LMFDB - Embedded newform 234.6.b.c.181.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [234,6,Mod(181,234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(234, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("234.181");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 234 = 2 \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	234.b (of order $2$, degree $1$, 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:	$37.5298138362$
Analytic rank:	$0$
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} - 2x^{5} + 2x^{4} - 2946x^{3} + 131769x^{2} - 1332936x + 6741792 $ x^6 - 2x^5 + 2x^4 - 2946x^3 + 131769x^2 - 1332936*x + 6741792
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.5
Root			$-15.0768 + 15.0768i$ of defining polynomial
Character	$\chi$	$=$	234.181
Dual form			234.6.b.c.181.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{2} -16.0000 q^{4} +9.73803i q^{5} +105.184i q^{7} -64.0000i q^{8} +O(q^{10})$	\(q+4.00000i q^{2} -16.0000 q^{4} +9.73803i q^{5} +105.184i q^{7} -64.0000i q^{8} -38.9521 q^{10} -269.350i q^{11} +(227.464 - 565.290i) q^{13} -420.736 q^{14} +256.000 q^{16} -1663.75 q^{17} -2.82742i q^{19} -155.809i q^{20} +1077.40 q^{22} -2151.14 q^{23} +3030.17 q^{25} +(2261.16 + 909.856i) q^{26} -1682.95i q^{28} -220.829 q^{29} -788.106i q^{31} +1024.00i q^{32} -6654.98i q^{34} -1024.29 q^{35} -980.445i q^{37} +11.3097 q^{38} +623.234 q^{40} -14809.7i q^{41} +14142.2 q^{43} +4309.61i q^{44} -8604.55i q^{46} -25181.2i q^{47} +5743.31 q^{49} +12120.7i q^{50} +(-3639.42 + 9044.65i) q^{52} +30900.9 q^{53} +2622.94 q^{55} +6731.78 q^{56} -883.317i q^{58} -25094.7i q^{59} +12060.8 q^{61} +3152.42 q^{62} -4096.00 q^{64} +(5504.81 + 2215.05i) q^{65} +14450.5i q^{67} +26619.9 q^{68} -4097.14i q^{70} +33547.5i q^{71} -25805.7i q^{73} +3921.78 q^{74} +45.2388i q^{76} +28331.4 q^{77} +15620.3 q^{79} +2492.94i q^{80} +59238.7 q^{82} -2819.78i q^{83} -16201.6i q^{85} +56568.8i q^{86} -17238.4 q^{88} +16227.2i q^{89} +(59459.6 + 23925.6i) q^{91} +34418.2 q^{92} +100725. q^{94} +27.5335 q^{95} -126289. i q^{97} +22973.2i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 96 q^{4}+O(q^{10}) $ 6 * q - 96 * q^4	$ 6 q - 96 q^{4} + 320 q^{10} + 530 q^{13} + 1360 q^{14} + 1536 q^{16} + 836 q^{17} - 1296 q^{22} + 416 q^{23} + 718 q^{25} + 1360 q^{26} - 18788 q^{29} - 6112 q^{35} - 528 q^{38} - 5120 q^{40} - 24200 q^{43} - 3038 q^{49} - 8480 q^{52} + 42396 q^{53} + 124656 q^{55} - 21760 q^{56} - 3196 q^{61} + 59344 q^{62} - 24576 q^{64} + 17168 q^{65} - 13376 q^{68} + 62240 q^{74} + 114024 q^{77} - 169328 q^{79} + 145120 q^{82} + 20736 q^{88} + 236152 q^{91} - 6656 q^{92} - 32688 q^{94} - 567168 q^{95}+O(q^{100}) $ 6 * q - 96 * q^4 + 320 * q^10 + 530 * q^13 + 1360 * q^14 + 1536 * q^16 + 836 * q^17 - 1296 * q^22 + 416 * q^23 + 718 * q^25 + 1360 * q^26 - 18788 * q^29 - 6112 * q^35 - 528 * q^38 - 5120 * q^40 - 24200 * q^43 - 3038 * q^49 - 8480 * q^52 + 42396 * q^53 + 124656 * q^55 - 21760 * q^56 - 3196 * q^61 + 59344 * q^62 - 24576 * q^64 + 17168 * q^65 - 13376 * q^68 + 62240 * q^74 + 114024 * q^77 - 169328 * q^79 + 145120 * q^82 + 20736 * q^88 + 236152 * q^91 - 6656 * q^92 - 32688 * q^94 - 567168 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/234\mathbb{Z}\right)^\times$.

$n$	$145$	$209$
$\chi(n)$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$	−16.0000			−0.500000
$5$			9.73803i			0.174199i	0.996200	+	0.0870996i	$0.0277598\pi$
$5$			9.73803i			0.174199i	−0.996200	+	0.0870996i	$0.972240\pi$
$6$		0			0
$7$			105.184i			0.811344i	0.914019	+	0.405672i	$0.132962\pi$
$7$			105.184i			0.811344i	−0.914019	+	0.405672i	$0.867038\pi$
$8$		−	64.0000i		−	0.353553i
$9$		0			0
$10$	−38.9521			−0.123177
$11$		−	269.350i		−	0.671175i	−0.942009	−	0.335588i	$-0.891065\pi$
$11$		−	269.350i		−	0.671175i	0.942009	−	0.335588i	$-0.108935\pi$
$12$		0			0
$13$	227.464	−	565.290i	0.373297	−	0.927712i
$13$	227.464	−	565.290i	0.373297	−	0.927712i
$14$	−420.736			−0.573707
$15$		0			0
$16$	256.000			0.250000
$17$	−1663.75			−1.39625			−0.698127	−	0.715974i	$-0.745983\pi$
$17$	−1663.75			−1.39625			−0.698127	+	0.715974i	$0.745983\pi$
$18$		0			0
$19$		−	2.82742i		−	0.00179683i	−1.00000		0.000898415i	$-0.999714\pi$
$19$		−	2.82742i		−	0.00179683i	1.00000		0.000898415i	$-0.000285974\pi$
$20$		−	155.809i		−	0.0870996i
$21$		0			0
$22$	1077.40			0.474592
$23$	−2151.14			−0.847908			−0.423954	−	0.905684i	$-0.639358\pi$
$23$	−2151.14			−0.847908			−0.423954	+	0.905684i	$0.639358\pi$
$24$		0			0
$25$	3030.17			0.969655
$26$	2261.16	+	909.856i	0.655991	+	0.263961i
$27$		0			0
$28$		−	1682.95i		−	0.405672i
$29$	−220.829			−0.0487598			−0.0243799	−	0.999703i	$-0.507761\pi$
$29$	−220.829			−0.0487598			−0.0243799	+	0.999703i	$0.507761\pi$
$30$		0			0
$31$		−	788.106i		−	0.147292i	−0.997284	−	0.0736462i	$-0.976536\pi$
$31$		−	788.106i		−	0.147292i	0.997284	−	0.0736462i	$-0.0234636\pi$
$32$			1024.00i			0.176777i
$33$		0			0
$34$		−	6654.98i		−	0.987301i
$35$	−1024.29			−0.141335
$36$		0			0
$37$		−	980.445i		−	0.117739i	−0.998266	−	0.0588693i	$-0.981250\pi$
$37$		−	980.445i		−	0.117739i	0.998266	−	0.0588693i	$-0.0187495\pi$
$38$	11.3097			0.00127055
$39$		0			0
$40$	623.234			0.0615887
$41$		−	14809.7i		−	1.37590i	−0.725759	−	0.687949i	$-0.758511\pi$
$41$		−	14809.7i		−	1.37590i	0.725759	−	0.687949i	$-0.241489\pi$
$42$		0			0
$43$	14142.2			1.16640			0.583198	−	0.812330i	$-0.301801\pi$
$43$	14142.2			1.16640			0.583198	+	0.812330i	$0.301801\pi$
$44$			4309.61i			0.335588i
$45$		0			0
$46$		−	8604.55i		−	0.599561i
$47$		−	25181.2i		−	1.66277i	−0.555699	−	0.831383i	$-0.687549\pi$
$47$		−	25181.2i		−	1.66277i	0.555699	−	0.831383i	$-0.312451\pi$
$48$		0			0
$49$	5743.31			0.341721
$50$			12120.7i			0.685649i
$51$		0			0
$52$	−3639.42	+	9044.65i	−0.186648	+	0.463856i
$53$	30900.9			1.51106			0.755530	−	0.655114i	$-0.227379\pi$
$53$	30900.9			1.51106			0.755530	+	0.655114i	$0.227379\pi$
$54$		0			0
$55$	2622.94			0.116918
$56$	6731.78			0.286853
$57$		0			0
$58$		−	883.317i		−	0.0344784i
$59$		−	25094.7i		−	0.938538i	−0.883055	−	0.469269i	$-0.844517\pi$
$59$		−	25094.7i		−	0.938538i	0.883055	−	0.469269i	$-0.155483\pi$
$60$		0			0
$61$	12060.8			0.415003			0.207501	−	0.978235i	$-0.433467\pi$
$61$	12060.8			0.415003			0.207501	+	0.978235i	$0.433467\pi$
$62$	3152.42			0.104151
$63$		0			0
$64$	−4096.00			−0.125000
$65$	5504.81	+	2215.05i	0.161607	+	0.0650280i
$66$		0			0
$67$			14450.5i			0.393276i	0.980476	+	0.196638i	$0.0630023\pi$
$67$			14450.5i			0.393276i	−0.980476	+	0.196638i	$0.936998\pi$
$68$	26619.9			0.698127
$69$		0			0
$70$		−	4097.14i		−	0.0999393i
$71$			33547.5i			0.789795i	0.918725	+	0.394897i	$0.129220\pi$
$71$			33547.5i			0.789795i	−0.918725	+	0.394897i	$0.870780\pi$
$72$		0			0
$73$		−	25805.7i		−	0.566772i	−0.959006	−	0.283386i	$-0.908542\pi$
$73$		−	25805.7i		−	0.566772i	0.959006	−	0.283386i	$-0.0914578\pi$
$74$	3921.78			0.0832538
$75$		0			0
$76$			45.2388i			0.000898415i
$77$	28331.4			0.544554
$78$		0			0
$79$	15620.3			0.281593			0.140797	−	0.990039i	$-0.455034\pi$
$79$	15620.3			0.281593			0.140797	+	0.990039i	$0.455034\pi$
$80$			2492.94i			0.0435498i
$81$		0			0
$82$	59238.7			0.972906
$83$		−	2819.78i		−	0.0449284i	−0.999748	−	0.0224642i	$-0.992849\pi$
$83$		−	2819.78i		−	0.0449284i	0.999748	−	0.0224642i	$-0.00715117\pi$
$84$		0			0
$85$		−	16201.6i		−	0.243226i
$86$			56568.8i			0.824766i
$87$		0			0
$88$	−17238.4			−0.237296
$89$			16227.2i			0.217154i	0.994088	+	0.108577i	$0.0346294\pi$
$89$			16227.2i			0.217154i	−0.994088	+	0.108577i	$0.965371\pi$
$90$		0			0
$91$	59459.6	+	23925.6i	0.752693	+	0.302872i
$92$	34418.2			0.423954
$93$		0			0
$94$	100725.			1.17575
$95$	27.5335			0.000313006
$96$		0			0
$97$		−	126289.i		−	1.36281i	−0.731906	−	0.681406i	$-0.761369\pi$
$97$		−	126289.i		−	1.36281i	0.731906	−	0.681406i	$-0.238631\pi$
$98$			22973.2i			0.241633i
$99$		0			0
$100$	−48482.7			−0.484827
$101$	−92398.8			−0.901287			−0.450643	−	0.892704i	$-0.648805\pi$
$101$	−92398.8			−0.901287			−0.450643	+	0.892704i	$0.648805\pi$
$102$		0			0
$103$	148091.			1.37542			0.687711	−	0.725985i	$-0.258616\pi$
$103$	148091.			1.37542			0.687711	+	0.725985i	$0.258616\pi$
$104$	−36178.6	−	14557.7i	−0.327996	−	0.131980i
$105$		0			0
$106$			123604.i			1.06848i
$107$	−88180.1			−0.744580			−0.372290	−	0.928117i	$-0.621427\pi$
$107$	−88180.1			−0.744580			−0.372290	+	0.928117i	$0.621427\pi$
$108$		0			0
$109$		−	61467.2i		−	0.495538i	−0.968819	−	0.247769i	$-0.920303\pi$
$109$		−	61467.2i		−	0.495538i	0.968819	−	0.247769i	$-0.0796974\pi$
$110$			10491.8i			0.0826736i
$111$		0			0
$112$			26927.1i			0.202836i
$113$	82256.1			0.605999			0.303000	−	0.952991i	$-0.402012\pi$
$113$	82256.1			0.605999			0.303000	+	0.952991i	$0.402012\pi$
$114$		0			0
$115$		−	20947.8i		−	0.147705i
$116$	3533.27			0.0243799
$117$		0			0
$118$	100379.			0.663647
$119$		−	175000.i		−	1.13284i
$120$		0			0
$121$	88501.4			0.549524
$122$			48243.1i			0.293451i
$123$		0			0
$124$			12609.7i			0.0736462i
$125$			59939.2i			0.343112i
$126$		0			0
$127$	−143150.			−0.787556			−0.393778	−	0.919206i	$-0.628832\pi$
$127$	−143150.			−0.787556			−0.393778	+	0.919206i	$0.628832\pi$
$128$		−	16384.0i		−	0.0883883i
$129$		0			0
$130$	−8860.20	+	22019.3i	−0.0459817	+	0.114273i
$131$	141329.			0.719539			0.359769	−	0.933041i	$-0.382855\pi$
$131$	141329.			0.719539			0.359769	+	0.933041i	$0.382855\pi$
$132$		0			0
$133$	297.400			0.00145785
$134$	−57802.2			−0.278088
$135$		0			0
$136$			106480.i			0.493650i
$137$		−	3958.33i		−	0.0180182i	−0.999959	−	0.00900909i	$-0.997132\pi$
$137$		−	3958.33i		−	0.0180182i	0.999959	−	0.00900909i	$-0.00286772\pi$
$138$		0			0
$139$	−445579.			−1.95608			−0.978042	−	0.208408i	$-0.933172\pi$
$139$	−445579.			−1.95608			−0.978042	+	0.208408i	$0.933172\pi$
$140$	16388.6			0.0706677
$141$		0			0
$142$	−134190.			−0.558469
$143$	−152261.	−	61267.5i	−0.622657	−	0.250547i
$144$		0			0
$145$		−	2150.44i		−	0.00849391i
$146$	103223.			0.400769
$147$		0			0
$148$			15687.1i			0.0588693i
$149$			49619.3i			0.183098i	0.995801	+	0.0915492i	$0.0291819\pi$
$149$			49619.3i			0.183098i	−0.995801	+	0.0915492i	$0.970818\pi$
$150$		0			0
$151$		−	402323.i		−	1.43593i	−0.696081	−	0.717964i	$-0.745074\pi$
$151$		−	402323.i		−	1.43593i	0.696081	−	0.717964i	$-0.254926\pi$
$152$	−180.955			−0.000635275
$153$		0			0
$154$			113325.i			0.385058i
$155$	7674.60			0.0256582
$156$		0			0
$157$	−562429.			−1.82104			−0.910519	−	0.413468i	$-0.864317\pi$
$157$	−562429.			−1.82104			−0.910519	+	0.413468i	$0.864317\pi$
$158$			62481.3i			0.199116i
$159$		0			0
$160$	−9971.74			−0.0307944
$161$		−	226265.i		−	0.687945i
$162$		0			0
$163$			313878.i			0.925319i	0.886536	+	0.462659i	$0.153105\pi$
$163$			313878.i			0.925319i	−0.886536	+	0.462659i	$0.846895\pi$
$164$			236955.i			0.687949i
$165$		0			0
$166$	11279.1			0.0317691
$167$		−	602228.i		−	1.67097i	−0.549510	−	0.835487i	$-0.685186\pi$
$167$		−	602228.i		−	1.67097i	0.549510	−	0.835487i	$-0.314814\pi$
$168$		0			0
$169$	−267813.	−	257166.i	−0.721299	−	0.692624i
$170$	64806.4			0.171987
$171$		0			0
$172$	−226275.			−0.583198
$173$	−304021.			−0.772304			−0.386152	−	0.922435i	$-0.626196\pi$
$173$	−304021.			−0.772304			−0.386152	+	0.922435i	$0.626196\pi$
$174$		0			0
$175$			318726.i			0.786723i
$176$		−	68953.7i		−	0.167794i
$177$		0			0
$178$	−64908.7			−0.153551
$179$	−406190.			−0.947539			−0.473769	−	0.880649i	$-0.657107\pi$
$179$	−406190.			−0.947539			−0.473769	+	0.880649i	$0.657107\pi$
$180$		0			0
$181$	−240967.			−0.546714			−0.273357	−	0.961913i	$-0.588134\pi$
$181$	−240967.			−0.546714			−0.273357	+	0.961913i	$0.588134\pi$
$182$	−95702.4	+	237838.i	−0.214163	+	0.532235i
$183$		0			0
$184$			137673.i			0.299781i
$185$	9547.60			0.0205100
$186$		0			0
$187$			448130.i			0.937131i
$188$			402899.i			0.831383i
$189$		0			0
$190$			110.134i			0.000221329i
$191$	−226140.			−0.448532			−0.224266	−	0.974528i	$-0.571999\pi$
$191$	−226140.			−0.448532			−0.224266	+	0.974528i	$0.571999\pi$
$192$		0			0
$193$			406075.i			0.784717i	0.919812	+	0.392359i	$0.128341\pi$
$193$			406075.i			0.784717i	−0.919812	+	0.392359i	$0.871659\pi$
$194$	505156.			0.963653
$195$		0			0
$196$	−91892.9			−0.170861
$197$		−	407390.i		−	0.747902i	−0.927449	−	0.373951i	$-0.878003\pi$
$197$		−	407390.i		−	0.747902i	0.927449	−	0.373951i	$-0.121997\pi$
$198$		0			0
$199$	202865.			0.363140			0.181570	−	0.983378i	$-0.441882\pi$
$199$	202865.			0.363140			0.181570	+	0.983378i	$0.441882\pi$
$200$		−	193931.i		−	0.342825i
$201$		0			0
$202$		−	369595.i		−	0.637306i
$203$		−	23227.7i		−	0.0395609i
$204$		0			0
$205$	144217.			0.239680
$206$			592364.i			0.972570i
$207$		0			0
$208$	58230.8	−	144714.i	0.0933242	−	0.231928i
$209$	−761.568			−0.00120599
$210$		0			0
$211$	657970.			1.01742			0.508710	−	0.860938i	$-0.330123\pi$
$211$	657970.			1.01742			0.508710	+	0.860938i	$0.330123\pi$
$212$	−494415.			−0.755530
$213$		0			0
$214$		−	352720.i		−	0.526497i
$215$			137717.i			0.203185i
$216$		0			0
$217$	82896.2			0.119505
$218$	245869.			0.350398
$219$		0			0
$220$	−41967.1			−0.0584591
$221$	−378442.	+	940499.i	−0.521217	+	1.29532i
$222$		0			0
$223$			978060.i			1.31705i	0.752558	+	0.658527i	$0.228820\pi$
$223$			978060.i			1.31705i	−0.752558	+	0.658527i	$0.771180\pi$
$224$	−107709.			−0.143427
$225$		0			0
$226$			329024.i			0.428506i
$227$			812137.i			1.04608i	0.852308	+	0.523040i	$0.175202\pi$
$227$			812137.i			1.04608i	−0.852308	+	0.523040i	$0.824798\pi$
$228$		0			0
$229$		−	1.36329e6i		−	1.71790i	−0.512058	−	0.858951i	$-0.671117\pi$
$229$		−	1.36329e6i		−	1.71790i	0.512058	−	0.858951i	$-0.328883\pi$
$230$	83791.4			0.104443
$231$		0			0
$232$			14133.1i			0.0172392i
$233$	−931926.			−1.12458			−0.562292	−	0.826939i	$-0.690080\pi$
$233$	−931926.			−1.12458			−0.562292	+	0.826939i	$0.690080\pi$
$234$		0			0
$235$	245215.			0.289653
$236$			401515.i			0.469269i
$237$		0			0
$238$	699998.			0.801041
$239$		−	885874.i		−	1.00318i	−0.865107	−	0.501588i	$-0.832749\pi$
$239$		−	885874.i		−	1.00318i	0.865107	−	0.501588i	$-0.167251\pi$
$240$		0			0
$241$		−	1.32690e6i		−	1.47161i	−0.677191	−	0.735807i	$-0.736803\pi$
$241$		−	1.32690e6i		−	1.47161i	0.677191	−	0.735807i	$-0.263197\pi$
$242$			354006.i			0.388572i
$243$		0			0
$244$	−192973.			−0.207501
$245$			55928.5i			0.0595275i
$246$		0			0
$247$	−1598.32	−	643.137i	−0.00166694	−	0.000670751i
$248$	−50438.8			−0.0520757
$249$		0			0
$250$	−239757.			−0.242617
$251$	547596.			0.548625			0.274313	−	0.961641i	$-0.411550\pi$
$251$	547596.			0.548625			0.274313	+	0.961641i	$0.411550\pi$
$252$		0			0
$253$			579409.i			0.569094i
$254$		−	572599.i		−	0.556886i
$255$		0			0
$256$	65536.0			0.0625000
$257$	1.65509e6			1.56311			0.781554	−	0.623837i	$-0.214427\pi$
$257$	1.65509e6			1.56311			0.781554	+	0.623837i	$0.214427\pi$
$258$		0			0
$259$	103127.			0.0955265
$260$	−88077.0	−	35440.8i	−0.0808033	−	0.0325140i
$261$		0			0
$262$			565317.i			0.508791i
$263$	668188.			0.595675			0.297838	−	0.954617i	$-0.403735\pi$
$263$	668188.			0.595675			0.297838	+	0.954617i	$0.403735\pi$
$264$		0			0
$265$			300914.i			0.263226i
$266$			1189.60i			0.00103085i
$267$		0			0
$268$		−	231209.i		−	0.196638i
$269$	384942.			0.324350			0.162175	−	0.986762i	$-0.448149\pi$
$269$	384942.			0.324350			0.162175	+	0.986762i	$0.448149\pi$
$270$		0			0
$271$			1.05344e6i			0.871334i	0.900108	+	0.435667i	$0.143487\pi$
$271$			1.05344e6i			0.871334i	−0.900108	+	0.435667i	$0.856513\pi$
$272$	−425919.			−0.349064
$273$		0			0
$274$	15833.3			0.0127408
$275$		−	816177.i		−	0.650808i
$276$		0			0
$277$	−1.62051e6			−1.26897			−0.634485	−	0.772936i	$-0.718788\pi$
$277$	−1.62051e6			−1.26897			−0.634485	+	0.772936i	$0.718788\pi$
$278$		−	1.78231e6i		−	1.38316i
$279$		0			0
$280$			65554.3i			0.0499696i
$281$			563009.i			0.425353i	0.977123	+	0.212677i	$0.0682181\pi$
$281$			563009.i			0.425353i	−0.977123	+	0.212677i	$0.931782\pi$
$282$		0			0
$283$	361892.			0.268604			0.134302	−	0.990940i	$-0.457121\pi$
$283$	361892.			0.268604			0.134302	+	0.990940i	$0.457121\pi$
$284$		−	536760.i		−	0.394897i
$285$		0			0
$286$	245070.	−	609045.i	0.177164	−	0.440285i
$287$	1.55774e6			1.11633
$288$		0			0
$289$	1.34819e6			0.949526
$290$	8601.77			0.00600610
$291$		0			0
$292$			412891.i			0.283386i
$293$			757318.i			0.515358i	0.966231	+	0.257679i	$0.0829577\pi$
$293$			757318.i			0.515358i	−0.966231	+	0.257679i	$0.917042\pi$
$294$		0			0
$295$	244373.			0.163493
$296$	−62748.5			−0.0416269
$297$		0			0
$298$	−198477.			−0.129470
$299$	−489306.	+	1.21602e6i	−0.316521	+	0.786614i
$300$		0			0
$301$			1.48753e6i			0.946348i
$302$	1.60929e6			1.01535
$303$		0			0
$304$		−	723.821i		−	0.000449207i
$305$			117448.i			0.0722932i
$306$		0			0
$307$		−	1.96280e6i		−	1.18858i	−0.804249	−	0.594292i	$-0.797432\pi$
$307$		−	1.96280e6i		−	1.18858i	0.804249	−	0.594292i	$-0.202568\pi$
$308$	−453302.			−0.272277
$309$		0			0
$310$			30698.4i			0.0181431i
$311$	−834419.			−0.489197			−0.244598	−	0.969624i	$-0.578656\pi$
$311$	−834419.			−0.489197			−0.244598	+	0.969624i	$0.578656\pi$
$312$		0			0
$313$	1.61834e6			0.933703			0.466851	−	0.884336i	$-0.345388\pi$
$313$	1.61834e6			0.933703			0.466851	+	0.884336i	$0.345388\pi$
$314$		−	2.24972e6i		−	1.28767i
$315$		0			0
$316$	−249925.			−0.140797
$317$		−	660304.i		−	0.369059i	−0.982827	−	0.184529i	$-0.940924\pi$
$317$		−	660304.i		−	0.369059i	0.982827	−	0.184529i	$-0.0590761\pi$
$318$		0			0
$319$			59480.4i			0.0327263i
$320$		−	39887.0i		−	0.0217749i
$321$		0			0
$322$	905062.			0.486450
$323$			4704.11i			0.00250883i
$324$		0			0
$325$	689255.	−	1.71293e6i	0.361969	−	0.899560i
$326$	−1.25551e6			−0.654299
$327$		0			0
$328$	−947819.			−0.486453
$329$	2.64866e6			1.34908
$330$		0			0
$331$		−	2.73497e6i		−	1.37209i	−0.727558	−	0.686046i	$-0.759345\pi$
$331$		−	2.73497e6i		−	1.37209i	0.727558	−	0.686046i	$-0.240655\pi$
$332$			45116.5i			0.0224642i
$333$		0			0
$334$	2.40891e6			1.18156
$335$	−140720.			−0.0685083
$336$		0			0
$337$	−880707.			−0.422432			−0.211216	−	0.977439i	$-0.567742\pi$
$337$	−880707.			−0.422432			−0.211216	+	0.977439i	$0.567742\pi$
$338$	1.02867e6	−	1.07125e6i	0.489759	−	0.510035i
$339$		0			0
$340$			259226.i			0.121613i
$341$	−212277.			−0.0988590
$342$		0			0
$343$			2.37193e6i			1.08860i
$344$		−	905101.i		−	0.412383i
$345$		0			0
$346$		−	1.21608e6i		−	0.546102i
$347$	33079.1			0.0147479			0.00737395	−	0.999973i	$-0.497653\pi$
$347$	33079.1			0.0147479			0.00737395	+	0.999973i	$0.497653\pi$
$348$		0			0
$349$		−	4.06448e6i		−	1.78624i	−0.449814	−	0.893122i	$-0.648510\pi$
$349$		−	4.06448e6i		−	1.78624i	0.449814	−	0.893122i	$-0.351490\pi$
$350$	−1.27490e6			−0.556297
$351$		0			0
$352$	275815.			0.118648
$353$		−	3.57770e6i		−	1.52815i	−0.645125	−	0.764077i	$-0.723195\pi$
$353$		−	3.57770e6i		−	1.52815i	0.645125	−	0.764077i	$-0.276805\pi$
$354$		0			0
$355$	−326686.			−0.137582
$356$		−	259635.i		−	0.108577i
$357$		0			0
$358$		−	1.62476e6i		−	0.670011i
$359$			3.31455e6i			1.35734i	0.734444	+	0.678669i	$0.237443\pi$
$359$			3.31455e6i			1.35734i	−0.734444	+	0.678669i	$0.762557\pi$
$360$		0			0
$361$	2.47609e6			0.999997
$362$		−	963867.i		−	0.386585i
$363$		0			0
$364$	−951353.	−	382809.i	−0.376347	−	0.151436i
$365$	251297.			0.0987313
$366$		0			0
$367$	192406.			0.0745681			0.0372840	−	0.999305i	$-0.488129\pi$
$367$	192406.			0.0745681			0.0372840	+	0.999305i	$0.488129\pi$
$368$	−550691.			−0.211977
$369$		0			0
$370$			38190.4i			0.0145027i
$371$			3.25029e6i			1.22599i
$372$		0			0
$373$	1.60161e6			0.596053			0.298027	−	0.954558i	$-0.403672\pi$
$373$	1.60161e6			0.596053			0.298027	+	0.954558i	$0.403672\pi$
$374$	−1.79252e6			−0.662652
$375$		0			0
$376$	−1.61160e6			−0.587877
$377$	−50230.7	+	124833.i	−0.0182019	+	0.0452350i
$378$		0			0
$379$			3.76006e6i			1.34461i	0.740274	+	0.672305i	$0.234696\pi$
$379$			3.76006e6i			1.34461i	−0.740274	+	0.672305i	$0.765304\pi$
$380$	−440.537			−0.000156503
$381$		0			0
$382$		−	904560.i		−	0.317160i
$383$			4.69693e6i			1.63613i	0.575128	+	0.818063i	$0.304952\pi$
$383$			4.69693e6i			1.63613i	−0.575128	+	0.818063i	$0.695048\pi$
$384$		0			0
$385$			275892.i			0.0948608i
$386$	−1.62430e6			−0.554879
$387$		0			0
$388$			2.02062e6i			0.681406i
$389$	−4.85881e6			−1.62801			−0.814003	−	0.580860i	$-0.802716\pi$
$389$	−4.85881e6			−1.62801			−0.814003	+	0.580860i	$0.802716\pi$
$390$		0			0
$391$	3.57894e6			1.18389
$392$		−	367572.i		−	0.120817i
$393$		0			0
$394$	1.62956e6			0.528846
$395$			152111.i			0.0490533i
$396$		0			0
$397$			3.91926e6i			1.24804i	0.781409	+	0.624019i	$0.214501\pi$
$397$			3.91926e6i			1.24804i	−0.781409	+	0.624019i	$0.785499\pi$
$398$			811460.i			0.256779i
$399$		0			0
$400$	775724.			0.242414
$401$		−	5.01898e6i		−	1.55867i	−0.626606	−	0.779336i	$-0.715557\pi$
$401$		−	5.01898e6i		−	1.55867i	0.626606	−	0.779336i	$-0.284443\pi$
$402$		0			0
$403$	−445509.	−	179266.i	−0.136645	−	0.0549838i
$404$	1.47838e6			0.450643
$405$		0			0
$406$	92910.9			0.0279738
$407$	−264083.			−0.0790232
$408$		0			0
$409$			2.97886e6i			0.880526i	0.897869	+	0.440263i	$0.145115\pi$
$409$			2.97886e6i			0.880526i	−0.897869	+	0.440263i	$0.854885\pi$
$410$			576868.i			0.169480i
$411$		0			0
$412$	−2.36946e6			−0.687711
$413$	2.63956e6			0.761477
$414$		0			0
$415$	27459.1			0.00782648
$416$	578857.	+	232923.i	0.163998	+	0.0659902i
$417$		0			0
$418$		−	3046.27i		−	0.000852762i
$419$	4.76887e6			1.32703			0.663515	−	0.748163i	$-0.269064\pi$
$419$	4.76887e6			1.32703			0.663515	+	0.748163i	$0.269064\pi$
$420$		0			0
$421$		−	651294.i		−	0.179090i	−0.995983	−	0.0895451i	$-0.971459\pi$
$421$		−	651294.i		−	0.179090i	0.995983	−	0.0895451i	$-0.0285413\pi$
$422$			2.63188e6i			0.719424i
$423$		0			0
$424$		−	1.97766e6i		−	0.534241i
$425$	−5.04143e6			−1.35388
$426$		0			0
$427$			1.26860e6i			0.336710i
$428$	1.41088e6			0.372290
$429$		0			0
$430$	−550869.			−0.143674
$431$			5.22621e6i			1.35517i	0.735444	+	0.677585i	$0.236973\pi$
$431$			5.22621e6i			1.35517i	−0.735444	+	0.677585i	$0.763027\pi$
$432$		0			0
$433$	−6.33506e6			−1.62379			−0.811897	−	0.583801i	$-0.801565\pi$
$433$	−6.33506e6			−1.62379			−0.811897	+	0.583801i	$0.801565\pi$
$434$			331585.i			0.0845026i
$435$		0			0
$436$			983475.i			0.247769i
$437$			6082.18i			0.00152355i
$438$		0			0
$439$	−7.44321e6			−1.84331			−0.921657	−	0.388006i	$-0.873164\pi$
$439$	−7.44321e6			−1.84331			−0.921657	+	0.388006i	$0.873164\pi$
$440$		−	167868.i		−	0.0413368i
$441$		0			0
$442$	−3.76200e6	−	1.51377e6i	−0.915931	−	0.368556i
$443$	−327031.			−0.0791734			−0.0395867	−	0.999216i	$-0.512604\pi$
$443$	−327031.			−0.0791734			−0.0395867	+	0.999216i	$0.512604\pi$
$444$		0			0
$445$	−158021.			−0.0378281
$446$	−3.91224e6			−0.931297
$447$		0			0
$448$		−	430834.i		−	0.101418i
$449$		−	5.82118e6i		−	1.36268i	−0.731965	−	0.681342i	$-0.761396\pi$
$449$		−	5.82118e6i		−	1.36268i	0.731965	−	0.681342i	$-0.238604\pi$
$450$		0			0
$451$	−3.98899e6			−0.923468
$452$	−1.31610e6			−0.303000
$453$		0			0
$454$	−3.24855e6			−0.739690
$455$	−232988.	+	579019.i	−0.0527601	+	0.131119i
$456$		0			0
$457$			8.15788e6i			1.82720i	0.406612	+	0.913601i	$0.366710\pi$
$457$			8.15788e6i			1.82720i	−0.406612	+	0.913601i	$0.633290\pi$
$458$	5.45315e6			1.21474
$459$		0			0
$460$			335165.i			0.0738524i
$461$			3.15011e6i			0.690357i	0.938537	+	0.345179i	$0.112182\pi$
$461$			3.15011e6i			0.690357i	−0.938537	+	0.345179i	$0.887818\pi$
$462$		0			0
$463$		−	202622.i		−	0.0439273i	−0.999759	−	0.0219637i	$-0.993008\pi$
$463$		−	202622.i		−	0.0439273i	0.999759	−	0.0219637i	$-0.00699182\pi$
$464$	−56532.3			−0.0121899
$465$		0			0
$466$		−	3.72771e6i		−	0.795201i
$467$	−91749.8			−0.0194676			−0.00973382	−	0.999953i	$-0.503098\pi$
$467$	−91749.8			−0.0194676			−0.00973382	+	0.999953i	$0.503098\pi$
$468$		0			0
$469$	−1.51997e6			−0.319082
$470$			980860.i			0.204815i
$471$		0			0
$472$	−1.60606e6			−0.331823
$473$		−	3.80921e6i		−	0.782856i
$474$		0			0
$475$		−	8567.58i		−	0.00174230i
$476$			2.79999e6i			0.566421i
$477$		0			0
$478$	3.54350e6			0.709353
$479$			2.82910e6i			0.563390i	0.959504	+	0.281695i	$0.0908967\pi$
$479$			2.82910e6i			0.563390i	−0.959504	+	0.281695i	$0.909103\pi$
$480$		0			0
$481$	−554236.	−	223016.i	−0.109228	−	0.0439514i
$482$	5.30758e6			1.04059
$483$		0			0
$484$	−1.41602e6			−0.274762
$485$	1.22981e6			0.237401
$486$		0			0
$487$			1.30176e6i			0.248719i	0.992237	+	0.124359i	$0.0396876\pi$
$487$			1.30176e6i			0.248719i	−0.992237	+	0.124359i	$0.960312\pi$
$488$		−	771890.i		−	0.146726i
$489$		0			0
$490$	−223714.			−0.0420923
$491$	6.07696e6			1.13758			0.568791	−	0.822482i	$-0.307412\pi$
$491$	6.07696e6			1.13758			0.568791	+	0.822482i	$0.307412\pi$
$492$		0			0
$493$	367404.			0.0680810
$494$	2572.55	−	6393.26i	0.000474292	−	0.00117870i
$495$		0			0
$496$		−	201755.i		−	0.0368231i
$497$	−3.52866e6			−0.640795
$498$		0			0
$499$		−	3.78708e6i		−	0.680853i	−0.940271	−	0.340426i	$-0.889429\pi$
$499$		−	3.78708e6i		−	0.680853i	0.940271	−	0.340426i	$-0.110571\pi$
$500$		−	959028.i		−	0.171556i
$501$		0			0
$502$			2.19038e6i			0.387937i
$503$	−2.28832e6			−0.403271			−0.201636	−	0.979461i	$-0.564626\pi$
$503$	−2.28832e6			−0.403271			−0.201636	+	0.979461i	$0.564626\pi$
$504$		0			0
$505$		−	899783.i		−	0.157003i
$506$	−2.31764e6			−0.402410
$507$		0			0
$508$	2.29040e6			0.393778
$509$		−	3.70029e6i		−	0.633055i	−0.948583	−	0.316528i	$-0.897483\pi$
$509$		−	3.70029e6i		−	0.633055i	0.948583	−	0.316528i	$-0.102517\pi$
$510$		0			0
$511$	2.71435e6			0.459847
$512$			262144.i			0.0441942i
$513$		0			0
$514$			6.62037e6i			1.10528i
$515$			1.44212e6i			0.239597i
$516$		0			0
$517$	−6.78256e6			−1.11601
$518$			412509.i			0.0675474i
$519$		0			0
$520$	141763.	−	352308.i	0.0229909	−	0.0571366i
$521$	−6.13756e6			−0.990608			−0.495304	−	0.868720i	$-0.664943\pi$
$521$	−6.13756e6			−0.990608			−0.495304	+	0.868720i	$0.664943\pi$
$522$		0			0
$523$	4.10572e6			0.656349			0.328175	−	0.944617i	$-0.393567\pi$
$523$	4.10572e6			0.656349			0.328175	+	0.944617i	$0.393567\pi$
$524$	−2.26127e6			−0.359769
$525$		0			0
$526$			2.67275e6i			0.421206i
$527$			1.31121e6i			0.205658i
$528$		0			0
$529$	−1.80895e6			−0.281053
$530$	−1.20366e6			−0.186129
$531$		0			0
$532$	−4758.40			−0.000728923
$533$	−8.37177e6	−	3.36867e6i	−1.27644	−	0.513618i
$534$		0			0
$535$		−	858701.i		−	0.129705i
$536$	924834.			0.139044
$537$		0			0
$538$			1.53977e6i			0.229350i
$539$		−	1.54696e6i		−	0.229355i
$540$		0			0
$541$		−	2.51266e6i		−	0.369098i	−0.982823	−	0.184549i	$-0.940918\pi$
$541$		−	2.51266e6i		−	0.369098i	0.982823	−	0.184549i	$-0.0590824\pi$
$542$	−4.21374e6			−0.616126
$543$		0			0
$544$		−	1.70368e6i		−	0.246825i
$545$	598569.			0.0863223
$546$		0			0
$547$	−2.29454e6			−0.327889			−0.163945	−	0.986470i	$-0.552422\pi$
$547$	−2.29454e6			−0.327889			−0.163945	+	0.986470i	$0.552422\pi$
$548$			63333.4i			0.00900909i
$549$		0			0
$550$	3.26471e6			0.460191
$551$			624.378i			8.76130e-5i
$552$		0			0
$553$			1.64301e6i			0.228469i
$554$		−	6.48202e6i		−	0.897297i
$555$		0			0
$556$	7.12926e6			0.978042
$557$		−	5.96324e6i		−	0.814412i	−0.913336	−	0.407206i	$-0.866503\pi$
$557$		−	5.96324e6i		−	0.814412i	0.913336	−	0.407206i	$-0.133497\pi$
$558$		0			0
$559$	3.21684e6	−	7.99445e6i	0.435412	−	1.08208i
$560$	−262217.			−0.0353339
$561$		0			0
$562$	−2.25204e6			−0.300770
$563$	7.37754e6			0.980935			0.490468	−	0.871459i	$-0.336826\pi$
$563$	7.37754e6			0.980935			0.490468	+	0.871459i	$0.336826\pi$
$564$		0			0
$565$			801012.i			0.105565i
$566$			1.44757e6i			0.189932i
$567$		0			0
$568$	2.14704e6			0.279235
$569$	−2.60845e6			−0.337756			−0.168878	−	0.985637i	$-0.554014\pi$
$569$	−2.60845e6			−0.337756			−0.168878	+	0.985637i	$0.554014\pi$
$570$		0			0
$571$	1.09867e7			1.41019			0.705095	−	0.709113i	$-0.250905\pi$
$571$	1.09867e7			1.41019			0.705095	+	0.709113i	$0.250905\pi$
$572$	2.43618e6	+	980280.i	0.311329	+	0.125274i
$573$		0			0
$574$			6.23097e6i			0.789362i
$575$	−6.51831e6			−0.822177
$576$		0			0
$577$		−	1.10048e7i		−	1.37607i	−0.725677	−	0.688036i	$-0.758473\pi$
$577$		−	1.10048e7i		−	1.37607i	0.725677	−	0.688036i	$-0.241527\pi$
$578$			5.39276e6i			0.671416i
$579$		0			0
$580$			34407.1i			0.00424696i
$581$	296596.			0.0364523
$582$		0			0
$583$		−	8.32318e6i		−	1.01419i
$584$	−1.65157e6			−0.200384
$585$		0			0
$586$	−3.02927e6			−0.364413
$587$			1.09851e7i			1.31586i	0.753080	+	0.657929i	$0.228567\pi$
$587$			1.09851e7i			1.31586i	−0.753080	+	0.657929i	$0.771433\pi$
$588$		0			0
$589$	−2228.31			−0.000264659
$590$			977492.i			0.115607i
$591$		0			0
$592$		−	250994.i		−	0.0294347i
$593$		−	3.90620e6i		−	0.456161i	−0.973642	−	0.228081i	$-0.926755\pi$
$593$		−	3.90620e6i		−	0.456161i	0.973642	−	0.228081i	$-0.0732449\pi$
$594$		0			0
$595$	1.70415e6			0.197340
$596$		−	793908.i		−	0.0915492i
$597$		0			0
$598$	−4.86407e6	−	1.95722e6i	−0.556220	−	0.223814i
$599$	1.69973e7			1.93559			0.967797	−	0.251733i	$-0.0810005\pi$
$599$	1.69973e7			1.93559			0.967797	+	0.251733i	$0.0810005\pi$
$600$		0			0
$601$	1.07099e7			1.20948			0.604739	−	0.796424i	$-0.293278\pi$
$601$	1.07099e7			1.20948			0.604739	+	0.796424i	$0.293278\pi$
$602$	−5.95014e6			−0.669169
$603$		0			0
$604$			6.43717e6i			0.717964i
$605$			861829.i			0.0957267i
$606$		0			0
$607$	−1.43102e7			−1.57642			−0.788212	−	0.615404i	$-0.788993\pi$
$607$	−1.43102e7			−1.57642			−0.788212	+	0.615404i	$0.788993\pi$
$608$	2895.28			0.000317638
$609$		0			0
$610$	−469793.			−0.0511190
$611$	−1.42347e7	−	5.72781e6i	−1.54257	−	0.620705i
$612$		0			0
$613$		−	1.67594e7i		−	1.80139i	−0.434450	−	0.900696i	$-0.643057\pi$
$613$		−	1.67594e7i		−	1.80139i	0.434450	−	0.900696i	$-0.356943\pi$
$614$	7.85120e6			0.840456
$615$		0			0
$616$		−	1.81321e6i		−	0.192529i
$617$		−	1.26793e7i		−	1.34085i	−0.741976	−	0.670427i	$-0.766111\pi$
$617$		−	1.26793e7i		−	1.34085i	0.741976	−	0.670427i	$-0.233889\pi$
$618$		0			0
$619$			5.43928e6i			0.570577i	0.958442	+	0.285289i	$0.0920895\pi$
$619$			5.43928e6i			0.570577i	−0.958442	+	0.285289i	$0.907911\pi$
$620$	−122794.			−0.0128291
$621$		0			0
$622$		−	3.33768e6i		−	0.345914i
$623$	−1.70684e6			−0.176187
$624$		0			0
$625$	8.88559e6			0.909885
$626$			6.47335e6i			0.660227i
$627$		0			0
$628$	8.99887e6			0.910519
$629$			1.63121e6i			0.164393i
$630$		0			0
$631$			5.28439e6i			0.528350i	0.964475	+	0.264175i	$0.0850996\pi$
$631$			5.28439e6i			0.528350i	−0.964475	+	0.264175i	$0.914900\pi$
$632$		−	999700.i		−	0.0995582i
$633$		0			0
$634$	2.64121e6			0.260964
$635$		−	1.39400e6i		−	0.137192i
$636$		0			0
$637$	1.30639e6	−	3.24664e6i	0.127563	−	0.317019i
$638$	−237922.			−0.0231410
$639$		0			0
$640$	159548.			0.0153972
$641$	−8.01855e6			−0.770816			−0.385408	−	0.922746i	$-0.625939\pi$
$641$	−8.01855e6			−0.770816			−0.385408	+	0.922746i	$0.625939\pi$
$642$		0			0
$643$			1.39227e7i			1.32799i	0.747735	+	0.663997i	$0.231141\pi$
$643$			1.39227e7i			1.32799i	−0.747735	+	0.663997i	$0.768859\pi$
$644$			3.62025e6i			0.343972i
$645$		0			0
$646$	−18816.5			−0.00177401
$647$	9.55517e6			0.897383			0.448692	−	0.893687i	$-0.351890\pi$
$647$	9.55517e6			0.897383			0.448692	+	0.893687i	$0.351890\pi$
$648$		0			0
$649$	−6.75926e6			−0.629923
$650$	6.85170e6	+	2.75702e6i	0.636085	+	0.255951i
$651$		0			0
$652$		−	5.02204e6i		−	0.462659i
$653$	1.87340e7			1.71928			0.859640	−	0.510901i	$-0.170688\pi$
$653$	1.87340e7			1.71928			0.859640	+	0.510901i	$0.170688\pi$
$654$		0			0
$655$			1.37627e6i			0.125343i
$656$		−	3.79128e6i		−	0.343974i
$657$		0			0
$658$			1.05946e7i			0.953941i
$659$	3.39787e6			0.304785			0.152392	−	0.988320i	$-0.451302\pi$
$659$	3.39787e6			0.304785			0.152392	+	0.988320i	$0.451302\pi$
$660$		0			0
$661$		−	1.81823e7i		−	1.61862i	−0.587382	−	0.809310i	$-0.699841\pi$
$661$		−	1.81823e7i		−	1.61862i	0.587382	−	0.809310i	$-0.300159\pi$
$662$	1.09399e7			0.970215
$663$		0			0
$664$	−180466.			−0.0158846
$665$			2896.09i			0.000253956i
$666$		0			0
$667$	475034.			0.0413438
$668$			9.63564e6i			0.835487i
$669$		0			0
$670$		−	562879.i		−	0.0484427i
$671$		−	3.24858e6i		−	0.278540i
$672$		0			0
$673$	686844.			0.0584548			0.0292274	−	0.999573i	$-0.490695\pi$
$673$	686844.			0.0584548			0.0292274	+	0.999573i	$0.490695\pi$
$674$		−	3.52283e6i		−	0.298704i
$675$		0			0
$676$	4.28501e6	+	4.11466e6i	0.360650	+	0.346312i
$677$	6.64858e6			0.557516			0.278758	−	0.960361i	$-0.410077\pi$
$677$	6.64858e6			0.557516			0.278758	+	0.960361i	$0.410077\pi$
$678$		0			0
$679$	1.32836e7			1.10571
$680$	−1.03690e6			−0.0859935
$681$		0			0
$682$		−	849106.i		−	0.0699038i
$683$			1.66853e7i			1.36862i	0.729193	+	0.684308i	$0.239896\pi$
$683$			1.66853e7i			1.36862i	−0.729193	+	0.684308i	$0.760104\pi$
$684$		0			0
$685$	38546.4			0.00313875
$686$	−9.48773e6			−0.769754
$687$		0			0
$688$	3.62040e6			0.291599
$689$	7.02885e6	−	1.74680e7i	0.564074	−	1.40183i
$690$		0			0
$691$			1.75290e7i			1.39657i	0.715822	+	0.698283i	$0.246052\pi$
$691$			1.75290e7i			1.39657i	−0.715822	+	0.698283i	$0.753948\pi$
$692$	4.86434e6			0.386152
$693$		0			0
$694$			132317.i			0.0104283i
$695$		−	4.33906e6i		−	0.340748i
$696$		0			0
$697$			2.46395e7i			1.92110i
$698$	1.62579e7			1.26307
$699$		0			0
$700$		−	5.09961e6i		−	0.393362i
$701$	−7.06379e6			−0.542929			−0.271464	−	0.962449i	$-0.587508\pi$
$701$	−7.06379e6			−0.542929			−0.271464	+	0.962449i	$0.587508\pi$
$702$		0			0
$703$	−2772.13			−0.000211556
$704$			1.10326e6i			0.0838969i
$705$		0			0
$706$	1.43108e7			1.08057
$707$		−	9.71889e6i		−	0.731253i
$708$		0			0
$709$		−	9.07731e6i		−	0.678175i	−0.940755	−	0.339087i	$-0.889882\pi$
$709$		−	9.07731e6i		−	0.678175i	0.940755	−	0.339087i	$-0.110118\pi$
$710$		−	1.30675e6i		−	0.0972849i
$711$		0			0
$712$	1.03854e6			0.0767755
$713$			1.69532e6i			0.124890i
$714$		0			0
$715$	596625.	−	1.48272e6i	0.0436452	−	0.108466i
$716$	6.49904e6			0.473769
$717$		0			0
$718$	−1.32582e7			−0.959783
$719$	1.03755e7			0.748492			0.374246	−	0.927330i	$-0.377902\pi$
$719$	1.03755e7			0.748492			0.374246	+	0.927330i	$0.377902\pi$
$720$		0			0
$721$			1.55768e7i			1.11594i
$722$			9.90436e6i			0.707104i
$723$		0			0
$724$	3.85547e6			0.273357
$725$	−669150.			−0.0472801
$726$		0			0
$727$	−2.94915e6			−0.206948			−0.103474	−	0.994632i	$-0.532996\pi$
$727$	−2.94915e6			−0.206948			−0.103474	+	0.994632i	$0.532996\pi$
$728$	1.53124e6	−	3.80541e6i	0.107081	−	0.266117i
$729$		0			0
$730$			1.00519e6i			0.0698136i
$731$	−2.35290e7			−1.62858
$732$		0			0
$733$		−	1.71605e7i		−	1.17970i	−0.807513	−	0.589849i	$-0.799187\pi$
$733$		−	1.71605e7i		−	1.17970i	0.807513	−	0.589849i	$-0.200813\pi$
$734$			769623.i			0.0527276i
$735$		0			0
$736$		−	2.20276e6i		−	0.149890i
$737$	3.89226e6			0.263957
$738$		0			0
$739$		−	2.20849e7i		−	1.48760i	−0.668405	−	0.743798i	$-0.733023\pi$
$739$		−	2.20849e7i		−	1.48760i	0.668405	−	0.743798i	$-0.266977\pi$
$740$	−152762.			−0.0102550
$741$		0			0
$742$	−1.30011e7			−0.866906
$743$			2.21459e7i			1.47171i	0.677140	+	0.735854i	$0.263219\pi$
$743$			2.21459e7i			1.47171i	−0.677140	+	0.735854i	$0.736781\pi$
$744$		0			0
$745$	−483194.			−0.0318956
$746$			6.40644e6i			0.421473i
$747$		0			0
$748$		−	7.17008e6i		−	0.468565i
$749$		−	9.27514e6i		−	0.604110i
$750$		0			0
$751$	6.38710e6			0.413242			0.206621	−	0.978421i	$-0.433753\pi$
$751$	6.38710e6			0.413242			0.206621	+	0.978421i	$0.433753\pi$
$752$		−	6.44638e6i		−	0.415692i
$753$		0			0
$754$	−499331.	−	200923.i	−0.0319860	−	0.0128707i
$755$	3.91783e6			0.250137
$756$		0			0
$757$	−2.47266e6			−0.156828			−0.0784141	−	0.996921i	$-0.524986\pi$
$757$	−2.47266e6			−0.156828			−0.0784141	+	0.996921i	$0.524986\pi$
$758$	−1.50402e7			−0.950783
$759$		0			0
$760$		−	1762.15i		−	0.000110664i
$761$		−	1.59009e7i		−	0.995315i	−0.867374	−	0.497657i	$-0.834194\pi$
$761$		−	1.59009e7i		−	0.995315i	0.867374	−	0.497657i	$-0.165806\pi$
$762$		0			0
$763$	6.46537e6			0.402052
$764$	3.61824e6			0.224266
$765$		0			0
$766$	−1.87877e7			−1.15692
$767$	−1.41858e7	−	5.70814e6i	−0.870693	−	0.350353i
$768$		0			0
$769$		−	4.80537e6i		−	0.293029i	−0.989209	−	0.146515i	$-0.953194\pi$
$769$		−	4.80537e6i		−	0.293029i	0.989209	−	0.146515i	$-0.0468055\pi$
$770$	−1.10357e6			−0.0670767
$771$		0			0
$772$		−	6.49720e6i		−	0.392359i
$773$			2.22008e7i			1.33635i	0.744004	+	0.668175i	$0.232924\pi$
$773$			2.22008e7i			1.33635i	−0.744004	+	0.668175i	$0.767076\pi$
$774$		0			0
$775$		−	2.38810e6i		−	0.142823i
$776$	−8.08249e6			−0.481827
$777$		0			0
$778$		−	1.94353e7i		−	1.15117i
$779$	−41873.2			−0.00247225
$780$		0			0
$781$	9.03603e6			0.530090
$782$			1.43158e7i			0.837140i
$783$		0			0
$784$	1.47029e6			0.0854303
$785$		−	5.47695e6i		−	0.317223i</

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:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	234.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{2} -16.0000 q^{4} +9.73803i q^{5} +105.184i q^{7} -64.0000i q^{8} +O(q^{10})\)	\(q+4.00000i q^{2} -16.0000 q^{4} +9.73803i q^{5} +105.184i q^{7} -64.0000i q^{8} -38.9521 q^{10} -269.350i q^{11} +(227.464 - 565.290i) q^{13} -420.736 q^{14} +256.000 q^{16} -1663.75 q^{17} -2.82742i q^{19} -155.809i q^{20} +1077.40 q^{22} -2151.14 q^{23} +3030.17 q^{25} +(2261.16 + 909.856i) q^{26} -1682.95i q^{28} -220.829 q^{29} -788.106i q^{31} +1024.00i q^{32} -6654.98i q^{34} -1024.29 q^{35} -980.445i q^{37} +11.3097 q^{38} +623.234 q^{40} -14809.7i q^{41} +14142.2 q^{43} +4309.61i q^{44} -8604.55i q^{46} -25181.2i q^{47} +5743.31 q^{49} +12120.7i q^{50} +(-3639.42 + 9044.65i) q^{52} +30900.9 q^{53} +2622.94 q^{55} +6731.78 q^{56} -883.317i q^{58} -25094.7i q^{59} +12060.8 q^{61} +3152.42 q^{62} -4096.00 q^{64} +(5504.81 + 2215.05i) q^{65} +14450.5i q^{67} +26619.9 q^{68} -4097.14i q^{70} +33547.5i q^{71} -25805.7i q^{73} +3921.78 q^{74} +45.2388i q^{76} +28331.4 q^{77} +15620.3 q^{79} +2492.94i q^{80} +59238.7 q^{82} -2819.78i q^{83} -16201.6i q^{85} +56568.8i q^{86} -17238.4 q^{88} +16227.2i q^{89} +(59459.6 + 23925.6i) q^{91} +34418.2 q^{92} +100725. q^{94} +27.5335 q^{95} -126289. i q^{97} +22973.2i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 96 q^{4}+O(q^{10}) \) 6 * q - 96 * q^4	\( 6 q - 96 q^{4} + 320 q^{10} + 530 q^{13} + 1360 q^{14} + 1536 q^{16} + 836 q^{17} - 1296 q^{22} + 416 q^{23} + 718 q^{25} + 1360 q^{26} - 18788 q^{29} - 6112 q^{35} - 528 q^{38} - 5120 q^{40} - 24200 q^{43} - 3038 q^{49} - 8480 q^{52} + 42396 q^{53} + 124656 q^{55} - 21760 q^{56} - 3196 q^{61} + 59344 q^{62} - 24576 q^{64} + 17168 q^{65} - 13376 q^{68} + 62240 q^{74} + 114024 q^{77} - 169328 q^{79} + 145120 q^{82} + 20736 q^{88} + 236152 q^{91} - 6656 q^{92} - 32688 q^{94} - 567168 q^{95}+O(q^{100}) \) 6 * q - 96 * q^4 + 320 * q^10 + 530 * q^13 + 1360 * q^14 + 1536 * q^16 + 836 * q^17 - 1296 * q^22 + 416 * q^23 + 718 * q^25 + 1360 * q^26 - 18788 * q^29 - 6112 * q^35 - 528 * q^38 - 5120 * q^40 - 24200 * q^43 - 3038 * q^49 - 8480 * q^52 + 42396 * q^53 + 124656 * q^55 - 21760 * q^56 - 3196 * q^61 + 59344 * q^62 - 24576 * q^64 + 17168 * q^65 - 13376 * q^68 + 62240 * q^74 + 114024 * q^77 - 169328 * q^79 + 145120 * q^82 + 20736 * q^88 + 236152 * q^91 - 6656 * q^92 - 32688 * q^94 - 567168 * q^95

Introduction

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