LMFDB - Embedded newform 336.3.m.b.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,3,Mod(127,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("336.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	336.m (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:	$9.15533688251$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$1.39564 + 0.228425i$ of defining polynomial
Character	$\chi$	$=$	336.127
Dual form			336.3.m.b.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} -2.00000 q^{5} -2.64575i q^{7} -3.00000 q^{9} -3.65480i q^{11} -16.3303 q^{13} +3.46410i q^{15} -4.33030 q^{17} +14.2378i q^{19} -4.58258 q^{21} -31.3676i q^{23} -21.0000 q^{25} +5.19615i q^{27} -50.6606 q^{29} +27.7128i q^{31} -6.33030 q^{33} +5.29150i q^{35} -10.6606 q^{37} +28.2849i q^{39} +8.33030 q^{41} -6.54680i q^{43} +6.00000 q^{45} -35.7852i q^{47} -7.00000 q^{49} +7.50030i q^{51} -26.6606 q^{53} +7.30960i q^{55} +24.6606 q^{57} -0.381401i q^{59} -24.3303 q^{61} +7.93725i q^{63} +32.6606 q^{65} +55.4256i q^{67} -54.3303 q^{69} -94.8656i q^{71} +94.6606 q^{73} +36.3731i q^{75} -9.66970 q^{77} -96.9948i q^{79} +9.00000 q^{81} -105.449i q^{83} +8.66061 q^{85} +87.7467i q^{87} +92.9909 q^{89} +43.2059i q^{91} +48.0000 q^{93} -28.4756i q^{95} +51.3212 q^{97} +10.9644i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{5} - 12 q^{9} + 8 q^{13} + 56 q^{17} - 84 q^{25} - 56 q^{29} + 48 q^{33} + 104 q^{37} - 40 q^{41} + 24 q^{45} - 28 q^{49} + 40 q^{53} - 48 q^{57} - 24 q^{61} - 16 q^{65} - 144 q^{69} + 232 q^{73}+ \cdots - 88 q^{97}+O(q^{100}) $ 4 * q - 8 * q^5 - 12 * q^9 + 8 * q^13 + 56 * q^17 - 84 * q^25 - 56 * q^29 + 48 * q^33 + 104 * q^37 - 40 * q^41 + 24 * q^45 - 28 * q^49 + 40 * q^53 - 48 * q^57 - 24 * q^61 - 16 * q^65 - 144 * q^69 + 232 * q^73 - 112 * q^77 + 36 * q^81 - 112 * q^85 + 152 * q^89 + 192 * q^93 - 88 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	−2.00000	−0.400000	−0.200000	−	0.979796i	$-0.564094\pi$
$5$	−2.00000	−0.400000	−0.200000	+	0.979796i	$0.564094\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 3.65480i	− 0.332255i	−0.986104	−	0.166127i	$-0.946874\pi$
$11$	− 3.65480i	− 0.332255i	0.986104	−	0.166127i	$-0.0531263\pi$
$12$	0	0
$13$	−16.3303	−1.25618	−0.628089	−	0.778142i	$-0.716162\pi$
$13$	−16.3303	−1.25618	−0.628089	+	0.778142i	$0.716162\pi$
$14$	0	0
$15$	3.46410i	0.230940i
$16$	0	0
$17$	−4.33030	−0.254724	−0.127362	−	0.991856i	$-0.540651\pi$
$17$	−4.33030	−0.254724	−0.127362	+	0.991856i	$0.540651\pi$
$18$	0	0
$19$	14.2378i	0.749358i	0.927155	+	0.374679i	$0.122247\pi$
$19$	14.2378i	0.749358i	−0.927155	+	0.374679i	$0.877753\pi$
$20$	0	0
$21$	−4.58258	−0.218218
$22$	0	0
$23$	− 31.3676i	− 1.36381i	−0.731441	−	0.681905i	$-0.761152\pi$
$23$	− 31.3676i	− 1.36381i	0.731441	−	0.681905i	$-0.238848\pi$
$24$	0	0
$25$	−21.0000	−0.840000
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−50.6606	−1.74692	−0.873459	−	0.486898i	$-0.838128\pi$
$29$	−50.6606	−1.74692	−0.873459	+	0.486898i	$0.838128\pi$
$30$	0	0
$31$	27.7128i	0.893962i	0.894544	+	0.446981i	$0.147501\pi$
$31$	27.7128i	0.893962i	−0.894544	+	0.446981i	$0.852499\pi$
$32$	0	0
$33$	−6.33030	−0.191827
$34$	0	0
$35$	5.29150i	0.151186i
$36$	0	0
$37$	−10.6606	−0.288124	−0.144062	−	0.989569i	$-0.546017\pi$
$37$	−10.6606	−0.288124	−0.144062	+	0.989569i	$0.546017\pi$
$38$	0	0
$39$	28.2849i	0.725254i
$40$	0	0
$41$	8.33030	0.203178	0.101589	−	0.994826i	$-0.467607\pi$
$41$	8.33030	0.203178	0.101589	+	0.994826i	$0.467607\pi$
$42$	0	0
$43$	− 6.54680i	− 0.152251i	−0.997098	−	0.0761256i	$-0.975745\pi$
$43$	− 6.54680i	− 0.152251i	0.997098	−	0.0761256i	$-0.0242550\pi$
$44$	0	0
$45$	6.00000	0.133333
$46$	0	0
$47$	− 35.7852i	− 0.761388i	−0.924701	−	0.380694i	$-0.875685\pi$
$47$	− 35.7852i	− 0.761388i	0.924701	−	0.380694i	$-0.124315\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	7.50030i	0.147065i
$52$	0	0
$53$	−26.6606	−0.503030	−0.251515	−	0.967853i	$-0.580929\pi$
$53$	−26.6606	−0.503030	−0.251515	+	0.967853i	$0.580929\pi$
$54$	0	0
$55$	7.30960i	0.132902i
$56$	0	0
$57$	24.6606	0.432642
$58$	0	0
$59$	− 0.381401i	− 0.00646442i	−0.999995	−	0.00323221i	$-0.998971\pi$
$59$	− 0.381401i	− 0.00646442i	0.999995	−	0.00323221i	$-0.00102885\pi$
$60$	0	0
$61$	−24.3303	−0.398857	−0.199429	−	0.979912i	$-0.563909\pi$
$61$	−24.3303	−0.398857	−0.199429	+	0.979912i	$0.563909\pi$
$62$	0	0
$63$	7.93725i	0.125988i
$64$	0	0
$65$	32.6606	0.502471
$66$	0	0
$67$	55.4256i	0.827248i	0.910448	+	0.413624i	$0.135737\pi$
$67$	55.4256i	0.827248i	−0.910448	+	0.413624i	$0.864263\pi$
$68$	0	0
$69$	−54.3303	−0.787396
$70$	0	0
$71$	− 94.8656i	− 1.33614i	−0.744100	−	0.668068i	$-0.767122\pi$
$71$	− 94.8656i	− 1.33614i	0.744100	−	0.668068i	$-0.232878\pi$
$72$	0	0
$73$	94.6606	1.29672	0.648360	−	0.761334i	$-0.275455\pi$
$73$	94.6606	1.29672	0.648360	+	0.761334i	$0.275455\pi$
$74$	0	0
$75$	36.3731i	0.484974i
$76$	0	0
$77$	−9.66970	−0.125580
$78$	0	0
$79$	− 96.9948i	− 1.22778i	−0.789390	−	0.613891i	$-0.789603\pi$
$79$	− 96.9948i	− 1.22778i	0.789390	−	0.613891i	$-0.210397\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	− 105.449i	− 1.27047i	−0.772321	−	0.635233i	$-0.780904\pi$
$83$	− 105.449i	− 1.27047i	0.772321	−	0.635233i	$-0.219096\pi$
$84$	0	0
$85$	8.66061	0.101889
$86$	0	0
$87$	87.7467i	1.00858i
$88$	0	0
$89$	92.9909	1.04484	0.522421	−	0.852688i	$-0.325029\pi$
$89$	92.9909	1.04484	0.522421	+	0.852688i	$0.325029\pi$
$90$	0	0
$91$	43.2059i	0.474790i
$92$	0	0
$93$	48.0000	0.516129
$94$	0	0
$95$	− 28.4756i	− 0.299743i
$96$	0	0
$97$	51.3212	0.529085	0.264542	−	0.964374i	$-0.414779\pi$
$97$	51.3212	0.529085	0.264542	+	0.964374i	$0.414779\pi$
$98$	0	0
$99$	10.9644i	0.110752i
$100$	0	0
$101$	74.6606	0.739214	0.369607	−	0.929188i	$-0.379492\pi$
$101$	74.6606	0.739214	0.369607	+	0.929188i	$0.379492\pi$
$102$	0	0
$103$	140.090i	1.36009i	0.733169	+	0.680047i	$0.238041\pi$
$103$	140.090i	1.36009i	−0.733169	+	0.680047i	$0.761959\pi$
$104$	0	0
$105$	9.16515	0.0872872
$106$	0	0
$107$	− 198.407i	− 1.85427i	−0.374723	−	0.927137i	$-0.622262\pi$
$107$	− 198.407i	− 1.85427i	0.374723	−	0.927137i	$-0.377738\pi$
$108$	0	0
$109$	−33.3394	−0.305866	−0.152933	−	0.988237i	$-0.548872\pi$
$109$	−33.3394	−0.305866	−0.152933	+	0.988237i	$0.548872\pi$
$110$	0	0
$111$	18.4647i	0.166349i
$112$	0	0
$113$	11.3212	0.100188	0.0500939	−	0.998745i	$-0.484048\pi$
$113$	11.3212	0.100188	0.0500939	+	0.998745i	$0.484048\pi$
$114$	0	0
$115$	62.7352i	0.545524i
$116$	0	0
$117$	48.9909	0.418726
$118$	0	0
$119$	11.4569i	0.0962765i
$120$	0	0
$121$	107.642	0.889607
$122$	0	0
$123$	− 14.4285i	− 0.117305i
$124$	0	0
$125$	92.0000	0.736000
$126$	0	0
$127$	84.6640i	0.666646i	0.942813	+	0.333323i	$0.108170\pi$
$127$	84.6640i	0.666646i	−0.942813	+	0.333323i	$0.891830\pi$
$128$	0	0
$129$	−11.3394	−0.0879023
$130$	0	0
$131$	160.111i	1.22223i	0.791544	+	0.611113i	$0.209278\pi$
$131$	160.111i	1.22223i	−0.791544	+	0.611113i	$0.790722\pi$
$132$	0	0
$133$	37.6697	0.283231
$134$	0	0
$135$	− 10.3923i	− 0.0769800i
$136$	0	0
$137$	193.303	1.41097	0.705486	−	0.708724i	$-0.250729\pi$
$137$	193.303	1.41097	0.705486	+	0.708724i	$0.250729\pi$
$138$	0	0
$139$	− 60.8282i	− 0.437613i	−0.975768	−	0.218807i	$-0.929784\pi$
$139$	− 60.8282i	− 0.437613i	0.975768	−	0.218807i	$-0.0702164\pi$
$140$	0	0
$141$	−61.9818	−0.439587
$142$	0	0
$143$	59.6840i	0.417371i
$144$	0	0
$145$	101.321	0.698767
$146$	0	0
$147$	12.1244i	0.0824786i
$148$	0	0
$149$	−131.982	−0.885784	−0.442892	−	0.896575i	$-0.646047\pi$
$149$	−131.982	−0.885784	−0.442892	+	0.896575i	$0.646047\pi$
$150$	0	0
$151$	62.7352i	0.415465i	0.978186	+	0.207733i	$0.0666084\pi$
$151$	62.7352i	0.415465i	−0.978186	+	0.207733i	$0.933392\pi$
$152$	0	0
$153$	12.9909	0.0849079
$154$	0	0
$155$	− 55.4256i	− 0.357585i
$156$	0	0
$157$	−86.3121	−0.549759	−0.274879	−	0.961479i	$-0.588638\pi$
$157$	−86.3121	−0.549759	−0.274879	+	0.961479i	$0.588638\pi$
$158$	0	0
$159$	46.1775i	0.290425i
$160$	0	0
$161$	−82.9909	−0.515471
$162$	0	0
$163$	− 165.514i	− 1.01542i	−0.861527	−	0.507712i	$-0.830491\pi$
$163$	− 165.514i	− 1.01542i	0.861527	−	0.507712i	$-0.169509\pi$
$164$	0	0
$165$	12.6606	0.0767309
$166$	0	0
$167$	− 270.581i	− 1.62025i	−0.586259	−	0.810124i	$-0.699400\pi$
$167$	− 270.581i	− 1.62025i	0.586259	−	0.810124i	$-0.300600\pi$
$168$	0	0
$169$	97.6788	0.577981
$170$	0	0
$171$	− 42.7134i	− 0.249786i
$172$	0	0
$173$	−332.642	−1.92279	−0.961394	−	0.275175i	$-0.911264\pi$
$173$	−332.642	−1.92279	−0.961394	+	0.275175i	$0.911264\pi$
$174$	0	0
$175$	55.5608i	0.317490i
$176$	0	0
$177$	−0.660606	−0.00373223
$178$	0	0
$179$	170.694i	0.953600i	0.879012	+	0.476800i	$0.158203\pi$
$179$	170.694i	0.953600i	−0.879012	+	0.476800i	$0.841797\pi$
$180$	0	0
$181$	−169.652	−0.937301	−0.468651	−	0.883384i	$-0.655260\pi$
$181$	−169.652	−0.937301	−0.468651	+	0.883384i	$0.655260\pi$
$182$	0	0
$183$	42.1413i	0.230280i
$184$	0	0
$185$	21.3212	0.115250
$186$	0	0
$187$	15.8264i	0.0846332i
$188$	0	0
$189$	13.7477	0.0727393
$190$	0	0
$191$	− 301.949i	− 1.58088i	−0.612537	−	0.790442i	$-0.709851\pi$
$191$	− 301.949i	− 1.58088i	0.612537	−	0.790442i	$-0.290149\pi$
$192$	0	0
$193$	−224.642	−1.16395	−0.581975	−	0.813207i	$-0.697720\pi$
$193$	−224.642	−1.16395	−0.581975	+	0.813207i	$0.697720\pi$
$194$	0	0
$195$	− 56.5698i	− 0.290102i
$196$	0	0
$197$	7.98182	0.0405168	0.0202584	−	0.999795i	$-0.493551\pi$
$197$	7.98182	0.0405168	0.0202584	+	0.999795i	$0.493551\pi$
$198$	0	0
$199$	369.102i	1.85478i	0.374093	+	0.927391i	$0.377954\pi$
$199$	369.102i	1.85478i	−0.374093	+	0.927391i	$0.622046\pi$
$200$	0	0
$201$	96.0000	0.477612
$202$	0	0
$203$	134.035i	0.660273i
$204$	0	0
$205$	−16.6606	−0.0812712
$206$	0	0
$207$	94.1028i	0.454603i
$208$	0	0
$209$	52.0364	0.248978
$210$	0	0
$211$	− 344.440i	− 1.63242i	−0.577757	−	0.816209i	$-0.696072\pi$
$211$	− 344.440i	− 1.63242i	0.577757	−	0.816209i	$-0.303928\pi$
$212$	0	0
$213$	−164.312	−0.771418
$214$	0	0
$215$	13.0936i	0.0609005i
$216$	0	0
$217$	73.3212	0.337886
$218$	0	0
$219$	− 163.957i	− 0.748662i
$220$	0	0
$221$	70.7152	0.319978
$222$	0	0
$223$	− 386.009i	− 1.73098i	−0.500923	−	0.865492i	$-0.667006\pi$
$223$	− 386.009i	− 1.73098i	0.500923	−	0.865492i	$-0.332994\pi$
$224$	0	0
$225$	63.0000	0.280000
$226$	0	0
$227$	268.230i	1.18163i	0.806807	+	0.590815i	$0.201194\pi$
$227$	268.230i	1.18163i	−0.806807	+	0.590815i	$0.798806\pi$
$228$	0	0
$229$	−89.6515	−0.391491	−0.195746	−	0.980655i	$-0.562713\pi$
$229$	−89.6515	−0.391491	−0.195746	+	0.980655i	$0.562713\pi$
$230$	0	0
$231$	16.7484i	0.0725039i
$232$	0	0
$233$	−298.661	−1.28181	−0.640903	−	0.767622i	$-0.721440\pi$
$233$	−298.661	−1.28181	−0.640903	+	0.767622i	$0.721440\pi$
$234$	0	0
$235$	71.5704i	0.304555i
$236$	0	0
$237$	−168.000	−0.708861
$238$	0	0
$239$	151.054i	0.632025i	0.948755	+	0.316013i	$0.102344\pi$
$239$	151.054i	0.632025i	−0.948755	+	0.316013i	$0.897656\pi$
$240$	0	0
$241$	−326.000	−1.35270	−0.676349	−	0.736582i	$-0.736439\pi$
$241$	−326.000	−1.35270	−0.676349	+	0.736582i	$0.736439\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	14.0000	0.0571429
$246$	0	0
$247$	− 232.508i	− 0.941327i
$248$	0	0
$249$	−182.642	−0.733504
$250$	0	0
$251$	379.844i	1.51332i	0.653807	+	0.756661i	$0.273171\pi$
$251$	379.844i	1.51332i	−0.653807	+	0.756661i	$0.726829\pi$
$252$	0	0
$253$	−114.642	−0.453132
$254$	0	0
$255$	− 15.0006i	− 0.0588259i
$256$	0	0
$257$	−53.6515	−0.208761	−0.104380	−	0.994537i	$-0.533286\pi$
$257$	−53.6515	−0.208761	−0.104380	+	0.994537i	$0.533286\pi$
$258$	0	0
$259$	28.2053i	0.108901i
$260$	0	0
$261$	151.982	0.582306
$262$	0	0
$263$	34.8632i	0.132560i	0.997801	+	0.0662799i	$0.0211130\pi$
$263$	34.8632i	0.132560i	−0.997801	+	0.0662799i	$0.978887\pi$
$264$	0	0
$265$	53.3212	0.201212
$266$	0	0
$267$	− 161.065i	− 0.603240i
$268$	0	0
$269$	−191.982	−0.713687	−0.356844	−	0.934164i	$-0.616147\pi$
$269$	−191.982	−0.713687	−0.356844	+	0.934164i	$0.616147\pi$
$270$	0	0
$271$	111.614i	0.411860i	0.978567	+	0.205930i	$0.0660219\pi$
$271$	111.614i	0.411860i	−0.978567	+	0.205930i	$0.933978\pi$
$272$	0	0
$273$	74.8348	0.274120
$274$	0	0
$275$	76.7508i	0.279094i
$276$	0	0
$277$	195.321	0.705131	0.352565	−	0.935787i	$-0.385309\pi$
$277$	195.321	0.705131	0.352565	+	0.935787i	$0.385309\pi$
$278$	0	0
$279$	− 83.1384i	− 0.297987i
$280$	0	0
$281$	−253.303	−0.901434	−0.450717	−	0.892667i	$-0.648832\pi$
$281$	−253.303	−0.901434	−0.450717	+	0.892667i	$0.648832\pi$
$282$	0	0
$283$	− 141.997i	− 0.501755i	−0.968019	−	0.250878i	$-0.919281\pi$
$283$	− 141.997i	− 0.501755i	0.968019	−	0.250878i	$-0.0807191\pi$
$284$	0	0
$285$	−49.3212	−0.173057
$286$	0	0
$287$	− 22.0399i	− 0.0767941i
$288$	0	0
$289$	−270.248	−0.935116
$290$	0	0
$291$	− 88.8909i	− 0.305467i
$292$	0	0
$293$	518.624	1.77005	0.885024	−	0.465545i	$-0.154142\pi$
$293$	518.624	1.77005	0.885024	+	0.465545i	$0.154142\pi$
$294$	0	0
$295$	0.762802i	0.00258577i
$296$	0	0
$297$	18.9909	0.0639425
$298$	0	0
$299$	512.243i	1.71319i
$300$	0	0
$301$	−17.3212	−0.0575456
$302$	0	0
$303$	− 129.316i	− 0.426785i
$304$	0	0
$305$	48.6606	0.159543
$306$	0	0
$307$	139.708i	0.455076i	0.973769	+	0.227538i	$0.0730676\pi$
$307$	139.708i	0.455076i	−0.973769	+	0.227538i	$0.926932\pi$
$308$	0	0
$309$	242.642	0.785251
$310$	0	0
$311$	299.057i	0.961598i	0.876831	+	0.480799i	$0.159653\pi$
$311$	299.057i	0.961598i	−0.876831	+	0.480799i	$0.840347\pi$
$312$	0	0
$313$	−66.6606	−0.212973	−0.106487	−	0.994314i	$-0.533960\pi$
$313$	−66.6606	−0.212973	−0.106487	+	0.994314i	$0.533960\pi$
$314$	0	0
$315$	− 15.8745i	− 0.0503953i
$316$	0	0
$317$	−85.3758	−0.269324	−0.134662	−	0.990892i	$-0.542995\pi$
$317$	−85.3758	−0.269324	−0.134662	+	0.990892i	$0.542995\pi$
$318$	0	0
$319$	185.154i	0.580422i
$320$	0	0
$321$	−343.652	−1.07057
$322$	0	0
$323$	− 61.6540i	− 0.190879i
$324$	0	0
$325$	342.936	1.05519
$326$	0	0
$327$	57.7455i	0.176592i
$328$	0	0
$329$	−94.6788	−0.287777
$330$	0	0
$331$	− 612.289i	− 1.84981i	−0.380192	−	0.924907i	$-0.624142\pi$
$331$	− 612.289i	− 1.84981i	0.380192	−	0.924907i	$-0.375858\pi$
$332$	0	0
$333$	31.9818	0.0960415
$334$	0	0
$335$	− 110.851i	− 0.330899i
$336$	0	0
$337$	445.964	1.32333	0.661667	−	0.749798i	$-0.269849\pi$
$337$	445.964	1.32333	0.661667	+	0.749798i	$0.269849\pi$
$338$	0	0
$339$	− 19.6089i	− 0.0578434i
$340$	0	0
$341$	101.285	0.297023
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	108.661	0.314958
$346$	0	0
$347$	259.173i	0.746895i	0.927651	+	0.373447i	$0.121824\pi$
$347$	259.173i	0.746895i	−0.927651	+	0.373447i	$0.878176\pi$
$348$	0	0
$349$	−332.991	−0.954129	−0.477064	−	0.878868i	$-0.658299\pi$
$349$	−332.991	−0.954129	−0.477064	+	0.878868i	$0.658299\pi$
$350$	0	0
$351$	− 84.8547i	− 0.241751i
$352$	0	0
$353$	75.0455	0.212593	0.106297	−	0.994334i	$-0.466101\pi$
$353$	75.0455	0.212593	0.106297	+	0.994334i	$0.466101\pi$
$354$	0	0
$355$	189.731i	0.534454i
$356$	0	0
$357$	19.8439	0.0555853
$358$	0	0
$359$	− 345.044i	− 0.961125i	−0.876961	−	0.480562i	$-0.840433\pi$
$359$	− 345.044i	− 0.961125i	0.876961	−	0.480562i	$-0.159567\pi$
$360$	0	0
$361$	158.285	0.438462
$362$	0	0
$363$	− 186.442i	− 0.513615i
$364$	0	0
$365$	−189.321	−0.518688
$366$	0	0
$367$	− 177.400i	− 0.483380i	−0.970354	−	0.241690i	$-0.922298\pi$
$367$	− 177.400i	− 0.483380i	0.970354	−	0.241690i	$-0.0777017\pi$
$368$	0	0
$369$	−24.9909	−0.0677260
$370$	0	0
$371$	70.5373i	0.190128i
$372$	0	0
$373$	635.248	1.70308	0.851540	−	0.524290i	$-0.175669\pi$
$373$	635.248	1.70308	0.851540	+	0.524290i	$0.175669\pi$
$374$	0	0
$375$	− 159.349i	− 0.424930i
$376$	0	0
$377$	827.303	2.19444
$378$	0	0
$379$	− 31.2084i	− 0.0823441i	−0.999152	−	0.0411720i	$-0.986891\pi$
$379$	− 31.2084i	− 0.0823441i	0.999152	−	0.0411720i	$-0.0131092\pi$
$380$	0	0
$381$	146.642	0.384888
$382$	0	0
$383$	− 28.4756i	− 0.0743489i	−0.999309	−	0.0371744i	$-0.988164\pi$
$383$	− 28.4756i	− 0.0743489i	0.999309	−	0.0371744i	$-0.0118357\pi$
$384$	0	0
$385$	19.3394	0.0502322
$386$	0	0
$387$	19.6404i	0.0507504i
$388$	0	0
$389$	405.267	1.04182	0.520908	−	0.853613i	$-0.325593\pi$
$389$	405.267	1.04182	0.520908	+	0.853613i	$0.325593\pi$
$390$	0	0
$391$	135.831i	0.347395i
$392$	0	0
$393$	277.321	0.705652
$394$	0	0
$395$	193.990i	0.491113i
$396$	0	0
$397$	490.936	1.23662	0.618308	−	0.785936i	$-0.287819\pi$
$397$	490.936	1.23662	0.618308	+	0.785936i	$0.287819\pi$
$398$	0	0
$399$	− 65.2458i	− 0.163523i
$400$	0	0
$401$	−248.018	−0.618499	−0.309250	−	0.950981i	$-0.600078\pi$
$401$	−248.018	−0.618499	−0.309250	+	0.950981i	$0.600078\pi$
$402$	0	0
$403$	− 452.559i	− 1.12297i
$404$	0	0
$405$	−18.0000	−0.0444444
$406$	0	0
$407$	38.9624i	0.0957307i
$408$	0	0
$409$	−369.964	−0.904557	−0.452278	−	0.891877i	$-0.649389\pi$
$409$	−369.964	−0.904557	−0.452278	+	0.891877i	$0.649389\pi$
$410$	0	0
$411$	− 334.811i	− 0.814624i
$412$	0	0
$413$	−1.00909	−0.00244332
$414$	0	0
$415$	210.897i	0.508186i
$416$	0	0
$417$	−105.358	−0.252656
$418$	0	0
$419$	68.9006i	0.164441i	0.996614	+	0.0822203i	$0.0262011\pi$
$419$	68.9006i	0.164441i	−0.996614	+	0.0822203i	$0.973799\pi$
$420$	0	0
$421$	296.606	0.704527	0.352264	−	0.935901i	$-0.385412\pi$
$421$	296.606	0.704527	0.352264	+	0.935901i	$0.385412\pi$
$422$	0	0
$423$	107.356i	0.253796i
$424$	0	0
$425$	90.9364	0.213968
$426$	0	0
$427$	64.3719i	0.150754i
$428$	0	0
$429$	103.376	0.240969
$430$	0	0
$431$	373.075i	0.865603i	0.901489	+	0.432802i	$0.142475\pi$
$431$	373.075i	0.865603i	−0.901489	+	0.432802i	$0.857525\pi$
$432$	0	0
$433$	−775.945	−1.79202	−0.896011	−	0.444032i	$-0.853548\pi$
$433$	−775.945	−1.79202	−0.896011	+	0.444032i	$0.853548\pi$
$434$	0	0
$435$	− 175.493i	− 0.403433i
$436$	0	0
$437$	446.606	1.02198
$438$	0	0
$439$	− 744.750i	− 1.69647i	−0.529620	−	0.848235i	$-0.677665\pi$
$439$	− 744.750i	− 1.69647i	0.529620	−	0.848235i	$-0.322335\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	737.156i	1.66401i	0.554770	+	0.832004i	$0.312806\pi$
$443$	737.156i	1.66401i	−0.554770	+	0.832004i	$0.687194\pi$
$444$	0	0
$445$	−185.982	−0.417937
$446$	0	0
$447$	228.599i	0.511408i
$448$	0	0
$449$	−851.285	−1.89596	−0.947979	−	0.318334i	$-0.896877\pi$
$449$	−851.285	−1.89596	−0.947979	+	0.318334i	$0.896877\pi$
$450$	0	0
$451$	− 30.4456i	− 0.0675069i
$452$	0	0
$453$	108.661	0.239869
$454$	0	0
$455$	− 86.4118i	− 0.189916i
$456$	0	0
$457$	601.927	1.31713	0.658564	−	0.752525i	$-0.271164\pi$
$457$	601.927	1.31713	0.658564	+	0.752525i	$0.271164\pi$
$458$	0	0
$459$	− 22.5009i	− 0.0490216i
$460$	0	0
$461$	758.000	1.64425	0.822126	−	0.569306i	$-0.192788\pi$
$461$	758.000	1.64425	0.822126	+	0.569306i	$0.192788\pi$
$462$	0	0
$463$	414.929i	0.896176i	0.893990	+	0.448088i	$0.147895\pi$
$463$	414.929i	0.896176i	−0.893990	+	0.448088i	$0.852105\pi$
$464$	0	0
$465$	−96.0000	−0.206452
$466$	0	0
$467$	− 45.7646i	− 0.0979971i	−0.998799	−	0.0489985i	$-0.984397\pi$
$467$	− 45.7646i	− 0.0979971i	0.998799	−	0.0489985i	$-0.0156030\pi$
$468$	0	0
$469$	146.642	0.312670
$470$	0	0
$471$	149.497i	0.317403i
$472$	0	0
$473$	−23.9273	−0.0505862
$474$	0	0
$475$	− 298.994i	− 0.629461i
$476$	0	0
$477$	79.9818	0.167677
$478$	0	0
$479$	− 140.534i	− 0.293391i	−0.989182	−	0.146695i	$-0.953136\pi$
$479$	− 140.534i	− 0.293391i	0.989182	−	0.146695i	$-0.0468637\pi$
$480$	0	0
$481$	174.091	0.361935
$482$	0	0
$483$	143.744i	0.297608i
$484$	0	0
$485$	−102.642	−0.211634
$486$	0	0
$487$	302.108i	0.620345i	0.950680	+	0.310173i	$0.100387\pi$
$487$	302.108i	0.620345i	−0.950680	+	0.310173i	$0.899613\pi$
$488$	0	0
$489$	−286.679	−0.586255
$490$	0	0
$491$	− 78.7208i	− 0.160328i	−0.996782	−	0.0801638i	$-0.974456\pi$
$491$	− 78.7208i	− 0.160328i	0.996782	−	0.0801638i	$-0.0255443\pi$
$492$	0	0
$493$	219.376	0.444981
$494$	0	0
$495$	− 21.9288i	− 0.0443006i
$496$	0	0
$497$	−250.991	−0.505012
$498$	0	0
$499$	616.991i	1.23646i	0.785999	+	0.618228i	$0.212149\pi$
$499$	616.991i	1.23646i	−0.785999	+	0.618228i	$0.787851\pi$
$500$	0	0
$501$	−468.661	−0.935450
$502$	0	0
$503$	− 309.418i	− 0.615145i	−0.951525	−	0.307572i	$-0.900483\pi$
$503$	− 309.418i	− 0.615145i	0.951525	−	0.307572i	$-0.0995166\pi$
$504$	0	0
$505$	−149.321	−0.295686
$506$	0	0
$507$	− 169.185i	− 0.333697i
$508$	0	0
$509$	−722.624	−1.41969	−0.709847	−	0.704356i	$-0.751236\pi$
$509$	−722.624	−1.41969	−0.709847	+	0.704356i	$0.751236\pi$
$510$	0	0
$511$	− 250.448i	− 0.490114i
$512$	0	0
$513$	−73.9818	−0.144214
$514$	0	0
$515$	− 280.179i	− 0.544038i
$516$	0	0
$517$	−130.788	−0.252975
$518$	0	0
$519$	576.154i	1.11012i
$520$	0	0
$521$	743.633	1.42732	0.713660	−	0.700493i	$-0.247036\pi$
$521$	743.633	1.42732	0.713660	+	0.700493i	$0.247036\pi$
$522$	0	0
$523$	694.283i	1.32750i	0.747954	+	0.663750i	$0.231036\pi$
$523$	694.283i	1.32750i	−0.747954	+	0.663750i	$0.768964\pi$
$524$	0	0
$525$	96.2341	0.183303
$526$	0	0
$527$	− 120.005i	− 0.227713i
$528$	0	0
$529$	−454.927	−0.859976
$530$	0	0
$531$	1.14420i	0.00215481i
$532$	0	0
$533$	−136.036	−0.255228
$534$	0	0
$535$	396.815i	0.741710i
$536$	0	0
$537$	295.652	0.550561
$538$	0	0
$539$	25.5836i	0.0474650i
$540$	0	0
$541$	−480.570	−0.888299	−0.444149	−	0.895953i	$-0.646494\pi$
$541$	−480.570	−0.888299	−0.444149	+	0.895953i	$0.646494\pi$
$542$	0	0
$543$	293.845i	0.541151i
$544$	0	0
$545$	66.6788	0.122346
$546$	0	0
$547$	− 724.347i	− 1.32422i	−0.749408	−	0.662109i	$-0.769662\pi$
$547$	− 724.347i	− 1.32422i	0.749408	−	0.662109i	$-0.230338\pi$
$548$	0	0
$549$	72.9909	0.132952
$550$	0	0
$551$	− 721.296i	− 1.30907i
$552$	0	0
$553$	−256.624	−0.464058
$554$	0	0
$555$	− 36.9294i	− 0.0665395i
$556$	0	0
$557$	449.303	0.806648	0.403324	−	0.915057i	$-0.367855\pi$
$557$	449.303	0.806648	0.403324	+	0.915057i	$0.367855\pi$
$558$	0	0
$559$	106.911i	0.191255i
$560$	0	0
$561$	27.4121	0.0488630
$562$	0	0
$563$	− 964.101i	− 1.71244i	−0.516615	−	0.856218i	$-0.672808\pi$
$563$	− 964.101i	− 1.71244i	0.516615	−	0.856218i	$-0.327192\pi$
$564$	0	0
$565$	−22.6424	−0.0400751
$566$	0	0
$567$	− 23.8118i	− 0.0419961i
$568$	0	0
$569$	−843.909	−1.48314	−0.741572	−	0.670873i	$-0.765919\pi$
$569$	−843.909	−1.48314	−0.741572	+	0.670873i	$0.765919\pi$
$570$	0	0
$571$	− 622.013i	− 1.08934i	−0.838651	−	0.544670i	$-0.816655\pi$
$571$	− 622.013i	− 1.08934i	0.838651	−	0.544670i	$-0.183345\pi$
$572$	0	0
$573$	−522.991	−0.912724
$574$	0	0
$575$	658.720i	1.14560i
$576$	0	0
$577$	−91.3576	−0.158332	−0.0791660	−	0.996861i	$-0.525226\pi$
$577$	−91.3576	−0.158332	−0.0791660	+	0.996861i	$0.525226\pi$
$578$	0	0
$579$	389.092i	0.672007i
$580$	0	0
$581$	−278.991	−0.480191
$582$	0	0
$583$	97.4392i	0.167134i
$584$	0	0
$585$	−97.9818	−0.167490
$586$	0	0
$587$	471.499i	0.803235i	0.915807	+	0.401618i	$0.131552\pi$
$587$	471.499i	0.803235i	−0.915807	+	0.401618i	$0.868448\pi$
$588$	0	0
$589$	−394.570	−0.669898
$590$	0	0
$591$	− 13.8249i	− 0.0233924i
$592$	0	0
$593$	−863.524	−1.45620	−0.728098	−	0.685473i	$-0.759595\pi$
$593$	−863.524	−1.45620	−0.728098	+	0.685473i	$0.759595\pi$
$594$	0	0
$595$	− 22.9138i	− 0.0385106i
$596$	0	0
$597$	639.303	1.07086
$598$	0	0
$599$	− 842.223i	− 1.40605i	−0.711166	−	0.703024i	$-0.751833\pi$
$599$	− 842.223i	− 1.40605i	0.711166	−	0.703024i	$-0.248167\pi$
$600$	0	0
$601$	−240.642	−0.400403	−0.200202	−	0.979755i	$-0.564160\pi$
$601$	−240.642	−0.400403	−0.200202	+	0.979755i	$0.564160\pi$
$602$	0	0
$603$	− 166.277i	− 0.275749i
$604$	0	0
$605$	−215.285	−0.355843
$606$	0	0
$607$	− 85.1084i	− 0.140212i	−0.997540	−	0.0701058i	$-0.977666\pi$
$607$	− 85.1084i	− 0.140212i	0.997540	−	0.0701058i	$-0.0223337\pi$
$608$	0	0
$609$	232.156	0.381209
$610$	0	0
$611$	584.383i	0.956438i
$612$	0	0
$613$	−147.982	−0.241406	−0.120703	−	0.992689i	$-0.538515\pi$
$613$	−147.982	−0.241406	−0.120703	+	0.992689i	$0.538515\pi$
$614$	0	0
$615$	28.8570i	0.0469220i
$616$	0	0
$617$	73.2303	0.118688	0.0593438	−	0.998238i	$-0.481099\pi$
$617$	73.2303	0.118688	0.0593438	+	0.998238i	$0.481099\pi$
$618$	0	0
$619$	− 748.946i	− 1.20993i	−0.796253	−	0.604964i	$-0.793187\pi$
$619$	− 748.946i	− 1.20993i	0.796253	−	0.604964i	$-0.206813\pi$
$620$	0	0
$621$	162.991	0.262465
$622$	0	0
$623$	− 246.031i	− 0.394913i
$624$	0	0
$625$	341.000	0.545600
$626$	0	0
$627$	− 90.1296i	− 0.143747i
$628$	0	0
$629$	46.1636	0.0733921
$630$	0	0
$631$	1214.66i	1.92498i	0.271321	+	0.962489i	$0.412539\pi$
$631$	1214.66i	1.92498i	−0.271321	+	0.962489i	$0.587461\pi$
$632$	0	0
$633$	−596.588	−0.942477
$634$	0	0
$635$	− 169.328i	− 0.266658i
$636$	0	0
$637$	114.312	0.179454
$638$	0	0
$639$	284.597i	0.445379i
$640$	0	0
$641$	−610.588	−0.952555	−0.476278	−	0.879295i	$-0.658014\pi$
$641$	−610.588	−0.952555	−0.476278	+	0.879295i	$0.658014\pi$
$642$	0	0
$643$	734.008i	1.14154i	0.821111	+	0.570768i	$0.193355\pi$
$643$	734.008i	1.14154i	−0.821111	+	0.570768i	$0.806645\pi$
$644$	0	0
$645$	22.6788	0.0351609
$646$	0	0
$647$	− 810.855i	− 1.25325i	−0.779319	−	0.626627i	$-0.784435\pi$
$647$	− 810.855i	− 1.25325i	0.779319	−	0.626627i	$-0.215565\pi$
$648$	0	0
$649$	−1.39394	−0.00214783
$650$	0	0
$651$	− 126.996i	− 0.195078i
$652$	0	0
$653$	−754.661	−1.15568	−0.577841	−	0.816149i	$-0.696105\pi$
$653$	−754.661	−1.15568	−0.577841	+	0.816149i	$0.696105\pi$
$654$	0	0
$655$	− 320.223i	− 0.488890i
$656$	0	0
$657$	−283.982	−0.432240
$658$	0	0
$659$	− 717.197i	− 1.08831i	−0.838984	−	0.544155i	$-0.816850\pi$
$659$	− 717.197i	− 1.08831i	0.838984	−	0.544155i	$-0.183150\pi$
$660$	0	0
$661$	−461.615	−0.698359	−0.349179	−	0.937056i	$-0.613540\pi$
$661$	−461.615	−0.698359	−0.349179	+	0.937056i	$0.613540\pi$
$662$	0	0
$663$	− 122.482i	− 0.184739i
$664$	0	0
$665$	−75.3394	−0.113292
$666$	0	0
$667$	1589.10i	2.38246i
$668$	0	0
$669$	−668.588	−0.999384
$670$	0	0
$671$	88.9224i	0.132522i
$672$	0	0
$673$	−575.248	−0.854753	−0.427376	−	0.904074i	$-0.640562\pi$
$673$	−575.248	−0.854753	−0.427376	+	0.904074i	$0.640562\pi$
$674$	0	0
$675$	− 109.119i	− 0.161658i
$676$	0	0
$677$	321.964	0.475574	0.237787	−	0.971317i	$-0.423578\pi$
$677$	321.964	0.475574	0.237787	+	0.971317i	$0.423578\pi$
$678$	0	0
$679$	− 135.783i	− 0.199975i
$680$	0	0
$681$	464.588	0.682214
$682$	0	0
$683$	− 1044.73i	− 1.52962i	−0.644257	−	0.764809i	$-0.722833\pi$
$683$	− 1044.73i	− 1.52962i	0.644257	−	0.764809i	$-0.277167\pi$
$684$	0	0
$685$	−386.606	−0.564388
$686$	0	0
$687$	155.281i	0.226028i
$688$	0	0
$689$	435.376	0.631895
$690$	0	0
$691$	− 544.914i	− 0.788587i	−0.918985	−	0.394294i	$-0.870989\pi$
$691$	− 544.914i	− 0.788587i	0.918985	−	0.394294i	$-0.129011\pi$
$692$	0	0
$693$	29.0091	0.0418602
$694$	0	0
$695$	121.656i	0.175045i
$696$	0	0
$697$	−36.0727	−0.0517543
$698$	0	0
$699$	517.295i	0.740051i
$700$	0	0
$701$	−368.018	−0.524990	−0.262495	−	0.964933i	$-0.584545\pi$
$701$	−368.018	−0.524990	−0.262495	+	0.964933i	$0.584545\pi$
$702$	0	0
$703$	− 151.784i	− 0.215908i
$704$	0	0
$705$	123.964	0.175835
$706$	0	0
$707$	− 197.533i	− 0.279397i
$708$	0	0
$709$	434.697	0.613113	0.306556	−	0.951852i	$-0.400823\pi$
$709$	434.697	0.613113	0.306556	+	0.951852i	$0.400823\pi$
$710$	0	0
$711$	290.985i	0.409261i
$712$	0	0
$713$	869.285	1.21919
$714$	0	0
$715$	− 119.368i	− 0.166948i
$716$	0	0
$717$	261.633	0.364900
$718$	0	0
$719$	− 752.060i	− 1.04598i	−0.852339	−	0.522990i	$-0.824816\pi$
$719$	− 752.060i	− 1.04598i	0.852339	−	0.522990i	$-0.175184\pi$
$720$	0	0
$721$	370.642	0.514067
$722$	0	0
$723$	564.649i	0.780980i
$724$	0	0
$725$	1063.87	1.46741
$726$	0	0
$727$	174.794i	0.240431i	0.992748	+	0.120216i	$0.0383586\pi$
$727$	174.794i	0.240431i	−0.992748	+	0.120216i	$0.961641\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	28.3496i	0.0387820i
$732$	0	0
$733$	416.294	0.567932	0.283966	−	0.958834i	$-0.408350\pi$
$733$	416.294	0.567932	0.283966	+	0.958834i	$0.408350\pi$
$734$	0	0
$735$	− 24.2487i	− 0.0329914i
$736$	0	0
$737$	202.570	0.274857
$738$	0	0
$739$	1280.77i	1.73311i	0.499084	+	0.866553i	$0.333670\pi$
$739$	1280.77i	1.73311i	−0.499084	+	0.866553i	$0.666330\pi$
$740$	0	0
$741$	−402.715	−0.543475
$742$	0	0
$743$	− 57.8732i	− 0.0778913i	−0.999241	−	0.0389457i	$-0.987600\pi$
$743$	− 57.8732i	− 0.0778913i	0.999241	−	0.0389457i	$-0.0123999\pi$
$744$	0	0
$745$	263.964	0.354314
$746$	0	0
$747$	316.346i	0.423489i
$748$	0	0
$749$	−524.936	−0.700850
$750$	0	0
$751$	399.866i	0.532444i	0.963912	+	0.266222i	$0.0857755\pi$
$751$	399.866i	0.532444i	−0.963912	+	0.266222i	$0.914224\pi$
$752$	0	0
$753$	657.909	0.873717
$754$	0	0
$755$	− 125.470i	− 0.166186i
$756$	0	0
$757$	846.661	1.11844	0.559221	−	0.829019i	$-0.311100\pi$
$757$	846.661	1.11844	0.559221	+	0.829019i	$0.311100\pi$
$758$	0	0
$759$	198.567i	0.261616i
$760$	0	0
$761$	−206.348	−0.271154	−0.135577	−	0.990767i	$-0.543289\pi$
$761$	−206.348	−0.271154	−0.135577	+	0.990767i	$0.543289\pi$
$762$	0	0
$763$	88.2077i	0.115606i
$764$	0	0
$765$	−25.9818	−0.0339632
$766$	0	0
$767$	6.22839i	0.00812046i
$768$	0	0
$769$	−203.982	−0.265256	−0.132628	−	0.991166i	$-0.542342\pi$
$769$	−203.982	−0.265256	−0.132628	+	0.991166i	$0.542342\pi$
$770$	0	0
$771$	92.9271i	0.120528i
$772$	0	0
$773$	−466.697	−0.603748	−0.301874	−	0.953348i	$-0.597612\pi$
$773$	−466.697	−0.603748	−0.301874	+	0.953348i	$0.597612\pi$
$774$	0	0
$775$	− 581.969i	− 0.750928i
$776$	0	0
$777$	48.8530	0.0628739
$778$	0	0
$779$	118.605i	0.152253i
$780$	0	0
$781$	−346.715	−0.443937
$782$	0	0
$783$	− 263.240i	− 0.336194i
$784$	0	0
$785$	172.624	0.219903
$786$	0	0
$787$	− 482.749i	− 0.613404i	−0.951806	−	0.306702i	$-0.900775\pi$
$787$	− 482.749i	− 0.613404i	0.951806	−	0.306702i	$-0.0992255\pi$
$788$	0	0
$789$	60.3849	0.0765334
$790$	0	0
$791$	− 29.9531i	− 0.0378674i
$792$	0	0
$793$	397.321	0.501036
$794$	0	0
$795$	− 92.3550i	− 0.116170i
$796$	0	0
$797$	1387.28	1.74063	0.870317	−	0.492492i	$-0.163914\pi$
$797$	1387.28	1.74063	0.870317	+	0.492492i	$0.163914\pi$
$798$	0	0
$799$	154.961i	0.193943i
$800$	0	0
$801$	−278.973	−0.348281
$802$	0	0
$803$	− 345.966i	− 0.430842i
$804$	0	0
$805$	165.982	0.206189
$806$	0	0
$807$	332.522i	0.412047i
$808$	0	0
$809$	976.606	1.20718	0.603588	−	0.797296i	$-0.293737\pi$
$809$	976.606	1.20718	0.603588	+	0.797296i	$0.293737\pi$
$810$	0	0
$811$	1258.58i	1.55189i	0.630801	+	0.775944i	$0.282726\pi$
$811$	1258.58i	1.55189i	−0.630801	+	0.775944i	$0.717274\pi$
$812$	0	0
$813$	193.321	0.237787
$814$	0	0
$815$	331.028i	0.406170i
$816$	0	0
$817$	93.2121	0.114091
$818$	0	0
$819$	− 129.618i	− 0.158263i
$820$	0	0
$821$	−984.018	−1.19856	−0.599280	−	0.800539i	$-0.704546\pi$
$821$	−984.018	−1.19856	−0.599280	+	0.800539i	$0.704546\pi$
$822$	0	0
$823$	138.438i	0.168212i	0.996457	+	0.0841058i	$0.0268034\pi$
$823$	138.438i	0.168212i	−0.996457	+	0.0841058i	$0.973197\pi$
$824$	0	0
$825$	132.936	0.161135
$826$	0	0
$827$	− 577.870i	− 0.698754i	−0.936982	−	0.349377i	$-0.886393\pi$
$827$	− 577.870i	− 0.698754i	0.936982	−	0.349377i	$-0.113607\pi$
$828$	0	0
$829$	−45.0636	−0.0543590	−0.0271795	−	0.999631i	$-0.508653\pi$
$829$	−45.0636	−0.0543590	−0.0271795	+	0.999631i	$0.508653\pi$
$830$	0	0
$831$	− 338.306i	− 0.407107i
$832$	0	0
$833$	30.3121	0.0363891
$834$	0	0
$835$	541.163i	0.648099i
$836$	0	0
$837$	−144.000	−0.172043
$838$	0	0
$839$	− 214.267i	− 0.255384i	−0.991814	−	0.127692i	$-0.959243\pi$
$839$	− 214.267i	− 0.255384i	0.991814	−	0.127692i	$-0.0407569\pi$
$840$	0	0
$841$	1725.50	2.05172
$842$	0	0
$843$	438.734i	0.520443i
$844$	0	0
$845$	−195.358	−0.231192
$846$	0	0
$847$	− 284.795i	− 0.336240i
$848$	0	0
$849$	−245.945	−0.289688
$850$	0	0
$851$	334.398i	0.392947i
$852$	0	0
$853$	−787.670	−0.923411	−0.461706	−	0.887033i	$-0.652762\pi$
$853$	−787.670	−0.923411	−0.461706	+	0.887033i	$0.652762\pi$
$854$	0	0
$855$	85.4268i	0.0999144i
$856$	0	0
$857$	1034.83	1.20750	0.603750	−	0.797174i	$-0.293673\pi$
$857$	1034.83	1.20750	0.603750	+	0.797174i	$0.293673\pi$
$858$	0	0
$859$	− 939.440i	− 1.09364i	−0.837249	−	0.546822i	$-0.815838\pi$
$859$	− 939.440i	− 1.09364i	0.837249	−	0.546822i	$-0.184162\pi$
$860$	0	0
$861$	−38.1742	−0.0443371
$862$	0	0
$863$	1259.44i	1.45937i	0.683781	+	0.729687i	$0.260334\pi$
$863$	1259.44i	1.45937i	−0.683781	+	0.729687i	$0.739666\pi$
$864$	0	0
$865$	665.285	0.769115
$866$	0	0
$867$	468.084i	0.539889i
$868$	0	0
$869$	−354.497	−0.407937
$870$	0	0
$871$	− 905.117i	− 1.03917i
$872$	0	0
$873$	−153.964	−0.176362
$874$	0	0
$875$	− 243.409i	− 0.278182i
$876$	0	0
$877$	−309.376	−0.352766	−0.176383	−	0.984322i	$-0.556440\pi$
$877$	−309.376	−0.352766	−0.176383	+	0.984322i	$0.556440\pi$
$878$	0	0
$879$	− 898.284i	− 1.02194i
$880$	0	0
$881$	1365.47	1.54991	0.774954	−	0.632017i	$-0.217773\pi$
$881$	1365.47	1.54991	0.774954	+	0.632017i	$0.217773\pi$
$882$	0	0
$883$	− 190.620i	− 0.215878i	−0.994158	−	0.107939i	$-0.965575\pi$
$883$	− 190.620i	− 0.215878i	0.994158	−	0.107939i	$-0.0344251\pi$
$884$	0	0
$885$	1.32121	0.00149289
$886$	0	0
$887$	− 1126.63i	− 1.27015i	−0.772449	−	0.635077i	$-0.780968\pi$
$887$	− 1126.63i	− 1.27015i	0.772449	−	0.635077i	$-0.219032\pi$
$888$	0	0
$889$	224.000	0.251969
$890$	0	0
$891$	− 32.8932i	− 0.0369172i
$892$	0	0
$893$	509.503	0.570552
$894$	0	0
$895$	− 341.389i	− 0.381440i
$896$	0	0
$897$	887.230	0.989108
$898$	0	0
$899$	− 1403.95i	− 1.56168i
$900$	0	0
$901$	115.448	0.128134
$902$	0	0
$903$	30.0012i	0.0332239i
$904$	0	0
$905$	339.303	0.374920
$906$	0	0
$907$	− 1526.69i	− 1.68323i	−0.540081	−	0.841613i	$-0.681607\pi$
$907$	− 1526.69i	− 1.68323i	0.540081	−	0.841613i	$-0.318393\pi$
$908$	0	0
$909$	−223.982	−0.246405
$910$	0	0
$911$	1389.93i	1.52572i	0.646563	+	0.762861i	$0.276206\pi$
$911$	1389.93i	1.52572i	−0.646563	+	0.762861i	$0.723794\pi$
$912$	0	0
$913$	−385.394	−0.422118
$914$	0	0
$915$	− 84.2826i	− 0.0921122i
$916$	0	0
$917$	423.615	0.461958
$918$	0	0
$919$	140.534i	0.152921i	0.997073	+	0.0764603i	$0.0243619\pi$
$919$	140.534i	0.152921i	−0.997073	+	0.0764603i	$0.975638\pi$
$920$	0	0
$921$	241.982	0.262738
$922$	0	0
$923$	1549.18i	1.67842i
$924$	0	0
$925$	223.873	0.242025
$926$	0	0
$927$	− 420.269i	− 0.453365i
$928$	0	0
$929$	290.900	0.313132	0.156566	−	0.987667i	$-0.449958\pi$
$929$	290.900	0.313132	0.156566	+	0.987667i	$0.449958\pi$
$930$	0	0
$931$	− 99.6647i	− 0.107051i
$932$	0	0
$933$	517.982	0.555179
$934$	0	0
$935$	− 31.6528i	− 0.0338533i
$936$	0	0
$937$	805.891	0.860076	0.430038	−	0.902811i	$-0.358500\pi$
$937$	805.891	0.860076	0.430038	+	0.902811i	$0.358500\pi$
$938$	0	0
$939$	115.460i	0.122960i
$940$	0	0
$941$	−1233.85	−1.31122	−0.655608	−	0.755101i	$-0.727588\pi$
$941$	−1233.85	−1.31122	−0.655608	+	0.755101i	$0.727588\pi$
$942$	0	0
$943$	− 261.302i	− 0.277096i
$944$	0	0
$945$	−27.4955	−0.0290957
$946$	0	0
$947$	− 145.907i	− 0.154073i	−0.997028	−	0.0770364i	$-0.975454\pi$
$947$	− 145.907i	− 0.154073i	0.997028	−	0.0770364i	$-0.0245458\pi$
$948$	0	0
$949$	−1545.84	−1.62891
$950$	0	0
$951$	147.875i	0.155494i
$952$	0	0
$953$	−1833.96	−1.92441	−0.962205	−	0.272324i	$-0.912208\pi$
$953$	−1833.96	−1.92441	−0.962205	+	0.272324i	$0.912208\pi$
$954$	0	0
$955$	603.898i	0.632354i
$956$	0	0
$957$	320.697	0.335107
$958$	0	0
$959$	− 511.432i	− 0.533297i
$960$	0	0
$961$	193.000	0.200832
$962$	0	0
$963$	595.222i	0.618091i
$964$	0	0
$965$	449.285	0.465580
$966$	0	0
$967$	1255.34i	1.29818i	0.760711	+	0.649091i	$0.224850\pi$
$967$	1255.34i	1.29818i	−0.760711	+	0.649091i	$0.775150\pi$
$968$	0	0
$969$	−106.788	−0.110204
$970$	0	0
$971$	1303.08i	1.34199i	0.741460	+	0.670997i	$0.234134\pi$
$971$	1303.08i	1.34199i	−0.741460	+	0.670997i	$0.765866\pi$
$972$	0	0
$973$	−160.936	−0.165402
$974$	0	0
$975$	− 593.983i	− 0.609214i
$976$	0	0
$977$	259.945	0.266065	0.133032	−	0.991112i	$-0.457529\pi$
$977$	259.945	0.266065	0.133032	+	0.991112i	$0.457529\pi$
$978$	0	0
$979$	− 339.863i	− 0.347154i
$980$	0	0
$981$	100.018	0.101955
$982$	0	0
$983$	− 153.946i	− 0.156608i	−0.996930	−	0.0783042i	$-0.975049\pi$
$983$	− 153.946i	− 0.156608i	0.996930	−	0.0783042i	$-0.0249505\pi$
$984$	0	0
$985$	−15.9636	−0.0162067
$986$	0	0
$987$	163.988i	0.166148i
$988$	0	0
$989$	−205.358	−0.207642
$990$	0	0
$991$	− 577.585i	− 0.582830i	−0.956597	−	0.291415i	$-0.905874\pi$
$991$	− 577.585i	− 0.582830i	0.956597	−	0.291415i	$-0.0941261\pi$
$992$	0	0
$993$	−1060.52	−1.06799
$994$	0	0
$995$	− 738.204i	− 0.741913i
$996$	0	0
$997$	567.670	0.569378	0.284689	−	0.958620i	$-0.408110\pi$
$997$	567.670	0.569378	0.284689	+	0.958620i	$0.408110\pi$
$998$	0	0
$999$	− 55.3941i	− 0.0554496i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.m.b.127.1	✓	4
3.2	odd	2		1008.3.m.e.127.1		4
4.3	odd	2	inner	336.3.m.b.127.4	yes	4
7.6	odd	2		2352.3.m.i.1471.3		4
8.3	odd	2		1344.3.m.b.127.2		4
8.5	even	2		1344.3.m.b.127.3		4
12.11	even	2		1008.3.m.e.127.4		4
28.27	even	2		2352.3.m.i.1471.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.m.b.127.1	✓	4	1.1	even	1	trivial
336.3.m.b.127.4	yes	4	4.3	odd	2	inner
1008.3.m.e.127.1		4	3.2	odd	2
1008.3.m.e.127.4		4	12.11	even	2
1344.3.m.b.127.2		4	8.3	odd	2
1344.3.m.b.127.3		4	8.5	even	2
2352.3.m.i.1471.2		4	28.27	even	2
2352.3.m.i.1471.3		4	7.6	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	336.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} -2.00000 q^{5} -2.64575i q^{7} -3.00000 q^{9} -3.65480i q^{11} -16.3303 q^{13} +3.46410i q^{15} -4.33030 q^{17} +14.2378i q^{19} -4.58258 q^{21} -31.3676i q^{23} -21.0000 q^{25} +5.19615i q^{27} -50.6606 q^{29} +27.7128i q^{31} -6.33030 q^{33} +5.29150i q^{35} -10.6606 q^{37} +28.2849i q^{39} +8.33030 q^{41} -6.54680i q^{43} +6.00000 q^{45} -35.7852i q^{47} -7.00000 q^{49} +7.50030i q^{51} -26.6606 q^{53} +7.30960i q^{55} +24.6606 q^{57} -0.381401i q^{59} -24.3303 q^{61} +7.93725i q^{63} +32.6606 q^{65} +55.4256i q^{67} -54.3303 q^{69} -94.8656i q^{71} +94.6606 q^{73} +36.3731i q^{75} -9.66970 q^{77} -96.9948i q^{79} +9.00000 q^{81} -105.449i q^{83} +8.66061 q^{85} +87.7467i q^{87} +92.9909 q^{89} +43.2059i q^{91} +48.0000 q^{93} -28.4756i q^{95} +51.3212 q^{97} +10.9644i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{5} - 12 q^{9} + 8 q^{13} + 56 q^{17} - 84 q^{25} - 56 q^{29} + 48 q^{33} + 104 q^{37} - 40 q^{41} + 24 q^{45} - 28 q^{49} + 40 q^{53} - 48 q^{57} - 24 q^{61} - 16 q^{65} - 144 q^{69} + 232 q^{73}+ \cdots - 88 q^{97}+O(q^{100}) \) 4 * q - 8 * q^5 - 12 * q^9 + 8 * q^13 + 56 * q^17 - 84 * q^25 - 56 * q^29 + 48 * q^33 + 104 * q^37 - 40 * q^41 + 24 * q^45 - 28 * q^49 + 40 * q^53 - 48 * q^57 - 24 * q^61 - 16 * q^65 - 144 * q^69 + 232 * q^73 - 112 * q^77 + 36 * q^81 - 112 * q^85 + 152 * q^89 + 192 * q^93 - 88 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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