LMFDB - Embedded newform 2001.4.a.h.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2001,4,Mod(1,2001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2001, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2001.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2001 = 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2001.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:	$118.062821921$
Analytic rank:	$0$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	2001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29985 q^{2} +3.00000 q^{3} -6.31038 q^{4} -11.8310 q^{5} -3.89956 q^{6} -23.8904 q^{7} +18.6014 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.29985 q^{2} +3.00000 q^{3} -6.31038 q^{4} -11.8310 q^{5} -3.89956 q^{6} -23.8904 q^{7} +18.6014 q^{8} +9.00000 q^{9} +15.3786 q^{10} +61.3461 q^{11} -18.9311 q^{12} +45.7194 q^{13} +31.0540 q^{14} -35.4931 q^{15} +26.3040 q^{16} +39.5737 q^{17} -11.6987 q^{18} -69.0493 q^{19} +74.6584 q^{20} -71.6712 q^{21} -79.7410 q^{22} +23.0000 q^{23} +55.8042 q^{24} +14.9734 q^{25} -59.4286 q^{26} +27.0000 q^{27} +150.758 q^{28} -29.0000 q^{29} +46.1358 q^{30} -158.956 q^{31} -183.002 q^{32} +184.038 q^{33} -51.4400 q^{34} +282.648 q^{35} -56.7934 q^{36} -40.8884 q^{37} +89.7539 q^{38} +137.158 q^{39} -220.074 q^{40} +205.688 q^{41} +93.1620 q^{42} -415.816 q^{43} -387.118 q^{44} -106.479 q^{45} -29.8966 q^{46} +39.3187 q^{47} +78.9119 q^{48} +227.751 q^{49} -19.4632 q^{50} +118.721 q^{51} -288.507 q^{52} -155.040 q^{53} -35.0960 q^{54} -725.789 q^{55} -444.395 q^{56} -207.148 q^{57} +37.6957 q^{58} +775.698 q^{59} +223.975 q^{60} +752.226 q^{61} +206.620 q^{62} -215.014 q^{63} +27.4446 q^{64} -540.908 q^{65} -239.223 q^{66} -927.608 q^{67} -249.725 q^{68} +69.0000 q^{69} -367.401 q^{70} +506.580 q^{71} +167.413 q^{72} -813.929 q^{73} +53.1489 q^{74} +44.9202 q^{75} +435.727 q^{76} -1465.58 q^{77} -178.286 q^{78} +665.696 q^{79} -311.203 q^{80} +81.0000 q^{81} -267.364 q^{82} -1083.97 q^{83} +452.273 q^{84} -468.198 q^{85} +540.500 q^{86} -87.0000 q^{87} +1141.12 q^{88} -1445.47 q^{89} +138.407 q^{90} -1092.26 q^{91} -145.139 q^{92} -476.869 q^{93} -51.1085 q^{94} +816.924 q^{95} -549.007 q^{96} -1280.20 q^{97} -296.043 q^{98} +552.115 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q + 6 q^{2} + 132 q^{3} + 210 q^{4} + 15 q^{5} + 18 q^{6} + 78 q^{7} + 12 q^{8} + 396 q^{9}+O(q^{10}) $ 44 * q + 6 * q^2 + 132 * q^3 + 210 * q^4 + 15 * q^5 + 18 * q^6 + 78 * q^7 + 12 * q^8 + 396 * q^9	\( 44 q + 6 q^{2} + 132 q^{3} + 210 q^{4} + 15 q^{5} + 18 q^{6} + 78 q^{7} + 12 q^{8} + 396 q^{9} + 214 q^{10} + 111 q^{11} + 630 q^{12} + 275 q^{13} + 104 q^{14} + 45 q^{15} + 1062 q^{16} - 58 q^{17} + 54 q^{18} + 331 q^{19} + 287 q^{20} + 234 q^{21} + 285 q^{22} + 1012 q^{23} + 36 q^{24} + 1903 q^{25} + 1084 q^{26} + 1188 q^{27} + 222 q^{28} - 1276 q^{29} + 642 q^{30} + 1394 q^{31} + 42 q^{32} + 333 q^{33} + 373 q^{34} + 567 q^{35} + 1890 q^{36} + 1229 q^{37} + 733 q^{38} + 825 q^{39} + 2483 q^{40} - 107 q^{41} + 312 q^{42} + 1165 q^{43} + 1639 q^{44} + 135 q^{45} + 138 q^{46} + 964 q^{47} + 3186 q^{48} + 4264 q^{49} + 495 q^{50} - 174 q^{51} + 2679 q^{52} - 380 q^{53} + 162 q^{54} + 1260 q^{55} + 2229 q^{56} + 993 q^{57} - 174 q^{58} + 897 q^{59} + 861 q^{60} + 2584 q^{61} + 3034 q^{62} + 702 q^{63} + 6866 q^{64} - 286 q^{65} + 855 q^{66} + 2277 q^{67} - 1554 q^{68} + 3036 q^{69} + 689 q^{70} + 4304 q^{71} + 108 q^{72} + 4712 q^{73} - 1005 q^{74} + 5709 q^{75} + 2877 q^{76} + 919 q^{77} + 3252 q^{78} + 3864 q^{79} + 2593 q^{80} + 3564 q^{81} + 3297 q^{82} - 540 q^{83} + 666 q^{84} + 6537 q^{85} + 3789 q^{86} - 3828 q^{87} + 1707 q^{88} - 331 q^{89} + 1926 q^{90} + 4311 q^{91} + 4830 q^{92} + 4182 q^{93} + 6189 q^{94} + 3267 q^{95} + 126 q^{96} + 5572 q^{97} + 2588 q^{98} + 999 q^{99}+O(q^{100}) \) 44 * q + 6 * q^2 + 132 * q^3 + 210 * q^4 + 15 * q^5 + 18 * q^6 + 78 * q^7 + 12 * q^8 + 396 * q^9 + 214 * q^10 + 111 * q^11 + 630 * q^12 + 275 * q^13 + 104 * q^14 + 45 * q^15 + 1062 * q^16 - 58 * q^17 + 54 * q^18 + 331 * q^19 + 287 * q^20 + 234 * q^21 + 285 * q^22 + 1012 * q^23 + 36 * q^24 + 1903 * q^25 + 1084 * q^26 + 1188 * q^27 + 222 * q^28 - 1276 * q^29 + 642 * q^30 + 1394 * q^31 + 42 * q^32 + 333 * q^33 + 373 * q^34 + 567 * q^35 + 1890 * q^36 + 1229 * q^37 + 733 * q^38 + 825 * q^39 + 2483 * q^40 - 107 * q^41 + 312 * q^42 + 1165 * q^43 + 1639 * q^44 + 135 * q^45 + 138 * q^46 + 964 * q^47 + 3186 * q^48 + 4264 * q^49 + 495 * q^50 - 174 * q^51 + 2679 * q^52 - 380 * q^53 + 162 * q^54 + 1260 * q^55 + 2229 * q^56 + 993 * q^57 - 174 * q^58 + 897 * q^59 + 861 * q^60 + 2584 * q^61 + 3034 * q^62 + 702 * q^63 + 6866 * q^64 - 286 * q^65 + 855 * q^66 + 2277 * q^67 - 1554 * q^68 + 3036 * q^69 + 689 * q^70 + 4304 * q^71 + 108 * q^72 + 4712 * q^73 - 1005 * q^74 + 5709 * q^75 + 2877 * q^76 + 919 * q^77 + 3252 * q^78 + 3864 * q^79 + 2593 * q^80 + 3564 * q^81 + 3297 * q^82 - 540 * q^83 + 666 * q^84 + 6537 * q^85 + 3789 * q^86 - 3828 * q^87 + 1707 * q^88 - 331 * q^89 + 1926 * q^90 + 4311 * q^91 + 4830 * q^92 + 4182 * q^93 + 6189 * q^94 + 3267 * q^95 + 126 * q^96 + 5572 * q^97 + 2588 * q^98 + 999 * 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 $

$2$	−1.29985	−0.459568	−0.229784	−	0.973242i	$-0.573802\pi$
$2$	−1.29985	−0.459568	−0.229784	+	0.973242i	$0.573802\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−6.31038	−0.788798
$5$	−11.8310	−1.05820	−0.529100	−	0.848559i	$-0.677470\pi$
$5$	−11.8310	−1.05820	−0.529100	+	0.848559i	$0.677470\pi$
$6$	−3.89956	−0.265331
$7$	−23.8904	−1.28996	−0.644980	−	0.764199i	$-0.723134\pi$
$7$	−23.8904	−1.28996	−0.644980	+	0.764199i	$0.723134\pi$
$8$	18.6014	0.822073
$9$	9.00000	0.333333
$10$	15.3786	0.486314
$11$	61.3461	1.68151	0.840753	−	0.541419i	$-0.182113\pi$
$11$	61.3461	1.68151	0.840753	+	0.541419i	$0.182113\pi$
$12$	−18.9311	−0.455413
$13$	45.7194	0.975407	0.487704	−	0.873009i	$-0.337835\pi$
$13$	45.7194	0.975407	0.487704	+	0.873009i	$0.337835\pi$
$14$	31.0540	0.592824
$15$	−35.4931	−0.610952
$16$	26.3040	0.411000
$17$	39.5737	0.564591	0.282295	−	0.959328i	$-0.408904\pi$
$17$	39.5737	0.564591	0.282295	+	0.959328i	$0.408904\pi$
$18$	−11.6987	−0.153189
$19$	−69.0493	−0.833736	−0.416868	−	0.908967i	$-0.636872\pi$
$19$	−69.0493	−0.833736	−0.416868	+	0.908967i	$0.636872\pi$
$20$	74.6584	0.834706
$21$	−71.6712	−0.744759
$22$	−79.7410	−0.772765
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	55.8042	0.474624
$25$	14.9734	0.119787
$26$	−59.4286	−0.448265
$27$	27.0000	0.192450
$28$	150.758	1.01752
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	46.1358	0.280774
$31$	−158.956	−0.920948	−0.460474	−	0.887673i	$-0.652320\pi$
$31$	−158.956	−0.920948	−0.460474	+	0.887673i	$0.652320\pi$
$32$	−183.002	−1.01096
$33$	184.038	0.970818
$34$	−51.4400	−0.259468
$35$	282.648	1.36504
$36$	−56.7934	−0.262933
$37$	−40.8884	−0.181676	−0.0908380	−	0.995866i	$-0.528955\pi$
$37$	−40.8884	−0.181676	−0.0908380	+	0.995866i	$0.528955\pi$
$38$	89.7539	0.383158
$39$	137.158	0.563152
$40$	−220.074	−0.869918
$41$	205.688	0.783488	0.391744	−	0.920074i	$-0.371872\pi$
$41$	205.688	0.783488	0.391744	+	0.920074i	$0.371872\pi$
$42$	93.1620	0.342267
$43$	−415.816	−1.47468	−0.737342	−	0.675520i	$-0.763919\pi$
$43$	−415.816	−1.47468	−0.737342	+	0.675520i	$0.763919\pi$
$44$	−387.118	−1.32637
$45$	−106.479	−0.352733
$46$	−29.8966	−0.0958264
$47$	39.3187	0.122026	0.0610129	−	0.998137i	$-0.480567\pi$
$47$	39.3187	0.122026	0.0610129	+	0.998137i	$0.480567\pi$
$48$	78.9119	0.237291
$49$	227.751	0.663998
$50$	−19.4632	−0.0550503
$51$	118.721	0.325967
$52$	−288.507	−0.769399
$53$	−155.040	−0.401817	−0.200909	−	0.979610i	$-0.564389\pi$
$53$	−155.040	−0.401817	−0.200909	+	0.979610i	$0.564389\pi$
$54$	−35.0960	−0.0884438
$55$	−725.789	−1.77937
$56$	−444.395	−1.06044
$57$	−207.148	−0.481358
$58$	37.6957	0.0853395
$59$	775.698	1.71165	0.855825	−	0.517266i	$-0.173050\pi$
$59$	775.698	1.71165	0.855825	+	0.517266i	$0.173050\pi$
$60$	223.975	0.481918
$61$	752.226	1.57890	0.789448	−	0.613817i	$-0.210367\pi$
$61$	752.226	1.57890	0.789448	+	0.613817i	$0.210367\pi$
$62$	206.620	0.423238
$63$	−215.014	−0.429987
$64$	27.4446	0.0536027
$65$	−540.908	−1.03218
$66$	−239.223	−0.446156
$67$	−927.608	−1.69142	−0.845712	−	0.533640i	$-0.820824\pi$
$67$	−927.608	−1.69142	−0.845712	+	0.533640i	$0.820824\pi$
$68$	−249.725	−0.445348
$69$	69.0000	0.120386
$70$	−367.401	−0.627326
$71$	506.580	0.846760	0.423380	−	0.905952i	$-0.360843\pi$
$71$	506.580	0.846760	0.423380	+	0.905952i	$0.360843\pi$
$72$	167.413	0.274024
$73$	−813.929	−1.30497	−0.652487	−	0.757800i	$-0.726274\pi$
$73$	−813.929	−1.30497	−0.652487	+	0.757800i	$0.726274\pi$
$74$	53.1489	0.0834924
$75$	44.9202	0.0691592
$76$	435.727	0.657649
$77$	−1465.58	−2.16908
$78$	−178.286	−0.258806
$79$	665.696	0.948059	0.474029	−	0.880509i	$-0.342799\pi$
$79$	665.696	0.948059	0.474029	+	0.880509i	$0.342799\pi$
$80$	−311.203	−0.434920
$81$	81.0000	0.111111
$82$	−267.364	−0.360066
$83$	−1083.97	−1.43351	−0.716753	−	0.697327i	$-0.754372\pi$
$83$	−1083.97	−1.43351	−0.716753	+	0.697327i	$0.754372\pi$
$84$	452.273	0.587464
$85$	−468.198	−0.597450
$86$	540.500	0.677717
$87$	−87.0000	−0.107211
$88$	1141.12	1.38232
$89$	−1445.47	−1.72156	−0.860781	−	0.508975i	$-0.830025\pi$
$89$	−1445.47	−1.72156	−0.860781	+	0.508975i	$0.830025\pi$
$90$	138.407	0.162105
$91$	−1092.26	−1.25824
$92$	−145.139	−0.164476
$93$	−476.869	−0.531709
$94$	−51.1085	−0.0560791
$95$	816.924	0.882260
$96$	−549.007	−0.583675
$97$	−1280.20	−1.34005	−0.670024	−	0.742339i	$-0.733716\pi$
$97$	−1280.20	−1.34005	−0.670024	+	0.742339i	$0.733716\pi$
$98$	−296.043	−0.305152
$99$	552.115	0.560502
$100$	−94.4879	−0.0944879
$101$	1824.02	1.79700	0.898500	−	0.438973i	$-0.144658\pi$
$101$	1824.02	1.79700	0.898500	+	0.438973i	$0.144658\pi$
$102$	−154.320	−0.149804
$103$	−1561.28	−1.49357	−0.746784	−	0.665067i	$-0.768403\pi$
$103$	−1561.28	−1.49357	−0.746784	+	0.665067i	$0.768403\pi$
$104$	850.445	0.801856
$105$	847.945	0.788104
$106$	201.529	0.184662
$107$	1148.63	1.03778	0.518890	−	0.854841i	$-0.326345\pi$
$107$	1148.63	1.03778	0.518890	+	0.854841i	$0.326345\pi$
$108$	−170.380	−0.151804
$109$	−517.227	−0.454508	−0.227254	−	0.973836i	$-0.572975\pi$
$109$	−517.227	−0.454508	−0.227254	+	0.973836i	$0.572975\pi$
$110$	943.418	0.817740
$111$	−122.665	−0.104891
$112$	−628.412	−0.530173
$113$	−103.095	−0.0858265	−0.0429132	−	0.999079i	$-0.513664\pi$
$113$	−103.095	−0.0858265	−0.0429132	+	0.999079i	$0.513664\pi$
$114$	269.262	0.221216
$115$	−272.114	−0.220650
$116$	183.001	0.146476
$117$	411.475	0.325136
$118$	−1008.29	−0.786619
$119$	−945.433	−0.728300
$120$	−660.221	−0.502247
$121$	2432.35	1.82746
$122$	−977.784	−0.725610
$123$	617.063	0.452347
$124$	1003.07	0.726441
$125$	1301.73	0.931441
$126$	279.486	0.197608
$127$	2259.18	1.57850	0.789250	−	0.614072i	$-0.210470\pi$
$127$	2259.18	1.57850	0.789250	+	0.614072i	$0.210470\pi$
$128$	1428.35	0.986321
$129$	−1247.45	−0.851409
$130$	703.101	0.474354
$131$	1348.42	0.899331	0.449666	−	0.893197i	$-0.351543\pi$
$131$	1348.42	0.899331	0.449666	+	0.893197i	$0.351543\pi$
$132$	−1161.35	−0.765779
$133$	1649.61	1.07549
$134$	1205.75	0.777323
$135$	−319.438	−0.203651
$136$	736.127	0.464135
$137$	1877.79	1.17102	0.585512	−	0.810664i	$-0.300893\pi$
$137$	1877.79	1.17102	0.585512	+	0.810664i	$0.300893\pi$
$138$	−89.6899	−0.0553254
$139$	−1005.93	−0.613829	−0.306915	−	0.951737i	$-0.599297\pi$
$139$	−1005.93	−0.613829	−0.306915	+	0.951737i	$0.599297\pi$
$140$	−1783.62	−1.07674
$141$	117.956	0.0704517
$142$	−658.480	−0.389144
$143$	2804.71	1.64015
$144$	236.736	0.137000
$145$	343.100	0.196503
$146$	1057.99	0.599724
$147$	683.254	0.383359
$148$	258.021	0.143306
$149$	−464.950	−0.255639	−0.127819	−	0.991797i	$-0.540798\pi$
$149$	−464.950	−0.255639	−0.127819	+	0.991797i	$0.540798\pi$
$150$	−58.3897	−0.0317833
$151$	964.595	0.519852	0.259926	−	0.965629i	$-0.416302\pi$
$151$	964.595	0.519852	0.259926	+	0.965629i	$0.416302\pi$
$152$	−1284.41	−0.685392
$153$	356.164	0.188197
$154$	1905.04	0.996837
$155$	1880.62	0.974547
$156$	−865.521	−0.444213
$157$	571.831	0.290682	0.145341	−	0.989382i	$-0.453572\pi$
$157$	571.831	0.290682	0.145341	+	0.989382i	$0.453572\pi$
$158$	−865.307	−0.435697
$159$	−465.119	−0.231989
$160$	2165.11	1.06979
$161$	−549.479	−0.268975
$162$	−105.288	−0.0510631
$163$	136.053	0.0653773	0.0326887	−	0.999466i	$-0.489593\pi$
$163$	136.053	0.0653773	0.0326887	+	0.999466i	$0.489593\pi$
$164$	−1297.97	−0.618014
$165$	−2177.37	−1.02732
$166$	1409.00	0.658793
$167$	−128.076	−0.0593463	−0.0296731	−	0.999560i	$-0.509447\pi$
$167$	−128.076	−0.0593463	−0.0296731	+	0.999560i	$0.509447\pi$
$168$	−1333.18	−0.612247
$169$	−106.733	−0.0485811
$170$	608.589	0.274569
$171$	−621.443	−0.277912
$172$	2623.96	1.16323
$173$	−1191.69	−0.523715	−0.261857	−	0.965107i	$-0.584335\pi$
$173$	−1191.69	−0.523715	−0.261857	+	0.965107i	$0.584335\pi$
$174$	113.087	0.0492708
$175$	−357.721	−0.154521
$176$	1613.65	0.691098
$177$	2327.10	0.988221
$178$	1878.89	0.791174
$179$	2294.07	0.957915	0.478958	−	0.877838i	$-0.341015\pi$
$179$	2294.07	0.957915	0.478958	+	0.877838i	$0.341015\pi$
$180$	671.925	0.278235
$181$	2180.32	0.895368	0.447684	−	0.894192i	$-0.352249\pi$
$181$	2180.32	0.895368	0.447684	+	0.894192i	$0.352249\pi$
$182$	1419.77	0.578245
$183$	2256.68	0.911576
$184$	427.832	0.171414
$185$	483.752	0.192249
$186$	619.859	0.244356
$187$	2427.70	0.949362
$188$	−248.116	−0.0962537
$189$	−645.041	−0.248253
$190$	−1061.88	−0.405458
$191$	−460.514	−0.174459	−0.0872295	−	0.996188i	$-0.527801\pi$
$191$	−460.514	−0.174459	−0.0872295	+	0.996188i	$0.527801\pi$
$192$	82.3337	0.0309475
$193$	1841.04	0.686639	0.343319	−	0.939219i	$-0.388449\pi$
$193$	1841.04	0.686639	0.343319	+	0.939219i	$0.388449\pi$
$194$	1664.07	0.615843
$195$	−1622.73	−0.595927
$196$	−1437.20	−0.523760
$197$	−2620.01	−0.947554	−0.473777	−	0.880645i	$-0.657110\pi$
$197$	−2620.01	−0.947554	−0.473777	+	0.880645i	$0.657110\pi$
$198$	−717.669	−0.257588
$199$	−1392.71	−0.496113	−0.248057	−	0.968746i	$-0.579792\pi$
$199$	−1392.71	−0.496113	−0.248057	+	0.968746i	$0.579792\pi$
$200$	278.526	0.0984739
$201$	−2782.82	−0.976544
$202$	−2370.96	−0.825843
$203$	692.822	0.239540
$204$	−749.176	−0.257122
$205$	−2433.50	−0.829087
$206$	2029.44	0.686395
$207$	207.000	0.0695048
$208$	1202.60	0.400892
$209$	−4235.91	−1.40193
$210$	−1102.20	−0.362187
$211$	5570.79	1.81758	0.908790	−	0.417254i	$-0.137008\pi$
$211$	5570.79	1.81758	0.908790	+	0.417254i	$0.137008\pi$
$212$	978.359	0.316953
$213$	1519.74	0.488877
$214$	−1493.05	−0.476930
$215$	4919.54	1.56051
$216$	502.238	0.158208
$217$	3797.53	1.18799
$218$	672.319	0.208877
$219$	−2441.79	−0.753427
$220$	4580.00	1.40356
$221$	1809.29	0.550706
$222$	159.447	0.0482043
$223$	−2672.60	−0.802560	−0.401280	−	0.915955i	$-0.631435\pi$
$223$	−2672.60	−0.802560	−0.401280	+	0.915955i	$0.631435\pi$
$224$	4372.00	1.30409
$225$	134.761	0.0399291
$226$	134.009	0.0394430
$227$	1914.15	0.559678	0.279839	−	0.960047i	$-0.409719\pi$
$227$	1914.15	0.559678	0.279839	+	0.960047i	$0.409719\pi$
$228$	1307.18	0.379694
$229$	1020.45	0.294467	0.147233	−	0.989102i	$-0.452963\pi$
$229$	1020.45	0.294467	0.147233	+	0.989102i	$0.452963\pi$
$230$	353.708	0.101404
$231$	−4396.75	−1.25232
$232$	−539.440	−0.152655
$233$	−4290.28	−1.20629	−0.603145	−	0.797631i	$-0.706086\pi$
$233$	−4290.28	−1.20629	−0.603145	+	0.797631i	$0.706086\pi$
$234$	−534.857	−0.149422
$235$	−465.181	−0.129128
$236$	−4894.95	−1.35015
$237$	1997.09	0.547362
$238$	1228.92	0.334703
$239$	4939.25	1.33679	0.668396	−	0.743805i	$-0.266981\pi$
$239$	4939.25	1.33679	0.668396	+	0.743805i	$0.266981\pi$
$240$	−933.610	−0.251101
$241$	1964.75	0.525148	0.262574	−	0.964912i	$-0.415429\pi$
$241$	1964.75	0.525148	0.262574	+	0.964912i	$0.415429\pi$
$242$	−3161.70	−0.839841
$243$	243.000	0.0641500
$244$	−4746.83	−1.24543
$245$	−2694.53	−0.702643
$246$	−802.091	−0.207884
$247$	−3156.89	−0.813232
$248$	−2956.81	−0.757087
$249$	−3251.90	−0.827635
$250$	−1692.06	−0.428060
$251$	313.237	0.0787702	0.0393851	−	0.999224i	$-0.487460\pi$
$251$	313.237	0.0787702	0.0393851	+	0.999224i	$0.487460\pi$
$252$	1356.82	0.339173
$253$	1410.96	0.350618
$254$	−2936.60	−0.725427
$255$	−1404.59	−0.344938
$256$	−2076.20	−0.506884
$257$	1373.62	0.333400	0.166700	−	0.986008i	$-0.446689\pi$
$257$	1373.62	0.333400	0.166700	+	0.986008i	$0.446689\pi$
$258$	1621.50	0.391280
$259$	976.840	0.234355
$260$	3413.34	0.814178
$261$	−261.000	−0.0618984
$262$	−1752.75	−0.413304
$263$	1445.97	0.339020	0.169510	−	0.985528i	$-0.445782\pi$
$263$	1445.97	0.339020	0.169510	+	0.985528i	$0.445782\pi$
$264$	3423.37	0.798083
$265$	1834.28	0.425203
$266$	−2144.26	−0.494259
$267$	−4336.40	−0.993945
$268$	5853.56	1.33419
$269$	−4404.06	−0.998217	−0.499109	−	0.866539i	$-0.666339\pi$
$269$	−4404.06	−0.998217	−0.499109	+	0.866539i	$0.666339\pi$
$270$	415.222	0.0935912
$271$	5018.54	1.12492	0.562462	−	0.826823i	$-0.309854\pi$
$271$	5018.54	1.12492	0.562462	+	0.826823i	$0.309854\pi$
$272$	1040.95	0.232047
$273$	−3276.77	−0.726443
$274$	−2440.85	−0.538165
$275$	918.561	0.201423
$276$	−435.416	−0.0949601
$277$	2368.84	0.513825	0.256913	−	0.966435i	$-0.417295\pi$
$277$	2368.84	0.513825	0.256913	+	0.966435i	$0.417295\pi$
$278$	1307.57	0.282096
$279$	−1430.61	−0.306983
$280$	5257.65	1.12216
$281$	5916.40	1.25603	0.628013	−	0.778203i	$-0.283869\pi$
$281$	5916.40	1.25603	0.628013	+	0.778203i	$0.283869\pi$
$282$	−153.325	−0.0323773
$283$	2686.32	0.564258	0.282129	−	0.959377i	$-0.408959\pi$
$283$	2686.32	0.564258	0.282129	+	0.959377i	$0.408959\pi$
$284$	−3196.71	−0.667923
$285$	2450.77	0.509373
$286$	−3645.71	−0.753761
$287$	−4913.96	−1.01067
$288$	−1647.02	−0.336985
$289$	−3346.92	−0.681237
$290$	−445.980	−0.0903063
$291$	−3840.60	−0.773678
$292$	5136.20	1.02936
$293$	−4453.96	−0.888065	−0.444033	−	0.896011i	$-0.646453\pi$
$293$	−4453.96	−0.888065	−0.444033	+	0.896011i	$0.646453\pi$
$294$	−888.130	−0.176180
$295$	−9177.32	−1.81127
$296$	−760.581	−0.149351
$297$	1656.35	0.323606
$298$	604.366	0.117483
$299$	1051.55	0.203386
$300$	−283.464	−0.0545526
$301$	9934.02	1.90228
$302$	−1253.83	−0.238907
$303$	5472.07	1.03750
$304$	−1816.27	−0.342665
$305$	−8899.62	−1.67079
$306$	−462.960	−0.0864892
$307$	7276.51	1.35274	0.676372	−	0.736560i	$-0.263551\pi$
$307$	7276.51	1.35274	0.676372	+	0.736560i	$0.263551\pi$
$308$	9248.40	1.71096
$309$	−4683.84	−0.862312
$310$	−2444.53	−0.447870
$311$	955.938	0.174297	0.0871483	−	0.996195i	$-0.472225\pi$
$311$	955.938	0.174297	0.0871483	+	0.996195i	$0.472225\pi$
$312$	2551.34	0.462952
$313$	8293.47	1.49768	0.748841	−	0.662750i	$-0.230611\pi$
$313$	8293.47	1.49768	0.748841	+	0.662750i	$0.230611\pi$
$314$	−743.296	−0.133588
$315$	2543.83	0.455012
$316$	−4200.79	−0.747826
$317$	572.228	0.101387	0.0506933	−	0.998714i	$-0.483857\pi$
$317$	572.228	0.101387	0.0506933	+	0.998714i	$0.483857\pi$
$318$	604.586	0.106615
$319$	−1779.04	−0.312248
$320$	−324.698	−0.0567223
$321$	3445.90	0.599163
$322$	714.242	0.123612
$323$	−2732.54	−0.470720
$324$	−511.141	−0.0876442
$325$	684.576	0.116841
$326$	−176.849	−0.0300453
$327$	−1551.68	−0.262410
$328$	3826.08	0.644085
$329$	−939.339	−0.157409
$330$	2830.26	0.472123
$331$	−10793.7	−1.79238	−0.896188	−	0.443675i	$-0.853674\pi$
$331$	−10793.7	−1.79238	−0.896188	+	0.443675i	$0.853674\pi$
$332$	6840.25	1.13075
$333$	−367.996	−0.0605587
$334$	166.480	0.0272736
$335$	10974.6	1.78986
$336$	−1885.24	−0.306096
$337$	−1601.68	−0.258899	−0.129450	−	0.991586i	$-0.541321\pi$
$337$	−1601.68	−0.258899	−0.129450	+	0.991586i	$0.541321\pi$
$338$	138.737	0.0223263
$339$	−309.286	−0.0495519
$340$	2954.51	0.471267
$341$	−9751.35	−1.54858
$342$	807.785	0.127719
$343$	2753.34	0.433429
$344$	−7734.76	−1.21230
$345$	−816.341	−0.127392
$346$	1549.02	0.240682
$347$	−1946.21	−0.301089	−0.150544	−	0.988603i	$-0.548103\pi$
$347$	−1946.21	−0.301089	−0.150544	+	0.988603i	$0.548103\pi$
$348$	549.003	0.0845680
$349$	−11097.5	−1.70211	−0.851053	−	0.525079i	$-0.824036\pi$
$349$	−11097.5	−1.70211	−0.851053	+	0.525079i	$0.824036\pi$
$350$	464.985	0.0710128
$351$	1234.42	0.187717
$352$	−11226.5	−1.69993
$353$	3266.91	0.492578	0.246289	−	0.969196i	$-0.420789\pi$
$353$	3266.91	0.492578	0.246289	+	0.969196i	$0.420789\pi$
$354$	−3024.88	−0.454154
$355$	−5993.37	−0.896042
$356$	9121.44	1.35796
$357$	−2836.30	−0.420484
$358$	−2981.95	−0.440227
$359$	−4716.06	−0.693326	−0.346663	−	0.937990i	$-0.612685\pi$
$359$	−4716.06	−0.693326	−0.346663	+	0.937990i	$0.612685\pi$
$360$	−1980.66	−0.289973
$361$	−2091.20	−0.304884
$362$	−2834.09	−0.411482
$363$	7297.05	1.05508
$364$	6892.55	0.992494
$365$	9629.62	1.38092
$366$	−2933.35	−0.418931
$367$	−7814.01	−1.11141	−0.555706	−	0.831379i	$-0.687552\pi$
$367$	−7814.01	−1.11141	−0.555706	+	0.831379i	$0.687552\pi$
$368$	604.991	0.0856993
$369$	1851.19	0.261163
$370$	−628.807	−0.0883516
$371$	3703.96	0.518328
$372$	3009.22	0.419411
$373$	7877.42	1.09351	0.546753	−	0.837294i	$-0.315864\pi$
$373$	7877.42	1.09351	0.546753	+	0.837294i	$0.315864\pi$
$374$	−3155.65	−0.436296
$375$	3905.19	0.537768
$376$	731.382	0.100314
$377$	−1325.86	−0.181129
$378$	838.458	0.114089
$379$	−2290.56	−0.310444	−0.155222	−	0.987880i	$-0.549609\pi$
$379$	−2290.56	−0.310444	−0.155222	+	0.987880i	$0.549609\pi$
$380$	−5155.10	−0.695924
$381$	6777.53	0.911347
$382$	598.601	0.0801757
$383$	−3659.39	−0.488214	−0.244107	−	0.969748i	$-0.578495\pi$
$383$	−3659.39	−0.488214	−0.244107	+	0.969748i	$0.578495\pi$
$384$	4285.04	0.569453
$385$	17339.4	2.29532
$386$	−2393.09	−0.315557
$387$	−3742.35	−0.491561
$388$	8078.56	1.05703
$389$	13385.3	1.74463	0.872313	−	0.488947i	$-0.162619\pi$
$389$	13385.3	1.74463	0.872313	+	0.488947i	$0.162619\pi$
$390$	2109.30	0.273869
$391$	910.196	0.117725
$392$	4236.49	0.545855
$393$	4045.27	0.519229
$394$	3405.63	0.435465
$395$	−7875.87	−1.00324
$396$	−3484.06	−0.442123
$397$	−2678.76	−0.338648	−0.169324	−	0.985560i	$-0.554158\pi$
$397$	−2678.76	−0.338648	−0.169324	+	0.985560i	$0.554158\pi$
$398$	1810.32	0.227998
$399$	4948.84	0.620933
$400$	393.860	0.0492325
$401$	2482.48	0.309150	0.154575	−	0.987981i	$-0.450599\pi$
$401$	2482.48	0.309150	0.154575	+	0.987981i	$0.450599\pi$
$402$	3617.26	0.448788
$403$	−7267.39	−0.898299
$404$	−11510.3	−1.41747
$405$	−958.314	−0.117578
$406$	−900.566	−0.110085
$407$	−2508.35	−0.305489
$408$	2208.38	0.267968
$409$	3824.62	0.462384	0.231192	−	0.972908i	$-0.425737\pi$
$409$	3824.62	0.462384	0.231192	+	0.972908i	$0.425737\pi$
$410$	3163.19	0.381022
$411$	5633.37	0.676091
$412$	9852.27	1.17812
$413$	−18531.7	−2.20796
$414$	−269.070	−0.0319421
$415$	12824.5	1.51694
$416$	−8366.77	−0.986093
$417$	−3017.80	−0.354394
$418$	5506.06	0.644282
$419$	−6078.79	−0.708755	−0.354377	−	0.935102i	$-0.615307\pi$
$419$	−6078.79	−0.708755	−0.354377	+	0.935102i	$0.615307\pi$
$420$	−5350.85	−0.621655
$421$	12042.4	1.39408	0.697042	−	0.717030i	$-0.254499\pi$
$421$	12042.4	1.39408	0.697042	+	0.717030i	$0.254499\pi$
$422$	−7241.22	−0.835301
$423$	353.868	0.0406753
$424$	−2883.95	−0.330323
$425$	592.554	0.0676308
$426$	−1975.44	−0.224672
$427$	−17971.0	−2.03671
$428$	−7248.31	−0.818598
$429$	8414.14	0.946942
$430$	−6394.68	−0.717160
$431$	−6074.37	−0.678867	−0.339434	−	0.940630i	$-0.610235\pi$
$431$	−6074.37	−0.678867	−0.339434	+	0.940630i	$0.610235\pi$
$432$	710.207	0.0790969
$433$	2239.69	0.248574	0.124287	−	0.992246i	$-0.460336\pi$
$433$	2239.69	0.248574	0.124287	+	0.992246i	$0.460336\pi$
$434$	−4936.23	−0.545960
$435$	1029.30	0.113451
$436$	3263.90	0.358515
$437$	−1588.13	−0.173846
$438$	3173.96	0.346251
$439$	12059.8	1.31113	0.655563	−	0.755140i	$-0.272431\pi$
$439$	12059.8	1.31113	0.655563	+	0.755140i	$0.272431\pi$
$440$	−13500.7	−1.46277
$441$	2049.76	0.221333
$442$	−2351.81	−0.253086
$443$	2516.71	0.269915	0.134958	−	0.990851i	$-0.456910\pi$
$443$	2516.71	0.269915	0.134958	+	0.990851i	$0.456910\pi$
$444$	774.064	0.0827375
$445$	17101.4	1.82176
$446$	3473.99	0.368830
$447$	−1394.85	−0.147593
$448$	−655.662	−0.0691453
$449$	−9926.02	−1.04329	−0.521646	−	0.853162i	$-0.674682\pi$
$449$	−9926.02	−1.04329	−0.521646	+	0.853162i	$0.674682\pi$
$450$	−175.169	−0.0183501
$451$	12618.1	1.31744
$452$	650.571	0.0676997
$453$	2893.78	0.300136
$454$	−2488.12	−0.257210
$455$	12922.5	1.33147
$456$	−3853.24	−0.395711
$457$	4302.58	0.440407	0.220204	−	0.975454i	$-0.429328\pi$
$457$	4302.58	0.440407	0.220204	+	0.975454i	$0.429328\pi$
$458$	−1326.43	−0.135327
$459$	1068.49	0.108656
$460$	1717.14	0.174048
$461$	−12281.9	−1.24084	−0.620419	−	0.784270i	$-0.713038\pi$
$461$	−12281.9	−1.24084	−0.620419	+	0.784270i	$0.713038\pi$
$462$	5715.13	0.575524
$463$	12265.5	1.23116	0.615580	−	0.788074i	$-0.288922\pi$
$463$	12265.5	1.23116	0.615580	+	0.788074i	$0.288922\pi$
$464$	−762.815	−0.0763207
$465$	5641.85	0.562655
$466$	5576.74	0.554372
$467$	17987.3	1.78234	0.891170	−	0.453669i	$-0.149885\pi$
$467$	17987.3	1.78234	0.891170	+	0.453669i	$0.149885\pi$
$468$	−2596.56	−0.256466
$469$	22160.9	2.18187
$470$	604.666	0.0593429
$471$	1715.49	0.167825
$472$	14429.1	1.40710
$473$	−25508.7	−2.47969
$474$	−2595.92	−0.251550
$475$	−1033.90	−0.0998710
$476$	5966.04	0.574481
$477$	−1395.36	−0.133939
$478$	−6420.30	−0.614346
$479$	8238.86	0.785894	0.392947	−	0.919561i	$-0.371456\pi$
$479$	8238.86	0.785894	0.392947	+	0.919561i	$0.371456\pi$
$480$	6495.33	0.617645
$481$	−1869.39	−0.177208
$482$	−2553.89	−0.241341
$483$	−1648.44	−0.155293
$484$	−15349.1	−1.44150
$485$	15146.1	1.41804
$486$	−315.864	−0.0294813
$487$	7621.90	0.709202	0.354601	−	0.935018i	$-0.384617\pi$
$487$	7621.90	0.709202	0.354601	+	0.935018i	$0.384617\pi$
$488$	13992.5	1.29797
$489$	408.159	0.0377456
$490$	3502.50	0.322912
$491$	14710.0	1.35204	0.676022	−	0.736881i	$-0.263702\pi$
$491$	14710.0	1.35204	0.676022	+	0.736881i	$0.263702\pi$
$492$	−3893.90	−0.356810
$493$	−1147.64	−0.104842
$494$	4103.50	0.373735
$495$	−6532.10	−0.593123
$496$	−4181.18	−0.378509
$497$	−12102.4	−1.09229
$498$	4227.00	0.380354
$499$	751.722	0.0674383	0.0337191	−	0.999431i	$-0.489265\pi$
$499$	751.722	0.0674383	0.0337191	+	0.999431i	$0.489265\pi$
$500$	−8214.40	−0.734719
$501$	−384.228	−0.0342636
$502$	−407.162	−0.0362002
$503$	12031.5	1.06652	0.533258	−	0.845953i	$-0.320968\pi$
$503$	12031.5	1.06652	0.533258	+	0.845953i	$0.320968\pi$
$504$	−3999.55	−0.353481
$505$	−21580.1	−1.90159
$506$	−1834.04	−0.161133
$507$	−320.198	−0.0280483
$508$	−14256.3	−1.24512
$509$	7676.84	0.668507	0.334254	−	0.942483i	$-0.391516\pi$
$509$	7676.84	0.668507	0.334254	+	0.942483i	$0.391516\pi$
$510$	1825.77	0.158522
$511$	19445.1	1.68337
$512$	−8728.02	−0.753374
$513$	−1864.33	−0.160453
$514$	−1785.50	−0.153220
$515$	18471.6	1.58049
$516$	7871.88	0.671590
$517$	2412.05	0.205187
$518$	−1269.75	−0.107702
$519$	−3575.08	−0.302367
$520$	−10061.7	−0.848524
$521$	19146.8	1.61005	0.805024	−	0.593242i	$-0.202152\pi$
$521$	19146.8	1.61005	0.805024	+	0.593242i	$0.202152\pi$
$522$	339.262	0.0284465
$523$	15491.6	1.29522	0.647609	−	0.761973i	$-0.275769\pi$
$523$	15491.6	1.29522	0.647609	+	0.761973i	$0.275769\pi$
$524$	−8509.07	−0.709391
$525$	−1073.16	−0.0892127
$526$	−1879.55	−0.155802
$527$	−6290.49	−0.519958
$528$	4840.94	0.399006
$529$	529.000	0.0434783
$530$	−2384.29	−0.195410
$531$	6981.29	0.570550
$532$	−10409.7	−0.848341
$533$	9403.93	0.764220
$534$	5636.68	0.456785
$535$	−13589.5	−1.09818
$536$	−17254.8	−1.39047
$537$	6882.21	0.553053
$538$	5724.63	0.458748
$539$	13971.7	1.11652
$540$	2015.78	0.160639
$541$	2163.30	0.171918	0.0859589	−	0.996299i	$-0.472605\pi$
$541$	2163.30	0.171918	0.0859589	+	0.996299i	$0.472605\pi$
$542$	−6523.36	−0.516979
$543$	6540.95	0.516941
$544$	−7242.09	−0.570776
$545$	6119.33	0.480960
$546$	4259.32	0.333850
$547$	−8869.72	−0.693313	−0.346656	−	0.937992i	$-0.612683\pi$
$547$	−8869.72	−0.693313	−0.346656	+	0.937992i	$0.612683\pi$
$548$	−11849.6	−0.923701
$549$	6770.04	0.526299
$550$	−1193.99	−0.0925675
$551$	2002.43	0.154821
$552$	1283.50	0.0989660
$553$	−15903.7	−1.22296
$554$	−3079.14	−0.236137
$555$	1451.26	0.110995
$556$	6347.83	0.484187
$557$	1578.72	0.120094	0.0600470	−	0.998196i	$-0.480875\pi$
$557$	1578.72	0.120094	0.0600470	+	0.998196i	$0.480875\pi$
$558$	1859.58	0.141079
$559$	−19010.9	−1.43842
$560$	7434.77	0.561029
$561$	7283.09	0.548115
$562$	−7690.46	−0.577228
$563$	−17357.7	−1.29936	−0.649681	−	0.760207i	$-0.725098\pi$
$563$	−17357.7	−1.29936	−0.649681	+	0.760207i	$0.725098\pi$
$564$	−744.347	−0.0555721
$565$	1219.72	0.0908216
$566$	−3491.82	−0.259314
$567$	−1935.12	−0.143329
$568$	9423.09	0.696099
$569$	−15709.3	−1.15741	−0.578706	−	0.815536i	$-0.696442\pi$
$569$	−15709.3	−1.15741	−0.578706	+	0.815536i	$0.696442\pi$
$570$	−3185.65	−0.234091
$571$	−5753.64	−0.421685	−0.210842	−	0.977520i	$-0.567621\pi$
$571$	−5753.64	−0.421685	−0.210842	+	0.977520i	$0.567621\pi$
$572$	−17698.8	−1.29375
$573$	−1381.54	−0.100724
$574$	6387.43	0.464471
$575$	344.388	0.0249774
$576$	247.001	0.0178676
$577$	−11310.3	−0.816042	−0.408021	−	0.912973i	$-0.633781\pi$
$577$	−11310.3	−0.816042	−0.408021	+	0.912973i	$0.633781\pi$
$578$	4350.50	0.313075
$579$	5523.13	0.396431
$580$	−2165.09	−0.155001
$581$	25896.4	1.84917
$582$	4992.22	0.355557
$583$	−9511.08	−0.675658
$584$	−15140.2	−1.07278
$585$	−4868.18	−0.344059
$586$	5789.49	0.408126
$587$	−24212.3	−1.70247	−0.851234	−	0.524786i	$-0.824145\pi$
$587$	−24212.3	−1.70247	−0.851234	+	0.524786i	$0.824145\pi$
$588$	−4311.59	−0.302393
$589$	10975.8	0.767827
$590$	11929.2	0.832400
$591$	−7860.03	−0.547070
$592$	−1075.53	−0.0746687
$593$	8578.05	0.594028	0.297014	−	0.954873i	$-0.404009\pi$
$593$	8578.05	0.594028	0.297014	+	0.954873i	$0.404009\pi$
$594$	−2153.01	−0.148719
$595$	11185.4	0.770687
$596$	2934.01	0.201647
$597$	−4178.13	−0.286431
$598$	−1366.86	−0.0934698
$599$	3085.87	0.210493	0.105246	−	0.994446i	$-0.466437\pi$
$599$	3085.87	0.210493	0.105246	+	0.994446i	$0.466437\pi$
$600$	835.579	0.0568540
$601$	−10852.9	−0.736603	−0.368301	−	0.929706i	$-0.620061\pi$
$601$	−10852.9	−0.736603	−0.368301	+	0.929706i	$0.620061\pi$
$602$	−12912.8	−0.874228
$603$	−8348.47	−0.563808
$604$	−6086.96	−0.410058
$605$	−28777.2	−1.93382
$606$	−7112.89	−0.476801
$607$	18115.3	1.21133	0.605664	−	0.795721i	$-0.292908\pi$
$607$	18115.3	1.21133	0.605664	+	0.795721i	$0.292908\pi$
$608$	12636.2	0.842870
$609$	2078.46	0.138298
$610$	11568.2	0.767840
$611$	1797.63	0.119025
$612$	−2247.53	−0.148449
$613$	−18722.5	−1.23360	−0.616798	−	0.787121i	$-0.711570\pi$
$613$	−18722.5	−1.23360	−0.616798	+	0.787121i	$0.711570\pi$
$614$	−9458.39	−0.621677
$615$	−7300.50	−0.478674
$616$	−27261.9	−1.78314
$617$	28715.7	1.87367	0.936833	−	0.349776i	$-0.113742\pi$
$617$	28715.7	1.87367	0.936833	+	0.349776i	$0.113742\pi$
$618$	6088.31	0.396291
$619$	26847.6	1.74329	0.871645	−	0.490138i	$-0.163054\pi$
$619$	26847.6	1.74329	0.871645	+	0.490138i	$0.163054\pi$
$620$	−11867.4	−0.768720
$621$	621.000	0.0401286
$622$	−1242.58	−0.0801011
$623$	34532.8	2.22075
$624$	3607.81	0.231455
$625$	−17272.5	−1.10544
$626$	−10780.3	−0.688286
$627$	−12707.7	−0.809406
$628$	−3608.47	−0.229289
$629$	−1618.11	−0.102573
$630$	−3306.61	−0.209109
$631$	14221.9	0.897252	0.448626	−	0.893720i	$-0.351914\pi$
$631$	14221.9	0.897252	0.448626	+	0.893720i	$0.351914\pi$
$632$	12382.9	0.779374
$633$	16712.4	1.04938
$634$	−743.812	−0.0465940
$635$	−26728.4	−1.67037
$636$	2935.08	0.182993
$637$	10412.7	0.647668
$638$	2312.49	0.143499
$639$	4559.22	0.282253
$640$	−16898.8	−1.04373
$641$	−4181.20	−0.257640	−0.128820	−	0.991668i	$-0.541119\pi$
$641$	−4181.20	−0.257640	−0.128820	+	0.991668i	$0.541119\pi$
$642$	−4479.16	−0.275356
$643$	17521.8	1.07464	0.537318	−	0.843380i	$-0.319437\pi$
$643$	17521.8	1.07464	0.537318	+	0.843380i	$0.319437\pi$
$644$	3467.42	0.212167
$645$	14758.6	0.900961
$646$	3551.90	0.216327
$647$	9867.51	0.599586	0.299793	−	0.954004i	$-0.403082\pi$
$647$	9867.51	0.599586	0.299793	+	0.954004i	$0.403082\pi$
$648$	1506.71	0.0913415
$649$	47586.1	2.87815
$650$	−889.848	−0.0536965
$651$	11392.6	0.685884
$652$	−858.547	−0.0515695
$653$	15304.0	0.917140	0.458570	−	0.888658i	$-0.348362\pi$
$653$	15304.0	0.917140	0.458570	+	0.888658i	$0.348362\pi$
$654$	2016.96	0.120595
$655$	−15953.3	−0.951673
$656$	5410.40	0.322013
$657$	−7325.36	−0.434992
$658$	1221.00	0.0723399
$659$	17549.7	1.03739	0.518696	−	0.854959i	$-0.326418\pi$
$659$	17549.7	1.03739	0.518696	+	0.854959i	$0.326418\pi$
$660$	13740.0	0.810347
$661$	2894.87	0.170344	0.0851720	−	0.996366i	$-0.472856\pi$
$661$	2894.87	0.170344	0.0851720	+	0.996366i	$0.472856\pi$
$662$	14030.2	0.823717
$663$	5427.87	0.317950
$664$	−20163.3	−1.17845
$665$	−19516.7	−1.13808
$666$	478.340	0.0278308
$667$	−667.000	−0.0387202
$668$	808.209	0.0468122
$669$	−8017.81	−0.463358
$670$	−14265.3	−0.822564
$671$	46146.2	2.65492
$672$	13116.0	0.752918
$673$	−13870.6	−0.794462	−0.397231	−	0.917719i	$-0.630029\pi$
$673$	−13870.6	−0.794462	−0.397231	+	0.917719i	$0.630029\pi$
$674$	2081.95	0.118982
$675$	404.282	0.0230531
$676$	673.523	0.0383206
$677$	10567.9	0.599937	0.299968	−	0.953949i	$-0.403024\pi$
$677$	10567.9	0.599937	0.299968	+	0.953949i	$0.403024\pi$
$678$	402.026	0.0227725
$679$	30584.5	1.72861
$680$	−8709.14	−0.491148
$681$	5742.46	0.323130
$682$	12675.3	0.711676
$683$	−29348.8	−1.64422	−0.822109	−	0.569330i	$-0.807203\pi$
$683$	−29348.8	−1.64422	−0.822109	+	0.569330i	$0.807203\pi$
$684$	3921.55	0.219216
$685$	−22216.2	−1.23918
$686$	−3578.93	−0.199190
$687$	3061.34	0.170011
$688$	−10937.6	−0.606094
$689$	−7088.32	−0.391935
$690$	1061.12	0.0585454
$691$	17.2346	0.000948823 0	0.000474411	−	1.00000i	$-0.499849\pi$
$691$	17.2346	0.000948823 0	0.000474411		1.00000i	$0.499849\pi$
$692$	7520.03	0.413105
$693$	−13190.3	−0.723025
$694$	2529.78	0.138371
$695$	11901.2	0.649554
$696$	−1618.32	−0.0881355
$697$	8139.83	0.442350
$698$	14425.1	0.782233
$699$	−12870.8	−0.696452
$700$	2257.36	0.121886
$701$	29851.4	1.60838	0.804189	−	0.594374i	$-0.202600\pi$
$701$	29851.4	1.60838	0.804189	+	0.594374i	$0.202600\pi$
$702$	−1604.57	−0.0862687
$703$	2823.31	0.151470
$704$	1683.62	0.0901332
$705$	−1395.54	−0.0745520
$706$	−4246.51	−0.226373
$707$	−43576.6	−2.31806
$708$	−14684.9	−0.779507
$709$	−22103.1	−1.17080	−0.585400	−	0.810744i	$-0.699063\pi$
$709$	−22103.1	−1.17080	−0.585400	+	0.810744i	$0.699063\pi$
$710$	7790.50	0.411792
$711$	5991.26	0.316020
$712$	−26887.7	−1.41525
$713$	−3655.99	−0.192031
$714$	3686.77	0.193241
$715$	−33182.6	−1.73561
$716$	−14476.5	−0.755601
$717$	14817.7	0.771798
$718$	6130.18	0.318630
$719$	1462.98	0.0758833	0.0379417	−	0.999280i	$-0.487920\pi$
$719$	1462.98	0.0758833	0.0379417	+	0.999280i	$0.487920\pi$
$720$	−2800.83	−0.144973
$721$	37299.6	1.92664
$722$	2718.25	0.140115
$723$	5894.25	0.303194
$724$	−13758.6	−0.706264
$725$	−434.229	−0.0222439
$726$	−9485.09	−0.484883
$727$	−23355.6	−1.19149	−0.595744	−	0.803174i	$-0.703143\pi$
$727$	−23355.6	−1.19149	−0.595744	+	0.803174i	$0.703143\pi$
$728$	−20317.5	−1.03436
$729$	729.000	0.0370370
$730$	−12517.1	−0.634628
$731$	−16455.4	−0.832593
$732$	−14240.5	−0.719049
$733$	33400.6	1.68306	0.841528	−	0.540214i	$-0.181657\pi$
$733$	33400.6	1.68306	0.841528	+	0.540214i	$0.181657\pi$
$734$	10157.1	0.510768
$735$	−8083.60	−0.405671
$736$	−4209.06	−0.210799
$737$	−56905.2	−2.84414
$738$	−2406.27	−0.120022
$739$	21886.8	1.08947	0.544736	−	0.838608i	$-0.316630\pi$
$739$	21886.8	1.08947	0.544736	+	0.838608i	$0.316630\pi$
$740$	−3052.66	−0.151646
$741$	−9470.68	−0.469520
$742$	−4814.60	−0.238207
$743$	−7969.43	−0.393499	−0.196750	−	0.980454i	$-0.563039\pi$
$743$	−7969.43	−0.393499	−0.196750	+	0.980454i	$0.563039\pi$
$744$	−8870.42	−0.437104
$745$	5500.84	0.270517
$746$	−10239.5	−0.502540
$747$	−9755.71	−0.477835
$748$	−15319.7	−0.748855
$749$	−27441.3	−1.33870
$750$	−5076.17	−0.247141
$751$	7917.68	0.384714	0.192357	−	0.981325i	$-0.438387\pi$
$751$	7917.68	0.384714	0.192357	+	0.981325i	$0.438387\pi$
$752$	1034.24	0.0501526
$753$	939.710	0.0454780
$754$	1723.43	0.0832408
$755$	−11412.2	−0.550107
$756$	4070.45	0.195821
$757$	−26956.0	−1.29423	−0.647115	−	0.762393i	$-0.724025\pi$
$757$	−26956.0	−1.29423	−0.647115	+	0.762393i	$0.724025\pi$
$758$	2977.39	0.142670
$759$	4232.88	0.202429
$760$	15195.9	0.725282
$761$	7116.74	0.339003	0.169502	−	0.985530i	$-0.445784\pi$
$761$	7116.74	0.339003	0.169502	+	0.985530i	$0.445784\pi$
$762$	−8809.79	−0.418826
$763$	12356.8	0.586297
$764$	2906.02	0.137613
$765$	−4213.78	−0.199150
$766$	4756.67	0.224367
$767$	35464.5	1.66956
$768$	−6228.59	−0.292650
$769$	−15374.9	−0.720978	−0.360489	−	0.932763i	$-0.617390\pi$
$769$	−15374.9	−0.720978	−0.360489	+	0.932763i	$0.617390\pi$
$770$	−22538.6	−1.05485
$771$	4120.85	0.192489
$772$	−11617.7	−0.541619
$773$	−14463.1	−0.672966	−0.336483	−	0.941689i	$-0.609238\pi$
$773$	−14463.1	−0.672966	−0.336483	+	0.941689i	$0.609238\pi$
$774$	4864.50	0.225906
$775$	−2380.12	−0.110318
$776$	−23813.5	−1.10162
$777$	2930.52	0.135305
$778$	−17398.9	−0.801774
$779$	−14202.6	−0.653223
$780$	10240.0	0.470066
$781$	31076.7	1.42383
$782$	−1183.12	−0.0541027
$783$	−783.000	−0.0357371
$784$	5990.76	0.272903
$785$	−6765.35	−0.307600
$786$	−5258.26	−0.238621
$787$	13225.1	0.599012	0.299506	−	0.954094i	$-0.403178\pi$
$787$	13225.1	0.599012	0.299506	+	0.954094i	$0.403178\pi$
$788$	16533.3	0.747428
$789$	4337.90	0.195733
$790$	10237.5	0.461054
$791$	2462.99	0.110713
$792$	10270.1	0.460774
$793$	34391.4	1.54007
$794$	3481.99	0.155631
$795$	5502.84	0.245491
$796$	8788.53	0.391333
$797$	11493.0	0.510793	0.255396	−	0.966836i	$-0.417794\pi$
$797$	11493.0	0.510793	0.255396	+	0.966836i	$0.417794\pi$
$798$	−6432.77	−0.285360
$799$	1555.99	0.0688947
$800$	−2740.17	−0.121100
$801$	−13009.2	−0.573854
$802$	−3226.86	−0.142075
$803$	−49931.4	−2.19432
$804$	17560.7	0.770296
$805$	6500.91	0.284630
$806$	9446.54	0.412829
$807$	−13212.2	−0.576321
$808$	33929.4	1.47727
$809$	−338.296	−0.0147019	−0.00735096	−	0.999973i	$-0.502340\pi$
$809$	−338.296	−0.0147019	−0.00735096	+	0.999973i	$0.502340\pi$
$810$	1245.67	0.0540349
$811$	−38019.6	−1.64617	−0.823087	−	0.567915i	$-0.807750\pi$
$811$	−38019.6	−1.64617	−0.823087	+	0.567915i	$0.807750\pi$
$812$	−4371.97	−0.188948
$813$	15055.6	0.649475
$814$	3260.48	0.140393
$815$	−1609.65	−0.0691823
$816$	3122.84	0.133972
$817$	28711.8	1.22950
$818$	−4971.44	−0.212497
$819$	−9830.30	−0.419412
$820$	15356.3	0.653982
$821$	35319.2	1.50140	0.750700	−	0.660643i	$-0.229716\pi$
$821$	35319.2	1.50140	0.750700	+	0.660643i	$0.229716\pi$
$822$	−7322.55	−0.310710
$823$	−2788.73	−0.118115	−0.0590577	−	0.998255i	$-0.518810\pi$
$823$	−2788.73	−0.118115	−0.0590577	+	0.998255i	$0.518810\pi$
$824$	−29042.0	−1.22782
$825$	2755.68	0.116292
$826$	24088.6	1.01471
$827$	−40046.5	−1.68386	−0.841930	−	0.539587i	$-0.818581\pi$
$827$	−40046.5	−1.68386	−0.841930	+	0.539587i	$0.818581\pi$
$828$	−1306.25	−0.0548252
$829$	34832.7	1.45933	0.729667	−	0.683802i	$-0.239675\pi$
$829$	34832.7	1.45933	0.729667	+	0.683802i	$0.239675\pi$
$830$	−16669.9	−0.697134
$831$	7106.51	0.296657
$832$	1254.75	0.0522844
$833$	9012.97	0.374887
$834$	3922.70	0.162868
$835$	1515.27	0.0628002
$836$	26730.2	1.10584
$837$	−4291.82	−0.177236
$838$	7901.53	0.325721
$839$	−27736.9	−1.14134	−0.570671	−	0.821179i	$-0.693317\pi$
$839$	−27736.9	−1.14134	−0.570671	+	0.821179i	$0.693317\pi$
$840$	15773.0	0.647879
$841$	841.000	0.0344828
$842$	−15653.3	−0.640676
$843$	17749.2	0.725166
$844$	−35153.8	−1.43370
$845$	1262.76	0.0514085
$846$	−459.976	−0.0186930
$847$	−58109.8	−2.35735
$848$	−4078.16	−0.165147
$849$	8058.95	0.325774
$850$	−770.233	−0.0310809
$851$	−940.433	−0.0378821
$852$	−9590.14	−0.385625
$853$	12883.4	0.517139	0.258569	−	0.965993i	$-0.416749\pi$
$853$	12883.4	0.517139	0.258569	+	0.965993i	$0.416749\pi$
$854$	23359.6	0.936008
$855$	7352.32	0.294087
$856$	21366.2	0.853131
$857$	−249.040	−0.00992655	−0.00496327	−	0.999988i	$-0.501580\pi$
$857$	−249.040	−0.00992655	−0.00496327	+	0.999988i	$0.501580\pi$
$858$	−10937.1	−0.435184
$859$	18264.6	0.725471	0.362735	−	0.931892i	$-0.381843\pi$
$859$	18264.6	0.725471	0.362735	+	0.931892i	$0.381843\pi$
$860$	−31044.2	−1.23093
$861$	−14741.9	−0.583510
$862$	7895.78	0.311985
$863$	37209.2	1.46769	0.733844	−	0.679318i	$-0.237724\pi$
$863$	37209.2	1.46769	0.733844	+	0.679318i	$0.237724\pi$
$864$	−4941.07	−0.194558
$865$	14099.0	0.554195
$866$	−2911.26	−0.114236
$867$	−10040.8	−0.393313
$868$	−23963.9	−0.937081
$869$	40837.9	1.59417
$870$	−1337.94	−0.0521384
$871$	−42409.7	−1.64983
$872$	−9621.14	−0.373639
$873$	−11521.8	−0.446683
$874$	2064.34	0.0798940
$875$	−31098.8	−1.20152
$876$	15408.6	0.594302
$877$	32393.9	1.24728	0.623639	−	0.781712i	$-0.285653\pi$
$877$	32393.9	1.24728	0.623639	+	0.781712i	$0.285653\pi$
$878$	−15676.0	−0.602551
$879$	−13361.9	−0.512725
$880$	−19091.1	−0.731320
$881$	−41117.5	−1.57240	−0.786199	−	0.617974i	$-0.787954\pi$
$881$	−41117.5	−1.57240	−0.786199	+	0.617974i	$0.787954\pi$
$882$	−2664.39	−0.101717
$883$	27141.8	1.03442	0.517210	−	0.855858i	$-0.326970\pi$
$883$	27141.8	1.03442	0.517210	+	0.855858i	$0.326970\pi$
$884$	−11417.3	−0.434395
$885$	−27531.9	−1.04574
$886$	−3271.35	−0.124044
$887$	−24297.9	−0.919779	−0.459889	−	0.887976i	$-0.652111\pi$
$887$	−24297.9	−0.919779	−0.459889	+	0.887976i	$0.652111\pi$
$888$	−2281.74	−0.0862278
$889$	−53972.6	−2.03620
$890$	−22229.3	−0.837221
$891$	4969.04	0.186834
$892$	16865.2	0.633057
$893$	−2714.93	−0.101737
$894$	1813.10	0.0678290
$895$	−27141.2	−1.01367
$896$	−34123.8	−1.27232
$897$	3154.64	0.117425
$898$	12902.4	0.479463
$899$	4609.73	0.171016
$900$	−850.392	−0.0314960
$901$	−6135.49	−0.226862
$902$	−16401.7	−0.605453
$903$	29802.1	1.09828
$904$	−1917.72	−0.0705556
$905$	−25795.4	−0.947478
$906$	−3761.49	−0.137933
$907$	−13682.5	−0.500906	−0.250453	−	0.968129i	$-0.580580\pi$
$907$	−13682.5	−0.500906	−0.250453	+	0.968129i	$0.580580\pi$
$908$	−12079.0	−0.441473
$909$	16416.2	0.599000
$910$	−16797.4	−0.611898
$911$	−14051.1	−0.511016	−0.255508	−	0.966807i	$-0.582243\pi$
$911$	−14051.1	−0.511016	−0.255508	+	0.966807i	$0.582243\pi$
$912$	−5448.81	−0.197838
$913$	−66497.3	−2.41045
$914$	−5592.72	−0.202397
$915$	−26698.8	−0.964630
$916$	−6439.40	−0.232275
$917$	−32214.4	−1.16010
$918$	−1388.88	−0.0499345
$919$	51826.7	1.86029	0.930144	−	0.367194i	$-0.119681\pi$
$919$	51826.7	1.86029	0.930144	+	0.367194i	$0.119681\pi$
$920$	−5061.70	−0.181390
$921$	21829.5	0.781007
$922$	15964.7	0.570249
$923$	23160.6	0.825936
$924$	27745.2	0.987824
$925$	−612.239	−0.0217625
$926$	−15943.4	−0.565801
$927$	−14051.5	−0.497856
$928$	5307.07	0.187730
$929$	2535.29	0.0895372	0.0447686	−	0.998997i	$-0.485745\pi$
$929$	2535.29	0.0895372	0.0447686	+	0.998997i	$0.485745\pi$
$930$	−7333.58	−0.258578
$931$	−15726.1	−0.553599
$932$	27073.3	0.951519
$933$	2867.81	0.100630
$934$	−23380.8	−0.819106
$935$	−28722.2	−1.00462
$936$	7654.01	0.267285
$937$	20093.2	0.700552	0.350276	−	0.936646i	$-0.386088\pi$
$937$	20093.2	0.700552	0.350276	+	0.936646i	$0.386088\pi$
$938$	−28806.0	−1.00272
$939$	24880.4	0.864687
$940$	2935.47	0.101856
$941$	12031.5	0.416808	0.208404	−	0.978043i	$-0.433173\pi$
$941$	12031.5	0.416808	0.208404	+	0.978043i	$0.433173\pi$
$942$	−2229.89	−0.0771271
$943$	4730.82	0.163369
$944$	20403.9	0.703487
$945$	7631.50	0.262701
$946$	33157.6	1.13958
$947$	10817.8	0.371206	0.185603	−	0.982625i	$-0.440576\pi$
$947$	10817.8	0.371206	0.185603	+	0.982625i	$0.440576\pi$
$948$	−12602.4	−0.431758
$949$	−37212.4	−1.27288
$950$	1343.92	0.0458975
$951$	1716.68	0.0585355
$952$	−17586.4	−0.598716
$953$	−32422.8	−1.10208	−0.551038	−	0.834480i	$-0.685768\pi$
$953$	−32422.8	−1.10208	−0.551038	+	0.834480i	$0.685768\pi$
$954$	1813.76	0.0615541
$955$	5448.36	0.184612
$956$	−31168.5	−1.05446
$957$	−5337.11	−0.180276
$958$	−10709.3	−0.361171
$959$	−44861.1	−1.51058
$960$	−974.093	−0.0327487
$961$	−4523.92	−0.151855
$962$	2429.94	0.0814390
$963$	10337.7	0.345927
$964$	−12398.3	−0.414236
$965$	−21781.5	−0.726601
$966$	2142.73	0.0713676
$967$	26595.2	0.884432	0.442216	−	0.896909i	$-0.354193\pi$
$967$	26595.2	0.884432	0.442216	+	0.896909i	$0.354193\pi$
$968$	45245.1	1.50231
$969$	−8197.61	−0.271770
$970$	−19687.7	−0.651685
$971$	−35700.0	−1.17989	−0.589943	−	0.807445i	$-0.700850\pi$
$971$	−35700.0	−1.17989	−0.589943	+	0.807445i	$0.700850\pi$
$972$	−1533.42	−0.0506014
$973$	24032.2	0.791815
$974$	−9907.35	−0.325926
$975$	2053.73	0.0674584
$976$	19786.5	0.648926
$977$	31671.0	1.03710	0.518549	−	0.855048i	$-0.326472\pi$
$977$	31671.0	1.03710	0.518549	+	0.855048i	$0.326472\pi$
$978$	−530.547	−0.0173467
$979$	−88673.8	−2.89482
$980$	17003.5	0.554243
$981$	−4655.04	−0.151503
$982$	−19120.9	−0.621356
$983$	55705.1	1.80744	0.903722	−	0.428119i	$-0.140824\pi$
$983$	55705.1	1.80744	0.903722	+	0.428119i	$0.140824\pi$
$984$	11478.2	0.371863
$985$	30997.5	1.00270
$986$	1491.76	0.0481819
$987$	−2818.02	−0.0908799
$988$	19921.2	0.641476
$989$	−9563.78	−0.307493
$990$	8490.77	0.272580
$991$	7336.54	0.235169	0.117585	−	0.993063i	$-0.462485\pi$
$991$	7336.54	0.235169	0.117585	+	0.993063i	$0.462485\pi$
$992$	29089.4	0.931037
$993$	−32381.1	−1.03483
$994$	15731.3	0.501980
$995$	16477.2	0.524987
$996$	20520.8	0.652837
$997$	−96.7613	−0.00307368	−0.00153684	−	0.999999i	$-0.500489\pi$
$997$	−96.7613	−0.00307368	−0.00153684	+	0.999999i	$0.500489\pi$
$998$	−977.128	−0.0309924
$999$	−1103.99	−0.0349636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2001.4.a.h.1.18	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.4.a.h.1.18	✓	44	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:	\( N \)	\(=\)	\( 2001 = 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29985 q^{2} +3.00000 q^{3} -6.31038 q^{4} -11.8310 q^{5} -3.89956 q^{6} -23.8904 q^{7} +18.6014 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.29985 q^{2} +3.00000 q^{3} -6.31038 q^{4} -11.8310 q^{5} -3.89956 q^{6} -23.8904 q^{7} +18.6014 q^{8} +9.00000 q^{9} +15.3786 q^{10} +61.3461 q^{11} -18.9311 q^{12} +45.7194 q^{13} +31.0540 q^{14} -35.4931 q^{15} +26.3040 q^{16} +39.5737 q^{17} -11.6987 q^{18} -69.0493 q^{19} +74.6584 q^{20} -71.6712 q^{21} -79.7410 q^{22} +23.0000 q^{23} +55.8042 q^{24} +14.9734 q^{25} -59.4286 q^{26} +27.0000 q^{27} +150.758 q^{28} -29.0000 q^{29} +46.1358 q^{30} -158.956 q^{31} -183.002 q^{32} +184.038 q^{33} -51.4400 q^{34} +282.648 q^{35} -56.7934 q^{36} -40.8884 q^{37} +89.7539 q^{38} +137.158 q^{39} -220.074 q^{40} +205.688 q^{41} +93.1620 q^{42} -415.816 q^{43} -387.118 q^{44} -106.479 q^{45} -29.8966 q^{46} +39.3187 q^{47} +78.9119 q^{48} +227.751 q^{49} -19.4632 q^{50} +118.721 q^{51} -288.507 q^{52} -155.040 q^{53} -35.0960 q^{54} -725.789 q^{55} -444.395 q^{56} -207.148 q^{57} +37.6957 q^{58} +775.698 q^{59} +223.975 q^{60} +752.226 q^{61} +206.620 q^{62} -215.014 q^{63} +27.4446 q^{64} -540.908 q^{65} -239.223 q^{66} -927.608 q^{67} -249.725 q^{68} +69.0000 q^{69} -367.401 q^{70} +506.580 q^{71} +167.413 q^{72} -813.929 q^{73} +53.1489 q^{74} +44.9202 q^{75} +435.727 q^{76} -1465.58 q^{77} -178.286 q^{78} +665.696 q^{79} -311.203 q^{80} +81.0000 q^{81} -267.364 q^{82} -1083.97 q^{83} +452.273 q^{84} -468.198 q^{85} +540.500 q^{86} -87.0000 q^{87} +1141.12 q^{88} -1445.47 q^{89} +138.407 q^{90} -1092.26 q^{91} -145.139 q^{92} -476.869 q^{93} -51.1085 q^{94} +816.924 q^{95} -549.007 q^{96} -1280.20 q^{97} -296.043 q^{98} +552.115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q + 6 q^{2} + 132 q^{3} + 210 q^{4} + 15 q^{5} + 18 q^{6} + 78 q^{7} + 12 q^{8} + 396 q^{9}+O(q^{10}) \) 44 * q + 6 * q^2 + 132 * q^3 + 210 * q^4 + 15 * q^5 + 18 * q^6 + 78 * q^7 + 12 * q^8 + 396 * q^9	\( 44 q + 6 q^{2} + 132 q^{3} + 210 q^{4} + 15 q^{5} + 18 q^{6} + 78 q^{7} + 12 q^{8} + 396 q^{9} + 214 q^{10} + 111 q^{11} + 630 q^{12} + 275 q^{13} + 104 q^{14} + 45 q^{15} + 1062 q^{16} - 58 q^{17} + 54 q^{18} + 331 q^{19} + 287 q^{20} + 234 q^{21} + 285 q^{22} + 1012 q^{23} + 36 q^{24} + 1903 q^{25} + 1084 q^{26} + 1188 q^{27} + 222 q^{28} - 1276 q^{29} + 642 q^{30} + 1394 q^{31} + 42 q^{32} + 333 q^{33} + 373 q^{34} + 567 q^{35} + 1890 q^{36} + 1229 q^{37} + 733 q^{38} + 825 q^{39} + 2483 q^{40} - 107 q^{41} + 312 q^{42} + 1165 q^{43} + 1639 q^{44} + 135 q^{45} + 138 q^{46} + 964 q^{47} + 3186 q^{48} + 4264 q^{49} + 495 q^{50} - 174 q^{51} + 2679 q^{52} - 380 q^{53} + 162 q^{54} + 1260 q^{55} + 2229 q^{56} + 993 q^{57} - 174 q^{58} + 897 q^{59} + 861 q^{60} + 2584 q^{61} + 3034 q^{62} + 702 q^{63} + 6866 q^{64} - 286 q^{65} + 855 q^{66} + 2277 q^{67} - 1554 q^{68} + 3036 q^{69} + 689 q^{70} + 4304 q^{71} + 108 q^{72} + 4712 q^{73} - 1005 q^{74} + 5709 q^{75} + 2877 q^{76} + 919 q^{77} + 3252 q^{78} + 3864 q^{79} + 2593 q^{80} + 3564 q^{81} + 3297 q^{82} - 540 q^{83} + 666 q^{84} + 6537 q^{85} + 3789 q^{86} - 3828 q^{87} + 1707 q^{88} - 331 q^{89} + 1926 q^{90} + 4311 q^{91} + 4830 q^{92} + 4182 q^{93} + 6189 q^{94} + 3267 q^{95} + 126 q^{96} + 5572 q^{97} + 2588 q^{98} + 999 q^{99}+O(q^{100}) \) 44 * q + 6 * q^2 + 132 * q^3 + 210 * q^4 + 15 * q^5 + 18 * q^6 + 78 * q^7 + 12 * q^8 + 396 * q^9 + 214 * q^10 + 111 * q^11 + 630 * q^12 + 275 * q^13 + 104 * q^14 + 45 * q^15 + 1062 * q^16 - 58 * q^17 + 54 * q^18 + 331 * q^19 + 287 * q^20 + 234 * q^21 + 285 * q^22 + 1012 * q^23 + 36 * q^24 + 1903 * q^25 + 1084 * q^26 + 1188 * q^27 + 222 * q^28 - 1276 * q^29 + 642 * q^30 + 1394 * q^31 + 42 * q^32 + 333 * q^33 + 373 * q^34 + 567 * q^35 + 1890 * q^36 + 1229 * q^37 + 733 * q^38 + 825 * q^39 + 2483 * q^40 - 107 * q^41 + 312 * q^42 + 1165 * q^43 + 1639 * q^44 + 135 * q^45 + 138 * q^46 + 964 * q^47 + 3186 * q^48 + 4264 * q^49 + 495 * q^50 - 174 * q^51 + 2679 * q^52 - 380 * q^53 + 162 * q^54 + 1260 * q^55 + 2229 * q^56 + 993 * q^57 - 174 * q^58 + 897 * q^59 + 861 * q^60 + 2584 * q^61 + 3034 * q^62 + 702 * q^63 + 6866 * q^64 - 286 * q^65 + 855 * q^66 + 2277 * q^67 - 1554 * q^68 + 3036 * q^69 + 689 * q^70 + 4304 * q^71 + 108 * q^72 + 4712 * q^73 - 1005 * q^74 + 5709 * q^75 + 2877 * q^76 + 919 * q^77 + 3252 * q^78 + 3864 * q^79 + 2593 * q^80 + 3564 * q^81 + 3297 * q^82 - 540 * q^83 + 666 * q^84 + 6537 * q^85 + 3789 * q^86 - 3828 * q^87 + 1707 * q^88 - 331 * q^89 + 1926 * q^90 + 4311 * q^91 + 4830 * q^92 + 4182 * q^93 + 6189 * q^94 + 3267 * q^95 + 126 * q^96 + 5572 * q^97 + 2588 * q^98 + 999 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more