LMFDB - Embedded newform 1600.3.b.j.1151.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(1151,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1600.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.b (of order $2$, degree $1$, not 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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 400)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1600.1151
Dual form			1600.3.b.j.1151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +3.46410i q^{7} +6.00000 q^{9} -12.1244i q^{11} +16.0000 q^{13} +3.00000 q^{17} +22.5167i q^{19} +6.00000 q^{21} +45.0333i q^{23} -25.9808i q^{27} -12.0000 q^{29} +31.1769i q^{31} -21.0000 q^{33} +50.0000 q^{37} -27.7128i q^{39} -63.0000 q^{41} -62.3538i q^{43} -13.8564i q^{47} +37.0000 q^{49} -5.19615i q^{51} +18.0000 q^{53} +39.0000 q^{57} +55.4256i q^{59} -26.0000 q^{61} +20.7846i q^{63} -32.9090i q^{67} +78.0000 q^{69} +27.7128i q^{71} +17.0000 q^{73} +42.0000 q^{77} +86.6025i q^{79} +9.00000 q^{81} +98.7269i q^{83} +20.7846i q^{87} +99.0000 q^{89} +55.4256i q^{91} +54.0000 q^{93} +134.000 q^{97} -72.7461i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{9} + 32 q^{13} + 6 q^{17} + 12 q^{21} - 24 q^{29} - 42 q^{33} + 100 q^{37} - 126 q^{41} + 74 q^{49} + 36 q^{53} + 78 q^{57} - 52 q^{61} + 156 q^{69} + 34 q^{73} + 84 q^{77} + 18 q^{81} + 198 q^{89}+ \cdots + 268 q^{97}+O(q^{100}) $ 2 * q + 12 * q^9 + 32 * q^13 + 6 * q^17 + 12 * q^21 - 24 * q^29 - 42 * q^33 + 100 * q^37 - 126 * q^41 + 74 * q^49 + 36 * q^53 + 78 * q^57 - 52 * q^61 + 156 * q^69 + 34 * q^73 + 84 * q^77 + 18 * q^81 + 198 * q^89 + 108 * q^93 + 268 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.73205i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.73205i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410i	0.494872i	0.968904	+	0.247436i	$0.0795879\pi$
$7$	3.46410i	0.494872i	−0.968904	+	0.247436i	$0.920412\pi$
$8$	0	0
$9$	6.00000	0.666667
$10$	0	0
$11$	− 12.1244i	− 1.10221i	−0.834435	−	0.551107i	$-0.814206\pi$
$11$	− 12.1244i	− 1.10221i	0.834435	−	0.551107i	$-0.185794\pi$
$12$	0	0
$13$	16.0000	1.23077	0.615385	−	0.788227i	$-0.289001\pi$
$13$	16.0000	1.23077	0.615385	+	0.788227i	$0.289001\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.176471	0.0882353	−	0.996100i	$-0.471877\pi$
$17$	3.00000	0.176471	0.0882353	+	0.996100i	$0.471877\pi$
$18$	0	0
$19$	22.5167i	1.18509i	0.805538	+	0.592544i	$0.201876\pi$
$19$	22.5167i	1.18509i	−0.805538	+	0.592544i	$0.798124\pi$
$20$	0	0
$21$	6.00000	0.285714
$22$	0	0
$23$	45.0333i	1.95797i	0.203931	+	0.978985i	$0.434628\pi$
$23$	45.0333i	1.95797i	−0.203931	+	0.978985i	$0.565372\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 25.9808i	− 0.962250i
$28$	0	0
$29$	−12.0000	−0.413793	−0.206897	−	0.978363i	$-0.566336\pi$
$29$	−12.0000	−0.413793	−0.206897	+	0.978363i	$0.566336\pi$
$30$	0	0
$31$	31.1769i	1.00571i	0.864372	+	0.502853i	$0.167716\pi$
$31$	31.1769i	1.00571i	−0.864372	+	0.502853i	$0.832284\pi$
$32$	0	0
$33$	−21.0000	−0.636364
$34$	0	0
$35$	0	0
$36$	0	0
$37$	50.0000	1.35135	0.675676	−	0.737199i	$-0.263852\pi$
$37$	50.0000	1.35135	0.675676	+	0.737199i	$0.263852\pi$
$38$	0	0
$39$	− 27.7128i	− 0.710585i
$40$	0	0
$41$	−63.0000	−1.53659	−0.768293	−	0.640099i	$-0.778893\pi$
$41$	−63.0000	−1.53659	−0.768293	+	0.640099i	$0.778893\pi$
$42$	0	0
$43$	− 62.3538i	− 1.45009i	−0.688702	−	0.725045i	$-0.741819\pi$
$43$	− 62.3538i	− 1.45009i	0.688702	−	0.725045i	$-0.258181\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 13.8564i	− 0.294817i	−0.989076	−	0.147409i	$-0.952907\pi$
$47$	− 13.8564i	− 0.294817i	0.989076	−	0.147409i	$-0.0470932\pi$
$48$	0	0
$49$	37.0000	0.755102
$50$	0	0
$51$	− 5.19615i	− 0.101885i
$52$	0	0
$53$	18.0000	0.339623	0.169811	−	0.985477i	$-0.445684\pi$
$53$	18.0000	0.339623	0.169811	+	0.985477i	$0.445684\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	39.0000	0.684211
$58$	0	0
$59$	55.4256i	0.939417i	0.882821	+	0.469709i	$0.155641\pi$
$59$	55.4256i	0.939417i	−0.882821	+	0.469709i	$0.844359\pi$
$60$	0	0
$61$	−26.0000	−0.426230	−0.213115	−	0.977027i	$-0.568361\pi$
$61$	−26.0000	−0.426230	−0.213115	+	0.977027i	$0.568361\pi$
$62$	0	0
$63$	20.7846i	0.329914i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 32.9090i	− 0.491179i	−0.969374	−	0.245589i	$-0.921019\pi$
$67$	− 32.9090i	− 0.491179i	0.969374	−	0.245589i	$-0.0789815\pi$
$68$	0	0
$69$	78.0000	1.13043
$70$	0	0
$71$	27.7128i	0.390321i	0.980771	+	0.195161i	$0.0625228\pi$
$71$	27.7128i	0.390321i	−0.980771	+	0.195161i	$0.937477\pi$
$72$	0	0
$73$	17.0000	0.232877	0.116438	−	0.993198i	$-0.462852\pi$
$73$	17.0000	0.232877	0.116438	+	0.993198i	$0.462852\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	42.0000	0.545455
$78$	0	0
$79$	86.6025i	1.09623i	0.836401	+	0.548117i	$0.184655\pi$
$79$	86.6025i	1.09623i	−0.836401	+	0.548117i	$0.815345\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	98.7269i	1.18948i	0.803918	+	0.594740i	$0.202745\pi$
$83$	98.7269i	1.18948i	−0.803918	+	0.594740i	$0.797255\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	20.7846i	0.238904i
$88$	0	0
$89$	99.0000	1.11236	0.556180	−	0.831062i	$-0.312267\pi$
$89$	99.0000	1.11236	0.556180	+	0.831062i	$0.312267\pi$
$90$	0	0
$91$	55.4256i	0.609073i
$92$	0	0
$93$	54.0000	0.580645
$94$	0	0
$95$	0	0
$96$	0	0
$97$	134.000	1.38144	0.690722	−	0.723121i	$-0.257293\pi$
$97$	134.000	1.38144	0.690722	+	0.723121i	$0.257293\pi$
$98$	0	0
$99$	− 72.7461i	− 0.734809i
$100$	0	0
$101$	−150.000	−1.48515	−0.742574	−	0.669764i	$-0.766395\pi$
$101$	−150.000	−1.48515	−0.742574	+	0.669764i	$0.766395\pi$
$102$	0	0
$103$	− 110.851i	− 1.07623i	−0.842873	−	0.538113i	$-0.819137\pi$
$103$	− 110.851i	− 1.07623i	0.842873	−	0.538113i	$-0.180863\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	164.545i	1.53780i	0.639368	+	0.768901i	$0.279196\pi$
$107$	164.545i	1.53780i	−0.639368	+	0.768901i	$0.720804\pi$
$108$	0	0
$109$	74.0000	0.678899	0.339450	−	0.940624i	$-0.389759\pi$
$109$	74.0000	0.678899	0.339450	+	0.940624i	$0.389759\pi$
$110$	0	0
$111$	− 86.6025i	− 0.780203i
$112$	0	0
$113$	201.000	1.77876	0.889381	−	0.457168i	$-0.151136\pi$
$113$	201.000	1.77876	0.889381	+	0.457168i	$0.151136\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	96.0000	0.820513
$118$	0	0
$119$	10.3923i	0.0873303i
$120$	0	0
$121$	−26.0000	−0.214876
$122$	0	0
$123$	109.119i	0.887148i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.3923i	0.0818292i	0.999163	+	0.0409146i	$0.0130272\pi$
$127$	10.3923i	0.0818292i	−0.999163	+	0.0409146i	$0.986973\pi$
$128$	0	0
$129$	−108.000	−0.837209
$130$	0	0
$131$	− 6.92820i	− 0.0528870i	−0.999650	−	0.0264435i	$-0.991582\pi$
$131$	− 6.92820i	− 0.0528870i	0.999650	−	0.0264435i	$-0.00841822\pi$
$132$	0	0
$133$	−78.0000	−0.586466
$134$	0	0
$135$	0	0
$136$	0	0
$137$	207.000	1.51095	0.755474	−	0.655178i	$-0.227406\pi$
$137$	207.000	1.51095	0.755474	+	0.655178i	$0.227406\pi$
$138$	0	0
$139$	− 178.401i	− 1.28346i	−0.766930	−	0.641731i	$-0.778217\pi$
$139$	− 178.401i	− 1.28346i	0.766930	−	0.641731i	$-0.221783\pi$
$140$	0	0
$141$	−24.0000	−0.170213
$142$	0	0
$143$	− 193.990i	− 1.35657i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 64.0859i	− 0.435958i
$148$	0	0
$149$	108.000	0.724832	0.362416	−	0.932016i	$-0.381952\pi$
$149$	108.000	0.724832	0.362416	+	0.932016i	$0.381952\pi$
$150$	0	0
$151$	− 266.736i	− 1.76646i	−0.468938	−	0.883231i	$-0.655363\pi$
$151$	− 266.736i	− 1.76646i	0.468938	−	0.883231i	$-0.344637\pi$
$152$	0	0
$153$	18.0000	0.117647
$154$	0	0
$155$	0	0
$156$	0	0
$157$	278.000	1.77070	0.885350	−	0.464925i	$-0.153919\pi$
$157$	278.000	1.77070	0.885350	+	0.464925i	$0.153919\pi$
$158$	0	0
$159$	− 31.1769i	− 0.196081i
$160$	0	0
$161$	−156.000	−0.968944
$162$	0	0
$163$	− 19.0526i	− 0.116887i	−0.998291	−	0.0584434i	$-0.981386\pi$
$163$	− 19.0526i	− 0.116887i	0.998291	−	0.0584434i	$-0.0186137\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 263.272i	− 1.57648i	−0.615370	−	0.788239i	$-0.710993\pi$
$167$	− 263.272i	− 1.57648i	0.615370	−	0.788239i	$-0.289007\pi$
$168$	0	0
$169$	87.0000	0.514793
$170$	0	0
$171$	135.100i	0.790058i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	96.0000	0.542373
$178$	0	0
$179$	− 12.1244i	− 0.0677338i	−0.999426	−	0.0338669i	$-0.989218\pi$
$179$	− 12.1244i	− 0.0677338i	0.999426	−	0.0338669i	$-0.0107822\pi$
$180$	0	0
$181$	−166.000	−0.917127	−0.458564	−	0.888662i	$-0.651636\pi$
$181$	−166.000	−0.917127	−0.458564	+	0.888662i	$0.651636\pi$
$182$	0	0
$183$	45.0333i	0.246084i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 36.3731i	− 0.194508i
$188$	0	0
$189$	90.0000	0.476190
$190$	0	0
$191$	301.377i	1.57789i	0.614464	+	0.788945i	$0.289372\pi$
$191$	301.377i	1.57789i	−0.614464	+	0.788945i	$0.710628\pi$
$192$	0	0
$193$	139.000	0.720207	0.360104	−	0.932912i	$-0.382741\pi$
$193$	139.000	0.720207	0.360104	+	0.932912i	$0.382741\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−126.000	−0.639594	−0.319797	−	0.947486i	$-0.603615\pi$
$197$	−126.000	−0.639594	−0.319797	+	0.947486i	$0.603615\pi$
$198$	0	0
$199$	− 76.2102i	− 0.382966i	−0.981496	−	0.191483i	$-0.938670\pi$
$199$	− 76.2102i	− 0.382966i	0.981496	−	0.191483i	$-0.0613297\pi$
$200$	0	0
$201$	−57.0000	−0.283582
$202$	0	0
$203$	− 41.5692i	− 0.204774i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	270.200i	1.30531i
$208$	0	0
$209$	273.000	1.30622
$210$	0	0
$211$	− 188.794i	− 0.894756i	−0.894345	−	0.447378i	$-0.852358\pi$
$211$	− 188.794i	− 0.894756i	0.894345	−	0.447378i	$-0.147642\pi$
$212$	0	0
$213$	48.0000	0.225352
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−108.000	−0.497696
$218$	0	0
$219$	− 29.4449i	− 0.134451i
$220$	0	0
$221$	48.0000	0.217195
$222$	0	0
$223$	145.492i	0.652432i	0.945295	+	0.326216i	$0.105774\pi$
$223$	145.492i	0.652432i	−0.945295	+	0.326216i	$0.894226\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	173.205i	0.763018i	0.924365	+	0.381509i	$0.124595\pi$
$227$	173.205i	0.763018i	−0.924365	+	0.381509i	$0.875405\pi$
$228$	0	0
$229$	140.000	0.611354	0.305677	−	0.952135i	$-0.401117\pi$
$229$	140.000	0.611354	0.305677	+	0.952135i	$0.401117\pi$
$230$	0	0
$231$	− 72.7461i	− 0.314918i
$232$	0	0
$233$	−366.000	−1.57082	−0.785408	−	0.618979i	$-0.787547\pi$
$233$	−366.000	−1.57082	−0.785408	+	0.618979i	$0.787547\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	150.000	0.632911
$238$	0	0
$239$	− 256.344i	− 1.07257i	−0.844038	−	0.536284i	$-0.819828\pi$
$239$	− 256.344i	− 1.07257i	0.844038	−	0.536284i	$-0.180172\pi$
$240$	0	0
$241$	73.0000	0.302905	0.151452	−	0.988465i	$-0.451605\pi$
$241$	73.0000	0.302905	0.151452	+	0.988465i	$0.451605\pi$
$242$	0	0
$243$	− 249.415i	− 1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	360.267i	1.45857i
$248$	0	0
$249$	171.000	0.686747
$250$	0	0
$251$	− 168.009i	− 0.669358i	−0.942332	−	0.334679i	$-0.891372\pi$
$251$	− 168.009i	− 0.669358i	0.942332	−	0.334679i	$-0.108628\pi$
$252$	0	0
$253$	546.000	2.15810
$254$	0	0
$255$	0	0
$256$	0	0
$257$	282.000	1.09728	0.548638	−	0.836060i	$-0.315146\pi$
$257$	282.000	1.09728	0.548638	+	0.836060i	$0.315146\pi$
$258$	0	0
$259$	173.205i	0.668745i
$260$	0	0
$261$	−72.0000	−0.275862
$262$	0	0
$263$	169.741i	0.645403i	0.946501	+	0.322701i	$0.104591\pi$
$263$	169.741i	0.645403i	−0.946501	+	0.322701i	$0.895409\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 171.473i	− 0.642221i
$268$	0	0
$269$	−288.000	−1.07063	−0.535316	−	0.844652i	$-0.679807\pi$
$269$	−288.000	−1.07063	−0.535316	+	0.844652i	$0.679807\pi$
$270$	0	0
$271$	169.741i	0.626350i	0.949695	+	0.313175i	$0.101393\pi$
$271$	169.741i	0.626350i	−0.949695	+	0.313175i	$0.898607\pi$
$272$	0	0
$273$	96.0000	0.351648
$274$	0	0
$275$	0	0
$276$	0	0
$277$	76.0000	0.274368	0.137184	−	0.990546i	$-0.456195\pi$
$277$	76.0000	0.274368	0.137184	+	0.990546i	$0.456195\pi$
$278$	0	0
$279$	187.061i	0.670471i
$280$	0	0
$281$	−354.000	−1.25979	−0.629893	−	0.776682i	$-0.716901\pi$
$281$	−354.000	−1.25979	−0.629893	+	0.776682i	$0.716901\pi$
$282$	0	0
$283$	− 316.965i	− 1.12002i	−0.828486	−	0.560009i	$-0.810798\pi$
$283$	− 316.965i	− 1.12002i	0.828486	−	0.560009i	$-0.189202\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 218.238i	− 0.760413i
$288$	0	0
$289$	−280.000	−0.968858
$290$	0	0
$291$	− 232.095i	− 0.797577i
$292$	0	0
$293$	−426.000	−1.45392	−0.726962	−	0.686677i	$-0.759069\pi$
$293$	−426.000	−1.45392	−0.726962	+	0.686677i	$0.759069\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−315.000	−1.06061
$298$	0	0
$299$	720.533i	2.40981i
$300$	0	0
$301$	216.000	0.717608
$302$	0	0
$303$	259.808i	0.857451i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 223.435i	− 0.727800i	−0.931438	−	0.363900i	$-0.881445\pi$
$307$	− 223.435i	− 0.727800i	0.931438	−	0.363900i	$-0.118555\pi$
$308$	0	0
$309$	−192.000	−0.621359
$310$	0	0
$311$	433.013i	1.39232i	0.717885	+	0.696162i	$0.245110\pi$
$311$	433.013i	1.39232i	−0.717885	+	0.696162i	$0.754890\pi$
$312$	0	0
$313$	−218.000	−0.696486	−0.348243	−	0.937404i	$-0.613222\pi$
$313$	−218.000	−0.696486	−0.348243	+	0.937404i	$0.613222\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.0000	0.0378549	0.0189274	−	0.999821i	$-0.493975\pi$
$317$	12.0000	0.0378549	0.0189274	+	0.999821i	$0.493975\pi$
$318$	0	0
$319$	145.492i	0.456089i
$320$	0	0
$321$	285.000	0.887850
$322$	0	0
$323$	67.5500i	0.209133i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 128.172i	− 0.391963i
$328$	0	0
$329$	48.0000	0.145897
$330$	0	0
$331$	465.922i	1.40762i	0.710389	+	0.703809i	$0.248519\pi$
$331$	465.922i	1.40762i	−0.710389	+	0.703809i	$0.751481\pi$
$332$	0	0
$333$	300.000	0.900901
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−67.0000	−0.198813	−0.0994065	−	0.995047i	$-0.531694\pi$
$337$	−67.0000	−0.198813	−0.0994065	+	0.995047i	$0.531694\pi$
$338$	0	0
$339$	− 348.142i	− 1.02697i
$340$	0	0
$341$	378.000	1.10850
$342$	0	0
$343$	297.913i	0.868550i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	192.258i	0.554057i	0.960862	+	0.277028i	$0.0893496\pi$
$347$	192.258i	0.554057i	−0.960862	+	0.277028i	$0.910650\pi$
$348$	0	0
$349$	−254.000	−0.727794	−0.363897	−	0.931439i	$-0.618554\pi$
$349$	−254.000	−0.727794	−0.363897	+	0.931439i	$0.618554\pi$
$350$	0	0
$351$	− 415.692i	− 1.18431i
$352$	0	0
$353$	114.000	0.322946	0.161473	−	0.986877i	$-0.448376\pi$
$353$	114.000	0.322946	0.161473	+	0.986877i	$0.448376\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	18.0000	0.0504202
$358$	0	0
$359$	− 370.659i	− 1.03248i	−0.856445	−	0.516238i	$-0.827332\pi$
$359$	− 370.659i	− 1.03248i	0.856445	−	0.516238i	$-0.172668\pi$
$360$	0	0
$361$	−146.000	−0.404432
$362$	0	0
$363$	45.0333i	0.124059i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	117.779i	0.320925i	0.987042	+	0.160462i	$0.0512986\pi$
$367$	117.779i	0.320925i	−0.987042	+	0.160462i	$0.948701\pi$
$368$	0	0
$369$	−378.000	−1.02439
$370$	0	0
$371$	62.3538i	0.168070i
$372$	0	0
$373$	−274.000	−0.734584	−0.367292	−	0.930106i	$-0.619715\pi$
$373$	−274.000	−0.734584	−0.367292	+	0.930106i	$0.619715\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−192.000	−0.509284
$378$	0	0
$379$	− 299.645i	− 0.790619i	−0.918548	−	0.395310i	$-0.870637\pi$
$379$	− 299.645i	− 0.790619i	0.918548	−	0.395310i	$-0.129363\pi$
$380$	0	0
$381$	18.0000	0.0472441
$382$	0	0
$383$	− 460.726i	− 1.20294i	−0.798896	−	0.601469i	$-0.794582\pi$
$383$	− 460.726i	− 1.20294i	0.798896	−	0.601469i	$-0.205418\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 374.123i	− 0.966726i
$388$	0	0
$389$	510.000	1.31105	0.655527	−	0.755172i	$-0.272447\pi$
$389$	510.000	1.31105	0.655527	+	0.755172i	$0.272447\pi$
$390$	0	0
$391$	135.100i	0.345524i
$392$	0	0
$393$	−12.0000	−0.0305344
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−532.000	−1.34005	−0.670025	−	0.742338i	$-0.733717\pi$
$397$	−532.000	−1.34005	−0.670025	+	0.742338i	$0.733717\pi$
$398$	0	0
$399$	135.100i	0.338596i
$400$	0	0
$401$	477.000	1.18953	0.594763	−	0.803901i	$-0.297246\pi$
$401$	477.000	1.18953	0.594763	+	0.803901i	$0.297246\pi$
$402$	0	0
$403$	498.831i	1.23779i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 606.218i	− 1.48948i
$408$	0	0
$409$	25.0000	0.0611247	0.0305623	−	0.999533i	$-0.490270\pi$
$409$	25.0000	0.0611247	0.0305623	+	0.999533i	$0.490270\pi$
$410$	0	0
$411$	− 358.535i	− 0.872347i
$412$	0	0
$413$	−192.000	−0.464891
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−309.000	−0.741007
$418$	0	0
$419$	57.1577i	0.136415i	0.997671	+	0.0682073i	$0.0217279\pi$
$419$	57.1577i	0.136415i	−0.997671	+	0.0682073i	$0.978272\pi$
$420$	0	0
$421$	−388.000	−0.921615	−0.460808	−	0.887500i	$-0.652440\pi$
$421$	−388.000	−0.921615	−0.460808	+	0.887500i	$0.652440\pi$
$422$	0	0
$423$	− 83.1384i	− 0.196545i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 90.0666i	− 0.210929i
$428$	0	0
$429$	−336.000	−0.783217
$430$	0	0
$431$	− 245.951i	− 0.570652i	−0.958430	−	0.285326i	$-0.907898\pi$
$431$	− 245.951i	− 0.570652i	0.958430	−	0.285326i	$-0.0921019\pi$
$432$	0	0
$433$	307.000	0.709007	0.354503	−	0.935055i	$-0.384650\pi$
$433$	307.000	0.709007	0.354503	+	0.935055i	$0.384650\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1014.00	−2.32037
$438$	0	0
$439$	561.184i	1.27832i	0.769072	+	0.639162i	$0.220719\pi$
$439$	561.184i	1.27832i	−0.769072	+	0.639162i	$0.779281\pi$
$440$	0	0
$441$	222.000	0.503401
$442$	0	0
$443$	479.778i	1.08302i	0.840694	+	0.541510i	$0.182147\pi$
$443$	479.778i	1.08302i	−0.840694	+	0.541510i	$0.817853\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 187.061i	− 0.418482i
$448$	0	0
$449$	−39.0000	−0.0868597	−0.0434298	−	0.999056i	$-0.513828\pi$
$449$	−39.0000	−0.0868597	−0.0434298	+	0.999056i	$0.513828\pi$
$450$	0	0
$451$	763.834i	1.69365i
$452$	0	0
$453$	−462.000	−1.01987
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−349.000	−0.763676	−0.381838	−	0.924229i	$-0.624709\pi$
$457$	−349.000	−0.763676	−0.381838	+	0.924229i	$0.624709\pi$
$458$	0	0
$459$	− 77.9423i	− 0.169809i
$460$	0	0
$461$	372.000	0.806941	0.403471	−	0.914993i	$-0.367804\pi$
$461$	372.000	0.806941	0.403471	+	0.914993i	$0.367804\pi$
$462$	0	0
$463$	− 422.620i	− 0.912787i	−0.889778	−	0.456394i	$-0.849141\pi$
$463$	− 422.620i	− 0.912787i	0.889778	−	0.456394i	$-0.150859\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	713.605i	1.52806i	0.645180	+	0.764031i	$0.276783\pi$
$467$	713.605i	1.52806i	−0.645180	+	0.764031i	$0.723217\pi$
$468$	0	0
$469$	114.000	0.243070
$470$	0	0
$471$	− 481.510i	− 1.02231i
$472$	0	0
$473$	−756.000	−1.59831
$474$	0	0
$475$	0	0
$476$	0	0
$477$	108.000	0.226415
$478$	0	0
$479$	758.638i	1.58380i	0.610653	+	0.791898i	$0.290907\pi$
$479$	758.638i	1.58380i	−0.610653	+	0.791898i	$0.709093\pi$
$480$	0	0
$481$	800.000	1.66320
$482$	0	0
$483$	270.200i	0.559420i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	523.079i	1.07408i	0.843555	+	0.537042i	$0.180459\pi$
$487$	523.079i	1.07408i	−0.843555	+	0.537042i	$0.819541\pi$
$488$	0	0
$489$	−33.0000	−0.0674847
$490$	0	0
$491$	− 478.046i	− 0.973617i	−0.873509	−	0.486809i	$-0.838161\pi$
$491$	− 478.046i	− 0.973617i	0.873509	−	0.486809i	$-0.161839\pi$
$492$	0	0
$493$	−36.0000	−0.0730223
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−96.0000	−0.193159
$498$	0	0
$499$	62.3538i	0.124958i	0.998046	+	0.0624788i	$0.0199006\pi$
$499$	62.3538i	0.124958i	−0.998046	+	0.0624788i	$0.980099\pi$
$500$	0	0
$501$	−456.000	−0.910180
$502$	0	0
$503$	− 561.184i	− 1.11567i	−0.829950	−	0.557837i	$-0.811631\pi$
$503$	− 561.184i	− 1.11567i	0.829950	−	0.557837i	$-0.188369\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 150.688i	− 0.297216i
$508$	0	0
$509$	−390.000	−0.766208	−0.383104	−	0.923705i	$-0.625145\pi$
$509$	−390.000	−0.766208	−0.383104	+	0.923705i	$0.625145\pi$
$510$	0	0
$511$	58.8897i	0.115244i
$512$	0	0
$513$	585.000	1.14035
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−168.000	−0.324952
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−183.000	−0.351248	−0.175624	−	0.984457i	$-0.556194\pi$
$521$	−183.000	−0.351248	−0.175624	+	0.984457i	$0.556194\pi$
$522$	0	0
$523$	185.329i	0.354358i	0.984179	+	0.177179i	$0.0566972\pi$
$523$	185.329i	0.354358i	−0.984179	+	0.177179i	$0.943303\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	93.5307i	0.177478i
$528$	0	0
$529$	−1499.00	−2.83365
$530$	0	0
$531$	332.554i	0.626278i
$532$	0	0
$533$	−1008.00	−1.89118
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−21.0000	−0.0391061
$538$	0	0
$539$	− 448.601i	− 0.832284i
$540$	0	0
$541$	968.000	1.78928	0.894640	−	0.446789i	$-0.147432\pi$
$541$	968.000	1.78928	0.894640	+	0.446789i	$0.147432\pi$
$542$	0	0
$543$	287.520i	0.529504i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	510.955i	0.934104i	0.884230	+	0.467052i	$0.154684\pi$
$547$	510.955i	0.934104i	−0.884230	+	0.467052i	$0.845316\pi$
$548$	0	0
$549$	−156.000	−0.284153
$550$	0	0
$551$	− 270.200i	− 0.490381i
$552$	0	0
$553$	−300.000	−0.542495
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−552.000	−0.991023	−0.495512	−	0.868601i	$-0.665019\pi$
$557$	−552.000	−0.991023	−0.495512	+	0.868601i	$0.665019\pi$
$558$	0	0
$559$	− 997.661i	− 1.78472i
$560$	0	0
$561$	−63.0000	−0.112299
$562$	0	0
$563$	− 401.836i	− 0.713740i	−0.934154	−	0.356870i	$-0.883844\pi$
$563$	− 401.836i	− 0.713740i	0.934154	−	0.356870i	$-0.116156\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	31.1769i	0.0549857i
$568$	0	0
$569$	117.000	0.205624	0.102812	−	0.994701i	$-0.467216\pi$
$569$	117.000	0.205624	0.102812	+	0.994701i	$0.467216\pi$
$570$	0	0
$571$	297.913i	0.521739i	0.965374	+	0.260869i	$0.0840092\pi$
$571$	297.913i	0.521739i	−0.965374	+	0.260869i	$0.915991\pi$
$572$	0	0
$573$	522.000	0.910995
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−491.000	−0.850953	−0.425477	−	0.904969i	$-0.639894\pi$
$577$	−491.000	−0.850953	−0.425477	+	0.904969i	$0.639894\pi$
$578$	0	0
$579$	− 240.755i	− 0.415812i
$580$	0	0
$581$	−342.000	−0.588640
$582$	0	0
$583$	− 218.238i	− 0.374337i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	244.219i	0.416046i	0.978124	+	0.208023i	$0.0667029\pi$
$587$	244.219i	0.416046i	−0.978124	+	0.208023i	$0.933297\pi$
$588$	0	0
$589$	−702.000	−1.19185
$590$	0	0
$591$	218.238i	0.369270i
$592$	0	0
$593$	57.0000	0.0961214	0.0480607	−	0.998844i	$-0.484696\pi$
$593$	57.0000	0.0961214	0.0480607	+	0.998844i	$0.484696\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−132.000	−0.221106
$598$	0	0
$599$	− 633.931i	− 1.05831i	−0.848524	−	0.529157i	$-0.822508\pi$
$599$	− 633.931i	− 1.05831i	0.848524	−	0.529157i	$-0.177492\pi$
$600$	0	0
$601$	263.000	0.437604	0.218802	−	0.975769i	$-0.429785\pi$
$601$	263.000	0.437604	0.218802	+	0.975769i	$0.429785\pi$
$602$	0	0
$603$	− 197.454i	− 0.327452i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 484.974i	− 0.798969i	−0.916740	−	0.399485i	$-0.869189\pi$
$607$	− 484.974i	− 0.798969i	0.916740	−	0.399485i	$-0.130811\pi$
$608$	0	0
$609$	−72.0000	−0.118227
$610$	0	0
$611$	− 221.703i	− 0.362852i
$612$	0	0
$613$	194.000	0.316476	0.158238	−	0.987401i	$-0.449419\pi$
$613$	194.000	0.316476	0.158238	+	0.987401i	$0.449419\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−726.000	−1.17666	−0.588331	−	0.808620i	$-0.700215\pi$
$617$	−726.000	−1.17666	−0.588331	+	0.808620i	$0.700215\pi$
$618$	0	0
$619$	− 408.764i	− 0.660362i	−0.943918	−	0.330181i	$-0.892890\pi$
$619$	− 408.764i	− 0.660362i	0.943918	−	0.330181i	$-0.107110\pi$
$620$	0	0
$621$	1170.00	1.88406
$622$	0	0
$623$	342.946i	0.550475i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 472.850i	− 0.754147i
$628$	0	0
$629$	150.000	0.238474
$630$	0	0
$631$	502.295i	0.796030i	0.917379	+	0.398015i	$0.130301\pi$
$631$	502.295i	0.796030i	−0.917379	+	0.398015i	$0.869699\pi$
$632$	0	0
$633$	−327.000	−0.516588
$634$	0	0
$635$	0	0
$636$	0	0
$637$	592.000	0.929356
$638$	0	0
$639$	166.277i	0.260214i
$640$	0	0
$641$	−174.000	−0.271451	−0.135725	−	0.990746i	$-0.543337\pi$
$641$	−174.000	−0.271451	−0.135725	+	0.990746i	$0.543337\pi$
$642$	0	0
$643$	− 976.877i	− 1.51925i	−0.650362	−	0.759624i	$-0.725383\pi$
$643$	− 976.877i	− 1.51925i	0.650362	−	0.759624i	$-0.274617\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	921.451i	1.42419i	0.702083	+	0.712095i	$0.252254\pi$
$647$	921.451i	1.42419i	−0.702083	+	0.712095i	$0.747746\pi$
$648$	0	0
$649$	672.000	1.03544
$650$	0	0
$651$	187.061i	0.287345i
$652$	0	0
$653$	222.000	0.339969	0.169985	−	0.985447i	$-0.445628\pi$
$653$	222.000	0.339969	0.169985	+	0.985447i	$0.445628\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	102.000	0.155251
$658$	0	0
$659$	857.365i	1.30101i	0.759502	+	0.650505i	$0.225443\pi$
$659$	857.365i	1.30101i	−0.759502	+	0.650505i	$0.774557\pi$
$660$	0	0
$661$	268.000	0.405446	0.202723	−	0.979236i	$-0.435021\pi$
$661$	268.000	0.405446	0.202723	+	0.979236i	$0.435021\pi$
$662$	0	0
$663$	− 83.1384i	− 0.125397i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 540.400i	− 0.810195i
$668$	0	0
$669$	252.000	0.376682
$670$	0	0
$671$	315.233i	0.469796i
$672$	0	0
$673$	550.000	0.817236	0.408618	−	0.912705i	$-0.366011\pi$
$673$	550.000	0.817236	0.408618	+	0.912705i	$0.366011\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−600.000	−0.886263	−0.443131	−	0.896457i	$-0.646133\pi$
$677$	−600.000	−0.886263	−0.443131	+	0.896457i	$0.646133\pi$
$678$	0	0
$679$	464.190i	0.683637i
$680$	0	0
$681$	300.000	0.440529
$682$	0	0
$683$	− 729.193i	− 1.06763i	−0.845600	−	0.533817i	$-0.820757\pi$
$683$	− 729.193i	− 1.06763i	0.845600	−	0.533817i	$-0.179243\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 242.487i	− 0.352965i
$688$	0	0
$689$	288.000	0.417997
$690$	0	0
$691$	− 1231.49i	− 1.78218i	−0.453824	−	0.891091i	$-0.649941\pi$
$691$	− 1231.49i	− 1.78218i	0.453824	−	0.891091i	$-0.350059\pi$
$692$	0	0
$693$	252.000	0.363636
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−189.000	−0.271162
$698$	0	0
$699$	633.931i	0.906911i
$700$	0	0
$701$	−1008.00	−1.43795	−0.718973	−	0.695038i	$-0.755387\pi$
$701$	−1008.00	−1.43795	−0.718973	+	0.695038i	$0.755387\pi$
$702$	0	0
$703$	1125.83i	1.60147i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 519.615i	− 0.734958i
$708$	0	0
$709$	−388.000	−0.547250	−0.273625	−	0.961837i	$-0.588223\pi$
$709$	−388.000	−0.547250	−0.273625	+	0.961837i	$0.588223\pi$
$710$	0	0
$711$	519.615i	0.730823i
$712$	0	0
$713$	−1404.00	−1.96914
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−444.000	−0.619247
$718$	0	0
$719$	− 973.413i	− 1.35384i	−0.736056	−	0.676921i	$-0.763314\pi$
$719$	− 973.413i	− 1.35384i	0.736056	−	0.676921i	$-0.236686\pi$
$720$	0	0
$721$	384.000	0.532594
$722$	0	0
$723$	− 126.440i	− 0.174882i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 187.061i	− 0.257306i	−0.991690	−	0.128653i	$-0.958935\pi$
$727$	− 187.061i	− 0.257306i	0.991690	−	0.128653i	$-0.0410653\pi$
$728$	0	0
$729$	−351.000	−0.481481
$730$	0	0
$731$	− 187.061i	− 0.255898i
$732$	0	0
$733$	976.000	1.33151	0.665757	−	0.746168i	$-0.268109\pi$
$733$	976.000	1.33151	0.665757	+	0.746168i	$0.268109\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−399.000	−0.541384
$738$	0	0
$739$	270.200i	0.365629i	0.983147	+	0.182815i	$0.0585208\pi$
$739$	270.200i	0.365629i	−0.983147	+	0.182815i	$0.941479\pi$
$740$	0	0
$741$	624.000	0.842105
$742$	0	0
$743$	− 38.1051i	− 0.0512855i	−0.999671	−	0.0256427i	$-0.991837\pi$
$743$	− 38.1051i	− 0.0512855i	0.999671	−	0.0256427i	$-0.00816323\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	592.361i	0.792987i
$748$	0	0
$749$	−570.000	−0.761015
$750$	0	0
$751$	− 921.451i	− 1.22697i	−0.789708	−	0.613483i	$-0.789768\pi$
$751$	− 921.451i	− 1.22697i	0.789708	−	0.613483i	$-0.210232\pi$
$752$	0	0
$753$	−291.000	−0.386454
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−392.000	−0.517834	−0.258917	−	0.965900i	$-0.583366\pi$
$757$	−392.000	−0.517834	−0.258917	+	0.965900i	$0.583366\pi$
$758$	0	0
$759$	− 945.700i	− 1.24598i
$760$	0	0
$761$	−1335.00	−1.75427	−0.877135	−	0.480243i	$-0.840548\pi$
$761$	−1335.00	−1.75427	−0.877135	+	0.480243i	$0.840548\pi$
$762$	0	0
$763$	256.344i	0.335968i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	886.810i	1.15621i
$768$	0	0
$769$	509.000	0.661899	0.330949	−	0.943649i	$-0.392631\pi$
$769$	509.000	0.661899	0.330949	+	0.943649i	$0.392631\pi$
$770$	0	0
$771$	− 488.438i	− 0.633513i
$772$	0	0
$773$	−684.000	−0.884864	−0.442432	−	0.896802i	$-0.645884\pi$
$773$	−684.000	−0.884864	−0.442432	+	0.896802i	$0.645884\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	300.000	0.386100
$778$	0	0
$779$	− 1418.55i	− 1.82099i
$780$	0	0
$781$	336.000	0.430218
$782$	0	0
$783$	311.769i	0.398173i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 6.92820i	− 0.00880331i	−0.999990	−	0.00440165i	$-0.998599\pi$
$787$	− 6.92820i	− 0.00880331i	0.999990	−	0.00440165i	$-0.00140109\pi$
$788$	0	0
$789$	294.000	0.372624
$790$	0	0
$791$	696.284i	0.880258i
$792$	0	0
$793$	−416.000	−0.524590
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1374.00	1.72396	0.861982	−	0.506938i	$-0.169223\pi$
$797$	1374.00	1.72396	0.861982	+	0.506938i	$0.169223\pi$
$798$	0	0
$799$	− 41.5692i	− 0.0520266i
$800$	0	0
$801$	594.000	0.741573
$802$	0	0
$803$	− 206.114i	− 0.256680i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	498.831i	0.618130i
$808$	0	0
$809$	546.000	0.674907	0.337454	−	0.941342i	$-0.390434\pi$
$809$	546.000	0.674907	0.337454	+	0.941342i	$0.390434\pi$
$810$	0	0
$811$	− 1309.43i	− 1.61459i	−0.590150	−	0.807294i	$-0.700931\pi$
$811$	− 1309.43i	− 1.61459i	0.590150	−	0.807294i	$-0.299069\pi$
$812$	0	0
$813$	294.000	0.361624
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1404.00	1.71848
$818$	0	0
$819$	332.554i	0.406049i
$820$	0	0
$821$	−1038.00	−1.26431	−0.632156	−	0.774841i	$-0.717830\pi$
$821$	−1038.00	−1.26431	−0.632156	+	0.774841i	$0.717830\pi$
$822$	0	0
$823$	381.051i	0.463003i	0.972835	+	0.231501i	$0.0743638\pi$
$823$	381.051i	0.463003i	−0.972835	+	0.231501i	$0.925636\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 102.191i	− 0.123568i	−0.998090	−	0.0617842i	$-0.980321\pi$
$827$	− 102.191i	− 0.123568i	0.998090	−	0.0617842i	$-0.0196790\pi$
$828$	0	0
$829$	104.000	0.125452	0.0627262	−	0.998031i	$-0.480021\pi$
$829$	104.000	0.125452	0.0627262	+	0.998031i	$0.480021\pi$
$830$	0	0
$831$	− 131.636i	− 0.158407i
$832$	0	0
$833$	111.000	0.133253
$834$	0	0
$835$	0	0
$836$	0	0
$837$	810.000	0.967742
$838$	0	0
$839$	1053.09i	1.25517i	0.778548	+	0.627585i	$0.215956\pi$
$839$	1053.09i	1.25517i	−0.778548	+	0.627585i	$0.784044\pi$
$840$	0	0
$841$	−697.000	−0.828775
$842$	0	0
$843$	613.146i	0.727338i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 90.0666i	− 0.106336i
$848$	0	0
$849$	−549.000	−0.646643
$850$	0	0
$851$	2251.67i	2.64591i
$852$	0	0
$853$	−1234.00	−1.44666	−0.723329	−	0.690503i	$-0.757389\pi$
$853$	−1234.00	−1.44666	−0.723329	+	0.690503i	$0.757389\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1185.00	−1.38273	−0.691365	−	0.722505i	$-0.742990\pi$
$857$	−1185.00	−1.38273	−0.691365	+	0.722505i	$0.742990\pi$
$858$	0	0
$859$	1692.21i	1.96998i	0.172609	+	0.984990i	$0.444780\pi$
$859$	1692.21i	1.96998i	−0.172609	+	0.984990i	$0.555220\pi$
$860$	0	0
$861$	−378.000	−0.439024
$862$	0	0
$863$	893.738i	1.03562i	0.855496	+	0.517809i	$0.173252\pi$
$863$	893.738i	1.03562i	−0.855496	+	0.517809i	$0.826748\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	484.974i	0.559371i
$868$	0	0
$869$	1050.00	1.20829
$870$	0	0
$871$	− 526.543i	− 0.604527i
$872$	0	0
$873$	804.000	0.920962
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.0000	0.0228050	0.0114025	−	0.999935i	$-0.496370\pi$
$877$	20.0000	0.0228050	0.0114025	+	0.999935i	$0.496370\pi$
$878$	0	0
$879$	737.854i	0.839424i
$880$	0	0
$881$	−330.000	−0.374574	−0.187287	−	0.982305i	$-0.559969\pi$
$881$	−330.000	−0.374574	−0.187287	+	0.982305i	$0.559969\pi$
$882$	0	0
$883$	95.2628i	0.107885i	0.998544	+	0.0539427i	$0.0171788\pi$
$883$	95.2628i	0.107885i	−0.998544	+	0.0539427i	$0.982821\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1212.44i	− 1.36689i	−0.730000	−	0.683447i	$-0.760480\pi$
$887$	− 1212.44i	− 1.36689i	0.730000	−	0.683447i	$-0.239520\pi$
$888$	0	0
$889$	−36.0000	−0.0404949
$890$	0	0
$891$	− 109.119i	− 0.122468i
$892$	0	0
$893$	312.000	0.349384
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1248.00	1.39130
$898$	0	0
$899$	− 374.123i	− 0.416155i
$900$	0	0
$901$	54.0000	0.0599334
$902$	0	0
$903$	− 374.123i	− 0.414311i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 297.913i	− 0.328459i	−0.986422	−	0.164230i	$-0.947486\pi$
$907$	− 297.913i	− 0.328459i	0.986422	−	0.164230i	$-0.0525138\pi$
$908$	0	0
$909$	−900.000	−0.990099
$910$	0	0
$911$	− 48.4974i	− 0.0532354i	−0.999646	−	0.0266177i	$-0.991526\pi$
$911$	− 48.4974i	− 0.0532354i	0.999646	−	0.0266177i	$-0.00847367\pi$
$912$	0	0
$913$	1197.00	1.31106
$914$	0	0
$915$	0	0
$916$	0	0
$917$	24.0000	0.0261723
$918$	0	0
$919$	758.638i	0.825504i	0.910843	+	0.412752i	$0.135432\pi$
$919$	758.638i	0.825504i	−0.910843	+	0.412752i	$0.864568\pi$
$920$	0	0
$921$	−387.000	−0.420195
$922$	0	0
$923$	443.405i	0.480395i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 665.108i	− 0.717484i
$928$	0	0
$929$	−258.000	−0.277718	−0.138859	−	0.990312i	$-0.544343\pi$
$929$	−258.000	−0.277718	−0.138859	+	0.990312i	$0.544343\pi$
$930$	0	0
$931$	833.116i	0.894862i
$932$	0	0
$933$	750.000	0.803859
$934$	0	0
$935$	0	0
$936$	0	0
$937$	481.000	0.513340	0.256670	−	0.966499i	$-0.417375\pi$
$937$	481.000	0.513340	0.256670	+	0.966499i	$0.417375\pi$
$938$	0	0
$939$	377.587i	0.402116i
$940$	0	0
$941$	24.0000	0.0255048	0.0127524	−	0.999919i	$-0.495941\pi$
$941$	24.0000	0.0255048	0.0127524	+	0.999919i	$0.495941\pi$
$942$	0	0
$943$	− 2837.10i	− 3.00859i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 665.108i	− 0.702331i	−0.936313	−	0.351166i	$-0.885785\pi$
$947$	− 665.108i	− 0.702331i	0.936313	−	0.351166i	$-0.114215\pi$
$948$	0	0
$949$	272.000	0.286617
$950$	0	0
$951$	− 20.7846i	− 0.0218555i
$952$	0	0
$953$	291.000	0.305352	0.152676	−	0.988276i	$-0.451211\pi$
$953$	291.000	0.305352	0.152676	+	0.988276i	$0.451211\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	252.000	0.263323
$958$	0	0
$959$	717.069i	0.747726i
$960$	0	0
$961$	−11.0000	−0.0114464
$962$	0	0
$963$	987.269i	1.02520i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 1094.66i	− 1.13201i	−0.824401	−	0.566006i	$-0.808488\pi$
$967$	− 1094.66i	− 1.13201i	0.824401	−	0.566006i	$-0.191512\pi$
$968$	0	0
$969$	117.000	0.120743
$970$	0	0
$971$	611.414i	0.629674i	0.949146	+	0.314837i	$0.101950\pi$
$971$	611.414i	0.629674i	−0.949146	+	0.314837i	$0.898050\pi$
$972$	0	0
$973$	618.000	0.635149
$974$	0	0
$975$	0	0
$976$	0	0
$977$	231.000	0.236438	0.118219	−	0.992988i	$-0.462282\pi$
$977$	231.000	0.236438	0.118219	+	0.992988i	$0.462282\pi$
$978$	0	0
$979$	− 1200.31i	− 1.22606i
$980$	0	0
$981$	444.000	0.452599
$982$	0	0
$983$	162.813i	0.165628i	0.996565	+	0.0828142i	$0.0263908\pi$
$983$	162.813i	0.165628i	−0.996565	+	0.0828142i	$0.973609\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 83.1384i	− 0.0842335i
$988$	0	0
$989$	2808.00	2.83923
$990$	0	0
$991$	723.997i	0.730572i	0.930895	+	0.365286i	$0.119029\pi$
$991$	723.997i	0.730572i	−0.930895	+	0.365286i	$0.880971\pi$
$992$	0	0
$993$	807.000	0.812689
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1100.00	1.10331	0.551655	−	0.834072i	$-0.313997\pi$
$997$	1100.00	1.10331	0.551655	+	0.834072i	$0.313997\pi$
$998$	0	0
$999$	− 1299.04i	− 1.30034i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.b.j.1151.1		2
4.3	odd	2	inner	1600.3.b.j.1151.2		2
5.2	odd	4		1600.3.h.h.1599.1		4
5.3	odd	4		1600.3.h.h.1599.3		4
5.4	even	2		1600.3.b.i.1151.2		2
8.3	odd	2		400.3.b.e.351.1	✓	2
8.5	even	2		400.3.b.e.351.2	yes	2
20.3	even	4		1600.3.h.h.1599.2		4
20.7	even	4		1600.3.h.h.1599.4		4
20.19	odd	2		1600.3.b.i.1151.1		2
24.5	odd	2		3600.3.e.j.3151.2		2
24.11	even	2		3600.3.e.j.3151.1		2
40.3	even	4		400.3.h.b.399.3		4
40.13	odd	4		400.3.h.b.399.2		4
40.19	odd	2		400.3.b.f.351.2	yes	2
40.27	even	4		400.3.h.b.399.1		4
40.29	even	2		400.3.b.f.351.1	yes	2
40.37	odd	4		400.3.h.b.399.4		4
120.29	odd	2		3600.3.e.u.3151.1		2
120.53	even	4		3600.3.j.f.1999.3		4
120.59	even	2		3600.3.e.u.3151.2		2
120.77	even	4		3600.3.j.f.1999.1		4
120.83	odd	4		3600.3.j.f.1999.2		4
120.107	odd	4		3600.3.j.f.1999.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
400.3.b.e.351.1	✓	2	8.3	odd	2
400.3.b.e.351.2	yes	2	8.5	even	2
400.3.b.f.351.1	yes	2	40.29	even	2
400.3.b.f.351.2	yes	2	40.19	odd	2
400.3.h.b.399.1		4	40.27	even	4
400.3.h.b.399.2		4	40.13	odd	4
400.3.h.b.399.3		4	40.3	even	4
400.3.h.b.399.4		4	40.37	odd	4
1600.3.b.i.1151.1		2	20.19	odd	2
1600.3.b.i.1151.2		2	5.4	even	2
1600.3.b.j.1151.1		2	1.1	even	1	trivial
1600.3.b.j.1151.2		2	4.3	odd	2	inner
1600.3.h.h.1599.1		4	5.2	odd	4
1600.3.h.h.1599.2		4	20.3	even	4
1600.3.h.h.1599.3		4	5.3	odd	4
1600.3.h.h.1599.4		4	20.7	even	4
3600.3.e.j.3151.1		2	24.11	even	2
3600.3.e.j.3151.2		2	24.5	odd	2
3600.3.e.u.3151.1		2	120.29	odd	2
3600.3.e.u.3151.2		2	120.59	even	2
3600.3.j.f.1999.1		4	120.77	even	4
3600.3.j.f.1999.2		4	120.83	odd	4
3600.3.j.f.1999.3		4	120.53	even	4
3600.3.j.f.1999.4		4	120.107	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +3.46410i q^{7} +6.00000 q^{9} -12.1244i q^{11} +16.0000 q^{13} +3.00000 q^{17} +22.5167i q^{19} +6.00000 q^{21} +45.0333i q^{23} -25.9808i q^{27} -12.0000 q^{29} +31.1769i q^{31} -21.0000 q^{33} +50.0000 q^{37} -27.7128i q^{39} -63.0000 q^{41} -62.3538i q^{43} -13.8564i q^{47} +37.0000 q^{49} -5.19615i q^{51} +18.0000 q^{53} +39.0000 q^{57} +55.4256i q^{59} -26.0000 q^{61} +20.7846i q^{63} -32.9090i q^{67} +78.0000 q^{69} +27.7128i q^{71} +17.0000 q^{73} +42.0000 q^{77} +86.6025i q^{79} +9.00000 q^{81} +98.7269i q^{83} +20.7846i q^{87} +99.0000 q^{89} +55.4256i q^{91} +54.0000 q^{93} +134.000 q^{97} -72.7461i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{9} + 32 q^{13} + 6 q^{17} + 12 q^{21} - 24 q^{29} - 42 q^{33} + 100 q^{37} - 126 q^{41} + 74 q^{49} + 36 q^{53} + 78 q^{57} - 52 q^{61} + 156 q^{69} + 34 q^{73} + 84 q^{77} + 18 q^{81} + 198 q^{89}+ \cdots + 268 q^{97}+O(q^{100}) \) 2 * q + 12 * q^9 + 32 * q^13 + 6 * q^17 + 12 * q^21 - 24 * q^29 - 42 * q^33 + 100 * q^37 - 126 * q^41 + 74 * q^49 + 36 * q^53 + 78 * q^57 - 52 * q^61 + 156 * q^69 + 34 * q^73 + 84 * q^77 + 18 * q^81 + 198 * q^89 + 108 * q^93 + 268 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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