LMFDB - Embedded newform 1008.3.f.i.433.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,3,Mod(433,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1008.433");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1008.f (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:	$27.4660106475$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-33})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 32x^{2} + 289 $ x^4 + 32*x^2 + 289
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.4
Root			$0.707107 + 4.06202i$ of defining polynomial
Character	$\chi$	$=$	1008.433
Dual form			1008.3.f.i.433.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.12404i q^{5} +(4.00000 + 5.74456i) q^{7} +O(q^{10})$	$q+8.12404i q^{5} +(4.00000 + 5.74456i) q^{7} +12.7279 q^{11} +22.9783i q^{13} +8.12404i q^{17} -11.4891i q^{19} +21.2132 q^{23} -41.0000 q^{25} +33.9411 q^{29} +(-46.6690 + 32.4962i) q^{35} +16.0000 q^{37} -56.8683i q^{41} -52.0000 q^{43} -32.4962i q^{47} +(-17.0000 + 45.9565i) q^{49} +16.9706 q^{53} +103.402i q^{55} +32.4962i q^{59} -22.9783i q^{61} -186.676 q^{65} +52.0000 q^{67} -89.0955 q^{71} +45.9565i q^{73} +(50.9117 + 73.1163i) q^{77} -104.000 q^{79} -162.481i q^{83} -66.0000 q^{85} -73.1163i q^{89} +(-132.000 + 91.9130i) q^{91} +93.3381 q^{95} +91.9130i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{7}+O(q^{10}) $ 4 * q + 16 * q^7	$ 4 q + 16 q^{7} - 164 q^{25} + 64 q^{37} - 208 q^{43} - 68 q^{49} + 208 q^{67} - 416 q^{79} - 264 q^{85} - 528 q^{91}+O(q^{100}) $ 4 * q + 16 * q^7 - 164 * q^25 + 64 * q^37 - 208 * q^43 - 68 * q^49 + 208 * q^67 - 416 * q^79 - 264 * q^85 - 528 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$			8.12404i			1.62481i	0.583095	+	0.812404i	$0.301841\pi$
$5$			8.12404i			1.62481i	−0.583095	+	0.812404i	$0.698159\pi$
$6$		0			0
$7$	4.00000	+	5.74456i	0.571429	+	0.820652i
$7$	4.00000	+	5.74456i	0.571429	+	0.820652i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	12.7279			1.15708			0.578542	−	0.815653i	$-0.303622\pi$
$11$	12.7279			1.15708			0.578542	+	0.815653i	$0.303622\pi$
$12$		0			0
$13$			22.9783i			1.76756i	0.467905	+	0.883779i	$0.345009\pi$
$13$			22.9783i			1.76756i	−0.467905	+	0.883779i	$0.654991\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			8.12404i			0.477885i	0.971034	+	0.238942i	$0.0768007\pi$
$17$			8.12404i			0.477885i	−0.971034	+	0.238942i	$0.923199\pi$
$18$		0			0
$19$		−	11.4891i		−	0.604691i	−0.953198	−	0.302345i	$-0.902230\pi$
$19$		−	11.4891i		−	0.604691i	0.953198	−	0.302345i	$-0.0977696\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	21.2132			0.922313			0.461157	−	0.887319i	$-0.347435\pi$
$23$	21.2132			0.922313			0.461157	+	0.887319i	$0.347435\pi$
$24$		0			0
$25$	−41.0000			−1.64000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	33.9411			1.17038			0.585192	−	0.810895i	$-0.301019\pi$
$29$	33.9411			1.17038			0.585192	+	0.810895i	$0.301019\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−46.6690	+	32.4962i	−1.33340	+	0.928462i
$36$		0			0
$37$	16.0000			0.432432			0.216216	−	0.976346i	$-0.430628\pi$
$37$	16.0000			0.432432			0.216216	+	0.976346i	$0.430628\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	56.8683i		−	1.38703i	−0.720442	−	0.693515i	$-0.756061\pi$
$41$		−	56.8683i		−	1.38703i	0.720442	−	0.693515i	$-0.243939\pi$
$42$		0			0
$43$	−52.0000			−1.20930			−0.604651	−	0.796490i	$-0.706687\pi$
$43$	−52.0000			−1.20930			−0.604651	+	0.796490i	$0.706687\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	32.4962i		−	0.691408i	−0.938344	−	0.345704i	$-0.887640\pi$
$47$		−	32.4962i		−	0.691408i	0.938344	−	0.345704i	$-0.112360\pi$
$48$		0			0
$49$	−17.0000	+	45.9565i	−0.346939	+	0.937888i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	16.9706			0.320199			0.160100	−	0.987101i	$-0.448818\pi$
$53$	16.9706			0.320199			0.160100	+	0.987101i	$0.448818\pi$
$54$		0			0
$55$			103.402i			1.88004i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			32.4962i			0.550782i	0.961332	+	0.275391i	$0.0888074\pi$
$59$			32.4962i			0.550782i	−0.961332	+	0.275391i	$0.911193\pi$
$60$		0			0
$61$		−	22.9783i		−	0.376693i	−0.982103	−	0.188346i	$-0.939687\pi$
$61$		−	22.9783i		−	0.376693i	0.982103	−	0.188346i	$-0.0603127\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−186.676			−2.87194
$66$		0			0
$67$	52.0000			0.776119			0.388060	−	0.921634i	$-0.373145\pi$
$67$	52.0000			0.776119			0.388060	+	0.921634i	$0.373145\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−89.0955			−1.25487			−0.627433	−	0.778671i	$-0.715894\pi$
$71$	−89.0955			−1.25487			−0.627433	+	0.778671i	$0.715894\pi$
$72$		0			0
$73$			45.9565i			0.629541i	0.949168	+	0.314771i	$0.101928\pi$
$73$			45.9565i			0.629541i	−0.949168	+	0.314771i	$0.898072\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	50.9117	+	73.1163i	0.661191	+	0.949563i
$78$		0			0
$79$	−104.000			−1.31646			−0.658228	−	0.752819i	$-0.728694\pi$
$79$	−104.000			−1.31646			−0.658228	+	0.752819i	$0.728694\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	162.481i		−	1.95760i	−0.204819	−	0.978800i	$-0.565661\pi$
$83$		−	162.481i		−	1.95760i	0.204819	−	0.978800i	$-0.434339\pi$
$84$		0			0
$85$	−66.0000			−0.776471
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	73.1163i		−	0.821532i	−0.911741	−	0.410766i	$-0.865261\pi$
$89$		−	73.1163i		−	0.821532i	0.911741	−	0.410766i	$-0.134739\pi$
$90$		0			0
$91$	−132.000	+	91.9130i	−1.45055	+	1.01003i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	93.3381			0.982506
$96$		0			0
$97$			91.9130i			0.947557i	0.880644	+	0.473778i	$0.157110\pi$
$97$			91.9130i			0.947557i	−0.880644	+	0.473778i	$0.842890\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			121.861i			1.20654i	0.797537	+	0.603270i	$0.206136\pi$
$101$			121.861i			1.20654i	−0.797537	+	0.603270i	$0.793864\pi$
$102$		0			0
$103$			91.9130i			0.892359i	0.894943	+	0.446180i	$0.147216\pi$
$103$			91.9130i			0.892359i	−0.894943	+	0.446180i	$0.852784\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	123.037			1.14987			0.574937	−	0.818197i	$-0.305026\pi$
$107$	123.037			1.14987			0.574937	+	0.818197i	$0.305026\pi$
$108$		0			0
$109$	−62.0000			−0.568807			−0.284404	−	0.958705i	$-0.591796\pi$
$109$	−62.0000			−0.568807			−0.284404	+	0.958705i	$0.591796\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	118.794			1.05127			0.525637	−	0.850709i	$-0.323827\pi$
$113$	118.794			1.05127			0.525637	+	0.850709i	$0.323827\pi$
$114$		0			0
$115$			172.337i			1.49858i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−46.6690	+	32.4962i	−0.392177	+	0.273077i
$120$		0			0
$121$	41.0000			0.338843
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	129.985i		−	1.03988i
$126$		0			0
$127$	64.0000			0.503937			0.251969	−	0.967735i	$-0.418922\pi$
$127$	64.0000			0.503937			0.251969	+	0.967735i	$0.418922\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			97.4885i			0.744187i	0.928195	+	0.372093i	$0.121360\pi$
$131$			97.4885i			0.744187i	−0.928195	+	0.372093i	$0.878640\pi$
$132$		0			0
$133$	66.0000	−	45.9565i	0.496241	−	0.345538i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−33.9411			−0.247745			−0.123873	−	0.992298i	$-0.539531\pi$
$137$	−33.9411			−0.247745			−0.123873	+	0.992298i	$0.539531\pi$
$138$		0			0
$139$		−	183.826i		−	1.32249i	−0.750170	−	0.661245i	$-0.770029\pi$
$139$		−	183.826i		−	1.32249i	0.750170	−	0.661245i	$-0.229971\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			292.465i			2.04521i
$144$		0			0
$145$			275.739i			1.90165i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	186.676			1.25286			0.626430	−	0.779478i	$-0.284515\pi$
$149$	186.676			1.25286			0.626430	+	0.779478i	$0.284515\pi$
$150$		0			0
$151$	32.0000			0.211921			0.105960	−	0.994370i	$-0.466208\pi$
$151$	32.0000			0.211921			0.105960	+	0.994370i	$0.466208\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			68.9348i			0.439075i	0.975604	+	0.219537i	$0.0704548\pi$
$157$			68.9348i			0.439075i	−0.975604	+	0.219537i	$0.929545\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	84.8528	+	121.861i	0.527036	+	0.756898i
$162$		0			0
$163$	124.000			0.760736			0.380368	−	0.924835i	$-0.375797\pi$
$163$	124.000			0.760736			0.380368	+	0.924835i	$0.375797\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	129.985i		−	0.778351i	−0.921164	−	0.389175i	$-0.872760\pi$
$167$		−	129.985i		−	0.778351i	0.921164	−	0.389175i	$-0.127240\pi$
$168$		0			0
$169$	−359.000			−2.12426
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	73.1163i		−	0.422638i	−0.977417	−	0.211319i	$-0.932224\pi$
$173$		−	73.1163i		−	0.422638i	0.977417	−	0.211319i	$-0.0677759\pi$
$174$		0			0
$175$	−164.000	−	235.527i	−0.937143	−	1.34587i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−156.978			−0.876970			−0.438485	−	0.898738i	$-0.644485\pi$
$179$	−156.978			−0.876970			−0.438485	+	0.898738i	$0.644485\pi$
$180$		0			0
$181$		−	252.761i		−	1.39647i	−0.715869	−	0.698234i	$-0.753969\pi$
$181$		−	252.761i		−	1.39647i	0.715869	−	0.698234i	$-0.246031\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			129.985i			0.702620i
$186$		0			0
$187$			103.402i			0.552953i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−12.7279			−0.0666383			−0.0333192	−	0.999445i	$-0.510608\pi$
$191$	−12.7279			−0.0666383			−0.0333192	+	0.999445i	$0.510608\pi$
$192$		0			0
$193$	−224.000			−1.16062			−0.580311	−	0.814395i	$-0.697069\pi$
$193$	−224.000			−1.16062			−0.580311	+	0.814395i	$0.697069\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−305.470			−1.55061			−0.775305	−	0.631587i	$-0.782404\pi$
$197$	−305.470			−1.55061			−0.775305	+	0.631587i	$0.782404\pi$
$198$		0			0
$199$		−	264.250i		−	1.32789i	−0.747782	−	0.663944i	$-0.768881\pi$
$199$		−	264.250i		−	1.32789i	0.747782	−	0.663944i	$-0.231119\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	135.765	+	194.977i	0.668791	+	0.960477i
$204$		0			0
$205$	462.000			2.25366
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	146.233i		−	0.699678i
$210$		0			0
$211$	−124.000			−0.587678			−0.293839	−	0.955855i	$-0.594933\pi$
$211$	−124.000			−0.587678			−0.293839	+	0.955855i	$0.594933\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	422.450i		−	1.96488i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−186.676			−0.844689
$222$		0			0
$223$			172.337i			0.772811i	0.922329	+	0.386406i	$0.126283\pi$
$223$			172.337i			0.772811i	−0.922329	+	0.386406i	$0.873717\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	389.954i		−	1.71786i	−0.512094	−	0.858929i	$-0.671130\pi$
$227$		−	389.954i		−	1.71786i	0.512094	−	0.858929i	$-0.328870\pi$
$228$		0			0
$229$		−	206.804i		−	0.903075i	−0.892252	−	0.451538i	$-0.850876\pi$
$229$		−	206.804i		−	0.903075i	0.892252	−	0.451538i	$-0.149124\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	186.676			0.801185			0.400593	−	0.916256i	$-0.368804\pi$
$233$	186.676			0.801185			0.400593	+	0.916256i	$0.368804\pi$
$234$		0			0
$235$	264.000			1.12340
$236$		0			0
$237$		0			0
$238$		0			0
$239$	258.801			1.08285			0.541425	−	0.840749i	$-0.317885\pi$
$239$	258.801			1.08285			0.541425	+	0.840749i	$0.317885\pi$
$240$		0			0
$241$			45.9565i			0.190691i	0.995444	+	0.0953454i	$0.0303956\pi$
$241$			45.9565i			0.190691i	−0.995444	+	0.0953454i	$0.969604\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−373.352	−	138.109i	−1.52389	−	0.563709i
$246$		0			0
$247$	264.000			1.06883
$248$		0			0
$249$		0			0
$250$		0			0
$251$			64.9923i			0.258933i	0.991584	+	0.129467i	$0.0413265\pi$
$251$			64.9923i			0.258933i	−0.991584	+	0.129467i	$0.958673\pi$
$252$		0			0
$253$	270.000			1.06719
$254$		0			0
$255$		0			0
$256$		0			0
$257$			268.093i			1.04316i	0.853201	+	0.521582i	$0.174658\pi$
$257$			268.093i			1.04316i	−0.853201	+	0.521582i	$0.825342\pi$
$258$		0			0
$259$	64.0000	+	91.9130i	0.247104	+	0.354876i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−428.507			−1.62930			−0.814652	−	0.579951i	$-0.803072\pi$
$263$	−428.507			−1.62930			−0.814652	+	0.579951i	$0.803072\pi$
$264$		0			0
$265$			137.870i			0.520262i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	203.101i		−	0.755022i	−0.926005	−	0.377511i	$-0.876780\pi$
$269$		−	203.101i		−	0.755022i	0.926005	−	0.377511i	$-0.123220\pi$
$270$		0			0
$271$		−	91.9130i		−	0.339162i	−0.985516	−	0.169581i	$-0.945759\pi$
$271$		−	91.9130i		−	0.339162i	0.985516	−	0.169581i	$-0.0542415\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−521.845			−1.89762
$276$		0			0
$277$	272.000			0.981949			0.490975	−	0.871174i	$-0.336641\pi$
$277$	272.000			0.981949			0.490975	+	0.871174i	$0.336641\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	50.9117			0.181180			0.0905902	−	0.995888i	$-0.471125\pi$
$281$	50.9117			0.181180			0.0905902	+	0.995888i	$0.471125\pi$
$282$		0			0
$283$			103.402i			0.365379i	0.983171	+	0.182689i	$0.0584802\pi$
$283$			103.402i			0.365379i	−0.983171	+	0.182689i	$0.941520\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	326.683	−	227.473i	1.13827	−	0.792589i
$288$		0			0
$289$	223.000			0.771626
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	8.12404i		−	0.0277271i	−0.999904	−	0.0138635i	$-0.995587\pi$
$293$		−	8.12404i		−	0.0277271i	0.999904	−	0.0138635i	$-0.00441305\pi$
$294$		0			0
$295$	−264.000			−0.894915
$296$		0			0
$297$		0			0
$298$		0			0
$299$			487.442i			1.63024i
$300$		0			0
$301$	−208.000	−	298.717i	−0.691030	−	0.992416i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	186.676			0.612053
$306$		0			0
$307$		−	379.141i		−	1.23499i	−0.786576	−	0.617494i	$-0.788148\pi$
$307$		−	379.141i		−	1.23499i	0.786576	−	0.617494i	$-0.211852\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			357.458i			1.14938i	0.818371	+	0.574691i	$0.194878\pi$
$311$			357.458i			1.14938i	−0.818371	+	0.574691i	$0.805122\pi$
$312$		0			0
$313$		−	183.826i		−	0.587304i	−0.955912	−	0.293652i	$-0.905129\pi$
$313$		−	183.826i		−	0.587304i	0.955912	−	0.293652i	$-0.0948706\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	356.382			1.12423			0.562116	−	0.827058i	$-0.309987\pi$
$317$	356.382			1.12423			0.562116	+	0.827058i	$0.309987\pi$
$318$		0			0
$319$	432.000			1.35423
$320$		0			0
$321$		0			0
$322$		0			0
$323$	93.3381			0.288972
$324$		0			0
$325$		−	942.108i		−	2.89879i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	186.676	−	129.985i	0.567405	−	0.395090i
$330$		0			0
$331$	−4.00000			−0.0120846			−0.00604230	−	0.999982i	$-0.501923\pi$
$331$	−4.00000			−0.0120846			−0.00604230	+	0.999982i	$0.501923\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			422.450i			1.26104i
$336$		0			0
$337$	82.0000			0.243323			0.121662	−	0.992572i	$-0.461178\pi$
$337$	82.0000			0.243323			0.121662	+	0.992572i	$0.461178\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−332.000	+	86.1684i	−0.967930	+	0.251220i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	555.786			1.60169			0.800844	−	0.598873i	$-0.204384\pi$
$347$	555.786			1.60169			0.800844	+	0.598873i	$0.204384\pi$
$348$		0			0
$349$		−	160.848i		−	0.460882i	−0.973086	−	0.230441i	$-0.925983\pi$
$349$		−	160.848i		−	0.460882i	0.973086	−	0.230441i	$-0.0740168\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	138.109i		−	0.391243i	−0.980680	−	0.195621i	$-0.937328\pi$
$353$		−	138.109i		−	0.391243i	0.980680	−	0.195621i	$-0.0626723\pi$
$354$		0			0
$355$		−	723.815i		−	2.03892i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−224.860			−0.626351			−0.313175	−	0.949695i	$-0.601393\pi$
$359$	−224.860			−0.626351			−0.313175	+	0.949695i	$0.601393\pi$
$360$		0			0
$361$	229.000			0.634349
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−373.352			−1.02288
$366$		0			0
$367$		−	195.315i		−	0.532194i	−0.963946	−	0.266097i	$-0.914266\pi$
$367$		−	195.315i		−	0.532194i	0.963946	−	0.266097i	$-0.0857341\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	67.8823	+	97.4885i	0.182971	+	0.262772i
$372$		0			0
$373$	530.000			1.42091			0.710456	−	0.703742i	$-0.248489\pi$
$373$	530.000			1.42091			0.710456	+	0.703742i	$0.248489\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			779.908i			2.06872i
$378$		0			0
$379$	−196.000			−0.517150			−0.258575	−	0.965991i	$-0.583253\pi$
$379$	−196.000			−0.517150			−0.258575	+	0.965991i	$0.583253\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			32.4962i			0.0848464i	0.999100	+	0.0424232i	$0.0135078\pi$
$383$			32.4962i			0.0848464i	−0.999100	+	0.0424232i	$0.986492\pi$
$384$		0			0
$385$	−594.000	+	413.609i	−1.54286	+	1.07431i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	644.881			1.65779			0.828896	−	0.559402i	$-0.188969\pi$
$389$	644.881			1.65779			0.828896	+	0.559402i	$0.188969\pi$
$390$		0			0
$391$			172.337i			0.440759i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	844.900i		−	2.13899i
$396$		0			0
$397$		−	758.282i		−	1.91003i	−0.296556	−	0.955015i	$-0.595838\pi$
$397$		−	758.282i		−	1.91003i	0.296556	−	0.955015i	$-0.404162\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	33.9411			0.0846412			0.0423206	−	0.999104i	$-0.486525\pi$
$401$	33.9411			0.0846412			0.0423206	+	0.999104i	$0.486525\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	203.647			0.500361
$408$		0			0
$409$			735.304i			1.79781i	0.438144	+	0.898905i	$0.355636\pi$
$409$			735.304i			1.79781i	−0.438144	+	0.898905i	$0.644364\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−186.676	+	129.985i	−0.452000	+	0.314733i
$414$		0			0
$415$	1320.00			3.18072
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	32.4962i		−	0.0775565i	−0.999248	−	0.0387782i	$-0.987653\pi$
$419$		−	32.4962i		−	0.0775565i	0.999248	−	0.0387782i	$-0.0123466\pi$
$420$		0			0
$421$	−496.000			−1.17815			−0.589074	−	0.808079i	$-0.700507\pi$
$421$	−496.000			−1.17815			−0.589074	+	0.808079i	$0.700507\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	333.086i		−	0.783731i
$426$		0			0
$427$	132.000	−	91.9130i	0.309133	−	0.215253i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	462.448			1.07296			0.536482	−	0.843912i	$-0.319753\pi$
$431$	462.448			1.07296			0.536482	+	0.843912i	$0.319753\pi$
$432$		0			0
$433$			137.870i			0.318405i	0.987246	+	0.159203i	$0.0508923\pi$
$433$			137.870i			0.318405i	−0.987246	+	0.159203i	$0.949108\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	243.721i		−	0.557714i
$438$		0			0
$439$		−	172.337i		−	0.392567i	−0.980547	−	0.196283i	$-0.937113\pi$
$439$		−	172.337i		−	0.392567i	0.980547	−	0.196283i	$-0.0628873\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	394.566			0.890667			0.445334	−	0.895365i	$-0.353085\pi$
$443$	394.566			0.890667			0.445334	+	0.895365i	$0.353085\pi$
$444$		0			0
$445$	594.000			1.33483
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−50.9117			−0.113389			−0.0566945	−	0.998392i	$-0.518056\pi$
$449$	−50.9117			−0.113389			−0.0566945	+	0.998392i	$0.518056\pi$
$450$		0			0
$451$		−	723.815i		−	1.60491i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−746.705	−	1072.37i	−1.64111	−	2.35686i
$456$		0			0
$457$	130.000			0.284464			0.142232	−	0.989833i	$-0.454572\pi$
$457$	130.000			0.284464			0.142232	+	0.989833i	$0.454572\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			723.039i			1.56842i	0.620499	+	0.784208i	$0.286930\pi$
$461$			723.039i			1.56842i	−0.620499	+	0.784208i	$0.713070\pi$
$462$		0			0
$463$	−80.0000			−0.172786			−0.0863931	−	0.996261i	$-0.527534\pi$
$463$	−80.0000			−0.172786			−0.0863931	+	0.996261i	$0.527534\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			194.977i			0.417509i	0.977968	+	0.208755i	$0.0669410\pi$
$467$			194.977i			0.417509i	−0.977968	+	0.208755i	$0.933059\pi$
$468$		0			0
$469$	208.000	+	298.717i	0.443497	+	0.636924i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−661.852			−1.39926
$474$		0			0
$475$			471.054i			0.991693i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			422.450i			0.881942i	0.897521	+	0.440971i	$0.145366\pi$
$479$			422.450i			0.881942i	−0.897521	+	0.440971i	$0.854634\pi$
$480$		0			0
$481$			367.652i			0.764349i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−746.705			−1.53960
$486$		0			0
$487$	−904.000			−1.85626			−0.928131	−	0.372253i	$-0.878585\pi$
$487$	−904.000			−1.85626			−0.928131	+	0.372253i	$0.878585\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−420.021			−0.855441			−0.427720	−	0.903911i	$-0.640683\pi$
$491$	−420.021			−0.855441			−0.427720	+	0.903911i	$0.640683\pi$
$492$		0			0
$493$			275.739i			0.559308i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−356.382	−	511.814i	−0.717066	−	1.02981i
$498$		0			0
$499$	716.000			1.43487			0.717435	−	0.696626i	$-0.245316\pi$
$499$	716.000			1.43487			0.717435	+	0.696626i	$0.245316\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			812.404i			1.61512i	0.589787	+	0.807558i	$0.299212\pi$
$503$			812.404i			1.61512i	−0.589787	+	0.807558i	$0.700788\pi$
$504$		0			0
$505$	−990.000			−1.96040
$506$		0			0
$507$		0			0
$508$		0			0
$509$			186.853i			0.367098i	0.983011	+	0.183549i	$0.0587586\pi$
$509$			186.853i			0.367098i	−0.983011	+	0.183549i	$0.941241\pi$
$510$		0			0
$511$	−264.000	+	183.826i	−0.516634	+	0.359738i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−746.705			−1.44991
$516$		0			0
$517$		−	413.609i		−	0.800016i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	593.055i		−	1.13830i	−0.822233	−	0.569150i	$-0.807272\pi$
$521$		−	593.055i		−	1.13830i	0.822233	−	0.569150i	$-0.192728\pi$
$522$		0			0
$523$			735.304i			1.40594i	0.711222	+	0.702968i	$0.248142\pi$
$523$			735.304i			1.40594i	−0.711222	+	0.702968i	$0.751858\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−79.0000			−0.149338
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1306.73			2.45166
$534$		0			0
$535$			999.554i			1.86833i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−216.375	+	584.931i	−0.401437	+	1.08521i
$540$		0			0
$541$	272.000			0.502773			0.251386	−	0.967887i	$-0.419114\pi$
$541$	272.000			0.502773			0.251386	+	0.967887i	$0.419114\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	503.690i		−	0.924203i
$546$		0			0
$547$	−644.000			−1.17733			−0.588665	−	0.808377i	$-0.700346\pi$
$547$	−644.000			−1.17733			−0.588665	+	0.808377i	$0.700346\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	389.954i		−	0.707720i
$552$		0			0
$553$	−416.000	−	597.435i	−0.752260	−	1.08035i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	152.735			0.274210			0.137105	−	0.990557i	$-0.456220\pi$
$557$	152.735			0.274210			0.137105	+	0.990557i	$0.456220\pi$
$558$		0			0
$559$		−	1194.87i		−	2.13751i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	877.396i		−	1.55843i	−0.626757	−	0.779215i	$-0.715618\pi$
$563$		−	877.396i		−	1.55843i	0.626757	−	0.779215i	$-0.284382\pi$
$564$		0			0
$565$			965.087i			1.70812i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−933.381			−1.64039			−0.820194	−	0.572085i	$-0.806135\pi$
$569$	−933.381			−1.64039			−0.820194	+	0.572085i	$0.806135\pi$
$570$		0			0
$571$	−260.000			−0.455342			−0.227671	−	0.973738i	$-0.573111\pi$
$571$	−260.000			−0.455342			−0.227671	+	0.973738i	$0.573111\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−869.741			−1.51259
$576$		0			0
$577$		−	505.522i		−	0.876120i	−0.898946	−	0.438060i	$-0.855666\pi$
$577$		−	505.522i		−	0.876120i	0.898946	−	0.438060i	$-0.144334\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	933.381	−	649.923i	1.60651	−	1.11863i
$582$		0			0
$583$	216.000			0.370497
$584$		0			0
$585$		0			0
$586$		0			0
$587$			194.977i			0.332158i	0.986112	+	0.166079i	$0.0531107\pi$
$587$			194.977i			0.332158i	−0.986112	+	0.166079i	$0.946889\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			138.109i			0.232898i	0.993197	+	0.116449i	$0.0371512\pi$
$593$			138.109i			0.232898i	−0.993197	+	0.116449i	$0.962849\pi$
$594$		0			0
$595$	−264.000	−	379.141i	−0.443697	−	0.637212i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	453.963			0.757867			0.378934	−	0.925424i	$-0.376291\pi$
$599$	453.963			0.757867			0.378934	+	0.925424i	$0.376291\pi$
$600$		0			0
$601$			827.217i			1.37640i	0.725521	+	0.688201i	$0.241599\pi$
$601$			827.217i			1.37640i	−0.725521	+	0.688201i	$0.758401\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			333.086i			0.550555i
$606$		0			0
$607$			356.163i			0.586759i	0.955996	+	0.293380i	$0.0947800\pi$
$607$			356.163i			0.586759i	−0.955996	+	0.293380i	$0.905220\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	746.705			1.22210
$612$		0			0
$613$	878.000			1.43230			0.716150	−	0.697946i	$-0.245903\pi$
$613$	878.000			1.43230			0.716150	+	0.697946i	$0.245903\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−101.823			−0.165030			−0.0825149	−	0.996590i	$-0.526295\pi$
$617$	−101.823			−0.165030			−0.0825149	+	0.996590i	$0.526295\pi$
$618$		0			0
$619$			735.304i			1.18789i	0.804506	+	0.593945i	$0.202430\pi$
$619$			735.304i			1.18789i	−0.804506	+	0.593945i	$0.797570\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	420.021	−	292.465i	0.674192	−	0.469447i
$624$		0			0
$625$	31.0000			0.0496000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			129.985i			0.206653i
$630$		0			0
$631$	16.0000			0.0253566			0.0126783	−	0.999920i	$-0.495964\pi$
$631$	16.0000			0.0253566			0.0126783	+	0.999920i	$0.495964\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			519.938i			0.818801i
$636$		0			0
$637$	−1056.00	−	390.630i	−1.65777	−	0.613234i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	67.8823			0.105901			0.0529503	−	0.998597i	$-0.483138\pi$
$641$	67.8823			0.105901			0.0529503	+	0.998597i	$0.483138\pi$
$642$		0			0
$643$			930.619i			1.44731i	0.690163	+	0.723654i	$0.257539\pi$
$643$			930.619i			1.44731i	−0.690163	+	0.723654i	$0.742461\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			584.931i			0.904066i	0.892001	+	0.452033i	$0.149301\pi$
$647$			584.931i			0.904066i	−0.892001	+	0.452033i	$0.850699\pi$
$648$		0			0
$649$			413.609i			0.637301i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−84.8528			−0.129943			−0.0649715	−	0.997887i	$-0.520696\pi$
$653$	−84.8528			−0.129943			−0.0649715	+	0.997887i	$0.520696\pi$
$654$		0			0
$655$	−792.000			−1.20916
$656$		0			0
$657$		0			0
$658$		0			0
$659$	700.036			1.06227			0.531135	−	0.847287i	$-0.321766\pi$
$659$	700.036			1.06227			0.531135	+	0.847287i	$0.321766\pi$
$660$		0			0
$661$		−	114.891i		−	0.173814i	−0.996216	−	0.0869072i	$-0.972302\pi$
$661$		−	114.891i		−	0.173814i	0.996216	−	0.0869072i	$-0.0276983\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	373.352	+	536.187i	0.561432	+	0.806296i
$666$		0			0
$667$	720.000			1.07946
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	292.465i		−	0.435865i
$672$		0			0
$673$	−368.000			−0.546805			−0.273403	−	0.961900i	$-0.588149\pi$
$673$	−368.000			−0.546805			−0.273403	+	0.961900i	$0.588149\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			1048.00i			1.54801i	0.633181	+	0.774004i	$0.281749\pi$
$677$			1048.00i			1.54801i	−0.633181	+	0.774004i	$0.718251\pi$
$678$		0			0
$679$	−528.000	+	367.652i	−0.777614	+	0.541461i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	258.801			0.378918			0.189459	−	0.981889i	$-0.439327\pi$
$683$	258.801			0.378918			0.189459	+	0.981889i	$0.439327\pi$
$684$		0			0
$685$		−	275.739i		−	0.402539i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			389.954i			0.565971i
$690$		0			0
$691$			1102.96i			1.59617i	0.602542	+	0.798087i	$0.294154\pi$
$691$			1102.96i			1.59617i	−0.602542	+	0.798087i	$0.705846\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	1493.41			2.14879
$696$		0			0
$697$	462.000			0.662841
$698$		0			0
$699$		0			0
$700$		0			0
$701$	390.323			0.556809			0.278404	−	0.960464i	$-0.410194\pi$
$701$	390.323			0.556809			0.278404	+	0.960464i	$0.410194\pi$
$702$		0			0
$703$		−	183.826i		−	0.261488i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−700.036	+	487.442i	−0.990150	+	0.689452i
$708$		0			0
$709$	238.000			0.335684			0.167842	−	0.985814i	$-0.446320\pi$
$709$	238.000			0.335684			0.167842	+	0.985814i	$0.446320\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−2376.00			−3.32308
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1137.37i		−	1.58187i	−0.611899	−	0.790936i	$-0.709594\pi$
$719$		−	1137.37i		−	1.58187i	0.611899	−	0.790936i	$-0.290406\pi$
$720$		0			0
$721$	−528.000	+	367.652i	−0.732316	+	0.509920i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1391.59			−1.91943
$726$		0			0
$727$			275.739i			0.379283i	0.981853	+	0.189642i	$0.0607326\pi$
$727$			275.739i			0.379283i	−0.981853	+	0.189642i	$0.939267\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	422.450i		−	0.577907i
$732$		0			0
$733$		−	206.804i		−	0.282134i	−0.990000	−	0.141067i	$-0.954947\pi$
$733$		−	206.804i		−	0.282134i	0.990000	−	0.141067i	$-0.0450533\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	661.852			0.898035
$738$		0			0
$739$	1348.00			1.82409			0.912043	−	0.410094i	$-0.134504\pi$
$739$	1348.00			1.82409			0.912043	+	0.410094i	$0.134504\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−182.434			−0.245536			−0.122768	−	0.992435i	$-0.539177\pi$
$743$	−182.434			−0.245536			−0.122768	+	0.992435i	$0.539177\pi$
$744$		0			0
$745$			1516.56i			2.03566i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	492.146	+	706.791i	0.657071	+	0.943647i
$750$		0			0
$751$	−232.000			−0.308921			−0.154461	−	0.987999i	$-0.549364\pi$
$751$	−232.000			−0.308921			−0.154461	+	0.987999i	$0.549364\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			259.969i			0.344330i
$756$		0			0
$757$	−434.000			−0.573316			−0.286658	−	0.958033i	$-0.592544\pi$
$757$	−434.000			−0.573316			−0.286658	+	0.958033i	$0.592544\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	56.8683i		−	0.0747283i	−0.999302	−	0.0373642i	$-0.988104\pi$
$761$		−	56.8683i		−	0.0747283i	0.999302	−	0.0373642i	$-0.0118962\pi$
$762$		0			0
$763$	−248.000	−	356.163i	−0.325033	−	0.466793i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−746.705			−0.973539
$768$		0			0
$769$		−	505.522i		−	0.657375i	−0.944439	−	0.328688i	$-0.893394\pi$
$769$		−	505.522i		−	0.657375i	0.944439	−	0.328688i	$-0.106606\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			8.12404i			0.0105098i	0.999986	+	0.00525488i	$0.00167269\pi$
$773$			8.12404i			0.0105098i	−0.999986	+	0.00525488i	$0.998327\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−653.367			−0.838725
$780$		0			0
$781$	−1134.00			−1.45198
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−560.029			−0.713412
$786$		0			0
$787$		−	919.130i		−	1.16789i	−0.811793	−	0.583945i	$-0.801508\pi$
$787$		−	919.130i		−	1.16789i	0.811793	−	0.583945i	$-0.198492\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	475.176	+	682.419i	0.600728	+	0.862730i
$792$		0			0
$793$	528.000			0.665826
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	706.791i		−	0.886815i	−0.896320	−	0.443407i	$-0.853770\pi$
$797$		−	706.791i		−	0.886815i	0.896320	−	0.443407i	$-0.146230\pi$
$798$		0			0
$799$	264.000			0.330413
$800$		0			0
$801$		0			0
$802$		0			0
$803$			584.931i			0.728432i
$804$		0			0
$805$	−990.000	+	689.348i	−1.22981	+	0.856332i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	899.440			1.11179			0.555896	−	0.831252i	$-0.312375\pi$
$809$	899.440			1.11179			0.555896	+	0.831252i	$0.312375\pi$
$810$		0			0
$811$			1102.96i			1.36000i	0.733214	+	0.679998i	$0.238019\pi$
$811$			1102.96i			1.36000i	−0.733214	+	0.679998i	$0.761981\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			1007.38i			1.23605i
$816$		0			0
$817$			597.435i			0.731254i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−186.676			−0.227377			−0.113688	−	0.993516i	$-0.536267\pi$
$821$	−186.676			−0.227377			−0.113688	+	0.993516i	$0.536267\pi$
$822$		0			0
$823$	424.000			0.515188			0.257594	−	0.966253i	$-0.417070\pi$
$823$	424.000			0.515188			0.257594	+	0.966253i	$0.417070\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1073.39			−1.29793			−0.648965	−	0.760818i	$-0.724798\pi$
$827$	−1073.39			−1.29793			−0.648965	+	0.760818i	$0.724798\pi$
$828$		0			0
$829$			850.195i			1.02557i	0.858518	+	0.512784i	$0.171386\pi$
$829$			850.195i			1.02557i	−0.858518	+	0.512784i	$0.828614\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−373.352	−	138.109i	−0.448202	−	0.165797i
$834$		0			0
$835$	1056.00			1.26467
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1299.85i		−	1.54928i	−0.632402	−	0.774640i	$-0.717931\pi$
$839$		−	1299.85i		−	1.54928i	0.632402	−	0.774640i	$-0.282069\pi$
$840$		0			0
$841$	311.000			0.369798
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	2916.53i		−	3.45151i
$846$		0			0
$847$	164.000	+	235.527i	0.193625	+	0.278072i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	339.411			0.398838
$852$		0			0
$853$			1401.67i			1.64323i	0.570044	+	0.821614i	$0.306926\pi$
$853$			1401.67i			1.64323i	−0.570044	+	0.821614i	$0.693074\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	186.853i		−	0.218031i	−0.994040	−	0.109016i	$-0.965230\pi$
$857$		−	186.853i		−	0.218031i	0.994040	−	0.109016i	$-0.0347699\pi$
$858$		0			0
$859$		−	471.054i		−	0.548375i	−0.961676	−	0.274188i	$-0.911591\pi$
$859$		−	471.054i		−	0.548375i	0.961676	−	0.274188i	$-0.0884089\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1166.73			−1.35194			−0.675971	−	0.736928i	$-0.736276\pi$
$863$	−1166.73			−1.35194			−0.675971	+	0.736928i	$0.736276\pi$
$864$		0			0
$865$	594.000			0.686705
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1323.70			−1.52325
$870$		0			0
$871$			1194.87i			1.37184i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	746.705	−	519.938i	0.853377	−	0.594215i
$876$		0			0
$877$	350.000			0.399088			0.199544	−	0.979889i	$-0.436054\pi$
$877$	350.000			0.399088			0.199544	+	0.979889i	$0.436054\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	268.093i		−	0.304306i	−0.988357	−	0.152153i	$-0.951379\pi$
$881$		−	268.093i		−	0.304306i	0.988357	−	0.152153i	$-0.0486206\pi$
$882$		0			0
$883$	−412.000			−0.466591			−0.233296	−	0.972406i	$-0.574951\pi$
$883$	−412.000			−0.466591			−0.233296	+	0.972406i	$0.574951\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			1039.88i			1.17235i	0.810183	+	0.586176i	$0.199367\pi$
$887$			1039.88i			1.17235i	−0.810183	+	0.586176i	$0.800633\pi$
$888$		0			0
$889$	256.000	+	367.652i	0.287964	+	0.413557i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−373.352			−0.418088
$894$		0			0
$895$		−	1275.29i		−	1.42491i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$			137.870i			0.153018i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2053.44			2.26899
$906$		0			0
$907$	−436.000			−0.480706			−0.240353	−	0.970686i	$-0.577263\pi$
$907$	−436.000			−0.480706			−0.240353	+	0.970686i	$0.577263\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−326.683			−0.358599			−0.179299	−	0.983795i	$-0.557383\pi$
$911$	−326.683			−0.358599			−0.179299	+	0.983795i	$0.557383\pi$
$912$		0			0
$913$		−	2068.04i		−	2.26511i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−560.029	+	389.954i	−0.610718	+	0.425250i
$918$		0			0
$919$	224.000			0.243743			0.121872	−	0.992546i	$-0.461110\pi$
$919$	224.000			0.243743			0.121872	+	0.992546i	$0.461110\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	2047.26i		−	2.21805i
$924$		0			0
$925$	−656.000			−0.709189
$926$		0			0
$927$		0			0
$928$		0			0
$929$			788.032i			0.848258i	0.905602	+	0.424129i	$0.139420\pi$
$929$			788.032i			0.848258i	−0.905602	+	0.424129i	$0.860580\pi$
$930$		0			0
$931$	528.000	+	195.315i	0.567132	+	0.209791i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−840.043			−0.898442
$936$		0			0
$937$		−	505.522i		−	0.539511i	−0.962929	−	0.269755i	$-0.913057\pi$
$937$		−	505.522i		−	0.539511i	0.962929	−	0.269755i	$-0.0869428\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			333.086i			0.353970i	0.984214	+	0.176985i	$0.0566344\pi$
$941$			333.086i			0.353970i	−0.984214	+	0.176985i	$0.943366\pi$
$942$		0			0
$943$		−	1206.36i		−	1.27928i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	46.6690			0.0492809			0.0246405	−	0.999696i	$-0.492156\pi$
$947$	46.6690			0.0492809			0.0246405	+	0.999696i	$0.492156\pi$
$948$		0			0
$949$	−1056.00			−1.11275
$950$		0			0
$951$		0			0
$952$		0			0
$953$	101.823			0.106845			0.0534225	−	0.998572i	$-0.482987\pi$
$953$	101.823			0.106845			0.0534225	+	0.998572i	$0.482987\pi$
$954$		0			0
$955$		−	103.402i		−	0.108274i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−135.765	−	194.977i	−0.141569	−	0.203313i
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1819.78i		−	1.88579i
$966$		0			0
$967$	664.000			0.686660			0.343330	−	0.939215i	$-0.388445\pi$
$967$	664.000			0.686660			0.343330	+	0.939215i	$0.388445\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	844.900i		−	0.870134i	−0.900398	−	0.435067i	$-0.856725\pi$
$971$		−	844.900i		−	0.870134i	0.900398	−	0.435067i	$-0.143275\pi$
$972$		0			0
$973$	1056.00	−	735.304i	1.08530	−	0.755708i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	424.264			0.434252			0.217126	−	0.976144i	$-0.430332\pi$
$977$	424.264			0.434252			0.217126	+	0.976144i	$0.430332\pi$
$978$		0			0
$979$		−	930.619i		−	0.950581i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1104.87i			1.12398i	0.827145	+	0.561988i	$0.189963\pi$
$983$			1104.87i			1.12398i	−0.827145	+	0.561988i	$0.810037\pi$
$984$		0			0
$985$		−	2481.65i		−	2.51944i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1103.09			−1.11536
$990$		0			0
$991$	−592.000			−0.597376			−0.298688	−	0.954351i	$-0.596549\pi$
$991$	−592.000			−0.597376			−0.298688	+	0.954351i	$0.596549\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2146.78			2.15756
$996$		0			0
$997$		−	344.674i		−	0.345711i	−0.984947	−	0.172855i	$-0.944701\pi$
$997$		−	344.674i		−	0.345711i	0.984947	−	0.172855i	$-0.0552993\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.3.f.i.433.4		4
3.2	odd	2	inner	1008.3.f.i.433.2		4
4.3	odd	2		126.3.c.a.55.2	yes	4
7.6	odd	2	inner	1008.3.f.i.433.1		4
12.11	even	2		126.3.c.a.55.3	yes	4
21.20	even	2	inner	1008.3.f.i.433.3		4
28.3	even	6		882.3.n.i.19.4		8
28.11	odd	6		882.3.n.i.19.3		8
28.19	even	6		882.3.n.i.325.3		8
28.23	odd	6		882.3.n.i.325.4		8
28.27	even	2		126.3.c.a.55.1	✓	4
84.11	even	6		882.3.n.i.19.2		8
84.23	even	6		882.3.n.i.325.1		8
84.47	odd	6		882.3.n.i.325.2		8
84.59	odd	6		882.3.n.i.19.1		8
84.83	odd	2		126.3.c.a.55.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.c.a.55.1	✓	4	28.27	even	2
126.3.c.a.55.2	yes	4	4.3	odd	2
126.3.c.a.55.3	yes	4	12.11	even	2
126.3.c.a.55.4	yes	4	84.83	odd	2
882.3.n.i.19.1		8	84.59	odd	6
882.3.n.i.19.2		8	84.11	even	6
882.3.n.i.19.3		8	28.11	odd	6
882.3.n.i.19.4		8	28.3	even	6
882.3.n.i.325.1		8	84.23	even	6
882.3.n.i.325.2		8	84.47	odd	6
882.3.n.i.325.3		8	28.19	even	6
882.3.n.i.325.4		8	28.23	odd	6
1008.3.f.i.433.1		4	7.6	odd	2	inner
1008.3.f.i.433.2		4	3.2	odd	2	inner
1008.3.f.i.433.3		4	21.20	even	2	inner
1008.3.f.i.433.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+8.12404i q^{5} +(4.00000 + 5.74456i) q^{7} +O(q^{10})\)	\(q+8.12404i q^{5} +(4.00000 + 5.74456i) q^{7} +12.7279 q^{11} +22.9783i q^{13} +8.12404i q^{17} -11.4891i q^{19} +21.2132 q^{23} -41.0000 q^{25} +33.9411 q^{29} +(-46.6690 + 32.4962i) q^{35} +16.0000 q^{37} -56.8683i q^{41} -52.0000 q^{43} -32.4962i q^{47} +(-17.0000 + 45.9565i) q^{49} +16.9706 q^{53} +103.402i q^{55} +32.4962i q^{59} -22.9783i q^{61} -186.676 q^{65} +52.0000 q^{67} -89.0955 q^{71} +45.9565i q^{73} +(50.9117 + 73.1163i) q^{77} -104.000 q^{79} -162.481i q^{83} -66.0000 q^{85} -73.1163i q^{89} +(-132.000 + 91.9130i) q^{91} +93.3381 q^{95} +91.9130i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{7}+O(q^{10}) \) 4 * q + 16 * q^7	\( 4 q + 16 q^{7} - 164 q^{25} + 64 q^{37} - 208 q^{43} - 68 q^{49} + 208 q^{67} - 416 q^{79} - 264 q^{85} - 528 q^{91}+O(q^{100}) \) 4 * q + 16 * q^7 - 164 * q^25 + 64 * q^37 - 208 * q^43 - 68 * q^49 + 208 * q^67 - 416 * q^79 - 264 * q^85 - 528 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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