LMFDB - Embedded newform 1600.3.b.w.1151.3

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:	$6$
Coefficient field:	6.0.1827904.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 14x^{2} + 1 $ x^6 + 9x^4 + 14x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 160)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.3
Root			$-0.273891i$ of defining polynomial
Character	$\chi$	$=$	1600.1151
Dual form			1600.3.b.w.1151.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} +O(q^{10})$	$q-0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} +17.2087i q^{11} +4.41325 q^{13} +27.0176 q^{17} -4.82650i q^{19} +5.50881 q^{21} -15.2653i q^{23} -9.69569i q^{27} -2.38225 q^{29} +38.0352i q^{31} +9.42663 q^{33} -16.5691 q^{37} -2.41749i q^{39} -13.3177 q^{41} -59.7918i q^{43} +62.4388i q^{47} -52.1351 q^{49} -14.7997i q^{51} -71.5952 q^{53} -2.64386 q^{57} +68.8265i q^{59} -40.9439 q^{61} +87.4917i q^{63} -51.0080i q^{67} -8.36206 q^{69} -40.4527i q^{71} +35.8441 q^{73} -173.061 q^{77} +126.800i q^{79} +72.9883 q^{81} +75.1490i q^{83} +1.30495i q^{87} +106.523 q^{89} +44.3822i q^{91} +20.8350 q^{93} +85.4351 q^{97} +149.715i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{9}+O(q^{10}) $ 6 * q - 18 * q^9	$ 6 q - 18 q^{9} + 80 q^{17} - 8 q^{21} + 44 q^{29} + 144 q^{33} - 208 q^{37} - 68 q^{41} + 62 q^{49} - 64 q^{53} + 400 q^{57} + 100 q^{61} - 184 q^{69} + 80 q^{73} - 400 q^{77} + 238 q^{81} - 76 q^{89} - 320 q^{93} + 208 q^{97}+O(q^{100}) $ 6 * q - 18 * q^9 + 80 * q^17 - 8 * q^21 + 44 * q^29 + 144 * q^33 - 208 * q^37 - 68 * q^41 + 62 * q^49 - 64 * q^53 + 400 * q^57 + 100 * q^61 - 184 * q^69 + 80 * q^73 - 400 * q^77 + 238 * q^81 - 76 * q^89 - 320 * q^93 + 208 * q^97

Display 100 coefficients 10 coefficients

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
$3$	− 0.547781i	− 0.182594i	−0.995824	−	0.0912969i	$-0.970899\pi$
$3$	− 0.547781i	− 0.182594i	0.995824	−	0.0912969i	$-0.0291012\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	10.0566i	1.43666i	0.695705	+	0.718328i	$0.255092\pi$
$7$	10.0566i	1.43666i	−0.695705	+	0.718328i	$0.744908\pi$
$8$	0	0
$9$	8.69994	0.966660
$10$	0	0
$11$	17.2087i	1.56443i	0.623008	+	0.782216i	$0.285911\pi$
$11$	17.2087i	1.56443i	−0.623008	+	0.782216i	$0.714089\pi$
$12$	0	0
$13$	4.41325	0.339481	0.169740	−	0.985489i	$-0.445707\pi$
$13$	4.41325	0.339481	0.169740	+	0.985489i	$0.445707\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	27.0176	1.58927	0.794636	−	0.607086i	$-0.207662\pi$
$17$	27.0176	1.58927	0.794636	+	0.607086i	$0.207662\pi$
$18$	0	0
$19$	− 4.82650i	− 0.254026i	−0.991901	−	0.127013i	$-0.959461\pi$
$19$	− 4.82650i	− 0.254026i	0.991901	−	0.127013i	$-0.0405390\pi$
$20$	0	0
$21$	5.50881	0.262324
$22$	0	0
$23$	− 15.2653i	− 0.663710i	−0.943330	−	0.331855i	$-0.892325\pi$
$23$	− 15.2653i	− 0.663710i	0.943330	−	0.331855i	$-0.107675\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 9.69569i	− 0.359100i
$28$	0	0
$29$	−2.38225	−0.0821465	−0.0410733	−	0.999156i	$-0.513078\pi$
$29$	−2.38225	−0.0821465	−0.0410733	+	0.999156i	$0.513078\pi$
$30$	0	0
$31$	38.0352i	1.22694i	0.789717	+	0.613472i	$0.210228\pi$
$31$	38.0352i	1.22694i	−0.789717	+	0.613472i	$0.789772\pi$
$32$	0	0
$33$	9.42663	0.285655
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−16.5691	−0.447814	−0.223907	−	0.974610i	$-0.571881\pi$
$37$	−16.5691	−0.447814	−0.223907	+	0.974610i	$0.571881\pi$
$38$	0	0
$39$	− 2.41749i	− 0.0619870i
$40$	0	0
$41$	−13.3177	−0.324822	−0.162411	−	0.986723i	$-0.551927\pi$
$41$	−13.3177	−0.324822	−0.162411	+	0.986723i	$0.551927\pi$
$42$	0	0
$43$	− 59.7918i	− 1.39051i	−0.718765	−	0.695253i	$-0.755292\pi$
$43$	− 59.7918i	− 1.39051i	0.718765	−	0.695253i	$-0.244708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	62.4388i	1.32849i	0.747517	+	0.664243i	$0.231246\pi$
$47$	62.4388i	1.32849i	−0.747517	+	0.664243i	$0.768754\pi$
$48$	0	0
$49$	−52.1351	−1.06398
$50$	0	0
$51$	− 14.7997i	− 0.290191i
$52$	0	0
$53$	−71.5952	−1.35085	−0.675427	−	0.737427i	$-0.736041\pi$
$53$	−71.5952	−1.35085	−0.675427	+	0.737427i	$0.736041\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.64386	−0.0463836
$58$	0	0
$59$	68.8265i	1.16655i	0.812274	+	0.583275i	$0.198229\pi$
$59$	68.8265i	1.16655i	−0.812274	+	0.583275i	$0.801771\pi$
$60$	0	0
$61$	−40.9439	−0.671212	−0.335606	−	0.942002i	$-0.608941\pi$
$61$	−40.9439	−0.671212	−0.335606	+	0.942002i	$0.608941\pi$
$62$	0	0
$63$	87.4917i	1.38876i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 51.0080i	− 0.761313i	−0.924717	−	0.380656i	$-0.875698\pi$
$67$	− 51.0080i	− 0.761313i	0.924717	−	0.380656i	$-0.124302\pi$
$68$	0	0
$69$	−8.36206	−0.121189
$70$	0	0
$71$	− 40.4527i	− 0.569757i	−0.958564	−	0.284878i	$-0.908047\pi$
$71$	− 40.4527i	− 0.569757i	0.958564	−	0.284878i	$-0.0919532\pi$
$72$	0	0
$73$	35.8441	0.491015	0.245508	−	0.969395i	$-0.421045\pi$
$73$	35.8441	0.491015	0.245508	+	0.969395i	$0.421045\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−173.061	−2.24755
$78$	0	0
$79$	126.800i	1.60506i	0.596612	+	0.802530i	$0.296513\pi$
$79$	126.800i	1.60506i	−0.596612	+	0.802530i	$0.703487\pi$
$80$	0	0
$81$	72.9883	0.901090
$82$	0	0
$83$	75.1490i	0.905409i	0.891661	+	0.452705i	$0.149541\pi$
$83$	75.1490i	0.905409i	−0.891661	+	0.452705i	$0.850459\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.30495i	0.0149994i
$88$	0	0
$89$	106.523	1.19689	0.598445	−	0.801164i	$-0.295785\pi$
$89$	106.523	1.19689	0.598445	+	0.801164i	$0.295785\pi$
$90$	0	0
$91$	44.3822i	0.487717i
$92$	0	0
$93$	20.8350	0.224032
$94$	0	0
$95$	0	0
$96$	0	0
$97$	85.4351	0.880774	0.440387	−	0.897808i	$-0.354841\pi$
$97$	85.4351	0.880774	0.440387	+	0.897808i	$0.354841\pi$
$98$	0	0
$99$	149.715i	1.51227i
$100$	0	0
$101$	83.2877	0.824631	0.412315	−	0.911041i	$-0.364720\pi$
$101$	83.2877	0.824631	0.412315	+	0.911041i	$0.364720\pi$
$102$	0	0
$103$	− 47.2653i	− 0.458887i	−0.973322	−	0.229443i	$-0.926309\pi$
$103$	− 47.2653i	− 0.458887i	0.973322	−	0.229443i	$-0.0736906\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 189.628i	− 1.77223i	−0.463467	−	0.886114i	$-0.653395\pi$
$107$	− 189.628i	− 1.77223i	0.463467	−	0.886114i	$-0.346605\pi$
$108$	0	0
$109$	84.5617	0.775795	0.387898	−	0.921702i	$-0.373201\pi$
$109$	84.5617	0.775795	0.387898	+	0.921702i	$0.373201\pi$
$110$	0	0
$111$	9.07626i	0.0817681i
$112$	0	0
$113$	−114.558	−1.01379	−0.506896	−	0.862007i	$-0.669207\pi$
$113$	−114.558	−1.01379	−0.506896	+	0.862007i	$0.669207\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	38.3950	0.328162
$118$	0	0
$119$	271.705i	2.28324i
$120$	0	0
$121$	−175.141	−1.44745
$122$	0	0
$123$	7.29518i	0.0593104i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	44.4121i	0.349701i	0.984595	+	0.174851i	$0.0559443\pi$
$127$	44.4121i	0.349701i	−0.984595	+	0.174851i	$0.944056\pi$
$128$	0	0
$129$	−32.7528	−0.253898
$130$	0	0
$131$	47.1200i	0.359695i	0.983695	+	0.179847i	$0.0575604\pi$
$131$	47.1200i	0.359695i	−0.983695	+	0.179847i	$0.942440\pi$
$132$	0	0
$133$	48.5381	0.364948
$134$	0	0
$135$	0	0
$136$	0	0
$137$	62.8617	0.458845	0.229422	−	0.973327i	$-0.426316\pi$
$137$	62.8617	0.458845	0.229422	+	0.973327i	$0.426316\pi$
$138$	0	0
$139$	128.320i	0.923167i	0.887097	+	0.461584i	$0.152719\pi$
$139$	128.320i	0.923167i	−0.887097	+	0.461584i	$0.847281\pi$
$140$	0	0
$141$	34.2028	0.242573
$142$	0	0
$143$	75.9465i	0.531094i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	28.5586i	0.194276i
$148$	0	0
$149$	−165.561	−1.11115	−0.555574	−	0.831467i	$-0.687501\pi$
$149$	−165.561	−1.11115	−0.555574	+	0.831467i	$0.687501\pi$
$150$	0	0
$151$	218.558i	1.44741i	0.690111	+	0.723704i	$0.257562\pi$
$151$	218.558i	1.44741i	−0.690111	+	0.723704i	$0.742438\pi$
$152$	0	0
$153$	235.052	1.53628
$154$	0	0
$155$	0	0
$156$	0	0
$157$	174.293	1.11014	0.555072	−	0.831802i	$-0.312691\pi$
$157$	174.293	1.11014	0.555072	+	0.831802i	$0.312691\pi$
$158$	0	0
$159$	39.2185i	0.246657i
$160$	0	0
$161$	153.517	0.953524
$162$	0	0
$163$	− 52.6353i	− 0.322916i	−0.986880	−	0.161458i	$-0.948380\pi$
$163$	− 52.6353i	− 0.322916i	0.986880	−	0.161458i	$-0.0516196\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 76.6812i	− 0.459169i	−0.973289	−	0.229584i	$-0.926263\pi$
$167$	− 76.6812i	− 0.459169i	0.973289	−	0.229584i	$-0.0737367\pi$
$168$	0	0
$169$	−149.523	−0.884753
$170$	0	0
$171$	− 41.9902i	− 0.245557i
$172$	0	0
$173$	9.72437	0.0562102	0.0281051	−	0.999605i	$-0.491053\pi$
$173$	9.72437	0.0562102	0.0281051	+	0.999605i	$0.491053\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	37.7019	0.213005
$178$	0	0
$179$	155.502i	0.868728i	0.900738	+	0.434364i	$0.143027\pi$
$179$	155.502i	0.868728i	−0.900738	+	0.434364i	$0.856973\pi$
$180$	0	0
$181$	−250.346	−1.38313	−0.691563	−	0.722316i	$-0.743077\pi$
$181$	−250.346	−1.38313	−0.691563	+	0.722316i	$0.743077\pi$
$182$	0	0
$183$	22.4283i	0.122559i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	464.939i	2.48631i
$188$	0	0
$189$	97.5056	0.515903
$190$	0	0
$191$	− 111.128i	− 0.581825i	−0.956750	−	0.290912i	$-0.906041\pi$
$191$	− 111.128i	− 0.581825i	0.956750	−	0.290912i	$-0.0939588\pi$
$192$	0	0
$193$	10.2701	0.0532130	0.0266065	−	0.999646i	$-0.491530\pi$
$193$	10.2701	0.0532130	0.0266065	+	0.999646i	$0.491530\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	163.213	0.828492	0.414246	−	0.910165i	$-0.364045\pi$
$197$	163.213	0.828492	0.414246	+	0.910165i	$0.364045\pi$
$198$	0	0
$199$	− 195.028i	− 0.980041i	−0.871711	−	0.490021i	$-0.836989\pi$
$199$	− 195.028i	− 0.980041i	0.871711	−	0.490021i	$-0.163011\pi$
$200$	0	0
$201$	−27.9412	−0.139011
$202$	0	0
$203$	− 23.9573i	− 0.118016i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 132.807i	− 0.641582i
$208$	0	0
$209$	83.0580	0.397407
$210$	0	0
$211$	297.038i	1.40776i	0.710317	+	0.703881i	$0.248551\pi$
$211$	297.038i	1.40776i	−0.710317	+	0.703881i	$0.751449\pi$
$212$	0	0
$213$	−22.1592	−0.104034
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−382.505	−1.76270
$218$	0	0
$219$	− 19.6347i	− 0.0896563i
$220$	0	0
$221$	119.236	0.539527
$222$	0	0
$223$	− 405.335i	− 1.81764i	−0.417184	−	0.908822i	$-0.636983\pi$
$223$	− 405.335i	− 1.81764i	0.417184	−	0.908822i	$-0.363017\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	36.6013i	0.161239i	0.996745	+	0.0806196i	$0.0256899\pi$
$227$	36.6013i	0.161239i	−0.996745	+	0.0806196i	$0.974310\pi$
$228$	0	0
$229$	−91.1467	−0.398021	−0.199010	−	0.979997i	$-0.563773\pi$
$229$	−91.1467	−0.398021	−0.199010	+	0.979997i	$0.563773\pi$
$230$	0	0
$231$	94.7997i	0.410388i
$232$	0	0
$233$	−338.802	−1.45409	−0.727043	−	0.686592i	$-0.759106\pi$
$233$	−338.802	−1.45409	−0.727043	+	0.686592i	$0.759106\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	69.4585	0.293074
$238$	0	0
$239$	30.7997i	0.128869i	0.997922	+	0.0644346i	$0.0205244\pi$
$239$	30.7997i	0.128869i	−0.997922	+	0.0644346i	$0.979476\pi$
$240$	0	0
$241$	240.282	0.997020	0.498510	−	0.866884i	$-0.333881\pi$
$241$	240.282	0.997020	0.498510	+	0.866884i	$0.333881\pi$
$242$	0	0
$243$	− 127.243i	− 0.523633i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 21.3005i	− 0.0862370i
$248$	0	0
$249$	41.1652	0.165322
$250$	0	0
$251$	− 86.0268i	− 0.342736i	−0.985207	−	0.171368i	$-0.945181\pi$
$251$	− 86.0268i	− 0.342736i	0.985207	−	0.171368i	$-0.0548187\pi$
$252$	0	0
$253$	262.697	1.03833
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−355.963	−1.38507	−0.692536	−	0.721383i	$-0.743507\pi$
$257$	−355.963	−1.38507	−0.692536	+	0.721383i	$0.743507\pi$
$258$	0	0
$259$	− 166.629i	− 0.643355i
$260$	0	0
$261$	−20.7254	−0.0794077
$262$	0	0
$263$	73.1254i	0.278043i	0.990289	+	0.139022i	$0.0443958\pi$
$263$	73.1254i	0.278043i	−0.990289	+	0.139022i	$0.955604\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 58.3514i	− 0.218545i
$268$	0	0
$269$	268.002	0.996290	0.498145	−	0.867094i	$-0.334015\pi$
$269$	268.002	0.996290	0.498145	+	0.867094i	$0.334015\pi$
$270$	0	0
$271$	229.412i	0.846538i	0.906004	+	0.423269i	$0.139117\pi$
$271$	229.412i	0.846538i	−0.906004	+	0.423269i	$0.860883\pi$
$272$	0	0
$273$	24.3118	0.0890541
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−126.327	−0.456053	−0.228026	−	0.973655i	$-0.573227\pi$
$277$	−126.327	−0.456053	−0.228026	+	0.973655i	$0.573227\pi$
$278$	0	0
$279$	330.904i	1.18604i
$280$	0	0
$281$	−458.746	−1.63255	−0.816275	−	0.577664i	$-0.803964\pi$
$281$	−458.746	−1.63255	−0.816275	+	0.577664i	$0.803964\pi$
$282$	0	0
$283$	465.558i	1.64508i	0.568706	+	0.822541i	$0.307444\pi$
$283$	465.558i	1.64508i	−0.568706	+	0.822541i	$0.692556\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 133.931i	− 0.466657i
$288$	0	0
$289$	440.952	1.52579
$290$	0	0
$291$	− 46.7997i	− 0.160824i
$292$	0	0
$293$	−165.218	−0.563884	−0.281942	−	0.959431i	$-0.590979\pi$
$293$	−165.218	−0.563884	−0.281942	+	0.959431i	$0.590979\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	166.851	0.561787
$298$	0	0
$299$	− 67.3697i	− 0.225317i
$300$	0	0
$301$	601.302	1.99768
$302$	0	0
$303$	− 45.6234i	− 0.150572i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	235.339i	0.766577i	0.923629	+	0.383288i	$0.125208\pi$
$307$	235.339i	0.766577i	−0.923629	+	0.383288i	$0.874792\pi$
$308$	0	0
$309$	−25.8911	−0.0837898
$310$	0	0
$311$	− 210.665i	− 0.677381i	−0.940898	−	0.338691i	$-0.890016\pi$
$311$	− 210.665i	− 0.677381i	0.940898	−	0.338691i	$-0.109984\pi$
$312$	0	0
$313$	−318.738	−1.01833	−0.509166	−	0.860668i	$-0.670046\pi$
$313$	−318.738	−1.01833	−0.509166	+	0.860668i	$0.670046\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−70.3950	−0.222066	−0.111033	−	0.993817i	$-0.535416\pi$
$317$	−70.3950	−0.222066	−0.111033	+	0.993817i	$0.535416\pi$
$318$	0	0
$319$	− 40.9955i	− 0.128513i
$320$	0	0
$321$	−103.875	−0.323598
$322$	0	0
$323$	− 130.401i	− 0.403717i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 46.3213i	− 0.141655i
$328$	0	0
$329$	−627.922	−1.90858
$330$	0	0
$331$	280.807i	0.848359i	0.905578	+	0.424180i	$0.139438\pi$
$331$	280.807i	0.848359i	−0.905578	+	0.424180i	$0.860562\pi$
$332$	0	0
$333$	−144.150	−0.432884
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−120.969	−0.358959	−0.179480	−	0.983762i	$-0.557441\pi$
$337$	−120.969	−0.358959	−0.179480	+	0.983762i	$0.557441\pi$
$338$	0	0
$339$	62.7530i	0.185112i
$340$	0	0
$341$	−654.539	−1.91947
$342$	0	0
$343$	− 31.5280i	− 0.0919182i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	214.333i	0.617675i	0.951115	+	0.308838i	$0.0999399\pi$
$347$	214.333i	0.617675i	−0.951115	+	0.308838i	$0.900060\pi$
$348$	0	0
$349$	−418.041	−1.19782	−0.598912	−	0.800815i	$-0.704400\pi$
$349$	−418.041	−1.19782	−0.598912	+	0.800815i	$0.704400\pi$
$350$	0	0
$351$	− 42.7895i	− 0.121907i
$352$	0	0
$353$	364.929	1.03379	0.516897	−	0.856048i	$-0.327087\pi$
$353$	364.929	1.03379	0.516897	+	0.856048i	$0.327087\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	148.835	0.416905
$358$	0	0
$359$	− 242.169i	− 0.674567i	−0.941403	−	0.337283i	$-0.890492\pi$
$359$	− 242.169i	− 0.674567i	0.941403	−	0.337283i	$-0.109508\pi$
$360$	0	0
$361$	337.705	0.935471
$362$	0	0
$363$	95.9389i	0.264295i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 224.564i	− 0.611891i	−0.952049	−	0.305945i	$-0.901028\pi$
$367$	− 224.564i	− 0.611891i	0.952049	−	0.305945i	$-0.0989725\pi$
$368$	0	0
$369$	−115.863	−0.313992
$370$	0	0
$371$	− 720.004i	− 1.94071i
$372$	0	0
$373$	572.174	1.53398	0.766989	−	0.641660i	$-0.221754\pi$
$373$	572.174	1.53398	0.766989	+	0.641660i	$0.221754\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.5135	−0.0278872
$378$	0	0
$379$	122.896i	0.324263i	0.986769	+	0.162132i	$0.0518369\pi$
$379$	122.896i	0.324263i	−0.986769	+	0.162132i	$0.948163\pi$
$380$	0	0
$381$	24.3281	0.0638533
$382$	0	0
$383$	289.717i	0.756442i	0.925715	+	0.378221i	$0.123464\pi$
$383$	289.717i	0.756442i	−0.925715	+	0.378221i	$0.876536\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 520.185i	− 1.34415i
$388$	0	0
$389$	111.590	0.286863	0.143432	−	0.989660i	$-0.454186\pi$
$389$	111.590	0.286863	0.143432	+	0.989660i	$0.454186\pi$
$390$	0	0
$391$	− 412.433i	− 1.05482i
$392$	0	0
$393$	25.8114	0.0656780
$394$	0	0
$395$	0	0
$396$	0	0
$397$	705.314	1.77661	0.888305	−	0.459253i	$-0.151883\pi$
$397$	705.314	1.77661	0.888305	+	0.459253i	$0.151883\pi$
$398$	0	0
$399$	− 26.5883i	− 0.0666373i
$400$	0	0
$401$	432.052	1.07744	0.538718	−	0.842486i	$-0.318909\pi$
$401$	432.052	1.07744	0.538718	+	0.842486i	$0.318909\pi$
$402$	0	0
$403$	167.859i	0.416524i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 285.134i	− 0.700575i
$408$	0	0
$409$	98.8102	0.241590	0.120795	−	0.992677i	$-0.461456\pi$
$409$	98.8102	0.241590	0.120795	+	0.992677i	$0.461456\pi$
$410$	0	0
$411$	− 34.4345i	− 0.0837822i
$412$	0	0
$413$	−692.160	−1.67593
$414$	0	0
$415$	0	0
$416$	0	0
$417$	70.2914	0.168565
$418$	0	0
$419$	− 204.980i	− 0.489213i	−0.969622	−	0.244607i	$-0.921341\pi$
$419$	− 204.980i	− 0.489213i	0.969622	−	0.244607i	$-0.0786588\pi$
$420$	0	0
$421$	−449.956	−1.06878	−0.534390	−	0.845238i	$-0.679459\pi$
$421$	−449.956	−1.06878	−0.534390	+	0.845238i	$0.679459\pi$
$422$	0	0
$423$	543.214i	1.28419i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 411.756i	− 0.964301i
$428$	0	0
$429$	41.6021	0.0969745
$430$	0	0
$431$	− 314.588i	− 0.729903i	−0.931026	−	0.364952i	$-0.881086\pi$
$431$	− 314.588i	− 0.729903i	0.931026	−	0.364952i	$-0.118914\pi$
$432$	0	0
$433$	−330.441	−0.763143	−0.381571	−	0.924339i	$-0.624617\pi$
$433$	−330.441	−0.763143	−0.381571	+	0.924339i	$0.624617\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−73.6781	−0.168600
$438$	0	0
$439$	− 664.815i	− 1.51439i	−0.653191	−	0.757193i	$-0.726570\pi$
$439$	− 664.815i	− 1.51439i	0.653191	−	0.757193i	$-0.273430\pi$
$440$	0	0
$441$	−453.572	−1.02851
$442$	0	0
$443$	− 312.860i	− 0.706229i	−0.935580	−	0.353115i	$-0.885123\pi$
$443$	− 312.860i	− 0.706229i	0.935580	−	0.353115i	$-0.114877\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	90.6912i	0.202889i
$448$	0	0
$449$	246.330	0.548620	0.274310	−	0.961641i	$-0.411551\pi$
$449$	246.330	0.548620	0.274310	+	0.961641i	$0.411551\pi$
$450$	0	0
$451$	− 229.181i	− 0.508161i
$452$	0	0
$453$	119.722	0.264287
$454$	0	0
$455$	0	0
$456$	0	0
$457$	707.094	1.54725	0.773626	−	0.633642i	$-0.218441\pi$
$457$	707.094	1.54725	0.773626	+	0.633642i	$0.218441\pi$
$458$	0	0
$459$	− 261.955i	− 0.570707i
$460$	0	0
$461$	611.288	1.32600	0.663002	−	0.748618i	$-0.269282\pi$
$461$	611.288	1.32600	0.663002	+	0.748618i	$0.269282\pi$
$462$	0	0
$463$	− 334.063i	− 0.721519i	−0.932659	−	0.360760i	$-0.882518\pi$
$463$	− 334.063i	− 0.721519i	0.932659	−	0.360760i	$-0.117482\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	128.864i	0.275939i	0.990436	+	0.137970i	$0.0440577\pi$
$467$	128.864i	0.275939i	−0.990436	+	0.137970i	$0.955942\pi$
$468$	0	0
$469$	512.966	1.09374
$470$	0	0
$471$	− 95.4742i	− 0.202705i
$472$	0	0
$473$	1028.94	2.17535
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−622.874	−1.30582
$478$	0	0
$479$	− 510.257i	− 1.06525i	−0.846350	−	0.532627i	$-0.821205\pi$
$479$	− 510.257i	− 1.06525i	0.846350	−	0.532627i	$-0.178795\pi$
$480$	0	0
$481$	−73.1237	−0.152024
$482$	0	0
$483$	− 84.0939i	− 0.174107i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	616.472i	1.26586i	0.774211	+	0.632928i	$0.218147\pi$
$487$	616.472i	1.26586i	−0.774211	+	0.632928i	$0.781853\pi$
$488$	0	0
$489$	−28.8326	−0.0589624
$490$	0	0
$491$	− 603.190i	− 1.22849i	−0.789114	−	0.614247i	$-0.789460\pi$
$491$	− 603.190i	− 1.22849i	0.789114	−	0.614247i	$-0.210540\pi$
$492$	0	0
$493$	−64.3627	−0.130553
$494$	0	0
$495$	0	0
$496$	0	0
$497$	406.817	0.818545
$498$	0	0
$499$	− 867.976i	− 1.73943i	−0.493553	−	0.869716i	$-0.664302\pi$
$499$	− 867.976i	− 1.73943i	0.493553	−	0.869716i	$-0.335698\pi$
$500$	0	0
$501$	−42.0045	−0.0838413
$502$	0	0
$503$	− 57.8408i	− 0.114992i	−0.998346	−	0.0574958i	$-0.981688\pi$
$503$	− 57.8408i	− 0.114992i	0.998346	−	0.0574958i	$-0.0183116\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	81.9060i	0.161550i
$508$	0	0
$509$	−168.498	−0.331038	−0.165519	−	0.986207i	$-0.552930\pi$
$509$	−168.498	−0.331038	−0.165519	+	0.986207i	$0.552930\pi$
$510$	0	0
$511$	360.470i	0.705420i
$512$	0	0
$513$	−46.7962	−0.0912207
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1074.49	−2.07833
$518$	0	0
$519$	− 5.32682i	− 0.0102636i
$520$	0	0
$521$	−87.1060	−0.167190	−0.0835951	−	0.996500i	$-0.526640\pi$
$521$	−87.1060	−0.167190	−0.0835951	+	0.996500i	$0.526640\pi$
$522$	0	0
$523$	− 243.469i	− 0.465524i	−0.972534	−	0.232762i	$-0.925224\pi$
$523$	− 243.469i	− 0.465524i	0.972534	−	0.232762i	$-0.0747763\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1027.62i	1.94995i
$528$	0	0
$529$	295.969	0.559488
$530$	0	0
$531$	598.786i	1.12766i
$532$	0	0
$533$	−58.7743	−0.110271
$534$	0	0
$535$	0	0
$536$	0	0
$537$	85.1812	0.158624
$538$	0	0
$539$	− 897.179i	− 1.66452i
$540$	0	0
$541$	812.616	1.50206	0.751032	−	0.660266i	$-0.229557\pi$
$541$	812.616	1.50206	0.751032	+	0.660266i	$0.229557\pi$
$542$	0	0
$543$	137.135i	0.252550i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 104.713i	− 0.191431i	−0.995409	−	0.0957155i	$-0.969486\pi$
$547$	− 104.713i	− 0.191431i	0.995409	−	0.0957155i	$-0.0305139\pi$
$548$	0	0
$549$	−356.210	−0.648833
$550$	0	0
$551$	11.4979i	0.0208674i
$552$	0	0
$553$	−1275.17	−2.30592
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−260.018	−0.466818	−0.233409	−	0.972379i	$-0.574988\pi$
$557$	−260.018	−0.466818	−0.233409	+	0.972379i	$0.574988\pi$
$558$	0	0
$559$	− 263.876i	− 0.472050i
$560$	0	0
$561$	254.685	0.453984
$562$	0	0
$563$	− 39.9073i	− 0.0708833i	−0.999372	−	0.0354416i	$-0.988716\pi$
$563$	− 39.9073i	− 0.0708833i	0.999372	−	0.0354416i	$-0.0112838\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	734.014i	1.29456i
$568$	0	0
$569$	211.588	0.371860	0.185930	−	0.982563i	$-0.440470\pi$
$569$	211.588	0.371860	0.185930	+	0.982563i	$0.440470\pi$
$570$	0	0
$571$	556.938i	0.975372i	0.873019	+	0.487686i	$0.162159\pi$
$571$	556.938i	0.975372i	−0.873019	+	0.487686i	$0.837841\pi$
$572$	0	0
$573$	−60.8741	−0.106237
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−516.233	−0.894684	−0.447342	−	0.894363i	$-0.647629\pi$
$577$	−516.233	−0.894684	−0.447342	+	0.894363i	$0.647629\pi$
$578$	0	0
$579$	− 5.62577i	− 0.00971636i
$580$	0	0
$581$	−755.742	−1.30076
$582$	0	0
$583$	− 1232.06i	− 2.11332i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 934.677i	− 1.59229i	−0.605103	−	0.796147i	$-0.706868\pi$
$587$	− 934.677i	− 1.59229i	0.605103	−	0.796147i	$-0.293132\pi$
$588$	0	0
$589$	183.577	0.311676
$590$	0	0
$591$	− 89.4050i	− 0.151277i
$592$	0	0
$593$	634.665	1.07026	0.535131	−	0.844769i	$-0.320262\pi$
$593$	634.665	1.07026	0.535131	+	0.844769i	$0.320262\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−106.833	−0.178949
$598$	0	0
$599$	− 914.029i	− 1.52593i	−0.646443	−	0.762963i	$-0.723744\pi$
$599$	− 914.029i	− 1.52593i	0.646443	−	0.762963i	$-0.276256\pi$
$600$	0	0
$601$	−598.569	−0.995955	−0.497977	−	0.867190i	$-0.665924\pi$
$601$	−598.569	−0.995955	−0.497977	+	0.867190i	$0.665924\pi$
$602$	0	0
$603$	− 443.766i	− 0.735930i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 201.830i	− 0.332504i	−0.986083	−	0.166252i	$-0.946834\pi$
$607$	− 201.830i	− 0.332504i	0.986083	−	0.166252i	$-0.0531664\pi$
$608$	0	0
$609$	−13.1234	−0.0215490
$610$	0	0
$611$	275.558i	0.450995i
$612$	0	0
$613$	−71.9205	−0.117325	−0.0586627	−	0.998278i	$-0.518684\pi$
$613$	−71.9205	−0.117325	−0.0586627	+	0.998278i	$0.518684\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	204.485	0.331418	0.165709	−	0.986175i	$-0.447009\pi$
$617$	204.485	0.331418	0.165709	+	0.986175i	$0.447009\pi$
$618$	0	0
$619$	− 287.512i	− 0.464479i	−0.972659	−	0.232239i	$-0.925395\pi$
$619$	− 287.512i	− 0.464479i	0.972659	−	0.232239i	$-0.0746053\pi$
$620$	0	0
$621$	−148.008	−0.238338
$622$	0	0
$623$	1071.26i	1.71952i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 45.4976i	− 0.0725640i
$628$	0	0
$629$	−447.658	−0.711699
$630$	0	0
$631$	274.203i	0.434554i	0.976110	+	0.217277i	$0.0697175\pi$
$631$	274.203i	0.434554i	−0.976110	+	0.217277i	$0.930283\pi$
$632$	0	0
$633$	162.712	0.257049
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−230.085	−0.361201
$638$	0	0
$639$	− 351.936i	− 0.550761i
$640$	0	0
$641$	681.328	1.06291	0.531457	−	0.847085i	$-0.321645\pi$
$641$	681.328	1.06291	0.531457	+	0.847085i	$0.321645\pi$
$642$	0	0
$643$	527.270i	0.820016i	0.912082	+	0.410008i	$0.134474\pi$
$643$	527.270i	0.820016i	−0.912082	+	0.410008i	$0.865526\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 551.627i	− 0.852592i	−0.904584	−	0.426296i	$-0.859818\pi$
$647$	− 551.627i	− 0.852592i	0.904584	−	0.426296i	$-0.140182\pi$
$648$	0	0
$649$	−1184.42	−1.82499
$650$	0	0
$651$	209.529i	0.321857i
$652$	0	0
$653$	699.422	1.07109	0.535545	−	0.844507i	$-0.320106\pi$
$653$	699.422	1.07109	0.535545	+	0.844507i	$0.320106\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	311.842	0.474645
$658$	0	0
$659$	38.3257i	0.0581574i	0.999577	+	0.0290787i	$0.00925734\pi$
$659$	38.3257i	0.0581574i	−0.999577	+	0.0290787i	$0.990743\pi$
$660$	0	0
$661$	−392.364	−0.593591	−0.296796	−	0.954941i	$-0.595918\pi$
$661$	−392.364	−0.593591	−0.296796	+	0.954941i	$0.595918\pi$
$662$	0	0
$663$	− 65.3150i	− 0.0985143i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	36.3658i	0.0545215i
$668$	0	0
$669$	−222.035	−0.331890
$670$	0	0
$671$	− 704.594i	− 1.05007i
$672$	0	0
$673$	427.290	0.634904	0.317452	−	0.948274i	$-0.397173\pi$
$673$	427.290	0.634904	0.317452	+	0.948274i	$0.397173\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	131.234	0.193847	0.0969233	−	0.995292i	$-0.469100\pi$
$677$	131.234	0.193847	0.0969233	+	0.995292i	$0.469100\pi$
$678$	0	0
$679$	859.186i	1.26537i
$680$	0	0
$681$	20.0495	0.0294413
$682$	0	0
$683$	− 896.229i	− 1.31219i	−0.754676	−	0.656097i	$-0.772206\pi$
$683$	− 896.229i	− 1.31219i	0.754676	−	0.656097i	$-0.227794\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	49.9285i	0.0726761i
$688$	0	0
$689$	−315.968	−0.458589
$690$	0	0
$691$	− 520.465i	− 0.753206i	−0.926375	−	0.376603i	$-0.877092\pi$
$691$	− 520.465i	− 0.753206i	0.926375	−	0.376603i	$-0.122908\pi$
$692$	0	0
$693$	−1505.62	−2.17262
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−359.812	−0.516230
$698$	0	0
$699$	185.589i	0.265507i
$700$	0	0
$701$	−80.7527	−0.115196	−0.0575982	−	0.998340i	$-0.518344\pi$
$701$	−80.7527	−0.115196	−0.0575982	+	0.998340i	$0.518344\pi$
$702$	0	0
$703$	79.9709i	0.113757i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	837.591i	1.18471i
$708$	0	0
$709$	−174.828	−0.246584	−0.123292	−	0.992370i	$-0.539345\pi$
$709$	−174.828	−0.246584	−0.123292	+	0.992370i	$0.539345\pi$
$710$	0	0
$711$	1103.15i	1.55155i
$712$	0	0
$713$	580.621	0.814335
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.8715	0.0235307
$718$	0	0
$719$	− 889.905i	− 1.23770i	−0.785510	−	0.618849i	$-0.787599\pi$
$719$	− 889.905i	− 1.23770i	0.785510	−	0.618849i	$-0.212401\pi$
$720$	0	0
$721$	475.328	0.659263
$722$	0	0
$723$	− 131.622i	− 0.182050i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 408.818i	− 0.562335i	−0.959659	−	0.281168i	$-0.909278\pi$
$727$	− 408.818i	− 0.562335i	0.959659	−	0.281168i	$-0.0907217\pi$
$728$	0	0
$729$	587.194	0.805478
$730$	0	0
$731$	− 1615.43i	− 2.20989i
$732$	0	0
$733$	388.369	0.529835	0.264917	−	0.964271i	$-0.414655\pi$
$733$	388.369	0.529835	0.264917	+	0.964271i	$0.414655\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	877.783	1.19102
$738$	0	0
$739$	109.561i	0.148256i	0.997249	+	0.0741280i	$0.0236173\pi$
$739$	109.561i	0.148256i	−0.997249	+	0.0741280i	$0.976383\pi$
$740$	0	0
$741$	−11.6680	−0.0157463
$742$	0	0
$743$	732.717i	0.986160i	0.869984	+	0.493080i	$0.164129\pi$
$743$	732.717i	0.986160i	−0.869984	+	0.493080i	$0.835871\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	653.791i	0.875222i
$748$	0	0
$749$	1907.02	2.54608
$750$	0	0
$751$	− 420.809i	− 0.560332i	−0.959952	−	0.280166i	$-0.909610\pi$
$751$	− 420.809i	− 0.560332i	0.959952	−	0.280166i	$-0.0903895\pi$
$752$	0	0
$753$	−47.1238	−0.0625814
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1306.99	1.72654	0.863269	−	0.504744i	$-0.168413\pi$
$757$	1306.99	1.72654	0.863269	+	0.504744i	$0.168413\pi$
$758$	0	0
$759$	− 143.901i	− 0.189592i
$760$	0	0
$761$	−1402.25	−1.84264	−0.921320	−	0.388804i	$-0.872888\pi$
$761$	−1402.25	−1.84264	−0.921320	+	0.388804i	$0.872888\pi$
$762$	0	0
$763$	850.402i	1.11455i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	303.748i	0.396021i
$768$	0	0
$769$	359.952	0.468078	0.234039	−	0.972227i	$-0.424806\pi$
$769$	359.952	0.468078	0.234039	+	0.972227i	$0.424806\pi$
$770$	0	0
$771$	194.990i	0.252905i
$772$	0	0
$773$	437.539	0.566027	0.283013	−	0.959116i	$-0.408666\pi$
$773$	437.539	0.566027	0.283013	+	0.959116i	$0.408666\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−91.2762	−0.117473
$778$	0	0
$779$	64.2778i	0.0825132i
$780$	0	0
$781$	696.141	0.891346
$782$	0	0
$783$	23.0975i	0.0294988i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 438.946i	− 0.557746i	−0.960328	−	0.278873i	$-0.910039\pi$
$787$	− 438.946i	− 0.557746i	0.960328	−	0.278873i	$-0.0899608\pi$
$788$	0	0
$789$	40.0567	0.0507690
$790$	0	0
$791$	− 1152.07i	− 1.45647i
$792$	0	0
$793$	−180.696	−0.227864
$794$	0	0
$795$	0	0
$796$	0	0
$797$	982.414	1.23264	0.616320	−	0.787496i	$-0.288623\pi$
$797$	982.414	1.23264	0.616320	+	0.787496i	$0.288623\pi$
$798$	0	0
$799$	1686.95i	2.11133i
$800$	0	0
$801$	926.745	1.15699
$802$	0	0
$803$	616.832i	0.768160i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 146.806i	− 0.181916i
$808$	0	0
$809$	449.408	0.555510	0.277755	−	0.960652i	$-0.410410\pi$
$809$	449.408	0.555510	0.277755	+	0.960652i	$0.410410\pi$
$810$	0	0
$811$	− 610.106i	− 0.752288i	−0.926561	−	0.376144i	$-0.877250\pi$
$811$	− 610.106i	− 0.752288i	0.926561	−	0.376144i	$-0.122750\pi$
$812$	0	0
$813$	125.667	0.154572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−288.585	−0.353225
$818$	0	0
$819$	386.123i	0.471456i
$820$	0	0
$821$	1099.76	1.33954	0.669770	−	0.742568i	$-0.266393\pi$
$821$	1099.76	1.33954	0.669770	+	0.742568i	$0.266393\pi$
$822$	0	0
$823$	236.092i	0.286867i	0.989660	+	0.143434i	$0.0458143\pi$
$823$	236.092i	0.286867i	−0.989660	+	0.143434i	$0.954186\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1266.81i	− 1.53182i	−0.642950	−	0.765908i	$-0.722290\pi$
$827$	− 1266.81i	− 1.53182i	0.642950	−	0.765908i	$-0.277710\pi$
$828$	0	0
$829$	−41.7642	−0.0503790	−0.0251895	−	0.999683i	$-0.508019\pi$
$829$	−41.7642	−0.0503790	−0.0251895	+	0.999683i	$0.508019\pi$
$830$	0	0
$831$	69.1993i	0.0832723i
$832$	0	0
$833$	−1408.57	−1.69095
$834$	0	0
$835$	0	0
$836$	0	0
$837$	368.778	0.440595
$838$	0	0
$839$	816.273i	0.972912i	0.873705	+	0.486456i	$0.161711\pi$
$839$	816.273i	0.972912i	−0.873705	+	0.486456i	$0.838289\pi$
$840$	0	0
$841$	−835.325	−0.993252
$842$	0	0
$843$	251.293i	0.298093i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1761.32i	− 2.07948i
$848$	0	0
$849$	255.024	0.300381
$850$	0	0
$851$	252.933i	0.297219i
$852$	0	0
$853$	1215.58	1.42507	0.712533	−	0.701639i	$-0.247548\pi$
$853$	1215.58	1.42507	0.712533	+	0.701639i	$0.247548\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−997.998	−1.16452	−0.582262	−	0.813001i	$-0.697832\pi$
$857$	−997.998	−1.16452	−0.582262	+	0.813001i	$0.697832\pi$
$858$	0	0
$859$	1110.42i	1.29269i	0.763044	+	0.646347i	$0.223704\pi$
$859$	1110.42i	1.29269i	−0.763044	+	0.646347i	$0.776296\pi$
$860$	0	0
$861$	−73.3646	−0.0852086
$862$	0	0
$863$	− 1372.33i	− 1.59019i	−0.606486	−	0.795094i	$-0.707422\pi$
$863$	− 1372.33i	− 1.59019i	0.606486	−	0.795094i	$-0.292578\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 241.545i	− 0.278599i
$868$	0	0
$869$	−2182.06	−2.51101
$870$	0	0
$871$	− 225.111i	− 0.258451i
$872$	0	0
$873$	743.280	0.851409
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−776.337	−0.885219	−0.442609	−	0.896714i	$-0.645947\pi$
$877$	−776.337	−0.885219	−0.442609	+	0.896714i	$0.645947\pi$
$878$	0	0
$879$	90.5034i	0.102962i
$880$	0	0
$881$	1047.38	1.18885	0.594427	−	0.804150i	$-0.297379\pi$
$881$	1047.38	1.18885	0.594427	+	0.804150i	$0.297379\pi$
$882$	0	0
$883$	1175.41i	1.33116i	0.746327	+	0.665579i	$0.231815\pi$
$883$	1175.41i	1.33116i	−0.746327	+	0.665579i	$0.768185\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 109.768i	− 0.123752i	−0.998084	−	0.0618762i	$-0.980292\pi$
$887$	− 109.768i	− 0.123752i	0.998084	−	0.0618762i	$-0.0197084\pi$
$888$	0	0
$889$	−446.634	−0.502401
$890$	0	0
$891$	1256.04i	1.40969i
$892$	0	0
$893$	301.361	0.337470
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−36.9039	−0.0411414
$898$	0	0
$899$	− 90.6094i	− 0.100789i
$900$	0	0
$901$	−1934.33	−2.14687
$902$	0	0
$903$	− 329.382i	− 0.364764i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1639.15i	1.80722i	0.428358	+	0.903609i	$0.359092\pi$
$907$	1639.15i	1.80722i	−0.428358	+	0.903609i	$0.640908\pi$
$908$	0	0
$909$	724.598	0.797137
$910$	0	0
$911$	6.82023i	0.00748654i	0.999993	+	0.00374327i	$0.00119152\pi$
$911$	6.82023i	0.00748654i	−0.999993	+	0.00374327i	$0.998808\pi$
$912$	0	0
$913$	−1293.22	−1.41645
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−473.867	−0.516757
$918$	0	0
$919$	72.4188i	0.0788017i	0.999223	+	0.0394009i	$0.0125449\pi$
$919$	72.4188i	0.0788017i	−0.999223	+	0.0394009i	$0.987455\pi$
$920$	0	0
$921$	128.914	0.139972
$922$	0	0
$923$	− 178.528i	− 0.193421i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 411.205i	− 0.443587i
$928$	0	0
$929$	368.537	0.396703	0.198352	−	0.980131i	$-0.436441\pi$
$929$	368.537	0.396703	0.198352	+	0.980131i	$0.436441\pi$
$930$	0	0
$931$	251.630i	0.270279i
$932$	0	0
$933$	−115.399	−0.123686
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−590.172	−0.629852	−0.314926	−	0.949116i	$-0.601980\pi$
$937$	−590.172	−0.629852	−0.314926	+	0.949116i	$0.601980\pi$
$938$	0	0
$939$	174.599i	0.185941i
$940$	0	0
$941$	189.017	0.200869	0.100434	−	0.994944i	$-0.467977\pi$
$941$	189.017	0.200869	0.100434	+	0.994944i	$0.467977\pi$
$942$	0	0
$943$	203.299i	0.215588i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	664.319i	0.701498i	0.936470	+	0.350749i	$0.114073\pi$
$947$	664.319i	0.701498i	−0.936470	+	0.350749i	$0.885927\pi$
$948$	0	0
$949$	158.189	0.166690
$950$	0	0
$951$	38.5610i	0.0405479i
$952$	0	0
$953$	173.943	0.182522	0.0912610	−	0.995827i	$-0.470910\pi$
$953$	173.943	0.182522	0.0912610	+	0.995827i	$0.470910\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−22.4566	−0.0234656
$958$	0	0
$959$	632.175i	0.659202i
$960$	0	0
$961$	−485.680	−0.505390
$962$	0	0
$963$	− 1649.76i	− 1.71314i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 255.364i	− 0.264078i	−0.991245	−	0.132039i	$-0.957848\pi$
$967$	− 255.364i	− 0.264078i	0.991245	−	0.132039i	$-0.0421524\pi$
$968$	0	0
$969$	−71.4309	−0.0737161
$970$	0	0
$971$	1623.60i	1.67209i	0.548659	+	0.836046i	$0.315139\pi$
$971$	1623.60i	1.67209i	−0.548659	+	0.836046i	$0.684861\pi$
$972$	0	0
$973$	−1290.46	−1.32627
$974$	0	0
$975$	0	0
$976$	0	0
$977$	104.946	0.107416	0.0537082	−	0.998557i	$-0.482896\pi$
$977$	104.946	0.107416	0.0537082	+	0.998557i	$0.482896\pi$
$978$	0	0
$979$	1833.13i	1.87245i
$980$	0	0
$981$	735.681	0.749930
$982$	0	0
$983$	− 889.933i	− 0.905323i	−0.891682	−	0.452662i	$-0.850475\pi$
$983$	− 889.933i	− 0.905323i	0.891682	−	0.452662i	$-0.149525\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	343.964i	0.348494i
$988$	0	0
$989$	−912.742	−0.922894
$990$	0	0
$991$	1509.57i	1.52328i	0.648001	+	0.761639i	$0.275605\pi$
$991$	1509.57i	1.52328i	−0.648001	+	0.761639i	$0.724395\pi$
$992$	0	0
$993$	153.821	0.154905
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1599.93	1.60474	0.802371	−	0.596825i	$-0.203571\pi$
$997$	1599.93	1.60474	0.802371	+	0.596825i	$0.203571\pi$
$998$	0	0
$999$	160.649i	0.160810i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.b.w.1151.3		6
4.3	odd	2	inner	1600.3.b.w.1151.4		6
5.2	odd	4		320.3.h.g.319.3		6
5.3	odd	4		320.3.h.f.319.4		6
5.4	even	2		1600.3.b.v.1151.4		6
8.3	odd	2		800.3.b.i.351.3		6
8.5	even	2		800.3.b.i.351.4		6
20.3	even	4		320.3.h.g.319.4		6
20.7	even	4		320.3.h.f.319.3		6
20.19	odd	2		1600.3.b.v.1151.3		6
40.3	even	4		160.3.h.a.159.3	✓	6
40.13	odd	4		160.3.h.b.159.3	yes	6
40.19	odd	2		800.3.b.h.351.4		6
40.27	even	4		160.3.h.b.159.4	yes	6
40.29	even	2		800.3.b.h.351.3		6
40.37	odd	4		160.3.h.a.159.4	yes	6
80.3	even	4		1280.3.e.g.639.3		6
80.13	odd	4		1280.3.e.f.639.4		6
80.27	even	4		1280.3.e.f.639.3		6
80.37	odd	4		1280.3.e.g.639.4		6
80.43	even	4		1280.3.e.i.639.4		6
80.53	odd	4		1280.3.e.h.639.3		6
80.67	even	4		1280.3.e.h.639.4		6
80.77	odd	4		1280.3.e.i.639.3		6
120.53	even	4		1440.3.j.b.1279.6		6
120.77	even	4		1440.3.j.a.1279.5		6
120.83	odd	4		1440.3.j.a.1279.6		6
120.107	odd	4		1440.3.j.b.1279.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.3.h.a.159.3	✓	6	40.3	even	4
160.3.h.a.159.4	yes	6	40.37	odd	4
160.3.h.b.159.3	yes	6	40.13	odd	4
160.3.h.b.159.4	yes	6	40.27	even	4
320.3.h.f.319.3		6	20.7	even	4
320.3.h.f.319.4		6	5.3	odd	4
320.3.h.g.319.3		6	5.2	odd	4
320.3.h.g.319.4		6	20.3	even	4
800.3.b.h.351.3		6	40.29	even	2
800.3.b.h.351.4		6	40.19	odd	2
800.3.b.i.351.3		6	8.3	odd	2
800.3.b.i.351.4		6	8.5	even	2
1280.3.e.f.639.3		6	80.27	even	4
1280.3.e.f.639.4		6	80.13	odd	4
1280.3.e.g.639.3		6	80.3	even	4
1280.3.e.g.639.4		6	80.37	odd	4
1280.3.e.h.639.3		6	80.53	odd	4
1280.3.e.h.639.4		6	80.67	even	4
1280.3.e.i.639.3		6	80.77	odd	4
1280.3.e.i.639.4		6	80.43	even	4
1440.3.j.a.1279.5		6	120.77	even	4
1440.3.j.a.1279.6		6	120.83	odd	4
1440.3.j.b.1279.5		6	120.107	odd	4
1440.3.j.b.1279.6		6	120.53	even	4
1600.3.b.v.1151.3		6	20.19	odd	2
1600.3.b.v.1151.4		6	5.4	even	2
1600.3.b.w.1151.3		6	1.1	even	1	trivial
1600.3.b.w.1151.4		6	4.3	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} +O(q^{10})\)	\(q-0.547781i q^{3} +10.0566i q^{7} +8.69994 q^{9} +17.2087i q^{11} +4.41325 q^{13} +27.0176 q^{17} -4.82650i q^{19} +5.50881 q^{21} -15.2653i q^{23} -9.69569i q^{27} -2.38225 q^{29} +38.0352i q^{31} +9.42663 q^{33} -16.5691 q^{37} -2.41749i q^{39} -13.3177 q^{41} -59.7918i q^{43} +62.4388i q^{47} -52.1351 q^{49} -14.7997i q^{51} -71.5952 q^{53} -2.64386 q^{57} +68.8265i q^{59} -40.9439 q^{61} +87.4917i q^{63} -51.0080i q^{67} -8.36206 q^{69} -40.4527i q^{71} +35.8441 q^{73} -173.061 q^{77} +126.800i q^{79} +72.9883 q^{81} +75.1490i q^{83} +1.30495i q^{87} +106.523 q^{89} +44.3822i q^{91} +20.8350 q^{93} +85.4351 q^{97} +149.715i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{9}+O(q^{10}) \) 6 * q - 18 * q^9	\( 6 q - 18 q^{9} + 80 q^{17} - 8 q^{21} + 44 q^{29} + 144 q^{33} - 208 q^{37} - 68 q^{41} + 62 q^{49} - 64 q^{53} + 400 q^{57} + 100 q^{61} - 184 q^{69} + 80 q^{73} - 400 q^{77} + 238 q^{81} - 76 q^{89} - 320 q^{93} + 208 q^{97}+O(q^{100}) \) 6 * q - 18 * q^9 + 80 * q^17 - 8 * q^21 + 44 * q^29 + 144 * q^33 - 208 * q^37 - 68 * q^41 + 62 * q^49 - 64 * q^53 + 400 * q^57 + 100 * q^61 - 184 * q^69 + 80 * q^73 - 400 * q^77 + 238 * q^81 - 76 * q^89 - 320 * q^93 + 208 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more