LMFDB - Embedded newform 1264.3.e.b.1105.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1264,3,Mod(1105,1264)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1264, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1264.1105");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1264 = 2^{4} \cdot 79 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1264.e (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:	$34.4415054151$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 77x^{6} + 2073x^{4} + 23519x^{2} + 95938 $ x^8 + 77x^6 + 2073x^4 + 23519*x^2 + 95938
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 79)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1105.1
Root			$-5.79987i$ of defining polynomial
Character	$\chi$	$=$	1264.1105
Dual form			1264.3.e.b.1105.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.79987i q^{3} +6.39881 q^{5} -7.72499i q^{7} -24.6385 q^{9} +O(q^{10})$	$q-5.79987i q^{3} +6.39881 q^{5} -7.72499i q^{7} -24.6385 q^{9} -5.61527 q^{11} +0.716990 q^{13} -37.1123i q^{15} -16.7851i q^{17} +18.2852 q^{19} -44.8040 q^{21} -22.8948 q^{23} +15.9448 q^{25} +90.7016i q^{27} +8.44567i q^{29} -32.3616 q^{31} +32.5679i q^{33} -49.4308i q^{35} +57.3597i q^{37} -4.15845i q^{39} -57.1558i q^{41} -8.33947i q^{43} -157.657 q^{45} -14.5253i q^{47} -10.6755 q^{49} -97.3517 q^{51} +13.5739i q^{53} -35.9311 q^{55} -106.052i q^{57} -76.3337i q^{59} -51.8236i q^{61} +190.333i q^{63} +4.58789 q^{65} -42.1742 q^{67} +132.787i q^{69} +74.6532i q^{71} -5.85752 q^{73} -92.4779i q^{75} +43.3779i q^{77} +(-59.1180 - 52.4028i) q^{79} +304.311 q^{81} +40.6315 q^{83} -107.405i q^{85} +48.9838 q^{87} +12.2414 q^{89} -5.53874i q^{91} +187.693i q^{93} +117.004 q^{95} +127.711 q^{97} +138.352 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} - 82 q^{9}+O(q^{10}) $ 8 * q - 2 * q^5 - 82 * q^9	$ 8 q - 2 q^{5} - 82 q^{9} - 28 q^{11} - 30 q^{13} + 32 q^{19} - 2 q^{21} - 60 q^{23} - 50 q^{25} - 36 q^{31} - 260 q^{45} - 294 q^{49} - 236 q^{51} - 140 q^{55} + 194 q^{65} - 132 q^{67} + 384 q^{73} - 210 q^{79} + 56 q^{81} - 96 q^{83} + 8 q^{87} + 102 q^{89} + 12 q^{95} + 470 q^{97} + 524 q^{99}+O(q^{100}) $ 8 * q - 2 * q^5 - 82 * q^9 - 28 * q^11 - 30 * q^13 + 32 * q^19 - 2 * q^21 - 60 * q^23 - 50 * q^25 - 36 * q^31 - 260 * q^45 - 294 * q^49 - 236 * q^51 - 140 * q^55 + 194 * q^65 - 132 * q^67 + 384 * q^73 - 210 * q^79 + 56 * q^81 - 96 * q^83 + 8 * q^87 + 102 * q^89 + 12 * q^95 + 470 * q^97 + 524 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1264\mathbb{Z}\right)^\times$.

$n$	$159$	$161$	$949$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		−	5.79987i		−	1.93329i	−0.256117	−	0.966646i	$-0.582443\pi$
$3$		−	5.79987i		−	1.93329i	0.256117	−	0.966646i	$-0.417557\pi$
$4$		0			0
$5$	6.39881			1.27976			0.639881	−	0.768474i	$-0.278983\pi$
$5$	6.39881			1.27976			0.639881	+	0.768474i	$0.278983\pi$
$6$		0			0
$7$		−	7.72499i		−	1.10357i	−0.833986	−	0.551785i	$-0.813947\pi$
$7$		−	7.72499i		−	1.10357i	0.833986	−	0.551785i	$-0.186053\pi$
$8$		0			0
$9$	−24.6385			−2.73762
$10$		0			0
$11$	−5.61527			−0.510479			−0.255240	−	0.966878i	$-0.582154\pi$
$11$	−5.61527			−0.510479			−0.255240	+	0.966878i	$0.582154\pi$
$12$		0			0
$13$	0.716990			0.0551531			0.0275765	−	0.999620i	$-0.491221\pi$
$13$	0.716990			0.0551531			0.0275765	+	0.999620i	$0.491221\pi$
$14$		0			0
$15$		−	37.1123i		−	2.47415i
$16$		0			0
$17$		−	16.7851i		−	0.987361i	−0.869643	−	0.493681i	$-0.835651\pi$
$17$		−	16.7851i		−	0.987361i	0.869643	−	0.493681i	$-0.164349\pi$
$18$		0			0
$19$	18.2852			0.962379			0.481189	−	0.876617i	$-0.340205\pi$
$19$	18.2852			0.962379			0.481189	+	0.876617i	$0.340205\pi$
$20$		0			0
$21$	−44.8040			−2.13352
$22$		0			0
$23$	−22.8948			−0.995425			−0.497713	−	0.867342i	$-0.665827\pi$
$23$	−22.8948			−0.995425			−0.497713	+	0.867342i	$0.665827\pi$
$24$		0			0
$25$	15.9448			0.637793
$26$		0			0
$27$			90.7016i			3.35932i
$28$		0			0
$29$			8.44567i			0.291230i	0.989341	+	0.145615i	$0.0465161\pi$
$29$			8.44567i			0.291230i	−0.989341	+	0.145615i	$0.953484\pi$
$30$		0			0
$31$	−32.3616			−1.04392			−0.521961	−	0.852969i	$-0.674800\pi$
$31$	−32.3616			−1.04392			−0.521961	+	0.852969i	$0.674800\pi$
$32$		0			0
$33$			32.5679i			0.986905i
$34$		0			0
$35$		−	49.4308i		−	1.41231i
$36$		0			0
$37$			57.3597i			1.55026i	0.631800	+	0.775132i	$0.282317\pi$
$37$			57.3597i			1.55026i	−0.631800	+	0.775132i	$0.717683\pi$
$38$		0			0
$39$		−	4.15845i		−	0.106627i
$40$		0			0
$41$		−	57.1558i		−	1.39404i	−0.717050	−	0.697022i	$-0.754508\pi$
$41$		−	57.1558i		−	1.39404i	0.717050	−	0.697022i	$-0.245492\pi$
$42$		0			0
$43$		−	8.33947i		−	0.193941i	−0.995287	−	0.0969706i	$-0.969085\pi$
$43$		−	8.33947i		−	0.193941i	0.995287	−	0.0969706i	$-0.0309153\pi$
$44$		0			0
$45$	−157.657			−3.50350
$46$		0			0
$47$		−	14.5253i		−	0.309050i	−0.987989	−	0.154525i	$-0.950615\pi$
$47$		−	14.5253i		−	0.309050i	0.987989	−	0.154525i	$-0.0493847\pi$
$48$		0			0
$49$	−10.6755			−0.217868
$50$		0			0
$51$	−97.3517			−1.90886
$52$		0			0
$53$			13.5739i			0.256111i	0.991767	+	0.128056i	$0.0408736\pi$
$53$			13.5739i			0.256111i	−0.991767	+	0.128056i	$0.959126\pi$
$54$		0			0
$55$	−35.9311			−0.653292
$56$		0			0
$57$		−	106.052i		−	1.86056i
$58$		0			0
$59$		−	76.3337i		−	1.29379i	−0.762578	−	0.646896i	$-0.776067\pi$
$59$		−	76.3337i		−	1.29379i	0.762578	−	0.646896i	$-0.223933\pi$
$60$		0			0
$61$		−	51.8236i		−	0.849567i	−0.905295	−	0.424784i	$-0.860350\pi$
$61$		−	51.8236i		−	0.849567i	0.905295	−	0.424784i	$-0.139650\pi$
$62$		0			0
$63$			190.333i			3.02115i
$64$		0			0
$65$	4.58789			0.0705829
$66$		0			0
$67$	−42.1742			−0.629466			−0.314733	−	0.949180i	$-0.601915\pi$
$67$	−42.1742			−0.629466			−0.314733	+	0.949180i	$0.601915\pi$
$68$		0			0
$69$			132.787i			1.92445i
$70$		0			0
$71$			74.6532i			1.05145i	0.850654	+	0.525727i	$0.176206\pi$
$71$			74.6532i			1.05145i	−0.850654	+	0.525727i	$0.823794\pi$
$72$		0			0
$73$	−5.85752			−0.0802401			−0.0401200	−	0.999195i	$-0.512774\pi$
$73$	−5.85752			−0.0802401			−0.0401200	+	0.999195i	$0.512774\pi$
$74$		0			0
$75$		−	92.4779i		−	1.23304i
$76$		0			0
$77$			43.3779i			0.563350i
$78$		0			0
$79$	−59.1180	−	52.4028i	−0.748330	−	0.663327i
$79$	−59.1180	−	52.4028i	−0.748330	−	0.663327i
$80$		0			0
$81$	304.311			3.75692
$82$		0			0
$83$	40.6315			0.489536			0.244768	−	0.969582i	$-0.421288\pi$
$83$	40.6315			0.489536			0.244768	+	0.969582i	$0.421288\pi$
$84$		0			0
$85$		−	107.405i		−	1.26359i
$86$		0			0
$87$	48.9838			0.563033
$88$		0			0
$89$	12.2414			0.137544			0.0687721	−	0.997632i	$-0.478092\pi$
$89$	12.2414			0.137544			0.0687721	+	0.997632i	$0.478092\pi$
$90$		0			0
$91$		−	5.53874i		−	0.0608653i
$92$		0			0
$93$			187.693i			2.01821i
$94$		0			0
$95$	117.004			1.23162
$96$		0			0
$97$	127.711			1.31661			0.658303	−	0.752753i	$-0.271275\pi$
$97$	127.711			1.31661			0.658303	+	0.752753i	$0.271275\pi$
$98$		0			0
$99$	138.352			1.39750
$100$		0			0
$101$	79.5433			0.787558			0.393779	−	0.919205i	$-0.371168\pi$
$101$	79.5433			0.787558			0.393779	+	0.919205i	$0.371168\pi$
$102$		0			0
$103$			14.4901i			0.140681i	0.997523	+	0.0703403i	$0.0224085\pi$
$103$			14.4901i			0.140681i	−0.997523	+	0.0703403i	$0.977591\pi$
$104$		0			0
$105$	−286.692			−2.73040
$106$		0			0
$107$			21.2636i			0.198726i	0.995051	+	0.0993628i	$0.0316804\pi$
$107$			21.2636i			0.198726i	−0.995051	+	0.0993628i	$0.968320\pi$
$108$		0			0
$109$		−	142.231i		−	1.30487i	−0.757844	−	0.652436i	$-0.773747\pi$
$109$		−	142.231i		−	1.30487i	0.757844	−	0.652436i	$-0.226253\pi$
$110$		0			0
$111$	332.679			2.99711
$112$		0			0
$113$			154.641i			1.36851i	0.729244	+	0.684254i	$0.239872\pi$
$113$			154.641i			1.36851i	−0.729244	+	0.684254i	$0.760128\pi$
$114$		0			0
$115$	−146.499			−1.27391
$116$		0			0
$117$	−17.6656			−0.150988
$118$		0			0
$119$	−129.665			−1.08962
$120$		0			0
$121$	−89.4687			−0.739411
$122$		0			0
$123$	−331.496			−2.69509
$124$		0			0
$125$	−57.9424			−0.463539
$126$		0			0
$127$		−	73.6366i		−	0.579816i	−0.957055	−	0.289908i	$-0.906375\pi$
$127$		−	73.6366i		−	0.579816i	0.957055	−	0.289908i	$-0.0936247\pi$
$128$		0			0
$129$	−48.3679			−0.374945
$130$		0			0
$131$	37.2976			0.284714			0.142357	−	0.989815i	$-0.454532\pi$
$131$	37.2976			0.284714			0.142357	+	0.989815i	$0.454532\pi$
$132$		0			0
$133$		−	141.253i		−	1.06205i
$134$		0			0
$135$			580.382i			4.29913i
$136$		0			0
$137$		−	153.111i		−	1.11760i	−0.829304	−	0.558798i	$-0.811263\pi$
$137$		−	153.111i		−	1.11760i	0.829304	−	0.558798i	$-0.188737\pi$
$138$		0			0
$139$			57.1969i			0.411488i	0.978606	+	0.205744i	$0.0659615\pi$
$139$			57.1969i			0.411488i	−0.978606	+	0.205744i	$0.934039\pi$
$140$		0			0
$141$	−84.2452			−0.597484
$142$		0			0
$143$	−4.02609			−0.0281545
$144$		0			0
$145$			54.0423i			0.372706i
$146$		0			0
$147$			61.9167i			0.421202i
$148$		0			0
$149$		−	17.0300i		−	0.114295i	−0.998366	−	0.0571477i	$-0.981799\pi$
$149$		−	17.0300i		−	0.114295i	0.998366	−	0.0571477i	$-0.0182006\pi$
$150$		0			0
$151$	91.2949			0.604602			0.302301	−	0.953213i	$-0.402245\pi$
$151$	91.2949			0.604602			0.302301	+	0.953213i	$0.402245\pi$
$152$		0			0
$153$			413.561i			2.70302i
$154$		0			0
$155$	−207.076			−1.33597
$156$		0			0
$157$			199.776i			1.27246i	0.771499	+	0.636231i	$0.219507\pi$
$157$			199.776i			1.27246i	−0.771499	+	0.636231i	$0.780493\pi$
$158$		0			0
$159$	78.7269			0.495138
$160$		0			0
$161$			176.862i			1.09852i
$162$		0			0
$163$	208.549			1.27944			0.639721	−	0.768607i	$-0.279050\pi$
$163$	208.549			1.27944			0.639721	+	0.768607i	$0.279050\pi$
$164$		0			0
$165$			208.396i			1.26300i
$166$		0			0
$167$	−157.415			−0.942606			−0.471303	−	0.881971i	$-0.656216\pi$
$167$	−157.415			−0.942606			−0.471303	+	0.881971i	$0.656216\pi$
$168$		0			0
$169$	−168.486			−0.996958
$170$		0			0
$171$	−450.521			−2.63462
$172$		0			0
$173$		−	115.266i		−	0.666278i	−0.942878	−	0.333139i	$-0.891892\pi$
$173$		−	115.266i		−	0.666278i	0.942878	−	0.333139i	$-0.108108\pi$
$174$		0			0
$175$		−	123.174i		−	0.703849i
$176$		0			0
$177$	−442.726			−2.50128
$178$		0			0
$179$	24.6611			0.137772			0.0688858	−	0.997625i	$-0.478056\pi$
$179$	24.6611			0.137772			0.0688858	+	0.997625i	$0.478056\pi$
$180$		0			0
$181$	173.834			0.960407			0.480203	−	0.877157i	$-0.340563\pi$
$181$	173.834			0.960407			0.480203	+	0.877157i	$0.340563\pi$
$182$		0			0
$183$	−300.570			−1.64246
$184$		0			0
$185$			367.034i			1.98397i
$186$		0			0
$187$			94.2531i			0.504027i
$188$		0			0
$189$	700.669			3.70724
$190$		0			0
$191$			367.616i			1.92469i	0.271826	+	0.962347i	$0.412373\pi$
$191$			367.616i			1.92469i	−0.271826	+	0.962347i	$0.587627\pi$
$192$		0			0
$193$		−	84.0672i		−	0.435581i	−0.975996	−	0.217791i	$-0.930115\pi$
$193$		−	84.0672i		−	0.435581i	0.975996	−	0.217791i	$-0.0698850\pi$
$194$		0			0
$195$		−	26.6092i		−	0.136457i
$196$		0			0
$197$		−	258.392i		−	1.31163i	−0.754920	−	0.655817i	$-0.772324\pi$
$197$		−	258.392i		−	1.31163i	0.754920	−	0.655817i	$-0.227676\pi$
$198$		0			0
$199$			16.1270i			0.0810400i	0.999179	+	0.0405200i	$0.0129015\pi$
$199$			16.1270i			0.0810400i	−0.999179	+	0.0405200i	$0.987099\pi$
$200$		0			0
$201$			244.605i			1.21694i
$202$		0			0
$203$	65.2428			0.321393
$204$		0			0
$205$		−	365.729i		−	1.78405i
$206$		0			0
$207$	564.094			2.72509
$208$		0			0
$209$	−102.676			−0.491274
$210$		0			0
$211$		−	110.640i		−	0.524361i	−0.965019	−	0.262181i	$-0.915558\pi$
$211$		−	110.640i		−	0.524361i	0.965019	−	0.262181i	$-0.0844417\pi$
$212$		0			0
$213$	432.979			2.03277
$214$		0			0
$215$		−	53.3627i		−	0.248199i
$216$		0			0
$217$			249.993i			1.15204i
$218$		0			0
$219$			33.9729i			0.155127i
$220$		0			0
$221$		−	12.0348i		−	0.0544560i
$222$		0			0
$223$	75.5373			0.338732			0.169366	−	0.985553i	$-0.445828\pi$
$223$	75.5373			0.338732			0.169366	+	0.985553i	$0.445828\pi$
$224$		0			0
$225$	−392.857			−1.74603
$226$		0			0
$227$			38.8968i			0.171352i	0.996323	+	0.0856758i	$0.0273049\pi$
$227$			38.8968i			0.171352i	−0.996323	+	0.0856758i	$0.972695\pi$
$228$		0			0
$229$		−	100.672i		−	0.439617i	−0.975543	−	0.219809i	$-0.929457\pi$
$229$		−	100.672i		−	0.439617i	0.975543	−	0.219809i	$-0.0705433\pi$
$230$		0			0
$231$	251.587			1.08912
$232$		0			0
$233$		−	437.608i		−	1.87815i	−0.343715	−	0.939074i	$-0.611685\pi$
$233$		−	437.608i		−	1.87815i	0.343715	−	0.939074i	$-0.388315\pi$
$234$		0			0
$235$		−	92.9450i		−	0.395511i
$236$		0			0
$237$	−303.930	+	342.877i	−1.28240	+	1.44674i
$238$		0			0
$239$	139.108			0.582041			0.291020	−	0.956717i	$-0.406005\pi$
$239$	139.108			0.582041			0.291020	+	0.956717i	$0.406005\pi$
$240$		0			0
$241$	−23.6947			−0.0983184			−0.0491592	−	0.998791i	$-0.515654\pi$
$241$	−23.6947			−0.0983184			−0.0491592	+	0.998791i	$0.515654\pi$
$242$		0			0
$243$		−	948.650i		−	3.90391i
$244$		0			0
$245$	−68.3107			−0.278819
$246$		0			0
$247$	13.1103			0.0530782
$248$		0			0
$249$		−	235.657i		−	0.946415i
$250$		0			0
$251$			220.942i			0.880246i	0.897937	+	0.440123i	$0.145065\pi$
$251$			220.942i			0.880246i	−0.897937	+	0.440123i	$0.854935\pi$
$252$		0			0
$253$	128.560			0.508144
$254$		0			0
$255$	−622.936			−2.44288
$256$		0			0
$257$	−87.3207			−0.339769			−0.169885	−	0.985464i	$-0.554339\pi$
$257$	−87.3207			−0.339769			−0.169885	+	0.985464i	$0.554339\pi$
$258$		0			0
$259$	443.104			1.71082
$260$		0			0
$261$		−	208.089i		−	0.797276i
$262$		0			0
$263$	208.424			0.792486			0.396243	−	0.918146i	$-0.370314\pi$
$263$	208.424			0.792486			0.396243	+	0.918146i	$0.370314\pi$
$264$		0			0
$265$			86.8569i			0.327762i
$266$		0			0
$267$		−	70.9988i		−	0.265913i
$268$		0			0
$269$	−242.999			−0.903343			−0.451671	−	0.892184i	$-0.649172\pi$
$269$	−242.999			−0.903343			−0.451671	+	0.892184i	$0.649172\pi$
$270$		0			0
$271$			220.990i			0.815461i	0.913102	+	0.407731i	$0.133680\pi$
$271$			220.990i			0.815461i	−0.913102	+	0.407731i	$0.866320\pi$
$272$		0			0
$273$	−32.1240			−0.117670
$274$		0			0
$275$	−89.5345			−0.325580
$276$		0			0
$277$	−320.138			−1.15573			−0.577866	−	0.816132i	$-0.696114\pi$
$277$	−320.138			−1.15573			−0.577866	+	0.816132i	$0.696114\pi$
$278$		0			0
$279$	797.343			2.85786
$280$		0			0
$281$	452.010			1.60858			0.804288	−	0.594240i	$-0.202547\pi$
$281$	452.010			1.60858			0.804288	+	0.594240i	$0.202547\pi$
$282$		0			0
$283$	−524.657			−1.85391			−0.926956	−	0.375169i	$-0.877585\pi$
$283$	−524.657			−1.85391			−0.926956	+	0.375169i	$0.877585\pi$
$284$		0			0
$285$		−	678.606i		−	2.38107i
$286$		0			0
$287$	−441.528			−1.53843
$288$		0			0
$289$	7.25891			0.0251173
$290$		0			0
$291$		−	740.706i		−	2.54538i
$292$		0			0
$293$			301.999i			1.03071i	0.856976	+	0.515357i	$0.172341\pi$
$293$			301.999i			1.03071i	−0.856976	+	0.515357i	$0.827659\pi$
$294$		0			0
$295$		−	488.445i		−	1.65575i
$296$		0			0
$297$		−	509.314i		−	1.71486i
$298$		0			0
$299$	−16.4153			−0.0549008
$300$		0			0
$301$	−64.4224			−0.214028
$302$		0			0
$303$		−	461.341i		−	1.52258i
$304$		0			0
$305$		−	331.610i		−	1.08724i
$306$		0			0
$307$		−	433.723i		−	1.41278i	−0.707824	−	0.706389i	$-0.750323\pi$
$307$		−	433.723i		−	1.41278i	0.707824	−	0.706389i	$-0.249677\pi$
$308$		0			0
$309$	84.0407			0.271976
$310$		0			0
$311$		−	148.313i		−	0.476892i	−0.971156	−	0.238446i	$-0.923362\pi$
$311$		−	148.313i		−	0.476892i	0.971156	−	0.238446i	$-0.0766380\pi$
$312$		0			0
$313$	−346.053			−1.10560			−0.552800	−	0.833314i	$-0.686441\pi$
$313$	−346.053			−1.10560			−0.552800	+	0.833314i	$0.686441\pi$
$314$		0			0
$315$			1217.90i			3.86636i
$316$		0			0
$317$	305.178			0.962706			0.481353	−	0.876527i	$-0.340145\pi$
$317$	305.178			0.962706			0.481353	+	0.876527i	$0.340145\pi$
$318$		0			0
$319$		−	47.4247i		−	0.148667i
$320$		0			0
$321$	123.326			0.384195
$322$		0			0
$323$		−	306.920i		−	0.950216i
$324$		0			0
$325$	11.4323			0.0351762
$326$		0			0
$327$	−824.922			−2.52270
$328$		0			0
$329$	−112.208			−0.341058
$330$		0			0
$331$		−	581.155i		−	1.75575i	−0.478885	−	0.877877i	$-0.658959\pi$
$331$		−	581.155i		−	1.75575i	0.478885	−	0.877877i	$-0.341041\pi$
$332$		0			0
$333$		−	1413.26i		−	4.24403i
$334$		0			0
$335$	−269.865			−0.805567
$336$		0			0
$337$	108.777			0.322779			0.161390	−	0.986891i	$-0.448402\pi$
$337$	108.777			0.322779			0.161390	+	0.986891i	$0.448402\pi$
$338$		0			0
$339$	896.900			2.64572
$340$		0			0
$341$	181.719			0.532901
$342$		0			0
$343$		−	296.056i		−	0.863138i
$344$		0			0
$345$			849.678i			2.46284i
$346$		0			0
$347$	229.092			0.660208			0.330104	−	0.943945i	$-0.392916\pi$
$347$	229.092			0.660208			0.330104	+	0.943945i	$0.392916\pi$
$348$		0			0
$349$		−	295.437i		−	0.846523i	−0.906007	−	0.423262i	$-0.860885\pi$
$349$		−	295.437i		−	0.846523i	0.906007	−	0.423262i	$-0.139115\pi$
$350$		0			0
$351$			65.0321i			0.185277i
$352$		0			0
$353$		−	298.377i		−	0.845260i	−0.906302	−	0.422630i	$-0.861107\pi$
$353$		−	298.377i		−	0.845260i	0.906302	−	0.422630i	$-0.138893\pi$
$354$		0			0
$355$			477.692i			1.34561i
$356$		0			0
$357$			752.041i			2.10656i
$358$		0			0
$359$			13.3730i			0.0372507i	0.999827	+	0.0186253i	$0.00592897\pi$
$359$			13.3730i			0.0372507i	−0.999827	+	0.0186253i	$0.994071\pi$
$360$		0			0
$361$	−26.6516			−0.0738271
$362$		0			0
$363$			518.907i			1.42950i
$364$		0			0
$365$	−37.4812			−0.102688
$366$		0			0
$367$	−432.610			−1.17877			−0.589387	−	0.807851i	$-0.700631\pi$
$367$	−432.610			−1.17877			−0.589387	+	0.807851i	$0.700631\pi$
$368$		0			0
$369$			1408.24i			3.81636i
$370$		0			0
$371$	104.858			0.282637
$372$		0			0
$373$		−	38.5002i		−	0.103218i	−0.998667	−	0.0516088i	$-0.983565\pi$
$373$		−	38.5002i		−	0.103218i	0.998667	−	0.0516088i	$-0.0164349\pi$
$374$		0			0
$375$			336.059i			0.896157i
$376$		0			0
$377$			6.05546i			0.0160622i
$378$		0			0
$379$			638.332i			1.68425i	0.539280	+	0.842127i	$0.318697\pi$
$379$			638.332i			1.68425i	−0.539280	+	0.842127i	$0.681303\pi$
$380$		0			0
$381$	−427.083			−1.12095
$382$		0			0
$383$	671.350			1.75287			0.876436	−	0.481519i	$-0.159915\pi$
$383$	671.350			1.75287			0.876436	+	0.481519i	$0.159915\pi$
$384$		0			0
$385$			277.567i			0.720954i
$386$		0			0
$387$			205.472i			0.530936i
$388$		0			0
$389$	587.616			1.51058			0.755290	−	0.655391i	$-0.227496\pi$
$389$	587.616			1.51058			0.755290	+	0.655391i	$0.227496\pi$
$390$		0			0
$391$			384.292i			0.982845i
$392$		0			0
$393$		−	216.321i		−	0.550436i
$394$		0			0
$395$	−378.285	−	335.316i	−0.957685	−	0.848901i
$396$		0			0
$397$	−159.890			−0.402746			−0.201373	−	0.979515i	$-0.564540\pi$
$397$	−159.890			−0.402746			−0.201373	+	0.979515i	$0.564540\pi$
$398$		0			0
$399$	−819.250			−2.05326
$400$		0			0
$401$		−	606.954i		−	1.51360i	−0.653646	−	0.756800i	$-0.726761\pi$
$401$		−	606.954i		−	1.51360i	0.653646	−	0.756800i	$-0.273239\pi$
$402$		0			0
$403$	−23.2030			−0.0575756
$404$		0			0
$405$	1947.23			4.80797
$406$		0			0
$407$		−	322.090i		−	0.791377i
$408$		0			0
$409$		−	621.424i		−	1.51937i	−0.650289	−	0.759687i	$-0.725352\pi$
$409$		−	621.424i		−	1.51937i	0.650289	−	0.759687i	$-0.274648\pi$
$410$		0			0
$411$	−888.022			−2.16064
$412$		0			0
$413$	−589.678			−1.42779
$414$		0			0
$415$	259.993			0.626490
$416$		0			0
$417$	331.735			0.795527
$418$		0			0
$419$		−	470.576i		−	1.12309i	−0.827445	−	0.561547i	$-0.810207\pi$
$419$		−	470.576i		−	1.12309i	0.827445	−	0.561547i	$-0.189793\pi$
$420$		0			0
$421$	582.850			1.38444			0.692221	−	0.721685i	$-0.256632\pi$
$421$	582.850			1.38444			0.692221	+	0.721685i	$0.256632\pi$
$422$		0			0
$423$			357.883i			0.846060i
$424$		0			0
$425$		−	267.636i		−	0.629732i
$426$		0			0
$427$	−400.337			−0.937557
$428$		0			0
$429$			23.3508i			0.0544308i
$430$		0			0
$431$	254.224			0.589847			0.294924	−	0.955521i	$-0.404706\pi$
$431$	254.224			0.589847			0.294924	+	0.955521i	$0.404706\pi$
$432$		0			0
$433$	−271.088			−0.626069			−0.313034	−	0.949742i	$-0.601346\pi$
$433$	−271.088			−0.626069			−0.313034	+	0.949742i	$0.601346\pi$
$434$		0			0
$435$	313.439			0.720548
$436$		0			0
$437$	−418.636			−0.957976
$438$		0			0
$439$	757.230			1.72490			0.862449	−	0.506144i	$-0.168930\pi$
$439$	757.230			1.72490			0.862449	+	0.506144i	$0.168930\pi$
$440$		0			0
$441$	263.029			0.596438
$442$		0			0
$443$			86.1383i			0.194443i	0.995263	+	0.0972216i	$0.0309956\pi$
$443$			86.1383i			0.194443i	−0.995263	+	0.0972216i	$0.969004\pi$
$444$		0			0
$445$	78.3307			0.176024
$446$		0			0
$447$	−98.7719			−0.220966
$448$		0			0
$449$		−	452.485i		−	1.00776i	−0.863773	−	0.503881i	$-0.831905\pi$
$449$		−	452.485i		−	1.00776i	0.863773	−	0.503881i	$-0.168095\pi$
$450$		0			0
$451$			320.945i			0.711630i
$452$		0			0
$453$		−	529.499i		−	1.16887i
$454$		0			0
$455$		−	35.4414i		−	0.0778932i
$456$		0			0
$457$	−772.113			−1.68952			−0.844762	−	0.535142i	$-0.820258\pi$
$457$	−772.113			−1.68952			−0.844762	+	0.535142i	$0.820258\pi$
$458$		0			0
$459$	1522.44			3.31686
$460$		0			0
$461$		−	99.6878i		−	0.216242i	−0.994138	−	0.108121i	$-0.965517\pi$
$461$		−	99.6878i		−	0.216242i	0.994138	−	0.108121i	$-0.0344835\pi$
$462$		0			0
$463$		−	611.798i		−	1.32138i	−0.750660	−	0.660689i	$-0.770264\pi$
$463$		−	611.798i		−	1.32138i	0.750660	−	0.660689i	$-0.229736\pi$
$464$		0			0
$465$			1201.01i			2.58283i
$466$		0			0
$467$	−594.094			−1.27215			−0.636075	−	0.771628i	$-0.719443\pi$
$467$	−594.094			−1.27215			−0.636075	+	0.771628i	$0.719443\pi$
$468$		0			0
$469$			325.796i			0.694660i
$470$		0			0
$471$	1158.68			2.46004
$472$		0			0
$473$			46.8284i			0.0990029i
$474$		0			0
$475$	291.554			0.613798
$476$		0			0
$477$		−	334.441i		−	0.701134i
$478$		0			0
$479$	−7.66178			−0.0159954			−0.00799768	−	0.999968i	$-0.502546\pi$
$479$	−7.66178			−0.0159954			−0.00799768	+	0.999968i	$0.502546\pi$
$480$		0			0
$481$			41.1264i			0.0855018i
$482$		0			0
$483$	1025.78			2.12376
$484$		0			0
$485$	817.197			1.68494
$486$		0			0
$487$	−672.701			−1.38132			−0.690659	−	0.723181i	$-0.742679\pi$
$487$	−672.701			−1.38132			−0.690659	+	0.723181i	$0.742679\pi$
$488$		0			0
$489$		−	1209.56i		−	2.47353i
$490$		0			0
$491$		−	729.442i		−	1.48563i	−0.669499	−	0.742813i	$-0.733491\pi$
$491$		−	729.442i		−	1.48563i	0.669499	−	0.742813i	$-0.266509\pi$
$492$		0			0
$493$	141.762			0.287549
$494$		0			0
$495$	885.289			1.78846
$496$		0			0
$497$	576.695			1.16035
$498$		0			0
$499$	15.1978			0.0304565			0.0152282	−	0.999884i	$-0.495153\pi$
$499$	15.1978			0.0304565			0.0152282	+	0.999884i	$0.495153\pi$
$500$		0			0
$501$			912.989i			1.82233i
$502$		0			0
$503$		−	452.158i		−	0.898922i	−0.893300	−	0.449461i	$-0.851616\pi$
$503$		−	452.158i		−	0.898922i	0.893300	−	0.449461i	$-0.148384\pi$
$504$		0			0
$505$	508.983			1.00789
$506$		0			0
$507$			977.197i			1.92741i
$508$		0			0
$509$		−	804.806i		−	1.58115i	−0.612364	−	0.790576i	$-0.709781\pi$
$509$		−	804.806i		−	1.58115i	0.612364	−	0.790576i	$-0.290219\pi$
$510$		0			0
$511$			45.2493i			0.0885506i
$512$		0			0
$513$			1658.50i			3.23294i
$514$		0			0
$515$			92.7194i			0.180038i
$516$		0			0
$517$			81.5637i			0.157764i
$518$		0			0
$519$	−668.529			−1.28811
$520$		0			0
$521$			771.747i			1.48128i	0.671902	+	0.740640i	$0.265478\pi$
$521$			771.747i			1.48128i	−0.671902	+	0.740640i	$0.734522\pi$
$522$		0			0
$523$	79.5893			0.152178			0.0760892	−	0.997101i	$-0.475757\pi$
$523$	79.5893			0.152178			0.0760892	+	0.997101i	$0.475757\pi$
$524$		0			0
$525$	−714.391			−1.36075
$526$		0			0
$527$			543.194i			1.03073i
$528$		0			0
$529$	−4.82887			−0.00912830
$530$		0			0
$531$			1880.75i			3.54191i
$532$		0			0
$533$		−	40.9801i		−	0.0768858i
$534$		0			0
$535$			136.062i			0.254322i
$536$		0			0
$537$		−	143.031i		−	0.266353i
$538$		0			0
$539$	59.9459			0.111217
$540$		0			0
$541$	−464.467			−0.858533			−0.429267	−	0.903178i	$-0.641228\pi$
$541$	−464.467			−0.858533			−0.429267	+	0.903178i	$0.641228\pi$
$542$		0			0
$543$		−	1008.21i		−	1.85675i
$544$		0			0
$545$		−	910.110i		−	1.66993i
$546$		0			0
$547$	678.614			1.24061			0.620306	−	0.784360i	$-0.287009\pi$
$547$	678.614			1.24061			0.620306	+	0.784360i	$0.287009\pi$
$548$		0			0
$549$			1276.86i			2.32579i
$550$		0			0
$551$			154.431i			0.280274i
$552$		0			0
$553$	−404.811	+	456.687i	−0.732028	+	0.825835i
$554$		0			0
$555$	2128.75			3.83559
$556$		0			0
$557$	−429.371			−0.770864			−0.385432	−	0.922736i	$-0.625948\pi$
$557$	−429.371			−0.770864			−0.385432	+	0.922736i	$0.625948\pi$
$558$		0			0
$559$		−	5.97932i		−	0.0106965i
$560$		0			0
$561$	546.656			0.974432
$562$		0			0
$563$	−500.284			−0.888604			−0.444302	−	0.895877i	$-0.646548\pi$
$563$	−500.284			−0.888604			−0.444302	+	0.895877i	$0.646548\pi$
$564$		0			0
$565$			989.521i			1.75136i
$566$		0			0
$567$		−	2350.80i		−	4.14603i
$568$		0			0
$569$	−694.114			−1.21988			−0.609942	−	0.792446i	$-0.708807\pi$
$569$	−694.114			−1.21988			−0.609942	+	0.792446i	$0.708807\pi$
$570$		0			0
$571$	−276.562			−0.484347			−0.242174	−	0.970233i	$-0.577860\pi$
$571$	−276.562			−0.484347			−0.242174	+	0.970233i	$0.577860\pi$
$572$		0			0
$573$	2132.13			3.72099
$574$		0			0
$575$	−365.053			−0.634875
$576$		0			0
$577$			541.450i			0.938388i	0.883095	+	0.469194i	$0.155455\pi$
$577$			541.450i			0.938388i	−0.883095	+	0.469194i	$0.844545\pi$
$578$		0			0
$579$	−487.579			−0.842105
$580$		0			0
$581$		−	313.878i		−	0.540237i
$582$		0			0
$583$		−	76.2211i		−	0.130739i
$584$		0			0
$585$	−113.039			−0.193229
$586$		0			0
$587$			495.305i			0.843790i	0.906645	+	0.421895i	$0.138635\pi$
$587$			495.305i			0.843790i	−0.906645	+	0.421895i	$0.861365\pi$
$588$		0			0
$589$	−591.738			−1.00465
$590$		0			0
$591$	−1498.64			−2.53577
$592$		0			0
$593$	−625.616			−1.05500			−0.527501	−	0.849554i	$-0.676871\pi$
$593$	−625.616			−1.05500			−0.527501	+	0.849554i	$0.676871\pi$
$594$		0			0
$595$	−829.703			−1.39446
$596$		0			0
$597$	93.5344			0.156674
$598$		0			0
$599$	−242.107			−0.404186			−0.202093	−	0.979366i	$-0.564774\pi$
$599$	−242.107			−0.404186			−0.202093	+	0.979366i	$0.564774\pi$
$600$		0			0
$601$			724.534i			1.20555i	0.797912	+	0.602773i	$0.205938\pi$
$601$			724.534i			1.20555i	−0.797912	+	0.602773i	$0.794062\pi$
$602$		0			0
$603$	1039.11			1.72324
$604$		0			0
$605$	−572.494			−0.946271
$606$		0			0
$607$			111.969i			0.184462i	0.995738	+	0.0922311i	$0.0293999\pi$
$607$			111.969i			0.184462i	−0.995738	+	0.0922311i	$0.970600\pi$
$608$		0			0
$609$		−	378.400i		−	0.621346i
$610$		0			0
$611$		−	10.4145i		−	0.0170451i
$612$		0			0
$613$		−	237.743i		−	0.387835i	−0.981018	−	0.193917i	$-0.937881\pi$
$613$		−	237.743i		−	0.387835i	0.981018	−	0.193917i	$-0.0621194\pi$
$614$		0			0
$615$	−2121.18			−3.44908
$616$		0			0
$617$	−162.306			−0.263056			−0.131528	−	0.991312i	$-0.541988\pi$
$617$	−162.306			−0.263056			−0.131528	+	0.991312i	$0.541988\pi$
$618$		0			0
$619$		−	109.480i		−	0.176865i	−0.996082	−	0.0884327i	$-0.971814\pi$
$619$		−	109.480i		−	0.176865i	0.996082	−	0.0884327i	$-0.0281858\pi$
$620$		0			0
$621$		−	2076.59i		−	3.34395i
$622$		0			0
$623$		−	94.5650i		−	0.151790i
$624$		0			0
$625$	−769.383			−1.23101
$626$		0			0
$627$			595.510i			0.949776i
$628$		0			0
$629$	962.792			1.53067
$630$		0			0
$631$			493.710i			0.782425i	0.920300	+	0.391213i	$0.127944\pi$
$631$			493.710i			0.782425i	−0.920300	+	0.391213i	$0.872056\pi$
$632$		0			0
$633$	−641.700			−1.01374
$634$		0			0
$635$		−	471.187i		−	0.742027i
$636$		0			0
$637$	−7.65424			−0.0120161
$638$		0			0
$639$		−	1839.34i		−	2.87847i
$640$		0			0
$641$	907.125			1.41517			0.707586	−	0.706628i	$-0.249784\pi$
$641$	907.125			1.41517			0.707586	+	0.706628i	$0.249784\pi$
$642$		0			0
$643$	968.751			1.50661			0.753306	−	0.657671i	$-0.228458\pi$
$643$	968.751			1.50661			0.753306	+	0.657671i	$0.228458\pi$
$644$		0			0
$645$	−309.497			−0.479840
$646$		0			0
$647$		−	811.942i		−	1.25493i	−0.778643	−	0.627467i	$-0.784092\pi$
$647$		−	811.942i		−	1.25493i	0.778643	−	0.627467i	$-0.215908\pi$
$648$		0			0
$649$			428.635i			0.660454i
$650$		0			0
$651$	1449.93			2.22723
$652$		0			0
$653$	348.523			0.533726			0.266863	−	0.963735i	$-0.414013\pi$
$653$	348.523			0.533726			0.266863	+	0.963735i	$0.414013\pi$
$654$		0			0
$655$	238.660			0.364367
$656$		0			0
$657$	144.321			0.219666
$658$		0			0
$659$		−	89.1660i		−	0.135305i	−0.997709	−	0.0676525i	$-0.978449\pi$
$659$		−	89.1660i		−	0.135305i	0.997709	−	0.0676525i	$-0.0215509\pi$
$660$		0			0
$661$			98.1421i			0.148475i	0.997241	+	0.0742376i	$0.0236523\pi$
$661$			98.1421i			0.148475i	−0.997241	+	0.0742376i	$0.976348\pi$
$662$		0			0
$663$	−69.8002			−0.105279
$664$		0			0
$665$		−	903.852i		−	1.35918i
$666$		0			0
$667$		−	193.362i		−	0.289898i
$668$		0			0
$669$		−	438.107i		−	0.654868i
$670$		0			0
$671$			291.004i			0.433686i
$672$		0			0
$673$		−	349.410i		−	0.519183i	−0.965718	−	0.259592i	$-0.916412\pi$
$673$		−	349.410i		−	0.519183i	0.965718	−	0.259592i	$-0.0835880\pi$
$674$		0			0
$675$			1446.22i			2.14255i
$676$		0			0
$677$	−561.067			−0.828755			−0.414378	−	0.910105i	$-0.636001\pi$
$677$	−561.067			−0.828755			−0.414378	+	0.910105i	$0.636001\pi$
$678$		0			0
$679$		−	986.565i		−	1.45297i
$680$		0			0
$681$	225.597			0.331273
$682$		0			0
$683$	341.339			0.499764			0.249882	−	0.968276i	$-0.419608\pi$
$683$	341.339			0.499764			0.249882	+	0.968276i	$0.419608\pi$
$684$		0			0
$685$		−	979.727i		−	1.43026i
$686$		0			0
$687$	−583.887			−0.849908
$688$		0			0
$689$			9.73235i			0.0141253i
$690$		0			0
$691$			566.537i			0.819879i	0.912113	+	0.409940i	$0.134450\pi$
$691$			566.537i			0.819879i	−0.912113	+	0.409940i	$0.865550\pi$
$692$		0			0
$693$		−	1068.77i		−	1.54223i
$694$		0			0
$695$			365.992i			0.526608i
$696$		0			0
$697$	−959.368			−1.37642
$698$		0			0
$699$	−2538.07			−3.63101
$700$		0			0
$701$			845.646i			1.20634i	0.797612	+	0.603171i	$0.206096\pi$
$701$			845.646i			1.20634i	−0.797612	+	0.603171i	$0.793904\pi$
$702$		0			0
$703$			1048.83i			1.49194i
$704$		0			0
$705$	−539.069			−0.764637
$706$		0			0
$707$		−	614.472i		−	0.869125i
$708$		0			0
$709$		−	6.44178i		−	0.00908572i	−0.999990	−	0.00454286i	$-0.998554\pi$
$709$		−	6.44178i		−	0.00908572i	0.999990	−	0.00454286i	$-0.00144604\pi$
$710$		0			0
$711$	1456.58	+	1291.13i	2.04864	+	1.81593i
$712$		0			0
$713$	740.912			1.03915
$714$		0			0
$715$	−25.7622			−0.0360311
$716$		0			0
$717$		−	806.807i		−	1.12525i
$718$		0			0
$719$	−50.0547			−0.0696171			−0.0348085	−	0.999394i	$-0.511082\pi$
$719$	−50.0547			−0.0696171			−0.0348085	+	0.999394i	$0.511082\pi$
$720$		0			0
$721$	111.936			0.155251
$722$		0			0
$723$			137.427i			0.190078i
$724$		0			0
$725$			134.665i			0.185744i
$726$		0			0
$727$	−538.942			−0.741323			−0.370661	−	0.928768i	$-0.620869\pi$
$727$	−538.942			−0.741323			−0.370661	+	0.928768i	$0.620869\pi$
$728$		0			0
$729$	−2763.25			−3.79047
$730$		0			0
$731$	−139.979			−0.191490
$732$		0			0
$733$	639.757			0.872792			0.436396	−	0.899755i	$-0.356255\pi$
$733$	639.757			0.872792			0.436396	+	0.899755i	$0.356255\pi$
$734$		0			0
$735$			396.193i			0.539039i
$736$		0			0
$737$	236.820			0.321329
$738$		0			0
$739$		−	1128.63i		−	1.52724i	−0.645667	−	0.763619i	$-0.723421\pi$
$739$		−	1128.63i		−	1.52724i	0.645667	−	0.763619i	$-0.276579\pi$
$740$		0			0
$741$		−	76.0381i		−	0.102616i
$742$		0			0
$743$	779.477			1.04909			0.524547	−	0.851381i	$-0.324235\pi$
$743$	779.477			1.04909			0.524547	+	0.851381i	$0.324235\pi$
$744$		0			0
$745$		−	108.972i		−	0.146271i
$746$		0			0
$747$	−1001.10			−1.34016
$748$		0			0
$749$	164.261			0.219308
$750$		0			0
$751$	−988.929			−1.31682			−0.658408	−	0.752661i	$-0.728770\pi$
$751$	−988.929			−1.31682			−0.658408	+	0.752661i	$0.728770\pi$
$752$		0			0
$753$	1281.43			1.70177
$754$		0			0
$755$	584.179			0.773747
$756$		0			0
$757$	−132.217			−0.174659			−0.0873295	−	0.996179i	$-0.527833\pi$
$757$	−132.217			−0.174659			−0.0873295	+	0.996179i	$0.527833\pi$
$758$		0			0
$759$		−	745.634i		−	0.982390i
$760$		0			0
$761$	100.415			0.131952			0.0659758	−	0.997821i	$-0.478984\pi$
$761$	100.415			0.131952			0.0659758	+	0.997821i	$0.478984\pi$
$762$		0			0
$763$	−1098.73			−1.44002
$764$		0			0
$765$			2646.30i			3.45922i
$766$		0			0
$767$		−	54.7305i		−	0.0713566i
$768$		0			0
$769$			414.587i			0.539125i	0.962983	+	0.269563i	$0.0868791\pi$
$769$			414.587i			0.539125i	−0.962983	+	0.269563i	$0.913121\pi$
$770$		0			0
$771$			506.449i			0.656873i
$772$		0			0
$773$	−1145.11			−1.48139			−0.740694	−	0.671843i	$-0.765503\pi$
$773$	−1145.11			−1.48139			−0.740694	+	0.671843i	$0.765503\pi$
$774$		0			0
$775$	−516.000			−0.665806
$776$		0			0
$777$		−	2569.95i		−	3.30752i
$778$		0			0
$779$		−	1045.10i		−	1.34160i
$780$		0			0
$781$		−	419.198i		−	0.536745i
$782$		0			0
$783$	−766.036			−0.978334
$784$		0			0
$785$			1278.33i			1.62845i
$786$		0			0
$787$	−798.997			−1.01524			−0.507622	−	0.861580i	$-0.669475\pi$
$787$	−798.997			−1.01524			−0.507622	+	0.861580i	$0.669475\pi$
$788$		0			0
$789$		−	1208.83i		−	1.53211i
$790$		0			0
$791$	1194.60			1.51024
$792$		0			0
$793$		−	37.1570i		−	0.0468563i
$794$		0			0
$795$	503.759			0.633659
$796$		0			0
$797$			369.855i			0.464060i	0.972709	+	0.232030i	$0.0745367\pi$
$797$			369.855i			0.464060i	−0.972709	+	0.232030i	$0.925463\pi$
$798$		0			0
$799$	−243.810			−0.305144
$800$		0			0
$801$	−301.611			−0.376543
$802$		0			0
$803$	32.8916			0.0409609
$804$		0			0
$805$			1131.71i			1.40585i
$806$		0			0
$807$			1409.36i			1.74642i
$808$		0			0
$809$	1209.25			1.49475			0.747374	−	0.664404i	$-0.231315\pi$
$809$	1209.25			1.49475			0.747374	+	0.664404i	$0.231315\pi$
$810$		0			0
$811$	−1049.68			−1.29431			−0.647153	−	0.762360i	$-0.724040\pi$
$811$	−1049.68			−1.29431			−0.647153	+	0.762360i	$0.724040\pi$
$812$		0			0
$813$	1281.71			1.57652
$814$		0			0
$815$	1334.47			1.63738
$816$		0			0
$817$		−	152.489i		−	0.186645i
$818$		0			0
$819$			136.467i			0.166626i
$820$		0			0
$821$	−207.870			−0.253192			−0.126596	−	0.991954i	$-0.540405\pi$
$821$	−207.870			−0.253192			−0.126596	+	0.991954i	$0.540405\pi$
$822$		0			0
$823$			1305.11i			1.58580i	0.609351	+	0.792901i	$0.291430\pi$
$823$			1305.11i			1.58580i	−0.609351	+	0.792901i	$0.708570\pi$
$824$		0			0
$825$			519.289i			0.629441i
$826$		0			0
$827$		−	343.902i		−	0.415843i	−0.978146	−	0.207921i	$-0.933330\pi$
$827$		−	343.902i		−	0.415843i	0.978146	−	0.207921i	$-0.0666698\pi$
$828$		0			0
$829$			27.5538i			0.0332374i	0.999862	+	0.0166187i	$0.00529014\pi$
$829$			27.5538i			0.0332374i	−0.999862	+	0.0166187i	$0.994710\pi$
$830$		0			0
$831$			1856.76i			2.23437i
$832$		0			0
$833$			179.190i			0.215114i
$834$		0			0
$835$	−1007.27			−1.20631
$836$		0			0
$837$		−	2935.25i		−	3.50687i
$838$		0			0
$839$	458.048			0.545945			0.272972	−	0.962022i	$-0.411993\pi$
$839$	458.048			0.545945			0.272972	+	0.962022i	$0.411993\pi$
$840$		0			0
$841$	769.671			0.915185
$842$		0			0
$843$		−	2621.60i		−	3.10984i
$844$		0			0
$845$	−1078.11			−1.27587
$846$		0			0
$847$			691.145i			0.815992i
$848$		0			0
$849$			3042.95i			3.58415i
$850$		0			0
$851$		−	1313.24i		−	1.54317i
$852$		0			0
$853$		−	633.466i		−	0.742633i	−0.928506	−	0.371317i	$-0.878906\pi$
$853$		−	633.466i		−	0.742633i	0.928506	−	0.371317i	$-0.121094\pi$
$854$		0			0
$855$	−2882.80			−3.37169
$856$		0			0
$857$	1187.28			1.38539			0.692695	−	0.721230i	$-0.256423\pi$
$857$	1187.28			1.38539			0.692695	+	0.721230i	$0.256423\pi$
$858$		0			0
$859$		−	646.718i		−	0.752873i	−0.926442	−	0.376436i	$-0.877149\pi$
$859$		−	646.718i		−	0.752873i	0.926442	−	0.376436i	$-0.122851\pi$
$860$		0			0
$861$			2560.81i			2.97422i
$862$		0			0
$863$	394.999			0.457704			0.228852	−	0.973461i	$-0.426503\pi$
$863$	394.999			0.457704			0.228852	+	0.973461i	$0.426503\pi$
$864$		0			0
$865$		−	737.566i		−	0.852678i
$866$		0			0
$867$		−	42.1007i		−	0.0485591i
$868$		0			0
$869$	331.964	+	294.256i	0.382007	+	0.338615i
$870$		0			0
$871$	−30.2385			−0.0347170
$872$		0			0
$873$	−3146.61			−3.60436
$874$		0			0
$875$			447.605i			0.511548i
$876$		0			0
$877$	732.466			0.835195			0.417597	−	0.908632i	$-0.362872\pi$
$877$	732.466			0.835195			0.417597	+	0.908632i	$0.362872\pi$
$878$		0			0
$879$	1751.56			1.99267
$880$		0			0
$881$			418.869i			0.475447i	0.971333	+	0.237724i	$0.0764012\pi$
$881$			418.869i			0.475447i	−0.971333	+	0.237724i	$0.923599\pi$
$882$		0			0
$883$			723.914i			0.819835i	0.912123	+	0.409918i	$0.134443\pi$
$883$			723.914i			0.819835i	−0.912123	+	0.409918i	$0.865557\pi$
$884$		0			0
$885$	−2832.92			−3.20104
$886$		0			0
$887$	1611.13			1.81638			0.908191	−	0.418556i	$-0.137464\pi$
$887$	1611.13			1.81638			0.908191	+	0.418556i	$0.137464\pi$
$888$		0			0
$889$	−568.842			−0.639868
$890$		0			0
$891$	−1708.79			−1.91783
$892$		0			0
$893$		−	265.599i		−	0.297423i
$894$		0			0
$895$	157.802			0.176315
$896$		0			0
$897$			95.2069i			0.106139i
$898$		0			0
$899$		−	273.316i		−	0.304022i
$900$		0			0
$901$	227.840			0.252874
$902$		0			0
$903$			373.642i			0.413778i
$904$		0			0
$905$	1112.33			1.22909
$906$		0			0
$907$	−413.826			−0.456258			−0.228129	−	0.973631i	$-0.573261\pi$
$907$	−413.826			−0.456258			−0.228129	+	0.973631i	$0.573261\pi$
$908$		0			0
$909$	−1959.83			−2.15603
$910$		0			0
$911$	882.425			0.968633			0.484317	−	0.874893i	$-0.339068\pi$
$911$	882.425			0.968633			0.484317	+	0.874893i	$0.339068\pi$
$912$		0			0
$913$	−228.157			−0.249898
$914$		0			0
$915$	−1923.29			−2.10196
$916$		0			0
$917$		−	288.124i		−	0.314202i
$918$		0			0
$919$	379.413			0.412854			0.206427	−	0.978462i	$-0.433816\pi$
$919$	379.413			0.412854			0.206427	+	0.978462i	$0.433816\pi$
$920$		0			0
$921$	−2515.54			−2.73131
$922$		0			0
$923$			53.5256i			0.0579909i
$924$		0			0
$925$			914.591i			0.988747i
$926$		0			0
$927$		−	357.015i		−	0.385129i
$928$		0			0
$929$			457.682i			0.492661i	0.969186	+	0.246330i	$0.0792248\pi$
$929$			457.682i			0.492661i	−0.969186	+	0.246330i	$0.920775\pi$
$930$		0			0
$931$	−195.204			−0.209671
$932$		0			0
$933$	−860.199			−0.921971
$934$		0			0
$935$			603.108i			0.645036i
$936$		0			0
$937$			1483.01i			1.58272i	0.611352	+	0.791359i	$0.290626\pi$
$937$			1483.01i			1.58272i	−0.611352	+	0.791359i	$0.709374\pi$
$938$		0			0
$939$			2007.06i			2.13745i
$940$		0			0
$941$	−127.076			−0.135044			−0.0675218	−	0.997718i	$-0.521509\pi$
$941$	−127.076			−0.135044			−0.0675218	+	0.997718i	$0.521509\pi$
$942$		0			0
$943$			1308.57i			1.38767i
$944$		0			0
$945$	4483.45			4.74439
$946$		0			0
$947$		−	1154.67i		−	1.21929i	−0.792674	−	0.609645i	$-0.791312\pi$
$947$		−	1154.67i		−	1.21929i	0.792674	−	0.609645i	$-0.208688\pi$
$948$		0			0
$949$	−4.19979			−0.00442549
$950$		0			0
$951$		−	1769.99i		−	1.86119i
$952$		0			0
$953$	604.411			0.634219			0.317110	−	0.948389i	$-0.397288\pi$
$953$	604.411			0.634219			0.317110	+	0.948389i	$0.397288\pi$
$954$		0			0
$955$			2352.31i			2.46315i
$956$		0			0
$957$	−275.058			−0.287416
$958$		0			0
$959$	−1182.78			−1.23335
$960$		0			0
$961$	86.2736			0.0897749
$962$		0			0
$963$		−	523.905i		−	0.544034i
$964$		0			0
$965$		−	537.930i		−	0.557441i
$966$		0			0
$967$	1822.30			1.88449			0.942243	−	0.334931i	$-0.108713\pi$
$967$	1822.30			1.88449			0.942243	+	0.334931i	$0.108713\pi$
$968$		0			0
$969$	−1780.10			−1.83704
$970$		0			0
$971$	1015.49			1.04581			0.522907	−	0.852389i	$-0.324847\pi$
$971$	1015.49			1.04581			0.522907	+	0.852389i	$0.324847\pi$
$972$		0			0
$973$	441.846			0.454106
$974$		0			0
$975$		−	66.3058i		−	0.0680059i
$976$		0			0
$977$		−	1808.32i		−	1.85089i	−0.378883	−	0.925445i	$-0.623692\pi$
$977$		−	1808.32i		−	1.85089i	0.378883	−	0.925445i	$-0.376308\pi$
$978$		0			0
$979$	−68.7390			−0.0702134
$980$		0			0
$981$			3504.37i			3.57224i
$982$		0			0
$983$			1529.34i			1.55579i	0.628397	+	0.777893i	$0.283711\pi$
$983$			1529.34i			1.55579i	−0.628397	+	0.777893i	$0.716289\pi$
$984$		0			0
$985$		−	1653.40i		−	1.67858i
$986$		0			0
$987$			650.793i			0.659365i
$988$		0			0
$989$			190.930i			0.193054i
$990$		0			0
$991$		−	477.812i		−	0.482151i	−0.970506	−	0.241076i	$-0.922500\pi$
$991$		−	477.812i		−	0.482151i	0.970506	−	0.241076i	$-0.0775002\pi$
$992$		0			0
$993$	−3370.63			−3.39439
$994$		0			0
$995$			103.193i			0.103712i
$996$		0			0
$997$	1538.67			1.54330			0.771649	−	0.636048i	$-0.219432\pi$
$997$	1538.67			1.54330			0.771649	+	0.636048i	$0.219432\pi$
$998$		0			0
$999$	−5202.62			−5.20783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1264.3.e.b.1105.1		8
4.3	odd	2		79.3.b.b.78.2	yes	8
12.11	even	2		711.3.d.b.631.8		8
79.78	odd	2	inner	1264.3.e.b.1105.8		8
316.315	even	2		79.3.b.b.78.1	✓	8
948.947	odd	2		711.3.d.b.631.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
79.3.b.b.78.1	✓	8	316.315	even	2
79.3.b.b.78.2	yes	8	4.3	odd	2
711.3.d.b.631.7		8	948.947	odd	2
711.3.d.b.631.8		8	12.11	even	2
1264.3.e.b.1105.1		8	1.1	even	1	trivial
1264.3.e.b.1105.8		8	79.78	odd	2	inner

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:	\( N \)	\(=\)	\( 1264 = 2^{4} \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1264.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.79987i q^{3} +6.39881 q^{5} -7.72499i q^{7} -24.6385 q^{9} +O(q^{10})\)	\(q-5.79987i q^{3} +6.39881 q^{5} -7.72499i q^{7} -24.6385 q^{9} -5.61527 q^{11} +0.716990 q^{13} -37.1123i q^{15} -16.7851i q^{17} +18.2852 q^{19} -44.8040 q^{21} -22.8948 q^{23} +15.9448 q^{25} +90.7016i q^{27} +8.44567i q^{29} -32.3616 q^{31} +32.5679i q^{33} -49.4308i q^{35} +57.3597i q^{37} -4.15845i q^{39} -57.1558i q^{41} -8.33947i q^{43} -157.657 q^{45} -14.5253i q^{47} -10.6755 q^{49} -97.3517 q^{51} +13.5739i q^{53} -35.9311 q^{55} -106.052i q^{57} -76.3337i q^{59} -51.8236i q^{61} +190.333i q^{63} +4.58789 q^{65} -42.1742 q^{67} +132.787i q^{69} +74.6532i q^{71} -5.85752 q^{73} -92.4779i q^{75} +43.3779i q^{77} +(-59.1180 - 52.4028i) q^{79} +304.311 q^{81} +40.6315 q^{83} -107.405i q^{85} +48.9838 q^{87} +12.2414 q^{89} -5.53874i q^{91} +187.693i q^{93} +117.004 q^{95} +127.711 q^{97} +138.352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} - 82 q^{9}+O(q^{10}) \) 8 * q - 2 * q^5 - 82 * q^9	\( 8 q - 2 q^{5} - 82 q^{9} - 28 q^{11} - 30 q^{13} + 32 q^{19} - 2 q^{21} - 60 q^{23} - 50 q^{25} - 36 q^{31} - 260 q^{45} - 294 q^{49} - 236 q^{51} - 140 q^{55} + 194 q^{65} - 132 q^{67} + 384 q^{73} - 210 q^{79} + 56 q^{81} - 96 q^{83} + 8 q^{87} + 102 q^{89} + 12 q^{95} + 470 q^{97} + 524 q^{99}+O(q^{100}) \) 8 * q - 2 * q^5 - 82 * q^9 - 28 * q^11 - 30 * q^13 + 32 * q^19 - 2 * q^21 - 60 * q^23 - 50 * q^25 - 36 * q^31 - 260 * q^45 - 294 * q^49 - 236 * q^51 - 140 * q^55 + 194 * q^65 - 132 * q^67 + 384 * q^73 - 210 * q^79 + 56 * q^81 - 96 * q^83 + 8 * q^87 + 102 * q^89 + 12 * q^95 + 470 * q^97 + 524 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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