LMFDB - Embedded newform 1400.3.p.a.1049.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1400,3,Mod(1049,1400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1400, 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("1400.1049");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1400 = 2^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1400.p (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:	$38.1472370104$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{13} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1049.6
Root			$-0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1400.1049
Dual form			1400.3.p.a.1049.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53073 q^{3} +(5.86030 + 3.82843i) q^{7} -6.65685 q^{9} +O(q^{10})$	$q+1.53073 q^{3} +(5.86030 + 3.82843i) q^{7} -6.65685 q^{9} -9.31371 q^{11} -0.262632 q^{13} +14.7821 q^{17} +25.4972i q^{19} +(8.97056 + 5.86030i) q^{21} -16.6274i q^{23} -23.9665 q^{27} -14.6863 q^{29} +8.65914i q^{31} -14.2568 q^{33} +43.9411i q^{37} -0.402020 q^{39} -22.9159i q^{41} +46.0000i q^{43} -86.1562 q^{47} +(19.6863 + 44.8715i) q^{49} +22.6274 q^{51} +60.6274i q^{53} +39.0294i q^{57} +58.1228i q^{59} -9.97230i q^{61} +(-39.0112 - 25.4853i) q^{63} -20.6274i q^{67} -25.4521i q^{69} -78.9706 q^{71} -21.4303 q^{73} +(-54.5811 - 35.6569i) q^{77} +90.2843 q^{79} +23.2254 q^{81} +71.8544 q^{83} -22.4808 q^{87} -2.01094i q^{89} +(-1.53911 - 1.00547i) q^{91} +13.2548i q^{93} -158.581 q^{97} +62.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^9	$ 8 q - 8 q^{9} + 16 q^{11} - 64 q^{21} - 208 q^{29} - 320 q^{39} + 248 q^{49} - 496 q^{71} + 496 q^{79} - 312 q^{81} + 576 q^{91} + 496 q^{99}+O(q^{100}) $ 8 * q - 8 * q^9 + 16 * q^11 - 64 * q^21 - 208 * q^29 - 320 * q^39 + 248 * q^49 - 496 * q^71 + 496 * q^79 - 312 * q^81 + 576 * q^91 + 496 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$.

$n$	$351$	$701$	$801$	$1177$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	1.53073			0.510245			0.255122	−	0.966909i	$-0.417884\pi$
$3$	1.53073			0.510245			0.255122	+	0.966909i	$0.417884\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	5.86030	+	3.82843i	0.837186	+	0.546918i
$7$	5.86030	+	3.82843i	0.837186	+	0.546918i
$8$		0			0
$9$	−6.65685			−0.739650
$10$		0			0
$11$	−9.31371			−0.846701			−0.423350	−	0.905966i	$-0.639146\pi$
$11$	−9.31371			−0.846701			−0.423350	+	0.905966i	$0.639146\pi$
$12$		0			0
$13$	−0.262632			−0.0202025			−0.0101012	−	0.999949i	$-0.503215\pi$
$13$	−0.262632			−0.0202025			−0.0101012	+	0.999949i	$0.503215\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.7821			0.869534			0.434767	−	0.900543i	$-0.356831\pi$
$17$	14.7821			0.869534			0.434767	+	0.900543i	$0.356831\pi$
$18$		0			0
$19$			25.4972i			1.34196i	0.741476	+	0.670979i	$0.234126\pi$
$19$			25.4972i			1.34196i	−0.741476	+	0.670979i	$0.765874\pi$
$20$		0			0
$21$	8.97056	+	5.86030i	0.427170	+	0.279062i
$22$		0			0
$23$		−	16.6274i		−	0.722931i	−0.932385	−	0.361466i	$-0.882276\pi$
$23$		−	16.6274i		−	0.722931i	0.932385	−	0.361466i	$-0.117724\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−23.9665			−0.887647
$28$		0			0
$29$	−14.6863			−0.506424			−0.253212	−	0.967411i	$-0.581487\pi$
$29$	−14.6863			−0.506424			−0.253212	+	0.967411i	$0.581487\pi$
$30$		0			0
$31$			8.65914i			0.279327i	0.990199	+	0.139664i	$0.0446021\pi$
$31$			8.65914i			0.279327i	−0.990199	+	0.139664i	$0.955398\pi$
$32$		0			0
$33$	−14.2568			−0.432024
$34$		0			0
$35$		0			0
$36$		0			0
$37$			43.9411i			1.18760i	0.804613	+	0.593799i	$0.202373\pi$
$37$			43.9411i			1.18760i	−0.804613	+	0.593799i	$0.797627\pi$
$38$		0			0
$39$	−0.402020			−0.0103082
$40$		0			0
$41$		−	22.9159i		−	0.558925i	−0.960157	−	0.279463i	$-0.909844\pi$
$41$		−	22.9159i		−	0.558925i	0.960157	−	0.279463i	$-0.0901563\pi$
$42$		0			0
$43$			46.0000i			1.06977i	0.844926	+	0.534884i	$0.179645\pi$
$43$			46.0000i			1.06977i	−0.844926	+	0.534884i	$0.820355\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−86.1562			−1.83311			−0.916556	−	0.399907i	$-0.869042\pi$
$47$	−86.1562			−1.83311			−0.916556	+	0.399907i	$0.869042\pi$
$48$		0			0
$49$	19.6863	+	44.8715i	0.401761	+	0.915745i
$50$		0			0
$51$	22.6274			0.443675
$52$		0			0
$53$			60.6274i			1.14391i	0.820284	+	0.571957i	$0.193815\pi$
$53$			60.6274i			1.14391i	−0.820284	+	0.571957i	$0.806185\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			39.0294i			0.684727i
$58$		0			0
$59$			58.1228i			0.985133i	0.870275	+	0.492566i	$0.163941\pi$
$59$			58.1228i			0.985133i	−0.870275	+	0.492566i	$0.836059\pi$
$60$		0			0
$61$		−	9.97230i		−	0.163480i	−0.996654	−	0.0817402i	$-0.973952\pi$
$61$		−	9.97230i		−	0.163480i	0.996654	−	0.0817402i	$-0.0260478\pi$
$62$		0			0
$63$	−39.0112	−	25.4853i	−0.619225	−	0.404528i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	20.6274i		−	0.307872i	−0.988081	−	0.153936i	$-0.950805\pi$
$67$		−	20.6274i		−	0.307872i	0.988081	−	0.153936i	$-0.0491949\pi$
$68$		0			0
$69$		−	25.4521i		−	0.368872i
$70$		0			0
$71$	−78.9706			−1.11226			−0.556131	−	0.831095i	$-0.687715\pi$
$71$	−78.9706			−1.11226			−0.556131	+	0.831095i	$0.687715\pi$
$72$		0			0
$73$	−21.4303			−0.293565			−0.146783	−	0.989169i	$-0.546892\pi$
$73$	−21.4303			−0.293565			−0.146783	+	0.989169i	$0.546892\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−54.5811	−	35.6569i	−0.708846	−	0.463076i
$78$		0			0
$79$	90.2843			1.14284			0.571419	−	0.820658i	$-0.306393\pi$
$79$	90.2843			1.14284			0.571419	+	0.820658i	$0.306393\pi$
$80$		0			0
$81$	23.2254			0.286733
$82$		0			0
$83$	71.8544			0.865715			0.432858	−	0.901462i	$-0.357505\pi$
$83$	71.8544			0.865715			0.432858	+	0.901462i	$0.357505\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−22.4808			−0.258400
$88$		0			0
$89$		−	2.01094i		−	0.0225948i	−0.999936	−	0.0112974i	$-0.996404\pi$
$89$		−	2.01094i		−	0.0225948i	0.999936	−	0.0112974i	$-0.00359615\pi$
$90$		0			0
$91$	−1.53911	−	1.00547i	−0.0169132	−	0.0110491i
$92$		0			0
$93$			13.2548i			0.142525i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−158.581			−1.63485			−0.817427	−	0.576032i	$-0.804601\pi$
$97$	−158.581			−1.63485			−0.817427	+	0.576032i	$0.804601\pi$
$98$		0			0
$99$	62.0000			0.626263
$100$		0			0
$101$		−	93.5022i		−	0.925764i	−0.886420	−	0.462882i	$-0.846815\pi$
$101$		−	93.5022i		−	0.925764i	0.886420	−	0.462882i	$-0.153185\pi$
$102$		0			0
$103$	−20.3797			−0.197862			−0.0989308	−	0.995094i	$-0.531542\pi$
$103$	−20.3797			−0.197862			−0.0989308	+	0.995094i	$0.531542\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			14.1177i			0.131942i	0.997822	+	0.0659708i	$0.0210144\pi$
$107$			14.1177i			0.131942i	−0.997822	+	0.0659708i	$0.978986\pi$
$108$		0			0
$109$	−23.2548			−0.213347			−0.106674	−	0.994294i	$-0.534020\pi$
$109$	−23.2548			−0.213347			−0.106674	+	0.994294i	$0.534020\pi$
$110$		0			0
$111$			67.2622i			0.605965i
$112$		0			0
$113$		−	13.7157i		−	0.121378i	−0.998157	−	0.0606891i	$-0.980670\pi$
$113$		−	13.7157i		−	0.121378i	0.998157	−	0.0606891i	$-0.0193298\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.74831			0.0149428
$118$		0			0
$119$	86.6274	+	56.5921i	0.727961	+	0.475564i
$120$		0			0
$121$	−34.2548			−0.283098
$122$		0			0
$123$		−	35.0782i		−	0.285189i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			177.765i			1.39972i	0.714280	+	0.699860i	$0.246754\pi$
$127$			177.765i			1.39972i	−0.714280	+	0.699860i	$0.753246\pi$
$128$		0			0
$129$			70.4138i			0.545843i
$130$		0			0
$131$			112.614i			0.859648i	0.902913	+	0.429824i	$0.141424\pi$
$131$			112.614i			0.859648i	−0.902913	+	0.429824i	$0.858576\pi$
$132$		0			0
$133$	−97.6142	+	149.421i	−0.733941	+	1.12347i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			121.765i			0.888792i	0.895830	+	0.444396i	$0.146582\pi$
$137$			121.765i			0.888792i	−0.895830	+	0.444396i	$0.853418\pi$
$138$		0			0
$139$			175.329i			1.26136i	0.776043	+	0.630679i	$0.217224\pi$
$139$			175.329i			1.26136i	−0.776043	+	0.630679i	$0.782776\pi$
$140$		0			0
$141$	−131.882			−0.935335
$142$		0			0
$143$	2.44608			0.0171055
$144$		0			0
$145$		0			0
$146$		0			0
$147$	30.1345	+	68.6863i	0.204996	+	0.467254i
$148$		0			0
$149$	237.765			1.59573			0.797867	−	0.602833i	$-0.205961\pi$
$149$	237.765			1.59573			0.797867	+	0.602833i	$0.205961\pi$
$150$		0			0
$151$	99.3726			0.658097			0.329048	−	0.944313i	$-0.393272\pi$
$151$	99.3726			0.658097			0.329048	+	0.944313i	$0.393272\pi$
$152$		0			0
$153$	−98.4021			−0.643151
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−96.6538			−0.615629			−0.307815	−	0.951446i	$-0.599598\pi$
$157$	−96.6538			−0.615629			−0.307815	+	0.951446i	$0.599598\pi$
$158$		0			0
$159$			92.8044i			0.583676i
$160$		0			0
$161$	63.6569	−	97.4417i	0.395384	−	0.605228i
$162$		0			0
$163$		−	170.569i		−	1.04643i	−0.852200	−	0.523216i	$-0.824732\pi$
$163$		−	170.569i		−	1.04643i	0.852200	−	0.523216i	$-0.175268\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−94.9055			−0.568296			−0.284148	−	0.958780i	$-0.591711\pi$
$167$	−94.9055			−0.568296			−0.284148	+	0.958780i	$0.591711\pi$
$168$		0			0
$169$	−168.931			−0.999592
$170$		0			0
$171$		−	169.731i		−	0.992580i
$172$		0			0
$173$	−182.720			−1.05618			−0.528092	−	0.849187i	$-0.677092\pi$
$173$	−182.720			−1.05618			−0.528092	+	0.849187i	$0.677092\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			88.9706i			0.502659i
$178$		0			0
$179$	204.059			1.13999			0.569997	−	0.821647i	$-0.306944\pi$
$179$	204.059			1.13999			0.569997	+	0.821647i	$0.306944\pi$
$180$		0			0
$181$			229.167i			1.26612i	0.774104	+	0.633059i	$0.218201\pi$
$181$			229.167i			1.26612i	−0.774104	+	0.633059i	$0.781799\pi$
$182$		0			0
$183$		−	15.2649i		−	0.0834149i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−137.676			−0.736235
$188$		0			0
$189$	−140.451	−	91.7539i	−0.743126	−	0.485470i
$190$		0			0
$191$	9.59798			0.0502512			0.0251256	−	0.999684i	$-0.492001\pi$
$191$	9.59798			0.0502512			0.0251256	+	0.999684i	$0.492001\pi$
$192$		0			0
$193$			3.42136i			0.0177272i	0.999961	+	0.00886362i	$0.00282141\pi$
$193$			3.42136i			0.0177272i	−0.999961	+	0.00886362i	$0.997179\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	146.451i		−	0.743405i	−0.928352	−	0.371703i	$-0.878774\pi$
$197$		−	146.451i		−	0.743405i	0.928352	−	0.371703i	$-0.121226\pi$
$198$		0			0
$199$			244.647i			1.22938i	0.788768	+	0.614691i	$0.210719\pi$
$199$			244.647i			1.22938i	−0.788768	+	0.614691i	$0.789281\pi$
$200$		0			0
$201$		−	31.5751i		−	0.157090i
$202$		0			0
$203$	−86.0661	−	56.2254i	−0.423971	−	0.276972i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			110.686i			0.534716i
$208$		0			0
$209$		−	237.474i		−	1.13624i
$210$		0			0
$211$	311.019			1.47403			0.737013	−	0.675879i	$-0.236236\pi$
$211$	311.019			1.47403			0.737013	+	0.675879i	$0.236236\pi$
$212$		0			0
$213$	−120.883			−0.567525
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−33.1509	+	50.7452i	−0.152769	+	0.233849i
$218$		0			0
$219$	−32.8040			−0.149790
$220$		0			0
$221$	−3.88225			−0.0175667
$222$		0			0
$223$	116.156			0.520877			0.260438	−	0.965490i	$-0.416133\pi$
$223$	116.156			0.520877			0.260438	+	0.965490i	$0.416133\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−59.1734			−0.260676			−0.130338	−	0.991470i	$-0.541606\pi$
$227$	−59.1734			−0.260676			−0.130338	+	0.991470i	$0.541606\pi$
$228$		0			0
$229$		−	128.664i		−	0.561852i	−0.959729	−	0.280926i	$-0.909359\pi$
$229$		−	128.664i		−	0.561852i	0.959729	−	0.280926i	$-0.0906415\pi$
$230$		0			0
$231$	−83.5492	−	54.5811i	−0.361685	−	0.236282i
$232$		0			0
$233$			88.9807i			0.381891i	0.981601	+	0.190946i	$0.0611554\pi$
$233$			88.9807i			0.381891i	−0.981601	+	0.190946i	$0.938845\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	138.201			0.583127
$238$		0			0
$239$	7.13708			0.0298623			0.0149311	−	0.999889i	$-0.495247\pi$
$239$	7.13708			0.0298623			0.0149311	+	0.999889i	$0.495247\pi$
$240$		0			0
$241$		−	367.451i		−	1.52469i	−0.647170	−	0.762346i	$-0.724047\pi$
$241$		−	367.451i		−	1.52469i	0.647170	−	0.762346i	$-0.275953\pi$
$242$		0			0
$243$	251.250			1.03395
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	6.69639i		−	0.0271109i
$248$		0			0
$249$	109.990			0.441727
$250$		0			0
$251$		−	176.995i		−	0.705159i	−0.935782	−	0.352579i	$-0.885305\pi$
$251$		−	176.995i		−	0.705159i	0.935782	−	0.352579i	$-0.114695\pi$
$252$		0			0
$253$			154.863i			0.612106i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−27.4631			−0.106860			−0.0534301	−	0.998572i	$-0.517015\pi$
$257$	−27.4631			−0.106860			−0.0534301	+	0.998572i	$0.517015\pi$
$258$		0			0
$259$	−168.225	+	257.508i	−0.649519	+	0.994240i
$260$		0			0
$261$	97.7645			0.374577
$262$		0			0
$263$		−	77.1472i		−	0.293335i	−0.989186	−	0.146668i	$-0.953145\pi$
$263$		−	77.1472i		−	0.293335i	0.989186	−	0.146668i	$-0.0468548\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	3.07821i		−	0.0115289i
$268$		0			0
$269$		−	34.3739i		−	0.127784i	−0.997957	−	0.0638920i	$-0.979649\pi$
$269$		−	34.3739i		−	0.127784i	0.997957	−	0.0638920i	$-0.0203513\pi$
$270$		0			0
$271$			202.927i			0.748809i	0.927266	+	0.374404i	$0.122153\pi$
$271$			202.927i			0.748809i	−0.927266	+	0.374404i	$0.877847\pi$
$272$		0			0
$273$	−2.35596	−	1.53911i	−0.00862989	−	0.00563775i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	254.098i		−	0.917320i	−0.888612	−	0.458660i	$-0.848330\pi$
$277$		−	254.098i		−	0.917320i	0.888612	−	0.458660i	$-0.151670\pi$
$278$		0			0
$279$		−	57.6426i		−	0.206604i
$280$		0			0
$281$	−221.529			−0.788359			−0.394180	−	0.919033i	$-0.628971\pi$
$281$	−221.529			−0.788359			−0.394180	+	0.919033i	$0.628971\pi$
$282$		0			0
$283$	283.786			1.00278			0.501388	−	0.865223i	$-0.332823\pi$
$283$	283.786			1.00278			0.501388	+	0.865223i	$0.332823\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	87.7320	−	134.294i	0.305687	−	0.467925i
$288$		0			0
$289$	−70.4903			−0.243911
$290$		0			0
$291$	−242.745			−0.834176
$292$		0			0
$293$	398.073			1.35861			0.679306	−	0.733855i	$-0.262281\pi$
$293$	398.073			1.35861			0.679306	+	0.733855i	$0.262281\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	223.217			0.751572
$298$		0			0
$299$			4.36690i			0.0146050i
$300$		0			0
$301$	−176.108	+	269.574i	−0.585075	+	0.895594i
$302$		0			0
$303$		−	143.127i		−	0.472366i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−167.720			−0.546320			−0.273160	−	0.961969i	$-0.588069\pi$
$307$	−167.720			−0.546320			−0.273160	+	0.961969i	$0.588069\pi$
$308$		0			0
$309$	−31.1960			−0.100958
$310$		0			0
$311$			284.101i			0.913508i	0.889593	+	0.456754i	$0.150988\pi$
$311$			284.101i			0.913508i	−0.889593	+	0.456754i	$0.849012\pi$
$312$		0			0
$313$	−514.311			−1.64317			−0.821583	−	0.570089i	$-0.806909\pi$
$313$	−514.311			−1.64317			−0.821583	+	0.570089i	$0.806909\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			18.5685i			0.0585758i	0.999571	+	0.0292879i	$0.00932397\pi$
$317$			18.5685i			0.0585758i	−0.999571	+	0.0292879i	$0.990676\pi$
$318$		0			0
$319$	136.784			0.428789
$320$		0			0
$321$			21.6105i			0.0673225i
$322$		0			0
$323$			376.902i			1.16688i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−35.5970			−0.108859
$328$		0			0
$329$	−504.902	−	329.843i	−1.53466	−	1.00256i
$330$		0			0
$331$	194.333			0.587109			0.293554	−	0.955942i	$-0.405162\pi$
$331$	194.333			0.587109			0.293554	+	0.955942i	$0.405162\pi$
$332$		0			0
$333$		−	292.510i		−	0.878407i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	125.265i		−	0.371706i	−0.982578	−	0.185853i	$-0.940495\pi$
$337$		−	125.265i		−	0.371706i	0.982578	−	0.185853i	$-0.0595048\pi$
$338$		0			0
$339$		−	20.9951i		−	0.0619325i
$340$		0			0
$341$		−	80.6487i		−	0.236506i
$342$		0			0
$343$	−56.4196	+	338.328i	−0.164489	+	0.986379i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	160.745i		−	0.463243i	−0.972806	−	0.231621i	$-0.925597\pi$
$347$		−	160.745i		−	0.463243i	0.972806	−	0.231621i	$-0.0744030\pi$
$348$		0			0
$349$		−	93.0671i		−	0.266668i	−0.991071	−	0.133334i	$-0.957432\pi$
$349$		−	93.0671i		−	0.266668i	0.991071	−	0.133334i	$-0.0425683\pi$
$350$		0			0
$351$	6.29437			0.0179327
$352$		0			0
$353$	189.631			0.537198			0.268599	−	0.963252i	$-0.413439\pi$
$353$	189.631			0.537198			0.268599	+	0.963252i	$0.413439\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	132.604	+	86.6274i	0.371438	+	0.242654i
$358$		0			0
$359$	494.431			1.37724			0.688622	−	0.725121i	$-0.258216\pi$
$359$	494.431			1.37724			0.688622	+	0.725121i	$0.258216\pi$
$360$		0			0
$361$	−289.108			−0.800852
$362$		0			0
$363$	−52.4350			−0.144449
$364$		0			0
$365$		0			0
$366$		0			0
$367$	718.994			1.95911			0.979556	−	0.201171i	$-0.0644746\pi$
$367$	718.994			1.95911			0.979556	+	0.201171i	$0.0644746\pi$
$368$		0			0
$369$			152.548i			0.413410i
$370$		0			0
$371$	−232.108	+	355.295i	−0.625627	+	0.957668i
$372$		0			0
$373$		−	132.843i		−	0.356147i	−0.984017	−	0.178073i	$-0.943014\pi$
$373$		−	132.843i		−	0.356147i	0.984017	−	0.178073i	$-0.0569864\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.85710			0.0102310
$378$		0			0
$379$	−324.607			−0.856483			−0.428242	−	0.903664i	$-0.640867\pi$
$379$	−324.607			−0.856483			−0.428242	+	0.903664i	$0.640867\pi$
$380$		0			0
$381$			272.110i			0.714200i
$382$		0			0
$383$	391.252			1.02155			0.510773	−	0.859715i	$-0.329359\pi$
$383$	391.252			1.02155			0.510773	+	0.859715i	$0.329359\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	306.215i		−	0.791254i
$388$		0			0
$389$	−223.352			−0.574171			−0.287085	−	0.957905i	$-0.592686\pi$
$389$	−223.352			−0.574171			−0.287085	+	0.957905i	$0.592686\pi$
$390$		0			0
$391$		−	245.788i		−	0.628613i
$392$		0			0
$393$			172.382i			0.438631i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	223.044			0.561824			0.280912	−	0.959733i	$-0.409363\pi$
$397$	223.044			0.561824			0.280912	+	0.959733i	$0.409363\pi$
$398$		0			0
$399$	−149.421	+	228.724i	−0.374490	+	0.573244i
$400$		0			0
$401$	136.225			0.339714			0.169857	−	0.985469i	$-0.445669\pi$
$401$	136.225			0.339714			0.169857	+	0.985469i	$0.445669\pi$
$402$		0			0
$403$		−	2.27417i		−	0.00564310i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	409.255i		−	1.00554i
$408$		0			0
$409$		−	671.587i		−	1.64202i	−0.570913	−	0.821010i	$-0.693411\pi$
$409$		−	671.587i		−	1.64202i	0.570913	−	0.821010i	$-0.306589\pi$
$410$		0			0
$411$			186.389i			0.453501i
$412$		0			0
$413$	−222.519	+	340.617i	−0.538787	+	0.824739i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			268.382i			0.643601i
$418$		0			0
$419$		−	10.8053i		−	0.0257882i	−0.999917	−	0.0128941i	$-0.995896\pi$
$419$		−	10.8053i		−	0.0257882i	0.999917	−	0.0128941i	$-0.00410443\pi$
$420$		0			0
$421$	421.647			1.00154			0.500768	−	0.865581i	$-0.333051\pi$
$421$	421.647			1.00154			0.500768	+	0.865581i	$0.333051\pi$
$422$		0			0
$423$	573.529			1.35586
$424$		0			0
$425$		0			0
$426$		0			0
$427$	38.1782	−	58.4407i	0.0894104	−	0.136863i
$428$		0			0
$429$	3.74430			0.00872797
$430$		0			0
$431$	793.529			1.84113			0.920567	−	0.390584i	$-0.127727\pi$
$431$	793.529			1.84113			0.920567	+	0.390584i	$0.127727\pi$
$432$		0			0
$433$	−254.627			−0.588053			−0.294027	−	0.955797i	$-0.594995\pi$
$433$	−254.627			−0.588053			−0.294027	+	0.955797i	$0.594995\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	423.953			0.970144
$438$		0			0
$439$		−	839.517i		−	1.91234i	−0.292814	−	0.956169i	$-0.594592\pi$
$439$		−	839.517i		−	1.91234i	0.292814	−	0.956169i	$-0.405408\pi$
$440$		0			0
$441$	−131.049	−	298.703i	−0.297163	−	0.677331i
$442$		0			0
$443$		−	774.215i		−	1.74766i	−0.486228	−	0.873832i	$-0.661627\pi$
$443$		−	774.215i		−	1.74766i	0.486228	−	0.873832i	$-0.338373\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	363.954			0.814215
$448$		0			0
$449$	−751.921			−1.67466			−0.837328	−	0.546700i	$-0.815884\pi$
$449$	−751.921			−1.67466			−0.837328	+	0.546700i	$0.815884\pi$
$450$		0			0
$451$			213.432i			0.473243i
$452$		0			0
$453$	152.113			0.335790
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	251.990i		−	0.551400i	−0.961244	−	0.275700i	$-0.911090\pi$
$457$		−	251.990i		−	0.551400i	0.961244	−	0.275700i	$-0.0889097\pi$
$458$		0			0
$459$	−354.274			−0.771839
$460$		0			0
$461$		−	93.4121i		−	0.202629i	−0.994854	−	0.101315i	$-0.967695\pi$
$461$		−	93.4121i		−	0.202629i	0.994854	−	0.101315i	$-0.0323049\pi$
$462$		0			0
$463$		−	167.990i		−	0.362829i	−0.983407	−	0.181415i	$-0.941932\pi$
$463$		−	167.990i		−	0.362829i	0.983407	−	0.181415i	$-0.0580676\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	364.179			0.779828			0.389914	−	0.920851i	$-0.372505\pi$
$467$	364.179			0.779828			0.389914	+	0.920851i	$0.372505\pi$
$468$		0			0
$469$	78.9706	−	120.883i	0.168381	−	0.257746i
$470$		0			0
$471$	−147.951			−0.314122
$472$		0			0
$473$		−	428.431i		−	0.905773i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	403.588i		−	0.846096i
$478$		0			0
$479$		−	137.061i		−	0.286139i	−0.989713	−	0.143069i	$-0.954303\pi$
$479$		−	137.061i		−	0.286139i	0.989713	−	0.143069i	$-0.0456972\pi$
$480$		0			0
$481$		−	11.5404i		−	0.0239924i
$482$		0			0
$483$	97.4417	−	149.157i	0.201743	−	0.308814i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			557.411i			1.14458i	0.820051	+	0.572291i	$0.193945\pi$
$487$			557.411i			1.14458i	−0.820051	+	0.572291i	$0.806055\pi$
$488$		0			0
$489$		−	261.095i		−	0.533937i
$490$		0			0
$491$	−537.647			−1.09500			−0.547502	−	0.836805i	$-0.684421\pi$
$491$	−537.647			−1.09500			−0.547502	+	0.836805i	$0.684421\pi$
$492$		0			0
$493$	−217.094			−0.440353
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−462.791	−	302.333i	−0.931170	−	0.608316i
$498$		0			0
$499$	633.980			1.27050			0.635250	−	0.772306i	$-0.280897\pi$
$499$	633.980			1.27050			0.635250	+	0.772306i	$0.280897\pi$
$500$		0			0
$501$	−145.275			−0.289970
$502$		0			0
$503$	91.1385			0.181190			0.0905949	−	0.995888i	$-0.471123\pi$
$503$	91.1385			0.181190			0.0905949	+	0.995888i	$0.471123\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−258.588			−0.510036
$508$		0			0
$509$			353.366i			0.694237i	0.937821	+	0.347118i	$0.112840\pi$
$509$			353.366i			0.694237i	−0.937821	+	0.347118i	$0.887160\pi$
$510$		0			0
$511$	−125.588	−	82.0442i	−0.245769	−	0.160556i
$512$		0			0
$513$		−	611.078i		−	1.19119i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	802.434			1.55210
$518$		0			0
$519$	−279.696			−0.538912
$520$		0			0
$521$		−	949.039i		−	1.82157i	−0.412878	−	0.910786i	$-0.635476\pi$
$521$		−	949.039i		−	1.82157i	0.412878	−	0.910786i	$-0.364524\pi$
$522$		0			0
$523$	−662.702			−1.26712			−0.633558	−	0.773695i	$-0.718406\pi$
$523$	−662.702			−1.26712			−0.633558	+	0.773695i	$0.718406\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			128.000i			0.242884i
$528$		0			0
$529$	252.529			0.477371
$530$		0			0
$531$		−	386.915i		−	0.728654i
$532$		0			0
$533$			6.01847i			0.0112917i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	312.360			0.581676
$538$		0			0
$539$	−183.352	−	417.920i	−0.340171	−	0.775362i
$540$		0			0
$541$	1026.90			1.89815			0.949077	−	0.315043i	$-0.102019\pi$
$541$	1026.90			1.89815			0.949077	+	0.315043i	$0.102019\pi$
$542$		0			0
$543$			350.794i			0.646029i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	258.686i		−	0.472918i	−0.971641	−	0.236459i	$-0.924013\pi$
$547$		−	258.686i		−	0.472918i	0.971641	−	0.236459i	$-0.0759869\pi$
$548$		0			0
$549$			66.3841i			0.120918i
$550$		0			0
$551$		−	374.459i		−	0.679600i
$552$		0			0
$553$	529.093	+	345.647i	0.956769	+	0.625039i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	741.529i		−	1.33129i	−0.746268	−	0.665645i	$-0.768156\pi$
$557$		−	741.529i		−	1.33129i	0.746268	−	0.665645i	$-0.231844\pi$
$558$		0			0
$559$		−	12.0811i		−	0.0216120i
$560$		0			0
$561$	−210.745			−0.375660
$562$		0			0
$563$	80.5135			0.143008			0.0715040	−	0.997440i	$-0.477220\pi$
$563$	80.5135			0.143008			0.0715040	+	0.997440i	$0.477220\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	136.108	+	88.9167i	0.240049	+	0.156820i
$568$		0			0
$569$	742.971			1.30575			0.652874	−	0.757467i	$-0.273563\pi$
$569$	742.971			1.30575			0.652874	+	0.757467i	$0.273563\pi$
$570$		0			0
$571$	597.882			1.04708			0.523540	−	0.852001i	$-0.324611\pi$
$571$	597.882			1.04708			0.523540	+	0.852001i	$0.324611\pi$
$572$		0			0
$573$	14.6920			0.0256404
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−830.767			−1.43980			−0.719902	−	0.694075i	$-0.755813\pi$
$577$	−830.767			−1.43980			−0.719902	+	0.694075i	$0.755813\pi$
$578$		0			0
$579$			5.23719i			0.00904523i
$580$		0			0
$581$	421.088	+	275.089i	0.724765	+	0.473475i
$582$		0			0
$583$		−	564.666i		−	0.968552i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	700.310			1.19303			0.596516	−	0.802601i	$-0.296551\pi$
$587$	700.310			1.19303			0.596516	+	0.802601i	$0.296551\pi$
$588$		0			0
$589$	−220.784			−0.374845
$590$		0			0
$591$		−	224.177i		−	0.379318i
$592$		0			0
$593$	417.050			0.703288			0.351644	−	0.936134i	$-0.385623\pi$
$593$	417.050			0.703288			0.351644	+	0.936134i	$0.385623\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			374.489i			0.627286i
$598$		0			0
$599$	−223.892			−0.373777			−0.186888	−	0.982381i	$-0.559840\pi$
$599$	−223.892			−0.373777			−0.186888	+	0.982381i	$0.559840\pi$
$600$		0			0
$601$			721.185i			1.19998i	0.800009	+	0.599988i	$0.204828\pi$
$601$			721.185i			1.19998i	−0.800009	+	0.599988i	$0.795172\pi$
$602$		0			0
$603$			137.314i			0.227718i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−320.494			−0.527996			−0.263998	−	0.964523i	$-0.585041\pi$
$607$	−320.494			−0.527996			−0.263998	+	0.964523i	$0.585041\pi$
$608$		0			0
$609$	−131.744	−	86.0661i	−0.216329	−	0.141324i
$610$		0			0
$611$	22.6274			0.0370334
$612$		0			0
$613$		−	195.373i		−	0.318715i	−0.987221	−	0.159358i	$-0.949058\pi$
$613$		−	195.373i		−	0.318715i	0.987221	−	0.159358i	$-0.0509423\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			838.284i			1.35865i	0.733840	+	0.679323i	$0.237726\pi$
$617$			838.284i			1.35865i	−0.733840	+	0.679323i	$0.762274\pi$
$618$		0			0
$619$		−	515.407i		−	0.832644i	−0.909217	−	0.416322i	$-0.863319\pi$
$619$		−	515.407i		−	0.832644i	0.909217	−	0.416322i	$-0.136681\pi$
$620$		0			0
$621$			398.501i			0.641708i
$622$		0			0
$623$	7.69873	−	11.7847i	0.0123575	−	0.0189161i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	363.509i		−	0.579759i
$628$		0			0
$629$			649.541i			1.03266i
$630$		0			0
$631$	−392.538			−0.622089			−0.311045	−	0.950395i	$-0.600679\pi$
$631$	−392.538			−0.622089			−0.311045	+	0.950395i	$0.600679\pi$
$632$		0			0
$633$	476.088			0.752113
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.17026	−	11.7847i	−0.00811657	−	0.0185003i
$638$		0			0
$639$	525.696			0.822685
$640$		0			0
$641$	−813.460			−1.26905			−0.634524	−	0.772903i	$-0.718804\pi$
$641$	−813.460			−1.26905			−0.634524	+	0.772903i	$0.718804\pi$
$642$		0			0
$643$	−906.299			−1.40948			−0.704742	−	0.709463i	$-0.748937\pi$
$643$	−906.299			−1.40948			−0.704742	+	0.709463i	$0.748937\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−114.325			−0.176700			−0.0883499	−	0.996089i	$-0.528159\pi$
$647$	−114.325			−0.176700			−0.0883499	+	0.996089i	$0.528159\pi$
$648$		0			0
$649$		−	541.339i		−	0.834113i
$650$		0			0
$651$	−50.7452	+	77.6773i	−0.0779496	+	0.119320i
$652$		0			0
$653$		−	415.685i		−	0.636578i	−0.947994	−	0.318289i	$-0.896892\pi$
$653$		−	415.685i		−	0.636578i	0.947994	−	0.318289i	$-0.103108\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	142.658			0.217136
$658$		0			0
$659$	−650.431			−0.986996			−0.493498	−	0.869747i	$-0.664282\pi$
$659$	−650.431			−0.986996			−0.493498	+	0.869747i	$0.664282\pi$
$660$		0			0
$661$			1023.81i			1.54888i	0.632645	+	0.774442i	$0.281969\pi$
$661$			1023.81i			1.54888i	−0.632645	+	0.774442i	$0.718031\pi$
$662$		0			0
$663$	−5.94269			−0.00896334
$664$		0			0
$665$		0			0
$666$		0			0
$667$			244.195i			0.366110i
$668$		0			0
$669$	177.803			0.265775
$670$		0			0
$671$			92.8791i			0.138419i
$672$		0			0
$673$			1187.21i			1.76406i	0.471190	+	0.882032i	$0.343824\pi$
$673$			1187.21i			1.76406i	−0.471190	+	0.882032i	$0.656176\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	905.826			1.33800			0.669000	−	0.743262i	$-0.266723\pi$
$677$	905.826			1.33800			0.669000	+	0.743262i	$0.266723\pi$
$678$		0			0
$679$	−929.332	−	607.116i	−1.36868	−	0.894132i
$680$		0			0
$681$	−90.5786			−0.133008
$682$		0			0
$683$			315.255i			0.461574i	0.973004	+	0.230787i	$0.0741300\pi$
$683$			315.255i			0.461574i	−0.973004	+	0.230787i	$0.925870\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	196.950i		−	0.286682i
$688$		0			0
$689$		−	15.9227i		−	0.0231099i
$690$		0			0
$691$			334.960i			0.484747i	0.970183	+	0.242374i	$0.0779260\pi$
$691$			334.960i			0.484747i	−0.970183	+	0.242374i	$0.922074\pi$
$692$		0			0
$693$	363.339	+	237.362i	0.524298	+	0.342514i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	338.745i		−	0.486005i
$698$		0			0
$699$			136.206i			0.194858i
$700$		0			0
$701$	−362.353			−0.516909			−0.258455	−	0.966023i	$-0.583213\pi$
$701$	−362.353			−0.516909			−0.258455	+	0.966023i	$0.583213\pi$
$702$		0			0
$703$	−1120.38			−1.59371
$704$		0			0
$705$		0			0
$706$		0			0
$707$	357.966	−	547.951i	0.506317	−	0.775037i
$708$		0			0
$709$	1117.96			1.57681			0.788406	−	0.615155i	$-0.210907\pi$
$709$	1117.96			1.57681			0.788406	+	0.615155i	$0.210907\pi$
$710$		0			0
$711$	−601.009			−0.845301
$712$		0			0
$713$	143.979			0.201934
$714$		0			0
$715$		0			0
$716$		0			0
$717$	10.9250			0.0152371
$718$		0			0
$719$			972.571i			1.35267i	0.736593	+	0.676336i	$0.236433\pi$
$719$			972.571i			1.35267i	−0.736593	+	0.676336i	$0.763567\pi$
$720$		0			0
$721$	−119.431	−	78.0224i	−0.165647	−	0.108214i
$722$		0			0
$723$		−	562.469i		−	0.777966i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−660.646			−0.908729			−0.454365	−	0.890816i	$-0.650134\pi$
$727$	−660.646			−0.908729			−0.454365	+	0.890816i	$0.650134\pi$
$728$		0			0
$729$	175.569			0.240835
$730$		0			0
$731$			679.975i			0.930199i
$732$		0			0
$733$	−258.222			−0.352280			−0.176140	−	0.984365i	$-0.556361\pi$
$733$	−258.222			−0.352280			−0.176140	+	0.984365i	$0.556361\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			192.118i			0.260675i
$738$		0			0
$739$	333.391			0.451138			0.225569	−	0.974227i	$-0.427576\pi$
$739$	333.391			0.451138			0.225569	+	0.974227i	$0.427576\pi$
$740$		0			0
$741$		−	10.2504i		−	0.0138332i
$742$		0			0
$743$			610.118i			0.821154i	0.911826	+	0.410577i	$0.134673\pi$
$743$			610.118i			0.821154i	−0.911826	+	0.410577i	$0.865327\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−478.324			−0.640327
$748$		0			0
$749$	−54.0488	+	82.7343i	−0.0721612	+	0.110460i
$750$		0			0
$751$	−18.1177			−0.0241248			−0.0120624	−	0.999927i	$-0.503840\pi$
$751$	−18.1177			−0.0241248			−0.0120624	+	0.999927i	$0.503840\pi$
$752$		0			0
$753$		−	270.932i		−	0.359803i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			107.137i			0.141529i	0.997493	+	0.0707643i	$0.0225438\pi$
$757$			107.137i			0.141529i	−0.997493	+	0.0707643i	$0.977456\pi$
$758$		0			0
$759$			237.054i			0.312324i
$760$		0			0
$761$		−	898.284i		−	1.18040i	−0.807257	−	0.590200i	$-0.799049\pi$
$761$		−	898.284i		−	1.18040i	0.807257	−	0.590200i	$-0.200951\pi$
$762$		0			0
$763$	−136.280	−	89.0294i	−0.178611	−	0.116683i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	15.2649i		−	0.0199021i
$768$		0			0
$769$			929.350i			1.20852i	0.796788	+	0.604259i	$0.206531\pi$
$769$			929.350i			1.20852i	−0.796788	+	0.604259i	$0.793469\pi$
$770$		0			0
$771$	−42.0387			−0.0545249
$772$		0			0
$773$	914.395			1.18292			0.591459	−	0.806335i	$-0.298552\pi$
$773$	914.395			1.18292			0.591459	+	0.806335i	$0.298552\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−257.508	+	394.177i	−0.331413	+	0.507306i
$778$		0			0
$779$	584.293			0.750055
$780$		0			0
$781$	735.509			0.941753
$782$		0			0
$783$	351.979			0.449526
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−761.720			−0.967878			−0.483939	−	0.875102i	$-0.660794\pi$
$787$	−761.720			−0.967878			−0.483939	+	0.875102i	$0.660794\pi$
$788$		0			0
$789$		−	118.092i		−	0.149673i
$790$		0			0
$791$	52.5097	−	80.3783i	0.0663839	−	0.101616i
$792$		0			0
$793$			2.61905i			0.00330271i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−395.267			−0.495943			−0.247972	−	0.968767i	$-0.579764\pi$
$797$	−395.267			−0.495943			−0.247972	+	0.968767i	$0.579764\pi$
$798$		0			0
$799$	−1273.57			−1.59395
$800$		0			0
$801$			13.3865i			0.0167123i
$802$		0			0
$803$	199.595			0.248562
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	52.6173i		−	0.0652011i
$808$		0			0
$809$	877.793			1.08503			0.542517	−	0.840045i	$-0.317471\pi$
$809$	877.793			1.08503			0.542517	+	0.840045i	$0.317471\pi$
$810$		0			0
$811$		−	851.717i		−	1.05021i	−0.851039	−	0.525103i	$-0.824027\pi$
$811$		−	851.717i		−	1.05021i	0.851039	−	0.525103i	$-0.175973\pi$
$812$		0			0
$813$			310.627i			0.382076i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1172.87			−1.43558
$818$		0			0
$819$	10.2456	+	6.69326i	0.0125099	+	0.00817248i
$820$		0			0
$821$	437.038			0.532324			0.266162	−	0.963928i	$-0.414244\pi$
$821$	437.038			0.532324			0.266162	+	0.963928i	$0.414244\pi$
$822$		0			0
$823$			1592.34i			1.93480i	0.253247	+	0.967402i	$0.418501\pi$
$823$			1592.34i			1.93480i	−0.253247	+	0.967402i	$0.581499\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1182.45i			1.42981i	0.699223	+	0.714904i	$0.253530\pi$
$827$			1182.45i			1.42981i	−0.699223	+	0.714904i	$0.746470\pi$
$828$		0			0
$829$			1048.05i			1.26423i	0.774873	+	0.632117i	$0.217814\pi$
$829$			1048.05i			1.26423i	−0.774873	+	0.632117i	$0.782186\pi$
$830$		0			0
$831$		−	388.956i		−	0.468057i
$832$		0			0
$833$	291.004	+	663.294i	0.349345	+	0.796271i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	207.529i		−	0.247944i
$838$		0			0
$839$		−	585.611i		−	0.697986i	−0.937125	−	0.348993i	$-0.886524\pi$
$839$		−	585.611i		−	0.697986i	0.937125	−	0.348993i	$-0.113476\pi$
$840$		0			0
$841$	−625.313			−0.743535
$842$		0			0
$843$	−339.102			−0.402256
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−200.744	−	131.142i	−0.237006	−	0.154831i
$848$		0			0
$849$	434.400			0.511661
$850$		0			0
$851$	730.627			0.858552
$852$		0			0
$853$	−901.189			−1.05649			−0.528247	−	0.849091i	$-0.677150\pi$
$853$	−901.189			−1.05649			−0.528247	+	0.849091i	$0.677150\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−965.997			−1.12718			−0.563592	−	0.826053i	$-0.690581\pi$
$857$	−965.997			−1.12718			−0.563592	+	0.826053i	$0.690581\pi$
$858$		0			0
$859$			250.830i			0.292003i	0.989284	+	0.146001i	$0.0466404\pi$
$859$			250.830i			0.292003i	−0.989284	+	0.146001i	$0.953360\pi$
$860$		0			0
$861$	134.294	−	205.569i	0.155975	−	0.238756i
$862$		0			0
$863$		−	1497.13i		−	1.73479i	−0.497617	−	0.867397i	$-0.665791\pi$
$863$		−	1497.13i		−	1.73479i	0.497617	−	0.867397i	$-0.334209\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−107.902			−0.124454
$868$		0			0
$869$	−840.881			−0.967643
$870$		0			0
$871$			5.41743i			0.00621978i
$872$		0			0
$873$	1055.65			1.20922
$874$		0			0
$875$		0			0
$876$		0			0
$877$			141.354i			0.161179i	0.996747	+	0.0805896i	$0.0256803\pi$
$877$			141.354i			0.161179i	−0.996747	+	0.0805896i	$0.974320\pi$
$878$		0			0
$879$	609.344			0.693224
$880$		0			0
$881$		−	1413.83i		−	1.60480i	−0.596788	−	0.802399i	$-0.703557\pi$
$881$		−	1413.83i		−	1.60480i	0.596788	−	0.802399i	$-0.296443\pi$
$882$		0			0
$883$			1386.04i			1.56969i	0.619692	+	0.784845i	$0.287258\pi$
$883$			1386.04i			1.56969i	−0.619692	+	0.784845i	$0.712742\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	303.595			0.342272			0.171136	−	0.985247i	$-0.445256\pi$
$887$	303.595			0.342272			0.171136	+	0.985247i	$0.445256\pi$
$888$		0			0
$889$	−680.558	+	1041.75i	−0.765533	+	1.17183i
$890$		0			0
$891$	−216.315			−0.242777
$892$		0			0
$893$		−	2196.74i		−	2.45996i
$894$		0			0
$895$		0			0
$896$		0			0
$897$			6.68456i			0.00745213i
$898$		0			0
$899$		−	127.171i		−	0.141458i
$900$		0			0
$901$			896.199i			0.994671i
$902$		0			0
$903$	−269.574	+	412.646i	−0.298531	+	0.456972i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1304.02i			1.43773i	0.695151	+	0.718864i	$0.255337\pi$
$907$			1304.02i			1.43773i	−0.695151	+	0.718864i	$0.744663\pi$
$908$		0			0
$909$			622.431i			0.684742i
$910$		0			0
$911$	−466.118			−0.511655			−0.255828	−	0.966722i	$-0.582348\pi$
$911$	−466.118			−0.511655			−0.255828	+	0.966722i	$0.582348\pi$
$912$		0			0
$913$	−669.231			−0.733002
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−431.134	+	659.951i	−0.470157	+	0.719685i
$918$		0			0
$919$	1178.34			1.28220			0.641100	−	0.767458i	$-0.278478\pi$
$919$	1178.34			1.28220			0.641100	+	0.767458i	$0.278478\pi$
$920$		0			0
$921$	−256.735			−0.278757
$922$		0			0
$923$	20.7402			0.0224705
$924$		0			0
$925$		0			0
$926$		0			0
$927$	135.665			0.146348
$928$		0			0
$929$			911.161i			0.980798i	0.871498	+	0.490399i	$0.163149\pi$
$929$			911.161i			0.980798i	−0.871498	+	0.490399i	$0.836851\pi$
$930$		0			0
$931$	−1144.10	+	501.945i	−1.22889	+	0.539147i
$932$		0			0
$933$			434.883i			0.466113i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1035.35			−1.10496			−0.552479	−	0.833527i	$-0.686318\pi$
$937$	−1035.35			−1.10496			−0.552479	+	0.833527i	$0.686318\pi$
$938$		0			0
$939$	−787.273			−0.838417
$940$		0			0
$941$			64.4479i			0.0684887i	0.999413	+	0.0342443i	$0.0109024\pi$
$941$			64.4479i			0.0684887i	−0.999413	+	0.0342443i	$0.989098\pi$
$942$		0			0
$943$	−381.033			−0.404065
$944$		0			0
$945$		0			0
$946$		0			0
$947$			132.274i			0.139677i	0.997558	+	0.0698385i	$0.0222484\pi$
$947$			132.274i			0.139677i	−0.997558	+	0.0698385i	$0.977752\pi$
$948$		0			0
$949$	5.62828			0.00593075
$950$		0			0
$951$			28.4235i			0.0298880i
$952$		0			0
$953$		−	1346.86i		−	1.41329i	−0.707570	−	0.706643i	$-0.750209\pi$
$953$		−	1346.86i		−	1.41329i	0.707570	−	0.706643i	$-0.249791\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	209.380			0.218787
$958$		0			0
$959$	−466.167	+	713.577i	−0.486096	+	0.744084i
$960$		0			0
$961$	886.019			0.921976
$962$		0			0
$963$		−	93.9798i		−	0.0975907i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	498.979i		−	0.516007i	−0.966144	−	0.258004i	$-0.916935\pi$
$967$		−	498.979i		−	0.516007i	0.966144	−	0.258004i	$-0.0830646\pi$
$968$		0			0
$969$			576.936i			0.595393i
$970$		0			0
$971$		−	1799.26i		−	1.85299i	−0.376304	−	0.926496i	$-0.622805\pi$
$971$		−	1799.26i		−	1.85299i	0.376304	−	0.926496i	$-0.377195\pi$
$972$		0			0
$973$	−671.234	+	1027.48i	−0.689860	+	1.05599i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			1381.61i			1.41413i	0.707148	+	0.707066i	$0.249982\pi$
$977$			1381.61i			1.41413i	−0.707148	+	0.707066i	$0.750018\pi$
$978$		0			0
$979$			18.7293i			0.0191310i
$980$		0			0
$981$	154.804			0.157802
$982$		0			0
$983$	−1306.67			−1.32927			−0.664636	−	0.747167i	$-0.731413\pi$
$983$	−1306.67			−1.32927			−0.664636	+	0.747167i	$0.731413\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−772.870	−	504.902i	−0.783050	−	0.511552i
$988$		0			0
$989$	764.861			0.773368
$990$		0			0
$991$	1033.99			1.04338			0.521689	−	0.853136i	$-0.325302\pi$
$991$	1033.99			1.04338			0.521689	+	0.853136i	$0.325302\pi$
$992$		0			0
$993$	297.472			0.299569
$994$		0			0
$995$		0			0
$996$		0			0
$997$	391.500			0.392678			0.196339	−	0.980536i	$-0.437095\pi$
$997$	391.500			0.392678			0.196339	+	0.980536i	$0.437095\pi$
$998$		0			0
$999$		−	1053.11i		−	1.05417i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1400.3.p.a.1049.6		8
5.2	odd	4		1400.3.f.a.601.2		4
5.3	odd	4		56.3.c.a.41.3	yes	4
5.4	even	2	inner	1400.3.p.a.1049.3		8
7.6	odd	2	inner	1400.3.p.a.1049.4		8
15.8	even	4		504.3.f.a.433.1		4
20.3	even	4		112.3.c.c.97.2		4
35.3	even	12		392.3.o.b.313.2		8
35.13	even	4		56.3.c.a.41.2	✓	4
35.18	odd	12		392.3.o.b.313.3		8
35.23	odd	12		392.3.o.b.129.2		8
35.27	even	4		1400.3.f.a.601.3		4
35.33	even	12		392.3.o.b.129.3		8
35.34	odd	2	inner	1400.3.p.a.1049.5		8
40.3	even	4		448.3.c.e.321.3		4
40.13	odd	4		448.3.c.f.321.2		4
60.23	odd	4		1008.3.f.h.433.1		4
105.83	odd	4		504.3.f.a.433.4		4
140.3	odd	12		784.3.s.f.705.3		8
140.23	even	12		784.3.s.f.129.3		8
140.83	odd	4		112.3.c.c.97.3		4
140.103	odd	12		784.3.s.f.129.2		8
140.123	even	12		784.3.s.f.705.2		8
280.13	even	4		448.3.c.f.321.3		4
280.83	odd	4		448.3.c.e.321.2		4
420.83	even	4		1008.3.f.h.433.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.c.a.41.2	✓	4	35.13	even	4
56.3.c.a.41.3	yes	4	5.3	odd	4
112.3.c.c.97.2		4	20.3	even	4
112.3.c.c.97.3		4	140.83	odd	4
392.3.o.b.129.2		8	35.23	odd	12
392.3.o.b.129.3		8	35.33	even	12
392.3.o.b.313.2		8	35.3	even	12
392.3.o.b.313.3		8	35.18	odd	12
448.3.c.e.321.2		4	280.83	odd	4
448.3.c.e.321.3		4	40.3	even	4
448.3.c.f.321.2		4	40.13	odd	4
448.3.c.f.321.3		4	280.13	even	4
504.3.f.a.433.1		4	15.8	even	4
504.3.f.a.433.4		4	105.83	odd	4
784.3.s.f.129.2		8	140.103	odd	12
784.3.s.f.129.3		8	140.23	even	12
784.3.s.f.705.2		8	140.123	even	12
784.3.s.f.705.3		8	140.3	odd	12
1008.3.f.h.433.1		4	60.23	odd	4
1008.3.f.h.433.4		4	420.83	even	4
1400.3.f.a.601.2		4	5.2	odd	4
1400.3.f.a.601.3		4	35.27	even	4
1400.3.p.a.1049.3		8	5.4	even	2	inner
1400.3.p.a.1049.4		8	7.6	odd	2	inner
1400.3.p.a.1049.5		8	35.34	odd	2	inner
1400.3.p.a.1049.6		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1400.p (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.53073 q^{3} +(5.86030 + 3.82843i) q^{7} -6.65685 q^{9} +O(q^{10})\)	\(q+1.53073 q^{3} +(5.86030 + 3.82843i) q^{7} -6.65685 q^{9} -9.31371 q^{11} -0.262632 q^{13} +14.7821 q^{17} +25.4972i q^{19} +(8.97056 + 5.86030i) q^{21} -16.6274i q^{23} -23.9665 q^{27} -14.6863 q^{29} +8.65914i q^{31} -14.2568 q^{33} +43.9411i q^{37} -0.402020 q^{39} -22.9159i q^{41} +46.0000i q^{43} -86.1562 q^{47} +(19.6863 + 44.8715i) q^{49} +22.6274 q^{51} +60.6274i q^{53} +39.0294i q^{57} +58.1228i q^{59} -9.97230i q^{61} +(-39.0112 - 25.4853i) q^{63} -20.6274i q^{67} -25.4521i q^{69} -78.9706 q^{71} -21.4303 q^{73} +(-54.5811 - 35.6569i) q^{77} +90.2843 q^{79} +23.2254 q^{81} +71.8544 q^{83} -22.4808 q^{87} -2.01094i q^{89} +(-1.53911 - 1.00547i) q^{91} +13.2548i q^{93} -158.581 q^{97} +62.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^9	\( 8 q - 8 q^{9} + 16 q^{11} - 64 q^{21} - 208 q^{29} - 320 q^{39} + 248 q^{49} - 496 q^{71} + 496 q^{79} - 312 q^{81} + 576 q^{91} + 496 q^{99}+O(q^{100}) \) 8 * q - 8 * q^9 + 16 * q^11 - 64 * q^21 - 208 * q^29 - 320 * q^39 + 248 * q^49 - 496 * q^71 + 496 * q^79 - 312 * q^81 + 576 * q^91 + 496 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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