LMFDB - Embedded newform 960.3.l.e.641.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,3,Mod(641,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, 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("960.641");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	960.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:	$26.1581053786$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.4
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	960.641
Dual form			960.3.l.e.641.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.581139 + 2.94317i) q^{3} -2.23607i q^{5} +11.4868 q^{7} +(-8.32456 + 3.42079i) q^{9} +O(q^{10})$	\(q+(0.581139 + 2.94317i) q^{3} -2.23607i q^{5} +11.4868 q^{7} +(-8.32456 + 3.42079i) q^{9} -8.48528i q^{11} +10.0000 q^{13} +(6.58114 - 1.29947i) q^{15} -3.55415i q^{17} +10.9737 q^{19} +(6.67544 + 33.8078i) q^{21} -17.6590i q^{23} -5.00000 q^{25} +(-14.9057 - 22.5127i) q^{27} -26.8328i q^{29} +8.00000 q^{31} +(24.9737 - 4.93113i) q^{33} -25.6853i q^{35} +59.9473 q^{37} +(5.81139 + 29.4317i) q^{39} +20.5247i q^{41} -42.4605 q^{43} +(7.64911 + 18.6143i) q^{45} -88.2952i q^{47} +82.9473 q^{49} +(10.4605 - 2.06546i) q^{51} +3.55415i q^{53} -18.9737 q^{55} +(6.37722 + 32.2974i) q^{57} +77.7445i q^{59} -21.9473 q^{61} +(-95.6228 + 39.2940i) q^{63} -22.3607i q^{65} +53.5395 q^{67} +(51.9737 - 10.2624i) q^{69} +69.2592i q^{71} +12.0527 q^{73} +(-2.90569 - 14.7159i) q^{75} -97.4690i q^{77} -9.02633 q^{79} +(57.5964 - 56.9530i) q^{81} +0.688486i q^{83} -7.94733 q^{85} +(78.9737 - 15.5936i) q^{87} +7.10831i q^{89} +114.868 q^{91} +(4.64911 + 23.5454i) q^{93} -24.5379i q^{95} +111.947 q^{97} +(29.0263 + 70.6362i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 20 q^{15} - 32 q^{19} + 52 q^{21} - 20 q^{25} - 28 q^{27} + 32 q^{31} + 24 q^{33} + 88 q^{37} - 40 q^{39} - 56 q^{43} - 20 q^{45} + 180 q^{49} - 72 q^{51} + 152 q^{57} + 64 q^{61} - 256 q^{63} + 328 q^{67} + 132 q^{69} + 200 q^{73} + 20 q^{75} - 112 q^{79} + 28 q^{81} + 120 q^{85} + 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 + 20 * q^15 - 32 * q^19 + 52 * q^21 - 20 * q^25 - 28 * q^27 + 32 * q^31 + 24 * q^33 + 88 * q^37 - 40 * q^39 - 56 * q^43 - 20 * q^45 + 180 * q^49 - 72 * q^51 + 152 * q^57 + 64 * q^61 - 256 * q^63 + 328 * q^67 + 132 * q^69 + 200 * q^73 + 20 * q^75 - 112 * q^79 + 28 * q^81 + 120 * q^85 + 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 $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		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	$-0.873962\pi$
$11$		−	8.48528i		−	0.771389i	0.922627	−	0.385695i	$-0.126038\pi$
$12$		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
$15$	6.58114	−	1.29947i	0.438743	−	0.0866311i
$16$		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	$-0.874583\pi$
$23$		−	17.6590i		−	0.767785i	0.923378	−	0.383892i	$-0.125417\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		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	$-0.458813\pi$
$31$	8.00000			0.258065			0.129032	+	0.991640i	$0.458813\pi$
$32$		0			0
$33$	24.9737	−	4.93113i	0.756778	−	0.149428i
$34$		0			0
$35$		−	25.6853i		−	0.733867i
$36$		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	$0.0805297\pi$
$41$			20.5247i			0.500603i	−0.968168	+	0.250301i	$0.919470\pi$
$42$		0			0
$43$	−42.4605			−0.987453			−0.493727	−	0.869617i	$-0.664366\pi$
$43$	−42.4605			−0.987453			−0.493727	+	0.869617i	$0.664366\pi$
$44$		0			0
$45$	7.64911	+	18.6143i	0.169980	+	0.413650i
$46$		0			0
$47$		−	88.2952i		−	1.87862i	−0.343067	−	0.939311i	$-0.611466\pi$
$47$		−	88.2952i		−	1.87862i	0.343067	−	0.939311i	$-0.388534\pi$
$48$		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	$0.0106749\pi$
$53$			3.55415i			0.0670595i	−0.999438	+	0.0335298i	$0.989325\pi$
$54$		0			0
$55$	−18.9737			−0.344976
$56$		0			0
$57$	6.37722	+	32.2974i	0.111881	+	0.566621i
$58$		0			0
$59$			77.7445i			1.31770i	0.752273	+	0.658852i	$0.228958\pi$
$59$			77.7445i			1.31770i	−0.752273	+	0.658852i	$0.771042\pi$
$60$		0			0
$61$	−21.9473			−0.359792			−0.179896	−	0.983686i	$-0.557576\pi$
$61$	−21.9473			−0.359792			−0.179896	+	0.983686i	$0.557576\pi$
$62$		0			0
$63$	−95.6228	+	39.2940i	−1.51782	+	0.623714i
$64$		0			0
$65$		−	22.3607i		−	0.344010i
$66$		0			0
$67$	53.5395			0.799097			0.399549	−	0.916712i	$-0.369167\pi$
$67$	53.5395			0.799097			0.399549	+	0.916712i	$0.369167\pi$
$68$		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	$-0.473693\pi$
$73$	12.0527			0.165105			0.0825525	+	0.996587i	$0.473693\pi$
$74$		0			0
$75$	−2.90569	−	14.7159i	−0.0387426	−	0.196212i
$76$		0			0
$77$		−	97.4690i		−	1.26583i
$78$		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
$83$			0.688486i			0.00829501i	0.999991	+	0.00414750i	$0.00132020\pi$
$83$			0.688486i			0.00829501i	−0.999991	+	0.00414750i	$0.998680\pi$
$84$		0			0
$85$	−7.94733			−0.0934980
$86$		0			0
$87$	78.9737	−	15.5936i	0.907743	−	0.179237i
$88$		0			0
$89$			7.10831i			0.0798686i	0.999202	+	0.0399343i	$0.0127149\pi$
$89$			7.10831i			0.0798686i	−0.999202	+	0.0399343i	$0.987285\pi$
$90$		0			0
$91$	114.868			1.26229
$92$		0			0
$93$	4.64911	+	23.5454i	0.0499904	+	0.253176i
$94$		0			0
$95$		−	24.5379i		−	0.258293i
$96$		0			0
$97$	111.947			1.15410			0.577048	−	0.816710i	$-0.304205\pi$
$97$	111.947			1.15410			0.577048	+	0.816710i	$0.304205\pi$
$98$		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
$105$	75.5964	−	14.9267i	0.719966	−	0.142160i
$106$		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	$-0.358349\pi$
$109$	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
$113$			195.738i			1.73220i	0.499874	+	0.866098i	$0.333380\pi$
$113$			195.738i			1.73220i	−0.499874	+	0.866098i	$0.666620\pi$
$114$		0			0
$115$	−39.4868			−0.343364
$116$		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
$125$			11.1803i			0.0894427i
$126$		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	$-0.873240\pi$
$131$		−	101.600i		−	0.775572i	0.921749	−	0.387786i	$-0.126760\pi$
$132$		0			0
$133$	126.053			0.947764
$134$		0			0
$135$	−50.3399	+	33.3301i	−0.372888	+	0.246890i
$136$		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
$145$	−60.0000			−0.413793
$146$		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	$-0.399308\pi$
$151$	93.9473			0.622168			0.311084	+	0.950382i	$0.399308\pi$
$152$		0			0
$153$	12.1580	+	29.5868i	0.0794641	+	0.193378i
$154$		0			0
$155$		−	17.8885i		−	0.115410i
$156$		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	$-0.662602\pi$
$163$	−159.381			−0.977801			−0.488900	+	0.872340i	$0.662602\pi$
$164$		0			0
$165$	−11.0263	−	55.8428i	−0.0668263	−	0.338441i
$166$		0			0
$167$			10.7742i			0.0645161i	0.999480	+	0.0322581i	$0.0102698\pi$
$167$			10.7742i			0.0645161i	−0.999480	+	0.0322581i	$0.989730\pi$
$168$		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	$0.305425\pi$
$173$			283.345i			1.63783i	−0.573913	+	0.818916i	$0.694575\pi$
$174$		0			0
$175$	−57.4342			−0.328195
$176$		0			0
$177$	−228.816	+	45.1804i	−1.29274	+	0.255256i
$178$		0			0
$179$		−	306.624i		−	1.71298i	−0.516163	−	0.856491i	$-0.672640\pi$
$179$		−	306.624i		−	1.71298i	0.516163	−	0.856491i	$-0.327360\pi$
$180$		0			0
$181$	−265.684			−1.46787			−0.733934	−	0.679221i	$-0.762318\pi$
$181$	−265.684			−1.46787			−0.733934	+	0.679221i	$0.762318\pi$
$182$		0			0
$183$	−12.7544	−	64.5948i	−0.0696964	−	0.352977i
$184$		0			0
$185$		−	134.046i		−	0.724575i
$186$		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	$-0.576649\pi$
$193$	−92.0527			−0.476957			−0.238478	+	0.971148i	$0.576649\pi$
$194$		0			0
$195$	65.8114	−	12.9947i	0.337494	−	0.0666393i
$196$		0			0
$197$			69.8360i			0.354497i	0.984166	+	0.177249i	$0.0567197\pi$
$197$			69.8360i			0.354497i	−0.984166	+	0.177249i	$0.943280\pi$
$198$		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
$203$		−	308.224i		−	1.51835i
$204$		0			0
$205$	45.8947			0.223876
$206$		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
$215$			94.9446i			0.441603i
$216$		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
$225$	41.6228	−	17.1039i	0.184990	−	0.0760175i
$226$		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	$-0.408620\pi$
$229$	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
$233$		−	185.876i		−	0.797751i	−0.917005	−	0.398875i	$-0.869401\pi$
$233$		−	185.876i		−	0.797751i	0.917005	−	0.398875i	$-0.130599\pi$
$234$		0			0
$235$	−197.434			−0.840145
$236$		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
$245$		−	185.476i		−	0.757044i
$246$		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	$-0.946080\pi$
$251$		−	84.6294i		−	0.337169i	0.985687	−	0.168584i	$-0.0539196\pi$
$252$		0			0
$253$	−149.842			−0.592261
$254$		0			0
$255$	−4.61850	−	23.3904i	−0.0181118	−	0.0917270i
$256$		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	$-0.822944\pi$
$263$		−	277.725i		−	1.05599i	0.849247	−	0.527995i	$-0.177056\pi$
$264$		0			0
$265$	7.94733			0.0299899
$266$		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	$-0.704030\pi$
$271$	−324.105			−1.19596			−0.597980	+	0.801511i	$0.704030\pi$
$272$		0			0
$273$	66.7544	+	338.078i	0.244522	+	1.23838i
$274$		0			0
$275$			42.4264i			0.154278i
$276$		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	$0.278845\pi$
$281$			431.726i			1.53639i	−0.640216	+	0.768195i	$0.721155\pi$
$282$		0			0
$283$	−141.540			−0.500140			−0.250070	−	0.968228i	$-0.580454\pi$
$283$	−141.540			−0.500140			−0.250070	+	0.968228i	$0.580454\pi$
$284$		0			0
$285$	72.2192	−	14.2599i	0.253401	−	0.0500348i
$286$		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	$-0.909146\pi$
$293$		−	164.998i		−	0.563133i	0.959542	−	0.281566i	$-0.0908540\pi$
$294$		0			0
$295$	173.842			0.589295
$296$		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
$305$			49.0757i			0.160904i
$306$		0			0
$307$	−159.381			−0.519158			−0.259579	−	0.965722i	$-0.583584\pi$
$307$	−159.381			−0.519158			−0.259579	+	0.965722i	$0.583584\pi$
$308$		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	$-0.795918\pi$
$313$	−501.684			−1.60282			−0.801412	+	0.598113i	$0.795918\pi$
$314$		0			0
$315$	87.8641	+	213.819i	0.278934	+	0.678791i
$316$		0			0
$317$			334.257i			1.05444i	0.849730	+	0.527219i	$0.176765\pi$
$317$			334.257i			1.05444i	−0.849730	+	0.527219i	$0.823235\pi$
$318$		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
$325$	−50.0000			−0.153846
$326$		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
$335$		−	119.718i		−	0.357367i
$336$		0			0
$337$	−129.684			−0.384819			−0.192409	−	0.981315i	$-0.561630\pi$
$337$	−129.684			−0.384819			−0.192409	+	0.981315i	$0.561630\pi$
$338$		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
$345$	−22.9473	−	116.217i	−0.0665140	−	0.336860i
$346$		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	$-0.760506\pi$
$349$	−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
$353$			637.679i			1.80646i	0.429159	+	0.903229i	$0.358810\pi$
$353$			637.679i			1.80646i	−0.429159	+	0.903229i	$0.641190\pi$
$354$		0			0
$355$	154.868			0.436249
$356$		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
$365$		−	26.9506i		−	0.0738372i
$366$		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
$375$	−32.9057	+	6.49733i	−0.0877485	+	0.0173262i
$376$		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	$0.143890\pi$
$383$			334.592i			0.873608i	−0.899557	+	0.436804i	$0.856110\pi$
$384$		0			0
$385$	−217.947			−0.566097
$386$		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
$395$			20.1835i			0.0510975i
$396$		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	$0.360387\pi$
$401$			726.086i			1.81069i	−0.424677	+	0.905345i	$0.639613\pi$
$402$		0			0
$403$	80.0000			0.198511
$404$		0			0
$405$	−127.351	−	128.790i	−0.314447	−	0.317999i
$406$		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
$415$	1.53950			0.00370964
$416$		0			0
$417$	−109.728	−	555.717i	−0.263137	−	1.33266i
$418$		0			0
$419$			580.907i			1.38641i	0.720740	+	0.693206i	$0.243802\pi$
$419$			580.907i			1.38641i	−0.720740	+	0.693206i	$0.756198\pi$
$420$		0			0
$421$	269.315			0.639704			0.319852	−	0.947468i	$-0.396367\pi$
$421$	269.315			0.639704			0.319852	+	0.947468i	$0.396367\pi$
$422$		0			0
$423$	302.039	+	735.019i	0.714041	+	1.73763i
$424$		0			0
$425$			17.7708i			0.0418136i
$426$		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	$-0.572493\pi$
$433$	−195.526			−0.451561			−0.225781	+	0.974178i	$0.572493\pi$
$434$		0			0
$435$	−34.8683	−	176.590i	−0.0801571	−	0.405955i
$436$		0			0
$437$		−	193.785i		−	0.443443i
$438$		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
$443$			216.728i			0.489228i	0.969621	+	0.244614i	$0.0786612\pi$
$443$			216.728i			0.489228i	−0.969621	+	0.244614i	$0.921339\pi$
$444$		0			0
$445$	15.8947			0.0357183
$446$		0			0
$447$	213.645	−	42.1848i	0.477953	−	0.0943732i
$448$		0			0
$449$		−	246.203i		−	0.548336i	−0.961682	−	0.274168i	$-0.911598\pi$
$449$		−	246.203i		−	0.548336i	0.961682	−	0.274168i	$-0.0884025\pi$
$450$		0			0
$451$	174.158			0.386160
$452$		0			0
$453$	54.5964	+	276.503i	0.120522	+	0.610383i
$454$		0			0
$455$		−	256.853i		−	0.564513i
$456$		0			0
$457$	553.052			1.21018			0.605090	−	0.796157i	$-0.293137\pi$
$457$	553.052			1.21018			0.605090	+	0.796157i	$0.293137\pi$
$458$		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
$465$	52.6491	−	10.3957i	0.113224	−	0.0223564i
$466$		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
$475$	−54.8683			−0.115512
$476$		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
$485$		−	250.322i		−	0.516128i
$486$		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	$0.258017\pi$
$491$			711.646i			1.44938i	−0.689074	+	0.724691i	$0.741983\pi$
$492$		0			0
$493$	−95.3680			−0.193444
$494$		0			0
$495$	157.947	−	64.9049i	0.319086	−	0.131121i
$496$		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	$0.0128449\pi$
$503$			40.5844i			0.0806847i	−0.999186	+	0.0403423i	$0.987155\pi$
$504$		0			0
$505$	347.684			0.688483
$506$		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
$515$		−	16.9766i		−	0.0329643i
$516$		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	$-0.826641\pi$
$521$		−	539.857i		−	1.03619i	0.855322	−	0.518097i	$-0.173359\pi$
$522$		0			0
$523$	266.644			0.509836			0.254918	−	0.966963i	$-0.417952\pi$
$523$	266.644			0.509836			0.254918	+	0.966963i	$0.417952\pi$
$524$		0			0
$525$	−33.3772	−	169.039i	−0.0635757	−	0.321979i
$526$		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
$535$	36.4078			0.0680520
$536$		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	$-0.398905\pi$
$541$	337.895			0.624574			0.312287	+	0.949988i	$0.398905\pi$
$542$		0			0
$543$	−154.399	−	781.954i	−0.284345	−	1.44006i
$544$		0			0
$545$		−	209.837i		−	0.385022i
$546$		0			0
$547$	214.566			0.392259			0.196130	−	0.980578i	$-0.437163\pi$
$547$	214.566			0.392259			0.196130	+	0.980578i	$0.437163\pi$
$548$		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
$555$	394.522	−	77.8995i	0.710850	−	0.140359i
$556$		0			0
$557$			674.821i			1.21153i	0.795644	+	0.605764i	$0.207133\pi$
$557$			674.821i			1.21153i	−0.795644	+	0.605764i	$0.792867\pi$
$558$		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
$565$	437.684			0.774662
$566$		0			0
$567$	661.601	−	654.210i	1.16684	−	1.15381i
$568$		0			0
$569$			183.122i			0.321831i	0.986968	+	0.160916i	$0.0514447\pi$
$569$			183.122i			0.321831i	−0.986968	+	0.160916i	$0.948555\pi$
$570$		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
$575$			88.2952i			0.153557i
$576$		0			0
$577$	−984.947			−1.70701			−0.853507	−	0.521082i	$-0.825529\pi$
$577$	−984.947			−1.70701			−0.853507	+	0.521082i	$0.825529\pi$
$578$		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
$585$	76.4911	+	186.143i	0.130754	+	0.318193i
$586$		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
$595$	−91.2897			−0.153428
$596$		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
$605$		−	109.567i		−	0.181103i
$606$		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
$615$	26.6712	+	135.076i	0.0433678	+	0.219636i
$616$		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
$625$	25.0000			0.0400000
$626$		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	$-0.837399\pi$
$631$	−1100.89			−1.74468			−0.872341	+	0.488898i	$0.837399\pi$
$632$		0			0
$633$	166.114	+	841.283i	0.262423	+	1.32904i
$634$		0			0
$635$			173.384i			0.273045i
$636$		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	$0.0846719\pi$
$641$			337.011i			0.525758i	−0.964829	+	0.262879i	$0.915328\pi$
$642$		0			0
$643$	599.381			0.932164			0.466082	−	0.884742i	$-0.345665\pi$
$643$	599.381			0.932164			0.466082	+	0.884742i	$0.345665\pi$
$644$		0			0
$645$	−279.438	+	55.1760i	−0.433238	+	0.0855441i
$646$		0			0
$647$		−	47.0224i		−	0.0726775i	−0.999340	−	0.0363388i	$-0.988430\pi$
$647$		−	47.0224i		−	0.0726775i	0.999340	−	0.0363388i	$-0.0115695\pi$
$648$		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	$-0.859148\pi$
$653$		−	559.228i		−	0.856399i	0.903684	−	0.428199i	$-0.140852\pi$
$654$		0			0
$655$	−227.184			−0.346846
$656$		0			0
$657$	−100.333	+	41.2296i	−0.152714	+	0.0627543i
$658$		0			0
$659$			665.759i			1.01026i	0.863044	+	0.505129i	$0.168555\pi$
$659$			665.759i			1.01026i	−0.863044	+	0.505129i	$0.831445\pi$
$660$		0			0
$661$	557.947			0.844096			0.422048	−	0.906574i	$-0.361311\pi$
$661$	557.947			0.844096			0.422048	+	0.906574i	$0.361311\pi$
$662$		0			0
$663$	104.605	−	20.6546i	0.157775	−	0.0311532i
$664$		0			0
$665$		−	281.862i		−	0.423853i
$666$		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	$-0.226484\pi$
$673$	1019.42			1.51474			0.757370	+	0.652985i	$0.226484\pi$
$674$		0			0
$675$	74.5285	+	112.563i	0.110413	+	0.166761i
$676$		0			0
$677$		−	624.616i		−	0.922624i	−0.887238	−	0.461312i	$-0.847379\pi$
$677$		−	624.616i		−	0.922624i	0.887238	−	0.461312i	$-0.152621\pi$
$678$		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
$685$	393.579			0.574567
$686$		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
$695$			422.205i			0.607489i
$696$		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
$705$	−114.737	−	581.083i	−0.162747	−	0.824232i
$706$		0			0
$707$			1786.08i			2.52627i
$708$		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
$713$		−	141.272i		−	0.198138i
$714$		0			0
$715$	−189.737			−0.265366
$716$		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
$725$			134.164i			0.185054i
$726$		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
$735$	545.888	−	107.787i	0.742705	−	0.146649i
$736$		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	$0.0985910\pi$
$743$			452.939i			0.609608i	−0.952415	+	0.304804i	$0.901409\pi$
$744$		0			0
$745$	−162.316			−0.217874
$746$		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	$-0.356513\pi$
$751$	654.369			0.871330			0.435665	+	0.900109i	$0.356513\pi$
$752$		0			0
$753$	249.079	−	49.1814i	0.330782	−	0.0653140i
$754$		0			0
$755$		−	210.073i		−	0.278242i
$756$		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	$-0.892850\pi$
$761$		−	502.715i		−	0.660598i	0.943876	−	0.330299i	$-0.107150\pi$
$762$		0			0
$763$	1077.95			1.41278
$764$		0			0
$765$	66.1580	−	27.1861i	0.0864811	−	0.0355374i
$766$		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	$0.216428\pi$
$773$			972.030i			1.25748i	−0.777617	+	0.628739i	$0.783572\pi$
$774$		0			0
$775$	−40.0000			−0.0516129
$776$		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
$785$		−	276.919i		−	0.352763i
$786$		0			0
$787$	−1492.70			−1.89669			−0.948346	−	0.317237i	$-0.897245\pi$
$787$	−1492.70			−1.89669			−0.948346	+	0.317237i	$0.897245\pi$
$788$		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
$795$	4.61850	+	23.3904i	0.00580944	+	0.0294219i
$796$		0			0
$797$			94.3618i			0.118396i	0.998246	+	0.0591981i	$0.0188544\pi$
$797$			94.3618i			0.118396i	−0.998246	+	0.0591981i	$0.981146\pi$
$798$		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
$805$	−453.579			−0.563452
$806$		0			0
$807$	843.354	−	166.523i	1.04505	−	0.206348i
$808$		0			0
$809$		−	1103.35i		−	1.36384i	−0.731427	−	0.681920i	$-0.761145\pi$
$809$		−	1103.35i		−	1.36384i	0.731427	−	0.681920i	$-0.238855\pi$
$810$		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
$815$			356.388i			0.437286i
$816$		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
$825$	−124.868	+	24.6556i	−0.151356	+	0.0298856i
$826$		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	$-0.461967\pi$
$829$	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
$833$		−	294.808i		−	0.353911i
$834$		0			0
$835$	24.0918			0.0288525
$836$		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
$845$			154.289i			0.182590i
$846$		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
$855$	83.9388	+	204.267i	0.0981740	+	0.238909i
$856$		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	$-0.947188\pi$
$863$		−	285.057i		−	0.330310i	0.986268	−	0.165155i	$-0.0528124\pi$
$864$		0			0
$865$	633.579			0.732461
$866$		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
$875$			128.427i			0.146773i
$876$		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	$-0.975176\pi$
$881$		−	137.271i		−	0.155813i	0.996961	−	0.0779065i	$-0.0248236\pi$
$882$		0			0
$883$	−1647.75			−1.86608			−0.933040	−	0.359772i	$-0.882854\pi$
$883$	−1647.75			−1.86608			−0.933040	+	0.359772i	$0.882854\pi$
$884$		0			0
$885$	101.026	+	511.647i	0.114154	+	0.578133i
$886$		0			0
$887$			514.419i			0.579954i	0.957034	+	0.289977i	$0.0936477\pi$
$887$			514.419i			0.579954i	−0.957034	+	0.289977i	$0.906352\pi$
$888$		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
$895$	−685.631			−0.766069
$896$		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
$905$			594.087i			0.656450i
$906$		0			0
$907$	−347.303			−0.382914			−0.191457	−	0.981501i	$-0.561321\pi$
$907$	−347.303			−0.382914			−0.191457	+	0.981501i	$0.561321\pi$
$908$		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
$915$	−144.438	+	28.5198i	−0.157856	+	0.0311692i
$916$		0			0
$917$		−	1167.06i		−	1.27270i
$918$		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
$923$			692.592i			0.750371i
$924$		0			0
$925$	−299.737			−0.324040
$926$		0			0
$927$	−63.2014	+	25.9712i	−0.0681785	+	0.0280164i
$928$		0			0
$929$			1025.96i			1.10437i	0.833723	+	0.552183i	$0.186205\pi$
$929$			1025.96i			1.10437i	−0.833723	+	0.552183i	$0.813795\pi$
$930$		0			0
$931$	910.236			0.977697
$932$		0			0
$933$	−421.157	+	83.1588i	−0.451401	+	0.0891305i
$934$		0			0
$935$			67.4353i			0.0721234i
$936$		0			0
$937$	−332.053			−0.354379			−0.177189	−	0.984177i	$-0.556700\pi$
$937$	−332.053			−0.354379			−0.177189	+	0.984177i	$0.556700\pi$
$938$		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
$945$	−578.246	+	382.858i	−0.611900	+	0.405141i
$946$		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
$955$	−356.921			−0.373739
$956$		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
$965$			205.836i			0.213302i
$966$		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	$0.183376\pi$
$971$			1057.91i			1.08950i	−0.838598	+	0.544751i	$0.816624\pi$
$972$		0			0
$973$	−2168.89			−2.22908
$974$		0			0
$975$	−29.0569	−	147.159i	−0.0298020	−	0.150932i
$976$		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	$-0.924706\pi$
$983$		−	460.718i		−	0.468685i	0.972154	−	0.234343i	$-0.0752937\pi$
$984$		0			0
$985$	156.158			0.158536
$986$		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	$-0.285275\pi$
$991$	1237.89			1.24914			0.624568	+	0.780971i	$0.285275\pi$
$992$		0			0
$993$	−226.307	−	1146.13i	−0.227903	−	1.15421i
$994$		0			0
$995$		−	332.997i		−	0.334671i
$996$		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	960.3.l.e.641.4		4
3.2	odd	2	inner	960.3.l.e.641.3		4
4.3	odd	2		960.3.l.f.641.1		4
8.3	odd	2		240.3.l.c.161.4		4
8.5	even	2		30.3.d.a.11.1	✓	4
12.11	even	2		960.3.l.f.641.2		4
24.5	odd	2		30.3.d.a.11.3	yes	4
24.11	even	2		240.3.l.c.161.3		4
40.3	even	4		1200.3.c.k.449.2		8
40.13	odd	4		150.3.b.b.149.3		8
40.19	odd	2		1200.3.l.u.401.1		4
40.27	even	4		1200.3.c.k.449.7		8
40.29	even	2		150.3.d.c.101.4		4
40.37	odd	4		150.3.b.b.149.6		8
72.5	odd	6		810.3.h.a.431.1		8
72.13	even	6		810.3.h.a.431.4		8
72.29	odd	6		810.3.h.a.701.4		8
72.61	even	6		810.3.h.a.701.1		8
120.29	odd	2		150.3.d.c.101.2		4
120.53	even	4		150.3.b.b.149.5		8
120.59	even	2		1200.3.l.u.401.2		4
120.77	even	4		150.3.b.b.149.4		8
120.83	odd	4		1200.3.c.k.449.8		8
120.107	odd	4		1200.3.c.k.449.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.3.d.a.11.1	✓	4	8.5	even	2
30.3.d.a.11.3	yes	4	24.5	odd	2
150.3.b.b.149.3		8	40.13	odd	4
150.3.b.b.149.4		8	120.77	even	4
150.3.b.b.149.5		8	120.53	even	4
150.3.b.b.149.6		8	40.37	odd	4
150.3.d.c.101.2		4	120.29	odd	2
150.3.d.c.101.4		4	40.29	even	2
240.3.l.c.161.3		4	24.11	even	2
240.3.l.c.161.4		4	8.3	odd	2
810.3.h.a.431.1		8	72.5	odd	6
810.3.h.a.431.4		8	72.13	even	6
810.3.h.a.701.1		8	72.61	even	6
810.3.h.a.701.4		8	72.29	odd	6
960.3.l.e.641.3		4	3.2	odd	2	inner
960.3.l.e.641.4		4	1.1	even	1	trivial
960.3.l.f.641.1		4	4.3	odd	2
960.3.l.f.641.2		4	12.11	even	2
1200.3.c.k.449.1		8	120.107	odd	4
1200.3.c.k.449.2		8	40.3	even	4
1200.3.c.k.449.7		8	40.27	even	4
1200.3.c.k.449.8		8	120.83	odd	4
1200.3.l.u.401.1		4	40.19	odd	2
1200.3.l.u.401.2		4	120.59	even	2

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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	960.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.581139 + 2.94317i) q^{3} -2.23607i q^{5} +11.4868 q^{7} +(-8.32456 + 3.42079i) q^{9} +O(q^{10})\)	\(q+(0.581139 + 2.94317i) q^{3} -2.23607i q^{5} +11.4868 q^{7} +(-8.32456 + 3.42079i) q^{9} -8.48528i q^{11} +10.0000 q^{13} +(6.58114 - 1.29947i) q^{15} -3.55415i q^{17} +10.9737 q^{19} +(6.67544 + 33.8078i) q^{21} -17.6590i q^{23} -5.00000 q^{25} +(-14.9057 - 22.5127i) q^{27} -26.8328i q^{29} +8.00000 q^{31} +(24.9737 - 4.93113i) q^{33} -25.6853i q^{35} +59.9473 q^{37} +(5.81139 + 29.4317i) q^{39} +20.5247i q^{41} -42.4605 q^{43} +(7.64911 + 18.6143i) q^{45} -88.2952i q^{47} +82.9473 q^{49} +(10.4605 - 2.06546i) q^{51} +3.55415i q^{53} -18.9737 q^{55} +(6.37722 + 32.2974i) q^{57} +77.7445i q^{59} -21.9473 q^{61} +(-95.6228 + 39.2940i) q^{63} -22.3607i q^{65} +53.5395 q^{67} +(51.9737 - 10.2624i) q^{69} +69.2592i q^{71} +12.0527 q^{73} +(-2.90569 - 14.7159i) q^{75} -97.4690i q^{77} -9.02633 q^{79} +(57.5964 - 56.9530i) q^{81} +0.688486i q^{83} -7.94733 q^{85} +(78.9737 - 15.5936i) q^{87} +7.10831i q^{89} +114.868 q^{91} +(4.64911 + 23.5454i) q^{93} -24.5379i q^{95} +111.947 q^{97} +(29.0263 + 70.6362i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 8 q^{7} - 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 8 * q^7 - 8 * q^9	\( 4 q - 4 q^{3} + 8 q^{7} - 8 q^{9} + 40 q^{13} + 20 q^{15} - 32 q^{19} + 52 q^{21} - 20 q^{25} - 28 q^{27} + 32 q^{31} + 24 q^{33} + 88 q^{37} - 40 q^{39} - 56 q^{43} - 20 q^{45} + 180 q^{49} - 72 q^{51} + 152 q^{57} + 64 q^{61} - 256 q^{63} + 328 q^{67} + 132 q^{69} + 200 q^{73} + 20 q^{75} - 112 q^{79} + 28 q^{81} + 120 q^{85} + 240 q^{87} + 80 q^{91} - 32 q^{93} + 296 q^{97} + 192 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^7 - 8 * q^9 + 40 * q^13 + 20 * q^15 - 32 * q^19 + 52 * q^21 - 20 * q^25 - 28 * q^27 + 32 * q^31 + 24 * q^33 + 88 * q^37 - 40 * q^39 - 56 * q^43 - 20 * q^45 + 180 * q^49 - 72 * q^51 + 152 * q^57 + 64 * q^61 - 256 * q^63 + 328 * q^67 + 132 * q^69 + 200 * q^73 + 20 * q^75 - 112 * q^79 + 28 * q^81 + 120 * q^85 + 240 * q^87 + 80 * q^91 - 32 * q^93 + 296 * q^97 + 192 * q^99

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