LMFDB - Embedded newform 338.8.a.l.1.3

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 = [5,40,27] 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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8883x^{3} + 76436x^{2} + 9553604x - 121499328 $ x^5 - 2x^4 - 8883x^3 + 76436x^2 + 9553604x - 121499328
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$13.5100$ 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)

$f(q)$	$=$	\(q+8.00000 q^{2} +18.5100 q^{3} +64.0000 q^{4} -6.61448 q^{5} +148.080 q^{6} -1604.67 q^{7} +512.000 q^{8} -1844.38 q^{9} -52.9158 q^{10} +1173.55 q^{11} +1184.64 q^{12} -12837.3 q^{14} -122.434 q^{15} +4096.00 q^{16} +28053.1 q^{17} -14755.1 q^{18} +13825.5 q^{19} -423.327 q^{20} -29702.3 q^{21} +9388.37 q^{22} -94424.9 q^{23} +9477.10 q^{24} -78081.2 q^{25} -74620.7 q^{27} -102699. q^{28} +159421. q^{29} -979.470 q^{30} +113805. q^{31} +32768.0 q^{32} +21722.3 q^{33} +224425. q^{34} +10614.0 q^{35} -118040. q^{36} +402800. q^{37} +110604. q^{38} -3386.61 q^{40} +583062. q^{41} -237618. q^{42} -88507.4 q^{43} +75107.0 q^{44} +12199.6 q^{45} -755399. q^{46} +112810. q^{47} +75816.8 q^{48} +1.75141e6 q^{49} -624650. q^{50} +519262. q^{51} +548744. q^{53} -596966. q^{54} -7762.40 q^{55} -821589. q^{56} +255910. q^{57} +1.27536e6 q^{58} +1.69495e6 q^{59} -7835.76 q^{60} +1.33588e6 q^{61} +910442. q^{62} +2.95961e6 q^{63} +262144. q^{64} +173778. q^{66} -719239. q^{67} +1.79540e6 q^{68} -1.74780e6 q^{69} +84912.2 q^{70} +1.85217e6 q^{71} -944323. q^{72} -1.50153e6 q^{73} +3.22240e6 q^{74} -1.44528e6 q^{75} +884833. q^{76} -1.88315e6 q^{77} -6.07672e6 q^{79} -27092.9 q^{80} +2.65244e6 q^{81} +4.66450e6 q^{82} -661334. q^{83} -1.90095e6 q^{84} -185557. q^{85} -708059. q^{86} +2.95087e6 q^{87} +600856. q^{88} -1.03091e7 q^{89} +97597.0 q^{90} -6.04319e6 q^{92} +2.10653e6 q^{93} +902482. q^{94} -91448.6 q^{95} +606534. q^{96} +6.14220e6 q^{97} +1.40113e7 q^{98} -2.16447e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 40 q^{2} + 27 q^{3} + 320 q^{4} + 71 q^{5} + 216 q^{6} + 237 q^{7} + 2560 q^{8} + 6980 q^{9} + 568 q^{10} + 1080 q^{11} + 1728 q^{12} + 1896 q^{14} + 27187 q^{15} + 20480 q^{16} + 3459 q^{17} + 55840 q^{18}+ \cdots - 41946120 q^{99}+O(q^{100}) $ 5 * q + 40 * q^2 + 27 * q^3 + 320 * q^4 + 71 * q^5 + 216 * q^6 + 237 * q^7 + 2560 * q^8 + 6980 * q^9 + 568 * q^10 + 1080 * q^11 + 1728 * q^12 + 1896 * q^14 + 27187 * q^15 + 20480 * q^16 + 3459 * q^17 + 55840 * q^18 - 37686 * q^19 + 4544 * q^20 + 112537 * q^21 + 8640 * q^22 - 47082 * q^23 + 13824 * q^24 + 165394 * q^25 - 26775 * q^27 + 15168 * q^28 - 65652 * q^29 + 217496 * q^30 + 263160 * q^31 + 163840 * q^32 - 347328 * q^33 + 27672 * q^34 + 436725 * q^35 + 446720 * q^36 + 1134087 * q^37 - 301488 * q^38 + 36352 * q^40 - 1116800 * q^41 + 900296 * q^42 - 1370387 * q^43 + 69120 * q^44 + 1441234 * q^45 - 376656 * q^46 + 515061 * q^47 + 110592 * q^48 + 1541952 * q^49 + 1323152 * q^50 - 551593 * q^51 - 1802958 * q^53 - 214200 * q^54 + 1999920 * q^55 + 121344 * q^56 + 3752874 * q^57 - 525216 * q^58 + 410726 * q^59 + 1739968 * q^60 + 5863828 * q^61 + 2105280 * q^62 + 1373518 * q^63 + 1310720 * q^64 - 2778624 * q^66 + 3031692 * q^67 + 221376 * q^68 - 1017898 * q^69 + 3493800 * q^70 + 4201215 * q^71 + 3573760 * q^72 - 736362 * q^73 + 9072696 * q^74 + 16427860 * q^75 - 2411904 * q^76 + 3573960 * q^77 - 14224614 * q^79 + 290816 * q^80 + 24655733 * q^81 - 8934400 * q^82 + 1634438 * q^83 + 7202368 * q^84 + 19129611 * q^85 - 10963096 * q^86 + 25078160 * q^87 + 552960 * q^88 - 7827414 * q^89 + 11529872 * q^90 - 3013248 * q^92 - 27372256 * q^93 + 4120488 * q^94 - 8273742 * q^95 + 884736 * q^96 + 22392876 * q^97 + 12335616 * q^98 - 41946120 * 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
$3$	18.5100	0.395805	0.197902	−	0.980222i	$-0.436587\pi$
$3$	18.5100	0.395805	0.197902	+	0.980222i	$0.436587\pi$
$4$	64.0000	0.500000
$5$	−6.61448	−0.0236647	−0.0118323	−	0.999930i	$-0.503766\pi$
$5$	−6.61448	−0.0236647	−0.0118323	+	0.999930i	$0.503766\pi$
$6$	148.080	0.279876
$7$	−1604.67	−1.76824	−0.884120	−	0.467260i	$-0.845241\pi$
$7$	−1604.67	−1.76824	−0.884120	+	0.467260i	$0.845241\pi$
$8$	512.000	0.353553
$9$	−1844.38	−0.843339
$10$	−52.9158	−0.0167335
$11$	1173.55	0.265843	0.132922	−	0.991127i	$-0.457564\pi$
$11$	1173.55	0.265843	0.132922	+	0.991127i	$0.457564\pi$
$12$	1184.64	0.197902
$13$	0	0
$13$	0	0
$14$	−12837.3	−1.25033
$15$	−122.434	−0.00936659
$16$	4096.00	0.250000
$17$	28053.1	1.38487	0.692437	−	0.721478i	$-0.256537\pi$
$17$	28053.1	1.38487	0.692437	+	0.721478i	$0.256537\pi$
$18$	−14755.1	−0.596330
$19$	13825.5	0.462428	0.231214	−	0.972903i	$-0.425730\pi$
$19$	13825.5	0.462428	0.231214	+	0.972903i	$0.425730\pi$
$20$	−423.327	−0.0118323
$21$	−29702.3	−0.699878
$22$	9388.37	0.187980
$23$	−94424.9	−1.61823	−0.809113	−	0.587653i	$-0.800052\pi$
$23$	−94424.9	−1.61823	−0.809113	+	0.587653i	$0.800052\pi$
$24$	9477.10	0.139938
$25$	−78081.2	−0.999440
$26$	0	0
$27$	−74620.7	−0.729602
$28$	−102699.	−0.884120
$29$	159421.	1.21381	0.606906	−	0.794774i	$-0.292410\pi$
$29$	159421.	1.21381	0.606906	+	0.794774i	$0.292410\pi$
$30$	−979.470	−0.00662318
$31$	113805.	0.686114	0.343057	−	0.939315i	$-0.388538\pi$
$31$	113805.	0.686114	0.343057	+	0.939315i	$0.388538\pi$
$32$	32768.0	0.176777
$33$	21722.3	0.105222
$34$	224425.	0.979254
$35$	10614.0	0.0418448
$36$	−118040.	−0.421669
$37$	402800.	1.30732	0.653662	−	0.756787i	$-0.273232\pi$
$37$	402800.	1.30732	0.653662	+	0.756787i	$0.273232\pi$
$38$	110604.	0.326986
$39$	0	0
$40$	−3386.61	−0.00836673
$41$	583062.	1.32121	0.660604	−	0.750734i	$-0.270300\pi$
$41$	583062.	1.32121	0.660604	+	0.750734i	$0.270300\pi$
$42$	−237618.	−0.494888
$43$	−88507.4	−0.169762	−0.0848809	−	0.996391i	$-0.527051\pi$
$43$	−88507.4	−0.169762	−0.0848809	+	0.996391i	$0.527051\pi$
$44$	75107.0	0.132922
$45$	12199.6	0.0199573
$46$	−755399.	−1.14426
$47$	112810.	0.158491	0.0792457	−	0.996855i	$-0.474749\pi$
$47$	112810.	0.158491	0.0792457	+	0.996855i	$0.474749\pi$
$48$	75816.8	0.0989512
$49$	1.75141e6	2.12667
$50$	−624650.	−0.706711
$51$	519262.	0.548140
$52$	0	0
$53$	548744.	0.506296	0.253148	−	0.967428i	$-0.418534\pi$
$53$	548744.	0.506296	0.253148	+	0.967428i	$0.418534\pi$
$54$	−596966.	−0.515907
$55$	−7762.40	−0.00629110
$56$	−821589.	−0.625167
$57$	255910.	0.183031
$58$	1.27536e6	0.858294
$59$	1.69495e6	1.07442	0.537211	−	0.843448i	$-0.319478\pi$
$59$	1.69495e6	1.07442	0.537211	+	0.843448i	$0.319478\pi$
$60$	−7835.76	−0.00468330
$61$	1.33588e6	0.753554	0.376777	−	0.926304i	$-0.377032\pi$
$61$	1.33588e6	0.753554	0.376777	+	0.926304i	$0.377032\pi$
$62$	910442.	0.485156
$63$	2.95961e6	1.49123
$64$	262144.	0.125000
$65$	0	0
$66$	173778.	0.0744033
$67$	−719239.	−0.292154	−0.146077	−	0.989273i	$-0.546665\pi$
$67$	−719239.	−0.292154	−0.146077	+	0.989273i	$0.546665\pi$
$68$	1.79540e6	0.692437
$69$	−1.74780e6	−0.640501
$70$	84912.2	0.0295888
$71$	1.85217e6	0.614153	0.307077	−	0.951685i	$-0.400649\pi$
$71$	1.85217e6	0.614153	0.307077	+	0.951685i	$0.400649\pi$
$72$	−944323.	−0.298165
$73$	−1.50153e6	−0.451756	−0.225878	−	0.974156i	$-0.572525\pi$
$73$	−1.50153e6	−0.451756	−0.225878	+	0.974156i	$0.572525\pi$
$74$	3.22240e6	0.924417
$75$	−1.44528e6	−0.395583
$76$	884833.	0.231214
$77$	−1.88315e6	−0.470075
$78$	0	0
$79$	−6.07672e6	−1.38667	−0.693337	−	0.720614i	$-0.743860\pi$
$79$	−6.07672e6	−1.38667	−0.693337	+	0.720614i	$0.743860\pi$
$80$	−27092.9	−0.00591617
$81$	2.65244e6	0.554559
$82$	4.66450e6	0.934235
$83$	−661334.	−0.126954	−0.0634771	−	0.997983i	$-0.520219\pi$
$83$	−661334.	−0.126954	−0.0634771	+	0.997983i	$0.520219\pi$
$84$	−1.90095e6	−0.349939
$85$	−185557.	−0.0327726
$86$	−708059.	−0.120040
$87$	2.95087e6	0.480432
$88$	600856.	0.0939898
$89$	−1.03091e7	−1.55008	−0.775041	−	0.631910i	$-0.782271\pi$
$89$	−1.03091e7	−1.55008	−0.775041	+	0.631910i	$0.782271\pi$
$90$	97597.0	0.0141120
$91$	0	0
$92$	−6.04319e6	−0.809113
$93$	2.10653e6	0.271567
$94$	902482.	0.112070
$95$	−91448.6	−0.0109432
$96$	606534.	0.0699691
$97$	6.14220e6	0.683318	0.341659	−	0.939824i	$-0.389011\pi$
$97$	6.14220e6	0.683318	0.341659	+	0.939824i	$0.389011\pi$
$98$	1.40113e7	1.50379
$99$	−2.16447e6	−0.224196
$100$	−4.99720e6	−0.499720
$101$	5.85424e6	0.565387	0.282693	−	0.959210i	$-0.408772\pi$
$101$	5.85424e6	0.565387	0.282693	+	0.959210i	$0.408772\pi$
$102$	4.15410e6	0.387593
$103$	−1.80551e6	−0.162806	−0.0814031	−	0.996681i	$-0.525940\pi$
$103$	−1.80551e6	−0.162806	−0.0814031	+	0.996681i	$0.525940\pi$
$104$	0	0
$105$	196465.	0.0165624
$106$	4.38996e6	0.358005
$107$	1.32024e7	1.04186	0.520931	−	0.853599i	$-0.325585\pi$
$107$	1.32024e7	1.04186	0.520931	+	0.853599i	$0.325585\pi$
$108$	−4.77572e6	−0.364801
$109$	2.36217e7	1.74710	0.873550	−	0.486734i	$-0.161812\pi$
$109$	2.36217e7	1.74710	0.873550	+	0.486734i	$0.161812\pi$
$110$	−62099.2	−0.00444848
$111$	7.45581e6	0.517445
$112$	−6.57271e6	−0.442060
$113$	1.03738e7	0.676340	0.338170	−	0.941085i	$-0.390192\pi$
$113$	1.03738e7	0.676340	0.338170	+	0.941085i	$0.390192\pi$
$114$	2.04728e6	0.129423
$115$	624571.	0.0382948
$116$	1.02029e7	0.606906
$117$	0	0
$118$	1.35596e7	0.759732
$119$	−4.50159e7	−2.44879
$120$	−62686.1	−0.00331159
$121$	−1.81100e7	−0.929327
$122$	1.06871e7	0.532843
$123$	1.07925e7	0.522941
$124$	7.28354e6	0.343057
$125$	1.03322e6	0.0473161
$126$	2.36769e7	1.05446
$127$	3.07208e7	1.33082	0.665409	−	0.746479i	$-0.268257\pi$
$127$	3.07208e7	1.33082	0.665409	+	0.746479i	$0.268257\pi$
$128$	2.09715e6	0.0883883
$129$	−1.63827e6	−0.0671925
$130$	0	0
$131$	1.62980e7	0.633411	0.316706	−	0.948524i	$-0.397423\pi$
$131$	1.62980e7	0.633411	0.316706	+	0.948524i	$0.397423\pi$
$132$	1.39023e6	0.0526110
$133$	−2.21853e7	−0.817683
$134$	−5.75391e6	−0.206584
$135$	493577.	0.0172658
$136$	1.43632e7	0.489627
$137$	3.22644e7	1.07202	0.536008	−	0.844213i	$-0.319932\pi$
$137$	3.22644e7	1.07202	0.536008	+	0.844213i	$0.319932\pi$
$138$	−1.39824e7	−0.452903
$139$	−3.77383e7	−1.19188	−0.595938	−	0.803030i	$-0.703220\pi$
$139$	−3.77383e7	−1.19188	−0.595938	+	0.803030i	$0.703220\pi$
$140$	679298.	0.0209224
$141$	2.08811e6	0.0627317
$142$	1.48174e7	0.434272
$143$	0	0
$144$	−7.55459e6	−0.210835
$145$	−1.05448e6	−0.0287245
$146$	−1.20122e7	−0.319439
$147$	3.24185e7	0.841748
$148$	2.57792e7	0.653662
$149$	1.69205e7	0.419045	0.209523	−	0.977804i	$-0.432809\pi$
$149$	1.69205e7	0.419045	0.209523	+	0.977804i	$0.432809\pi$
$150$	−1.15622e7	−0.279720
$151$	−3.87829e7	−0.916687	−0.458343	−	0.888775i	$-0.651557\pi$
$151$	−3.87829e7	−0.916687	−0.458343	+	0.888775i	$0.651557\pi$
$152$	7.07867e6	0.163493
$153$	−5.17407e7	−1.16792
$154$	−1.50652e7	−0.332393
$155$	−752763.	−0.0162367
$156$	0	0
$157$	−4.86546e7	−1.00340	−0.501701	−	0.865041i	$-0.667292\pi$
$157$	−4.86546e7	−1.00340	−0.501701	+	0.865041i	$0.667292\pi$
$158$	−4.86137e7	−0.980526
$159$	1.01572e7	0.200394
$160$	−216743.	−0.00418336
$161$	1.51520e8	2.86141
$162$	2.12195e7	0.392132
$163$	8.82607e7	1.59629	0.798143	−	0.602468i	$-0.205816\pi$
$163$	8.82607e7	1.59629	0.798143	+	0.602468i	$0.205816\pi$
$164$	3.73160e7	0.660604
$165$	−143682.	−0.00249005
$166$	−5.29067e6	−0.0897702
$167$	7.18201e7	1.19327	0.596634	−	0.802513i	$-0.296504\pi$
$167$	7.18201e7	1.19327	0.596634	+	0.802513i	$0.296504\pi$
$168$	−1.52076e7	−0.247444
$169$	0	0
$170$	−1.48446e6	−0.0231737
$171$	−2.54995e7	−0.389983
$172$	−5.66447e6	−0.0848809
$173$	6.00357e7	0.881553	0.440776	−	0.897617i	$-0.354703\pi$
$173$	6.00357e7	0.881553	0.440776	+	0.897617i	$0.354703\pi$
$174$	2.36069e7	0.339717
$175$	1.25294e8	1.76725
$176$	4.80685e6	0.0664609
$177$	3.13735e7	0.425262
$178$	−8.24726e7	−1.09607
$179$	−1.34440e8	−1.75203	−0.876015	−	0.482284i	$-0.839807\pi$
$179$	−1.34440e8	−1.75203	−0.876015	+	0.482284i	$0.839807\pi$
$180$	780776.	0.00997867
$181$	−1.55862e8	−1.95374	−0.976868	−	0.213843i	$-0.931402\pi$
$181$	−1.55862e8	−1.95374	−0.976868	+	0.213843i	$0.931402\pi$
$182$	0	0
$183$	2.47272e7	0.298260
$184$	−4.83455e7	−0.572129
$185$	−2.66431e6	−0.0309374
$186$	1.68522e7	0.192027
$187$	3.29217e7	0.368160
$188$	7.21985e6	0.0792457
$189$	1.19741e8	1.29011
$190$	−731589.	−0.00773801
$191$	−2.23323e7	−0.231909	−0.115954	−	0.993255i	$-0.536993\pi$
$191$	−2.23323e7	−0.231909	−0.115954	+	0.993255i	$0.536993\pi$
$192$	4.85227e6	0.0494756
$193$	2.69730e6	0.0270072	0.0135036	−	0.999909i	$-0.495702\pi$
$193$	2.69730e6	0.0270072	0.0135036	+	0.999909i	$0.495702\pi$
$194$	4.91376e7	0.483179
$195$	0	0
$196$	1.12090e8	1.06334
$197$	7.30124e7	0.680401	0.340200	−	0.940353i	$-0.389505\pi$
$197$	7.30124e7	0.680401	0.340200	+	0.940353i	$0.389505\pi$
$198$	−1.73157e7	−0.158531
$199$	−1.10593e8	−0.994817	−0.497409	−	0.867516i	$-0.665715\pi$
$199$	−1.10593e8	−0.994817	−0.497409	+	0.867516i	$0.665715\pi$
$200$	−3.99776e7	−0.353355
$201$	−1.33131e7	−0.115636
$202$	4.68339e7	0.399789
$203$	−2.55817e8	−2.14631
$204$	3.32328e7	0.274070
$205$	−3.85665e6	−0.0312660
$206$	−1.44441e7	−0.115121
$207$	1.74156e8	1.36471
$208$	0	0
$209$	1.62249e7	0.122933
$210$	1.57172e6	0.0117114
$211$	−7.79932e7	−0.571568	−0.285784	−	0.958294i	$-0.592254\pi$
$211$	−7.79932e7	−0.571568	−0.285784	+	0.958294i	$0.592254\pi$
$212$	3.51196e7	0.253148
$213$	3.42836e7	0.243085
$214$	1.05619e8	0.736708
$215$	585430.	0.00401736
$216$	−3.82058e7	−0.257953
$217$	−1.82619e8	−1.21322
$218$	1.88973e8	1.23539
$219$	−2.77932e7	−0.178807
$220$	−496793.	−0.00314555
$221$	0	0
$222$	5.96464e7	0.365889
$223$	−2.81334e8	−1.69885	−0.849425	−	0.527709i	$-0.823051\pi$
$223$	−2.81334e8	−1.69885	−0.849425	+	0.527709i	$0.823051\pi$
$224$	−5.25817e7	−0.312584
$225$	1.44012e8	0.842866
$226$	8.29908e7	0.478245
$227$	2.87156e8	1.62940	0.814700	−	0.579882i	$-0.196901\pi$
$227$	2.87156e8	1.62940	0.814700	+	0.579882i	$0.196901\pi$
$228$	1.63782e7	0.0915156
$229$	1.77698e8	0.977819	0.488909	−	0.872335i	$-0.337395\pi$
$229$	1.77698e8	0.977819	0.488909	+	0.872335i	$0.337395\pi$
$230$	4.99657e6	0.0270785
$231$	−3.48570e7	−0.186058
$232$	8.16233e7	0.429147
$233$	1.38051e8	0.714979	0.357490	−	0.933917i	$-0.383633\pi$
$233$	1.38051e8	0.714979	0.357490	+	0.933917i	$0.383633\pi$
$234$	0	0
$235$	−746181.	−0.00375065
$236$	1.08477e8	0.537211
$237$	−1.12480e8	−0.548852
$238$	−3.60127e8	−1.73156
$239$	6.38380e7	0.302473	0.151237	−	0.988498i	$-0.451674\pi$
$239$	6.38380e7	0.302473	0.151237	+	0.988498i	$0.451674\pi$
$240$	−501489.	−0.00234165
$241$	1.77813e8	0.818285	0.409143	−	0.912470i	$-0.365828\pi$
$241$	1.77813e8	0.818285	0.409143	+	0.912470i	$0.365828\pi$
$242$	−1.44880e8	−0.657134
$243$	2.12292e8	0.949099
$244$	8.54966e7	0.376777
$245$	−1.15846e7	−0.0503271
$246$	8.63396e7	0.369775
$247$	0	0
$248$	5.82683e7	0.242578
$249$	−1.22413e7	−0.0502491
$250$	8.26578e6	0.0334575
$251$	2.57165e8	1.02649	0.513244	−	0.858243i	$-0.328444\pi$
$251$	2.57165e8	1.02649	0.513244	+	0.858243i	$0.328444\pi$
$252$	1.89415e8	0.745613
$253$	−1.10812e8	−0.430195
$254$	2.45766e8	0.941031
$255$	−3.43465e6	−0.0129716
$256$	1.67772e7	0.0625000
$257$	−2.24906e8	−0.826486	−0.413243	−	0.910621i	$-0.635604\pi$
$257$	−2.24906e8	−0.826486	−0.413243	+	0.910621i	$0.635604\pi$
$258$	−1.31061e7	−0.0475123
$259$	−6.46359e8	−2.31166
$260$	0	0
$261$	−2.94032e8	−1.02365
$262$	1.30384e8	0.447890
$263$	4.78020e8	1.62032	0.810161	−	0.586208i	$-0.199380\pi$
$263$	4.78020e8	1.62032	0.810161	+	0.586208i	$0.199380\pi$
$264$	1.11218e7	0.0372016
$265$	−3.62966e6	−0.0119813
$266$	−1.77483e8	−0.578189
$267$	−1.90821e8	−0.613530
$268$	−4.60313e7	−0.146077
$269$	1.76243e8	0.552049	0.276025	−	0.961151i	$-0.410983\pi$
$269$	1.76243e8	0.552049	0.276025	+	0.961151i	$0.410983\pi$
$270$	3.94862e6	0.0122088
$271$	−1.88337e8	−0.574834	−0.287417	−	0.957806i	$-0.592797\pi$
$271$	−1.88337e8	−0.574834	−0.287417	+	0.957806i	$0.592797\pi$
$272$	1.14906e8	0.346219
$273$	0	0
$274$	2.58115e8	0.758030
$275$	−9.16320e7	−0.265695
$276$	−1.11859e8	−0.320251
$277$	3.15057e7	0.0890656	0.0445328	−	0.999008i	$-0.485820\pi$
$277$	3.15057e7	0.0890656	0.0445328	+	0.999008i	$0.485820\pi$
$278$	−3.01907e8	−0.842784
$279$	−2.09900e8	−0.578627
$280$	5.43438e6	0.0147944
$281$	2.67569e8	0.719388	0.359694	−	0.933070i	$-0.382881\pi$
$281$	2.67569e8	0.719388	0.359694	+	0.933070i	$0.382881\pi$
$282$	1.67049e7	0.0443580
$283$	−1.31279e8	−0.344303	−0.172152	−	0.985070i	$-0.555072\pi$
$283$	−1.31279e8	−0.344303	−0.172152	+	0.985070i	$0.555072\pi$
$284$	1.18539e8	0.307077
$285$	−1.69271e6	−0.00433137
$286$	0	0
$287$	−9.35620e8	−2.33621
$288$	−6.04367e7	−0.149083
$289$	3.76640e8	0.917876
$290$	−8.43587e6	−0.0203113
$291$	1.13692e8	0.270460
$292$	−9.60978e7	−0.225878
$293$	−5.02530e8	−1.16715	−0.583574	−	0.812060i	$-0.698346\pi$
$293$	−5.02530e8	−1.16715	−0.583574	+	0.812060i	$0.698346\pi$
$294$	2.59348e8	0.595206
$295$	−1.12112e7	−0.0254259
$296$	2.06233e8	0.462209
$297$	−8.75709e7	−0.193960
$298$	1.35364e8	0.296310
$299$	0	0
$300$	−9.24980e7	−0.197792
$301$	1.42025e8	0.300180
$302$	−3.10263e8	−0.648196
$303$	1.08362e8	0.223783
$304$	5.66293e7	0.115607
$305$	−8.83618e6	−0.0178326
$306$	−4.13926e8	−0.825843
$307$	−4.04464e7	−0.0797802	−0.0398901	−	0.999204i	$-0.512701\pi$
$307$	−4.04464e7	−0.0797802	−0.0398901	+	0.999204i	$0.512701\pi$
$308$	−1.20522e8	−0.235038
$309$	−3.34200e7	−0.0644394
$310$	−6.02210e6	−0.0114811
$311$	−5.80479e7	−0.109427	−0.0547136	−	0.998502i	$-0.517425\pi$
$311$	−5.80479e7	−0.109427	−0.0547136	+	0.998502i	$0.517425\pi$
$312$	0	0
$313$	1.87305e8	0.345259	0.172629	−	0.984987i	$-0.444774\pi$
$313$	1.87305e8	0.345259	0.172629	+	0.984987i	$0.444774\pi$
$314$	−3.89237e8	−0.709513
$315$	−1.95763e7	−0.0352894
$316$	−3.88910e8	−0.693337
$317$	6.09607e7	0.107484	0.0537419	−	0.998555i	$-0.482885\pi$
$317$	6.09607e7	0.107484	0.0537419	+	0.998555i	$0.482885\pi$
$318$	8.12579e7	0.141700
$319$	1.87087e8	0.322684
$320$	−1.73395e6	−0.00295808
$321$	2.44376e8	0.412374
$322$	1.21216e9	2.02332
$323$	3.87849e8	0.640404
$324$	1.69756e8	0.277279
$325$	0	0
$326$	7.06085e8	1.12874
$327$	4.37236e8	0.691511
$328$	2.98528e8	0.467118
$329$	−1.81023e8	−0.280251
$330$	−1.14945e6	−0.00176073
$331$	−4.54468e8	−0.688820	−0.344410	−	0.938819i	$-0.611921\pi$
$331$	−4.54468e8	−0.688820	−0.344410	+	0.938819i	$0.611921\pi$
$332$	−4.23254e7	−0.0634771
$333$	−7.42916e8	−1.10252
$334$	5.74561e8	0.843769
$335$	4.75739e6	0.00691372
$336$	−1.21661e8	−0.174969
$337$	2.05246e8	0.292125	0.146063	−	0.989275i	$-0.453340\pi$
$337$	2.05246e8	0.292125	0.146063	+	0.989275i	$0.453340\pi$
$338$	0	0
$339$	1.92019e8	0.267699
$340$	−1.18756e7	−0.0163863
$341$	1.33556e8	0.182399
$342$	−2.03996e8	−0.275760
$343$	−1.48891e9	−1.99223
$344$	−4.53158e7	−0.0600198
$345$	1.15608e7	0.0151573
$346$	4.80286e8	0.623352
$347$	2.99289e8	0.384537	0.192268	−	0.981342i	$-0.438416\pi$
$347$	2.99289e8	0.384537	0.192268	+	0.981342i	$0.438416\pi$
$348$	1.88856e8	0.240216
$349$	1.03098e8	0.129826	0.0649128	−	0.997891i	$-0.479323\pi$
$349$	1.03098e8	0.129826	0.0649128	+	0.997891i	$0.479323\pi$
$350$	1.00235e9	1.24963
$351$	0	0
$352$	3.84548e7	0.0469949
$353$	3.79231e8	0.458873	0.229436	−	0.973324i	$-0.426312\pi$
$353$	3.79231e8	0.458873	0.229436	+	0.973324i	$0.426312\pi$
$354$	2.50988e8	0.300705
$355$	−1.22511e7	−0.0145337
$356$	−6.59781e8	−0.775041
$357$	−8.33242e8	−0.969243
$358$	−1.07552e9	−1.23887
$359$	1.29204e9	1.47382	0.736912	−	0.675988i	$-0.236283\pi$
$359$	1.29204e9	1.47382	0.736912	+	0.675988i	$0.236283\pi$
$360$	6.24621e6	0.00705598
$361$	−7.02727e8	−0.786161
$362$	−1.24690e9	−1.38150
$363$	−3.35215e8	−0.367832
$364$	0	0
$365$	9.93183e6	0.0106907
$366$	1.97817e8	0.210902
$367$	−3.21664e8	−0.339681	−0.169841	−	0.985472i	$-0.554325\pi$
$367$	−3.21664e8	−0.339681	−0.169841	+	0.985472i	$0.554325\pi$
$368$	−3.86764e8	−0.404556
$369$	−1.07539e9	−1.11423
$370$	−2.13145e7	−0.0218760
$371$	−8.80551e8	−0.895253
$372$	1.34818e8	0.135784
$373$	−5.95386e8	−0.594043	−0.297021	−	0.954871i	$-0.595993\pi$
$373$	−5.95386e8	−0.594043	−0.297021	+	0.954871i	$0.595993\pi$
$374$	2.63373e8	0.260328
$375$	1.91249e7	0.0187279
$376$	5.77588e7	0.0560352
$377$	0	0
$378$	9.57930e8	0.912247
$379$	1.84954e9	1.74512	0.872562	−	0.488503i	$-0.162457\pi$
$379$	1.84954e9	1.74512	0.872562	+	0.488503i	$0.162457\pi$
$380$	−5.85271e6	−0.00547160
$381$	5.68640e8	0.526744
$382$	−1.78659e8	−0.163984
$383$	−1.94010e9	−1.76453	−0.882263	−	0.470757i	$-0.843981\pi$
$383$	−1.94010e9	−1.76453	−0.882263	+	0.470757i	$0.843981\pi$
$384$	3.88182e7	0.0349845
$385$	1.24561e7	0.0111242
$386$	2.15784e7	0.0190969
$387$	1.63241e8	0.143167
$388$	3.93101e8	0.341659
$389$	−1.63159e9	−1.40536	−0.702680	−	0.711506i	$-0.748014\pi$
$389$	−1.63159e9	−1.40536	−0.702680	+	0.711506i	$0.748014\pi$
$390$	0	0
$391$	−2.64891e9	−2.24104
$392$	8.96721e8	0.751893
$393$	3.01676e8	0.250707
$394$	5.84099e8	0.481116
$395$	4.01943e7	0.0328152
$396$	−1.38526e8	−0.112098
$397$	−1.46137e9	−1.17218	−0.586090	−	0.810246i	$-0.699334\pi$
$397$	−1.46137e9	−1.17218	−0.586090	+	0.810246i	$0.699334\pi$
$398$	−8.84747e8	−0.703442
$399$	−4.10650e8	−0.323643
$400$	−3.19821e8	−0.249860
$401$	2.15957e8	0.167248	0.0836240	−	0.996497i	$-0.473351\pi$
$401$	2.15957e8	0.167248	0.0836240	+	0.996497i	$0.473351\pi$
$402$	−1.06505e8	−0.0817669
$403$	0	0
$404$	3.74671e8	0.282693
$405$	−1.75445e7	−0.0131234
$406$	−2.04653e9	−1.51767
$407$	4.72704e8	0.347543
$408$	2.65862e8	0.193797
$409$	2.10034e9	1.51795	0.758975	−	0.651120i	$-0.225700\pi$
$409$	2.10034e9	1.51795	0.758975	+	0.651120i	$0.225700\pi$
$410$	−3.08532e7	−0.0221084
$411$	5.97213e8	0.424309
$412$	−1.15553e8	−0.0814031
$413$	−2.71983e9	−1.89984
$414$	1.39324e9	0.964997
$415$	4.37438e6	0.00300433
$416$	0	0
$417$	−6.98535e8	−0.471750
$418$	1.29799e8	0.0869270
$419$	4.33365e8	0.287809	0.143905	−	0.989592i	$-0.454034\pi$
$419$	4.33365e8	0.287809	0.143905	+	0.989592i	$0.454034\pi$
$420$	1.25738e7	0.00828119
$421$	−2.60776e9	−1.70326	−0.851629	−	0.524145i	$-0.824385\pi$
$421$	−2.60776e9	−1.70326	−0.851629	+	0.524145i	$0.824385\pi$
$422$	−6.23946e8	−0.404160
$423$	−2.08065e8	−0.133662
$424$	2.80957e8	0.179003
$425$	−2.19042e9	−1.38410
$426$	2.74269e8	0.171887
$427$	−2.14365e9	−1.33246
$428$	8.44954e8	0.520931
$429$	0	0
$430$	4.68344e6	0.00284070
$431$	−6.82317e8	−0.410503	−0.205251	−	0.978709i	$-0.565801\pi$
$431$	−6.82317e8	−0.410503	−0.205251	+	0.978709i	$0.565801\pi$
$432$	−3.05646e8	−0.182401
$433$	2.49592e9	1.47749	0.738743	−	0.673987i	$-0.235420\pi$
$433$	2.49592e9	1.47749	0.738743	+	0.673987i	$0.235420\pi$
$434$	−1.46096e9	−0.857873
$435$	−1.95185e7	−0.0113693
$436$	1.51179e9	0.873550
$437$	−1.30547e9	−0.748312
$438$	−2.22346e8	−0.126436
$439$	−1.51778e9	−0.856216	−0.428108	−	0.903727i	$-0.640820\pi$
$439$	−1.51778e9	−0.856216	−0.428108	+	0.903727i	$0.640820\pi$
$440$	−3.97435e6	−0.00222424
$441$	−3.23026e9	−1.79351
$442$	0	0
$443$	2.56533e9	1.40194	0.700971	−	0.713190i	$-0.252750\pi$
$443$	2.56533e9	1.40194	0.700971	+	0.713190i	$0.252750\pi$
$444$	4.77172e8	0.258722
$445$	6.81892e7	0.0366822
$446$	−2.25067e9	−1.20127
$447$	3.13197e8	0.165860
$448$	−4.20653e8	−0.221030
$449$	−2.22454e9	−1.15978	−0.579892	−	0.814693i	$-0.696905\pi$
$449$	−2.22454e9	−1.15978	−0.579892	+	0.814693i	$0.696905\pi$
$450$	1.15209e9	0.595996
$451$	6.84251e8	0.351235
$452$	6.63926e8	0.338170
$453$	−7.17870e8	−0.362829
$454$	2.29725e9	1.15216
$455$	0	0
$456$	1.31026e8	0.0647113
$457$	−1.58220e9	−0.775453	−0.387726	−	0.921775i	$-0.626739\pi$
$457$	−1.58220e9	−0.775453	−0.387726	+	0.921775i	$0.626739\pi$
$458$	1.42158e9	0.691422
$459$	−2.09335e9	−1.01041
$460$	3.99726e7	0.0191474
$461$	−1.33292e9	−0.633652	−0.316826	−	0.948484i	$-0.602617\pi$
$461$	−1.33292e9	−0.633652	−0.316826	+	0.948484i	$0.602617\pi$
$462$	−2.78856e8	−0.131563
$463$	2.03271e9	0.951790	0.475895	−	0.879502i	$-0.342124\pi$
$463$	2.03271e9	0.951790	0.475895	+	0.879502i	$0.342124\pi$
$464$	6.52987e8	0.303453
$465$	−1.39336e7	−0.00642655
$466$	1.10441e9	0.505567
$467$	−3.10675e7	−0.0141155	−0.00705777	−	0.999975i	$-0.502247\pi$
$467$	−3.10675e7	−0.0141155	−0.00705777	+	0.999975i	$0.502247\pi$
$468$	0	0
$469$	1.15414e9	0.516598
$470$	−5.96945e6	−0.00265211
$471$	−9.00595e8	−0.397152
$472$	8.67815e8	0.379866
$473$	−1.03867e8	−0.0451300
$474$	−8.99838e8	−0.388097
$475$	−1.07951e9	−0.462169
$476$	−2.88102e9	−1.22440
$477$	−1.01209e9	−0.426979
$478$	5.10704e8	0.213881
$479$	2.95620e9	1.22902	0.614512	−	0.788908i	$-0.289353\pi$
$479$	2.95620e9	1.22902	0.614512	+	0.788908i	$0.289353\pi$
$480$	−4.01191e6	−0.00165580
$481$	0	0
$482$	1.42251e9	0.578615
$483$	2.80464e9	1.13256
$484$	−1.15904e9	−0.464664
$485$	−4.06274e7	−0.0161705
$486$	1.69834e9	0.671114
$487$	3.14356e8	0.123330	0.0616652	−	0.998097i	$-0.480359\pi$
$487$	3.14356e8	0.123330	0.0616652	+	0.998097i	$0.480359\pi$
$488$	6.83973e8	0.266422
$489$	1.63370e9	0.631817
$490$	−9.26772e7	−0.0355866
$491$	1.74300e9	0.664525	0.332262	−	0.943187i	$-0.392188\pi$
$491$	1.74300e9	0.664525	0.332262	+	0.943187i	$0.392188\pi$
$492$	6.90717e8	0.261470
$493$	4.47225e9	1.68098
$494$	0	0
$495$	1.43168e7	0.00530553
$496$	4.66146e8	0.171529
$497$	−2.97211e9	−1.08597
$498$	−9.79301e7	−0.0355315
$499$	3.47846e9	1.25324	0.626621	−	0.779324i	$-0.284437\pi$
$499$	3.47846e9	1.25324	0.626621	+	0.779324i	$0.284437\pi$
$500$	6.61263e7	0.0236581
$501$	1.32939e9	0.472302
$502$	2.05732e9	0.725837
$503$	−2.26923e9	−0.795043	−0.397521	−	0.917593i	$-0.630130\pi$
$503$	−2.26923e9	−0.795043	−0.397521	+	0.917593i	$0.630130\pi$
$504$	1.51532e9	0.527228
$505$	−3.87227e7	−0.0133797
$506$	−8.86496e8	−0.304194
$507$	0	0
$508$	1.96613e9	0.665409
$509$	−3.74316e9	−1.25813	−0.629065	−	0.777352i	$-0.716562\pi$
$509$	−3.74316e9	−1.25813	−0.629065	+	0.777352i	$0.716562\pi$
$510$	−2.74772e7	−0.00917227
$511$	2.40945e9	0.798812
$512$	1.34218e8	0.0441942
$513$	−1.03167e9	−0.337388
$514$	−1.79925e9	−0.584414
$515$	1.19425e7	0.00385275
$516$	−1.04849e8	−0.0335962
$517$	1.32388e8	0.0421339
$518$	−5.17087e9	−1.63459
$519$	1.11126e9	0.348923
$520$	0	0
$521$	−1.68654e9	−0.522475	−0.261238	−	0.965275i	$-0.584131\pi$
$521$	−1.68654e9	−0.522475	−0.261238	+	0.965275i	$0.584131\pi$
$522$	−2.35226e9	−0.723833
$523$	4.11403e9	1.25751	0.628755	−	0.777604i	$-0.283565\pi$
$523$	4.11403e9	1.25751	0.628755	+	0.777604i	$0.283565\pi$
$524$	1.04307e9	0.316706
$525$	2.31919e9	0.699486
$526$	3.82416e9	1.14574
$527$	3.19260e9	0.950182
$528$	8.89745e7	0.0263055
$529$	5.51124e9	1.61865
$530$	−2.90373e7	−0.00847208
$531$	−3.12614e9	−0.906102
$532$	−1.41986e9	−0.408842
$533$	0	0
$534$	−1.52657e9	−0.433831
$535$	−8.73271e7	−0.0246553
$536$	−3.68250e8	−0.103292
$537$	−2.48847e9	−0.693462
$538$	1.40994e9	0.390358
$539$	2.05536e9	0.565362
$540$	3.15889e7	0.00863290
$541$	−5.21199e9	−1.41519	−0.707593	−	0.706620i	$-0.750219\pi$
$541$	−5.21199e9	−1.41519	−0.707593	+	0.706620i	$0.750219\pi$
$542$	−1.50669e9	−0.406469
$543$	−2.88500e9	−0.773298
$544$	9.19245e8	0.244813
$545$	−1.56245e8	−0.0413446
$546$	0	0
$547$	−5.06174e9	−1.32234	−0.661172	−	0.750235i	$-0.729941\pi$
$547$	−5.06174e9	−1.32234	−0.661172	+	0.750235i	$0.729941\pi$
$548$	2.06492e9	0.536008
$549$	−2.46388e9	−0.635501
$550$	−7.33056e8	−0.187874
$551$	2.20407e9	0.561300
$552$	−8.94874e8	−0.226451
$553$	9.75110e9	2.45197
$554$	2.52046e8	0.0629789
$555$	−4.93163e7	−0.0122452
$556$	−2.41525e9	−0.595938
$557$	5.64001e9	1.38289	0.691444	−	0.722430i	$-0.256975\pi$
$557$	5.64001e9	1.38289	0.691444	+	0.722430i	$0.256975\pi$
$558$	−1.67920e9	−0.409151
$559$	0	0
$560$	4.34750e7	0.0104612
$561$	6.09379e8	0.145719
$562$	2.14055e9	0.508684
$563$	−1.03881e9	−0.245334	−0.122667	−	0.992448i	$-0.539145\pi$
$563$	−1.03881e9	−0.245334	−0.122667	+	0.992448i	$0.539145\pi$
$564$	1.33639e8	0.0313658
$565$	−6.86176e7	−0.0160054
$566$	−1.05023e9	−0.243459
$567$	−4.25627e9	−0.980593
$568$	9.48311e8	0.217136
$569$	1.36295e9	0.310160	0.155080	−	0.987902i	$-0.450436\pi$
$569$	1.36295e9	0.310160	0.155080	+	0.987902i	$0.450436\pi$
$570$	−1.35417e7	−0.00306274
$571$	−7.59504e9	−1.70727	−0.853637	−	0.520868i	$-0.825608\pi$
$571$	−7.59504e9	−1.70727	−0.853637	+	0.520868i	$0.825608\pi$
$572$	0	0
$573$	−4.13370e8	−0.0917906
$574$	−7.48496e9	−1.65195
$575$	7.37281e9	1.61732
$576$	−4.83494e8	−0.105417
$577$	−9.14980e8	−0.198288	−0.0991439	−	0.995073i	$-0.531610\pi$
$577$	−9.14980e8	−0.198288	−0.0991439	+	0.995073i	$0.531610\pi$
$578$	3.01312e9	0.649036
$579$	4.99269e7	0.0106896
$580$	−6.74870e7	−0.0143622
$581$	1.06122e9	0.224486
$582$	9.09534e8	0.191244
$583$	6.43977e8	0.134595
$584$	−7.68782e8	−0.159720
$585$	0	0
$586$	−4.02024e9	−0.825298
$587$	−3.69625e9	−0.754272	−0.377136	−	0.926158i	$-0.623091\pi$
$587$	−3.69625e9	−0.754272	−0.377136	+	0.926158i	$0.623091\pi$
$588$	2.07478e9	0.420874
$589$	1.57342e9	0.317278
$590$	−8.96898e7	−0.0179788
$591$	1.35146e9	0.269306
$592$	1.64987e9	0.326831
$593$	4.56755e9	0.899480	0.449740	−	0.893159i	$-0.351517\pi$
$593$	4.56755e9	0.899480	0.449740	+	0.893159i	$0.351517\pi$
$594$	−7.00567e8	−0.137150
$595$	2.97757e8	0.0579498
$596$	1.08291e9	0.209523
$597$	−2.04708e9	−0.393753
$598$	0	0
$599$	1.63940e9	0.311668	0.155834	−	0.987783i	$-0.450194\pi$
$599$	1.63940e9	0.311668	0.155834	+	0.987783i	$0.450194\pi$
$600$	−7.39984e8	−0.139860
$601$	−4.92573e9	−0.925572	−0.462786	−	0.886470i	$-0.653150\pi$
$601$	−4.92573e9	−0.925572	−0.462786	+	0.886470i	$0.653150\pi$
$602$	1.13620e9	0.212259
$603$	1.32655e9	0.246384
$604$	−2.48211e9	−0.458343
$605$	1.19788e8	0.0219922
$606$	8.66894e8	0.158238
$607$	7.26751e9	1.31894	0.659470	−	0.751731i	$-0.270781\pi$
$607$	7.26751e9	1.31894	0.659470	+	0.751731i	$0.270781\pi$
$608$	4.53035e8	0.0817464
$609$	−4.73515e9	−0.849520
$610$	−7.06894e7	−0.0126096
$611$	0	0
$612$	−3.31140e9	−0.583959
$613$	3.16466e9	0.554900	0.277450	−	0.960740i	$-0.410511\pi$
$613$	3.16466e9	0.554900	0.277450	+	0.960740i	$0.410511\pi$
$614$	−3.23571e8	−0.0564131
$615$	−7.13865e7	−0.0123752
$616$	−9.64172e8	−0.166197
$617$	1.11473e10	1.91060	0.955301	−	0.295634i	$-0.0955309\pi$
$617$	1.11473e10	1.91060	0.955301	+	0.295634i	$0.0955309\pi$
$618$	−2.67360e8	−0.0455656
$619$	−8.69140e9	−1.47290	−0.736449	−	0.676493i	$-0.763499\pi$
$619$	−8.69140e9	−1.47290	−0.736449	+	0.676493i	$0.763499\pi$
$620$	−4.81768e7	−0.00811834
$621$	7.04605e9	1.18066
$622$	−4.64384e8	−0.0773767
$623$	1.65426e10	2.74092
$624$	0	0
$625$	6.09326e9	0.998320
$626$	1.49844e9	0.244135
$627$	3.00322e8	0.0486576
$628$	−3.11390e9	−0.501701
$629$	1.12998e10	1.81048
$630$	−1.56610e8	−0.0249534
$631$	3.41348e9	0.540872	0.270436	−	0.962738i	$-0.412832\pi$
$631$	3.41348e9	0.540872	0.270436	+	0.962738i	$0.412832\pi$
$632$	−3.11128e9	−0.490263
$633$	−1.44365e9	−0.226229
$634$	4.87686e8	0.0760025
$635$	−2.03202e8	−0.0314934
$636$	6.50063e8	0.100197
$637$	0	0
$638$	1.49670e9	0.228172
$639$	−3.41611e9	−0.517939
$640$	−1.38716e7	−0.00209168
$641$	1.93523e9	0.290221	0.145110	−	0.989415i	$-0.453646\pi$
$641$	1.93523e9	0.290221	0.145110	+	0.989415i	$0.453646\pi$
$642$	1.95501e9	0.291592
$643$	7.98936e9	1.18515	0.592575	−	0.805515i	$-0.298111\pi$
$643$	7.98936e9	1.18515	0.592575	+	0.805515i	$0.298111\pi$
$644$	9.69730e9	1.43071
$645$	1.08363e7	0.00159009
$646$	3.10279e9	0.452834
$647$	8.47178e9	1.22973	0.614865	−	0.788632i	$-0.289211\pi$
$647$	8.47178e9	1.22973	0.614865	+	0.788632i	$0.289211\pi$
$648$	1.35805e9	0.196066
$649$	1.98910e9	0.285628
$650$	0	0
$651$	−3.38028e9	−0.480196
$652$	5.64868e9	0.798143
$653$	8.72038e9	1.22557	0.612787	−	0.790248i	$-0.290048\pi$
$653$	8.72038e9	1.22557	0.612787	+	0.790248i	$0.290048\pi$
$654$	3.49789e9	0.488972
$655$	−1.07803e8	−0.0149895
$656$	2.38822e9	0.330302
$657$	2.76939e9	0.380983
$658$	−1.44818e9	−0.198167
$659$	−4.03937e9	−0.549813	−0.274907	−	0.961471i	$-0.588647\pi$
$659$	−4.03937e9	−0.549813	−0.274907	+	0.961471i	$0.588647\pi$
$660$	−9.19563e6	−0.00124502
$661$	3.29803e9	0.444170	0.222085	−	0.975027i	$-0.428714\pi$
$661$	3.29803e9	0.444170	0.222085	+	0.975027i	$0.428714\pi$
$662$	−3.63575e9	−0.487069
$663$	0	0
$664$	−3.38603e8	−0.0448851
$665$	1.46744e8	0.0193502
$666$	−5.94333e9	−0.779597
$667$	−1.50533e10	−1.96422
$668$	4.59649e9	0.596634
$669$	−5.20748e9	−0.672413
$670$	3.80591e7	0.00488874
$671$	1.56772e9	0.200327
$672$	−9.73284e8	−0.123722
$673$	5.27449e9	0.667003	0.333502	−	0.942750i	$-0.391770\pi$
$673$	5.27449e9	0.667003	0.333502	+	0.942750i	$0.391770\pi$
$674$	1.64196e9	0.206564
$675$	5.82648e9	0.729194
$676$	0	0
$677$	1.02661e10	1.27158	0.635791	−	0.771861i	$-0.280674\pi$
$677$	1.02661e10	1.27158	0.635791	+	0.771861i	$0.280674\pi$
$678$	1.53616e9	0.189292
$679$	−9.85617e9	−1.20827
$680$	−9.50051e7	−0.0115869
$681$	5.31525e9	0.644925
$682$	1.06845e9	0.128976
$683$	3.81750e9	0.458465	0.229233	−	0.973372i	$-0.426378\pi$
$683$	3.81750e9	0.458465	0.229233	+	0.973372i	$0.426378\pi$
$684$	−1.63197e9	−0.194992
$685$	−2.13412e8	−0.0253689
$686$	−1.19113e10	−1.40872
$687$	3.28918e9	0.387025
$688$	−3.62526e8	−0.0424404
$689$	0	0
$690$	9.24863e7	0.0107178
$691$	−1.41493e10	−1.63140	−0.815700	−	0.578475i	$-0.803648\pi$
$691$	−1.41493e10	−1.63140	−0.815700	+	0.578475i	$0.803648\pi$
$692$	3.84229e9	0.440776
$693$	3.47325e9	0.396432
$694$	2.39431e9	0.271909
$695$	2.49619e8	0.0282054
$696$	1.51084e9	0.169859
$697$	1.63567e10	1.82971
$698$	8.24782e8	0.0918006
$699$	2.55532e9	0.282992
$700$	8.01883e9	0.883625
$701$	−2.63084e9	−0.288457	−0.144228	−	0.989544i	$-0.546070\pi$
$701$	−2.63084e9	−0.288457	−0.144228	+	0.989544i	$0.546070\pi$
$702$	0	0
$703$	5.56892e9	0.604543
$704$	3.07638e8	0.0332304
$705$	−1.38118e7	−0.00148453
$706$	3.03385e9	0.324472
$707$	−9.39409e9	−0.999740
$708$	2.00790e9	0.212631
$709$	−1.48321e10	−1.56293	−0.781466	−	0.623947i	$-0.785528\pi$
$709$	−1.48321e10	−1.56293	−0.781466	+	0.623947i	$0.785528\pi$
$710$	−9.80092e7	−0.0102769
$711$	1.12078e10	1.16944
$712$	−5.27825e9	−0.548037
$713$	−1.07461e10	−1.11029
$714$	−6.66594e9	−0.685358
$715$	0	0
$716$	−8.60413e9	−0.876015
$717$	1.18164e9	0.119720
$718$	1.03363e10	1.04215
$719$	4.09811e9	0.411180	0.205590	−	0.978638i	$-0.434089\pi$
$719$	4.09811e9	0.411180	0.205590	+	0.978638i	$0.434089\pi$
$720$	4.99697e7	0.00498933
$721$	2.89725e9	0.287880
$722$	−5.62181e9	−0.555899
$723$	3.29132e9	0.323881
$724$	−9.97518e9	−0.976868
$725$	−1.24478e10	−1.21313
$726$	−2.68172e9	−0.260097
$727$	−1.07923e10	−1.04171	−0.520853	−	0.853646i	$-0.674386\pi$
$727$	−1.07923e10	−1.04171	−0.520853	+	0.853646i	$0.674386\pi$
$728$	0	0
$729$	−1.87136e9	−0.178901
$730$	7.94546e7	0.00755943
$731$	−2.48291e9	−0.235099
$732$	1.58254e9	0.149130
$733$	4.31755e9	0.404924	0.202462	−	0.979290i	$-0.435106\pi$
$733$	4.31755e9	0.404924	0.202462	+	0.979290i	$0.435106\pi$
$734$	−2.57331e9	−0.240191
$735$	−2.14431e8	−0.0199197
$736$	−3.09411e9	−0.286065
$737$	−8.44060e8	−0.0776671
$738$	−8.60311e9	−0.787877
$739$	1.59219e10	1.45124	0.725619	−	0.688097i	$-0.241554\pi$
$739$	1.59219e10	1.45124	0.725619	+	0.688097i	$0.241554\pi$
$740$	−1.70516e8	−0.0154687
$741$	0	0
$742$	−7.04441e9	−0.633040
$743$	−7.46206e9	−0.667418	−0.333709	−	0.942676i	$-0.608300\pi$
$743$	−7.46206e9	−0.667418	−0.333709	+	0.942676i	$0.608300\pi$
$744$	1.07854e9	0.0960136
$745$	−1.11920e8	−0.00991657
$746$	−4.76309e9	−0.420052
$747$	1.21975e9	0.107065
$748$	2.10699e9	0.184080
$749$	−2.11855e10	−1.84226
$750$	1.52999e8	0.0132427
$751$	7.78292e9	0.670507	0.335253	−	0.942128i	$-0.391178\pi$
$751$	7.78292e9	0.670507	0.335253	+	0.942128i	$0.391178\pi$
$752$	4.62071e8	0.0396229
$753$	4.76011e9	0.406289
$754$	0	0
$755$	2.56529e8	0.0216931
$756$	7.66344e9	0.645056
$757$	−1.36187e10	−1.14104	−0.570518	−	0.821285i	$-0.693257\pi$
$757$	−1.36187e10	−1.14104	−0.570518	+	0.821285i	$0.693257\pi$
$758$	1.47963e10	1.23399
$759$	−2.05113e9	−0.170273
$760$	−4.68217e7	−0.00386901
$761$	7.93048e9	0.652309	0.326154	−	0.945317i	$-0.394247\pi$
$761$	7.93048e9	0.652309	0.326154	+	0.945317i	$0.394247\pi$
$762$	4.54912e9	0.372464
$763$	−3.79049e10	−3.08929
$764$	−1.42927e9	−0.115954
$765$	3.42238e8	0.0276384
$766$	−1.55208e10	−1.24771
$767$	0	0
$768$	3.10546e8	0.0247378
$769$	1.80037e10	1.42764	0.713820	−	0.700329i	$-0.246964\pi$
$769$	1.80037e10	1.42764	0.713820	+	0.700329i	$0.246964\pi$
$770$	9.96484e7	0.00786598
$771$	−4.16300e9	−0.327127
$772$	1.72627e8	0.0135036
$773$	−1.79933e10	−1.40114	−0.700571	−	0.713583i	$-0.747071\pi$
$773$	−1.79933e10	−1.40114	−0.700571	+	0.713583i	$0.747071\pi$
$774$	1.30593e9	0.101234
$775$	−8.88606e9	−0.685730
$776$	3.14480e9	0.241589
$777$	−1.19641e10	−0.914967
$778$	−1.30527e10	−0.993740
$779$	8.06114e9	0.610963
$780$	0	0
$781$	2.17361e9	0.163269
$782$	−2.11913e10	−1.58465
$783$	−1.18961e10	−0.885600
$784$	7.17377e9	0.531669
$785$	3.21825e8	0.0237452
$786$	2.41341e9	0.177277
$787$	−4.61318e9	−0.337356	−0.168678	−	0.985671i	$-0.553950\pi$
$787$	−4.61318e9	−0.337356	−0.168678	+	0.985671i	$0.553950\pi$
$788$	4.67279e9	0.340200
$789$	8.84813e9	0.641331
$790$	3.21554e8	0.0232038
$791$	−1.66465e10	−1.19593
$792$	−1.10821e9	−0.0792653
$793$	0	0
$794$	−1.16910e10	−0.828857
$795$	−6.71848e7	−0.00474227
$796$	−7.07798e9	−0.497409
$797$	−2.80281e9	−0.196106	−0.0980528	−	0.995181i	$-0.531261\pi$
$797$	−2.80281e9	−0.196106	−0.0980528	+	0.995181i	$0.531261\pi$
$798$	−3.28520e9	−0.228850
$799$	3.16468e9	0.219491
$800$	−2.55857e9	−0.176678
$801$	1.90139e10	1.30724
$802$	1.72765e9	0.118262
$803$	−1.76211e9	−0.120096
$804$	−8.52037e8	−0.0578179
$805$	−1.00223e9	−0.0677144
$806$	0	0
$807$	3.26224e9	0.218504
$808$	2.99737e9	0.199894
$809$	8.36628e9	0.555537	0.277768	−	0.960648i	$-0.410405\pi$
$809$	8.36628e9	0.555537	0.277768	+	0.960648i	$0.410405\pi$
$810$	−1.40356e8	−0.00927968
$811$	−4.54358e9	−0.299106	−0.149553	−	0.988754i	$-0.547783\pi$
$811$	−4.54358e9	−0.299106	−0.149553	+	0.988754i	$0.547783\pi$
$812$	−1.63723e10	−1.07316
$813$	−3.48611e9	−0.227522
$814$	3.78163e9	0.245750
$815$	−5.83798e8	−0.0377756
$816$	2.12690e9	0.137035
$817$	−1.22366e9	−0.0785025
$818$	1.68027e10	1.07335
$819$	0	0
$820$	−2.46826e8	−0.0156330
$821$	1.50748e10	0.950717	0.475358	−	0.879792i	$-0.342318\pi$
$821$	1.50748e10	0.950717	0.475358	+	0.879792i	$0.342318\pi$
$822$	4.77770e9	0.300032
$823$	−1.67438e10	−1.04702	−0.523508	−	0.852021i	$-0.675377\pi$
$823$	−1.67438e10	−1.04702	−0.523508	+	0.852021i	$0.675377\pi$
$824$	−9.24424e8	−0.0575607
$825$	−1.69610e9	−0.105163
$826$	−2.17586e10	−1.34339
$827$	1.05833e10	0.650656	0.325328	−	0.945601i	$-0.394525\pi$
$827$	1.05833e10	0.650656	0.325328	+	0.945601i	$0.394525\pi$
$828$	1.11460e10	0.682356
$829$	2.53716e10	1.54670	0.773352	−	0.633976i	$-0.218578\pi$
$829$	2.53716e10	1.54670	0.773352	+	0.633976i	$0.218578\pi$
$830$	3.49950e7	0.00212438
$831$	5.83169e8	0.0352526
$832$	0	0
$833$	4.91325e10	2.94518
$834$	−5.58828e9	−0.333578
$835$	−4.75053e8	−0.0282383
$836$	1.03839e9	0.0614667
$837$	−8.49223e9	−0.500591
$838$	3.46692e9	0.203512
$839$	−8.37014e9	−0.489290	−0.244645	−	0.969613i	$-0.578671\pi$
$839$	−8.37014e9	−0.489290	−0.244645	+	0.969613i	$0.578671\pi$
$840$	1.00590e8	0.00585569
$841$	8.16504e9	0.473339
$842$	−2.08621e10	−1.20439
$843$	4.95268e9	0.284737
$844$	−4.99156e9	−0.285784
$845$	0	0
$846$	−1.66452e9	−0.0945133
$847$	2.90604e10	1.64327
$848$	2.24766e9	0.126574
$849$	−2.42996e9	−0.136277
$850$	−1.75234e10	−0.978705
$851$	−3.80343e10	−2.11554
$852$	2.19415e9	0.121542
$853$	−8.05268e9	−0.444241	−0.222121	−	0.975019i	$-0.571298\pi$
$853$	−8.05268e9	−0.444241	−0.222121	+	0.975019i	$0.571298\pi$
$854$	−1.71492e10	−0.942195
$855$	1.68666e8	0.00922883
$856$	6.75964e9	0.368354
$857$	2.21962e10	1.20461	0.602304	−	0.798267i	$-0.294249\pi$
$857$	2.21962e10	1.20461	0.602304	+	0.798267i	$0.294249\pi$
$858$	0	0
$859$	−4.25915e8	−0.0229270	−0.0114635	−	0.999934i	$-0.503649\pi$
$859$	−4.25915e8	−0.0229270	−0.0114635	+	0.999934i	$0.503649\pi$
$860$	3.74675e7	0.00200868
$861$	−1.73183e10	−0.924685
$862$	−5.45854e9	−0.290269
$863$	−1.84502e10	−0.977156	−0.488578	−	0.872520i	$-0.662484\pi$
$863$	−1.84502e10	−0.977156	−0.488578	+	0.872520i	$0.662484\pi$
$864$	−2.44517e9	−0.128977
$865$	−3.97105e8	−0.0208617
$866$	1.99674e10	1.04474
$867$	6.97159e9	0.363300
$868$	−1.16876e10	−0.606608
$869$	−7.13131e9	−0.368638
$870$	−1.56148e8	−0.00803930
$871$	0	0
$872$	1.20943e10	0.617693
$873$	−1.13286e10	−0.576268
$874$	−1.04438e10	−0.529137
$875$	−1.65798e9	−0.0836662
$876$	−1.77877e9	−0.0894035
$877$	−1.81523e10	−0.908725	−0.454363	−	0.890817i	$-0.650133\pi$
$877$	−1.81523e10	−0.908725	−0.454363	+	0.890817i	$0.650133\pi$
$878$	−1.21423e10	−0.605436
$879$	−9.30182e9	−0.461962
$880$	−3.17948e7	−0.00157277
$881$	2.80509e10	1.38207	0.691037	−	0.722819i	$-0.257154\pi$
$881$	2.80509e10	1.38207	0.691037	+	0.722819i	$0.257154\pi$
$882$	−2.58421e10	−1.26820
$883$	2.12689e9	0.103964	0.0519819	−	0.998648i	$-0.483446\pi$
$883$	2.12689e9	0.103964	0.0519819	+	0.998648i	$0.483446\pi$
$884$	0	0
$885$	−2.07519e8	−0.0100637
$886$	2.05226e10	0.991323
$887$	2.61618e10	1.25874	0.629368	−	0.777107i	$-0.283314\pi$
$887$	2.61618e10	1.25874	0.629368	+	0.777107i	$0.283314\pi$
$888$	3.81737e9	0.182944
$889$	−4.92965e10	−2.35321
$890$	5.45514e8	0.0259382
$891$	3.11276e9	0.147426
$892$	−1.80054e10	−0.849425
$893$	1.55966e9	0.0732909
$894$	2.50558e9	0.117281
$895$	8.89248e8	0.0414612
$896$	−3.36523e9	−0.156292
$897$	0	0
$898$	−1.77963e10	−0.820091
$899$	1.81429e10	0.832814
$900$	9.21674e9	0.421433
$901$	1.53940e10	0.701156
$902$	5.47400e9	0.248360
$903$	2.62887e9	0.118812
$904$	5.31141e9	0.239122
$905$	1.03095e9	0.0462345
$906$	−5.74296e9	−0.256559
$907$	1.57121e10	0.699213	0.349607	−	0.936897i	$-0.386315\pi$
$907$	1.57121e10	0.699213	0.349607	+	0.936897i	$0.386315\pi$
$908$	1.83780e10	0.814700
$909$	−1.07974e10	−0.476812
$910$	0	0
$911$	−3.17054e10	−1.38937	−0.694687	−	0.719312i	$-0.744457\pi$
$911$	−3.17054e10	−1.38937	−0.694687	+	0.719312i	$0.744457\pi$
$912$	1.04821e9	0.0457578
$913$	−7.76106e8	−0.0337500
$914$	−1.26576e10	−0.548328
$915$	−1.63557e8	−0.00705823
$916$	1.13727e10	0.488909
$917$	−2.61529e10	−1.12002
$918$	−1.67468e10	−0.714466
$919$	−7.15392e9	−0.304047	−0.152023	−	0.988377i	$-0.548579\pi$
$919$	−7.15392e9	−0.304047	−0.152023	+	0.988377i	$0.548579\pi$
$920$	3.19781e8	0.0135393
$921$	−7.48660e8	−0.0315774
$922$	−1.06634e10	−0.448059
$923$	0	0
$924$	−2.23085e9	−0.0930290
$925$	−3.14511e10	−1.30659
$926$	1.62617e10	0.673017
$927$	3.33006e9	0.137301
$928$	5.22389e9	0.214574
$929$	3.59995e10	1.47313	0.736566	−	0.676365i	$-0.236446\pi$
$929$	3.59995e10	1.47313	0.736566	+	0.676365i	$0.236446\pi$
$930$	−1.11469e8	−0.00454426
$931$	2.42141e10	0.983433
$932$	8.83526e9	0.357490
$933$	−1.07446e9	−0.0433118
$934$	−2.48540e8	−0.00998119
$935$	−2.17760e8	−0.00871238
$936$	0	0
$937$	−1.38764e10	−0.551047	−0.275523	−	0.961294i	$-0.588851\pi$
$937$	−1.38764e10	−0.551047	−0.275523	+	0.961294i	$0.588851\pi$
$938$	9.23310e9	0.365290
$939$	3.46701e9	0.136655
$940$	−4.77556e7	−0.00187532
$941$	−1.24319e9	−0.0486379	−0.0243190	−	0.999704i	$-0.507742\pi$
$941$	−1.24319e9	−0.0486379	−0.0243190	+	0.999704i	$0.507742\pi$
$942$	−7.20476e9	−0.280829
$943$	−5.50556e10	−2.13801
$944$	6.94252e9	0.268606
$945$	−7.92026e8	−0.0305301
$946$	−8.30940e8	−0.0319118
$947$	−1.08695e10	−0.415896	−0.207948	−	0.978140i	$-0.566679\pi$
$947$	−1.08695e10	−0.415896	−0.207948	+	0.978140i	$0.566679\pi$
$948$	−7.19870e9	−0.274426
$949$	0	0
$950$	−8.63611e9	−0.326803
$951$	1.12838e9	0.0425426
$952$	−2.30481e10	−0.865778
$953$	4.97595e9	0.186230	0.0931152	−	0.995655i	$-0.470318\pi$
$953$	4.97595e9	0.186230	0.0931152	+	0.995655i	$0.470318\pi$
$954$	−8.09675e9	−0.301920
$955$	1.47717e8	0.00548804
$956$	4.08563e9	0.151237
$957$	3.46298e9	0.127720
$958$	2.36496e10	0.869051
$959$	−5.17736e10	−1.89558
$960$	−3.20953e7	−0.00117082
$961$	−1.45610e10	−0.529247
$962$	0	0
$963$	−2.43503e10	−0.878642
$964$	1.13801e10	0.409143
$965$	−1.78412e7	−0.000639116 0
$966$	2.24371e10	0.800841
$967$	1.95659e10	0.695838	0.347919	−	0.937525i	$-0.386888\pi$
$967$	1.95659e10	0.695838	0.347919	+	0.937525i	$0.386888\pi$
$968$	−9.27230e9	−0.328567
$969$	7.17907e9	0.253475
$970$	−3.25019e8	−0.0114343
$971$	−6.28877e9	−0.220444	−0.110222	−	0.993907i	$-0.535156\pi$
$971$	−6.28877e9	−0.220444	−0.110222	+	0.993907i	$0.535156\pi$
$972$	1.35867e10	0.474550
$973$	6.05574e10	2.10752
$974$	2.51485e9	0.0872078
$975$	0	0
$976$	5.47178e9	0.188389
$977$	−1.75153e9	−0.0600878	−0.0300439	−	0.999549i	$-0.509565\pi$
$977$	−1.75153e9	−0.0600878	−0.0300439	+	0.999549i	$0.509565\pi$
$978$	1.30696e10	0.446762
$979$	−1.20982e10	−0.412079
$980$	−7.41417e8	−0.0251635
$981$	−4.35674e10	−1.47340
$982$	1.39440e10	0.469890
$983$	3.03137e10	1.01789	0.508946	−	0.860799i	$-0.330035\pi$
$983$	3.03137e10	1.01789	0.508946	+	0.860799i	$0.330035\pi$
$984$	5.52574e9	0.184887
$985$	−4.82939e8	−0.0161015
$986$	3.57780e10	1.18863
$987$	−3.35072e9	−0.110925
$988$	0	0
$989$	8.35730e9	0.274713
$990$	1.14535e8	0.00375157
$991$	4.08606e10	1.33367	0.666833	−	0.745207i	$-0.267649\pi$
$991$	4.08606e10	1.33367	0.666833	+	0.745207i	$0.267649\pi$
$992$	3.72917e9	0.121289
$993$	−8.41219e9	−0.272638
$994$	−2.37769e10	−0.767897
$995$	7.31517e8	0.0235420
$996$	−7.83441e8	−0.0251246
$997$	5.95647e10	1.90351	0.951757	−	0.306853i	$-0.0992759\pi$
$997$	5.95647e10	1.90351	0.951757	+	0.306853i	$0.0992759\pi$
$998$	2.78277e10	0.886176
$999$	−3.00572e10	−0.953826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.a.l.1.3		5
13.5	odd	4		26.8.b.a.25.3	✓	10
13.8	odd	4		26.8.b.a.25.8	yes	10
13.12	even	2		338.8.a.k.1.3		5
39.5	even	4		234.8.b.c.181.8		10
39.8	even	4		234.8.b.c.181.3		10
52.31	even	4		208.8.f.c.129.6		10
52.47	even	4		208.8.f.c.129.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.b.a.25.3	✓	10	13.5	odd	4
26.8.b.a.25.8	yes	10	13.8	odd	4
208.8.f.c.129.5		10	52.47	even	4
208.8.f.c.129.6		10	52.31	even	4
234.8.b.c.181.3		10	39.8	even	4
234.8.b.c.181.8		10	39.5	even	4
338.8.a.k.1.3		5	13.12	even	2
338.8.a.l.1.3		5	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+8.00000 q^{2} +18.5100 q^{3} +64.0000 q^{4} -6.61448 q^{5} +148.080 q^{6} -1604.67 q^{7} +512.000 q^{8} -1844.38 q^{9} -52.9158 q^{10} +1173.55 q^{11} +1184.64 q^{12} -12837.3 q^{14} -122.434 q^{15} +4096.00 q^{16} +28053.1 q^{17} -14755.1 q^{18} +13825.5 q^{19} -423.327 q^{20} -29702.3 q^{21} +9388.37 q^{22} -94424.9 q^{23} +9477.10 q^{24} -78081.2 q^{25} -74620.7 q^{27} -102699. q^{28} +159421. q^{29} -979.470 q^{30} +113805. q^{31} +32768.0 q^{32} +21722.3 q^{33} +224425. q^{34} +10614.0 q^{35} -118040. q^{36} +402800. q^{37} +110604. q^{38} -3386.61 q^{40} +583062. q^{41} -237618. q^{42} -88507.4 q^{43} +75107.0 q^{44} +12199.6 q^{45} -755399. q^{46} +112810. q^{47} +75816.8 q^{48} +1.75141e6 q^{49} -624650. q^{50} +519262. q^{51} +548744. q^{53} -596966. q^{54} -7762.40 q^{55} -821589. q^{56} +255910. q^{57} +1.27536e6 q^{58} +1.69495e6 q^{59} -7835.76 q^{60} +1.33588e6 q^{61} +910442. q^{62} +2.95961e6 q^{63} +262144. q^{64} +173778. q^{66} -719239. q^{67} +1.79540e6 q^{68} -1.74780e6 q^{69} +84912.2 q^{70} +1.85217e6 q^{71} -944323. q^{72} -1.50153e6 q^{73} +3.22240e6 q^{74} -1.44528e6 q^{75} +884833. q^{76} -1.88315e6 q^{77} -6.07672e6 q^{79} -27092.9 q^{80} +2.65244e6 q^{81} +4.66450e6 q^{82} -661334. q^{83} -1.90095e6 q^{84} -185557. q^{85} -708059. q^{86} +2.95087e6 q^{87} +600856. q^{88} -1.03091e7 q^{89} +97597.0 q^{90} -6.04319e6 q^{92} +2.10653e6 q^{93} +902482. q^{94} -91448.6 q^{95} +606534. q^{96} +6.14220e6 q^{97} +1.40113e7 q^{98} -2.16447e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 40 q^{2} + 27 q^{3} + 320 q^{4} + 71 q^{5} + 216 q^{6} + 237 q^{7} + 2560 q^{8} + 6980 q^{9} + 568 q^{10} + 1080 q^{11} + 1728 q^{12} + 1896 q^{14} + 27187 q^{15} + 20480 q^{16} + 3459 q^{17} + 55840 q^{18}+ \cdots - 41946120 q^{99}+O(q^{100}) \) 5 * q + 40 * q^2 + 27 * q^3 + 320 * q^4 + 71 * q^5 + 216 * q^6 + 237 * q^7 + 2560 * q^8 + 6980 * q^9 + 568 * q^10 + 1080 * q^11 + 1728 * q^12 + 1896 * q^14 + 27187 * q^15 + 20480 * q^16 + 3459 * q^17 + 55840 * q^18 - 37686 * q^19 + 4544 * q^20 + 112537 * q^21 + 8640 * q^22 - 47082 * q^23 + 13824 * q^24 + 165394 * q^25 - 26775 * q^27 + 15168 * q^28 - 65652 * q^29 + 217496 * q^30 + 263160 * q^31 + 163840 * q^32 - 347328 * q^33 + 27672 * q^34 + 436725 * q^35 + 446720 * q^36 + 1134087 * q^37 - 301488 * q^38 + 36352 * q^40 - 1116800 * q^41 + 900296 * q^42 - 1370387 * q^43 + 69120 * q^44 + 1441234 * q^45 - 376656 * q^46 + 515061 * q^47 + 110592 * q^48 + 1541952 * q^49 + 1323152 * q^50 - 551593 * q^51 - 1802958 * q^53 - 214200 * q^54 + 1999920 * q^55 + 121344 * q^56 + 3752874 * q^57 - 525216 * q^58 + 410726 * q^59 + 1739968 * q^60 + 5863828 * q^61 + 2105280 * q^62 + 1373518 * q^63 + 1310720 * q^64 - 2778624 * q^66 + 3031692 * q^67 + 221376 * q^68 - 1017898 * q^69 + 3493800 * q^70 + 4201215 * q^71 + 3573760 * q^72 - 736362 * q^73 + 9072696 * q^74 + 16427860 * q^75 - 2411904 * q^76 + 3573960 * q^77 - 14224614 * q^79 + 290816 * q^80 + 24655733 * q^81 - 8934400 * q^82 + 1634438 * q^83 + 7202368 * q^84 + 19129611 * q^85 - 10963096 * q^86 + 25078160 * q^87 + 552960 * q^88 - 7827414 * q^89 + 11529872 * q^90 - 3013248 * q^92 - 27372256 * q^93 + 4120488 * q^94 - 8273742 * q^95 + 884736 * q^96 + 22392876 * q^97 + 12335616 * q^98 - 41946120 * q^99

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