LMFDB - Embedded newform 528.6.o.b.175.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(175,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("528.175");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.o (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:	$84.6826568613$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.18
Character	$\chi$	$=$	528.175
Dual form			528.6.o.b.175.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000i q^{3} -84.4799 q^{5} +98.5983 q^{7} -81.0000 q^{9} +O(q^{10})$	$q+9.00000i q^{3} -84.4799 q^{5} +98.5983 q^{7} -81.0000 q^{9} +(-362.700 + 171.754i) q^{11} -28.0828i q^{13} -760.320i q^{15} +334.573i q^{17} -2603.43 q^{19} +887.384i q^{21} -1429.78i q^{23} +4011.86 q^{25} -729.000i q^{27} +2770.19i q^{29} -4566.13i q^{31} +(-1545.78 - 3264.30i) q^{33} -8329.58 q^{35} -5083.07 q^{37} +252.745 q^{39} +4622.84i q^{41} -8788.85 q^{43} +6842.88 q^{45} +621.893i q^{47} -7085.38 q^{49} -3011.16 q^{51} +8700.63 q^{53} +(30640.9 - 14509.8i) q^{55} -23430.9i q^{57} +43386.5i q^{59} -43577.7i q^{61} -7986.46 q^{63} +2372.43i q^{65} -20172.7i q^{67} +12868.0 q^{69} +23684.4i q^{71} -77957.3i q^{73} +36106.8i q^{75} +(-35761.6 + 16934.6i) q^{77} +37516.9 q^{79} +6561.00 q^{81} +85313.3 q^{83} -28264.7i q^{85} -24931.7 q^{87} +48357.9 q^{89} -2768.92i q^{91} +41095.2 q^{93} +219938. q^{95} +151635. q^{97} +(29378.7 - 13912.1i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) $ 40 * q + 88 * q^5 - 3240 * q^9	$ 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) $ 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$.

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$			9.00000i			0.577350i
$3$			9.00000i			0.577350i
$4$		0			0
$5$	−84.4799			−1.51122			−0.755612	−	0.655020i	$-0.772660\pi$
$5$	−84.4799			−1.51122			−0.755612	+	0.655020i	$0.772660\pi$
$6$		0			0
$7$	98.5983			0.760544			0.380272	−	0.924875i	$-0.375830\pi$
$7$	98.5983			0.760544			0.380272	+	0.924875i	$0.375830\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	−362.700	+	171.754i	−0.903788	+	0.427981i
$11$	−362.700	+	171.754i	−0.903788	+	0.427981i
$12$		0			0
$13$		−	28.0828i		−	0.0460874i	−0.999734	−	0.0230437i	$-0.992664\pi$
$13$		−	28.0828i		−	0.0460874i	0.999734	−	0.0230437i	$-0.00733568\pi$
$14$		0			0
$15$		−	760.320i		−	0.872505i
$16$		0			0
$17$			334.573i			0.280782i	0.990096	+	0.140391i	$0.0448359\pi$
$17$			334.573i			0.280782i	−0.990096	+	0.140391i	$0.955164\pi$
$18$		0			0
$19$	−2603.43			−1.65448			−0.827241	−	0.561847i	$-0.810091\pi$
$19$	−2603.43			−1.65448			−0.827241	+	0.561847i	$0.810091\pi$
$20$		0			0
$21$			887.384i			0.439100i
$22$		0			0
$23$		−	1429.78i		−	0.563573i	−0.959477	−	0.281786i	$-0.909073\pi$
$23$		−	1429.78i		−	0.563573i	0.959477	−	0.281786i	$-0.0909269\pi$
$24$		0			0
$25$	4011.86			1.28380
$26$		0			0
$27$		−	729.000i		−	0.192450i
$28$		0			0
$29$			2770.19i			0.611666i	0.952085	+	0.305833i	$0.0989349\pi$
$29$			2770.19i			0.611666i	−0.952085	+	0.305833i	$0.901065\pi$
$30$		0			0
$31$		−	4566.13i		−	0.853383i	−0.904397	−	0.426691i	$-0.859679\pi$
$31$		−	4566.13i		−	0.853383i	0.904397	−	0.426691i	$-0.140321\pi$
$32$		0			0
$33$	−1545.78	−	3264.30i	−0.247095	−	0.521802i
$34$		0			0
$35$	−8329.58			−1.14935
$36$		0			0
$37$	−5083.07			−0.610411			−0.305205	−	0.952287i	$-0.598725\pi$
$37$	−5083.07			−0.610411			−0.305205	+	0.952287i	$0.598725\pi$
$38$		0			0
$39$	252.745			0.0266086
$40$		0			0
$41$			4622.84i			0.429486i	0.976671	+	0.214743i	$0.0688914\pi$
$41$			4622.84i			0.429486i	−0.976671	+	0.214743i	$0.931109\pi$
$42$		0			0
$43$	−8788.85			−0.724871			−0.362435	−	0.932009i	$-0.618055\pi$
$43$	−8788.85			−0.724871			−0.362435	+	0.932009i	$0.618055\pi$
$44$		0			0
$45$	6842.88			0.503741
$46$		0			0
$47$			621.893i			0.0410650i	0.999789	+	0.0205325i	$0.00653615\pi$
$47$			621.893i			0.0410650i	−0.999789	+	0.0205325i	$0.993464\pi$
$48$		0			0
$49$	−7085.38			−0.421573
$50$		0			0
$51$	−3011.16			−0.162109
$52$		0			0
$53$	8700.63			0.425462			0.212731	−	0.977111i	$-0.431764\pi$
$53$	8700.63			0.425462			0.212731	+	0.977111i	$0.431764\pi$
$54$		0			0
$55$	30640.9	−	14509.8i	1.36582	−	0.646775i
$56$		0			0
$57$		−	23430.9i		−	0.955216i
$58$		0			0
$59$			43386.5i			1.62265i	0.584595	+	0.811325i	$0.301254\pi$
$59$			43386.5i			1.62265i	−0.584595	+	0.811325i	$0.698746\pi$
$60$		0			0
$61$		−	43577.7i		−	1.49948i	−0.661734	−	0.749739i	$-0.730179\pi$
$61$		−	43577.7i		−	1.49948i	0.661734	−	0.749739i	$-0.269821\pi$
$62$		0			0
$63$	−7986.46			−0.253515
$64$		0			0
$65$			2372.43i			0.0696483i
$66$		0			0
$67$		−	20172.7i		−	0.549007i	−0.961586	−	0.274503i	$-0.911487\pi$
$67$		−	20172.7i		−	0.549007i	0.961586	−	0.274503i	$-0.0885135\pi$
$68$		0			0
$69$	12868.0			0.325379
$70$		0			0
$71$			23684.4i			0.557591i	0.960350	+	0.278796i	$0.0899352\pi$
$71$			23684.4i			0.557591i	−0.960350	+	0.278796i	$0.910065\pi$
$72$		0			0
$73$		−	77957.3i		−	1.71218i	−0.516827	−	0.856090i	$-0.672887\pi$
$73$		−	77957.3i		−	1.71218i	0.516827	−	0.856090i	$-0.327113\pi$
$74$		0			0
$75$			36106.8i			0.741200i
$76$		0			0
$77$	−35761.6	+	16934.6i	−0.687370	+	0.325498i
$78$		0			0
$79$	37516.9			0.676330			0.338165	−	0.941087i	$-0.390194\pi$
$79$	37516.9			0.676330			0.338165	+	0.941087i	$0.390194\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$	85313.3			1.35932			0.679660	−	0.733527i	$-0.262127\pi$
$83$	85313.3			1.35932			0.679660	+	0.733527i	$0.262127\pi$
$84$		0			0
$85$		−	28264.7i		−	0.424324i
$86$		0			0
$87$	−24931.7			−0.353146
$88$		0			0
$89$	48357.9			0.647131			0.323566	−	0.946206i	$-0.395118\pi$
$89$	48357.9			0.647131			0.323566	+	0.946206i	$0.395118\pi$
$90$		0			0
$91$		−	2768.92i		−	0.0350515i
$92$		0			0
$93$	41095.2			0.492701
$94$		0			0
$95$	219938.			2.50029
$96$		0			0
$97$	151635.			1.63633			0.818163	−	0.574986i	$-0.194992\pi$
$97$	151635.			1.63633			0.818163	+	0.574986i	$0.194992\pi$
$98$		0			0
$99$	29378.7	−	13912.1i	0.301263	−	0.142660i
$100$		0			0
$101$			101993.i			0.994873i	0.867501	+	0.497436i	$0.165725\pi$
$101$			101993.i			0.994873i	−0.867501	+	0.497436i	$0.834275\pi$
$102$		0			0
$103$			58035.5i			0.539015i	0.962998	+	0.269507i	$0.0868608\pi$
$103$			58035.5i			0.539015i	−0.962998	+	0.269507i	$0.913139\pi$
$104$		0			0
$105$		−	74966.2i		−	0.663578i
$106$		0			0
$107$	108834.			0.918977			0.459489	−	0.888184i	$-0.348033\pi$
$107$	108834.			0.918977			0.459489	+	0.888184i	$0.348033\pi$
$108$		0			0
$109$			99511.7i			0.802246i	0.916024	+	0.401123i	$0.131380\pi$
$109$			99511.7i			0.802246i	−0.916024	+	0.401123i	$0.868620\pi$
$110$		0			0
$111$		−	45747.7i		−	0.352421i
$112$		0			0
$113$	−81536.6			−0.600698			−0.300349	−	0.953829i	$-0.597103\pi$
$113$	−81536.6			−0.600698			−0.300349	+	0.953829i	$0.597103\pi$
$114$		0			0
$115$			120788.i			0.851684i
$116$		0			0
$117$			2274.71i			0.0153625i
$118$		0			0
$119$			32988.3i			0.213547i
$120$		0			0
$121$	102052.	−	124590.i	0.633664	−	0.773608i
$122$		0			0
$123$	−41605.6			−0.247964
$124$		0			0
$125$	−74922.0			−0.428879
$126$		0			0
$127$	−2659.37			−0.0146308			−0.00731541	−	0.999973i	$-0.502329\pi$
$127$	−2659.37			−0.0146308			−0.00731541	+	0.999973i	$0.502329\pi$
$128$		0			0
$129$		−	79099.6i		−	0.418504i
$130$		0			0
$131$	−56107.0			−0.285653			−0.142827	−	0.989748i	$-0.545619\pi$
$131$	−56107.0			−0.285653			−0.142827	+	0.989748i	$0.545619\pi$
$132$		0			0
$133$	−256694.			−1.25831
$134$		0			0
$135$			61585.9i			0.290835i
$136$		0			0
$137$	−125238.			−0.570077			−0.285038	−	0.958516i	$-0.592006\pi$
$137$	−125238.			−0.570077			−0.285038	+	0.958516i	$0.592006\pi$
$138$		0			0
$139$	171861.			0.754469			0.377235	−	0.926118i	$-0.376875\pi$
$139$	171861.			0.754469			0.377235	+	0.926118i	$0.376875\pi$
$140$		0			0
$141$	−5597.04			−0.0237089
$142$		0			0
$143$	4823.33	+	10185.6i	0.0197245	+	0.0416532i
$144$		0			0
$145$		−	234025.i		−	0.924364i
$146$		0			0
$147$		−	63768.4i		−	0.243395i
$148$		0			0
$149$			116076.i			0.428329i	0.976798	+	0.214165i	$0.0687029\pi$
$149$			116076.i			0.428329i	−0.976798	+	0.214165i	$0.931297\pi$
$150$		0			0
$151$	−49294.7			−0.175937			−0.0879686	−	0.996123i	$-0.528038\pi$
$151$	−49294.7			−0.175937			−0.0879686	+	0.996123i	$0.528038\pi$
$152$		0			0
$153$		−	27100.4i		−	0.0935939i
$154$		0			0
$155$			385746.i			1.28965i
$156$		0			0
$157$	51458.0			0.166611			0.0833056	−	0.996524i	$-0.473452\pi$
$157$	51458.0			0.166611			0.0833056	+	0.996524i	$0.473452\pi$
$158$		0			0
$159$			78305.7i			0.245641i
$160$		0			0
$161$		−	140974.i		−	0.428622i
$162$		0			0
$163$			144348.i			0.425542i	0.977102	+	0.212771i	$0.0682488\pi$
$163$			144348.i			0.425542i	−0.977102	+	0.212771i	$0.931751\pi$
$164$		0			0
$165$	130588.	+	275768.i	0.373416	+	0.788559i
$166$		0			0
$167$	651889.			1.80877			0.904383	−	0.426723i	$-0.140332\pi$
$167$	651889.			1.80877			0.904383	+	0.426723i	$0.140332\pi$
$168$		0			0
$169$	370504.			0.997876
$170$		0			0
$171$	210878.			0.551494
$172$		0			0
$173$			322571.i			0.819428i	0.912214	+	0.409714i	$0.134371\pi$
$173$			322571.i			0.819428i	−0.912214	+	0.409714i	$0.865629\pi$
$174$		0			0
$175$	395563.			0.976383
$176$		0			0
$177$	−390479.			−0.936838
$178$		0			0
$179$		−	628094.i		−	1.46518i	−0.680668	−	0.732592i	$-0.738310\pi$
$179$		−	628094.i		−	1.46518i	0.680668	−	0.732592i	$-0.261690\pi$
$180$		0			0
$181$	117000.			0.265453			0.132727	−	0.991153i	$-0.457627\pi$
$181$	117000.			0.265453			0.132727	+	0.991153i	$0.457627\pi$
$182$		0			0
$183$	392200.			0.865724
$184$		0			0
$185$	429418.			0.922467
$186$		0			0
$187$	−57464.2	−	121350.i	−0.120169	−	0.253767i
$188$		0			0
$189$		−	71878.1i		−	0.146367i
$190$		0			0
$191$			791774.i			1.57043i	0.619225	+	0.785213i	$0.287447\pi$
$191$			791774.i			1.57043i	−0.619225	+	0.785213i	$0.712553\pi$
$192$		0			0
$193$		−	523798.i		−	1.01221i	−0.862472	−	0.506105i	$-0.831085\pi$
$193$		−	523798.i		−	1.01221i	0.862472	−	0.506105i	$-0.168915\pi$
$194$		0			0
$195$	−21351.9			−0.0402115
$196$		0			0
$197$		−	1.07723e6i		−	1.97762i	−0.149183	−	0.988810i	$-0.547665\pi$
$197$		−	1.07723e6i		−	1.97762i	0.149183	−	0.988810i	$-0.452335\pi$
$198$		0			0
$199$			287574.i			0.514774i	0.966308	+	0.257387i	$0.0828616\pi$
$199$			287574.i			0.514774i	−0.966308	+	0.257387i	$0.917138\pi$
$200$		0			0
$201$	181555.			0.316969
$202$		0			0
$203$			273136.i			0.465199i
$204$		0			0
$205$		−	390537.i		−	0.649050i
$206$		0			0
$207$			115812.i			0.187858i
$208$		0			0
$209$	944266.	−	447149.i	1.49530	−	0.708087i
$210$		0			0
$211$	1.12196e6			1.73489			0.867446	−	0.497531i	$-0.165760\pi$
$211$	1.12196e6			1.73489			0.867446	+	0.497531i	$0.165760\pi$
$212$		0			0
$213$	−213159.			−0.321925
$214$		0			0
$215$	742481.			1.09544
$216$		0			0
$217$		−	450212.i		−	0.649035i
$218$		0			0
$219$	701615.			0.988528
$220$		0			0
$221$	9395.75			0.0129405
$222$		0			0
$223$			811595.i			1.09289i	0.837495	+	0.546446i	$0.184020\pi$
$223$			811595.i			1.09289i	−0.837495	+	0.546446i	$0.815980\pi$
$224$		0			0
$225$	−324961.			−0.427932
$226$		0			0
$227$	−633295.			−0.815720			−0.407860	−	0.913044i	$-0.633725\pi$
$227$	−633295.			−0.815720			−0.407860	+	0.913044i	$0.633725\pi$
$228$		0			0
$229$	734813.			0.925951			0.462976	−	0.886371i	$-0.346782\pi$
$229$	734813.			0.925951			0.462976	+	0.886371i	$0.346782\pi$
$230$		0			0
$231$	−152412.	−	321855.i	−0.187927	−	0.396853i
$232$		0			0
$233$		−	1.23320e6i		−	1.48814i	−0.668101	−	0.744071i	$-0.732892\pi$
$233$		−	1.23320e6i		−	1.48814i	0.668101	−	0.744071i	$-0.267108\pi$
$234$		0			0
$235$		−	52537.5i		−	0.0620583i
$236$		0			0
$237$			337652.i			0.390480i
$238$		0			0
$239$	265732.			0.300919			0.150459	−	0.988616i	$-0.451925\pi$
$239$	265732.			0.300919			0.150459	+	0.988616i	$0.451925\pi$
$240$		0			0
$241$			134448.i			0.149112i	0.997217	+	0.0745561i	$0.0237540\pi$
$241$			134448.i			0.149112i	−0.997217	+	0.0745561i	$0.976246\pi$
$242$		0			0
$243$			59049.0i			0.0641500i
$244$		0			0
$245$	598573.			0.637091
$246$		0			0
$247$			73111.6i			0.0762507i
$248$		0			0
$249$			767820.i			0.784804i
$250$		0			0
$251$		−	1.61927e6i		−	1.62231i	−0.584831	−	0.811155i	$-0.698839\pi$
$251$		−	1.61927e6i		−	1.62231i	0.584831	−	0.811155i	$-0.301161\pi$
$252$		0			0
$253$	245570.	+	518582.i	0.241198	+	0.509350i
$254$		0			0
$255$	254383.			0.244984
$256$		0			0
$257$	−465342.			−0.439480			−0.219740	−	0.975558i	$-0.570521\pi$
$257$	−465342.			−0.439480			−0.219740	+	0.975558i	$0.570521\pi$
$258$		0			0
$259$	−501182.			−0.464244
$260$		0			0
$261$		−	224385.i		−	0.203889i
$262$		0			0
$263$	−253938.			−0.226380			−0.113190	−	0.993573i	$-0.536107\pi$
$263$	−253938.			−0.226380			−0.113190	+	0.993573i	$0.536107\pi$
$264$		0			0
$265$	−735029.			−0.642969
$266$		0			0
$267$			435221.i			0.373621i
$268$		0			0
$269$	−1.90717e6			−1.60697			−0.803486	−	0.595324i	$-0.797024\pi$
$269$	−1.90717e6			−1.60697			−0.803486	+	0.595324i	$0.797024\pi$
$270$		0			0
$271$	−462178.			−0.382284			−0.191142	−	0.981562i	$-0.561219\pi$
$271$	−462178.			−0.382284			−0.191142	+	0.981562i	$0.561219\pi$
$272$		0			0
$273$	24920.2			0.0202370
$274$		0			0
$275$	−1.45510e6	+	689052.i	−1.16028	+	0.549440i
$276$		0			0
$277$		−	1.40413e6i		−	1.09953i	−0.835319	−	0.549766i	$-0.814717\pi$
$277$		−	1.40413e6i		−	1.09953i	0.835319	−	0.549766i	$-0.185283\pi$
$278$		0			0
$279$			369856.i			0.284461i
$280$		0			0
$281$			574517.i			0.434048i	0.976166	+	0.217024i	$0.0696349\pi$
$281$			574517.i			0.434048i	−0.976166	+	0.217024i	$0.930365\pi$
$282$		0			0
$283$	−1.44945e6			−1.07582			−0.537908	−	0.843004i	$-0.680785\pi$
$283$	−1.44945e6			−1.07582			−0.537908	+	0.843004i	$0.680785\pi$
$284$		0			0
$285$			1.97944e6i			1.44354i
$286$		0			0
$287$			455804.i			0.326643i
$288$		0			0
$289$	1.30792e6			0.921162
$290$		0			0
$291$			1.36471e6i			0.944734i
$292$		0			0
$293$		−	1.28608e6i		−	0.875185i	−0.899173	−	0.437593i	$-0.855831\pi$
$293$		−	1.28608e6i		−	0.875185i	0.899173	−	0.437593i	$-0.144169\pi$
$294$		0			0
$295$		−	3.66529e6i		−	2.45219i
$296$		0			0
$297$	125209.	+	264409.i	0.0823650	+	0.173934i
$298$		0			0
$299$	−40152.2			−0.0259736
$300$		0			0
$301$	−866565.			−0.551296
$302$		0			0
$303$	−917938.			−0.574390
$304$		0			0
$305$			3.68144e6i			2.26605i
$306$		0			0
$307$	−1.42091e6			−0.860439			−0.430219	−	0.902724i	$-0.641564\pi$
$307$	−1.42091e6			−0.860439			−0.430219	+	0.902724i	$0.641564\pi$
$308$		0			0
$309$	−522319.			−0.311200
$310$		0			0
$311$		−	604784.i		−	0.354568i	−0.984160	−	0.177284i	$-0.943269\pi$
$311$		−	604784.i		−	0.354568i	0.984160	−	0.177284i	$-0.0567311\pi$
$312$		0			0
$313$	−2.37715e6			−1.37150			−0.685749	−	0.727838i	$-0.740525\pi$
$313$	−2.37715e6			−1.37150			−0.685749	+	0.727838i	$0.740525\pi$
$314$		0			0
$315$	674696.			0.383117
$316$		0			0
$317$	1.71817e6			0.960323			0.480161	−	0.877180i	$-0.340578\pi$
$317$	1.71817e6			0.960323			0.480161	+	0.877180i	$0.340578\pi$
$318$		0			0
$319$	−475790.	−	1.00475e6i	−0.261782	−	0.552816i
$320$		0			0
$321$			979505.i			0.530572i
$322$		0			0
$323$		−	871038.i		−	0.464548i
$324$		0			0
$325$		−	112664.i		−	0.0591668i
$326$		0			0
$327$	−895605.			−0.463177
$328$		0			0
$329$			61317.6i			0.0312317i
$330$		0			0
$331$		−	800874.i		−	0.401786i	−0.979613	−	0.200893i	$-0.935616\pi$
$331$		−	800874.i		−	0.401786i	0.979613	−	0.200893i	$-0.0643843\pi$
$332$		0			0
$333$	411729.			0.203470
$334$		0			0
$335$			1.70419e6i			0.829672i
$336$		0			0
$337$			710583.i			0.340832i	0.985372	+	0.170416i	$0.0545111\pi$
$337$			710583.i			0.340832i	−0.985372	+	0.170416i	$0.945489\pi$
$338$		0			0
$339$		−	733829.i		−	0.346813i
$340$		0			0
$341$	784250.	+	1.65614e6i	0.365232	+	0.771277i
$342$		0			0
$343$	−2.35575e6			−1.08117
$344$		0			0
$345$	−1.08709e6			−0.491720
$346$		0			0
$347$	−1.21598e6			−0.542129			−0.271065	−	0.962561i	$-0.587376\pi$
$347$	−1.21598e6			−0.542129			−0.271065	+	0.962561i	$0.587376\pi$
$348$		0			0
$349$		−	3.12935e6i		−	1.37528i	−0.726052	−	0.687640i	$-0.758647\pi$
$349$		−	3.12935e6i		−	1.37528i	0.726052	−	0.687640i	$-0.241353\pi$
$350$		0			0
$351$	−20472.4			−0.00886952
$352$		0			0
$353$	−4.32937e6			−1.84922			−0.924609	−	0.380919i	$-0.875608\pi$
$353$	−4.32937e6			−1.84922			−0.924609	+	0.380919i	$0.875608\pi$
$354$		0			0
$355$		−	2.00085e6i		−	0.842645i
$356$		0			0
$357$	−296895.			−0.123291
$358$		0			0
$359$	2.87405e6			1.17695			0.588476	−	0.808515i	$-0.299728\pi$
$359$	2.87405e6			1.17695			0.588476	+	0.808515i	$0.299728\pi$
$360$		0			0
$361$	4.30175e6			1.73731
$362$		0			0
$363$	1.12131e6	+	918470.i	0.446643	+	0.365846i
$364$		0			0
$365$			6.58583e6i			2.58749i
$366$		0			0
$367$			1.85121e6i			0.717450i	0.933443	+	0.358725i	$0.116788\pi$
$367$			1.85121e6i			0.717450i	−0.933443	+	0.358725i	$0.883212\pi$
$368$		0			0
$369$		−	374450.i		−	0.143162i
$370$		0			0
$371$	857867.			0.323583
$372$		0			0
$373$		−	34313.3i		−	0.0127700i	−0.999980	−	0.00638498i	$-0.997968\pi$
$373$		−	34313.3i		−	0.0127700i	0.999980	−	0.00638498i	$-0.00203242\pi$
$374$		0			0
$375$		−	674298.i		−	0.247613i
$376$		0			0
$377$	77794.7			0.0281901
$378$		0			0
$379$			2.57547e6i			0.920997i	0.887661	+	0.460498i	$0.152329\pi$
$379$			2.57547e6i			0.920997i	−0.887661	+	0.460498i	$0.847671\pi$
$380$		0			0
$381$		−	23934.3i		−	0.00844711i
$382$		0			0
$383$			709425.i			0.247121i	0.992337	+	0.123561i	$0.0394313\pi$
$383$			709425.i			0.247121i	−0.992337	+	0.123561i	$0.960569\pi$
$384$		0			0
$385$	3.02114e6	−	1.43064e6i	1.03877	−	0.491901i
$386$		0			0
$387$	711897.			0.241624
$388$		0			0
$389$	−1.49858e6			−0.502119			−0.251060	−	0.967972i	$-0.580779\pi$
$389$	−1.49858e6			−0.502119			−0.251060	+	0.967972i	$0.580779\pi$
$390$		0			0
$391$	478366.			0.158241
$392$		0			0
$393$		−	504963.i		−	0.164922i
$394$		0			0
$395$	−3.16942e6			−1.02209
$396$		0			0
$397$	−4.06788e6			−1.29537			−0.647683	−	0.761910i	$-0.724262\pi$
$397$	−4.06788e6			−1.29537			−0.647683	+	0.761910i	$0.724262\pi$
$398$		0			0
$399$		−	2.31024e6i		−	0.726483i
$400$		0			0
$401$	−2.56858e6			−0.797688			−0.398844	−	0.917019i	$-0.630588\pi$
$401$	−2.56858e6			−0.797688			−0.398844	+	0.917019i	$0.630588\pi$
$402$		0			0
$403$	−128230.			−0.0393302
$404$		0			0
$405$	−554273.			−0.167914
$406$		0			0
$407$	1.84363e6	−	873037.i	0.551682	−	0.261244i
$408$		0			0
$409$		−	3.95418e6i		−	1.16882i	−0.811458	−	0.584411i	$-0.801326\pi$
$409$		−	3.95418e6i		−	1.16882i	0.811458	−	0.584411i	$-0.198674\pi$
$410$		0			0
$411$		−	1.12714e6i		−	0.329134i
$412$		0			0
$413$			4.27784e6i			1.23410i
$414$		0			0
$415$	−7.20727e6			−2.05424
$416$		0			0
$417$			1.54675e6i			0.435593i
$418$		0			0
$419$		−	568740.i		−	0.158263i	−0.996864	−	0.0791314i	$-0.974785\pi$
$419$		−	568740.i		−	0.158263i	0.996864	−	0.0791314i	$-0.0252147\pi$
$420$		0			0
$421$	5.11971e6			1.40780			0.703899	−	0.710300i	$-0.251441\pi$
$421$	5.11971e6			1.40780			0.703899	+	0.710300i	$0.251441\pi$
$422$		0			0
$423$		−	50373.4i		−	0.0136883i
$424$		0			0
$425$			1.34226e6i			0.360466i
$426$		0			0
$427$		−	4.29669e6i		−	1.14042i
$428$		0			0
$429$	−91670.8	+	43409.9i	−0.0240485	+	0.0113880i
$430$		0			0
$431$	1.35850e6			0.352262			0.176131	−	0.984367i	$-0.443642\pi$
$431$	1.35850e6			0.352262			0.176131	+	0.984367i	$0.443642\pi$
$432$		0			0
$433$	5.78095e6			1.48177			0.740883	−	0.671634i	$-0.234407\pi$
$433$	5.78095e6			1.48177			0.740883	+	0.671634i	$0.234407\pi$
$434$		0			0
$435$	2.10623e6			0.533682
$436$		0			0
$437$			3.72233e6i			0.932421i
$438$		0			0
$439$	−6.45438e6			−1.59843			−0.799214	−	0.601047i	$-0.794751\pi$
$439$	−6.45438e6			−1.59843			−0.799214	+	0.601047i	$0.794751\pi$
$440$		0			0
$441$	573916.			0.140524
$442$		0			0
$443$		−	181260.i		−	0.0438827i	−0.999759	−	0.0219413i	$-0.993015\pi$
$443$		−	181260.i		−	0.0438827i	0.999759	−	0.0219413i	$-0.00698471\pi$
$444$		0			0
$445$	−4.08527e6			−0.977959
$446$		0			0
$447$	−1.04469e6			−0.247296
$448$		0			0
$449$	−2.72343e6			−0.637529			−0.318764	−	0.947834i	$-0.603268\pi$
$449$	−2.72343e6			−0.637529			−0.318764	+	0.947834i	$0.603268\pi$
$450$		0			0
$451$	−793991.	−	1.67671e6i	−0.183812	−	0.388164i
$452$		0			0
$453$		−	443652.i		−	0.101577i
$454$		0			0
$455$			233918.i			0.0529706i
$456$		0			0
$457$		−	8.23680e6i		−	1.84488i	−0.386141	−	0.922440i	$-0.626192\pi$
$457$		−	8.23680e6i		−	1.84488i	0.386141	−	0.922440i	$-0.373808\pi$
$458$		0			0
$459$	243904.			0.0540365
$460$		0			0
$461$			5.98166e6i			1.31090i	0.755239	+	0.655450i	$0.227521\pi$
$461$			5.98166e6i			1.31090i	−0.755239	+	0.655450i	$0.772479\pi$
$462$		0			0
$463$		−	170486.i		−	0.0369603i	−0.999829	−	0.0184801i	$-0.994117\pi$
$463$		−	170486.i		−	0.0369603i	0.999829	−	0.0184801i	$-0.00588275\pi$
$464$		0			0
$465$	−3.47172e6			−0.744581
$466$		0			0
$467$			3.17902e6i			0.674529i	0.941410	+	0.337265i	$0.109502\pi$
$467$			3.17902e6i			0.674529i	−0.941410	+	0.337265i	$0.890498\pi$
$468$		0			0
$469$		−	1.98900e6i		−	0.417544i
$470$		0			0
$471$			463122.i			0.0961930i
$472$		0			0
$473$	3.18772e6	−	1.50952e6i	0.655129	−	0.310231i
$474$		0			0
$475$	−1.04446e7			−2.12402
$476$		0			0
$477$	−704751.			−0.141821
$478$		0			0
$479$	−2.28965e6			−0.455965			−0.227982	−	0.973665i	$-0.573213\pi$
$479$	−2.28965e6			−0.455965			−0.227982	+	0.973665i	$0.573213\pi$
$480$		0			0
$481$			142747.i			0.0281322i
$482$		0			0
$483$	1.26877e6			0.247465
$484$		0			0
$485$	−1.28101e7			−2.47286
$486$		0			0
$487$		−	6.22855e6i		−	1.19005i	−0.803708	−	0.595024i	$-0.797142\pi$
$487$		−	6.22855e6i		−	1.19005i	0.803708	−	0.595024i	$-0.202858\pi$
$488$		0			0
$489$	−1.29913e6			−0.245687
$490$		0			0
$491$	−6.45165e6			−1.20772			−0.603861	−	0.797090i	$-0.706372\pi$
$491$	−6.45165e6			−1.20772			−0.603861	+	0.797090i	$0.706372\pi$
$492$		0			0
$493$	−926831.			−0.171745
$494$		0			0
$495$	−2.48191e6	+	1.17529e6i	−0.455275	+	0.215592i
$496$		0			0
$497$			2.33524e6i			0.424073i
$498$		0			0
$499$			7.08566e6i			1.27388i	0.770913	+	0.636941i	$0.219800\pi$
$499$			7.08566e6i			1.27388i	−0.770913	+	0.636941i	$0.780200\pi$
$500$		0			0
$501$			5.86700e6i			1.04429i
$502$		0			0
$503$	3.01138e6			0.530696			0.265348	−	0.964153i	$-0.414513\pi$
$503$	3.01138e6			0.530696			0.265348	+	0.964153i	$0.414513\pi$
$504$		0			0
$505$		−	8.61637e6i		−	1.50347i
$506$		0			0
$507$			3.33454e6i			0.576124i
$508$		0			0
$509$	2.19143e6			0.374916			0.187458	−	0.982273i	$-0.439975\pi$
$509$	2.19143e6			0.374916			0.187458	+	0.982273i	$0.439975\pi$
$510$		0			0
$511$		−	7.68645e6i		−	1.30219i
$512$		0			0
$513$			1.89790e6i			0.318405i
$514$		0			0
$515$		−	4.90283e6i		−	0.814571i
$516$		0			0
$517$	−106813.	−	225561.i	−0.0175750	−	0.0371140i
$518$		0			0
$519$	−2.90314e6			−0.473097
$520$		0			0
$521$	5.73872e6			0.926235			0.463117	−	0.886297i	$-0.346731\pi$
$521$	5.73872e6			0.926235			0.463117	+	0.886297i	$0.346731\pi$
$522$		0			0
$523$	−4.59132e6			−0.733979			−0.366989	−	0.930225i	$-0.619612\pi$
$523$	−4.59132e6			−0.733979			−0.366989	+	0.930225i	$0.619612\pi$
$524$		0			0
$525$			3.56006e6i			0.563715i
$526$		0			0
$527$	1.52770e6			0.239614
$528$		0			0
$529$	4.39207e6			0.682386
$530$		0			0
$531$		−	3.51431e6i		−	0.540884i
$532$		0			0
$533$	129822.			0.0197939
$534$		0			0
$535$	−9.19428e6			−1.38878
$536$		0			0
$537$	5.65285e6			0.845925
$538$		0			0
$539$	2.56987e6	−	1.21694e6i	0.381013	−	0.180425i
$540$		0			0
$541$			2.37417e6i			0.348753i	0.984679	+	0.174377i	$0.0557910\pi$
$541$			2.37417e6i			0.348753i	−0.984679	+	0.174377i	$0.944209\pi$
$542$		0			0
$543$			1.05300e6i			0.153260i
$544$		0			0
$545$		−	8.40674e6i		−	1.21237i
$546$		0			0
$547$	7.13123e6			1.01905			0.509526	−	0.860455i	$-0.329821\pi$
$547$	7.13123e6			1.01905			0.509526	+	0.860455i	$0.329821\pi$
$548$		0			0
$549$			3.52980e6i			0.499826i
$550$		0			0
$551$		−	7.21200e6i		−	1.01199i
$552$		0			0
$553$	3.69910e6			0.514379
$554$		0			0
$555$			3.86476e6i			0.532587i
$556$		0			0
$557$			1.00494e7i			1.37246i	0.727384	+	0.686231i	$0.240736\pi$
$557$			1.00494e7i			1.37246i	−0.727384	+	0.686231i	$0.759264\pi$
$558$		0			0
$559$			246815.i			0.0334074i
$560$		0			0
$561$	1.09215e6	−	517178.i	0.146513	−	0.0693798i
$562$		0			0
$563$	7.19051e6			0.956067			0.478034	−	0.878342i	$-0.341350\pi$
$563$	7.19051e6			0.956067			0.478034	+	0.878342i	$0.341350\pi$
$564$		0			0
$565$	6.88820e6			0.907789
$566$		0			0
$567$	646903.			0.0845049
$568$		0			0
$569$		−	8.91977e6i		−	1.15498i	−0.816399	−	0.577488i	$-0.804033\pi$
$569$		−	8.91977e6i		−	1.15498i	0.816399	−	0.577488i	$-0.195967\pi$
$570$		0			0
$571$	1.24566e7			1.59886			0.799429	−	0.600760i	$-0.205135\pi$
$571$	1.24566e7			1.59886			0.799429	+	0.600760i	$0.205135\pi$
$572$		0			0
$573$	−7.12596e6			−0.906686
$574$		0			0
$575$		−	5.73608e6i		−	0.723512i
$576$		0			0
$577$	1.94920e6			0.243734			0.121867	−	0.992546i	$-0.461112\pi$
$577$	1.94920e6			0.243734			0.121867	+	0.992546i	$0.461112\pi$
$578$		0			0
$579$	4.71418e6			0.584399
$580$		0			0
$581$	8.41175e6			1.03382
$582$		0			0
$583$	−3.15572e6	+	1.49437e6i	−0.384528	+	0.182090i
$584$		0			0
$585$		−	192167.i		−	0.0232161i
$586$		0			0
$587$			4.19559e6i			0.502572i	0.967913	+	0.251286i	$0.0808534\pi$
$587$			4.19559e6i			0.502572i	−0.967913	+	0.251286i	$0.919147\pi$
$588$		0			0
$589$			1.18876e7i			1.41191i
$590$		0			0
$591$	9.69506e6			1.14178
$592$		0			0
$593$		−	1.52096e7i		−	1.77616i	−0.459689	−	0.888080i	$-0.652039\pi$
$593$		−	1.52096e7i		−	1.77616i	0.459689	−	0.888080i	$-0.347961\pi$
$594$		0			0
$595$		−	2.78685e6i		−	0.322717i
$596$		0			0
$597$	−2.58817e6			−0.297205
$598$		0			0
$599$			1.69447e7i			1.92960i	0.262986	+	0.964800i	$0.415293\pi$
$599$			1.69447e7i			1.92960i	−0.262986	+	0.964800i	$0.584707\pi$
$600$		0			0
$601$		−	1.71737e7i		−	1.93944i	−0.244216	−	0.969721i	$-0.578531\pi$
$601$		−	1.71737e7i		−	1.93944i	0.244216	−	0.969721i	$-0.421469\pi$
$602$		0			0
$603$			1.63399e6i			0.183002i
$604$		0			0
$605$	−8.62137e6	+	1.05254e7i	−0.957608	+	1.16909i
$606$		0			0
$607$	1.01635e7			1.11962			0.559809	−	0.828622i	$-0.310875\pi$
$607$	1.01635e7			1.11962			0.559809	+	0.828622i	$0.310875\pi$
$608$		0			0
$609$	−2.45822e6			−0.268583
$610$		0			0
$611$	17464.5			0.00189258
$612$		0			0
$613$		−	6.28709e6i		−	0.675770i	−0.941187	−	0.337885i	$-0.890289\pi$
$613$		−	6.28709e6i		−	0.675770i	0.941187	−	0.337885i	$-0.109711\pi$
$614$		0			0
$615$	3.51484e6			0.374729
$616$		0			0
$617$	5.24976e6			0.555171			0.277586	−	0.960701i	$-0.410466\pi$
$617$	5.24976e6			0.555171			0.277586	+	0.960701i	$0.410466\pi$
$618$		0			0
$619$			8.10161e6i			0.849855i	0.905227	+	0.424927i	$0.139700\pi$
$619$			8.10161e6i			0.849855i	−0.905227	+	0.424927i	$0.860300\pi$
$620$		0			0
$621$	−1.04231e6			−0.108460
$622$		0			0
$623$	4.76800e6			0.492171
$624$		0			0
$625$	−6.20766e6			−0.635664
$626$		0			0
$627$	4.02434e6	+	8.49839e6i	0.408814	+	0.863312i
$628$		0			0
$629$		−	1.70066e6i		−	0.171392i
$630$		0			0
$631$			1.91990e7i			1.91958i	0.280725	+	0.959788i	$0.409425\pi$
$631$			1.91990e7i			1.91958i	−0.280725	+	0.959788i	$0.590575\pi$
$632$		0			0
$633$			1.00977e7i			1.00164i
$634$		0			0
$635$	224663.			0.0221104
$636$		0			0
$637$			198977.i			0.0194292i
$638$		0			0
$639$		−	1.91843e6i		−	0.185864i
$640$		0			0
$641$	4.20853e6			0.404563			0.202281	−	0.979327i	$-0.435164\pi$
$641$	4.20853e6			0.404563			0.202281	+	0.979327i	$0.435164\pi$
$642$		0			0
$643$		−	1.11859e7i		−	1.06695i	−0.845816	−	0.533475i	$-0.820886\pi$
$643$		−	1.11859e7i		−	1.06695i	0.845816	−	0.533475i	$-0.179114\pi$
$644$		0			0
$645$			6.68233e6i			0.632454i
$646$		0			0
$647$			6.02783e6i			0.566109i	0.959104	+	0.283055i	$0.0913477\pi$
$647$			6.02783e6i			0.566109i	−0.959104	+	0.283055i	$0.908652\pi$
$648$		0			0
$649$	−7.45180e6	−	1.57363e7i	−0.694464	−	1.46653i
$650$		0			0
$651$	4.05191e6			0.374720
$652$		0			0
$653$	−1.49553e7			−1.37250			−0.686248	−	0.727367i	$-0.740744\pi$
$653$	−1.49553e7			−1.37250			−0.686248	+	0.727367i	$0.740744\pi$
$654$		0			0
$655$	4.73992e6			0.431686
$656$		0			0
$657$			6.31454e6i			0.570727i
$658$		0			0
$659$	1.76171e7			1.58023			0.790115	−	0.612958i	$-0.210021\pi$
$659$	1.76171e7			1.58023			0.790115	+	0.612958i	$0.210021\pi$
$660$		0			0
$661$	−2.67167e6			−0.237837			−0.118918	−	0.992904i	$-0.537943\pi$
$661$	−2.67167e6			−0.237837			−0.118918	+	0.992904i	$0.537943\pi$
$662$		0			0
$663$			84561.8i			0.00747120i
$664$		0			0
$665$	2.16855e7			1.90158
$666$		0			0
$667$	3.96076e6			0.344718
$668$		0			0
$669$	−7.30435e6			−0.630981
$670$		0			0
$671$	7.48464e6	+	1.58057e7i	0.641748	+	1.35521i
$672$		0			0
$673$		−	1.68940e7i		−	1.43779i	−0.695120	−	0.718893i	$-0.744649\pi$
$673$		−	1.68940e7i		−	1.43779i	0.695120	−	0.718893i	$-0.255351\pi$
$674$		0			0
$675$		−	2.92465e6i		−	0.247067i
$676$		0			0
$677$		−	1.98219e7i		−	1.66216i	−0.556149	−	0.831082i	$-0.687722\pi$
$677$		−	1.98219e7i		−	1.66216i	0.556149	−	0.831082i	$-0.312278\pi$
$678$		0			0
$679$	1.49509e7			1.24450
$680$		0			0
$681$		−	5.69965e6i		−	0.470956i
$682$		0			0
$683$			1.43898e6i			0.118033i	0.998257	+	0.0590165i	$0.0187964\pi$
$683$			1.43898e6i			0.118033i	−0.998257	+	0.0590165i	$0.981204\pi$
$684$		0			0
$685$	1.05801e7			0.861513
$686$		0			0
$687$			6.61332e6i			0.534598i
$688$		0			0
$689$		−	244338.i		−	0.0196084i
$690$		0			0
$691$		−	1.36294e7i		−	1.08588i	−0.839772	−	0.542939i	$-0.817311\pi$
$691$		−	1.36294e7i		−	1.08588i	0.839772	−	0.542939i	$-0.182689\pi$
$692$		0			0
$693$	2.89669e6	−	1.37170e6i	0.229123	−	0.108499i
$694$		0			0
$695$	−1.45188e7			−1.14017
$696$		0			0
$697$	−1.54668e6			−0.120592
$698$		0			0
$699$	1.10988e7			0.859179
$700$		0			0
$701$		−	8.89409e6i		−	0.683607i	−0.939771	−	0.341804i	$-0.888962\pi$
$701$		−	8.89409e6i		−	0.683607i	0.939771	−	0.341804i	$-0.111038\pi$
$702$		0			0
$703$	1.32334e7			1.00991
$704$		0			0
$705$	472838.			0.0358294
$706$		0			0
$707$			1.00563e7i			0.756644i
$708$		0			0
$709$	1.32807e7			0.992214			0.496107	−	0.868261i	$-0.334762\pi$
$709$	1.32807e7			0.992214			0.496107	+	0.868261i	$0.334762\pi$
$710$		0			0
$711$	−3.03887e6			−0.225443
$712$		0			0
$713$	−6.52856e6			−0.480943
$714$		0			0
$715$	−407474.	−	860483.i	−0.0298082	−	0.0629473i
$716$		0			0
$717$			2.39159e6i			0.173736i
$718$		0			0
$719$		−	1.04490e7i		−	0.753791i	−0.926256	−	0.376895i	$-0.876992\pi$
$719$		−	1.04490e7i		−	0.753791i	0.926256	−	0.376895i	$-0.123008\pi$
$720$		0			0
$721$			5.72220e6i			0.409944i
$722$		0			0
$723$	−1.21004e6			−0.0860900
$724$		0			0
$725$			1.11136e7i			0.785254i
$726$		0			0
$727$		−	7.05963e6i		−	0.495388i	−0.968838	−	0.247694i	$-0.920327\pi$
$727$		−	7.05963e6i		−	0.495388i	0.968838	−	0.247694i	$-0.0796728\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$		−	2.94051e6i		−	0.203531i
$732$		0			0
$733$			1.87940e7i			1.29199i	0.763341	+	0.645996i	$0.223558\pi$
$733$			1.87940e7i			1.29199i	−0.763341	+	0.645996i	$0.776442\pi$
$734$		0			0
$735$			5.38715e6i			0.367825i
$736$		0			0
$737$	3.46474e6	+	7.31666e6i	0.234965	+	0.496186i
$738$		0			0
$739$	−7.93053e6			−0.534184			−0.267092	−	0.963671i	$-0.586063\pi$
$739$	−7.93053e6			−0.534184			−0.267092	+	0.963671i	$0.586063\pi$
$740$		0			0
$741$	−658005.			−0.0440234
$742$		0			0
$743$	−3.88178e6			−0.257964			−0.128982	−	0.991647i	$-0.541171\pi$
$743$	−3.88178e6			−0.257964			−0.128982	+	0.991647i	$0.541171\pi$
$744$		0			0
$745$		−	9.80612e6i		−	0.647301i
$746$		0			0
$747$	−6.91038e6			−0.453107
$748$		0			0
$749$	1.07308e7			0.698922
$750$		0			0
$751$			1.21003e7i			0.782880i	0.920204	+	0.391440i	$0.128023\pi$
$751$			1.21003e7i			0.782880i	−0.920204	+	0.391440i	$0.871977\pi$
$752$		0			0
$753$	1.45734e7			0.936642
$754$		0			0
$755$	4.16441e6			0.265880
$756$		0			0
$757$	−2.03914e7			−1.29332			−0.646661	−	0.762777i	$-0.723835\pi$
$757$	−2.03914e7			−1.29332			−0.646661	+	0.762777i	$0.723835\pi$
$758$		0			0
$759$	−4.66724e6	+	2.21013e6i	−0.294073	+	0.139256i
$760$		0			0
$761$			1.83726e7i			1.15003i	0.818144	+	0.575014i	$0.195003\pi$
$761$			1.83726e7i			1.15003i	−0.818144	+	0.575014i	$0.804997\pi$
$762$		0			0
$763$			9.81168e6i			0.610143i
$764$		0			0
$765$			2.28944e6i			0.141441i
$766$		0			0
$767$	1.21842e6			0.0747837
$768$		0			0
$769$			2.92187e6i			0.178174i	0.996024	+	0.0890872i	$0.0283950\pi$
$769$			2.92187e6i			0.178174i	−0.996024	+	0.0890872i	$0.971605\pi$
$770$		0			0
$771$		−	4.18808e6i		−	0.253734i
$772$		0			0
$773$	−9.10094e6			−0.547820			−0.273910	−	0.961755i	$-0.588317\pi$
$773$	−9.10094e6			−0.547820			−0.273910	+	0.961755i	$0.588317\pi$
$774$		0			0
$775$		−	1.83187e7i		−	1.09557i
$776$		0			0
$777$		−	4.51064e6i		−	0.268031i
$778$		0			0
$779$		−	1.20352e7i		−	0.710577i
$780$		0			0
$781$	−4.06788e6	−	8.59033e6i	−0.238639	−	0.503944i
$782$		0			0
$783$	2.01947e6			0.117715
$784$		0			0
$785$	−4.34717e6			−0.251787
$786$		0			0
$787$	3.63720e6			0.209329			0.104665	−	0.994508i	$-0.466623\pi$
$787$	3.63720e6			0.209329			0.104665	+	0.994508i	$0.466623\pi$
$788$		0			0
$789$		−	2.28544e6i		−	0.130701i
$790$		0			0
$791$	−8.03936e6			−0.456857
$792$		0			0
$793$	−1.22378e6			−0.0691070
$794$		0			0
$795$		−	6.61526e6i		−	0.371218i
$796$		0			0
$797$	1.89379e7			1.05605			0.528026	−	0.849228i	$-0.322932\pi$
$797$	1.89379e7			1.05605			0.528026	+	0.849228i	$0.322932\pi$
$798$		0			0
$799$	−208069.			−0.0115303
$800$		0			0
$801$	−3.91699e6			−0.215710
$802$		0			0
$803$	1.33895e7	+	2.82751e7i	0.732781	+	1.54745i
$804$		0			0
$805$			1.19095e7i			0.647743i
$806$		0			0
$807$		−	1.71645e7i		−	0.927785i
$808$		0			0
$809$		−	7.54191e6i		−	0.405145i	−0.979267	−	0.202572i	$-0.935070\pi$
$809$		−	7.54191e6i		−	0.405145i	0.979267	−	0.202572i	$-0.0649301\pi$
$810$		0			0
$811$	−1.24138e7			−0.662755			−0.331377	−	0.943498i	$-0.607513\pi$
$811$	−1.24138e7			−0.662755			−0.331377	+	0.943498i	$0.607513\pi$
$812$		0			0
$813$		−	4.15961e6i		−	0.220712i
$814$		0			0
$815$		−	1.21945e7i		−	0.643089i
$816$		0			0
$817$	2.28812e7			1.19929
$818$		0			0
$819$			224282.i			0.0116838i
$820$		0			0
$821$		−	6.06178e6i		−	0.313865i	−0.987609	−	0.156932i	$-0.949840\pi$
$821$		−	6.06178e6i		−	0.313865i	0.987609	−	0.156932i	$-0.0501605\pi$
$822$		0			0
$823$			1.63422e7i			0.841031i	0.907285	+	0.420516i	$0.138151\pi$
$823$			1.63422e7i			0.841031i	−0.907285	+	0.420516i	$0.861849\pi$
$824$		0			0
$825$	−6.20147e6	−	1.30959e7i	−0.317220	−	0.669887i
$826$		0			0
$827$	2.43775e7			1.23944			0.619721	−	0.784822i	$-0.287246\pi$
$827$	2.43775e7			1.23944			0.619721	+	0.784822i	$0.287246\pi$
$828$		0			0
$829$	3.23096e7			1.63285			0.816423	−	0.577454i	$-0.195954\pi$
$829$	3.23096e7			1.63285			0.816423	+	0.577454i	$0.195954\pi$
$830$		0			0
$831$	1.26372e7			0.634815
$832$		0			0
$833$		−	2.37058e6i		−	0.118370i
$834$		0			0
$835$	−5.50715e7			−2.73345
$836$		0			0
$837$	−3.32871e6			−0.164234
$838$		0			0
$839$		−	3.17138e7i		−	1.55541i	−0.628632	−	0.777703i	$-0.716385\pi$
$839$		−	3.17138e7i		−	1.55541i	0.628632	−	0.777703i	$-0.283615\pi$
$840$		0			0
$841$	1.28372e7			0.625865
$842$		0			0
$843$	−5.17066e6			−0.250597
$844$		0			0
$845$	−3.13002e7			−1.50801
$846$		0			0
$847$	1.00622e7	−	1.22844e7i	0.481929	−	0.588363i
$848$		0			0
$849$		−	1.30451e7i		−	0.621123i
$850$		0			0
$851$			7.26768e6i			0.344011i
$852$		0			0
$853$		−	3.32149e7i		−	1.56301i	−0.623901	−	0.781503i	$-0.714453\pi$
$853$		−	3.32149e7i		−	1.56301i	0.623901	−	0.781503i	$-0.285547\pi$
$854$		0			0
$855$	−1.78150e7			−0.833431
$856$		0			0
$857$			6.54933e6i			0.304610i	0.988334	+	0.152305i	$0.0486697\pi$
$857$			6.54933e6i			0.304610i	−0.988334	+	0.152305i	$0.951330\pi$
$858$		0			0
$859$			1.99695e7i			0.923386i	0.887040	+	0.461693i	$0.152758\pi$
$859$			1.99695e7i			0.923386i	−0.887040	+	0.461693i	$0.847242\pi$
$860$		0			0
$861$	−4.10224e6			−0.188587
$862$		0			0
$863$		−	2.90556e7i		−	1.32801i	−0.747727	−	0.664006i	$-0.768855\pi$
$863$		−	2.90556e7i		−	1.32801i	0.747727	−	0.664006i	$-0.231145\pi$
$864$		0			0
$865$		−	2.72508e7i		−	1.23834i
$866$		0			0
$867$			1.17713e7i			0.531833i
$868$		0			0
$869$	−1.36074e7	+	6.44367e6i	−0.611259	+	0.289457i
$870$		0			0
$871$	−566507.			−0.0253023
$872$		0			0
$873$	−1.22824e7			−0.545442
$874$		0			0
$875$	−7.38718e6			−0.326181
$876$		0			0
$877$			1.06559e7i			0.467835i	0.972257	+	0.233917i	$0.0751545\pi$
$877$			1.06559e7i			0.467835i	−0.972257	+	0.233917i	$0.924846\pi$
$878$		0			0
$879$	1.15747e7			0.505288
$880$		0			0
$881$	−2.31416e7			−1.00451			−0.502254	−	0.864720i	$-0.667496\pi$
$881$	−2.31416e7			−1.00451			−0.502254	+	0.864720i	$0.667496\pi$
$882$		0			0
$883$			4.23206e7i			1.82663i	0.407256	+	0.913314i	$0.366486\pi$
$883$			4.23206e7i			1.82663i	−0.407256	+	0.913314i	$0.633514\pi$
$884$		0			0
$885$	3.29876e7			1.41577
$886$		0			0
$887$	1.97319e7			0.842094			0.421047	−	0.907039i	$-0.361663\pi$
$887$	1.97319e7			0.842094			0.421047	+	0.907039i	$0.361663\pi$
$888$		0			0
$889$	−262209.			−0.0111274
$890$		0			0
$891$	−2.37968e6	+	1.12688e6i	−0.100421	+	0.0475535i
$892$		0			0
$893$		−	1.61906e6i		−	0.0679412i
$894$		0			0
$895$			5.30614e7i			2.21422i
$896$		0			0
$897$		−	361370.i		−	0.0149959i
$898$		0			0
$899$	1.26490e7			0.521985
$900$		0			0
$901$			2.91100e6i			0.119462i
$902$		0			0
$903$		−	7.79908e6i		−	0.318291i
$904$		0			0
$905$	−9.88413e6			−0.401159
$906$		0			0
$907$		−	4.67471e7i		−	1.88685i	−0.331592	−	0.943423i	$-0.607586\pi$
$907$		−	4.67471e7i		−	1.88685i	0.331592	−	0.943423i	$-0.392414\pi$
$908$		0			0
$909$		−	8.26144e6i		−	0.331624i
$910$		0			0
$911$			3.69357e7i			1.47452i	0.675609	+	0.737260i	$0.263881\pi$
$911$			3.69357e7i			1.47452i	−0.675609	+	0.737260i	$0.736119\pi$
$912$		0			0
$913$	−3.09432e7	+	1.46529e7i	−1.22854	+	0.581763i
$914$		0			0
$915$	−3.31330e7			−1.30830
$916$		0			0
$917$	−5.53205e6			−0.217252
$918$		0			0
$919$	7.61041e6			0.297248			0.148624	−	0.988894i	$-0.452516\pi$
$919$	7.61041e6			0.297248			0.148624	+	0.988894i	$0.452516\pi$
$920$		0			0
$921$		−	1.27882e7i		−	0.496774i
$922$		0			0
$923$	665123.			0.0256979
$924$		0			0
$925$	−2.03926e7			−0.783643
$926$		0			0
$927$		−	4.70087e6i		−	0.179672i
$928$		0			0
$929$	4.33447e7			1.64777			0.823885	−	0.566757i	$-0.191802\pi$
$929$	4.33447e7			1.64777			0.823885	+	0.566757i	$0.191802\pi$
$930$		0			0
$931$	1.84463e7			0.697485
$932$		0			0
$933$	5.44306e6			0.204710
$934$		0			0
$935$	4.85457e6	+	1.02516e7i	0.181603	+	0.383499i
$936$		0			0
$937$		−	4.41138e6i		−	0.164144i	−0.996626	−	0.0820721i	$-0.973846\pi$
$937$		−	4.41138e6i		−	0.164144i	0.996626	−	0.0820721i	$-0.0261538\pi$
$938$		0			0
$939$		−	2.13943e7i		−	0.791835i
$940$		0			0
$941$		−	912832.i		−	0.0336060i	−0.999859	−	0.0168030i	$-0.994651\pi$
$941$		−	912832.i		−	0.0336060i	0.999859	−	0.0168030i	$-0.00534881\pi$
$942$		0			0
$943$	6.60965e6			0.242047
$944$		0			0
$945$			6.07226e6i			0.221193i
$946$		0			0
$947$			2.58674e7i			0.937299i	0.883384	+	0.468650i	$0.155259\pi$
$947$			2.58674e7i			0.937299i	−0.883384	+	0.468650i	$0.844741\pi$
$948$		0			0
$949$	−2.18926e6			−0.0789099
$950$		0			0
$951$			1.54635e7i			0.554443i
$952$		0			0
$953$		−	2.15605e7i		−	0.768999i	−0.923125	−	0.384499i	$-0.874374\pi$
$953$		−	2.15605e7i		−	0.768999i	0.923125	−	0.384499i	$-0.125626\pi$
$954$		0			0
$955$		−	6.68890e7i		−	2.37327i
$956$		0			0
$957$	9.04274e6	−	4.28211e6i	0.319169	−	0.151140i
$958$		0			0
$959$	−1.23482e7			−0.433568
$960$		0			0
$961$	7.77963e6			0.271738
$962$		0			0
$963$	−8.81555e6			−0.306326
$964$		0			0
$965$			4.42504e7i			1.52967i
$966$		0			0
$967$	3.76768e7			1.29571			0.647856	−	0.761763i	$-0.275666\pi$
$967$	3.76768e7			1.29571			0.647856	+	0.761763i	$0.275666\pi$
$968$		0			0
$969$	7.83934e6			0.268207
$970$		0			0
$971$		−	2.26910e7i		−	0.772336i	−0.922428	−	0.386168i	$-0.873798\pi$
$971$		−	2.26910e7i		−	0.772336i	0.922428	−	0.386168i	$-0.126202\pi$
$972$		0			0
$973$	1.69452e7			0.573807
$974$		0			0
$975$	1.01398e6			0.0341600
$976$		0			0
$977$	−1.08035e7			−0.362099			−0.181050	−	0.983474i	$-0.557950\pi$
$977$	−1.08035e7			−0.362099			−0.181050	+	0.983474i	$0.557950\pi$
$978$		0			0
$979$	−1.75394e7	+	8.30565e6i	−0.584869	+	0.276960i
$980$		0			0
$981$		−	8.06044e6i		−	0.267415i
$982$		0			0
$983$		−	4.10042e6i		−	0.135346i	−0.997708	−	0.0676728i	$-0.978443\pi$
$983$		−	4.10042e6i		−	0.135346i	0.997708	−	0.0676728i	$-0.0215574\pi$
$984$		0			0
$985$			9.10043e7i			2.98862i
$986$		0			0
$987$	−551859.			−0.0180316
$988$		0			0
$989$			1.25661e7i			0.408517i
$990$		0			0
$991$		−	6.06756e6i		−	0.196259i	−0.995174	−	0.0981296i	$-0.968714\pi$
$991$		−	6.06756e6i		−	0.196259i	0.995174	−	0.0981296i	$-0.0312860\pi$
$992$		0			0
$993$	7.20787e6			0.231971
$994$		0			0
$995$		−	2.42942e7i		−	0.777939i
$996$		0			0
$997$			3.62350e7i			1.15449i	0.816571	+	0.577245i	$0.195872\pi$
$997$			3.62350e7i			1.15449i	−0.816571	+	0.577245i	$0.804128\pi$
$998$		0			0
$999$			3.70556e6i			0.117474i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.o.b.175.18	yes	40
4.3	odd	2	inner	528.6.o.b.175.23	yes	40
11.10	odd	2	inner	528.6.o.b.175.24	yes	40
44.43	even	2	inner	528.6.o.b.175.17	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.6.o.b.175.17	✓	40	44.43	even	2	inner
528.6.o.b.175.18	yes	40	1.1	even	1	trivial
528.6.o.b.175.23	yes	40	4.3	odd	2	inner
528.6.o.b.175.24	yes	40	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.00000i q^{3} -84.4799 q^{5} +98.5983 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q+9.00000i q^{3} -84.4799 q^{5} +98.5983 q^{7} -81.0000 q^{9} +(-362.700 + 171.754i) q^{11} -28.0828i q^{13} -760.320i q^{15} +334.573i q^{17} -2603.43 q^{19} +887.384i q^{21} -1429.78i q^{23} +4011.86 q^{25} -729.000i q^{27} +2770.19i q^{29} -4566.13i q^{31} +(-1545.78 - 3264.30i) q^{33} -8329.58 q^{35} -5083.07 q^{37} +252.745 q^{39} +4622.84i q^{41} -8788.85 q^{43} +6842.88 q^{45} +621.893i q^{47} -7085.38 q^{49} -3011.16 q^{51} +8700.63 q^{53} +(30640.9 - 14509.8i) q^{55} -23430.9i q^{57} +43386.5i q^{59} -43577.7i q^{61} -7986.46 q^{63} +2372.43i q^{65} -20172.7i q^{67} +12868.0 q^{69} +23684.4i q^{71} -77957.3i q^{73} +36106.8i q^{75} +(-35761.6 + 16934.6i) q^{77} +37516.9 q^{79} +6561.00 q^{81} +85313.3 q^{83} -28264.7i q^{85} -24931.7 q^{87} +48357.9 q^{89} -2768.92i q^{91} +41095.2 q^{93} +219938. q^{95} +151635. q^{97} +(29378.7 - 13912.1i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) 40 * q + 88 * q^5 - 3240 * q^9	\( 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) \) 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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