LMFDB - Embedded newform 1900.3.e.g.1101.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 89x^{12} + 3026x^{10} + 49092x^{8} + 390297x^{6} + 1440495x^{4} + 1994425x^{2} + 151875 $ x^14 + 89x^12 + 3026x^10 + 49092x^8 + 390297x^6 + 1440495x^4 + 1994425x^2 + 151875
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.3
Root			$-4.58743i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.g.1101.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.58743i q^{3} -1.52935 q^{7} -12.0445 q^{9} +O(q^{10})$	$q-4.58743i q^{3} -1.52935 q^{7} -12.0445 q^{9} +19.7663 q^{11} +16.2291i q^{13} +15.9769 q^{17} +(-18.9164 + 1.78065i) q^{19} +7.01577i q^{21} -37.5766 q^{23} +13.9665i q^{27} -31.5765i q^{29} -38.9409i q^{31} -90.6764i q^{33} +39.1052i q^{37} +74.4499 q^{39} -69.4666i q^{41} +16.5777 q^{43} -52.0477 q^{47} -46.6611 q^{49} -73.2930i q^{51} -16.8503i q^{53} +(8.16859 + 86.7776i) q^{57} -92.5930i q^{59} +51.5327 q^{61} +18.4202 q^{63} -48.7310i q^{67} +172.380i q^{69} +5.51515i q^{71} +62.8328 q^{73} -30.2295 q^{77} -146.361i q^{79} -44.3302 q^{81} -157.433 q^{83} -144.855 q^{87} -87.6554i q^{89} -24.8199i q^{91} -178.639 q^{93} +115.339i q^{97} -238.075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) $ 14 * q - 4 * q^7 - 52 * q^9	$ 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) $ 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	4.58743i		−	1.52914i	−0.644539	−	0.764572i	$-0.722951\pi$
$3$		−	4.58743i		−	1.52914i	0.644539	−	0.764572i	$-0.277049\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.52935			−0.218478			−0.109239	−	0.994016i	$-0.534841\pi$
$7$	−1.52935			−0.218478			−0.109239	+	0.994016i	$0.534841\pi$
$8$		0			0
$9$	−12.0445			−1.33828
$10$		0			0
$11$	19.7663			1.79693			0.898467	−	0.439042i	$-0.144682\pi$
$11$	19.7663			1.79693			0.898467	+	0.439042i	$0.144682\pi$
$12$		0			0
$13$			16.2291i			1.24839i	0.781267	+	0.624197i	$0.214574\pi$
$13$			16.2291i			1.24839i	−0.781267	+	0.624197i	$0.785426\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	15.9769			0.939819			0.469909	−	0.882715i	$-0.344287\pi$
$17$	15.9769			0.939819			0.469909	+	0.882715i	$0.344287\pi$
$18$		0			0
$19$	−18.9164	+	1.78065i	−0.995599	+	0.0937182i
$19$	−18.9164	+	1.78065i	−0.995599	+	0.0937182i
$20$		0			0
$21$			7.01577i			0.334084i
$22$		0			0
$23$	−37.5766			−1.63377			−0.816883	−	0.576804i	$-0.804300\pi$
$23$	−37.5766			−1.63377			−0.816883	+	0.576804i	$0.804300\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			13.9665i			0.517279i
$28$		0			0
$29$		−	31.5765i		−	1.08885i	−0.838811	−	0.544423i	$-0.816749\pi$
$29$		−	31.5765i		−	1.08885i	0.838811	−	0.544423i	$-0.183251\pi$
$30$		0			0
$31$		−	38.9409i		−	1.25616i	−0.778149	−	0.628080i	$-0.783841\pi$
$31$		−	38.9409i		−	1.25616i	0.778149	−	0.628080i	$-0.216159\pi$
$32$		0			0
$33$		−	90.6764i		−	2.74777i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			39.1052i			1.05690i	0.848965	+	0.528449i	$0.177226\pi$
$37$			39.1052i			1.05690i	−0.848965	+	0.528449i	$0.822774\pi$
$38$		0			0
$39$	74.4499			1.90897
$40$		0			0
$41$		−	69.4666i		−	1.69431i	−0.531348	−	0.847154i	$-0.678314\pi$
$41$		−	69.4666i		−	1.69431i	0.531348	−	0.847154i	$-0.321686\pi$
$42$		0			0
$43$	16.5777			0.385528			0.192764	−	0.981245i	$-0.438255\pi$
$43$	16.5777			0.385528			0.192764	+	0.981245i	$0.438255\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−52.0477			−1.10740			−0.553699	−	0.832717i	$-0.686784\pi$
$47$	−52.0477			−1.10740			−0.553699	+	0.832717i	$0.686784\pi$
$48$		0			0
$49$	−46.6611			−0.952267
$50$		0			0
$51$		−	73.2930i		−	1.43712i
$52$		0			0
$53$		−	16.8503i		−	0.317931i	−0.987284	−	0.158965i	$-0.949184\pi$
$53$		−	16.8503i		−	0.317931i	0.987284	−	0.158965i	$-0.0508158\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.16859	+	86.7776i	0.143309	+	1.52241i
$58$		0			0
$59$		−	92.5930i		−	1.56937i	−0.619893	−	0.784686i	$-0.712824\pi$
$59$		−	92.5930i		−	1.56937i	0.619893	−	0.784686i	$-0.287176\pi$
$60$		0			0
$61$	51.5327			0.844798			0.422399	−	0.906410i	$-0.361188\pi$
$61$	51.5327			0.844798			0.422399	+	0.906410i	$0.361188\pi$
$62$		0			0
$63$	18.4202			0.292385
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	48.7310i		−	0.727328i	−0.931530	−	0.363664i	$-0.881526\pi$
$67$		−	48.7310i		−	0.727328i	0.931530	−	0.363664i	$-0.118474\pi$
$68$		0			0
$69$			172.380i			2.49826i
$70$		0			0
$71$			5.51515i			0.0776782i	0.999245	+	0.0388391i	$0.0123660\pi$
$71$			5.51515i			0.0776782i	−0.999245	+	0.0388391i	$0.987634\pi$
$72$		0			0
$73$	62.8328			0.860723			0.430362	−	0.902657i	$-0.358386\pi$
$73$	62.8328			0.860723			0.430362	+	0.902657i	$0.358386\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−30.2295			−0.392590
$78$		0			0
$79$		−	146.361i		−	1.85268i	−0.376692	−	0.926338i	$-0.622939\pi$
$79$		−	146.361i		−	1.85268i	0.376692	−	0.926338i	$-0.377061\pi$
$80$		0			0
$81$	−44.3302			−0.547287
$82$		0			0
$83$	−157.433			−1.89678			−0.948391	−	0.317103i	$-0.897290\pi$
$83$	−157.433			−1.89678			−0.948391	+	0.317103i	$0.897290\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−144.855			−1.66500
$88$		0			0
$89$		−	87.6554i		−	0.984893i	−0.870343	−	0.492446i	$-0.836103\pi$
$89$		−	87.6554i		−	0.984893i	0.870343	−	0.492446i	$-0.163897\pi$
$90$		0			0
$91$		−	24.8199i		−	0.272746i
$92$		0			0
$93$	−178.639			−1.92085
$94$		0			0
$95$		0			0
$96$		0			0
$97$			115.339i			1.18906i	0.804073	+	0.594531i	$0.202662\pi$
$97$			115.339i			1.18906i	−0.804073	+	0.594531i	$0.797338\pi$
$98$		0			0
$99$	−238.075			−2.40480
$100$		0			0
$101$	−105.886			−1.04837			−0.524186	−	0.851604i	$-0.675631\pi$
$101$	−105.886			−1.04837			−0.524186	+	0.851604i	$0.675631\pi$
$102$		0			0
$103$		−	72.5992i		−	0.704847i	−0.935841	−	0.352423i	$-0.885358\pi$
$103$		−	72.5992i		−	0.704847i	0.935841	−	0.352423i	$-0.114642\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			2.35545i			0.0220135i	0.999939	+	0.0110068i	$0.00350363\pi$
$107$			2.35545i			0.0220135i	−0.999939	+	0.0110068i	$0.996496\pi$
$108$		0			0
$109$			182.099i			1.67063i	0.549772	+	0.835315i	$0.314715\pi$
$109$			182.099i			1.67063i	−0.549772	+	0.835315i	$0.685285\pi$
$110$		0			0
$111$	179.393			1.61615
$112$		0			0
$113$		−	86.4522i		−	0.765064i	−0.923942	−	0.382532i	$-0.875052\pi$
$113$		−	86.4522i		−	0.765064i	0.923942	−	0.382532i	$-0.124948\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	195.472i		−	1.67070i
$118$		0			0
$119$	−24.4342			−0.205330
$120$		0			0
$121$	269.705			2.22897
$122$		0			0
$123$	−318.673			−2.59084
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	119.187i		−	0.938480i	−0.883071	−	0.469240i	$-0.844528\pi$
$127$		−	119.187i		−	0.938480i	0.883071	−	0.469240i	$-0.155472\pi$
$128$		0			0
$129$		−	76.0491i		−	0.589528i
$130$		0			0
$131$	111.536			0.851423			0.425712	−	0.904859i	$-0.360024\pi$
$131$	111.536			0.851423			0.425712	+	0.904859i	$0.360024\pi$
$132$		0			0
$133$	28.9297	−	2.72322i	0.217516	−	0.0204754i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	82.0424			0.598849			0.299425	−	0.954120i	$-0.403205\pi$
$137$	82.0424			0.598849			0.299425	+	0.954120i	$0.403205\pi$
$138$		0			0
$139$	−169.587			−1.22005			−0.610027	−	0.792381i	$-0.708841\pi$
$139$	−169.587			−1.22005			−0.610027	+	0.792381i	$0.708841\pi$
$140$		0			0
$141$			238.765i			1.69337i
$142$		0			0
$143$			320.789i			2.24328i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			214.055i			1.45615i
$148$		0			0
$149$	67.4940			0.452980			0.226490	−	0.974013i	$-0.427275\pi$
$149$	67.4940			0.452980			0.226490	+	0.974013i	$0.427275\pi$
$150$		0			0
$151$			152.293i			1.00856i	0.863540	+	0.504280i	$0.168242\pi$
$151$			152.293i			1.00856i	−0.863540	+	0.504280i	$0.831758\pi$
$152$		0			0
$153$	−192.434			−1.25774
$154$		0			0
$155$		0			0
$156$		0			0
$157$	50.8926			0.324157			0.162078	−	0.986778i	$-0.448180\pi$
$157$	50.8926			0.324157			0.162078	+	0.986778i	$0.448180\pi$
$158$		0			0
$159$	−77.2997			−0.486162
$160$		0			0
$161$	57.4676			0.356942
$162$		0			0
$163$	56.1081			0.344222			0.172111	−	0.985078i	$-0.444941\pi$
$163$	56.1081			0.344222			0.172111	+	0.985078i	$0.444941\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	172.992i		−	1.03588i	−0.855418	−	0.517939i	$-0.826699\pi$
$167$		−	172.992i		−	1.03588i	0.855418	−	0.517939i	$-0.173301\pi$
$168$		0			0
$169$	−94.3840			−0.558485
$170$		0			0
$171$	227.839	−	21.4470i	1.33239	−	0.125421i
$172$		0			0
$173$			85.4671i			0.494030i	0.969012	+	0.247015i	$0.0794496\pi$
$173$			85.4671i			0.494030i	−0.969012	+	0.247015i	$0.920550\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−424.764			−2.39980
$178$		0			0
$179$			181.685i			1.01500i	0.861652	+	0.507500i	$0.169430\pi$
$179$			181.685i			1.01500i	−0.861652	+	0.507500i	$0.830570\pi$
$180$		0			0
$181$		−	98.1169i		−	0.542083i	−0.962568	−	0.271041i	$-0.912632\pi$
$181$		−	98.1169i		−	0.542083i	0.962568	−	0.271041i	$-0.0873680\pi$
$182$		0			0
$183$		−	236.403i		−	1.29182i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	315.804			1.68879
$188$		0			0
$189$		−	21.3596i		−	0.113014i
$190$		0			0
$191$	−150.077			−0.785741			−0.392871	−	0.919594i	$-0.628518\pi$
$191$	−150.077			−0.785741			−0.392871	+	0.919594i	$0.628518\pi$
$192$		0			0
$193$			20.0780i			0.104031i	0.998646	+	0.0520156i	$0.0165645\pi$
$193$			20.0780i			0.104031i	−0.998646	+	0.0520156i	$0.983435\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−109.441			−0.555536			−0.277768	−	0.960648i	$-0.589595\pi$
$197$	−109.441			−0.555536			−0.277768	+	0.960648i	$0.589595\pi$
$198$		0			0
$199$	29.0917			0.146189			0.0730947	−	0.997325i	$-0.476712\pi$
$199$	29.0917			0.146189			0.0730947	+	0.997325i	$0.476712\pi$
$200$		0			0
$201$	−223.550			−1.11219
$202$		0			0
$203$			48.2914i			0.237889i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	452.592			2.18643
$208$		0			0
$209$	−373.906	+	35.1967i	−1.78902	+	0.168405i
$210$		0			0
$211$			134.671i			0.638249i	0.947713	+	0.319125i	$0.103389\pi$
$211$			134.671i			0.638249i	−0.947713	+	0.319125i	$0.896611\pi$
$212$		0			0
$213$	25.3004			0.118781
$214$		0			0
$215$		0			0
$216$		0			0
$217$			59.5541i			0.274443i
$218$		0			0
$219$		−	288.241i		−	1.31617i
$220$		0			0
$221$			259.291i			1.17326i
$222$		0			0
$223$			201.645i			0.904236i	0.891958	+	0.452118i	$0.149331\pi$
$223$			201.645i			0.904236i	−0.891958	+	0.452118i	$0.850669\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	435.566i		−	1.91879i	−0.282064	−	0.959396i	$-0.591019\pi$
$227$		−	435.566i		−	1.91879i	0.282064	−	0.959396i	$-0.408981\pi$
$228$		0			0
$229$	336.632			1.47001			0.735003	−	0.678063i	$-0.237181\pi$
$229$	336.632			1.47001			0.735003	+	0.678063i	$0.237181\pi$
$230$		0			0
$231$			138.676i			0.600327i
$232$		0			0
$233$	−270.736			−1.16196			−0.580979	−	0.813919i	$-0.697330\pi$
$233$	−270.736			−1.16196			−0.580979	+	0.813919i	$0.697330\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−671.423			−2.83301
$238$		0			0
$239$	111.441			0.466279			0.233139	−	0.972443i	$-0.425100\pi$
$239$	111.441			0.466279			0.233139	+	0.972443i	$0.425100\pi$
$240$		0			0
$241$			269.049i			1.11639i	0.829711	+	0.558194i	$0.188505\pi$
$241$			269.049i			1.11639i	−0.829711	+	0.558194i	$0.811495\pi$
$242$		0			0
$243$			329.061i			1.35416i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−28.8983	−	306.996i	−0.116997	−	1.24290i
$248$		0			0
$249$			722.213i			2.90045i
$250$		0			0
$251$	−69.8084			−0.278121			−0.139061	−	0.990284i	$-0.544408\pi$
$251$	−69.8084			−0.278121			−0.139061	+	0.990284i	$0.544408\pi$
$252$		0			0
$253$	−742.749			−2.93577
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	205.201i		−	0.798448i	−0.916853	−	0.399224i	$-0.869280\pi$
$257$		−	205.201i		−	0.798448i	0.916853	−	0.399224i	$-0.130720\pi$
$258$		0			0
$259$		−	59.8054i		−	0.230909i
$260$		0			0
$261$			380.324i			1.45718i
$262$		0			0
$263$	369.721			1.40578			0.702892	−	0.711297i	$-0.251892\pi$
$263$	369.721			1.40578			0.702892	+	0.711297i	$0.251892\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−402.113			−1.50604
$268$		0			0
$269$		−	56.1803i		−	0.208849i	−0.994533	−	0.104424i	$-0.966700\pi$
$269$		−	56.1803i		−	0.208849i	0.994533	−	0.104424i	$-0.0333000\pi$
$270$		0			0
$271$	−108.318			−0.399697			−0.199849	−	0.979827i	$-0.564045\pi$
$271$	−108.318			−0.399697			−0.199849	+	0.979827i	$0.564045\pi$
$272$		0			0
$273$	−113.860			−0.417068
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−186.564			−0.673517			−0.336758	−	0.941591i	$-0.609331\pi$
$277$	−186.564			−0.673517			−0.336758	+	0.941591i	$0.609331\pi$
$278$		0			0
$279$			469.025i			1.68109i
$280$		0			0
$281$		−	136.711i		−	0.486516i	−0.969962	−	0.243258i	$-0.921784\pi$
$281$		−	136.711i		−	0.486516i	0.969962	−	0.243258i	$-0.0782161\pi$
$282$		0			0
$283$	−100.842			−0.356332			−0.178166	−	0.984000i	$-0.557016\pi$
$283$	−100.842			−0.356332			−0.178166	+	0.984000i	$0.557016\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			106.238i			0.370169i
$288$		0			0
$289$	−33.7382			−0.116741
$290$		0			0
$291$	529.110			1.81825
$292$		0			0
$293$		−	345.837i		−	1.18033i	−0.807283	−	0.590165i	$-0.799063\pi$
$293$		−	345.837i		−	1.18033i	0.807283	−	0.590165i	$-0.200937\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			276.066i			0.929515i
$298$		0			0
$299$		−	609.835i		−	2.03958i
$300$		0			0
$301$	−25.3531			−0.0842294
$302$		0			0
$303$			485.743i			1.60311i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			212.148i			0.691035i	0.938412	+	0.345518i	$0.112297\pi$
$307$			212.148i			0.691035i	−0.938412	+	0.345518i	$0.887703\pi$
$308$		0			0
$309$	−333.044			−1.07781
$310$		0			0
$311$	534.396			1.71832			0.859158	−	0.511711i	$-0.170988\pi$
$311$	534.396			1.71832			0.859158	+	0.511711i	$0.170988\pi$
$312$		0			0
$313$	−242.134			−0.773590			−0.386795	−	0.922166i	$-0.626418\pi$
$313$	−242.134			−0.773590			−0.386795	+	0.922166i	$0.626418\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			448.993i			1.41638i	0.706021	+	0.708191i	$0.250488\pi$
$317$			448.993i			1.41638i	−0.706021	+	0.708191i	$0.749512\pi$
$318$		0			0
$319$		−	624.150i		−	1.95658i
$320$		0			0
$321$	10.8055			0.0336619
$322$		0			0
$323$	−302.225	+	28.4492i	−0.935682	+	0.0880781i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	835.365			2.55463
$328$		0			0
$329$	79.5989			0.241942
$330$		0			0
$331$		−	444.084i		−	1.34164i	−0.741618	−	0.670822i	$-0.765941\pi$
$331$		−	444.084i		−	1.34164i	0.741618	−	0.670822i	$-0.234059\pi$
$332$		0			0
$333$		−	471.004i		−	1.41443i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	517.271i		−	1.53493i	−0.641092	−	0.767464i	$-0.721518\pi$
$337$		−	517.271i		−	1.53493i	0.641092	−	0.767464i	$-0.278482\pi$
$338$		0			0
$339$	−396.593			−1.16989
$340$		0			0
$341$		−	769.717i		−	2.25723i
$342$		0			0
$343$	146.299			0.426527
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.8683			−0.0889576			−0.0444788	−	0.999010i	$-0.514163\pi$
$347$	−30.8683			−0.0889576			−0.0444788	+	0.999010i	$0.514163\pi$
$348$		0			0
$349$	−156.386			−0.448097			−0.224049	−	0.974578i	$-0.571927\pi$
$349$	−156.386			−0.448097			−0.224049	+	0.974578i	$0.571927\pi$
$350$		0			0
$351$	−226.664			−0.645767
$352$		0			0
$353$	569.039			1.61201			0.806005	−	0.591909i	$-0.201626\pi$
$353$	569.039			1.61201			0.806005	+	0.591909i	$0.201626\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			112.090i			0.313978i
$358$		0			0
$359$	−339.420			−0.945459			−0.472729	−	0.881208i	$-0.656731\pi$
$359$	−339.420			−0.945459			−0.472729	+	0.881208i	$0.656731\pi$
$360$		0			0
$361$	354.659	−	67.3667i	0.982434	−	0.186611i
$362$		0			0
$363$		−	1237.25i		−	3.40841i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	72.4587			0.197435			0.0987176	−	0.995115i	$-0.468526\pi$
$367$	72.4587			0.197435			0.0987176	+	0.995115i	$0.468526\pi$
$368$		0			0
$369$			836.692i			2.26746i
$370$		0			0
$371$			25.7700i			0.0694608i
$372$		0			0
$373$			480.589i			1.28844i	0.764840	+	0.644221i	$0.222818\pi$
$373$			480.589i			1.28844i	−0.764840	+	0.644221i	$0.777182\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	512.459			1.35931
$378$		0			0
$379$		−	343.159i		−	0.905433i	−0.891654	−	0.452717i	$-0.850455\pi$
$379$		−	343.159i		−	0.905433i	0.891654	−	0.452717i	$-0.149545\pi$
$380$		0			0
$381$	−546.762			−1.43507
$382$		0			0
$383$			380.989i			0.994751i	0.867536	+	0.497375i	$0.165703\pi$
$383$			380.989i			0.994751i	−0.867536	+	0.497375i	$0.834297\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−199.671			−0.515945
$388$		0			0
$389$	153.297			0.394081			0.197040	−	0.980395i	$-0.436867\pi$
$389$	153.297			0.394081			0.197040	+	0.980395i	$0.436867\pi$
$390$		0			0
$391$	−600.358			−1.53544
$392$		0			0
$393$		−	511.666i		−	1.30195i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−410.383			−1.03371			−0.516855	−	0.856073i	$-0.672897\pi$
$397$	−410.383			−1.03371			−0.516855	+	0.856073i	$0.672897\pi$
$398$		0			0
$399$	−12.4926	−	132.713i	−0.0313098	−	0.332614i
$400$		0			0
$401$			96.2006i			0.239902i	0.992780	+	0.119951i	$0.0382737\pi$
$401$			96.2006i			0.239902i	−0.992780	+	0.119951i	$0.961726\pi$
$402$		0			0
$403$	631.977			1.56818
$404$		0			0
$405$		0			0
$406$		0			0
$407$			772.964i			1.89918i
$408$		0			0
$409$		−	804.488i		−	1.96696i	−0.181008	−	0.983482i	$-0.557936\pi$
$409$		−	804.488i		−	1.96696i	0.181008	−	0.983482i	$-0.442064\pi$
$410$		0			0
$411$		−	376.364i		−	0.915727i
$412$		0			0
$413$			141.607i			0.342873i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			777.970i			1.86564i
$418$		0			0
$419$	105.005			0.250609			0.125304	−	0.992118i	$-0.460009\pi$
$419$	105.005			0.250609			0.125304	+	0.992118i	$0.460009\pi$
$420$		0			0
$421$		−	228.201i		−	0.542045i	−0.962573	−	0.271023i	$-0.912638\pi$
$421$		−	228.201i		−	0.542045i	0.962573	−	0.271023i	$-0.0873618\pi$
$422$		0			0
$423$	626.890			1.48201
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−78.8113			−0.184570
$428$		0			0
$429$	1471.60			3.43030
$430$		0			0
$431$			182.455i			0.423329i	0.977342	+	0.211665i	$0.0678884\pi$
$431$			182.455i			0.423329i	−0.977342	+	0.211665i	$0.932112\pi$
$432$		0			0
$433$			29.9156i			0.0690890i	0.999403	+	0.0345445i	$0.0109980\pi$
$433$			29.9156i			0.0690890i	−0.999403	+	0.0345445i	$0.989002\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	710.813	−	66.9106i	1.62657	−	0.153113i
$438$		0			0
$439$			497.696i			1.13370i	0.823820	+	0.566852i	$0.191839\pi$
$439$			497.696i			1.13370i	−0.823820	+	0.566852i	$0.808161\pi$
$440$		0			0
$441$	562.011			1.27440
$442$		0			0
$443$	348.602			0.786912			0.393456	−	0.919343i	$-0.371279\pi$
$443$	348.602			0.786912			0.393456	+	0.919343i	$0.371279\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	309.624i		−	0.692672i
$448$		0			0
$449$		−	285.726i		−	0.636360i	−0.948030	−	0.318180i	$-0.896928\pi$
$449$		−	285.726i		−	0.636360i	0.948030	−	0.318180i	$-0.103072\pi$
$450$		0			0
$451$		−	1373.10i		−	3.04456i
$452$		0			0
$453$	698.632			1.54223
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−394.636			−0.863535			−0.431768	−	0.901985i	$-0.642110\pi$
$457$	−394.636			−0.863535			−0.431768	+	0.901985i	$0.642110\pi$
$458$		0			0
$459$			223.142i			0.486148i
$460$		0			0
$461$	−15.9653			−0.0346318			−0.0173159	−	0.999850i	$-0.505512\pi$
$461$	−15.9653			−0.0346318			−0.0173159	+	0.999850i	$0.505512\pi$
$462$		0			0
$463$	−291.052			−0.628623			−0.314311	−	0.949320i	$-0.601774\pi$
$463$	−291.052			−0.628623			−0.314311	+	0.949320i	$0.601774\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	592.490			1.26871			0.634357	−	0.773040i	$-0.281265\pi$
$467$	592.490			1.26871			0.634357	+	0.773040i	$0.281265\pi$
$468$		0			0
$469$			74.5265i			0.158905i
$470$		0			0
$471$		−	233.466i		−	0.495682i
$472$		0			0
$473$	327.679			0.692768
$474$		0			0
$475$		0			0
$476$		0			0
$477$			202.954i			0.425480i
$478$		0			0
$479$	198.971			0.415389			0.207695	−	0.978194i	$-0.433404\pi$
$479$	198.971			0.415389			0.207695	+	0.978194i	$0.433404\pi$
$480$		0			0
$481$	−634.643			−1.31942
$482$		0			0
$483$		−	263.629i		−	0.545815i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	136.234i		−	0.279741i	−0.990170	−	0.139871i	$-0.955331\pi$
$487$		−	136.234i		−	0.279741i	0.990170	−	0.139871i	$-0.0446687\pi$
$488$		0			0
$489$		−	257.392i		−	0.526364i
$490$		0			0
$491$	220.115			0.448299			0.224149	−	0.974555i	$-0.428040\pi$
$491$	220.115			0.448299			0.224149	+	0.974555i	$0.428040\pi$
$492$		0			0
$493$		−	504.495i		−	1.02332i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	8.43457i		−	0.0169710i
$498$		0			0
$499$	929.974			1.86367			0.931837	−	0.362877i	$-0.118205\pi$
$499$	929.974			1.86367			0.931837	+	0.362877i	$0.118205\pi$
$500$		0			0
$501$	−793.587			−1.58401
$502$		0			0
$503$	−606.816			−1.20639			−0.603196	−	0.797593i	$-0.706107\pi$
$503$	−606.816			−1.20639			−0.603196	+	0.797593i	$0.706107\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			432.980i			0.854004i
$508$		0			0
$509$		−	342.999i		−	0.673869i	−0.941528	−	0.336935i	$-0.890610\pi$
$509$		−	342.999i		−	0.673869i	0.941528	−	0.336935i	$-0.109390\pi$
$510$		0			0
$511$	−96.0931			−0.188049
$512$		0			0
$513$	−24.8694	−	264.196i	−0.0484784	−	0.515002i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−1028.79			−1.98992
$518$		0			0
$519$	392.074			0.755442
$520$		0			0
$521$			303.047i			0.581664i	0.956774	+	0.290832i	$0.0939320\pi$
$521$			303.047i			0.581664i	−0.956774	+	0.290832i	$0.906068\pi$
$522$		0			0
$523$			839.474i			1.60511i	0.596576	+	0.802556i	$0.296527\pi$
$523$			839.474i			1.60511i	−0.596576	+	0.802556i	$0.703473\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	622.156i		−	1.18056i
$528$		0			0
$529$	883.001			1.66919
$530$		0			0
$531$			1115.24i			2.10026i
$532$		0			0
$533$	1127.38			2.11516
$534$		0			0
$535$		0			0
$536$		0			0
$537$	833.468			1.55208
$538$		0			0
$539$	−922.316			−1.71116
$540$		0			0
$541$	−673.205			−1.24437			−0.622186	−	0.782869i	$-0.713755\pi$
$541$	−673.205			−1.24437			−0.622186	+	0.782869i	$0.713755\pi$
$542$		0			0
$543$	−450.105			−0.828922
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	393.199i		−	0.718828i	−0.933178	−	0.359414i	$-0.882977\pi$
$547$		−	393.199i		−	0.718828i	0.933178	−	0.359414i	$-0.117023\pi$
$548$		0			0
$549$	−620.686			−1.13058
$550$		0			0
$551$	56.2266	+	597.313i	0.102045	+	1.08405i
$552$		0			0
$553$			223.837i			0.404769i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	62.2937			0.111838			0.0559189	−	0.998435i	$-0.482191\pi$
$557$	62.2937			0.111838			0.0559189	+	0.998435i	$0.482191\pi$
$558$		0			0
$559$			269.042i			0.481291i
$560$		0			0
$561$		−	1448.73i		−	2.58240i
$562$		0			0
$563$		−	763.894i		−	1.35683i	−0.734680	−	0.678414i	$-0.762668\pi$
$563$		−	763.894i		−	1.35683i	0.734680	−	0.678414i	$-0.237332\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	67.7962			0.119570
$568$		0			0
$569$			344.276i			0.605055i	0.953141	+	0.302527i	$0.0978304\pi$
$569$			344.276i			0.605055i	−0.953141	+	0.302527i	$0.902170\pi$
$570$		0			0
$571$	−394.946			−0.691675			−0.345837	−	0.938294i	$-0.612405\pi$
$571$	−394.946			−0.691675			−0.345837	+	0.938294i	$0.612405\pi$
$572$		0			0
$573$			688.466i			1.20151i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−491.043			−0.851027			−0.425514	−	0.904952i	$-0.639907\pi$
$577$	−491.043			−0.851027			−0.425514	+	0.904952i	$0.639907\pi$
$578$		0			0
$579$	92.1065			0.159079
$580$		0			0
$581$	240.769			0.414405
$582$		0			0
$583$		−	333.068i		−	0.571300i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	99.3437			0.169240			0.0846198	−	0.996413i	$-0.473032\pi$
$587$	99.3437			0.169240			0.0846198	+	0.996413i	$0.473032\pi$
$588$		0			0
$589$	69.3400	+	736.621i	0.117725	+	1.25063i
$590$		0			0
$591$			502.052i			0.849495i
$592$		0			0
$593$	−879.094			−1.48245			−0.741226	−	0.671256i	$-0.765755\pi$
$593$	−879.094			−1.48245			−0.741226	+	0.671256i	$0.765755\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	133.456i		−	0.223545i
$598$		0			0
$599$		−	14.4131i		−	0.0240620i	−0.999928	−	0.0120310i	$-0.996170\pi$
$599$		−	14.4131i		−	0.0240620i	0.999928	−	0.0120310i	$-0.00382967\pi$
$600$		0			0
$601$			485.996i			0.808645i	0.914617	+	0.404322i	$0.132493\pi$
$601$			485.996i			0.808645i	−0.914617	+	0.404322i	$0.867507\pi$
$602$		0			0
$603$			586.941i			0.973368i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			249.106i			0.410389i	0.978721	+	0.205195i	$0.0657827\pi$
$607$			249.106i			0.410389i	−0.978721	+	0.205195i	$0.934217\pi$
$608$		0			0
$609$	221.533			0.363766
$610$		0			0
$611$		−	844.688i		−	1.38247i
$612$		0			0
$613$	439.644			0.717200			0.358600	−	0.933491i	$-0.383254\pi$
$613$	439.644			0.717200			0.358600	+	0.933491i	$0.383254\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	31.6889			0.0513597			0.0256798	−	0.999670i	$-0.491825\pi$
$617$	31.6889			0.0513597			0.0256798	+	0.999670i	$0.491825\pi$
$618$		0			0
$619$	−590.945			−0.954677			−0.477339	−	0.878719i	$-0.658398\pi$
$619$	−590.945			−0.954677			−0.477339	+	0.878719i	$0.658398\pi$
$620$		0			0
$621$		−	524.814i		−	0.845112i
$622$		0			0
$623$			134.055i			0.215177i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	161.462	+	1715.27i	0.257516	+	2.73568i
$628$		0			0
$629$			624.781i			0.993293i
$630$		0			0
$631$	−144.511			−0.229019			−0.114509	−	0.993422i	$-0.536530\pi$
$631$	−144.511			−0.229019			−0.114509	+	0.993422i	$0.536530\pi$
$632$		0			0
$633$	617.792			0.975975
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	757.268i		−	1.18880i
$638$		0			0
$639$		−	66.4273i		−	0.103955i
$640$		0			0
$641$			41.9005i			0.0653674i	0.999466	+	0.0326837i	$0.0104054\pi$
$641$			41.9005i			0.0653674i	−0.999466	+	0.0326837i	$0.989595\pi$
$642$		0			0
$643$	847.463			1.31798			0.658992	−	0.752150i	$-0.270983\pi$
$643$	847.463			1.31798			0.658992	+	0.752150i	$0.270983\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	828.509			1.28054			0.640269	−	0.768150i	$-0.278823\pi$
$647$	828.509			1.28054			0.640269	+	0.768150i	$0.278823\pi$
$648$		0			0
$649$		−	1830.22i		−	2.82006i
$650$		0			0
$651$	273.200			0.419663
$652$		0			0
$653$	376.647			0.576795			0.288397	−	0.957511i	$-0.406878\pi$
$653$	376.647			0.576795			0.288397	+	0.957511i	$0.406878\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−756.791			−1.15189
$658$		0			0
$659$		−	656.057i		−	0.995534i	−0.867311	−	0.497767i	$-0.834154\pi$
$659$		−	656.057i		−	0.995534i	0.867311	−	0.497767i	$-0.165846\pi$
$660$		0			0
$661$			837.563i			1.26712i	0.773696	+	0.633558i	$0.218406\pi$
$661$			837.563i			1.26712i	−0.773696	+	0.633558i	$0.781594\pi$
$662$		0			0
$663$	1189.48			1.79409
$664$		0			0
$665$		0			0
$666$		0			0
$667$			1186.54i			1.77892i
$668$		0			0
$669$	925.030			1.38271
$670$		0			0
$671$	1018.61			1.51805
$672$		0			0
$673$			383.937i			0.570486i	0.958455	+	0.285243i	$0.0920743\pi$
$673$			383.937i			0.570486i	−0.958455	+	0.285243i	$0.907926\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	597.994i		−	0.883300i	−0.897187	−	0.441650i	$-0.854393\pi$
$677$		−	597.994i		−	0.883300i	0.897187	−	0.441650i	$-0.145607\pi$
$678$		0			0
$679$		−	176.393i		−	0.259784i
$680$		0			0
$681$	−1998.13			−2.93411
$682$		0			0
$683$		−	814.772i		−	1.19293i	−0.802639	−	0.596466i	$-0.796571\pi$
$683$		−	814.772i		−	1.19293i	0.802639	−	0.596466i	$-0.203429\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		−	1544.27i		−	2.24785i
$688$		0			0
$689$	273.466			0.396903
$690$		0			0
$691$	183.046			0.264900			0.132450	−	0.991190i	$-0.457716\pi$
$691$	183.046			0.264900			0.132450	+	0.991190i	$0.457716\pi$
$692$		0			0
$693$	364.099			0.525396
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	1109.86i		−	1.59234i
$698$		0			0
$699$			1241.98i			1.77680i
$700$		0			0
$701$	20.2332			0.0288633			0.0144317	−	0.999896i	$-0.495406\pi$
$701$	20.2332			0.0288633			0.0144317	+	0.999896i	$0.495406\pi$
$702$		0			0
$703$	−69.6325	−	739.729i	−0.0990506	−	1.05225i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	161.936			0.229046
$708$		0			0
$709$	348.032			0.490877			0.245438	−	0.969412i	$-0.421068\pi$
$709$	348.032			0.490877			0.245438	+	0.969412i	$0.421068\pi$
$710$		0			0
$711$			1762.85i			2.47940i
$712$		0			0
$713$			1463.27i			2.05227i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	511.226i		−	0.713007i
$718$		0			0
$719$	−1282.72			−1.78403			−0.892014	−	0.452007i	$-0.850708\pi$
$719$	−1282.72			−1.78403			−0.892014	+	0.452007i	$0.850708\pi$
$720$		0			0
$721$			111.029i			0.153993i
$722$		0			0
$723$	1234.25			1.70712
$724$		0			0
$725$		0			0
$726$		0			0
$727$	203.759			0.280274			0.140137	−	0.990132i	$-0.455246\pi$
$727$	203.759			0.280274			0.140137	+	0.990132i	$0.455246\pi$
$728$		0			0
$729$	1110.57			1.52342
$730$		0			0
$731$	264.861			0.362327
$732$		0			0
$733$	−350.631			−0.478350			−0.239175	−	0.970976i	$-0.576877\pi$
$733$	−350.631			−0.478350			−0.239175	+	0.970976i	$0.576877\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	963.229i		−	1.30696i
$738$		0			0
$739$	1288.92			1.74415			0.872073	−	0.489375i	$-0.162775\pi$
$739$	1288.92			1.74415			0.872073	+	0.489375i	$0.162775\pi$
$740$		0			0
$741$	−1408.32	+	132.569i	−1.90057	+	0.178905i
$742$		0			0
$743$			1222.62i			1.64551i	0.568393	+	0.822757i	$0.307565\pi$
$743$			1222.62i			1.64551i	−0.568393	+	0.822757i	$0.692435\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1896.20			2.53843
$748$		0			0
$749$		−	3.60229i		−	0.00480947i
$750$		0			0
$751$			1256.47i			1.67306i	0.547919	+	0.836531i	$0.315420\pi$
$751$			1256.47i			1.67306i	−0.547919	+	0.836531i	$0.684580\pi$
$752$		0			0
$753$			320.241i			0.425287i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	680.666			0.899162			0.449581	−	0.893240i	$-0.351573\pi$
$757$	680.666			0.899162			0.449581	+	0.893240i	$0.351573\pi$
$758$		0			0
$759$			3407.31i			4.48921i
$760$		0			0
$761$	656.481			0.862656			0.431328	−	0.902195i	$-0.358045\pi$
$761$	656.481			0.862656			0.431328	+	0.902195i	$0.358045\pi$
$762$		0			0
$763$		−	278.492i		−	0.364996i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1502.70			1.95919
$768$		0			0
$769$	−429.894			−0.559030			−0.279515	−	0.960141i	$-0.590174\pi$
$769$	−429.894			−0.559030			−0.279515	+	0.960141i	$0.590174\pi$
$770$		0			0
$771$	−941.346			−1.22094
$772$		0			0
$773$			1410.12i			1.82422i	0.409943	+	0.912111i	$0.365549\pi$
$773$			1410.12i			1.82422i	−0.409943	+	0.912111i	$0.634451\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−274.353			−0.353093
$778$		0			0
$779$	123.695	+	1314.06i	0.158787	+	1.68685i
$780$		0			0
$781$			109.014i			0.139582i
$782$		0			0
$783$	441.014			0.563236
$784$		0			0
$785$		0			0
$786$		0			0
$787$			696.889i			0.885501i	0.896645	+	0.442750i	$0.145997\pi$
$787$			696.889i			0.885501i	−0.896645	+	0.442750i	$0.854003\pi$
$788$		0			0
$789$		−	1696.07i		−	2.14964i
$790$		0			0
$791$			132.215i			0.167150i
$792$		0			0
$793$			836.330i			1.05464i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			1145.32i			1.43704i	0.695506	+	0.718520i	$0.255180\pi$
$797$			1145.32i			1.43704i	−0.695506	+	0.718520i	$0.744820\pi$
$798$		0			0
$799$	−831.562			−1.04075
$800$		0			0
$801$			1055.77i			1.31806i
$802$		0			0
$803$	1241.97			1.54666
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−257.723			−0.319360
$808$		0			0
$809$	355.323			0.439212			0.219606	−	0.975589i	$-0.429523\pi$
$809$	355.323			0.439212			0.219606	+	0.975589i	$0.429523\pi$
$810$		0			0
$811$		−	108.258i		−	0.133487i	−0.997770	−	0.0667437i	$-0.978739\pi$
$811$		−	108.258i		−	0.133487i	0.997770	−	0.0667437i	$-0.0212610\pi$
$812$		0			0
$813$			496.901i			0.611195i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−313.590	+	29.5190i	−0.383831	+	0.0361310i
$818$		0			0
$819$			298.944i			0.365011i
$820$		0			0
$821$	−470.491			−0.573071			−0.286535	−	0.958070i	$-0.592504\pi$
$821$	−470.491			−0.573071			−0.286535	+	0.958070i	$0.592504\pi$
$822$		0			0
$823$	986.856			1.19910			0.599548	−	0.800339i	$-0.295347\pi$
$823$	986.856			1.19910			0.599548	+	0.800339i	$0.295347\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1180.91i		−	1.42794i	−0.700174	−	0.713972i	$-0.746894\pi$
$827$		−	1180.91i		−	1.42794i	0.700174	−	0.713972i	$-0.253106\pi$
$828$		0			0
$829$			357.631i			0.431401i	0.976460	+	0.215701i	$0.0692035\pi$
$829$			357.631i			0.431401i	−0.976460	+	0.215701i	$0.930797\pi$
$830$		0			0
$831$			855.850i			1.02990i
$832$		0			0
$833$	−745.500			−0.894959
$834$		0			0
$835$		0			0
$836$		0			0
$837$	543.869			0.649784
$838$		0			0
$839$		−	1334.94i		−	1.59111i	−0.605884	−	0.795553i	$-0.707181\pi$
$839$		−	1334.94i		−	1.59111i	0.605884	−	0.795553i	$-0.292819\pi$
$840$		0			0
$841$	−156.076			−0.185584
$842$		0			0
$843$	−627.152			−0.743952
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−412.473			−0.486981
$848$		0			0
$849$			462.605i			0.544882i
$850$		0			0
$851$		−	1469.44i		−	1.72672i
$852$		0			0
$853$	−1346.94			−1.57906			−0.789529	−	0.613713i	$-0.789675\pi$
$853$	−1346.94			−1.57906			−0.789529	+	0.613713i	$0.789675\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	10.3502i		−	0.0120773i	−0.999982	−	0.00603863i	$-0.998078\pi$
$857$		−	10.3502i		−	0.0120773i	0.999982	−	0.00603863i	$-0.00192217\pi$
$858$		0			0
$859$	1026.42			1.19490			0.597449	−	0.801907i	$-0.296181\pi$
$859$	1026.42			1.19490			0.597449	+	0.801907i	$0.296181\pi$
$860$		0			0
$861$	487.362			0.566041
$862$		0			0
$863$		−	1145.76i		−	1.32764i	−0.747891	−	0.663822i	$-0.768933\pi$
$863$		−	1145.76i		−	1.32764i	0.747891	−	0.663822i	$-0.231067\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			154.772i			0.178514i
$868$		0			0
$869$		−	2893.02i		−	3.32914i
$870$		0			0
$871$	790.860			0.907991
$872$		0			0
$873$		−	1389.20i		−	1.59130i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1242.23i		−	1.41645i	−0.705987	−	0.708225i	$-0.749496\pi$
$877$		−	1242.23i		−	1.41645i	0.705987	−	0.708225i	$-0.250504\pi$
$878$		0			0
$879$	−1586.50			−1.80489
$880$		0			0
$881$	−1102.73			−1.25168			−0.625842	−	0.779950i	$-0.715244\pi$
$881$	−1102.73			−1.25168			−0.625842	+	0.779950i	$0.715244\pi$
$882$		0			0
$883$	597.580			0.676761			0.338381	−	0.941009i	$-0.390121\pi$
$883$	597.580			0.676761			0.338381	+	0.941009i	$0.390121\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	327.216i		−	0.368902i	−0.982842	−	0.184451i	$-0.940949\pi$
$887$		−	327.216i		−	0.368902i	0.982842	−	0.184451i	$-0.0590507\pi$
$888$		0			0
$889$			182.278i			0.205037i
$890$		0			0
$891$	−876.243			−0.983438
$892$		0			0
$893$	984.554	−	92.6785i	1.10252	−	0.103783i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−2797.57			−3.11881
$898$		0			0
$899$	−1229.62			−1.36776
$900$		0			0
$901$		−	269.216i		−	0.298797i
$902$		0			0
$903$			116.305i			0.128799i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			253.226i			0.279190i	0.990209	+	0.139595i	$0.0445801\pi$
$907$			253.226i			0.279190i	−0.990209	+	0.139595i	$0.955420\pi$
$908$		0			0
$909$	1275.34			1.40302
$910$		0			0
$911$			1122.92i			1.23263i	0.787500	+	0.616314i	$0.211375\pi$
$911$			1122.92i			1.23263i	−0.787500	+	0.616314i	$0.788625\pi$
$912$		0			0
$913$	−3111.86			−3.40839
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−170.578			−0.186017
$918$		0			0
$919$	967.325			1.05258			0.526292	−	0.850304i	$-0.323582\pi$
$919$	967.325			1.05258			0.526292	+	0.850304i	$0.323582\pi$
$920$		0			0
$921$	973.213			1.05669
$922$		0			0
$923$	−89.5060			−0.0969729
$924$		0			0
$925$		0			0
$926$		0			0
$927$			874.423i			0.943282i
$928$		0			0
$929$	374.642			0.403275			0.201637	−	0.979460i	$-0.435374\pi$
$929$	374.642			0.403275			0.201637	+	0.979460i	$0.435374\pi$
$930$		0			0
$931$	882.659	−	83.0869i	0.948076	−	0.0892448i
$932$		0			0
$933$		−	2451.51i		−	2.62755i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−310.678			−0.331567			−0.165783	−	0.986162i	$-0.553015\pi$
$937$	−310.678			−0.331567			−0.165783	+	0.986162i	$0.553015\pi$
$938$		0			0
$939$			1110.77i			1.18293i
$940$		0			0
$941$			118.112i			0.125518i	0.998029	+	0.0627589i	$0.0199899\pi$
$941$			118.112i			0.125518i	−0.998029	+	0.0627589i	$0.980010\pi$
$942$		0			0
$943$			2610.32i			2.76810i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−553.081			−0.584034			−0.292017	−	0.956413i	$-0.594326\pi$
$947$	−553.081			−0.584034			−0.292017	+	0.956413i	$0.594326\pi$
$948$		0			0
$949$			1019.72i			1.07452i
$950$		0			0
$951$	2059.72			2.16585
$952$		0			0
$953$			375.463i			0.393980i	0.980406	+	0.196990i	$0.0631166\pi$
$953$			375.463i			0.393980i	−0.980406	+	0.196990i	$0.936883\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−2863.24			−2.99189
$958$		0			0
$959$	−125.471			−0.130835
$960$		0			0
$961$	−555.396			−0.577936
$962$		0			0
$963$		−	28.3702i		−	0.0294603i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1280.32			−1.32401			−0.662004	−	0.749500i	$-0.730294\pi$
$967$	−1280.32			−1.32401			−0.662004	+	0.749500i	$0.730294\pi$
$968$		0			0
$969$	130.509	+	1386.44i	0.134684	+	1.43079i
$970$		0			0
$971$			487.041i			0.501587i	0.968041	+	0.250793i	$0.0806915\pi$
$971$			487.041i			0.501587i	−0.968041	+	0.250793i	$0.919309\pi$
$972$		0			0
$973$	259.358			0.266555
$974$		0			0
$975$		0			0
$976$		0			0
$977$			85.6576i			0.0876741i	0.999039	+	0.0438371i	$0.0139582\pi$
$977$			85.6576i			0.0876741i	−0.999039	+	0.0438371i	$0.986042\pi$
$978$		0			0
$979$		−	1732.62i		−	1.76979i
$980$		0			0
$981$		−	2193.29i		−	2.23577i
$982$		0			0
$983$			220.950i			0.224771i	0.993665	+	0.112386i	$0.0358492\pi$
$983$			220.950i			0.224771i	−0.993665	+	0.112386i	$0.964151\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	365.155i		−	0.369964i
$988$		0			0
$989$	−622.934			−0.629862
$990$		0			0
$991$		−	1569.41i		−	1.58367i	−0.610737	−	0.791834i	$-0.709127\pi$
$991$		−	1569.41i		−	1.58367i	0.610737	−	0.791834i	$-0.290873\pi$
$992$		0			0
$993$	−2037.21			−2.05157
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1821.95			1.82743			0.913715	−	0.406355i	$-0.133200\pi$
$997$	1821.95			1.82743			0.913715	+	0.406355i	$0.133200\pi$
$998$		0			0
$999$	−546.164			−0.546711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.g.1101.3	✓	14
5.2	odd	4		1900.3.g.d.949.6		28
5.3	odd	4		1900.3.g.d.949.23		28
5.4	even	2		1900.3.e.h.1101.12	yes	14
19.18	odd	2	inner	1900.3.e.g.1101.12	yes	14
95.18	even	4		1900.3.g.d.949.5		28
95.37	even	4		1900.3.g.d.949.24		28
95.94	odd	2		1900.3.e.h.1101.3	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.3.e.g.1101.3	✓	14	1.1	even	1	trivial
1900.3.e.g.1101.12	yes	14	19.18	odd	2	inner
1900.3.e.h.1101.3	yes	14	95.94	odd	2
1900.3.e.h.1101.12	yes	14	5.4	even	2
1900.3.g.d.949.5		28	95.18	even	4
1900.3.g.d.949.6		28	5.2	odd	4
1900.3.g.d.949.23		28	5.3	odd	4
1900.3.g.d.949.24		28	95.37	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.58743i q^{3} -1.52935 q^{7} -12.0445 q^{9} +O(q^{10})\)	\(q-4.58743i q^{3} -1.52935 q^{7} -12.0445 q^{9} +19.7663 q^{11} +16.2291i q^{13} +15.9769 q^{17} +(-18.9164 + 1.78065i) q^{19} +7.01577i q^{21} -37.5766 q^{23} +13.9665i q^{27} -31.5765i q^{29} -38.9409i q^{31} -90.6764i q^{33} +39.1052i q^{37} +74.4499 q^{39} -69.4666i q^{41} +16.5777 q^{43} -52.0477 q^{47} -46.6611 q^{49} -73.2930i q^{51} -16.8503i q^{53} +(8.16859 + 86.7776i) q^{57} -92.5930i q^{59} +51.5327 q^{61} +18.4202 q^{63} -48.7310i q^{67} +172.380i q^{69} +5.51515i q^{71} +62.8328 q^{73} -30.2295 q^{77} -146.361i q^{79} -44.3302 q^{81} -157.433 q^{83} -144.855 q^{87} -87.6554i q^{89} -24.8199i q^{91} -178.639 q^{93} +115.339i q^{97} -238.075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 4 q^{7} - 52 q^{9}+O(q^{10}) \) 14 * q - 4 * q^7 - 52 * q^9	\( 14 q - 4 q^{7} - 52 q^{9} + 4 q^{11} + 6 q^{17} - 29 q^{19} + 28 q^{23} - 56 q^{39} - 22 q^{43} - 84 q^{47} + 138 q^{49} + 5 q^{57} - 50 q^{61} - 234 q^{63} + 204 q^{73} - 68 q^{77} + 66 q^{81} - 256 q^{83} - 18 q^{87} - 118 q^{93} + 92 q^{99}+O(q^{100}) \) 14 * q - 4 * q^7 - 52 * q^9 + 4 * q^11 + 6 * q^17 - 29 * q^19 + 28 * q^23 - 56 * q^39 - 22 * q^43 - 84 * q^47 + 138 * q^49 + 5 * q^57 - 50 * q^61 - 234 * q^63 + 204 * q^73 - 68 * q^77 + 66 * q^81 - 256 * q^83 - 18 * q^87 - 118 * q^93 + 92 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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