LMFDB - Embedded newform 338.6.b.f.337.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("338.337");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$54.2097310968$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 366x^{10} + 54661x^{8} + 4259913x^{6} + 182592336x^{4} + 4080040448x^{2} + 37134831616 $ x^12 + 366x^10 + 54661x^8 + 4259913x^6 + 182592336x^4 + 4080040448*x^2 + 37134831616
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 13^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$-8.91319i$ of defining polynomial
Character	$\chi$	$=$	338.337
Dual form			338.6.b.f.337.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00000i q^{2} +7.64039 q^{3} -16.0000 q^{4} +7.89924i q^{5} -30.5616i q^{6} -19.7414i q^{7} +64.0000i q^{8} -184.624 q^{9} +31.5970 q^{10} -191.688i q^{11} -122.246 q^{12} -78.9657 q^{14} +60.3533i q^{15} +256.000 q^{16} -626.910 q^{17} +738.498i q^{18} +166.142i q^{19} -126.388i q^{20} -150.832i q^{21} -766.750 q^{22} +2094.88 q^{23} +488.985i q^{24} +3062.60 q^{25} -3267.22 q^{27} +315.863i q^{28} -2637.10 q^{29} +241.413 q^{30} +4239.82i q^{31} -1024.00i q^{32} -1464.57i q^{33} +2507.64i q^{34} +155.942 q^{35} +2953.99 q^{36} +3248.76i q^{37} +664.568 q^{38} -505.551 q^{40} +13951.1i q^{41} -603.329 q^{42} -3448.84 q^{43} +3067.00i q^{44} -1458.39i q^{45} -8379.51i q^{46} +18989.0i q^{47} +1955.94 q^{48} +16417.3 q^{49} -12250.4i q^{50} -4789.84 q^{51} -16815.3 q^{53} +13068.9i q^{54} +1514.19 q^{55} +1263.45 q^{56} +1269.39i q^{57} +10548.4i q^{58} +1190.77i q^{59} -965.653i q^{60} +16179.7 q^{61} +16959.3 q^{62} +3644.75i q^{63} -4096.00 q^{64} -5858.27 q^{66} +41861.8i q^{67} +10030.6 q^{68} +16005.7 q^{69} -623.769i q^{70} +65659.8i q^{71} -11816.0i q^{72} +61713.3i q^{73} +12995.0 q^{74} +23399.5 q^{75} -2658.27i q^{76} -3784.19 q^{77} -82077.3 q^{79} +2022.21i q^{80} +19900.9 q^{81} +55804.2 q^{82} -83347.9i q^{83} +2413.32i q^{84} -4952.12i q^{85} +13795.4i q^{86} -20148.5 q^{87} +12268.0 q^{88} +48879.0i q^{89} -5833.57 q^{90} -33518.0 q^{92} +32393.9i q^{93} +75955.8 q^{94} -1312.40 q^{95} -7823.76i q^{96} +44969.4i q^{97} -65669.1i q^{98} +35390.2i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 72 q^{3} - 192 q^{4} + 588 q^{9} + 992 q^{10} + 1152 q^{12} - 928 q^{14} + 3072 q^{16} + 1960 q^{17} + 1664 q^{22} + 10816 q^{23} + 5044 q^{25} - 30792 q^{27} + 6556 q^{29} - 9744 q^{30} + 13436 q^{35}+ \cdots + 501812 q^{95}+O(q^{100}) $ 12 * q - 72 * q^3 - 192 * q^4 + 588 * q^9 + 992 * q^10 + 1152 * q^12 - 928 * q^14 + 3072 * q^16 + 1960 * q^17 + 1664 * q^22 + 10816 * q^23 + 5044 * q^25 - 30792 * q^27 + 6556 * q^29 - 9744 * q^30 + 13436 * q^35 - 9408 * q^36 + 432 * q^38 - 15872 * q^40 + 34080 * q^42 + 32100 * q^43 - 18432 * q^48 + 123320 * q^49 - 39336 * q^51 - 44652 * q^53 - 75156 * q^55 + 14848 * q^56 - 136408 * q^61 + 5536 * q^62 - 49152 * q^64 - 198016 * q^66 - 31360 * q^68 + 60736 * q^69 - 75392 * q^74 + 406184 * q^75 + 313512 * q^77 - 132412 * q^79 + 61380 * q^81 + 277424 * q^82 - 743548 * q^87 - 26624 * q^88 + 402000 * q^90 - 173056 * q^92 - 372224 * q^94 + 501812 * q^95

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	7.64039	0.490131	0.245066	−	0.969506i	$-0.421190\pi$
$3$	7.64039	0.490131	0.245066	+	0.969506i	$0.421190\pi$
$4$	−16.0000	−0.500000
$5$	7.89924i	0.141306i	0.997501	+	0.0706529i	$0.0225083\pi$
$5$	7.89924i	0.141306i	−0.997501	+	0.0706529i	$0.977492\pi$
$6$	− 30.5616i	− 0.346575i
$7$	− 19.7414i	− 0.152277i	−0.997097	−	0.0761384i	$-0.975741\pi$
$7$	− 19.7414i	− 0.152277i	0.997097	−	0.0761384i	$-0.0242591\pi$
$8$	64.0000i	0.353553i
$9$	−184.624	−0.759771
$10$	31.5970	0.0999184
$11$	− 191.688i	− 0.477653i	−0.971062	−	0.238826i	$-0.923237\pi$
$11$	− 191.688i	− 0.477653i	0.971062	−	0.238826i	$-0.0767627\pi$
$12$	−122.246	−0.245066
$13$	0	0
$13$	0	0
$14$	−78.9657	−0.107676
$15$	60.3533i	0.0692585i
$16$	256.000	0.250000
$17$	−626.910	−0.526118	−0.263059	−	0.964780i	$-0.584731\pi$
$17$	−626.910	−0.526118	−0.263059	+	0.964780i	$0.584731\pi$
$18$	738.498i	0.537239i
$19$	166.142i	0.105583i	0.998606	+	0.0527917i	$0.0168119\pi$
$19$	166.142i	0.105583i	−0.998606	+	0.0527917i	$0.983188\pi$
$20$	− 126.388i	− 0.0706529i
$21$	− 150.832i	− 0.0746356i
$22$	−766.750	−0.337752
$23$	2094.88	0.825731	0.412866	−	0.910792i	$-0.364528\pi$
$23$	2094.88	0.825731	0.412866	+	0.910792i	$0.364528\pi$
$24$	488.985i	0.173288i
$25$	3062.60	0.980033
$26$	0	0
$27$	−3267.22	−0.862519
$28$	315.863i	0.0761384i
$29$	−2637.10	−0.582280	−0.291140	−	0.956680i	$-0.594035\pi$
$29$	−2637.10	−0.582280	−0.291140	+	0.956680i	$0.594035\pi$
$30$	241.413	0.0489731
$31$	4239.82i	0.792397i	0.918165	+	0.396199i	$0.129671\pi$
$31$	4239.82i	0.792397i	−0.918165	+	0.396199i	$0.870329\pi$
$32$	− 1024.00i	− 0.176777i
$33$	− 1464.57i	− 0.234113i
$34$	2507.64i	0.372022i
$35$	155.942	0.0215176
$36$	2953.99	0.379886
$37$	3248.76i	0.390133i	0.980790	+	0.195067i	$0.0624923\pi$
$37$	3248.76i	0.390133i	−0.980790	+	0.195067i	$0.937508\pi$
$38$	664.568	0.0746587
$39$	0	0
$40$	−505.551	−0.0499592
$41$	13951.1i	1.29613i	0.761587	+	0.648063i	$0.224421\pi$
$41$	13951.1i	1.29613i	−0.761587	+	0.648063i	$0.775579\pi$
$42$	−603.329	−0.0527753
$43$	−3448.84	−0.284448	−0.142224	−	0.989835i	$-0.545425\pi$
$43$	−3448.84	−0.284448	−0.142224	+	0.989835i	$0.545425\pi$
$44$	3067.00i	0.238826i
$45$	− 1458.39i	− 0.107360i
$46$	− 8379.51i	− 0.583880i
$47$	18989.0i	1.25388i	0.779067	+	0.626941i	$0.215693\pi$
$47$	18989.0i	1.25388i	−0.779067	+	0.626941i	$0.784307\pi$
$48$	1955.94	0.122533
$49$	16417.3	0.976812
$50$	− 12250.4i	− 0.692988i
$51$	−4789.84	−0.257867
$52$	0	0
$53$	−16815.3	−0.822271	−0.411135	−	0.911574i	$-0.634868\pi$
$53$	−16815.3	−0.822271	−0.411135	+	0.911574i	$0.634868\pi$
$54$	13068.9i	0.609893i
$55$	1514.19	0.0674952
$56$	1263.45	0.0538379
$57$	1269.39i	0.0517497i
$58$	10548.4i	0.411734i
$59$	1190.77i	0.0445346i	0.999752	+	0.0222673i	$0.00708849\pi$
$59$	1190.77i	0.0445346i	−0.999752	+	0.0222673i	$0.992912\pi$
$60$	− 965.653i	− 0.0346292i
$61$	16179.7	0.556731	0.278366	−	0.960475i	$-0.410207\pi$
$61$	16179.7	0.556731	0.278366	+	0.960475i	$0.410207\pi$
$62$	16959.3	0.560309
$63$	3644.75i	0.115695i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	−5858.27	−0.165543
$67$	41861.8i	1.13928i	0.821894	+	0.569640i	$0.192917\pi$
$67$	41861.8i	1.13928i	−0.821894	+	0.569640i	$0.807083\pi$
$68$	10030.6	0.263059
$69$	16005.7	0.404717
$70$	− 623.769i	− 0.0152152i
$71$	65659.8i	1.54580i	0.634527	+	0.772901i	$0.281195\pi$
$71$	65659.8i	1.54580i	−0.634527	+	0.772901i	$0.718805\pi$
$72$	− 11816.0i	− 0.268620i
$73$	61713.3i	1.35541i	0.735332	+	0.677707i	$0.237026\pi$
$73$	61713.3i	1.35541i	−0.735332	+	0.677707i	$0.762974\pi$
$74$	12995.0	0.275866
$75$	23399.5	0.480345
$76$	− 2658.27i	− 0.0527917i
$77$	−3784.19	−0.0727354
$78$	0	0
$79$	−82077.3	−1.47964	−0.739819	−	0.672806i	$-0.765089\pi$
$79$	−82077.3	−1.47964	−0.739819	+	0.672806i	$0.765089\pi$
$80$	2022.21i	0.0353265i
$81$	19900.9	0.337024
$82$	55804.2	0.916500
$83$	− 83347.9i	− 1.32800i	−0.747731	−	0.664002i	$-0.768857\pi$
$83$	− 83347.9i	− 1.32800i	0.747731	−	0.664002i	$-0.231143\pi$
$84$	2413.32i	0.0373178i
$85$	− 4952.12i	− 0.0743436i
$86$	13795.4i	0.201135i
$87$	−20148.5	−0.285394
$88$	12268.0	0.168876
$89$	48879.0i	0.654104i	0.945006	+	0.327052i	$0.106055\pi$
$89$	48879.0i	0.654104i	−0.945006	+	0.327052i	$0.893945\pi$
$90$	−5833.57	−0.0759151
$91$	0	0
$92$	−33518.0	−0.412866
$93$	32393.9i	0.388379i
$94$	75955.8	0.886628
$95$	−1312.40	−0.0149196
$96$	− 7823.76i	− 0.0866438i
$97$	44969.4i	0.485275i	0.970117	+	0.242638i	$0.0780126\pi$
$97$	44969.4i	0.485275i	−0.970117	+	0.242638i	$0.921987\pi$
$98$	− 65669.1i	− 0.690710i
$99$	35390.2i	0.362907i
$100$	−49001.6	−0.490016
$101$	196960.	1.92121	0.960604	−	0.277921i	$-0.0896455\pi$
$101$	196960.	1.92121	0.960604	+	0.277921i	$0.0896455\pi$
$102$	19159.4i	0.182339i
$103$	−3601.43	−0.0334489	−0.0167244	−	0.999860i	$-0.505324\pi$
$103$	−3601.43	−0.0334489	−0.0167244	+	0.999860i	$0.505324\pi$
$104$	0	0
$105$	1191.46	0.0105464
$106$	67261.2i	0.581433i
$107$	113378.	0.957344	0.478672	−	0.877994i	$-0.341118\pi$
$107$	113378.	0.957344	0.478672	+	0.877994i	$0.341118\pi$
$108$	52275.5	0.431260
$109$	− 226249.i	− 1.82398i	−0.410210	−	0.911991i	$-0.634544\pi$
$109$	− 226249.i	− 1.82398i	0.410210	−	0.911991i	$-0.365456\pi$
$110$	− 6056.75i	− 0.0477263i
$111$	24821.8i	0.191217i
$112$	− 5053.81i	− 0.0380692i
$113$	−117866.	−0.868345	−0.434173	−	0.900830i	$-0.642959\pi$
$113$	−117866.	−0.868345	−0.434173	+	0.900830i	$0.642959\pi$
$114$	5077.56	0.0365926
$115$	16547.9i	0.116681i
$116$	42193.6	0.291140
$117$	0	0
$118$	4763.08	0.0314907
$119$	12376.1i	0.0801155i
$120$	−3862.61	−0.0244866
$121$	124307.	0.771848
$122$	− 64718.8i	− 0.393669i
$123$	106592.i	0.635272i
$124$	− 67837.1i	− 0.396199i
$125$	48877.4i	0.279790i
$126$	14579.0	0.0818090
$127$	251718.	1.38486	0.692429	−	0.721486i	$-0.256541\pi$
$127$	251718.	1.38486	0.692429	+	0.721486i	$0.256541\pi$
$128$	16384.0i	0.0883883i
$129$	−26350.5	−0.139417
$130$	0	0
$131$	−2258.42	−0.0114981	−0.00574907	−	0.999983i	$-0.501830\pi$
$131$	−2258.42	−0.0114981	−0.00574907	+	0.999983i	$0.501830\pi$
$132$	23433.1i	0.117056i
$133$	3279.88	0.0160779
$134$	167447.	0.805593
$135$	− 25808.5i	− 0.121879i
$136$	− 40122.3i	− 0.186011i
$137$	276784.i	1.25991i	0.776631	+	0.629956i	$0.216927\pi$
$137$	276784.i	1.25991i	−0.776631	+	0.629956i	$0.783073\pi$
$138$	− 64022.7i	− 0.286178i
$139$	−210899.	−0.925842	−0.462921	−	0.886399i	$-0.653199\pi$
$139$	−210899.	−0.925842	−0.462921	+	0.886399i	$0.653199\pi$
$140$	−2495.08	−0.0107588
$141$	145083.i	0.614567i
$142$	262639.	1.09305
$143$	0	0
$144$	−47263.8	−0.189943
$145$	− 20831.1i	− 0.0822796i
$146$	246853.	0.958422
$147$	125434.	0.478766
$148$	− 51980.1i	− 0.195067i
$149$	171949.i	0.634504i	0.948341	+	0.317252i	$0.102760\pi$
$149$	171949.i	0.634504i	−0.948341	+	0.317252i	$0.897240\pi$
$150$	− 93597.9i	− 0.339655i
$151$	345547.i	1.23329i	0.787242	+	0.616644i	$0.211508\pi$
$151$	345547.i	1.23329i	−0.787242	+	0.616644i	$0.788492\pi$
$152$	−10633.1	−0.0373294
$153$	115743.	0.399729
$154$	15136.7i	0.0514317i
$155$	−33491.3	−0.111970
$156$	0	0
$157$	153393.	0.496656	0.248328	−	0.968676i	$-0.420119\pi$
$157$	153393.	0.496656	0.248328	+	0.968676i	$0.420119\pi$
$158$	328309.i	1.04626i
$159$	−128475.	−0.403021
$160$	8088.82	0.0249796
$161$	− 41355.9i	− 0.125740i
$162$	− 79603.6i	− 0.238312i
$163$	257757.i	0.759873i	0.925013	+	0.379937i	$0.124054\pi$
$163$	257757.i	0.759873i	−0.925013	+	0.379937i	$0.875946\pi$
$164$	− 223217.i	− 0.648063i
$165$	11569.0	0.0330815
$166$	−333392.	−0.939041
$167$	− 274189.i	− 0.760779i	−0.924826	−	0.380390i	$-0.875790\pi$
$167$	− 274189.i	− 0.760779i	0.924826	−	0.380390i	$-0.124210\pi$
$168$	9653.26	0.0263877
$169$	0	0
$170$	−19808.5	−0.0525688
$171$	− 30673.9i	− 0.0802192i
$172$	55181.5	0.142224
$173$	−227782.	−0.578635	−0.289317	−	0.957233i	$-0.593428\pi$
$173$	−227782.	−0.578635	−0.289317	+	0.957233i	$0.593428\pi$
$174$	80594.0i	0.201804i
$175$	− 60460.1i	− 0.149236i
$176$	− 49072.0i	− 0.119413i
$177$	9097.95i	0.0218278i
$178$	195516.	0.462522
$179$	81060.2	0.189093	0.0945465	−	0.995520i	$-0.469860\pi$
$179$	81060.2	0.189093	0.0945465	+	0.995520i	$0.469860\pi$
$180$	23334.3i	0.0536801i
$181$	−213265.	−0.483863	−0.241932	−	0.970293i	$-0.577781\pi$
$181$	−213265.	−0.483863	−0.241932	+	0.970293i	$0.577781\pi$
$182$	0	0
$183$	123619.	0.272872
$184$	134072.i	0.291940i
$185$	−25662.7	−0.0551281
$186$	129575.	0.274625
$187$	120171.i	0.251302i
$188$	− 303823.i	− 0.626941i
$189$	64499.6i	0.131342i
$190$	5249.59i	0.0105497i
$191$	356122.	0.706343	0.353172	−	0.935559i	$-0.385103\pi$
$191$	356122.	0.706343	0.353172	+	0.935559i	$0.385103\pi$
$192$	−31295.0	−0.0612664
$193$	− 331586.i	− 0.640772i	−0.947287	−	0.320386i	$-0.896187\pi$
$193$	− 331586.i	− 0.640772i	0.947287	−	0.320386i	$-0.103813\pi$
$194$	179878.	0.343141
$195$	0	0
$196$	−262676.	−0.488406
$197$	− 525162.i	− 0.964113i	−0.876140	−	0.482056i	$-0.839890\pi$
$197$	− 525162.i	− 0.964113i	0.876140	−	0.482056i	$-0.160110\pi$
$198$	141561.	0.256614
$199$	−774344.	−1.38612	−0.693061	−	0.720879i	$-0.743738\pi$
$199$	−774344.	−1.38612	−0.693061	+	0.720879i	$0.743738\pi$
$200$	196007.i	0.346494i
$201$	319840.i	0.558397i
$202$	− 787839.i	− 1.35850i
$203$	52060.2i	0.0886677i
$204$	76637.5	0.128933
$205$	−110203.	−0.183150
$206$	14405.7i	0.0236519i
$207$	−386765.	−0.627367
$208$	0	0
$209$	31847.4	0.0504322
$210$	− 4765.84i	− 0.00745747i
$211$	−1.21664e6	−1.88129	−0.940647	−	0.339386i	$-0.889781\pi$
$211$	−1.21664e6	−1.88129	−0.940647	+	0.339386i	$0.889781\pi$
$212$	269045.	0.411135
$213$	501667.i	0.757646i
$214$	− 453511.i	− 0.676945i
$215$	− 27243.2i	− 0.0401941i
$216$	− 209102.i	− 0.304947i
$217$	83700.1	0.120664
$218$	−904997.	−1.28975
$219$	471514.i	0.664331i
$220$	−24227.0	−0.0337476
$221$	0	0
$222$	99287.1	0.135211
$223$	− 759944.i	− 1.02334i	−0.859183	−	0.511669i	$-0.829027\pi$
$223$	− 759944.i	− 1.02334i	0.859183	−	0.511669i	$-0.170973\pi$
$224$	−20215.2	−0.0269190
$225$	−565431.	−0.744601
$226$	471464.i	0.614013i
$227$	− 50248.5i	− 0.0647230i	−0.999476	−	0.0323615i	$-0.989697\pi$
$227$	− 50248.5i	− 0.0647230i	0.999476	−	0.0323615i	$-0.0103028\pi$
$228$	− 20310.3i	− 0.0258749i
$229$	− 372379.i	− 0.469241i	−0.972087	−	0.234621i	$-0.924615\pi$
$229$	− 372379.i	− 0.469241i	0.972087	−	0.234621i	$-0.0753848\pi$
$230$	66191.7	0.0825057
$231$	−28912.7	−0.0356499
$232$	− 168775.i	− 0.205867i
$233$	−737473.	−0.889931	−0.444966	−	0.895548i	$-0.646784\pi$
$233$	−737473.	−0.889931	−0.444966	+	0.895548i	$0.646784\pi$
$234$	0	0
$235$	−149998.	−0.177181
$236$	− 19052.3i	− 0.0222673i
$237$	−627103.	−0.725217
$238$	49504.4	0.0566502
$239$	897518.i	1.01636i	0.861250	+	0.508181i	$0.169682\pi$
$239$	897518.i	1.01636i	−0.861250	+	0.508181i	$0.830318\pi$
$240$	15450.4i	0.0173146i
$241$	448304.i	0.497199i	0.968606	+	0.248599i	$0.0799702\pi$
$241$	448304.i	0.497199i	−0.968606	+	0.248599i	$0.920030\pi$
$242$	− 497227.i	− 0.545779i
$243$	945985.	1.02770
$244$	−258875.	−0.278366
$245$	129684.i	0.138029i
$246$	426366.	0.449205
$247$	0	0
$248$	−271348.	−0.280155
$249$	− 636811.i	− 0.650897i
$250$	195509.	0.197842
$251$	−1.83575e6	−1.83920	−0.919600	−	0.392855i	$-0.871487\pi$
$251$	−1.83575e6	−1.83920	−0.919600	+	0.392855i	$0.871487\pi$
$252$	− 58316.0i	− 0.0578477i
$253$	− 401562.i	− 0.394413i
$254$	− 1.00687e6i	− 0.979242i
$255$	− 37836.1i	− 0.0364381i
$256$	65536.0	0.0625000
$257$	−1.51773e6	−1.43338	−0.716689	−	0.697393i	$-0.754343\pi$
$257$	−1.51773e6	−1.43338	−0.716689	+	0.697393i	$0.754343\pi$
$258$	105402.i	0.0985825i
$259$	64135.1	0.0594082
$260$	0	0
$261$	486874.	0.442400
$262$	9033.70i	0.00813041i
$263$	−220646.	−0.196701	−0.0983504	−	0.995152i	$-0.531357\pi$
$263$	−220646.	−0.196701	−0.0983504	+	0.995152i	$0.531357\pi$
$264$	93732.4	0.0827713
$265$	− 132828.i	− 0.116192i
$266$	− 13119.5i	− 0.0113688i
$267$	373455.i	0.320597i
$268$	− 669788.i	− 0.569640i
$269$	−1.45268e6	−1.22402	−0.612011	−	0.790849i	$-0.709639\pi$
$269$	−1.45268e6	−1.22402	−0.612011	+	0.790849i	$0.709639\pi$
$270$	−103234.	−0.0861815
$271$	− 831510.i	− 0.687771i	−0.939012	−	0.343886i	$-0.888257\pi$
$271$	− 831510.i	− 0.687771i	0.939012	−	0.343886i	$-0.111743\pi$
$272$	−160489.	−0.131530
$273$	0	0
$274$	1.10714e6	0.890892
$275$	− 587063.i	− 0.468115i
$276$	−256091.	−0.202358
$277$	−674969.	−0.528548	−0.264274	−	0.964448i	$-0.585132\pi$
$277$	−674969.	−0.528548	−0.264274	+	0.964448i	$0.585132\pi$
$278$	843595.i	0.654669i
$279$	− 782774.i	− 0.602041i
$280$	9980.31i	0.00760762i
$281$	− 674842.i	− 0.509843i	−0.966962	−	0.254921i	$-0.917950\pi$
$281$	− 674842.i	− 0.509843i	0.966962	−	0.254921i	$-0.0820496\pi$
$282$	580332.	0.434564
$283$	−805117.	−0.597576	−0.298788	−	0.954320i	$-0.596582\pi$
$283$	−805117.	−0.597576	−0.298788	+	0.954320i	$0.596582\pi$
$284$	− 1.05056e6i	− 0.772901i
$285$	−10027.2	−0.00731254
$286$	0	0
$287$	275414.	0.197370
$288$	189055.i	0.134310i
$289$	−1.02684e6	−0.723200
$290$	−83324.4	−0.0581805
$291$	343584.i	0.237849i
$292$	− 987414.i	− 0.677707i
$293$	2.06384e6i	1.40445i	0.711955	+	0.702225i	$0.247810\pi$
$293$	2.06384e6i	1.40445i	−0.711955	+	0.702225i	$0.752190\pi$
$294$	− 501738.i	− 0.338539i
$295$	−9406.18	−0.00629301
$296$	−207920.	−0.137933
$297$	626285.i	0.411985i
$298$	687797.	0.448662
$299$	0	0
$300$	−374392.	−0.240172
$301$	68085.1i	0.0433148i
$302$	1.38219e6	0.872066
$303$	1.50485e6	0.941644
$304$	42532.4i	0.0263959i
$305$	127807.i	0.0786694i
$306$	− 462972.i	− 0.282651i
$307$	772333.i	0.467691i	0.972274	+	0.233845i	$0.0751309\pi$
$307$	772333.i	0.467691i	−0.972274	+	0.233845i	$0.924869\pi$
$308$	60547.0	0.0363677
$309$	−27516.3	−0.0163943
$310$	133965.i	0.0791750i
$311$	−15325.1	−0.00898470	−0.00449235	−	0.999990i	$-0.501430\pi$
$311$	−15325.1	−0.00898470	−0.00449235	+	0.999990i	$0.501430\pi$
$312$	0	0
$313$	2.83907e6	1.63800	0.819001	−	0.573792i	$-0.194528\pi$
$313$	2.83907e6	1.63800	0.819001	+	0.573792i	$0.194528\pi$
$314$	− 613571.i	− 0.351189i
$315$	−28790.7	−0.0163485
$316$	1.31324e6	0.739819
$317$	− 651243.i	− 0.363995i	−0.983299	−	0.181997i	$-0.941744\pi$
$317$	− 651243.i	− 0.363995i	0.983299	−	0.181997i	$-0.0582562\pi$
$318$	513902.i	0.284979i
$319$	505500.i	0.278128i
$320$	− 32355.3i	− 0.0176632i
$321$	866250.	0.469224
$322$	−165423.	−0.0889114
$323$	− 104156.i	− 0.0555493i
$324$	−318414.	−0.168512
$325$	0	0
$326$	1.03103e6	0.537311
$327$	− 1.72863e6i	− 0.893991i
$328$	−892867.	−0.458250
$329$	374869.	0.190937
$330$	− 46275.9i	− 0.0233922i
$331$	− 929149.i	− 0.466139i	−0.972460	−	0.233069i	$-0.925123\pi$
$331$	− 929149.i	− 0.466139i	0.972460	−	0.233069i	$-0.0748769\pi$
$332$	1.33357e6i	0.664002i
$333$	− 599800.i	− 0.296412i
$334$	−1.09676e6	−0.537952
$335$	−330676.	−0.160987
$336$	− 38613.1i	− 0.0186589i
$337$	1.72888e6	0.829258	0.414629	−	0.909990i	$-0.363911\pi$
$337$	1.72888e6	0.829258	0.414629	+	0.909990i	$0.363911\pi$
$338$	0	0
$339$	−900543.	−0.425603
$340$	79233.8i	0.0371718i
$341$	812721.	0.378491
$342$	−122696.	−0.0567236
$343$	− 655895.i	− 0.301022i
$344$	− 220726.i	− 0.100567i
$345$	126433.i	0.0571889i
$346$	911129.i	0.409157i
$347$	−101581.	−0.0452886	−0.0226443	−	0.999744i	$-0.507209\pi$
$347$	−101581.	−0.0452886	−0.0226443	+	0.999744i	$0.507209\pi$
$348$	322376.	0.142697
$349$	798134.i	0.350762i	0.984501	+	0.175381i	$0.0561157\pi$
$349$	798134.i	0.350762i	−0.984501	+	0.175381i	$0.943884\pi$
$350$	−241841.	−0.105526
$351$	0	0
$352$	−196288.	−0.0844379
$353$	1.37965e6i	0.589296i	0.955606	+	0.294648i	$0.0952024\pi$
$353$	1.37965e6i	0.589296i	−0.955606	+	0.294648i	$0.904798\pi$
$354$	36391.8	0.0154346
$355$	−518663.	−0.218431
$356$	− 782064.i	− 0.327052i
$357$	94558.3i	0.0392671i
$358$	− 324241.i	− 0.133709i
$359$	− 953405.i	− 0.390428i	−0.980761	−	0.195214i	$-0.937460\pi$
$359$	− 953405.i	− 0.390428i	0.980761	−	0.195214i	$-0.0625402\pi$
$360$	93337.1	0.0379575
$361$	2.44850e6	0.988852
$362$	853059.i	0.342143i
$363$	949753.	0.378307
$364$	0	0
$365$	−487489.	−0.191528
$366$	− 494477.i	− 0.192949i
$367$	2.19900e6	0.852237	0.426119	−	0.904667i	$-0.359881\pi$
$367$	2.19900e6	0.852237	0.426119	+	0.904667i	$0.359881\pi$
$368$	536288.	0.206433
$369$	− 2.57570e6i	− 0.984760i
$370$	102651.i	0.0389815i
$371$	331958.i	0.125213i
$372$	− 518302.i	− 0.194189i
$373$	−1.83392e6	−0.682511	−0.341255	−	0.939971i	$-0.610852\pi$
$373$	−1.83392e6	−0.682511	−0.341255	+	0.939971i	$0.610852\pi$
$374$	480684.	0.177697
$375$	373442.i	0.137134i
$376$	−1.21529e6	−0.443314
$377$	0	0
$378$	257998.	0.0928725
$379$	2.31121e6i	0.826497i	0.910618	+	0.413248i	$0.135606\pi$
$379$	2.31121e6i	0.826497i	−0.910618	+	0.413248i	$0.864394\pi$
$380$	20998.3	0.00745978
$381$	1.92322e6	0.678762
$382$	− 1.42449e6i	− 0.499460i
$383$	3.63010e6i	1.26451i	0.774761	+	0.632254i	$0.217870\pi$
$383$	3.63010e6i	1.26451i	−0.774761	+	0.632254i	$0.782130\pi$
$384$	125180.i	0.0433219i
$385$	− 29892.2i	− 0.0102779i
$386$	−1.32635e6	−0.453094
$387$	636741.	0.216115
$388$	− 719511.i	− 0.242638i
$389$	−1.90596e6	−0.638618	−0.319309	−	0.947651i	$-0.603451\pi$
$389$	−1.90596e6	−0.638618	−0.319309	+	0.947651i	$0.603451\pi$
$390$	0	0
$391$	−1.31330e6	−0.434432
$392$	1.05071e6i	0.345355i
$393$	−17255.2	−0.00563560
$394$	−2.10065e6	−0.681731
$395$	− 648348.i	− 0.209082i
$396$	− 566243.i	− 0.181453i
$397$	− 3.89970e6i	− 1.24181i	−0.783886	−	0.620905i	$-0.786765\pi$
$397$	− 3.89970e6i	− 1.24181i	0.783886	−	0.620905i	$-0.213235\pi$
$398$	3.09738e6i	0.980136i
$399$	25059.6	0.00788028
$400$	784026.	0.245008
$401$	− 281948.i	− 0.0875604i	−0.999041	−	0.0437802i	$-0.986060\pi$
$401$	− 281948.i	− 0.0875604i	0.999041	−	0.0437802i	$-0.0139401\pi$
$402$	1.27936e6	0.394846
$403$	0	0
$404$	−3.15136e6	−0.960604
$405$	157202.i	0.0476234i
$406$	208241.	0.0626975
$407$	622747.	0.186348
$408$	− 306550.i	− 0.0911697i
$409$	− 667978.i	− 0.197448i	−0.995115	−	0.0987242i	$-0.968524\pi$
$409$	− 667978.i	− 0.197448i	0.995115	−	0.0987242i	$-0.0314762\pi$
$410$	440811.i	0.129507i
$411$	2.11474e6i	0.617522i
$412$	57622.8	0.0167244
$413$	23507.5	0.00678159
$414$	1.54706e6i	0.443615i
$415$	658385.	0.187655
$416$	0	0
$417$	−1.61135e6	−0.453784
$418$	− 127390.i	− 0.0356610i
$419$	4.71505e6	1.31205	0.656027	−	0.754738i	$-0.272236\pi$
$419$	4.71505e6	1.31205	0.656027	+	0.754738i	$0.272236\pi$
$420$	−19063.4	−0.00527322
$421$	282812.i	0.0777665i	0.999244	+	0.0388833i	$0.0123800\pi$
$421$	282812.i	0.0777665i	−0.999244	+	0.0388833i	$0.987620\pi$
$422$	4.86657e6i	1.33028i
$423$	− 3.50582e6i	− 0.952663i
$424$	− 1.07618e6i	− 0.290717i
$425$	−1.91998e6	−0.515613
$426$	2.00667e6	0.535737
$427$	− 319410.i	− 0.0847772i
$428$	−1.81404e6	−0.478672
$429$	0	0
$430$	−108973.	−0.0284215
$431$	933504.i	0.242060i	0.992649	+	0.121030i	$0.0386197\pi$
$431$	933504.i	0.242060i	−0.992649	+	0.121030i	$0.961380\pi$
$432$	−836408.	−0.215630
$433$	6.10852e6	1.56573	0.782864	−	0.622193i	$-0.213758\pi$
$433$	6.10852e6	1.56573	0.782864	+	0.622193i	$0.213758\pi$
$434$	− 334800.i	− 0.0853221i
$435$	− 159158.i	− 0.0403278i
$436$	3.61999e6i	0.911991i
$437$	348047.i	0.0871835i
$438$	1.88606e6	0.469753
$439$	5.77404e6	1.42994	0.714971	−	0.699154i	$-0.246440\pi$
$439$	5.77404e6	1.42994	0.714971	+	0.699154i	$0.246440\pi$
$440$	96907.9i	0.0238631i
$441$	−3.03103e6	−0.742153
$442$	0	0
$443$	−7.75717e6	−1.87799	−0.938997	−	0.343925i	$-0.888243\pi$
$443$	−7.75717e6	−1.87799	−0.938997	+	0.343925i	$0.888243\pi$
$444$	− 397149.i	− 0.0956083i
$445$	−386107.	−0.0924288
$446$	−3.03977e6	−0.723609
$447$	1.31376e6i	0.310990i
$448$	80860.9i	0.0190346i
$449$	− 5.82572e6i	− 1.36375i	−0.731470	−	0.681874i	$-0.761165\pi$
$449$	− 5.82572e6i	− 1.36375i	0.731470	−	0.681874i	$-0.238835\pi$
$450$	2.26172e6i	0.526512i
$451$	2.67424e6	0.619098
$452$	1.88586e6	0.434173
$453$	2.64011e6i	0.604473i
$454$	−200994.	−0.0457660
$455$	0	0
$456$	−81241.0	−0.0182963
$457$	7.98941e6i	1.78947i	0.446599	+	0.894734i	$0.352635\pi$
$457$	7.98941e6i	1.78947i	−0.446599	+	0.894734i	$0.647365\pi$
$458$	−1.48952e6	−0.331804
$459$	2.04825e6	0.453787
$460$	− 264767.i	− 0.0583404i
$461$	6.13769e6i	1.34509i	0.740055	+	0.672547i	$0.234800\pi$
$461$	6.13769e6i	1.34509i	−0.740055	+	0.672547i	$0.765200\pi$
$462$	115651.i	0.0252083i
$463$	4.55816e6i	0.988181i	0.869410	+	0.494091i	$0.164499\pi$
$463$	4.55816e6i	0.988181i	−0.869410	+	0.494091i	$0.835501\pi$
$464$	−675098.	−0.145570
$465$	−255887.	−0.0548802
$466$	2.94989e6i	0.629277i
$467$	4.63782e6	0.984060	0.492030	−	0.870578i	$-0.336255\pi$
$467$	4.63782e6	0.984060	0.492030	+	0.870578i	$0.336255\pi$
$468$	0	0
$469$	826411.	0.173486
$470$	599993.i	0.125286i
$471$	1.17198e6	0.243427
$472$	−76209.3	−0.0157454
$473$	661101.i	0.135867i
$474$	2.50841e6i	0.512806i
$475$	508827.i	0.103475i
$476$	− 198018.i	− 0.0400578i
$477$	3.10451e6	0.624737
$478$	3.59007e6	0.718676
$479$	− 6.87591e6i	− 1.36928i	−0.728883	−	0.684638i	$-0.759960\pi$
$479$	− 6.87591e6i	− 1.36928i	0.728883	−	0.684638i	$-0.240040\pi$
$480$	61801.8	0.0122433
$481$	0	0
$482$	1.79322e6	0.351572
$483$	− 315975.i	− 0.0616290i
$484$	−1.98891e6	−0.385924
$485$	−355224.	−0.0685722
$486$	− 3.78394e6i	− 0.726697i
$487$	− 6.16885e6i	− 1.17864i	−0.807899	−	0.589321i	$-0.799395\pi$
$487$	− 6.16885e6i	− 1.17864i	0.807899	−	0.589321i	$-0.200605\pi$
$488$	1.03550e6i	0.196834i
$489$	1.96936e6i	0.372438i
$490$	518736.	0.0976014
$491$	−250672.	−0.0469248	−0.0234624	−	0.999725i	$-0.507469\pi$
$491$	−250672.	−0.0469248	−0.0234624	+	0.999725i	$0.507469\pi$
$492$	− 1.70546e6i	− 0.317636i
$493$	1.65323e6	0.306348
$494$	0	0
$495$	−279556.	−0.0512809
$496$	1.08539e6i	0.198099i
$497$	1.29622e6	0.235390
$498$	−2.54724e6	−0.460253
$499$	324783.i	0.0583904i	0.999574	+	0.0291952i	$0.00929444\pi$
$499$	324783.i	0.0583904i	−0.999574	+	0.0291952i	$0.990706\pi$
$500$	− 782038.i	− 0.139895i
$501$	− 2.09491e6i	− 0.372882i
$502$	7.34300e6i	1.30051i
$503$	1.54529e6	0.272327	0.136163	−	0.990686i	$-0.456523\pi$
$503$	1.54529e6	0.272327	0.136163	+	0.990686i	$0.456523\pi$
$504$	−233264.	−0.0409045
$505$	1.55583e6i	0.271478i
$506$	−1.60625e6	−0.278892
$507$	0	0
$508$	−4.02749e6	−0.692429
$509$	9.63563e6i	1.64849i	0.566236	+	0.824243i	$0.308399\pi$
$509$	9.63563e6i	1.64849i	−0.566236	+	0.824243i	$0.691601\pi$
$510$	−151344.	−0.0257656
$511$	1.21831e6	0.206398
$512$	− 262144.i	− 0.0441942i
$513$	− 542823.i	− 0.0910677i
$514$	6.07091e6i	1.01355i
$515$	− 28448.5i	− 0.00472652i
$516$	421608.	0.0697084
$517$	3.63995e6	0.598920
$518$	− 256540.i	− 0.0420079i
$519$	−1.74035e6	−0.283607
$520$	0	0
$521$	−2.86040e6	−0.461670	−0.230835	−	0.972993i	$-0.574146\pi$
$521$	−2.86040e6	−0.461670	−0.230835	+	0.972993i	$0.574146\pi$
$522$	− 1.94749e6i	− 0.312824i
$523$	5.00224e6	0.799670	0.399835	−	0.916587i	$-0.369068\pi$
$523$	5.00224e6	0.799670	0.399835	+	0.916587i	$0.369068\pi$
$524$	36134.8	0.00574907
$525$	− 461939.i	− 0.0731453i
$526$	882583.i	0.139088i
$527$	− 2.65799e6i	− 0.416894i
$528$	− 374930.i	− 0.0585282i
$529$	−2.04784e6	−0.318168
$530$	−531312.	−0.0821599
$531$	− 219845.i	− 0.0338361i
$532$	−52478.1	−0.00803895
$533$	0	0
$534$	1.49382e6	0.226696
$535$	895597.i	0.135278i
$536$	−2.67915e6	−0.402796
$537$	619332.	0.0926804
$538$	5.81072e6i	0.865514i
$539$	− 3.14699e6i	− 0.466577i
$540$	412937.i	0.0609395i
$541$	− 7.10054e6i	− 1.04303i	−0.853241	−	0.521516i	$-0.825367\pi$
$541$	− 7.10054e6i	− 1.04303i	0.853241	−	0.521516i	$-0.174633\pi$
$542$	−3.32604e6	−0.486328
$543$	−1.62943e6	−0.237156
$544$	641956.i	0.0930054i
$545$	1.78720e6	0.257739
$546$	0	0
$547$	6.22951e6	0.890196	0.445098	−	0.895482i	$-0.353169\pi$
$547$	6.22951e6	0.890196	0.445098	+	0.895482i	$0.353169\pi$
$548$	− 4.42855e6i	− 0.629956i
$549$	−2.98717e6	−0.422988
$550$	−2.34825e6	−0.331008
$551$	− 438134.i	− 0.0614791i
$552$	1.02436e6i	0.143089i
$553$	1.62032e6i	0.225314i
$554$	2.69988e6i	0.373740i
$555$	−196073.	−0.0270200
$556$	3.37438e6	0.462921
$557$	− 740978.i	− 0.101197i	−0.998719	−	0.0505985i	$-0.983887\pi$
$557$	− 740978.i	− 0.101197i	0.998719	−	0.0505985i	$-0.0161129\pi$
$558$	−3.13110e6	−0.425707
$559$	0	0
$560$	39921.2	0.00537940
$561$	918153.i	0.123171i
$562$	−2.69937e6	−0.360513
$563$	4.43102e6	0.589159	0.294580	−	0.955627i	$-0.404820\pi$
$563$	4.43102e6	0.589159	0.294580	+	0.955627i	$0.404820\pi$
$564$	− 2.32133e6i	− 0.307283i
$565$	− 931052.i	− 0.122702i
$566$	3.22047e6i	0.422550i
$567$	− 392872.i	− 0.0513208i
$568$	−4.20223e6	−0.546523
$569$	−1.13478e7	−1.46936	−0.734682	−	0.678411i	$-0.762669\pi$
$569$	−1.13478e7	−1.46936	−0.734682	+	0.678411i	$0.762669\pi$
$570$	40108.9i	0.00517075i
$571$	1.42572e7	1.82997	0.914987	−	0.403484i	$-0.132201\pi$
$571$	1.42572e7	1.82997	0.914987	+	0.403484i	$0.132201\pi$
$572$	0	0
$573$	2.72091e6	0.346201
$574$	− 1.10165e6i	− 0.139562i
$575$	6.41577e6	0.809244
$576$	756222.	0.0949714
$577$	1.06731e7i	1.33460i	0.744788	+	0.667301i	$0.232550\pi$
$577$	1.06731e7i	1.33460i	−0.744788	+	0.667301i	$0.767450\pi$
$578$	4.10736e6i	0.511380i
$579$	− 2.53345e6i	− 0.314062i
$580$	333298.i	0.0411398i
$581$	−1.64541e6	−0.202224
$582$	1.37434e6	0.168184
$583$	3.22328e6i	0.392760i
$584$	−3.94965e6	−0.479211
$585$	0	0
$586$	8.25535e6	0.993096
$587$	− 1.19910e7i	− 1.43636i	−0.695860	−	0.718178i	$-0.744976\pi$
$587$	− 1.19910e7i	− 1.43636i	0.695860	−	0.718178i	$-0.255024\pi$
$588$	−2.00695e6	−0.239383
$589$	−704412.	−0.0836640
$590$	37624.7i	0.00444983i
$591$	− 4.01244e6i	− 0.472542i
$592$	831682.i	0.0975333i
$593$	− 1.09751e7i	− 1.28165i	−0.767686	−	0.640826i	$-0.778592\pi$
$593$	− 1.09751e7i	− 1.28165i	0.767686	−	0.640826i	$-0.221408\pi$
$594$	2.50514e6	0.291317
$595$	−97761.8	−0.0113208
$596$	− 2.75119e6i	− 0.317252i
$597$	−5.91629e6	−0.679382
$598$	0	0
$599$	−4.11161e6	−0.468214	−0.234107	−	0.972211i	$-0.575217\pi$
$599$	−4.11161e6	−0.468214	−0.234107	+	0.972211i	$0.575217\pi$
$600$	1.49757e6i	0.169828i
$601$	−5.22300e6	−0.589840	−0.294920	−	0.955522i	$-0.595293\pi$
$601$	−5.22300e6	−0.589840	−0.294920	+	0.955522i	$0.595293\pi$
$602$	272340.	0.0306282
$603$	− 7.72870e6i	− 0.865592i
$604$	− 5.52875e6i	− 0.616644i
$605$	981930.i	0.109067i
$606$	− 6.01940e6i	− 0.665843i
$607$	1.17012e6	0.128902	0.0644508	−	0.997921i	$-0.479470\pi$
$607$	1.17012e6	0.128902	0.0644508	+	0.997921i	$0.479470\pi$
$608$	170130.	0.0186647
$609$	397760.i	0.0434588i
$610$	511229.	0.0556277
$611$	0	0
$612$	−1.85189e6	−0.199865
$613$	− 1.16250e6i	− 0.124951i	−0.998046	−	0.0624757i	$-0.980100\pi$
$613$	− 1.16250e6i	− 0.124951i	0.998046	−	0.0624757i	$-0.0198996\pi$
$614$	3.08933e6	0.330707
$615$	−841992.	−0.0897677
$616$	− 242188.i	− 0.0257158i
$617$	8.51152e6i	0.900107i	0.893002	+	0.450053i	$0.148595\pi$
$617$	8.51152e6i	0.900107i	−0.893002	+	0.450053i	$0.851405\pi$
$618$	110065.i	0.0115925i
$619$	− 6.03620e6i	− 0.633194i	−0.948560	−	0.316597i	$-0.897460\pi$
$619$	− 6.03620e6i	− 0.633194i	0.948560	−	0.316597i	$-0.102540\pi$
$620$	535861.	0.0559852
$621$	−6.84442e6	−0.712209
$622$	61300.6i	0.00635314i
$623$	964941.	0.0996049
$624$	0	0
$625$	9.18454e6	0.940497
$626$	− 1.13563e7i	− 1.15824i
$627$	243327.	0.0247184
$628$	−2.45428e6	−0.248328
$629$	− 2.03668e6i	− 0.205256i
$630$	115163.i	0.0115601i
$631$	6.04074e6i	0.603972i	0.953312	+	0.301986i	$0.0976495\pi$
$631$	6.04074e6i	0.603972i	−0.953312	+	0.301986i	$0.902350\pi$
$632$	− 5.25295e6i	− 0.523131i
$633$	−9.29562e6	−0.922081
$634$	−2.60497e6	−0.257383
$635$	1.98838e6i	0.195688i
$636$	2.05561e6	0.201510
$637$	0	0
$638$	2.02200e6	0.196666
$639$	− 1.21224e7i	− 1.17446i
$640$	−129421.	−0.0124898
$641$	−5.48840e6	−0.527595	−0.263797	−	0.964578i	$-0.584975\pi$
$641$	−5.48840e6	−0.527595	−0.263797	+	0.964578i	$0.584975\pi$
$642$	− 3.46500e6i	− 0.331792i
$643$	− 1.97963e7i	− 1.88823i	−0.329610	−	0.944117i	$-0.606917\pi$
$643$	− 1.97963e7i	− 1.88823i	0.329610	−	0.944117i	$-0.393083\pi$
$644$	661694.i	0.0628698i
$645$	− 208149.i	− 0.0197004i
$646$	−416625.	−0.0392793
$647$	−1.89495e7	−1.77966	−0.889832	−	0.456288i	$-0.849179\pi$
$647$	−1.89495e7	−1.77966	−0.889832	+	0.456288i	$0.849179\pi$
$648$	1.27366e6i	0.119156i
$649$	228256.	0.0212721
$650$	0	0
$651$	639501.	0.0591410
$652$	− 4.12411e6i	− 0.379937i
$653$	−8.52088e6	−0.781990	−0.390995	−	0.920393i	$-0.627869\pi$
$653$	−8.52088e6	−0.781990	−0.390995	+	0.920393i	$0.627869\pi$
$654$	−6.91453e6	−0.632147
$655$	− 17839.8i	− 0.00162475i
$656$	3.57147e6i	0.324032i
$657$	− 1.13938e7i	− 1.02980i
$658$	− 1.49948e6i	− 0.135013i
$659$	7.47936e6	0.670889	0.335445	−	0.942060i	$-0.391113\pi$
$659$	7.47936e6	0.670889	0.335445	+	0.942060i	$0.391113\pi$
$660$	−185104.	−0.0165407
$661$	4.66740e6i	0.415500i	0.978182	+	0.207750i	$0.0666141\pi$
$661$	4.66740e6i	0.415500i	−0.978182	+	0.207750i	$0.933386\pi$
$662$	−3.71660e6	−0.329610
$663$	0	0
$664$	5.33427e6	0.469520
$665$	25908.6i	0.00227190i
$666$	−2.39920e6	−0.209595
$667$	−5.52440e6	−0.480807
$668$	4.38702e6i	0.380390i
$669$	− 5.80627e6i	− 0.501570i
$670$	1.32270e6i	0.113835i
$671$	− 3.10145e6i	− 0.265924i
$672$	−154452.	−0.0131938
$673$	−5.63150e6	−0.479277	−0.239638	−	0.970862i	$-0.577029\pi$
$673$	−5.63150e6	−0.479277	−0.239638	+	0.970862i	$0.577029\pi$
$674$	− 6.91552e6i	− 0.586374i
$675$	−1.00062e7	−0.845297
$676$	0	0
$677$	9.19599e6	0.771129	0.385564	−	0.922681i	$-0.374007\pi$
$677$	9.19599e6	0.771129	0.385564	+	0.922681i	$0.374007\pi$
$678$	3.60217e6i	0.300947i
$679$	887761.	0.0738961
$680$	316935.	0.0262844
$681$	− 383918.i	− 0.0317228i
$682$	− 3.25088e6i	− 0.267633i
$683$	8.08689e6i	0.663330i	0.943397	+	0.331665i	$0.107610\pi$
$683$	8.08689e6i	0.663330i	−0.943397	+	0.331665i	$0.892390\pi$
$684$	490782.i	0.0401096i
$685$	−2.18639e6	−0.178033
$686$	−2.62358e6	−0.212855
$687$	− 2.84512e6i	− 0.229990i
$688$	−882904.	−0.0711119
$689$	0	0
$690$	505731.	0.0404386
$691$	− 1.37327e7i	− 1.09411i	−0.837097	−	0.547055i	$-0.815749\pi$
$691$	− 1.37327e7i	− 1.09411i	0.837097	−	0.547055i	$-0.184251\pi$
$692$	3.64451e6	0.289317
$693$	698653.	0.0552623
$694$	406325.i	0.0320239i
$695$	− 1.66594e6i	− 0.130827i
$696$	− 1.28950e6i	− 0.100902i
$697$	− 8.74606e6i	− 0.681915i
$698$	3.19253e6	0.248026
$699$	−5.63459e6	−0.436183
$700$	967362.i	0.0746181i
$701$	−1.28324e7	−0.986306	−0.493153	−	0.869943i	$-0.664156\pi$
$701$	−1.28324e7	−0.986306	−0.493153	+	0.869943i	$0.664156\pi$
$702$	0	0
$703$	−539755.	−0.0411916
$704$	785152.i	0.0597066i
$705$	−1.14605e6	−0.0868419
$706$	5.51862e6	0.416695
$707$	− 3.88827e6i	− 0.292555i
$708$	− 145567.i	− 0.0109139i
$709$	1.63596e7i	1.22224i	0.791537	+	0.611121i	$0.209281\pi$
$709$	1.63596e7i	1.22224i	−0.791537	+	0.611121i	$0.790719\pi$
$710$	2.07465e6i	0.154454i
$711$	1.51535e7	1.12419
$712$	−3.12825e6	−0.231261
$713$	8.88189e6i	0.654307i
$714$	378233.	0.0277661
$715$	0	0
$716$	−1.29696e6	−0.0945465
$717$	6.85739e6i	0.498151i
$718$	−3.81362e6	−0.276074
$719$	−5.07377e6	−0.366023	−0.183012	−	0.983111i	$-0.558585\pi$
$719$	−5.07377e6	−0.366023	−0.183012	+	0.983111i	$0.558585\pi$
$720$	− 373348.i	− 0.0268400i
$721$	71097.3i	0.00509348i
$722$	− 9.79398e6i	− 0.699224i
$723$	3.42522e6i	0.243693i
$724$	3.41223e6	0.241932
$725$	−8.07640e6	−0.570654
$726$	− 3.79901e6i	− 0.267503i
$727$	2.26620e7	1.59024	0.795119	−	0.606454i	$-0.207408\pi$
$727$	2.26620e7	1.59024	0.795119	+	0.606454i	$0.207408\pi$
$728$	0	0
$729$	2.39178e6	0.166687
$730$	1.94995e6i	0.135431i
$731$	2.16212e6	0.149653
$732$	−1.97791e6	−0.136436
$733$	2.62116e7i	1.80191i	0.433910	+	0.900956i	$0.357134\pi$
$733$	2.62116e7i	1.80191i	−0.433910	+	0.900956i	$0.642866\pi$
$734$	− 8.79601e6i	− 0.602623i
$735$	990837.i	0.0676525i
$736$	− 2.14515e6i	− 0.145970i
$737$	8.02438e6	0.544180
$738$	−1.03028e7	−0.696330
$739$	− 2.80735e7i	− 1.89097i	−0.325663	−	0.945486i	$-0.605587\pi$
$739$	− 2.80735e7i	− 1.89097i	0.325663	−	0.945486i	$-0.394413\pi$
$740$	410603.	0.0275641
$741$	0	0
$742$	1.32783e6	0.0885387
$743$	− 2.74094e7i	− 1.82149i	−0.412968	−	0.910746i	$-0.635508\pi$
$743$	− 2.74094e7i	− 1.82149i	0.412968	−	0.910746i	$-0.364492\pi$
$744$	−2.07321e6	−0.137313
$745$	−1.35827e6	−0.0896592
$746$	7.33570e6i	0.482608i
$747$	1.53881e7i	1.00898i
$748$	− 1.92274e6i	− 0.125651i
$749$	− 2.23824e6i	− 0.145781i
$750$	1.49377e6	0.0969684
$751$	1.45703e7	0.942688	0.471344	−	0.881949i	$-0.343769\pi$
$751$	1.45703e7	0.942688	0.471344	+	0.881949i	$0.343769\pi$
$752$	4.86117e6i	0.313470i
$753$	−1.40258e7	−0.901450
$754$	0	0
$755$	−2.72956e6	−0.174271
$756$	− 1.03199e6i	− 0.0656708i
$757$	−2.54940e7	−1.61696	−0.808478	−	0.588527i	$-0.799708\pi$
$757$	−2.54940e7	−1.61696	−0.808478	+	0.588527i	$0.799708\pi$
$758$	9.24483e6	0.584421
$759$	− 3.06809e6i	− 0.193314i
$760$	− 83993.4i	− 0.00527486i
$761$	− 8.76773e6i	− 0.548814i	−0.961614	−	0.274407i	$-0.911518\pi$
$761$	− 8.76773e6i	− 0.548814i	0.961614	−	0.274407i	$-0.0884816\pi$
$762$	− 7.69290e6i	− 0.479957i
$763$	−4.46648e6	−0.277750
$764$	−5.69796e6	−0.353172
$765$	914281.i	0.0564841i
$766$	1.45204e7	0.894142
$767$	0	0
$768$	500721.	0.0306332
$769$	2.85273e7i	1.73958i	0.493419	+	0.869792i	$0.335747\pi$
$769$	2.85273e7i	1.73958i	−0.493419	+	0.869792i	$0.664253\pi$
$770$	−119569.	−0.00726760
$771$	−1.15960e7	−0.702544
$772$	5.30538e6i	0.320386i
$773$	1.13022e7i	0.680319i	0.940368	+	0.340159i	$0.110481\pi$
$773$	1.13022e7i	0.680319i	−0.940368	+	0.340159i	$0.889519\pi$
$774$	− 2.54696e6i	− 0.152817i
$775$	1.29849e7i	0.776575i
$776$	−2.87804e6	−0.171571
$777$	490017.	0.0291178
$778$	7.62386e6i	0.451571i
$779$	−2.31786e6	−0.136849
$780$	0	0
$781$	1.25862e7	0.738356
$782$	5.25320e6i	0.307190i
$783$	8.61599e6	0.502228
$784$	4.20282e6	0.244203
$785$	1.21169e6i	0.0701804i
$786$	69021.0i	0.00398497i
$787$	9.70280e6i	0.558419i	0.960230	+	0.279210i	$0.0900724\pi$
$787$	9.70280e6i	0.558419i	−0.960230	+	0.279210i	$0.909928\pi$
$788$	8.40259e6i	0.482056i
$789$	−1.68582e6	−0.0964092
$790$	−2.59339e6	−0.147843
$791$	2.32684e6i	0.132229i
$792$	−2.26497e6	−0.128307
$793$	0	0
$794$	−1.55988e7	−0.878092
$795$	− 1.01486e6i	− 0.0569492i
$796$	1.23895e7	0.693061
$797$	−1.69795e7	−0.946846	−0.473423	−	0.880835i	$-0.656982\pi$
$797$	−1.69795e7	−0.946846	−0.473423	+	0.880835i	$0.656982\pi$
$798$	− 100238.i	− 0.00557220i
$799$	− 1.19044e7i	− 0.659690i
$800$	− 3.13610e6i	− 0.173247i
$801$	− 9.02425e6i	− 0.496970i
$802$	−1.12779e6	−0.0619146
$803$	1.18297e7	0.647417
$804$	− 5.11745e6i	− 0.279199i
$805$	326680.	0.0177678
$806$	0	0
$807$	−1.10990e7	−0.599932
$808$	1.26054e7i	0.679249i
$809$	−1.88752e7	−1.01396	−0.506979	−	0.861958i	$-0.669238\pi$
$809$	−1.88752e7	−1.01396	−0.506979	+	0.861958i	$0.669238\pi$
$810$	628808.	0.0336748
$811$	2.52570e7i	1.34843i	0.738534	+	0.674217i	$0.235519\pi$
$811$	2.52570e7i	1.34843i	−0.738534	+	0.674217i	$0.764481\pi$
$812$	− 832963.i	− 0.0443339i
$813$	− 6.35306e6i	− 0.337098i
$814$	− 2.49099e6i	− 0.131768i
$815$	−2.03608e6	−0.107375
$816$	−1.22620e6	−0.0644667
$817$	− 572998.i	− 0.0300330i
$818$	−2.67191e6	−0.139617
$819$	0	0
$820$	1.76324e6	0.0915752
$821$	1.46338e7i	0.757702i	0.925458	+	0.378851i	$0.123681\pi$
$821$	1.46338e7i	0.757702i	−0.925458	+	0.378851i	$0.876319\pi$
$822$	8.45896e6	0.436654
$823$	3.53653e6	0.182003	0.0910013	−	0.995851i	$-0.470993\pi$
$823$	3.53653e6	0.182003	0.0910013	+	0.995851i	$0.470993\pi$
$824$	− 230491.i	− 0.0118260i
$825$	− 4.48539e6i	− 0.229438i
$826$	− 94030.0i	− 0.00479531i
$827$	2.34804e7i	1.19383i	0.802306	+	0.596914i	$0.203606\pi$
$827$	2.34804e7i	1.19383i	−0.802306	+	0.596914i	$0.796394\pi$
$828$	6.18825e6	0.313683
$829$	2.78441e7	1.40717	0.703586	−	0.710610i	$-0.251581\pi$
$829$	2.78441e7	1.40717	0.703586	+	0.710610i	$0.251581\pi$
$830$	− 2.63354e6i	− 0.132692i
$831$	−5.15703e6	−0.259058
$832$	0	0
$833$	−1.02922e7	−0.513918
$834$	6.44540e6i	0.320874i
$835$	2.16588e6	0.107503
$836$	−509558.	−0.0252161
$837$	− 1.38524e7i	− 0.683458i
$838$	− 1.88602e7i	− 0.927762i
$839$	1.37068e7i	0.672250i	0.941817	+	0.336125i	$0.109116\pi$
$839$	1.37068e7i	0.672250i	−0.941817	+	0.336125i	$0.890884\pi$
$840$	76253.4i	0.00372873i
$841$	−1.35568e7	−0.660950
$842$	1.13125e6	0.0549892
$843$	− 5.15606e6i	− 0.249890i
$844$	1.94663e7	0.940647
$845$	0	0
$846$	−1.40233e7	−0.673634
$847$	− 2.45399e6i	− 0.117534i
$848$	−4.30471e6	−0.205568
$849$	−6.15141e6	−0.292891
$850$	7.67991e6i	0.364593i
$851$	6.80575e6i	0.322145i
$852$	− 8.02667e6i	− 0.378823i
$853$	2.64722e7i	1.24571i	0.782337	+	0.622856i	$0.214028\pi$
$853$	2.64722e7i	1.24571i	−0.782337	+	0.622856i	$0.785972\pi$
$854$	−1.27764e6	−0.0599465
$855$	242300.	0.0113355
$856$	7.25617e6i	0.338472i
$857$	−8.80546e6	−0.409544	−0.204772	−	0.978810i	$-0.565645\pi$
$857$	−8.80546e6	−0.409544	−0.204772	+	0.978810i	$0.565645\pi$
$858$	0	0
$859$	−2.99287e6	−0.138390	−0.0691951	−	0.997603i	$-0.522043\pi$
$859$	−2.99287e6	−0.138390	−0.0691951	+	0.997603i	$0.522043\pi$
$860$	435892.i	0.0200971i
$861$	2.10427e6	0.0967372
$862$	3.73402e6	0.171162
$863$	− 1.71927e7i	− 0.785811i	−0.919579	−	0.392905i	$-0.871470\pi$
$863$	− 1.71927e7i	− 0.785811i	0.919579	−	0.392905i	$-0.128530\pi$
$864$	3.34563e6i	0.152473i
$865$	− 1.79931e6i	− 0.0817645i
$866$	− 2.44341e7i	− 1.10714i
$867$	−7.84546e6	−0.354463
$868$	−1.33920e6	−0.0603318
$869$	1.57332e7i	0.706753i
$870$	−636631.	−0.0285161
$871$	0	0
$872$	1.44799e7	0.644875
$873$	− 8.30245e6i	− 0.368698i
$874$	1.39219e6	0.0616481
$875$	964909.	0.0426055
$876$	− 7.54423e6i	− 0.332165i
$877$	2.94683e7i	1.29377i	0.762589	+	0.646883i	$0.223928\pi$
$877$	2.94683e7i	1.29377i	−0.762589	+	0.646883i	$0.776072\pi$
$878$	− 2.30962e7i	− 1.01112i
$879$	1.57685e7i	0.688365i
$880$	387632.	0.0168738
$881$	2.82805e7	1.22758	0.613788	−	0.789471i	$-0.289645\pi$
$881$	2.82805e7	1.22758	0.613788	+	0.789471i	$0.289645\pi$
$882$	1.21241e7i	0.524782i
$883$	1.96720e7	0.849076	0.424538	−	0.905410i	$-0.360437\pi$
$883$	1.96720e7	0.849076	0.424538	+	0.905410i	$0.360437\pi$
$884$	0	0
$885$	−71866.9	−0.00308440
$886$	3.10287e7i	1.32794i
$887$	1.24458e7	0.531144	0.265572	−	0.964091i	$-0.414439\pi$
$887$	1.24458e7	0.531144	0.265572	+	0.964091i	$0.414439\pi$
$888$	−1.58859e6	−0.0676053
$889$	− 4.96927e6i	− 0.210881i
$890$	1.54443e6i	0.0653570i
$891$	− 3.81476e6i	− 0.160980i
$892$	1.21591e7i	0.511669i
$893$	−3.15487e6	−0.132389
$894$	5.25504e6	0.219903
$895$	640314.i	0.0267199i
$896$	323444.	0.0134595
$897$	0	0
$898$	−2.33029e7	−0.964315
$899$	− 1.11808e7i	− 0.461397i
$900$	9.04690e6	0.372300
$901$	1.05417e7	0.432611
$902$	− 1.06970e7i	− 0.437769i
$903$	520197.i	0.0212299i
$904$	− 7.54343e6i	− 0.307006i
$905$	− 1.68463e6i	− 0.0683727i
$906$	1.05605e7	0.427427
$907$	4.12557e7	1.66520	0.832599	−	0.553876i	$-0.186852\pi$
$907$	4.12557e7	1.66520	0.832599	+	0.553876i	$0.186852\pi$
$908$	803976.i	0.0323615i
$909$	−3.63636e7	−1.45968
$910$	0	0
$911$	−3.01353e7	−1.20304	−0.601519	−	0.798859i	$-0.705438\pi$
$911$	−3.01353e7	−1.20304	−0.601519	+	0.798859i	$0.705438\pi$
$912$	324964.i	0.0129374i
$913$	−1.59768e7	−0.634325
$914$	3.19576e7	1.26535
$915$	976498.i	0.0385584i
$916$	5.95806e6i	0.234621i
$917$	44584.5i	0.00175090i
$918$	− 8.19301e6i	− 0.320876i
$919$	−1.66097e7	−0.648742	−0.324371	−	0.945930i	$-0.605153\pi$
$919$	−1.66097e7	−0.648742	−0.324371	+	0.945930i	$0.605153\pi$
$920$	−1.05907e6	−0.0412529
$921$	5.90093e6i	0.229230i
$922$	2.45508e7	0.951125
$923$	0	0
$924$	462603.	0.0178250
$925$	9.94965e6i	0.382343i
$926$	1.82326e7	0.698750
$927$	664911.	0.0254135
$928$	2.70039e6i	0.102934i
$929$	1.90297e7i	0.723422i	0.932290	+	0.361711i	$0.117807\pi$
$929$	1.90297e7i	0.723422i	−0.932290	+	0.361711i	$0.882193\pi$
$930$	1.02355e6i	0.0388062i
$931$	2.72760e6i	0.103135i
$932$	1.17996e7	0.444966
$933$	−117090.	−0.00440368
$934$	− 1.85513e7i	− 0.695835i
$935$	−949259.	−0.0355104
$936$	0	0
$937$	3.19208e7	1.18775	0.593875	−	0.804557i	$-0.297597\pi$
$937$	3.19208e7	1.18775	0.593875	+	0.804557i	$0.297597\pi$
$938$	− 3.30564e6i	− 0.122673i
$939$	2.16916e7	0.802836
$940$	2.39997e6	0.0885904
$941$	− 3.88808e7i	− 1.43140i	−0.698408	−	0.715700i	$-0.746108\pi$
$941$	− 3.88808e7i	− 1.43140i	0.698408	−	0.715700i	$-0.253892\pi$
$942$	− 4.68792e6i	− 0.172129i
$943$	2.92257e7i	1.07025i
$944$	304837.i	0.0111337i
$945$	−509497.	−0.0185593
$946$	2.64440e6	0.0960727
$947$	6.30671e6i	0.228522i	0.993451	+	0.114261i	$0.0364500\pi$
$947$	6.30671e6i	0.228522i	−0.993451	+	0.114261i	$0.963550\pi$
$948$	1.00336e7	0.362608
$949$	0	0
$950$	2.03531e6	0.0731680
$951$	− 4.97575e6i	− 0.178405i
$952$	−792071.	−0.0283251
$953$	3.86241e7	1.37761	0.688805	−	0.724946i	$-0.258135\pi$
$953$	3.86241e7	1.37761	0.688805	+	0.724946i	$0.258135\pi$
$954$	− 1.24181e7i	− 0.441756i
$955$	2.81310e6i	0.0998105i
$956$	− 1.43603e7i	− 0.508181i
$957$	3.86222e6i	0.136319i
$958$	−2.75036e7	−0.968225
$959$	5.46412e6	0.191855
$960$	− 247207.i	− 0.00865731i
$961$	1.06531e7	0.372107
$962$	0	0
$963$	−2.09323e7	−0.727363
$964$	− 7.17286e6i	− 0.248599i
$965$	2.61928e6	0.0905448
$966$	−1.26390e6	−0.0435783
$967$	− 1.80045e7i	− 0.619176i	−0.950871	−	0.309588i	$-0.899809\pi$
$967$	− 1.80045e7i	− 0.619176i	0.950871	−	0.309588i	$-0.100191\pi$
$968$	7.95564e6i	0.272889i
$969$	− 795794.i	− 0.0272265i
$970$	1.42090e6i	0.0484879i
$971$	5.15346e7	1.75409	0.877043	−	0.480411i	$-0.159513\pi$
$971$	5.15346e7	1.75409	0.877043	+	0.480411i	$0.159513\pi$
$972$	−1.51358e7	−0.513852
$973$	4.16344e6i	0.140984i
$974$	−2.46754e7	−0.833426
$975$	0	0
$976$	4.14200e6	0.139183
$977$	2.83425e7i	0.949951i	0.879999	+	0.474975i	$0.157543\pi$
$977$	2.83425e7i	0.949951i	−0.879999	+	0.474975i	$0.842457\pi$
$978$	7.87745e6	0.263353
$979$	9.36949e6	0.312435
$980$	− 2.07494e6i	− 0.0690146i
$981$	4.17711e7i	1.38581i
$982$	1.00269e6i	0.0331809i
$983$	3.16627e7i	1.04512i	0.852603	+	0.522558i	$0.175022\pi$
$983$	3.16627e7i	1.04512i	−0.852603	+	0.522558i	$0.824978\pi$
$984$	−6.82186e6	−0.224603
$985$	4.14838e6	0.136235
$986$	− 6.61291e6i	− 0.216621i
$987$	2.86415e6	0.0935842
$988$	0	0
$989$	−7.22490e6	−0.234877
$990$	1.11822e6i	0.0362611i
$991$	2.51605e7	0.813832	0.406916	−	0.913466i	$-0.366604\pi$
$991$	2.51605e7	0.813832	0.406916	+	0.913466i	$0.366604\pi$
$992$	4.34157e6	0.140077
$993$	− 7.09906e6i	− 0.228469i
$994$	− 5.18487e6i	− 0.166446i
$995$	− 6.11673e6i	− 0.195867i
$996$	1.01890e7i	0.325448i
$997$	−5.62891e7	−1.79344	−0.896719	−	0.442600i	$-0.854056\pi$
$997$	−5.62891e7	−1.79344	−0.896719	+	0.442600i	$0.854056\pi$
$998$	1.29913e6	0.0412883
$999$	− 1.06144e7i	− 0.336497i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.6.b.f.337.4		12
13.5	odd	4		338.6.a.l.1.4	✓	6
13.8	odd	4		338.6.a.n.1.4	yes	6
13.12	even	2	inner	338.6.b.f.337.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
338.6.a.l.1.4	✓	6	13.5	odd	4
338.6.a.n.1.4	yes	6	13.8	odd	4
338.6.b.f.337.4		12	1.1	even	1	trivial
338.6.b.f.337.10		12	13.12	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} +7.64039 q^{3} -16.0000 q^{4} +7.89924i q^{5} -30.5616i q^{6} -19.7414i q^{7} +64.0000i q^{8} -184.624 q^{9} +31.5970 q^{10} -191.688i q^{11} -122.246 q^{12} -78.9657 q^{14} +60.3533i q^{15} +256.000 q^{16} -626.910 q^{17} +738.498i q^{18} +166.142i q^{19} -126.388i q^{20} -150.832i q^{21} -766.750 q^{22} +2094.88 q^{23} +488.985i q^{24} +3062.60 q^{25} -3267.22 q^{27} +315.863i q^{28} -2637.10 q^{29} +241.413 q^{30} +4239.82i q^{31} -1024.00i q^{32} -1464.57i q^{33} +2507.64i q^{34} +155.942 q^{35} +2953.99 q^{36} +3248.76i q^{37} +664.568 q^{38} -505.551 q^{40} +13951.1i q^{41} -603.329 q^{42} -3448.84 q^{43} +3067.00i q^{44} -1458.39i q^{45} -8379.51i q^{46} +18989.0i q^{47} +1955.94 q^{48} +16417.3 q^{49} -12250.4i q^{50} -4789.84 q^{51} -16815.3 q^{53} +13068.9i q^{54} +1514.19 q^{55} +1263.45 q^{56} +1269.39i q^{57} +10548.4i q^{58} +1190.77i q^{59} -965.653i q^{60} +16179.7 q^{61} +16959.3 q^{62} +3644.75i q^{63} -4096.00 q^{64} -5858.27 q^{66} +41861.8i q^{67} +10030.6 q^{68} +16005.7 q^{69} -623.769i q^{70} +65659.8i q^{71} -11816.0i q^{72} +61713.3i q^{73} +12995.0 q^{74} +23399.5 q^{75} -2658.27i q^{76} -3784.19 q^{77} -82077.3 q^{79} +2022.21i q^{80} +19900.9 q^{81} +55804.2 q^{82} -83347.9i q^{83} +2413.32i q^{84} -4952.12i q^{85} +13795.4i q^{86} -20148.5 q^{87} +12268.0 q^{88} +48879.0i q^{89} -5833.57 q^{90} -33518.0 q^{92} +32393.9i q^{93} +75955.8 q^{94} -1312.40 q^{95} -7823.76i q^{96} +44969.4i q^{97} -65669.1i q^{98} +35390.2i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 72 q^{3} - 192 q^{4} + 588 q^{9} + 992 q^{10} + 1152 q^{12} - 928 q^{14} + 3072 q^{16} + 1960 q^{17} + 1664 q^{22} + 10816 q^{23} + 5044 q^{25} - 30792 q^{27} + 6556 q^{29} - 9744 q^{30} + 13436 q^{35}+ \cdots + 501812 q^{95}+O(q^{100}) \) 12 * q - 72 * q^3 - 192 * q^4 + 588 * q^9 + 992 * q^10 + 1152 * q^12 - 928 * q^14 + 3072 * q^16 + 1960 * q^17 + 1664 * q^22 + 10816 * q^23 + 5044 * q^25 - 30792 * q^27 + 6556 * q^29 - 9744 * q^30 + 13436 * q^35 - 9408 * q^36 + 432 * q^38 - 15872 * q^40 + 34080 * q^42 + 32100 * q^43 - 18432 * q^48 + 123320 * q^49 - 39336 * q^51 - 44652 * q^53 - 75156 * q^55 + 14848 * q^56 - 136408 * q^61 + 5536 * q^62 - 49152 * q^64 - 198016 * q^66 - 31360 * q^68 + 60736 * q^69 - 75392 * q^74 + 406184 * q^75 + 313512 * q^77 - 132412 * q^79 + 61380 * q^81 + 277424 * q^82 - 743548 * q^87 - 26624 * q^88 + 402000 * q^90 - 173056 * q^92 - 372224 * q^94 + 501812 * q^95

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