LMFDB - Embedded newform 304.3.e.f.113.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,3,Mod(113,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("304.113");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	304.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.28340003655$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			113.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	304.113
Dual form			304.3.e.f.113.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.65685i q^{3} +7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} +O(q^{10})$	$q-5.65685i q^{3} +7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} -3.00000 q^{11} -11.3137i q^{13} -39.5980i q^{15} -17.0000 q^{17} +19.0000 q^{19} +62.2254i q^{21} -2.00000 q^{23} +24.0000 q^{25} +79.1960i q^{27} -39.5980i q^{29} -5.65685i q^{31} +16.9706i q^{33} -77.0000 q^{35} -39.5980i q^{37} -64.0000 q^{39} +39.5980i q^{41} +21.0000 q^{43} -161.000 q^{45} +5.00000 q^{47} +72.0000 q^{49} +96.1665i q^{51} +5.65685i q^{53} -21.0000 q^{55} -107.480i q^{57} -33.9411i q^{59} +23.0000 q^{61} +253.000 q^{63} -79.1960i q^{65} -39.5980i q^{67} +11.3137i q^{69} -90.5097i q^{71} +39.0000 q^{73} -135.765i q^{75} +33.0000 q^{77} +96.1665i q^{79} +241.000 q^{81} +6.00000 q^{83} -119.000 q^{85} -224.000 q^{87} +118.794i q^{89} +124.451i q^{91} -32.0000 q^{93} +133.000 q^{95} -169.706i q^{97} +69.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{5} - 22 q^{7} - 46 q^{9}+O(q^{10}) $ 2 * q + 14 * q^5 - 22 * q^7 - 46 * q^9	$ 2 q + 14 q^{5} - 22 q^{7} - 46 q^{9} - 6 q^{11} - 34 q^{17} + 38 q^{19} - 4 q^{23} + 48 q^{25} - 154 q^{35} - 128 q^{39} + 42 q^{43} - 322 q^{45} + 10 q^{47} + 144 q^{49} - 42 q^{55} + 46 q^{61} + 506 q^{63} + 78 q^{73} + 66 q^{77} + 482 q^{81} + 12 q^{83} - 238 q^{85} - 448 q^{87} - 64 q^{93} + 266 q^{95} + 138 q^{99}+O(q^{100}) $ 2 * q + 14 * q^5 - 22 * q^7 - 46 * q^9 - 6 * q^11 - 34 * q^17 + 38 * q^19 - 4 * q^23 + 48 * q^25 - 154 * q^35 - 128 * q^39 + 42 * q^43 - 322 * q^45 + 10 * q^47 + 144 * q^49 - 42 * q^55 + 46 * q^61 + 506 * q^63 + 78 * q^73 + 66 * q^77 + 482 * q^81 + 12 * q^83 - 238 * q^85 - 448 * q^87 - 64 * q^93 + 266 * q^95 + 138 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/304\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 5.65685i	− 1.88562i	−0.333333	−	0.942809i	$-0.608173\pi$
$3$	− 5.65685i	− 1.88562i	0.333333	−	0.942809i	$-0.391827\pi$
$4$	0	0
$5$	7.00000	1.40000	0.700000	−	0.714143i	$-0.253183\pi$
$5$	7.00000	1.40000	0.700000	+	0.714143i	$0.253183\pi$
$6$	0	0
$7$	−11.0000	−1.57143	−0.785714	−	0.618590i	$-0.787704\pi$
$7$	−11.0000	−1.57143	−0.785714	+	0.618590i	$0.787704\pi$
$8$	0	0
$9$	−23.0000	−2.55556
$10$	0	0
$11$	−3.00000	−0.272727	−0.136364	−	0.990659i	$-0.543542\pi$
$11$	−3.00000	−0.272727	−0.136364	+	0.990659i	$0.543542\pi$
$12$	0	0
$13$	− 11.3137i	− 0.870285i	−0.900362	−	0.435143i	$-0.856698\pi$
$13$	− 11.3137i	− 0.870285i	0.900362	−	0.435143i	$-0.143302\pi$
$14$	0	0
$15$	− 39.5980i	− 2.63987i
$16$	0	0
$17$	−17.0000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	−17.0000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$18$	0	0
$19$	19.0000	1.00000
$19$	19.0000	1.00000
$20$	0	0
$21$	62.2254i	2.96311i
$22$	0	0
$23$	−2.00000	−0.0869565	−0.0434783	−	0.999054i	$-0.513844\pi$
$23$	−2.00000	−0.0869565	−0.0434783	+	0.999054i	$0.513844\pi$
$24$	0	0
$25$	24.0000	0.960000
$26$	0	0
$27$	79.1960i	2.93318i
$28$	0	0
$29$	− 39.5980i	− 1.36545i	−0.730677	−	0.682724i	$-0.760795\pi$
$29$	− 39.5980i	− 1.36545i	0.730677	−	0.682724i	$-0.239205\pi$
$30$	0	0
$31$	− 5.65685i	− 0.182479i	−0.995829	−	0.0912396i	$-0.970917\pi$
$31$	− 5.65685i	− 0.182479i	0.995829	−	0.0912396i	$-0.0290829\pi$
$32$	0	0
$33$	16.9706i	0.514259i
$34$	0	0
$35$	−77.0000	−2.20000
$36$	0	0
$37$	− 39.5980i	− 1.07022i	−0.844784	−	0.535108i	$-0.820271\pi$
$37$	− 39.5980i	− 1.07022i	0.844784	−	0.535108i	$-0.179729\pi$
$38$	0	0
$39$	−64.0000	−1.64103
$40$	0	0
$41$	39.5980i	0.965804i	0.875674	+	0.482902i	$0.160417\pi$
$41$	39.5980i	0.965804i	−0.875674	+	0.482902i	$0.839583\pi$
$42$	0	0
$43$	21.0000	0.488372	0.244186	−	0.969728i	$-0.421479\pi$
$43$	21.0000	0.488372	0.244186	+	0.969728i	$0.421479\pi$
$44$	0	0
$45$	−161.000	−3.57778
$46$	0	0
$47$	5.00000	0.106383	0.0531915	−	0.998584i	$-0.483061\pi$
$47$	5.00000	0.106383	0.0531915	+	0.998584i	$0.483061\pi$
$48$	0	0
$49$	72.0000	1.46939
$50$	0	0
$51$	96.1665i	1.88562i
$52$	0	0
$53$	5.65685i	0.106733i	0.998575	+	0.0533665i	$0.0169952\pi$
$53$	5.65685i	0.106733i	−0.998575	+	0.0533665i	$0.983005\pi$
$54$	0	0
$55$	−21.0000	−0.381818
$56$	0	0
$57$	− 107.480i	− 1.88562i
$58$	0	0
$59$	− 33.9411i	− 0.575273i	−0.957740	−	0.287637i	$-0.907130\pi$
$59$	− 33.9411i	− 0.575273i	0.957740	−	0.287637i	$-0.0928695\pi$
$60$	0	0
$61$	23.0000	0.377049	0.188525	−	0.982068i	$-0.439629\pi$
$61$	23.0000	0.377049	0.188525	+	0.982068i	$0.439629\pi$
$62$	0	0
$63$	253.000	4.01587
$64$	0	0
$65$	− 79.1960i	− 1.21840i
$66$	0	0
$67$	− 39.5980i	− 0.591015i	−0.955340	−	0.295507i	$-0.904511\pi$
$67$	− 39.5980i	− 0.591015i	0.955340	−	0.295507i	$-0.0954887\pi$
$68$	0	0
$69$	11.3137i	0.163967i
$70$	0	0
$71$	− 90.5097i	− 1.27478i	−0.770540	−	0.637392i	$-0.780013\pi$
$71$	− 90.5097i	− 1.27478i	0.770540	−	0.637392i	$-0.219987\pi$
$72$	0	0
$73$	39.0000	0.534247	0.267123	−	0.963662i	$-0.413927\pi$
$73$	39.0000	0.534247	0.267123	+	0.963662i	$0.413927\pi$
$74$	0	0
$75$	− 135.765i	− 1.81019i
$76$	0	0
$77$	33.0000	0.428571
$78$	0	0
$79$	96.1665i	1.21730i	0.793440	+	0.608649i	$0.208288\pi$
$79$	96.1665i	1.21730i	−0.793440	+	0.608649i	$0.791712\pi$
$80$	0	0
$81$	241.000	2.97531
$82$	0	0
$83$	6.00000	0.0722892	0.0361446	−	0.999347i	$-0.488492\pi$
$83$	6.00000	0.0722892	0.0361446	+	0.999347i	$0.488492\pi$
$84$	0	0
$85$	−119.000	−1.40000
$86$	0	0
$87$	−224.000	−2.57471
$88$	0	0
$89$	118.794i	1.33476i	0.744716	+	0.667382i	$0.232585\pi$
$89$	118.794i	1.33476i	−0.744716	+	0.667382i	$0.767415\pi$
$90$	0	0
$91$	124.451i	1.36759i
$92$	0	0
$93$	−32.0000	−0.344086
$94$	0	0
$95$	133.000	1.40000
$96$	0	0
$97$	− 169.706i	− 1.74954i	−0.484536	−	0.874771i	$-0.661012\pi$
$97$	− 169.706i	− 1.74954i	0.484536	−	0.874771i	$-0.338988\pi$
$98$	0	0
$99$	69.0000	0.696970
$100$	0	0
$101$	122.000	1.20792	0.603960	−	0.797014i	$-0.293589\pi$
$101$	122.000	1.20792	0.603960	+	0.797014i	$0.293589\pi$
$102$	0	0
$103$	− 101.823i	− 0.988576i	−0.869298	−	0.494288i	$-0.835429\pi$
$103$	− 101.823i	− 0.988576i	0.869298	−	0.494288i	$-0.164571\pi$
$104$	0	0
$105$	435.578i	4.14836i
$106$	0	0
$107$	− 158.392i	− 1.48030i	−0.672443	−	0.740149i	$-0.734755\pi$
$107$	− 158.392i	− 1.48030i	0.672443	−	0.740149i	$-0.265245\pi$
$108$	0	0
$109$	− 118.794i	− 1.08985i	−0.838484	−	0.544926i	$-0.816558\pi$
$109$	− 118.794i	− 1.08985i	0.838484	−	0.544926i	$-0.183442\pi$
$110$	0	0
$111$	−224.000	−2.01802
$112$	0	0
$113$	50.9117i	0.450546i	0.974296	+	0.225273i	$0.0723274\pi$
$113$	50.9117i	0.450546i	−0.974296	+	0.225273i	$0.927673\pi$
$114$	0	0
$115$	−14.0000	−0.121739
$116$	0	0
$117$	260.215i	2.22406i
$118$	0	0
$119$	187.000	1.57143
$120$	0	0
$121$	−112.000	−0.925620
$122$	0	0
$123$	224.000	1.82114
$124$	0	0
$125$	−7.00000	−0.0560000
$126$	0	0
$127$	39.5980i	0.311795i	0.987773	+	0.155898i	$0.0498270\pi$
$127$	39.5980i	0.311795i	−0.987773	+	0.155898i	$0.950173\pi$
$128$	0	0
$129$	− 118.794i	− 0.920883i
$130$	0	0
$131$	149.000	1.13740	0.568702	−	0.822543i	$-0.307446\pi$
$131$	149.000	1.13740	0.568702	+	0.822543i	$0.307446\pi$
$132$	0	0
$133$	−209.000	−1.57143
$134$	0	0
$135$	554.372i	4.10646i
$136$	0	0
$137$	95.0000	0.693431	0.346715	−	0.937970i	$-0.387297\pi$
$137$	95.0000	0.693431	0.346715	+	0.937970i	$0.387297\pi$
$138$	0	0
$139$	−155.000	−1.11511	−0.557554	−	0.830141i	$-0.688260\pi$
$139$	−155.000	−1.11511	−0.557554	+	0.830141i	$0.688260\pi$
$140$	0	0
$141$	− 28.2843i	− 0.200598i
$142$	0	0
$143$	33.9411i	0.237351i
$144$	0	0
$145$	− 277.186i	− 1.91163i
$146$	0	0
$147$	− 407.294i	− 2.77070i
$148$	0	0
$149$	63.0000	0.422819	0.211409	−	0.977398i	$-0.432195\pi$
$149$	63.0000	0.422819	0.211409	+	0.977398i	$0.432195\pi$
$150$	0	0
$151$	124.451i	0.824177i	0.911144	+	0.412089i	$0.135201\pi$
$151$	124.451i	0.824177i	−0.911144	+	0.412089i	$0.864799\pi$
$152$	0	0
$153$	391.000	2.55556
$154$	0	0
$155$	− 39.5980i	− 0.255471i
$156$	0	0
$157$	−150.000	−0.955414	−0.477707	−	0.878519i	$-0.658532\pi$
$157$	−150.000	−0.955414	−0.477707	+	0.878519i	$0.658532\pi$
$158$	0	0
$159$	32.0000	0.201258
$160$	0	0
$161$	22.0000	0.136646
$162$	0	0
$163$	166.000	1.01840	0.509202	−	0.860647i	$-0.329940\pi$
$163$	166.000	1.01840	0.509202	+	0.860647i	$0.329940\pi$
$164$	0	0
$165$	118.794i	0.719963i
$166$	0	0
$167$	− 209.304i	− 1.25332i	−0.779295	−	0.626658i	$-0.784423\pi$
$167$	− 209.304i	− 1.25332i	0.779295	−	0.626658i	$-0.215577\pi$
$168$	0	0
$169$	41.0000	0.242604
$170$	0	0
$171$	−437.000	−2.55556
$172$	0	0
$173$	56.5685i	0.326986i	0.986545	+	0.163493i	$0.0522761\pi$
$173$	56.5685i	0.326986i	−0.986545	+	0.163493i	$0.947724\pi$
$174$	0	0
$175$	−264.000	−1.50857
$176$	0	0
$177$	−192.000	−1.08475
$178$	0	0
$179$	73.5391i	0.410833i	0.978675	+	0.205416i	$0.0658549\pi$
$179$	73.5391i	0.410833i	−0.978675	+	0.205416i	$0.934145\pi$
$180$	0	0
$181$	− 79.1960i	− 0.437547i	−0.975776	−	0.218773i	$-0.929794\pi$
$181$	− 79.1960i	− 0.437547i	0.975776	−	0.218773i	$-0.0702055\pi$
$182$	0	0
$183$	− 130.108i	− 0.710971i
$184$	0	0
$185$	− 277.186i	− 1.49830i
$186$	0	0
$187$	51.0000	0.272727
$188$	0	0
$189$	− 871.156i	− 4.60929i
$190$	0	0
$191$	301.000	1.57592	0.787958	−	0.615729i	$-0.211138\pi$
$191$	301.000	1.57592	0.787958	+	0.615729i	$0.211138\pi$
$192$	0	0
$193$	152.735i	0.791373i	0.918386	+	0.395687i	$0.129493\pi$
$193$	152.735i	0.791373i	−0.918386	+	0.395687i	$0.870507\pi$
$194$	0	0
$195$	−448.000	−2.29744
$196$	0	0
$197$	90.0000	0.456853	0.228426	−	0.973561i	$-0.426642\pi$
$197$	90.0000	0.456853	0.228426	+	0.973561i	$0.426642\pi$
$198$	0	0
$199$	−147.000	−0.738693	−0.369347	−	0.929292i	$-0.620419\pi$
$199$	−147.000	−0.738693	−0.369347	+	0.929292i	$0.620419\pi$
$200$	0	0
$201$	−224.000	−1.11443
$202$	0	0
$203$	435.578i	2.14570i
$204$	0	0
$205$	277.186i	1.35213i
$206$	0	0
$207$	46.0000	0.222222
$208$	0	0
$209$	−57.0000	−0.272727
$210$	0	0
$211$	328.098i	1.55496i	0.628905	+	0.777482i	$0.283504\pi$
$211$	328.098i	1.55496i	−0.628905	+	0.777482i	$0.716496\pi$
$212$	0	0
$213$	−512.000	−2.40376
$214$	0	0
$215$	147.000	0.683721
$216$	0	0
$217$	62.2254i	0.286753i
$218$	0	0
$219$	− 220.617i	− 1.00739i
$220$	0	0
$221$	192.333i	0.870285i
$222$	0	0
$223$	− 356.382i	− 1.59812i	−0.601248	−	0.799062i	$-0.705330\pi$
$223$	− 356.382i	− 1.59812i	0.601248	−	0.799062i	$-0.294670\pi$
$224$	0	0
$225$	−552.000	−2.45333
$226$	0	0
$227$	316.784i	1.39552i	0.716330	+	0.697762i	$0.245821\pi$
$227$	316.784i	1.39552i	−0.716330	+	0.697762i	$0.754179\pi$
$228$	0	0
$229$	−257.000	−1.12227	−0.561135	−	0.827724i	$-0.689635\pi$
$229$	−257.000	−1.12227	−0.561135	+	0.827724i	$0.689635\pi$
$230$	0	0
$231$	− 186.676i	− 0.808122i
$232$	0	0
$233$	−177.000	−0.759657	−0.379828	−	0.925057i	$-0.624017\pi$
$233$	−177.000	−0.759657	−0.379828	+	0.925057i	$0.624017\pi$
$234$	0	0
$235$	35.0000	0.148936
$236$	0	0
$237$	544.000	2.29536
$238$	0	0
$239$	−363.000	−1.51883	−0.759414	−	0.650607i	$-0.774514\pi$
$239$	−363.000	−1.51883	−0.759414	+	0.650607i	$0.774514\pi$
$240$	0	0
$241$	356.382i	1.47876i	0.673287	+	0.739381i	$0.264882\pi$
$241$	356.382i	1.47876i	−0.673287	+	0.739381i	$0.735118\pi$
$242$	0	0
$243$	− 650.538i	− 2.67711i
$244$	0	0
$245$	504.000	2.05714
$246$	0	0
$247$	− 214.960i	− 0.870285i
$248$	0	0
$249$	− 33.9411i	− 0.136310i
$250$	0	0
$251$	133.000	0.529880	0.264940	−	0.964265i	$-0.414648\pi$
$251$	133.000	0.529880	0.264940	+	0.964265i	$0.414648\pi$
$252$	0	0
$253$	6.00000	0.0237154
$254$	0	0
$255$	673.166i	2.63987i
$256$	0	0
$257$	− 135.765i	− 0.528267i	−0.964486	−	0.264133i	$-0.914914\pi$
$257$	− 135.765i	− 0.528267i	0.964486	−	0.264133i	$-0.0850859\pi$
$258$	0	0
$259$	435.578i	1.68177i
$260$	0	0
$261$	910.754i	3.48948i
$262$	0	0
$263$	101.000	0.384030	0.192015	−	0.981392i	$-0.438498\pi$
$263$	101.000	0.384030	0.192015	+	0.981392i	$0.438498\pi$
$264$	0	0
$265$	39.5980i	0.149426i
$266$	0	0
$267$	672.000	2.51685
$268$	0	0
$269$	356.382i	1.32484i	0.749133	+	0.662420i	$0.230470\pi$
$269$	356.382i	1.32484i	−0.749133	+	0.662420i	$0.769530\pi$
$270$	0	0
$271$	142.000	0.523985	0.261993	−	0.965070i	$-0.415620\pi$
$271$	142.000	0.523985	0.261993	+	0.965070i	$0.415620\pi$
$272$	0	0
$273$	704.000	2.57875
$274$	0	0
$275$	−72.0000	−0.261818
$276$	0	0
$277$	199.000	0.718412	0.359206	−	0.933258i	$-0.383048\pi$
$277$	199.000	0.718412	0.359206	+	0.933258i	$0.383048\pi$
$278$	0	0
$279$	130.108i	0.466336i
$280$	0	0
$281$	463.862i	1.65075i	0.564582	+	0.825377i	$0.309038\pi$
$281$	463.862i	1.65075i	−0.564582	+	0.825377i	$0.690962\pi$
$282$	0	0
$283$	−427.000	−1.50883	−0.754417	−	0.656395i	$-0.772080\pi$
$283$	−427.000	−1.50883	−0.754417	+	0.656395i	$0.772080\pi$
$284$	0	0
$285$	− 752.362i	− 2.63987i
$286$	0	0
$287$	− 435.578i	− 1.51769i
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−960.000	−3.29897
$292$	0	0
$293$	− 237.588i	− 0.810880i	−0.914122	−	0.405440i	$-0.867118\pi$
$293$	− 237.588i	− 0.810880i	0.914122	−	0.405440i	$-0.132882\pi$
$294$	0	0
$295$	− 237.588i	− 0.805383i
$296$	0	0
$297$	− 237.588i	− 0.799959i
$298$	0	0
$299$	22.6274i	0.0756770i
$300$	0	0
$301$	−231.000	−0.767442
$302$	0	0
$303$	− 690.136i	− 2.27768i
$304$	0	0
$305$	161.000	0.527869
$306$	0	0
$307$	526.087i	1.71364i	0.515616	+	0.856820i	$0.327563\pi$
$307$	526.087i	1.71364i	−0.515616	+	0.856820i	$0.672437\pi$
$308$	0	0
$309$	−576.000	−1.86408
$310$	0	0
$311$	−235.000	−0.755627	−0.377814	−	0.925882i	$-0.623324\pi$
$311$	−235.000	−0.755627	−0.377814	+	0.925882i	$0.623324\pi$
$312$	0	0
$313$	530.000	1.69329	0.846645	−	0.532158i	$-0.178619\pi$
$313$	530.000	1.69329	0.846645	+	0.532158i	$0.178619\pi$
$314$	0	0
$315$	1771.00	5.62222
$316$	0	0
$317$	− 214.960i	− 0.678109i	−0.940767	−	0.339054i	$-0.889893\pi$
$317$	− 214.960i	− 0.678109i	0.940767	−	0.339054i	$-0.110107\pi$
$318$	0	0
$319$	118.794i	0.372395i
$320$	0	0
$321$	−896.000	−2.79128
$322$	0	0
$323$	−323.000	−1.00000
$324$	0	0
$325$	− 271.529i	− 0.835474i
$326$	0	0
$327$	−672.000	−2.05505
$328$	0	0
$329$	−55.0000	−0.167173
$330$	0	0
$331$	− 96.1665i	− 0.290533i	−0.989393	−	0.145267i	$-0.953596\pi$
$331$	− 96.1665i	− 0.290533i	0.989393	−	0.145267i	$-0.0464040\pi$
$332$	0	0
$333$	910.754i	2.73500i
$334$	0	0
$335$	− 277.186i	− 0.827420i
$336$	0	0
$337$	− 169.706i	− 0.503578i	−0.967782	−	0.251789i	$-0.918981\pi$
$337$	− 169.706i	− 0.503578i	0.967782	−	0.251789i	$-0.0810188\pi$
$338$	0	0
$339$	288.000	0.849558
$340$	0	0
$341$	16.9706i	0.0497670i
$342$	0	0
$343$	−253.000	−0.737609
$344$	0	0
$345$	79.1960i	0.229554i
$346$	0	0
$347$	253.000	0.729107	0.364553	−	0.931183i	$-0.381222\pi$
$347$	253.000	0.729107	0.364553	+	0.931183i	$0.381222\pi$
$348$	0	0
$349$	351.000	1.00573	0.502865	−	0.864365i	$-0.332279\pi$
$349$	351.000	1.00573	0.502865	+	0.864365i	$0.332279\pi$
$350$	0	0
$351$	896.000	2.55271
$352$	0	0
$353$	98.0000	0.277620	0.138810	−	0.990319i	$-0.455672\pi$
$353$	98.0000	0.277620	0.138810	+	0.990319i	$0.455672\pi$
$354$	0	0
$355$	− 633.568i	− 1.78470i
$356$	0	0
$357$	− 1057.83i	− 2.96311i
$358$	0	0
$359$	61.0000	0.169916	0.0849582	−	0.996385i	$-0.472924\pi$
$359$	61.0000	0.169916	0.0849582	+	0.996385i	$0.472924\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	633.568i	1.74537i
$364$	0	0
$365$	273.000	0.747945
$366$	0	0
$367$	686.000	1.86921	0.934605	−	0.355688i	$-0.115753\pi$
$367$	686.000	1.86921	0.934605	+	0.355688i	$0.115753\pi$
$368$	0	0
$369$	− 910.754i	− 2.46817i
$370$	0	0
$371$	− 62.2254i	− 0.167723i
$372$	0	0
$373$	135.765i	0.363980i	0.983300	+	0.181990i	$0.0582538\pi$
$373$	135.765i	0.363980i	−0.983300	+	0.181990i	$0.941746\pi$
$374$	0	0
$375$	39.5980i	0.105595i
$376$	0	0
$377$	−448.000	−1.18833
$378$	0	0
$379$	− 79.1960i	− 0.208960i	−0.994527	−	0.104480i	$-0.966682\pi$
$379$	− 79.1960i	− 0.208960i	0.994527	−	0.104480i	$-0.0333179\pi$
$380$	0	0
$381$	224.000	0.587927
$382$	0	0
$383$	16.9706i	0.0443096i	0.999755	+	0.0221548i	$0.00705266\pi$
$383$	16.9706i	0.0443096i	−0.999755	+	0.0221548i	$0.992947\pi$
$384$	0	0
$385$	231.000	0.600000
$386$	0	0
$387$	−483.000	−1.24806
$388$	0	0
$389$	599.000	1.53985	0.769923	−	0.638137i	$-0.220295\pi$
$389$	599.000	1.53985	0.769923	+	0.638137i	$0.220295\pi$
$390$	0	0
$391$	34.0000	0.0869565
$392$	0	0
$393$	− 842.871i	− 2.14471i
$394$	0	0
$395$	673.166i	1.70422i
$396$	0	0
$397$	−569.000	−1.43325	−0.716625	−	0.697459i	$-0.754314\pi$
$397$	−569.000	−1.43325	−0.716625	+	0.697459i	$0.754314\pi$
$398$	0	0
$399$	1182.28i	2.96311i
$400$	0	0
$401$	− 16.9706i	− 0.0423206i	−0.999776	−	0.0211603i	$-0.993264\pi$
$401$	− 16.9706i	− 0.0423206i	0.999776	−	0.0211603i	$-0.00673604\pi$
$402$	0	0
$403$	−64.0000	−0.158809
$404$	0	0
$405$	1687.00	4.16543
$406$	0	0
$407$	118.794i	0.291877i
$408$	0	0
$409$	593.970i	1.45225i	0.687563	+	0.726124i	$0.258680\pi$
$409$	593.970i	1.45225i	−0.687563	+	0.726124i	$0.741320\pi$
$410$	0	0
$411$	− 537.401i	− 1.30755i
$412$	0	0
$413$	373.352i	0.904001i
$414$	0	0
$415$	42.0000	0.101205
$416$	0	0
$417$	876.812i	2.10267i
$418$	0	0
$419$	230.000	0.548926	0.274463	−	0.961598i	$-0.411500\pi$
$419$	230.000	0.548926	0.274463	+	0.961598i	$0.411500\pi$
$420$	0	0
$421$	− 593.970i	− 1.41085i	−0.708782	−	0.705427i	$-0.750755\pi$
$421$	− 593.970i	− 1.41085i	0.708782	−	0.705427i	$-0.249245\pi$
$422$	0	0
$423$	−115.000	−0.271868
$424$	0	0
$425$	−408.000	−0.960000
$426$	0	0
$427$	−253.000	−0.592506
$428$	0	0
$429$	192.000	0.447552
$430$	0	0
$431$	475.176i	1.10250i	0.834341	+	0.551248i	$0.185848\pi$
$431$	475.176i	1.10250i	−0.834341	+	0.551248i	$0.814152\pi$
$432$	0	0
$433$	593.970i	1.37175i	0.727717	+	0.685877i	$0.240581\pi$
$433$	593.970i	1.37175i	−0.727717	+	0.685877i	$0.759419\pi$
$434$	0	0
$435$	−1568.00	−3.60460
$436$	0	0
$437$	−38.0000	−0.0869565
$438$	0	0
$439$	− 118.794i	− 0.270601i	−0.990805	−	0.135301i	$-0.956800\pi$
$439$	− 118.794i	− 0.270601i	0.990805	−	0.135301i	$-0.0432000\pi$
$440$	0	0
$441$	−1656.00	−3.75510
$442$	0	0
$443$	653.000	1.47404	0.737020	−	0.675871i	$-0.236232\pi$
$443$	653.000	1.47404	0.737020	+	0.675871i	$0.236232\pi$
$444$	0	0
$445$	831.558i	1.86867i
$446$	0	0
$447$	− 356.382i	− 0.797275i
$448$	0	0
$449$	− 147.078i	− 0.327568i	−0.986496	−	0.163784i	$-0.947630\pi$
$449$	− 147.078i	− 0.327568i	0.986496	−	0.163784i	$-0.0523701\pi$
$450$	0	0
$451$	− 118.794i	− 0.263401i
$452$	0	0
$453$	704.000	1.55408
$454$	0	0
$455$	871.156i	1.91463i
$456$	0	0
$457$	−817.000	−1.78775	−0.893873	−	0.448320i	$-0.852022\pi$
$457$	−817.000	−1.78775	−0.893873	+	0.448320i	$0.852022\pi$
$458$	0	0
$459$	− 1346.33i	− 2.93318i
$460$	0	0
$461$	463.000	1.00434	0.502169	−	0.864769i	$-0.332535\pi$
$461$	463.000	1.00434	0.502169	+	0.864769i	$0.332535\pi$
$462$	0	0
$463$	29.0000	0.0626350	0.0313175	−	0.999509i	$-0.490030\pi$
$463$	29.0000	0.0626350	0.0313175	+	0.999509i	$0.490030\pi$
$464$	0	0
$465$	−224.000	−0.481720
$466$	0	0
$467$	−19.0000	−0.0406852	−0.0203426	−	0.999793i	$-0.506476\pi$
$467$	−19.0000	−0.0406852	−0.0203426	+	0.999793i	$0.506476\pi$
$468$	0	0
$469$	435.578i	0.928737i
$470$	0	0
$471$	848.528i	1.80155i
$472$	0	0
$473$	−63.0000	−0.133192
$474$	0	0
$475$	456.000	0.960000
$476$	0	0
$477$	− 130.108i	− 0.272762i
$478$	0	0
$479$	206.000	0.430063	0.215031	−	0.976607i	$-0.431015\pi$
$479$	206.000	0.430063	0.215031	+	0.976607i	$0.431015\pi$
$480$	0	0
$481$	−448.000	−0.931393
$482$	0	0
$483$	− 124.451i	− 0.257662i
$484$	0	0
$485$	− 1187.94i	− 2.44936i
$486$	0	0
$487$	− 45.2548i	− 0.0929257i	−0.998920	−	0.0464629i	$-0.985205\pi$
$487$	− 45.2548i	− 0.0929257i	0.998920	−	0.0464629i	$-0.0147949\pi$
$488$	0	0
$489$	− 939.038i	− 1.92032i
$490$	0	0
$491$	246.000	0.501018	0.250509	−	0.968114i	$-0.419402\pi$
$491$	246.000	0.501018	0.250509	+	0.968114i	$0.419402\pi$
$492$	0	0
$493$	673.166i	1.36545i
$494$	0	0
$495$	483.000	0.975758
$496$	0	0
$497$	995.606i	2.00323i
$498$	0	0
$499$	21.0000	0.0420842	0.0210421	−	0.999779i	$-0.493302\pi$
$499$	21.0000	0.0420842	0.0210421	+	0.999779i	$0.493302\pi$
$500$	0	0
$501$	−1184.00	−2.36327
$502$	0	0
$503$	−386.000	−0.767396	−0.383698	−	0.923459i	$-0.625350\pi$
$503$	−386.000	−0.767396	−0.383698	+	0.923459i	$0.625350\pi$
$504$	0	0
$505$	854.000	1.69109
$506$	0	0
$507$	− 231.931i	− 0.457458i
$508$	0	0
$509$	− 475.176i	− 0.933548i	−0.884377	−	0.466774i	$-0.845416\pi$
$509$	− 475.176i	− 0.933548i	0.884377	−	0.466774i	$-0.154584\pi$
$510$	0	0
$511$	−429.000	−0.839530
$512$	0	0
$513$	1504.72i	2.93318i
$514$	0	0
$515$	− 712.764i	− 1.38401i
$516$	0	0
$517$	−15.0000	−0.0290135
$518$	0	0
$519$	320.000	0.616570
$520$	0	0
$521$	− 786.303i	− 1.50922i	−0.656175	−	0.754609i	$-0.727826\pi$
$521$	− 786.303i	− 1.50922i	0.656175	−	0.754609i	$-0.272174\pi$
$522$	0	0
$523$	− 395.980i	− 0.757132i	−0.925574	−	0.378566i	$-0.876417\pi$
$523$	− 395.980i	− 0.757132i	0.925574	−	0.378566i	$-0.123583\pi$
$524$	0	0
$525$	1493.41i	2.84459i
$526$	0	0
$527$	96.1665i	0.182479i
$528$	0	0
$529$	−525.000	−0.992439
$530$	0	0
$531$	780.646i	1.47014i
$532$	0	0
$533$	448.000	0.840525
$534$	0	0
$535$	− 1108.74i	− 2.07242i
$536$	0	0
$537$	416.000	0.774674
$538$	0	0
$539$	−216.000	−0.400742
$540$	0	0
$541$	7.00000	0.0129390	0.00646950	−	0.999979i	$-0.497941\pi$
$541$	7.00000	0.0129390	0.00646950	+	0.999979i	$0.497941\pi$
$542$	0	0
$543$	−448.000	−0.825046
$544$	0	0
$545$	− 831.558i	− 1.52579i
$546$	0	0
$547$	− 712.764i	− 1.30304i	−0.758631	−	0.651521i	$-0.774131\pi$
$547$	− 712.764i	− 1.30304i	0.758631	−	0.651521i	$-0.225869\pi$
$548$	0	0
$549$	−529.000	−0.963570
$550$	0	0
$551$	− 752.362i	− 1.36545i
$552$	0	0
$553$	− 1057.83i	− 1.91290i
$554$	0	0
$555$	−1568.00	−2.82523
$556$	0	0
$557$	−1001.00	−1.79713	−0.898564	−	0.438843i	$-0.855388\pi$
$557$	−1001.00	−1.79713	−0.898564	+	0.438843i	$0.855388\pi$
$558$	0	0
$559$	− 237.588i	− 0.425023i
$560$	0	0
$561$	− 288.500i	− 0.514259i
$562$	0	0
$563$	322.441i	0.572719i	0.958122	+	0.286359i	$0.0924451\pi$
$563$	322.441i	0.572719i	−0.958122	+	0.286359i	$0.907555\pi$
$564$	0	0
$565$	356.382i	0.630764i
$566$	0	0
$567$	−2651.00	−4.67549
$568$	0	0
$569$	− 712.764i	− 1.25266i	−0.779558	−	0.626330i	$-0.784556\pi$
$569$	− 712.764i	− 1.25266i	0.779558	−	0.626330i	$-0.215444\pi$
$570$	0	0
$571$	−746.000	−1.30648	−0.653240	−	0.757151i	$-0.726591\pi$
$571$	−746.000	−1.30648	−0.653240	+	0.757151i	$0.726591\pi$
$572$	0	0
$573$	− 1702.71i	− 2.97158i
$574$	0	0
$575$	−48.0000	−0.0834783
$576$	0	0
$577$	−25.0000	−0.0433276	−0.0216638	−	0.999765i	$-0.506896\pi$
$577$	−25.0000	−0.0433276	−0.0216638	+	0.999765i	$0.506896\pi$
$578$	0	0
$579$	864.000	1.49223
$580$	0	0
$581$	−66.0000	−0.113597
$582$	0	0
$583$	− 16.9706i	− 0.0291090i
$584$	0	0
$585$	1821.51i	3.11369i
$586$	0	0
$587$	869.000	1.48041	0.740204	−	0.672382i	$-0.234729\pi$
$587$	869.000	1.48041	0.740204	+	0.672382i	$0.234729\pi$
$588$	0	0
$589$	− 107.480i	− 0.182479i
$590$	0	0
$591$	− 509.117i	− 0.861450i
$592$	0	0
$593$	898.000	1.51433	0.757167	−	0.653221i	$-0.226583\pi$
$593$	898.000	1.51433	0.757167	+	0.653221i	$0.226583\pi$
$594$	0	0
$595$	1309.00	2.20000
$596$	0	0
$597$	831.558i	1.39289i
$598$	0	0
$599$	277.186i	0.462748i	0.972865	+	0.231374i	$0.0743220\pi$
$599$	277.186i	0.462748i	−0.972865	+	0.231374i	$0.925678\pi$
$600$	0	0
$601$	− 475.176i	− 0.790642i	−0.918543	−	0.395321i	$-0.870633\pi$
$601$	− 475.176i	− 0.790642i	0.918543	−	0.395321i	$-0.129367\pi$
$602$	0	0
$603$	910.754i	1.51037i
$604$	0	0
$605$	−784.000	−1.29587
$606$	0	0
$607$	452.548i	0.745549i	0.927922	+	0.372775i	$0.121594\pi$
$607$	452.548i	0.745549i	−0.927922	+	0.372775i	$0.878406\pi$
$608$	0	0
$609$	2464.00	4.04598
$610$	0	0
$611$	− 56.5685i	− 0.0925835i
$612$	0	0
$613$	−585.000	−0.954323	−0.477162	−	0.878816i	$-0.658334\pi$
$613$	−585.000	−0.954323	−0.477162	+	0.878816i	$0.658334\pi$
$614$	0	0
$615$	1568.00	2.54959
$616$	0	0
$617$	−873.000	−1.41491	−0.707455	−	0.706758i	$-0.750157\pi$
$617$	−873.000	−1.41491	−0.707455	+	0.706758i	$0.750157\pi$
$618$	0	0
$619$	−970.000	−1.56704	−0.783522	−	0.621364i	$-0.786579\pi$
$619$	−970.000	−1.56704	−0.783522	+	0.621364i	$0.786579\pi$
$620$	0	0
$621$	− 158.392i	− 0.255059i
$622$	0	0
$623$	− 1306.73i	− 2.09749i
$624$	0	0
$625$	−649.000	−1.03840
$626$	0	0
$627$	322.441i	0.514259i
$628$	0	0
$629$	673.166i	1.07022i
$630$	0	0
$631$	−259.000	−0.410460	−0.205230	−	0.978714i	$-0.565794\pi$
$631$	−259.000	−0.410460	−0.205230	+	0.978714i	$0.565794\pi$
$632$	0	0
$633$	1856.00	2.93207
$634$	0	0
$635$	277.186i	0.436513i
$636$	0	0
$637$	− 814.587i	− 1.27879i
$638$	0	0
$639$	2081.72i	3.25778i
$640$	0	0
$641$	1063.49i	1.65911i	0.558426	+	0.829554i	$0.311405\pi$
$641$	1063.49i	1.65911i	−0.558426	+	0.829554i	$0.688595\pi$
$642$	0	0
$643$	645.000	1.00311	0.501555	−	0.865126i	$-0.332761\pi$
$643$	645.000	1.00311	0.501555	+	0.865126i	$0.332761\pi$
$644$	0	0
$645$	− 831.558i	− 1.28924i
$646$	0	0
$647$	29.0000	0.0448223	0.0224111	−	0.999749i	$-0.492866\pi$
$647$	29.0000	0.0448223	0.0224111	+	0.999749i	$0.492866\pi$
$648$	0	0
$649$	101.823i	0.156893i
$650$	0	0
$651$	352.000	0.540707
$652$	0	0
$653$	135.000	0.206738	0.103369	−	0.994643i	$-0.467038\pi$
$653$	135.000	0.206738	0.103369	+	0.994643i	$0.467038\pi$
$654$	0	0
$655$	1043.00	1.59237
$656$	0	0
$657$	−897.000	−1.36530
$658$	0	0
$659$	− 780.646i	− 1.18459i	−0.805721	−	0.592296i	$-0.798222\pi$
$659$	− 780.646i	− 1.18459i	0.805721	−	0.592296i	$-0.201778\pi$
$660$	0	0
$661$	− 995.606i	− 1.50621i	−0.657899	−	0.753106i	$-0.728555\pi$
$661$	− 995.606i	− 1.50621i	0.657899	−	0.753106i	$-0.271445\pi$
$662$	0	0
$663$	1088.00	1.64103
$664$	0	0
$665$	−1463.00	−2.20000
$666$	0	0
$667$	79.1960i	0.118735i
$668$	0	0
$669$	−2016.00	−3.01345
$670$	0	0
$671$	−69.0000	−0.102832
$672$	0	0
$673$	− 395.980i	− 0.588380i	−0.955747	−	0.294190i	$-0.904950\pi$
$673$	− 395.980i	− 0.588380i	0.955747	−	0.294190i	$-0.0950499\pi$
$674$	0	0
$675$	1900.70i	2.81586i
$676$	0	0
$677$	356.382i	0.526413i	0.964739	+	0.263207i	$0.0847801\pi$
$677$	356.382i	0.526413i	−0.964739	+	0.263207i	$0.915220\pi$
$678$	0	0
$679$	1866.76i	2.74928i
$680$	0	0
$681$	1792.00	2.63142
$682$	0	0
$683$	− 520.431i	− 0.761977i	−0.924580	−	0.380989i	$-0.875584\pi$
$683$	− 520.431i	− 0.761977i	0.924580	−	0.380989i	$-0.124416\pi$
$684$	0	0
$685$	665.000	0.970803
$686$	0	0
$687$	1453.81i	2.11617i
$688$	0	0
$689$	64.0000	0.0928882
$690$	0	0
$691$	−835.000	−1.20839	−0.604197	−	0.796835i	$-0.706506\pi$
$691$	−835.000	−1.20839	−0.604197	+	0.796835i	$0.706506\pi$
$692$	0	0
$693$	−759.000	−1.09524
$694$	0	0
$695$	−1085.00	−1.56115
$696$	0	0
$697$	− 673.166i	− 0.965804i
$698$	0	0
$699$	1001.26i	1.43242i
$700$	0	0
$701$	1002.00	1.42939	0.714693	−	0.699438i	$-0.246566\pi$
$701$	1002.00	1.42939	0.714693	+	0.699438i	$0.246566\pi$
$702$	0	0
$703$	− 752.362i	− 1.07022i
$704$	0	0
$705$	− 197.990i	− 0.280837i
$706$	0	0
$707$	−1342.00	−1.89816
$708$	0	0
$709$	250.000	0.352609	0.176305	−	0.984336i	$-0.443586\pi$
$709$	250.000	0.352609	0.176305	+	0.984336i	$0.443586\pi$
$710$	0	0
$711$	− 2211.83i	− 3.11087i
$712$	0	0
$713$	11.3137i	0.0158678i
$714$	0	0
$715$	237.588i	0.332291i
$716$	0	0
$717$	2053.44i	2.86393i
$718$	0	0
$719$	−1171.00	−1.62865	−0.814325	−	0.580409i	$-0.802893\pi$
$719$	−1171.00	−1.62865	−0.814325	+	0.580409i	$0.802893\pi$
$720$	0	0
$721$	1120.06i	1.55348i
$722$	0	0
$723$	2016.00	2.78838
$724$	0	0
$725$	− 950.352i	− 1.31083i
$726$	0	0
$727$	−91.0000	−0.125172	−0.0625860	−	0.998040i	$-0.519935\pi$
$727$	−91.0000	−0.125172	−0.0625860	+	0.998040i	$0.519935\pi$
$728$	0	0
$729$	−1511.00	−2.07270
$730$	0	0
$731$	−357.000	−0.488372
$732$	0	0
$733$	682.000	0.930423	0.465211	−	0.885200i	$-0.345978\pi$
$733$	682.000	0.930423	0.465211	+	0.885200i	$0.345978\pi$
$734$	0	0
$735$	− 2851.05i	− 3.87899i
$736$	0	0
$737$	118.794i	0.161186i
$738$	0	0
$739$	221.000	0.299053	0.149526	−	0.988758i	$-0.452225\pi$
$739$	221.000	0.299053	0.149526	+	0.988758i	$0.452225\pi$
$740$	0	0
$741$	−1216.00	−1.64103
$742$	0	0
$743$	288.500i	0.388290i	0.980973	+	0.194145i	$0.0621933\pi$
$743$	288.500i	0.388290i	−0.980973	+	0.194145i	$0.937807\pi$
$744$	0	0
$745$	441.000	0.591946
$746$	0	0
$747$	−138.000	−0.184739
$748$	0	0
$749$	1742.31i	2.32618i
$750$	0	0
$751$	− 927.724i	− 1.23532i	−0.786446	−	0.617659i	$-0.788081\pi$
$751$	− 927.724i	− 1.23532i	0.786446	−	0.617659i	$-0.211919\pi$
$752$	0	0
$753$	− 752.362i	− 0.999152i
$754$	0	0
$755$	871.156i	1.15385i
$756$	0	0
$757$	895.000	1.18230	0.591149	−	0.806562i	$-0.298674\pi$
$757$	895.000	1.18230	0.591149	+	0.806562i	$0.298674\pi$
$758$	0	0
$759$	− 33.9411i	− 0.0447182i
$760$	0	0
$761$	95.0000	0.124836	0.0624179	−	0.998050i	$-0.480119\pi$
$761$	95.0000	0.124836	0.0624179	+	0.998050i	$0.480119\pi$
$762$	0	0
$763$	1306.73i	1.71263i
$764$	0	0
$765$	2737.00	3.57778
$766$	0	0
$767$	−384.000	−0.500652
$768$	0	0
$769$	679.000	0.882965	0.441482	−	0.897270i	$-0.354453\pi$
$769$	679.000	0.882965	0.441482	+	0.897270i	$0.354453\pi$
$770$	0	0
$771$	−768.000	−0.996109
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 135.765i	− 0.175180i
$776$	0	0
$777$	2464.00	3.17117
$778$	0	0
$779$	752.362i	0.965804i
$780$	0	0
$781$	271.529i	0.347668i
$782$	0	0
$783$	3136.00	4.00511
$784$	0	0
$785$	−1050.00	−1.33758
$786$	0	0
$787$	1391.59i	1.76822i	0.467282	+	0.884108i	$0.345233\pi$
$787$	1391.59i	1.76822i	−0.467282	+	0.884108i	$0.654767\pi$
$788$	0	0
$789$	− 571.342i	− 0.724135i
$790$	0	0
$791$	− 560.029i	− 0.708001i
$792$	0	0
$793$	− 260.215i	− 0.328140i
$794$	0	0
$795$	224.000	0.281761
$796$	0	0
$797$	− 356.382i	− 0.447154i	−0.974686	−	0.223577i	$-0.928227\pi$
$797$	− 356.382i	− 0.447154i	0.974686	−	0.223577i	$-0.0717734\pi$
$798$	0	0
$799$	−85.0000	−0.106383
$800$	0	0
$801$	− 2732.26i	− 3.41106i
$802$	0	0
$803$	−117.000	−0.145704
$804$	0	0
$805$	154.000	0.191304
$806$	0	0
$807$	2016.00	2.49814
$808$	0	0
$809$	455.000	0.562423	0.281211	−	0.959646i	$-0.409264\pi$
$809$	455.000	0.562423	0.281211	+	0.959646i	$0.409264\pi$
$810$	0	0
$811$	− 475.176i	− 0.585913i	−0.956126	−	0.292957i	$-0.905361\pi$
$811$	− 475.176i	− 0.585913i	0.956126	−	0.292957i	$-0.0946392\pi$
$812$	0	0
$813$	− 803.273i	− 0.988036i
$814$	0	0
$815$	1162.00	1.42577
$816$	0	0
$817$	399.000	0.488372
$818$	0	0
$819$	− 2862.37i	− 3.49496i
$820$	0	0
$821$	831.000	1.01218	0.506090	−	0.862481i	$-0.331090\pi$
$821$	831.000	1.01218	0.506090	+	0.862481i	$0.331090\pi$
$822$	0	0
$823$	109.000	0.132442	0.0662211	−	0.997805i	$-0.478906\pi$
$823$	109.000	0.132442	0.0662211	+	0.997805i	$0.478906\pi$
$824$	0	0
$825$	407.294i	0.493689i
$826$	0	0
$827$	− 1238.85i	− 1.49801i	−0.662567	−	0.749003i	$-0.730533\pi$
$827$	− 1238.85i	− 1.49801i	0.662567	−	0.749003i	$-0.269467\pi$
$828$	0	0
$829$	− 667.509i	− 0.805198i	−0.915377	−	0.402599i	$-0.868107\pi$
$829$	− 667.509i	− 0.805198i	0.915377	−	0.402599i	$-0.131893\pi$
$830$	0	0
$831$	− 1125.71i	− 1.35465i
$832$	0	0
$833$	−1224.00	−1.46939
$834$	0	0
$835$	− 1465.13i	− 1.75464i
$836$	0	0
$837$	448.000	0.535245
$838$	0	0
$839$	− 316.784i	− 0.377573i	−0.982018	−	0.188787i	$-0.939545\pi$
$839$	− 316.784i	− 0.377573i	0.982018	−	0.188787i	$-0.0604554\pi$
$840$	0	0
$841$	−727.000	−0.864447
$842$	0	0
$843$	2624.00	3.11269
$844$	0	0
$845$	287.000	0.339645
$846$	0	0
$847$	1232.00	1.45455
$848$	0	0
$849$	2415.48i	2.84508i
$850$	0	0
$851$	79.1960i	0.0930622i
$852$	0	0
$853$	−1126.00	−1.32005	−0.660023	−	0.751245i	$-0.729454\pi$
$853$	−1126.00	−1.32005	−0.660023	+	0.751245i	$0.729454\pi$
$854$	0	0
$855$	−3059.00	−3.57778
$856$	0	0
$857$	− 203.647i	− 0.237627i	−0.992917	−	0.118814i	$-0.962091\pi$
$857$	− 203.647i	− 0.237627i	0.992917	−	0.118814i	$-0.0379091\pi$
$858$	0	0
$859$	−651.000	−0.757858	−0.378929	−	0.925426i	$-0.623708\pi$
$859$	−651.000	−0.757858	−0.378929	+	0.925426i	$0.623708\pi$
$860$	0	0
$861$	−2464.00	−2.86179
$862$	0	0
$863$	− 39.5980i	− 0.0458841i	−0.999737	−	0.0229421i	$-0.992697\pi$
$863$	− 39.5980i	− 0.0458841i	0.999737	−	0.0229421i	$-0.00730332\pi$
$864$	0	0
$865$	395.980i	0.457780i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 288.500i	− 0.331990i
$870$	0	0
$871$	−448.000	−0.514351
$872$	0	0
$873$	3903.23i	4.47105i
$874$	0	0
$875$	77.0000	0.0880000
$876$	0	0
$877$	1029.55i	1.17394i	0.809608	+	0.586971i	$0.199680\pi$
$877$	1029.55i	1.17394i	−0.809608	+	0.586971i	$0.800320\pi$
$878$	0	0
$879$	−1344.00	−1.52901
$880$	0	0
$881$	391.000	0.443814	0.221907	−	0.975068i	$-0.428772\pi$
$881$	391.000	0.443814	0.221907	+	0.975068i	$0.428772\pi$
$882$	0	0
$883$	−995.000	−1.12684	−0.563420	−	0.826171i	$-0.690515\pi$
$883$	−995.000	−1.12684	−0.563420	+	0.826171i	$0.690515\pi$
$884$	0	0
$885$	−1344.00	−1.51864
$886$	0	0
$887$	475.176i	0.535711i	0.963459	+	0.267856i	$0.0863150\pi$
$887$	475.176i	0.535711i	−0.963459	+	0.267856i	$0.913685\pi$
$888$	0	0
$889$	− 435.578i	− 0.489964i
$890$	0	0
$891$	−723.000	−0.811448
$892$	0	0
$893$	95.0000	0.106383
$894$	0	0
$895$	514.774i	0.575166i
$896$	0	0
$897$	128.000	0.142698
$898$	0	0
$899$	−224.000	−0.249166
$900$	0	0
$901$	− 96.1665i	− 0.106733i
$902$	0	0
$903$	1306.73i	1.44710i
$904$	0	0
$905$	− 554.372i	− 0.612565i
$906$	0	0
$907$	627.911i	0.692294i	0.938180	+	0.346147i	$0.112510\pi$
$907$	627.911i	0.692294i	−0.938180	+	0.346147i	$0.887490\pi$
$908$	0	0
$909$	−2806.00	−3.08691
$910$	0	0
$911$	803.273i	0.881749i	0.897569	+	0.440874i	$0.145332\pi$
$911$	803.273i	0.881749i	−0.897569	+	0.440874i	$0.854668\pi$
$912$	0	0
$913$	−18.0000	−0.0197152
$914$	0	0
$915$	− 910.754i	− 0.995359i
$916$	0	0
$917$	−1639.00	−1.78735
$918$	0	0
$919$	−1090.00	−1.18607	−0.593036	−	0.805176i	$-0.702071\pi$
$919$	−1090.00	−1.18607	−0.593036	+	0.805176i	$0.702071\pi$
$920$	0	0
$921$	2976.00	3.23127
$922$	0	0
$923$	−1024.00	−1.10943
$924$	0	0
$925$	− 950.352i	− 1.02741i
$926$	0	0
$927$	2341.94i	2.52636i
$928$	0	0
$929$	226.000	0.243272	0.121636	−	0.992575i	$-0.461186\pi$
$929$	226.000	0.243272	0.121636	+	0.992575i	$0.461186\pi$
$930$	0	0
$931$	1368.00	1.46939
$932$	0	0
$933$	1329.36i	1.42482i
$934$	0	0
$935$	357.000	0.381818
$936$	0	0
$937$	623.000	0.664888	0.332444	−	0.943123i	$-0.392127\pi$
$937$	623.000	0.664888	0.332444	+	0.943123i	$0.392127\pi$
$938$	0	0
$939$	− 2998.13i	− 3.19290i
$940$	0	0
$941$	− 1549.98i	− 1.64716i	−0.567200	−	0.823580i	$-0.691973\pi$
$941$	− 1549.98i	− 1.64716i	0.567200	−	0.823580i	$-0.308027\pi$
$942$	0	0
$943$	− 79.1960i	− 0.0839830i
$944$	0	0
$945$	− 6098.09i	− 6.45300i
$946$	0	0
$947$	−602.000	−0.635692	−0.317846	−	0.948142i	$-0.602959\pi$
$947$	−602.000	−0.635692	−0.317846	+	0.948142i	$0.602959\pi$
$948$	0	0
$949$	− 441.235i	− 0.464947i
$950$	0	0
$951$	−1216.00	−1.27865
$952$	0	0
$953$	277.186i	0.290856i	0.989369	+	0.145428i	$0.0464559\pi$
$953$	277.186i	0.290856i	−0.989369	+	0.145428i	$0.953544\pi$
$954$	0	0
$955$	2107.00	2.20628
$956$	0	0
$957$	672.000	0.702194
$958$	0	0
$959$	−1045.00	−1.08968
$960$	0	0
$961$	929.000	0.966701
$962$	0	0
$963$	3643.01i	3.78298i
$964$	0	0
$965$	1069.15i	1.10792i
$966$	0	0
$967$	−770.000	−0.796277	−0.398139	−	0.917325i	$-0.630344\pi$
$967$	−770.000	−0.796277	−0.398139	+	0.917325i	$0.630344\pi$
$968$	0	0
$969$	1827.16i	1.88562i
$970$	0	0
$971$	712.764i	0.734051i	0.930211	+	0.367026i	$0.119624\pi$
$971$	712.764i	0.734051i	−0.930211	+	0.367026i	$0.880376\pi$
$972$	0	0
$973$	1705.00	1.75231
$974$	0	0
$975$	−1536.00	−1.57538
$976$	0	0
$977$	1340.67i	1.37224i	0.727490	+	0.686118i	$0.240687\pi$
$977$	1340.67i	1.37224i	−0.727490	+	0.686118i	$0.759313\pi$
$978$	0	0
$979$	− 356.382i	− 0.364026i
$980$	0	0
$981$	2732.26i	2.78518i
$982$	0	0
$983$	− 554.372i	− 0.563959i	−0.959420	−	0.281980i	$-0.909009\pi$
$983$	− 554.372i	− 0.563959i	0.959420	−	0.281980i	$-0.0909910\pi$
$984$	0	0
$985$	630.000	0.639594
$986$	0	0
$987$	311.127i	0.315225i
$988$	0	0
$989$	−42.0000	−0.0424671
$990$	0	0
$991$	639.225i	0.645030i	0.946564	+	0.322515i	$0.104528\pi$
$991$	639.225i	0.645030i	−0.946564	+	0.322515i	$0.895472\pi$
$992$	0	0
$993$	−544.000	−0.547835
$994$	0	0
$995$	−1029.00	−1.03417
$996$	0	0
$997$	−473.000	−0.474423	−0.237212	−	0.971458i	$-0.576233\pi$
$997$	−473.000	−0.474423	−0.237212	+	0.971458i	$0.576233\pi$
$998$	0	0
$999$	3136.00	3.13914

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.3.e.f.113.1		2
3.2	odd	2		2736.3.o.b.721.1		2
4.3	odd	2		152.3.e.a.113.2	yes	2
8.3	odd	2		1216.3.e.d.1025.1		2
8.5	even	2		1216.3.e.c.1025.2		2
12.11	even	2		1368.3.o.a.721.1		2
19.18	odd	2	inner	304.3.e.f.113.2		2
57.56	even	2		2736.3.o.b.721.2		2
76.75	even	2		152.3.e.a.113.1	✓	2
152.37	odd	2		1216.3.e.c.1025.1		2
152.75	even	2		1216.3.e.d.1025.2		2
228.227	odd	2		1368.3.o.a.721.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.3.e.a.113.1	✓	2	76.75	even	2
152.3.e.a.113.2	yes	2	4.3	odd	2
304.3.e.f.113.1		2	1.1	even	1	trivial
304.3.e.f.113.2		2	19.18	odd	2	inner
1216.3.e.c.1025.1		2	152.37	odd	2
1216.3.e.c.1025.2		2	8.5	even	2
1216.3.e.d.1025.1		2	8.3	odd	2
1216.3.e.d.1025.2		2	152.75	even	2
1368.3.o.a.721.1		2	12.11	even	2
1368.3.o.a.721.2		2	228.227	odd	2
2736.3.o.b.721.1		2	3.2	odd	2
2736.3.o.b.721.2		2	57.56	even	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	304.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.65685i q^{3} +7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} +O(q^{10})\)	\(q-5.65685i q^{3} +7.00000 q^{5} -11.0000 q^{7} -23.0000 q^{9} -3.00000 q^{11} -11.3137i q^{13} -39.5980i q^{15} -17.0000 q^{17} +19.0000 q^{19} +62.2254i q^{21} -2.00000 q^{23} +24.0000 q^{25} +79.1960i q^{27} -39.5980i q^{29} -5.65685i q^{31} +16.9706i q^{33} -77.0000 q^{35} -39.5980i q^{37} -64.0000 q^{39} +39.5980i q^{41} +21.0000 q^{43} -161.000 q^{45} +5.00000 q^{47} +72.0000 q^{49} +96.1665i q^{51} +5.65685i q^{53} -21.0000 q^{55} -107.480i q^{57} -33.9411i q^{59} +23.0000 q^{61} +253.000 q^{63} -79.1960i q^{65} -39.5980i q^{67} +11.3137i q^{69} -90.5097i q^{71} +39.0000 q^{73} -135.765i q^{75} +33.0000 q^{77} +96.1665i q^{79} +241.000 q^{81} +6.00000 q^{83} -119.000 q^{85} -224.000 q^{87} +118.794i q^{89} +124.451i q^{91} -32.0000 q^{93} +133.000 q^{95} -169.706i q^{97} +69.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{5} - 22 q^{7} - 46 q^{9}+O(q^{10}) \) 2 * q + 14 * q^5 - 22 * q^7 - 46 * q^9	\( 2 q + 14 q^{5} - 22 q^{7} - 46 q^{9} - 6 q^{11} - 34 q^{17} + 38 q^{19} - 4 q^{23} + 48 q^{25} - 154 q^{35} - 128 q^{39} + 42 q^{43} - 322 q^{45} + 10 q^{47} + 144 q^{49} - 42 q^{55} + 46 q^{61} + 506 q^{63} + 78 q^{73} + 66 q^{77} + 482 q^{81} + 12 q^{83} - 238 q^{85} - 448 q^{87} - 64 q^{93} + 266 q^{95} + 138 q^{99}+O(q^{100}) \) 2 * q + 14 * q^5 - 22 * q^7 - 46 * q^9 - 6 * q^11 - 34 * q^17 + 38 * q^19 - 4 * q^23 + 48 * q^25 - 154 * q^35 - 128 * q^39 + 42 * q^43 - 322 * q^45 + 10 * q^47 + 144 * q^49 - 42 * q^55 + 46 * q^61 + 506 * q^63 + 78 * q^73 + 66 * q^77 + 482 * q^81 + 12 * q^83 - 238 * q^85 - 448 * q^87 - 64 * q^93 + 266 * q^95 + 138 * q^99

Introduction

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