LMFDB - Embedded newform 4140.3.c.a.4049.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,3,Mod(4049,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1, 0]))

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

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

chi := DirichletCharacter("4140.4049");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	4140.c (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:	$112.806829445$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.8
Character	$\chi$	$=$	4140.4049
Dual form			4140.3.c.a.4049.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.79317 + 1.42321i) q^{5} +6.33514i q^{7} +O(q^{10})$	$q+(-4.79317 + 1.42321i) q^{5} +6.33514i q^{7} +7.95078i q^{11} +13.5343i q^{13} -7.31662 q^{17} -31.1068 q^{19} -4.79583 q^{23} +(20.9489 - 13.6434i) q^{25} +48.5946i q^{29} -36.6246 q^{31} +(-9.01625 - 30.3654i) q^{35} +53.1116i q^{37} -12.8489i q^{41} +47.2824i q^{43} +5.53624 q^{47} +8.86599 q^{49} +7.85364 q^{53} +(-11.3156 - 38.1094i) q^{55} -86.8361i q^{59} -44.4340 q^{61} +(-19.2621 - 64.8720i) q^{65} +59.9578i q^{67} +37.3888i q^{71} +9.12506i q^{73} -50.3693 q^{77} -104.229 q^{79} -42.0578 q^{83} +(35.0698 - 10.4131i) q^{85} +74.0762i q^{89} -85.7414 q^{91} +(149.100 - 44.2715i) q^{95} +52.8702i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q+O(q^{10}) $ 88 * q	$ 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) $ 88 * q - 16 * q^19 - 48 * q^25 + 272 * q^31 - 600 * q^49 + 112 * q^55 + 448 * q^61 - 32 * q^79 - 264 * q^85 - 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−4.79317	+	1.42321i	−0.958634	+	0.284642i
$5$	−4.79317	+	1.42321i	−0.958634	+	0.284642i
$6$		0			0
$7$			6.33514i			0.905020i	0.891759	+	0.452510i	$0.149471\pi$
$7$			6.33514i			0.905020i	−0.891759	+	0.452510i	$0.850529\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			7.95078i			0.722798i	0.932411	+	0.361399i	$0.117701\pi$
$11$			7.95078i			0.722798i	−0.932411	+	0.361399i	$0.882299\pi$
$12$		0			0
$13$			13.5343i			1.04110i	0.853832	+	0.520548i	$0.174272\pi$
$13$			13.5343i			1.04110i	−0.853832	+	0.520548i	$0.825728\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.31662			−0.430389			−0.215195	−	0.976571i	$-0.569039\pi$
$17$	−7.31662			−0.430389			−0.215195	+	0.976571i	$0.569039\pi$
$18$		0			0
$19$	−31.1068			−1.63720			−0.818599	−	0.574365i	$-0.805249\pi$
$19$	−31.1068			−1.63720			−0.818599	+	0.574365i	$0.805249\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.79583			−0.208514
$23$	−4.79583			−0.208514
$24$		0			0
$25$	20.9489	−	13.6434i	0.837958	−	0.545735i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			48.5946i			1.67568i	0.545919	+	0.837838i	$0.316181\pi$
$29$			48.5946i			1.67568i	−0.545919	+	0.837838i	$0.683819\pi$
$30$		0			0
$31$	−36.6246			−1.18144			−0.590719	−	0.806878i	$-0.701156\pi$
$31$	−36.6246			−1.18144			−0.590719	+	0.806878i	$0.701156\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−9.01625	−	30.3654i	−0.257607	−	0.867583i
$36$		0			0
$37$			53.1116i			1.43545i	0.696327	+	0.717725i	$0.254816\pi$
$37$			53.1116i			1.43545i	−0.696327	+	0.717725i	$0.745184\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	12.8489i		−	0.313388i	−0.987647	−	0.156694i	$-0.949916\pi$
$41$		−	12.8489i		−	0.313388i	0.987647	−	0.156694i	$-0.0500837\pi$
$42$		0			0
$43$			47.2824i			1.09959i	0.835299	+	0.549796i	$0.185294\pi$
$43$			47.2824i			1.09959i	−0.835299	+	0.549796i	$0.814706\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.53624			0.117792			0.0588962	−	0.998264i	$-0.481242\pi$
$47$	5.53624			0.117792			0.0588962	+	0.998264i	$0.481242\pi$
$48$		0			0
$49$	8.86599			0.180939
$50$		0			0
$51$		0			0
$52$		0			0
$53$	7.85364			0.148182			0.0740910	−	0.997251i	$-0.476394\pi$
$53$	7.85364			0.148182			0.0740910	+	0.997251i	$0.476394\pi$
$54$		0			0
$55$	−11.3156	−	38.1094i	−0.205739	−	0.692899i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	86.8361i		−	1.47180i	−0.677091	−	0.735900i	$-0.736760\pi$
$59$		−	86.8361i		−	1.47180i	0.677091	−	0.735900i	$-0.263240\pi$
$60$		0			0
$61$	−44.4340			−0.728426			−0.364213	−	0.931316i	$-0.618662\pi$
$61$	−44.4340			−0.728426			−0.364213	+	0.931316i	$0.618662\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−19.2621	−	64.8720i	−0.296340	−	0.998030i
$66$		0			0
$67$			59.9578i			0.894893i	0.894311	+	0.447446i	$0.147666\pi$
$67$			59.9578i			0.894893i	−0.894311	+	0.447446i	$0.852334\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			37.3888i			0.526603i	0.964714	+	0.263301i	$0.0848114\pi$
$71$			37.3888i			0.526603i	−0.964714	+	0.263301i	$0.915189\pi$
$72$		0			0
$73$			9.12506i			0.125001i	0.998045	+	0.0625004i	$0.0199075\pi$
$73$			9.12506i			0.125001i	−0.998045	+	0.0625004i	$0.980093\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−50.3693			−0.654147
$78$		0			0
$79$	−104.229			−1.31936			−0.659678	−	0.751549i	$-0.729307\pi$
$79$	−104.229			−1.31936			−0.659678	+	0.751549i	$0.729307\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−42.0578			−0.506721			−0.253360	−	0.967372i	$-0.581536\pi$
$83$	−42.0578			−0.506721			−0.253360	+	0.967372i	$0.581536\pi$
$84$		0			0
$85$	35.0698	−	10.4131i	0.412586	−	0.122507i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			74.0762i			0.832317i	0.909292	+	0.416158i	$0.136624\pi$
$89$			74.0762i			0.832317i	−0.909292	+	0.416158i	$0.863376\pi$
$90$		0			0
$91$	−85.7414			−0.942213
$92$		0			0
$93$		0			0
$94$		0			0
$95$	149.100	−	44.2715i	1.56947	−	0.466016i
$96$		0			0
$97$			52.8702i			0.545053i	0.962148	+	0.272527i	$0.0878593\pi$
$97$			52.8702i			0.545053i	−0.962148	+	0.272527i	$0.912141\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	7.07949i		−	0.0700939i	−0.999386	−	0.0350470i	$-0.988842\pi$
$101$		−	7.07949i		−	0.0700939i	0.999386	−	0.0350470i	$-0.0111581\pi$
$102$		0			0
$103$		−	32.1711i		−	0.312341i	−0.987730	−	0.156170i	$-0.950085\pi$
$103$		−	32.1711i		−	0.312341i	0.987730	−	0.156170i	$-0.0499149\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	126.686			1.18398			0.591990	−	0.805945i	$-0.298342\pi$
$107$	126.686			1.18398			0.591990	+	0.805945i	$0.298342\pi$
$108$		0			0
$109$	−36.8513			−0.338086			−0.169043	−	0.985609i	$-0.554068\pi$
$109$	−36.8513			−0.338086			−0.169043	+	0.985609i	$0.554068\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	196.529			1.73919			0.869596	−	0.493764i	$-0.164379\pi$
$113$	196.529			1.73919			0.869596	+	0.493764i	$0.164379\pi$
$114$		0			0
$115$	22.9872	−	6.82548i	0.199889	−	0.0593520i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	46.3518i		−	0.389511i
$120$		0			0
$121$	57.7850			0.477562
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−80.9944	+	95.2098i	−0.647955	+	0.761679i
$126$		0			0
$127$			26.8214i			0.211193i	0.994409	+	0.105596i	$0.0336751\pi$
$127$			26.8214i			0.211193i	−0.994409	+	0.105596i	$0.966325\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	132.658i		−	1.01265i	−0.862341	−	0.506327i	$-0.831003\pi$
$131$		−	132.658i		−	1.01265i	0.862341	−	0.506327i	$-0.168997\pi$
$132$		0			0
$133$		−	197.066i		−	1.48170i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	85.4570			0.623774			0.311887	−	0.950119i	$-0.399039\pi$
$137$	85.4570			0.623774			0.311887	+	0.950119i	$0.399039\pi$
$138$		0			0
$139$	104.168			0.749411			0.374706	−	0.927144i	$-0.377744\pi$
$139$	104.168			0.749411			0.374706	+	0.927144i	$0.377744\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−107.608			−0.752503
$144$		0			0
$145$	−69.1604	−	232.922i	−0.476968	−	1.60636i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	47.2866i		−	0.317360i	−0.987330	−	0.158680i	$-0.949276\pi$
$149$		−	47.2866i		−	0.317360i	0.987330	−	0.158680i	$-0.0507237\pi$
$150$		0			0
$151$	27.0094			0.178870			0.0894352	−	0.995993i	$-0.471494\pi$
$151$	27.0094			0.178870			0.0894352	+	0.995993i	$0.471494\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	175.548	−	52.1245i	1.13257	−	0.336287i
$156$		0			0
$157$			124.573i			0.793458i	0.917936	+	0.396729i	$0.129855\pi$
$157$			124.573i			0.793458i	−0.917936	+	0.396729i	$0.870145\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	30.3823i		−	0.188710i
$162$		0			0
$163$			141.851i			0.870250i	0.900370	+	0.435125i	$0.143296\pi$
$163$			141.851i			0.870250i	−0.900370	+	0.435125i	$0.856704\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	307.114			1.83900			0.919502	−	0.393085i	$-0.128592\pi$
$167$	307.114			1.83900			0.919502	+	0.393085i	$0.128592\pi$
$168$		0			0
$169$	−14.1760			−0.0838817
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−222.822			−1.28799			−0.643996	−	0.765029i	$-0.722724\pi$
$173$	−222.822			−1.28799			−0.643996	+	0.765029i	$0.722724\pi$
$174$		0			0
$175$	86.4328	+	132.714i	0.493902	+	0.758368i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			89.3788i			0.499323i	0.968333	+	0.249661i	$0.0803193\pi$
$179$			89.3788i			0.499323i	−0.968333	+	0.249661i	$0.919681\pi$
$180$		0			0
$181$	−295.777			−1.63413			−0.817065	−	0.576546i	$-0.804400\pi$
$181$	−295.777			−1.63413			−0.817065	+	0.576546i	$0.804400\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−75.5891	−	254.573i	−0.408590	−	1.37607i
$186$		0			0
$187$		−	58.1729i		−	0.311085i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			23.2634i			0.121798i	0.998144	+	0.0608990i	$0.0193968\pi$
$191$			23.2634i			0.121798i	−0.998144	+	0.0608990i	$0.980603\pi$
$192$		0			0
$193$			52.7112i			0.273115i	0.990632	+	0.136557i	$0.0436038\pi$
$193$			52.7112i			0.273115i	−0.990632	+	0.136557i	$0.956396\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−165.422			−0.839703			−0.419852	−	0.907593i	$-0.637918\pi$
$197$	−165.422			−0.839703			−0.419852	+	0.907593i	$0.637918\pi$
$198$		0			0
$199$	35.4590			0.178186			0.0890930	−	0.996023i	$-0.471603\pi$
$199$	35.4590			0.178186			0.0890930	+	0.996023i	$0.471603\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−307.854			−1.51652
$204$		0			0
$205$	18.2867	+	61.5870i	0.0892036	+	0.300425i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	247.323i		−	1.18336i
$210$		0			0
$211$	166.398			0.788615			0.394307	−	0.918979i	$-0.370985\pi$
$211$	166.398			0.788615			0.394307	+	0.918979i	$0.370985\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−67.2929	−	226.633i	−0.312990	−	1.05411i
$216$		0			0
$217$		−	232.022i		−	1.06922i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	99.0250i		−	0.448077i
$222$		0			0
$223$		−	111.535i		−	0.500157i	−0.968226	−	0.250078i	$-0.919544\pi$
$223$		−	111.535i		−	0.500157i	0.968226	−	0.250078i	$-0.0804564\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	157.335			0.693108			0.346554	−	0.938030i	$-0.387352\pi$
$227$	157.335			0.693108			0.346554	+	0.938030i	$0.387352\pi$
$228$		0			0
$229$	−256.692			−1.12093			−0.560463	−	0.828180i	$-0.689377\pi$
$229$	−256.692			−1.12093			−0.560463	+	0.828180i	$0.689377\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	185.834			0.797573			0.398786	−	0.917044i	$-0.369431\pi$
$233$	185.834			0.797573			0.398786	+	0.917044i	$0.369431\pi$
$234$		0			0
$235$	−26.5361	+	7.87924i	−0.112920	+	0.0335287i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			24.6842i			0.103281i	0.998666	+	0.0516406i	$0.0164450\pi$
$239$			24.6842i			0.103281i	−0.998666	+	0.0516406i	$0.983555\pi$
$240$		0			0
$241$	85.6682			0.355470			0.177735	−	0.984078i	$-0.443123\pi$
$241$	85.6682			0.355470			0.177735	+	0.984078i	$0.443123\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−42.4962	+	12.6182i	−0.173454	+	0.0515028i
$246$		0			0
$247$		−	421.007i		−	1.70448i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			24.3213i			0.0968977i	0.998826	+	0.0484488i	$0.0154278\pi$
$251$			24.3213i			0.0968977i	−0.998826	+	0.0484488i	$0.984572\pi$
$252$		0			0
$253$		−	38.1306i		−	0.150714i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	252.685			0.983208			0.491604	−	0.870819i	$-0.336411\pi$
$257$	252.685			0.983208			0.491604	+	0.870819i	$0.336411\pi$
$258$		0			0
$259$	−336.470			−1.29911
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−59.0935			−0.224690			−0.112345	−	0.993669i	$-0.535836\pi$
$263$	−59.0935			−0.224690			−0.112345	+	0.993669i	$0.535836\pi$
$264$		0			0
$265$	−37.6438	+	11.1774i	−0.142052	+	0.0421788i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	231.052i		−	0.858931i	−0.903083	−	0.429465i	$-0.858702\pi$
$269$		−	231.052i		−	0.858931i	0.903083	−	0.429465i	$-0.141298\pi$
$270$		0			0
$271$	−499.479			−1.84310			−0.921548	−	0.388265i	$-0.873075\pi$
$271$	−499.479			−1.84310			−0.921548	+	0.388265i	$0.873075\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	108.476	+	166.560i	0.394457	+	0.605674i
$276$		0			0
$277$		−	479.538i		−	1.73118i	−0.500750	−	0.865592i	$-0.666942\pi$
$277$		−	479.538i		−	1.73118i	0.500750	−	0.865592i	$-0.333058\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	413.605i		−	1.47190i	−0.677035	−	0.735951i	$-0.736735\pi$
$281$		−	413.605i		−	1.47190i	0.677035	−	0.735951i	$-0.263265\pi$
$282$		0			0
$283$		−	415.951i		−	1.46979i	−0.678180	−	0.734896i	$-0.737231\pi$
$283$		−	415.951i		−	1.46979i	0.678180	−	0.734896i	$-0.262769\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	81.3997			0.283623
$288$		0			0
$289$	−235.467			−0.814765
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−323.733			−1.10489			−0.552445	−	0.833549i	$-0.686305\pi$
$293$	−323.733			−1.10489			−0.552445	+	0.833549i	$0.686305\pi$
$294$		0			0
$295$	123.586	+	416.220i	0.418936	+	1.41092i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	64.9080i		−	0.217084i
$300$		0			0
$301$	−299.541			−0.995152
$302$		0			0
$303$		0			0
$304$		0			0
$305$	212.980	−	63.2390i	0.698294	−	0.207341i
$306$		0			0
$307$		−	141.504i		−	0.460927i	−0.973081	−	0.230463i	$-0.925976\pi$
$307$		−	141.504i		−	0.460927i	0.973081	−	0.230463i	$-0.0740242\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	468.870i		−	1.50762i	−0.657093	−	0.753810i	$-0.728214\pi$
$311$		−	468.870i		−	1.50762i	0.657093	−	0.753810i	$-0.271786\pi$
$312$		0			0
$313$		−	516.666i		−	1.65069i	−0.564630	−	0.825344i	$-0.690981\pi$
$313$		−	516.666i		−	1.65069i	0.564630	−	0.825344i	$-0.309019\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−271.925			−0.857809			−0.428904	−	0.903350i	$-0.641100\pi$
$317$	−271.925			−0.857809			−0.428904	+	0.903350i	$0.641100\pi$
$318$		0			0
$319$	−386.365			−1.21118
$320$		0			0
$321$		0			0
$322$		0			0
$323$	227.596			0.704633
$324$		0			0
$325$	184.653	+	283.528i	0.568163	+	0.872395i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			35.0729i			0.106604i
$330$		0			0
$331$	−44.1458			−0.133371			−0.0666856	−	0.997774i	$-0.521242\pi$
$331$	−44.1458			−0.133371			−0.0666856	+	0.997774i	$0.521242\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−85.3326	−	287.388i	−0.254724	−	0.857874i
$336$		0			0
$337$		−	131.891i		−	0.391369i	−0.980667	−	0.195684i	$-0.937307\pi$
$337$		−	131.891i		−	0.391369i	0.980667	−	0.195684i	$-0.0626928\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	291.194i		−	0.853941i
$342$		0			0
$343$			366.589i			1.06877i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.45484			0.0243655			0.0121828	−	0.999926i	$-0.496122\pi$
$347$	8.45484			0.0243655			0.0121828	+	0.999926i	$0.496122\pi$
$348$		0			0
$349$	366.910			1.05132			0.525659	−	0.850695i	$-0.323819\pi$
$349$	366.910			1.05132			0.525659	+	0.850695i	$0.323819\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−61.0243			−0.172873			−0.0864367	−	0.996257i	$-0.527548\pi$
$353$	−61.0243			−0.172873			−0.0864367	+	0.996257i	$0.527548\pi$
$354$		0			0
$355$	−53.2122	−	179.211i	−0.149893	−	0.504819i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	506.093i		−	1.40973i	−0.709342	−	0.704864i	$-0.751008\pi$
$359$		−	506.093i		−	1.40973i	0.709342	−	0.704864i	$-0.248992\pi$
$360$		0			0
$361$	606.631			1.68042
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−12.9869	−	43.7379i	−0.0355805	−	0.119830i
$366$		0			0
$367$			131.251i			0.357633i	0.983882	+	0.178816i	$0.0572268\pi$
$367$			131.251i			0.357633i	−0.983882	+	0.178816i	$0.942773\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			49.7539i			0.134108i
$372$		0			0
$373$			576.158i			1.54466i	0.635223	+	0.772329i	$0.280908\pi$
$373$			576.158i			1.54466i	−0.635223	+	0.772329i	$0.719092\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−657.692			−1.74454
$378$		0			0
$379$	−347.761			−0.917574			−0.458787	−	0.888546i	$-0.651716\pi$
$379$	−347.761			−0.917574			−0.458787	+	0.888546i	$0.651716\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	338.986			0.885080			0.442540	−	0.896749i	$-0.354077\pi$
$383$	338.986			0.885080			0.442540	+	0.896749i	$0.354077\pi$
$384$		0			0
$385$	241.429	−	71.6862i	0.627088	−	0.186198i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	191.912i		−	0.493347i	−0.969099	−	0.246673i	$-0.920662\pi$
$389$		−	191.912i		−	0.493347i	0.969099	−	0.246673i	$-0.0793375\pi$
$390$		0			0
$391$	35.0893			0.0897424
$392$		0			0
$393$		0			0
$394$		0			0
$395$	499.587	−	148.340i	1.26478	−	0.375544i
$396$		0			0
$397$			711.622i			1.79250i	0.443551	+	0.896249i	$0.353718\pi$
$397$			711.622i			1.79250i	−0.443551	+	0.896249i	$0.646282\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.9980i		−	0.0797956i	−0.999204	−	0.0398978i	$-0.987297\pi$
$401$		−	31.9980i		−	0.0797956i	0.999204	−	0.0398978i	$-0.0127032\pi$
$402$		0			0
$403$		−	495.686i		−	1.22999i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−422.279			−1.03754
$408$		0			0
$409$	−11.0225			−0.0269499			−0.0134749	−	0.999909i	$-0.504289\pi$
$409$	−11.0225			−0.0269499			−0.0134749	+	0.999909i	$0.504289\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	550.119			1.33201
$414$		0			0
$415$	201.590	−	59.8572i	0.485760	−	0.144234i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			672.499i			1.60501i	0.596646	+	0.802505i	$0.296500\pi$
$419$			672.499i			1.60501i	−0.596646	+	0.802505i	$0.703500\pi$
$420$		0			0
$421$	349.010			0.829003			0.414502	−	0.910049i	$-0.363956\pi$
$421$	349.010			0.829003			0.414502	+	0.910049i	$0.363956\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−153.275	+	99.8235i	−0.360648	+	0.234879i
$426$		0			0
$427$		−	281.496i		−	0.659240i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			101.441i			0.235362i	0.993051	+	0.117681i	$0.0375460\pi$
$431$			101.441i			0.235362i	−0.993051	+	0.117681i	$0.962454\pi$
$432$		0			0
$433$			345.888i			0.798817i	0.916773	+	0.399408i	$0.130784\pi$
$433$			345.888i			0.798817i	−0.916773	+	0.399408i	$0.869216\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	149.183			0.341379
$438$		0			0
$439$	−433.031			−0.986404			−0.493202	−	0.869915i	$-0.664174\pi$
$439$	−433.031			−0.986404			−0.493202	+	0.869915i	$0.664174\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	90.5705			0.204448			0.102224	−	0.994761i	$-0.467404\pi$
$443$	90.5705			0.204448			0.102224	+	0.994761i	$0.467404\pi$
$444$		0			0
$445$	−105.426	−	355.060i	−0.236913	−	0.797887i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	64.3437i		−	0.143304i	−0.997430	−	0.0716522i	$-0.977173\pi$
$449$		−	64.3437i		−	0.143304i	0.997430	−	0.0716522i	$-0.0228272\pi$
$450$		0			0
$451$	102.159			0.226517
$452$		0			0
$453$		0			0
$454$		0			0
$455$	410.973	−	122.028i	0.903237	−	0.268194i
$456$		0			0
$457$			423.759i			0.927262i	0.886028	+	0.463631i	$0.153454\pi$
$457$			423.759i			0.927262i	−0.886028	+	0.463631i	$0.846546\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			232.822i			0.505037i	0.967592	+	0.252518i	$0.0812588\pi$
$461$			232.822i			0.505037i	−0.967592	+	0.252518i	$0.918741\pi$
$462$		0			0
$463$		−	265.109i		−	0.572591i	−0.958141	−	0.286295i	$-0.907576\pi$
$463$		−	265.109i		−	0.572591i	0.958141	−	0.286295i	$-0.0924238\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−109.265			−0.233972			−0.116986	−	0.993134i	$-0.537323\pi$
$467$	−109.265			−0.233972			−0.116986	+	0.993134i	$0.537323\pi$
$468$		0			0
$469$	−379.841			−0.809896
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−375.932			−0.794783
$474$		0			0
$475$	−651.654	+	424.402i	−1.37190	+	0.893477i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	556.612i		−	1.16203i	−0.813893	−	0.581014i	$-0.802656\pi$
$479$		−	556.612i		−	1.16203i	0.813893	−	0.581014i	$-0.197344\pi$
$480$		0			0
$481$	−718.826			−1.49444
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−75.2455	−	253.416i	−0.155145	−	0.522507i
$486$		0			0
$487$			309.042i			0.634584i	0.948328	+	0.317292i	$0.102774\pi$
$487$			309.042i			0.634584i	−0.948328	+	0.317292i	$0.897226\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	446.180i		−	0.908717i	−0.890819	−	0.454358i	$-0.849869\pi$
$491$		−	446.180i		−	0.908717i	0.890819	−	0.454358i	$-0.150131\pi$
$492$		0			0
$493$		−	355.548i		−	0.721193i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−236.863			−0.476586
$498$		0			0
$499$	−622.520			−1.24753			−0.623767	−	0.781610i	$-0.714399\pi$
$499$	−622.520			−1.24753			−0.623767	+	0.781610i	$0.714399\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−137.982			−0.274318			−0.137159	−	0.990549i	$-0.543797\pi$
$503$	−137.982			−0.274318			−0.137159	+	0.990549i	$0.543797\pi$
$504$		0			0
$505$	10.0756	+	33.9332i	0.0199517	+	0.0671944i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			235.529i			0.462728i	0.972867	+	0.231364i	$0.0743188\pi$
$509$			235.529i			0.462728i	−0.972867	+	0.231364i	$0.925681\pi$
$510$		0			0
$511$	−57.8085			−0.113128
$512$		0			0
$513$		0			0
$514$		0			0
$515$	45.7863	+	154.202i	0.0889054	+	0.299420i
$516$		0			0
$517$			44.0175i			0.0851401i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	332.575i		−	0.638340i	−0.947698	−	0.319170i	$-0.896596\pi$
$521$		−	332.575i		−	0.638340i	0.947698	−	0.319170i	$-0.103404\pi$
$522$		0			0
$523$			86.3820i			0.165166i	0.996584	+	0.0825831i	$0.0263170\pi$
$523$			86.3820i			0.165166i	−0.996584	+	0.0825831i	$0.973683\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	267.968			0.508478
$528$		0			0
$529$	23.0000			0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$	173.901			0.326267
$534$		0			0
$535$	−607.227	+	180.301i	−1.13500	+	0.337011i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			70.4916i			0.130782i
$540$		0			0
$541$	707.990			1.30867			0.654335	−	0.756205i	$-0.272949\pi$
$541$	707.990			1.30867			0.654335	+	0.756205i	$0.272949\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	176.635	−	52.4472i	0.324100	−	0.0962335i
$546$		0			0
$547$			292.328i			0.534421i	0.963638	+	0.267210i	$0.0861019\pi$
$547$			292.328i			0.534421i	−0.963638	+	0.267210i	$0.913898\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	1511.62i		−	2.74341i
$552$		0			0
$553$		−	660.306i		−	1.19404i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	196.226			0.352290			0.176145	−	0.984364i	$-0.443637\pi$
$557$	196.226			0.352290			0.176145	+	0.984364i	$0.443637\pi$
$558$		0			0
$559$	−639.932			−1.14478
$560$		0			0
$561$		0			0
$562$		0			0
$563$	528.570			0.938845			0.469423	−	0.882974i	$-0.344462\pi$
$563$	528.570			0.938845			0.469423	+	0.882974i	$0.344462\pi$
$564$		0			0
$565$	−941.995	+	279.702i	−1.66725	+	0.495048i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	218.941i		−	0.384782i	−0.981318	−	0.192391i	$-0.938376\pi$
$569$		−	218.941i		−	0.384782i	0.981318	−	0.192391i	$-0.0616241\pi$
$570$		0			0
$571$	92.5282			0.162046			0.0810230	−	0.996712i	$-0.474181\pi$
$571$	92.5282			0.162046			0.0810230	+	0.996712i	$0.474181\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−100.468	+	65.4314i	−0.174726	+	0.113794i
$576$		0			0
$577$			331.328i			0.574225i	0.957897	+	0.287113i	$0.0926954\pi$
$577$			331.328i			0.574225i	−0.957897	+	0.287113i	$0.907305\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	266.442i		−	0.458593i
$582$		0			0
$583$			62.4426i			0.107106i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	396.885			0.676124			0.338062	−	0.941124i	$-0.390229\pi$
$587$	396.885			0.676124			0.338062	+	0.941124i	$0.390229\pi$
$588$		0			0
$589$	1139.27			1.93425
$590$		0			0
$591$		0			0
$592$		0			0
$593$	224.890			0.379241			0.189620	−	0.981857i	$-0.439274\pi$
$593$	224.890			0.379241			0.189620	+	0.981857i	$0.439274\pi$
$594$		0			0
$595$	65.9684	+	222.172i	0.110871	+	0.373398i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	421.747i		−	0.704085i	−0.935984	−	0.352043i	$-0.885487\pi$
$599$		−	421.747i		−	0.704085i	0.935984	−	0.352043i	$-0.114513\pi$
$600$		0			0
$601$	−1001.24			−1.66596			−0.832979	−	0.553305i	$-0.813366\pi$
$601$	−1001.24			−1.66596			−0.832979	+	0.553305i	$0.813366\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−276.973	+	82.2403i	−0.457807	+	0.135934i
$606$		0			0
$607$		−	213.971i		−	0.352506i	−0.984345	−	0.176253i	$-0.943602\pi$
$607$		−	213.971i		−	0.352506i	0.984345	−	0.176253i	$-0.0563977\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			74.9289i			0.122633i
$612$		0			0
$613$			412.716i			0.673273i	0.941635	+	0.336636i	$0.109289\pi$
$613$			412.716i			0.673273i	−0.941635	+	0.336636i	$0.890711\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	544.393			0.882323			0.441161	−	0.897428i	$-0.354567\pi$
$617$	544.393			0.882323			0.441161	+	0.897428i	$0.354567\pi$
$618$		0			0
$619$	−911.501			−1.47254			−0.736269	−	0.676689i	$-0.763414\pi$
$619$	−911.501			−1.47254			−0.736269	+	0.676689i	$0.763414\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−469.283			−0.753263
$624$		0			0
$625$	252.716	−	571.629i	0.404346	−	0.914606i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	388.598i		−	0.617802i
$630$		0			0
$631$	721.117			1.14282			0.571408	−	0.820666i	$-0.306397\pi$
$631$	721.117			1.14282			0.571408	+	0.820666i	$0.306397\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−38.1726	−	128.560i	−0.0601143	−	0.202456i
$636$		0			0
$637$			119.995i			0.188375i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			762.419i			1.18942i	0.803940	+	0.594711i	$0.202733\pi$
$641$			762.419i			1.18942i	−0.803940	+	0.594711i	$0.797267\pi$
$642$		0			0
$643$		−	266.048i		−	0.413761i	−0.978366	−	0.206881i	$-0.933669\pi$
$643$		−	266.048i		−	0.413761i	0.978366	−	0.206881i	$-0.0663312\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	620.696			0.959345			0.479673	−	0.877447i	$-0.340755\pi$
$647$	620.696			0.959345			0.479673	+	0.877447i	$0.340755\pi$
$648$		0			0
$649$	690.415			1.06381
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−616.336			−0.943853			−0.471927	−	0.881638i	$-0.656441\pi$
$653$	−616.336			−0.943853			−0.471927	+	0.881638i	$0.656441\pi$
$654$		0			0
$655$	188.800	+	635.851i	0.288244	+	0.970765i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	346.244i		−	0.525409i	−0.964876	−	0.262704i	$-0.915386\pi$
$659$		−	346.244i		−	0.525409i	0.964876	−	0.262704i	$-0.0846144\pi$
$660$		0			0
$661$	−751.351			−1.13669			−0.568344	−	0.822791i	$-0.692416\pi$
$661$	−751.351			−1.13669			−0.568344	+	0.822791i	$0.692416\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	280.466	+	944.570i	0.421754	+	1.42041i
$666$		0			0
$667$		−	233.052i		−	0.349403i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	353.285i		−	0.526505i
$672$		0			0
$673$			104.503i			0.155279i	0.996981	+	0.0776395i	$0.0247383\pi$
$673$			104.503i			0.155279i	−0.996981	+	0.0776395i	$0.975262\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	246.783			0.364525			0.182262	−	0.983250i	$-0.441658\pi$
$677$	246.783			0.364525			0.182262	+	0.983250i	$0.441658\pi$
$678$		0			0
$679$	−334.940			−0.493284
$680$		0			0
$681$		0			0
$682$		0			0
$683$	469.518			0.687436			0.343718	−	0.939073i	$-0.388314\pi$
$683$	469.518			0.687436			0.343718	+	0.939073i	$0.388314\pi$
$684$		0			0
$685$	−409.610	+	121.623i	−0.597971	+	0.177552i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			106.293i			0.154272i
$690$		0			0
$691$	−491.904			−0.711872			−0.355936	−	0.934510i	$-0.615838\pi$
$691$	−491.904			−0.711872			−0.355936	+	0.934510i	$0.615838\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−499.296	+	148.253i	−0.718411	+	0.213314i
$696$		0			0
$697$			94.0107i			0.134879i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1257.27i			1.79354i	0.442494	+	0.896772i	$0.354094\pi$
$701$			1257.27i			1.79354i	−0.442494	+	0.896772i	$0.645906\pi$
$702$		0			0
$703$		−	1652.13i		−	2.35012i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	44.8496			0.0634364
$708$		0			0
$709$	695.815			0.981403			0.490702	−	0.871328i	$-0.336741\pi$
$709$	695.815			0.981403			0.490702	+	0.871328i	$0.336741\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	175.645			0.246347
$714$		0			0
$715$	515.783	−	153.149i	0.721375	−	0.214194i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			122.969i			0.171028i	0.996337	+	0.0855141i	$0.0272533\pi$
$719$			122.969i			0.171028i	−0.996337	+	0.0855141i	$0.972747\pi$
$720$		0			0
$721$	203.808			0.282675
$722$		0			0
$723$		0			0
$724$		0			0
$725$	662.995	+	1018.01i	0.914476	+	1.40415i
$726$		0			0
$727$			1228.63i			1.69000i	0.534769	+	0.844998i	$0.320398\pi$
$727$			1228.63i			1.69000i	−0.534769	+	0.844998i	$0.679602\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	345.948i		−	0.473253i
$732$		0			0
$733$		−	919.541i		−	1.25449i	−0.778822	−	0.627245i	$-0.784183\pi$
$733$		−	919.541i		−	1.25449i	0.778822	−	0.627245i	$-0.215817\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−476.712			−0.646827
$738$		0			0
$739$	641.326			0.867830			0.433915	−	0.900954i	$-0.357132\pi$
$739$	641.326			0.867830			0.433915	+	0.900954i	$0.357132\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1423.65			−1.91608			−0.958040	−	0.286633i	$-0.907464\pi$
$743$	−1423.65			−1.91608			−0.958040	+	0.286633i	$0.907464\pi$
$744$		0			0
$745$	67.2988	+	226.653i	0.0903340	+	0.304232i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			802.573i			1.07153i
$750$		0			0
$751$	690.276			0.919142			0.459571	−	0.888141i	$-0.348003\pi$
$751$	690.276			0.919142			0.459571	+	0.888141i	$0.348003\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−129.461	+	38.4401i	−0.171471	+	0.0509141i
$756$		0			0
$757$			206.960i			0.273395i	0.990613	+	0.136698i	$0.0436489\pi$
$757$			206.960i			0.273395i	−0.990613	+	0.136698i	$0.956351\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			1040.71i			1.36756i	0.729689	+	0.683780i	$0.239665\pi$
$761$			1040.71i			1.36756i	−0.729689	+	0.683780i	$0.760335\pi$
$762$		0			0
$763$		−	233.458i		−	0.305974i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1175.26			1.53228
$768$		0			0
$769$	498.007			0.647604			0.323802	−	0.946125i	$-0.395039\pi$
$769$	498.007			0.647604			0.323802	+	0.946125i	$0.395039\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	649.594			0.840354			0.420177	−	0.907442i	$-0.361968\pi$
$773$	649.594			0.840354			0.420177	+	0.907442i	$0.361968\pi$
$774$		0			0
$775$	−767.245	+	499.683i	−0.989994	+	0.644752i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			399.688i			0.513079i
$780$		0			0
$781$	−297.270			−0.380628
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−177.294	−	597.099i	−0.225852	−	0.760635i
$786$		0			0
$787$			406.506i			0.516526i	0.966075	+	0.258263i	$0.0831502\pi$
$787$			406.506i			0.516526i	−0.966075	+	0.258263i	$0.916850\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1245.04i			1.57400i
$792$		0			0
$793$		−	601.381i		−	0.758362i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1365.88			1.71378			0.856889	−	0.515502i	$-0.172394\pi$
$797$	1365.88			1.71378			0.856889	+	0.515502i	$0.172394\pi$
$798$		0			0
$799$	−40.5066			−0.0506966
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−72.5514			−0.0903504
$804$		0			0
$805$	43.2404	+	145.627i	0.0537148	+	0.180904i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			203.797i			0.251912i	0.992036	+	0.125956i	$0.0401998\pi$
$809$			203.797i			0.251912i	−0.992036	+	0.125956i	$0.959800\pi$
$810$		0			0
$811$	1536.96			1.89514			0.947570	−	0.319549i	$-0.103531\pi$
$811$	1536.96			1.89514			0.947570	+	0.319549i	$0.103531\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−201.884	−	679.915i	−0.247710	−	0.834251i
$816$		0			0
$817$		−	1470.80i		−	1.80025i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	75.1242i		−	0.0915033i	−0.998953	−	0.0457517i	$-0.985432\pi$
$821$		−	75.1242i		−	0.0915033i	0.998953	−	0.0457517i	$-0.0145683\pi$
$822$		0			0
$823$		−	287.518i		−	0.349353i	−0.984626	−	0.174677i	$-0.944112\pi$
$823$		−	287.518i		−	0.349353i	0.984626	−	0.174677i	$-0.0558880\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−408.629			−0.494110			−0.247055	−	0.969001i	$-0.579463\pi$
$827$	−408.629			−0.494110			−0.247055	+	0.969001i	$0.579463\pi$
$828$		0			0
$829$	500.767			0.604062			0.302031	−	0.953298i	$-0.402335\pi$
$829$	500.767			0.604062			0.302031	+	0.953298i	$0.402335\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−64.8691			−0.0778741
$834$		0			0
$835$	−1472.05	+	437.088i	−1.76293	+	0.523458i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			592.090i			0.705709i	0.935678	+	0.352854i	$0.114789\pi$
$839$			592.090i			0.705709i	−0.935678	+	0.352854i	$0.885211\pi$
$840$		0			0
$841$	−1520.44			−1.80789
$842$		0			0
$843$		0			0
$844$		0			0
$845$	67.9480	−	20.1755i	0.0804118	−	0.0238763i
$846$		0			0
$847$			366.076i			0.432204i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	254.714i		−	0.299312i
$852$		0			0
$853$		−	984.615i		−	1.15430i	−0.816639	−	0.577149i	$-0.804165\pi$
$853$		−	984.615i		−	1.15430i	0.816639	−	0.577149i	$-0.195835\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−784.085			−0.914918			−0.457459	−	0.889231i	$-0.651240\pi$
$857$	−784.085			−0.914918			−0.457459	+	0.889231i	$0.651240\pi$
$858$		0			0
$859$	969.542			1.12869			0.564343	−	0.825540i	$-0.309129\pi$
$859$	969.542			1.12869			0.564343	+	0.825540i	$0.309129\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1271.18			−1.47298			−0.736490	−	0.676449i	$-0.763518\pi$
$863$	−1271.18			−1.47298			−0.736490	+	0.676449i	$0.763518\pi$
$864$		0			0
$865$	1068.03	−	317.123i	1.23471	−	0.366617i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	828.703i		−	0.953628i
$870$		0			0
$871$	−811.484			−0.931669
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−603.168	−	513.111i	−0.689334	−	0.586412i
$876$		0			0
$877$			206.249i			0.235176i	0.993062	+	0.117588i	$0.0375162\pi$
$877$			206.249i			0.235176i	−0.993062	+	0.117588i	$0.962484\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			597.747i			0.678486i	0.940699	+	0.339243i	$0.110171\pi$
$881$			597.747i			0.678486i	−0.940699	+	0.339243i	$0.889829\pi$
$882$		0			0
$883$		−	1408.59i		−	1.59524i	−0.603163	−	0.797618i	$-0.706093\pi$
$883$		−	1408.59i		−	1.59524i	0.603163	−	0.797618i	$-0.293907\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1099.09			1.23911			0.619557	−	0.784951i	$-0.287312\pi$
$887$	1099.09			1.23911			0.619557	+	0.784951i	$0.287312\pi$
$888$		0			0
$889$	−169.918			−0.191133
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−172.215			−0.192849
$894$		0			0
$895$	−127.205	−	428.408i	−0.142128	−	0.478668i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1779.76i		−	1.97971i
$900$		0			0
$901$	−57.4621			−0.0637759
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1417.71	−	420.954i	1.56653	−	0.465142i
$906$		0			0
$907$		−	7.82274i		−	0.00862485i	−0.999991	−	0.00431242i	$-0.998627\pi$
$907$		−	7.82274i		−	0.00862485i	0.999991	−	0.00431242i	$-0.00137269\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1561.67i		−	1.71423i	−0.515123	−	0.857116i	$-0.672254\pi$
$911$		−	1561.67i		−	1.71423i	0.515123	−	0.857116i	$-0.327746\pi$
$912$		0			0
$913$		−	334.393i		−	0.366257i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	840.405			0.916473
$918$		0			0
$919$	1342.42			1.46073			0.730367	−	0.683055i	$-0.239349\pi$
$919$	1342.42			1.46073			0.730367	+	0.683055i	$0.239349\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−506.030			−0.548244
$924$		0			0
$925$	724.622	+	1112.63i	0.783376	+	1.20285i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	402.168i		−	0.432904i	−0.976293	−	0.216452i	$-0.930552\pi$
$929$		−	402.168i		−	0.432904i	0.976293	−	0.216452i	$-0.0694485\pi$
$930$		0			0
$931$	−275.792			−0.296232
$932$		0			0
$933$		0			0
$934$		0			0
$935$	82.7923	+	278.832i	0.0885479	+	0.298216i
$936$		0			0
$937$			1432.64i			1.52896i	0.644645	+	0.764482i	$0.277005\pi$
$937$			1432.64i			1.52896i	−0.644645	+	0.764482i	$0.722995\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			543.444i			0.577518i	0.957402	+	0.288759i	$0.0932426\pi$
$941$			543.444i			0.577518i	−0.957402	+	0.288759i	$0.906757\pi$
$942$		0			0
$943$			61.6213i			0.0653460i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1665.82			−1.75905			−0.879524	−	0.475855i	$-0.842139\pi$
$947$	−1665.82			−1.75905			−0.879524	+	0.475855i	$0.842139\pi$
$948$		0			0
$949$	−123.501			−0.130138
$950$		0			0
$951$		0			0
$952$		0			0
$953$	207.156			0.217373			0.108686	−	0.994076i	$-0.465336\pi$
$953$	207.156			0.217373			0.108686	+	0.994076i	$0.465336\pi$
$954$		0			0
$955$	−33.1088	−	111.505i	−0.0346689	−	0.116760i
$956$		0			0
$957$		0			0
$958$		0			0
$959$			541.382i			0.564528i
$960$		0			0
$961$	380.358			0.395794
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−75.0192	−	252.654i	−0.0777401	−	0.261817i
$966$		0			0
$967$			1575.66i			1.62943i	0.579864	+	0.814714i	$0.303106\pi$
$967$			1575.66i			1.62943i	−0.579864	+	0.814714i	$0.696894\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	1506.58i		−	1.55157i	−0.630996	−	0.775786i	$-0.717354\pi$
$971$		−	1506.58i		−	1.55157i	0.630996	−	0.775786i	$-0.282646\pi$
$972$		0			0
$973$			659.920i			0.678232i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	606.942			0.621230			0.310615	−	0.950536i	$-0.399465\pi$
$977$	606.942			0.621230			0.310615	+	0.950536i	$0.399465\pi$
$978$		0			0
$979$	−588.964			−0.601597
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−976.674			−0.993565			−0.496782	−	0.867875i	$-0.665485\pi$
$983$	−976.674			−0.993565			−0.496782	+	0.867875i	$0.665485\pi$
$984$		0			0
$985$	792.893	−	235.430i	0.804968	−	0.239015i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	226.759i		−	0.229281i
$990$		0			0
$991$	−754.844			−0.761699			−0.380850	−	0.924637i	$-0.624368\pi$
$991$	−754.844			−0.761699			−0.380850	+	0.924637i	$0.624368\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−169.961	+	50.4657i	−0.170815	+	0.0507193i
$996$		0			0
$997$		−	260.070i		−	0.260852i	−0.991458	−	0.130426i	$-0.958365\pi$
$997$		−	260.070i		−	0.260852i	0.991458	−	0.130426i	$-0.0416345\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.3.c.a.4049.8	yes	88
3.2	odd	2	inner	4140.3.c.a.4049.81	yes	88
5.4	even	2	inner	4140.3.c.a.4049.82	yes	88
15.14	odd	2	inner	4140.3.c.a.4049.7	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4140.3.c.a.4049.7	✓	88	15.14	odd	2	inner
4140.3.c.a.4049.8	yes	88	1.1	even	1	trivial
4140.3.c.a.4049.81	yes	88	3.2	odd	2	inner
4140.3.c.a.4049.82	yes	88	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	4140.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.79317 + 1.42321i) q^{5} +6.33514i q^{7} +O(q^{10})\)	\(q+(-4.79317 + 1.42321i) q^{5} +6.33514i q^{7} +7.95078i q^{11} +13.5343i q^{13} -7.31662 q^{17} -31.1068 q^{19} -4.79583 q^{23} +(20.9489 - 13.6434i) q^{25} +48.5946i q^{29} -36.6246 q^{31} +(-9.01625 - 30.3654i) q^{35} +53.1116i q^{37} -12.8489i q^{41} +47.2824i q^{43} +5.53624 q^{47} +8.86599 q^{49} +7.85364 q^{53} +(-11.3156 - 38.1094i) q^{55} -86.8361i q^{59} -44.4340 q^{61} +(-19.2621 - 64.8720i) q^{65} +59.9578i q^{67} +37.3888i q^{71} +9.12506i q^{73} -50.3693 q^{77} -104.229 q^{79} -42.0578 q^{83} +(35.0698 - 10.4131i) q^{85} +74.0762i q^{89} -85.7414 q^{91} +(149.100 - 44.2715i) q^{95} +52.8702i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q+O(q^{10}) \) 88 * q	\( 88 q - 16 q^{19} - 48 q^{25} + 272 q^{31} - 600 q^{49} + 112 q^{55} + 448 q^{61} - 32 q^{79} - 264 q^{85} - 16 q^{91}+O(q^{100}) \) 88 * q - 16 * q^19 - 48 * q^25 + 272 * q^31 - 600 * q^49 + 112 * q^55 + 448 * q^61 - 32 * q^79 - 264 * q^85 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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