LMFDB - Embedded newform 768.3.b.c.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,3,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("768.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.b (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:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.3.b.c.127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +2.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +2.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +6.92820 q^{11} +2.00000i q^{13} -3.46410i q^{15} +10.0000 q^{17} -20.7846 q^{19} -12.0000i q^{21} +27.7128i q^{23} +21.0000 q^{25} -5.19615 q^{27} -26.0000i q^{29} +6.92820i q^{31} -12.0000 q^{33} -13.8564 q^{35} -26.0000i q^{37} -3.46410i q^{39} -58.0000 q^{41} -48.4974 q^{43} +6.00000i q^{45} +69.2820i q^{47} +1.00000 q^{49} -17.3205 q^{51} +74.0000i q^{53} +13.8564i q^{55} +36.0000 q^{57} -90.0666 q^{59} +26.0000i q^{61} +20.7846i q^{63} -4.00000 q^{65} -6.92820 q^{67} -48.0000i q^{69} +46.0000 q^{73} -36.3731 q^{75} +48.0000i q^{77} +117.779i q^{79} +9.00000 q^{81} -48.4974 q^{83} +20.0000i q^{85} +45.0333i q^{87} -82.0000 q^{89} -13.8564 q^{91} -12.0000i q^{93} -41.5692i q^{95} +2.00000 q^{97} +20.7846 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} + 40 q^{17} + 84 q^{25} - 48 q^{33} - 232 q^{41} + 4 q^{49} + 144 q^{57} - 16 q^{65} + 184 q^{73} + 36 q^{81} - 328 q^{89} + 8 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 + 40 * q^17 + 84 * q^25 - 48 * q^33 - 232 * q^41 + 4 * q^49 + 144 * q^57 - 16 * q^65 + 184 * q^73 + 36 * q^81 - 328 * q^89 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	2.00000i	0.400000i	0.979796	+	0.200000i	$0.0640942\pi$
$5$	2.00000i	0.400000i	−0.979796	+	0.200000i	$0.935906\pi$
$6$	0	0
$7$	6.92820i	0.989743i	0.868966	+	0.494872i	$0.164785\pi$
$7$	6.92820i	0.989743i	−0.868966	+	0.494872i	$0.835215\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	6.92820	0.629837	0.314918	−	0.949119i	$-0.398023\pi$
$11$	6.92820	0.629837	0.314918	+	0.949119i	$0.398023\pi$
$12$	0	0
$13$	2.00000i	0.153846i	0.997037	+	0.0769231i	$0.0245096\pi$
$13$	2.00000i	0.153846i	−0.997037	+	0.0769231i	$0.975490\pi$
$14$	0	0
$15$	− 3.46410i	− 0.230940i
$16$	0	0
$17$	10.0000	0.588235	0.294118	−	0.955769i	$-0.404974\pi$
$17$	10.0000	0.588235	0.294118	+	0.955769i	$0.404974\pi$
$18$	0	0
$19$	−20.7846	−1.09393	−0.546963	−	0.837157i	$-0.684216\pi$
$19$	−20.7846	−1.09393	−0.546963	+	0.837157i	$0.684216\pi$
$20$	0	0
$21$	− 12.0000i	− 0.571429i
$22$	0	0
$23$	27.7128i	1.20490i	0.798155	+	0.602452i	$0.205810\pi$
$23$	27.7128i	1.20490i	−0.798155	+	0.602452i	$0.794190\pi$
$24$	0	0
$25$	21.0000	0.840000
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	− 26.0000i	− 0.896552i	−0.893895	−	0.448276i	$-0.852038\pi$
$29$	− 26.0000i	− 0.896552i	0.893895	−	0.448276i	$-0.147962\pi$
$30$	0	0
$31$	6.92820i	0.223490i	0.993737	+	0.111745i	$0.0356441\pi$
$31$	6.92820i	0.223490i	−0.993737	+	0.111745i	$0.964356\pi$
$32$	0	0
$33$	−12.0000	−0.363636
$34$	0	0
$35$	−13.8564	−0.395897
$36$	0	0
$37$	− 26.0000i	− 0.702703i	−0.936244	−	0.351351i	$-0.885722\pi$
$37$	− 26.0000i	− 0.702703i	0.936244	−	0.351351i	$-0.114278\pi$
$38$	0	0
$39$	− 3.46410i	− 0.0888231i
$40$	0	0
$41$	−58.0000	−1.41463	−0.707317	−	0.706896i	$-0.750095\pi$
$41$	−58.0000	−1.41463	−0.707317	+	0.706896i	$0.750095\pi$
$42$	0	0
$43$	−48.4974	−1.12785	−0.563924	−	0.825827i	$-0.690709\pi$
$43$	−48.4974	−1.12785	−0.563924	+	0.825827i	$0.690709\pi$
$44$	0	0
$45$	6.00000i	0.133333i
$46$	0	0
$47$	69.2820i	1.47409i	0.675846	+	0.737043i	$0.263778\pi$
$47$	69.2820i	1.47409i	−0.675846	+	0.737043i	$0.736222\pi$
$48$	0	0
$49$	1.00000	0.0204082
$50$	0	0
$51$	−17.3205	−0.339618
$52$	0	0
$53$	74.0000i	1.39623i	0.715987	+	0.698113i	$0.245977\pi$
$53$	74.0000i	1.39623i	−0.715987	+	0.698113i	$0.754023\pi$
$54$	0	0
$55$	13.8564i	0.251935i
$56$	0	0
$57$	36.0000	0.631579
$58$	0	0
$59$	−90.0666	−1.52655	−0.763277	−	0.646072i	$-0.776411\pi$
$59$	−90.0666	−1.52655	−0.763277	+	0.646072i	$0.776411\pi$
$60$	0	0
$61$	26.0000i	0.426230i	0.977027	+	0.213115i	$0.0683608\pi$
$61$	26.0000i	0.426230i	−0.977027	+	0.213115i	$0.931639\pi$
$62$	0	0
$63$	20.7846i	0.329914i
$64$	0	0
$65$	−4.00000	−0.0615385
$66$	0	0
$67$	−6.92820	−0.103406	−0.0517030	−	0.998663i	$-0.516465\pi$
$67$	−6.92820	−0.103406	−0.0517030	+	0.998663i	$0.516465\pi$
$68$	0	0
$69$	− 48.0000i	− 0.695652i
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	46.0000	0.630137	0.315068	−	0.949069i	$-0.397973\pi$
$73$	46.0000	0.630137	0.315068	+	0.949069i	$0.397973\pi$
$74$	0	0
$75$	−36.3731	−0.484974
$76$	0	0
$77$	48.0000i	0.623377i
$78$	0	0
$79$	117.779i	1.49088i	0.666573	+	0.745440i	$0.267760\pi$
$79$	117.779i	1.49088i	−0.666573	+	0.745440i	$0.732240\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	−48.4974	−0.584306	−0.292153	−	0.956372i	$-0.594372\pi$
$83$	−48.4974	−0.584306	−0.292153	+	0.956372i	$0.594372\pi$
$84$	0	0
$85$	20.0000i	0.235294i
$86$	0	0
$87$	45.0333i	0.517624i
$88$	0	0
$89$	−82.0000	−0.921348	−0.460674	−	0.887569i	$-0.652392\pi$
$89$	−82.0000	−0.921348	−0.460674	+	0.887569i	$0.652392\pi$
$90$	0	0
$91$	−13.8564	−0.152268
$92$	0	0
$93$	− 12.0000i	− 0.129032i
$94$	0	0
$95$	− 41.5692i	− 0.437571i
$96$	0	0
$97$	2.00000	0.0206186	0.0103093	−	0.999947i	$-0.496718\pi$
$97$	2.00000	0.0206186	0.0103093	+	0.999947i	$0.496718\pi$
$98$	0	0
$99$	20.7846	0.209946
$100$	0	0
$101$	74.0000i	0.732673i	0.930482	+	0.366337i	$0.119388\pi$
$101$	74.0000i	0.732673i	−0.930482	+	0.366337i	$0.880612\pi$
$102$	0	0
$103$	76.2102i	0.739905i	0.929051	+	0.369953i	$0.120626\pi$
$103$	76.2102i	0.739905i	−0.929051	+	0.369953i	$0.879374\pi$
$104$	0	0
$105$	24.0000	0.228571
$106$	0	0
$107$	20.7846	0.194249	0.0971243	−	0.995272i	$-0.469036\pi$
$107$	20.7846	0.194249	0.0971243	+	0.995272i	$0.469036\pi$
$108$	0	0
$109$	− 46.0000i	− 0.422018i	−0.977484	−	0.211009i	$-0.932325\pi$
$109$	− 46.0000i	− 0.422018i	0.977484	−	0.211009i	$-0.0676750\pi$
$110$	0	0
$111$	45.0333i	0.405706i
$112$	0	0
$113$	−110.000	−0.973451	−0.486726	−	0.873555i	$-0.661809\pi$
$113$	−110.000	−0.973451	−0.486726	+	0.873555i	$0.661809\pi$
$114$	0	0
$115$	−55.4256	−0.481962
$116$	0	0
$117$	6.00000i	0.0512821i
$118$	0	0
$119$	69.2820i	0.582202i
$120$	0	0
$121$	−73.0000	−0.603306
$122$	0	0
$123$	100.459	0.816739
$124$	0	0
$125$	92.0000i	0.736000i
$126$	0	0
$127$	− 145.492i	− 1.14561i	−0.819692	−	0.572804i	$-0.805856\pi$
$127$	− 145.492i	− 1.14561i	0.819692	−	0.572804i	$-0.194144\pi$
$128$	0	0
$129$	84.0000	0.651163
$130$	0	0
$131$	−117.779	−0.899080	−0.449540	−	0.893260i	$-0.648412\pi$
$131$	−117.779	−0.899080	−0.449540	+	0.893260i	$0.648412\pi$
$132$	0	0
$133$	− 144.000i	− 1.08271i
$134$	0	0
$135$	− 10.3923i	− 0.0769800i
$136$	0	0
$137$	−10.0000	−0.0729927	−0.0364964	−	0.999334i	$-0.511620\pi$
$137$	−10.0000	−0.0729927	−0.0364964	+	0.999334i	$0.511620\pi$
$138$	0	0
$139$	48.4974	0.348902	0.174451	−	0.984666i	$-0.444185\pi$
$139$	48.4974	0.348902	0.174451	+	0.984666i	$0.444185\pi$
$140$	0	0
$141$	− 120.000i	− 0.851064i
$142$	0	0
$143$	13.8564i	0.0968979i
$144$	0	0
$145$	52.0000	0.358621
$146$	0	0
$147$	−1.73205	−0.0117827
$148$	0	0
$149$	2.00000i	0.0134228i	0.999977	+	0.00671141i	$0.00213632\pi$
$149$	2.00000i	0.0134228i	−0.999977	+	0.00671141i	$0.997864\pi$
$150$	0	0
$151$	90.0666i	0.596468i	0.954493	+	0.298234i	$0.0963975\pi$
$151$	90.0666i	0.596468i	−0.954493	+	0.298234i	$0.903602\pi$
$152$	0	0
$153$	30.0000	0.196078
$154$	0	0
$155$	−13.8564	−0.0893962
$156$	0	0
$157$	− 214.000i	− 1.36306i	−0.731791	−	0.681529i	$-0.761315\pi$
$157$	− 214.000i	− 1.36306i	0.731791	−	0.681529i	$-0.238685\pi$
$158$	0	0
$159$	− 128.172i	− 0.806112i
$160$	0	0
$161$	−192.000	−1.19255
$162$	0	0
$163$	−20.7846	−0.127513	−0.0637565	−	0.997965i	$-0.520308\pi$
$163$	−20.7846	−0.127513	−0.0637565	+	0.997965i	$0.520308\pi$
$164$	0	0
$165$	− 24.0000i	− 0.145455i
$166$	0	0
$167$	− 96.9948i	− 0.580807i	−0.956904	−	0.290404i	$-0.906210\pi$
$167$	− 96.9948i	− 0.580807i	0.956904	−	0.290404i	$-0.0937896\pi$
$168$	0	0
$169$	165.000	0.976331
$170$	0	0
$171$	−62.3538	−0.364642
$172$	0	0
$173$	334.000i	1.93064i	0.261077	+	0.965318i	$0.415922\pi$
$173$	334.000i	1.93064i	−0.261077	+	0.965318i	$0.584078\pi$
$174$	0	0
$175$	145.492i	0.831384i
$176$	0	0
$177$	156.000	0.881356
$178$	0	0
$179$	187.061	1.04504	0.522518	−	0.852628i	$-0.324993\pi$
$179$	187.061	1.04504	0.522518	+	0.852628i	$0.324993\pi$
$180$	0	0
$181$	− 2.00000i	− 0.0110497i	−0.999985	−	0.00552486i	$-0.998241\pi$
$181$	− 2.00000i	− 0.0110497i	0.999985	−	0.00552486i	$-0.00175863\pi$
$182$	0	0
$183$	− 45.0333i	− 0.246084i
$184$	0	0
$185$	52.0000	0.281081
$186$	0	0
$187$	69.2820	0.370492
$188$	0	0
$189$	− 36.0000i	− 0.190476i
$190$	0	0
$191$	− 221.703i	− 1.16075i	−0.814351	−	0.580373i	$-0.802907\pi$
$191$	− 221.703i	− 1.16075i	0.814351	−	0.580373i	$-0.197093\pi$
$192$	0	0
$193$	290.000	1.50259	0.751295	−	0.659966i	$-0.229429\pi$
$193$	290.000	1.50259	0.751295	+	0.659966i	$0.229429\pi$
$194$	0	0
$195$	6.92820	0.0355292
$196$	0	0
$197$	26.0000i	0.131980i	0.997820	+	0.0659898i	$0.0210205\pi$
$197$	26.0000i	0.131980i	−0.997820	+	0.0659898i	$0.978980\pi$
$198$	0	0
$199$	− 394.908i	− 1.98446i	−0.124416	−	0.992230i	$-0.539706\pi$
$199$	− 394.908i	− 1.98446i	0.124416	−	0.992230i	$-0.460294\pi$
$200$	0	0
$201$	12.0000	0.0597015
$202$	0	0
$203$	180.133	0.887356
$204$	0	0
$205$	− 116.000i	− 0.565854i
$206$	0	0
$207$	83.1384i	0.401635i
$208$	0	0
$209$	−144.000	−0.688995
$210$	0	0
$211$	242.487	1.14923	0.574614	−	0.818425i	$-0.305152\pi$
$211$	242.487	1.14923	0.574614	+	0.818425i	$0.305152\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 96.9948i	− 0.451139i
$216$	0	0
$217$	−48.0000	−0.221198
$218$	0	0
$219$	−79.6743	−0.363810
$220$	0	0
$221$	20.0000i	0.0904977i
$222$	0	0
$223$	− 339.482i	− 1.52234i	−0.648552	−	0.761170i	$-0.724625\pi$
$223$	− 339.482i	− 1.52234i	0.648552	−	0.761170i	$-0.275375\pi$
$224$	0	0
$225$	63.0000	0.280000
$226$	0	0
$227$	284.056	1.25135	0.625675	−	0.780084i	$-0.284824\pi$
$227$	284.056	1.25135	0.625675	+	0.780084i	$0.284824\pi$
$228$	0	0
$229$	142.000i	0.620087i	0.950722	+	0.310044i	$0.100344\pi$
$229$	142.000i	0.620087i	−0.950722	+	0.310044i	$0.899656\pi$
$230$	0	0
$231$	− 83.1384i	− 0.359907i
$232$	0	0
$233$	−82.0000	−0.351931	−0.175966	−	0.984396i	$-0.556305\pi$
$233$	−82.0000	−0.351931	−0.175966	+	0.984396i	$0.556305\pi$
$234$	0	0
$235$	−138.564	−0.589634
$236$	0	0
$237$	− 204.000i	− 0.860759i
$238$	0	0
$239$	387.979i	1.62334i	0.584113	+	0.811672i	$0.301442\pi$
$239$	387.979i	1.62334i	−0.584113	+	0.811672i	$0.698558\pi$
$240$	0	0
$241$	−46.0000	−0.190871	−0.0954357	−	0.995436i	$-0.530424\pi$
$241$	−46.0000	−0.190871	−0.0954357	+	0.995436i	$0.530424\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	2.00000i	0.00816327i
$246$	0	0
$247$	− 41.5692i	− 0.168296i
$248$	0	0
$249$	84.0000	0.337349
$250$	0	0
$251$	145.492	0.579650	0.289825	−	0.957080i	$-0.406403\pi$
$251$	145.492	0.579650	0.289825	+	0.957080i	$0.406403\pi$
$252$	0	0
$253$	192.000i	0.758893i
$254$	0	0
$255$	− 34.6410i	− 0.135847i
$256$	0	0
$257$	−254.000	−0.988327	−0.494163	−	0.869369i	$-0.664526\pi$
$257$	−254.000	−0.988327	−0.494163	+	0.869369i	$0.664526\pi$
$258$	0	0
$259$	180.133	0.695495
$260$	0	0
$261$	− 78.0000i	− 0.298851i
$262$	0	0
$263$	− 152.420i	− 0.579546i	−0.957095	−	0.289773i	$-0.906420\pi$
$263$	− 152.420i	− 0.579546i	0.957095	−	0.289773i	$-0.0935797\pi$
$264$	0	0
$265$	−148.000	−0.558491
$266$	0	0
$267$	142.028	0.531941
$268$	0	0
$269$	262.000i	0.973978i	0.873408	+	0.486989i	$0.161905\pi$
$269$	262.000i	0.973978i	−0.873408	+	0.486989i	$0.838095\pi$
$270$	0	0
$271$	20.7846i	0.0766960i	0.999264	+	0.0383480i	$0.0122095\pi$
$271$	20.7846i	0.0766960i	−0.999264	+	0.0383480i	$0.987790\pi$
$272$	0	0
$273$	24.0000	0.0879121
$274$	0	0
$275$	145.492	0.529063
$276$	0	0
$277$	− 290.000i	− 1.04693i	−0.852047	−	0.523466i	$-0.824639\pi$
$277$	− 290.000i	− 1.04693i	0.852047	−	0.523466i	$-0.175361\pi$
$278$	0	0
$279$	20.7846i	0.0744968i
$280$	0	0
$281$	−226.000	−0.804270	−0.402135	−	0.915580i	$-0.631732\pi$
$281$	−226.000	−0.804270	−0.402135	+	0.915580i	$0.631732\pi$
$282$	0	0
$283$	297.913	1.05270	0.526348	−	0.850269i	$-0.323561\pi$
$283$	297.913	1.05270	0.526348	+	0.850269i	$0.323561\pi$
$284$	0	0
$285$	72.0000i	0.252632i
$286$	0	0
$287$	− 401.836i	− 1.40012i
$288$	0	0
$289$	−189.000	−0.653979
$290$	0	0
$291$	−3.46410	−0.0119041
$292$	0	0
$293$	362.000i	1.23549i	0.786377	+	0.617747i	$0.211955\pi$
$293$	362.000i	1.23549i	−0.786377	+	0.617747i	$0.788045\pi$
$294$	0	0
$295$	− 180.133i	− 0.610621i
$296$	0	0
$297$	−36.0000	−0.121212
$298$	0	0
$299$	−55.4256	−0.185370
$300$	0	0
$301$	− 336.000i	− 1.11628i
$302$	0	0
$303$	− 128.172i	− 0.423009i
$304$	0	0
$305$	−52.0000	−0.170492
$306$	0	0
$307$	−145.492	−0.473916	−0.236958	−	0.971520i	$-0.576150\pi$
$307$	−145.492	−0.473916	−0.236958	+	0.971520i	$0.576150\pi$
$308$	0	0
$309$	− 132.000i	− 0.427184i
$310$	0	0
$311$	235.559i	0.757424i	0.925515	+	0.378712i	$0.123633\pi$
$311$	235.559i	0.757424i	−0.925515	+	0.378712i	$0.876367\pi$
$312$	0	0
$313$	478.000	1.52716	0.763578	−	0.645715i	$-0.223441\pi$
$313$	478.000	1.52716	0.763578	+	0.645715i	$0.223441\pi$
$314$	0	0
$315$	−41.5692	−0.131966
$316$	0	0
$317$	− 170.000i	− 0.536278i	−0.963380	−	0.268139i	$-0.913591\pi$
$317$	− 170.000i	− 0.536278i	0.963380	−	0.268139i	$-0.0864086\pi$
$318$	0	0
$319$	− 180.133i	− 0.564681i
$320$	0	0
$321$	−36.0000	−0.112150
$322$	0	0
$323$	−207.846	−0.643486
$324$	0	0
$325$	42.0000i	0.129231i
$326$	0	0
$327$	79.6743i	0.243652i
$328$	0	0
$329$	−480.000	−1.45897
$330$	0	0
$331$	408.764	1.23494	0.617468	−	0.786596i	$-0.288158\pi$
$331$	408.764	1.23494	0.617468	+	0.786596i	$0.288158\pi$
$332$	0	0
$333$	− 78.0000i	− 0.234234i
$334$	0	0
$335$	− 13.8564i	− 0.0413624i
$336$	0	0
$337$	338.000	1.00297	0.501484	−	0.865167i	$-0.332788\pi$
$337$	338.000	1.00297	0.501484	+	0.865167i	$0.332788\pi$
$338$	0	0
$339$	190.526	0.562022
$340$	0	0
$341$	48.0000i	0.140762i
$342$	0	0
$343$	346.410i	1.00994i
$344$	0	0
$345$	96.0000	0.278261
$346$	0	0
$347$	200.918	0.579014	0.289507	−	0.957176i	$-0.406509\pi$
$347$	200.918	0.579014	0.289507	+	0.957176i	$0.406509\pi$
$348$	0	0
$349$	506.000i	1.44986i	0.688824	+	0.724928i	$0.258127\pi$
$349$	506.000i	1.44986i	−0.688824	+	0.724928i	$0.741873\pi$
$350$	0	0
$351$	− 10.3923i	− 0.0296077i
$352$	0	0
$353$	178.000	0.504249	0.252125	−	0.967695i	$-0.418871\pi$
$353$	178.000	0.504249	0.252125	+	0.967695i	$0.418871\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 120.000i	− 0.336134i
$358$	0	0
$359$	− 166.277i	− 0.463167i	−0.972815	−	0.231583i	$-0.925609\pi$
$359$	− 166.277i	− 0.463167i	0.972815	−	0.231583i	$-0.0743906\pi$
$360$	0	0
$361$	71.0000	0.196676
$362$	0	0
$363$	126.440	0.348319
$364$	0	0
$365$	92.0000i	0.252055i
$366$	0	0
$367$	200.918i	0.547460i	0.961807	+	0.273730i	$0.0882575\pi$
$367$	200.918i	0.547460i	−0.961807	+	0.273730i	$0.911742\pi$
$368$	0	0
$369$	−174.000	−0.471545
$370$	0	0
$371$	−512.687	−1.38191
$372$	0	0
$373$	310.000i	0.831099i	0.909571	+	0.415550i	$0.136411\pi$
$373$	310.000i	0.831099i	−0.909571	+	0.415550i	$0.863589\pi$
$374$	0	0
$375$	− 159.349i	− 0.424930i
$376$	0	0
$377$	52.0000	0.137931
$378$	0	0
$379$	−436.477	−1.15165	−0.575827	−	0.817572i	$-0.695320\pi$
$379$	−436.477	−1.15165	−0.575827	+	0.817572i	$0.695320\pi$
$380$	0	0
$381$	252.000i	0.661417i
$382$	0	0
$383$	− 609.682i	− 1.59186i	−0.605390	−	0.795929i	$-0.706983\pi$
$383$	− 609.682i	− 1.59186i	0.605390	−	0.795929i	$-0.293017\pi$
$384$	0	0
$385$	−96.0000	−0.249351
$386$	0	0
$387$	−145.492	−0.375949
$388$	0	0
$389$	578.000i	1.48586i	0.669368	+	0.742931i	$0.266565\pi$
$389$	578.000i	1.48586i	−0.669368	+	0.742931i	$0.733435\pi$
$390$	0	0
$391$	277.128i	0.708768i
$392$	0	0
$393$	204.000	0.519084
$394$	0	0
$395$	−235.559	−0.596352
$396$	0	0
$397$	26.0000i	0.0654912i	0.999464	+	0.0327456i	$0.0104251\pi$
$397$	26.0000i	0.0654912i	−0.999464	+	0.0327456i	$0.989575\pi$
$398$	0	0
$399$	249.415i	0.625101i
$400$	0	0
$401$	250.000	0.623441	0.311721	−	0.950174i	$-0.399095\pi$
$401$	250.000	0.623441	0.311721	+	0.950174i	$0.399095\pi$
$402$	0	0
$403$	−13.8564	−0.0343831
$404$	0	0
$405$	18.0000i	0.0444444i
$406$	0	0
$407$	− 180.133i	− 0.442588i
$408$	0	0
$409$	−290.000	−0.709046	−0.354523	−	0.935047i	$-0.615357\pi$
$409$	−290.000	−0.709046	−0.354523	+	0.935047i	$0.615357\pi$
$410$	0	0
$411$	17.3205	0.0421424
$412$	0	0
$413$	− 624.000i	− 1.51090i
$414$	0	0
$415$	− 96.9948i	− 0.233723i
$416$	0	0
$417$	−84.0000	−0.201439
$418$	0	0
$419$	339.482	0.810219	0.405110	−	0.914268i	$-0.367233\pi$
$419$	339.482	0.810219	0.405110	+	0.914268i	$0.367233\pi$
$420$	0	0
$421$	− 674.000i	− 1.60095i	−0.599366	−	0.800475i	$-0.704581\pi$
$421$	− 674.000i	− 1.60095i	0.599366	−	0.800475i	$-0.295419\pi$
$422$	0	0
$423$	207.846i	0.491362i
$424$	0	0
$425$	210.000	0.494118
$426$	0	0
$427$	−180.133	−0.421858
$428$	0	0
$429$	− 24.0000i	− 0.0559441i
$430$	0	0
$431$	540.400i	1.25383i	0.779088	+	0.626914i	$0.215682\pi$
$431$	540.400i	1.25383i	−0.779088	+	0.626914i	$0.784318\pi$
$432$	0	0
$433$	−334.000	−0.771363	−0.385681	−	0.922632i	$-0.626034\pi$
$433$	−334.000	−0.771363	−0.385681	+	0.922632i	$0.626034\pi$
$434$	0	0
$435$	−90.0666	−0.207050
$436$	0	0
$437$	− 576.000i	− 1.31808i
$438$	0	0
$439$	− 117.779i	− 0.268290i	−0.990962	−	0.134145i	$-0.957171\pi$
$439$	− 117.779i	− 0.268290i	0.990962	−	0.134145i	$-0.0428288\pi$
$440$	0	0
$441$	3.00000	0.00680272
$442$	0	0
$443$	−76.2102	−0.172032	−0.0860161	−	0.996294i	$-0.527414\pi$
$443$	−76.2102	−0.172032	−0.0860161	+	0.996294i	$0.527414\pi$
$444$	0	0
$445$	− 164.000i	− 0.368539i
$446$	0	0
$447$	− 3.46410i	− 0.00774967i
$448$	0	0
$449$	394.000	0.877506	0.438753	−	0.898608i	$-0.355420\pi$
$449$	394.000	0.877506	0.438753	+	0.898608i	$0.355420\pi$
$450$	0	0
$451$	−401.836	−0.890988
$452$	0	0
$453$	− 156.000i	− 0.344371i
$454$	0	0
$455$	− 27.7128i	− 0.0609073i
$456$	0	0
$457$	478.000	1.04595	0.522976	−	0.852347i	$-0.324822\pi$
$457$	478.000	1.04595	0.522976	+	0.852347i	$0.324822\pi$
$458$	0	0
$459$	−51.9615	−0.113206
$460$	0	0
$461$	142.000i	0.308026i	0.988069	+	0.154013i	$0.0492198\pi$
$461$	142.000i	0.308026i	−0.988069	+	0.154013i	$0.950780\pi$
$462$	0	0
$463$	630.466i	1.36170i	0.732423	+	0.680849i	$0.238389\pi$
$463$	630.466i	1.36170i	−0.732423	+	0.680849i	$0.761611\pi$
$464$	0	0
$465$	24.0000	0.0516129
$466$	0	0
$467$	−20.7846	−0.0445067	−0.0222533	−	0.999752i	$-0.507084\pi$
$467$	−20.7846	−0.0445067	−0.0222533	+	0.999752i	$0.507084\pi$
$468$	0	0
$469$	− 48.0000i	− 0.102345i
$470$	0	0
$471$	370.659i	0.786962i
$472$	0	0
$473$	−336.000	−0.710359
$474$	0	0
$475$	−436.477	−0.918899
$476$	0	0
$477$	222.000i	0.465409i
$478$	0	0
$479$	734.390i	1.53317i	0.642141	+	0.766586i	$0.278046\pi$
$479$	734.390i	1.53317i	−0.642141	+	0.766586i	$0.721954\pi$
$480$	0	0
$481$	52.0000	0.108108
$482$	0	0
$483$	332.554	0.688517
$484$	0	0
$485$	4.00000i	0.00824742i
$486$	0	0
$487$	− 103.923i	− 0.213394i	−0.994292	−	0.106697i	$-0.965972\pi$
$487$	− 103.923i	− 0.213394i	0.994292	−	0.106697i	$-0.0340275\pi$
$488$	0	0
$489$	36.0000	0.0736196
$490$	0	0
$491$	−921.451	−1.87668	−0.938341	−	0.345711i	$-0.887638\pi$
$491$	−921.451	−1.87668	−0.938341	+	0.345711i	$0.887638\pi$
$492$	0	0
$493$	− 260.000i	− 0.527383i
$494$	0	0
$495$	41.5692i	0.0839782i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	76.2102	0.152726	0.0763630	−	0.997080i	$-0.475669\pi$
$499$	76.2102	0.152726	0.0763630	+	0.997080i	$0.475669\pi$
$500$	0	0
$501$	168.000i	0.335329i
$502$	0	0
$503$	581.969i	1.15700i	0.815684	+	0.578498i	$0.196361\pi$
$503$	581.969i	1.15700i	−0.815684	+	0.578498i	$0.803639\pi$
$504$	0	0
$505$	−148.000	−0.293069
$506$	0	0
$507$	−285.788	−0.563685
$508$	0	0
$509$	− 842.000i	− 1.65422i	−0.562037	−	0.827112i	$-0.689982\pi$
$509$	− 842.000i	− 1.65422i	0.562037	−	0.827112i	$-0.310018\pi$
$510$	0	0
$511$	318.697i	0.623674i
$512$	0	0
$513$	108.000	0.210526
$514$	0	0
$515$	−152.420	−0.295962
$516$	0	0
$517$	480.000i	0.928433i
$518$	0	0
$519$	− 578.505i	− 1.11465i
$520$	0	0
$521$	326.000	0.625720	0.312860	−	0.949799i	$-0.398713\pi$
$521$	326.000	0.625720	0.312860	+	0.949799i	$0.398713\pi$
$522$	0	0
$523$	311.769	0.596117	0.298058	−	0.954548i	$-0.403661\pi$
$523$	311.769	0.596117	0.298058	+	0.954548i	$0.403661\pi$
$524$	0	0
$525$	− 252.000i	− 0.480000i
$526$	0	0
$527$	69.2820i	0.131465i
$528$	0	0
$529$	−239.000	−0.451796
$530$	0	0
$531$	−270.200	−0.508851
$532$	0	0
$533$	− 116.000i	− 0.217636i
$534$	0	0
$535$	41.5692i	0.0776995i
$536$	0	0
$537$	−324.000	−0.603352
$538$	0	0
$539$	6.92820	0.0128538
$540$	0	0
$541$	530.000i	0.979667i	0.871816	+	0.489834i	$0.162942\pi$
$541$	530.000i	0.979667i	−0.871816	+	0.489834i	$0.837058\pi$
$542$	0	0
$543$	3.46410i	0.00637956i
$544$	0	0
$545$	92.0000	0.168807
$546$	0	0
$547$	339.482	0.620625	0.310313	−	0.950635i	$-0.399566\pi$
$547$	339.482	0.620625	0.310313	+	0.950635i	$0.399566\pi$
$548$	0	0
$549$	78.0000i	0.142077i
$550$	0	0
$551$	540.400i	0.980762i
$552$	0	0
$553$	−816.000	−1.47559
$554$	0	0
$555$	−90.0666	−0.162282
$556$	0	0
$557$	766.000i	1.37522i	0.726078	+	0.687612i	$0.241341\pi$
$557$	766.000i	1.37522i	−0.726078	+	0.687612i	$0.758659\pi$
$558$	0	0
$559$	− 96.9948i	− 0.173515i
$560$	0	0
$561$	−120.000	−0.213904
$562$	0	0
$563$	−491.902	−0.873717	−0.436858	−	0.899530i	$-0.643909\pi$
$563$	−491.902	−0.873717	−0.436858	+	0.899530i	$0.643909\pi$
$564$	0	0
$565$	− 220.000i	− 0.389381i
$566$	0	0
$567$	62.3538i	0.109971i
$568$	0	0
$569$	422.000	0.741652	0.370826	−	0.928702i	$-0.379075\pi$
$569$	422.000	0.741652	0.370826	+	0.928702i	$0.379075\pi$
$570$	0	0
$571$	−284.056	−0.497472	−0.248736	−	0.968571i	$-0.580015\pi$
$571$	−284.056	−0.497472	−0.248736	+	0.968571i	$0.580015\pi$
$572$	0	0
$573$	384.000i	0.670157i
$574$	0	0
$575$	581.969i	1.01212i
$576$	0	0
$577$	−46.0000	−0.0797227	−0.0398614	−	0.999205i	$-0.512692\pi$
$577$	−46.0000	−0.0797227	−0.0398614	+	0.999205i	$0.512692\pi$
$578$	0	0
$579$	−502.295	−0.867521
$580$	0	0
$581$	− 336.000i	− 0.578313i
$582$	0	0
$583$	512.687i	0.879395i
$584$	0	0
$585$	−12.0000	−0.0205128
$586$	0	0
$587$	630.466	1.07405	0.537024	−	0.843567i	$-0.319548\pi$
$587$	630.466	1.07405	0.537024	+	0.843567i	$0.319548\pi$
$588$	0	0
$589$	− 144.000i	− 0.244482i
$590$	0	0
$591$	− 45.0333i	− 0.0761985i
$592$	0	0
$593$	82.0000	0.138280	0.0691400	−	0.997607i	$-0.477974\pi$
$593$	82.0000	0.138280	0.0691400	+	0.997607i	$0.477974\pi$
$594$	0	0
$595$	−138.564	−0.232881
$596$	0	0
$597$	684.000i	1.14573i
$598$	0	0
$599$	− 55.4256i	− 0.0925303i	−0.998929	−	0.0462651i	$-0.985268\pi$
$599$	− 55.4256i	− 0.0925303i	0.998929	−	0.0462651i	$-0.0147319\pi$
$600$	0	0
$601$	334.000	0.555740	0.277870	−	0.960619i	$-0.410371\pi$
$601$	334.000	0.555740	0.277870	+	0.960619i	$0.410371\pi$
$602$	0	0
$603$	−20.7846	−0.0344687
$604$	0	0
$605$	− 146.000i	− 0.241322i
$606$	0	0
$607$	− 367.195i	− 0.604934i	−0.953160	−	0.302467i	$-0.902190\pi$
$607$	− 367.195i	− 0.604934i	0.953160	−	0.302467i	$-0.0978102\pi$
$608$	0	0
$609$	−312.000	−0.512315
$610$	0	0
$611$	−138.564	−0.226782
$612$	0	0
$613$	214.000i	0.349103i	0.984648	+	0.174551i	$0.0558475\pi$
$613$	214.000i	0.349103i	−0.984648	+	0.174551i	$0.944152\pi$
$614$	0	0
$615$	200.918i	0.326696i
$616$	0	0
$617$	1118.00	1.81199	0.905997	−	0.423285i	$-0.139123\pi$
$617$	1118.00	1.81199	0.905997	+	0.423285i	$0.139123\pi$
$618$	0	0
$619$	−672.036	−1.08568	−0.542840	−	0.839836i	$-0.682651\pi$
$619$	−672.036	−1.08568	−0.542840	+	0.839836i	$0.682651\pi$
$620$	0	0
$621$	− 144.000i	− 0.231884i
$622$	0	0
$623$	− 568.113i	− 0.911898i
$624$	0	0
$625$	341.000	0.545600
$626$	0	0
$627$	249.415	0.397792
$628$	0	0
$629$	− 260.000i	− 0.413355i
$630$	0	0
$631$	145.492i	0.230574i	0.993332	+	0.115287i	$0.0367788\pi$
$631$	145.492i	0.230574i	−0.993332	+	0.115287i	$0.963221\pi$
$632$	0	0
$633$	−420.000	−0.663507
$634$	0	0
$635$	290.985	0.458243
$636$	0	0
$637$	2.00000i	0.00313972i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.0000	0.0156006	0.00780031	−	0.999970i	$-0.497517\pi$
$641$	10.0000	0.0156006	0.00780031	+	0.999970i	$0.497517\pi$
$642$	0	0
$643$	−1212.44	−1.88559	−0.942796	−	0.333370i	$-0.891814\pi$
$643$	−1212.44	−1.88559	−0.942796	+	0.333370i	$0.891814\pi$
$644$	0	0
$645$	168.000i	0.260465i
$646$	0	0
$647$	332.554i	0.513993i	0.966412	+	0.256997i	$0.0827330\pi$
$647$	332.554i	0.513993i	−0.966412	+	0.256997i	$0.917267\pi$
$648$	0	0
$649$	−624.000	−0.961479
$650$	0	0
$651$	83.1384	0.127709
$652$	0	0
$653$	670.000i	1.02603i	0.858379	+	0.513017i	$0.171472\pi$
$653$	670.000i	1.02603i	−0.858379	+	0.513017i	$0.828528\pi$
$654$	0	0
$655$	− 235.559i	− 0.359632i
$656$	0	0
$657$	138.000	0.210046
$658$	0	0
$659$	824.456	1.25107	0.625536	−	0.780195i	$-0.284880\pi$
$659$	824.456	1.25107	0.625536	+	0.780195i	$0.284880\pi$
$660$	0	0
$661$	1222.00i	1.84871i	0.381529	+	0.924357i	$0.375398\pi$
$661$	1222.00i	1.84871i	−0.381529	+	0.924357i	$0.624602\pi$
$662$	0	0
$663$	− 34.6410i	− 0.0522489i
$664$	0	0
$665$	288.000	0.433083
$666$	0	0
$667$	720.533	1.08026
$668$	0	0
$669$	588.000i	0.878924i
$670$	0	0
$671$	180.133i	0.268455i
$672$	0	0
$673$	−334.000	−0.496285	−0.248143	−	0.968724i	$-0.579820\pi$
$673$	−334.000	−0.496285	−0.248143	+	0.968724i	$0.579820\pi$
$674$	0	0
$675$	−109.119	−0.161658
$676$	0	0
$677$	− 1006.00i	− 1.48597i	−0.669309	−	0.742984i	$-0.733410\pi$
$677$	− 1006.00i	− 1.48597i	0.669309	−	0.742984i	$-0.266590\pi$
$678$	0	0
$679$	13.8564i	0.0204071i
$680$	0	0
$681$	−492.000	−0.722467
$682$	0	0
$683$	−187.061	−0.273882	−0.136941	−	0.990579i	$-0.543727\pi$
$683$	−187.061	−0.273882	−0.136941	+	0.990579i	$0.543727\pi$
$684$	0	0
$685$	− 20.0000i	− 0.0291971i
$686$	0	0
$687$	− 245.951i	− 0.358008i
$688$	0	0
$689$	−148.000	−0.214804
$690$	0	0
$691$	−990.733	−1.43377	−0.716884	−	0.697193i	$-0.754432\pi$
$691$	−990.733	−1.43377	−0.716884	+	0.697193i	$0.754432\pi$
$692$	0	0
$693$	144.000i	0.207792i
$694$	0	0
$695$	96.9948i	0.139561i
$696$	0	0
$697$	−580.000	−0.832138
$698$	0	0
$699$	142.028	0.203188
$700$	0	0
$701$	− 1034.00i	− 1.47504i	−0.675328	−	0.737518i	$-0.735998\pi$
$701$	− 1034.00i	− 1.47504i	0.675328	−	0.737518i	$-0.264002\pi$
$702$	0	0
$703$	540.400i	0.768705i
$704$	0	0
$705$	240.000	0.340426
$706$	0	0
$707$	−512.687	−0.725158
$708$	0	0
$709$	− 530.000i	− 0.747532i	−0.927523	−	0.373766i	$-0.878066\pi$
$709$	− 530.000i	− 0.747532i	0.927523	−	0.373766i	$-0.121934\pi$
$710$	0	0
$711$	353.338i	0.496960i
$712$	0	0
$713$	−192.000	−0.269285
$714$	0	0
$715$	−27.7128	−0.0387592
$716$	0	0
$717$	− 672.000i	− 0.937238i
$718$	0	0
$719$	− 706.677i	− 0.982861i	−0.870917	−	0.491430i	$-0.836474\pi$
$719$	− 706.677i	− 0.982861i	0.870917	−	0.491430i	$-0.163526\pi$
$720$	0	0
$721$	−528.000	−0.732316
$722$	0	0
$723$	79.6743	0.110200
$724$	0	0
$725$	− 546.000i	− 0.753103i
$726$	0	0
$727$	− 242.487i	− 0.333545i	−0.985995	−	0.166772i	$-0.946665\pi$
$727$	− 242.487i	− 0.333545i	0.985995	−	0.166772i	$-0.0533345\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	−484.974	−0.663439
$732$	0	0
$733$	194.000i	0.264666i	0.991205	+	0.132333i	$0.0422468\pi$
$733$	194.000i	0.264666i	−0.991205	+	0.132333i	$0.957753\pi$
$734$	0	0
$735$	− 3.46410i	− 0.00471306i
$736$	0	0
$737$	−48.0000	−0.0651289
$738$	0	0
$739$	1351.00	1.82815	0.914073	−	0.405550i	$-0.132920\pi$
$739$	1351.00	1.82815	0.914073	+	0.405550i	$0.132920\pi$
$740$	0	0
$741$	72.0000i	0.0971660i
$742$	0	0
$743$	− 678.964i	− 0.913814i	−0.889514	−	0.456907i	$-0.848957\pi$
$743$	− 678.964i	− 0.913814i	0.889514	−	0.456907i	$-0.151043\pi$
$744$	0	0
$745$	−4.00000	−0.00536913
$746$	0	0
$747$	−145.492	−0.194769
$748$	0	0
$749$	144.000i	0.192256i
$750$	0	0
$751$	− 658.179i	− 0.876404i	−0.898877	−	0.438202i	$-0.855615\pi$
$751$	− 658.179i	− 0.876404i	0.898877	−	0.438202i	$-0.144385\pi$
$752$	0	0
$753$	−252.000	−0.334661
$754$	0	0
$755$	−180.133	−0.238587
$756$	0	0
$757$	1006.00i	1.32893i	0.747319	+	0.664465i	$0.231341\pi$
$757$	1006.00i	1.32893i	−0.747319	+	0.664465i	$0.768659\pi$
$758$	0	0
$759$	− 332.554i	− 0.438147i
$760$	0	0
$761$	758.000	0.996058	0.498029	−	0.867160i	$-0.334057\pi$
$761$	758.000	0.996058	0.498029	+	0.867160i	$0.334057\pi$
$762$	0	0
$763$	318.697	0.417690
$764$	0	0
$765$	60.0000i	0.0784314i
$766$	0	0
$767$	− 180.133i	− 0.234854i
$768$	0	0
$769$	2.00000	0.00260078	0.00130039	−	0.999999i	$-0.499586\pi$
$769$	2.00000	0.00260078	0.00130039	+	0.999999i	$0.499586\pi$
$770$	0	0
$771$	439.941	0.570611
$772$	0	0
$773$	− 262.000i	− 0.338939i	−0.985535	−	0.169470i	$-0.945795\pi$
$773$	− 262.000i	− 0.338939i	0.985535	−	0.169470i	$-0.0542055\pi$
$774$	0	0
$775$	145.492i	0.187732i
$776$	0	0
$777$	−312.000	−0.401544
$778$	0	0
$779$	1205.51	1.54751
$780$	0	0
$781$	0	0
$782$	0	0
$783$	135.100i	0.172541i
$784$	0	0
$785$	428.000	0.545223
$786$	0	0
$787$	1447.99	1.83989	0.919946	−	0.392046i	$-0.128233\pi$
$787$	1447.99	1.83989	0.919946	+	0.392046i	$0.128233\pi$
$788$	0	0
$789$	264.000i	0.334601i
$790$	0	0
$791$	− 762.102i	− 0.963467i
$792$	0	0
$793$	−52.0000	−0.0655738
$794$	0	0
$795$	256.344	0.322445
$796$	0	0
$797$	− 866.000i	− 1.08657i	−0.839547	−	0.543287i	$-0.817179\pi$
$797$	− 866.000i	− 1.08657i	0.839547	−	0.543287i	$-0.182821\pi$
$798$	0	0
$799$	692.820i	0.867109i
$800$	0	0
$801$	−246.000	−0.307116
$802$	0	0
$803$	318.697	0.396883
$804$	0	0
$805$	− 384.000i	− 0.477019i
$806$	0	0
$807$	− 453.797i	− 0.562326i
$808$	0	0
$809$	−10.0000	−0.0123609	−0.00618047	−	0.999981i	$-0.501967\pi$
$809$	−10.0000	−0.0123609	−0.00618047	+	0.999981i	$0.501967\pi$
$810$	0	0
$811$	−436.477	−0.538196	−0.269098	−	0.963113i	$-0.586726\pi$
$811$	−436.477	−0.538196	−0.269098	+	0.963113i	$0.586726\pi$
$812$	0	0
$813$	− 36.0000i	− 0.0442804i
$814$	0	0
$815$	− 41.5692i	− 0.0510052i
$816$	0	0
$817$	1008.00	1.23378
$818$	0	0
$819$	−41.5692	−0.0507561
$820$	0	0
$821$	− 838.000i	− 1.02071i	−0.859965	−	0.510353i	$-0.829515\pi$
$821$	− 838.000i	− 1.02071i	0.859965	−	0.510353i	$-0.170485\pi$
$822$	0	0
$823$	− 879.882i	− 1.06912i	−0.845132	−	0.534558i	$-0.820478\pi$
$823$	− 879.882i	− 1.06912i	0.845132	−	0.534558i	$-0.179522\pi$
$824$	0	0
$825$	−252.000	−0.305455
$826$	0	0
$827$	−727.461	−0.879639	−0.439819	−	0.898086i	$-0.644958\pi$
$827$	−727.461	−0.879639	−0.439819	+	0.898086i	$0.644958\pi$
$828$	0	0
$829$	1298.00i	1.56574i	0.622184	+	0.782871i	$0.286246\pi$
$829$	1298.00i	1.56574i	−0.622184	+	0.782871i	$0.713754\pi$
$830$	0	0
$831$	502.295i	0.604446i
$832$	0	0
$833$	10.0000	0.0120048
$834$	0	0
$835$	193.990	0.232323
$836$	0	0
$837$	− 36.0000i	− 0.0430108i
$838$	0	0
$839$	− 193.990i	− 0.231215i	−0.993295	−	0.115608i	$-0.963118\pi$
$839$	− 193.990i	− 0.231215i	0.993295	−	0.115608i	$-0.0368815\pi$
$840$	0	0
$841$	165.000	0.196195
$842$	0	0
$843$	391.443	0.464346
$844$	0	0
$845$	330.000i	0.390533i
$846$	0	0
$847$	− 505.759i	− 0.597118i
$848$	0	0
$849$	−516.000	−0.607774
$850$	0	0
$851$	720.533	0.846690
$852$	0	0
$853$	− 506.000i	− 0.593200i	−0.955002	−	0.296600i	$-0.904147\pi$
$853$	− 506.000i	− 0.593200i	0.955002	−	0.296600i	$-0.0958529\pi$
$854$	0	0
$855$	− 124.708i	− 0.145857i
$856$	0	0
$857$	998.000	1.16453	0.582264	−	0.813000i	$-0.302167\pi$
$857$	998.000	1.16453	0.582264	+	0.813000i	$0.302167\pi$
$858$	0	0
$859$	505.759	0.588776	0.294388	−	0.955686i	$-0.404884\pi$
$859$	505.759	0.588776	0.294388	+	0.955686i	$0.404884\pi$
$860$	0	0
$861$	696.000i	0.808362i
$862$	0	0
$863$	166.277i	0.192673i	0.995349	+	0.0963365i	$0.0307125\pi$
$863$	166.277i	0.192673i	−0.995349	+	0.0963365i	$0.969287\pi$
$864$	0	0
$865$	−668.000	−0.772254
$866$	0	0
$867$	327.358	0.377575
$868$	0	0
$869$	816.000i	0.939010i
$870$	0	0
$871$	− 13.8564i	− 0.0159086i
$872$	0	0
$873$	6.00000	0.00687285
$874$	0	0
$875$	−637.395	−0.728451
$876$	0	0
$877$	− 646.000i	− 0.736602i	−0.929707	−	0.368301i	$-0.879940\pi$
$877$	− 646.000i	− 0.736602i	0.929707	−	0.368301i	$-0.120060\pi$
$878$	0	0
$879$	− 627.002i	− 0.713313i
$880$	0	0
$881$	898.000	1.01930	0.509648	−	0.860383i	$-0.329776\pi$
$881$	898.000	1.01930	0.509648	+	0.860383i	$0.329776\pi$
$882$	0	0
$883$	727.461	0.823852	0.411926	−	0.911217i	$-0.364856\pi$
$883$	727.461	0.823852	0.411926	+	0.911217i	$0.364856\pi$
$884$	0	0
$885$	312.000i	0.352542i
$886$	0	0
$887$	− 845.241i	− 0.952921i	−0.879196	−	0.476460i	$-0.841920\pi$
$887$	− 845.241i	− 0.952921i	0.879196	−	0.476460i	$-0.158080\pi$
$888$	0	0
$889$	1008.00	1.13386
$890$	0	0
$891$	62.3538	0.0699819
$892$	0	0
$893$	− 1440.00i	− 1.61254i
$894$	0	0
$895$	374.123i	0.418014i
$896$	0	0
$897$	96.0000	0.107023
$898$	0	0
$899$	180.133	0.200371
$900$	0	0
$901$	740.000i	0.821310i
$902$	0	0
$903$	581.969i	0.644484i
$904$	0	0
$905$	4.00000	0.00441989
$906$	0	0
$907$	1364.86	1.50480	0.752401	−	0.658705i	$-0.228895\pi$
$907$	1364.86	1.50480	0.752401	+	0.658705i	$0.228895\pi$
$908$	0	0
$909$	222.000i	0.244224i
$910$	0	0
$911$	− 387.979i	− 0.425883i	−0.977065	−	0.212941i	$-0.931696\pi$
$911$	− 387.979i	− 0.425883i	0.977065	−	0.212941i	$-0.0683044\pi$
$912$	0	0
$913$	−336.000	−0.368018
$914$	0	0
$915$	90.0666	0.0984335
$916$	0	0
$917$	− 816.000i	− 0.889858i
$918$	0	0
$919$	− 602.754i	− 0.655880i	−0.944699	−	0.327940i	$-0.893646\pi$
$919$	− 602.754i	− 0.655880i	0.944699	−	0.327940i	$-0.106354\pi$
$920$	0	0
$921$	252.000	0.273616
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 546.000i	− 0.590270i
$926$	0	0
$927$	228.631i	0.246635i
$928$	0	0
$929$	1594.00	1.71582	0.857912	−	0.513797i	$-0.171762\pi$
$929$	1594.00	1.71582	0.857912	+	0.513797i	$0.171762\pi$
$930$	0	0
$931$	−20.7846	−0.0223250
$932$	0	0
$933$	− 408.000i	− 0.437299i
$934$	0	0
$935$	138.564i	0.148197i
$936$	0	0
$937$	−674.000	−0.719317	−0.359658	−	0.933084i	$-0.617107\pi$
$937$	−674.000	−0.719317	−0.359658	+	0.933084i	$0.617107\pi$
$938$	0	0
$939$	−827.920	−0.881704
$940$	0	0
$941$	430.000i	0.456961i	0.973549	+	0.228480i	$0.0733757\pi$
$941$	430.000i	0.456961i	−0.973549	+	0.228480i	$0.926624\pi$
$942$	0	0
$943$	− 1607.34i	− 1.70450i
$944$	0	0
$945$	72.0000	0.0761905
$946$	0	0
$947$	76.2102	0.0804754	0.0402377	−	0.999190i	$-0.487188\pi$
$947$	76.2102	0.0804754	0.0402377	+	0.999190i	$0.487188\pi$
$948$	0	0
$949$	92.0000i	0.0969442i
$950$	0	0
$951$	294.449i	0.309620i
$952$	0	0
$953$	−730.000	−0.766002	−0.383001	−	0.923748i	$-0.625109\pi$
$953$	−730.000	−0.766002	−0.383001	+	0.923748i	$0.625109\pi$
$954$	0	0
$955$	443.405	0.464298
$956$	0	0
$957$	312.000i	0.326019i
$958$	0	0
$959$	− 69.2820i	− 0.0722440i
$960$	0	0
$961$	913.000	0.950052
$962$	0	0
$963$	62.3538	0.0647496
$964$	0	0
$965$	580.000i	0.601036i
$966$	0	0
$967$	− 921.451i	− 0.952897i	−0.879202	−	0.476448i	$-0.841924\pi$
$967$	− 921.451i	− 0.952897i	0.879202	−	0.476448i	$-0.158076\pi$
$968$	0	0
$969$	360.000	0.371517
$970$	0	0
$971$	1475.71	1.51978	0.759890	−	0.650051i	$-0.225253\pi$
$971$	1475.71	1.51978	0.759890	+	0.650051i	$0.225253\pi$
$972$	0	0
$973$	336.000i	0.345324i
$974$	0	0
$975$	− 72.7461i	− 0.0746114i
$976$	0	0
$977$	346.000	0.354145	0.177073	−	0.984198i	$-0.443337\pi$
$977$	346.000	0.354145	0.177073	+	0.984198i	$0.443337\pi$
$978$	0	0
$979$	−568.113	−0.580299
$980$	0	0
$981$	− 138.000i	− 0.140673i
$982$	0	0
$983$	− 734.390i	− 0.747090i	−0.927612	−	0.373545i	$-0.878142\pi$
$983$	− 734.390i	− 0.747090i	0.927612	−	0.373545i	$-0.121858\pi$
$984$	0	0
$985$	−52.0000	−0.0527919
$986$	0	0
$987$	831.384	0.842335
$988$	0	0
$989$	− 1344.00i	− 1.35895i
$990$	0	0
$991$	976.877i	0.985748i	0.870101	+	0.492874i	$0.164054\pi$
$991$	976.877i	0.985748i	−0.870101	+	0.492874i	$0.835946\pi$
$992$	0	0
$993$	−708.000	−0.712991
$994$	0	0
$995$	789.815	0.793784
$996$	0	0
$997$	− 458.000i	− 0.459378i	−0.973264	−	0.229689i	$-0.926229\pi$
$997$	− 458.000i	− 0.459378i	0.973264	−	0.229689i	$-0.0737709\pi$
$998$	0	0
$999$	135.100i	0.135235i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.3.b.c.127.2		4
3.2	odd	2		2304.3.b.l.127.2		4
4.3	odd	2	inner	768.3.b.c.127.4		4
8.3	odd	2	inner	768.3.b.c.127.1		4
8.5	even	2	inner	768.3.b.c.127.3		4
12.11	even	2		2304.3.b.l.127.1		4
16.3	odd	4		192.3.g.b.127.2		2
16.5	even	4		12.3.d.a.7.1	✓	2
16.11	odd	4		12.3.d.a.7.2	yes	2
16.13	even	4		192.3.g.b.127.1		2
24.5	odd	2		2304.3.b.l.127.4		4
24.11	even	2		2304.3.b.l.127.3		4
48.5	odd	4		36.3.d.c.19.2		2
48.11	even	4		36.3.d.c.19.1		2
48.29	odd	4		576.3.g.e.127.1		2
48.35	even	4		576.3.g.e.127.2		2
80.27	even	4		300.3.f.a.199.1		4
80.37	odd	4		300.3.f.a.199.3		4
80.43	even	4		300.3.f.a.199.4		4
80.53	odd	4		300.3.f.a.199.2		4
80.59	odd	4		300.3.c.b.151.1		2
80.69	even	4		300.3.c.b.151.2		2
112.27	even	4		588.3.g.b.295.2		2
112.69	odd	4		588.3.g.b.295.1		2
144.5	odd	12		324.3.f.a.55.1		2
144.11	even	12		324.3.f.a.271.1		2
144.43	odd	12		324.3.f.j.271.1		2
144.59	even	12		324.3.f.g.55.1		2
144.85	even	12		324.3.f.j.55.1		2
144.101	odd	12		324.3.f.g.271.1		2
144.133	even	12		324.3.f.d.271.1		2
144.139	odd	12		324.3.f.d.55.1		2
240.53	even	4		900.3.f.c.199.3		4
240.59	even	4		900.3.c.e.451.2		2
240.107	odd	4		900.3.f.c.199.4		4
240.149	odd	4		900.3.c.e.451.1		2
240.197	even	4		900.3.f.c.199.2		4
240.203	odd	4		900.3.f.c.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.3.d.a.7.1	✓	2	16.5	even	4
12.3.d.a.7.2	yes	2	16.11	odd	4
36.3.d.c.19.1		2	48.11	even	4
36.3.d.c.19.2		2	48.5	odd	4
192.3.g.b.127.1		2	16.13	even	4
192.3.g.b.127.2		2	16.3	odd	4
300.3.c.b.151.1		2	80.59	odd	4
300.3.c.b.151.2		2	80.69	even	4
300.3.f.a.199.1		4	80.27	even	4
300.3.f.a.199.2		4	80.53	odd	4
300.3.f.a.199.3		4	80.37	odd	4
300.3.f.a.199.4		4	80.43	even	4
324.3.f.a.55.1		2	144.5	odd	12
324.3.f.a.271.1		2	144.11	even	12
324.3.f.d.55.1		2	144.139	odd	12
324.3.f.d.271.1		2	144.133	even	12
324.3.f.g.55.1		2	144.59	even	12
324.3.f.g.271.1		2	144.101	odd	12
324.3.f.j.55.1		2	144.85	even	12
324.3.f.j.271.1		2	144.43	odd	12
576.3.g.e.127.1		2	48.29	odd	4
576.3.g.e.127.2		2	48.35	even	4
588.3.g.b.295.1		2	112.69	odd	4
588.3.g.b.295.2		2	112.27	even	4
768.3.b.c.127.1		4	8.3	odd	2	inner
768.3.b.c.127.2		4	1.1	even	1	trivial
768.3.b.c.127.3		4	8.5	even	2	inner
768.3.b.c.127.4		4	4.3	odd	2	inner
900.3.c.e.451.1		2	240.149	odd	4
900.3.c.e.451.2		2	240.59	even	4
900.3.f.c.199.1		4	240.203	odd	4
900.3.f.c.199.2		4	240.197	even	4
900.3.f.c.199.3		4	240.53	even	4
900.3.f.c.199.4		4	240.107	odd	4
2304.3.b.l.127.1		4	12.11	even	2
2304.3.b.l.127.2		4	3.2	odd	2
2304.3.b.l.127.3		4	24.11	even	2
2304.3.b.l.127.4		4	24.5	odd	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +2.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +2.00000i q^{5} +6.92820i q^{7} +3.00000 q^{9} +6.92820 q^{11} +2.00000i q^{13} -3.46410i q^{15} +10.0000 q^{17} -20.7846 q^{19} -12.0000i q^{21} +27.7128i q^{23} +21.0000 q^{25} -5.19615 q^{27} -26.0000i q^{29} +6.92820i q^{31} -12.0000 q^{33} -13.8564 q^{35} -26.0000i q^{37} -3.46410i q^{39} -58.0000 q^{41} -48.4974 q^{43} +6.00000i q^{45} +69.2820i q^{47} +1.00000 q^{49} -17.3205 q^{51} +74.0000i q^{53} +13.8564i q^{55} +36.0000 q^{57} -90.0666 q^{59} +26.0000i q^{61} +20.7846i q^{63} -4.00000 q^{65} -6.92820 q^{67} -48.0000i q^{69} +46.0000 q^{73} -36.3731 q^{75} +48.0000i q^{77} +117.779i q^{79} +9.00000 q^{81} -48.4974 q^{83} +20.0000i q^{85} +45.0333i q^{87} -82.0000 q^{89} -13.8564 q^{91} -12.0000i q^{93} -41.5692i q^{95} +2.00000 q^{97} +20.7846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} + 40 q^{17} + 84 q^{25} - 48 q^{33} - 232 q^{41} + 4 q^{49} + 144 q^{57} - 16 q^{65} + 184 q^{73} + 36 q^{81} - 328 q^{89} + 8 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 + 40 * q^17 + 84 * q^25 - 48 * q^33 - 232 * q^41 + 4 * q^49 + 144 * q^57 - 16 * q^65 + 184 * q^73 + 36 * q^81 - 328 * q^89 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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