LMFDB - Embedded newform 1200.3.e.m.751.2

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			751.2
Root			$-1.65831 + 2.87228i$ of defining polynomial
Character	$\chi$	$=$	1200.751
Dual form			1200.3.e.m.751.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +9.75707i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +9.75707i q^{7} -3.00000 q^{9} +11.4891i q^{11} +1.00000 q^{13} +7.89975 q^{17} -9.75707i q^{19} +16.8997 q^{21} -9.29548i q^{23} +5.19615i q^{27} -43.8997 q^{29} +14.4916i q^{31} +19.8997 q^{33} -61.7995 q^{37} -1.73205i q^{39} -65.6992 q^{41} -47.8622i q^{43} -56.3488i q^{47} -46.2005 q^{49} -13.6828i q^{51} -29.6992 q^{53} -16.8997 q^{57} +97.9180i q^{59} +2.79950 q^{61} -29.2712i q^{63} +40.0108i q^{67} -16.1003 q^{69} +21.8814i q^{71} +109.799 q^{73} -112.100 q^{77} +94.4539i q^{79} +9.00000 q^{81} -72.7461i q^{83} +76.0366i q^{87} -96.0000 q^{89} +9.75707i q^{91} +25.1003 q^{93} +44.5990 q^{97} -34.4674i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	9.75707i	1.39387i	0.717135	+	0.696934i	$0.245453\pi$
$7$	9.75707i	1.39387i	−0.717135	+	0.696934i	$0.754547\pi$
$8$	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	$-0.487754\pi$
$13$	1.00000	0.0769231	0.0384615	+	0.999260i	$0.487754\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.89975	0.464691	0.232346	−	0.972633i	$-0.425360\pi$
$17$	7.89975	0.464691	0.232346	+	0.972633i	$0.425360\pi$
$18$	0	0
$19$	− 9.75707i	− 0.513530i	−0.966474	−	0.256765i	$-0.917343\pi$
$19$	− 9.75707i	− 0.513530i	0.966474	−	0.256765i	$-0.0826567\pi$
$20$	0	0
$21$	16.8997	0.804750
$22$	0	0
$23$	− 9.29548i	− 0.404151i	−0.979370	−	0.202076i	$-0.935231\pi$
$23$	− 9.29548i	− 0.404151i	0.979370	−	0.202076i	$-0.0647687\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−43.8997	−1.51378	−0.756892	−	0.653540i	$-0.773283\pi$
$29$	−43.8997	−1.51378	−0.756892	+	0.653540i	$0.773283\pi$
$30$	0	0
$31$	14.4916i	0.467472i	0.972300	+	0.233736i	$0.0750952\pi$
$31$	14.4916i	0.467472i	−0.972300	+	0.233736i	$0.924905\pi$
$32$	0	0
$33$	19.8997	0.603023
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−61.7995	−1.67026	−0.835128	−	0.550055i	$-0.814607\pi$
$37$	−61.7995	−1.67026	−0.835128	+	0.550055i	$0.814607\pi$
$38$	0	0
$39$	− 1.73205i	− 0.0444116i
$40$	0	0
$41$	−65.6992	−1.60242	−0.801210	−	0.598383i	$-0.795810\pi$
$41$	−65.6992	−1.60242	−0.801210	+	0.598383i	$0.795810\pi$
$42$	0	0
$43$	− 47.8622i	− 1.11307i	−0.830823	−	0.556537i	$-0.812130\pi$
$43$	− 47.8622i	− 1.11307i	0.830823	−	0.556537i	$-0.187870\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 56.3488i	− 1.19891i	−0.800408	−	0.599455i	$-0.795384\pi$
$47$	− 56.3488i	− 1.19891i	0.800408	−	0.599455i	$-0.204616\pi$
$48$	0	0
$49$	−46.2005	−0.942867
$50$	0	0
$51$	− 13.6828i	− 0.268290i
$52$	0	0
$53$	−29.6992	−0.560363	−0.280182	−	0.959947i	$-0.590395\pi$
$53$	−29.6992	−0.560363	−0.280182	+	0.959947i	$0.590395\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−16.8997	−0.296487
$58$	0	0
$59$	97.9180i	1.65963i	0.558040	+	0.829814i	$0.311553\pi$
$59$	97.9180i	1.65963i	−0.558040	+	0.829814i	$0.688447\pi$
$60$	0	0
$61$	2.79950	0.0458934	0.0229467	−	0.999737i	$-0.492695\pi$
$61$	2.79950	0.0458934	0.0229467	+	0.999737i	$0.492695\pi$
$62$	0	0
$63$	− 29.2712i	− 0.464623i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	40.0108i	0.597176i	0.954382	+	0.298588i	$0.0965157\pi$
$67$	40.0108i	0.597176i	−0.954382	+	0.298588i	$0.903484\pi$
$68$	0	0
$69$	−16.1003	−0.233337
$70$	0	0
$71$	21.8814i	0.308189i	0.988056	+	0.154095i	$0.0492460\pi$
$71$	21.8814i	0.308189i	−0.988056	+	0.154095i	$0.950754\pi$
$72$	0	0
$73$	109.799	1.50410	0.752051	−	0.659105i	$-0.229065\pi$
$73$	109.799	1.50410	0.752051	+	0.659105i	$0.229065\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−112.100	−1.45585
$78$	0	0
$79$	94.4539i	1.19562i	0.801638	+	0.597810i	$0.203962\pi$
$79$	94.4539i	1.19562i	−0.801638	+	0.597810i	$0.796038\pi$
$80$	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
$87$	76.0366i	0.873984i
$88$	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
$91$	9.75707i	0.107221i
$92$	0	0
$93$	25.1003	0.269895
$94$	0	0
$95$	0	0
$96$	0	0
$97$	44.5990	0.459783	0.229892	−	0.973216i	$-0.426163\pi$
$97$	44.5990	0.459783	0.229892	+	0.973216i	$0.426163\pi$
$98$	0	0
$99$	− 34.4674i	− 0.348155i
$100$	0	0
$101$	−94.1003	−0.931686	−0.465843	−	0.884867i	$-0.654249\pi$
$101$	−94.1003	−0.931686	−0.465843	+	0.884867i	$0.654249\pi$
$102$	0	0
$103$	57.6193i	0.559410i	0.960086	+	0.279705i	$0.0902367\pi$
$103$	57.6193i	0.559410i	−0.960086	+	0.279705i	$0.909763\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	95.2035i	0.889752i	0.895592	+	0.444876i	$0.146752\pi$
$107$	95.2035i	0.889752i	−0.895592	+	0.444876i	$0.853248\pi$
$108$	0	0
$109$	−214.398	−1.96696	−0.983479	−	0.181020i	$-0.942060\pi$
$109$	−214.398	−1.96696	−0.983479	+	0.181020i	$0.942060\pi$
$110$	0	0
$111$	107.040i	0.964323i
$112$	0	0
$113$	187.599	1.66017	0.830084	−	0.557638i	$-0.188292\pi$
$113$	187.599	1.66017	0.830084	+	0.557638i	$0.188292\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.00000	−0.0256410
$118$	0	0
$119$	77.0784i	0.647718i
$120$	0	0
$121$	−11.0000	−0.0909091
$122$	0	0
$123$	113.794i	0.925158i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	152.073i	1.19743i	0.800963	+	0.598713i	$0.204321\pi$
$127$	152.073i	1.19743i	−0.800963	+	0.598713i	$0.795679\pi$
$128$	0	0
$129$	−82.8997	−0.642634
$130$	0	0
$131$	− 165.180i	− 1.26092i	−0.776223	−	0.630458i	$-0.782867\pi$
$131$	− 165.180i	− 1.26092i	0.776223	−	0.630458i	$-0.217133\pi$
$132$	0	0
$133$	95.2005	0.715793
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−197.398	−1.44086	−0.720432	−	0.693525i	$-0.756057\pi$
$137$	−197.398	−1.44086	−0.720432	+	0.693525i	$0.756057\pi$
$138$	0	0
$139$	− 59.4656i	− 0.427810i	−0.976854	−	0.213905i	$-0.931382\pi$
$139$	− 59.4656i	− 0.427810i	0.976854	−	0.213905i	$-0.0686183\pi$
$140$	0	0
$141$	−97.5990	−0.692191
$142$	0	0
$143$	11.4891i	0.0803435i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	80.0216i	0.544365i
$148$	0	0
$149$	77.3985	0.519453	0.259726	−	0.965682i	$-0.416368\pi$
$149$	77.3985	0.519453	0.259726	+	0.965682i	$0.416368\pi$
$150$	0	0
$151$	156.520i	1.03655i	0.855213	+	0.518277i	$0.173427\pi$
$151$	156.520i	1.03655i	−0.855213	+	0.518277i	$0.826573\pi$
$152$	0	0
$153$	−23.6992	−0.154897
$154$	0	0
$155$	0	0
$156$	0	0
$157$	58.3985	0.371965	0.185982	−	0.982553i	$-0.440453\pi$
$157$	58.3985	0.371965	0.185982	+	0.982553i	$0.440453\pi$
$158$	0	0
$159$	51.4406i	0.323526i
$160$	0	0
$161$	90.6967	0.563334
$162$	0	0
$163$	− 25.8071i	− 0.158326i	−0.996862	−	0.0791630i	$-0.974775\pi$
$163$	− 25.8071i	− 0.158326i	0.996862	−	0.0791630i	$-0.0252247\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 298.662i	− 1.78840i	−0.447671	−	0.894198i	$-0.647746\pi$
$167$	− 298.662i	− 1.78840i	0.447671	−	0.894198i	$-0.352254\pi$
$168$	0	0
$169$	−168.000	−0.994083
$170$	0	0
$171$	29.2712i	0.171177i
$172$	0	0
$173$	83.0977	0.480334	0.240167	−	0.970732i	$-0.422798\pi$
$173$	83.0977	0.480334	0.240167	+	0.970732i	$0.422798\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	169.599	0.958186
$178$	0	0
$179$	325.507i	1.81847i	0.416279	+	0.909237i	$0.363334\pi$
$179$	325.507i	1.81847i	−0.416279	+	0.909237i	$0.636666\pi$
$180$	0	0
$181$	−249.797	−1.38009	−0.690047	−	0.723765i	$-0.742410\pi$
$181$	−249.797	−1.38009	−0.690047	+	0.723765i	$0.742410\pi$
$182$	0	0
$183$	− 4.84887i	− 0.0264966i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	90.7612i	0.485354i
$188$	0	0
$189$	−50.6992	−0.268250
$190$	0	0
$191$	− 199.702i	− 1.04556i	−0.852467	−	0.522781i	$-0.824894\pi$
$191$	− 199.702i	− 1.04556i	0.852467	−	0.522781i	$-0.175106\pi$
$192$	0	0
$193$	−127.997	−0.663199	−0.331600	−	0.943420i	$-0.607588\pi$
$193$	−127.997	−0.663199	−0.331600	+	0.943420i	$0.607588\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−341.398	−1.73299	−0.866494	−	0.499188i	$-0.833632\pi$
$197$	−341.398	−1.73299	−0.866494	+	0.499188i	$0.833632\pi$
$198$	0	0
$199$	208.363i	1.04705i	0.852011	+	0.523524i	$0.175383\pi$
$199$	208.363i	1.04705i	−0.852011	+	0.523524i	$0.824617\pi$
$200$	0	0
$201$	69.3008	0.344780
$202$	0	0
$203$	− 428.333i	− 2.11002i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	27.8865i	0.134717i
$208$	0	0
$209$	112.100	0.536365
$210$	0	0
$211$	− 40.0108i	− 0.189625i	−0.995495	−	0.0948123i	$-0.969775\pi$
$211$	− 40.0108i	− 0.189625i	0.995495	−	0.0948123i	$-0.0302251\pi$
$212$	0	0
$213$	37.8997	0.177933
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−141.396	−0.651594
$218$	0	0
$219$	− 190.178i	− 0.868394i
$220$	0	0
$221$	7.89975	0.0357455
$222$	0	0
$223$	124.996i	0.560518i	0.959924	+	0.280259i	$0.0904204\pi$
$223$	124.996i	0.560518i	−0.959924	+	0.280259i	$0.909580\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 227.534i	− 1.00235i	−0.865345	−	0.501176i	$-0.832901\pi$
$227$	− 227.534i	− 1.00235i	0.865345	−	0.501176i	$-0.167099\pi$
$228$	0	0
$229$	371.195	1.62094	0.810470	−	0.585779i	$-0.199212\pi$
$229$	371.195	1.62094	0.810470	+	0.585779i	$0.199212\pi$
$230$	0	0
$231$	194.163i	0.840534i
$232$	0	0
$233$	−248.897	−1.06823	−0.534114	−	0.845412i	$-0.679355\pi$
$233$	−248.897	−1.06823	−0.534114	+	0.845412i	$0.679355\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	163.599	0.690291
$238$	0	0
$239$	− 62.4088i	− 0.261125i	−0.991440	−	0.130562i	$-0.958322\pi$
$239$	− 62.4088i	− 0.261125i	0.991440	−	0.130562i	$-0.0416783\pi$
$240$	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
$247$	− 9.75707i	− 0.0395023i
$248$	0	0
$249$	−126.000	−0.506024
$250$	0	0
$251$	− 397.101i	− 1.58208i	−0.611767	−	0.791038i	$-0.709541\pi$
$251$	− 397.101i	− 1.58208i	0.611767	−	0.791038i	$-0.290459\pi$
$252$	0	0
$253$	106.797	0.422122
$254$	0	0
$255$	0	0
$256$	0	0
$257$	126.997	0.494154	0.247077	−	0.968996i	$-0.420530\pi$
$257$	126.997	0.494154	0.247077	+	0.968996i	$0.420530\pi$
$258$	0	0
$259$	− 602.982i	− 2.32812i
$260$	0	0
$261$	131.699	0.504595
$262$	0	0
$263$	124.763i	0.474383i	0.971463	+	0.237191i	$0.0762268\pi$
$263$	124.763i	0.474383i	−0.971463	+	0.237191i	$0.923773\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	166.277i	0.622760i
$268$	0	0
$269$	24.9023	0.0925735	0.0462867	−	0.998928i	$-0.485261\pi$
$269$	24.9023	0.0925735	0.0462867	+	0.998928i	$0.485261\pi$
$270$	0	0
$271$	488.320i	1.80192i	0.433905	+	0.900959i	$0.357135\pi$
$271$	488.320i	1.80192i	−0.433905	+	0.900959i	$0.642865\pi$
$272$	0	0
$273$	16.8997	0.0619038
$274$	0	0
$275$	0	0
$276$	0	0
$277$	181.000	0.653430	0.326715	−	0.945123i	$-0.394058\pi$
$277$	181.000	0.653430	0.326715	+	0.945123i	$0.394058\pi$
$278$	0	0
$279$	− 43.4749i	− 0.155824i
$280$	0	0
$281$	−218.897	−0.778994	−0.389497	−	0.921028i	$-0.627351\pi$
$281$	−218.897	−0.778994	−0.389497	+	0.921028i	$0.627351\pi$
$282$	0	0
$283$	46.8204i	0.165443i	0.996573	+	0.0827215i	$0.0263612\pi$
$283$	46.8204i	0.165443i	−0.996573	+	0.0827215i	$0.973639\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 641.032i	− 2.23356i
$288$	0	0
$289$	−226.594	−0.784062
$290$	0	0
$291$	− 77.2477i	− 0.265456i
$292$	0	0
$293$	−322.401	−1.10034	−0.550172	−	0.835051i	$-0.685438\pi$
$293$	−322.401	−1.10034	−0.550172	+	0.835051i	$0.685438\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−59.6992	−0.201008
$298$	0	0
$299$	− 9.29548i	− 0.0310886i
$300$	0	0
$301$	466.995	1.55148
$302$	0	0
$303$	162.986i	0.537909i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	496.459i	1.61713i	0.588407	+	0.808565i	$0.299755\pi$
$307$	496.459i	1.61713i	−0.588407	+	0.808565i	$0.700245\pi$
$308$	0	0
$309$	99.7995	0.322976
$310$	0	0
$311$	421.807i	1.35629i	0.734927	+	0.678147i	$0.237217\pi$
$311$	421.807i	1.35629i	−0.734927	+	0.678147i	$0.762783\pi$
$312$	0	0
$313$	−301.602	−0.963583	−0.481792	−	0.876286i	$-0.660014\pi$
$313$	−301.602	−0.963583	−0.481792	+	0.876286i	$0.660014\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	194.596	0.613869	0.306935	−	0.951731i	$-0.400697\pi$
$317$	194.596	0.613869	0.306935	+	0.951731i	$0.400697\pi$
$318$	0	0
$319$	− 504.370i	− 1.58110i
$320$	0	0
$321$	164.897	0.513699
$322$	0	0
$323$	− 77.0784i	− 0.238633i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	371.349i	1.13562i
$328$	0	0
$329$	549.799	1.67112
$330$	0	0
$331$	− 322.043i	− 0.972939i	−0.873698	−	0.486469i	$-0.838284\pi$
$331$	− 322.043i	− 0.972939i	0.873698	−	0.486469i	$-0.161716\pi$
$332$	0	0
$333$	185.398	0.556752
$334$	0	0
$335$	0	0
$336$	0	0
$337$	623.195	1.84924	0.924622	−	0.380885i	$-0.124381\pi$
$337$	623.195	1.84924	0.924622	+	0.380885i	$0.124381\pi$
$338$	0	0
$339$	− 324.931i	− 0.958498i
$340$	0	0
$341$	−166.496	−0.488259
$342$	0	0
$343$	27.3149i	0.0796353i
$344$	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
$349$	237.599	0.680799	0.340400	−	0.940281i	$-0.389438\pi$
$349$	237.599	0.680799	0.340400	+	0.940281i	$0.389438\pi$
$350$	0	0
$351$	5.19615i	0.0148039i
$352$	0	0
$353$	−235.298	−0.666567	−0.333284	−	0.942827i	$-0.608157\pi$
$353$	−235.298	−0.666567	−0.333284	+	0.942827i	$0.608157\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	133.504	0.373960
$358$	0	0
$359$	330.415i	0.920376i	0.887821	+	0.460188i	$0.152218\pi$
$359$	330.415i	0.920376i	−0.887821	+	0.460188i	$0.847782\pi$
$360$	0	0
$361$	265.799	0.736287
$362$	0	0
$363$	19.0526i	0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 316.792i	− 0.863193i	−0.902067	−	0.431596i	$-0.857951\pi$
$367$	− 316.792i	− 0.863193i	0.902067	−	0.431596i	$-0.142049\pi$
$368$	0	0
$369$	197.098	0.534140
$370$	0	0
$371$	− 289.778i	− 0.781072i
$372$	0	0
$373$	−103.396	−0.277201	−0.138601	−	0.990348i	$-0.544260\pi$
$373$	−103.396	−0.277201	−0.138601	+	0.990348i	$0.544260\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−43.8997	−0.116445
$378$	0	0
$379$	321.289i	0.847728i	0.905726	+	0.423864i	$0.139327\pi$
$379$	321.289i	0.847728i	−0.905726	+	0.423864i	$0.860673\pi$
$380$	0	0
$381$	263.398	0.691335
$382$	0	0
$383$	313.963i	0.819746i	0.912143	+	0.409873i	$0.134427\pi$
$383$	313.963i	0.819746i	−0.912143	+	0.409873i	$0.865573\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	143.587i	0.371025i
$388$	0	0
$389$	394.100	1.01311	0.506556	−	0.862207i	$-0.330918\pi$
$389$	394.100	1.01311	0.506556	+	0.862207i	$0.330918\pi$
$390$	0	0
$391$	− 73.4320i	− 0.187806i
$392$	0	0
$393$	−286.100	−0.727990
$394$	0	0
$395$	0	0
$396$	0	0
$397$	192.203	0.484139	0.242069	−	0.970259i	$-0.422174\pi$
$397$	192.203	0.484139	0.242069	+	0.970259i	$0.422174\pi$
$398$	0	0
$399$	− 164.892i	− 0.413263i
$400$	0	0
$401$	−711.895	−1.77530	−0.887649	−	0.460520i	$-0.847663\pi$
$401$	−711.895	−1.77530	−0.887649	+	0.460520i	$0.847663\pi$
$402$	0	0
$403$	14.4916i	0.0359594i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 710.022i	− 1.74453i
$408$	0	0
$409$	741.992	1.81416	0.907081	−	0.420956i	$-0.138305\pi$
$409$	741.992	1.81416	0.907081	+	0.420956i	$0.138305\pi$
$410$	0	0
$411$	341.904i	0.831884i
$412$	0	0
$413$	−955.393	−2.31330
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−102.997	−0.246996
$418$	0	0
$419$	− 169.567i	− 0.404695i	−0.979314	−	0.202348i	$-0.935143\pi$
$419$	− 169.567i	− 0.404695i	0.979314	−	0.202348i	$-0.0648571\pi$
$420$	0	0
$421$	229.799	0.545842	0.272921	−	0.962036i	$-0.412010\pi$
$421$	229.799	0.545842	0.272921	+	0.962036i	$0.412010\pi$
$422$	0	0
$423$	169.046i	0.399637i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	27.3149i	0.0639693i
$428$	0	0
$429$	19.8997	0.0463864
$430$	0	0
$431$	− 460.086i	− 1.06748i	−0.845647	−	0.533742i	$-0.820785\pi$
$431$	− 460.086i	− 1.06748i	0.845647	−	0.533742i	$-0.179215\pi$
$432$	0	0
$433$	559.992	1.29329	0.646643	−	0.762793i	$-0.276173\pi$
$433$	559.992	1.29329	0.646643	+	0.762793i	$0.276173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−90.6967	−0.207544
$438$	0	0
$439$	− 123.378i	− 0.281043i	−0.990078	−	0.140521i	$-0.955122\pi$
$439$	− 123.378i	− 0.281043i	0.990078	−	0.140521i	$-0.0448779\pi$
$440$	0	0
$441$	138.602	0.314289
$442$	0	0
$443$	− 264.771i	− 0.597677i	−0.954304	−	0.298838i	$-0.903401\pi$
$443$	− 264.771i	− 0.597677i	0.954304	−	0.298838i	$-0.0965991\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 134.058i	− 0.299906i
$448$	0	0
$449$	704.201	1.56838	0.784188	−	0.620524i	$-0.213080\pi$
$449$	704.201	1.56838	0.784188	+	0.620524i	$0.213080\pi$
$450$	0	0
$451$	− 754.827i	− 1.67367i
$452$	0	0
$453$	271.100	0.598455
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−26.0000	−0.0568928	−0.0284464	−	0.999595i	$-0.509056\pi$
$457$	−26.0000	−0.0568928	−0.0284464	+	0.999595i	$0.509056\pi$
$458$	0	0
$459$	41.0483i	0.0894298i
$460$	0	0
$461$	469.298	1.01800	0.509000	−	0.860766i	$-0.330015\pi$
$461$	469.298	1.01800	0.509000	+	0.860766i	$0.330015\pi$
$462$	0	0
$463$	756.673i	1.63428i	0.576437	+	0.817142i	$0.304443\pi$
$463$	756.673i	1.63428i	−0.576437	+	0.817142i	$0.695557\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 526.827i	− 1.12811i	−0.825737	−	0.564055i	$-0.809241\pi$
$467$	− 526.827i	− 1.12811i	0.825737	−	0.564055i	$-0.190759\pi$
$468$	0	0
$469$	−390.388	−0.832385
$470$	0	0
$471$	− 101.149i	− 0.214754i
$472$	0	0
$473$	549.895	1.16257
$474$	0	0
$475$	0	0
$476$	0	0
$477$	89.0977	0.186788
$478$	0	0
$479$	507.139i	1.05875i	0.848389	+	0.529373i	$0.177573\pi$
$479$	507.139i	1.05875i	−0.848389	+	0.529373i	$0.822427\pi$
$480$	0	0
$481$	−61.7995	−0.128481
$482$	0	0
$483$	− 157.091i	− 0.325241i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 53.1727i	− 0.109184i	−0.998509	−	0.0545920i	$-0.982614\pi$
$487$	− 53.1727i	− 0.109184i	0.998509	−	0.0545920i	$-0.0173858\pi$
$488$	0	0
$489$	−44.6992	−0.0914095
$490$	0	0
$491$	− 884.032i	− 1.80047i	−0.435402	−	0.900236i	$-0.643394\pi$
$491$	− 884.032i	− 1.80047i	0.435402	−	0.900236i	$-0.356606\pi$
$492$	0	0
$493$	−346.797	−0.703442
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−213.499	−0.429575
$498$	0	0
$499$	− 526.484i	− 1.05508i	−0.849531	−	0.527539i	$-0.823115\pi$
$499$	− 526.484i	− 1.05508i	0.849531	−	0.527539i	$-0.176885\pi$
$500$	0	0
$501$	−517.298	−1.03253
$502$	0	0
$503$	315.060i	0.626361i	0.949694	+	0.313181i	$0.101394\pi$
$503$	315.060i	0.626361i	−0.949694	+	0.313181i	$0.898606\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	290.985i	0.573934i
$508$	0	0
$509$	−278.201	−0.546563	−0.273281	−	0.961934i	$-0.588109\pi$
$509$	−278.201	−0.546563	−0.273281	+	0.961934i	$0.588109\pi$
$510$	0	0
$511$	1071.32i	2.09652i
$512$	0	0
$513$	50.6992	0.0988289
$514$	0	0
$515$	0	0
$516$	0	0
$517$	647.398	1.25222
$518$	0	0
$519$	− 143.930i	− 0.277321i
$520$	0	0
$521$	209.494	0.402099	0.201050	−	0.979581i	$-0.435565\pi$
$521$	209.494	0.402099	0.201050	+	0.979581i	$0.435565\pi$
$522$	0	0
$523$	79.9623i	0.152892i	0.997074	+	0.0764458i	$0.0243572\pi$
$523$	79.9623i	0.152892i	−0.997074	+	0.0764458i	$0.975643\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	114.480i	0.217230i
$528$	0	0
$529$	442.594	0.836662
$530$	0	0
$531$	− 293.754i	− 0.553209i
$532$	0	0
$533$	−65.6992	−0.123263
$534$	0	0
$535$	0	0
$536$	0	0
$537$	563.794	1.04990
$538$	0	0
$539$	− 530.803i	− 0.984793i
$540$	0	0
$541$	−420.995	−0.778179	−0.389090	−	0.921200i	$-0.627210\pi$
$541$	−420.995	−0.778179	−0.389090	+	0.921200i	$0.627210\pi$
$542$	0	0
$543$	432.661i	0.796798i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	609.097i	1.11352i	0.830672	+	0.556762i	$0.187956\pi$
$547$	609.097i	1.11352i	−0.830672	+	0.556762i	$0.812044\pi$
$548$	0	0
$549$	−8.39849	−0.0152978
$550$	0	0
$551$	428.333i	0.777374i
$552$	0	0
$553$	−921.594	−1.66654
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.59899	0.00287073	0.00143536	−	0.999999i	$-0.499543\pi$
$557$	1.59899	0.00287073	0.00143536	+	0.999999i	$0.499543\pi$
$558$	0	0
$559$	− 47.8622i	− 0.0856211i
$560$	0	0
$561$	157.203	0.280219
$562$	0	0
$563$	139.542i	0.247855i	0.992291	+	0.123927i	$0.0395490\pi$
$563$	139.542i	0.247855i	−0.992291	+	0.123927i	$0.960451\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	87.8137i	0.154874i
$568$	0	0
$569$	275.098	0.483476	0.241738	−	0.970342i	$-0.422283\pi$
$569$	275.098	0.483476	0.241738	+	0.970342i	$0.422283\pi$
$570$	0	0
$571$	305.467i	0.534969i	0.963562	+	0.267485i	$0.0861925\pi$
$571$	305.467i	0.534969i	−0.963562	+	0.267485i	$0.913808\pi$
$572$	0	0
$573$	−345.895	−0.603656
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−583.396	−1.01108	−0.505542	−	0.862802i	$-0.668708\pi$
$577$	−583.396	−1.01108	−0.505542	+	0.862802i	$0.668708\pi$
$578$	0	0
$579$	221.698i	0.382898i
$580$	0	0
$581$	709.789	1.22167
$582$	0	0
$583$	− 341.218i	− 0.585280i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	775.776i	1.32160i	0.750564	+	0.660798i	$0.229782\pi$
$587$	775.776i	1.32160i	−0.750564	+	0.660798i	$0.770218\pi$
$588$	0	0
$589$	141.396	0.240061
$590$	0	0
$591$	591.320i	1.00054i
$592$	0	0
$593$	1101.69	1.85782	0.928912	−	0.370301i	$-0.120746\pi$
$593$	1101.69	1.85782	0.928912	+	0.370301i	$0.120746\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	360.895	0.604514
$598$	0	0
$599$	718.797i	1.19999i	0.800002	+	0.599997i	$0.204832\pi$
$599$	718.797i	1.19999i	−0.800002	+	0.599997i	$0.795168\pi$
$600$	0	0
$601$	−72.3985	−0.120463	−0.0602317	−	0.998184i	$-0.519184\pi$
$601$	−72.3985	−0.120463	−0.0602317	+	0.998184i	$0.519184\pi$
$602$	0	0
$603$	− 120.032i	− 0.199059i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 122.861i	− 0.202407i	−0.994866	−	0.101204i	$-0.967731\pi$
$607$	− 122.861i	− 0.202407i	0.994866	−	0.101204i	$-0.0322694\pi$
$608$	0	0
$609$	−741.895	−1.21822
$610$	0	0
$611$	− 56.3488i	− 0.0922239i
$612$	0	0
$613$	908.396	1.48189	0.740943	−	0.671568i	$-0.234379\pi$
$613$	908.396	1.48189	0.740943	+	0.671568i	$0.234379\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	157.298	0.254940	0.127470	−	0.991842i	$-0.459314\pi$
$617$	157.298	0.254940	0.127470	+	0.991842i	$0.459314\pi$
$618$	0	0
$619$	203.628i	0.328963i	0.986380	+	0.164482i	$0.0525951\pi$
$619$	203.628i	0.328963i	−0.986380	+	0.164482i	$0.947405\pi$
$620$	0	0
$621$	48.3008	0.0777790
$622$	0	0
$623$	− 936.679i	− 1.50350i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 194.163i	− 0.309670i
$628$	0	0
$629$	−488.201	−0.776153
$630$	0	0
$631$	206.288i	0.326922i	0.986550	+	0.163461i	$0.0522658\pi$
$631$	206.288i	0.326922i	−0.986550	+	0.163461i	$0.947734\pi$
$632$	0	0
$633$	−69.3008	−0.109480
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−46.2005	−0.0725283
$638$	0	0
$639$	− 65.6443i	− 0.102730i
$640$	0	0
$641$	−114.206	−0.178168	−0.0890839	−	0.996024i	$-0.528394\pi$
$641$	−114.206	−0.178168	−0.0890839	+	0.996024i	$0.528394\pi$
$642$	0	0
$643$	27.3655i	0.0425591i	0.999774	+	0.0212796i	$0.00677401\pi$
$643$	27.3655i	0.0425591i	−0.999774	+	0.0212796i	$0.993226\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	486.821i	0.752427i	0.926533	+	0.376214i	$0.122774\pi$
$647$	486.821i	0.752427i	−0.926533	+	0.376214i	$0.877226\pi$
$648$	0	0
$649$	−1124.99	−1.73342
$650$	0	0
$651$	244.905i	0.376198i
$652$	0	0
$653$	−383.604	−0.587449	−0.293724	−	0.955890i	$-0.594895\pi$
$653$	−383.604	−0.587449	−0.293724	+	0.955890i	$0.594895\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−329.398	−0.501368
$658$	0	0
$659$	− 772.321i	− 1.17196i	−0.810326	−	0.585980i	$-0.800710\pi$
$659$	− 772.321i	− 1.17196i	0.810326	−	0.585980i	$-0.199290\pi$
$660$	0	0
$661$	−265.198	−0.401207	−0.200604	−	0.979672i	$-0.564290\pi$
$661$	−265.198	−0.401207	−0.200604	+	0.979672i	$0.564290\pi$
$662$	0	0
$663$	− 13.6828i	− 0.0206377i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	408.069i	0.611798i
$668$	0	0
$669$	216.499	0.323615
$670$	0	0
$671$	32.1638i	0.0479341i
$672$	0	0
$673$	−514.190	−0.764027	−0.382014	−	0.924157i	$-0.624769\pi$
$673$	−514.190	−0.764027	−0.382014	+	0.924157i	$0.624769\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−751.298	−1.10975	−0.554873	−	0.831935i	$-0.687233\pi$
$677$	−751.298	−1.10975	−0.554873	+	0.831935i	$0.687233\pi$
$678$	0	0
$679$	435.156i	0.640877i
$680$	0	0
$681$	−394.100	−0.578708
$682$	0	0
$683$	− 643.171i	− 0.941685i	−0.882217	−	0.470843i	$-0.843950\pi$
$683$	− 643.171i	− 0.941685i	0.882217	−	0.470843i	$-0.156050\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 642.929i	− 0.935851i
$688$	0	0
$689$	−29.6992	−0.0431049
$690$	0	0
$691$	65.4794i	0.0947603i	0.998877	+	0.0473801i	$0.0150872\pi$
$691$	65.4794i	0.0947603i	−0.998877	+	0.0473801i	$0.984913\pi$
$692$	0	0
$693$	336.301	0.485282
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−519.008	−0.744631
$698$	0	0
$699$	431.103i	0.616742i
$700$	0	0
$701$	194.105	0.276898	0.138449	−	0.990370i	$-0.455788\pi$
$701$	194.105	0.276898	0.138449	+	0.990370i	$0.455788\pi$
$702$	0	0
$703$	602.982i	0.857727i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 918.143i	− 1.29865i
$708$	0	0
$709$	−448.003	−0.631879	−0.315940	−	0.948779i	$-0.602320\pi$
$709$	−448.003	−0.631879	−0.315940	+	0.948779i	$0.602320\pi$
$710$	0	0
$711$	− 283.362i	− 0.398540i
$712$	0	0
$713$	134.707	0.188930
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−108.095	−0.150760
$718$	0	0
$719$	− 129.616i	− 0.180272i	−0.995929	−	0.0901362i	$-0.971270\pi$
$719$	− 129.616i	− 0.180272i	0.995929	−	0.0901362i	$-0.0287302\pi$
$720$	0	0
$721$	−562.195	−0.779744
$722$	0	0
$723$	− 334.286i	− 0.462359i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 481.213i	− 0.661917i	−0.943645	−	0.330958i	$-0.892628\pi$
$727$	− 481.213i	− 0.661917i	0.943645	−	0.330958i	$-0.107372\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 378.099i	− 0.517236i
$732$	0	0
$733$	776.797	1.05975	0.529875	−	0.848076i	$-0.322239\pi$
$733$	776.797	1.05975	0.529875	+	0.848076i	$0.322239\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−459.689	−0.623730
$738$	0	0
$739$	629.077i	0.851255i	0.904898	+	0.425627i	$0.139947\pi$
$739$	629.077i	0.851255i	−0.904898	+	0.425627i	$0.860053\pi$
$740$	0	0
$741$	−16.8997	−0.0228067
$742$	0	0
$743$	− 335.789i	− 0.451937i	−0.974135	−	0.225969i	$-0.927445\pi$
$743$	− 335.789i	− 0.451937i	0.974135	−	0.225969i	$-0.0725547\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	218.238i	0.292153i
$748$	0	0
$749$	−928.907	−1.24020
$750$	0	0
$751$	− 947.190i	− 1.26124i	−0.776092	−	0.630619i	$-0.782801\pi$
$751$	− 947.190i	− 1.26124i	0.776092	−	0.630619i	$-0.217199\pi$
$752$	0	0
$753$	−687.799	−0.913412
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−278.799	−0.368295	−0.184148	−	0.982899i	$-0.558952\pi$
$757$	−278.799	−0.368295	−0.184148	+	0.982899i	$0.558952\pi$
$758$	0	0
$759$	− 184.978i	− 0.243713i
$760$	0	0
$761$	−147.198	−0.193427	−0.0967135	−	0.995312i	$-0.530833\pi$
$761$	−147.198	−0.193427	−0.0967135	+	0.995312i	$0.530833\pi$
$762$	0	0
$763$	− 2091.90i	− 2.74168i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	97.9180i	0.127664i
$768$	0	0
$769$	−816.398	−1.06164	−0.530818	−	0.847486i	$-0.678115\pi$
$769$	−816.398	−1.06164	−0.530818	+	0.847486i	$0.678115\pi$
$770$	0	0
$771$	− 219.966i	− 0.285300i
$772$	0	0
$773$	926.090	1.19805	0.599023	−	0.800732i	$-0.295556\pi$
$773$	926.090	1.19805	0.599023	+	0.800732i	$0.295556\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1044.40	−1.34414
$778$	0	0
$779$	641.032i	0.822891i
$780$	0	0
$781$	−251.398	−0.321893
$782$	0	0
$783$	− 228.110i	− 0.291328i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 811.692i	− 1.03138i	−0.856777	−	0.515688i	$-0.827537\pi$
$787$	− 811.692i	− 1.03138i	0.856777	−	0.515688i	$-0.172463\pi$
$788$	0	0
$789$	216.095	0.273885
$790$	0	0
$791$	1830.42i	2.31405i
$792$	0	0
$793$	2.79950	0.00353026
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−949.393	−1.19121	−0.595604	−	0.803278i	$-0.703087\pi$
$797$	−949.393	−1.19121	−0.595604	+	0.803278i	$0.703087\pi$
$798$	0	0
$799$	− 445.141i	− 0.557123i
$800$	0	0
$801$	288.000	0.359551
$802$	0	0
$803$	1261.50i	1.57098i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 43.1320i	− 0.0534473i
$808$	0	0
$809$	−1028.90	−1.27181	−0.635907	−	0.771766i	$-0.719374\pi$
$809$	−1028.90	−1.27181	−0.635907	+	0.771766i	$0.719374\pi$
$810$	0	0
$811$	123.378i	0.152131i	0.997103	+	0.0760653i	$0.0242357\pi$
$811$	123.378i	0.152131i	−0.997103	+	0.0760653i	$0.975764\pi$
$812$	0	0
$813$	845.794	1.04034
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−466.995	−0.571597
$818$	0	0
$819$	− 29.2712i	− 0.0357402i
$820$	0	0
$821$	−527.193	−0.642135	−0.321068	−	0.947056i	$-0.604042\pi$
$821$	−527.193	−0.642135	−0.321068	+	0.947056i	$0.604042\pi$
$822$	0	0
$823$	− 804.535i	− 0.977564i	−0.872406	−	0.488782i	$-0.837441\pi$
$823$	− 804.535i	− 0.977564i	0.872406	−	0.488782i	$-0.162559\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1462.60i	1.76856i	0.466956	+	0.884281i	$0.345351\pi$
$827$	1462.60i	1.76856i	−0.466956	+	0.884281i	$0.654649\pi$
$828$	0	0
$829$	563.995	0.680332	0.340166	−	0.940365i	$-0.389517\pi$
$829$	563.995	0.680332	0.340166	+	0.940365i	$0.389517\pi$
$830$	0	0
$831$	− 313.501i	− 0.377258i
$832$	0	0
$833$	−364.972	−0.438142
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−75.3008	−0.0899651
$838$	0	0
$839$	1079.35i	1.28647i	0.765670	+	0.643234i	$0.222408\pi$
$839$	1079.35i	1.28647i	−0.765670	+	0.643234i	$0.777592\pi$
$840$	0	0
$841$	1086.19	1.29154
$842$	0	0
$843$	379.141i	0.449752i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 107.328i	− 0.126715i
$848$	0	0
$849$	81.0952	0.0955185
$850$	0	0
$851$	574.456i	0.675037i
$852$	0	0
$853$	498.789	0.584747	0.292374	−	0.956304i	$-0.405555\pi$
$853$	498.789	0.584747	0.292374	+	0.956304i	$0.405555\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1251.39	1.46020	0.730098	−	0.683342i	$-0.239474\pi$
$857$	1251.39	1.46020	0.730098	+	0.683342i	$0.239474\pi$
$858$	0	0
$859$	1026.87i	1.19543i	0.801709	+	0.597714i	$0.203924\pi$
$859$	1026.87i	1.19543i	−0.801709	+	0.597714i	$0.796076\pi$
$860$	0	0
$861$	−1110.30	−1.28955
$862$	0	0
$863$	133.372i	0.154545i	0.997010	+	0.0772725i	$0.0246211\pi$
$863$	133.372i	0.154545i	−0.997010	+	0.0772725i	$0.975379\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	392.472i	0.452679i
$868$	0	0
$869$	−1085.19	−1.24878
$870$	0	0
$871$	40.0108i	0.0459366i
$872$	0	0
$873$	−133.797	−0.153261
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−813.201	−0.927253	−0.463626	−	0.886031i	$-0.653452\pi$
$877$	−813.201	−0.927253	−0.463626	+	0.886031i	$0.653452\pi$
$878$	0	0
$879$	558.415i	0.635284i
$880$	0	0
$881$	−1671.39	−1.89715	−0.948575	−	0.316554i	$-0.897474\pi$
$881$	−1671.39	−1.89715	−0.948575	+	0.316554i	$0.897474\pi$
$882$	0	0
$883$	192.321i	0.217804i	0.994052	+	0.108902i	$0.0347335\pi$
$883$	192.321i	0.217804i	−0.994052	+	0.108902i	$0.965266\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1701.79i	1.91859i	0.282408	+	0.959294i	$0.408867\pi$
$887$	1701.79i	1.91859i	−0.282408	+	0.959294i	$0.591133\pi$
$888$	0	0
$889$	−1483.79	−1.66905
$890$	0	0
$891$	103.402i	0.116052i
$892$	0	0
$893$	−549.799	−0.615677
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−16.1003	−0.0179490
$898$	0	0
$899$	− 636.179i	− 0.707652i
$900$	0	0
$901$	−234.617	−0.260396
$902$	0	0
$903$	− 808.859i	− 0.895746i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 45.4993i	− 0.0501646i	−0.999685	−	0.0250823i	$-0.992015\pi$
$907$	− 45.4993i	− 0.0501646i	0.999685	−	0.0250823i	$-0.00798478\pi$
$908$	0	0
$909$	282.301	0.310562
$910$	0	0
$911$	421.697i	0.462895i	0.972847	+	0.231447i	$0.0743461\pi$
$911$	421.697i	0.462895i	−0.972847	+	0.231447i	$0.925654\pi$
$912$	0	0
$913$	835.789	0.915432
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1611.67	1.75755
$918$	0	0
$919$	1585.92i	1.72571i	0.505455	+	0.862853i	$0.331325\pi$
$919$	1585.92i	1.72571i	−0.505455	+	0.862853i	$0.668675\pi$
$920$	0	0
$921$	859.892	0.933651
$922$	0	0
$923$	21.8814i	0.0237069i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 172.858i	− 0.186470i
$928$	0	0
$929$	−485.494	−0.522598	−0.261299	−	0.965258i	$-0.584151\pi$
$929$	−485.494	−0.522598	−0.261299	+	0.965258i	$0.584151\pi$
$930$	0	0
$931$	450.782i	0.484191i
$932$	0	0
$933$	730.591	0.783056
$934$	0	0
$935$	0	0
$936$	0	0
$937$	972.393	1.03777	0.518887	−	0.854843i	$-0.326347\pi$
$937$	972.393	1.03777	0.518887	+	0.854843i	$0.326347\pi$
$938$	0	0
$939$	522.389i	0.556325i
$940$	0	0
$941$	360.807	0.383429	0.191715	−	0.981451i	$-0.438595\pi$
$941$	360.807	0.383429	0.191715	+	0.981451i	$0.438595\pi$
$942$	0	0
$943$	610.706i	0.647621i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	719.318i	0.759575i	0.925074	+	0.379788i	$0.124003\pi$
$947$	719.318i	0.759575i	−0.925074	+	0.379788i	$0.875997\pi$
$948$	0	0
$949$	109.799	0.115700
$950$	0	0
$951$	− 337.051i	− 0.354417i
$952$	0	0
$953$	1349.19	1.41573	0.707866	−	0.706347i	$-0.249658\pi$
$953$	1349.19	1.41573	0.707866	+	0.706347i	$0.249658\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−873.594	−0.912846
$958$	0	0
$959$	− 1926.03i	− 2.00838i
$960$	0	0
$961$	750.992	0.781470
$962$	0	0
$963$	− 285.610i	− 0.296584i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1584.69i	1.63877i	0.573241	+	0.819387i	$0.305686\pi$
$967$	1584.69i	1.63877i	−0.573241	+	0.819387i	$0.694314\pi$
$968$	0	0
$969$	−133.504	−0.137775
$970$	0	0
$971$	930.344i	0.958130i	0.877779	+	0.479065i	$0.159024\pi$
$971$	930.344i	0.958130i	−0.877779	+	0.479065i	$0.840976\pi$
$972$	0	0
$973$	580.211	0.596311
$974$	0	0
$975$	0	0
$976$	0	0
$977$	116.596	0.119341	0.0596707	−	0.998218i	$-0.480995\pi$
$977$	116.596	0.119341	0.0596707	+	0.998218i	$0.480995\pi$
$978$	0	0
$979$	− 1102.96i	− 1.12661i
$980$	0	0
$981$	643.195	0.655653
$982$	0	0
$983$	648.930i	0.660153i	0.943954	+	0.330076i	$0.107074\pi$
$983$	648.930i	0.660153i	−0.943954	+	0.330076i	$0.892926\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 952.281i	− 0.964823i
$988$	0	0
$989$	−444.902	−0.449851
$990$	0	0
$991$	− 1086.29i	− 1.09615i	−0.836428	−	0.548077i	$-0.815360\pi$
$991$	− 1086.29i	− 1.09615i	0.836428	−	0.548077i	$-0.184640\pi$
$992$	0	0
$993$	−557.794	−0.561727
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−779.203	−0.781548	−0.390774	−	0.920487i	$-0.627793\pi$
$997$	−779.203	−0.781548	−0.390774	+	0.920487i	$0.627793\pi$
$998$	0	0
$999$	− 321.120i	− 0.321441i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.e.m.751.2	yes	4
3.2	odd	2		3600.3.e.bf.3151.3		4
4.3	odd	2	inner	1200.3.e.m.751.3	yes	4
5.2	odd	4		1200.3.j.d.799.2		8
5.3	odd	4		1200.3.j.d.799.8		8
5.4	even	2		1200.3.e.k.751.3	yes	4
12.11	even	2		3600.3.e.bf.3151.2		4
15.2	even	4		3600.3.j.j.1999.3		8
15.8	even	4		3600.3.j.j.1999.5		8
15.14	odd	2		3600.3.e.bc.3151.2		4
20.3	even	4		1200.3.j.d.799.1		8
20.7	even	4		1200.3.j.d.799.7		8
20.19	odd	2		1200.3.e.k.751.2	✓	4
60.23	odd	4		3600.3.j.j.1999.4		8
60.47	odd	4		3600.3.j.j.1999.6		8
60.59	even	2		3600.3.e.bc.3151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1200.3.e.k.751.2	✓	4	20.19	odd	2
1200.3.e.k.751.3	yes	4	5.4	even	2
1200.3.e.m.751.2	yes	4	1.1	even	1	trivial
1200.3.e.m.751.3	yes	4	4.3	odd	2	inner
1200.3.j.d.799.1		8	20.3	even	4
1200.3.j.d.799.2		8	5.2	odd	4
1200.3.j.d.799.7		8	20.7	even	4
1200.3.j.d.799.8		8	5.3	odd	4
3600.3.e.bc.3151.2		4	15.14	odd	2
3600.3.e.bc.3151.3		4	60.59	even	2
3600.3.e.bf.3151.2		4	12.11	even	2
3600.3.e.bf.3151.3		4	3.2	odd	2
3600.3.j.j.1999.3		8	15.2	even	4
3600.3.j.j.1999.4		8	60.23	odd	4
3600.3.j.j.1999.5		8	15.8	even	4
3600.3.j.j.1999.6		8	60.47	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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +9.75707i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +9.75707i q^{7} -3.00000 q^{9} +11.4891i q^{11} +1.00000 q^{13} +7.89975 q^{17} -9.75707i q^{19} +16.8997 q^{21} -9.29548i q^{23} +5.19615i q^{27} -43.8997 q^{29} +14.4916i q^{31} +19.8997 q^{33} -61.7995 q^{37} -1.73205i q^{39} -65.6992 q^{41} -47.8622i q^{43} -56.3488i q^{47} -46.2005 q^{49} -13.6828i q^{51} -29.6992 q^{53} -16.8997 q^{57} +97.9180i q^{59} +2.79950 q^{61} -29.2712i q^{63} +40.0108i q^{67} -16.1003 q^{69} +21.8814i q^{71} +109.799 q^{73} -112.100 q^{77} +94.4539i q^{79} +9.00000 q^{81} -72.7461i q^{83} +76.0366i q^{87} -96.0000 q^{89} +9.75707i q^{91} +25.1003 q^{93} +44.5990 q^{97} -34.4674i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9}+O(q^{10}) \) 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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Database

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