LMFDB - Embedded newform 1200.3.e.n.751.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(751,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1200.751");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.e (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:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.4
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	1200.751
Dual form			1200.3.e.n.751.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +4.28187i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+1.73205i q^{3} +4.28187i q^{7} -3.00000 q^{9} -11.2101i q^{11} -5.41641 q^{13} +1.41641 q^{17} -8.56373i q^{19} -7.41641 q^{21} +24.0557i q^{23} -5.19615i q^{27} -25.4164 q^{29} -50.1329i q^{31} +19.4164 q^{33} -60.2492 q^{37} -9.38149i q^{39} -30.0000 q^{41} -37.9121i q^{43} -39.9337i q^{47} +30.6656 q^{49} +2.45329i q^{51} +16.2492 q^{53} +14.8328 q^{57} +81.7415i q^{59} +94.4984 q^{61} -12.8456i q^{63} -129.614i q^{67} -41.6656 q^{69} -102.288i q^{71} -24.8328 q^{73} +48.0000 q^{77} +91.7022i q^{79} +9.00000 q^{81} -88.0450i q^{83} -44.0225i q^{87} -20.8328 q^{89} -23.1923i q^{91} +86.8328 q^{93} +14.0000 q^{97} +33.6302i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} + 32 q^{13} - 48 q^{17} + 24 q^{21} - 48 q^{29} + 24 q^{33} - 80 q^{37} - 120 q^{41} - 92 q^{49} - 96 q^{53} - 48 q^{57} + 56 q^{61} + 48 q^{69} + 8 q^{73} + 192 q^{77} + 36 q^{81} + 24 q^{89} + 240 q^{93} + 56 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 + 32 * q^13 - 48 * q^17 + 24 * q^21 - 48 * q^29 + 24 * q^33 - 80 * q^37 - 120 * q^41 - 92 * q^49 - 96 * q^53 - 48 * q^57 + 56 * q^61 + 48 * q^69 + 8 * q^73 + 192 * q^77 + 36 * q^81 + 24 * q^89 + 240 * q^93 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.28187i	0.611695i	0.952081	+	0.305848i	$0.0989398\pi$
$7$	4.28187i	0.611695i	−0.952081	+	0.305848i	$0.901060\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 11.2101i	− 1.01910i	−0.860442	−	0.509549i	$-0.829812\pi$
$11$	− 11.2101i	− 1.01910i	0.860442	−	0.509549i	$-0.170188\pi$
$12$	0	0
$13$	−5.41641	−0.416647	−0.208323	−	0.978060i	$-0.566801\pi$
$13$	−5.41641	−0.416647	−0.208323	+	0.978060i	$0.566801\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.41641	0.0833181	0.0416591	−	0.999132i	$-0.486736\pi$
$17$	1.41641	0.0833181	0.0416591	+	0.999132i	$0.486736\pi$
$18$	0	0
$19$	− 8.56373i	− 0.450723i	−0.974275	−	0.225361i	$-0.927644\pi$
$19$	− 8.56373i	− 0.450723i	0.974275	−	0.225361i	$-0.0723563\pi$
$20$	0	0
$21$	−7.41641	−0.353162
$22$	0	0
$23$	24.0557i	1.04590i	0.852364	+	0.522949i	$0.175168\pi$
$23$	24.0557i	1.04590i	−0.852364	+	0.522949i	$0.824832\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	−25.4164	−0.876428	−0.438214	−	0.898871i	$-0.644389\pi$
$29$	−25.4164	−0.876428	−0.438214	+	0.898871i	$0.644389\pi$
$30$	0	0
$31$	− 50.1329i	− 1.61719i	−0.588364	−	0.808596i	$-0.700228\pi$
$31$	− 50.1329i	− 1.61719i	0.588364	−	0.808596i	$-0.299772\pi$
$32$	0	0
$33$	19.4164	0.588376
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−60.2492	−1.62836	−0.814179	−	0.580614i	$-0.802812\pi$
$37$	−60.2492	−1.62836	−0.814179	+	0.580614i	$0.802812\pi$
$38$	0	0
$39$	− 9.38149i	− 0.240551i
$40$	0	0
$41$	−30.0000	−0.731707	−0.365854	−	0.930672i	$-0.619223\pi$
$41$	−30.0000	−0.731707	−0.365854	+	0.930672i	$0.619223\pi$
$42$	0	0
$43$	− 37.9121i	− 0.881676i	−0.897587	−	0.440838i	$-0.854681\pi$
$43$	− 37.9121i	− 0.881676i	0.897587	−	0.440838i	$-0.145319\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 39.9337i	− 0.849653i	−0.905275	−	0.424827i	$-0.860335\pi$
$47$	− 39.9337i	− 0.849653i	0.905275	−	0.424827i	$-0.139665\pi$
$48$	0	0
$49$	30.6656	0.625829
$50$	0	0
$51$	2.45329i	0.0481037i
$52$	0	0
$53$	16.2492	0.306589	0.153295	−	0.988181i	$-0.451012\pi$
$53$	16.2492	0.306589	0.153295	+	0.988181i	$0.451012\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	14.8328	0.260225
$58$	0	0
$59$	81.7415i	1.38545i	0.721202	+	0.692725i	$0.243590\pi$
$59$	81.7415i	1.38545i	−0.721202	+	0.692725i	$0.756410\pi$
$60$	0	0
$61$	94.4984	1.54915	0.774577	−	0.632479i	$-0.217963\pi$
$61$	94.4984	1.54915	0.774577	+	0.632479i	$0.217963\pi$
$62$	0	0
$63$	− 12.8456i	− 0.203898i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 129.614i	− 1.93454i	−0.253748	−	0.967270i	$-0.581663\pi$
$67$	− 129.614i	− 1.93454i	0.253748	−	0.967270i	$-0.418337\pi$
$68$	0	0
$69$	−41.6656	−0.603850
$70$	0	0
$71$	− 102.288i	− 1.44067i	−0.693627	−	0.720335i	$-0.743988\pi$
$71$	− 102.288i	− 1.44067i	0.693627	−	0.720335i	$-0.256012\pi$
$72$	0	0
$73$	−24.8328	−0.340176	−0.170088	−	0.985429i	$-0.554405\pi$
$73$	−24.8328	−0.340176	−0.170088	+	0.985429i	$0.554405\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	48.0000	0.623377
$78$	0	0
$79$	91.7022i	1.16079i	0.814336	+	0.580393i	$0.197101\pi$
$79$	91.7022i	1.16079i	−0.814336	+	0.580393i	$0.802899\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 88.0450i	− 1.06078i	−0.847753	−	0.530392i	$-0.822045\pi$
$83$	− 88.0450i	− 1.06078i	0.847753	−	0.530392i	$-0.177955\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 44.0225i	− 0.506006i
$88$	0	0
$89$	−20.8328	−0.234077	−0.117038	−	0.993127i	$-0.537340\pi$
$89$	−20.8328	−0.234077	−0.117038	+	0.993127i	$0.537340\pi$
$90$	0	0
$91$	− 23.1923i	− 0.254861i
$92$	0	0
$93$	86.8328	0.933686
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000	0.144330	0.0721649	−	0.997393i	$-0.477009\pi$
$97$	14.0000	0.144330	0.0721649	+	0.997393i	$0.477009\pi$
$98$	0	0
$99$	33.6302i	0.339699i
$100$	0	0
$101$	−132.748	−1.31433	−0.657167	−	0.753745i	$-0.728245\pi$
$101$	−132.748	−1.31433	−0.657167	+	0.753745i	$0.728245\pi$
$102$	0	0
$103$	62.9785i	0.611442i	0.952121	+	0.305721i	$0.0988975\pi$
$103$	62.9785i	0.611442i	−0.952121	+	0.305721i	$0.901102\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 174.455i	− 1.63042i	−0.579168	−	0.815208i	$-0.696623\pi$
$107$	− 174.455i	− 1.63042i	0.579168	−	0.815208i	$-0.303377\pi$
$108$	0	0
$109$	96.8328	0.888374	0.444187	−	0.895934i	$-0.353493\pi$
$109$	96.8328	0.888374	0.444187	+	0.895934i	$0.353493\pi$
$110$	0	0
$111$	− 104.355i	− 0.940133i
$112$	0	0
$113$	−211.082	−1.86798	−0.933991	−	0.357296i	$-0.883699\pi$
$113$	−211.082	−1.86798	−0.933991	+	0.357296i	$0.883699\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.2492	0.138882
$118$	0	0
$119$	6.06487i	0.0509653i
$120$	0	0
$121$	−4.66563	−0.0385589
$122$	0	0
$123$	− 51.9615i	− 0.422451i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 44.6017i	− 0.351194i	−0.984462	−	0.175597i	$-0.943814\pi$
$127$	− 44.6017i	− 0.351194i	0.984462	−	0.175597i	$-0.0561856\pi$
$128$	0	0
$129$	65.6656	0.509036
$130$	0	0
$131$	− 161.609i	− 1.23366i	−0.787098	−	0.616828i	$-0.788417\pi$
$131$	− 161.609i	− 1.23366i	0.787098	−	0.616828i	$-0.211583\pi$
$132$	0	0
$133$	36.6687	0.275705
$134$	0	0
$135$	0	0
$136$	0	0
$137$	106.584	0.777982	0.388991	−	0.921241i	$-0.372824\pi$
$137$	106.584	0.777982	0.388991	+	0.921241i	$0.372824\pi$
$138$	0	0
$139$	− 24.4418i	− 0.175840i	−0.996128	−	0.0879200i	$-0.971978\pi$
$139$	− 24.4418i	− 0.175840i	0.996128	−	0.0879200i	$-0.0280220\pi$
$140$	0	0
$141$	69.1672	0.490547
$142$	0	0
$143$	60.7183i	0.424604i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	53.1144i	0.361323i
$148$	0	0
$149$	91.0820	0.611289	0.305644	−	0.952146i	$-0.401128\pi$
$149$	91.0820	0.611289	0.305644	+	0.952146i	$0.401128\pi$
$150$	0	0
$151$	− 251.914i	− 1.66831i	−0.551533	−	0.834153i	$-0.685957\pi$
$151$	− 251.914i	− 1.66831i	0.551533	−	0.834153i	$-0.314043\pi$
$152$	0	0
$153$	−4.24922	−0.0277727
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−21.4164	−0.136410	−0.0682051	−	0.997671i	$-0.521727\pi$
$157$	−21.4164	−0.136410	−0.0682051	+	0.997671i	$0.521727\pi$
$158$	0	0
$159$	28.1445i	0.177009i
$160$	0	0
$161$	−103.003	−0.639771
$162$	0	0
$163$	14.7197i	0.0903052i	0.998980	+	0.0451526i	$0.0143774\pi$
$163$	14.7197i	0.0903052i	−0.998980	+	0.0451526i	$0.985623\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	78.7091i	0.471312i	0.971837	+	0.235656i	$0.0757239\pi$
$167$	78.7091i	0.471312i	−0.971837	+	0.235656i	$0.924276\pi$
$168$	0	0
$169$	−139.663	−0.826405
$170$	0	0
$171$	25.6912i	0.150241i
$172$	0	0
$173$	−146.085	−0.844423	−0.422211	−	0.906497i	$-0.638746\pi$
$173$	−146.085	−0.844423	−0.422211	+	0.906497i	$0.638746\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−141.580	−0.799890
$178$	0	0
$179$	− 238.682i	− 1.33342i	−0.745316	−	0.666711i	$-0.767702\pi$
$179$	− 238.682i	− 1.33342i	0.745316	−	0.666711i	$-0.232298\pi$
$180$	0	0
$181$	115.167	0.636283	0.318141	−	0.948043i	$-0.396941\pi$
$181$	115.167	0.636283	0.318141	+	0.948043i	$0.396941\pi$
$182$	0	0
$183$	163.676i	0.894405i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 15.8780i	− 0.0849092i
$188$	0	0
$189$	22.2492	0.117721
$190$	0	0
$191$	− 6.06487i	− 0.0317532i	−0.999874	−	0.0158766i	$-0.994946\pi$
$191$	− 6.06487i	− 0.0317532i	0.999874	−	0.0158766i	$-0.00505390\pi$
$192$	0	0
$193$	358.827	1.85921	0.929603	−	0.368563i	$-0.120150\pi$
$193$	358.827	1.85921	0.929603	+	0.368563i	$0.120150\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	141.915	0.720380	0.360190	−	0.932879i	$-0.382712\pi$
$197$	141.915	0.720380	0.360190	+	0.932879i	$0.382712\pi$
$198$	0	0
$199$	124.708i	0.626672i	0.949642	+	0.313336i	$0.101447\pi$
$199$	124.708i	0.626672i	−0.949642	+	0.313336i	$0.898553\pi$
$200$	0	0
$201$	224.498	1.11691
$202$	0	0
$203$	− 108.830i	− 0.536107i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 72.1670i	− 0.348633i
$208$	0	0
$209$	−96.0000	−0.459330
$210$	0	0
$211$	− 102.765i	− 0.487037i	−0.969896	−	0.243518i	$-0.921698\pi$
$211$	− 102.765i	− 0.487037i	0.969896	−	0.243518i	$-0.0783016\pi$
$212$	0	0
$213$	177.167	0.831771
$214$	0	0
$215$	0	0
$216$	0	0
$217$	214.663	0.989228
$218$	0	0
$219$	− 43.0117i	− 0.196400i
$220$	0	0
$221$	−7.67184	−0.0347142
$222$	0	0
$223$	− 39.7862i	− 0.178414i	−0.996013	−	0.0892068i	$-0.971567\pi$
$223$	− 39.7862i	− 0.178414i	0.996013	−	0.0892068i	$-0.0284332\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.2425i	0.0627423i	0.999508	+	0.0313711i	$0.00998739\pi$
$227$	14.2425i	0.0627423i	−0.999508	+	0.0313711i	$0.990013\pi$
$228$	0	0
$229$	−45.0031	−0.196520	−0.0982601	−	0.995161i	$-0.531328\pi$
$229$	−45.0031	−0.196520	−0.0982601	+	0.995161i	$0.531328\pi$
$230$	0	0
$231$	83.1384i	0.359907i
$232$	0	0
$233$	−393.915	−1.69062	−0.845311	−	0.534275i	$-0.820585\pi$
$233$	−393.915	−1.69062	−0.845311	+	0.534275i	$0.820585\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−158.833	−0.670181
$238$	0	0
$239$	329.760i	1.37975i	0.723929	+	0.689874i	$0.242334\pi$
$239$	329.760i	1.37975i	−0.723929	+	0.689874i	$0.757666\pi$
$240$	0	0
$241$	−210.328	−0.872731	−0.436365	−	0.899770i	$-0.643735\pi$
$241$	−210.328	−0.872731	−0.436365	+	0.899770i	$0.643735\pi$
$242$	0	0
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	46.3847i	0.187792i
$248$	0	0
$249$	152.498	0.612444
$250$	0	0
$251$	254.083i	1.01228i	0.862450	+	0.506142i	$0.168929\pi$
$251$	254.083i	1.01228i	−0.862450	+	0.506142i	$0.831071\pi$
$252$	0	0
$253$	269.666	1.06587
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−407.410	−1.58525	−0.792627	−	0.609707i	$-0.791287\pi$
$257$	−407.410	−1.58525	−0.792627	+	0.609707i	$0.791287\pi$
$258$	0	0
$259$	− 257.979i	− 0.996058i
$260$	0	0
$261$	76.2492	0.292143
$262$	0	0
$263$	138.950i	0.528328i	0.964478	+	0.264164i	$0.0850960\pi$
$263$	138.950i	0.528328i	−0.964478	+	0.264164i	$0.914904\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 36.0835i	− 0.135144i
$268$	0	0
$269$	−55.0820	−0.204766	−0.102383	−	0.994745i	$-0.532647\pi$
$269$	−55.0820	−0.204766	−0.102383	+	0.994745i	$0.532647\pi$
$270$	0	0
$271$	406.061i	1.49838i	0.662354	+	0.749191i	$0.269557\pi$
$271$	406.061i	1.49838i	−0.662354	+	0.749191i	$0.730443\pi$
$272$	0	0
$273$	40.1703	0.147144
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−350.584	−1.26564	−0.632822	−	0.774297i	$-0.718104\pi$
$277$	−350.584	−1.26564	−0.632822	+	0.774297i	$0.718104\pi$
$278$	0	0
$279$	150.399i	0.539064i
$280$	0	0
$281$	−457.161	−1.62691	−0.813454	−	0.581630i	$-0.802415\pi$
$281$	−457.161	−1.62691	−0.813454	+	0.581630i	$0.802415\pi$
$282$	0	0
$283$	− 339.959i	− 1.20127i	−0.799524	−	0.600635i	$-0.794915\pi$
$283$	− 339.959i	− 1.20127i	0.799524	−	0.600635i	$-0.205085\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 128.456i	− 0.447582i
$288$	0	0
$289$	−286.994	−0.993058
$290$	0	0
$291$	24.2487i	0.0833289i
$292$	0	0
$293$	−575.410	−1.96386	−0.981929	−	0.189251i	$-0.939394\pi$
$293$	−575.410	−1.96386	−0.981929	+	0.189251i	$0.939394\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−58.2492	−0.196125
$298$	0	0
$299$	− 130.295i	− 0.435770i
$300$	0	0
$301$	162.334	0.539317
$302$	0	0
$303$	− 229.926i	− 0.758831i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	358.336i	1.16722i	0.812035	+	0.583609i	$0.198360\pi$
$307$	358.336i	1.16722i	−0.812035	+	0.583609i	$0.801640\pi$
$308$	0	0
$309$	−109.082	−0.353016
$310$	0	0
$311$	364.787i	1.17295i	0.809968	+	0.586474i	$0.199485\pi$
$311$	364.787i	1.17295i	−0.809968	+	0.586474i	$0.800515\pi$
$312$	0	0
$313$	265.502	0.848248	0.424124	−	0.905604i	$-0.360582\pi$
$313$	265.502	0.848248	0.424124	+	0.905604i	$0.360582\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	459.580	1.44978	0.724890	−	0.688864i	$-0.241890\pi$
$317$	459.580	1.44978	0.724890	+	0.688864i	$0.241890\pi$
$318$	0	0
$319$	284.920i	0.893165i
$320$	0	0
$321$	302.164	0.941321
$322$	0	0
$323$	− 12.1297i	− 0.0375534i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	167.719i	0.512903i
$328$	0	0
$329$	170.991	0.519729
$330$	0	0
$331$	583.219i	1.76199i	0.473126	+	0.880995i	$0.343125\pi$
$331$	583.219i	1.76199i	−0.473126	+	0.880995i	$0.656875\pi$
$332$	0	0
$333$	180.748	0.542786
$334$	0	0
$335$	0	0
$336$	0	0
$337$	123.495	0.366455	0.183228	−	0.983071i	$-0.441346\pi$
$337$	123.495	0.366455	0.183228	+	0.983071i	$0.441346\pi$
$338$	0	0
$339$	− 365.605i	− 1.07848i
$340$	0	0
$341$	−561.994	−1.64808
$342$	0	0
$343$	341.117i	0.994512i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	487.859i	1.40593i	0.711222	+	0.702967i	$0.248142\pi$
$347$	487.859i	1.40593i	−0.711222	+	0.702967i	$0.751858\pi$
$348$	0	0
$349$	−12.8328	−0.0367702	−0.0183851	−	0.999831i	$-0.505852\pi$
$349$	−12.8328	−0.0367702	−0.0183851	+	0.999831i	$0.505852\pi$
$350$	0	0
$351$	28.1445i	0.0801837i
$352$	0	0
$353$	5.74457	0.0162736	0.00813678	−	0.999967i	$-0.497410\pi$
$353$	5.74457	0.0162736	0.00813678	+	0.999967i	$0.497410\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−10.5047	−0.0294248
$358$	0	0
$359$	44.8403i	0.124903i	0.998048	+	0.0624516i	$0.0198919\pi$
$359$	44.8403i	0.124903i	−0.998048	+	0.0624516i	$0.980108\pi$
$360$	0	0
$361$	287.663	0.796849
$362$	0	0
$363$	− 8.08111i	− 0.0222620i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 83.8543i	− 0.228486i	−0.993453	−	0.114243i	$-0.963556\pi$
$367$	− 83.8543i	− 0.228486i	0.993453	−	0.114243i	$-0.0364442\pi$
$368$	0	0
$369$	90.0000	0.243902
$370$	0	0
$371$	69.5770i	0.187539i
$372$	0	0
$373$	−110.413	−0.296014	−0.148007	−	0.988986i	$-0.547286\pi$
$373$	−110.413	−0.296014	−0.148007	+	0.988986i	$0.547286\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	137.666	0.365161
$378$	0	0
$379$	− 662.791i	− 1.74879i	−0.485216	−	0.874394i	$-0.661259\pi$
$379$	− 662.791i	− 1.74879i	0.485216	−	0.874394i	$-0.338741\pi$
$380$	0	0
$381$	77.2523	0.202762
$382$	0	0
$383$	84.7740i	0.221342i	0.993857	+	0.110671i	$0.0353000\pi$
$383$	84.7740i	0.221342i	−0.993857	+	0.110671i	$0.964700\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	113.736i	0.293892i
$388$	0	0
$389$	−379.240	−0.974910	−0.487455	−	0.873148i	$-0.662075\pi$
$389$	−379.240	−0.974910	−0.487455	+	0.873148i	$0.662075\pi$
$390$	0	0
$391$	34.0726i	0.0871423i
$392$	0	0
$393$	279.915	0.712252
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−94.2430	−0.237388	−0.118694	−	0.992931i	$-0.537871\pi$
$397$	−94.2430	−0.237388	−0.118694	+	0.992931i	$0.537871\pi$
$398$	0	0
$399$	63.5121i	0.159178i
$400$	0	0
$401$	246.158	0.613860	0.306930	−	0.951732i	$-0.400698\pi$
$401$	246.158	0.613860	0.306930	+	0.951732i	$0.400698\pi$
$402$	0	0
$403$	271.541i	0.673798i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	675.398i	1.65945i
$408$	0	0
$409$	584.663	1.42949	0.714746	−	0.699384i	$-0.246542\pi$
$409$	584.663	1.42949	0.714746	+	0.699384i	$0.246542\pi$
$410$	0	0
$411$	184.608i	0.449168i
$412$	0	0
$413$	−350.006	−0.847473
$414$	0	0
$415$	0	0
$416$	0	0
$417$	42.3344	0.101521
$418$	0	0
$419$	− 645.039i	− 1.53947i	−0.638362	−	0.769736i	$-0.720388\pi$
$419$	− 645.039i	− 1.53947i	0.638362	−	0.769736i	$-0.279612\pi$
$420$	0	0
$421$	468.991	1.11399	0.556996	−	0.830515i	$-0.311954\pi$
$421$	468.991	1.11399	0.556996	+	0.830515i	$0.311954\pi$
$422$	0	0
$423$	119.801i	0.283218i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	404.630i	0.947610i
$428$	0	0
$429$	−105.167	−0.245145
$430$	0	0
$431$	− 150.876i	− 0.350061i	−0.984563	−	0.175030i	$-0.943998\pi$
$431$	− 150.876i	− 0.350061i	0.984563	−	0.175030i	$-0.0560023\pi$
$432$	0	0
$433$	25.5016	0.0588950	0.0294475	−	0.999566i	$-0.490625\pi$
$433$	25.5016	0.0588950	0.0294475	+	0.999566i	$0.490625\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	206.006	0.471410
$438$	0	0
$439$	− 44.0681i	− 0.100383i	−0.998740	−	0.0501914i	$-0.984017\pi$
$439$	− 44.0681i	− 0.100383i	0.998740	−	0.0501914i	$-0.0159831\pi$
$440$	0	0
$441$	−91.9969	−0.208610
$442$	0	0
$443$	95.0644i	0.214592i	0.994227	+	0.107296i	$0.0342193\pi$
$443$	95.0644i	0.214592i	−0.994227	+	0.107296i	$0.965781\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	157.759i	0.352928i
$448$	0	0
$449$	−194.498	−0.433181	−0.216591	−	0.976263i	$-0.569494\pi$
$449$	−194.498	−0.433181	−0.216591	+	0.976263i	$0.569494\pi$
$450$	0	0
$451$	336.302i	0.745681i
$452$	0	0
$453$	436.328	0.963197
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−289.659	−0.633828	−0.316914	−	0.948454i	$-0.602647\pi$
$457$	−289.659	−0.633828	−0.316914	+	0.948454i	$0.602647\pi$
$458$	0	0
$459$	− 7.35987i	− 0.0160346i
$460$	0	0
$461$	−640.249	−1.38883	−0.694413	−	0.719576i	$-0.744336\pi$
$461$	−640.249	−1.38883	−0.694413	+	0.719576i	$0.744336\pi$
$462$	0	0
$463$	514.175i	1.11053i	0.831674	+	0.555265i	$0.187383\pi$
$463$	514.175i	1.11053i	−0.831674	+	0.555265i	$0.812617\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	69.3732i	0.148551i	0.997238	+	0.0742754i	$0.0236644\pi$
$467$	69.3732i	0.148551i	−0.997238	+	0.0742754i	$0.976336\pi$
$468$	0	0
$469$	554.991	1.18335
$470$	0	0
$471$	− 37.0943i	− 0.0787565i
$472$	0	0
$473$	−424.997	−0.898514
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−48.7477	−0.102196
$478$	0	0
$479$	502.579i	1.04923i	0.851341	+	0.524613i	$0.175790\pi$
$479$	502.579i	1.04923i	−0.851341	+	0.524613i	$0.824210\pi$
$480$	0	0
$481$	326.334	0.678450
$482$	0	0
$483$	− 178.407i	− 0.369372i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	580.368i	1.19172i	0.803088	+	0.595861i	$0.203189\pi$
$487$	580.368i	1.19172i	−0.803088	+	0.595861i	$0.796811\pi$
$488$	0	0
$489$	−25.4953	−0.0521377
$490$	0	0
$491$	619.348i	1.26140i	0.776027	+	0.630700i	$0.217232\pi$
$491$	619.348i	1.26140i	−0.776027	+	0.630700i	$0.782768\pi$
$492$	0	0
$493$	−36.0000	−0.0730223
$494$	0	0
$495$	0	0
$496$	0	0
$497$	437.981	0.881250
$498$	0	0
$499$	− 86.7044i	− 0.173756i	−0.996219	−	0.0868782i	$-0.972311\pi$
$499$	− 86.7044i	− 0.173756i	0.996219	−	0.0868782i	$-0.0276891\pi$
$500$	0	0
$501$	−136.328	−0.272112
$502$	0	0
$503$	442.542i	0.879805i	0.898046	+	0.439902i	$0.144987\pi$
$503$	442.542i	0.879805i	−0.898046	+	0.439902i	$0.855013\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 241.903i	− 0.477125i
$508$	0	0
$509$	861.246	1.69204	0.846018	−	0.533155i	$-0.178994\pi$
$509$	861.246	1.69204	0.846018	+	0.533155i	$0.178994\pi$
$510$	0	0
$511$	− 106.331i	− 0.208084i
$512$	0	0
$513$	−44.4984	−0.0867416
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−447.659	−0.865879
$518$	0	0
$519$	− 253.027i	− 0.487528i
$520$	0	0
$521$	−331.495	−0.636267	−0.318134	−	0.948046i	$-0.603056\pi$
$521$	−331.495	−0.636267	−0.318134	+	0.948046i	$0.603056\pi$
$522$	0	0
$523$	− 892.671i	− 1.70683i	−0.521234	−	0.853414i	$-0.674528\pi$
$523$	− 892.671i	− 1.70683i	0.521234	−	0.853414i	$-0.325472\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 71.0087i	− 0.134741i
$528$	0	0
$529$	−49.6749	−0.0939035
$530$	0	0
$531$	− 245.225i	− 0.461817i
$532$	0	0
$533$	162.492	0.304863
$534$	0	0
$535$	0	0
$536$	0	0
$537$	413.410	0.769851
$538$	0	0
$539$	− 343.764i	− 0.637781i
$540$	0	0
$541$	−243.495	−0.450084	−0.225042	−	0.974349i	$-0.572252\pi$
$541$	−243.495	−0.450084	−0.225042	+	0.974349i	$0.572252\pi$
$542$	0	0
$543$	199.475i	0.367358i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 130.864i	− 0.239239i	−0.992820	−	0.119619i	$-0.961833\pi$
$547$	− 130.864i	− 0.239239i	0.992820	−	0.119619i	$-0.0381674\pi$
$548$	0	0
$549$	−283.495	−0.516385
$550$	0	0
$551$	217.659i	0.395026i
$552$	0	0
$553$	−392.656	−0.710048
$554$	0	0
$555$	0	0
$556$	0	0
$557$	717.915	1.28890	0.644448	−	0.764648i	$-0.277087\pi$
$557$	717.915	1.28890	0.644448	+	0.764648i	$0.277087\pi$
$558$	0	0
$559$	205.347i	0.367347i
$560$	0	0
$561$	27.5016	0.0490224
$562$	0	0
$563$	− 777.482i	− 1.38096i	−0.723351	−	0.690481i	$-0.757399\pi$
$563$	− 777.482i	− 1.38096i	0.723351	−	0.690481i	$-0.242601\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	38.5368i	0.0679661i
$568$	0	0
$569$	−964.152	−1.69447	−0.847233	−	0.531221i	$-0.821733\pi$
$569$	−964.152	−1.69447	−0.847233	+	0.531221i	$0.821733\pi$
$570$	0	0
$571$	45.1352i	0.0790459i	0.999219	+	0.0395230i	$0.0125838\pi$
$571$	45.1352i	0.0790459i	−0.999219	+	0.0395230i	$0.987416\pi$
$572$	0	0
$573$	10.5047	0.0183327
$574$	0	0
$575$	0	0
$576$	0	0
$577$	286.000	0.495667	0.247834	−	0.968803i	$-0.420281\pi$
$577$	286.000	0.495667	0.247834	+	0.968803i	$0.420281\pi$
$578$	0	0
$579$	621.506i	1.07341i
$580$	0	0
$581$	376.997	0.648876
$582$	0	0
$583$	− 182.155i	− 0.312444i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 719.080i	− 1.22501i	−0.790467	−	0.612504i	$-0.790162\pi$
$587$	− 719.080i	− 1.22501i	0.790467	−	0.612504i	$-0.209838\pi$
$588$	0	0
$589$	−429.325	−0.728905
$590$	0	0
$591$	245.804i	0.415912i
$592$	0	0
$593$	360.590	0.608077	0.304039	−	0.952660i	$-0.401665\pi$
$593$	360.590	0.608077	0.304039	+	0.952660i	$0.401665\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−216.000	−0.361809
$598$	0	0
$599$	− 58.8790i	− 0.0982954i	−0.998792	−	0.0491477i	$-0.984349\pi$
$599$	− 58.8790i	− 0.0982954i	0.998792	−	0.0491477i	$-0.0156505\pi$
$600$	0	0
$601$	570.997	0.950078	0.475039	−	0.879965i	$-0.342434\pi$
$601$	570.997	0.950078	0.475039	+	0.879965i	$0.342434\pi$
$602$	0	0
$603$	388.843i	0.644847i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 667.073i	− 1.09897i	−0.835505	−	0.549483i	$-0.814825\pi$
$607$	− 667.073i	− 1.09897i	0.835505	−	0.549483i	$-0.185175\pi$
$608$	0	0
$609$	188.498	0.309521
$610$	0	0
$611$	216.297i	0.354005i
$612$	0	0
$613$	−293.246	−0.478379	−0.239189	−	0.970973i	$-0.576882\pi$
$613$	−293.246	−0.478379	−0.239189	+	0.970973i	$0.576882\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−900.906	−1.46014	−0.730069	−	0.683373i	$-0.760512\pi$
$617$	−900.906	−1.46014	−0.730069	+	0.683373i	$0.760512\pi$
$618$	0	0
$619$	604.277i	0.976214i	0.872784	+	0.488107i	$0.162312\pi$
$619$	604.277i	0.976214i	−0.872784	+	0.488107i	$0.837688\pi$
$620$	0	0
$621$	124.997	0.201283
$622$	0	0
$623$	− 89.2033i	− 0.143183i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 166.277i	− 0.265194i
$628$	0	0
$629$	−85.3375	−0.135672
$630$	0	0
$631$	− 162.529i	− 0.257573i	−0.991672	−	0.128787i	$-0.958892\pi$
$631$	− 162.529i	− 0.257573i	0.991672	−	0.128787i	$-0.0411082\pi$
$632$	0	0
$633$	177.994	0.281191
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−166.098	−0.260750
$638$	0	0
$639$	306.863i	0.480223i
$640$	0	0
$641$	−507.325	−0.791459	−0.395729	−	0.918367i	$-0.629508\pi$
$641$	−507.325	−0.791459	−0.395729	+	0.918367i	$0.629508\pi$
$642$	0	0
$643$	− 357.087i	− 0.555345i	−0.960676	−	0.277672i	$-0.910437\pi$
$643$	− 357.087i	− 0.555345i	0.960676	−	0.277672i	$-0.0895630\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 130.569i	− 0.201806i	−0.994896	−	0.100903i	$-0.967827\pi$
$647$	− 130.569i	− 0.201806i	0.994896	−	0.100903i	$-0.0321732\pi$
$648$	0	0
$649$	916.328	1.41191
$650$	0	0
$651$	371.806i	0.571131i
$652$	0	0
$653$	−1052.74	−1.61215	−0.806076	−	0.591812i	$-0.798413\pi$
$653$	−1052.74	−1.61215	−0.806076	+	0.591812i	$0.798413\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	74.4984	0.113392
$658$	0	0
$659$	0.442426i	0 0.000671360i	1.00000		0.000335680i	$0.000106850\pi$
$659$	0.442426i	0 0.000671360i	−1.00000		0.000335680i	$0.999893\pi$
$660$	0	0
$661$	−255.508	−0.386547	−0.193274	−	0.981145i	$-0.561911\pi$
$661$	−255.508	−0.386547	−0.193274	+	0.981145i	$0.561911\pi$
$662$	0	0
$663$	− 13.2880i	− 0.0200423i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 611.409i	− 0.916655i
$668$	0	0
$669$	68.9117	0.103007
$670$	0	0
$671$	− 1059.33i	− 1.57874i
$672$	0	0
$673$	532.492	0.791222	0.395611	−	0.918418i	$-0.370533\pi$
$673$	532.492	0.791222	0.395611	+	0.918418i	$0.370533\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	201.246	0.297262	0.148631	−	0.988893i	$-0.452513\pi$
$677$	201.246	0.297262	0.148631	+	0.988893i	$0.452513\pi$
$678$	0	0
$679$	59.9461i	0.0882859i
$680$	0	0
$681$	−24.6687	−0.0362243
$682$	0	0
$683$	− 619.586i	− 0.907154i	−0.891217	−	0.453577i	$-0.850148\pi$
$683$	− 619.586i	− 0.907154i	0.891217	−	0.453577i	$-0.149852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 77.9477i	− 0.113461i
$688$	0	0
$689$	−88.0124	−0.127739
$690$	0	0
$691$	− 835.133i	− 1.20859i	−0.796762	−	0.604293i	$-0.793456\pi$
$691$	− 835.133i	− 1.20859i	0.796762	−	0.604293i	$-0.206544\pi$
$692$	0	0
$693$	−144.000	−0.207792
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−42.4922	−0.0609645
$698$	0	0
$699$	− 682.281i	− 0.976081i
$700$	0	0
$701$	383.921	0.547676	0.273838	−	0.961776i	$-0.411707\pi$
$701$	383.921	0.547676	0.273838	+	0.961776i	$0.411707\pi$
$702$	0	0
$703$	515.958i	0.733938i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 568.408i	− 0.803971i
$708$	0	0
$709$	−1127.99	−1.59096	−0.795482	−	0.605977i	$-0.792782\pi$
$709$	−1127.99	−1.59096	−0.795482	+	0.605977i	$0.792782\pi$
$710$	0	0
$711$	− 275.107i	− 0.386929i
$712$	0	0
$713$	1205.98	1.69142
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−571.161	−0.796598
$718$	0	0
$719$	480.159i	0.667815i	0.942606	+	0.333907i	$0.108367\pi$
$719$	480.159i	0.667815i	−0.942606	+	0.333907i	$0.891633\pi$
$720$	0	0
$721$	−269.666	−0.374016
$722$	0	0
$723$	− 364.299i	− 0.503871i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	626.753i	0.862109i	0.902326	+	0.431054i	$0.141858\pi$
$727$	626.753i	0.862109i	−0.902326	+	0.431054i	$0.858142\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 53.6990i	− 0.0734596i
$732$	0	0
$733$	721.076	0.983732	0.491866	−	0.870671i	$-0.336315\pi$
$733$	721.076	0.983732	0.491866	+	0.870671i	$0.336315\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1452.98	−1.97149
$738$	0	0
$739$	358.245i	0.484770i	0.970180	+	0.242385i	$0.0779297\pi$
$739$	358.245i	0.484770i	−0.970180	+	0.242385i	$0.922070\pi$
$740$	0	0
$741$	−80.3406	−0.108422
$742$	0	0
$743$	847.059i	1.14005i	0.821627	+	0.570026i	$0.193067\pi$
$743$	847.059i	1.14005i	−0.821627	+	0.570026i	$0.806933\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	264.135i	0.353594i
$748$	0	0
$749$	746.991	0.997317
$750$	0	0
$751$	− 1184.81i	− 1.57765i	−0.614619	−	0.788824i	$-0.710690\pi$
$751$	− 1184.81i	− 1.57765i	0.614619	−	0.788824i	$-0.289310\pi$
$752$	0	0
$753$	−440.085	−0.584442
$754$	0	0
$755$	0	0
$756$	0	0
$757$	462.741	0.611283	0.305642	−	0.952147i	$-0.401129\pi$
$757$	462.741	0.611283	0.305642	+	0.952147i	$0.401129\pi$
$758$	0	0
$759$	467.075i	0.615382i
$760$	0	0
$761$	390.158	0.512691	0.256346	−	0.966585i	$-0.417482\pi$
$761$	390.158	0.512691	0.256346	+	0.966585i	$0.417482\pi$
$762$	0	0
$763$	414.625i	0.543414i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 442.745i	− 0.577243i
$768$	0	0
$769$	−579.981	−0.754202	−0.377101	−	0.926172i	$-0.623079\pi$
$769$	−579.981	−0.754202	−0.377101	+	0.926172i	$0.623079\pi$
$770$	0	0
$771$	− 705.655i	− 0.915247i
$772$	0	0
$773$	744.906	0.963655	0.481828	−	0.876266i	$-0.339973\pi$
$773$	744.906	0.963655	0.481828	+	0.876266i	$0.339973\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	446.833	0.575074
$778$	0	0
$779$	256.912i	0.329797i
$780$	0	0
$781$	−1146.65	−1.46818
$782$	0	0
$783$	132.068i	0.168669i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1393.82i	− 1.77105i	−0.464589	−	0.885526i	$-0.653798\pi$
$787$	− 1393.82i	− 1.77105i	0.464589	−	0.885526i	$-0.346202\pi$
$788$	0	0
$789$	−240.669	−0.305030
$790$	0	0
$791$	− 903.825i	− 1.14264i
$792$	0	0
$793$	−511.842	−0.645450
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−406.073	−0.509502	−0.254751	−	0.967007i	$-0.581993\pi$
$797$	−406.073	−0.509502	−0.254751	+	0.967007i	$0.581993\pi$
$798$	0	0
$799$	− 56.5624i	− 0.0707915i
$800$	0	0
$801$	62.4984	0.0780255
$802$	0	0
$803$	278.378i	0.346672i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 95.4049i	− 0.118222i
$808$	0	0
$809$	−123.325	−0.152441	−0.0762207	−	0.997091i	$-0.524285\pi$
$809$	−123.325	−0.152441	−0.0762207	+	0.997091i	$0.524285\pi$
$810$	0	0
$811$	− 183.404i	− 0.226146i	−0.993587	−	0.113073i	$-0.963931\pi$
$811$	− 183.404i	− 0.226146i	0.993587	−	0.113073i	$-0.0360694\pi$
$812$	0	0
$813$	−703.319	−0.865091
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−324.669	−0.397391
$818$	0	0
$819$	69.5770i	0.0849536i
$820$	0	0
$821$	746.912	0.909759	0.454879	−	0.890553i	$-0.349682\pi$
$821$	746.912	0.909759	0.454879	+	0.890553i	$0.349682\pi$
$822$	0	0
$823$	52.0982i	0.0633028i	0.999499	+	0.0316514i	$0.0100766\pi$
$823$	52.0982i	0.0633028i	−0.999499	+	0.0316514i	$0.989923\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1327.69i	1.60543i	0.596360	+	0.802717i	$0.296613\pi$
$827$	1327.69i	1.60543i	−0.596360	+	0.802717i	$0.703387\pi$
$828$	0	0
$829$	−798.486	−0.963192	−0.481596	−	0.876393i	$-0.659943\pi$
$829$	−798.486	−0.963192	−0.481596	+	0.876393i	$0.659943\pi$
$830$	0	0
$831$	− 607.229i	− 0.730720i
$832$	0	0
$833$	43.4350	0.0521429
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−260.498	−0.311229
$838$	0	0
$839$	− 608.138i	− 0.724836i	−0.932016	−	0.362418i	$-0.881951\pi$
$839$	− 608.138i	− 0.724836i	0.932016	−	0.362418i	$-0.118049\pi$
$840$	0	0
$841$	−195.006	−0.231874
$842$	0	0
$843$	− 791.826i	− 0.939295i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 19.9776i	− 0.0235863i
$848$	0	0
$849$	588.827	0.693553
$850$	0	0
$851$	− 1449.34i	− 1.70310i
$852$	0	0
$853$	661.404	0.775386	0.387693	−	0.921789i	$-0.373272\pi$
$853$	661.404	0.775386	0.387693	+	0.921789i	$0.373272\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	291.070	0.339638	0.169819	−	0.985475i	$-0.445682\pi$
$857$	291.070	0.339638	0.169819	+	0.985475i	$0.445682\pi$
$858$	0	0
$859$	− 1024.97i	− 1.19321i	−0.802535	−	0.596604i	$-0.796516\pi$
$859$	− 1024.97i	− 1.19321i	0.802535	−	0.596604i	$-0.203484\pi$
$860$	0	0
$861$	222.492	0.258411
$862$	0	0
$863$	− 301.956i	− 0.349891i	−0.984578	−	0.174946i	$-0.944025\pi$
$863$	− 301.956i	− 0.349891i	0.984578	−	0.174946i	$-0.0559749\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 497.088i	− 0.573342i
$868$	0	0
$869$	1027.99	1.18295
$870$	0	0
$871$	702.044i	0.806020i
$872$	0	0
$873$	−42.0000	−0.0481100
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1358.23	1.54872	0.774362	−	0.632743i	$-0.218071\pi$
$877$	1358.23	1.54872	0.774362	+	0.632743i	$0.218071\pi$
$878$	0	0
$879$	− 996.640i	− 1.13383i
$880$	0	0
$881$	1176.49	1.33541	0.667703	−	0.744428i	$-0.267278\pi$
$881$	1176.49	1.33541	0.667703	+	0.744428i	$0.267278\pi$
$882$	0	0
$883$	506.418i	0.573520i	0.958002	+	0.286760i	$0.0925783\pi$
$883$	506.418i	0.573520i	−0.958002	+	0.286760i	$0.907422\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1497.24i	− 1.68798i	−0.536355	−	0.843992i	$-0.680199\pi$
$887$	− 1497.24i	− 1.68798i	0.536355	−	0.843992i	$-0.319801\pi$
$888$	0	0
$889$	190.978	0.214824
$890$	0	0
$891$	− 100.891i	− 0.113233i
$892$	0	0
$893$	−341.981	−0.382958
$894$	0	0
$895$	0	0
$896$	0	0
$897$	225.678	0.251592
$898$	0	0
$899$	1274.20i	1.41735i
$900$	0	0
$901$	23.0155	0.0255444
$902$	0	0
$903$	281.171i	0.311375i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 199.374i	− 0.219816i	−0.993942	−	0.109908i	$-0.964944\pi$
$907$	− 199.374i	− 0.219816i	0.993942	−	0.109908i	$-0.0350557\pi$
$908$	0	0
$909$	398.243	0.438111
$910$	0	0
$911$	− 1385.82i	− 1.52121i	−0.649215	−	0.760605i	$-0.724902\pi$
$911$	− 1385.82i	− 1.52121i	0.649215	−	0.760605i	$-0.275098\pi$
$912$	0	0
$913$	−986.991	−1.08104
$914$	0	0
$915$	0	0
$916$	0	0
$917$	691.988	0.754621
$918$	0	0
$919$	483.135i	0.525718i	0.964834	+	0.262859i	$0.0846654\pi$
$919$	483.135i	0.525718i	−0.964834	+	0.262859i	$0.915335\pi$
$920$	0	0
$921$	−620.656	−0.673894
$922$	0	0
$923$	554.031i	0.600250i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 188.936i	− 0.203814i
$928$	0	0
$929$	−1149.15	−1.23697	−0.618487	−	0.785795i	$-0.712254\pi$
$929$	−1149.15	−1.23697	−0.618487	+	0.785795i	$0.712254\pi$
$930$	0	0
$931$	− 262.612i	− 0.282075i
$932$	0	0
$933$	−631.830	−0.677202
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1349.00	1.43970	0.719852	−	0.694127i	$-0.244210\pi$
$937$	1349.00	1.43970	0.719852	+	0.694127i	$0.244210\pi$
$938$	0	0
$939$	459.862i	0.489736i
$940$	0	0
$941$	1160.07	1.23280	0.616401	−	0.787432i	$-0.288590\pi$
$941$	1160.07	1.23280	0.616401	+	0.787432i	$0.288590\pi$
$942$	0	0
$943$	− 721.670i	− 0.765292i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 620.063i	− 0.654766i	−0.944892	−	0.327383i	$-0.893833\pi$
$947$	− 620.063i	− 0.654766i	0.944892	−	0.327383i	$-0.106167\pi$
$948$	0	0
$949$	134.505	0.141733
$950$	0	0
$951$	796.017i	0.837031i
$952$	0	0
$953$	545.587	0.572494	0.286247	−	0.958156i	$-0.407592\pi$
$953$	545.587	0.572494	0.286247	+	0.958156i	$0.407592\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−493.495	−0.515669
$958$	0	0
$959$	456.377i	0.475888i
$960$	0	0
$961$	−1552.31	−1.61531
$962$	0	0
$963$	523.364i	0.543472i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 181.621i	− 0.187819i	−0.995581	−	0.0939097i	$-0.970064\pi$
$967$	− 181.621i	− 0.187819i	0.995581	−	0.0939097i	$-0.0299365\pi$
$968$	0	0
$969$	21.0093	0.0216814
$970$	0	0
$971$	337.222i	0.347293i	0.984808	+	0.173647i	$0.0555550\pi$
$971$	337.222i	0.347293i	−0.984808	+	0.173647i	$0.944445\pi$
$972$	0	0
$973$	104.656	0.107560
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−940.249	−0.962384	−0.481192	−	0.876615i	$-0.659796\pi$
$977$	−940.249	−0.962384	−0.481192	+	0.876615i	$0.659796\pi$
$978$	0	0
$979$	233.537i	0.238547i
$980$	0	0
$981$	−290.498	−0.296125
$982$	0	0
$983$	− 1651.87i	− 1.68043i	−0.542250	−	0.840217i	$-0.682427\pi$
$983$	− 1651.87i	− 1.68043i	0.542250	−	0.840217i	$-0.317573\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	296.165i	0.300065i
$988$	0	0
$989$	912.000	0.922144
$990$	0	0
$991$	− 970.903i	− 0.979720i	−0.871801	−	0.489860i	$-0.837048\pi$
$991$	− 970.903i	− 0.979720i	0.871801	−	0.489860i	$-0.162952\pi$
$992$	0	0
$993$	−1010.16	−1.01729
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1285.08	−1.28894	−0.644471	−	0.764628i	$-0.722922\pi$
$997$	−1285.08	−1.28894	−0.644471	+	0.764628i	$0.722922\pi$
$998$	0	0
$999$	313.064i	0.313378i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.e.n.751.4		4
3.2	odd	2		3600.3.e.bh.3151.3		4
4.3	odd	2	inner	1200.3.e.n.751.1		4
5.2	odd	4		1200.3.j.f.799.5		8
5.3	odd	4		1200.3.j.f.799.3		8
5.4	even	2		240.3.e.a.31.2	✓	4
12.11	even	2		3600.3.e.bh.3151.2		4
15.2	even	4		3600.3.j.l.1999.4		8
15.8	even	4		3600.3.j.l.1999.6		8
15.14	odd	2		720.3.e.a.271.1		4
20.3	even	4		1200.3.j.f.799.6		8
20.7	even	4		1200.3.j.f.799.4		8
20.19	odd	2		240.3.e.a.31.4	yes	4
40.19	odd	2		960.3.e.b.511.1		4
40.29	even	2		960.3.e.b.511.3		4
60.23	odd	4		3600.3.j.l.1999.3		8
60.47	odd	4		3600.3.j.l.1999.5		8
60.59	even	2		720.3.e.a.271.2		4
120.29	odd	2		2880.3.e.f.2431.3		4
120.59	even	2		2880.3.e.f.2431.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.3.e.a.31.2	✓	4	5.4	even	2
240.3.e.a.31.4	yes	4	20.19	odd	2
720.3.e.a.271.1		4	15.14	odd	2
720.3.e.a.271.2		4	60.59	even	2
960.3.e.b.511.1		4	40.19	odd	2
960.3.e.b.511.3		4	40.29	even	2
1200.3.e.n.751.1		4	4.3	odd	2	inner
1200.3.e.n.751.4		4	1.1	even	1	trivial
1200.3.j.f.799.3		8	5.3	odd	4
1200.3.j.f.799.4		8	20.7	even	4
1200.3.j.f.799.5		8	5.2	odd	4
1200.3.j.f.799.6		8	20.3	even	4
2880.3.e.f.2431.3		4	120.29	odd	2
2880.3.e.f.2431.4		4	120.59	even	2
3600.3.e.bh.3151.2		4	12.11	even	2
3600.3.e.bh.3151.3		4	3.2	odd	2
3600.3.j.l.1999.3		8	60.23	odd	4
3600.3.j.l.1999.4		8	15.2	even	4
3600.3.j.l.1999.5		8	60.47	odd	4
3600.3.j.l.1999.6		8	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +4.28187i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+1.73205i q^{3} +4.28187i q^{7} -3.00000 q^{9} -11.2101i q^{11} -5.41641 q^{13} +1.41641 q^{17} -8.56373i q^{19} -7.41641 q^{21} +24.0557i q^{23} -5.19615i q^{27} -25.4164 q^{29} -50.1329i q^{31} +19.4164 q^{33} -60.2492 q^{37} -9.38149i q^{39} -30.0000 q^{41} -37.9121i q^{43} -39.9337i q^{47} +30.6656 q^{49} +2.45329i q^{51} +16.2492 q^{53} +14.8328 q^{57} +81.7415i q^{59} +94.4984 q^{61} -12.8456i q^{63} -129.614i q^{67} -41.6656 q^{69} -102.288i q^{71} -24.8328 q^{73} +48.0000 q^{77} +91.7022i q^{79} +9.00000 q^{81} -88.0450i q^{83} -44.0225i q^{87} -20.8328 q^{89} -23.1923i q^{91} +86.8328 q^{93} +14.0000 q^{97} +33.6302i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} + 32 q^{13} - 48 q^{17} + 24 q^{21} - 48 q^{29} + 24 q^{33} - 80 q^{37} - 120 q^{41} - 92 q^{49} - 96 q^{53} - 48 q^{57} + 56 q^{61} + 48 q^{69} + 8 q^{73} + 192 q^{77} + 36 q^{81} + 24 q^{89} + 240 q^{93} + 56 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 + 32 * q^13 - 48 * q^17 + 24 * q^21 - 48 * q^29 + 24 * q^33 - 80 * q^37 - 120 * q^41 - 92 * q^49 - 96 * q^53 - 48 * q^57 + 56 * q^61 + 48 * q^69 + 8 * q^73 + 192 * q^77 + 36 * q^81 + 24 * q^89 + 240 * q^93 + 56 * q^97

Introduction

L-functions

Modular forms

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Representations

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Database

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