LMFDB - Embedded newform 171.6.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,6,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("171.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	171.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:	$27.4256331880$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 117x^{4} + 2916x^{2} - 1216 $ x^6 - 117x^4 + 2916x^2 - 1216
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.651309$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.651309 q^{2} -31.5758 q^{4} -26.1790 q^{5} +104.778 q^{7} -41.4075 q^{8} -17.0506 q^{10} +666.012 q^{11} +59.2525 q^{13} +68.2428 q^{14} +983.456 q^{16} -1622.03 q^{17} -361.000 q^{19} +826.624 q^{20} +433.780 q^{22} -3075.44 q^{23} -2439.66 q^{25} +38.5917 q^{26} -3308.45 q^{28} +4576.41 q^{29} -10093.8 q^{31} +1965.57 q^{32} -1056.44 q^{34} -2742.99 q^{35} -12600.4 q^{37} -235.123 q^{38} +1084.01 q^{40} -2740.11 q^{41} -754.986 q^{43} -21029.9 q^{44} -2003.06 q^{46} +20823.5 q^{47} -5828.58 q^{49} -1588.97 q^{50} -1870.95 q^{52} +39977.9 q^{53} -17435.6 q^{55} -4338.59 q^{56} +2980.66 q^{58} -32031.4 q^{59} +9691.67 q^{61} -6574.19 q^{62} -30190.4 q^{64} -1551.17 q^{65} -16096.6 q^{67} +51216.8 q^{68} -1786.53 q^{70} -71074.9 q^{71} -61898.9 q^{73} -8206.77 q^{74} +11398.9 q^{76} +69783.4 q^{77} -61222.3 q^{79} -25745.9 q^{80} -1784.66 q^{82} -88873.1 q^{83} +42463.1 q^{85} -491.730 q^{86} -27577.9 q^{88} +49104.1 q^{89} +6208.36 q^{91} +97109.5 q^{92} +13562.5 q^{94} +9450.63 q^{95} +102195. q^{97} -3796.21 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 42 q^{4} - 10 q^{7} - 788 q^{10} - 1256 q^{13} - 606 q^{16} - 2166 q^{19} - 2524 q^{22} - 3944 q^{25} - 9632 q^{28} - 18136 q^{31} - 14072 q^{34} - 23764 q^{37} - 34284 q^{40} - 24606 q^{43} - 45640 q^{46}+ \cdots - 16492 q^{97}+O(q^{100}) $ 6 * q + 42 * q^4 - 10 * q^7 - 788 * q^10 - 1256 * q^13 - 606 * q^16 - 2166 * q^19 - 2524 * q^22 - 3944 * q^25 - 9632 * q^28 - 18136 * q^31 - 14072 * q^34 - 23764 * q^37 - 34284 * q^40 - 24606 * q^43 - 45640 * q^46 - 63256 * q^49 - 89128 * q^52 - 63402 * q^55 - 78572 * q^58 - 42334 * q^61 - 109662 * q^64 - 129696 * q^67 - 15096 * q^70 - 16998 * q^73 - 15162 * q^76 - 129004 * q^79 + 211396 * q^82 - 11754 * q^85 + 55044 * q^88 + 23932 * q^91 + 235368 * q^94 - 16492 * q^97

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 $

$2$	0.651309	0.115136	0.0575681	−	0.998342i	$-0.481665\pi$
$2$	0.651309	0.115136	0.0575681	+	0.998342i	$0.481665\pi$
$3$	0	0
$3$	0	0
$4$	−31.5758	−0.986744
$5$	−26.1790	−0.468305	−0.234152	−	0.972200i	$-0.575231\pi$
$5$	−26.1790	−0.468305	−0.234152	+	0.972200i	$0.575231\pi$
$6$	0	0
$7$	104.778	0.808211	0.404106	−	0.914712i	$-0.367583\pi$
$7$	104.778	0.808211	0.404106	+	0.914712i	$0.367583\pi$
$8$	−41.4075	−0.228746
$9$	0	0
$10$	−17.0506	−0.0539189
$11$	666.012	1.65959	0.829794	−	0.558070i	$-0.188458\pi$
$11$	666.012	1.65959	0.829794	+	0.558070i	$0.188458\pi$
$12$	0	0
$13$	59.2525	0.0972408	0.0486204	−	0.998817i	$-0.484518\pi$
$13$	59.2525	0.0972408	0.0486204	+	0.998817i	$0.484518\pi$
$14$	68.2428	0.0930544
$15$	0	0
$16$	983.456	0.960407
$17$	−1622.03	−1.36124	−0.680621	−	0.732635i	$-0.738290\pi$
$17$	−1622.03	−1.36124	−0.680621	+	0.732635i	$0.738290\pi$
$18$	0	0
$19$	−361.000	−0.229416
$19$	−361.000	−0.229416
$20$	826.624	0.462097
$21$	0	0
$22$	433.780	0.191079
$23$	−3075.44	−1.21224	−0.606119	−	0.795374i	$-0.707274\pi$
$23$	−3075.44	−1.21224	−0.606119	+	0.795374i	$0.707274\pi$
$24$	0	0
$25$	−2439.66	−0.780691
$26$	38.5917	0.0111959
$27$	0	0
$28$	−3308.45	−0.797497
$29$	4576.41	1.01049	0.505243	−	0.862977i	$-0.331403\pi$
$29$	4576.41	1.01049	0.505243	+	0.862977i	$0.331403\pi$
$30$	0	0
$31$	−10093.8	−1.88647	−0.943237	−	0.332120i	$-0.892236\pi$
$31$	−10093.8	−1.88647	−0.943237	+	0.332120i	$0.892236\pi$
$32$	1965.57	0.339324
$33$	0	0
$34$	−1056.44	−0.156728
$35$	−2742.99	−0.378489
$36$	0	0
$37$	−12600.4	−1.51315	−0.756573	−	0.653910i	$-0.773128\pi$
$37$	−12600.4	−1.51315	−0.756573	+	0.653910i	$0.773128\pi$
$38$	−235.123	−0.0264141
$39$	0	0
$40$	1084.01	0.107123
$41$	−2740.11	−0.254571	−0.127285	−	0.991866i	$-0.540626\pi$
$41$	−2740.11	−0.254571	−0.127285	+	0.991866i	$0.540626\pi$
$42$	0	0
$43$	−754.986	−0.0622684	−0.0311342	−	0.999515i	$-0.509912\pi$
$43$	−754.986	−0.0622684	−0.0311342	+	0.999515i	$0.509912\pi$
$44$	−21029.9	−1.63759
$45$	0	0
$46$	−2003.06	−0.139573
$47$	20823.5	1.37502	0.687510	−	0.726175i	$-0.258704\pi$
$47$	20823.5	1.37502	0.687510	+	0.726175i	$0.258704\pi$
$48$	0	0
$49$	−5828.58	−0.346795
$50$	−1588.97	−0.0898858
$51$	0	0
$52$	−1870.95	−0.0959518
$53$	39977.9	1.95492	0.977462	−	0.211111i	$-0.0677081\pi$
$53$	39977.9	1.95492	0.977462	+	0.211111i	$0.0677081\pi$
$54$	0	0
$55$	−17435.6	−0.777193
$56$	−4338.59	−0.184875
$57$	0	0
$58$	2980.66	0.116344
$59$	−32031.4	−1.19797	−0.598985	−	0.800760i	$-0.704429\pi$
$59$	−32031.4	−1.19797	−0.598985	+	0.800760i	$0.704429\pi$
$60$	0	0
$61$	9691.67	0.333483	0.166742	−	0.986001i	$-0.446675\pi$
$61$	9691.67	0.333483	0.166742	+	0.986001i	$0.446675\pi$
$62$	−6574.19	−0.217202
$63$	0	0
$64$	−30190.4	−0.921338
$65$	−1551.17	−0.0455383
$66$	0	0
$67$	−16096.6	−0.438074	−0.219037	−	0.975717i	$-0.570292\pi$
$67$	−16096.6	−0.438074	−0.219037	+	0.975717i	$0.570292\pi$
$68$	51216.8	1.34320
$69$	0	0
$70$	−1786.53	−0.0435778
$71$	−71074.9	−1.67329	−0.836643	−	0.547748i	$-0.815485\pi$
$71$	−71074.9	−1.67329	−0.836643	+	0.547748i	$0.815485\pi$
$72$	0	0
$73$	−61898.9	−1.35949	−0.679745	−	0.733448i	$-0.737910\pi$
$73$	−61898.9	−1.35949	−0.679745	+	0.733448i	$0.737910\pi$
$74$	−8206.77	−0.174218
$75$	0	0
$76$	11398.9	0.226375
$77$	69783.4	1.34130
$78$	0	0
$79$	−61222.3	−1.10368	−0.551838	−	0.833951i	$-0.686074\pi$
$79$	−61222.3	−1.10368	−0.551838	+	0.833951i	$0.686074\pi$
$80$	−25745.9	−0.449763
$81$	0	0
$82$	−1784.66	−0.0293103
$83$	−88873.1	−1.41604	−0.708019	−	0.706193i	$-0.750411\pi$
$83$	−88873.1	−1.41604	−0.708019	+	0.706193i	$0.750411\pi$
$84$	0	0
$85$	42463.1	0.637476
$86$	−491.730	−0.00716936
$87$	0	0
$88$	−27577.9	−0.379625
$89$	49104.1	0.657117	0.328559	−	0.944484i	$-0.393437\pi$
$89$	49104.1	0.657117	0.328559	+	0.944484i	$0.393437\pi$
$90$	0	0
$91$	6208.36	0.0785911
$92$	97109.5	1.19617
$93$	0	0
$94$	13562.5	0.158315
$95$	9450.63	0.107436
$96$	0	0
$97$	102195.	1.10281	0.551404	−	0.834238i	$-0.314092\pi$
$97$	102195.	1.10281	0.551404	+	0.834238i	$0.314092\pi$
$98$	−3796.21	−0.0399287
$99$	0	0
$100$	77034.1	0.770341
$101$	25539.6	0.249121	0.124560	−	0.992212i	$-0.460248\pi$
$101$	25539.6	0.249121	0.124560	+	0.992212i	$0.460248\pi$
$102$	0	0
$103$	−16349.9	−0.151853	−0.0759263	−	0.997113i	$-0.524191\pi$
$103$	−16349.9	−0.151853	−0.0759263	+	0.997113i	$0.524191\pi$
$104$	−2453.50	−0.0222435
$105$	0	0
$106$	26038.0	0.225083
$107$	112847.	0.952862	0.476431	−	0.879212i	$-0.341930\pi$
$107$	112847.	0.952862	0.476431	+	0.879212i	$0.341930\pi$
$108$	0	0
$109$	−119910.	−0.966694	−0.483347	−	0.875429i	$-0.660579\pi$
$109$	−119910.	−0.966694	−0.483347	+	0.875429i	$0.660579\pi$
$110$	−11355.9	−0.0894831
$111$	0	0
$112$	103045.	0.776211
$113$	−224892.	−1.65683	−0.828416	−	0.560114i	$-0.810757\pi$
$113$	−224892.	−1.65683	−0.828416	+	0.560114i	$0.810757\pi$
$114$	0	0
$115$	80512.1	0.567697
$116$	−144504.	−0.997091
$117$	0	0
$118$	−20862.3	−0.137930
$119$	−169953.	−1.10017
$120$	0	0
$121$	282521.	1.75423
$122$	6312.28	0.0383960
$123$	0	0
$124$	318720.	1.86147
$125$	145677.	0.833906
$126$	0	0
$127$	15307.7	0.0842171	0.0421085	−	0.999113i	$-0.486592\pi$
$127$	15307.7	0.0842171	0.0421085	+	0.999113i	$0.486592\pi$
$128$	−82561.7	−0.445403
$129$	0	0
$130$	−1010.29	−0.00524312
$131$	321750.	1.63810	0.819049	−	0.573723i	$-0.194501\pi$
$131$	321750.	1.63810	0.819049	+	0.573723i	$0.194501\pi$
$132$	0	0
$133$	−37824.8	−0.185416
$134$	−10483.9	−0.0504382
$135$	0	0
$136$	67164.0	0.311379
$137$	65909.6	0.300018	0.150009	−	0.988685i	$-0.452070\pi$
$137$	65909.6	0.300018	0.150009	+	0.988685i	$0.452070\pi$
$138$	0	0
$139$	123516.	0.542233	0.271117	−	0.962547i	$-0.412607\pi$
$139$	123516.	0.542233	0.271117	+	0.962547i	$0.412607\pi$
$140$	86612.0	0.373472
$141$	0	0
$142$	−46291.7	−0.192656
$143$	39462.9	0.161380
$144$	0	0
$145$	−119806.	−0.473216
$146$	−40315.3	−0.156527
$147$	0	0
$148$	397868.	1.49309
$149$	−389588.	−1.43760	−0.718802	−	0.695215i	$-0.755309\pi$
$149$	−389588.	−1.43760	−0.718802	+	0.695215i	$0.755309\pi$
$150$	0	0
$151$	−136823.	−0.488335	−0.244168	−	0.969733i	$-0.578515\pi$
$151$	−136823.	−0.488335	−0.244168	+	0.969733i	$0.578515\pi$
$152$	14948.1	0.0524780
$153$	0	0
$154$	45450.5	0.154432
$155$	264246.	0.883445
$156$	0	0
$157$	109944.	0.355978	0.177989	−	0.984033i	$-0.443041\pi$
$157$	109944.	0.355978	0.177989	+	0.984033i	$0.443041\pi$
$158$	−39874.6	−0.127073
$159$	0	0
$160$	−51456.8	−0.158907
$161$	−322238.	−0.979744
$162$	0	0
$163$	−242251.	−0.714161	−0.357081	−	0.934074i	$-0.616228\pi$
$163$	−242251.	−0.714161	−0.357081	+	0.934074i	$0.616228\pi$
$164$	86521.2	0.251196
$165$	0	0
$166$	−57883.9	−0.163037
$167$	−241920.	−0.671245	−0.335622	−	0.941997i	$-0.608947\pi$
$167$	−241920.	−0.671245	−0.335622	+	0.941997i	$0.608947\pi$
$168$	0	0
$169$	−367782.	−0.990544
$170$	27656.6	0.0733967
$171$	0	0
$172$	23839.3	0.0614430
$173$	520255.	1.32160	0.660801	−	0.750561i	$-0.270217\pi$
$173$	520255.	1.32160	0.660801	+	0.750561i	$0.270217\pi$
$174$	0	0
$175$	−255622.	−0.630963
$176$	654994.	1.59388
$177$	0	0
$178$	31982.0	0.0756580
$179$	−21203.7	−0.0494628	−0.0247314	−	0.999694i	$-0.507873\pi$
$179$	−21203.7	−0.0494628	−0.0247314	+	0.999694i	$0.507873\pi$
$180$	0	0
$181$	637689.	1.44681	0.723406	−	0.690423i	$-0.242575\pi$
$181$	637689.	1.44681	0.723406	+	0.690423i	$0.242575\pi$
$182$	4043.56	0.00904869
$183$	0	0
$184$	127346.	0.277295
$185$	329867.	0.708613
$186$	0	0
$187$	−1.08029e6	−2.25910
$188$	−657518.	−1.35679
$189$	0	0
$190$	6155.28	0.0123698
$191$	−725313.	−1.43861	−0.719303	−	0.694696i	$-0.755539\pi$
$191$	−725313.	−1.43861	−0.719303	+	0.694696i	$0.755539\pi$
$192$	0	0
$193$	−509164.	−0.983931	−0.491965	−	0.870615i	$-0.663721\pi$
$193$	−509164.	−0.983931	−0.491965	+	0.870615i	$0.663721\pi$
$194$	66560.5	0.126973
$195$	0	0
$196$	184042.	0.342198
$197$	168522.	0.309380	0.154690	−	0.987963i	$-0.450562\pi$
$197$	168522.	0.309380	0.154690	+	0.987963i	$0.450562\pi$
$198$	0	0
$199$	398703.	0.713702	0.356851	−	0.934161i	$-0.383850\pi$
$199$	398703.	0.713702	0.356851	+	0.934161i	$0.383850\pi$
$200$	101020.	0.178580
$201$	0	0
$202$	16634.2	0.0286829
$203$	479507.	0.816686
$204$	0	0
$205$	71733.5	0.119217
$206$	−10648.8	−0.0174838
$207$	0	0
$208$	58272.3	0.0933907
$209$	−240430.	−0.380736
$210$	0	0
$211$	645289.	0.997811	0.498905	−	0.866656i	$-0.333736\pi$
$211$	645289.	0.997811	0.498905	+	0.866656i	$0.333736\pi$
$212$	−1.26233e6	−1.92901
$213$	0	0
$214$	73498.2	0.109709
$215$	19764.8	0.0291606
$216$	0	0
$217$	−1.05761e6	−1.52467
$218$	−78098.4	−0.111302
$219$	0	0
$220$	550541.	0.766890
$221$	−96109.2	−0.132368
$222$	0	0
$223$	−481247.	−0.648046	−0.324023	−	0.946049i	$-0.605036\pi$
$223$	−481247.	−0.648046	−0.324023	+	0.946049i	$0.605036\pi$
$224$	205949.	0.274245
$225$	0	0
$226$	−146474.	−0.190761
$227$	726154.	0.935328	0.467664	−	0.883906i	$-0.345096\pi$
$227$	726154.	0.935328	0.467664	+	0.883906i	$0.345096\pi$
$228$	0	0
$229$	−1.26607e6	−1.59540	−0.797698	−	0.603057i	$-0.793949\pi$
$229$	−1.26607e6	−1.59540	−0.797698	+	0.603057i	$0.793949\pi$
$230$	52438.3	0.0653625
$231$	0	0
$232$	−189498.	−0.231145
$233$	173092.	0.208875	0.104438	−	0.994531i	$-0.466696\pi$
$233$	173092.	0.208875	0.104438	+	0.994531i	$0.466696\pi$
$234$	0	0
$235$	−545139.	−0.643928
$236$	1.01142e6	1.18209
$237$	0	0
$238$	−110692.	−0.126670
$239$	−1.25124e6	−1.41693	−0.708463	−	0.705748i	$-0.750611\pi$
$239$	−1.25124e6	−1.41693	−0.708463	+	0.705748i	$0.750611\pi$
$240$	0	0
$241$	−711877.	−0.789518	−0.394759	−	0.918785i	$-0.629172\pi$
$241$	−711877.	−0.789518	−0.394759	+	0.918785i	$0.629172\pi$
$242$	184008.	0.201976
$243$	0	0
$244$	−306022.	−0.329063
$245$	152587.	0.162406
$246$	0	0
$247$	−21390.2	−0.0223086
$248$	417959.	0.431524
$249$	0	0
$250$	94881.0	0.0960128
$251$	885324.	0.886989	0.443494	−	0.896277i	$-0.353739\pi$
$251$	885324.	0.886989	0.443494	+	0.896277i	$0.353739\pi$
$252$	0	0
$253$	−2.04828e6	−2.01182
$254$	9970.03	0.00969644
$255$	0	0
$256$	912320.	0.870056
$257$	−892654.	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$257$	−892654.	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$258$	0	0
$259$	−1.32025e6	−1.22294
$260$	48979.6	0.0449347
$261$	0	0
$262$	209559.	0.188605
$263$	−132173.	−0.117830	−0.0589148	−	0.998263i	$-0.518764\pi$
$263$	−132173.	−0.117830	−0.0589148	+	0.998263i	$0.518764\pi$
$264$	0	0
$265$	−1.04658e6	−0.915501
$266$	−24635.7	−0.0213481
$267$	0	0
$268$	508264.	0.432267
$269$	669484.	0.564104	0.282052	−	0.959399i	$-0.408985\pi$
$269$	669484.	0.564104	0.282052	+	0.959399i	$0.408985\pi$
$270$	0	0
$271$	1.70117e6	1.40710	0.703550	−	0.710646i	$-0.251597\pi$
$271$	1.70117e6	1.40710	0.703550	+	0.710646i	$0.251597\pi$
$272$	−1.59519e6	−1.30735
$273$	0	0
$274$	42927.5	0.0345429
$275$	−1.62484e6	−1.29562
$276$	0	0
$277$	795334.	0.622802	0.311401	−	0.950279i	$-0.399202\pi$
$277$	795334.	0.622802	0.311401	+	0.950279i	$0.399202\pi$
$278$	80447.1	0.0624307
$279$	0	0
$280$	113580.	0.0865780
$281$	−464075.	−0.350608	−0.175304	−	0.984514i	$-0.556091\pi$
$281$	−464075.	−0.350608	−0.175304	+	0.984514i	$0.556091\pi$
$282$	0	0
$283$	−1.49463e6	−1.10934	−0.554672	−	0.832069i	$-0.687156\pi$
$283$	−1.49463e6	−1.10934	−0.554672	+	0.832069i	$0.687156\pi$
$284$	2.24425e6	1.65110
$285$	0	0
$286$	25702.5	0.0185807
$287$	−287103.	−0.205747
$288$	0	0
$289$	1.21111e6	0.852981
$290$	−78030.8	−0.0544843
$291$	0	0
$292$	1.95451e6	1.34147
$293$	923428.	0.628397	0.314198	−	0.949357i	$-0.398264\pi$
$293$	923428.	0.628397	0.314198	+	0.949357i	$0.398264\pi$
$294$	0	0
$295$	838551.	0.561015
$296$	521752.	0.346126
$297$	0	0
$298$	−253742.	−0.165520
$299$	−182228.	−0.117879
$300$	0	0
$301$	−79105.9	−0.0503260
$302$	−89114.4	−0.0562251
$303$	0	0
$304$	−355028.	−0.220332
$305$	−253719.	−0.156172
$306$	0	0
$307$	2.40441e6	1.45601	0.728003	−	0.685574i	$-0.240449\pi$
$307$	2.40441e6	1.45601	0.728003	+	0.685574i	$0.240449\pi$
$308$	−2.20347e6	−1.32352
$309$	0	0
$310$	172106.	0.101717
$311$	344292.	0.201849	0.100924	−	0.994894i	$-0.467820\pi$
$311$	344292.	0.201849	0.100924	+	0.994894i	$0.467820\pi$
$312$	0	0
$313$	778240.	0.449007	0.224503	−	0.974473i	$-0.427924\pi$
$313$	778240.	0.449007	0.224503	+	0.974473i	$0.427924\pi$
$314$	71607.6	0.0409859
$315$	0	0
$316$	1.93314e6	1.08905
$317$	568016.	0.317477	0.158739	−	0.987321i	$-0.449257\pi$
$317$	568016.	0.317477	0.158739	+	0.987321i	$0.449257\pi$
$318$	0	0
$319$	3.04795e6	1.67699
$320$	790356.	0.431467
$321$	0	0
$322$	−209877.	−0.112804
$323$	585551.	0.312290
$324$	0	0
$325$	−144556.	−0.0759150
$326$	−157780.	−0.0822259
$327$	0	0
$328$	113461.	0.0582321
$329$	2.18184e6	1.11131
$330$	0	0
$331$	−1.22041e6	−0.612260	−0.306130	−	0.951990i	$-0.599034\pi$
$331$	−1.22041e6	−0.612260	−0.306130	+	0.951990i	$0.599034\pi$
$332$	2.80624e6	1.39727
$333$	0	0
$334$	−157565.	−0.0772846
$335$	421394.	0.205152
$336$	0	0
$337$	2.85370e6	1.36878	0.684391	−	0.729115i	$-0.260068\pi$
$337$	2.85370e6	1.36878	0.684391	+	0.729115i	$0.260068\pi$
$338$	−239540.	−0.114048
$339$	0	0
$340$	−1.34081e6	−0.629026
$341$	−6.72260e6	−3.13077
$342$	0	0
$343$	−2.37171e6	−1.08849
$344$	31262.1	0.0142437
$345$	0	0
$346$	338847.	0.152164
$347$	−2.14302e6	−0.955440	−0.477720	−	0.878512i	$-0.658537\pi$
$347$	−2.14302e6	−0.955440	−0.477720	+	0.878512i	$0.658537\pi$
$348$	0	0
$349$	−1.05371e6	−0.463083	−0.231542	−	0.972825i	$-0.574377\pi$
$349$	−1.05371e6	−0.463083	−0.231542	+	0.972825i	$0.574377\pi$
$350$	−166489.	−0.0726467
$351$	0	0
$352$	1.30910e6	0.563138
$353$	2.17660e6	0.929699	0.464850	−	0.885390i	$-0.346108\pi$
$353$	2.17660e6	0.929699	0.464850	+	0.885390i	$0.346108\pi$
$354$	0	0
$355$	1.86067e6	0.783608
$356$	−1.55050e6	−0.648406
$357$	0	0
$358$	−13810.2	−0.00569497
$359$	1.35843e6	0.556290	0.278145	−	0.960539i	$-0.410280\pi$
$359$	1.35843e6	0.556290	0.278145	+	0.960539i	$0.410280\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	415332.	0.166581
$363$	0	0
$364$	−196034.	−0.0775493
$365$	1.62045e6	0.636656
$366$	0	0
$367$	309670.	0.120015	0.0600074	−	0.998198i	$-0.480888\pi$
$367$	309670.	0.120015	0.0600074	+	0.998198i	$0.480888\pi$
$368$	−3.02456e6	−1.16424
$369$	0	0
$370$	214845.	0.0815871
$371$	4.18880e6	1.57999
$372$	0	0
$373$	2.01892e6	0.751357	0.375679	−	0.926750i	$-0.377410\pi$
$373$	2.01892e6	0.751357	0.375679	+	0.926750i	$0.377410\pi$
$374$	−703602.	−0.260105
$375$	0	0
$376$	−862248.	−0.314530
$377$	271164.	0.0982605
$378$	0	0
$379$	2.82488e6	1.01019	0.505094	−	0.863064i	$-0.331458\pi$
$379$	2.82488e6	1.01019	0.505094	+	0.863064i	$0.331458\pi$
$380$	−298411.	−0.106012
$381$	0	0
$382$	−472403.	−0.165636
$383$	3.09625e6	1.07855	0.539274	−	0.842130i	$-0.318699\pi$
$383$	3.09625e6	1.07855	0.539274	+	0.842130i	$0.318699\pi$
$384$	0	0
$385$	−1.82686e6	−0.628136
$386$	−331623.	−0.113286
$387$	0	0
$388$	−3.22689e6	−1.08819
$389$	1.48523e6	0.497644	0.248822	−	0.968549i	$-0.419957\pi$
$389$	1.48523e6	0.497644	0.248822	+	0.968549i	$0.419957\pi$
$390$	0	0
$391$	4.98845e6	1.65015
$392$	241347.	0.0793280
$393$	0	0
$394$	109760.	0.0356208
$395$	1.60274e6	0.516857
$396$	0	0
$397$	−450478.	−0.143449	−0.0717244	−	0.997424i	$-0.522850\pi$
$397$	−450478.	−0.143449	−0.0717244	+	0.997424i	$0.522850\pi$
$398$	259679.	0.0821730
$399$	0	0
$400$	−2.39930e6	−0.749780
$401$	−608993.	−0.189126	−0.0945631	−	0.995519i	$-0.530145\pi$
$401$	−608993.	−0.189126	−0.0945631	+	0.995519i	$0.530145\pi$
$402$	0	0
$403$	−598084.	−0.183442
$404$	−806432.	−0.245818
$405$	0	0
$406$	312308.	0.0940302
$407$	−8.39203e6	−2.51120
$408$	0	0
$409$	−350421.	−0.103581	−0.0517907	−	0.998658i	$-0.516493\pi$
$409$	−350421.	−0.103581	−0.0517907	+	0.998658i	$0.516493\pi$
$410$	46720.7	0.0137262
$411$	0	0
$412$	516262.	0.149840
$413$	−3.35618e6	−0.968213
$414$	0	0
$415$	2.32661e6	0.663138
$416$	116465.	0.0329961
$417$	0	0
$418$	−156594.	−0.0438365
$419$	−2.70802e6	−0.753559	−0.376779	−	0.926303i	$-0.622969\pi$
$419$	−2.70802e6	−0.753559	−0.376779	+	0.926303i	$0.622969\pi$
$420$	0	0
$421$	−4.72890e6	−1.30033	−0.650167	−	0.759791i	$-0.725301\pi$
$421$	−4.72890e6	−1.30033	−0.650167	+	0.759791i	$0.725301\pi$
$422$	420283.	0.114884
$423$	0	0
$424$	−1.65538e6	−0.447182
$425$	3.95719e6	1.06271
$426$	0	0
$427$	1.01547e6	0.269525
$428$	−3.56323e6	−0.940231
$429$	0	0
$430$	12873.0	0.00335744
$431$	−1.05099e6	−0.272523	−0.136262	−	0.990673i	$-0.543509\pi$
$431$	−1.05099e6	−0.272523	−0.136262	+	0.990673i	$0.543509\pi$
$432$	0	0
$433$	−982966.	−0.251953	−0.125976	−	0.992033i	$-0.540206\pi$
$433$	−982966.	−0.251953	−0.125976	+	0.992033i	$0.540206\pi$
$434$	−688830.	−0.175545
$435$	0	0
$436$	3.78625e6	0.953879
$437$	1.11023e6	0.278106
$438$	0	0
$439$	−4.02905e6	−0.997794	−0.498897	−	0.866661i	$-0.666261\pi$
$439$	−4.02905e6	−0.997794	−0.498897	+	0.866661i	$0.666261\pi$
$440$	721963.	0.177780
$441$	0	0
$442$	−62596.8	−0.0152404
$443$	2.56728e6	0.621533	0.310767	−	0.950486i	$-0.399414\pi$
$443$	2.56728e6	0.621533	0.310767	+	0.950486i	$0.399414\pi$
$444$	0	0
$445$	−1.28550e6	−0.307731
$446$	−313441.	−0.0746136
$447$	0	0
$448$	−3.16329e6	−0.744636
$449$	−336148.	−0.0786891	−0.0393446	−	0.999226i	$-0.512527\pi$
$449$	−336148.	−0.0786891	−0.0393446	+	0.999226i	$0.512527\pi$
$450$	0	0
$451$	−1.82495e6	−0.422483
$452$	7.10115e6	1.63487
$453$	0	0
$454$	472951.	0.107690
$455$	−162529.	−0.0368046
$456$	0	0
$457$	−693109.	−0.155243	−0.0776214	−	0.996983i	$-0.524733\pi$
$457$	−693109.	−0.155243	−0.0776214	+	0.996983i	$0.524733\pi$
$458$	−824602.	−0.183688
$459$	0	0
$460$	−2.54223e6	−0.560171
$461$	1.04251e6	0.228470	0.114235	−	0.993454i	$-0.463558\pi$
$461$	1.04251e6	0.228470	0.114235	+	0.993454i	$0.463558\pi$
$462$	0	0
$463$	6.36632e6	1.38018	0.690091	−	0.723723i	$-0.257571\pi$
$463$	6.36632e6	1.38018	0.690091	+	0.723723i	$0.257571\pi$
$464$	4.50070e6	0.970478
$465$	0	0
$466$	112736.	0.0240491
$467$	2.08396e6	0.442178	0.221089	−	0.975254i	$-0.429039\pi$
$467$	2.08396e6	0.442178	0.221089	+	0.975254i	$0.429039\pi$
$468$	0	0
$469$	−1.68657e6	−0.354056
$470$	−355054.	−0.0741395
$471$	0	0
$472$	1.32634e6	0.274031
$473$	−502830.	−0.103340
$474$	0	0
$475$	880717.	0.179103
$476$	5.36639e6	1.08559
$477$	0	0
$478$	−814946.	−0.163139
$479$	−6.65036e6	−1.32436	−0.662181	−	0.749344i	$-0.730369\pi$
$479$	−6.65036e6	−1.32436	−0.662181	+	0.749344i	$0.730369\pi$
$480$	0	0
$481$	−746607.	−0.147139
$482$	−463652.	−0.0909022
$483$	0	0
$484$	−8.92082e6	−1.73098
$485$	−2.67537e6	−0.516451
$486$	0	0
$487$	−4.54164e6	−0.867740	−0.433870	−	0.900975i	$-0.642852\pi$
$487$	−4.54164e6	−0.867740	−0.433870	+	0.900975i	$0.642852\pi$
$488$	−401308.	−0.0762831
$489$	0	0
$490$	99381.1	0.0186988
$491$	7.40910e6	1.38695	0.693477	−	0.720479i	$-0.256078\pi$
$491$	7.40910e6	1.38695	0.693477	+	0.720479i	$0.256078\pi$
$492$	0	0
$493$	−7.42306e6	−1.37552
$494$	−13931.6	−0.00256853
$495$	0	0
$496$	−9.92682e6	−1.81178
$497$	−7.44708e6	−1.35237
$498$	0	0
$499$	6.19453e6	1.11367	0.556835	−	0.830623i	$-0.312015\pi$
$499$	6.19453e6	1.11367	0.556835	+	0.830623i	$0.312015\pi$
$500$	−4.59988e6	−0.822851
$501$	0	0
$502$	576620.	0.102125
$503$	3.56610e6	0.628454	0.314227	−	0.949348i	$-0.398255\pi$
$503$	3.56610e6	0.628454	0.314227	+	0.949348i	$0.398255\pi$
$504$	0	0
$505$	−668601.	−0.116665
$506$	−1.33406e6	−0.231633
$507$	0	0
$508$	−483352.	−0.0831006
$509$	−9.21155e6	−1.57593	−0.787967	−	0.615717i	$-0.788866\pi$
$509$	−9.21155e6	−1.57593	−0.787967	+	0.615717i	$0.788866\pi$
$510$	0	0
$511$	−6.48564e6	−1.09875
$512$	3.23618e6	0.545578
$513$	0	0
$514$	−581394.	−0.0970650
$515$	428025.	0.0711133
$516$	0	0
$517$	1.38687e7	2.28197
$518$	−859888.	−0.140805
$519$	0	0
$520$	64230.2	0.0104167
$521$	3.11211e6	0.502297	0.251148	−	0.967949i	$-0.419192\pi$
$521$	3.11211e6	0.502297	0.251148	+	0.967949i	$0.419192\pi$
$522$	0	0
$523$	−1.17678e7	−1.88122	−0.940612	−	0.339484i	$-0.889748\pi$
$523$	−1.17678e7	−1.88122	−0.940612	+	0.339484i	$0.889748\pi$
$524$	−1.01595e7	−1.61638
$525$	0	0
$526$	−86085.7	−0.0135665
$527$	1.63724e7	2.56795
$528$	0	0
$529$	3.02200e6	0.469521
$530$	−681648.	−0.105407
$531$	0	0
$532$	1.19435e6	0.182958
$533$	−162359.	−0.0247547
$534$	0	0
$535$	−2.95422e6	−0.446230
$536$	666521.	0.100208
$537$	0	0
$538$	436041.	0.0649489
$539$	−3.88190e6	−0.575537
$540$	0	0
$541$	791282.	0.116235	0.0581176	−	0.998310i	$-0.481490\pi$
$541$	791282.	0.116235	0.0581176	+	0.998310i	$0.481490\pi$
$542$	1.10799e6	0.162008
$543$	0	0
$544$	−3.18821e6	−0.461902
$545$	3.13913e6	0.452707
$546$	0	0
$547$	−3.23625e6	−0.462460	−0.231230	−	0.972899i	$-0.574275\pi$
$547$	−3.23625e6	−0.462460	−0.231230	+	0.972899i	$0.574275\pi$
$548$	−2.08115e6	−0.296041
$549$	0	0
$550$	−1.05827e6	−0.149173
$551$	−1.65209e6	−0.231821
$552$	0	0
$553$	−6.41475e6	−0.892004
$554$	518008.	0.0717071
$555$	0	0
$556$	−3.90012e6	−0.535045
$557$	42741.4	0.00583728	0.00291864	−	0.999996i	$-0.499071\pi$
$557$	42741.4	0.00583728	0.00291864	+	0.999996i	$0.499071\pi$
$558$	0	0
$559$	−44734.9	−0.00605503
$560$	−2.69761e6	−0.363504
$561$	0	0
$562$	−302256.	−0.0403677
$563$	−3.19178e6	−0.424386	−0.212193	−	0.977228i	$-0.568061\pi$
$563$	−3.19178e6	−0.424386	−0.212193	+	0.977228i	$0.568061\pi$
$564$	0	0
$565$	5.88746e6	0.775902
$566$	−973464.	−0.127726
$567$	0	0
$568$	2.94303e6	0.382758
$569$	1.13198e7	1.46574	0.732870	−	0.680368i	$-0.238180\pi$
$569$	1.13198e7	1.46574	0.732870	+	0.680368i	$0.238180\pi$
$570$	0	0
$571$	1.35992e7	1.74551	0.872754	−	0.488159i	$-0.162332\pi$
$571$	1.35992e7	1.74551	0.872754	+	0.488159i	$0.162332\pi$
$572$	−1.24607e6	−0.159240
$573$	0	0
$574$	−186993.	−0.0236889
$575$	7.50303e6	0.946383
$576$	0	0
$577$	1.09106e7	1.36430	0.682152	−	0.731211i	$-0.261044\pi$
$577$	1.09106e7	1.36430	0.682152	+	0.731211i	$0.261044\pi$
$578$	788808.	0.0982091
$579$	0	0
$580$	3.78297e6	0.466942
$581$	−9.31194e6	−1.14446
$582$	0	0
$583$	2.66257e7	3.24437
$584$	2.56308e6	0.310978
$585$	0	0
$586$	601437.	0.0723513
$587$	−4.76976e6	−0.571348	−0.285674	−	0.958327i	$-0.592218\pi$
$587$	−4.76976e6	−0.571348	−0.285674	+	0.958327i	$0.592218\pi$
$588$	0	0
$589$	3.64387e6	0.432787
$590$	546156.	0.0645932
$591$	0	0
$592$	−1.23920e7	−1.45323
$593$	−83908.0	−0.00979866	−0.00489933	−	0.999988i	$-0.501560\pi$
$593$	−83908.0	−0.00979866	−0.00489933	+	0.999988i	$0.501560\pi$
$594$	0	0
$595$	4.44919e6	0.515216
$596$	1.23015e7	1.41855
$597$	0	0
$598$	−118687.	−0.0135722
$599$	−1.02532e7	−1.16760	−0.583800	−	0.811897i	$-0.698435\pi$
$599$	−1.02532e7	−1.16760	−0.583800	+	0.811897i	$0.698435\pi$
$600$	0	0
$601$	−8.66133e6	−0.978134	−0.489067	−	0.872246i	$-0.662663\pi$
$601$	−8.66133e6	−0.978134	−0.489067	+	0.872246i	$0.662663\pi$
$602$	−51522.4	−0.00579435
$603$	0	0
$604$	4.32031e6	0.481862
$605$	−7.39613e6	−0.821516
$606$	0	0
$607$	1.46859e7	1.61781	0.808905	−	0.587940i	$-0.200061\pi$
$607$	1.46859e7	1.61781	0.808905	+	0.587940i	$0.200061\pi$
$608$	−709572.	−0.0778462
$609$	0	0
$610$	−165249.	−0.0179810
$611$	1.23384e6	0.133708
$612$	0	0
$613$	−338556.	−0.0363898	−0.0181949	−	0.999834i	$-0.505792\pi$
$613$	−338556.	−0.0363898	−0.0181949	+	0.999834i	$0.505792\pi$
$614$	1.56602e6	0.167639
$615$	0	0
$616$	−2.88955e6	−0.306817
$617$	−5.99200e6	−0.633664	−0.316832	−	0.948482i	$-0.602619\pi$
$617$	−5.99200e6	−0.633664	−0.316832	+	0.948482i	$0.602619\pi$
$618$	0	0
$619$	8.64654e6	0.907018	0.453509	−	0.891252i	$-0.350172\pi$
$619$	8.64654e6	0.907018	0.453509	+	0.891252i	$0.350172\pi$
$620$	−8.34379e6	−0.871734
$621$	0	0
$622$	224241.	0.0232401
$623$	5.14503e6	0.531089
$624$	0	0
$625$	3.81024e6	0.390168
$626$	506875.	0.0516970
$627$	0	0
$628$	−3.47157e6	−0.351259
$629$	2.04382e7	2.05976
$630$	0	0
$631$	1.23119e7	1.23098	0.615489	−	0.788146i	$-0.288959\pi$
$631$	1.23119e7	1.23098	0.615489	+	0.788146i	$0.288959\pi$
$632$	2.53506e6	0.252462
$633$	0	0
$634$	369954.	0.0365532
$635$	−400740.	−0.0394393
$636$	0	0
$637$	−345358.	−0.0337226
$638$	1.98516e6	0.193082
$639$	0	0
$640$	2.16138e6	0.208585
$641$	−9.97174e6	−0.958574	−0.479287	−	0.877658i	$-0.659105\pi$
$641$	−9.97174e6	−0.958574	−0.479287	+	0.877658i	$0.659105\pi$
$642$	0	0
$643$	1.12498e7	1.07304	0.536520	−	0.843888i	$-0.319739\pi$
$643$	1.12498e7	1.07304	0.536520	+	0.843888i	$0.319739\pi$
$644$	1.01749e7	0.966756
$645$	0	0
$646$	381375.	0.0359560
$647$	−9.23633e6	−0.867439	−0.433719	−	0.901048i	$-0.642799\pi$
$647$	−9.23633e6	−0.867439	−0.433719	+	0.901048i	$0.642799\pi$
$648$	0	0
$649$	−2.13333e7	−1.98814
$650$	−94150.6	−0.00874057
$651$	0	0
$652$	7.64926e6	0.704694
$653$	−6.03428e6	−0.553787	−0.276893	−	0.960901i	$-0.589305\pi$
$653$	−6.03428e6	−0.553787	−0.276893	+	0.960901i	$0.589305\pi$
$654$	0	0
$655$	−8.42310e6	−0.767130
$656$	−2.69478e6	−0.244492
$657$	0	0
$658$	1.42105e6	0.127952
$659$	−9.48721e6	−0.850991	−0.425495	−	0.904961i	$-0.639900\pi$
$659$	−9.48721e6	−0.850991	−0.425495	+	0.904961i	$0.639900\pi$
$660$	0	0
$661$	−1.05339e7	−0.937745	−0.468872	−	0.883266i	$-0.655340\pi$
$661$	−1.05339e7	−0.937745	−0.468872	+	0.883266i	$0.655340\pi$
$662$	−794864.	−0.0704933
$663$	0	0
$664$	3.68001e6	0.323914
$665$	990218.	0.0868314
$666$	0	0
$667$	−1.40745e7	−1.22495
$668$	7.63882e6	0.662347
$669$	0	0
$670$	274458.	0.0236205
$671$	6.45477e6	0.553445
$672$	0	0
$673$	−8.45056e6	−0.719197	−0.359598	−	0.933107i	$-0.617086\pi$
$673$	−8.45056e6	−0.719197	−0.359598	+	0.933107i	$0.617086\pi$
$674$	1.85864e6	0.157596
$675$	0	0
$676$	1.16130e7	0.977413
$677$	2.17879e7	1.82702	0.913511	−	0.406814i	$-0.133360\pi$
$677$	2.17879e7	1.82702	0.913511	+	0.406814i	$0.133360\pi$
$678$	0	0
$679$	1.07078e7	0.891302
$680$	−1.75829e6	−0.145820
$681$	0	0
$682$	−4.37849e6	−0.360465
$683$	−1.17335e7	−0.962446	−0.481223	−	0.876598i	$-0.659807\pi$
$683$	−1.17335e7	−0.962446	−0.481223	+	0.876598i	$0.659807\pi$
$684$	0	0
$685$	−1.72545e6	−0.140500
$686$	−1.54472e6	−0.125325
$687$	0	0
$688$	−742496.	−0.0598030
$689$	2.36879e6	0.190098
$690$	0	0
$691$	1.45733e7	1.16108	0.580541	−	0.814231i	$-0.302841\pi$
$691$	1.45733e7	1.16108	0.580541	+	0.814231i	$0.302841\pi$
$692$	−1.64275e7	−1.30408
$693$	0	0
$694$	−1.39577e6	−0.110006
$695$	−3.23353e6	−0.253930
$696$	0	0
$697$	4.44453e6	0.346533
$698$	−686293.	−0.0533177
$699$	0	0
$700$	8.07148e6	0.622598
$701$	−7.05513e6	−0.542263	−0.271132	−	0.962542i	$-0.587398\pi$
$701$	−7.05513e6	−0.542263	−0.271132	+	0.962542i	$0.587398\pi$
$702$	0	0
$703$	4.54875e6	0.347139
$704$	−2.01072e7	−1.52904
$705$	0	0
$706$	1.41764e6	0.107042
$707$	2.67598e6	0.201342
$708$	0	0
$709$	−6.85574e6	−0.512199	−0.256100	−	0.966650i	$-0.582437\pi$
$709$	−6.85574e6	−0.512199	−0.256100	+	0.966650i	$0.582437\pi$
$710$	1.21187e6	0.0902217
$711$	0	0
$712$	−2.03328e6	−0.150313
$713$	3.10429e7	2.28686
$714$	0	0
$715$	−1.03310e6	−0.0755749
$716$	669524.	0.0488071
$717$	0	0
$718$	884758.	0.0640492
$719$	5.08273e6	0.366670	0.183335	−	0.983051i	$-0.441311\pi$
$719$	5.08273e6	0.366670	0.183335	+	0.983051i	$0.441311\pi$
$720$	0	0
$721$	−1.71311e6	−0.122729
$722$	84879.3	0.00605980
$723$	0	0
$724$	−2.01355e7	−1.42763
$725$	−1.11649e7	−0.788877
$726$	0	0
$727$	1.60828e7	1.12856	0.564282	−	0.825582i	$-0.309153\pi$
$727$	1.60828e7	1.12856	0.564282	+	0.825582i	$0.309153\pi$
$728$	−257073.	−0.0179774
$729$	0	0
$730$	1.05542e6	0.0733022
$731$	1.22461e6	0.0847625
$732$	0	0
$733$	−1.20642e7	−0.829348	−0.414674	−	0.909970i	$-0.636104\pi$
$733$	−1.20642e7	−0.829348	−0.414674	+	0.909970i	$0.636104\pi$
$734$	201691.	0.0138181
$735$	0	0
$736$	−6.04501e6	−0.411341
$737$	−1.07205e7	−0.727023
$738$	0	0
$739$	−1.32586e7	−0.893074	−0.446537	−	0.894765i	$-0.647343\pi$
$739$	−1.32586e7	−0.893074	−0.446537	+	0.894765i	$0.647343\pi$
$740$	−1.04158e7	−0.699220
$741$	0	0
$742$	2.72820e6	0.181914
$743$	−1.26802e7	−0.842661	−0.421331	−	0.906907i	$-0.638437\pi$
$743$	−1.26802e7	−0.842661	−0.421331	+	0.906907i	$0.638437\pi$
$744$	0	0
$745$	1.01990e7	0.673237
$746$	1.31494e6	0.0865085
$747$	0	0
$748$	3.41110e7	2.22915
$749$	1.18239e7	0.770114
$750$	0	0
$751$	3.25560e6	0.210635	0.105318	−	0.994439i	$-0.466414\pi$
$751$	3.25560e6	0.210635	0.105318	+	0.994439i	$0.466414\pi$
$752$	2.04790e7	1.32058
$753$	0	0
$754$	176612.	0.0113133
$755$	3.58191e6	0.228690
$756$	0	0
$757$	8.66051e6	0.549293	0.274646	−	0.961545i	$-0.411439\pi$
$757$	8.66051e6	0.549293	0.274646	+	0.961545i	$0.411439\pi$
$758$	1.83987e6	0.116309
$759$	0	0
$760$	−391327.	−0.0245757
$761$	2.45364e7	1.53585	0.767926	−	0.640539i	$-0.221289\pi$
$761$	2.45364e7	1.53585	0.767926	+	0.640539i	$0.221289\pi$
$762$	0	0
$763$	−1.25639e7	−0.781292
$764$	2.29023e7	1.41954
$765$	0	0
$766$	2.01662e6	0.124180
$767$	−1.89794e6	−0.116492
$768$	0	0
$769$	−2.14448e7	−1.30769	−0.653846	−	0.756628i	$-0.726846\pi$
$769$	−2.14448e7	−1.30769	−0.653846	+	0.756628i	$0.726846\pi$
$770$	−1.18985e6	−0.0723213
$771$	0	0
$772$	1.60773e7	0.970888
$773$	2.10403e7	1.26650	0.633248	−	0.773949i	$-0.281721\pi$
$773$	2.10403e7	1.26650	0.633248	+	0.773949i	$0.281721\pi$
$774$	0	0
$775$	2.46254e7	1.47275
$776$	−4.23164e6	−0.252263
$777$	0	0
$778$	967341.	0.0572968
$779$	989180.	0.0584026
$780$	0	0
$781$	−4.73367e7	−2.77697
$782$	3.24902e6	0.189992
$783$	0	0
$784$	−5.73216e6	−0.333064
$785$	−2.87823e6	−0.166706
$786$	0	0
$787$	−2.21485e7	−1.27470	−0.637348	−	0.770576i	$-0.719969\pi$
$787$	−2.21485e7	−1.27470	−0.637348	+	0.770576i	$0.719969\pi$
$788$	−5.32123e6	−0.305278
$789$	0	0
$790$	1.04388e6	0.0595090
$791$	−2.35637e7	−1.33907
$792$	0	0
$793$	574256.	0.0324282
$794$	−293400.	−0.0165162
$795$	0	0
$796$	−1.25894e7	−0.704241
$797$	−1.39498e7	−0.777894	−0.388947	−	0.921260i	$-0.627161\pi$
$797$	−1.39498e7	−0.777894	−0.388947	+	0.921260i	$0.627161\pi$
$798$	0	0
$799$	−3.37762e7	−1.87173
$800$	−4.79533e6	−0.264907
$801$	0	0
$802$	−396643.	−0.0217753
$803$	−4.12254e7	−2.25619
$804$	0	0
$805$	8.43589e6	0.458819
$806$	−389538.	−0.0211209
$807$	0	0
$808$	−1.05753e6	−0.0569855
$809$	2.30945e7	1.24061	0.620307	−	0.784359i	$-0.287008\pi$
$809$	2.30945e7	1.24061	0.620307	+	0.784359i	$0.287008\pi$
$810$	0	0
$811$	−6.43405e6	−0.343504	−0.171752	−	0.985140i	$-0.554943\pi$
$811$	−6.43405e6	−0.343504	−0.171752	+	0.985140i	$0.554943\pi$
$812$	−1.51408e7	−0.805860
$813$	0	0
$814$	−5.46580e6	−0.289130
$815$	6.34189e6	0.334445
$816$	0	0
$817$	272550.	0.0142854
$818$	−228233.	−0.0119260
$819$	0	0
$820$	−2.26504e6	−0.117636
$821$	−3.20040e7	−1.65709	−0.828545	−	0.559922i	$-0.810831\pi$
$821$	−3.20040e7	−1.65709	−0.828545	+	0.559922i	$0.810831\pi$
$822$	0	0
$823$	−2.47454e7	−1.27349	−0.636744	−	0.771075i	$-0.719719\pi$
$823$	−2.47454e7	−1.27349	−0.636744	+	0.771075i	$0.719719\pi$
$824$	677009.	0.0347357
$825$	0	0
$826$	−2.18591e6	−0.111476
$827$	−2.86416e7	−1.45624	−0.728120	−	0.685449i	$-0.759606\pi$
$827$	−2.86416e7	−1.45624	−0.728120	+	0.685449i	$0.759606\pi$
$828$	0	0
$829$	−2.72449e7	−1.37689	−0.688444	−	0.725290i	$-0.741706\pi$
$829$	−2.72449e7	−1.37689	−0.688444	+	0.725290i	$0.741706\pi$
$830$	1.51534e6	0.0763512
$831$	0	0
$832$	−1.78886e6	−0.0895917
$833$	9.45411e6	0.472072
$834$	0	0
$835$	6.33324e6	0.314347
$836$	7.59178e6	0.375688
$837$	0	0
$838$	−1.76376e6	−0.0867620
$839$	2.28624e7	1.12129	0.560644	−	0.828057i	$-0.310554\pi$
$839$	2.28624e7	1.12129	0.560644	+	0.828057i	$0.310554\pi$
$840$	0	0
$841$	432425.	0.0210824
$842$	−3.07998e6	−0.149716
$843$	0	0
$844$	−2.03755e7	−0.984583
$845$	9.62818e6	0.463877
$846$	0	0
$847$	2.96020e7	1.41779
$848$	3.93165e7	1.87752
$849$	0	0
$850$	2.57735e6	0.122356
$851$	3.87518e7	1.83429
$852$	0	0
$853$	−3.04928e7	−1.43491	−0.717456	−	0.696604i	$-0.754694\pi$
$853$	−3.04928e7	−1.43491	−0.717456	+	0.696604i	$0.754694\pi$
$854$	661387.	0.0310321
$855$	0	0
$856$	−4.67271e6	−0.217964
$857$	−3.67912e7	−1.71116	−0.855582	−	0.517668i	$-0.826800\pi$
$857$	−3.67912e7	−1.71116	−0.855582	+	0.517668i	$0.826800\pi$
$858$	0	0
$859$	1.21414e7	0.561420	0.280710	−	0.959793i	$-0.409430\pi$
$859$	1.21414e7	0.561420	0.280710	+	0.959793i	$0.409430\pi$
$860$	−624090.	−0.0287741
$861$	0	0
$862$	−684517.	−0.0313773
$863$	3.31868e7	1.51684	0.758418	−	0.651769i	$-0.225973\pi$
$863$	3.31868e7	1.51684	0.758418	+	0.651769i	$0.225973\pi$
$864$	0	0
$865$	−1.36198e7	−0.618913
$866$	−640215.	−0.0290089
$867$	0	0
$868$	3.33948e7	1.50446
$869$	−4.07748e7	−1.83165
$870$	0	0
$871$	−953766.	−0.0425987
$872$	4.96517e6	0.221128
$873$	0	0
$874$	723106.	0.0320201
$875$	1.52638e7	0.673972
$876$	0	0
$877$	−2.34716e7	−1.03049	−0.515244	−	0.857043i	$-0.672299\pi$
$877$	−2.34716e7	−1.03049	−0.515244	+	0.857043i	$0.672299\pi$
$878$	−2.62416e6	−0.114882
$879$	0	0
$880$	−1.71471e7	−0.746421
$881$	−2.09622e7	−0.909909	−0.454954	−	0.890515i	$-0.650344\pi$
$881$	−2.09622e7	−0.909909	−0.454954	+	0.890515i	$0.650344\pi$
$882$	0	0
$883$	2.84258e7	1.22690	0.613452	−	0.789732i	$-0.289781\pi$
$883$	2.84258e7	1.22690	0.613452	+	0.789732i	$0.289781\pi$
$884$	3.03472e6	0.130614
$885$	0	0
$886$	1.67209e6	0.0715610
$887$	1.66822e7	0.711940	0.355970	−	0.934497i	$-0.384151\pi$
$887$	1.66822e7	0.711940	0.355970	+	0.934497i	$0.384151\pi$
$888$	0	0
$889$	1.60391e6	0.0680652
$890$	−837257.	−0.0354310
$891$	0	0
$892$	1.51958e7	0.639455
$893$	−7.51728e6	−0.315451
$894$	0	0
$895$	555092.	0.0231637
$896$	−8.65064e6	−0.359980
$897$	0	0
$898$	−218936.	−0.00905997
$899$	−4.61935e7	−1.90626
$900$	0	0
$901$	−6.48451e7	−2.66113
$902$	−1.18860e6	−0.0486431
$903$	0	0
$904$	9.31222e6	0.378994
$905$	−1.66941e7	−0.677549
$906$	0	0
$907$	3.46166e7	1.39722	0.698611	−	0.715501i	$-0.253802\pi$
$907$	3.46166e7	1.39722	0.698611	+	0.715501i	$0.253802\pi$
$908$	−2.29289e7	−0.922929
$909$	0	0
$910$	−105857.	−0.00423754
$911$	3.10235e7	1.23850	0.619249	−	0.785195i	$-0.287437\pi$
$911$	3.10235e7	1.23850	0.619249	+	0.785195i	$0.287437\pi$
$912$	0	0
$913$	−5.91906e7	−2.35004
$914$	−451429.	−0.0178741
$915$	0	0
$916$	3.99771e7	1.57425
$917$	3.37123e7	1.32393
$918$	0	0
$919$	1.35357e7	0.528679	0.264340	−	0.964430i	$-0.414846\pi$
$919$	1.35357e7	0.528679	0.264340	+	0.964430i	$0.414846\pi$
$920$	−3.33380e6	−0.129859
$921$	0	0
$922$	678999.	0.0263052
$923$	−4.21137e6	−0.162712
$924$	0	0
$925$	3.07407e7	1.18130
$926$	4.14644e6	0.158909
$927$	0	0
$928$	8.99528e6	0.342882
$929$	−8.96239e6	−0.340710	−0.170355	−	0.985383i	$-0.554491\pi$
$929$	−8.96239e6	−0.340710	−0.170355	+	0.985383i	$0.554491\pi$
$930$	0	0
$931$	2.10412e6	0.0795602
$932$	−5.46552e6	−0.206107
$933$	0	0
$934$	1.35730e6	0.0509107
$935$	2.82809e7	1.05795
$936$	0	0
$937$	−2.81081e6	−0.104588	−0.0522941	−	0.998632i	$-0.516653\pi$
$937$	−2.81081e6	−0.104588	−0.0522941	+	0.998632i	$0.516653\pi$
$938$	−1.09848e6	−0.0407647
$939$	0	0
$940$	1.72132e7	0.635392
$941$	2.85941e7	1.05270	0.526348	−	0.850269i	$-0.323561\pi$
$941$	2.85941e7	1.05270	0.526348	+	0.850269i	$0.323561\pi$
$942$	0	0
$943$	8.42705e6	0.308600
$944$	−3.15015e7	−1.15054
$945$	0	0
$946$	−327498.	−0.0118982
$947$	1.37193e7	0.497114	0.248557	−	0.968617i	$-0.420044\pi$
$947$	1.37193e7	0.497114	0.248557	+	0.968617i	$0.420044\pi$
$948$	0	0
$949$	−3.66767e6	−0.132198
$950$	573619.	0.0206212
$951$	0	0
$952$	7.03731e6	0.251660
$953$	−5.00552e7	−1.78532	−0.892661	−	0.450728i	$-0.851164\pi$
$953$	−5.00552e7	−1.78532	−0.892661	+	0.450728i	$0.851164\pi$
$954$	0	0
$955$	1.89880e7	0.673706
$956$	3.95090e7	1.39814
$957$	0	0
$958$	−4.33144e6	−0.152482
$959$	6.90587e6	0.242478
$960$	0	0
$961$	7.32559e7	2.55879
$962$	−486272.	−0.0169411
$963$	0	0
$964$	2.24781e7	0.779052
$965$	1.33294e7	0.460780
$966$	0	0
$967$	−5.43686e7	−1.86974	−0.934872	−	0.354984i	$-0.884486\pi$
$967$	−5.43686e7	−1.86974	−0.934872	+	0.354984i	$0.884486\pi$
$968$	−1.16985e7	−0.401274
$969$	0	0
$970$	−1.74249e6	−0.0594622
$971$	3.23373e7	1.10067	0.550333	−	0.834945i	$-0.314501\pi$
$971$	3.23373e7	1.10067	0.550333	+	0.834945i	$0.314501\pi$
$972$	0	0
$973$	1.29418e7	0.438239
$974$	−2.95801e6	−0.0999084
$975$	0	0
$976$	9.53134e6	0.320280
$977$	3.20023e6	0.107262	0.0536309	−	0.998561i	$-0.482921\pi$
$977$	3.20023e6	0.107262	0.0536309	+	0.998561i	$0.482921\pi$
$978$	0	0
$979$	3.27039e7	1.09054
$980$	−4.81804e6	−0.160253
$981$	0	0
$982$	4.82562e6	0.159689
$983$	2.95731e7	0.976142	0.488071	−	0.872804i	$-0.337701\pi$
$983$	2.95731e7	0.976142	0.488071	+	0.872804i	$0.337701\pi$
$984$	0	0
$985$	−4.41175e6	−0.144884
$986$	−4.83471e6	−0.158372
$987$	0	0
$988$	675412.	0.0220128
$989$	2.32192e6	0.0754842
$990$	0	0
$991$	−1.28280e7	−0.414931	−0.207465	−	0.978242i	$-0.566521\pi$
$991$	−1.28280e7	−0.414931	−0.207465	+	0.978242i	$0.566521\pi$
$992$	−1.98401e7	−0.640126
$993$	0	0
$994$	−4.85035e6	−0.155707
$995$	−1.04377e7	−0.334230
$996$	0	0
$997$	1.08720e7	0.346395	0.173197	−	0.984887i	$-0.444590\pi$
$997$	1.08720e7	0.346395	0.173197	+	0.984887i	$0.444590\pi$
$998$	4.03455e6	0.128224
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.6.a.k.1.4	yes	6
3.2	odd	2	inner	171.6.a.k.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.6.a.k.1.3	✓	6	3.2	odd	2	inner
171.6.a.k.1.4	yes	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+0.651309 q^{2} -31.5758 q^{4} -26.1790 q^{5} +104.778 q^{7} -41.4075 q^{8} -17.0506 q^{10} +666.012 q^{11} +59.2525 q^{13} +68.2428 q^{14} +983.456 q^{16} -1622.03 q^{17} -361.000 q^{19} +826.624 q^{20} +433.780 q^{22} -3075.44 q^{23} -2439.66 q^{25} +38.5917 q^{26} -3308.45 q^{28} +4576.41 q^{29} -10093.8 q^{31} +1965.57 q^{32} -1056.44 q^{34} -2742.99 q^{35} -12600.4 q^{37} -235.123 q^{38} +1084.01 q^{40} -2740.11 q^{41} -754.986 q^{43} -21029.9 q^{44} -2003.06 q^{46} +20823.5 q^{47} -5828.58 q^{49} -1588.97 q^{50} -1870.95 q^{52} +39977.9 q^{53} -17435.6 q^{55} -4338.59 q^{56} +2980.66 q^{58} -32031.4 q^{59} +9691.67 q^{61} -6574.19 q^{62} -30190.4 q^{64} -1551.17 q^{65} -16096.6 q^{67} +51216.8 q^{68} -1786.53 q^{70} -71074.9 q^{71} -61898.9 q^{73} -8206.77 q^{74} +11398.9 q^{76} +69783.4 q^{77} -61222.3 q^{79} -25745.9 q^{80} -1784.66 q^{82} -88873.1 q^{83} +42463.1 q^{85} -491.730 q^{86} -27577.9 q^{88} +49104.1 q^{89} +6208.36 q^{91} +97109.5 q^{92} +13562.5 q^{94} +9450.63 q^{95} +102195. q^{97} -3796.21 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 42 q^{4} - 10 q^{7} - 788 q^{10} - 1256 q^{13} - 606 q^{16} - 2166 q^{19} - 2524 q^{22} - 3944 q^{25} - 9632 q^{28} - 18136 q^{31} - 14072 q^{34} - 23764 q^{37} - 34284 q^{40} - 24606 q^{43} - 45640 q^{46}+ \cdots - 16492 q^{97}+O(q^{100}) \) 6 * q + 42 * q^4 - 10 * q^7 - 788 * q^10 - 1256 * q^13 - 606 * q^16 - 2166 * q^19 - 2524 * q^22 - 3944 * q^25 - 9632 * q^28 - 18136 * q^31 - 14072 * q^34 - 23764 * q^37 - 34284 * q^40 - 24606 * q^43 - 45640 * q^46 - 63256 * q^49 - 89128 * q^52 - 63402 * q^55 - 78572 * q^58 - 42334 * q^61 - 109662 * q^64 - 129696 * q^67 - 15096 * q^70 - 16998 * q^73 - 15162 * q^76 - 129004 * q^79 + 211396 * q^82 - 11754 * q^85 + 55044 * q^88 + 23932 * q^91 + 235368 * q^94 - 16492 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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