LMFDB - Embedded newform 2028.2.b.f.337.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2028,2,Mod(337,2028)] 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("2028.337"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2028, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2028 = 2^{2} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2028.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,0,0,0,0,6,0,0,0,0,0,0,0,-26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$16.1936615299$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.153664.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 5x^{4} + 6x^{2} + 1 $ x^6 + 5x^4 + 6x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.3
Root			$-1.80194i$ of defining polynomial
Character	$\chi$	$=$	2028.337
Dual form			2028.2.b.f.337.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.554958i q^{5} +1.04892i q^{7} +1.00000 q^{9} -2.91185i q^{11} +0.554958i q^{15} -2.75302 q^{17} +4.63102i q^{19} -1.04892i q^{21} +5.76271 q^{23} +4.69202 q^{25} -1.00000 q^{27} -2.80194 q^{29} -4.18598i q^{31} +2.91185i q^{33} +0.582105 q^{35} -0.466812i q^{37} -3.89977i q^{41} -9.19567 q^{43} -0.554958i q^{45} -11.5211i q^{47} +5.89977 q^{49} +2.75302 q^{51} +5.62565 q^{53} -1.61596 q^{55} -4.63102i q^{57} -3.10992i q^{59} +10.9051 q^{61} +1.04892i q^{63} +8.04892i q^{67} -5.76271 q^{69} +13.6920i q^{71} -9.36658i q^{73} -4.69202 q^{75} +3.05429 q^{77} -3.60925 q^{79} +1.00000 q^{81} +1.65519i q^{83} +1.52781i q^{85} +2.80194 q^{87} -17.9705i q^{89} +4.18598i q^{93} +2.57002 q^{95} -1.31767i q^{97} -2.91185i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{9} - 26 q^{17} + 18 q^{25} - 6 q^{27} - 8 q^{29} - 8 q^{35} + 18 q^{43} - 10 q^{49} + 26 q^{51} + 10 q^{53} - 30 q^{55} + 14 q^{61} - 18 q^{75} - 6 q^{77} + 2 q^{79} + 6 q^{81} + 8 q^{87}+ \cdots - 34 q^{95}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^9 - 26 * q^17 + 18 * q^25 - 6 * q^27 - 8 * q^29 - 8 * q^35 + 18 * q^43 - 10 * q^49 + 26 * q^51 + 10 * q^53 - 30 * q^55 + 14 * q^61 - 18 * q^75 - 6 * q^77 + 2 * q^79 + 6 * q^81 + 8 * q^87 - 34 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2028\mathbb{Z}\right)^\times$.

$n$	$677$	$1015$	$1861$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	− 0.554958i	− 0.248185i	−0.992271	−	0.124092i	$-0.960398\pi$
$5$	− 0.554958i	− 0.248185i	0.992271	−	0.124092i	$-0.0396019\pi$
$6$	0	0
$7$	1.04892i	0.396453i	0.980156	+	0.198227i	$0.0635182\pi$
$7$	1.04892i	0.396453i	−0.980156	+	0.198227i	$0.936482\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	− 2.91185i	− 0.877957i	−0.898498	−	0.438979i	$-0.855340\pi$
$11$	− 2.91185i	− 0.877957i	0.898498	−	0.438979i	$-0.144660\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.554958i	0.143290i
$16$	0	0
$17$	−2.75302	−0.667706	−0.333853	−	0.942625i	$-0.608349\pi$
$17$	−2.75302	−0.667706	−0.333853	+	0.942625i	$0.608349\pi$
$18$	0	0
$19$	4.63102i	1.06243i	0.847237	+	0.531215i	$0.178264\pi$
$19$	4.63102i	1.06243i	−0.847237	+	0.531215i	$0.821736\pi$
$20$	0	0
$21$	− 1.04892i	− 0.228893i
$22$	0	0
$23$	5.76271	1.20161	0.600804	−	0.799396i	$-0.294847\pi$
$23$	5.76271	1.20161	0.600804	+	0.799396i	$0.294847\pi$
$24$	0	0
$25$	4.69202	0.938404
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.80194	−0.520307	−0.260153	−	0.965567i	$-0.583773\pi$
$29$	−2.80194	−0.520307	−0.260153	+	0.965567i	$0.583773\pi$
$30$	0	0
$31$	− 4.18598i	− 0.751824i	−0.926655	−	0.375912i	$-0.877329\pi$
$31$	− 4.18598i	− 0.751824i	0.926655	−	0.375912i	$-0.122671\pi$
$32$	0	0
$33$	2.91185i	0.506889i
$34$	0	0
$35$	0.582105	0.0983937
$36$	0	0
$37$	− 0.466812i	− 0.0767434i	−0.999264	−	0.0383717i	$-0.987783\pi$
$37$	− 0.466812i	− 0.0767434i	0.999264	−	0.0383717i	$-0.0122171\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.89977i	− 0.609042i	−0.952506	−	0.304521i	$-0.901504\pi$
$41$	− 3.89977i	− 0.609042i	0.952506	−	0.304521i	$-0.0984964\pi$
$42$	0	0
$43$	−9.19567	−1.40233	−0.701163	−	0.713001i	$-0.747336\pi$
$43$	−9.19567	−1.40233	−0.701163	+	0.713001i	$0.747336\pi$
$44$	0	0
$45$	− 0.554958i	− 0.0827283i
$46$	0	0
$47$	− 11.5211i	− 1.68053i	−0.542179	−	0.840263i	$-0.682401\pi$
$47$	− 11.5211i	− 1.68053i	0.542179	−	0.840263i	$-0.317599\pi$
$48$	0	0
$49$	5.89977	0.842825
$50$	0	0
$51$	2.75302	0.385500
$52$	0	0
$53$	5.62565	0.772742	0.386371	−	0.922343i	$-0.373728\pi$
$53$	5.62565	0.772742	0.386371	+	0.922343i	$0.373728\pi$
$54$	0	0
$55$	−1.61596	−0.217896
$56$	0	0
$57$	− 4.63102i	− 0.613394i
$58$	0	0
$59$	− 3.10992i	− 0.404877i	−0.979295	−	0.202438i	$-0.935113\pi$
$59$	− 3.10992i	− 0.404877i	0.979295	−	0.202438i	$-0.0648866\pi$
$60$	0	0
$61$	10.9051	1.39626	0.698131	−	0.715970i	$-0.254015\pi$
$61$	10.9051	1.39626	0.698131	+	0.715970i	$0.254015\pi$
$62$	0	0
$63$	1.04892i	0.132151i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.04892i	0.983332i	0.870784	+	0.491666i	$0.163612\pi$
$67$	8.04892i	0.983332i	−0.870784	+	0.491666i	$0.836388\pi$
$68$	0	0
$69$	−5.76271	−0.693749
$70$	0	0
$71$	13.6920i	1.62494i	0.583000	+	0.812472i	$0.301879\pi$
$71$	13.6920i	1.62494i	−0.583000	+	0.812472i	$0.698121\pi$
$72$	0	0
$73$	− 9.36658i	− 1.09628i	−0.836388	−	0.548138i	$-0.815337\pi$
$73$	− 9.36658i	− 1.09628i	0.836388	−	0.548138i	$-0.184663\pi$
$74$	0	0
$75$	−4.69202	−0.541788
$76$	0	0
$77$	3.05429	0.348069
$78$	0	0
$79$	−3.60925	−0.406073	−0.203036	−	0.979171i	$-0.565081\pi$
$79$	−3.60925	−0.406073	−0.203036	+	0.979171i	$0.565081\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.65519i	0.181680i	0.995865	+	0.0908401i	$0.0289552\pi$
$83$	1.65519i	0.181680i	−0.995865	+	0.0908401i	$0.971045\pi$
$84$	0	0
$85$	1.52781i	0.165714i
$86$	0	0
$87$	2.80194	0.300399
$88$	0	0
$89$	− 17.9705i	− 1.90486i	−0.304750	−	0.952432i	$-0.598573\pi$
$89$	− 17.9705i	− 1.90486i	0.304750	−	0.952432i	$-0.401427\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.18598i	0.434066i
$94$	0	0
$95$	2.57002	0.263679
$96$	0	0
$97$	− 1.31767i	− 0.133789i	−0.997760	−	0.0668944i	$-0.978691\pi$
$97$	− 1.31767i	− 0.133789i	0.997760	−	0.0668944i	$-0.0213091\pi$
$98$	0	0
$99$	− 2.91185i	− 0.292652i
$100$	0	0
$101$	15.0248	1.49502	0.747509	−	0.664251i	$-0.231249\pi$
$101$	15.0248	1.49502	0.747509	+	0.664251i	$0.231249\pi$
$102$	0	0
$103$	9.20775	0.907267	0.453633	−	0.891188i	$-0.350128\pi$
$103$	9.20775	0.907267	0.453633	+	0.891188i	$0.350128\pi$
$104$	0	0
$105$	−0.582105	−0.0568077
$106$	0	0
$107$	7.22952	0.698904	0.349452	−	0.936954i	$-0.386368\pi$
$107$	7.22952	0.698904	0.349452	+	0.936954i	$0.386368\pi$
$108$	0	0
$109$	− 15.5036i	− 1.48498i	−0.669857	−	0.742490i	$-0.733645\pi$
$109$	− 15.5036i	− 1.48498i	0.669857	−	0.742490i	$-0.266355\pi$
$110$	0	0
$111$	0.466812i	0.0443078i
$112$	0	0
$113$	13.8267	1.30071	0.650353	−	0.759632i	$-0.274621\pi$
$113$	13.8267	1.30071	0.650353	+	0.759632i	$0.274621\pi$
$114$	0	0
$115$	− 3.19806i	− 0.298221i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 2.88769i	− 0.264714i
$120$	0	0
$121$	2.52111	0.229191
$122$	0	0
$123$	3.89977i	0.351631i
$124$	0	0
$125$	− 5.37867i	− 0.481083i
$126$	0	0
$127$	−7.17629	−0.636793	−0.318396	−	0.947958i	$-0.603144\pi$
$127$	−7.17629	−0.636793	−0.318396	+	0.947958i	$0.603144\pi$
$128$	0	0
$129$	9.19567	0.809634
$130$	0	0
$131$	13.1304	1.14720	0.573602	−	0.819134i	$-0.305545\pi$
$131$	13.1304	1.14720	0.573602	+	0.819134i	$0.305545\pi$
$132$	0	0
$133$	−4.85756	−0.421204
$134$	0	0
$135$	0.554958i	0.0477632i
$136$	0	0
$137$	− 21.8116i	− 1.86349i	−0.363109	−	0.931747i	$-0.618285\pi$
$137$	− 21.8116i	− 1.86349i	0.363109	−	0.931747i	$-0.381715\pi$
$138$	0	0
$139$	−2.96615	−0.251585	−0.125793	−	0.992057i	$-0.540147\pi$
$139$	−2.96615	−0.251585	−0.125793	+	0.992057i	$0.540147\pi$
$140$	0	0
$141$	11.5211i	0.970252i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.55496i	0.129132i
$146$	0	0
$147$	−5.89977	−0.486605
$148$	0	0
$149$	− 5.69633i	− 0.466662i	−0.972397	−	0.233331i	$-0.925037\pi$
$149$	− 5.69633i	− 0.466662i	0.972397	−	0.233331i	$-0.0749625\pi$
$150$	0	0
$151$	− 19.1468i	− 1.55814i	−0.626937	−	0.779070i	$-0.715691\pi$
$151$	− 19.1468i	− 1.55814i	0.626937	−	0.779070i	$-0.284309\pi$
$152$	0	0
$153$	−2.75302	−0.222569
$154$	0	0
$155$	−2.32304	−0.186591
$156$	0	0
$157$	−8.38404	−0.669119	−0.334560	−	0.942375i	$-0.608588\pi$
$157$	−8.38404	−0.669119	−0.334560	+	0.942375i	$0.608588\pi$
$158$	0	0
$159$	−5.62565	−0.446143
$160$	0	0
$161$	6.04461i	0.476382i
$162$	0	0
$163$	1.97046i	0.154338i	0.997018	+	0.0771692i	$0.0245882\pi$
$163$	1.97046i	0.154338i	−0.997018	+	0.0771692i	$0.975412\pi$
$164$	0	0
$165$	1.61596	0.125802
$166$	0	0
$167$	− 24.7995i	− 1.91905i	−0.281630	−	0.959523i	$-0.590875\pi$
$167$	− 24.7995i	− 1.91905i	0.281630	−	0.959523i	$-0.409125\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	4.63102i	0.354143i
$172$	0	0
$173$	2.57135	0.195496	0.0977481	−	0.995211i	$-0.468836\pi$
$173$	2.57135	0.195496	0.0977481	+	0.995211i	$0.468836\pi$
$174$	0	0
$175$	4.92154i	0.372034i
$176$	0	0
$177$	3.10992i	0.233756i
$178$	0	0
$179$	3.00538	0.224632	0.112316	−	0.993673i	$-0.464173\pi$
$179$	3.00538	0.224632	0.112316	+	0.993673i	$0.464173\pi$
$180$	0	0
$181$	14.6843	1.09147	0.545736	−	0.837957i	$-0.316250\pi$
$181$	14.6843	1.09147	0.545736	+	0.837957i	$0.316250\pi$
$182$	0	0
$183$	−10.9051	−0.806132
$184$	0	0
$185$	−0.259061	−0.0190466
$186$	0	0
$187$	8.01639i	0.586217i
$188$	0	0
$189$	− 1.04892i	− 0.0762975i
$190$	0	0
$191$	−17.0804	−1.23589	−0.617946	−	0.786220i	$-0.712035\pi$
$191$	−17.0804	−1.23589	−0.617946	+	0.786220i	$0.712035\pi$
$192$	0	0
$193$	14.9758i	1.07798i	0.842311	+	0.538992i	$0.181195\pi$
$193$	14.9758i	1.07798i	−0.842311	+	0.538992i	$0.818805\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.192685i	0.0137283i	0.999976	+	0.00686414i	$0.00218494\pi$
$197$	0.192685i	0.0137283i	−0.999976	+	0.00686414i	$0.997815\pi$
$198$	0	0
$199$	−11.8726	−0.841628	−0.420814	−	0.907147i	$-0.638256\pi$
$199$	−11.8726	−0.841628	−0.420814	+	0.907147i	$0.638256\pi$
$200$	0	0
$201$	− 8.04892i	− 0.567727i
$202$	0	0
$203$	− 2.93900i	− 0.206277i
$204$	0	0
$205$	−2.16421	−0.151155
$206$	0	0
$207$	5.76271	0.400536
$208$	0	0
$209$	13.4849	0.932767
$210$	0	0
$211$	6.03385	0.415387	0.207694	−	0.978194i	$-0.433404\pi$
$211$	6.03385	0.415387	0.207694	+	0.978194i	$0.433404\pi$
$212$	0	0
$213$	− 13.6920i	− 0.938162i
$214$	0	0
$215$	5.10321i	0.348036i
$216$	0	0
$217$	4.39075	0.298063
$218$	0	0
$219$	9.36658i	0.632935i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.52648i	0.570976i	0.958382	+	0.285488i	$0.0921556\pi$
$223$	8.52648i	0.570976i	−0.958382	+	0.285488i	$0.907844\pi$
$224$	0	0
$225$	4.69202	0.312801
$226$	0	0
$227$	11.0000i	0.730096i	0.930989	+	0.365048i	$0.118947\pi$
$227$	11.0000i	0.730096i	−0.930989	+	0.365048i	$0.881053\pi$
$228$	0	0
$229$	17.9119i	1.18365i	0.806067	+	0.591824i	$0.201592\pi$
$229$	17.9119i	1.18365i	−0.806067	+	0.591824i	$0.798408\pi$
$230$	0	0
$231$	−3.05429	−0.200958
$232$	0	0
$233$	−22.8159	−1.49472	−0.747361	−	0.664418i	$-0.768679\pi$
$233$	−22.8159	−1.49472	−0.747361	+	0.664418i	$0.768679\pi$
$234$	0	0
$235$	−6.39373	−0.417081
$236$	0	0
$237$	3.60925	0.234446
$238$	0	0
$239$	8.34481i	0.539781i	0.962891	+	0.269891i	$0.0869875\pi$
$239$	8.34481i	0.539781i	−0.962891	+	0.269891i	$0.913012\pi$
$240$	0	0
$241$	3.75541i	0.241907i	0.992658	+	0.120954i	$0.0385953\pi$
$241$	3.75541i	0.241907i	−0.992658	+	0.120954i	$0.961405\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	− 3.27413i	− 0.209176i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 1.65519i	− 0.104893i
$250$	0	0
$251$	30.3086	1.91306	0.956530	−	0.291634i	$-0.0941989\pi$
$251$	30.3086	1.91306	0.956530	+	0.291634i	$0.0941989\pi$
$252$	0	0
$253$	− 16.7802i	− 1.05496i
$254$	0	0
$255$	− 1.52781i	− 0.0956752i
$256$	0	0
$257$	20.8412	1.30004	0.650018	−	0.759919i	$-0.274761\pi$
$257$	20.8412	1.30004	0.650018	+	0.759919i	$0.274761\pi$
$258$	0	0
$259$	0.489647	0.0304252
$260$	0	0
$261$	−2.80194	−0.173436
$262$	0	0
$263$	8.58642	0.529461	0.264731	−	0.964322i	$-0.414717\pi$
$263$	8.58642	0.529461	0.264731	+	0.964322i	$0.414717\pi$
$264$	0	0
$265$	− 3.12200i	− 0.191783i
$266$	0	0
$267$	17.9705i	1.09977i
$268$	0	0
$269$	14.8267	0.903999	0.452000	−	0.892018i	$-0.350711\pi$
$269$	14.8267	0.903999	0.452000	+	0.892018i	$0.350711\pi$
$270$	0	0
$271$	7.23490i	0.439489i	0.975557	+	0.219744i	$0.0705223\pi$
$271$	7.23490i	0.439489i	−0.975557	+	0.219744i	$0.929478\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 13.6625i	− 0.823879i
$276$	0	0
$277$	−11.2054	−0.673265	−0.336632	−	0.941636i	$-0.609288\pi$
$277$	−11.2054	−0.673265	−0.336632	+	0.941636i	$0.609288\pi$
$278$	0	0
$279$	− 4.18598i	− 0.250608i
$280$	0	0
$281$	12.3002i	0.733769i	0.930267	+	0.366884i	$0.119576\pi$
$281$	12.3002i	0.733769i	−0.930267	+	0.366884i	$0.880424\pi$
$282$	0	0
$283$	−24.3521	−1.44758	−0.723791	−	0.690019i	$-0.757602\pi$
$283$	−24.3521	−1.44758	−0.723791	+	0.690019i	$0.757602\pi$
$284$	0	0
$285$	−2.57002	−0.152235
$286$	0	0
$287$	4.09054	0.241457
$288$	0	0
$289$	−9.42088	−0.554169
$290$	0	0
$291$	1.31767i	0.0772430i
$292$	0	0
$293$	− 6.20237i	− 0.362347i	−0.983451	−	0.181173i	$-0.942011\pi$
$293$	− 6.20237i	− 0.362347i	0.983451	−	0.181173i	$-0.0579895\pi$
$294$	0	0
$295$	−1.72587	−0.100484
$296$	0	0
$297$	2.91185i	0.168963i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 9.64550i	− 0.555957i
$302$	0	0
$303$	−15.0248	−0.863150
$304$	0	0
$305$	− 6.05190i	− 0.346531i
$306$	0	0
$307$	26.4795i	1.51126i	0.654996	+	0.755632i	$0.272670\pi$
$307$	26.4795i	1.51126i	−0.654996	+	0.755632i	$0.727330\pi$
$308$	0	0
$309$	−9.20775	−0.523811
$310$	0	0
$311$	−15.5308	−0.880671	−0.440335	−	0.897833i	$-0.645140\pi$
$311$	−15.5308	−0.880671	−0.440335	+	0.897833i	$0.645140\pi$
$312$	0	0
$313$	16.9681	0.959092	0.479546	−	0.877517i	$-0.340801\pi$
$313$	16.9681	0.959092	0.479546	+	0.877517i	$0.340801\pi$
$314$	0	0
$315$	0.582105	0.0327979
$316$	0	0
$317$	25.5066i	1.43260i	0.697795	+	0.716298i	$0.254165\pi$
$317$	25.5066i	1.43260i	−0.697795	+	0.716298i	$0.745835\pi$
$318$	0	0
$319$	8.15883i	0.456807i
$320$	0	0
$321$	−7.22952	−0.403513
$322$	0	0
$323$	− 12.7493i	− 0.709390i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.5036i	0.857354i
$328$	0	0
$329$	12.0847	0.666250
$330$	0	0
$331$	10.6136i	0.583374i	0.956514	+	0.291687i	$0.0942166\pi$
$331$	10.6136i	0.583374i	−0.956514	+	0.291687i	$0.905783\pi$
$332$	0	0
$333$	− 0.466812i	− 0.0255811i
$334$	0	0
$335$	4.46681	0.244048
$336$	0	0
$337$	−29.2717	−1.59453	−0.797266	−	0.603628i	$-0.793721\pi$
$337$	−29.2717	−1.59453	−0.797266	+	0.603628i	$0.793721\pi$
$338$	0	0
$339$	−13.8267	−0.750963
$340$	0	0
$341$	−12.1890	−0.660069
$342$	0	0
$343$	13.5308i	0.730594i
$344$	0	0
$345$	3.19806i	0.172178i
$346$	0	0
$347$	−34.8146	−1.86895	−0.934473	−	0.356034i	$-0.884129\pi$
$347$	−34.8146	−1.86895	−0.934473	+	0.356034i	$0.884129\pi$
$348$	0	0
$349$	30.1957i	1.61634i	0.588951	+	0.808169i	$0.299541\pi$
$349$	30.1957i	1.61634i	−0.588951	+	0.808169i	$0.700459\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 24.9691i	− 1.32897i	−0.747300	−	0.664486i	$-0.768650\pi$
$353$	− 24.9691i	− 1.32897i	0.747300	−	0.664486i	$-0.231350\pi$
$354$	0	0
$355$	7.59850	0.403286
$356$	0	0
$357$	2.88769i	0.152833i
$358$	0	0
$359$	14.1661i	0.747660i	0.927497	+	0.373830i	$0.121956\pi$
$359$	14.1661i	0.747660i	−0.927497	+	0.373830i	$0.878044\pi$
$360$	0	0
$361$	−2.44637	−0.128756
$362$	0	0
$363$	−2.52111	−0.132324
$364$	0	0
$365$	−5.19806	−0.272079
$366$	0	0
$367$	8.78687	0.458671	0.229335	−	0.973347i	$-0.426345\pi$
$367$	8.78687	0.458671	0.229335	+	0.973347i	$0.426345\pi$
$368$	0	0
$369$	− 3.89977i	− 0.203014i
$370$	0	0
$371$	5.90084i	0.306356i
$372$	0	0
$373$	−18.3260	−0.948886	−0.474443	−	0.880286i	$-0.657350\pi$
$373$	−18.3260	−0.948886	−0.474443	+	0.880286i	$0.657350\pi$
$374$	0	0
$375$	5.37867i	0.277753i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 4.77586i	− 0.245319i	−0.992449	−	0.122660i	$-0.960858\pi$
$379$	− 4.77586i	− 0.245319i	0.992449	−	0.122660i	$-0.0391423\pi$
$380$	0	0
$381$	7.17629	0.367653
$382$	0	0
$383$	6.19029i	0.316309i	0.987414	+	0.158155i	$0.0505544\pi$
$383$	6.19029i	0.316309i	−0.987414	+	0.158155i	$0.949446\pi$
$384$	0	0
$385$	− 1.69501i	− 0.0863855i
$386$	0	0
$387$	−9.19567	−0.467442
$388$	0	0
$389$	−24.8780	−1.26136	−0.630682	−	0.776041i	$-0.717225\pi$
$389$	−24.8780	−1.26136	−0.630682	+	0.776041i	$0.717225\pi$
$390$	0	0
$391$	−15.8649	−0.802320
$392$	0	0
$393$	−13.1304	−0.662339
$394$	0	0
$395$	2.00298i	0.100781i
$396$	0	0
$397$	4.62027i	0.231885i	0.993256	+	0.115942i	$0.0369888\pi$
$397$	4.62027i	0.231885i	−0.993256	+	0.115942i	$0.963011\pi$
$398$	0	0
$399$	4.85756	0.243182
$400$	0	0
$401$	1.64848i	0.0823212i	0.999153	+	0.0411606i	$0.0131055\pi$
$401$	1.64848i	0.0823212i	−0.999153	+	0.0411606i	$0.986894\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 0.554958i	− 0.0275761i
$406$	0	0
$407$	−1.35929	−0.0673774
$408$	0	0
$409$	− 34.8049i	− 1.72099i	−0.509457	−	0.860496i	$-0.670154\pi$
$409$	− 34.8049i	− 1.72099i	0.509457	−	0.860496i	$-0.329846\pi$
$410$	0	0
$411$	21.8116i	1.07589i
$412$	0	0
$413$	3.26205	0.160515
$414$	0	0
$415$	0.918559	0.0450903
$416$	0	0
$417$	2.96615	0.145253
$418$	0	0
$419$	21.6353	1.05696	0.528478	−	0.848947i	$-0.322763\pi$
$419$	21.6353	1.05696	0.528478	+	0.848947i	$0.322763\pi$
$420$	0	0
$421$	− 35.1008i	− 1.71071i	−0.518043	−	0.855355i	$-0.673339\pi$
$421$	− 35.1008i	− 1.71071i	0.518043	−	0.855355i	$-0.326661\pi$
$422$	0	0
$423$	− 11.5211i	− 0.560175i
$424$	0	0
$425$	−12.9172	−0.626578
$426$	0	0
$427$	11.4386i	0.553553i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 20.0925i	− 0.967820i	−0.875118	−	0.483910i	$-0.839216\pi$
$431$	− 20.0925i	− 0.967820i	0.875118	−	0.483910i	$-0.160784\pi$
$432$	0	0
$433$	18.6939	0.898373	0.449187	−	0.893438i	$-0.351714\pi$
$433$	18.6939	0.898373	0.449187	+	0.893438i	$0.351714\pi$
$434$	0	0
$435$	− 1.55496i	− 0.0745545i
$436$	0	0
$437$	26.6872i	1.27662i
$438$	0	0
$439$	−1.35019	−0.0644411	−0.0322206	−	0.999481i	$-0.510258\pi$
$439$	−1.35019	−0.0644411	−0.0322206	+	0.999481i	$0.510258\pi$
$440$	0	0
$441$	5.89977	0.280942
$442$	0	0
$443$	−0.400436	−0.0190253	−0.00951265	−	0.999955i	$-0.503028\pi$
$443$	−0.400436	−0.0190253	−0.00951265	+	0.999955i	$0.503028\pi$
$444$	0	0
$445$	−9.97285	−0.472759
$446$	0	0
$447$	5.69633i	0.269427i
$448$	0	0
$449$	− 35.7904i	− 1.68906i	−0.535512	−	0.844528i	$-0.679881\pi$
$449$	− 35.7904i	− 1.68906i	0.535512	−	0.844528i	$-0.320119\pi$
$450$	0	0
$451$	−11.3556	−0.534713
$452$	0	0
$453$	19.1468i	0.899593i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.17092i	0.241885i	0.992660	+	0.120943i	$0.0385917\pi$
$457$	5.17092i	0.241885i	−0.992660	+	0.120943i	$0.961408\pi$
$458$	0	0
$459$	2.75302	0.128500
$460$	0	0
$461$	− 20.8780i	− 0.972386i	−0.873852	−	0.486193i	$-0.838385\pi$
$461$	− 20.8780i	− 0.972386i	0.873852	−	0.486193i	$-0.161615\pi$
$462$	0	0
$463$	30.7198i	1.42767i	0.700315	+	0.713834i	$0.253043\pi$
$463$	30.7198i	1.42767i	−0.700315	+	0.713834i	$0.746957\pi$
$464$	0	0
$465$	2.32304	0.107729
$466$	0	0
$467$	−15.2741	−0.706802	−0.353401	−	0.935472i	$-0.614975\pi$
$467$	−15.2741	−0.706802	−0.353401	+	0.935472i	$0.614975\pi$
$468$	0	0
$469$	−8.44265	−0.389845
$470$	0	0
$471$	8.38404	0.386316
$472$	0	0
$473$	26.7764i	1.23118i
$474$	0	0
$475$	21.7289i	0.996988i
$476$	0	0
$477$	5.62565	0.257581
$478$	0	0
$479$	− 3.05861i	− 0.139751i	−0.997556	−	0.0698756i	$-0.977740\pi$
$479$	− 3.05861i	− 0.139751i	0.997556	−	0.0698756i	$-0.0222602\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 6.04461i	− 0.275039i
$484$	0	0
$485$	−0.731250	−0.0332044
$486$	0	0
$487$	15.1521i	0.686608i	0.939224	+	0.343304i	$0.111546\pi$
$487$	15.1521i	0.686608i	−0.939224	+	0.343304i	$0.888454\pi$
$488$	0	0
$489$	− 1.97046i	− 0.0891073i
$490$	0	0
$491$	−23.4795	−1.05961	−0.529807	−	0.848118i	$-0.677736\pi$
$491$	−23.4795	−1.05961	−0.529807	+	0.848118i	$0.677736\pi$
$492$	0	0
$493$	7.71379	0.347412
$494$	0	0
$495$	−1.61596	−0.0726319
$496$	0	0
$497$	−14.3618	−0.644215
$498$	0	0
$499$	38.0689i	1.70420i	0.523381	+	0.852099i	$0.324670\pi$
$499$	38.0689i	1.70420i	−0.523381	+	0.852099i	$0.675330\pi$
$500$	0	0
$501$	24.7995i	1.10796i
$502$	0	0
$503$	−11.1274	−0.496145	−0.248073	−	0.968741i	$-0.579797\pi$
$503$	−11.1274	−0.496145	−0.248073	+	0.968741i	$0.579797\pi$
$504$	0	0
$505$	− 8.33811i	− 0.371041i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	26.7633i	1.18626i	0.805106	+	0.593131i	$0.202108\pi$
$509$	26.7633i	1.18626i	−0.805106	+	0.593131i	$0.797892\pi$
$510$	0	0
$511$	9.82477	0.434622
$512$	0	0
$513$	− 4.63102i	− 0.204465i
$514$	0	0
$515$	− 5.10992i	− 0.225170i
$516$	0	0
$517$	−33.5478	−1.47543
$518$	0	0
$519$	−2.57135	−0.112870
$520$	0	0
$521$	20.5797	0.901614	0.450807	−	0.892622i	$-0.351136\pi$
$521$	20.5797	0.901614	0.450807	+	0.892622i	$0.351136\pi$
$522$	0	0
$523$	−6.77479	−0.296241	−0.148120	−	0.988969i	$-0.547322\pi$
$523$	−6.77479	−0.296241	−0.148120	+	0.988969i	$0.547322\pi$
$524$	0	0
$525$	− 4.92154i	− 0.214794i
$526$	0	0
$527$	11.5241i	0.501997i
$528$	0	0
$529$	10.2088	0.443862
$530$	0	0
$531$	− 3.10992i	− 0.134959i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 4.01208i	− 0.173457i
$536$	0	0
$537$	−3.00538	−0.129692
$538$	0	0
$539$	− 17.1793i	− 0.739964i
$540$	0	0
$541$	− 1.94331i	− 0.0835495i	−0.999127	−	0.0417748i	$-0.986699\pi$
$541$	− 1.94331i	− 0.0835495i	0.999127	−	0.0417748i	$-0.0133012\pi$
$542$	0	0
$543$	−14.6843	−0.630162
$544$	0	0
$545$	−8.60388	−0.368550
$546$	0	0
$547$	−39.1323	−1.67318	−0.836588	−	0.547833i	$-0.815453\pi$
$547$	−39.1323	−1.67318	−0.836588	+	0.547833i	$0.815453\pi$
$548$	0	0
$549$	10.9051	0.465420
$550$	0	0
$551$	− 12.9758i	− 0.552789i
$552$	0	0
$553$	− 3.78581i	− 0.160989i
$554$	0	0
$555$	0.259061	0.0109965
$556$	0	0
$557$	24.4989i	1.03805i	0.854759	+	0.519025i	$0.173705\pi$
$557$	24.4989i	1.03805i	−0.854759	+	0.519025i	$0.826295\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	− 8.01639i	− 0.338452i
$562$	0	0
$563$	21.8267	0.919885	0.459943	−	0.887949i	$-0.347870\pi$
$563$	21.8267	0.919885	0.459943	+	0.887949i	$0.347870\pi$
$564$	0	0
$565$	− 7.67324i	− 0.322815i
$566$	0	0
$567$	1.04892i	0.0440504i
$568$	0	0
$569$	11.5627	0.484735	0.242367	−	0.970185i	$-0.422076\pi$
$569$	11.5627	0.484735	0.242367	+	0.970185i	$0.422076\pi$
$570$	0	0
$571$	−24.7614	−1.03623	−0.518116	−	0.855310i	$-0.673366\pi$
$571$	−24.7614	−1.03623	−0.518116	+	0.855310i	$0.673366\pi$
$572$	0	0
$573$	17.0804	0.713543
$574$	0	0
$575$	27.0388	1.12759
$576$	0	0
$577$	13.5090i	0.562388i	0.959651	+	0.281194i	$0.0907304\pi$
$577$	13.5090i	0.562388i	−0.959651	+	0.281194i	$0.909270\pi$
$578$	0	0
$579$	− 14.9758i	− 0.622375i
$580$	0	0
$581$	−1.73615	−0.0720278
$582$	0	0
$583$	− 16.3811i	− 0.678434i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.70304i	− 0.276664i	−0.990386	−	0.138332i	$-0.955826\pi$
$587$	− 6.70304i	− 0.276664i	0.990386	−	0.138332i	$-0.0441741\pi$
$588$	0	0
$589$	19.3854	0.798760
$590$	0	0
$591$	− 0.192685i	− 0.00792602i
$592$	0	0
$593$	16.0194i	0.657837i	0.944358	+	0.328918i	$0.106684\pi$
$593$	16.0194i	0.657837i	−0.944358	+	0.328918i	$0.893316\pi$
$594$	0	0
$595$	−1.60255	−0.0656980
$596$	0	0
$597$	11.8726	0.485914
$598$	0	0
$599$	19.8243	0.809999	0.404999	−	0.914317i	$-0.367272\pi$
$599$	19.8243	0.809999	0.404999	+	0.914317i	$0.367272\pi$
$600$	0	0
$601$	29.1909	1.19072	0.595360	−	0.803459i	$-0.297009\pi$
$601$	29.1909	1.19072	0.595360	+	0.803459i	$0.297009\pi$
$602$	0	0
$603$	8.04892i	0.327777i
$604$	0	0
$605$	− 1.39911i	− 0.0568818i
$606$	0	0
$607$	−6.36227	−0.258237	−0.129118	−	0.991629i	$-0.541215\pi$
$607$	−6.36227	−0.258237	−0.129118	+	0.991629i	$0.541215\pi$
$608$	0	0
$609$	2.93900i	0.119094i
$610$	0	0
$611$	0	0
$612$	0	0
$613$	30.7711i	1.24283i	0.783481	+	0.621416i	$0.213442\pi$
$613$	30.7711i	1.24283i	−0.783481	+	0.621416i	$0.786558\pi$
$614$	0	0
$615$	2.16421	0.0872694
$616$	0	0
$617$	12.0242i	0.484075i	0.970267	+	0.242037i	$0.0778156\pi$
$617$	12.0242i	0.484075i	−0.970267	+	0.242037i	$0.922184\pi$
$618$	0	0
$619$	− 9.02715i	− 0.362832i	−0.983406	−	0.181416i	$-0.941932\pi$
$619$	− 9.02715i	− 0.362832i	0.983406	−	0.181416i	$-0.0580680\pi$
$620$	0	0
$621$	−5.76271	−0.231250
$622$	0	0
$623$	18.8495	0.755190
$624$	0	0
$625$	20.4752	0.819007
$626$	0	0
$627$	−13.4849	−0.538533
$628$	0	0
$629$	1.28514i	0.0512420i
$630$	0	0
$631$	− 15.5526i	− 0.619138i	−0.950877	−	0.309569i	$-0.899815\pi$
$631$	− 15.5526i	− 0.619138i	0.950877	−	0.309569i	$-0.100185\pi$
$632$	0	0
$633$	−6.03385	−0.239824
$634$	0	0
$635$	3.98254i	0.158042i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	13.6920i	0.541648i
$640$	0	0
$641$	19.4209	0.767079	0.383539	−	0.923525i	$-0.374705\pi$
$641$	19.4209	0.767079	0.383539	+	0.923525i	$0.374705\pi$
$642$	0	0
$643$	− 26.8267i	− 1.05794i	−0.848640	−	0.528971i	$-0.822578\pi$
$643$	− 26.8267i	− 1.05794i	0.848640	−	0.528971i	$-0.177422\pi$
$644$	0	0
$645$	− 5.10321i	− 0.200939i
$646$	0	0
$647$	−5.08815	−0.200036	−0.100018	−	0.994986i	$-0.531890\pi$
$647$	−5.08815	−0.200036	−0.100018	+	0.994986i	$0.531890\pi$
$648$	0	0
$649$	−9.05562	−0.355464
$650$	0	0
$651$	−4.39075	−0.172087
$652$	0	0
$653$	−19.7157	−0.771535	−0.385768	−	0.922596i	$-0.626063\pi$
$653$	−19.7157	−0.771535	−0.385768	+	0.922596i	$0.626063\pi$
$654$	0	0
$655$	− 7.28680i	− 0.284719i
$656$	0	0
$657$	− 9.36658i	− 0.365425i
$658$	0	0
$659$	9.42519	0.367153	0.183577	−	0.983005i	$-0.441232\pi$
$659$	9.42519	0.367153	0.183577	+	0.983005i	$0.441232\pi$
$660$	0	0
$661$	29.9946i	1.16666i	0.812237	+	0.583328i	$0.198250\pi$
$661$	29.9946i	1.16666i	−0.812237	+	0.583328i	$0.801750\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.69574i	0.104536i
$666$	0	0
$667$	−16.1468	−0.625205
$668$	0	0
$669$	− 8.52648i	− 0.329653i
$670$	0	0
$671$	− 31.7542i	− 1.22586i
$672$	0	0
$673$	47.7294	1.83984	0.919918	−	0.392112i	$-0.128255\pi$
$673$	47.7294	1.83984	0.919918	+	0.392112i	$0.128255\pi$
$674$	0	0
$675$	−4.69202	−0.180596
$676$	0	0
$677$	−42.9124	−1.64926	−0.824630	−	0.565673i	$-0.808616\pi$
$677$	−42.9124	−1.64926	−0.824630	+	0.565673i	$0.808616\pi$
$678$	0	0
$679$	1.38212	0.0530411
$680$	0	0
$681$	− 11.0000i	− 0.421521i
$682$	0	0
$683$	− 0.970460i	− 0.0371336i	−0.999828	−	0.0185668i	$-0.994090\pi$
$683$	− 0.970460i	− 0.0371336i	0.999828	−	0.0185668i	$-0.00591034\pi$
$684$	0	0
$685$	−12.1045	−0.462491
$686$	0	0
$687$	− 17.9119i	− 0.683380i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1.16660i	0.0443797i	0.999754	+	0.0221898i	$0.00706383\pi$
$691$	1.16660i	0.0443797i	−0.999754	+	0.0221898i	$0.992936\pi$
$692$	0	0
$693$	3.05429	0.116023
$694$	0	0
$695$	1.64609i	0.0624397i
$696$	0	0
$697$	10.7362i	0.406661i
$698$	0	0
$699$	22.8159	0.862978
$700$	0	0
$701$	17.3274	0.654445	0.327223	−	0.944947i	$-0.393887\pi$
$701$	17.3274	0.654445	0.327223	+	0.944947i	$0.393887\pi$
$702$	0	0
$703$	2.16182	0.0815345
$704$	0	0
$705$	6.39373	0.240802
$706$	0	0
$707$	15.7597i	0.592705i
$708$	0	0
$709$	− 16.4601i	− 0.618172i	−0.951034	−	0.309086i	$-0.899977\pi$
$709$	− 16.4601i	− 0.618172i	0.951034	−	0.309086i	$-0.100023\pi$
$710$	0	0
$711$	−3.60925	−0.135358
$712$	0	0
$713$	− 24.1226i	− 0.903398i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 8.34481i	− 0.311643i
$718$	0	0
$719$	−14.1715	−0.528508	−0.264254	−	0.964453i	$-0.585126\pi$
$719$	−14.1715	−0.528508	−0.264254	+	0.964453i	$0.585126\pi$
$720$	0	0
$721$	9.65817i	0.359689i
$722$	0	0
$723$	− 3.75541i	− 0.139665i
$724$	0	0
$725$	−13.1468	−0.488258
$726$	0	0
$727$	37.8810	1.40493	0.702464	−	0.711719i	$-0.252083\pi$
$727$	37.8810	1.40493	0.702464	+	0.711719i	$0.252083\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	25.3159	0.936341
$732$	0	0
$733$	0.377338i	0.0139373i	0.999976	+	0.00696865i	$0.00221821\pi$
$733$	0.377338i	0.0139373i	−0.999976	+	0.00696865i	$0.997782\pi$
$734$	0	0
$735$	3.27413i	0.120768i
$736$	0	0
$737$	23.4373	0.863323
$738$	0	0
$739$	− 1.12631i	− 0.0414320i	−0.999785	−	0.0207160i	$-0.993405\pi$
$739$	− 1.12631i	− 0.0414320i	0.999785	−	0.0207160i	$-0.00659457\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 22.7047i	− 0.832954i	−0.909146	−	0.416477i	$-0.863265\pi$
$743$	− 22.7047i	− 0.832954i	0.909146	−	0.416477i	$-0.136735\pi$
$744$	0	0
$745$	−3.16123	−0.115818
$746$	0	0
$747$	1.65519i	0.0605601i
$748$	0	0
$749$	7.58317i	0.277083i
$750$	0	0
$751$	−13.9282	−0.508249	−0.254124	−	0.967172i	$-0.581787\pi$
$751$	−13.9282	−0.508249	−0.254124	+	0.967172i	$0.581787\pi$
$752$	0	0
$753$	−30.3086	−1.10451
$754$	0	0
$755$	−10.6256	−0.386707
$756$	0	0
$757$	1.16660	0.0424009	0.0212005	−	0.999775i	$-0.493251\pi$
$757$	1.16660	0.0424009	0.0212005	+	0.999775i	$0.493251\pi$
$758$	0	0
$759$	16.7802i	0.609081i
$760$	0	0
$761$	− 0.821789i	− 0.0297898i	−0.999889	−	0.0148949i	$-0.995259\pi$
$761$	− 0.821789i	− 0.0297898i	0.999889	−	0.0148949i	$-0.00474137\pi$
$762$	0	0
$763$	16.2620	0.588726
$764$	0	0
$765$	1.52781i	0.0552381i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	− 33.7017i	− 1.21531i	−0.794199	−	0.607657i	$-0.792109\pi$
$769$	− 33.7017i	− 1.21531i	0.794199	−	0.607657i	$-0.207891\pi$
$770$	0	0
$771$	−20.8412	−0.750576
$772$	0	0
$773$	− 0.764037i	− 0.0274805i	−0.999906	−	0.0137403i	$-0.995626\pi$
$773$	− 0.764037i	− 0.0274805i	0.999906	−	0.0137403i	$-0.00437380\pi$
$774$	0	0
$775$	− 19.6407i	− 0.705515i
$776$	0	0
$777$	−0.489647	−0.0175660
$778$	0	0
$779$	18.0599	0.647064
$780$	0	0
$781$	39.8692	1.42663
$782$	0	0
$783$	2.80194	0.100133
$784$	0	0
$785$	4.65279i	0.166065i
$786$	0	0
$787$	− 12.1739i	− 0.433953i	−0.976177	−	0.216976i	$-0.930381\pi$
$787$	− 12.1739i	− 0.433953i	0.976177	−	0.216976i	$-0.0696194\pi$
$788$	0	0
$789$	−8.58642	−0.305685
$790$	0	0
$791$	14.5031i	0.515669i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	3.12200i	0.110726i
$796$	0	0
$797$	−38.2887	−1.35626	−0.678128	−	0.734944i	$-0.737209\pi$
$797$	−38.2887	−1.35626	−0.678128	+	0.734944i	$0.737209\pi$
$798$	0	0
$799$	31.7178i	1.12210i
$800$	0	0
$801$	− 17.9705i	− 0.634955i
$802$	0	0
$803$	−27.2741	−0.962483
$804$	0	0
$805$	3.35450	0.118231
$806$	0	0
$807$	−14.8267	−0.521924
$808$	0	0
$809$	−29.9638	−1.05347	−0.526735	−	0.850030i	$-0.676584\pi$
$809$	−29.9638	−1.05347	−0.526735	+	0.850030i	$0.676584\pi$
$810$	0	0
$811$	− 46.3521i	− 1.62764i	−0.581115	−	0.813821i	$-0.697383\pi$
$811$	− 46.3521i	− 1.62764i	0.581115	−	0.813821i	$-0.302617\pi$
$812$	0	0
$813$	− 7.23490i	− 0.253739i
$814$	0	0
$815$	1.09352	0.0383044
$816$	0	0
$817$	− 42.5854i	− 1.48987i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.3105i	1.68605i	0.537876	+	0.843024i	$0.319227\pi$
$821$	48.3105i	1.68605i	−0.537876	+	0.843024i	$0.680773\pi$
$822$	0	0
$823$	5.31229	0.185175	0.0925874	−	0.995705i	$-0.470486\pi$
$823$	5.31229	0.185175	0.0925874	+	0.995705i	$0.470486\pi$
$824$	0	0
$825$	13.6625i	0.475667i
$826$	0	0
$827$	− 22.0968i	− 0.768380i	−0.923254	−	0.384190i	$-0.874481\pi$
$827$	− 22.0968i	− 0.768380i	0.923254	−	0.384190i	$-0.125519\pi$
$828$	0	0
$829$	1.61655	0.0561450	0.0280725	−	0.999606i	$-0.491063\pi$
$829$	1.61655	0.0561450	0.0280725	+	0.999606i	$0.491063\pi$
$830$	0	0
$831$	11.2054	0.388710
$832$	0	0
$833$	−16.2422	−0.562759
$834$	0	0
$835$	−13.7627	−0.476278
$836$	0	0
$837$	4.18598i	0.144689i
$838$	0	0
$839$	41.0823i	1.41832i	0.705048	+	0.709159i	$0.250925\pi$
$839$	41.0823i	1.41832i	−0.705048	+	0.709159i	$0.749075\pi$
$840$	0	0
$841$	−21.1491	−0.729281
$842$	0	0
$843$	− 12.3002i	− 0.423642i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.64443i	0.0908638i
$848$	0	0
$849$	24.3521	0.835762
$850$	0	0
$851$	− 2.69010i	− 0.0922155i
$852$	0	0
$853$	− 35.0441i	− 1.19989i	−0.800042	−	0.599944i	$-0.795189\pi$
$853$	− 35.0441i	− 1.19989i	0.800042	−	0.599944i	$-0.204811\pi$
$854$	0	0
$855$	2.57002	0.0878930
$856$	0	0
$857$	3.76079	0.128466	0.0642331	−	0.997935i	$-0.479540\pi$
$857$	3.76079	0.128466	0.0642331	+	0.997935i	$0.479540\pi$
$858$	0	0
$859$	7.11662	0.242816	0.121408	−	0.992603i	$-0.461259\pi$
$859$	7.11662	0.242816	0.121408	+	0.992603i	$0.461259\pi$
$860$	0	0
$861$	−4.09054	−0.139405
$862$	0	0
$863$	− 57.5900i	− 1.96039i	−0.198044	−	0.980193i	$-0.563459\pi$
$863$	− 57.5900i	− 1.96039i	0.198044	−	0.980193i	$-0.436541\pi$
$864$	0	0
$865$	− 1.42699i	− 0.0485192i
$866$	0	0
$867$	9.42088	0.319950
$868$	0	0
$869$	10.5096i	0.356514i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 1.31767i	− 0.0445963i
$874$	0	0
$875$	5.64178	0.190727
$876$	0	0
$877$	19.2150i	0.648846i	0.945912	+	0.324423i	$0.105170\pi$
$877$	19.2150i	0.648846i	−0.945912	+	0.324423i	$0.894830\pi$
$878$	0	0
$879$	6.20237i	0.209201i
$880$	0	0
$881$	11.1444	0.375463	0.187731	−	0.982220i	$-0.439887\pi$
$881$	11.1444	0.375463	0.187731	+	0.982220i	$0.439887\pi$
$882$	0	0
$883$	4.28919	0.144343	0.0721714	−	0.997392i	$-0.477007\pi$
$883$	4.28919	0.144343	0.0721714	+	0.997392i	$0.477007\pi$
$884$	0	0
$885$	1.72587	0.0580146
$886$	0	0
$887$	−33.1739	−1.11387	−0.556935	−	0.830556i	$-0.688023\pi$
$887$	−33.1739	−1.11387	−0.556935	+	0.830556i	$0.688023\pi$
$888$	0	0
$889$	− 7.52734i	− 0.252459i
$890$	0	0
$891$	− 2.91185i	− 0.0975508i
$892$	0	0
$893$	53.3545	1.78544
$894$	0	0
$895$	− 1.66786i	− 0.0557504i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.7289i	0.391179i
$900$	0	0
$901$	−15.4875	−0.515964
$902$	0	0
$903$	9.64550i	0.320982i
$904$	0	0
$905$	− 8.14914i	− 0.270887i
$906$	0	0
$907$	40.6282	1.34904	0.674518	−	0.738259i	$-0.264352\pi$
$907$	40.6282	1.34904	0.674518	+	0.738259i	$0.264352\pi$
$908$	0	0
$909$	15.0248	0.498340
$910$	0	0
$911$	46.2731	1.53309	0.766547	−	0.642188i	$-0.221973\pi$
$911$	46.2731	1.53309	0.766547	+	0.642188i	$0.221973\pi$
$912$	0	0
$913$	4.81966	0.159507
$914$	0	0
$915$	6.05190i	0.200070i
$916$	0	0
$917$	13.7727i	0.454813i
$918$	0	0
$919$	50.4674	1.66477	0.832383	−	0.554201i	$-0.186976\pi$
$919$	50.4674	1.66477	0.832383	+	0.554201i	$0.186976\pi$
$920$	0	0
$921$	− 26.4795i	− 0.872529i
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 2.19029i	− 0.0720164i
$926$	0	0
$927$	9.20775	0.302422
$928$	0	0
$929$	17.9250i	0.588100i	0.955790	+	0.294050i	$0.0950033\pi$
$929$	17.9250i	0.588100i	−0.955790	+	0.294050i	$0.904997\pi$
$930$	0	0
$931$	27.3220i	0.895442i
$932$	0	0
$933$	15.5308	0.508455
$934$	0	0
$935$	4.44876	0.145490
$936$	0	0
$937$	5.85325	0.191217	0.0956086	−	0.995419i	$-0.469520\pi$
$937$	5.85325	0.191217	0.0956086	+	0.995419i	$0.469520\pi$
$938$	0	0
$939$	−16.9681	−0.553732
$940$	0	0
$941$	− 56.7958i	− 1.85149i	−0.378147	−	0.925745i	$-0.623439\pi$
$941$	− 56.7958i	− 1.85149i	0.378147	−	0.925745i	$-0.376561\pi$
$942$	0	0
$943$	− 22.4733i	− 0.731830i
$944$	0	0
$945$	−0.582105	−0.0189359
$946$	0	0
$947$	6.14974i	0.199840i	0.994995	+	0.0999198i	$0.0318586\pi$
$947$	6.14974i	0.199840i	−0.994995	+	0.0999198i	$0.968141\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 25.5066i	− 0.827109i
$952$	0	0
$953$	−48.9778	−1.58655	−0.793273	−	0.608867i	$-0.791624\pi$
$953$	−48.9778	−1.58655	−0.793273	+	0.608867i	$0.791624\pi$
$954$	0	0
$955$	9.47889i	0.306730i
$956$	0	0
$957$	− 8.15883i	− 0.263738i
$958$	0	0
$959$	22.8786	0.738788
$960$	0	0
$961$	13.4776	0.434760
$962$	0	0
$963$	7.22952	0.232968
$964$	0	0
$965$	8.31096	0.267539
$966$	0	0
$967$	54.2616i	1.74493i	0.488673	+	0.872467i	$0.337481\pi$
$967$	54.2616i	1.74493i	−0.488673	+	0.872467i	$0.662519\pi$
$968$	0	0
$969$	12.7493i	0.409567i
$970$	0	0
$971$	−36.3142	−1.16538	−0.582689	−	0.812695i	$-0.698000\pi$
$971$	−36.3142	−1.16538	−0.582689	+	0.812695i	$0.698000\pi$
$972$	0	0
$973$	− 3.11124i	− 0.0997419i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	49.7375i	1.59124i	0.605794	+	0.795621i	$0.292856\pi$
$977$	49.7375i	1.59124i	−0.605794	+	0.795621i	$0.707144\pi$
$978$	0	0
$979$	−52.3274	−1.67239
$980$	0	0
$981$	− 15.5036i	− 0.494993i
$982$	0	0
$983$	10.9981i	0.350784i	0.984499	+	0.175392i	$0.0561193\pi$
$983$	10.9981i	0.350784i	−0.984499	+	0.175392i	$0.943881\pi$
$984$	0	0
$985$	0.106932	0.00340715
$986$	0	0
$987$	−12.0847	−0.384660
$988$	0	0
$989$	−52.9920	−1.68505
$990$	0	0
$991$	6.73078	0.213810	0.106905	−	0.994269i	$-0.465906\pi$
$991$	6.73078	0.213810	0.106905	+	0.994269i	$0.465906\pi$
$992$	0	0
$993$	− 10.6136i	− 0.336811i
$994$	0	0
$995$	6.58881i	0.208879i
$996$	0	0
$997$	43.3497	1.37290	0.686450	−	0.727177i	$-0.259168\pi$
$997$	43.3497	1.37290	0.686450	+	0.727177i	$0.259168\pi$
$998$	0	0
$999$	0.466812i	0.0147693i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2028.2.b.f.337.3		6
3.2	odd	2		6084.2.b.r.4393.4		6
13.2	odd	12		2028.2.i.l.529.2		6
13.3	even	3		2028.2.q.j.1837.3		12
13.4	even	6		2028.2.q.j.361.4		12
13.5	odd	4		2028.2.a.i.1.2	✓	3
13.6	odd	12		2028.2.i.l.2005.2		6
13.7	odd	12		2028.2.i.m.2005.2		6
13.8	odd	4		2028.2.a.j.1.2	yes	3
13.9	even	3		2028.2.q.j.361.3		12
13.10	even	6		2028.2.q.j.1837.4		12
13.11	odd	12		2028.2.i.m.529.2		6
13.12	even	2	inner	2028.2.b.f.337.4		6
39.5	even	4		6084.2.a.bb.1.2		3
39.8	even	4		6084.2.a.y.1.2		3
39.38	odd	2		6084.2.b.r.4393.3		6
52.31	even	4		8112.2.a.ch.1.2		3
52.47	even	4		8112.2.a.co.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2028.2.a.i.1.2	✓	3	13.5	odd	4
2028.2.a.j.1.2	yes	3	13.8	odd	4
2028.2.b.f.337.3		6	1.1	even	1	trivial
2028.2.b.f.337.4		6	13.12	even	2	inner
2028.2.i.l.529.2		6	13.2	odd	12
2028.2.i.l.2005.2		6	13.6	odd	12
2028.2.i.m.529.2		6	13.11	odd	12
2028.2.i.m.2005.2		6	13.7	odd	12
2028.2.q.j.361.3		12	13.9	even	3
2028.2.q.j.361.4		12	13.4	even	6
2028.2.q.j.1837.3		12	13.3	even	3
2028.2.q.j.1837.4		12	13.10	even	6
6084.2.a.y.1.2		3	39.8	even	4
6084.2.a.bb.1.2		3	39.5	even	4
6084.2.b.r.4393.3		6	39.38	odd	2
6084.2.b.r.4393.4		6	3.2	odd	2
8112.2.a.ch.1.2		3	52.31	even	4
8112.2.a.co.1.2		3	52.47	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:	\( N \)	\(=\)	\( 2028 = 2^{2} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2028.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.554958i q^{5} +1.04892i q^{7} +1.00000 q^{9} -2.91185i q^{11} +0.554958i q^{15} -2.75302 q^{17} +4.63102i q^{19} -1.04892i q^{21} +5.76271 q^{23} +4.69202 q^{25} -1.00000 q^{27} -2.80194 q^{29} -4.18598i q^{31} +2.91185i q^{33} +0.582105 q^{35} -0.466812i q^{37} -3.89977i q^{41} -9.19567 q^{43} -0.554958i q^{45} -11.5211i q^{47} +5.89977 q^{49} +2.75302 q^{51} +5.62565 q^{53} -1.61596 q^{55} -4.63102i q^{57} -3.10992i q^{59} +10.9051 q^{61} +1.04892i q^{63} +8.04892i q^{67} -5.76271 q^{69} +13.6920i q^{71} -9.36658i q^{73} -4.69202 q^{75} +3.05429 q^{77} -3.60925 q^{79} +1.00000 q^{81} +1.65519i q^{83} +1.52781i q^{85} +2.80194 q^{87} -17.9705i q^{89} +4.18598i q^{93} +2.57002 q^{95} -1.31767i q^{97} -2.91185i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{9} - 26 q^{17} + 18 q^{25} - 6 q^{27} - 8 q^{29} - 8 q^{35} + 18 q^{43} - 10 q^{49} + 26 q^{51} + 10 q^{53} - 30 q^{55} + 14 q^{61} - 18 q^{75} - 6 q^{77} + 2 q^{79} + 6 q^{81} + 8 q^{87}+ \cdots - 34 q^{95}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^9 - 26 * q^17 + 18 * q^25 - 6 * q^27 - 8 * q^29 - 8 * q^35 + 18 * q^43 - 10 * q^49 + 26 * q^51 + 10 * q^53 - 30 * q^55 + 14 * q^61 - 18 * q^75 - 6 * q^77 + 2 * q^79 + 6 * q^81 + 8 * q^87 - 34 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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