LMFDB - Embedded newform 74.4.a.c.1.2

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			1.2
Root			$-3.73537$ 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)

$f(q)$	$=$	$q-2.00000 q^{2} +5.42375 q^{3} +4.00000 q^{4} -1.04699 q^{5} -10.8475 q^{6} +25.1006 q^{7} -8.00000 q^{8} +2.41712 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +5.42375 q^{3} +4.00000 q^{4} -1.04699 q^{5} -10.8475 q^{6} +25.1006 q^{7} -8.00000 q^{8} +2.41712 q^{9} +2.09398 q^{10} -25.3134 q^{11} +21.6950 q^{12} +84.9366 q^{13} -50.2012 q^{14} -5.67862 q^{15} +16.0000 q^{16} +84.8377 q^{17} -4.83423 q^{18} -0.837748 q^{19} -4.18796 q^{20} +136.140 q^{21} +50.6267 q^{22} +62.2096 q^{23} -43.3900 q^{24} -123.904 q^{25} -169.873 q^{26} -133.332 q^{27} +100.402 q^{28} -167.914 q^{29} +11.3572 q^{30} -191.714 q^{31} -32.0000 q^{32} -137.294 q^{33} -169.675 q^{34} -26.2801 q^{35} +9.66847 q^{36} -37.0000 q^{37} +1.67550 q^{38} +460.675 q^{39} +8.37592 q^{40} -223.124 q^{41} -272.279 q^{42} +314.482 q^{43} -101.253 q^{44} -2.53070 q^{45} -124.419 q^{46} -357.420 q^{47} +86.7801 q^{48} +287.041 q^{49} +247.808 q^{50} +460.139 q^{51} +339.746 q^{52} -123.648 q^{53} +266.663 q^{54} +26.5029 q^{55} -200.805 q^{56} -4.54374 q^{57} +335.829 q^{58} -333.733 q^{59} -22.7145 q^{60} -60.3667 q^{61} +383.427 q^{62} +60.6711 q^{63} +64.0000 q^{64} -88.9278 q^{65} +274.587 q^{66} +519.193 q^{67} +339.351 q^{68} +337.410 q^{69} +52.5602 q^{70} +262.807 q^{71} -19.3369 q^{72} -957.634 q^{73} +74.0000 q^{74} -672.024 q^{75} -3.35099 q^{76} -635.381 q^{77} -921.351 q^{78} +169.806 q^{79} -16.7518 q^{80} -788.420 q^{81} +446.248 q^{82} +2.57382 q^{83} +544.558 q^{84} -88.8243 q^{85} -628.963 q^{86} -910.727 q^{87} +202.507 q^{88} +1378.90 q^{89} +5.06140 q^{90} +2131.96 q^{91} +248.839 q^{92} -1039.81 q^{93} +714.839 q^{94} +0.877114 q^{95} -173.560 q^{96} +1124.41 q^{97} -574.082 q^{98} -61.1854 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	5.42375	1.04380	0.521901	−	0.853006i	$-0.325223\pi$
$3$	5.42375	1.04380	0.521901	+	0.853006i	$0.325223\pi$
$4$	4.00000	0.500000
$5$	−1.04699	−0.0936457	−0.0468228	−	0.998903i	$-0.514910\pi$
$5$	−1.04699	−0.0936457	−0.0468228	+	0.998903i	$0.514910\pi$
$6$	−10.8475	−0.738080
$7$	25.1006	1.35531	0.677653	−	0.735382i	$-0.262997\pi$
$7$	25.1006	1.35531	0.677653	+	0.735382i	$0.262997\pi$
$8$	−8.00000	−0.353553
$9$	2.41712	0.0895229
$10$	2.09398	0.0662175
$11$	−25.3134	−0.693843	−0.346921	−	0.937894i	$-0.612773\pi$
$11$	−25.3134	−0.693843	−0.346921	+	0.937894i	$0.612773\pi$
$12$	21.6950	0.521901
$13$	84.9366	1.81209	0.906045	−	0.423181i	$-0.139086\pi$
$13$	84.9366	1.81209	0.906045	+	0.423181i	$0.139086\pi$
$14$	−50.2012	−0.958346
$15$	−5.67862	−0.0977476
$16$	16.0000	0.250000
$17$	84.8377	1.21036	0.605182	−	0.796087i	$-0.293100\pi$
$17$	84.8377	1.21036	0.605182	+	0.796087i	$0.293100\pi$
$18$	−4.83423	−0.0633022
$19$	−0.837748	−0.0101154	−0.00505770	−	0.999987i	$-0.501610\pi$
$19$	−0.837748	−0.0101154	−0.00505770	+	0.999987i	$0.501610\pi$
$20$	−4.18796	−0.0468228
$21$	136.140	1.41467
$22$	50.6267	0.490621
$23$	62.2096	0.563983	0.281991	−	0.959417i	$-0.409005\pi$
$23$	62.2096	0.563983	0.281991	+	0.959417i	$0.409005\pi$
$24$	−43.3900	−0.369040
$25$	−123.904	−0.991230
$26$	−169.873	−1.28134
$27$	−133.332	−0.950358
$28$	100.402	0.677653
$29$	−167.914	−1.07520	−0.537602	−	0.843199i	$-0.680670\pi$
$29$	−167.914	−1.07520	−0.537602	+	0.843199i	$0.680670\pi$
$30$	11.3572	0.0691180
$31$	−191.714	−1.11073	−0.555367	−	0.831605i	$-0.687422\pi$
$31$	−191.714	−1.11073	−0.555367	+	0.831605i	$0.687422\pi$
$32$	−32.0000	−0.176777
$33$	−137.294	−0.724234
$34$	−169.675	−0.855856
$35$	−26.2801	−0.126919
$36$	9.66847	0.0447614
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	1.67550	0.00715267
$39$	460.675	1.89146
$40$	8.37592	0.0331087
$41$	−223.124	−0.849905	−0.424953	−	0.905216i	$-0.639709\pi$
$41$	−223.124	−0.849905	−0.424953	+	0.905216i	$0.639709\pi$
$42$	−272.279	−1.00032
$43$	314.482	1.11530	0.557651	−	0.830076i	$-0.311703\pi$
$43$	314.482	1.11530	0.557651	+	0.830076i	$0.311703\pi$
$44$	−101.253	−0.346921
$45$	−2.53070	−0.00838343
$46$	−124.419	−0.398796
$47$	−357.420	−1.10926	−0.554628	−	0.832099i	$-0.687139\pi$
$47$	−357.420	−1.10926	−0.554628	+	0.832099i	$0.687139\pi$
$48$	86.7801	0.260951
$49$	287.041	0.836854
$50$	247.808	0.700906
$51$	460.139	1.26338
$52$	339.746	0.906045
$53$	−123.648	−0.320459	−0.160229	−	0.987080i	$-0.551223\pi$
$53$	−123.648	−0.320459	−0.160229	+	0.987080i	$0.551223\pi$
$54$	266.663	0.672005
$55$	26.5029	0.0649754
$56$	−200.805	−0.479173
$57$	−4.54374	−0.0105585
$58$	335.829	0.760284
$59$	−333.733	−0.736412	−0.368206	−	0.929744i	$-0.620028\pi$
$59$	−333.733	−0.736412	−0.368206	+	0.929744i	$0.620028\pi$
$60$	−22.7145	−0.0488738
$61$	−60.3667	−0.126708	−0.0633538	−	0.997991i	$-0.520180\pi$
$61$	−60.3667	−0.126708	−0.0633538	+	0.997991i	$0.520180\pi$
$62$	383.427	0.785408
$63$	60.6711	0.121331
$64$	64.0000	0.125000
$65$	−88.9278	−0.169694
$66$	274.587	0.512111
$67$	519.193	0.946710	0.473355	−	0.880872i	$-0.343043\pi$
$67$	519.193	0.946710	0.473355	+	0.880872i	$0.343043\pi$
$68$	339.351	0.605182
$69$	337.410	0.588686
$70$	52.5602	0.0897450
$71$	262.807	0.439289	0.219644	−	0.975580i	$-0.429510\pi$
$71$	262.807	0.439289	0.219644	+	0.975580i	$0.429510\pi$
$72$	−19.3369	−0.0316511
$73$	−957.634	−1.53538	−0.767689	−	0.640823i	$-0.778593\pi$
$73$	−957.634	−1.53538	−0.767689	+	0.640823i	$0.778593\pi$
$74$	74.0000	0.116248
$75$	−672.024	−1.03465
$76$	−3.35099	−0.00505770
$77$	−635.381	−0.940369
$78$	−921.351	−1.33747
$79$	169.806	0.241831	0.120916	−	0.992663i	$-0.461417\pi$
$79$	169.806	0.241831	0.120916	+	0.992663i	$0.461417\pi$
$80$	−16.7518	−0.0234114
$81$	−788.420	−1.08151
$82$	446.248	0.600974
$83$	2.57382	0.00340378	0.00170189	−	0.999999i	$-0.499458\pi$
$83$	2.57382	0.00340378	0.00170189	+	0.999999i	$0.499458\pi$
$84$	544.558	0.707336
$85$	−88.8243	−0.113345
$86$	−628.963	−0.788637
$87$	−910.727	−1.12230
$88$	202.507	0.245310
$89$	1378.90	1.64228	0.821140	−	0.570727i	$-0.193338\pi$
$89$	1378.90	1.64228	0.821140	+	0.570727i	$0.193338\pi$
$90$	5.06140	0.00592798
$91$	2131.96	2.45594
$92$	248.839	0.281991
$93$	−1039.81	−1.15939
$94$	714.839	0.784362
$95$	0.877114	0.000947263 0
$96$	−173.560	−0.184520
$97$	1124.41	1.17697	0.588487	−	0.808506i	$-0.299724\pi$
$97$	1124.41	1.17697	0.588487	+	0.808506i	$0.299724\pi$
$98$	−574.082	−0.591745
$99$	−61.1854	−0.0621148
$100$	−495.615	−0.495615
$101$	−1878.12	−1.85030	−0.925151	−	0.379600i	$-0.876061\pi$
$101$	−1878.12	−1.85030	−0.925151	+	0.379600i	$0.876061\pi$
$102$	−920.278	−0.893344
$103$	482.932	0.461987	0.230994	−	0.972955i	$-0.425802\pi$
$103$	482.932	0.461987	0.230994	+	0.972955i	$0.425802\pi$
$104$	−679.493	−0.640671
$105$	−142.537	−0.132478
$106$	247.295	0.226599
$107$	1772.62	1.60155	0.800773	−	0.598968i	$-0.204422\pi$
$107$	1772.62	1.60155	0.800773	+	0.598968i	$0.204422\pi$
$108$	−533.326	−0.475179
$109$	−1277.87	−1.12292	−0.561459	−	0.827505i	$-0.689760\pi$
$109$	−1277.87	−1.12292	−0.561459	+	0.827505i	$0.689760\pi$
$110$	−53.0057	−0.0459445
$111$	−200.679	−0.171600
$112$	401.610	0.338827
$113$	−1615.01	−1.34449	−0.672246	−	0.740328i	$-0.734670\pi$
$113$	−1615.01	−1.34449	−0.672246	+	0.740328i	$0.734670\pi$
$114$	9.08748	0.00746597
$115$	−65.1329	−0.0528146
$116$	−671.658	−0.537602
$117$	205.302	0.162224
$118$	667.466	0.520722
$119$	2129.48	1.64041
$120$	45.4290	0.0345590
$121$	−690.233	−0.518582
$122$	120.733	0.0895958
$123$	−1210.17	−0.887133
$124$	−766.854	−0.555367
$125$	260.600	0.186470
$126$	−121.342	−0.0857939
$127$	191.504	0.133805	0.0669026	−	0.997760i	$-0.478688\pi$
$127$	191.504	0.133805	0.0669026	+	0.997760i	$0.478688\pi$
$128$	−128.000	−0.0883883
$129$	1705.67	1.16415
$130$	177.856	0.119992
$131$	−879.357	−0.586487	−0.293243	−	0.956038i	$-0.594735\pi$
$131$	−879.357	−0.586487	−0.293243	+	0.956038i	$0.594735\pi$
$132$	−549.174	−0.362117
$133$	−21.0280	−0.0137095
$134$	−1038.39	−0.669425
$135$	139.597	0.0889969
$136$	−678.702	−0.427928
$137$	2568.28	1.60163	0.800815	−	0.598911i	$-0.204400\pi$
$137$	2568.28	1.60163	0.800815	+	0.598911i	$0.204400\pi$
$138$	−674.820	−0.416264
$139$	1827.40	1.11509	0.557547	−	0.830146i	$-0.311743\pi$
$139$	1827.40	1.11509	0.557547	+	0.830146i	$0.311743\pi$
$140$	−105.120	−0.0634593
$141$	−1938.56	−1.15784
$142$	−525.615	−0.310624
$143$	−2150.03	−1.25731
$144$	38.6739	0.0223807
$145$	175.805	0.100688
$146$	1915.27	1.08568
$147$	1556.84	0.873510
$148$	−148.000	−0.0821995
$149$	1829.62	1.00596	0.502981	−	0.864298i	$-0.332237\pi$
$149$	1829.62	1.00596	0.502981	+	0.864298i	$0.332237\pi$
$150$	1344.05	0.731607
$151$	−1838.54	−0.990850	−0.495425	−	0.868651i	$-0.664988\pi$
$151$	−1838.54	−0.990850	−0.495425	+	0.868651i	$0.664988\pi$
$152$	6.70198	0.00357633
$153$	205.063	0.108355
$154$	1270.76	0.664941
$155$	200.722	0.104015
$156$	1842.70	0.945732
$157$	−1865.44	−0.948270	−0.474135	−	0.880452i	$-0.657239\pi$
$157$	−1865.44	−0.948270	−0.474135	+	0.880452i	$0.657239\pi$
$158$	−339.612	−0.171000
$159$	−670.635	−0.334496
$160$	33.5037	0.0165544
$161$	1561.50	0.764369
$162$	1576.84	0.764742
$163$	394.342	0.189493	0.0947463	−	0.995501i	$-0.469796\pi$
$163$	394.342	0.189493	0.0947463	+	0.995501i	$0.469796\pi$
$164$	−892.496	−0.424953
$165$	143.745	0.0678214
$166$	−5.14764	−0.00240683
$167$	3526.46	1.63405	0.817023	−	0.576605i	$-0.195623\pi$
$167$	3526.46	1.63405	0.817023	+	0.576605i	$0.195623\pi$
$168$	−1089.12	−0.500162
$169$	5017.23	2.28367
$170$	177.649	0.0801472
$171$	−2.02493	−0.000905559 0
$172$	1257.93	0.557651
$173$	1163.71	0.511417	0.255709	−	0.966754i	$-0.417691\pi$
$173$	1163.71	0.511417	0.255709	+	0.966754i	$0.417691\pi$
$174$	1821.45	0.793586
$175$	−3110.06	−1.34342
$176$	−405.014	−0.173461
$177$	−1810.09	−0.768669
$178$	−2757.80	−1.16127
$179$	−664.012	−0.277266	−0.138633	−	0.990344i	$-0.544271\pi$
$179$	−664.012	−0.277266	−0.138633	+	0.990344i	$0.544271\pi$
$180$	−10.1228	−0.00419171
$181$	3567.28	1.46494	0.732469	−	0.680800i	$-0.238368\pi$
$181$	3567.28	1.46494	0.732469	+	0.680800i	$0.238368\pi$
$182$	−4263.92	−1.73661
$183$	−327.414	−0.132258
$184$	−497.677	−0.199398
$185$	38.7387	0.0153953
$186$	2079.61	0.819811
$187$	−2147.53	−0.839802
$188$	−1429.68	−0.554628
$189$	−3346.70	−1.28803
$190$	−1.75423	−0.000669816 0
$191$	405.343	0.153558	0.0767791	−	0.997048i	$-0.475536\pi$
$191$	405.343	0.153558	0.0767791	+	0.997048i	$0.475536\pi$
$192$	347.120	0.130475
$193$	−1665.01	−0.620985	−0.310492	−	0.950576i	$-0.600494\pi$
$193$	−1665.01	−0.620985	−0.310492	+	0.950576i	$0.600494\pi$
$194$	−2248.82	−0.832247
$195$	−482.323	−0.177127
$196$	1148.16	0.418427
$197$	−3170.05	−1.14648	−0.573240	−	0.819387i	$-0.694314\pi$
$197$	−3170.05	−1.14648	−0.573240	+	0.819387i	$0.694314\pi$
$198$	122.371	0.0439218
$199$	−2024.27	−0.721090	−0.360545	−	0.932742i	$-0.617409\pi$
$199$	−2024.27	−0.721090	−0.360545	+	0.932742i	$0.617409\pi$
$200$	991.230	0.350453
$201$	2815.98	0.988177
$202$	3756.25	1.30836
$203$	−4214.76	−1.45723
$204$	1840.56	0.631690
$205$	233.609	0.0795899
$206$	−965.864	−0.326674
$207$	150.368	0.0504894
$208$	1358.99	0.453023
$209$	21.2062	0.00701849
$210$	285.074	0.0936760
$211$	5647.11	1.84248	0.921240	−	0.388996i	$-0.127178\pi$
$211$	5647.11	1.84248	0.921240	+	0.388996i	$0.127178\pi$
$212$	−494.591	−0.160229
$213$	1425.40	0.458530
$214$	−3545.24	−1.13246
$215$	−329.259	−0.104443
$216$	1066.65	0.336002
$217$	−4812.13	−1.50539
$218$	2555.75	0.794023
$219$	−5193.97	−1.60263
$220$	106.011	0.0324877
$221$	7205.83	2.19329
$222$	401.358	0.121340
$223$	−6334.47	−1.90218	−0.951092	−	0.308907i	$-0.900037\pi$
$223$	−6334.47	−1.90218	−0.951092	+	0.308907i	$0.900037\pi$
$224$	−803.220	−0.239587
$225$	−299.490	−0.0887378
$226$	3230.02	0.950699
$227$	−1322.60	−0.386715	−0.193357	−	0.981128i	$-0.561938\pi$
$227$	−1322.60	−0.386715	−0.193357	+	0.981128i	$0.561938\pi$
$228$	−18.1750	−0.00527924
$229$	−2769.81	−0.799277	−0.399638	−	0.916673i	$-0.630864\pi$
$229$	−2769.81	−0.799277	−0.399638	+	0.916673i	$0.630864\pi$
$230$	130.266	0.0373455
$231$	−3446.15	−0.981559
$232$	1343.32	0.380142
$233$	−701.247	−0.197168	−0.0985842	−	0.995129i	$-0.531431\pi$
$233$	−701.247	−0.197168	−0.0985842	+	0.995129i	$0.531431\pi$
$234$	−410.604	−0.114709
$235$	374.215	0.103877
$236$	−1334.93	−0.368206
$237$	920.986	0.252424
$238$	−4258.96	−1.15995
$239$	1841.74	0.498461	0.249231	−	0.968444i	$-0.419822\pi$
$239$	1841.74	0.498461	0.249231	+	0.968444i	$0.419822\pi$
$240$	−90.8579	−0.0244369
$241$	−6191.89	−1.65500	−0.827500	−	0.561466i	$-0.810238\pi$
$241$	−6191.89	−1.65500	−0.827500	+	0.561466i	$0.810238\pi$
$242$	1380.47	0.366693
$243$	−676.244	−0.178523
$244$	−241.467	−0.0633538
$245$	−300.529	−0.0783678
$246$	2420.34	0.627298
$247$	−71.1555	−0.0183300
$248$	1533.71	0.392704
$249$	13.9598	0.00355287
$250$	−521.200	−0.131854
$251$	4462.00	1.12207	0.561034	−	0.827793i	$-0.310404\pi$
$251$	4462.00	1.12207	0.561034	+	0.827793i	$0.310404\pi$
$252$	242.685	0.0606654
$253$	−1574.74	−0.391315
$254$	−383.009	−0.0946145
$255$	−481.761	−0.118310
$256$	256.000	0.0625000
$257$	−2713.94	−0.658719	−0.329359	−	0.944205i	$-0.606833\pi$
$257$	−2713.94	−0.658719	−0.329359	+	0.944205i	$0.606833\pi$
$258$	−3411.34	−0.823181
$259$	−928.723	−0.222811
$260$	−355.711	−0.0848472
$261$	−405.869	−0.0962554
$262$	1758.71	0.414709
$263$	4994.70	1.17105	0.585526	−	0.810654i	$-0.300888\pi$
$263$	4994.70	1.17105	0.585526	+	0.810654i	$0.300888\pi$
$264$	1098.35	0.256056
$265$	129.458	0.0300096
$266$	42.0560	0.00969405
$267$	7478.81	1.71422
$268$	2076.77	0.473355
$269$	5940.98	1.34657	0.673287	−	0.739382i	$-0.264882\pi$
$269$	5940.98	1.34657	0.673287	+	0.739382i	$0.264882\pi$
$270$	−279.194	−0.0629303
$271$	−7734.73	−1.73377	−0.866885	−	0.498508i	$-0.833881\pi$
$271$	−7734.73	−1.73377	−0.866885	+	0.498508i	$0.833881\pi$
$272$	1357.40	0.302591
$273$	11563.2	2.56351
$274$	−5136.57	−1.13252
$275$	3136.42	0.687758
$276$	1349.64	0.294343
$277$	5244.73	1.13764	0.568818	−	0.822463i	$-0.307401\pi$
$277$	5244.73	1.13764	0.568818	+	0.822463i	$0.307401\pi$
$278$	−3654.80	−0.788490
$279$	−463.394	−0.0994361
$280$	210.241	0.0448725
$281$	−5790.99	−1.22940	−0.614700	−	0.788761i	$-0.710723\pi$
$281$	−5790.99	−1.22940	−0.614700	+	0.788761i	$0.710723\pi$
$282$	3877.11	0.818719
$283$	7297.64	1.53286	0.766431	−	0.642327i	$-0.222031\pi$
$283$	7297.64	1.53286	0.766431	+	0.642327i	$0.222031\pi$
$284$	1051.23	0.219644
$285$	4.75725	0.000988755 0
$286$	4300.06	0.889049
$287$	−5600.55	−1.15188
$288$	−77.3478	−0.0158256
$289$	2284.44	0.464979
$290$	−351.610	−0.0711973
$291$	6098.53	1.22853
$292$	−3830.54	−0.767689
$293$	−3441.47	−0.686188	−0.343094	−	0.939301i	$-0.611475\pi$
$293$	−3441.47	−0.686188	−0.343094	+	0.939301i	$0.611475\pi$
$294$	−3113.68	−0.617665
$295$	349.415	0.0689618
$296$	296.000	0.0581238
$297$	3375.07	0.659399
$298$	−3659.24	−0.711322
$299$	5283.88	1.02199
$300$	−2688.10	−0.517324
$301$	7893.68	1.51158
$302$	3677.08	0.700636
$303$	−10186.5	−1.93135
$304$	−13.4040	−0.00252885
$305$	63.2034	0.0118656
$306$	−410.126	−0.0766187
$307$	1599.67	0.297387	0.148694	−	0.988883i	$-0.452493\pi$
$307$	1599.67	0.297387	0.148694	+	0.988883i	$0.452493\pi$
$308$	−2541.53	−0.470185
$309$	2619.30	0.482223
$310$	−401.445	−0.0735501
$311$	4517.47	0.823673	0.411837	−	0.911258i	$-0.364887\pi$
$311$	4517.47	0.823673	0.411837	+	0.911258i	$0.364887\pi$
$312$	−3685.40	−0.668733
$313$	2759.12	0.498257	0.249129	−	0.968470i	$-0.419856\pi$
$313$	2759.12	0.498257	0.249129	+	0.968470i	$0.419856\pi$
$314$	3730.88	0.670528
$315$	−63.5221	−0.0113621
$316$	679.224	0.120916
$317$	−1138.17	−0.201659	−0.100830	−	0.994904i	$-0.532150\pi$
$317$	−1138.17	−0.201659	−0.100830	+	0.994904i	$0.532150\pi$
$318$	1341.27	0.236524
$319$	4250.48	0.746023
$320$	−67.0074	−0.0117057
$321$	9614.24	1.67170
$322$	−3123.00	−0.540491
$323$	−71.0726	−0.0122433
$324$	−3153.68	−0.540754
$325$	−10524.0	−1.79620
$326$	−788.685	−0.133991
$327$	−6930.87	−1.17210
$328$	1784.99	0.300487
$329$	−8971.45	−1.50338
$330$	−287.490	−0.0479570
$331$	−7862.53	−1.30563	−0.652815	−	0.757517i	$-0.726412\pi$
$331$	−7862.53	−1.30563	−0.652815	+	0.757517i	$0.726412\pi$
$332$	10.2953	0.00170189
$333$	−89.4333	−0.0147175
$334$	−7052.92	−1.15545
$335$	−543.590	−0.0886553
$336$	2178.23	0.353668
$337$	−3250.32	−0.525389	−0.262695	−	0.964879i	$-0.584611\pi$
$337$	−3250.32	−0.525389	−0.262695	+	0.964879i	$0.584611\pi$
$338$	−10034.5	−1.61480
$339$	−8759.43	−1.40338
$340$	−355.297	−0.0566726
$341$	4852.92	0.770675
$342$	4.04987	0.000640327 0
$343$	−1404.60	−0.221112
$344$	−2515.85	−0.394319
$345$	−353.265	−0.0551279
$346$	−2327.42	−0.361626
$347$	11392.1	1.76241	0.881207	−	0.472731i	$-0.156732\pi$
$347$	11392.1	1.76241	0.881207	+	0.472731i	$0.156732\pi$
$348$	−3642.91	−0.561150
$349$	8112.23	1.24423	0.622117	−	0.782924i	$-0.286273\pi$
$349$	8112.23	1.24423	0.622117	+	0.782924i	$0.286273\pi$
$350$	6220.12	0.949942
$351$	−11324.7	−1.72213
$352$	810.028	0.122655
$353$	−10619.6	−1.60121	−0.800603	−	0.599195i	$-0.795487\pi$
$353$	−10619.6	−1.60121	−0.800603	+	0.599195i	$0.795487\pi$
$354$	3620.17	0.543531
$355$	−275.157	−0.0411375
$356$	5515.59	0.821140
$357$	11549.8	1.71227
$358$	1328.02	0.196057
$359$	−3723.15	−0.547354	−0.273677	−	0.961822i	$-0.588240\pi$
$359$	−3723.15	−0.547354	−0.273677	+	0.961822i	$0.588240\pi$
$360$	20.2456	0.00296399
$361$	−6858.30	−0.999898
$362$	−7134.56	−1.03587
$363$	−3743.66	−0.541297
$364$	8527.85	1.22797
$365$	1002.63	0.143781
$366$	654.828	0.0935203
$367$	−469.720	−0.0668097	−0.0334049	−	0.999442i	$-0.510635\pi$
$367$	−469.720	−0.0668097	−0.0334049	+	0.999442i	$0.510635\pi$
$368$	995.354	0.140996
$369$	−539.317	−0.0760859
$370$	−77.4773	−0.0108861
$371$	−3103.63	−0.434320
$372$	−4159.23	−0.579694
$373$	2734.10	0.379534	0.189767	−	0.981829i	$-0.439227\pi$
$373$	2734.10	0.379534	0.189767	+	0.981829i	$0.439227\pi$
$374$	4295.06	0.593829
$375$	1413.43	0.194638
$376$	2859.36	0.392181
$377$	−14262.1	−1.94837
$378$	6693.41	0.910772
$379$	−7906.49	−1.07158	−0.535790	−	0.844351i	$-0.679986\pi$
$379$	−7906.49	−1.07158	−0.535790	+	0.844351i	$0.679986\pi$
$380$	3.50846	0.000473632 0
$381$	1038.67	0.139666
$382$	−810.686	−0.108582
$383$	14257.1	1.90210	0.951048	−	0.309042i	$-0.100009\pi$
$383$	14257.1	1.90210	0.951048	+	0.309042i	$0.100009\pi$
$384$	−694.241	−0.0922599
$385$	665.238	0.0880615
$386$	3330.02	0.439103
$387$	760.139	0.0998450
$388$	4497.64	0.588487
$389$	5199.84	0.677744	0.338872	−	0.940832i	$-0.389955\pi$
$389$	5199.84	0.677744	0.338872	+	0.940832i	$0.389955\pi$
$390$	964.645	0.125248
$391$	5277.72	0.682624
$392$	−2296.33	−0.295873
$393$	−4769.42	−0.612176
$394$	6340.10	0.810684
$395$	−177.785	−0.0226464
$396$	−244.742	−0.0310574
$397$	1171.82	0.148141	0.0740704	−	0.997253i	$-0.476401\pi$
$397$	1171.82	0.148141	0.0740704	+	0.997253i	$0.476401\pi$
$398$	4048.55	0.509888
$399$	−114.051	−0.0143100
$400$	−1982.46	−0.247808
$401$	8339.51	1.03854	0.519271	−	0.854610i	$-0.326203\pi$
$401$	8339.51	1.03854	0.519271	+	0.854610i	$0.326203\pi$
$402$	−5631.95	−0.698747
$403$	−16283.5	−2.01275
$404$	−7512.50	−0.925151
$405$	825.468	0.101279
$406$	8429.51	1.03042
$407$	936.595	0.114067
$408$	−3681.11	−0.446672
$409$	9113.91	1.10184	0.550922	−	0.834557i	$-0.314276\pi$
$409$	9113.91	1.10184	0.550922	+	0.834557i	$0.314276\pi$
$410$	−467.217	−0.0562786
$411$	13929.7	1.67179
$412$	1931.73	0.230994
$413$	−8376.90	−0.998064
$414$	−300.736	−0.0357014
$415$	−2.69477	−0.000318749 0
$416$	−2717.97	−0.320335
$417$	9911.37	1.16394
$418$	−42.4124	−0.00496282
$419$	9931.99	1.15802	0.579009	−	0.815321i	$-0.303440\pi$
$419$	9931.99	1.15802	0.579009	+	0.815321i	$0.303440\pi$
$420$	−570.148	−0.0662389
$421$	−13696.5	−1.58557	−0.792787	−	0.609498i	$-0.791371\pi$
$421$	−13696.5	−1.58557	−0.792787	+	0.609498i	$0.791371\pi$
$422$	−11294.2	−1.30283
$423$	−863.925	−0.0993037
$424$	989.182	0.113299
$425$	−10511.7	−1.19975
$426$	−2850.81	−0.324230
$427$	−1515.24	−0.171728
$428$	7090.47	0.800773
$429$	−11661.2	−1.31238
$430$	658.518	0.0738525
$431$	568.024	0.0634820	0.0317410	−	0.999496i	$-0.489895\pi$
$431$	568.024	0.0634820	0.0317410	+	0.999496i	$0.489895\pi$
$432$	−2133.30	−0.237589
$433$	−2925.55	−0.324695	−0.162347	−	0.986734i	$-0.551906\pi$
$433$	−2925.55	−0.324695	−0.162347	+	0.986734i	$0.551906\pi$
$434$	9624.26	1.06447
$435$	953.522	0.105099
$436$	−5111.49	−0.561459
$437$	−52.1160	−0.00570491
$438$	10387.9	1.13323
$439$	−7551.91	−0.821032	−0.410516	−	0.911853i	$-0.634651\pi$
$439$	−7551.91	−0.821032	−0.410516	+	0.911853i	$0.634651\pi$
$440$	−212.023	−0.0229723
$441$	693.812	0.0749176
$442$	−14411.7	−1.55089
$443$	9530.59	1.02215	0.511074	−	0.859536i	$-0.329248\pi$
$443$	9530.59	1.02215	0.511074	+	0.859536i	$0.329248\pi$
$444$	−802.716	−0.0858000
$445$	−1443.69	−0.153792
$446$	12668.9	1.34505
$447$	9923.41	1.05002
$448$	1606.44	0.169413
$449$	6271.47	0.659174	0.329587	−	0.944125i	$-0.393091\pi$
$449$	6271.47	0.659174	0.329587	+	0.944125i	$0.393091\pi$
$450$	598.980	0.0627471
$451$	5648.02	0.589700
$452$	−6460.05	−0.672246
$453$	−9971.79	−1.03425
$454$	2645.21	0.273449
$455$	−2232.14	−0.229988
$456$	36.3499	0.00373298
$457$	9688.56	0.991711	0.495855	−	0.868405i	$-0.334855\pi$
$457$	9688.56	0.991711	0.495855	+	0.868405i	$0.334855\pi$
$458$	5539.63	0.565174
$459$	−11311.5	−1.15028
$460$	−260.532	−0.0264073
$461$	6755.66	0.682522	0.341261	−	0.939969i	$-0.389146\pi$
$461$	6755.66	0.682522	0.341261	+	0.939969i	$0.389146\pi$
$462$	6892.31	0.694067
$463$	2869.08	0.287986	0.143993	−	0.989579i	$-0.454006\pi$
$463$	2869.08	0.287986	0.143993	+	0.989579i	$0.454006\pi$
$464$	−2686.63	−0.268801
$465$	1088.67	0.108572
$466$	1402.49	0.139419
$467$	3216.08	0.318678	0.159339	−	0.987224i	$-0.449064\pi$
$467$	3216.08	0.318678	0.159339	+	0.987224i	$0.449064\pi$
$468$	821.207	0.0811118
$469$	13032.1	1.28308
$470$	−748.430	−0.0734521
$471$	−10117.7	−0.989806
$472$	2669.86	0.260361
$473$	−7960.59	−0.773844
$474$	−1841.97	−0.178491
$475$	103.800	0.0100267
$476$	8517.92	0.820206
$477$	−298.871	−0.0286884
$478$	−3683.48	−0.352465
$479$	−19576.0	−1.86733	−0.933665	−	0.358147i	$-0.883409\pi$
$479$	−19576.0	−1.86733	−0.933665	+	0.358147i	$0.883409\pi$
$480$	181.716	0.0172795
$481$	−3142.65	−0.297906
$482$	12383.8	1.17026
$483$	8469.19	0.797850
$484$	−2760.93	−0.259291
$485$	−1177.25	−0.110219
$486$	1352.49	0.126235
$487$	14527.2	1.35172	0.675862	−	0.737028i	$-0.263771\pi$
$487$	14527.2	1.35172	0.675862	+	0.737028i	$0.263771\pi$
$488$	482.934	0.0447979
$489$	2138.82	0.197793
$490$	601.059	0.0554144
$491$	1116.06	0.102580	0.0512902	−	0.998684i	$-0.483667\pi$
$491$	1116.06	0.102580	0.0512902	+	0.998684i	$0.483667\pi$
$492$	−4840.68	−0.443566
$493$	−14245.5	−1.30139
$494$	142.311	0.0129613
$495$	64.0605	0.00581678
$496$	−3067.42	−0.277684
$497$	6596.63	0.595371
$498$	−27.9195	−0.00251226
$499$	−135.662	−0.0121704	−0.00608522	−	0.999981i	$-0.501937\pi$
$499$	−135.662	−0.0121704	−0.00608522	+	0.999981i	$0.501937\pi$
$500$	1042.40	0.0932351
$501$	19126.7	1.70562
$502$	−8924.00	−0.793422
$503$	−9265.74	−0.821350	−0.410675	−	0.911782i	$-0.634707\pi$
$503$	−9265.74	−0.821350	−0.410675	+	0.911782i	$0.634707\pi$
$504$	−485.369	−0.0428969
$505$	1966.38	0.173273
$506$	3149.47	0.276702
$507$	27212.2	2.38370
$508$	766.017	0.0669026
$509$	4538.53	0.395220	0.197610	−	0.980281i	$-0.436682\pi$
$509$	4538.53	0.395220	0.197610	+	0.980281i	$0.436682\pi$
$510$	963.523	0.0836578
$511$	−24037.2	−2.08091
$512$	−512.000	−0.0441942
$513$	111.698	0.00961325
$514$	5427.87	0.465784
$515$	−505.625	−0.0432631
$516$	6822.68	0.582077
$517$	9047.50	0.769649
$518$	1857.45	0.157551
$519$	6311.67	0.533818
$520$	711.423	0.0599960
$521$	−8549.25	−0.718905	−0.359452	−	0.933163i	$-0.617036\pi$
$521$	−8549.25	−0.718905	−0.359452	+	0.933163i	$0.617036\pi$
$522$	811.738	0.0680628
$523$	16633.7	1.39071	0.695356	−	0.718665i	$-0.255247\pi$
$523$	16633.7	1.39071	0.695356	+	0.718665i	$0.255247\pi$
$524$	−3517.43	−0.293243
$525$	−16868.2	−1.40227
$526$	−9989.40	−0.828058
$527$	−16264.5	−1.34439
$528$	−2196.70	−0.181059
$529$	−8296.96	−0.681923
$530$	−258.916	−0.0212200
$531$	−806.672	−0.0659257
$532$	−84.1119	−0.00685473
$533$	−18951.4	−1.54011
$534$	−14957.6	−1.21213
$535$	−1855.91	−0.149978
$536$	−4153.55	−0.334712
$537$	−3601.44	−0.289411
$538$	−11882.0	−0.952171
$539$	−7265.98	−0.580645
$540$	558.387	0.0444985
$541$	−16356.5	−1.29986	−0.649929	−	0.759995i	$-0.725201\pi$
$541$	−16356.5	−1.29986	−0.649929	+	0.759995i	$0.725201\pi$
$542$	15469.5	1.22596
$543$	19348.0	1.52911
$544$	−2714.81	−0.213964
$545$	1337.92	0.105156
$546$	−23126.5	−1.81268
$547$	12816.6	1.00183	0.500913	−	0.865498i	$-0.332998\pi$
$547$	12816.6	1.00183	0.500913	+	0.865498i	$0.332998\pi$
$548$	10273.1	0.800815
$549$	−145.913	−0.0113432
$550$	−6272.85	−0.486318
$551$	140.670	0.0108761
$552$	−2699.28	−0.208132
$553$	4262.23	0.327755
$554$	−10489.5	−0.804431
$555$	210.109	0.0160696
$556$	7309.60	0.557547
$557$	−16751.0	−1.27426	−0.637128	−	0.770758i	$-0.719878\pi$
$557$	−16751.0	−1.27426	−0.637128	+	0.770758i	$0.719878\pi$
$558$	926.788	0.0703120
$559$	26711.0	2.02103
$560$	−420.482	−0.0317296
$561$	−11647.7	−0.876587
$562$	11582.0	0.869318
$563$	18146.6	1.35841	0.679207	−	0.733947i	$-0.262324\pi$
$563$	18146.6	1.35841	0.679207	+	0.733947i	$0.262324\pi$
$564$	−7754.23	−0.578922
$565$	1690.90	0.125906
$566$	−14595.3	−1.08390
$567$	−19789.8	−1.46578
$568$	−2102.46	−0.155312
$569$	−15059.3	−1.10952	−0.554762	−	0.832009i	$-0.687191\pi$
$569$	−15059.3	−1.10952	−0.554762	+	0.832009i	$0.687191\pi$
$570$	−9.51450	−0.000699156 0
$571$	11397.3	0.835313	0.417656	−	0.908605i	$-0.362852\pi$
$571$	11397.3	0.835313	0.417656	+	0.908605i	$0.362852\pi$
$572$	−8600.13	−0.628653
$573$	2198.48	0.160284
$574$	11201.1	0.814503
$575$	−7708.01	−0.559037
$576$	154.696	0.0111904
$577$	8920.62	0.643623	0.321811	−	0.946804i	$-0.395708\pi$
$577$	8920.62	0.643623	0.321811	+	0.946804i	$0.395708\pi$
$578$	−4568.89	−0.328790
$579$	−9030.61	−0.648185
$580$	703.219	0.0503441
$581$	64.6045	0.00461316
$582$	−12197.1	−0.868701
$583$	3129.94	0.222348
$584$	7661.07	0.542838
$585$	−214.949	−0.0151915
$586$	6882.95	0.485208
$587$	17855.1	1.25546	0.627732	−	0.778430i	$-0.283983\pi$
$587$	17855.1	1.25546	0.627732	+	0.778430i	$0.283983\pi$
$588$	6227.36	0.436755
$589$	160.608	0.0112355
$590$	−698.830	−0.0487634
$591$	−17193.6	−1.19670
$592$	−592.000	−0.0410997
$593$	−7102.52	−0.491847	−0.245924	−	0.969289i	$-0.579091\pi$
$593$	−7102.52	−0.491847	−0.245924	+	0.969289i	$0.579091\pi$
$594$	−6750.14	−0.466265
$595$	−2229.55	−0.153618
$596$	7318.48	0.502981
$597$	−10979.2	−0.752675
$598$	−10567.8	−0.722655
$599$	13260.9	0.904548	0.452274	−	0.891879i	$-0.350613\pi$
$599$	13260.9	0.904548	0.452274	+	0.891879i	$0.350613\pi$
$600$	5376.19	0.365803
$601$	10939.2	0.742461	0.371231	−	0.928541i	$-0.378936\pi$
$601$	10939.2	0.742461	0.371231	+	0.928541i	$0.378936\pi$
$602$	−15787.4	−1.06885
$603$	1254.95	0.0847522
$604$	−7354.16	−0.495425
$605$	722.668	0.0485630
$606$	20373.0	1.36567
$607$	17647.1	1.18002	0.590010	−	0.807396i	$-0.299124\pi$
$607$	17647.1	1.18002	0.590010	+	0.807396i	$0.299124\pi$
$608$	26.8079	0.00178817
$609$	−22859.8	−1.52106
$610$	−126.407	−0.00839026
$611$	−30358.0	−2.01007
$612$	820.251	0.0541776
$613$	19542.2	1.28760	0.643801	−	0.765193i	$-0.277356\pi$
$613$	19542.2	1.28760	0.643801	+	0.765193i	$0.277356\pi$
$614$	−3199.34	−0.210284
$615$	1267.04	0.0830762
$616$	5083.05	0.332471
$617$	23046.4	1.50375	0.751873	−	0.659308i	$-0.229150\pi$
$617$	23046.4	1.50375	0.751873	+	0.659308i	$0.229150\pi$
$618$	−5238.61	−0.340983
$619$	27107.1	1.76014	0.880069	−	0.474845i	$-0.157496\pi$
$619$	27107.1	1.76014	0.880069	+	0.474845i	$0.157496\pi$
$620$	802.889	0.0520077
$621$	−8294.51	−0.535986
$622$	−9034.95	−0.582425
$623$	34611.2	2.22579
$624$	7370.81	0.472866
$625$	15215.1	0.973768
$626$	−5518.23	−0.352321
$627$	115.017	0.00732592
$628$	−7461.76	−0.474135
$629$	−3139.00	−0.198983
$630$	127.044	0.00803423
$631$	−8405.89	−0.530322	−0.265161	−	0.964204i	$-0.585425\pi$
$631$	−8405.89	−0.530322	−0.265161	+	0.964204i	$0.585425\pi$
$632$	−1358.45	−0.0855002
$633$	30628.5	1.92318
$634$	2276.34	0.142595
$635$	−200.503	−0.0125303
$636$	−2682.54	−0.167248
$637$	24380.3	1.51646
$638$	−8500.96	−0.527518
$639$	635.236	0.0393264
$640$	134.015	0.00827719
$641$	5098.12	0.314140	0.157070	−	0.987587i	$-0.449795\pi$
$641$	5098.12	0.314140	0.157070	+	0.987587i	$0.449795\pi$
$642$	−19228.5	−1.18207
$643$	−6770.39	−0.415238	−0.207619	−	0.978210i	$-0.566571\pi$
$643$	−6770.39	−0.415238	−0.207619	+	0.978210i	$0.566571\pi$
$644$	6246.00	0.382185
$645$	−1785.82	−0.109018
$646$	142.145	0.00865732
$647$	−4462.54	−0.271160	−0.135580	−	0.990766i	$-0.543290\pi$
$647$	−4462.54	−0.271160	−0.135580	+	0.990766i	$0.543290\pi$
$648$	6307.36	0.382371
$649$	8447.91	0.510954
$650$	21047.9	1.27010
$651$	−26099.8	−1.57132
$652$	1577.37	0.0947463
$653$	−16729.5	−1.00257	−0.501283	−	0.865283i	$-0.667138\pi$
$653$	−16729.5	−1.00257	−0.501283	+	0.865283i	$0.667138\pi$
$654$	13861.7	0.828803
$655$	920.678	0.0549220
$656$	−3569.98	−0.212476
$657$	−2314.71	−0.137451
$658$	17942.9	1.06305
$659$	−11738.3	−0.693866	−0.346933	−	0.937890i	$-0.612777\pi$
$659$	−11738.3	−0.693866	−0.346933	+	0.937890i	$0.612777\pi$
$660$	574.980	0.0339107
$661$	−16479.9	−0.969731	−0.484865	−	0.874589i	$-0.661131\pi$
$661$	−16479.9	−0.969731	−0.484865	+	0.874589i	$0.661131\pi$
$662$	15725.1	0.923220
$663$	39082.7	2.28936
$664$	−20.5906	−0.00120342
$665$	22.0161	0.00128383
$666$	178.867	0.0104068
$667$	−10445.9	−0.606397
$668$	14105.8	0.817023
$669$	−34356.6	−1.98550
$670$	1087.18	0.0626887
$671$	1528.09	0.0879152
$672$	−4356.47	−0.250081
$673$	−2113.10	−0.121031	−0.0605157	−	0.998167i	$-0.519275\pi$
$673$	−2113.10	−0.121031	−0.0605157	+	0.998167i	$0.519275\pi$
$674$	6500.64	0.371506
$675$	16520.3	0.942024
$676$	20068.9	1.14184
$677$	13043.5	0.740477	0.370239	−	0.928937i	$-0.379276\pi$
$677$	13043.5	0.740477	0.370239	+	0.928937i	$0.379276\pi$
$678$	17518.9	0.992342
$679$	28223.4	1.59516
$680$	710.595	0.0400736
$681$	−7173.47	−0.403654
$682$	−9705.83	−0.544950
$683$	12304.5	0.689339	0.344669	−	0.938724i	$-0.387991\pi$
$683$	12304.5	0.689339	0.344669	+	0.938724i	$0.387991\pi$
$684$	−8.09974	−0.000452780 0
$685$	−2688.97	−0.149986
$686$	2809.21	0.156350
$687$	−15022.8	−0.834287
$688$	5031.70	0.278825
$689$	−10502.2	−0.580700
$690$	706.530	0.0389813
$691$	−21060.4	−1.15944	−0.579722	−	0.814815i	$-0.696839\pi$
$691$	−21060.4	−1.15944	−0.579722	+	0.814815i	$0.696839\pi$
$692$	4654.84	0.255709
$693$	−1535.79	−0.0841845
$694$	−22784.1	−1.24621
$695$	−1913.27	−0.104424
$696$	7285.81	0.396793
$697$	−18929.3	−1.02869
$698$	−16224.5	−0.879807
$699$	−3803.39	−0.205805
$700$	−12440.2	−0.671710
$701$	−6646.99	−0.358136	−0.179068	−	0.983837i	$-0.557308\pi$
$701$	−6646.99	−0.358136	−0.179068	+	0.983837i	$0.557308\pi$
$702$	22649.5	1.21773
$703$	30.9967	0.00166296
$704$	−1620.06	−0.0867303
$705$	2029.65	0.108427
$706$	21239.3	1.13222
$707$	−47142.1	−2.50772
$708$	−7240.34	−0.384334
$709$	14189.2	0.751602	0.375801	−	0.926700i	$-0.377368\pi$
$709$	14189.2	0.751602	0.375801	+	0.926700i	$0.377368\pi$
$710$	550.314	0.0290886
$711$	410.441	0.0216494
$712$	−11031.2	−0.580634
$713$	−11926.4	−0.626435
$714$	−23099.6	−1.21076
$715$	2251.06	0.117741
$716$	−2656.05	−0.138633
$717$	9989.14	0.520295
$718$	7446.29	0.387038
$719$	−17967.4	−0.931949	−0.465974	−	0.884798i	$-0.654296\pi$
$719$	−17967.4	−0.931949	−0.465974	+	0.884798i	$0.654296\pi$
$720$	−40.4912	−0.00209586
$721$	12121.9	0.626134
$722$	13716.6	0.707034
$723$	−33583.3	−1.72749
$724$	14269.1	0.732469
$725$	20805.2	1.06578
$726$	7487.31	0.382755
$727$	−19431.8	−0.991316	−0.495658	−	0.868518i	$-0.665073\pi$
$727$	−19431.8	−0.991316	−0.495658	+	0.868518i	$0.665073\pi$
$728$	−17055.7	−0.868305
$729$	17619.6	0.895166
$730$	−2005.27	−0.101669
$731$	26679.9	1.34992
$732$	−1309.66	−0.0661288
$733$	−11122.1	−0.560440	−0.280220	−	0.959936i	$-0.590407\pi$
$733$	−11122.1	−0.560440	−0.280220	+	0.959936i	$0.590407\pi$
$734$	939.440	0.0472416
$735$	−1630.00	−0.0818005
$736$	−1990.71	−0.0996990
$737$	−13142.5	−0.656868
$738$	1078.63	0.0538009
$739$	−17295.6	−0.860935	−0.430467	−	0.902606i	$-0.641651\pi$
$739$	−17295.6	−0.860935	−0.430467	+	0.902606i	$0.641651\pi$
$740$	154.955	0.00769763
$741$	−385.930	−0.0191329
$742$	6207.27	0.307111
$743$	−17032.9	−0.841020	−0.420510	−	0.907288i	$-0.638149\pi$
$743$	−17032.9	−0.841020	−0.420510	+	0.907288i	$0.638149\pi$
$744$	8318.46	0.409905
$745$	−1915.59	−0.0942039
$746$	−5468.19	−0.268371
$747$	6.22123	0.000304716 0
$748$	−8590.12	−0.419901
$749$	44493.8	2.17058
$750$	−2826.86	−0.137630
$751$	−14499.9	−0.704537	−0.352268	−	0.935899i	$-0.614590\pi$
$751$	−14499.9	−0.704537	−0.352268	+	0.935899i	$0.614590\pi$
$752$	−5718.71	−0.277314
$753$	24200.8	1.17122
$754$	28524.2	1.37770
$755$	1924.93	0.0927888
$756$	−13386.8	−0.644013
$757$	26710.7	1.28246	0.641228	−	0.767351i	$-0.278425\pi$
$757$	26710.7	1.28246	0.641228	+	0.767351i	$0.278425\pi$
$758$	15813.0	0.757722
$759$	−8540.98	−0.408456
$760$	−7.01691	−0.000334908 0
$761$	−5605.30	−0.267006	−0.133503	−	0.991048i	$-0.542623\pi$
$761$	−5605.30	−0.267006	−0.133503	+	0.991048i	$0.542623\pi$
$762$	−2077.34	−0.0987589
$763$	−32075.4	−1.52190
$764$	1621.37	0.0767791
$765$	−214.699	−0.0101470
$766$	−28514.2	−1.34499
$767$	−28346.1	−1.33445
$768$	1388.48	0.0652376
$769$	18151.9	0.851204	0.425602	−	0.904910i	$-0.360062\pi$
$769$	18151.9	0.851204	0.425602	+	0.904910i	$0.360062\pi$
$770$	−1330.48	−0.0622689
$771$	−14719.7	−0.687572
$772$	−6660.04	−0.310492
$773$	−33298.4	−1.54937	−0.774684	−	0.632349i	$-0.782091\pi$
$773$	−33298.4	−1.54937	−0.774684	+	0.632349i	$0.782091\pi$
$774$	−1520.28	−0.0706011
$775$	23754.0	1.10099
$776$	−8995.28	−0.416123
$777$	−5037.17	−0.232571
$778$	−10399.7	−0.479238
$779$	186.922	0.00859713
$780$	−1929.29	−0.0885637
$781$	−6652.54	−0.304797
$782$	−10555.4	−0.482688
$783$	22388.3	1.02183
$784$	4592.66	0.209214
$785$	1953.10	0.0888014
$786$	9538.83	0.432874
$787$	497.339	0.0225263	0.0112632	−	0.999937i	$-0.496415\pi$
$787$	497.339	0.0225263	0.0112632	+	0.999937i	$0.496415\pi$
$788$	−12680.2	−0.573240
$789$	27090.0	1.22235
$790$	355.570	0.0160135
$791$	−40537.8	−1.82220
$792$	489.483	0.0219609
$793$	−5127.34	−0.229606
$794$	−2343.64	−0.104751
$795$	702.148	0.0313241
$796$	−8097.09	−0.360545
$797$	21437.1	0.952747	0.476374	−	0.879243i	$-0.341951\pi$
$797$	21437.1	0.952747	0.476374	+	0.879243i	$0.341951\pi$
$798$	228.101	0.0101187
$799$	−30322.7	−1.34260
$800$	3964.92	0.175226
$801$	3332.96	0.147022
$802$	−16679.0	−0.734360
$803$	24240.9	1.06531
$804$	11263.9	0.494089
$805$	−1634.88	−0.0715799
$806$	32567.0	1.42323
$807$	32222.4	1.40556
$808$	15025.0	0.654180
$809$	−40500.4	−1.76010	−0.880049	−	0.474884i	$-0.842490\pi$
$809$	−40500.4	−1.76010	−0.880049	+	0.474884i	$0.842490\pi$
$810$	−1650.94	−0.0716148
$811$	−26180.7	−1.13358	−0.566788	−	0.823864i	$-0.691814\pi$
$811$	−26180.7	−1.13358	−0.566788	+	0.823864i	$0.691814\pi$
$812$	−16859.0	−0.728615
$813$	−41951.3	−1.80971
$814$	−1873.19	−0.0806576
$815$	−412.873	−0.0177452
$816$	7362.23	0.315845
$817$	−263.456	−0.0112817
$818$	−18227.8	−0.779121
$819$	5153.20	0.219863
$820$	934.435	0.0397950
$821$	2619.49	0.111353	0.0556764	−	0.998449i	$-0.482268\pi$
$821$	2619.49	0.111353	0.0556764	+	0.998449i	$0.482268\pi$
$822$	−27859.5	−1.18213
$823$	−36777.6	−1.55770	−0.778850	−	0.627210i	$-0.784197\pi$
$823$	−36777.6	−1.55770	−0.778850	+	0.627210i	$0.784197\pi$
$824$	−3863.46	−0.163337
$825$	17011.2	0.717883
$826$	16753.8	0.705738
$827$	24047.1	1.01112	0.505562	−	0.862790i	$-0.331285\pi$
$827$	24047.1	1.01112	0.505562	+	0.862790i	$0.331285\pi$
$828$	601.472	0.0252447
$829$	25838.5	1.08252	0.541259	−	0.840856i	$-0.317948\pi$
$829$	25838.5	1.08252	0.541259	+	0.840856i	$0.317948\pi$
$830$	5.38953	0.000225390 0
$831$	28446.1	1.18747
$832$	5435.94	0.226511
$833$	24351.9	1.01290
$834$	−19822.7	−0.823028
$835$	−3692.17	−0.153021
$836$	84.8249	0.00350925
$837$	25561.5	1.05560
$838$	−19864.0	−0.818842
$839$	19796.3	0.814592	0.407296	−	0.913296i	$-0.366472\pi$
$839$	19796.3	0.814592	0.407296	+	0.913296i	$0.366472\pi$
$840$	1140.30	0.0468380
$841$	3806.25	0.156064
$842$	27393.0	1.12117
$843$	−31408.9	−1.28325
$844$	22588.4	0.921240
$845$	−5252.99	−0.213856
$846$	1727.85	0.0702184
$847$	−17325.3	−0.702838
$848$	−1978.36	−0.0801147
$849$	39580.6	1.60000
$850$	21023.4	0.848351
$851$	−2301.76	−0.0927182
$852$	5701.61	0.229265
$853$	35837.2	1.43850	0.719252	−	0.694749i	$-0.244485\pi$
$853$	35837.2	1.43850	0.719252	+	0.694749i	$0.244485\pi$
$854$	3030.48	0.121430
$855$	2.12009	8.48017e−5 0
$856$	−14180.9	−0.566232
$857$	−17095.8	−0.681423	−0.340712	−	0.940168i	$-0.610668\pi$
$857$	−17095.8	−0.681423	−0.340712	+	0.940168i	$0.610668\pi$
$858$	23322.5	0.927992
$859$	−5453.02	−0.216594	−0.108297	−	0.994119i	$-0.534540\pi$
$859$	−5453.02	−0.216594	−0.108297	+	0.994119i	$0.534540\pi$
$860$	−1317.04	−0.0522216
$861$	−30376.0	−1.20234
$862$	−1136.05	−0.0448886
$863$	49898.6	1.96822	0.984108	−	0.177573i	$-0.0568246\pi$
$863$	49898.6	1.96822	0.984108	+	0.177573i	$0.0568246\pi$
$864$	4266.61	0.168001
$865$	−1218.39	−0.0478920
$866$	5851.09	0.229594
$867$	12390.3	0.485346
$868$	−19248.5	−0.752693
$869$	−4298.36	−0.167793
$870$	−1907.04	−0.0743159
$871$	44098.5	1.71552
$872$	10223.0	0.397011
$873$	2717.83	0.105366
$874$	104.232	0.00403398
$875$	6541.22	0.252724
$876$	−20775.9	−0.801315
$877$	−27631.7	−1.06392	−0.531959	−	0.846770i	$-0.678544\pi$
$877$	−27631.7	−1.06392	−0.531959	+	0.846770i	$0.678544\pi$
$878$	15103.8	0.580557
$879$	−18665.7	−0.716244
$880$	424.046	0.0162438
$881$	3908.73	0.149476	0.0747380	−	0.997203i	$-0.476188\pi$
$881$	3908.73	0.149476	0.0747380	+	0.997203i	$0.476188\pi$
$882$	−1387.62	−0.0529747
$883$	22038.5	0.839927	0.419964	−	0.907541i	$-0.362043\pi$
$883$	22038.5	0.839927	0.419964	+	0.907541i	$0.362043\pi$
$884$	28823.3	1.09664
$885$	1895.14	0.0719825
$886$	−19061.2	−0.722768
$887$	−14295.5	−0.541147	−0.270574	−	0.962699i	$-0.587213\pi$
$887$	−14295.5	−0.541147	−0.270574	+	0.962699i	$0.587213\pi$
$888$	1605.43	0.0606698
$889$	4806.88	0.181347
$890$	2887.39	0.108748
$891$	19957.6	0.750397
$892$	−25337.9	−0.951092
$893$	299.428	0.0112206
$894$	−19846.8	−0.742479
$895$	695.215	0.0259648
$896$	−3212.88	−0.119793
$897$	28658.4	1.06675
$898$	−12542.9	−0.466106
$899$	32191.5	1.19427
$900$	−1197.96	−0.0443689
$901$	−10490.0	−0.387872
$902$	−11296.0	−0.416981
$903$	42813.4	1.57779
$904$	12920.1	0.475350
$905$	−3734.91	−0.137185
$906$	19943.6	0.731326
$907$	−33095.8	−1.21161	−0.605803	−	0.795615i	$-0.707148\pi$
$907$	−33095.8	−1.21161	−0.605803	+	0.795615i	$0.707148\pi$
$908$	−5290.41	−0.193357
$909$	−4539.65	−0.165644
$910$	4464.29	0.162626
$911$	6252.85	0.227405	0.113703	−	0.993515i	$-0.463729\pi$
$911$	6252.85	0.227405	0.113703	+	0.993515i	$0.463729\pi$
$912$	−72.6998	−0.00263962
$913$	−65.1521	−0.00236169
$914$	−19377.1	−0.701245
$915$	342.800	0.0123854
$916$	−11079.3	−0.399638
$917$	−22072.4	−0.794869
$918$	22623.1	0.813370
$919$	−33150.4	−1.18992	−0.594958	−	0.803757i	$-0.702831\pi$
$919$	−33150.4	−1.18992	−0.594958	+	0.803757i	$0.702831\pi$
$920$	521.063	0.0186728
$921$	8676.21	0.310413
$922$	−13511.3	−0.482616
$923$	22322.0	0.796031
$924$	−13784.6	−0.490780
$925$	4584.44	0.162957
$926$	−5738.17	−0.203637
$927$	1167.30	0.0413584
$928$	5373.26	0.190071
$929$	680.168	0.0240211	0.0120105	−	0.999928i	$-0.496177\pi$
$929$	680.168	0.0240211	0.0120105	+	0.999928i	$0.496177\pi$
$930$	−2177.34	−0.0767717
$931$	−240.468	−0.00846511
$932$	−2804.99	−0.0985842
$933$	24501.7	0.859752
$934$	−6432.16	−0.225339
$935$	2248.44	0.0786438
$936$	−1642.41	−0.0573547
$937$	−24844.5	−0.866206	−0.433103	−	0.901345i	$-0.642581\pi$
$937$	−24844.5	−0.866206	−0.433103	+	0.901345i	$0.642581\pi$
$938$	−26064.1	−0.907275
$939$	14964.8	0.520082
$940$	1496.86	0.0519385
$941$	13694.7	0.474427	0.237213	−	0.971458i	$-0.423766\pi$
$941$	13694.7	0.474427	0.237213	+	0.971458i	$0.423766\pi$
$942$	20235.4	0.699899
$943$	−13880.5	−0.479332
$944$	−5339.73	−0.184103
$945$	3503.97	0.120618
$946$	15921.2	0.547190
$947$	−12805.0	−0.439394	−0.219697	−	0.975568i	$-0.570507\pi$
$947$	−12805.0	−0.439394	−0.219697	+	0.975568i	$0.570507\pi$
$948$	3683.94	0.126212
$949$	−81338.2	−2.78224
$950$	−207.600	−0.00708994
$951$	−6173.15	−0.210492
$952$	−17035.8	−0.579973
$953$	25643.2	0.871630	0.435815	−	0.900036i	$-0.356460\pi$
$953$	25643.2	0.871630	0.435815	+	0.900036i	$0.356460\pi$
$954$	597.742	0.0202858
$955$	−424.390	−0.0143801
$956$	7366.96	0.249231
$957$	23053.6	0.778700
$958$	39152.0	1.32040
$959$	64465.5	2.17070
$960$	−363.432	−0.0122184
$961$	6963.09	0.233731
$962$	6285.31	0.210651
$963$	4284.62	0.143375
$964$	−24767.6	−0.827500
$965$	1743.25	0.0581525
$966$	−16938.4	−0.564165
$967$	−49771.2	−1.65515	−0.827577	−	0.561351i	$-0.810282\pi$
$967$	−49771.2	−1.65515	−0.827577	+	0.561351i	$0.810282\pi$
$968$	5521.86	0.183347
$969$	−385.481	−0.0127796
$970$	2354.49	0.0779363
$971$	−32058.7	−1.05954	−0.529769	−	0.848142i	$-0.677721\pi$
$971$	−32058.7	−1.05954	−0.529769	+	0.848142i	$0.677721\pi$
$972$	−2704.98	−0.0892615
$973$	45868.9	1.51129
$974$	−29054.4	−0.955814
$975$	−57079.4	−1.87488
$976$	−965.867	−0.0316769
$977$	1683.92	0.0551417	0.0275709	−	0.999620i	$-0.491223\pi$
$977$	1683.92	0.0551417	0.0275709	+	0.999620i	$0.491223\pi$
$978$	−4277.63	−0.139861
$979$	−34904.6	−1.13948
$980$	−1202.12	−0.0391839
$981$	−3088.77	−0.100527
$982$	−2232.12	−0.0725353
$983$	−101.059	−0.00327903	−0.00163952	−	0.999999i	$-0.500522\pi$
$983$	−101.059	−0.00327903	−0.00163952	+	0.999999i	$0.500522\pi$
$984$	9681.36	0.313649
$985$	3319.01	0.107363
$986$	28491.0	0.920220
$987$	−48659.0	−1.56923
$988$	−284.622	−0.00916501
$989$	19563.8	0.629011
$990$	−128.121	−0.00411309
$991$	36883.5	1.18228	0.591141	−	0.806568i	$-0.298678\pi$
$991$	36883.5	1.18228	0.591141	+	0.806568i	$0.298678\pi$
$992$	6134.83	0.196352
$993$	−42644.4	−1.36282
$994$	−13193.3	−0.420991
$995$	2119.39	0.0675270
$996$	55.8391	0.00177644
$997$	26013.8	0.826343	0.413172	−	0.910653i	$-0.364421\pi$
$997$	26013.8	0.826343	0.413172	+	0.910653i	$0.364421\pi$
$998$	271.323	0.00860580
$999$	4933.27	0.156238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.4.a.c.1.2	✓	3
3.2	odd	2		666.4.a.n.1.2		3
4.3	odd	2		592.4.a.c.1.2		3
5.4	even	2		1850.4.a.i.1.2		3
8.3	odd	2		2368.4.a.f.1.2		3
8.5	even	2		2368.4.a.e.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.4.a.c.1.2	✓	3	1.1	even	1	trivial
592.4.a.c.1.2		3	4.3	odd	2
666.4.a.n.1.2		3	3.2	odd	2
1850.4.a.i.1.2		3	5.4	even	2
2368.4.a.e.1.2		3	8.5	even	2
2368.4.a.f.1.2		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)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +5.42375 q^{3} +4.00000 q^{4} -1.04699 q^{5} -10.8475 q^{6} +25.1006 q^{7} -8.00000 q^{8} +2.41712 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +5.42375 q^{3} +4.00000 q^{4} -1.04699 q^{5} -10.8475 q^{6} +25.1006 q^{7} -8.00000 q^{8} +2.41712 q^{9} +2.09398 q^{10} -25.3134 q^{11} +21.6950 q^{12} +84.9366 q^{13} -50.2012 q^{14} -5.67862 q^{15} +16.0000 q^{16} +84.8377 q^{17} -4.83423 q^{18} -0.837748 q^{19} -4.18796 q^{20} +136.140 q^{21} +50.6267 q^{22} +62.2096 q^{23} -43.3900 q^{24} -123.904 q^{25} -169.873 q^{26} -133.332 q^{27} +100.402 q^{28} -167.914 q^{29} +11.3572 q^{30} -191.714 q^{31} -32.0000 q^{32} -137.294 q^{33} -169.675 q^{34} -26.2801 q^{35} +9.66847 q^{36} -37.0000 q^{37} +1.67550 q^{38} +460.675 q^{39} +8.37592 q^{40} -223.124 q^{41} -272.279 q^{42} +314.482 q^{43} -101.253 q^{44} -2.53070 q^{45} -124.419 q^{46} -357.420 q^{47} +86.7801 q^{48} +287.041 q^{49} +247.808 q^{50} +460.139 q^{51} +339.746 q^{52} -123.648 q^{53} +266.663 q^{54} +26.5029 q^{55} -200.805 q^{56} -4.54374 q^{57} +335.829 q^{58} -333.733 q^{59} -22.7145 q^{60} -60.3667 q^{61} +383.427 q^{62} +60.6711 q^{63} +64.0000 q^{64} -88.9278 q^{65} +274.587 q^{66} +519.193 q^{67} +339.351 q^{68} +337.410 q^{69} +52.5602 q^{70} +262.807 q^{71} -19.3369 q^{72} -957.634 q^{73} +74.0000 q^{74} -672.024 q^{75} -3.35099 q^{76} -635.381 q^{77} -921.351 q^{78} +169.806 q^{79} -16.7518 q^{80} -788.420 q^{81} +446.248 q^{82} +2.57382 q^{83} +544.558 q^{84} -88.8243 q^{85} -628.963 q^{86} -910.727 q^{87} +202.507 q^{88} +1378.90 q^{89} +5.06140 q^{90} +2131.96 q^{91} +248.839 q^{92} -1039.81 q^{93} +714.839 q^{94} +0.877114 q^{95} -173.560 q^{96} +1124.41 q^{97} -574.082 q^{98} -61.1854 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{2} + 4 q^{3} + 12 q^{4} + 7 q^{5} - 8 q^{6} + 7 q^{7} - 24 q^{8} + 43 q^{9}+O(q^{10}) \) 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