LMFDB - Embedded newform 471.4.a.c.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, 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("471.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29561 q^{2} -3.00000 q^{3} -6.32139 q^{4} -17.0439 q^{5} -3.88684 q^{6} +13.0789 q^{7} -18.5550 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+1.29561 q^{2} -3.00000 q^{3} -6.32139 q^{4} -17.0439 q^{5} -3.88684 q^{6} +13.0789 q^{7} -18.5550 q^{8} +9.00000 q^{9} -22.0823 q^{10} -58.1406 q^{11} +18.9642 q^{12} -76.5545 q^{13} +16.9452 q^{14} +51.1318 q^{15} +26.5310 q^{16} +31.4366 q^{17} +11.6605 q^{18} -95.5862 q^{19} +107.741 q^{20} -39.2366 q^{21} -75.3277 q^{22} -84.0752 q^{23} +55.6649 q^{24} +165.496 q^{25} -99.1850 q^{26} -27.0000 q^{27} -82.6766 q^{28} +242.496 q^{29} +66.2470 q^{30} +59.8467 q^{31} +182.814 q^{32} +174.422 q^{33} +40.7297 q^{34} -222.915 q^{35} -56.8925 q^{36} -350.655 q^{37} -123.843 q^{38} +229.664 q^{39} +316.250 q^{40} +428.368 q^{41} -50.8355 q^{42} -416.188 q^{43} +367.529 q^{44} -153.395 q^{45} -108.929 q^{46} +448.930 q^{47} -79.5931 q^{48} -171.943 q^{49} +214.418 q^{50} -94.3099 q^{51} +483.931 q^{52} +288.679 q^{53} -34.9816 q^{54} +990.944 q^{55} -242.678 q^{56} +286.758 q^{57} +314.180 q^{58} -119.355 q^{59} -323.224 q^{60} +514.231 q^{61} +77.5382 q^{62} +117.710 q^{63} +24.6078 q^{64} +1304.79 q^{65} +225.983 q^{66} +978.181 q^{67} -198.723 q^{68} +252.226 q^{69} -288.812 q^{70} -285.867 q^{71} -166.995 q^{72} -1068.09 q^{73} -454.313 q^{74} -496.487 q^{75} +604.237 q^{76} -760.413 q^{77} +297.555 q^{78} +374.903 q^{79} -452.193 q^{80} +81.0000 q^{81} +554.999 q^{82} -1366.02 q^{83} +248.030 q^{84} -535.804 q^{85} -539.219 q^{86} -727.487 q^{87} +1078.80 q^{88} -136.973 q^{89} -198.741 q^{90} -1001.25 q^{91} +531.472 q^{92} -179.540 q^{93} +581.640 q^{94} +1629.16 q^{95} -548.441 q^{96} -972.977 q^{97} -222.772 q^{98} -523.265 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) $ 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * 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 $

$2$	1.29561	0.458068	0.229034	−	0.973418i	$-0.426443\pi$
$2$	1.29561	0.458068	0.229034	+	0.973418i	$0.426443\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−6.32139	−0.790173
$5$	−17.0439	−1.52446	−0.762228	−	0.647309i	$-0.775894\pi$
$5$	−17.0439	−1.52446	−0.762228	+	0.647309i	$0.775894\pi$
$6$	−3.88684	−0.264466
$7$	13.0789	0.706193	0.353096	−	0.935587i	$-0.385129\pi$
$7$	13.0789	0.706193	0.353096	+	0.935587i	$0.385129\pi$
$8$	−18.5550	−0.820022
$9$	9.00000	0.333333
$10$	−22.0823	−0.698305
$11$	−58.1406	−1.59364	−0.796820	−	0.604217i	$-0.793486\pi$
$11$	−58.1406	−1.59364	−0.796820	+	0.604217i	$0.793486\pi$
$12$	18.9642	0.456207
$13$	−76.5545	−1.63326	−0.816631	−	0.577160i	$-0.804161\pi$
$13$	−76.5545	−1.63326	−0.816631	+	0.577160i	$0.804161\pi$
$14$	16.9452	0.323485
$15$	51.1318	0.880145
$16$	26.5310	0.414547
$17$	31.4366	0.448500	0.224250	−	0.974532i	$-0.428007\pi$
$17$	31.4366	0.448500	0.224250	+	0.974532i	$0.428007\pi$
$18$	11.6605	0.152689
$19$	−95.5862	−1.15416	−0.577078	−	0.816689i	$-0.695807\pi$
$19$	−95.5862	−1.15416	−0.577078	+	0.816689i	$0.695807\pi$
$20$	107.741	1.20458
$21$	−39.2366	−0.407720
$22$	−75.3277	−0.729996
$23$	−84.0752	−0.762213	−0.381106	−	0.924531i	$-0.624457\pi$
$23$	−84.0752	−0.762213	−0.381106	+	0.924531i	$0.624457\pi$
$24$	55.6649	0.473440
$25$	165.496	1.32396
$26$	−99.1850	−0.748146
$27$	−27.0000	−0.192450
$28$	−82.6766	−0.558015
$29$	242.496	1.55277	0.776384	−	0.630260i	$-0.217052\pi$
$29$	242.496	1.55277	0.776384	+	0.630260i	$0.217052\pi$
$30$	66.2470	0.403167
$31$	59.8467	0.346735	0.173368	−	0.984857i	$-0.444535\pi$
$31$	59.8467	0.346735	0.173368	+	0.984857i	$0.444535\pi$
$32$	182.814	1.00991
$33$	174.422	0.920089
$34$	40.7297	0.205444
$35$	−222.915	−1.07656
$36$	−56.8925	−0.263391
$37$	−350.655	−1.55804	−0.779018	−	0.627001i	$-0.784282\pi$
$37$	−350.655	−1.55804	−0.779018	+	0.627001i	$0.784282\pi$
$38$	−123.843	−0.528683
$39$	229.664	0.942964
$40$	316.250	1.25009
$41$	428.368	1.63170	0.815852	−	0.578261i	$-0.196268\pi$
$41$	428.368	1.63170	0.815852	+	0.578261i	$0.196268\pi$
$42$	−50.8355	−0.186764
$43$	−416.188	−1.47600	−0.738002	−	0.674799i	$-0.764230\pi$
$43$	−416.188	−1.47600	−0.738002	+	0.674799i	$0.764230\pi$
$44$	367.529	1.25925
$45$	−153.395	−0.508152
$46$	−108.929	−0.349146
$47$	448.930	1.39326	0.696630	−	0.717431i	$-0.254682\pi$
$47$	448.930	1.39326	0.696630	+	0.717431i	$0.254682\pi$
$48$	−79.5931	−0.239339
$49$	−171.943	−0.501292
$50$	214.418	0.606466
$51$	−94.3099	−0.258942
$52$	483.931	1.29056
$53$	288.679	0.748173	0.374086	−	0.927394i	$-0.377956\pi$
$53$	288.679	0.748173	0.374086	+	0.927394i	$0.377956\pi$
$54$	−34.9816	−0.0881553
$55$	990.944	2.42943
$56$	−242.678	−0.579093
$57$	286.758	0.666352
$58$	314.180	0.711274
$59$	−119.355	−0.263367	−0.131684	−	0.991292i	$-0.542038\pi$
$59$	−119.355	−0.263367	−0.131684	+	0.991292i	$0.542038\pi$
$60$	−323.224	−0.695467
$61$	514.231	1.07935	0.539676	−	0.841873i	$-0.318547\pi$
$61$	514.231	1.07935	0.539676	+	0.841873i	$0.318547\pi$
$62$	77.5382	0.158828
$63$	117.710	0.235398
$64$	24.6078	0.0480621
$65$	1304.79	2.48983
$66$	225.983	0.421464
$67$	978.181	1.78364	0.891820	−	0.452391i	$-0.149429\pi$
$67$	978.181	1.78364	0.891820	+	0.452391i	$0.149429\pi$
$68$	−198.723	−0.354393
$69$	252.226	0.440064
$70$	−288.812	−0.493138
$71$	−285.867	−0.477834	−0.238917	−	0.971040i	$-0.576792\pi$
$71$	−285.867	−0.477834	−0.238917	+	0.971040i	$0.576792\pi$
$72$	−166.995	−0.273341
$73$	−1068.09	−1.71247	−0.856234	−	0.516587i	$-0.827202\pi$
$73$	−1068.09	−1.71247	−0.856234	+	0.516587i	$0.827202\pi$
$74$	−454.313	−0.713687
$75$	−496.487	−0.764391
$76$	604.237	0.911983
$77$	−760.413	−1.12542
$78$	297.555	0.431942
$79$	374.903	0.533923	0.266962	−	0.963707i	$-0.413980\pi$
$79$	374.903	0.533923	0.266962	+	0.963707i	$0.413980\pi$
$80$	−452.193	−0.631959
$81$	81.0000	0.111111
$82$	554.999	0.747432
$83$	−1366.02	−1.80650	−0.903252	−	0.429111i	$-0.858827\pi$
$83$	−1366.02	−1.80650	−0.903252	+	0.429111i	$0.858827\pi$
$84$	248.030	0.322170
$85$	−535.804	−0.683719
$86$	−539.219	−0.676110
$87$	−727.487	−0.896491
$88$	1078.80	1.30682
$89$	−136.973	−0.163136	−0.0815679	−	0.996668i	$-0.525993\pi$
$89$	−136.973	−0.163136	−0.0815679	+	0.996668i	$0.525993\pi$
$90$	−198.741	−0.232768
$91$	−1001.25	−1.15340
$92$	531.472	0.602280
$93$	−179.540	−0.200188
$94$	581.640	0.638208
$95$	1629.16	1.75946
$96$	−548.441	−0.583073
$97$	−972.977	−1.01846	−0.509231	−	0.860630i	$-0.670070\pi$
$97$	−972.977	−1.01846	−0.509231	+	0.860630i	$0.670070\pi$
$98$	−222.772	−0.229626
$99$	−523.265	−0.531213
$100$	−1046.16	−1.04616
$101$	−65.1152	−0.0641505	−0.0320753	−	0.999485i	$-0.510212\pi$
$101$	−65.1152	−0.0641505	−0.0320753	+	0.999485i	$0.510212\pi$
$102$	−122.189	−0.118613
$103$	−0.465769	−0.000445569 0	−0.000222784	−	1.00000i	$-0.500071\pi$
$103$	−0.465769	−0.000445569 0	−0.000222784		1.00000i	$0.500071\pi$
$104$	1420.47	1.33931
$105$	668.746	0.621552
$106$	374.017	0.342714
$107$	40.9221	0.0369728	0.0184864	−	0.999829i	$-0.494115\pi$
$107$	40.9221	0.0369728	0.0184864	+	0.999829i	$0.494115\pi$
$108$	170.677	0.152069
$109$	−1515.90	−1.33208	−0.666039	−	0.745917i	$-0.732011\pi$
$109$	−1515.90	−1.33208	−0.666039	+	0.745917i	$0.732011\pi$
$110$	1283.88	1.11285
$111$	1051.97	0.899533
$112$	346.996	0.292750
$113$	−1797.17	−1.49613	−0.748067	−	0.663623i	$-0.769018\pi$
$113$	−1797.17	−1.49613	−0.748067	+	0.663623i	$0.769018\pi$
$114$	371.528	0.305235
$115$	1432.97	1.16196
$116$	−1532.91	−1.22696
$117$	−688.991	−0.544421
$118$	−154.638	−0.120640
$119$	411.156	0.316728
$120$	−948.749	−0.721738
$121$	2049.33	1.53969
$122$	666.244	0.494417
$123$	−1285.10	−0.942065
$124$	−378.314	−0.273981
$125$	−690.203	−0.493869
$126$	152.506	0.107828
$127$	1262.33	0.881999	0.441000	−	0.897507i	$-0.354624\pi$
$127$	1262.33	0.881999	0.441000	+	0.897507i	$0.354624\pi$
$128$	−1430.63	−0.987897
$129$	1248.57	0.852171
$130$	1690.50	1.14051
$131$	−100.921	−0.0673091	−0.0336546	−	0.999434i	$-0.510715\pi$
$131$	−100.921	−0.0673091	−0.0336546	+	0.999434i	$0.510715\pi$
$132$	−1102.59	−0.727029
$133$	−1250.16	−0.815057
$134$	1267.34	0.817029
$135$	460.186	0.293382
$136$	−583.306	−0.367780
$137$	2631.13	1.64082	0.820411	−	0.571775i	$-0.193745\pi$
$137$	2631.13	1.64082	0.820411	+	0.571775i	$0.193745\pi$
$138$	326.787	0.201579
$139$	−19.2049	−0.0117190	−0.00585949	−	0.999983i	$-0.501865\pi$
$139$	−19.2049	−0.0117190	−0.00585949	+	0.999983i	$0.501865\pi$
$140$	1409.13	0.850668
$141$	−1346.79	−0.804399
$142$	−370.373	−0.218880
$143$	4450.92	2.60283
$144$	238.779	0.138182
$145$	−4133.08	−2.36713
$146$	−1383.83	−0.784428
$147$	515.829	0.289421
$148$	2216.63	1.23112
$149$	−1454.92	−0.799943	−0.399971	−	0.916528i	$-0.630980\pi$
$149$	−1454.92	−0.799943	−0.399971	+	0.916528i	$0.630980\pi$
$150$	−643.255	−0.350143
$151$	−2745.05	−1.47940	−0.739698	−	0.672939i	$-0.765032\pi$
$151$	−2745.05	−1.47940	−0.739698	+	0.672939i	$0.765032\pi$
$152$	1773.60	0.946433
$153$	282.930	0.149500
$154$	−985.201	−0.515518
$155$	−1020.02	−0.528582
$156$	−1451.79	−0.745105
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	485.730	0.244573
$159$	−866.038	−0.431958
$160$	−3115.86	−1.53957
$161$	−1099.61	−0.538269
$162$	104.945	0.0508965
$163$	817.476	0.392820	0.196410	−	0.980522i	$-0.437072\pi$
$163$	817.476	0.392820	0.196410	+	0.980522i	$0.437072\pi$
$164$	−2707.88	−1.28933
$165$	−2972.83	−1.40263
$166$	−1769.83	−0.827502
$167$	2830.65	1.31163	0.655816	−	0.754921i	$-0.272325\pi$
$167$	2830.65	1.31163	0.655816	+	0.754921i	$0.272325\pi$
$168$	728.034	0.334340
$169$	3663.59	1.66754
$170$	−694.195	−0.313190
$171$	−860.275	−0.384719
$172$	2630.89	1.16630
$173$	2635.48	1.15822	0.579110	−	0.815250i	$-0.303400\pi$
$173$	2635.48	1.15822	0.579110	+	0.815250i	$0.303400\pi$
$174$	−942.541	−0.410654
$175$	2164.49	0.934974
$176$	−1542.53	−0.660639
$177$	358.064	0.152055
$178$	−177.464	−0.0747274
$179$	3657.05	1.52705	0.763523	−	0.645781i	$-0.223468\pi$
$179$	3657.05	1.52705	0.763523	+	0.645781i	$0.223468\pi$
$180$	969.671	0.401528
$181$	1513.78	0.621648	0.310824	−	0.950468i	$-0.399395\pi$
$181$	1513.78	0.621648	0.310824	+	0.950468i	$0.399395\pi$
$182$	−1297.23	−0.528335
$183$	−1542.69	−0.623165
$184$	1560.01	0.625031
$185$	5976.54	2.37516
$186$	−232.615	−0.0916997
$187$	−1827.74	−0.714748
$188$	−2837.86	−1.10092
$189$	−353.129	−0.135907
$190$	2110.77	0.805953
$191$	−3935.78	−1.49101	−0.745506	−	0.666499i	$-0.767792\pi$
$191$	−3935.78	−1.49101	−0.745506	+	0.666499i	$0.767792\pi$
$192$	−73.8233	−0.0277486
$193$	−3733.30	−1.39238	−0.696189	−	0.717859i	$-0.745122\pi$
$193$	−3733.30	−1.39238	−0.696189	+	0.717859i	$0.745122\pi$
$194$	−1260.60	−0.466526
$195$	−3914.37	−1.43751
$196$	1086.92	0.396108
$197$	64.7286	0.0234098	0.0117049	−	0.999931i	$-0.496274\pi$
$197$	64.7286	0.0234098	0.0117049	+	0.999931i	$0.496274\pi$
$198$	−677.949	−0.243332
$199$	−2025.58	−0.721556	−0.360778	−	0.932652i	$-0.617489\pi$
$199$	−2025.58	−0.721556	−0.360778	+	0.932652i	$0.617489\pi$
$200$	−3070.77	−1.08568
$201$	−2934.54	−1.02978
$202$	−84.3641	−0.0293853
$203$	3171.57	1.09655
$204$	596.169	0.204609
$205$	−7301.08	−2.48746
$206$	−0.603457	−0.000204101 0
$207$	−756.677	−0.254071
$208$	−2031.07	−0.677064
$209$	5557.43	1.83931
$210$	866.436	0.284713
$211$	−521.692	−0.170212	−0.0851061	−	0.996372i	$-0.527123\pi$
$211$	−521.692	−0.170212	−0.0851061	+	0.996372i	$0.527123\pi$
$212$	−1824.85	−0.591186
$213$	857.601	0.275877
$214$	53.0192	0.0169361
$215$	7093.49	2.25010
$216$	500.984	0.157813
$217$	782.728	0.244862
$218$	−1964.01	−0.610183
$219$	3204.26	0.988694
$220$	−6264.14	−1.91967
$221$	−2406.62	−0.732518
$222$	1362.94	0.412048
$223$	−2183.71	−0.655750	−0.327875	−	0.944721i	$-0.606333\pi$
$223$	−2183.71	−0.655750	−0.327875	+	0.944721i	$0.606333\pi$
$224$	2391.00	0.713193
$225$	1489.46	0.441321
$226$	−2328.43	−0.685332
$227$	1321.10	0.386274	0.193137	−	0.981172i	$-0.438134\pi$
$227$	1321.10	0.386274	0.193137	+	0.981172i	$0.438134\pi$
$228$	−1812.71	−0.526534
$229$	814.033	0.234903	0.117452	−	0.993079i	$-0.462528\pi$
$229$	814.033	0.234903	0.117452	+	0.993079i	$0.462528\pi$
$230$	1856.58	0.532257
$231$	2281.24	0.649760
$232$	−4499.50	−1.27330
$233$	−207.226	−0.0582653	−0.0291326	−	0.999576i	$-0.509275\pi$
$233$	−207.226	−0.0582653	−0.0291326	+	0.999576i	$0.509275\pi$
$234$	−892.665	−0.249382
$235$	−7651.54	−2.12396
$236$	754.488	0.208106
$237$	−1124.71	−0.308261
$238$	532.699	0.145083
$239$	−1943.07	−0.525887	−0.262943	−	0.964811i	$-0.584693\pi$
$239$	−1943.07	−0.525887	−0.262943	+	0.964811i	$0.584693\pi$
$240$	1356.58	0.364861
$241$	7055.07	1.88571	0.942857	−	0.333196i	$-0.108127\pi$
$241$	7055.07	1.88571	0.942857	+	0.333196i	$0.108127\pi$
$242$	2655.13	0.705283
$243$	−243.000	−0.0641500
$244$	−3250.65	−0.852876
$245$	2930.59	0.764197
$246$	−1665.00	−0.431530
$247$	7317.55	1.88504
$248$	−1110.45	−0.284330
$249$	4098.05	1.04299
$250$	−894.236	−0.226226
$251$	−4562.80	−1.14742	−0.573708	−	0.819060i	$-0.694495\pi$
$251$	−4562.80	−1.14742	−0.573708	+	0.819060i	$0.694495\pi$
$252$	−744.089	−0.186005
$253$	4888.18	1.21469
$254$	1635.49	0.404016
$255$	1607.41	0.394745
$256$	−2050.40	−0.500587
$257$	115.964	0.0281464	0.0140732	−	0.999901i	$-0.495520\pi$
$257$	115.964	0.0281464	0.0140732	+	0.999901i	$0.495520\pi$
$258$	1617.66	0.390353
$259$	−4586.17	−1.10027
$260$	−8248.08	−1.96740
$261$	2182.46	0.517590
$262$	−130.754	−0.0308322
$263$	−2480.64	−0.581607	−0.290803	−	0.956783i	$-0.593923\pi$
$263$	−2480.64	−0.581607	−0.290803	+	0.956783i	$0.593923\pi$
$264$	−3236.39	−0.754493
$265$	−4920.23	−1.14056
$266$	−1619.72	−0.373352
$267$	410.919	0.0941865
$268$	−6183.46	−1.40938
$269$	5230.52	1.18554	0.592770	−	0.805372i	$-0.298034\pi$
$269$	5230.52	1.18554	0.592770	+	0.805372i	$0.298034\pi$
$270$	596.223	0.134389
$271$	8462.85	1.89698	0.948490	−	0.316808i	$-0.102611\pi$
$271$	8462.85	1.89698	0.948490	+	0.316808i	$0.102611\pi$
$272$	834.046	0.185925
$273$	3003.74	0.665914
$274$	3408.92	0.751608
$275$	−9622.00	−2.10992
$276$	−1594.42	−0.347727
$277$	−3139.31	−0.680949	−0.340474	−	0.940254i	$-0.610588\pi$
$277$	−3139.31	−0.680949	−0.340474	+	0.940254i	$0.610588\pi$
$278$	−24.8821	−0.00536810
$279$	538.621	0.115578
$280$	4136.19	0.882802
$281$	3616.91	0.767854	0.383927	−	0.923364i	$-0.374572\pi$
$281$	3616.91	0.767854	0.383927	+	0.923364i	$0.374572\pi$
$282$	−1744.92	−0.368470
$283$	2764.01	0.580576	0.290288	−	0.956939i	$-0.406249\pi$
$283$	2764.01	0.580576	0.290288	+	0.956939i	$0.406249\pi$
$284$	1807.08	0.377571
$285$	−4887.49	−1.01582
$286$	5766.67	1.19227
$287$	5602.57	1.15230
$288$	1645.32	0.336638
$289$	−3924.74	−0.798847
$290$	−5354.87	−1.08431
$291$	2918.93	0.588010
$292$	6751.80	1.35315
$293$	2120.30	0.422761	0.211381	−	0.977404i	$-0.432204\pi$
$293$	2120.30	0.422761	0.211381	+	0.977404i	$0.432204\pi$
$294$	668.315	0.132575
$295$	2034.28	0.401492
$296$	6506.40	1.27762
$297$	1569.80	0.306696
$298$	−1885.01	−0.366428
$299$	6436.34	1.24489
$300$	3138.48	0.604001
$301$	−5443.27	−1.04234
$302$	−3556.52	−0.677665
$303$	195.346	0.0370373
$304$	−2536.00	−0.478452
$305$	−8764.51	−1.64542
$306$	366.568	0.0684813
$307$	3139.91	0.583726	0.291863	−	0.956460i	$-0.405725\pi$
$307$	3139.91	0.583726	0.291863	+	0.956460i	$0.405725\pi$
$308$	4806.86	0.889274
$309$	1.39731	0.000257249 0
$310$	−1321.56	−0.242127
$311$	−2788.58	−0.508444	−0.254222	−	0.967146i	$-0.581819\pi$
$311$	−2788.58	−0.508444	−0.254222	+	0.967146i	$0.581819\pi$
$312$	−4261.40	−0.773251
$313$	6826.40	1.23275	0.616376	−	0.787452i	$-0.288600\pi$
$313$	6826.40	1.23275	0.616376	+	0.787452i	$0.288600\pi$
$314$	−203.411	−0.0365578
$315$	−2006.24	−0.358853
$316$	−2369.91	−0.421892
$317$	1202.99	0.213144	0.106572	−	0.994305i	$-0.466012\pi$
$317$	1202.99	0.213144	0.106572	+	0.994305i	$0.466012\pi$
$318$	−1122.05	−0.197866
$319$	−14098.8	−2.47455
$320$	−419.413	−0.0732685
$321$	−122.766	−0.0213463
$322$	−1424.67	−0.246564
$323$	−3004.91	−0.517639
$324$	−512.032	−0.0877970
$325$	−12669.4	−2.16238
$326$	1059.13	0.179938
$327$	4547.69	0.769075
$328$	−7948.36	−1.33803
$329$	5871.50	0.983910
$330$	−3851.64	−0.642502
$331$	4436.21	0.736665	0.368332	−	0.929694i	$-0.379929\pi$
$331$	4436.21	0.736665	0.368332	+	0.929694i	$0.379929\pi$
$332$	8635.12	1.42745
$333$	−3155.90	−0.519345
$334$	3667.43	0.600817
$335$	−16672.0	−2.71908
$336$	−1040.99	−0.169019
$337$	−1650.92	−0.266859	−0.133429	−	0.991058i	$-0.542599\pi$
$337$	−1650.92	−0.266859	−0.133429	+	0.991058i	$0.542599\pi$
$338$	4746.60	0.763849
$339$	5391.50	0.863794
$340$	3387.02	0.540256
$341$	−3479.52	−0.552571
$342$	−1114.58	−0.176228
$343$	−6734.87	−1.06020
$344$	7722.37	1.21035
$345$	−4298.92	−0.670858
$346$	3414.57	0.530544
$347$	3598.19	0.556659	0.278330	−	0.960486i	$-0.410219\pi$
$347$	3598.19	0.556659	0.278330	+	0.960486i	$0.410219\pi$
$348$	4598.72	0.708384
$349$	2684.36	0.411721	0.205861	−	0.978581i	$-0.434001\pi$
$349$	2684.36	0.411721	0.205861	+	0.978581i	$0.434001\pi$
$350$	2804.35	0.428282
$351$	2066.97	0.314321
$352$	−10628.9	−1.60944
$353$	−3706.39	−0.558843	−0.279421	−	0.960169i	$-0.590143\pi$
$353$	−3706.39	−0.558843	−0.279421	+	0.960169i	$0.590143\pi$
$354$	463.913	0.0696517
$355$	4872.30	0.728436
$356$	865.858	0.128906
$357$	−1233.47	−0.182863
$358$	4738.13	0.699491
$359$	−1780.91	−0.261819	−0.130909	−	0.991394i	$-0.541790\pi$
$359$	−1780.91	−0.261819	−0.130909	+	0.991394i	$0.541790\pi$
$360$	2846.25	0.416696
$361$	2277.71	0.332076
$362$	1961.27	0.284757
$363$	−6147.98	−0.888940
$364$	6329.27	0.911384
$365$	18204.4	2.61058
$366$	−1998.73	−0.285452
$367$	−8641.10	−1.22905	−0.614526	−	0.788897i	$-0.710653\pi$
$367$	−8641.10	−1.22905	−0.614526	+	0.788897i	$0.710653\pi$
$368$	−2230.60	−0.315973
$369$	3855.31	0.543901
$370$	7743.28	1.08798
$371$	3775.60	0.528354
$372$	1134.94	0.158183
$373$	5646.46	0.783814	0.391907	−	0.920005i	$-0.371815\pi$
$373$	5646.46	0.783814	0.391907	+	0.920005i	$0.371815\pi$
$374$	−2368.05	−0.327404
$375$	2070.61	0.285135
$376$	−8329.89	−1.14250
$377$	−18564.1	−2.53608
$378$	−457.519	−0.0622546
$379$	3679.86	0.498738	0.249369	−	0.968408i	$-0.419777\pi$
$379$	3679.86	0.498738	0.249369	+	0.968408i	$0.419777\pi$
$380$	−10298.6	−1.39028
$381$	−3787.00	−0.509223
$382$	−5099.25	−0.682985
$383$	6094.43	0.813083	0.406542	−	0.913632i	$-0.366735\pi$
$383$	6094.43	0.813083	0.406542	+	0.913632i	$0.366735\pi$
$384$	4291.88	0.570363
$385$	12960.4	1.71565
$386$	−4836.91	−0.637804
$387$	−3745.70	−0.492001
$388$	6150.56	0.804762
$389$	−4136.14	−0.539102	−0.269551	−	0.962986i	$-0.586875\pi$
$389$	−4136.14	−0.539102	−0.269551	+	0.962986i	$0.586875\pi$
$390$	−5071.51	−0.658476
$391$	−2643.04	−0.341853
$392$	3190.40	0.411070
$393$	302.763	0.0388609
$394$	83.8632	0.0107233
$395$	−6389.83	−0.813942
$396$	3307.76	0.419751
$397$	304.547	0.0385008	0.0192504	−	0.999815i	$-0.493872\pi$
$397$	304.547	0.0385008	0.0192504	+	0.999815i	$0.493872\pi$
$398$	−2624.37	−0.330522
$399$	3750.48	0.470573
$400$	4390.76	0.548846
$401$	15431.0	1.92167	0.960833	−	0.277128i	$-0.0893828\pi$
$401$	15431.0	1.92167	0.960833	+	0.277128i	$0.0893828\pi$
$402$	−3802.03	−0.471712
$403$	−4581.54	−0.566309
$404$	411.618	0.0506900
$405$	−1380.56	−0.169384
$406$	4109.12	0.502297
$407$	20387.3	2.48295
$408$	1749.92	0.212338
$409$	−5988.47	−0.723988	−0.361994	−	0.932180i	$-0.617904\pi$
$409$	−5988.47	−0.723988	−0.361994	+	0.932180i	$0.617904\pi$
$410$	−9459.37	−1.13943
$411$	−7893.39	−0.947328
$412$	2.94431	0.000352077 0
$413$	−1561.03	−0.185988
$414$	−980.361	−0.116382
$415$	23282.3	2.75393
$416$	−13995.2	−1.64945
$417$	57.6147	0.00676596
$418$	7200.28	0.842530
$419$	2259.80	0.263480	0.131740	−	0.991284i	$-0.457944\pi$
$419$	2259.80	0.263480	0.131740	+	0.991284i	$0.457944\pi$
$420$	−4227.40	−0.491134
$421$	−5858.36	−0.678192	−0.339096	−	0.940752i	$-0.610121\pi$
$421$	−5858.36	−0.678192	−0.339096	+	0.940752i	$0.610121\pi$
$422$	−675.911	−0.0779688
$423$	4040.37	0.464420
$424$	−5356.44	−0.613518
$425$	5202.62	0.593798
$426$	1111.12	0.126371
$427$	6725.56	0.762231
$428$	−258.684	−0.0292149
$429$	−13352.8	−1.50275
$430$	9190.41	1.03070
$431$	−6455.58	−0.721472	−0.360736	−	0.932668i	$-0.617474\pi$
$431$	−6455.58	−0.721472	−0.360736	+	0.932668i	$0.617474\pi$
$432$	−716.337	−0.0797796
$433$	13032.3	1.44641	0.723203	−	0.690636i	$-0.242669\pi$
$433$	13032.3	1.44641	0.723203	+	0.690636i	$0.242669\pi$
$434$	1014.11	0.112163
$435$	12399.2	1.36666
$436$	9582.56	1.05257
$437$	8036.43	0.879713
$438$	4151.49	0.452890
$439$	15774.4	1.71497	0.857486	−	0.514507i	$-0.172025\pi$
$439$	15774.4	1.71497	0.857486	+	0.514507i	$0.172025\pi$
$440$	−18386.9	−1.99219
$441$	−1547.49	−0.167097
$442$	−3118.04	−0.335544
$443$	8906.02	0.955164	0.477582	−	0.878587i	$-0.341513\pi$
$443$	8906.02	0.955164	0.477582	+	0.878587i	$0.341513\pi$
$444$	−6649.88	−0.710787
$445$	2334.56	0.248693
$446$	−2829.25	−0.300378
$447$	4364.75	0.461847
$448$	321.842	0.0339411
$449$	−9081.24	−0.954499	−0.477250	−	0.878768i	$-0.658366\pi$
$449$	−9081.24	−0.954499	−0.477250	+	0.878768i	$0.658366\pi$
$450$	1929.76	0.202155
$451$	−24905.6	−2.60035
$452$	11360.6	1.18221
$453$	8235.14	0.854130
$454$	1711.63	0.176940
$455$	17065.2	1.75830
$456$	−5320.80	−0.546424
$457$	902.671	0.0923965	0.0461982	−	0.998932i	$-0.485289\pi$
$457$	902.671	0.0923965	0.0461982	+	0.998932i	$0.485289\pi$
$458$	1054.67	0.107602
$459$	−848.789	−0.0863139
$460$	−9058.37	−0.918149
$461$	10096.2	1.02001	0.510007	−	0.860170i	$-0.329643\pi$
$461$	10096.2	1.02001	0.510007	+	0.860170i	$0.329643\pi$
$462$	2955.60	0.297634
$463$	18172.9	1.82412	0.912058	−	0.410061i	$-0.134492\pi$
$463$	18172.9	1.82412	0.912058	+	0.410061i	$0.134492\pi$
$464$	6433.65	0.643696
$465$	3060.07	0.305177
$466$	−268.484	−0.0266895
$467$	−13346.7	−1.32251	−0.661257	−	0.750160i	$-0.729977\pi$
$467$	−13346.7	−1.32251	−0.661257	+	0.750160i	$0.729977\pi$
$468$	4355.38	0.430187
$469$	12793.5	1.25959
$470$	−9913.43	−0.972920
$471$	471.000	0.0460776
$472$	2214.63	0.215967
$473$	24197.4	2.35222
$474$	−1457.19	−0.141205
$475$	−15819.1	−1.52806
$476$	−2599.07	−0.250270
$477$	2598.11	0.249391
$478$	−2517.47	−0.240892
$479$	−20632.6	−1.96812	−0.984058	−	0.177845i	$-0.943087\pi$
$479$	−20632.6	−1.96812	−0.984058	+	0.177845i	$0.943087\pi$
$480$	9347.59	0.888869
$481$	26844.2	2.54468
$482$	9140.65	0.863786
$483$	3298.83	0.310770
$484$	−12954.6	−1.21662
$485$	16583.4	1.55260
$486$	−314.834	−0.0293851
$487$	−4043.33	−0.376224	−0.188112	−	0.982148i	$-0.560237\pi$
$487$	−4043.33	−0.376224	−0.188112	+	0.982148i	$0.560237\pi$
$488$	−9541.54	−0.885093
$489$	−2452.43	−0.226795
$490$	3796.91	0.350055
$491$	13939.7	1.28124	0.640620	−	0.767858i	$-0.278677\pi$
$491$	13939.7	1.28124	0.640620	+	0.767858i	$0.278677\pi$
$492$	8123.64	0.744394
$493$	7623.24	0.696417
$494$	9480.72	0.863477
$495$	8918.49	0.809811
$496$	1587.80	0.143738
$497$	−3738.82	−0.337443
$498$	5309.49	0.477759
$499$	−1090.23	−0.0978068	−0.0489034	−	0.998804i	$-0.515573\pi$
$499$	−1090.23	−0.0978068	−0.0489034	+	0.998804i	$0.515573\pi$
$500$	4363.04	0.390242
$501$	−8491.96	−0.757271
$502$	−5911.62	−0.525595
$503$	−3314.67	−0.293824	−0.146912	−	0.989150i	$-0.546933\pi$
$503$	−3314.67	−0.293824	−0.146912	+	0.989150i	$0.546933\pi$
$504$	−2184.10	−0.193031
$505$	1109.82	0.0977946
$506$	6333.19	0.556412
$507$	−10990.8	−0.962757
$508$	−7979.69	−0.696932
$509$	−1704.60	−0.148438	−0.0742190	−	0.997242i	$-0.523646\pi$
$509$	−1704.60	−0.148438	−0.0742190	+	0.997242i	$0.523646\pi$
$510$	2082.58	0.180820
$511$	−13969.4	−1.20933
$512$	8788.49	0.758594
$513$	2580.83	0.222117
$514$	150.244	0.0128930
$515$	7.93853	0.000679250 0
$516$	−7892.66	−0.673363
$517$	−26101.1	−2.22036
$518$	−5941.91	−0.504001
$519$	−7906.45	−0.668698
$520$	−24210.3	−2.04172
$521$	9494.57	0.798396	0.399198	−	0.916865i	$-0.369289\pi$
$521$	9494.57	0.798396	0.399198	+	0.916865i	$0.369289\pi$
$522$	2827.62	0.237091
$523$	−7090.57	−0.592828	−0.296414	−	0.955060i	$-0.595791\pi$
$523$	−7090.57	−0.592828	−0.296414	+	0.955060i	$0.595791\pi$
$524$	637.960	0.0531859
$525$	−6493.48	−0.539807
$526$	−3213.94	−0.266416
$527$	1881.38	0.155511
$528$	4627.59	0.381420
$529$	−5098.36	−0.419032
$530$	−6374.71	−0.522453
$531$	−1074.19	−0.0877891
$532$	7902.74	0.644036
$533$	−32793.5	−2.66500
$534$	532.392	0.0431439
$535$	−697.474	−0.0563634
$536$	−18150.1	−1.46262
$537$	−10971.2	−0.881640
$538$	6776.73	0.543059
$539$	9996.87	0.798879
$540$	−2909.01	−0.231822
$541$	5329.69	0.423552	0.211776	−	0.977318i	$-0.432075\pi$
$541$	5329.69	0.423552	0.211776	+	0.977318i	$0.432075\pi$
$542$	10964.6	0.868946
$543$	−4541.33	−0.358908
$544$	5747.05	0.452946
$545$	25836.8	2.03069
$546$	3891.68	0.305034
$547$	−16214.3	−1.26741	−0.633705	−	0.773575i	$-0.718467\pi$
$547$	−16214.3	−1.26741	−0.633705	+	0.773575i	$0.718467\pi$
$548$	−16632.4	−1.29653
$549$	4628.08	0.359784
$550$	−12466.4	−0.966489
$551$	−23179.2	−1.79214
$552$	−4680.04	−0.360862
$553$	4903.31	0.377053
$554$	−4067.33	−0.311921
$555$	−17929.6	−1.37130
$556$	121.402	0.00926003
$557$	−6520.37	−0.496008	−0.248004	−	0.968759i	$-0.579775\pi$
$557$	−6520.37	−0.496008	−0.248004	+	0.968759i	$0.579775\pi$
$558$	697.844	0.0529428
$559$	31861.1	2.41070
$560$	−5914.17	−0.446285
$561$	5483.23	0.412660
$562$	4686.12	0.351729
$563$	21583.6	1.61571	0.807853	−	0.589384i	$-0.200630\pi$
$563$	21583.6	1.61571	0.807853	+	0.589384i	$0.200630\pi$
$564$	8513.59	0.635615
$565$	30630.8	2.28079
$566$	3581.08	0.265944
$567$	1059.39	0.0784658
$568$	5304.26	0.391834
$569$	6629.78	0.488462	0.244231	−	0.969717i	$-0.421464\pi$
$569$	6629.78	0.488462	0.244231	+	0.969717i	$0.421464\pi$
$570$	−6332.30	−0.465317
$571$	−12705.9	−0.931215	−0.465608	−	0.884991i	$-0.654164\pi$
$571$	−12705.9	−0.931215	−0.465608	+	0.884991i	$0.654164\pi$
$572$	−28136.0	−2.05669
$573$	11807.3	0.860836
$574$	7258.76	0.527831
$575$	−13914.1	−1.00914
$576$	221.470	0.0160207
$577$	−9022.72	−0.650989	−0.325495	−	0.945544i	$-0.605531\pi$
$577$	−9022.72	−0.650989	−0.325495	+	0.945544i	$0.605531\pi$
$578$	−5084.94	−0.365927
$579$	11199.9	0.803889
$580$	26126.8	1.87044
$581$	−17866.0	−1.27574
$582$	3781.81	0.269349
$583$	−16784.0	−1.19232
$584$	19818.3	1.40426
$585$	11743.1	0.829945
$586$	2747.08	0.193654
$587$	−10142.8	−0.713185	−0.356592	−	0.934260i	$-0.616062\pi$
$587$	−10142.8	−0.713185	−0.356592	+	0.934260i	$0.616062\pi$
$588$	−3260.76	−0.228693
$589$	−5720.52	−0.400187
$590$	2635.63	0.183911
$591$	−194.186	−0.0135156
$592$	−9303.24	−0.645880
$593$	11314.4	0.783521	0.391760	−	0.920067i	$-0.371866\pi$
$593$	11314.4	0.783521	0.391760	+	0.920067i	$0.371866\pi$
$594$	2033.85	0.140488
$595$	−7007.71	−0.482837
$596$	9197.09	0.632093
$597$	6076.75	0.416591
$598$	8339.00	0.570246
$599$	18862.1	1.28662	0.643309	−	0.765607i	$-0.277561\pi$
$599$	18862.1	1.28662	0.643309	+	0.765607i	$0.277561\pi$
$600$	9212.30	0.626817
$601$	1673.16	0.113560	0.0567799	−	0.998387i	$-0.481917\pi$
$601$	1673.16	0.113560	0.0567799	+	0.998387i	$0.481917\pi$
$602$	−7052.38	−0.477464
$603$	8803.63	0.594546
$604$	17352.5	1.16898
$605$	−34928.6	−2.34719
$606$	253.092	0.0169656
$607$	−7661.05	−0.512278	−0.256139	−	0.966640i	$-0.582450\pi$
$607$	−7661.05	−0.512278	−0.256139	+	0.966640i	$0.582450\pi$
$608$	−17474.5	−1.16560
$609$	−9514.70	−0.633096
$610$	−11355.4	−0.753717
$611$	−34367.6	−2.27556
$612$	−1788.51	−0.118131
$613$	−1944.29	−0.128106	−0.0640532	−	0.997946i	$-0.520403\pi$
$613$	−1944.29	−0.128106	−0.0640532	+	0.997946i	$0.520403\pi$
$614$	4068.10	0.267387
$615$	21903.2	1.43614
$616$	14109.4	0.922867
$617$	−18988.7	−1.23899	−0.619496	−	0.785000i	$-0.712663\pi$
$617$	−18988.7	−1.23899	−0.619496	+	0.785000i	$0.712663\pi$
$618$	1.81037	0.000117838 0
$619$	−29167.6	−1.89393	−0.946966	−	0.321334i	$-0.895869\pi$
$619$	−29167.6	−1.89393	−0.946966	+	0.321334i	$0.895869\pi$
$620$	6447.96	0.417672
$621$	2270.03	0.146688
$622$	−3612.93	−0.232902
$623$	−1791.45	−0.115205
$624$	6093.21	0.390903
$625$	−8923.17	−0.571083
$626$	8844.38	0.564684
$627$	−16672.3	−1.06193
$628$	992.458	0.0630627
$629$	−11023.4	−0.698780
$630$	−2599.31	−0.164379
$631$	8364.94	0.527738	0.263869	−	0.964558i	$-0.415001\pi$
$631$	8364.94	0.527738	0.263869	+	0.964558i	$0.415001\pi$
$632$	−6956.33	−0.437829
$633$	1565.08	0.0982720
$634$	1558.61	0.0976347
$635$	−21515.1	−1.34457
$636$	5474.56	0.341321
$637$	13163.0	0.818741
$638$	−18266.6	−1.13352
$639$	−2572.80	−0.159278
$640$	24383.5	1.50601
$641$	11060.4	0.681530	0.340765	−	0.940149i	$-0.389314\pi$
$641$	11060.4	0.681530	0.340765	+	0.940149i	$0.389314\pi$
$642$	−159.058	−0.00977805
$643$	3511.14	0.215343	0.107672	−	0.994187i	$-0.465660\pi$
$643$	3511.14	0.215343	0.107672	+	0.994187i	$0.465660\pi$
$644$	6951.05	0.425326
$645$	−21280.5	−1.29910
$646$	−3893.20	−0.237114
$647$	−16326.8	−0.992074	−0.496037	−	0.868301i	$-0.665212\pi$
$647$	−16326.8	−0.992074	−0.496037	+	0.868301i	$0.665212\pi$
$648$	−1502.95	−0.0911135
$649$	6939.36	0.419713
$650$	−16414.7	−0.990518
$651$	−2348.18	−0.141371
$652$	−5167.58	−0.310396
$653$	28779.2	1.72468	0.862341	−	0.506328i	$-0.168998\pi$
$653$	28779.2	1.72468	0.862341	+	0.506328i	$0.168998\pi$
$654$	5892.04	0.352289
$655$	1720.09	0.102610
$656$	11365.0	0.676418
$657$	−9612.79	−0.570823
$658$	7607.20	0.450698
$659$	1221.78	0.0722212	0.0361106	−	0.999348i	$-0.488503\pi$
$659$	1221.78	0.0722212	0.0361106	+	0.999348i	$0.488503\pi$
$660$	18792.4	1.10832
$661$	9977.32	0.587099	0.293550	−	0.955944i	$-0.405163\pi$
$661$	9977.32	0.587099	0.293550	+	0.955944i	$0.405163\pi$
$662$	5747.61	0.337443
$663$	7219.85	0.422920
$664$	25346.4	1.48137
$665$	21307.6	1.24252
$666$	−4088.82	−0.237896
$667$	−20387.9	−1.18354
$668$	−17893.6	−1.03642
$669$	6551.14	0.378598
$670$	−21600.5	−1.24552
$671$	−29897.7	−1.72010
$672$	−7172.99	−0.411762
$673$	27263.6	1.56157	0.780783	−	0.624802i	$-0.214820\pi$
$673$	27263.6	1.56157	0.780783	+	0.624802i	$0.214820\pi$
$674$	−2138.96	−0.122240
$675$	−4468.38	−0.254797
$676$	−23159.0	−1.31765
$677$	25343.5	1.43874	0.719371	−	0.694626i	$-0.244430\pi$
$677$	25343.5	1.43874	0.719371	+	0.694626i	$0.244430\pi$
$678$	6985.30	0.395677
$679$	−12725.4	−0.719231
$680$	9941.83	0.560664
$681$	−3963.29	−0.223015
$682$	−4508.12	−0.253115
$683$	−30448.8	−1.70584	−0.852921	−	0.522039i	$-0.825171\pi$
$683$	−30448.8	−1.70584	−0.852921	+	0.522039i	$0.825171\pi$
$684$	5438.13	0.303994
$685$	−44844.8	−2.50136
$686$	−8725.79	−0.485645
$687$	−2442.10	−0.135621
$688$	−11041.9	−0.611873
$689$	−22099.7	−1.22196
$690$	−5569.73	−0.307299
$691$	18434.6	1.01489	0.507444	−	0.861685i	$-0.330591\pi$
$691$	18434.6	1.01489	0.507444	+	0.861685i	$0.330591\pi$
$692$	−16659.9	−0.915194
$693$	−6843.72	−0.375139
$694$	4661.86	0.254988
$695$	327.327	0.0178651
$696$	13498.5	0.735143
$697$	13466.5	0.731820
$698$	3477.90	0.188596
$699$	621.677	0.0336395
$700$	−13682.6	−0.738791
$701$	8215.66	0.442655	0.221328	−	0.975200i	$-0.428961\pi$
$701$	8215.66	0.442655	0.221328	+	0.975200i	$0.428961\pi$
$702$	2678.00	0.143981
$703$	33517.8	1.79822
$704$	−1430.71	−0.0765936
$705$	22954.6	1.22627
$706$	−4802.05	−0.255988
$707$	−851.633	−0.0453026
$708$	−2263.46	−0.120150
$709$	−16511.3	−0.874604	−0.437302	−	0.899315i	$-0.644066\pi$
$709$	−16511.3	−0.874604	−0.437302	+	0.899315i	$0.644066\pi$
$710$	6312.61	0.333674
$711$	3374.13	0.177974
$712$	2541.53	0.133775
$713$	−5031.63	−0.264286
$714$	−1598.10	−0.0837637
$715$	−75861.2	−3.96790
$716$	−23117.7	−1.20663
$717$	5829.22	0.303621
$718$	−2307.37	−0.119931
$719$	−13298.7	−0.689789	−0.344895	−	0.938641i	$-0.612085\pi$
$719$	−13298.7	−0.689789	−0.344895	+	0.938641i	$0.612085\pi$
$720$	−4069.74	−0.210653
$721$	−6.09173	−0.000314657 0
$722$	2951.03	0.152114
$723$	−21165.2	−1.08872
$724$	−9569.17	−0.491209
$725$	40131.9	2.05581
$726$	−7965.40	−0.407195
$727$	−17478.5	−0.891668	−0.445834	−	0.895116i	$-0.647093\pi$
$727$	−17478.5	−0.891668	−0.445834	+	0.895116i	$0.647093\pi$
$728$	18578.1	0.945811
$729$	729.000	0.0370370
$730$	23585.9	1.19583
$731$	−13083.6	−0.661988
$732$	9751.96	0.492408
$733$	23711.7	1.19483	0.597416	−	0.801932i	$-0.296194\pi$
$733$	23711.7	1.19483	0.597416	+	0.801932i	$0.296194\pi$
$734$	−11195.5	−0.562990
$735$	−8791.76	−0.441209
$736$	−15370.1	−0.769768
$737$	−56872.0	−2.84248
$738$	4994.99	0.249144
$739$	−10771.9	−0.536201	−0.268101	−	0.963391i	$-0.586396\pi$
$739$	−10771.9	−0.536201	−0.268101	+	0.963391i	$0.586396\pi$
$740$	−37780.0	−1.87679
$741$	−21952.7	−1.08833
$742$	4891.72	0.242022
$743$	−36275.9	−1.79116	−0.895581	−	0.444899i	$-0.853240\pi$
$743$	−36275.9	−1.79116	−0.895581	+	0.444899i	$0.853240\pi$
$744$	3331.36	0.164158
$745$	24797.5	1.21948
$746$	7315.63	0.359041
$747$	−12294.1	−0.602168
$748$	11553.9	0.564775
$749$	535.215	0.0261099
$750$	2682.71	0.130612
$751$	2945.64	0.143127	0.0715633	−	0.997436i	$-0.477201\pi$
$751$	2945.64	0.143127	0.0715633	+	0.997436i	$0.477201\pi$
$752$	11910.6	0.577572
$753$	13688.4	0.662460
$754$	−24051.9	−1.16170
$755$	46786.4	2.25527
$756$	2232.27	0.107390
$757$	19282.5	0.925805	0.462903	−	0.886409i	$-0.346808\pi$
$757$	19282.5	0.925805	0.462903	+	0.886409i	$0.346808\pi$
$758$	4767.68	0.228456
$759$	−14664.5	−0.701303
$760$	−30229.1	−1.44280
$761$	20744.0	0.988136	0.494068	−	0.869423i	$-0.335509\pi$
$761$	20744.0	0.988136	0.494068	+	0.869423i	$0.335509\pi$
$762$	−4906.48	−0.233259
$763$	−19826.2	−0.940703
$764$	24879.6	1.17816
$765$	−4822.23	−0.227906
$766$	7896.03	0.372448
$767$	9137.15	0.430148
$768$	6151.21	0.289014
$769$	5338.36	0.250333	0.125166	−	0.992136i	$-0.460054\pi$
$769$	5338.36	0.250333	0.125166	+	0.992136i	$0.460054\pi$
$770$	16791.7	0.785884
$771$	−347.892	−0.0162503
$772$	23599.6	1.10022
$773$	−3523.66	−0.163955	−0.0819775	−	0.996634i	$-0.526124\pi$
$773$	−3523.66	−0.163955	−0.0819775	+	0.996634i	$0.526124\pi$
$774$	−4852.97	−0.225370
$775$	9904.37	0.459065
$776$	18053.6	0.835162
$777$	13758.5	0.635243
$778$	−5358.84	−0.246946
$779$	−40946.1	−1.88324
$780$	24744.2	1.13588
$781$	16620.5	0.761495
$782$	−3424.36	−0.156592
$783$	−6547.38	−0.298830
$784$	−4561.83	−0.207809
$785$	2675.90	0.121665
$786$	392.263	0.0178010
$787$	36950.8	1.67364	0.836818	−	0.547480i	$-0.184413\pi$
$787$	36950.8	1.67364	0.836818	+	0.547480i	$0.184413\pi$
$788$	−409.174	−0.0184978
$789$	7441.91	0.335791
$790$	−8278.75	−0.372841
$791$	−23504.9	−1.05656
$792$	9709.17	0.435607
$793$	−39366.7	−1.76287
$794$	394.576	0.0176360
$795$	14760.7	0.658500
$796$	12804.5	0.570155
$797$	−41807.9	−1.85811	−0.929054	−	0.369944i	$-0.879377\pi$
$797$	−41807.9	−1.85811	−0.929054	+	0.369944i	$0.879377\pi$
$798$	4859.17	0.215555
$799$	14112.9	0.624878
$800$	30254.9	1.33709
$801$	−1232.76	−0.0543786
$802$	19992.6	0.880254
$803$	62099.2	2.72906
$804$	18550.4	0.813708
$805$	18741.7	0.820567
$806$	−5935.90	−0.259408
$807$	−15691.6	−0.684472
$808$	1208.21	0.0526048
$809$	14491.2	0.629767	0.314884	−	0.949130i	$-0.398035\pi$
$809$	14491.2	0.629767	0.314884	+	0.949130i	$0.398035\pi$
$810$	−1788.67	−0.0775894
$811$	−17074.2	−0.739278	−0.369639	−	0.929175i	$-0.620519\pi$
$811$	−17074.2	−0.739278	−0.369639	+	0.929175i	$0.620519\pi$
$812$	−20048.7	−0.866467
$813$	−25388.5	−1.09522
$814$	26414.0	1.13736
$815$	−13933.0	−0.598836
$816$	−2502.14	−0.107344
$817$	39781.8	1.70354
$818$	−7758.75	−0.331636
$819$	−9011.22	−0.384466
$820$	46152.9	1.96552
$821$	11772.9	0.500458	0.250229	−	0.968187i	$-0.419494\pi$
$821$	11772.9	0.500458	0.250229	+	0.968187i	$0.419494\pi$
$822$	−10226.8	−0.433941
$823$	−11047.4	−0.467906	−0.233953	−	0.972248i	$-0.575166\pi$
$823$	−11047.4	−0.467906	−0.233953	+	0.972248i	$0.575166\pi$
$824$	8.64233	0.000365376 0
$825$	28866.0	1.21816
$826$	−2022.49	−0.0851953
$827$	34047.3	1.43161	0.715804	−	0.698301i	$-0.246061\pi$
$827$	34047.3	1.43161	0.715804	+	0.698301i	$0.246061\pi$
$828$	4783.25	0.200760
$829$	37685.5	1.57886	0.789428	−	0.613844i	$-0.210377\pi$
$829$	37685.5	1.57886	0.789428	+	0.613844i	$0.210377\pi$
$830$	30164.8	1.26149
$831$	9417.93	0.393146
$832$	−1883.84	−0.0784979
$833$	−5405.31	−0.224830
$834$	74.6464	0.00309927
$835$	−48245.4	−1.99952
$836$	−35130.7	−1.45337
$837$	−1615.86	−0.0667292
$838$	2927.82	0.120692
$839$	3387.89	0.139408	0.0697038	−	0.997568i	$-0.477795\pi$
$839$	3387.89	0.139408	0.0697038	+	0.997568i	$0.477795\pi$
$840$	−12408.6	−0.509686
$841$	34415.1	1.41109
$842$	−7590.16	−0.310658
$843$	−10850.7	−0.443320
$844$	3297.82	0.134497
$845$	−62442.0	−2.54210
$846$	5234.76	0.212736
$847$	26802.9	1.08732
$848$	7658.95	0.310153
$849$	−8292.02	−0.335196
$850$	6740.59	0.272000
$851$	29481.4	1.18756
$852$	−5421.23	−0.217991
$853$	−28694.4	−1.15179	−0.575895	−	0.817524i	$-0.695346\pi$
$853$	−28694.4	−1.15179	−0.575895	+	0.817524i	$0.695346\pi$
$854$	8713.72	0.349154
$855$	14662.5	0.586487
$856$	−759.309	−0.0303185
$857$	−8422.44	−0.335712	−0.167856	−	0.985812i	$-0.553684\pi$
$857$	−8422.44	−0.335712	−0.167856	+	0.985812i	$0.553684\pi$
$858$	−17300.0	−0.688360
$859$	22162.3	0.880288	0.440144	−	0.897927i	$-0.354927\pi$
$859$	22162.3	0.880288	0.440144	+	0.897927i	$0.354927\pi$
$860$	−44840.7	−1.77797
$861$	−16807.7	−0.665279
$862$	−8363.94	−0.330484
$863$	7717.65	0.304417	0.152208	−	0.988348i	$-0.451362\pi$
$863$	7717.65	0.304417	0.152208	+	0.988348i	$0.451362\pi$
$864$	−4935.97	−0.194358
$865$	−44919.0	−1.76565
$866$	16884.9	0.662553
$867$	11774.2	0.461215
$868$	−4947.92	−0.193483
$869$	−21797.1	−0.850882
$870$	16064.6	0.626024
$871$	−74884.2	−2.91315
$872$	28127.4	1.09233
$873$	−8756.79	−0.339488
$874$	10412.1	0.402969
$875$	−9027.08	−0.348767
$876$	−20255.4	−0.781240
$877$	−5091.49	−0.196040	−0.0980202	−	0.995184i	$-0.531251\pi$
$877$	−5091.49	−0.196040	−0.0980202	+	0.995184i	$0.531251\pi$
$878$	20437.6	0.785575
$879$	−6360.89	−0.244081
$880$	26290.7	1.00711
$881$	16155.9	0.617829	0.308915	−	0.951090i	$-0.400034\pi$
$881$	16155.9	0.617829	0.308915	+	0.951090i	$0.400034\pi$
$882$	−2004.95	−0.0765420
$883$	−18664.4	−0.711334	−0.355667	−	0.934613i	$-0.615746\pi$
$883$	−18664.4	−0.711334	−0.355667	+	0.934613i	$0.615746\pi$
$884$	15213.2	0.578816
$885$	−6102.83	−0.231801
$886$	11538.8	0.437531
$887$	41225.4	1.56056	0.780279	−	0.625432i	$-0.215077\pi$
$887$	41225.4	1.56056	0.780279	+	0.625432i	$0.215077\pi$
$888$	−19519.2	−0.737636
$889$	16509.9	0.622861
$890$	3024.68	0.113919
$891$	−4709.39	−0.177071
$892$	13804.1	0.518156
$893$	−42911.5	−1.60804
$894$	5655.03	0.211558
$895$	−62330.6	−2.32791
$896$	−18711.0	−0.697646
$897$	−19309.0	−0.718739
$898$	−11765.8	−0.437226
$899$	14512.6	0.538400
$900$	−9415.45	−0.348720
$901$	9075.11	0.335556
$902$	−32268.0	−1.19114
$903$	16329.8	0.601797
$904$	33346.4	1.22686
$905$	−25800.7	−0.947674
$906$	10669.6	0.391250
$907$	11142.7	0.407924	0.203962	−	0.978979i	$-0.434618\pi$
$907$	11142.7	0.407924	0.203962	+	0.978979i	$0.434618\pi$
$908$	−8351.16	−0.305223
$909$	−586.037	−0.0213835
$910$	22109.9	0.805423
$911$	−3162.46	−0.115013	−0.0575066	−	0.998345i	$-0.518315\pi$
$911$	−3162.46	−0.115013	−0.0575066	+	0.998345i	$0.518315\pi$
$912$	7607.99	0.276234
$913$	79421.0	2.87892
$914$	1169.51	0.0423239
$915$	26293.5	0.949987
$916$	−5145.82	−0.185614
$917$	−1319.93	−0.0475332
$918$	−1099.70	−0.0395377
$919$	−41515.8	−1.49019	−0.745093	−	0.666961i	$-0.767595\pi$
$919$	−41515.8	−1.49019	−0.745093	+	0.666961i	$0.767595\pi$
$920$	−26588.8	−0.952832
$921$	−9419.72	−0.337014
$922$	13080.8	0.467236
$923$	21884.4	0.780427
$924$	−14420.6	−0.513423
$925$	−58031.9	−2.06278
$926$	23545.0	0.835570
$927$	−4.19192	−0.000148523 0
$928$	44331.5	1.56816
$929$	−14540.7	−0.513527	−0.256763	−	0.966474i	$-0.582656\pi$
$929$	−14540.7	−0.513527	−0.256763	+	0.966474i	$0.582656\pi$
$930$	3964.67	0.139792
$931$	16435.4	0.578569
$932$	1309.95	0.0460397
$933$	8365.75	0.293550
$934$	−17292.2	−0.605802
$935$	31151.9	1.08960
$936$	12784.2	0.446437
$937$	28381.6	0.989528	0.494764	−	0.869027i	$-0.335255\pi$
$937$	28381.6	0.989528	0.494764	+	0.869027i	$0.335255\pi$
$938$	16575.4	0.576980
$939$	−20479.2	−0.711729
$940$	48368.3	1.67830
$941$	−38844.6	−1.34569	−0.672847	−	0.739782i	$-0.734929\pi$
$941$	−38844.6	−1.34569	−0.672847	+	0.739782i	$0.734929\pi$
$942$	610.234	0.0211067
$943$	−36015.1	−1.24371
$944$	−3166.60	−0.109178
$945$	6018.71	0.207184
$946$	31350.5	1.07748
$947$	5194.45	0.178244	0.0891219	−	0.996021i	$-0.471594\pi$
$947$	5194.45	0.178244	0.0891219	+	0.996021i	$0.471594\pi$
$948$	7109.73	0.243579
$949$	81766.9	2.79691
$950$	−20495.4	−0.699957
$951$	−3608.97	−0.123059
$952$	−7628.99	−0.259724
$953$	−12991.5	−0.441590	−0.220795	−	0.975320i	$-0.570865\pi$
$953$	−12991.5	−0.441590	−0.220795	+	0.975320i	$0.570865\pi$
$954$	3366.15	0.114238
$955$	67081.2	2.27298
$956$	12282.9	0.415542
$957$	42296.5	1.42868
$958$	−26731.9	−0.901532
$959$	34412.2	1.15874
$960$	1258.24	0.0423016
$961$	−26209.4	−0.879775
$962$	34779.7	1.16564
$963$	368.299	0.0123243
$964$	−44597.9	−1.49004
$965$	63630.1	2.12262
$966$	4274.00	0.142354
$967$	−44230.3	−1.47089	−0.735445	−	0.677584i	$-0.763027\pi$
$967$	−44230.3	−1.47089	−0.735445	+	0.677584i	$0.763027\pi$
$968$	−38025.2	−1.26258
$969$	9014.72	0.298859
$970$	21485.6	0.711197
$971$	−17876.8	−0.590826	−0.295413	−	0.955370i	$-0.595457\pi$
$971$	−17876.8	−0.590826	−0.295413	+	0.955370i	$0.595457\pi$
$972$	1536.10	0.0506896
$973$	−251.178	−0.00827586
$974$	−5238.60	−0.172336
$975$	38008.3	1.24845
$976$	13643.1	0.447443
$977$	−34107.0	−1.11687	−0.558433	−	0.829550i	$-0.688597\pi$
$977$	−34107.0	−1.11687	−0.558433	+	0.829550i	$0.688597\pi$
$978$	−3177.40	−0.103887
$979$	7963.68	0.259980
$980$	−18525.4	−0.603848
$981$	−13643.1	−0.444026
$982$	18060.4	0.586896
$983$	−6171.90	−0.200257	−0.100129	−	0.994974i	$-0.531925\pi$
$983$	−6171.90	−0.200257	−0.100129	+	0.994974i	$0.531925\pi$
$984$	23845.1	0.772514
$985$	−1103.23	−0.0356871
$986$	9876.78	0.319007
$987$	−17614.5	−0.568061
$988$	−46257.1	−1.48951
$989$	34991.1	1.12503
$990$	11554.9	0.370949
$991$	−343.343	−0.0110057	−0.00550285	−	0.999985i	$-0.501752\pi$
$991$	−343.343	−0.0110057	−0.00550285	+	0.999985i	$0.501752\pi$
$992$	10940.8	0.350172
$993$	−13308.6	−0.425314
$994$	−4844.06	−0.154572
$995$	34523.9	1.09998
$996$	−25905.4	−0.824139
$997$	39610.7	1.25826	0.629128	−	0.777301i	$-0.283412\pi$
$997$	39610.7	1.25826	0.629128	+	0.777301i	$0.283412\pi$
$998$	−1412.52	−0.0448022
$999$	9467.69	0.299844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.12	✓	22
3.2	odd	2		1413.4.a.e.1.11		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.12	✓	22	1.1	even	1	trivial
1413.4.a.e.1.11		22	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q+1.29561 q^{2} -3.00000 q^{3} -6.32139 q^{4} -17.0439 q^{5} -3.88684 q^{6} +13.0789 q^{7} -18.5550 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+1.29561 q^{2} -3.00000 q^{3} -6.32139 q^{4} -17.0439 q^{5} -3.88684 q^{6} +13.0789 q^{7} -18.5550 q^{8} +9.00000 q^{9} -22.0823 q^{10} -58.1406 q^{11} +18.9642 q^{12} -76.5545 q^{13} +16.9452 q^{14} +51.1318 q^{15} +26.5310 q^{16} +31.4366 q^{17} +11.6605 q^{18} -95.5862 q^{19} +107.741 q^{20} -39.2366 q^{21} -75.3277 q^{22} -84.0752 q^{23} +55.6649 q^{24} +165.496 q^{25} -99.1850 q^{26} -27.0000 q^{27} -82.6766 q^{28} +242.496 q^{29} +66.2470 q^{30} +59.8467 q^{31} +182.814 q^{32} +174.422 q^{33} +40.7297 q^{34} -222.915 q^{35} -56.8925 q^{36} -350.655 q^{37} -123.843 q^{38} +229.664 q^{39} +316.250 q^{40} +428.368 q^{41} -50.8355 q^{42} -416.188 q^{43} +367.529 q^{44} -153.395 q^{45} -108.929 q^{46} +448.930 q^{47} -79.5931 q^{48} -171.943 q^{49} +214.418 q^{50} -94.3099 q^{51} +483.931 q^{52} +288.679 q^{53} -34.9816 q^{54} +990.944 q^{55} -242.678 q^{56} +286.758 q^{57} +314.180 q^{58} -119.355 q^{59} -323.224 q^{60} +514.231 q^{61} +77.5382 q^{62} +117.710 q^{63} +24.6078 q^{64} +1304.79 q^{65} +225.983 q^{66} +978.181 q^{67} -198.723 q^{68} +252.226 q^{69} -288.812 q^{70} -285.867 q^{71} -166.995 q^{72} -1068.09 q^{73} -454.313 q^{74} -496.487 q^{75} +604.237 q^{76} -760.413 q^{77} +297.555 q^{78} +374.903 q^{79} -452.193 q^{80} +81.0000 q^{81} +554.999 q^{82} -1366.02 q^{83} +248.030 q^{84} -535.804 q^{85} -539.219 q^{86} -727.487 q^{87} +1078.80 q^{88} -136.973 q^{89} -198.741 q^{90} -1001.25 q^{91} +531.472 q^{92} -179.540 q^{93} +581.640 q^{94} +1629.16 q^{95} -548.441 q^{96} -972.977 q^{97} -222.772 q^{98} -523.265 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more