LMFDB - Embedded newform 350.6.c.k.99.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [350,6,Mod(99,350)] 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("350.99"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-64,0,40,0,0,-182,0,830] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1129})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 565x^{2} + 79524 $ x^4 + 565*x^2 + 79524
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.2
Root			$17.3003i$ of defining polynomial
Character	$\chi$	$=$	350.99
Dual form			350.6.c.k.99.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000i q^{2} +19.3003i q^{3} -16.0000 q^{4} +77.2012 q^{6} -49.0000i q^{7} +64.0000i q^{8} -129.501 q^{9} -10.9039 q^{11} -308.805i q^{12} +29.6967i q^{13} -196.000 q^{14} +256.000 q^{16} +432.519i q^{17} +518.006i q^{18} +956.234 q^{19} +945.715 q^{21} +43.6155i q^{22} +979.429i q^{23} -1235.22 q^{24} +118.787 q^{26} +2190.56i q^{27} +784.000i q^{28} -996.928 q^{29} +4790.53 q^{31} -1024.00i q^{32} -210.448i q^{33} +1730.08 q^{34} +2072.02 q^{36} +1889.95i q^{37} -3824.94i q^{38} -573.156 q^{39} -1928.56 q^{41} -3782.86i q^{42} -18079.7i q^{43} +174.462 q^{44} +3917.72 q^{46} +28563.5i q^{47} +4940.88i q^{48} -2401.00 q^{49} -8347.75 q^{51} -475.148i q^{52} -287.102i q^{53} +8762.22 q^{54} +3136.00 q^{56} +18455.6i q^{57} +3987.71i q^{58} -11271.3 q^{59} -32884.4 q^{61} -19162.1i q^{62} +6345.57i q^{63} -4096.00 q^{64} -841.792 q^{66} +37022.2i q^{67} -6920.31i q^{68} -18903.3 q^{69} -63930.2 q^{71} -8288.10i q^{72} +49142.9i q^{73} +7559.81 q^{74} -15299.7 q^{76} +534.290i q^{77} +2292.62i q^{78} -71237.6 q^{79} -73747.2 q^{81} +7714.26i q^{82} +94396.8i q^{83} -15131.4 q^{84} -72319.0 q^{86} -19241.0i q^{87} -697.848i q^{88} -78631.8 q^{89} +1455.14 q^{91} -15670.9i q^{92} +92458.7i q^{93} +114254. q^{94} +19763.5 q^{96} -93414.6i q^{97} +9604.00i q^{98} +1412.07 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 64 q^{4} + 40 q^{6} - 182 q^{9} + 830 q^{11} - 784 q^{14} + 1024 q^{16} - 3836 q^{19} + 490 q^{21} - 640 q^{24} + 3432 q^{26} + 2262 q^{29} - 10944 q^{31} - 10552 q^{34} + 2912 q^{36} + 10274 q^{39}+ \cdots + 35620 q^{99}+O(q^{100}) $ 4 * q - 64 * q^4 + 40 * q^6 - 182 * q^9 + 830 * q^11 - 784 * q^14 + 1024 * q^16 - 3836 * q^19 + 490 * q^21 - 640 * q^24 + 3432 * q^26 + 2262 * q^29 - 10944 * q^31 - 10552 * q^34 + 2912 * q^36 + 10274 * q^39 - 15644 * q^41 - 13280 * q^44 - 10672 * q^46 - 9604 * q^49 - 66790 * q^51 - 14680 * q^54 + 12544 * q^56 - 13904 * q^59 - 90276 * q^61 - 16384 * q^64 - 50408 * q^66 - 103972 * q^69 - 124544 * q^71 + 73248 * q^74 + 61376 * q^76 - 124138 * q^79 - 243916 * q^81 - 7840 * q^84 + 16624 * q^86 - 128380 * q^89 + 42042 * q^91 + 68056 * q^94 + 10240 * q^96 + 35620 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	19.3003i	1.23811i	0.785346	+	0.619057i	$0.212485\pi$
$3$	19.3003i	1.23811i	−0.785346	+	0.619057i	$0.787515\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	77.2012	0.875479
$7$	− 49.0000i	− 0.377964i
$7$	− 49.0000i	− 0.377964i
$8$	64.0000i	0.353553i
$9$	−129.501	−0.532928
$10$	0	0
$11$	−10.9039	−0.0271706	−0.0135853	−	0.999908i	$-0.504324\pi$
$11$	−10.9039	−0.0271706	−0.0135853	+	0.999908i	$0.504324\pi$
$12$	− 308.805i	− 0.619057i
$13$	29.6967i	0.0487360i	0.999703	+	0.0243680i	$0.00775735\pi$
$13$	29.6967i	0.0487360i	−0.999703	+	0.0243680i	$0.992243\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	432.519i	0.362980i	0.983393	+	0.181490i	$0.0580921\pi$
$17$	432.519i	0.362980i	−0.983393	+	0.181490i	$0.941908\pi$
$18$	518.006i	0.376837i
$19$	956.234	0.607687	0.303844	−	0.952722i	$-0.401730\pi$
$19$	956.234	0.607687	0.303844	+	0.952722i	$0.401730\pi$
$20$	0	0
$21$	945.715	0.467963
$22$	43.6155i	0.0192125i
$23$	979.429i	0.386059i	0.981193	+	0.193029i	$0.0618313\pi$
$23$	979.429i	0.386059i	−0.981193	+	0.193029i	$0.938169\pi$
$24$	−1235.22	−0.437740
$25$	0	0
$26$	118.787	0.0344616
$27$	2190.56i	0.578289i
$28$	784.000i	0.188982i
$29$	−996.928	−0.220125	−0.110062	−	0.993925i	$-0.535105\pi$
$29$	−996.928	−0.220125	−0.110062	+	0.993925i	$0.535105\pi$
$30$	0	0
$31$	4790.53	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$31$	4790.53	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$32$	− 1024.00i	− 0.176777i
$33$	− 210.448i	− 0.0336403i
$34$	1730.08	0.256666
$35$	0	0
$36$	2072.02	0.266464
$37$	1889.95i	0.226959i	0.993540	+	0.113479i	$0.0361996\pi$
$37$	1889.95i	0.226959i	−0.993540	+	0.113479i	$0.963800\pi$
$38$	− 3824.94i	− 0.429700i
$39$	−573.156	−0.0603408
$40$	0	0
$41$	−1928.56	−0.179174	−0.0895869	−	0.995979i	$-0.528555\pi$
$41$	−1928.56	−0.179174	−0.0895869	+	0.995979i	$0.528555\pi$
$42$	− 3782.86i	− 0.330900i
$43$	− 18079.7i	− 1.49115i	−0.666422	−	0.745574i	$-0.732175\pi$
$43$	− 18079.7i	− 1.49115i	0.666422	−	0.745574i	$-0.267825\pi$
$44$	174.462	0.0135853
$45$	0	0
$46$	3917.72	0.272985
$47$	28563.5i	1.88611i	0.332635	+	0.943055i	$0.392062\pi$
$47$	28563.5i	1.88611i	−0.332635	+	0.943055i	$0.607938\pi$
$48$	4940.88i	0.309529i
$49$	−2401.00	−0.142857
$50$	0	0
$51$	−8347.75	−0.449411
$52$	− 475.148i	− 0.0243680i
$53$	− 287.102i	− 0.0140393i	−0.999975	−	0.00701966i	$-0.997766\pi$
$53$	− 287.102i	− 0.0140393i	0.999975	−	0.00701966i	$-0.00223445\pi$
$54$	8762.22	0.408912
$55$	0	0
$56$	3136.00	0.133631
$57$	18455.6i	0.752387i
$58$	3987.71i	0.155652i
$59$	−11271.3	−0.421546	−0.210773	−	0.977535i	$-0.567598\pi$
$59$	−11271.3	−0.421546	−0.210773	+	0.977535i	$0.567598\pi$
$60$	0	0
$61$	−32884.4	−1.13153	−0.565764	−	0.824567i	$-0.691419\pi$
$61$	−32884.4	−1.13153	−0.565764	+	0.824567i	$0.691419\pi$
$62$	− 19162.1i	− 0.633089i
$63$	6345.57i	0.201428i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−841.792	−0.0237873
$67$	37022.2i	1.00757i	0.863829	+	0.503785i	$0.168059\pi$
$67$	37022.2i	1.00757i	−0.863829	+	0.503785i	$0.831941\pi$
$68$	− 6920.31i	− 0.181490i
$69$	−18903.3	−0.477985
$70$	0	0
$71$	−63930.2	−1.50508	−0.752541	−	0.658546i	$-0.771172\pi$
$71$	−63930.2	−1.50508	−0.752541	+	0.658546i	$0.771172\pi$
$72$	− 8288.10i	− 0.188418i
$73$	49142.9i	1.07933i	0.841880	+	0.539664i	$0.181449\pi$
$73$	49142.9i	1.07933i	−0.841880	+	0.539664i	$0.818551\pi$
$74$	7559.81	0.160484
$75$	0	0
$76$	−15299.7	−0.303844
$77$	534.290i	0.0102695i
$78$	2292.62i	0.0426674i
$79$	−71237.6	−1.28423	−0.642113	−	0.766610i	$-0.721942\pi$
$79$	−71237.6	−1.28423	−0.642113	+	0.766610i	$0.721942\pi$
$80$	0	0
$81$	−73747.2	−1.24892
$82$	7714.26i	0.126695i
$83$	94396.8i	1.50405i	0.659135	+	0.752025i	$0.270923\pi$
$83$	94396.8i	1.50405i	−0.659135	+	0.752025i	$0.729077\pi$
$84$	−15131.4	−0.233982
$85$	0	0
$86$	−72319.0	−1.05440
$87$	− 19241.0i	− 0.272540i
$88$	− 697.848i	− 0.00960625i
$89$	−78631.8	−1.05226	−0.526130	−	0.850404i	$-0.676358\pi$
$89$	−78631.8	−1.05226	−0.526130	+	0.850404i	$0.676358\pi$
$90$	0	0
$91$	1455.14	0.0184205
$92$	− 15670.9i	− 0.193029i
$93$	92458.7i	1.10851i
$94$	114254.	1.33368
$95$	0	0
$96$	19763.5	0.218870
$97$	− 93414.6i	− 1.00806i	−0.863687	−	0.504029i	$-0.831851\pi$
$97$	− 93414.6i	− 1.00806i	0.863687	−	0.504029i	$-0.168149\pi$
$98$	9604.00i	0.101015i
$99$	1412.07	0.0144800
$100$	0	0
$101$	−14190.4	−0.138417	−0.0692086	−	0.997602i	$-0.522047\pi$
$101$	−14190.4	−0.138417	−0.0692086	+	0.997602i	$0.522047\pi$
$102$	33391.0i	0.317782i
$103$	197607.i	1.83531i	0.397374	+	0.917657i	$0.369921\pi$
$103$	197607.i	1.83531i	−0.397374	+	0.917657i	$0.630079\pi$
$104$	−1900.59	−0.0172308
$105$	0	0
$106$	−1148.41	−0.00992730
$107$	− 163104.i	− 1.37723i	−0.725128	−	0.688615i	$-0.758219\pi$
$107$	− 163104.i	− 1.37723i	0.725128	−	0.688615i	$-0.241781\pi$
$108$	− 35048.9i	− 0.289144i
$109$	67208.1	0.541820	0.270910	−	0.962605i	$-0.412675\pi$
$109$	67208.1	0.541820	0.270910	+	0.962605i	$0.412675\pi$
$110$	0	0
$111$	−36476.6	−0.281001
$112$	− 12544.0i	− 0.0944911i
$113$	55975.9i	0.412387i	0.978511	+	0.206193i	$0.0661076\pi$
$113$	55975.9i	0.412387i	−0.978511	+	0.206193i	$0.933892\pi$
$114$	73822.4	0.532018
$115$	0	0
$116$	15950.8	0.110062
$117$	− 3845.77i	− 0.0259728i
$118$	45085.4i	0.298078i
$119$	21193.4	0.137194
$120$	0	0
$121$	−160932.	−0.999262
$122$	131538.i	0.800111i
$123$	− 37221.9i	− 0.221838i
$124$	−76648.5	−0.447661
$125$	0	0
$126$	25382.3	0.142431
$127$	86091.4i	0.473642i	0.971553	+	0.236821i	$0.0761055\pi$
$127$	86091.4i	0.473642i	−0.971553	+	0.236821i	$0.923894\pi$
$128$	16384.0i	0.0883883i
$129$	348944.	1.84621
$130$	0	0
$131$	221094.	1.12564	0.562820	−	0.826580i	$-0.309717\pi$
$131$	221094.	1.12564	0.562820	+	0.826580i	$0.309717\pi$
$132$	3367.17i	0.0168201i
$133$	− 46855.5i	− 0.229684i
$134$	148089.	0.712459
$135$	0	0
$136$	−27681.2	−0.128333
$137$	425173.i	1.93537i	0.252158	+	0.967686i	$0.418860\pi$
$137$	425173.i	1.93537i	−0.252158	+	0.967686i	$0.581140\pi$
$138$	75613.1i	0.337986i
$139$	14492.4	0.0636216	0.0318108	−	0.999494i	$-0.489873\pi$
$139$	14492.4	0.0636216	0.0318108	+	0.999494i	$0.489873\pi$
$140$	0	0
$141$	−551285.	−2.33522
$142$	255721.i	1.06425i
$143$	− 323.809i	− 0.00132419i
$144$	−33152.4	−0.133232
$145$	0	0
$146$	196572.	0.763201
$147$	− 46340.0i	− 0.176874i
$148$	− 30239.2i	− 0.113479i
$149$	36977.8	0.136451	0.0682253	−	0.997670i	$-0.478266\pi$
$149$	36977.8	0.136451	0.0682253	+	0.997670i	$0.478266\pi$
$150$	0	0
$151$	81428.5	0.290626	0.145313	−	0.989386i	$-0.453581\pi$
$151$	81428.5	0.290626	0.145313	+	0.989386i	$0.453581\pi$
$152$	61199.0i	0.214850i
$153$	− 56011.9i	− 0.193442i
$154$	2137.16	0.00726164
$155$	0	0
$156$	9170.49	0.0301704
$157$	113780.i	0.368397i	0.982889	+	0.184199i	$0.0589690\pi$
$157$	113780.i	0.368397i	−0.982889	+	0.184199i	$0.941031\pi$
$158$	284950.i	0.908085i
$159$	5541.15	0.0173823
$160$	0	0
$161$	47992.0	0.145917
$162$	294989.i	0.883117i
$163$	440567.i	1.29880i	0.760446	+	0.649401i	$0.224980\pi$
$163$	440567.i	1.29880i	−0.760446	+	0.649401i	$0.775020\pi$
$164$	30857.0	0.0895869
$165$	0	0
$166$	377587.	1.06352
$167$	− 621094.i	− 1.72332i	−0.507484	−	0.861661i	$-0.669424\pi$
$167$	− 621094.i	− 1.72332i	0.507484	−	0.861661i	$-0.330576\pi$
$168$	60525.7i	0.165450i
$169$	370411.	0.997625
$170$	0	0
$171$	−123834.	−0.323854
$172$	289276.i	0.745574i
$173$	506925.i	1.28774i	0.765134	+	0.643871i	$0.222673\pi$
$173$	506925.i	1.28774i	−0.765134	+	0.643871i	$0.777327\pi$
$174$	−76964.0	−0.192715
$175$	0	0
$176$	−2791.39	−0.00679265
$177$	− 217540.i	− 0.521923i
$178$	314527.i	0.744061i
$179$	−800840.	−1.86816	−0.934078	−	0.357070i	$-0.883776\pi$
$179$	−800840.	−1.86816	−0.934078	+	0.357070i	$0.883776\pi$
$180$	0	0
$181$	−91559.5	−0.207734	−0.103867	−	0.994591i	$-0.533122\pi$
$181$	−91559.5	−0.207734	−0.103867	+	0.994591i	$0.533122\pi$
$182$	− 5820.56i	− 0.0130253i
$183$	− 634678.i	− 1.40096i
$184$	−62683.5	−0.136492
$185$	0	0
$186$	369835.	0.783837
$187$	− 4716.13i	− 0.00986239i
$188$	− 457016.i	− 0.943055i
$189$	107337.	0.218573
$190$	0	0
$191$	409991.	0.813187	0.406594	−	0.913609i	$-0.366717\pi$
$191$	409991.	0.813187	0.406594	+	0.913609i	$0.366717\pi$
$192$	− 79054.0i	− 0.154764i
$193$	− 65112.4i	− 0.125826i	−0.998019	−	0.0629130i	$-0.979961\pi$
$193$	− 65112.4i	− 0.125826i	0.998019	−	0.0629130i	$-0.0200391\pi$
$194$	−373658.	−0.712804
$195$	0	0
$196$	38416.0	0.0714286
$197$	− 183218.i	− 0.336360i	−0.985756	−	0.168180i	$-0.946211\pi$
$197$	− 183218.i	− 0.336360i	0.985756	−	0.168180i	$-0.0537889\pi$
$198$	− 5648.27i	− 0.0102389i
$199$	563040.	1.00787	0.503937	−	0.863740i	$-0.331884\pi$
$199$	563040.	1.00787	0.503937	+	0.863740i	$0.331884\pi$
$200$	0	0
$201$	−714539.	−1.24749
$202$	56761.5i	0.0978758i
$203$	48849.5i	0.0831993i
$204$	133564.	0.224706
$205$	0	0
$206$	790430.	1.29776
$207$	− 126838.i	− 0.205742i
$208$	7602.36i	0.0121840i
$209$	−10426.6	−0.0165112
$210$	0	0
$211$	−564811.	−0.873367	−0.436684	−	0.899615i	$-0.643847\pi$
$211$	−564811.	−0.873367	−0.436684	+	0.899615i	$0.643847\pi$
$212$	4593.63i	0.00701966i
$213$	− 1.23387e6i	− 1.86346i
$214$	−652418.	−0.973848
$215$	0	0
$216$	−140196.	−0.204456
$217$	− 234736.i	− 0.338400i
$218$	− 268832.i	− 0.383125i
$219$	−948473.	−1.33633
$220$	0	0
$221$	−12844.4	−0.0176902
$222$	145907.i	0.198698i
$223$	− 1.17657e6i	− 1.58437i	−0.610280	−	0.792186i	$-0.708943\pi$
$223$	− 1.17657e6i	− 1.58437i	0.610280	−	0.792186i	$-0.291057\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	223903.	0.291601
$227$	− 984829.i	− 1.26852i	−0.773121	−	0.634258i	$-0.781306\pi$
$227$	− 984829.i	− 1.26852i	0.773121	−	0.634258i	$-0.218694\pi$
$228$	− 295290.i	− 0.376193i
$229$	−174251.	−0.219577	−0.109789	−	0.993955i	$-0.535017\pi$
$229$	−174251.	−0.219577	−0.109789	+	0.993955i	$0.535017\pi$
$230$	0	0
$231$	−10311.9	−0.0127148
$232$	− 63803.4i	− 0.0778258i
$233$	895160.i	1.08022i	0.841596	+	0.540108i	$0.181617\pi$
$233$	895160.i	1.08022i	−0.841596	+	0.540108i	$0.818383\pi$
$234$	−15383.1	−0.0183655
$235$	0	0
$236$	180341.	0.210773
$237$	− 1.37491e6i	− 1.59002i
$238$	− 84773.8i	− 0.0970106i
$239$	−250128.	−0.283249	−0.141624	−	0.989920i	$-0.545232\pi$
$239$	−250128.	−0.283249	−0.141624	+	0.989920i	$0.545232\pi$
$240$	0	0
$241$	−824888.	−0.914855	−0.457427	−	0.889247i	$-0.651229\pi$
$241$	−824888.	−0.914855	−0.457427	+	0.889247i	$0.651229\pi$
$242$	643728.i	0.706585i
$243$	− 891039.i	− 0.968012i
$244$	526150.	0.565764
$245$	0	0
$246$	−148888.	−0.156863
$247$	28397.0i	0.0296163i
$248$	306594.i	0.316544i
$249$	−1.82189e6	−1.86219
$250$	0	0
$251$	729516.	0.730887	0.365444	−	0.930834i	$-0.380917\pi$
$251$	729516.	0.730887	0.365444	+	0.930834i	$0.380917\pi$
$252$	− 101529.i	− 0.100714i
$253$	− 10679.6i	− 0.0104894i
$254$	344366.	0.334916
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 118644.i	− 0.112050i	−0.998429	−	0.0560252i	$-0.982157\pi$
$257$	− 118644.i	− 0.112050i	0.998429	−	0.0560252i	$-0.0178427\pi$
$258$	− 1.39578e6i	− 1.30547i
$259$	92607.7	0.0857823
$260$	0	0
$261$	129104.	0.117311
$262$	− 884377.i	− 0.795947i
$263$	− 766194.i	− 0.683045i	−0.939874	−	0.341523i	$-0.889057\pi$
$263$	− 766194.i	− 0.683045i	0.939874	−	0.341523i	$-0.110943\pi$
$264$	13468.7	0.0118936
$265$	0	0
$266$	−187422.	−0.162411
$267$	− 1.51762e6i	− 1.30282i
$268$	− 592355.i	− 0.503785i
$269$	774789.	0.652834	0.326417	−	0.945226i	$-0.394159\pi$
$269$	774789.	0.652834	0.326417	+	0.945226i	$0.394159\pi$
$270$	0	0
$271$	−1.63371e6	−1.35130	−0.675648	−	0.737224i	$-0.736136\pi$
$271$	−1.63371e6	−1.35130	−0.675648	+	0.737224i	$0.736136\pi$
$272$	110725.i	0.0907451i
$273$	28084.6i	0.0228067i
$274$	1.70069e6	1.36851
$275$	0	0
$276$	302452.	0.238993
$277$	2.14606e6i	1.68051i	0.542189	+	0.840257i	$0.317596\pi$
$277$	2.14606e6i	1.68051i	−0.542189	+	0.840257i	$0.682404\pi$
$278$	− 57969.8i	− 0.0449873i
$279$	−620381.	−0.477143
$280$	0	0
$281$	913760.	0.690345	0.345173	−	0.938539i	$-0.387820\pi$
$281$	913760.	0.690345	0.345173	+	0.938539i	$0.387820\pi$
$282$	2.20514e6i	1.65125i
$283$	− 1.11656e6i	− 0.828738i	−0.910109	−	0.414369i	$-0.864002\pi$
$283$	− 1.11656e6i	− 0.828738i	0.910109	−	0.414369i	$-0.135998\pi$
$284$	1.02288e6	0.752541
$285$	0	0
$286$	−1295.24	−0.000936341 0
$287$	94499.7i	0.0677213i
$288$	132610.i	0.0942092i
$289$	1.23278e6	0.868245
$290$	0	0
$291$	1.80293e6	1.24809
$292$	− 786287.i	− 0.539664i
$293$	396556.i	0.269858i	0.990855	+	0.134929i	$0.0430807\pi$
$293$	396556.i	0.269858i	−0.990855	+	0.134929i	$0.956919\pi$
$294$	−185360.	−0.125068
$295$	0	0
$296$	−120957.	−0.0802420
$297$	− 23885.5i	− 0.0157124i
$298$	− 147911.i	− 0.0964852i
$299$	−29085.8	−0.0188150
$300$	0	0
$301$	−885907.	−0.563601
$302$	− 325714.i	− 0.205504i
$303$	− 273878.i	− 0.171376i
$304$	244796.	0.151922
$305$	0	0
$306$	−224048.	−0.136784
$307$	− 1.43537e6i	− 0.869197i	−0.900624	−	0.434598i	$-0.856890\pi$
$307$	− 1.43537e6i	− 0.869197i	0.900624	−	0.434598i	$-0.143110\pi$
$308$	− 8548.63i	− 0.00513476i
$309$	−3.81388e6	−2.27233
$310$	0	0
$311$	1.61306e6	0.945691	0.472846	−	0.881145i	$-0.343227\pi$
$311$	1.61306e6	0.945691	0.472846	+	0.881145i	$0.343227\pi$
$312$	− 36682.0i	− 0.0213337i
$313$	2.21419e6i	1.27748i	0.769423	+	0.638740i	$0.220544\pi$
$313$	2.21419e6i	1.27748i	−0.769423	+	0.638740i	$0.779456\pi$
$314$	455120.	0.260496
$315$	0	0
$316$	1.13980e6	0.642113
$317$	2.33004e6i	1.30231i	0.758944	+	0.651155i	$0.225715\pi$
$317$	2.33004e6i	1.30231i	−0.758944	+	0.651155i	$0.774285\pi$
$318$	− 22164.6i	− 0.0122911i
$319$	10870.4	0.00598091
$320$	0	0
$321$	3.14796e6	1.70517
$322$	− 191968.i	− 0.103179i
$323$	413590.i	0.220579i
$324$	1.17996e6	0.624458
$325$	0	0
$326$	1.76227e6	0.918391
$327$	1.29714e6i	0.670836i
$328$	− 123428.i	− 0.0633475i
$329$	1.39961e6	0.712883
$330$	0	0
$331$	2.63604e6	1.32246	0.661230	−	0.750184i	$-0.270035\pi$
$331$	2.63604e6	1.32246	0.661230	+	0.750184i	$0.270035\pi$
$332$	− 1.51035e6i	− 0.752025i
$333$	− 244752.i	− 0.120953i
$334$	−2.48438e6	−1.21857
$335$	0	0
$336$	242103.	0.116991
$337$	− 1.13473e6i	− 0.544275i	−0.962258	−	0.272138i	$-0.912269\pi$
$337$	− 1.13473e6i	− 0.544275i	0.962258	−	0.272138i	$-0.0877306\pi$
$338$	− 1.48164e6i	− 0.705427i
$339$	−1.08035e6	−0.510582
$340$	0	0
$341$	−52235.3	−0.0243264
$342$	495335.i	0.228999i
$343$	117649.i	0.0539949i
$344$	1.15710e6	0.527201
$345$	0	0
$346$	2.02770e6	0.910571
$347$	2.62759e6i	1.17148i	0.810499	+	0.585740i	$0.199196\pi$
$347$	2.62759e6i	1.17148i	−0.810499	+	0.585740i	$0.800804\pi$
$348$	307856.i	0.136270i
$349$	3.82656e6	1.68169	0.840844	−	0.541278i	$-0.182059\pi$
$349$	3.82656e6	1.68169	0.840844	+	0.541278i	$0.182059\pi$
$350$	0	0
$351$	−65052.3	−0.0281835
$352$	11165.6i	0.00480313i
$353$	1.49143e6i	0.637040i	0.947916	+	0.318520i	$0.103186\pi$
$353$	1.49143e6i	0.637040i	−0.947916	+	0.318520i	$0.896814\pi$
$354$	−870161.	−0.369055
$355$	0	0
$356$	1.25811e6	0.526130
$357$	409040.i	0.169862i
$358$	3.20336e6i	1.32099i
$359$	406372.	0.166413	0.0832066	−	0.996532i	$-0.473484\pi$
$359$	406372.	0.166413	0.0832066	+	0.996532i	$0.473484\pi$
$360$	0	0
$361$	−1.56172e6	−0.630716
$362$	366238.i	0.146890i
$363$	− 3.10604e6i	− 1.23720i
$364$	−23282.2	−0.00921024
$365$	0	0
$366$	−2.53871e6	−0.990629
$367$	− 2.09427e6i	− 0.811647i	−0.913951	−	0.405824i	$-0.866985\pi$
$367$	− 2.09427e6i	− 0.811647i	0.913951	−	0.405824i	$-0.133015\pi$
$368$	250734.i	0.0965147i
$369$	249752.	0.0954867
$370$	0	0
$371$	−14068.0	−0.00530637
$372$	− 1.47934e6i	− 0.554256i
$373$	2.11130e6i	0.785736i	0.919595	+	0.392868i	$0.128517\pi$
$373$	2.11130e6i	0.785736i	−0.919595	+	0.392868i	$0.871483\pi$
$374$	−18864.5	−0.00697376
$375$	0	0
$376$	−1.82807e6	−0.666841
$377$	− 29605.5i	− 0.0107280i
$378$	− 429349.i	− 0.154554i
$379$	−1.21058e6	−0.432907	−0.216453	−	0.976293i	$-0.569449\pi$
$379$	−1.21058e6	−0.432907	−0.216453	+	0.976293i	$0.569449\pi$
$380$	0	0
$381$	−1.66159e6	−0.586424
$382$	− 1.63996e6i	− 0.575010i
$383$	− 994517.i	− 0.346430i	−0.984884	−	0.173215i	$-0.944584\pi$
$383$	− 994517.i	− 0.346430i	0.984884	−	0.173215i	$-0.0554155\pi$
$384$	−316216.	−0.109435
$385$	0	0
$386$	−260449.	−0.0889724
$387$	2.34135e6i	0.794675i
$388$	1.49463e6i	0.504029i
$389$	2.49121e6	0.834711	0.417355	−	0.908743i	$-0.362957\pi$
$389$	2.49121e6	0.834711	0.417355	+	0.908743i	$0.362957\pi$
$390$	0	0
$391$	−423622.	−0.140132
$392$	− 153664.i	− 0.0505076i
$393$	4.26719e6i	1.39367i
$394$	−732874.	−0.237842
$395$	0	0
$396$	−22593.1	−0.00723998
$397$	− 2.30834e6i	− 0.735062i	−0.930011	−	0.367531i	$-0.880203\pi$
$397$	− 2.30834e6i	− 0.735062i	0.930011	−	0.367531i	$-0.119797\pi$
$398$	− 2.25216e6i	− 0.712675i
$399$	904324.	0.284375
$400$	0	0
$401$	−633811.	−0.196833	−0.0984167	−	0.995145i	$-0.531378\pi$
$401$	−633811.	−0.196833	−0.0984167	+	0.995145i	$0.531378\pi$
$402$	2.85816e6i	0.882106i
$403$	142263.i	0.0436345i
$404$	227046.	0.0692086
$405$	0	0
$406$	195398.	0.0588308
$407$	− 20607.8i	− 0.00616660i
$408$	− 534256.i	− 0.158891i
$409$	−942669.	−0.278645	−0.139322	−	0.990247i	$-0.544492\pi$
$409$	−942669.	−0.278645	−0.139322	+	0.990247i	$0.544492\pi$
$410$	0	0
$411$	−8.20597e6	−2.39621
$412$	− 3.16172e6i	− 0.917657i
$413$	552296.i	0.159330i
$414$	−507350.	−0.145481
$415$	0	0
$416$	30409.4	0.00861540
$417$	279709.i	0.0787709i
$418$	41706.6i	0.0116752i
$419$	3.93604e6	1.09528	0.547639	−	0.836715i	$-0.315527\pi$
$419$	3.93604e6	1.09528	0.547639	+	0.836715i	$0.315527\pi$
$420$	0	0
$421$	−5.76142e6	−1.58425	−0.792125	−	0.610358i	$-0.791025\pi$
$421$	−5.76142e6	−1.58425	−0.792125	+	0.610358i	$0.791025\pi$
$422$	2.25924e6i	0.617564i
$423$	− 3.69902e6i	− 1.00516i
$424$	18374.5	0.00496365
$425$	0	0
$426$	−4.93549e6	−1.31767
$427$	1.61133e6i	0.427677i
$428$	2.60967e6i	0.688615i
$429$	6249.61	0.00163949
$430$	0	0
$431$	1.93127e6	0.500785	0.250392	−	0.968144i	$-0.419440\pi$
$431$	1.93127e6	0.500785	0.250392	+	0.968144i	$0.419440\pi$
$432$	560782.i	0.144572i
$433$	1.40458e6i	0.360021i	0.983665	+	0.180010i	$0.0576132\pi$
$433$	1.40458e6i	0.360021i	−0.983665	+	0.180010i	$0.942387\pi$
$434$	−938945.	−0.239285
$435$	0	0
$436$	−1.07533e6	−0.270910
$437$	936563.i	0.234603i
$438$	3.79389e6i	0.944930i
$439$	3.66469e6	0.907562	0.453781	−	0.891113i	$-0.350075\pi$
$439$	3.66469e6	0.907562	0.453781	+	0.891113i	$0.350075\pi$
$440$	0	0
$441$	310933.	0.0761326
$442$	51377.6i	0.0125089i
$443$	4.68414e6i	1.13402i	0.823711	+	0.567009i	$0.191900\pi$
$443$	4.68414e6i	1.13402i	−0.823711	+	0.567009i	$0.808100\pi$
$444$	583626.	0.140500
$445$	0	0
$446$	−4.70630e6	−1.12032
$447$	713683.i	0.168942i
$448$	200704.i	0.0472456i
$449$	873525.	0.204484	0.102242	−	0.994760i	$-0.467398\pi$
$449$	873525.	0.204484	0.102242	+	0.994760i	$0.467398\pi$
$450$	0	0
$451$	21028.8	0.00486826
$452$	− 895614.i	− 0.206193i
$453$	1.57160e6i	0.359828i
$454$	−3.93932e6	−0.896977
$455$	0	0
$456$	−1.18116e6	−0.266009
$457$	− 5.20487e6i	− 1.16579i	−0.812548	−	0.582894i	$-0.801920\pi$
$457$	− 5.20487e6i	− 1.16579i	0.812548	−	0.582894i	$-0.198080\pi$
$458$	697005.i	0.155264i
$459$	−947457.	−0.209908
$460$	0	0
$461$	−1.93474e6	−0.424004	−0.212002	−	0.977269i	$-0.567998\pi$
$461$	−1.93474e6	−0.424004	−0.212002	+	0.977269i	$0.567998\pi$
$462$	41247.8i	0.00899075i
$463$	− 2.35881e6i	− 0.511375i	−0.966759	−	0.255688i	$-0.917698\pi$
$463$	− 2.35881e6i	− 0.511375i	0.966759	−	0.255688i	$-0.0823019\pi$
$464$	−255213.	−0.0550312
$465$	0	0
$466$	3.58064e6	0.763828
$467$	3.37402e6i	0.715905i	0.933740	+	0.357953i	$0.116525\pi$
$467$	3.37402e6i	0.715905i	−0.933740	+	0.357953i	$0.883475\pi$
$468$	61532.3i	0.0129864i
$469$	1.81409e6	0.380825
$470$	0	0
$471$	−2.19599e6	−0.456118
$472$	− 721366.i	− 0.149039i
$473$	197139.i	0.0405154i
$474$	−5.49963e6	−1.12431
$475$	0	0
$476$	−339095.	−0.0685969
$477$	37180.1i	0.00748195i
$478$	1.00051e6i	0.200287i
$479$	−4.74768e6	−0.945459	−0.472729	−	0.881208i	$-0.656731\pi$
$479$	−4.74768e6	−0.945459	−0.472729	+	0.881208i	$0.656731\pi$
$480$	0	0
$481$	−56125.4	−0.0110611
$482$	3.29955e6i	0.646900i
$483$	926260.i	0.180661i
$484$	2.57491e6	0.499631
$485$	0	0
$486$	−3.56415e6	−0.684488
$487$	− 5.24868e6i	− 1.00283i	−0.865207	−	0.501416i	$-0.832813\pi$
$487$	− 5.24868e6i	− 1.00283i	0.865207	−	0.501416i	$-0.167187\pi$
$488$	− 2.10460e6i	− 0.400055i
$489$	−8.50307e6	−1.60807
$490$	0	0
$491$	−4.42605e6	−0.828537	−0.414269	−	0.910155i	$-0.635963\pi$
$491$	−4.42605e6	−0.828537	−0.414269	+	0.910155i	$0.635963\pi$
$492$	595550.i	0.110919i
$493$	− 431191.i	− 0.0799009i
$494$	113588.	0.0209419
$495$	0	0
$496$	1.22638e6	0.223831
$497$	3.13258e6i	0.568867i
$498$	7.28754e6i	1.31676i
$499$	45902.0	0.00825240	0.00412620	−	0.999991i	$-0.498687\pi$
$499$	45902.0	0.00825240	0.00412620	+	0.999991i	$0.498687\pi$
$500$	0	0
$501$	1.19873e7	2.13367
$502$	− 2.91806e6i	− 0.516815i
$503$	− 2.04449e6i	− 0.360301i	−0.983639	−	0.180150i	$-0.942342\pi$
$503$	− 2.04449e6i	− 0.360301i	0.983639	−	0.180150i	$-0.0576584\pi$
$504$	−406117.	−0.0712155
$505$	0	0
$506$	−42718.3	−0.00741716
$507$	7.14904e6i	1.23517i
$508$	− 1.37746e6i	− 0.236821i
$509$	−398969.	−0.0682566	−0.0341283	−	0.999417i	$-0.510865\pi$
$509$	−398969.	−0.0682566	−0.0341283	+	0.999417i	$0.510865\pi$
$510$	0	0
$511$	2.40800e6	0.407948
$512$	− 262144.i	− 0.0441942i
$513$	2.09468e6i	0.351419i
$514$	−474576.	−0.0792315
$515$	0	0
$516$	−5.58311e6	−0.923107
$517$	− 311453.i	− 0.0512467i
$518$	− 370431.i	− 0.0606572i
$519$	−9.78381e6	−1.59437
$520$	0	0
$521$	1.02501e7	1.65437	0.827187	−	0.561927i	$-0.189940\pi$
$521$	1.02501e7	1.65437	0.827187	+	0.561927i	$0.189940\pi$
$522$	− 516414.i	− 0.0829511i
$523$	1.51839e6i	0.242733i	0.992608	+	0.121367i	$0.0387277\pi$
$523$	1.51839e6i	0.242733i	−0.992608	+	0.121367i	$0.961272\pi$
$524$	−3.53751e6	−0.562820
$525$	0	0
$526$	−3.06478e6	−0.482986
$527$	2.07200e6i	0.324985i
$528$	− 53874.7i	− 0.00841007i
$529$	5.47706e6	0.850959
$530$	0	0
$531$	1.45966e6	0.224654
$532$	749687.i	0.114842i
$533$	− 57272.1i	− 0.00873222i
$534$	−6.07047e6	−0.921232
$535$	0	0
$536$	−2.36942e6	−0.356230
$537$	− 1.54564e7i	− 2.31299i
$538$	− 3.09916e6i	− 0.461624i
$539$	26180.2	0.00388151
$540$	0	0
$541$	3.59686e6	0.528360	0.264180	−	0.964473i	$-0.414899\pi$
$541$	3.59686e6	0.528360	0.264180	+	0.964473i	$0.414899\pi$
$542$	6.53483e6i	0.955511i
$543$	− 1.76713e6i	− 0.257198i
$544$	442900.	0.0641665
$545$	0	0
$546$	112339.	0.0161268
$547$	7.64729e6i	1.09280i	0.837526	+	0.546398i	$0.184001\pi$
$547$	7.64729e6i	1.09280i	−0.837526	+	0.546398i	$0.815999\pi$
$548$	− 6.80277e6i	− 0.967686i
$549$	4.25858e6	0.603023
$550$	0	0
$551$	−953296.	−0.133767
$552$	− 1.20981e6i	− 0.168993i
$553$	3.49064e6i	0.485392i
$554$	8.58423e6	1.18830
$555$	0	0
$556$	−231879.	−0.0318108
$557$	8.89209e6i	1.21441i	0.794544	+	0.607206i	$0.207710\pi$
$557$	8.89209e6i	1.21441i	−0.794544	+	0.607206i	$0.792290\pi$
$558$	2.48152e6i	0.337391i
$559$	536909.	0.0726727
$560$	0	0
$561$	91022.8	0.0122108
$562$	− 3.65504e6i	− 0.488148i
$563$	− 3.33731e6i	− 0.443737i	−0.975077	−	0.221869i	$-0.928784\pi$
$563$	− 3.33731e6i	− 0.443737i	0.975077	−	0.221869i	$-0.0712156\pi$
$564$	8.82055e6	1.16761
$565$	0	0
$566$	−4.46625e6	−0.586006
$567$	3.61361e6i	0.472046i
$568$	− 4.09153e6i	− 0.532127i
$569$	1.17027e7	1.51532	0.757661	−	0.652648i	$-0.226342\pi$
$569$	1.17027e7	1.51532	0.757661	+	0.652648i	$0.226342\pi$
$570$	0	0
$571$	685949.	0.0880444	0.0440222	−	0.999031i	$-0.485983\pi$
$571$	685949.	0.0880444	0.0440222	+	0.999031i	$0.485983\pi$
$572$	5180.95i	0 0.000662093i
$573$	7.91294e6i	1.00682i
$574$	377999.	0.0478862
$575$	0	0
$576$	530438.	0.0666160
$577$	1.10785e7i	1.38530i	0.721276	+	0.692648i	$0.243556\pi$
$577$	1.10785e7i	1.38530i	−0.721276	+	0.692648i	$0.756444\pi$
$578$	− 4.93114e6i	− 0.613942i
$579$	1.25669e6	0.155787
$580$	0	0
$581$	4.62544e6	0.568477
$582$	− 7.21172e6i	− 0.882533i
$583$	3130.52i	0 0.000381457i
$584$	−3.14515e6	−0.381600
$585$	0	0
$586$	1.58622e6	0.190819
$587$	8.07328e6i	0.967063i	0.875327	+	0.483531i	$0.160646\pi$
$587$	8.07328e6i	0.967063i	−0.875327	+	0.483531i	$0.839354\pi$
$588$	741440.i	0.0884368i
$589$	4.58087e6	0.544076
$590$	0	0
$591$	3.53617e6	0.416452
$592$	483828.i	0.0567396i
$593$	1.33847e7i	1.56304i	0.623879	+	0.781521i	$0.285556\pi$
$593$	1.33847e7i	1.56304i	−0.623879	+	0.781521i	$0.714444\pi$
$594$	−95542.1	−0.0111104
$595$	0	0
$596$	−591645.	−0.0682253
$597$	1.08668e7i	1.24786i
$598$	116343.i	0.0133042i
$599$	2.30111e6	0.262042	0.131021	−	0.991380i	$-0.458174\pi$
$599$	2.30111e6	0.262042	0.131021	+	0.991380i	$0.458174\pi$
$600$	0	0
$601$	5.21404e6	0.588828	0.294414	−	0.955678i	$-0.404876\pi$
$601$	5.21404e6	0.588828	0.294414	+	0.955678i	$0.404876\pi$
$602$	3.54363e6i	0.398526i
$603$	− 4.79443e6i	− 0.536962i
$604$	−1.30286e6	−0.145313
$605$	0	0
$606$	−1.09551e6	−0.121181
$607$	− 1.75294e7i	− 1.93106i	−0.260296	−	0.965529i	$-0.583820\pi$
$607$	− 1.75294e7i	− 1.93106i	0.260296	−	0.965529i	$-0.416180\pi$
$608$	− 979184.i	− 0.107425i
$609$	−942809.	−0.103010
$610$	0	0
$611$	−848243.	−0.0919216
$612$	896190.i	0.0967212i
$613$	− 1.70871e7i	− 1.83661i	−0.395876	−	0.918304i	$-0.629559\pi$
$613$	− 1.70871e7i	− 1.83661i	0.395876	−	0.918304i	$-0.370441\pi$
$614$	−5.74148e6	−0.614615
$615$	0	0
$616$	−34194.5	−0.00363082
$617$	4.74820e6i	0.502130i	0.967970	+	0.251065i	$0.0807809\pi$
$617$	4.74820e6i	0.502130i	−0.967970	+	0.251065i	$0.919219\pi$
$618$	1.52555e7i	1.60678i
$619$	−1.68570e6	−0.176829	−0.0884144	−	0.996084i	$-0.528180\pi$
$619$	−1.68570e6	−0.176829	−0.0884144	+	0.996084i	$0.528180\pi$
$620$	0	0
$621$	−2.14549e6	−0.223253
$622$	− 6.45224e6i	− 0.668705i
$623$	3.85296e6i	0.397717i
$624$	−146728.	−0.0150852
$625$	0	0
$626$	8.85676e6	0.903315
$627$	− 201237.i	− 0.0204428i
$628$	− 1.82048e6i	− 0.184199i
$629$	−817441.	−0.0823815
$630$	0	0
$631$	−8.18052e6	−0.817913	−0.408957	−	0.912554i	$-0.634107\pi$
$631$	−8.18052e6	−0.817913	−0.408957	+	0.912554i	$0.634107\pi$
$632$	− 4.55921e6i	− 0.454043i
$633$	− 1.09010e7i	− 1.08133i
$634$	9.32015e6	0.920873
$635$	0	0
$636$	−88658.4	−0.00869115
$637$	− 71301.8i	− 0.00696229i
$638$	− 43481.5i	− 0.00422914i
$639$	8.27905e6	0.802100
$640$	0	0
$641$	7.88556e6	0.758032	0.379016	−	0.925390i	$-0.376263\pi$
$641$	7.88556e6	0.758032	0.379016	+	0.925390i	$0.376263\pi$
$642$	− 1.25919e7i	− 1.20574i
$643$	7.75540e6i	0.739736i	0.929084	+	0.369868i	$0.120597\pi$
$643$	7.75540e6i	0.739736i	−0.929084	+	0.369868i	$0.879403\pi$
$644$	−767872.	−0.0729583
$645$	0	0
$646$	1.65436e6	0.155973
$647$	7.60702e6i	0.714420i	0.934024	+	0.357210i	$0.116272\pi$
$647$	7.60702e6i	0.714420i	−0.934024	+	0.357210i	$0.883728\pi$
$648$	− 4.71982e6i	− 0.441558i
$649$	122901.	0.0114537
$650$	0	0
$651$	4.53048e6	0.418978
$652$	− 7.04907e6i	− 0.649401i
$653$	− 1.23713e7i	− 1.13535i	−0.823252	−	0.567677i	$-0.807842\pi$
$653$	− 1.23713e7i	− 1.13535i	0.823252	−	0.567677i	$-0.192158\pi$
$654$	5.18854e6	0.474352
$655$	0	0
$656$	−493713.	−0.0447935
$657$	− 6.36408e6i	− 0.575204i
$658$	− 5.59845e6i	− 0.504084i
$659$	−1.94708e7	−1.74650	−0.873252	−	0.487269i	$-0.837993\pi$
$659$	−1.94708e7	−1.74650	−0.873252	+	0.487269i	$0.837993\pi$
$660$	0	0
$661$	−3.17932e6	−0.283029	−0.141515	−	0.989936i	$-0.545197\pi$
$661$	−3.17932e6	−0.283029	−0.141515	+	0.989936i	$0.545197\pi$
$662$	− 1.05442e7i	− 0.935120i
$663$	− 247901.i	− 0.0219025i
$664$	−6.04139e6	−0.531762
$665$	0	0
$666$	−979007.	−0.0855264
$667$	− 976420.i	− 0.0849811i
$668$	9.93751e6i	0.861661i
$669$	2.27082e7	1.96163
$670$	0	0
$671$	358567.	0.0307443
$672$	− 968412.i	− 0.0827250i
$673$	− 1.79934e7i	− 1.53136i	−0.643224	−	0.765678i	$-0.722403\pi$
$673$	− 1.79934e7i	− 1.53136i	0.643224	−	0.765678i	$-0.277597\pi$
$674$	−4.53893e6	−0.384861
$675$	0	0
$676$	−5.92658e6	−0.498812
$677$	− 1.47781e7i	− 1.23921i	−0.784913	−	0.619606i	$-0.787292\pi$
$677$	− 1.47781e7i	− 1.23921i	0.784913	−	0.619606i	$-0.212708\pi$
$678$	4.32140e6i	0.361036i
$679$	−4.57731e6	−0.381010
$680$	0	0
$681$	1.90075e7	1.57057
$682$	208941.i	0.0172014i
$683$	− 882971.i	− 0.0724260i	−0.999344	−	0.0362130i	$-0.988471\pi$
$683$	− 882971.i	− 0.0724260i	0.999344	−	0.0362130i	$-0.0115295\pi$
$684$	1.98134e6	0.161927
$685$	0	0
$686$	470596.	0.0381802
$687$	− 3.36310e6i	− 0.271862i
$688$	− 4.62841e6i	− 0.372787i
$689$	8525.98	0.000684221 0
$690$	0	0
$691$	−1.56459e7	−1.24654	−0.623270	−	0.782007i	$-0.714196\pi$
$691$	−1.56459e7	−1.24654	−0.623270	+	0.782007i	$0.714196\pi$
$692$	− 8.11081e6i	− 0.643871i
$693$	− 69191.3i	− 0.00547291i
$694$	1.05104e7	0.828361
$695$	0	0
$696$	1.23142e6	0.0963573
$697$	− 834142.i	− 0.0650366i
$698$	− 1.53063e7i	− 1.18913i
$699$	−1.72769e7	−1.33743
$700$	0	0
$701$	−1.39490e7	−1.07213	−0.536067	−	0.844175i	$-0.680091\pi$
$701$	−1.39490e7	−1.07213	−0.536067	+	0.844175i	$0.680091\pi$
$702$	260209.i	0.0199287i
$703$	1.80724e6i	0.137920i
$704$	44662.2	0.00339632
$705$	0	0
$706$	5.96573e6	0.450455
$707$	695328.i	0.0523168i
$708$	3.48064e6i	0.260961i
$709$	7.66584e6	0.572722	0.286361	−	0.958122i	$-0.407554\pi$
$709$	7.66584e6	0.572722	0.286361	+	0.958122i	$0.407554\pi$
$710$	0	0
$711$	9.22538e6	0.684400
$712$	− 5.03244e6i	− 0.372030i
$713$	4.69199e6i	0.345647i
$714$	1.63616e6	0.120110
$715$	0	0
$716$	1.28134e7	0.934078
$717$	− 4.82755e6i	− 0.350694i
$718$	− 1.62549e6i	− 0.117672i
$719$	−1.46181e7	−1.05456	−0.527278	−	0.849693i	$-0.676787\pi$
$719$	−1.46181e7	−1.05456	−0.527278	+	0.849693i	$0.676787\pi$
$720$	0	0
$721$	9.68276e6	0.693683
$722$	6.24686e6i	0.445984i
$723$	− 1.59206e7i	− 1.13270i
$724$	1.46495e6	0.103867
$725$	0	0
$726$	−1.24242e7	−0.874833
$727$	− 1.17581e7i	− 0.825089i	−0.910938	−	0.412544i	$-0.864640\pi$
$727$	− 1.17581e7i	− 0.825089i	0.910938	−	0.412544i	$-0.135360\pi$
$728$	93128.9i	0.00651263i
$729$	−723267.	−0.0504057
$730$	0	0
$731$	7.81984e6	0.541258
$732$	1.01549e7i	0.700480i
$733$	− 9.42125e6i	− 0.647662i	−0.946115	−	0.323831i	$-0.895029\pi$
$733$	− 9.42125e6i	− 0.647662i	0.946115	−	0.323831i	$-0.104971\pi$
$734$	−8.37707e6	−0.573921
$735$	0	0
$736$	1.00294e6	0.0682462
$737$	− 403685.i	− 0.0273762i
$738$	− 999008.i	− 0.0675193i
$739$	−8.69064e6	−0.585384	−0.292692	−	0.956207i	$-0.594551\pi$
$739$	−8.69064e6	−0.585384	−0.292692	+	0.956207i	$0.594551\pi$
$740$	0	0
$741$	−548071.	−0.0366683
$742$	56272.0i	0.00375217i
$743$	1.42528e7i	0.947172i	0.880748	+	0.473586i	$0.157041\pi$
$743$	1.42528e7i	0.947172i	−0.880748	+	0.473586i	$0.842959\pi$
$744$	−5.91736e6	−0.391918
$745$	0	0
$746$	8.44518e6	0.555600
$747$	− 1.22245e7i	− 0.801550i
$748$	75458.1i	0.00493119i
$749$	−7.99212e6	−0.520544
$750$	0	0
$751$	1.37921e7	0.892343	0.446172	−	0.894947i	$-0.352787\pi$
$751$	1.37921e7	0.892343	0.446172	+	0.894947i	$0.352787\pi$
$752$	7.31226e6i	0.471528i
$753$	1.40799e7i	0.904922i
$754$	−118422.	−0.00758584
$755$	0	0
$756$	−1.71740e6	−0.109286
$757$	− 2.62790e6i	− 0.166674i	−0.996521	−	0.0833371i	$-0.973442\pi$
$757$	− 2.62790e6i	− 0.166674i	0.996521	−	0.0833371i	$-0.0265578\pi$
$758$	4.84231e6i	0.306111i
$759$	206119.	0.0129871
$760$	0	0
$761$	−7.93120e6	−0.496452	−0.248226	−	0.968702i	$-0.579848\pi$
$761$	−7.93120e6	−0.496452	−0.248226	+	0.968702i	$0.579848\pi$
$762$	6.64636e6i	0.414664i
$763$	− 3.29320e6i	− 0.204789i
$764$	−6.55985e6	−0.406594
$765$	0	0
$766$	−3.97807e6	−0.244963
$767$	− 334722.i	− 0.0205445i
$768$	1.26486e6i	0.0773822i
$769$	−3.06343e6	−0.186806	−0.0934032	−	0.995628i	$-0.529775\pi$
$769$	−3.06343e6	−0.186806	−0.0934032	+	0.995628i	$0.529775\pi$
$770$	0	0
$771$	2.28986e6	0.138731
$772$	1.04180e6i	0.0629130i
$773$	1.17714e7i	0.708562i	0.935139	+	0.354281i	$0.115274\pi$
$773$	1.17714e7i	0.708562i	−0.935139	+	0.354281i	$0.884726\pi$
$774$	9.36541e6	0.561920
$775$	0	0
$776$	5.97853e6	0.356402
$777$	1.78736e6i	0.106208i
$778$	− 9.96483e6i	− 0.590230i
$779$	−1.84416e6	−0.108882
$780$	0	0
$781$	697086.	0.0408939
$782$	1.69449e6i	0.0990881i
$783$	− 2.18382e6i	− 0.127296i
$784$	−614656.	−0.0357143
$785$	0	0
$786$	1.70687e7	0.985474
$787$	− 1.28179e7i	− 0.737701i	−0.929489	−	0.368851i	$-0.879751\pi$
$787$	− 1.28179e7i	− 0.737701i	0.929489	−	0.368851i	$-0.120249\pi$
$788$	2.93150e6i	0.168180i
$789$	1.47878e7	0.845689
$790$	0	0
$791$	2.74282e6	0.155868
$792$	90372.3i	0.00511944i
$793$	− 976559.i	− 0.0551462i
$794$	−9.23337e6	−0.519767
$795$	0	0
$796$	−9.00864e6	−0.503937
$797$	3.33470e7i	1.85957i	0.368108	+	0.929783i	$0.380006\pi$
$797$	3.33470e7i	1.85957i	−0.368108	+	0.929783i	$0.619994\pi$
$798$	− 3.61730e6i	− 0.201084i
$799$	−1.23543e7	−0.684621
$800$	0	0
$801$	1.01829e7	0.560779
$802$	2.53524e6i	0.139182i
$803$	− 535848.i	− 0.0293260i
$804$	1.14326e7	0.623743
$805$	0	0
$806$	569053.	0.0308542
$807$	1.49537e7i	0.808284i
$808$	− 908184.i	− 0.0489379i
$809$	−1.95295e7	−1.04911	−0.524554	−	0.851377i	$-0.675768\pi$
$809$	−1.95295e7	−1.04911	−0.524554	+	0.851377i	$0.675768\pi$
$810$	0	0
$811$	−2.21959e7	−1.18501	−0.592504	−	0.805568i	$-0.701861\pi$
$811$	−2.21959e7	−1.18501	−0.592504	+	0.805568i	$0.701861\pi$
$812$	− 781591.i	− 0.0415996i
$813$	− 3.15310e7i	− 1.67306i
$814$	−82431.2	−0.00436044
$815$	0	0
$816$	−2.13702e6	−0.112353
$817$	− 1.72885e7i	− 0.906152i
$818$	3.77068e6i	0.197032i
$819$	−188443.	−0.00981679
$820$	0	0
$821$	2.37914e7	1.23186	0.615930	−	0.787801i	$-0.288780\pi$
$821$	2.37914e7	1.23186	0.615930	+	0.787801i	$0.288780\pi$
$822$	3.28239e7i	1.69438i
$823$	− 1.75911e7i	− 0.905300i	−0.891688	−	0.452650i	$-0.850479\pi$
$823$	− 1.75911e7i	− 0.905300i	0.891688	−	0.452650i	$-0.149521\pi$
$824$	−1.26469e7	−0.648881
$825$	0	0
$826$	2.20918e6	0.112663
$827$	1.50096e6i	0.0763145i	0.999272	+	0.0381572i	$0.0121488\pi$
$827$	1.50096e6i	0.0763145i	−0.999272	+	0.0381572i	$0.987851\pi$
$828$	2.02940e6i	0.102871i
$829$	3.12166e7	1.57761	0.788805	−	0.614644i	$-0.210700\pi$
$829$	3.12166e7	1.57761	0.788805	+	0.614644i	$0.210700\pi$
$830$	0	0
$831$	−4.14195e7	−2.08067
$832$	− 121638.i	− 0.00609200i
$833$	− 1.03848e6i	− 0.0518543i
$834$	1.11883e6	0.0556994
$835$	0	0
$836$	166826.	0.00825561
$837$	1.04939e7i	0.517755i
$838$	− 1.57442e7i	− 0.774478i
$839$	2.72358e7	1.33578	0.667890	−	0.744260i	$-0.267198\pi$
$839$	2.72358e7	1.33578	0.667890	+	0.744260i	$0.267198\pi$
$840$	0	0
$841$	−1.95173e7	−0.951545
$842$	2.30457e7i	1.12023i
$843$	1.76358e7i	0.854727i
$844$	9.03697e6	0.436684
$845$	0	0
$846$	−1.47961e7	−0.710756
$847$	7.88567e6i	0.377685i
$848$	− 73498.1i	− 0.00350983i
$849$	2.15500e7	1.02607
$850$	0	0
$851$	−1.85107e6	−0.0876193
$852$	1.97419e7i	0.931732i
$853$	2.58892e6i	0.121828i	0.998143	+	0.0609140i	$0.0194015\pi$
$853$	2.58892e6i	0.121828i	−0.998143	+	0.0609140i	$0.980598\pi$
$854$	6.44534e6	0.302413
$855$	0	0
$856$	1.04387e7	0.486924
$857$	− 3.19020e7i	− 1.48377i	−0.670529	−	0.741883i	$-0.733933\pi$
$857$	− 3.19020e7i	− 1.48377i	0.670529	−	0.741883i	$-0.266067\pi$
$858$	− 24998.5i	− 0.00115930i
$859$	−2.89610e7	−1.33915	−0.669577	−	0.742742i	$-0.733525\pi$
$859$	−2.89610e7	−1.33915	−0.669577	+	0.742742i	$0.733525\pi$
$860$	0	0
$861$	−1.82387e6	−0.0838468
$862$	− 7.72510e6i	− 0.354108i
$863$	− 3.43636e7i	− 1.57062i	−0.619101	−	0.785311i	$-0.712503\pi$
$863$	− 3.43636e7i	− 1.57062i	0.619101	−	0.785311i	$-0.287497\pi$
$864$	2.24313e6	0.102228
$865$	0	0
$866$	5.61833e6	0.254573
$867$	2.37931e7i	1.07499i
$868$	3.75578e6i	0.169200i
$869$	776766.	0.0348932
$870$	0	0
$871$	−1.09944e6	−0.0491049
$872$	4.30132e6i	0.191562i
$873$	1.20973e7i	0.537222i
$874$	3.74625e6	0.165889
$875$	0	0
$876$	1.51756e7	0.668166
$877$	− 1.55161e7i	− 0.681213i	−0.940206	−	0.340607i	$-0.889368\pi$
$877$	− 1.55161e7i	− 0.681213i	0.940206	−	0.340607i	$-0.110632\pi$
$878$	− 1.46588e7i	− 0.641743i
$879$	−7.65365e6	−0.334115
$880$	0	0
$881$	−2.13678e7	−0.927515	−0.463757	−	0.885962i	$-0.653499\pi$
$881$	−2.13678e7	−0.927515	−0.463757	+	0.885962i	$0.653499\pi$
$882$	− 1.24373e6i	− 0.0538339i
$883$	1.70291e7i	0.735003i	0.930023	+	0.367502i	$0.119787\pi$
$883$	1.70291e7i	0.735003i	−0.930023	+	0.367502i	$0.880213\pi$
$884$	205511.	0.00884511
$885$	0	0
$886$	1.87365e7	0.801872
$887$	− 3.88954e7i	− 1.65993i	−0.557817	−	0.829964i	$-0.688361\pi$
$887$	− 3.88954e7i	− 1.65993i	0.557817	−	0.829964i	$-0.311639\pi$
$888$	− 2.33451e6i	− 0.0993488i
$889$	4.21848e6	0.179020
$890$	0	0
$891$	804130.	0.0339338
$892$	1.88252e7i	0.792186i
$893$	2.73134e7i	1.14617i
$894$	2.85473e6	0.119460
$895$	0	0
$896$	802816.	0.0334077
$897$	− 561365.i	− 0.0232951i
$898$	− 3.49410e6i	− 0.144592i
$899$	−4.77582e6	−0.197083
$900$	0	0
$901$	124177.	0.00509600
$902$	− 84115.3i	− 0.00344238i
$903$	− 1.70983e7i	− 0.697803i
$904$	−3.58245e6	−0.145801
$905$	0	0
$906$	6.28638e6	0.254437
$907$	9.89966e6i	0.399578i	0.979839	+	0.199789i	$0.0640257\pi$
$907$	9.89966e6i	0.399578i	−0.979839	+	0.199789i	$0.935974\pi$
$908$	1.57573e7i	0.634258i
$909$	1.83767e6	0.0737664
$910$	0	0
$911$	8.31180e6	0.331817	0.165909	−	0.986141i	$-0.446944\pi$
$911$	8.31180e6	0.331817	0.165909	+	0.986141i	$0.446944\pi$
$912$	4.72463e6i	0.188097i
$913$	− 1.02929e6i	− 0.0408659i
$914$	−2.08195e7	−0.824337
$915$	0	0
$916$	2.78802e6	0.109789
$917$	− 1.08336e7i	− 0.425452i
$918$	3.78983e6i	0.148427i
$919$	1.40148e7	0.547393	0.273697	−	0.961816i	$-0.411754\pi$
$919$	1.40148e7	0.547393	0.273697	+	0.961816i	$0.411754\pi$
$920$	0	0
$921$	2.77031e7	1.07617
$922$	7.73896e6i	0.299816i
$923$	− 1.89852e6i	− 0.0733517i
$924$	164991.	0.00635742
$925$	0	0
$926$	−9.43522e6	−0.361597
$927$	− 2.55905e7i	− 0.978090i
$928$	1.02085e6i	0.0389129i
$929$	−2.55744e7	−0.972225	−0.486112	−	0.873896i	$-0.661585\pi$
$929$	−2.55744e7	−0.972225	−0.486112	+	0.873896i	$0.661585\pi$
$930$	0	0
$931$	−2.29592e6	−0.0868125
$932$	− 1.43226e7i	− 0.540108i
$933$	3.11325e7i	1.17087i
$934$	1.34961e7	0.506221
$935$	0	0
$936$	246129.	0.00918277
$937$	− 4.55745e7i	− 1.69579i	−0.530162	−	0.847896i	$-0.677869\pi$
$937$	− 4.55745e7i	− 1.69579i	0.530162	−	0.847896i	$-0.322131\pi$
$938$	− 7.25635e6i	− 0.269284i
$939$	−4.27345e7	−1.58167
$940$	0	0
$941$	−1.44078e7	−0.530425	−0.265213	−	0.964190i	$-0.585442\pi$
$941$	−1.44078e7	−0.530425	−0.265213	+	0.964190i	$0.585442\pi$
$942$	8.78395e6i	0.322524i
$943$	− 1.88889e6i	− 0.0691716i
$944$	−2.88546e6	−0.105387
$945$	0	0
$946$	788556.	0.0286487
$947$	4.58169e7i	1.66016i	0.557641	+	0.830082i	$0.311707\pi$
$947$	4.58169e7i	1.66016i	−0.557641	+	0.830082i	$0.688293\pi$
$948$	2.19985e7i	0.795010i
$949$	−1.45938e6	−0.0526022
$950$	0	0
$951$	−4.49704e7	−1.61241
$952$	1.35638e6i	0.0485053i
$953$	2.52608e7i	0.900980i	0.892781	+	0.450490i	$0.148751\pi$
$953$	2.52608e7i	0.900980i	−0.892781	+	0.450490i	$0.851249\pi$
$954$	148720.	0.00529054
$955$	0	0
$956$	4.00205e6	0.141624
$957$	209801.i	0.00740506i
$958$	1.89907e7i	0.668540i
$959$	2.08335e7	0.731502
$960$	0	0
$961$	−5.67994e6	−0.198397
$962$	224502.i	0.00782135i
$963$	2.11223e7i	0.733964i
$964$	1.31982e7	0.457427
$965$	0	0
$966$	3.70504e6	0.127747
$967$	− 5.68247e7i	− 1.95421i	−0.212758	−	0.977105i	$-0.568245\pi$
$967$	− 5.68247e7i	− 1.95421i	0.212758	−	0.977105i	$-0.431755\pi$
$968$	− 1.02997e7i	− 0.353292i
$969$	−7.98240e6	−0.273102
$970$	0	0
$971$	−2.46332e7	−0.838441	−0.419221	−	0.907884i	$-0.637697\pi$
$971$	−2.46332e7	−0.838441	−0.419221	+	0.907884i	$0.637697\pi$
$972$	1.42566e7i	0.484006i
$973$	− 710130.i	− 0.0240467i
$974$	−2.09947e7	−0.709109
$975$	0	0
$976$	−8.41840e6	−0.282882
$977$	4.16806e7i	1.39700i	0.715608	+	0.698502i	$0.246150\pi$
$977$	4.16806e7i	1.39700i	−0.715608	+	0.698502i	$0.753850\pi$
$978$	3.40123e7i	1.13707i
$979$	857391.	0.0285905
$980$	0	0
$981$	−8.70355e6	−0.288751
$982$	1.77042e7i	0.585864i
$983$	3.85482e6i	0.127239i	0.997974	+	0.0636196i	$0.0202644\pi$
$983$	3.85482e6i	0.127239i	−0.997974	+	0.0636196i	$0.979736\pi$
$984$	2.38220e6	0.0784315
$985$	0	0
$986$	−1.72476e6	−0.0564985
$987$	2.70129e7i	0.882631i
$988$	− 454352.i	− 0.0148081i
$989$	1.77078e7	0.575671
$990$	0	0
$991$	−3.85510e7	−1.24696	−0.623478	−	0.781841i	$-0.714281\pi$
$991$	−3.85510e7	−1.24696	−0.623478	+	0.781841i	$0.714281\pi$
$992$	− 4.90551e6i	− 0.158272i
$993$	5.08764e7i	1.63736i
$994$	1.25303e7	0.402250
$995$	0	0
$996$	2.91502e7	0.931093
$997$	− 3.80257e6i	− 0.121155i	−0.998164	−	0.0605773i	$-0.980706\pi$
$997$	− 3.80257e6i	− 0.121155i	0.998164	−	0.0605773i	$-0.0192942\pi$
$998$	− 183608.i	− 0.00583533i
$999$	−4.14004e6	−0.131248

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.c.k.99.2		4
5.2	odd	4		70.6.a.h.1.2	✓	2
5.3	odd	4		350.6.a.p.1.1		2
5.4	even	2	inner	350.6.c.k.99.3		4
15.2	even	4		630.6.a.s.1.2		2
20.7	even	4		560.6.a.k.1.1		2
35.27	even	4		490.6.a.u.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.h.1.2	✓	2	5.2	odd	4
350.6.a.p.1.1		2	5.3	odd	4
350.6.c.k.99.2		4	1.1	even	1	trivial
350.6.c.k.99.3		4	5.4	even	2	inner
490.6.a.u.1.1		2	35.27	even	4
560.6.a.k.1.1		2	20.7	even	4
630.6.a.s.1.2		2	15.2	even	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +19.3003i q^{3} -16.0000 q^{4} +77.2012 q^{6} -49.0000i q^{7} +64.0000i q^{8} -129.501 q^{9} -10.9039 q^{11} -308.805i q^{12} +29.6967i q^{13} -196.000 q^{14} +256.000 q^{16} +432.519i q^{17} +518.006i q^{18} +956.234 q^{19} +945.715 q^{21} +43.6155i q^{22} +979.429i q^{23} -1235.22 q^{24} +118.787 q^{26} +2190.56i q^{27} +784.000i q^{28} -996.928 q^{29} +4790.53 q^{31} -1024.00i q^{32} -210.448i q^{33} +1730.08 q^{34} +2072.02 q^{36} +1889.95i q^{37} -3824.94i q^{38} -573.156 q^{39} -1928.56 q^{41} -3782.86i q^{42} -18079.7i q^{43} +174.462 q^{44} +3917.72 q^{46} +28563.5i q^{47} +4940.88i q^{48} -2401.00 q^{49} -8347.75 q^{51} -475.148i q^{52} -287.102i q^{53} +8762.22 q^{54} +3136.00 q^{56} +18455.6i q^{57} +3987.71i q^{58} -11271.3 q^{59} -32884.4 q^{61} -19162.1i q^{62} +6345.57i q^{63} -4096.00 q^{64} -841.792 q^{66} +37022.2i q^{67} -6920.31i q^{68} -18903.3 q^{69} -63930.2 q^{71} -8288.10i q^{72} +49142.9i q^{73} +7559.81 q^{74} -15299.7 q^{76} +534.290i q^{77} +2292.62i q^{78} -71237.6 q^{79} -73747.2 q^{81} +7714.26i q^{82} +94396.8i q^{83} -15131.4 q^{84} -72319.0 q^{86} -19241.0i q^{87} -697.848i q^{88} -78631.8 q^{89} +1455.14 q^{91} -15670.9i q^{92} +92458.7i q^{93} +114254. q^{94} +19763.5 q^{96} -93414.6i q^{97} +9604.00i q^{98} +1412.07 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 64 q^{4} + 40 q^{6} - 182 q^{9} + 830 q^{11} - 784 q^{14} + 1024 q^{16} - 3836 q^{19} + 490 q^{21} - 640 q^{24} + 3432 q^{26} + 2262 q^{29} - 10944 q^{31} - 10552 q^{34} + 2912 q^{36} + 10274 q^{39}+ \cdots + 35620 q^{99}+O(q^{100}) \) 4 * q - 64 * q^4 + 40 * q^6 - 182 * q^9 + 830 * q^11 - 784 * q^14 + 1024 * q^16 - 3836 * q^19 + 490 * q^21 - 640 * q^24 + 3432 * q^26 + 2262 * q^29 - 10944 * q^31 - 10552 * q^34 + 2912 * q^36 + 10274 * q^39 - 15644 * q^41 - 13280 * q^44 - 10672 * q^46 - 9604 * q^49 - 66790 * q^51 - 14680 * q^54 + 12544 * q^56 - 13904 * q^59 - 90276 * q^61 - 16384 * q^64 - 50408 * q^66 - 103972 * q^69 - 124544 * q^71 + 73248 * q^74 + 61376 * q^76 - 124138 * q^79 - 243916 * q^81 - 7840 * q^84 + 16624 * q^86 - 128380 * q^89 + 42042 * q^91 + 68056 * q^94 + 10240 * q^96 + 35620 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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