LMFDB - Embedded newform 338.8.a.j.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [338,8,Mod(1,338)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("338.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 338 = 2 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	338.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,32,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$105.586138614$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6981x^{2} - 35424x + 7188480 $ x^4 - 6981x^2 - 35424x + 7188480
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$79.2712$ of defining polynomial
Character	$\chi$	$=$	338.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.00000 q^{2} +79.2712 q^{3} +64.0000 q^{4} -132.638 q^{5} +634.170 q^{6} -1508.92 q^{7} +512.000 q^{8} +4096.92 q^{9} -1061.10 q^{10} +2926.23 q^{11} +5073.36 q^{12} -12071.3 q^{14} -10514.4 q^{15} +4096.00 q^{16} -21738.9 q^{17} +32775.4 q^{18} -53478.1 q^{19} -8488.82 q^{20} -119614. q^{21} +23409.8 q^{22} -28447.9 q^{23} +40586.9 q^{24} -60532.2 q^{25} +151402. q^{27} -96570.7 q^{28} +152648. q^{29} -84114.8 q^{30} -83326.6 q^{31} +32768.0 q^{32} +231966. q^{33} -173911. q^{34} +200139. q^{35} +262203. q^{36} -112459. q^{37} -427825. q^{38} -67910.5 q^{40} -110130. q^{41} -956910. q^{42} +105935. q^{43} +187279. q^{44} -543407. q^{45} -227584. q^{46} -413100. q^{47} +324695. q^{48} +1.45329e6 q^{49} -484258. q^{50} -1.72327e6 q^{51} -889607. q^{53} +1.21122e6 q^{54} -388128. q^{55} -772566. q^{56} -4.23928e6 q^{57} +1.22119e6 q^{58} -2.41070e6 q^{59} -672919. q^{60} +2.38459e6 q^{61} -666613. q^{62} -6.18192e6 q^{63} +262144. q^{64} +1.85573e6 q^{66} +369763. q^{67} -1.39129e6 q^{68} -2.25510e6 q^{69} +1.60112e6 q^{70} +3.18718e6 q^{71} +2.09762e6 q^{72} -5.84239e6 q^{73} -899671. q^{74} -4.79846e6 q^{75} -3.42260e6 q^{76} -4.41544e6 q^{77} +1.07694e6 q^{79} -543284. q^{80} +3.04184e6 q^{81} -881042. q^{82} -647944. q^{83} -7.65528e6 q^{84} +2.88340e6 q^{85} +847478. q^{86} +1.21006e7 q^{87} +1.49823e6 q^{88} +1.16258e7 q^{89} -4.34725e6 q^{90} -1.82067e6 q^{92} -6.60540e6 q^{93} -3.30480e6 q^{94} +7.09322e6 q^{95} +2.59756e6 q^{96} -62366.7 q^{97} +1.16263e7 q^{98} +1.19885e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 32 q^{2} + 256 q^{4} - 278 q^{5} - 548 q^{7} + 2048 q^{8} + 5214 q^{9} - 2224 q^{10} - 7392 q^{11} - 4384 q^{14} + 15528 q^{15} + 16384 q^{16} - 28316 q^{17} + 41712 q^{18} - 99888 q^{19} - 17792 q^{20}+ \cdots + 10357656 q^{99}+O(q^{100}) $ 4 * q + 32 * q^2 + 256 * q^4 - 278 * q^5 - 548 * q^7 + 2048 * q^8 + 5214 * q^9 - 2224 * q^10 - 7392 * q^11 - 4384 * q^14 + 15528 * q^15 + 16384 * q^16 - 28316 * q^17 + 41712 * q^18 - 99888 * q^19 - 17792 * q^20 - 91074 * q^21 - 59136 * q^22 + 33388 * q^23 + 86878 * q^25 + 106272 * q^27 - 35072 * q^28 - 93140 * q^29 + 124224 * q^30 - 311160 * q^31 + 131072 * q^32 + 238638 * q^33 - 226528 * q^34 - 141544 * q^35 + 333696 * q^36 - 9636 * q^37 - 799104 * q^38 - 142336 * q^40 + 82892 * q^41 - 728592 * q^42 + 569264 * q^43 - 473088 * q^44 - 2303394 * q^45 + 267104 * q^46 + 574200 * q^47 + 717798 * q^49 + 695024 * q^50 - 2729928 * q^51 + 1235350 * q^53 + 850176 * q^54 + 1092512 * q^55 - 280576 * q^56 - 3528462 * q^57 - 745120 * q^58 + 231504 * q^59 + 993792 * q^60 - 685684 * q^61 - 2489280 * q^62 - 5951712 * q^63 + 1048576 * q^64 + 1909104 * q^66 + 3271056 * q^67 - 1812224 * q^68 - 5600034 * q^69 - 1132352 * q^70 - 175012 * q^71 + 2669568 * q^72 - 7137890 * q^73 - 77088 * q^74 - 22200960 * q^75 - 6392832 * q^76 - 13915206 * q^77 - 7053952 * q^79 - 1138688 * q^80 - 3758004 * q^81 + 663136 * q^82 - 657288 * q^83 - 5828736 * q^84 + 11814998 * q^85 + 4554112 * q^86 + 7182900 * q^87 - 3784704 * q^88 - 11452234 * q^89 - 18427152 * q^90 + 2136832 * q^92 + 2984688 * q^93 + 4593600 * q^94 + 23334088 * q^95 - 428002 * q^97 + 5742384 * q^98 + 10357656 * q^99

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 $

$2$	8.00000	0.707107
$2$	8.00000	0.707107
$3$	79.2712	1.69508	0.847542	−	0.530729i	$-0.178082\pi$
$3$	79.2712	1.69508	0.847542	+	0.530729i	$0.178082\pi$
$4$	64.0000	0.500000
$5$	−132.638	−0.474539	−0.237270	−	0.971444i	$-0.576252\pi$
$5$	−132.638	−0.474539	−0.237270	+	0.971444i	$0.576252\pi$
$6$	634.170	1.19860
$7$	−1508.92	−1.66273	−0.831366	−	0.555725i	$-0.812441\pi$
$7$	−1508.92	−1.66273	−0.831366	+	0.555725i	$0.812441\pi$
$8$	512.000	0.353553
$9$	4096.92	1.87331
$10$	−1061.10	−0.335550
$11$	2926.23	0.662879	0.331439	−	0.943477i	$-0.392466\pi$
$11$	2926.23	0.662879	0.331439	+	0.943477i	$0.392466\pi$
$12$	5073.36	0.847542
$13$	0	0
$13$	0	0
$14$	−12071.3	−1.17573
$15$	−10514.4	−0.804383
$16$	4096.00	0.250000
$17$	−21738.9	−1.07317	−0.536583	−	0.843848i	$-0.680285\pi$
$17$	−21738.9	−1.07317	−0.536583	+	0.843848i	$0.680285\pi$
$18$	32775.4	1.32463
$19$	−53478.1	−1.78870	−0.894352	−	0.447363i	$-0.852363\pi$
$19$	−53478.1	−1.78870	−0.894352	+	0.447363i	$0.852363\pi$
$20$	−8488.82	−0.237270
$21$	−119614.	−2.81847
$22$	23409.8	0.468726
$23$	−28447.9	−0.487532	−0.243766	−	0.969834i	$-0.578383\pi$
$23$	−28447.9	−0.487532	−0.243766	+	0.969834i	$0.578383\pi$
$24$	40586.9	0.599302
$25$	−60532.2	−0.774813
$26$	0	0
$27$	151402.	1.48033
$28$	−96570.7	−0.831366
$29$	152648.	1.16225	0.581124	−	0.813815i	$-0.302613\pi$
$29$	152648.	1.16225	0.581124	+	0.813815i	$0.302613\pi$
$30$	−84114.8	−0.568785
$31$	−83326.6	−0.502363	−0.251182	−	0.967940i	$-0.580819\pi$
$31$	−83326.6	−0.502363	−0.251182	+	0.967940i	$0.580819\pi$
$32$	32768.0	0.176777
$33$	231966.	1.12363
$34$	−173911.	−0.758843
$35$	200139.	0.789032
$36$	262203.	0.936654
$37$	−112459.	−0.364996	−0.182498	−	0.983206i	$-0.558418\pi$
$37$	−112459.	−0.364996	−0.182498	+	0.983206i	$0.558418\pi$
$38$	−427825.	−1.26481
$39$	0	0
$40$	−67910.5	−0.167775
$41$	−110130.	−0.249553	−0.124777	−	0.992185i	$-0.539821\pi$
$41$	−110130.	−0.249553	−0.124777	+	0.992185i	$0.539821\pi$
$42$	−956910.	−1.99296
$43$	105935.	0.203188	0.101594	−	0.994826i	$-0.467606\pi$
$43$	105935.	0.203188	0.101594	+	0.994826i	$0.467606\pi$
$44$	187279.	0.331439
$45$	−543407.	−0.888958
$46$	−227584.	−0.344737
$47$	−413100.	−0.580381	−0.290190	−	0.956969i	$-0.593719\pi$
$47$	−413100.	−0.580381	−0.290190	+	0.956969i	$0.593719\pi$
$48$	324695.	0.423771
$49$	1.45329e6	1.76468
$50$	−484258.	−0.547875
$51$	−1.72327e6	−1.81911
$52$	0	0
$53$	−889607.	−0.820791	−0.410395	−	0.911908i	$-0.634609\pi$
$53$	−889607.	−0.820791	−0.410395	+	0.911908i	$0.634609\pi$
$54$	1.21122e6	1.04675
$55$	−388128.	−0.314562
$56$	−772566.	−0.587865
$57$	−4.23928e6	−3.03200
$58$	1.22119e6	0.821834
$59$	−2.41070e6	−1.52813	−0.764066	−	0.645138i	$-0.776800\pi$
$59$	−2.41070e6	−1.52813	−0.764066	+	0.645138i	$0.776800\pi$
$60$	−672919.	−0.402192
$61$	2.38459e6	1.34511	0.672557	−	0.740045i	$-0.265196\pi$
$61$	2.38459e6	1.34511	0.672557	+	0.740045i	$0.265196\pi$
$62$	−666613.	−0.355224
$63$	−6.18192e6	−3.11481
$64$	262144.	0.125000
$65$	0	0
$66$	1.85573e6	0.794530
$67$	369763.	0.150197	0.0750986	−	0.997176i	$-0.476073\pi$
$67$	369763.	0.150197	0.0750986	+	0.997176i	$0.476073\pi$
$68$	−1.39129e6	−0.536583
$69$	−2.25510e6	−0.826408
$70$	1.60112e6	0.557930
$71$	3.18718e6	1.05682	0.528412	−	0.848988i	$-0.322788\pi$
$71$	3.18718e6	1.05682	0.528412	+	0.848988i	$0.322788\pi$
$72$	2.09762e6	0.662314
$73$	−5.84239e6	−1.75776	−0.878882	−	0.477038i	$-0.841710\pi$
$73$	−5.84239e6	−1.75776	−0.878882	+	0.477038i	$0.841710\pi$
$74$	−899671.	−0.258091
$75$	−4.79846e6	−1.31337
$76$	−3.42260e6	−0.894352
$77$	−4.41544e6	−1.10219
$78$	0	0
$79$	1.07694e6	0.245751	0.122876	−	0.992422i	$-0.460788\pi$
$79$	1.07694e6	0.245751	0.122876	+	0.992422i	$0.460788\pi$
$80$	−543284.	−0.118635
$81$	3.04184e6	0.635973
$82$	−881042.	−0.176461
$83$	−647944.	−0.124384	−0.0621920	−	0.998064i	$-0.519809\pi$
$83$	−647944.	−0.124384	−0.0621920	+	0.998064i	$0.519809\pi$
$84$	−7.65528e6	−1.40924
$85$	2.88340e6	0.509259
$86$	847478.	0.143676
$87$	1.21006e7	1.97011
$88$	1.49823e6	0.234363
$89$	1.16258e7	1.74807	0.874034	−	0.485865i	$-0.161495\pi$
$89$	1.16258e7	1.74807	0.874034	+	0.485865i	$0.161495\pi$
$90$	−4.34725e6	−0.628588
$91$	0	0
$92$	−1.82067e6	−0.243766
$93$	−6.60540e6	−0.851548
$94$	−3.30480e6	−0.410391
$95$	7.09322e6	0.848811
$96$	2.59756e6	0.299651
$97$	−62366.7	−0.00693828	−0.00346914	−	0.999994i	$-0.501104\pi$
$97$	−62366.7	−0.00693828	−0.00346914	+	0.999994i	$0.501104\pi$
$98$	1.16263e7	1.24782
$99$	1.19885e7	1.24178
$100$	−3.87406e6	−0.387406
$101$	−1.48059e7	−1.42991	−0.714955	−	0.699171i	$-0.753553\pi$
$101$	−1.48059e7	−1.42991	−0.714955	+	0.699171i	$0.753553\pi$
$102$	−1.37862e7	−1.28630
$103$	−1.24059e7	−1.11866	−0.559332	−	0.828944i	$-0.688942\pi$
$103$	−1.24059e7	−1.11866	−0.559332	+	0.828944i	$0.688942\pi$
$104$	0	0
$105$	1.58653e7	1.33747
$106$	−7.11685e6	−0.580387
$107$	569302.	0.0449262	0.0224631	−	0.999748i	$-0.492849\pi$
$107$	569302.	0.0449262	0.0224631	+	0.999748i	$0.492849\pi$
$108$	9.68972e6	0.740164
$109$	3.87174e6	0.286361	0.143181	−	0.989697i	$-0.454267\pi$
$109$	3.87174e6	0.286361	0.143181	+	0.989697i	$0.454267\pi$
$110$	−3.10503e6	−0.222429
$111$	−8.91475e6	−0.618698
$112$	−6.18053e6	−0.415683
$113$	−2.78061e7	−1.81287	−0.906433	−	0.422349i	$-0.861206\pi$
$113$	−2.78061e7	−1.81287	−0.906433	+	0.422349i	$0.861206\pi$
$114$	−3.39142e7	−2.14395
$115$	3.77327e6	0.231353
$116$	9.76949e6	0.581124
$117$	0	0
$118$	−1.92856e7	−1.08055
$119$	3.28022e7	1.78439
$120$	−5.38335e6	−0.284393
$121$	−1.09244e7	−0.560592
$122$	1.90767e7	0.951140
$123$	−8.73016e6	−0.423014
$124$	−5.33290e6	−0.251182
$125$	1.83912e7	0.842218
$126$	−4.94554e7	−2.20250
$127$	1.60372e7	0.694730	0.347365	−	0.937730i	$-0.387076\pi$
$127$	1.60372e7	0.694730	0.347365	+	0.937730i	$0.387076\pi$
$128$	2.09715e6	0.0883883
$129$	8.39757e6	0.344421
$130$	0	0
$131$	2.10431e7	0.817824	0.408912	−	0.912574i	$-0.365908\pi$
$131$	2.10431e7	0.817824	0.408912	+	0.912574i	$0.365908\pi$
$132$	1.48458e7	0.561817
$133$	8.06941e7	2.97414
$134$	2.95811e6	0.106206
$135$	−2.00816e7	−0.702474
$136$	−1.11303e7	−0.379421
$137$	−4.38736e7	−1.45775	−0.728873	−	0.684649i	$-0.759955\pi$
$137$	−4.38736e7	−1.45775	−0.728873	+	0.684649i	$0.759955\pi$
$138$	−1.80408e7	−0.584359
$139$	1.39014e7	0.439043	0.219521	−	0.975608i	$-0.429550\pi$
$139$	1.39014e7	0.439043	0.219521	+	0.975608i	$0.429550\pi$
$140$	1.28089e7	0.394516
$141$	−3.27470e7	−0.983794
$142$	2.54974e7	0.747287
$143$	0	0
$144$	1.67810e7	0.468327
$145$	−2.02469e7	−0.551532
$146$	−4.67391e7	−1.24293
$147$	1.15204e8	2.99128
$148$	−7.19737e6	−0.182498
$149$	−4.07753e7	−1.00982	−0.504912	−	0.863171i	$-0.668475\pi$
$149$	−4.07753e7	−1.00982	−0.504912	+	0.863171i	$0.668475\pi$
$150$	−3.83877e7	−0.928694
$151$	6.68855e7	1.58093	0.790465	−	0.612507i	$-0.209839\pi$
$151$	6.68855e7	1.58093	0.790465	+	0.612507i	$0.209839\pi$
$152$	−2.73808e7	−0.632403
$153$	−8.90627e7	−2.01037
$154$	−3.53235e7	−0.779366
$155$	1.10523e7	0.238391
$156$	0	0
$157$	7.74485e7	1.59722	0.798609	−	0.601850i	$-0.205569\pi$
$157$	7.74485e7	1.59722	0.798609	+	0.601850i	$0.205569\pi$
$158$	8.61550e6	0.173772
$159$	−7.05202e7	−1.39131
$160$	−4.34627e6	−0.0838875
$161$	4.29256e7	0.810636
$162$	2.43347e7	0.449701
$163$	3.07976e7	0.557006	0.278503	−	0.960435i	$-0.410162\pi$
$163$	3.07976e7	0.557006	0.278503	+	0.960435i	$0.410162\pi$
$164$	−7.04834e6	−0.124777
$165$	−3.07674e7	−0.533209
$166$	−5.18355e6	−0.0879527
$167$	−4.07252e7	−0.676637	−0.338318	−	0.941032i	$-0.609858\pi$
$167$	−4.07252e7	−0.676637	−0.338318	+	0.941032i	$0.609858\pi$
$168$	−6.12422e7	−0.996480
$169$	0	0
$170$	2.30672e7	0.360101
$171$	−2.19096e8	−3.35079
$172$	6.77982e6	0.101594
$173$	2.94018e7	0.431730	0.215865	−	0.976423i	$-0.430743\pi$
$173$	2.94018e7	0.431730	0.215865	+	0.976423i	$0.430743\pi$
$174$	9.68049e7	1.39308
$175$	9.13381e7	1.28831
$176$	1.19858e7	0.165720
$177$	−1.91099e8	−2.59031
$178$	9.30065e7	1.23607
$179$	4.50362e7	0.586917	0.293458	−	0.955972i	$-0.405194\pi$
$179$	4.50362e7	0.586917	0.293458	+	0.955972i	$0.405194\pi$
$180$	−3.47780e7	−0.444479
$181$	−7.76931e7	−0.973885	−0.486942	−	0.873434i	$-0.661888\pi$
$181$	−7.76931e7	−0.973885	−0.486942	+	0.873434i	$0.661888\pi$
$182$	0	0
$183$	1.89029e8	2.28008
$184$	−1.45653e7	−0.172369
$185$	1.49163e7	0.173205
$186$	−5.28432e7	−0.602135
$187$	−6.36131e7	−0.711379
$188$	−2.64384e7	−0.290190
$189$	−2.28453e8	−2.46139
$190$	5.67458e7	0.600200
$191$	1.65531e7	0.171895	0.0859476	−	0.996300i	$-0.472608\pi$
$191$	1.65531e7	0.171895	0.0859476	+	0.996300i	$0.472608\pi$
$192$	2.07805e7	0.211885
$193$	5.80414e7	0.581149	0.290574	−	0.956852i	$-0.406154\pi$
$193$	5.80414e7	0.581149	0.290574	+	0.956852i	$0.406154\pi$
$194$	−498934.	−0.00490611
$195$	0	0
$196$	9.30105e7	0.882340
$197$	−3.87910e7	−0.361492	−0.180746	−	0.983530i	$-0.557851\pi$
$197$	−3.87910e7	−0.361492	−0.180746	+	0.983530i	$0.557851\pi$
$198$	9.59083e7	0.878068
$199$	2.83672e7	0.255170	0.127585	−	0.991828i	$-0.459277\pi$
$199$	2.83672e7	0.255170	0.127585	+	0.991828i	$0.459277\pi$
$200$	−3.09925e7	−0.273938
$201$	2.93116e7	0.254597
$202$	−1.18447e8	−1.01110
$203$	−2.30334e8	−1.93251
$204$	−1.10289e8	−0.909553
$205$	1.46074e7	0.118423
$206$	−9.92476e7	−0.791015
$207$	−1.16549e8	−0.913298
$208$	0	0
$209$	−1.56489e8	−1.18569
$210$	1.26922e8	0.945737
$211$	−6.52934e7	−0.478499	−0.239249	−	0.970958i	$-0.576901\pi$
$211$	−6.52934e7	−0.478499	−0.239249	+	0.970958i	$0.576901\pi$
$212$	−5.69348e7	−0.410395
$213$	2.52652e8	1.79140
$214$	4.55441e6	0.0317676
$215$	−1.40509e7	−0.0964208
$216$	7.75178e7	0.523375
$217$	1.25733e8	0.835296
$218$	3.09740e7	0.202488
$219$	−4.63134e8	−2.97956
$220$	−2.48402e7	−0.157281
$221$	0	0
$222$	−7.13180e7	−0.437485
$223$	1.04880e8	0.633325	0.316663	−	0.948538i	$-0.397438\pi$
$223$	1.04880e8	0.633325	0.316663	+	0.948538i	$0.397438\pi$
$224$	−4.94442e7	−0.293932
$225$	−2.47996e8	−1.45146
$226$	−2.22449e8	−1.28189
$227$	−4.98470e7	−0.282845	−0.141423	−	0.989949i	$-0.545168\pi$
$227$	−4.98470e7	−0.282845	−0.141423	+	0.989949i	$0.545168\pi$
$228$	−2.71314e8	−1.51600
$229$	−1.15632e8	−0.636287	−0.318144	−	0.948043i	$-0.603059\pi$
$229$	−1.15632e8	−0.636287	−0.318144	+	0.948043i	$0.603059\pi$
$230$	3.01862e7	0.163591
$231$	−3.50017e8	−1.86830
$232$	7.81559e7	0.410917
$233$	7.04962e7	0.365107	0.182553	−	0.983196i	$-0.441564\pi$
$233$	7.04962e7	0.365107	0.182553	+	0.983196i	$0.441564\pi$
$234$	0	0
$235$	5.47927e7	0.275413
$236$	−1.54285e8	−0.764066
$237$	8.53701e7	0.416569
$238$	2.62418e8	1.26175
$239$	3.08705e8	1.46269	0.731343	−	0.682010i	$-0.238894\pi$
$239$	3.08705e8	1.46269	0.731343	+	0.682010i	$0.238894\pi$
$240$	−4.30668e7	−0.201096
$241$	−1.23764e8	−0.569554	−0.284777	−	0.958594i	$-0.591919\pi$
$241$	−1.23764e8	−0.569554	−0.284777	+	0.958594i	$0.591919\pi$
$242$	−8.73948e7	−0.396398
$243$	−8.99857e7	−0.402301
$244$	1.52614e8	0.672557
$245$	−1.92761e8	−0.837410
$246$	−6.98413e7	−0.299116
$247$	0	0
$248$	−4.26632e7	−0.177612
$249$	−5.13633e7	−0.210841
$250$	1.47129e8	0.595538
$251$	3.36476e8	1.34306	0.671531	−	0.740977i	$-0.265637\pi$
$251$	3.36476e8	1.34306	0.671531	+	0.740977i	$0.265637\pi$
$252$	−3.95643e8	−1.55740
$253$	−8.32452e7	−0.323175
$254$	1.28298e8	0.491249
$255$	2.28571e8	0.863237
$256$	1.67772e7	0.0625000
$257$	2.70499e8	0.994032	0.497016	−	0.867741i	$-0.334429\pi$
$257$	2.70499e8	0.994032	0.497016	+	0.867741i	$0.334429\pi$
$258$	6.71806e7	0.243543
$259$	1.69691e8	0.606890
$260$	0	0
$261$	6.25388e8	2.17725
$262$	1.68345e8	0.578289
$263$	−4.02987e8	−1.36598	−0.682992	−	0.730426i	$-0.739321\pi$
$263$	−4.02987e8	−1.36598	−0.682992	+	0.730426i	$0.739321\pi$
$264$	1.18766e8	0.397265
$265$	1.17995e8	0.389497
$266$	6.45553e8	2.10303
$267$	9.21592e8	2.96312
$268$	2.36649e7	0.0750986
$269$	−1.45527e8	−0.455839	−0.227920	−	0.973680i	$-0.573192\pi$
$269$	−1.45527e8	−0.455839	−0.227920	+	0.973680i	$0.573192\pi$
$270$	−1.60653e8	−0.496724
$271$	3.56078e8	1.08681	0.543403	−	0.839472i	$-0.317135\pi$
$271$	3.56078e8	1.08681	0.543403	+	0.839472i	$0.317135\pi$
$272$	−8.90426e7	−0.268291
$273$	0	0
$274$	−3.50989e8	−1.03078
$275$	−1.77131e8	−0.513607
$276$	−1.44327e8	−0.413204
$277$	4.86354e8	1.37491	0.687454	−	0.726228i	$-0.258728\pi$
$277$	4.86354e8	1.37491	0.687454	+	0.726228i	$0.258728\pi$
$278$	1.11211e8	0.310450
$279$	−3.41383e8	−0.941081
$280$	1.02471e8	0.278965
$281$	4.89344e7	0.131566	0.0657828	−	0.997834i	$-0.479046\pi$
$281$	4.89344e7	0.131566	0.0657828	+	0.997834i	$0.479046\pi$
$282$	−2.61976e8	−0.695647
$283$	−2.75406e8	−0.722306	−0.361153	−	0.932507i	$-0.617617\pi$
$283$	−2.75406e8	−0.722306	−0.361153	+	0.932507i	$0.617617\pi$
$284$	2.03980e8	0.528412
$285$	5.62288e8	1.43880
$286$	0	0
$287$	1.66178e8	0.414940
$288$	1.34248e8	0.331157
$289$	6.22422e7	0.151685
$290$	−1.61975e8	−0.389992
$291$	−4.94389e6	−0.0117610
$292$	−3.73913e8	−0.878882
$293$	3.06879e8	0.712738	0.356369	−	0.934345i	$-0.384015\pi$
$293$	3.06879e8	0.712738	0.356369	+	0.934345i	$0.384015\pi$
$294$	9.21632e8	2.11515
$295$	3.19750e8	0.725159
$296$	−5.75789e7	−0.129045
$297$	4.43037e8	0.981278
$298$	−3.26203e8	−0.714053
$299$	0	0
$300$	−3.07102e8	−0.656686
$301$	−1.59847e8	−0.337848
$302$	5.35084e8	1.11789
$303$	−1.17368e9	−2.42382
$304$	−2.19046e8	−0.447176
$305$	−3.16287e8	−0.638310
$306$	−7.12502e8	−1.42155
$307$	1.55157e8	0.306046	0.153023	−	0.988223i	$-0.451099\pi$
$307$	1.55157e8	0.306046	0.153023	+	0.988223i	$0.451099\pi$
$308$	−2.82588e8	−0.551095
$309$	−9.83434e8	−1.89623
$310$	8.84180e7	0.168568
$311$	−5.61165e8	−1.05786	−0.528931	−	0.848665i	$-0.677407\pi$
$311$	−5.61165e8	−1.05786	−0.528931	+	0.848665i	$0.677407\pi$
$312$	0	0
$313$	−3.88240e8	−0.715640	−0.357820	−	0.933791i	$-0.616480\pi$
$313$	−3.88240e8	−0.715640	−0.357820	+	0.933791i	$0.616480\pi$
$314$	6.19588e8	1.12940
$315$	8.19956e8	1.47810
$316$	6.89240e7	0.122876
$317$	−6.56778e7	−0.115801	−0.0579003	−	0.998322i	$-0.518441\pi$
$317$	−6.56778e7	−0.115801	−0.0579003	+	0.998322i	$0.518441\pi$
$318$	−5.64162e8	−0.983804
$319$	4.46684e8	0.770429
$320$	−3.47702e7	−0.0593174
$321$	4.51292e7	0.0761536
$322$	3.43405e8	0.573206
$323$	1.16256e9	1.91958
$324$	1.94678e8	0.317987
$325$	0	0
$326$	2.46381e8	0.393863
$327$	3.06918e8	0.485406
$328$	−5.63867e7	−0.0882304
$329$	6.23334e8	0.965018
$330$	−2.46139e8	−0.377035
$331$	−6.82173e8	−1.03394	−0.516971	−	0.856003i	$-0.672941\pi$
$331$	−6.82173e8	−1.03394	−0.516971	+	0.856003i	$0.672941\pi$
$332$	−4.14684e7	−0.0621920
$333$	−4.60735e8	−0.683749
$334$	−3.25802e8	−0.478454
$335$	−4.90446e7	−0.0712745
$336$	−4.89938e8	−0.704618
$337$	5.06834e8	0.721375	0.360687	−	0.932687i	$-0.382542\pi$
$337$	5.06834e8	0.721375	0.360687	+	0.932687i	$0.382542\pi$
$338$	0	0
$339$	−2.20422e9	−3.07296
$340$	1.84538e8	0.254630
$341$	−2.43833e8	−0.333006
$342$	−1.75277e9	−2.36937
$343$	−9.50235e8	−1.27146
$344$	5.42386e7	0.0718379
$345$	2.99112e8	0.392163
$346$	2.35214e8	0.305279
$347$	−4.84709e8	−0.622770	−0.311385	−	0.950284i	$-0.600793\pi$
$347$	−4.84709e8	−0.622770	−0.311385	+	0.950284i	$0.600793\pi$
$348$	7.74439e8	0.985054
$349$	−2.92464e8	−0.368284	−0.184142	−	0.982900i	$-0.558951\pi$
$349$	−2.92464e8	−0.368284	−0.184142	+	0.982900i	$0.558951\pi$
$350$	7.30705e8	0.910970
$351$	0	0
$352$	9.58867e7	0.117181
$353$	−7.90376e8	−0.956361	−0.478181	−	0.878261i	$-0.658704\pi$
$353$	−7.90376e8	−0.956361	−0.478181	+	0.878261i	$0.658704\pi$
$354$	−1.52879e9	−1.83163
$355$	−4.22740e8	−0.501504
$356$	7.44052e8	0.874034
$357$	2.60027e9	3.02469
$358$	3.60290e8	0.415013
$359$	1.31770e8	0.150309	0.0751544	−	0.997172i	$-0.476055\pi$
$359$	1.31770e8	0.150309	0.0751544	+	0.997172i	$0.476055\pi$
$360$	−2.78224e8	−0.314294
$361$	1.96604e9	2.19947
$362$	−6.21545e8	−0.688640
$363$	−8.65986e8	−0.950250
$364$	0	0
$365$	7.74922e8	0.834128
$366$	1.51223e9	1.61226
$367$	−8.73299e8	−0.922214	−0.461107	−	0.887345i	$-0.652548\pi$
$367$	−8.73299e8	−0.922214	−0.461107	+	0.887345i	$0.652548\pi$
$368$	−1.16523e8	−0.121883
$369$	−4.51195e8	−0.467490
$370$	1.19330e8	0.122474
$371$	1.34234e9	1.36476
$372$	−4.22746e8	−0.425774
$373$	1.26136e9	1.25851	0.629255	−	0.777199i	$-0.283360\pi$
$373$	1.26136e9	1.25851	0.629255	+	0.777199i	$0.283360\pi$
$374$	−5.08905e8	−0.503021
$375$	1.45789e9	1.42763
$376$	−2.11507e8	−0.205196
$377$	0	0
$378$	−1.82762e9	−1.74047
$379$	−4.71021e8	−0.444430	−0.222215	−	0.974998i	$-0.571329\pi$
$379$	−4.71021e8	−0.444430	−0.222215	+	0.974998i	$0.571329\pi$
$380$	4.53966e8	0.424405
$381$	1.27129e9	1.17763
$382$	1.32425e8	0.121548
$383$	−8.33063e8	−0.757674	−0.378837	−	0.925463i	$-0.623676\pi$
$383$	−8.33063e8	−0.757674	−0.378837	+	0.925463i	$0.623676\pi$
$384$	1.66244e8	0.149826
$385$	5.85654e8	0.523032
$386$	4.64331e8	0.410934
$387$	4.34006e8	0.380634
$388$	−3.99147e6	−0.00346914
$389$	−1.21290e8	−0.104472	−0.0522362	−	0.998635i	$-0.516635\pi$
$389$	−1.21290e8	−0.104472	−0.0522362	+	0.998635i	$0.516635\pi$
$390$	0	0
$391$	6.18428e8	0.523203
$392$	7.44084e8	0.623908
$393$	1.66811e9	1.38628
$394$	−3.10328e8	−0.255614
$395$	−1.42842e8	−0.116619
$396$	7.67266e8	0.620888
$397$	7.91465e8	0.634841	0.317420	−	0.948285i	$-0.397183\pi$
$397$	7.91465e8	0.634841	0.317420	+	0.948285i	$0.397183\pi$
$398$	2.26937e8	0.180433
$399$	6.39672e9	5.04141
$400$	−2.47940e8	−0.193703
$401$	1.51079e9	1.17003	0.585017	−	0.811021i	$-0.301088\pi$
$401$	1.51079e9	1.17003	0.585017	+	0.811021i	$0.301088\pi$
$402$	2.34493e8	0.180027
$403$	0	0
$404$	−9.47575e8	−0.714955
$405$	−4.03463e8	−0.301794
$406$	−1.84267e9	−1.36649
$407$	−3.29080e8	−0.241948
$408$	−8.82315e8	−0.643151
$409$	2.33328e9	1.68630	0.843151	−	0.537677i	$-0.180698\pi$
$409$	2.33328e9	1.68630	0.843151	+	0.537677i	$0.180698\pi$
$410$	1.16859e8	0.0837376
$411$	−3.47792e9	−2.47100
$412$	−7.93981e8	−0.559332
$413$	3.63755e9	2.54088
$414$	−9.32392e8	−0.645799
$415$	8.59418e7	0.0590250
$416$	0	0
$417$	1.10198e9	0.744214
$418$	−1.25191e9	−0.838412
$419$	1.01986e9	0.677313	0.338656	−	0.940910i	$-0.390028\pi$
$419$	1.01986e9	0.677313	0.338656	+	0.940910i	$0.390028\pi$
$420$	1.01538e9	0.668737
$421$	1.73659e9	1.13426	0.567128	−	0.823630i	$-0.308055\pi$
$421$	1.73659e9	1.13426	0.567128	+	0.823630i	$0.308055\pi$
$422$	−5.22347e8	−0.338350
$423$	−1.69244e9	−1.08723
$424$	−4.55479e8	−0.290193
$425$	1.31591e9	0.831502
$426$	2.02121e9	1.26671
$427$	−3.59815e9	−2.23657
$428$	3.64353e7	0.0224631
$429$	0	0
$430$	−1.12408e8	−0.0681798
$431$	2.40870e8	0.144915	0.0724573	−	0.997372i	$-0.476916\pi$
$431$	2.40870e8	0.144915	0.0724573	+	0.997372i	$0.476916\pi$
$432$	6.20142e8	0.370082
$433$	−2.21033e9	−1.30842	−0.654212	−	0.756311i	$-0.727000\pi$
$433$	−2.21033e9	−1.30842	−0.654212	+	0.756311i	$0.727000\pi$
$434$	1.00586e9	0.590643
$435$	−1.60500e9	−0.934893
$436$	2.47792e8	0.143181
$437$	1.52134e9	0.872052
$438$	−3.70507e9	−2.10687
$439$	−2.30612e9	−1.30094	−0.650468	−	0.759534i	$-0.725427\pi$
$439$	−2.30612e9	−1.30094	−0.650468	+	0.759534i	$0.725427\pi$
$440$	−1.98722e8	−0.111214
$441$	5.95401e9	3.30579
$442$	0	0
$443$	1.10170e9	0.602073	0.301036	−	0.953613i	$-0.402667\pi$
$443$	1.10170e9	0.602073	0.301036	+	0.953613i	$0.402667\pi$
$444$	−5.70544e8	−0.309349
$445$	−1.54202e9	−0.829527
$446$	8.39042e8	0.447829
$447$	−3.23231e9	−1.71173
$448$	−3.95554e8	−0.207842
$449$	−1.80705e9	−0.942126	−0.471063	−	0.882100i	$-0.656130\pi$
$449$	−1.80705e9	−0.942126	−0.471063	+	0.882100i	$0.656130\pi$
$450$	−1.98397e9	−1.02634
$451$	−3.22267e8	−0.165424
$452$	−1.77959e9	−0.906433
$453$	5.30209e9	2.67981
$454$	−3.98776e8	−0.200002
$455$	0	0
$456$	−2.17051e9	−1.07198
$457$	−1.80207e9	−0.883212	−0.441606	−	0.897209i	$-0.645591\pi$
$457$	−1.80207e9	−0.883212	−0.441606	+	0.897209i	$0.645591\pi$
$458$	−9.25055e8	−0.449923
$459$	−3.29131e9	−1.58864
$460$	2.41489e8	0.115677
$461$	7.81569e8	0.371547	0.185774	−	0.982593i	$-0.440521\pi$
$461$	7.81569e8	0.371547	0.185774	+	0.982593i	$0.440521\pi$
$462$	−2.80014e9	−1.32109
$463$	7.80510e8	0.365464	0.182732	−	0.983163i	$-0.441506\pi$
$463$	7.80510e8	0.365464	0.182732	+	0.983163i	$0.441506\pi$
$464$	6.25247e8	0.290562
$465$	8.76126e8	0.404093
$466$	5.63969e8	0.258169
$467$	−2.48897e9	−1.13087	−0.565433	−	0.824794i	$-0.691291\pi$
$467$	−2.48897e9	−1.13087	−0.565433	+	0.824794i	$0.691291\pi$
$468$	0	0
$469$	−5.57942e8	−0.249738
$470$	4.38342e8	0.194747
$471$	6.13944e9	2.70742
$472$	−1.23428e9	−0.540276
$473$	3.09989e8	0.134689
$474$	6.82961e8	0.294558
$475$	3.23715e9	1.38591
$476$	2.09934e9	0.892194
$477$	−3.64465e9	−1.53759
$478$	2.46964e9	1.03427
$479$	3.46691e9	1.44135	0.720673	−	0.693275i	$-0.243833\pi$
$479$	3.46691e9	1.44135	0.720673	+	0.693275i	$0.243833\pi$
$480$	−3.44534e8	−0.142196
$481$	0	0
$482$	−9.90112e8	−0.402735
$483$	3.40276e9	1.37410
$484$	−6.99159e8	−0.280296
$485$	8.27218e6	0.00329249
$486$	−7.19885e8	−0.284470
$487$	1.00101e9	0.392724	0.196362	−	0.980531i	$-0.437087\pi$
$487$	1.00101e9	0.392724	0.196362	+	0.980531i	$0.437087\pi$
$488$	1.22091e9	0.475570
$489$	2.44136e9	0.944172
$490$	−1.54209e9	−0.592138
$491$	−1.47873e8	−0.0563771	−0.0281885	−	0.999603i	$-0.508974\pi$
$491$	−1.47873e8	−0.0563771	−0.0281885	+	0.999603i	$0.508974\pi$
$492$	−5.58730e8	−0.211507
$493$	−3.31841e9	−1.24729
$494$	0	0
$495$	−1.59013e9	−0.589271
$496$	−3.41306e8	−0.125591
$497$	−4.80919e9	−1.75721
$498$	−4.10906e8	−0.149087
$499$	3.44602e9	1.24155	0.620777	−	0.783988i	$-0.286817\pi$
$499$	3.44602e9	1.24155	0.620777	+	0.783988i	$0.286817\pi$
$500$	1.17704e9	0.421109
$501$	−3.22834e9	−1.14696
$502$	2.69181e9	0.949688
$503$	−5.67817e9	−1.98939	−0.994696	−	0.102861i	$-0.967200\pi$
$503$	−5.67817e9	−1.98939	−0.994696	+	0.102861i	$0.967200\pi$
$504$	−3.16514e9	−1.10125
$505$	1.96382e9	0.678548
$506$	−6.65962e8	−0.228519
$507$	0	0
$508$	1.02638e9	0.347365
$509$	−3.65083e9	−1.22710	−0.613549	−	0.789656i	$-0.710259\pi$
$509$	−3.65083e9	−1.22710	−0.613549	+	0.789656i	$0.710259\pi$
$510$	1.82857e9	0.610401
$511$	8.81569e9	2.92269
$512$	1.34218e8	0.0441942
$513$	−8.09669e9	−2.64787
$514$	2.16400e9	0.702887
$515$	1.64550e9	0.530850
$516$	5.37445e8	0.172211
$517$	−1.20883e9	−0.384722
$518$	1.35753e9	0.429136
$519$	2.33071e9	0.731819
$520$	0	0
$521$	3.14626e9	0.974680	0.487340	−	0.873212i	$-0.337967\pi$
$521$	3.14626e9	0.974680	0.487340	+	0.873212i	$0.337967\pi$
$522$	5.00311e9	1.53955
$523$	−6.43264e9	−1.96623	−0.983113	−	0.183000i	$-0.941419\pi$
$523$	−6.43264e9	−1.96623	−0.983113	+	0.183000i	$0.941419\pi$
$524$	1.34676e9	0.408912
$525$	7.24048e9	2.18379
$526$	−3.22389e9	−0.965896
$527$	1.81143e9	0.539119
$528$	9.50132e8	0.280909
$529$	−2.59554e9	−0.762312
$530$	9.43963e8	0.275416
$531$	−9.87645e9	−2.86266
$532$	5.16442e9	1.48707
$533$	0	0
$534$	7.37274e9	2.09524
$535$	−7.55109e7	−0.0213192
$536$	1.89319e8	0.0531028
$537$	3.57008e9	0.994873
$538$	−1.16422e9	−0.322327
$539$	4.25266e9	1.16977
$540$	−1.28522e9	−0.351237
$541$	−1.42344e9	−0.386500	−0.193250	−	0.981150i	$-0.561903\pi$
$541$	−1.42344e9	−0.386500	−0.193250	+	0.981150i	$0.561903\pi$
$542$	2.84862e9	0.768488
$543$	−6.15883e9	−1.65082
$544$	−7.12341e8	−0.189711
$545$	−5.13539e8	−0.135890
$546$	0	0
$547$	−1.52035e9	−0.397181	−0.198591	−	0.980083i	$-0.563636\pi$
$547$	−1.52035e9	−0.397181	−0.198591	+	0.980083i	$0.563636\pi$
$548$	−2.80791e9	−0.728873
$549$	9.76948e9	2.51981
$550$	−1.41705e9	−0.363175
$551$	−8.16335e9	−2.07892
$552$	−1.15461e9	−0.292179
$553$	−1.62501e9	−0.408618
$554$	3.89084e9	0.972207
$555$	1.18243e9	0.293596
$556$	8.89689e8	0.219521
$557$	8.30213e8	0.203562	0.101781	−	0.994807i	$-0.467546\pi$
$557$	8.30213e8	0.203562	0.101781	+	0.994807i	$0.467546\pi$
$558$	−2.73106e9	−0.665445
$559$	0	0
$560$	8.19771e8	0.197258
$561$	−5.04269e9	−1.20585
$562$	3.91475e8	0.0930309
$563$	−3.23721e9	−0.764524	−0.382262	−	0.924054i	$-0.624855\pi$
$563$	−3.23721e9	−0.764524	−0.382262	+	0.924054i	$0.624855\pi$
$564$	−2.09581e9	−0.491897
$565$	3.68814e9	0.860276
$566$	−2.20325e9	−0.510747
$567$	−4.58989e9	−1.05745
$568$	1.63184e9	0.373644
$569$	−4.71974e9	−1.07405	−0.537026	−	0.843565i	$-0.680452\pi$
$569$	−4.71974e9	−1.07405	−0.537026	+	0.843565i	$0.680452\pi$
$570$	4.49830e9	1.01739
$571$	3.95218e9	0.888403	0.444201	−	0.895927i	$-0.353487\pi$
$571$	3.95218e9	0.888403	0.444201	+	0.895927i	$0.353487\pi$
$572$	0	0
$573$	1.31219e9	0.291377
$574$	1.32942e9	0.293407
$575$	1.72202e9	0.377746
$576$	1.07398e9	0.234163
$577$	2.12731e9	0.461016	0.230508	−	0.973070i	$-0.425961\pi$
$577$	2.12731e9	0.461016	0.230508	+	0.973070i	$0.425961\pi$
$578$	4.97938e8	0.107257
$579$	4.60101e9	0.985096
$580$	−1.29580e9	−0.275766
$581$	9.77694e8	0.206817
$582$	−3.95511e7	−0.00831626
$583$	−2.60319e9	−0.544085
$584$	−2.99131e9	−0.621464
$585$	0	0
$586$	2.45503e9	0.503982
$587$	5.30118e9	1.08178	0.540890	−	0.841093i	$-0.318087\pi$
$587$	5.30118e9	1.08178	0.540890	+	0.841093i	$0.318087\pi$
$588$	7.37305e9	1.49564
$589$	4.45615e9	0.898580
$590$	2.55800e9	0.512765
$591$	−3.07501e9	−0.612760
$592$	−4.60631e8	−0.0912489
$593$	−7.20568e9	−1.41900	−0.709501	−	0.704704i	$-0.751080\pi$
$593$	−7.20568e9	−1.41900	−0.709501	+	0.704704i	$0.751080\pi$
$594$	3.54429e9	0.693868
$595$	−4.35082e9	−0.846762
$596$	−2.60962e9	−0.504912
$597$	2.24870e9	0.432535
$598$	0	0
$599$	−1.75875e9	−0.334357	−0.167179	−	0.985927i	$-0.553466\pi$
$599$	−1.75875e9	−0.334357	−0.167179	+	0.985927i	$0.553466\pi$
$600$	−2.45681e9	−0.464347
$601$	−7.01266e9	−1.31772	−0.658858	−	0.752267i	$-0.728960\pi$
$601$	−7.01266e9	−1.31772	−0.658858	+	0.752267i	$0.728960\pi$
$602$	−1.27877e9	−0.238895
$603$	1.51489e9	0.281366
$604$	4.28067e9	0.790465
$605$	1.44898e9	0.266023
$606$	−9.38942e9	−1.71390
$607$	1.59340e9	0.289177	0.144588	−	0.989492i	$-0.453814\pi$
$607$	1.59340e9	0.289177	0.144588	+	0.989492i	$0.453814\pi$
$608$	−1.75237e9	−0.316201
$609$	−1.82588e10	−3.27576
$610$	−2.53029e9	−0.451353
$611$	0	0
$612$	−5.70001e9	−1.00518
$613$	1.08040e9	0.189441	0.0947205	−	0.995504i	$-0.469804\pi$
$613$	1.08040e9	0.189441	0.0947205	+	0.995504i	$0.469804\pi$
$614$	1.24125e9	0.216407
$615$	1.15795e9	0.200737
$616$	−2.26070e9	−0.389683
$617$	−6.51043e9	−1.11587	−0.557933	−	0.829886i	$-0.688405\pi$
$617$	−6.51043e9	−1.11587	−0.557933	+	0.829886i	$0.688405\pi$
$618$	−7.86747e9	−1.34084
$619$	2.46380e9	0.417531	0.208766	−	0.977966i	$-0.433055\pi$
$619$	2.46380e9	0.417531	0.208766	+	0.977966i	$0.433055\pi$
$620$	7.07344e8	0.119196
$621$	−4.30707e9	−0.721708
$622$	−4.48932e9	−0.748022
$623$	−1.75424e10	−2.90657
$624$	0	0
$625$	2.28972e9	0.375147
$626$	−3.10592e9	−0.506034
$627$	−1.24051e10	−2.00985
$628$	4.95671e9	0.798609
$629$	2.44473e9	0.391701
$630$	6.55965e9	1.04517
$631$	3.12429e9	0.495049	0.247524	−	0.968882i	$-0.420383\pi$
$631$	3.12429e9	0.495049	0.247524	+	0.968882i	$0.420383\pi$
$632$	5.51392e8	0.0868861
$633$	−5.17589e9	−0.811095
$634$	−5.25422e8	−0.0818834
$635$	−2.12714e9	−0.329677
$636$	−4.51329e9	−0.695654
$637$	0	0
$638$	3.57347e9	0.544776
$639$	1.30576e10	1.97976
$640$	−2.78162e8	−0.0419437
$641$	−8.30127e9	−1.24492	−0.622460	−	0.782651i	$-0.713867\pi$
$641$	−8.30127e9	−1.24492	−0.622460	+	0.782651i	$0.713867\pi$
$642$	3.61034e8	0.0538487
$643$	−7.75399e9	−1.15024	−0.575118	−	0.818071i	$-0.695044\pi$
$643$	−7.75399e9	−1.15024	−0.575118	+	0.818071i	$0.695044\pi$
$644$	2.74724e9	0.405318
$645$	−1.11384e9	−0.163441
$646$	9.30046e9	1.35735
$647$	9.75460e9	1.41594	0.707969	−	0.706243i	$-0.249612\pi$
$647$	9.75460e9	1.41594	0.707969	+	0.706243i	$0.249612\pi$
$648$	1.55742e9	0.224850
$649$	−7.05426e9	−1.01297
$650$	0	0
$651$	9.96701e9	1.41590
$652$	1.97105e9	0.278503
$653$	−2.80079e9	−0.393626	−0.196813	−	0.980441i	$-0.563059\pi$
$653$	−2.80079e9	−0.393626	−0.196813	+	0.980441i	$0.563059\pi$
$654$	2.45534e9	0.343234
$655$	−2.79111e9	−0.388090
$656$	−4.51094e8	−0.0623883
$657$	−2.39358e10	−3.29283
$658$	4.98668e9	0.682371
$659$	−9.29358e9	−1.26498	−0.632491	−	0.774568i	$-0.717967\pi$
$659$	−9.29358e9	−1.26498	−0.632491	+	0.774568i	$0.717967\pi$
$660$	−1.96911e9	−0.266604
$661$	−4.77913e9	−0.643641	−0.321820	−	0.946801i	$-0.604295\pi$
$661$	−4.77913e9	−0.643641	−0.321820	+	0.946801i	$0.604295\pi$
$662$	−5.45738e9	−0.731108
$663$	0	0
$664$	−3.31747e8	−0.0439763
$665$	−1.07031e10	−1.41134
$666$	−3.68588e9	−0.483483
$667$	−4.34253e9	−0.566634
$668$	−2.60641e9	−0.338318
$669$	8.31399e9	1.07354
$670$	−3.92357e8	−0.0503987
$671$	6.97786e9	0.891648
$672$	−3.91950e9	−0.498240
$673$	2.96698e9	0.375199	0.187599	−	0.982246i	$-0.439929\pi$
$673$	2.96698e9	0.375199	0.187599	+	0.982246i	$0.439929\pi$
$674$	4.05467e9	0.510089
$675$	−9.16469e9	−1.14698
$676$	0	0
$677$	−7.59327e8	−0.0940521	−0.0470260	−	0.998894i	$-0.514974\pi$
$677$	−7.59327e8	−0.0940521	−0.0470260	+	0.998894i	$0.514974\pi$
$678$	−1.76338e10	−2.17291
$679$	9.41063e7	0.0115365
$680$	1.47630e9	0.180050
$681$	−3.95143e9	−0.479446
$682$	−1.95066e9	−0.235471
$683$	6.60843e9	0.793644	0.396822	−	0.917896i	$-0.370113\pi$
$683$	6.60843e9	0.793644	0.396822	+	0.917896i	$0.370113\pi$
$684$	−1.40221e10	−1.67540
$685$	5.81930e9	0.691757
$686$	−7.60188e9	−0.899056
$687$	−9.16628e9	−1.07856
$688$	4.33909e8	0.0507971
$689$	0	0
$690$	2.39289e9	0.277301
$691$	−8.68411e9	−1.00127	−0.500636	−	0.865658i	$-0.666901\pi$
$691$	−8.68411e9	−1.00127	−0.500636	+	0.865658i	$0.666901\pi$
$692$	1.88171e9	0.215865
$693$	−1.80897e10	−2.06474
$694$	−3.87767e9	−0.440365
$695$	−1.84385e9	−0.208343
$696$	6.19551e9	0.696538
$697$	2.39411e9	0.267812
$698$	−2.33971e9	−0.260416
$699$	5.58831e9	0.618886
$700$	5.84564e9	0.644153
$701$	7.50161e9	0.822510	0.411255	−	0.911520i	$-0.365091\pi$
$701$	7.50161e9	0.822510	0.411255	+	0.911520i	$0.365091\pi$
$702$	0	0
$703$	6.01409e9	0.652869
$704$	7.67094e8	0.0828598
$705$	4.34348e9	0.466849
$706$	−6.32301e9	−0.676250
$707$	2.23408e10	2.37756
$708$	−1.22303e10	−1.29516
$709$	−7.21866e9	−0.760668	−0.380334	−	0.924849i	$-0.624191\pi$
$709$	−7.21866e9	−0.760668	−0.380334	+	0.924849i	$0.624191\pi$
$710$	−3.38192e9	−0.354617
$711$	4.41213e9	0.460367
$712$	5.95242e9	0.618035
$713$	2.37047e9	0.244918
$714$	2.08022e10	2.13878
$715$	0	0
$716$	2.88232e9	0.293458
$717$	2.44714e10	2.47937
$718$	1.05416e9	0.106284
$719$	1.47862e10	1.48356	0.741781	−	0.670642i	$-0.233981\pi$
$719$	1.47862e10	1.48356	0.741781	+	0.670642i	$0.233981\pi$
$720$	−2.22579e9	−0.222239
$721$	1.87196e10	1.86004
$722$	1.57283e10	1.55526
$723$	−9.81092e9	−0.965441
$724$	−4.97236e9	−0.486942
$725$	−9.24014e9	−0.900525
$726$	−6.92789e9	−0.671928
$727$	−5.23529e9	−0.505325	−0.252662	−	0.967554i	$-0.581306\pi$
$727$	−5.23529e9	−0.505325	−0.252662	+	0.967554i	$0.581306\pi$
$728$	0	0
$729$	−1.37858e10	−1.31791
$730$	6.19938e9	0.589818
$731$	−2.30291e9	−0.218055
$732$	1.20979e10	1.14004
$733$	−2.06458e10	−1.93628	−0.968138	−	0.250417i	$-0.919432\pi$
$733$	−2.06458e10	−1.93628	−0.968138	+	0.250417i	$0.919432\pi$
$734$	−6.98639e9	−0.652104
$735$	−1.52804e10	−1.41948
$736$	−9.32182e8	−0.0861844
$737$	1.08201e9	0.0995625
$738$	−3.60956e9	−0.330565
$739$	8.68404e9	0.791528	0.395764	−	0.918352i	$-0.370480\pi$
$739$	8.68404e9	0.791528	0.395764	+	0.918352i	$0.370480\pi$
$740$	9.54642e8	0.0866024
$741$	0	0
$742$	1.07387e10	0.965028
$743$	−1.02783e10	−0.919307	−0.459653	−	0.888098i	$-0.652026\pi$
$743$	−1.02783e10	−0.919307	−0.459653	+	0.888098i	$0.652026\pi$
$744$	−3.38197e9	−0.301068
$745$	5.40835e9	0.479201
$746$	1.00909e10	0.889901
$747$	−2.65458e9	−0.233009
$748$	−4.07124e9	−0.355689
$749$	−8.59030e8	−0.0747002
$750$	1.16631e10	1.00949
$751$	8.82545e9	0.760321	0.380161	−	0.924921i	$-0.375869\pi$
$751$	8.82545e9	0.760321	0.380161	+	0.924921i	$0.375869\pi$
$752$	−1.69206e9	−0.145095
$753$	2.66729e10	2.27660
$754$	0	0
$755$	−8.87154e9	−0.750213
$756$	−1.46210e10	−1.23070
$757$	1.62553e10	1.36194	0.680972	−	0.732309i	$-0.261557\pi$
$757$	1.62553e10	1.36194	0.680972	+	0.732309i	$0.261557\pi$
$758$	−3.76817e9	−0.314259
$759$	−6.59895e9	−0.547808
$760$	3.63173e9	0.300100
$761$	2.91445e8	0.0239723	0.0119862	−	0.999928i	$-0.496185\pi$
$761$	2.91445e8	0.0239723	0.0119862	+	0.999928i	$0.496185\pi$
$762$	1.01703e10	0.832707
$763$	−5.84214e9	−0.476142
$764$	1.05940e9	0.0859476
$765$	1.18131e10	0.953999
$766$	−6.66451e9	−0.535756
$767$	0	0
$768$	1.32995e9	0.105943
$769$	−4.44661e9	−0.352603	−0.176302	−	0.984336i	$-0.556413\pi$
$769$	−4.44661e9	−0.352603	−0.176302	+	0.984336i	$0.556413\pi$
$770$	4.68523e9	0.369840
$771$	2.14428e10	1.68497
$772$	3.71465e9	0.290574
$773$	−5.13452e9	−0.399827	−0.199913	−	0.979814i	$-0.564066\pi$
$773$	−5.13452e9	−0.399827	−0.199913	+	0.979814i	$0.564066\pi$
$774$	3.47205e9	0.269149
$775$	5.04395e9	0.389237
$776$	−3.19318e7	−0.00245305
$777$	1.34516e10	1.02873
$778$	−9.70321e8	−0.0738732
$779$	5.88956e9	0.446377
$780$	0	0
$781$	9.32642e9	0.700546
$782$	4.94742e9	0.369960
$783$	2.31112e10	1.72051
$784$	5.95267e9	0.441170
$785$	−1.02726e10	−0.757943
$786$	1.33449e10	0.980248
$787$	−2.14114e10	−1.56579	−0.782895	−	0.622153i	$-0.786258\pi$
$787$	−2.14114e10	−1.56579	−0.782895	+	0.622153i	$0.786258\pi$
$788$	−2.48262e9	−0.180746
$789$	−3.19452e10	−2.31546
$790$	−1.14274e9	−0.0824618
$791$	4.19571e10	3.01431
$792$	6.13813e9	0.439034
$793$	0	0
$794$	6.33172e9	0.448900
$795$	9.35364e9	0.660231
$796$	1.81550e9	0.127585
$797$	4.49892e9	0.314778	0.157389	−	0.987537i	$-0.449692\pi$
$797$	4.49892e9	0.314778	0.157389	+	0.987537i	$0.449692\pi$
$798$	5.11738e10	3.56482
$799$	8.98036e9	0.622845
$800$	−1.98352e9	−0.136969
$801$	4.76301e10	3.27467
$802$	1.20863e10	0.827338
$803$	−1.70962e10	−1.16518
$804$	1.87594e9	0.127298
$805$	−5.69356e9	−0.384679
$806$	0	0
$807$	−1.15361e10	−0.772685
$808$	−7.58060e9	−0.505550
$809$	−2.47066e10	−1.64056	−0.820282	−	0.571959i	$-0.806184\pi$
$809$	−2.47066e10	−1.64056	−0.820282	+	0.571959i	$0.806184\pi$
$810$	−3.22770e9	−0.213401
$811$	−1.66270e9	−0.109456	−0.0547282	−	0.998501i	$-0.517429\pi$
$811$	−1.66270e9	−0.109456	−0.0547282	+	0.998501i	$0.517429\pi$
$812$	−1.47414e10	−0.966254
$813$	2.82267e10	1.84223
$814$	−2.63264e9	−0.171083
$815$	−4.08492e9	−0.264321
$816$	−7.05852e9	−0.454776
$817$	−5.66519e9	−0.363444
$818$	1.86662e10	1.19240
$819$	0	0
$820$	9.34876e8	0.0592114
$821$	1.48863e10	0.938830	0.469415	−	0.882978i	$-0.344465\pi$
$821$	1.48863e10	0.938830	0.469415	+	0.882978i	$0.344465\pi$
$822$	−2.78233e10	−1.74726
$823$	−9.38235e9	−0.586695	−0.293347	−	0.956006i	$-0.594769\pi$
$823$	−9.38235e9	−0.586695	−0.293347	+	0.956006i	$0.594769\pi$
$824$	−6.35184e9	−0.395507
$825$	−1.40414e10	−0.870606
$826$	2.91004e10	1.79667
$827$	5.95599e9	0.366172	0.183086	−	0.983097i	$-0.441391\pi$
$827$	5.95599e9	0.366172	0.183086	+	0.983097i	$0.441391\pi$
$828$	−7.45914e9	−0.456649
$829$	2.05374e10	1.25200	0.626002	−	0.779822i	$-0.284690\pi$
$829$	2.05374e10	1.25200	0.626002	+	0.779822i	$0.284690\pi$
$830$	6.87535e8	0.0417370
$831$	3.85539e10	2.33058
$832$	0	0
$833$	−3.15929e10	−1.89379
$834$	8.81584e9	0.526239
$835$	5.40170e9	0.321091
$836$	−1.00153e10	−0.592847
$837$	−1.26158e10	−0.743663
$838$	8.15884e9	0.478933
$839$	1.37499e10	0.803769	0.401884	−	0.915690i	$-0.368355\pi$
$839$	1.37499e10	0.803769	0.401884	+	0.915690i	$0.368355\pi$
$840$	8.12303e9	0.472869
$841$	6.05162e9	0.350821
$842$	1.38927e10	0.802040
$843$	3.87909e9	0.223015
$844$	−4.17878e9	−0.239249
$845$	0	0
$846$	−1.35395e10	−0.768789
$847$	1.64839e10	0.932115
$848$	−3.64383e9	−0.205198
$849$	−2.18318e10	−1.22437
$850$	1.05272e10	0.587961
$851$	3.19922e9	0.177947
$852$	1.61697e10	0.895702
$853$	−9.49367e9	−0.523736	−0.261868	−	0.965104i	$-0.584339\pi$
$853$	−9.49367e9	−0.523736	−0.261868	+	0.965104i	$0.584339\pi$
$854$	−2.87852e10	−1.58149
$855$	2.90604e10	1.59008
$856$	2.91483e8	0.0158838
$857$	−1.03739e9	−0.0562998	−0.0281499	−	0.999604i	$-0.508962\pi$
$857$	−1.03739e9	−0.0562998	−0.0281499	+	0.999604i	$0.508962\pi$
$858$	0	0
$859$	−3.21586e10	−1.73110	−0.865549	−	0.500824i	$-0.833030\pi$
$859$	−3.21586e10	−1.73110	−0.865549	+	0.500824i	$0.833030\pi$
$860$	−8.99260e8	−0.0482104
$861$	1.31731e10	0.703358
$862$	1.92696e9	0.102470
$863$	1.58573e10	0.839828	0.419914	−	0.907564i	$-0.362060\pi$
$863$	1.58573e10	0.839828	0.419914	+	0.907564i	$0.362060\pi$
$864$	4.96114e9	0.261688
$865$	−3.89979e9	−0.204873
$866$	−1.76826e10	−0.925196
$867$	4.93401e9	0.257119
$868$	8.04691e9	0.417648
$869$	3.15136e9	0.162903
$870$	−1.28400e10	−0.661069
$871$	0	0
$872$	1.98233e9	0.101244
$873$	−2.55512e8	−0.0129975
$874$	1.21707e10	0.616634
$875$	−2.77508e10	−1.40038
$876$	−2.96405e10	−1.48978
$877$	2.88105e10	1.44229	0.721145	−	0.692784i	$-0.243616\pi$
$877$	2.88105e10	1.44229	0.721145	+	0.692784i	$0.243616\pi$
$878$	−1.84489e10	−0.919900
$879$	2.43266e10	1.20815
$880$	−1.58977e9	−0.0786405
$881$	−3.30491e10	−1.62833	−0.814167	−	0.580631i	$-0.802806\pi$
$881$	−3.30491e10	−1.62833	−0.814167	+	0.580631i	$0.802806\pi$
$882$	4.76321e10	2.33754
$883$	−1.91851e9	−0.0937783	−0.0468891	−	0.998900i	$-0.514931\pi$
$883$	−1.91851e9	−0.0937783	−0.0468891	+	0.998900i	$0.514931\pi$
$884$	0	0
$885$	2.53469e10	1.22920
$886$	8.81357e9	0.425730
$887$	2.25153e10	1.08329	0.541646	−	0.840606i	$-0.317801\pi$
$887$	2.25153e10	1.08329	0.541646	+	0.840606i	$0.317801\pi$
$888$	−4.56435e9	−0.218743
$889$	−2.41989e10	−1.15515
$890$	−1.23362e10	−0.586564
$891$	8.90112e9	0.421573
$892$	6.71234e9	0.316663
$893$	2.20918e10	1.03813
$894$	−2.58585e10	−1.21038
$895$	−5.97350e9	−0.278515
$896$	−3.16443e9	−0.146966
$897$	0	0
$898$	−1.44564e10	−0.666184
$899$	−1.27197e10	−0.583871
$900$	−1.58717e10	−0.725731
$901$	1.93391e10	0.880845
$902$	−2.57813e9	−0.116972
$903$	−1.26712e10	−0.572680
$904$	−1.42367e10	−0.640945
$905$	1.03050e10	0.462146
$906$	4.24167e10	1.89491
$907$	3.55964e10	1.58409	0.792047	−	0.610460i	$-0.209015\pi$
$907$	3.55964e10	1.58409	0.792047	+	0.610460i	$0.209015\pi$
$908$	−3.19021e9	−0.141423
$909$	−6.06585e10	−2.67866
$910$	0	0
$911$	−2.01254e10	−0.881923	−0.440962	−	0.897526i	$-0.645363\pi$
$911$	−2.01254e10	−0.881923	−0.440962	+	0.897526i	$0.645363\pi$
$912$	−1.73641e10	−0.758001
$913$	−1.89603e9	−0.0824514
$914$	−1.44166e10	−0.624525
$915$	−2.50724e10	−1.08199
$916$	−7.40044e9	−0.318144
$917$	−3.17523e10	−1.35982
$918$	−2.63305e10	−1.12334
$919$	1.00604e10	0.427573	0.213786	−	0.976880i	$-0.431420\pi$
$919$	1.00604e10	0.427573	0.213786	+	0.976880i	$0.431420\pi$
$920$	1.93191e9	0.0817957
$921$	1.22995e10	0.518773
$922$	6.25255e9	0.262724
$923$	0	0
$924$	−2.24011e10	−0.934152
$925$	6.80738e9	0.282803
$926$	6.24408e9	0.258422
$927$	−5.08262e10	−2.09560
$928$	5.00198e9	0.205458
$929$	−2.49287e10	−1.02010	−0.510052	−	0.860144i	$-0.670374\pi$
$929$	−2.49287e10	−1.02010	−0.510052	+	0.860144i	$0.670374\pi$
$930$	7.00900e9	0.285737
$931$	−7.77192e10	−3.15649
$932$	4.51175e9	0.182553
$933$	−4.44842e10	−1.79316
$934$	−1.99118e10	−0.799643
$935$	8.43750e9	0.337577
$936$	0	0
$937$	3.64214e9	0.144633	0.0723166	−	0.997382i	$-0.476961\pi$
$937$	3.64214e9	0.144633	0.0723166	+	0.997382i	$0.476961\pi$
$938$	−4.46354e9	−0.176591
$939$	−3.07762e10	−1.21307
$940$	3.50673e9	0.137707
$941$	3.92765e10	1.53663	0.768314	−	0.640073i	$-0.221096\pi$
$941$	3.92765e10	1.53663	0.768314	+	0.640073i	$0.221096\pi$
$942$	4.91155e10	1.91443
$943$	3.13298e9	0.121665
$944$	−9.87422e9	−0.382033
$945$	3.03015e10	1.16803
$946$	2.47991e9	0.0952396
$947$	−8.03964e9	−0.307618	−0.153809	−	0.988101i	$-0.549154\pi$
$947$	−8.03964e9	−0.307618	−0.153809	+	0.988101i	$0.549154\pi$
$948$	5.46369e9	0.208284
$949$	0	0
$950$	2.58972e10	0.979987
$951$	−5.20636e9	−0.196292
$952$	1.67948e10	0.630876
$953$	−1.55196e10	−0.580839	−0.290420	−	0.956899i	$-0.593795\pi$
$953$	−1.55196e10	−0.580839	−0.290420	+	0.956899i	$0.593795\pi$
$954$	−2.91572e10	−1.08724
$955$	−2.19557e9	−0.0815710
$956$	1.97571e10	0.731343
$957$	3.54092e10	1.30594
$958$	2.77353e10	1.01919
$959$	6.62017e10	2.42384
$960$	−2.75627e9	−0.100548
$961$	−2.05693e10	−0.747631
$962$	0	0
$963$	2.33239e9	0.0841605
$964$	−7.92089e9	−0.284777
$965$	−7.69848e9	−0.275778
$966$	2.72221e10	0.971632
$967$	1.65901e10	0.590006	0.295003	−	0.955496i	$-0.404679\pi$
$967$	1.65901e10	0.590006	0.295003	+	0.955496i	$0.404679\pi$
$968$	−5.59327e9	−0.198199
$969$	9.21573e10	3.25384
$970$	6.61775e7	0.00232814
$971$	5.50167e9	0.192854	0.0964268	−	0.995340i	$-0.469259\pi$
$971$	5.50167e9	0.192854	0.0964268	+	0.995340i	$0.469259\pi$
$972$	−5.75908e9	−0.201151
$973$	−2.09761e10	−0.730010
$974$	8.00809e9	0.277698
$975$	0	0
$976$	9.76728e9	0.336279
$977$	2.72406e10	0.934513	0.467256	−	0.884122i	$-0.345243\pi$
$977$	2.72406e10	0.934513	0.467256	+	0.884122i	$0.345243\pi$
$978$	1.95309e10	0.667630
$979$	3.40198e10	1.15876
$980$	−1.23367e10	−0.418705
$981$	1.58622e10	0.536442
$982$	−1.18298e9	−0.0398646
$983$	−3.12589e10	−1.04963	−0.524816	−	0.851216i	$-0.675866\pi$
$983$	−3.12589e10	−1.04963	−0.524816	+	0.851216i	$0.675866\pi$
$984$	−4.46984e9	−0.149558
$985$	5.14515e9	0.171542
$986$	−2.65473e10	−0.881964
$987$	4.94125e10	1.63579
$988$	0	0
$989$	−3.01363e9	−0.0990609
$990$	−1.27211e10	−0.416678
$991$	−5.40942e10	−1.76560	−0.882802	−	0.469746i	$-0.844346\pi$
$991$	−5.40942e10	−1.76560	−0.882802	+	0.469746i	$0.844346\pi$
$992$	−2.73045e9	−0.0888061
$993$	−5.40767e10	−1.75262
$994$	−3.84735e10	−1.24254
$995$	−3.76256e9	−0.121088
$996$	−3.28725e9	−0.105421
$997$	3.54238e10	1.13204	0.566021	−	0.824391i	$-0.308482\pi$
$997$	3.54238e10	1.13204	0.566021	+	0.824391i	$0.308482\pi$
$998$	2.75681e10	0.877911
$999$	−1.70265e10	−0.540313

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.a.j.1.4		4
13.4	even	6		26.8.c.b.3.1	✓	8
13.5	odd	4		338.8.b.h.337.4		8
13.8	odd	4		338.8.b.h.337.8		8
13.10	even	6		26.8.c.b.9.1	yes	8
13.12	even	2		338.8.a.i.1.4		4
39.17	odd	6		234.8.h.b.55.2		8
39.23	odd	6		234.8.h.b.217.2		8
52.23	odd	6		208.8.i.b.113.4		8
52.43	odd	6		208.8.i.b.81.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.c.b.3.1	✓	8	13.4	even	6
26.8.c.b.9.1	yes	8	13.10	even	6
208.8.i.b.81.4		8	52.43	odd	6
208.8.i.b.113.4		8	52.23	odd	6
234.8.h.b.55.2		8	39.17	odd	6
234.8.h.b.217.2		8	39.23	odd	6
338.8.a.i.1.4		4	13.12	even	2
338.8.a.j.1.4		4	1.1	even	1	trivial
338.8.b.h.337.4		8	13.5	odd	4
338.8.b.h.337.8		8	13.8	odd	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:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.a (trivial)

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +79.2712 q^{3} +64.0000 q^{4} -132.638 q^{5} +634.170 q^{6} -1508.92 q^{7} +512.000 q^{8} +4096.92 q^{9} -1061.10 q^{10} +2926.23 q^{11} +5073.36 q^{12} -12071.3 q^{14} -10514.4 q^{15} +4096.00 q^{16} -21738.9 q^{17} +32775.4 q^{18} -53478.1 q^{19} -8488.82 q^{20} -119614. q^{21} +23409.8 q^{22} -28447.9 q^{23} +40586.9 q^{24} -60532.2 q^{25} +151402. q^{27} -96570.7 q^{28} +152648. q^{29} -84114.8 q^{30} -83326.6 q^{31} +32768.0 q^{32} +231966. q^{33} -173911. q^{34} +200139. q^{35} +262203. q^{36} -112459. q^{37} -427825. q^{38} -67910.5 q^{40} -110130. q^{41} -956910. q^{42} +105935. q^{43} +187279. q^{44} -543407. q^{45} -227584. q^{46} -413100. q^{47} +324695. q^{48} +1.45329e6 q^{49} -484258. q^{50} -1.72327e6 q^{51} -889607. q^{53} +1.21122e6 q^{54} -388128. q^{55} -772566. q^{56} -4.23928e6 q^{57} +1.22119e6 q^{58} -2.41070e6 q^{59} -672919. q^{60} +2.38459e6 q^{61} -666613. q^{62} -6.18192e6 q^{63} +262144. q^{64} +1.85573e6 q^{66} +369763. q^{67} -1.39129e6 q^{68} -2.25510e6 q^{69} +1.60112e6 q^{70} +3.18718e6 q^{71} +2.09762e6 q^{72} -5.84239e6 q^{73} -899671. q^{74} -4.79846e6 q^{75} -3.42260e6 q^{76} -4.41544e6 q^{77} +1.07694e6 q^{79} -543284. q^{80} +3.04184e6 q^{81} -881042. q^{82} -647944. q^{83} -7.65528e6 q^{84} +2.88340e6 q^{85} +847478. q^{86} +1.21006e7 q^{87} +1.49823e6 q^{88} +1.16258e7 q^{89} -4.34725e6 q^{90} -1.82067e6 q^{92} -6.60540e6 q^{93} -3.30480e6 q^{94} +7.09322e6 q^{95} +2.59756e6 q^{96} -62366.7 q^{97} +1.16263e7 q^{98} +1.19885e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 32 q^{2} + 256 q^{4} - 278 q^{5} - 548 q^{7} + 2048 q^{8} + 5214 q^{9} - 2224 q^{10} - 7392 q^{11} - 4384 q^{14} + 15528 q^{15} + 16384 q^{16} - 28316 q^{17} + 41712 q^{18} - 99888 q^{19} - 17792 q^{20}+ \cdots + 10357656 q^{99}+O(q^{100}) \) 4 * q + 32 * q^2 + 256 * q^4 - 278 * q^5 - 548 * q^7 + 2048 * q^8 + 5214 * q^9 - 2224 * q^10 - 7392 * q^11 - 4384 * q^14 + 15528 * q^15 + 16384 * q^16 - 28316 * q^17 + 41712 * q^18 - 99888 * q^19 - 17792 * q^20 - 91074 * q^21 - 59136 * q^22 + 33388 * q^23 + 86878 * q^25 + 106272 * q^27 - 35072 * q^28 - 93140 * q^29 + 124224 * q^30 - 311160 * q^31 + 131072 * q^32 + 238638 * q^33 - 226528 * q^34 - 141544 * q^35 + 333696 * q^36 - 9636 * q^37 - 799104 * q^38 - 142336 * q^40 + 82892 * q^41 - 728592 * q^42 + 569264 * q^43 - 473088 * q^44 - 2303394 * q^45 + 267104 * q^46 + 574200 * q^47 + 717798 * q^49 + 695024 * q^50 - 2729928 * q^51 + 1235350 * q^53 + 850176 * q^54 + 1092512 * q^55 - 280576 * q^56 - 3528462 * q^57 - 745120 * q^58 + 231504 * q^59 + 993792 * q^60 - 685684 * q^61 - 2489280 * q^62 - 5951712 * q^63 + 1048576 * q^64 + 1909104 * q^66 + 3271056 * q^67 - 1812224 * q^68 - 5600034 * q^69 - 1132352 * q^70 - 175012 * q^71 + 2669568 * q^72 - 7137890 * q^73 - 77088 * q^74 - 22200960 * q^75 - 6392832 * q^76 - 13915206 * q^77 - 7053952 * q^79 - 1138688 * q^80 - 3758004 * q^81 + 663136 * q^82 - 657288 * q^83 - 5828736 * q^84 + 11814998 * q^85 + 4554112 * q^86 + 7182900 * q^87 - 3784704 * q^88 - 11452234 * q^89 - 18427152 * q^90 + 2136832 * q^92 + 2984688 * q^93 + 4593600 * q^94 + 23334088 * q^95 - 428002 * q^97 + 5742384 * q^98 + 10357656 * q^99

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