LMFDB - Embedded newform 1200.3.c.g.449.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1200.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.g.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.82843 - 1.00000i) q^{3} -1.00000i q^{7} +(7.00000 + 5.65685i) q^{9} +O(q^{10})$	$q+(-2.82843 - 1.00000i) q^{3} -1.00000i q^{7} +(7.00000 + 5.65685i) q^{9} +8.48528i q^{11} -15.0000i q^{13} +19.7990 q^{17} -23.0000 q^{19} +(-1.00000 + 2.82843i) q^{21} +2.82843 q^{23} +(-14.1421 - 23.0000i) q^{27} +25.4558i q^{29} -33.0000 q^{31} +(8.48528 - 24.0000i) q^{33} +66.0000i q^{37} +(-15.0000 + 42.4264i) q^{39} -36.7696i q^{41} -7.00000i q^{43} +45.2548 q^{47} +48.0000 q^{49} +(-56.0000 - 19.7990i) q^{51} -36.7696 q^{53} +(65.0538 + 23.0000i) q^{57} +101.823i q^{59} +39.0000 q^{61} +(5.65685 - 7.00000i) q^{63} -113.000i q^{67} +(-8.00000 - 2.82843i) q^{69} +25.4558i q^{71} -58.0000i q^{73} +8.48528 q^{77} +70.0000 q^{79} +(17.0000 + 79.1960i) q^{81} +152.735 q^{83} +(25.4558 - 72.0000i) q^{87} +90.5097i q^{89} -15.0000 q^{91} +(93.3381 + 33.0000i) q^{93} -1.00000i q^{97} +(-48.0000 + 59.3970i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{9}+O(q^{10}) $ 4 * q + 28 * q^9	$ 4 q + 28 q^{9} - 92 q^{19} - 4 q^{21} - 132 q^{31} - 60 q^{39} + 192 q^{49} - 224 q^{51} + 156 q^{61} - 32 q^{69} + 280 q^{79} + 68 q^{81} - 60 q^{91} - 192 q^{99}+O(q^{100}) $ 4 * q + 28 * q^9 - 92 * q^19 - 4 * q^21 - 132 * q^31 - 60 * q^39 + 192 * q^49 - 224 * q^51 + 156 * q^61 - 32 * q^69 + 280 * q^79 + 68 * q^81 - 60 * q^91 - 192 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$3$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	1.00000i		−	0.142857i	−0.997446	−	0.0714286i	$-0.977244\pi$
$7$		−	1.00000i		−	0.142857i	0.997446	−	0.0714286i	$-0.0227558\pi$
$8$		0			0
$9$	7.00000	+	5.65685i	0.777778	+	0.628539i
$10$		0			0
$11$			8.48528i			0.771389i	0.922627	+	0.385695i	$0.126038\pi$
$11$			8.48528i			0.771389i	−0.922627	+	0.385695i	$0.873962\pi$
$12$		0			0
$13$		−	15.0000i		−	1.15385i	−0.816798	−	0.576923i	$-0.804253\pi$
$13$		−	15.0000i		−	1.15385i	0.816798	−	0.576923i	$-0.195747\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.7990			1.16465			0.582323	−	0.812957i	$-0.302144\pi$
$17$	19.7990			1.16465			0.582323	+	0.812957i	$0.302144\pi$
$18$		0			0
$19$	−23.0000			−1.21053			−0.605263	−	0.796025i	$-0.706932\pi$
$19$	−23.0000			−1.21053			−0.605263	+	0.796025i	$0.706932\pi$
$20$		0			0
$21$	−1.00000	+	2.82843i	−0.0476190	+	0.134687i
$22$		0			0
$23$	2.82843			0.122975			0.0614875	−	0.998108i	$-0.480416\pi$
$23$	2.82843			0.122975			0.0614875	+	0.998108i	$0.480416\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−14.1421	−	23.0000i	−0.523783	−	0.851852i
$28$		0			0
$29$			25.4558i			0.877788i	0.898539	+	0.438894i	$0.144630\pi$
$29$			25.4558i			0.877788i	−0.898539	+	0.438894i	$0.855370\pi$
$30$		0			0
$31$	−33.0000			−1.06452			−0.532258	−	0.846582i	$-0.678656\pi$
$31$	−33.0000			−1.06452			−0.532258	+	0.846582i	$0.678656\pi$
$32$		0			0
$33$	8.48528	−	24.0000i	0.257130	−	0.727273i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			66.0000i			1.78378i	0.452249	+	0.891892i	$0.350622\pi$
$37$			66.0000i			1.78378i	−0.452249	+	0.891892i	$0.649378\pi$
$38$		0			0
$39$	−15.0000	+	42.4264i	−0.384615	+	1.08786i
$40$		0			0
$41$		−	36.7696i		−	0.896818i	−0.893828	−	0.448409i	$-0.851991\pi$
$41$		−	36.7696i		−	0.896818i	0.893828	−	0.448409i	$-0.148009\pi$
$42$		0			0
$43$		−	7.00000i		−	0.162791i	−0.996682	−	0.0813953i	$-0.974062\pi$
$43$		−	7.00000i		−	0.162791i	0.996682	−	0.0813953i	$-0.0259376\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	45.2548			0.962869			0.481434	−	0.876482i	$-0.340116\pi$
$47$	45.2548			0.962869			0.481434	+	0.876482i	$0.340116\pi$
$48$		0			0
$49$	48.0000			0.979592
$50$		0			0
$51$	−56.0000	−	19.7990i	−1.09804	−	0.388215i
$52$		0			0
$53$	−36.7696			−0.693765			−0.346883	−	0.937909i	$-0.612760\pi$
$53$	−36.7696			−0.693765			−0.346883	+	0.937909i	$0.612760\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	65.0538	+	23.0000i	1.14130	+	0.403509i
$58$		0			0
$59$			101.823i			1.72582i	0.505358	+	0.862910i	$0.331361\pi$
$59$			101.823i			1.72582i	−0.505358	+	0.862910i	$0.668639\pi$
$60$		0			0
$61$	39.0000			0.639344			0.319672	−	0.947528i	$-0.396427\pi$
$61$	39.0000			0.639344			0.319672	+	0.947528i	$0.396427\pi$
$62$		0			0
$63$	5.65685	−	7.00000i	0.0897913	−	0.111111i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	113.000i		−	1.68657i	−0.537469	−	0.843284i	$-0.680619\pi$
$67$		−	113.000i		−	1.68657i	0.537469	−	0.843284i	$-0.319381\pi$
$68$		0			0
$69$	−8.00000	−	2.82843i	−0.115942	−	0.0409917i
$70$		0			0
$71$			25.4558i			0.358533i	0.983801	+	0.179267i	$0.0573724\pi$
$71$			25.4558i			0.358533i	−0.983801	+	0.179267i	$0.942628\pi$
$72$		0			0
$73$		−	58.0000i		−	0.794521i	−0.917706	−	0.397260i	$-0.869961\pi$
$73$		−	58.0000i		−	0.794521i	0.917706	−	0.397260i	$-0.130039\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.48528			0.110198
$78$		0			0
$79$	70.0000			0.886076			0.443038	−	0.896503i	$-0.353901\pi$
$79$	70.0000			0.886076			0.443038	+	0.896503i	$0.353901\pi$
$80$		0			0
$81$	17.0000	+	79.1960i	0.209877	+	0.977728i
$82$		0			0
$83$	152.735			1.84018			0.920091	−	0.391705i	$-0.128115\pi$
$83$	152.735			1.84018			0.920091	+	0.391705i	$0.128115\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	25.4558	−	72.0000i	0.292596	−	0.827586i
$88$		0			0
$89$			90.5097i			1.01696i	0.861073	+	0.508481i	$0.169793\pi$
$89$			90.5097i			1.01696i	−0.861073	+	0.508481i	$0.830207\pi$
$90$		0			0
$91$	−15.0000			−0.164835
$92$		0			0
$93$	93.3381	+	33.0000i	1.00364	+	0.354839i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.00000i		−	0.0103093i	−0.999987	−	0.00515464i	$-0.998359\pi$
$97$		−	1.00000i		−	0.0103093i	0.999987	−	0.00515464i	$-0.00164078\pi$
$98$		0			0
$99$	−48.0000	+	59.3970i	−0.484848	+	0.599969i
$100$		0			0
$101$		−	189.505i		−	1.87628i	−0.346252	−	0.938142i	$-0.612546\pi$
$101$		−	189.505i		−	1.87628i	0.346252	−	0.938142i	$-0.387454\pi$
$102$		0			0
$103$		−	26.0000i		−	0.252427i	−0.992003	−	0.126214i	$-0.959718\pi$
$103$		−	26.0000i		−	0.252427i	0.992003	−	0.126214i	$-0.0402825\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	48.0833			0.449376			0.224688	−	0.974431i	$-0.427864\pi$
$107$	48.0833			0.449376			0.224688	+	0.974431i	$0.427864\pi$
$108$		0			0
$109$	145.000			1.33028			0.665138	−	0.746721i	$-0.268373\pi$
$109$	145.000			1.33028			0.665138	+	0.746721i	$0.268373\pi$
$110$		0			0
$111$	66.0000	−	186.676i	0.594595	−	1.68177i
$112$		0			0
$113$	152.735			1.35164			0.675819	−	0.737068i	$-0.263790\pi$
$113$	152.735			1.35164			0.675819	+	0.737068i	$0.263790\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	84.8528	−	105.000i	0.725238	−	0.897436i
$118$		0			0
$119$		−	19.7990i		−	0.166378i
$120$		0			0
$121$	49.0000			0.404959
$122$		0			0
$123$	−36.7696	+	104.000i	−0.298939	+	0.845528i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			226.000i			1.77953i	0.456421	+	0.889764i	$0.349131\pi$
$127$			226.000i			1.77953i	−0.456421	+	0.889764i	$0.650869\pi$
$128$		0			0
$129$	−7.00000	+	19.7990i	−0.0542636	+	0.153481i
$130$		0			0
$131$			104.652i			0.798869i	0.916762	+	0.399434i	$0.130793\pi$
$131$			104.652i			0.798869i	−0.916762	+	0.399434i	$0.869207\pi$
$132$		0			0
$133$			23.0000i			0.172932i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16.9706			−0.123873			−0.0619364	−	0.998080i	$-0.519728\pi$
$137$	−16.9706			−0.123873			−0.0619364	+	0.998080i	$0.519728\pi$
$138$		0			0
$139$	102.000			0.733813			0.366906	−	0.930258i	$-0.380417\pi$
$139$	102.000			0.733813			0.366906	+	0.930258i	$0.380417\pi$
$140$		0			0
$141$	−128.000	−	45.2548i	−0.907801	−	0.320956i
$142$		0			0
$143$	127.279			0.890064
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−135.765	−	48.0000i	−0.923568	−	0.326531i
$148$		0			0
$149$			288.500i			1.93624i	0.250489	+	0.968119i	$0.419408\pi$
$149$			288.500i			1.93624i	−0.250489	+	0.968119i	$0.580592\pi$
$150$		0			0
$151$	207.000			1.37086			0.685430	−	0.728138i	$-0.259614\pi$
$151$	207.000			1.37086			0.685430	+	0.728138i	$0.259614\pi$
$152$		0			0
$153$	138.593	+	112.000i	0.905836	+	0.732026i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	89.0000i		−	0.566879i	−0.958990	−	0.283439i	$-0.908524\pi$
$157$		−	89.0000i		−	0.566879i	0.958990	−	0.283439i	$-0.0914755\pi$
$158$		0			0
$159$	104.000	+	36.7696i	0.654088	+	0.231255i
$160$		0			0
$161$		−	2.82843i		−	0.0175679i
$162$		0			0
$163$			17.0000i			0.104294i	0.998639	+	0.0521472i	$0.0166065\pi$
$163$			17.0000i			0.104294i	−0.998639	+	0.0521472i	$0.983393\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	268.701			1.60899			0.804493	−	0.593962i	$-0.202437\pi$
$167$	268.701			1.60899			0.804493	+	0.593962i	$0.202437\pi$
$168$		0			0
$169$	−56.0000			−0.331361
$170$		0			0
$171$	−161.000	−	130.108i	−0.941520	−	0.760863i
$172$		0			0
$173$	−110.309			−0.637622			−0.318811	−	0.947818i	$-0.603284\pi$
$173$	−110.309			−0.637622			−0.318811	+	0.947818i	$0.603284\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	101.823	−	288.000i	0.575273	−	1.62712i
$178$		0			0
$179$		−	73.5391i		−	0.410833i	−0.978675	−	0.205416i	$-0.934145\pi$
$179$		−	73.5391i		−	0.410833i	0.978675	−	0.205416i	$-0.0658549\pi$
$180$		0			0
$181$	−145.000			−0.801105			−0.400552	−	0.916274i	$-0.631182\pi$
$181$	−145.000			−0.801105			−0.400552	+	0.916274i	$0.631182\pi$
$182$		0			0
$183$	−110.309	−	39.0000i	−0.602780	−	0.213115i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			168.000i			0.898396i
$188$		0			0
$189$	−23.0000	+	14.1421i	−0.121693	+	0.0748261i
$190$		0			0
$191$		−	53.7401i		−	0.281362i	−0.990055	−	0.140681i	$-0.955071\pi$
$191$		−	53.7401i		−	0.281362i	0.990055	−	0.140681i	$-0.0449292\pi$
$192$		0			0
$193$			57.0000i			0.295337i	0.989037	+	0.147668i	$0.0471768\pi$
$193$			57.0000i			0.295337i	−0.989037	+	0.147668i	$0.952823\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−226.274			−1.14860			−0.574300	−	0.818645i	$-0.694726\pi$
$197$	−226.274			−1.14860			−0.574300	+	0.818645i	$0.694726\pi$
$198$		0			0
$199$	225.000			1.13065			0.565327	−	0.824867i	$-0.308750\pi$
$199$	225.000			1.13065			0.565327	+	0.824867i	$0.308750\pi$
$200$		0			0
$201$	−113.000	+	319.612i	−0.562189	+	1.59011i
$202$		0			0
$203$	25.4558			0.125398
$204$		0			0
$205$		0			0
$206$		0			0
$207$	19.7990	+	16.0000i	0.0956473	+	0.0772947i
$208$		0			0
$209$		−	195.161i		−	0.933787i
$210$		0			0
$211$	231.000			1.09479			0.547393	−	0.836875i	$-0.315620\pi$
$211$	231.000			1.09479			0.547393	+	0.836875i	$0.315620\pi$
$212$		0			0
$213$	25.4558	−	72.0000i	0.119511	−	0.338028i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			33.0000i			0.152074i
$218$		0			0
$219$	−58.0000	+	164.049i	−0.264840	+	0.749081i
$220$		0			0
$221$		−	296.985i		−	1.34382i
$222$		0			0
$223$		−	327.000i		−	1.46637i	−0.680030	−	0.733184i	$-0.738033\pi$
$223$		−	327.000i		−	1.46637i	0.680030	−	0.733184i	$-0.261967\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	76.3675			0.336421			0.168210	−	0.985751i	$-0.446201\pi$
$227$	76.3675			0.336421			0.168210	+	0.985751i	$0.446201\pi$
$228$		0			0
$229$	−183.000			−0.799127			−0.399563	−	0.916706i	$-0.630838\pi$
$229$	−183.000			−0.799127			−0.399563	+	0.916706i	$0.630838\pi$
$230$		0			0
$231$	−24.0000	−	8.48528i	−0.103896	−	0.0367328i
$232$		0			0
$233$	−98.9949			−0.424871			−0.212436	−	0.977175i	$-0.568140\pi$
$233$	−98.9949			−0.424871			−0.212436	+	0.977175i	$0.568140\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−197.990	−	70.0000i	−0.835400	−	0.295359i
$238$		0			0
$239$		−	8.48528i		−	0.0355033i	−0.999842	−	0.0177516i	$-0.994349\pi$
$239$		−	8.48528i		−	0.0355033i	0.999842	−	0.0177516i	$-0.00565082\pi$
$240$		0			0
$241$	215.000			0.892116			0.446058	−	0.895004i	$-0.352827\pi$
$241$	215.000			0.892116			0.446058	+	0.895004i	$0.352827\pi$
$242$		0			0
$243$	31.1127	−	241.000i	0.128036	−	0.991770i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			345.000i			1.39676i
$248$		0			0
$249$	−432.000	−	152.735i	−1.73494	−	0.613394i
$250$		0			0
$251$			164.049i			0.653581i	0.945097	+	0.326790i	$0.105967\pi$
$251$			164.049i			0.653581i	−0.945097	+	0.326790i	$0.894033\pi$
$252$		0			0
$253$			24.0000i			0.0948617i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	152.735			0.594300			0.297150	−	0.954831i	$-0.403964\pi$
$257$	152.735			0.594300			0.297150	+	0.954831i	$0.403964\pi$
$258$		0			0
$259$	66.0000			0.254826
$260$		0			0
$261$	−144.000	+	178.191i	−0.551724	+	0.682724i
$262$		0			0
$263$	−59.3970			−0.225844			−0.112922	−	0.993604i	$-0.536021\pi$
$263$	−59.3970			−0.225844			−0.112922	+	0.993604i	$0.536021\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	90.5097	−	256.000i	0.338988	−	0.958801i
$268$		0			0
$269$			8.48528i			0.0315438i	0.999876	+	0.0157719i	$0.00502056\pi$
$269$			8.48528i			0.0315438i	−0.999876	+	0.0157719i	$0.994979\pi$
$270$		0			0
$271$	−86.0000			−0.317343			−0.158672	−	0.987331i	$-0.550721\pi$
$271$	−86.0000			−0.317343			−0.158672	+	0.987331i	$0.550721\pi$
$272$		0			0
$273$	42.4264	+	15.0000i	0.155408	+	0.0549451i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			319.000i			1.15162i	0.817582	+	0.575812i	$0.195314\pi$
$277$			319.000i			1.15162i	−0.817582	+	0.575812i	$0.804686\pi$
$278$		0			0
$279$	−231.000	−	186.676i	−0.827957	−	0.669090i
$280$		0			0
$281$			427.092i			1.51990i	0.649980	+	0.759951i	$0.274777\pi$
$281$			427.092i			1.51990i	−0.649980	+	0.759951i	$0.725223\pi$
$282$		0			0
$283$			1.00000i			0.00353357i	0.999998	+	0.00176678i	$0.000562385\pi$
$283$			1.00000i			0.00353357i	−0.999998	+	0.00176678i	$0.999438\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−36.7696			−0.128117
$288$		0			0
$289$	103.000			0.356401
$290$		0			0
$291$	−1.00000	+	2.82843i	−0.00343643	+	0.00971968i
$292$		0			0
$293$	−486.489			−1.66037			−0.830187	−	0.557485i	$-0.811766\pi$
$293$	−486.489			−1.66037			−0.830187	+	0.557485i	$0.811766\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	195.161	−	120.000i	0.657109	−	0.404040i
$298$		0			0
$299$		−	42.4264i		−	0.141894i
$300$		0			0
$301$	−7.00000			−0.0232558
$302$		0			0
$303$	−189.505	+	536.000i	−0.625428	+	1.76898i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			215.000i			0.700326i	0.936689	+	0.350163i	$0.113874\pi$
$307$			215.000i			0.700326i	−0.936689	+	0.350163i	$0.886126\pi$
$308$		0			0
$309$	−26.0000	+	73.5391i	−0.0841424	+	0.237991i
$310$		0			0
$311$			214.960i			0.691191i	0.938384	+	0.345596i	$0.112323\pi$
$311$			214.960i			0.691191i	−0.938384	+	0.345596i	$0.887677\pi$
$312$		0			0
$313$		−	535.000i		−	1.70927i	−0.519233	−	0.854633i	$-0.673782\pi$
$313$		−	535.000i		−	1.70927i	0.519233	−	0.854633i	$-0.326218\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−469.519			−1.48113			−0.740566	−	0.671984i	$-0.765443\pi$
$317$	−469.519			−1.48113			−0.740566	+	0.671984i	$0.765443\pi$
$318$		0			0
$319$	−216.000			−0.677116
$320$		0			0
$321$	−136.000	−	48.0833i	−0.423676	−	0.149792i
$322$		0			0
$323$	−455.377			−1.40984
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−410.122	−	145.000i	−1.25420	−	0.443425i
$328$		0			0
$329$		−	45.2548i		−	0.137553i
$330$		0			0
$331$	−94.0000			−0.283988			−0.141994	−	0.989868i	$-0.545351\pi$
$331$	−94.0000			−0.283988			−0.141994	+	0.989868i	$0.545351\pi$
$332$		0			0
$333$	−373.352	+	462.000i	−1.12118	+	1.38739i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			295.000i			0.875371i	0.899128	+	0.437685i	$0.144202\pi$
$337$			295.000i			0.875371i	−0.899128	+	0.437685i	$0.855798\pi$
$338$		0			0
$339$	−432.000	−	152.735i	−1.27434	−	0.450546i
$340$		0			0
$341$		−	280.014i		−	0.821156i
$342$		0			0
$343$		−	97.0000i		−	0.282799i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−441.235			−1.27157			−0.635785	−	0.771866i	$-0.719323\pi$
$347$	−441.235			−1.27157			−0.635785	+	0.771866i	$0.719323\pi$
$348$		0			0
$349$	374.000			1.07163			0.535817	−	0.844334i	$-0.320004\pi$
$349$	374.000			1.07163			0.535817	+	0.844334i	$0.320004\pi$
$350$		0			0
$351$	−345.000	+	212.132i	−0.982906	+	0.604365i
$352$		0			0
$353$	280.014			0.793242			0.396621	−	0.917983i	$-0.370183\pi$
$353$	280.014			0.793242			0.396621	+	0.917983i	$0.370183\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−19.7990	+	56.0000i	−0.0554594	+	0.156863i
$358$		0			0
$359$		−	370.524i		−	1.03210i	−0.856558	−	0.516050i	$-0.827402\pi$
$359$		−	370.524i		−	1.03210i	0.856558	−	0.516050i	$-0.172598\pi$
$360$		0			0
$361$	168.000			0.465374
$362$		0			0
$363$	−138.593	−	49.0000i	−0.381799	−	0.134986i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	241.000i		−	0.656676i	−0.944560	−	0.328338i	$-0.893512\pi$
$367$		−	241.000i		−	0.656676i	0.944560	−	0.328338i	$-0.106488\pi$
$368$		0			0
$369$	208.000	−	257.387i	0.563686	−	0.697525i
$370$		0			0
$371$			36.7696i			0.0991093i
$372$		0			0
$373$		−	47.0000i		−	0.126005i	−0.998013	−	0.0630027i	$-0.979932\pi$
$373$		−	47.0000i		−	0.126005i	0.998013	−	0.0630027i	$-0.0200677\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	381.838			1.01283
$378$		0			0
$379$	73.0000			0.192612			0.0963061	−	0.995352i	$-0.469297\pi$
$379$	73.0000			0.192612			0.0963061	+	0.995352i	$0.469297\pi$
$380$		0			0
$381$	226.000	−	639.225i	0.593176	−	1.67775i
$382$		0			0
$383$	−113.137			−0.295397			−0.147699	−	0.989032i	$-0.547187\pi$
$383$	−113.137			−0.295397			−0.147699	+	0.989032i	$0.547187\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	39.5980	−	49.0000i	0.102320	−	0.126615i
$388$		0			0
$389$		−	534.573i		−	1.37422i	−0.726552	−	0.687111i	$-0.758878\pi$
$389$		−	534.573i		−	1.37422i	0.726552	−	0.687111i	$-0.241122\pi$
$390$		0			0
$391$	56.0000			0.143223
$392$		0			0
$393$	104.652	−	296.000i	0.266290	−	0.753181i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			695.000i			1.75063i	0.483553	+	0.875315i	$0.339346\pi$
$397$			695.000i			1.75063i	−0.483553	+	0.875315i	$0.660654\pi$
$398$		0			0
$399$	23.0000	−	65.0538i	0.0576441	−	0.163042i
$400$		0			0
$401$			721.249i			1.79863i	0.437306	+	0.899313i	$0.355933\pi$
$401$			721.249i			1.79863i	−0.437306	+	0.899313i	$0.644067\pi$
$402$		0			0
$403$			495.000i			1.22829i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−560.029			−1.37599
$408$		0			0
$409$	425.000			1.03912			0.519560	−	0.854434i	$-0.326096\pi$
$409$	425.000			1.03912			0.519560	+	0.854434i	$0.326096\pi$
$410$		0			0
$411$	48.0000	+	16.9706i	0.116788	+	0.0412909i
$412$		0			0
$413$	101.823			0.246546
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−288.500	−	102.000i	−0.691845	−	0.244604i
$418$		0			0
$419$		−	659.024i		−	1.57285i	−0.617687	−	0.786424i	$-0.711930\pi$
$419$		−	659.024i		−	1.57285i	0.617687	−	0.786424i	$-0.288070\pi$
$420$		0			0
$421$	306.000			0.726841			0.363420	−	0.931625i	$-0.381609\pi$
$421$	306.000			0.726841			0.363420	+	0.931625i	$0.381609\pi$
$422$		0			0
$423$	316.784	+	256.000i	0.748898	+	0.605201i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	39.0000i		−	0.0913349i
$428$		0			0
$429$	−360.000	−	127.279i	−0.839161	−	0.296688i
$430$		0			0
$431$			104.652i			0.242812i	0.992603	+	0.121406i	$0.0387402\pi$
$431$			104.652i			0.242812i	−0.992603	+	0.121406i	$0.961260\pi$
$432$		0			0
$433$			281.000i			0.648961i	0.945892	+	0.324480i	$0.105189\pi$
$433$			281.000i			0.648961i	−0.945892	+	0.324480i	$0.894811\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−65.0538			−0.148865
$438$		0			0
$439$	−671.000			−1.52847			−0.764237	−	0.644936i	$-0.776884\pi$
$439$	−671.000			−1.52847			−0.764237	+	0.644936i	$0.776884\pi$
$440$		0			0
$441$	336.000	+	271.529i	0.761905	+	0.615712i
$442$		0			0
$443$	424.264			0.957707			0.478853	−	0.877895i	$-0.341053\pi$
$443$	424.264			0.957707			0.478853	+	0.877895i	$0.341053\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	288.500	−	816.000i	0.645413	−	1.82550i
$448$		0			0
$449$		−	605.283i		−	1.34807i	−0.738699	−	0.674035i	$-0.764560\pi$
$449$		−	605.283i		−	1.34807i	0.738699	−	0.674035i	$-0.235440\pi$
$450$		0			0
$451$	312.000			0.691796
$452$		0			0
$453$	−585.484	−	207.000i	−1.29246	−	0.456954i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			202.000i			0.442013i	0.975272	+	0.221007i	$0.0709342\pi$
$457$			202.000i			0.442013i	−0.975272	+	0.221007i	$0.929066\pi$
$458$		0			0
$459$	−280.000	−	455.377i	−0.610022	−	0.992106i
$460$		0			0
$461$		−	336.583i		−	0.730115i	−0.930985	−	0.365057i	$-0.881049\pi$
$461$		−	336.583i		−	0.730115i	0.930985	−	0.365057i	$-0.118951\pi$
$462$		0			0
$463$			166.000i			0.358531i	0.983801	+	0.179266i	$0.0573722\pi$
$463$			166.000i			0.358531i	−0.983801	+	0.179266i	$0.942628\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−121.622			−0.260433			−0.130217	−	0.991486i	$-0.541567\pi$
$467$	−121.622			−0.260433			−0.130217	+	0.991486i	$0.541567\pi$
$468$		0			0
$469$	−113.000			−0.240938
$470$		0			0
$471$	−89.0000	+	251.730i	−0.188960	+	0.534459i
$472$		0			0
$473$	59.3970			0.125575
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−257.387	−	208.000i	−0.539595	−	0.436059i
$478$		0			0
$479$			797.616i			1.66517i	0.553897	+	0.832585i	$0.313140\pi$
$479$			797.616i			1.66517i	−0.553897	+	0.832585i	$0.686860\pi$
$480$		0			0
$481$	990.000			2.05821
$482$		0			0
$483$	−2.82843	+	8.00000i	−0.00585596	+	0.0165631i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	369.000i		−	0.757700i	−0.925458	−	0.378850i	$-0.876320\pi$
$487$		−	369.000i		−	0.757700i	0.925458	−	0.378850i	$-0.123680\pi$
$488$		0			0
$489$	17.0000	−	48.0833i	0.0347648	−	0.0983298i
$490$		0			0
$491$		−	588.313i		−	1.19819i	−0.800677	−	0.599097i	$-0.795527\pi$
$491$		−	588.313i		−	1.19819i	0.800677	−	0.599097i	$-0.204473\pi$
$492$		0			0
$493$			504.000i			1.02231i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	25.4558			0.0512190
$498$		0			0
$499$	785.000			1.57315			0.786573	−	0.617497i	$-0.211853\pi$
$499$	785.000			1.57315			0.786573	+	0.617497i	$0.211853\pi$
$500$		0			0
$501$	−760.000	−	268.701i	−1.51697	−	0.536328i
$502$		0			0
$503$	−280.014			−0.556688			−0.278344	−	0.960481i	$-0.589786\pi$
$503$	−280.014			−0.556688			−0.278344	+	0.960481i	$0.589786\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	158.392	+	56.0000i	0.312410	+	0.110454i
$508$		0			0
$509$		−	220.617i		−	0.433433i	−0.976235	−	0.216716i	$-0.930465\pi$
$509$		−	220.617i		−	0.433433i	0.976235	−	0.216716i	$-0.0695347\pi$
$510$		0			0
$511$	−58.0000			−0.113503
$512$		0			0
$513$	325.269	+	529.000i	0.634053	+	1.03119i
$514$		0			0
$515$		0			0
$516$		0			0
$517$			384.000i			0.742747i
$518$		0			0
$519$	312.000	+	110.309i	0.601156	+	0.212541i
$520$		0			0
$521$			659.024i			1.26492i	0.774593	+	0.632460i	$0.217955\pi$
$521$			659.024i			1.26492i	−0.774593	+	0.632460i	$0.782045\pi$
$522$		0			0
$523$			129.000i			0.246654i	0.992366	+	0.123327i	$0.0393564\pi$
$523$			129.000i			0.246654i	−0.992366	+	0.123327i	$0.960644\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−653.367			−1.23978
$528$		0			0
$529$	−521.000			−0.984877
$530$		0			0
$531$	−576.000	+	712.764i	−1.08475	+	1.34230i
$532$		0			0
$533$	−551.543			−1.03479
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−73.5391	+	208.000i	−0.136944	+	0.387337i
$538$		0			0
$539$			407.294i			0.755647i
$540$		0			0
$541$	119.000			0.219963			0.109982	−	0.993934i	$-0.464921\pi$
$541$	119.000			0.219963			0.109982	+	0.993934i	$0.464921\pi$
$542$		0			0
$543$	410.122	+	145.000i	0.755289	+	0.267035i
$544$		0			0
$545$		0			0
$546$		0			0
$547$			650.000i			1.18830i	0.804354	+	0.594150i	$0.202511\pi$
$547$			650.000i			1.18830i	−0.804354	+	0.594150i	$0.797489\pi$
$548$		0			0
$549$	273.000	+	220.617i	0.497268	+	0.401853i
$550$		0			0
$551$		−	585.484i		−	1.06259i
$552$		0			0
$553$		−	70.0000i		−	0.126582i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−107.480			−0.192963			−0.0964814	−	0.995335i	$-0.530759\pi$
$557$	−107.480			−0.192963			−0.0964814	+	0.995335i	$0.530759\pi$
$558$		0			0
$559$	−105.000			−0.187835
$560$		0			0
$561$	168.000	−	475.176i	0.299465	−	0.847016i
$562$		0			0
$563$	−789.131			−1.40165			−0.700827	−	0.713331i	$-0.747185\pi$
$563$	−789.131			−1.40165			−0.700827	+	0.713331i	$0.747185\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	79.1960	−	17.0000i	0.139675	−	0.0299824i
$568$		0			0
$569$		−	523.259i		−	0.919612i	−0.888019	−	0.459806i	$-0.847919\pi$
$569$		−	523.259i		−	0.919612i	0.888019	−	0.459806i	$-0.152081\pi$
$570$		0			0
$571$	503.000			0.880911			0.440455	−	0.897775i	$-0.354817\pi$
$571$	503.000			0.880911			0.440455	+	0.897775i	$0.354817\pi$
$572$		0			0
$573$	−53.7401	+	152.000i	−0.0937873	+	0.265271i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	353.000i		−	0.611785i	−0.952066	−	0.305893i	$-0.901045\pi$
$577$		−	353.000i		−	0.611785i	0.952066	−	0.305893i	$-0.0989548\pi$
$578$		0			0
$579$	57.0000	−	161.220i	0.0984456	−	0.278446i
$580$		0			0
$581$		−	152.735i		−	0.262883i
$582$		0			0
$583$		−	312.000i		−	0.535163i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1077.63			1.83583			0.917914	−	0.396780i	$-0.129872\pi$
$587$	1077.63			1.83583			0.917914	+	0.396780i	$0.129872\pi$
$588$		0			0
$589$	759.000			1.28862
$590$		0			0
$591$	640.000	+	226.274i	1.08291	+	0.382867i
$592$		0			0
$593$	726.906			1.22581			0.612905	−	0.790156i	$-0.290001\pi$
$593$	726.906			1.22581			0.612905	+	0.790156i	$0.290001\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−636.396	−	225.000i	−1.06599	−	0.376884i
$598$		0			0
$599$		−	871.156i		−	1.45435i	−0.686452	−	0.727175i	$-0.740833\pi$
$599$		−	871.156i		−	1.45435i	0.686452	−	0.727175i	$-0.259167\pi$
$600$		0			0
$601$	−1033.00			−1.71880			−0.859401	−	0.511302i	$-0.829163\pi$
$601$	−1033.00			−1.71880			−0.859401	+	0.511302i	$0.829163\pi$
$602$		0			0
$603$	639.225	−	791.000i	1.06007	−	1.31177i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			330.000i			0.543657i	0.962346	+	0.271829i	$0.0876284\pi$
$607$			330.000i			0.543657i	−0.962346	+	0.271829i	$0.912372\pi$
$608$		0			0
$609$	−72.0000	−	25.4558i	−0.118227	−	0.0417994i
$610$		0			0
$611$		−	678.823i		−	1.11100i
$612$		0			0
$613$		−	770.000i		−	1.25612i	−0.778166	−	0.628059i	$-0.783850\pi$
$613$		−	770.000i		−	1.25612i	0.778166	−	0.628059i	$-0.216150\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	630.739			1.02227			0.511134	−	0.859501i	$-0.329226\pi$
$617$	630.739			1.02227			0.511134	+	0.859501i	$0.329226\pi$
$618$		0			0
$619$	401.000			0.647819			0.323910	−	0.946088i	$-0.395003\pi$
$619$	401.000			0.647819			0.323910	+	0.946088i	$0.395003\pi$
$620$		0			0
$621$	−40.0000	−	65.0538i	−0.0644122	−	0.104757i
$622$		0			0
$623$	90.5097			0.145280
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−195.161	+	552.000i	−0.311262	+	0.880383i
$628$		0			0
$629$			1306.73i			2.07748i
$630$		0			0
$631$	−889.000			−1.40887			−0.704437	−	0.709766i	$-0.748801\pi$
$631$	−889.000			−1.40887			−0.704437	+	0.709766i	$0.748801\pi$
$632$		0			0
$633$	−653.367	−	231.000i	−1.03217	−	0.364929i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	720.000i		−	1.13030i
$638$		0			0
$639$	−144.000	+	178.191i	−0.225352	+	0.278859i
$640$		0			0
$641$			678.823i			1.05901i	0.848308	+	0.529503i	$0.177621\pi$
$641$			678.823i			1.05901i	−0.848308	+	0.529503i	$0.822379\pi$
$642$		0			0
$643$		−	42.0000i		−	0.0653188i	−0.999467	−	0.0326594i	$-0.989602\pi$
$643$		−	42.0000i		−	0.0653188i	0.999467	−	0.0326594i	$-0.0103977\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−458.205			−0.708200			−0.354100	−	0.935208i	$-0.615213\pi$
$647$	−458.205			−0.708200			−0.354100	+	0.935208i	$0.615213\pi$
$648$		0			0
$649$	−864.000			−1.33128
$650$		0			0
$651$	33.0000	−	93.3381i	0.0506912	−	0.143376i
$652$		0			0
$653$	1001.26			1.53333			0.766664	−	0.642049i	$-0.221915\pi$
$653$	1001.26			1.53333			0.766664	+	0.642049i	$0.221915\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	328.098	−	406.000i	0.499387	−	0.617960i
$658$		0			0
$659$			376.181i			0.570836i	0.958403	+	0.285418i	$0.0921324\pi$
$659$			376.181i			0.570836i	−0.958403	+	0.285418i	$0.907868\pi$
$660$		0			0
$661$	−974.000			−1.47352			−0.736762	−	0.676152i	$-0.763646\pi$
$661$	−974.000			−1.47352			−0.736762	+	0.676152i	$0.763646\pi$
$662$		0			0
$663$	−296.985	+	840.000i	−0.447941	+	1.26697i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			72.0000i			0.107946i
$668$		0			0
$669$	−327.000	+	924.896i	−0.488789	+	1.38250i
$670$		0			0
$671$			330.926i			0.493183i
$672$		0			0
$673$		−	26.0000i		−	0.0386330i	−0.999813	−	0.0193165i	$-0.993851\pi$
$673$		−	26.0000i		−	0.0386330i	0.999813	−	0.0193165i	$-0.00614901\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	93.3381			0.137870			0.0689351	−	0.997621i	$-0.478040\pi$
$677$	93.3381			0.137870			0.0689351	+	0.997621i	$0.478040\pi$
$678$		0			0
$679$	−1.00000			−0.00147275
$680$		0			0
$681$	−216.000	−	76.3675i	−0.317181	−	0.112140i
$682$		0			0
$683$	−16.9706			−0.0248471			−0.0124235	−	0.999923i	$-0.503955\pi$
$683$	−16.9706			−0.0248471			−0.0124235	+	0.999923i	$0.503955\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	517.602	+	183.000i	0.753424	+	0.266376i
$688$		0			0
$689$			551.543i			0.800498i
$690$		0			0
$691$	722.000			1.04486			0.522431	−	0.852681i	$-0.325025\pi$
$691$	722.000			1.04486			0.522431	+	0.852681i	$0.325025\pi$
$692$		0			0
$693$	59.3970	+	48.0000i	0.0857099	+	0.0692641i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	728.000i		−	1.04448i
$698$		0			0
$699$	280.000	+	98.9949i	0.400572	+	0.141624i
$700$		0			0
$701$			65.0538i			0.0928015i	0.998923	+	0.0464007i	$0.0147751\pi$
$701$			65.0538i			0.0928015i	−0.998923	+	0.0464007i	$0.985225\pi$
$702$		0			0
$703$		−	1518.00i		−	2.15932i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−189.505			−0.268040
$708$		0			0
$709$	−103.000			−0.145275			−0.0726375	−	0.997358i	$-0.523142\pi$
$709$	−103.000			−0.145275			−0.0726375	+	0.997358i	$0.523142\pi$
$710$		0			0
$711$	490.000	+	395.980i	0.689170	+	0.556934i
$712$		0			0
$713$	−93.3381			−0.130909
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−8.48528	+	24.0000i	−0.0118344	+	0.0334728i
$718$		0			0
$719$			670.337i			0.932319i	0.884701	+	0.466159i	$0.154363\pi$
$719$			670.337i			0.932319i	−0.884701	+	0.466159i	$0.845637\pi$
$720$		0			0
$721$	−26.0000			−0.0360610
$722$		0			0
$723$	−608.112	−	215.000i	−0.841095	−	0.297372i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			895.000i			1.23109i	0.788103	+	0.615543i	$0.211063\pi$
$727$			895.000i			1.23109i	−0.788103	+	0.615543i	$0.788937\pi$
$728$		0			0
$729$	−329.000	+	650.538i	−0.451303	+	0.892371i
$730$		0			0
$731$		−	138.593i		−	0.189594i
$732$		0			0
$733$			406.000i			0.553888i	0.960886	+	0.276944i	$0.0893217\pi$
$733$			406.000i			0.553888i	−0.960886	+	0.276944i	$0.910678\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	958.837			1.30100
$738$		0			0
$739$	−578.000			−0.782138			−0.391069	−	0.920361i	$-0.627895\pi$
$739$	−578.000			−0.782138			−0.391069	+	0.920361i	$0.627895\pi$
$740$		0			0
$741$	345.000	−	975.807i	0.465587	−	1.31688i
$742$		0			0
$743$	−492.146			−0.662377			−0.331189	−	0.943565i	$-0.607450\pi$
$743$	−492.146			−0.662377			−0.331189	+	0.943565i	$0.607450\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1069.15	+	864.000i	1.43125	+	1.15663i
$748$		0			0
$749$		−	48.0833i		−	0.0641966i
$750$		0			0
$751$	−30.0000			−0.0399467			−0.0199734	−	0.999801i	$-0.506358\pi$
$751$	−30.0000			−0.0399467			−0.0199734	+	0.999801i	$0.506358\pi$
$752$		0			0
$753$	164.049	−	464.000i	0.217860	−	0.616202i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	897.000i		−	1.18494i	−0.805592	−	0.592470i	$-0.798153\pi$
$757$		−	897.000i		−	1.18494i	0.805592	−	0.592470i	$-0.201847\pi$
$758$		0			0
$759$	24.0000	−	67.8823i	0.0316206	−	0.0894364i
$760$		0			0
$761$			164.049i			0.215570i	0.994174	+	0.107785i	$0.0343758\pi$
$761$			164.049i			0.215570i	−0.994174	+	0.107785i	$0.965624\pi$
$762$		0			0
$763$		−	145.000i		−	0.190039i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1527.35			1.99133
$768$		0			0
$769$	465.000			0.604681			0.302341	−	0.953200i	$-0.402232\pi$
$769$	465.000			0.604681			0.302341	+	0.953200i	$0.402232\pi$
$770$		0			0
$771$	−432.000	−	152.735i	−0.560311	−	0.198100i
$772$		0			0
$773$	−172.534			−0.223201			−0.111600	−	0.993753i	$-0.535598\pi$
$773$	−172.534			−0.223201			−0.111600	+	0.993753i	$0.535598\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−186.676	−	66.0000i	−0.240252	−	0.0849421i
$778$		0			0
$779$			845.700i			1.08562i
$780$		0			0
$781$	−216.000			−0.276569
$782$		0			0
$783$	585.484	−	360.000i	0.747745	−	0.459770i
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	801.000i		−	1.01779i	−0.860829	−	0.508895i	$-0.830054\pi$
$787$		−	801.000i		−	1.01779i	0.860829	−	0.508895i	$-0.169946\pi$
$788$		0			0
$789$	168.000	+	59.3970i	0.212928	+	0.0752813i
$790$		0			0
$791$		−	152.735i		−	0.193091i
$792$		0			0
$793$		−	585.000i		−	0.737705i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	197.990			0.248419			0.124209	−	0.992256i	$-0.460361\pi$
$797$	197.990			0.248419			0.124209	+	0.992256i	$0.460361\pi$
$798$		0			0
$799$	896.000			1.12140
$800$		0			0
$801$	−512.000	+	633.568i	−0.639201	+	0.790971i
$802$		0			0
$803$	492.146			0.612885
$804$		0			0
$805$		0			0
$806$		0			0
$807$	8.48528	−	24.0000i	0.0105146	−	0.0297398i
$808$		0			0
$809$			670.337i			0.828600i	0.910140	+	0.414300i	$0.135974\pi$
$809$			670.337i			0.828600i	−0.910140	+	0.414300i	$0.864026\pi$
$810$		0			0
$811$	879.000			1.08385			0.541924	−	0.840428i	$-0.317696\pi$
$811$	879.000			1.08385			0.541924	+	0.840428i	$0.317696\pi$
$812$		0			0
$813$	243.245	+	86.0000i	0.299194	+	0.105781i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			161.000i			0.197062i
$818$		0			0
$819$	−105.000	−	84.8528i	−0.128205	−	0.103605i
$820$		0			0
$821$			842.871i			1.02664i	0.858197	+	0.513320i	$0.171585\pi$
$821$			842.871i			1.02664i	−0.858197	+	0.513320i	$0.828415\pi$
$822$		0			0
$823$			521.000i			0.633050i	0.948584	+	0.316525i	$0.102516\pi$
$823$			521.000i			0.633050i	−0.948584	+	0.316525i	$0.897484\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−172.534			−0.208626			−0.104313	−	0.994544i	$-0.533264\pi$
$827$	−172.534			−0.208626			−0.104313	+	0.994544i	$0.533264\pi$
$828$		0			0
$829$	−818.000			−0.986731			−0.493366	−	0.869822i	$-0.664233\pi$
$829$	−818.000			−0.986731			−0.493366	+	0.869822i	$0.664233\pi$
$830$		0			0
$831$	319.000	−	902.268i	0.383875	−	1.08576i
$832$		0			0
$833$	950.352			1.14088
$834$		0			0
$835$		0			0
$836$		0			0
$837$	466.690	+	759.000i	0.557575	+	0.906810i
$838$		0			0
$839$			1309.56i			1.56086i	0.625243	+	0.780430i	$0.285000\pi$
$839$			1309.56i			1.56086i	−0.625243	+	0.780430i	$0.715000\pi$
$840$		0			0
$841$	193.000			0.229489
$842$		0			0
$843$	427.092	−	1208.00i	0.506634	−	1.43298i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	49.0000i		−	0.0578512i
$848$		0			0
$849$	1.00000	−	2.82843i	0.00117786	−	0.00333148i
$850$		0			0
$851$			186.676i			0.219361i
$852$		0			0
$853$			233.000i			0.273154i	0.990629	+	0.136577i	$0.0436100\pi$
$853$			233.000i			0.273154i	−0.990629	+	0.136577i	$0.956390\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−808.930			−0.943909			−0.471955	−	0.881623i	$-0.656451\pi$
$857$	−808.930			−0.943909			−0.471955	+	0.881623i	$0.656451\pi$
$858$		0			0
$859$	1230.00			1.43190			0.715949	−	0.698153i	$-0.245994\pi$
$859$	1230.00			1.43190			0.715949	+	0.698153i	$0.245994\pi$
$860$		0			0
$861$	104.000	+	36.7696i	0.120790	+	0.0427056i
$862$		0			0
$863$	1284.11			1.48796			0.743978	−	0.668204i	$-0.232937\pi$
$863$	1284.11			1.48796			0.743978	+	0.668204i	$0.232937\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−291.328	−	103.000i	−0.336018	−	0.118800i
$868$		0			0
$869$			593.970i			0.683509i
$870$		0			0
$871$	−1695.00			−1.94604
$872$		0			0
$873$	5.65685	−	7.00000i	0.00647979	−	0.00801833i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	113.000i		−	0.128848i	−0.997923	−	0.0644242i	$-0.979479\pi$
$877$		−	113.000i		−	0.128848i	0.997923	−	0.0644242i	$-0.0205211\pi$
$878$		0			0
$879$	1376.00	+	486.489i	1.56542	+	0.553458i
$880$		0			0
$881$		−	1295.42i		−	1.47040i	−0.677852	−	0.735198i	$-0.737089\pi$
$881$		−	1295.42i		−	1.47040i	0.677852	−	0.735198i	$-0.262911\pi$
$882$		0			0
$883$			793.000i			0.898075i	0.893513	+	0.449037i	$0.148233\pi$
$883$			793.000i			0.898075i	−0.893513	+	0.449037i	$0.851767\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	67.8823			0.0765302			0.0382651	−	0.999268i	$-0.487817\pi$
$887$	67.8823			0.0765302			0.0382651	+	0.999268i	$0.487817\pi$
$888$		0			0
$889$	226.000			0.254218
$890$		0			0
$891$	−672.000	+	144.250i	−0.754209	+	0.161897i
$892$		0			0
$893$	−1040.86			−1.16558
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−42.4264	+	120.000i	−0.0472981	+	0.133779i
$898$		0			0
$899$		−	840.043i		−	0.934419i
$900$		0			0
$901$	−728.000			−0.807991
$902$		0			0
$903$	19.7990	+	7.00000i	0.0219258	+	0.00775194i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	1678.00i		−	1.85006i	−0.379900	−	0.925028i	$-0.624042\pi$
$907$		−	1678.00i		−	1.85006i	0.379900	−	0.925028i	$-0.375958\pi$
$908$		0			0
$909$	1072.00	−	1326.53i	1.17932	−	1.45933i
$910$		0			0
$911$		−	1702.71i		−	1.86906i	−0.355885	−	0.934530i	$-0.615821\pi$
$911$		−	1702.71i		−	1.86906i	0.355885	−	0.934530i	$-0.384179\pi$
$912$		0			0
$913$			1296.00i			1.41950i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	104.652			0.114124
$918$		0			0
$919$	−1079.00			−1.17410			−0.587051	−	0.809550i	$-0.699711\pi$
$919$	−1079.00			−1.17410			−0.587051	+	0.809550i	$0.699711\pi$
$920$		0			0
$921$	215.000	−	608.112i	0.233442	−	0.660273i
$922$		0			0
$923$	381.838			0.413692
$924$		0			0
$925$		0			0
$926$		0			0
$927$	147.078	−	182.000i	0.158660	−	0.196332i
$928$		0			0
$929$		−	144.250i		−	0.155274i	−0.996982	−	0.0776371i	$-0.975262\pi$
$929$		−	144.250i		−	0.155274i	0.996982	−	0.0776371i	$-0.0247376\pi$
$930$		0			0
$931$	−1104.00			−1.18582
$932$		0			0
$933$	214.960	−	608.000i	0.230397	−	0.651661i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	73.0000i		−	0.0779082i	−0.999241	−	0.0389541i	$-0.987597\pi$
$937$		−	73.0000i		−	0.0779082i	0.999241	−	0.0389541i	$-0.0124026\pi$
$938$		0			0
$939$	−535.000	+	1513.21i	−0.569755	+	1.61151i
$940$		0			0
$941$		−	1368.96i		−	1.45479i	−0.686218	−	0.727396i	$-0.740731\pi$
$941$		−	1368.96i		−	1.45479i	0.686218	−	0.727396i	$-0.259269\pi$
$942$		0			0
$943$		−	104.000i		−	0.110286i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−596.798			−0.630199			−0.315099	−	0.949059i	$-0.602038\pi$
$947$	−596.798			−0.630199			−0.315099	+	0.949059i	$0.602038\pi$
$948$		0			0
$949$	−870.000			−0.916754
$950$		0			0
$951$	1328.00	+	469.519i	1.39642	+	0.493711i
$952$		0			0
$953$	763.675			0.801338			0.400669	−	0.916223i	$-0.368778\pi$
$953$	763.675			0.801338			0.400669	+	0.916223i	$0.368778\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	610.940	+	216.000i	0.638391	+	0.225705i
$958$		0			0
$959$			16.9706i			0.0176961i
$960$		0			0
$961$	128.000			0.133195
$962$		0			0
$963$	336.583	+	272.000i	0.349515	+	0.282451i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			10.0000i			0.0103413i	0.999987	+	0.00517063i	$0.00164587\pi$
$967$			10.0000i			0.0103413i	−0.999987	+	0.00517063i	$0.998354\pi$
$968$		0			0
$969$	1288.00	+	455.377i	1.32921	+	0.469945i
$970$		0			0
$971$			486.489i			0.501019i	0.968114	+	0.250510i	$0.0805981\pi$
$971$			486.489i			0.501019i	−0.968114	+	0.250510i	$0.919402\pi$
$972$		0			0
$973$		−	102.000i		−	0.104830i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−820.244			−0.839554			−0.419777	−	0.907627i	$-0.637892\pi$
$977$	−820.244			−0.839554			−0.419777	+	0.907627i	$0.637892\pi$
$978$		0			0
$979$	−768.000			−0.784474
$980$		0			0
$981$	1015.00	+	820.244i	1.03466	+	0.836130i
$982$		0			0
$983$	−135.765			−0.138112			−0.0690562	−	0.997613i	$-0.521999\pi$
$983$	−135.765			−0.138112			−0.0690562	+	0.997613i	$0.521999\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−45.2548	+	128.000i	−0.0458509	+	0.129686i
$988$		0			0
$989$		−	19.7990i		−	0.0200192i
$990$		0			0
$991$	79.0000			0.0797175			0.0398587	−	0.999205i	$-0.487309\pi$
$991$	79.0000			0.0797175			0.0398587	+	0.999205i	$0.487309\pi$
$992$		0			0
$993$	265.872	+	94.0000i	0.267746	+	0.0946626i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			378.000i			0.379137i	0.981867	+	0.189569i	$0.0607090\pi$
$997$			378.000i			0.379137i	−0.981867	+	0.189569i	$0.939291\pi$
$998$		0			0
$999$	1518.00	−	933.381i	1.51952	−	0.934315i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.c.g.449.1		4
3.2	odd	2	inner	1200.3.c.g.449.3		4
4.3	odd	2		600.3.c.b.449.4		4
5.2	odd	4		1200.3.l.j.401.2		2
5.3	odd	4		1200.3.l.o.401.1		2
5.4	even	2	inner	1200.3.c.g.449.4		4
12.11	even	2		600.3.c.b.449.2		4
15.2	even	4		1200.3.l.j.401.1		2
15.8	even	4		1200.3.l.o.401.2		2
15.14	odd	2	inner	1200.3.c.g.449.2		4
20.3	even	4		600.3.l.a.401.2	yes	2
20.7	even	4		600.3.l.c.401.1	yes	2
20.19	odd	2		600.3.c.b.449.1		4
60.23	odd	4		600.3.l.a.401.1	✓	2
60.47	odd	4		600.3.l.c.401.2	yes	2
60.59	even	2		600.3.c.b.449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.c.b.449.1		4	20.19	odd	2
600.3.c.b.449.2		4	12.11	even	2
600.3.c.b.449.3		4	60.59	even	2
600.3.c.b.449.4		4	4.3	odd	2
600.3.l.a.401.1	✓	2	60.23	odd	4
600.3.l.a.401.2	yes	2	20.3	even	4
600.3.l.c.401.1	yes	2	20.7	even	4
600.3.l.c.401.2	yes	2	60.47	odd	4
1200.3.c.g.449.1		4	1.1	even	1	trivial
1200.3.c.g.449.2		4	15.14	odd	2	inner
1200.3.c.g.449.3		4	3.2	odd	2	inner
1200.3.c.g.449.4		4	5.4	even	2	inner
1200.3.l.j.401.1		2	15.2	even	4
1200.3.l.j.401.2		2	5.2	odd	4
1200.3.l.o.401.1		2	5.3	odd	4
1200.3.l.o.401.2		2	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-2.82843 - 1.00000i) q^{3} -1.00000i q^{7} +(7.00000 + 5.65685i) q^{9} +O(q^{10})\)	\(q+(-2.82843 - 1.00000i) q^{3} -1.00000i q^{7} +(7.00000 + 5.65685i) q^{9} +8.48528i q^{11} -15.0000i q^{13} +19.7990 q^{17} -23.0000 q^{19} +(-1.00000 + 2.82843i) q^{21} +2.82843 q^{23} +(-14.1421 - 23.0000i) q^{27} +25.4558i q^{29} -33.0000 q^{31} +(8.48528 - 24.0000i) q^{33} +66.0000i q^{37} +(-15.0000 + 42.4264i) q^{39} -36.7696i q^{41} -7.00000i q^{43} +45.2548 q^{47} +48.0000 q^{49} +(-56.0000 - 19.7990i) q^{51} -36.7696 q^{53} +(65.0538 + 23.0000i) q^{57} +101.823i q^{59} +39.0000 q^{61} +(5.65685 - 7.00000i) q^{63} -113.000i q^{67} +(-8.00000 - 2.82843i) q^{69} +25.4558i q^{71} -58.0000i q^{73} +8.48528 q^{77} +70.0000 q^{79} +(17.0000 + 79.1960i) q^{81} +152.735 q^{83} +(25.4558 - 72.0000i) q^{87} +90.5097i q^{89} -15.0000 q^{91} +(93.3381 + 33.0000i) q^{93} -1.00000i q^{97} +(-48.0000 + 59.3970i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{9}+O(q^{10}) \) 4 * q + 28 * q^9	\( 4 q + 28 q^{9} - 92 q^{19} - 4 q^{21} - 132 q^{31} - 60 q^{39} + 192 q^{49} - 224 q^{51} + 156 q^{61} - 32 q^{69} + 280 q^{79} + 68 q^{81} - 60 q^{91} - 192 q^{99}+O(q^{100}) \) 4 * q + 28 * q^9 - 92 * q^19 - 4 * q^21 - 132 * q^31 - 60 * q^39 + 192 * q^49 - 224 * q^51 + 156 * q^61 - 32 * q^69 + 280 * q^79 + 68 * q^81 - 60 * q^91 - 192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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