LMFDB - Embedded newform 338.4.b.f.337.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [338,4,Mod(337,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.337"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	338.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.9426455819$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$-1.24698i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.4.b.f.337.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000i q^{2} -8.74094 q^{3} -4.00000 q^{4} +14.1685i q^{5} +17.4819i q^{6} +28.6504i q^{7} +8.00000i q^{8} +49.4040 q^{9} +28.3370 q^{10} +9.49349i q^{11} +34.9638 q^{12} +57.3008 q^{14} -123.846i q^{15} +16.0000 q^{16} -30.6104 q^{17} -98.8080i q^{18} +153.860i q^{19} -56.6741i q^{20} -250.431i q^{21} +18.9870 q^{22} +36.0030 q^{23} -69.9275i q^{24} -75.7470 q^{25} -195.832 q^{27} -114.602i q^{28} -49.2567 q^{29} -247.692 q^{30} +166.984i q^{31} -32.0000i q^{32} -82.9820i q^{33} +61.2209i q^{34} -405.934 q^{35} -197.616 q^{36} +23.8808i q^{37} +307.720 q^{38} -113.348 q^{40} +125.474i q^{41} -500.863 q^{42} +434.774 q^{43} -37.9739i q^{44} +699.982i q^{45} -72.0061i q^{46} +186.017i q^{47} -139.855 q^{48} -477.845 q^{49} +151.494i q^{50} +267.564 q^{51} -400.631 q^{53} +391.664i q^{54} -134.509 q^{55} -229.203 q^{56} -1344.88i q^{57} +98.5134i q^{58} +408.208i q^{59} +495.385i q^{60} -603.762 q^{61} +333.968 q^{62} +1415.44i q^{63} -64.0000 q^{64} -165.964 q^{66} -287.548i q^{67} +122.442 q^{68} -314.700 q^{69} +811.868i q^{70} -961.803i q^{71} +395.232i q^{72} -963.198i q^{73} +47.7616 q^{74} +662.100 q^{75} -615.440i q^{76} -271.992 q^{77} -1043.18 q^{79} +226.696i q^{80} +377.848 q^{81} +250.948 q^{82} +313.304i q^{83} +1001.73i q^{84} -433.705i q^{85} -869.549i q^{86} +430.550 q^{87} -75.9479 q^{88} -675.754i q^{89} +1399.96 q^{90} -144.012 q^{92} -1459.60i q^{93} +372.034 q^{94} -2179.97 q^{95} +279.710i q^{96} -272.435i q^{97} +955.691i q^{98} +469.016i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{3} - 24 q^{4} + 18 q^{9} + 48 q^{10} + 96 q^{12} + 108 q^{14} + 96 q^{16} + 180 q^{17} - 328 q^{22} - 38 q^{23} + 122 q^{25} - 138 q^{27} - 202 q^{29} - 360 q^{30} - 916 q^{35} - 72 q^{36} + 520 q^{38}+ \cdots - 3658 q^{95}+O(q^{100}) $ 6 * q - 24 * q^3 - 24 * q^4 + 18 * q^9 + 48 * q^10 + 96 * q^12 + 108 * q^14 + 96 * q^16 + 180 * q^17 - 328 * q^22 - 38 * q^23 + 122 * q^25 - 138 * q^27 - 202 * q^29 - 360 * q^30 - 916 * q^35 - 72 * q^36 + 520 * q^38 - 192 * q^40 - 936 * q^42 + 2410 * q^43 - 384 * q^48 + 368 * q^49 + 414 * q^51 - 2190 * q^53 - 1052 * q^55 - 432 * q^56 - 2216 * q^61 + 2076 * q^62 - 384 * q^64 + 584 * q^66 - 720 * q^68 + 628 * q^69 - 336 * q^74 + 1010 * q^75 - 960 * q^77 - 3922 * q^79 + 1830 * q^81 + 748 * q^82 + 3272 * q^87 + 1312 * q^88 + 2580 * q^90 + 152 * q^92 + 2144 * q^94 - 3658 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/338\mathbb{Z}\right)^\times$.

$n$	$171$
$\chi(n)$	$-1$

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 $

$2$	− 2.00000i	− 0.707107i
$2$	− 2.00000i	− 0.707107i
$3$	−8.74094	−1.68219	−0.841097	−	0.540884i	$-0.818090\pi$
$3$	−8.74094	−1.68219	−0.841097	+	0.540884i	$0.818090\pi$
$4$	−4.00000	−0.500000
$5$	14.1685i	1.26727i	0.773632	+	0.633636i	$0.218438\pi$
$5$	14.1685i	1.26727i	−0.773632	+	0.633636i	$0.781562\pi$
$6$	17.4819i	1.18949i
$7$	28.6504i	1.54698i	0.633811	+	0.773488i	$0.281490\pi$
$7$	28.6504i	1.54698i	−0.633811	+	0.773488i	$0.718510\pi$
$8$	8.00000i	0.353553i
$9$	49.4040	1.82978
$10$	28.3370	0.896096
$11$	9.49349i	0.260218i	0.991500	+	0.130109i	$0.0415327\pi$
$11$	9.49349i	0.260218i	−0.991500	+	0.130109i	$0.958467\pi$
$12$	34.9638	0.841097
$13$	0	0
$13$	0	0
$14$	57.3008	1.09388
$15$	− 123.846i	− 2.13180i
$16$	16.0000	0.250000
$17$	−30.6104	−0.436713	−0.218356	−	0.975869i	$-0.570070\pi$
$17$	−30.6104	−0.436713	−0.218356	+	0.975869i	$0.570070\pi$
$18$	− 98.8080i	− 1.29385i
$19$	153.860i	1.85778i	0.370351	+	0.928892i	$0.379237\pi$
$19$	153.860i	1.85778i	−0.370351	+	0.928892i	$0.620763\pi$
$20$	− 56.6741i	− 0.633636i
$21$	− 250.431i	− 2.60231i
$22$	18.9870	0.184002
$23$	36.0030	0.326398	0.163199	−	0.986593i	$-0.447819\pi$
$23$	36.0030	0.326398	0.163199	+	0.986593i	$0.447819\pi$
$24$	− 69.9275i	− 0.594746i
$25$	−75.7470	−0.605976
$26$	0	0
$27$	−195.832	−1.39585
$28$	− 114.602i	− 0.773488i
$29$	−49.2567	−0.315405	−0.157702	−	0.987487i	$-0.550409\pi$
$29$	−49.2567	−0.315405	−0.157702	+	0.987487i	$0.550409\pi$
$30$	−247.692	−1.50741
$31$	166.984i	0.967459i	0.875217	+	0.483730i	$0.160718\pi$
$31$	166.984i	0.967459i	−0.875217	+	0.483730i	$0.839282\pi$
$32$	− 32.0000i	− 0.176777i
$33$	− 82.9820i	− 0.437737i
$34$	61.2209i	0.308803i
$35$	−405.934	−1.96044
$36$	−197.616	−0.914889
$37$	23.8808i	0.106107i	0.998592	+	0.0530537i	$0.0168955\pi$
$37$	23.8808i	0.106107i	−0.998592	+	0.0530537i	$0.983105\pi$
$38$	307.720	1.31365
$39$	0	0
$40$	−113.348	−0.448048
$41$	125.474i	0.477946i	0.971026	+	0.238973i	$0.0768107\pi$
$41$	125.474i	0.477946i	−0.971026	+	0.238973i	$0.923189\pi$
$42$	−500.863	−1.84011
$43$	434.774	1.54192	0.770959	−	0.636885i	$-0.219777\pi$
$43$	434.774	1.54192	0.770959	+	0.636885i	$0.219777\pi$
$44$	− 37.9739i	− 0.130109i
$45$	699.982i	2.31883i
$46$	− 72.0061i	− 0.230798i
$47$	186.017i	0.577306i	0.957434	+	0.288653i	$0.0932073\pi$
$47$	186.017i	0.577306i	−0.957434	+	0.288653i	$0.906793\pi$
$48$	−139.855	−0.420549
$49$	−477.845	−1.39314
$50$	151.494i	0.428490i
$51$	267.564	0.734636
$52$	0	0
$53$	−400.631	−1.03832	−0.519159	−	0.854678i	$-0.673755\pi$
$53$	−400.631	−1.03832	−0.519159	+	0.854678i	$0.673755\pi$
$54$	391.664i	0.987014i
$55$	−134.509	−0.329766
$56$	−229.203	−0.546939
$57$	− 1344.88i	− 3.12515i
$58$	98.5134i	0.223025i
$59$	408.208i	0.900748i	0.892840	+	0.450374i	$0.148709\pi$
$59$	408.208i	0.900748i	−0.892840	+	0.450374i	$0.851291\pi$
$60$	495.385i	1.06590i
$61$	−603.762	−1.26728	−0.633638	−	0.773630i	$-0.718439\pi$
$61$	−603.762	−1.26728	−0.633638	+	0.773630i	$0.718439\pi$
$62$	333.968	0.684097
$63$	1415.44i	2.83062i
$64$	−64.0000	−0.125000
$65$	0	0
$66$	−165.964	−0.309527
$67$	− 287.548i	− 0.524322i	−0.965024	−	0.262161i	$-0.915565\pi$
$67$	− 287.548i	− 0.524322i	0.965024	−	0.262161i	$-0.0844351\pi$
$68$	122.442	0.218356
$69$	−314.700	−0.549065
$70$	811.868i	1.38624i
$71$	− 961.803i	− 1.60768i	−0.594848	−	0.803838i	$-0.702788\pi$
$71$	− 961.803i	− 1.60768i	0.594848	−	0.803838i	$-0.297212\pi$
$72$	395.232i	0.646924i
$73$	− 963.198i	− 1.54430i	−0.635441	−	0.772149i	$-0.719182\pi$
$73$	− 963.198i	− 1.54430i	0.635441	−	0.772149i	$-0.280818\pi$
$74$	47.7616	0.0750293
$75$	662.100	1.01937
$76$	− 615.440i	− 0.928892i
$77$	−271.992	−0.402550
$78$	0	0
$79$	−1043.18	−1.48566	−0.742828	−	0.669482i	$-0.766516\pi$
$79$	−1043.18	−1.48566	−0.742828	+	0.669482i	$0.766516\pi$
$80$	226.696i	0.316818i
$81$	377.848	0.518310
$82$	250.948	0.337959
$83$	313.304i	0.414332i	0.978306	+	0.207166i	$0.0664240\pi$
$83$	313.304i	0.414332i	−0.978306	+	0.207166i	$0.933576\pi$
$84$	1001.73i	1.30116i
$85$	− 433.705i	− 0.553434i
$86$	− 869.549i	− 1.09030i
$87$	430.550	0.530572
$88$	−75.9479	−0.0920008
$89$	− 675.754i	− 0.804829i	−0.915458	−	0.402414i	$-0.868171\pi$
$89$	− 675.754i	− 0.804829i	0.915458	−	0.402414i	$-0.131829\pi$
$90$	1399.96	1.63966
$91$	0	0
$92$	−144.012	−0.163199
$93$	− 1459.60i	− 1.62745i
$94$	372.034	0.408217
$95$	−2179.97	−2.35432
$96$	279.710i	0.297373i
$97$	− 272.435i	− 0.285171i	−0.989783	−	0.142585i	$-0.954458\pi$
$97$	− 272.435i	− 0.285171i	0.989783	−	0.142585i	$-0.0455415\pi$
$98$	955.691i	0.985095i
$99$	469.016i	0.476140i
$100$	302.988	0.302988
$101$	661.561	0.651760	0.325880	−	0.945411i	$-0.394339\pi$
$101$	661.561	0.651760	0.325880	+	0.945411i	$0.394339\pi$
$102$	− 535.128i	− 0.519466i
$103$	315.353	0.301676	0.150838	−	0.988558i	$-0.451803\pi$
$103$	315.353	0.301676	0.150838	+	0.988558i	$0.451803\pi$
$104$	0	0
$105$	3548.24	3.29784
$106$	801.262i	0.734202i
$107$	2121.32	1.91660	0.958299	−	0.285769i	$-0.0922490\pi$
$107$	2121.32	1.91660	0.958299	+	0.285769i	$0.0922490\pi$
$108$	783.328	0.697924
$109$	884.432i	0.777185i	0.921410	+	0.388593i	$0.127039\pi$
$109$	884.432i	0.777185i	−0.921410	+	0.388593i	$0.872961\pi$
$110$	269.017i	0.233180i
$111$	− 208.740i	− 0.178493i
$112$	458.406i	0.386744i
$113$	−132.205	−0.110060	−0.0550299	−	0.998485i	$-0.517525\pi$
$113$	−132.205	−0.110060	−0.0550299	+	0.998485i	$0.517525\pi$
$114$	−2689.76	−2.20982
$115$	510.110i	0.413635i
$116$	197.027	0.157702
$117$	0	0
$118$	816.415	0.636925
$119$	− 877.001i	− 0.675585i
$120$	990.769	0.753704
$121$	1240.87	0.932287
$122$	1207.52i	0.896099i
$123$	− 1096.76i	− 0.803998i
$124$	− 667.937i	− 0.483730i
$125$	697.842i	0.499335i
$126$	2830.89	2.00155
$127$	1872.81	1.30854	0.654271	−	0.756260i	$-0.272976\pi$
$127$	1872.81	1.30854	0.654271	+	0.756260i	$0.272976\pi$
$128$	128.000i	0.0883883i
$129$	−3800.34	−2.59381
$130$	0	0
$131$	−1850.21	−1.23400	−0.617000	−	0.786963i	$-0.711652\pi$
$131$	−1850.21	−1.23400	−0.617000	+	0.786963i	$0.711652\pi$
$132$	331.928i	0.218868i
$133$	−4408.15	−2.87395
$134$	−575.096	−0.370751
$135$	− 2774.65i	− 1.76892i
$136$	− 244.883i	− 0.154401i
$137$	− 967.375i	− 0.603273i	−0.953423	−	0.301637i	$-0.902467\pi$
$137$	− 967.375i	− 0.603273i	0.953423	−	0.301637i	$-0.0975329\pi$
$138$	629.401i	0.388247i
$139$	−103.597	−0.0632157	−0.0316078	−	0.999500i	$-0.510063\pi$
$139$	−103.597	−0.0632157	−0.0316078	+	0.999500i	$0.510063\pi$
$140$	1623.74	0.980219
$141$	− 1625.96i	− 0.971140i
$142$	−1923.61	−1.13680
$143$	0	0
$144$	790.464	0.457445
$145$	− 697.894i	− 0.399703i
$146$	−1926.40	−1.09198
$147$	4176.82	2.34352
$148$	− 95.5231i	− 0.0530537i
$149$	− 1917.44i	− 1.05424i	−0.849789	−	0.527122i	$-0.823271\pi$
$149$	− 1917.44i	− 1.05424i	0.849789	−	0.527122i	$-0.176729\pi$
$150$	− 1324.20i	− 0.720803i
$151$	2035.28i	1.09688i	0.836191	+	0.548439i	$0.184778\pi$
$151$	2035.28i	1.09688i	−0.836191	+	0.548439i	$0.815222\pi$
$152$	−1230.88	−0.656826
$153$	−1512.28	−0.799088
$154$	543.984i	0.284646i
$155$	−2365.92	−1.22603
$156$	0	0
$157$	−615.393	−0.312826	−0.156413	−	0.987692i	$-0.549993\pi$
$157$	−615.393	−0.312826	−0.156413	+	0.987692i	$0.549993\pi$
$158$	2086.36i	1.05052i
$159$	3501.89	1.74665
$160$	453.393	0.224024
$161$	1031.50i	0.504930i
$162$	− 755.696i	− 0.366501i
$163$	1160.09i	0.557456i	0.960370	+	0.278728i	$0.0899128\pi$
$163$	1160.09i	0.557456i	−0.960370	+	0.278728i	$0.910087\pi$
$164$	− 501.897i	− 0.238973i
$165$	1175.73	0.554731
$166$	626.608	0.292977
$167$	− 2029.26i	− 0.940295i	−0.882588	−	0.470147i	$-0.844201\pi$
$167$	− 2029.26i	− 0.940295i	0.882588	−	0.470147i	$-0.155799\pi$
$168$	2003.45	0.920057
$169$	0	0
$170$	−867.409	−0.391337
$171$	7601.30i	3.39933i
$172$	−1739.10	−0.770959
$173$	3073.60	1.35076	0.675380	−	0.737469i	$-0.263979\pi$
$173$	3073.60	1.35076	0.675380	+	0.737469i	$0.263979\pi$
$174$	− 861.099i	− 0.375171i
$175$	− 2170.18i	− 0.937431i
$176$	151.896i	0.0650544i
$177$	− 3568.12i	− 1.51523i
$178$	−1351.51	−0.569100
$179$	−2313.89	−0.966190	−0.483095	−	0.875568i	$-0.660487\pi$
$179$	−2313.89	−0.966190	−0.483095	+	0.875568i	$0.660487\pi$
$180$	− 2799.93i	− 1.15941i
$181$	1670.01	0.685805	0.342903	−	0.939371i	$-0.388590\pi$
$181$	1670.01	0.685805	0.342903	+	0.939371i	$0.388590\pi$
$182$	0	0
$183$	5277.44	2.13180
$184$	288.024i	0.115399i
$185$	−338.355	−0.134467
$186$	−2919.20	−1.15078
$187$	− 290.600i	− 0.113640i
$188$	− 744.068i	− 0.288653i
$189$	− 5610.67i	− 2.15934i
$190$	4359.94i	1.66475i
$191$	3825.21	1.44912	0.724561	−	0.689211i	$-0.242043\pi$
$191$	3825.21	1.44912	0.724561	+	0.689211i	$0.242043\pi$
$192$	559.420	0.210274
$193$	908.832i	0.338959i	0.985534	+	0.169480i	$0.0542087\pi$
$193$	908.832i	0.338959i	−0.985534	+	0.169480i	$0.945791\pi$
$194$	−544.869	−0.201646
$195$	0	0
$196$	1911.38	0.696568
$197$	− 1337.67i	− 0.483781i	−0.970304	−	0.241890i	$-0.922233\pi$
$197$	− 1337.67i	− 0.483781i	0.970304	−	0.241890i	$-0.0777674\pi$
$198$	938.033	0.336682
$199$	−1144.58	−0.407725	−0.203862	−	0.979000i	$-0.565349\pi$
$199$	−1144.58	−0.407725	−0.203862	+	0.979000i	$0.565349\pi$
$200$	− 605.976i	− 0.214245i
$201$	2513.44i	0.882011i
$202$	− 1323.12i	− 0.460864i
$203$	− 1411.22i	− 0.487924i
$204$	−1070.26	−0.367318
$205$	−1777.78	−0.605687
$206$	− 630.706i	− 0.213317i
$207$	1778.69	0.597236
$208$	0	0
$209$	−1460.67	−0.483428
$210$	− 7096.49i	− 2.33192i
$211$	−11.2252	−0.00366245	−0.00183123	−	0.999998i	$-0.500583\pi$
$211$	−11.2252	−0.00366245	−0.00183123	+	0.999998i	$0.500583\pi$
$212$	1602.52	0.519159
$213$	8407.06i	2.70442i
$214$	− 4242.64i	− 1.35524i
$215$	6160.11i	1.95403i
$216$	− 1566.66i	− 0.493507i
$217$	−4784.16	−1.49664
$218$	1768.86	0.549553
$219$	8419.25i	2.59781i
$220$	538.035	0.164883
$221$	0	0
$222$	−417.481	−0.126214
$223$	− 2739.74i	− 0.822720i	−0.911473	−	0.411360i	$-0.865054\pi$
$223$	− 2739.74i	− 0.822720i	0.911473	−	0.411360i	$-0.134946\pi$
$224$	916.813	0.273469
$225$	−3742.21	−1.10880
$226$	264.409i	0.0778241i
$227$	4529.75i	1.32445i	0.749305	+	0.662225i	$0.230388\pi$
$227$	4529.75i	1.32445i	−0.749305	+	0.662225i	$0.769612\pi$
$228$	5379.52i	1.56258i
$229$	867.892i	0.250445i	0.992129	+	0.125222i	$0.0399645\pi$
$229$	867.892i	0.250445i	−0.992129	+	0.125222i	$0.960036\pi$
$230$	1020.22	0.292484
$231$	2377.47	0.677168
$232$	− 394.053i	− 0.111512i
$233$	−535.138	−0.150464	−0.0752319	−	0.997166i	$-0.523970\pi$
$233$	−535.138	−0.150464	−0.0752319	+	0.997166i	$0.523970\pi$
$234$	0	0
$235$	−2635.59	−0.731603
$236$	− 1632.83i	− 0.450374i
$237$	9118.36	2.49916
$238$	−1754.00	−0.477710
$239$	− 1146.73i	− 0.310359i	−0.987886	−	0.155180i	$-0.950404\pi$
$239$	− 1146.73i	− 0.310359i	0.987886	−	0.155180i	$-0.0495956\pi$
$240$	− 1981.54i	− 0.532949i
$241$	3353.03i	0.896215i	0.893980	+	0.448107i	$0.147902\pi$
$241$	3353.03i	0.896215i	−0.893980	+	0.448107i	$0.852098\pi$
$242$	− 2481.75i	− 0.659226i
$243$	1984.72	0.523950
$244$	2415.05	0.633638
$245$	− 6770.36i	− 1.76548i
$246$	−2193.52	−0.568512
$247$	0	0
$248$	−1335.87	−0.342048
$249$	− 2738.57i	− 0.696987i
$250$	1395.68	0.353083
$251$	−5357.96	−1.34738	−0.673688	−	0.739016i	$-0.735291\pi$
$251$	−5357.96	−1.34738	−0.673688	+	0.739016i	$0.735291\pi$
$252$	− 5661.78i	− 1.41531i
$253$	341.794i	0.0849345i
$254$	− 3745.62i	− 0.925279i
$255$	3790.99i	0.930983i
$256$	256.000	0.0625000
$257$	6292.11	1.52720	0.763601	−	0.645689i	$-0.223430\pi$
$257$	6292.11	1.52720	0.763601	+	0.645689i	$0.223430\pi$
$258$	7600.67i	1.83410i
$259$	−684.194	−0.164146
$260$	0	0
$261$	−2433.48	−0.577121
$262$	3700.43i	0.872569i
$263$	−4797.76	−1.12488	−0.562438	−	0.826840i	$-0.690136\pi$
$263$	−4797.76	−1.12488	−0.562438	+	0.826840i	$0.690136\pi$
$264$	663.856	0.154763
$265$	− 5676.35i	− 1.31583i
$266$	8816.30i	2.03219i
$267$	5906.72i	1.35388i
$268$	1150.19i	0.262161i
$269$	6246.43	1.41581	0.707903	−	0.706310i	$-0.249642\pi$
$269$	6246.43	1.41581	0.707903	+	0.706310i	$0.249642\pi$
$270$	−5549.30	−1.25081
$271$	− 1831.15i	− 0.410460i	−0.978714	−	0.205230i	$-0.934206\pi$
$271$	− 1831.15i	− 0.410460i	0.978714	−	0.205230i	$-0.0657942\pi$
$272$	−489.767	−0.109178
$273$	0	0
$274$	−1934.75	−0.426578
$275$	− 719.103i	− 0.157686i
$276$	1258.80	0.274532
$277$	1601.42	0.347365	0.173683	−	0.984802i	$-0.444433\pi$
$277$	1601.42	0.347365	0.173683	+	0.984802i	$0.444433\pi$
$278$	207.194i	0.0447002i
$279$	8249.69i	1.77024i
$280$	− 3247.47i	− 0.693120i
$281$	− 4516.64i	− 0.958861i	−0.877580	−	0.479431i	$-0.840843\pi$
$281$	− 4516.64i	− 0.958861i	0.877580	−	0.479431i	$-0.159157\pi$
$282$	−3251.93	−0.686700
$283$	−7083.02	−1.48778	−0.743890	−	0.668302i	$-0.767022\pi$
$283$	−7083.02	−1.48778	−0.743890	+	0.668302i	$0.767022\pi$
$284$	3847.21i	0.803838i
$285$	19055.0	3.96042
$286$	0	0
$287$	−3594.89	−0.739371
$288$	− 1580.93i	− 0.323462i
$289$	−3976.00	−0.809282
$290$	−1395.79	−0.282633
$291$	2381.34i	0.479713i
$292$	3852.79i	0.772149i
$293$	− 6923.52i	− 1.38047i	−0.723588	−	0.690233i	$-0.757508\pi$
$293$	− 6923.52i	− 1.38047i	0.723588	−	0.690233i	$-0.242492\pi$
$294$	− 8353.64i	− 1.65712i
$295$	−5783.70	−1.14149
$296$	−191.046	−0.0375146
$297$	− 1859.13i	− 0.363224i
$298$	−3834.87	−0.745463
$299$	0	0
$300$	−2648.40	−0.509685
$301$	12456.5i	2.38531i
$302$	4070.56	0.775610
$303$	−5782.66	−1.09639
$304$	2461.76i	0.464446i
$305$	− 8554.41i	− 1.60598i
$306$	3024.56i	0.565040i
$307$	8778.50i	1.63197i	0.578071	+	0.815986i	$0.303805\pi$
$307$	8778.50i	1.63197i	−0.578071	+	0.815986i	$0.696195\pi$
$308$	1087.97	0.201275
$309$	−2756.48	−0.507478
$310$	4731.84i	0.866936i
$311$	2197.35	0.400644	0.200322	−	0.979730i	$-0.435801\pi$
$311$	2197.35	0.400644	0.200322	+	0.979730i	$0.435801\pi$
$312$	0	0
$313$	−5052.55	−0.912419	−0.456209	−	0.889873i	$-0.650793\pi$
$313$	−5052.55	−0.912419	−0.456209	+	0.889873i	$0.650793\pi$
$314$	1230.79i	0.221201i
$315$	−20054.8	−3.58717
$316$	4172.72	0.742828
$317$	− 1889.91i	− 0.334852i	−0.985885	−	0.167426i	$-0.946455\pi$
$317$	− 1889.91i	− 0.334852i	0.985885	−	0.167426i	$-0.0535455\pi$
$318$	− 7003.78i	− 1.23507i
$319$	− 467.618i	− 0.0820738i
$320$	− 906.785i	− 0.158409i
$321$	−18542.3	−3.22409
$322$	2063.00	0.357039
$323$	− 4709.72i	− 0.811318i
$324$	−1511.39	−0.259155
$325$	0	0
$326$	2320.18	0.394181
$327$	− 7730.76i	− 1.30738i
$328$	−1003.79	−0.168979
$329$	−5329.46	−0.893078
$330$	− 2351.46i	− 0.392254i
$331$	2549.39i	0.423344i	0.977341	+	0.211672i	$0.0678909\pi$
$331$	2549.39i	0.423344i	−0.977341	+	0.211672i	$0.932109\pi$
$332$	− 1253.22i	− 0.207166i
$333$	1179.81i	0.194153i
$334$	−4058.53	−0.664889
$335$	4074.13	0.664458
$336$	− 4006.90i	− 0.650579i
$337$	−140.649	−0.0227349	−0.0113674	−	0.999935i	$-0.503618\pi$
$337$	−140.649	−0.0227349	−0.0113674	+	0.999935i	$0.503618\pi$
$338$	0	0
$339$	1155.59	0.185142
$340$	1734.82i	0.276717i
$341$	−1585.26	−0.251750
$342$	15202.6	2.40369
$343$	− 3863.38i	− 0.608171i
$344$	3478.19i	0.545150i
$345$	− 4458.84i	− 0.695814i
$346$	− 6147.20i	− 0.955132i
$347$	140.513	0.0217382	0.0108691	−	0.999941i	$-0.496540\pi$
$347$	140.513	0.0217382	0.0108691	+	0.999941i	$0.496540\pi$
$348$	−1722.20	−0.265286
$349$	12273.2i	1.88244i	0.337794	+	0.941220i	$0.390319\pi$
$349$	12273.2i	1.88244i	−0.337794	+	0.941220i	$0.609681\pi$
$350$	−4340.37	−0.662864
$351$	0	0
$352$	303.792	0.0460004
$353$	− 5887.97i	− 0.887776i	−0.896082	−	0.443888i	$-0.853599\pi$
$353$	− 5887.97i	− 0.887776i	0.896082	−	0.443888i	$-0.146401\pi$
$354$	−7136.24	−1.07143
$355$	13627.3	2.03736
$356$	2703.02i	0.402414i
$357$	7665.81i	1.13646i
$358$	4627.77i	0.683199i
$359$	− 5902.88i	− 0.867806i	−0.900960	−	0.433903i	$-0.857136\pi$
$359$	− 5902.88i	− 0.867806i	0.900960	−	0.433903i	$-0.142864\pi$
$360$	−5599.85	−0.819828
$361$	−16813.9	−2.45136
$362$	− 3340.02i	− 0.484937i
$363$	−10846.4	−1.56829
$364$	0	0
$365$	13647.1	1.95704
$366$	− 10554.9i	− 1.50741i
$367$	−3318.93	−0.472062	−0.236031	−	0.971746i	$-0.575847\pi$
$367$	−3318.93	−0.472062	−0.236031	+	0.971746i	$0.575847\pi$
$368$	576.049	0.0815995
$369$	6198.93i	0.874535i
$370$	676.711i	0.0950825i
$371$	− 11478.2i	− 1.60625i
$372$	5838.39i	0.813727i
$373$	242.215	0.0336232	0.0168116	−	0.999859i	$-0.494648\pi$
$373$	242.215	0.0336232	0.0168116	+	0.999859i	$0.494648\pi$
$374$	−581.199	−0.0803559
$375$	− 6099.79i	− 0.839979i
$376$	−1488.14	−0.204108
$377$	0	0
$378$	−11221.3	−1.52689
$379$	12904.3i	1.74894i	0.485081	+	0.874469i	$0.338790\pi$
$379$	12904.3i	1.74894i	−0.485081	+	0.874469i	$0.661210\pi$
$380$	8719.87	1.17716
$381$	−16370.1	−2.20122
$382$	− 7650.41i	− 1.02468i
$383$	4591.02i	0.612507i	0.951950	+	0.306253i	$0.0990755\pi$
$383$	4591.02i	0.612507i	−0.951950	+	0.306253i	$0.900925\pi$
$384$	− 1118.84i	− 0.148686i
$385$	− 3853.73i	− 0.510141i
$386$	1817.66	0.239680
$387$	21479.6	2.82137
$388$	1089.74i	0.142585i
$389$	13973.1	1.82125	0.910623	−	0.413239i	$-0.135603\pi$
$389$	13973.1	1.82125	0.910623	+	0.413239i	$0.135603\pi$
$390$	0	0
$391$	−1102.07	−0.142542
$392$	− 3822.76i	− 0.492548i
$393$	16172.6	2.07583
$394$	−2675.33	−0.342085
$395$	− 14780.3i	− 1.88273i
$396$	− 1876.07i	− 0.238070i
$397$	2617.73i	0.330932i	0.986215	+	0.165466i	$0.0529129\pi$
$397$	2617.73i	0.330932i	−0.986215	+	0.165466i	$0.947087\pi$
$398$	2289.16i	0.288305i
$399$	38531.4	4.83454
$400$	−1211.95	−0.151494
$401$	7889.19i	0.982463i	0.871029	+	0.491231i	$0.163453\pi$
$401$	7889.19i	0.982463i	−0.871029	+	0.491231i	$0.836547\pi$
$402$	5026.87	0.623676
$403$	0	0
$404$	−2646.24	−0.325880
$405$	5353.55i	0.656839i
$406$	−2822.45	−0.345014
$407$	−226.712	−0.0276110
$408$	2140.51i	0.259733i
$409$	− 7731.54i	− 0.934719i	−0.884067	−	0.467359i	$-0.845205\pi$
$409$	− 7731.54i	− 0.934719i	0.884067	−	0.467359i	$-0.154795\pi$
$410$	3555.57i	0.428285i
$411$	8455.76i	1.01482i
$412$	−1261.41	−0.150838
$413$	−11695.3	−1.39344
$414$	− 3557.39i	− 0.422309i
$415$	−4439.05	−0.525071
$416$	0	0
$417$	905.535	0.106341
$418$	2921.33i	0.341835i
$419$	2756.41	0.321383	0.160691	−	0.987005i	$-0.448628\pi$
$419$	2756.41	0.321383	0.160691	+	0.987005i	$0.448628\pi$
$420$	−14193.0	−1.64892
$421$	11815.2i	1.36778i	0.729585	+	0.683890i	$0.239713\pi$
$421$	11815.2i	1.36778i	−0.729585	+	0.683890i	$0.760287\pi$
$422$	22.4505i	0.00258974i
$423$	9189.98i	1.05634i
$424$	− 3205.05i	− 0.367101i
$425$	2318.65	0.264638
$426$	16814.1	1.91232
$427$	− 17298.0i	− 1.96044i
$428$	−8485.29	−0.958299
$429$	0	0
$430$	12320.2	1.38171
$431$	− 10601.2i	− 1.18479i	−0.805649	−	0.592393i	$-0.798183\pi$
$431$	− 10601.2i	− 1.18479i	0.805649	−	0.592393i	$-0.201817\pi$
$432$	−3133.31	−0.348962
$433$	9229.72	1.02437	0.512185	−	0.858875i	$-0.328836\pi$
$433$	9229.72	1.02437	0.512185	+	0.858875i	$0.328836\pi$
$434$	9568.32i	1.05828i
$435$	6100.25i	0.672379i
$436$	− 3537.73i	− 0.388593i
$437$	5539.42i	0.606377i
$438$	16838.5	1.83693
$439$	−7201.66	−0.782953	−0.391476	−	0.920188i	$-0.628036\pi$
$439$	−7201.66	−0.782953	−0.391476	+	0.920188i	$0.628036\pi$
$440$	− 1076.07i	− 0.116590i
$441$	−23607.5	−2.54913
$442$	0	0
$443$	−3544.70	−0.380167	−0.190083	−	0.981768i	$-0.560876\pi$
$443$	−3544.70	−0.380167	−0.190083	+	0.981768i	$0.560876\pi$
$444$	834.962i	0.0892467i
$445$	9574.43	1.01994
$446$	−5479.48	−0.581751
$447$	16760.2i	1.77344i
$448$	− 1833.63i	− 0.193372i
$449$	18325.4i	1.92612i	0.269289	+	0.963059i	$0.413211\pi$
$449$	18325.4i	1.92612i	−0.269289	+	0.963059i	$0.586789\pi$
$450$	7484.41i	0.784041i
$451$	−1191.19	−0.124370
$452$	528.818	0.0550299
$453$	− 17790.2i	− 1.84516i
$454$	9059.50	0.936527
$455$	0	0
$456$	10759.0	1.10491
$457$	− 5383.84i	− 0.551084i	−0.961289	−	0.275542i	$-0.911143\pi$
$457$	− 5383.84i	− 0.551084i	0.961289	−	0.275542i	$-0.0888573\pi$
$458$	1735.78	0.177091
$459$	5994.51	0.609585
$460$	− 2040.44i	− 0.206817i
$461$	− 10621.1i	− 1.07305i	−0.843885	−	0.536525i	$-0.819737\pi$
$461$	− 10621.1i	− 1.07305i	0.843885	−	0.536525i	$-0.180263\pi$
$462$	− 4754.93i	− 0.478830i
$463$	− 10407.3i	− 1.04464i	−0.852751	−	0.522318i	$-0.825067\pi$
$463$	− 10407.3i	− 1.04464i	0.852751	−	0.522318i	$-0.174933\pi$
$464$	−788.107	−0.0788512
$465$	20680.3	2.06243
$466$	1070.28i	0.106394i
$467$	15784.0	1.56402	0.782010	−	0.623266i	$-0.214195\pi$
$467$	15784.0	1.56402	0.782010	+	0.623266i	$0.214195\pi$
$468$	0	0
$469$	8238.36	0.811113
$470$	5271.17i	0.517321i
$471$	5379.11	0.526234
$472$	−3265.66	−0.318462
$473$	4127.52i	0.401234i
$474$	− 18236.7i	− 1.76717i
$475$	− 11654.4i	− 1.12577i
$476$	3508.00i	0.337792i
$477$	−19792.8	−1.89989
$478$	−2293.46	−0.219457
$479$	18263.7i	1.74215i	0.491149	+	0.871075i	$0.336577\pi$
$479$	18263.7i	1.74215i	−0.491149	+	0.871075i	$0.663423\pi$
$480$	−3963.08	−0.376852
$481$	0	0
$482$	6706.06	0.633720
$483$	− 9016.29i	− 0.849390i
$484$	−4963.49	−0.466143
$485$	3860.00	0.361389
$486$	− 3969.44i	− 0.370489i
$487$	− 5369.01i	− 0.499575i	−0.968301	−	0.249788i	$-0.919639\pi$
$487$	− 5369.01i	− 0.499575i	0.968301	−	0.249788i	$-0.0803608\pi$
$488$	− 4830.09i	− 0.448049i
$489$	− 10140.3i	− 0.937750i
$490$	−13540.7	−1.24838
$491$	1491.33	0.137073	0.0685366	−	0.997649i	$-0.478167\pi$
$491$	1491.33	0.137073	0.0685366	+	0.997649i	$0.478167\pi$
$492$	4387.05i	0.401999i
$493$	1507.77	0.137741
$494$	0	0
$495$	−6645.27	−0.603399
$496$	2671.75i	0.241865i
$497$	27556.0	2.48704
$498$	−5477.14	−0.492844
$499$	− 10067.4i	− 0.903163i	−0.892230	−	0.451582i	$-0.850860\pi$
$499$	− 10067.4i	− 0.903163i	0.892230	−	0.451582i	$-0.149140\pi$
$500$	− 2791.37i	− 0.249668i
$501$	17737.7i	1.58176i
$502$	10715.9i	0.952738i
$503$	4683.07	0.415124	0.207562	−	0.978222i	$-0.433447\pi$
$503$	4683.07	0.415124	0.207562	+	0.978222i	$0.433447\pi$
$504$	−11323.6	−1.00078
$505$	9373.34i	0.825957i
$506$	683.589	0.0600577
$507$	0	0
$508$	−7491.23	−0.654271
$509$	11545.0i	1.00535i	0.864476	+	0.502673i	$0.167650\pi$
$509$	11545.0i	1.00535i	−0.864476	+	0.502673i	$0.832350\pi$
$510$	7581.97	0.658305
$511$	27596.0	2.38899
$512$	− 512.000i	− 0.0441942i
$513$	− 30130.7i	− 2.59318i
$514$	− 12584.2i	− 1.07989i
$515$	4468.09i	0.382306i
$516$	15201.3	1.29690
$517$	−1765.95	−0.150225
$518$	1368.39i	0.116069i
$519$	−26866.2	−2.27224
$520$	0	0
$521$	19889.6	1.67252	0.836258	−	0.548336i	$-0.184739\pi$
$521$	19889.6	1.67252	0.836258	+	0.548336i	$0.184739\pi$
$522$	4866.96i	0.408086i
$523$	−5090.07	−0.425570	−0.212785	−	0.977099i	$-0.568253\pi$
$523$	−5090.07	−0.425570	−0.212785	+	0.977099i	$0.568253\pi$
$524$	7400.86	0.617000
$525$	18969.4i	1.57694i
$526$	9595.51i	0.795407i
$527$	− 5111.46i	− 0.422502i
$528$	− 1327.71i	− 0.109434i
$529$	−10870.8	−0.893464
$530$	−11352.7	−0.930433
$531$	20167.1i	1.64817i
$532$	17632.6	1.43697
$533$	0	0
$534$	11813.4	0.957337
$535$	30056.0i	2.42885i
$536$	2300.38	0.185376
$537$	20225.5	1.62532
$538$	− 12492.9i	− 1.00113i
$539$	− 4536.42i	− 0.362518i
$540$	11098.6i	0.884459i
$541$	5958.49i	0.473523i	0.971568	+	0.236761i	$0.0760859\pi$
$541$	5958.49i	0.473523i	−0.971568	+	0.236761i	$0.923914\pi$
$542$	−3662.30	−0.290239
$543$	−14597.4	−1.15366
$544$	979.534i	0.0772007i
$545$	−12531.1	−0.984905
$546$	0	0
$547$	2475.52	0.193502	0.0967509	−	0.995309i	$-0.469155\pi$
$547$	2475.52	0.193502	0.0967509	+	0.995309i	$0.469155\pi$
$548$	3869.50i	0.301637i
$549$	−29828.3	−2.31883
$550$	−1438.21	−0.111501
$551$	− 7578.63i	− 0.585954i
$552$	− 2517.60i	− 0.194124i
$553$	− 29887.5i	− 2.29827i
$554$	− 3202.85i	− 0.245624i
$555$	2957.54	0.226199
$556$	414.388	0.0316078
$557$	− 23311.5i	− 1.77332i	−0.462417	−	0.886662i	$-0.653018\pi$
$557$	− 23311.5i	− 1.77332i	0.462417	−	0.886662i	$-0.346982\pi$
$558$	16499.4	1.25175
$559$	0	0
$560$	−6494.94	−0.490110
$561$	2540.11i	0.191165i
$562$	−9033.27	−0.678017
$563$	−24626.2	−1.84347	−0.921733	−	0.387825i	$-0.873227\pi$
$563$	−24626.2	−1.84347	−0.921733	+	0.387825i	$0.873227\pi$
$564$	6503.85i	0.485570i
$565$	− 1873.14i	− 0.139476i
$566$	14166.0i	1.05202i
$567$	10825.5i	0.801813i
$568$	7694.42	0.568399
$569$	13198.6	0.972431	0.486215	−	0.873839i	$-0.338377\pi$
$569$	13198.6	0.972431	0.486215	+	0.873839i	$0.338377\pi$
$570$	− 38109.9i	− 2.80044i
$571$	−20934.2	−1.53427	−0.767137	−	0.641483i	$-0.778319\pi$
$571$	−20934.2	−1.53427	−0.767137	+	0.641483i	$0.778319\pi$
$572$	0	0
$573$	−33435.9	−2.43770
$574$	7189.77i	0.522814i
$575$	−2727.12	−0.197789
$576$	−3161.86	−0.228722
$577$	− 17083.2i	− 1.23255i	−0.787531	−	0.616275i	$-0.788641\pi$
$577$	− 17083.2i	− 1.23255i	0.787531	−	0.616275i	$-0.211359\pi$
$578$	7952.00i	0.572249i
$579$	− 7944.04i	− 0.570195i
$580$	2791.58i	0.199852i
$581$	−8976.28	−0.640962
$582$	4762.67	0.339208
$583$	− 3803.38i	− 0.270189i
$584$	7705.58	0.545992
$585$	0	0
$586$	−13847.0	−0.976136
$587$	− 4458.82i	− 0.313518i	−0.987637	−	0.156759i	$-0.949895\pi$
$587$	− 4458.82i	− 0.313518i	0.987637	−	0.156759i	$-0.0501046\pi$
$588$	−16707.3	−1.17176
$589$	−25692.2	−1.79733
$590$	11567.4i	0.807156i
$591$	11692.5i	0.813813i
$592$	382.092i	0.0265269i
$593$	− 3217.47i	− 0.222809i	−0.993775	−	0.111404i	$-0.964465\pi$
$593$	− 3217.47i	− 0.222809i	0.993775	−	0.111404i	$-0.0355349\pi$
$594$	−3718.26	−0.256838
$595$	12425.8	0.856149
$596$	7669.74i	0.527122i
$597$	10004.7	0.685872
$598$	0	0
$599$	−4495.37	−0.306637	−0.153319	−	0.988177i	$-0.548996\pi$
$599$	−4495.37	−0.306637	−0.153319	+	0.988177i	$0.548996\pi$
$600$	5296.80i	0.360402i
$601$	−10918.5	−0.741054	−0.370527	−	0.928822i	$-0.620823\pi$
$601$	−10918.5	−0.741054	−0.370527	+	0.928822i	$0.620823\pi$
$602$	24912.9	1.68667
$603$	− 14206.0i	− 0.959392i
$604$	− 8141.11i	− 0.548439i
$605$	17581.3i	1.18146i
$606$	11565.3i	0.775263i
$607$	12487.0	0.834980	0.417490	−	0.908682i	$-0.362910\pi$
$607$	12487.0	0.834980	0.417490	+	0.908682i	$0.362910\pi$
$608$	4923.52	0.328413
$609$	12335.4i	0.820782i
$610$	−17108.8	−1.13560
$611$	0	0
$612$	6049.11	0.399544
$613$	− 961.468i	− 0.0633496i	−0.999498	−	0.0316748i	$-0.989916\pi$
$613$	− 961.468i	− 0.0633496i	0.999498	−	0.0316748i	$-0.0100841\pi$
$614$	17557.0	1.15398
$615$	15539.5	1.01888
$616$	− 2175.94i	− 0.142323i
$617$	15433.5i	1.00701i	0.863991	+	0.503507i	$0.167957\pi$
$617$	15433.5i	1.00701i	−0.863991	+	0.503507i	$0.832043\pi$
$618$	5512.96i	0.358841i
$619$	24569.6i	1.59537i	0.603073	+	0.797686i	$0.293943\pi$
$619$	24569.6i	1.59537i	−0.603073	+	0.797686i	$0.706057\pi$
$620$	9463.67	0.613017
$621$	−7050.55	−0.455602
$622$	− 4394.70i	− 0.283298i
$623$	19360.6	1.24505
$624$	0	0
$625$	−19355.8	−1.23877
$626$	10105.1i	0.645177i
$627$	12767.6	0.813220
$628$	2461.57	0.156413
$629$	− 731.001i	− 0.0463385i
$630$	40109.5i	2.53651i
$631$	13322.2i	0.840489i	0.907411	+	0.420245i	$0.138056\pi$
$631$	13322.2i	0.840489i	−0.907411	+	0.420245i	$0.861944\pi$
$632$	− 8345.43i	− 0.525259i
$633$	98.1191	0.00616096
$634$	−3779.82	−0.236776
$635$	26534.9i	1.65828i
$636$	−14007.6	−0.873327
$637$	0	0
$638$	−935.235	−0.0580350
$639$	− 47516.9i	− 2.94169i
$640$	−1813.57	−0.112012
$641$	−14894.6	−0.917784	−0.458892	−	0.888492i	$-0.651754\pi$
$641$	−14894.6	−0.917784	−0.458892	+	0.888492i	$0.651754\pi$
$642$	37084.7i	2.27978i
$643$	− 9247.56i	− 0.567167i	−0.958948	−	0.283583i	$-0.908477\pi$
$643$	− 9247.56i	− 0.567167i	0.958948	−	0.283583i	$-0.0915233\pi$
$644$	− 4126.01i	− 0.252465i
$645$	− 53845.1i	− 3.28705i
$646$	−9419.44	−0.573689
$647$	−2734.14	−0.166136	−0.0830681	−	0.996544i	$-0.526472\pi$
$647$	−2734.14	−0.166136	−0.0830681	+	0.996544i	$0.526472\pi$
$648$	3022.78i	0.183250i
$649$	−3875.31	−0.234390
$650$	0	0
$651$	41818.1	2.51763
$652$	− 4640.37i	− 0.278728i
$653$	−6562.78	−0.393295	−0.196647	−	0.980474i	$-0.563005\pi$
$653$	−6562.78	−0.393295	−0.196647	+	0.980474i	$0.563005\pi$
$654$	−15461.5	−0.924455
$655$	− 26214.8i	− 1.56381i
$656$	2007.59i	0.119486i
$657$	− 47585.8i	− 2.82572i
$658$	10658.9i	0.631501i
$659$	−4953.84	−0.292829	−0.146414	−	0.989223i	$-0.546773\pi$
$659$	−4953.84	−0.292829	−0.146414	+	0.989223i	$0.546773\pi$
$660$	−4702.93	−0.277365
$661$	− 9347.27i	− 0.550025i	−0.961441	−	0.275013i	$-0.911318\pi$
$661$	− 9347.27i	− 0.550025i	0.961441	−	0.275013i	$-0.0886821\pi$
$662$	5098.78	0.299350
$663$	0	0
$664$	−2506.43	−0.146489
$665$	− 62456.9i	− 3.64207i
$666$	2359.61	0.137287
$667$	−1773.39	−0.102947
$668$	8117.06i	0.470147i
$669$	23947.9i	1.38398i
$670$	− 8148.25i	− 0.469843i
$671$	− 5731.80i	− 0.329767i
$672$	−8013.80	−0.460029
$673$	−15419.0	−0.883150	−0.441575	−	0.897224i	$-0.645580\pi$
$673$	−15419.0	−0.883150	−0.441575	+	0.897224i	$0.645580\pi$
$674$	281.298i	0.0160760i
$675$	14833.7	0.845851
$676$	0	0
$677$	26079.0	1.48050	0.740248	−	0.672334i	$-0.234708\pi$
$677$	26079.0	1.48050	0.740248	+	0.672334i	$0.234708\pi$
$678$	− 2311.18i	− 0.130915i
$679$	7805.36	0.441152
$680$	3469.64	0.195668
$681$	− 39594.3i	− 2.22798i
$682$	3170.52i	0.178014i
$683$	14890.0i	0.834190i	0.908863	+	0.417095i	$0.136952\pi$
$683$	14890.0i	0.834190i	−0.908863	+	0.417095i	$0.863048\pi$
$684$	− 30405.2i	− 1.69967i
$685$	13706.3	0.764510
$686$	−7726.75	−0.430042
$687$	− 7586.19i	− 0.421297i
$688$	6956.39	0.385479
$689$	0	0
$690$	−8917.68	−0.492015
$691$	− 4071.03i	− 0.224123i	−0.993701	−	0.112062i	$-0.964255\pi$
$691$	− 4071.03i	− 0.224123i	0.993701	−	0.112062i	$-0.0357454\pi$
$692$	−12294.4	−0.675380
$693$	−13437.5	−0.736578
$694$	− 281.026i	− 0.0153712i
$695$	− 1467.82i	− 0.0801114i
$696$	3444.40i	0.187586i
$697$	− 3840.82i	− 0.208725i
$698$	24546.5	1.33109
$699$	4677.61	0.253109
$700$	8680.73i	0.468715i
$701$	−18378.1	−0.990200	−0.495100	−	0.868836i	$-0.664869\pi$
$701$	−18378.1	−0.990200	−0.495100	+	0.868836i	$0.664869\pi$
$702$	0	0
$703$	−3674.29	−0.197125
$704$	− 607.583i	− 0.0325272i
$705$	23037.5	1.23070
$706$	−11775.9	−0.627753
$707$	18954.0i	1.00826i
$708$	14272.5i	0.757616i
$709$	− 14946.4i	− 0.791710i	−0.918313	−	0.395855i	$-0.870448\pi$
$709$	− 14946.4i	− 0.791710i	0.918313	−	0.395855i	$-0.129552\pi$
$710$	− 27254.7i	− 1.44063i
$711$	−51537.2	−2.71842
$712$	5406.03	0.284550
$713$	6011.94i	0.315777i
$714$	15331.6	0.803602
$715$	0	0
$716$	9255.55	0.483095
$717$	10023.5i	0.522084i
$718$	−11805.8	−0.613631
$719$	−15005.8	−0.778335	−0.389168	−	0.921167i	$-0.627237\pi$
$719$	−15005.8	−0.778335	−0.389168	+	0.921167i	$0.627237\pi$
$720$	11199.7i	0.579706i
$721$	9034.99i	0.466686i
$722$	33627.8i	1.73337i
$723$	− 29308.6i	− 1.50761i
$724$	−6680.03	−0.342903
$725$	3731.05	0.191128
$726$	21692.8i	1.10895i
$727$	30390.1	1.55035	0.775177	−	0.631744i	$-0.217661\pi$
$727$	30390.1	1.55035	0.775177	+	0.631744i	$0.217661\pi$
$728$	0	0
$729$	−27550.2	−1.39970
$730$	− 27294.2i	− 1.38384i
$731$	−13308.6	−0.673375
$732$	−21109.8	−1.06590
$733$	31143.3i	1.56931i	0.619932	+	0.784655i	$0.287160\pi$
$733$	31143.3i	1.56931i	−0.619932	+	0.784655i	$0.712840\pi$
$734$	6637.86i	0.333798i
$735$	59179.3i	2.96988i
$736$	− 1152.10i	− 0.0576995i
$737$	2729.83	0.136438
$738$	12397.9	0.618389
$739$	− 3884.81i	− 0.193376i	−0.995315	−	0.0966881i	$-0.969175\pi$
$739$	− 3884.81i	− 0.193376i	0.995315	−	0.0966881i	$-0.0308249\pi$
$740$	1353.42	0.0672334
$741$	0	0
$742$	−22956.5	−1.13579
$743$	11614.8i	0.573495i	0.958006	+	0.286748i	$0.0925741\pi$
$743$	11614.8i	0.573495i	−0.958006	+	0.286748i	$0.907426\pi$
$744$	11676.8	0.575392
$745$	27167.2	1.33601
$746$	− 484.431i	− 0.0237752i
$747$	15478.5i	0.758136i
$748$	1162.40i	0.0568202i
$749$	60776.7i	2.96493i
$750$	−12199.6	−0.593955
$751$	34410.2	1.67196	0.835982	−	0.548757i	$-0.184899\pi$
$751$	34410.2	1.67196	0.835982	+	0.548757i	$0.184899\pi$
$752$	2976.27i	0.144326i
$753$	46833.6	2.26655
$754$	0	0
$755$	−28836.9	−1.39004
$756$	22442.7i	1.07967i
$757$	−29456.8	−1.41430	−0.707150	−	0.707063i	$-0.750020\pi$
$757$	−29456.8	−1.41430	−0.707150	+	0.707063i	$0.750020\pi$
$758$	25808.5	1.23669
$759$	− 2987.60i	− 0.142876i
$760$	− 17439.7i	− 0.832376i
$761$	− 32950.1i	− 1.56957i	−0.619769	−	0.784784i	$-0.712774\pi$
$761$	− 32950.1i	− 1.56957i	0.619769	−	0.784784i	$-0.287226\pi$
$762$	32740.2i	1.55650i
$763$	−25339.3	−1.20229
$764$	−15300.8	−0.724561
$765$	− 21426.7i	− 1.01266i
$766$	9182.04	0.433108
$767$	0	0
$768$	−2237.68	−0.105137
$769$	10223.7i	0.479424i	0.970844	+	0.239712i	$0.0770530\pi$
$769$	10223.7i	0.479424i	−0.970844	+	0.239712i	$0.922947\pi$
$770$	−7707.45	−0.360724
$771$	−54998.9	−2.56905
$772$	− 3635.33i	− 0.169480i
$773$	10140.1i	0.471815i	0.971776	+	0.235908i	$0.0758062\pi$
$773$	10140.1i	0.471815i	−0.971776	+	0.235908i	$0.924194\pi$
$774$	− 42959.2i	− 1.99501i
$775$	− 12648.6i	− 0.586257i
$776$	2179.48	0.100823
$777$	5980.50	0.276125
$778$	− 27946.2i	− 1.28781i
$779$	−19305.4	−0.887920
$780$	0	0
$781$	9130.86	0.418346
$782$	2204.14i	0.100793i
$783$	9646.04	0.440257
$784$	−7645.53	−0.348284
$785$	− 8719.21i	− 0.396435i
$786$	− 32345.2i	− 1.46783i
$787$	9088.53i	0.411653i	0.978588	+	0.205827i	$0.0659883\pi$
$787$	9088.53i	0.411653i	−0.978588	+	0.205827i	$0.934012\pi$
$788$	5350.67i	0.241890i
$789$	41936.9	1.89226
$790$	−29560.6	−1.33129
$791$	− 3787.71i	− 0.170260i
$792$	−3752.13	−0.168341
$793$	0	0
$794$	5235.46	0.234004
$795$	49616.6i	2.21348i
$796$	4578.32	0.203862
$797$	21032.4	0.934764	0.467382	−	0.884055i	$-0.345197\pi$
$797$	21032.4	0.934764	0.467382	+	0.884055i	$0.345197\pi$
$798$	− 77062.7i	− 3.41853i
$799$	− 5694.06i	− 0.252117i
$800$	2423.90i	0.107122i
$801$	− 33385.0i	− 1.47266i
$802$	15778.4	0.694706
$803$	9144.10	0.401854
$804$	− 10053.7i	− 0.441005i
$805$	−14614.9	−0.639883
$806$	0	0
$807$	−54599.7	−2.38166
$808$	5292.49i	0.230432i
$809$	−43644.9	−1.89675	−0.948376	−	0.317147i	$-0.897275\pi$
$809$	−43644.9	−1.89675	−0.948376	+	0.317147i	$0.897275\pi$
$810$	10707.1	0.464456
$811$	28688.1i	1.24214i	0.783756	+	0.621069i	$0.213301\pi$
$811$	28688.1i	1.24214i	−0.783756	+	0.621069i	$0.786699\pi$
$812$	5644.89i	0.243962i
$813$	16006.0i	0.690473i
$814$	453.424i	0.0195239i
$815$	−16436.8	−0.706448
$816$	4281.02	0.183659
$817$	66894.3i	2.86455i
$818$	−15463.1	−0.660946
$819$	0	0
$820$	7111.14	0.302843
$821$	16042.2i	0.681947i	0.940073	+	0.340973i	$0.110757\pi$
$821$	16042.2i	0.681947i	−0.940073	+	0.340973i	$0.889243\pi$
$822$	16911.5	0.717588
$823$	12138.9	0.514137	0.257068	−	0.966393i	$-0.417243\pi$
$823$	12138.9	0.514137	0.257068	+	0.966393i	$0.417243\pi$
$824$	2522.82i	0.106659i
$825$	6285.64i	0.265258i
$826$	23390.6i	0.985307i
$827$	17136.2i	0.720538i	0.932848	+	0.360269i	$0.117315\pi$
$827$	17136.2i	0.720538i	−0.932848	+	0.360269i	$0.882685\pi$
$828$	−7114.78	−0.298618
$829$	−10947.3	−0.458642	−0.229321	−	0.973351i	$-0.573651\pi$
$829$	−10947.3	−0.458642	−0.229321	+	0.973351i	$0.573651\pi$
$830$	8878.11i	0.371281i
$831$	−13997.9	−0.584336
$832$	0	0
$833$	14627.1	0.608400
$834$	− 1811.07i	− 0.0751945i
$835$	28751.7	1.19161
$836$	5842.67	0.241714
$837$	− 32700.9i	− 1.35043i
$838$	− 5512.82i	− 0.227252i
$839$	− 13628.7i	− 0.560805i	−0.959883	−	0.280402i	$-0.909532\pi$
$839$	− 13628.7i	− 0.560805i	0.959883	−	0.280402i	$-0.0904679\pi$
$840$	28385.9i	1.16596i
$841$	−21962.8	−0.900520
$842$	23630.3	0.967167
$843$	39479.6i	1.61299i
$844$	44.9010	0.00183123
$845$	0	0
$846$	18380.0	0.746946
$847$	35551.5i	1.44223i
$848$	−6410.09	−0.259580
$849$	61912.3	2.50274
$850$	− 4637.30i	− 0.187127i
$851$	859.780i	0.0346332i
$852$	− 33628.2i	− 1.35221i
$853$	6645.59i	0.266753i	0.991065	+	0.133377i	$0.0425820\pi$
$853$	6645.59i	0.266753i	−0.991065	+	0.133377i	$0.957418\pi$
$854$	−34596.0	−1.38624
$855$	−107699.	−4.30788
$856$	16970.6i	0.677619i
$857$	−15552.6	−0.619914	−0.309957	−	0.950751i	$-0.600315\pi$
$857$	−15552.6	−0.619914	−0.309957	+	0.950751i	$0.600315\pi$
$858$	0	0
$859$	−48006.5	−1.90682	−0.953412	−	0.301672i	$-0.902455\pi$
$859$	−48006.5	−1.90682	−0.953412	+	0.301672i	$0.902455\pi$
$860$	− 24640.4i	− 0.977014i
$861$	31422.7	1.24377
$862$	−21202.4	−0.837770
$863$	3572.64i	0.140920i	0.997515	+	0.0704601i	$0.0224468\pi$
$863$	3572.64i	0.140920i	−0.997515	+	0.0704601i	$0.977553\pi$
$864$	6266.63i	0.246753i
$865$	43548.4i	1.71178i
$866$	− 18459.4i	− 0.724339i
$867$	34754.0	1.36137
$868$	19136.6	0.748318
$869$	− 9903.40i	− 0.386594i
$870$	12200.5	0.475443
$871$	0	0
$872$	−7075.45	−0.274777
$873$	− 13459.4i	− 0.521799i
$874$	11078.8	0.428773
$875$	−19993.4	−0.772459
$876$	− 33677.0i	− 1.29891i
$877$	2699.84i	0.103953i	0.998648	+	0.0519767i	$0.0165521\pi$
$877$	2699.84i	0.103953i	−0.998648	+	0.0519767i	$0.983448\pi$
$878$	14403.3i	0.553631i
$879$	60518.1i	2.32221i
$880$	−2152.14	−0.0824416
$881$	−3982.46	−0.152295	−0.0761477	−	0.997097i	$-0.524262\pi$
$881$	−3982.46	−0.152295	−0.0761477	+	0.997097i	$0.524262\pi$
$882$	47215.0i	1.80251i
$883$	−21896.7	−0.834523	−0.417261	−	0.908787i	$-0.637010\pi$
$883$	−21896.7	−0.834523	−0.417261	+	0.908787i	$0.637010\pi$
$884$	0	0
$885$	50555.0	1.92021
$886$	7089.41i	0.268819i
$887$	−21219.7	−0.803254	−0.401627	−	0.915803i	$-0.631555\pi$
$887$	−21219.7	−0.803254	−0.401627	+	0.915803i	$0.631555\pi$
$888$	1669.92	0.0631069
$889$	53656.7i	2.02428i
$890$	− 19148.9i	− 0.721204i
$891$	3587.09i	0.134873i
$892$	10959.0i	0.411360i
$893$	−28620.6	−1.07251
$894$	33520.4	1.25401
$895$	− 32784.4i	− 1.22442i
$896$	−3667.25	−0.136735
$897$	0	0
$898$	36650.7	1.36197
$899$	− 8225.08i	− 0.305141i
$900$	14968.8	0.554401
$901$	12263.5	0.453447
$902$	2382.37i	0.0879428i
$903$	− 108881.i	− 4.01256i
$904$	− 1057.64i	− 0.0389120i
$905$	23661.5i	0.869101i
$906$	−35580.5	−1.30473
$907$	42873.6	1.56957	0.784783	−	0.619771i	$-0.212775\pi$
$907$	42873.6	1.56957	0.784783	+	0.619771i	$0.212775\pi$
$908$	− 18119.0i	− 0.662225i
$909$	32683.8	1.19258
$910$	0	0
$911$	28943.9	1.05264	0.526319	−	0.850287i	$-0.323572\pi$
$911$	28943.9	1.05264	0.526319	+	0.850287i	$0.323572\pi$
$912$	− 21518.1i	− 0.781288i
$913$	−2974.35	−0.107817
$914$	−10767.7	−0.389675
$915$	74773.6i	2.70157i
$916$	− 3471.57i	− 0.125222i
$917$	− 53009.4i	− 1.90897i
$918$	− 11989.0i	− 0.431042i
$919$	2735.02	0.0981720	0.0490860	−	0.998795i	$-0.484369\pi$
$919$	2735.02	0.0981720	0.0490860	+	0.998795i	$0.484369\pi$
$920$	−4080.88	−0.146242
$921$	− 76732.3i	− 2.74529i
$922$	−21242.3	−0.758760
$923$	0	0
$924$	−9509.87	−0.338584
$925$	− 1808.90i	− 0.0642986i
$926$	−20814.5	−0.738669
$927$	15579.7	0.552001
$928$	1576.21i	0.0557562i
$929$	15091.8i	0.532988i	0.963837	+	0.266494i	$0.0858652\pi$
$929$	15091.8i	0.532988i	−0.963837	+	0.266494i	$0.914135\pi$
$930$	− 41360.7i	− 1.45836i
$931$	− 73521.3i	− 2.58814i
$932$	2140.55	0.0752319
$933$	−19206.9	−0.673961
$934$	− 31568.0i	− 1.10593i
$935$	4117.37	0.144013
$936$	0	0
$937$	−20974.5	−0.731278	−0.365639	−	0.930757i	$-0.619150\pi$
$937$	−20974.5	−0.731278	−0.365639	+	0.930757i	$0.619150\pi$
$938$	− 16476.7i	− 0.573544i
$939$	44164.0	1.53487
$940$	10542.3	0.365801
$941$	− 886.927i	− 0.0307258i	−0.999882	−	0.0153629i	$-0.995110\pi$
$941$	− 886.927i	− 0.0307258i	0.999882	−	0.0153629i	$-0.00489036\pi$
$942$	− 10758.2i	− 0.372104i
$943$	4517.45i	0.156000i
$944$	6531.32i	0.225187i
$945$	79494.9	2.73647
$946$	8255.05	0.283715
$947$	− 43341.3i	− 1.48723i	−0.668610	−	0.743613i	$-0.733110\pi$
$947$	− 43341.3i	− 1.48723i	0.668610	−	0.743613i	$-0.266890\pi$
$948$	−36473.5	−1.24958
$949$	0	0
$950$	−23308.9	−0.796041
$951$	16519.6i	0.563286i
$952$	7016.01	0.238855
$953$	−16014.9	−0.544357	−0.272178	−	0.962247i	$-0.587744\pi$
$953$	−16014.9	−0.544357	−0.272178	+	0.962247i	$0.587744\pi$
$954$	39585.5i	1.34343i
$955$	54197.5i	1.83643i
$956$	4586.92i	0.155180i
$957$	4087.42i	0.138064i
$958$	36527.4	1.23189
$959$	27715.7	0.933249
$960$	7926.16i	0.266475i
$961$	1907.30	0.0640227
$962$	0	0
$963$	104802.	3.50695
$964$	− 13412.1i	− 0.448107i
$965$	−12876.8	−0.429553
$966$	−18032.6	−0.600610
$967$	7986.48i	0.265592i	0.991143	+	0.132796i	$0.0423956\pi$
$967$	7986.48i	0.265592i	−0.991143	+	0.132796i	$0.957604\pi$
$968$	9926.99i	0.329613i
$969$	41167.4i	1.36479i
$970$	− 7720.00i	− 0.255540i
$971$	25245.4	0.834361	0.417180	−	0.908824i	$-0.363018\pi$
$971$	25245.4	0.834361	0.417180	+	0.908824i	$0.363018\pi$
$972$	−7938.88	−0.261975
$973$	− 2968.09i	− 0.0977932i
$974$	−10738.0	−0.353253
$975$	0	0
$976$	−9660.19	−0.316819
$977$	− 5231.76i	− 0.171319i	−0.996324	−	0.0856595i	$-0.972700\pi$
$977$	− 5231.76i	− 0.171319i	0.996324	−	0.0856595i	$-0.0272997\pi$
$978$	−20280.6	−0.663089
$979$	6415.26	0.209431
$980$	27081.5i	0.882740i
$981$	43694.5i	1.42208i
$982$	− 2982.67i	− 0.0969254i
$983$	− 22720.8i	− 0.737212i	−0.929586	−	0.368606i	$-0.879835\pi$
$983$	− 22720.8i	− 0.737212i	0.929586	−	0.368606i	$-0.120165\pi$
$984$	8774.10	0.284256
$985$	18952.8	0.613081
$986$	− 3015.54i	− 0.0973978i
$987$	46584.5	1.50233
$988$	0	0
$989$	15653.2	0.503279
$990$	13290.5i	0.426668i
$991$	19005.1	0.609200	0.304600	−	0.952480i	$-0.401477\pi$
$991$	19005.1	0.609200	0.304600	+	0.952480i	$0.401477\pi$
$992$	5343.49	0.171024
$993$	− 22284.0i	− 0.712148i
$994$	− 55112.1i	− 1.75860i
$995$	− 16217.0i	− 0.516698i
$996$	10954.3i	0.348494i
$997$	49243.2	1.56424	0.782121	−	0.623127i	$-0.214138\pi$
$997$	49243.2	1.56424	0.782121	+	0.623127i	$0.214138\pi$
$998$	−20134.8	−0.638633
$999$	− 4676.62i	− 0.148110i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.4.b.f.337.1		6
13.2	odd	12		338.4.c.l.191.3		6
13.3	even	3		338.4.e.h.147.6		12
13.4	even	6		338.4.e.h.23.6		12
13.5	odd	4		338.4.a.j.1.1	✓	3
13.6	odd	12		338.4.c.l.315.3		6
13.7	odd	12		338.4.c.k.315.3		6
13.8	odd	4		338.4.a.k.1.1	yes	3
13.9	even	3		338.4.e.h.23.3		12
13.10	even	6		338.4.e.h.147.3		12
13.11	odd	12		338.4.c.k.191.3		6
13.12	even	2	inner	338.4.b.f.337.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.4.a.j.1.1	✓	3	13.5	odd	4
338.4.a.k.1.1	yes	3	13.8	odd	4
338.4.b.f.337.1		6	1.1	even	1	trivial
338.4.b.f.337.4		6	13.12	even	2	inner
338.4.c.k.191.3		6	13.11	odd	12
338.4.c.k.315.3		6	13.7	odd	12
338.4.c.l.191.3		6	13.2	odd	12
338.4.c.l.315.3		6	13.6	odd	12
338.4.e.h.23.3		12	13.9	even	3
338.4.e.h.23.6		12	13.4	even	6
338.4.e.h.147.3		12	13.10	even	6
338.4.e.h.147.6		12	13.3	even	3

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	338.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{2} -8.74094 q^{3} -4.00000 q^{4} +14.1685i q^{5} +17.4819i q^{6} +28.6504i q^{7} +8.00000i q^{8} +49.4040 q^{9} +28.3370 q^{10} +9.49349i q^{11} +34.9638 q^{12} +57.3008 q^{14} -123.846i q^{15} +16.0000 q^{16} -30.6104 q^{17} -98.8080i q^{18} +153.860i q^{19} -56.6741i q^{20} -250.431i q^{21} +18.9870 q^{22} +36.0030 q^{23} -69.9275i q^{24} -75.7470 q^{25} -195.832 q^{27} -114.602i q^{28} -49.2567 q^{29} -247.692 q^{30} +166.984i q^{31} -32.0000i q^{32} -82.9820i q^{33} +61.2209i q^{34} -405.934 q^{35} -197.616 q^{36} +23.8808i q^{37} +307.720 q^{38} -113.348 q^{40} +125.474i q^{41} -500.863 q^{42} +434.774 q^{43} -37.9739i q^{44} +699.982i q^{45} -72.0061i q^{46} +186.017i q^{47} -139.855 q^{48} -477.845 q^{49} +151.494i q^{50} +267.564 q^{51} -400.631 q^{53} +391.664i q^{54} -134.509 q^{55} -229.203 q^{56} -1344.88i q^{57} +98.5134i q^{58} +408.208i q^{59} +495.385i q^{60} -603.762 q^{61} +333.968 q^{62} +1415.44i q^{63} -64.0000 q^{64} -165.964 q^{66} -287.548i q^{67} +122.442 q^{68} -314.700 q^{69} +811.868i q^{70} -961.803i q^{71} +395.232i q^{72} -963.198i q^{73} +47.7616 q^{74} +662.100 q^{75} -615.440i q^{76} -271.992 q^{77} -1043.18 q^{79} +226.696i q^{80} +377.848 q^{81} +250.948 q^{82} +313.304i q^{83} +1001.73i q^{84} -433.705i q^{85} -869.549i q^{86} +430.550 q^{87} -75.9479 q^{88} -675.754i q^{89} +1399.96 q^{90} -144.012 q^{92} -1459.60i q^{93} +372.034 q^{94} -2179.97 q^{95} +279.710i q^{96} -272.435i q^{97} +955.691i q^{98} +469.016i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{3} - 24 q^{4} + 18 q^{9} + 48 q^{10} + 96 q^{12} + 108 q^{14} + 96 q^{16} + 180 q^{17} - 328 q^{22} - 38 q^{23} + 122 q^{25} - 138 q^{27} - 202 q^{29} - 360 q^{30} - 916 q^{35} - 72 q^{36} + 520 q^{38}+ \cdots - 3658 q^{95}+O(q^{100}) \) 6 * q - 24 * q^3 - 24 * q^4 + 18 * q^9 + 48 * q^10 + 96 * q^12 + 108 * q^14 + 96 * q^16 + 180 * q^17 - 328 * q^22 - 38 * q^23 + 122 * q^25 - 138 * q^27 - 202 * q^29 - 360 * q^30 - 916 * q^35 - 72 * q^36 + 520 * q^38 - 192 * q^40 - 936 * q^42 + 2410 * q^43 - 384 * q^48 + 368 * q^49 + 414 * q^51 - 2190 * q^53 - 1052 * q^55 - 432 * q^56 - 2216 * q^61 + 2076 * q^62 - 384 * q^64 + 584 * q^66 - 720 * q^68 + 628 * q^69 - 336 * q^74 + 1010 * q^75 - 960 * q^77 - 3922 * q^79 + 1830 * q^81 + 748 * q^82 + 3272 * q^87 + 1312 * q^88 + 2580 * q^90 + 152 * q^92 + 2144 * q^94 - 3658 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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