LMFDB - Embedded newform 338.4.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, 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("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.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:	$19.9426455819$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6681389953.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951 $ x^6 - x^5 - 107x^4 + 85x^3 + 3703x^2 - 1659x - 41951
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.40469$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} -1.24585 q^{3} +4.00000 q^{4} -20.5002 q^{5} +2.49171 q^{6} -21.2545 q^{7} -8.00000 q^{8} -25.4478 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} -1.24585 q^{3} +4.00000 q^{4} -20.5002 q^{5} +2.49171 q^{6} -21.2545 q^{7} -8.00000 q^{8} -25.4478 q^{9} +41.0004 q^{10} -53.0579 q^{11} -4.98341 q^{12} +42.5090 q^{14} +25.5403 q^{15} +16.0000 q^{16} -69.2150 q^{17} +50.8957 q^{18} +46.0603 q^{19} -82.0009 q^{20} +26.4800 q^{21} +106.116 q^{22} -87.2424 q^{23} +9.96683 q^{24} +295.259 q^{25} +65.3423 q^{27} -85.0181 q^{28} -162.062 q^{29} -51.0805 q^{30} -28.3466 q^{31} -32.0000 q^{32} +66.1024 q^{33} +138.430 q^{34} +435.722 q^{35} -101.791 q^{36} -111.972 q^{37} -92.1206 q^{38} +164.002 q^{40} -84.2751 q^{41} -52.9600 q^{42} -328.359 q^{43} -212.232 q^{44} +521.686 q^{45} +174.485 q^{46} -63.2021 q^{47} -19.9337 q^{48} +108.754 q^{49} -590.518 q^{50} +86.2318 q^{51} +721.631 q^{53} -130.685 q^{54} +1087.70 q^{55} +170.036 q^{56} -57.3844 q^{57} +324.123 q^{58} -819.081 q^{59} +102.161 q^{60} -397.828 q^{61} +56.6933 q^{62} +540.882 q^{63} +64.0000 q^{64} -132.205 q^{66} +77.8816 q^{67} -276.860 q^{68} +108.691 q^{69} -871.444 q^{70} +721.069 q^{71} +203.583 q^{72} -57.1470 q^{73} +223.945 q^{74} -367.849 q^{75} +184.241 q^{76} +1127.72 q^{77} -419.705 q^{79} -328.003 q^{80} +605.685 q^{81} +168.550 q^{82} +917.600 q^{83} +105.920 q^{84} +1418.92 q^{85} +656.717 q^{86} +201.905 q^{87} +424.463 q^{88} -378.804 q^{89} -1043.37 q^{90} -348.970 q^{92} +35.3158 q^{93} +126.404 q^{94} -944.246 q^{95} +39.8673 q^{96} -346.444 q^{97} -217.509 q^{98} +1350.21 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) $ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * 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
$3$	−1.24585	−0.239765	−0.119882	−	0.992788i	$-0.538252\pi$
$3$	−1.24585	−0.239765	−0.119882	+	0.992788i	$0.538252\pi$
$4$	4.00000	0.500000
$5$	−20.5002	−1.83359	−0.916797	−	0.399352i	$-0.869235\pi$
$5$	−20.5002	−1.83359	−0.916797	+	0.399352i	$0.869235\pi$
$6$	2.49171	0.169539
$7$	−21.2545	−1.14764	−0.573818	−	0.818983i	$-0.694538\pi$
$7$	−21.2545	−1.14764	−0.573818	+	0.818983i	$0.694538\pi$
$8$	−8.00000	−0.353553
$9$	−25.4478	−0.942513
$10$	41.0004	1.29655
$11$	−53.0579	−1.45432	−0.727162	−	0.686466i	$-0.759161\pi$
$11$	−53.0579	−1.45432	−0.727162	+	0.686466i	$0.759161\pi$
$12$	−4.98341	−0.119882
$13$	0	0
$13$	0	0
$14$	42.5090	0.811501
$15$	25.5403	0.439631
$16$	16.0000	0.250000
$17$	−69.2150	−0.987477	−0.493738	−	0.869611i	$-0.664370\pi$
$17$	−69.2150	−0.987477	−0.493738	+	0.869611i	$0.664370\pi$
$18$	50.8957	0.666457
$19$	46.0603	0.556156	0.278078	−	0.960559i	$-0.410303\pi$
$19$	46.0603	0.556156	0.278078	+	0.960559i	$0.410303\pi$
$20$	−82.0009	−0.916797
$21$	26.4800	0.275163
$22$	106.116	1.02836
$23$	−87.2424	−0.790926	−0.395463	−	0.918482i	$-0.629416\pi$
$23$	−87.2424	−0.790926	−0.395463	+	0.918482i	$0.629416\pi$
$24$	9.96683	0.0847696
$25$	295.259	2.36207
$26$	0	0
$27$	65.3423	0.465746
$28$	−85.0181	−0.573818
$29$	−162.062	−1.03773	−0.518864	−	0.854857i	$-0.673645\pi$
$29$	−162.062	−1.03773	−0.518864	+	0.854857i	$0.673645\pi$
$30$	−51.0805	−0.310866
$31$	−28.3466	−0.164232	−0.0821162	−	0.996623i	$-0.526168\pi$
$31$	−28.3466	−0.164232	−0.0821162	+	0.996623i	$0.526168\pi$
$32$	−32.0000	−0.176777
$33$	66.1024	0.348695
$34$	138.430	0.698251
$35$	435.722	2.10430
$36$	−101.791	−0.471256
$37$	−111.972	−0.497517	−0.248759	−	0.968566i	$-0.580023\pi$
$37$	−111.972	−0.497517	−0.248759	+	0.968566i	$0.580023\pi$
$38$	−92.1206	−0.393261
$39$	0	0
$40$	164.002	0.648274
$41$	−84.2751	−0.321014	−0.160507	−	0.987035i	$-0.551313\pi$
$41$	−84.2751	−0.321014	−0.160507	+	0.987035i	$0.551313\pi$
$42$	−52.9600	−0.194569
$43$	−328.359	−1.16452	−0.582258	−	0.813004i	$-0.697831\pi$
$43$	−328.359	−1.16452	−0.582258	+	0.813004i	$0.697831\pi$
$44$	−212.232	−0.727162
$45$	521.686	1.72819
$46$	174.485	0.559269
$47$	−63.2021	−0.196148	−0.0980742	−	0.995179i	$-0.531268\pi$
$47$	−63.2021	−0.196148	−0.0980742	+	0.995179i	$0.531268\pi$
$48$	−19.9337	−0.0599412
$49$	108.754	0.317068
$50$	−590.518	−1.67024
$51$	86.2318	0.236762
$52$	0	0
$53$	721.631	1.87026	0.935129	−	0.354308i	$-0.115284\pi$
$53$	721.631	1.87026	0.935129	+	0.354308i	$0.115284\pi$
$54$	−130.685	−0.329332
$55$	1087.70	2.66664
$56$	170.036	0.405751
$57$	−57.3844	−0.133346
$58$	324.123	0.733784
$59$	−819.081	−1.80738	−0.903689	−	0.428190i	$-0.859151\pi$
$59$	−819.081	−1.80738	−0.903689	+	0.428190i	$0.859151\pi$
$60$	102.161	0.219816
$61$	−397.828	−0.835027	−0.417514	−	0.908671i	$-0.637098\pi$
$61$	−397.828	−0.835027	−0.417514	+	0.908671i	$0.637098\pi$
$62$	56.6933	0.116130
$63$	540.882	1.08166
$64$	64.0000	0.125000
$65$	0	0
$66$	−132.205	−0.246565
$67$	77.8816	0.142011	0.0710056	−	0.997476i	$-0.477379\pi$
$67$	77.8816	0.142011	0.0710056	+	0.997476i	$0.477379\pi$
$68$	−276.860	−0.493738
$69$	108.691	0.189636
$70$	−871.444	−1.48796
$71$	721.069	1.20528	0.602642	−	0.798011i	$-0.294115\pi$
$71$	721.069	1.20528	0.602642	+	0.798011i	$0.294115\pi$
$72$	203.583	0.333229
$73$	−57.1470	−0.0916240	−0.0458120	−	0.998950i	$-0.514588\pi$
$73$	−57.1470	−0.0916240	−0.0458120	+	0.998950i	$0.514588\pi$
$74$	223.945	0.351798
$75$	−367.849	−0.566341
$76$	184.241	0.278078
$77$	1127.72	1.66903
$78$	0	0
$79$	−419.705	−0.597728	−0.298864	−	0.954296i	$-0.596608\pi$
$79$	−419.705	−0.597728	−0.298864	+	0.954296i	$0.596608\pi$
$80$	−328.003	−0.458399
$81$	605.685	0.830843
$82$	168.550	0.226991
$83$	917.600	1.21349	0.606745	−	0.794896i	$-0.292475\pi$
$83$	917.600	1.21349	0.606745	+	0.794896i	$0.292475\pi$
$84$	105.920	0.137581
$85$	1418.92	1.81063
$86$	656.717	0.823438
$87$	201.905	0.248810
$88$	424.463	0.514181
$89$	−378.804	−0.451159	−0.225580	−	0.974225i	$-0.572428\pi$
$89$	−378.804	−0.451159	−0.225580	+	0.974225i	$0.572428\pi$
$90$	−1043.37	−1.22201
$91$	0	0
$92$	−348.970	−0.395463
$93$	35.3158	0.0393771
$94$	126.404	0.138698
$95$	−944.246	−1.01976
$96$	39.8673	0.0423848
$97$	−346.444	−0.362640	−0.181320	−	0.983424i	$-0.558037\pi$
$97$	−346.444	−0.362640	−0.181320	+	0.983424i	$0.558037\pi$
$98$	−217.509	−0.224201
$99$	1350.21	1.37072
$100$	1181.04	1.18104
$101$	−519.293	−0.511600	−0.255800	−	0.966730i	$-0.582339\pi$
$101$	−519.293	−0.511600	−0.255800	+	0.966730i	$0.582339\pi$
$102$	−172.464	−0.167416
$103$	−386.911	−0.370130	−0.185065	−	0.982726i	$-0.559250\pi$
$103$	−386.911	−0.370130	−0.185065	+	0.982726i	$0.559250\pi$
$104$	0	0
$105$	−542.846	−0.504537
$106$	−1443.26	−1.32247
$107$	−1542.78	−1.39389	−0.696944	−	0.717125i	$-0.745457\pi$
$107$	−1542.78	−1.39389	−0.696944	+	0.717125i	$0.745457\pi$
$108$	261.369	0.232873
$109$	535.630	0.470679	0.235340	−	0.971913i	$-0.424380\pi$
$109$	535.630	0.470679	0.235340	+	0.971913i	$0.424380\pi$
$110$	−2175.40	−1.88560
$111$	139.501	0.119287
$112$	−340.072	−0.286909
$113$	−997.890	−0.830739	−0.415370	−	0.909653i	$-0.636348\pi$
$113$	−997.890	−0.830739	−0.415370	+	0.909653i	$0.636348\pi$
$114$	114.769	0.0942902
$115$	1788.49	1.45024
$116$	−648.247	−0.518864
$117$	0	0
$118$	1638.16	1.27801
$119$	1471.13	1.13326
$120$	−204.322	−0.155433
$121$	1484.14	1.11506
$122$	795.656	0.590454
$123$	104.994	0.0769678
$124$	−113.387	−0.0821162
$125$	−3490.34	−2.49749
$126$	−1081.76	−0.764850
$127$	−612.391	−0.427881	−0.213941	−	0.976847i	$-0.568630\pi$
$127$	−612.391	−0.427881	−0.213941	+	0.976847i	$0.568630\pi$
$128$	−128.000	−0.0883883
$129$	409.087	0.279210
$130$	0	0
$131$	−2115.00	−1.41060	−0.705298	−	0.708911i	$-0.749187\pi$
$131$	−2115.00	−1.41060	−0.705298	+	0.708911i	$0.749187\pi$
$132$	264.409	0.174348
$133$	−978.990	−0.638264
$134$	−155.763	−0.100417
$135$	−1339.53	−0.853989
$136$	553.720	0.349126
$137$	−2925.55	−1.82443	−0.912214	−	0.409713i	$-0.865629\pi$
$137$	−2925.55	−1.82443	−0.912214	+	0.409713i	$0.865629\pi$
$138$	−217.382	−0.134093
$139$	−2162.95	−1.31985	−0.659923	−	0.751333i	$-0.729411\pi$
$139$	−2162.95	−1.31985	−0.659923	+	0.751333i	$0.729411\pi$
$140$	1742.89	1.05215
$141$	78.7406	0.0470294
$142$	−1442.14	−0.852265
$143$	0	0
$144$	−407.166	−0.235628
$145$	3322.30	1.90277
$146$	114.294	0.0647879
$147$	−135.492	−0.0760218
$148$	−447.889	−0.248759
$149$	−1185.39	−0.651751	−0.325876	−	0.945413i	$-0.605659\pi$
$149$	−1185.39	−0.651751	−0.325876	+	0.945413i	$0.605659\pi$
$150$	735.699	0.400464
$151$	−1757.32	−0.947077	−0.473538	−	0.880773i	$-0.657024\pi$
$151$	−1757.32	−0.947077	−0.473538	+	0.880773i	$0.657024\pi$
$152$	−368.482	−0.196631
$153$	1761.37	0.930710
$154$	−2255.44	−1.18019
$155$	581.112	0.301136
$156$	0	0
$157$	509.980	0.259241	0.129620	−	0.991564i	$-0.458624\pi$
$157$	509.980	0.259241	0.129620	+	0.991564i	$0.458624\pi$
$158$	839.411	0.422658
$159$	−899.047	−0.448422
$160$	656.007	0.324137
$161$	1854.29	0.907695
$162$	−1211.37	−0.587495
$163$	1763.88	0.847591	0.423796	−	0.905758i	$-0.360697\pi$
$163$	1763.88	0.847591	0.423796	+	0.905758i	$0.360697\pi$
$164$	−337.101	−0.160507
$165$	−1355.11	−0.639366
$166$	−1835.20	−0.858067
$167$	−397.025	−0.183968	−0.0919842	−	0.995760i	$-0.529321\pi$
$167$	−397.025	−0.183968	−0.0919842	+	0.995760i	$0.529321\pi$
$168$	−211.840	−0.0972847
$169$	0	0
$170$	−2837.84	−1.28031
$171$	−1172.14	−0.524184
$172$	−1313.43	−0.582258
$173$	−12.3237	−0.00541593	−0.00270796	−	0.999996i	$-0.500862\pi$
$173$	−12.3237	−0.00541593	−0.00270796	+	0.999996i	$0.500862\pi$
$174$	−403.810	−0.175936
$175$	−6275.58	−2.71080
$176$	−848.926	−0.363581
$177$	1020.46	0.433345
$178$	757.609	0.319018
$179$	−278.459	−0.116274	−0.0581368	−	0.998309i	$-0.518516\pi$
$179$	−278.459	−0.116274	−0.0581368	+	0.998309i	$0.518516\pi$
$180$	2086.75	0.864093
$181$	−3482.28	−1.43003	−0.715017	−	0.699107i	$-0.753581\pi$
$181$	−3482.28	−1.43003	−0.715017	+	0.699107i	$0.753581\pi$
$182$	0	0
$183$	495.636	0.200210
$184$	697.939	0.279635
$185$	2295.46	0.912245
$186$	−70.6315	−0.0278438
$187$	3672.40	1.43611
$188$	−252.808	−0.0980742
$189$	−1388.82	−0.534507
$190$	1888.49	0.721082
$191$	3784.38	1.43366	0.716828	−	0.697250i	$-0.245593\pi$
$191$	3784.38	1.43366	0.716828	+	0.697250i	$0.245593\pi$
$192$	−79.7346	−0.0299706
$193$	1990.44	0.742356	0.371178	−	0.928562i	$-0.378954\pi$
$193$	1990.44	0.742356	0.371178	+	0.928562i	$0.378954\pi$
$194$	692.888	0.256425
$195$	0	0
$196$	435.018	0.158534
$197$	−4088.79	−1.47875	−0.739377	−	0.673292i	$-0.764880\pi$
$197$	−4088.79	−1.47875	−0.739377	+	0.673292i	$0.764880\pi$
$198$	−2700.42	−0.969244
$199$	3461.38	1.23302	0.616510	−	0.787347i	$-0.288546\pi$
$199$	3461.38	1.23302	0.616510	+	0.787347i	$0.288546\pi$
$200$	−2362.07	−0.835118
$201$	−97.0291	−0.0340493
$202$	1038.59	0.361756
$203$	3444.54	1.19093
$204$	344.927	0.118381
$205$	1727.66	0.588609
$206$	773.821	0.261722
$207$	2220.13	0.745458
$208$	0	0
$209$	−2443.86	−0.808830
$210$	1085.69	0.356761
$211$	−197.608	−0.0644736	−0.0322368	−	0.999480i	$-0.510263\pi$
$211$	−197.608	−0.0644736	−0.0322368	+	0.999480i	$0.510263\pi$
$212$	2886.52	0.935129
$213$	−898.347	−0.288985
$214$	3085.56	0.985628
$215$	6731.42	2.13525
$216$	−522.739	−0.164666
$217$	602.494	0.188479
$218$	−1071.26	−0.332821
$219$	71.1968	0.0219682
$220$	4350.79	1.33332
$221$	0	0
$222$	−279.002	−0.0843487
$223$	3622.66	1.08785	0.543927	−	0.839132i	$-0.316937\pi$
$223$	3622.66	1.08785	0.543927	+	0.839132i	$0.316937\pi$
$224$	680.145	0.202875
$225$	−7513.70	−2.22628
$226$	1995.78	0.587421
$227$	5788.77	1.69257	0.846287	−	0.532727i	$-0.178833\pi$
$227$	5788.77	1.69257	0.846287	+	0.532727i	$0.178833\pi$
$228$	−229.538	−0.0666732
$229$	−988.340	−0.285203	−0.142601	−	0.989780i	$-0.545547\pi$
$229$	−988.340	−0.285203	−0.142601	+	0.989780i	$0.545547\pi$
$230$	−3576.98	−1.02547
$231$	−1404.97	−0.400175
$232$	1296.49	0.366892
$233$	−6121.48	−1.72116	−0.860582	−	0.509312i	$-0.829900\pi$
$233$	−6121.48	−1.72116	−0.860582	+	0.509312i	$0.829900\pi$
$234$	0	0
$235$	1295.66	0.359657
$236$	−3276.32	−0.903689
$237$	522.891	0.143314
$238$	−2942.26	−0.801338
$239$	−224.263	−0.0606961	−0.0303480	−	0.999539i	$-0.509662\pi$
$239$	−224.263	−0.0606961	−0.0303480	+	0.999539i	$0.509662\pi$
$240$	408.644	0.109908
$241$	−5636.03	−1.50642	−0.753212	−	0.657777i	$-0.771497\pi$
$241$	−5636.03	−1.50642	−0.753212	+	0.657777i	$0.771497\pi$
$242$	−2968.28	−0.788464
$243$	−2518.84	−0.664953
$244$	−1591.31	−0.417514
$245$	−2229.49	−0.581375
$246$	−209.989	−0.0544244
$247$	0	0
$248$	226.773	0.0580649
$249$	−1143.20	−0.290952
$250$	6980.68	1.76599
$251$	2416.44	0.607666	0.303833	−	0.952725i	$-0.401734\pi$
$251$	2416.44	0.607666	0.303833	+	0.952725i	$0.401734\pi$
$252$	2163.53	0.540831
$253$	4628.90	1.15026
$254$	1224.78	0.302558
$255$	−1767.77	−0.434126
$256$	256.000	0.0625000
$257$	−907.064	−0.220160	−0.110080	−	0.993923i	$-0.535111\pi$
$257$	−907.064	−0.220160	−0.110080	+	0.993923i	$0.535111\pi$
$258$	−818.174	−0.197431
$259$	2379.92	0.570969
$260$	0	0
$261$	4124.12	0.978072
$262$	4229.99	0.997442
$263$	3034.33	0.711425	0.355712	−	0.934596i	$-0.384238\pi$
$263$	3034.33	0.711425	0.355712	+	0.934596i	$0.384238\pi$
$264$	−528.819	−0.123282
$265$	−14793.6	−3.42929
$266$	1957.98	0.451321
$267$	471.935	0.108172
$268$	311.526	0.0710056
$269$	−4047.61	−0.917424	−0.458712	−	0.888585i	$-0.651689\pi$
$269$	−4047.61	−0.917424	−0.458712	+	0.888585i	$0.651689\pi$
$270$	2679.06	0.603862
$271$	−3613.26	−0.809926	−0.404963	−	0.914333i	$-0.632716\pi$
$271$	−3613.26	−0.809926	−0.404963	+	0.914333i	$0.632716\pi$
$272$	−1107.44	−0.246869
$273$	0	0
$274$	5851.10	1.29007
$275$	−15665.8	−3.43521
$276$	434.765	0.0948180
$277$	4858.36	1.05383	0.526914	−	0.849918i	$-0.323349\pi$
$277$	4858.36	1.05383	0.526914	+	0.849918i	$0.323349\pi$
$278$	4325.89	0.933272
$279$	721.361	0.154791
$280$	−3485.78	−0.743982
$281$	4570.79	0.970358	0.485179	−	0.874415i	$-0.338754\pi$
$281$	4570.79	0.970358	0.485179	+	0.874415i	$0.338754\pi$
$282$	−157.481	−0.0332548
$283$	867.036	0.182120	0.0910600	−	0.995845i	$-0.470974\pi$
$283$	867.036	0.182120	0.0910600	+	0.995845i	$0.470974\pi$
$284$	2884.28	0.602642
$285$	1176.39	0.244503
$286$	0	0
$287$	1791.23	0.368407
$288$	814.331	0.166614
$289$	−122.284	−0.0248898
$290$	−6644.60	−1.34546
$291$	431.619	0.0869482
$292$	−228.588	−0.0458120
$293$	−1149.40	−0.229176	−0.114588	−	0.993413i	$-0.536555\pi$
$293$	−1149.40	−0.229176	−0.114588	+	0.993413i	$0.536555\pi$
$294$	270.984	0.0537555
$295$	16791.3	3.31400
$296$	895.779	0.175899
$297$	−3466.93	−0.677345
$298$	2370.78	0.460858
$299$	0	0
$300$	−1471.40	−0.283171
$301$	6979.10	1.33644
$302$	3514.64	0.669685
$303$	646.963	0.122664
$304$	736.965	0.139039
$305$	8155.56	1.53110
$306$	−3522.75	−0.658111
$307$	6457.99	1.20058	0.600288	−	0.799784i	$-0.295053\pi$
$307$	6457.99	1.20058	0.600288	+	0.799784i	$0.295053\pi$
$308$	4510.88	0.834517
$309$	482.034	0.0887442
$310$	−1162.22	−0.212935
$311$	4856.58	0.885504	0.442752	−	0.896644i	$-0.354002\pi$
$311$	4856.58	0.885504	0.442752	+	0.896644i	$0.354002\pi$
$312$	0	0
$313$	2594.40	0.468511	0.234255	−	0.972175i	$-0.424735\pi$
$313$	2594.40	0.468511	0.234255	+	0.972175i	$0.424735\pi$
$314$	−1019.96	−0.183311
$315$	−11088.2	−1.98333
$316$	−1678.82	−0.298864
$317$	−2690.19	−0.476643	−0.238322	−	0.971186i	$-0.576597\pi$
$317$	−2690.19	−0.476643	−0.238322	+	0.971186i	$0.576597\pi$
$318$	1798.09	0.317082
$319$	8598.65	1.50919
$320$	−1312.01	−0.229199
$321$	1922.08	0.334205
$322$	−3708.59	−0.641837
$323$	−3188.06	−0.549191
$324$	2422.74	0.415422
$325$	0	0
$326$	−3527.75	−0.599338
$327$	−667.317	−0.112852
$328$	674.201	0.113496
$329$	1343.33	0.225107
$330$	2710.23	0.452100
$331$	−10641.6	−1.76711	−0.883556	−	0.468326i	$-0.844857\pi$
$331$	−10641.6	−1.76711	−0.883556	+	0.468326i	$0.844857\pi$
$332$	3670.40	0.606745
$333$	2849.46	0.468916
$334$	794.050	0.130085
$335$	−1596.59	−0.260391
$336$	423.680	0.0687906
$337$	670.259	0.108342	0.0541711	−	0.998532i	$-0.482748\pi$
$337$	670.259	0.108342	0.0541711	+	0.998532i	$0.482748\pi$
$338$	0	0
$339$	1243.22	0.199182
$340$	5675.69	0.905316
$341$	1504.01	0.238847
$342$	2344.27	0.370654
$343$	4978.78	0.783757
$344$	2626.87	0.411719
$345$	−2228.19	−0.347716
$346$	24.6474	0.00382964
$347$	1047.94	0.162122	0.0810609	−	0.996709i	$-0.474169\pi$
$347$	1047.94	0.162122	0.0810609	+	0.996709i	$0.474169\pi$
$348$	807.621	0.124405
$349$	−8329.53	−1.27756	−0.638782	−	0.769388i	$-0.720561\pi$
$349$	−8329.53	−1.27756	−0.638782	+	0.769388i	$0.720561\pi$
$350$	12551.2	1.91682
$351$	0	0
$352$	1697.85	0.257090
$353$	2824.64	0.425893	0.212947	−	0.977064i	$-0.431694\pi$
$353$	2824.64	0.425893	0.212947	+	0.977064i	$0.431694\pi$
$354$	−2040.91	−0.306421
$355$	−14782.1	−2.21000
$356$	−1515.22	−0.225580
$357$	−1832.81	−0.271717
$358$	556.918	0.0822179
$359$	8753.05	1.28682	0.643410	−	0.765522i	$-0.277519\pi$
$359$	8753.05	1.28682	0.643410	+	0.765522i	$0.277519\pi$
$360$	−4173.49	−0.611006
$361$	−4737.45	−0.690691
$362$	6964.57	1.01119
$363$	−1849.02	−0.267351
$364$	0	0
$365$	1171.53	0.168001
$366$	−991.271	−0.141570
$367$	7378.20	1.04942	0.524712	−	0.851280i	$-0.324173\pi$
$367$	7378.20	1.04942	0.524712	+	0.851280i	$0.324173\pi$
$368$	−1395.88	−0.197731
$369$	2144.62	0.302560
$370$	−4590.91	−0.645055
$371$	−15337.9	−2.14637
$372$	141.263	0.0196886
$373$	2626.42	0.364587	0.182293	−	0.983244i	$-0.441648\pi$
$373$	2626.42	0.364587	0.182293	+	0.983244i	$0.441648\pi$
$374$	−7344.80	−1.01548
$375$	4348.46	0.598809
$376$	505.617	0.0693489
$377$	0	0
$378$	2777.64	0.377953
$379$	−503.491	−0.0682391	−0.0341195	−	0.999418i	$-0.510863\pi$
$379$	−503.491	−0.0682391	−0.0341195	+	0.999418i	$0.510863\pi$
$380$	−3776.98	−0.509882
$381$	762.949	0.102591
$382$	−7568.76	−1.01375
$383$	−264.479	−0.0352852	−0.0176426	−	0.999844i	$-0.505616\pi$
$383$	−264.479	−0.0352852	−0.0176426	+	0.999844i	$0.505616\pi$
$384$	159.469	0.0211924
$385$	−23118.5	−3.06033
$386$	−3980.87	−0.524925
$387$	8356.02	1.09757
$388$	−1385.78	−0.181320
$389$	−5074.42	−0.661397	−0.330698	−	0.943736i	$-0.607284\pi$
$389$	−5074.42	−0.661397	−0.330698	+	0.943736i	$0.607284\pi$
$390$	0	0
$391$	6038.48	0.781021
$392$	−870.036	−0.112101
$393$	2634.98	0.338211
$394$	8177.59	1.04564
$395$	8604.05	1.09599
$396$	5400.84	0.685359
$397$	1633.19	0.206467	0.103234	−	0.994657i	$-0.467081\pi$
$397$	1633.19	0.206467	0.103234	+	0.994657i	$0.467081\pi$
$398$	−6922.77	−0.871877
$399$	1219.68	0.153033
$400$	4724.14	0.590518
$401$	−3512.59	−0.437432	−0.218716	−	0.975789i	$-0.570187\pi$
$401$	−3512.59	−0.437432	−0.218716	+	0.975789i	$0.570187\pi$
$402$	194.058	0.0240765
$403$	0	0
$404$	−2077.17	−0.255800
$405$	−12416.7	−1.52343
$406$	−6889.09	−0.842117
$407$	5941.02	0.723551
$408$	−689.854	−0.0837080
$409$	1502.01	0.181589	0.0907944	−	0.995870i	$-0.471059\pi$
$409$	1502.01	0.181589	0.0907944	+	0.995870i	$0.471059\pi$
$410$	−3455.32	−0.416210
$411$	3644.81	0.437434
$412$	−1547.64	−0.185065
$413$	17409.2	2.07421
$414$	−4440.26	−0.527118
$415$	−18811.0	−2.22505
$416$	0	0
$417$	2694.71	0.316452
$418$	4887.73	0.571929
$419$	−13040.3	−1.52043	−0.760214	−	0.649673i	$-0.774906\pi$
$419$	−13040.3	−1.52043	−0.760214	+	0.649673i	$0.774906\pi$
$420$	−2171.38	−0.252268
$421$	−13790.6	−1.59647	−0.798234	−	0.602348i	$-0.794232\pi$
$421$	−13790.6	−1.59647	−0.798234	+	0.602348i	$0.794232\pi$
$422$	395.217	0.0455897
$423$	1608.36	0.184872
$424$	−5773.05	−0.661236
$425$	−20436.3	−2.33249
$426$	1796.69	0.204343
$427$	8455.64	0.958308
$428$	−6171.12	−0.696944
$429$	0	0
$430$	−13462.8	−1.50985
$431$	13366.5	1.49383	0.746915	−	0.664920i	$-0.231534\pi$
$431$	13366.5	1.49383	0.746915	+	0.664920i	$0.231534\pi$
$432$	1045.48	0.116436
$433$	12375.5	1.37351	0.686753	−	0.726891i	$-0.259036\pi$
$433$	12375.5	1.37351	0.686753	+	0.726891i	$0.259036\pi$
$434$	−1204.99	−0.133275
$435$	−4139.10	−0.456218
$436$	2142.52	0.235340
$437$	−4018.41	−0.439878
$438$	−142.394	−0.0155339
$439$	−12339.8	−1.34156	−0.670782	−	0.741655i	$-0.734041\pi$
$439$	−12339.8	−1.34156	−0.670782	+	0.741655i	$0.734041\pi$
$440$	−8701.59	−0.942800
$441$	−2767.57	−0.298841
$442$	0	0
$443$	−15065.3	−1.61575	−0.807873	−	0.589356i	$-0.799381\pi$
$443$	−15065.3	−1.61575	−0.807873	+	0.589356i	$0.799381\pi$
$444$	558.005	0.0596435
$445$	7765.57	0.827244
$446$	−7245.33	−0.769229
$447$	1476.82	0.156267
$448$	−1360.29	−0.143454
$449$	4223.62	0.443931	0.221966	−	0.975055i	$-0.428753\pi$
$449$	4223.62	0.443931	0.221966	+	0.975055i	$0.428753\pi$
$450$	15027.4	1.57422
$451$	4471.46	0.466858
$452$	−3991.56	−0.415370
$453$	2189.36	0.227076
$454$	−11577.5	−1.19683
$455$	0	0
$456$	459.075	0.0471451
$457$	−5141.66	−0.526295	−0.263148	−	0.964756i	$-0.584761\pi$
$457$	−5141.66	−0.526295	−0.263148	+	0.964756i	$0.584761\pi$
$458$	1976.68	0.201669
$459$	−4522.67	−0.459913
$460$	7153.95	0.725119
$461$	−11630.3	−1.17500	−0.587501	−	0.809223i	$-0.699888\pi$
$461$	−11630.3	−1.17500	−0.587501	+	0.809223i	$0.699888\pi$
$462$	2809.95	0.282967
$463$	1028.85	0.103272	0.0516359	−	0.998666i	$-0.483556\pi$
$463$	1028.85	0.103272	0.0516359	+	0.998666i	$0.483556\pi$
$464$	−2592.99	−0.259432
$465$	−723.981	−0.0722017
$466$	12243.0	1.21705
$467$	14089.0	1.39606	0.698030	−	0.716068i	$-0.254060\pi$
$467$	14089.0	1.39606	0.698030	+	0.716068i	$0.254060\pi$
$468$	0	0
$469$	−1655.34	−0.162977
$470$	−2591.31	−0.254316
$471$	−635.360	−0.0621568
$472$	6552.65	0.639004
$473$	17422.0	1.69358
$474$	−1045.78	−0.101338
$475$	13599.7	1.31368
$476$	5884.53	0.566632
$477$	−18364.0	−1.76274
$478$	448.526	0.0429186
$479$	14176.7	1.35230	0.676149	−	0.736765i	$-0.263648\pi$
$479$	14176.7	1.35230	0.676149	+	0.736765i	$0.263648\pi$
$480$	−817.289	−0.0777166
$481$	0	0
$482$	11272.1	1.06520
$483$	−2310.18	−0.217633
$484$	5936.56	0.557528
$485$	7102.18	0.664934
$486$	5037.68	0.470193
$487$	−5713.18	−0.531600	−0.265800	−	0.964028i	$-0.585636\pi$
$487$	−5713.18	−0.531600	−0.265800	+	0.964028i	$0.585636\pi$
$488$	3182.63	0.295227
$489$	−2197.53	−0.203222
$490$	4458.98	0.411094
$491$	−14868.4	−1.36660	−0.683301	−	0.730137i	$-0.739456\pi$
$491$	−14868.4	−1.36660	−0.683301	+	0.730137i	$0.739456\pi$
$492$	419.978	0.0384839
$493$	11217.1	1.02473
$494$	0	0
$495$	−27679.6	−2.51334
$496$	−453.546	−0.0410581
$497$	−15326.0	−1.38323
$498$	2286.39	0.205734
$499$	−6429.79	−0.576828	−0.288414	−	0.957506i	$-0.593128\pi$
$499$	−6429.79	−0.576828	−0.288414	+	0.957506i	$0.593128\pi$
$500$	−13961.4	−1.24874
$501$	494.635	0.0441091
$502$	−4832.87	−0.429685
$503$	11799.0	1.04590	0.522952	−	0.852362i	$-0.324831\pi$
$503$	11799.0	1.04590	0.522952	+	0.852362i	$0.324831\pi$
$504$	−4327.05	−0.382425
$505$	10645.6	0.938067
$506$	−9257.79	−0.813358
$507$	0	0
$508$	−2449.56	−0.213941
$509$	9183.02	0.799667	0.399833	−	0.916588i	$-0.369068\pi$
$509$	9183.02	0.799667	0.399833	+	0.916588i	$0.369068\pi$
$510$	3535.54	0.306973
$511$	1214.63	0.105151
$512$	−512.000	−0.0441942
$513$	3009.69	0.259027
$514$	1814.13	0.155677
$515$	7931.75	0.678669
$516$	1636.35	0.139605
$517$	3353.37	0.285263
$518$	−4759.84	−0.403736
$519$	15.3536	0.00129855
$520$	0	0
$521$	−2688.15	−0.226046	−0.113023	−	0.993592i	$-0.536053\pi$
$521$	−2688.15	−0.226046	−0.113023	+	0.993592i	$0.536053\pi$
$522$	−8248.24	−0.691601
$523$	15920.5	1.33108	0.665538	−	0.746364i	$-0.268202\pi$
$523$	15920.5	1.33108	0.665538	+	0.746364i	$0.268202\pi$
$524$	−8459.98	−0.705298
$525$	7818.46	0.649953
$526$	−6068.66	−0.503053
$527$	1962.01	0.162176
$528$	1057.64	0.0871738
$529$	−4555.77	−0.374436
$530$	29587.2	2.42488
$531$	20843.9	1.70348
$532$	−3915.96	−0.319132
$533$	0	0
$534$	−943.870	−0.0764892
$535$	31627.3	2.55583
$536$	−623.053	−0.0502086
$537$	346.919	0.0278783
$538$	8095.22	0.648717
$539$	−5770.28	−0.461120
$540$	−5358.13	−0.426995
$541$	1967.63	0.156368	0.0781838	−	0.996939i	$-0.475088\pi$
$541$	1967.63	0.156368	0.0781838	+	0.996939i	$0.475088\pi$
$542$	7226.52	0.572704
$543$	4338.42	0.342871
$544$	2214.88	0.174563
$545$	−10980.5	−0.863035
$546$	0	0
$547$	−6075.05	−0.474863	−0.237432	−	0.971404i	$-0.576306\pi$
$547$	−6075.05	−0.474863	−0.237432	+	0.971404i	$0.576306\pi$
$548$	−11702.2	−0.912214
$549$	10123.9	0.787024
$550$	31331.6	2.42906
$551$	−7464.61	−0.577138
$552$	−869.530	−0.0670465
$553$	8920.63	0.685974
$554$	−9716.72	−0.745169
$555$	−2859.80	−0.218724
$556$	−8651.78	−0.659923
$557$	−9690.09	−0.737132	−0.368566	−	0.929602i	$-0.620151\pi$
$557$	−9690.09	−0.737132	−0.368566	+	0.929602i	$0.620151\pi$
$558$	−1442.72	−0.109454
$559$	0	0
$560$	6971.55	0.526075
$561$	−4575.28	−0.344329
$562$	−9141.58	−0.686147
$563$	−6132.79	−0.459088	−0.229544	−	0.973298i	$-0.573723\pi$
$563$	−6132.79	−0.459088	−0.229544	+	0.973298i	$0.573723\pi$
$564$	314.962	0.0235147
$565$	20457.0	1.52324
$566$	−1734.07	−0.128778
$567$	−12873.5	−0.953506
$568$	−5768.56	−0.426132
$569$	12081.1	0.890098	0.445049	−	0.895506i	$-0.353186\pi$
$569$	12081.1	0.890098	0.445049	+	0.895506i	$0.353186\pi$
$570$	−2352.79	−0.172890
$571$	712.570	0.0522244	0.0261122	−	0.999659i	$-0.491687\pi$
$571$	712.570	0.0522244	0.0261122	+	0.999659i	$0.491687\pi$
$572$	0	0
$573$	−4714.79	−0.343740
$574$	−3582.45	−0.260503
$575$	−25759.1	−1.86822
$576$	−1628.66	−0.117814
$577$	−1413.94	−0.102016	−0.0510078	−	0.998698i	$-0.516243\pi$
$577$	−1413.94	−0.102016	−0.0510078	+	0.998698i	$0.516243\pi$
$578$	244.568	0.0175998
$579$	−2479.79	−0.177991
$580$	13289.2	0.951386
$581$	−19503.1	−1.39265
$582$	−863.237	−0.0614817
$583$	−38288.2	−2.71996
$584$	457.176	0.0323940
$585$	0	0
$586$	2298.79	0.162052
$587$	−7352.50	−0.516985	−0.258492	−	0.966013i	$-0.583226\pi$
$587$	−7352.50	−0.516985	−0.258492	+	0.966013i	$0.583226\pi$
$588$	−541.968	−0.0380109
$589$	−1305.65	−0.0913388
$590$	−33582.7	−2.34335
$591$	5094.04	0.354553
$592$	−1791.56	−0.124379
$593$	−15534.1	−1.07573	−0.537866	−	0.843031i	$-0.680769\pi$
$593$	−15534.1	−1.07573	−0.537866	+	0.843031i	$0.680769\pi$
$594$	6933.85	0.478955
$595$	−30158.5	−2.07795
$596$	−4741.56	−0.325876
$597$	−4312.38	−0.295635
$598$	0	0
$599$	20257.5	1.38180	0.690901	−	0.722949i	$-0.257214\pi$
$599$	20257.5	1.38180	0.690901	+	0.722949i	$0.257214\pi$
$600$	2942.79	0.200232
$601$	25021.6	1.69826	0.849128	−	0.528187i	$-0.177128\pi$
$601$	25021.6	1.69826	0.849128	+	0.528187i	$0.177128\pi$
$602$	−13958.2	−0.945007
$603$	−1981.92	−0.133847
$604$	−7029.28	−0.473538
$605$	−30425.2	−2.04456
$606$	−1293.93	−0.0867363
$607$	−4754.85	−0.317946	−0.158973	−	0.987283i	$-0.550818\pi$
$607$	−4754.85	−0.317946	−0.158973	+	0.987283i	$0.550818\pi$
$608$	−1473.93	−0.0983154
$609$	−4291.40	−0.285544
$610$	−16311.1	−1.08265
$611$	0	0
$612$	7045.49	0.465355
$613$	11141.9	0.734125	0.367063	−	0.930196i	$-0.380363\pi$
$613$	11141.9	0.734125	0.367063	+	0.930196i	$0.380363\pi$
$614$	−12916.0	−0.848936
$615$	−2152.41	−0.141128
$616$	−9021.76	−0.590093
$617$	−8032.61	−0.524118	−0.262059	−	0.965052i	$-0.584401\pi$
$617$	−8032.61	−0.524118	−0.262059	+	0.965052i	$0.584401\pi$
$618$	−964.068	−0.0627516
$619$	−5142.70	−0.333930	−0.166965	−	0.985963i	$-0.553397\pi$
$619$	−5142.70	−0.333930	−0.166965	+	0.985963i	$0.553397\pi$
$620$	2324.45	0.150568
$621$	−5700.62	−0.368371
$622$	−9713.17	−0.626146
$623$	8051.30	0.517767
$624$	0	0
$625$	34645.4	2.21731
$626$	−5188.79	−0.331287
$627$	3044.70	0.193929
$628$	2039.92	0.129620
$629$	7750.17	0.491287
$630$	22176.4	1.40243
$631$	−9537.79	−0.601733	−0.300866	−	0.953666i	$-0.597276\pi$
$631$	−9537.79	−0.601733	−0.300866	+	0.953666i	$0.597276\pi$
$632$	3357.64	0.211329
$633$	246.191	0.0154585
$634$	5380.37	0.337038
$635$	12554.1	0.784561
$636$	−3596.19	−0.224211
$637$	0	0
$638$	−17197.3	−1.06716
$639$	−18349.7	−1.13600
$640$	2624.03	0.162068
$641$	−25332.4	−1.56095	−0.780476	−	0.625186i	$-0.785023\pi$
$641$	−25332.4	−1.56095	−0.780476	+	0.625186i	$0.785023\pi$
$642$	−3844.15	−0.236319
$643$	−813.455	−0.0498904	−0.0249452	−	0.999689i	$-0.507941\pi$
$643$	−813.455	−0.0498904	−0.0249452	+	0.999689i	$0.507941\pi$
$644$	7417.18	0.453847
$645$	−8386.37	−0.511958
$646$	6376.13	0.388337
$647$	18110.9	1.10048	0.550241	−	0.835006i	$-0.314536\pi$
$647$	18110.9	1.10048	0.550241	+	0.835006i	$0.314536\pi$
$648$	−4845.48	−0.293748
$649$	43458.7	2.62851
$650$	0	0
$651$	−750.619	−0.0451906
$652$	7055.50	0.423796
$653$	−11793.7	−0.706773	−0.353387	−	0.935477i	$-0.614970\pi$
$653$	−11793.7	−0.706773	−0.353387	+	0.935477i	$0.614970\pi$
$654$	1334.63	0.0797986
$655$	43357.9	2.58646
$656$	−1348.40	−0.0802534
$657$	1454.27	0.0863568
$658$	−2686.66	−0.159175
$659$	8729.73	0.516027	0.258014	−	0.966141i	$-0.416932\pi$
$659$	8729.73	0.516027	0.258014	+	0.966141i	$0.416932\pi$
$660$	−5420.45	−0.319683
$661$	28673.9	1.68727	0.843636	−	0.536916i	$-0.180411\pi$
$661$	28673.9	1.68727	0.843636	+	0.536916i	$0.180411\pi$
$662$	21283.2	1.24954
$663$	0	0
$664$	−7340.80	−0.429034
$665$	20069.5	1.17032
$666$	−5698.91	−0.331574
$667$	14138.7	0.820766
$668$	−1588.10	−0.0919842
$669$	−4513.31	−0.260829
$670$	3193.18	0.184124
$671$	21107.9	1.21440
$672$	−847.361	−0.0486423
$673$	13868.6	0.794346	0.397173	−	0.917744i	$-0.369991\pi$
$673$	13868.6	0.794346	0.397173	+	0.917744i	$0.369991\pi$
$674$	−1340.52	−0.0766095
$675$	19292.9	1.10012
$676$	0	0
$677$	15320.7	0.869753	0.434876	−	0.900490i	$-0.356792\pi$
$677$	15320.7	0.869753	0.434876	+	0.900490i	$0.356792\pi$
$678$	−2486.45	−0.140843
$679$	7363.50	0.416178
$680$	−11351.4	−0.640155
$681$	−7211.96	−0.405819
$682$	−3008.03	−0.168890
$683$	7588.13	0.425112	0.212556	−	0.977149i	$-0.431821\pi$
$683$	7588.13	0.425112	0.212556	+	0.977149i	$0.431821\pi$
$684$	−4688.54	−0.262092
$685$	59974.4	3.34526
$686$	−9957.55	−0.554200
$687$	1231.33	0.0683815
$688$	−5253.74	−0.291129
$689$	0	0
$690$	4456.39	0.245872
$691$	−18737.6	−1.03156	−0.515782	−	0.856720i	$-0.672498\pi$
$691$	−18737.6	−1.03156	−0.515782	+	0.856720i	$0.672498\pi$
$692$	−49.2949	−0.00270796
$693$	−28698.0	−1.57309
$694$	−2095.88	−0.114637
$695$	44340.8	2.42006
$696$	−1615.24	−0.0879678
$697$	5833.10	0.316994
$698$	16659.1	0.903374
$699$	7626.47	0.412674
$700$	−25102.3	−1.35540
$701$	−20758.4	−1.11845	−0.559226	−	0.829015i	$-0.688902\pi$
$701$	−20758.4	−1.11845	−0.559226	+	0.829015i	$0.688902\pi$
$702$	0	0
$703$	−5157.48	−0.276697
$704$	−3395.71	−0.181790
$705$	−1614.20	−0.0862330
$706$	−5649.27	−0.301152
$707$	11037.3	0.587131
$708$	4081.82	0.216673
$709$	−11658.7	−0.617564	−0.308782	−	0.951133i	$-0.599921\pi$
$709$	−11658.7	−0.617564	−0.308782	+	0.951133i	$0.599921\pi$
$710$	29564.2	1.56271
$711$	10680.6	0.563367
$712$	3030.43	0.159509
$713$	2473.03	0.129896
$714$	3665.63	0.192133
$715$	0	0
$716$	−1113.84	−0.0581368
$717$	279.399	0.0145528
$718$	−17506.1	−0.909919
$719$	−11213.2	−0.581614	−0.290807	−	0.956782i	$-0.593924\pi$
$719$	−11213.2	−0.581614	−0.290807	+	0.956782i	$0.593924\pi$
$720$	8346.98	0.432047
$721$	8223.60	0.424775
$722$	9474.90	0.488392
$723$	7021.66	0.361187
$724$	−13929.1	−0.715017
$725$	−47850.1	−2.45119
$726$	3698.04	0.189046
$727$	31125.2	1.58785	0.793927	−	0.608014i	$-0.208033\pi$
$727$	31125.2	1.58785	0.793927	+	0.608014i	$0.208033\pi$
$728$	0	0
$729$	−13215.4	−0.671411
$730$	−2343.05	−0.118795
$731$	22727.3	1.14993
$732$	1982.54	0.100105
$733$	−38636.1	−1.94687	−0.973437	−	0.228955i	$-0.926469\pi$
$733$	−38636.1	−1.94687	−0.973437	+	0.228955i	$0.926469\pi$
$734$	−14756.4	−0.742055
$735$	2777.62	0.139393
$736$	2791.76	0.139817
$737$	−4132.23	−0.206530
$738$	−4289.24	−0.213942
$739$	−23806.6	−1.18503	−0.592517	−	0.805558i	$-0.701866\pi$
$739$	−23806.6	−1.18503	−0.592517	+	0.805558i	$0.701866\pi$
$740$	9181.83	0.456123
$741$	0	0
$742$	30675.8	1.51772
$743$	−12575.0	−0.620903	−0.310451	−	0.950589i	$-0.600480\pi$
$743$	−12575.0	−0.620903	−0.310451	+	0.950589i	$0.600480\pi$
$744$	−282.526	−0.0139219
$745$	24300.8	1.19505
$746$	−5252.84	−0.257802
$747$	−23351.0	−1.14373
$748$	14689.6	0.718055
$749$	32791.0	1.59968
$750$	−8696.91	−0.423422
$751$	17709.8	0.860506	0.430253	−	0.902708i	$-0.358424\pi$
$751$	17709.8	0.860506	0.430253	+	0.902708i	$0.358424\pi$
$752$	−1011.23	−0.0490371
$753$	−3010.53	−0.145697
$754$	0	0
$755$	36025.4	1.73656
$756$	−5555.28	−0.267253
$757$	−5976.41	−0.286943	−0.143472	−	0.989654i	$-0.545827\pi$
$757$	−5976.41	−0.286943	−0.143472	+	0.989654i	$0.545827\pi$
$758$	1006.98	0.0482523
$759$	−5766.93	−0.275792
$760$	7553.97	0.360541
$761$	−14868.4	−0.708249	−0.354125	−	0.935198i	$-0.615221\pi$
$761$	−14868.4	−0.708249	−0.354125	+	0.935198i	$0.615221\pi$
$762$	−1525.90	−0.0725426
$763$	−11384.6	−0.540168
$764$	15137.5	0.716828
$765$	−36108.5	−1.70654
$766$	528.957	0.0249504
$767$	0	0
$768$	−318.939	−0.0149853
$769$	−6592.53	−0.309145	−0.154573	−	0.987981i	$-0.549400\pi$
$769$	−6592.53	−0.309145	−0.154573	+	0.987981i	$0.549400\pi$
$770$	46237.0	2.16398
$771$	1130.07	0.0527866
$772$	7961.74	0.371178
$773$	33862.6	1.57562	0.787810	−	0.615918i	$-0.211215\pi$
$773$	33862.6	1.57562	0.787810	+	0.615918i	$0.211215\pi$
$774$	−16712.0	−0.776101
$775$	−8369.59	−0.387929
$776$	2771.55	0.128213
$777$	−2965.03	−0.136898
$778$	10148.8	0.467678
$779$	−3881.74	−0.178534
$780$	0	0
$781$	−38258.4	−1.75287
$782$	−12077.0	−0.552265
$783$	−10589.5	−0.483317
$784$	1740.07	0.0792671
$785$	−10454.7	−0.475343
$786$	−5269.95	−0.239151
$787$	15674.1	0.709939	0.354970	−	0.934878i	$-0.384491\pi$
$787$	15674.1	0.709939	0.354970	+	0.934878i	$0.384491\pi$
$788$	−16355.2	−0.739377
$789$	−3780.33	−0.170574
$790$	−17208.1	−0.774983
$791$	21209.7	0.953386
$792$	−10801.7	−0.484622
$793$	0	0
$794$	−3266.38	−0.145994
$795$	18430.7	0.822224
$796$	13845.5	0.616510
$797$	18387.2	0.817200	0.408600	−	0.912714i	$-0.366017\pi$
$797$	18387.2	0.817200	0.408600	+	0.912714i	$0.366017\pi$
$798$	−2439.36	−0.108211
$799$	4374.53	0.193692
$800$	−9448.28	−0.417559
$801$	9639.76	0.425224
$802$	7025.18	0.309311
$803$	3032.10	0.133251
$804$	−388.116	−0.0170246
$805$	−38013.4	−1.66434
$806$	0	0
$807$	5042.73	0.219966
$808$	4154.35	0.180878
$809$	17954.5	0.780280	0.390140	−	0.920756i	$-0.372427\pi$
$809$	17954.5	0.780280	0.390140	+	0.920756i	$0.372427\pi$
$810$	24833.3	1.07723
$811$	−42308.9	−1.83189	−0.915947	−	0.401300i	$-0.868559\pi$
$811$	−42308.9	−1.83189	−0.915947	+	0.401300i	$0.868559\pi$
$812$	13778.2	0.595467
$813$	4501.59	0.194192
$814$	−11882.0	−0.511628
$815$	−36159.8	−1.55414
$816$	1379.71	0.0591905
$817$	−15124.3	−0.647653
$818$	−3004.03	−0.128403
$819$	0	0
$820$	6910.63	0.294305
$821$	6656.89	0.282981	0.141490	−	0.989940i	$-0.454811\pi$
$821$	6656.89	0.282981	0.141490	+	0.989940i	$0.454811\pi$
$822$	−7289.62	−0.309312
$823$	−31429.9	−1.33120	−0.665600	−	0.746308i	$-0.731824\pi$
$823$	−31429.9	−1.33120	−0.665600	+	0.746308i	$0.731824\pi$
$824$	3095.28	0.130861
$825$	19517.3	0.823643
$826$	−34818.3	−1.46669
$827$	−31475.2	−1.32346	−0.661730	−	0.749742i	$-0.730178\pi$
$827$	−31475.2	−1.32346	−0.661730	+	0.749742i	$0.730178\pi$
$828$	8880.52	0.372729
$829$	17919.1	0.750732	0.375366	−	0.926877i	$-0.377517\pi$
$829$	17919.1	0.750732	0.375366	+	0.926877i	$0.377517\pi$
$830$	37622.0	1.57335
$831$	−6052.81	−0.252671
$832$	0	0
$833$	−7527.44	−0.313098
$834$	−5389.43	−0.223766
$835$	8139.10	0.337324
$836$	−9775.45	−0.404415
$837$	−1852.24	−0.0764906
$838$	26080.6	1.07510
$839$	48376.2	1.99062	0.995310	−	0.0967360i	$-0.0308403\pi$
$839$	48376.2	1.99062	0.995310	+	0.0967360i	$0.0308403\pi$
$840$	4342.77	0.178381
$841$	1875.00	0.0768788
$842$	27581.2	1.12887
$843$	−5694.54	−0.232657
$844$	−790.434	−0.0322368
$845$	0	0
$846$	−3216.71	−0.130725
$847$	−31544.7	−1.27968
$848$	11546.1	0.467564
$849$	−1080.20	−0.0436659
$850$	40872.7	1.64932
$851$	9768.73	0.393499
$852$	−3593.39	−0.144492
$853$	−36926.5	−1.48222	−0.741112	−	0.671381i	$-0.765701\pi$
$853$	−36926.5	−1.48222	−0.741112	+	0.671381i	$0.765701\pi$
$854$	−16911.3	−0.677626
$855$	24029.0	0.961141
$856$	12342.2	0.492814
$857$	17315.0	0.690161	0.345080	−	0.938573i	$-0.387852\pi$
$857$	17315.0	0.690161	0.345080	+	0.938573i	$0.387852\pi$
$858$	0	0
$859$	37342.8	1.48326	0.741630	−	0.670809i	$-0.234053\pi$
$859$	37342.8	1.48326	0.741630	+	0.670809i	$0.234053\pi$
$860$	26925.7	1.06763
$861$	−2231.61	−0.0883310
$862$	−26733.0	−1.05630
$863$	13285.9	0.524053	0.262026	−	0.965061i	$-0.415609\pi$
$863$	13285.9	0.524053	0.262026	+	0.965061i	$0.415609\pi$
$864$	−2090.96	−0.0823330
$865$	252.639	0.00993062
$866$	−24751.0	−0.971215
$867$	152.348	0.00596770
$868$	2409.98	0.0942395
$869$	22268.7	0.869290
$870$	8278.20	0.322595
$871$	0	0
$872$	−4285.04	−0.166410
$873$	8816.25	0.341793
$874$	8036.82	0.311041
$875$	74185.5	2.86620
$876$	284.787	0.0109841
$877$	−44228.8	−1.70296	−0.851482	−	0.524384i	$-0.824296\pi$
$877$	−44228.8	−1.70296	−0.851482	+	0.524384i	$0.824296\pi$
$878$	24679.6	0.948628
$879$	1431.98	0.0549482
$880$	17403.2	0.666660
$881$	−49187.9	−1.88102	−0.940511	−	0.339762i	$-0.889653\pi$
$881$	−49187.9	−1.88102	−0.940511	+	0.339762i	$0.889653\pi$
$882$	5535.13	0.211312
$883$	−19458.9	−0.741612	−0.370806	−	0.928710i	$-0.620918\pi$
$883$	−19458.9	−0.741612	−0.370806	+	0.928710i	$0.620918\pi$
$884$	0	0
$885$	−20919.6	−0.794580
$886$	30130.7	1.14251
$887$	13529.3	0.512141	0.256071	−	0.966658i	$-0.417572\pi$
$887$	13529.3	0.512141	0.256071	+	0.966658i	$0.417572\pi$
$888$	−1116.01	−0.0421743
$889$	13016.1	0.491052
$890$	−15531.1	−0.584950
$891$	−32136.4	−1.20832
$892$	14490.7	0.543927
$893$	−2911.11	−0.109089
$894$	−2953.65	−0.110497
$895$	5708.47	0.213199
$896$	2720.58	0.101438
$897$	0	0
$898$	−8447.24	−0.313907
$899$	4593.90	0.170429
$900$	−30054.8	−1.11314
$901$	−49947.7	−1.84684
$902$	−8942.92	−0.330118
$903$	−8694.94	−0.320431
$904$	7983.12	0.293711
$905$	71387.6	2.62210
$906$	−4378.73	−0.160567
$907$	−49365.3	−1.80722	−0.903609	−	0.428357i	$-0.859092\pi$
$907$	−49365.3	−1.80722	−0.903609	+	0.428357i	$0.859092\pi$
$908$	23155.1	0.846287
$909$	13214.9	0.482190
$910$	0	0
$911$	−38465.7	−1.39893	−0.699465	−	0.714667i	$-0.746578\pi$
$911$	−38465.7	−1.39893	−0.699465	+	0.714667i	$0.746578\pi$
$912$	−918.150	−0.0333366
$913$	−48685.9	−1.76481
$914$	10283.3	0.372147
$915$	−10160.6	−0.367104
$916$	−3953.36	−0.142601
$917$	44953.2	1.61885
$918$	9045.34	0.325208
$919$	−14358.4	−0.515386	−0.257693	−	0.966227i	$-0.582962\pi$
$919$	−14358.4	−0.515386	−0.257693	+	0.966227i	$0.582962\pi$
$920$	−14307.9	−0.512736
$921$	−8045.71	−0.287856
$922$	23260.6	0.830852
$923$	0	0
$924$	−5619.90	−0.200088
$925$	−33060.8	−1.17517
$926$	−2057.71	−0.0730243
$927$	9846.04	0.348853
$928$	5185.97	0.183446
$929$	−48022.5	−1.69598	−0.847992	−	0.530009i	$-0.822188\pi$
$929$	−48022.5	−1.69598	−0.847992	+	0.530009i	$0.822188\pi$
$930$	1447.96	0.0510543
$931$	5009.26	0.176339
$932$	−24485.9	−0.860582
$933$	−6050.59	−0.212312
$934$	−28178.0	−0.987164
$935$	−75285.0	−2.63324
$936$	0	0
$937$	−38908.9	−1.35656	−0.678281	−	0.734803i	$-0.737275\pi$
$937$	−38908.9	−1.35656	−0.678281	+	0.734803i	$0.737275\pi$
$938$	3310.67	0.115242
$939$	−3232.24	−0.112332
$940$	5182.63	0.179828
$941$	19540.6	0.676946	0.338473	−	0.940976i	$-0.390090\pi$
$941$	19540.6	0.676946	0.338473	+	0.940976i	$0.390090\pi$
$942$	1270.72	0.0439515
$943$	7352.36	0.253898
$944$	−13105.3	−0.451844
$945$	28471.1	0.980069
$946$	−34844.0	−1.19754
$947$	−8232.59	−0.282496	−0.141248	−	0.989974i	$-0.545111\pi$
$947$	−8232.59	−0.282496	−0.141248	+	0.989974i	$0.545111\pi$
$948$	2091.57	0.0716571
$949$	0	0
$950$	−27199.4	−0.928911
$951$	3351.58	0.114282
$952$	−11769.1	−0.400669
$953$	16636.2	0.565478	0.282739	−	0.959197i	$-0.408757\pi$
$953$	16636.2	0.565478	0.282739	+	0.959197i	$0.408757\pi$
$954$	36727.9	1.24645
$955$	−77580.7	−2.62874
$956$	−897.052	−0.0303480
$957$	−10712.7	−0.361851
$958$	−28353.4	−0.956219
$959$	62181.2	2.09378
$960$	1634.58	0.0549539
$961$	−28987.5	−0.973028
$962$	0	0
$963$	39260.4	1.31376
$964$	−22544.1	−0.753212
$965$	−40804.4	−1.36118
$966$	4620.36	0.153890
$967$	9036.53	0.300512	0.150256	−	0.988647i	$-0.451990\pi$
$967$	9036.53	0.300512	0.150256	+	0.988647i	$0.451990\pi$
$968$	−11873.1	−0.394232
$969$	3971.86	0.131677
$970$	−14204.4	−0.470180
$971$	42397.3	1.40123	0.700614	−	0.713540i	$-0.252909\pi$
$971$	42397.3	1.40123	0.700614	+	0.713540i	$0.252909\pi$
$972$	−10075.4	−0.332476
$973$	45972.3	1.51470
$974$	11426.4	0.375898
$975$	0	0
$976$	−6365.25	−0.208757
$977$	−33074.7	−1.08307	−0.541533	−	0.840680i	$-0.682156\pi$
$977$	−33074.7	−1.08307	−0.541533	+	0.840680i	$0.682156\pi$
$978$	4395.06	0.143700
$979$	20098.6	0.656132
$980$	−8917.96	−0.290687
$981$	−13630.6	−0.443621
$982$	29736.8	0.966334
$983$	17120.9	0.555517	0.277759	−	0.960651i	$-0.410408\pi$
$983$	17120.9	0.555517	0.277759	+	0.960651i	$0.410408\pi$
$984$	−839.956	−0.0272122
$985$	83821.2	2.71144
$986$	−22434.2	−0.724595
$987$	−1673.59	−0.0539727
$988$	0	0
$989$	28646.8	0.921046
$990$	55359.2	1.77720
$991$	−43698.5	−1.40074	−0.700368	−	0.713782i	$-0.746981\pi$
$991$	−43698.5	−1.40074	−0.700368	+	0.713782i	$0.746981\pi$
$992$	907.092	0.0290325
$993$	13257.8	0.423691
$994$	30652.0	0.978090
$995$	−70959.1	−2.26086
$996$	−4572.78	−0.145476
$997$	−2611.53	−0.0829570	−0.0414785	−	0.999139i	$-0.513207\pi$
$997$	−2611.53	−0.0829570	−0.0414785	+	0.999139i	$0.513207\pi$
$998$	12859.6	0.407879
$999$	−7316.54	−0.231717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.a.n.1.3	✓	6
13.2	odd	12		338.4.e.i.147.2		24
13.3	even	3		338.4.c.p.191.4		12
13.4	even	6		338.4.c.o.315.4		12
13.5	odd	4		338.4.b.h.337.9		12
13.6	odd	12		338.4.e.i.23.8		24
13.7	odd	12		338.4.e.i.23.2		24
13.8	odd	4		338.4.b.h.337.3		12
13.9	even	3		338.4.c.p.315.4		12
13.10	even	6		338.4.c.o.191.4		12
13.11	odd	12		338.4.e.i.147.8		24
13.12	even	2		338.4.a.o.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.n.1.3	✓	6	1.1	even	1	trivial
338.4.a.o.1.3	yes	6	13.12	even	2
338.4.b.h.337.3		12	13.8	odd	4
338.4.b.h.337.9		12	13.5	odd	4
338.4.c.o.191.4		12	13.10	even	6
338.4.c.o.315.4		12	13.4	even	6
338.4.c.p.191.4		12	13.3	even	3
338.4.c.p.315.4		12	13.9	even	3
338.4.e.i.23.2		24	13.7	odd	12
338.4.e.i.23.8		24	13.6	odd	12
338.4.e.i.147.2		24	13.2	odd	12
338.4.e.i.147.8		24	13.11	odd	12

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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -1.24585 q^{3} +4.00000 q^{4} -20.5002 q^{5} +2.49171 q^{6} -21.2545 q^{7} -8.00000 q^{8} -25.4478 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} -1.24585 q^{3} +4.00000 q^{4} -20.5002 q^{5} +2.49171 q^{6} -21.2545 q^{7} -8.00000 q^{8} -25.4478 q^{9} +41.0004 q^{10} -53.0579 q^{11} -4.98341 q^{12} +42.5090 q^{14} +25.5403 q^{15} +16.0000 q^{16} -69.2150 q^{17} +50.8957 q^{18} +46.0603 q^{19} -82.0009 q^{20} +26.4800 q^{21} +106.116 q^{22} -87.2424 q^{23} +9.96683 q^{24} +295.259 q^{25} +65.3423 q^{27} -85.0181 q^{28} -162.062 q^{29} -51.0805 q^{30} -28.3466 q^{31} -32.0000 q^{32} +66.1024 q^{33} +138.430 q^{34} +435.722 q^{35} -101.791 q^{36} -111.972 q^{37} -92.1206 q^{38} +164.002 q^{40} -84.2751 q^{41} -52.9600 q^{42} -328.359 q^{43} -212.232 q^{44} +521.686 q^{45} +174.485 q^{46} -63.2021 q^{47} -19.9337 q^{48} +108.754 q^{49} -590.518 q^{50} +86.2318 q^{51} +721.631 q^{53} -130.685 q^{54} +1087.70 q^{55} +170.036 q^{56} -57.3844 q^{57} +324.123 q^{58} -819.081 q^{59} +102.161 q^{60} -397.828 q^{61} +56.6933 q^{62} +540.882 q^{63} +64.0000 q^{64} -132.205 q^{66} +77.8816 q^{67} -276.860 q^{68} +108.691 q^{69} -871.444 q^{70} +721.069 q^{71} +203.583 q^{72} -57.1470 q^{73} +223.945 q^{74} -367.849 q^{75} +184.241 q^{76} +1127.72 q^{77} -419.705 q^{79} -328.003 q^{80} +605.685 q^{81} +168.550 q^{82} +917.600 q^{83} +105.920 q^{84} +1418.92 q^{85} +656.717 q^{86} +201.905 q^{87} +424.463 q^{88} -378.804 q^{89} -1043.37 q^{90} -348.970 q^{92} +35.3158 q^{93} +126.404 q^{94} -944.246 q^{95} +39.8673 q^{96} -346.444 q^{97} -217.509 q^{98} +1350.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9	\( 6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100}) \) 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more