LMFDB - Embedded newform 1200.3.e.k.751.1

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:	$\Q(\sqrt{-3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 121 $ x^4 + 11*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.1
Root			$1.65831 - 2.87228i$ of defining polynomial
Character	$\chi$	$=$	1200.751
Dual form			1200.3.e.k.751.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -13.2212i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} -13.2212i q^{7} -3.00000 q^{9} +11.4891i q^{11} -1.00000 q^{13} +31.8997 q^{17} -13.2212i q^{19} -22.8997 q^{21} -32.2737i q^{23} +5.19615i q^{27} -4.10025 q^{29} -37.4699i q^{31} +19.8997 q^{33} -17.7995 q^{37} +1.73205i q^{39} +53.6992 q^{41} -24.8839i q^{43} +35.5642i q^{47} -125.799 q^{49} -55.2520i q^{51} -89.6992 q^{53} -22.8997 q^{57} -6.00502i q^{59} -76.7995 q^{61} +39.6635i q^{63} +108.946i q^{67} -55.8997 q^{69} +1.09682i q^{71} -30.2005 q^{73} +151.900 q^{77} -2.54092i q^{79} +9.00000 q^{81} -72.7461i q^{83} +7.10184i q^{87} -96.0000 q^{89} +13.2212i q^{91} -64.8997 q^{93} +114.599 q^{97} -34.4674i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9}+O(q^{10}) $ 4 * q - 12 * q^9	$ 4 q - 12 q^{9} - 4 q^{13} + 48 q^{17} - 12 q^{21} - 96 q^{29} + 88 q^{37} - 24 q^{41} - 344 q^{49} - 120 q^{53} - 12 q^{57} - 148 q^{61} - 144 q^{69} - 280 q^{73} + 528 q^{77} + 36 q^{81} - 384 q^{89} - 180 q^{93} + 140 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 - 4 * q^13 + 48 * q^17 - 12 * q^21 - 96 * q^29 + 88 * q^37 - 24 * q^41 - 344 * q^49 - 120 * q^53 - 12 * q^57 - 148 * q^61 - 144 * q^69 - 280 * q^73 + 528 * q^77 + 36 * q^81 - 384 * q^89 - 180 * q^93 + 140 * 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
$7$	− 13.2212i	− 1.88874i	−0.328886	−	0.944370i	$-0.606673\pi$
$7$	− 13.2212i	− 1.88874i	0.328886	−	0.944370i	$-0.393327\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	11.4891i	1.04447i	0.852803	+	0.522233i	$0.174901\pi$
$11$	11.4891i	1.04447i	−0.852803	+	0.522233i	$0.825099\pi$
$12$	0	0
$13$	−1.00000	−0.0769231	−0.0384615	−	0.999260i	$-0.512246\pi$
$13$	−1.00000	−0.0769231	−0.0384615	+	0.999260i	$0.512246\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	31.8997	1.87646	0.938228	−	0.346018i	$-0.112466\pi$
$17$	31.8997	1.87646	0.938228	+	0.346018i	$0.112466\pi$
$18$	0	0
$19$	− 13.2212i	− 0.695851i	−0.937522	−	0.347926i	$-0.886886\pi$
$19$	− 13.2212i	− 0.695851i	0.937522	−	0.347926i	$-0.113114\pi$
$20$	0	0
$21$	−22.8997	−1.09046
$22$	0	0
$23$	− 32.2737i	− 1.40321i	−0.712568	−	0.701603i	$-0.752468\pi$
$23$	− 32.2737i	− 1.40321i	0.712568	−	0.701603i	$-0.247532\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−4.10025	−0.141388	−0.0706940	−	0.997498i	$-0.522521\pi$
$29$	−4.10025	−0.141388	−0.0706940	+	0.997498i	$0.522521\pi$
$30$	0	0
$31$	− 37.4699i	− 1.20871i	−0.796717	−	0.604353i	$-0.793432\pi$
$31$	− 37.4699i	− 1.20871i	0.796717	−	0.604353i	$-0.206568\pi$
$32$	0	0
$33$	19.8997	0.603023
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−17.7995	−0.481067	−0.240534	−	0.970641i	$-0.577322\pi$
$37$	−17.7995	−0.481067	−0.240534	+	0.970641i	$0.577322\pi$
$38$	0	0
$39$	1.73205i	0.0444116i
$40$	0	0
$41$	53.6992	1.30974	0.654869	−	0.755743i	$-0.272724\pi$
$41$	53.6992	1.30974	0.654869	+	0.755743i	$0.272724\pi$
$42$	0	0
$43$	− 24.8839i	− 0.578696i	−0.957224	−	0.289348i	$-0.906561\pi$
$43$	− 24.8839i	− 0.578696i	0.957224	−	0.289348i	$-0.0934385\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	35.5642i	0.756685i	0.925666	+	0.378343i	$0.123506\pi$
$47$	35.5642i	0.756685i	−0.925666	+	0.378343i	$0.876494\pi$
$48$	0	0
$49$	−125.799	−2.56734
$50$	0	0
$51$	− 55.2520i	− 1.08337i
$52$	0	0
$53$	−89.6992	−1.69244	−0.846219	−	0.532835i	$-0.821127\pi$
$53$	−89.6992	−1.69244	−0.846219	+	0.532835i	$0.821127\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−22.8997	−0.401750
$58$	0	0
$59$	− 6.00502i	− 0.101780i	−0.998704	−	0.0508900i	$-0.983794\pi$
$59$	− 6.00502i	− 0.101780i	0.998704	−	0.0508900i	$-0.0162058\pi$
$60$	0	0
$61$	−76.7995	−1.25901	−0.629504	−	0.776997i	$-0.716742\pi$
$61$	−76.7995	−1.25901	−0.629504	+	0.776997i	$0.716742\pi$
$62$	0	0
$63$	39.6635i	0.629580i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	108.946i	1.62605i	0.582227	+	0.813027i	$0.302182\pi$
$67$	108.946i	1.62605i	−0.582227	+	0.813027i	$0.697818\pi$
$68$	0	0
$69$	−55.8997	−0.810141
$70$	0	0
$71$	1.09682i	0.0154482i	0.999970	+	0.00772409i	$0.00245868\pi$
$71$	1.09682i	0.0154482i	−0.999970	+	0.00772409i	$0.997541\pi$
$72$	0	0
$73$	−30.2005	−0.413706	−0.206853	−	0.978372i	$-0.566322\pi$
$73$	−30.2005	−0.413706	−0.206853	+	0.978372i	$0.566322\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	151.900	1.97272
$78$	0	0
$79$	− 2.54092i	− 0.0321636i	−0.999871	−	0.0160818i	$-0.994881\pi$
$79$	− 2.54092i	− 0.0321636i	0.999871	−	0.0160818i	$-0.00511921\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 72.7461i	− 0.876459i	−0.898863	−	0.438230i	$-0.855606\pi$
$83$	− 72.7461i	− 0.876459i	0.898863	−	0.438230i	$-0.144394\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.10184i	0.0816304i
$88$	0	0
$89$	−96.0000	−1.07865	−0.539326	−	0.842097i	$-0.681321\pi$
$89$	−96.0000	−1.07865	−0.539326	+	0.842097i	$0.681321\pi$
$90$	0	0
$91$	13.2212i	0.145288i
$92$	0	0
$93$	−64.8997	−0.697847
$94$	0	0
$95$	0	0
$96$	0	0
$97$	114.599	1.18143	0.590716	−	0.806879i	$-0.298845\pi$
$97$	114.599	1.18143	0.590716	+	0.806879i	$0.298845\pi$
$98$	0	0
$99$	− 34.4674i	− 0.348155i
$100$	0	0
$101$	−133.900	−1.32574	−0.662870	−	0.748734i	$-0.730662\pi$
$101$	−133.900	−1.32574	−0.662870	+	0.748734i	$0.730662\pi$
$102$	0	0
$103$	11.6628i	0.113231i	0.998396	+	0.0566154i	$0.0180309\pi$
$103$	11.6628i	0.113231i	−0.998396	+	0.0566154i	$0.981969\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 157.557i	− 1.47250i	−0.676711	−	0.736249i	$-0.736595\pi$
$107$	− 157.557i	− 1.47250i	0.676711	−	0.736249i	$-0.263405\pi$
$108$	0	0
$109$	24.3985	0.223839	0.111920	−	0.993717i	$-0.464300\pi$
$109$	24.3985	0.223839	0.111920	+	0.993717i	$0.464300\pi$
$110$	0	0
$111$	30.8296i	0.277744i
$112$	0	0
$113$	−28.4010	−0.251336	−0.125668	−	0.992072i	$-0.540107\pi$
$113$	−28.4010	−0.251336	−0.125668	+	0.992072i	$0.540107\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.00000	0.0256410
$118$	0	0
$119$	− 421.752i	− 3.54414i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	− 93.0098i	− 0.756177i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.2037i	0.111840i	0.998435	+	0.0559200i	$0.0178092\pi$
$127$	14.2037i	0.111840i	−0.998435	+	0.0559200i	$0.982191\pi$
$128$	0	0
$129$	−43.1003	−0.334110
$130$	0	0
$131$	188.158i	1.43632i	0.695876	+	0.718161i	$0.255016\pi$
$131$	188.158i	1.43632i	−0.695876	+	0.718161i	$0.744984\pi$
$132$	0	0
$133$	−174.799	−1.31428
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−41.3985	−0.302179	−0.151089	−	0.988520i	$-0.548278\pi$
$137$	−41.3985	−0.302179	−0.151089	+	0.988520i	$0.548278\pi$
$138$	0	0
$139$	− 170.317i	− 1.22530i	−0.790354	−	0.612651i	$-0.790103\pi$
$139$	− 170.317i	− 1.22530i	0.790354	−	0.612651i	$-0.209897\pi$
$140$	0	0
$141$	61.5990	0.436872
$142$	0	0
$143$	− 11.4891i	− 0.0803435i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	217.891i	1.48225i
$148$	0	0
$149$	−161.398	−1.08321	−0.541606	−	0.840633i	$-0.682183\pi$
$149$	−161.398	−1.08321	−0.541606	+	0.840633i	$0.682183\pi$
$150$	0	0
$151$	− 179.498i	− 1.18873i	−0.804196	−	0.594364i	$-0.797404\pi$
$151$	− 179.498i	− 1.18873i	0.804196	−	0.594364i	$-0.202596\pi$
$152$	0	0
$153$	−95.6992	−0.625485
$154$	0	0
$155$	0	0
$156$	0	0
$157$	180.398	1.14903	0.574517	−	0.818492i	$-0.305190\pi$
$157$	180.398	1.14903	0.574517	+	0.818492i	$0.305190\pi$
$158$	0	0
$159$	155.364i	0.977130i
$160$	0	0
$161$	−426.697	−2.65029
$162$	0	0
$163$	43.1276i	0.264587i	0.991211	+	0.132293i	$0.0422341\pi$
$163$	43.1276i	0.264587i	−0.991211	+	0.132293i	$0.957766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 137.815i	− 0.825237i	−0.910904	−	0.412618i	$-0.864614\pi$
$167$	− 137.815i	− 0.825237i	0.910904	−	0.412618i	$-0.135386\pi$
$168$	0	0
$169$	−168.000	−0.994083
$170$	0	0
$171$	39.6635i	0.231950i
$172$	0	0
$173$	275.098	1.59016	0.795080	−	0.606504i	$-0.207429\pi$
$173$	275.098	1.59016	0.795080	+	0.606504i	$0.207429\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.4010	−0.0587627
$178$	0	0
$179$	180.015i	1.00567i	0.864383	+	0.502834i	$0.167709\pi$
$179$	180.015i	1.00567i	−0.864383	+	0.502834i	$0.832291\pi$
$180$	0	0
$181$	227.797	1.25855	0.629273	−	0.777184i	$-0.283353\pi$
$181$	227.797	1.25855	0.629273	+	0.777184i	$0.283353\pi$
$182$	0	0
$183$	133.021i	0.726889i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	366.500i	1.95989i
$188$	0	0
$189$	68.6992	0.363488
$190$	0	0
$191$	− 282.841i	− 1.48084i	−0.672143	−	0.740421i	$-0.734626\pi$
$191$	− 282.841i	− 1.48084i	0.672143	−	0.740421i	$-0.265374\pi$
$192$	0	0
$193$	−269.997	−1.39895	−0.699475	−	0.714657i	$-0.746583\pi$
$193$	−269.997	−1.39895	−0.699475	+	0.714657i	$0.746583\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	102.602	0.520820	0.260410	−	0.965498i	$-0.416142\pi$
$197$	102.602	0.520820	0.260410	+	0.965498i	$0.416142\pi$
$198$	0	0
$199$	274.181i	1.37779i	0.724860	+	0.688896i	$0.241904\pi$
$199$	274.181i	1.37779i	−0.724860	+	0.688896i	$0.758096\pi$
$200$	0	0
$201$	188.699	0.938802
$202$	0	0
$203$	54.2101i	0.267045i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	96.8212i	0.467735i
$208$	0	0
$209$	151.900	0.726793
$210$	0	0
$211$	108.946i	0.516330i	0.966101	+	0.258165i	$0.0831178\pi$
$211$	108.946i	0.516330i	−0.966101	+	0.258165i	$0.916882\pi$
$212$	0	0
$213$	1.89975	0.00891901
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−495.396	−2.28293
$218$	0	0
$219$	52.3088i	0.238853i
$220$	0	0
$221$	−31.8997	−0.144343
$222$	0	0
$223$	10.1044i	0.0453110i	0.999743	+	0.0226555i	$0.00721209\pi$
$223$	10.1044i	0.0453110i	−0.999743	+	0.0226555i	$0.992788\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 250.512i	− 1.10358i	−0.833984	−	0.551789i	$-0.813945\pi$
$227$	− 250.512i	− 1.10358i	0.833984	−	0.551789i	$-0.186055\pi$
$228$	0	0
$229$	−345.195	−1.50740	−0.753702	−	0.657216i	$-0.771734\pi$
$229$	−345.195	−1.50740	−0.753702	+	0.657216i	$0.771734\pi$
$230$	0	0
$231$	− 263.098i	− 1.13895i
$232$	0	0
$233$	−188.897	−0.810718	−0.405359	−	0.914158i	$-0.632853\pi$
$233$	−188.897	−0.810718	−0.405359	+	0.914158i	$0.632853\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.40101	−0.0185696
$238$	0	0
$239$	− 374.178i	− 1.56560i	−0.622275	−	0.782799i	$-0.713791\pi$
$239$	− 374.178i	− 1.56560i	0.622275	−	0.782799i	$-0.286209\pi$
$240$	0	0
$241$	193.000	0.800830	0.400415	−	0.916334i	$-0.368866\pi$
$241$	193.000	0.800830	0.400415	+	0.916334i	$0.368866\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	13.2212i	0.0535270i
$248$	0	0
$249$	−126.000	−0.506024
$250$	0	0
$251$	351.145i	1.39898i	0.714641	+	0.699491i	$0.246590\pi$
$251$	351.145i	1.39898i	−0.714641	+	0.699491i	$0.753410\pi$
$252$	0	0
$253$	370.797	1.46560
$254$	0	0
$255$	0	0
$256$	0	0
$257$	270.997	1.05446	0.527232	−	0.849721i	$-0.323230\pi$
$257$	270.997	1.05446	0.527232	+	0.849721i	$0.323230\pi$
$258$	0	0
$259$	235.330i	0.908611i
$260$	0	0
$261$	12.3008	0.0471293
$262$	0	0
$263$	− 311.824i	− 1.18564i	−0.805334	−	0.592822i	$-0.798014\pi$
$263$	− 311.824i	− 1.18564i	0.805334	−	0.592822i	$-0.201986\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	166.277i	0.622760i
$268$	0	0
$269$	383.098	1.42416	0.712078	−	0.702101i	$-0.247754\pi$
$269$	383.098	1.42416	0.712078	+	0.702101i	$0.247754\pi$
$270$	0	0
$271$	17.2018i	0.0634754i	0.999496	+	0.0317377i	$0.0101041\pi$
$271$	17.2018i	0.0634754i	−0.999496	+	0.0317377i	$0.989896\pi$
$272$	0	0
$273$	22.8997	0.0838819
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−181.000	−0.653430	−0.326715	−	0.945123i	$-0.605942\pi$
$277$	−181.000	−0.653430	−0.326715	+	0.945123i	$0.605942\pi$
$278$	0	0
$279$	112.410i	0.402902i
$280$	0	0
$281$	218.897	0.778994	0.389497	−	0.921028i	$-0.372649\pi$
$281$	218.897	0.778994	0.389497	+	0.921028i	$0.372649\pi$
$282$	0	0
$283$	− 389.766i	− 1.37727i	−0.725110	−	0.688633i	$-0.758211\pi$
$283$	− 389.766i	− 1.37727i	0.725110	−	0.688633i	$-0.241789\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 709.967i	− 2.47375i
$288$	0	0
$289$	728.594	2.52109
$290$	0	0
$291$	− 198.491i	− 0.682101i
$292$	0	0
$293$	481.599	1.64368	0.821841	−	0.569717i	$-0.192947\pi$
$293$	481.599	1.64368	0.821841	+	0.569717i	$0.192947\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−59.6992	−0.201008
$298$	0	0
$299$	32.2737i	0.107939i
$300$	0	0
$301$	−328.995	−1.09301
$302$	0	0
$303$	231.921i	0.765416i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 215.867i	− 0.703149i	−0.936160	−	0.351575i	$-0.885646\pi$
$307$	− 215.867i	− 0.703149i	0.936160	−	0.351575i	$-0.114354\pi$
$308$	0	0
$309$	20.2005	0.0653738
$310$	0	0
$311$	359.453i	1.15580i	0.816108	+	0.577899i	$0.196127\pi$
$311$	359.453i	1.15580i	−0.816108	+	0.577899i	$0.803873\pi$
$312$	0	0
$313$	540.398	1.72651	0.863256	−	0.504766i	$-0.168421\pi$
$313$	540.398	1.72651	0.863256	+	0.504766i	$0.168421\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	362.596	1.14384	0.571919	−	0.820310i	$-0.306199\pi$
$317$	362.596	1.14384	0.571919	+	0.820310i	$0.306199\pi$
$318$	0	0
$319$	− 47.1083i	− 0.147675i
$320$	0	0
$321$	−272.897	−0.850147
$322$	0	0
$323$	− 421.752i	− 1.30573i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 42.2594i	− 0.129234i
$328$	0	0
$329$	470.201	1.42918
$330$	0	0
$331$	− 183.479i	− 0.554316i	−0.960824	−	0.277158i	$-0.910607\pi$
$331$	− 183.479i	− 0.554316i	0.960824	−	0.277158i	$-0.0893926\pi$
$332$	0	0
$333$	53.3985	0.160356
$334$	0	0
$335$	0	0
$336$	0	0
$337$	93.1955	0.276544	0.138272	−	0.990394i	$-0.455845\pi$
$337$	93.1955	0.276544	0.138272	+	0.990394i	$0.455845\pi$
$338$	0	0
$339$	49.1920i	0.145109i
$340$	0	0
$341$	430.496	1.26245
$342$	0	0
$343$	1015.38i	2.96029i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 218.238i	− 0.628929i	−0.949269	−	0.314465i	$-0.898175\pi$
$347$	− 218.238i	− 0.628929i	0.949269	−	0.314465i	$-0.101825\pi$
$348$	0	0
$349$	78.4010	0.224645	0.112322	−	0.993672i	$-0.464171\pi$
$349$	78.4010	0.224645	0.112322	+	0.993672i	$0.464171\pi$
$350$	0	0
$351$	− 5.19615i	− 0.0148039i
$352$	0	0
$353$	−43.2982	−0.122658	−0.0613290	−	0.998118i	$-0.519534\pi$
$353$	−43.2982	−0.122658	−0.0613290	+	0.998118i	$0.519534\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−730.496	−2.04621
$358$	0	0
$359$	60.2152i	0.167730i	0.996477	+	0.0838651i	$0.0267265\pi$
$359$	60.2152i	0.167730i	−0.996477	+	0.0838651i	$0.973274\pi$
$360$	0	0
$361$	186.201	0.515791
$362$	0	0
$363$	19.0526i	0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 247.857i	− 0.675359i	−0.941261	−	0.337680i	$-0.890358\pi$
$367$	− 247.857i	− 0.675359i	0.941261	−	0.337680i	$-0.109642\pi$
$368$	0	0
$369$	−161.098	−0.436579
$370$	0	0
$371$	1185.93i	3.19658i
$372$	0	0
$373$	−533.396	−1.43002	−0.715008	−	0.699116i	$-0.753577\pi$
$373$	−533.396	−1.43002	−0.715008	+	0.699116i	$0.753577\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.10025	0.0108760
$378$	0	0
$379$	712.732i	1.88056i	0.340401	+	0.940280i	$0.389437\pi$
$379$	712.732i	1.88056i	−0.340401	+	0.940280i	$0.610563\pi$
$380$	0	0
$381$	24.6015	0.0645709
$382$	0	0
$383$	268.006i	0.699755i	0.936795	+	0.349878i	$0.113777\pi$
$383$	268.006i	0.699755i	−0.936795	+	0.349878i	$0.886223\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	74.6518i	0.192899i
$388$	0	0
$389$	433.900	1.11542	0.557712	−	0.830035i	$-0.311679\pi$
$389$	433.900	1.11542	0.557712	+	0.830035i	$0.311679\pi$
$390$	0	0
$391$	− 1029.52i	− 2.63305i
$392$	0	0
$393$	325.900	0.829261
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−669.797	−1.68715	−0.843573	−	0.537014i	$-0.819552\pi$
$397$	−669.797	−1.68715	−0.843573	+	0.537014i	$0.819552\pi$
$398$	0	0
$399$	302.762i	0.758801i
$400$	0	0
$401$	123.895	0.308964	0.154482	−	0.987996i	$-0.450629\pi$
$401$	123.895	0.308964	0.154482	+	0.987996i	$0.450629\pi$
$402$	0	0
$403$	37.4699i	0.0929774i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 204.501i	− 0.502459i
$408$	0	0
$409$	−451.992	−1.10512	−0.552558	−	0.833474i	$-0.686348\pi$
$409$	−451.992	−1.10512	−0.552558	+	0.833474i	$0.686348\pi$
$410$	0	0
$411$	71.7043i	0.174463i
$412$	0	0
$413$	−79.3935	−0.192236
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−294.997	−0.707428
$418$	0	0
$419$	100.633i	0.240173i	0.992763	+	0.120087i	$0.0383172\pi$
$419$	100.633i	0.240173i	−0.992763	+	0.120087i	$0.961683\pi$
$420$	0	0
$421$	150.201	0.356771	0.178385	−	0.983961i	$-0.442913\pi$
$421$	150.201	0.356771	0.178385	+	0.983961i	$0.442913\pi$
$422$	0	0
$423$	− 106.693i	− 0.252228i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1015.38i	2.37794i
$428$	0	0
$429$	−19.8997	−0.0463864
$430$	0	0
$431$	− 252.240i	− 0.585243i	−0.956228	−	0.292622i	$-0.905472\pi$
$431$	− 252.240i	− 0.585243i	0.956228	−	0.292622i	$-0.0945276\pi$
$432$	0	0
$433$	633.992	1.46419	0.732093	−	0.681205i	$-0.238544\pi$
$433$	633.992	1.46419	0.732093	+	0.681205i	$0.238544\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−426.697	−0.976423
$438$	0	0
$439$	− 175.339i	− 0.399406i	−0.979856	−	0.199703i	$-0.936002\pi$
$439$	− 175.339i	− 0.399406i	0.979856	−	0.199703i	$-0.0639978\pi$
$440$	0	0
$441$	377.398	0.855779
$442$	0	0
$443$	56.9247i	0.128498i	0.997934	+	0.0642491i	$0.0204652\pi$
$443$	56.9247i	0.128498i	−0.997934	+	0.0642491i	$0.979535\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	279.550i	0.625392i
$448$	0	0
$449$	783.799	1.74566	0.872828	−	0.488028i	$-0.162284\pi$
$449$	783.799	1.74566	0.872828	+	0.488028i	$0.162284\pi$
$450$	0	0
$451$	616.957i	1.36798i
$452$	0	0
$453$	−310.900	−0.686313
$454$	0	0
$455$	0	0
$456$	0	0
$457$	26.0000	0.0568928	0.0284464	−	0.999595i	$-0.490944\pi$
$457$	26.0000	0.0568928	0.0284464	+	0.999595i	$0.490944\pi$
$458$	0	0
$459$	165.756i	0.361124i
$460$	0	0
$461$	190.702	0.413670	0.206835	−	0.978376i	$-0.433684\pi$
$461$	190.702	0.413670	0.206835	+	0.978376i	$0.433684\pi$
$462$	0	0
$463$	434.978i	0.939477i	0.882806	+	0.469738i	$0.155652\pi$
$463$	434.978i	0.939477i	−0.882806	+	0.469738i	$0.844348\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	277.412i	0.594029i	0.954873	+	0.297015i	$0.0959910\pi$
$467$	277.412i	0.594029i	−0.954873	+	0.297015i	$0.904009\pi$
$468$	0	0
$469$	1440.39	3.07119
$470$	0	0
$471$	− 312.459i	− 0.663396i
$472$	0	0
$473$	285.895	0.604429
$474$	0	0
$475$	0	0
$476$	0	0
$477$	269.098	0.564146
$478$	0	0
$479$	320.078i	0.668221i	0.942534	+	0.334110i	$0.108436\pi$
$479$	320.078i	0.668221i	−0.942534	+	0.334110i	$0.891564\pi$
$480$	0	0
$481$	17.7995	0.0370052
$482$	0	0
$483$	739.060i	1.53015i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	153.632i	0.315465i	0.987482	+	0.157733i	$0.0504184\pi$
$487$	153.632i	0.315465i	−0.987482	+	0.157733i	$0.949582\pi$
$488$	0	0
$489$	74.6992	0.152759
$490$	0	0
$491$	− 218.924i	− 0.445874i	−0.974833	−	0.222937i	$-0.928436\pi$
$491$	− 218.924i	− 0.445874i	0.974833	−	0.222937i	$-0.0715645\pi$
$492$	0	0
$493$	−130.797	−0.265308
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.5013	0.0291776
$498$	0	0
$499$	273.723i	0.548544i	0.961652	+	0.274272i	$0.0884368\pi$
$499$	273.723i	0.548544i	−0.961652	+	0.274272i	$0.911563\pi$
$500$	0	0
$501$	−238.702	−0.476451
$502$	0	0
$503$	246.125i	0.489314i	0.969610	+	0.244657i	$0.0786753\pi$
$503$	246.125i	0.489314i	−0.969610	+	0.244657i	$0.921325\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	290.985i	0.573934i
$508$	0	0
$509$	−357.799	−0.702946	−0.351473	−	0.936198i	$-0.614319\pi$
$509$	−357.799	−0.702946	−0.351473	+	0.936198i	$0.614319\pi$
$510$	0	0
$511$	399.286i	0.781382i
$512$	0	0
$513$	68.6992	0.133917
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−408.602	−0.790332
$518$	0	0
$519$	− 476.483i	− 0.918079i
$520$	0	0
$521$	−785.494	−1.50767	−0.753833	−	0.657066i	$-0.771797\pi$
$521$	−785.494	−1.50767	−0.753833	+	0.657066i	$0.771797\pi$
$522$	0	0
$523$	− 34.9290i	− 0.0667858i	−0.999442	−	0.0333929i	$-0.989369\pi$
$523$	− 34.9290i	− 0.0667858i	0.999442	−	0.0333929i	$-0.0106313\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1195.28i	− 2.26808i
$528$	0	0
$529$	−512.594	−0.968987
$530$	0	0
$531$	18.0151i	0.0339267i
$532$	0	0
$533$	−53.6992	−0.100749
$534$	0	0
$535$	0	0
$536$	0	0
$537$	311.794	0.580623
$538$	0	0
$539$	− 1445.33i	− 2.68150i
$540$	0	0
$541$	374.995	0.693152	0.346576	−	0.938022i	$-0.387344\pi$
$541$	374.995	0.693152	0.346576	+	0.938022i	$0.387344\pi$
$542$	0	0
$543$	− 394.556i	− 0.726622i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 539.815i	− 0.986865i	−0.869784	−	0.493433i	$-0.835742\pi$
$547$	− 539.815i	− 0.986865i	0.869784	−	0.493433i	$-0.164258\pi$
$548$	0	0
$549$	230.398	0.419669
$550$	0	0
$551$	54.2101i	0.0983850i
$552$	0	0
$553$	−33.5940	−0.0607486
$554$	0	0
$555$	0	0
$556$	0	0
$557$	157.599	0.282943	0.141471	−	0.989942i	$-0.454817\pi$
$557$	157.599	0.282943	0.141471	+	0.989942i	$0.454817\pi$
$558$	0	0
$559$	24.8839i	0.0445151i
$560$	0	0
$561$	634.797	1.13155
$562$	0	0
$563$	− 388.958i	− 0.690866i	−0.938443	−	0.345433i	$-0.887732\pi$
$563$	− 388.958i	− 0.690866i	0.938443	−	0.345433i	$-0.112268\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 118.991i	− 0.209860i
$568$	0	0
$569$	−83.0977	−0.146042	−0.0730209	−	0.997330i	$-0.523264\pi$
$569$	−83.0977	−0.146042	−0.0730209	+	0.997330i	$0.523264\pi$
$570$	0	0
$571$	1050.25i	1.83932i	0.392720	+	0.919658i	$0.371534\pi$
$571$	1050.25i	1.83932i	−0.392720	+	0.919658i	$0.628466\pi$
$572$	0	0
$573$	−489.895	−0.854965
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−53.3960	−0.0925407	−0.0462703	−	0.998929i	$-0.514734\pi$
$577$	−53.3960	−0.0925407	−0.0462703	+	0.998929i	$0.514734\pi$
$578$	0	0
$579$	467.649i	0.807685i
$580$	0	0
$581$	−961.789	−1.65540
$582$	0	0
$583$	− 1030.57i	− 1.76769i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 671.853i	− 1.14455i	−0.820060	−	0.572277i	$-0.806060\pi$
$587$	− 671.853i	− 1.14455i	0.820060	−	0.572277i	$-0.193940\pi$
$588$	0	0
$589$	−495.396	−0.841080
$590$	0	0
$591$	− 177.711i	− 0.300695i
$592$	0	0
$593$	609.689	1.02814	0.514072	−	0.857747i	$-0.328136\pi$
$593$	609.689	1.02814	0.514072	+	0.857747i	$0.328136\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	474.895	0.795469
$598$	0	0
$599$	− 29.4492i	− 0.0491640i	−0.999698	−	0.0245820i	$-0.992175\pi$
$599$	− 29.4492i	− 0.0491640i	0.999698	−	0.0245820i	$-0.00782548\pi$
$600$	0	0
$601$	166.398	0.276869	0.138435	−	0.990372i	$-0.455793\pi$
$601$	166.398	0.276869	0.138435	+	0.990372i	$0.455793\pi$
$602$	0	0
$603$	− 326.837i	− 0.542018i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 306.687i	− 0.505251i	−0.967564	−	0.252625i	$-0.918706\pi$
$607$	− 306.687i	− 0.505251i	0.967564	−	0.252625i	$-0.0812940\pi$
$608$	0	0
$609$	93.8947	0.154179
$610$	0	0
$611$	− 35.5642i	− 0.0582065i
$612$	0	0
$613$	−271.604	−0.443073	−0.221537	−	0.975152i	$-0.571107\pi$
$613$	−271.604	−0.443073	−0.221537	+	0.975152i	$0.571107\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	121.298	0.196594	0.0982968	−	0.995157i	$-0.468661\pi$
$617$	121.298	0.196594	0.0982968	+	0.995157i	$0.468661\pi$
$618$	0	0
$619$	324.872i	0.524833i	0.964955	+	0.262417i	$0.0845194\pi$
$619$	324.872i	0.524833i	−0.964955	+	0.262417i	$0.915481\pi$
$620$	0	0
$621$	167.699	0.270047
$622$	0	0
$623$	1269.23i	2.03729i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 263.098i	− 0.419614i
$628$	0	0
$629$	−567.799	−0.902702
$630$	0	0
$631$	− 275.222i	− 0.436169i	−0.975930	−	0.218084i	$-0.930019\pi$
$631$	− 275.222i	− 0.436169i	0.975930	−	0.218084i	$-0.0699808\pi$
$632$	0	0
$633$	188.699	0.298103
$634$	0	0
$635$	0	0
$636$	0	0
$637$	125.799	0.197487
$638$	0	0
$639$	− 3.29046i	− 0.00514939i
$640$	0	0
$641$	−989.794	−1.54414	−0.772071	−	0.635537i	$-0.780779\pi$
$641$	−989.794	−1.54414	−0.772071	+	0.635537i	$0.780779\pi$
$642$	0	0
$643$	− 110.504i	− 0.171857i	−0.996301	−	0.0859284i	$-0.972614\pi$
$643$	− 110.504i	− 0.171857i	0.996301	−	0.0859284i	$-0.0273856\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	302.995i	0.468307i	0.972200	+	0.234153i	$0.0752318\pi$
$647$	302.995i	0.468307i	−0.972200	+	0.234153i	$0.924768\pi$
$648$	0	0
$649$	68.9925	0.106306
$650$	0	0
$651$	858.051i	1.31805i
$652$	0	0
$653$	1020.40	1.56263	0.781314	−	0.624138i	$-0.214550\pi$
$653$	1020.40	1.56263	0.781314	+	0.624138i	$0.214550\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	90.6015	0.137902
$658$	0	0
$659$	703.386i	1.06735i	0.845688	+	0.533677i	$0.179190\pi$
$659$	703.386i	1.06735i	−0.845688	+	0.533677i	$0.820810\pi$
$660$	0	0
$661$	53.1980	0.0804811	0.0402405	−	0.999190i	$-0.487188\pi$
$661$	53.1980	0.0804811	0.0402405	+	0.999190i	$0.487188\pi$
$662$	0	0
$663$	55.2520i	0.0833363i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	132.330i	0.198396i
$668$	0	0
$669$	17.5013	0.0261603
$670$	0	0
$671$	− 882.359i	− 1.31499i
$672$	0	0
$673$	−998.190	−1.48320	−0.741598	−	0.670845i	$-0.765932\pi$
$673$	−998.190	−1.48320	−0.741598	+	0.670845i	$0.765932\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	472.702	0.698230	0.349115	−	0.937080i	$-0.386482\pi$
$677$	472.702	0.698230	0.349115	+	0.937080i	$0.386482\pi$
$678$	0	0
$679$	− 1515.13i	− 2.23142i
$680$	0	0
$681$	−433.900	−0.637151
$682$	0	0
$683$	− 1102.74i	− 1.61455i	−0.590177	−	0.807274i	$-0.700942\pi$
$683$	− 1102.74i	− 1.61455i	0.590177	−	0.807274i	$-0.299058\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	597.896i	0.870300i
$688$	0	0
$689$	89.6992	0.130188
$690$	0	0
$691$	− 1306.30i	− 1.89046i	−0.326411	−	0.945228i	$-0.605839\pi$
$691$	− 1306.30i	− 1.89046i	0.326411	−	0.945228i	$-0.394161\pi$
$692$	0	0
$693$	−455.699	−0.657575
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1712.99	2.45766
$698$	0	0
$699$	327.180i	0.468068i
$700$	0	0
$701$	1029.89	1.46918	0.734590	−	0.678512i	$-0.237375\pi$
$701$	1029.89	1.46918	0.734590	+	0.678512i	$0.237375\pi$
$702$	0	0
$703$	235.330i	0.334751i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1770.31i	2.50398i
$708$	0	0
$709$	−845.997	−1.19323	−0.596613	−	0.802529i	$-0.703487\pi$
$709$	−845.997	−1.19323	−0.596613	+	0.802529i	$0.703487\pi$
$710$	0	0
$711$	7.62276i	0.0107212i
$712$	0	0
$713$	−1209.29	−1.69606
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−648.095	−0.903899
$718$	0	0
$719$	244.507i	0.340066i	0.985438	+	0.170033i	$0.0543874\pi$
$719$	244.507i	0.340066i	−0.985438	+	0.170033i	$0.945613\pi$
$720$	0	0
$721$	154.195	0.213863
$722$	0	0
$723$	− 334.286i	− 0.462359i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	782.590i	1.07647i	0.842796	+	0.538233i	$0.180908\pi$
$727$	782.590i	1.07647i	−0.842796	+	0.538233i	$0.819092\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 793.791i	− 1.08590i
$732$	0	0
$733$	−299.203	−0.408190	−0.204095	−	0.978951i	$-0.565425\pi$
$733$	−299.203	−0.408190	−0.204095	+	0.978951i	$0.565425\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1251.69	−1.69836
$738$	0	0
$739$	− 77.5994i	− 0.105006i	−0.998621	−	0.0525029i	$-0.983280\pi$
$739$	− 77.5994i	− 0.105006i	0.998621	−	0.0525029i	$-0.0167199\pi$
$740$	0	0
$741$	22.8997	0.0309038
$742$	0	0
$743$	− 703.441i	− 0.946758i	−0.880859	−	0.473379i	$-0.843034\pi$
$743$	− 703.441i	− 0.946758i	0.880859	−	0.473379i	$-0.156966\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	218.238i	0.292153i
$748$	0	0
$749$	−2083.09	−2.78117
$750$	0	0
$751$	− 753.200i	− 1.00293i	−0.865178	−	0.501465i	$-0.832795\pi$
$751$	− 753.200i	− 1.00293i	0.865178	−	0.501465i	$-0.167205\pi$
$752$	0	0
$753$	608.201	0.807703
$754$	0	0
$755$	0	0
$756$	0	0
$757$	199.201	0.263145	0.131572	−	0.991307i	$-0.457997\pi$
$757$	199.201	0.263145	0.131572	+	0.991307i	$0.457997\pi$
$758$	0	0
$759$	− 642.239i	− 0.846165i
$760$	0	0
$761$	171.198	0.224965	0.112482	−	0.993654i	$-0.464120\pi$
$761$	171.198	0.224965	0.112482	+	0.993654i	$0.464120\pi$
$762$	0	0
$763$	− 322.577i	− 0.422774i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.00502i	0.00782923i
$768$	0	0
$769$	−577.602	−0.751107	−0.375554	−	0.926801i	$-0.622547\pi$
$769$	−577.602	−0.751107	−0.375554	+	0.926801i	$0.622547\pi$
$770$	0	0
$771$	− 469.381i	− 0.608796i
$772$	0	0
$773$	626.090	0.809949	0.404974	−	0.914328i	$-0.367280\pi$
$773$	626.090	0.809949	0.404974	+	0.914328i	$0.367280\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	407.604	0.524587
$778$	0	0
$779$	− 709.967i	− 0.911383i
$780$	0	0
$781$	−12.6015	−0.0161351
$782$	0	0
$783$	− 21.3055i	− 0.0272101i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 99.3665i	− 0.126260i	−0.998005	−	0.0631299i	$-0.979892\pi$
$787$	− 99.3665i	− 0.126260i	0.998005	−	0.0631299i	$-0.0201082\pi$
$788$	0	0
$789$	−540.095	−0.684531
$790$	0	0
$791$	375.495i	0.474709i
$792$	0	0
$793$	76.7995	0.0968468
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−85.3935	−0.107144	−0.0535718	−	0.998564i	$-0.517061\pi$
$797$	−85.3935	−0.107144	−0.0535718	+	0.998564i	$0.517061\pi$
$798$	0	0
$799$	1134.49i	1.41989i
$800$	0	0
$801$	288.000	0.359551
$802$	0	0
$803$	− 346.977i	− 0.432101i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 663.545i	− 0.822236i
$808$	0	0
$809$	−591.103	−0.730659	−0.365329	−	0.930878i	$-0.619044\pi$
$809$	−591.103	−0.730659	−0.365329	+	0.930878i	$0.619044\pi$
$810$	0	0
$811$	175.339i	0.216201i	0.994140	+	0.108101i	$0.0344769\pi$
$811$	175.339i	0.216201i	−0.994140	+	0.108101i	$0.965523\pi$
$812$	0	0
$813$	29.7945	0.0366476
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−328.995	−0.402687
$818$	0	0
$819$	− 39.6635i	− 0.0484292i
$820$	0	0
$821$	587.193	0.715217	0.357608	−	0.933872i	$-0.383592\pi$
$821$	587.193	0.715217	0.357608	+	0.933872i	$0.383592\pi$
$822$	0	0
$823$	− 459.862i	− 0.558763i	−0.960180	−	0.279381i	$-0.909871\pi$
$823$	− 459.862i	− 0.558763i	0.960180	−	0.279381i	$-0.0901294\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1301.75i	1.57407i	0.616911	+	0.787033i	$0.288384\pi$
$827$	1301.75i	1.57407i	−0.616911	+	0.787033i	$0.711616\pi$
$828$	0	0
$829$	−231.995	−0.279849	−0.139925	−	0.990162i	$-0.544686\pi$
$829$	−231.995	−0.279849	−0.139925	+	0.990162i	$0.544686\pi$
$830$	0	0
$831$	313.501i	0.377258i
$832$	0	0
$833$	−4012.97	−4.81749
$834$	0	0
$835$	0	0
$836$	0	0
$837$	194.699	0.232616
$838$	0	0
$839$	414.239i	0.493730i	0.969050	+	0.246865i	$0.0794004\pi$
$839$	414.239i	0.493730i	−0.969050	+	0.246865i	$0.920600\pi$
$840$	0	0
$841$	−824.188	−0.980009
$842$	0	0
$843$	− 379.141i	− 0.449752i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	145.433i	0.171704i
$848$	0	0
$849$	−675.095	−0.795165
$850$	0	0
$851$	574.456i	0.675037i
$852$	0	0
$853$	1172.79	1.37490	0.687450	−	0.726232i	$-0.258730\pi$
$853$	1172.79	1.37490	0.687450	+	0.726232i	$0.258730\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	579.388	0.676066	0.338033	−	0.941134i	$-0.390238\pi$
$857$	579.388	0.676066	0.338033	+	0.941134i	$0.390238\pi$
$858$	0	0
$859$	− 705.178i	− 0.820929i	−0.911877	−	0.410464i	$-0.865367\pi$
$859$	− 705.178i	− 0.820929i	0.911877	−	0.410464i	$-0.134633\pi$
$860$	0	0
$861$	−1229.70	−1.42822
$862$	0	0
$863$	822.720i	0.953325i	0.879086	+	0.476663i	$0.158154\pi$
$863$	822.720i	0.953325i	−0.879086	+	0.476663i	$0.841846\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1261.96i	− 1.45555i
$868$	0	0
$869$	29.1930	0.0335937
$870$	0	0
$871$	− 108.946i	− 0.125081i
$872$	0	0
$873$	−343.797	−0.393811
$874$	0	0
$875$	0	0
$876$	0	0
$877$	892.799	1.01802	0.509008	−	0.860762i	$-0.330012\pi$
$877$	892.799	1.01802	0.509008	+	0.860762i	$0.330012\pi$
$878$	0	0
$879$	− 834.154i	− 0.948981i
$880$	0	0
$881$	159.388	0.180918	0.0904588	−	0.995900i	$-0.471167\pi$
$881$	159.388	0.180918	0.0904588	+	0.995900i	$0.471167\pi$
$882$	0	0
$883$	1134.43i	1.28474i	0.766393	+	0.642372i	$0.222050\pi$
$883$	1134.43i	1.28474i	−0.766393	+	0.642372i	$0.777950\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	231.180i	0.260632i	0.991473	+	0.130316i	$0.0415991\pi$
$887$	231.180i	0.260632i	−0.991473	+	0.130316i	$0.958401\pi$
$888$	0	0
$889$	187.789	0.211237
$890$	0	0
$891$	103.402i	0.116052i
$892$	0	0
$893$	470.201	0.526540
$894$	0	0
$895$	0	0
$896$	0	0
$897$	55.8997	0.0623186
$898$	0	0
$899$	153.636i	0.170896i
$900$	0	0
$901$	−2861.38	−3.17579
$902$	0	0
$903$	569.836i	0.631048i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 688.890i	− 0.759526i	−0.925084	−	0.379763i	$-0.876006\pi$
$907$	− 688.890i	− 0.759526i	0.925084	−	0.379763i	$-0.123994\pi$
$908$	0	0
$909$	401.699	0.441913
$910$	0	0
$911$	− 513.610i	− 0.563787i	−0.959446	−	0.281894i	$-0.909037\pi$
$911$	− 513.610i	− 0.563787i	0.959446	−	0.281894i	$-0.0909625\pi$
$912$	0	0
$913$	835.789	0.915432
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2487.67	2.71284
$918$	0	0
$919$	− 1562.95i	− 1.70070i	−0.526216	−	0.850351i	$-0.676390\pi$
$919$	− 1562.95i	− 1.70070i	0.526216	−	0.850351i	$-0.323610\pi$
$920$	0	0
$921$	−373.892	−0.405963
$922$	0	0
$923$	− 1.09682i	− 0.00118832i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 34.9883i	− 0.0377436i
$928$	0	0
$929$	509.494	0.548432	0.274216	−	0.961668i	$-0.411582\pi$
$929$	509.494	0.548432	0.274216	+	0.961668i	$0.411582\pi$
$930$	0	0
$931$	1663.22i	1.78648i
$932$	0	0
$933$	622.591	0.667301
$934$	0	0
$935$	0	0
$936$	0	0
$937$	62.3935	0.0665885	0.0332943	−	0.999446i	$-0.489400\pi$
$937$	62.3935	0.0665885	0.0332943	+	0.999446i	$0.489400\pi$
$938$	0	0
$939$	− 935.998i	− 0.996803i
$940$	0	0
$941$	1475.19	1.56769	0.783843	−	0.620959i	$-0.213257\pi$
$941$	1475.19	1.56769	0.783843	+	0.620959i	$0.213257\pi$
$942$	0	0
$943$	− 1733.08i	− 1.83783i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	236.774i	0.250026i	0.992155	+	0.125013i	$0.0398972\pi$
$947$	236.774i	0.250026i	−0.992155	+	0.125013i	$0.960103\pi$
$948$	0	0
$949$	30.2005	0.0318235
$950$	0	0
$951$	− 628.036i	− 0.660395i
$952$	0	0
$953$	−234.807	−0.246387	−0.123194	−	0.992383i	$-0.539314\pi$
$953$	−234.807	−0.246387	−0.123194	+	0.992383i	$0.539314\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−81.5940	−0.0852602
$958$	0	0
$959$	547.337i	0.570737i
$960$	0	0
$961$	−442.992	−0.460970
$962$	0	0
$963$	472.672i	0.490833i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 988.869i	− 1.02262i	−0.859398	−	0.511308i	$-0.829161\pi$
$967$	− 988.869i	− 1.02262i	0.859398	−	0.511308i	$-0.170839\pi$
$968$	0	0
$969$	−730.496	−0.753866
$970$	0	0
$971$	− 1252.04i	− 1.28943i	−0.764422	−	0.644717i	$-0.776975\pi$
$971$	− 1252.04i	− 1.28943i	0.764422	−	0.644717i	$-0.223025\pi$
$972$	0	0
$973$	−2251.79	−2.31427
$974$	0	0
$975$	0	0
$976$	0	0
$977$	440.596	0.450969	0.225484	−	0.974247i	$-0.427604\pi$
$977$	440.596	0.450969	0.225484	+	0.974247i	$0.427604\pi$
$978$	0	0
$979$	− 1102.96i	− 1.12661i
$980$	0	0
$981$	−73.1955	−0.0746131
$982$	0	0
$983$	− 1189.33i	− 1.20990i	−0.796264	−	0.604949i	$-0.793193\pi$
$983$	− 1189.33i	− 1.20990i	0.796264	−	0.604949i	$-0.206807\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 814.411i	− 0.825138i
$988$	0	0
$989$	−803.098	−0.812030
$990$	0	0
$991$	1660.74i	1.67583i	0.545803	+	0.837913i	$0.316225\pi$
$991$	1660.74i	1.67583i	−0.545803	+	0.837913i	$0.683775\pi$
$992$	0	0
$993$	−317.794	−0.320035
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1256.80	1.26058	0.630289	−	0.776360i	$-0.282936\pi$
$997$	1256.80	1.26058	0.630289	+	0.776360i	$0.282936\pi$
$998$	0	0
$999$	− 92.4889i	− 0.0925815i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.e.k.751.1	✓	4
3.2	odd	2		3600.3.e.bc.3151.1		4
4.3	odd	2	inner	1200.3.e.k.751.4	yes	4
5.2	odd	4		1200.3.j.d.799.4		8
5.3	odd	4		1200.3.j.d.799.6		8
5.4	even	2		1200.3.e.m.751.4	yes	4
12.11	even	2		3600.3.e.bc.3151.4		4
15.2	even	4		3600.3.j.j.1999.7		8
15.8	even	4		3600.3.j.j.1999.1		8
15.14	odd	2		3600.3.e.bf.3151.4		4
20.3	even	4		1200.3.j.d.799.3		8
20.7	even	4		1200.3.j.d.799.5		8
20.19	odd	2		1200.3.e.m.751.1	yes	4
60.23	odd	4		3600.3.j.j.1999.8		8
60.47	odd	4		3600.3.j.j.1999.2		8
60.59	even	2		3600.3.e.bf.3151.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1200.3.e.k.751.1	✓	4	1.1	even	1	trivial
1200.3.e.k.751.4	yes	4	4.3	odd	2	inner
1200.3.e.m.751.1	yes	4	20.19	odd	2
1200.3.e.m.751.4	yes	4	5.4	even	2
1200.3.j.d.799.3		8	20.3	even	4
1200.3.j.d.799.4		8	5.2	odd	4
1200.3.j.d.799.5		8	20.7	even	4
1200.3.j.d.799.6		8	5.3	odd	4
3600.3.e.bc.3151.1		4	3.2	odd	2
3600.3.e.bc.3151.4		4	12.11	even	2
3600.3.e.bf.3151.1		4	60.59	even	2
3600.3.e.bf.3151.4		4	15.14	odd	2
3600.3.j.j.1999.1		8	15.8	even	4
3600.3.j.j.1999.2		8	60.47	odd	4
3600.3.j.j.1999.7		8	15.2	even	4
3600.3.j.j.1999.8		8	60.23	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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -13.2212i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} -13.2212i q^{7} -3.00000 q^{9} +11.4891i q^{11} -1.00000 q^{13} +31.8997 q^{17} -13.2212i q^{19} -22.8997 q^{21} -32.2737i q^{23} +5.19615i q^{27} -4.10025 q^{29} -37.4699i q^{31} +19.8997 q^{33} -17.7995 q^{37} +1.73205i q^{39} +53.6992 q^{41} -24.8839i q^{43} +35.5642i q^{47} -125.799 q^{49} -55.2520i q^{51} -89.6992 q^{53} -22.8997 q^{57} -6.00502i q^{59} -76.7995 q^{61} +39.6635i q^{63} +108.946i q^{67} -55.8997 q^{69} +1.09682i q^{71} -30.2005 q^{73} +151.900 q^{77} -2.54092i q^{79} +9.00000 q^{81} -72.7461i q^{83} +7.10184i q^{87} -96.0000 q^{89} +13.2212i q^{91} -64.8997 q^{93} +114.599 q^{97} -34.4674i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 4 * q - 12 * q^9	\( 4 q - 12 q^{9} - 4 q^{13} + 48 q^{17} - 12 q^{21} - 96 q^{29} + 88 q^{37} - 24 q^{41} - 344 q^{49} - 120 q^{53} - 12 q^{57} - 148 q^{61} - 144 q^{69} - 280 q^{73} + 528 q^{77} + 36 q^{81} - 384 q^{89} - 180 q^{93} + 140 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 - 4 * q^13 + 48 * q^17 - 12 * q^21 - 96 * q^29 + 88 * q^37 - 24 * q^41 - 344 * q^49 - 120 * q^53 - 12 * q^57 - 148 * q^61 - 144 * q^69 - 280 * q^73 + 528 * q^77 + 36 * q^81 - 384 * q^89 - 180 * q^93 + 140 * q^97

Introduction

L-functions

Modular forms

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