LMFDB - Embedded newform 2013.4.a.f.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2013,4,Mod(1,2013)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2013, 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("2013.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2013 = 3 \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2013.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:	$118.770844842$
Analytic rank:	$0$
Dimension:	$38$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	2013.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.08981 q^{2} +3.00000 q^{3} +8.72654 q^{4} -14.1681 q^{5} -12.2694 q^{6} -18.6667 q^{7} -2.97140 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.08981 q^{2} +3.00000 q^{3} +8.72654 q^{4} -14.1681 q^{5} -12.2694 q^{6} -18.6667 q^{7} -2.97140 q^{8} +9.00000 q^{9} +57.9450 q^{10} +11.0000 q^{11} +26.1796 q^{12} +63.5300 q^{13} +76.3434 q^{14} -42.5044 q^{15} -57.6598 q^{16} +65.4898 q^{17} -36.8083 q^{18} -27.2937 q^{19} -123.639 q^{20} -56.0002 q^{21} -44.9879 q^{22} +43.8234 q^{23} -8.91420 q^{24} +75.7363 q^{25} -259.825 q^{26} +27.0000 q^{27} -162.896 q^{28} -169.239 q^{29} +173.835 q^{30} +341.308 q^{31} +259.589 q^{32} +33.0000 q^{33} -267.841 q^{34} +264.473 q^{35} +78.5388 q^{36} +6.98484 q^{37} +111.626 q^{38} +190.590 q^{39} +42.0992 q^{40} +255.704 q^{41} +229.030 q^{42} -301.688 q^{43} +95.9919 q^{44} -127.513 q^{45} -179.229 q^{46} +83.8562 q^{47} -172.980 q^{48} +5.44692 q^{49} -309.747 q^{50} +196.469 q^{51} +554.397 q^{52} +260.694 q^{53} -110.425 q^{54} -155.850 q^{55} +55.4663 q^{56} -81.8811 q^{57} +692.154 q^{58} -66.5333 q^{59} -370.917 q^{60} -61.0000 q^{61} -1395.89 q^{62} -168.001 q^{63} -600.390 q^{64} -900.102 q^{65} -134.964 q^{66} -443.837 q^{67} +571.499 q^{68} +131.470 q^{69} -1081.64 q^{70} -482.651 q^{71} -26.7426 q^{72} -177.933 q^{73} -28.5667 q^{74} +227.209 q^{75} -238.179 q^{76} -205.334 q^{77} -779.476 q^{78} +211.528 q^{79} +816.933 q^{80} +81.0000 q^{81} -1045.78 q^{82} -948.157 q^{83} -488.688 q^{84} -927.869 q^{85} +1233.84 q^{86} -507.716 q^{87} -32.6854 q^{88} +680.357 q^{89} +521.505 q^{90} -1185.90 q^{91} +382.427 q^{92} +1023.93 q^{93} -342.956 q^{94} +386.701 q^{95} +778.767 q^{96} -1478.33 q^{97} -22.2769 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) $ 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * 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 $

$2$	−4.08981	−1.44597	−0.722983	−	0.690866i	$-0.757229\pi$
$2$	−4.08981	−1.44597	−0.722983	+	0.690866i	$0.757229\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	8.72654	1.09082
$5$	−14.1681	−1.26724	−0.633619	−	0.773645i	$-0.718431\pi$
$5$	−14.1681	−1.26724	−0.633619	+	0.773645i	$0.718431\pi$
$6$	−12.2694	−0.834829
$7$	−18.6667	−1.00791	−0.503954	−	0.863730i	$-0.668122\pi$
$7$	−18.6667	−1.00791	−0.503954	+	0.863730i	$0.668122\pi$
$8$	−2.97140	−0.131319
$9$	9.00000	0.333333
$10$	57.9450	1.83238
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	26.1796	0.629784
$13$	63.5300	1.35539	0.677694	−	0.735344i	$-0.262979\pi$
$13$	63.5300	1.35539	0.677694	+	0.735344i	$0.262979\pi$
$14$	76.3434	1.45740
$15$	−42.5044	−0.731640
$16$	−57.6598	−0.900935
$17$	65.4898	0.934330	0.467165	−	0.884170i	$-0.345275\pi$
$17$	65.4898	0.934330	0.467165	+	0.884170i	$0.345275\pi$
$18$	−36.8083	−0.481989
$19$	−27.2937	−0.329558	−0.164779	−	0.986331i	$-0.552691\pi$
$19$	−27.2937	−0.329558	−0.164779	+	0.986331i	$0.552691\pi$
$20$	−123.639	−1.38232
$21$	−56.0002	−0.581916
$22$	−44.9879	−0.435975
$23$	43.8234	0.397296	0.198648	−	0.980071i	$-0.436345\pi$
$23$	43.8234	0.397296	0.198648	+	0.980071i	$0.436345\pi$
$24$	−8.91420	−0.0758168
$25$	75.7363	0.605891
$26$	−259.825	−1.95984
$27$	27.0000	0.192450
$28$	−162.896	−1.09944
$29$	−169.239	−1.08368	−0.541842	−	0.840480i	$-0.682273\pi$
$29$	−169.239	−1.08368	−0.541842	+	0.840480i	$0.682273\pi$
$30$	173.835	1.05793
$31$	341.308	1.97744	0.988722	−	0.149760i	$-0.0478501\pi$
$31$	341.308	1.97744	0.988722	+	0.149760i	$0.0478501\pi$
$32$	259.589	1.43404
$33$	33.0000	0.174078
$34$	−267.841	−1.35101
$35$	264.473	1.27726
$36$	78.5388	0.363606
$37$	6.98484	0.0310352	0.0155176	−	0.999880i	$-0.495060\pi$
$37$	6.98484	0.0310352	0.0155176	+	0.999880i	$0.495060\pi$
$38$	111.626	0.476530
$39$	190.590	0.782534
$40$	42.0992	0.166412
$41$	255.704	0.974008	0.487004	−	0.873400i	$-0.338090\pi$
$41$	255.704	0.974008	0.487004	+	0.873400i	$0.338090\pi$
$42$	229.030	0.841431
$43$	−301.688	−1.06993	−0.534964	−	0.844875i	$-0.679675\pi$
$43$	−301.688	−1.06993	−0.534964	+	0.844875i	$0.679675\pi$
$44$	95.9919	0.328894
$45$	−127.513	−0.422412
$46$	−179.229	−0.574477
$47$	83.8562	0.260249	0.130124	−	0.991498i	$-0.458462\pi$
$47$	83.8562	0.260249	0.130124	+	0.991498i	$0.458462\pi$
$48$	−172.980	−0.520155
$49$	5.44692	0.0158802
$50$	−309.747	−0.876097
$51$	196.469	0.539436
$52$	554.397	1.47848
$53$	260.694	0.675642	0.337821	−	0.941210i	$-0.390310\pi$
$53$	260.694	0.675642	0.337821	+	0.941210i	$0.390310\pi$
$54$	−110.425	−0.278276
$55$	−155.850	−0.382086
$56$	55.4663	0.132357
$57$	−81.8811	−0.190270
$58$	692.154	1.56697
$59$	−66.5333	−0.146812	−0.0734060	−	0.997302i	$-0.523387\pi$
$59$	−66.5333	−0.146812	−0.0734060	+	0.997302i	$0.523387\pi$
$60$	−370.917	−0.798085
$61$	−61.0000	−0.128037
$61$	−61.0000	−0.128037
$62$	−1395.89	−2.85932
$63$	−168.001	−0.335970
$64$	−600.390	−1.17264
$65$	−900.102	−1.71760
$66$	−134.964	−0.251710
$67$	−443.837	−0.809303	−0.404652	−	0.914471i	$-0.632607\pi$
$67$	−443.837	−0.809303	−0.404652	+	0.914471i	$0.632607\pi$
$68$	571.499	1.01918
$69$	131.470	0.229379
$70$	−1081.64	−1.84687
$71$	−482.651	−0.806762	−0.403381	−	0.915032i	$-0.632165\pi$
$71$	−482.651	−0.806762	−0.403381	+	0.915032i	$0.632165\pi$
$72$	−26.7426	−0.0437728
$73$	−177.933	−0.285280	−0.142640	−	0.989775i	$-0.545559\pi$
$73$	−177.933	−0.285280	−0.142640	+	0.989775i	$0.545559\pi$
$74$	−28.5667	−0.0448758
$75$	227.209	0.349811
$76$	−238.179	−0.359487
$77$	−205.334	−0.303896
$78$	−779.476	−1.13152
$79$	211.528	0.301250	0.150625	−	0.988591i	$-0.451871\pi$
$79$	211.528	0.301250	0.150625	+	0.988591i	$0.451871\pi$
$80$	816.933	1.14170
$81$	81.0000	0.111111
$82$	−1045.78	−1.40838
$83$	−948.157	−1.25390	−0.626951	−	0.779059i	$-0.715697\pi$
$83$	−948.157	−1.25390	−0.626951	+	0.779059i	$0.715697\pi$
$84$	−488.688	−0.634764
$85$	−927.869	−1.18402
$86$	1233.84	1.54708
$87$	−507.716	−0.625665
$88$	−32.6854	−0.0395940
$89$	680.357	0.810311	0.405156	−	0.914248i	$-0.367217\pi$
$89$	680.357	0.810311	0.405156	+	0.914248i	$0.367217\pi$
$90$	521.505	0.610794
$91$	−1185.90	−1.36611
$92$	382.427	0.433378
$93$	1023.93	1.14168
$94$	−342.956	−0.376310
$95$	386.701	0.417628
$96$	778.767	0.827943
$97$	−1478.33	−1.54744	−0.773719	−	0.633529i	$-0.781606\pi$
$97$	−1478.33	−1.54744	−0.773719	+	0.633529i	$0.781606\pi$
$98$	−22.2769	−0.0229623
$99$	99.0000	0.100504
$100$	660.916	0.660916
$101$	61.2332	0.0603260	0.0301630	−	0.999545i	$-0.490397\pi$
$101$	61.2332	0.0603260	0.0301630	+	0.999545i	$0.490397\pi$
$102$	−803.523	−0.780006
$103$	−32.5120	−0.0311019	−0.0155510	−	0.999879i	$-0.504950\pi$
$103$	−32.5120	−0.0311019	−0.0155510	+	0.999879i	$0.504950\pi$
$104$	−188.773	−0.177988
$105$	793.419	0.737426
$106$	−1066.19	−0.976955
$107$	953.362	0.861355	0.430677	−	0.902506i	$-0.358275\pi$
$107$	953.362	0.861355	0.430677	+	0.902506i	$0.358275\pi$
$108$	235.617	0.209928
$109$	−702.711	−0.617500	−0.308750	−	0.951143i	$-0.599911\pi$
$109$	−702.711	−0.617500	−0.308750	+	0.951143i	$0.599911\pi$
$110$	637.395	0.552484
$111$	20.9545	0.0179182
$112$	1076.32	0.908060
$113$	−640.681	−0.533364	−0.266682	−	0.963785i	$-0.585927\pi$
$113$	−640.681	−0.533364	−0.266682	+	0.963785i	$0.585927\pi$
$114$	334.878	0.275124
$115$	−620.897	−0.503469
$116$	−1476.87	−1.18210
$117$	571.770	0.451796
$118$	272.109	0.212285
$119$	−1222.48	−0.941720
$120$	126.298	0.0960779
$121$	121.000	0.0909091
$122$	249.478	0.185137
$123$	767.113	0.562344
$124$	2978.44	2.15703
$125$	697.975	0.499430
$126$	687.090	0.485801
$127$	628.392	0.439061	0.219531	−	0.975606i	$-0.429547\pi$
$127$	628.392	0.439061	0.219531	+	0.975606i	$0.429547\pi$
$128$	378.771	0.261554
$129$	−905.063	−0.617723
$130$	3681.24	2.48359
$131$	1491.41	0.994692	0.497346	−	0.867552i	$-0.334308\pi$
$131$	1491.41	0.994692	0.497346	+	0.867552i	$0.334308\pi$
$132$	287.976	0.189887
$133$	509.484	0.332164
$134$	1815.21	1.17022
$135$	−382.540	−0.243880
$136$	−194.596	−0.122695
$137$	1598.67	0.996961	0.498480	−	0.866901i	$-0.333892\pi$
$137$	1598.67	0.996961	0.498480	+	0.866901i	$0.333892\pi$
$138$	−537.688	−0.331674
$139$	−703.767	−0.429444	−0.214722	−	0.976675i	$-0.568885\pi$
$139$	−703.767	−0.429444	−0.214722	+	0.976675i	$0.568885\pi$
$140$	2307.93	1.39326
$141$	251.569	0.150255
$142$	1973.95	1.16655
$143$	698.830	0.408665
$144$	−518.939	−0.300312
$145$	2397.80	1.37328
$146$	727.710	0.412505
$147$	16.3408	0.00916846
$148$	60.9535	0.0338537
$149$	3029.69	1.66578	0.832891	−	0.553437i	$-0.186684\pi$
$149$	3029.69	1.66578	0.832891	+	0.553437i	$0.186684\pi$
$150$	−929.241	−0.505815
$151$	−3567.68	−1.92274	−0.961369	−	0.275264i	$-0.911235\pi$
$151$	−3567.68	−1.92274	−0.961369	+	0.275264i	$0.911235\pi$
$152$	81.1004	0.0432771
$153$	589.408	0.311443
$154$	839.777	0.439423
$155$	−4835.71	−2.50589
$156$	1663.19	0.853601
$157$	1146.41	0.582761	0.291380	−	0.956607i	$-0.405885\pi$
$157$	1146.41	0.582761	0.291380	+	0.956607i	$0.405885\pi$
$158$	−865.110	−0.435598
$159$	782.081	0.390082
$160$	−3677.89	−1.81727
$161$	−818.040	−0.400438
$162$	−331.275	−0.160663
$163$	1384.89	0.665478	0.332739	−	0.943019i	$-0.392027\pi$
$163$	1384.89	0.665478	0.332739	+	0.943019i	$0.392027\pi$
$164$	2231.41	1.06246
$165$	−467.549	−0.220598
$166$	3877.78	1.81310
$167$	134.032	0.0621058	0.0310529	−	0.999518i	$-0.490114\pi$
$167$	134.032	0.0621058	0.0310529	+	0.999518i	$0.490114\pi$
$168$	166.399	0.0764164
$169$	1839.06	0.837076
$170$	3794.81	1.71205
$171$	−245.643	−0.109853
$172$	−2632.69	−1.16710
$173$	−3601.80	−1.58289	−0.791444	−	0.611241i	$-0.790670\pi$
$173$	−3601.80	−1.58289	−0.791444	+	0.611241i	$0.790670\pi$
$174$	2076.46	0.904691
$175$	−1413.75	−0.610682
$176$	−634.258	−0.271642
$177$	−199.600	−0.0847619
$178$	−2782.53	−1.17168
$179$	1241.97	0.518597	0.259299	−	0.965797i	$-0.416509\pi$
$179$	1241.97	0.518597	0.259299	+	0.965797i	$0.416509\pi$
$180$	−1112.75	−0.460775
$181$	−2169.25	−0.890824	−0.445412	−	0.895326i	$-0.646943\pi$
$181$	−2169.25	−0.890824	−0.445412	+	0.895326i	$0.646943\pi$
$182$	4850.09	1.97534
$183$	−183.000	−0.0739221
$184$	−130.217	−0.0521724
$185$	−98.9622	−0.0393289
$186$	−4187.66	−1.65083
$187$	720.388	0.281711
$188$	731.774	0.283884
$189$	−504.002	−0.193972
$190$	−1581.53	−0.603876
$191$	2526.65	0.957184	0.478592	−	0.878037i	$-0.341147\pi$
$191$	2526.65	0.957184	0.478592	+	0.878037i	$0.341147\pi$
$192$	−1801.17	−0.677023
$193$	−3743.31	−1.39611	−0.698054	−	0.716045i	$-0.745951\pi$
$193$	−3743.31	−1.39611	−0.698054	+	0.716045i	$0.745951\pi$
$194$	6046.08	2.23754
$195$	−2700.31	−0.991656
$196$	47.5328	0.0173224
$197$	258.469	0.0934779	0.0467390	−	0.998907i	$-0.485117\pi$
$197$	258.469	0.0934779	0.0467390	+	0.998907i	$0.485117\pi$
$198$	−404.891	−0.145325
$199$	1147.05	0.408604	0.204302	−	0.978908i	$-0.434508\pi$
$199$	1147.05	0.408604	0.204302	+	0.978908i	$0.434508\pi$
$200$	−225.043	−0.0795647
$201$	−1331.51	−0.467251
$202$	−250.432	−0.0872294
$203$	3159.13	1.09225
$204$	1714.50	0.588426
$205$	−3622.86	−1.23430
$206$	132.968	0.0449723
$207$	394.411	0.132432
$208$	−3663.13	−1.22112
$209$	−300.231	−0.0993655
$210$	−3244.93	−1.06629
$211$	2992.38	0.976322	0.488161	−	0.872754i	$-0.337668\pi$
$211$	2992.38	0.976322	0.488161	+	0.872754i	$0.337668\pi$
$212$	2274.95	0.737002
$213$	−1447.95	−0.465784
$214$	−3899.07	−1.24549
$215$	4274.35	1.35585
$216$	−80.2278	−0.0252723
$217$	−6371.11	−1.99308
$218$	2873.96	0.892884
$219$	−533.798	−0.164706
$220$	−1360.03	−0.416786
$221$	4160.57	1.26638
$222$	−85.7000	−0.0259090
$223$	−2927.24	−0.879025	−0.439513	−	0.898236i	$-0.644849\pi$
$223$	−2927.24	−0.879025	−0.439513	+	0.898236i	$0.644849\pi$
$224$	−4845.68	−1.44538
$225$	681.627	0.201964
$226$	2620.26	0.771227
$227$	1958.73	0.572710	0.286355	−	0.958124i	$-0.407556\pi$
$227$	1958.73	0.572710	0.286355	+	0.958124i	$0.407556\pi$
$228$	−714.538	−0.207550
$229$	−3698.96	−1.06740	−0.533699	−	0.845675i	$-0.679198\pi$
$229$	−3698.96	−1.06740	−0.533699	+	0.845675i	$0.679198\pi$
$230$	2539.35	0.727999
$231$	−616.002	−0.175454
$232$	502.876	0.142308
$233$	4852.38	1.36433	0.682167	−	0.731196i	$-0.261038\pi$
$233$	4852.38	1.36433	0.682167	+	0.731196i	$0.261038\pi$
$234$	−2338.43	−0.653282
$235$	−1188.09	−0.329797
$236$	−580.606	−0.160145
$237$	634.585	0.173927
$238$	4999.71	1.36169
$239$	−1313.62	−0.355528	−0.177764	−	0.984073i	$-0.556886\pi$
$239$	−1313.62	−0.355528	−0.177764	+	0.984073i	$0.556886\pi$
$240$	2450.80	0.659160
$241$	5660.65	1.51301	0.756503	−	0.653990i	$-0.226906\pi$
$241$	5660.65	1.51301	0.756503	+	0.653990i	$0.226906\pi$
$242$	−494.867	−0.131451
$243$	243.000	0.0641500
$244$	−532.319	−0.139665
$245$	−77.1728	−0.0201240
$246$	−3137.35	−0.813130
$247$	−1733.97	−0.446679
$248$	−1014.16	−0.259675
$249$	−2844.47	−0.723940
$250$	−2854.58	−0.722159
$251$	1567.61	0.394210	0.197105	−	0.980382i	$-0.436846\pi$
$251$	1567.61	0.394210	0.197105	+	0.980382i	$0.436846\pi$
$252$	−1466.06	−0.366481
$253$	482.058	0.119789
$254$	−2570.00	−0.634867
$255$	−2783.61	−0.683593
$256$	3254.02	0.794439
$257$	−1831.19	−0.444462	−0.222231	−	0.974994i	$-0.571334\pi$
$257$	−1831.19	−0.444462	−0.222231	+	0.974994i	$0.571334\pi$
$258$	3701.53	0.893207
$259$	−130.384	−0.0312806
$260$	−7854.77	−1.87359
$261$	−1523.15	−0.361228
$262$	−6099.56	−1.43829
$263$	−2730.22	−0.640124	−0.320062	−	0.947397i	$-0.603704\pi$
$263$	−2730.22	−0.640124	−0.320062	+	0.947397i	$0.603704\pi$
$264$	−98.0562	−0.0228596
$265$	−3693.55	−0.856199
$266$	−2083.69	−0.480298
$267$	2041.07	0.467833
$268$	−3873.16	−0.882802
$269$	−4511.68	−1.02261	−0.511304	−	0.859400i	$-0.670838\pi$
$269$	−4511.68	−1.02261	−0.511304	+	0.859400i	$0.670838\pi$
$270$	1564.52	0.352642
$271$	4775.90	1.07054	0.535268	−	0.844682i	$-0.320210\pi$
$271$	4775.90	1.07054	0.535268	+	0.844682i	$0.320210\pi$
$272$	−3776.13	−0.841771
$273$	−3557.69	−0.788723
$274$	−6538.26	−1.44157
$275$	833.100	0.182683
$276$	1147.28	0.250211
$277$	4917.95	1.06675	0.533377	−	0.845878i	$-0.320923\pi$
$277$	4917.95	1.06675	0.533377	+	0.845878i	$0.320923\pi$
$278$	2878.27	0.620961
$279$	3071.78	0.659148
$280$	−785.855	−0.167728
$281$	1781.93	0.378295	0.189148	−	0.981949i	$-0.439428\pi$
$281$	1781.93	0.378295	0.189148	+	0.981949i	$0.439428\pi$
$282$	−1028.87	−0.217263
$283$	−5127.34	−1.07699	−0.538496	−	0.842628i	$-0.681007\pi$
$283$	−5127.34	−1.07699	−0.538496	+	0.842628i	$0.681007\pi$
$284$	−4211.87	−0.880030
$285$	1160.10	0.241118
$286$	−2858.08	−0.590915
$287$	−4773.17	−0.981711
$288$	2336.30	0.478013
$289$	−624.083	−0.127027
$290$	−9806.54	−1.98572
$291$	−4434.98	−0.893414
$292$	−1552.74	−0.311188
$293$	2253.75	0.449370	0.224685	−	0.974431i	$-0.427865\pi$
$293$	2253.75	0.449370	0.224685	+	0.974431i	$0.427865\pi$
$294$	−66.8306	−0.0132573
$295$	942.654	0.186046
$296$	−20.7548	−0.00407549
$297$	297.000	0.0580259
$298$	−12390.8	−2.40866
$299$	2784.10	0.538491
$300$	1982.75	0.381580
$301$	5631.52	1.07839
$302$	14591.1	2.78021
$303$	183.700	0.0348293
$304$	1573.75	0.296910
$305$	864.257	0.162253
$306$	−2410.57	−0.450337
$307$	5352.97	0.995147	0.497573	−	0.867422i	$-0.334224\pi$
$307$	5352.97	0.995147	0.497573	+	0.867422i	$0.334224\pi$
$308$	−1791.86	−0.331495
$309$	−97.5359	−0.0179567
$310$	19777.1	3.62343
$311$	5944.69	1.08390	0.541950	−	0.840411i	$-0.317686\pi$
$311$	5944.69	1.08390	0.541950	+	0.840411i	$0.317686\pi$
$312$	−566.319	−0.102761
$313$	−9259.46	−1.67213	−0.836064	−	0.548632i	$-0.815149\pi$
$313$	−9259.46	−1.67213	−0.836064	+	0.548632i	$0.815149\pi$
$314$	−4688.60	−0.842652
$315$	2380.26	0.425753
$316$	1845.91	0.328609
$317$	7502.78	1.32933	0.664665	−	0.747141i	$-0.268574\pi$
$317$	7502.78	1.32933	0.664665	+	0.747141i	$0.268574\pi$
$318$	−3198.56	−0.564045
$319$	−1861.63	−0.326743
$320$	8506.42	1.48601
$321$	2860.09	0.497304
$322$	3345.63	0.579020
$323$	−1787.46	−0.307916
$324$	706.850	0.121202
$325$	4811.53	0.821217
$326$	−5663.94	−0.962259
$327$	−2108.13	−0.356514
$328$	−759.800	−0.127905
$329$	−1565.32	−0.262307
$330$	1912.19	0.318977
$331$	−787.685	−0.130801	−0.0654005	−	0.997859i	$-0.520832\pi$
$331$	−787.685	−0.130801	−0.0654005	+	0.997859i	$0.520832\pi$
$332$	−8274.13	−1.36778
$333$	62.8636	0.0103451
$334$	−548.163	−0.0898029
$335$	6288.35	1.02558
$336$	3228.96	0.524269
$337$	895.764	0.144793	0.0723967	−	0.997376i	$-0.476935\pi$
$337$	895.764	0.144793	0.0723967	+	0.997376i	$0.476935\pi$
$338$	−7521.39	−1.21038
$339$	−1922.04	−0.307938
$340$	−8097.09	−1.29155
$341$	3754.39	0.596222
$342$	1004.63	0.158843
$343$	6301.01	0.991903
$344$	896.434	0.140501
$345$	−1862.69	−0.290678
$346$	14730.7	2.28880
$347$	2655.05	0.410750	0.205375	−	0.978683i	$-0.434159\pi$
$347$	2655.05	0.410750	0.205375	+	0.978683i	$0.434159\pi$
$348$	−4430.60	−0.682486
$349$	11337.0	1.73884	0.869421	−	0.494072i	$-0.164492\pi$
$349$	11337.0	1.73884	0.869421	+	0.494072i	$0.164492\pi$
$350$	5781.97	0.883026
$351$	1715.31	0.260845
$352$	2855.48	0.432379
$353$	5732.59	0.864349	0.432174	−	0.901790i	$-0.357746\pi$
$353$	5732.59	0.864349	0.432174	+	0.901790i	$0.357746\pi$
$354$	816.326	0.122563
$355$	6838.26	1.02236
$356$	5937.16	0.883902
$357$	−3667.44	−0.543702
$358$	−5079.41	−0.749874
$359$	5813.00	0.854591	0.427296	−	0.904112i	$-0.359466\pi$
$359$	5813.00	0.854591	0.427296	+	0.904112i	$0.359466\pi$
$360$	378.893	0.0554706
$361$	−6114.05	−0.891392
$362$	8871.83	1.28810
$363$	363.000	0.0524864
$364$	−10348.8	−1.49017
$365$	2520.97	0.361517
$366$	748.435	0.106889
$367$	1705.95	0.242643	0.121321	−	0.992613i	$-0.461287\pi$
$367$	1705.95	0.242643	0.121321	+	0.992613i	$0.461287\pi$
$368$	−2526.85	−0.357938
$369$	2301.34	0.324669
$370$	404.737	0.0568683
$371$	−4866.30	−0.680986
$372$	8935.32	1.24536
$373$	2227.89	0.309265	0.154632	−	0.987972i	$-0.450581\pi$
$373$	2227.89	0.309265	0.154632	+	0.987972i	$0.450581\pi$
$374$	−2946.25	−0.407345
$375$	2093.92	0.288346
$376$	−249.170	−0.0341755
$377$	−10751.7	−1.46881
$378$	2061.27	0.280477
$379$	2443.43	0.331162	0.165581	−	0.986196i	$-0.447050\pi$
$379$	2443.43	0.331162	0.165581	+	0.986196i	$0.447050\pi$
$380$	3374.56	0.455556
$381$	1885.18	0.253492
$382$	−10333.5	−1.38406
$383$	−7151.37	−0.954094	−0.477047	−	0.878878i	$-0.658293\pi$
$383$	−7151.37	−0.954094	−0.477047	+	0.878878i	$0.658293\pi$
$384$	1136.31	0.151008
$385$	2909.20	0.385108
$386$	15309.4	2.01873
$387$	−2715.19	−0.356643
$388$	−12900.7	−1.68797
$389$	−9766.07	−1.27290	−0.636452	−	0.771316i	$-0.719599\pi$
$389$	−9766.07	−1.27290	−0.636452	+	0.771316i	$0.719599\pi$
$390$	11043.7	1.43390
$391$	2869.99	0.371206
$392$	−16.1850	−0.00208537
$393$	4474.22	0.574286
$394$	−1057.09	−0.135166
$395$	−2996.96	−0.381756
$396$	863.927	0.109631
$397$	−14683.9	−1.85633	−0.928167	−	0.372165i	$-0.878616\pi$
$397$	−14683.9	−1.85633	−0.928167	+	0.372165i	$0.878616\pi$
$398$	−4691.22	−0.590828
$399$	1528.45	0.191775
$400$	−4366.94	−0.545868
$401$	6244.45	0.777638	0.388819	−	0.921314i	$-0.372883\pi$
$401$	6244.45	0.777638	0.388819	+	0.921314i	$0.372883\pi$
$402$	5445.63	0.675630
$403$	21683.3	2.68020
$404$	534.354	0.0658047
$405$	−1147.62	−0.140804
$406$	−12920.3	−1.57936
$407$	76.8332	0.00935745
$408$	−583.789	−0.0708379
$409$	9843.29	1.19002	0.595011	−	0.803717i	$-0.297148\pi$
$409$	9843.29	1.19002	0.595011	+	0.803717i	$0.297148\pi$
$410$	14816.8	1.78475
$411$	4796.01	0.575596
$412$	−283.717	−0.0339265
$413$	1241.96	0.147973
$414$	−1613.06	−0.191492
$415$	13433.6	1.58899
$416$	16491.7	1.94368
$417$	−2111.30	−0.247940
$418$	1227.89	0.143679
$419$	13265.8	1.54672	0.773361	−	0.633966i	$-0.218574\pi$
$419$	13265.8	1.54672	0.773361	+	0.633966i	$0.218574\pi$
$420$	6923.80	0.804397
$421$	−1901.57	−0.220136	−0.110068	−	0.993924i	$-0.535107\pi$
$421$	−1901.57	−0.220136	−0.110068	+	0.993924i	$0.535107\pi$
$422$	−12238.3	−1.41173
$423$	754.706	0.0867495
$424$	−774.625	−0.0887243
$425$	4959.96	0.566102
$426$	5921.85	0.673508
$427$	1138.67	0.129050
$428$	8319.55	0.939581
$429$	2096.49	0.235943
$430$	−17481.3	−1.96052
$431$	−6643.46	−0.742469	−0.371235	−	0.928539i	$-0.621065\pi$
$431$	−6643.46	−0.742469	−0.371235	+	0.928539i	$0.621065\pi$
$432$	−1556.82	−0.173385
$433$	8790.62	0.975636	0.487818	−	0.872945i	$-0.337793\pi$
$433$	8790.62	0.975636	0.487818	+	0.872945i	$0.337793\pi$
$434$	26056.6	2.88193
$435$	7193.40	0.792866
$436$	−6132.24	−0.673580
$437$	−1196.10	−0.130932
$438$	2183.13	0.238160
$439$	539.400	0.0586428	0.0293214	−	0.999570i	$-0.490665\pi$
$439$	539.400	0.0586428	0.0293214	+	0.999570i	$0.490665\pi$
$440$	463.091	0.0501750
$441$	49.0223	0.00529341
$442$	−17015.9	−1.83114
$443$	5521.01	0.592124	0.296062	−	0.955169i	$-0.404326\pi$
$443$	5521.01	0.592124	0.296062	+	0.955169i	$0.404326\pi$
$444$	182.860	0.0195454
$445$	−9639.40	−1.02686
$446$	11971.9	1.27104
$447$	9089.06	0.961740
$448$	11207.3	1.18191
$449$	−2391.76	−0.251390	−0.125695	−	0.992069i	$-0.540116\pi$
$449$	−2391.76	−0.251390	−0.125695	+	0.992069i	$0.540116\pi$
$450$	−2787.72	−0.292032
$451$	2812.75	0.293674
$452$	−5590.92	−0.581803
$453$	−10703.0	−1.11009
$454$	−8010.81	−0.828119
$455$	16802.0	1.73118
$456$	243.301	0.0249860
$457$	360.687	0.0369195	0.0184598	−	0.999830i	$-0.494124\pi$
$457$	360.687	0.0369195	0.0184598	+	0.999830i	$0.494124\pi$
$458$	15128.0	1.54342
$459$	1768.23	0.179812
$460$	−5418.28	−0.549192
$461$	13204.1	1.33400	0.667000	−	0.745058i	$-0.267578\pi$
$461$	13204.1	1.33400	0.667000	+	0.745058i	$0.267578\pi$
$462$	2519.33	0.253701
$463$	19539.6	1.96130	0.980650	−	0.195770i	$-0.0627206\pi$
$463$	19539.6	1.96130	0.980650	+	0.195770i	$0.0627206\pi$
$464$	9758.28	0.976329
$465$	−14507.1	−1.44678
$466$	−19845.3	−1.97278
$467$	17070.6	1.69151	0.845753	−	0.533575i	$-0.179152\pi$
$467$	17070.6	1.69151	0.845753	+	0.533575i	$0.179152\pi$
$468$	4989.57	0.492827
$469$	8284.99	0.815704
$470$	4859.05	0.476875
$471$	3439.23	0.336457
$472$	197.697	0.0192791
$473$	−3318.56	−0.322596
$474$	−2595.33	−0.251493
$475$	−2067.12	−0.199676
$476$	−10668.0	−1.02724
$477$	2346.24	0.225214
$478$	5372.47	0.514082
$479$	−7667.15	−0.731359	−0.365679	−	0.930741i	$-0.619163\pi$
$479$	−7667.15	−0.731359	−0.365679	+	0.930741i	$0.619163\pi$
$480$	−11033.7	−1.04920
$481$	443.747	0.0420647
$482$	−23151.0	−2.18776
$483$	−2454.12	−0.231193
$484$	1055.91	0.0991652
$485$	20945.2	1.96097
$486$	−993.824	−0.0927588
$487$	5354.33	0.498209	0.249105	−	0.968477i	$-0.419864\pi$
$487$	5354.33	0.498209	0.249105	+	0.968477i	$0.419864\pi$
$488$	181.255	0.0168136
$489$	4154.67	0.384214
$490$	315.622	0.0290987
$491$	−1900.06	−0.174640	−0.0873201	−	0.996180i	$-0.527830\pi$
$491$	−1900.06	−0.174640	−0.0873201	+	0.996180i	$0.527830\pi$
$492$	6694.24	0.613414
$493$	−11083.4	−1.01252
$494$	7091.59	0.645882
$495$	−1402.65	−0.127362
$496$	−19679.8	−1.78155
$497$	9009.51	0.813143
$498$	11633.3	1.04679
$499$	−1269.05	−0.113849	−0.0569245	−	0.998378i	$-0.518129\pi$
$499$	−1269.05	−0.113849	−0.0569245	+	0.998378i	$0.518129\pi$
$500$	6090.90	0.544787
$501$	402.095	0.0358568
$502$	−6411.22	−0.570014
$503$	5006.29	0.443776	0.221888	−	0.975072i	$-0.428778\pi$
$503$	5006.29	0.443776	0.221888	+	0.975072i	$0.428778\pi$
$504$	499.197	0.0441190
$505$	−867.561	−0.0764474
$506$	−1971.52	−0.173211
$507$	5517.17	0.483286
$508$	5483.69	0.478935
$509$	−20228.2	−1.76149	−0.880744	−	0.473593i	$-0.842957\pi$
$509$	−20228.2	−1.76149	−0.880744	+	0.473593i	$0.842957\pi$
$510$	11384.4	0.988453
$511$	3321.42	0.287536
$512$	−16338.5	−1.41029
$513$	−736.930	−0.0634235
$514$	7489.23	0.642676
$515$	460.634	0.0394135
$516$	−7898.06	−0.673823
$517$	922.418	0.0784679
$518$	533.246	0.0452307
$519$	−10805.4	−0.913881
$520$	2674.56	0.225552
$521$	−7795.39	−0.655513	−0.327757	−	0.944762i	$-0.606293\pi$
$521$	−7795.39	−0.655513	−0.327757	+	0.944762i	$0.606293\pi$
$522$	6229.39	0.522323
$523$	11212.5	0.937454	0.468727	−	0.883343i	$-0.344713\pi$
$523$	11212.5	0.937454	0.468727	+	0.883343i	$0.344713\pi$
$524$	13014.8	1.08503
$525$	−4241.25	−0.352578
$526$	11166.1	0.925598
$527$	22352.2	1.84759
$528$	−1902.77	−0.156833
$529$	−10246.5	−0.842156
$530$	15105.9	1.23803
$531$	−598.800	−0.0489373
$532$	4446.03	0.362331
$533$	16244.9	1.32016
$534$	−8347.59	−0.676471
$535$	−13507.4	−1.09154
$536$	1318.82	0.106277
$537$	3725.90	0.299412
$538$	18451.9	1.47866
$539$	59.9162	0.00478807
$540$	−3338.25	−0.266028
$541$	4419.57	0.351224	0.175612	−	0.984459i	$-0.443810\pi$
$541$	4419.57	0.351224	0.175612	+	0.984459i	$0.443810\pi$
$542$	−19532.5	−1.54796
$543$	−6507.76	−0.514318
$544$	17000.4	1.33987
$545$	9956.12	0.782520
$546$	14550.3	1.14047
$547$	−241.745	−0.0188963	−0.00944814	−	0.999955i	$-0.503007\pi$
$547$	−241.745	−0.0188963	−0.00944814	+	0.999955i	$0.503007\pi$
$548$	13950.9	1.08750
$549$	−549.000	−0.0426790
$550$	−3407.22	−0.264153
$551$	4619.15	0.357137
$552$	−390.651	−0.0301217
$553$	−3948.54	−0.303633
$554$	−20113.5	−1.54249
$555$	−296.887	−0.0227066
$556$	−6141.45	−0.468445
$557$	1427.70	0.108606	0.0543032	−	0.998524i	$-0.482706\pi$
$557$	1427.70	0.108606	0.0543032	+	0.998524i	$0.482706\pi$
$558$	−12563.0	−0.953106
$559$	−19166.2	−1.45017
$560$	−15249.5	−1.15073
$561$	2161.16	0.162646
$562$	−7287.75	−0.547002
$563$	23467.8	1.75675	0.878375	−	0.477972i	$-0.158628\pi$
$563$	23467.8	1.75675	0.878375	+	0.477972i	$0.158628\pi$
$564$	2195.32	0.163900
$565$	9077.26	0.675899
$566$	20969.8	1.55729
$567$	−1512.01	−0.111990
$568$	1434.15	0.105943
$569$	13960.3	1.02855	0.514277	−	0.857624i	$-0.328061\pi$
$569$	13960.3	1.02855	0.514277	+	0.857624i	$0.328061\pi$
$570$	−4744.60	−0.348648
$571$	20291.6	1.48718	0.743588	−	0.668638i	$-0.233122\pi$
$571$	20291.6	1.48718	0.743588	+	0.668638i	$0.233122\pi$
$572$	6098.36	0.445779
$573$	7579.96	0.552630
$574$	19521.3	1.41952
$575$	3319.03	0.240718
$576$	−5403.51	−0.390879
$577$	6554.10	0.472878	0.236439	−	0.971646i	$-0.424020\pi$
$577$	6554.10	0.472878	0.236439	+	0.971646i	$0.424020\pi$
$578$	2552.38	0.183676
$579$	−11229.9	−0.806044
$580$	20924.5	1.49800
$581$	17699.0	1.26382
$582$	18138.2	1.29185
$583$	2867.63	0.203714
$584$	528.709	0.0374625
$585$	−8100.92	−0.572533
$586$	−9217.39	−0.649773
$587$	17289.7	1.21571	0.607856	−	0.794047i	$-0.292030\pi$
$587$	17289.7	1.21571	0.607856	+	0.794047i	$0.292030\pi$
$588$	142.598	0.0100011
$589$	−9315.56	−0.651683
$590$	−3855.28	−0.269016
$591$	775.407	0.0539695
$592$	−402.745	−0.0279607
$593$	−1273.07	−0.0881596	−0.0440798	−	0.999028i	$-0.514036\pi$
$593$	−1273.07	−0.0881596	−0.0440798	+	0.999028i	$0.514036\pi$
$594$	−1214.67	−0.0839034
$595$	17320.3	1.19338
$596$	26438.7	1.81706
$597$	3441.15	0.235908
$598$	−11386.4	−0.778639
$599$	−14433.0	−0.984504	−0.492252	−	0.870453i	$-0.663826\pi$
$599$	−14433.0	−0.984504	−0.492252	+	0.870453i	$0.663826\pi$
$600$	−675.129	−0.0459367
$601$	−25497.5	−1.73056	−0.865279	−	0.501290i	$-0.832859\pi$
$601$	−25497.5	−1.73056	−0.865279	+	0.501290i	$0.832859\pi$
$602$	−23031.8	−1.55932
$603$	−3994.53	−0.269768
$604$	−31133.5	−2.09735
$605$	−1714.35	−0.115203
$606$	−751.296	−0.0503619
$607$	−18612.4	−1.24457	−0.622286	−	0.782790i	$-0.713796\pi$
$607$	−18612.4	−1.24457	−0.622286	+	0.782790i	$0.713796\pi$
$608$	−7085.14	−0.472599
$609$	9477.40	0.630614
$610$	−3534.65	−0.234612
$611$	5327.38	0.352738
$612$	5143.49	0.339728
$613$	24591.1	1.62027	0.810134	−	0.586244i	$-0.199394\pi$
$613$	24591.1	1.62027	0.810134	+	0.586244i	$0.199394\pi$
$614$	−21892.6	−1.43895
$615$	−10868.6	−0.712623
$616$	610.130	0.0399072
$617$	2926.13	0.190926	0.0954631	−	0.995433i	$-0.469567\pi$
$617$	2926.13	0.190926	0.0954631	+	0.995433i	$0.469567\pi$
$618$	398.903	0.0259648
$619$	4460.68	0.289645	0.144822	−	0.989458i	$-0.453739\pi$
$619$	4460.68	0.289645	0.144822	+	0.989458i	$0.453739\pi$
$620$	−42199.0	−2.73347
$621$	1183.23	0.0764597
$622$	−24312.7	−1.56728
$623$	−12700.0	−0.816720
$624$	−10989.4	−0.705012
$625$	−19356.0	−1.23879
$626$	37869.4	2.41784
$627$	−900.692	−0.0573687
$628$	10004.2	0.635686
$629$	457.436	0.0289971
$630$	−9734.80	−0.615625
$631$	−11304.4	−0.713190	−0.356595	−	0.934259i	$-0.616062\pi$
$631$	−11304.4	−0.713190	−0.356595	+	0.934259i	$0.616062\pi$
$632$	−628.535	−0.0395598
$633$	8977.14	0.563680
$634$	−30684.9	−1.92217
$635$	−8903.15	−0.556395
$636$	6824.86	0.425508
$637$	346.043	0.0215239
$638$	7613.69	0.472459
$639$	−4343.86	−0.268921
$640$	−5366.48	−0.331451
$641$	21054.9	1.29738	0.648688	−	0.761055i	$-0.275318\pi$
$641$	21054.9	1.29738	0.648688	+	0.761055i	$0.275318\pi$
$642$	−11697.2	−0.719084
$643$	28927.5	1.77417	0.887085	−	0.461606i	$-0.152727\pi$
$643$	28927.5	1.77417	0.887085	+	0.461606i	$0.152727\pi$
$644$	−7138.66	−0.436805
$645$	12823.1	0.782802
$646$	7310.37	0.445236
$647$	−28879.7	−1.75484	−0.877418	−	0.479726i	$-0.840736\pi$
$647$	−28879.7	−1.75484	−0.877418	+	0.479726i	$0.840736\pi$
$648$	−240.683	−0.0145909
$649$	−731.867	−0.0442655
$650$	−19678.2	−1.18745
$651$	−19113.3	−1.15071
$652$	12085.3	0.725915
$653$	−1021.53	−0.0612183	−0.0306092	−	0.999531i	$-0.509745\pi$
$653$	−1021.53	−0.0612183	−0.0306092	+	0.999531i	$0.509745\pi$
$654$	8621.87	0.515507
$655$	−21130.5	−1.26051
$656$	−14743.9	−0.877518
$657$	−1601.39	−0.0950933
$658$	6401.86	0.379287
$659$	−19918.6	−1.17742	−0.588710	−	0.808344i	$-0.700364\pi$
$659$	−19918.6	−1.17742	−0.588710	+	0.808344i	$0.700364\pi$
$660$	−4080.08	−0.240632
$661$	−6642.80	−0.390885	−0.195442	−	0.980715i	$-0.562614\pi$
$661$	−6642.80	−0.390885	−0.195442	+	0.980715i	$0.562614\pi$
$662$	3221.48	0.189134
$663$	12481.7	0.731145
$664$	2817.35	0.164660
$665$	−7218.44	−0.420931
$666$	−257.100	−0.0149586
$667$	−7416.62	−0.430544
$668$	1169.63	0.0677461
$669$	−8781.73	−0.507505
$670$	−25718.1	−1.48295
$671$	−671.000	−0.0386046
$672$	−14537.0	−0.834491
$673$	30095.4	1.72376	0.861881	−	0.507111i	$-0.169286\pi$
$673$	30095.4	1.72376	0.861881	+	0.507111i	$0.169286\pi$
$674$	−3663.50	−0.209366
$675$	2044.88	0.116604
$676$	16048.6	0.913097
$677$	23447.4	1.33111	0.665553	−	0.746351i	$-0.268196\pi$
$677$	23447.4	1.33111	0.665553	+	0.746351i	$0.268196\pi$
$678$	7860.78	0.445268
$679$	27595.6	1.55968
$680$	2757.07	0.155484
$681$	5876.18	0.330654
$682$	−15354.7	−0.862117
$683$	−14229.6	−0.797192	−0.398596	−	0.917127i	$-0.630502\pi$
$683$	−14229.6	−0.797192	−0.398596	+	0.917127i	$0.630502\pi$
$684$	−2143.61	−0.119829
$685$	−22650.2	−1.26339
$686$	−25769.9	−1.43426
$687$	−11096.9	−0.616262
$688$	17395.3	0.963936
$689$	16561.9	0.915757
$690$	7618.05	0.420310
$691$	10097.0	0.555875	0.277937	−	0.960599i	$-0.410349\pi$
$691$	10097.0	0.555875	0.277937	+	0.960599i	$0.410349\pi$
$692$	−31431.2	−1.72664
$693$	−1848.01	−0.101299
$694$	−10858.6	−0.593931
$695$	9971.07	0.544207
$696$	1508.63	0.0821615
$697$	16746.0	0.910045
$698$	−46366.2	−2.51431
$699$	14557.1	0.787699
$700$	−12337.1	−0.666143
$701$	−19249.1	−1.03713	−0.518565	−	0.855038i	$-0.673534\pi$
$701$	−19249.1	−1.03713	−0.518565	+	0.855038i	$0.673534\pi$
$702$	−7015.29	−0.377172
$703$	−190.642	−0.0102279
$704$	−6604.29	−0.353564
$705$	−3564.26	−0.190408
$706$	−23445.2	−1.24982
$707$	−1143.02	−0.0608032
$708$	−1741.82	−0.0924597
$709$	−5130.06	−0.271740	−0.135870	−	0.990727i	$-0.543383\pi$
$709$	−5130.06	−0.271740	−0.135870	+	0.990727i	$0.543383\pi$
$710$	−27967.2	−1.47830
$711$	1903.75	0.100417
$712$	−2021.61	−0.106409
$713$	14957.3	0.785632
$714$	14999.1	0.786175
$715$	−9901.12	−0.517875
$716$	10838.1	0.565695
$717$	−3940.87	−0.205264
$718$	−23774.0	−1.23571
$719$	−11904.5	−0.617472	−0.308736	−	0.951148i	$-0.599906\pi$
$719$	−11904.5	−0.617472	−0.308736	+	0.951148i	$0.599906\pi$
$720$	7352.40	0.380566
$721$	606.892	0.0313479
$722$	25005.3	1.28892
$723$	16982.0	0.873535
$724$	−18930.1	−0.971727
$725$	−12817.5	−0.656594
$726$	−1484.60	−0.0758935
$727$	−32913.6	−1.67909	−0.839544	−	0.543291i	$-0.817178\pi$
$727$	−32913.6	−1.67909	−0.839544	+	0.543291i	$0.817178\pi$
$728$	3523.77	0.179395
$729$	729.000	0.0370370
$730$	−10310.3	−0.522742
$731$	−19757.5	−0.999667
$732$	−1596.96	−0.0806355
$733$	37305.3	1.87981	0.939907	−	0.341431i	$-0.110912\pi$
$733$	37305.3	1.87981	0.939907	+	0.341431i	$0.110912\pi$
$734$	−6977.01	−0.350853
$735$	−231.518	−0.0116186
$736$	11376.1	0.569739
$737$	−4882.21	−0.244014
$738$	−9412.04	−0.469461
$739$	12364.6	0.615478	0.307739	−	0.951471i	$-0.400428\pi$
$739$	12364.6	0.615478	0.307739	+	0.951471i	$0.400428\pi$
$740$	−863.598	−0.0429006
$741$	−5201.90	−0.257890
$742$	19902.2	0.984682
$743$	18905.3	0.933472	0.466736	−	0.884397i	$-0.345430\pi$
$743$	18905.3	0.933472	0.466736	+	0.884397i	$0.345430\pi$
$744$	−3042.49	−0.149924
$745$	−42925.0	−2.11094
$746$	−9111.65	−0.447187
$747$	−8533.42	−0.417967
$748$	6286.49	0.307295
$749$	−17796.2	−0.868167
$750$	−8563.75	−0.416939
$751$	−21424.4	−1.04100	−0.520498	−	0.853863i	$-0.674254\pi$
$751$	−21424.4	−1.04100	−0.520498	+	0.853863i	$0.674254\pi$
$752$	−4835.13	−0.234467
$753$	4702.83	0.227597
$754$	43972.5	2.12385
$755$	50547.3	2.43656
$756$	−4398.19	−0.211588
$757$	6544.61	0.314224	0.157112	−	0.987581i	$-0.449782\pi$
$757$	6544.61	0.314224	0.157112	+	0.987581i	$0.449782\pi$
$758$	−9993.15	−0.478849
$759$	1446.17	0.0691604
$760$	−1149.04	−0.0548423
$761$	24211.5	1.15331	0.576653	−	0.816990i	$-0.304359\pi$
$761$	24211.5	1.15331	0.576653	+	0.816990i	$0.304359\pi$
$762$	−7710.01	−0.366541
$763$	13117.3	0.622384
$764$	22048.9	1.04411
$765$	−8350.82	−0.394673
$766$	29247.7	1.37959
$767$	−4226.86	−0.198987
$768$	9762.07	0.458670
$769$	−35978.9	−1.68717	−0.843585	−	0.536996i	$-0.819559\pi$
$769$	−35978.9	−1.68717	−0.843585	+	0.536996i	$0.819559\pi$
$770$	−11898.1	−0.556853
$771$	−5493.58	−0.256610
$772$	−32666.1	−1.52290
$773$	24320.8	1.13164	0.565821	−	0.824528i	$-0.308559\pi$
$773$	24320.8	1.13164	0.565821	+	0.824528i	$0.308559\pi$
$774$	11104.6	0.515693
$775$	25849.4	1.19812
$776$	4392.70	0.203207
$777$	−391.152	−0.0180599
$778$	39941.4	1.84058
$779$	−6979.12	−0.320992
$780$	−23564.3	−1.08172
$781$	−5309.16	−0.243248
$782$	−11737.7	−0.536751
$783$	−4569.44	−0.208555
$784$	−314.069	−0.0143071
$785$	−16242.5	−0.738496
$786$	−18298.7	−0.830398
$787$	8750.99	0.396365	0.198182	−	0.980165i	$-0.436496\pi$
$787$	8750.99	0.396365	0.198182	+	0.980165i	$0.436496\pi$
$788$	2255.54	0.101967
$789$	−8190.66	−0.369576
$790$	12257.0	0.552006
$791$	11959.4	0.537583
$792$	−294.169	−0.0131980
$793$	−3875.33	−0.173540
$794$	60054.4	2.68419
$795$	−11080.6	−0.494327
$796$	10009.8	0.445712
$797$	−9308.87	−0.413723	−0.206861	−	0.978370i	$-0.566325\pi$
$797$	−9308.87	−0.413723	−0.206861	+	0.978370i	$0.566325\pi$
$798$	−6251.08	−0.277300
$799$	5491.73	0.243158
$800$	19660.3	0.868871
$801$	6123.21	0.270104
$802$	−25538.6	−1.12444
$803$	−1957.26	−0.0860151
$804$	−11619.5	−0.509686
$805$	11590.1	0.507451
$806$	−88680.6	−3.87548
$807$	−13535.0	−0.590403
$808$	−181.948	−0.00792193
$809$	−20403.3	−0.886704	−0.443352	−	0.896348i	$-0.646211\pi$
$809$	−20403.3	−0.886704	−0.443352	+	0.896348i	$0.646211\pi$
$810$	4693.55	0.203598
$811$	20268.3	0.877581	0.438790	−	0.898589i	$-0.355407\pi$
$811$	20268.3	0.877581	0.438790	+	0.898589i	$0.355407\pi$
$812$	27568.3	1.19145
$813$	14327.7	0.618075
$814$	−314.233	−0.0135306
$815$	−19621.3	−0.843319
$816$	−11328.4	−0.485997
$817$	8234.17	0.352603
$818$	−40257.2	−1.72073
$819$	−10673.1	−0.455369
$820$	−31615.0	−1.34639
$821$	4750.77	0.201952	0.100976	−	0.994889i	$-0.467803\pi$
$821$	4750.77	0.201952	0.100976	+	0.994889i	$0.467803\pi$
$822$	−19614.8	−0.832292
$823$	−18027.3	−0.763541	−0.381770	−	0.924257i	$-0.624685\pi$
$823$	−18027.3	−0.763541	−0.381770	+	0.924257i	$0.624685\pi$
$824$	96.6060	0.00408426
$825$	2499.30	0.105472
$826$	−5079.38	−0.213964
$827$	−29872.5	−1.25607	−0.628034	−	0.778186i	$-0.716140\pi$
$827$	−29872.5	−1.25607	−0.628034	+	0.778186i	$0.716140\pi$
$828$	3441.84	0.144459
$829$	2534.96	0.106204	0.0531019	−	0.998589i	$-0.483089\pi$
$829$	2534.96	0.106204	0.0531019	+	0.998589i	$0.483089\pi$
$830$	−54941.0	−2.29763
$831$	14753.8	0.615890
$832$	−38142.8	−1.58938
$833$	356.718	0.0148374
$834$	8634.81	0.358512
$835$	−1898.98	−0.0787028
$836$	−2619.97	−0.108390
$837$	9215.33	0.380559
$838$	−54254.6	−2.23651
$839$	16979.3	0.698677	0.349338	−	0.936997i	$-0.386406\pi$
$839$	16979.3	0.698677	0.349338	+	0.936997i	$0.386406\pi$
$840$	−2357.56	−0.0968377
$841$	4252.74	0.174371
$842$	7777.07	0.318308
$843$	5345.79	0.218409
$844$	26113.1	1.06499
$845$	−26056.0	−1.06077
$846$	−3086.60	−0.125437
$847$	−2258.67	−0.0916281
$848$	−15031.6	−0.608710
$849$	−15382.0	−0.621802
$850$	−20285.3	−0.818564
$851$	306.100	0.0123302
$852$	−12635.6	−0.508085
$853$	16625.1	0.667328	0.333664	−	0.942692i	$-0.391715\pi$
$853$	16625.1	0.667328	0.333664	+	0.942692i	$0.391715\pi$
$854$	−4656.95	−0.186601
$855$	3480.31	0.139209
$856$	−2832.82	−0.113112
$857$	−17420.1	−0.694351	−0.347175	−	0.937800i	$-0.612859\pi$
$857$	−17420.1	−0.694351	−0.347175	+	0.937800i	$0.612859\pi$
$858$	−8574.24	−0.341165
$859$	−11692.3	−0.464421	−0.232210	−	0.972666i	$-0.574596\pi$
$859$	−11692.3	−0.464421	−0.232210	+	0.972666i	$0.574596\pi$
$860$	37300.3	1.47899
$861$	−14319.5	−0.566791
$862$	27170.5	1.07359
$863$	−13136.5	−0.518158	−0.259079	−	0.965856i	$-0.583419\pi$
$863$	−13136.5	−0.518158	−0.259079	+	0.965856i	$0.583419\pi$
$864$	7008.90	0.275981
$865$	51030.8	2.00590
$866$	−35952.0	−1.41074
$867$	−1872.25	−0.0733390
$868$	−55597.7	−2.17409
$869$	2326.81	0.0908304
$870$	−29419.6	−1.14646
$871$	−28197.0	−1.09692
$872$	2088.04	0.0810892
$873$	−13305.0	−0.515813
$874$	4891.83	0.189323
$875$	−13028.9	−0.503380
$876$	−4658.21	−0.179665
$877$	−13595.9	−0.523491	−0.261746	−	0.965137i	$-0.584298\pi$
$877$	−13595.9	−0.523491	−0.261746	+	0.965137i	$0.584298\pi$
$878$	−2206.04	−0.0847954
$879$	6761.24	0.259444
$880$	8986.26	0.344235
$881$	13921.9	0.532394	0.266197	−	0.963919i	$-0.414233\pi$
$881$	13921.9	0.532394	0.266197	+	0.963919i	$0.414233\pi$
$882$	−200.492	−0.00765410
$883$	14191.0	0.540846	0.270423	−	0.962742i	$-0.412837\pi$
$883$	14191.0	0.540846	0.270423	+	0.962742i	$0.412837\pi$
$884$	36307.3	1.38139
$885$	2827.96	0.107413
$886$	−22579.9	−0.856191
$887$	3882.56	0.146971	0.0734857	−	0.997296i	$-0.476588\pi$
$887$	3882.56	0.146971	0.0734857	+	0.997296i	$0.476588\pi$
$888$	−62.2643	−0.00235299
$889$	−11730.0	−0.442534
$890$	39423.3	1.48480
$891$	891.000	0.0335013
$892$	−25544.7	−0.958856
$893$	−2288.74	−0.0857670
$894$	−37172.5	−1.39064
$895$	−17596.4	−0.657186
$896$	−7070.41	−0.263623
$897$	8352.30	0.310898
$898$	9781.85	0.363502
$899$	−57762.6	−2.14293
$900$	5948.24	0.220305
$901$	17072.8	0.631273
$902$	−11503.6	−0.424643
$903$	16894.6	0.622609
$904$	1903.72	0.0700406
$905$	30734.3	1.12889
$906$	43773.3	1.60516
$907$	20883.9	0.764539	0.382270	−	0.924051i	$-0.375143\pi$
$907$	20883.9	0.764539	0.382270	+	0.924051i	$0.375143\pi$
$908$	17092.9	0.624722
$909$	551.099	0.0201087
$910$	−68716.8	−2.50323
$911$	13622.6	0.495429	0.247714	−	0.968833i	$-0.420321\pi$
$911$	13622.6	0.495429	0.247714	+	0.968833i	$0.420321\pi$
$912$	4721.25	0.171421
$913$	−10429.7	−0.378065
$914$	−1475.14	−0.0533844
$915$	2592.77	0.0936769
$916$	−32279.1	−1.16434
$917$	−27839.7	−1.00256
$918$	−7231.70	−0.260002
$919$	−1891.81	−0.0679056	−0.0339528	−	0.999423i	$-0.510810\pi$
$919$	−1891.81	−0.0679056	−0.0339528	+	0.999423i	$0.510810\pi$
$920$	1844.93	0.0661148
$921$	16058.9	0.574548
$922$	−54002.1	−1.92892
$923$	−30662.8	−1.09348
$924$	−5375.57	−0.191389
$925$	529.006	0.0188039
$926$	−79913.2	−2.83597
$927$	−292.608	−0.0103673
$928$	−43932.5	−1.55405
$929$	−7253.34	−0.256162	−0.128081	−	0.991764i	$-0.540882\pi$
$929$	−7253.34	−0.256162	−0.128081	+	0.991764i	$0.540882\pi$
$930$	59331.3	2.09199
$931$	−148.667	−0.00523346
$932$	42344.5	1.48824
$933$	17834.1	0.625789
$934$	−69815.5	−2.44586
$935$	−10206.6	−0.356995
$936$	−1698.96	−0.0593292
$937$	16586.9	0.578305	0.289152	−	0.957283i	$-0.406627\pi$
$937$	16586.9	0.578305	0.289152	+	0.957283i	$0.406627\pi$
$938$	−33884.0	−1.17948
$939$	−27778.4	−0.965403
$940$	−10367.9	−0.359748
$941$	32063.8	1.11078	0.555392	−	0.831588i	$-0.312568\pi$
$941$	32063.8	1.11078	0.555392	+	0.831588i	$0.312568\pi$
$942$	−14065.8	−0.486506
$943$	11205.8	0.386970
$944$	3836.30	0.132268
$945$	7140.77	0.245809
$946$	13572.3	0.466462
$947$	−6750.43	−0.231636	−0.115818	−	0.993270i	$-0.536949\pi$
$947$	−6750.43	−0.231636	−0.115818	+	0.993270i	$0.536949\pi$
$948$	5537.73	0.189723
$949$	−11304.1	−0.386665
$950$	8454.14	0.288725
$951$	22508.3	0.767489
$952$	3632.48	0.123665
$953$	19362.3	0.658141	0.329070	−	0.944305i	$-0.393265\pi$
$953$	19362.3	0.658141	0.329070	+	0.944305i	$0.393265\pi$
$954$	−9595.69	−0.325652
$955$	−35798.0	−1.21298
$956$	−11463.4	−0.387816
$957$	−5584.88	−0.188645
$958$	31357.2	1.05752
$959$	−29842.0	−1.00485
$960$	25519.3	0.857948
$961$	86700.4	2.91029
$962$	−1814.84	−0.0608241
$963$	8580.26	0.287118
$964$	49397.9	1.65041
$965$	53035.7	1.76920
$966$	10036.9	0.334298
$967$	−32364.1	−1.07628	−0.538139	−	0.842856i	$-0.680872\pi$
$967$	−32364.1	−1.07628	−0.538139	+	0.842856i	$0.680872\pi$
$968$	−359.539	−0.0119380
$969$	−5362.38	−0.177775
$970$	−85661.7	−2.83550
$971$	903.170	0.0298498	0.0149249	−	0.999889i	$-0.495249\pi$
$971$	903.170	0.0298498	0.0149249	+	0.999889i	$0.495249\pi$
$972$	2120.55	0.0699760
$973$	13137.0	0.432840
$974$	−21898.2	−0.720394
$975$	14434.6	0.474130
$976$	3517.25	0.115353
$977$	−25577.1	−0.837548	−0.418774	−	0.908091i	$-0.637540\pi$
$977$	−25577.1	−0.837548	−0.418774	+	0.908091i	$0.637540\pi$
$978$	−16991.8	−0.555561
$979$	7483.93	0.244318
$980$	−673.451	−0.0219516
$981$	−6324.40	−0.205833
$982$	7770.87	0.252524
$983$	−13577.4	−0.440541	−0.220271	−	0.975439i	$-0.570694\pi$
$983$	−13577.4	−0.440541	−0.220271	+	0.975439i	$0.570694\pi$
$984$	−2279.40	−0.0738462
$985$	−3662.02	−0.118459
$986$	45329.0	1.46407
$987$	−4695.96	−0.151443
$988$	−15131.5	−0.487245
$989$	−13221.0	−0.425079
$990$	5736.56	0.184161
$991$	−5095.05	−0.163319	−0.0816597	−	0.996660i	$-0.526022\pi$
$991$	−5095.05	−0.163319	−0.0816597	+	0.996660i	$0.526022\pi$
$992$	88599.9	2.83573
$993$	−2363.06	−0.0755179
$994$	−36847.2	−1.17578
$995$	−16251.6	−0.517798
$996$	−24822.4	−0.789686
$997$	40499.6	1.28650	0.643248	−	0.765658i	$-0.277586\pi$
$997$	40499.6	1.28650	0.643248	+	0.765658i	$0.277586\pi$
$998$	5190.19	0.164622
$999$	188.591	0.00597272

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2013.4.a.f.1.6	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.4.a.f.1.6	✓	38	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:	\( N \)	\(=\)	\( 2013 = 3 \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2013.a (trivial)

\(f(q)\)	\(=\)	\(q-4.08981 q^{2} +3.00000 q^{3} +8.72654 q^{4} -14.1681 q^{5} -12.2694 q^{6} -18.6667 q^{7} -2.97140 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.08981 q^{2} +3.00000 q^{3} +8.72654 q^{4} -14.1681 q^{5} -12.2694 q^{6} -18.6667 q^{7} -2.97140 q^{8} +9.00000 q^{9} +57.9450 q^{10} +11.0000 q^{11} +26.1796 q^{12} +63.5300 q^{13} +76.3434 q^{14} -42.5044 q^{15} -57.6598 q^{16} +65.4898 q^{17} -36.8083 q^{18} -27.2937 q^{19} -123.639 q^{20} -56.0002 q^{21} -44.9879 q^{22} +43.8234 q^{23} -8.91420 q^{24} +75.7363 q^{25} -259.825 q^{26} +27.0000 q^{27} -162.896 q^{28} -169.239 q^{29} +173.835 q^{30} +341.308 q^{31} +259.589 q^{32} +33.0000 q^{33} -267.841 q^{34} +264.473 q^{35} +78.5388 q^{36} +6.98484 q^{37} +111.626 q^{38} +190.590 q^{39} +42.0992 q^{40} +255.704 q^{41} +229.030 q^{42} -301.688 q^{43} +95.9919 q^{44} -127.513 q^{45} -179.229 q^{46} +83.8562 q^{47} -172.980 q^{48} +5.44692 q^{49} -309.747 q^{50} +196.469 q^{51} +554.397 q^{52} +260.694 q^{53} -110.425 q^{54} -155.850 q^{55} +55.4663 q^{56} -81.8811 q^{57} +692.154 q^{58} -66.5333 q^{59} -370.917 q^{60} -61.0000 q^{61} -1395.89 q^{62} -168.001 q^{63} -600.390 q^{64} -900.102 q^{65} -134.964 q^{66} -443.837 q^{67} +571.499 q^{68} +131.470 q^{69} -1081.64 q^{70} -482.651 q^{71} -26.7426 q^{72} -177.933 q^{73} -28.5667 q^{74} +227.209 q^{75} -238.179 q^{76} -205.334 q^{77} -779.476 q^{78} +211.528 q^{79} +816.933 q^{80} +81.0000 q^{81} -1045.78 q^{82} -948.157 q^{83} -488.688 q^{84} -927.869 q^{85} +1233.84 q^{86} -507.716 q^{87} -32.6854 q^{88} +680.357 q^{89} +521.505 q^{90} -1185.90 q^{91} +382.427 q^{92} +1023.93 q^{93} -342.956 q^{94} +386.701 q^{95} +778.767 q^{96} -1478.33 q^{97} -22.2769 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9}+O(q^{10}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9	\( 38 q + 14 q^{2} + 114 q^{3} + 170 q^{4} + 35 q^{5} + 42 q^{6} + 105 q^{7} + 147 q^{8} + 342 q^{9} + 99 q^{10} + 418 q^{11} + 510 q^{12} + 209 q^{13} + 128 q^{14} + 105 q^{15} + 798 q^{16} + 512 q^{17} + 126 q^{18} + 487 q^{19} + 328 q^{20} + 315 q^{21} + 154 q^{22} + 417 q^{23} + 441 q^{24} + 925 q^{25} + 177 q^{26} + 1026 q^{27} + 902 q^{28} + 626 q^{29} + 297 q^{30} + 300 q^{31} + 1625 q^{32} + 1254 q^{33} - 180 q^{34} + 1086 q^{35} + 1530 q^{36} + 554 q^{37} + 845 q^{38} + 627 q^{39} + 329 q^{40} + 1378 q^{41} + 384 q^{42} + 1979 q^{43} + 1870 q^{44} + 315 q^{45} + 937 q^{46} + 1345 q^{47} + 2394 q^{48} + 2635 q^{49} + 800 q^{50} + 1536 q^{51} + 2006 q^{52} + 1497 q^{53} + 378 q^{54} + 385 q^{55} + 415 q^{56} + 1461 q^{57} + 1241 q^{58} + 2827 q^{59} + 984 q^{60} - 2318 q^{61} + 509 q^{62} + 945 q^{63} + 1003 q^{64} + 2810 q^{65} + 462 q^{66} + 369 q^{67} + 3936 q^{68} + 1251 q^{69} + 922 q^{70} + 965 q^{71} + 1323 q^{72} + 3081 q^{73} + 722 q^{74} + 2775 q^{75} + 2210 q^{76} + 1155 q^{77} + 531 q^{78} + 3795 q^{79} + 3793 q^{80} + 3078 q^{81} - 1678 q^{82} + 3869 q^{83} + 2706 q^{84} + 3553 q^{85} + 3305 q^{86} + 1878 q^{87} + 1617 q^{88} + 2849 q^{89} + 891 q^{90} + 1252 q^{91} + 4519 q^{92} + 900 q^{93} + 340 q^{94} + 1504 q^{95} + 4875 q^{96} + 2562 q^{97} + 6164 q^{98} + 3762 q^{99}+O(q^{100}) \) 38 * q + 14 * q^2 + 114 * q^3 + 170 * q^4 + 35 * q^5 + 42 * q^6 + 105 * q^7 + 147 * q^8 + 342 * q^9 + 99 * q^10 + 418 * q^11 + 510 * q^12 + 209 * q^13 + 128 * q^14 + 105 * q^15 + 798 * q^16 + 512 * q^17 + 126 * q^18 + 487 * q^19 + 328 * q^20 + 315 * q^21 + 154 * q^22 + 417 * q^23 + 441 * q^24 + 925 * q^25 + 177 * q^26 + 1026 * q^27 + 902 * q^28 + 626 * q^29 + 297 * q^30 + 300 * q^31 + 1625 * q^32 + 1254 * q^33 - 180 * q^34 + 1086 * q^35 + 1530 * q^36 + 554 * q^37 + 845 * q^38 + 627 * q^39 + 329 * q^40 + 1378 * q^41 + 384 * q^42 + 1979 * q^43 + 1870 * q^44 + 315 * q^45 + 937 * q^46 + 1345 * q^47 + 2394 * q^48 + 2635 * q^49 + 800 * q^50 + 1536 * q^51 + 2006 * q^52 + 1497 * q^53 + 378 * q^54 + 385 * q^55 + 415 * q^56 + 1461 * q^57 + 1241 * q^58 + 2827 * q^59 + 984 * q^60 - 2318 * q^61 + 509 * q^62 + 945 * q^63 + 1003 * q^64 + 2810 * q^65 + 462 * q^66 + 369 * q^67 + 3936 * q^68 + 1251 * q^69 + 922 * q^70 + 965 * q^71 + 1323 * q^72 + 3081 * q^73 + 722 * q^74 + 2775 * q^75 + 2210 * q^76 + 1155 * q^77 + 531 * q^78 + 3795 * q^79 + 3793 * q^80 + 3078 * q^81 - 1678 * q^82 + 3869 * q^83 + 2706 * q^84 + 3553 * q^85 + 3305 * q^86 + 1878 * q^87 + 1617 * q^88 + 2849 * q^89 + 891 * q^90 + 1252 * q^91 + 4519 * q^92 + 900 * q^93 + 340 * q^94 + 1504 * q^95 + 4875 * q^96 + 2562 * q^97 + 6164 * q^98 + 3762 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more