LMFDB - Embedded newform 600.4.f.f.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,4,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, 4); 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, 4, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,-18,0,32,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$35.4011460034$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.4.f.f.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +20.0000i q^{7} -9.00000 q^{9} +16.0000 q^{11} -58.0000i q^{13} +38.0000i q^{17} -4.00000 q^{19} +60.0000 q^{21} +80.0000i q^{23} +27.0000i q^{27} -82.0000 q^{29} -8.00000 q^{31} -48.0000i q^{33} +426.000i q^{37} -174.000 q^{39} -246.000 q^{41} +524.000i q^{43} -464.000i q^{47} -57.0000 q^{49} +114.000 q^{51} +702.000i q^{53} +12.0000i q^{57} +592.000 q^{59} +574.000 q^{61} -180.000i q^{63} -172.000i q^{67} +240.000 q^{69} +768.000 q^{71} +558.000i q^{73} +320.000i q^{77} -408.000 q^{79} +81.0000 q^{81} -164.000i q^{83} +246.000i q^{87} +510.000 q^{89} +1160.00 q^{91} +24.0000i q^{93} +514.000i q^{97} -144.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9} + 32 q^{11} - 8 q^{19} + 120 q^{21} - 164 q^{29} - 16 q^{31} - 348 q^{39} - 492 q^{41} - 114 q^{49} + 228 q^{51} + 1184 q^{59} + 1148 q^{61} + 480 q^{69} + 1536 q^{71} - 816 q^{79} + 162 q^{81}+ \cdots - 288 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 32 * q^11 - 8 * q^19 + 120 * q^21 - 164 * q^29 - 16 * q^31 - 348 * q^39 - 492 * q^41 - 114 * q^49 + 228 * q^51 + 1184 * q^59 + 1148 * q^61 + 480 * q^69 + 1536 * q^71 - 816 * q^79 + 162 * q^81 + 1020 * q^89 + 2320 * q^91 - 288 * 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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	20.0000i	1.07990i	0.841698	+	0.539949i	$0.181557\pi$
$7$	20.0000i	1.07990i	−0.841698	+	0.539949i	$0.818443\pi$
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	16.0000	0.438562	0.219281	−	0.975662i	$-0.429629\pi$
$11$	16.0000	0.438562	0.219281	+	0.975662i	$0.429629\pi$
$12$	0	0
$13$	− 58.0000i	− 1.23741i	−0.785624	−	0.618704i	$-0.787658\pi$
$13$	− 58.0000i	− 1.23741i	0.785624	−	0.618704i	$-0.212342\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	38.0000i	0.542138i	0.962560	+	0.271069i	$0.0873772\pi$
$17$	38.0000i	0.542138i	−0.962560	+	0.271069i	$0.912623\pi$
$18$	0	0
$19$	−4.00000	−0.0482980	−0.0241490	−	0.999708i	$-0.507688\pi$
$19$	−4.00000	−0.0482980	−0.0241490	+	0.999708i	$0.507688\pi$
$20$	0	0
$21$	60.0000	0.623480
$22$	0	0
$23$	80.0000i	0.725268i	0.931932	+	0.362634i	$0.118122\pi$
$23$	80.0000i	0.725268i	−0.931932	+	0.362634i	$0.881878\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	−82.0000	−0.525070	−0.262535	−	0.964923i	$-0.584558\pi$
$29$	−82.0000	−0.525070	−0.262535	+	0.964923i	$0.584558\pi$
$30$	0	0
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	0	0
$33$	− 48.0000i	− 0.253204i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	426.000i	1.89281i	0.322982	+	0.946405i	$0.395315\pi$
$37$	426.000i	1.89281i	−0.322982	+	0.946405i	$0.604685\pi$
$38$	0	0
$39$	−174.000	−0.714418
$40$	0	0
$41$	−246.000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−246.000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	524.000i	1.85835i	0.369634	+	0.929177i	$0.379483\pi$
$43$	524.000i	1.85835i	−0.369634	+	0.929177i	$0.620517\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 464.000i	− 1.44003i	−0.693959	−	0.720014i	$-0.744135\pi$
$47$	− 464.000i	− 1.44003i	0.693959	−	0.720014i	$-0.255865\pi$
$48$	0	0
$49$	−57.0000	−0.166181
$50$	0	0
$51$	114.000	0.313004
$52$	0	0
$53$	702.000i	1.81938i	0.415288	+	0.909690i	$0.363681\pi$
$53$	702.000i	1.81938i	−0.415288	+	0.909690i	$0.636319\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	12.0000i	0.0278849i
$58$	0	0
$59$	592.000	1.30630	0.653151	−	0.757228i	$-0.273447\pi$
$59$	592.000	1.30630	0.653151	+	0.757228i	$0.273447\pi$
$60$	0	0
$61$	574.000	1.20481	0.602403	−	0.798192i	$-0.294210\pi$
$61$	574.000	1.20481	0.602403	+	0.798192i	$0.294210\pi$
$62$	0	0
$63$	− 180.000i	− 0.359966i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 172.000i	− 0.313629i	−0.987628	−	0.156815i	$-0.949878\pi$
$67$	− 172.000i	− 0.313629i	0.987628	−	0.156815i	$-0.0501225\pi$
$68$	0	0
$69$	240.000	0.418733
$70$	0	0
$71$	768.000	1.28373	0.641865	−	0.766818i	$-0.278161\pi$
$71$	768.000	1.28373	0.641865	+	0.766818i	$0.278161\pi$
$72$	0	0
$73$	558.000i	0.894643i	0.894373	+	0.447322i	$0.147622\pi$
$73$	558.000i	0.894643i	−0.894373	+	0.447322i	$0.852378\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	320.000i	0.473602i
$78$	0	0
$79$	−408.000	−0.581058	−0.290529	−	0.956866i	$-0.593831\pi$
$79$	−408.000	−0.581058	−0.290529	+	0.956866i	$0.593831\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 164.000i	− 0.216884i	−0.994103	−	0.108442i	$-0.965414\pi$
$83$	− 164.000i	− 0.216884i	0.994103	−	0.108442i	$-0.0345861\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	246.000i	0.303149i
$88$	0	0
$89$	510.000	0.607415	0.303707	−	0.952765i	$-0.401776\pi$
$89$	510.000	0.607415	0.303707	+	0.952765i	$0.401776\pi$
$90$	0	0
$91$	1160.00	1.33628
$92$	0	0
$93$	24.0000i	0.0267600i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	514.000i	0.538029i	0.963136	+	0.269014i	$0.0866979\pi$
$97$	514.000i	0.538029i	−0.963136	+	0.269014i	$0.913302\pi$
$98$	0	0
$99$	−144.000	−0.146187
$100$	0	0
$101$	666.000	0.656133	0.328067	−	0.944655i	$-0.393603\pi$
$101$	666.000	0.656133	0.328067	+	0.944655i	$0.393603\pi$
$102$	0	0
$103$	1100.00i	1.05229i	0.850394	+	0.526147i	$0.176364\pi$
$103$	1100.00i	1.05229i	−0.850394	+	0.526147i	$0.823636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1212.00i	1.09503i	0.836795	+	0.547516i	$0.184427\pi$
$107$	1212.00i	1.09503i	−0.836795	+	0.547516i	$0.815573\pi$
$108$	0	0
$109$	−2078.00	−1.82602	−0.913011	−	0.407936i	$-0.866249\pi$
$109$	−2078.00	−1.82602	−0.913011	+	0.407936i	$0.866249\pi$
$110$	0	0
$111$	1278.00	1.09281
$112$	0	0
$113$	1458.00i	1.21378i	0.794786	+	0.606890i	$0.207583\pi$
$113$	1458.00i	1.21378i	−0.794786	+	0.606890i	$0.792417\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	522.000i	0.412469i
$118$	0	0
$119$	−760.000	−0.585455
$120$	0	0
$121$	−1075.00	−0.807663
$122$	0	0
$123$	738.000i	0.541002i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2436.00i	− 1.70205i	−0.525127	−	0.851024i	$-0.675982\pi$
$127$	− 2436.00i	− 1.70205i	0.525127	−	0.851024i	$-0.324018\pi$
$128$	0	0
$129$	1572.00	1.07292
$130$	0	0
$131$	2544.00	1.69672	0.848360	−	0.529420i	$-0.177590\pi$
$131$	2544.00	1.69672	0.848360	+	0.529420i	$0.177590\pi$
$132$	0	0
$133$	− 80.0000i	− 0.0521570i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	694.000i	0.432791i	0.976306	+	0.216396i	$0.0694301\pi$
$137$	694.000i	0.432791i	−0.976306	+	0.216396i	$0.930570\pi$
$138$	0	0
$139$	−516.000	−0.314867	−0.157434	−	0.987530i	$-0.550322\pi$
$139$	−516.000	−0.314867	−0.157434	+	0.987530i	$0.550322\pi$
$140$	0	0
$141$	−1392.00	−0.831401
$142$	0	0
$143$	− 928.000i	− 0.542680i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	171.000i	0.0959445i
$148$	0	0
$149$	−770.000	−0.423361	−0.211681	−	0.977339i	$-0.567894\pi$
$149$	−770.000	−0.423361	−0.211681	+	0.977339i	$0.567894\pi$
$150$	0	0
$151$	−424.000	−0.228507	−0.114254	−	0.993452i	$-0.536448\pi$
$151$	−424.000	−0.228507	−0.114254	+	0.993452i	$0.536448\pi$
$152$	0	0
$153$	− 342.000i	− 0.180713i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	922.000i	0.468685i	0.972154	+	0.234343i	$0.0752938\pi$
$157$	922.000i	0.468685i	−0.972154	+	0.234343i	$0.924706\pi$
$158$	0	0
$159$	2106.00	1.05042
$160$	0	0
$161$	−1600.00	−0.783215
$162$	0	0
$163$	3788.00i	1.82024i	0.414345	+	0.910120i	$0.364011\pi$
$163$	3788.00i	1.82024i	−0.414345	+	0.910120i	$0.635989\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 48.0000i	− 0.0222416i	−0.999938	−	0.0111208i	$-0.996460\pi$
$167$	− 48.0000i	− 0.0222416i	0.999938	−	0.0111208i	$-0.00353994\pi$
$168$	0	0
$169$	−1167.00	−0.531179
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	− 3242.00i	− 1.42477i	−0.701790	−	0.712384i	$-0.747616\pi$
$173$	− 3242.00i	− 1.42477i	0.701790	−	0.712384i	$-0.252384\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1776.00i	− 0.754194i
$178$	0	0
$179$	2728.00	1.13911	0.569554	−	0.821954i	$-0.307116\pi$
$179$	2728.00	1.13911	0.569554	+	0.821954i	$0.307116\pi$
$180$	0	0
$181$	−4090.00	−1.67960	−0.839799	−	0.542897i	$-0.817327\pi$
$181$	−4090.00	−1.67960	−0.839799	+	0.542897i	$0.817327\pi$
$182$	0	0
$183$	− 1722.00i	− 0.695595i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	608.000i	0.237761i
$188$	0	0
$189$	−540.000	−0.207827
$190$	0	0
$191$	−1480.00	−0.560676	−0.280338	−	0.959901i	$-0.590446\pi$
$191$	−1480.00	−0.560676	−0.280338	+	0.959901i	$0.590446\pi$
$192$	0	0
$193$	1622.00i	0.604944i	0.953158	+	0.302472i	$0.0978118\pi$
$193$	1622.00i	0.604944i	−0.953158	+	0.302472i	$0.902188\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2530.00i	0.915000i	0.889210	+	0.457500i	$0.151255\pi$
$197$	2530.00i	0.915000i	−0.889210	+	0.457500i	$0.848745\pi$
$198$	0	0
$199$	2440.00	0.869181	0.434590	−	0.900628i	$-0.356893\pi$
$199$	2440.00	0.869181	0.434590	+	0.900628i	$0.356893\pi$
$200$	0	0
$201$	−516.000	−0.181074
$202$	0	0
$203$	− 1640.00i	− 0.567022i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 720.000i	− 0.241756i
$208$	0	0
$209$	−64.0000	−0.0211817
$210$	0	0
$211$	−148.000	−0.0482879	−0.0241439	−	0.999708i	$-0.507686\pi$
$211$	−148.000	−0.0482879	−0.0241439	+	0.999708i	$0.507686\pi$
$212$	0	0
$213$	− 2304.00i	− 0.741162i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 160.000i	− 0.0500530i
$218$	0	0
$219$	1674.00	0.516523
$220$	0	0
$221$	2204.00	0.670847
$222$	0	0
$223$	676.000i	0.202997i	0.994836	+	0.101498i	$0.0323637\pi$
$223$	676.000i	0.202997i	−0.994836	+	0.101498i	$0.967636\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 6276.00i	− 1.83503i	−0.397696	−	0.917517i	$-0.630190\pi$
$227$	− 6276.00i	− 1.83503i	0.397696	−	0.917517i	$-0.369810\pi$
$228$	0	0
$229$	−6190.00	−1.78623	−0.893115	−	0.449828i	$-0.851485\pi$
$229$	−6190.00	−1.78623	−0.893115	+	0.449828i	$0.851485\pi$
$230$	0	0
$231$	960.000	0.273434
$232$	0	0
$233$	− 5406.00i	− 1.52000i	−0.649926	−	0.759998i	$-0.725200\pi$
$233$	− 5406.00i	− 1.52000i	0.649926	−	0.759998i	$-0.274800\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1224.00i	0.335474i
$238$	0	0
$239$	600.000	0.162388	0.0811941	−	0.996698i	$-0.474127\pi$
$239$	600.000	0.162388	0.0811941	+	0.996698i	$0.474127\pi$
$240$	0	0
$241$	−1054.00	−0.281718	−0.140859	−	0.990030i	$-0.544986\pi$
$241$	−1054.00	−0.281718	−0.140859	+	0.990030i	$0.544986\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	232.000i	0.0597644i
$248$	0	0
$249$	−492.000	−0.125218
$250$	0	0
$251$	−2232.00	−0.561285	−0.280643	−	0.959812i	$-0.590548\pi$
$251$	−2232.00	−0.561285	−0.280643	+	0.959812i	$0.590548\pi$
$252$	0	0
$253$	1280.00i	0.318075i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3630.00i	0.881063i	0.897737	+	0.440531i	$0.145210\pi$
$257$	3630.00i	0.881063i	−0.897737	+	0.440531i	$0.854790\pi$
$258$	0	0
$259$	−8520.00	−2.04404
$260$	0	0
$261$	738.000	0.175023
$262$	0	0
$263$	− 6960.00i	− 1.63183i	−0.578170	−	0.815916i	$-0.696233\pi$
$263$	− 6960.00i	− 1.63183i	0.578170	−	0.815916i	$-0.303767\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1530.00i	− 0.350691i
$268$	0	0
$269$	2062.00	0.467369	0.233685	−	0.972312i	$-0.424922\pi$
$269$	2062.00	0.467369	0.233685	+	0.972312i	$0.424922\pi$
$270$	0	0
$271$	−2544.00	−0.570247	−0.285124	−	0.958491i	$-0.592035\pi$
$271$	−2544.00	−0.570247	−0.285124	+	0.958491i	$0.592035\pi$
$272$	0	0
$273$	− 3480.00i	− 0.771499i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 694.000i	− 0.150536i	−0.997163	−	0.0752679i	$-0.976019\pi$
$277$	− 694.000i	− 0.150536i	0.997163	−	0.0752679i	$-0.0239812\pi$
$278$	0	0
$279$	72.0000	0.0154499
$280$	0	0
$281$	−1982.00	−0.420769	−0.210385	−	0.977619i	$-0.567472\pi$
$281$	−1982.00	−0.420769	−0.210385	+	0.977619i	$0.567472\pi$
$282$	0	0
$283$	− 5228.00i	− 1.09814i	−0.835778	−	0.549068i	$-0.814983\pi$
$283$	− 5228.00i	− 1.09814i	0.835778	−	0.549068i	$-0.185017\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 4920.00i	− 1.01191i
$288$	0	0
$289$	3469.00	0.706086
$290$	0	0
$291$	1542.00	0.310631
$292$	0	0
$293$	7454.00i	1.48624i	0.669160	+	0.743118i	$0.266654\pi$
$293$	7454.00i	1.48624i	−0.669160	+	0.743118i	$0.733346\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	432.000i	0.0844013i
$298$	0	0
$299$	4640.00	0.897452
$300$	0	0
$301$	−10480.0	−2.00683
$302$	0	0
$303$	− 1998.00i	− 0.378819i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1316.00i	− 0.244652i	−0.992490	−	0.122326i	$-0.960965\pi$
$307$	− 1316.00i	− 0.244652i	0.992490	−	0.122326i	$-0.0390353\pi$
$308$	0	0
$309$	3300.00	0.607542
$310$	0	0
$311$	−832.000	−0.151699	−0.0758495	−	0.997119i	$-0.524167\pi$
$311$	−832.000	−0.151699	−0.0758495	+	0.997119i	$0.524167\pi$
$312$	0	0
$313$	− 6770.00i	− 1.22257i	−0.791412	−	0.611283i	$-0.790654\pi$
$313$	− 6770.00i	− 1.22257i	0.791412	−	0.611283i	$-0.209346\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6582.00i	− 1.16619i	−0.812404	−	0.583095i	$-0.801842\pi$
$317$	− 6582.00i	− 1.16619i	0.812404	−	0.583095i	$-0.198158\pi$
$318$	0	0
$319$	−1312.00	−0.230276
$320$	0	0
$321$	3636.00	0.632217
$322$	0	0
$323$	− 152.000i	− 0.0261842i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6234.00i	1.05425i
$328$	0	0
$329$	9280.00	1.55508
$330$	0	0
$331$	11292.0	1.87512	0.937560	−	0.347825i	$-0.113080\pi$
$331$	11292.0	1.87512	0.937560	+	0.347825i	$0.113080\pi$
$332$	0	0
$333$	− 3834.00i	− 0.630937i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8006.00i	− 1.29411i	−0.762444	−	0.647054i	$-0.776001\pi$
$337$	− 8006.00i	− 1.29411i	0.762444	−	0.647054i	$-0.223999\pi$
$338$	0	0
$339$	4374.00	0.700776
$340$	0	0
$341$	−128.000	−0.0203272
$342$	0	0
$343$	5720.00i	0.900440i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 316.000i	− 0.0488869i	−0.999701	−	0.0244435i	$-0.992219\pi$
$347$	− 316.000i	− 0.0488869i	0.999701	−	0.0244435i	$-0.00778137\pi$
$348$	0	0
$349$	−4926.00	−0.755538	−0.377769	−	0.925900i	$-0.623309\pi$
$349$	−4926.00	−0.755538	−0.377769	+	0.925900i	$0.623309\pi$
$350$	0	0
$351$	1566.00	0.238139
$352$	0	0
$353$	− 2438.00i	− 0.367597i	−0.982964	−	0.183798i	$-0.941161\pi$
$353$	− 2438.00i	− 0.367597i	0.982964	−	0.183798i	$-0.0588393\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2280.00i	0.338012i
$358$	0	0
$359$	3336.00	0.490438	0.245219	−	0.969468i	$-0.421140\pi$
$359$	3336.00	0.490438	0.245219	+	0.969468i	$0.421140\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	3225.00i	0.466305i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	44.0000i	0.00625826i	0.999995	+	0.00312913i	$0.000996035\pi$
$367$	44.0000i	0.00625826i	−0.999995	+	0.00312913i	$0.999004\pi$
$368$	0	0
$369$	2214.00	0.312348
$370$	0	0
$371$	−14040.0	−1.96475
$372$	0	0
$373$	11966.0i	1.66106i	0.556973	+	0.830531i	$0.311963\pi$
$373$	11966.0i	1.66106i	−0.556973	+	0.830531i	$0.688037\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4756.00i	0.649725i
$378$	0	0
$379$	−12676.0	−1.71800	−0.859001	−	0.511975i	$-0.828914\pi$
$379$	−12676.0	−1.71800	−0.859001	+	0.511975i	$0.828914\pi$
$380$	0	0
$381$	−7308.00	−0.982678
$382$	0	0
$383$	− 6672.00i	− 0.890139i	−0.895496	−	0.445070i	$-0.853179\pi$
$383$	− 6672.00i	− 0.890139i	0.895496	−	0.445070i	$-0.146821\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4716.00i	− 0.619452i
$388$	0	0
$389$	−354.000	−0.0461401	−0.0230701	−	0.999734i	$-0.507344\pi$
$389$	−354.000	−0.0461401	−0.0230701	+	0.999734i	$0.507344\pi$
$390$	0	0
$391$	−3040.00	−0.393195
$392$	0	0
$393$	− 7632.00i	− 0.979602i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5054.00i	− 0.638924i	−0.947599	−	0.319462i	$-0.896498\pi$
$397$	− 5054.00i	− 0.638924i	0.947599	−	0.319462i	$-0.103502\pi$
$398$	0	0
$399$	−240.000	−0.0301129
$400$	0	0
$401$	10266.0	1.27845	0.639226	−	0.769019i	$-0.279255\pi$
$401$	10266.0	1.27845	0.639226	+	0.769019i	$0.279255\pi$
$402$	0	0
$403$	464.000i	0.0573536i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6816.00i	0.830114i
$408$	0	0
$409$	1526.00	0.184489	0.0922443	−	0.995736i	$-0.470596\pi$
$409$	1526.00	0.184489	0.0922443	+	0.995736i	$0.470596\pi$
$410$	0	0
$411$	2082.00	0.249872
$412$	0	0
$413$	11840.0i	1.41067i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1548.00i	0.181789i
$418$	0	0
$419$	−2064.00	−0.240652	−0.120326	−	0.992734i	$-0.538394\pi$
$419$	−2064.00	−0.240652	−0.120326	+	0.992734i	$0.538394\pi$
$420$	0	0
$421$	4590.00	0.531361	0.265680	−	0.964061i	$-0.414403\pi$
$421$	4590.00	0.531361	0.265680	+	0.964061i	$0.414403\pi$
$422$	0	0
$423$	4176.00i	0.480010i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11480.0i	1.30107i
$428$	0	0
$429$	−2784.00	−0.313317
$430$	0	0
$431$	−5536.00	−0.618700	−0.309350	−	0.950948i	$-0.600111\pi$
$431$	−5536.00	−0.618700	−0.309350	+	0.950948i	$0.600111\pi$
$432$	0	0
$433$	− 1850.00i	− 0.205324i	−0.994716	−	0.102662i	$-0.967264\pi$
$433$	− 1850.00i	− 0.205324i	0.994716	−	0.102662i	$-0.0327360\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 320.000i	− 0.0350290i
$438$	0	0
$439$	−11704.0	−1.27244	−0.636220	−	0.771507i	$-0.719503\pi$
$439$	−11704.0	−1.27244	−0.636220	+	0.771507i	$0.719503\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	− 6948.00i	− 0.745168i	−0.927998	−	0.372584i	$-0.878472\pi$
$443$	− 6948.00i	− 0.745168i	0.927998	−	0.372584i	$-0.121528\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2310.00i	0.244428i
$448$	0	0
$449$	−12090.0	−1.27074	−0.635370	−	0.772208i	$-0.719152\pi$
$449$	−12090.0	−1.27074	−0.635370	+	0.772208i	$0.719152\pi$
$450$	0	0
$451$	−3936.00	−0.410951
$452$	0	0
$453$	1272.00i	0.131929i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11626.0i	1.19002i	0.803717	+	0.595012i	$0.202853\pi$
$457$	11626.0i	1.19002i	−0.803717	+	0.595012i	$0.797147\pi$
$458$	0	0
$459$	−1026.00	−0.104335
$460$	0	0
$461$	16314.0	1.64820	0.824098	−	0.566447i	$-0.191682\pi$
$461$	16314.0	1.64820	0.824098	+	0.566447i	$0.191682\pi$
$462$	0	0
$463$	15756.0i	1.58152i	0.612127	+	0.790760i	$0.290314\pi$
$463$	15756.0i	1.58152i	−0.612127	+	0.790760i	$0.709686\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5684.00i	0.563221i	0.959529	+	0.281610i	$0.0908686\pi$
$467$	5684.00i	0.563221i	−0.959529	+	0.281610i	$0.909131\pi$
$468$	0	0
$469$	3440.00	0.338688
$470$	0	0
$471$	2766.00	0.270596
$472$	0	0
$473$	8384.00i	0.815004i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6318.00i	− 0.606460i
$478$	0	0
$479$	3368.00	0.321269	0.160634	−	0.987014i	$-0.448646\pi$
$479$	3368.00	0.321269	0.160634	+	0.987014i	$0.448646\pi$
$480$	0	0
$481$	24708.0	2.34218
$482$	0	0
$483$	4800.00i	0.452190i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 5588.00i	− 0.519952i	−0.965615	−	0.259976i	$-0.916285\pi$
$487$	− 5588.00i	− 0.519952i	0.965615	−	0.259976i	$-0.0837146\pi$
$488$	0	0
$489$	11364.0	1.05092
$490$	0	0
$491$	10584.0	0.972809	0.486405	−	0.873734i	$-0.338308\pi$
$491$	10584.0	0.972809	0.486405	+	0.873734i	$0.338308\pi$
$492$	0	0
$493$	− 3116.00i	− 0.284660i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15360.0i	1.38630i
$498$	0	0
$499$	12220.0	1.09628	0.548139	−	0.836388i	$-0.315337\pi$
$499$	12220.0	1.09628	0.548139	+	0.836388i	$0.315337\pi$
$500$	0	0
$501$	−144.000	−0.0128412
$502$	0	0
$503$	− 16152.0i	− 1.43177i	−0.698216	−	0.715887i	$-0.746023\pi$
$503$	− 16152.0i	− 1.43177i	0.698216	−	0.715887i	$-0.253977\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3501.00i	0.306676i
$508$	0	0
$509$	−10642.0	−0.926716	−0.463358	−	0.886171i	$-0.653356\pi$
$509$	−10642.0	−0.926716	−0.463358	+	0.886171i	$0.653356\pi$
$510$	0	0
$511$	−11160.0	−0.966124
$512$	0	0
$513$	− 108.000i	− 0.00929496i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7424.00i	− 0.631542i
$518$	0	0
$519$	−9726.00	−0.822590
$520$	0	0
$521$	22882.0	1.92414	0.962072	−	0.272797i	$-0.0879487\pi$
$521$	22882.0	1.92414	0.962072	+	0.272797i	$0.0879487\pi$
$522$	0	0
$523$	10052.0i	0.840427i	0.907425	+	0.420213i	$0.138045\pi$
$523$	10052.0i	0.840427i	−0.907425	+	0.420213i	$0.861955\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 304.000i	− 0.0251280i
$528$	0	0
$529$	5767.00	0.473987
$530$	0	0
$531$	−5328.00	−0.435434
$532$	0	0
$533$	14268.0i	1.15950i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 8184.00i	− 0.657664i
$538$	0	0
$539$	−912.000	−0.0728806
$540$	0	0
$541$	−6530.00	−0.518940	−0.259470	−	0.965751i	$-0.583548\pi$
$541$	−6530.00	−0.518940	−0.259470	+	0.965751i	$0.583548\pi$
$542$	0	0
$543$	12270.0i	0.969717i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16652.0i	1.30162i	0.759239	+	0.650812i	$0.225571\pi$
$547$	16652.0i	1.30162i	−0.759239	+	0.650812i	$0.774429\pi$
$548$	0	0
$549$	−5166.00	−0.401602
$550$	0	0
$551$	328.000	0.0253598
$552$	0	0
$553$	− 8160.00i	− 0.627484i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12886.0i	− 0.980247i	−0.871653	−	0.490123i	$-0.836952\pi$
$557$	− 12886.0i	− 0.980247i	0.871653	−	0.490123i	$-0.163048\pi$
$558$	0	0
$559$	30392.0	2.29954
$560$	0	0
$561$	1824.00	0.137272
$562$	0	0
$563$	11108.0i	0.831521i	0.909474	+	0.415761i	$0.136485\pi$
$563$	11108.0i	0.831521i	−0.909474	+	0.415761i	$0.863515\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1620.00i	0.119989i
$568$	0	0
$569$	9214.00	0.678859	0.339430	−	0.940631i	$-0.389766\pi$
$569$	9214.00	0.678859	0.339430	+	0.940631i	$0.389766\pi$
$570$	0	0
$571$	−4052.00	−0.296972	−0.148486	−	0.988915i	$-0.547440\pi$
$571$	−4052.00	−0.296972	−0.148486	+	0.988915i	$0.547440\pi$
$572$	0	0
$573$	4440.00i	0.323706i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8446.00i	− 0.609379i	−0.952452	−	0.304689i	$-0.901447\pi$
$577$	− 8446.00i	− 0.609379i	0.952452	−	0.304689i	$-0.0985527\pi$
$578$	0	0
$579$	4866.00	0.349264
$580$	0	0
$581$	3280.00	0.234212
$582$	0	0
$583$	11232.0i	0.797911i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2172.00i	0.152722i	0.997080	+	0.0763612i	$0.0243302\pi$
$587$	2172.00i	0.152722i	−0.997080	+	0.0763612i	$0.975670\pi$
$588$	0	0
$589$	32.0000	0.00223860
$590$	0	0
$591$	7590.00	0.528276
$592$	0	0
$593$	1218.00i	0.0843461i	0.999110	+	0.0421731i	$0.0134281\pi$
$593$	1218.00i	0.0843461i	−0.999110	+	0.0421731i	$0.986572\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 7320.00i	− 0.501822i
$598$	0	0
$599$	−21240.0	−1.44882	−0.724410	−	0.689370i	$-0.757888\pi$
$599$	−21240.0	−1.44882	−0.724410	+	0.689370i	$0.757888\pi$
$600$	0	0
$601$	17626.0	1.19631	0.598153	−	0.801382i	$-0.295902\pi$
$601$	17626.0	1.19631	0.598153	+	0.801382i	$0.295902\pi$
$602$	0	0
$603$	1548.00i	0.104543i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2580.00i	0.172519i	0.996273	+	0.0862594i	$0.0274914\pi$
$607$	2580.00i	0.172519i	−0.996273	+	0.0862594i	$0.972509\pi$
$608$	0	0
$609$	−4920.00	−0.327370
$610$	0	0
$611$	−26912.0	−1.78190
$612$	0	0
$613$	14166.0i	0.933376i	0.884422	+	0.466688i	$0.154553\pi$
$613$	14166.0i	0.933376i	−0.884422	+	0.466688i	$0.845447\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21426.0i	− 1.39802i	−0.715112	−	0.699010i	$-0.753624\pi$
$617$	− 21426.0i	− 1.39802i	0.715112	−	0.699010i	$-0.246376\pi$
$618$	0	0
$619$	−3668.00	−0.238173	−0.119087	−	0.992884i	$-0.537997\pi$
$619$	−3668.00	−0.238173	−0.119087	+	0.992884i	$0.537997\pi$
$620$	0	0
$621$	−2160.00	−0.139578
$622$	0	0
$623$	10200.0i	0.655946i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	192.000i	0.0122293i
$628$	0	0
$629$	−16188.0	−1.02617
$630$	0	0
$631$	20032.0	1.26381	0.631903	−	0.775048i	$-0.282274\pi$
$631$	20032.0	1.26381	0.631903	+	0.775048i	$0.282274\pi$
$632$	0	0
$633$	444.000i	0.0278790i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3306.00i	0.205633i
$638$	0	0
$639$	−6912.00	−0.427910
$640$	0	0
$641$	7458.00	0.459553	0.229776	−	0.973243i	$-0.426201\pi$
$641$	7458.00	0.459553	0.229776	+	0.973243i	$0.426201\pi$
$642$	0	0
$643$	− 7092.00i	− 0.434963i	−0.976064	−	0.217481i	$-0.930216\pi$
$643$	− 7092.00i	− 0.434963i	0.976064	−	0.217481i	$-0.0697842\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3384.00i	0.205624i	0.994701	+	0.102812i	$0.0327840\pi$
$647$	3384.00i	0.205624i	−0.994701	+	0.102812i	$0.967216\pi$
$648$	0	0
$649$	9472.00	0.572894
$650$	0	0
$651$	−480.000	−0.0288981
$652$	0	0
$653$	29398.0i	1.76177i	0.473335	+	0.880883i	$0.343050\pi$
$653$	29398.0i	1.76177i	−0.473335	+	0.880883i	$0.656950\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 5022.00i	− 0.298214i
$658$	0	0
$659$	6624.00	0.391554	0.195777	−	0.980648i	$-0.437277\pi$
$659$	6624.00	0.391554	0.195777	+	0.980648i	$0.437277\pi$
$660$	0	0
$661$	8646.00	0.508760	0.254380	−	0.967104i	$-0.418129\pi$
$661$	8646.00	0.508760	0.254380	+	0.967104i	$0.418129\pi$
$662$	0	0
$663$	− 6612.00i	− 0.387313i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 6560.00i	− 0.380816i
$668$	0	0
$669$	2028.00	0.117200
$670$	0	0
$671$	9184.00	0.528382
$672$	0	0
$673$	− 28698.0i	− 1.64372i	−0.569686	−	0.821862i	$-0.692935\pi$
$673$	− 28698.0i	− 1.64372i	0.569686	−	0.821862i	$-0.307065\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19426.0i	1.10281i	0.834238	+	0.551405i	$0.185908\pi$
$677$	19426.0i	1.10281i	−0.834238	+	0.551405i	$0.814092\pi$
$678$	0	0
$679$	−10280.0	−0.581016
$680$	0	0
$681$	−18828.0	−1.05946
$682$	0	0
$683$	− 8604.00i	− 0.482025i	−0.970522	−	0.241012i	$-0.922521\pi$
$683$	− 8604.00i	− 0.482025i	0.970522	−	0.241012i	$-0.0774794\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18570.0i	1.03128i
$688$	0	0
$689$	40716.0	2.25132
$690$	0	0
$691$	12980.0	0.714591	0.357296	−	0.933991i	$-0.383699\pi$
$691$	12980.0	0.714591	0.357296	+	0.933991i	$0.383699\pi$
$692$	0	0
$693$	− 2880.00i	− 0.157867i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 9348.00i	− 0.508007i
$698$	0	0
$699$	−16218.0	−0.877570
$700$	0	0
$701$	−19630.0	−1.05765	−0.528827	−	0.848730i	$-0.677368\pi$
$701$	−19630.0	−1.05765	−0.528827	+	0.848730i	$0.677368\pi$
$702$	0	0
$703$	− 1704.00i	− 0.0914190i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13320.0i	0.708558i
$708$	0	0
$709$	−8030.00	−0.425350	−0.212675	−	0.977123i	$-0.568218\pi$
$709$	−8030.00	−0.425350	−0.212675	+	0.977123i	$0.568218\pi$
$710$	0	0
$711$	3672.00	0.193686
$712$	0	0
$713$	− 640.000i	− 0.0336160i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1800.00i	− 0.0937549i
$718$	0	0
$719$	−22720.0	−1.17846	−0.589230	−	0.807965i	$-0.700569\pi$
$719$	−22720.0	−1.17846	−0.589230	+	0.807965i	$0.700569\pi$
$720$	0	0
$721$	−22000.0	−1.13637
$722$	0	0
$723$	3162.00i	0.162650i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27116.0i	1.38332i	0.722221	+	0.691662i	$0.243121\pi$
$727$	27116.0i	1.38332i	−0.722221	+	0.691662i	$0.756879\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	−19912.0	−1.00749
$732$	0	0
$733$	− 30882.0i	− 1.55614i	−0.628176	−	0.778071i	$-0.716198\pi$
$733$	− 30882.0i	− 1.55614i	0.628176	−	0.778071i	$-0.283802\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2752.00i	− 0.137546i
$738$	0	0
$739$	13836.0	0.688722	0.344361	−	0.938837i	$-0.388096\pi$
$739$	13836.0	0.688722	0.344361	+	0.938837i	$0.388096\pi$
$740$	0	0
$741$	696.000	0.0345050
$742$	0	0
$743$	− 32712.0i	− 1.61519i	−0.589737	−	0.807595i	$-0.700769\pi$
$743$	− 32712.0i	− 1.61519i	0.589737	−	0.807595i	$-0.299231\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1476.00i	0.0722945i
$748$	0	0
$749$	−24240.0	−1.18252
$750$	0	0
$751$	−8472.00	−0.411648	−0.205824	−	0.978589i	$-0.565987\pi$
$751$	−8472.00	−0.411648	−0.205824	+	0.978589i	$0.565987\pi$
$752$	0	0
$753$	6696.00i	0.324058i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9866.00i	0.473693i	0.971547	+	0.236847i	$0.0761139\pi$
$757$	9866.00i	0.473693i	−0.971547	+	0.236847i	$0.923886\pi$
$758$	0	0
$759$	3840.00	0.183641
$760$	0	0
$761$	−3774.00	−0.179773	−0.0898866	−	0.995952i	$-0.528650\pi$
$761$	−3774.00	−0.179773	−0.0898866	+	0.995952i	$0.528650\pi$
$762$	0	0
$763$	− 41560.0i	− 1.97192i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 34336.0i	− 1.61643i
$768$	0	0
$769$	28670.0	1.34443	0.672215	−	0.740356i	$-0.265343\pi$
$769$	28670.0	1.34443	0.672215	+	0.740356i	$0.265343\pi$
$770$	0	0
$771$	10890.0	0.508682
$772$	0	0
$773$	3246.00i	0.151036i	0.997144	+	0.0755178i	$0.0240610\pi$
$773$	3246.00i	0.151036i	−0.997144	+	0.0755178i	$0.975939\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	25560.0i	1.18013i
$778$	0	0
$779$	984.000	0.0452573
$780$	0	0
$781$	12288.0	0.562995
$782$	0	0
$783$	− 2214.00i	− 0.101050i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 19372.0i	− 0.877430i	−0.898626	−	0.438715i	$-0.855434\pi$
$787$	− 19372.0i	− 0.877430i	0.898626	−	0.438715i	$-0.144566\pi$
$788$	0	0
$789$	−20880.0	−0.942139
$790$	0	0
$791$	−29160.0	−1.31076
$792$	0	0
$793$	− 33292.0i	− 1.49084i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 11814.0i	− 0.525061i	−0.964924	−	0.262530i	$-0.915443\pi$
$797$	− 11814.0i	− 0.525061i	0.964924	−	0.262530i	$-0.0845570\pi$
$798$	0	0
$799$	17632.0	0.780695
$800$	0	0
$801$	−4590.00	−0.202472
$802$	0	0
$803$	8928.00i	0.392357i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 6186.00i	− 0.269836i
$808$	0	0
$809$	30054.0	1.30611	0.653055	−	0.757311i	$-0.273487\pi$
$809$	30054.0	1.30611	0.653055	+	0.757311i	$0.273487\pi$
$810$	0	0
$811$	2852.00	0.123486	0.0617431	−	0.998092i	$-0.480334\pi$
$811$	2852.00	0.123486	0.0617431	+	0.998092i	$0.480334\pi$
$812$	0	0
$813$	7632.00i	0.329232i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2096.00i	− 0.0897549i
$818$	0	0
$819$	−10440.0	−0.445425
$820$	0	0
$821$	2170.00	0.0922455	0.0461227	−	0.998936i	$-0.485313\pi$
$821$	2170.00	0.0922455	0.0461227	+	0.998936i	$0.485313\pi$
$822$	0	0
$823$	− 19804.0i	− 0.838790i	−0.907804	−	0.419395i	$-0.862242\pi$
$823$	− 19804.0i	− 0.838790i	0.907804	−	0.419395i	$-0.137758\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5508.00i	0.231598i	0.993273	+	0.115799i	$0.0369429\pi$
$827$	5508.00i	0.231598i	−0.993273	+	0.115799i	$0.963057\pi$
$828$	0	0
$829$	−33262.0	−1.39353	−0.696765	−	0.717299i	$-0.745378\pi$
$829$	−33262.0	−1.39353	−0.696765	+	0.717299i	$0.745378\pi$
$830$	0	0
$831$	−2082.00	−0.0869119
$832$	0	0
$833$	− 2166.00i	− 0.0900930i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 216.000i	− 0.00892001i
$838$	0	0
$839$	4600.00	0.189284	0.0946422	−	0.995511i	$-0.469829\pi$
$839$	4600.00	0.189284	0.0946422	+	0.995511i	$0.469829\pi$
$840$	0	0
$841$	−17665.0	−0.724302
$842$	0	0
$843$	5946.00i	0.242931i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 21500.0i	− 0.872195i
$848$	0	0
$849$	−15684.0	−0.634009
$850$	0	0
$851$	−34080.0	−1.37279
$852$	0	0
$853$	4198.00i	0.168507i	0.996444	+	0.0842537i	$0.0268506\pi$
$853$	4198.00i	0.168507i	−0.996444	+	0.0842537i	$0.973149\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 5826.00i	− 0.232220i	−0.993236	−	0.116110i	$-0.962958\pi$
$857$	− 5826.00i	− 0.232220i	0.993236	−	0.116110i	$-0.0370425\pi$
$858$	0	0
$859$	3004.00	0.119319	0.0596596	−	0.998219i	$-0.480998\pi$
$859$	3004.00	0.119319	0.0596596	+	0.998219i	$0.480998\pi$
$860$	0	0
$861$	−14760.0	−0.584227
$862$	0	0
$863$	36936.0i	1.45691i	0.685092	+	0.728457i	$0.259762\pi$
$863$	36936.0i	1.45691i	−0.685092	+	0.728457i	$0.740238\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 10407.0i	− 0.407659i
$868$	0	0
$869$	−6528.00	−0.254830
$870$	0	0
$871$	−9976.00	−0.388087
$872$	0	0
$873$	− 4626.00i	− 0.179343i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5434.00i	0.209228i	0.994513	+	0.104614i	$0.0333607\pi$
$877$	5434.00i	0.209228i	−0.994513	+	0.104614i	$0.966639\pi$
$878$	0	0
$879$	22362.0	0.858079
$880$	0	0
$881$	−4758.00	−0.181954	−0.0909768	−	0.995853i	$-0.528999\pi$
$881$	−4758.00	−0.181954	−0.0909768	+	0.995853i	$0.528999\pi$
$882$	0	0
$883$	− 15476.0i	− 0.589818i	−0.955525	−	0.294909i	$-0.904711\pi$
$883$	− 15476.0i	− 0.589818i	0.955525	−	0.294909i	$-0.0952893\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 27440.0i	− 1.03872i	−0.854555	−	0.519360i	$-0.826170\pi$
$887$	− 27440.0i	− 1.03872i	0.854555	−	0.519360i	$-0.173830\pi$
$888$	0	0
$889$	48720.0	1.83804
$890$	0	0
$891$	1296.00	0.0487291
$892$	0	0
$893$	1856.00i	0.0695506i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 13920.0i	− 0.518144i
$898$	0	0
$899$	656.000	0.0243368
$900$	0	0
$901$	−26676.0	−0.986356
$902$	0	0
$903$	31440.0i	1.15865i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 48924.0i	− 1.79106i	−0.444997	−	0.895532i	$-0.646795\pi$
$907$	− 48924.0i	− 1.79106i	0.444997	−	0.895532i	$-0.353205\pi$
$908$	0	0
$909$	−5994.00	−0.218711
$910$	0	0
$911$	−3440.00	−0.125107	−0.0625534	−	0.998042i	$-0.519924\pi$
$911$	−3440.00	−0.125107	−0.0625534	+	0.998042i	$0.519924\pi$
$912$	0	0
$913$	− 2624.00i	− 0.0951169i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	50880.0i	1.83229i
$918$	0	0
$919$	27184.0	0.975753	0.487877	−	0.872913i	$-0.337772\pi$
$919$	27184.0	0.975753	0.487877	+	0.872913i	$0.337772\pi$
$920$	0	0
$921$	−3948.00	−0.141250
$922$	0	0
$923$	− 44544.0i	− 1.58850i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 9900.00i	− 0.350764i
$928$	0	0
$929$	−42490.0	−1.50059	−0.750297	−	0.661101i	$-0.770089\pi$
$929$	−42490.0	−1.50059	−0.750297	+	0.661101i	$0.770089\pi$
$930$	0	0
$931$	228.000	0.00802621
$932$	0	0
$933$	2496.00i	0.0875835i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37354.0i	1.30235i	0.758928	+	0.651175i	$0.225724\pi$
$937$	37354.0i	1.30235i	−0.758928	+	0.651175i	$0.774276\pi$
$938$	0	0
$939$	−20310.0	−0.705849
$940$	0	0
$941$	−24470.0	−0.847714	−0.423857	−	0.905729i	$-0.639324\pi$
$941$	−24470.0	−0.847714	−0.423857	+	0.905729i	$0.639324\pi$
$942$	0	0
$943$	− 19680.0i	− 0.679607i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34100.0i	− 1.17012i	−0.810991	−	0.585059i	$-0.801071\pi$
$947$	− 34100.0i	− 1.17012i	0.810991	−	0.585059i	$-0.198929\pi$
$948$	0	0
$949$	32364.0	1.10704
$950$	0	0
$951$	−19746.0	−0.673300
$952$	0	0
$953$	− 1878.00i	− 0.0638346i	−0.999491	−	0.0319173i	$-0.989839\pi$
$953$	− 1878.00i	− 0.0638346i	0.999491	−	0.0319173i	$-0.0101613\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3936.00i	0.132950i
$958$	0	0
$959$	−13880.0	−0.467371
$960$	0	0
$961$	−29727.0	−0.997852
$962$	0	0
$963$	− 10908.0i	− 0.365011i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38484.0i	1.27980i	0.768460	+	0.639898i	$0.221023\pi$
$967$	38484.0i	1.27980i	−0.768460	+	0.639898i	$0.778977\pi$
$968$	0	0
$969$	−456.000	−0.0151175
$970$	0	0
$971$	45272.0	1.49624	0.748119	−	0.663564i	$-0.230957\pi$
$971$	45272.0	1.49624	0.748119	+	0.663564i	$0.230957\pi$
$972$	0	0
$973$	− 10320.0i	− 0.340025i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 25354.0i	− 0.830242i	−0.909766	−	0.415121i	$-0.863739\pi$
$977$	− 25354.0i	− 0.830242i	0.909766	−	0.415121i	$-0.136261\pi$
$978$	0	0
$979$	8160.00	0.266389
$980$	0	0
$981$	18702.0	0.608674
$982$	0	0
$983$	18744.0i	0.608180i	0.952643	+	0.304090i	$0.0983523\pi$
$983$	18744.0i	0.608180i	−0.952643	+	0.304090i	$0.901648\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 27840.0i	− 0.897829i
$988$	0	0
$989$	−41920.0	−1.34780
$990$	0	0
$991$	59600.0	1.91045	0.955225	−	0.295880i	$-0.0956127\pi$
$991$	59600.0	1.91045	0.955225	+	0.295880i	$0.0956127\pi$
$992$	0	0
$993$	− 33876.0i	− 1.08260i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 17886.0i	− 0.568160i	−0.958801	−	0.284080i	$-0.908312\pi$
$997$	− 17886.0i	− 0.568160i	0.958801	−	0.284080i	$-0.0916881\pi$
$998$	0	0
$999$	−11502.0	−0.364271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.4.f.f.49.1		2
3.2	odd	2		1800.4.f.k.649.2		2
4.3	odd	2		1200.4.f.h.49.2		2
5.2	odd	4		600.4.a.a.1.1		1
5.3	odd	4		120.4.a.e.1.1	✓	1
5.4	even	2	inner	600.4.f.f.49.2		2
15.2	even	4		1800.4.a.e.1.1		1
15.8	even	4		360.4.a.m.1.1		1
15.14	odd	2		1800.4.f.k.649.1		2
20.3	even	4		240.4.a.a.1.1		1
20.7	even	4		1200.4.a.bj.1.1		1
20.19	odd	2		1200.4.f.h.49.1		2
40.3	even	4		960.4.a.bd.1.1		1
40.13	odd	4		960.4.a.q.1.1		1
60.23	odd	4		720.4.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.e.1.1	✓	1	5.3	odd	4
240.4.a.a.1.1		1	20.3	even	4
360.4.a.m.1.1		1	15.8	even	4
600.4.a.a.1.1		1	5.2	odd	4
600.4.f.f.49.1		2	1.1	even	1	trivial
600.4.f.f.49.2		2	5.4	even	2	inner
720.4.a.s.1.1		1	60.23	odd	4
960.4.a.q.1.1		1	40.13	odd	4
960.4.a.bd.1.1		1	40.3	even	4
1200.4.a.bj.1.1		1	20.7	even	4
1200.4.f.h.49.1		2	20.19	odd	2
1200.4.f.h.49.2		2	4.3	odd	2
1800.4.a.e.1.1		1	15.2	even	4
1800.4.f.k.649.1		2	15.14	odd	2
1800.4.f.k.649.2		2	3.2	odd	2

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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +20.0000i q^{7} -9.00000 q^{9} +16.0000 q^{11} -58.0000i q^{13} +38.0000i q^{17} -4.00000 q^{19} +60.0000 q^{21} +80.0000i q^{23} +27.0000i q^{27} -82.0000 q^{29} -8.00000 q^{31} -48.0000i q^{33} +426.000i q^{37} -174.000 q^{39} -246.000 q^{41} +524.000i q^{43} -464.000i q^{47} -57.0000 q^{49} +114.000 q^{51} +702.000i q^{53} +12.0000i q^{57} +592.000 q^{59} +574.000 q^{61} -180.000i q^{63} -172.000i q^{67} +240.000 q^{69} +768.000 q^{71} +558.000i q^{73} +320.000i q^{77} -408.000 q^{79} +81.0000 q^{81} -164.000i q^{83} +246.000i q^{87} +510.000 q^{89} +1160.00 q^{91} +24.0000i q^{93} +514.000i q^{97} -144.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9} + 32 q^{11} - 8 q^{19} + 120 q^{21} - 164 q^{29} - 16 q^{31} - 348 q^{39} - 492 q^{41} - 114 q^{49} + 228 q^{51} + 1184 q^{59} + 1148 q^{61} + 480 q^{69} + 1536 q^{71} - 816 q^{79} + 162 q^{81}+ \cdots - 288 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 32 * q^11 - 8 * q^19 + 120 * q^21 - 164 * q^29 - 16 * q^31 - 348 * q^39 - 492 * q^41 - 114 * q^49 + 228 * q^51 + 1184 * q^59 + 1148 * q^61 + 480 * q^69 + 1536 * q^71 - 816 * q^79 + 162 * q^81 + 1020 * q^89 + 2320 * q^91 - 288 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

Properties

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