LMFDB - Embedded newform 784.4.f.j.783.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(783,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("784.783");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$46.2574974445$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.5
Character	$\chi$	$=$	784.783
Dual form			784.4.f.j.783.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.68961 q^{3} +20.4139i q^{5} +5.37166 q^{9} +O(q^{10})$	$q-5.68961 q^{3} +20.4139i q^{5} +5.37166 q^{9} -27.4118i q^{11} +33.2667i q^{13} -116.147i q^{15} +51.6329i q^{17} -123.055 q^{19} +203.652i q^{23} -291.726 q^{25} +123.057 q^{27} -27.1005 q^{29} -132.591 q^{31} +155.963i q^{33} -98.1248 q^{37} -189.274i q^{39} +475.182i q^{41} -288.305i q^{43} +109.656i q^{45} +568.917 q^{47} -293.771i q^{51} -133.525 q^{53} +559.581 q^{55} +700.134 q^{57} -307.068 q^{59} -735.787i q^{61} -679.101 q^{65} +130.251i q^{67} -1158.70i q^{69} +708.596i q^{71} +562.738i q^{73} +1659.81 q^{75} -787.673i q^{79} -845.180 q^{81} +300.047 q^{83} -1054.03 q^{85} +154.191 q^{87} -1134.02i q^{89} +754.392 q^{93} -2512.03i q^{95} +234.071i q^{97} -147.247i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 88 q^{9}+O(q^{10}) $ 24 * q + 88 * q^9	$ 24 q + 88 q^{9} - 856 q^{25} + 896 q^{29} - 2496 q^{53} + 4416 q^{57} - 4416 q^{65} - 4904 q^{81} - 2432 q^{85} - 11968 q^{93}+O(q^{100}) $ 24 * q + 88 * q^9 - 856 * q^25 + 896 * q^29 - 2496 * q^53 + 4416 * q^57 - 4416 * q^65 - 4904 * q^81 - 2432 * q^85 - 11968 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−5.68961	−1.09497	−0.547483	−	0.836817i	$-0.684414\pi$
$3$	−5.68961	−1.09497	−0.547483	+	0.836817i	$0.684414\pi$
$4$	0	0
$5$	20.4139i	1.82587i	0.408103	+	0.912936i	$0.366190\pi$
$5$	20.4139i	1.82587i	−0.408103	+	0.912936i	$0.633810\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.37166	0.198951
$10$	0	0
$11$	− 27.4118i	− 0.751362i	−0.926749	−	0.375681i	$-0.877409\pi$
$11$	− 27.4118i	− 0.751362i	0.926749	−	0.375681i	$-0.122591\pi$
$12$	0	0
$13$	33.2667i	0.709732i	0.934917	+	0.354866i	$0.115473\pi$
$13$	33.2667i	0.709732i	−0.934917	+	0.354866i	$0.884527\pi$
$14$	0	0
$15$	− 116.147i	− 1.99927i
$16$	0	0
$17$	51.6329i	0.736637i	0.929700	+	0.368318i	$0.120066\pi$
$17$	51.6329i	0.736637i	−0.929700	+	0.368318i	$0.879934\pi$
$18$	0	0
$19$	−123.055	−1.48583	−0.742914	−	0.669387i	$-0.766557\pi$
$19$	−123.055	−1.48583	−0.742914	+	0.669387i	$0.766557\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	203.652i	1.84628i	0.384469	+	0.923138i	$0.374385\pi$
$23$	203.652i	1.84628i	−0.384469	+	0.923138i	$0.625615\pi$
$24$	0	0
$25$	−291.726	−2.33381
$26$	0	0
$27$	123.057	0.877122
$28$	0	0
$29$	−27.1005	−0.173532	−0.0867662	−	0.996229i	$-0.527653\pi$
$29$	−27.1005	−0.173532	−0.0867662	+	0.996229i	$0.527653\pi$
$30$	0	0
$31$	−132.591	−0.768196	−0.384098	−	0.923292i	$-0.625488\pi$
$31$	−132.591	−0.768196	−0.384098	+	0.923292i	$0.625488\pi$
$32$	0	0
$33$	155.963i	0.822715i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−98.1248	−0.435990	−0.217995	−	0.975950i	$-0.569952\pi$
$37$	−98.1248	−0.435990	−0.217995	+	0.975950i	$0.569952\pi$
$38$	0	0
$39$	− 189.274i	− 0.777132i
$40$	0	0
$41$	475.182i	1.81002i	0.425388	+	0.905011i	$0.360138\pi$
$41$	475.182i	1.81002i	−0.425388	+	0.905011i	$0.639862\pi$
$42$	0	0
$43$	− 288.305i	− 1.02247i	−0.859442	−	0.511233i	$-0.829189\pi$
$43$	− 288.305i	− 1.02247i	0.859442	−	0.511233i	$-0.170811\pi$
$44$	0	0
$45$	109.656i	0.363258i
$46$	0	0
$47$	568.917	1.76564	0.882821	−	0.469710i	$-0.155642\pi$
$47$	568.917	1.76564	0.882821	+	0.469710i	$0.155642\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 293.771i	− 0.806592i
$52$	0	0
$53$	−133.525	−0.346059	−0.173029	−	0.984917i	$-0.555356\pi$
$53$	−133.525	−0.346059	−0.173029	+	0.984917i	$0.555356\pi$
$54$	0	0
$55$	559.581	1.37189
$56$	0	0
$57$	700.134	1.62693
$58$	0	0
$59$	−307.068	−0.677573	−0.338786	−	0.940863i	$-0.610016\pi$
$59$	−307.068	−0.677573	−0.338786	+	0.940863i	$0.610016\pi$
$60$	0	0
$61$	− 735.787i	− 1.54439i	−0.635385	−	0.772195i	$-0.719159\pi$
$61$	− 735.787i	− 1.54439i	0.635385	−	0.772195i	$-0.280841\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−679.101	−1.29588
$66$	0	0
$67$	130.251i	0.237504i	0.992924	+	0.118752i	$0.0378893\pi$
$67$	130.251i	0.237504i	−0.992924	+	0.118752i	$0.962111\pi$
$68$	0	0
$69$	− 1158.70i	− 2.02161i
$70$	0	0
$71$	708.596i	1.18443i	0.805778	+	0.592217i	$0.201747\pi$
$71$	708.596i	1.18443i	−0.805778	+	0.592217i	$0.798253\pi$
$72$	0	0
$73$	562.738i	0.902240i	0.892463	+	0.451120i	$0.148975\pi$
$73$	562.738i	0.902240i	−0.892463	+	0.451120i	$0.851025\pi$
$74$	0	0
$75$	1659.81	2.55544
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 787.673i	− 1.12177i	−0.827893	−	0.560887i	$-0.810460\pi$
$79$	− 787.673i	− 1.12177i	0.827893	−	0.560887i	$-0.189540\pi$
$80$	0	0
$81$	−845.180	−1.15937
$82$	0	0
$83$	300.047	0.396801	0.198400	−	0.980121i	$-0.436425\pi$
$83$	300.047	0.396801	0.198400	+	0.980121i	$0.436425\pi$
$84$	0	0
$85$	−1054.03	−1.34500
$86$	0	0
$87$	154.191	0.190012
$88$	0	0
$89$	− 1134.02i	− 1.35062i	−0.737532	−	0.675312i	$-0.764009\pi$
$89$	− 1134.02i	− 1.35062i	0.737532	−	0.675312i	$-0.235991\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	754.392	0.841149
$94$	0	0
$95$	− 2512.03i	− 2.71293i
$96$	0	0
$97$	234.071i	0.245013i	0.992468	+	0.122507i	$0.0390933\pi$
$97$	234.071i	0.245013i	−0.992468	+	0.122507i	$0.960907\pi$
$98$	0	0
$99$	− 147.247i	− 0.149484i
$100$	0	0
$101$	− 548.705i	− 0.540576i	−0.962780	−	0.270288i	$-0.912881\pi$
$101$	− 548.705i	− 0.540576i	0.962780	−	0.270288i	$-0.0871189\pi$
$102$	0	0
$103$	407.887	0.390197	0.195099	−	0.980784i	$-0.437497\pi$
$103$	407.887	0.390197	0.195099	+	0.980784i	$0.437497\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 431.721i	− 0.390056i	−0.980798	−	0.195028i	$-0.937520\pi$
$107$	− 431.721i	− 0.390056i	0.980798	−	0.195028i	$-0.0624798\pi$
$108$	0	0
$109$	1417.95	1.24601	0.623004	−	0.782219i	$-0.285912\pi$
$109$	1417.95	1.24601	0.623004	+	0.782219i	$0.285912\pi$
$110$	0	0
$111$	558.292	0.477394
$112$	0	0
$113$	−705.689	−0.587483	−0.293742	−	0.955885i	$-0.594901\pi$
$113$	−705.689	−0.587483	−0.293742	+	0.955885i	$0.594901\pi$
$114$	0	0
$115$	−4157.32	−3.37106
$116$	0	0
$117$	178.697i	0.141202i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	579.591	0.435456
$122$	0	0
$123$	− 2703.60i	− 1.98191i
$124$	0	0
$125$	− 3403.52i	− 2.43536i
$126$	0	0
$127$	− 1941.75i	− 1.35671i	−0.734735	−	0.678355i	$-0.762693\pi$
$127$	− 1941.75i	− 1.35671i	0.734735	−	0.678355i	$-0.237307\pi$
$128$	0	0
$129$	1640.34i	1.11957i
$130$	0	0
$131$	−1042.75	−0.695463	−0.347731	−	0.937594i	$-0.613048\pi$
$131$	−1042.75	−0.695463	−0.347731	+	0.937594i	$0.613048\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2512.06i	1.60151i
$136$	0	0
$137$	−2026.12	−1.26353	−0.631763	−	0.775162i	$-0.717668\pi$
$137$	−2026.12	−1.26353	−0.631763	+	0.775162i	$0.717668\pi$
$138$	0	0
$139$	−995.762	−0.607622	−0.303811	−	0.952732i	$-0.598259\pi$
$139$	−995.762	−0.607622	−0.303811	+	0.952732i	$0.598259\pi$
$140$	0	0
$141$	−3236.92	−1.93332
$142$	0	0
$143$	911.900	0.533265
$144$	0	0
$145$	− 553.226i	− 0.316848i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	925.945	0.509103	0.254551	−	0.967059i	$-0.418072\pi$
$149$	925.945	0.509103	0.254551	+	0.967059i	$0.418072\pi$
$150$	0	0
$151$	− 2101.38i	− 1.13250i	−0.824233	−	0.566251i	$-0.808393\pi$
$151$	− 2101.38i	− 1.13250i	0.824233	−	0.566251i	$-0.191607\pi$
$152$	0	0
$153$	277.355i	0.146554i
$154$	0	0
$155$	− 2706.70i	− 1.40263i
$156$	0	0
$157$	2459.33i	1.25016i	0.780560	+	0.625081i	$0.214934\pi$
$157$	2459.33i	1.25016i	−0.780560	+	0.625081i	$0.785066\pi$
$158$	0	0
$159$	759.707	0.378922
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 2902.94i	− 1.39494i	−0.716612	−	0.697472i	$-0.754308\pi$
$163$	− 2902.94i	− 1.39494i	0.716612	−	0.697472i	$-0.245692\pi$
$164$	0	0
$165$	−3183.80	−1.50217
$166$	0	0
$167$	1040.40	0.482089	0.241044	−	0.970514i	$-0.422510\pi$
$167$	1040.40	0.482089	0.241044	+	0.970514i	$0.422510\pi$
$168$	0	0
$169$	1090.33	0.496281
$170$	0	0
$171$	−661.010	−0.295606
$172$	0	0
$173$	− 144.134i	− 0.0633428i	−0.999498	−	0.0316714i	$-0.989917\pi$
$173$	− 144.134i	− 0.0633428i	0.999498	−	0.0316714i	$-0.0100830\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1747.10	0.741919
$178$	0	0
$179$	− 717.436i	− 0.299574i	−0.988718	−	0.149787i	$-0.952141\pi$
$179$	− 717.436i	− 0.299574i	0.988718	−	0.149787i	$-0.0478587\pi$
$180$	0	0
$181$	473.409i	0.194410i	0.995264	+	0.0972050i	$0.0309903\pi$
$181$	473.409i	0.194410i	−0.995264	+	0.0972050i	$0.969010\pi$
$182$	0	0
$183$	4186.34i	1.69106i
$184$	0	0
$185$	− 2003.11i	− 0.796061i
$186$	0	0
$187$	1415.35	0.553481
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2529.99i	0.958450i	0.877692	+	0.479225i	$0.159082\pi$
$191$	2529.99i	0.958450i	−0.877692	+	0.479225i	$0.840918\pi$
$192$	0	0
$193$	3252.98	1.21324	0.606619	−	0.794993i	$-0.292525\pi$
$193$	3252.98	1.21324	0.606619	+	0.794993i	$0.292525\pi$
$194$	0	0
$195$	3863.82	1.41894
$196$	0	0
$197$	5178.04	1.87269	0.936346	−	0.351078i	$-0.114185\pi$
$197$	5178.04	1.87269	0.936346	+	0.351078i	$0.114185\pi$
$198$	0	0
$199$	−2039.63	−0.726561	−0.363280	−	0.931680i	$-0.618343\pi$
$199$	−2039.63	−0.726561	−0.363280	+	0.931680i	$0.618343\pi$
$200$	0	0
$201$	− 741.080i	− 0.260058i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9700.30	−3.30487
$206$	0	0
$207$	1093.95i	0.367318i
$208$	0	0
$209$	3373.16i	1.11639i
$210$	0	0
$211$	− 1313.78i	− 0.428647i	−0.976763	−	0.214323i	$-0.931245\pi$
$211$	− 1313.78i	− 0.428647i	0.976763	−	0.214323i	$-0.0687546\pi$
$212$	0	0
$213$	− 4031.63i	− 1.29692i
$214$	0	0
$215$	5885.41	1.86689
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 3201.76i	− 0.987922i
$220$	0	0
$221$	−1717.66	−0.522815
$222$	0	0
$223$	769.389	0.231041	0.115520	−	0.993305i	$-0.463146\pi$
$223$	769.389	0.231041	0.115520	+	0.993305i	$0.463146\pi$
$224$	0	0
$225$	−1567.05	−0.464312
$226$	0	0
$227$	3693.41	1.07991	0.539956	−	0.841693i	$-0.318441\pi$
$227$	3693.41	1.07991	0.539956	+	0.841693i	$0.318441\pi$
$228$	0	0
$229$	224.251i	0.0647115i	0.999476	+	0.0323557i	$0.0103009\pi$
$229$	224.251i	0.0647115i	−0.999476	+	0.0323557i	$0.989699\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2826.69	0.794775	0.397387	−	0.917651i	$-0.369917\pi$
$233$	2826.69	0.794775	0.397387	+	0.917651i	$0.369917\pi$
$234$	0	0
$235$	11613.8i	3.22383i
$236$	0	0
$237$	4481.55i	1.22830i
$238$	0	0
$239$	635.796i	0.172076i	0.996292	+	0.0860381i	$0.0274207\pi$
$239$	635.796i	0.172076i	−0.996292	+	0.0860381i	$0.972579\pi$
$240$	0	0
$241$	6316.97i	1.68843i	0.536004	+	0.844216i	$0.319933\pi$
$241$	6316.97i	1.68843i	−0.536004	+	0.844216i	$0.680067\pi$
$242$	0	0
$243$	1486.21	0.392348
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 4093.63i	− 1.05454i
$248$	0	0
$249$	−1707.15	−0.434483
$250$	0	0
$251$	−2027.17	−0.509775	−0.254888	−	0.966971i	$-0.582039\pi$
$251$	−2027.17	−0.509775	−0.254888	+	0.966971i	$0.582039\pi$
$252$	0	0
$253$	5582.47	1.38722
$254$	0	0
$255$	5997.01	1.47273
$256$	0	0
$257$	− 5246.80i	− 1.27349i	−0.771075	−	0.636744i	$-0.780281\pi$
$257$	− 5246.80i	− 1.27349i	0.771075	−	0.636744i	$-0.219719\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−145.575	−0.0345244
$262$	0	0
$263$	631.404i	0.148038i	0.997257	+	0.0740191i	$0.0235826\pi$
$263$	631.404i	0.148038i	−0.997257	+	0.0740191i	$0.976417\pi$
$264$	0	0
$265$	− 2725.77i	− 0.631859i
$266$	0	0
$267$	6452.11i	1.47889i
$268$	0	0
$269$	− 1164.99i	− 0.264056i	−0.991246	−	0.132028i	$-0.957851\pi$
$269$	− 1164.99i	− 0.264056i	0.991246	−	0.132028i	$-0.0421488\pi$
$270$	0	0
$271$	−7955.89	−1.78334	−0.891672	−	0.452683i	$-0.850467\pi$
$271$	−7955.89	−1.78334	−0.891672	+	0.452683i	$0.850467\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7996.74i	1.75353i
$276$	0	0
$277$	−3600.97	−0.781087	−0.390544	−	0.920584i	$-0.627713\pi$
$277$	−3600.97	−0.781087	−0.390544	+	0.920584i	$0.627713\pi$
$278$	0	0
$279$	−712.235	−0.152833
$280$	0	0
$281$	1629.95	0.346031	0.173016	−	0.984919i	$-0.444649\pi$
$281$	1629.95	0.346031	0.173016	+	0.984919i	$0.444649\pi$
$282$	0	0
$283$	−3562.61	−0.748323	−0.374162	−	0.927364i	$-0.622069\pi$
$283$	−3562.61	−0.748323	−0.374162	+	0.927364i	$0.622069\pi$
$284$	0	0
$285$	14292.4i	2.97057i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2247.04	0.457366
$290$	0	0
$291$	− 1331.77i	− 0.268281i
$292$	0	0
$293$	2256.82i	0.449982i	0.974361	+	0.224991i	$0.0722352\pi$
$293$	2256.82i	0.449982i	−0.974361	+	0.224991i	$0.927765\pi$
$294$	0	0
$295$	− 6268.44i	− 1.23716i
$296$	0	0
$297$	− 3373.21i	− 0.659036i
$298$	0	0
$299$	−6774.82	−1.31036
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3121.92i	0.591912i
$304$	0	0
$305$	15020.2	2.81986
$306$	0	0
$307$	−7898.99	−1.46847	−0.734234	−	0.678897i	$-0.762458\pi$
$307$	−7898.99	−1.46847	−0.734234	+	0.678897i	$0.762458\pi$
$308$	0	0
$309$	−2320.72	−0.427253
$310$	0	0
$311$	6032.25	1.09986	0.549932	−	0.835209i	$-0.314654\pi$
$311$	6032.25	1.09986	0.549932	+	0.835209i	$0.314654\pi$
$312$	0	0
$313$	648.749i	0.117155i	0.998283	+	0.0585774i	$0.0186564\pi$
$313$	648.749i	0.117155i	−0.998283	+	0.0585774i	$0.981344\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5400.21	−0.956801	−0.478401	−	0.878142i	$-0.658783\pi$
$317$	−5400.21	−0.956801	−0.478401	+	0.878142i	$0.658783\pi$
$318$	0	0
$319$	742.875i	0.130386i
$320$	0	0
$321$	2456.32i	0.427098i
$322$	0	0
$323$	− 6353.68i	− 1.09452i
$324$	0	0
$325$	− 9704.75i	− 1.65638i
$326$	0	0
$327$	−8067.57	−1.36434
$328$	0	0
$329$	0	0
$330$	0	0
$331$	4604.90i	0.764678i	0.924022	+	0.382339i	$0.124881\pi$
$331$	4604.90i	0.764678i	−0.924022	+	0.382339i	$0.875119\pi$
$332$	0	0
$333$	−527.093	−0.0867403
$334$	0	0
$335$	−2658.94	−0.433651
$336$	0	0
$337$	−8621.08	−1.39353	−0.696766	−	0.717299i	$-0.745378\pi$
$337$	−8621.08	−1.39353	−0.696766	+	0.717299i	$0.745378\pi$
$338$	0	0
$339$	4015.09	0.643274
$340$	0	0
$341$	3634.57i	0.577193i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	23653.5	3.69120
$346$	0	0
$347$	− 2071.61i	− 0.320489i	−0.987077	−	0.160244i	$-0.948772\pi$
$347$	− 2071.61i	− 0.320489i	0.987077	−	0.160244i	$-0.0512283\pi$
$348$	0	0
$349$	− 2795.68i	− 0.428794i	−0.976747	−	0.214397i	$-0.931221\pi$
$349$	− 2795.68i	− 0.428794i	0.976747	−	0.214397i	$-0.0687787\pi$
$350$	0	0
$351$	4093.69i	0.622521i
$352$	0	0
$353$	− 348.273i	− 0.0525118i	−0.999655	−	0.0262559i	$-0.991642\pi$
$353$	− 348.273i	− 0.0525118i	0.999655	−	0.0262559i	$-0.00835848\pi$
$354$	0	0
$355$	−14465.2	−2.16263
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 4852.06i	− 0.713321i	−0.934234	−	0.356660i	$-0.883915\pi$
$359$	− 4852.06i	− 0.713321i	0.934234	−	0.356660i	$-0.116085\pi$
$360$	0	0
$361$	8283.51	1.20768
$362$	0	0
$363$	−3297.65	−0.476809
$364$	0	0
$365$	−11487.7	−1.64737
$366$	0	0
$367$	−9356.53	−1.33081	−0.665404	−	0.746483i	$-0.731741\pi$
$367$	−9356.53	−1.33081	−0.665404	+	0.746483i	$0.731741\pi$
$368$	0	0
$369$	2552.52i	0.360105i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−4573.24	−0.634835	−0.317418	−	0.948286i	$-0.602816\pi$
$373$	−4573.24	−0.634835	−0.317418	+	0.948286i	$0.602816\pi$
$374$	0	0
$375$	19364.7i	2.66664i
$376$	0	0
$377$	− 901.544i	− 0.123161i
$378$	0	0
$379$	11843.8i	1.60521i	0.596514	+	0.802603i	$0.296552\pi$
$379$	11843.8i	1.60521i	−0.596514	+	0.802603i	$0.703448\pi$
$380$	0	0
$381$	11047.8i	1.48555i
$382$	0	0
$383$	−10479.3	−1.39808	−0.699041	−	0.715082i	$-0.746389\pi$
$383$	−10479.3	−1.39808	−0.699041	+	0.715082i	$0.746389\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1548.68i	− 0.203420i
$388$	0	0
$389$	−1599.54	−0.208483	−0.104242	−	0.994552i	$-0.533242\pi$
$389$	−1599.54	−0.208483	−0.104242	+	0.994552i	$0.533242\pi$
$390$	0	0
$391$	−10515.1	−1.36003
$392$	0	0
$393$	5932.85	0.761508
$394$	0	0
$395$	16079.4	2.04821
$396$	0	0
$397$	11.6116i	0.00146793i	1.00000		0.000733966i	$0.000233629\pi$
$397$	11.6116i	0.00146793i	−1.00000		0.000733966i	$0.999766\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−7462.34	−0.929306	−0.464653	−	0.885493i	$-0.653821\pi$
$401$	−7462.34	−0.929306	−0.464653	+	0.885493i	$0.653821\pi$
$402$	0	0
$403$	− 4410.87i	− 0.545213i
$404$	0	0
$405$	− 17253.4i	− 2.11686i
$406$	0	0
$407$	2689.78i	0.327586i
$408$	0	0
$409$	− 5640.51i	− 0.681920i	−0.940078	−	0.340960i	$-0.889248\pi$
$409$	− 5640.51i	− 0.681920i	0.940078	−	0.340960i	$-0.110752\pi$
$410$	0	0
$411$	11527.8	1.38352
$412$	0	0
$413$	0	0
$414$	0	0
$415$	6125.12i	0.724507i
$416$	0	0
$417$	5665.50	0.665325
$418$	0	0
$419$	−8037.03	−0.937076	−0.468538	−	0.883443i	$-0.655219\pi$
$419$	−8037.03	−0.937076	−0.468538	+	0.883443i	$0.655219\pi$
$420$	0	0
$421$	9815.24	1.13626	0.568130	−	0.822939i	$-0.307667\pi$
$421$	9815.24	1.13626	0.568130	+	0.822939i	$0.307667\pi$
$422$	0	0
$423$	3056.03	0.351275
$424$	0	0
$425$	− 15062.7i	− 1.71917i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5188.36	−0.583907
$430$	0	0
$431$	4053.33i	0.452998i	0.974011	+	0.226499i	$0.0727280\pi$
$431$	4053.33i	0.452998i	−0.974011	+	0.226499i	$0.927272\pi$
$432$	0	0
$433$	13331.5i	1.47961i	0.672822	+	0.739805i	$0.265082\pi$
$433$	13331.5i	1.47961i	−0.672822	+	0.739805i	$0.734918\pi$
$434$	0	0
$435$	3147.64i	0.346938i
$436$	0	0
$437$	− 25060.4i	− 2.74325i
$438$	0	0
$439$	11361.5	1.23521	0.617604	−	0.786489i	$-0.288104\pi$
$439$	11361.5	1.23521	0.617604	+	0.786489i	$0.288104\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12669.9i	− 1.35884i	−0.733750	−	0.679419i	$-0.762232\pi$
$443$	− 12669.9i	− 1.35884i	0.733750	−	0.679419i	$-0.237768\pi$
$444$	0	0
$445$	23149.7	2.46607
$446$	0	0
$447$	−5268.26	−0.557450
$448$	0	0
$449$	8911.65	0.936674	0.468337	−	0.883550i	$-0.344853\pi$
$449$	8911.65	0.936674	0.468337	+	0.883550i	$0.344853\pi$
$450$	0	0
$451$	13025.6	1.35998
$452$	0	0
$453$	11956.0i	1.24005i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13833.7	1.41600	0.707999	−	0.706213i	$-0.249598\pi$
$457$	13833.7	1.41600	0.707999	+	0.706213i	$0.249598\pi$
$458$	0	0
$459$	6353.78i	0.646120i
$460$	0	0
$461$	− 852.971i	− 0.0861753i	−0.999071	−	0.0430877i	$-0.986281\pi$
$461$	− 852.971i	− 0.0861753i	0.999071	−	0.0430877i	$-0.0137195\pi$
$462$	0	0
$463$	6647.58i	0.667255i	0.942705	+	0.333628i	$0.108273\pi$
$463$	6647.58i	0.667255i	−0.942705	+	0.333628i	$0.891727\pi$
$464$	0	0
$465$	15400.1i	1.53583i
$466$	0	0
$467$	7396.09	0.732870	0.366435	−	0.930444i	$-0.380578\pi$
$467$	7396.09	0.732870	0.366435	+	0.930444i	$0.380578\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 13992.6i	− 1.36889i
$472$	0	0
$473$	−7902.96	−0.768242
$474$	0	0
$475$	35898.3	3.46764
$476$	0	0
$477$	−717.253	−0.0688485
$478$	0	0
$479$	−10779.1	−1.02820	−0.514102	−	0.857729i	$-0.671875\pi$
$479$	−10779.1	−1.02820	−0.514102	+	0.857729i	$0.671875\pi$
$480$	0	0
$481$	− 3264.28i	− 0.309436i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4778.29	−0.447363
$486$	0	0
$487$	5643.59i	0.525124i	0.964915	+	0.262562i	$0.0845675\pi$
$487$	5643.59i	0.525124i	−0.964915	+	0.262562i	$0.915433\pi$
$488$	0	0
$489$	16516.6i	1.52742i
$490$	0	0
$491$	2800.93i	0.257442i	0.991681	+	0.128721i	$0.0410872\pi$
$491$	2800.93i	0.257442i	−0.991681	+	0.128721i	$0.958913\pi$
$492$	0	0
$493$	− 1399.28i	− 0.127830i
$494$	0	0
$495$	3005.88	0.272938
$496$	0	0
$497$	0	0
$498$	0	0
$499$	17704.6i	1.58831i	0.607712	+	0.794157i	$0.292087\pi$
$499$	17704.6i	1.58831i	−0.607712	+	0.794157i	$0.707913\pi$
$500$	0	0
$501$	−5919.49	−0.527871
$502$	0	0
$503$	−8247.73	−0.731110	−0.365555	−	0.930790i	$-0.619121\pi$
$503$	−8247.73	−0.731110	−0.365555	+	0.930790i	$0.619121\pi$
$504$	0	0
$505$	11201.2	0.987022
$506$	0	0
$507$	−6203.55	−0.543410
$508$	0	0
$509$	− 409.106i	− 0.0356253i	−0.999841	−	0.0178127i	$-0.994330\pi$
$509$	− 409.106i	− 0.0356253i	0.999841	−	0.0178127i	$-0.00567025\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−15142.7	−1.30325
$514$	0	0
$515$	8326.55i	0.712450i
$516$	0	0
$517$	− 15595.1i	− 1.32664i
$518$	0	0
$519$	820.066i	0.0693582i
$520$	0	0
$521$	10656.7i	0.896123i	0.894003	+	0.448062i	$0.147886\pi$
$521$	10656.7i	0.896123i	−0.894003	+	0.448062i	$0.852114\pi$
$522$	0	0
$523$	−3241.19	−0.270989	−0.135494	−	0.990778i	$-0.543262\pi$
$523$	−3241.19	−0.270989	−0.135494	+	0.990778i	$0.543262\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 6846.07i	− 0.565882i
$528$	0	0
$529$	−29307.1	−2.40874
$530$	0	0
$531$	−1649.46	−0.134803
$532$	0	0
$533$	−15807.7	−1.28463
$534$	0	0
$535$	8813.09	0.712193
$536$	0	0
$537$	4081.93i	0.328023i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−15351.8	−1.22001	−0.610004	−	0.792399i	$-0.708832\pi$
$541$	−15351.8	−1.22001	−0.610004	+	0.792399i	$0.708832\pi$
$542$	0	0
$543$	− 2693.51i	− 0.212872i
$544$	0	0
$545$	28945.8i	2.27505i
$546$	0	0
$547$	− 3418.07i	− 0.267178i	−0.991037	−	0.133589i	$-0.957350\pi$
$547$	− 3418.07i	− 0.267178i	0.991037	−	0.133589i	$-0.0426502\pi$
$548$	0	0
$549$	− 3952.40i	− 0.307257i
$550$	0	0
$551$	3334.85	0.257839
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11396.9i	0.871660i
$556$	0	0
$557$	−13943.5	−1.06069	−0.530345	−	0.847782i	$-0.677938\pi$
$557$	−13943.5	−1.06069	−0.530345	+	0.847782i	$0.677938\pi$
$558$	0	0
$559$	9590.94	0.725677
$560$	0	0
$561$	−8052.81	−0.606042
$562$	0	0
$563$	−10616.9	−0.794761	−0.397380	−	0.917654i	$-0.630081\pi$
$563$	−10616.9	−0.794761	−0.397380	+	0.917654i	$0.630081\pi$
$564$	0	0
$565$	− 14405.8i	− 1.07267i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	421.063	0.0310227	0.0155113	−	0.999880i	$-0.495062\pi$
$569$	421.063	0.0310227	0.0155113	+	0.999880i	$0.495062\pi$
$570$	0	0
$571$	− 25824.9i	− 1.89271i	−0.323126	−	0.946356i	$-0.604734\pi$
$571$	− 25824.9i	− 1.89271i	0.323126	−	0.946356i	$-0.395266\pi$
$572$	0	0
$573$	− 14394.7i	− 1.04947i
$574$	0	0
$575$	− 59410.5i	− 4.30885i
$576$	0	0
$577$	8500.59i	0.613318i	0.951820	+	0.306659i	$0.0992110\pi$
$577$	8500.59i	0.613318i	−0.951820	+	0.306659i	$0.900789\pi$
$578$	0	0
$579$	−18508.2	−1.32845
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3660.17i	0.260015i
$584$	0	0
$585$	−3647.90	−0.257816
$586$	0	0
$587$	−20752.1	−1.45916	−0.729582	−	0.683893i	$-0.760285\pi$
$587$	−20752.1	−1.45916	−0.729582	+	0.683893i	$0.760285\pi$
$588$	0	0
$589$	16316.0	1.14141
$590$	0	0
$591$	−29461.1	−2.05053
$592$	0	0
$593$	4423.58i	0.306332i	0.988201	+	0.153166i	$0.0489469\pi$
$593$	4423.58i	0.306332i	−0.988201	+	0.153166i	$0.951053\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	11604.7	0.795559
$598$	0	0
$599$	− 19047.4i	− 1.29926i	−0.760250	−	0.649630i	$-0.774924\pi$
$599$	− 19047.4i	− 1.29926i	0.760250	−	0.649630i	$-0.225076\pi$
$600$	0	0
$601$	13205.5i	0.896282i	0.893963	+	0.448141i	$0.147914\pi$
$601$	13205.5i	0.896282i	−0.893963	+	0.448141i	$0.852086\pi$
$602$	0	0
$603$	699.667i	0.0472515i
$604$	0	0
$605$	11831.7i	0.795086i
$606$	0	0
$607$	−8202.94	−0.548512	−0.274256	−	0.961657i	$-0.588432\pi$
$607$	−8202.94	−0.548512	−0.274256	+	0.961657i	$0.588432\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18926.0i	1.25313i
$612$	0	0
$613$	−13977.1	−0.920932	−0.460466	−	0.887677i	$-0.652318\pi$
$613$	−13977.1	−0.920932	−0.460466	+	0.887677i	$0.652318\pi$
$614$	0	0
$615$	55190.9	3.61872
$616$	0	0
$617$	17469.6	1.13987	0.569935	−	0.821690i	$-0.306968\pi$
$617$	17469.6	1.13987	0.569935	+	0.821690i	$0.306968\pi$
$618$	0	0
$619$	11475.8	0.745156	0.372578	−	0.928001i	$-0.378474\pi$
$619$	11475.8	0.745156	0.372578	+	0.928001i	$0.378474\pi$
$620$	0	0
$621$	25060.7i	1.61941i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	33013.2	2.11285
$626$	0	0
$627$	− 19192.0i	− 1.22241i
$628$	0	0
$629$	− 5066.47i	− 0.321166i
$630$	0	0
$631$	25620.4i	1.61638i	0.588924	+	0.808188i	$0.299552\pi$
$631$	25620.4i	1.61638i	−0.588924	+	0.808188i	$0.700448\pi$
$632$	0	0
$633$	7474.90i	0.469353i
$634$	0	0
$635$	39638.5	2.47718
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3806.34i	0.235644i
$640$	0	0
$641$	13393.0	0.825258	0.412629	−	0.910899i	$-0.364611\pi$
$641$	13393.0	0.825258	0.412629	+	0.910899i	$0.364611\pi$
$642$	0	0
$643$	6636.01	0.406997	0.203498	−	0.979075i	$-0.434769\pi$
$643$	6636.01	0.406997	0.203498	+	0.979075i	$0.434769\pi$
$644$	0	0
$645$	−33485.7	−2.04418
$646$	0	0
$647$	−10046.4	−0.610458	−0.305229	−	0.952279i	$-0.598733\pi$
$647$	−10046.4	−0.610458	−0.305229	+	0.952279i	$0.598733\pi$
$648$	0	0
$649$	8417.29i	0.509102i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8981.76	−0.538260	−0.269130	−	0.963104i	$-0.586736\pi$
$653$	−8981.76	−0.538260	−0.269130	+	0.963104i	$0.586736\pi$
$654$	0	0
$655$	− 21286.6i	− 1.26983i
$656$	0	0
$657$	3022.84i	0.179501i
$658$	0	0
$659$	− 30917.7i	− 1.82759i	−0.406177	−	0.913794i	$-0.633138\pi$
$659$	− 30917.7i	− 1.82759i	0.406177	−	0.913794i	$-0.366862\pi$
$660$	0	0
$661$	3506.27i	0.206321i	0.994665	+	0.103160i	$0.0328955\pi$
$661$	3506.27i	0.206321i	−0.994665	+	0.103160i	$0.967105\pi$
$662$	0	0
$663$	9772.79	0.572464
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5519.07i	− 0.320389i
$668$	0	0
$669$	−4377.52	−0.252982
$670$	0	0
$671$	−20169.3	−1.16040
$672$	0	0
$673$	−3063.54	−0.175469	−0.0877345	−	0.996144i	$-0.527963\pi$
$673$	−3063.54	−0.175469	−0.0877345	+	0.996144i	$0.527963\pi$
$674$	0	0
$675$	−35898.8	−2.04703
$676$	0	0
$677$	− 6003.94i	− 0.340842i	−0.985371	−	0.170421i	$-0.945487\pi$
$677$	− 6003.94i	− 0.340842i	0.985371	−	0.170421i	$-0.0545128\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−21014.0	−1.18247
$682$	0	0
$683$	1856.95i	0.104033i	0.998646	+	0.0520164i	$0.0165648\pi$
$683$	1856.95i	0.104033i	−0.998646	+	0.0520164i	$0.983435\pi$
$684$	0	0
$685$	− 41360.9i	− 2.30703i
$686$	0	0
$687$	− 1275.90i	− 0.0708569i
$688$	0	0
$689$	− 4441.94i	− 0.245609i
$690$	0	0
$691$	9303.94	0.512212	0.256106	−	0.966649i	$-0.417560\pi$
$691$	9303.94	0.512212	0.256106	+	0.966649i	$0.417560\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 20327.4i	− 1.10944i
$696$	0	0
$697$	−24535.0	−1.33333
$698$	0	0
$699$	−16082.8	−0.870251
$700$	0	0
$701$	−28304.2	−1.52502	−0.762508	−	0.646979i	$-0.776032\pi$
$701$	−28304.2	−1.52502	−0.762508	+	0.646979i	$0.776032\pi$
$702$	0	0
$703$	12074.7	0.647806
$704$	0	0
$705$	− 66078.0i	− 3.52999i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5950.28	−0.315187	−0.157593	−	0.987504i	$-0.550374\pi$
$709$	−5950.28	−0.315187	−0.157593	+	0.987504i	$0.550374\pi$
$710$	0	0
$711$	− 4231.11i	− 0.223177i
$712$	0	0
$713$	− 27002.4i	− 1.41830i
$714$	0	0
$715$	18615.4i	0.973674i
$716$	0	0
$717$	− 3617.43i	− 0.188418i
$718$	0	0
$719$	11142.8	0.577966	0.288983	−	0.957334i	$-0.406683\pi$
$719$	11142.8	0.577966	0.288983	+	0.957334i	$0.406683\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 35941.1i	− 1.84877i
$724$	0	0
$725$	7905.92	0.404991
$726$	0	0
$727$	−5517.58	−0.281480	−0.140740	−	0.990047i	$-0.544948\pi$
$727$	−5517.58	−0.281480	−0.140740	+	0.990047i	$0.544948\pi$
$728$	0	0
$729$	14363.9	0.729762
$730$	0	0
$731$	14886.0	0.753186
$732$	0	0
$733$	− 20181.2i	− 1.01693i	−0.861083	−	0.508464i	$-0.830213\pi$
$733$	− 20181.2i	− 1.01693i	0.861083	−	0.508464i	$-0.169787\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3570.43	0.178451
$738$	0	0
$739$	− 16392.3i	− 0.815968i	−0.912989	−	0.407984i	$-0.866232\pi$
$739$	− 16392.3i	− 0.815968i	0.912989	−	0.407984i	$-0.133768\pi$
$740$	0	0
$741$	23291.1i	1.15468i
$742$	0	0
$743$	20914.6i	1.03268i	0.856383	+	0.516341i	$0.172706\pi$
$743$	20914.6i	1.03268i	−0.856383	+	0.516341i	$0.827294\pi$
$744$	0	0
$745$	18902.1i	0.929556i
$746$	0	0
$747$	1611.75	0.0789437
$748$	0	0
$749$	0	0
$750$	0	0
$751$	18466.3i	0.897266i	0.893716	+	0.448633i	$0.148089\pi$
$751$	18466.3i	0.897266i	−0.893716	+	0.448633i	$0.851911\pi$
$752$	0	0
$753$	11533.8	0.558187
$754$	0	0
$755$	42897.3	2.06780
$756$	0	0
$757$	21663.1	1.04010	0.520051	−	0.854135i	$-0.325913\pi$
$757$	21663.1	1.04010	0.520051	+	0.854135i	$0.325913\pi$
$758$	0	0
$759$	−31762.1	−1.51896
$760$	0	0
$761$	− 26900.4i	− 1.28139i	−0.767795	−	0.640696i	$-0.778646\pi$
$761$	− 26900.4i	− 1.28139i	0.767795	−	0.640696i	$-0.221354\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5661.88	−0.267589
$766$	0	0
$767$	− 10215.1i	− 0.480895i
$768$	0	0
$769$	− 30388.7i	− 1.42502i	−0.701660	−	0.712512i	$-0.747558\pi$
$769$	− 30388.7i	− 1.42502i	0.701660	−	0.712512i	$-0.252442\pi$
$770$	0	0
$771$	29852.3i	1.39443i
$772$	0	0
$773$	801.174i	0.0372784i	0.999826	+	0.0186392i	$0.00593339\pi$
$773$	801.174i	0.0372784i	−0.999826	+	0.0186392i	$0.994067\pi$
$774$	0	0
$775$	38680.3	1.79282
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 58473.5i	− 2.68938i
$780$	0	0
$781$	19423.9	0.889939
$782$	0	0
$783$	−3334.90	−0.152209
$784$	0	0
$785$	−50204.3	−2.28264
$786$	0	0
$787$	−33591.9	−1.52150	−0.760752	−	0.649043i	$-0.775169\pi$
$787$	−33591.9	−1.52150	−0.760752	+	0.649043i	$0.775169\pi$
$788$	0	0
$789$	− 3592.44i	− 0.162097i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	24477.2	1.09610
$794$	0	0
$795$	15508.5i	0.691864i
$796$	0	0
$797$	2563.22i	0.113919i	0.998376	+	0.0569597i	$0.0181406\pi$
$797$	2563.22i	0.113919i	−0.998376	+	0.0569597i	$0.981859\pi$
$798$	0	0
$799$	29374.9i	1.30064i
$800$	0	0
$801$	− 6091.56i	− 0.268707i
$802$	0	0
$803$	15425.7	0.677908
$804$	0	0
$805$	0	0
$806$	0	0
$807$	6628.37i	0.289132i
$808$	0	0
$809$	34904.7	1.51692	0.758458	−	0.651722i	$-0.225953\pi$
$809$	34904.7	1.51692	0.758458	+	0.651722i	$0.225953\pi$
$810$	0	0
$811$	21556.7	0.933363	0.466682	−	0.884425i	$-0.345449\pi$
$811$	21556.7	0.933363	0.466682	+	0.884425i	$0.345449\pi$
$812$	0	0
$813$	45265.9	1.95270
$814$	0	0
$815$	59260.2	2.54699
$816$	0	0
$817$	35477.3i	1.51921i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25349.0	−1.07757	−0.538785	−	0.842443i	$-0.681117\pi$
$821$	−25349.0	−1.07757	−0.538785	+	0.842443i	$0.681117\pi$
$822$	0	0
$823$	− 16954.9i	− 0.718116i	−0.933315	−	0.359058i	$-0.883098\pi$
$823$	− 16954.9i	− 0.718116i	0.933315	−	0.359058i	$-0.116902\pi$
$824$	0	0
$825$	− 45498.3i	− 1.92006i
$826$	0	0
$827$	− 38797.2i	− 1.63133i	−0.578523	−	0.815666i	$-0.696371\pi$
$827$	− 38797.2i	− 1.63133i	0.578523	−	0.815666i	$-0.303629\pi$
$828$	0	0
$829$	− 29945.2i	− 1.25457i	−0.778789	−	0.627285i	$-0.784166\pi$
$829$	− 29945.2i	− 1.25457i	0.778789	−	0.627285i	$-0.215834\pi$
$830$	0	0
$831$	20488.1	0.855264
$832$	0	0
$833$	0	0
$834$	0	0
$835$	21238.7i	0.880232i
$836$	0	0
$837$	−16316.3	−0.673802
$838$	0	0
$839$	−36918.6	−1.51916	−0.759578	−	0.650416i	$-0.774595\pi$
$839$	−36918.6	−1.51916	−0.759578	+	0.650416i	$0.774595\pi$
$840$	0	0
$841$	−23654.6	−0.969887
$842$	0	0
$843$	−9273.79	−0.378893
$844$	0	0
$845$	22257.8i	0.906145i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	20269.9	0.819388
$850$	0	0
$851$	− 19983.3i	− 0.804957i
$852$	0	0
$853$	21921.6i	0.879930i	0.898015	+	0.439965i	$0.145009\pi$
$853$	21921.6i	0.879930i	−0.898015	+	0.439965i	$0.854991\pi$
$854$	0	0
$855$	− 13493.8i	− 0.539739i
$856$	0	0
$857$	− 7836.91i	− 0.312373i	−0.987728	−	0.156187i	$-0.950080\pi$
$857$	− 7836.91i	− 0.312373i	0.987728	−	0.156187i	$-0.0499201\pi$
$858$	0	0
$859$	5866.31	0.233010	0.116505	−	0.993190i	$-0.462831\pi$
$859$	5866.31	0.233010	0.116505	+	0.993190i	$0.462831\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 472.119i	− 0.0186224i	−0.999957	−	0.00931120i	$-0.997036\pi$
$863$	− 472.119i	− 0.0186224i	0.999957	−	0.00931120i	$-0.00296389\pi$
$864$	0	0
$865$	2942.33	0.115656
$866$	0	0
$867$	−12784.8	−0.500801
$868$	0	0
$869$	−21591.6	−0.842858
$870$	0	0
$871$	−4333.03	−0.168564
$872$	0	0
$873$	1257.35i	0.0487455i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23962.5	0.922640	0.461320	−	0.887234i	$-0.347376\pi$
$877$	23962.5	0.922640	0.461320	+	0.887234i	$0.347376\pi$
$878$	0	0
$879$	− 12840.4i	− 0.492715i
$880$	0	0
$881$	24460.0i	0.935390i	0.883890	+	0.467695i	$0.154915\pi$
$881$	24460.0i	0.935390i	−0.883890	+	0.467695i	$0.845085\pi$
$882$	0	0
$883$	− 17519.7i	− 0.667708i	−0.942625	−	0.333854i	$-0.891651\pi$
$883$	− 17519.7i	− 0.667708i	0.942625	−	0.333854i	$-0.108349\pi$
$884$	0	0
$885$	35665.0i	1.35465i
$886$	0	0
$887$	−7452.05	−0.282092	−0.141046	−	0.990003i	$-0.545046\pi$
$887$	−7452.05	−0.282092	−0.141046	+	0.990003i	$0.545046\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	23167.9i	0.871106i
$892$	0	0
$893$	−70008.1	−2.62344
$894$	0	0
$895$	14645.6	0.546983
$896$	0	0
$897$	38546.1	1.43480
$898$	0	0
$899$	3593.29	0.133307
$900$	0	0
$901$	− 6894.30i	− 0.254919i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9664.11	−0.354968
$906$	0	0
$907$	37014.8i	1.35508i	0.735486	+	0.677540i	$0.236954\pi$
$907$	37014.8i	1.35508i	−0.735486	+	0.677540i	$0.763046\pi$
$908$	0	0
$909$	− 2947.46i	− 0.107548i
$910$	0	0
$911$	− 12342.5i	− 0.448876i	−0.974488	−	0.224438i	$-0.927945\pi$
$911$	− 12342.5i	− 0.448876i	0.974488	−	0.224438i	$-0.0720546\pi$
$912$	0	0
$913$	− 8224.84i	− 0.298141i
$914$	0	0
$915$	−85459.3	−3.08765
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16482.6i	0.591633i	0.955245	+	0.295817i	$0.0955918\pi$
$919$	16482.6i	0.591633i	−0.955245	+	0.295817i	$0.904408\pi$
$920$	0	0
$921$	44942.2	1.60792
$922$	0	0
$923$	−23572.6	−0.840631
$924$	0	0
$925$	28625.5	1.01752
$926$	0	0
$927$	2191.03	0.0776299
$928$	0	0
$929$	− 21750.4i	− 0.768145i	−0.923303	−	0.384073i	$-0.874521\pi$
$929$	− 21750.4i	− 0.768145i	0.923303	−	0.384073i	$-0.125479\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−34321.2	−1.20431
$934$	0	0
$935$	28892.8i	1.01058i
$936$	0	0
$937$	20698.0i	0.721638i	0.932636	+	0.360819i	$0.117503\pi$
$937$	20698.0i	0.721638i	−0.932636	+	0.360819i	$0.882497\pi$
$938$	0	0
$939$	− 3691.13i	− 0.128281i
$940$	0	0
$941$	40305.4i	1.39630i	0.715951	+	0.698151i	$0.245994\pi$
$941$	40305.4i	1.39630i	−0.715951	+	0.698151i	$0.754006\pi$
$942$	0	0
$943$	−96771.7	−3.34180
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7361.90i	0.252618i	0.991991	+	0.126309i	$0.0403131\pi$
$947$	7361.90i	0.252618i	−0.991991	+	0.126309i	$0.959687\pi$
$948$	0	0
$949$	−18720.4	−0.640348
$950$	0	0
$951$	30725.1	1.04767
$952$	0	0
$953$	−38052.9	−1.29345	−0.646723	−	0.762725i	$-0.723861\pi$
$953$	−38052.9	−1.29345	−0.646723	+	0.762725i	$0.723861\pi$
$954$	0	0
$955$	−51647.0	−1.75001
$956$	0	0
$957$	− 4226.67i	− 0.142768i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−12210.6	−0.409874
$962$	0	0
$963$	− 2319.06i	− 0.0776019i
$964$	0	0
$965$	66405.9i	2.21522i
$966$	0	0
$967$	6095.61i	0.202711i	0.994850	+	0.101356i	$0.0323180\pi$
$967$	6095.61i	0.202711i	−0.994850	+	0.101356i	$0.967682\pi$
$968$	0	0
$969$	36150.0i	1.19846i
$970$	0	0
$971$	22135.7	0.731585	0.365792	−	0.930697i	$-0.380798\pi$
$971$	22135.7	0.731585	0.365792	+	0.930697i	$0.380798\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	55216.2i	1.81368i
$976$	0	0
$977$	−49659.8	−1.62616	−0.813080	−	0.582153i	$-0.802211\pi$
$977$	−49659.8	−1.62616	−0.813080	+	0.582153i	$0.802211\pi$
$978$	0	0
$979$	−31085.5	−1.01481
$980$	0	0
$981$	7616.74	0.247894
$982$	0	0
$983$	50074.5	1.62475	0.812375	−	0.583135i	$-0.198174\pi$
$983$	50074.5	1.62475	0.812375	+	0.583135i	$0.198174\pi$
$984$	0	0
$985$	105704.i	3.41930i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	58713.8	1.88776
$990$	0	0
$991$	6833.25i	0.219037i	0.993985	+	0.109518i	$0.0349308\pi$
$991$	6833.25i	0.219037i	−0.993985	+	0.109518i	$0.965069\pi$
$992$	0	0
$993$	− 26200.1i	− 0.837297i
$994$	0	0
$995$	− 41636.7i	− 1.32661i
$996$	0	0
$997$	− 29362.7i	− 0.932724i	−0.884594	−	0.466362i	$-0.845564\pi$
$997$	− 29362.7i	− 0.932724i	0.884594	−	0.466362i	$-0.154436\pi$
$998$	0	0
$999$	−12074.9	−0.382416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.f.j.783.5	✓	24
4.3	odd	2	inner	784.4.f.j.783.19	yes	24
7.6	odd	2	inner	784.4.f.j.783.20	yes	24
28.27	even	2	inner	784.4.f.j.783.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.4.f.j.783.5	✓	24	1.1	even	1	trivial
784.4.f.j.783.6	yes	24	28.27	even	2	inner
784.4.f.j.783.19	yes	24	4.3	odd	2	inner
784.4.f.j.783.20	yes	24	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.68961 q^{3} +20.4139i q^{5} +5.37166 q^{9} +O(q^{10})\)	\(q-5.68961 q^{3} +20.4139i q^{5} +5.37166 q^{9} -27.4118i q^{11} +33.2667i q^{13} -116.147i q^{15} +51.6329i q^{17} -123.055 q^{19} +203.652i q^{23} -291.726 q^{25} +123.057 q^{27} -27.1005 q^{29} -132.591 q^{31} +155.963i q^{33} -98.1248 q^{37} -189.274i q^{39} +475.182i q^{41} -288.305i q^{43} +109.656i q^{45} +568.917 q^{47} -293.771i q^{51} -133.525 q^{53} +559.581 q^{55} +700.134 q^{57} -307.068 q^{59} -735.787i q^{61} -679.101 q^{65} +130.251i q^{67} -1158.70i q^{69} +708.596i q^{71} +562.738i q^{73} +1659.81 q^{75} -787.673i q^{79} -845.180 q^{81} +300.047 q^{83} -1054.03 q^{85} +154.191 q^{87} -1134.02i q^{89} +754.392 q^{93} -2512.03i q^{95} +234.071i q^{97} -147.247i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 88 q^{9}+O(q^{10}) \) 24 * q + 88 * q^9	\( 24 q + 88 q^{9} - 856 q^{25} + 896 q^{29} - 2496 q^{53} + 4416 q^{57} - 4416 q^{65} - 4904 q^{81} - 2432 q^{85} - 11968 q^{93}+O(q^{100}) \) 24 * q + 88 * q^9 - 856 * q^25 + 896 * q^29 - 2496 * q^53 + 4416 * q^57 - 4416 * q^65 - 4904 * q^81 - 2432 * q^85 - 11968 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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