LMFDB - Embedded newform 77.4.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,4,Mod(1,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("77.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	77.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:	$4.54314707044$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 42x^{3} + 18x^{2} + 368x + 352 $ x^5 - x^4 - 42x^3 + 18x^2 + 368*x + 352
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.41547$ of defining polynomial
Character	$\chi$	$=$	77.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.41547 q^{2} -5.03332 q^{3} +21.3273 q^{4} +5.67299 q^{5} -27.2578 q^{6} +7.00000 q^{7} +72.1734 q^{8} -1.66567 q^{9} +O(q^{10})$	\(q+5.41547 q^{2} -5.03332 q^{3} +21.3273 q^{4} +5.67299 q^{5} -27.2578 q^{6} +7.00000 q^{7} +72.1734 q^{8} -1.66567 q^{9} +30.7219 q^{10} +11.0000 q^{11} -107.347 q^{12} -77.6253 q^{13} +37.9083 q^{14} -28.5540 q^{15} +220.234 q^{16} +45.4552 q^{17} -9.02036 q^{18} -71.6423 q^{19} +120.989 q^{20} -35.2333 q^{21} +59.5701 q^{22} -140.231 q^{23} -363.272 q^{24} -92.8172 q^{25} -420.377 q^{26} +144.284 q^{27} +149.291 q^{28} +2.75206 q^{29} -154.633 q^{30} -67.3849 q^{31} +615.284 q^{32} -55.3665 q^{33} +246.161 q^{34} +39.7109 q^{35} -35.5241 q^{36} +152.642 q^{37} -387.976 q^{38} +390.713 q^{39} +409.439 q^{40} +65.5861 q^{41} -190.804 q^{42} +502.130 q^{43} +234.600 q^{44} -9.44931 q^{45} -759.416 q^{46} +351.461 q^{47} -1108.51 q^{48} +49.0000 q^{49} -502.648 q^{50} -228.791 q^{51} -1655.53 q^{52} -695.945 q^{53} +781.363 q^{54} +62.4029 q^{55} +505.214 q^{56} +360.599 q^{57} +14.9037 q^{58} +887.694 q^{59} -608.979 q^{60} -130.309 q^{61} -364.921 q^{62} -11.6597 q^{63} +1570.18 q^{64} -440.367 q^{65} -299.836 q^{66} +236.245 q^{67} +969.435 q^{68} +705.828 q^{69} +215.053 q^{70} +428.848 q^{71} -120.217 q^{72} -418.070 q^{73} +826.630 q^{74} +467.179 q^{75} -1527.93 q^{76} +77.0000 q^{77} +2115.89 q^{78} -1244.14 q^{79} +1249.39 q^{80} -681.253 q^{81} +355.179 q^{82} +533.380 q^{83} -751.429 q^{84} +257.867 q^{85} +2719.27 q^{86} -13.8520 q^{87} +793.907 q^{88} -630.298 q^{89} -51.1724 q^{90} -543.377 q^{91} -2990.75 q^{92} +339.170 q^{93} +1903.32 q^{94} -406.426 q^{95} -3096.92 q^{96} +561.108 q^{97} +265.358 q^{98} -18.3223 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9}+O(q^{10}) $ 5 * q + q^2 + 2 * q^3 + 45 * q^4 - 24 * q^5 + 4 * q^6 + 35 * q^7 + 57 * q^8 + 63 * q^9	\( 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9} - 10 q^{10} + 55 q^{11} + 24 q^{12} - 50 q^{13} + 7 q^{14} - 146 q^{15} + 433 q^{16} + 222 q^{17} + 245 q^{18} + 160 q^{19} - 430 q^{20} + 14 q^{21} + 11 q^{22} + 54 q^{23} - 220 q^{24} + 125 q^{25} - 1026 q^{26} + 110 q^{27} + 315 q^{28} + 14 q^{29} - 1080 q^{30} - 34 q^{31} - 583 q^{32} + 22 q^{33} - 750 q^{34} - 168 q^{35} - 411 q^{36} + 1044 q^{37} - 156 q^{38} - 124 q^{39} + 1158 q^{40} - 114 q^{41} + 28 q^{42} + 672 q^{43} + 495 q^{44} - 530 q^{45} - 1224 q^{46} - 292 q^{47} - 1652 q^{48} + 245 q^{49} + 1143 q^{50} + 768 q^{51} - 914 q^{52} - 710 q^{53} + 1608 q^{54} - 264 q^{55} + 399 q^{56} + 2012 q^{57} - 810 q^{58} + 270 q^{59} - 3068 q^{60} + 138 q^{61} - 480 q^{62} + 441 q^{63} + 3001 q^{64} - 196 q^{65} + 44 q^{66} + 1942 q^{67} + 3130 q^{68} - 1306 q^{69} - 70 q^{70} - 278 q^{71} + 3565 q^{72} - 338 q^{73} + 462 q^{74} + 3960 q^{75} - 1332 q^{76} + 385 q^{77} + 280 q^{78} + 576 q^{79} - 1602 q^{80} - 1439 q^{81} + 1386 q^{82} + 1644 q^{83} + 168 q^{84} + 360 q^{85} + 2020 q^{86} - 1956 q^{87} + 627 q^{88} - 3656 q^{89} - 6162 q^{90} - 350 q^{91} - 748 q^{92} - 1442 q^{93} + 976 q^{94} - 1276 q^{95} - 4508 q^{96} + 692 q^{97} + 49 q^{98} + 693 q^{99}+O(q^{100}) \) 5 * q + q^2 + 2 * q^3 + 45 * q^4 - 24 * q^5 + 4 * q^6 + 35 * q^7 + 57 * q^8 + 63 * q^9 - 10 * q^10 + 55 * q^11 + 24 * q^12 - 50 * q^13 + 7 * q^14 - 146 * q^15 + 433 * q^16 + 222 * q^17 + 245 * q^18 + 160 * q^19 - 430 * q^20 + 14 * q^21 + 11 * q^22 + 54 * q^23 - 220 * q^24 + 125 * q^25 - 1026 * q^26 + 110 * q^27 + 315 * q^28 + 14 * q^29 - 1080 * q^30 - 34 * q^31 - 583 * q^32 + 22 * q^33 - 750 * q^34 - 168 * q^35 - 411 * q^36 + 1044 * q^37 - 156 * q^38 - 124 * q^39 + 1158 * q^40 - 114 * q^41 + 28 * q^42 + 672 * q^43 + 495 * q^44 - 530 * q^45 - 1224 * q^46 - 292 * q^47 - 1652 * q^48 + 245 * q^49 + 1143 * q^50 + 768 * q^51 - 914 * q^52 - 710 * q^53 + 1608 * q^54 - 264 * q^55 + 399 * q^56 + 2012 * q^57 - 810 * q^58 + 270 * q^59 - 3068 * q^60 + 138 * q^61 - 480 * q^62 + 441 * q^63 + 3001 * q^64 - 196 * q^65 + 44 * q^66 + 1942 * q^67 + 3130 * q^68 - 1306 * q^69 - 70 * q^70 - 278 * q^71 + 3565 * q^72 - 338 * q^73 + 462 * q^74 + 3960 * q^75 - 1332 * q^76 + 385 * q^77 + 280 * q^78 + 576 * q^79 - 1602 * q^80 - 1439 * q^81 + 1386 * q^82 + 1644 * q^83 + 168 * q^84 + 360 * q^85 + 2020 * q^86 - 1956 * q^87 + 627 * q^88 - 3656 * q^89 - 6162 * q^90 - 350 * q^91 - 748 * q^92 - 1442 * q^93 + 976 * q^94 - 1276 * q^95 - 4508 * q^96 + 692 * q^97 + 49 * q^98 + 693 * 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 $

$2$	5.41547	1.91466	0.957328	−	0.289003i	$-0.0933238\pi$
$2$	5.41547	1.91466	0.957328	+	0.289003i	$0.0933238\pi$
$3$	−5.03332	−0.968663	−0.484332	−	0.874884i	$-0.660937\pi$
$3$	−5.03332	−0.968663	−0.484332	+	0.874884i	$0.660937\pi$
$4$	21.3273	2.66591
$5$	5.67299	0.507408	0.253704	−	0.967282i	$-0.418351\pi$
$5$	5.67299	0.507408	0.253704	+	0.967282i	$0.418351\pi$
$6$	−27.2578	−1.85466
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	72.1734	3.18964
$9$	−1.66567	−0.0616913
$10$	30.7219	0.971511
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−107.347	−2.58237
$13$	−77.6253	−1.65611	−0.828053	−	0.560650i	$-0.810551\pi$
$13$	−77.6253	−1.65611	−0.828053	+	0.560650i	$0.810551\pi$
$14$	37.9083	0.723672
$15$	−28.5540	−0.491507
$16$	220.234	3.44116
$17$	45.4552	0.648500	0.324250	−	0.945971i	$-0.394888\pi$
$17$	45.4552	0.648500	0.324250	+	0.945971i	$0.394888\pi$
$18$	−9.02036	−0.118118
$19$	−71.6423	−0.865046	−0.432523	−	0.901623i	$-0.642377\pi$
$19$	−71.6423	−0.865046	−0.432523	+	0.901623i	$0.642377\pi$
$20$	120.989	1.35270
$21$	−35.2333	−0.366120
$22$	59.5701	0.577291
$23$	−140.231	−1.27131	−0.635656	−	0.771972i	$-0.719270\pi$
$23$	−140.231	−1.27131	−0.635656	+	0.771972i	$0.719270\pi$
$24$	−363.272	−3.08969
$25$	−92.8172	−0.742537
$26$	−420.377	−3.17087
$27$	144.284	1.02842
$28$	149.291	1.00762
$29$	2.75206	0.0176222	0.00881111	−	0.999961i	$-0.497195\pi$
$29$	2.75206	0.0176222	0.00881111	+	0.999961i	$0.497195\pi$
$30$	−154.633	−0.941067
$31$	−67.3849	−0.390409	−0.195205	−	0.980763i	$-0.562537\pi$
$31$	−67.3849	−0.390409	−0.195205	+	0.980763i	$0.562537\pi$
$32$	615.284	3.39900
$33$	−55.3665	−0.292063
$34$	246.161	1.24165
$35$	39.7109	0.191782
$36$	−35.5241	−0.164463
$37$	152.642	0.678223	0.339112	−	0.940746i	$-0.389874\pi$
$37$	152.642	0.678223	0.339112	+	0.940746i	$0.389874\pi$
$38$	−387.976	−1.65627
$39$	390.713	1.60421
$40$	409.439	1.61845
$41$	65.5861	0.249825	0.124913	−	0.992168i	$-0.460135\pi$
$41$	65.5861	0.249825	0.124913	+	0.992168i	$0.460135\pi$
$42$	−190.804	−0.700995
$43$	502.130	1.78079	0.890396	−	0.455186i	$-0.150427\pi$
$43$	502.130	1.78079	0.890396	+	0.455186i	$0.150427\pi$
$44$	234.600	0.803802
$45$	−9.44931	−0.0313027
$46$	−759.416	−2.43413
$47$	351.461	1.09076	0.545381	−	0.838188i	$-0.316385\pi$
$47$	351.461	1.09076	0.545381	+	0.838188i	$0.316385\pi$
$48$	−1108.51	−3.33333
$49$	49.0000	0.142857
$50$	−502.648	−1.42170
$51$	−228.791	−0.628178
$52$	−1655.53	−4.41503
$53$	−695.945	−1.80369	−0.901843	−	0.432064i	$-0.857785\pi$
$53$	−695.945	−1.80369	−0.901843	+	0.432064i	$0.857785\pi$
$54$	781.363	1.96907
$55$	62.4029	0.152989
$56$	505.214	1.20557
$57$	360.599	0.837938
$58$	14.9037	0.0337405
$59$	887.694	1.95878	0.979389	−	0.201985i	$-0.0647392\pi$
$59$	887.694	1.95878	0.979389	+	0.201985i	$0.0647392\pi$
$60$	−608.979	−1.31031
$61$	−130.309	−0.273515	−0.136757	−	0.990605i	$-0.543668\pi$
$61$	−130.309	−0.273515	−0.136757	+	0.990605i	$0.543668\pi$
$62$	−364.921	−0.747499
$63$	−11.6597	−0.0233171
$64$	1570.18	3.06675
$65$	−440.367	−0.840321
$66$	−299.836	−0.559200
$67$	236.245	0.430775	0.215388	−	0.976529i	$-0.430899\pi$
$67$	236.245	0.430775	0.215388	+	0.976529i	$0.430899\pi$
$68$	969.435	1.72884
$69$	705.828	1.23147
$70$	215.053	0.367197
$71$	428.848	0.716829	0.358414	−	0.933563i	$-0.383318\pi$
$71$	428.848	0.716829	0.358414	+	0.933563i	$0.383318\pi$
$72$	−120.217	−0.196773
$73$	−418.070	−0.670293	−0.335146	−	0.942166i	$-0.608786\pi$
$73$	−418.070	−0.670293	−0.335146	+	0.942166i	$0.608786\pi$
$74$	826.630	1.29856
$75$	467.179	0.719269
$76$	−1527.93	−2.30613
$77$	77.0000	0.113961
$78$	2115.89	3.07151
$79$	−1244.14	−1.77185	−0.885927	−	0.463824i	$-0.846477\pi$
$79$	−1244.14	−1.77185	−0.885927	+	0.463824i	$0.846477\pi$
$80$	1249.39	1.74607
$81$	−681.253	−0.934503
$82$	355.179	0.478329
$83$	533.380	0.705375	0.352687	−	0.935741i	$-0.385268\pi$
$83$	533.380	0.705375	0.352687	+	0.935741i	$0.385268\pi$
$84$	−751.429	−0.976043
$85$	257.867	0.329054
$86$	2719.27	3.40961
$87$	−13.8520	−0.0170700
$88$	793.907	0.961713
$89$	−630.298	−0.750691	−0.375345	−	0.926885i	$-0.622476\pi$
$89$	−630.298	−0.750691	−0.375345	+	0.926885i	$0.622476\pi$
$90$	−51.1724	−0.0599338
$91$	−543.377	−0.625949
$92$	−2990.75	−3.38920
$93$	339.170	0.378175
$94$	1903.32	2.08844
$95$	−406.426	−0.438931
$96$	−3096.92	−3.29248
$97$	561.108	0.587339	0.293669	−	0.955907i	$-0.405124\pi$
$97$	561.108	0.587339	0.293669	+	0.955907i	$0.405124\pi$
$98$	265.358	0.273522
$99$	−18.3223	−0.0186006
$100$	−1979.54	−1.97954
$101$	121.392	0.119593	0.0597967	−	0.998211i	$-0.480955\pi$
$101$	121.392	0.119593	0.0597967	+	0.998211i	$0.480955\pi$
$102$	−1239.01	−1.20275
$103$	1462.42	1.39900	0.699499	−	0.714634i	$-0.253407\pi$
$103$	1462.42	1.39900	0.699499	+	0.714634i	$0.253407\pi$
$104$	−5602.48	−5.28238
$105$	−199.878	−0.185772
$106$	−3768.86	−3.45344
$107$	−112.660	−0.101787	−0.0508937	−	0.998704i	$-0.516207\pi$
$107$	−112.660	−0.101787	−0.0508937	+	0.998704i	$0.516207\pi$
$108$	3077.17	2.74168
$109$	800.629	0.703544	0.351772	−	0.936086i	$-0.385579\pi$
$109$	800.629	0.703544	0.351772	+	0.936086i	$0.385579\pi$
$110$	337.941	0.292922
$111$	−768.298	−0.656970
$112$	1541.64	1.30064
$113$	−1030.72	−0.858069	−0.429035	−	0.903288i	$-0.641146\pi$
$113$	−1030.72	−0.858069	−0.429035	+	0.903288i	$0.641146\pi$
$114$	1952.81	1.60436
$115$	−795.529	−0.645074
$116$	58.6939	0.0469792
$117$	129.298	0.102167
$118$	4807.27	3.75039
$119$	318.186	0.245110
$120$	−2060.84	−1.56773
$121$	121.000	0.0909091
$122$	−705.686	−0.523687
$123$	−330.116	−0.241996
$124$	−1437.14	−1.04080
$125$	−1235.67	−0.884177
$126$	−63.1425	−0.0446443
$127$	1875.12	1.31016	0.655078	−	0.755561i	$-0.272636\pi$
$127$	1875.12	1.31016	0.655078	+	0.755561i	$0.272636\pi$
$128$	3580.96	2.47278
$129$	−2527.38	−1.72499
$130$	−2384.79	−1.60893
$131$	−1602.24	−1.06861	−0.534306	−	0.845291i	$-0.679427\pi$
$131$	−1602.24	−1.06861	−0.534306	+	0.845291i	$0.679427\pi$
$132$	−1180.82	−0.778613
$133$	−501.496	−0.326957
$134$	1279.38	0.824787
$135$	818.519	0.521829
$136$	3280.65	2.06848
$137$	−602.184	−0.375533	−0.187767	−	0.982214i	$-0.560125\pi$
$137$	−602.184	−0.375533	−0.187767	+	0.982214i	$0.560125\pi$
$138$	3822.39	2.35785
$139$	−571.452	−0.348704	−0.174352	−	0.984683i	$-0.555783\pi$
$139$	−571.452	−0.348704	−0.174352	+	0.984683i	$0.555783\pi$
$140$	846.926	0.511274
$141$	−1769.02	−1.05658
$142$	2322.41	1.37248
$143$	−853.878	−0.499335
$144$	−366.837	−0.212290
$145$	15.6124	0.00894165
$146$	−2264.04	−1.28338
$147$	−246.633	−0.138380
$148$	3255.44	1.80808
$149$	−1509.58	−0.829999	−0.415000	−	0.909822i	$-0.636218\pi$
$149$	−1509.58	−0.829999	−0.415000	+	0.909822i	$0.636218\pi$
$150$	2529.99	1.37715
$151$	1771.22	0.954568	0.477284	−	0.878749i	$-0.341621\pi$
$151$	1771.22	0.954568	0.477284	+	0.878749i	$0.341621\pi$
$152$	−5170.67	−2.75919
$153$	−75.7132	−0.0400068
$154$	416.991	0.218195
$155$	−382.274	−0.198097
$156$	8332.84	4.27667
$157$	−143.100	−0.0727427	−0.0363714	−	0.999338i	$-0.511580\pi$
$157$	−143.100	−0.0727427	−0.0363714	+	0.999338i	$0.511580\pi$
$158$	−6737.59	−3.39249
$159$	3502.91	1.74716
$160$	3490.50	1.72468
$161$	−981.617	−0.480511
$162$	−3689.30	−1.78925
$163$	−2201.89	−1.05807	−0.529035	−	0.848600i	$-0.677446\pi$
$163$	−2201.89	−1.05807	−0.529035	+	0.848600i	$0.677446\pi$
$164$	1398.77	0.666011
$165$	−314.094	−0.148195
$166$	2888.50	1.35055
$167$	−2685.56	−1.24440	−0.622199	−	0.782859i	$-0.713761\pi$
$167$	−2685.56	−1.24440	−0.622199	+	0.782859i	$0.713761\pi$
$168$	−2542.90	−1.16779
$169$	3828.68	1.74269
$170$	1396.47	0.630025
$171$	119.332	0.0533658
$172$	10709.1	4.74743
$173$	2176.94	0.956702	0.478351	−	0.878169i	$-0.341235\pi$
$173$	2176.94	0.956702	0.478351	+	0.878169i	$0.341235\pi$
$174$	−75.0150	−0.0326832
$175$	−649.720	−0.280653
$176$	2422.58	1.03755
$177$	−4468.05	−1.89740
$178$	−3413.36	−1.43732
$179$	377.780	0.157746	0.0788732	−	0.996885i	$-0.474868\pi$
$179$	377.780	0.157746	0.0788732	+	0.996885i	$0.474868\pi$
$180$	−201.528	−0.0834500
$181$	607.385	0.249429	0.124714	−	0.992193i	$-0.460199\pi$
$181$	607.385	0.249429	0.124714	+	0.992193i	$0.460199\pi$
$182$	−2942.64	−1.19848
$183$	655.889	0.264944
$184$	−10120.9	−4.05503
$185$	865.939	0.344136
$186$	1836.76	0.724075
$187$	500.007	0.195530
$188$	7495.70	2.90787
$189$	1009.98	0.388707
$190$	−2200.99	−0.840402
$191$	−900.572	−0.341168	−0.170584	−	0.985343i	$-0.554565\pi$
$191$	−900.572	−0.341168	−0.170584	+	0.985343i	$0.554565\pi$
$192$	−7903.20	−2.97065
$193$	141.661	0.0528340	0.0264170	−	0.999651i	$-0.491590\pi$
$193$	141.661	0.0528340	0.0264170	+	0.999651i	$0.491590\pi$
$194$	3038.66	1.12455
$195$	2216.51	0.813988
$196$	1045.04	0.380844
$197$	3071.07	1.11068	0.555342	−	0.831622i	$-0.312587\pi$
$197$	3071.07	1.11068	0.555342	+	0.831622i	$0.312587\pi$
$198$	−99.2239	−0.0356138
$199$	−5571.12	−1.98455	−0.992277	−	0.124039i	$-0.960415\pi$
$199$	−5571.12	−1.98455	−0.992277	+	0.124039i	$0.960415\pi$
$200$	−6698.93	−2.36843
$201$	−1189.10	−0.417276
$202$	657.393	0.228980
$203$	19.2644	0.00666057
$204$	−4879.48	−1.67467
$205$	372.069	0.126763
$206$	7919.70	2.67860
$207$	233.578	0.0784290
$208$	−17095.7	−5.69892
$209$	−788.065	−0.260821
$210$	−1082.43	−0.355690
$211$	4089.28	1.33421	0.667103	−	0.744966i	$-0.267534\pi$
$211$	4089.28	1.33421	0.667103	+	0.744966i	$0.267534\pi$
$212$	−14842.6	−4.80846
$213$	−2158.53	−0.694366
$214$	−610.107	−0.194888
$215$	2848.58	0.903588
$216$	10413.4	3.28030
$217$	−471.694	−0.147561
$218$	4335.78	1.34705
$219$	2104.28	0.649288
$220$	1330.88	0.407855
$221$	−3528.47	−1.07398
$222$	−4160.69	−1.25787
$223$	4646.08	1.39518	0.697588	−	0.716499i	$-0.254257\pi$
$223$	4646.08	1.39518	0.697588	+	0.716499i	$0.254257\pi$
$224$	4306.99	1.28470
$225$	154.602	0.0458081
$226$	−5581.82	−1.64291
$227$	536.550	0.156881	0.0784407	−	0.996919i	$-0.475006\pi$
$227$	536.550	0.156881	0.0784407	+	0.996919i	$0.475006\pi$
$228$	7690.59	2.23387
$229$	5834.41	1.68362	0.841810	−	0.539774i	$-0.181490\pi$
$229$	5834.41	1.68362	0.841810	+	0.539774i	$0.181490\pi$
$230$	−4308.16	−1.23509
$231$	−387.566	−0.110389
$232$	198.625	0.0562086
$233$	2850.45	0.801457	0.400728	−	0.916197i	$-0.368757\pi$
$233$	2850.45	0.801457	0.400728	+	0.916197i	$0.368757\pi$
$234$	700.208	0.195615
$235$	1993.83	0.553461
$236$	18932.1	5.22192
$237$	6262.15	1.71633
$238$	1723.13	0.469301
$239$	−456.260	−0.123486	−0.0617428	−	0.998092i	$-0.519666\pi$
$239$	−456.260	−0.123486	−0.0617428	+	0.998092i	$0.519666\pi$
$240$	−6288.57	−1.69136
$241$	−550.254	−0.147075	−0.0735373	−	0.997292i	$-0.523429\pi$
$241$	−550.254	−0.147075	−0.0735373	+	0.997292i	$0.523429\pi$
$242$	655.271	0.174060
$243$	−466.692	−0.123203
$244$	−2779.14	−0.729165
$245$	277.977	0.0724868
$246$	−1787.73	−0.463340
$247$	5561.25	1.43261
$248$	−4863.40	−1.24527
$249$	−2684.67	−0.683271
$250$	−6691.75	−1.69289
$251$	−1829.79	−0.460140	−0.230070	−	0.973174i	$-0.573895\pi$
$251$	−1829.79	−0.460140	−0.230070	+	0.973174i	$0.573895\pi$
$252$	−248.669	−0.0621613
$253$	−1542.54	−0.383315
$254$	10154.6	2.50850
$255$	−1297.93	−0.318742
$256$	6831.17	1.66777
$257$	−7086.02	−1.71990	−0.859948	−	0.510381i	$-0.829504\pi$
$257$	−7086.02	−1.71990	−0.859948	+	0.510381i	$0.829504\pi$
$258$	−13686.9	−3.30276
$259$	1068.50	0.256344
$260$	−9391.83	−2.24022
$261$	−4.58401	−0.00108714
$262$	−8676.87	−2.04603
$263$	800.431	0.187668	0.0938340	−	0.995588i	$-0.470088\pi$
$263$	800.431	0.187668	0.0938340	+	0.995588i	$0.470088\pi$
$264$	−3995.99	−0.931576
$265$	−3948.09	−0.915204
$266$	−2715.83	−0.626009
$267$	3172.49	0.727167
$268$	5038.46	1.14841
$269$	−6401.38	−1.45093	−0.725463	−	0.688261i	$-0.758375\pi$
$269$	−6401.38	−1.45093	−0.725463	+	0.688261i	$0.758375\pi$
$270$	4432.66	0.999123
$271$	−6153.45	−1.37932	−0.689660	−	0.724134i	$-0.742240\pi$
$271$	−6153.45	−1.37932	−0.689660	+	0.724134i	$0.742240\pi$
$272$	10010.8	2.23159
$273$	2734.99	0.606334
$274$	−3261.11	−0.719017
$275$	−1020.99	−0.223883
$276$	15053.4	3.28300
$277$	3132.77	0.679531	0.339766	−	0.940510i	$-0.389652\pi$
$277$	3132.77	0.679531	0.339766	+	0.940510i	$0.389652\pi$
$278$	−3094.68	−0.667649
$279$	112.241	0.0240849
$280$	2866.07	0.611716
$281$	402.095	0.0853628	0.0426814	−	0.999089i	$-0.486410\pi$
$281$	402.095	0.0853628	0.0426814	+	0.999089i	$0.486410\pi$
$282$	−9580.04	−2.02299
$283$	3798.55	0.797880	0.398940	−	0.916977i	$-0.369378\pi$
$283$	3798.55	0.797880	0.398940	+	0.916977i	$0.369378\pi$
$284$	9146.15	1.91100
$285$	2045.67	0.425176
$286$	−4624.15	−0.956054
$287$	459.103	0.0944250
$288$	−1024.86	−0.209689
$289$	−2846.83	−0.579448
$290$	84.5484	0.0171202
$291$	−2824.24	−0.568934
$292$	−8916.29	−1.78694
$293$	2897.02	0.577630	0.288815	−	0.957385i	$-0.406739\pi$
$293$	2897.02	0.577630	0.288815	+	0.957385i	$0.406739\pi$
$294$	−1335.63	−0.264951
$295$	5035.88	0.993899
$296$	11016.7	2.16329
$297$	1587.12	0.310081
$298$	−8175.09	−1.58916
$299$	10885.5	2.10543
$300$	9963.65	1.91750
$301$	3514.91	0.673076
$302$	9591.97	1.82767
$303$	−611.004	−0.115846
$304$	−15778.1	−2.97676
$305$	−739.243	−0.138783
$306$	−410.022	−0.0765993
$307$	5598.35	1.04077	0.520383	−	0.853933i	$-0.325789\pi$
$307$	5598.35	1.04077	0.520383	+	0.853933i	$0.325789\pi$
$308$	1642.20	0.303808
$309$	−7360.85	−1.35516
$310$	−2070.19	−0.379287
$311$	−7654.57	−1.39566	−0.697831	−	0.716263i	$-0.745851\pi$
$311$	−7654.57	−1.39566	−0.697831	+	0.716263i	$0.745851\pi$
$312$	28199.1	5.11685
$313$	−5071.00	−0.915750	−0.457875	−	0.889017i	$-0.651389\pi$
$313$	−5071.00	−0.915750	−0.457875	+	0.889017i	$0.651389\pi$
$314$	−774.952	−0.139277
$315$	−66.1452	−0.0118313
$316$	−26534.1	−4.72360
$317$	−8289.31	−1.46869	−0.734344	−	0.678777i	$-0.762510\pi$
$317$	−8289.31	−1.46869	−0.734344	+	0.678777i	$0.762510\pi$
$318$	18969.9	3.34522
$319$	30.2726	0.00531330
$320$	8907.59	1.55609
$321$	567.054	0.0985978
$322$	−5315.92	−0.920014
$323$	−3256.51	−0.560982
$324$	−14529.3	−2.49130
$325$	7204.96	1.22972
$326$	−11924.3	−2.02584
$327$	−4029.82	−0.681498
$328$	4733.57	0.796853
$329$	2460.23	0.412269
$330$	−1700.96	−0.283743
$331$	−499.328	−0.0829171	−0.0414586	−	0.999140i	$-0.513200\pi$
$331$	−499.328	−0.0829171	−0.0414586	+	0.999140i	$0.513200\pi$
$332$	11375.5	1.88046
$333$	−254.251	−0.0418405
$334$	−14543.5	−2.38260
$335$	1340.22	0.218579
$336$	−7759.57	−1.25988
$337$	−510.083	−0.0824510	−0.0412255	−	0.999150i	$-0.513126\pi$
$337$	−510.083	−0.0824510	−0.0412255	+	0.999150i	$0.513126\pi$
$338$	20734.1	3.33664
$339$	5187.94	0.831180
$340$	5499.60	0.877228
$341$	−741.234	−0.117713
$342$	646.239	0.102177
$343$	343.000	0.0539949
$344$	36240.4	5.68009
$345$	4004.16	0.624859
$346$	11789.1	1.83176
$347$	2173.68	0.336281	0.168140	−	0.985763i	$-0.446224\pi$
$347$	2173.68	0.336281	0.168140	+	0.985763i	$0.446224\pi$
$348$	−295.425	−0.0455070
$349$	914.762	0.140304	0.0701520	−	0.997536i	$-0.477652\pi$
$349$	914.762	0.140304	0.0701520	+	0.997536i	$0.477652\pi$
$350$	−3518.54	−0.537354
$351$	−11200.0	−1.70317
$352$	6768.13	1.02484
$353$	1365.71	0.205918	0.102959	−	0.994686i	$-0.467169\pi$
$353$	1365.71	0.205918	0.102959	+	0.994686i	$0.467169\pi$
$354$	−24196.6	−3.63286
$355$	2432.85	0.363724
$356$	−13442.5	−2.00127
$357$	−1601.53	−0.237429
$358$	2045.85	0.302030
$359$	−8039.01	−1.18185	−0.590923	−	0.806728i	$-0.701236\pi$
$359$	−8039.01	−1.18185	−0.590923	+	0.806728i	$0.701236\pi$
$360$	−681.988	−0.0998443
$361$	−1726.38	−0.251696
$362$	3289.27	0.477570
$363$	−609.032	−0.0880603
$364$	−11588.7	−1.66872
$365$	−2371.71	−0.340112
$366$	3551.94	0.507276
$367$	−7072.33	−1.00592	−0.502960	−	0.864310i	$-0.667756\pi$
$367$	−7072.33	−1.00592	−0.502960	+	0.864310i	$0.667756\pi$
$368$	−30883.7	−4.37479
$369$	−109.245	−0.0154120
$370$	4689.46	0.658901
$371$	−4871.61	−0.681729
$372$	7233.57	1.00818
$373$	2483.30	0.344720	0.172360	−	0.985034i	$-0.444861\pi$
$373$	2483.30	0.344720	0.172360	+	0.985034i	$0.444861\pi$
$374$	2707.77	0.374373
$375$	6219.55	0.856470
$376$	25366.1	3.47914
$377$	−213.629	−0.0291842
$378$	5469.54	0.744240
$379$	5917.58	0.802021	0.401010	−	0.916073i	$-0.368659\pi$
$379$	5917.58	0.802021	0.401010	+	0.916073i	$0.368659\pi$
$380$	−8667.96	−1.17015
$381$	−9438.07	−1.26910
$382$	−4877.02	−0.653220
$383$	−6554.93	−0.874520	−0.437260	−	0.899335i	$-0.644051\pi$
$383$	−6554.93	−0.874520	−0.437260	+	0.899335i	$0.644051\pi$
$384$	−18024.1	−2.39529
$385$	436.820	0.0578245
$386$	767.159	0.101159
$387$	−836.380	−0.109859
$388$	11966.9	1.56579
$389$	−4384.19	−0.571433	−0.285716	−	0.958314i	$-0.592232\pi$
$389$	−4384.19	−0.571433	−0.285716	+	0.958314i	$0.592232\pi$
$390$	12003.4	1.55851
$391$	−6374.23	−0.824446
$392$	3536.50	0.455663
$393$	8064.58	1.03513
$394$	16631.3	2.12658
$395$	−7057.99	−0.899053
$396$	−390.765	−0.0495876
$397$	3968.04	0.501638	0.250819	−	0.968034i	$-0.419300\pi$
$397$	3968.04	0.501638	0.250819	+	0.968034i	$0.419300\pi$
$398$	−30170.2	−3.79974
$399$	2524.19	0.316711
$400$	−20441.5	−2.55519
$401$	2383.87	0.296870	0.148435	−	0.988922i	$-0.452576\pi$
$401$	2383.87	0.296870	0.148435	+	0.988922i	$0.452576\pi$
$402$	−6439.52	−0.798940
$403$	5230.77	0.646559
$404$	2588.95	0.318825
$405$	−3864.74	−0.474174
$406$	104.326	0.0127527
$407$	1679.07	0.204492
$408$	−16512.6	−2.00366
$409$	−278.299	−0.0336455	−0.0168227	−	0.999858i	$-0.505355\pi$
$409$	−278.299	−0.0336455	−0.0168227	+	0.999858i	$0.505355\pi$
$410$	2014.93	0.242708
$411$	3030.99	0.363765
$412$	31189.5	3.72960
$413$	6213.86	0.740348
$414$	1264.93	0.150165
$415$	3025.86	0.357913
$416$	−47761.6	−5.62910
$417$	2876.30	0.337777
$418$	−4267.74	−0.499383
$419$	−11342.4	−1.32247	−0.661234	−	0.750179i	$-0.729967\pi$
$419$	−11342.4	−1.32247	−0.661234	+	0.750179i	$0.729967\pi$
$420$	−4262.85	−0.495252
$421$	−554.327	−0.0641716	−0.0320858	−	0.999485i	$-0.510215\pi$
$421$	−554.327	−0.0641716	−0.0320858	+	0.999485i	$0.510215\pi$
$422$	22145.3	2.55454
$423$	−585.416	−0.0672906
$424$	−50228.7	−5.75311
$425$	−4219.02	−0.481536
$426$	−11689.4	−1.32947
$427$	−912.165	−0.103379
$428$	−2402.73	−0.271356
$429$	4297.84	0.483687
$430$	15426.4	1.73006
$431$	7007.67	0.783173	0.391586	−	0.920141i	$-0.371926\pi$
$431$	7007.67	0.783173	0.391586	+	0.920141i	$0.371926\pi$
$432$	31776.2	3.53896
$433$	6629.96	0.735832	0.367916	−	0.929859i	$-0.380071\pi$
$433$	6629.96	0.735832	0.367916	+	0.929859i	$0.380071\pi$
$434$	−2554.44	−0.282528
$435$	−78.5822	−0.00866144
$436$	17075.2	1.87559
$437$	10046.5	1.09974
$438$	11395.7	1.24316
$439$	−11055.1	−1.20189	−0.600946	−	0.799289i	$-0.705210\pi$
$439$	−11055.1	−1.20189	−0.600946	+	0.799289i	$0.705210\pi$
$440$	4503.83	0.487981
$441$	−81.6176	−0.00881305
$442$	−19108.3	−2.05631
$443$	11571.6	1.24105	0.620523	−	0.784189i	$-0.286920\pi$
$443$	11571.6	1.24105	0.620523	+	0.784189i	$0.286920\pi$
$444$	−16385.7	−1.75142
$445$	−3575.68	−0.380906
$446$	25160.7	2.67128
$447$	7598.22	0.803990
$448$	10991.2	1.15912
$449$	−5009.88	−0.526572	−0.263286	−	0.964718i	$-0.584806\pi$
$449$	−5009.88	−0.526572	−0.263286	+	0.964718i	$0.584806\pi$
$450$	837.244	0.0877068
$451$	721.447	0.0753251
$452$	−21982.4	−2.28753
$453$	−8915.12	−0.924655
$454$	2905.67	0.300374
$455$	−3082.57	−0.317611
$456$	26025.6	2.67272
$457$	−8804.21	−0.901189	−0.450595	−	0.892729i	$-0.648788\pi$
$457$	−8804.21	−0.901189	−0.450595	+	0.892729i	$0.648788\pi$
$458$	31596.1	3.22355
$459$	6558.43	0.666931
$460$	−16966.5	−1.71971
$461$	−485.657	−0.0490657	−0.0245329	−	0.999699i	$-0.507810\pi$
$461$	−485.657	−0.0490657	−0.0245329	+	0.999699i	$0.507810\pi$
$462$	−2098.85	−0.211358
$463$	−3594.38	−0.360788	−0.180394	−	0.983594i	$-0.557737\pi$
$463$	−3594.38	−0.360788	−0.180394	+	0.983594i	$0.557737\pi$
$464$	606.097	0.0606409
$465$	1924.11	0.191889
$466$	15436.5	1.53451
$467$	−5424.49	−0.537506	−0.268753	−	0.963209i	$-0.586612\pi$
$467$	−5424.49	−0.537506	−0.268753	+	0.963209i	$0.586612\pi$
$468$	2757.57	0.272369
$469$	1653.72	0.162818
$470$	10797.5	1.05969
$471$	720.267	0.0704632
$472$	64067.8	6.24780
$473$	5523.43	0.536929
$474$	33912.5	3.28618
$475$	6649.64	0.642329
$476$	6786.04	0.653441
$477$	1159.21	0.111272
$478$	−2470.86	−0.236432
$479$	10354.0	0.987653	0.493827	−	0.869560i	$-0.335598\pi$
$479$	10354.0	0.987653	0.493827	+	0.869560i	$0.335598\pi$
$480$	−17568.8	−1.67063
$481$	−11848.9	−1.12321
$482$	−2979.88	−0.281597
$483$	4940.80	0.465453
$484$	2580.60	0.242355
$485$	3183.16	0.298020
$486$	−2527.35	−0.235891
$487$	−368.694	−0.0343062	−0.0171531	−	0.999853i	$-0.505460\pi$
$487$	−368.694	−0.0343062	−0.0171531	+	0.999853i	$0.505460\pi$
$488$	−9404.86	−0.872414
$489$	11082.8	1.02491
$490$	1505.37	0.138787
$491$	−2040.40	−0.187539	−0.0937697	−	0.995594i	$-0.529892\pi$
$491$	−2040.40	−0.187539	−0.0937697	+	0.995594i	$0.529892\pi$
$492$	−7040.47	−0.645140
$493$	125.095	0.0114280
$494$	30116.8	2.74295
$495$	−103.942	−0.00943811
$496$	−14840.5	−1.34346
$497$	3001.93	0.270936
$498$	−14538.8	−1.30823
$499$	−5427.28	−0.486891	−0.243445	−	0.969915i	$-0.578278\pi$
$499$	−5427.28	−0.486891	−0.243445	+	0.969915i	$0.578278\pi$
$500$	−26353.6	−2.35713
$501$	13517.3	1.20540
$502$	−9909.15	−0.881010
$503$	−11323.6	−1.00376	−0.501882	−	0.864936i	$-0.667359\pi$
$503$	−11323.6	−1.00376	−0.501882	+	0.864936i	$0.667359\pi$
$504$	−841.517	−0.0743733
$505$	688.654	0.0606826
$506$	−8353.58	−0.733917
$507$	−19271.0	−1.68808
$508$	39991.1	3.49276
$509$	−17246.8	−1.50187	−0.750933	−	0.660378i	$-0.770396\pi$
$509$	−17246.8	−1.50187	−0.750933	+	0.660378i	$0.770396\pi$
$510$	−7028.88	−0.610282
$511$	−2926.49	−0.253347
$512$	8346.25	0.720421
$513$	−10336.8	−0.889632
$514$	−38374.1	−3.29301
$515$	8296.31	0.709862
$516$	−53902.1	−4.59866
$517$	3866.07	0.328877
$518$	5786.41	0.490811
$519$	−10957.2	−0.926722
$520$	−31782.8	−2.68032
$521$	12465.8	1.04824	0.524122	−	0.851643i	$-0.324393\pi$
$521$	12465.8	1.04824	0.524122	+	0.851643i	$0.324393\pi$
$522$	−24.8245	−0.00208150
$523$	15814.3	1.32221	0.661103	−	0.750296i	$-0.270089\pi$
$523$	15814.3	1.32221	0.661103	+	0.750296i	$0.270089\pi$
$524$	−34171.4	−2.84882
$525$	3270.25	0.271858
$526$	4334.71	0.359320
$527$	−3062.99	−0.253180
$528$	−12193.6	−1.00504
$529$	7497.75	0.616236
$530$	−21380.7	−1.75230
$531$	−1478.60	−0.120840
$532$	−10695.5	−0.871636
$533$	−5091.14	−0.413737
$534$	17180.5	1.39227
$535$	−639.120	−0.0516477
$536$	17050.6	1.37402
$537$	−1901.49	−0.152803
$538$	−34666.5	−2.77803
$539$	539.000	0.0430730
$540$	17456.8	1.39115
$541$	9999.08	0.794628	0.397314	−	0.917683i	$-0.369942\pi$
$541$	9999.08	0.794628	0.397314	+	0.917683i	$0.369942\pi$
$542$	−33323.8	−2.64092
$543$	−3057.17	−0.241612
$544$	27967.9	2.20425
$545$	4541.96	0.356984
$546$	14811.2	1.16092
$547$	−1738.98	−0.135929	−0.0679647	−	0.997688i	$-0.521651\pi$
$547$	−1738.98	−0.135929	−0.0679647	+	0.997688i	$0.521651\pi$
$548$	−12842.9	−1.00114
$549$	217.052	0.0168735
$550$	−5529.13	−0.428660
$551$	−197.164	−0.0152440
$552$	50942.0	3.92796
$553$	−8708.97	−0.669698
$554$	16965.4	1.30107
$555$	−4358.55	−0.333352
$556$	−12187.5	−0.929614
$557$	18282.6	1.39077	0.695385	−	0.718637i	$-0.255234\pi$
$557$	18282.6	1.39077	0.695385	+	0.718637i	$0.255234\pi$
$558$	607.836	0.0461142
$559$	−38978.0	−2.94918
$560$	8745.71	0.659953
$561$	−2516.70	−0.189403
$562$	2177.53	0.163440
$563$	940.407	0.0703968	0.0351984	−	0.999380i	$-0.488794\pi$
$563$	940.407	0.0703968	0.0351984	+	0.999380i	$0.488794\pi$
$564$	−37728.3	−2.81675
$565$	−5847.25	−0.435391
$566$	20570.9	1.52767
$567$	−4768.77	−0.353209
$568$	30951.4	2.28643
$569$	21714.7	1.59987	0.799935	−	0.600086i	$-0.204867\pi$
$569$	21714.7	1.59987	0.799935	+	0.600086i	$0.204867\pi$
$570$	11078.3	0.814066
$571$	−7170.55	−0.525531	−0.262766	−	0.964860i	$-0.584635\pi$
$571$	−7170.55	−0.525531	−0.262766	+	0.964860i	$0.584635\pi$
$572$	−18210.9	−1.33118
$573$	4532.87	0.330477
$574$	2486.25	0.180791
$575$	13015.9	0.943997
$576$	−2615.39	−0.189192
$577$	4297.68	0.310078	0.155039	−	0.987908i	$-0.450450\pi$
$577$	4297.68	0.310078	0.155039	+	0.987908i	$0.450450\pi$
$578$	−15416.9	−1.10944
$579$	−713.024	−0.0511783
$580$	332.970	0.0238376
$581$	3733.66	0.266607
$582$	−15294.6	−1.08931
$583$	−7655.39	−0.543832
$584$	−30173.5	−2.13799
$585$	733.505	0.0518405
$586$	15688.7	1.10596
$587$	4580.53	0.322076	0.161038	−	0.986948i	$-0.448516\pi$
$587$	4580.53	0.322076	0.161038	+	0.986948i	$0.448516\pi$
$588$	−5260.00	−0.368910
$589$	4827.61	0.337722
$590$	27271.6	1.90297
$591$	−15457.7	−1.07588
$592$	33617.1	2.33387
$593$	−13480.0	−0.933483	−0.466742	−	0.884394i	$-0.654572\pi$
$593$	−13480.0	−0.933483	−0.466742	+	0.884394i	$0.654572\pi$
$594$	8594.99	0.593698
$595$	1805.07	0.124371
$596$	−32195.3	−2.21270
$597$	28041.3	1.92237
$598$	58949.9	4.03117
$599$	−8350.68	−0.569615	−0.284808	−	0.958585i	$-0.591930\pi$
$599$	−8350.68	−0.569615	−0.284808	+	0.958585i	$0.591930\pi$
$600$	33717.9	2.29421
$601$	−16076.0	−1.09110	−0.545551	−	0.838078i	$-0.683680\pi$
$601$	−16076.0	−1.09110	−0.545551	+	0.838078i	$0.683680\pi$
$602$	19034.9	1.28871
$603$	−393.506	−0.0265751
$604$	37775.3	2.54479
$605$	686.432	0.0461280
$606$	−3308.87	−0.221805
$607$	3420.75	0.228738	0.114369	−	0.993438i	$-0.463515\pi$
$607$	3420.75	0.228738	0.114369	+	0.993438i	$0.463515\pi$
$608$	−44080.4	−2.94029
$609$	−96.9639	−0.00645185
$610$	−4003.35	−0.265723
$611$	−27282.2	−1.80642
$612$	−1614.75	−0.106655
$613$	11172.4	0.736135	0.368068	−	0.929799i	$-0.380020\pi$
$613$	11172.4	0.736135	0.368068	+	0.929799i	$0.380020\pi$
$614$	30317.7	1.99271
$615$	−1872.74	−0.122791
$616$	5557.35	0.363493
$617$	17418.5	1.13653	0.568267	−	0.822844i	$-0.307614\pi$
$617$	17418.5	1.13653	0.568267	+	0.822844i	$0.307614\pi$
$618$	−39862.4	−2.59466
$619$	153.847	0.00998974	0.00499487	−	0.999988i	$-0.498410\pi$
$619$	153.847	0.00998974	0.00499487	+	0.999988i	$0.498410\pi$
$620$	−8152.86	−0.528108
$621$	−20233.0	−1.30745
$622$	−41453.1	−2.67221
$623$	−4412.09	−0.283734
$624$	86048.4	5.52034
$625$	4592.18	0.293899
$626$	−27461.8	−1.75335
$627$	3966.59	0.252648
$628$	−3051.93	−0.193925
$629$	6938.39	0.439828
$630$	−358.207	−0.0226529
$631$	−4580.70	−0.288993	−0.144497	−	0.989505i	$-0.546156\pi$
$631$	−4580.70	−0.288993	−0.144497	+	0.989505i	$0.546156\pi$
$632$	−89793.7	−5.65158
$633$	−20582.6	−1.29240
$634$	−44890.5	−2.81203
$635$	10637.5	0.664783
$636$	74707.6	4.65778
$637$	−3803.64	−0.236587
$638$	163.940	0.0101731
$639$	−714.317	−0.0442221
$640$	20314.8	1.25471
$641$	−9484.96	−0.584451	−0.292226	−	0.956349i	$-0.594396\pi$
$641$	−9484.96	−0.584451	−0.292226	+	0.956349i	$0.594396\pi$
$642$	3070.86	0.188781
$643$	25911.5	1.58919	0.794595	−	0.607140i	$-0.207683\pi$
$643$	25911.5	1.58919	0.794595	+	0.607140i	$0.207683\pi$
$644$	−20935.2	−1.28100
$645$	−14337.8	−0.875272
$646$	−17635.5	−1.07409
$647$	8152.23	0.495359	0.247679	−	0.968842i	$-0.420332\pi$
$647$	8152.23	0.495359	0.247679	+	0.968842i	$0.420332\pi$
$648$	−49168.3	−2.98073
$649$	9764.63	0.590594
$650$	39018.2	2.35449
$651$	2374.19	0.142937
$652$	−46960.3	−2.82072
$653$	24960.6	1.49584	0.747920	−	0.663788i	$-0.231052\pi$
$653$	24960.6	1.49584	0.747920	+	0.663788i	$0.231052\pi$
$654$	−21823.4	−1.30483
$655$	−9089.48	−0.542222
$656$	14444.3	0.859688
$657$	696.365	0.0413512
$658$	13323.3	0.789354
$659$	−8524.55	−0.503899	−0.251949	−	0.967740i	$-0.581072\pi$
$659$	−8524.55	−0.503899	−0.251949	+	0.967740i	$0.581072\pi$
$660$	−6698.77	−0.395074
$661$	4589.06	0.270036	0.135018	−	0.990843i	$-0.456891\pi$
$661$	4589.06	0.270036	0.135018	+	0.990843i	$0.456891\pi$
$662$	−2704.10	−0.158758
$663$	17759.9	1.04033
$664$	38495.9	2.24989
$665$	−2844.98	−0.165900
$666$	−1376.89	−0.0801101
$667$	−385.924	−0.0224033
$668$	−57275.6	−3.31745
$669$	−23385.2	−1.35146
$670$	7257.90	0.418503
$671$	−1433.40	−0.0824678
$672$	−21678.5	−1.24444
$673$	33968.2	1.94558	0.972791	−	0.231685i	$-0.0744240\pi$
$673$	33968.2	1.94558	0.972791	+	0.231685i	$0.0744240\pi$
$674$	−2762.34	−0.157865
$675$	−13392.0	−0.763641
$676$	81655.3	4.64584
$677$	−17434.7	−0.989763	−0.494882	−	0.868960i	$-0.664789\pi$
$677$	−17434.7	−0.989763	−0.494882	+	0.868960i	$0.664789\pi$
$678$	28095.1	1.59142
$679$	3927.75	0.221993
$680$	18611.1	1.04956
$681$	−2700.63	−0.151965
$682$	−4014.13	−0.225380
$683$	16257.5	0.910798	0.455399	−	0.890288i	$-0.349497\pi$
$683$	16257.5	0.910798	0.455399	+	0.890288i	$0.349497\pi$
$684$	2545.03	0.142268
$685$	−3416.19	−0.190549
$686$	1857.50	0.103382
$687$	−29366.5	−1.63086
$688$	110586.	6.12799
$689$	54022.9	2.98709
$690$	21684.4	1.19639
$691$	5946.17	0.327356	0.163678	−	0.986514i	$-0.447664\pi$
$691$	5946.17	0.327356	0.163678	+	0.986514i	$0.447664\pi$
$692$	46428.1	2.55048
$693$	−128.256	−0.00703038
$694$	11771.5	0.643862
$695$	−3241.84	−0.176935
$696$	−999.745	−0.0544472
$697$	2981.23	0.162012
$698$	4953.86	0.268634
$699$	−14347.3	−0.776342
$700$	−13856.8	−0.748195
$701$	5922.72	0.319113	0.159556	−	0.987189i	$-0.448994\pi$
$701$	5922.72	0.319113	0.159556	+	0.987189i	$0.448994\pi$
$702$	−60653.5	−3.26099
$703$	−10935.7	−0.586694
$704$	17271.9	0.924660
$705$	−10035.6	−0.536118
$706$	7395.93	0.394263
$707$	849.742	0.0452020
$708$	−95291.3	−5.05828
$709$	−33739.1	−1.78716	−0.893582	−	0.448899i	$-0.851816\pi$
$709$	−33739.1	−1.78716	−0.893582	+	0.448899i	$0.851816\pi$
$710$	13175.0	0.696407
$711$	2072.32	0.109308
$712$	−45490.8	−2.39444
$713$	9449.45	0.496332
$714$	−8673.05	−0.454595
$715$	−4844.04	−0.253366
$716$	8057.01	0.420537
$717$	2296.51	0.119616
$718$	−43535.0	−2.26283
$719$	−3766.02	−0.195339	−0.0976696	−	0.995219i	$-0.531139\pi$
$719$	−3766.02	−0.195339	−0.0976696	+	0.995219i	$0.531139\pi$
$720$	−2081.06	−0.107717
$721$	10237.0	0.528772
$722$	−9349.16	−0.481911
$723$	2769.60	0.142466
$724$	12953.9	0.664954
$725$	−255.438	−0.0130852
$726$	−3298.19	−0.168605
$727$	6693.20	0.341454	0.170727	−	0.985318i	$-0.445388\pi$
$727$	6693.20	0.341454	0.170727	+	0.985318i	$0.445388\pi$
$728$	−39217.3	−1.99655
$729$	20742.8	1.05384
$730$	−12843.9	−0.651197
$731$	22824.4	1.15484
$732$	13988.3	0.706316
$733$	38633.8	1.94676	0.973379	−	0.229202i	$-0.0736116\pi$
$733$	38633.8	1.94676	0.973379	+	0.229202i	$0.0736116\pi$
$734$	−38300.0	−1.92599
$735$	−1399.15	−0.0702153
$736$	−86281.9	−4.32119
$737$	2598.70	0.129884
$738$	−591.610	−0.0295088
$739$	3475.70	0.173012	0.0865058	−	0.996251i	$-0.472430\pi$
$739$	3475.70	0.173012	0.0865058	+	0.996251i	$0.472430\pi$
$740$	18468.1	0.917434
$741$	−27991.6	−1.38771
$742$	−26382.0	−1.30528
$743$	20111.4	0.993022	0.496511	−	0.868030i	$-0.334614\pi$
$743$	20111.4	0.993022	0.496511	+	0.868030i	$0.334614\pi$
$744$	24479.0	1.20624
$745$	−8563.85	−0.421148
$746$	13448.2	0.660021
$747$	−888.433	−0.0435155
$748$	10663.8	0.521265
$749$	−788.620	−0.0384720
$750$	33681.8	1.63985
$751$	11561.3	0.561754	0.280877	−	0.959744i	$-0.409375\pi$
$751$	11561.3	0.561754	0.280877	+	0.959744i	$0.409375\pi$
$752$	77403.7	3.75349
$753$	9209.91	0.445721
$754$	−1156.90	−0.0558778
$755$	10048.1	0.484355
$756$	21540.2	1.03626
$757$	−30012.3	−1.44097	−0.720486	−	0.693469i	$-0.756081\pi$
$757$	−30012.3	−1.44097	−0.720486	+	0.693469i	$0.756081\pi$
$758$	32046.5	1.53559
$759$	7764.11	0.371303
$760$	−29333.1	−1.40003
$761$	23356.1	1.11256	0.556280	−	0.830995i	$-0.312228\pi$
$761$	23356.1	1.11256	0.556280	+	0.830995i	$0.312228\pi$
$762$	−51111.6	−2.42989
$763$	5604.40	0.265915
$764$	−19206.7	−0.909523
$765$	−429.520	−0.0202998
$766$	−35498.0	−1.67441
$767$	−68907.4	−3.24394
$768$	−34383.5	−1.61550
$769$	20347.6	0.954167	0.477084	−	0.878858i	$-0.341694\pi$
$769$	20347.6	0.954167	0.477084	+	0.878858i	$0.341694\pi$
$770$	2365.59	0.110714
$771$	35666.2	1.66600
$772$	3021.24	0.140851
$773$	23612.2	1.09867	0.549334	−	0.835603i	$-0.314881\pi$
$773$	23612.2	1.09867	0.549334	+	0.835603i	$0.314881\pi$
$774$	−4529.39	−0.210343
$775$	6254.48	0.289893
$776$	40497.0	1.87340
$777$	−5378.09	−0.248311
$778$	−23742.4	−1.09410
$779$	−4698.74	−0.216110
$780$	47272.1	2.17002
$781$	4717.32	0.216132
$782$	−34519.4	−1.57853
$783$	397.076	0.0181231
$784$	10791.5	0.491594
$785$	−811.804	−0.0369102
$786$	43673.5	1.98191
$787$	34821.8	1.57721	0.788605	−	0.614901i	$-0.210804\pi$
$787$	34821.8	1.57721	0.788605	+	0.614901i	$0.210804\pi$
$788$	65497.5	2.96098
$789$	−4028.83	−0.181787
$790$	−38222.3	−1.72138
$791$	−7215.03	−0.324320
$792$	−1322.38	−0.0593294
$793$	10115.3	0.452969
$794$	21488.8	0.960464
$795$	19872.0	0.886525
$796$	−118817.	−5.29064
$797$	−5188.07	−0.230578	−0.115289	−	0.993332i	$-0.536779\pi$
$797$	−5188.07	−0.230578	−0.115289	+	0.993332i	$0.536779\pi$
$798$	13669.7	0.606392
$799$	15975.7	0.707360
$800$	−57108.9	−2.52388
$801$	1049.87	0.0463111
$802$	12909.8	0.568404
$803$	−4598.77	−0.202101
$804$	−25360.2	−1.11242
$805$	−5568.71	−0.243815
$806$	28327.1	1.23794
$807$	32220.2	1.40546
$808$	8761.25	0.381460
$809$	−30778.8	−1.33761	−0.668805	−	0.743438i	$-0.733194\pi$
$809$	−30778.8	−1.33761	−0.668805	+	0.743438i	$0.733194\pi$
$810$	−20929.4	−0.907880
$811$	12441.7	0.538700	0.269350	−	0.963042i	$-0.413191\pi$
$811$	12441.7	0.538700	0.269350	+	0.963042i	$0.413191\pi$
$812$	410.857	0.0177565
$813$	30972.3	1.33610
$814$	9092.92	0.391532
$815$	−12491.3	−0.536873
$816$	−50387.5	−2.16166
$817$	−35973.7	−1.54047
$818$	−1507.12	−0.0644196
$819$	905.084	0.0386156
$820$	7935.22	0.337939
$821$	−2392.56	−0.101707	−0.0508533	−	0.998706i	$-0.516194\pi$
$821$	−2392.56	−0.101707	−0.0508533	+	0.998706i	$0.516194\pi$
$822$	16414.2	0.696486
$823$	−20913.9	−0.885798	−0.442899	−	0.896571i	$-0.646050\pi$
$823$	−20913.9	−0.885798	−0.442899	+	0.896571i	$0.646050\pi$
$824$	105548.	4.46230
$825$	5138.97	0.216868
$826$	33650.9	1.41751
$827$	1763.94	0.0741694	0.0370847	−	0.999312i	$-0.488193\pi$
$827$	1763.94	0.0741694	0.0370847	+	0.999312i	$0.488193\pi$
$828$	4981.58	0.209085
$829$	−8282.73	−0.347010	−0.173505	−	0.984833i	$-0.555509\pi$
$829$	−8282.73	−0.347010	−0.173505	+	0.984833i	$0.555509\pi$
$830$	16386.4	0.685279
$831$	−15768.3	−0.658237
$832$	−121885.	−5.07886
$833$	2227.30	0.0926429
$834$	15576.5	0.646727
$835$	−15235.1	−0.631418
$836$	−16807.3	−0.695325
$837$	−9722.53	−0.401505
$838$	−61424.6	−2.53207
$839$	18086.0	0.744217	0.372109	−	0.928189i	$-0.378635\pi$
$839$	18086.0	0.744217	0.372109	+	0.928189i	$0.378635\pi$
$840$	−14425.9	−0.592547
$841$	−24381.4	−0.999689
$842$	−3001.94	−0.122867
$843$	−2023.87	−0.0826878
$844$	87213.1	3.55687
$845$	21720.1	0.884252
$846$	−3170.30	−0.128838
$847$	847.000	0.0343604
$848$	−153271.	−6.20677
$849$	−19119.3	−0.772877
$850$	−22848.0	−0.921975
$851$	−21405.2	−0.862234
$852$	−46035.5	−1.85112
$853$	−44805.6	−1.79849	−0.899246	−	0.437443i	$-0.855884\pi$
$853$	−44805.6	−1.79849	−0.899246	+	0.437443i	$0.855884\pi$
$854$	−4939.80	−0.197935
$855$	676.970	0.0270782
$856$	−8131.06	−0.324666
$857$	22554.8	0.899017	0.449509	−	0.893276i	$-0.351599\pi$
$857$	22554.8	0.899017	0.449509	+	0.893276i	$0.351599\pi$
$858$	23274.8	0.926095
$859$	39517.6	1.56964	0.784821	−	0.619722i	$-0.212755\pi$
$859$	39517.6	1.56964	0.784821	+	0.619722i	$0.212755\pi$
$860$	60752.4	2.40888
$861$	−2310.81	−0.0914660
$862$	37949.8	1.49951
$863$	1259.51	0.0496805	0.0248403	−	0.999691i	$-0.492092\pi$
$863$	1259.51	0.0496805	0.0248403	+	0.999691i	$0.492092\pi$
$864$	88775.4	3.49560
$865$	12349.7	0.485438
$866$	35904.3	1.40887
$867$	14329.0	0.561290
$868$	−10060.0	−0.393384
$869$	−13685.5	−0.534234
$870$	−425.559	−0.0165837
$871$	−18338.6	−0.713409
$872$	57784.1	2.24406
$873$	−934.618	−0.0362337
$874$	54406.3	2.10563
$875$	−8649.72	−0.334187
$876$	44878.5	1.73094
$877$	−8834.84	−0.340173	−0.170086	−	0.985429i	$-0.554405\pi$
$877$	−8834.84	−0.340173	−0.170086	+	0.985429i	$0.554405\pi$
$878$	−59868.5	−2.30121
$879$	−14581.6	−0.559529
$880$	13743.3	0.526460
$881$	−48905.3	−1.87022	−0.935109	−	0.354361i	$-0.884698\pi$
$881$	−48905.3	−1.87022	−0.935109	+	0.354361i	$0.884698\pi$
$882$	−441.998	−0.0168740
$883$	−22475.3	−0.856572	−0.428286	−	0.903643i	$-0.640882\pi$
$883$	−22475.3	−0.856572	−0.428286	+	0.903643i	$0.640882\pi$
$884$	−75252.6	−2.86314
$885$	−25347.2	−0.962753
$886$	62665.6	2.37618
$887$	27402.6	1.03730	0.518652	−	0.854985i	$-0.326434\pi$
$887$	27402.6	1.03730	0.518652	+	0.854985i	$0.326434\pi$
$888$	−55450.7	−2.09550
$889$	13125.8	0.495192
$890$	−19364.0	−0.729305
$891$	−7493.78	−0.281763
$892$	99088.1	3.71941
$893$	−25179.5	−0.943560
$894$	41147.9	1.53936
$895$	2143.14	0.0800417
$896$	25066.7	0.934621
$897$	−54790.1	−2.03945
$898$	−27130.8	−1.00821
$899$	−185.447	−0.00687987
$900$	3297.25	0.122120
$901$	−31634.3	−1.16969
$902$	3906.97	0.144222
$903$	−17691.7	−0.651984
$904$	−74390.4	−2.73693
$905$	3445.69	0.126562
$906$	−48279.5	−1.77040
$907$	−24498.1	−0.896853	−0.448426	−	0.893820i	$-0.648015\pi$
$907$	−24498.1	−0.896853	−0.448426	+	0.893820i	$0.648015\pi$
$908$	11443.2	0.418232
$909$	−202.198	−0.00737788
$910$	−16693.6	−0.608117
$911$	14259.3	0.518585	0.259293	−	0.965799i	$-0.416511\pi$
$911$	14259.3	0.518585	0.259293	+	0.965799i	$0.416511\pi$
$912$	79416.2	2.88348
$913$	5867.18	0.212678
$914$	−47678.9	−1.72547
$915$	3720.85	0.134434
$916$	124432.	4.48838
$917$	−11215.7	−0.403898
$918$	35517.0	1.27694
$919$	−35303.6	−1.26720	−0.633600	−	0.773661i	$-0.718424\pi$
$919$	−35303.6	−1.26720	−0.633600	+	0.773661i	$0.718424\pi$
$920$	−57416.0	−2.05756
$921$	−28178.3	−1.00815
$922$	−2630.06	−0.0939440
$923$	−33289.4	−1.18714
$924$	−8265.72	−0.294288
$925$	−14167.8	−0.503606
$926$	−19465.2	−0.690785
$927$	−2435.91	−0.0863061
$928$	1693.30	0.0598978
$929$	24408.1	0.862005	0.431002	−	0.902351i	$-0.358160\pi$
$929$	24408.1	0.862005	0.431002	+	0.902351i	$0.358160\pi$
$930$	10419.9	0.367401
$931$	−3510.47	−0.123578
$932$	60792.4	2.13661
$933$	38527.9	1.35193
$934$	−29376.1	−1.02914
$935$	2836.54	0.0992135
$936$	9331.85	0.325877
$937$	−2380.23	−0.0829869	−0.0414935	−	0.999139i	$-0.513212\pi$
$937$	−2380.23	−0.0829869	−0.0414935	+	0.999139i	$0.513212\pi$
$938$	8955.64	0.311740
$939$	25524.0	0.887054
$940$	42523.0	1.47548
$941$	−3998.86	−0.138532	−0.0692662	−	0.997598i	$-0.522066\pi$
$941$	−3998.86	−0.138532	−0.0692662	+	0.997598i	$0.522066\pi$
$942$	3900.58	0.134913
$943$	−9197.21	−0.317606
$944$	195501.	6.74047
$945$	5729.63	0.197233
$946$	29911.9	1.02803
$947$	20329.1	0.697579	0.348790	−	0.937201i	$-0.386593\pi$
$947$	20329.1	0.697579	0.348790	+	0.937201i	$0.386593\pi$
$948$	133555.	4.57558
$949$	32452.8	1.11008
$950$	36010.9	1.22984
$951$	41722.8	1.42266
$952$	22964.6	0.781813
$953$	−53307.8	−1.81197	−0.905986	−	0.423307i	$-0.860869\pi$
$953$	−53307.8	−1.81197	−0.905986	+	0.423307i	$0.860869\pi$
$954$	6277.67	0.213047
$955$	−5108.94	−0.173111
$956$	−9730.79	−0.329201
$957$	−152.372	−0.00514680
$958$	56071.7	1.89102
$959$	−4215.29	−0.141938
$960$	−44834.8	−1.50733
$961$	−25250.3	−0.847581
$962$	−64167.3	−2.15056
$963$	187.654	0.00627941
$964$	−11735.4	−0.392087
$965$	803.640	0.0268084
$966$	26756.7	0.891183
$967$	11020.3	0.366483	0.183241	−	0.983068i	$-0.441341\pi$
$967$	11020.3	0.366483	0.183241	+	0.983068i	$0.441341\pi$
$968$	8732.98	0.289967
$969$	16391.1	0.543403
$970$	17238.3	0.570606
$971$	38567.0	1.27464	0.637319	−	0.770600i	$-0.280043\pi$
$971$	38567.0	1.27464	0.637319	+	0.770600i	$0.280043\pi$
$972$	−9953.26	−0.328447
$973$	−4000.16	−0.131798
$974$	−1996.65	−0.0656846
$975$	−36264.9	−1.19119
$976$	−28698.6	−0.941208
$977$	9439.72	0.309113	0.154556	−	0.987984i	$-0.450605\pi$
$977$	9439.72	0.309113	0.154556	+	0.987984i	$0.450605\pi$
$978$	60018.7	1.96236
$979$	−6933.28	−0.226342
$980$	5928.48	0.193243
$981$	−1333.58	−0.0434026
$982$	−11049.7	−0.359074
$983$	−25415.8	−0.824657	−0.412328	−	0.911035i	$-0.635284\pi$
$983$	−25415.8	−0.824657	−0.412328	+	0.911035i	$0.635284\pi$
$984$	−23825.6	−0.771882
$985$	17422.2	0.563569
$986$	677.449	0.0218807
$987$	−12383.1	−0.399350
$988$	118606.	3.81920
$989$	−70414.2	−2.26394
$990$	−562.896	−0.0180707
$991$	−28122.5	−0.901454	−0.450727	−	0.892662i	$-0.648835\pi$
$991$	−28122.5	−0.901454	−0.450727	+	0.892662i	$0.648835\pi$
$992$	−41460.9	−1.32700
$993$	2513.28	0.0803188
$994$	16256.9	0.518749
$995$	−31604.9	−1.00698
$996$	−57256.8	−1.82154
$997$	−56082.5	−1.78150	−0.890748	−	0.454497i	$-0.849819\pi$
$997$	−56082.5	−1.78150	−0.890748	+	0.454497i	$0.849819\pi$
$998$	−29391.3	−0.932228
$999$	22023.8	0.697499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.4.a.e.1.5	✓	5
3.2	odd	2		693.4.a.o.1.1		5
4.3	odd	2		1232.4.a.y.1.4		5
5.4	even	2		1925.4.a.r.1.1		5
7.6	odd	2		539.4.a.h.1.5		5
11.10	odd	2		847.4.a.f.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.4.a.e.1.5	✓	5	1.1	even	1	trivial
539.4.a.h.1.5		5	7.6	odd	2
693.4.a.o.1.1		5	3.2	odd	2
847.4.a.f.1.1		5	11.10	odd	2
1232.4.a.y.1.4		5	4.3	odd	2
1925.4.a.r.1.1		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)

\(f(q)\)	\(=\)	\(q+5.41547 q^{2} -5.03332 q^{3} +21.3273 q^{4} +5.67299 q^{5} -27.2578 q^{6} +7.00000 q^{7} +72.1734 q^{8} -1.66567 q^{9} +O(q^{10})\)	\(q+5.41547 q^{2} -5.03332 q^{3} +21.3273 q^{4} +5.67299 q^{5} -27.2578 q^{6} +7.00000 q^{7} +72.1734 q^{8} -1.66567 q^{9} +30.7219 q^{10} +11.0000 q^{11} -107.347 q^{12} -77.6253 q^{13} +37.9083 q^{14} -28.5540 q^{15} +220.234 q^{16} +45.4552 q^{17} -9.02036 q^{18} -71.6423 q^{19} +120.989 q^{20} -35.2333 q^{21} +59.5701 q^{22} -140.231 q^{23} -363.272 q^{24} -92.8172 q^{25} -420.377 q^{26} +144.284 q^{27} +149.291 q^{28} +2.75206 q^{29} -154.633 q^{30} -67.3849 q^{31} +615.284 q^{32} -55.3665 q^{33} +246.161 q^{34} +39.7109 q^{35} -35.5241 q^{36} +152.642 q^{37} -387.976 q^{38} +390.713 q^{39} +409.439 q^{40} +65.5861 q^{41} -190.804 q^{42} +502.130 q^{43} +234.600 q^{44} -9.44931 q^{45} -759.416 q^{46} +351.461 q^{47} -1108.51 q^{48} +49.0000 q^{49} -502.648 q^{50} -228.791 q^{51} -1655.53 q^{52} -695.945 q^{53} +781.363 q^{54} +62.4029 q^{55} +505.214 q^{56} +360.599 q^{57} +14.9037 q^{58} +887.694 q^{59} -608.979 q^{60} -130.309 q^{61} -364.921 q^{62} -11.6597 q^{63} +1570.18 q^{64} -440.367 q^{65} -299.836 q^{66} +236.245 q^{67} +969.435 q^{68} +705.828 q^{69} +215.053 q^{70} +428.848 q^{71} -120.217 q^{72} -418.070 q^{73} +826.630 q^{74} +467.179 q^{75} -1527.93 q^{76} +77.0000 q^{77} +2115.89 q^{78} -1244.14 q^{79} +1249.39 q^{80} -681.253 q^{81} +355.179 q^{82} +533.380 q^{83} -751.429 q^{84} +257.867 q^{85} +2719.27 q^{86} -13.8520 q^{87} +793.907 q^{88} -630.298 q^{89} -51.1724 q^{90} -543.377 q^{91} -2990.75 q^{92} +339.170 q^{93} +1903.32 q^{94} -406.426 q^{95} -3096.92 q^{96} +561.108 q^{97} +265.358 q^{98} -18.3223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9}+O(q^{10}) \) 5 * q + q^2 + 2 * q^3 + 45 * q^4 - 24 * q^5 + 4 * q^6 + 35 * q^7 + 57 * q^8 + 63 * q^9	\( 5 q + q^{2} + 2 q^{3} + 45 q^{4} - 24 q^{5} + 4 q^{6} + 35 q^{7} + 57 q^{8} + 63 q^{9} - 10 q^{10} + 55 q^{11} + 24 q^{12} - 50 q^{13} + 7 q^{14} - 146 q^{15} + 433 q^{16} + 222 q^{17} + 245 q^{18} + 160 q^{19} - 430 q^{20} + 14 q^{21} + 11 q^{22} + 54 q^{23} - 220 q^{24} + 125 q^{25} - 1026 q^{26} + 110 q^{27} + 315 q^{28} + 14 q^{29} - 1080 q^{30} - 34 q^{31} - 583 q^{32} + 22 q^{33} - 750 q^{34} - 168 q^{35} - 411 q^{36} + 1044 q^{37} - 156 q^{38} - 124 q^{39} + 1158 q^{40} - 114 q^{41} + 28 q^{42} + 672 q^{43} + 495 q^{44} - 530 q^{45} - 1224 q^{46} - 292 q^{47} - 1652 q^{48} + 245 q^{49} + 1143 q^{50} + 768 q^{51} - 914 q^{52} - 710 q^{53} + 1608 q^{54} - 264 q^{55} + 399 q^{56} + 2012 q^{57} - 810 q^{58} + 270 q^{59} - 3068 q^{60} + 138 q^{61} - 480 q^{62} + 441 q^{63} + 3001 q^{64} - 196 q^{65} + 44 q^{66} + 1942 q^{67} + 3130 q^{68} - 1306 q^{69} - 70 q^{70} - 278 q^{71} + 3565 q^{72} - 338 q^{73} + 462 q^{74} + 3960 q^{75} - 1332 q^{76} + 385 q^{77} + 280 q^{78} + 576 q^{79} - 1602 q^{80} - 1439 q^{81} + 1386 q^{82} + 1644 q^{83} + 168 q^{84} + 360 q^{85} + 2020 q^{86} - 1956 q^{87} + 627 q^{88} - 3656 q^{89} - 6162 q^{90} - 350 q^{91} - 748 q^{92} - 1442 q^{93} + 976 q^{94} - 1276 q^{95} - 4508 q^{96} + 692 q^{97} + 49 q^{98} + 693 q^{99}+O(q^{100}) \) 5 * q + q^2 + 2 * q^3 + 45 * q^4 - 24 * q^5 + 4 * q^6 + 35 * q^7 + 57 * q^8 + 63 * q^9 - 10 * q^10 + 55 * q^11 + 24 * q^12 - 50 * q^13 + 7 * q^14 - 146 * q^15 + 433 * q^16 + 222 * q^17 + 245 * q^18 + 160 * q^19 - 430 * q^20 + 14 * q^21 + 11 * q^22 + 54 * q^23 - 220 * q^24 + 125 * q^25 - 1026 * q^26 + 110 * q^27 + 315 * q^28 + 14 * q^29 - 1080 * q^30 - 34 * q^31 - 583 * q^32 + 22 * q^33 - 750 * q^34 - 168 * q^35 - 411 * q^36 + 1044 * q^37 - 156 * q^38 - 124 * q^39 + 1158 * q^40 - 114 * q^41 + 28 * q^42 + 672 * q^43 + 495 * q^44 - 530 * q^45 - 1224 * q^46 - 292 * q^47 - 1652 * q^48 + 245 * q^49 + 1143 * q^50 + 768 * q^51 - 914 * q^52 - 710 * q^53 + 1608 * q^54 - 264 * q^55 + 399 * q^56 + 2012 * q^57 - 810 * q^58 + 270 * q^59 - 3068 * q^60 + 138 * q^61 - 480 * q^62 + 441 * q^63 + 3001 * q^64 - 196 * q^65 + 44 * q^66 + 1942 * q^67 + 3130 * q^68 - 1306 * q^69 - 70 * q^70 - 278 * q^71 + 3565 * q^72 - 338 * q^73 + 462 * q^74 + 3960 * q^75 - 1332 * q^76 + 385 * q^77 + 280 * q^78 + 576 * q^79 - 1602 * q^80 - 1439 * q^81 + 1386 * q^82 + 1644 * q^83 + 168 * q^84 + 360 * q^85 + 2020 * q^86 - 1956 * q^87 + 627 * q^88 - 3656 * q^89 - 6162 * q^90 - 350 * q^91 - 748 * q^92 - 1442 * q^93 + 976 * q^94 - 1276 * q^95 - 4508 * q^96 + 692 * q^97 + 49 * q^98 + 693 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more