LMFDB - Embedded newform 6223.2.a.n.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6223,2,Mod(1,6223)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6223, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6223.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.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:	$49.6909051778$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6223.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.26367 q^{2} -2.09890 q^{3} +3.12419 q^{4} +3.74304 q^{5} +4.75121 q^{6} -2.54480 q^{8} +1.40538 q^{9} +O(q^{10})$	\(q-2.26367 q^{2} -2.09890 q^{3} +3.12419 q^{4} +3.74304 q^{5} +4.75121 q^{6} -2.54480 q^{8} +1.40538 q^{9} -8.47301 q^{10} -3.91073 q^{11} -6.55737 q^{12} +4.91696 q^{13} -7.85627 q^{15} -0.487802 q^{16} -0.561032 q^{17} -3.18131 q^{18} +7.13571 q^{19} +11.6940 q^{20} +8.85260 q^{22} +0.855423 q^{23} +5.34128 q^{24} +9.01037 q^{25} -11.1304 q^{26} +3.34695 q^{27} +1.72568 q^{29} +17.7840 q^{30} -7.32042 q^{31} +6.19382 q^{32} +8.20823 q^{33} +1.26999 q^{34} +4.39067 q^{36} -6.94388 q^{37} -16.1529 q^{38} -10.3202 q^{39} -9.52530 q^{40} +6.52441 q^{41} -6.92915 q^{43} -12.2179 q^{44} +5.26039 q^{45} -1.93639 q^{46} -2.00307 q^{47} +1.02385 q^{48} -20.3965 q^{50} +1.17755 q^{51} +15.3615 q^{52} +3.22678 q^{53} -7.57638 q^{54} -14.6380 q^{55} -14.9771 q^{57} -3.90636 q^{58} +7.27472 q^{59} -24.5445 q^{60} +13.1925 q^{61} +16.5710 q^{62} -13.0452 q^{64} +18.4044 q^{65} -18.5807 q^{66} -15.4796 q^{67} -1.75277 q^{68} -1.79545 q^{69} +5.21793 q^{71} -3.57641 q^{72} +14.6228 q^{73} +15.7186 q^{74} -18.9119 q^{75} +22.2933 q^{76} +23.3615 q^{78} +1.71068 q^{79} -1.82586 q^{80} -11.2410 q^{81} -14.7691 q^{82} -7.94664 q^{83} -2.09997 q^{85} +15.6853 q^{86} -3.62202 q^{87} +9.95204 q^{88} -11.1336 q^{89} -11.9078 q^{90} +2.67251 q^{92} +15.3648 q^{93} +4.53430 q^{94} +26.7093 q^{95} -13.0002 q^{96} +1.72668 q^{97} -5.49606 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 6 q^{2} + 46 q^{4} + 30 q^{8} + 66 q^{9}+O(q^{10}) $ 34 * q + 6 * q^2 + 46 * q^4 + 30 * q^8 + 66 * q^9	$ 34 q + 6 q^{2} + 46 q^{4} + 30 q^{8} + 66 q^{9} + 16 q^{11} + 12 q^{15} + 54 q^{16} + 48 q^{18} + 30 q^{22} + 46 q^{23} + 80 q^{25} + 28 q^{29} - 2 q^{30} + 36 q^{32} + 98 q^{36} + 52 q^{37} - 2 q^{39} + 10 q^{43} + 16 q^{44} - 36 q^{46} + 74 q^{50} + 56 q^{51} + 64 q^{53} + 44 q^{57} + 32 q^{58} - 132 q^{60} + 74 q^{64} + 108 q^{65} + 14 q^{67} + 6 q^{71} + 172 q^{72} - 6 q^{74} + 52 q^{78} + 46 q^{81} - 2 q^{85} + 96 q^{86} - 170 q^{88} + 130 q^{92} + 6 q^{93} + 14 q^{95} + 2 q^{99}+O(q^{100}) $ 34 * q + 6 * q^2 + 46 * q^4 + 30 * q^8 + 66 * q^9 + 16 * q^11 + 12 * q^15 + 54 * q^16 + 48 * q^18 + 30 * q^22 + 46 * q^23 + 80 * q^25 + 28 * q^29 - 2 * q^30 + 36 * q^32 + 98 * q^36 + 52 * q^37 - 2 * q^39 + 10 * q^43 + 16 * q^44 - 36 * q^46 + 74 * q^50 + 56 * q^51 + 64 * q^53 + 44 * q^57 + 32 * q^58 - 132 * q^60 + 74 * q^64 + 108 * q^65 + 14 * q^67 + 6 * q^71 + 172 * q^72 - 6 * q^74 + 52 * q^78 + 46 * q^81 - 2 * q^85 + 96 * q^86 - 170 * q^88 + 130 * q^92 + 6 * q^93 + 14 * q^95 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.26367	−1.60066	−0.800328	−	0.599563i	$-0.795341\pi$
$2$	−2.26367	−1.60066	−0.800328	+	0.599563i	$0.795341\pi$
$3$	−2.09890	−1.21180	−0.605900	−	0.795541i	$-0.707187\pi$
$3$	−2.09890	−1.21180	−0.605900	+	0.795541i	$0.707187\pi$
$4$	3.12419	1.56210
$5$	3.74304	1.67394	0.836970	−	0.547249i	$-0.184325\pi$
$5$	3.74304	1.67394	0.836970	+	0.547249i	$0.184325\pi$
$6$	4.75121	1.93967
$7$	0	0
$7$	0	0
$8$	−2.54480	−0.899723
$9$	1.40538	0.468460
$10$	−8.47301	−2.67940
$11$	−3.91073	−1.17913	−0.589565	−	0.807721i	$-0.700701\pi$
$11$	−3.91073	−1.17913	−0.589565	+	0.807721i	$0.700701\pi$
$12$	−6.55737	−1.89295
$13$	4.91696	1.36372	0.681859	−	0.731484i	$-0.261172\pi$
$13$	4.91696	1.36372	0.681859	+	0.731484i	$0.261172\pi$
$14$	0	0
$15$	−7.85627	−2.02848
$16$	−0.487802	−0.121950
$17$	−0.561032	−0.136070	−0.0680352	−	0.997683i	$-0.521673\pi$
$17$	−0.561032	−0.136070	−0.0680352	+	0.997683i	$0.521673\pi$
$18$	−3.18131	−0.749842
$19$	7.13571	1.63704	0.818522	−	0.574475i	$-0.194794\pi$
$19$	7.13571	1.63704	0.818522	+	0.574475i	$0.194794\pi$
$20$	11.6940	2.61486
$21$	0	0
$22$	8.85260	1.88738
$23$	0.855423	0.178368	0.0891840	−	0.996015i	$-0.471574\pi$
$23$	0.855423	0.178368	0.0891840	+	0.996015i	$0.471574\pi$
$24$	5.34128	1.09028
$25$	9.01037	1.80207
$26$	−11.1304	−2.18284
$27$	3.34695	0.644121
$28$	0	0
$29$	1.72568	0.320450	0.160225	−	0.987081i	$-0.448778\pi$
$29$	1.72568	0.320450	0.160225	+	0.987081i	$0.448778\pi$
$30$	17.7840	3.24690
$31$	−7.32042	−1.31479	−0.657393	−	0.753548i	$-0.728341\pi$
$31$	−7.32042	−1.31479	−0.657393	+	0.753548i	$0.728341\pi$
$32$	6.19382	1.09492
$33$	8.20823	1.42887
$34$	1.26999	0.217802
$35$	0	0
$36$	4.39067	0.731779
$37$	−6.94388	−1.14157	−0.570783	−	0.821101i	$-0.693360\pi$
$37$	−6.94388	−1.14157	−0.570783	+	0.821101i	$0.693360\pi$
$38$	−16.1529	−2.62034
$39$	−10.3202	−1.65255
$40$	−9.52530	−1.50608
$41$	6.52441	1.01894	0.509471	−	0.860488i	$-0.329841\pi$
$41$	6.52441	1.01894	0.509471	+	0.860488i	$0.329841\pi$
$42$	0	0
$43$	−6.92915	−1.05669	−0.528343	−	0.849031i	$-0.677186\pi$
$43$	−6.92915	−1.05669	−0.528343	+	0.849031i	$0.677186\pi$
$44$	−12.2179	−1.84192
$45$	5.26039	0.784173
$46$	−1.93639	−0.285506
$47$	−2.00307	−0.292178	−0.146089	−	0.989271i	$-0.546669\pi$
$47$	−2.00307	−0.292178	−0.146089	+	0.989271i	$0.546669\pi$
$48$	1.02385	0.147780
$49$	0	0
$50$	−20.3965	−2.88450
$51$	1.17755	0.164890
$52$	15.3615	2.13026
$53$	3.22678	0.443232	0.221616	−	0.975134i	$-0.428867\pi$
$53$	3.22678	0.443232	0.221616	+	0.975134i	$0.428867\pi$
$54$	−7.57638	−1.03102
$55$	−14.6380	−1.97379
$56$	0	0
$57$	−14.9771	−1.98377
$58$	−3.90636	−0.512930
$59$	7.27472	0.947087	0.473544	−	0.880770i	$-0.342975\pi$
$59$	7.27472	0.947087	0.473544	+	0.880770i	$0.342975\pi$
$60$	−24.5445	−3.16868
$61$	13.1925	1.68912	0.844561	−	0.535459i	$-0.179861\pi$
$61$	13.1925	1.68912	0.844561	+	0.535459i	$0.179861\pi$
$62$	16.5710	2.10452
$63$	0	0
$64$	−13.0452	−1.63064
$65$	18.4044	2.28278
$66$	−18.5807	−2.28713
$67$	−15.4796	−1.89113	−0.945566	−	0.325430i	$-0.894491\pi$
$67$	−15.4796	−1.89113	−0.945566	+	0.325430i	$0.894491\pi$
$68$	−1.75277	−0.212555
$69$	−1.79545	−0.216146
$70$	0	0
$71$	5.21793	0.619254	0.309627	−	0.950858i	$-0.399796\pi$
$71$	5.21793	0.619254	0.309627	+	0.950858i	$0.399796\pi$
$72$	−3.57641	−0.421484
$73$	14.6228	1.71147	0.855736	−	0.517413i	$-0.173105\pi$
$73$	14.6228	1.71147	0.855736	+	0.517413i	$0.173105\pi$
$74$	15.7186	1.82725
$75$	−18.9119	−2.18375
$76$	22.2933	2.55722
$77$	0	0
$78$	23.3615	2.64517
$79$	1.71068	0.192466	0.0962331	−	0.995359i	$-0.469321\pi$
$79$	1.71068	0.192466	0.0962331	+	0.995359i	$0.469321\pi$
$80$	−1.82586	−0.204138
$81$	−11.2410	−1.24901
$82$	−14.7691	−1.63097
$83$	−7.94664	−0.872258	−0.436129	−	0.899884i	$-0.643651\pi$
$83$	−7.94664	−0.872258	−0.436129	+	0.899884i	$0.643651\pi$
$84$	0	0
$85$	−2.09997	−0.227773
$86$	15.6853	1.69139
$87$	−3.62202	−0.388321
$88$	9.95204	1.06089
$89$	−11.1336	−1.18016	−0.590082	−	0.807344i	$-0.700904\pi$
$89$	−11.1336	−1.18016	−0.590082	+	0.807344i	$0.700904\pi$
$90$	−11.9078	−1.25519
$91$	0	0
$92$	2.67251	0.278628
$93$	15.3648	1.59326
$94$	4.53430	0.467677
$95$	26.7093	2.74031
$96$	−13.0002	−1.32683
$97$	1.72668	0.175318	0.0876589	−	0.996151i	$-0.472061\pi$
$97$	1.72668	0.175318	0.0876589	+	0.996151i	$0.472061\pi$
$98$	0	0
$99$	−5.49606	−0.552375
$100$	28.1501	2.81501
$101$	3.76385	0.374517	0.187259	−	0.982311i	$-0.440040\pi$
$101$	3.76385	0.374517	0.187259	+	0.982311i	$0.440040\pi$
$102$	−2.66558	−0.263932
$103$	6.66634	0.656854	0.328427	−	0.944529i	$-0.393482\pi$
$103$	6.66634	0.656854	0.328427	+	0.944529i	$0.393482\pi$
$104$	−12.5127	−1.22697
$105$	0	0
$106$	−7.30435	−0.709461
$107$	14.7957	1.43035	0.715176	−	0.698944i	$-0.246347\pi$
$107$	14.7957	1.43035	0.715176	+	0.698944i	$0.246347\pi$
$108$	10.4565	1.00618
$109$	−2.87124	−0.275015	−0.137508	−	0.990501i	$-0.543909\pi$
$109$	−2.87124	−0.275015	−0.137508	+	0.990501i	$0.543909\pi$
$110$	33.1357	3.15936
$111$	14.5745	1.38335
$112$	0	0
$113$	13.4763	1.26774	0.633871	−	0.773439i	$-0.281465\pi$
$113$	13.4763	1.26774	0.633871	+	0.773439i	$0.281465\pi$
$114$	33.9033	3.17533
$115$	3.20188	0.298577
$116$	5.39134	0.500574
$117$	6.91018	0.638847
$118$	−16.4675	−1.51596
$119$	0	0
$120$	19.9926	1.82507
$121$	4.29383	0.390348
$122$	−29.8634	−2.70370
$123$	−13.6941	−1.23475
$124$	−22.8704	−2.05382
$125$	15.0110	1.34262
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	17.1423	1.51518
$129$	14.5436	1.28049
$130$	−41.6614	−3.65395
$131$	18.1348	1.58445	0.792224	−	0.610230i	$-0.208923\pi$
$131$	18.1348	1.58445	0.792224	+	0.610230i	$0.208923\pi$
$132$	25.6441	2.23203
$133$	0	0
$134$	35.0406	3.02705
$135$	12.5278	1.07822
$136$	1.42772	0.122426
$137$	13.7048	1.17088	0.585442	−	0.810715i	$-0.300921\pi$
$137$	13.7048	1.17088	0.585442	+	0.810715i	$0.300921\pi$
$138$	4.06430	0.345976
$139$	−3.69582	−0.313476	−0.156738	−	0.987640i	$-0.550098\pi$
$139$	−3.69582	−0.313476	−0.156738	+	0.987640i	$0.550098\pi$
$140$	0	0
$141$	4.20425	0.354062
$142$	−11.8117	−0.991212
$143$	−19.2289	−1.60800
$144$	−0.685546	−0.0571289
$145$	6.45928	0.536414
$146$	−33.1012	−2.73948
$147$	0	0
$148$	−21.6940	−1.78324
$149$	−1.11364	−0.0912333	−0.0456167	−	0.998959i	$-0.514525\pi$
$149$	−1.11364	−0.0912333	−0.0456167	+	0.998959i	$0.514525\pi$
$150$	42.8102	3.49543
$151$	−13.4324	−1.09311	−0.546556	−	0.837423i	$-0.684061\pi$
$151$	−13.4324	−1.09311	−0.546556	+	0.837423i	$0.684061\pi$
$152$	−18.1590	−1.47289
$153$	−0.788463	−0.0637434
$154$	0	0
$155$	−27.4006	−2.20087
$156$	−32.2423	−2.58145
$157$	8.47896	0.676695	0.338347	−	0.941021i	$-0.390132\pi$
$157$	8.47896	0.676695	0.338347	+	0.941021i	$0.390132\pi$
$158$	−3.87241	−0.308072
$159$	−6.77268	−0.537108
$160$	23.1837	1.83284
$161$	0	0
$162$	25.4460	1.99923
$163$	7.00001	0.548283	0.274142	−	0.961689i	$-0.411606\pi$
$163$	7.00001	0.548283	0.274142	+	0.961689i	$0.411606\pi$
$164$	20.3835	1.59169
$165$	30.7238	2.39184
$166$	17.9886	1.39618
$167$	3.19454	0.247201	0.123600	−	0.992332i	$-0.460556\pi$
$167$	3.19454	0.247201	0.123600	+	0.992332i	$0.460556\pi$
$168$	0	0
$169$	11.1765	0.859727
$170$	4.75363	0.364587
$171$	10.0284	0.766889
$172$	−21.6480	−1.65064
$173$	7.84021	0.596080	0.298040	−	0.954553i	$-0.403667\pi$
$173$	7.84021	0.596080	0.298040	+	0.954553i	$0.403667\pi$
$174$	8.19905	0.621568
$175$	0	0
$176$	1.90766	0.143795
$177$	−15.2689	−1.14768
$178$	25.2029	1.88903
$179$	−3.49905	−0.261532	−0.130766	−	0.991413i	$-0.541744\pi$
$179$	−3.49905	−0.261532	−0.130766	+	0.991413i	$0.541744\pi$
$180$	16.4345	1.22495
$181$	−21.2030	−1.57601	−0.788003	−	0.615671i	$-0.788885\pi$
$181$	−21.2030	−1.57601	−0.788003	+	0.615671i	$0.788885\pi$
$182$	0	0
$183$	−27.6897	−2.04688
$184$	−2.17688	−0.160482
$185$	−25.9912	−1.91091
$186$	−34.7809	−2.55026
$187$	2.19405	0.160445
$188$	−6.25799	−0.456411
$189$	0	0
$190$	−60.4609	−4.38630
$191$	8.93447	0.646475	0.323238	−	0.946318i	$-0.395229\pi$
$191$	8.93447	0.646475	0.323238	+	0.946318i	$0.395229\pi$
$192$	27.3805	1.97602
$193$	17.2840	1.24413	0.622065	−	0.782965i	$-0.286294\pi$
$193$	17.2840	1.24413	0.622065	+	0.782965i	$0.286294\pi$
$194$	−3.90863	−0.280623
$195$	−38.6289	−2.76627
$196$	0	0
$197$	−22.7833	−1.62325	−0.811623	−	0.584182i	$-0.801416\pi$
$197$	−22.7833	−1.62325	−0.811623	+	0.584182i	$0.801416\pi$
$198$	12.4413	0.884162
$199$	−16.7823	−1.18967	−0.594833	−	0.803849i	$-0.702782\pi$
$199$	−16.7823	−1.18967	−0.594833	+	0.803849i	$0.702782\pi$
$200$	−22.9296	−1.62137
$201$	32.4901	2.29167
$202$	−8.52011	−0.599473
$203$	0	0
$204$	3.67889	0.257574
$205$	24.4211	1.70565
$206$	−15.0904	−1.05140
$207$	1.20219	0.0835582
$208$	−2.39850	−0.166306
$209$	−27.9059	−1.93029
$210$	0	0
$211$	27.9010	1.92078	0.960392	−	0.278653i	$-0.0898880\pi$
$211$	27.9010	1.92078	0.960392	+	0.278653i	$0.0898880\pi$
$212$	10.0811	0.692371
$213$	−10.9519	−0.750412
$214$	−33.4925	−2.28950
$215$	−25.9361	−1.76883
$216$	−8.51732	−0.579530
$217$	0	0
$218$	6.49954	0.440205
$219$	−30.6918	−2.07396
$220$	−45.7321	−3.08325
$221$	−2.75857	−0.185562
$222$	−32.9918	−2.21427
$223$	10.2128	0.683899	0.341949	−	0.939718i	$-0.388913\pi$
$223$	10.2128	0.683899	0.341949	+	0.939718i	$0.388913\pi$
$224$	0	0
$225$	12.6630	0.844198
$226$	−30.5058	−2.02922
$227$	−16.0449	−1.06494	−0.532468	−	0.846450i	$-0.678735\pi$
$227$	−16.0449	−1.06494	−0.532468	+	0.846450i	$0.678735\pi$
$228$	−46.7915	−3.09884
$229$	−16.2559	−1.07422	−0.537110	−	0.843512i	$-0.680484\pi$
$229$	−16.2559	−1.07422	−0.537110	+	0.843512i	$0.680484\pi$
$230$	−7.24800	−0.477919
$231$	0	0
$232$	−4.39150	−0.288316
$233$	−0.225807	−0.0147931	−0.00739655	−	0.999973i	$-0.502354\pi$
$233$	−0.225807	−0.0147931	−0.00739655	+	0.999973i	$0.502354\pi$
$234$	−15.6424	−1.02257
$235$	−7.49759	−0.489089
$236$	22.7276	1.47944
$237$	−3.59054	−0.233231
$238$	0	0
$239$	−19.0744	−1.23382	−0.616909	−	0.787035i	$-0.711615\pi$
$239$	−19.0744	−1.23382	−0.616909	+	0.787035i	$0.711615\pi$
$240$	3.83230	0.247374
$241$	−24.8631	−1.60157	−0.800785	−	0.598952i	$-0.795584\pi$
$241$	−24.8631	−1.60157	−0.800785	+	0.598952i	$0.795584\pi$
$242$	−9.71980	−0.624813
$243$	13.5530	0.869424
$244$	41.2158	2.63857
$245$	0	0
$246$	30.9989	1.97641
$247$	35.0860	2.23247
$248$	18.6290	1.18294
$249$	16.6792	1.05700
$250$	−33.9798	−2.14907
$251$	4.02268	0.253909	0.126955	−	0.991909i	$-0.459480\pi$
$251$	4.02268	0.253909	0.126955	+	0.991909i	$0.459480\pi$
$252$	0	0
$253$	−3.34533	−0.210319
$254$	2.26367	0.142035
$255$	4.40762	0.276016
$256$	−12.7141	−0.794630
$257$	6.03878	0.376689	0.188344	−	0.982103i	$-0.439688\pi$
$257$	6.03878	0.376689	0.188344	+	0.982103i	$0.439688\pi$
$258$	−32.9219	−2.04963
$259$	0	0
$260$	57.4988	3.56593
$261$	2.42523	0.150118
$262$	−41.0513	−2.53615
$263$	−25.1152	−1.54867	−0.774334	−	0.632777i	$-0.781915\pi$
$263$	−25.1152	−1.54867	−0.774334	+	0.632777i	$0.781915\pi$
$264$	−20.8883	−1.28559
$265$	12.0780	0.741943
$266$	0	0
$267$	23.3684	1.43012
$268$	−48.3612	−2.95413
$269$	−8.68453	−0.529505	−0.264753	−	0.964316i	$-0.585290\pi$
$269$	−8.68453	−0.529505	−0.264753	+	0.964316i	$0.585290\pi$
$270$	−28.3587	−1.72586
$271$	−9.52309	−0.578487	−0.289243	−	0.957256i	$-0.593404\pi$
$271$	−9.52309	−0.578487	−0.289243	+	0.957256i	$0.593404\pi$
$272$	0.273673	0.0165938
$273$	0	0
$274$	−31.0232	−1.87418
$275$	−35.2371	−2.12488
$276$	−5.60932	−0.337642
$277$	27.9300	1.67815	0.839077	−	0.544013i	$-0.183096\pi$
$277$	27.9300	1.67815	0.839077	+	0.544013i	$0.183096\pi$
$278$	8.36612	0.501766
$279$	−10.2880	−0.615924
$280$	0	0
$281$	−0.978923	−0.0583976	−0.0291988	−	0.999574i	$-0.509296\pi$
$281$	−0.978923	−0.0583976	−0.0291988	+	0.999574i	$0.509296\pi$
$282$	−9.51703	−0.566731
$283$	13.8023	0.820460	0.410230	−	0.911982i	$-0.365448\pi$
$283$	13.8023	0.820460	0.410230	+	0.911982i	$0.365448\pi$
$284$	16.3018	0.967335
$285$	−56.0601	−3.32071
$286$	43.5278	2.57386
$287$	0	0
$288$	8.70467	0.512927
$289$	−16.6852	−0.981485
$290$	−14.6217	−0.858613
$291$	−3.62413	−0.212450
$292$	45.6845	2.67349
$293$	8.59811	0.502307	0.251154	−	0.967947i	$-0.419190\pi$
$293$	8.59811	0.502307	0.251154	+	0.967947i	$0.419190\pi$
$294$	0	0
$295$	27.2296	1.58537
$296$	17.6708	1.02709
$297$	−13.0890	−0.759502
$298$	2.52092	0.146033
$299$	4.20608	0.243244
$300$	−59.0843	−3.41123
$301$	0	0
$302$	30.4064	1.74969
$303$	−7.89995	−0.453840
$304$	−3.48081	−0.199638
$305$	49.3800	2.82749
$306$	1.78482	0.102031
$307$	6.06753	0.346292	0.173146	−	0.984896i	$-0.444607\pi$
$307$	6.06753	0.346292	0.173146	+	0.984896i	$0.444607\pi$
$308$	0	0
$309$	−13.9920	−0.795976
$310$	62.0259	3.52284
$311$	1.67249	0.0948380	0.0474190	−	0.998875i	$-0.484900\pi$
$311$	1.67249	0.0948380	0.0474190	+	0.998875i	$0.484900\pi$
$312$	26.2628	1.48684
$313$	−12.6201	−0.713328	−0.356664	−	0.934233i	$-0.616086\pi$
$313$	−12.6201	−0.713328	−0.356664	+	0.934233i	$0.616086\pi$
$314$	−19.1936	−1.08315
$315$	0	0
$316$	5.34449	0.300651
$317$	−1.70410	−0.0957119	−0.0478559	−	0.998854i	$-0.515239\pi$
$317$	−1.70410	−0.0957119	−0.0478559	+	0.998854i	$0.515239\pi$
$318$	15.3311	0.859725
$319$	−6.74865	−0.377852
$320$	−48.8286	−2.72960
$321$	−31.0546	−1.73330
$322$	0	0
$323$	−4.00336	−0.222753
$324$	−35.1192	−1.95107
$325$	44.3036	2.45752
$326$	−15.8457	−0.877613
$327$	6.02645	0.333264
$328$	−16.6033	−0.916765
$329$	0	0
$330$	−69.5484	−3.82851
$331$	−29.9165	−1.64436	−0.822180	−	0.569228i	$-0.807242\pi$
$331$	−29.9165	−1.64436	−0.822180	+	0.569228i	$0.807242\pi$
$332$	−24.8269	−1.36255
$333$	−9.75878	−0.534778
$334$	−7.23137	−0.395683
$335$	−57.9407	−3.16564
$336$	0	0
$337$	20.1292	1.09651	0.548254	−	0.836312i	$-0.315293\pi$
$337$	20.1292	1.09651	0.548254	+	0.836312i	$0.315293\pi$
$338$	−25.2998	−1.37613
$339$	−28.2853	−1.53625
$340$	−6.56070	−0.355804
$341$	28.6282	1.55030
$342$	−22.7009	−1.22752
$343$	0	0
$344$	17.6333	0.950724
$345$	−6.72043	−0.361816
$346$	−17.7476	−0.954118
$347$	31.2534	1.67777	0.838886	−	0.544307i	$-0.183207\pi$
$347$	31.2534	1.67777	0.838886	+	0.544307i	$0.183207\pi$
$348$	−11.3159	−0.606595
$349$	26.8543	1.43748	0.718740	−	0.695279i	$-0.244719\pi$
$349$	26.8543	1.43748	0.718740	+	0.695279i	$0.244719\pi$
$350$	0	0
$351$	16.4568	0.878399
$352$	−24.2224	−1.29106
$353$	15.2670	0.812581	0.406291	−	0.913744i	$-0.366822\pi$
$353$	15.2670	0.812581	0.406291	+	0.913744i	$0.366822\pi$
$354$	34.5637	1.83704
$355$	19.5309	1.03659
$356$	−34.7836	−1.84353
$357$	0	0
$358$	7.92070	0.418622
$359$	−11.1658	−0.589308	−0.294654	−	0.955604i	$-0.595204\pi$
$359$	−11.1658	−0.589308	−0.294654	+	0.955604i	$0.595204\pi$
$360$	−13.3867	−0.705538
$361$	31.9184	1.67991
$362$	47.9966	2.52264
$363$	−9.01231	−0.473024
$364$	0	0
$365$	54.7338	2.86490
$366$	62.6802	3.27635
$367$	17.8610	0.932338	0.466169	−	0.884696i	$-0.345634\pi$
$367$	17.8610	0.932338	0.466169	+	0.884696i	$0.345634\pi$
$368$	−0.417277	−0.0217521
$369$	9.16927	0.477333
$370$	58.8355	3.05871
$371$	0	0
$372$	48.0027	2.48882
$373$	−1.19238	−0.0617391	−0.0308695	−	0.999523i	$-0.509828\pi$
$373$	−1.19238	−0.0617391	−0.0308695	+	0.999523i	$0.509828\pi$
$374$	−4.96659	−0.256816
$375$	−31.5065	−1.62699
$376$	5.09743	0.262880
$377$	8.48507	0.437003
$378$	0	0
$379$	30.8748	1.58593	0.792967	−	0.609264i	$-0.208535\pi$
$379$	30.8748	1.58593	0.792967	+	0.609264i	$0.208535\pi$
$380$	83.4449	4.28063
$381$	2.09890	0.107530
$382$	−20.2247	−1.03478
$383$	6.78915	0.346910	0.173455	−	0.984842i	$-0.444507\pi$
$383$	6.78915	0.346910	0.173455	+	0.984842i	$0.444507\pi$
$384$	−35.9799	−1.83609
$385$	0	0
$386$	−39.1253	−1.99142
$387$	−9.73808	−0.495014
$388$	5.39448	0.273863
$389$	−16.9880	−0.861325	−0.430663	−	0.902513i	$-0.641720\pi$
$389$	−16.9880	−0.861325	−0.430663	+	0.902513i	$0.641720\pi$
$390$	87.4431	4.42785
$391$	−0.479920	−0.0242706
$392$	0	0
$393$	−38.0632	−1.92003
$394$	51.5739	2.59826
$395$	6.40314	0.322177
$396$	−17.1708	−0.862863
$397$	−30.1352	−1.51244	−0.756222	−	0.654316i	$-0.772957\pi$
$397$	−30.1352	−1.51244	−0.756222	+	0.654316i	$0.772957\pi$
$398$	37.9896	1.90424
$399$	0	0
$400$	−4.39527	−0.219764
$401$	23.1923	1.15817	0.579084	−	0.815268i	$-0.303410\pi$
$401$	23.1923	1.15817	0.579084	+	0.815268i	$0.303410\pi$
$402$	−73.5468	−3.66818
$403$	−35.9942	−1.79300
$404$	11.7590	0.585032
$405$	−42.0757	−2.09076
$406$	0	0
$407$	27.1557	1.34606
$408$	−2.99663	−0.148355
$409$	−1.05309	−0.0520720	−0.0260360	−	0.999661i	$-0.508288\pi$
$409$	−1.05309	−0.0520720	−0.0260360	+	0.999661i	$0.508288\pi$
$410$	−55.2814	−2.73015
$411$	−28.7651	−1.41888
$412$	20.8269	1.02607
$413$	0	0
$414$	−2.72137	−0.133748
$415$	−29.7446	−1.46011
$416$	30.4548	1.49317
$417$	7.75716	0.379870
$418$	63.1696	3.08973
$419$	−10.2222	−0.499389	−0.249695	−	0.968325i	$-0.580330\pi$
$419$	−10.2222	−0.499389	−0.249695	+	0.968325i	$0.580330\pi$
$420$	0	0
$421$	−9.53999	−0.464951	−0.232475	−	0.972602i	$-0.574682\pi$
$421$	−9.53999	−0.464951	−0.232475	+	0.972602i	$0.574682\pi$
$422$	−63.1586	−3.07451
$423$	−2.81508	−0.136874
$424$	−8.21151	−0.398786
$425$	−5.05511	−0.245209
$426$	24.7915	1.20115
$427$	0	0
$428$	46.2246	2.23435
$429$	40.3595	1.94858
$430$	58.7107	2.83128
$431$	23.2020	1.11760	0.558800	−	0.829303i	$-0.311262\pi$
$431$	23.2020	1.11760	0.558800	+	0.829303i	$0.311262\pi$
$432$	−1.63265	−0.0785508
$433$	−33.6619	−1.61769	−0.808844	−	0.588024i	$-0.799906\pi$
$433$	−33.6619	−1.61769	−0.808844	+	0.588024i	$0.799906\pi$
$434$	0	0
$435$	−13.5574	−0.650026
$436$	−8.97032	−0.429601
$437$	6.10405	0.291996
$438$	69.4761	3.31970
$439$	−20.5168	−0.979211	−0.489606	−	0.871944i	$-0.662859\pi$
$439$	−20.5168	−0.979211	−0.489606	+	0.871944i	$0.662859\pi$
$440$	37.2509	1.77587
$441$	0	0
$442$	6.24449	0.297020
$443$	16.6599	0.791538	0.395769	−	0.918350i	$-0.370478\pi$
$443$	16.6599	0.791538	0.395769	+	0.918350i	$0.370478\pi$
$444$	45.5336	2.16093
$445$	−41.6737	−1.97552
$446$	−23.1184	−1.09469
$447$	2.33743	0.110557
$448$	0	0
$449$	−20.1075	−0.948929	−0.474465	−	0.880275i	$-0.657358\pi$
$449$	−20.1075	−0.948929	−0.474465	+	0.880275i	$0.657358\pi$
$450$	−28.6648	−1.35127
$451$	−25.5152	−1.20147
$452$	42.1025	1.98033
$453$	28.1932	1.32463
$454$	36.3203	1.70459
$455$	0	0
$456$	38.1138	1.78484
$457$	−25.5224	−1.19389	−0.596944	−	0.802283i	$-0.703618\pi$
$457$	−25.5224	−1.19389	−0.596944	+	0.802283i	$0.703618\pi$
$458$	36.7980	1.71946
$459$	−1.87775	−0.0876457
$460$	10.0033	0.466407
$461$	38.9326	1.81327	0.906635	−	0.421916i	$-0.138642\pi$
$461$	38.9326	1.81327	0.906635	+	0.421916i	$0.138642\pi$
$462$	0	0
$463$	−32.0158	−1.48790	−0.743951	−	0.668234i	$-0.767051\pi$
$463$	−32.0158	−1.48790	−0.743951	+	0.668234i	$0.767051\pi$
$464$	−0.841787	−0.0390790
$465$	57.5112	2.66702
$466$	0.511152	0.0236787
$467$	40.1531	1.85806	0.929031	−	0.370001i	$-0.120643\pi$
$467$	40.1531	1.85806	0.929031	+	0.370001i	$0.120643\pi$
$468$	21.5888	0.997940
$469$	0	0
$470$	16.9721	0.782863
$471$	−17.7965	−0.820019
$472$	−18.5127	−0.852116
$473$	27.0981	1.24597
$474$	8.12779	0.373322
$475$	64.2954	2.95007
$476$	0	0
$477$	4.53484	0.207636
$478$	43.1780	1.97492
$479$	−4.01649	−0.183518	−0.0917590	−	0.995781i	$-0.529249\pi$
$479$	−4.01649	−0.183518	−0.0917590	+	0.995781i	$0.529249\pi$
$480$	−48.6603	−2.22103
$481$	−34.1427	−1.55678
$482$	56.2817	2.56356
$483$	0	0
$484$	13.4148	0.609761
$485$	6.46304	0.293471
$486$	−30.6794	−1.39165
$487$	−13.2424	−0.600070	−0.300035	−	0.953928i	$-0.596998\pi$
$487$	−13.2424	−0.600070	−0.300035	+	0.953928i	$0.596998\pi$
$488$	−33.5722	−1.51974
$489$	−14.6923	−0.664410
$490$	0	0
$491$	28.1287	1.26943	0.634716	−	0.772745i	$-0.281117\pi$
$491$	28.1287	1.26943	0.634716	+	0.772745i	$0.281117\pi$
$492$	−42.7830	−1.92880
$493$	−0.968159	−0.0436037
$494$	−79.4230	−3.57341
$495$	−20.5720	−0.924642
$496$	3.57091	0.160339
$497$	0	0
$498$	−37.7562	−1.69190
$499$	−18.4340	−0.825221	−0.412611	−	0.910907i	$-0.635383\pi$
$499$	−18.4340	−0.825221	−0.412611	+	0.910907i	$0.635383\pi$
$500$	46.8972	2.09731
$501$	−6.70501	−0.299558
$502$	−9.10600	−0.406421
$503$	−3.23880	−0.144411	−0.0722054	−	0.997390i	$-0.523004\pi$
$503$	−3.23880	−0.144411	−0.0722054	+	0.997390i	$0.523004\pi$
$504$	0	0
$505$	14.0883	0.626919
$506$	7.57272	0.336648
$507$	−23.4582	−1.04182
$508$	−3.12419	−0.138614
$509$	−23.7432	−1.05240	−0.526200	−	0.850361i	$-0.676384\pi$
$509$	−23.7432	−1.05240	−0.526200	+	0.850361i	$0.676384\pi$
$510$	−9.97739	−0.441806
$511$	0	0
$512$	−5.50408	−0.243248
$513$	23.8829	1.05445
$514$	−13.6698	−0.602949
$515$	24.9524	1.09953
$516$	45.4370	2.00025
$517$	7.83349	0.344516
$518$	0	0
$519$	−16.4558	−0.722330
$520$	−46.8355	−2.05387
$521$	16.2610	0.712405	0.356203	−	0.934409i	$-0.384071\pi$
$521$	16.2610	0.712405	0.356203	+	0.934409i	$0.384071\pi$
$522$	−5.48991	−0.240287
$523$	27.4928	1.20218	0.601088	−	0.799183i	$-0.294734\pi$
$523$	27.4928	1.20218	0.601088	+	0.799183i	$0.294734\pi$
$524$	56.6567	2.47506
$525$	0	0
$526$	56.8524	2.47888
$527$	4.10699	0.178903
$528$	−4.00399	−0.174251
$529$	−22.2683	−0.968185
$530$	−27.3405	−1.18760
$531$	10.2237	0.443672
$532$	0	0
$533$	32.0802	1.38955
$534$	−52.8983	−2.28913
$535$	55.3808	2.39432
$536$	39.3925	1.70150
$537$	7.34416	0.316924
$538$	19.6589	0.847555
$539$	0	0
$540$	39.1392	1.68428
$541$	−15.9708	−0.686640	−0.343320	−	0.939218i	$-0.611552\pi$
$541$	−15.9708	−0.686640	−0.343320	+	0.939218i	$0.611552\pi$
$542$	21.5571	0.925958
$543$	44.5030	1.90980
$544$	−3.47493	−0.148987
$545$	−10.7472	−0.460359
$546$	0	0
$547$	28.3003	1.21003	0.605016	−	0.796213i	$-0.293167\pi$
$547$	28.3003	1.21003	0.605016	+	0.796213i	$0.293167\pi$
$548$	42.8166	1.82903
$549$	18.5404	0.791286
$550$	79.7652	3.40120
$551$	12.3139	0.524591
$552$	4.56906	0.194472
$553$	0	0
$554$	−63.2244	−2.68615
$555$	54.5530	2.31565
$556$	−11.5465	−0.489679
$557$	−15.4910	−0.656376	−0.328188	−	0.944612i	$-0.606438\pi$
$557$	−15.4910	−0.656376	−0.328188	+	0.944612i	$0.606438\pi$
$558$	23.2885	0.985882
$559$	−34.0703	−1.44102
$560$	0	0
$561$	−4.60508	−0.194427
$562$	2.21596	0.0934745
$563$	29.6519	1.24968	0.624839	−	0.780754i	$-0.285165\pi$
$563$	29.6519	1.24968	0.624839	+	0.780754i	$0.285165\pi$
$564$	13.1349	0.553079
$565$	50.4423	2.12212
$566$	−31.2438	−1.31327
$567$	0	0
$568$	−13.2786	−0.557157
$569$	10.8405	0.454456	0.227228	−	0.973842i	$-0.427034\pi$
$569$	10.8405	0.454456	0.227228	+	0.973842i	$0.427034\pi$
$570$	126.901	5.31531
$571$	42.5729	1.78162	0.890811	−	0.454374i	$-0.150137\pi$
$571$	42.5729	1.78162	0.890811	+	0.454374i	$0.150137\pi$
$572$	−60.0748	−2.51185
$573$	−18.7525	−0.783399
$574$	0	0
$575$	7.70767	0.321432
$576$	−18.3334	−0.763891
$577$	3.95362	0.164591	0.0822956	−	0.996608i	$-0.473775\pi$
$577$	3.95362	0.164591	0.0822956	+	0.996608i	$0.473775\pi$
$578$	37.7699	1.57102
$579$	−36.2774	−1.50764
$580$	20.1800	0.837930
$581$	0	0
$582$	8.20383	0.340060
$583$	−12.6191	−0.522628
$584$	−37.2122	−1.53985
$585$	25.8651	1.06939
$586$	−19.4633	−0.804021
$587$	−4.41708	−0.182312	−0.0911561	−	0.995837i	$-0.529056\pi$
$587$	−4.41708	−0.182312	−0.0911561	+	0.995837i	$0.529056\pi$
$588$	0	0
$589$	−52.2364	−2.15236
$590$	−61.6387	−2.53763
$591$	47.8199	1.96705
$592$	3.38724	0.139215
$593$	38.1474	1.56653	0.783263	−	0.621690i	$-0.213554\pi$
$593$	38.1474	1.56653	0.783263	+	0.621690i	$0.213554\pi$
$594$	29.6292	1.21570
$595$	0	0
$596$	−3.47924	−0.142515
$597$	35.2244	1.44164
$598$	−9.52116	−0.389349
$599$	10.6234	0.434059	0.217030	−	0.976165i	$-0.430363\pi$
$599$	10.6234	0.434059	0.217030	+	0.976165i	$0.430363\pi$
$600$	48.1269	1.96477
$601$	14.1136	0.575707	0.287854	−	0.957674i	$-0.407058\pi$
$601$	14.1136	0.575707	0.287854	+	0.957674i	$0.407058\pi$
$602$	0	0
$603$	−21.7547	−0.885919
$604$	−41.9653	−1.70755
$605$	16.0720	0.653419
$606$	17.8829	0.726441
$607$	27.3262	1.10914	0.554568	−	0.832139i	$-0.312884\pi$
$607$	27.3262	1.10914	0.554568	+	0.832139i	$0.312884\pi$
$608$	44.1973	1.79244
$609$	0	0
$610$	−111.780	−4.52583
$611$	−9.84903	−0.398449
$612$	−2.46331	−0.0995734
$613$	4.40664	0.177983	0.0889913	−	0.996032i	$-0.471636\pi$
$613$	4.40664	0.177983	0.0889913	+	0.996032i	$0.471636\pi$
$614$	−13.7349	−0.554294
$615$	−51.2575	−2.06690
$616$	0	0
$617$	−2.46888	−0.0993933	−0.0496967	−	0.998764i	$-0.515825\pi$
$617$	−2.46888	−0.0993933	−0.0496967	+	0.998764i	$0.515825\pi$
$618$	31.6732	1.27408
$619$	7.92380	0.318484	0.159242	−	0.987240i	$-0.449095\pi$
$619$	7.92380	0.318484	0.159242	+	0.987240i	$0.449095\pi$
$620$	−85.6049	−3.43797
$621$	2.86306	0.114891
$622$	−3.78595	−0.151803
$623$	0	0
$624$	5.03421	0.201530
$625$	11.1349	0.445394
$626$	28.5676	1.14179
$627$	58.5716	2.33912
$628$	26.4899	1.05706
$629$	3.89574	0.155333
$630$	0	0
$631$	20.2597	0.806525	0.403262	−	0.915084i	$-0.367876\pi$
$631$	20.2597	0.806525	0.403262	+	0.915084i	$0.367876\pi$
$632$	−4.35333	−0.173166
$633$	−58.5614	−2.32761
$634$	3.85752	0.153202
$635$	−3.74304	−0.148538
$636$	−21.1592	−0.839015
$637$	0	0
$638$	15.2767	0.604811
$639$	7.33317	0.290096
$640$	64.1642	2.53631
$641$	15.1915	0.600029	0.300014	−	0.953935i	$-0.403009\pi$
$641$	15.1915	0.600029	0.300014	+	0.953935i	$0.403009\pi$
$642$	70.2974	2.77442
$643$	7.18006	0.283154	0.141577	−	0.989927i	$-0.454783\pi$
$643$	7.18006	0.283154	0.141577	+	0.989927i	$0.454783\pi$
$644$	0	0
$645$	54.4373	2.14347
$646$	9.06229	0.356551
$647$	−38.4142	−1.51022	−0.755109	−	0.655600i	$-0.772416\pi$
$647$	−38.4142	−1.51022	−0.755109	+	0.655600i	$0.772416\pi$
$648$	28.6062	1.12376
$649$	−28.4495	−1.11674
$650$	−100.289	−3.93364
$651$	0	0
$652$	21.8694	0.856472
$653$	22.7462	0.890128	0.445064	−	0.895499i	$-0.353181\pi$
$653$	22.7462	0.890128	0.445064	+	0.895499i	$0.353181\pi$
$654$	−13.6419	−0.533440
$655$	67.8795	2.65227
$656$	−3.18262	−0.124260
$657$	20.5506	0.801755
$658$	0	0
$659$	36.1010	1.40630	0.703148	−	0.711044i	$-0.251777\pi$
$659$	36.1010	1.40630	0.703148	+	0.711044i	$0.251777\pi$
$660$	95.9870	3.73629
$661$	−39.4314	−1.53370	−0.766852	−	0.641824i	$-0.778178\pi$
$661$	−39.4314	−1.53370	−0.766852	+	0.641824i	$0.778178\pi$
$662$	67.7210	2.63205
$663$	5.78996	0.224863
$664$	20.2226	0.784790
$665$	0	0
$666$	22.0906	0.855995
$667$	1.47618	0.0571580
$668$	9.98035	0.386151
$669$	−21.4356	−0.828749
$670$	131.159	5.06710
$671$	−51.5922	−1.99170
$672$	0	0
$673$	37.9522	1.46295	0.731474	−	0.681869i	$-0.238833\pi$
$673$	37.9522	1.46295	0.731474	+	0.681869i	$0.238833\pi$
$674$	−45.5659	−1.75513
$675$	30.1572	1.16075
$676$	34.9174	1.34298
$677$	−41.4217	−1.59197	−0.795983	−	0.605319i	$-0.793046\pi$
$677$	−41.4217	−1.59197	−0.795983	+	0.605319i	$0.793046\pi$
$678$	64.0286	2.45900
$679$	0	0
$680$	5.34400	0.204933
$681$	33.6766	1.29049
$682$	−64.8047	−2.48150
$683$	−4.25205	−0.162700	−0.0813500	−	0.996686i	$-0.525923\pi$
$683$	−4.25205	−0.162700	−0.0813500	+	0.996686i	$0.525923\pi$
$684$	31.3306	1.19795
$685$	51.2978	1.95999
$686$	0	0
$687$	34.1195	1.30174
$688$	3.38005	0.128863
$689$	15.8659	0.604443
$690$	15.2128	0.579143
$691$	−41.6368	−1.58394	−0.791970	−	0.610560i	$-0.790944\pi$
$691$	−41.6368	−1.58394	−0.791970	+	0.610560i	$0.790944\pi$
$692$	24.4943	0.931135
$693$	0	0
$694$	−70.7474	−2.68554
$695$	−13.8336	−0.524739
$696$	9.21732	0.349382
$697$	−3.66040	−0.138648
$698$	−60.7893	−2.30091
$699$	0.473946	0.0179263
$700$	0	0
$701$	33.8970	1.28027	0.640136	−	0.768262i	$-0.278878\pi$
$701$	33.8970	1.28027	0.640136	+	0.768262i	$0.278878\pi$
$702$	−37.2527	−1.40601
$703$	−49.5495	−1.86880
$704$	51.0161	1.92274
$705$	15.7367	0.592678
$706$	−34.5595	−1.30066
$707$	0	0
$708$	−47.7030	−1.79279
$709$	−15.9074	−0.597415	−0.298708	−	0.954345i	$-0.596556\pi$
$709$	−15.9074	−0.597415	−0.298708	+	0.954345i	$0.596556\pi$
$710$	−44.2115	−1.65923
$711$	2.40415	0.0901626
$712$	28.3329	1.06182
$713$	−6.26205	−0.234516
$714$	0	0
$715$	−71.9746	−2.69170
$716$	−10.9317	−0.408538
$717$	40.0352	1.49514
$718$	25.2757	0.943279
$719$	−6.55673	−0.244525	−0.122262	−	0.992498i	$-0.539015\pi$
$719$	−6.55673	−0.244525	−0.122262	+	0.992498i	$0.539015\pi$
$720$	−2.56603	−0.0956302
$721$	0	0
$722$	−72.2526	−2.68896
$723$	52.1851	1.94078
$724$	−66.2423	−2.46187
$725$	15.5490	0.577474
$726$	20.4009	0.757148
$727$	−7.62561	−0.282818	−0.141409	−	0.989951i	$-0.545163\pi$
$727$	−7.62561	−0.282818	−0.141409	+	0.989951i	$0.545163\pi$
$728$	0	0
$729$	5.27681	0.195437
$730$	−123.899	−4.58572
$731$	3.88748	0.143784
$732$	−86.5079	−3.19742
$733$	32.6350	1.20540	0.602701	−	0.797967i	$-0.294091\pi$
$733$	32.6350	1.20540	0.602701	+	0.797967i	$0.294091\pi$
$734$	−40.4314	−1.49235
$735$	0	0
$736$	5.29834	0.195299
$737$	60.5365	2.22989
$738$	−20.7562	−0.764046
$739$	−12.2203	−0.449531	−0.224766	−	0.974413i	$-0.572162\pi$
$739$	−12.2203	−0.449531	−0.224766	+	0.974413i	$0.572162\pi$
$740$	−81.2017	−2.98503
$741$	−73.6419	−2.70530
$742$	0	0
$743$	−47.0649	−1.72664	−0.863322	−	0.504654i	$-0.831620\pi$
$743$	−47.0649	−1.72664	−0.863322	+	0.504654i	$0.831620\pi$
$744$	−39.1004	−1.43349
$745$	−4.16842	−0.152719
$746$	2.69915	0.0988229
$747$	−11.1680	−0.408617
$748$	6.85463	0.250630
$749$	0	0
$750$	71.3203	2.60425
$751$	−48.5981	−1.77337	−0.886684	−	0.462376i	$-0.846997\pi$
$751$	−48.5981	−1.77337	−0.886684	+	0.462376i	$0.846997\pi$
$752$	0.977103	0.0356313
$753$	−8.44319	−0.307687
$754$	−19.2074	−0.699492
$755$	−50.2780	−1.82980
$756$	0	0
$757$	−22.9773	−0.835125	−0.417562	−	0.908648i	$-0.637115\pi$
$757$	−22.9773	−0.835125	−0.417562	+	0.908648i	$0.637115\pi$
$758$	−69.8904	−2.53853
$759$	7.02151	0.254865
$760$	−67.9698	−2.46552
$761$	−0.307687	−0.0111537	−0.00557683	−	0.999984i	$-0.501775\pi$
$761$	−0.307687	−0.0111537	−0.00557683	+	0.999984i	$0.501775\pi$
$762$	−4.75121	−0.172118
$763$	0	0
$764$	27.9130	1.00986
$765$	−2.95125	−0.106703
$766$	−15.3684	−0.555283
$767$	35.7695	1.29156
$768$	26.6856	0.962932
$769$	41.4619	1.49515	0.747577	−	0.664176i	$-0.231217\pi$
$769$	41.4619	1.49515	0.747577	+	0.664176i	$0.231217\pi$
$770$	0	0
$771$	−12.6748	−0.456472
$772$	53.9986	1.94345
$773$	−10.3325	−0.371635	−0.185817	−	0.982584i	$-0.559493\pi$
$773$	−10.3325	−0.371635	−0.185817	+	0.982584i	$0.559493\pi$
$774$	22.0438	0.792347
$775$	−65.9596	−2.36934
$776$	−4.39406	−0.157738
$777$	0	0
$778$	38.4552	1.37868
$779$	46.5563	1.66805
$780$	−120.684	−4.32119
$781$	−20.4059	−0.730181
$782$	1.08638	0.0388488
$783$	5.77575	0.206408
$784$	0	0
$785$	31.7371	1.13275
$786$	86.1625	3.07331
$787$	36.8002	1.31179	0.655893	−	0.754853i	$-0.272292\pi$
$787$	36.8002	1.31179	0.655893	+	0.754853i	$0.272292\pi$
$788$	−71.1796	−2.53567
$789$	52.7142	1.87668
$790$	−14.4946	−0.515694
$791$	0	0
$792$	13.9864	0.496984
$793$	64.8668	2.30349
$794$	68.2161	2.42090
$795$	−25.3504	−0.899087
$796$	−52.4311	−1.85837
$797$	12.2212	0.432896	0.216448	−	0.976294i	$-0.430553\pi$
$797$	12.2212	0.432896	0.216448	+	0.976294i	$0.430553\pi$
$798$	0	0
$799$	1.12379	0.0397568
$800$	55.8086	1.97313
$801$	−15.6470	−0.552859
$802$	−52.4996	−1.85383
$803$	−57.1860	−2.01805
$804$	101.505	3.57982
$805$	0	0
$806$	81.4789	2.86997
$807$	18.2280	0.641655
$808$	−9.57825	−0.336962
$809$	19.8974	0.699555	0.349778	−	0.936833i	$-0.386257\pi$
$809$	19.8974	0.699555	0.349778	+	0.936833i	$0.386257\pi$
$810$	95.2455	3.34658
$811$	−26.3415	−0.924977	−0.462488	−	0.886625i	$-0.653043\pi$
$811$	−26.3415	−0.924977	−0.462488	+	0.886625i	$0.653043\pi$
$812$	0	0
$813$	19.9880	0.701010
$814$	−61.4714	−2.15457
$815$	26.2014	0.917793
$816$	−0.574411	−0.0201084
$817$	−49.4444	−1.72984
$818$	2.38385	0.0833493
$819$	0	0
$820$	76.2964	2.66439
$821$	2.70927	0.0945541	0.0472771	−	0.998882i	$-0.484946\pi$
$821$	2.70927	0.0945541	0.0472771	+	0.998882i	$0.484946\pi$
$822$	65.1146	2.27113
$823$	41.0802	1.43197	0.715983	−	0.698118i	$-0.245979\pi$
$823$	41.0802	1.43197	0.715983	+	0.698118i	$0.245979\pi$
$824$	−16.9645	−0.590987
$825$	73.9592	2.57493
$826$	0	0
$827$	−5.20974	−0.181161	−0.0905803	−	0.995889i	$-0.528872\pi$
$827$	−5.20974	−0.181161	−0.0905803	+	0.995889i	$0.528872\pi$
$828$	3.75588	0.130526
$829$	2.13885	0.0742853	0.0371426	−	0.999310i	$-0.488174\pi$
$829$	2.13885	0.0742853	0.0371426	+	0.999310i	$0.488174\pi$
$830$	67.3320	2.33713
$831$	−58.6224	−2.03359
$832$	−64.1425	−2.22374
$833$	0	0
$834$	−17.5596	−0.608041
$835$	11.9573	0.413799
$836$	−87.1833	−3.01530
$837$	−24.5011	−0.846881
$838$	23.1398	0.799350
$839$	−7.16223	−0.247268	−0.123634	−	0.992328i	$-0.539455\pi$
$839$	−7.16223	−0.247268	−0.123634	+	0.992328i	$0.539455\pi$
$840$	0	0
$841$	−26.0220	−0.897312
$842$	21.5954	0.744225
$843$	2.05466	0.0707663
$844$	87.1681	3.00045
$845$	41.8339	1.43913
$846$	6.37240	0.219088
$847$	0	0
$848$	−1.57403	−0.0540523
$849$	−28.9696	−0.994234
$850$	11.4431	0.392494
$851$	−5.93995	−0.203619
$852$	−34.2159	−1.17222
$853$	23.9118	0.818725	0.409363	−	0.912372i	$-0.365751\pi$
$853$	23.9118	0.818725	0.409363	+	0.912372i	$0.365751\pi$
$854$	0	0
$855$	37.5366	1.28373
$856$	−37.6521	−1.28692
$857$	6.35009	0.216915	0.108458	−	0.994101i	$-0.465409\pi$
$857$	6.35009	0.216915	0.108458	+	0.994101i	$0.465409\pi$
$858$	−91.3606	−3.11900
$859$	27.2697	0.930429	0.465214	−	0.885198i	$-0.345977\pi$
$859$	27.2697	0.930429	0.465214	+	0.885198i	$0.345977\pi$
$860$	−81.0294	−2.76308
$861$	0	0
$862$	−52.5216	−1.78889
$863$	41.7696	1.42185	0.710927	−	0.703266i	$-0.248276\pi$
$863$	41.7696	1.42185	0.710927	+	0.703266i	$0.248276\pi$
$864$	20.7304	0.705263
$865$	29.3462	0.997802
$866$	76.1994	2.58936
$867$	35.0206	1.18936
$868$	0	0
$869$	−6.69000	−0.226943
$870$	30.6894	1.04047
$871$	−76.1124	−2.57897
$872$	7.30675	0.247438
$873$	2.42664	0.0821293
$874$	−13.8175	−0.467385
$875$	0	0
$876$	−95.8872	−3.23973
$877$	−2.08436	−0.0703838	−0.0351919	−	0.999381i	$-0.511204\pi$
$877$	−2.08436	−0.0703838	−0.0351919	+	0.999381i	$0.511204\pi$
$878$	46.4431	1.56738
$879$	−18.0466	−0.608696
$880$	7.14046	0.240705
$881$	50.8345	1.71266	0.856329	−	0.516431i	$-0.172740\pi$
$881$	50.8345	1.71266	0.856329	+	0.516431i	$0.172740\pi$
$882$	0	0
$883$	13.6277	0.458610	0.229305	−	0.973355i	$-0.426355\pi$
$883$	13.6277	0.458610	0.229305	+	0.973355i	$0.426355\pi$
$884$	−8.61831	−0.289865
$885$	−57.1521	−1.92115
$886$	−37.7126	−1.26698
$887$	−32.8327	−1.10241	−0.551207	−	0.834368i	$-0.685833\pi$
$887$	−32.8327	−1.10241	−0.551207	+	0.834368i	$0.685833\pi$
$888$	−37.0892	−1.24463
$889$	0	0
$890$	94.3354	3.16213
$891$	43.9607	1.47274
$892$	31.9067	1.06832
$893$	−14.2934	−0.478309
$894$	−5.29116	−0.176963
$895$	−13.0971	−0.437788
$896$	0	0
$897$	−8.82813	−0.294763
$898$	45.5166	1.51891
$899$	−12.6327	−0.421323
$900$	39.5616	1.31872
$901$	−1.81033	−0.0603107
$902$	57.7580	1.92313
$903$	0	0
$904$	−34.2944	−1.14062
$905$	−79.3637	−2.63814
$906$	−63.8201	−2.12028
$907$	7.77855	0.258283	0.129141	−	0.991626i	$-0.458778\pi$
$907$	7.77855	0.258283	0.129141	+	0.991626i	$0.458778\pi$
$908$	−50.1273	−1.66353
$909$	5.28964	0.175446
$910$	0	0
$911$	−21.5690	−0.714613	−0.357306	−	0.933987i	$-0.616305\pi$
$911$	−21.5690	−0.714613	−0.357306	+	0.933987i	$0.616305\pi$
$912$	7.30587	0.241922
$913$	31.0772	1.02851
$914$	57.7742	1.91100
$915$	−103.644	−3.42635
$916$	−50.7866	−1.67804
$917$	0	0
$918$	4.25060	0.140291
$919$	−41.8629	−1.38093	−0.690465	−	0.723366i	$-0.742594\pi$
$919$	−41.8629	−1.38093	−0.690465	+	0.723366i	$0.742594\pi$
$920$	−8.14816	−0.268637
$921$	−12.7351	−0.419637
$922$	−88.1304	−2.90242
$923$	25.6563	0.844488
$924$	0	0
$925$	−62.5669	−2.05719
$926$	72.4733	2.38162
$927$	9.36873	0.307710
$928$	10.6885	0.350868
$929$	16.6855	0.547434	0.273717	−	0.961810i	$-0.411747\pi$
$929$	16.6855	0.547434	0.273717	+	0.961810i	$0.411747\pi$
$930$	−130.186	−4.26897
$931$	0	0
$932$	−0.705465	−0.0231083
$933$	−3.51038	−0.114925
$934$	−90.8932	−2.97412
$935$	8.21241	0.268575
$936$	−17.5850	−0.574785
$937$	−9.91371	−0.323867	−0.161933	−	0.986802i	$-0.551773\pi$
$937$	−9.91371	−0.323867	−0.161933	+	0.986802i	$0.551773\pi$
$938$	0	0
$939$	26.4883	0.864411
$940$	−23.4239	−0.764004
$941$	20.5022	0.668353	0.334177	−	0.942510i	$-0.391542\pi$
$941$	20.5022	0.668353	0.334177	+	0.942510i	$0.391542\pi$
$942$	40.2853	1.31257
$943$	5.58113	0.181747
$944$	−3.54862	−0.115498
$945$	0	0
$946$	−61.3410	−1.99437
$947$	25.5489	0.830229	0.415114	−	0.909769i	$-0.363742\pi$
$947$	25.5489	0.830229	0.415114	+	0.909769i	$0.363742\pi$
$948$	−11.2175	−0.364329
$949$	71.8998	2.33397
$950$	−145.543	−4.72205
$951$	3.57674	0.115984
$952$	0	0
$953$	42.7039	1.38331	0.691657	−	0.722226i	$-0.256881\pi$
$953$	42.7039	1.38331	0.691657	+	0.722226i	$0.256881\pi$
$954$	−10.2654	−0.332354
$955$	33.4421	1.08216
$956$	−59.5920	−1.92734
$957$	14.1647	0.457881
$958$	9.09199	0.293749
$959$	0	0
$960$	102.486	3.30773
$961$	22.5885	0.728662
$962$	77.2879	2.49186
$963$	20.7935	0.670062
$964$	−77.6770	−2.50181
$965$	64.6948	2.08260
$966$	0	0
$967$	−29.1511	−0.937436	−0.468718	−	0.883348i	$-0.655284\pi$
$967$	−29.1511	−0.937436	−0.468718	+	0.883348i	$0.655284\pi$
$968$	−10.9269	−0.351205
$969$	8.40266	0.269932
$970$	−14.6302	−0.469747
$971$	29.9921	0.962492	0.481246	−	0.876586i	$-0.340184\pi$
$971$	29.9921	0.962492	0.481246	+	0.876586i	$0.340184\pi$
$972$	42.3421	1.35812
$973$	0	0
$974$	29.9764	0.960505
$975$	−92.9887	−2.97802
$976$	−6.43531	−0.205989
$977$	42.6162	1.36341	0.681706	−	0.731626i	$-0.261238\pi$
$977$	42.6162	1.36341	0.681706	+	0.731626i	$0.261238\pi$
$978$	33.2586	1.06349
$979$	43.5407	1.39157
$980$	0	0
$981$	−4.03519	−0.128834
$982$	−63.6741	−2.03192
$983$	−11.4872	−0.366383	−0.183192	−	0.983077i	$-0.558643\pi$
$983$	−11.4872	−0.366383	−0.183192	+	0.983077i	$0.558643\pi$
$984$	34.8487	1.11094
$985$	−85.2790	−2.71721
$986$	2.19159	0.0697945
$987$	0	0
$988$	109.615	3.48733
$989$	−5.92735	−0.188479
$990$	46.5681	1.48003
$991$	−3.05243	−0.0969637	−0.0484819	−	0.998824i	$-0.515438\pi$
$991$	−3.05243	−0.0969637	−0.0484819	+	0.998824i	$0.515438\pi$
$992$	−45.3414	−1.43959
$993$	62.7917	1.99264
$994$	0	0
$995$	−62.8168	−1.99143
$996$	52.1091	1.65114
$997$	34.6601	1.09770	0.548849	−	0.835922i	$-0.315066\pi$
$997$	34.6601	1.09770	0.548849	+	0.835922i	$0.315066\pi$
$998$	41.7286	1.32089
$999$	−23.2408	−0.735307

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.n.1.5	✓	34
7.6	odd	2	inner	6223.2.a.n.1.6	yes	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6223.2.a.n.1.5	✓	34	1.1	even	1	trivial
6223.2.a.n.1.6	yes	34	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q-2.26367 q^{2} -2.09890 q^{3} +3.12419 q^{4} +3.74304 q^{5} +4.75121 q^{6} -2.54480 q^{8} +1.40538 q^{9} +O(q^{10})\)	\(q-2.26367 q^{2} -2.09890 q^{3} +3.12419 q^{4} +3.74304 q^{5} +4.75121 q^{6} -2.54480 q^{8} +1.40538 q^{9} -8.47301 q^{10} -3.91073 q^{11} -6.55737 q^{12} +4.91696 q^{13} -7.85627 q^{15} -0.487802 q^{16} -0.561032 q^{17} -3.18131 q^{18} +7.13571 q^{19} +11.6940 q^{20} +8.85260 q^{22} +0.855423 q^{23} +5.34128 q^{24} +9.01037 q^{25} -11.1304 q^{26} +3.34695 q^{27} +1.72568 q^{29} +17.7840 q^{30} -7.32042 q^{31} +6.19382 q^{32} +8.20823 q^{33} +1.26999 q^{34} +4.39067 q^{36} -6.94388 q^{37} -16.1529 q^{38} -10.3202 q^{39} -9.52530 q^{40} +6.52441 q^{41} -6.92915 q^{43} -12.2179 q^{44} +5.26039 q^{45} -1.93639 q^{46} -2.00307 q^{47} +1.02385 q^{48} -20.3965 q^{50} +1.17755 q^{51} +15.3615 q^{52} +3.22678 q^{53} -7.57638 q^{54} -14.6380 q^{55} -14.9771 q^{57} -3.90636 q^{58} +7.27472 q^{59} -24.5445 q^{60} +13.1925 q^{61} +16.5710 q^{62} -13.0452 q^{64} +18.4044 q^{65} -18.5807 q^{66} -15.4796 q^{67} -1.75277 q^{68} -1.79545 q^{69} +5.21793 q^{71} -3.57641 q^{72} +14.6228 q^{73} +15.7186 q^{74} -18.9119 q^{75} +22.2933 q^{76} +23.3615 q^{78} +1.71068 q^{79} -1.82586 q^{80} -11.2410 q^{81} -14.7691 q^{82} -7.94664 q^{83} -2.09997 q^{85} +15.6853 q^{86} -3.62202 q^{87} +9.95204 q^{88} -11.1336 q^{89} -11.9078 q^{90} +2.67251 q^{92} +15.3648 q^{93} +4.53430 q^{94} +26.7093 q^{95} -13.0002 q^{96} +1.72668 q^{97} -5.49606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 6 q^{2} + 46 q^{4} + 30 q^{8} + 66 q^{9}+O(q^{10}) \) 34 * q + 6 * q^2 + 46 * q^4 + 30 * q^8 + 66 * q^9	\( 34 q + 6 q^{2} + 46 q^{4} + 30 q^{8} + 66 q^{9} + 16 q^{11} + 12 q^{15} + 54 q^{16} + 48 q^{18} + 30 q^{22} + 46 q^{23} + 80 q^{25} + 28 q^{29} - 2 q^{30} + 36 q^{32} + 98 q^{36} + 52 q^{37} - 2 q^{39} + 10 q^{43} + 16 q^{44} - 36 q^{46} + 74 q^{50} + 56 q^{51} + 64 q^{53} + 44 q^{57} + 32 q^{58} - 132 q^{60} + 74 q^{64} + 108 q^{65} + 14 q^{67} + 6 q^{71} + 172 q^{72} - 6 q^{74} + 52 q^{78} + 46 q^{81} - 2 q^{85} + 96 q^{86} - 170 q^{88} + 130 q^{92} + 6 q^{93} + 14 q^{95} + 2 q^{99}+O(q^{100}) \) 34 * q + 6 * q^2 + 46 * q^4 + 30 * q^8 + 66 * q^9 + 16 * q^11 + 12 * q^15 + 54 * q^16 + 48 * q^18 + 30 * q^22 + 46 * q^23 + 80 * q^25 + 28 * q^29 - 2 * q^30 + 36 * q^32 + 98 * q^36 + 52 * q^37 - 2 * q^39 + 10 * q^43 + 16 * q^44 - 36 * q^46 + 74 * q^50 + 56 * q^51 + 64 * q^53 + 44 * q^57 + 32 * q^58 - 132 * q^60 + 74 * q^64 + 108 * q^65 + 14 * q^67 + 6 * q^71 + 172 * q^72 - 6 * q^74 + 52 * q^78 + 46 * q^81 - 2 * q^85 + 96 * q^86 - 170 * q^88 + 130 * q^92 + 6 * q^93 + 14 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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