LMFDB - Embedded newform 1200.3.c.k.449.2

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:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$1.14412 + 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1200.449
Dual form			1200.3.c.k.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.94317 + 0.581139i) q^{3} +11.4868i q^{7} +(8.32456 - 3.42079i) q^{9} +O(q^{10})$	$q+(-2.94317 + 0.581139i) q^{3} +11.4868i q^{7} +(8.32456 - 3.42079i) q^{9} -8.48528i q^{11} -10.0000i q^{13} -3.55415 q^{17} -10.9737 q^{19} +(-6.67544 - 33.8078i) q^{21} -17.6590 q^{23} +(-22.5127 + 14.9057i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(4.93113 + 24.9737i) q^{33} +59.9473i q^{37} +(5.81139 + 29.4317i) q^{39} +20.5247i q^{41} -42.4605i q^{43} +88.2952 q^{47} -82.9473 q^{49} +(10.4605 - 2.06546i) q^{51} +3.55415 q^{53} +(32.2974 - 6.37722i) q^{57} -77.7445i q^{59} +21.9473 q^{61} +(39.2940 + 95.6228i) q^{63} -53.5395i q^{67} +(51.9737 - 10.2624i) q^{69} -69.2592i q^{71} +12.0527i q^{73} +97.4690 q^{77} -9.02633 q^{79} +(57.5964 - 56.9530i) q^{81} -0.688486 q^{83} +(15.5936 + 78.9737i) q^{87} -7.10831i q^{89} +114.868 q^{91} +(23.5454 - 4.64911i) q^{93} -111.947i q^{97} +(-29.0263 - 70.6362i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{9}+O(q^{10}) $ 8 * q + 16 * q^9	$ 8 q + 16 q^{9} + 64 q^{19} - 104 q^{21} - 64 q^{31} - 80 q^{39} - 360 q^{49} - 144 q^{51} - 128 q^{61} + 264 q^{69} - 224 q^{79} + 56 q^{81} + 160 q^{91} - 384 q^{99}+O(q^{100}) $ 8 * q + 16 * q^9 + 64 * q^19 - 104 * q^21 - 64 * q^31 - 80 * q^39 - 360 * q^49 - 144 * q^51 - 128 * q^61 + 264 * q^69 - 224 * q^79 + 56 * q^81 + 160 * q^91 - 384 * 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
$3$	−2.94317	+	0.581139i	−0.981058	+	0.193713i
$3$	−2.94317	+	0.581139i	−0.981058	+	0.193713i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$			11.4868i			1.64098i	0.571664	+	0.820488i	$0.306298\pi$
$7$			11.4868i			1.64098i	−0.571664	+	0.820488i	$0.693702\pi$
$8$		0			0
$9$	8.32456	−	3.42079i	0.924951	−	0.380087i
$10$		0			0
$11$		−	8.48528i		−	0.771389i	−0.922627	−	0.385695i	$-0.873962\pi$
$11$		−	8.48528i		−	0.771389i	0.922627	−	0.385695i	$-0.126038\pi$
$12$		0			0
$13$		−	10.0000i		−	0.769231i	−0.923077	−	0.384615i	$-0.874334\pi$
$13$		−	10.0000i		−	0.769231i	0.923077	−	0.384615i	$-0.125666\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.55415			−0.209068			−0.104534	−	0.994521i	$-0.533335\pi$
$17$	−3.55415			−0.209068			−0.104534	+	0.994521i	$0.533335\pi$
$18$		0			0
$19$	−10.9737			−0.577561			−0.288781	−	0.957395i	$-0.593250\pi$
$19$	−10.9737			−0.577561			−0.288781	+	0.957395i	$0.593250\pi$
$20$		0			0
$21$	−6.67544	−	33.8078i	−0.317878	−	1.60989i
$22$		0			0
$23$	−17.6590			−0.767785			−0.383892	−	0.923378i	$-0.625417\pi$
$23$	−17.6590			−0.767785			−0.383892	+	0.923378i	$0.625417\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−22.5127	+	14.9057i	−0.833803	+	0.552063i
$28$		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
$33$	4.93113	+	24.9737i	0.149428	+	0.756778i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			59.9473i			1.62020i	0.586293	+	0.810099i	$0.300587\pi$
$37$			59.9473i			1.62020i	−0.586293	+	0.810099i	$0.699413\pi$
$38$		0			0
$39$	5.81139	+	29.4317i	0.149010	+	0.754660i
$40$		0			0
$41$			20.5247i			0.500603i	0.968168	+	0.250301i	$0.0805297\pi$
$41$			20.5247i			0.500603i	−0.968168	+	0.250301i	$0.919470\pi$
$42$		0			0
$43$		−	42.4605i		−	0.987453i	−0.869617	−	0.493727i	$-0.835634\pi$
$43$		−	42.4605i		−	0.987453i	0.869617	−	0.493727i	$-0.164366\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	88.2952			1.87862			0.939311	−	0.343067i	$-0.111466\pi$
$47$	88.2952			1.87862			0.939311	+	0.343067i	$0.111466\pi$
$48$		0			0
$49$	−82.9473			−1.69280
$50$		0			0
$51$	10.4605	−	2.06546i	0.205108	−	0.0404992i
$52$		0			0
$53$	3.55415			0.0670595			0.0335298	−	0.999438i	$-0.489325\pi$
$53$	3.55415			0.0670595			0.0335298	+	0.999438i	$0.489325\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	32.2974	−	6.37722i	0.566621	−	0.111881i
$58$		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
$63$	39.2940	+	95.6228i	0.623714	+	1.51782i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	53.5395i		−	0.799097i	−0.916712	−	0.399549i	$-0.869167\pi$
$67$		−	53.5395i		−	0.799097i	0.916712	−	0.399549i	$-0.130833\pi$
$68$		0			0
$69$	51.9737	−	10.2624i	0.753242	−	0.148730i
$70$		0			0
$71$		−	69.2592i		−	0.975482i	−0.872988	−	0.487741i	$-0.837821\pi$
$71$		−	69.2592i		−	0.975482i	0.872988	−	0.487741i	$-0.162179\pi$
$72$		0			0
$73$			12.0527i			0.165105i	0.996587	+	0.0825525i	$0.0263072\pi$
$73$			12.0527i			0.165105i	−0.996587	+	0.0825525i	$0.973693\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	97.4690			1.26583
$78$		0			0
$79$	−9.02633			−0.114257			−0.0571287	−	0.998367i	$-0.518195\pi$
$79$	−9.02633			−0.114257			−0.0571287	+	0.998367i	$0.518195\pi$
$80$		0			0
$81$	57.5964	−	56.9530i	0.711067	−	0.703124i
$82$		0			0
$83$	−0.688486			−0.00829501			−0.00414750	−	0.999991i	$-0.501320\pi$
$83$	−0.688486			−0.00829501			−0.00414750	+	0.999991i	$0.501320\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	15.5936	+	78.9737i	0.179237	+	0.907743i
$88$		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
$93$	23.5454	−	4.64911i	0.253176	−	0.0499904i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	111.947i		−	1.15410i	−0.816710	−	0.577048i	$-0.804205\pi$
$97$		−	111.947i		−	1.15410i	0.816710	−	0.577048i	$-0.195795\pi$
$98$		0			0
$99$	−29.0263	−	70.6362i	−0.293195	−	0.713497i
$100$		0			0
$101$		−	155.489i		−	1.53950i	−0.638348	−	0.769748i	$-0.720382\pi$
$101$		−	155.489i		−	1.53950i	0.638348	−	0.769748i	$-0.279618\pi$
$102$		0			0
$103$		−	7.59217i		−	0.0737104i	−0.999321	−	0.0368552i	$-0.988266\pi$
$103$		−	7.59217i		−	0.0737104i	0.999321	−	0.0368552i	$-0.0117340\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	16.2821			0.152169			0.0760845	−	0.997101i	$-0.475758\pi$
$107$	16.2821			0.152169			0.0760845	+	0.997101i	$0.475758\pi$
$108$		0			0
$109$	93.8420			0.860936			0.430468	−	0.902606i	$-0.358349\pi$
$109$	93.8420			0.860936			0.430468	+	0.902606i	$0.358349\pi$
$110$		0			0
$111$	−34.8377	−	176.435i	−0.313853	−	1.58951i
$112$		0			0
$113$	−195.738			−1.73220			−0.866098	−	0.499874i	$-0.833380\pi$
$113$	−195.738			−1.73220			−0.866098	+	0.499874i	$0.833380\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−34.2079	−	83.2456i	−0.292375	−	0.711500i
$118$		0			0
$119$		−	40.8260i		−	0.343075i
$120$		0			0
$121$	49.0000			0.404959
$122$		0			0
$123$	−11.9277	−	60.4078i	−0.0969733	−	0.491121i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	77.5395i		−	0.610547i	−0.952265	−	0.305274i	$-0.901252\pi$
$127$		−	77.5395i		−	0.610547i	0.952265	−	0.305274i	$-0.0987479\pi$
$128$		0			0
$129$	24.6754	+	124.969i	0.191283	+	0.968749i
$130$		0			0
$131$		−	101.600i		−	0.775572i	−0.921749	−	0.387786i	$-0.873240\pi$
$131$		−	101.600i		−	0.775572i	0.921749	−	0.387786i	$-0.126760\pi$
$132$		0			0
$133$		−	126.053i		−	0.947764i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	176.014			1.28477			0.642386	−	0.766381i	$-0.277945\pi$
$137$	176.014			1.28477			0.642386	+	0.766381i	$0.277945\pi$
$138$		0			0
$139$	188.816			1.35839			0.679193	−	0.733960i	$-0.262330\pi$
$139$	188.816			1.35839			0.679193	+	0.733960i	$0.262330\pi$
$140$		0			0
$141$	−259.868	+	51.3118i	−1.84304	+	0.363913i
$142$		0			0
$143$	−84.8528			−0.593376
$144$		0			0
$145$		0			0
$146$		0			0
$147$	244.128	−	48.2039i	1.66074	−	0.327918i
$148$		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
$153$	−29.5868	+	12.1580i	−0.193378	+	0.0794641i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			123.842i			0.788803i	0.918938	+	0.394401i	$0.129048\pi$
$157$			123.842i			0.788803i	−0.918938	+	0.394401i	$0.870952\pi$
$158$		0			0
$159$	−10.4605	+	2.06546i	−0.0657893	+	0.0129903i
$160$		0			0
$161$		−	202.847i		−	1.25992i
$162$		0			0
$163$		−	159.381i		−	0.977801i	−0.872340	−	0.488900i	$-0.837398\pi$
$163$		−	159.381i		−	0.977801i	0.872340	−	0.488900i	$-0.162602\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−10.7742			−0.0645161			−0.0322581	−	0.999480i	$-0.510270\pi$
$167$	−10.7742			−0.0645161			−0.0322581	+	0.999480i	$0.510270\pi$
$168$		0			0
$169$	69.0000			0.408284
$170$		0			0
$171$	−91.3509	+	37.5386i	−0.534216	+	0.219524i
$172$		0			0
$173$	283.345			1.63783			0.818916	−	0.573913i	$-0.194575\pi$
$173$	283.345			1.63783			0.818916	+	0.573913i	$0.194575\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	45.1804	+	228.816i	0.255256	+	1.29274i
$178$		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
$183$	−64.5948	+	12.7544i	−0.352977	+	0.0696964i
$184$		0			0
$185$		0			0
$186$		0			0
$187$			30.1580i			0.161273i
$188$		0			0
$189$	−171.219	−	258.599i	−0.905922	−	1.36825i
$190$		0			0
$191$			159.620i			0.835706i	0.908515	+	0.417853i	$0.137217\pi$
$191$			159.620i			0.835706i	−0.908515	+	0.417853i	$0.862783\pi$
$192$		0			0
$193$		−	92.0527i		−	0.476957i	−0.971148	−	0.238478i	$-0.923351\pi$
$193$		−	92.0527i		−	0.476957i	0.971148	−	0.238478i	$-0.0766486\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−69.8360			−0.354497			−0.177249	−	0.984166i	$-0.556720\pi$
$197$	−69.8360			−0.354497			−0.177249	+	0.984166i	$0.556720\pi$
$198$		0			0
$199$	148.921			0.748347			0.374173	−	0.927359i	$-0.377926\pi$
$199$	148.921			0.748347			0.374173	+	0.927359i	$0.377926\pi$
$200$		0			0
$201$	31.1139	+	157.576i	0.154795	+	0.783961i
$202$		0			0
$203$	308.224			1.51835
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−147.004	+	60.4078i	−0.710163	+	0.291825i
$208$		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
$213$	40.2492	+	203.842i	0.188963	+	0.957005i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	91.8947i		−	0.423478i
$218$		0			0
$219$	−7.00427	−	35.4731i	−0.0319830	−	0.161978i
$220$		0			0
$221$			35.5415i			0.160821i
$222$		0			0
$223$			71.3815i			0.320096i	0.987109	+	0.160048i	$0.0511650\pi$
$223$			71.3815i			0.320096i	−0.987109	+	0.160048i	$0.948835\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	29.1217			0.128290			0.0641448	−	0.997941i	$-0.479568\pi$
$227$	29.1217			0.128290			0.0641448	+	0.997941i	$0.479568\pi$
$228$		0			0
$229$	129.684			0.566306			0.283153	−	0.959075i	$-0.408620\pi$
$229$	129.684			0.566306			0.283153	+	0.959075i	$0.408620\pi$
$230$		0			0
$231$	−286.868	+	56.6430i	−1.24185	+	0.245208i
$232$		0			0
$233$	185.876			0.797751			0.398875	−	0.917005i	$-0.369401\pi$
$233$	185.876			0.797751			0.398875	+	0.917005i	$0.369401\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	26.5661	−	5.24555i	0.112093	−	0.0221331i
$238$		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
$243$	−136.419	+	201.094i	−0.561394	+	0.827549i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			109.737i			0.444278i
$248$		0			0
$249$	2.02633	−	0.400106i	0.00813789	−	0.00160685i
$250$		0			0
$251$		−	84.6294i		−	0.337169i	−0.985687	−	0.168584i	$-0.946080\pi$
$251$		−	84.6294i		−	0.337169i	0.985687	−	0.168584i	$-0.0539196\pi$
$252$		0			0
$253$			149.842i			0.592261i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−292.054			−1.13640			−0.568198	−	0.822892i	$-0.692359\pi$
$257$	−292.054			−1.13640			−0.568198	+	0.822892i	$0.692359\pi$
$258$		0			0
$259$	−688.605			−2.65871
$260$		0			0
$261$	−91.7893	−	223.371i	−0.351683	−	0.855829i
$262$		0			0
$263$	−277.725			−1.05599			−0.527995	−	0.849247i	$-0.677056\pi$
$263$	−277.725			−1.05599			−0.527995	+	0.849247i	$0.677056\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	4.13091	+	20.9210i	0.0154716	+	0.0783558i
$268$		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
$273$	−338.078	+	66.7544i	−1.23838	+	0.244522i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	415.842i		−	1.50123i	−0.660737	−	0.750617i	$-0.729756\pi$
$277$		−	415.842i		−	1.50123i	0.660737	−	0.750617i	$-0.270244\pi$
$278$		0			0
$279$	−66.5964	+	27.3663i	−0.238697	+	0.0980871i
$280$		0			0
$281$			431.726i			1.53639i	0.640216	+	0.768195i	$0.278845\pi$
$281$			431.726i			1.53639i	−0.640216	+	0.768195i	$0.721155\pi$
$282$		0			0
$283$		−	141.540i		−	0.500140i	−0.968228	−	0.250070i	$-0.919546\pi$
$283$		−	141.540i		−	0.500140i	0.968228	−	0.250070i	$-0.0804536\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−235.764			−0.821477
$288$		0			0
$289$	−276.368			−0.956291
$290$		0			0
$291$	65.0569	+	329.481i	0.223563	+	1.13224i
$292$		0			0
$293$	−164.998			−0.563133			−0.281566	−	0.959542i	$-0.590854\pi$
$293$	−164.998			−0.563133			−0.281566	+	0.959542i	$0.590854\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	126.479	+	191.026i	0.425855	+	0.643186i
$298$		0			0
$299$			176.590i			0.590604i
$300$		0			0
$301$	487.737			1.62039
$302$		0			0
$303$	90.3607	+	457.631i	0.298220	+	1.51033i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			159.381i			0.519158i	0.965722	+	0.259579i	$0.0835838\pi$
$307$			159.381i			0.519158i	−0.965722	+	0.259579i	$0.916416\pi$
$308$		0			0
$309$	4.41210	+	22.3451i	0.0142787	+	0.0723142i
$310$		0			0
$311$		−	143.096i		−	0.460117i	−0.973177	−	0.230058i	$-0.926108\pi$
$311$		−	143.096i		−	0.460117i	0.973177	−	0.230058i	$-0.0738917\pi$
$312$		0			0
$313$		−	501.684i		−	1.60282i	−0.598113	−	0.801412i	$-0.704082\pi$
$313$		−	501.684i		−	1.60282i	0.598113	−	0.801412i	$-0.295918\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−334.257			−1.05444			−0.527219	−	0.849730i	$-0.676765\pi$
$317$	−334.257			−1.05444			−0.527219	+	0.849730i	$0.676765\pi$
$318$		0			0
$319$	−227.684			−0.713743
$320$		0			0
$321$	−47.9210	+	9.46215i	−0.149287	+	0.0294771i
$322$		0			0
$323$	39.0021			0.120750
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−276.193	+	54.5352i	−0.844628	+	0.166774i
$328$		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
$333$	205.067	+	499.035i	0.615817	+	1.49860i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			129.684i			0.384819i	0.981315	+	0.192409i	$0.0616302\pi$
$337$			129.684i			0.384819i	−0.981315	+	0.192409i	$0.938370\pi$
$338$		0			0
$339$	576.092	−	113.751i	1.69939	−	0.335549i
$340$		0			0
$341$			67.8823i			0.199068i
$342$		0			0
$343$		−	389.947i		−	1.13687i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−346.985			−0.999956			−0.499978	−	0.866038i	$-0.666659\pi$
$347$	−346.985			−0.999956			−0.499978	+	0.866038i	$0.666659\pi$
$348$		0			0
$349$	−509.579			−1.46011			−0.730055	−	0.683388i	$-0.760506\pi$
$349$	−509.579			−1.46011			−0.730055	+	0.683388i	$0.760506\pi$
$350$		0			0
$351$	149.057	+	225.127i	0.424664	+	0.641387i
$352$		0			0
$353$	−637.679			−1.80646			−0.903229	−	0.429159i	$-0.858810\pi$
$353$	−637.679			−1.80646			−0.903229	+	0.429159i	$0.858810\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	23.7256	+	120.158i	0.0664582	+	0.336577i
$358$		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
$363$	−144.216	+	28.4758i	−0.397288	+	0.0784457i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	505.828i		−	1.37828i	−0.724629	−	0.689140i	$-0.757989\pi$
$367$		−	505.828i		−	1.37828i	0.724629	−	0.689140i	$-0.242011\pi$
$368$		0			0
$369$	70.2107	+	170.859i	0.190273	+	0.463033i
$370$		0			0
$371$			40.8260i			0.110043i
$372$		0			0
$373$		−	416.053i		−	1.11542i	−0.830035	−	0.557711i	$-0.811680\pi$
$373$		−	416.053i		−	1.11542i	0.830035	−	0.557711i	$-0.188320\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−268.328			−0.711746
$378$		0			0
$379$	82.7630			0.218372			0.109186	−	0.994021i	$-0.465176\pi$
$379$	82.7630			0.218372			0.109186	+	0.994021i	$0.465176\pi$
$380$		0			0
$381$	45.0612	+	228.212i	0.118271	+	0.598982i
$382$		0			0
$383$	334.592			0.873608			0.436804	−	0.899557i	$-0.356110\pi$
$383$	334.592			0.873608			0.436804	+	0.899557i	$0.356110\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−145.248	−	353.465i	−0.375319	−	0.913346i
$388$		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
$393$	59.0437	+	299.026i	0.150238	+	0.760881i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			214.000i			0.539043i	0.962994	+	0.269521i	$0.0868655\pi$
$397$			214.000i			0.539043i	−0.962994	+	0.269521i	$0.913135\pi$
$398$		0			0
$399$	73.2541	+	370.995i	0.183594	+	0.929812i
$400$		0			0
$401$			726.086i			1.81069i	0.424677	+	0.905345i	$0.360387\pi$
$401$			726.086i			1.81069i	−0.424677	+	0.905345i	$0.639613\pi$
$402$		0			0
$403$			80.0000i			0.198511i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	508.670			1.24980
$408$		0			0
$409$	346.158			0.846352			0.423176	−	0.906047i	$-0.360915\pi$
$409$	346.158			0.846352			0.423176	+	0.906047i	$0.360915\pi$
$410$		0			0
$411$	−518.039	+	102.288i	−1.26044	+	0.248877i
$412$		0			0
$413$	893.038			2.16232
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−555.717	+	109.728i	−1.33266	+	0.263137i
$418$		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
$423$	735.019	−	302.039i	1.73763	−	0.714041i
$424$		0			0
$425$		0			0
$426$		0			0
$427$			252.105i			0.590411i
$428$		0			0
$429$	249.737	−	49.3113i	0.582137	−	0.114945i
$430$		0			0
$431$			776.068i			1.80062i	0.435247	+	0.900311i	$0.356661\pi$
$431$			776.068i			1.80062i	−0.435247	+	0.900311i	$0.643339\pi$
$432$		0			0
$433$		−	195.526i		−	0.451561i	−0.974178	−	0.225781i	$-0.927507\pi$
$433$		−	195.526i		−	0.451561i	0.974178	−	0.225781i	$-0.0724932\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	193.785			0.443443
$438$		0			0
$439$	598.763			1.36392			0.681962	−	0.731387i	$-0.261127\pi$
$439$	598.763			1.36392			0.681962	+	0.731387i	$0.261127\pi$
$440$		0			0
$441$	−690.500	+	283.745i	−1.56576	+	0.643413i
$442$		0			0
$443$	−216.728			−0.489228			−0.244614	−	0.969621i	$-0.578661\pi$
$443$	−216.728			−0.489228			−0.244614	+	0.969621i	$0.578661\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	42.1848	+	213.645i	0.0943732	+	0.477953i
$448$		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
$453$	276.503	−	54.5964i	0.610383	−	0.120522i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	553.052i		−	1.21018i	−0.796157	−	0.605090i	$-0.793137\pi$
$457$		−	553.052i		−	1.21018i	0.796157	−	0.605090i	$-0.206863\pi$
$458$		0			0
$459$	80.0135	−	52.9771i	0.174321	−	0.115419i
$460$		0			0
$461$		−	124.749i		−	0.270605i	−0.990804	−	0.135302i	$-0.956799\pi$
$461$		−	124.749i		−	0.270605i	0.990804	−	0.135302i	$-0.0432006\pi$
$462$		0			0
$463$		−	669.723i		−	1.44649i	−0.690593	−	0.723243i	$-0.742651\pi$
$463$		−	669.723i		−	1.44649i	0.690593	−	0.723243i	$-0.257349\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−486.880			−1.04257			−0.521285	−	0.853383i	$-0.674547\pi$
$467$	−486.880			−1.04257			−0.521285	+	0.853383i	$0.674547\pi$
$468$		0			0
$469$	614.999			1.31130
$470$		0			0
$471$	−71.9694	−	364.489i	−0.152801	−	0.773861i
$472$		0			0
$473$	−360.289			−0.761711
$474$		0			0
$475$		0			0
$476$		0			0
$477$	29.5868	−	12.1580i	0.0620267	−	0.0254885i
$478$		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
$483$	117.882	+	597.013i	0.244062	+	1.23605i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	263.381i		−	0.540824i	−0.962745	−	0.270412i	$-0.912840\pi$
$487$		−	263.381i		−	0.540824i	0.962745	−	0.270412i	$-0.0871600\pi$
$488$		0			0
$489$	92.6228	+	469.088i	0.189413	+	0.959279i
$490$		0			0
$491$			711.646i			1.44938i	0.689074	+	0.724691i	$0.258017\pi$
$491$			711.646i			1.44938i	−0.689074	+	0.724691i	$0.741983\pi$
$492$		0			0
$493$			95.3680i			0.193444i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	795.569			1.60074
$498$		0			0
$499$	23.0790			0.0462505			0.0231253	−	0.999733i	$-0.492638\pi$
$499$	23.0790			0.0462505			0.0231253	+	0.999733i	$0.492638\pi$
$500$		0			0
$501$	31.7103	−	6.26130i	0.0632941	−	0.0124976i
$502$		0			0
$503$	40.5844			0.0806847			0.0403423	−	0.999186i	$-0.487155\pi$
$503$	40.5844			0.0806847			0.0403423	+	0.999186i	$0.487155\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−203.079	+	40.0986i	−0.400550	+	0.0790899i
$508$		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
$513$	247.047	−	163.570i	0.481572	−	0.318850i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	749.210i		−	1.44915i
$518$		0			0
$519$	−833.934	+	164.663i	−1.60681	+	0.317269i
$520$		0			0
$521$		−	539.857i		−	1.03619i	−0.855322	−	0.518097i	$-0.826641\pi$
$521$		−	539.857i		−	1.03619i	0.855322	−	0.518097i	$-0.173359\pi$
$522$		0			0
$523$			266.644i			0.509836i	0.966963	+	0.254918i	$0.0820485\pi$
$523$			266.644i			0.509836i	−0.966963	+	0.254918i	$0.917952\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	28.4332			0.0539530
$528$		0			0
$529$	−217.158			−0.410507
$530$		0			0
$531$	−265.947	−	647.188i	−0.500842	−	1.21881i
$532$		0			0
$533$	205.247			0.385079
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−178.191	−	902.447i	−0.331827	−	1.68053i
$538$		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
$543$	−781.954	+	154.399i	−1.44006	+	0.284345i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	214.566i		−	0.392259i	−0.980578	−	0.196130i	$-0.937163\pi$
$547$		−	214.566i		−	0.392259i	0.980578	−	0.196130i	$-0.0628373\pi$
$548$		0			0
$549$	182.702	−	75.0771i	0.332790	−	0.136753i
$550$		0			0
$551$			294.454i			0.534400i
$552$		0			0
$553$		−	103.684i		−	0.187494i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−674.821			−1.21153			−0.605764	−	0.795644i	$-0.707133\pi$
$557$	−674.821			−1.21153			−0.605764	+	0.795644i	$0.707133\pi$
$558$		0			0
$559$	−424.605			−0.759580
$560$		0			0
$561$	−17.5260	−	88.7603i	−0.0312406	−	0.158218i
$562$		0			0
$563$	−718.513			−1.27622			−0.638111	−	0.769944i	$-0.720284\pi$
$563$	−718.513			−1.27622			−0.638111	+	0.769944i	$0.720284\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	654.210	+	661.601i	1.15381	+	1.16684i
$568$		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
$573$	−92.7613	−	469.789i	−0.161887	−	0.819877i
$574$		0			0
$575$		0			0
$576$		0			0
$577$			984.947i			1.70701i	0.521082	+	0.853507i	$0.325529\pi$
$577$			984.947i			1.70701i	−0.521082	+	0.853507i	$0.674471\pi$
$578$		0			0
$579$	53.4954	+	270.927i	0.0923927	+	0.467922i
$580$		0			0
$581$		−	7.90852i		−	0.0136119i
$582$		0			0
$583$		−	30.1580i		−	0.0517290i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−378.172			−0.644245			−0.322122	−	0.946698i	$-0.604396\pi$
$587$	−378.172			−0.644245			−0.322122	+	0.946698i	$0.604396\pi$
$588$		0			0
$589$	87.7893			0.149048
$590$		0			0
$591$	205.540	−	40.5844i	0.347783	−	0.0686707i
$592$		0			0
$593$	−265.221			−0.447253			−0.223626	−	0.974675i	$-0.571790\pi$
$593$	−265.221			−0.447253			−0.223626	+	0.974675i	$0.571790\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−438.301	+	86.5438i	−0.734172	+	0.144964i
$598$		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
$603$	−183.147	−	445.693i	−0.303727	−	0.739125i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	835.487i		−	1.37642i	−0.725512	−	0.688210i	$-0.758397\pi$
$607$		−	835.487i		−	1.37642i	0.725512	−	0.688210i	$-0.241603\pi$
$608$		0			0
$609$	−907.157	+	179.121i	−1.48959	+	0.294123i
$610$		0			0
$611$		−	882.952i		−	1.44509i
$612$		0			0
$613$			833.263i			1.35932i	0.733528	+	0.679660i	$0.237872\pi$
$613$			833.263i			1.35932i	−0.733528	+	0.679660i	$0.762128\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	455.098			0.737598			0.368799	−	0.929509i	$-0.379769\pi$
$617$	455.098			0.737598			0.368799	+	0.929509i	$0.379769\pi$
$618$		0			0
$619$	336.710			0.543959			0.271979	−	0.962303i	$-0.412322\pi$
$619$	336.710			0.543959			0.271979	+	0.962303i	$0.412322\pi$
$620$		0			0
$621$	397.552	−	263.220i	0.640181	−	0.423865i
$622$		0			0
$623$	81.6520			0.131063
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−54.1125	−	274.053i	−0.0863039	−	0.437086i
$628$		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
$633$	−841.283	+	166.114i	−1.32904	+	0.262423i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			829.473i			1.30216i
$638$		0			0
$639$	−236.921	−	576.552i	−0.370768	−	0.902273i
$640$		0			0
$641$			337.011i			0.525758i	0.964829	+	0.262879i	$0.0846719\pi$
$641$			337.011i			0.525758i	−0.964829	+	0.262879i	$0.915328\pi$
$642$		0			0
$643$			599.381i			0.932164i	0.884742	+	0.466082i	$0.154335\pi$
$643$			599.381i			0.932164i	−0.884742	+	0.466082i	$0.845665\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	47.0224			0.0726775			0.0363388	−	0.999340i	$-0.488430\pi$
$647$	47.0224			0.0726775			0.0363388	+	0.999340i	$0.488430\pi$
$648$		0			0
$649$	−659.684			−1.01646
$650$		0			0
$651$	53.4036	+	270.462i	0.0820331	+	0.415456i
$652$		0			0
$653$	−559.228			−0.856399			−0.428199	−	0.903684i	$-0.640852\pi$
$653$	−559.228			−0.856399			−0.428199	+	0.903684i	$0.640852\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	41.2296	+	100.333i	0.0627543	+	0.152714i
$658$		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
$663$	−20.6546	−	104.605i	−0.0311532	−	0.157775i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			473.842i			0.710408i
$668$		0			0
$669$	−41.4826	−	210.088i	−0.0620068	−	0.314033i
$670$		0			0
$671$		−	186.229i		−	0.277540i
$672$		0			0
$673$			1019.42i			1.51474i	0.652985	+	0.757370i	$0.273516\pi$
$673$			1019.42i			1.51474i	−0.652985	+	0.757370i	$0.726484\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	624.616			0.922624			0.461312	−	0.887238i	$-0.347379\pi$
$677$	624.616			0.922624			0.461312	+	0.887238i	$0.347379\pi$
$678$		0			0
$679$	1285.92			1.89384
$680$		0			0
$681$	−85.7103	+	16.9238i	−0.125860	+	0.0248513i
$682$		0			0
$683$	525.882			0.769959			0.384980	−	0.922925i	$-0.374208\pi$
$683$	525.882			0.769959			0.384980	+	0.922925i	$0.374208\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−381.683	+	75.3644i	−0.555579	+	0.109701i
$688$		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
$693$	811.386	−	333.421i	1.17083	−	0.481126i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	72.9480i		−	0.104660i
$698$		0			0
$699$	−547.065	+	108.020i	−0.782640	+	0.154535i
$700$		0			0
$701$		−	606.045i		−	0.864544i	−0.901743	−	0.432272i	$-0.857712\pi$
$701$		−	606.045i		−	0.864544i	0.901743	−	0.432272i	$-0.142288\pi$
$702$		0			0
$703$		−	657.842i		−	0.935764i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1786.08			2.52627
$708$		0			0
$709$	−489.473			−0.690371			−0.345186	−	0.938534i	$-0.612184\pi$
$709$	−489.473			−0.690371			−0.345186	+	0.938534i	$0.612184\pi$
$710$		0			0
$711$	−75.1402	+	30.8772i	−0.105682	+	0.0434278i
$712$		0			0
$713$	141.272			0.198138
$714$		0			0
$715$		0			0
$716$		0			0
$717$	175.920	+	890.947i	0.245356	+	1.24260i
$718$		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
$723$	699.080	−	138.036i	0.966916	−	0.190921i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	150.172i		−	0.206563i	−0.994652	−	0.103282i	$-0.967066\pi$
$727$		−	150.172i		−	0.206563i	0.994652	−	0.103282i	$-0.0329343\pi$
$728$		0			0
$729$	284.641	−	671.134i	0.390453	−	0.920623i
$730$		0			0
$731$			150.911i			0.206445i
$732$		0			0
$733$		−	675.526i		−	0.921591i	−0.887506	−	0.460795i	$-0.847564\pi$
$733$		−	675.526i		−	0.921591i	0.887506	−	0.460795i	$-0.152436\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−454.298			−0.616415
$738$		0			0
$739$	−936.921			−1.26782			−0.633911	−	0.773406i	$-0.718552\pi$
$739$	−936.921			−1.26782			−0.633911	+	0.773406i	$0.718552\pi$
$740$		0			0
$741$	−63.7722	−	322.974i	−0.0860624	−	0.435863i
$742$		0			0
$743$	452.939			0.609608			0.304804	−	0.952415i	$-0.401409\pi$
$743$	452.939			0.609608			0.304804	+	0.952415i	$0.401409\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.73134	+	2.35516i	−0.00767247	+	0.00315283i
$748$		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
$753$	49.1814	+	249.079i	0.0653140	+	0.330782i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	123.315i		−	0.162900i	−0.996677	−	0.0814500i	$-0.974045\pi$
$757$		−	123.315i		−	0.162900i	0.996677	−	0.0814500i	$-0.0259551\pi$
$758$		0			0
$759$	−87.0790	−	441.011i	−0.114729	−	0.581042i
$760$		0			0
$761$		−	502.715i		−	0.660598i	−0.943876	−	0.330299i	$-0.892850\pi$
$761$		−	502.715i		−	0.660598i	0.943876	−	0.330299i	$-0.107150\pi$
$762$		0			0
$763$			1077.95i			1.41278i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−777.445			−1.01362
$768$		0			0
$769$	286.316			0.372323			0.186161	−	0.982519i	$-0.440395\pi$
$769$	286.316			0.372323			0.186161	+	0.982519i	$0.440395\pi$
$770$		0			0
$771$	859.565	−	169.724i	1.11487	−	0.220135i
$772$		0			0
$773$	972.030			1.25748			0.628739	−	0.777617i	$-0.283572\pi$
$773$	972.030			1.25748			0.628739	+	0.777617i	$0.283572\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	2026.68	−	400.175i	2.60835	−	0.515026i
$778$		0			0
$779$		−	225.231i		−	0.289129i
$780$		0			0
$781$	−587.684			−0.752476
$782$		0			0
$783$	399.962	+	604.078i	0.510807	+	0.771492i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			1492.70i			1.89669i	0.317237	+	0.948346i	$0.397245\pi$
$787$			1492.70i			1.89669i	−0.317237	+	0.948346i	$0.602755\pi$
$788$		0			0
$789$	817.394	−	161.397i	1.03599	−	0.204559i
$790$		0			0
$791$		−	2248.41i		−	2.84249i
$792$		0			0
$793$		−	219.473i		−	0.276763i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−94.3618			−0.118396			−0.0591981	−	0.998246i	$-0.518854\pi$
$797$	−94.3618			−0.118396			−0.0591981	+	0.998246i	$0.518854\pi$
$798$		0			0
$799$	−313.815			−0.392760
$800$		0			0
$801$	−24.3160	−	59.1735i	−0.0303571	−	0.0738746i
$802$		0			0
$803$	102.270			0.127360
$804$		0			0
$805$		0			0
$806$		0			0
$807$	166.523	+	843.354i	0.206348	+	1.04505i
$808$		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
$813$	−953.899	+	188.350i	−1.17331	+	0.231673i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			465.947i			0.570315i
$818$		0			0
$819$	956.228	−	392.940i	1.16756	−	0.479780i
$820$		0			0
$821$			337.011i			0.410488i	0.978711	+	0.205244i	$0.0657988\pi$
$821$			337.011i			0.410488i	−0.978711	+	0.205244i	$0.934201\pi$
$822$		0			0
$823$			901.512i			1.09540i	0.836675	+	0.547699i	$0.184496\pi$
$823$			901.512i			1.09540i	−0.836675	+	0.547699i	$0.815504\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	531.354			0.642508			0.321254	−	0.946993i	$-0.395896\pi$
$827$	531.354			0.642508			0.321254	+	0.946993i	$0.395896\pi$
$828$		0			0
$829$	197.631			0.238397			0.119199	−	0.992870i	$-0.461967\pi$
$829$	197.631			0.238397			0.119199	+	0.992870i	$0.461967\pi$
$830$		0			0
$831$	241.662	+	1223.90i	0.290809	+	1.47280i
$832$		0			0
$833$	294.808			0.353911
$834$		0			0
$835$		0			0
$836$		0			0
$837$	180.101	−	119.246i	0.215175	−	0.142468i
$838$		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
$843$	−250.893	−	1270.64i	−0.297619	−	1.50729i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			562.855i			0.664528i
$848$		0			0
$849$	82.2541	+	416.575i	0.0968835	+	0.490666i
$850$		0			0
$851$		−	1058.61i		−	1.24396i
$852$		0			0
$853$		−	1196.42i		−	1.40260i	−0.712865	−	0.701301i	$-0.752603\pi$
$853$		−	1196.42i		−	1.40260i	0.712865	−	0.701301i	$-0.247397\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1266.04			−1.47729			−0.738645	−	0.674095i	$-0.764534\pi$
$857$	−1266.04			−1.47729			−0.738645	+	0.674095i	$0.764534\pi$
$858$		0			0
$859$	−470.868			−0.548159			−0.274079	−	0.961707i	$-0.588373\pi$
$859$	−470.868			−0.548159			−0.274079	+	0.961707i	$0.588373\pi$
$860$		0			0
$861$	693.895	−	137.012i	0.805917	−	0.159131i
$862$		0			0
$863$	−285.057			−0.330310			−0.165155	−	0.986268i	$-0.552812\pi$
$863$	−285.057			−0.330310			−0.165155	+	0.986268i	$0.552812\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	813.399	−	160.608i	0.938177	−	0.185246i
$868$		0			0
$869$			76.5910i			0.0881369i
$870$		0			0
$871$	−535.395			−0.614690
$872$		0			0
$873$	−382.948	−	931.912i	−0.438657	−	1.06748i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	244.579i		−	0.278882i	−0.990230	−	0.139441i	$-0.955469\pi$
$877$		−	244.579i		−	0.278882i	0.990230	−	0.139441i	$-0.0445305\pi$
$878$		0			0
$879$	485.618	−	95.8867i	0.552466	−	0.109086i
$880$		0			0
$881$		−	137.271i		−	0.155813i	−0.996961	−	0.0779065i	$-0.975176\pi$
$881$		−	137.271i		−	0.155813i	0.996961	−	0.0779065i	$-0.0248236\pi$
$882$		0			0
$883$		−	1647.75i		−	1.86608i	−0.359772	−	0.933040i	$-0.617146\pi$
$883$		−	1647.75i		−	1.86608i	0.359772	−	0.933040i	$-0.382854\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−514.419			−0.579954			−0.289977	−	0.957034i	$-0.593648\pi$
$887$	−514.419			−0.579954			−0.289977	+	0.957034i	$0.593648\pi$
$888$		0			0
$889$	890.683			1.00189
$890$		0			0
$891$	−483.263	−	488.722i	−0.542382	−	0.548510i
$892$		0			0
$893$	−968.923			−1.08502
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−102.624	−	519.737i	−0.114408	−	0.579417i
$898$		0			0
$899$			214.663i			0.238779i
$900$		0			0
$901$	−12.6320			−0.0140200
$902$		0			0
$903$	−1435.49	+	283.443i	−1.58969	+	0.313890i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			347.303i			0.382914i	0.981501	+	0.191457i	$0.0613213\pi$
$907$			347.303i			0.382914i	−0.981501	+	0.191457i	$0.938679\pi$
$908$		0			0
$909$	−531.895	−	1294.38i	−0.585143	−	1.42396i
$910$		0			0
$911$		−	304.540i		−	0.334292i	−0.985932	−	0.167146i	$-0.946545\pi$
$911$		−	304.540i		−	0.334292i	0.985932	−	0.167146i	$-0.0534551\pi$
$912$		0			0
$913$			5.84200i			0.00639868i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1167.06			1.27270
$918$		0			0
$919$	244.289			0.265820			0.132910	−	0.991128i	$-0.457568\pi$
$919$	244.289			0.265820			0.132910	+	0.991128i	$0.457568\pi$
$920$		0			0
$921$	−92.6228	−	469.088i	−0.100568	−	0.509324i
$922$		0			0
$923$	−692.592			−0.750371
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−25.9712	−	63.2014i	−0.0280164	−	0.0681785i
$928$		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
$933$	83.1588	+	421.157i	0.0891305	+	0.451401i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			332.053i			0.354379i	0.984177	+	0.177189i	$0.0567005\pi$
$937$			332.053i			0.354379i	−0.984177	+	0.177189i	$0.943300\pi$
$938$		0			0
$939$	291.548	+	1476.54i	0.310488	+	1.57246i
$940$		0			0
$941$		−	1636.99i		−	1.73963i	−0.493381	−	0.869813i	$-0.664239\pi$
$941$		−	1636.99i		−	1.73963i	0.493381	−	0.869813i	$-0.335761\pi$
$942$		0			0
$943$		−	362.447i		−	0.384355i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1209.32			1.27700			0.638500	−	0.769622i	$-0.279555\pi$
$947$	1209.32			1.27700			0.638500	+	0.769622i	$0.279555\pi$
$948$		0			0
$949$	120.527			0.127004
$950$		0			0
$951$	983.776	−	194.250i	1.03446	−	0.204258i
$952$		0			0
$953$	−1205.26			−1.26470			−0.632352	−	0.774681i	$-0.717910\pi$
$953$	−1205.26			−1.26470			−0.632352	+	0.774681i	$0.717910\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	670.114	−	132.316i	0.700223	−	0.138261i
$958$		0			0
$959$			2021.84i			2.10828i
$960$		0			0
$961$	−897.000			−0.933403
$962$		0			0
$963$	135.541	−	55.6975i	0.140749	−	0.0578375i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	1845.93i		−	1.90893i	−0.298325	−	0.954464i	$-0.596428\pi$
$967$		−	1845.93i		−	1.90893i	0.298325	−	0.954464i	$-0.403572\pi$
$968$		0			0
$969$	−114.790	+	22.6656i	−0.118462	+	0.0233908i
$970$		0			0
$971$			1057.91i			1.08950i	0.838598	+	0.544751i	$0.183376\pi$
$971$			1057.91i			1.08950i	−0.838598	+	0.544751i	$0.816624\pi$
$972$		0			0
$973$			2168.89i			2.22908i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	964.028			0.986722			0.493361	−	0.869825i	$-0.335768\pi$
$977$	964.028			0.986722			0.493361	+	0.869825i	$0.335768\pi$
$978$		0			0
$979$	−60.3160			−0.0616098
$980$		0			0
$981$	781.193	−	321.013i	0.796323	−	0.327231i
$982$		0			0
$983$	−460.718			−0.468685			−0.234343	−	0.972154i	$-0.575294\pi$
$983$	−460.718			−0.468685			−0.234343	+	0.972154i	$0.575294\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−589.410	−	2985.06i	−0.597173	−	3.02438i
$988$		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
$993$	1146.13	−	226.307i	1.15421	−	0.227903i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			9.36798i			0.00939617i	0.999989	+	0.00469809i	$0.00149545\pi$
$997$			9.36798i			0.00939617i	−0.999989	+	0.00469809i	$0.998505\pi$
$998$		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.c.k.449.2		8
3.2	odd	2	inner	1200.3.c.k.449.8		8
4.3	odd	2		150.3.b.b.149.3		8
5.2	odd	4		240.3.l.c.161.4		4
5.3	odd	4		1200.3.l.u.401.1		4
5.4	even	2	inner	1200.3.c.k.449.7		8
12.11	even	2		150.3.b.b.149.5		8
15.2	even	4		240.3.l.c.161.3		4
15.8	even	4		1200.3.l.u.401.2		4
15.14	odd	2	inner	1200.3.c.k.449.1		8
20.3	even	4		150.3.d.c.101.4		4
20.7	even	4		30.3.d.a.11.1	✓	4
20.19	odd	2		150.3.b.b.149.6		8
40.27	even	4		960.3.l.e.641.4		4
40.37	odd	4		960.3.l.f.641.1		4
60.23	odd	4		150.3.d.c.101.2		4
60.47	odd	4		30.3.d.a.11.3	yes	4
60.59	even	2		150.3.b.b.149.4		8
120.77	even	4		960.3.l.f.641.2		4
120.107	odd	4		960.3.l.e.641.3		4
180.7	even	12		810.3.h.a.701.1		8
180.47	odd	12		810.3.h.a.701.4		8
180.67	even	12		810.3.h.a.431.4		8
180.167	odd	12		810.3.h.a.431.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.3.d.a.11.1	✓	4	20.7	even	4
30.3.d.a.11.3	yes	4	60.47	odd	4
150.3.b.b.149.3		8	4.3	odd	2
150.3.b.b.149.4		8	60.59	even	2
150.3.b.b.149.5		8	12.11	even	2
150.3.b.b.149.6		8	20.19	odd	2
150.3.d.c.101.2		4	60.23	odd	4
150.3.d.c.101.4		4	20.3	even	4
240.3.l.c.161.3		4	15.2	even	4
240.3.l.c.161.4		4	5.2	odd	4
810.3.h.a.431.1		8	180.167	odd	12
810.3.h.a.431.4		8	180.67	even	12
810.3.h.a.701.1		8	180.7	even	12
810.3.h.a.701.4		8	180.47	odd	12
960.3.l.e.641.3		4	120.107	odd	4
960.3.l.e.641.4		4	40.27	even	4
960.3.l.f.641.1		4	40.37	odd	4
960.3.l.f.641.2		4	120.77	even	4
1200.3.c.k.449.1		8	15.14	odd	2	inner
1200.3.c.k.449.2		8	1.1	even	1	trivial
1200.3.c.k.449.7		8	5.4	even	2	inner
1200.3.c.k.449.8		8	3.2	odd	2	inner
1200.3.l.u.401.1		4	5.3	odd	4
1200.3.l.u.401.2		4	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)

\(f(q)\)	\(=\)	\(q+(-2.94317 + 0.581139i) q^{3} +11.4868i q^{7} +(8.32456 - 3.42079i) q^{9} +O(q^{10})\)	\(q+(-2.94317 + 0.581139i) q^{3} +11.4868i q^{7} +(8.32456 - 3.42079i) q^{9} -8.48528i q^{11} -10.0000i q^{13} -3.55415 q^{17} -10.9737 q^{19} +(-6.67544 - 33.8078i) q^{21} -17.6590 q^{23} +(-22.5127 + 14.9057i) q^{27} -26.8328i q^{29} -8.00000 q^{31} +(4.93113 + 24.9737i) q^{33} +59.9473i q^{37} +(5.81139 + 29.4317i) q^{39} +20.5247i q^{41} -42.4605i q^{43} +88.2952 q^{47} -82.9473 q^{49} +(10.4605 - 2.06546i) q^{51} +3.55415 q^{53} +(32.2974 - 6.37722i) q^{57} -77.7445i q^{59} +21.9473 q^{61} +(39.2940 + 95.6228i) q^{63} -53.5395i q^{67} +(51.9737 - 10.2624i) q^{69} -69.2592i q^{71} +12.0527i q^{73} +97.4690 q^{77} -9.02633 q^{79} +(57.5964 - 56.9530i) q^{81} -0.688486 q^{83} +(15.5936 + 78.9737i) q^{87} -7.10831i q^{89} +114.868 q^{91} +(23.5454 - 4.64911i) q^{93} -111.947i q^{97} +(-29.0263 - 70.6362i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{9}+O(q^{10}) \) 8 * q + 16 * q^9	\( 8 q + 16 q^{9} + 64 q^{19} - 104 q^{21} - 64 q^{31} - 80 q^{39} - 360 q^{49} - 144 q^{51} - 128 q^{61} + 264 q^{69} - 224 q^{79} + 56 q^{81} + 160 q^{91} - 384 q^{99}+O(q^{100}) \) 8 * q + 16 * q^9 + 64 * q^19 - 104 * q^21 - 64 * q^31 - 80 * q^39 - 360 * q^49 - 144 * q^51 - 128 * q^61 + 264 * q^69 - 224 * q^79 + 56 * q^81 + 160 * q^91 - 384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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