LMFDB - Embedded newform 338.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,8,Mod(1,338)]

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([0]))

N = Newforms(chi, 8, names="a")

//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);

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

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:	yes
Analytic conductor:	$105.586138614$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-8.00000 q^{2} -87.0000 q^{3} +64.0000 q^{4} -321.000 q^{5} +696.000 q^{6} +181.000 q^{7} -512.000 q^{8} +5382.00 q^{9} +O(q^{10})$

\(q-8.00000 q^{2} -87.0000 q^{3} +64.0000 q^{4} -321.000 q^{5} +696.000 q^{6} +181.000 q^{7} -512.000 q^{8} +5382.00 q^{9} +2568.00 q^{10} -7782.00 q^{11} -5568.00 q^{12} -1448.00 q^{14} +27927.0 q^{15} +4096.00 q^{16} +9069.00 q^{17} -43056.0 q^{18} +37150.0 q^{19} -20544.0 q^{20} -15747.0 q^{21} +62256.0 q^{22} +19008.0 q^{23} +44544.0 q^{24} +24916.0 q^{25} -277965. q^{27} +11584.0 q^{28} +174750. q^{29} -223416. q^{30} -29012.0 q^{31} -32768.0 q^{32} +677034. q^{33} -72552.0 q^{34} -58101.0 q^{35} +344448. q^{36} -323669. q^{37} -297200. q^{38} +164352. q^{40} -795312. q^{41} +125976. q^{42} -314137. q^{43} -498048. q^{44} -1.72762e6 q^{45} -152064. q^{46} +447441. q^{47} -356352. q^{48} -790782. q^{49} -199328. q^{50} -789003. q^{51} -1.46923e6 q^{53} +2.22372e6 q^{54} +2.49802e6 q^{55} -92672.0 q^{56} -3.23205e6 q^{57} -1.39800e6 q^{58} -1.62777e6 q^{59} +1.78733e6 q^{60} -2.39961e6 q^{61} +232096. q^{62} +974142. q^{63} +262144. q^{64} -5.41627e6 q^{66} +64066.0 q^{67} +580416. q^{68} -1.65370e6 q^{69} +464808. q^{70} +322383. q^{71} -2.75558e6 q^{72} +4.45478e6 q^{73} +2.58935e6 q^{74} -2.16769e6 q^{75} +2.37760e6 q^{76} -1.40854e6 q^{77} +753560. q^{79} -1.31482e6 q^{80} +1.24125e7 q^{81} +6.36250e6 q^{82} +1.21909e6 q^{83} -1.00781e6 q^{84} -2.91115e6 q^{85} +2.51310e6 q^{86} -1.52032e7 q^{87} +3.98438e6 q^{88} -3.39033e6 q^{89} +1.38210e7 q^{90} +1.21651e6 q^{92} +2.52404e6 q^{93} -3.57953e6 q^{94} -1.19252e7 q^{95} +2.85082e6 q^{96} -1.62877e6 q^{97} +6.32626e6 q^{98} -4.18827e7 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$3$	−87.0000	−1.86035	−0.930175	−	0.367115i	$-0.880345\pi$
$3$	−87.0000	−1.86035	−0.930175	+	0.367115i	$0.880345\pi$
$4$	64.0000	0.500000
$5$	−321.000	−1.14844	−0.574222	−	0.818699i	$-0.694695\pi$
$5$	−321.000	−1.14844	−0.574222	+	0.818699i	$0.694695\pi$
$6$	696.000	1.31547
$7$	181.000	0.199451	0.0997253	−	0.995015i	$-0.468204\pi$
$7$	181.000	0.199451	0.0997253	+	0.995015i	$0.468204\pi$
$8$	−512.000	−0.353553
$9$	5382.00	2.46091
$10$	2568.00	0.812073
$11$	−7782.00	−1.76286	−0.881428	−	0.472318i	$-0.843417\pi$
$11$	−7782.00	−1.76286	−0.881428	+	0.472318i	$0.843417\pi$
$12$	−5568.00	−0.930175
$13$	0	0
$13$	0	0
$14$	−1448.00	−0.141033
$15$	27927.0	2.13651
$16$	4096.00	0.250000
$17$	9069.00	0.447701	0.223851	−	0.974623i	$-0.428137\pi$
$17$	9069.00	0.447701	0.223851	+	0.974623i	$0.428137\pi$
$18$	−43056.0	−1.74012
$19$	37150.0	1.24257	0.621286	−	0.783584i	$-0.286611\pi$
$19$	37150.0	1.24257	0.621286	+	0.783584i	$0.286611\pi$
$20$	−20544.0	−0.574222
$21$	−15747.0	−0.371048
$22$	62256.0	1.24653
$23$	19008.0	0.325753	0.162877	−	0.986646i	$-0.447923\pi$
$23$	19008.0	0.325753	0.162877	+	0.986646i	$0.447923\pi$
$24$	44544.0	0.657733
$25$	24916.0	0.318925
$26$	0	0
$27$	−277965.	−2.71780
$28$	11584.0	0.0997253
$29$	174750.	1.33053	0.665264	−	0.746608i	$-0.268319\pi$
$29$	174750.	1.33053	0.665264	+	0.746608i	$0.268319\pi$
$30$	−223416.	−1.51074
$31$	−29012.0	−0.174909	−0.0874544	−	0.996169i	$-0.527873\pi$
$31$	−29012.0	−0.174909	−0.0874544	+	0.996169i	$0.527873\pi$
$32$	−32768.0	−0.176777
$33$	677034.	3.27953
$34$	−72552.0	−0.316572
$35$	−58101.0	−0.229058
$36$	344448.	1.23045
$37$	−323669.	−1.05050	−0.525249	−	0.850949i	$-0.676028\pi$
$37$	−323669.	−1.05050	−0.525249	+	0.850949i	$0.676028\pi$
$38$	−297200.	−0.878630
$39$	0	0
$40$	164352.	0.406036
$41$	−795312.	−1.80216	−0.901081	−	0.433650i	$-0.857225\pi$
$41$	−795312.	−1.80216	−0.901081	+	0.433650i	$0.857225\pi$
$42$	125976.	0.262371
$43$	−314137.	−0.602531	−0.301266	−	0.953540i	$-0.597409\pi$
$43$	−314137.	−0.602531	−0.301266	+	0.953540i	$0.597409\pi$
$44$	−498048.	−0.881428
$45$	−1.72762e6	−2.82621
$46$	−152064.	−0.230342
$47$	447441.	0.628627	0.314314	−	0.949319i	$-0.398226\pi$
$47$	447441.	0.628627	0.314314	+	0.949319i	$0.398226\pi$
$48$	−356352.	−0.465088
$49$	−790782.	−0.960219
$50$	−199328.	−0.225514
$51$	−789003.	−0.832881
$52$	0	0
$53$	−1.46923e6	−1.35558	−0.677790	−	0.735256i	$-0.737062\pi$
$53$	−1.46923e6	−1.35558	−0.677790	+	0.735256i	$0.737062\pi$
$54$	2.22372e6	1.92177
$55$	2.49802e6	2.02454
$56$	−92672.0	−0.0705165
$57$	−3.23205e6	−2.31162
$58$	−1.39800e6	−0.940826
$59$	−1.62777e6	−1.03184	−0.515918	−	0.856638i	$-0.672549\pi$
$59$	−1.62777e6	−1.03184	−0.515918	+	0.856638i	$0.672549\pi$
$60$	1.78733e6	1.06825
$61$	−2.39961e6	−1.35359	−0.676793	−	0.736173i	$-0.736631\pi$
$61$	−2.39961e6	−1.35359	−0.676793	+	0.736173i	$0.736631\pi$
$62$	232096.	0.123679
$63$	974142.	0.490829
$64$	262144.	0.125000
$65$	0	0
$66$	−5.41627e6	−2.31898
$67$	64066.0	0.0260235	0.0130118	−	0.999915i	$-0.495858\pi$
$67$	64066.0	0.0260235	0.0130118	+	0.999915i	$0.495858\pi$
$68$	580416.	0.223851
$69$	−1.65370e6	−0.606016
$70$	464808.	0.161968
$71$	322383.	0.106898	0.0534488	−	0.998571i	$-0.482979\pi$
$71$	322383.	0.106898	0.0534488	+	0.998571i	$0.482979\pi$
$72$	−2.75558e6	−0.870061
$73$	4.45478e6	1.34028	0.670141	−	0.742233i	$-0.266233\pi$
$73$	4.45478e6	1.34028	0.670141	+	0.742233i	$0.266233\pi$
$74$	2.58935e6	0.742814
$75$	−2.16769e6	−0.593312
$76$	2.37760e6	0.621286
$77$	−1.40854e6	−0.351603
$78$	0	0
$79$	753560.	0.171958	0.0859791	−	0.996297i	$-0.472598\pi$
$79$	753560.	0.171958	0.0859791	+	0.996297i	$0.472598\pi$
$80$	−1.31482e6	−0.287111
$81$	1.24125e7	2.59515
$82$	6.36250e6	1.27432
$83$	1.21909e6	0.234025	0.117013	−	0.993130i	$-0.462668\pi$
$83$	1.21909e6	0.234025	0.117013	+	0.993130i	$0.462668\pi$
$84$	−1.00781e6	−0.185524
$85$	−2.91115e6	−0.514160
$86$	2.51310e6	0.426054
$87$	−1.52032e7	−2.47525
$88$	3.98438e6	0.623264
$89$	−3.39033e6	−0.509773	−0.254887	−	0.966971i	$-0.582038\pi$
$89$	−3.39033e6	−0.509773	−0.254887	+	0.966971i	$0.582038\pi$
$90$	1.38210e7	1.99843
$91$	0	0
$92$	1.21651e6	0.162877
$93$	2.52404e6	0.325392
$94$	−3.57953e6	−0.444507
$95$	−1.19252e7	−1.42702
$96$	2.85082e6	0.328867
$97$	−1.62877e6	−0.181201	−0.0906003	−	0.995887i	$-0.528879\pi$
$97$	−1.62877e6	−0.181201	−0.0906003	+	0.995887i	$0.528879\pi$
$98$	6.32626e6	0.678978
$99$	−4.18827e7	−4.33822
$100$	1.59462e6	0.159462
$101$	−1.53503e7	−1.48249	−0.741244	−	0.671236i	$-0.765764\pi$
$101$	−1.53503e7	−1.48249	−0.741244	+	0.671236i	$0.765764\pi$
$102$	6.31202e6	0.588936
$103$	6.87643e6	0.620058	0.310029	−	0.950727i	$-0.399661\pi$
$103$	6.87643e6	0.620058	0.310029	+	0.950727i	$0.399661\pi$
$104$	0	0
$105$	5.05479e6	0.426128
$106$	1.17539e7	0.958539
$107$	−1.52027e7	−1.19971	−0.599857	−	0.800107i	$-0.704776\pi$
$107$	−1.52027e7	−1.19971	−0.599857	+	0.800107i	$0.704776\pi$
$108$	−1.77898e7	−1.35890
$109$	−6.73260e6	−0.497955	−0.248978	−	0.968509i	$-0.580095\pi$
$109$	−6.73260e6	−0.497955	−0.248978	+	0.968509i	$0.580095\pi$
$110$	−1.99842e7	−1.43157
$111$	2.81592e7	1.95429
$112$	741376.	0.0498627
$113$	−1.15292e7	−0.751667	−0.375833	−	0.926687i	$-0.622644\pi$
$113$	−1.15292e7	−0.751667	−0.375833	+	0.926687i	$0.622644\pi$
$114$	2.58564e7	1.63456
$115$	−6.10157e6	−0.374110
$116$	1.11840e7	0.665264
$117$	0	0
$118$	1.30222e7	0.729619
$119$	1.64149e6	0.0892943
$120$	−1.42986e7	−0.755370
$121$	4.10724e7	2.10766
$122$	1.91969e7	0.957130
$123$	6.91921e7	3.35266
$124$	−1.85677e6	−0.0874544
$125$	1.70801e7	0.782177
$126$	−7.79314e6	−0.347069
$127$	2.06699e7	0.895418	0.447709	−	0.894179i	$-0.352240\pi$
$127$	2.06699e7	0.895418	0.447709	+	0.894179i	$0.352240\pi$
$128$	−2.09715e6	−0.0883883
$129$	2.73299e7	1.12092
$130$	0	0
$131$	−1.90949e7	−0.742107	−0.371054	−	0.928611i	$-0.621003\pi$
$131$	−1.90949e7	−0.742107	−0.371054	+	0.928611i	$0.621003\pi$
$132$	4.33302e7	1.63977
$133$	6.72415e6	0.247832
$134$	−512528.	−0.0184014
$135$	8.92268e7	3.12124
$136$	−4.64333e6	−0.158286
$137$	−2.96901e7	−0.986482	−0.493241	−	0.869893i	$-0.664188\pi$
$137$	−2.96901e7	−0.986482	−0.493241	+	0.869893i	$0.664188\pi$
$138$	1.32296e7	0.428518
$139$	1.55652e7	0.491591	0.245795	−	0.969322i	$-0.420951\pi$
$139$	1.55652e7	0.491591	0.245795	+	0.969322i	$0.420951\pi$
$140$	−3.71846e6	−0.114529
$141$	−3.89274e7	−1.16947
$142$	−2.57906e6	−0.0755880
$143$	0	0
$144$	2.20447e7	0.615226
$145$	−5.60948e7	−1.52804
$146$	−3.56383e7	−0.947723
$147$	6.87980e7	1.78635
$148$	−2.07148e7	−0.525249
$149$	2.49675e6	0.0618334	0.0309167	−	0.999522i	$-0.490157\pi$
$149$	2.49675e6	0.0618334	0.0309167	+	0.999522i	$0.490157\pi$
$150$	1.73415e7	0.419535
$151$	−2.39802e7	−0.566804	−0.283402	−	0.959001i	$-0.591463\pi$
$151$	−2.39802e7	−0.566804	−0.283402	+	0.959001i	$0.591463\pi$
$152$	−1.90208e7	−0.439315
$153$	4.88094e7	1.10175
$154$	1.12683e7	0.248621
$155$	9.31285e6	0.200873
$156$	0	0
$157$	1.70550e7	0.351725	0.175863	−	0.984415i	$-0.443729\pi$
$157$	1.70550e7	0.351725	0.175863	+	0.984415i	$0.443729\pi$
$158$	−6.02848e6	−0.121593
$159$	1.27823e8	2.52185
$160$	1.05185e7	0.203018
$161$	3.44045e6	0.0649717
$162$	−9.93002e7	−1.83505
$163$	7.34586e7	1.32857	0.664287	−	0.747477i	$-0.268735\pi$
$163$	7.34586e7	1.32857	0.664287	+	0.747477i	$0.268735\pi$
$164$	−5.09000e7	−0.901081
$165$	−2.17328e8	−3.76636
$166$	−9.75274e6	−0.165481
$167$	−4.66860e7	−0.775674	−0.387837	−	0.921728i	$-0.626778\pi$
$167$	−4.66860e7	−0.775674	−0.387837	+	0.921728i	$0.626778\pi$
$168$	8.06246e6	0.131185
$169$	0	0
$170$	2.32892e7	0.363566
$171$	1.99941e8	3.05785
$172$	−2.01048e7	−0.301266
$173$	−7.80931e7	−1.14670	−0.573352	−	0.819309i	$-0.694357\pi$
$173$	−7.80931e7	−1.14670	−0.573352	+	0.819309i	$0.694357\pi$
$174$	1.21626e8	1.75027
$175$	4.50980e6	0.0636098
$176$	−3.18751e7	−0.440714
$177$	1.41616e8	1.91958
$178$	2.71226e7	0.360464
$179$	−5.56163e7	−0.724797	−0.362399	−	0.932023i	$-0.618042\pi$
$179$	−5.56163e7	−0.724797	−0.362399	+	0.932023i	$0.618042\pi$
$180$	−1.10568e8	−1.41311
$181$	−1.19435e8	−1.49711	−0.748557	−	0.663070i	$-0.769253\pi$
$181$	−1.19435e8	−1.49711	−0.748557	+	0.663070i	$0.769253\pi$
$182$	0	0
$183$	2.08766e8	2.51815
$184$	−9.73210e6	−0.115171
$185$	1.03898e8	1.20644
$186$	−2.01924e7	−0.230087
$187$	−7.05750e7	−0.789233
$188$	2.86362e7	0.314314
$189$	−5.03117e7	−0.542066
$190$	9.54012e7	1.00906
$191$	1.05485e8	1.09540	0.547700	−	0.836675i	$-0.315504\pi$
$191$	1.05485e8	1.09540	0.547700	+	0.836675i	$0.315504\pi$
$192$	−2.28065e7	−0.232544
$193$	−2.12059e7	−0.212327	−0.106164	−	0.994349i	$-0.533857\pi$
$193$	−2.12059e7	−0.212327	−0.106164	+	0.994349i	$0.533857\pi$
$194$	1.30302e7	0.128128
$195$	0	0
$196$	−5.06100e7	−0.480110
$197$	1.66535e8	1.55194	0.775969	−	0.630771i	$-0.217261\pi$
$197$	1.66535e8	1.55194	0.775969	+	0.630771i	$0.217261\pi$
$198$	3.35062e8	3.06759
$199$	−1.26351e8	−1.13656	−0.568279	−	0.822836i	$-0.692391\pi$
$199$	−1.26351e8	−1.13656	−0.568279	+	0.822836i	$0.692391\pi$
$200$	−1.27570e7	−0.112757
$201$	−5.57374e6	−0.0484129
$202$	1.22802e8	1.04828
$203$	3.16298e7	0.265375
$204$	−5.04962e7	−0.416441
$205$	2.55295e8	2.06968
$206$	−5.50114e7	−0.438448
$207$	1.02301e8	0.801648
$208$	0	0
$209$	−2.89101e8	−2.19047
$210$	−4.04383e7	−0.301318
$211$	1.08571e8	0.795655	0.397828	−	0.917460i	$-0.369764\pi$
$211$	1.08571e8	0.795655	0.397828	+	0.917460i	$0.369764\pi$
$212$	−9.40308e7	−0.677790
$213$	−2.80473e7	−0.198867
$214$	1.21622e8	0.848326
$215$	1.00838e8	0.691974
$216$	1.42318e8	0.960886
$217$	−5.25117e6	−0.0348857
$218$	5.38608e7	0.352108
$219$	−3.87566e8	−2.49340
$220$	1.59873e8	1.01227
$221$	0	0
$222$	−2.25274e8	−1.38189
$223$	1.25603e8	0.758459	0.379229	−	0.925303i	$-0.376189\pi$
$223$	1.25603e8	0.758459	0.379229	+	0.925303i	$0.376189\pi$
$224$	−5.93101e6	−0.0352582
$225$	1.34098e8	0.784844
$226$	9.22338e7	0.531509
$227$	−1.90774e8	−1.08250	−0.541252	−	0.840861i	$-0.682049\pi$
$227$	−1.90774e8	−1.08250	−0.541252	+	0.840861i	$0.682049\pi$
$228$	−2.06851e8	−1.15581
$229$	5.28911e7	0.291044	0.145522	−	0.989355i	$-0.453514\pi$
$229$	5.28911e7	0.291044	0.145522	+	0.989355i	$0.453514\pi$
$230$	4.88125e7	0.264536
$231$	1.22543e8	0.654104
$232$	−8.94720e7	−0.470413
$233$	1.51254e8	0.783359	0.391680	−	0.920102i	$-0.371894\pi$
$233$	1.51254e8	0.783359	0.391680	+	0.920102i	$0.371894\pi$
$234$	0	0
$235$	−1.43629e8	−0.721944
$236$	−1.04177e8	−0.515918
$237$	−6.55597e7	−0.319903
$238$	−1.31319e7	−0.0631406
$239$	−2.61917e8	−1.24100	−0.620498	−	0.784208i	$-0.713070\pi$
$239$	−2.61917e8	−1.24100	−0.620498	+	0.784208i	$0.713070\pi$
$240$	1.14389e8	0.534127
$241$	1.31752e8	0.606312	0.303156	−	0.952941i	$-0.401960\pi$
$241$	1.31752e8	0.606312	0.303156	+	0.952941i	$0.401960\pi$
$242$	−3.28579e8	−1.49034
$243$	−4.71980e8	−2.11009
$244$	−1.53575e8	−0.676793
$245$	2.53841e8	1.10276
$246$	−5.53537e8	−2.37069
$247$	0	0
$248$	1.48541e7	0.0618396
$249$	−1.06061e8	−0.435370
$250$	−1.36641e8	−0.553083
$251$	2.47061e8	0.986159	0.493080	−	0.869984i	$-0.335871\pi$
$251$	2.47061e8	0.986159	0.493080	+	0.869984i	$0.335871\pi$
$252$	6.23451e7	0.245415
$253$	−1.47920e8	−0.574256
$254$	−1.65359e8	−0.633156
$255$	2.53270e8	0.956518
$256$	1.67772e7	0.0625000
$257$	2.27286e8	0.835231	0.417616	−	0.908624i	$-0.362866\pi$
$257$	2.27286e8	0.835231	0.417616	+	0.908624i	$0.362866\pi$
$258$	−2.18639e8	−0.792610
$259$	−5.85841e7	−0.209522
$260$	0	0
$261$	9.40504e8	3.27430
$262$	1.52759e8	0.524749
$263$	−4.25872e8	−1.44356	−0.721779	−	0.692124i	$-0.756675\pi$
$263$	−4.25872e8	−1.44356	−0.721779	+	0.692124i	$0.756675\pi$
$264$	−3.46641e8	−1.15949
$265$	4.71623e8	1.55681
$266$	−5.37932e7	−0.175243
$267$	2.94959e8	0.948357
$268$	4.10022e6	0.0130118
$269$	−5.14154e8	−1.61050	−0.805250	−	0.592936i	$-0.797969\pi$
$269$	−5.14154e8	−1.61050	−0.805250	+	0.592936i	$0.797969\pi$
$270$	−7.13814e8	−2.20705
$271$	−4.57096e7	−0.139513	−0.0697565	−	0.997564i	$-0.522222\pi$
$271$	−4.57096e7	−0.139513	−0.0697565	+	0.997564i	$0.522222\pi$
$272$	3.71466e7	0.111925
$273$	0	0
$274$	2.37521e8	0.697548
$275$	−1.93896e8	−0.562218
$276$	−1.05837e8	−0.303008
$277$	−2.73964e8	−0.774487	−0.387244	−	0.921977i	$-0.626573\pi$
$277$	−2.73964e8	−0.774487	−0.387244	+	0.921977i	$0.626573\pi$
$278$	−1.24522e8	−0.347607
$279$	−1.56143e8	−0.430434
$280$	2.97477e7	0.0809842
$281$	−4.21707e8	−1.13381	−0.566903	−	0.823784i	$-0.691859\pi$
$281$	−4.21707e8	−1.13381	−0.566903	+	0.823784i	$0.691859\pi$
$282$	3.11419e8	0.826938
$283$	3.81957e8	1.00176	0.500878	−	0.865518i	$-0.333010\pi$
$283$	3.81957e8	1.00176	0.500878	+	0.865518i	$0.333010\pi$
$284$	2.06325e7	0.0534488
$285$	1.03749e9	2.65477
$286$	0	0
$287$	−1.43951e8	−0.359443
$288$	−1.76357e8	−0.435031
$289$	−3.28092e8	−0.799564
$290$	4.48758e8	1.08049
$291$	1.41703e8	0.337097
$292$	2.85106e8	0.670141
$293$	4.04833e8	0.940240	0.470120	−	0.882602i	$-0.344211\pi$
$293$	4.04833e8	0.940240	0.470120	+	0.882602i	$0.344211\pi$
$294$	−5.50384e8	−1.26314
$295$	5.22514e8	1.18501
$296$	1.65719e8	0.371407
$297$	2.16312e9	4.79108
$298$	−1.99740e7	−0.0437228
$299$	0	0
$300$	−1.38732e8	−0.296656
$301$	−5.68588e7	−0.120175
$302$	1.91841e8	0.400791
$303$	1.33547e9	2.75795
$304$	1.52166e8	0.310643
$305$	7.70274e8	1.55452
$306$	−3.90475e8	−0.779055
$307$	4.75520e7	0.0937960	0.0468980	−	0.998900i	$-0.485066\pi$
$307$	4.75520e7	0.0937960	0.0468980	+	0.998900i	$0.485066\pi$
$308$	−9.01467e7	−0.175801
$309$	−5.98249e8	−1.15353
$310$	−7.45028e7	−0.142039
$311$	−3.02841e8	−0.570892	−0.285446	−	0.958395i	$-0.592142\pi$
$311$	−3.02841e8	−0.570892	−0.285446	+	0.958395i	$0.592142\pi$
$312$	0	0
$313$	−6.31685e8	−1.16438	−0.582191	−	0.813052i	$-0.697804\pi$
$313$	−6.31685e8	−1.16438	−0.582191	+	0.813052i	$0.697804\pi$
$314$	−1.36440e8	−0.248707
$315$	−3.12700e8	−0.563690
$316$	4.82278e7	0.0859791
$317$	−7.93332e8	−1.39877	−0.699387	−	0.714743i	$-0.746544\pi$
$317$	−7.93332e8	−1.39877	−0.699387	+	0.714743i	$0.746544\pi$
$318$	−1.02259e9	−1.78322
$319$	−1.35990e9	−2.34553
$320$	−8.41482e7	−0.143556
$321$	1.32264e9	2.23189
$322$	−2.75236e7	−0.0459420
$323$	3.36913e8	0.556300
$324$	7.94401e8	1.29757
$325$	0	0
$326$	−5.87669e8	−0.939444
$327$	5.85737e8	0.926372
$328$	4.07200e8	0.637161
$329$	8.09868e7	0.125380
$330$	1.73862e9	2.66322
$331$	1.21628e9	1.84346	0.921731	−	0.387829i	$-0.126775\pi$
$331$	1.21628e9	1.84346	0.921731	+	0.387829i	$0.126775\pi$
$332$	7.80219e7	0.117013
$333$	−1.74199e9	−2.58518
$334$	3.73488e8	0.548484
$335$	−2.05652e7	−0.0298866
$336$	−6.44997e7	−0.0927620
$337$	−1.51221e8	−0.215232	−0.107616	−	0.994193i	$-0.534322\pi$
$337$	−1.51221e8	−0.215232	−0.107616	+	0.994193i	$0.534322\pi$
$338$	0	0
$339$	1.00304e9	1.39836
$340$	−1.86314e8	−0.257080
$341$	2.25771e8	0.308339
$342$	−1.59953e9	−2.16223
$343$	−2.92193e8	−0.390967
$344$	1.60838e8	0.213027
$345$	5.30836e8	0.695975
$346$	6.24745e8	0.810842
$347$	−5.97234e8	−0.767347	−0.383673	−	0.923469i	$-0.625341\pi$
$347$	−5.97234e8	−0.767347	−0.383673	+	0.923469i	$0.625341\pi$
$348$	−9.73008e8	−1.23762
$349$	−1.19600e8	−0.150606	−0.0753029	−	0.997161i	$-0.523992\pi$
$349$	−1.19600e8	−0.150606	−0.0753029	+	0.997161i	$0.523992\pi$
$350$	−3.60784e7	−0.0449789
$351$	0	0
$352$	2.55001e8	0.311632
$353$	4.66414e8	0.564366	0.282183	−	0.959361i	$-0.408942\pi$
$353$	4.66414e8	0.564366	0.282183	+	0.959361i	$0.408942\pi$
$354$	−1.13293e9	−1.35735
$355$	−1.03485e8	−0.122766
$356$	−2.16981e8	−0.254887
$357$	−1.42810e8	−0.166119
$358$	4.44931e8	0.512509
$359$	7.70102e8	0.878451	0.439225	−	0.898377i	$-0.355253\pi$
$359$	7.70102e8	0.878451	0.439225	+	0.898377i	$0.355253\pi$
$360$	8.84542e8	0.999217
$361$	4.86251e8	0.543983
$362$	9.55477e8	1.05862
$363$	−3.57329e9	−3.92099
$364$	0	0
$365$	−1.42999e9	−1.53924
$366$	−1.67013e9	−1.78060
$367$	−8.55319e8	−0.903227	−0.451613	−	0.892214i	$-0.649151\pi$
$367$	−8.55319e8	−0.903227	−0.451613	+	0.892214i	$0.649151\pi$
$368$	7.78568e7	0.0814384
$369$	−4.28037e9	−4.43495
$370$	−8.31182e8	−0.853081
$371$	−2.65931e8	−0.270371
$372$	1.61539e8	0.162696
$373$	−5.29609e8	−0.528414	−0.264207	−	0.964466i	$-0.585110\pi$
$373$	−5.29609e8	−0.528414	−0.264207	+	0.964466i	$0.585110\pi$
$374$	5.64600e8	0.558072
$375$	−1.48597e9	−1.45512
$376$	−2.29090e8	−0.222253
$377$	0	0
$378$	4.02493e8	0.383299
$379$	−1.98358e9	−1.87159	−0.935797	−	0.352540i	$-0.885318\pi$
$379$	−1.98358e9	−1.87159	−0.935797	+	0.352540i	$0.885318\pi$
$380$	−7.63210e8	−0.713512
$381$	−1.79828e9	−1.66579
$382$	−8.43877e8	−0.774564
$383$	−8.98756e8	−0.817422	−0.408711	−	0.912664i	$-0.634022\pi$
$383$	−8.98756e8	−0.817422	−0.408711	+	0.912664i	$0.634022\pi$
$384$	1.82452e8	0.164433
$385$	4.52142e8	0.403796
$386$	1.69647e8	0.150138
$387$	−1.69069e9	−1.48277
$388$	−1.04242e8	−0.0906003
$389$	1.82475e9	1.57174	0.785868	−	0.618395i	$-0.212217\pi$
$389$	1.82475e9	1.57174	0.785868	+	0.618395i	$0.212217\pi$
$390$	0	0
$391$	1.72384e8	0.145840
$392$	4.04880e8	0.339489
$393$	1.66125e9	1.38058
$394$	−1.33228e9	−1.09739
$395$	−2.41893e8	−0.197485
$396$	−2.68049e9	−2.16911
$397$	−4.93083e8	−0.395506	−0.197753	−	0.980252i	$-0.563364\pi$
$397$	−4.93083e8	−0.395506	−0.197753	+	0.980252i	$0.563364\pi$
$398$	1.01081e9	0.803668
$399$	−5.85001e8	−0.461054
$400$	1.02056e8	0.0797312
$401$	5.68280e8	0.440105	0.220053	−	0.975488i	$-0.429377\pi$
$401$	5.68280e8	0.440105	0.220053	+	0.975488i	$0.429377\pi$
$402$	4.45899e7	0.0342331
$403$	0	0
$404$	−9.82417e8	−0.741244
$405$	−3.98442e9	−2.98039
$406$	−2.53038e8	−0.187648
$407$	2.51879e9	1.85188
$408$	4.03970e8	0.294468
$409$	1.28472e9	0.928489	0.464245	−	0.885707i	$-0.346326\pi$
$409$	1.28472e9	0.928489	0.464245	+	0.885707i	$0.346326\pi$
$410$	−2.04236e9	−1.46349
$411$	2.58304e9	1.83520
$412$	4.40091e8	0.310029
$413$	−2.94626e8	−0.205801
$414$	−8.18408e8	−0.566851
$415$	−3.91329e8	−0.268765
$416$	0	0
$417$	−1.35418e9	−0.914532
$418$	2.31281e9	1.54890
$419$	2.74847e8	0.182533	0.0912667	−	0.995826i	$-0.470908\pi$
$419$	2.74847e8	0.182533	0.0912667	+	0.995826i	$0.470908\pi$
$420$	3.23506e8	0.213064
$421$	−7.51368e8	−0.490756	−0.245378	−	0.969428i	$-0.578912\pi$
$421$	−7.51368e8	−0.490756	−0.245378	+	0.969428i	$0.578912\pi$
$422$	−8.68568e8	−0.562613
$423$	2.40813e9	1.54699
$424$	7.52247e8	0.479270
$425$	2.25963e8	0.142783
$426$	2.24379e8	0.140620
$427$	−4.34329e8	−0.269974
$428$	−9.72974e8	−0.599857
$429$	0	0
$430$	−8.06704e8	−0.489299
$431$	1.30756e8	0.0786668	0.0393334	−	0.999226i	$-0.487477\pi$
$431$	1.30756e8	0.0786668	0.0393334	+	0.999226i	$0.487477\pi$
$432$	−1.13854e9	−0.679449
$433$	1.66736e9	0.987010	0.493505	−	0.869743i	$-0.335716\pi$
$433$	1.66736e9	0.987010	0.493505	+	0.869743i	$0.335716\pi$
$434$	4.20094e7	0.0246679
$435$	4.88024e9	2.84269
$436$	−4.30887e8	−0.248978
$437$	7.06147e8	0.404772
$438$	3.10053e9	1.76310
$439$	2.31478e9	1.30582	0.652910	−	0.757436i	$-0.273548\pi$
$439$	2.31478e9	1.30582	0.652910	+	0.757436i	$0.273548\pi$
$440$	−1.27899e9	−0.715784
$441$	−4.25599e9	−2.36301
$442$	0	0
$443$	−6.90047e8	−0.377108	−0.188554	−	0.982063i	$-0.560380\pi$
$443$	−6.90047e8	−0.377108	−0.188554	+	0.982063i	$0.560380\pi$
$444$	1.80219e9	0.977147
$445$	1.08830e9	0.585446
$446$	−1.00482e9	−0.536311
$447$	−2.17217e8	−0.115032
$448$	4.74481e7	0.0249313
$449$	−2.63806e9	−1.37538	−0.687690	−	0.726004i	$-0.741375\pi$
$449$	−2.63806e9	−1.37538	−0.687690	+	0.726004i	$0.741375\pi$
$450$	−1.07278e9	−0.554968
$451$	6.18912e9	3.17695
$452$	−7.37870e8	−0.375833
$453$	2.08627e9	1.05445
$454$	1.52619e9	0.765445
$455$	0	0
$456$	1.65481e9	0.817280
$457$	6.16222e8	0.302016	0.151008	−	0.988533i	$-0.451748\pi$
$457$	6.16222e8	0.302016	0.151008	+	0.988533i	$0.451748\pi$
$458$	−4.23129e8	−0.205799
$459$	−2.52086e9	−1.21676
$460$	−3.90500e8	−0.187055
$461$	−1.23621e9	−0.587679	−0.293839	−	0.955855i	$-0.594933\pi$
$461$	−1.23621e9	−0.587679	−0.293839	+	0.955855i	$0.594933\pi$
$462$	−9.80345e8	−0.462522
$463$	−6.78469e7	−0.0317685	−0.0158843	−	0.999874i	$-0.505056\pi$
$463$	−6.78469e7	−0.0317685	−0.0158843	+	0.999874i	$0.505056\pi$
$464$	7.15776e8	0.332632
$465$	−8.10218e8	−0.373694
$466$	−1.21003e9	−0.553919
$467$	−1.17502e9	−0.533869	−0.266934	−	0.963715i	$-0.586011\pi$
$467$	−1.17502e9	−0.533869	−0.266934	+	0.963715i	$0.586011\pi$
$468$	0	0
$469$	1.15959e7	0.00519040
$470$	1.14903e9	0.510491
$471$	−1.48379e9	−0.654332
$472$	8.33418e8	0.364809
$473$	2.44461e9	1.06218
$474$	5.24478e8	0.226205
$475$	9.25629e8	0.396287
$476$	1.05055e8	0.0446471
$477$	−7.90741e9	−3.33595
$478$	2.09533e9	0.877517
$479$	−3.96154e8	−0.164699	−0.0823494	−	0.996604i	$-0.526242\pi$
$479$	−3.96154e8	−0.164699	−0.0823494	+	0.996604i	$0.526242\pi$
$480$	−9.15112e8	−0.377685
$481$	0	0
$482$	−1.05401e9	−0.428728
$483$	−2.99319e8	−0.120870
$484$	2.62863e9	1.05383
$485$	5.22836e8	0.208099
$486$	3.77584e9	1.49206
$487$	3.03665e9	1.19136	0.595680	−	0.803222i	$-0.296883\pi$
$487$	3.03665e9	1.19136	0.595680	+	0.803222i	$0.296883\pi$
$488$	1.22860e9	0.478565
$489$	−6.39090e9	−2.47162
$490$	−2.03073e9	−0.779768
$491$	−2.91974e9	−1.11316	−0.556582	−	0.830793i	$-0.687887\pi$
$491$	−2.91974e9	−1.11316	−0.556582	+	0.830793i	$0.687887\pi$
$492$	4.42830e9	1.67633
$493$	1.58481e9	0.595679
$494$	0	0
$495$	1.34444e10	4.98221
$496$	−1.18833e8	−0.0437272
$497$	5.83513e7	0.0213208
$498$	8.48488e8	0.307853
$499$	1.62343e9	0.584898	0.292449	−	0.956281i	$-0.405530\pi$
$499$	1.62343e9	0.584898	0.292449	+	0.956281i	$0.405530\pi$
$500$	1.09313e9	0.391089
$501$	4.06168e9	1.44303
$502$	−1.97649e9	−0.697320
$503$	4.75888e9	1.66731	0.833655	−	0.552285i	$-0.186244\pi$
$503$	4.75888e9	1.66731	0.833655	+	0.552285i	$0.186244\pi$
$504$	−4.98761e8	−0.173534
$505$	4.92744e9	1.70256
$506$	1.18336e9	0.406061
$507$	0	0
$508$	1.32288e9	0.447709
$509$	9.19375e8	0.309016	0.154508	−	0.987992i	$-0.450621\pi$
$509$	9.19375e8	0.309016	0.154508	+	0.987992i	$0.450621\pi$
$510$	−2.02616e9	−0.676360
$511$	8.06316e8	0.267320
$512$	−1.34218e8	−0.0441942
$513$	−1.03264e10	−3.37706
$514$	−1.81829e9	−0.590598
$515$	−2.20733e9	−0.712103
$516$	1.74911e9	0.560460
$517$	−3.48199e9	−1.10818
$518$	4.68673e8	0.148155
$519$	6.79410e9	2.13327
$520$	0	0
$521$	−1.46089e9	−0.452569	−0.226284	−	0.974061i	$-0.572658\pi$
$521$	−1.46089e9	−0.452569	−0.226284	+	0.974061i	$0.572658\pi$
$522$	−7.52404e9	−2.31528
$523$	2.12856e9	0.650624	0.325312	−	0.945607i	$-0.394531\pi$
$523$	2.12856e9	0.650624	0.325312	+	0.945607i	$0.394531\pi$
$524$	−1.22207e9	−0.371054
$525$	−3.92352e8	−0.118336
$526$	3.40698e9	1.02075
$527$	−2.63110e8	−0.0783069
$528$	2.77313e9	0.819883
$529$	−3.04352e9	−0.893885
$530$	−3.77299e9	−1.10083
$531$	−8.76066e9	−2.53925
$532$	4.30346e8	0.123916
$533$	0	0
$534$	−2.35967e9	−0.670590
$535$	4.88007e9	1.37781
$536$	−3.28018e7	−0.00920070
$537$	4.83862e9	1.34838
$538$	4.11323e9	1.13879
$539$	6.15387e9	1.69273
$540$	5.71051e9	1.56062
$541$	1.72479e7	0.00468324	0.00234162	−	0.999997i	$-0.499255\pi$
$541$	1.72479e7	0.00468324	0.00234162	+	0.999997i	$0.499255\pi$
$542$	3.65677e8	0.0986507
$543$	1.03908e10	2.78516
$544$	−2.97173e8	−0.0791431
$545$	2.16117e9	0.571874
$546$	0	0
$547$	7.51154e8	0.196234	0.0981168	−	0.995175i	$-0.468718\pi$
$547$	7.51154e8	0.196234	0.0981168	+	0.995175i	$0.468718\pi$
$548$	−1.90016e9	−0.493241
$549$	−1.29147e10	−3.33105
$550$	1.55117e9	0.397549
$551$	6.49196e9	1.65328
$552$	8.46692e8	0.214259
$553$	1.36394e8	0.0342972
$554$	2.19171e9	0.547645
$555$	−9.03910e9	−2.24440
$556$	9.96175e8	0.245795
$557$	−3.00701e9	−0.737295	−0.368647	−	0.929569i	$-0.620179\pi$
$557$	−3.00701e9	−0.737295	−0.368647	+	0.929569i	$0.620179\pi$
$558$	1.24914e9	0.304363
$559$	0	0
$560$	−2.37982e8	−0.0572645
$561$	6.14002e9	1.46825
$562$	3.37366e9	0.801722
$563$	2.82880e9	0.668070	0.334035	−	0.942561i	$-0.391590\pi$
$563$	2.82880e9	0.668070	0.334035	+	0.942561i	$0.391590\pi$
$564$	−2.49135e9	−0.584734
$565$	3.70088e9	0.863248
$566$	−3.05566e9	−0.708349
$567$	2.24667e9	0.517604
$568$	−1.65060e8	−0.0377940
$569$	7.67290e9	1.74609	0.873045	−	0.487639i	$-0.162142\pi$
$569$	7.67290e9	1.74609	0.873045	+	0.487639i	$0.162142\pi$
$570$	−8.29990e9	−1.87720
$571$	−3.09363e9	−0.695411	−0.347706	−	0.937604i	$-0.613039\pi$
$571$	−3.09363e9	−0.695411	−0.347706	+	0.937604i	$0.613039\pi$
$572$	0	0
$573$	−9.17717e9	−2.03783
$574$	1.15161e9	0.254164
$575$	4.73603e8	0.103891
$576$	1.41086e9	0.307613
$577$	−3.71815e9	−0.805770	−0.402885	−	0.915251i	$-0.631993\pi$
$577$	−3.71815e9	−0.805770	−0.402885	+	0.915251i	$0.631993\pi$
$578$	2.62474e9	0.565377
$579$	1.84491e9	0.395003
$580$	−3.59006e9	−0.764019
$581$	2.20656e8	0.0466765
$582$	−1.13363e9	−0.238363
$583$	1.14336e10	2.38969
$584$	−2.28085e9	−0.473862
$585$	0	0
$586$	−3.23866e9	−0.664850
$587$	2.74853e9	0.560876	0.280438	−	0.959872i	$-0.409520\pi$
$587$	2.74853e9	0.560876	0.280438	+	0.959872i	$0.409520\pi$
$588$	4.40307e9	0.893173
$589$	−1.07780e9	−0.217337
$590$	−4.18011e9	−0.837927
$591$	−1.44886e10	−2.88715
$592$	−1.32575e9	−0.262624
$593$	9.11262e9	1.79453	0.897267	−	0.441488i	$-0.145549\pi$
$593$	9.11262e9	1.79453	0.897267	+	0.441488i	$0.145549\pi$
$594$	−1.73050e10	−3.38781
$595$	−5.26918e8	−0.102550
$596$	1.59792e8	0.0309167
$597$	1.09925e10	2.11440
$598$	0	0
$599$	−5.52493e9	−1.05035	−0.525174	−	0.850995i	$-0.676000\pi$
$599$	−5.52493e9	−1.05035	−0.525174	+	0.850995i	$0.676000\pi$
$600$	1.10986e9	0.209767
$601$	−1.78219e9	−0.334883	−0.167441	−	0.985882i	$-0.553551\pi$
$601$	−1.78219e9	−0.334883	−0.167441	+	0.985882i	$0.553551\pi$
$602$	4.54870e8	0.0849767
$603$	3.44803e8	0.0640414
$604$	−1.53473e9	−0.283402
$605$	−1.31842e10	−2.42053
$606$	−1.06838e10	−1.95016
$607$	9.53705e9	1.73083	0.865414	−	0.501058i	$-0.167056\pi$
$607$	9.53705e9	1.73083	0.865414	+	0.501058i	$0.167056\pi$
$608$	−1.21733e9	−0.219658
$609$	−2.75179e9	−0.493690
$610$	−6.16219e9	−1.09921
$611$	0	0
$612$	3.12380e9	0.550875
$613$	1.18627e9	0.208004	0.104002	−	0.994577i	$-0.466835\pi$
$613$	1.18627e9	0.208004	0.104002	+	0.994577i	$0.466835\pi$
$614$	−3.80416e8	−0.0663238
$615$	−2.22107e10	−3.85034
$616$	7.21174e8	0.124310
$617$	−1.32256e9	−0.226682	−0.113341	−	0.993556i	$-0.536155\pi$
$617$	−1.32256e9	−0.226682	−0.113341	+	0.993556i	$0.536155\pi$
$618$	4.78599e9	0.815666
$619$	3.59450e9	0.609147	0.304573	−	0.952489i	$-0.401486\pi$
$619$	3.59450e9	0.609147	0.304573	+	0.952489i	$0.401486\pi$
$620$	5.96023e8	0.100437
$621$	−5.28356e9	−0.885332
$622$	2.42273e9	0.403681
$623$	−6.13650e8	−0.101675
$624$	0	0
$625$	−7.42927e9	−1.21721
$626$	5.05348e9	0.823343
$627$	2.51518e10	4.07505
$628$	1.09152e9	0.175863
$629$	−2.93535e9	−0.470309
$630$	2.50160e9	0.398589
$631$	−7.49102e6	−0.00118697	−0.000593483	−	1.00000i	$-0.500189\pi$
$631$	−7.49102e6	−0.00118697	−0.000593483		1.00000i	$0.500189\pi$
$632$	−3.85823e8	−0.0607964
$633$	−9.44567e9	−1.48020
$634$	6.34666e9	0.989083
$635$	−6.63505e9	−1.02834
$636$	8.18068e9	1.26093
$637$	0	0
$638$	1.08792e10	1.65854
$639$	1.73507e9	0.263065
$640$	6.73186e8	0.101509
$641$	4.06396e9	0.609462	0.304731	−	0.952438i	$-0.401433\pi$
$641$	4.06396e9	0.609462	0.304731	+	0.952438i	$0.401433\pi$
$642$	−1.05811e10	−1.57818
$643$	−1.56544e6	−0.000232219 0	−0.000116109	−	1.00000i	$-0.500037\pi$
$643$	−1.56544e6	−0.000232219 0	−0.000116109		1.00000i	$0.500037\pi$
$644$	2.20189e8	0.0324859
$645$	−8.77290e9	−1.28731
$646$	−2.69531e9	−0.393364
$647$	1.31025e10	1.90191	0.950956	−	0.309325i	$-0.100103\pi$
$647$	1.31025e10	1.90191	0.950956	+	0.309325i	$0.100103\pi$
$648$	−6.35521e9	−0.917524
$649$	1.26673e10	1.81898
$650$	0	0
$651$	4.56852e8	0.0648996
$652$	4.70135e9	0.664287
$653$	7.63326e9	1.07279	0.536394	−	0.843968i	$-0.319786\pi$
$653$	7.63326e9	1.07279	0.536394	+	0.843968i	$0.319786\pi$
$654$	−4.68589e9	−0.655044
$655$	6.12945e9	0.852269
$656$	−3.25760e9	−0.450541
$657$	2.39756e10	3.29831
$658$	−6.47895e8	−0.0886571
$659$	−9.25900e9	−1.26027	−0.630137	−	0.776484i	$-0.717001\pi$
$659$	−9.25900e9	−1.26027	−0.630137	+	0.776484i	$0.717001\pi$
$660$	−1.39090e10	−1.88318
$661$	−4.79962e9	−0.646401	−0.323201	−	0.946330i	$-0.604759\pi$
$661$	−4.79962e9	−0.646401	−0.323201	+	0.946330i	$0.604759\pi$
$662$	−9.73021e9	−1.30352
$663$	0	0
$664$	−6.24175e8	−0.0827405
$665$	−2.15845e9	−0.284621
$666$	1.39359e10	1.82799
$667$	3.32165e9	0.433424
$668$	−2.98791e9	−0.387837
$669$	−1.09274e10	−1.41100
$670$	1.64521e8	0.0211330
$671$	1.86737e10	2.38618
$672$	5.15998e8	0.0655927
$673$	−1.08997e10	−1.37836	−0.689182	−	0.724589i	$-0.742030\pi$
$673$	−1.08997e10	−1.37836	−0.689182	+	0.724589i	$0.742030\pi$
$674$	1.20977e9	0.152192
$675$	−6.92578e9	−0.866773
$676$	0	0
$677$	3.44099e9	0.426210	0.213105	−	0.977029i	$-0.431642\pi$
$677$	3.44099e9	0.426210	0.213105	+	0.977029i	$0.431642\pi$
$678$	−8.02434e9	−0.988793
$679$	−2.94808e8	−0.0361406
$680$	1.49051e9	0.181783
$681$	1.65974e10	2.01384
$682$	−1.80617e9	−0.218029
$683$	−5.53553e9	−0.664794	−0.332397	−	0.943140i	$-0.607857\pi$
$683$	−5.53553e9	−0.664794	−0.332397	+	0.943140i	$0.607857\pi$
$684$	1.27962e10	1.52892
$685$	9.53051e9	1.13292
$686$	2.33754e9	0.276455
$687$	−4.60153e9	−0.541444
$688$	−1.28671e9	−0.150633
$689$	0	0
$690$	−4.24669e9	−0.492129
$691$	−4.21595e8	−0.0486097	−0.0243048	−	0.999705i	$-0.507737\pi$
$691$	−4.21595e8	−0.0486097	−0.0243048	+	0.999705i	$0.507737\pi$
$692$	−4.99796e9	−0.573352
$693$	−7.58077e9	−0.865261
$694$	4.77788e9	0.542596
$695$	−4.99644e9	−0.564565
$696$	7.78406e9	0.875133
$697$	−7.21268e9	−0.806830
$698$	9.56799e8	0.106494
$699$	−1.31591e10	−1.45732
$700$	2.88627e8	0.0318049
$701$	−5.14995e9	−0.564663	−0.282332	−	0.959317i	$-0.591108\pi$
$701$	−5.14995e9	−0.564663	−0.282332	+	0.959317i	$0.591108\pi$
$702$	0	0
$703$	−1.20243e10	−1.30532
$704$	−2.04000e9	−0.220357
$705$	1.24957e10	1.34307
$706$	−3.73132e9	−0.399067
$707$	−2.77840e9	−0.295683
$708$	9.06342e9	0.959789
$709$	−1.05683e10	−1.11363	−0.556817	−	0.830635i	$-0.687978\pi$
$709$	−1.05683e10	−1.11363	−0.556817	+	0.830635i	$0.687978\pi$
$710$	8.27880e8	0.0868086
$711$	4.05566e9	0.423173
$712$	1.73585e9	0.180232
$713$	−5.51460e8	−0.0569772
$714$	1.14248e9	0.117464
$715$	0	0
$716$	−3.55944e9	−0.362399
$717$	2.27868e10	2.30869
$718$	−6.16081e9	−0.621159
$719$	1.53690e10	1.54204	0.771020	−	0.636811i	$-0.219747\pi$
$719$	1.53690e10	1.54204	0.771020	+	0.636811i	$0.219747\pi$
$720$	−7.07634e9	−0.706553
$721$	1.24463e9	0.123671
$722$	−3.89001e9	−0.384654
$723$	−1.14624e10	−1.12795
$724$	−7.64381e9	−0.748557
$725$	4.35407e9	0.424339
$726$	2.85864e10	2.77256
$727$	4.88599e9	0.471609	0.235804	−	0.971801i	$-0.424228\pi$
$727$	4.88599e9	0.471609	0.235804	+	0.971801i	$0.424228\pi$
$728$	0	0
$729$	1.39161e10	1.33036
$730$	1.14399e10	1.08841
$731$	−2.84891e9	−0.269754
$732$	1.33610e10	1.25907
$733$	−3.59889e9	−0.337524	−0.168762	−	0.985657i	$-0.553977\pi$
$733$	−3.59889e9	−0.337524	−0.168762	+	0.985657i	$0.553977\pi$
$734$	6.84255e9	0.638678
$735$	−2.20842e10	−2.05152
$736$	−6.22854e8	−0.0575856
$737$	−4.98562e8	−0.0458757
$738$	3.42430e10	3.13598
$739$	−2.78886e9	−0.254198	−0.127099	−	0.991890i	$-0.540567\pi$
$739$	−2.78886e9	−0.254198	−0.127099	+	0.991890i	$0.540567\pi$
$740$	6.64946e9	0.603219
$741$	0	0
$742$	2.12745e9	0.191181
$743$	3.08130e9	0.275597	0.137798	−	0.990460i	$-0.455997\pi$
$743$	3.08130e9	0.275597	0.137798	+	0.990460i	$0.455997\pi$
$744$	−1.29231e9	−0.115043
$745$	−8.01457e8	−0.0710122
$746$	4.23687e9	0.373645
$747$	6.56115e9	0.575915
$748$	−4.51680e9	−0.394616
$749$	−2.75169e9	−0.239284
$750$	1.18877e10	1.02893
$751$	−6.41281e8	−0.0552470	−0.0276235	−	0.999618i	$-0.508794\pi$
$751$	−6.41281e8	−0.0552470	−0.0276235	+	0.999618i	$0.508794\pi$
$752$	1.83272e9	0.157157
$753$	−2.14943e10	−1.83460
$754$	0	0
$755$	7.69763e9	0.650943
$756$	−3.21995e9	−0.271033
$757$	−1.60219e10	−1.34239	−0.671195	−	0.741280i	$-0.734219\pi$
$757$	−1.60219e10	−1.34239	−0.671195	+	0.741280i	$0.734219\pi$
$758$	1.58686e10	1.32342
$759$	1.28691e10	1.06832
$760$	6.10568e9	0.504529
$761$	5.73623e9	0.471824	0.235912	−	0.971774i	$-0.424192\pi$
$761$	5.73623e9	0.471824	0.235912	+	0.971774i	$0.424192\pi$
$762$	1.43863e10	1.17789
$763$	−1.21860e9	−0.0993175
$764$	6.75102e9	0.547700
$765$	−1.56678e10	−1.26530
$766$	7.19005e9	0.578005
$767$	0	0
$768$	−1.45962e9	−0.116272
$769$	−2.45874e10	−1.94971	−0.974857	−	0.222832i	$-0.928470\pi$
$769$	−2.45874e10	−1.94971	−0.974857	+	0.222832i	$0.928470\pi$
$770$	−3.61714e9	−0.285527
$771$	−1.97739e10	−1.55382
$772$	−1.35718e9	−0.106164
$773$	1.31517e10	1.02413	0.512065	−	0.858947i	$-0.328881\pi$
$773$	1.31517e10	1.02413	0.512065	+	0.858947i	$0.328881\pi$
$774$	1.35255e10	1.04848
$775$	−7.22863e8	−0.0557828
$776$	8.33932e8	0.0640641
$777$	5.09682e9	0.389785
$778$	−1.45980e10	−1.11138
$779$	−2.95458e10	−2.23932
$780$	0	0
$781$	−2.50878e9	−0.188445
$782$	−1.37907e9	−0.103125
$783$	−4.85744e10	−3.61611
$784$	−3.23904e9	−0.240055
$785$	−5.47466e9	−0.403937
$786$	−1.32900e10	−0.976218
$787$	7.38863e9	0.540322	0.270161	−	0.962815i	$-0.412923\pi$
$787$	7.38863e9	0.540322	0.270161	+	0.962815i	$0.412923\pi$
$788$	1.06583e10	0.775969
$789$	3.70509e10	2.68552
$790$	1.93514e9	0.139643
$791$	−2.08679e9	−0.149920
$792$	2.14440e10	1.53379
$793$	0	0
$794$	3.94467e9	0.279665
$795$	−4.10312e10	−2.89621
$796$	−8.08645e9	−0.568279
$797$	−5.22399e9	−0.365509	−0.182754	−	0.983159i	$-0.558501\pi$
$797$	−5.22399e9	−0.365509	−0.182754	+	0.983159i	$0.558501\pi$
$798$	4.68001e9	0.326014
$799$	4.05784e9	0.281437
$800$	−8.16447e8	−0.0563785
$801$	−1.82468e10	−1.25450
$802$	−4.54624e9	−0.311202
$803$	−3.46671e10	−2.36273
$804$	−3.56719e8	−0.0242064
$805$	−1.10438e9	−0.0746164
$806$	0	0
$807$	4.47314e10	2.99609
$808$	7.85934e9	0.524139
$809$	−7.92102e9	−0.525970	−0.262985	−	0.964800i	$-0.584707\pi$
$809$	−7.92102e9	−0.525970	−0.262985	+	0.964800i	$0.584707\pi$
$810$	3.18754e10	2.10745
$811$	8.16607e9	0.537576	0.268788	−	0.963199i	$-0.413377\pi$
$811$	8.16607e9	0.537576	0.268788	+	0.963199i	$0.413377\pi$
$812$	2.02430e9	0.132687
$813$	3.97674e9	0.259543
$814$	−2.01503e10	−1.30947
$815$	−2.35802e10	−1.52579
$816$	−3.23176e9	−0.208220
$817$	−1.16702e10	−0.748688
$818$	−1.02778e10	−0.656541
$819$	0	0
$820$	1.63389e10	1.03484
$821$	−2.63749e10	−1.66338	−0.831688	−	0.555244i	$-0.812625\pi$
$821$	−2.63749e10	−1.66338	−0.831688	+	0.555244i	$0.812625\pi$
$822$	−2.06643e10	−1.29768
$823$	2.04085e10	1.27618	0.638090	−	0.769962i	$-0.279725\pi$
$823$	2.04085e10	1.27618	0.638090	+	0.769962i	$0.279725\pi$
$824$	−3.52073e9	−0.219224
$825$	1.68690e10	1.04592
$826$	2.35701e9	0.145523
$827$	2.55307e10	1.56962	0.784809	−	0.619738i	$-0.212761\pi$
$827$	2.55307e10	1.56962	0.784809	+	0.619738i	$0.212761\pi$
$828$	6.54727e9	0.400824
$829$	8.48208e9	0.517085	0.258542	−	0.966000i	$-0.416758\pi$
$829$	8.48208e9	0.517085	0.258542	+	0.966000i	$0.416758\pi$
$830$	3.13063e9	0.190046
$831$	2.38349e10	1.44082
$832$	0	0
$833$	−7.17160e9	−0.429891
$834$	1.08334e10	0.646672
$835$	1.49862e10	0.890819
$836$	−1.85025e10	−1.09524
$837$	8.06432e9	0.475367
$838$	−2.19878e9	−0.129071
$839$	2.29323e10	1.34055	0.670273	−	0.742115i	$-0.266177\pi$
$839$	2.29323e10	1.34055	0.670273	+	0.742115i	$0.266177\pi$
$840$	−2.58805e9	−0.150659
$841$	1.32877e10	0.770306
$842$	6.01094e9	0.347017
$843$	3.66885e10	2.10928
$844$	6.94854e9	0.397828
$845$	0	0
$846$	−1.92650e10	−1.09389
$847$	7.43410e9	0.420374
$848$	−6.01797e9	−0.338895
$849$	−3.32303e10	−1.86362
$850$	−1.80771e9	−0.100963
$851$	−6.15230e9	−0.342203
$852$	−1.79503e9	−0.0994335
$853$	−2.47175e10	−1.36358	−0.681792	−	0.731546i	$-0.738799\pi$
$853$	−2.47175e10	−1.36358	−0.681792	+	0.731546i	$0.738799\pi$
$854$	3.47463e9	0.190900
$855$	−6.41812e10	−3.51177
$856$	7.78379e9	0.424163
$857$	1.19081e10	0.646265	0.323133	−	0.946354i	$-0.395264\pi$
$857$	1.19081e10	0.646265	0.323133	+	0.946354i	$0.395264\pi$
$858$	0	0
$859$	−4.94214e9	−0.266035	−0.133018	−	0.991114i	$-0.542467\pi$
$859$	−4.94214e9	−0.266035	−0.133018	+	0.991114i	$0.542467\pi$
$860$	6.45363e9	0.345987
$861$	1.25238e10	0.668689
$862$	−1.04605e9	−0.0556259
$863$	2.05387e10	1.08776	0.543881	−	0.839162i	$-0.316954\pi$
$863$	2.05387e10	1.08776	0.543881	+	0.839162i	$0.316954\pi$
$864$	9.10836e9	0.480443
$865$	2.50679e10	1.31693
$866$	−1.33389e10	−0.697921
$867$	2.85440e10	1.48747
$868$	−3.36075e8	−0.0174428
$869$	−5.86420e9	−0.303138
$870$	−3.90419e10	−2.01008
$871$	0	0
$872$	3.44709e9	0.176054
$873$	−8.76606e9	−0.445918
$874$	−5.64918e9	−0.286217
$875$	3.09150e9	0.156006
$876$	−2.48042e10	−1.24670
$877$	1.42584e10	0.713791	0.356895	−	0.934144i	$-0.383835\pi$
$877$	1.42584e10	0.713791	0.356895	+	0.934144i	$0.383835\pi$
$878$	−1.85182e10	−0.923354
$879$	−3.52204e10	−1.74918
$880$	1.02319e10	0.506136
$881$	1.78398e10	0.878971	0.439486	−	0.898250i	$-0.355161\pi$
$881$	1.78398e10	0.878971	0.439486	+	0.898250i	$0.355161\pi$
$882$	3.40479e10	1.67090
$883$	3.79954e10	1.85724	0.928622	−	0.371028i	$-0.120995\pi$
$883$	3.79954e10	1.85724	0.928622	+	0.371028i	$0.120995\pi$
$884$	0	0
$885$	−4.54587e10	−2.20453
$886$	5.52038e9	0.266656
$887$	8.45195e7	0.00406653	0.00203327	−	0.999998i	$-0.499353\pi$
$887$	8.45195e7	0.00406653	0.00203327	+	0.999998i	$0.499353\pi$
$888$	−1.44175e10	−0.690947
$889$	3.74126e9	0.178592
$890$	−8.70637e9	−0.413973
$891$	−9.65942e10	−4.57488
$892$	8.03857e9	0.379229
$893$	1.66224e10	0.781114
$894$	1.73774e9	0.0813398
$895$	1.78528e10	0.832390
$896$	−3.79585e8	−0.0176291
$897$	0	0
$898$	2.11045e10	0.972541
$899$	−5.06985e9	−0.232721
$900$	8.58227e9	0.392422
$901$	−1.33245e10	−0.606894
$902$	−4.95129e10	−2.24645
$903$	4.94672e9	0.223568
$904$	5.90296e9	0.265754
$905$	3.83385e10	1.71935
$906$	−1.66902e10	−0.745611
$907$	1.82024e10	0.810033	0.405017	−	0.914309i	$-0.367266\pi$
$907$	1.82024e10	0.810033	0.405017	+	0.914309i	$0.367266\pi$
$908$	−1.22096e10	−0.541252
$909$	−8.26151e10	−3.64826
$910$	0	0
$911$	−3.66963e10	−1.60808	−0.804040	−	0.594575i	$-0.797320\pi$
$911$	−3.66963e10	−1.60808	−0.804040	+	0.594575i	$0.797320\pi$
$912$	−1.32385e10	−0.577905
$913$	−9.48697e9	−0.412553
$914$	−4.92978e9	−0.213558
$915$	−6.70139e10	−2.89195
$916$	3.38503e9	0.145522
$917$	−3.45617e9	−0.148014
$918$	2.01669e10	0.860380
$919$	1.33474e10	0.567275	0.283638	−	0.958932i	$-0.408459\pi$
$919$	1.33474e10	0.567275	0.283638	+	0.958932i	$0.408459\pi$
$920$	3.12400e9	0.132268
$921$	−4.13702e9	−0.174493
$922$	9.88970e9	0.415552
$923$	0	0
$924$	7.84276e9	0.327052
$925$	−8.06454e9	−0.335030
$926$	5.42775e8	0.0224637
$927$	3.70089e10	1.52591
$928$	−5.72621e9	−0.235206
$929$	−2.71771e10	−1.11211	−0.556055	−	0.831146i	$-0.687686\pi$
$929$	−2.71771e10	−1.11211	−0.556055	+	0.831146i	$0.687686\pi$
$930$	6.48174e9	0.264242
$931$	−2.93776e10	−1.19314
$932$	9.68025e9	0.391680
$933$	2.63472e10	1.06206
$934$	9.40012e9	0.377502
$935$	2.26546e10	0.906390
$936$	0	0
$937$	4.04333e10	1.60565	0.802825	−	0.596214i	$-0.203329\pi$
$937$	4.04333e10	1.60565	0.802825	+	0.596214i	$0.203329\pi$
$938$	−9.27676e7	−0.00367017
$939$	5.49566e10	2.16616
$940$	−9.19223e9	−0.360972
$941$	8.49843e9	0.332487	0.166244	−	0.986085i	$-0.446836\pi$
$941$	8.49843e9	0.332487	0.166244	+	0.986085i	$0.446836\pi$
$942$	1.18703e10	0.462683
$943$	−1.51173e10	−0.587061
$944$	−6.66735e9	−0.257959
$945$	1.61500e10	0.622533
$946$	−1.95569e10	−0.751072
$947$	4.40082e9	0.168387	0.0841935	−	0.996449i	$-0.473169\pi$
$947$	4.40082e9	0.168387	0.0841935	+	0.996449i	$0.473169\pi$
$948$	−4.19582e9	−0.159951
$949$	0	0
$950$	−7.40504e9	−0.280217
$951$	6.90199e10	2.60221
$952$	−8.40442e8	−0.0315703
$953$	−1.73133e10	−0.647970	−0.323985	−	0.946062i	$-0.605023\pi$
$953$	−1.73133e10	−0.647970	−0.323985	+	0.946062i	$0.605023\pi$
$954$	6.32593e10	2.35887
$955$	−3.38606e10	−1.25801
$956$	−1.67627e10	−0.620498
$957$	1.18312e11	4.36351
$958$	3.16924e9	0.116460
$959$	−5.37390e9	−0.196754
$960$	7.32090e9	0.267064
$961$	−2.66709e10	−0.969407
$962$	0	0
$963$	−8.18210e10	−2.95238
$964$	8.43211e9	0.303156
$965$	6.80708e9	0.243846
$966$	2.39455e9	0.0854681
$967$	−1.40918e10	−0.501158	−0.250579	−	0.968096i	$-0.580621\pi$
$967$	−1.40918e10	−0.501158	−0.250579	+	0.968096i	$0.580621\pi$
$968$	−2.10290e10	−0.745171
$969$	−2.93115e10	−1.03491
$970$	−4.18269e9	−0.147148
$971$	−7.27843e9	−0.255135	−0.127568	−	0.991830i	$-0.540717\pi$
$971$	−7.27843e9	−0.255135	−0.127568	+	0.991830i	$0.540717\pi$
$972$	−3.02067e10	−1.05505
$973$	2.81731e9	0.0980481
$974$	−2.42932e10	−0.842419
$975$	0	0
$976$	−9.82879e9	−0.338397
$977$	−2.43791e10	−0.836348	−0.418174	−	0.908367i	$-0.637330\pi$
$977$	−2.43791e10	−0.836348	−0.418174	+	0.908367i	$0.637330\pi$
$978$	5.11272e10	1.74770
$979$	2.63835e10	0.898657
$980$	1.62458e10	0.551379
$981$	−3.62349e10	−1.22542
$982$	2.33579e10	0.787126
$983$	−4.06556e10	−1.36516	−0.682579	−	0.730811i	$-0.739142\pi$
$983$	−4.06556e10	−1.36516	−0.682579	+	0.730811i	$0.739142\pi$
$984$	−3.54264e10	−1.18534
$985$	−5.34578e10	−1.78231
$986$	−1.26785e10	−0.421209
$987$	−7.04585e9	−0.233251
$988$	0	0
$989$	−5.97112e9	−0.196277
$990$	−1.07555e11	−3.52295
$991$	−4.86636e10	−1.58835	−0.794175	−	0.607689i	$-0.792097\pi$
$991$	−4.86636e10	−1.58835	−0.794175	+	0.607689i	$0.792097\pi$
$992$	9.50665e8	0.0309198
$993$	−1.05816e11	−3.42949
$994$	−4.66811e8	−0.0150761
$995$	4.05586e10	1.30527
$996$	−6.78790e9	−0.217685
$997$	−1.76682e10	−0.564622	−0.282311	−	0.959323i	$-0.591101\pi$
$997$	−1.76682e10	−0.564622	−0.282311	+	0.959323i	$0.591101\pi$
$998$	−1.29874e10	−0.413586
$999$	8.99687e10	2.85504

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	338.8.a.a.1.1		1
13.5	odd	4		338.8.b.a.337.2		2
13.8	odd	4		338.8.b.a.337.1		2
13.12	even	2		26.8.a.b.1.1	✓	1
39.38	odd	2		234.8.a.a.1.1		1
52.51	odd	2		208.8.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.8.a.b.1.1	✓	1	13.12	even	2
208.8.a.e.1.1		1	52.51	odd	2
234.8.a.a.1.1		1	39.38	odd	2
338.8.a.a.1.1		1	1.1	even	1	trivial
338.8.b.a.337.1		2	13.8	odd	4
338.8.b.a.337.2		2	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more