LMFDB - Embedded newform 1200.3.l.u.401.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(401,1200)]

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, 0]))

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

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

chi := DirichletCharacter("1200.401");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.l (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:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.2
Root			$-1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1200.401
Dual form			1200.3.l.u.401.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.581139 + 2.94317i) q^{3} +11.4868 q^{7} +(-8.32456 - 3.42079i) q^{9} +O(q^{10})$	$q+(-0.581139 + 2.94317i) q^{3} +11.4868 q^{7} +(-8.32456 - 3.42079i) q^{9} +8.48528i q^{11} +10.0000 q^{13} -3.55415i q^{17} +10.9737 q^{19} +(-6.67544 + 33.8078i) q^{21} +17.6590i q^{23} +(14.9057 - 22.5127i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(-24.9737 - 4.93113i) q^{33} +59.9473 q^{37} +(-5.81139 + 29.4317i) q^{39} -20.5247i q^{41} +42.4605 q^{43} +88.2952i q^{47} +82.9473 q^{49} +(10.4605 + 2.06546i) q^{51} -3.55415i q^{53} +(-6.37722 + 32.2974i) q^{57} -77.7445i q^{59} +21.9473 q^{61} +(-95.6228 - 39.2940i) q^{63} -53.5395 q^{67} +(-51.9737 - 10.2624i) q^{69} +69.2592i q^{71} -12.0527 q^{73} +97.4690i q^{77} +9.02633 q^{79} +(57.5964 + 56.9530i) q^{81} +0.688486i q^{83} +(78.9737 + 15.5936i) q^{87} -7.10831i q^{89} +114.868 q^{91} +(4.64911 - 23.5454i) q^{93} -111.947 q^{97} +(29.0263 - 70.6362i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 8 q^{7} - 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 8 * q^7 - 8 * q^9	$ 4 q + 4 q^{3} + 8 q^{7} - 8 q^{9} + 40 q^{13} - 32 q^{19} - 52 q^{21} + 28 q^{27} - 32 q^{31} - 24 q^{33} + 88 q^{37} + 40 q^{39} + 56 q^{43} + 180 q^{49} - 72 q^{51} - 152 q^{57} - 64 q^{61} - 256 q^{63} - 328 q^{67} - 132 q^{69} - 200 q^{73} + 112 q^{79} + 28 q^{81} + 240 q^{87} + 80 q^{91} - 32 q^{93} - 296 q^{97} + 192 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 8 * q^7 - 8 * q^9 + 40 * q^13 - 32 * q^19 - 52 * q^21 + 28 * q^27 - 32 * q^31 - 24 * q^33 + 88 * q^37 + 40 * q^39 + 56 * q^43 + 180 * q^49 - 72 * q^51 - 152 * q^57 - 64 * q^61 - 256 * q^63 - 328 * q^67 - 132 * q^69 - 200 * q^73 + 112 * q^79 + 28 * q^81 + 240 * q^87 + 80 * q^91 - 32 * q^93 - 296 * q^97 + 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
$3$	−0.581139	+	2.94317i	−0.193713	+	0.981058i
$3$	−0.581139	+	2.94317i	−0.193713	+	0.981058i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	11.4868			1.64098			0.820488	−	0.571664i	$-0.193702\pi$
$7$	11.4868			1.64098			0.820488	+	0.571664i	$0.193702\pi$
$8$		0			0
$9$	−8.32456	−	3.42079i	−0.924951	−	0.380087i
$10$		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
$13$	10.0000			0.769231			0.384615	−	0.923077i	$-0.374334\pi$
$13$	10.0000			0.769231			0.384615	+	0.923077i	$0.374334\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.55415i		−	0.209068i	−0.994521	−	0.104534i	$-0.966665\pi$
$17$		−	3.55415i		−	0.209068i	0.994521	−	0.104534i	$-0.0333351\pi$
$18$		0			0
$19$	10.9737			0.577561			0.288781	−	0.957395i	$-0.406750\pi$
$19$	10.9737			0.577561			0.288781	+	0.957395i	$0.406750\pi$
$20$		0			0
$21$	−6.67544	+	33.8078i	−0.317878	+	1.60989i
$22$		0			0
$23$			17.6590i			0.767785i	0.923378	+	0.383892i	$0.125417\pi$
$23$			17.6590i			0.767785i	−0.923378	+	0.383892i	$0.874583\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	14.9057	−	22.5127i	0.552063	−	0.833803i
$28$		0			0
$29$		−	26.8328i		−	0.925270i	−0.886549	−	0.462635i	$-0.846904\pi$
$29$		−	26.8328i		−	0.925270i	0.886549	−	0.462635i	$-0.153096\pi$
$30$		0			0
$31$	−8.00000			−0.258065			−0.129032	−	0.991640i	$-0.541187\pi$
$31$	−8.00000			−0.258065			−0.129032	+	0.991640i	$0.541187\pi$
$32$		0			0
$33$	−24.9737	−	4.93113i	−0.756778	−	0.149428i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	59.9473			1.62020			0.810099	−	0.586293i	$-0.199413\pi$
$37$	59.9473			1.62020			0.810099	+	0.586293i	$0.199413\pi$
$38$		0			0
$39$	−5.81139	+	29.4317i	−0.149010	+	0.754660i
$40$		0			0
$41$		−	20.5247i		−	0.500603i	−0.968168	−	0.250301i	$-0.919470\pi$
$41$		−	20.5247i		−	0.500603i	0.968168	−	0.250301i	$-0.0805297\pi$
$42$		0			0
$43$	42.4605			0.987453			0.493727	−	0.869617i	$-0.335634\pi$
$43$	42.4605			0.987453			0.493727	+	0.869617i	$0.335634\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			88.2952i			1.87862i	0.343067	+	0.939311i	$0.388534\pi$
$47$			88.2952i			1.87862i	−0.343067	+	0.939311i	$0.611466\pi$
$48$		0			0
$49$	82.9473			1.69280
$50$		0			0
$51$	10.4605	+	2.06546i	0.205108	+	0.0404992i
$52$		0			0
$53$		−	3.55415i		−	0.0670595i	−0.999438	−	0.0335298i	$-0.989325\pi$
$53$		−	3.55415i		−	0.0670595i	0.999438	−	0.0335298i	$-0.0106749\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.37722	+	32.2974i	−0.111881	+	0.566621i
$58$		0			0
$59$		−	77.7445i		−	1.31770i	−0.752273	−	0.658852i	$-0.771042\pi$
$59$		−	77.7445i		−	1.31770i	0.752273	−	0.658852i	$-0.228958\pi$
$60$		0			0
$61$	21.9473			0.359792			0.179896	−	0.983686i	$-0.442424\pi$
$61$	21.9473			0.359792			0.179896	+	0.983686i	$0.442424\pi$
$62$		0			0
$63$	−95.6228	−	39.2940i	−1.51782	−	0.623714i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−53.5395			−0.799097			−0.399549	−	0.916712i	$-0.630833\pi$
$67$	−53.5395			−0.799097			−0.399549	+	0.916712i	$0.630833\pi$
$68$		0			0
$69$	−51.9737	−	10.2624i	−0.753242	−	0.148730i
$70$		0			0
$71$			69.2592i			0.975482i	0.872988	+	0.487741i	$0.162179\pi$
$71$			69.2592i			0.975482i	−0.872988	+	0.487741i	$0.837821\pi$
$72$		0			0
$73$	−12.0527			−0.165105			−0.0825525	−	0.996587i	$-0.526307\pi$
$73$	−12.0527			−0.165105			−0.0825525	+	0.996587i	$0.526307\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			97.4690i			1.26583i
$78$		0			0
$79$	9.02633			0.114257			0.0571287	−	0.998367i	$-0.481805\pi$
$79$	9.02633			0.114257			0.0571287	+	0.998367i	$0.481805\pi$
$80$		0			0
$81$	57.5964	+	56.9530i	0.711067	+	0.703124i
$82$		0			0
$83$			0.688486i			0.00829501i	0.999991	+	0.00414750i	$0.00132020\pi$
$83$			0.688486i			0.00829501i	−0.999991	+	0.00414750i	$0.998680\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	78.9737	+	15.5936i	0.907743	+	0.179237i
$88$		0			0
$89$		−	7.10831i		−	0.0798686i	−0.999202	−	0.0399343i	$-0.987285\pi$
$89$		−	7.10831i		−	0.0798686i	0.999202	−	0.0399343i	$-0.0127149\pi$
$90$		0			0
$91$	114.868			1.26229
$92$		0			0
$93$	4.64911	−	23.5454i	0.0499904	−	0.253176i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−111.947			−1.15410			−0.577048	−	0.816710i	$-0.695795\pi$
$97$	−111.947			−1.15410			−0.577048	+	0.816710i	$0.695795\pi$
$98$		0			0
$99$	29.0263	−	70.6362i	0.293195	−	0.713497i
$100$		0			0
$101$			155.489i			1.53950i	0.638348	+	0.769748i	$0.279618\pi$
$101$			155.489i			1.53950i	−0.638348	+	0.769748i	$0.720382\pi$
$102$		0			0
$103$	7.59217			0.0737104			0.0368552	−	0.999321i	$-0.488266\pi$
$103$	7.59217			0.0737104			0.0368552	+	0.999321i	$0.488266\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			16.2821i			0.152169i	0.997101	+	0.0760845i	$0.0242419\pi$
$107$			16.2821i			0.152169i	−0.997101	+	0.0760845i	$0.975758\pi$
$108$		0			0
$109$	−93.8420			−0.860936			−0.430468	−	0.902606i	$-0.641651\pi$
$109$	−93.8420			−0.860936			−0.430468	+	0.902606i	$0.641651\pi$
$110$		0			0
$111$	−34.8377	+	176.435i	−0.313853	+	1.58951i
$112$		0			0
$113$			195.738i			1.73220i	0.499874	+	0.866098i	$0.333380\pi$
$113$			195.738i			1.73220i	−0.499874	+	0.866098i	$0.666620\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−83.2456	−	34.2079i	−0.711500	−	0.292375i
$118$		0			0
$119$		−	40.8260i		−	0.343075i
$120$		0			0
$121$	49.0000			0.404959
$122$		0			0
$123$	60.4078	+	11.9277i	0.491121	+	0.0969733i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−77.5395			−0.610547			−0.305274	−	0.952265i	$-0.598748\pi$
$127$	−77.5395			−0.610547			−0.305274	+	0.952265i	$0.598748\pi$
$128$		0			0
$129$	−24.6754	+	124.969i	−0.191283	+	0.968749i
$130$		0			0
$131$			101.600i			0.775572i	0.921749	+	0.387786i	$0.126760\pi$
$131$			101.600i			0.775572i	−0.921749	+	0.387786i	$0.873240\pi$
$132$		0			0
$133$	126.053			0.947764
$134$		0			0
$135$		0			0
$136$		0			0
$137$			176.014i			1.28477i	0.766381	+	0.642386i	$0.222055\pi$
$137$			176.014i			1.28477i	−0.766381	+	0.642386i	$0.777945\pi$
$138$		0			0
$139$	−188.816			−1.35839			−0.679193	−	0.733960i	$-0.737670\pi$
$139$	−188.816			−1.35839			−0.679193	+	0.733960i	$0.737670\pi$
$140$		0			0
$141$	−259.868	−	51.3118i	−1.84304	−	0.363913i
$142$		0			0
$143$			84.8528i			0.593376i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−48.2039	+	244.128i	−0.327918	+	1.66074i
$148$		0			0
$149$		−	72.5899i		−	0.487181i	−0.969878	−	0.243590i	$-0.921675\pi$
$149$		−	72.5899i		−	0.487181i	0.969878	−	0.243590i	$-0.0783252\pi$
$150$		0			0
$151$	−93.9473			−0.622168			−0.311084	−	0.950382i	$-0.600692\pi$
$151$	−93.9473			−0.622168			−0.311084	+	0.950382i	$0.600692\pi$
$152$		0			0
$153$	−12.1580	+	29.5868i	−0.0794641	+	0.193378i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	123.842			0.788803			0.394401	−	0.918938i	$-0.370952\pi$
$157$	123.842			0.788803			0.394401	+	0.918938i	$0.370952\pi$
$158$		0			0
$159$	10.4605	+	2.06546i	0.0657893	+	0.0129903i
$160$		0			0
$161$			202.847i			1.25992i
$162$		0			0
$163$	159.381			0.977801			0.488900	−	0.872340i	$-0.337398\pi$
$163$	159.381			0.977801			0.488900	+	0.872340i	$0.337398\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	10.7742i		−	0.0645161i	−0.999480	−	0.0322581i	$-0.989730\pi$
$167$		−	10.7742i		−	0.0645161i	0.999480	−	0.0322581i	$-0.0102698\pi$
$168$		0			0
$169$	−69.0000			−0.408284
$170$		0			0
$171$	−91.3509	−	37.5386i	−0.534216	−	0.219524i
$172$		0			0
$173$		−	283.345i		−	1.63783i	−0.573913	−	0.818916i	$-0.694575\pi$
$173$		−	283.345i		−	1.63783i	0.573913	−	0.818916i	$-0.305425\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	228.816	+	45.1804i	1.29274	+	0.255256i
$178$		0			0
$179$			306.624i			1.71298i	0.516163	+	0.856491i	$0.327360\pi$
$179$			306.624i			1.71298i	−0.516163	+	0.856491i	$0.672640\pi$
$180$		0			0
$181$	265.684			1.46787			0.733934	−	0.679221i	$-0.237682\pi$
$181$	265.684			1.46787			0.733934	+	0.679221i	$0.237682\pi$
$182$		0			0
$183$	−12.7544	+	64.5948i	−0.0696964	+	0.352977i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	30.1580			0.161273
$188$		0			0
$189$	171.219	−	258.599i	0.905922	−	1.36825i
$190$		0			0
$191$		−	159.620i		−	0.835706i	−0.908515	−	0.417853i	$-0.862783\pi$
$191$		−	159.620i		−	0.835706i	0.908515	−	0.417853i	$-0.137217\pi$
$192$		0			0
$193$	92.0527			0.476957			0.238478	−	0.971148i	$-0.423351\pi$
$193$	92.0527			0.476957			0.238478	+	0.971148i	$0.423351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	69.8360i		−	0.354497i	−0.984166	−	0.177249i	$-0.943280\pi$
$197$		−	69.8360i		−	0.354497i	0.984166	−	0.177249i	$-0.0567197\pi$
$198$		0			0
$199$	−148.921			−0.748347			−0.374173	−	0.927359i	$-0.622074\pi$
$199$	−148.921			−0.748347			−0.374173	+	0.927359i	$0.622074\pi$
$200$		0			0
$201$	31.1139	−	157.576i	0.154795	−	0.783961i
$202$		0			0
$203$		−	308.224i		−	1.51835i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	60.4078	−	147.004i	0.291825	−	0.710163i
$208$		0			0
$209$			93.1146i			0.445525i
$210$		0			0
$211$	285.842			1.35470			0.677351	−	0.735660i	$-0.263128\pi$
$211$	285.842			1.35470			0.677351	+	0.735660i	$0.263128\pi$
$212$		0			0
$213$	−203.842	−	40.2492i	−0.957005	−	0.188963i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−91.8947			−0.423478
$218$		0			0
$219$	7.00427	−	35.4731i	0.0319830	−	0.161978i
$220$		0			0
$221$		−	35.5415i		−	0.160821i
$222$		0			0
$223$	−71.3815			−0.320096			−0.160048	−	0.987109i	$-0.551165\pi$
$223$	−71.3815			−0.320096			−0.160048	+	0.987109i	$0.551165\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			29.1217i			0.128290i	0.997941	+	0.0641448i	$0.0204319\pi$
$227$			29.1217i			0.128290i	−0.997941	+	0.0641448i	$0.979568\pi$
$228$		0			0
$229$	−129.684			−0.566306			−0.283153	−	0.959075i	$-0.591380\pi$
$229$	−129.684			−0.566306			−0.283153	+	0.959075i	$0.591380\pi$
$230$		0			0
$231$	−286.868	−	56.6430i	−1.24185	−	0.245208i
$232$		0			0
$233$		−	185.876i		−	0.797751i	−0.917005	−	0.398875i	$-0.869401\pi$
$233$		−	185.876i		−	0.797751i	0.917005	−	0.398875i	$-0.130599\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−5.24555	+	26.5661i	−0.0221331	+	0.112093i
$238$		0			0
$239$		−	302.716i		−	1.26659i	−0.773908	−	0.633297i	$-0.781701\pi$
$239$		−	302.716i		−	1.26659i	0.773908	−	0.633297i	$-0.218299\pi$
$240$		0			0
$241$	−237.526			−0.985585			−0.492793	−	0.870147i	$-0.664024\pi$
$241$	−237.526			−0.985585			−0.492793	+	0.870147i	$0.664024\pi$
$242$		0			0
$243$	−201.094	+	136.419i	−0.827549	+	0.561394i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	109.737			0.444278
$248$		0			0
$249$	−2.02633	−	0.400106i	−0.00813789	−	0.00160685i
$250$		0			0
$251$			84.6294i			0.337169i	0.985687	+	0.168584i	$0.0539196\pi$
$251$			84.6294i			0.337169i	−0.985687	+	0.168584i	$0.946080\pi$
$252$		0			0
$253$	−149.842			−0.592261
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	292.054i		−	1.13640i	−0.822892	−	0.568198i	$-0.807641\pi$
$257$		−	292.054i		−	1.13640i	0.822892	−	0.568198i	$-0.192359\pi$
$258$		0			0
$259$	688.605			2.65871
$260$		0			0
$261$	−91.7893	+	223.371i	−0.351683	+	0.855829i
$262$		0			0
$263$			277.725i			1.05599i	0.849247	+	0.527995i	$0.177056\pi$
$263$			277.725i			1.05599i	−0.849247	+	0.527995i	$0.822944\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	20.9210	+	4.13091i	0.0783558	+	0.0154716i
$268$		0			0
$269$		−	286.546i		−	1.06523i	−0.846359	−	0.532613i	$-0.821210\pi$
$269$		−	286.546i		−	1.06523i	0.846359	−	0.532613i	$-0.178790\pi$
$270$		0			0
$271$	324.105			1.19596			0.597980	−	0.801511i	$-0.295970\pi$
$271$	324.105			1.19596			0.597980	+	0.801511i	$0.295970\pi$
$272$		0			0
$273$	−66.7544	+	338.078i	−0.244522	+	1.23838i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−415.842			−1.50123			−0.750617	−	0.660737i	$-0.770244\pi$
$277$	−415.842			−1.50123			−0.750617	+	0.660737i	$0.770244\pi$
$278$		0			0
$279$	66.5964	+	27.3663i	0.238697	+	0.0980871i
$280$		0			0
$281$		−	431.726i		−	1.53639i	−0.640216	−	0.768195i	$-0.721155\pi$
$281$		−	431.726i		−	1.53639i	0.640216	−	0.768195i	$-0.278845\pi$
$282$		0			0
$283$	141.540			0.500140			0.250070	−	0.968228i	$-0.419546\pi$
$283$	141.540			0.500140			0.250070	+	0.968228i	$0.419546\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	235.764i		−	0.821477i
$288$		0			0
$289$	276.368			0.956291
$290$		0			0
$291$	65.0569	−	329.481i	0.223563	−	1.13224i
$292$		0			0
$293$			164.998i			0.563133i	0.959542	+	0.281566i	$0.0908540\pi$
$293$			164.998i			0.563133i	−0.959542	+	0.281566i	$0.909146\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	191.026	+	126.479i	0.643186	+	0.425855i
$298$		0			0
$299$			176.590i			0.590604i
$300$		0			0
$301$	487.737			1.62039
$302$		0			0
$303$	−457.631	−	90.3607i	−1.51033	−	0.298220i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	159.381			0.519158			0.259579	−	0.965722i	$-0.416416\pi$
$307$	159.381			0.519158			0.259579	+	0.965722i	$0.416416\pi$
$308$		0			0
$309$	−4.41210	+	22.3451i	−0.0142787	+	0.0723142i
$310$		0			0
$311$			143.096i			0.460117i	0.973177	+	0.230058i	$0.0738917\pi$
$311$			143.096i			0.460117i	−0.973177	+	0.230058i	$0.926108\pi$
$312$		0			0
$313$	501.684			1.60282			0.801412	−	0.598113i	$-0.204082\pi$
$313$	501.684			1.60282			0.801412	+	0.598113i	$0.204082\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	334.257i		−	1.05444i	−0.849730	−	0.527219i	$-0.823235\pi$
$317$		−	334.257i		−	1.05444i	0.849730	−	0.527219i	$-0.176765\pi$
$318$		0			0
$319$	227.684			0.713743
$320$		0			0
$321$	−47.9210	−	9.46215i	−0.149287	−	0.0294771i
$322$		0			0
$323$		−	39.0021i		−	0.120750i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	54.5352	−	276.193i	0.166774	−	0.844628i
$328$		0			0
$329$			1014.23i			3.08277i
$330$		0			0
$331$	−389.421			−1.17650			−0.588249	−	0.808680i	$-0.700182\pi$
$331$	−389.421			−1.17650			−0.588249	+	0.808680i	$0.700182\pi$
$332$		0			0
$333$	−499.035	−	205.067i	−1.49860	−	0.615817i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	129.684			0.384819			0.192409	−	0.981315i	$-0.438370\pi$
$337$	129.684			0.384819			0.192409	+	0.981315i	$0.438370\pi$
$338$		0			0
$339$	−576.092	−	113.751i	−1.69939	−	0.335549i
$340$		0			0
$341$		−	67.8823i		−	0.199068i
$342$		0			0
$343$	389.947			1.13687
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	346.985i		−	0.999956i	−0.866038	−	0.499978i	$-0.833341\pi$
$347$		−	346.985i		−	0.999956i	0.866038	−	0.499978i	$-0.166659\pi$
$348$		0			0
$349$	509.579			1.46011			0.730055	−	0.683388i	$-0.239494\pi$
$349$	509.579			1.46011			0.730055	+	0.683388i	$0.239494\pi$
$350$		0			0
$351$	149.057	−	225.127i	0.424664	−	0.641387i
$352$		0			0
$353$			637.679i			1.80646i	0.429159	+	0.903229i	$0.358810\pi$
$353$			637.679i			1.80646i	−0.429159	+	0.903229i	$0.641190\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	120.158	+	23.7256i	0.336577	+	0.0664582i
$358$		0			0
$359$			166.952i			0.465046i	0.972591	+	0.232523i	$0.0746982\pi$
$359$			166.952i			0.465046i	−0.972591	+	0.232523i	$0.925302\pi$
$360$		0			0
$361$	−240.579			−0.666423
$362$		0			0
$363$	−28.4758	+	144.216i	−0.0784457	+	0.397288i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−505.828			−1.37828			−0.689140	−	0.724629i	$-0.742011\pi$
$367$	−505.828			−1.37828			−0.689140	+	0.724629i	$0.742011\pi$
$368$		0			0
$369$	−70.2107	+	170.859i	−0.190273	+	0.463033i
$370$		0			0
$371$		−	40.8260i		−	0.110043i
$372$		0			0
$373$	416.053			1.11542			0.557711	−	0.830035i	$-0.311680\pi$
$373$	416.053			1.11542			0.557711	+	0.830035i	$0.311680\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	268.328i		−	0.711746i
$378$		0			0
$379$	−82.7630			−0.218372			−0.109186	−	0.994021i	$-0.534824\pi$
$379$	−82.7630			−0.218372			−0.109186	+	0.994021i	$0.534824\pi$
$380$		0			0
$381$	45.0612	−	228.212i	0.118271	−	0.598982i
$382$		0			0
$383$		−	334.592i		−	0.873608i	−0.899557	−	0.436804i	$-0.856110\pi$
$383$		−	334.592i		−	0.873608i	0.899557	−	0.436804i	$-0.143890\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−353.465	−	145.248i	−0.913346	−	0.375319i
$388$		0			0
$389$		−	542.964i		−	1.39580i	−0.716197	−	0.697898i	$-0.754119\pi$
$389$		−	542.964i		−	1.39580i	0.716197	−	0.697898i	$-0.245881\pi$
$390$		0			0
$391$	62.7630			0.160519
$392$		0			0
$393$	−299.026	−	59.0437i	−0.760881	−	0.150238i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	214.000			0.539043			0.269521	−	0.962994i	$-0.413135\pi$
$397$	214.000			0.539043			0.269521	+	0.962994i	$0.413135\pi$
$398$		0			0
$399$	−73.2541	+	370.995i	−0.183594	+	0.929812i
$400$		0			0
$401$		−	726.086i		−	1.81069i	−0.424677	−	0.905345i	$-0.639613\pi$
$401$		−	726.086i		−	1.81069i	0.424677	−	0.905345i	$-0.360387\pi$
$402$		0			0
$403$	−80.0000			−0.198511
$404$		0			0
$405$		0			0
$406$		0			0
$407$			508.670i			1.24980i
$408$		0			0
$409$	−346.158			−0.846352			−0.423176	−	0.906047i	$-0.639085\pi$
$409$	−346.158			−0.846352			−0.423176	+	0.906047i	$0.639085\pi$
$410$		0			0
$411$	−518.039	−	102.288i	−1.26044	−	0.248877i
$412$		0			0
$413$		−	893.038i		−	2.16232i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	109.728	−	555.717i	0.263137	−	1.33266i
$418$		0			0
$419$		−	580.907i		−	1.38641i	−0.720740	−	0.693206i	$-0.756198\pi$
$419$		−	580.907i		−	1.38641i	0.720740	−	0.693206i	$-0.243802\pi$
$420$		0			0
$421$	−269.315			−0.639704			−0.319852	−	0.947468i	$-0.603633\pi$
$421$	−269.315			−0.639704			−0.319852	+	0.947468i	$0.603633\pi$
$422$		0			0
$423$	302.039	−	735.019i	0.714041	−	1.73763i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	252.105			0.590411
$428$		0			0
$429$	−249.737	−	49.3113i	−0.582137	−	0.114945i
$430$		0			0
$431$		−	776.068i		−	1.80062i	−0.435247	−	0.900311i	$-0.643339\pi$
$431$		−	776.068i		−	1.80062i	0.435247	−	0.900311i	$-0.356661\pi$
$432$		0			0
$433$	195.526			0.451561			0.225781	−	0.974178i	$-0.427507\pi$
$433$	195.526			0.451561			0.225781	+	0.974178i	$0.427507\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			193.785i			0.443443i
$438$		0			0
$439$	−598.763			−1.36392			−0.681962	−	0.731387i	$-0.738873\pi$
$439$	−598.763			−1.36392			−0.681962	+	0.731387i	$0.738873\pi$
$440$		0			0
$441$	−690.500	−	283.745i	−1.56576	−	0.643413i
$442$		0			0
$443$			216.728i			0.489228i	0.969621	+	0.244614i	$0.0786612\pi$
$443$			216.728i			0.489228i	−0.969621	+	0.244614i	$0.921339\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	213.645	+	42.1848i	0.477953	+	0.0943732i
$448$		0			0
$449$			246.203i			0.548336i	0.961682	+	0.274168i	$0.0884025\pi$
$449$			246.203i			0.548336i	−0.961682	+	0.274168i	$0.911598\pi$
$450$		0			0
$451$	174.158			0.386160
$452$		0			0
$453$	54.5964	−	276.503i	0.120522	−	0.610383i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−553.052			−1.21018			−0.605090	−	0.796157i	$-0.706863\pi$
$457$	−553.052			−1.21018			−0.605090	+	0.796157i	$0.706863\pi$
$458$		0			0
$459$	−80.0135	−	52.9771i	−0.174321	−	0.115419i
$460$		0			0
$461$			124.749i			0.270605i	0.990804	+	0.135302i	$0.0432006\pi$
$461$			124.749i			0.270605i	−0.990804	+	0.135302i	$0.956799\pi$
$462$		0			0
$463$	669.723			1.44649			0.723243	−	0.690593i	$-0.242651\pi$
$463$	669.723			1.44649			0.723243	+	0.690593i	$0.242651\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	486.880i		−	1.04257i	−0.853383	−	0.521285i	$-0.825453\pi$
$467$		−	486.880i		−	1.04257i	0.853383	−	0.521285i	$-0.174547\pi$
$468$		0			0
$469$	−614.999			−1.31130
$470$		0			0
$471$	−71.9694	+	364.489i	−0.152801	+	0.773861i
$472$		0			0
$473$			360.289i			0.761711i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−12.1580	+	29.5868i	−0.0254885	+	0.0620267i
$478$		0			0
$479$			520.580i			1.08680i	0.839472	+	0.543402i	$0.182864\pi$
$479$			520.580i			1.08680i	−0.839472	+	0.543402i	$0.817136\pi$
$480$		0			0
$481$	599.473			1.24631
$482$		0			0
$483$	−597.013	−	117.882i	−1.23605	−	0.244062i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−263.381			−0.540824			−0.270412	−	0.962745i	$-0.587160\pi$
$487$	−263.381			−0.540824			−0.270412	+	0.962745i	$0.587160\pi$
$488$		0			0
$489$	−92.6228	+	469.088i	−0.189413	+	0.959279i
$490$		0			0
$491$		−	711.646i		−	1.44938i	−0.689074	−	0.724691i	$-0.741983\pi$
$491$		−	711.646i		−	1.44938i	0.689074	−	0.724691i	$-0.258017\pi$
$492$		0			0
$493$	−95.3680			−0.193444
$494$		0			0
$495$		0			0
$496$		0			0
$497$			795.569i			1.60074i
$498$		0			0
$499$	−23.0790			−0.0462505			−0.0231253	−	0.999733i	$-0.507362\pi$
$499$	−23.0790			−0.0462505			−0.0231253	+	0.999733i	$0.507362\pi$
$500$		0			0
$501$	31.7103	+	6.26130i	0.0632941	+	0.0124976i
$502$		0			0
$503$		−	40.5844i		−	0.0806847i	−0.999186	−	0.0403423i	$-0.987155\pi$
$503$		−	40.5844i		−	0.0806847i	0.999186	−	0.0403423i	$-0.0128449\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	40.0986	−	203.079i	0.0790899	−	0.400550i
$508$		0			0
$509$		−	484.591i		−	0.952045i	−0.879433	−	0.476023i	$-0.842078\pi$
$509$		−	484.591i		−	0.952045i	0.879433	−	0.476023i	$-0.157922\pi$
$510$		0			0
$511$	−138.447			−0.270933
$512$		0			0
$513$	163.570	−	247.047i	0.318850	−	0.481572i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−749.210			−1.44915
$518$		0			0
$519$	833.934	+	164.663i	1.60681	+	0.317269i
$520$		0			0
$521$			539.857i			1.03619i	0.855322	+	0.518097i	$0.173359\pi$
$521$			539.857i			1.03619i	−0.855322	+	0.518097i	$0.826641\pi$
$522$		0			0
$523$	−266.644			−0.509836			−0.254918	−	0.966963i	$-0.582048\pi$
$523$	−266.644			−0.509836			−0.254918	+	0.966963i	$0.582048\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			28.4332i			0.0539530i
$528$		0			0
$529$	217.158			0.410507
$530$		0			0
$531$	−265.947	+	647.188i	−0.500842	+	1.21881i
$532$		0			0
$533$		−	205.247i		−	0.385079i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−902.447	−	178.191i	−1.68053	−	0.331827i
$538$		0			0
$539$			703.831i			1.30581i
$540$		0			0
$541$	−337.895			−0.624574			−0.312287	−	0.949988i	$-0.601095\pi$
$541$	−337.895			−0.624574			−0.312287	+	0.949988i	$0.601095\pi$
$542$		0			0
$543$	−154.399	+	781.954i	−0.284345	+	1.44006i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−214.566			−0.392259			−0.196130	−	0.980578i	$-0.562837\pi$
$547$	−214.566			−0.392259			−0.196130	+	0.980578i	$0.562837\pi$
$548$		0			0
$549$	−182.702	−	75.0771i	−0.332790	−	0.136753i
$550$		0			0
$551$		−	294.454i		−	0.534400i
$552$		0			0
$553$	103.684			0.187494
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	674.821i		−	1.21153i	−0.795644	−	0.605764i	$-0.792867\pi$
$557$		−	674.821i		−	1.21153i	0.795644	−	0.605764i	$-0.207133\pi$
$558$		0			0
$559$	424.605			0.759580
$560$		0			0
$561$	−17.5260	+	88.7603i	−0.0312406	+	0.158218i
$562$		0			0
$563$			718.513i			1.27622i	0.769944	+	0.638111i	$0.220284\pi$
$563$			718.513i			1.27622i	−0.769944	+	0.638111i	$0.779716\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	661.601	+	654.210i	1.16684	+	1.15381i
$568$		0			0
$569$		−	183.122i		−	0.321831i	−0.986968	−	0.160916i	$-0.948555\pi$
$569$		−	183.122i		−	0.321831i	0.986968	−	0.160916i	$-0.0514447\pi$
$570$		0			0
$571$	295.895			0.518204			0.259102	−	0.965850i	$-0.416573\pi$
$571$	295.895			0.518204			0.259102	+	0.965850i	$0.416573\pi$
$572$		0			0
$573$	469.789	+	92.7613i	0.819877	+	0.161887i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	984.947			1.70701			0.853507	−	0.521082i	$-0.174471\pi$
$577$	984.947			1.70701			0.853507	+	0.521082i	$0.174471\pi$
$578$		0			0
$579$	−53.4954	+	270.927i	−0.0923927	+	0.467922i
$580$		0			0
$581$			7.90852i			0.0136119i
$582$		0			0
$583$	30.1580			0.0517290
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	378.172i		−	0.644245i	−0.946698	−	0.322122i	$-0.895604\pi$
$587$		−	378.172i		−	0.644245i	0.946698	−	0.322122i	$-0.104396\pi$
$588$		0			0
$589$	−87.7893			−0.149048
$590$		0			0
$591$	205.540	+	40.5844i	0.347783	+	0.0686707i
$592$		0			0
$593$			265.221i			0.447253i	0.974675	+	0.223626i	$0.0717895\pi$
$593$			265.221i			0.447253i	−0.974675	+	0.223626i	$0.928210\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	86.5438	−	438.301i	0.144964	−	0.734172i
$598$		0			0
$599$		−	382.061i		−	0.637832i	−0.947783	−	0.318916i	$-0.896681\pi$
$599$		−	382.061i		−	0.637832i	0.947783	−	0.318916i	$-0.103319\pi$
$600$		0			0
$601$	1021.53			1.69971			0.849855	−	0.527016i	$-0.176689\pi$
$601$	1021.53			1.69971			0.849855	+	0.527016i	$0.176689\pi$
$602$		0			0
$603$	445.693	+	183.147i	0.739125	+	0.303727i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−835.487			−1.37642			−0.688210	−	0.725512i	$-0.741603\pi$
$607$	−835.487			−1.37642			−0.688210	+	0.725512i	$0.741603\pi$
$608$		0			0
$609$	907.157	+	179.121i	1.48959	+	0.294123i
$610$		0			0
$611$			882.952i			1.44509i
$612$		0			0
$613$	−833.263			−1.35932			−0.679660	−	0.733528i	$-0.737872\pi$
$613$	−833.263			−1.35932			−0.679660	+	0.733528i	$0.737872\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			455.098i			0.737598i	0.929509	+	0.368799i	$0.120231\pi$
$617$			455.098i			0.737598i	−0.929509	+	0.368799i	$0.879769\pi$
$618$		0			0
$619$	−336.710			−0.543959			−0.271979	−	0.962303i	$-0.587678\pi$
$619$	−336.710			−0.543959			−0.271979	+	0.962303i	$0.587678\pi$
$620$		0			0
$621$	397.552	+	263.220i	0.640181	+	0.423865i
$622$		0			0
$623$		−	81.6520i		−	0.131063i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−274.053	−	54.1125i	−0.437086	−	0.0863039i
$628$		0			0
$629$		−	213.062i		−	0.338731i
$630$		0			0
$631$	1100.89			1.74468			0.872341	−	0.488898i	$-0.162601\pi$
$631$	1100.89			1.74468			0.872341	+	0.488898i	$0.162601\pi$
$632$		0			0
$633$	−166.114	+	841.283i	−0.262423	+	1.32904i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	829.473			1.30216
$638$		0			0
$639$	236.921	−	576.552i	0.370768	−	0.902273i
$640$		0			0
$641$		−	337.011i		−	0.525758i	−0.964829	−	0.262879i	$-0.915328\pi$
$641$		−	337.011i		−	0.525758i	0.964829	−	0.262879i	$-0.0846719\pi$
$642$		0			0
$643$	−599.381			−0.932164			−0.466082	−	0.884742i	$-0.654335\pi$
$643$	−599.381			−0.932164			−0.466082	+	0.884742i	$0.654335\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			47.0224i			0.0726775i	0.999340	+	0.0363388i	$0.0115695\pi$
$647$			47.0224i			0.0726775i	−0.999340	+	0.0363388i	$0.988430\pi$
$648$		0			0
$649$	659.684			1.01646
$650$		0			0
$651$	53.4036	−	270.462i	0.0820331	−	0.415456i
$652$		0			0
$653$			559.228i			0.856399i	0.903684	+	0.428199i	$0.140852\pi$
$653$			559.228i			0.856399i	−0.903684	+	0.428199i	$0.859148\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	100.333	+	41.2296i	0.152714	+	0.0627543i
$658$		0			0
$659$		−	665.759i		−	1.01026i	−0.863044	−	0.505129i	$-0.831445\pi$
$659$		−	665.759i		−	1.01026i	0.863044	−	0.505129i	$-0.168555\pi$
$660$		0			0
$661$	−557.947			−0.844096			−0.422048	−	0.906574i	$-0.638689\pi$
$661$	−557.947			−0.844096			−0.422048	+	0.906574i	$0.638689\pi$
$662$		0			0
$663$	104.605	+	20.6546i	0.157775	+	0.0311532i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	473.842			0.710408
$668$		0			0
$669$	41.4826	−	210.088i	0.0620068	−	0.314033i
$670$		0			0
$671$			186.229i			0.277540i
$672$		0			0
$673$	−1019.42			−1.51474			−0.757370	−	0.652985i	$-0.773516\pi$
$673$	−1019.42			−1.51474			−0.757370	+	0.652985i	$0.773516\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			624.616i			0.922624i	0.887238	+	0.461312i	$0.152621\pi$
$677$			624.616i			0.922624i	−0.887238	+	0.461312i	$0.847379\pi$
$678$		0			0
$679$	−1285.92			−1.89384
$680$		0			0
$681$	−85.7103	−	16.9238i	−0.125860	−	0.0248513i
$682$		0			0
$683$		−	525.882i		−	0.769959i	−0.922925	−	0.384980i	$-0.874208\pi$
$683$		−	525.882i		−	0.769959i	0.922925	−	0.384980i	$-0.125792\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	75.3644	−	381.683i	0.109701	−	0.555579i
$688$		0			0
$689$		−	35.5415i		−	0.0515843i
$690$		0			0
$691$	−932.000			−1.34877			−0.674385	−	0.738380i	$-0.735591\pi$
$691$	−932.000			−1.34877			−0.674385	+	0.738380i	$0.735591\pi$
$692$		0			0
$693$	333.421	−	811.386i	0.481126	−	1.17083i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−72.9480			−0.104660
$698$		0			0
$699$	547.065	+	108.020i	0.782640	+	0.154535i
$700$		0			0
$701$			606.045i			0.864544i	0.901743	+	0.432272i	$0.142288\pi$
$701$			606.045i			0.864544i	−0.901743	+	0.432272i	$0.857712\pi$
$702$		0			0
$703$	657.842			0.935764
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1786.08i			2.52627i
$708$		0			0
$709$	489.473			0.690371			0.345186	−	0.938534i	$-0.387816\pi$
$709$	489.473			0.690371			0.345186	+	0.938534i	$0.387816\pi$
$710$		0			0
$711$	−75.1402	−	30.8772i	−0.105682	−	0.0434278i
$712$		0			0
$713$		−	141.272i		−	0.198138i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	890.947	+	175.920i	1.24260	+	0.245356i
$718$		0			0
$719$		−	107.778i		−	0.149900i	−0.997187	−	0.0749500i	$-0.976120\pi$
$719$		−	107.778i		−	0.149900i	0.997187	−	0.0749500i	$-0.0238797\pi$
$720$		0			0
$721$	87.2100			0.120957
$722$		0			0
$723$	138.036	−	699.080i	0.190921	−	0.966916i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−150.172			−0.206563			−0.103282	−	0.994652i	$-0.532934\pi$
$727$	−150.172			−0.206563			−0.103282	+	0.994652i	$0.532934\pi$
$728$		0			0
$729$	−284.641	−	671.134i	−0.390453	−	0.920623i
$730$		0			0
$731$		−	150.911i		−	0.206445i
$732$		0			0
$733$	675.526			0.921591			0.460795	−	0.887506i	$-0.347564\pi$
$733$	675.526			0.921591			0.460795	+	0.887506i	$0.347564\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	454.298i		−	0.616415i
$738$		0			0
$739$	936.921			1.26782			0.633911	−	0.773406i	$-0.281448\pi$
$739$	936.921			1.26782			0.633911	+	0.773406i	$0.281448\pi$
$740$		0			0
$741$	−63.7722	+	322.974i	−0.0860624	+	0.435863i
$742$		0			0
$743$		−	452.939i		−	0.609608i	−0.952415	−	0.304804i	$-0.901409\pi$
$743$		−	452.939i		−	0.609608i	0.952415	−	0.304804i	$-0.0985910\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	2.35516	−	5.73134i	0.00315283	−	0.00767247i
$748$		0			0
$749$			187.029i			0.249706i
$750$		0			0
$751$	−654.369			−0.871330			−0.435665	−	0.900109i	$-0.643487\pi$
$751$	−654.369			−0.871330			−0.435665	+	0.900109i	$0.643487\pi$
$752$		0			0
$753$	−249.079	−	49.1814i	−0.330782	−	0.0653140i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−123.315			−0.162900			−0.0814500	−	0.996677i	$-0.525955\pi$
$757$	−123.315			−0.162900			−0.0814500	+	0.996677i	$0.525955\pi$
$758$		0			0
$759$	87.0790	−	441.011i	0.114729	−	0.581042i
$760$		0			0
$761$			502.715i			0.660598i	0.943876	+	0.330299i	$0.107150\pi$
$761$			502.715i			0.660598i	−0.943876	+	0.330299i	$0.892850\pi$
$762$		0			0
$763$	−1077.95			−1.41278
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	777.445i		−	1.01362i
$768$		0			0
$769$	−286.316			−0.372323			−0.186161	−	0.982519i	$-0.559605\pi$
$769$	−286.316			−0.372323			−0.186161	+	0.982519i	$0.559605\pi$
$770$		0			0
$771$	859.565	+	169.724i	1.11487	+	0.220135i
$772$		0			0
$773$		−	972.030i		−	1.25748i	−0.777617	−	0.628739i	$-0.783572\pi$
$773$		−	972.030i		−	1.25748i	0.777617	−	0.628739i	$-0.216428\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−400.175	+	2026.68i	−0.515026	+	2.60835i
$778$		0			0
$779$		−	225.231i		−	0.289129i
$780$		0			0
$781$	−587.684			−0.752476
$782$		0			0
$783$	−604.078	−	399.962i	−0.771492	−	0.510807i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1492.70			1.89669			0.948346	−	0.317237i	$-0.102755\pi$
$787$	1492.70			1.89669			0.948346	+	0.317237i	$0.102755\pi$
$788$		0			0
$789$	−817.394	−	161.397i	−1.03599	−	0.204559i
$790$		0			0
$791$			2248.41i			2.84249i
$792$		0			0
$793$	219.473			0.276763
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	94.3618i		−	0.118396i	−0.998246	−	0.0591981i	$-0.981146\pi$
$797$		−	94.3618i		−	0.118396i	0.998246	−	0.0591981i	$-0.0188544\pi$
$798$		0			0
$799$	313.815			0.392760
$800$		0			0
$801$	−24.3160	+	59.1735i	−0.0303571	+	0.0738746i
$802$		0			0
$803$		−	102.270i		−	0.127360i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	843.354	+	166.523i	1.04505	+	0.206348i
$808$		0			0
$809$			1103.35i			1.36384i	0.731427	+	0.681920i	$0.238855\pi$
$809$			1103.35i			1.36384i	−0.731427	+	0.681920i	$0.761145\pi$
$810$		0			0
$811$	10.1580			0.0125253			0.00626264	−	0.999980i	$-0.498007\pi$
$811$	10.1580			0.0125253			0.00626264	+	0.999980i	$0.498007\pi$
$812$		0			0
$813$	−188.350	+	953.899i	−0.231673	+	1.17331i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	465.947			0.570315
$818$		0			0
$819$	−956.228	−	392.940i	−1.16756	−	0.479780i
$820$		0			0
$821$		−	337.011i		−	0.410488i	−0.978711	−	0.205244i	$-0.934201\pi$
$821$		−	337.011i		−	0.410488i	0.978711	−	0.205244i	$-0.0657988\pi$
$822$		0			0
$823$	−901.512			−1.09540			−0.547699	−	0.836675i	$-0.684496\pi$
$823$	−901.512			−1.09540			−0.547699	+	0.836675i	$0.684496\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			531.354i			0.642508i	0.946993	+	0.321254i	$0.104104\pi$
$827$			531.354i			0.642508i	−0.946993	+	0.321254i	$0.895896\pi$
$828$		0			0
$829$	−197.631			−0.238397			−0.119199	−	0.992870i	$-0.538033\pi$
$829$	−197.631			−0.238397			−0.119199	+	0.992870i	$0.538033\pi$
$830$		0			0
$831$	241.662	−	1223.90i	0.290809	−	1.47280i
$832$		0			0
$833$		−	294.808i		−	0.353911i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−119.246	+	180.101i	−0.142468	+	0.215175i
$838$		0			0
$839$		−	943.950i		−	1.12509i	−0.826767	−	0.562545i	$-0.809822\pi$
$839$		−	943.950i		−	1.12509i	0.826767	−	0.562545i	$-0.190178\pi$
$840$		0			0
$841$	121.000			0.143876
$842$		0			0
$843$	1270.64	+	250.893i	1.50729	+	0.297619i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	562.855			0.664528
$848$		0			0
$849$	−82.2541	+	416.575i	−0.0968835	+	0.490666i
$850$		0			0
$851$			1058.61i			1.24396i
$852$		0			0
$853$	1196.42			1.40260			0.701301	−	0.712865i	$-0.252603\pi$
$853$	1196.42			1.40260			0.701301	+	0.712865i	$0.252603\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	1266.04i		−	1.47729i	−0.674095	−	0.738645i	$-0.735466\pi$
$857$		−	1266.04i		−	1.47729i	0.674095	−	0.738645i	$-0.264534\pi$
$858$		0			0
$859$	470.868			0.548159			0.274079	−	0.961707i	$-0.411627\pi$
$859$	470.868			0.548159			0.274079	+	0.961707i	$0.411627\pi$
$860$		0			0
$861$	693.895	+	137.012i	0.805917	+	0.159131i
$862$		0			0
$863$			285.057i			0.330310i	0.986268	+	0.165155i	$0.0528124\pi$
$863$			285.057i			0.330310i	−0.986268	+	0.165155i	$0.947188\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−160.608	+	813.399i	−0.185246	+	0.938177i
$868$		0			0
$869$			76.5910i			0.0881369i
$870$		0			0
$871$	−535.395			−0.614690
$872$		0			0
$873$	931.912	+	382.948i	1.06748	+	0.438657i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−244.579			−0.278882			−0.139441	−	0.990230i	$-0.544531\pi$
$877$	−244.579			−0.278882			−0.139441	+	0.990230i	$0.544531\pi$
$878$		0			0
$879$	−485.618	−	95.8867i	−0.552466	−	0.109086i
$880$		0			0
$881$			137.271i			0.155813i	0.996961	+	0.0779065i	$0.0248236\pi$
$881$			137.271i			0.155813i	−0.996961	+	0.0779065i	$0.975176\pi$
$882$		0			0
$883$	1647.75			1.86608			0.933040	−	0.359772i	$-0.117146\pi$
$883$	1647.75			1.86608			0.933040	+	0.359772i	$0.117146\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	514.419i		−	0.579954i	−0.957034	−	0.289977i	$-0.906352\pi$
$887$		−	514.419i		−	0.579954i	0.957034	−	0.289977i	$-0.0936477\pi$
$888$		0			0
$889$	−890.683			−1.00189
$890$		0			0
$891$	−483.263	+	488.722i	−0.542382	+	0.548510i
$892$		0			0
$893$			968.923i			1.08502i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−519.737	−	102.624i	−0.579417	−	0.114408i
$898$		0			0
$899$			214.663i			0.238779i
$900$		0			0
$901$	−12.6320			−0.0140200
$902$		0			0
$903$	−283.443	+	1435.49i	−0.313890	+	1.58969i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	347.303			0.382914			0.191457	−	0.981501i	$-0.438679\pi$
$907$	347.303			0.382914			0.191457	+	0.981501i	$0.438679\pi$
$908$		0			0
$909$	531.895	−	1294.38i	0.585143	−	1.42396i
$910$		0			0
$911$			304.540i			0.334292i	0.985932	+	0.167146i	$0.0534551\pi$
$911$			304.540i			0.334292i	−0.985932	+	0.167146i	$0.946545\pi$
$912$		0			0
$913$	−5.84200			−0.00639868
$914$		0			0
$915$		0			0
$916$		0			0
$917$			1167.06i			1.27270i
$918$		0			0
$919$	−244.289			−0.265820			−0.132910	−	0.991128i	$-0.542432\pi$
$919$	−244.289			−0.265820			−0.132910	+	0.991128i	$0.542432\pi$
$920$		0			0
$921$	−92.6228	+	469.088i	−0.100568	+	0.509324i
$922$		0			0
$923$			692.592i			0.750371i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−63.2014	−	25.9712i	−0.0681785	−	0.0280164i
$928$		0			0
$929$		−	1025.96i		−	1.10437i	−0.833723	−	0.552183i	$-0.813795\pi$
$929$		−	1025.96i		−	1.10437i	0.833723	−	0.552183i	$-0.186205\pi$
$930$		0			0
$931$	910.236			0.977697
$932$		0			0
$933$	−421.157	−	83.1588i	−0.451401	−	0.0891305i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	332.053			0.354379			0.177189	−	0.984177i	$-0.443300\pi$
$937$	332.053			0.354379			0.177189	+	0.984177i	$0.443300\pi$
$938$		0			0
$939$	−291.548	+	1476.54i	−0.310488	+	1.57246i
$940$		0			0
$941$			1636.99i			1.73963i	0.493381	+	0.869813i	$0.335761\pi$
$941$			1636.99i			1.73963i	−0.493381	+	0.869813i	$0.664239\pi$
$942$		0			0
$943$	362.447			0.384355
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1209.32i			1.27700i	0.769622	+	0.638500i	$0.220445\pi$
$947$			1209.32i			1.27700i	−0.769622	+	0.638500i	$0.779555\pi$
$948$		0			0
$949$	−120.527			−0.127004
$950$		0			0
$951$	983.776	+	194.250i	1.03446	+	0.204258i
$952$		0			0
$953$			1205.26i			1.26470i	0.774681	+	0.632352i	$0.217910\pi$
$953$			1205.26i			1.26470i	−0.774681	+	0.632352i	$0.782090\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−132.316	+	670.114i	−0.138261	+	0.700223i
$958$		0			0
$959$			2021.84i			2.10828i
$960$		0			0
$961$	−897.000			−0.933403
$962$		0			0
$963$	55.6975	−	135.541i	0.0578375	−	0.140749i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1845.93			−1.90893			−0.954464	−	0.298325i	$-0.903572\pi$
$967$	−1845.93			−1.90893			−0.954464	+	0.298325i	$0.903572\pi$
$968$		0			0
$969$	114.790	+	22.6656i	0.118462	+	0.0233908i
$970$		0			0
$971$		−	1057.91i		−	1.08950i	−0.838598	−	0.544751i	$-0.816624\pi$
$971$		−	1057.91i		−	1.08950i	0.838598	−	0.544751i	$-0.183376\pi$
$972$		0			0
$973$	−2168.89			−2.22908
$974$		0			0
$975$		0			0
$976$		0			0
$977$			964.028i			0.986722i	0.869825	+	0.493361i	$0.164232\pi$
$977$			964.028i			0.986722i	−0.869825	+	0.493361i	$0.835768\pi$
$978$		0			0
$979$	60.3160			0.0616098
$980$		0			0
$981$	781.193	+	321.013i	0.796323	+	0.327231i
$982$		0			0
$983$			460.718i			0.468685i	0.972154	+	0.234343i	$0.0752937\pi$
$983$			460.718i			0.468685i	−0.972154	+	0.234343i	$0.924706\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−2985.06	−	589.410i	−3.02438	−	0.597173i
$988$		0			0
$989$			749.812i			0.758152i
$990$		0			0
$991$	−1237.89			−1.24914			−0.624568	−	0.780971i	$-0.714725\pi$
$991$	−1237.89			−1.24914			−0.624568	+	0.780971i	$0.714725\pi$
$992$		0			0
$993$	226.307	−	1146.13i	0.227903	−	1.15421i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.36798			0.00939617			0.00469809	−	0.999989i	$-0.498505\pi$
$997$	9.36798			0.00939617			0.00469809	+	0.999989i	$0.498505\pi$
$998$		0			0
$999$	893.557	−	1349.57i	0.894451	−	1.35093i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.l.u.401.2		4
3.2	odd	2	inner	1200.3.l.u.401.1		4
4.3	odd	2		150.3.d.c.101.2		4
5.2	odd	4		1200.3.c.k.449.8		8
5.3	odd	4		1200.3.c.k.449.1		8
5.4	even	2		240.3.l.c.161.3		4
12.11	even	2		150.3.d.c.101.4		4
15.2	even	4		1200.3.c.k.449.2		8
15.8	even	4		1200.3.c.k.449.7		8
15.14	odd	2		240.3.l.c.161.4		4
20.3	even	4		150.3.b.b.149.4		8
20.7	even	4		150.3.b.b.149.5		8
20.19	odd	2		30.3.d.a.11.3	yes	4
40.19	odd	2		960.3.l.e.641.3		4
40.29	even	2		960.3.l.f.641.2		4
60.23	odd	4		150.3.b.b.149.6		8
60.47	odd	4		150.3.b.b.149.3		8
60.59	even	2		30.3.d.a.11.1	✓	4
120.29	odd	2		960.3.l.f.641.1		4
120.59	even	2		960.3.l.e.641.4		4
180.59	even	6		810.3.h.a.431.4		8
180.79	odd	6		810.3.h.a.701.4		8
180.119	even	6		810.3.h.a.701.1		8
180.139	odd	6		810.3.h.a.431.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.3.d.a.11.1	✓	4	60.59	even	2
30.3.d.a.11.3	yes	4	20.19	odd	2
150.3.b.b.149.3		8	60.47	odd	4
150.3.b.b.149.4		8	20.3	even	4
150.3.b.b.149.5		8	20.7	even	4
150.3.b.b.149.6		8	60.23	odd	4
150.3.d.c.101.2		4	4.3	odd	2
150.3.d.c.101.4		4	12.11	even	2
240.3.l.c.161.3		4	5.4	even	2
240.3.l.c.161.4		4	15.14	odd	2
810.3.h.a.431.1		8	180.139	odd	6
810.3.h.a.431.4		8	180.59	even	6
810.3.h.a.701.1		8	180.119	even	6
810.3.h.a.701.4		8	180.79	odd	6
960.3.l.e.641.3		4	40.19	odd	2
960.3.l.e.641.4		4	120.59	even	2
960.3.l.f.641.1		4	120.29	odd	2
960.3.l.f.641.2		4	40.29	even	2
1200.3.c.k.449.1		8	5.3	odd	4
1200.3.c.k.449.2		8	15.2	even	4
1200.3.c.k.449.7		8	15.8	even	4
1200.3.c.k.449.8		8	5.2	odd	4
1200.3.l.u.401.1		4	3.2	odd	2	inner
1200.3.l.u.401.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-0.581139 + 2.94317i) q^{3} +11.4868 q^{7} +(-8.32456 - 3.42079i) q^{9} +O(q^{10})\)	\(q+(-0.581139 + 2.94317i) q^{3} +11.4868 q^{7} +(-8.32456 - 3.42079i) q^{9} +8.48528i q^{11} +10.0000 q^{13} -3.55415i q^{17} +10.9737 q^{19} +(-6.67544 + 33.8078i) q^{21} +17.6590i q^{23} +(14.9057 - 22.5127i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(-24.9737 - 4.93113i) q^{33} +59.9473 q^{37} +(-5.81139 + 29.4317i) q^{39} -20.5247i q^{41} +42.4605 q^{43} +88.2952i q^{47} +82.9473 q^{49} +(10.4605 + 2.06546i) q^{51} -3.55415i q^{53} +(-6.37722 + 32.2974i) q^{57} -77.7445i q^{59} +21.9473 q^{61} +(-95.6228 - 39.2940i) q^{63} -53.5395 q^{67} +(-51.9737 - 10.2624i) q^{69} +69.2592i q^{71} -12.0527 q^{73} +97.4690i q^{77} +9.02633 q^{79} +(57.5964 + 56.9530i) q^{81} +0.688486i q^{83} +(78.9737 + 15.5936i) q^{87} -7.10831i q^{89} +114.868 q^{91} +(4.64911 - 23.5454i) q^{93} -111.947 q^{97} +(29.0263 - 70.6362i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 8 q^{7} - 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 8 * q^7 - 8 * q^9	\( 4 q + 4 q^{3} + 8 q^{7} - 8 q^{9} + 40 q^{13} - 32 q^{19} - 52 q^{21} + 28 q^{27} - 32 q^{31} - 24 q^{33} + 88 q^{37} + 40 q^{39} + 56 q^{43} + 180 q^{49} - 72 q^{51} - 152 q^{57} - 64 q^{61} - 256 q^{63} - 328 q^{67} - 132 q^{69} - 200 q^{73} + 112 q^{79} + 28 q^{81} + 240 q^{87} + 80 q^{91} - 32 q^{93} - 296 q^{97} + 192 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 8 * q^7 - 8 * q^9 + 40 * q^13 - 32 * q^19 - 52 * q^21 + 28 * q^27 - 32 * q^31 - 24 * q^33 + 88 * q^37 + 40 * q^39 + 56 * q^43 + 180 * q^49 - 72 * q^51 - 152 * q^57 - 64 * q^61 - 256 * q^63 - 328 * q^67 - 132 * q^69 - 200 * q^73 + 112 * q^79 + 28 * q^81 + 240 * q^87 + 80 * q^91 - 32 * q^93 - 296 * q^97 + 192 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more