LMFDB - Embedded newform 798.4.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [798,4,Mod(1,798)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(798, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("798.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 798 = 2 \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	798.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:	$47.0835241846$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 310x^{2} - 1444x - 1712 $ x^4 - 310x^2 - 1444x - 1712
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$19.6913$ of defining polynomial
Character	$\chi$	$=$	798.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +19.6913 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +19.6913 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -39.3826 q^{10} -56.8324 q^{11} -12.0000 q^{12} -6.27616 q^{13} +14.0000 q^{14} -59.0739 q^{15} +16.0000 q^{16} +33.3440 q^{17} -18.0000 q^{18} -19.0000 q^{19} +78.7652 q^{20} +21.0000 q^{21} +113.665 q^{22} -20.2681 q^{23} +24.0000 q^{24} +262.747 q^{25} +12.5523 q^{26} -27.0000 q^{27} -28.0000 q^{28} +199.154 q^{29} +118.148 q^{30} -143.252 q^{31} -32.0000 q^{32} +170.497 q^{33} -66.6881 q^{34} -137.839 q^{35} +36.0000 q^{36} +323.948 q^{37} +38.0000 q^{38} +18.8285 q^{39} -157.530 q^{40} +496.658 q^{41} -42.0000 q^{42} -435.150 q^{43} -227.330 q^{44} +177.222 q^{45} +40.5361 q^{46} +559.878 q^{47} -48.0000 q^{48} +49.0000 q^{49} -525.494 q^{50} -100.032 q^{51} -25.1046 q^{52} -709.463 q^{53} +54.0000 q^{54} -1119.10 q^{55} +56.0000 q^{56} +57.0000 q^{57} -398.309 q^{58} -256.402 q^{59} -236.296 q^{60} +261.074 q^{61} +286.504 q^{62} -63.0000 q^{63} +64.0000 q^{64} -123.586 q^{65} -340.994 q^{66} +774.153 q^{67} +133.376 q^{68} +60.8042 q^{69} +275.678 q^{70} +130.497 q^{71} -72.0000 q^{72} -674.281 q^{73} -647.897 q^{74} -788.241 q^{75} -76.0000 q^{76} +397.827 q^{77} -37.6569 q^{78} +927.944 q^{79} +315.061 q^{80} +81.0000 q^{81} -993.315 q^{82} +82.8695 q^{83} +84.0000 q^{84} +656.587 q^{85} +870.299 q^{86} -597.463 q^{87} +454.659 q^{88} -1122.12 q^{89} -354.443 q^{90} +43.9331 q^{91} -81.0722 q^{92} +429.756 q^{93} -1119.76 q^{94} -374.135 q^{95} +96.0000 q^{96} +830.799 q^{97} -98.0000 q^{98} -511.491 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) $ 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} - 66 q^{11} - 48 q^{12} + 10 q^{13} + 56 q^{14} + 64 q^{16} - 8 q^{17} - 72 q^{18} - 76 q^{19} + 84 q^{21} + 132 q^{22} - 250 q^{23} + 96 q^{24} + 120 q^{25} - 20 q^{26} - 108 q^{27} - 112 q^{28} - 66 q^{29} - 52 q^{31} - 128 q^{32} + 198 q^{33} + 16 q^{34} + 144 q^{36} + 288 q^{37} + 152 q^{38} - 30 q^{39} - 196 q^{41} - 168 q^{42} - 280 q^{43} - 264 q^{44} + 500 q^{46} + 498 q^{47} - 192 q^{48} + 196 q^{49} - 240 q^{50} + 24 q^{51} + 40 q^{52} - 250 q^{53} + 216 q^{54} - 1432 q^{55} + 224 q^{56} + 228 q^{57} + 132 q^{58} - 872 q^{59} + 600 q^{61} + 104 q^{62} - 252 q^{63} + 256 q^{64} - 908 q^{65} - 396 q^{66} + 1174 q^{67} - 32 q^{68} + 750 q^{69} - 1766 q^{71} - 288 q^{72} + 556 q^{73} - 576 q^{74} - 360 q^{75} - 304 q^{76} + 462 q^{77} + 60 q^{78} + 26 q^{79} + 324 q^{81} + 392 q^{82} + 674 q^{83} + 336 q^{84} + 2052 q^{85} + 560 q^{86} + 198 q^{87} + 528 q^{88} - 324 q^{89} - 70 q^{91} - 1000 q^{92} + 156 q^{93} - 996 q^{94} + 384 q^{96} + 2902 q^{97} - 392 q^{98} - 594 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 - 66 * q^11 - 48 * q^12 + 10 * q^13 + 56 * q^14 + 64 * q^16 - 8 * q^17 - 72 * q^18 - 76 * q^19 + 84 * q^21 + 132 * q^22 - 250 * q^23 + 96 * q^24 + 120 * q^25 - 20 * q^26 - 108 * q^27 - 112 * q^28 - 66 * q^29 - 52 * q^31 - 128 * q^32 + 198 * q^33 + 16 * q^34 + 144 * q^36 + 288 * q^37 + 152 * q^38 - 30 * q^39 - 196 * q^41 - 168 * q^42 - 280 * q^43 - 264 * q^44 + 500 * q^46 + 498 * q^47 - 192 * q^48 + 196 * q^49 - 240 * q^50 + 24 * q^51 + 40 * q^52 - 250 * q^53 + 216 * q^54 - 1432 * q^55 + 224 * q^56 + 228 * q^57 + 132 * q^58 - 872 * q^59 + 600 * q^61 + 104 * q^62 - 252 * q^63 + 256 * q^64 - 908 * q^65 - 396 * q^66 + 1174 * q^67 - 32 * q^68 + 750 * q^69 - 1766 * q^71 - 288 * q^72 + 556 * q^73 - 576 * q^74 - 360 * q^75 - 304 * q^76 + 462 * q^77 + 60 * q^78 + 26 * q^79 + 324 * q^81 + 392 * q^82 + 674 * q^83 + 336 * q^84 + 2052 * q^85 + 560 * q^86 + 198 * q^87 + 528 * q^88 - 324 * q^89 - 70 * q^91 - 1000 * q^92 + 156 * q^93 - 996 * q^94 + 384 * q^96 + 2902 * q^97 - 392 * q^98 - 594 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	19.6913	1.76124	0.880622	−	0.473820i	$-0.157125\pi$
$5$	19.6913	1.76124	0.880622	+	0.473820i	$0.157125\pi$
$6$	6.00000	0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	−39.3826	−1.24539
$11$	−56.8324	−1.55778	−0.778891	−	0.627159i	$-0.784218\pi$
$11$	−56.8324	−1.55778	−0.778891	+	0.627159i	$0.784218\pi$
$12$	−12.0000	−0.288675
$13$	−6.27616	−0.133899	−0.0669497	−	0.997756i	$-0.521327\pi$
$13$	−6.27616	−0.133899	−0.0669497	+	0.997756i	$0.521327\pi$
$14$	14.0000	0.267261
$15$	−59.0739	−1.01685
$16$	16.0000	0.250000
$17$	33.3440	0.475713	0.237856	−	0.971300i	$-0.423555\pi$
$17$	33.3440	0.475713	0.237856	+	0.971300i	$0.423555\pi$
$18$	−18.0000	−0.235702
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	78.7652	0.880622
$21$	21.0000	0.218218
$22$	113.665	1.10152
$23$	−20.2681	−0.183747	−0.0918735	−	0.995771i	$-0.529286\pi$
$23$	−20.2681	−0.183747	−0.0918735	+	0.995771i	$0.529286\pi$
$24$	24.0000	0.204124
$25$	262.747	2.10198
$26$	12.5523	0.0946812
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	199.154	1.27524	0.637622	−	0.770350i	$-0.279918\pi$
$29$	199.154	1.27524	0.637622	+	0.770350i	$0.279918\pi$
$30$	118.148	0.719024
$31$	−143.252	−0.829963	−0.414981	−	0.909830i	$-0.636212\pi$
$31$	−143.252	−0.829963	−0.414981	+	0.909830i	$0.636212\pi$
$32$	−32.0000	−0.176777
$33$	170.497	0.899386
$34$	−66.6881	−0.336380
$35$	−137.839	−0.665687
$36$	36.0000	0.166667
$37$	323.948	1.43937	0.719686	−	0.694300i	$-0.244286\pi$
$37$	323.948	1.43937	0.719686	+	0.694300i	$0.244286\pi$
$38$	38.0000	0.162221
$39$	18.8285	0.0773069
$40$	−157.530	−0.622693
$41$	496.658	1.89183	0.945913	−	0.324420i	$-0.105169\pi$
$41$	496.658	1.89183	0.945913	+	0.324420i	$0.105169\pi$
$42$	−42.0000	−0.154303
$43$	−435.150	−1.54325	−0.771624	−	0.636079i	$-0.780555\pi$
$43$	−435.150	−1.54325	−0.771624	+	0.636079i	$0.780555\pi$
$44$	−227.330	−0.778891
$45$	177.222	0.587081
$46$	40.5361	0.129929
$47$	559.878	1.73759	0.868794	−	0.495174i	$-0.164896\pi$
$47$	559.878	1.73759	0.868794	+	0.495174i	$0.164896\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	−525.494	−1.48632
$51$	−100.032	−0.274653
$52$	−25.1046	−0.0669497
$53$	−709.463	−1.83872	−0.919361	−	0.393414i	$-0.871294\pi$
$53$	−709.463	−1.83872	−0.919361	+	0.393414i	$0.871294\pi$
$54$	54.0000	0.136083
$55$	−1119.10	−2.74363
$56$	56.0000	0.133631
$57$	57.0000	0.132453
$58$	−398.309	−0.901733
$59$	−256.402	−0.565775	−0.282888	−	0.959153i	$-0.591292\pi$
$59$	−256.402	−0.565775	−0.282888	+	0.959153i	$0.591292\pi$
$60$	−236.296	−0.508427
$61$	261.074	0.547984	0.273992	−	0.961732i	$-0.411656\pi$
$61$	261.074	0.547984	0.273992	+	0.961732i	$0.411656\pi$
$62$	286.504	0.586872
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−123.586	−0.235829
$66$	−340.994	−0.635962
$67$	774.153	1.41161	0.705805	−	0.708406i	$-0.250585\pi$
$67$	774.153	1.41161	0.705805	+	0.708406i	$0.250585\pi$
$68$	133.376	0.237856
$69$	60.8042	0.106086
$70$	275.678	0.470712
$71$	130.497	0.218129	0.109065	−	0.994035i	$-0.465214\pi$
$71$	130.497	0.218129	0.109065	+	0.994035i	$0.465214\pi$
$72$	−72.0000	−0.117851
$73$	−674.281	−1.08108	−0.540539	−	0.841319i	$-0.681780\pi$
$73$	−674.281	−1.08108	−0.540539	+	0.841319i	$0.681780\pi$
$74$	−647.897	−1.01779
$75$	−788.241	−1.21358
$76$	−76.0000	−0.114708
$77$	397.827	0.588786
$78$	−37.6569	−0.0546642
$79$	927.944	1.32154	0.660772	−	0.750587i	$-0.270229\pi$
$79$	927.944	1.32154	0.660772	+	0.750587i	$0.270229\pi$
$80$	315.061	0.440311
$81$	81.0000	0.111111
$82$	−993.315	−1.33772
$83$	82.8695	0.109592	0.0547959	−	0.998498i	$-0.482549\pi$
$83$	82.8695	0.109592	0.0547959	+	0.998498i	$0.482549\pi$
$84$	84.0000	0.109109
$85$	656.587	0.837846
$86$	870.299	1.09124
$87$	−597.463	−0.736262
$88$	454.659	0.550759
$89$	−1122.12	−1.33646	−0.668228	−	0.743957i	$-0.732947\pi$
$89$	−1122.12	−1.33646	−0.668228	+	0.743957i	$0.732947\pi$
$90$	−354.443	−0.415129
$91$	43.9331	0.0506092
$92$	−81.0722	−0.0918735
$93$	429.756	0.479179
$94$	−1119.76	−1.22866
$95$	−374.135	−0.404057
$96$	96.0000	0.102062
$97$	830.799	0.869638	0.434819	−	0.900518i	$-0.356812\pi$
$97$	830.799	0.869638	0.434819	+	0.900518i	$0.356812\pi$
$98$	−98.0000	−0.101015
$99$	−511.491	−0.519261
$100$	1050.99	1.05099
$101$	−1137.16	−1.12032	−0.560158	−	0.828385i	$-0.689260\pi$
$101$	−1137.16	−1.12032	−0.560158	+	0.828385i	$0.689260\pi$
$102$	200.064	0.194209
$103$	1354.92	1.29615	0.648077	−	0.761575i	$-0.275574\pi$
$103$	1354.92	1.29615	0.648077	+	0.761575i	$0.275574\pi$
$104$	50.2093	0.0473406
$105$	413.517	0.384335
$106$	1418.93	1.30017
$107$	1566.43	1.41526	0.707630	−	0.706583i	$-0.249764\pi$
$107$	1566.43	1.41526	0.707630	+	0.706583i	$0.249764\pi$
$108$	−108.000	−0.0962250
$109$	113.073	0.0993621	0.0496811	−	0.998765i	$-0.484180\pi$
$109$	113.073	0.0993621	0.0496811	+	0.998765i	$0.484180\pi$
$110$	2238.21	1.94004
$111$	−971.845	−0.831022
$112$	−112.000	−0.0944911
$113$	1983.21	1.65101	0.825507	−	0.564392i	$-0.190889\pi$
$113$	1983.21	1.65101	0.825507	+	0.564392i	$0.190889\pi$
$114$	−114.000	−0.0936586
$115$	−399.104	−0.323623
$116$	796.618	0.637622
$117$	−56.4854	−0.0446332
$118$	512.805	0.400064
$119$	−233.408	−0.179803
$120$	472.591	0.359512
$121$	1898.92	1.42669
$122$	−522.147	−0.387483
$123$	−1489.97	−1.09225
$124$	−573.008	−0.414981
$125$	2712.42	1.94085
$126$	126.000	0.0890871
$127$	548.819	0.383463	0.191732	−	0.981447i	$-0.438590\pi$
$127$	548.819	0.383463	0.191732	+	0.981447i	$0.438590\pi$
$128$	−128.000	−0.0883883
$129$	1305.45	0.890995
$130$	247.171	0.166757
$131$	−632.343	−0.421741	−0.210871	−	0.977514i	$-0.567630\pi$
$131$	−632.343	−0.421741	−0.210871	+	0.977514i	$0.567630\pi$
$132$	681.989	0.449693
$133$	133.000	0.0867110
$134$	−1548.31	−0.998159
$135$	−531.665	−0.338951
$136$	−266.752	−0.168190
$137$	960.743	0.599137	0.299569	−	0.954075i	$-0.403157\pi$
$137$	960.743	0.599137	0.299569	+	0.954075i	$0.403157\pi$
$138$	−121.608	−0.0750144
$139$	2480.46	1.51360	0.756799	−	0.653647i	$-0.226762\pi$
$139$	2480.46	1.51360	0.756799	+	0.653647i	$0.226762\pi$
$140$	−551.356	−0.332844
$141$	−1679.63	−1.00320
$142$	−260.995	−0.154241
$143$	356.689	0.208586
$144$	144.000	0.0833333
$145$	3921.61	2.24601
$146$	1348.56	0.764437
$147$	−147.000	−0.0824786
$148$	1295.79	0.719686
$149$	1855.25	1.02005	0.510027	−	0.860158i	$-0.329635\pi$
$149$	1855.25	1.02005	0.510027	+	0.860158i	$0.329635\pi$
$150$	1576.48	0.858129
$151$	3100.14	1.67077	0.835385	−	0.549666i	$-0.185245\pi$
$151$	3100.14	1.67077	0.835385	+	0.549666i	$0.185245\pi$
$152$	152.000	0.0811107
$153$	300.096	0.158571
$154$	−795.653	−0.416335
$155$	−2820.82	−1.46177
$156$	75.3139	0.0386534
$157$	−2382.75	−1.21124	−0.605618	−	0.795756i	$-0.707074\pi$
$157$	−2382.75	−1.21124	−0.605618	+	0.795756i	$0.707074\pi$
$158$	−1855.89	−0.934472
$159$	2128.39	1.06159
$160$	−630.121	−0.311347
$161$	141.876	0.0694499
$162$	−162.000	−0.0785674
$163$	2742.58	1.31789	0.658944	−	0.752192i	$-0.271003\pi$
$163$	2742.58	1.31789	0.658944	+	0.752192i	$0.271003\pi$
$164$	1986.63	0.945913
$165$	3357.31	1.58404
$166$	−165.739	−0.0774931
$167$	−2905.77	−1.34644	−0.673218	−	0.739444i	$-0.735089\pi$
$167$	−2905.77	−1.34644	−0.673218	+	0.739444i	$0.735089\pi$
$168$	−168.000	−0.0771517
$169$	−2157.61	−0.982071
$170$	−1313.17	−0.592447
$171$	−171.000	−0.0764719
$172$	−1740.60	−0.771624
$173$	−3738.89	−1.64313	−0.821567	−	0.570112i	$-0.806900\pi$
$173$	−3738.89	−1.64313	−0.821567	+	0.570112i	$0.806900\pi$
$174$	1194.93	0.520616
$175$	−1839.23	−0.794473
$176$	−909.318	−0.389446
$177$	769.207	0.326650
$178$	2244.24	0.945016
$179$	−3404.07	−1.42141	−0.710705	−	0.703490i	$-0.751624\pi$
$179$	−3404.07	−1.42141	−0.710705	+	0.703490i	$0.751624\pi$
$180$	708.887	0.293541
$181$	4582.07	1.88167	0.940836	−	0.338863i	$-0.110042\pi$
$181$	4582.07	1.88167	0.940836	+	0.338863i	$0.110042\pi$
$182$	−87.8662	−0.0357861
$183$	−783.221	−0.316379
$184$	162.144	0.0649644
$185$	6378.96	2.53508
$186$	−859.513	−0.338831
$187$	−1895.02	−0.741057
$188$	2239.51	0.868794
$189$	189.000	0.0727393
$190$	748.269	0.285711
$191$	3318.28	1.25708	0.628540	−	0.777777i	$-0.283653\pi$
$191$	3318.28	1.25708	0.628540	+	0.777777i	$0.283653\pi$
$192$	−192.000	−0.0721688
$193$	1693.97	0.631786	0.315893	−	0.948795i	$-0.397696\pi$
$193$	1693.97	0.631786	0.315893	+	0.948795i	$0.397696\pi$
$194$	−1661.60	−0.614927
$195$	370.757	0.136156
$196$	196.000	0.0714286
$197$	−3227.24	−1.16716	−0.583582	−	0.812054i	$-0.698349\pi$
$197$	−3227.24	−1.16716	−0.583582	+	0.812054i	$0.698349\pi$
$198$	1022.98	0.367173
$199$	2458.03	0.875605	0.437802	−	0.899071i	$-0.355757\pi$
$199$	2458.03	0.875605	0.437802	+	0.899071i	$0.355757\pi$
$200$	−2101.98	−0.743161
$201$	−2322.46	−0.814994
$202$	2274.33	0.792184
$203$	−1394.08	−0.481997
$204$	−400.129	−0.137326
$205$	9779.83	3.33197
$206$	−2709.83	−0.916519
$207$	−182.413	−0.0612490
$208$	−100.419	−0.0334749
$209$	1079.82	0.357380
$210$	−827.034	−0.271766
$211$	−1541.33	−0.502887	−0.251444	−	0.967872i	$-0.580905\pi$
$211$	−1541.33	−0.502887	−0.251444	+	0.967872i	$0.580905\pi$
$212$	−2837.85	−0.919361
$213$	−391.492	−0.125937
$214$	−3132.87	−1.00074
$215$	−8568.66	−2.71804
$216$	216.000	0.0680414
$217$	1002.76	0.313696
$218$	−226.147	−0.0702596
$219$	2022.84	0.624160
$220$	−4476.41	−1.37182
$221$	−209.272	−0.0636977
$222$	1943.69	0.587621
$223$	2868.67	0.861438	0.430719	−	0.902486i	$-0.358260\pi$
$223$	2868.67	0.861438	0.430719	+	0.902486i	$0.358260\pi$
$224$	224.000	0.0668153
$225$	2364.72	0.700659
$226$	−3966.42	−1.16744
$227$	2402.75	0.702539	0.351269	−	0.936274i	$-0.385750\pi$
$227$	2402.75	0.702539	0.351269	+	0.936274i	$0.385750\pi$
$228$	228.000	0.0662266
$229$	−3172.87	−0.915587	−0.457794	−	0.889058i	$-0.651360\pi$
$229$	−3172.87	−0.915587	−0.457794	+	0.889058i	$0.651360\pi$
$230$	798.209	0.228836
$231$	−1193.48	−0.339936
$232$	−1593.24	−0.450867
$233$	−4095.09	−1.15141	−0.575704	−	0.817658i	$-0.695272\pi$
$233$	−4095.09	−1.15141	−0.575704	+	0.817658i	$0.695272\pi$
$234$	112.971	0.0315604
$235$	11024.7	3.06031
$236$	−1025.61	−0.282888
$237$	−2783.83	−0.762993
$238$	466.817	0.127140
$239$	−5644.53	−1.52767	−0.763837	−	0.645409i	$-0.776687\pi$
$239$	−5644.53	−1.52767	−0.763837	+	0.645409i	$0.776687\pi$
$240$	−945.182	−0.254214
$241$	−1294.84	−0.346091	−0.173045	−	0.984914i	$-0.555361\pi$
$241$	−1294.84	−0.346091	−0.173045	+	0.984914i	$0.555361\pi$
$242$	−3797.84	−1.00882
$243$	−243.000	−0.0641500
$244$	1044.29	0.273992
$245$	964.874	0.251606
$246$	2979.95	0.772335
$247$	119.247	0.0307186
$248$	1146.02	0.293436
$249$	−248.609	−0.0632728
$250$	−5424.84	−1.37239
$251$	−987.037	−0.248212	−0.124106	−	0.992269i	$-0.539606\pi$
$251$	−987.037	−0.248212	−0.124106	+	0.992269i	$0.539606\pi$
$252$	−252.000	−0.0629941
$253$	1151.88	0.286238
$254$	−1097.64	−0.271149
$255$	−1969.76	−0.483731
$256$	256.000	0.0625000
$257$	3787.18	0.919214	0.459607	−	0.888122i	$-0.347990\pi$
$257$	3787.18	0.919214	0.459607	+	0.888122i	$0.347990\pi$
$258$	−2610.90	−0.630029
$259$	−2267.64	−0.544032
$260$	−494.343	−0.117915
$261$	1792.39	0.425081
$262$	1264.69	0.298216
$263$	7035.34	1.64950	0.824748	−	0.565500i	$-0.191317\pi$
$263$	7035.34	1.64950	0.824748	+	0.565500i	$0.191317\pi$
$264$	−1363.98	−0.317981
$265$	−13970.3	−3.23844
$266$	−266.000	−0.0613139
$267$	3366.36	0.771603
$268$	3096.61	0.705805
$269$	−122.994	−0.0278777	−0.0139388	−	0.999903i	$-0.504437\pi$
$269$	−122.994	−0.0278777	−0.0139388	+	0.999903i	$0.504437\pi$
$270$	1063.33	0.239675
$271$	1422.57	0.318875	0.159437	−	0.987208i	$-0.449032\pi$
$271$	1422.57	0.318875	0.159437	+	0.987208i	$0.449032\pi$
$272$	533.505	0.118928
$273$	−131.799	−0.0292193
$274$	−1921.49	−0.423654
$275$	−14932.5	−3.27442
$276$	243.217	0.0530432
$277$	5038.60	1.09292	0.546462	−	0.837484i	$-0.315974\pi$
$277$	5038.60	1.09292	0.546462	+	0.837484i	$0.315974\pi$
$278$	−4960.93	−1.07028
$279$	−1289.27	−0.276654
$280$	1102.71	0.235356
$281$	2222.09	0.471738	0.235869	−	0.971785i	$-0.424206\pi$
$281$	2222.09	0.471738	0.235869	+	0.971785i	$0.424206\pi$
$282$	3359.27	0.709367
$283$	1554.44	0.326508	0.163254	−	0.986584i	$-0.447801\pi$
$283$	1554.44	0.326508	0.163254	+	0.986584i	$0.447801\pi$
$284$	521.989	0.109065
$285$	1122.40	0.233282
$286$	−713.378	−0.147493
$287$	−3476.60	−0.715043
$288$	−288.000	−0.0589256
$289$	−3801.17	−0.773697
$290$	−7843.22	−1.58817
$291$	−2492.40	−0.502086
$292$	−2697.13	−0.540539
$293$	−648.115	−0.129226	−0.0646131	−	0.997910i	$-0.520581\pi$
$293$	−648.115	−0.129226	−0.0646131	+	0.997910i	$0.520581\pi$
$294$	294.000	0.0583212
$295$	−5048.89	−0.996468
$296$	−2591.59	−0.508895
$297$	1534.47	0.299795
$298$	−3710.50	−0.721287
$299$	127.206	0.0246036
$300$	−3152.97	−0.606789
$301$	3046.05	0.583293
$302$	−6200.29	−1.18141
$303$	3411.49	0.646815
$304$	−304.000	−0.0573539
$305$	5140.88	0.965134
$306$	−600.193	−0.112127
$307$	6908.82	1.28439	0.642194	−	0.766542i	$-0.278024\pi$
$307$	6908.82	1.28439	0.642194	+	0.766542i	$0.278024\pi$
$308$	1591.31	0.294393
$309$	−4064.75	−0.748335
$310$	5641.64	1.03362
$311$	8276.58	1.50907	0.754537	−	0.656258i	$-0.227862\pi$
$311$	8276.58	1.50907	0.754537	+	0.656258i	$0.227862\pi$
$312$	−150.628	−0.0273321
$313$	−1466.78	−0.264879	−0.132440	−	0.991191i	$-0.542281\pi$
$313$	−1466.78	−0.264879	−0.132440	+	0.991191i	$0.542281\pi$
$314$	4765.49	0.856473
$315$	−1240.55	−0.221896
$316$	3711.78	0.660772
$317$	8873.59	1.57221	0.786105	−	0.618093i	$-0.212095\pi$
$317$	8873.59	1.57221	0.786105	+	0.618093i	$0.212095\pi$
$318$	−4256.78	−0.750655
$319$	−11318.4	−1.98655
$320$	1260.24	0.220155
$321$	−4699.30	−0.817101
$322$	−283.753	−0.0491085
$323$	−633.537	−0.109136
$324$	324.000	0.0555556
$325$	−1649.04	−0.281454
$326$	−5485.17	−0.931887
$327$	−339.220	−0.0573667
$328$	−3973.26	−0.668862
$329$	−3919.15	−0.656746
$330$	−6714.62	−1.12008
$331$	−4241.44	−0.704322	−0.352161	−	0.935940i	$-0.614553\pi$
$331$	−4241.44	−0.704322	−0.352161	+	0.935940i	$0.614553\pi$
$332$	331.478	0.0547959
$333$	2915.53	0.479791
$334$	5811.53	0.952075
$335$	15244.1	2.48619
$336$	336.000	0.0545545
$337$	4657.53	0.752855	0.376427	−	0.926446i	$-0.377152\pi$
$337$	4657.53	0.752855	0.376427	+	0.926446i	$0.377152\pi$
$338$	4315.22	0.694429
$339$	−5949.62	−0.953213
$340$	2626.35	0.418923
$341$	8141.36	1.29290
$342$	342.000	0.0540738
$343$	−343.000	−0.0539949
$344$	3481.20	0.545621
$345$	1197.31	0.186844
$346$	7477.77	1.16187
$347$	1451.97	0.224628	0.112314	−	0.993673i	$-0.464174\pi$
$347$	1451.97	0.224628	0.112314	+	0.993673i	$0.464174\pi$
$348$	−2389.85	−0.368131
$349$	965.468	0.148081	0.0740406	−	0.997255i	$-0.476411\pi$
$349$	965.468	0.148081	0.0740406	+	0.997255i	$0.476411\pi$
$350$	3678.46	0.561777
$351$	169.456	0.0257690
$352$	1818.64	0.275380
$353$	−2979.16	−0.449192	−0.224596	−	0.974452i	$-0.572106\pi$
$353$	−2979.16	−0.449192	−0.224596	+	0.974452i	$0.572106\pi$
$354$	−1538.41	−0.230977
$355$	2569.66	0.384179
$356$	−4488.48	−0.668228
$357$	700.225	0.103809
$358$	6808.15	1.00509
$359$	−581.336	−0.0854644	−0.0427322	−	0.999087i	$-0.513606\pi$
$359$	−581.336	−0.0854644	−0.0427322	+	0.999087i	$0.513606\pi$
$360$	−1417.77	−0.207564
$361$	361.000	0.0526316
$362$	−9164.14	−1.33054
$363$	−5696.76	−0.823697
$364$	175.732	0.0253046
$365$	−13277.5	−1.90404
$366$	1566.44	0.223714
$367$	6712.40	0.954726	0.477363	−	0.878706i	$-0.341593\pi$
$367$	6712.40	0.954726	0.477363	+	0.878706i	$0.341593\pi$
$368$	−324.289	−0.0459368
$369$	4469.92	0.630609
$370$	−12757.9	−1.79258
$371$	4966.24	0.694972
$372$	1719.03	0.239590
$373$	−5607.99	−0.778473	−0.389237	−	0.921138i	$-0.627261\pi$
$373$	−5607.99	−0.778473	−0.389237	+	0.921138i	$0.627261\pi$
$374$	3790.04	0.524006
$375$	−8137.26	−1.12055
$376$	−4479.02	−0.614330
$377$	−1249.92	−0.170754
$378$	−378.000	−0.0514344
$379$	−2653.82	−0.359677	−0.179839	−	0.983696i	$-0.557558\pi$
$379$	−2653.82	−0.359677	−0.179839	+	0.983696i	$0.557558\pi$
$380$	−1496.54	−0.202028
$381$	−1646.46	−0.221393
$382$	−6636.56	−0.888889
$383$	−2525.78	−0.336975	−0.168488	−	0.985704i	$-0.553888\pi$
$383$	−2525.78	−0.336975	−0.168488	+	0.985704i	$0.553888\pi$
$384$	384.000	0.0510310
$385$	7833.72	1.03700
$386$	−3387.94	−0.446740
$387$	−3916.35	−0.514416
$388$	3323.20	0.434819
$389$	3821.81	0.498132	0.249066	−	0.968487i	$-0.419876\pi$
$389$	3821.81	0.498132	0.249066	+	0.968487i	$0.419876\pi$
$390$	−741.514	−0.0962770
$391$	−675.819	−0.0874108
$392$	−392.000	−0.0505076
$393$	1897.03	0.243492
$394$	6454.48	0.825309
$395$	18272.4	2.32756
$396$	−2045.97	−0.259630
$397$	14241.4	1.80039	0.900197	−	0.435483i	$-0.143422\pi$
$397$	14241.4	1.80039	0.900197	+	0.435483i	$0.143422\pi$
$398$	−4916.07	−0.619146
$399$	−399.000	−0.0500626
$400$	4203.95	0.525494
$401$	−6197.97	−0.771850	−0.385925	−	0.922530i	$-0.626118\pi$
$401$	−6197.97	−0.771850	−0.385925	+	0.922530i	$0.626118\pi$
$402$	4644.92	0.576288
$403$	899.073	0.111132
$404$	−4548.65	−0.560158
$405$	1594.99	0.195694
$406$	2788.16	0.340823
$407$	−18410.8	−2.24223
$408$	800.257	0.0971045
$409$	−6382.42	−0.771615	−0.385807	−	0.922579i	$-0.626077\pi$
$409$	−6382.42	−0.771615	−0.385807	+	0.922579i	$0.626077\pi$
$410$	−19559.7	−2.35606
$411$	−2882.23	−0.345912
$412$	5419.66	0.648077
$413$	1794.82	0.213843
$414$	364.825	0.0433096
$415$	1631.81	0.193018
$416$	200.837	0.0236703
$417$	−7441.39	−0.873877
$418$	−2159.63	−0.252706
$419$	210.199	0.0245082	0.0122541	−	0.999925i	$-0.496099\pi$
$419$	210.199	0.0245082	0.0122541	+	0.999925i	$0.496099\pi$
$420$	1654.07	0.192167
$421$	1705.02	0.197381	0.0986906	−	0.995118i	$-0.468535\pi$
$421$	1705.02	0.197381	0.0986906	+	0.995118i	$0.468535\pi$
$422$	3082.65	0.355595
$423$	5038.90	0.579196
$424$	5675.71	0.650087
$425$	8761.05	0.999938
$426$	782.984	0.0890510
$427$	−1827.51	−0.207119
$428$	6265.73	0.707630
$429$	−1070.07	−0.120427
$430$	17137.3	1.92194
$431$	−17426.6	−1.94759	−0.973796	−	0.227424i	$-0.926970\pi$
$431$	−17426.6	−1.94759	−0.973796	+	0.227424i	$0.926970\pi$
$432$	−432.000	−0.0481125
$433$	−3379.22	−0.375046	−0.187523	−	0.982260i	$-0.560046\pi$
$433$	−3379.22	−0.375046	−0.187523	+	0.982260i	$0.560046\pi$
$434$	−2005.53	−0.221817
$435$	−11764.8	−1.29674
$436$	452.294	0.0496811
$437$	385.093	0.0421545
$438$	−4045.69	−0.441348
$439$	−197.880	−0.0215132	−0.0107566	−	0.999942i	$-0.503424\pi$
$439$	−197.880	−0.0215132	−0.0107566	+	0.999942i	$0.503424\pi$
$440$	8952.83	0.970021
$441$	441.000	0.0476190
$442$	418.545	0.0450411
$443$	−1710.93	−0.183496	−0.0917482	−	0.995782i	$-0.529245\pi$
$443$	−1710.93	−0.183496	−0.0917482	+	0.995782i	$0.529245\pi$
$444$	−3887.38	−0.415511
$445$	−22096.0	−2.35382
$446$	−5737.35	−0.609129
$447$	−5565.75	−0.588929
$448$	−448.000	−0.0472456
$449$	6242.58	0.656137	0.328069	−	0.944654i	$-0.393602\pi$
$449$	6242.58	0.656137	0.328069	+	0.944654i	$0.393602\pi$
$450$	−4729.45	−0.495441
$451$	−28226.2	−2.94705
$452$	7932.83	0.825507
$453$	−9300.43	−0.964619
$454$	−4805.50	−0.496770
$455$	865.100	0.0891352
$456$	−456.000	−0.0468293
$457$	−16920.3	−1.73194	−0.865972	−	0.500093i	$-0.833299\pi$
$457$	−16920.3	−1.73194	−0.865972	+	0.500093i	$0.833299\pi$
$458$	6345.75	0.647418
$459$	−900.289	−0.0915510
$460$	−1596.42	−0.161812
$461$	−12960.0	−1.30934	−0.654672	−	0.755913i	$-0.727193\pi$
$461$	−12960.0	−1.30934	−0.654672	+	0.755913i	$0.727193\pi$
$462$	2386.96	0.240371
$463$	−12316.4	−1.23626	−0.618132	−	0.786075i	$-0.712110\pi$
$463$	−12316.4	−1.23626	−0.618132	+	0.786075i	$0.712110\pi$
$464$	3186.47	0.318811
$465$	8462.46	0.843951
$466$	8190.17	0.814168
$467$	−7708.39	−0.763815	−0.381908	−	0.924201i	$-0.624733\pi$
$467$	−7708.39	−0.763815	−0.381908	+	0.924201i	$0.624733\pi$
$468$	−225.942	−0.0223166
$469$	−5419.07	−0.533539
$470$	−22049.4	−2.16397
$471$	7148.24	0.699307
$472$	2051.22	0.200032
$473$	24730.6	2.40405
$474$	5567.67	0.539518
$475$	−4992.20	−0.482227
$476$	−933.633	−0.0899013
$477$	−6385.17	−0.612908
$478$	11289.1	1.08023
$479$	−6454.64	−0.615699	−0.307850	−	0.951435i	$-0.599609\pi$
$479$	−6454.64	−0.615699	−0.307850	+	0.951435i	$0.599609\pi$
$480$	1890.36	0.179756
$481$	−2033.15	−0.192731
$482$	2589.68	0.244723
$483$	−425.629	−0.0400969
$484$	7595.68	0.713343
$485$	16359.5	1.53164
$486$	486.000	0.0453609
$487$	17597.0	1.63737	0.818684	−	0.574245i	$-0.194704\pi$
$487$	17597.0	1.63737	0.818684	+	0.574245i	$0.194704\pi$
$488$	−2088.59	−0.193742
$489$	−8227.75	−0.760883
$490$	−1929.75	−0.177912
$491$	−10073.6	−0.925895	−0.462948	−	0.886386i	$-0.653208\pi$
$491$	−10073.6	−0.925895	−0.462948	+	0.886386i	$0.653208\pi$
$492$	−5959.89	−0.546123
$493$	6640.62	0.606650
$494$	−238.494	−0.0217214
$495$	−10071.9	−0.914544
$496$	−2292.03	−0.207491
$497$	−913.482	−0.0824452
$498$	497.217	0.0447406
$499$	−8818.96	−0.791164	−0.395582	−	0.918431i	$-0.629457\pi$
$499$	−8818.96	−0.791164	−0.395582	+	0.918431i	$0.629457\pi$
$500$	10849.7	0.970425
$501$	8717.30	0.777366
$502$	1974.07	0.175512
$503$	17080.4	1.51407	0.757034	−	0.653376i	$-0.226648\pi$
$503$	17080.4	1.51407	0.757034	+	0.653376i	$0.226648\pi$
$504$	504.000	0.0445435
$505$	−22392.2	−1.97315
$506$	−2303.76	−0.202401
$507$	6472.83	0.566999
$508$	2195.28	0.191732
$509$	9582.81	0.834481	0.417240	−	0.908796i	$-0.362997\pi$
$509$	9582.81	0.834481	0.417240	+	0.908796i	$0.362997\pi$
$510$	3939.52	0.342049
$511$	4719.97	0.408609
$512$	−512.000	−0.0441942
$513$	513.000	0.0441511
$514$	−7574.37	−0.649982
$515$	26680.1	2.28284
$516$	5221.79	0.445497
$517$	−31819.2	−2.70678
$518$	4535.28	0.384688
$519$	11216.7	0.948664
$520$	988.685	0.0833783
$521$	−984.349	−0.0827738	−0.0413869	−	0.999143i	$-0.513178\pi$
$521$	−984.349	−0.0827738	−0.0413869	+	0.999143i	$0.513178\pi$
$522$	−3584.78	−0.300578
$523$	3247.80	0.271542	0.135771	−	0.990740i	$-0.456649\pi$
$523$	3247.80	0.271542	0.135771	+	0.990740i	$0.456649\pi$
$524$	−2529.37	−0.210871
$525$	5517.69	0.458689
$526$	−14070.7	−1.16637
$527$	−4776.60	−0.394824
$528$	2727.95	0.224847
$529$	−11756.2	−0.966237
$530$	27940.5	2.28992
$531$	−2307.62	−0.188592
$532$	532.000	0.0433555
$533$	−3117.10	−0.253315
$534$	−6732.72	−0.545606
$535$	30845.1	2.49262
$536$	−6193.23	−0.499080
$537$	10212.2	0.820651
$538$	245.989	0.0197125
$539$	−2784.79	−0.222540
$540$	−2126.66	−0.169476
$541$	16088.3	1.27854	0.639272	−	0.768981i	$-0.279236\pi$
$541$	16088.3	1.27854	0.639272	+	0.768981i	$0.279236\pi$
$542$	−2845.14	−0.225479
$543$	−13746.2	−1.08638
$544$	−1067.01	−0.0840949
$545$	2226.56	0.175001
$546$	263.599	0.0206611
$547$	10057.1	0.786123	0.393061	−	0.919512i	$-0.371416\pi$
$547$	10057.1	0.786123	0.393061	+	0.919512i	$0.371416\pi$
$548$	3842.97	0.299569
$549$	2349.66	0.182661
$550$	29865.1	2.31537
$551$	−3783.93	−0.292561
$552$	−486.433	−0.0375072
$553$	−6495.61	−0.499496
$554$	−10077.2	−0.772815
$555$	−19136.9	−1.46363
$556$	9921.86	0.756799
$557$	−802.449	−0.0610428	−0.0305214	−	0.999534i	$-0.509717\pi$
$557$	−802.449	−0.0610428	−0.0305214	+	0.999534i	$0.509717\pi$
$558$	2578.54	0.195624
$559$	2731.07	0.206640
$560$	−2205.43	−0.166422
$561$	5685.06	0.427850
$562$	−4444.17	−0.333569
$563$	2213.61	0.165706	0.0828530	−	0.996562i	$-0.473597\pi$
$563$	2213.61	0.165706	0.0828530	+	0.996562i	$0.473597\pi$
$564$	−6718.54	−0.501598
$565$	39051.9	2.90784
$566$	−3108.88	−0.230876
$567$	−567.000	−0.0419961
$568$	−1043.98	−0.0771204
$569$	−20665.0	−1.52253	−0.761266	−	0.648440i	$-0.775422\pi$
$569$	−20665.0	−1.52253	−0.761266	+	0.648440i	$0.775422\pi$
$570$	−2244.81	−0.164956
$571$	−10306.7	−0.755378	−0.377689	−	0.925932i	$-0.623281\pi$
$571$	−10306.7	−0.755378	−0.377689	+	0.925932i	$0.623281\pi$
$572$	1426.76	0.104293
$573$	−9954.84	−0.725775
$574$	6953.21	0.505612
$575$	−5325.37	−0.386232
$576$	576.000	0.0416667
$577$	16752.7	1.20871	0.604354	−	0.796716i	$-0.293431\pi$
$577$	16752.7	1.20871	0.604354	+	0.796716i	$0.293431\pi$
$578$	7602.35	0.547087
$579$	−5081.91	−0.364762
$580$	15686.4	1.12301
$581$	−580.087	−0.0414218
$582$	4984.79	0.355028
$583$	40320.5	2.86433
$584$	5394.25	0.382219
$585$	−1112.27	−0.0786098
$586$	1296.23	0.0913768
$587$	−7075.50	−0.497508	−0.248754	−	0.968567i	$-0.580021\pi$
$587$	−7075.50	−0.497508	−0.248754	+	0.968567i	$0.580021\pi$
$588$	−588.000	−0.0412393
$589$	2721.79	0.190406
$590$	10097.8	0.704609
$591$	9681.71	0.673862
$592$	5183.17	0.359843
$593$	−22649.9	−1.56850	−0.784248	−	0.620447i	$-0.786951\pi$
$593$	−22649.9	−1.56850	−0.784248	+	0.620447i	$0.786951\pi$
$594$	−3068.95	−0.211987
$595$	−4596.11	−0.316676
$596$	7421.00	0.510027
$597$	−7374.10	−0.505531
$598$	−254.411	−0.0173974
$599$	−11271.7	−0.768864	−0.384432	−	0.923153i	$-0.625603\pi$
$599$	−11271.7	−0.768864	−0.384432	+	0.923153i	$0.625603\pi$
$600$	6305.93	0.429064
$601$	−2255.03	−0.153053	−0.0765263	−	0.997068i	$-0.524383\pi$
$601$	−2255.03	−0.153053	−0.0765263	+	0.997068i	$0.524383\pi$
$602$	−6092.09	−0.412450
$603$	6967.38	0.470537
$604$	12400.6	0.835385
$605$	37392.2	2.51274
$606$	−6822.98	−0.457367
$607$	15634.5	1.04544	0.522721	−	0.852504i	$-0.324917\pi$
$607$	15634.5	1.04544	0.522721	+	0.852504i	$0.324917\pi$
$608$	608.000	0.0405554
$609$	4182.24	0.278281
$610$	−10281.8	−0.682453
$611$	−3513.88	−0.232662
$612$	1200.39	0.0792855
$613$	3517.45	0.231759	0.115880	−	0.993263i	$-0.463031\pi$
$613$	3517.45	0.231759	0.115880	+	0.993263i	$0.463031\pi$
$614$	−13817.6	−0.908200
$615$	−29339.5	−1.92371
$616$	−3182.61	−0.208167
$617$	−3227.56	−0.210595	−0.105297	−	0.994441i	$-0.533579\pi$
$617$	−3227.56	−0.210595	−0.105297	+	0.994441i	$0.533579\pi$
$618$	8129.50	0.529153
$619$	−371.174	−0.0241014	−0.0120507	−	0.999927i	$-0.503836\pi$
$619$	−371.174	−0.0241014	−0.0120507	+	0.999927i	$0.503836\pi$
$620$	−11283.3	−0.730883
$621$	547.238	0.0353621
$622$	−16553.2	−1.06708
$623$	7854.84	0.505133
$624$	301.256	0.0193267
$625$	20567.7	1.31633
$626$	2933.56	0.187298
$627$	−3239.45	−0.206333
$628$	−9530.99	−0.605618
$629$	10801.7	0.684728
$630$	2481.10	0.156904
$631$	4101.36	0.258752	0.129376	−	0.991596i	$-0.458703\pi$
$631$	4101.36	0.258752	0.129376	+	0.991596i	$0.458703\pi$
$632$	−7423.56	−0.467236
$633$	4623.98	0.290342
$634$	−17747.2	−1.11172
$635$	10807.0	0.675372
$636$	8513.56	0.530793
$637$	−307.532	−0.0191285
$638$	22636.8	1.40470
$639$	1174.48	0.0727098
$640$	−2520.49	−0.155673
$641$	−8951.03	−0.551551	−0.275776	−	0.961222i	$-0.588935\pi$
$641$	−8951.03	−0.551551	−0.275776	+	0.961222i	$0.588935\pi$
$642$	9398.60	0.577778
$643$	−9008.89	−0.552529	−0.276264	−	0.961082i	$-0.589097\pi$
$643$	−9008.89	−0.552529	−0.276264	+	0.961082i	$0.589097\pi$
$644$	567.506	0.0347249
$645$	25706.0	1.56926
$646$	1267.07	0.0771708
$647$	−446.489	−0.0271303	−0.0135651	−	0.999908i	$-0.504318\pi$
$647$	−446.489	−0.0271303	−0.0135651	+	0.999908i	$0.504318\pi$
$648$	−648.000	−0.0392837
$649$	14572.0	0.881355
$650$	3298.08	0.199018
$651$	−3008.29	−0.181113
$652$	10970.3	0.658944
$653$	−19450.4	−1.16562	−0.582812	−	0.812607i	$-0.698047\pi$
$653$	−19450.4	−1.16562	−0.582812	+	0.812607i	$0.698047\pi$
$654$	678.441	0.0405644
$655$	−12451.7	−0.742789
$656$	7946.52	0.472957
$657$	−6068.53	−0.360359
$658$	7838.29	0.464390
$659$	3908.39	0.231031	0.115515	−	0.993306i	$-0.463148\pi$
$659$	3908.39	0.231031	0.115515	+	0.993306i	$0.463148\pi$
$660$	13429.2	0.792019
$661$	−7329.06	−0.431267	−0.215633	−	0.976474i	$-0.569182\pi$
$661$	−7329.06	−0.431267	−0.215633	+	0.976474i	$0.569182\pi$
$662$	8482.87	0.498031
$663$	627.817	0.0367759
$664$	−662.956	−0.0387465
$665$	2618.94	0.152719
$666$	−5831.07	−0.339263
$667$	−4036.47	−0.234322
$668$	−11623.1	−0.673218
$669$	−8606.02	−0.497351
$670$	−30488.2	−1.75800
$671$	−14837.4	−0.853640
$672$	−672.000	−0.0385758
$673$	−14114.9	−0.808453	−0.404227	−	0.914659i	$-0.632459\pi$
$673$	−14114.9	−0.808453	−0.404227	+	0.914659i	$0.632459\pi$
$674$	−9315.07	−0.532349
$675$	−7094.17	−0.404526
$676$	−8630.44	−0.491035
$677$	24774.3	1.40643	0.703216	−	0.710976i	$-0.251747\pi$
$677$	24774.3	1.40643	0.703216	+	0.710976i	$0.251747\pi$
$678$	11899.2	0.674023
$679$	−5815.59	−0.328692
$680$	−5252.70	−0.296223
$681$	−7208.26	−0.405611
$682$	−16282.7	−0.914219
$683$	6312.73	0.353660	0.176830	−	0.984241i	$-0.443416\pi$
$683$	6312.73	0.353660	0.176830	+	0.984241i	$0.443416\pi$
$684$	−684.000	−0.0382360
$685$	18918.3	1.05523
$686$	686.000	0.0381802
$687$	9518.62	0.528615
$688$	−6962.39	−0.385812
$689$	4452.70	0.246204
$690$	−2394.63	−0.132119
$691$	−4728.00	−0.260292	−0.130146	−	0.991495i	$-0.541545\pi$
$691$	−4728.00	−0.260292	−0.130146	+	0.991495i	$0.541545\pi$
$692$	−14955.5	−0.821567
$693$	3580.44	0.196262
$694$	−2903.94	−0.158836
$695$	48843.5	2.66581
$696$	4779.71	0.260308
$697$	16560.6	0.899966
$698$	−1930.94	−0.104709
$699$	12285.3	0.664766
$700$	−7356.92	−0.397236
$701$	−13400.5	−0.722013	−0.361006	−	0.932563i	$-0.617567\pi$
$701$	−13400.5	−0.722013	−0.361006	+	0.932563i	$0.617567\pi$
$702$	−338.913	−0.0182214
$703$	−6155.02	−0.330215
$704$	−3637.27	−0.194723
$705$	−33074.2	−1.76687
$706$	5958.33	0.317627
$707$	7960.15	0.423440
$708$	3076.83	0.163325
$709$	8925.96	0.472809	0.236404	−	0.971655i	$-0.424031\pi$
$709$	8925.96	0.472809	0.236404	+	0.971655i	$0.424031\pi$
$710$	−5139.32	−0.271656
$711$	8351.50	0.440514
$712$	8976.96	0.472508
$713$	2903.44	0.152503
$714$	−1400.45	−0.0734041
$715$	7023.67	0.367371
$716$	−13616.3	−0.710705
$717$	16933.6	0.882004
$718$	1162.67	0.0604325
$719$	−11188.3	−0.580322	−0.290161	−	0.956978i	$-0.593709\pi$
$719$	−11188.3	−0.580322	−0.290161	+	0.956978i	$0.593709\pi$
$720$	2835.55	0.146770
$721$	−9484.41	−0.489900
$722$	−722.000	−0.0372161
$723$	3884.52	0.199816
$724$	18328.3	0.940836
$725$	52327.3	2.68053
$726$	11393.5	0.582442
$727$	−1720.02	−0.0877467	−0.0438734	−	0.999037i	$-0.513970\pi$
$727$	−1720.02	−0.0877467	−0.0438734	+	0.999037i	$0.513970\pi$
$728$	−351.465	−0.0178931
$729$	729.000	0.0370370
$730$	26554.9	1.34636
$731$	−14509.6	−0.734143
$732$	−3132.88	−0.158189
$733$	−26478.0	−1.33423	−0.667113	−	0.744956i	$-0.732470\pi$
$733$	−26478.0	−1.33423	−0.667113	+	0.744956i	$0.732470\pi$
$734$	−13424.8	−0.675094
$735$	−2894.62	−0.145265
$736$	648.578	0.0324822
$737$	−43997.0	−2.19898
$738$	−8939.84	−0.445908
$739$	−9007.33	−0.448363	−0.224181	−	0.974547i	$-0.571971\pi$
$739$	−9007.33	−0.448363	−0.224181	+	0.974547i	$0.571971\pi$
$740$	25515.8	1.26754
$741$	−357.741	−0.0177354
$742$	−9932.49	−0.491419
$743$	3536.91	0.174639	0.0873193	−	0.996180i	$-0.472170\pi$
$743$	3536.91	0.174639	0.0873193	+	0.996180i	$0.472170\pi$
$744$	−3438.05	−0.169415
$745$	36532.3	1.79656
$746$	11216.0	0.550464
$747$	745.826	0.0365306
$748$	−7580.08	−0.370529
$749$	−10965.0	−0.534918
$750$	16274.5	0.792349
$751$	26455.7	1.28546	0.642732	−	0.766091i	$-0.277801\pi$
$751$	26455.7	1.28546	0.642732	+	0.766091i	$0.277801\pi$
$752$	8958.05	0.434397
$753$	2961.11	0.143305
$754$	2499.85	0.120742
$755$	61045.9	2.94263
$756$	756.000	0.0363696
$757$	4606.00	0.221146	0.110573	−	0.993868i	$-0.464731\pi$
$757$	4606.00	0.221146	0.110573	+	0.993868i	$0.464731\pi$
$758$	5307.64	0.254330
$759$	−3455.65	−0.165260
$760$	2993.08	0.142856
$761$	−16426.4	−0.782467	−0.391234	−	0.920291i	$-0.627952\pi$
$761$	−16426.4	−0.782467	−0.391234	+	0.920291i	$0.627952\pi$
$762$	3292.91	0.156548
$763$	−791.514	−0.0375553
$764$	13273.1	0.628540
$765$	5909.29	0.279282
$766$	5051.57	0.238278
$767$	1609.22	0.0757570
$768$	−768.000	−0.0360844
$769$	−167.992	−0.00787769	−0.00393885	−	0.999992i	$-0.501254\pi$
$769$	−167.992	−0.00787769	−0.00393885	+	0.999992i	$0.501254\pi$
$770$	−15667.4	−0.733267
$771$	−11361.5	−0.530708
$772$	6775.88	0.315893
$773$	14066.0	0.654485	0.327243	−	0.944940i	$-0.393881\pi$
$773$	14066.0	0.654485	0.327243	+	0.944940i	$0.393881\pi$
$774$	7832.69	0.363747
$775$	−37639.1	−1.74456
$776$	−6646.39	−0.307463
$777$	6802.91	0.314097
$778$	−7643.62	−0.352233
$779$	−9436.49	−0.434015
$780$	1483.03	0.0680781
$781$	−7416.48	−0.339798
$782$	1351.64	0.0618088
$783$	−5377.17	−0.245421
$784$	784.000	0.0357143
$785$	−46919.4	−2.13328
$786$	−3794.06	−0.172175
$787$	17808.6	0.806617	0.403308	−	0.915064i	$-0.367860\pi$
$787$	17808.6	0.806617	0.403308	+	0.915064i	$0.367860\pi$
$788$	−12909.0	−0.583582
$789$	−21106.0	−0.952337
$790$	−36544.9	−1.64583
$791$	−13882.5	−0.624024
$792$	4091.93	0.183586
$793$	−1638.54	−0.0733748
$794$	−28482.8	−1.27307
$795$	41910.8	1.86971
$796$	9832.13	0.437802
$797$	27287.2	1.21275	0.606375	−	0.795179i	$-0.292623\pi$
$797$	27287.2	1.21275	0.606375	+	0.795179i	$0.292623\pi$
$798$	798.000	0.0353996
$799$	18668.6	0.826593
$800$	−8407.91	−0.371581
$801$	−10099.1	−0.445485
$802$	12395.9	0.545781
$803$	38321.0	1.68408
$804$	−9289.84	−0.407497
$805$	2793.73	0.122318
$806$	−1798.15	−0.0785819
$807$	368.983	0.0160952
$808$	9097.31	0.396092
$809$	6033.41	0.262205	0.131102	−	0.991369i	$-0.458148\pi$
$809$	6033.41	0.262205	0.131102	+	0.991369i	$0.458148\pi$
$810$	−3189.99	−0.138376
$811$	−18955.3	−0.820726	−0.410363	−	0.911922i	$-0.634598\pi$
$811$	−18955.3	−0.820726	−0.410363	+	0.911922i	$0.634598\pi$
$812$	−5576.33	−0.240998
$813$	−4267.71	−0.184102
$814$	36821.5	1.58550
$815$	54005.0	2.32112
$816$	−1600.51	−0.0686632
$817$	8267.84	0.354045
$818$	12764.8	0.545614
$819$	395.398	0.0168697
$820$	39119.3	1.66598
$821$	14070.4	0.598127	0.299063	−	0.954233i	$-0.403326\pi$
$821$	14070.4	0.598127	0.299063	+	0.954233i	$0.403326\pi$
$822$	5764.46	0.244597
$823$	−2299.64	−0.0974003	−0.0487001	−	0.998813i	$-0.515508\pi$
$823$	−2299.64	−0.0974003	−0.0487001	+	0.998813i	$0.515508\pi$
$824$	−10839.3	−0.458260
$825$	44797.6	1.89049
$826$	−3589.63	−0.151210
$827$	−9746.62	−0.409822	−0.204911	−	0.978781i	$-0.565691\pi$
$827$	−9746.62	−0.409822	−0.204911	+	0.978781i	$0.565691\pi$
$828$	−729.650	−0.0306245
$829$	8416.82	0.352628	0.176314	−	0.984334i	$-0.443583\pi$
$829$	8416.82	0.352628	0.176314	+	0.984334i	$0.443583\pi$
$830$	−3263.62	−0.136484
$831$	−15115.8	−0.631000
$832$	−401.674	−0.0167374
$833$	1633.86	0.0679590
$834$	14882.8	0.617924
$835$	−57218.3	−2.37140
$836$	4319.26	0.178690
$837$	3867.81	0.159726
$838$	−420.399	−0.0173299
$839$	−6110.08	−0.251423	−0.125711	−	0.992067i	$-0.540121\pi$
$839$	−6110.08	−0.251423	−0.125711	+	0.992067i	$0.540121\pi$
$840$	−3308.14	−0.135883
$841$	15273.5	0.626246
$842$	−3410.04	−0.139570
$843$	−6666.26	−0.272358
$844$	−6165.30	−0.251444
$845$	−42486.1	−1.72967
$846$	−10077.8	−0.409553
$847$	−13292.4	−0.539237
$848$	−11351.4	−0.459681
$849$	−4663.31	−0.188509
$850$	−17522.1	−0.707063
$851$	−6565.80	−0.264480
$852$	−1565.97	−0.0629685
$853$	18256.5	0.732815	0.366408	−	0.930454i	$-0.380588\pi$
$853$	18256.5	0.732815	0.366408	+	0.930454i	$0.380588\pi$
$854$	3655.03	0.146455
$855$	−3367.21	−0.134686
$856$	−12531.5	−0.500370
$857$	3747.36	0.149367	0.0746834	−	0.997207i	$-0.476205\pi$
$857$	3747.36	0.149367	0.0746834	+	0.997207i	$0.476205\pi$
$858$	2140.13	0.0851550
$859$	−7801.24	−0.309866	−0.154933	−	0.987925i	$-0.549516\pi$
$859$	−7801.24	−0.309866	−0.154933	+	0.987925i	$0.549516\pi$
$860$	−34274.6	−1.35902
$861$	10429.8	0.412830
$862$	34853.3	1.37716
$863$	−39970.3	−1.57660	−0.788299	−	0.615292i	$-0.789038\pi$
$863$	−39970.3	−1.57660	−0.788299	+	0.615292i	$0.789038\pi$
$864$	864.000	0.0340207
$865$	−73623.5	−2.89396
$866$	6758.43	0.265197
$867$	11403.5	0.446694
$868$	4011.06	0.156848
$869$	−52737.3	−2.05868
$870$	23529.7	0.916931
$871$	−4858.71	−0.189014
$872$	−904.587	−0.0351298
$873$	7477.19	0.289879
$874$	−770.186	−0.0298077
$875$	−18986.9	−0.733572
$876$	8091.38	0.312080
$877$	−23769.3	−0.915201	−0.457600	−	0.889158i	$-0.651291\pi$
$877$	−23769.3	−0.915201	−0.457600	+	0.889158i	$0.651291\pi$
$878$	395.760	0.0152121
$879$	1944.35	0.0746088
$880$	−17905.7	−0.685908
$881$	−37180.3	−1.42184	−0.710918	−	0.703275i	$-0.751720\pi$
$881$	−37180.3	−1.42184	−0.710918	+	0.703275i	$0.751720\pi$
$882$	−882.000	−0.0336718
$883$	−44878.4	−1.71040	−0.855198	−	0.518302i	$-0.826564\pi$
$883$	−44878.4	−1.71040	−0.855198	+	0.518302i	$0.826564\pi$
$884$	−837.090	−0.0318488
$885$	15146.7	0.575311
$886$	3421.87	0.129752
$887$	4168.71	0.157803	0.0789017	−	0.996882i	$-0.474859\pi$
$887$	4168.71	0.157803	0.0789017	+	0.996882i	$0.474859\pi$
$888$	7774.76	0.293811
$889$	−3841.73	−0.144935
$890$	44192.0	1.66440
$891$	−4603.42	−0.173087
$892$	11474.7	0.430719
$893$	−10637.7	−0.398630
$894$	11131.5	0.416435
$895$	−67030.6	−2.50345
$896$	896.000	0.0334077
$897$	−381.617	−0.0142049
$898$	−12485.2	−0.463959
$899$	−28529.3	−1.05840
$900$	9458.90	0.350330
$901$	−23656.4	−0.874704
$902$	56452.5	2.08388
$903$	−9138.14	−0.336764
$904$	−15865.7	−0.583721
$905$	90226.9	3.31408
$906$	18600.9	0.682089
$907$	−18092.3	−0.662345	−0.331172	−	0.943570i	$-0.607444\pi$
$907$	−18092.3	−0.662345	−0.331172	+	0.943570i	$0.607444\pi$
$908$	9611.01	0.351269
$909$	−10234.5	−0.373439
$910$	−1730.20	−0.0630281
$911$	−25213.1	−0.916957	−0.458479	−	0.888705i	$-0.651605\pi$
$911$	−25213.1	−0.916957	−0.458479	+	0.888705i	$0.651605\pi$
$912$	912.000	0.0331133
$913$	−4709.67	−0.170720
$914$	33840.6	1.22467
$915$	−15422.6	−0.557220
$916$	−12691.5	−0.457794
$917$	4426.40	0.159403
$918$	1800.58	0.0647363
$919$	−15381.6	−0.552115	−0.276058	−	0.961141i	$-0.589028\pi$
$919$	−15381.6	−0.552115	−0.276058	+	0.961141i	$0.589028\pi$
$920$	3192.83	0.114418
$921$	−20726.5	−0.741542
$922$	25920.0	0.925846
$923$	−819.022	−0.0292074
$924$	−4773.92	−0.169968
$925$	85116.5	3.02553
$926$	24632.7	0.874170
$927$	12194.2	0.432051
$928$	−6372.94	−0.225433
$929$	49121.5	1.73479	0.867397	−	0.497616i	$-0.165791\pi$
$929$	49121.5	1.73479	0.867397	+	0.497616i	$0.165791\pi$
$930$	−16924.9	−0.596763
$931$	−931.000	−0.0327737
$932$	−16380.3	−0.575704
$933$	−24829.7	−0.871264
$934$	15416.8	0.540099
$935$	−37315.4	−1.30518
$936$	451.883	0.0157802
$937$	−14935.1	−0.520712	−0.260356	−	0.965513i	$-0.583840\pi$
$937$	−14935.1	−0.520712	−0.260356	+	0.965513i	$0.583840\pi$
$938$	10838.1	0.377269
$939$	4400.34	0.152928
$940$	44098.9	1.53016
$941$	−45493.8	−1.57604	−0.788020	−	0.615649i	$-0.788894\pi$
$941$	−45493.8	−1.57604	−0.788020	+	0.615649i	$0.788894\pi$
$942$	−14296.5	−0.494485
$943$	−10066.3	−0.347618
$944$	−4102.44	−0.141444
$945$	3721.65	0.128112
$946$	−49461.2	−1.69992
$947$	45860.9	1.57368	0.786842	−	0.617155i	$-0.211715\pi$
$947$	45860.9	1.57368	0.786842	+	0.617155i	$0.211715\pi$
$948$	−11135.3	−0.381497
$949$	4231.90	0.144756
$950$	9984.39	0.340986
$951$	−26620.8	−0.907716
$952$	1867.27	0.0635698
$953$	42841.5	1.45622	0.728108	−	0.685463i	$-0.240400\pi$
$953$	42841.5	1.45622	0.728108	+	0.685463i	$0.240400\pi$
$954$	12770.3	0.433391
$955$	65341.2	2.21402
$956$	−22578.1	−0.763837
$957$	33955.3	1.14694
$958$	12909.3	0.435365
$959$	−6725.20	−0.226453
$960$	−3780.73	−0.127107
$961$	−9269.83	−0.311162
$962$	4066.30	0.136282
$963$	14097.9	0.471753
$964$	−5179.36	−0.173045
$965$	33356.5	1.11273
$966$	851.258	0.0283528
$967$	−24239.4	−0.806089	−0.403044	−	0.915180i	$-0.632048\pi$
$967$	−24239.4	−0.806089	−0.403044	+	0.915180i	$0.632048\pi$
$968$	−15191.4	−0.504410
$969$	1900.61	0.0630097
$970$	−32719.0	−1.08304
$971$	−17282.5	−0.571188	−0.285594	−	0.958351i	$-0.592191\pi$
$971$	−17282.5	−0.571188	−0.285594	+	0.958351i	$0.592191\pi$
$972$	−972.000	−0.0320750
$973$	−17363.2	−0.572086
$974$	−35194.1	−1.15779
$975$	4947.13	0.162497
$976$	4177.18	0.136996
$977$	−8904.58	−0.291589	−0.145795	−	0.989315i	$-0.546574\pi$
$977$	−8904.58	−0.291589	−0.145795	+	0.989315i	$0.546574\pi$
$978$	16455.5	0.538025
$979$	63772.8	2.08191
$980$	3859.49	0.125803
$981$	1017.66	0.0331207
$982$	20147.2	0.654707
$983$	−26880.5	−0.872183	−0.436091	−	0.899902i	$-0.643638\pi$
$983$	−26880.5	−0.872183	−0.436091	+	0.899902i	$0.643638\pi$
$984$	11919.8	0.386167
$985$	−63548.5	−2.05566
$986$	−13281.2	−0.428966
$987$	11757.4	0.379173
$988$	476.988	0.0153593
$989$	8819.64	0.283567
$990$	20143.9	0.646681
$991$	−7510.21	−0.240736	−0.120368	−	0.992729i	$-0.538408\pi$
$991$	−7510.21	−0.240736	−0.120368	+	0.992729i	$0.538408\pi$
$992$	4584.07	0.146718
$993$	12724.3	0.406640
$994$	1826.96	0.0582975
$995$	48401.9	1.54215
$996$	−994.434	−0.0316364
$997$	−12238.6	−0.388767	−0.194384	−	0.980926i	$-0.562271\pi$
$997$	−12238.6	−0.388767	−0.194384	+	0.980926i	$0.562271\pi$
$998$	17637.9	0.559437
$999$	−8746.60	−0.277007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	798.4.a.k.1.4	✓	4
3.2	odd	2		2394.4.a.x.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.4.a.k.1.4	✓	4	1.1	even	1	trivial
2394.4.a.x.1.1		4	3.2	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:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	798.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +19.6913 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +19.6913 q^{5} +6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} -39.3826 q^{10} -56.8324 q^{11} -12.0000 q^{12} -6.27616 q^{13} +14.0000 q^{14} -59.0739 q^{15} +16.0000 q^{16} +33.3440 q^{17} -18.0000 q^{18} -19.0000 q^{19} +78.7652 q^{20} +21.0000 q^{21} +113.665 q^{22} -20.2681 q^{23} +24.0000 q^{24} +262.747 q^{25} +12.5523 q^{26} -27.0000 q^{27} -28.0000 q^{28} +199.154 q^{29} +118.148 q^{30} -143.252 q^{31} -32.0000 q^{32} +170.497 q^{33} -66.6881 q^{34} -137.839 q^{35} +36.0000 q^{36} +323.948 q^{37} +38.0000 q^{38} +18.8285 q^{39} -157.530 q^{40} +496.658 q^{41} -42.0000 q^{42} -435.150 q^{43} -227.330 q^{44} +177.222 q^{45} +40.5361 q^{46} +559.878 q^{47} -48.0000 q^{48} +49.0000 q^{49} -525.494 q^{50} -100.032 q^{51} -25.1046 q^{52} -709.463 q^{53} +54.0000 q^{54} -1119.10 q^{55} +56.0000 q^{56} +57.0000 q^{57} -398.309 q^{58} -256.402 q^{59} -236.296 q^{60} +261.074 q^{61} +286.504 q^{62} -63.0000 q^{63} +64.0000 q^{64} -123.586 q^{65} -340.994 q^{66} +774.153 q^{67} +133.376 q^{68} +60.8042 q^{69} +275.678 q^{70} +130.497 q^{71} -72.0000 q^{72} -674.281 q^{73} -647.897 q^{74} -788.241 q^{75} -76.0000 q^{76} +397.827 q^{77} -37.6569 q^{78} +927.944 q^{79} +315.061 q^{80} +81.0000 q^{81} -993.315 q^{82} +82.8695 q^{83} +84.0000 q^{84} +656.587 q^{85} +870.299 q^{86} -597.463 q^{87} +454.659 q^{88} -1122.12 q^{89} -354.443 q^{90} +43.9331 q^{91} -81.0722 q^{92} +429.756 q^{93} -1119.76 q^{94} -374.135 q^{95} +96.0000 q^{96} +830.799 q^{97} -98.0000 q^{98} -511.491 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9}+O(q^{10}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9	\( 4 q - 8 q^{2} - 12 q^{3} + 16 q^{4} + 24 q^{6} - 28 q^{7} - 32 q^{8} + 36 q^{9} - 66 q^{11} - 48 q^{12} + 10 q^{13} + 56 q^{14} + 64 q^{16} - 8 q^{17} - 72 q^{18} - 76 q^{19} + 84 q^{21} + 132 q^{22} - 250 q^{23} + 96 q^{24} + 120 q^{25} - 20 q^{26} - 108 q^{27} - 112 q^{28} - 66 q^{29} - 52 q^{31} - 128 q^{32} + 198 q^{33} + 16 q^{34} + 144 q^{36} + 288 q^{37} + 152 q^{38} - 30 q^{39} - 196 q^{41} - 168 q^{42} - 280 q^{43} - 264 q^{44} + 500 q^{46} + 498 q^{47} - 192 q^{48} + 196 q^{49} - 240 q^{50} + 24 q^{51} + 40 q^{52} - 250 q^{53} + 216 q^{54} - 1432 q^{55} + 224 q^{56} + 228 q^{57} + 132 q^{58} - 872 q^{59} + 600 q^{61} + 104 q^{62} - 252 q^{63} + 256 q^{64} - 908 q^{65} - 396 q^{66} + 1174 q^{67} - 32 q^{68} + 750 q^{69} - 1766 q^{71} - 288 q^{72} + 556 q^{73} - 576 q^{74} - 360 q^{75} - 304 q^{76} + 462 q^{77} + 60 q^{78} + 26 q^{79} + 324 q^{81} + 392 q^{82} + 674 q^{83} + 336 q^{84} + 2052 q^{85} + 560 q^{86} + 198 q^{87} + 528 q^{88} - 324 q^{89} - 70 q^{91} - 1000 q^{92} + 156 q^{93} - 996 q^{94} + 384 q^{96} + 2902 q^{97} - 392 q^{98} - 594 q^{99}+O(q^{100}) \) 4 * q - 8 * q^2 - 12 * q^3 + 16 * q^4 + 24 * q^6 - 28 * q^7 - 32 * q^8 + 36 * q^9 - 66 * q^11 - 48 * q^12 + 10 * q^13 + 56 * q^14 + 64 * q^16 - 8 * q^17 - 72 * q^18 - 76 * q^19 + 84 * q^21 + 132 * q^22 - 250 * q^23 + 96 * q^24 + 120 * q^25 - 20 * q^26 - 108 * q^27 - 112 * q^28 - 66 * q^29 - 52 * q^31 - 128 * q^32 + 198 * q^33 + 16 * q^34 + 144 * q^36 + 288 * q^37 + 152 * q^38 - 30 * q^39 - 196 * q^41 - 168 * q^42 - 280 * q^43 - 264 * q^44 + 500 * q^46 + 498 * q^47 - 192 * q^48 + 196 * q^49 - 240 * q^50 + 24 * q^51 + 40 * q^52 - 250 * q^53 + 216 * q^54 - 1432 * q^55 + 224 * q^56 + 228 * q^57 + 132 * q^58 - 872 * q^59 + 600 * q^61 + 104 * q^62 - 252 * q^63 + 256 * q^64 - 908 * q^65 - 396 * q^66 + 1174 * q^67 - 32 * q^68 + 750 * q^69 - 1766 * q^71 - 288 * q^72 + 556 * q^73 - 576 * q^74 - 360 * q^75 - 304 * q^76 + 462 * q^77 + 60 * q^78 + 26 * q^79 + 324 * q^81 + 392 * q^82 + 674 * q^83 + 336 * q^84 + 2052 * q^85 + 560 * q^86 + 198 * q^87 + 528 * q^88 - 324 * q^89 - 70 * q^91 - 1000 * q^92 + 156 * q^93 - 996 * q^94 + 384 * q^96 + 2902 * q^97 - 392 * q^98 - 594 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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