LMFDB - Embedded newform 600.6.f.k.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,6,Mod(49,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.49"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-324,0,128] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$96.2302918878$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{1489})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 745x^{2} + 138384 $ x^4 + 745*x^2 + 138384
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-19.7938i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.6.f.k.49.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +162.350i q^{7} -81.0000 q^{9} -585.401 q^{11} -1045.05i q^{13} +310.249i q^{17} -239.548 q^{19} +1461.15 q^{21} -4804.86i q^{23} +729.000i q^{27} +3969.21 q^{29} +1157.45 q^{31} +5268.61i q^{33} +7168.26i q^{37} -9405.46 q^{39} +17563.1 q^{41} -9789.99i q^{43} +15565.9i q^{47} -9550.60 q^{49} +2792.24 q^{51} +18151.3i q^{53} +2155.93i q^{57} -8148.61 q^{59} -55069.7 q^{61} -13150.4i q^{63} +61162.8i q^{67} -43243.7 q^{69} -67880.9 q^{71} +52131.1i q^{73} -95040.0i q^{77} +11749.9 q^{79} +6561.00 q^{81} +41872.8i q^{83} -35722.9i q^{87} +76344.7 q^{89} +169664. q^{91} -10417.0i q^{93} +33431.7i q^{97} +47417.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{9} + 128 q^{11} - 5280 q^{19} + 288 q^{21} - 3880 q^{29} + 12656 q^{31} - 20952 q^{39} + 4808 q^{41} - 28324 q^{49} + 38952 q^{51} - 10368 q^{59} - 107912 q^{61} - 720 q^{69} - 143104 q^{71}+ \cdots - 10368 q^{99}+O(q^{100}) $ 4 * q - 324 * q^9 + 128 * q^11 - 5280 * q^19 + 288 * q^21 - 3880 * q^29 + 12656 * q^31 - 20952 * q^39 + 4808 * q^41 - 28324 * q^49 + 38952 * q^51 - 10368 * q^59 - 107912 * q^61 - 720 * q^69 - 143104 * q^71 - 58576 * q^79 + 26244 * q^81 + 279448 * q^89 + 304512 * q^91 - 10368 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	162.350i	1.25230i	0.779703	+	0.626149i	$0.215370\pi$
$7$	162.350i	1.25230i	−0.779703	+	0.626149i	$0.784630\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−585.401	−1.45872	−0.729360	−	0.684130i	$-0.760182\pi$
$11$	−585.401	−1.45872	−0.729360	+	0.684130i	$0.760182\pi$
$12$	0	0
$13$	− 1045.05i	− 1.71506i	−0.514435	−	0.857529i	$-0.671998\pi$
$13$	− 1045.05i	− 1.71506i	0.514435	−	0.857529i	$-0.328002\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	310.249i	0.260368i	0.991490	+	0.130184i	$0.0415568\pi$
$17$	310.249i	0.260368i	−0.991490	+	0.130184i	$0.958443\pi$
$18$	0	0
$19$	−239.548	−0.152233	−0.0761165	−	0.997099i	$-0.524252\pi$
$19$	−239.548	−0.152233	−0.0761165	+	0.997099i	$0.524252\pi$
$20$	0	0
$21$	1461.15	0.723015
$22$	0	0
$23$	− 4804.86i	− 1.89392i	−0.321356	−	0.946959i	$-0.604139\pi$
$23$	− 4804.86i	− 1.89392i	0.321356	−	0.946959i	$-0.395861\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	729.000i	0.192450i
$28$	0	0
$29$	3969.21	0.876413	0.438207	−	0.898874i	$-0.355614\pi$
$29$	3969.21	0.876413	0.438207	+	0.898874i	$0.355614\pi$
$30$	0	0
$31$	1157.45	0.216320	0.108160	−	0.994133i	$-0.465504\pi$
$31$	1157.45	0.216320	0.108160	+	0.994133i	$0.465504\pi$
$32$	0	0
$33$	5268.61i	0.842192i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7168.26i	0.860814i	0.902635	+	0.430407i	$0.141630\pi$
$37$	7168.26i	0.860814i	−0.902635	+	0.430407i	$0.858370\pi$
$38$	0	0
$39$	−9405.46	−0.990190
$40$	0	0
$41$	17563.1	1.63171	0.815854	−	0.578259i	$-0.196268\pi$
$41$	17563.1	1.63171	0.815854	+	0.578259i	$0.196268\pi$
$42$	0	0
$43$	− 9789.99i	− 0.807442i	−0.914882	−	0.403721i	$-0.867717\pi$
$43$	− 9789.99i	− 0.807442i	0.914882	−	0.403721i	$-0.132283\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	15565.9i	1.02785i	0.857834	+	0.513926i	$0.171810\pi$
$47$	15565.9i	1.02785i	−0.857834	+	0.513926i	$0.828190\pi$
$48$	0	0
$49$	−9550.60	−0.568252
$50$	0	0
$51$	2792.24	0.150324
$52$	0	0
$53$	18151.3i	0.887602i	0.896125	+	0.443801i	$0.146370\pi$
$53$	18151.3i	0.887602i	−0.896125	+	0.443801i	$0.853630\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2155.93i	0.0878918i
$58$	0	0
$59$	−8148.61	−0.304757	−0.152378	−	0.988322i	$-0.548693\pi$
$59$	−8148.61	−0.304757	−0.152378	+	0.988322i	$0.548693\pi$
$60$	0	0
$61$	−55069.7	−1.89491	−0.947455	−	0.319890i	$-0.896354\pi$
$61$	−55069.7	−1.89491	−0.947455	+	0.319890i	$0.896354\pi$
$62$	0	0
$63$	− 13150.4i	− 0.417433i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	61162.8i	1.66456i	0.554352	+	0.832282i	$0.312966\pi$
$67$	61162.8i	1.66456i	−0.554352	+	0.832282i	$0.687034\pi$
$68$	0	0
$69$	−43243.7	−1.09345
$70$	0	0
$71$	−67880.9	−1.59809	−0.799045	−	0.601271i	$-0.794661\pi$
$71$	−67880.9	−1.59809	−0.799045	+	0.601271i	$0.794661\pi$
$72$	0	0
$73$	52131.1i	1.14496i	0.819919	+	0.572479i	$0.194018\pi$
$73$	52131.1i	1.14496i	−0.819919	+	0.572479i	$0.805982\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 95040.0i	− 1.82675i
$78$	0	0
$79$	11749.9	0.211820	0.105910	−	0.994376i	$-0.466225\pi$
$79$	11749.9	0.211820	0.105910	+	0.994376i	$0.466225\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	41872.8i	0.667170i	0.942720	+	0.333585i	$0.108258\pi$
$83$	41872.8i	0.667170i	−0.942720	+	0.333585i	$0.891742\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 35722.9i	− 0.505997i
$88$	0	0
$89$	76344.7	1.02165	0.510827	−	0.859683i	$-0.329339\pi$
$89$	76344.7	1.02165	0.510827	+	0.859683i	$0.329339\pi$
$90$	0	0
$91$	169664.	2.14777
$92$	0	0
$93$	− 10417.0i	− 0.124892i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	33431.7i	0.360769i	0.983596	+	0.180384i	$0.0577342\pi$
$97$	33431.7i	0.360769i	−0.983596	+	0.180384i	$0.942266\pi$
$98$	0	0
$99$	47417.5	0.486240
$100$	0	0
$101$	−5866.57	−0.0572243	−0.0286122	−	0.999591i	$-0.509109\pi$
$101$	−5866.57	−0.0572243	−0.0286122	+	0.999591i	$0.509109\pi$
$102$	0	0
$103$	177537.i	1.64891i	0.565928	+	0.824455i	$0.308518\pi$
$103$	177537.i	1.64891i	−0.565928	+	0.824455i	$0.691482\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 50695.8i	− 0.428068i	−0.976826	−	0.214034i	$-0.931340\pi$
$107$	− 50695.8i	− 0.428068i	0.976826	−	0.214034i	$-0.0686603\pi$
$108$	0	0
$109$	99979.0	0.806014	0.403007	−	0.915197i	$-0.367965\pi$
$109$	99979.0	0.806014	0.403007	+	0.915197i	$0.367965\pi$
$110$	0	0
$111$	64514.3	0.496991
$112$	0	0
$113$	98637.1i	0.726681i	0.931656	+	0.363341i	$0.118364\pi$
$113$	98637.1i	0.726681i	−0.931656	+	0.363341i	$0.881636\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	84649.1i	0.571686i
$118$	0	0
$119$	−50369.0	−0.326059
$120$	0	0
$121$	181643.	1.12786
$122$	0	0
$123$	− 158068.i	− 0.942067i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	358979.i	1.97497i	0.157719	+	0.987484i	$0.449586\pi$
$127$	358979.i	1.97497i	−0.157719	+	0.987484i	$0.550414\pi$
$128$	0	0
$129$	−88109.9	−0.466177
$130$	0	0
$131$	−65053.6	−0.331202	−0.165601	−	0.986193i	$-0.552956\pi$
$131$	−65053.6	−0.331202	−0.165601	+	0.986193i	$0.552956\pi$
$132$	0	0
$133$	− 38890.7i	− 0.190641i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 42571.0i	− 0.193782i	−0.995295	−	0.0968909i	$-0.969110\pi$
$137$	− 42571.0i	− 0.193782i	0.995295	−	0.0968909i	$-0.0308898\pi$
$138$	0	0
$139$	−91861.1	−0.403269	−0.201634	−	0.979461i	$-0.564625\pi$
$139$	−91861.1	−0.403269	−0.201634	+	0.979461i	$0.564625\pi$
$140$	0	0
$141$	140093.	0.593431
$142$	0	0
$143$	611774.i	2.50179i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	85955.4i	0.328080i
$148$	0	0
$149$	27586.1	0.101795	0.0508973	−	0.998704i	$-0.483792\pi$
$149$	27586.1	0.101795	0.0508973	+	0.998704i	$0.483792\pi$
$150$	0	0
$151$	195938.	0.699319	0.349660	−	0.936877i	$-0.386297\pi$
$151$	195938.	0.699319	0.349660	+	0.936877i	$0.386297\pi$
$152$	0	0
$153$	− 25130.1i	− 0.0867894i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 547638.i	− 1.77314i	−0.462590	−	0.886572i	$-0.653080\pi$
$157$	− 547638.i	− 1.77314i	0.462590	−	0.886572i	$-0.346920\pi$
$158$	0	0
$159$	163362.	0.512457
$160$	0	0
$161$	780070.	2.37175
$162$	0	0
$163$	− 141584.i	− 0.417392i	−0.977981	−	0.208696i	$-0.933078\pi$
$163$	− 141584.i	− 0.417392i	0.977981	−	0.208696i	$-0.0669219\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	291788.i	0.809611i	0.914403	+	0.404805i	$0.132661\pi$
$167$	291788.i	0.809611i	−0.914403	+	0.404805i	$0.867339\pi$
$168$	0	0
$169$	−720838.	−1.94143
$170$	0	0
$171$	19403.4	0.0507444
$172$	0	0
$173$	22476.3i	0.0570965i	0.999592	+	0.0285482i	$0.00908842\pi$
$173$	22476.3i	0.0570965i	−0.999592	+	0.0285482i	$0.990912\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	73337.5i	0.175951i
$178$	0	0
$179$	68168.6	0.159020	0.0795100	−	0.996834i	$-0.474664\pi$
$179$	68168.6	0.159020	0.0795100	+	0.996834i	$0.474664\pi$
$180$	0	0
$181$	262075.	0.594605	0.297303	−	0.954783i	$-0.403913\pi$
$181$	262075.	0.594605	0.297303	+	0.954783i	$0.403913\pi$
$182$	0	0
$183$	495628.i	1.09403i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 181620.i	− 0.379804i
$188$	0	0
$189$	−118353.	−0.241005
$190$	0	0
$191$	336486.	0.667396	0.333698	−	0.942680i	$-0.391703\pi$
$191$	336486.	0.667396	0.333698	+	0.942680i	$0.391703\pi$
$192$	0	0
$193$	963901.i	1.86268i	0.364145	+	0.931342i	$0.381361\pi$
$193$	963901.i	1.86268i	−0.364145	+	0.931342i	$0.618639\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	809195.i	1.48555i	0.669540	+	0.742776i	$0.266491\pi$
$197$	809195.i	1.48555i	−0.669540	+	0.742776i	$0.733509\pi$
$198$	0	0
$199$	−490997.	−0.878913	−0.439457	−	0.898264i	$-0.644829\pi$
$199$	−490997.	−0.878913	−0.439457	+	0.898264i	$0.644829\pi$
$200$	0	0
$201$	550465.	0.961036
$202$	0	0
$203$	644402.i	1.09753i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	389193.i	0.631306i
$208$	0	0
$209$	140232.	0.222065
$210$	0	0
$211$	−297359.	−0.459806	−0.229903	−	0.973214i	$-0.573841\pi$
$211$	−297359.	−0.459806	−0.229903	+	0.973214i	$0.573841\pi$
$212$	0	0
$213$	610928.i	0.922658i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	187912.i	0.270897i
$218$	0	0
$219$	469180.	0.661042
$220$	0	0
$221$	324226.	0.446547
$222$	0	0
$223$	95392.2i	0.128455i	0.997935	+	0.0642275i	$0.0204583\pi$
$223$	95392.2i	0.128455i	−0.997935	+	0.0642275i	$0.979542\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 496175.i	− 0.639102i	−0.947569	−	0.319551i	$-0.896468\pi$
$227$	− 496175.i	− 0.639102i	0.947569	−	0.319551i	$-0.103532\pi$
$228$	0	0
$229$	605872.	0.763470	0.381735	−	0.924272i	$-0.375327\pi$
$229$	605872.	0.763470	0.381735	+	0.924272i	$0.375327\pi$
$230$	0	0
$231$	−855360.	−1.05468
$232$	0	0
$233$	− 1.11618e6i	− 1.34693i	−0.739220	−	0.673464i	$-0.764806\pi$
$233$	− 1.11618e6i	− 1.34693i	0.739220	−	0.673464i	$-0.235194\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 105749.i	− 0.122294i
$238$	0	0
$239$	1.13721e6	1.28779	0.643896	−	0.765113i	$-0.277317\pi$
$239$	1.13721e6	1.28779	0.643896	+	0.765113i	$0.277317\pi$
$240$	0	0
$241$	203852.	0.226086	0.113043	−	0.993590i	$-0.463940\pi$
$241$	203852.	0.226086	0.113043	+	0.993590i	$0.463940\pi$
$242$	0	0
$243$	− 59049.0i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	250340.i	0.261089i
$248$	0	0
$249$	376855.	0.385191
$250$	0	0
$251$	−391896.	−0.392633	−0.196317	−	0.980541i	$-0.562898\pi$
$251$	−391896.	−0.392633	−0.196317	+	0.980541i	$0.562898\pi$
$252$	0	0
$253$	2.81277e6i	2.76269i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	417652.i	0.394441i	0.980359	+	0.197220i	$0.0631914\pi$
$257$	417652.i	0.394441i	−0.980359	+	0.197220i	$0.936809\pi$
$258$	0	0
$259$	−1.16377e6	−1.07800
$260$	0	0
$261$	−321506.	−0.292138
$262$	0	0
$263$	− 1.17895e6i	− 1.05101i	−0.850791	−	0.525505i	$-0.823876\pi$
$263$	− 1.17895e6i	− 1.05101i	0.850791	−	0.525505i	$-0.176124\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 687102.i	− 0.589852i
$268$	0	0
$269$	929355.	0.783070	0.391535	−	0.920163i	$-0.371944\pi$
$269$	929355.	0.783070	0.391535	+	0.920163i	$0.371944\pi$
$270$	0	0
$271$	−1.08603e6	−0.898293	−0.449147	−	0.893458i	$-0.648272\pi$
$271$	−1.08603e6	−0.898293	−0.449147	+	0.893458i	$0.648272\pi$
$272$	0	0
$273$	− 1.52698e6i	− 1.24001i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.33219e6i	− 1.04320i	−0.853192	−	0.521598i	$-0.825336\pi$
$277$	− 1.33219e6i	− 1.04320i	0.853192	−	0.521598i	$-0.174664\pi$
$278$	0	0
$279$	−93753.2	−0.0721067
$280$	0	0
$281$	1.21255e6	0.916079	0.458040	−	0.888932i	$-0.348552\pi$
$281$	1.21255e6	0.916079	0.458040	+	0.888932i	$0.348552\pi$
$282$	0	0
$283$	− 29554.8i	− 0.0219362i	−0.999940	−	0.0109681i	$-0.996509\pi$
$283$	− 29554.8i	− 0.0219362i	0.999940	−	0.0109681i	$-0.00349133\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.85138e6i	2.04338i
$288$	0	0
$289$	1.32360e6	0.932208
$290$	0	0
$291$	300885.	0.208290
$292$	0	0
$293$	1.41049e6i	0.959844i	0.877311	+	0.479922i	$0.159335\pi$
$293$	1.41049e6i	0.959844i	−0.877311	+	0.479922i	$0.840665\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 426757.i	− 0.280731i
$298$	0	0
$299$	−5.02132e6	−3.24818
$300$	0	0
$301$	1.58941e6	1.01116
$302$	0	0
$303$	52799.1i	0.0330385i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.25330e6i	1.36450i	0.731119	+	0.682250i	$0.238999\pi$
$307$	2.25330e6i	1.36450i	−0.731119	+	0.682250i	$0.761001\pi$
$308$	0	0
$309$	1.59784e6	0.951998
$310$	0	0
$311$	1.00991e6	0.592081	0.296041	−	0.955175i	$-0.404334\pi$
$311$	1.00991e6	0.592081	0.296041	+	0.955175i	$0.404334\pi$
$312$	0	0
$313$	970607.i	0.559993i	0.960001	+	0.279996i	$0.0903333\pi$
$313$	970607.i	0.559993i	−0.960001	+	0.279996i	$0.909667\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.23575e6i	− 1.24961i	−0.780780	−	0.624806i	$-0.785178\pi$
$317$	− 2.23575e6i	− 1.24961i	0.780780	−	0.624806i	$-0.214822\pi$
$318$	0	0
$319$	−2.32358e6	−1.27844
$320$	0	0
$321$	−456262.	−0.247145
$322$	0	0
$323$	− 74319.5i	− 0.0396366i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 899811.i	− 0.465352i
$328$	0	0
$329$	−2.52713e6	−1.28718
$330$	0	0
$331$	−2.12265e6	−1.06490	−0.532449	−	0.846462i	$-0.678728\pi$
$331$	−2.12265e6	−1.06490	−0.532449	+	0.846462i	$0.678728\pi$
$332$	0	0
$333$	− 580629.i	− 0.286938i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	603103.i	0.289279i	0.989484	+	0.144639i	$0.0462022\pi$
$337$	603103.i	0.289279i	−0.989484	+	0.144639i	$0.953798\pi$
$338$	0	0
$339$	887734.	0.419550
$340$	0	0
$341$	−677570.	−0.315550
$342$	0	0
$343$	1.17808e6i	0.540678i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.25150e6i	0.557965i	0.960296	+	0.278982i	$0.0899971\pi$
$347$	1.25150e6i	0.557965i	−0.960296	+	0.278982i	$0.910003\pi$
$348$	0	0
$349$	−911762.	−0.400699	−0.200349	−	0.979725i	$-0.564208\pi$
$349$	−911762.	−0.400699	−0.200349	+	0.979725i	$0.564208\pi$
$350$	0	0
$351$	761842.	0.330063
$352$	0	0
$353$	− 2.80269e6i	− 1.19712i	−0.801078	−	0.598560i	$-0.795740\pi$
$353$	− 2.80269e6i	− 1.19712i	0.801078	−	0.598560i	$-0.204260\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	453321.i	0.188250i
$358$	0	0
$359$	3.65390e6	1.49631	0.748153	−	0.663526i	$-0.230941\pi$
$359$	3.65390e6	1.49631	0.748153	+	0.663526i	$0.230941\pi$
$360$	0	0
$361$	−2.41872e6	−0.976825
$362$	0	0
$363$	− 1.63479e6i	− 0.651172i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.32839e6i	1.28994i	0.764208	+	0.644970i	$0.223130\pi$
$367$	3.32839e6i	1.28994i	−0.764208	+	0.644970i	$0.776870\pi$
$368$	0	0
$369$	−1.42261e6	−0.543902
$370$	0	0
$371$	−2.94687e6	−1.11154
$372$	0	0
$373$	− 3.38609e6i	− 1.26016i	−0.776530	−	0.630080i	$-0.783022\pi$
$373$	− 3.38609e6i	− 1.26016i	0.776530	−	0.630080i	$-0.216978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.14802e6i	− 1.50310i
$378$	0	0
$379$	302178.	0.108060	0.0540300	−	0.998539i	$-0.482793\pi$
$379$	302178.	0.108060	0.0540300	+	0.998539i	$0.482793\pi$
$380$	0	0
$381$	3.23081e6	1.14025
$382$	0	0
$383$	− 3.56094e6i	− 1.24042i	−0.784436	−	0.620209i	$-0.787048\pi$
$383$	− 3.56094e6i	− 1.24042i	0.784436	−	0.620209i	$-0.212952\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	792989.i	0.269147i
$388$	0	0
$389$	4.22731e6	1.41641	0.708207	−	0.706005i	$-0.249504\pi$
$389$	4.22731e6	1.41641	0.708207	+	0.706005i	$0.249504\pi$
$390$	0	0
$391$	1.49070e6	0.493116
$392$	0	0
$393$	585482.i	0.191220i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.79565e6i	− 1.84555i	−0.385340	−	0.922775i	$-0.625916\pi$
$397$	− 5.79565e6i	− 1.84555i	0.385340	−	0.922775i	$-0.374084\pi$
$398$	0	0
$399$	−350016.	−0.110067
$400$	0	0
$401$	4.94036e6	1.53426	0.767128	−	0.641495i	$-0.221685\pi$
$401$	4.94036e6	1.53426	0.767128	+	0.641495i	$0.221685\pi$
$402$	0	0
$403$	− 1.20959e6i	− 0.371002i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.19631e6i	− 1.25569i
$408$	0	0
$409$	1.58766e6	0.469299	0.234649	−	0.972080i	$-0.424606\pi$
$409$	1.58766e6	0.469299	0.234649	+	0.972080i	$0.424606\pi$
$410$	0	0
$411$	−383139.	−0.111880
$412$	0	0
$413$	− 1.32293e6i	− 0.381647i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	826750.i	0.232827i
$418$	0	0
$419$	5.39675e6	1.50175	0.750875	−	0.660445i	$-0.229632\pi$
$419$	5.39675e6	1.50175	0.750875	+	0.660445i	$0.229632\pi$
$420$	0	0
$421$	−3.97269e6	−1.09239	−0.546197	−	0.837657i	$-0.683925\pi$
$421$	−3.97269e6	−1.09239	−0.546197	+	0.837657i	$0.683925\pi$
$422$	0	0
$423$	− 1.26084e6i	− 0.342618i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.94059e6i	− 2.37299i
$428$	0	0
$429$	5.50596e6	1.44441
$430$	0	0
$431$	−4.11081e6	−1.06594	−0.532972	−	0.846133i	$-0.678925\pi$
$431$	−4.11081e6	−1.06594	−0.532972	+	0.846133i	$0.678925\pi$
$432$	0	0
$433$	− 1.77432e6i	− 0.454792i	−0.973802	−	0.227396i	$-0.926979\pi$
$433$	− 1.77432e6i	− 0.454792i	0.973802	−	0.227396i	$-0.0730211\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.15100e6i	0.288317i
$438$	0	0
$439$	−5.97904e6	−1.48071	−0.740355	−	0.672216i	$-0.765343\pi$
$439$	−5.97904e6	−1.48071	−0.740355	+	0.672216i	$0.765343\pi$
$440$	0	0
$441$	773599.	0.189417
$442$	0	0
$443$	− 4.92048e6i	− 1.19124i	−0.803267	−	0.595619i	$-0.796907\pi$
$443$	− 4.92048e6i	− 1.19124i	0.803267	−	0.595619i	$-0.203093\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 248275.i	− 0.0587711i
$448$	0	0
$449$	4.14322e6	0.969888	0.484944	−	0.874545i	$-0.338840\pi$
$449$	4.14322e6	0.969888	0.484944	+	0.874545i	$0.338840\pi$
$450$	0	0
$451$	−1.02815e7	−2.38020
$452$	0	0
$453$	− 1.76344e6i	− 0.403752i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.38768e6i	1.65469i	0.561691	+	0.827347i	$0.310151\pi$
$457$	7.38768e6i	1.65469i	−0.561691	+	0.827347i	$0.689849\pi$
$458$	0	0
$459$	−226171.	−0.0501079
$460$	0	0
$461$	−4.74779e6	−1.04049	−0.520246	−	0.854016i	$-0.674160\pi$
$461$	−4.74779e6	−1.04049	−0.520246	+	0.854016i	$0.674160\pi$
$462$	0	0
$463$	6.92369e6i	1.50101i	0.660862	+	0.750507i	$0.270191\pi$
$463$	6.92369e6i	1.50101i	−0.660862	+	0.750507i	$0.729809\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.92098e6i	1.25632i	0.778083	+	0.628161i	$0.216192\pi$
$467$	5.92098e6i	1.25632i	−0.778083	+	0.628161i	$0.783808\pi$
$468$	0	0
$469$	−9.92980e6	−2.08453
$470$	0	0
$471$	−4.92874e6	−1.02373
$472$	0	0
$473$	5.73107e6i	1.17783i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.47026e6i	− 0.295867i
$478$	0	0
$479$	4.07288e6	0.811079	0.405539	−	0.914078i	$-0.367084\pi$
$479$	4.07288e6	0.811079	0.405539	+	0.914078i	$0.367084\pi$
$480$	0	0
$481$	7.49119e6	1.47635
$482$	0	0
$483$	− 7.02063e6i	− 1.36933i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.62701e6i	− 0.692989i	−0.938052	−	0.346495i	$-0.887372\pi$
$487$	− 3.62701e6i	− 0.692989i	0.938052	−	0.346495i	$-0.112628\pi$
$488$	0	0
$489$	−1.27425e6	−0.240981
$490$	0	0
$491$	−2.81431e6	−0.526826	−0.263413	−	0.964683i	$-0.584848\pi$
$491$	−2.81431e6	−0.526826	−0.263413	+	0.964683i	$0.584848\pi$
$492$	0	0
$493$	1.23144e6i	0.228190i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.10205e7i	− 2.00129i
$498$	0	0
$499$	−2.07684e6	−0.373380	−0.186690	−	0.982419i	$-0.559776\pi$
$499$	−2.07684e6	−0.373380	−0.186690	+	0.982419i	$0.559776\pi$
$500$	0	0
$501$	2.62609e6	0.467429
$502$	0	0
$503$	1.62288e6i	0.286000i	0.989723	+	0.143000i	$0.0456750\pi$
$503$	1.62288e6i	0.286000i	−0.989723	+	0.143000i	$0.954325\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.48754e6i	1.12088i
$508$	0	0
$509$	4.29837e6	0.735377	0.367688	−	0.929949i	$-0.380149\pi$
$509$	4.29837e6	0.735377	0.367688	+	0.929949i	$0.380149\pi$
$510$	0	0
$511$	−8.46350e6	−1.43383
$512$	0	0
$513$	− 174631.i	− 0.0292973i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 9.11232e6i	− 1.49935i
$518$	0	0
$519$	202287.	0.0329647
$520$	0	0
$521$	−4.85382e6	−0.783410	−0.391705	−	0.920091i	$-0.628115\pi$
$521$	−4.85382e6	−0.783410	−0.391705	+	0.920091i	$0.628115\pi$
$522$	0	0
$523$	3.13717e6i	0.501514i	0.968050	+	0.250757i	$0.0806795\pi$
$523$	3.13717e6i	0.501514i	−0.968050	+	0.250757i	$0.919320\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	359096.i	0.0563228i
$528$	0	0
$529$	−1.66503e7	−2.58692
$530$	0	0
$531$	660037.	0.101586
$532$	0	0
$533$	− 1.83544e7i	− 2.79847i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 613518.i	− 0.0918103i
$538$	0	0
$539$	5.59093e6	0.828920
$540$	0	0
$541$	1.13491e7	1.66713	0.833566	−	0.552421i	$-0.186296\pi$
$541$	1.13491e7	1.66713	0.833566	+	0.552421i	$0.186296\pi$
$542$	0	0
$543$	− 2.35867e6i	− 0.343296i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.26889e7i	1.81325i	0.421943	+	0.906623i	$0.361348\pi$
$547$	1.26889e7i	1.81325i	−0.421943	+	0.906623i	$0.638652\pi$
$548$	0	0
$549$	4.46065e6	0.631637
$550$	0	0
$551$	−950817.	−0.133419
$552$	0	0
$553$	1.90760e6i	0.265261i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.91482e6i	0.807799i	0.914803	+	0.403900i	$0.132346\pi$
$557$	5.91482e6i	0.807799i	−0.914803	+	0.403900i	$0.867654\pi$
$558$	0	0
$559$	−1.02310e7	−1.38481
$560$	0	0
$561$	−1.63458e6	−0.219280
$562$	0	0
$563$	− 1.19890e6i	− 0.159409i	−0.996819	−	0.0797043i	$-0.974602\pi$
$563$	− 1.19890e6i	− 0.159409i	0.996819	−	0.0797043i	$-0.0253976\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.06518e6i	0.139144i
$568$	0	0
$569$	−76146.2	−0.00985979	−0.00492989	−	0.999988i	$-0.501569\pi$
$569$	−76146.2	−0.00985979	−0.00492989	+	0.999988i	$0.501569\pi$
$570$	0	0
$571$	−9.47302e6	−1.21590	−0.607950	−	0.793975i	$-0.708008\pi$
$571$	−9.47302e6	−1.21590	−0.607950	+	0.793975i	$0.708008\pi$
$572$	0	0
$573$	− 3.02837e6i	− 0.385321i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 4.73955e6i	− 0.592649i	−0.955087	−	0.296325i	$-0.904239\pi$
$577$	− 4.73955e6i	− 0.592649i	0.955087	−	0.296325i	$-0.0957610\pi$
$578$	0	0
$579$	8.67511e6	1.07542
$580$	0	0
$581$	−6.79805e6	−0.835496
$582$	0	0
$583$	− 1.06258e7i	− 1.29476i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.58535e6i	0.189902i	0.995482	+	0.0949511i	$0.0302695\pi$
$587$	1.58535e6i	0.189902i	−0.995482	+	0.0949511i	$0.969731\pi$
$588$	0	0
$589$	−277264.	−0.0329311
$590$	0	0
$591$	7.28276e6	0.857684
$592$	0	0
$593$	− 7.56532e6i	− 0.883467i	−0.897146	−	0.441734i	$-0.854364\pi$
$593$	− 7.56532e6i	− 0.883467i	0.897146	−	0.441734i	$-0.145636\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.41897e6i	0.507441i
$598$	0	0
$599$	−1.25418e7	−1.42821	−0.714105	−	0.700038i	$-0.753166\pi$
$599$	−1.25418e7	−1.42821	−0.714105	+	0.700038i	$0.753166\pi$
$600$	0	0
$601$	−6.32087e6	−0.713824	−0.356912	−	0.934138i	$-0.616170\pi$
$601$	−6.32087e6	−0.713824	−0.356912	+	0.934138i	$0.616170\pi$
$602$	0	0
$603$	− 4.95419e6i	− 0.554855i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	950320.i	0.104688i	0.998629	+	0.0523441i	$0.0166693\pi$
$607$	950320.i	0.104688i	−0.998629	+	0.0523441i	$0.983331\pi$
$608$	0	0
$609$	5.79962e6	0.633660
$610$	0	0
$611$	1.62672e7	1.76283
$612$	0	0
$613$	5.24076e6i	0.563304i	0.959517	+	0.281652i	$0.0908824\pi$
$613$	5.24076e6i	0.563304i	−0.959517	+	0.281652i	$0.909118\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	4.76539e6i	0.503948i	0.967734	+	0.251974i	$0.0810797\pi$
$617$	4.76539e6i	0.503948i	−0.967734	+	0.251974i	$0.918920\pi$
$618$	0	0
$619$	1.24730e7	1.30841	0.654206	−	0.756316i	$-0.273003\pi$
$619$	1.24730e7	1.30841	0.654206	+	0.756316i	$0.273003\pi$
$620$	0	0
$621$	3.50274e6	0.364485
$622$	0	0
$623$	1.23946e7i	1.27942i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.26209e6i	− 0.128209i
$628$	0	0
$629$	−2.22394e6	−0.224129
$630$	0	0
$631$	−61030.3	−0.00610200	−0.00305100	−	0.999995i	$-0.500971\pi$
$631$	−61030.3	−0.00610200	−0.00305100	+	0.999995i	$0.500971\pi$
$632$	0	0
$633$	2.67623e6i	0.265469i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.98087e6i	0.974585i
$638$	0	0
$639$	5.49835e6	0.532697
$640$	0	0
$641$	−6.76769e6	−0.650572	−0.325286	−	0.945616i	$-0.605461\pi$
$641$	−6.76769e6	−0.650572	−0.325286	+	0.945616i	$0.605461\pi$
$642$	0	0
$643$	1.01829e7i	0.971276i	0.874160	+	0.485638i	$0.161413\pi$
$643$	1.01829e7i	0.971276i	−0.874160	+	0.485638i	$0.838587\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.67748e6i	0.721037i	0.932752	+	0.360519i	$0.117400\pi$
$647$	7.67748e6i	0.721037i	−0.932752	+	0.360519i	$0.882600\pi$
$648$	0	0
$649$	4.77020e6	0.444555
$650$	0	0
$651$	1.69121e6	0.156403
$652$	0	0
$653$	2.26141e6i	0.207537i	0.994601	+	0.103768i	$0.0330901\pi$
$653$	2.26141e6i	0.207537i	−0.994601	+	0.103768i	$0.966910\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 4.22262e6i	− 0.381653i
$658$	0	0
$659$	−7.39789e6	−0.663581	−0.331791	−	0.943353i	$-0.607653\pi$
$659$	−7.39789e6	−0.663581	−0.331791	+	0.943353i	$0.607653\pi$
$660$	0	0
$661$	−5.25362e6	−0.467687	−0.233843	−	0.972274i	$-0.575130\pi$
$661$	−5.25362e6	−0.467687	−0.233843	+	0.972274i	$0.575130\pi$
$662$	0	0
$663$	− 2.91803e6i	− 0.257814i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.90715e7i	− 1.65985i
$668$	0	0
$669$	858530.	0.0741635
$670$	0	0
$671$	3.22379e7	2.76414
$672$	0	0
$673$	2.17636e7i	1.85222i	0.377249	+	0.926112i	$0.376870\pi$
$673$	2.17636e7i	1.85222i	−0.377249	+	0.926112i	$0.623130\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.65690e6i	0.474358i	0.971466	+	0.237179i	$0.0762229\pi$
$677$	5.65690e6i	0.474358i	−0.971466	+	0.237179i	$0.923777\pi$
$678$	0	0
$679$	−5.42764e6	−0.451790
$680$	0	0
$681$	−4.46557e6	−0.368986
$682$	0	0
$683$	3.74261e6i	0.306989i	0.988149	+	0.153495i	$0.0490528\pi$
$683$	3.74261e6i	0.306989i	−0.988149	+	0.153495i	$0.950947\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 5.45285e6i	− 0.440790i
$688$	0	0
$689$	1.89690e7	1.52229
$690$	0	0
$691$	8.06260e6	0.642362	0.321181	−	0.947018i	$-0.395920\pi$
$691$	8.06260e6	0.642362	0.321181	+	0.947018i	$0.395920\pi$
$692$	0	0
$693$	7.69824e6i	0.608917i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.44894e6i	0.424844i
$698$	0	0
$699$	−1.00456e7	−0.777649
$700$	0	0
$701$	6.25082e6	0.480443	0.240221	−	0.970718i	$-0.422780\pi$
$701$	6.25082e6	0.480443	0.240221	+	0.970718i	$0.422780\pi$
$702$	0	0
$703$	− 1.71714e6i	− 0.131044i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 952439.i	− 0.0716619i
$708$	0	0
$709$	−1.05968e7	−0.791697	−0.395849	−	0.918316i	$-0.629550\pi$
$709$	−1.05968e7	−0.791697	−0.395849	+	0.918316i	$0.629550\pi$
$710$	0	0
$711$	−951741.	−0.0706065
$712$	0	0
$713$	− 5.56137e6i	− 0.409692i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.02349e7i	− 0.743507i
$718$	0	0
$719$	−6.08563e6	−0.439019	−0.219509	−	0.975610i	$-0.570446\pi$
$719$	−6.08563e6	−0.439019	−0.219509	+	0.975610i	$0.570446\pi$
$720$	0	0
$721$	−2.88232e7	−2.06493
$722$	0	0
$723$	− 1.83467e6i	− 0.130531i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 3.58779e6i	− 0.251763i	−0.992045	−	0.125881i	$-0.959824\pi$
$727$	− 3.58779e6i	− 0.251763i	0.992045	−	0.125881i	$-0.0401758\pi$
$728$	0	0
$729$	−531441.	−0.0370370
$730$	0	0
$731$	3.03733e6	0.210232
$732$	0	0
$733$	1.82918e7i	1.25747i	0.777620	+	0.628734i	$0.216427\pi$
$733$	1.82918e7i	1.25747i	−0.777620	+	0.628734i	$0.783573\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 3.58048e7i	− 2.42813i
$738$	0	0
$739$	2.48989e6	0.167714	0.0838568	−	0.996478i	$-0.473276\pi$
$739$	2.48989e6	0.167714	0.0838568	+	0.996478i	$0.473276\pi$
$740$	0	0
$741$	2.25306e6	0.150740
$742$	0	0
$743$	2.84643e7i	1.89159i	0.324759	+	0.945797i	$0.394717\pi$
$743$	2.84643e7i	1.89159i	−0.324759	+	0.945797i	$0.605283\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 3.39169e6i	− 0.222390i
$748$	0	0
$749$	8.23047e6	0.536068
$750$	0	0
$751$	1.15017e7	0.744156	0.372078	−	0.928202i	$-0.378645\pi$
$751$	1.15017e7	0.744156	0.372078	+	0.928202i	$0.378645\pi$
$752$	0	0
$753$	3.52707e6i	0.226687i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00041e7i	1.26876i	0.773022	+	0.634379i	$0.218744\pi$
$757$	2.00041e7i	1.26876i	−0.773022	+	0.634379i	$0.781256\pi$
$758$	0	0
$759$	2.53149e7	1.59504
$760$	0	0
$761$	2.55313e7	1.59813	0.799063	−	0.601248i	$-0.205330\pi$
$761$	2.55313e7	1.59813	0.799063	+	0.601248i	$0.205330\pi$
$762$	0	0
$763$	1.62316e7i	1.00937i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.51571e6i	0.522676i
$768$	0	0
$769$	−1.13207e7	−0.690330	−0.345165	−	0.938542i	$-0.612177\pi$
$769$	−1.13207e7	−0.690330	−0.345165	+	0.938542i	$0.612177\pi$
$770$	0	0
$771$	3.75887e6	0.227730
$772$	0	0
$773$	− 2.37503e7i	− 1.42962i	−0.699319	−	0.714810i	$-0.746513\pi$
$773$	− 2.37503e7i	− 1.42962i	0.699319	−	0.714810i	$-0.253487\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.04739e7i	0.622381i
$778$	0	0
$779$	−4.20722e6	−0.248400
$780$	0	0
$781$	3.97375e7	2.33117
$782$	0	0
$783$	2.89355e6i	0.168666i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.66771e7i	− 0.959808i	−0.877321	−	0.479904i	$-0.840671\pi$
$787$	− 1.66771e7i	− 0.959808i	0.877321	−	0.479904i	$-0.159329\pi$
$788$	0	0
$789$	−1.06106e7	−0.606801
$790$	0	0
$791$	−1.60138e7	−0.910022
$792$	0	0
$793$	5.75507e7i	3.24988i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.03514e7i	1.69252i	0.532771	+	0.846259i	$0.321151\pi$
$797$	3.03514e7i	1.69252i	−0.532771	+	0.846259i	$0.678849\pi$
$798$	0	0
$799$	−4.82931e6	−0.267620
$800$	0	0
$801$	−6.18392e6	−0.340551
$802$	0	0
$803$	− 3.05176e7i	− 1.67017i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 8.36419e6i	− 0.452106i
$808$	0	0
$809$	−1.09466e7	−0.588042	−0.294021	−	0.955799i	$-0.594994\pi$
$809$	−1.09466e7	−0.588042	−0.294021	+	0.955799i	$0.594994\pi$
$810$	0	0
$811$	−1.24677e7	−0.665634	−0.332817	−	0.942991i	$-0.607999\pi$
$811$	−1.24677e7	−0.665634	−0.332817	+	0.942991i	$0.607999\pi$
$812$	0	0
$813$	9.77426e6i	0.518630i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.34517e6i	0.122919i
$818$	0	0
$819$	−1.37428e7	−0.715922
$820$	0	0
$821$	−2.12322e6	−0.109935	−0.0549676	−	0.998488i	$-0.517506\pi$
$821$	−2.12322e6	−0.109935	−0.0549676	+	0.998488i	$0.517506\pi$
$822$	0	0
$823$	2.17058e7i	1.11706i	0.829484	+	0.558531i	$0.188635\pi$
$823$	2.17058e7i	1.11706i	−0.829484	+	0.558531i	$0.811365\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.95925e6i	0.404677i	0.979316	+	0.202338i	$0.0648541\pi$
$827$	7.95925e6i	0.404677i	−0.979316	+	0.202338i	$0.935146\pi$
$828$	0	0
$829$	2.15416e7	1.08866	0.544329	−	0.838872i	$-0.316784\pi$
$829$	2.15416e7	1.08866	0.544329	+	0.838872i	$0.316784\pi$
$830$	0	0
$831$	−1.19897e7	−0.602289
$832$	0	0
$833$	− 2.96306e6i	− 0.147955i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	843779.i	0.0416308i
$838$	0	0
$839$	2.79514e7	1.37088	0.685439	−	0.728130i	$-0.259610\pi$
$839$	2.79514e7	1.37088	0.685439	+	0.728130i	$0.259610\pi$
$840$	0	0
$841$	−4.75654e6	−0.231900
$842$	0	0
$843$	− 1.09129e7i	− 0.528899i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.94898e7i	1.41242i
$848$	0	0
$849$	−265993.	−0.0126649
$850$	0	0
$851$	3.44425e7	1.63031
$852$	0	0
$853$	− 2.67662e7i	− 1.25955i	−0.776778	−	0.629774i	$-0.783147\pi$
$853$	− 2.67662e7i	− 1.25955i	0.776778	−	0.629774i	$-0.216853\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.97359e7i	1.84812i	0.382245	+	0.924061i	$0.375151\pi$
$857$	3.97359e7i	1.84812i	−0.382245	+	0.924061i	$0.624849\pi$
$858$	0	0
$859$	−2.81141e7	−1.29999	−0.649997	−	0.759937i	$-0.725230\pi$
$859$	−2.81141e7	−1.29999	−0.649997	+	0.759937i	$0.725230\pi$
$860$	0	0
$861$	2.56624e7	1.17975
$862$	0	0
$863$	6.04052e6i	0.276088i	0.990426	+	0.138044i	$0.0440815\pi$
$863$	6.04052e6i	0.276088i	−0.990426	+	0.138044i	$0.955919\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.19124e7i	− 0.538211i
$868$	0	0
$869$	−6.87840e6	−0.308985
$870$	0	0
$871$	6.39183e7	2.85483
$872$	0	0
$873$	− 2.70797e6i	− 0.120256i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.52617e7i	1.54812i	0.633114	+	0.774059i	$0.281776\pi$
$877$	3.52617e7i	1.54812i	−0.633114	+	0.774059i	$0.718224\pi$
$878$	0	0
$879$	1.26944e7	0.554166
$880$	0	0
$881$	−1.24085e7	−0.538617	−0.269308	−	0.963054i	$-0.586795\pi$
$881$	−1.24085e7	−0.538617	−0.269308	+	0.963054i	$0.586795\pi$
$882$	0	0
$883$	1.24560e7i	0.537623i	0.963193	+	0.268812i	$0.0866309\pi$
$883$	1.24560e7i	0.537623i	−0.963193	+	0.268812i	$0.913369\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 3.54632e7i	− 1.51345i	−0.653732	−	0.756726i	$-0.726798\pi$
$887$	− 3.54632e7i	− 1.51345i	0.653732	−	0.756726i	$-0.273202\pi$
$888$	0	0
$889$	−5.82804e7	−2.47325
$890$	0	0
$891$	−3.84082e6	−0.162080
$892$	0	0
$893$	− 3.72879e6i	− 0.156473i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.51919e7i	1.87534i
$898$	0	0
$899$	4.59415e6	0.189586
$900$	0	0
$901$	−5.63142e6	−0.231103
$902$	0	0
$903$	− 1.43047e7i	− 0.583792i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.62354e7i	− 1.05894i	−0.848330	−	0.529469i	$-0.822391\pi$
$907$	− 2.62354e7i	− 1.05894i	0.848330	−	0.529469i	$-0.177609\pi$
$908$	0	0
$909$	475192.	0.0190748
$910$	0	0
$911$	2.34607e7	0.936581	0.468290	−	0.883575i	$-0.344870\pi$
$911$	2.34607e7	0.936581	0.468290	+	0.883575i	$0.344870\pi$
$912$	0	0
$913$	− 2.45124e7i	− 0.973213i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.05615e7i	− 0.414764i
$918$	0	0
$919$	3.36115e7	1.31280	0.656401	−	0.754412i	$-0.272078\pi$
$919$	3.36115e7	1.31280	0.656401	+	0.754412i	$0.272078\pi$
$920$	0	0
$921$	2.02797e7	0.787795
$922$	0	0
$923$	7.09389e7i	2.74082i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1.43805e7i	− 0.549637i
$928$	0	0
$929$	−4.54932e6	−0.172945	−0.0864724	−	0.996254i	$-0.527559\pi$
$929$	−4.54932e6	−0.172945	−0.0864724	+	0.996254i	$0.527559\pi$
$930$	0	0
$931$	2.28783e6	0.0865067
$932$	0	0
$933$	− 9.08918e6i	− 0.341838i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.90582e7i	− 1.82542i	−0.408610	−	0.912709i	$-0.633986\pi$
$937$	− 4.90582e7i	− 1.82542i	0.408610	−	0.912709i	$-0.366014\pi$
$938$	0	0
$939$	8.73546e6	0.323312
$940$	0	0
$941$	−3.63415e7	−1.33792	−0.668958	−	0.743301i	$-0.733259\pi$
$941$	−3.63415e7	−1.33792	−0.668958	+	0.743301i	$0.733259\pi$
$942$	0	0
$943$	− 8.43883e7i	− 3.09032i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.46706e7i	− 0.531586i	−0.964030	−	0.265793i	$-0.914366\pi$
$947$	− 1.46706e7i	− 0.531586i	0.964030	−	0.265793i	$-0.0856338\pi$
$948$	0	0
$949$	5.44797e7	1.96367
$950$	0	0
$951$	−2.01218e7	−0.721464
$952$	0	0
$953$	2.95120e7i	1.05261i	0.850296	+	0.526304i	$0.176423\pi$
$953$	2.95120e7i	1.05261i	−0.850296	+	0.526304i	$0.823577\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.09122e7i	0.738108i
$958$	0	0
$959$	6.91142e6	0.242673
$960$	0	0
$961$	−2.72895e7	−0.953206
$962$	0	0
$963$	4.10636e6i	0.142689i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.67297e6i	0.126314i	0.998004	+	0.0631569i	$0.0201169\pi$
$967$	3.67297e6i	0.126314i	−0.998004	+	0.0631569i	$0.979883\pi$
$968$	0	0
$969$	−668876.	−0.0228842
$970$	0	0
$971$	3.02080e7	1.02819	0.514096	−	0.857733i	$-0.328127\pi$
$971$	3.02080e7	1.02819	0.514096	+	0.857733i	$0.328127\pi$
$972$	0	0
$973$	− 1.49137e7i	− 0.505013i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.41688e7i	1.14523i	0.819824	+	0.572615i	$0.194071\pi$
$977$	3.41688e7i	1.14523i	−0.819824	+	0.572615i	$0.805929\pi$
$978$	0	0
$979$	−4.46923e7	−1.49031
$980$	0	0
$981$	−8.09830e6	−0.268671
$982$	0	0
$983$	3.84158e7i	1.26802i	0.773326	+	0.634009i	$0.218592\pi$
$983$	3.84158e7i	1.26802i	−0.773326	+	0.634009i	$0.781408\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.27442e7i	0.743153i
$988$	0	0
$989$	−4.70395e7	−1.52923
$990$	0	0
$991$	−3.67819e7	−1.18974	−0.594868	−	0.803824i	$-0.702796\pi$
$991$	−3.67819e7	−1.18974	−0.594868	+	0.803824i	$0.702796\pi$
$992$	0	0
$993$	1.91038e7i	0.614819i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5.01540e6i	− 0.159796i	−0.996803	−	0.0798982i	$-0.974540\pi$
$997$	− 5.01540e6i	− 0.159796i	0.996803	−	0.0798982i	$-0.0254595\pi$
$998$	0	0
$999$	−5.22566e6	−0.165664

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.6.f.k.49.2		4
5.2	odd	4		600.6.a.k.1.1		2
5.3	odd	4		120.6.a.g.1.2	✓	2
5.4	even	2	inner	600.6.f.k.49.3		4
15.8	even	4		360.6.a.n.1.2		2
20.3	even	4		240.6.a.o.1.1		2
40.3	even	4		960.6.a.bn.1.1		2
40.13	odd	4		960.6.a.bi.1.2		2
60.23	odd	4		720.6.a.bf.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.a.g.1.2	✓	2	5.3	odd	4
240.6.a.o.1.1		2	20.3	even	4
360.6.a.n.1.2		2	15.8	even	4
600.6.a.k.1.1		2	5.2	odd	4
600.6.f.k.49.2		4	1.1	even	1	trivial
600.6.f.k.49.3		4	5.4	even	2	inner
720.6.a.bf.1.1		2	60.23	odd	4
960.6.a.bi.1.2		2	40.13	odd	4
960.6.a.bn.1.1		2	40.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} +162.350i q^{7} -81.0000 q^{9} -585.401 q^{11} -1045.05i q^{13} +310.249i q^{17} -239.548 q^{19} +1461.15 q^{21} -4804.86i q^{23} +729.000i q^{27} +3969.21 q^{29} +1157.45 q^{31} +5268.61i q^{33} +7168.26i q^{37} -9405.46 q^{39} +17563.1 q^{41} -9789.99i q^{43} +15565.9i q^{47} -9550.60 q^{49} +2792.24 q^{51} +18151.3i q^{53} +2155.93i q^{57} -8148.61 q^{59} -55069.7 q^{61} -13150.4i q^{63} +61162.8i q^{67} -43243.7 q^{69} -67880.9 q^{71} +52131.1i q^{73} -95040.0i q^{77} +11749.9 q^{79} +6561.00 q^{81} +41872.8i q^{83} -35722.9i q^{87} +76344.7 q^{89} +169664. q^{91} -10417.0i q^{93} +33431.7i q^{97} +47417.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{9} + 128 q^{11} - 5280 q^{19} + 288 q^{21} - 3880 q^{29} + 12656 q^{31} - 20952 q^{39} + 4808 q^{41} - 28324 q^{49} + 38952 q^{51} - 10368 q^{59} - 107912 q^{61} - 720 q^{69} - 143104 q^{71}+ \cdots - 10368 q^{99}+O(q^{100}) \) 4 * q - 324 * q^9 + 128 * q^11 - 5280 * q^19 + 288 * q^21 - 3880 * q^29 + 12656 * q^31 - 20952 * q^39 + 4808 * q^41 - 28324 * q^49 + 38952 * q^51 - 10368 * q^59 - 107912 * q^61 - 720 * q^69 - 143104 * q^71 - 58576 * q^79 + 26244 * q^81 + 279448 * q^89 + 304512 * q^91 - 10368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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