi$
$113$	−151.010	−0.125716	−0.0628578	+	0.998022i	$0.520021\pi$
$114$	0	0
$115$	−273.874	−0.222077
$116$	0	0
$117$	1828.87	1.44512
$118$	0	0
$119$	1563.67	1.20455
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−102.854	−0.0753987
$124$	0	0
$125$	380.351	0.272157
$126$	0	0
$127$	−695.749	−0.486124	−0.243062	−	0.970011i	$-0.578152\pi$
$127$	−695.749	−0.486124	−0.243062	+	0.970011i	$0.578152\pi$
$128$	0	0
$129$	−111.580	−0.0761553
$130$	0	0
$131$	1665.68	1.11093	0.555463	−	0.831541i	$-0.312541\pi$
$131$	1665.68	1.11093	0.555463	+	0.831541i	$0.312541\pi$
$132$	0	0
$133$	1558.68	1.01620
$134$	0	0
$135$	44.2102	0.0281853
$136$	0	0
$137$	−1605.48	−1.00121	−0.500603	−	0.865677i	$-0.666888\pi$
$137$	−1605.48	−1.00121	−0.500603	+	0.865677i	$0.666888\pi$
$138$	0	0
$139$	−1069.30	−0.652495	−0.326248	−	0.945284i	$-0.605784\pi$
$139$	−1069.30	−0.652495	−0.326248	+	0.945284i	$0.605784\pi$
$140$	0	0
$141$	−274.656	−0.164044
$142$	0	0
$143$	0	0
$144$	0	0
$145$	173.820	0.0995517
$146$	0	0
$147$	243.828	0.136807
$148$	0	0
$149$	355.172	0.195281	0.0976403	−	0.995222i	$-0.468871\pi$
$149$	355.172	0.195281	0.0976403	+	0.995222i	$0.468871\pi$
$150$	0	0
$151$	1879.55	1.01295	0.506476	−	0.862254i	$-0.330948\pi$
$151$	1879.55	1.01295	0.506476	+	0.862254i	$0.330948\pi$
$152$	0	0
$153$	1478.66	0.781322
$154$	0	0
$155$	108.761	0.0563607
$156$	0	0
$157$	2499.99	1.27083	0.635417	−	0.772169i	$-0.280828\pi$
$157$	2499.99	1.27083	0.635417	+	0.772169i	$0.280828\pi$
$158$	0	0
$159$	−201.202	−0.100355
$160$	0	0
$161$	−5037.18	−2.46575
$162$	0	0
$163$	−1863.02	−0.895235	−0.447617	−	0.894225i	$-0.647727\pi$
$163$	−1863.02	−0.895235	−0.447617	+	0.894225i	$0.647727\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2647.27	−1.22666	−0.613328	−	0.789828i	$-0.710170\pi$
$167$	−2647.27	−1.22666	−0.613328	+	0.789828i	$0.710170\pi$
$168$	0	0
$169$	2490.33	1.13352
$170$	0	0
$171$	1473.93	0.659148
$172$	0	0
$173$	2109.38	0.927015	0.463507	−	0.886093i	$-0.346591\pi$
$173$	2109.38	0.927015	0.463507	+	0.886093i	$0.346591\pi$
$174$	0	0
$175$	3464.45	1.49650
$176$	0	0
$177$	271.437	0.115268
$178$	0	0
$179$	−1391.25	−0.580931	−0.290465	−	0.956886i	$-0.593810\pi$
$179$	−1391.25	−0.580931	−0.290465	+	0.956886i	$0.593810\pi$
$180$	0	0
$181$	3701.40	1.52002	0.760008	−	0.649913i	$-0.225195\pi$
$181$	3701.40	1.52002	0.760008	+	0.649913i	$0.225195\pi$
$182$	0	0
$183$	251.174	0.101461
$184$	0	0
$185$	323.559	0.128586
$186$	0	0
$187$	0	0
$188$	0	0
$189$	813.128	0.312944
$190$	0	0
$191$	3533.03	1.33844	0.669218	−	0.743066i	$-0.266629\pi$
$191$	3533.03	1.33844	0.669218	+	0.743066i	$0.266629\pi$
$192$	0	0
$193$	−2605.66	−0.971811	−0.485906	−	0.874011i	$-0.661510\pi$
$193$	−2605.66	−0.971811	−0.485906	+	0.874011i	$0.661510\pi$
$194$	0	0
$195$	56.3518	0.0206945
$196$	0	0
$197$	719.202	0.260107	0.130053	−	0.991507i	$-0.458485\pi$
$197$	719.202	0.260107	0.130053	+	0.991507i	$0.458485\pi$
$198$	0	0
$199$	−1035.15	−0.368744	−0.184372	−	0.982857i	$-0.559025\pi$
$199$	−1035.15	−0.368744	−0.184372	+	0.982857i	$0.559025\pi$
$200$	0	0
$201$	155.354	0.0545166
$202$	0	0
$203$	3196.96	1.10533
$204$	0	0
$205$	294.782	0.100432
$206$	0	0
$207$	−4763.30	−1.59938
$208$	0	0
$209$	0	0
$210$	0	0
$211$	356.297	0.116249	0.0581244	−	0.998309i	$-0.481488\pi$
$211$	356.297	0.116249	0.0581244	+	0.998309i	$0.481488\pi$
$212$	0	0
$213$	211.149	0.0679236
$214$	0	0
$215$	319.790	0.101439
$216$	0	0
$217$	2000.37	0.625779
$218$	0	0
$219$	−155.163	−0.0478765
$220$	0	0
$221$	3789.75	1.15351
$222$	0	0
$223$	292.544	0.0878485	0.0439242	−	0.999035i	$-0.486014\pi$
$223$	292.544	0.0878485	0.0439242	+	0.999035i	$0.486014\pi$
$224$	0	0
$225$	3276.09	0.970692
$226$	0	0
$227$	−5604.04	−1.63856	−0.819280	−	0.573394i	$-0.805626\pi$
$227$	−5604.04	−1.63856	−0.819280	+	0.573394i	$0.805626\pi$
$228$	0	0
$229$	−5654.38	−1.63167	−0.815833	−	0.578287i	$-0.803721\pi$
$229$	−5654.38	−1.63167	−0.815833	+	0.578287i	$0.803721\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2553.08	−0.717845	−0.358923	−	0.933367i	$-0.616856\pi$
$233$	−2553.08	−0.717845	−0.358923	+	0.933367i	$0.616856\pi$
$234$	0	0
$235$	787.171	0.218508
$236$	0	0
$237$	−90.8814	−0.0249088
$238$	0	0
$239$	−5297.27	−1.43369	−0.716845	−	0.697233i	$-0.754414\pi$
$239$	−5297.27	−1.43369	−0.716845	+	0.697233i	$0.754414\pi$
$240$	0	0
$241$	−4145.14	−1.10793	−0.553966	−	0.832539i	$-0.686886\pi$
$241$	−4145.14	−1.10793	−0.553966	+	0.832539i	$0.686886\pi$
$242$	0	0
$243$	1155.43	0.305025
$244$	0	0
$245$	−698.818	−0.182228
$246$	0	0
$247$	3777.64	0.973139
$248$	0	0
$249$	−162.554	−0.0413714
$250$	0	0
$251$	1788.13	0.449665	0.224832	−	0.974397i	$-0.427817\pi$
$251$	1788.13	0.449665	0.224832	+	0.974397i	$0.427817\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	45.5609	0.0111888
$256$	0	0
$257$	5167.01	1.25412	0.627061	−	0.778970i	$-0.284258\pi$
$257$	5167.01	1.25412	0.627061	+	0.778970i	$0.284258\pi$
$258$	0	0
$259$	5950.99	1.42771
$260$	0	0
$261$	3023.14	0.716963
$262$	0	0
$263$	−57.6791	−0.0135234	−0.00676169	−	0.999977i	$-0.502152\pi$
$263$	−57.6791	−0.0135234	−0.00676169	+	0.999977i	$0.502152\pi$
$264$	0	0
$265$	576.651	0.133673
$266$	0	0
$267$	−614.505	−0.140851
$268$	0	0
$269$	−3028.06	−0.686335	−0.343167	−	0.939274i	$-0.611500\pi$
$269$	−3028.06	−0.686335	−0.343167	+	0.939274i	$0.611500\pi$
$270$	0	0
$271$	−1487.84	−0.333504	−0.166752	−	0.985999i	$-0.553328\pi$
$271$	−1487.84	−0.333504	−0.166752	+	0.985999i	$0.553328\pi$
$272$	0	0
$273$	1036.44	0.229774
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7460.46	−1.61825	−0.809126	−	0.587635i	$-0.800059\pi$
$277$	−7460.46	−1.61825	−0.809126	+	0.587635i	$0.800059\pi$
$278$	0	0
$279$	1891.61	0.405906
$280$	0	0
$281$	−900.155	−0.191099	−0.0955493	−	0.995425i	$-0.530461\pi$
$281$	−900.155	−0.191099	−0.0955493	+	0.995425i	$0.530461\pi$
$282$	0	0
$283$	6486.92	1.36257	0.681285	−	0.732018i	$-0.261422\pi$
$283$	6486.92	1.36257	0.681285	+	0.732018i	$0.261422\pi$
$284$	0	0
$285$	45.4153	0.00943920
$286$	0	0
$287$	5421.72	1.11510
$288$	0	0
$289$	−1848.95	−0.376339
$290$	0	0
$291$	343.832	0.0692639
$292$	0	0
$293$	−6129.38	−1.22212	−0.611062	−	0.791583i	$-0.709257\pi$
$293$	−6129.38	−1.22212	−0.611062	+	0.791583i	$0.709257\pi$
$294$	0	0
$295$	−777.944	−0.153538
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−12208.2	−2.36127
$300$	0	0
$301$	5881.67	1.12629
$302$	0	0
$303$	−593.545	−0.112535
$304$	0	0
$305$	−719.872	−0.135147
$306$	0	0
$307$	5377.67	0.999740	0.499870	−	0.866101i	$-0.333381\pi$
$307$	5377.67	0.999740	0.499870	+	0.866101i	$0.333381\pi$
$308$	0	0
$309$	159.857	0.0294303
$310$	0	0
$311$	6066.41	1.10609	0.553046	−	0.833151i	$-0.313465\pi$
$311$	6066.41	1.10609	0.553046	+	0.833151i	$0.313465\pi$
$312$	0	0
$313$	3241.18	0.585311	0.292655	−	0.956218i	$-0.405461\pi$
$313$	3241.18	0.585311	0.292655	+	0.956218i	$0.405461\pi$
$314$	0	0
$315$	−1158.99	−0.207307
$316$	0	0
$317$	6519.32	1.15508	0.577542	−	0.816361i	$-0.304012\pi$
$317$	6519.32	1.15508	0.577542	+	0.816361i	$0.304012\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−320.535	−0.0557336
$322$	0	0
$323$	3054.25	0.526140
$324$	0	0
$325$	8396.51	1.43309
$326$	0	0
$327$	820.008	0.138675
$328$	0	0
$329$	14477.9	2.42612
$330$	0	0
$331$	−5879.55	−0.976343	−0.488171	−	0.872748i	$-0.662336\pi$
$331$	−5879.55	−0.976343	−0.488171	+	0.872748i	$0.662336\pi$
$332$	0	0
$333$	5627.43	0.926070
$334$	0	0
$335$	−445.249	−0.0726166
$336$	0	0
$337$	−1342.66	−0.217031	−0.108516	−	0.994095i	$-0.534610\pi$
$337$	−1342.66	−0.217031	−0.108516	+	0.994095i	$0.534610\pi$
$338$	0	0
$339$	−80.9262	−0.0129655
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−3163.56	−0.498007
$344$	0	0
$345$	−146.769	−0.0229037
$346$	0	0
$347$	−5650.03	−0.874091	−0.437046	−	0.899439i	$-0.643975\pi$
$347$	−5650.03	−0.874091	−0.437046	+	0.899439i	$0.643975\pi$
$348$	0	0
$349$	1249.55	0.191653	0.0958266	−	0.995398i	$-0.469451\pi$
$349$	1249.55	0.191653	0.0958266	+	0.995398i	$0.469451\pi$
$350$	0	0
$351$	1970.71	0.299683
$352$	0	0
$353$	5984.25	0.902293	0.451147	−	0.892450i	$-0.351015\pi$
$353$	5984.25	0.902293	0.451147	+	0.892450i	$0.351015\pi$
$354$	0	0
$355$	−605.160	−0.0904748
$356$	0	0
$357$	837.971	0.124230
$358$	0	0
$359$	−2176.01	−0.319904	−0.159952	−	0.987125i	$-0.551134\pi$
$359$	−2176.01	−0.319904	−0.159952	+	0.987125i	$0.551134\pi$
$360$	0	0
$361$	−3814.51	−0.556132
$362$	0	0
$363$	0	0
$364$	0	0
$365$	444.701	0.0637719
$366$	0	0
$367$	8464.39	1.20392	0.601958	−	0.798528i	$-0.294387\pi$
$367$	8464.39	1.20392	0.601958	+	0.798528i	$0.294387\pi$
$368$	0	0
$369$	5126.94	0.723301
$370$	0	0
$371$	10605.9	1.48419
$372$	0	0
$373$	4248.93	0.589816	0.294908	−	0.955526i	$-0.404711\pi$
$373$	4248.93	0.589816	0.294908	+	0.955526i	$0.404711\pi$
$374$	0	0
$375$	203.830	0.0280686
$376$	0	0
$377$	7748.20	1.05850
$378$	0	0
$379$	−4852.93	−0.657727	−0.328863	−	0.944378i	$-0.606666\pi$
$379$	−4852.93	−0.657727	−0.328863	+	0.944378i	$0.606666\pi$
$380$	0	0
$381$	−372.851	−0.0501357
$382$	0	0
$383$	8181.81	1.09157	0.545785	−	0.837926i	$-0.316232\pi$
$383$	8181.81	1.09157	0.545785	+	0.837926i	$0.316232\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5561.88	0.730559
$388$	0	0
$389$	1254.82	0.163552	0.0817761	−	0.996651i	$-0.473941\pi$
$389$	1254.82	0.163552	0.0817761	+	0.996651i	$0.473941\pi$
$390$	0	0
$391$	−9870.44	−1.27665
$392$	0	0
$393$	892.636	0.114574
$394$	0	0
$395$	260.469	0.0331787
$396$	0	0
$397$	−11519.3	−1.45627	−0.728133	−	0.685436i	$-0.759612\pi$
$397$	−11519.3	−1.45627	−0.728133	+	0.685436i	$0.759612\pi$
$398$	0	0
$399$	835.292	0.104804
$400$	0	0
$401$	−1500.66	−0.186882	−0.0934409	−	0.995625i	$-0.529787\pi$
$401$	−1500.66	−0.186882	−0.0934409	+	0.995625i	$0.529787\pi$
$402$	0	0
$403$	4848.13	0.599262
$404$	0	0
$405$	−1084.07	−0.133007
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−91.3936	−0.0110492	−0.00552460	−	0.999985i	$-0.501759\pi$
$409$	−91.3936	−0.0110492	−0.00552460	+	0.999985i	$0.501759\pi$
$410$	0	0
$411$	−860.372	−0.103258
$412$	0	0
$413$	−14308.2	−1.70475
$414$	0	0
$415$	465.885	0.0551070
$416$	0	0
$417$	−573.036	−0.0672942
$418$	0	0
$419$	1880.83	0.219295	0.109648	−	0.993971i	$-0.465028\pi$
$419$	1880.83	0.219295	0.109648	+	0.993971i	$0.465028\pi$
$420$	0	0
$421$	−7279.83	−0.842748	−0.421374	−	0.906887i	$-0.638452\pi$
$421$	−7279.83	−0.842748	−0.421374	+	0.906887i	$0.638452\pi$
$422$	0	0
$423$	13690.7	1.57368
$424$	0	0
$425$	6788.65	0.774819
$426$	0	0
$427$	−13240.1	−1.50055
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6870.61	0.767855	0.383928	−	0.923363i	$-0.374571\pi$
$431$	6870.61	0.767855	0.383928	+	0.923363i	$0.374571\pi$
$432$	0	0
$433$	1121.32	0.124451	0.0622255	−	0.998062i	$-0.480180\pi$
$433$	1121.32	0.124451	0.0622255	+	0.998062i	$0.480180\pi$
$434$	0	0
$435$	93.1500	0.0102671
$436$	0	0
$437$	−9838.89	−1.07702
$438$	0	0
$439$	7114.94	0.773525	0.386763	−	0.922179i	$-0.373593\pi$
$439$	7114.94	0.773525	0.386763	+	0.922179i	$0.373593\pi$
$440$	0	0
$441$	−12154.1	−1.31239
$442$	0	0
$443$	14057.4	1.50764	0.753822	−	0.657079i	$-0.228208\pi$
$443$	14057.4	1.50764	0.753822	+	0.657079i	$0.228208\pi$
$444$	0	0
$445$	1761.19	0.187614
$446$	0	0
$447$	190.336	0.0201400
$448$	0	0
$449$	−15323.9	−1.61064	−0.805320	−	0.592840i	$-0.798007\pi$
$449$	−15323.9	−1.61064	−0.805320	+	0.592840i	$0.798007\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	1007.25	0.104470
$454$	0	0
$455$	−2970.46	−0.306060
$456$	0	0
$457$	7212.20	0.738233	0.369117	−	0.929383i	$-0.379660\pi$
$457$	7212.20	0.738233	0.369117	+	0.929383i	$0.379660\pi$
$458$	0	0
$459$	1593.34	0.162028
$460$	0	0
$461$	10159.6	1.02642	0.513212	−	0.858262i	$-0.328456\pi$
$461$	10159.6	1.02642	0.513212	+	0.858262i	$0.328456\pi$
$462$	0	0
$463$	10292.0	1.03306	0.516532	−	0.856268i	$-0.327223\pi$
$463$	10292.0	1.03306	0.516532	+	0.856268i	$0.327223\pi$
$464$	0	0
$465$	58.2850	0.00581269
$466$	0	0
$467$	−18023.0	−1.78588	−0.892938	−	0.450180i	$-0.851360\pi$
$467$	−18023.0	−1.78588	−0.892938	+	0.450180i	$0.851360\pi$
$468$	0	0
$469$	−8189.16	−0.806269
$470$	0	0
$471$	1339.74	0.131066
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6766.95	0.653661
$476$	0	0
$477$	10029.3	0.962704
$478$	0	0
$479$	17083.8	1.62960	0.814801	−	0.579741i	$-0.196846\pi$
$479$	17083.8	1.62960	0.814801	+	0.579741i	$0.196846\pi$
$480$	0	0
$481$	14422.9	1.36721
$482$	0	0
$483$	−2699.42	−0.254302
$484$	0	0
$485$	−985.432	−0.0922601
$486$	0	0
$487$	−461.804	−0.0429699	−0.0214850	−	0.999769i	$-0.506839\pi$
$487$	−461.804	−0.0429699	−0.0214850	+	0.999769i	$0.506839\pi$
$488$	0	0
$489$	−998.391	−0.0923288
$490$	0	0
$491$	−12542.7	−1.15284	−0.576422	−	0.817152i	$-0.695552\pi$
$491$	−12542.7	−1.15284	−0.576422	+	0.817152i	$0.695552\pi$
$492$	0	0
$493$	6264.49	0.572289
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11130.3	−1.00455
$498$	0	0
$499$	18277.0	1.63966	0.819829	−	0.572608i	$-0.194068\pi$
$499$	18277.0	1.63966	0.819829	+	0.572608i	$0.194068\pi$
$500$	0	0
$501$	−1418.67	−0.126510
$502$	0	0
$503$	−2655.18	−0.235365	−0.117683	−	0.993051i	$-0.537547\pi$
$503$	−2655.18	−0.235365	−0.117683	+	0.993051i	$0.537547\pi$
$504$	0	0
$505$	1701.11	0.149898
$506$	0	0
$507$	1334.57	0.116904
$508$	0	0
$509$	−4887.16	−0.425579	−0.212789	−	0.977098i	$-0.568255\pi$
$509$	−4887.16	−0.425579	−0.212789	+	0.977098i	$0.568255\pi$
$510$	0	0
$511$	8179.08	0.708065
$512$	0	0
$513$	1588.25	0.136692
$514$	0	0
$515$	−458.155	−0.0392014
$516$	0	0
$517$	0	0
$518$	0	0
$519$	1130.42	0.0956064
$520$	0	0
$521$	−21941.1	−1.84502	−0.922512	−	0.385969i	$-0.873867\pi$
$521$	−21941.1	−1.84502	−0.922512	+	0.385969i	$0.873867\pi$
$522$	0	0
$523$	−6102.28	−0.510199	−0.255099	−	0.966915i	$-0.582108\pi$
$523$	−6102.28	−0.510199	−0.255099	+	0.966915i	$0.582108\pi$
$524$	0	0
$525$	1856.59	0.154340
$526$	0	0
$527$	3919.76	0.323999
$528$	0	0
$529$	19629.4	1.61333
$530$	0	0
$531$	−13530.2	−1.10577
$532$	0	0
$533$	13140.2	1.06785
$534$	0	0
$535$	918.660	0.0742377
$536$	0	0
$537$	−745.566	−0.0599135
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2099.35	0.166836	0.0834179	−	0.996515i	$-0.473416\pi$
$541$	2099.35	0.166836	0.0834179	+	0.996515i	$0.473416\pi$
$542$	0	0
$543$	1983.58	0.156765
$544$	0	0
$545$	−2350.16	−0.184716
$546$	0	0
$547$	−9029.06	−0.705767	−0.352884	−	0.935667i	$-0.614799\pi$
$547$	−9029.06	−0.705767	−0.352884	+	0.935667i	$0.614799\pi$
$548$	0	0
$549$	−12520.2	−0.973315
$550$	0	0
$551$	6244.47	0.482801
$552$	0	0
$553$	4790.62	0.368386
$554$	0	0
$555$	173.394	0.0132616
$556$	0	0
$557$	−7894.54	−0.600543	−0.300271	−	0.953854i	$-0.597077\pi$
$557$	−7894.54	−0.600543	−0.300271	+	0.953854i	$0.597077\pi$
$558$	0	0
$559$	14254.9	1.07857
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22377.6	−1.67514	−0.837569	−	0.546332i	$-0.816024\pi$
$563$	−22377.6	−1.67514	−0.837569	+	0.546332i	$0.816024\pi$
$564$	0	0
$565$	231.936	0.0172702
$566$	0	0
$567$	−19938.5	−1.47679
$568$	0	0
$569$	−16920.5	−1.24665	−0.623325	−	0.781963i	$-0.714219\pi$
$569$	−16920.5	−1.24665	−0.623325	+	0.781963i	$0.714219\pi$
$570$	0	0
$571$	−16320.0	−1.19609	−0.598047	−	0.801461i	$-0.704056\pi$
$571$	−16320.0	−1.19609	−0.598047	+	0.801461i	$0.704056\pi$
$572$	0	0
$573$	1893.35	0.138038
$574$	0	0
$575$	−21868.8	−1.58607
$576$	0	0
$577$	830.262	0.0599034	0.0299517	−	0.999551i	$-0.490465\pi$
$577$	830.262	0.0599034	0.0299517	+	0.999551i	$0.490465\pi$
$578$	0	0
$579$	−1396.37	−0.100226
$580$	0	0
$581$	8568.70	0.611858
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−2808.96	−0.198523
$586$	0	0
$587$	−21919.4	−1.54124	−0.770621	−	0.637294i	$-0.780054\pi$
$587$	−21919.4	−1.54124	−0.770621	+	0.637294i	$0.780054\pi$
$588$	0	0
$589$	3907.23	0.273336
$590$	0	0
$591$	385.419	0.0268258
$592$	0	0
$593$	−8236.51	−0.570376	−0.285188	−	0.958472i	$-0.592056\pi$
$593$	−8236.51	−0.570376	−0.285188	+	0.958472i	$0.592056\pi$
$594$	0	0
$595$	−2401.64	−0.165475
$596$	0	0
$597$	−554.737	−0.0380299
$598$	0	0
$599$	10922.0	0.745009	0.372505	−	0.928030i	$-0.378499\pi$
$599$	10922.0	0.745009	0.372505	+	0.928030i	$0.378499\pi$
$600$	0	0
$601$	−1386.44	−0.0940997	−0.0470498	−	0.998893i	$-0.514982\pi$
$601$	−1386.44	−0.0940997	−0.0470498	+	0.998893i	$0.514982\pi$
$602$	0	0
$603$	−7743.91	−0.522979
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1417.78	−0.0948035	−0.0474017	−	0.998876i	$-0.515094\pi$
$607$	−1417.78	−0.0948035	−0.0474017	+	0.998876i	$0.515094\pi$
$608$	0	0
$609$	1713.24	0.113997
$610$	0	0
$611$	35088.9	2.32331
$612$	0	0
$613$	−15424.9	−1.01632	−0.508162	−	0.861261i	$-0.669675\pi$
$613$	−15424.9	−1.01632	−0.508162	+	0.861261i	$0.669675\pi$
$614$	0	0
$615$	157.973	0.0103579
$616$	0	0
$617$	15169.5	0.989793	0.494897	−	0.868952i	$-0.335206\pi$
$617$	15169.5	0.989793	0.494897	+	0.868952i	$0.335206\pi$
$618$	0	0
$619$	2081.56	0.135162	0.0675809	−	0.997714i	$-0.478472\pi$
$619$	2081.56	0.135162	0.0675809	+	0.997714i	$0.478472\pi$
$620$	0	0
$621$	−5132.74	−0.331674
$622$	0	0
$623$	32392.3	2.08310
$624$	0	0
$625$	14745.9	0.943741
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11661.1	0.739200
$630$	0	0
$631$	−25249.7	−1.59298	−0.796492	−	0.604649i	$-0.793313\pi$
$631$	−25249.7	−1.59298	−0.796492	+	0.604649i	$0.793313\pi$
$632$	0	0
$633$	190.939	0.0119892
$634$	0	0
$635$	1068.60	0.0667812
$636$	0	0
$637$	−31150.5	−1.93756
$638$	0	0
$639$	−10525.1	−0.651592
$640$	0	0
$641$	−2626.57	−0.161846	−0.0809231	−	0.996720i	$-0.525787\pi$
$641$	−2626.57	−0.161846	−0.0809231	+	0.996720i	$0.525787\pi$
$642$	0	0
$643$	−9229.61	−0.566066	−0.283033	−	0.959110i	$-0.591341\pi$
$643$	−9229.61	−0.566066	−0.283033	+	0.959110i	$0.591341\pi$
$644$	0	0
$645$	171.375	0.0104618
$646$	0	0
$647$	−316.901	−0.0192561	−0.00962803	−	0.999954i	$-0.503065\pi$
$647$	−316.901	−0.0192561	−0.00962803	+	0.999954i	$0.503065\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	1071.99	0.0645389
$652$	0	0
$653$	−5022.66	−0.300998	−0.150499	−	0.988610i	$-0.548088\pi$
$653$	−5022.66	−0.300998	−0.150499	+	0.988610i	$0.548088\pi$
$654$	0	0
$655$	−2558.32	−0.152613
$656$	0	0
$657$	7734.38	0.459280
$658$	0	0
$659$	−24927.5	−1.47350	−0.736752	−	0.676163i	$-0.763642\pi$
$659$	−24927.5	−1.47350	−0.736752	+	0.676163i	$0.763642\pi$
$660$	0	0
$661$	−16440.5	−0.967418	−0.483709	−	0.875229i	$-0.660711\pi$
$661$	−16440.5	−0.967418	−0.483709	+	0.875229i	$0.660711\pi$
$662$	0	0
$663$	2030.92	0.118966
$664$	0	0
$665$	−2393.97	−0.139600
$666$	0	0
$667$	−20180.3	−1.17149
$668$	0	0
$669$	156.774	0.00906014
$670$	0	0
$671$	0	0
$672$	0	0
$673$	11777.4	0.674572	0.337286	−	0.941402i	$-0.390491\pi$
$673$	11777.4	0.674572	0.337286	+	0.941402i	$0.390491\pi$
$674$	0	0
$675$	3530.17	0.201298
$676$	0	0
$677$	2818.49	0.160005	0.0800025	−	0.996795i	$-0.474507\pi$
$677$	2818.49	0.160005	0.0800025	+	0.996795i	$0.474507\pi$
$678$	0	0
$679$	−18124.4	−1.02437
$680$	0	0
$681$	−3003.19	−0.168991
$682$	0	0
$683$	−15803.2	−0.885346	−0.442673	−	0.896683i	$-0.645970\pi$
$683$	−15803.2	−0.885346	−0.442673	+	0.896683i	$0.645970\pi$
$684$	0	0
$685$	2465.85	0.137540
$686$	0	0
$687$	−3030.17	−0.168280
$688$	0	0
$689$	25704.8	1.42130
$690$	0	0
$691$	−3300.72	−0.181716	−0.0908578	−	0.995864i	$-0.528961\pi$
$691$	−3300.72	−0.181716	−0.0908578	+	0.995864i	$0.528961\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1642.34	0.0896365
$696$	0	0
$697$	10624.0	0.577348
$698$	0	0
$699$	−1368.19	−0.0740340
$700$	0	0
$701$	29773.2	1.60416	0.802082	−	0.597214i	$-0.203726\pi$
$701$	29773.2	1.60416	0.802082	+	0.597214i	$0.203726\pi$
$702$	0	0
$703$	11623.8	0.623612
$704$	0	0
$705$	421.844	0.0225355
$706$	0	0
$707$	31287.4	1.66433
$708$	0	0
$709$	−24002.5	−1.27141	−0.635707	−	0.771931i	$-0.719291\pi$
$709$	−24002.5	−1.27141	−0.635707	+	0.771931i	$0.719291\pi$
$710$	0	0
$711$	4530.15	0.238951
$712$	0	0
$713$	−12627.0	−0.663233
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2838.80	−0.147862
$718$	0	0
$719$	20668.7	1.07206	0.536032	−	0.844198i	$-0.319923\pi$
$719$	20668.7	1.07206	0.536032	+	0.844198i	$0.319923\pi$
$720$	0	0
$721$	−8426.52	−0.435257
$722$	0	0
$723$	−2221.37	−0.114265
$724$	0	0
$725$	13879.5	0.710995
$726$	0	0
$727$	−21928.9	−1.11870	−0.559351	−	0.828931i	$-0.688950\pi$
$727$	−21928.9	−1.11870	−0.559351	+	0.828931i	$0.688950\pi$
$728$	0	0
$729$	−18438.0	−0.936745
$730$	0	0
$731$	11525.2	0.583141
$732$	0	0
$733$	25124.0	1.26600	0.633000	−	0.774152i	$-0.281823\pi$
$733$	25124.0	1.26600	0.633000	+	0.774152i	$0.281823\pi$
$734$	0	0
$735$	−374.495	−0.0187938
$736$	0	0
$737$	0	0
$738$	0	0
$739$	37038.1	1.84367	0.921833	−	0.387588i	$-0.126692\pi$
$739$	37038.1	1.84367	0.921833	+	0.387588i	$0.126692\pi$
$740$	0	0
$741$	2024.43	0.100363
$742$	0	0
$743$	−24798.0	−1.22443	−0.612213	−	0.790693i	$-0.709721\pi$
$743$	−24798.0	−1.22443	−0.612213	+	0.790693i	$0.709721\pi$
$744$	0	0
$745$	−545.508	−0.0268267
$746$	0	0
$747$	8102.82	0.396876
$748$	0	0
$749$	16896.3	0.824268
$750$	0	0
$751$	−13967.2	−0.678654	−0.339327	−	0.940668i	$-0.610199\pi$
$751$	−13967.2	−0.678654	−0.339327	+	0.940668i	$0.610199\pi$
$752$	0	0
$753$	958.256	0.0463756
$754$	0	0
$755$	−2886.80	−0.139154
$756$	0	0
$757$	−8515.45	−0.408850	−0.204425	−	0.978882i	$-0.565532\pi$
$757$	−8515.45	−0.408850	−0.204425	+	0.978882i	$0.565532\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31658.0	1.50802	0.754009	−	0.656865i	$-0.228118\pi$
$761$	31658.0	1.50802	0.754009	+	0.656865i	$0.228118\pi$
$762$	0	0
$763$	−43224.9	−2.05091
$764$	0	0
$765$	−2271.07	−0.107334
$766$	0	0
$767$	−34677.6	−1.63251
$768$	0	0
$769$	606.519	0.0284416	0.0142208	−	0.999899i	$-0.495473\pi$
$769$	606.519	0.0284416	0.0142208	+	0.999899i	$0.495473\pi$
$770$	0	0
$771$	2768.99	0.129342
$772$	0	0
$773$	−4699.76	−0.218679	−0.109339	−	0.994004i	$-0.534874\pi$
$773$	−4699.76	−0.218679	−0.109339	+	0.994004i	$0.534874\pi$
$774$	0	0
$775$	8684.55	0.402527
$776$	0	0
$777$	3189.12	0.147245
$778$	0	0
$779$	10590.0	0.487068
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3257.60	0.148681
$784$	0	0
$785$	−3839.73	−0.174581
$786$	0	0
$787$	24078.1	1.09059	0.545293	−	0.838245i	$-0.316418\pi$
$787$	24078.1	1.09059	0.545293	+	0.838245i	$0.316418\pi$
$788$	0	0
$789$	−30.9101	−0.00139472
$790$	0	0
$791$	4265.85	0.191752
$792$	0	0
$793$	−32088.9	−1.43696
$794$	0	0
$795$	309.026	0.0137862
$796$	0	0
$797$	−18977.7	−0.843443	−0.421722	−	0.906725i	$-0.638574\pi$
$797$	−18977.7	−0.843443	−0.421722	+	0.906725i	$0.638574\pi$
$798$	0	0
$799$	28369.7	1.25613
$800$	0	0
$801$	30631.1	1.35118
$802$	0	0
$803$	0	0
$804$	0	0
$805$	7736.59	0.338732
$806$	0	0
$807$	−1622.73	−0.0707842
$808$	0	0
$809$	−9120.70	−0.396374	−0.198187	−	0.980164i	$-0.563505\pi$
$809$	−9120.70	−0.396374	−0.198187	+	0.980164i	$0.563505\pi$
$810$	0	0
$811$	39874.2	1.72648	0.863239	−	0.504796i	$-0.168432\pi$
$811$	39874.2	1.72648	0.863239	+	0.504796i	$0.168432\pi$
$812$	0	0
$813$	−797.329	−0.0343955
$814$	0	0
$815$	2861.41	0.122983
$816$	0	0
$817$	11488.4	0.491956
$818$	0	0
$819$	−51663.2	−2.20422
$820$	0	0
$821$	−4913.94	−0.208889	−0.104444	−	0.994531i	$-0.533306\pi$
$821$	−4913.94	−0.208889	−0.104444	+	0.994531i	$0.533306\pi$
$822$	0	0
$823$	−2777.58	−0.117643	−0.0588215	−	0.998269i	$-0.518734\pi$
$823$	−2777.58	−0.117643	−0.0588215	+	0.998269i	$0.518734\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	20306.5	0.853842	0.426921	−	0.904289i	$-0.359598\pi$
$827$	20306.5	0.853842	0.426921	+	0.904289i	$0.359598\pi$
$828$	0	0
$829$	−35572.7	−1.49034	−0.745170	−	0.666875i	$-0.767632\pi$
$829$	−35572.7	−1.49034	−0.745170	+	0.666875i	$0.767632\pi$
$830$	0	0
$831$	−3998.05	−0.166896
$832$	0	0
$833$	−25185.4	−1.04757
$834$	0	0
$835$	4065.93	0.168512
$836$	0	0
$837$	2038.32	0.0841751
$838$	0	0
$839$	−7096.72	−0.292021	−0.146011	−	0.989283i	$-0.546643\pi$
$839$	−7096.72	−0.292021	−0.146011	+	0.989283i	$0.546643\pi$
$840$	0	0
$841$	−11581.2	−0.474851
$842$	0	0
$843$	−482.391	−0.0197087
$844$	0	0
$845$	−3824.90	−0.155717
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3476.33	0.140527
$850$	0	0
$851$	−37564.6	−1.51316
$852$	0	0
$853$	−24157.6	−0.969682	−0.484841	−	0.874602i	$-0.661123\pi$
$853$	−24157.6	−0.969682	−0.484841	+	0.874602i	$0.661123\pi$
$854$	0	0
$855$	−2263.81	−0.0905504
$856$	0	0
$857$	−28806.8	−1.14822	−0.574108	−	0.818779i	$-0.694651\pi$
$857$	−28806.8	−1.14822	−0.574108	+	0.818779i	$0.694651\pi$
$858$	0	0
$859$	−11244.4	−0.446628	−0.223314	−	0.974747i	$-0.571688\pi$
$859$	−11244.4	−0.446628	−0.223314	+	0.974747i	$0.571688\pi$
$860$	0	0
$861$	2905.49	0.115005
$862$	0	0
$863$	1291.92	0.0509589	0.0254794	−	0.999675i	$-0.491889\pi$
$863$	1291.92	0.0509589	0.0254794	+	0.999675i	$0.491889\pi$
$864$	0	0
$865$	−3239.80	−0.127349
$866$	0	0
$867$	−990.851	−0.0388132
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−19847.4	−0.772105
$872$	0	0
$873$	−17138.9	−0.664450
$874$	0	0
$875$	−10744.4	−0.415118
$876$	0	0
$877$	−26823.1	−1.03278	−0.516391	−	0.856353i	$-0.672725\pi$
$877$	−26823.1	−1.03278	−0.516391	+	0.856353i	$0.672725\pi$
$878$	0	0
$879$	−3284.73	−0.126042
$880$	0	0
$881$	−28515.7	−1.09049	−0.545243	−	0.838278i	$-0.683563\pi$
$881$	−28515.7	−1.09049	−0.545243	+	0.838278i	$0.683563\pi$
$882$	0	0
$883$	−41686.4	−1.58874	−0.794371	−	0.607433i	$-0.792199\pi$
$883$	−41686.4	−1.58874	−0.794371	+	0.607433i	$0.792199\pi$
$884$	0	0
$885$	−416.899	−0.0158349
$886$	0	0
$887$	49179.3	1.86164	0.930822	−	0.365473i	$-0.119093\pi$
$887$	49179.3	1.86164	0.930822	+	0.365473i	$0.119093\pi$
$888$	0	0
$889$	19654.0	0.741478
$890$	0	0
$891$	0	0
$892$	0	0
$893$	28279.0	1.05971
$894$	0	0
$895$	2136.81	0.0798053
$896$	0	0
$897$	−6542.35	−0.243526
$898$	0	0
$899$	8014.01	0.297310
$900$	0	0
$901$	20782.5	0.768442
$902$	0	0
$903$	3151.98	0.116159
$904$	0	0
$905$	−5684.98	−0.208812
$906$	0	0
$907$	−52977.8	−1.93947	−0.969734	−	0.244163i	$-0.921487\pi$
$907$	−52977.8	−1.93947	−0.969734	+	0.244163i	$0.921487\pi$
$908$	0	0
$909$	29586.3	1.07955
$910$	0	0
$911$	31469.8	1.14450	0.572251	−	0.820078i	$-0.306070\pi$
$911$	31469.8	1.14450	0.572251	+	0.820078i	$0.306070\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−385.778	−0.0139382
$916$	0	0
$917$	−47053.4	−1.69448
$918$	0	0
$919$	−42860.9	−1.53847	−0.769234	−	0.638967i	$-0.779362\pi$
$919$	−42860.9	−1.53847	−0.769234	+	0.638967i	$0.779362\pi$
$920$	0	0
$921$	2881.89	0.103107
$922$	0	0
$923$	−26975.6	−0.961984
$924$	0	0
$925$	25836.1	0.918361
$926$	0	0
$927$	−7968.37	−0.282325
$928$	0	0
$929$	−20968.3	−0.740525	−0.370262	−	0.928927i	$-0.620732\pi$
$929$	−20968.3	−0.740525	−0.370262	+	0.928927i	$0.620732\pi$
$930$	0	0
$931$	−25104.9	−0.883760
$932$	0	0
$933$	3250.98	0.114075
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17126.8	−0.597127	−0.298563	−	0.954390i	$-0.596507\pi$
$937$	−17126.8	−0.597127	−0.298563	+	0.954390i	$0.596507\pi$
$938$	0	0
$939$	1736.94	0.0603652
$940$	0	0
$941$	44021.2	1.52503	0.762513	−	0.646973i	$-0.223966\pi$
$941$	44021.2	1.52503	0.762513	+	0.646973i	$0.223966\pi$
$942$	0	0
$943$	−34223.7	−1.18184
$944$	0	0
$945$	−1248.88	−0.0429906
$946$	0	0
$947$	−8692.03	−0.298261	−0.149130	−	0.988818i	$-0.547647\pi$
$947$	−8692.03	−0.298261	−0.149130	+	0.988818i	$0.547647\pi$
$948$	0	0
$949$	19823.0	0.678062
$950$	0	0
$951$	3493.69	0.119128
$952$	0	0
$953$	−57906.0	−1.96827	−0.984133	−	0.177431i	$-0.943221\pi$
$953$	−57906.0	−1.96827	−0.984133	+	0.177431i	$0.943221\pi$
$954$	0	0
$955$	−5426.38	−0.183867
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45352.6	1.52713
$960$	0	0
$961$	−24776.6	−0.831679
$962$	0	0
$963$	15977.6	0.534654
$964$	0	0
$965$	4002.03	0.133503
$966$	0	0
$967$	−56564.3	−1.88106	−0.940530	−	0.339711i	$-0.889671\pi$
$967$	−56564.3	−1.88106	−0.940530	+	0.339711i	$0.889671\pi$
$968$	0	0
$969$	1636.77	0.0542628
$970$	0	0
$971$	30894.7	1.02107	0.510534	−	0.859857i	$-0.329448\pi$
$971$	30894.7	1.02107	0.510534	+	0.859857i	$0.329448\pi$
$972$	0	0
$973$	30206.3	0.995242
$974$	0	0
$975$	4499.67	0.147800
$976$	0	0
$977$	−29998.9	−0.982342	−0.491171	−	0.871063i	$-0.663431\pi$
$977$	−29998.9	−0.982342	−0.491171	+	0.871063i	$0.663431\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−40874.8	−1.33031
$982$	0	0
$983$	24485.9	0.794485	0.397242	−	0.917714i	$-0.369967\pi$
$983$	24485.9	0.794485	0.397242	+	0.917714i	$0.369967\pi$
$984$	0	0
$985$	−1104.62	−0.0357321
$986$	0	0
$987$	7758.68	0.250214
$988$	0	0
$989$	−37127.1	−1.19370
$990$	0	0
$991$	52661.1	1.68803	0.844014	−	0.536321i	$-0.180186\pi$
$991$	52661.1	1.68803	0.844014	+	0.536321i	$0.180186\pi$
$992$	0	0
$993$	−3150.84	−0.100694
$994$	0	0
$995$	1589.89	0.0506562
$996$	0	0
$997$	54748.6	1.73912	0.869561	−	0.493826i	$-0.164402\pi$
$997$	54748.6	1.73912	0.869561	+	0.493826i	$0.164402\pi$
$998$	0	0
$999$	6063.88	0.192045

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1936.4.a.y.1.1		2
4.3	odd	2		121.4.a.e.1.1	yes	2
11.10	odd	2		1936.4.a.z.1.1		2
12.11	even	2		1089.4.a.k.1.2		2
44.3	odd	10		121.4.c.d.9.1		8
44.7	even	10		121.4.c.g.27.2		8
44.15	odd	10		121.4.c.d.27.1		8
44.19	even	10		121.4.c.g.9.2		8
44.27	odd	10		121.4.c.d.3.2		8
44.31	odd	10		121.4.c.d.81.2		8
44.35	even	10		121.4.c.g.81.1		8
44.39	even	10		121.4.c.g.3.1		8
44.43	even	2		121.4.a.b.1.2	✓	2
132.131	odd	2		1089.4.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
121.4.a.b.1.2	✓	2	44.43	even	2
121.4.a.e.1.1	yes	2	4.3	odd	2
121.4.c.d.3.2		8	44.27	odd	10
121.4.c.d.9.1		8	44.3	odd	10
121.4.c.d.27.1		8	44.15	odd	10
121.4.c.d.81.2		8	44.31	odd	10
121.4.c.g.3.1		8	44.39	even	10
121.4.c.g.9.2		8	44.19	even	10
121.4.c.g.27.2		8	44.7	even	10
121.4.c.g.81.1		8	44.35	even	10
1089.4.a.k.1.2		2	12.11	even	2
1089.4.a.x.1.1		2	132.131	odd	2
1936.4.a.y.1.1		2	1.1	even	1	trivial
1936.4.a.z.1.1		2	11.10	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1936 = 2^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1936.a (trivial)

\(f(q)\)	\(=\)	\(q+0.535898 q^{3} -1.53590 q^{5} -28.2487 q^{7} -26.7128 q^{9} +O(q^{10})\)	\(q+0.535898 q^{3} -1.53590 q^{5} -28.2487 q^{7} -26.7128 q^{9} -68.4641 q^{13} -0.823085 q^{15} -55.3538 q^{17} -55.1769 q^{19} -15.1384 q^{21} +178.315 q^{23} -122.641 q^{25} -28.7846 q^{27} -113.172 q^{29} -70.8128 q^{31} +43.3872 q^{35} -210.664 q^{37} -36.6898 q^{39} -191.928 q^{41} -208.210 q^{43} +41.0282 q^{45} -512.515 q^{47} +454.990 q^{49} -29.6640 q^{51} -375.449 q^{53} -29.5692 q^{57} +506.508 q^{59} +468.697 q^{61} +754.603 q^{63} +105.154 q^{65} +289.895 q^{67} +95.5589 q^{69} +394.010 q^{71} -289.538 q^{73} -65.7231 q^{75} -169.587 q^{79} +705.820 q^{81} -303.331 q^{83} +85.0179 q^{85} -60.6486 q^{87} -1146.68 q^{89} +1934.02 q^{91} -37.9485 q^{93} +84.7461 q^{95} +641.600 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{3} - 10 q^{5} - 8 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 8 * q^3 - 10 * q^5 - 8 * q^7 + 2 * q^9	\( 2 q + 8 q^{3} - 10 q^{5} - 8 q^{7} + 2 q^{9} - 130 q^{13} - 64 q^{15} + 14 q^{17} - 48 q^{19} + 136 q^{21} + 128 q^{23} - 176 q^{25} - 16 q^{27} + 30 q^{29} + 184 q^{31} - 128 q^{35} + 126 q^{37} - 496 q^{39} - 370 q^{41} - 264 q^{43} - 202 q^{45} - 256 q^{47} + 522 q^{49} + 488 q^{51} - 162 q^{53} + 24 q^{57} + 1304 q^{59} + 300 q^{61} + 1336 q^{63} + 626 q^{65} + 656 q^{67} - 280 q^{69} + 1176 q^{71} + 668 q^{73} - 464 q^{75} + 416 q^{79} + 26 q^{81} - 960 q^{83} - 502 q^{85} + 1008 q^{87} - 1074 q^{89} + 688 q^{91} + 1864 q^{93} + 24 q^{95} - 338 q^{97}+O(q^{100}) \) 2 * q + 8 * q^3 - 10 * q^5 - 8 * q^7 + 2 * q^9 - 130 * q^13 - 64 * q^15 + 14 * q^17 - 48 * q^19 + 136 * q^21 + 128 * q^23 - 176 * q^25 - 16 * q^27 + 30 * q^29 + 184 * q^31 - 128 * q^35 + 126 * q^37 - 496 * q^39 - 370 * q^41 - 264 * q^43 - 202 * q^45 - 256 * q^47 + 522 * q^49 + 488 * q^51 - 162 * q^53 + 24 * q^57 + 1304 * q^59 + 300 * q^61 + 1336 * q^63 + 626 * q^65 + 656 * q^67 - 280 * q^69 + 1176 * q^71 + 668 * q^73 - 464 * q^75 + 416 * q^79 + 26 * q^81 - 960 * q^83 - 502 * q^85 + 1008 * q^87 - 1074 * q^89 + 688 * q^91 + 1864 * q^93 + 24 * q^95 - 338 * q^97

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