LMFDB - Embedded newform 605.4.a.t.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,4,Mod(1,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("605.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 605 = 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	605.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:	$35.6961555535$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 58 x^{10} + 264 x^{9} + 1280 x^{8} - 5073 x^{7} - 13326 x^{6} + 43166 x^{5} + \cdots + 123904 $ x^12 - 5x^11 - 58x^10 + 264x^9 + 1280x^8 - 5073x^7 - 13326x^6 + 43166x^5 + 65591x^4 - 154592x^3 - 137872x^2 + 173184*x + 123904
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11^{2} $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.661955$ of defining polynomial
Character	$\chi$	$=$	605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.66196 q^{2} +1.98058 q^{3} -5.23790 q^{4} +5.00000 q^{5} +3.29164 q^{6} -8.81834 q^{7} -22.0008 q^{8} -23.0773 q^{9} +8.30978 q^{10} -10.3741 q^{12} -10.3377 q^{13} -14.6557 q^{14} +9.90292 q^{15} +5.33888 q^{16} +48.5862 q^{17} -38.3534 q^{18} +97.1627 q^{19} -26.1895 q^{20} -17.4655 q^{21} +196.403 q^{23} -43.5745 q^{24} +25.0000 q^{25} -17.1808 q^{26} -99.1823 q^{27} +46.1896 q^{28} +169.043 q^{29} +16.4582 q^{30} +40.8279 q^{31} +184.879 q^{32} +80.7481 q^{34} -44.0917 q^{35} +120.877 q^{36} +145.892 q^{37} +161.480 q^{38} -20.4747 q^{39} -110.004 q^{40} +157.333 q^{41} -29.0268 q^{42} -266.051 q^{43} -115.386 q^{45} +326.414 q^{46} +483.044 q^{47} +10.5741 q^{48} -265.237 q^{49} +41.5489 q^{50} +96.2290 q^{51} +54.1480 q^{52} -344.896 q^{53} -164.837 q^{54} +194.010 q^{56} +192.439 q^{57} +280.942 q^{58} -772.298 q^{59} -51.8706 q^{60} +800.490 q^{61} +67.8541 q^{62} +203.503 q^{63} +264.550 q^{64} -51.6887 q^{65} +128.194 q^{67} -254.490 q^{68} +388.994 q^{69} -73.2784 q^{70} -288.991 q^{71} +507.719 q^{72} -25.4865 q^{73} +242.466 q^{74} +49.5146 q^{75} -508.929 q^{76} -34.0281 q^{78} +214.115 q^{79} +26.6944 q^{80} +426.648 q^{81} +261.481 q^{82} +518.134 q^{83} +91.4824 q^{84} +242.931 q^{85} -442.166 q^{86} +334.804 q^{87} +6.39690 q^{89} -191.767 q^{90} +91.1616 q^{91} -1028.74 q^{92} +80.8631 q^{93} +802.797 q^{94} +485.814 q^{95} +366.169 q^{96} -325.571 q^{97} -440.812 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} - 3 q^{3} + 47 q^{4} + 60 q^{5} + 30 q^{6} + 97 q^{7} + 105 q^{8} + 131 q^{9} + 35 q^{10} + 5 q^{12} + 205 q^{13} + 109 q^{14} - 15 q^{15} + 115 q^{16} + 231 q^{17} + 241 q^{18} + 276 q^{19}+ \cdots + 3889 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 - 3 * q^3 + 47 * q^4 + 60 * q^5 + 30 * q^6 + 97 * q^7 + 105 * q^8 + 131 * q^9 + 35 * q^10 + 5 * q^12 + 205 * q^13 + 109 * q^14 - 15 * q^15 + 115 * q^16 + 231 * q^17 + 241 * q^18 + 276 * q^19 + 235 * q^20 + 100 * q^21 - 91 * q^23 + 320 * q^24 + 300 * q^25 - 462 * q^26 - 366 * q^27 + 866 * q^28 + 140 * q^29 + 150 * q^30 + 420 * q^31 + 252 * q^32 + 523 * q^34 + 485 * q^35 - 81 * q^36 - 235 * q^37 - 1044 * q^38 + 352 * q^39 + 525 * q^40 + 306 * q^41 - 558 * q^42 + 1829 * q^43 + 655 * q^45 - 959 * q^46 - 157 * q^47 - 659 * q^48 - 71 * q^49 + 175 * q^50 + 872 * q^51 + 1154 * q^52 - 771 * q^53 - 1392 * q^54 + 4326 * q^56 + 546 * q^57 - 598 * q^58 - 556 * q^59 + 25 * q^60 - 6 * q^61 + 1315 * q^62 + 3743 * q^63 + 583 * q^64 + 1025 * q^65 + 1983 * q^67 + 1534 * q^68 + 558 * q^69 + 545 * q^70 - 488 * q^71 + 1142 * q^72 + 2295 * q^73 + 3299 * q^74 - 75 * q^75 - 1524 * q^76 + 5032 * q^78 + 1534 * q^79 + 575 * q^80 - 1980 * q^81 + 1132 * q^82 + 4225 * q^83 + 727 * q^84 + 1155 * q^85 + 3612 * q^86 + 1922 * q^87 + 2284 * q^89 + 1205 * q^90 + 500 * q^91 - 3335 * q^92 - 6318 * q^93 + 1716 * q^94 + 1380 * q^95 + 5247 * q^96 - 765 * q^97 + 3889 * q^98

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 $

$2$	1.66196	0.587590	0.293795	−	0.955868i	$-0.405082\pi$
$2$	1.66196	0.587590	0.293795	+	0.955868i	$0.405082\pi$
$3$	1.98058	0.381164	0.190582	−	0.981671i	$-0.438963\pi$
$3$	1.98058	0.381164	0.190582	+	0.981671i	$0.438963\pi$
$4$	−5.23790	−0.654738
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	3.29164	0.223968
$7$	−8.81834	−0.476145	−0.238073	−	0.971247i	$-0.576516\pi$
$7$	−8.81834	−0.476145	−0.238073	+	0.971247i	$0.576516\pi$
$8$	−22.0008	−0.972307
$9$	−23.0773	−0.854714
$10$	8.30978	0.262778
$11$	0	0
$11$	0	0
$12$	−10.3741	−0.249562
$13$	−10.3377	−0.220552	−0.110276	−	0.993901i	$-0.535173\pi$
$13$	−10.3377	−0.220552	−0.110276	+	0.993901i	$0.535173\pi$
$14$	−14.6557	−0.279778
$15$	9.90292	0.170462
$16$	5.33888	0.0834200
$17$	48.5862	0.693169	0.346585	−	0.938019i	$-0.387341\pi$
$17$	48.5862	0.693169	0.346585	+	0.938019i	$0.387341\pi$
$18$	−38.3534	−0.502222
$19$	97.1627	1.17319	0.586596	−	0.809879i	$-0.300468\pi$
$19$	97.1627	1.17319	0.586596	+	0.809879i	$0.300468\pi$
$20$	−26.1895	−0.292808
$21$	−17.4655	−0.181489
$22$	0	0
$23$	196.403	1.78056	0.890281	−	0.455411i	$-0.150508\pi$
$23$	196.403	1.78056	0.890281	+	0.455411i	$0.150508\pi$
$24$	−43.5745	−0.370608
$25$	25.0000	0.200000
$26$	−17.1808	−0.129594
$27$	−99.1823	−0.706950
$28$	46.1896	0.311750
$29$	169.043	1.08243	0.541216	−	0.840884i	$-0.317964\pi$
$29$	169.043	1.08243	0.541216	+	0.840884i	$0.317964\pi$
$30$	16.4582	0.100161
$31$	40.8279	0.236545	0.118273	−	0.992981i	$-0.462264\pi$
$31$	40.8279	0.236545	0.118273	+	0.992981i	$0.462264\pi$
$32$	184.879	1.02132
$33$	0	0
$34$	80.7481	0.407299
$35$	−44.0917	−0.212939
$36$	120.877	0.559614
$37$	145.892	0.648229	0.324114	−	0.946018i	$-0.394934\pi$
$37$	145.892	0.648229	0.324114	+	0.946018i	$0.394934\pi$
$38$	161.480	0.689356
$39$	−20.4747	−0.0840662
$40$	−110.004	−0.434829
$41$	157.333	0.599301	0.299650	−	0.954049i	$-0.403130\pi$
$41$	157.333	0.599301	0.299650	+	0.954049i	$0.403130\pi$
$42$	−29.0268	−0.106641
$43$	−266.051	−0.943546	−0.471773	−	0.881720i	$-0.656386\pi$
$43$	−266.051	−0.943546	−0.471773	+	0.881720i	$0.656386\pi$
$44$	0	0
$45$	−115.386	−0.382240
$46$	326.414	1.04624
$47$	483.044	1.49913	0.749566	−	0.661930i	$-0.230262\pi$
$47$	483.044	1.49913	0.749566	+	0.661930i	$0.230262\pi$
$48$	10.5741	0.0317967
$49$	−265.237	−0.773286
$50$	41.5489	0.117518
$51$	96.2290	0.264211
$52$	54.1480	0.144404
$53$	−344.896	−0.893871	−0.446935	−	0.894566i	$-0.647485\pi$
$53$	−344.896	−0.893871	−0.446935	+	0.894566i	$0.647485\pi$
$54$	−164.837	−0.415397
$55$	0	0
$56$	194.010	0.462960
$57$	192.439	0.447178
$58$	280.942	0.636026
$59$	−772.298	−1.70415	−0.852073	−	0.523424i	$-0.824655\pi$
$59$	−772.298	−1.70415	−0.852073	+	0.523424i	$0.824655\pi$
$60$	−51.8706	−0.111608
$61$	800.490	1.68020	0.840101	−	0.542431i	$-0.182496\pi$
$61$	800.490	1.68020	0.840101	+	0.542431i	$0.182496\pi$
$62$	67.8541	0.138992
$63$	203.503	0.406968
$64$	264.550	0.516700
$65$	−51.6887	−0.0986337
$66$	0	0
$67$	128.194	0.233751	0.116876	−	0.993147i	$-0.462712\pi$
$67$	128.194	0.233751	0.116876	+	0.993147i	$0.462712\pi$
$68$	−254.490	−0.453844
$69$	388.994	0.678686
$70$	−73.2784	−0.125121
$71$	−288.991	−0.483055	−0.241528	−	0.970394i	$-0.577648\pi$
$71$	−288.991	−0.483055	−0.241528	+	0.970394i	$0.577648\pi$
$72$	507.719	0.831045
$73$	−25.4865	−0.0408626	−0.0204313	−	0.999791i	$-0.506504\pi$
$73$	−25.4865	−0.0408626	−0.0204313	+	0.999791i	$0.506504\pi$
$74$	242.466	0.380893
$75$	49.5146	0.0762327
$76$	−508.929	−0.768134
$77$	0	0
$78$	−34.0281	−0.0493965
$79$	214.115	0.304934	0.152467	−	0.988309i	$-0.451278\pi$
$79$	214.115	0.304934	0.152467	+	0.988309i	$0.451278\pi$
$80$	26.6944	0.0373066
$81$	426.648	0.585251
$82$	261.481	0.352143
$83$	518.134	0.685212	0.342606	−	0.939479i	$-0.388690\pi$
$83$	518.134	0.685212	0.342606	+	0.939479i	$0.388690\pi$
$84$	91.4824	0.118828
$85$	242.931	0.309995
$86$	−442.166	−0.554418
$87$	334.804	0.412584
$88$	0	0
$89$	6.39690	0.00761876	0.00380938	−	0.999993i	$-0.498787\pi$
$89$	6.39690	0.00761876	0.00380938	+	0.999993i	$0.498787\pi$
$90$	−191.767	−0.224600
$91$	91.1616	0.105015
$92$	−1028.74	−1.16580
$93$	80.8631	0.0901625
$94$	802.797	0.880874
$95$	485.814	0.524668
$96$	366.169	0.389292
$97$	−325.571	−0.340791	−0.170395	−	0.985376i	$-0.554504\pi$
$97$	−325.571	−0.340791	−0.170395	+	0.985376i	$0.554504\pi$
$98$	−440.812	−0.454375
$99$	0	0
$100$	−130.948	−0.130948
$101$	1255.66	1.23706	0.618528	−	0.785763i	$-0.287730\pi$
$101$	1255.66	1.23706	0.618528	+	0.785763i	$0.287730\pi$
$102$	159.928	0.155248
$103$	−1462.82	−1.39938	−0.699688	−	0.714449i	$-0.746677\pi$
$103$	−1462.82	−1.39938	−0.699688	+	0.714449i	$0.746677\pi$
$104$	227.438	0.214444
$105$	−87.3273	−0.0811645
$106$	−573.202	−0.525230
$107$	1571.15	1.41952	0.709762	−	0.704442i	$-0.248803\pi$
$107$	1571.15	1.41952	0.709762	+	0.704442i	$0.248803\pi$
$108$	519.507	0.462867
$109$	1292.93	1.13615	0.568074	−	0.822977i	$-0.307689\pi$
$109$	1292.93	1.13615	0.568074	+	0.822977i	$0.307689\pi$
$110$	0	0
$111$	288.951	0.247081
$112$	−47.0800	−0.0397201
$113$	999.873	0.832390	0.416195	−	0.909275i	$-0.363363\pi$
$113$	999.873	0.832390	0.416195	+	0.909275i	$0.363363\pi$
$114$	319.825	0.262758
$115$	982.017	0.796292
$116$	−885.432	−0.708709
$117$	238.567	0.188509
$118$	−1283.52	−1.00134
$119$	−428.449	−0.330049
$120$	−217.872	−0.165741
$121$	0	0
$122$	1330.38	0.987269
$123$	311.612	0.228432
$124$	−213.853	−0.154875
$125$	125.000	0.0894427
$126$	338.213	0.239130
$127$	−646.206	−0.451508	−0.225754	−	0.974184i	$-0.572485\pi$
$127$	−646.206	−0.451508	−0.225754	+	0.974184i	$0.572485\pi$
$128$	−1039.36	−0.717717
$129$	−526.937	−0.359645
$130$	−85.9042	−0.0579561
$131$	367.577	0.245155	0.122578	−	0.992459i	$-0.460884\pi$
$131$	367.577	0.245155	0.122578	+	0.992459i	$0.460884\pi$
$132$	0	0
$133$	−856.814	−0.558610
$134$	213.052	0.137350
$135$	−495.911	−0.316157
$136$	−1068.94	−0.673974
$137$	−791.385	−0.493523	−0.246761	−	0.969076i	$-0.579366\pi$
$137$	−791.385	−0.493523	−0.246761	+	0.969076i	$0.579366\pi$
$138$	646.490	0.398789
$139$	−1909.61	−1.16526	−0.582629	−	0.812739i	$-0.697976\pi$
$139$	−1909.61	−1.16526	−0.582629	+	0.812739i	$0.697976\pi$
$140$	230.948	0.139419
$141$	956.709	0.571414
$142$	−480.290	−0.283838
$143$	0	0
$144$	−123.207	−0.0713003
$145$	845.216	0.484078
$146$	−42.3574	−0.0240105
$147$	−525.324	−0.294748
$148$	−764.167	−0.424420
$149$	1969.24	1.08273	0.541363	−	0.840789i	$-0.317908\pi$
$149$	1969.24	1.08273	0.541363	+	0.840789i	$0.317908\pi$
$150$	82.2911	0.0447936
$151$	2653.58	1.43010	0.715049	−	0.699074i	$-0.246404\pi$
$151$	2653.58	1.43010	0.715049	+	0.699074i	$0.246404\pi$
$152$	−2137.66	−1.14070
$153$	−1121.24	−0.592462
$154$	0	0
$155$	204.139	0.105786
$156$	107.245	0.0550414
$157$	−2060.15	−1.04725	−0.523625	−	0.851949i	$-0.675421\pi$
$157$	−2060.15	−1.04725	−0.523625	+	0.851949i	$0.675421\pi$
$158$	355.850	0.179176
$159$	−683.096	−0.340711
$160$	924.397	0.456750
$161$	−1731.95	−0.847807
$162$	709.070	0.343887
$163$	−219.965	−0.105699	−0.0528496	−	0.998602i	$-0.516830\pi$
$163$	−219.965	−0.105699	−0.0528496	+	0.998602i	$0.516830\pi$
$164$	−824.096	−0.392385
$165$	0	0
$166$	861.115	0.402623
$167$	−990.902	−0.459151	−0.229576	−	0.973291i	$-0.573734\pi$
$167$	−990.902	−0.459151	−0.229576	+	0.973291i	$0.573734\pi$
$168$	384.254	0.176463
$169$	−2090.13	−0.951357
$170$	403.740	0.182150
$171$	−2242.25	−1.00274
$172$	1393.55	0.617775
$173$	1556.90	0.684213	0.342107	−	0.939661i	$-0.388860\pi$
$173$	1556.90	0.684213	0.342107	+	0.939661i	$0.388860\pi$
$174$	556.430	0.242430
$175$	−220.458	−0.0952291
$176$	0	0
$177$	−1529.60	−0.649558
$178$	10.6314	0.00447671
$179$	−1344.51	−0.561417	−0.280708	−	0.959793i	$-0.590569\pi$
$179$	−1344.51	−0.561417	−0.280708	+	0.959793i	$0.590569\pi$
$180$	604.383	0.250267
$181$	−1489.71	−0.611765	−0.305882	−	0.952069i	$-0.598951\pi$
$181$	−1489.71	−0.611765	−0.305882	+	0.952069i	$0.598951\pi$
$182$	151.506	0.0617055
$183$	1585.44	0.640432
$184$	−4321.03	−1.73125
$185$	729.459	0.289897
$186$	134.391	0.0529786
$187$	0	0
$188$	−2530.14	−0.981538
$189$	874.623	0.336611
$190$	807.401	0.308289
$191$	4194.91	1.58918	0.794589	−	0.607148i	$-0.207686\pi$
$191$	4194.91	1.58918	0.794589	+	0.607148i	$0.207686\pi$
$192$	523.964	0.196947
$193$	−3372.22	−1.25771	−0.628854	−	0.777523i	$-0.716476\pi$
$193$	−3372.22	−1.25771	−0.628854	+	0.777523i	$0.716476\pi$
$194$	−541.084	−0.200245
$195$	−102.374	−0.0375956
$196$	1389.29	0.506300
$197$	5256.04	1.90090	0.950450	−	0.310878i	$-0.100623\pi$
$197$	5256.04	1.90090	0.950450	+	0.310878i	$0.100623\pi$
$198$	0	0
$199$	3413.08	1.21581	0.607907	−	0.794008i	$-0.292009\pi$
$199$	3413.08	1.21581	0.607907	+	0.794008i	$0.292009\pi$
$200$	−550.020	−0.194461
$201$	253.898	0.0890975
$202$	2086.85	0.726881
$203$	−1490.68	−0.515395
$204$	−504.039	−0.172989
$205$	786.666	0.268015
$206$	−2431.14	−0.822259
$207$	−4532.46	−1.52187
$208$	−55.1919	−0.0183984
$209$	0	0
$210$	−145.134	−0.0476914
$211$	1088.49	0.355142	0.177571	−	0.984108i	$-0.443176\pi$
$211$	1088.49	0.355142	0.177571	+	0.984108i	$0.443176\pi$
$212$	1806.53	0.585251
$213$	−572.371	−0.184123
$214$	2611.19	0.834098
$215$	−1330.26	−0.421966
$216$	2182.09	0.687372
$217$	−360.034	−0.112630
$218$	2148.79	0.667590
$219$	−50.4782	−0.0155753
$220$	0	0
$221$	−502.271	−0.152880
$222$	480.224	0.145182
$223$	−195.453	−0.0586929	−0.0293464	−	0.999569i	$-0.509343\pi$
$223$	−195.453	−0.0586929	−0.0293464	+	0.999569i	$0.509343\pi$
$224$	−1630.33	−0.486299
$225$	−576.932	−0.170943
$226$	1661.74	0.489104
$227$	1935.65	0.565962	0.282981	−	0.959126i	$-0.408677\pi$
$227$	1935.65	0.565962	0.282981	+	0.959126i	$0.408677\pi$
$228$	−1007.98	−0.292785
$229$	3483.24	1.00515	0.502575	−	0.864534i	$-0.332386\pi$
$229$	3483.24	1.00515	0.502575	+	0.864534i	$0.332386\pi$
$230$	1632.07	0.467893
$231$	0	0
$232$	−3719.09	−1.05246
$233$	−943.151	−0.265184	−0.132592	−	0.991171i	$-0.542330\pi$
$233$	−943.151	−0.265184	−0.132592	+	0.991171i	$0.542330\pi$
$234$	396.487	0.110766
$235$	2415.22	0.670432
$236$	4045.22	1.11577
$237$	424.073	0.116230
$238$	−712.064	−0.193934
$239$	−402.723	−0.108996	−0.0544979	−	0.998514i	$-0.517356\pi$
$239$	−402.723	−0.108996	−0.0544979	+	0.998514i	$0.517356\pi$
$240$	52.8705	0.0142199
$241$	−4356.62	−1.16446	−0.582229	−	0.813025i	$-0.697819\pi$
$241$	−4356.62	−1.16446	−0.582229	+	0.813025i	$0.697819\pi$
$242$	0	0
$243$	3522.93	0.930026
$244$	−4192.89	−1.10009
$245$	−1326.18	−0.345824
$246$	517.885	0.134224
$247$	−1004.44	−0.258749
$248$	−898.246	−0.229995
$249$	1026.21	0.261178
$250$	207.744	0.0525556
$251$	5231.68	1.31562	0.657810	−	0.753184i	$-0.271483\pi$
$251$	5231.68	1.31562	0.657810	+	0.753184i	$0.271483\pi$
$252$	−1065.93	−0.266458
$253$	0	0
$254$	−1073.96	−0.265301
$255$	481.145	0.118159
$256$	−3843.78	−0.938423
$257$	7412.38	1.79911	0.899555	−	0.436807i	$-0.143891\pi$
$257$	7412.38	1.79911	0.899555	+	0.436807i	$0.143891\pi$
$258$	−875.746	−0.211324
$259$	−1286.52	−0.308651
$260$	270.740	0.0645792
$261$	−3901.06	−0.925170
$262$	610.896	0.144051
$263$	−774.208	−0.181520	−0.0907599	−	0.995873i	$-0.528930\pi$
$263$	−774.208	−0.181520	−0.0907599	+	0.995873i	$0.528930\pi$
$264$	0	0
$265$	−1724.48	−0.399751
$266$	−1423.99	−0.328234
$267$	12.6696	0.00290399
$268$	−671.465	−0.153046
$269$	−3400.24	−0.770692	−0.385346	−	0.922772i	$-0.625918\pi$
$269$	−3400.24	−0.770692	−0.385346	+	0.922772i	$0.625918\pi$
$270$	−824.183	−0.185771
$271$	1107.64	0.248281	0.124140	−	0.992265i	$-0.460383\pi$
$271$	1107.64	0.248281	0.124140	+	0.992265i	$0.460383\pi$
$272$	259.396	0.0578242
$273$	180.553	0.0400278
$274$	−1315.25	−0.289989
$275$	0	0
$276$	−2037.51	−0.444361
$277$	−7648.53	−1.65905	−0.829523	−	0.558473i	$-0.811387\pi$
$277$	−7648.53	−1.65905	−0.829523	+	0.558473i	$0.811387\pi$
$278$	−3173.68	−0.684693
$279$	−942.197	−0.202179
$280$	970.052	0.207042
$281$	−4933.48	−1.04736	−0.523678	−	0.851917i	$-0.675440\pi$
$281$	−4933.48	−1.04736	−0.523678	+	0.851917i	$0.675440\pi$
$282$	1590.01	0.335757
$283$	−5768.82	−1.21173	−0.605867	−	0.795566i	$-0.707174\pi$
$283$	−5768.82	−1.21173	−0.605867	+	0.795566i	$0.707174\pi$
$284$	1513.71	0.316275
$285$	962.195	0.199984
$286$	0	0
$287$	−1387.42	−0.285354
$288$	−4266.52	−0.872940
$289$	−2552.38	−0.519516
$290$	1404.71	0.284440
$291$	−644.820	−0.129897
$292$	133.496	0.0267543
$293$	−6038.18	−1.20394	−0.601969	−	0.798519i	$-0.705617\pi$
$293$	−6038.18	−1.20394	−0.601969	+	0.798519i	$0.705617\pi$
$294$	−873.065	−0.173191
$295$	−3861.49	−0.762117
$296$	−3209.74	−0.630278
$297$	0	0
$298$	3272.78	0.636199
$299$	−2030.37	−0.392706
$300$	−259.353	−0.0499125
$301$	2346.13	0.449265
$302$	4410.12	0.840312
$303$	2486.94	0.471520
$304$	518.740	0.0978677
$305$	4002.45	0.751409
$306$	−1863.45	−0.348125
$307$	9098.53	1.69147	0.845734	−	0.533605i	$-0.179163\pi$
$307$	9098.53	1.69147	0.845734	+	0.533605i	$0.179163\pi$
$308$	0	0
$309$	−2897.23	−0.533391
$310$	339.271	0.0621589
$311$	3814.96	0.695584	0.347792	−	0.937572i	$-0.386932\pi$
$311$	3814.96	0.695584	0.347792	+	0.937572i	$0.386932\pi$
$312$	450.461	0.0817382
$313$	2589.83	0.467687	0.233843	−	0.972274i	$-0.424870\pi$
$313$	2589.83	0.467687	0.233843	+	0.972274i	$0.424870\pi$
$314$	−3423.88	−0.615353
$315$	1017.52	0.182002
$316$	−1121.51	−0.199652
$317$	−6968.91	−1.23474	−0.617370	−	0.786673i	$-0.711802\pi$
$317$	−6968.91	−1.23474	−0.617370	+	0.786673i	$0.711802\pi$
$318$	−1135.28	−0.200198
$319$	0	0
$320$	1322.75	0.231075
$321$	3111.80	0.541071
$322$	−2878.43	−0.498163
$323$	4720.77	0.813221
$324$	−2234.74	−0.383186
$325$	−258.443	−0.0441103
$326$	−365.572	−0.0621078
$327$	2560.76	0.433059
$328$	−3461.46	−0.582704
$329$	−4259.64	−0.713804
$330$	0	0
$331$	−2383.56	−0.395808	−0.197904	−	0.980221i	$-0.563413\pi$
$331$	−2383.56	−0.395808	−0.197904	+	0.980221i	$0.563413\pi$
$332$	−2713.93	−0.448634
$333$	−3366.79	−0.554050
$334$	−1646.83	−0.269793
$335$	640.968	0.104537
$336$	−93.2460	−0.0151398
$337$	9836.49	1.58999	0.794997	−	0.606613i	$-0.207472\pi$
$337$	9836.49	1.58999	0.794997	+	0.606613i	$0.207472\pi$
$338$	−3473.70	−0.559008
$339$	1980.33	0.317277
$340$	−1272.45	−0.202965
$341$	0	0
$342$	−3726.52	−0.589203
$343$	5363.64	0.844342
$344$	5853.35	0.917417
$345$	1944.97	0.303517
$346$	2587.50	0.402037
$347$	−9537.02	−1.47543	−0.737715	−	0.675112i	$-0.764095\pi$
$347$	−9537.02	−1.47543	−0.737715	+	0.675112i	$0.764095\pi$
$348$	−1753.67	−0.270134
$349$	−10464.6	−1.60504	−0.802519	−	0.596626i	$-0.796508\pi$
$349$	−10464.6	−1.60504	−0.802519	+	0.596626i	$0.796508\pi$
$350$	−366.392	−0.0559556
$351$	1025.32	0.155919
$352$	0	0
$353$	8203.85	1.23696	0.618480	−	0.785800i	$-0.287749\pi$
$353$	8203.85	1.23696	0.618480	+	0.785800i	$0.287749\pi$
$354$	−2542.13	−0.381674
$355$	−1444.96	−0.216029
$356$	−33.5063	−0.00498829
$357$	−848.580	−0.125803
$358$	−2234.52	−0.329883
$359$	−12956.3	−1.90475	−0.952375	−	0.304930i	$-0.901367\pi$
$359$	−12956.3	−1.90475	−0.952375	+	0.304930i	$0.901367\pi$
$360$	2538.59	0.371655
$361$	2581.60	0.376381
$362$	−2475.84	−0.359467
$363$	0	0
$364$	−477.496	−0.0687571
$365$	−127.433	−0.0182743
$366$	2634.93	0.376311
$367$	−1259.31	−0.179116	−0.0895581	−	0.995982i	$-0.528545\pi$
$367$	−1259.31	−0.179116	−0.0895581	+	0.995982i	$0.528545\pi$
$368$	1048.57	0.148535
$369$	−3630.82	−0.512231
$370$	1212.33	0.170340
$371$	3041.41	0.425612
$372$	−423.553	−0.0590328
$373$	14100.6	1.95738	0.978691	−	0.205339i	$-0.0658297\pi$
$373$	14100.6	1.95738	0.978691	+	0.205339i	$0.0658297\pi$
$374$	0	0
$375$	247.573	0.0340923
$376$	−10627.4	−1.45762
$377$	−1747.52	−0.238732
$378$	1453.58	0.197789
$379$	−11911.1	−1.61434	−0.807168	−	0.590322i	$-0.799001\pi$
$379$	−11911.1	−1.61434	−0.807168	+	0.590322i	$0.799001\pi$
$380$	−2544.65	−0.343520
$381$	−1279.86	−0.172098
$382$	6971.75	0.933785
$383$	8578.16	1.14445	0.572224	−	0.820097i	$-0.306081\pi$
$383$	8578.16	1.14445	0.572224	+	0.820097i	$0.306081\pi$
$384$	−2058.55	−0.273567
$385$	0	0
$386$	−5604.48	−0.739017
$387$	6139.75	0.806462
$388$	1705.31	0.223129
$389$	−9962.68	−1.29853	−0.649265	−	0.760563i	$-0.724923\pi$
$389$	−9962.68	−1.29853	−0.649265	+	0.760563i	$0.724923\pi$
$390$	−170.141	−0.0220908
$391$	9542.49	1.23423
$392$	5835.43	0.751871
$393$	728.016	0.0934442
$394$	8735.30	1.11695
$395$	1070.57	0.136371
$396$	0	0
$397$	9843.58	1.24442	0.622211	−	0.782850i	$-0.286235\pi$
$397$	9843.58	1.24442	0.622211	+	0.782850i	$0.286235\pi$
$398$	5672.39	0.714400
$399$	−1696.99	−0.212922
$400$	133.472	0.0166840
$401$	4986.05	0.620926	0.310463	−	0.950585i	$-0.399516\pi$
$401$	4986.05	0.620926	0.310463	+	0.950585i	$0.399516\pi$
$402$	421.967	0.0523528
$403$	−422.068	−0.0521704
$404$	−6577.01	−0.809947
$405$	2133.24	0.261732
$406$	−2477.44	−0.302841
$407$	0	0
$408$	−2117.12	−0.256894
$409$	12051.2	1.45696	0.728478	−	0.685069i	$-0.240228\pi$
$409$	12051.2	1.45696	0.728478	+	0.685069i	$0.240228\pi$
$410$	1307.40	0.157483
$411$	−1567.41	−0.188113
$412$	7662.10	0.916224
$413$	6810.38	0.811421
$414$	−7532.74	−0.894237
$415$	2590.67	0.306436
$416$	−1911.23	−0.225255
$417$	−3782.14	−0.444154
$418$	0	0
$419$	811.934	0.0946673	0.0473336	−	0.998879i	$-0.484928\pi$
$419$	811.934	0.0946673	0.0473336	+	0.998879i	$0.484928\pi$
$420$	457.412	0.0531415
$421$	−13342.8	−1.54463	−0.772315	−	0.635240i	$-0.780901\pi$
$421$	−13342.8	−1.54463	−0.772315	+	0.635240i	$0.780901\pi$
$422$	1809.02	0.208678
$423$	−11147.3	−1.28133
$424$	7588.00	0.869117
$425$	1214.65	0.138634
$426$	−951.255	−0.108189
$427$	−7058.99	−0.800020
$428$	−8229.55	−0.929416
$429$	0	0
$430$	−2210.83	−0.247943
$431$	8448.33	0.944181	0.472090	−	0.881550i	$-0.343500\pi$
$431$	8448.33	0.944181	0.472090	+	0.881550i	$0.343500\pi$
$432$	−529.522	−0.0589737
$433$	−7630.81	−0.846913	−0.423456	−	0.905916i	$-0.639183\pi$
$433$	−7630.81	−0.846913	−0.423456	+	0.905916i	$0.639183\pi$
$434$	−598.360	−0.0661802
$435$	1674.02	0.184513
$436$	−6772.24	−0.743880
$437$	19083.1	2.08894
$438$	−83.8925	−0.00915191
$439$	−1738.52	−0.189009	−0.0945043	−	0.995524i	$-0.530127\pi$
$439$	−1738.52	−0.189009	−0.0945043	+	0.995524i	$0.530127\pi$
$440$	0	0
$441$	6120.95	0.660938
$442$	−834.752	−0.0898305
$443$	1631.42	0.174968	0.0874841	−	0.996166i	$-0.472117\pi$
$443$	1631.42	0.174968	0.0874841	+	0.996166i	$0.472117\pi$
$444$	−1513.50	−0.161773
$445$	31.9845	0.00340721
$446$	−324.834	−0.0344873
$447$	3900.24	0.412696
$448$	−2332.89	−0.246024
$449$	2261.67	0.237717	0.118858	−	0.992911i	$-0.462077\pi$
$449$	2261.67	0.237717	0.118858	+	0.992911i	$0.462077\pi$
$450$	−958.835	−0.100444
$451$	0	0
$452$	−5237.24	−0.544998
$453$	5255.63	0.545102
$454$	3216.96	0.332554
$455$	455.808	0.0469640
$456$	−4233.81	−0.434795
$457$	6364.73	0.651486	0.325743	−	0.945458i	$-0.394386\pi$
$457$	6364.73	0.651486	0.325743	+	0.945458i	$0.394386\pi$
$458$	5788.99	0.590616
$459$	−4818.89	−0.490036
$460$	−5143.71	−0.521363
$461$	2491.39	0.251704	0.125852	−	0.992049i	$-0.459834\pi$
$461$	2491.39	0.251704	0.125852	+	0.992049i	$0.459834\pi$
$462$	0	0
$463$	−12869.2	−1.29175	−0.645877	−	0.763442i	$-0.723508\pi$
$463$	−12869.2	−1.29175	−0.645877	+	0.763442i	$0.723508\pi$
$464$	902.501	0.0902965
$465$	404.315	0.0403219
$466$	−1567.47	−0.155819
$467$	12857.1	1.27400	0.636998	−	0.770866i	$-0.280176\pi$
$467$	12857.1	1.27400	0.636998	+	0.770866i	$0.280176\pi$
$468$	−1249.59	−0.123424
$469$	−1130.45	−0.111300
$470$	4013.99	0.393939
$471$	−4080.31	−0.399174
$472$	16991.2	1.65695
$473$	0	0
$474$	704.790	0.0682955
$475$	2429.07	0.234639
$476$	2244.18	0.216096
$477$	7959.27	0.764004
$478$	−669.308	−0.0640448
$479$	5115.67	0.487977	0.243989	−	0.969778i	$-0.421544\pi$
$479$	5115.67	0.487977	0.243989	+	0.969778i	$0.421544\pi$
$480$	1830.85	0.174097
$481$	−1508.19	−0.142968
$482$	−7240.50	−0.684224
$483$	−3430.28	−0.323153
$484$	0	0
$485$	−1627.85	−0.152406
$486$	5854.96	0.546474
$487$	3968.67	0.369276	0.184638	−	0.982807i	$-0.440889\pi$
$487$	3968.67	0.369276	0.184638	+	0.982807i	$0.440889\pi$
$488$	−17611.4	−1.63367
$489$	−435.659	−0.0402887
$490$	−2204.06	−0.203203
$491$	−42.2164	−0.00388024	−0.00194012	−	0.999998i	$-0.500618\pi$
$491$	−42.2164	−0.00388024	−0.00194012	+	0.999998i	$0.500618\pi$
$492$	−1632.19	−0.149563
$493$	8213.16	0.750309
$494$	−1669.34	−0.152039
$495$	0	0
$496$	217.975	0.0197326
$497$	2548.42	0.230005
$498$	1705.51	0.153465
$499$	−158.607	−0.0142289	−0.00711447	−	0.999975i	$-0.502265\pi$
$499$	−158.607	−0.0142289	−0.00711447	+	0.999975i	$0.502265\pi$
$500$	−654.738	−0.0585616
$501$	−1962.56	−0.175012
$502$	8694.82	0.773045
$503$	8842.46	0.783828	0.391914	−	0.920002i	$-0.371813\pi$
$503$	8842.46	0.783828	0.391914	+	0.920002i	$0.371813\pi$
$504$	−4477.24	−0.395698
$505$	6278.29	0.553228
$506$	0	0
$507$	−4139.68	−0.362623
$508$	3384.76	0.295619
$509$	11018.9	0.959539	0.479770	−	0.877394i	$-0.340720\pi$
$509$	11018.9	0.959539	0.479770	+	0.877394i	$0.340720\pi$
$510$	799.642	0.0694289
$511$	224.749	0.0194565
$512$	1926.73	0.166309
$513$	−9636.82	−0.829388
$514$	12319.0	1.05714
$515$	−7314.08	−0.625820
$516$	2760.05	0.235473
$517$	0	0
$518$	−2138.14	−0.181360
$519$	3083.57	0.260797
$520$	1137.19	0.0959022
$521$	−21660.3	−1.82141	−0.910704	−	0.413060i	$-0.864460\pi$
$521$	−21660.3	−1.82141	−0.910704	+	0.413060i	$0.864460\pi$
$522$	−6483.38	−0.543621
$523$	13269.4	1.10943	0.554715	−	0.832040i	$-0.312827\pi$
$523$	13269.4	1.10943	0.554715	+	0.832040i	$0.312827\pi$
$524$	−1925.33	−0.160512
$525$	−436.636	−0.0362979
$526$	−1286.70	−0.106659
$527$	1983.67	0.163966
$528$	0	0
$529$	26407.3	2.17040
$530$	−2866.01	−0.234890
$531$	17822.5	1.45656
$532$	4487.91	0.365743
$533$	−1626.47	−0.132177
$534$	21.0563	0.00170636
$535$	7855.76	0.634830
$536$	−2820.36	−0.227278
$537$	−2662.92	−0.213992
$538$	−5651.04	−0.452851
$539$	0	0
$540$	2597.54	0.207000
$541$	−6801.60	−0.540524	−0.270262	−	0.962787i	$-0.587110\pi$
$541$	−6801.60	−0.540524	−0.270262	+	0.962787i	$0.587110\pi$
$542$	1840.84	0.145887
$543$	−2950.50	−0.233183
$544$	8982.59	0.707951
$545$	6464.65	0.508101
$546$	300.071	0.0235199
$547$	24734.6	1.93341	0.966704	−	0.255898i	$-0.0823712\pi$
$547$	24734.6	1.93341	0.966704	+	0.255898i	$0.0823712\pi$
$548$	4145.20	0.323128
$549$	−18473.1	−1.43609
$550$	0	0
$551$	16424.7	1.26990
$552$	−8558.17	−0.659891
$553$	−1888.14	−0.145193
$554$	−12711.5	−0.974838
$555$	1444.76	0.110498
$556$	10002.3	0.762938
$557$	−19430.2	−1.47807	−0.739033	−	0.673670i	$-0.764717\pi$
$557$	−19430.2	−1.47807	−0.739033	+	0.673670i	$0.764717\pi$
$558$	−1565.89	−0.118798
$559$	2750.37	0.208101
$560$	−235.400	−0.0177633
$561$	0	0
$562$	−8199.22	−0.615415
$563$	−12323.1	−0.922483	−0.461242	−	0.887275i	$-0.652596\pi$
$563$	−12323.1	−0.922483	−0.461242	+	0.887275i	$0.652596\pi$
$564$	−5011.15	−0.374127
$565$	4999.36	0.372256
$566$	−9587.53	−0.712003
$567$	−3762.32	−0.278664
$568$	6358.04	0.469678
$569$	−19071.5	−1.40513	−0.702564	−	0.711620i	$-0.747962\pi$
$569$	−19071.5	−1.40513	−0.702564	+	0.711620i	$0.747962\pi$
$570$	1599.13	0.117509
$571$	4453.84	0.326423	0.163211	−	0.986591i	$-0.447815\pi$
$571$	4453.84	0.326423	0.163211	+	0.986591i	$0.447815\pi$
$572$	0	0
$573$	8308.37	0.605737
$574$	−2305.83	−0.167671
$575$	4910.09	0.356113
$576$	−6105.10	−0.441631
$577$	3184.19	0.229739	0.114870	−	0.993381i	$-0.463355\pi$
$577$	3184.19	0.229739	0.114870	+	0.993381i	$0.463355\pi$
$578$	−4241.95	−0.305262
$579$	−6678.97	−0.479393
$580$	−4427.16	−0.316944
$581$	−4569.08	−0.326260
$582$	−1071.66	−0.0763262
$583$	0	0
$584$	560.724	0.0397310
$585$	1192.83	0.0843036
$586$	−10035.2	−0.707422
$587$	23915.2	1.68158	0.840788	−	0.541365i	$-0.182092\pi$
$587$	23915.2	1.68158	0.840788	+	0.541365i	$0.182092\pi$
$588$	2751.60	0.192983
$589$	3966.95	0.277513
$590$	−6417.62	−0.447812
$591$	10410.0	0.724554
$592$	778.899	0.0540752
$593$	−6201.48	−0.429451	−0.214725	−	0.976674i	$-0.568886\pi$
$593$	−6201.48	−0.429451	−0.214725	+	0.976674i	$0.568886\pi$
$594$	0	0
$595$	−2142.25	−0.147603
$596$	−10314.7	−0.708902
$597$	6759.90	0.463424
$598$	−3374.38	−0.230750
$599$	−8540.80	−0.582584	−0.291292	−	0.956634i	$-0.594085\pi$
$599$	−8540.80	−0.582584	−0.291292	+	0.956634i	$0.594085\pi$
$600$	−1089.36	−0.0741216
$601$	8219.57	0.557875	0.278938	−	0.960309i	$-0.410018\pi$
$601$	8219.57	0.557875	0.278938	+	0.960309i	$0.410018\pi$
$602$	3899.16	0.263984
$603$	−2958.36	−0.199790
$604$	−13899.2	−0.936340
$605$	0	0
$606$	4133.18	0.277061
$607$	24499.1	1.63820	0.819100	−	0.573651i	$-0.194473\pi$
$607$	24499.1	1.63820	0.819100	+	0.573651i	$0.194473\pi$
$608$	17963.4	1.19821
$609$	−2952.42	−0.196450
$610$	6651.89	0.441520
$611$	−4993.58	−0.330636
$612$	5872.93	0.387907
$613$	−9816.60	−0.646800	−0.323400	−	0.946262i	$-0.604826\pi$
$613$	−9816.60	−0.646800	−0.323400	+	0.946262i	$0.604826\pi$
$614$	15121.3	0.993889
$615$	1558.06	0.102158
$616$	0	0
$617$	−4396.62	−0.286874	−0.143437	−	0.989659i	$-0.545815\pi$
$617$	−4396.62	−0.286874	−0.143437	+	0.989659i	$0.545815\pi$
$618$	−4815.07	−0.313415
$619$	−7477.57	−0.485540	−0.242770	−	0.970084i	$-0.578056\pi$
$619$	−7477.57	−0.485540	−0.242770	+	0.970084i	$0.578056\pi$
$620$	−1069.26	−0.0692623
$621$	−19479.7	−1.25877
$622$	6340.29	0.408718
$623$	−56.4100	−0.00362764
$624$	−109.312	−0.00701281
$625$	625.000	0.0400000
$626$	4304.19	0.274808
$627$	0	0
$628$	10790.9	0.685674
$629$	7088.33	0.449332
$630$	1691.07	0.106942
$631$	2168.51	0.136810	0.0684050	−	0.997658i	$-0.478209\pi$
$631$	2168.51	0.136810	0.0684050	+	0.997658i	$0.478209\pi$
$632$	−4710.70	−0.296490
$633$	2155.85	0.135367
$634$	−11582.0	−0.725521
$635$	−3231.03	−0.201920
$636$	3577.99	0.223077
$637$	2741.95	0.170549
$638$	0	0
$639$	6669.13	0.412874
$640$	−5196.82	−0.320973
$641$	21985.0	1.35469	0.677344	−	0.735666i	$-0.263131\pi$
$641$	21985.0	1.35469	0.677344	+	0.735666i	$0.263131\pi$
$642$	5171.67	0.317928
$643$	28532.7	1.74995	0.874976	−	0.484166i	$-0.160877\pi$
$643$	28532.7	1.74995	0.874976	+	0.484166i	$0.160877\pi$
$644$	9071.79	0.555091
$645$	−2634.69	−0.160838
$646$	7845.70	0.477841
$647$	−28479.4	−1.73051	−0.865255	−	0.501331i	$-0.832844\pi$
$647$	−28479.4	−1.73051	−0.865255	+	0.501331i	$0.832844\pi$
$648$	−9386.60	−0.569044
$649$	0	0
$650$	−429.521	−0.0259188
$651$	−713.078	−0.0429304
$652$	1152.16	0.0692053
$653$	2776.35	0.166381	0.0831905	−	0.996534i	$-0.473489\pi$
$653$	2776.35	0.166381	0.0831905	+	0.996534i	$0.473489\pi$
$654$	4255.86	0.254461
$655$	1837.88	0.109637
$656$	839.983	0.0499937
$657$	588.159	0.0349258
$658$	−7079.33	−0.419424
$659$	−6085.26	−0.359709	−0.179854	−	0.983693i	$-0.557563\pi$
$659$	−6085.26	−0.359709	−0.179854	+	0.983693i	$0.557563\pi$
$660$	0	0
$661$	−18103.8	−1.06529	−0.532645	−	0.846339i	$-0.678802\pi$
$661$	−18103.8	−1.06529	−0.532645	+	0.846339i	$0.678802\pi$
$662$	−3961.37	−0.232573
$663$	−994.790	−0.0582722
$664$	−11399.4	−0.666236
$665$	−4284.07	−0.249818
$666$	−5595.45	−0.325554
$667$	33200.6	1.92734
$668$	5190.25	0.300624
$669$	−387.112	−0.0223716
$670$	1065.26	0.0614247
$671$	0	0
$672$	−3229.00	−0.185359
$673$	−5229.86	−0.299548	−0.149774	−	0.988720i	$-0.547855\pi$
$673$	−5229.86	−0.299548	−0.149774	+	0.988720i	$0.547855\pi$
$674$	16347.8	0.934264
$675$	−2479.56	−0.141390
$676$	10947.9	0.622890
$677$	7280.29	0.413300	0.206650	−	0.978415i	$-0.433744\pi$
$677$	7280.29	0.413300	0.206650	+	0.978415i	$0.433744\pi$
$678$	3291.22	0.186429
$679$	2870.99	0.162266
$680$	−5344.68	−0.301410
$681$	3833.71	0.215724
$682$	0	0
$683$	−4696.64	−0.263122	−0.131561	−	0.991308i	$-0.541999\pi$
$683$	−4696.64	−0.263122	−0.131561	+	0.991308i	$0.541999\pi$
$684$	11744.7	0.656535
$685$	−3956.93	−0.220710
$686$	8914.13	0.496127
$687$	6898.86	0.383126
$688$	−1420.42	−0.0787106
$689$	3565.45	0.197145
$690$	3232.45	0.178344
$691$	−12179.0	−0.670495	−0.335248	−	0.942130i	$-0.608820\pi$
$691$	−12179.0	−0.670495	−0.335248	+	0.942130i	$0.608820\pi$
$692$	−8154.89	−0.447980
$693$	0	0
$694$	−15850.1	−0.866948
$695$	−9548.03	−0.521119
$696$	−7365.96	−0.401158
$697$	7644.22	0.415417
$698$	−17391.7	−0.943105
$699$	−1867.99	−0.101078
$700$	1154.74	0.0623501
$701$	−12610.0	−0.679422	−0.339711	−	0.940530i	$-0.610329\pi$
$701$	−12610.0	−0.679422	−0.339711	+	0.940530i	$0.610329\pi$
$702$	1704.04	0.0916164
$703$	14175.2	0.760497
$704$	0	0
$705$	4783.54	0.255544
$706$	13634.4	0.726826
$707$	−11072.8	−0.589018
$708$	8011.90	0.425291
$709$	3325.26	0.176140	0.0880698	−	0.996114i	$-0.471930\pi$
$709$	3325.26	0.176140	0.0880698	+	0.996114i	$0.471930\pi$
$710$	−2401.45	−0.126936
$711$	−4941.19	−0.260632
$712$	−140.737	−0.00740778
$713$	8018.74	0.421184
$714$	−1410.30	−0.0739205
$715$	0	0
$716$	7042.43	0.367581
$717$	−797.627	−0.0415452
$718$	−21532.7	−1.11921
$719$	19063.2	0.988787	0.494393	−	0.869238i	$-0.335390\pi$
$719$	19063.2	0.988787	0.494393	+	0.869238i	$0.335390\pi$
$720$	−616.034	−0.0318865
$721$	12899.6	0.666306
$722$	4290.50	0.221158
$723$	−8628.64	−0.443849
$724$	7802.97	0.400546
$725$	4226.08	0.216486
$726$	0	0
$727$	8017.43	0.409009	0.204505	−	0.978866i	$-0.434442\pi$
$727$	8017.43	0.409009	0.204505	+	0.978866i	$0.434442\pi$
$728$	−2005.63	−0.102106
$729$	−4542.02	−0.230759
$730$	−211.787	−0.0107378
$731$	−12926.4	−0.654037
$732$	−8304.38	−0.419315
$733$	−8712.10	−0.439003	−0.219501	−	0.975612i	$-0.570443\pi$
$733$	−8712.10	−0.439003	−0.219501	+	0.975612i	$0.570443\pi$
$734$	−2092.92	−0.105247
$735$	−2626.62	−0.131815
$736$	36310.9	1.81853
$737$	0	0
$738$	−6034.27	−0.300982
$739$	−22769.9	−1.13343	−0.566715	−	0.823914i	$-0.691786\pi$
$739$	−22769.9	−1.13343	−0.566715	+	0.823914i	$0.691786\pi$
$740$	−3820.84	−0.189806
$741$	−1989.38	−0.0986259
$742$	5054.69	0.250086
$743$	−8449.10	−0.417184	−0.208592	−	0.978003i	$-0.566888\pi$
$743$	−8449.10	−0.417184	−0.208592	+	0.978003i	$0.566888\pi$
$744$	−1779.05	−0.0876656
$745$	9846.18	0.484210
$746$	23434.6	1.15014
$747$	−11957.1	−0.585660
$748$	0	0
$749$	−13855.0	−0.675900
$750$	411.455	0.0200323
$751$	−9469.75	−0.460128	−0.230064	−	0.973176i	$-0.573894\pi$
$751$	−9469.75	−0.460128	−0.230064	+	0.973176i	$0.573894\pi$
$752$	2578.91	0.125058
$753$	10361.8	0.501467
$754$	−2904.30	−0.140277
$755$	13267.9	0.639560
$756$	−4581.19	−0.220392
$757$	38387.6	1.84309	0.921545	−	0.388271i	$-0.126927\pi$
$757$	38387.6	1.84309	0.921545	+	0.388271i	$0.126927\pi$
$758$	−19795.8	−0.948567
$759$	0	0
$760$	−10688.3	−0.510138
$761$	−1865.95	−0.0888841	−0.0444420	−	0.999012i	$-0.514151\pi$
$761$	−1865.95	−0.0888841	−0.0444420	+	0.999012i	$0.514151\pi$
$762$	−2127.08	−0.101123
$763$	−11401.5	−0.540972
$764$	−21972.5	−1.04050
$765$	−5606.19	−0.264957
$766$	14256.5	0.672466
$767$	7983.80	0.375852
$768$	−7612.93	−0.357693
$769$	29983.8	1.40604	0.703019	−	0.711171i	$-0.251835\pi$
$769$	29983.8	1.40604	0.703019	+	0.711171i	$0.251835\pi$
$770$	0	0
$771$	14680.8	0.685755
$772$	17663.4	0.823470
$773$	6987.07	0.325107	0.162553	−	0.986700i	$-0.448027\pi$
$773$	6987.07	0.325107	0.162553	+	0.986700i	$0.448027\pi$
$774$	10204.0	0.473869
$775$	1020.70	0.0473091
$776$	7162.82	0.331353
$777$	−2548.07	−0.117647
$778$	−16557.5	−0.763003
$779$	15286.9	0.703095
$780$	536.224	0.0246152
$781$	0	0
$782$	15859.2	0.725222
$783$	−16766.1	−0.765225
$784$	−1416.07	−0.0645075
$785$	−10300.8	−0.468344
$786$	1209.93	0.0549069
$787$	−17890.6	−0.810332	−0.405166	−	0.914243i	$-0.632786\pi$
$787$	−17890.6	−0.810332	−0.405166	+	0.914243i	$0.632786\pi$
$788$	−27530.6	−1.24459
$789$	−1533.38	−0.0691888
$790$	1779.25	0.0801301
$791$	−8817.21	−0.396339
$792$	0	0
$793$	−8275.25	−0.370571
$794$	16359.6	0.731210
$795$	−3415.48	−0.152371
$796$	−17877.4	−0.796040
$797$	−2164.72	−0.0962087	−0.0481043	−	0.998842i	$-0.515318\pi$
$797$	−2164.72	−0.0962087	−0.0481043	+	0.998842i	$0.515318\pi$
$798$	−2820.32	−0.125111
$799$	23469.3	1.03915
$800$	4621.99	0.204265
$801$	−147.623	−0.00651186
$802$	8286.59	0.364850
$803$	0	0
$804$	−1329.89	−0.0583355
$805$	−8659.76	−0.379151
$806$	−701.458	−0.0306548
$807$	−6734.46	−0.293760
$808$	−27625.5	−1.20280
$809$	−24725.8	−1.07455	−0.537277	−	0.843406i	$-0.680547\pi$
$809$	−24725.8	−1.07455	−0.537277	+	0.843406i	$0.680547\pi$
$810$	3545.35	0.153791
$811$	−26049.1	−1.12788	−0.563938	−	0.825817i	$-0.690714\pi$
$811$	−26049.1	−1.12788	−0.563938	+	0.825817i	$0.690714\pi$
$812$	7808.04	0.337449
$813$	2193.77	0.0946357
$814$	0	0
$815$	−1099.82	−0.0472702
$816$	513.755	0.0220405
$817$	−25850.3	−1.10696
$818$	20028.6	0.856093
$819$	−2103.76	−0.0897575
$820$	−4120.48	−0.175480
$821$	−12835.8	−0.545645	−0.272822	−	0.962064i	$-0.587957\pi$
$821$	−12835.8	−0.545645	−0.272822	+	0.962064i	$0.587957\pi$
$822$	−2604.96	−0.110533
$823$	7513.40	0.318227	0.159113	−	0.987260i	$-0.449136\pi$
$823$	7513.40	0.318227	0.159113	+	0.987260i	$0.449136\pi$
$824$	32183.2	1.36062
$825$	0	0
$826$	11318.5	0.476783
$827$	−10033.8	−0.421896	−0.210948	−	0.977497i	$-0.567655\pi$
$827$	−10033.8	−0.421896	−0.210948	+	0.977497i	$0.567655\pi$
$828$	23740.6	0.996428
$829$	1442.57	0.0604374	0.0302187	−	0.999543i	$-0.490380\pi$
$829$	1442.57	0.0604374	0.0302187	+	0.999543i	$0.490380\pi$
$830$	4305.58	0.180059
$831$	−15148.6	−0.632368
$832$	−2734.85	−0.113959
$833$	−12886.9	−0.536018
$834$	−6285.74	−0.260980
$835$	−4954.51	−0.205339
$836$	0	0
$837$	−4049.40	−0.167226
$838$	1349.40	0.0556255
$839$	−33081.1	−1.36125	−0.680624	−	0.732633i	$-0.738291\pi$
$839$	−33081.1	−1.36125	−0.680624	+	0.732633i	$0.738291\pi$
$840$	1921.27	0.0789168
$841$	4186.59	0.171659
$842$	−22175.2	−0.907609
$843$	−9771.18	−0.399214
$844$	−5701.42	−0.232525
$845$	−10450.7	−0.425460
$846$	−18526.4	−0.752896
$847$	0	0
$848$	−1841.36	−0.0745667
$849$	−11425.6	−0.461869
$850$	2018.70	0.0814599
$851$	28653.6	1.15421
$852$	2998.03	0.120552
$853$	−3320.28	−0.133276	−0.0666378	−	0.997777i	$-0.521227\pi$
$853$	−3320.28	−0.133276	−0.0666378	+	0.997777i	$0.521227\pi$
$854$	−11731.7	−0.470084
$855$	−11211.3	−0.448441
$856$	−34566.6	−1.38021
$857$	−14357.0	−0.572259	−0.286130	−	0.958191i	$-0.592369\pi$
$857$	−14357.0	−0.572259	−0.286130	+	0.958191i	$0.592369\pi$
$858$	0	0
$859$	−36024.5	−1.43090	−0.715448	−	0.698666i	$-0.753778\pi$
$859$	−36024.5	−1.43090	−0.715448	+	0.698666i	$0.753778\pi$
$860$	6967.76	0.276278
$861$	−2747.90	−0.108767
$862$	14040.8	0.554791
$863$	24955.0	0.984332	0.492166	−	0.870501i	$-0.336205\pi$
$863$	24955.0	0.984332	0.492166	+	0.870501i	$0.336205\pi$
$864$	−18336.8	−0.722025
$865$	7784.50	0.305989
$866$	−12682.1	−0.497637
$867$	−5055.21	−0.198021
$868$	1885.82	0.0737431
$869$	0	0
$870$	2782.15	0.108418
$871$	−1325.23	−0.0515542
$872$	−28445.5	−1.10469
$873$	7513.29	0.291279
$874$	31715.2	1.22744
$875$	−1102.29	−0.0425877
$876$	264.400	0.0101978
$877$	−24077.3	−0.927062	−0.463531	−	0.886081i	$-0.653418\pi$
$877$	−24077.3	−0.927062	−0.463531	+	0.886081i	$0.653418\pi$
$878$	−2889.33	−0.111060
$879$	−11959.1	−0.458898
$880$	0	0
$881$	−37851.9	−1.44752	−0.723759	−	0.690052i	$-0.757587\pi$
$881$	−37851.9	−1.44752	−0.723759	+	0.690052i	$0.757587\pi$
$882$	10172.7	0.388361
$883$	−34000.9	−1.29583	−0.647917	−	0.761711i	$-0.724359\pi$
$883$	−34000.9	−1.29583	−0.647917	+	0.761711i	$0.724359\pi$
$884$	2630.85	0.100096
$885$	−7648.00	−0.290491
$886$	2711.34	0.102810
$887$	−14809.0	−0.560582	−0.280291	−	0.959915i	$-0.590431\pi$
$887$	−14809.0	−0.560582	−0.280291	+	0.959915i	$0.590431\pi$
$888$	−6357.15	−0.240239
$889$	5698.46	0.214983
$890$	53.1568	0.00200204
$891$	0	0
$892$	1023.77	0.0384285
$893$	46933.8	1.75877
$894$	6482.02	0.242496
$895$	−6722.56	−0.251073
$896$	9165.47	0.341737
$897$	−4021.31	−0.149685
$898$	3758.79	0.139680
$899$	6901.67	0.256044
$900$	3021.92	0.111923
$901$	−16757.2	−0.619604
$902$	0	0
$903$	4646.71	0.171243
$904$	−21998.0	−0.809339
$905$	−7448.56	−0.273590
$906$	8734.62	0.320296
$907$	16421.9	0.601191	0.300595	−	0.953752i	$-0.402815\pi$
$907$	16421.9	0.601191	0.300595	+	0.953752i	$0.402815\pi$
$908$	−10138.7	−0.370557
$909$	−28977.2	−1.05733
$910$	757.532	0.0275956
$911$	−23657.8	−0.860394	−0.430197	−	0.902735i	$-0.641556\pi$
$911$	−23657.8	−0.860394	−0.430197	+	0.902735i	$0.641556\pi$
$912$	1027.41	0.0373036
$913$	0	0
$914$	10577.9	0.382807
$915$	7927.19	0.286410
$916$	−18244.9	−0.658109
$917$	−3241.41	−0.116729
$918$	−8008.78	−0.287940
$919$	11134.2	0.399654	0.199827	−	0.979831i	$-0.435962\pi$
$919$	11134.2	0.399654	0.199827	+	0.979831i	$0.435962\pi$
$920$	−21605.2	−0.774240
$921$	18020.4	0.644726
$922$	4140.58	0.147899
$923$	2987.51	0.106539
$924$	0	0
$925$	3647.29	0.129646
$926$	−21388.0	−0.759021
$927$	33757.8	1.19607
$928$	31252.6	1.10551
$929$	386.886	0.0136634	0.00683172	−	0.999977i	$-0.497825\pi$
$929$	386.886	0.0136634	0.00683172	+	0.999977i	$0.497825\pi$
$930$	671.954	0.0236927
$931$	−25771.1	−0.907213
$932$	4940.13	0.173626
$933$	7555.85	0.265131
$934$	21367.9	0.748587
$935$	0	0
$936$	−5248.66	−0.183288
$937$	−40709.8	−1.41935	−0.709676	−	0.704528i	$-0.751159\pi$
$937$	−40709.8	−1.41935	−0.709676	+	0.704528i	$0.751159\pi$
$938$	−1878.76	−0.0653985
$939$	5129.38	0.178265
$940$	−12650.7	−0.438957
$941$	−33033.8	−1.14439	−0.572195	−	0.820117i	$-0.693908\pi$
$941$	−33033.8	−1.14439	−0.572195	+	0.820117i	$0.693908\pi$
$942$	−6781.29	−0.234550
$943$	30900.8	1.06709
$944$	−4123.20	−0.142160
$945$	4373.11	0.150537
$946$	0	0
$947$	−11956.8	−0.410288	−0.205144	−	0.978732i	$-0.565766\pi$
$947$	−11956.8	−0.410288	−0.205144	+	0.978732i	$0.565766\pi$
$948$	−2221.25	−0.0761002
$949$	263.473	0.00901231
$950$	4037.00	0.137871
$951$	−13802.5	−0.470638
$952$	9426.23	0.320910
$953$	46066.0	1.56582	0.782909	−	0.622136i	$-0.213735\pi$
$953$	46066.0	1.56582	0.782909	+	0.622136i	$0.213735\pi$
$954$	13228.0	0.448921
$955$	20974.5	0.710702
$956$	2109.43	0.0713637
$957$	0	0
$958$	8502.02	0.286730
$959$	6978.70	0.234989
$960$	2619.82	0.0880775
$961$	−28124.1	−0.944046
$962$	−2506.54	−0.0840065
$963$	−36257.9	−1.21329
$964$	22819.5	0.762415
$965$	−16861.1	−0.562465
$966$	−5700.96	−0.189881
$967$	14540.7	0.483555	0.241778	−	0.970332i	$-0.422270\pi$
$967$	14540.7	0.483555	0.241778	+	0.970332i	$0.422270\pi$
$968$	0	0
$969$	9349.88	0.309970
$970$	−2705.42	−0.0895524
$971$	−30945.6	−1.02275	−0.511375	−	0.859357i	$-0.670864\pi$
$971$	−30945.6	−1.02275	−0.511375	+	0.859357i	$0.670864\pi$
$972$	−18452.8	−0.608923
$973$	16839.6	0.554832
$974$	6595.75	0.216983
$975$	−511.869	−0.0168132
$976$	4273.72	0.140162
$977$	4588.59	0.150258	0.0751290	−	0.997174i	$-0.476063\pi$
$977$	4588.59	0.150258	0.0751290	+	0.997174i	$0.476063\pi$
$978$	−724.046	−0.0236732
$979$	0	0
$980$	6946.43	0.226424
$981$	−29837.3	−0.971083
$982$	−70.1617	−0.00227999
$983$	−1752.33	−0.0568571	−0.0284285	−	0.999596i	$-0.509050\pi$
$983$	−1752.33	−0.0568571	−0.0284285	+	0.999596i	$0.509050\pi$
$984$	−6855.71	−0.222106
$985$	26280.2	0.850108
$986$	13649.9	0.440874
$987$	−8436.58	−0.272076
$988$	5261.17	0.169413
$989$	−52253.4	−1.68004
$990$	0	0
$991$	−23938.4	−0.767335	−0.383668	−	0.923471i	$-0.625339\pi$
$991$	−23938.4	−0.767335	−0.383668	+	0.923471i	$0.625339\pi$
$992$	7548.24	0.241589
$993$	−4720.84	−0.150867
$994$	4235.36	0.135148
$995$	17065.4	0.543729
$996$	−5375.18	−0.171003
$997$	−31948.4	−1.01486	−0.507430	−	0.861693i	$-0.669404\pi$
$997$	−31948.4	−1.01486	−0.507430	+	0.861693i	$0.669404\pi$
$998$	−263.598	−0.00836078
$999$	−14469.9	−0.458265

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	605.4.a.t.1.7		12
11.7	odd	10		55.4.g.b.16.3	✓	24
11.8	odd	10		55.4.g.b.31.3	yes	24
11.10	odd	2		605.4.a.p.1.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.4.g.b.16.3	✓	24	11.7	odd	10
55.4.g.b.31.3	yes	24	11.8	odd	10
605.4.a.p.1.6		12	11.10	odd	2
605.4.a.t.1.7		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 605 = 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	605.a (trivial)

\(f(q)\)	\(=\)	\(q+1.66196 q^{2} +1.98058 q^{3} -5.23790 q^{4} +5.00000 q^{5} +3.29164 q^{6} -8.81834 q^{7} -22.0008 q^{8} -23.0773 q^{9} +8.30978 q^{10} -10.3741 q^{12} -10.3377 q^{13} -14.6557 q^{14} +9.90292 q^{15} +5.33888 q^{16} +48.5862 q^{17} -38.3534 q^{18} +97.1627 q^{19} -26.1895 q^{20} -17.4655 q^{21} +196.403 q^{23} -43.5745 q^{24} +25.0000 q^{25} -17.1808 q^{26} -99.1823 q^{27} +46.1896 q^{28} +169.043 q^{29} +16.4582 q^{30} +40.8279 q^{31} +184.879 q^{32} +80.7481 q^{34} -44.0917 q^{35} +120.877 q^{36} +145.892 q^{37} +161.480 q^{38} -20.4747 q^{39} -110.004 q^{40} +157.333 q^{41} -29.0268 q^{42} -266.051 q^{43} -115.386 q^{45} +326.414 q^{46} +483.044 q^{47} +10.5741 q^{48} -265.237 q^{49} +41.5489 q^{50} +96.2290 q^{51} +54.1480 q^{52} -344.896 q^{53} -164.837 q^{54} +194.010 q^{56} +192.439 q^{57} +280.942 q^{58} -772.298 q^{59} -51.8706 q^{60} +800.490 q^{61} +67.8541 q^{62} +203.503 q^{63} +264.550 q^{64} -51.6887 q^{65} +128.194 q^{67} -254.490 q^{68} +388.994 q^{69} -73.2784 q^{70} -288.991 q^{71} +507.719 q^{72} -25.4865 q^{73} +242.466 q^{74} +49.5146 q^{75} -508.929 q^{76} -34.0281 q^{78} +214.115 q^{79} +26.6944 q^{80} +426.648 q^{81} +261.481 q^{82} +518.134 q^{83} +91.4824 q^{84} +242.931 q^{85} -442.166 q^{86} +334.804 q^{87} +6.39690 q^{89} -191.767 q^{90} +91.1616 q^{91} -1028.74 q^{92} +80.8631 q^{93} +802.797 q^{94} +485.814 q^{95} +366.169 q^{96} -325.571 q^{97} -440.812 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} - 3 q^{3} + 47 q^{4} + 60 q^{5} + 30 q^{6} + 97 q^{7} + 105 q^{8} + 131 q^{9} + 35 q^{10} + 5 q^{12} + 205 q^{13} + 109 q^{14} - 15 q^{15} + 115 q^{16} + 231 q^{17} + 241 q^{18} + 276 q^{19}+ \cdots + 3889 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 - 3 * q^3 + 47 * q^4 + 60 * q^5 + 30 * q^6 + 97 * q^7 + 105 * q^8 + 131 * q^9 + 35 * q^10 + 5 * q^12 + 205 * q^13 + 109 * q^14 - 15 * q^15 + 115 * q^16 + 231 * q^17 + 241 * q^18 + 276 * q^19 + 235 * q^20 + 100 * q^21 - 91 * q^23 + 320 * q^24 + 300 * q^25 - 462 * q^26 - 366 * q^27 + 866 * q^28 + 140 * q^29 + 150 * q^30 + 420 * q^31 + 252 * q^32 + 523 * q^34 + 485 * q^35 - 81 * q^36 - 235 * q^37 - 1044 * q^38 + 352 * q^39 + 525 * q^40 + 306 * q^41 - 558 * q^42 + 1829 * q^43 + 655 * q^45 - 959 * q^46 - 157 * q^47 - 659 * q^48 - 71 * q^49 + 175 * q^50 + 872 * q^51 + 1154 * q^52 - 771 * q^53 - 1392 * q^54 + 4326 * q^56 + 546 * q^57 - 598 * q^58 - 556 * q^59 + 25 * q^60 - 6 * q^61 + 1315 * q^62 + 3743 * q^63 + 583 * q^64 + 1025 * q^65 + 1983 * q^67 + 1534 * q^68 + 558 * q^69 + 545 * q^70 - 488 * q^71 + 1142 * q^72 + 2295 * q^73 + 3299 * q^74 - 75 * q^75 - 1524 * q^76 + 5032 * q^78 + 1534 * q^79 + 575 * q^80 - 1980 * q^81 + 1132 * q^82 + 4225 * q^83 + 727 * q^84 + 1155 * q^85 + 3612 * q^86 + 1922 * q^87 + 2284 * q^89 + 1205 * q^90 + 500 * q^91 - 3335 * q^92 - 6318 * q^93 + 1716 * q^94 + 1380 * q^95 + 5247 * q^96 - 765 * q^97 + 3889 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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