LMFDB - Embedded newform 74.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [74,4,Mod(1,74)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(74, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("74.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 74 = 2 \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	74.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:	$4.36614134042$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.15629.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 26x - 45 $ x^3 - 26*x - 45
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			1.3
Root			$5.80916$ of defining polynomial
Character	$\chi$	$=$	74.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +6.12805 q^{3} +4.00000 q^{4} +18.7464 q^{5} -12.2561 q^{6} -21.9178 q^{7} -8.00000 q^{8} +10.5531 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +6.12805 q^{3} +4.00000 q^{4} +18.7464 q^{5} -12.2561 q^{6} -21.9178 q^{7} -8.00000 q^{8} +10.5531 q^{9} -37.4928 q^{10} +57.7703 q^{11} +24.5122 q^{12} -18.6447 q^{13} +43.8355 q^{14} +114.879 q^{15} +16.0000 q^{16} -44.5600 q^{17} -21.1061 q^{18} +128.560 q^{19} +74.9855 q^{20} -134.313 q^{21} -115.541 q^{22} -112.094 q^{23} -49.0244 q^{24} +226.427 q^{25} +37.2894 q^{26} -100.788 q^{27} -87.6711 q^{28} -127.235 q^{29} -229.758 q^{30} +38.8369 q^{31} -32.0000 q^{32} +354.019 q^{33} +89.1199 q^{34} -410.879 q^{35} +42.2122 q^{36} -37.0000 q^{37} -257.120 q^{38} -114.256 q^{39} -149.971 q^{40} -438.665 q^{41} +268.627 q^{42} -256.005 q^{43} +231.081 q^{44} +197.832 q^{45} +224.187 q^{46} +275.094 q^{47} +98.0489 q^{48} +137.388 q^{49} -452.854 q^{50} -273.066 q^{51} -74.5788 q^{52} +0.358276 q^{53} +201.576 q^{54} +1082.98 q^{55} +175.342 q^{56} +787.822 q^{57} +254.469 q^{58} +698.000 q^{59} +459.515 q^{60} -494.971 q^{61} -77.6738 q^{62} -231.299 q^{63} +64.0000 q^{64} -349.520 q^{65} -708.039 q^{66} -525.144 q^{67} -178.240 q^{68} -686.915 q^{69} +821.758 q^{70} -1014.99 q^{71} -84.4244 q^{72} -14.4727 q^{73} +74.0000 q^{74} +1387.56 q^{75} +514.240 q^{76} -1266.19 q^{77} +228.511 q^{78} +980.168 q^{79} +299.942 q^{80} -902.566 q^{81} +877.330 q^{82} -273.659 q^{83} -537.253 q^{84} -835.338 q^{85} +512.010 q^{86} -779.701 q^{87} -462.162 q^{88} +559.670 q^{89} -395.663 q^{90} +408.650 q^{91} -448.374 q^{92} +237.995 q^{93} -550.189 q^{94} +2410.03 q^{95} -196.098 q^{96} +77.1825 q^{97} -274.777 q^{98} +609.653 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{2} + 4 q^{3} + 12 q^{4} + 7 q^{5} - 8 q^{6} + 7 q^{7} - 24 q^{8} + 43 q^{9}+O(q^{10}) $ 3 * q - 6 * q^2 + 4 * q^3 + 12 * q^4 + 7 * q^5 - 8 * q^6 + 7 * q^7 - 24 * q^8 + 43 * q^9	\( 3 q - 6 q^{2} + 4 q^{3} + 12 q^{4} + 7 q^{5} - 8 q^{6} + 7 q^{7} - 24 q^{8} + 43 q^{9} - 14 q^{10} + 74 q^{11} + 16 q^{12} + 107 q^{13} - 14 q^{14} + 190 q^{15} + 48 q^{16} + 24 q^{17} - 86 q^{18} + 228 q^{19} + 28 q^{20} - 27 q^{21} - 148 q^{22} - 149 q^{23} - 32 q^{24} + 92 q^{25} - 214 q^{26} - 257 q^{27} + 28 q^{28} - 325 q^{29} - 380 q^{30} + 23 q^{31} - 96 q^{32} - 97 q^{33} - 48 q^{34} - 478 q^{35} + 172 q^{36} - 111 q^{37} - 456 q^{38} + 39 q^{39} - 56 q^{40} - 464 q^{41} + 54 q^{42} + 588 q^{43} + 296 q^{44} - 126 q^{45} + 298 q^{46} + 155 q^{47} + 64 q^{48} + 96 q^{49} - 184 q^{50} + 310 q^{51} + 428 q^{52} - 579 q^{53} + 514 q^{54} + 665 q^{55} - 56 q^{56} + 26 q^{57} + 650 q^{58} + 1258 q^{59} + 760 q^{60} + 291 q^{61} - 46 q^{62} - 56 q^{63} + 192 q^{64} - 874 q^{65} + 194 q^{66} - 127 q^{67} + 96 q^{68} + 399 q^{69} + 956 q^{70} - 201 q^{71} - 344 q^{72} + 42 q^{73} + 222 q^{74} + 795 q^{75} + 912 q^{76} - 1743 q^{77} - 78 q^{78} + 413 q^{79} + 112 q^{80} - 2329 q^{81} + 928 q^{82} - 1037 q^{83} - 108 q^{84} - 750 q^{85} - 1176 q^{86} - 1465 q^{87} - 592 q^{88} + 660 q^{89} + 252 q^{90} + 2696 q^{91} - 596 q^{92} - 2130 q^{93} - 310 q^{94} + 1338 q^{95} - 128 q^{96} + 1384 q^{97} - 192 q^{98} + 1796 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 4 * q^3 + 12 * q^4 + 7 * q^5 - 8 * q^6 + 7 * q^7 - 24 * q^8 + 43 * q^9 - 14 * q^10 + 74 * q^11 + 16 * q^12 + 107 * q^13 - 14 * q^14 + 190 * q^15 + 48 * q^16 + 24 * q^17 - 86 * q^18 + 228 * q^19 + 28 * q^20 - 27 * q^21 - 148 * q^22 - 149 * q^23 - 32 * q^24 + 92 * q^25 - 214 * q^26 - 257 * q^27 + 28 * q^28 - 325 * q^29 - 380 * q^30 + 23 * q^31 - 96 * q^32 - 97 * q^33 - 48 * q^34 - 478 * q^35 + 172 * q^36 - 111 * q^37 - 456 * q^38 + 39 * q^39 - 56 * q^40 - 464 * q^41 + 54 * q^42 + 588 * q^43 + 296 * q^44 - 126 * q^45 + 298 * q^46 + 155 * q^47 + 64 * q^48 + 96 * q^49 - 184 * q^50 + 310 * q^51 + 428 * q^52 - 579 * q^53 + 514 * q^54 + 665 * q^55 - 56 * q^56 + 26 * q^57 + 650 * q^58 + 1258 * q^59 + 760 * q^60 + 291 * q^61 - 46 * q^62 - 56 * q^63 + 192 * q^64 - 874 * q^65 + 194 * q^66 - 127 * q^67 + 96 * q^68 + 399 * q^69 + 956 * q^70 - 201 * q^71 - 344 * q^72 + 42 * q^73 + 222 * q^74 + 795 * q^75 + 912 * q^76 - 1743 * q^77 - 78 * q^78 + 413 * q^79 + 112 * q^80 - 2329 * q^81 + 928 * q^82 - 1037 * q^83 - 108 * q^84 - 750 * q^85 - 1176 * q^86 - 1465 * q^87 - 592 * q^88 + 660 * q^89 + 252 * q^90 + 2696 * q^91 - 596 * q^92 - 2130 * q^93 - 310 * q^94 + 1338 * q^95 - 128 * q^96 + 1384 * q^97 - 192 * q^98 + 1796 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	6.12805	1.17934	0.589672	−	0.807643i	$-0.299257\pi$
$3$	6.12805	1.17934	0.589672	+	0.807643i	$0.299257\pi$
$4$	4.00000	0.500000
$5$	18.7464	1.67673	0.838364	−	0.545111i	$-0.183513\pi$
$5$	18.7464	1.67673	0.838364	+	0.545111i	$0.183513\pi$
$6$	−12.2561	−0.833923
$7$	−21.9178	−1.18345	−0.591724	−	0.806141i	$-0.701552\pi$
$7$	−21.9178	−1.18345	−0.591724	+	0.806141i	$0.701552\pi$
$8$	−8.00000	−0.353553
$9$	10.5531	0.390854
$10$	−37.4928	−1.18563
$11$	57.7703	1.58349	0.791745	−	0.610852i	$-0.209173\pi$
$11$	57.7703	1.58349	0.791745	+	0.610852i	$0.209173\pi$
$12$	24.5122	0.589672
$13$	−18.6447	−0.397777	−0.198889	−	0.980022i	$-0.563733\pi$
$13$	−18.6447	−0.397777	−0.198889	+	0.980022i	$0.563733\pi$
$14$	43.8355	0.836824
$15$	114.879	1.97744
$16$	16.0000	0.250000
$17$	−44.5600	−0.635728	−0.317864	−	0.948136i	$-0.602966\pi$
$17$	−44.5600	−0.635728	−0.317864	+	0.948136i	$0.602966\pi$
$18$	−21.1061	−0.276375
$19$	128.560	1.55230	0.776149	−	0.630549i	$-0.217170\pi$
$19$	128.560	1.55230	0.776149	+	0.630549i	$0.217170\pi$
$20$	74.9855	0.838364
$21$	−134.313	−1.39569
$22$	−115.541	−1.11970
$23$	−112.094	−1.01622	−0.508111	−	0.861291i	$-0.669656\pi$
$23$	−112.094	−1.01622	−0.508111	+	0.861291i	$0.669656\pi$
$24$	−49.0244	−0.416961
$25$	226.427	1.81141
$26$	37.2894	0.281271
$27$	−100.788	−0.718393
$28$	−87.6711	−0.591724
$29$	−127.235	−0.814721	−0.407360	−	0.913268i	$-0.633551\pi$
$29$	−127.235	−0.814721	−0.407360	+	0.913268i	$0.633551\pi$
$30$	−229.758	−1.39826
$31$	38.8369	0.225010	0.112505	−	0.993651i	$-0.464113\pi$
$31$	38.8369	0.225010	0.112505	+	0.993651i	$0.464113\pi$
$32$	−32.0000	−0.176777
$33$	354.019	1.86748
$34$	89.1199	0.449528
$35$	−410.879	−1.98432
$36$	42.2122	0.195427
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	−257.120	−1.09764
$39$	−114.256	−0.469117
$40$	−149.971	−0.592813
$41$	−438.665	−1.67093	−0.835463	−	0.549546i	$-0.814801\pi$
$41$	−438.665	−1.67093	−0.835463	+	0.549546i	$0.814801\pi$
$42$	268.627	0.986904
$43$	−256.005	−0.907915	−0.453958	−	0.891023i	$-0.649988\pi$
$43$	−256.005	−0.907915	−0.453958	+	0.891023i	$0.649988\pi$
$44$	231.081	0.791745
$45$	197.832	0.655355
$46$	224.187	0.718578
$47$	275.094	0.853758	0.426879	−	0.904309i	$-0.359613\pi$
$47$	275.094	0.853758	0.426879	+	0.904309i	$0.359613\pi$
$48$	98.0489	0.294836
$49$	137.388	0.400549
$50$	−452.854	−1.28086
$51$	−273.066	−0.749743
$52$	−74.5788	−0.198889
$53$	0.358276	0.000928548 0	0.000464274	−	1.00000i	$-0.499852\pi$
$53$	0.358276	0.000928548 0	0.000464274		1.00000i	$0.499852\pi$
$54$	201.576	0.507981
$55$	1082.98	2.65508
$56$	175.342	0.418412
$57$	787.822	1.83070
$58$	254.469	0.576094
$59$	698.000	1.54020	0.770101	−	0.637922i	$-0.220206\pi$
$59$	698.000	1.54020	0.770101	+	0.637922i	$0.220206\pi$
$60$	459.515	0.988720
$61$	−494.971	−1.03893	−0.519463	−	0.854493i	$-0.673868\pi$
$61$	−494.971	−1.03893	−0.519463	+	0.854493i	$0.673868\pi$
$62$	−77.6738	−0.159106
$63$	−231.299	−0.462555
$64$	64.0000	0.125000
$65$	−349.520	−0.666964
$66$	−708.039	−1.32051
$67$	−525.144	−0.957561	−0.478781	−	0.877935i	$-0.658921\pi$
$67$	−525.144	−0.957561	−0.478781	+	0.877935i	$0.658921\pi$
$68$	−178.240	−0.317864
$69$	−686.915	−1.19848
$70$	821.758	1.40313
$71$	−1014.99	−1.69659	−0.848294	−	0.529526i	$-0.822370\pi$
$71$	−1014.99	−1.69659	−0.848294	+	0.529526i	$0.822370\pi$
$72$	−84.4244	−0.138188
$73$	−14.4727	−0.0232041	−0.0116021	−	0.999933i	$-0.503693\pi$
$73$	−14.4727	−0.0232041	−0.0116021	+	0.999933i	$0.503693\pi$
$74$	74.0000	0.116248
$75$	1387.56	2.13628
$76$	514.240	0.776149
$77$	−1266.19	−1.87398
$78$	228.511	0.331716
$79$	980.168	1.39592	0.697959	−	0.716138i	$-0.254092\pi$
$79$	980.168	1.39592	0.697959	+	0.716138i	$0.254092\pi$
$80$	299.942	0.419182
$81$	−902.566	−1.23809
$82$	877.330	1.18152
$83$	−273.659	−0.361904	−0.180952	−	0.983492i	$-0.557918\pi$
$83$	−273.659	−0.361904	−0.180952	+	0.983492i	$0.557918\pi$
$84$	−537.253	−0.697847
$85$	−835.338	−1.06594
$86$	512.010	0.641993
$87$	−779.701	−0.960836
$88$	−462.162	−0.559848
$89$	559.670	0.666572	0.333286	−	0.942826i	$-0.391843\pi$
$89$	559.670	0.666572	0.333286	+	0.942826i	$0.391843\pi$
$90$	−395.663	−0.463406
$91$	408.650	0.470749
$92$	−448.374	−0.508111
$93$	237.995	0.265364
$94$	−550.189	−0.603698
$95$	2410.03	2.60278
$96$	−196.098	−0.208481
$97$	77.1825	0.0807907	0.0403953	−	0.999184i	$-0.487138\pi$
$97$	77.1825	0.0807907	0.0403953	+	0.999184i	$0.487138\pi$
$98$	−274.777	−0.283231
$99$	609.653	0.618913
$100$	905.707	0.905707
$101$	−1786.86	−1.76038	−0.880192	−	0.474617i	$-0.842586\pi$
$101$	−1786.86	−1.76038	−0.880192	+	0.474617i	$0.842586\pi$
$102$	546.132	0.530148
$103$	405.434	0.387850	0.193925	−	0.981016i	$-0.437878\pi$
$103$	405.434	0.387850	0.193925	+	0.981016i	$0.437878\pi$
$104$	149.158	0.140636
$105$	−2517.89	−2.34020
$106$	−0.716553	−0.000656582 0
$107$	1283.22	1.15938	0.579691	−	0.814837i	$-0.303173\pi$
$107$	1283.22	1.15938	0.579691	+	0.814837i	$0.303173\pi$
$108$	−403.151	−0.359197
$109$	1054.88	0.926965	0.463483	−	0.886106i	$-0.346600\pi$
$109$	1054.88	0.926965	0.463483	+	0.886106i	$0.346600\pi$
$110$	−2165.97	−1.87743
$111$	−226.738	−0.193883
$112$	−350.684	−0.295862
$113$	1565.33	1.30314	0.651568	−	0.758591i	$-0.274112\pi$
$113$	1565.33	1.30314	0.651568	+	0.758591i	$0.274112\pi$
$114$	−1575.64	−1.29450
$115$	−2101.35	−1.70393
$116$	−508.939	−0.407360
$117$	−196.758	−0.155473
$118$	−1396.00	−1.08909
$119$	976.655	0.752351
$120$	−919.031	−0.699130
$121$	2006.40	1.50744
$122$	989.941	0.734632
$123$	−2688.16	−1.97060
$124$	155.348	0.112505
$125$	1901.39	1.36052
$126$	462.599	0.327076
$127$	1514.19	1.05797	0.528986	−	0.848631i	$-0.322572\pi$
$127$	1514.19	1.05797	0.528986	+	0.848631i	$0.322572\pi$
$128$	−128.000	−0.0883883
$129$	−1568.81	−1.07075
$130$	699.041	0.471615
$131$	1487.35	0.991987	0.495993	−	0.868326i	$-0.334804\pi$
$131$	1487.35	0.991987	0.495993	+	0.868326i	$0.334804\pi$
$132$	1416.08	0.933740
$133$	−2817.75	−1.83706
$134$	1050.29	0.677098
$135$	−1889.41	−1.20455
$136$	356.480	0.224764
$137$	608.789	0.379652	0.189826	−	0.981818i	$-0.439208\pi$
$137$	608.789	0.379652	0.189826	+	0.981818i	$0.439208\pi$
$138$	1373.83	0.847451
$139$	1184.95	0.723067	0.361534	−	0.932359i	$-0.382253\pi$
$139$	1184.95	0.723067	0.361534	+	0.932359i	$0.382253\pi$
$140$	−1643.52	−0.992160
$141$	1685.79	1.00687
$142$	2029.99	1.19967
$143$	−1077.11	−0.629876
$144$	168.849	0.0977135
$145$	−2385.19	−1.36606
$146$	28.9454	0.0164078
$147$	841.923	0.472385
$148$	−148.000	−0.0821995
$149$	−747.940	−0.411233	−0.205616	−	0.978633i	$-0.565920\pi$
$149$	−747.940	−0.411233	−0.205616	+	0.978633i	$0.565920\pi$
$150$	−2775.11	−1.51058
$151$	1272.79	0.685946	0.342973	−	0.939345i	$-0.388566\pi$
$151$	1272.79	0.685946	0.342973	+	0.939345i	$0.388566\pi$
$152$	−1028.48	−0.548821
$153$	−470.244	−0.248477
$154$	2532.39	1.32510
$155$	728.051	0.377280
$156$	−457.023	−0.234558
$157$	1995.46	1.01436	0.507181	−	0.861839i	$-0.330687\pi$
$157$	1995.46	1.01436	0.507181	+	0.861839i	$0.330687\pi$
$158$	−1960.34	−0.987063
$159$	2.19554	0.00109508
$160$	−599.884	−0.296406
$161$	2456.84	1.20265
$162$	1805.13	0.875460
$163$	3323.18	1.59688	0.798440	−	0.602075i	$-0.205659\pi$
$163$	3323.18	1.59688	0.798440	+	0.602075i	$0.205659\pi$
$164$	−1754.66	−0.835463
$165$	6636.58	3.13126
$166$	547.319	0.255905
$167$	−3663.64	−1.69761	−0.848804	−	0.528707i	$-0.822677\pi$
$167$	−3663.64	−1.69761	−0.848804	+	0.528707i	$0.822677\pi$
$168$	1074.51	0.493452
$169$	−1849.38	−0.841773
$170$	1670.68	0.753735
$171$	1356.70	0.606722
$172$	−1024.02	−0.453958
$173$	−3094.32	−1.35987	−0.679933	−	0.733274i	$-0.737991\pi$
$173$	−3094.32	−1.35987	−0.679933	+	0.733274i	$0.737991\pi$
$174$	1559.40	0.679414
$175$	−4962.77	−2.14372
$176$	924.324	0.395872
$177$	4277.38	1.81643
$178$	−1119.34	−0.471338
$179$	1247.66	0.520977	0.260488	−	0.965477i	$-0.416116\pi$
$179$	1247.66	0.520977	0.260488	+	0.965477i	$0.416116\pi$
$180$	791.326	0.327678
$181$	2860.98	1.17489	0.587445	−	0.809264i	$-0.300134\pi$
$181$	2860.98	1.17489	0.587445	+	0.809264i	$0.300134\pi$
$182$	−817.300	−0.332870
$183$	−3033.21	−1.22525
$184$	896.748	0.359289
$185$	−693.616	−0.275652
$186$	−475.989	−0.187641
$187$	−2574.24	−1.00667
$188$	1100.38	0.426879
$189$	2209.04	0.850181
$190$	−4820.07	−1.84044
$191$	−1176.94	−0.445865	−0.222932	−	0.974834i	$-0.571563\pi$
$191$	−1176.94	−0.445865	−0.222932	+	0.974834i	$0.571563\pi$
$192$	392.195	0.147418
$193$	1983.29	0.739690	0.369845	−	0.929094i	$-0.379411\pi$
$193$	1983.29	0.739690	0.369845	+	0.929094i	$0.379411\pi$
$194$	−154.365	−0.0571276
$195$	−2141.88	−0.786581
$196$	549.553	0.200275
$197$	−2747.71	−0.993737	−0.496869	−	0.867826i	$-0.665517\pi$
$197$	−2747.71	−0.993737	−0.496869	+	0.867826i	$0.665517\pi$
$198$	−1219.31	−0.437638
$199$	1669.49	0.594707	0.297354	−	0.954767i	$-0.403896\pi$
$199$	1669.49	0.594707	0.297354	+	0.954767i	$0.403896\pi$
$200$	−1811.41	−0.640432
$201$	−3218.11	−1.12929
$202$	3573.71	1.24478
$203$	2788.70	0.964179
$204$	−1092.26	−0.374871
$205$	−8223.39	−2.80169
$206$	−810.867	−0.274251
$207$	−1182.93	−0.397195
$208$	−298.315	−0.0994444
$209$	7426.94	2.45805
$210$	5035.78	1.65477
$211$	1864.44	0.608308	0.304154	−	0.952623i	$-0.401626\pi$
$211$	1864.44	0.608308	0.304154	+	0.952623i	$0.401626\pi$
$212$	1.43311	0.000464274 0
$213$	−6219.94	−2.00086
$214$	−2566.45	−0.819806
$215$	−4799.16	−1.52233
$216$	806.302	0.253990
$217$	−851.218	−0.266288
$218$	−2109.76	−0.655463
$219$	−88.6896	−0.0273657
$220$	4331.93	1.32754
$221$	830.807	0.252878
$222$	453.476	0.137096
$223$	−2200.54	−0.660803	−0.330401	−	0.943841i	$-0.607184\pi$
$223$	−2200.54	−0.660803	−0.330401	+	0.943841i	$0.607184\pi$
$224$	701.368	0.209206
$225$	2389.49	0.707998
$226$	−3130.67	−0.921456
$227$	−2020.06	−0.590643	−0.295321	−	0.955398i	$-0.595427\pi$
$227$	−2020.06	−0.590643	−0.295321	+	0.955398i	$0.595427\pi$
$228$	3151.29	0.915348
$229$	−1993.14	−0.575154	−0.287577	−	0.957758i	$-0.592850\pi$
$229$	−1993.14	−0.575154	−0.287577	+	0.957758i	$0.592850\pi$
$230$	4202.70	1.20486
$231$	−7759.31	−2.21007
$232$	1017.88	0.288047
$233$	−2822.20	−0.793512	−0.396756	−	0.917924i	$-0.629864\pi$
$233$	−2822.20	−0.793512	−0.396756	+	0.917924i	$0.629864\pi$
$234$	393.517	0.109936
$235$	5157.02	1.43152
$236$	2792.00	0.770101
$237$	6006.52	1.64627
$238$	−1953.31	−0.531993
$239$	3010.33	0.814735	0.407368	−	0.913264i	$-0.366447\pi$
$239$	3010.33	0.814735	0.407368	+	0.913264i	$0.366447\pi$
$240$	1838.06	0.494360
$241$	2988.16	0.798689	0.399344	−	0.916801i	$-0.369238\pi$
$241$	2988.16	0.798689	0.399344	+	0.916801i	$0.369238\pi$
$242$	−4012.80	−1.06592
$243$	−2809.70	−0.741738
$244$	−1979.88	−0.519463
$245$	2575.53	0.671612
$246$	5376.33	1.39342
$247$	−2396.96	−0.617469
$248$	−310.695	−0.0795531
$249$	−1677.00	−0.426809
$250$	−3802.77	−0.962034
$251$	687.024	0.172767	0.0863836	−	0.996262i	$-0.472469\pi$
$251$	687.024	0.172767	0.0863836	+	0.996262i	$0.472469\pi$
$252$	−925.197	−0.231278
$253$	−6475.67	−1.60918
$254$	−3028.38	−0.748099
$255$	−5119.00	−1.25711
$256$	256.000	0.0625000
$257$	−2247.34	−0.545468	−0.272734	−	0.962089i	$-0.587928\pi$
$257$	−2247.34	−0.545468	−0.272734	+	0.962089i	$0.587928\pi$
$258$	3137.62	0.757131
$259$	810.957	0.194558
$260$	−1398.08	−0.333482
$261$	−1342.71	−0.318437
$262$	−2974.70	−0.701441
$263$	582.840	0.136652	0.0683259	−	0.997663i	$-0.478234\pi$
$263$	582.840	0.136652	0.0683259	+	0.997663i	$0.478234\pi$
$264$	−2832.15	−0.660254
$265$	6.71638	0.00155692
$266$	5635.49	1.29900
$267$	3429.69	0.786118
$268$	−2100.58	−0.478781
$269$	−1121.78	−0.254260	−0.127130	−	0.991886i	$-0.540577\pi$
$269$	−1121.78	−0.254260	−0.127130	+	0.991886i	$0.540577\pi$
$270$	3778.81	0.851745
$271$	−52.7185	−0.0118171	−0.00590853	−	0.999983i	$-0.501881\pi$
$271$	−52.7185	−0.0118171	−0.00590853	+	0.999983i	$0.501881\pi$
$272$	−712.959	−0.158932
$273$	2504.23	0.555175
$274$	−1217.58	−0.268454
$275$	13080.7	2.86836
$276$	−2747.66	−0.599238
$277$	−3642.45	−0.790084	−0.395042	−	0.918663i	$-0.629270\pi$
$277$	−3642.45	−0.790084	−0.395042	+	0.918663i	$0.629270\pi$
$278$	−2369.90	−0.511286
$279$	409.848	0.0879460
$280$	3287.03	0.701563
$281$	6839.66	1.45203	0.726014	−	0.687680i	$-0.241371\pi$
$281$	6839.66	1.45203	0.726014	+	0.687680i	$0.241371\pi$
$282$	−3371.59	−0.711968
$283$	5729.18	1.20341	0.601704	−	0.798719i	$-0.294489\pi$
$283$	5729.18	1.20341	0.601704	+	0.798719i	$0.294489\pi$
$284$	−4059.98	−0.848294
$285$	14768.8	3.06958
$286$	2154.22	0.445390
$287$	9614.56	1.97745
$288$	−337.698	−0.0690939
$289$	−2927.41	−0.595850
$290$	4770.38	0.965953
$291$	472.979	0.0952801
$292$	−57.8909	−0.0116021
$293$	4372.85	0.871894	0.435947	−	0.899972i	$-0.356414\pi$
$293$	4372.85	0.871894	0.435947	+	0.899972i	$0.356414\pi$
$294$	−1683.85	−0.334027
$295$	13085.0	2.58250
$296$	296.000	0.0581238
$297$	−5822.54	−1.13757
$298$	1495.88	0.290785
$299$	2089.95	0.404230
$300$	5550.22	1.06814
$301$	5611.05	1.07447
$302$	−2545.57	−0.485037
$303$	−10950.0	−2.07610
$304$	2056.96	0.388075
$305$	−9278.91	−1.74200
$306$	940.487	0.175700
$307$	−5480.70	−1.01889	−0.509446	−	0.860502i	$-0.670150\pi$
$307$	−5480.70	−1.01889	−0.509446	+	0.860502i	$0.670150\pi$
$308$	−5064.78	−0.936989
$309$	2484.52	0.457409
$310$	−1456.10	−0.266778
$311$	−4414.78	−0.804949	−0.402475	−	0.915431i	$-0.631850\pi$
$311$	−4414.78	−0.804949	−0.402475	+	0.915431i	$0.631850\pi$
$312$	914.045	0.165858
$313$	−2574.12	−0.464849	−0.232424	−	0.972614i	$-0.574666\pi$
$313$	−2574.12	−0.464849	−0.232424	+	0.972614i	$0.574666\pi$
$314$	−3990.92	−0.717263
$315$	−4336.03	−0.775579
$316$	3920.67	0.697959
$317$	8967.38	1.58883	0.794414	−	0.607377i	$-0.207778\pi$
$317$	8967.38	1.58883	0.794414	+	0.607377i	$0.207778\pi$
$318$	−4.39107	−0.000774337 0
$319$	−7350.38	−1.29010
$320$	1199.77	0.209591
$321$	7863.66	1.36731
$322$	−4913.68	−0.850400
$323$	−5728.63	−0.986840
$324$	−3610.26	−0.619044
$325$	−4221.66	−0.720540
$326$	−6646.36	−1.12916
$327$	6464.37	1.09321
$328$	3509.32	0.590762
$329$	−6029.45	−1.01038
$330$	−13273.2	−2.21413
$331$	−9429.00	−1.56575	−0.782877	−	0.622177i	$-0.786249\pi$
$331$	−9429.00	−1.56575	−0.782877	+	0.622177i	$0.786249\pi$
$332$	−1094.64	−0.180952
$333$	−390.463	−0.0642560
$334$	7327.27	1.20039
$335$	−9844.56	−1.60557
$336$	−2149.01	−0.348923
$337$	7683.47	1.24197	0.620987	−	0.783821i	$-0.286732\pi$
$337$	7683.47	1.24197	0.620987	+	0.783821i	$0.286732\pi$
$338$	3698.75	0.595223
$339$	9592.46	1.53685
$340$	−3341.35	−0.532971
$341$	2243.62	0.356301
$342$	−2713.40	−0.429017
$343$	4506.55	0.709419
$344$	2048.04	0.320997
$345$	−12877.2	−2.00952
$346$	6188.64	0.961570
$347$	−3186.70	−0.493000	−0.246500	−	0.969143i	$-0.579281\pi$
$347$	−3186.70	−0.493000	−0.246500	+	0.969143i	$0.579281\pi$
$348$	−3118.81	−0.480418
$349$	−1658.03	−0.254304	−0.127152	−	0.991883i	$-0.540584\pi$
$349$	−1658.03	−0.254304	−0.127152	+	0.991883i	$0.540584\pi$
$350$	9925.54	1.51584
$351$	1879.16	0.285761
$352$	−1848.65	−0.279924
$353$	2121.24	0.319837	0.159918	−	0.987130i	$-0.448877\pi$
$353$	2121.24	0.319837	0.159918	+	0.987130i	$0.448877\pi$
$354$	−8554.77	−1.28441
$355$	−19027.5	−2.84471
$356$	2238.68	0.333286
$357$	5984.99	0.887281
$358$	−2495.33	−0.368386
$359$	−6278.89	−0.923084	−0.461542	−	0.887118i	$-0.652704\pi$
$359$	−6278.89	−0.923084	−0.461542	+	0.887118i	$0.652704\pi$
$360$	−1582.65	−0.231703
$361$	9668.66	1.40963
$362$	−5721.96	−0.830773
$363$	12295.3	1.77779
$364$	1634.60	0.235374
$365$	−271.311	−0.0389070
$366$	6066.41	0.866384
$367$	−9201.10	−1.30870	−0.654351	−	0.756191i	$-0.727058\pi$
$367$	−9201.10	−1.30870	−0.654351	+	0.756191i	$0.727058\pi$
$368$	−1793.50	−0.254056
$369$	−4629.26	−0.653088
$370$	1387.23	0.194916
$371$	−7.85262	−0.00109889
$372$	951.978	0.132682
$373$	528.312	0.0733377	0.0366688	−	0.999327i	$-0.488325\pi$
$373$	528.312	0.0733377	0.0366688	+	0.999327i	$0.488325\pi$
$374$	5148.48	0.711822
$375$	11651.8	1.60452
$376$	−2200.75	−0.301849
$377$	2372.25	0.324077
$378$	−4418.09	−0.601169
$379$	−2504.25	−0.339406	−0.169703	−	0.985495i	$-0.554281\pi$
$379$	−2504.25	−0.339406	−0.169703	+	0.985495i	$0.554281\pi$
$380$	9640.14	1.30139
$381$	9279.03	1.24771
$382$	2353.87	0.315274
$383$	−5155.87	−0.687866	−0.343933	−	0.938994i	$-0.611759\pi$
$383$	−5155.87	−0.687866	−0.343933	+	0.938994i	$0.611759\pi$
$384$	−784.391	−0.104240
$385$	−23736.6	−3.14215
$386$	−3966.57	−0.523040
$387$	−2701.63	−0.354862
$388$	308.730	0.0403953
$389$	10222.9	1.33245	0.666224	−	0.745752i	$-0.267910\pi$
$389$	10222.9	1.33245	0.666224	+	0.745752i	$0.267910\pi$
$390$	4283.76	0.556197
$391$	4994.88	0.646041
$392$	−1099.11	−0.141616
$393$	9114.55	1.16989
$394$	5495.42	0.702678
$395$	18374.6	2.34057
$396$	2438.61	0.309457
$397$	6440.88	0.814254	0.407127	−	0.913372i	$-0.366531\pi$
$397$	6440.88	0.814254	0.407127	+	0.913372i	$0.366531\pi$
$398$	−3338.97	−0.420522
$399$	−17267.3	−2.16653
$400$	3622.83	0.452854
$401$	−9165.65	−1.14142	−0.570712	−	0.821150i	$-0.693333\pi$
$401$	−9165.65	−1.14142	−0.570712	+	0.821150i	$0.693333\pi$
$402$	6436.23	0.798532
$403$	−724.102	−0.0895039
$404$	−7147.42	−0.880192
$405$	−16919.8	−2.07593
$406$	−5577.40	−0.681778
$407$	−2137.50	−0.260324
$408$	2184.53	0.265074
$409$	5305.56	0.641426	0.320713	−	0.947176i	$-0.396078\pi$
$409$	5305.56	0.641426	0.320713	+	0.947176i	$0.396078\pi$
$410$	16446.8	1.98109
$411$	3730.69	0.447741
$412$	1621.73	0.193925
$413$	−15298.6	−1.82275
$414$	2365.86	0.280859
$415$	−5130.12	−0.606814
$416$	596.630	0.0703178
$417$	7261.45	0.852745
$418$	−14853.9	−1.73810
$419$	−1097.10	−0.127916	−0.0639582	−	0.997953i	$-0.520372\pi$
$419$	−1097.10	−0.127916	−0.0639582	+	0.997953i	$0.520372\pi$
$420$	−10071.6	−1.17010
$421$	6390.76	0.739826	0.369913	−	0.929066i	$-0.379387\pi$
$421$	6390.76	0.739826	0.369913	+	0.929066i	$0.379387\pi$
$422$	−3728.87	−0.430139
$423$	2903.08	0.333695
$424$	−2.86621	−0.000328291 0
$425$	−10089.6	−1.15157
$426$	12439.9	1.41482
$427$	10848.6	1.22952
$428$	5132.89	0.579691
$429$	−6600.58	−0.742841
$430$	9598.33	1.07645
$431$	14874.6	1.66238	0.831190	−	0.555989i	$-0.187660\pi$
$431$	14874.6	1.66238	0.831190	+	0.555989i	$0.187660\pi$
$432$	−1612.60	−0.179598
$433$	9227.03	1.02407	0.512036	−	0.858964i	$-0.328892\pi$
$433$	9227.03	1.02407	0.512036	+	0.858964i	$0.328892\pi$
$434$	1702.44	0.188294
$435$	−14616.6	−1.61106
$436$	4219.52	0.463483
$437$	−14410.7	−1.57748
$438$	177.379	0.0193505
$439$	2282.58	0.248159	0.124079	−	0.992272i	$-0.460402\pi$
$439$	2282.58	0.248159	0.124079	+	0.992272i	$0.460402\pi$
$440$	−8663.87	−0.938713
$441$	1449.87	0.156556
$442$	−1661.61	−0.178812
$443$	7122.03	0.763833	0.381916	−	0.924197i	$-0.375264\pi$
$443$	7122.03	0.763833	0.381916	+	0.924197i	$0.375264\pi$
$444$	−906.952	−0.0969415
$445$	10491.8	1.11766
$446$	4401.08	0.467258
$447$	−4583.42	−0.484985
$448$	−1402.74	−0.147931
$449$	2321.34	0.243988	0.121994	−	0.992531i	$-0.461071\pi$
$449$	2321.34	0.243988	0.121994	+	0.992531i	$0.461071\pi$
$450$	−4778.99	−0.500630
$451$	−25341.8	−2.64590
$452$	6261.34	0.651568
$453$	7799.70	0.808966
$454$	4040.11	0.417647
$455$	7660.71	0.789318
$456$	−6302.58	−0.647249
$457$	7882.87	0.806882	0.403441	−	0.915006i	$-0.367814\pi$
$457$	7882.87	0.806882	0.403441	+	0.915006i	$0.367814\pi$
$458$	3986.27	0.406695
$459$	4491.10	0.456703
$460$	−8405.39	−0.851964
$461$	−17965.7	−1.81507	−0.907534	−	0.419979i	$-0.862038\pi$
$461$	−17965.7	−1.81507	−0.907534	+	0.419979i	$0.862038\pi$
$462$	15518.6	1.56275
$463$	−15536.2	−1.55946	−0.779729	−	0.626118i	$-0.784643\pi$
$463$	−15536.2	−1.55946	−0.779729	+	0.626118i	$0.784643\pi$
$464$	−2035.76	−0.203680
$465$	4461.54	0.444944
$466$	5644.40	0.561098
$467$	8320.10	0.824428	0.412214	−	0.911087i	$-0.364756\pi$
$467$	8320.10	0.824428	0.412214	+	0.911087i	$0.364756\pi$
$468$	−787.034	−0.0777364
$469$	11510.0	1.13322
$470$	−10314.0	−1.01224
$471$	12228.3	1.19628
$472$	−5584.00	−0.544544
$473$	−14789.5	−1.43767
$474$	−12013.0	−1.16409
$475$	29109.4	2.81186
$476$	3906.62	0.376176
$477$	3.78091	0.000362927 0
$478$	−6020.65	−0.576105
$479$	12756.6	1.21684	0.608418	−	0.793616i	$-0.291804\pi$
$479$	12756.6	1.21684	0.608418	+	0.793616i	$0.291804\pi$
$480$	−3676.12	−0.349565
$481$	689.854	0.0653942
$482$	−5976.31	−0.564758
$483$	15055.7	1.41834
$484$	8025.61	0.753720
$485$	1446.89	0.135464
$486$	5619.40	0.524488
$487$	−10017.3	−0.932087	−0.466044	−	0.884762i	$-0.654321\pi$
$487$	−10017.3	−0.932087	−0.466044	+	0.884762i	$0.654321\pi$
$488$	3959.77	0.367316
$489$	20364.6	1.88327
$490$	−5151.07	−0.474901
$491$	−15622.5	−1.43591	−0.717957	−	0.696088i	$-0.754923\pi$
$491$	−15622.5	−1.43591	−0.717957	+	0.696088i	$0.754923\pi$
$492$	−10752.7	−0.985299
$493$	5669.57	0.517941
$494$	4793.92	0.436617
$495$	11428.8	1.03775
$496$	621.390	0.0562525
$497$	22246.4	2.00782
$498$	3354.00	0.301800
$499$	−13165.1	−1.18107	−0.590534	−	0.807013i	$-0.701083\pi$
$499$	−13165.1	−1.18107	−0.590534	+	0.807013i	$0.701083\pi$
$500$	7605.54	0.680261
$501$	−22451.0	−2.00207
$502$	−1374.05	−0.122165
$503$	7491.62	0.664085	0.332042	−	0.943264i	$-0.392262\pi$
$503$	7491.62	0.664085	0.332042	+	0.943264i	$0.392262\pi$
$504$	1850.39	0.163538
$505$	−33497.1	−2.95168
$506$	12951.3	1.13786
$507$	−11333.1	−0.992741
$508$	6056.75	0.528986
$509$	−19209.6	−1.67279	−0.836397	−	0.548124i	$-0.815342\pi$
$509$	−19209.6	−1.67279	−0.836397	+	0.548124i	$0.815342\pi$
$510$	10238.0	0.888914
$511$	317.209	0.0274609
$512$	−512.000	−0.0441942
$513$	−12957.3	−1.11516
$514$	4494.69	0.385704
$515$	7600.41	0.650319
$516$	−6275.25	−0.535373
$517$	15892.3	1.35192
$518$	−1621.91	−0.137573
$519$	−18962.2	−1.60375
$520$	2796.16	0.235807
$521$	−22521.1	−1.89380	−0.946899	−	0.321530i	$-0.895803\pi$
$521$	−22521.1	−1.89380	−0.946899	+	0.321530i	$0.895803\pi$
$522$	2685.43	0.225169
$523$	6652.48	0.556200	0.278100	−	0.960552i	$-0.410295\pi$
$523$	6652.48	0.556200	0.278100	+	0.960552i	$0.410295\pi$
$524$	5949.40	0.495993
$525$	−30412.1	−2.52818
$526$	−1165.68	−0.0966275
$527$	−1730.57	−0.143045
$528$	5664.31	0.466870
$529$	397.967	0.0327087
$530$	−13.4328	−0.00110091
$531$	7366.03	0.601994
$532$	−11271.0	−0.918532
$533$	8178.78	0.664657
$534$	−6859.38	−0.555870
$535$	24055.8	1.94397
$536$	4201.16	0.338549
$537$	7645.76	0.614411
$538$	2243.56	0.179789
$539$	7936.96	0.634265
$540$	−7557.63	−0.602275
$541$	2547.01	0.202411	0.101205	−	0.994866i	$-0.467730\pi$
$541$	2547.01	0.202411	0.101205	+	0.994866i	$0.467730\pi$
$542$	105.437	0.00835592
$543$	17532.3	1.38560
$544$	1425.92	0.112382
$545$	19775.2	1.55427
$546$	−5008.46	−0.392568
$547$	−12621.3	−0.986556	−0.493278	−	0.869872i	$-0.664201\pi$
$547$	−12621.3	−0.986556	−0.493278	+	0.869872i	$0.664201\pi$
$548$	2435.15	0.189826
$549$	−5223.45	−0.406068
$550$	−26161.5	−2.02823
$551$	−16357.3	−1.26469
$552$	5495.32	0.423726
$553$	−21483.1	−1.65200
$554$	7284.89	0.558674
$555$	−4250.52	−0.325089
$556$	4739.81	0.361534
$557$	−7719.22	−0.587206	−0.293603	−	0.955927i	$-0.594854\pi$
$557$	−7719.22	−0.587206	−0.293603	+	0.955927i	$0.594854\pi$
$558$	−819.696	−0.0621872
$559$	4773.13	0.361148
$560$	−6574.06	−0.496080
$561$	−15775.1	−1.18721
$562$	−13679.3	−1.02674
$563$	16041.0	1.20080	0.600399	−	0.799701i	$-0.295008\pi$
$563$	16041.0	1.20080	0.600399	+	0.799701i	$0.295008\pi$
$564$	6743.17	0.503437
$565$	29344.4	2.18500
$566$	−11458.4	−0.850938
$567$	19782.2	1.46521
$568$	8119.96	0.599834
$569$	−22137.9	−1.63105	−0.815527	−	0.578719i	$-0.803553\pi$
$569$	−22137.9	−1.63105	−0.815527	+	0.578719i	$0.803553\pi$
$570$	−29537.6	−2.17052
$571$	2259.08	0.165569	0.0827843	−	0.996567i	$-0.473619\pi$
$571$	2259.08	0.165569	0.0827843	+	0.996567i	$0.473619\pi$
$572$	−4308.43	−0.314938
$573$	−7212.33	−0.525828
$574$	−19229.1	−1.39827
$575$	−25381.0	−1.84080
$576$	675.395	0.0488567
$577$	−3558.04	−0.256713	−0.128356	−	0.991728i	$-0.540970\pi$
$577$	−3558.04	−0.256713	−0.128356	+	0.991728i	$0.540970\pi$
$578$	5854.82	0.421329
$579$	12153.7	0.872349
$580$	−9540.76	−0.683032
$581$	5998.00	0.428294
$582$	−945.957	−0.0673732
$583$	20.6977	0.00147035
$584$	115.782	0.00820391
$585$	−3688.51	−0.260686
$586$	−8745.71	−0.616522
$587$	−3589.31	−0.252379	−0.126190	−	0.992006i	$-0.540275\pi$
$587$	−3589.31	−0.252379	−0.126190	+	0.992006i	$0.540275\pi$
$588$	3367.69	0.236193
$589$	4992.87	0.349283
$590$	−26170.0	−1.82610
$591$	−16838.1	−1.17196
$592$	−592.000	−0.0410997
$593$	−13372.9	−0.926069	−0.463035	−	0.886340i	$-0.653239\pi$
$593$	−13372.9	−0.926069	−0.463035	+	0.886340i	$0.653239\pi$
$594$	11645.1	0.804382
$595$	18308.7	1.26149
$596$	−2991.76	−0.205616
$597$	10230.7	0.701365
$598$	−4179.90	−0.285834
$599$	18874.1	1.28743	0.643717	−	0.765264i	$-0.277391\pi$
$599$	18874.1	1.28743	0.643717	+	0.765264i	$0.277391\pi$
$600$	−11100.4	−0.755290
$601$	9262.10	0.628634	0.314317	−	0.949318i	$-0.398225\pi$
$601$	9262.10	0.628634	0.314317	+	0.949318i	$0.398225\pi$
$602$	−11222.1	−0.759766
$603$	−5541.88	−0.374266
$604$	5091.14	0.342973
$605$	37612.8	2.52757
$606$	21899.9	1.46802
$607$	14170.9	0.947580	0.473790	−	0.880638i	$-0.342886\pi$
$607$	14170.9	0.947580	0.473790	+	0.880638i	$0.342886\pi$
$608$	−4113.92	−0.274410
$609$	17089.3	1.13710
$610$	18557.8	1.23178
$611$	−5129.05	−0.339606
$612$	−1880.97	−0.124238
$613$	−5229.34	−0.344553	−0.172277	−	0.985049i	$-0.555112\pi$
$613$	−5229.34	−0.344553	−0.172277	+	0.985049i	$0.555112\pi$
$614$	10961.4	0.720466
$615$	−50393.4	−3.30416
$616$	10129.6	0.662551
$617$	22237.1	1.45094	0.725470	−	0.688253i	$-0.241622\pi$
$617$	22237.1	1.45094	0.725470	+	0.688253i	$0.241622\pi$
$618$	−4969.04	−0.323437
$619$	−7885.16	−0.512005	−0.256003	−	0.966676i	$-0.582406\pi$
$619$	−7885.16	−0.512005	−0.256003	+	0.966676i	$0.582406\pi$
$620$	2912.20	0.188640
$621$	11297.7	0.730048
$622$	8829.56	0.569185
$623$	−12266.7	−0.788853
$624$	−1828.09	−0.117279
$625$	7340.76	0.469808
$626$	5148.24	0.328698
$627$	45512.7	2.89889
$628$	7981.83	0.507181
$629$	1648.72	0.104513
$630$	8672.05	0.548417
$631$	3424.85	0.216072	0.108036	−	0.994147i	$-0.465544\pi$
$631$	3424.85	0.216072	0.108036	+	0.994147i	$0.465544\pi$
$632$	−7841.35	−0.493532
$633$	11425.4	0.717405
$634$	−17934.8	−1.12347
$635$	28385.5	1.77393
$636$	8.78215	0.000547539 0
$637$	−2561.56	−0.159329
$638$	14700.8	0.912240
$639$	−10711.3	−0.663118
$640$	−2399.54	−0.148203
$641$	−9634.06	−0.593639	−0.296819	−	0.954934i	$-0.595926\pi$
$641$	−9634.06	−0.593639	−0.296819	+	0.954934i	$0.595926\pi$
$642$	−15727.3	−0.966834
$643$	12056.9	0.739470	0.369735	−	0.929137i	$-0.379449\pi$
$643$	12056.9	0.739470	0.369735	+	0.929137i	$0.379449\pi$
$644$	9827.36	0.601323
$645$	−29409.5	−1.79535
$646$	11457.3	0.697801
$647$	−16899.0	−1.02684	−0.513422	−	0.858137i	$-0.671622\pi$
$647$	−16899.0	−1.02684	−0.513422	+	0.858137i	$0.671622\pi$
$648$	7220.52	0.437730
$649$	40323.7	2.43889
$650$	8443.32	0.509499
$651$	−5216.31	−0.314045
$652$	13292.7	0.798440
$653$	−15988.9	−0.958182	−0.479091	−	0.877765i	$-0.659034\pi$
$653$	−15988.9	−0.958182	−0.479091	+	0.877765i	$0.659034\pi$
$654$	−12928.7	−0.773017
$655$	27882.4	1.66329
$656$	−7018.64	−0.417732
$657$	−152.731	−0.00906943
$658$	12058.9	0.714445
$659$	−22424.5	−1.32554	−0.662772	−	0.748822i	$-0.730620\pi$
$659$	−22424.5	−1.32554	−0.662772	+	0.748822i	$0.730620\pi$
$660$	26546.3	1.56563
$661$	5630.87	0.331339	0.165670	−	0.986181i	$-0.447021\pi$
$661$	5630.87	0.331339	0.165670	+	0.986181i	$0.447021\pi$
$662$	18858.0	1.10716
$663$	5091.23	0.298231
$664$	2189.28	0.127952
$665$	−52822.6	−3.08026
$666$	780.926	0.0454358
$667$	14262.2	0.827938
$668$	−14654.5	−0.848804
$669$	−13485.0	−0.779314
$670$	19689.1	1.13531
$671$	−28594.6	−1.64513
$672$	4298.02	0.246726
$673$	29088.7	1.66610	0.833050	−	0.553197i	$-0.186592\pi$
$673$	29088.7	1.66610	0.833050	+	0.553197i	$0.186592\pi$
$674$	−15366.9	−0.878209
$675$	−22821.1	−1.30131
$676$	−7397.50	−0.420887
$677$	−12811.2	−0.727291	−0.363646	−	0.931537i	$-0.618468\pi$
$677$	−12811.2	−0.727291	−0.363646	+	0.931537i	$0.618468\pi$
$678$	−19184.9	−1.08671
$679$	−1691.67	−0.0956116
$680$	6682.70	0.376868
$681$	−12379.0	−0.696571
$682$	−4487.23	−0.251943
$683$	−14164.5	−0.793544	−0.396772	−	0.917917i	$-0.629870\pi$
$683$	−14164.5	−0.793544	−0.396772	+	0.917917i	$0.629870\pi$
$684$	5426.80	0.303361
$685$	11412.6	0.636573
$686$	−9013.09	−0.501635
$687$	−12214.1	−0.678304
$688$	−4096.08	−0.226979
$689$	−6.67995	−0.000369355 0
$690$	25754.4	1.42094
$691$	−13318.1	−0.733204	−0.366602	−	0.930378i	$-0.619479\pi$
$691$	−13318.1	−0.733204	−0.366602	+	0.930378i	$0.619479\pi$
$692$	−12377.3	−0.679933
$693$	−13362.2	−0.732451
$694$	6373.40	0.348604
$695$	22213.6	1.21239
$696$	6237.61	0.339707
$697$	19546.9	1.06226
$698$	3316.05	0.179820
$699$	−17294.6	−0.935824
$700$	−19851.1	−1.07186
$701$	−20777.7	−1.11949	−0.559745	−	0.828665i	$-0.689101\pi$
$701$	−20777.7	−1.11949	−0.559745	+	0.828665i	$0.689101\pi$
$702$	−3758.31	−0.202063
$703$	−4756.72	−0.255196
$704$	3697.30	0.197936
$705$	31602.5	1.68825
$706$	−4242.48	−0.226159
$707$	39163.9	2.08332
$708$	17109.5	0.908214
$709$	16081.4	0.851833	0.425917	−	0.904762i	$-0.359952\pi$
$709$	16081.4	0.851833	0.425917	+	0.904762i	$0.359952\pi$
$710$	38055.0	2.01152
$711$	10343.8	0.545600
$712$	−4477.36	−0.235669
$713$	−4353.36	−0.228660
$714$	−11970.0	−0.627403
$715$	−20191.9	−1.05613
$716$	4990.66	0.260488
$717$	18447.4	0.960854
$718$	12557.8	0.652719
$719$	−2335.59	−0.121145	−0.0605723	−	0.998164i	$-0.519293\pi$
$719$	−2335.59	−0.121145	−0.0605723	+	0.998164i	$0.519293\pi$
$720$	3165.31	0.163839
$721$	−8886.20	−0.459000
$722$	−19337.3	−0.996760
$723$	18311.6	0.941929
$724$	11443.9	0.587445
$725$	−28809.4	−1.47580
$726$	−24590.7	−1.25709
$727$	−14356.4	−0.732392	−0.366196	−	0.930538i	$-0.619340\pi$
$727$	−14356.4	−0.732392	−0.366196	+	0.930538i	$0.619340\pi$
$728$	−3269.20	−0.166435
$729$	7151.27	0.363322
$730$	542.622	0.0275114
$731$	11407.6	0.577187
$732$	−12132.8	−0.612626
$733$	−17167.8	−0.865087	−0.432544	−	0.901613i	$-0.642384\pi$
$733$	−17167.8	−0.865087	−0.432544	+	0.901613i	$0.642384\pi$
$734$	18402.2	0.925392
$735$	15783.0	0.792062
$736$	3586.99	0.179645
$737$	−30337.7	−1.51629
$738$	9258.52	0.461803
$739$	4163.21	0.207234	0.103617	−	0.994617i	$-0.466958\pi$
$739$	4163.21	0.207234	0.103617	+	0.994617i	$0.466958\pi$
$740$	−2774.46	−0.137826
$741$	−14688.7	−0.728209
$742$	15.7052	0.000777031 0
$743$	28806.1	1.42233	0.711166	−	0.703024i	$-0.248167\pi$
$743$	28806.1	1.42233	0.711166	+	0.703024i	$0.248167\pi$
$744$	−1903.96	−0.0938205
$745$	−14021.2	−0.689525
$746$	−1056.62	−0.0518576
$747$	−2887.94	−0.141452
$748$	−10297.0	−0.503334
$749$	−28125.4	−1.37207
$750$	−23303.6	−1.13457
$751$	−3865.88	−0.187840	−0.0939200	−	0.995580i	$-0.529940\pi$
$751$	−3865.88	−0.187840	−0.0939200	+	0.995580i	$0.529940\pi$
$752$	4401.51	0.213439
$753$	4210.12	0.203752
$754$	−4744.50	−0.229157
$755$	23860.1	1.15014
$756$	8836.17	0.425091
$757$	−22540.5	−1.08223	−0.541114	−	0.840949i	$-0.681997\pi$
$757$	−22540.5	−1.08223	−0.541114	+	0.840949i	$0.681997\pi$
$758$	5008.50	0.239996
$759$	−39683.3	−1.89778
$760$	−19280.3	−0.920222
$761$	40558.8	1.93200	0.966001	−	0.258538i	$-0.0832408\pi$
$761$	40558.8	1.93200	0.966001	+	0.258538i	$0.0832408\pi$
$762$	−18558.1	−0.882267
$763$	−23120.6	−1.09702
$764$	−4707.75	−0.222932
$765$	−8815.37	−0.416628
$766$	10311.7	0.486395
$767$	−13014.0	−0.612657
$768$	1568.78	0.0737090
$769$	38115.1	1.78734	0.893671	−	0.448722i	$-0.148121\pi$
$769$	38115.1	1.78734	0.893671	+	0.448722i	$0.148121\pi$
$770$	47473.1	2.22184
$771$	−13771.8	−0.643295
$772$	7933.15	0.369845
$773$	19667.0	0.915099	0.457549	−	0.889184i	$-0.348727\pi$
$773$	19667.0	0.915099	0.457549	+	0.889184i	$0.348727\pi$
$774$	5403.27	0.250926
$775$	8793.71	0.407586
$776$	−617.460	−0.0285638
$777$	4969.59	0.229451
$778$	−20445.8	−0.942183
$779$	−56394.8	−2.59378
$780$	−8567.52	−0.393290
$781$	−58636.5	−2.68653
$782$	−9989.77	−0.456820
$783$	12823.7	0.585290
$784$	2198.21	0.100137
$785$	37407.6	1.70081
$786$	−18229.1	−0.827240
$787$	−42414.0	−1.92109	−0.960543	−	0.278132i	$-0.910285\pi$
$787$	−42414.0	−1.92109	−0.960543	+	0.278132i	$0.910285\pi$
$788$	−10990.8	−0.496869
$789$	3571.67	0.161160
$790$	−36749.2	−1.65504
$791$	−34308.6	−1.54219
$792$	−4877.22	−0.218819
$793$	9228.57	0.413261
$794$	−12881.8	−0.575764
$795$	41.1584	0.00183615
$796$	6677.95	0.297354
$797$	−20257.9	−0.900343	−0.450171	−	0.892942i	$-0.648637\pi$
$797$	−20257.9	−0.900343	−0.450171	+	0.892942i	$0.648637\pi$
$798$	34534.6	1.53197
$799$	−12258.2	−0.542758
$800$	−7245.66	−0.320216
$801$	5906.23	0.260532
$802$	18331.3	0.807108
$803$	−836.092	−0.0367435
$804$	−12872.5	−0.564647
$805$	46056.9	2.01651
$806$	1448.20	0.0632888
$807$	−6874.32	−0.299861
$808$	14294.8	0.622390
$809$	22120.5	0.961327	0.480664	−	0.876905i	$-0.340396\pi$
$809$	22120.5	0.961327	0.480664	+	0.876905i	$0.340396\pi$
$810$	33839.7	1.46791
$811$	−325.141	−0.0140780	−0.00703898	−	0.999975i	$-0.502241\pi$
$811$	−325.141	−0.0140780	−0.00703898	+	0.999975i	$0.502241\pi$
$812$	11154.8	0.482090
$813$	−323.062	−0.0139364
$814$	4275.00	0.184077
$815$	62297.6	2.67753
$816$	−4369.05	−0.187436
$817$	−32912.0	−1.40936
$818$	−10611.1	−0.453557
$819$	4312.50	0.183994
$820$	−32893.5	−1.40084
$821$	28457.1	1.20969	0.604847	−	0.796341i	$-0.293234\pi$
$821$	28457.1	1.20969	0.604847	+	0.796341i	$0.293234\pi$
$822$	−7461.38	−0.316600
$823$	−3812.08	−0.161459	−0.0807295	−	0.996736i	$-0.525725\pi$
$823$	−3812.08	−0.161459	−0.0807295	+	0.996736i	$0.525725\pi$
$824$	−3243.47	−0.137126
$825$	80159.5	3.38278
$826$	30597.2	1.28888
$827$	27739.8	1.16639	0.583197	−	0.812330i	$-0.301801\pi$
$827$	27739.8	1.16639	0.583197	+	0.812330i	$0.301801\pi$
$828$	−4731.72	−0.198597
$829$	3649.24	0.152887	0.0764434	−	0.997074i	$-0.475644\pi$
$829$	3649.24	0.152887	0.0764434	+	0.997074i	$0.475644\pi$
$830$	10260.2	0.429082
$831$	−22321.1	−0.931782
$832$	−1193.26	−0.0497222
$833$	−6122.02	−0.254640
$834$	−14522.9	−0.602982
$835$	−68679.9	−2.84643
$836$	29707.8	1.22902
$837$	−3914.28	−0.161646
$838$	2194.21	0.0904506
$839$	−10275.5	−0.422826	−0.211413	−	0.977397i	$-0.567807\pi$
$839$	−10275.5	−0.422826	−0.211413	+	0.977397i	$0.567807\pi$
$840$	20143.1	0.827385
$841$	−8200.32	−0.336230
$842$	−12781.5	−0.523136
$843$	41913.8	1.71244
$844$	7457.74	0.304154
$845$	−34669.1	−1.41142
$846$	−5806.17	−0.235958
$847$	−43975.8	−1.78398
$848$	5.73242	0.000232137 0
$849$	35108.7	1.41923
$850$	20179.1	0.814281
$851$	4147.46	0.167066
$852$	−24879.8	−1.00043
$853$	13200.6	0.529870	0.264935	−	0.964266i	$-0.414649\pi$
$853$	13200.6	0.529870	0.264935	+	0.964266i	$0.414649\pi$
$854$	−21697.3	−0.869398
$855$	25433.2	1.01731
$856$	−10265.8	−0.409903
$857$	−11275.5	−0.449431	−0.224716	−	0.974424i	$-0.572145\pi$
$857$	−11275.5	−0.449431	−0.224716	+	0.974424i	$0.572145\pi$
$858$	13201.2	0.525268
$859$	35615.6	1.41465	0.707327	−	0.706886i	$-0.249901\pi$
$859$	35615.6	1.41465	0.707327	+	0.706886i	$0.249901\pi$
$860$	−19196.7	−0.761163
$861$	58918.5	2.33210
$862$	−29749.3	−1.17548
$863$	−16687.2	−0.658216	−0.329108	−	0.944292i	$-0.606748\pi$
$863$	−16687.2	−0.658216	−0.329108	+	0.944292i	$0.606748\pi$
$864$	3225.21	0.126995
$865$	−58007.3	−2.28012
$866$	−18454.1	−0.724128
$867$	−17939.3	−0.702712
$868$	−3404.87	−0.133144
$869$	56624.6	2.21042
$870$	29233.2	1.13919
$871$	9791.15	0.380896
$872$	−8439.04	−0.327732
$873$	814.511	0.0315774
$874$	28821.5	1.11545
$875$	−41674.1	−1.61011
$876$	−354.758	−0.0136828
$877$	4058.29	0.156258	0.0781292	−	0.996943i	$-0.475105\pi$
$877$	4058.29	0.156258	0.0781292	+	0.996943i	$0.475105\pi$
$878$	−4565.16	−0.175475
$879$	26797.1	1.02826
$880$	17327.7	0.663770
$881$	7089.26	0.271105	0.135552	−	0.990770i	$-0.456719\pi$
$881$	7089.26	0.271105	0.135552	+	0.990770i	$0.456719\pi$
$882$	−2899.73	−0.110702
$883$	−13613.9	−0.518851	−0.259425	−	0.965763i	$-0.583533\pi$
$883$	−13613.9	−0.518851	−0.259425	+	0.965763i	$0.583533\pi$
$884$	3323.23	0.126439
$885$	80185.5	3.04566
$886$	−14244.1	−0.540111
$887$	34658.8	1.31198	0.655991	−	0.754768i	$-0.272251\pi$
$887$	34658.8	1.31198	0.655991	+	0.754768i	$0.272251\pi$
$888$	1813.90	0.0685480
$889$	−33187.6	−1.25205
$890$	−20983.6	−0.790305
$891$	−52141.4	−1.96050
$892$	−8802.16	−0.330401
$893$	35366.1	1.32529
$894$	9166.84	0.342936
$895$	23389.2	0.873536
$896$	2805.47	0.104603
$897$	12807.3	0.476727
$898$	−4642.68	−0.172526
$899$	−4941.40	−0.183320
$900$	9557.98	0.353999
$901$	−15.9648	−0.000590304 0
$902$	50683.6	1.87093
$903$	34384.8	1.26717
$904$	−12522.7	−0.460728
$905$	53633.1	1.96997
$906$	−15599.4	−0.572026
$907$	−28527.3	−1.04436	−0.522180	−	0.852836i	$-0.674881\pi$
$907$	−28527.3	−1.04436	−0.522180	+	0.852836i	$0.674881\pi$
$908$	−8080.23	−0.295321
$909$	−18856.8	−0.688053
$910$	−15321.4	−0.558132
$911$	25045.5	0.910862	0.455431	−	0.890271i	$-0.349485\pi$
$911$	25045.5	0.910862	0.455431	+	0.890271i	$0.349485\pi$
$912$	12605.2	0.457674
$913$	−15809.4	−0.573071
$914$	−15765.7	−0.570552
$915$	−56861.7	−2.05441
$916$	−7972.55	−0.287577
$917$	−32599.4	−1.17396
$918$	−8982.20	−0.322938
$919$	26149.1	0.938608	0.469304	−	0.883037i	$-0.344505\pi$
$919$	26149.1	0.938608	0.469304	+	0.883037i	$0.344505\pi$
$920$	16810.8	0.602430
$921$	−33586.0	−1.20163
$922$	35931.4	1.28345
$923$	18924.3	0.674864
$924$	−31037.2	−1.10503
$925$	−8377.79	−0.297795
$926$	31072.4	1.10270
$927$	4278.56	0.151593
$928$	4071.51	0.144024
$929$	−31005.3	−1.09499	−0.547497	−	0.836807i	$-0.684419\pi$
$929$	−31005.3	−1.09499	−0.547497	+	0.836807i	$0.684419\pi$
$930$	−8923.07	−0.314623
$931$	17662.6	0.621772
$932$	−11288.8	−0.396756
$933$	−27054.0	−0.949313
$934$	−16640.2	−0.582959
$935$	−48257.7	−1.68791
$936$	1574.07	0.0549680
$937$	39648.0	1.38233	0.691166	−	0.722696i	$-0.257098\pi$
$937$	39648.0	1.38233	0.691166	+	0.722696i	$0.257098\pi$
$938$	−23020.0	−0.801310
$939$	−15774.3	−0.548217
$940$	20628.1	0.715760
$941$	19549.2	0.677242	0.338621	−	0.940923i	$-0.390040\pi$
$941$	19549.2	0.677242	0.338621	+	0.940923i	$0.390040\pi$
$942$	−24456.6	−0.845900
$943$	49171.5	1.69803
$944$	11168.0	0.385050
$945$	41411.6	1.42552
$946$	29578.9	1.01659
$947$	15025.4	0.515587	0.257793	−	0.966200i	$-0.417005\pi$
$947$	15025.4	0.515587	0.257793	+	0.966200i	$0.417005\pi$
$948$	24026.1	0.823134
$949$	269.839	0.00923009
$950$	−58218.9	−1.98828
$951$	54952.6	1.87377
$952$	−7813.24	−0.265996
$953$	1441.58	0.0490003	0.0245002	−	0.999700i	$-0.492201\pi$
$953$	1441.58	0.0490003	0.0245002	+	0.999700i	$0.492201\pi$
$954$	−7.56182	−0.000256628 0
$955$	−22063.3	−0.747594
$956$	12041.3	0.407368
$957$	−45043.5	−1.52147
$958$	−25513.2	−0.860433
$959$	−13343.3	−0.449298
$960$	7352.25	0.247180
$961$	−28282.7	−0.949370
$962$	−1379.71	−0.0462407
$963$	13541.9	0.453149
$964$	11952.6	0.399344
$965$	37179.5	1.24026
$966$	−30111.3	−1.00291
$967$	15366.4	0.511012	0.255506	−	0.966807i	$-0.417758\pi$
$967$	15366.4	0.511012	0.255506	+	0.966807i	$0.417758\pi$
$968$	−16051.2	−0.532960
$969$	−35105.3	−1.16382
$970$	−2893.79	−0.0957875
$971$	22937.8	0.758094	0.379047	−	0.925377i	$-0.376252\pi$
$971$	22937.8	0.758094	0.379047	+	0.925377i	$0.376252\pi$
$972$	−11238.8	−0.370869
$973$	−25971.5	−0.855712
$974$	20034.6	0.659085
$975$	−25870.6	−0.849765
$976$	−7919.53	−0.259732
$977$	−6065.55	−0.198623	−0.0993113	−	0.995056i	$-0.531664\pi$
$977$	−6065.55	−0.198623	−0.0993113	+	0.995056i	$0.531664\pi$
$978$	−40729.2	−1.33167
$979$	32332.3	1.05551
$980$	10302.1	0.335806
$981$	11132.2	0.362308
$982$	31245.0	1.01534
$983$	−50064.8	−1.62443	−0.812216	−	0.583356i	$-0.801739\pi$
$983$	−50064.8	−1.62443	−0.812216	+	0.583356i	$0.801739\pi$
$984$	21505.3	0.696712
$985$	−51509.6	−1.66623
$986$	−11339.1	−0.366239
$987$	−36948.8	−1.19158
$988$	−9587.84	−0.308735
$989$	28696.5	0.922644
$990$	−22857.6	−0.733799
$991$	−839.793	−0.0269192	−0.0134596	−	0.999909i	$-0.504284\pi$
$991$	−839.793	−0.0269192	−0.0134596	+	0.999909i	$0.504284\pi$
$992$	−1242.78	−0.0397765
$993$	−57781.4	−1.84656
$994$	−44492.8	−1.41975
$995$	31296.8	0.997162
$996$	−6708.00	−0.213405
$997$	22126.0	0.702847	0.351423	−	0.936217i	$-0.385698\pi$
$997$	22126.0	0.702847	0.351423	+	0.936217i	$0.385698\pi$
$998$	26330.3	0.835141
$999$	3729.15	0.118103

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.a.c.1.3	✓	3
3.2	odd	2		666.4.a.n.1.1		3
4.3	odd	2		592.4.a.c.1.1		3
5.4	even	2		1850.4.a.i.1.1		3
8.3	odd	2		2368.4.a.f.1.3		3
8.5	even	2		2368.4.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.a.c.1.3	✓	3	1.1	even	1	trivial
592.4.a.c.1.1		3	4.3	odd	2
666.4.a.n.1.1		3	3.2	odd	2
1850.4.a.i.1.1		3	5.4	even	2
2368.4.a.e.1.1		3	8.5	even	2
2368.4.a.f.1.3		3	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	74.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +6.12805 q^{3} +4.00000 q^{4} +18.7464 q^{5} -12.2561 q^{6} -21.9178 q^{7} -8.00000 q^{8} +10.5531 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +6.12805 q^{3} +4.00000 q^{4} +18.7464 q^{5} -12.2561 q^{6} -21.9178 q^{7} -8.00000 q^{8} +10.5531 q^{9} -37.4928 q^{10} +57.7703 q^{11} +24.5122 q^{12} -18.6447 q^{13} +43.8355 q^{14} +114.879 q^{15} +16.0000 q^{16} -44.5600 q^{17} -21.1061 q^{18} +128.560 q^{19} +74.9855 q^{20} -134.313 q^{21} -115.541 q^{22} -112.094 q^{23} -49.0244 q^{24} +226.427 q^{25} +37.2894 q^{26} -100.788 q^{27} -87.6711 q^{28} -127.235 q^{29} -229.758 q^{30} +38.8369 q^{31} -32.0000 q^{32} +354.019 q^{33} +89.1199 q^{34} -410.879 q^{35} +42.2122 q^{36} -37.0000 q^{37} -257.120 q^{38} -114.256 q^{39} -149.971 q^{40} -438.665 q^{41} +268.627 q^{42} -256.005 q^{43} +231.081 q^{44} +197.832 q^{45} +224.187 q^{46} +275.094 q^{47} +98.0489 q^{48} +137.388 q^{49} -452.854 q^{50} -273.066 q^{51} -74.5788 q^{52} +0.358276 q^{53} +201.576 q^{54} +1082.98 q^{55} +175.342 q^{56} +787.822 q^{57} +254.469 q^{58} +698.000 q^{59} +459.515 q^{60} -494.971 q^{61} -77.6738 q^{62} -231.299 q^{63} +64.0000 q^{64} -349.520 q^{65} -708.039 q^{66} -525.144 q^{67} -178.240 q^{68} -686.915 q^{69} +821.758 q^{70} -1014.99 q^{71} -84.4244 q^{72} -14.4727 q^{73} +74.0000 q^{74} +1387.56 q^{75} +514.240 q^{76} -1266.19 q^{77} +228.511 q^{78} +980.168 q^{79} +299.942 q^{80} -902.566 q^{81} +877.330 q^{82} -273.659 q^{83} -537.253 q^{84} -835.338 q^{85} +512.010 q^{86} -779.701 q^{87} -462.162 q^{88} +559.670 q^{89} -395.663 q^{90} +408.650 q^{91} -448.374 q^{92} +237.995 q^{93} -550.189 q^{94} +2410.03 q^{95} -196.098 q^{96} +77.1825 q^{97} -274.777 q^{98} +609.653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 4 q^{3} + 12 q^{4} + 7 q^{5} - 8 q^{6} + 7 q^{7} - 24 q^{8} + 43 q^{9}+O(q^{10}) \) 3 * q - 6 * q^2 + 4 * q^3 + 12 * q^4 + 7 * q^5 - 8 * q^6 + 7 * q^7 - 24 * q^8 + 43 * q^9	\( 3 q - 6 q^{2} + 4 q^{3} + 12 q^{4} + 7 q^{5} - 8 q^{6} + 7 q^{7} - 24 q^{8} + 43 q^{9} - 14 q^{10} + 74 q^{11} + 16 q^{12} + 107 q^{13} - 14 q^{14} + 190 q^{15} + 48 q^{16} + 24 q^{17} - 86 q^{18} + 228 q^{19} + 28 q^{20} - 27 q^{21} - 148 q^{22} - 149 q^{23} - 32 q^{24} + 92 q^{25} - 214 q^{26} - 257 q^{27} + 28 q^{28} - 325 q^{29} - 380 q^{30} + 23 q^{31} - 96 q^{32} - 97 q^{33} - 48 q^{34} - 478 q^{35} + 172 q^{36} - 111 q^{37} - 456 q^{38} + 39 q^{39} - 56 q^{40} - 464 q^{41} + 54 q^{42} + 588 q^{43} + 296 q^{44} - 126 q^{45} + 298 q^{46} + 155 q^{47} + 64 q^{48} + 96 q^{49} - 184 q^{50} + 310 q^{51} + 428 q^{52} - 579 q^{53} + 514 q^{54} + 665 q^{55} - 56 q^{56} + 26 q^{57} + 650 q^{58} + 1258 q^{59} + 760 q^{60} + 291 q^{61} - 46 q^{62} - 56 q^{63} + 192 q^{64} - 874 q^{65} + 194 q^{66} - 127 q^{67} + 96 q^{68} + 399 q^{69} + 956 q^{70} - 201 q^{71} - 344 q^{72} + 42 q^{73} + 222 q^{74} + 795 q^{75} + 912 q^{76} - 1743 q^{77} - 78 q^{78} + 413 q^{79} + 112 q^{80} - 2329 q^{81} + 928 q^{82} - 1037 q^{83} - 108 q^{84} - 750 q^{85} - 1176 q^{86} - 1465 q^{87} - 592 q^{88} + 660 q^{89} + 252 q^{90} + 2696 q^{91} - 596 q^{92} - 2130 q^{93} - 310 q^{94} + 1338 q^{95} - 128 q^{96} + 1384 q^{97} - 192 q^{98} + 1796 q^{99}+O(q^{100}) \) 3 * q - 6 * q^2 + 4 * q^3 + 12 * q^4 + 7 * q^5 - 8 * q^6 + 7 * q^7 - 24 * q^8 + 43 * q^9 - 14 * q^10 + 74 * q^11 + 16 * q^12 + 107 * q^13 - 14 * q^14 + 190 * q^15 + 48 * q^16 + 24 * q^17 - 86 * q^18 + 228 * q^19 + 28 * q^20 - 27 * q^21 - 148 * q^22 - 149 * q^23 - 32 * q^24 + 92 * q^25 - 214 * q^26 - 257 * q^27 + 28 * q^28 - 325 * q^29 - 380 * q^30 + 23 * q^31 - 96 * q^32 - 97 * q^33 - 48 * q^34 - 478 * q^35 + 172 * q^36 - 111 * q^37 - 456 * q^38 + 39 * q^39 - 56 * q^40 - 464 * q^41 + 54 * q^42 + 588 * q^43 + 296 * q^44 - 126 * q^45 + 298 * q^46 + 155 * q^47 + 64 * q^48 + 96 * q^49 - 184 * q^50 + 310 * q^51 + 428 * q^52 - 579 * q^53 + 514 * q^54 + 665 * q^55 - 56 * q^56 + 26 * q^57 + 650 * q^58 + 1258 * q^59 + 760 * q^60 + 291 * q^61 - 46 * q^62 - 56 * q^63 + 192 * q^64 - 874 * q^65 + 194 * q^66 - 127 * q^67 + 96 * q^68 + 399 * q^69 + 956 * q^70 - 201 * q^71 - 344 * q^72 + 42 * q^73 + 222 * q^74 + 795 * q^75 + 912 * q^76 - 1743 * q^77 - 78 * q^78 + 413 * q^79 + 112 * q^80 - 2329 * q^81 + 928 * q^82 - 1037 * q^83 - 108 * q^84 - 750 * q^85 - 1176 * q^86 - 1465 * q^87 - 592 * q^88 + 660 * q^89 + 252 * q^90 + 2696 * q^91 - 596 * q^92 - 2130 * q^93 - 310 * q^94 + 1338 * q^95 - 128 * q^96 + 1384 * q^97 - 192 * q^98 + 1796 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more