LMFDB - Embedded newform 338.6.b.g.337.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,6,Mod(337,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	338.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$54.2097310968$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 2164x^{10} + 1693662x^{8} + 565924828x^{6} + 68470588153x^{4} + 28868949144x^{2} + 2602224144 $ x^12 + 2164x^10 + 1693662x^8 + 565924828x^6 + 68470588153x^4 + 28868949144*x^2 + 2602224144
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{20}\cdot 13^{2} $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.8
Root			$19.3520i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.6.b.g.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -19.3520 q^{3} -16.0000 q^{4} +52.2356i q^{5} -77.4082i q^{6} -19.1214i q^{7} -64.0000i q^{8} +131.501 q^{9} +O(q^{10})$	\(q+4.00000i q^{2} -19.3520 q^{3} -16.0000 q^{4} +52.2356i q^{5} -77.4082i q^{6} -19.1214i q^{7} -64.0000i q^{8} +131.501 q^{9} -208.942 q^{10} +381.988i q^{11} +309.633 q^{12} +76.4855 q^{14} -1010.86i q^{15} +256.000 q^{16} +1057.42 q^{17} +526.006i q^{18} +2343.73i q^{19} -835.769i q^{20} +370.038i q^{21} -1527.95 q^{22} +2747.96 q^{23} +1238.53i q^{24} +396.447 q^{25} +2157.72 q^{27} +305.942i q^{28} -121.718 q^{29} +4043.46 q^{30} +5556.97i q^{31} +1024.00i q^{32} -7392.25i q^{33} +4229.66i q^{34} +998.816 q^{35} -2104.02 q^{36} +8146.46i q^{37} -9374.92 q^{38} +3343.08 q^{40} +13485.3i q^{41} -1480.15 q^{42} +22019.7 q^{43} -6111.81i q^{44} +6869.05i q^{45} +10991.8i q^{46} +3489.22i q^{47} -4954.12 q^{48} +16441.4 q^{49} +1585.79i q^{50} -20463.2 q^{51} -28937.9 q^{53} +8630.90i q^{54} -19953.4 q^{55} -1223.77 q^{56} -45355.9i q^{57} -486.873i q^{58} -34446.8i q^{59} +16173.8i q^{60} +5305.93 q^{61} -22227.9 q^{62} -2514.49i q^{63} -4096.00 q^{64} +29569.0 q^{66} -56900.3i q^{67} -16918.7 q^{68} -53178.6 q^{69} +3995.26i q^{70} +2393.52i q^{71} -8416.09i q^{72} +47969.6i q^{73} -32585.8 q^{74} -7672.05 q^{75} -37499.7i q^{76} +7304.14 q^{77} -79330.2 q^{79} +13372.3i q^{80} -73711.2 q^{81} -53941.1 q^{82} -22090.3i q^{83} -5920.60i q^{84} +55234.7i q^{85} +88078.6i q^{86} +2355.50 q^{87} +24447.2 q^{88} +124339. i q^{89} -27476.2 q^{90} -43967.3 q^{92} -107539. i q^{93} -13956.9 q^{94} -122426. q^{95} -19816.5i q^{96} -94093.5i q^{97} +65765.5i q^{98} +50232.0i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 192 q^{4} + 1412 q^{9}+O(q^{10}) $ 12 * q - 192 * q^4 + 1412 * q^9	$ 12 q - 192 q^{4} + 1412 q^{9} + 736 q^{10} - 2272 q^{14} + 3072 q^{16} + 3820 q^{17} - 4320 q^{22} - 11840 q^{23} - 4672 q^{25} - 2736 q^{27} - 15844 q^{29} + 23392 q^{30} - 33728 q^{35} - 22592 q^{36} + 26496 q^{38} - 11776 q^{40} - 47296 q^{42} + 83840 q^{43} - 9364 q^{49} + 133264 q^{51} - 187812 q^{53} - 34944 q^{55} + 36352 q^{56} - 55868 q^{61} + 8672 q^{62} - 49152 q^{64} + 6336 q^{66} - 61120 q^{68} + 154696 q^{69} - 192416 q^{74} - 384160 q^{75} - 151944 q^{77} + 30464 q^{79} + 144620 q^{81} + 174080 q^{82} + 243872 q^{87} + 69120 q^{88} + 282016 q^{90} + 189440 q^{92} - 316512 q^{94} - 465120 q^{95}+O(q^{100}) $ 12 * q - 192 * q^4 + 1412 * q^9 + 736 * q^10 - 2272 * q^14 + 3072 * q^16 + 3820 * q^17 - 4320 * q^22 - 11840 * q^23 - 4672 * q^25 - 2736 * q^27 - 15844 * q^29 + 23392 * q^30 - 33728 * q^35 - 22592 * q^36 + 26496 * q^38 - 11776 * q^40 - 47296 * q^42 + 83840 * q^43 - 9364 * q^49 + 133264 * q^51 - 187812 * q^53 - 34944 * q^55 + 36352 * q^56 - 55868 * q^61 + 8672 * q^62 - 49152 * q^64 + 6336 * q^66 - 61120 * q^68 + 154696 * q^69 - 192416 * q^74 - 384160 * q^75 - 151944 * q^77 + 30464 * q^79 + 144620 * q^81 + 174080 * q^82 + 243872 * q^87 + 69120 * q^88 + 282016 * q^90 + 189440 * q^92 - 316512 * q^94 - 465120 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	4.00000i	0.707107i
$2$	4.00000i	0.707107i
$3$	−19.3520	−1.24143	−0.620717	−	0.784035i	$-0.713158\pi$
$3$	−19.3520	−1.24143	−0.620717	+	0.784035i	$0.713158\pi$
$4$	−16.0000	−0.500000
$5$	52.2356i	0.934418i	0.884147	+	0.467209i	$0.154740\pi$
$5$	52.2356i	0.934418i	−0.884147	+	0.467209i	$0.845260\pi$
$6$	− 77.4082i	− 0.877826i
$7$	− 19.1214i	− 0.147494i	−0.997277	−	0.0737469i	$-0.976504\pi$
$7$	− 19.1214i	− 0.147494i	0.997277	−	0.0737469i	$-0.0234957\pi$
$8$	− 64.0000i	− 0.353553i
$9$	131.501	0.541158
$10$	−208.942	−0.660733
$11$	381.988i	0.951849i	0.879486	+	0.475924i	$0.157886\pi$
$11$	381.988i	0.951849i	−0.879486	+	0.475924i	$0.842114\pi$
$12$	309.633	0.620717
$13$	0	0
$13$	0	0
$14$	76.4855	0.104294
$15$	− 1010.86i	− 1.16002i
$16$	256.000	0.250000
$17$	1057.42	0.887409	0.443704	−	0.896173i	$-0.353664\pi$
$17$	1057.42	0.887409	0.443704	+	0.896173i	$0.353664\pi$
$18$	526.006i	0.382657i
$19$	2343.73i	1.48944i	0.667376	+	0.744721i	$0.267417\pi$
$19$	2343.73i	1.48944i	−0.667376	+	0.744721i	$0.732583\pi$
$20$	− 835.769i	− 0.467209i
$21$	370.038i	0.183104i
$22$	−1527.95	−0.673059
$23$	2747.96	1.08315	0.541577	−	0.840651i	$-0.317827\pi$
$23$	2747.96	1.08315	0.541577	+	0.840651i	$0.317827\pi$
$24$	1238.53i	0.438913i
$25$	396.447	0.126863
$26$	0	0
$27$	2157.72	0.569622
$28$	305.942i	0.0737469i
$29$	−121.718	−0.0268757	−0.0134379	−	0.999910i	$-0.504278\pi$
$29$	−121.718	−0.0268757	−0.0134379	+	0.999910i	$0.504278\pi$
$30$	4043.46	0.820257
$31$	5556.97i	1.03856i	0.854603	+	0.519282i	$0.173801\pi$
$31$	5556.97i	1.03856i	−0.854603	+	0.519282i	$0.826199\pi$
$32$	1024.00i	0.176777i
$33$	− 7392.25i	− 1.18166i
$34$	4229.66i	0.627493i
$35$	998.816	0.137821
$36$	−2104.02	−0.270579
$37$	8146.46i	0.978283i	0.872204	+	0.489142i	$0.162690\pi$
$37$	8146.46i	0.978283i	−0.872204	+	0.489142i	$0.837310\pi$
$38$	−9374.92	−1.05319
$39$	0	0
$40$	3343.08	0.330367
$41$	13485.3i	1.25285i	0.779481	+	0.626426i	$0.215483\pi$
$41$	13485.3i	1.25285i	−0.779481	+	0.626426i	$0.784517\pi$
$42$	−1480.15	−0.129474
$43$	22019.7	1.81610	0.908049	−	0.418863i	$-0.137571\pi$
$43$	22019.7	1.81610	0.908049	+	0.418863i	$0.137571\pi$
$44$	− 6111.81i	− 0.475924i
$45$	6869.05i	0.505668i
$46$	10991.8i	0.765906i
$47$	3489.22i	0.230401i	0.993342	+	0.115200i	$0.0367510\pi$
$47$	3489.22i	0.230401i	−0.993342	+	0.115200i	$0.963249\pi$
$48$	−4954.12	−0.310358
$49$	16441.4	0.978246
$50$	1585.79i	0.0897056i
$51$	−20463.2	−1.10166
$52$	0	0
$53$	−28937.9	−1.41507	−0.707534	−	0.706679i	$-0.750192\pi$
$53$	−28937.9	−1.41507	−0.707534	+	0.706679i	$0.750192\pi$
$54$	8630.90i	0.402783i
$55$	−19953.4	−0.889425
$56$	−1223.77	−0.0521470
$57$	− 45355.9i	− 1.84904i
$58$	− 486.873i	− 0.0190040i
$59$	− 34446.8i	− 1.28830i	−0.764897	−	0.644152i	$-0.777210\pi$
$59$	− 34446.8i	− 1.28830i	0.764897	−	0.644152i	$-0.222790\pi$
$60$	16173.8i	0.580009i
$61$	5305.93	0.182573	0.0912866	−	0.995825i	$-0.470902\pi$
$61$	5305.93	0.182573	0.0912866	+	0.995825i	$0.470902\pi$
$62$	−22227.9	−0.734376
$63$	− 2514.49i	− 0.0798175i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	29569.0	0.835558
$67$	− 56900.3i	− 1.54856i	−0.632844	−	0.774279i	$-0.718113\pi$
$67$	− 56900.3i	− 1.54856i	0.632844	−	0.774279i	$-0.281887\pi$
$68$	−16918.7	−0.443704
$69$	−53178.6	−1.34466
$70$	3995.26i	0.0974541i
$71$	2393.52i	0.0563496i	0.999603	+	0.0281748i	$0.00896950\pi$
$71$	2393.52i	0.0563496i	−0.999603	+	0.0281748i	$0.991031\pi$
$72$	− 8416.09i	− 0.191328i
$73$	47969.6i	1.05356i	0.850002	+	0.526780i	$0.176601\pi$
$73$	47969.6i	1.05356i	−0.850002	+	0.526780i	$0.823399\pi$
$74$	−32585.8	−0.691751
$75$	−7672.05	−0.157492
$76$	− 37499.7i	− 0.744721i
$77$	7304.14	0.140392
$78$	0	0
$79$	−79330.2	−1.43011	−0.715057	−	0.699066i	$-0.753599\pi$
$79$	−79330.2	−1.43011	−0.715057	+	0.699066i	$0.753599\pi$
$80$	13372.3i	0.233605i
$81$	−73711.2	−1.24831
$82$	−53941.1	−0.885901
$83$	− 22090.3i	− 0.351971i	−0.984393	−	0.175986i	$-0.943689\pi$
$83$	− 22090.3i	− 0.351971i	0.984393	−	0.175986i	$-0.0563112\pi$
$84$	− 5920.60i	− 0.0915520i
$85$	55234.7i	0.829211i
$86$	88078.6i	1.28418i
$87$	2355.50	0.0333645
$88$	24447.2	0.336529
$89$	124339.i	1.66392i	0.554835	+	0.831960i	$0.312781\pi$
$89$	124339.i	1.66392i	−0.554835	+	0.831960i	$0.687219\pi$
$90$	−27476.2	−0.357561
$91$	0	0
$92$	−43967.3	−0.541577
$93$	− 107539.i	− 1.28931i
$94$	−13956.9	−0.162918
$95$	−122426.	−1.39176
$96$	− 19816.5i	− 0.219457i
$97$	− 94093.5i	− 1.01538i	−0.861539	−	0.507692i	$-0.830499\pi$
$97$	− 94093.5i	− 1.01538i	0.861539	−	0.507692i	$-0.169501\pi$
$98$	65765.5i	0.691724i
$99$	50232.0i	0.515101i
$100$	−6343.15	−0.0634315
$101$	57335.4	0.559267	0.279634	−	0.960107i	$-0.409787\pi$
$101$	57335.4	0.559267	0.279634	+	0.960107i	$0.409787\pi$
$102$	− 81852.6i	− 0.778991i
$103$	−160258.	−1.48842	−0.744211	−	0.667945i	$-0.767174\pi$
$103$	−160258.	−1.48842	−0.744211	+	0.667945i	$0.767174\pi$
$104$	0	0
$105$	−19329.1	−0.171096
$106$	− 115752.i	− 1.00060i
$107$	18055.2	0.152455	0.0762276	−	0.997090i	$-0.475712\pi$
$107$	18055.2	0.152455	0.0762276	+	0.997090i	$0.475712\pi$
$108$	−34523.6	−0.284811
$109$	− 25954.4i	− 0.209240i	−0.994512	−	0.104620i	$-0.966637\pi$
$109$	− 25954.4i	− 0.209240i	0.994512	−	0.104620i	$-0.0333627\pi$
$110$	− 79813.4i	− 0.628918i
$111$	− 157651.i	− 1.21447i
$112$	− 4895.07i	− 0.0368735i
$113$	−109573.	−0.807246	−0.403623	−	0.914925i	$-0.632249\pi$
$113$	−109573.	−0.807246	−0.403623	+	0.914925i	$0.632249\pi$
$114$	181424.	1.30747
$115$	143541.i	1.01212i
$116$	1947.49	0.0134379
$117$	0	0
$118$	137787.	0.910969
$119$	− 20219.2i	− 0.130887i
$120$	−64695.3	−0.410128
$121$	15136.2	0.0939836
$122$	21223.7i	0.129099i
$123$	− 260968.i	− 1.55533i
$124$	− 88911.5i	− 0.519282i
$125$	183945.i	1.05296i
$126$	10058.0	0.0564395
$127$	58353.4	0.321038	0.160519	−	0.987033i	$-0.448683\pi$
$127$	58353.4	0.321038	0.160519	+	0.987033i	$0.448683\pi$
$128$	− 16384.0i	− 0.0883883i
$129$	−426125.	−2.25457
$130$	0	0
$131$	29397.6	0.149670	0.0748349	−	0.997196i	$-0.476157\pi$
$131$	29397.6	0.149670	0.0748349	+	0.997196i	$0.476157\pi$
$132$	118276.i	0.590829i
$133$	44815.3	0.219684
$134$	227601.	1.09500
$135$	112710.i	0.532265i
$136$	− 67674.6i	− 0.313746i
$137$	− 229924.i	− 1.04660i	−0.852148	−	0.523302i	$-0.824700\pi$
$137$	− 229924.i	− 1.04660i	0.852148	−	0.523302i	$-0.175300\pi$
$138$	− 212714.i	− 0.950822i
$139$	169311.	0.743273	0.371637	−	0.928378i	$-0.378797\pi$
$139$	169311.	0.743273	0.371637	+	0.928378i	$0.378797\pi$
$140$	−15981.1	−0.0689105
$141$	− 67523.5i	− 0.286027i
$142$	−9574.07	−0.0398452
$143$	0	0
$144$	33664.4	0.135290
$145$	− 6358.02i	− 0.0251132i
$146$	−191878.	−0.744979
$147$	−318174.	−1.21443
$148$	− 130343.i	− 0.489142i
$149$	492678.i	1.81801i	0.416780	+	0.909007i	$0.363158\pi$
$149$	492678.i	1.81801i	−0.416780	+	0.909007i	$0.636842\pi$
$150$	− 30688.2i	− 0.111364i
$151$	− 36336.4i	− 0.129688i	−0.997895	−	0.0648440i	$-0.979345\pi$
$151$	− 36336.4i	− 0.129688i	0.997895	−	0.0648440i	$-0.0206550\pi$
$152$	149999.	0.526597
$153$	139052.	0.480229
$154$	29216.5i	0.0992721i
$155$	−290271.	−0.970453
$156$	0	0
$157$	368197.	1.19215	0.596075	−	0.802929i	$-0.296726\pi$
$157$	368197.	1.19215	0.596075	+	0.802929i	$0.296726\pi$
$158$	− 317321.i	− 1.01124i
$159$	560007.	1.75671
$160$	−53489.2	−0.165183
$161$	− 52544.7i	− 0.159759i
$162$	− 294845.i	− 0.882686i
$163$	− 226017.i	− 0.666303i	−0.942873	−	0.333151i	$-0.891888\pi$
$163$	− 226017.i	− 0.666303i	0.942873	−	0.333151i	$-0.108112\pi$
$164$	− 215764.i	− 0.626426i
$165$	386138.	1.10416
$166$	88361.4	0.248881
$167$	161239.i	0.447382i	0.974660	+	0.223691i	$0.0718106\pi$
$167$	161239.i	0.447382i	−0.974660	+	0.223691i	$0.928189\pi$
$168$	23682.4	0.0647370
$169$	0	0
$170$	−220939.	−0.586340
$171$	308204.i	0.806024i
$172$	−352314.	−0.908049
$173$	300682.	0.763822	0.381911	−	0.924199i	$-0.375266\pi$
$173$	300682.	0.763822	0.381911	+	0.924199i	$0.375266\pi$
$174$	9421.98i	0.0235922i
$175$	− 7580.61i	− 0.0187115i
$176$	97788.9i	0.237962i
$177$	666616.i	1.59935i
$178$	−497356.	−1.17657
$179$	−277007.	−0.646186	−0.323093	−	0.946367i	$-0.604723\pi$
$179$	−277007.	−0.646186	−0.323093	+	0.946367i	$0.604723\pi$
$180$	− 109905.i	− 0.252834i
$181$	585036.	1.32735	0.663676	−	0.748020i	$-0.268995\pi$
$181$	585036.	1.32735	0.663676	+	0.748020i	$0.268995\pi$
$182$	0	0
$183$	−102681.	−0.226652
$184$	− 175869.i	− 0.382953i
$185$	−425535.	−0.914126
$186$	430155.	0.911679
$187$	403920.i	0.844679i
$188$	− 55827.5i	− 0.115200i
$189$	− 41258.6i	− 0.0840157i
$190$	− 489704.i	− 0.984124i
$191$	−648564.	−1.28638	−0.643190	−	0.765707i	$-0.722389\pi$
$191$	−648564.	−1.28638	−0.643190	+	0.765707i	$0.722389\pi$
$192$	79266.0	0.155179
$193$	658457.i	1.27243i	0.771512	+	0.636215i	$0.219501\pi$
$193$	658457.i	1.27243i	−0.771512	+	0.636215i	$0.780499\pi$
$194$	376374.	0.717985
$195$	0	0
$196$	−263062.	−0.489123
$197$	639300.i	1.17365i	0.809713	+	0.586826i	$0.199623\pi$
$197$	639300.i	1.17365i	−0.809713	+	0.586826i	$0.800377\pi$
$198$	−200928.	−0.364231
$199$	−826294.	−1.47911	−0.739557	−	0.673093i	$-0.764965\pi$
$199$	−826294.	−1.47911	−0.739557	+	0.673093i	$0.764965\pi$
$200$	− 25372.6i	− 0.0448528i
$201$	1.10114e6i	1.92243i
$202$	229342.i	0.395462i
$203$	2327.42i	0.00396401i
$204$	327411.	0.550830
$205$	−704411.	−1.17069
$206$	− 641031.i	− 1.05247i
$207$	361360.	0.586158
$208$	0	0
$209$	−895277.	−1.41772
$210$	− 77316.5i	− 0.120983i
$211$	448279.	0.693175	0.346587	−	0.938018i	$-0.387340\pi$
$211$	448279.	0.693175	0.346587	+	0.938018i	$0.387340\pi$
$212$	463006.	0.707534
$213$	− 46319.4i	− 0.0699543i
$214$	72220.7i	0.107802i
$215$	1.15021e6i	1.69700i
$216$	− 138094.i	− 0.201392i
$217$	106257.	0.153182
$218$	103818.	0.147955
$219$	− 928310.i	− 1.30792i
$220$	319254.	0.444712
$221$	0	0
$222$	630603.	0.858763
$223$	− 195080.i	− 0.262695i	−0.991336	−	0.131347i	$-0.958070\pi$
$223$	− 195080.i	− 0.262695i	0.991336	−	0.131347i	$-0.0419303\pi$
$224$	19580.3	0.0260735
$225$	52133.3	0.0686529
$226$	− 438290.i	− 0.570809i
$227$	− 563534.i	− 0.725864i	−0.931816	−	0.362932i	$-0.881776\pi$
$227$	− 563534.i	− 0.725864i	0.931816	−	0.362932i	$-0.118224\pi$
$228$	725695.i	0.924522i
$229$	27368.8i	0.0344879i	0.999851	+	0.0172440i	$0.00548919\pi$
$229$	27368.8i	0.0344879i	−0.999851	+	0.0172440i	$0.994511\pi$
$230$	−574164.	−0.715676
$231$	−141350.	−0.174287
$232$	7789.97i	0.00950201i
$233$	489880.	0.591153	0.295577	−	0.955319i	$-0.404488\pi$
$233$	489880.	0.591153	0.295577	+	0.955319i	$0.404488\pi$
$234$	0	0
$235$	−182261.	−0.215290
$236$	551149.i	0.644152i
$237$	1.53520e6	1.77539
$238$	80877.0	0.0925513
$239$	97635.3i	0.110564i	0.998471	+	0.0552818i	$0.0176057\pi$
$239$	97635.3i	0.110564i	−0.998471	+	0.0552818i	$0.982394\pi$
$240$	− 258781.i	− 0.290005i
$241$	1.11350e6i	1.23494i	0.786594	+	0.617470i	$0.211842\pi$
$241$	1.11350e6i	1.23494i	−0.786594	+	0.617470i	$0.788158\pi$
$242$	60544.6i	0.0664564i
$243$	902136.	0.980068
$244$	−84894.9	−0.0912866
$245$	858824.i	0.914090i
$246$	1.04387e6	1.09979
$247$	0	0
$248$	355646.	0.367188
$249$	427493.i	0.436949i
$250$	−735779.	−0.744556
$251$	1.21731e6	1.21960	0.609800	−	0.792555i	$-0.291250\pi$
$251$	1.21731e6	1.21960	0.609800	+	0.792555i	$0.291250\pi$
$252$	40231.8i	0.0399088i
$253$	1.04969e6i	1.03100i
$254$	233414.i	0.227008i
$255$	− 1.06890e6i	− 1.02941i
$256$	65536.0	0.0625000
$257$	−1.87353e6	−1.76941	−0.884704	−	0.466154i	$-0.845639\pi$
$257$	−1.87353e6	−1.76941	−0.884704	+	0.466154i	$0.845639\pi$
$258$	− 1.70450e6i	− 1.59422i
$259$	155772.	0.144291
$260$	0	0
$261$	−16006.1	−0.0145440
$262$	117591.i	0.105833i
$263$	831123.	0.740927	0.370464	−	0.928847i	$-0.379199\pi$
$263$	831123.	0.740927	0.370464	+	0.928847i	$0.379199\pi$
$264$	−473104.	−0.417779
$265$	− 1.51159e6i	− 1.32226i
$266$	179261.i	0.155340i
$267$	− 2.40621e6i	− 2.06565i
$268$	910405.i	0.774279i
$269$	−810878.	−0.683242	−0.341621	−	0.939838i	$-0.610976\pi$
$269$	−810878.	−0.683242	−0.341621	+	0.939838i	$0.610976\pi$
$270$	−450840.	−0.376368
$271$	140436.i	0.116160i	0.998312	+	0.0580799i	$0.0184978\pi$
$271$	140436.i	0.116160i	−0.998312	+	0.0580799i	$0.981502\pi$
$272$	270699.	0.221852
$273$	0	0
$274$	919694.	0.740060
$275$	151438.i	0.120754i
$276$	850857.	0.672332
$277$	1.38608e6	1.08540	0.542698	−	0.839928i	$-0.317403\pi$
$277$	1.38608e6	1.08540	0.542698	+	0.839928i	$0.317403\pi$
$278$	677245.i	0.525574i
$279$	730749.i	0.562028i
$280$	− 63924.2i	− 0.0487271i
$281$	− 1.39045e6i	− 1.05049i	−0.850953	−	0.525243i	$-0.823975\pi$
$281$	− 1.39045e6i	− 1.05049i	0.850953	−	0.525243i	$-0.176025\pi$
$282$	270094.	0.202252
$283$	−912951.	−0.677612	−0.338806	−	0.940856i	$-0.610023\pi$
$283$	−912951.	−0.677612	−0.338806	+	0.940856i	$0.610023\pi$
$284$	− 38296.3i	− 0.0281748i
$285$	2.36919e6	1.72778
$286$	0	0
$287$	257857.	0.184788
$288$	134658.i	0.0956642i
$289$	−301728.	−0.212506
$290$	25432.1	0.0177577
$291$	1.82090e6i	1.26053i
$292$	− 767513.i	− 0.526780i
$293$	68805.6i	0.0468225i	0.999726	+	0.0234112i	$0.00745271\pi$
$293$	68805.6i	0.0468225i	−0.999726	+	0.0234112i	$0.992547\pi$
$294$	− 1.27270e6i	− 0.858730i
$295$	1.79935e6	1.20382
$296$	521374.	0.345875
$297$	824225.i	0.542194i
$298$	−1.97071e6	−1.28553
$299$	0	0
$300$	122753.	0.0787460
$301$	− 421046.i	− 0.267863i
$302$	145346.	0.0917033
$303$	−1.10956e6	−0.694294
$304$	599995.i	0.372360i
$305$	277158.i	0.170600i
$306$	556207.i	0.339573i
$307$	− 2.66005e6i	− 1.61081i	−0.592726	−	0.805404i	$-0.701948\pi$
$307$	− 2.66005e6i	− 1.61081i	0.592726	−	0.805404i	$-0.298052\pi$
$308$	−116866.	−0.0701959
$309$	3.10132e6	1.84778
$310$	− 1.16108e6i	− 0.686214i
$311$	−2.21314e6	−1.29750	−0.648751	−	0.761000i	$-0.724709\pi$
$311$	−2.21314e6	−1.29750	−0.648751	+	0.761000i	$0.724709\pi$
$312$	0	0
$313$	−635019.	−0.366375	−0.183188	−	0.983078i	$-0.558642\pi$
$313$	−635019.	−0.366375	−0.183188	+	0.983078i	$0.558642\pi$
$314$	1.47279e6i	0.842978i
$315$	131346.	0.0745830
$316$	1.26928e6	0.715057
$317$	2.29568e6i	1.28311i	0.767078	+	0.641554i	$0.221710\pi$
$317$	2.29568e6i	1.28311i	−0.767078	+	0.641554i	$0.778290\pi$
$318$	2.24003e6i	1.24218i
$319$	− 46494.9i	− 0.0255817i
$320$	− 213957.i	− 0.116802i
$321$	−349405.	−0.189263
$322$	210179.	0.112966
$323$	2.47830e6i	1.32174i
$324$	1.17938e6	0.624153
$325$	0	0
$326$	904067.	0.471147
$327$	502272.i	0.259758i
$328$	863057.	0.442950
$329$	66718.7	0.0339827
$330$	1.54455e6i	0.780761i
$331$	− 1.92131e6i	− 0.963892i	−0.876201	−	0.481946i	$-0.839930\pi$
$331$	− 1.92131e6i	− 0.963892i	0.876201	−	0.481946i	$-0.160070\pi$
$332$	353445.i	0.175986i
$333$	1.07127e6i	0.529406i
$334$	−644955.	−0.316347
$335$	2.97222e6	1.44700
$336$	94729.6i	0.0457760i
$337$	574677.	0.275644	0.137822	−	0.990457i	$-0.455990\pi$
$337$	574677.	0.275644	0.137822	+	0.990457i	$0.455990\pi$
$338$	0	0
$339$	2.12045e6	1.00214
$340$	− 883755.i	− 0.414605i
$341$	−2.12269e6	−0.988557
$342$	−1.23282e6	−0.569945
$343$	− 635755.i	− 0.291779i
$344$	− 1.40926e6i	− 0.642088i
$345$	− 2.77781e6i	− 1.25648i
$346$	1.20273e6i	0.540104i
$347$	2.63825e6	1.17623	0.588114	−	0.808778i	$-0.299871\pi$
$347$	2.63825e6	1.17623	0.588114	+	0.808778i	$0.299871\pi$
$348$	−37687.9	−0.0166822
$349$	− 1.24694e6i	− 0.548004i	−0.961729	−	0.274002i	$-0.911653\pi$
$349$	− 1.24694e6i	− 0.548004i	0.961729	−	0.274002i	$-0.0883474\pi$
$350$	30322.4	0.0132310
$351$	0	0
$352$	−391156.	−0.168265
$353$	1.50991e6i	0.644933i	0.946581	+	0.322467i	$0.104512\pi$
$353$	1.50991e6i	0.644933i	−0.946581	+	0.322467i	$0.895488\pi$
$354$	−2.66646e6	−1.13091
$355$	−125027.	−0.0526540
$356$	− 1.98942e6i	− 0.831960i
$357$	391284.i	0.162488i
$358$	− 1.10803e6i	− 0.456923i
$359$	576783.i	0.236198i	0.993002	+	0.118099i	$0.0376801\pi$
$359$	576783.i	0.236198i	−0.993002	+	0.118099i	$0.962320\pi$
$360$	439619.	0.178781
$361$	−3.01697e6	−1.21844
$362$	2.34014e6i	0.938579i
$363$	−292915.	−0.116674
$364$	0	0
$365$	−2.50572e6	−0.984465
$366$	− 410722.i	− 0.160267i
$367$	2.88039e6	1.11631	0.558157	−	0.829735i	$-0.311509\pi$
$367$	2.88039e6	1.11631	0.558157	+	0.829735i	$0.311509\pi$
$368$	703477.	0.270789
$369$	1.77333e6i	0.677992i
$370$	− 1.70214e6i	− 0.646384i
$371$	553332.i	0.208714i
$372$	1.72062e6i	0.644655i
$373$	−2.71423e6	−1.01012	−0.505061	−	0.863084i	$-0.668530\pi$
$373$	−2.71423e6	−1.01012	−0.505061	+	0.863084i	$0.668530\pi$
$374$	−1.61568e6	−0.597278
$375$	− 3.55971e6i	− 1.30718i
$376$	223310.	0.0814589
$377$	0	0
$378$	165035.	0.0594081
$379$	− 2.97459e6i	− 1.06372i	−0.846831	−	0.531861i	$-0.821493\pi$
$379$	− 2.97459e6i	− 1.06372i	0.846831	−	0.531861i	$-0.178507\pi$
$380$	1.95882e6	0.695881
$381$	−1.12926e6	−0.398548
$382$	− 2.59425e6i	− 0.909608i
$383$	− 4.05276e6i	− 1.41174i	−0.708342	−	0.705869i	$-0.750557\pi$
$383$	− 4.05276e6i	− 1.41174i	0.708342	−	0.705869i	$-0.249443\pi$
$384$	317064.i	0.109728i
$385$	381536.i	0.131185i
$386$	−2.63383e6	−0.899744
$387$	2.89562e6	0.982797
$388$	1.50550e6i	0.507692i
$389$	−4.80279e6	−1.60924	−0.804618	−	0.593793i	$-0.797630\pi$
$389$	−4.80279e6	−1.60924	−0.804618	+	0.593793i	$0.797630\pi$
$390$	0	0
$391$	2.90573e6	0.961201
$392$	− 1.05225e6i	− 0.345862i
$393$	−568904.	−0.185805
$394$	−2.55720e6	−0.829898
$395$	− 4.14385e6i	− 1.33632i
$396$	− 803712.i	− 0.257551i
$397$	4.61361e6i	1.46915i	0.678530	+	0.734573i	$0.262617\pi$
$397$	4.61361e6i	1.46915i	−0.678530	+	0.734573i	$0.737383\pi$
$398$	− 3.30518e6i	− 1.04589i
$399$	−867268.	−0.272723
$400$	101490.	0.0317157
$401$	− 2.10739e6i	− 0.654461i	−0.944945	−	0.327230i	$-0.893885\pi$
$401$	− 2.10739e6i	− 0.654461i	0.944945	−	0.327230i	$-0.106115\pi$
$402$	−4.40455e6	−1.35937
$403$	0	0
$404$	−917367.	−0.279634
$405$	− 3.85035e6i	− 1.16644i
$406$	−9309.68	−0.00280298
$407$	−3.11185e6	−0.931178
$408$	1.30964e6i	0.389495i
$409$	2.17339e6i	0.642435i	0.947006	+	0.321217i	$0.104092\pi$
$409$	2.17339e6i	0.642435i	−0.947006	+	0.321217i	$0.895908\pi$
$410$	− 2.81764e6i	− 0.827802i
$411$	4.44949e6i	1.29929i
$412$	2.56412e6	0.744211
$413$	−658670.	−0.190017
$414$	1.44544e6i	0.414476i
$415$	1.15390e6	0.328888
$416$	0	0
$417$	−3.27652e6	−0.922725
$418$	− 3.58111e6i	− 1.00248i
$419$	−1.84207e6	−0.512590	−0.256295	−	0.966599i	$-0.582502\pi$
$419$	−1.84207e6	−0.512590	−0.256295	+	0.966599i	$0.582502\pi$
$420$	309266.	0.0855478
$421$	− 2.27354e6i	− 0.625170i	−0.949890	−	0.312585i	$-0.898805\pi$
$421$	− 2.27354e6i	− 0.625170i	0.949890	−	0.312585i	$-0.101195\pi$
$422$	1.79312e6i	0.490149i
$423$	458838.i	0.124683i
$424$	1.85203e6i	0.500302i
$425$	419209.	0.112579
$426$	185278.	0.0494651
$427$	− 101457.i	− 0.0269284i
$428$	−288883.	−0.0762276
$429$	0	0
$430$	−4.60084e6	−1.19996
$431$	− 358055.i	− 0.0928445i	−0.998922	−	0.0464222i	$-0.985218\pi$
$431$	− 358055.i	− 0.0928445i	0.998922	−	0.0464222i	$-0.0147820\pi$
$432$	552377.	0.142405
$433$	−3.84975e6	−0.986763	−0.493382	−	0.869813i	$-0.664239\pi$
$433$	−3.84975e6	−0.986763	−0.493382	+	0.869813i	$0.664239\pi$
$434$	425027.i	0.108316i
$435$	123041.i	0.0311764i
$436$	415271.i	0.104620i
$437$	6.44047e6i	1.61330i
$438$	3.71324e6	0.924842
$439$	−5.07937e6	−1.25791	−0.628954	−	0.777443i	$-0.716517\pi$
$439$	−5.07937e6	−1.25791	−0.628954	+	0.777443i	$0.716517\pi$
$440$	1.27701e6i	0.314459i
$441$	2.16206e6	0.529386
$442$	0	0
$443$	4.53588e6	1.09813	0.549063	−	0.835781i	$-0.314985\pi$
$443$	4.53588e6	1.09813	0.549063	+	0.835781i	$0.314985\pi$
$444$	2.52241e6i	0.607237i
$445$	−6.49492e6	−1.55480
$446$	780321.	0.185753
$447$	− 9.53432e6i	− 2.25694i
$448$	78321.2i	0.0184367i
$449$	− 1.15562e6i	− 0.270520i	−0.990810	−	0.135260i	$-0.956813\pi$
$449$	− 1.15562e6i	− 0.270520i	0.990810	−	0.135260i	$-0.0431870\pi$
$450$	208533.i	0.0485450i
$451$	−5.15121e6	−1.19253
$452$	1.75316e6	0.403623
$453$	703184.i	0.160999i
$454$	2.25414e6	0.513264
$455$	0	0
$456$	−2.90278e6	−0.653736
$457$	1.53521e6i	0.343857i	0.985109	+	0.171928i	$0.0549997\pi$
$457$	1.53521e6i	0.343857i	−0.985109	+	0.171928i	$0.945000\pi$
$458$	−109475.	−0.0243866
$459$	2.28161e6	0.505487
$460$	− 2.29666e6i	− 0.506060i
$461$	− 6.41152e6i	− 1.40510i	−0.711632	−	0.702552i	$-0.752044\pi$
$461$	− 6.41152e6i	− 1.40510i	0.711632	−	0.702552i	$-0.247956\pi$
$462$	− 565400.i	− 0.123240i
$463$	3.67970e6i	0.797736i	0.917008	+	0.398868i	$0.130597\pi$
$463$	3.67970e6i	0.797736i	−0.917008	+	0.398868i	$0.869403\pi$
$464$	−31159.9	−0.00671894
$465$	5.61734e6	1.20475
$466$	1.95952e6i	0.418008i
$467$	−5.57471e6	−1.18285	−0.591425	−	0.806360i	$-0.701435\pi$
$467$	−5.57471e6	−1.18285	−0.591425	+	0.806360i	$0.701435\pi$
$468$	0	0
$469$	−1.08801e6	−0.228403
$470$	− 729045.i	− 0.152233i
$471$	−7.12536e6	−1.47998
$472$	−2.20459e6	−0.455485
$473$	8.41124e6i	1.72865i
$474$	6.14080e6i	1.25539i
$475$	929164.i	0.188955i
$476$	323508.i	0.0654437i
$477$	−3.80538e6	−0.765776
$478$	−390541.	−0.0781803
$479$	− 1.51593e6i	− 0.301884i	−0.988543	−	0.150942i	$-0.951769\pi$
$479$	− 1.51593e6i	− 0.301884i	0.988543	−	0.150942i	$-0.0482307\pi$
$480$	1.03513e6	0.205064
$481$	0	0
$482$	−4.45398e6	−0.873234
$483$	1.01685e6i	0.198330i
$484$	−242178.	−0.0469918
$485$	4.91503e6	0.948793
$486$	3.60854e6i	0.693013i
$487$	− 602882.i	− 0.115189i	−0.998340	−	0.0575943i	$-0.981657\pi$
$487$	− 602882.i	− 0.115189i	0.998340	−	0.0575943i	$-0.0183430\pi$
$488$	− 339579.i	− 0.0645493i
$489$	4.37388e6i	0.827171i
$490$	−3.43530e6	−0.646359
$491$	6.06738e6	1.13579	0.567895	−	0.823101i	$-0.307758\pi$
$491$	6.06738e6	1.13579	0.567895	+	0.823101i	$0.307758\pi$
$492$	4.17548e6i	0.777667i
$493$	−128707.	−0.0238498
$494$	0	0
$495$	−2.62390e6	−0.481320
$496$	1.42258e6i	0.259641i
$497$	45767.3	0.00831122
$498$	−1.70997e6	−0.308970
$499$	294588.i	0.0529618i	0.999649	+	0.0264809i	$0.00843012\pi$
$499$	294588.i	0.0529618i	−0.999649	+	0.0264809i	$0.991570\pi$
$500$	− 2.94312e6i	− 0.526481i
$501$	− 3.12030e6i	− 0.555395i
$502$	4.86924e6i	0.862387i
$503$	−4.49541e6	−0.792227	−0.396113	−	0.918202i	$-0.629641\pi$
$503$	−4.49541e6	−0.792227	−0.396113	+	0.918202i	$0.629641\pi$
$504$	−160927.	−0.0282198
$505$	2.99495e6i	0.522590i
$506$	−4.19875e6	−0.729027
$507$	0	0
$508$	−933654.	−0.160519
$509$	953239.i	0.163082i	0.996670	+	0.0815412i	$0.0259842\pi$
$509$	953239.i	0.163082i	−0.996670	+	0.0815412i	$0.974016\pi$
$510$	4.27562e6	0.727903
$511$	917245.	0.155394
$512$	262144.i	0.0441942i
$513$	5.05712e6i	0.848418i
$514$	− 7.49412e6i	− 1.25116i
$515$	− 8.37116e6i	− 1.39081i
$516$	6.81800e6	1.12728
$517$	−1.33284e6	−0.219307
$518$	623086.i	0.102029i
$519$	−5.81881e6	−0.948235
$520$	0	0
$521$	3.66026e6	0.590769	0.295384	−	0.955378i	$-0.404552\pi$
$521$	3.66026e6	0.590769	0.295384	+	0.955378i	$0.404552\pi$
$522$	− 64024.5i	− 0.0102842i
$523$	−4.22181e6	−0.674908	−0.337454	−	0.941342i	$-0.609566\pi$
$523$	−4.22181e6	−0.674908	−0.337454	+	0.941342i	$0.609566\pi$
$524$	−470362.	−0.0748349
$525$	146700.i	0.0232291i
$526$	3.32449e6i	0.523915i
$527$	5.87602e6i	0.921631i
$528$	− 1.89242e6i	− 0.295414i
$529$	1.11493e6	0.173224
$530$	6.04635e6	0.934983
$531$	− 4.52980e6i	− 0.697177i
$532$	−717045.	−0.109842
$533$	0	0
$534$	9.62486e6	1.46063
$535$	943122.i	0.142457i
$536$	−3.64162e6	−0.547498
$537$	5.36064e6	0.802197
$538$	− 3.24351e6i	− 0.483125i
$539$	6.28041e6i	0.931142i
$540$	− 1.80336e6i	− 0.266132i
$541$	− 1.03047e7i	− 1.51371i	−0.653582	−	0.756856i	$-0.726734\pi$
$541$	− 1.03047e7i	− 1.51371i	0.653582	−	0.756856i	$-0.273266\pi$
$542$	−561744.	−0.0821373
$543$	−1.13216e7	−1.64782
$544$	1.08279e6i	0.156873i
$545$	1.35574e6	0.195518
$546$	0	0
$547$	−5.03080e6	−0.718901	−0.359450	−	0.933164i	$-0.617036\pi$
$547$	−5.03080e6	−0.718901	−0.359450	+	0.933164i	$0.617036\pi$
$548$	3.67878e6i	0.523302i
$549$	697737.	0.0988010
$550$	−605752.	−0.0853862
$551$	− 285275.i	− 0.0400299i
$552$	3.40343e6i	0.475411i
$553$	1.51690e6i	0.210933i
$554$	5.54431e6i	0.767491i
$555$	8.23497e6	1.13483
$556$	−2.70898e6	−0.371637
$557$	3.16081e6i	0.431678i	0.976429	+	0.215839i	$0.0692487\pi$
$557$	3.16081e6i	0.431678i	−0.976429	+	0.215839i	$0.930751\pi$
$558$	−2.92300e6	−0.397414
$559$	0	0
$560$	255697.	0.0344552
$561$	− 7.81668e6i	− 1.04861i
$562$	5.56180e6	0.742805
$563$	1.25890e7	1.67387	0.836933	−	0.547305i	$-0.184346\pi$
$563$	1.25890e7	1.67387	0.836933	+	0.547305i	$0.184346\pi$
$564$	1.08038e6i	0.143014i
$565$	− 5.72359e6i	− 0.754305i
$566$	− 3.65180e6i	− 0.479144i
$567$	1.40946e6i	0.184117i
$568$	153185.	0.0199226
$569$	1.10666e7	1.43296	0.716478	−	0.697609i	$-0.245753\pi$
$569$	1.10666e7	1.43296	0.716478	+	0.697609i	$0.245753\pi$
$570$	9.47677e6i	1.22172i
$571$	409116.	0.0525117	0.0262559	−	0.999655i	$-0.491642\pi$
$571$	409116.	0.0525117	0.0262559	+	0.999655i	$0.491642\pi$
$572$	0	0
$573$	1.25510e7	1.59696
$574$	1.03143e6i	0.130665i
$575$	1.08942e6	0.137412
$576$	−538630.	−0.0676448
$577$	1.63753e6i	0.204762i	0.994745	+	0.102381i	$0.0326461\pi$
$577$	1.63753e6i	0.204762i	−0.994745	+	0.102381i	$0.967354\pi$
$578$	− 1.20691e6i	− 0.150264i
$579$	− 1.27425e7i	− 1.57964i
$580$	101728.i	0.0125566i
$581$	−422398.	−0.0519136
$582$	−7.28361e6	−0.891331
$583$	− 1.10539e7i	− 1.34693i
$584$	3.07005e6	0.372489
$585$	0	0
$586$	−275222.	−0.0331085
$587$	1.12561e7i	1.34832i	0.738584	+	0.674161i	$0.235495\pi$
$587$	1.12561e7i	1.34832i	−0.738584	+	0.674161i	$0.764505\pi$
$588$	5.09079e6	0.607214
$589$	−1.30240e7	−1.54688
$590$	7.19739e6i	0.851226i
$591$	− 1.23718e7i	− 1.45701i
$592$	2.08549e6i	0.244571i
$593$	− 7.47990e6i	− 0.873492i	−0.899585	−	0.436746i	$-0.856131\pi$
$593$	− 7.47990e6i	− 0.873492i	0.899585	−	0.436746i	$-0.143869\pi$
$594$	−3.29690e6	−0.383389
$595$	1.05616e6	0.122303
$596$	− 7.88285e6i	− 0.909007i
$597$	1.59905e7	1.83622
$598$	0	0
$599$	7.02078e6	0.799499	0.399750	−	0.916624i	$-0.369097\pi$
$599$	7.02078e6	0.799499	0.399750	+	0.916624i	$0.369097\pi$
$600$	491011.i	0.0556818i
$601$	7.77451e6	0.877984	0.438992	−	0.898491i	$-0.355336\pi$
$601$	7.77451e6	0.877984	0.438992	+	0.898491i	$0.355336\pi$
$602$	1.68418e6	0.189408
$603$	− 7.48247e6i	− 0.838015i
$604$	581383.i	0.0648440i
$605$	790645.i	0.0878200i
$606$	− 4.43823e6i	− 0.490940i
$607$	1.46735e7	1.61644	0.808222	−	0.588878i	$-0.200430\pi$
$607$	1.46735e7	1.61644	0.808222	+	0.588878i	$0.200430\pi$
$608$	−2.39998e6	−0.263299
$609$	− 45040.3i	− 0.00492106i
$610$	−1.10863e6	−0.120632
$611$	0	0
$612$	−2.22483e6	−0.240114
$613$	− 1.35799e7i	− 1.45964i	−0.683639	−	0.729820i	$-0.739604\pi$
$613$	− 1.35799e7i	− 1.45964i	0.683639	−	0.729820i	$-0.260396\pi$
$614$	1.06402e7	1.13901
$615$	1.36318e7	1.45333
$616$	− 467465.i	− 0.0496360i
$617$	− 5.57160e6i	− 0.589206i	−0.955620	−	0.294603i	$-0.904813\pi$
$617$	− 5.57160e6i	− 0.589206i	0.955620	−	0.294603i	$-0.0951875\pi$
$618$	1.24053e7i	1.30658i
$619$	− 2.82758e6i	− 0.296611i	−0.988942	−	0.148306i	$-0.952618\pi$
$619$	− 2.82758e6i	− 0.296611i	0.988942	−	0.148306i	$-0.0473820\pi$
$620$	4.64434e6	0.485227
$621$	5.92933e6	0.616988
$622$	− 8.85257e6i	− 0.917473i
$623$	2.37753e6	0.245418
$624$	0	0
$625$	−8.36956e6	−0.857043
$626$	− 2.54008e6i	− 0.259066i
$627$	1.73254e7	1.76001
$628$	−5.89115e6	−0.596075
$629$	8.61420e6i	0.868137i
$630$	525383.i	0.0527381i
$631$	8.52815e6i	0.852671i	0.904565	+	0.426336i	$0.140196\pi$
$631$	8.52815e6i	0.852671i	−0.904565	+	0.426336i	$0.859804\pi$
$632$	5.07713e6i	0.505622i
$633$	−8.67512e6	−0.860531
$634$	−9.18273e6	−0.907295
$635$	3.04812e6i	0.299984i
$636$	−8.96012e6	−0.878357
$637$	0	0
$638$	185980.	0.0180890
$639$	314751.i	0.0304940i
$640$	855827.	0.0825917
$641$	−1.73936e7	−1.67203	−0.836016	−	0.548704i	$-0.815121\pi$
$641$	−1.73936e7	−1.67203	−0.836016	+	0.548704i	$0.815121\pi$
$642$	− 1.39762e6i	− 0.133829i
$643$	3.04461e6i	0.290405i	0.989402	+	0.145203i	$0.0463834\pi$
$643$	3.04461e6i	0.290405i	−0.989402	+	0.145203i	$0.953617\pi$
$644$	840716.i	0.0798793i
$645$	− 2.22589e7i	− 2.10671i
$646$	−9.91319e6	−0.934614
$647$	872410.	0.0819332	0.0409666	−	0.999161i	$-0.486956\pi$
$647$	872410.	0.0819332	0.0409666	+	0.999161i	$0.486956\pi$
$648$	4.71752e6i	0.441343i
$649$	1.31583e7	1.22627
$650$	0	0
$651$	−2.05629e6	−0.190165
$652$	3.61627e6i	0.333151i
$653$	−1.96438e7	−1.80278	−0.901391	−	0.433005i	$-0.857453\pi$
$653$	−1.96438e7	−1.80278	−0.901391	+	0.433005i	$0.857453\pi$
$654$	−2.00909e6	−0.183677
$655$	1.53560e6i	0.139854i
$656$	3.45223e6i	0.313213i
$657$	6.30807e6i	0.570142i
$658$	266875.i	0.0240294i
$659$	−9.62362e6	−0.863227	−0.431613	−	0.902059i	$-0.642056\pi$
$659$	−9.62362e6	−0.863227	−0.431613	+	0.902059i	$0.642056\pi$
$660$	−6.17821e6	−0.552081
$661$	− 1.59973e7i	− 1.42410i	−0.702126	−	0.712052i	$-0.747766\pi$
$661$	− 1.59973e7i	− 1.42410i	0.702126	−	0.712052i	$-0.252234\pi$
$662$	7.68526e6	0.681575
$663$	0	0
$664$	−1.41378e6	−0.124441
$665$	2.34095e6i	0.205276i
$666$	−4.28509e6	−0.374347
$667$	−334476.	−0.0291106
$668$	− 2.57982e6i	− 0.223691i
$669$	3.77520e6i	0.326118i
$670$	1.18889e7i	1.02318i
$671$	2.02680e6i	0.173782i
$672$	−378919.	−0.0323685
$673$	−5.09751e6	−0.433831	−0.216915	−	0.976190i	$-0.569600\pi$
$673$	−5.09751e6	−0.433831	−0.216915	+	0.976190i	$0.569600\pi$
$674$	2.29871e6i	0.194910i
$675$	855423.	0.0722639
$676$	0	0
$677$	2.72355e6	0.228383	0.114191	−	0.993459i	$-0.463572\pi$
$677$	2.72355e6	0.228383	0.114191	+	0.993459i	$0.463572\pi$
$678$	8.48181e6i	0.708622i
$679$	−1.79920e6	−0.149763
$680$	3.53502e6	0.293170
$681$	1.09055e7i	0.901113i
$682$	− 8.49078e6i	− 0.699015i
$683$	1.23503e7i	1.01304i	0.862230	+	0.506518i	$0.169068\pi$
$683$	1.23503e7i	1.01304i	−0.862230	+	0.506518i	$0.830932\pi$
$684$	− 4.93126e6i	− 0.403012i
$685$	1.20102e7	0.977965
$686$	2.54302e6	0.206319
$687$	− 529642.i	− 0.0428145i
$688$	5.63703e6	0.454025
$689$	0	0
$690$	1.11113e7	0.888465
$691$	6.72129e6i	0.535497i	0.963489	+	0.267749i	$0.0862797\pi$
$691$	6.72129e6i	0.535497i	−0.963489	+	0.267749i	$0.913720\pi$
$692$	−4.81091e6	−0.381911
$693$	960505.	0.0759742
$694$	1.05530e7i	0.831719i
$695$	8.84406e6i	0.694528i
$696$	− 150752.i	− 0.0117961i
$697$	1.42595e7i	1.11179i
$698$	4.98778e6	0.387497
$699$	−9.48018e6	−0.733878
$700$	121290.i	0.00935575i
$701$	1.81950e7	1.39848	0.699241	−	0.714886i	$-0.253521\pi$
$701$	1.81950e7	1.39848	0.699241	+	0.714886i	$0.253521\pi$
$702$	0	0
$703$	−1.90931e7	−1.45710
$704$	− 1.56462e6i	− 0.118981i
$705$	3.52713e6	0.267269
$706$	−6.03964e6	−0.456037
$707$	− 1.09633e6i	− 0.0824885i
$708$	− 1.06658e7i	− 0.799673i
$709$	1.74271e7i	1.30199i	0.759080	+	0.650997i	$0.225649\pi$
$709$	1.74271e7i	1.30199i	−0.759080	+	0.650997i	$0.774351\pi$
$710$	− 500107.i	− 0.0372320i
$711$	−1.04320e7	−0.773918
$712$	7.95770e6	0.588285
$713$	1.52703e7i	1.12493i
$714$	−1.56513e6	−0.114896
$715$	0	0
$716$	4.43211e6	0.323093
$717$	− 1.88944e6i	− 0.137257i
$718$	−2.30713e6	−0.167017
$719$	4.60218e6	0.332003	0.166001	−	0.986126i	$-0.446914\pi$
$719$	4.60218e6	0.332003	0.166001	+	0.986126i	$0.446914\pi$
$720$	1.75848e6i	0.126417i
$721$	3.06435e6i	0.219533i
$722$	− 1.20679e7i	− 0.861564i
$723$	− 2.15484e7i	− 1.53310i
$724$	−9.36057e6	−0.663676
$725$	−48254.8	−0.00340954
$726$	− 1.17166e6i	− 0.0825013i
$727$	2.47173e7	1.73446	0.867231	−	0.497905i	$-0.165897\pi$
$727$	2.47173e7	1.73446	0.867231	+	0.497905i	$0.165897\pi$
$728$	0	0
$729$	453661.	0.0316164
$730$	− 1.00229e7i	− 0.696122i
$731$	2.32839e7	1.61162
$732$	1.64289e6	0.113326
$733$	6.19827e6i	0.426099i	0.977041	+	0.213050i	$0.0683396\pi$
$733$	6.19827e6i	0.426099i	−0.977041	+	0.213050i	$0.931660\pi$
$734$	1.15216e7i	0.789353i
$735$	− 1.66200e7i	− 1.13478i
$736$	2.81391e6i	0.191476i
$737$	2.17352e7	1.47399
$738$	−7.09333e6	−0.479413
$739$	− 1.98238e7i	− 1.33529i	−0.744480	−	0.667644i	$-0.767303\pi$
$739$	− 1.98238e7i	− 1.33529i	0.744480	−	0.667644i	$-0.232697\pi$
$740$	6.80856e6	0.457063
$741$	0	0
$742$	−2.21333e6	−0.147583
$743$	1.81952e7i	1.20916i	0.796543	+	0.604581i	$0.206660\pi$
$743$	1.81952e7i	1.20916i	−0.796543	+	0.604581i	$0.793340\pi$
$744$	−6.88247e6	−0.455840
$745$	−2.57353e7	−1.69879
$746$	− 1.08569e7i	− 0.714264i
$747$	− 2.90491e6i	− 0.190472i
$748$	− 6.46272e6i	− 0.422339i
$749$	− 345240.i	− 0.0224862i
$750$	1.42388e7	0.924317
$751$	−1.52179e7	−0.984592	−0.492296	−	0.870428i	$-0.663842\pi$
$751$	−1.52179e7	−0.984592	−0.492296	+	0.870428i	$0.663842\pi$
$752$	893240.i	0.0576002i
$753$	−2.35575e7	−1.51405
$754$	0	0
$755$	1.89805e6	0.121183
$756$	660138.i	0.0420079i
$757$	−3.35941e6	−0.213071	−0.106535	−	0.994309i	$-0.533976\pi$
$757$	−3.35941e6	−0.213071	−0.106535	+	0.994309i	$0.533976\pi$
$758$	1.18983e7	0.752165
$759$	− 2.03136e7i	− 1.27992i
$760$	7.83526e6i	0.492062i
$761$	− 1.83089e7i	− 1.14605i	−0.819540	−	0.573023i	$-0.805771\pi$
$761$	− 1.83089e7i	− 1.14605i	0.819540	−	0.573023i	$-0.194229\pi$
$762$	− 4.51703e6i	− 0.281816i
$763$	−496285.	−0.0308617
$764$	1.03770e7	0.643190
$765$	7.26345e6i	0.448734i
$766$	1.62111e7	0.998250
$767$	0	0
$768$	−1.26826e6	−0.0775896
$769$	− 1.74442e7i	− 1.06374i	−0.846826	−	0.531869i	$-0.821490\pi$
$769$	− 1.74442e7i	− 1.06374i	0.846826	−	0.531869i	$-0.178510\pi$
$770$	−1.52614e6	−0.0927616
$771$	3.62566e7	2.19660
$772$	− 1.05353e7i	− 0.636215i
$773$	2.89508e7i	1.74265i	0.490702	+	0.871327i	$0.336740\pi$
$773$	2.89508e7i	1.74265i	−0.490702	+	0.871327i	$0.663260\pi$
$774$	1.15825e7i	0.694942i
$775$	2.20304e6i	0.131755i
$776$	−6.02199e6	−0.358993
$777$	−3.01450e6	−0.179128
$778$	− 1.92112e7i	− 1.13790i
$779$	−3.16058e7	−1.86605
$780$	0	0
$781$	−914295.	−0.0536363
$782$	1.16229e7i	0.679671i
$783$	−262634.	−0.0153090
$784$	4.20899e6	0.244561
$785$	1.92330e7i	1.11397i
$786$	− 2.27562e6i	− 0.131384i
$787$	− 5.63429e6i	− 0.324267i	−0.986769	−	0.162133i	$-0.948163\pi$
$787$	− 5.63429e6i	− 0.324267i	0.986769	−	0.162133i	$-0.0518375\pi$
$788$	− 1.02288e7i	− 0.586826i
$789$	−1.60839e7	−0.919813
$790$	1.65754e7	0.944924
$791$	2.09518e6i	0.119064i
$792$	3.21485e6	0.182116
$793$	0	0
$794$	−1.84544e7	−1.03884
$795$	2.92523e7i	1.64150i
$796$	1.32207e7	0.739557
$797$	2.76037e7	1.53929	0.769646	−	0.638471i	$-0.220433\pi$
$797$	2.76037e7	1.53929	0.769646	+	0.638471i	$0.220433\pi$
$798$	− 3.46907e6i	− 0.192844i
$799$	3.68956e6i	0.204459i
$800$	405961.i	0.0224264i
$801$	1.63508e7i	0.900444i
$802$	8.42955e6	0.462774
$803$	−1.83238e7	−1.00283
$804$	− 1.76182e7i	− 0.961217i
$805$	2.74470e6	0.149281
$806$	0	0
$807$	1.56921e7	0.848200
$808$	− 3.66947e6i	− 0.197731i
$809$	1.41851e7	0.762013	0.381007	−	0.924572i	$-0.375578\pi$
$809$	1.41851e7	0.762013	0.381007	+	0.924572i	$0.375578\pi$
$810$	1.54014e7	0.824797
$811$	− 2.59109e7i	− 1.38334i	−0.722212	−	0.691671i	$-0.756875\pi$
$811$	− 2.59109e7i	− 1.38334i	0.722212	−	0.691671i	$-0.243125\pi$
$812$	− 37238.7i	− 0.00198200i
$813$	− 2.71772e6i	− 0.144205i
$814$	− 1.24474e7i	− 0.658442i
$815$	1.18061e7	0.622605
$816$	−5.23857e6	−0.275415
$817$	5.16081e7i	2.70497i
$818$	−8.69355e6	−0.454270
$819$	0	0
$820$	1.12706e7	0.585344
$821$	9.04298e6i	0.468224i	0.972210	+	0.234112i	$0.0752183\pi$
$821$	9.04298e6i	0.468224i	−0.972210	+	0.234112i	$0.924782\pi$
$822$	−1.77980e7	−0.918736
$823$	−3.03689e7	−1.56289	−0.781446	−	0.623973i	$-0.785518\pi$
$823$	−3.03689e7	−1.56289	−0.781446	+	0.623973i	$0.785518\pi$
$824$	1.02565e7i	0.526237i
$825$	− 2.93063e6i	− 0.149909i
$826$	− 2.63468e6i	− 0.134362i
$827$	2.58958e7i	1.31664i	0.752740	+	0.658318i	$0.228732\pi$
$827$	2.58958e7i	1.31664i	−0.752740	+	0.658318i	$0.771268\pi$
$828$	−5.78177e6	−0.293079
$829$	1.38901e7	0.701969	0.350984	−	0.936381i	$-0.385847\pi$
$829$	1.38901e7	0.701969	0.350984	+	0.936381i	$0.385847\pi$
$830$	4.61561e6i	0.232559i
$831$	−2.68234e7	−1.34745
$832$	0	0
$833$	1.73854e7	0.868103
$834$	− 1.31061e7i	− 0.652465i
$835$	−8.42239e6	−0.418041
$836$	1.43244e7	0.708862
$837$	1.19904e7i	0.591589i
$838$	− 7.36826e6i	− 0.362456i
$839$	− 3.77438e6i	− 0.185115i	−0.995707	−	0.0925574i	$-0.970496\pi$
$839$	− 3.77438e6i	− 0.185115i	0.995707	−	0.0925574i	$-0.0295042\pi$
$840$	1.23706e6i	0.0604914i
$841$	−2.04963e7	−0.999278
$842$	9.09417e6	0.442062
$843$	2.69081e7i	1.30411i
$844$	−7.17247e6	−0.346587
$845$	0	0
$846$	−1.83535e6	−0.0881643
$847$	− 289424.i	− 0.0138620i
$848$	−7.40810e6	−0.353767
$849$	1.76675e7	0.841211
$850$	1.67684e6i	0.0796056i
$851$	2.23861e7i	1.05963i
$852$	741111.i	0.0349771i
$853$	2.15044e7i	1.01194i	0.862551	+	0.505970i	$0.168866\pi$
$853$	2.15044e7i	1.01194i	−0.862551	+	0.505970i	$0.831134\pi$
$854$	405827.	0.0190413
$855$	−1.60992e7	−0.753163
$856$	− 1.15553e6i	− 0.0539011i
$857$	3.57752e6	0.166391	0.0831954	−	0.996533i	$-0.473487\pi$
$857$	3.57752e6	0.166391	0.0831954	+	0.996533i	$0.473487\pi$
$858$	0	0
$859$	3.10648e7	1.43643	0.718217	−	0.695819i	$-0.244959\pi$
$859$	3.10648e7	1.43643	0.718217	+	0.695819i	$0.244959\pi$
$860$	− 1.84033e7i	− 0.848498i
$861$	−4.99006e6	−0.229402
$862$	1.43222e6	0.0656510
$863$	2.54355e6i	0.116255i	0.998309	+	0.0581277i	$0.0185131\pi$
$863$	2.54355e6i	0.116255i	−0.998309	+	0.0581277i	$0.981487\pi$
$864$	2.20951e6i	0.100696i
$865$	1.57063e7i	0.713729i
$866$	− 1.53990e7i	− 0.697747i
$867$	5.83906e6	0.263812
$868$	−1.70011e6	−0.0765910
$869$	− 3.03032e7i	− 1.36125i
$870$	−492163.	−0.0220450
$871$	0	0
$872$	−1.66108e6	−0.0739777
$873$	− 1.23734e7i	− 0.549484i
$874$	−2.57619e7	−1.14077
$875$	3.51728e6	0.155305
$876$	1.48530e7i	0.653962i
$877$	− 4.32299e7i	− 1.89795i	−0.315345	−	0.948977i	$-0.602120\pi$
$877$	− 4.32299e7i	− 1.89795i	0.315345	−	0.948977i	$-0.397880\pi$
$878$	− 2.03175e7i	− 0.889475i
$879$	− 1.33153e6i	− 0.0581270i
$880$	−5.10806e6	−0.222356
$881$	3.74303e6	0.162474	0.0812369	−	0.996695i	$-0.474113\pi$
$881$	3.74303e6	0.162474	0.0812369	+	0.996695i	$0.474113\pi$
$882$	8.64826e6i	0.374332i
$883$	−2.87001e7	−1.23874	−0.619371	−	0.785098i	$-0.712612\pi$
$883$	−2.87001e7	−1.23874	−0.619371	+	0.785098i	$0.712612\pi$
$884$	0	0
$885$	−3.48210e7	−1.49446
$886$	1.81435e7i	0.776492i
$887$	1.08628e7	0.463590	0.231795	−	0.972765i	$-0.425540\pi$
$887$	1.08628e7	0.463590	0.231795	+	0.972765i	$0.425540\pi$
$888$	−1.00896e7	−0.429382
$889$	− 1.11580e6i	− 0.0473512i
$890$	− 2.59797e7i	− 1.09941i
$891$	− 2.81568e7i	− 1.18820i
$892$	3.12128e6i	0.131347i
$893$	−8.17779e6	−0.343168
$894$	3.81373e7	1.59590
$895$	− 1.44696e7i	− 0.603808i
$896$	−313285.	−0.0130367
$897$	0	0
$898$	4.62249e6	0.191287
$899$	− 676384.i	− 0.0279122i
$900$	−834133.	−0.0343265
$901$	−3.05994e7	−1.25574
$902$	− 2.06049e7i	− 0.843244i
$903$	8.14810e6i	0.332535i
$904$	7.01265e6i	0.285405i
$905$	3.05597e7i	1.24030i
$906$	−2.81274e6	−0.113844
$907$	2.58463e7	1.04323	0.521615	−	0.853181i	$-0.325330\pi$
$907$	2.58463e7	1.04323	0.521615	+	0.853181i	$0.325330\pi$
$908$	9.01654e6i	0.362932i
$909$	7.53969e6	0.302652
$910$	0	0
$911$	−4.52408e7	−1.80607	−0.903034	−	0.429569i	$-0.858666\pi$
$911$	−4.52408e7	−1.80607	−0.903034	+	0.429569i	$0.858666\pi$
$912$	− 1.16111e7i	− 0.462261i
$913$	8.43825e6	0.335023
$914$	−6.14084e6	−0.243143
$915$	− 5.36358e6i	− 0.211788i
$916$	− 437901.i	− 0.0172440i
$917$	− 562123.i	− 0.0220754i
$918$	9.12645e6i	0.357433i
$919$	−1.19556e7	−0.466963	−0.233481	−	0.972361i	$-0.575012\pi$
$919$	−1.19556e7	−0.466963	−0.233481	+	0.972361i	$0.575012\pi$
$920$	9.18663e6	0.357838
$921$	5.14774e7i	1.99971i
$922$	2.56461e7	0.993559
$923$	0	0
$924$	2.26160e6	0.0871436
$925$	3.22964e6i	0.124108i
$926$	−1.47188e7	−0.564085
$927$	−2.10741e7	−0.805472
$928$	− 124639.i	− 0.00475101i
$929$	4.45911e7i	1.69515i	0.530673	+	0.847576i	$0.321939\pi$
$929$	4.45911e7i	1.69515i	−0.530673	+	0.847576i	$0.678061\pi$
$930$	2.24694e7i	0.851890i
$931$	3.85341e7i	1.45704i
$932$	−7.83808e6	−0.295577
$933$	4.28288e7	1.61076
$934$	− 2.22988e7i	− 0.836402i
$935$	−2.10990e7	−0.789283
$936$	0	0
$937$	−4.58884e7	−1.70747	−0.853737	−	0.520705i	$-0.825669\pi$
$937$	−4.58884e7	−1.70747	−0.853737	+	0.520705i	$0.825669\pi$
$938$	− 4.35205e6i	− 0.161505i
$939$	1.22889e7	0.454830
$940$	2.91618e6	0.107645
$941$	2.52187e7i	0.928430i	0.885723	+	0.464215i	$0.153663\pi$
$941$	2.52187e7i	0.928430i	−0.885723	+	0.464215i	$0.846337\pi$
$942$	− 2.85014e7i	− 1.04650i
$943$	3.70570e7i	1.35703i
$944$	− 8.81838e6i	− 0.322076i
$945$	2.15517e6	0.0785058
$946$	−3.36450e7	−1.22234
$947$	− 1.99538e7i	− 0.723022i	−0.932368	−	0.361511i	$-0.882261\pi$
$947$	− 1.99538e7i	− 0.723022i	0.932368	−	0.361511i	$-0.117739\pi$
$948$	−2.45632e7	−0.887696
$949$	0	0
$950$	−3.71666e6	−0.133611
$951$	− 4.44261e7i	− 1.59289i
$952$	−1.29403e6	−0.0462757
$953$	−4.03421e7	−1.43889	−0.719443	−	0.694552i	$-0.755603\pi$
$953$	−4.03421e7	−1.43889	−0.719443	+	0.694552i	$0.755603\pi$
$954$	− 1.52215e7i	− 0.541485i
$955$	− 3.38781e7i	− 1.20202i
$956$	− 1.56217e6i	− 0.0552818i
$957$	899771.i	0.0317579i
$958$	6.06372e6	0.213464
$959$	−4.39646e6	−0.154368
$960$	4.14050e6i	0.145002i
$961$	−2.25072e6	−0.0786163
$962$	0	0
$963$	2.37428e6	0.0825024
$964$	− 1.78159e7i	− 0.617470i
$965$	−3.43949e7	−1.18898
$966$	−4.06739e6	−0.140240
$967$	− 3.93424e7i	− 1.35299i	−0.736447	−	0.676496i	$-0.763498\pi$
$967$	− 3.93424e7i	− 1.35299i	0.736447	−	0.676496i	$-0.236502\pi$
$968$	− 968714.i	− 0.0332282i
$969$	− 4.79601e7i	− 1.64086i
$970$	1.96601e7i	0.670898i
$971$	2.55227e7	0.868719	0.434359	−	0.900740i	$-0.356975\pi$
$971$	2.55227e7	0.868719	0.434359	+	0.900740i	$0.356975\pi$
$972$	−1.44342e7	−0.490034
$973$	− 3.23746e6i	− 0.109628i
$974$	2.41153e6	0.0814507
$975$	0	0
$976$	1.35832e6	0.0456433
$977$	− 1.53803e7i	− 0.515500i	−0.966212	−	0.257750i	$-0.917019\pi$
$977$	− 1.53803e7i	− 0.515500i	0.966212	−	0.257750i	$-0.0829811\pi$
$978$	−1.74955e7	−0.584898
$979$	−4.74960e7	−1.58380
$980$	− 1.37412e7i	− 0.457045i
$981$	− 3.41305e6i	− 0.113232i
$982$	2.42695e7i	0.803124i
$983$	− 1.23631e7i	− 0.408077i	−0.978963	−	0.204038i	$-0.934593\pi$
$983$	− 1.23631e7i	− 0.408077i	0.978963	−	0.204038i	$-0.0654067\pi$
$984$	−1.67019e7	−0.549894
$985$	−3.33942e7	−1.09668
$986$	− 514827.i	− 0.0168643i
$987$	−1.29114e6	−0.0421873
$988$	0	0
$989$	6.05091e7	1.96712
$990$	− 1.04956e7i	− 0.340344i
$991$	−2.73319e6	−0.0884068	−0.0442034	−	0.999023i	$-0.514075\pi$
$991$	−2.73319e6	−0.0884068	−0.0442034	+	0.999023i	$0.514075\pi$
$992$	−5.69033e6	−0.183594
$993$	3.71814e7i	1.19661i
$994$	183069.i	0.00587692i
$995$	− 4.31619e7i	− 1.38211i
$996$	− 6.83989e6i	− 0.218475i
$997$	8.43065e6	0.268611	0.134305	−	0.990940i	$-0.457120\pi$
$997$	8.43065e6	0.268611	0.134305	+	0.990940i	$0.457120\pi$
$998$	−1.17835e6	−0.0374497
$999$	1.75778e7i	0.557251i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.6.b.g.337.8		12
13.5	odd	4		338.6.a.o.1.2		6
13.8	odd	4		338.6.a.m.1.2		6
13.9	even	3		26.6.e.a.23.6	yes	12
13.10	even	6		26.6.e.a.17.6	✓	12
13.12	even	2	inner	338.6.b.g.337.2		12
39.23	odd	6		234.6.l.c.199.3		12
39.35	odd	6		234.6.l.c.127.1		12
52.23	odd	6		208.6.w.c.17.2		12
52.35	odd	6		208.6.w.c.49.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.6.e.a.17.6	✓	12	13.10	even	6
26.6.e.a.23.6	yes	12	13.9	even	3
208.6.w.c.17.2		12	52.23	odd	6
208.6.w.c.49.2		12	52.35	odd	6
234.6.l.c.127.1		12	39.35	odd	6
234.6.l.c.199.3		12	39.23	odd	6
338.6.a.m.1.2		6	13.8	odd	4
338.6.a.o.1.2		6	13.5	odd	4
338.6.b.g.337.2		12	13.12	even	2	inner
338.6.b.g.337.8		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	338.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -19.3520 q^{3} -16.0000 q^{4} +52.2356i q^{5} -77.4082i q^{6} -19.1214i q^{7} -64.0000i q^{8} +131.501 q^{9} +O(q^{10})\)	\(q+4.00000i q^{2} -19.3520 q^{3} -16.0000 q^{4} +52.2356i q^{5} -77.4082i q^{6} -19.1214i q^{7} -64.0000i q^{8} +131.501 q^{9} -208.942 q^{10} +381.988i q^{11} +309.633 q^{12} +76.4855 q^{14} -1010.86i q^{15} +256.000 q^{16} +1057.42 q^{17} +526.006i q^{18} +2343.73i q^{19} -835.769i q^{20} +370.038i q^{21} -1527.95 q^{22} +2747.96 q^{23} +1238.53i q^{24} +396.447 q^{25} +2157.72 q^{27} +305.942i q^{28} -121.718 q^{29} +4043.46 q^{30} +5556.97i q^{31} +1024.00i q^{32} -7392.25i q^{33} +4229.66i q^{34} +998.816 q^{35} -2104.02 q^{36} +8146.46i q^{37} -9374.92 q^{38} +3343.08 q^{40} +13485.3i q^{41} -1480.15 q^{42} +22019.7 q^{43} -6111.81i q^{44} +6869.05i q^{45} +10991.8i q^{46} +3489.22i q^{47} -4954.12 q^{48} +16441.4 q^{49} +1585.79i q^{50} -20463.2 q^{51} -28937.9 q^{53} +8630.90i q^{54} -19953.4 q^{55} -1223.77 q^{56} -45355.9i q^{57} -486.873i q^{58} -34446.8i q^{59} +16173.8i q^{60} +5305.93 q^{61} -22227.9 q^{62} -2514.49i q^{63} -4096.00 q^{64} +29569.0 q^{66} -56900.3i q^{67} -16918.7 q^{68} -53178.6 q^{69} +3995.26i q^{70} +2393.52i q^{71} -8416.09i q^{72} +47969.6i q^{73} -32585.8 q^{74} -7672.05 q^{75} -37499.7i q^{76} +7304.14 q^{77} -79330.2 q^{79} +13372.3i q^{80} -73711.2 q^{81} -53941.1 q^{82} -22090.3i q^{83} -5920.60i q^{84} +55234.7i q^{85} +88078.6i q^{86} +2355.50 q^{87} +24447.2 q^{88} +124339. i q^{89} -27476.2 q^{90} -43967.3 q^{92} -107539. i q^{93} -13956.9 q^{94} -122426. q^{95} -19816.5i q^{96} -94093.5i q^{97} +65765.5i q^{98} +50232.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 192 q^{4} + 1412 q^{9}+O(q^{10}) \) 12 * q - 192 * q^4 + 1412 * q^9	\( 12 q - 192 q^{4} + 1412 q^{9} + 736 q^{10} - 2272 q^{14} + 3072 q^{16} + 3820 q^{17} - 4320 q^{22} - 11840 q^{23} - 4672 q^{25} - 2736 q^{27} - 15844 q^{29} + 23392 q^{30} - 33728 q^{35} - 22592 q^{36} + 26496 q^{38} - 11776 q^{40} - 47296 q^{42} + 83840 q^{43} - 9364 q^{49} + 133264 q^{51} - 187812 q^{53} - 34944 q^{55} + 36352 q^{56} - 55868 q^{61} + 8672 q^{62} - 49152 q^{64} + 6336 q^{66} - 61120 q^{68} + 154696 q^{69} - 192416 q^{74} - 384160 q^{75} - 151944 q^{77} + 30464 q^{79} + 144620 q^{81} + 174080 q^{82} + 243872 q^{87} + 69120 q^{88} + 282016 q^{90} + 189440 q^{92} - 316512 q^{94} - 465120 q^{95}+O(q^{100}) \) 12 * q - 192 * q^4 + 1412 * q^9 + 736 * q^10 - 2272 * q^14 + 3072 * q^16 + 3820 * q^17 - 4320 * q^22 - 11840 * q^23 - 4672 * q^25 - 2736 * q^27 - 15844 * q^29 + 23392 * q^30 - 33728 * q^35 - 22592 * q^36 + 26496 * q^38 - 11776 * q^40 - 47296 * q^42 + 83840 * q^43 - 9364 * q^49 + 133264 * q^51 - 187812 * q^53 - 34944 * q^55 + 36352 * q^56 - 55868 * q^61 + 8672 * q^62 - 49152 * q^64 + 6336 * q^66 - 61120 * q^68 + 154696 * q^69 - 192416 * q^74 - 384160 * q^75 - 151944 * q^77 + 30464 * q^79 + 144620 * q^81 + 174080 * q^82 + 243872 * q^87 + 69120 * q^88 + 282016 * q^90 + 189440 * q^92 - 316512 * q^94 - 465120 * q^95

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