LMFDB - Embedded newform 1344.4.c.d.673.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, 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:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 722x^{3} + 11881x^{2} + 54936x + 127008 $ x^6 - 2x^5 + 2x^4 + 722x^3 + 11881x^2 + 54936*x + 127008
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.1
Root			$8.80977 + 8.80977i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.d.673.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -15.6195i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -15.6195i q^{5} +7.00000 q^{7} -9.00000 q^{9} -13.3667i q^{11} -7.74714i q^{13} -46.8586 q^{15} -25.3667 q^{17} -71.9701i q^{19} -21.0000i q^{21} +23.6172 q^{23} -118.970 q^{25} +27.0000i q^{27} +9.49195i q^{29} -74.7586 q^{31} -40.1000 q^{33} -109.337i q^{35} -34.2529i q^{37} -23.2414 q^{39} -406.514 q^{41} -141.940i q^{43} +140.576i q^{45} -127.044 q^{47} +49.0000 q^{49} +76.1000i q^{51} -106.499i q^{53} -208.781 q^{55} -215.910 q^{57} -65.5265i q^{59} +35.5470i q^{61} -63.0000 q^{63} -121.007 q^{65} -606.021i q^{67} -70.8516i q^{69} -921.470 q^{71} +705.561 q^{73} +356.910i q^{75} -93.5668i q^{77} -294.343 q^{79} +81.0000 q^{81} -291.740i q^{83} +396.216i q^{85} +28.4758 q^{87} +0.646871 q^{89} -54.2300i q^{91} +224.276i q^{93} -1124.14 q^{95} +1353.90 q^{97} +120.300i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 42 q^{7} - 54 q^{9}+O(q^{10}) $ 6 * q + 42 * q^7 - 54 * q^9	$ 6 q + 42 q^{7} - 54 q^{9} + 24 q^{15} - 16 q^{17} - 60 q^{23} - 138 q^{25} - 552 q^{31} + 168 q^{33} - 36 q^{39} - 272 q^{41} - 1576 q^{47} + 294 q^{49} - 1632 q^{55} + 432 q^{57} - 378 q^{63} - 664 q^{65} - 2548 q^{71} + 444 q^{73} - 3528 q^{79} + 486 q^{81} - 336 q^{87} + 16 q^{89} - 4776 q^{95} + 1548 q^{97}+O(q^{100}) $ 6 * q + 42 * q^7 - 54 * q^9 + 24 * q^15 - 16 * q^17 - 60 * q^23 - 138 * q^25 - 552 * q^31 + 168 * q^33 - 36 * q^39 - 272 * q^41 - 1576 * q^47 + 294 * q^49 - 1632 * q^55 + 432 * q^57 - 378 * q^63 - 664 * q^65 - 2548 * q^71 + 444 * q^73 - 3528 * q^79 + 486 * q^81 - 336 * q^87 + 16 * q^89 - 4776 * q^95 + 1548 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	− 15.6195i	− 1.39705i	−0.715584	−	0.698527i	$-0.753839\pi$
$5$	− 15.6195i	− 1.39705i	0.715584	−	0.698527i	$-0.246161\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 13.3667i	− 0.366382i	−0.983077	−	0.183191i	$-0.941357\pi$
$11$	− 13.3667i	− 0.366382i	0.983077	−	0.183191i	$-0.0586427\pi$
$12$	0	0
$13$	− 7.74714i	− 0.165282i	−0.996579	−	0.0826411i	$-0.973664\pi$
$13$	− 7.74714i	− 0.165282i	0.996579	−	0.0826411i	$-0.0263355\pi$
$14$	0	0
$15$	−46.8586	−0.806590
$16$	0	0
$17$	−25.3667	−0.361901	−0.180951	−	0.983492i	$-0.557917\pi$
$17$	−25.3667	−0.361901	−0.180951	+	0.983492i	$0.557917\pi$
$18$	0	0
$19$	− 71.9701i	− 0.869004i	−0.900671	−	0.434502i	$-0.856924\pi$
$19$	− 71.9701i	− 0.869004i	0.900671	−	0.434502i	$-0.143076\pi$
$20$	0	0
$21$	− 21.0000i	− 0.218218i
$22$	0	0
$23$	23.6172	0.214110	0.107055	−	0.994253i	$-0.465858\pi$
$23$	23.6172	0.214110	0.107055	+	0.994253i	$0.465858\pi$
$24$	0	0
$25$	−118.970	−0.951761
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	9.49195i	0.0607797i	0.999538	+	0.0303898i	$0.00967487\pi$
$29$	9.49195i	0.0607797i	−0.999538	+	0.0303898i	$0.990325\pi$
$30$	0	0
$31$	−74.7586	−0.433130	−0.216565	−	0.976268i	$-0.569485\pi$
$31$	−74.7586	−0.433130	−0.216565	+	0.976268i	$0.569485\pi$
$32$	0	0
$33$	−40.1000	−0.211531
$34$	0	0
$35$	− 109.337i	− 0.528037i
$36$	0	0
$37$	− 34.2529i	− 0.152193i	−0.997100	−	0.0760964i	$-0.975754\pi$
$37$	− 34.2529i	− 0.152193i	0.997100	−	0.0760964i	$-0.0242457\pi$
$38$	0	0
$39$	−23.2414	−0.0954258
$40$	0	0
$41$	−406.514	−1.54846	−0.774229	−	0.632905i	$-0.781862\pi$
$41$	−406.514	−1.54846	−0.774229	+	0.632905i	$0.781862\pi$
$42$	0	0
$43$	− 141.940i	− 0.503388i	−0.967807	−	0.251694i	$-0.919012\pi$
$43$	− 141.940i	− 0.503388i	0.967807	−	0.251694i	$-0.0809876\pi$
$44$	0	0
$45$	140.576i	0.465685i
$46$	0	0
$47$	−127.044	−0.394281	−0.197141	−	0.980375i	$-0.563166\pi$
$47$	−127.044	−0.394281	−0.197141	+	0.980375i	$0.563166\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	76.1000i	0.208944i
$52$	0	0
$53$	− 106.499i	− 0.276014i	−0.990431	−	0.138007i	$-0.955930\pi$
$53$	− 106.499i	− 0.276014i	0.990431	−	0.138007i	$-0.0440696\pi$
$54$	0	0
$55$	−208.781	−0.511856
$56$	0	0
$57$	−215.910	−0.501720
$58$	0	0
$59$	− 65.5265i	− 0.144590i	−0.997383	−	0.0722951i	$-0.976968\pi$
$59$	− 65.5265i	− 0.144590i	0.997383	−	0.0722951i	$-0.0230323\pi$
$60$	0	0
$61$	35.5470i	0.0746120i	0.999304	+	0.0373060i	$0.0118776\pi$
$61$	35.5470i	0.0746120i	−0.999304	+	0.0373060i	$0.988122\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−121.007	−0.230908
$66$	0	0
$67$	− 606.021i	− 1.10503i	−0.833502	−	0.552517i	$-0.813668\pi$
$67$	− 606.021i	− 1.10503i	0.833502	−	0.552517i	$-0.186332\pi$
$68$	0	0
$69$	− 70.8516i	− 0.123616i
$70$	0	0
$71$	−921.470	−1.54026	−0.770129	−	0.637888i	$-0.779808\pi$
$71$	−921.470	−1.54026	−0.770129	+	0.637888i	$0.779808\pi$
$72$	0	0
$73$	705.561	1.13123	0.565614	−	0.824670i	$-0.308639\pi$
$73$	705.561	1.13123	0.565614	+	0.824670i	$0.308639\pi$
$74$	0	0
$75$	356.910i	0.549499i
$76$	0	0
$77$	− 93.5668i	− 0.138480i
$78$	0	0
$79$	−294.343	−0.419191	−0.209596	−	0.977788i	$-0.567215\pi$
$79$	−294.343	−0.419191	−0.209596	+	0.977788i	$0.567215\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	− 291.740i	− 0.385815i	−0.981217	−	0.192907i	$-0.938208\pi$
$83$	− 291.740i	− 0.385815i	0.981217	−	0.192907i	$-0.0617917\pi$
$84$	0	0
$85$	396.216i	0.505596i
$86$	0	0
$87$	28.4758	0.0350912
$88$	0	0
$89$	0.646871	0.000770429 0	0.000385214	−	1.00000i	$-0.499877\pi$
$89$	0.646871	0.000770429 0	0.000385214		1.00000i	$0.499877\pi$
$90$	0	0
$91$	− 54.2300i	− 0.0624708i
$92$	0	0
$93$	224.276i	0.250068i
$94$	0	0
$95$	−1124.14	−1.21405
$96$	0	0
$97$	1353.90	1.41719	0.708597	−	0.705613i	$-0.249328\pi$
$97$	1353.90	1.41719	0.708597	+	0.705613i	$0.249328\pi$
$98$	0	0
$99$	120.300i	0.122127i
$100$	0	0
$101$	1146.50i	1.12952i	0.825255	+	0.564760i	$0.191031\pi$
$101$	1146.50i	1.12952i	−0.825255	+	0.564760i	$0.808969\pi$
$102$	0	0
$103$	305.554	0.292302	0.146151	−	0.989262i	$-0.453311\pi$
$103$	305.554	0.292302	0.146151	+	0.989262i	$0.453311\pi$
$104$	0	0
$105$	−328.010	−0.304862
$106$	0	0
$107$	1528.22i	1.38074i	0.723458	+	0.690368i	$0.242552\pi$
$107$	1528.22i	1.38074i	−0.723458	+	0.690368i	$0.757448\pi$
$108$	0	0
$109$	293.191i	0.257638i	0.991668	+	0.128819i	$0.0411187\pi$
$109$	293.191i	0.257638i	−0.991668	+	0.128819i	$0.958881\pi$
$110$	0	0
$111$	−102.759	−0.0878686
$112$	0	0
$113$	212.793	0.177150	0.0885748	−	0.996070i	$-0.471769\pi$
$113$	212.793	0.177150	0.0885748	+	0.996070i	$0.471769\pi$
$114$	0	0
$115$	− 368.890i	− 0.299123i
$116$	0	0
$117$	69.7243i	0.0550941i
$118$	0	0
$119$	−177.567	−0.136786
$120$	0	0
$121$	1152.33	0.865764
$122$	0	0
$123$	1219.54i	0.894003i
$124$	0	0
$125$	− 94.1840i	− 0.0673926i
$126$	0	0
$127$	−1886.23	−1.31792	−0.658961	−	0.752177i	$-0.729004\pi$
$127$	−1886.23	−1.31792	−0.658961	+	0.752177i	$0.729004\pi$
$128$	0	0
$129$	−425.821	−0.290631
$130$	0	0
$131$	1107.68i	0.738767i	0.929277	+	0.369384i	$0.120431\pi$
$131$	1107.68i	0.738767i	−0.929277	+	0.369384i	$0.879569\pi$
$132$	0	0
$133$	− 503.791i	− 0.328453i
$134$	0	0
$135$	421.728	0.268863
$136$	0	0
$137$	−770.260	−0.480348	−0.240174	−	0.970730i	$-0.577205\pi$
$137$	−770.260	−0.480348	−0.240174	+	0.970730i	$0.577205\pi$
$138$	0	0
$139$	833.201i	0.508426i	0.967148	+	0.254213i	$0.0818163\pi$
$139$	833.201i	0.508426i	−0.967148	+	0.254213i	$0.918184\pi$
$140$	0	0
$141$	381.131i	0.227638i
$142$	0	0
$143$	−103.554	−0.0605565
$144$	0	0
$145$	148.260	0.0849125
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	1247.39i	0.685838i	0.939365	+	0.342919i	$0.111416\pi$
$149$	1247.39i	0.685838i	−0.939365	+	0.342919i	$0.888584\pi$
$150$	0	0
$151$	−1887.55	−1.01726	−0.508632	−	0.860984i	$-0.669849\pi$
$151$	−1887.55	−1.01726	−0.508632	+	0.860984i	$0.669849\pi$
$152$	0	0
$153$	228.300	0.120634
$154$	0	0
$155$	1167.69i	0.605107i
$156$	0	0
$157$	3920.85i	1.99311i	0.0829520	+	0.996554i	$0.473565\pi$
$157$	3920.85i	1.99311i	−0.0829520	+	0.996554i	$0.526435\pi$
$158$	0	0
$159$	−319.496	−0.159357
$160$	0	0
$161$	165.320	0.0809260
$162$	0	0
$163$	− 2552.50i	− 1.22655i	−0.789870	−	0.613274i	$-0.789852\pi$
$163$	− 2552.50i	− 1.22655i	0.789870	−	0.613274i	$-0.210148\pi$
$164$	0	0
$165$	626.344i	0.295520i
$166$	0	0
$167$	−1623.67	−0.752354	−0.376177	−	0.926548i	$-0.622761\pi$
$167$	−1623.67	−0.752354	−0.376177	+	0.926548i	$0.622761\pi$
$168$	0	0
$169$	2136.98	0.972682
$170$	0	0
$171$	647.731i	0.289668i
$172$	0	0
$173$	1973.48i	0.867289i	0.901084	+	0.433645i	$0.142773\pi$
$173$	1973.48i	0.867289i	−0.901084	+	0.433645i	$0.857227\pi$
$174$	0	0
$175$	−832.791	−0.359732
$176$	0	0
$177$	−196.579	−0.0834792
$178$	0	0
$179$	− 814.995i	− 0.340310i	−0.985417	−	0.170155i	$-0.945573\pi$
$179$	− 814.995i	− 0.340310i	0.985417	−	0.170155i	$-0.0544269\pi$
$180$	0	0
$181$	1335.83i	0.548574i	0.961648	+	0.274287i	$0.0884418\pi$
$181$	1335.83i	0.548574i	−0.961648	+	0.274287i	$0.911558\pi$
$182$	0	0
$183$	106.641	0.0430773
$184$	0	0
$185$	−535.014	−0.212622
$186$	0	0
$187$	339.068i	0.132594i
$188$	0	0
$189$	189.000i	0.0727393i
$190$	0	0
$191$	−111.893	−0.0423890	−0.0211945	−	0.999775i	$-0.506747\pi$
$191$	−111.893	−0.0423890	−0.0211945	+	0.999775i	$0.506747\pi$
$192$	0	0
$193$	−1515.23	−0.565123	−0.282561	−	0.959249i	$-0.591184\pi$
$193$	−1515.23	−0.565123	−0.282561	+	0.959249i	$0.591184\pi$
$194$	0	0
$195$	363.020i	0.133315i
$196$	0	0
$197$	− 2099.84i	− 0.759428i	−0.925104	−	0.379714i	$-0.876022\pi$
$197$	− 2099.84i	− 0.759428i	0.925104	−	0.379714i	$-0.123978\pi$
$198$	0	0
$199$	4567.75	1.62713	0.813566	−	0.581472i	$-0.197523\pi$
$199$	4567.75	1.62713	0.813566	+	0.581472i	$0.197523\pi$
$200$	0	0
$201$	−1818.06	−0.637991
$202$	0	0
$203$	66.4436i	0.0229726i
$204$	0	0
$205$	6349.56i	2.16328i
$206$	0	0
$207$	−212.555	−0.0713700
$208$	0	0
$209$	−962.002	−0.318388
$210$	0	0
$211$	− 3093.08i	− 1.00918i	−0.863360	−	0.504589i	$-0.831644\pi$
$211$	− 3093.08i	− 1.00918i	0.863360	−	0.504589i	$-0.168356\pi$
$212$	0	0
$213$	2764.41i	0.889269i
$214$	0	0
$215$	−2217.04	−0.703260
$216$	0	0
$217$	−523.310	−0.163708
$218$	0	0
$219$	− 2116.68i	− 0.653115i
$220$	0	0
$221$	196.519i	0.0598159i
$222$	0	0
$223$	−4006.78	−1.20320	−0.601601	−	0.798797i	$-0.705470\pi$
$223$	−4006.78	−1.20320	−0.601601	+	0.798797i	$0.705470\pi$
$224$	0	0
$225$	1070.73	0.317254
$226$	0	0
$227$	− 4194.83i	− 1.22652i	−0.789881	−	0.613261i	$-0.789858\pi$
$227$	− 4194.83i	− 1.22652i	0.789881	−	0.613261i	$-0.210142\pi$
$228$	0	0
$229$	4438.14i	1.28070i	0.768083	+	0.640351i	$0.221211\pi$
$229$	4438.14i	1.28070i	−0.768083	+	0.640351i	$0.778789\pi$
$230$	0	0
$231$	−280.700	−0.0799512
$232$	0	0
$233$	−3390.24	−0.953229	−0.476614	−	0.879112i	$-0.658136\pi$
$233$	−3390.24	−0.953229	−0.476614	+	0.879112i	$0.658136\pi$
$234$	0	0
$235$	1984.36i	0.550832i
$236$	0	0
$237$	883.028i	0.242020i
$238$	0	0
$239$	−5283.23	−1.42989	−0.714945	−	0.699181i	$-0.753548\pi$
$239$	−5283.23	−1.42989	−0.714945	+	0.699181i	$0.753548\pi$
$240$	0	0
$241$	−4582.43	−1.22481	−0.612407	−	0.790543i	$-0.709798\pi$
$241$	−4582.43	−1.22481	−0.612407	+	0.790543i	$0.709798\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 765.358i	− 0.199579i
$246$	0	0
$247$	−557.563	−0.143631
$248$	0	0
$249$	−875.220	−0.222750
$250$	0	0
$251$	− 687.240i	− 0.172822i	−0.996260	−	0.0864108i	$-0.972460\pi$
$251$	− 687.240i	− 0.172822i	0.996260	−	0.0864108i	$-0.0275398\pi$
$252$	0	0
$253$	− 315.684i	− 0.0784461i
$254$	0	0
$255$	1188.65	0.291906
$256$	0	0
$257$	3107.94	0.754350	0.377175	−	0.926142i	$-0.376895\pi$
$257$	3107.94	0.754350	0.377175	+	0.926142i	$0.376895\pi$
$258$	0	0
$259$	− 239.770i	− 0.0575235i
$260$	0	0
$261$	− 85.4275i	− 0.0202599i
$262$	0	0
$263$	−6499.44	−1.52385	−0.761925	−	0.647665i	$-0.775746\pi$
$263$	−6499.44	−1.52385	−0.761925	+	0.647665i	$0.775746\pi$
$264$	0	0
$265$	−1663.46	−0.385606
$266$	0	0
$267$	− 1.94061i	0 0.000444807i
$268$	0	0
$269$	5787.54i	1.31179i	0.754851	+	0.655897i	$0.227709\pi$
$269$	5787.54i	1.31179i	−0.754851	+	0.655897i	$0.772291\pi$
$270$	0	0
$271$	1272.04	0.285133	0.142567	−	0.989785i	$-0.454465\pi$
$271$	1272.04	0.285133	0.142567	+	0.989785i	$0.454465\pi$
$272$	0	0
$273$	−162.690	−0.0360676
$274$	0	0
$275$	1590.24i	0.348708i
$276$	0	0
$277$	− 4286.24i	− 0.929730i	−0.885382	−	0.464865i	$-0.846103\pi$
$277$	− 4286.24i	− 0.929730i	0.885382	−	0.464865i	$-0.153897\pi$
$278$	0	0
$279$	672.827	0.144377
$280$	0	0
$281$	4276.87	0.907960	0.453980	−	0.891012i	$-0.350004\pi$
$281$	4276.87	0.907960	0.453980	+	0.891012i	$0.350004\pi$
$282$	0	0
$283$	− 2544.93i	− 0.534560i	−0.963619	−	0.267280i	$-0.913875\pi$
$283$	− 2544.93i	− 0.534560i	0.963619	−	0.267280i	$-0.0861248\pi$
$284$	0	0
$285$	3372.42i	0.700930i
$286$	0	0
$287$	−2845.60	−0.585262
$288$	0	0
$289$	−4269.53	−0.869027
$290$	0	0
$291$	− 4061.70i	− 0.818218i
$292$	0	0
$293$	− 3696.67i	− 0.737071i	−0.929614	−	0.368536i	$-0.879859\pi$
$293$	− 3696.67i	− 0.737071i	0.929614	−	0.368536i	$-0.120141\pi$
$294$	0	0
$295$	−1023.49	−0.202000
$296$	0	0
$297$	360.900	0.0705103
$298$	0	0
$299$	− 182.966i	− 0.0353886i
$300$	0	0
$301$	− 993.582i	− 0.190263i
$302$	0	0
$303$	3439.51	0.652128
$304$	0	0
$305$	555.229	0.104237
$306$	0	0
$307$	2853.13i	0.530413i	0.964192	+	0.265206i	$0.0854401\pi$
$307$	2853.13i	0.530413i	−0.964192	+	0.265206i	$0.914560\pi$
$308$	0	0
$309$	− 916.662i	− 0.168761i
$310$	0	0
$311$	4295.78	0.783252	0.391626	−	0.920124i	$-0.371913\pi$
$311$	4295.78	0.783252	0.391626	+	0.920124i	$0.371913\pi$
$312$	0	0
$313$	3462.76	0.625325	0.312663	−	0.949864i	$-0.398779\pi$
$313$	3462.76	0.625325	0.312663	+	0.949864i	$0.398779\pi$
$314$	0	0
$315$	984.031i	0.176012i
$316$	0	0
$317$	− 12.6028i	− 0.00223295i	−0.999999	−	0.00111648i	$-0.999645\pi$
$317$	− 12.6028i	− 0.00223295i	0.999999	−	0.00111648i	$-0.000355385\pi$
$318$	0	0
$319$	126.876	0.0222686
$320$	0	0
$321$	4584.67	0.797169
$322$	0	0
$323$	1825.64i	0.314494i
$324$	0	0
$325$	921.678i	0.157309i
$326$	0	0
$327$	879.572	0.148748
$328$	0	0
$329$	−889.306	−0.149024
$330$	0	0
$331$	4298.31i	0.713765i	0.934149	+	0.356883i	$0.116160\pi$
$331$	4298.31i	0.713765i	−0.934149	+	0.356883i	$0.883840\pi$
$332$	0	0
$333$	308.276i	0.0507310i
$334$	0	0
$335$	−9465.77	−1.54379
$336$	0	0
$337$	−2888.55	−0.466912	−0.233456	−	0.972367i	$-0.575003\pi$
$337$	−2888.55	−0.466912	−0.233456	+	0.972367i	$0.575003\pi$
$338$	0	0
$339$	− 638.379i	− 0.102277i
$340$	0	0
$341$	999.274i	0.158691i
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−1106.67	−0.172699
$346$	0	0
$347$	− 6581.98i	− 1.01827i	−0.860687	−	0.509134i	$-0.829966\pi$
$347$	− 6581.98i	− 1.01827i	0.860687	−	0.509134i	$-0.170034\pi$
$348$	0	0
$349$	3708.89i	0.568861i	0.958697	+	0.284430i	$0.0918045\pi$
$349$	3708.89i	0.568861i	−0.958697	+	0.284430i	$0.908196\pi$
$350$	0	0
$351$	209.173	0.0318086
$352$	0	0
$353$	11185.6	1.68655	0.843274	−	0.537484i	$-0.180625\pi$
$353$	11185.6	1.68655	0.843274	+	0.537484i	$0.180625\pi$
$354$	0	0
$355$	14392.9i	2.15183i
$356$	0	0
$357$	532.700i	0.0789734i
$358$	0	0
$359$	1248.88	0.183603	0.0918015	−	0.995777i	$-0.470737\pi$
$359$	1248.88	0.183603	0.0918015	+	0.995777i	$0.470737\pi$
$360$	0	0
$361$	1679.30	0.244832
$362$	0	0
$363$	− 3457.00i	− 0.499849i
$364$	0	0
$365$	− 11020.5i	− 1.58039i
$366$	0	0
$367$	5872.81	0.835309	0.417654	−	0.908606i	$-0.362852\pi$
$367$	5872.81	0.835309	0.417654	+	0.908606i	$0.362852\pi$
$368$	0	0
$369$	3658.62	0.516153
$370$	0	0
$371$	− 745.491i	− 0.104323i
$372$	0	0
$373$	7954.92i	1.10426i	0.833757	+	0.552131i	$0.186185\pi$
$373$	7954.92i	1.10426i	−0.833757	+	0.552131i	$0.813815\pi$
$374$	0	0
$375$	−282.552	−0.0389091
$376$	0	0
$377$	73.5354	0.0100458
$378$	0	0
$379$	1706.16i	0.231239i	0.993294	+	0.115620i	$0.0368853\pi$
$379$	1706.16i	0.231239i	−0.993294	+	0.115620i	$0.963115\pi$
$380$	0	0
$381$	5658.70i	0.760903i
$382$	0	0
$383$	−6696.81	−0.893449	−0.446724	−	0.894672i	$-0.647410\pi$
$383$	−6696.81	−0.893449	−0.446724	+	0.894672i	$0.647410\pi$
$384$	0	0
$385$	−1461.47	−0.193463
$386$	0	0
$387$	1277.46i	0.167796i
$388$	0	0
$389$	− 10007.1i	− 1.30432i	−0.758081	−	0.652160i	$-0.773863\pi$
$389$	− 10007.1i	− 1.30432i	0.758081	−	0.652160i	$-0.226137\pi$
$390$	0	0
$391$	−599.090	−0.0774867
$392$	0	0
$393$	3323.04	0.426527
$394$	0	0
$395$	4597.50i	0.585633i
$396$	0	0
$397$	− 7583.54i	− 0.958707i	−0.877622	−	0.479354i	$-0.840871\pi$
$397$	− 7583.54i	− 0.958707i	0.877622	−	0.479354i	$-0.159129\pi$
$398$	0	0
$399$	−1511.37	−0.189632
$400$	0	0
$401$	−4654.03	−0.579579	−0.289790	−	0.957090i	$-0.593585\pi$
$401$	−4654.03	−0.579579	−0.289790	+	0.957090i	$0.593585\pi$
$402$	0	0
$403$	579.165i	0.0715888i
$404$	0	0
$405$	− 1265.18i	− 0.155228i
$406$	0	0
$407$	−457.847	−0.0557608
$408$	0	0
$409$	631.125	0.0763011	0.0381505	−	0.999272i	$-0.487853\pi$
$409$	631.125	0.0763011	0.0381505	+	0.999272i	$0.487853\pi$
$410$	0	0
$411$	2310.78i	0.277329i
$412$	0	0
$413$	− 458.685i	− 0.0546500i
$414$	0	0
$415$	−4556.85	−0.539004
$416$	0	0
$417$	2499.60	0.293540
$418$	0	0
$419$	10883.6i	1.26897i	0.772934	+	0.634486i	$0.218788\pi$
$419$	10883.6i	1.26897i	−0.772934	+	0.634486i	$0.781212\pi$
$420$	0	0
$421$	− 446.347i	− 0.0516713i	−0.999666	−	0.0258356i	$-0.991775\pi$
$421$	− 446.347i	− 0.0516713i	0.999666	−	0.0258356i	$-0.00822465\pi$
$422$	0	0
$423$	1143.39	0.131427
$424$	0	0
$425$	3017.88	0.344444
$426$	0	0
$427$	248.829i	0.0282007i
$428$	0	0
$429$	310.661i	0.0349623i
$430$	0	0
$431$	−10094.7	−1.12817	−0.564087	−	0.825715i	$-0.690772\pi$
$431$	−10094.7	−1.12817	−0.564087	+	0.825715i	$0.690772\pi$
$432$	0	0
$433$	−13157.6	−1.46031	−0.730154	−	0.683282i	$-0.760552\pi$
$433$	−13157.6	−1.46031	−0.730154	+	0.683282i	$0.760552\pi$
$434$	0	0
$435$	− 444.780i	− 0.0490242i
$436$	0	0
$437$	− 1699.73i	− 0.186062i
$438$	0	0
$439$	12868.8	1.39908	0.699538	−	0.714595i	$-0.253389\pi$
$439$	12868.8	1.39908	0.699538	+	0.714595i	$0.253389\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 9934.79i	− 1.06550i	−0.846273	−	0.532750i	$-0.821159\pi$
$443$	− 9934.79i	− 1.06550i	0.846273	−	0.532750i	$-0.178841\pi$
$444$	0	0
$445$	− 10.1038i	− 0.00107633i
$446$	0	0
$447$	3742.16	0.395969
$448$	0	0
$449$	4998.49	0.525375	0.262687	−	0.964881i	$-0.415391\pi$
$449$	4998.49	0.525375	0.262687	+	0.964881i	$0.415391\pi$
$450$	0	0
$451$	5433.74i	0.567328i
$452$	0	0
$453$	5662.66i	0.587318i
$454$	0	0
$455$	−847.047	−0.0872752
$456$	0	0
$457$	1960.32	0.200657	0.100328	−	0.994954i	$-0.468011\pi$
$457$	1960.32	0.200657	0.100328	+	0.994954i	$0.468011\pi$
$458$	0	0
$459$	− 684.900i	− 0.0696480i
$460$	0	0
$461$	− 18568.0i	− 1.87591i	−0.346752	−	0.937957i	$-0.612716\pi$
$461$	− 18568.0i	− 1.87591i	0.346752	−	0.937957i	$-0.387284\pi$
$462$	0	0
$463$	−7696.28	−0.772520	−0.386260	−	0.922390i	$-0.626233\pi$
$463$	−7696.28	−0.772520	−0.386260	+	0.922390i	$0.626233\pi$
$464$	0	0
$465$	3503.08	0.349358
$466$	0	0
$467$	10624.0i	1.05272i	0.850262	+	0.526359i	$0.176443\pi$
$467$	10624.0i	1.05272i	−0.850262	+	0.526359i	$0.823557\pi$
$468$	0	0
$469$	− 4242.15i	− 0.417663i
$470$	0	0
$471$	11762.5	1.15072
$472$	0	0
$473$	−1897.27	−0.184432
$474$	0	0
$475$	8562.29i	0.827084i
$476$	0	0
$477$	958.489i	0.0920046i
$478$	0	0
$479$	−11097.2	−1.05855	−0.529274	−	0.848451i	$-0.677536\pi$
$479$	−11097.2	−1.05855	−0.529274	+	0.848451i	$0.677536\pi$
$480$	0	0
$481$	−265.362	−0.0251548
$482$	0	0
$483$	− 495.961i	− 0.0467226i
$484$	0	0
$485$	− 21147.3i	− 1.97990i
$486$	0	0
$487$	12174.8	1.13284	0.566422	−	0.824116i	$-0.308327\pi$
$487$	12174.8	1.13284	0.566422	+	0.824116i	$0.308327\pi$
$488$	0	0
$489$	−7657.51	−0.708148
$490$	0	0
$491$	− 17789.7i	− 1.63511i	−0.575854	−	0.817553i	$-0.695330\pi$
$491$	− 17789.7i	− 1.63511i	0.575854	−	0.817553i	$-0.304670\pi$
$492$	0	0
$493$	− 240.779i	− 0.0219962i
$494$	0	0
$495$	1879.03	0.170619
$496$	0	0
$497$	−6450.29	−0.582163
$498$	0	0
$499$	− 8148.36i	− 0.731004i	−0.930811	−	0.365502i	$-0.880897\pi$
$499$	− 8148.36i	− 0.731004i	0.930811	−	0.365502i	$-0.119103\pi$
$500$	0	0
$501$	4871.00i	0.434372i
$502$	0	0
$503$	13339.1	1.18243	0.591216	−	0.806513i	$-0.298648\pi$
$503$	13339.1	1.18243	0.591216	+	0.806513i	$0.298648\pi$
$504$	0	0
$505$	17907.9	1.57800
$506$	0	0
$507$	− 6410.95i	− 0.561578i
$508$	0	0
$509$	− 17169.3i	− 1.49512i	−0.664192	−	0.747562i	$-0.731224\pi$
$509$	− 17169.3i	− 1.49512i	0.664192	−	0.747562i	$-0.268776\pi$
$510$	0	0
$511$	4938.93	0.427564
$512$	0	0
$513$	1943.19	0.167240
$514$	0	0
$515$	− 4772.61i	− 0.408362i
$516$	0	0
$517$	1698.15i	0.144458i
$518$	0	0
$519$	5920.45	0.500730
$520$	0	0
$521$	−12337.0	−1.03742	−0.518708	−	0.854952i	$-0.673587\pi$
$521$	−12337.0	−1.03742	−0.518708	+	0.854952i	$0.673587\pi$
$522$	0	0
$523$	− 2797.74i	− 0.233913i	−0.993137	−	0.116956i	$-0.962686\pi$
$523$	− 2797.74i	− 0.233913i	0.993137	−	0.116956i	$-0.0373138\pi$
$524$	0	0
$525$	2498.37i	0.207691i
$526$	0	0
$527$	1896.38	0.156750
$528$	0	0
$529$	−11609.2	−0.954157
$530$	0	0
$531$	589.738i	0.0481967i
$532$	0	0
$533$	3149.32i	0.255933i
$534$	0	0
$535$	23870.1	1.92896
$536$	0	0
$537$	−2444.98	−0.196478
$538$	0	0
$539$	− 654.967i	− 0.0523403i
$540$	0	0
$541$	13646.6i	1.08450i	0.840217	+	0.542250i	$0.182427\pi$
$541$	13646.6i	1.08450i	−0.840217	+	0.542250i	$0.817573\pi$
$542$	0	0
$543$	4007.50	0.316719
$544$	0	0
$545$	4579.51	0.359935
$546$	0	0
$547$	− 7329.24i	− 0.572899i	−0.958095	−	0.286450i	$-0.907525\pi$
$547$	− 7329.24i	− 0.572899i	0.958095	−	0.286450i	$-0.0924751\pi$
$548$	0	0
$549$	− 319.923i	− 0.0248707i
$550$	0	0
$551$	683.136	0.0528178
$552$	0	0
$553$	−2060.40	−0.158439
$554$	0	0
$555$	1605.04i	0.122757i
$556$	0	0
$557$	− 825.578i	− 0.0628023i	−0.999507	−	0.0314011i	$-0.990003\pi$
$557$	− 825.578i	− 0.0628023i	0.999507	−	0.0314011i	$-0.00999693\pi$
$558$	0	0
$559$	−1099.63	−0.0832011
$560$	0	0
$561$	1017.21	0.0765534
$562$	0	0
$563$	− 7496.18i	− 0.561148i	−0.959832	−	0.280574i	$-0.909475\pi$
$563$	− 7496.18i	− 0.561148i	0.959832	−	0.280574i	$-0.0905248\pi$
$564$	0	0
$565$	− 3323.73i	− 0.247487i
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	905.513	0.0667154	0.0333577	−	0.999443i	$-0.489380\pi$
$569$	905.513	0.0667154	0.0333577	+	0.999443i	$0.489380\pi$
$570$	0	0
$571$	− 24158.1i	− 1.77055i	−0.465065	−	0.885277i	$-0.653969\pi$
$571$	− 24158.1i	− 1.77055i	0.465065	−	0.885277i	$-0.346031\pi$
$572$	0	0
$573$	335.679i	0.0244733i
$574$	0	0
$575$	−2809.74	−0.203781
$576$	0	0
$577$	5227.62	0.377173	0.188586	−	0.982057i	$-0.439609\pi$
$577$	5227.62	0.377173	0.188586	+	0.982057i	$0.439609\pi$
$578$	0	0
$579$	4545.69i	0.326274i
$580$	0	0
$581$	− 2042.18i	− 0.145824i
$582$	0	0
$583$	−1423.53	−0.101127
$584$	0	0
$585$	1089.06	0.0769695
$586$	0	0
$587$	− 11007.8i	− 0.774002i	−0.922079	−	0.387001i	$-0.873511\pi$
$587$	− 11007.8i	− 0.774002i	0.922079	−	0.387001i	$-0.126489\pi$
$588$	0	0
$589$	5380.38i	0.376392i
$590$	0	0
$591$	−6299.52	−0.438456
$592$	0	0
$593$	−15552.3	−1.07699	−0.538496	−	0.842628i	$-0.681007\pi$
$593$	−15552.3	−1.07699	−0.538496	+	0.842628i	$0.681007\pi$
$594$	0	0
$595$	2773.51i	0.191097i
$596$	0	0
$597$	− 13703.3i	− 0.939425i
$598$	0	0
$599$	5854.56	0.399350	0.199675	−	0.979862i	$-0.436011\pi$
$599$	5854.56	0.399350	0.199675	+	0.979862i	$0.436011\pi$
$600$	0	0
$601$	4846.77	0.328958	0.164479	−	0.986381i	$-0.447406\pi$
$601$	4846.77	0.328958	0.164479	+	0.986381i	$0.447406\pi$
$602$	0	0
$603$	5454.19i	0.368344i
$604$	0	0
$605$	− 17998.9i	− 1.20952i
$606$	0	0
$607$	17579.6	1.17551	0.587754	−	0.809040i	$-0.300012\pi$
$607$	17579.6	1.17551	0.587754	+	0.809040i	$0.300012\pi$
$608$	0	0
$609$	199.331	0.0132632
$610$	0	0
$611$	984.225i	0.0651677i
$612$	0	0
$613$	− 8251.78i	− 0.543697i	−0.962340	−	0.271848i	$-0.912365\pi$
$613$	− 8251.78i	− 0.543697i	0.962340	−	0.271848i	$-0.0876350\pi$
$614$	0	0
$615$	19048.7	1.24897
$616$	0	0
$617$	−9569.90	−0.624424	−0.312212	−	0.950012i	$-0.601070\pi$
$617$	−9569.90	−0.624424	−0.312212	+	0.950012i	$0.601070\pi$
$618$	0	0
$619$	− 19931.9i	− 1.29423i	−0.762391	−	0.647117i	$-0.775975\pi$
$619$	− 19931.9i	− 1.29423i	0.762391	−	0.647117i	$-0.224025\pi$
$620$	0	0
$621$	637.665i	0.0412055i
$622$	0	0
$623$	4.52810	0.000291195 0
$624$	0	0
$625$	−16342.4	−1.04591
$626$	0	0
$627$	2886.01i	0.183821i
$628$	0	0
$629$	868.881i	0.0550788i
$630$	0	0
$631$	20939.7	1.32107	0.660537	−	0.750794i	$-0.270329\pi$
$631$	20939.7	1.32107	0.660537	+	0.750794i	$0.270329\pi$
$632$	0	0
$633$	−9279.25	−0.582649
$634$	0	0
$635$	29462.1i	1.84121i
$636$	0	0
$637$	− 379.610i	− 0.0236118i
$638$	0	0
$639$	8293.23	0.513420
$640$	0	0
$641$	11964.1	0.737214	0.368607	−	0.929585i	$-0.379835\pi$
$641$	11964.1	0.737214	0.368607	+	0.929585i	$0.379835\pi$
$642$	0	0
$643$	8277.59i	0.507677i	0.967247	+	0.253838i	$0.0816932\pi$
$643$	8277.59i	0.507677i	−0.967247	+	0.253838i	$0.918307\pi$
$644$	0	0
$645$	6651.12i	0.406028i
$646$	0	0
$647$	−12733.4	−0.773725	−0.386862	−	0.922137i	$-0.626441\pi$
$647$	−12733.4	−0.773725	−0.386862	+	0.922137i	$0.626441\pi$
$648$	0	0
$649$	−875.872	−0.0529753
$650$	0	0
$651$	1569.93i	0.0945168i
$652$	0	0
$653$	− 22687.1i	− 1.35959i	−0.733400	−	0.679797i	$-0.762068\pi$
$653$	− 22687.1i	− 1.35959i	0.733400	−	0.679797i	$-0.237932\pi$
$654$	0	0
$655$	17301.5	1.03210
$656$	0	0
$657$	−6350.05	−0.377076
$658$	0	0
$659$	− 11843.4i	− 0.700083i	−0.936734	−	0.350041i	$-0.886168\pi$
$659$	− 11843.4i	− 0.700083i	0.936734	−	0.350041i	$-0.113832\pi$
$660$	0	0
$661$	− 27322.4i	− 1.60774i	−0.594805	−	0.803870i	$-0.702771\pi$
$661$	− 27322.4i	− 1.60774i	0.594805	−	0.803870i	$-0.297229\pi$
$662$	0	0
$663$	589.558	0.0345347
$664$	0	0
$665$	−7868.98	−0.458866
$666$	0	0
$667$	224.173i	0.0130135i
$668$	0	0
$669$	12020.3i	0.694669i
$670$	0	0
$671$	475.146	0.0273365
$672$	0	0
$673$	31900.7	1.82717	0.913583	−	0.406653i	$-0.133304\pi$
$673$	31900.7	1.82717	0.913583	+	0.406653i	$0.133304\pi$
$674$	0	0
$675$	− 3212.19i	− 0.183166i
$676$	0	0
$677$	11279.2i	0.640320i	0.947364	+	0.320160i	$0.103737\pi$
$677$	11279.2i	0.640320i	−0.947364	+	0.320160i	$0.896263\pi$
$678$	0	0
$679$	9477.31	0.535649
$680$	0	0
$681$	−12584.5	−0.708132
$682$	0	0
$683$	4906.49i	0.274878i	0.990510	+	0.137439i	$0.0438871\pi$
$683$	4906.49i	0.274878i	−0.990510	+	0.137439i	$0.956113\pi$
$684$	0	0
$685$	12031.1i	0.671073i
$686$	0	0
$687$	13314.4	0.739413
$688$	0	0
$689$	−825.060	−0.0456202
$690$	0	0
$691$	− 450.536i	− 0.0248035i	−0.999923	−	0.0124017i	$-0.996052\pi$
$691$	− 450.536i	− 0.0248035i	0.999923	−	0.0124017i	$-0.00394770\pi$
$692$	0	0
$693$	842.101i	0.0461598i
$694$	0	0
$695$	13014.2	0.710298
$696$	0	0
$697$	10311.9	0.560389
$698$	0	0
$699$	10170.7i	0.550347i
$700$	0	0
$701$	− 22019.5i	− 1.18640i	−0.805057	−	0.593198i	$-0.797865\pi$
$701$	− 22019.5i	− 1.18640i	0.805057	−	0.593198i	$-0.202135\pi$
$702$	0	0
$703$	−2465.18	−0.132256
$704$	0	0
$705$	5953.09	0.318023
$706$	0	0
$707$	8025.53i	0.426918i
$708$	0	0
$709$	10311.6i	0.546206i	0.961985	+	0.273103i	$0.0880499\pi$
$709$	10311.6i	0.546206i	−0.961985	+	0.273103i	$0.911950\pi$
$710$	0	0
$711$	2649.08	0.139730
$712$	0	0
$713$	−1765.59	−0.0927375
$714$	0	0
$715$	1617.46i	0.0846008i
$716$	0	0
$717$	15849.7i	0.825547i
$718$	0	0
$719$	−20881.3	−1.08309	−0.541544	−	0.840672i	$-0.682160\pi$
$719$	−20881.3	−1.08309	−0.541544	+	0.840672i	$0.682160\pi$
$720$	0	0
$721$	2138.88	0.110480
$722$	0	0
$723$	13747.3i	0.707146i
$724$	0	0
$725$	− 1129.26i	− 0.0578477i
$726$	0	0
$727$	17648.4	0.900334	0.450167	−	0.892944i	$-0.351364\pi$
$727$	17648.4	0.900334	0.450167	+	0.892944i	$0.351364\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	3600.55i	0.182177i
$732$	0	0
$733$	− 30985.8i	− 1.56137i	−0.624923	−	0.780687i	$-0.714870\pi$
$733$	− 30985.8i	− 1.56137i	0.624923	−	0.780687i	$-0.285130\pi$
$734$	0	0
$735$	−2296.07	−0.115227
$736$	0	0
$737$	−8100.49	−0.404865
$738$	0	0
$739$	− 1896.07i	− 0.0943815i	−0.998886	−	0.0471908i	$-0.984973\pi$
$739$	− 1896.07i	− 0.0943815i	0.998886	−	0.0471908i	$-0.0150269\pi$
$740$	0	0
$741$	1672.69i	0.0829254i
$742$	0	0
$743$	9997.93	0.493659	0.246829	−	0.969059i	$-0.420611\pi$
$743$	9997.93	0.493659	0.246829	+	0.969059i	$0.420611\pi$
$744$	0	0
$745$	19483.6	0.958153
$746$	0	0
$747$	2625.66i	0.128605i
$748$	0	0
$749$	10697.6i	0.521869i
$750$	0	0
$751$	−35412.2	−1.72065	−0.860326	−	0.509744i	$-0.829740\pi$
$751$	−35412.2	−1.72065	−0.860326	+	0.509744i	$0.829740\pi$
$752$	0	0
$753$	−2061.72	−0.0997786
$754$	0	0
$755$	29482.7i	1.42117i
$756$	0	0
$757$	29416.3i	1.41235i	0.708035	+	0.706177i	$0.249582\pi$
$757$	29416.3i	1.41235i	−0.708035	+	0.706177i	$0.750418\pi$
$758$	0	0
$759$	−947.051	−0.0452909
$760$	0	0
$761$	848.266	0.0404069	0.0202034	−	0.999796i	$-0.493569\pi$
$761$	848.266	0.0404069	0.0202034	+	0.999796i	$0.493569\pi$
$762$	0	0
$763$	2052.34i	0.0973781i
$764$	0	0
$765$	− 3565.94i	− 0.168532i
$766$	0	0
$767$	−507.643	−0.0238982
$768$	0	0
$769$	3380.24	0.158510	0.0792552	−	0.996854i	$-0.474746\pi$
$769$	3380.24	0.158510	0.0792552	+	0.996854i	$0.474746\pi$
$770$	0	0
$771$	− 9323.82i	− 0.435524i
$772$	0	0
$773$	− 20128.7i	− 0.936582i	−0.883574	−	0.468291i	$-0.844870\pi$
$773$	− 20128.7i	− 0.936582i	0.883574	−	0.468291i	$-0.155130\pi$
$774$	0	0
$775$	8894.04	0.412236
$776$	0	0
$777$	−719.310	−0.0332112
$778$	0	0
$779$	29256.8i	1.34562i
$780$	0	0
$781$	12317.0i	0.564324i
$782$	0	0
$783$	−256.283	−0.0116971
$784$	0	0
$785$	61241.9	2.78448
$786$	0	0
$787$	− 31312.6i	− 1.41826i	−0.705076	−	0.709131i	$-0.749087\pi$
$787$	− 31312.6i	− 1.41826i	0.705076	−	0.709131i	$-0.250913\pi$
$788$	0	0
$789$	19498.3i	0.879795i
$790$	0	0
$791$	1489.55	0.0669562
$792$	0	0
$793$	275.388	0.0123320
$794$	0	0
$795$	4990.38i	0.222630i
$796$	0	0
$797$	− 19940.7i	− 0.886244i	−0.896461	−	0.443122i	$-0.853871\pi$
$797$	− 19940.7i	− 0.886244i	0.896461	−	0.443122i	$-0.146129\pi$
$798$	0	0
$799$	3222.68	0.142691
$800$	0	0
$801$	−5.82184	−0.000256810 0
$802$	0	0
$803$	− 9431.01i	− 0.414462i
$804$	0	0
$805$	− 2582.23i	− 0.113058i
$806$	0	0
$807$	17362.6	0.757364
$808$	0	0
$809$	−24087.6	−1.04681	−0.523407	−	0.852083i	$-0.675339\pi$
$809$	−24087.6	−1.04681	−0.523407	+	0.852083i	$0.675339\pi$
$810$	0	0
$811$	5952.49i	0.257731i	0.991662	+	0.128866i	$0.0411336\pi$
$811$	5952.49i	0.257731i	−0.991662	+	0.128866i	$0.958866\pi$
$812$	0	0
$813$	− 3816.13i	− 0.164622i
$814$	0	0
$815$	−39868.9	−1.71356
$816$	0	0
$817$	−10215.5	−0.437446
$818$	0	0
$819$	488.070i	0.0208236i
$820$	0	0
$821$	− 31555.8i	− 1.34142i	−0.741720	−	0.670710i	$-0.765990\pi$
$821$	− 31555.8i	− 1.34142i	0.741720	−	0.670710i	$-0.234010\pi$
$822$	0	0
$823$	39793.7	1.68544	0.842722	−	0.538349i	$-0.180952\pi$
$823$	39793.7	1.68544	0.842722	+	0.538349i	$0.180952\pi$
$824$	0	0
$825$	4770.71	0.201327
$826$	0	0
$827$	− 775.434i	− 0.0326052i	−0.999867	−	0.0163026i	$-0.994810\pi$
$827$	− 775.434i	− 0.0326052i	0.999867	−	0.0163026i	$-0.00518950\pi$
$828$	0	0
$829$	− 37882.7i	− 1.58712i	−0.608495	−	0.793558i	$-0.708226\pi$
$829$	− 37882.7i	− 1.58712i	0.608495	−	0.793558i	$-0.291774\pi$
$830$	0	0
$831$	−12858.7	−0.536780
$832$	0	0
$833$	−1242.97	−0.0517002
$834$	0	0
$835$	25360.9i	1.05108i
$836$	0	0
$837$	− 2018.48i	− 0.0833560i
$838$	0	0
$839$	−455.725	−0.0187525	−0.00937627	−	0.999956i	$-0.502985\pi$
$839$	−455.725	−0.0187525	−0.00937627	+	0.999956i	$0.502985\pi$
$840$	0	0
$841$	24298.9	0.996306
$842$	0	0
$843$	− 12830.6i	− 0.524211i
$844$	0	0
$845$	− 33378.7i	− 1.35889i
$846$	0	0
$847$	8066.32	0.327228
$848$	0	0
$849$	−7634.79	−0.308628
$850$	0	0
$851$	− 808.957i	− 0.0325860i
$852$	0	0
$853$	33752.9i	1.35484i	0.735597	+	0.677420i	$0.236902\pi$
$853$	33752.9i	1.35484i	−0.735597	+	0.677420i	$0.763098\pi$
$854$	0	0
$855$	10117.3	0.404682
$856$	0	0
$857$	29740.1	1.18542	0.592708	−	0.805417i	$-0.298059\pi$
$857$	29740.1	1.18542	0.592708	+	0.805417i	$0.298059\pi$
$858$	0	0
$859$	19320.9i	0.767430i	0.923452	+	0.383715i	$0.125355\pi$
$859$	19320.9i	0.767430i	−0.923452	+	0.383715i	$0.874645\pi$
$860$	0	0
$861$	8536.79i	0.337901i
$862$	0	0
$863$	−22903.7	−0.903418	−0.451709	−	0.892165i	$-0.649185\pi$
$863$	−22903.7	−0.903418	−0.451709	+	0.892165i	$0.649185\pi$
$864$	0	0
$865$	30824.9	1.21165
$866$	0	0
$867$	12808.6i	0.501733i
$868$	0	0
$869$	3934.38i	0.153584i
$870$	0	0
$871$	−4694.93	−0.182642
$872$	0	0
$873$	−12185.1	−0.472398
$874$	0	0
$875$	− 659.288i	− 0.0254720i
$876$	0	0
$877$	28372.3i	1.09243i	0.837644	+	0.546217i	$0.183933\pi$
$877$	28372.3i	1.09243i	−0.837644	+	0.546217i	$0.816067\pi$
$878$	0	0
$879$	−11090.0	−0.425548
$880$	0	0
$881$	−44790.3	−1.71285	−0.856427	−	0.516269i	$-0.827321\pi$
$881$	−44790.3	−1.71285	−0.856427	+	0.516269i	$0.827321\pi$
$882$	0	0
$883$	− 50005.6i	− 1.90580i	−0.303281	−	0.952901i	$-0.598082\pi$
$883$	− 50005.6i	− 1.90580i	0.303281	−	0.952901i	$-0.401918\pi$
$884$	0	0
$885$	3070.48i	0.116625i
$886$	0	0
$887$	−5305.70	−0.200843	−0.100422	−	0.994945i	$-0.532019\pi$
$887$	−5305.70	−0.200843	−0.100422	+	0.994945i	$0.532019\pi$
$888$	0	0
$889$	−13203.6	−0.498128
$890$	0	0
$891$	− 1082.70i	− 0.0407092i
$892$	0	0
$893$	9143.35i	0.342632i
$894$	0	0
$895$	−12729.8	−0.475432
$896$	0	0
$897$	−548.897	−0.0204316
$898$	0	0
$899$	− 709.604i	− 0.0263255i
$900$	0	0
$901$	2701.52i	0.0998897i
$902$	0	0
$903$	−2980.74	−0.109848
$904$	0	0
$905$	20865.1	0.766387
$906$	0	0
$907$	18224.3i	0.667175i	0.942719	+	0.333587i	$0.108259\pi$
$907$	18224.3i	0.667175i	−0.942719	+	0.333587i	$0.891741\pi$
$908$	0	0
$909$	− 10318.5i	− 0.376507i
$910$	0	0
$911$	7287.74	0.265042	0.132521	−	0.991180i	$-0.457693\pi$
$911$	7287.74	0.265042	0.132521	+	0.991180i	$0.457693\pi$
$912$	0	0
$913$	−3899.60	−0.141356
$914$	0	0
$915$	− 1665.69i	− 0.0601813i
$916$	0	0
$917$	7753.76i	0.279228i
$918$	0	0
$919$	−40520.1	−1.45444	−0.727222	−	0.686403i	$-0.759189\pi$
$919$	−40520.1	−1.45444	−0.727222	+	0.686403i	$0.759189\pi$
$920$	0	0
$921$	8559.39	0.306234
$922$	0	0
$923$	7138.76i	0.254578i
$924$	0	0
$925$	4075.07i	0.144851i
$926$	0	0
$927$	−2749.98	−0.0974340
$928$	0	0
$929$	−13417.8	−0.473869	−0.236935	−	0.971526i	$-0.576143\pi$
$929$	−13417.8	−0.473869	−0.236935	+	0.971526i	$0.576143\pi$
$930$	0	0
$931$	− 3526.54i	− 0.124143i
$932$	0	0
$933$	− 12887.3i	− 0.452211i
$934$	0	0
$935$	5296.09	0.185241
$936$	0	0
$937$	−29216.3	−1.01863	−0.509314	−	0.860581i	$-0.670101\pi$
$937$	−29216.3	−1.01863	−0.509314	+	0.860581i	$0.670101\pi$
$938$	0	0
$939$	− 10388.3i	− 0.361032i
$940$	0	0
$941$	− 9792.02i	− 0.339225i	−0.985511	−	0.169612i	$-0.945748\pi$
$941$	− 9792.02i	− 0.339225i	0.985511	−	0.169612i	$-0.0542516\pi$
$942$	0	0
$943$	−9600.72	−0.331540
$944$	0	0
$945$	2952.09	0.101621
$946$	0	0
$947$	− 17608.5i	− 0.604222i	−0.953273	−	0.302111i	$-0.902309\pi$
$947$	− 17608.5i	− 0.604222i	0.953273	−	0.302111i	$-0.0976913\pi$
$948$	0	0
$949$	− 5466.08i	− 0.186972i
$950$	0	0
$951$	−37.8085	−0.00128919
$952$	0	0
$953$	2510.99	0.0853503	0.0426751	−	0.999089i	$-0.486412\pi$
$953$	2510.99	0.0853503	0.0426751	+	0.999089i	$0.486412\pi$
$954$	0	0
$955$	1747.72i	0.0592198i
$956$	0	0
$957$	− 380.627i	− 0.0128568i
$958$	0	0
$959$	−5391.82	−0.181555
$960$	0	0
$961$	−24202.2	−0.812398
$962$	0	0
$963$	− 13754.0i	− 0.460246i
$964$	0	0
$965$	23667.2i	0.789507i
$966$	0	0
$967$	29783.0	0.990442	0.495221	−	0.868767i	$-0.335087\pi$
$967$	29783.0	0.990442	0.495221	+	0.868767i	$0.335087\pi$
$968$	0	0
$969$	5476.93	0.181573
$970$	0	0
$971$	36466.0i	1.20520i	0.798043	+	0.602600i	$0.205869\pi$
$971$	36466.0i	1.20520i	−0.798043	+	0.602600i	$0.794131\pi$
$972$	0	0
$973$	5832.40i	0.192167i
$974$	0	0
$975$	2765.03	0.0908225
$976$	0	0
$977$	−35245.9	−1.15416	−0.577082	−	0.816687i	$-0.695809\pi$
$977$	−35245.9	−1.15416	−0.577082	+	0.816687i	$0.695809\pi$
$978$	0	0
$979$	− 8.64652i	0 0.000282272i
$980$	0	0
$981$	− 2638.72i	− 0.0858794i
$982$	0	0
$983$	−35994.4	−1.16790	−0.583949	−	0.811790i	$-0.698493\pi$
$983$	−35994.4	−1.16790	−0.583949	+	0.811790i	$0.698493\pi$
$984$	0	0
$985$	−32798.5	−1.06096
$986$	0	0
$987$	2667.92i	0.0860392i
$988$	0	0
$989$	− 3352.23i	− 0.107780i
$990$	0	0
$991$	−20472.2	−0.656227	−0.328114	−	0.944638i	$-0.606413\pi$
$991$	−20472.2	−0.656227	−0.328114	+	0.944638i	$0.606413\pi$
$992$	0	0
$993$	12894.9	0.412092
$994$	0	0
$995$	− 71346.2i	− 2.27319i
$996$	0	0
$997$	− 45849.5i	− 1.45644i	−0.685344	−	0.728219i	$-0.740348\pi$
$997$	− 45849.5i	− 1.45644i	0.685344	−	0.728219i	$-0.259652\pi$
$998$	0	0
$999$	924.827	0.0292895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.d.673.1	yes	6
4.3	odd	2		1344.4.c.a.673.4	yes	6
8.3	odd	2		1344.4.c.a.673.3	✓	6
8.5	even	2	inner	1344.4.c.d.673.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.a.673.3	✓	6	8.3	odd	2
1344.4.c.a.673.4	yes	6	4.3	odd	2
1344.4.c.d.673.1	yes	6	1.1	even	1	trivial
1344.4.c.d.673.6	yes	6	8.5	even	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -15.6195i q^{5} +7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -15.6195i q^{5} +7.00000 q^{7} -9.00000 q^{9} -13.3667i q^{11} -7.74714i q^{13} -46.8586 q^{15} -25.3667 q^{17} -71.9701i q^{19} -21.0000i q^{21} +23.6172 q^{23} -118.970 q^{25} +27.0000i q^{27} +9.49195i q^{29} -74.7586 q^{31} -40.1000 q^{33} -109.337i q^{35} -34.2529i q^{37} -23.2414 q^{39} -406.514 q^{41} -141.940i q^{43} +140.576i q^{45} -127.044 q^{47} +49.0000 q^{49} +76.1000i q^{51} -106.499i q^{53} -208.781 q^{55} -215.910 q^{57} -65.5265i q^{59} +35.5470i q^{61} -63.0000 q^{63} -121.007 q^{65} -606.021i q^{67} -70.8516i q^{69} -921.470 q^{71} +705.561 q^{73} +356.910i q^{75} -93.5668i q^{77} -294.343 q^{79} +81.0000 q^{81} -291.740i q^{83} +396.216i q^{85} +28.4758 q^{87} +0.646871 q^{89} -54.2300i q^{91} +224.276i q^{93} -1124.14 q^{95} +1353.90 q^{97} +120.300i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 42 q^{7} - 54 q^{9}+O(q^{10}) \) 6 * q + 42 * q^7 - 54 * q^9	\( 6 q + 42 q^{7} - 54 q^{9} + 24 q^{15} - 16 q^{17} - 60 q^{23} - 138 q^{25} - 552 q^{31} + 168 q^{33} - 36 q^{39} - 272 q^{41} - 1576 q^{47} + 294 q^{49} - 1632 q^{55} + 432 q^{57} - 378 q^{63} - 664 q^{65} - 2548 q^{71} + 444 q^{73} - 3528 q^{79} + 486 q^{81} - 336 q^{87} + 16 q^{89} - 4776 q^{95} + 1548 q^{97}+O(q^{100}) \) 6 * q + 42 * q^7 - 54 * q^9 + 24 * q^15 - 16 * q^17 - 60 * q^23 - 138 * q^25 - 552 * q^31 + 168 * q^33 - 36 * q^39 - 272 * q^41 - 1576 * q^47 + 294 * q^49 - 1632 * q^55 + 432 * q^57 - 378 * q^63 - 664 * q^65 - 2548 * q^71 + 444 * q^73 - 3528 * q^79 + 486 * q^81 - 336 * q^87 + 16 * q^89 - 4776 * q^95 + 1548 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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