LMFDB - Embedded newform 693.4.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.509800.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 5x + 15 $ x^4 - x^3 - 10x^2 + 5x + 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.11082$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.65527 q^{2} +5.36103 q^{4} +10.0822 q^{5} -7.00000 q^{7} -9.64616 q^{8} +O(q^{10})$	\(q+3.65527 q^{2} +5.36103 q^{4} +10.0822 q^{5} -7.00000 q^{7} -9.64616 q^{8} +36.8533 q^{10} -11.0000 q^{11} -84.5724 q^{13} -25.5869 q^{14} -78.1476 q^{16} +38.2525 q^{17} -127.283 q^{19} +54.0511 q^{20} -40.2080 q^{22} -140.378 q^{23} -23.3486 q^{25} -309.135 q^{26} -37.5272 q^{28} +116.806 q^{29} +338.709 q^{31} -208.482 q^{32} +139.823 q^{34} -70.5756 q^{35} -75.3416 q^{37} -465.256 q^{38} -97.2548 q^{40} +22.4446 q^{41} +181.844 q^{43} -58.9713 q^{44} -513.121 q^{46} -300.530 q^{47} +49.0000 q^{49} -85.3455 q^{50} -453.395 q^{52} +31.8596 q^{53} -110.905 q^{55} +67.5231 q^{56} +426.958 q^{58} +68.3030 q^{59} -145.315 q^{61} +1238.07 q^{62} -136.877 q^{64} -852.679 q^{65} -668.020 q^{67} +205.073 q^{68} -257.973 q^{70} -727.608 q^{71} -416.982 q^{73} -275.394 q^{74} -682.370 q^{76} +77.0000 q^{77} +458.805 q^{79} -787.902 q^{80} +82.0412 q^{82} -355.737 q^{83} +385.671 q^{85} +664.690 q^{86} +106.108 q^{88} +1245.97 q^{89} +592.007 q^{91} -752.571 q^{92} -1098.52 q^{94} -1283.30 q^{95} -935.338 q^{97} +179.108 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	$ 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.65527	1.29233	0.646167	−	0.763196i	$-0.276371\pi$
$2$	3.65527	1.29233	0.646167	+	0.763196i	$0.276371\pi$
$3$	0	0
$3$	0	0
$4$	5.36103	0.670129
$5$	10.0822	0.901782	0.450891	−	0.892579i	$-0.351106\pi$
$5$	10.0822	0.901782	0.450891	+	0.892579i	$0.351106\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−9.64616	−0.426304
$9$	0	0
$10$	36.8533	1.16540
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−84.5724	−1.80432	−0.902161	−	0.431400i	$-0.858020\pi$
$13$	−84.5724	−1.80432	−0.902161	+	0.431400i	$0.858020\pi$
$14$	−25.5869	−0.488457
$15$	0	0
$16$	−78.1476	−1.22106
$17$	38.2525	0.545741	0.272871	−	0.962051i	$-0.412027\pi$
$17$	38.2525	0.545741	0.272871	+	0.962051i	$0.412027\pi$
$18$	0	0
$19$	−127.283	−1.53688	−0.768442	−	0.639919i	$-0.778968\pi$
$19$	−127.283	−1.53688	−0.768442	+	0.639919i	$0.778968\pi$
$20$	54.0511	0.604310
$21$	0	0
$22$	−40.2080	−0.389654
$23$	−140.378	−1.27265	−0.636323	−	0.771423i	$-0.719545\pi$
$23$	−140.378	−1.27265	−0.636323	+	0.771423i	$0.719545\pi$
$24$	0	0
$25$	−23.3486	−0.186789
$26$	−309.135	−2.33179
$27$	0	0
$28$	−37.5272	−0.253285
$29$	116.806	0.747943	0.373971	−	0.927440i	$-0.377996\pi$
$29$	116.806	0.747943	0.373971	+	0.927440i	$0.377996\pi$
$30$	0	0
$31$	338.709	1.96238	0.981192	−	0.193034i	$-0.0618327\pi$
$31$	338.709	1.96238	0.981192	+	0.193034i	$0.0618327\pi$
$32$	−208.482	−1.15171
$33$	0	0
$34$	139.823	0.705280
$35$	−70.5756	−0.340842
$36$	0	0
$37$	−75.3416	−0.334759	−0.167379	−	0.985893i	$-0.553531\pi$
$37$	−75.3416	−0.334759	−0.167379	+	0.985893i	$0.553531\pi$
$38$	−465.256	−1.98617
$39$	0	0
$40$	−97.2548	−0.384434
$41$	22.4446	0.0854941	0.0427471	−	0.999086i	$-0.486389\pi$
$41$	22.4446	0.0854941	0.0427471	+	0.999086i	$0.486389\pi$
$42$	0	0
$43$	181.844	0.644906	0.322453	−	0.946585i	$-0.395492\pi$
$43$	181.844	0.644906	0.322453	+	0.946585i	$0.395492\pi$
$44$	−58.9713	−0.202051
$45$	0	0
$46$	−513.121	−1.64468
$47$	−300.530	−0.932698	−0.466349	−	0.884601i	$-0.654431\pi$
$47$	−300.530	−0.932698	−0.466349	+	0.884601i	$0.654431\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−85.3455	−0.241394
$51$	0	0
$52$	−453.395	−1.20913
$53$	31.8596	0.0825708	0.0412854	−	0.999147i	$-0.486855\pi$
$53$	31.8596	0.0825708	0.0412854	+	0.999147i	$0.486855\pi$
$54$	0	0
$55$	−110.905	−0.271898
$56$	67.5231	0.161128
$57$	0	0
$58$	426.958	0.966593
$59$	68.3030	0.150717	0.0753584	−	0.997157i	$-0.475990\pi$
$59$	68.3030	0.150717	0.0753584	+	0.997157i	$0.475990\pi$
$60$	0	0
$61$	−145.315	−0.305012	−0.152506	−	0.988303i	$-0.548734\pi$
$61$	−145.315	−0.305012	−0.152506	+	0.988303i	$0.548734\pi$
$62$	1238.07	2.53606
$63$	0	0
$64$	−136.877	−0.267337
$65$	−852.679	−1.62710
$66$	0	0
$67$	−668.020	−1.21808	−0.609042	−	0.793138i	$-0.708446\pi$
$67$	−668.020	−1.21808	−0.609042	+	0.793138i	$0.708446\pi$
$68$	205.073	0.365717
$69$	0	0
$70$	−257.973	−0.440481
$71$	−727.608	−1.21621	−0.608107	−	0.793855i	$-0.708071\pi$
$71$	−727.608	−1.21621	−0.608107	+	0.793855i	$0.708071\pi$
$72$	0	0
$73$	−416.982	−0.668548	−0.334274	−	0.942476i	$-0.608491\pi$
$73$	−416.982	−0.668548	−0.334274	+	0.942476i	$0.608491\pi$
$74$	−275.394	−0.432621
$75$	0	0
$76$	−682.370	−1.02991
$77$	77.0000	0.113961
$78$	0	0
$79$	458.805	0.653413	0.326707	−	0.945126i	$-0.394061\pi$
$79$	458.805	0.653413	0.326707	+	0.945126i	$0.394061\pi$
$80$	−787.902	−1.10113
$81$	0	0
$82$	82.0412	0.110487
$83$	−355.737	−0.470449	−0.235224	−	0.971941i	$-0.575582\pi$
$83$	−355.737	−0.470449	−0.235224	+	0.971941i	$0.575582\pi$
$84$	0	0
$85$	385.671	0.492140
$86$	664.690	0.833435
$87$	0	0
$88$	106.108	0.128536
$89$	1245.97	1.48396	0.741980	−	0.670422i	$-0.233887\pi$
$89$	1245.97	1.48396	0.741980	+	0.670422i	$0.233887\pi$
$90$	0	0
$91$	592.007	0.681969
$92$	−752.571	−0.852837
$93$	0	0
$94$	−1098.52	−1.20536
$95$	−1283.30	−1.38594
$96$	0	0
$97$	−935.338	−0.979063	−0.489532	−	0.871986i	$-0.662832\pi$
$97$	−935.338	−0.979063	−0.489532	+	0.871986i	$0.662832\pi$
$98$	179.108	0.184619
$99$	0	0
$100$	−125.173	−0.125173
$101$	533.395	0.525493	0.262747	−	0.964865i	$-0.415372\pi$
$101$	533.395	0.525493	0.262747	+	0.964865i	$0.415372\pi$
$102$	0	0
$103$	−738.096	−0.706086	−0.353043	−	0.935607i	$-0.614853\pi$
$103$	−738.096	−0.706086	−0.353043	+	0.935607i	$0.614853\pi$
$104$	815.800	0.769190
$105$	0	0
$106$	116.456	0.106709
$107$	2039.07	1.84228	0.921141	−	0.389229i	$-0.127259\pi$
$107$	2039.07	1.84228	0.921141	+	0.389229i	$0.127259\pi$
$108$	0	0
$109$	1488.69	1.30817	0.654085	−	0.756421i	$-0.273054\pi$
$109$	1488.69	1.30817	0.654085	+	0.756421i	$0.273054\pi$
$110$	−405.387	−0.351383
$111$	0	0
$112$	547.033	0.461516
$113$	532.743	0.443507	0.221753	−	0.975103i	$-0.428822\pi$
$113$	532.743	0.443507	0.221753	+	0.975103i	$0.428822\pi$
$114$	0	0
$115$	−1415.32	−1.14765
$116$	626.201	0.501218
$117$	0	0
$118$	249.666	0.194776
$119$	−267.768	−0.206271
$120$	0	0
$121$	121.000	0.0909091
$122$	−531.167	−0.394177
$123$	0	0
$124$	1815.83	1.31505
$125$	−1495.68	−1.07023
$126$	0	0
$127$	−2257.44	−1.57729	−0.788645	−	0.614849i	$-0.789217\pi$
$127$	−2257.44	−1.57729	−0.788645	+	0.614849i	$0.789217\pi$
$128$	1167.53	0.806220
$129$	0	0
$130$	−3116.78	−2.10276
$131$	−1174.21	−0.783142	−0.391571	−	0.920148i	$-0.628068\pi$
$131$	−1174.21	−0.783142	−0.391571	+	0.920148i	$0.628068\pi$
$132$	0	0
$133$	890.984	0.580888
$134$	−2441.79	−1.57417
$135$	0	0
$136$	−368.990	−0.232652
$137$	−2690.08	−1.67758	−0.838792	−	0.544451i	$-0.816738\pi$
$137$	−2690.08	−1.67758	−0.838792	+	0.544451i	$0.816738\pi$
$138$	0	0
$139$	17.7500	0.0108312	0.00541559	−	0.999985i	$-0.498276\pi$
$139$	17.7500	0.0108312	0.00541559	+	0.999985i	$0.498276\pi$
$140$	−378.358	−0.228408
$141$	0	0
$142$	−2659.61	−1.57175
$143$	930.297	0.544023
$144$	0	0
$145$	1177.67	0.674482
$146$	−1524.18	−0.863988
$147$	0	0
$148$	−403.908	−0.224332
$149$	−1517.86	−0.834550	−0.417275	−	0.908780i	$-0.637015\pi$
$149$	−1517.86	−0.834550	−0.417275	+	0.908780i	$0.637015\pi$
$150$	0	0
$151$	1948.86	1.05031	0.525153	−	0.851008i	$-0.324008\pi$
$151$	1948.86	1.05031	0.525153	+	0.851008i	$0.324008\pi$
$152$	1227.80	0.655180
$153$	0	0
$154$	281.456	0.147275
$155$	3414.94	1.76964
$156$	0	0
$157$	−1554.20	−0.790055	−0.395027	−	0.918669i	$-0.629265\pi$
$157$	−1554.20	−0.790055	−0.395027	+	0.918669i	$0.629265\pi$
$158$	1677.06	0.844428
$159$	0	0
$160$	−2101.96	−1.03859
$161$	982.647	0.481015
$162$	0	0
$163$	−3472.71	−1.66873	−0.834367	−	0.551209i	$-0.814167\pi$
$163$	−3472.71	−1.66873	−0.834367	+	0.551209i	$0.814167\pi$
$164$	120.326	0.0572921
$165$	0	0
$166$	−1300.32	−0.607977
$167$	2228.90	1.03280	0.516400	−	0.856347i	$-0.327272\pi$
$167$	2228.90	1.03280	0.516400	+	0.856347i	$0.327272\pi$
$168$	0	0
$169$	4955.50	2.25558
$170$	1409.73	0.636009
$171$	0	0
$172$	974.872	0.432170
$173$	1008.04	0.443005	0.221502	−	0.975160i	$-0.428904\pi$
$173$	1008.04	0.443005	0.221502	+	0.975160i	$0.428904\pi$
$174$	0	0
$175$	163.440	0.0705995
$176$	859.624	0.368162
$177$	0	0
$178$	4554.36	1.91777
$179$	746.246	0.311603	0.155802	−	0.987788i	$-0.450204\pi$
$179$	746.246	0.311603	0.155802	+	0.987788i	$0.450204\pi$
$180$	0	0
$181$	−2787.76	−1.14482	−0.572410	−	0.819968i	$-0.693991\pi$
$181$	−2787.76	−1.14482	−0.572410	+	0.819968i	$0.693991\pi$
$182$	2163.95	0.881333
$183$	0	0
$184$	1354.11	0.542534
$185$	−759.611	−0.301880
$186$	0	0
$187$	−420.778	−0.164547
$188$	−1611.15	−0.625028
$189$	0	0
$190$	−4690.82	−1.79109
$191$	911.917	0.345466	0.172733	−	0.984969i	$-0.444740\pi$
$191$	911.917	0.345466	0.172733	+	0.984969i	$0.444740\pi$
$192$	0	0
$193$	4454.66	1.66142	0.830709	−	0.556707i	$-0.187935\pi$
$193$	4454.66	1.66142	0.830709	+	0.556707i	$0.187935\pi$
$194$	−3418.92	−1.26528
$195$	0	0
$196$	262.690	0.0957327
$197$	1377.46	0.498171	0.249086	−	0.968481i	$-0.419870\pi$
$197$	1377.46	0.498171	0.249086	+	0.968481i	$0.419870\pi$
$198$	0	0
$199$	−94.3572	−0.0336121	−0.0168060	−	0.999859i	$-0.505350\pi$
$199$	−94.3572	−0.0336121	−0.0168060	+	0.999859i	$0.505350\pi$
$200$	225.224	0.0796288
$201$	0	0
$202$	1949.71	0.679113
$203$	−817.643	−0.282696
$204$	0	0
$205$	226.292	0.0770971
$206$	−2697.95	−0.912499
$207$	0	0
$208$	6609.13	2.20318
$209$	1400.12	0.463388
$210$	0	0
$211$	1174.19	0.383101	0.191550	−	0.981483i	$-0.438648\pi$
$211$	1174.19	0.383101	0.191550	+	0.981483i	$0.438648\pi$
$212$	170.800	0.0553330
$213$	0	0
$214$	7453.35	2.38084
$215$	1833.40	0.581565
$216$	0	0
$217$	−2370.96	−0.741712
$218$	5441.56	1.69059
$219$	0	0
$220$	−594.563	−0.182206
$221$	−3235.11	−0.984692
$222$	0	0
$223$	88.5875	0.0266021	0.0133010	−	0.999912i	$-0.495766\pi$
$223$	88.5875	0.0266021	0.0133010	+	0.999912i	$0.495766\pi$
$224$	1459.37	0.435305
$225$	0	0
$226$	1947.32	0.573159
$227$	−883.312	−0.258271	−0.129135	−	0.991627i	$-0.541220\pi$
$227$	−883.312	−0.258271	−0.129135	+	0.991627i	$0.541220\pi$
$228$	0	0
$229$	1240.26	0.357898	0.178949	−	0.983858i	$-0.442730\pi$
$229$	1240.26	0.357898	0.178949	+	0.983858i	$0.442730\pi$
$230$	−5173.40	−1.48315
$231$	0	0
$232$	−1126.73	−0.318851
$233$	5479.93	1.54078	0.770392	−	0.637571i	$-0.220061\pi$
$233$	5479.93	1.54078	0.770392	+	0.637571i	$0.220061\pi$
$234$	0	0
$235$	−3030.01	−0.841090
$236$	366.174	0.101000
$237$	0	0
$238$	−978.764	−0.266571
$239$	−594.006	−0.160766	−0.0803830	−	0.996764i	$-0.525614\pi$
$239$	−594.006	−0.160766	−0.0803830	+	0.996764i	$0.525614\pi$
$240$	0	0
$241$	308.785	0.0825336	0.0412668	−	0.999148i	$-0.486861\pi$
$241$	308.785	0.0825336	0.0412668	+	0.999148i	$0.486861\pi$
$242$	442.288	0.117485
$243$	0	0
$244$	−779.039	−0.204397
$245$	494.029	0.128826
$246$	0	0
$247$	10764.7	2.77303
$248$	−3267.24	−0.836573
$249$	0	0
$250$	−5467.14	−1.38309
$251$	−3487.59	−0.877031	−0.438515	−	0.898724i	$-0.644495\pi$
$251$	−3487.59	−0.877031	−0.438515	+	0.898724i	$0.644495\pi$
$252$	0	0
$253$	1544.16	0.383717
$254$	−8251.58	−2.03839
$255$	0	0
$256$	5362.66	1.30924
$257$	−451.445	−0.109574	−0.0547868	−	0.998498i	$-0.517448\pi$
$257$	−451.445	−0.109574	−0.0547868	+	0.998498i	$0.517448\pi$
$258$	0	0
$259$	527.391	0.126527
$260$	−4571.24	−1.09037
$261$	0	0
$262$	−4292.08	−1.01208
$263$	5878.61	1.37829	0.689146	−	0.724622i	$-0.257986\pi$
$263$	5878.61	1.37829	0.689146	+	0.724622i	$0.257986\pi$
$264$	0	0
$265$	321.216	0.0744609
$266$	3256.79	0.750701
$267$	0	0
$268$	−3581.27	−0.816272
$269$	52.8516	0.0119792	0.00598962	−	0.999982i	$-0.498093\pi$
$269$	52.8516	0.0119792	0.00598962	+	0.999982i	$0.498093\pi$
$270$	0	0
$271$	−6822.19	−1.52922	−0.764610	−	0.644493i	$-0.777069\pi$
$271$	−6822.19	−1.52922	−0.764610	+	0.644493i	$0.777069\pi$
$272$	−2989.34	−0.666381
$273$	0	0
$274$	−9832.98	−2.16800
$275$	256.835	0.0563189
$276$	0	0
$277$	469.032	0.101738	0.0508689	−	0.998705i	$-0.483801\pi$
$277$	469.032	0.101738	0.0508689	+	0.998705i	$0.483801\pi$
$278$	64.8810	0.0139975
$279$	0	0
$280$	680.784	0.145302
$281$	2305.05	0.489352	0.244676	−	0.969605i	$-0.421318\pi$
$281$	2305.05	0.489352	0.244676	+	0.969605i	$0.421318\pi$
$282$	0	0
$283$	−7370.80	−1.54823	−0.774114	−	0.633046i	$-0.781805\pi$
$283$	−7370.80	−1.54823	−0.774114	+	0.633046i	$0.781805\pi$
$284$	−3900.73	−0.815020
$285$	0	0
$286$	3400.49	0.703060
$287$	−157.112	−0.0323137
$288$	0	0
$289$	−3449.74	−0.702167
$290$	4304.69	0.871656
$291$	0	0
$292$	−2235.45	−0.448013
$293$	−1758.90	−0.350702	−0.175351	−	0.984506i	$-0.556106\pi$
$293$	−1758.90	−0.350702	−0.175351	+	0.984506i	$0.556106\pi$
$294$	0	0
$295$	688.646	0.135914
$296$	726.757	0.142709
$297$	0	0
$298$	−5548.20	−1.07852
$299$	11872.1	2.29626
$300$	0	0
$301$	−1272.91	−0.243752
$302$	7123.63	1.35735
$303$	0	0
$304$	9946.89	1.87662
$305$	−1465.10	−0.275054
$306$	0	0
$307$	3468.10	0.644739	0.322369	−	0.946614i	$-0.395521\pi$
$307$	3468.10	0.644739	0.322369	+	0.946614i	$0.395521\pi$
$308$	412.799	0.0763682
$309$	0	0
$310$	12482.5	2.28697
$311$	−1983.98	−0.361741	−0.180870	−	0.983507i	$-0.557891\pi$
$311$	−1983.98	−0.361741	−0.180870	+	0.983507i	$0.557891\pi$
$312$	0	0
$313$	−10094.2	−1.82287	−0.911436	−	0.411443i	$-0.865025\pi$
$313$	−10094.2	−1.82287	−0.911436	+	0.411443i	$0.865025\pi$
$314$	−5681.03	−1.02102
$315$	0	0
$316$	2459.67	0.437871
$317$	3051.34	0.540633	0.270316	−	0.962772i	$-0.412872\pi$
$317$	3051.34	0.540633	0.270316	+	0.962772i	$0.412872\pi$
$318$	0	0
$319$	−1284.87	−0.225513
$320$	−1380.02	−0.241080
$321$	0	0
$322$	3591.84	0.621632
$323$	−4868.91	−0.838741
$324$	0	0
$325$	1974.65	0.337027
$326$	−12693.7	−2.15656
$327$	0	0
$328$	−216.504	−0.0364465
$329$	2103.71	0.352527
$330$	0	0
$331$	−26.9826	−0.00448066	−0.00224033	−	0.999997i	$-0.500713\pi$
$331$	−26.9826	−0.00448066	−0.00224033	+	0.999997i	$0.500713\pi$
$332$	−1907.12	−0.315261
$333$	0	0
$334$	8147.25	1.33472
$335$	−6735.13	−1.09845
$336$	0	0
$337$	6818.62	1.10218	0.551089	−	0.834447i	$-0.314213\pi$
$337$	6818.62	1.10218	0.551089	+	0.834447i	$0.314213\pi$
$338$	18113.7	2.91496
$339$	0	0
$340$	2067.59	0.329797
$341$	−3725.80	−0.591681
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−1754.10	−0.274926
$345$	0	0
$346$	3684.66	0.572510
$347$	−11907.0	−1.84208	−0.921038	−	0.389473i	$-0.872657\pi$
$347$	−11907.0	−1.84208	−0.921038	+	0.389473i	$0.872657\pi$
$348$	0	0
$349$	−4352.72	−0.667609	−0.333805	−	0.942642i	$-0.608333\pi$
$349$	−4352.72	−0.667609	−0.333805	+	0.942642i	$0.608333\pi$
$350$	597.419	0.0912382
$351$	0	0
$352$	2293.30	0.347253
$353$	1326.31	0.199978	0.0999891	−	0.994989i	$-0.468119\pi$
$353$	1326.31	0.199978	0.0999891	+	0.994989i	$0.468119\pi$
$354$	0	0
$355$	−7335.91	−1.09676
$356$	6679.68	0.994444
$357$	0	0
$358$	2727.73	0.402696
$359$	8292.92	1.21917	0.609587	−	0.792719i	$-0.291335\pi$
$359$	8292.92	1.21917	0.609587	+	0.792719i	$0.291335\pi$
$360$	0	0
$361$	9342.06	1.36201
$362$	−10190.0	−1.47949
$363$	0	0
$364$	3173.77	0.457007
$365$	−4204.10	−0.602885
$366$	0	0
$367$	−11027.5	−1.56848	−0.784241	−	0.620457i	$-0.786947\pi$
$367$	−11027.5	−1.56848	−0.784241	+	0.620457i	$0.786947\pi$
$368$	10970.2	1.55397
$369$	0	0
$370$	−2776.59	−0.390130
$371$	−223.017	−0.0312088
$372$	0	0
$373$	8245.72	1.14463	0.572316	−	0.820034i	$-0.306045\pi$
$373$	8245.72	1.14463	0.572316	+	0.820034i	$0.306045\pi$
$374$	−1538.06	−0.212650
$375$	0	0
$376$	2898.96	0.397613
$377$	−9878.58	−1.34953
$378$	0	0
$379$	10163.4	1.37747	0.688734	−	0.725014i	$-0.258167\pi$
$379$	10163.4	1.37747	0.688734	+	0.725014i	$0.258167\pi$
$380$	−6879.81	−0.928755
$381$	0	0
$382$	3333.31	0.446458
$383$	−14338.9	−1.91301	−0.956506	−	0.291714i	$-0.905774\pi$
$383$	−14338.9	−1.91301	−0.956506	+	0.291714i	$0.905774\pi$
$384$	0	0
$385$	776.332	0.102768
$386$	16283.0	2.14711
$387$	0	0
$388$	−5014.37	−0.656098
$389$	2382.91	0.310587	0.155294	−	0.987868i	$-0.450368\pi$
$389$	2382.91	0.310587	0.155294	+	0.987868i	$0.450368\pi$
$390$	0	0
$391$	−5369.82	−0.694535
$392$	−472.662	−0.0609006
$393$	0	0
$394$	5034.98	0.643804
$395$	4625.78	0.589236
$396$	0	0
$397$	−9868.22	−1.24754	−0.623768	−	0.781609i	$-0.714399\pi$
$397$	−9868.22	−1.24754	−0.623768	+	0.781609i	$0.714399\pi$
$398$	−344.901	−0.0434380
$399$	0	0
$400$	1824.64	0.228080
$401$	5879.12	0.732143	0.366072	−	0.930587i	$-0.380702\pi$
$401$	5879.12	0.732143	0.366072	+	0.930587i	$0.380702\pi$
$402$	0	0
$403$	−28645.4	−3.54077
$404$	2859.55	0.352148
$405$	0	0
$406$	−2988.71	−0.365338
$407$	828.758	0.100934
$408$	0	0
$409$	5680.84	0.686796	0.343398	−	0.939190i	$-0.388422\pi$
$409$	5680.84	0.686796	0.343398	+	0.939190i	$0.388422\pi$
$410$	827.158	0.0996352
$411$	0	0
$412$	−3956.96	−0.473168
$413$	−478.121	−0.0569656
$414$	0	0
$415$	−3586.62	−0.424242
$416$	17631.8	2.07805
$417$	0	0
$418$	5117.81	0.598853
$419$	1098.50	0.128079	0.0640395	−	0.997947i	$-0.479602\pi$
$419$	1098.50	0.128079	0.0640395	+	0.997947i	$0.479602\pi$
$420$	0	0
$421$	−5265.06	−0.609509	−0.304754	−	0.952431i	$-0.598574\pi$
$421$	−5265.06	−0.609509	−0.304754	+	0.952431i	$0.598574\pi$
$422$	4291.97	0.495095
$423$	0	0
$424$	−307.323	−0.0352003
$425$	−893.143	−0.101938
$426$	0	0
$427$	1017.21	0.115284
$428$	10931.5	1.23457
$429$	0	0
$430$	6701.56	0.751577
$431$	−4273.45	−0.477598	−0.238799	−	0.971069i	$-0.576754\pi$
$431$	−4273.45	−0.477598	−0.238799	+	0.971069i	$0.576754\pi$
$432$	0	0
$433$	−8560.19	−0.950061	−0.475031	−	0.879969i	$-0.657563\pi$
$433$	−8560.19	−0.950061	−0.475031	+	0.879969i	$0.657563\pi$
$434$	−8666.52	−0.958539
$435$	0	0
$436$	7980.90	0.876642
$437$	17867.8	1.95591
$438$	0	0
$439$	12664.2	1.37684	0.688419	−	0.725314i	$-0.258305\pi$
$439$	12664.2	1.37684	0.688419	+	0.725314i	$0.258305\pi$
$440$	1069.80	0.115911
$441$	0	0
$442$	−11825.2	−1.27255
$443$	−12368.9	−1.32656	−0.663279	−	0.748372i	$-0.730836\pi$
$443$	−12368.9	−1.32656	−0.663279	+	0.748372i	$0.730836\pi$
$444$	0	0
$445$	12562.2	1.33821
$446$	323.812	0.0343788
$447$	0	0
$448$	958.136	0.101044
$449$	2092.69	0.219956	0.109978	−	0.993934i	$-0.464922\pi$
$449$	2092.69	0.219956	0.109978	+	0.993934i	$0.464922\pi$
$450$	0	0
$451$	−246.891	−0.0257775
$452$	2856.05	0.297207
$453$	0	0
$454$	−3228.75	−0.333772
$455$	5968.75	0.614988
$456$	0	0
$457$	7825.71	0.801031	0.400515	−	0.916290i	$-0.368831\pi$
$457$	7825.71	0.801031	0.400515	+	0.916290i	$0.368831\pi$
$458$	4533.49	0.462525
$459$	0	0
$460$	−7587.60	−0.769073
$461$	−4775.60	−0.482477	−0.241238	−	0.970466i	$-0.577554\pi$
$461$	−4775.60	−0.482477	−0.241238	+	0.970466i	$0.577554\pi$
$462$	0	0
$463$	11518.3	1.15615	0.578077	−	0.815982i	$-0.303803\pi$
$463$	11518.3	1.15615	0.578077	+	0.815982i	$0.303803\pi$
$464$	−9128.12	−0.913280
$465$	0	0
$466$	20030.7	1.99121
$467$	7420.17	0.735256	0.367628	−	0.929973i	$-0.380170\pi$
$467$	7420.17	0.735256	0.367628	+	0.929973i	$0.380170\pi$
$468$	0	0
$469$	4676.14	0.460392
$470$	−11075.5	−1.08697
$471$	0	0
$472$	−658.861	−0.0642512
$473$	−2000.29	−0.194447
$474$	0	0
$475$	2971.89	0.287073
$476$	−1435.51	−0.138228
$477$	0	0
$478$	−2171.26	−0.207764
$479$	−10159.2	−0.969076	−0.484538	−	0.874770i	$-0.661012\pi$
$479$	−10159.2	−0.969076	−0.484538	+	0.874770i	$0.661012\pi$
$480$	0	0
$481$	6371.82	0.604013
$482$	1128.69	0.106661
$483$	0	0
$484$	648.685	0.0609208
$485$	−9430.29	−0.882902
$486$	0	0
$487$	−12344.9	−1.14866	−0.574331	−	0.818623i	$-0.694738\pi$
$487$	−12344.9	−1.14866	−0.574331	+	0.818623i	$0.694738\pi$
$488$	1401.73	0.130028
$489$	0	0
$490$	1805.81	0.166486
$491$	−15344.9	−1.41040	−0.705200	−	0.709009i	$-0.749143\pi$
$491$	−15344.9	−1.41040	−0.705200	+	0.709009i	$0.749143\pi$
$492$	0	0
$493$	4468.13	0.408183
$494$	39347.8	3.58369
$495$	0	0
$496$	−26469.3	−2.39618
$497$	5093.26	0.459686
$498$	0	0
$499$	−5022.76	−0.450601	−0.225300	−	0.974289i	$-0.572336\pi$
$499$	−5022.76	−0.450601	−0.225300	+	0.974289i	$0.572336\pi$
$500$	−8018.41	−0.717188
$501$	0	0
$502$	−12748.1	−1.13342
$503$	−8735.90	−0.774383	−0.387191	−	0.921999i	$-0.626555\pi$
$503$	−8735.90	−0.774383	−0.387191	+	0.921999i	$0.626555\pi$
$504$	0	0
$505$	5377.82	0.473881
$506$	5644.33	0.495891
$507$	0	0
$508$	−12102.2	−1.05699
$509$	21805.5	1.89884	0.949420	−	0.314008i	$-0.101672\pi$
$509$	21805.5	1.89884	0.949420	+	0.314008i	$0.101672\pi$
$510$	0	0
$511$	2918.87	0.252687
$512$	10261.7	0.885760
$513$	0	0
$514$	−1650.16	−0.141606
$515$	−7441.66	−0.636735
$516$	0	0
$517$	3305.83	0.281219
$518$	1927.76	0.163515
$519$	0	0
$520$	8225.08	0.693642
$521$	4732.28	0.397936	0.198968	−	0.980006i	$-0.436241\pi$
$521$	4732.28	0.397936	0.198968	+	0.980006i	$0.436241\pi$
$522$	0	0
$523$	−7511.23	−0.627998	−0.313999	−	0.949423i	$-0.601669\pi$
$523$	−7511.23	−0.627998	−0.313999	+	0.949423i	$0.601669\pi$
$524$	−6295.00	−0.524806
$525$	0	0
$526$	21487.9	1.78121
$527$	12956.5	1.07095
$528$	0	0
$529$	7539.02	0.619629
$530$	1174.13	0.0962283
$531$	0	0
$532$	4776.59	0.389270
$533$	−1898.20	−0.154259
$534$	0	0
$535$	20558.4	1.66134
$536$	6443.82	0.519274
$537$	0	0
$538$	193.187	0.0154812
$539$	−539.000	−0.0430730
$540$	0	0
$541$	−598.410	−0.0475557	−0.0237779	−	0.999717i	$-0.507569\pi$
$541$	−598.410	−0.0475557	−0.0237779	+	0.999717i	$0.507569\pi$
$542$	−24937.0	−1.97626
$543$	0	0
$544$	−7974.95	−0.628535
$545$	15009.3	1.17968
$546$	0	0
$547$	−14042.3	−1.09763	−0.548816	−	0.835943i	$-0.684921\pi$
$547$	−14042.3	−1.09763	−0.548816	+	0.835943i	$0.684921\pi$
$548$	−14421.6	−1.12420
$549$	0	0
$550$	938.801	0.0727829
$551$	−14867.5	−1.14950
$552$	0	0
$553$	−3211.64	−0.246967
$554$	1714.44	0.131479
$555$	0	0
$556$	95.1581	0.00725828
$557$	−4965.87	−0.377757	−0.188878	−	0.982000i	$-0.560485\pi$
$557$	−4965.87	−0.377757	−0.188878	+	0.982000i	$0.560485\pi$
$558$	0	0
$559$	−15379.0	−1.16362
$560$	5515.32	0.416187
$561$	0	0
$562$	8425.60	0.632406
$563$	15362.4	1.14999	0.574997	−	0.818155i	$-0.305003\pi$
$563$	15362.4	1.14999	0.574997	+	0.818155i	$0.305003\pi$
$564$	0	0
$565$	5371.24	0.399947
$566$	−26942.3	−2.00083
$567$	0	0
$568$	7018.62	0.518477
$569$	−17735.3	−1.30668	−0.653341	−	0.757064i	$-0.726633\pi$
$569$	−17735.3	−1.30668	−0.653341	+	0.757064i	$0.726633\pi$
$570$	0	0
$571$	−19818.8	−1.45252	−0.726262	−	0.687418i	$-0.758744\pi$
$571$	−19818.8	−1.45252	−0.726262	+	0.687418i	$0.758744\pi$
$572$	4987.35	0.364566
$573$	0	0
$574$	−574.288	−0.0417602
$575$	3277.63	0.237716
$576$	0	0
$577$	6579.03	0.474677	0.237339	−	0.971427i	$-0.423725\pi$
$577$	6579.03	0.474677	0.237339	+	0.971427i	$0.423725\pi$
$578$	−12609.8	−0.907434
$579$	0	0
$580$	6313.50	0.451989
$581$	2490.16	0.177813
$582$	0	0
$583$	−350.455	−0.0248960
$584$	4022.27	0.285005
$585$	0	0
$586$	−6429.25	−0.453225
$587$	−13901.5	−0.977470	−0.488735	−	0.872432i	$-0.662541\pi$
$587$	−13901.5	−0.977470	−0.488735	+	0.872432i	$0.662541\pi$
$588$	0	0
$589$	−43112.0	−3.01596
$590$	2517.19	0.175646
$591$	0	0
$592$	5887.76	0.408760
$593$	−23928.0	−1.65700	−0.828502	−	0.559986i	$-0.810807\pi$
$593$	−23928.0	−1.65700	−0.828502	+	0.559986i	$0.810807\pi$
$594$	0	0
$595$	−2699.70	−0.186011
$596$	−8137.30	−0.559256
$597$	0	0
$598$	43395.9	2.96754
$599$	−24078.9	−1.64247	−0.821233	−	0.570594i	$-0.806713\pi$
$599$	−24078.9	−1.64247	−0.821233	+	0.570594i	$0.806713\pi$
$600$	0	0
$601$	−11806.6	−0.801336	−0.400668	−	0.916223i	$-0.631222\pi$
$601$	−11806.6	−0.801336	−0.400668	+	0.916223i	$0.631222\pi$
$602$	−4652.83	−0.315009
$603$	0	0
$604$	10447.9	0.703840
$605$	1219.95	0.0819802
$606$	0	0
$607$	1957.71	0.130908	0.0654539	−	0.997856i	$-0.479150\pi$
$607$	1957.71	0.130908	0.0654539	+	0.997856i	$0.479150\pi$
$608$	26536.2	1.77004
$609$	0	0
$610$	−5355.35	−0.355462
$611$	25416.6	1.68289
$612$	0	0
$613$	−16029.2	−1.05614	−0.528069	−	0.849201i	$-0.677084\pi$
$613$	−16029.2	−1.05614	−0.528069	+	0.849201i	$0.677084\pi$
$614$	12676.9	0.833218
$615$	0	0
$616$	−742.754	−0.0485819
$617$	7153.99	0.466789	0.233394	−	0.972382i	$-0.425017\pi$
$617$	7153.99	0.466789	0.233394	+	0.972382i	$0.425017\pi$
$618$	0	0
$619$	−11035.9	−0.716590	−0.358295	−	0.933608i	$-0.616642\pi$
$619$	−11035.9	−0.716590	−0.358295	+	0.933608i	$0.616642\pi$
$620$	18307.6	1.18589
$621$	0	0
$622$	−7252.00	−0.467490
$623$	−8721.78	−0.560884
$624$	0	0
$625$	−12161.3	−0.778321
$626$	−36897.1	−2.35576
$627$	0	0
$628$	−8332.11	−0.529438
$629$	−2882.01	−0.182692
$630$	0	0
$631$	−4311.46	−0.272007	−0.136004	−	0.990708i	$-0.543426\pi$
$631$	−4311.46	−0.272007	−0.136004	+	0.990708i	$0.543426\pi$
$632$	−4425.71	−0.278553
$633$	0	0
$634$	11153.5	0.698678
$635$	−22760.1	−1.42237
$636$	0	0
$637$	−4144.05	−0.257760
$638$	−4696.54	−0.291439
$639$	0	0
$640$	11771.3	0.727035
$641$	−2692.13	−0.165886	−0.0829429	−	0.996554i	$-0.526432\pi$
$641$	−2692.13	−0.165886	−0.0829429	+	0.996554i	$0.526432\pi$
$642$	0	0
$643$	19694.3	1.20788	0.603941	−	0.797029i	$-0.293596\pi$
$643$	19694.3	1.20788	0.603941	+	0.797029i	$0.293596\pi$
$644$	5268.00	0.322342
$645$	0	0
$646$	−17797.2	−1.08393
$647$	21225.5	1.28974	0.644870	−	0.764292i	$-0.276911\pi$
$647$	21225.5	1.28974	0.644870	+	0.764292i	$0.276911\pi$
$648$	0	0
$649$	−751.333	−0.0454428
$650$	7217.88	0.435552
$651$	0	0
$652$	−18617.3	−1.11827
$653$	12929.7	0.774850	0.387425	−	0.921901i	$-0.373365\pi$
$653$	12929.7	0.774850	0.387425	+	0.921901i	$0.373365\pi$
$654$	0	0
$655$	−11838.7	−0.706224
$656$	−1753.99	−0.104393
$657$	0	0
$658$	7689.64	0.455582
$659$	20835.3	1.23161	0.615803	−	0.787900i	$-0.288832\pi$
$659$	20835.3	1.23161	0.615803	+	0.787900i	$0.288832\pi$
$660$	0	0
$661$	−1451.06	−0.0853850	−0.0426925	−	0.999088i	$-0.513594\pi$
$661$	−1451.06	−0.0853850	−0.0426925	+	0.999088i	$0.513594\pi$
$662$	−98.6288	−0.00579051
$663$	0	0
$664$	3431.50	0.200554
$665$	8983.10	0.523834
$666$	0	0
$667$	−16397.0	−0.951867
$668$	11949.2	0.692109
$669$	0	0
$670$	−24618.7	−1.41956
$671$	1598.47	0.0919645
$672$	0	0
$673$	−28986.0	−1.66022	−0.830111	−	0.557598i	$-0.811723\pi$
$673$	−28986.0	−1.66022	−0.830111	+	0.557598i	$0.811723\pi$
$674$	24923.9	1.42438
$675$	0	0
$676$	26566.6	1.51153
$677$	−24818.6	−1.40895	−0.704474	−	0.709730i	$-0.748817\pi$
$677$	−24818.6	−1.40895	−0.704474	+	0.709730i	$0.748817\pi$
$678$	0	0
$679$	6547.36	0.370051
$680$	−3720.24	−0.209801
$681$	0	0
$682$	−13618.8	−0.764650
$683$	7450.70	0.417413	0.208706	−	0.977978i	$-0.433075\pi$
$683$	7450.70	0.417413	0.208706	+	0.977978i	$0.433075\pi$
$684$	0	0
$685$	−27122.0	−1.51282
$686$	−1253.76	−0.0697795
$687$	0	0
$688$	−14210.7	−0.787467
$689$	−2694.44	−0.148984
$690$	0	0
$691$	−28469.1	−1.56731	−0.783657	−	0.621193i	$-0.786648\pi$
$691$	−28469.1	−1.56731	−0.783657	+	0.621193i	$0.786648\pi$
$692$	5404.13	0.296870
$693$	0	0
$694$	−43523.3	−2.38058
$695$	178.959	0.00976736
$696$	0	0
$697$	858.563	0.0466577
$698$	−15910.4	−0.862775
$699$	0	0
$700$	876.208	0.0473108
$701$	−20045.7	−1.08005	−0.540027	−	0.841648i	$-0.681586\pi$
$701$	−20045.7	−1.08005	−0.540027	+	0.841648i	$0.681586\pi$
$702$	0	0
$703$	9589.73	0.514486
$704$	1505.64	0.0806052
$705$	0	0
$706$	4848.02	0.258439
$707$	−3733.77	−0.198618
$708$	0	0
$709$	15326.9	0.811868	0.405934	−	0.913902i	$-0.366946\pi$
$709$	15326.9	0.811868	0.405934	+	0.913902i	$0.366946\pi$
$710$	−26814.8	−1.41738
$711$	0	0
$712$	−12018.8	−0.632618
$713$	−47547.3	−2.49742
$714$	0	0
$715$	9379.47	0.490591
$716$	4000.64	0.208814
$717$	0	0
$718$	30312.9	1.57558
$719$	11023.4	0.571770	0.285885	−	0.958264i	$-0.407712\pi$
$719$	11023.4	0.571770	0.285885	+	0.958264i	$0.407712\pi$
$720$	0	0
$721$	5166.68	0.266875
$722$	34147.8	1.76018
$723$	0	0
$724$	−14945.3	−0.767177
$725$	−2727.26	−0.139707
$726$	0	0
$727$	28755.9	1.46698	0.733492	−	0.679699i	$-0.237889\pi$
$727$	28755.9	1.46698	0.733492	+	0.679699i	$0.237889\pi$
$728$	−5710.60	−0.290726
$729$	0	0
$730$	−15367.2	−0.779129
$731$	6956.00	0.351952
$732$	0	0
$733$	21500.9	1.08343	0.541714	−	0.840563i	$-0.317775\pi$
$733$	21500.9	1.08343	0.541714	+	0.840563i	$0.317775\pi$
$734$	−40308.7	−2.02700
$735$	0	0
$736$	29266.3	1.46572
$737$	7348.21	0.367266
$738$	0	0
$739$	−2598.63	−0.129353	−0.0646767	−	0.997906i	$-0.520602\pi$
$739$	−2598.63	−0.129353	−0.0646767	+	0.997906i	$0.520602\pi$
$740$	−4072.30	−0.202298
$741$	0	0
$742$	−815.189	−0.0403322
$743$	−29920.5	−1.47736	−0.738678	−	0.674058i	$-0.764550\pi$
$743$	−29920.5	−1.47736	−0.738678	+	0.674058i	$0.764550\pi$
$744$	0	0
$745$	−15303.4	−0.752583
$746$	30140.4	1.47925
$747$	0	0
$748$	−2255.80	−0.110268
$749$	−14273.5	−0.696317
$750$	0	0
$751$	17763.1	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$751$	17763.1	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$752$	23485.7	1.13888
$753$	0	0
$754$	−36108.9	−1.74404
$755$	19648.9	0.947147
$756$	0	0
$757$	6472.04	0.310740	0.155370	−	0.987856i	$-0.450343\pi$
$757$	6472.04	0.310740	0.155370	+	0.987856i	$0.450343\pi$
$758$	37150.1	1.78015
$759$	0	0
$760$	12378.9	0.590830
$761$	33720.2	1.60625	0.803126	−	0.595809i	$-0.203169\pi$
$761$	33720.2	1.60625	0.803126	+	0.595809i	$0.203169\pi$
$762$	0	0
$763$	−10420.8	−0.494441
$764$	4888.81	0.231507
$765$	0	0
$766$	−52412.6	−2.47225
$767$	−5776.55	−0.271941
$768$	0	0
$769$	−9361.49	−0.438991	−0.219495	−	0.975614i	$-0.570441\pi$
$769$	−9361.49	−0.438991	−0.219495	+	0.975614i	$0.570441\pi$
$770$	2837.71	0.132810
$771$	0	0
$772$	23881.6	1.11336
$773$	−34886.0	−1.62324	−0.811618	−	0.584188i	$-0.801413\pi$
$773$	−34886.0	−1.62324	−0.811618	+	0.584188i	$0.801413\pi$
$774$	0	0
$775$	−7908.38	−0.366551
$776$	9022.42	0.417379
$777$	0	0
$778$	8710.20	0.401383
$779$	−2856.83	−0.131395
$780$	0	0
$781$	8003.69	0.366702
$782$	−19628.2	−0.897572
$783$	0	0
$784$	−3829.23	−0.174437
$785$	−15669.8	−0.712458
$786$	0	0
$787$	−9526.64	−0.431497	−0.215748	−	0.976449i	$-0.569219\pi$
$787$	−9526.64	−0.431497	−0.215748	+	0.976449i	$0.569219\pi$
$788$	7384.58	0.333839
$789$	0	0
$790$	16908.5	0.761490
$791$	−3729.20	−0.167630
$792$	0	0
$793$	12289.7	0.550339
$794$	−36071.1	−1.61223
$795$	0	0
$796$	−505.852	−0.0225244
$797$	33267.9	1.47856	0.739278	−	0.673400i	$-0.235167\pi$
$797$	33267.9	1.47856	0.739278	+	0.673400i	$0.235167\pi$
$798$	0	0
$799$	−11496.0	−0.509012
$800$	4867.75	0.215126
$801$	0	0
$802$	21489.8	0.946174
$803$	4586.80	0.201575
$804$	0	0
$805$	9907.27	0.433771
$806$	−104707.	−4.57586
$807$	0	0
$808$	−5145.22	−0.224020
$809$	−17949.6	−0.780069	−0.390035	−	0.920800i	$-0.627537\pi$
$809$	−17949.6	−0.780069	−0.390035	+	0.920800i	$0.627537\pi$
$810$	0	0
$811$	10877.9	0.470991	0.235495	−	0.971875i	$-0.424329\pi$
$811$	10877.9	0.470991	0.235495	+	0.971875i	$0.424329\pi$
$812$	−4383.41	−0.189443
$813$	0	0
$814$	3029.34	0.130440
$815$	−35012.7	−1.50484
$816$	0	0
$817$	−23145.7	−0.991147
$818$	20765.0	0.887570
$819$	0	0
$820$	1213.16	0.0516650
$821$	4519.04	0.192102	0.0960509	−	0.995376i	$-0.469379\pi$
$821$	4519.04	0.192102	0.0960509	+	0.995376i	$0.469379\pi$
$822$	0	0
$823$	36987.7	1.56660	0.783300	−	0.621644i	$-0.213535\pi$
$823$	36987.7	1.56660	0.783300	+	0.621644i	$0.213535\pi$
$824$	7119.80	0.301007
$825$	0	0
$826$	−1747.66	−0.0736186
$827$	−15325.3	−0.644392	−0.322196	−	0.946673i	$-0.604421\pi$
$827$	−15325.3	−0.644392	−0.322196	+	0.946673i	$0.604421\pi$
$828$	0	0
$829$	26546.3	1.11217	0.556087	−	0.831124i	$-0.312302\pi$
$829$	26546.3	1.11217	0.556087	+	0.831124i	$0.312302\pi$
$830$	−13110.1	−0.548263
$831$	0	0
$832$	11576.0	0.482362
$833$	1874.37	0.0779630
$834$	0	0
$835$	22472.3	0.931361
$836$	7506.07	0.310530
$837$	0	0
$838$	4015.31	0.165521
$839$	−11906.5	−0.489938	−0.244969	−	0.969531i	$-0.578778\pi$
$839$	−11906.5	−0.489938	−0.244969	+	0.969531i	$0.578778\pi$
$840$	0	0
$841$	−10745.3	−0.440581
$842$	−19245.2	−0.787689
$843$	0	0
$844$	6294.85	0.256727
$845$	49962.5	2.03404
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−2489.75	−0.100824
$849$	0	0
$850$	−3264.68	−0.131738
$851$	10576.3	0.426030
$852$	0	0
$853$	−6859.53	−0.275341	−0.137670	−	0.990478i	$-0.543961\pi$
$853$	−6859.53	−0.275341	−0.137670	+	0.990478i	$0.543961\pi$
$854$	3718.17	0.148985
$855$	0	0
$856$	−19669.2	−0.785372
$857$	−5193.59	−0.207013	−0.103506	−	0.994629i	$-0.533006\pi$
$857$	−5193.59	−0.207013	−0.103506	+	0.994629i	$0.533006\pi$
$858$	0	0
$859$	5265.73	0.209155	0.104578	−	0.994517i	$-0.466651\pi$
$859$	5265.73	0.209155	0.104578	+	0.994517i	$0.466651\pi$
$860$	9828.89	0.389724
$861$	0	0
$862$	−15620.6	−0.617216
$863$	16016.7	0.631767	0.315884	−	0.948798i	$-0.397699\pi$
$863$	16016.7	0.631767	0.315884	+	0.948798i	$0.397699\pi$
$864$	0	0
$865$	10163.3	0.399494
$866$	−31289.8	−1.22780
$867$	0	0
$868$	−12710.8	−0.497042
$869$	−5046.86	−0.197011
$870$	0	0
$871$	56496.0	2.19781
$872$	−14360.1	−0.557678
$873$	0	0
$874$	65311.7	2.52769
$875$	10469.8	0.404507
$876$	0	0
$877$	−11195.5	−0.431065	−0.215533	−	0.976497i	$-0.569149\pi$
$877$	−11195.5	−0.431065	−0.215533	+	0.976497i	$0.569149\pi$
$878$	46291.3	1.77933
$879$	0	0
$880$	8666.92	0.332002
$881$	−45542.5	−1.74162	−0.870809	−	0.491621i	$-0.836405\pi$
$881$	−45542.5	−1.74162	−0.870809	+	0.491621i	$0.836405\pi$
$882$	0	0
$883$	10394.9	0.396167	0.198083	−	0.980185i	$-0.436528\pi$
$883$	10394.9	0.396167	0.198083	+	0.980185i	$0.436528\pi$
$884$	−17343.5	−0.659871
$885$	0	0
$886$	−45211.8	−1.71436
$887$	4020.51	0.152193	0.0760967	−	0.997100i	$-0.475754\pi$
$887$	4020.51	0.152193	0.0760967	+	0.997100i	$0.475754\pi$
$888$	0	0
$889$	15802.1	0.596159
$890$	45918.1	1.72941
$891$	0	0
$892$	474.920	0.0178268
$893$	38252.5	1.43345
$894$	0	0
$895$	7523.82	0.280998
$896$	−8172.72	−0.304723
$897$	0	0
$898$	7649.36	0.284257
$899$	39563.3	1.46775
$900$	0	0
$901$	1218.71	0.0450623
$902$	−902.453	−0.0333131
$903$	0	0
$904$	−5138.93	−0.189069
$905$	−28106.8	−1.03238
$906$	0	0
$907$	−23907.5	−0.875231	−0.437615	−	0.899162i	$-0.644177\pi$
$907$	−23907.5	−0.875231	−0.437615	+	0.899162i	$0.644177\pi$
$908$	−4735.46	−0.173075
$909$	0	0
$910$	21817.4	0.794770
$911$	−40571.4	−1.47551	−0.737755	−	0.675069i	$-0.764114\pi$
$911$	−40571.4	−1.47551	−0.737755	+	0.675069i	$0.764114\pi$
$912$	0	0
$913$	3913.11	0.141846
$914$	28605.1	1.03520
$915$	0	0
$916$	6649.07	0.239838
$917$	8219.50	0.296000
$918$	0	0
$919$	20551.0	0.737667	0.368834	−	0.929495i	$-0.379757\pi$
$919$	20551.0	0.737667	0.368834	+	0.929495i	$0.379757\pi$
$920$	13652.5	0.489248
$921$	0	0
$922$	−17456.1	−0.623521
$923$	61535.6	2.19444
$924$	0	0
$925$	1759.12	0.0625292
$926$	42102.4	1.49414
$927$	0	0
$928$	−24351.9	−0.861413
$929$	−34068.4	−1.20317	−0.601587	−	0.798807i	$-0.705465\pi$
$929$	−34068.4	−1.20317	−0.601587	+	0.798807i	$0.705465\pi$
$930$	0	0
$931$	−6236.89	−0.219555
$932$	29378.1	1.03252
$933$	0	0
$934$	27122.8	0.950197
$935$	−4242.38	−0.148386
$936$	0	0
$937$	15597.9	0.543824	0.271912	−	0.962322i	$-0.412344\pi$
$937$	15597.9	0.543824	0.271912	+	0.962322i	$0.412344\pi$
$938$	17092.6	0.594981
$939$	0	0
$940$	−16244.0	−0.563639
$941$	22855.3	0.791775	0.395887	−	0.918299i	$-0.370437\pi$
$941$	22855.3	0.791775	0.395887	+	0.918299i	$0.370437\pi$
$942$	0	0
$943$	−3150.73	−0.108804
$944$	−5337.71	−0.184034
$945$	0	0
$946$	−7311.59	−0.251290
$947$	−36670.7	−1.25833	−0.629164	−	0.777272i	$-0.716603\pi$
$947$	−36670.7	−1.25833	−0.629164	+	0.777272i	$0.716603\pi$
$948$	0	0
$949$	35265.1	1.20628
$950$	10863.1	0.370994
$951$	0	0
$952$	2582.93	0.0879341
$953$	19922.8	0.677192	0.338596	−	0.940932i	$-0.390048\pi$
$953$	19922.8	0.677192	0.338596	+	0.940932i	$0.390048\pi$
$954$	0	0
$955$	9194.16	0.311535
$956$	−3184.49	−0.107734
$957$	0	0
$958$	−37134.8	−1.25237
$959$	18830.6	0.634068
$960$	0	0
$961$	84932.7	2.85095
$962$	23290.8	0.780587
$963$	0	0
$964$	1655.41	0.0553081
$965$	44913.0	1.49824
$966$	0	0
$967$	24523.8	0.815545	0.407772	−	0.913084i	$-0.366306\pi$
$967$	24523.8	0.815545	0.407772	+	0.913084i	$0.366306\pi$
$968$	−1167.19	−0.0387549
$969$	0	0
$970$	−34470.3	−1.14100
$971$	−4493.82	−0.148520	−0.0742602	−	0.997239i	$-0.523660\pi$
$971$	−4493.82	−0.148520	−0.0742602	+	0.997239i	$0.523660\pi$
$972$	0	0
$973$	−124.250	−0.00409380
$974$	−45123.8	−1.48446
$975$	0	0
$976$	11356.0	0.372436
$977$	18285.3	0.598771	0.299385	−	0.954132i	$-0.403218\pi$
$977$	18285.3	0.598771	0.299385	+	0.954132i	$0.403218\pi$
$978$	0	0
$979$	−13705.7	−0.447431
$980$	2648.51	0.0863300
$981$	0	0
$982$	−56089.9	−1.82271
$983$	25850.8	0.838772	0.419386	−	0.907808i	$-0.362245\pi$
$983$	25850.8	0.838772	0.419386	+	0.907808i	$0.362245\pi$
$984$	0	0
$985$	13887.8	0.449242
$986$	16332.2	0.527509
$987$	0	0
$988$	57709.7	1.85829
$989$	−25526.9	−0.820738
$990$	0	0
$991$	−26842.1	−0.860412	−0.430206	−	0.902731i	$-0.641559\pi$
$991$	−26842.1	−0.860412	−0.430206	+	0.902731i	$0.641559\pi$
$992$	−70614.6	−2.26010
$993$	0	0
$994$	18617.2	0.594068
$995$	−951.331	−0.0303108
$996$	0	0
$997$	−10468.1	−0.332526	−0.166263	−	0.986081i	$-0.553170\pi$
$997$	−10468.1	−0.332526	−0.166263	+	0.986081i	$0.553170\pi$
$998$	−18359.6	−0.582327
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.m.1.3		4
3.2	odd	2		77.4.a.c.1.2	✓	4
12.11	even	2		1232.4.a.w.1.1		4
15.14	odd	2		1925.4.a.q.1.3		4
21.20	even	2		539.4.a.f.1.2		4
33.32	even	2		847.4.a.e.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.c.1.2	✓	4	3.2	odd	2
539.4.a.f.1.2		4	21.20	even	2
693.4.a.m.1.3		4	1.1	even	1	trivial
847.4.a.e.1.3		4	33.32	even	2
1232.4.a.w.1.1		4	12.11	even	2
1925.4.a.q.1.3		4	15.14	odd	2

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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+3.65527 q^{2} +5.36103 q^{4} +10.0822 q^{5} -7.00000 q^{7} -9.64616 q^{8} +O(q^{10})\)	\(q+3.65527 q^{2} +5.36103 q^{4} +10.0822 q^{5} -7.00000 q^{7} -9.64616 q^{8} +36.8533 q^{10} -11.0000 q^{11} -84.5724 q^{13} -25.5869 q^{14} -78.1476 q^{16} +38.2525 q^{17} -127.283 q^{19} +54.0511 q^{20} -40.2080 q^{22} -140.378 q^{23} -23.3486 q^{25} -309.135 q^{26} -37.5272 q^{28} +116.806 q^{29} +338.709 q^{31} -208.482 q^{32} +139.823 q^{34} -70.5756 q^{35} -75.3416 q^{37} -465.256 q^{38} -97.2548 q^{40} +22.4446 q^{41} +181.844 q^{43} -58.9713 q^{44} -513.121 q^{46} -300.530 q^{47} +49.0000 q^{49} -85.3455 q^{50} -453.395 q^{52} +31.8596 q^{53} -110.905 q^{55} +67.5231 q^{56} +426.958 q^{58} +68.3030 q^{59} -145.315 q^{61} +1238.07 q^{62} -136.877 q^{64} -852.679 q^{65} -668.020 q^{67} +205.073 q^{68} -257.973 q^{70} -727.608 q^{71} -416.982 q^{73} -275.394 q^{74} -682.370 q^{76} +77.0000 q^{77} +458.805 q^{79} -787.902 q^{80} +82.0412 q^{82} -355.737 q^{83} +385.671 q^{85} +664.690 q^{86} +106.108 q^{88} +1245.97 q^{89} +592.007 q^{91} -752.571 q^{92} -1098.52 q^{94} -1283.30 q^{95} -935.338 q^{97} +179.108 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8	\( 4 q + 4 q^{2} + 22 q^{4} + 18 q^{5} - 28 q^{7} + 60 q^{8} - 92 q^{10} - 44 q^{11} - 134 q^{13} - 28 q^{14} - 6 q^{16} + 74 q^{17} - 164 q^{19} - 116 q^{20} - 44 q^{22} - 194 q^{23} + 38 q^{25} - 734 q^{26} - 154 q^{28} + 108 q^{29} - 412 q^{31} + 4 q^{32} - 346 q^{34} - 126 q^{35} + 286 q^{37} - 224 q^{38} - 540 q^{40} + 18 q^{41} - 496 q^{43} - 242 q^{44} - 284 q^{46} - 62 q^{47} + 196 q^{49} - 212 q^{50} - 822 q^{52} + 828 q^{53} - 198 q^{55} - 420 q^{56} + 1388 q^{58} + 1224 q^{59} - 350 q^{61} + 878 q^{62} - 718 q^{64} + 396 q^{65} - 1498 q^{67} - 1058 q^{68} + 644 q^{70} - 2326 q^{71} - 1630 q^{73} + 1156 q^{74} - 3152 q^{76} + 308 q^{77} - 1020 q^{79} - 3072 q^{80} + 2118 q^{82} + 1920 q^{83} + 2008 q^{85} - 1056 q^{86} - 660 q^{88} - 1550 q^{89} + 938 q^{91} - 2592 q^{92} - 1042 q^{94} - 2332 q^{95} - 2202 q^{97} + 196 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 22 * q^4 + 18 * q^5 - 28 * q^7 + 60 * q^8 - 92 * q^10 - 44 * q^11 - 134 * q^13 - 28 * q^14 - 6 * q^16 + 74 * q^17 - 164 * q^19 - 116 * q^20 - 44 * q^22 - 194 * q^23 + 38 * q^25 - 734 * q^26 - 154 * q^28 + 108 * q^29 - 412 * q^31 + 4 * q^32 - 346 * q^34 - 126 * q^35 + 286 * q^37 - 224 * q^38 - 540 * q^40 + 18 * q^41 - 496 * q^43 - 242 * q^44 - 284 * q^46 - 62 * q^47 + 196 * q^49 - 212 * q^50 - 822 * q^52 + 828 * q^53 - 198 * q^55 - 420 * q^56 + 1388 * q^58 + 1224 * q^59 - 350 * q^61 + 878 * q^62 - 718 * q^64 + 396 * q^65 - 1498 * q^67 - 1058 * q^68 + 644 * q^70 - 2326 * q^71 - 1630 * q^73 + 1156 * q^74 - 3152 * q^76 + 308 * q^77 - 1020 * q^79 - 3072 * q^80 + 2118 * q^82 + 1920 * q^83 + 2008 * q^85 - 1056 * q^86 - 660 * q^88 - 1550 * q^89 + 938 * q^91 - 2592 * q^92 - 1042 * q^94 - 2332 * q^95 - 2202 * q^97 + 196 * q^98

Introduction

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