LMFDB - Embedded newform 1200.3.c.l.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(449,1200)]

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([0, 0, 1, 1]))

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

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

chi := DirichletCharacter("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 18x^{10} + 75x^{8} + 1270x^{6} + 14397x^{4} - 7740x^{2} + 39204 $ x^12 + 18x^10 + 75x^8 + 1270x^6 + 14397x^4 - 7740*x^2 + 39204
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$0.994180 - 0.788754i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.l.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.40839 - 1.78875i) q^{3} -12.2014i q^{7} +(2.60072 + 8.61605i) q^{9} +O(q^{10})$	$q+(-2.40839 - 1.78875i) q^{3} -12.2014i q^{7} +(2.60072 + 8.61605i) q^{9} +8.79351i q^{11} +6.20144i q^{13} +16.6156 q^{17} -8.93396 q^{19} +(-21.8254 + 29.3859i) q^{21} +40.1210 q^{23} +(9.14843 - 25.4029i) q^{27} +4.94701i q^{29} +50.6043 q^{31} +(15.7294 - 21.1782i) q^{33} +41.3407i q^{37} +(11.0928 - 14.9355i) q^{39} +10.8654i q^{41} +39.5422i q^{43} -78.2993 q^{47} -99.8751 q^{49} +(-40.0170 - 29.7213i) q^{51} +69.7316 q^{53} +(21.5165 + 15.9807i) q^{57} -96.0731i q^{59} +26.7287 q^{61} +(105.128 - 31.7325i) q^{63} -66.5311i q^{67} +(-96.6273 - 71.7667i) q^{69} +34.7444i q^{71} -70.5344i q^{73} +107.293 q^{77} +31.2164 q^{79} +(-67.4725 + 44.8158i) q^{81} -71.5420 q^{83} +(8.84898 - 11.9143i) q^{87} -30.4885i q^{89} +75.6665 q^{91} +(-121.875 - 90.5187i) q^{93} -108.194i q^{97} +(-75.7653 + 22.8694i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 32 q^{9}+O(q^{10}) $ 12 * q - 32 * q^9	$ 12 q - 32 q^{9} + 100 q^{19} - 36 q^{21} + 228 q^{31} - 12 q^{39} - 152 q^{49} + 12 q^{51} + 124 q^{61} - 312 q^{69} - 152 q^{79} - 448 q^{81} + 620 q^{91} - 500 q^{99}+O(q^{100}) $ 12 * q - 32 * q^9 + 100 * q^19 - 36 * q^21 + 228 * q^31 - 12 * q^39 - 152 * q^49 + 12 * q^51 + 124 * q^61 - 312 * q^69 - 152 * q^79 - 448 * q^81 + 620 * q^91 - 500 * q^99

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
$3$	−2.40839	−	1.78875i	−0.802798	−	0.596251i
$3$	−2.40839	−	1.78875i	−0.802798	−	0.596251i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	12.2014i		−	1.74306i	−0.490340	−	0.871531i	$-0.663127\pi$
$7$		−	12.2014i		−	1.74306i	0.490340	−	0.871531i	$-0.336873\pi$
$8$		0			0
$9$	2.60072	+	8.61605i	0.288969	+	0.957339i
$10$		0			0
$11$			8.79351i			0.799410i	0.916644	+	0.399705i	$0.130887\pi$
$11$			8.79351i			0.799410i	−0.916644	+	0.399705i	$0.869113\pi$
$12$		0			0
$13$			6.20144i			0.477034i	0.971138	+	0.238517i	$0.0766612\pi$
$13$			6.20144i			0.477034i	−0.971138	+	0.238517i	$0.923339\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	16.6156			0.977391			0.488695	−	0.872455i	$-0.337473\pi$
$17$	16.6156			0.977391			0.488695	+	0.872455i	$0.337473\pi$
$18$		0			0
$19$	−8.93396			−0.470209			−0.235104	−	0.971970i	$-0.575543\pi$
$19$	−8.93396			−0.470209			−0.235104	+	0.971970i	$0.575543\pi$
$20$		0			0
$21$	−21.8254	+	29.3859i	−1.03930	+	1.39933i
$22$		0			0
$23$	40.1210			1.74439			0.872197	−	0.489156i	$-0.162695\pi$
$23$	40.1210			1.74439			0.872197	+	0.489156i	$0.162695\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	9.14843	−	25.4029i	0.338831	−	0.940847i
$28$		0			0
$29$			4.94701i			0.170587i	0.996356	+	0.0852933i	$0.0271827\pi$
$29$			4.94701i			0.170587i	−0.996356	+	0.0852933i	$0.972817\pi$
$30$		0			0
$31$	50.6043			1.63240			0.816199	−	0.577771i	$-0.196077\pi$
$31$	50.6043			1.63240			0.816199	+	0.577771i	$0.196077\pi$
$32$		0			0
$33$	15.7294	−	21.1782i	0.476649	−	0.641764i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			41.3407i			1.11732i	0.829398	+	0.558658i	$0.188684\pi$
$37$			41.3407i			1.11732i	−0.829398	+	0.558658i	$0.811316\pi$
$38$		0			0
$39$	11.0928	−	14.9355i	0.284432	−	0.382962i
$40$		0			0
$41$			10.8654i			0.265010i	0.991182	+	0.132505i	$0.0423020\pi$
$41$			10.8654i			0.265010i	−0.991182	+	0.132505i	$0.957698\pi$
$42$		0			0
$43$			39.5422i			0.919585i	0.888026	+	0.459792i	$0.152076\pi$
$43$			39.5422i			0.919585i	−0.888026	+	0.459792i	$0.847924\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−78.2993			−1.66594			−0.832972	−	0.553316i	$-0.813362\pi$
$47$	−78.2993			−1.66594			−0.832972	+	0.553316i	$0.813362\pi$
$48$		0			0
$49$	−99.8751			−2.03827
$50$		0			0
$51$	−40.0170	−	29.7213i	−0.784647	−	0.582770i
$52$		0			0
$53$	69.7316			1.31569			0.657845	−	0.753153i	$-0.271468\pi$
$53$	69.7316			1.31569			0.657845	+	0.753153i	$0.271468\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	21.5165	+	15.9807i	0.377482	+	0.280362i
$58$		0			0
$59$		−	96.0731i		−	1.62836i	−0.580614	−	0.814179i	$-0.697187\pi$
$59$		−	96.0731i		−	1.62836i	0.580614	−	0.814179i	$-0.302813\pi$
$60$		0			0
$61$	26.7287			0.438175			0.219087	−	0.975705i	$-0.429692\pi$
$61$	26.7287			0.438175			0.219087	+	0.975705i	$0.429692\pi$
$62$		0			0
$63$	105.128	−	31.7325i	1.66870	−	0.503691i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	66.5311i		−	0.993001i	−0.868036	−	0.496501i	$-0.834618\pi$
$67$		−	66.5311i		−	0.993001i	0.868036	−	0.496501i	$-0.165382\pi$
$68$		0			0
$69$	−96.6273	−	71.7667i	−1.40040	−	1.04010i
$70$		0			0
$71$			34.7444i			0.489357i	0.969604	+	0.244679i	$0.0786824\pi$
$71$			34.7444i			0.489357i	−0.969604	+	0.244679i	$0.921318\pi$
$72$		0			0
$73$		−	70.5344i		−	0.966225i	−0.875558	−	0.483112i	$-0.839506\pi$
$73$		−	70.5344i		−	0.966225i	0.875558	−	0.483112i	$-0.160494\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	107.293			1.39342
$78$		0			0
$79$	31.2164			0.395144			0.197572	−	0.980288i	$-0.436694\pi$
$79$	31.2164			0.395144			0.197572	+	0.980288i	$0.436694\pi$
$80$		0			0
$81$	−67.4725	+	44.8158i	−0.832994	+	0.553282i
$82$		0			0
$83$	−71.5420			−0.861951			−0.430976	−	0.902364i	$-0.641831\pi$
$83$	−71.5420			−0.861951			−0.430976	+	0.902364i	$0.641831\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	8.84898	−	11.9143i	0.101712	−	0.136947i
$88$		0			0
$89$		−	30.4885i		−	0.342568i	−0.985222	−	0.171284i	$-0.945208\pi$
$89$		−	30.4885i		−	0.342568i	0.985222	−	0.171284i	$-0.0547916\pi$
$90$		0			0
$91$	75.6665			0.831500
$92$		0			0
$93$	−121.875	−	90.5187i	−1.31049	−	0.973319i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	108.194i		−	1.11540i	−0.830043	−	0.557699i	$-0.811684\pi$
$97$		−	108.194i		−	1.11540i	0.830043	−	0.557699i	$-0.188316\pi$
$98$		0			0
$99$	−75.7653	+	22.8694i	−0.765306	+	0.231005i
$100$		0			0
$101$			30.7501i			0.304456i	0.988345	+	0.152228i	$0.0486448\pi$
$101$			30.7501i			0.304456i	−0.988345	+	0.152228i	$0.951355\pi$
$102$		0			0
$103$		−	49.7358i		−	0.482872i	−0.970417	−	0.241436i	$-0.922382\pi$
$103$		−	49.7358i		−	0.482872i	0.970417	−	0.241436i	$-0.0776184\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	188.098			1.75792			0.878961	−	0.476894i	$-0.158238\pi$
$107$	188.098			1.75792			0.878961	+	0.476894i	$0.158238\pi$
$108$		0			0
$109$	72.7585			0.667509			0.333755	−	0.942660i	$-0.391684\pi$
$109$	72.7585			0.667509			0.333755	+	0.942660i	$0.391684\pi$
$110$		0			0
$111$	73.9484	−	99.5647i	0.666201	−	0.896979i
$112$		0			0
$113$	79.0278			0.699361			0.349681	−	0.936869i	$-0.386290\pi$
$113$	79.0278			0.699361			0.349681	+	0.936869i	$0.386290\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−53.4319	+	16.1282i	−0.456683	+	0.137848i
$118$		0			0
$119$		−	202.735i		−	1.70365i
$120$		0			0
$121$	43.6742			0.360944
$122$		0			0
$123$	19.4355	−	26.1681i	0.158012	−	0.212749i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	194.410i		−	1.53078i	−0.643565	−	0.765392i	$-0.722545\pi$
$127$		−	194.410i		−	1.53078i	0.643565	−	0.765392i	$-0.277455\pi$
$128$		0			0
$129$	70.7312	−	95.2331i	0.548304	−	0.738241i
$130$		0			0
$131$			221.663i			1.69209i	0.533114	+	0.846044i	$0.321022\pi$
$131$			221.663i			1.69209i	−0.533114	+	0.846044i	$0.678978\pi$
$132$		0			0
$133$			109.007i			0.819603i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	254.898			1.86057			0.930285	−	0.366839i	$-0.119560\pi$
$137$	254.898			1.86057			0.930285	+	0.366839i	$0.119560\pi$
$138$		0			0
$139$	−103.142			−0.742029			−0.371015	−	0.928627i	$-0.620990\pi$
$139$	−103.142			−0.742029			−0.371015	+	0.928627i	$0.620990\pi$
$140$		0			0
$141$	188.576	+	140.058i	1.33742	+	0.993321i
$142$		0			0
$143$	−54.5324			−0.381345
$144$		0			0
$145$		0			0
$146$		0			0
$147$	240.539	+	178.652i	1.63632	+	1.21532i
$148$		0			0
$149$		−	130.351i		−	0.874836i	−0.899258	−	0.437418i	$-0.855893\pi$
$149$		−	130.351i		−	0.874836i	0.899258	−	0.437418i	$-0.144107\pi$
$150$		0			0
$151$	−82.4800			−0.546225			−0.273113	−	0.961982i	$-0.588053\pi$
$151$	−82.4800			−0.546225			−0.273113	+	0.961982i	$0.588053\pi$
$152$		0			0
$153$	43.2126	+	143.161i	0.282435	+	0.935694i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	189.015i		−	1.20392i	−0.798527	−	0.601958i	$-0.794387\pi$
$157$		−	189.015i		−	1.20392i	0.798527	−	0.601958i	$-0.205613\pi$
$158$		0			0
$159$	−167.941	−	124.733i	−1.05623	−	0.784482i
$160$		0			0
$161$		−	489.534i		−	3.04059i
$162$		0			0
$163$		−	222.963i		−	1.36787i	−0.729543	−	0.683935i	$-0.760267\pi$
$163$		−	222.963i		−	1.36787i	0.729543	−	0.683935i	$-0.239733\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−96.2445			−0.576314			−0.288157	−	0.957583i	$-0.593043\pi$
$167$	−96.2445			−0.576314			−0.288157	+	0.957583i	$0.593043\pi$
$168$		0			0
$169$	130.542			0.772439
$170$		0			0
$171$	−23.2347	−	76.9754i	−0.135876	−	0.450149i
$172$		0			0
$173$	−61.4068			−0.354952			−0.177476	−	0.984125i	$-0.556793\pi$
$173$	−61.4068			−0.354952			−0.177476	+	0.984125i	$0.556793\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−171.851	+	231.382i	−0.970911	+	1.30724i
$178$		0			0
$179$			172.095i			0.961425i	0.876878	+	0.480713i	$0.159622\pi$
$179$			172.095i			0.961425i	−0.876878	+	0.480713i	$0.840378\pi$
$180$		0			0
$181$	101.512			0.560841			0.280421	−	0.959877i	$-0.409526\pi$
$181$	101.512			0.560841			0.280421	+	0.959877i	$0.409526\pi$
$182$		0			0
$183$	−64.3731	−	47.8110i	−0.351766	−	0.261262i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			146.110i			0.781336i
$188$		0			0
$189$	−309.952	−	111.624i	−1.63996	−	0.590603i
$190$		0			0
$191$			46.5031i			0.243472i	0.992563	+	0.121736i	$0.0388461\pi$
$191$			46.5031i			0.243472i	−0.992563	+	0.121736i	$0.961154\pi$
$192$		0			0
$193$		−	241.951i		−	1.25363i	−0.779167	−	0.626816i	$-0.784358\pi$
$193$		−	241.951i		−	1.25363i	0.779167	−	0.626816i	$-0.215642\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−375.594			−1.90657			−0.953285	−	0.302074i	$-0.902321\pi$
$197$	−375.594			−1.90657			−0.953285	+	0.302074i	$0.902321\pi$
$198$		0			0
$199$	−75.0149			−0.376959			−0.188480	−	0.982077i	$-0.560356\pi$
$199$	−75.0149			−0.376959			−0.188480	+	0.982077i	$0.560356\pi$
$200$		0			0
$201$	−119.008	+	160.233i	−0.592078	+	0.797179i
$202$		0			0
$203$	60.3606			0.297343
$204$		0			0
$205$		0			0
$206$		0			0
$207$	104.344	+	345.685i	0.504075	+	1.66997i
$208$		0			0
$209$		−	78.5609i		−	0.375889i
$210$		0			0
$211$	−64.6919			−0.306597			−0.153298	−	0.988180i	$-0.548990\pi$
$211$	−64.6919			−0.306597			−0.153298	+	0.988180i	$0.548990\pi$
$212$		0			0
$213$	62.1491	−	83.6781i	0.291780	−	0.392855i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	617.446i		−	2.84537i
$218$		0			0
$219$	−126.169	+	169.875i	−0.576113	+	0.775683i
$220$		0			0
$221$			103.041i			0.466248i
$222$		0			0
$223$			250.487i			1.12326i	0.827389	+	0.561629i	$0.189825\pi$
$223$			250.487i			1.12326i	−0.827389	+	0.561629i	$0.810175\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−191.981			−0.845733			−0.422866	−	0.906192i	$-0.638976\pi$
$227$	−191.981			−0.845733			−0.422866	+	0.906192i	$0.638976\pi$
$228$		0			0
$229$	112.598			0.491693			0.245846	−	0.969309i	$-0.420934\pi$
$229$	112.598			0.491693			0.245846	+	0.969309i	$0.420934\pi$
$230$		0			0
$231$	−258.405	−	191.922i	−1.11864	−	0.830829i
$232$		0			0
$233$	402.723			1.72843			0.864213	−	0.503125i	$-0.167817\pi$
$233$	402.723			1.72843			0.864213	+	0.503125i	$0.167817\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−75.1813	−	55.8384i	−0.317221	−	0.235605i
$238$		0			0
$239$		−	99.9740i		−	0.418301i	−0.977883	−	0.209151i	$-0.932930\pi$
$239$		−	99.9740i		−	0.418301i	0.977883	−	0.209151i	$-0.0670699\pi$
$240$		0			0
$241$	1.82898			0.00758914			0.00379457	−	0.999993i	$-0.498792\pi$
$241$	1.82898			0.00758914			0.00379457	+	0.999993i	$0.498792\pi$
$242$		0			0
$243$	242.665	+	12.7575i	0.998621	+	0.0525001i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	55.4034i		−	0.224305i
$248$		0			0
$249$	172.301	+	127.971i	0.691973	+	0.513940i
$250$		0			0
$251$			349.307i			1.39166i	0.718206	+	0.695831i	$0.244964\pi$
$251$			349.307i			1.39166i	−0.718206	+	0.695831i	$0.755036\pi$
$252$		0			0
$253$			352.805i			1.39448i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	12.7334			0.0495463			0.0247731	−	0.999693i	$-0.492114\pi$
$257$	12.7334			0.0495463			0.0247731	+	0.999693i	$0.492114\pi$
$258$		0			0
$259$	504.416			1.94755
$260$		0			0
$261$	−42.6237	+	12.8658i	−0.163309	+	0.0492942i
$262$		0			0
$263$	−188.806			−0.717892			−0.358946	−	0.933358i	$-0.616864\pi$
$263$	−188.806			−0.717892			−0.358946	+	0.933358i	$0.616864\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−54.5365	+	73.4284i	−0.204257	+	0.275013i
$268$		0			0
$269$			113.044i			0.420237i	0.977676	+	0.210118i	$0.0673849\pi$
$269$			113.044i			0.420237i	−0.977676	+	0.210118i	$0.932615\pi$
$270$		0			0
$271$	−22.6737			−0.0836667			−0.0418334	−	0.999125i	$-0.513320\pi$
$271$	−22.6737			−0.0836667			−0.0418334	+	0.999125i	$0.513320\pi$
$272$		0			0
$273$	−182.235	−	135.349i	−0.667526	−	0.495783i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			341.284i			1.23207i	0.787719	+	0.616035i	$0.211262\pi$
$277$			341.284i			1.23207i	−0.787719	+	0.616035i	$0.788738\pi$
$278$		0			0
$279$	131.608	+	436.009i	0.471712	+	1.56276i
$280$		0			0
$281$		−	5.77214i		−	0.0205414i	−0.999947	−	0.0102707i	$-0.996731\pi$
$281$		−	5.77214i		−	0.0205414i	0.999947	−	0.0102707i	$-0.00326933\pi$
$282$		0			0
$283$			14.9406i			0.0527937i	0.999652	+	0.0263968i	$0.00840335\pi$
$283$			14.9406i			0.0527937i	−0.999652	+	0.0263968i	$0.991597\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	132.573			0.461928
$288$		0			0
$289$	−12.9205			−0.0447075
$290$		0			0
$291$	−193.532	+	260.573i	−0.665058	+	0.895440i
$292$		0			0
$293$	6.23602			0.0212833			0.0106417	−	0.999943i	$-0.496613\pi$
$293$	6.23602			0.0212833			0.0106417	+	0.999943i	$0.496613\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	223.380	+	80.4468i	0.752123	+	0.270865i
$298$		0			0
$299$			248.808i			0.832134i
$300$		0			0
$301$	482.471			1.60289
$302$		0			0
$303$	55.0043	−	74.0583i	0.181532	−	0.244417i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	283.513i		−	0.923496i	−0.887011	−	0.461748i	$-0.847222\pi$
$307$		−	283.513i		−	0.923496i	0.887011	−	0.461748i	$-0.152778\pi$
$308$		0			0
$309$	−88.9652	+	119.784i	−0.287913	+	0.387649i
$310$		0			0
$311$		−	108.620i		−	0.349260i	−0.984634	−	0.174630i	$-0.944127\pi$
$311$		−	108.620i		−	0.349260i	0.984634	−	0.174630i	$-0.0558729\pi$
$312$		0			0
$313$			387.197i			1.23705i	0.785765	+	0.618525i	$0.212270\pi$
$313$			387.197i			1.23705i	−0.785765	+	0.618525i	$0.787730\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	118.772			0.374675			0.187338	−	0.982296i	$-0.440014\pi$
$317$	118.772			0.374675			0.187338	+	0.982296i	$0.440014\pi$
$318$		0			0
$319$	−43.5016			−0.136369
$320$		0			0
$321$	−453.013	−	336.460i	−1.41126	−	1.04816i
$322$		0			0
$323$	−148.444			−0.459577
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−175.231	−	130.147i	−0.535875	−	0.398003i
$328$		0			0
$329$			955.365i			2.90384i
$330$		0			0
$331$	494.645			1.49439			0.747197	−	0.664602i	$-0.231399\pi$
$331$	494.645			1.49439			0.747197	+	0.664602i	$0.231399\pi$
$332$		0			0
$333$	−356.193	+	107.516i	−1.06965	+	0.322870i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			14.6414i			0.0434463i	0.999764	+	0.0217231i	$0.00691523\pi$
$337$			14.6414i			0.0434463i	−0.999764	+	0.0217231i	$0.993085\pi$
$338$		0			0
$339$	−190.330	−	141.361i	−0.561446	−	0.416995i
$340$		0			0
$341$			444.989i			1.30495i
$342$		0			0
$343$			620.750i			1.80977i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−146.105			−0.421053			−0.210526	−	0.977588i	$-0.567518\pi$
$347$	−146.105			−0.421053			−0.210526	+	0.977588i	$0.567518\pi$
$348$		0			0
$349$	594.189			1.70255			0.851273	−	0.524723i	$-0.175831\pi$
$349$	594.189			1.70255			0.851273	+	0.524723i	$0.175831\pi$
$350$		0			0
$351$	157.534	+	56.7334i	0.448816	+	0.161634i
$352$		0			0
$353$	−81.0080			−0.229484			−0.114742	−	0.993395i	$-0.536604\pi$
$353$	−81.0080			−0.229484			−0.114742	+	0.993395i	$0.536604\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−362.643	+	488.265i	−1.01581	+	1.36769i
$358$		0			0
$359$			290.075i			0.808009i	0.914757	+	0.404005i	$0.132382\pi$
$359$			290.075i			0.808009i	−0.914757	+	0.404005i	$0.867618\pi$
$360$		0			0
$361$	−281.184			−0.778904
$362$		0			0
$363$	−105.185	−	78.1224i	−0.289765	−	0.215213i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	168.056i		−	0.457918i	−0.973436	−	0.228959i	$-0.926468\pi$
$367$		−	168.056i		−	0.457918i	0.973436	−	0.228959i	$-0.0735322\pi$
$368$		0			0
$369$	−93.6167	+	28.2578i	−0.253704	+	0.0765795i
$370$		0			0
$371$		−	850.826i		−	2.29333i
$372$		0			0
$373$			237.510i			0.636756i	0.947964	+	0.318378i	$0.103138\pi$
$373$			237.510i			0.636756i	−0.947964	+	0.318378i	$0.896862\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−30.6786			−0.0813756
$378$		0			0
$379$	136.385			0.359854			0.179927	−	0.983680i	$-0.442414\pi$
$379$	136.385			0.359854			0.179927	+	0.983680i	$0.442414\pi$
$380$		0			0
$381$	−347.751	+	468.215i	−0.912732	+	1.22891i
$382$		0			0
$383$	202.859			0.529658			0.264829	−	0.964295i	$-0.414685\pi$
$383$	202.859			0.529658			0.264829	+	0.964295i	$0.414685\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−340.697	+	102.838i	−0.880354	+	0.265731i
$388$		0			0
$389$		−	60.7715i		−	0.156225i	−0.996945	−	0.0781125i	$-0.975111\pi$
$389$		−	60.7715i		−	0.156225i	0.996945	−	0.0781125i	$-0.0248893\pi$
$390$		0			0
$391$	666.637			1.70495
$392$		0			0
$393$	396.501	−	533.853i	1.00891	−	1.35840i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			456.360i			1.14952i	0.818321	+	0.574761i	$0.194905\pi$
$397$			456.360i			1.14952i	−0.818321	+	0.574761i	$0.805095\pi$
$398$		0			0
$399$	194.987	−	262.532i	0.488689	−	0.657976i
$400$		0			0
$401$		−	479.363i		−	1.19542i	−0.801712	−	0.597710i	$-0.796077\pi$
$401$		−	479.363i		−	1.19542i	0.801712	−	0.597710i	$-0.203923\pi$
$402$		0			0
$403$			313.820i			0.778709i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−363.530			−0.893194
$408$		0			0
$409$	45.2033			0.110522			0.0552608	−	0.998472i	$-0.482401\pi$
$409$	45.2033			0.110522			0.0552608	+	0.998472i	$0.482401\pi$
$410$		0			0
$411$	−613.895	−	455.950i	−1.49366	−	1.10937i
$412$		0			0
$413$	−1172.23			−2.83833
$414$		0			0
$415$		0			0
$416$		0			0
$417$	248.407	+	184.496i	0.595699	+	0.442436i
$418$		0			0
$419$		−	199.218i		−	0.475461i	−0.971331	−	0.237730i	$-0.923597\pi$
$419$		−	199.218i		−	0.475461i	0.971331	−	0.237730i	$-0.0764034\pi$
$420$		0			0
$421$	−73.7967			−0.175289			−0.0876445	−	0.996152i	$-0.527934\pi$
$421$	−73.7967			−0.175289			−0.0876445	+	0.996152i	$0.527934\pi$
$422$		0			0
$423$	−203.635	−	674.631i	−0.481406	−	1.59487i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	326.128i		−	0.763766i
$428$		0			0
$429$	131.335	+	97.5450i	0.306143	+	0.227378i
$430$		0			0
$431$			42.2198i			0.0979577i	0.998800	+	0.0489789i	$0.0155967\pi$
$431$			42.2198i			0.0979577i	−0.998800	+	0.0489789i	$0.984403\pi$
$432$		0			0
$433$		−	434.287i		−	1.00297i	−0.865166	−	0.501486i	$-0.832787\pi$
$433$		−	434.287i		−	1.00297i	0.865166	−	0.501486i	$-0.167213\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−358.440			−0.820228
$438$		0			0
$439$	739.972			1.68558			0.842792	−	0.538239i	$-0.180910\pi$
$439$	739.972			1.68558			0.842792	+	0.538239i	$0.180910\pi$
$440$		0			0
$441$	−259.747	−	860.529i	−0.588996	−	1.95131i
$442$		0			0
$443$	−127.665			−0.288184			−0.144092	−	0.989564i	$-0.546026\pi$
$443$	−127.665			−0.288184			−0.144092	+	0.989564i	$0.546026\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−233.165	+	313.935i	−0.521622	+	0.702316i
$448$		0			0
$449$		−	200.120i		−	0.445701i	−0.974853	−	0.222850i	$-0.928464\pi$
$449$		−	200.120i		−	0.445701i	0.974853	−	0.222850i	$-0.0715361\pi$
$450$		0			0
$451$	−95.5449			−0.211851
$452$		0			0
$453$	198.644	+	147.536i	0.438508	+	0.325687i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			364.615i			0.797845i	0.916985	+	0.398923i	$0.130616\pi$
$457$			364.615i			0.797845i	−0.916985	+	0.398923i	$0.869384\pi$
$458$		0			0
$459$	152.007	−	422.085i	0.331170	−	0.919575i
$460$		0			0
$461$			39.0715i			0.0847538i	0.999102	+	0.0423769i	$0.0134930\pi$
$461$			39.0715i			0.0847538i	−0.999102	+	0.0423769i	$0.986507\pi$
$462$		0			0
$463$		−	403.683i		−	0.871887i	−0.899974	−	0.435943i	$-0.856415\pi$
$463$		−	403.683i		−	0.871887i	0.899974	−	0.435943i	$-0.143585\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−535.689			−1.14709			−0.573543	−	0.819175i	$-0.694432\pi$
$467$	−535.689			−1.14709			−0.573543	+	0.819175i	$0.694432\pi$
$468$		0			0
$469$	−811.775			−1.73086
$470$		0			0
$471$	−338.101	+	455.222i	−0.717837	+	0.966502i
$472$		0			0
$473$	−347.714			−0.735125
$474$		0			0
$475$		0			0
$476$		0			0
$477$	181.352	+	600.811i	0.380194	+	1.25956i
$478$		0			0
$479$			768.476i			1.60433i	0.597099	+	0.802167i	$0.296320\pi$
$479$			768.476i			1.60433i	−0.597099	+	0.802167i	$0.703680\pi$
$480$		0			0
$481$	−256.372			−0.532998
$482$		0			0
$483$	−875.657	+	1178.99i	−1.81295	+	2.44098i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			130.785i			0.268553i	0.990944	+	0.134276i	$0.0428710\pi$
$487$			130.785i			0.268553i	−0.990944	+	0.134276i	$0.957129\pi$
$488$		0			0
$489$	−398.825	+	536.982i	−0.815594	+	1.09812i
$490$		0			0
$491$		−	576.106i		−	1.17333i	−0.809829	−	0.586666i	$-0.800440\pi$
$491$		−	576.106i		−	1.17333i	0.809829	−	0.586666i	$-0.199560\pi$
$492$		0			0
$493$			82.1977i			0.166730i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	423.931			0.852980
$498$		0			0
$499$	569.803			1.14189			0.570945	−	0.820988i	$-0.306577\pi$
$499$	569.803			1.14189			0.570945	+	0.820988i	$0.306577\pi$
$500$		0			0
$501$	231.795	+	172.158i	0.462664	+	0.343628i
$502$		0			0
$503$	167.894			0.333784			0.166892	−	0.985975i	$-0.446627\pi$
$503$	167.894			0.333784			0.166892	+	0.985975i	$0.446627\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−314.397	−	233.508i	−0.620112	−	0.460568i
$508$		0			0
$509$		−	422.910i		−	0.830865i	−0.909624	−	0.415433i	$-0.863630\pi$
$509$		−	422.910i		−	0.830865i	0.909624	−	0.415433i	$-0.136370\pi$
$510$		0			0
$511$	−860.621			−1.68419
$512$		0			0
$513$	−81.7317	+	226.948i	−0.159321	+	0.442394i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	688.526i		−	1.33177i
$518$		0			0
$519$	147.892	+	109.842i	0.284955	+	0.211641i
$520$		0			0
$521$			412.630i			0.791995i	0.918251	+	0.395998i	$0.129601\pi$
$521$			412.630i			0.791995i	−0.918251	+	0.395998i	$0.870399\pi$
$522$		0			0
$523$			38.1647i			0.0729727i	0.999334	+	0.0364864i	$0.0116165\pi$
$523$			38.1647i			0.0729727i	−0.999334	+	0.0364864i	$0.988383\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	840.823			1.59549
$528$		0			0
$529$	1080.70			2.04291
$530$		0			0
$531$	827.771	−	249.859i	1.55889	−	0.470545i
$532$		0			0
$533$	−67.3811			−0.126419
$534$		0			0
$535$		0			0
$536$		0			0
$537$	307.836	−	414.473i	0.573251	−	0.771830i
$538$		0			0
$539$		−	878.253i		−	1.62941i
$540$		0			0
$541$	671.986			1.24212			0.621059	−	0.783764i	$-0.286703\pi$
$541$	671.986			1.24212			0.621059	+	0.783764i	$0.286703\pi$
$542$		0			0
$543$	−244.482	−	181.580i	−0.450242	−	0.334402i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			56.8126i			0.103862i	0.998651	+	0.0519311i	$0.0165376\pi$
$547$			56.8126i			0.103862i	−0.998651	+	0.0519311i	$0.983462\pi$
$548$		0			0
$549$	69.5137	+	230.295i	0.126619	+	0.419481i
$550$		0			0
$551$		−	44.1964i		−	0.0802113i
$552$		0			0
$553$		−	380.885i		−	0.688761i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−240.349			−0.431507			−0.215753	−	0.976448i	$-0.569221\pi$
$557$	−240.349			−0.431507			−0.215753	+	0.976448i	$0.569221\pi$
$558$		0			0
$559$	−245.218			−0.438673
$560$		0			0
$561$	261.354	−	351.890i	0.465872	−	0.627255i
$562$		0			0
$563$	156.807			0.278521			0.139261	−	0.990256i	$-0.455527\pi$
$563$	156.807			0.278521			0.139261	+	0.990256i	$0.455527\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	546.818	+	823.262i	0.964405	+	1.45196i
$568$		0			0
$569$		−	457.623i		−	0.804258i	−0.915583	−	0.402129i	$-0.868270\pi$
$569$		−	457.623i		−	0.804258i	0.915583	−	0.402129i	$-0.131730\pi$
$570$		0			0
$571$	609.680			1.06774			0.533870	−	0.845566i	$-0.320737\pi$
$571$	609.680			1.06774			0.533870	+	0.845566i	$0.320737\pi$
$572$		0			0
$573$	83.1827	−	111.998i	0.145170	−	0.195459i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	565.696i		−	0.980409i	−0.871608	−	0.490204i	$-0.836922\pi$
$577$		−	565.696i		−	0.980409i	0.871608	−	0.490204i	$-0.163078\pi$
$578$		0			0
$579$	−432.791	+	582.714i	−0.747480	+	1.00641i
$580$		0			0
$581$			872.915i			1.50244i
$582$		0			0
$583$			613.185i			1.05178i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	555.622			0.946545			0.473272	−	0.880916i	$-0.343073\pi$
$587$	555.622			0.946545			0.473272	+	0.880916i	$0.343073\pi$
$588$		0			0
$589$	−452.097			−0.767567
$590$		0			0
$591$	904.578	+	671.845i	1.53059	+	1.13679i
$592$		0			0
$593$	166.238			0.280333			0.140167	−	0.990128i	$-0.455236\pi$
$593$	166.238			0.280333			0.140167	+	0.990128i	$0.455236\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	180.665	+	134.183i	0.302622	+	0.224763i
$598$		0			0
$599$			629.119i			1.05028i	0.851015	+	0.525141i	$0.175987\pi$
$599$			629.119i			1.05028i	−0.851015	+	0.525141i	$0.824013\pi$
$600$		0			0
$601$	−616.381			−1.02559			−0.512796	−	0.858511i	$-0.671390\pi$
$601$	−616.381			−1.02559			−0.512796	+	0.858511i	$0.671390\pi$
$602$		0			0
$603$	573.235	−	173.029i	0.950638	−	0.286946i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	835.637i		−	1.37667i	−0.725394	−	0.688334i	$-0.758343\pi$
$607$		−	835.637i		−	1.37667i	0.725394	−	0.688334i	$-0.241657\pi$
$608$		0			0
$609$	−145.372	−	107.970i	−0.238706	−	0.177291i
$610$		0			0
$611$		−	485.569i		−	0.794711i
$612$		0			0
$613$			111.889i			0.182527i	0.995827	+	0.0912634i	$0.0290905\pi$
$613$			111.889i			0.182527i	−0.995827	+	0.0912634i	$0.970909\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−749.519			−1.21478			−0.607390	−	0.794404i	$-0.707783\pi$
$617$	−749.519			−1.21478			−0.607390	+	0.794404i	$0.707783\pi$
$618$		0			0
$619$	−195.232			−0.315398			−0.157699	−	0.987487i	$-0.550408\pi$
$619$	−195.232			−0.315398			−0.157699	+	0.987487i	$0.550408\pi$
$620$		0			0
$621$	367.045	−	1019.19i	0.591054	−	1.64121i
$622$		0			0
$623$	−372.004			−0.597117
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−140.526	+	189.205i	−0.224124	+	0.301763i
$628$		0			0
$629$			686.902i			1.09205i
$630$		0			0
$631$	−273.937			−0.434132			−0.217066	−	0.976157i	$-0.569649\pi$
$631$	−273.937			−0.434132			−0.217066	+	0.976157i	$0.569649\pi$
$632$		0			0
$633$	155.804	+	115.718i	0.246135	+	0.182809i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	619.369i		−	0.972323i
$638$		0			0
$639$	−299.359	+	90.3603i	−0.468480	+	0.141409i
$640$		0			0
$641$		−	22.1793i		−	0.0346010i	−0.999850	−	0.0173005i	$-0.994493\pi$
$641$		−	22.1793i		−	0.0346010i	0.999850	−	0.0173005i	$-0.00550720\pi$
$642$		0			0
$643$		−	1188.17i		−	1.84785i	−0.382572	−	0.923926i	$-0.624961\pi$
$643$		−	1188.17i		−	1.84785i	0.382572	−	0.923926i	$-0.375039\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	333.016			0.514708			0.257354	−	0.966317i	$-0.417149\pi$
$647$	333.016			0.514708			0.257354	+	0.966317i	$0.417149\pi$
$648$		0			0
$649$	844.820			1.30173
$650$		0			0
$651$	−1104.46	+	1487.05i	−1.69656	+	2.28426i
$652$		0			0
$653$	−71.5403			−0.109556			−0.0547781	−	0.998499i	$-0.517445\pi$
$653$	−71.5403			−0.109556			−0.0547781	+	0.998499i	$0.517445\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	607.728	−	183.440i	0.925004	−	0.279209i
$658$		0			0
$659$			724.319i			1.09912i	0.835455	+	0.549559i	$0.185204\pi$
$659$			724.319i			1.09912i	−0.835455	+	0.549559i	$0.814796\pi$
$660$		0			0
$661$	545.047			0.824579			0.412289	−	0.911053i	$-0.364729\pi$
$661$	545.047			0.824579			0.412289	+	0.911053i	$0.364729\pi$
$662$		0			0
$663$	184.315	−	248.163i	0.278001	−	0.374303i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			198.479i			0.297570i
$668$		0			0
$669$	448.059	−	603.270i	0.669744	−	0.901749i
$670$		0			0
$671$			235.039i			0.350281i
$672$		0			0
$673$			1106.67i			1.64439i	0.569208	+	0.822193i	$0.307250\pi$
$673$			1106.67i			1.64439i	−0.569208	+	0.822193i	$0.692750\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1011.02			−1.49338			−0.746690	−	0.665172i	$-0.768358\pi$
$677$	−1011.02			−1.49338			−0.746690	+	0.665172i	$0.768358\pi$
$678$		0			0
$679$	−1320.12			−1.94421
$680$		0			0
$681$	462.367	+	343.407i	0.678953	+	0.504269i
$682$		0			0
$683$	1073.43			1.57163			0.785816	−	0.618460i	$-0.212243\pi$
$683$	1073.43			1.57163			0.785816	+	0.618460i	$0.212243\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−271.180	−	201.410i	−0.394730	−	0.293172i
$688$		0			0
$689$			432.436i			0.627629i
$690$		0			0
$691$	−685.207			−0.991617			−0.495809	−	0.868432i	$-0.665128\pi$
$691$	−685.207			−0.991617			−0.495809	+	0.868432i	$0.665128\pi$
$692$		0			0
$693$	279.040	+	924.445i	0.402655	+	1.33398i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			180.535i			0.259018i
$698$		0			0
$699$	−969.917	−	720.373i	−1.38758	−	1.03058i
$700$		0			0
$701$			475.846i			0.678810i	0.940640	+	0.339405i	$0.110226\pi$
$701$			475.846i			0.678810i	−0.940640	+	0.339405i	$0.889774\pi$
$702$		0			0
$703$		−	369.336i		−	0.525372i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	375.195			0.530686
$708$		0			0
$709$	−296.954			−0.418835			−0.209418	−	0.977826i	$-0.567157\pi$
$709$	−296.954			−0.418835			−0.209418	+	0.977826i	$0.567157\pi$
$710$		0			0
$711$	81.1850	+	268.962i	0.114184	+	0.378287i
$712$		0			0
$713$	2030.30			2.84754
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−178.829	+	240.777i	−0.249413	+	0.335811i
$718$		0			0
$719$		−	1335.46i		−	1.85739i	−0.370844	−	0.928695i	$-0.620932\pi$
$719$		−	1335.46i		−	1.85739i	0.370844	−	0.928695i	$-0.379068\pi$
$720$		0			0
$721$	−606.849			−0.841677
$722$		0			0
$723$	−4.40491	−	3.27160i	−0.00609254	−	0.00452503i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			941.614i			1.29521i	0.761978	+	0.647603i	$0.224228\pi$
$727$			941.614i			1.29521i	−0.761978	+	0.647603i	$0.775772\pi$
$728$		0			0
$729$	−561.612	−	464.793i	−0.770387	−	0.637576i
$730$		0			0
$731$			657.018i			0.898794i
$732$		0			0
$733$		−	1073.15i		−	1.46405i	−0.681278	−	0.732025i	$-0.738575\pi$
$733$		−	1073.15i		−	1.46405i	0.681278	−	0.732025i	$-0.261425\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	585.042			0.793815
$738$		0			0
$739$	−76.4461			−0.103445			−0.0517226	−	0.998661i	$-0.516471\pi$
$739$	−76.4461			−0.103445			−0.0517226	+	0.998661i	$0.516471\pi$
$740$		0			0
$741$	−99.1031	+	133.433i	−0.133742	+	0.180072i
$742$		0			0
$743$	78.2278			0.105286			0.0526432	−	0.998613i	$-0.483235\pi$
$743$	78.2278			0.105286			0.0526432	+	0.998613i	$0.483235\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−186.061	−	616.409i	−0.249077	−	0.825179i
$748$		0			0
$749$		−	2295.06i		−	3.06417i
$750$		0			0
$751$	−1150.75			−1.53229			−0.766143	−	0.642670i	$-0.777827\pi$
$751$	−1150.75			−1.53229			−0.766143	+	0.642670i	$0.777827\pi$
$752$		0			0
$753$	624.824	−	841.269i	0.829780	−	1.11722i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	257.571i		−	0.340252i	−0.985422	−	0.170126i	$-0.945582\pi$
$757$		−	257.571i		−	0.340252i	0.985422	−	0.170126i	$-0.0544175\pi$
$758$		0			0
$759$	631.081	−	849.693i	0.831463	−	1.11949i
$760$		0			0
$761$		−	1001.67i		−	1.31626i	−0.752906	−	0.658128i	$-0.771348\pi$
$761$		−	1001.67i		−	1.31626i	0.752906	−	0.658128i	$-0.228652\pi$
$762$		0			0
$763$		−	887.759i		−	1.16351i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	595.792			0.776782
$768$		0			0
$769$	132.122			0.171810			0.0859050	−	0.996303i	$-0.472622\pi$
$769$	132.122			0.171810			0.0859050	+	0.996303i	$0.472622\pi$
$770$		0			0
$771$	−30.6670	−	22.7769i	−0.0397757	−	0.0295420i
$772$		0			0
$773$	254.447			0.329168			0.164584	−	0.986363i	$-0.447372\pi$
$773$	254.447			0.329168			0.164584	+	0.986363i	$0.447372\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1214.83	−	902.276i	−1.56349	−	1.16123i
$778$		0			0
$779$		−	97.0710i		−	0.124610i
$780$		0			0
$781$	−305.525			−0.391197
$782$		0			0
$783$	125.668	+	45.2574i	0.160496	+	0.0578000i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1278.73i			1.62481i	0.583091	+	0.812407i	$0.301843\pi$
$787$			1278.73i			1.62481i	−0.583091	+	0.812407i	$0.698157\pi$
$788$		0			0
$789$	454.718	+	337.727i	0.576323	+	0.428044i
$790$		0			0
$791$		−	964.253i		−	1.21903i
$792$		0			0
$793$			165.756i			0.209024i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−747.744			−0.938199			−0.469099	−	0.883145i	$-0.655421\pi$
$797$	−747.744			−0.938199			−0.469099	+	0.883145i	$0.655421\pi$
$798$		0			0
$799$	−1300.99			−1.62828
$800$		0			0
$801$	262.691	−	79.2922i	0.327954	−	0.0989915i
$802$		0			0
$803$	620.245			0.772410
$804$		0			0
$805$		0			0
$806$		0			0
$807$	202.207	−	272.254i	0.250567	−	0.337365i
$808$		0			0
$809$		−	340.146i		−	0.420453i	−0.977653	−	0.210226i	$-0.932580\pi$
$809$		−	340.146i		−	0.420453i	0.977653	−	0.210226i	$-0.0674201\pi$
$810$		0			0
$811$	1112.71			1.37202			0.686011	−	0.727591i	$-0.259360\pi$
$811$	1112.71			1.37202			0.686011	+	0.727591i	$0.259360\pi$
$812$		0			0
$813$	54.6071	+	40.5576i	0.0671675	+	0.0498864i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	353.268i		−	0.432397i
$818$		0			0
$819$	196.787	+	651.946i	0.240278	+	0.796027i
$820$		0			0
$821$			1283.05i			1.56279i	0.624036	+	0.781395i	$0.285492\pi$
$821$			1283.05i			1.56279i	−0.624036	+	0.781395i	$0.714508\pi$
$822$		0			0
$823$			1430.32i			1.73794i	0.494867	+	0.868968i	$0.335217\pi$
$823$			1430.32i			1.73794i	−0.494867	+	0.868968i	$0.664783\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	144.113			0.174260			0.0871300	−	0.996197i	$-0.472230\pi$
$827$	144.113			0.174260			0.0871300	+	0.996197i	$0.472230\pi$
$828$		0			0
$829$	−1116.06			−1.34627			−0.673137	−	0.739518i	$-0.735054\pi$
$829$	−1116.06			−1.34627			−0.673137	+	0.739518i	$0.735054\pi$
$830$		0			0
$831$	610.472	−	821.945i	0.734624	−	0.989104i
$832$		0			0
$833$	−1659.49			−1.99218
$834$		0			0
$835$		0			0
$836$		0			0
$837$	462.950	−	1285.50i	0.553106	−	1.53584i
$838$		0			0
$839$			949.194i			1.13134i	0.824632	+	0.565670i	$0.191382\pi$
$839$			949.194i			1.13134i	−0.824632	+	0.565670i	$0.808618\pi$
$840$		0			0
$841$	816.527			0.970900
$842$		0			0
$843$	−10.3249	+	13.9016i	−0.0122479	+	0.0164906i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	532.888i		−	0.629148i
$848$		0			0
$849$	26.7251	−	35.9829i	0.0314783	−	0.0423827i
$850$		0			0
$851$			1658.63i			1.94904i
$852$		0			0
$853$		−	666.717i		−	0.781614i	−0.920473	−	0.390807i	$-0.872196\pi$
$853$		−	666.717i		−	0.781614i	0.920473	−	0.390807i	$-0.127804\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	788.585			0.920169			0.460085	−	0.887875i	$-0.347819\pi$
$857$	788.585			0.920169			0.460085	+	0.887875i	$0.347819\pi$
$858$		0			0
$859$	406.487			0.473209			0.236605	−	0.971606i	$-0.423965\pi$
$859$	406.487			0.473209			0.236605	+	0.971606i	$0.423965\pi$
$860$		0			0
$861$	−319.289	−	237.141i	−0.370835	−	0.275425i
$862$		0			0
$863$	1031.38			1.19512			0.597558	−	0.801826i	$-0.296138\pi$
$863$	1031.38			1.19512			0.597558	+	0.801826i	$0.296138\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	31.1176	+	23.1116i	0.0358911	+	0.0266569i
$868$		0			0
$869$			274.501i			0.315882i
$870$		0			0
$871$	412.588			0.473695
$872$		0			0
$873$	932.202	−	281.381i	1.06781	−	0.322316i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			725.461i			0.827208i	0.910457	+	0.413604i	$0.135730\pi$
$877$			725.461i			0.827208i	−0.910457	+	0.413604i	$0.864270\pi$
$878$		0			0
$879$	−15.0188	−	11.1547i	−0.0170862	−	0.0126902i
$880$		0			0
$881$			282.083i			0.320185i	0.987102	+	0.160092i	$0.0511792\pi$
$881$			282.083i			0.320185i	−0.987102	+	0.160092i	$0.948821\pi$
$882$		0			0
$883$		−	454.628i		−	0.514868i	−0.966296	−	0.257434i	$-0.917123\pi$
$883$		−	454.628i		−	0.514868i	0.966296	−	0.257434i	$-0.0828770\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	309.991			0.349482			0.174741	−	0.984614i	$-0.444091\pi$
$887$	309.991			0.349482			0.174741	+	0.984614i	$0.444091\pi$
$888$		0			0
$889$	−2372.08			−2.66825
$890$		0			0
$891$	−394.088	−	593.320i	−0.442299	−	0.665904i
$892$		0			0
$893$	699.523			0.783341
$894$		0			0
$895$		0			0
$896$		0			0
$897$	445.057	−	599.228i	0.496161	−	0.668036i
$898$		0			0
$899$			250.340i			0.278465i
$900$		0			0
$901$	1158.64			1.28594
$902$		0			0
$903$	−1161.98	−	863.022i	−1.28680	−	0.955728i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	87.5518i		−	0.0965290i	−0.998835	−	0.0482645i	$-0.984631\pi$
$907$		−	87.5518i		−	0.0965290i	0.998835	−	0.0482645i	$-0.0153690\pi$
$908$		0			0
$909$	−264.944	+	79.9723i	−0.291468	+	0.0879784i
$910$		0			0
$911$			685.361i			0.752317i	0.926555	+	0.376158i	$0.122755\pi$
$911$			685.361i			0.752317i	−0.926555	+	0.376158i	$0.877245\pi$
$912$		0			0
$913$		−	629.105i		−	0.689052i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2704.61			2.94941
$918$		0			0
$919$	−285.772			−0.310960			−0.155480	−	0.987839i	$-0.549692\pi$
$919$	−285.772			−0.310960			−0.155480	+	0.987839i	$0.549692\pi$
$920$		0			0
$921$	−507.135	+	682.811i	−0.550635	+	0.741380i
$922$		0			0
$923$	−215.465			−0.233440
$924$		0			0
$925$		0			0
$926$		0			0
$927$	428.526	−	129.349i	0.462272	−	0.139535i
$928$		0			0
$929$		−	1319.46i		−	1.42031i	−0.704048	−	0.710153i	$-0.748626\pi$
$929$		−	1319.46i		−	1.42031i	0.704048	−	0.710153i	$-0.251374\pi$
$930$		0			0
$931$	892.281			0.958411
$932$		0			0
$933$	−194.294	+	261.599i	−0.208246	+	0.280385i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			820.446i			0.875609i	0.899070	+	0.437805i	$0.144244\pi$
$937$			820.446i			0.875609i	−0.899070	+	0.437805i	$0.855756\pi$
$938$		0			0
$939$	692.600	−	932.522i	0.737593	−	0.993102i
$940$		0			0
$941$		−	1134.15i		−	1.20526i	−0.798020	−	0.602631i	$-0.794119\pi$
$941$		−	1134.15i		−	1.20526i	0.798020	−	0.602631i	$-0.205881\pi$
$942$		0			0
$943$			435.931i			0.462281i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−513.273			−0.541999			−0.270999	−	0.962580i	$-0.587354\pi$
$947$	−513.273			−0.541999			−0.270999	+	0.962580i	$0.587354\pi$
$948$		0			0
$949$	437.415			0.460922
$950$		0			0
$951$	−286.050	−	212.454i	−0.300788	−	0.223401i
$952$		0			0
$953$	−742.919			−0.779559			−0.389779	−	0.920908i	$-0.627449\pi$
$953$	−742.919			−0.779559			−0.389779	+	0.920908i	$0.627449\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	104.769	+	77.8136i	0.109476	+	0.0813099i
$958$		0			0
$959$		−	3110.12i		−	3.24309i
$960$		0			0
$961$	1599.80			1.66472
$962$		0			0
$963$	489.189	+	1620.66i	0.507985	+	1.68293i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	152.272i		−	0.157468i	−0.996896	−	0.0787340i	$-0.974912\pi$
$967$		−	152.272i		−	0.157468i	0.996896	−	0.0787340i	$-0.0250878\pi$
$968$		0			0
$969$	357.510	+	265.529i	0.368948	+	0.274024i
$970$		0			0
$971$		−	52.6817i		−	0.0542551i	−0.999632	−	0.0271275i	$-0.991364\pi$
$971$		−	52.6817i		−	0.0542551i	0.999632	−	0.0271275i	$-0.00863602\pi$
$972$		0			0
$973$			1258.48i			1.29340i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	386.394			0.395490			0.197745	−	0.980253i	$-0.436638\pi$
$977$	386.394			0.395490			0.197745	+	0.980253i	$0.436638\pi$
$978$		0			0
$979$	268.101			0.273852
$980$		0			0
$981$	189.225	+	626.891i	0.192889	+	0.639032i
$982$		0			0
$983$	−1706.21			−1.73572			−0.867860	−	0.496809i	$-0.834505\pi$
$983$	−1706.21			−1.73572			−0.867860	+	0.496809i	$0.834505\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	1708.91	−	2300.89i	1.73142	−	2.33120i
$988$		0			0
$989$			1586.47i			1.60412i
$990$		0			0
$991$	−217.282			−0.219255			−0.109627	−	0.993973i	$-0.534966\pi$
$991$	−217.282			−0.219255			−0.109627	+	0.993973i	$0.534966\pi$
$992$		0			0
$993$	−1191.30	−	884.798i	−1.19970	−	0.891035i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			730.791i			0.732990i	0.930420	+	0.366495i	$0.119442\pi$
$997$			730.791i			0.732990i	−0.930420	+	0.366495i	$0.880558\pi$
$998$		0			0
$999$	1050.17	+	378.203i	1.05122	+	0.378581i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.l.449.1		12
3.2	odd	2	inner	1200.3.c.l.449.11		12
4.3	odd	2		600.3.c.c.449.12		12
5.2	odd	4		1200.3.l.v.401.4		6
5.3	odd	4		1200.3.l.w.401.3		6
5.4	even	2	inner	1200.3.c.l.449.12		12
12.11	even	2		600.3.c.c.449.2		12
15.2	even	4		1200.3.l.v.401.3		6
15.8	even	4		1200.3.l.w.401.4		6
15.14	odd	2	inner	1200.3.c.l.449.2		12
20.3	even	4		600.3.l.d.401.4	yes	6
20.7	even	4		600.3.l.e.401.3	yes	6
20.19	odd	2		600.3.c.c.449.1		12
60.23	odd	4		600.3.l.d.401.3	✓	6
60.47	odd	4		600.3.l.e.401.4	yes	6
60.59	even	2		600.3.c.c.449.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.c.449.1		12	20.19	odd	2
600.3.c.c.449.2		12	12.11	even	2
600.3.c.c.449.11		12	60.59	even	2
600.3.c.c.449.12		12	4.3	odd	2
600.3.l.d.401.3	✓	6	60.23	odd	4
600.3.l.d.401.4	yes	6	20.3	even	4
600.3.l.e.401.3	yes	6	20.7	even	4
600.3.l.e.401.4	yes	6	60.47	odd	4
1200.3.c.l.449.1		12	1.1	even	1	trivial
1200.3.c.l.449.2		12	15.14	odd	2	inner
1200.3.c.l.449.11		12	3.2	odd	2	inner
1200.3.c.l.449.12		12	5.4	even	2	inner
1200.3.l.v.401.3		6	15.2	even	4
1200.3.l.v.401.4		6	5.2	odd	4
1200.3.l.w.401.3		6	5.3	odd	4
1200.3.l.w.401.4		6	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.40839 - 1.78875i) q^{3} -12.2014i q^{7} +(2.60072 + 8.61605i) q^{9} +O(q^{10})\)	\(q+(-2.40839 - 1.78875i) q^{3} -12.2014i q^{7} +(2.60072 + 8.61605i) q^{9} +8.79351i q^{11} +6.20144i q^{13} +16.6156 q^{17} -8.93396 q^{19} +(-21.8254 + 29.3859i) q^{21} +40.1210 q^{23} +(9.14843 - 25.4029i) q^{27} +4.94701i q^{29} +50.6043 q^{31} +(15.7294 - 21.1782i) q^{33} +41.3407i q^{37} +(11.0928 - 14.9355i) q^{39} +10.8654i q^{41} +39.5422i q^{43} -78.2993 q^{47} -99.8751 q^{49} +(-40.0170 - 29.7213i) q^{51} +69.7316 q^{53} +(21.5165 + 15.9807i) q^{57} -96.0731i q^{59} +26.7287 q^{61} +(105.128 - 31.7325i) q^{63} -66.5311i q^{67} +(-96.6273 - 71.7667i) q^{69} +34.7444i q^{71} -70.5344i q^{73} +107.293 q^{77} +31.2164 q^{79} +(-67.4725 + 44.8158i) q^{81} -71.5420 q^{83} +(8.84898 - 11.9143i) q^{87} -30.4885i q^{89} +75.6665 q^{91} +(-121.875 - 90.5187i) q^{93} -108.194i q^{97} +(-75.7653 + 22.8694i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 32 q^{9}+O(q^{10}) \) 12 * q - 32 * q^9	\( 12 q - 32 q^{9} + 100 q^{19} - 36 q^{21} + 228 q^{31} - 12 q^{39} - 152 q^{49} + 12 q^{51} + 124 q^{61} - 312 q^{69} - 152 q^{79} - 448 q^{81} + 620 q^{91} - 500 q^{99}+O(q^{100}) \) 12 * q - 32 * q^9 + 100 * q^19 - 36 * q^21 + 228 * q^31 - 12 * q^39 - 152 * q^49 + 12 * q^51 + 124 * q^61 - 312 * q^69 - 152 * q^79 - 448 * q^81 + 620 * q^91 - 500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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