LMFDB - Embedded newform 2300.3.d.b.1149.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,3,Mod(1149,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2300.1149");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2300.d (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:	$62.6704608029$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1149.3
Character	$\chi$	$=$	2300.1149
Dual form			2300.3.d.b.1149.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.14043i q^{3} +7.88014 q^{7} -17.4240 q^{9} +O(q^{10})$	$q-5.14043i q^{3} +7.88014 q^{7} -17.4240 q^{9} -3.98380i q^{11} +13.5122i q^{13} +8.06277 q^{17} +2.96508i q^{19} -40.5073i q^{21} +(-22.4934 - 4.80075i) q^{23} +43.3029i q^{27} -0.688711 q^{29} -36.6485 q^{31} -20.4784 q^{33} -3.63322 q^{37} +69.4587 q^{39} -32.4636 q^{41} -42.5054 q^{43} +59.3870i q^{47} +13.0966 q^{49} -41.4461i q^{51} -57.5176 q^{53} +15.2418 q^{57} +25.3686 q^{59} -31.8220i q^{61} -137.303 q^{63} -108.405 q^{67} +(-24.6779 + 115.626i) q^{69} -67.6191 q^{71} -102.151i q^{73} -31.3929i q^{77} +48.5263i q^{79} +65.7793 q^{81} -73.0389 q^{83} +3.54027i q^{87} +92.9465i q^{89} +106.478i q^{91} +188.389i q^{93} -138.539 q^{97} +69.4136i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 128 q^{9}+O(q^{10}) $ 32 * q - 128 * q^9	$ 32 q - 128 q^{9} - 180 q^{29} + 20 q^{31} - 40 q^{39} + 372 q^{41} - 4 q^{49} + 180 q^{59} + 464 q^{69} - 476 q^{71} + 1408 q^{81}+O(q^{100}) $ 32 * q - 128 * q^9 - 180 * q^29 + 20 * q^31 - 40 * q^39 + 372 * q^41 - 4 * q^49 + 180 * q^59 + 464 * q^69 - 476 * q^71 + 1408 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$		−	5.14043i		−	1.71348i	−0.515752	−	0.856738i	$-0.672488\pi$
$3$		−	5.14043i		−	1.71348i	0.515752	−	0.856738i	$-0.327512\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	7.88014			1.12573			0.562867	−	0.826547i	$-0.309698\pi$
$7$	7.88014			1.12573			0.562867	+	0.826547i	$0.309698\pi$
$8$		0			0
$9$	−17.4240			−1.93600
$10$		0			0
$11$		−	3.98380i		−	0.362163i	−0.983468	−	0.181082i	$-0.942040\pi$
$11$		−	3.98380i		−	0.362163i	0.983468	−	0.181082i	$-0.0579598\pi$
$12$		0			0
$13$			13.5122i			1.03940i	0.854348	+	0.519702i	$0.173957\pi$
$13$			13.5122i			1.03940i	−0.854348	+	0.519702i	$0.826043\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.06277			0.474280			0.237140	−	0.971475i	$-0.423790\pi$
$17$	8.06277			0.474280			0.237140	+	0.971475i	$0.423790\pi$
$18$		0			0
$19$			2.96508i			0.156057i	0.996951	+	0.0780284i	$0.0248625\pi$
$19$			2.96508i			0.156057i	−0.996951	+	0.0780284i	$0.975138\pi$
$20$		0			0
$21$		−	40.5073i		−	1.92892i
$22$		0			0
$23$	−22.4934	−	4.80075i	−0.977974	−	0.208728i
$23$	−22.4934	−	4.80075i	−0.977974	−	0.208728i
$24$		0			0
$25$		0			0
$26$		0			0
$27$			43.3029i			1.60381i
$28$		0			0
$29$	−0.688711			−0.0237487			−0.0118743	−	0.999929i	$-0.503780\pi$
$29$	−0.688711			−0.0237487			−0.0118743	+	0.999929i	$0.503780\pi$
$30$		0			0
$31$	−36.6485			−1.18221			−0.591105	−	0.806595i	$-0.701308\pi$
$31$	−36.6485			−1.18221			−0.591105	+	0.806595i	$0.701308\pi$
$32$		0			0
$33$	−20.4784			−0.620558
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.63322			−0.0981952			−0.0490976	−	0.998794i	$-0.515635\pi$
$37$	−3.63322			−0.0981952			−0.0490976	+	0.998794i	$0.515635\pi$
$38$		0			0
$39$	69.4587			1.78099
$40$		0			0
$41$	−32.4636			−0.791796			−0.395898	−	0.918294i	$-0.629567\pi$
$41$	−32.4636			−0.791796			−0.395898	+	0.918294i	$0.629567\pi$
$42$		0			0
$43$	−42.5054			−0.988497			−0.494248	−	0.869321i	$-0.664557\pi$
$43$	−42.5054			−0.988497			−0.494248	+	0.869321i	$0.664557\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			59.3870i			1.26355i	0.775151	+	0.631777i	$0.217674\pi$
$47$			59.3870i			1.26355i	−0.775151	+	0.631777i	$0.782326\pi$
$48$		0			0
$49$	13.0966			0.267279
$50$		0			0
$51$		−	41.4461i		−	0.812668i
$52$		0			0
$53$	−57.5176			−1.08524			−0.542618	−	0.839979i	$-0.682567\pi$
$53$	−57.5176			−1.08524			−0.542618	+	0.839979i	$0.682567\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	15.2418			0.267399
$58$		0			0
$59$	25.3686			0.429977			0.214989	−	0.976617i	$-0.431029\pi$
$59$	25.3686			0.429977			0.214989	+	0.976617i	$0.431029\pi$
$60$		0			0
$61$		−	31.8220i		−	0.521672i	−0.965383	−	0.260836i	$-0.916002\pi$
$61$		−	31.8220i		−	0.521672i	0.965383	−	0.260836i	$-0.0839981\pi$
$62$		0			0
$63$	−137.303			−2.17942
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−108.405			−1.61799			−0.808993	−	0.587819i	$-0.799987\pi$
$67$	−108.405			−1.61799			−0.808993	+	0.587819i	$0.799987\pi$
$68$		0			0
$69$	−24.6779	+	115.626i	−0.357651	+	1.67573i
$70$		0			0
$71$	−67.6191			−0.952382			−0.476191	−	0.879342i	$-0.657983\pi$
$71$	−67.6191			−0.952382			−0.476191	+	0.879342i	$0.657983\pi$
$72$		0			0
$73$		−	102.151i		−	1.39933i	−0.714473	−	0.699663i	$-0.753333\pi$
$73$		−	102.151i		−	1.39933i	0.714473	−	0.699663i	$-0.246667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	31.3929i		−	0.407700i
$78$		0			0
$79$			48.5263i			0.614257i	0.951668	+	0.307128i	$0.0993681\pi$
$79$			48.5263i			0.614257i	−0.951668	+	0.307128i	$0.900632\pi$
$80$		0			0
$81$	65.7793			0.812091
$82$		0			0
$83$	−73.0389			−0.879987			−0.439993	−	0.898001i	$-0.645019\pi$
$83$	−73.0389			−0.879987			−0.439993	+	0.898001i	$0.645019\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			3.54027i			0.0406928i
$88$		0			0
$89$			92.9465i			1.04434i	0.852840	+	0.522172i	$0.174878\pi$
$89$			92.9465i			1.04434i	−0.852840	+	0.522172i	$0.825122\pi$
$90$		0			0
$91$			106.478i			1.17009i
$92$		0			0
$93$			188.389i			2.02569i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−138.539			−1.42824			−0.714119	−	0.700025i	$-0.753172\pi$
$97$	−138.539			−1.42824			−0.714119	+	0.700025i	$0.753172\pi$
$98$		0			0
$99$			69.4136i			0.701148i
$100$		0			0
$101$	71.2312			0.705259			0.352630	−	0.935763i	$-0.385288\pi$
$101$	71.2312			0.705259			0.352630	+	0.935763i	$0.385288\pi$
$102$		0			0
$103$	−173.582			−1.68526			−0.842630	−	0.538492i	$-0.818994\pi$
$103$	−173.582			−1.68526			−0.842630	+	0.538492i	$0.818994\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−88.3635			−0.825827			−0.412914	−	0.910770i	$-0.635489\pi$
$107$	−88.3635			−0.825827			−0.412914	+	0.910770i	$0.635489\pi$
$108$		0			0
$109$		−	142.645i		−	1.30867i	−0.756203	−	0.654337i	$-0.772948\pi$
$109$		−	142.645i		−	1.30867i	0.756203	−	0.654337i	$-0.227052\pi$
$110$		0			0
$111$			18.6763i			0.168255i
$112$		0			0
$113$	91.5083			0.809808			0.404904	−	0.914359i	$-0.367305\pi$
$113$	91.5083			0.809808			0.404904	+	0.914359i	$0.367305\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	235.437i		−	2.01228i
$118$		0			0
$119$	63.5358			0.533914
$120$		0			0
$121$	105.129			0.868838
$122$		0			0
$123$			166.877i			1.35672i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	101.492i		−	0.799147i	−0.916701	−	0.399574i	$-0.869158\pi$
$127$		−	101.492i		−	0.799147i	0.916701	−	0.399574i	$-0.130842\pi$
$128$		0			0
$129$			218.496i			1.69377i
$130$		0			0
$131$	−43.9095			−0.335187			−0.167594	−	0.985856i	$-0.553600\pi$
$131$	−43.9095			−0.335187			−0.167594	+	0.985856i	$0.553600\pi$
$132$		0			0
$133$			23.3652i			0.175678i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	215.195			1.57077			0.785384	−	0.619009i	$-0.212465\pi$
$137$	215.195			1.57077			0.785384	+	0.619009i	$0.212465\pi$
$138$		0			0
$139$	−63.6545			−0.457946			−0.228973	−	0.973433i	$-0.573537\pi$
$139$	−63.6545			−0.457946			−0.228973	+	0.973433i	$0.573537\pi$
$140$		0			0
$141$	305.275			2.16507
$142$		0			0
$143$	53.8300			0.376434
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	67.3223i		−	0.457975i
$148$		0			0
$149$		−	97.7394i		−	0.655969i	−0.944683	−	0.327985i	$-0.893631\pi$
$149$		−	97.7394i		−	0.655969i	0.944683	−	0.327985i	$-0.106369\pi$
$150$		0			0
$151$	−209.737			−1.38899			−0.694493	−	0.719500i	$-0.744371\pi$
$151$	−209.737			−1.38899			−0.694493	+	0.719500i	$0.744371\pi$
$152$		0			0
$153$	−140.486			−0.918206
$154$		0			0
$155$		0			0
$156$		0			0
$157$	287.459			1.83095			0.915474	−	0.402378i	$-0.131816\pi$
$157$	287.459			1.83095			0.915474	+	0.402378i	$0.131816\pi$
$158$		0			0
$159$			295.665i			1.85953i
$160$		0			0
$161$	−177.251	−	37.8306i	−1.10094	−	0.234973i
$162$		0			0
$163$			112.054i			0.687446i	0.939071	+	0.343723i	$0.111688\pi$
$163$			112.054i			0.687446i	−0.939071	+	0.343723i	$0.888312\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			192.709i			1.15394i	0.816764	+	0.576972i	$0.195766\pi$
$167$			192.709i			1.15394i	−0.816764	+	0.576972i	$0.804234\pi$
$168$		0			0
$169$	−13.5807			−0.0803593
$170$		0			0
$171$		−	51.6635i		−	0.302125i
$172$		0			0
$173$		−	303.454i		−	1.75407i	−0.480429	−	0.877034i	$-0.659519\pi$
$173$		−	303.454i		−	1.75407i	0.480429	−	0.877034i	$-0.340481\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	130.406i		−	0.736755i
$178$		0			0
$179$	94.6527			0.528786			0.264393	−	0.964415i	$-0.414828\pi$
$179$	94.6527			0.528786			0.264393	+	0.964415i	$0.414828\pi$
$180$		0			0
$181$		−	251.014i		−	1.38682i	−0.720545	−	0.693408i	$-0.756108\pi$
$181$		−	251.014i		−	1.38682i	0.720545	−	0.693408i	$-0.243892\pi$
$182$		0			0
$183$	−163.578			−0.893871
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	32.1204i		−	0.171767i
$188$		0			0
$189$			341.233i			1.80546i
$190$		0			0
$191$			74.3125i			0.389071i	0.980895	+	0.194535i	$0.0623199\pi$
$191$			74.3125i			0.389071i	−0.980895	+	0.194535i	$0.937680\pi$
$192$		0			0
$193$		−	279.864i		−	1.45007i	−0.688710	−	0.725037i	$-0.741823\pi$
$193$		−	279.864i		−	1.45007i	0.688710	−	0.725037i	$-0.258177\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			239.461i			1.21554i	0.794113	+	0.607770i	$0.207936\pi$
$197$			239.461i			1.21554i	−0.794113	+	0.607770i	$0.792064\pi$
$198$		0			0
$199$		−	302.424i		−	1.51972i	−0.650087	−	0.759860i	$-0.725267\pi$
$199$		−	302.424i		−	1.51972i	0.650087	−	0.759860i	$-0.274733\pi$
$200$		0			0
$201$			557.248i			2.77238i
$202$		0			0
$203$	−5.42714			−0.0267347
$204$		0			0
$205$		0			0
$206$		0			0
$207$	391.925	+	83.6483i	1.89336	+	0.404098i
$208$		0			0
$209$	11.8123			0.0565180
$210$		0			0
$211$	313.131			1.48403			0.742016	−	0.670382i	$-0.233870\pi$
$211$	313.131			1.48403			0.742016	+	0.670382i	$0.233870\pi$
$212$		0			0
$213$			347.591i			1.63188i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−288.795			−1.33085
$218$		0			0
$219$	−525.099			−2.39771
$220$		0			0
$221$			108.946i			0.492969i
$222$		0			0
$223$		−	42.5933i		−	0.191001i	−0.995429	−	0.0955007i	$-0.969555\pi$
$223$		−	42.5933i		−	0.191001i	0.995429	−	0.0955007i	$-0.0304452\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−246.716			−1.08686			−0.543428	−	0.839456i	$-0.682874\pi$
$227$	−246.716			−1.08686			−0.543428	+	0.839456i	$0.682874\pi$
$228$		0			0
$229$			192.536i			0.840767i	0.907347	+	0.420383i	$0.138104\pi$
$229$			192.536i			0.840767i	−0.907347	+	0.420383i	$0.861896\pi$
$230$		0			0
$231$	−161.373			−0.698584
$232$		0			0
$233$			246.508i			1.05797i	0.848630	+	0.528986i	$0.177428\pi$
$233$			246.508i			1.05797i	−0.848630	+	0.528986i	$0.822572\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	249.446			1.05251
$238$		0			0
$239$	36.1050			0.151067			0.0755335	−	0.997143i	$-0.475934\pi$
$239$	36.1050			0.151067			0.0755335	+	0.997143i	$0.475934\pi$
$240$		0			0
$241$			137.672i			0.571253i	0.958341	+	0.285627i	$0.0922017\pi$
$241$			137.672i			0.571253i	−0.958341	+	0.285627i	$0.907798\pi$
$242$		0			0
$243$			51.5919i			0.212312i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−40.0648			−0.162206
$248$		0			0
$249$			375.451i			1.50784i
$250$		0			0
$251$		−	60.0114i		−	0.239089i	−0.992829	−	0.119545i	$-0.961857\pi$
$251$		−	60.0114i		−	0.239089i	0.992829	−	0.119545i	$-0.0381435\pi$
$252$		0			0
$253$	−19.1252	+	89.6091i	−0.0755938	+	0.354186i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			162.546i			0.632475i	0.948680	+	0.316237i	$0.102420\pi$
$257$			162.546i			0.632475i	−0.948680	+	0.316237i	$0.897580\pi$
$258$		0			0
$259$	−28.6303			−0.110542
$260$		0			0
$261$	12.0001			0.0459774
$262$		0			0
$263$	416.956			1.58539			0.792693	−	0.609621i	$-0.208678\pi$
$263$	416.956			1.58539			0.792693	+	0.609621i	$0.208678\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	477.785			1.78946
$268$		0			0
$269$	354.383			1.31741			0.658704	−	0.752402i	$-0.271105\pi$
$269$	354.383			1.31741			0.658704	+	0.752402i	$0.271105\pi$
$270$		0			0
$271$	430.921			1.59011			0.795056	−	0.606535i	$-0.207441\pi$
$271$	430.921			1.59011			0.795056	+	0.606535i	$0.207441\pi$
$272$		0			0
$273$	547.344			2.00492
$274$		0			0
$275$		0			0
$276$		0			0
$277$			17.8119i			0.0643029i	0.999483	+	0.0321515i	$0.0102359\pi$
$277$			17.8119i			0.0643029i	−0.999483	+	0.0321515i	$0.989764\pi$
$278$		0			0
$279$	638.563			2.28876
$280$		0			0
$281$			537.350i			1.91228i	0.292915	+	0.956138i	$0.405375\pi$
$281$			537.350i			1.91228i	−0.292915	+	0.956138i	$0.594625\pi$
$282$		0			0
$283$	227.251			0.803007			0.401503	−	0.915858i	$-0.368488\pi$
$283$	227.251			0.803007			0.401503	+	0.915858i	$0.368488\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−255.818			−0.891352
$288$		0			0
$289$	−223.992			−0.775058
$290$		0			0
$291$			712.150i			2.44725i
$292$		0			0
$293$	−570.580			−1.94737			−0.973685	−	0.227896i	$-0.926815\pi$
$293$	−570.580			−1.94737			−0.973685	+	0.227896i	$0.926815\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	172.510			0.580841
$298$		0			0
$299$	64.8690	−	303.936i	0.216953	−	1.01651i
$300$		0			0
$301$	−334.948			−1.11279
$302$		0			0
$303$		−	366.159i		−	1.20844i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			17.4675i			0.0568973i	0.999595	+	0.0284487i	$0.00905671\pi$
$307$			17.4675i			0.0568973i	−0.999595	+	0.0284487i	$0.990943\pi$
$308$		0			0
$309$			892.285i			2.88765i
$310$		0			0
$311$	−41.5565			−0.133622			−0.0668111	−	0.997766i	$-0.521282\pi$
$311$	−41.5565			−0.133622			−0.0668111	+	0.997766i	$0.521282\pi$
$312$		0			0
$313$	193.750			0.619011			0.309505	−	0.950898i	$-0.399836\pi$
$313$	193.750			0.619011			0.309505	+	0.950898i	$0.399836\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	292.088i		−	0.921413i	−0.887553	−	0.460707i	$-0.847596\pi$
$317$		−	292.088i		−	0.921413i	0.887553	−	0.460707i	$-0.152404\pi$
$318$		0			0
$319$			2.74369i			0.00860090i
$320$		0			0
$321$			454.226i			1.41503i
$322$		0			0
$323$			23.9067i			0.0740146i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−733.259			−2.24238
$328$		0			0
$329$			467.978i			1.42243i
$330$		0			0
$331$	562.886			1.70056			0.850281	−	0.526329i	$-0.176432\pi$
$331$	562.886			1.70056			0.850281	+	0.526329i	$0.176432\pi$
$332$		0			0
$333$	63.3052			0.190106
$334$		0			0
$335$		0			0
$336$		0			0
$337$	178.192			0.528759			0.264379	−	0.964419i	$-0.414833\pi$
$337$	178.192			0.528759			0.264379	+	0.964419i	$0.414833\pi$
$338$		0			0
$339$		−	470.392i		−	1.38759i
$340$		0			0
$341$			146.000i			0.428153i
$342$		0			0
$343$	−282.924			−0.824850
$344$		0			0
$345$		0			0
$346$		0			0
$347$			462.453i			1.33272i	0.745631	+	0.666359i	$0.232148\pi$
$347$			462.453i			1.33272i	−0.745631	+	0.666359i	$0.767852\pi$
$348$		0			0
$349$	−159.441			−0.456850			−0.228425	−	0.973562i	$-0.573358\pi$
$349$	−159.441			−0.456850			−0.228425	+	0.973562i	$0.573358\pi$
$350$		0			0
$351$	−585.119			−1.66701
$352$		0			0
$353$		−	474.031i		−	1.34286i	−0.741067	−	0.671432i	$-0.765680\pi$
$353$		−	474.031i		−	1.34286i	0.741067	−	0.671432i	$-0.234320\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		−	326.601i		−	0.914848i
$358$		0			0
$359$		−	462.027i		−	1.28698i	−0.765453	−	0.643492i	$-0.777485\pi$
$359$		−	462.027i		−	1.28698i	0.765453	−	0.643492i	$-0.222515\pi$
$360$		0			0
$361$	352.208			0.975646
$362$		0			0
$363$		−	540.410i		−	1.48873i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−215.737			−0.587839			−0.293919	−	0.955830i	$-0.594960\pi$
$367$	−215.737			−0.587839			−0.293919	+	0.955830i	$0.594960\pi$
$368$		0			0
$369$	565.646			1.53292
$370$		0			0
$371$	−453.247			−1.22169
$372$		0			0
$373$	−137.933			−0.369795			−0.184897	−	0.982758i	$-0.559195\pi$
$373$	−137.933			−0.369795			−0.184897	+	0.982758i	$0.559195\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	9.30604i		−	0.0246844i
$378$		0			0
$379$		−	15.4281i		−	0.0407073i	−0.999793	−	0.0203536i	$-0.993521\pi$
$379$		−	15.4281i		−	0.0407073i	0.999793	−	0.0203536i	$-0.00647921\pi$
$380$		0			0
$381$	−521.711			−1.36932
$382$		0			0
$383$	−713.842			−1.86382			−0.931908	−	0.362694i	$-0.881857\pi$
$383$	−713.842			−1.86382			−0.931908	+	0.362694i	$0.881857\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	740.613			1.91373
$388$		0			0
$389$		−	165.475i		−	0.425385i	−0.977119	−	0.212692i	$-0.931777\pi$
$389$		−	165.475i		−	0.425385i	0.977119	−	0.212692i	$-0.0682232\pi$
$390$		0			0
$391$	−181.359	−	38.7074i	−0.463834	−	0.0989958i
$392$		0			0
$393$			225.714i			0.574335i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			92.3570i			0.232637i	0.993212	+	0.116319i	$0.0371094\pi$
$397$			92.3570i			0.232637i	−0.993212	+	0.116319i	$0.962891\pi$
$398$		0			0
$399$	120.107			0.301021
$400$		0			0
$401$		−	162.145i		−	0.404352i	−0.979349	−	0.202176i	$-0.935199\pi$
$401$		−	162.145i		−	0.404352i	0.979349	−	0.202176i	$-0.0648013\pi$
$402$		0			0
$403$		−	495.203i		−	1.22879i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			14.4740i			0.0355627i
$408$		0			0
$409$	91.8175			0.224493			0.112246	−	0.993680i	$-0.464195\pi$
$409$	91.8175			0.224493			0.112246	+	0.993680i	$0.464195\pi$
$410$		0			0
$411$		−	1106.20i		−	2.69147i
$412$		0			0
$413$	199.909			0.484040
$414$		0			0
$415$		0			0
$416$		0			0
$417$			327.211i			0.784679i
$418$		0			0
$419$			210.857i			0.503238i	0.967826	+	0.251619i	$0.0809629\pi$
$419$			210.857i			0.503238i	−0.967826	+	0.251619i	$0.919037\pi$
$420$		0			0
$421$		−	557.635i		−	1.32455i	−0.749261	−	0.662274i	$-0.769591\pi$
$421$		−	557.635i		−	1.32455i	0.749261	−	0.662274i	$-0.230409\pi$
$422$		0			0
$423$		−	1034.76i		−	2.44624i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	250.762i		−	0.587264i
$428$		0			0
$429$		−	276.709i		−	0.645010i
$430$		0			0
$431$		−	162.258i		−	0.376469i	−0.982124	−	0.188234i	$-0.939724\pi$
$431$		−	162.258i		−	0.376469i	0.982124	−	0.188234i	$-0.0602765\pi$
$432$		0			0
$433$	−465.859			−1.07589			−0.537944	−	0.842981i	$-0.680799\pi$
$433$	−465.859			−1.07589			−0.537944	+	0.842981i	$0.680799\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	14.2346	−	66.6947i	0.0325735	−	0.152619i
$438$		0			0
$439$	112.072			0.255289			0.127644	−	0.991820i	$-0.459258\pi$
$439$	112.072			0.255289			0.127644	+	0.991820i	$0.459258\pi$
$440$		0			0
$441$	−228.196			−0.517451
$442$		0			0
$443$			765.565i			1.72814i	0.503373	+	0.864069i	$0.332092\pi$
$443$			765.565i			1.72814i	−0.503373	+	0.864069i	$0.667908\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−502.422			−1.12399
$448$		0			0
$449$	167.225			0.372439			0.186220	−	0.982508i	$-0.440376\pi$
$449$	167.225			0.372439			0.186220	+	0.982508i	$0.440376\pi$
$450$		0			0
$451$			129.329i			0.286759i
$452$		0			0
$453$			1078.14i			2.37999i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−283.250			−0.619804			−0.309902	−	0.950768i	$-0.600296\pi$
$457$	−283.250			−0.619804			−0.309902	+	0.950768i	$0.600296\pi$
$458$		0			0
$459$			349.141i			0.760656i
$460$		0			0
$461$	−98.1109			−0.212822			−0.106411	−	0.994322i	$-0.533936\pi$
$461$	−98.1109			−0.212822			−0.106411	+	0.994322i	$0.533936\pi$
$462$		0			0
$463$			749.068i			1.61786i	0.587907	+	0.808929i	$0.299952\pi$
$463$			749.068i			1.61786i	−0.587907	+	0.808929i	$0.700048\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−71.8733			−0.153904			−0.0769522	−	0.997035i	$-0.524519\pi$
$467$	−71.8733			−0.153904			−0.0769522	+	0.997035i	$0.524519\pi$
$468$		0			0
$469$	−854.247			−1.82142
$470$		0			0
$471$		−	1477.66i		−	3.13728i
$472$		0			0
$473$			169.333i			0.357997i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	1002.18			2.10102
$478$		0			0
$479$		−	624.966i		−	1.30473i	−0.757905	−	0.652365i	$-0.773777\pi$
$479$		−	624.966i		−	1.30473i	0.757905	−	0.652365i	$-0.226223\pi$
$480$		0			0
$481$		−	49.0930i		−	0.102064i
$482$		0			0
$483$	−194.466	+	911.146i	−0.402620	+	1.88643i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	879.504i		−	1.80596i	−0.429679	−	0.902982i	$-0.641373\pi$
$487$		−	879.504i		−	1.80596i	0.429679	−	0.902982i	$-0.358627\pi$
$488$		0			0
$489$	576.004			1.17792
$490$		0			0
$491$	716.334			1.45893			0.729464	−	0.684019i	$-0.239770\pi$
$491$	716.334			1.45893			0.729464	+	0.684019i	$0.239770\pi$
$492$		0			0
$493$	−5.55292			−0.0112635
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−532.848			−1.07213
$498$		0			0
$499$	34.6000			0.0693387			0.0346693	−	0.999399i	$-0.488962\pi$
$499$	34.6000			0.0693387			0.0346693	+	0.999399i	$0.488962\pi$
$500$		0			0
$501$	990.604			1.97725
$502$		0			0
$503$	−287.726			−0.572020			−0.286010	−	0.958227i	$-0.592329\pi$
$503$	−287.726			−0.572020			−0.286010	+	0.958227i	$0.592329\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			69.8107i			0.137694i
$508$		0			0
$509$	−284.848			−0.559624			−0.279812	−	0.960055i	$-0.590272\pi$
$509$	−284.848			−0.559624			−0.279812	+	0.960055i	$0.590272\pi$
$510$		0			0
$511$		−	804.963i		−	1.57527i
$512$		0			0
$513$	−128.396			−0.250285
$514$		0			0
$515$		0			0
$516$		0			0
$517$	236.586			0.457613
$518$		0			0
$519$	−1559.88			−3.00555
$520$		0			0
$521$			195.190i			0.374645i	0.982298	+	0.187323i	$0.0599810\pi$
$521$			195.190i			0.374645i	−0.982298	+	0.187323i	$0.940019\pi$
$522$		0			0
$523$	−0.505041			−0.000965661			−0.000482831	−	1.00000i	$-0.500154\pi$
$523$	−0.505041			−0.000965661			−0.000482831		1.00000i	$0.500154\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−295.488			−0.560699
$528$		0			0
$529$	482.906	+	215.970i	0.912865	+	0.408262i
$530$		0			0
$531$	−442.023			−0.832435
$532$		0			0
$533$		−	438.657i		−	0.822995i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	486.555i		−	0.906062i
$538$		0			0
$539$		−	52.1744i		−	0.0967985i
$540$		0			0
$541$	579.191			1.07059			0.535296	−	0.844664i	$-0.320200\pi$
$541$	579.191			1.07059			0.535296	+	0.844664i	$0.320200\pi$
$542$		0			0
$543$	−1290.32			−2.37628
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	623.934i		−	1.14065i	−0.821420	−	0.570324i	$-0.806818\pi$
$547$		−	623.934i		−	1.14065i	0.821420	−	0.570324i	$-0.193182\pi$
$548$		0			0
$549$			554.465i			1.00996i
$550$		0			0
$551$		−	2.04208i		−	0.00370614i
$552$		0			0
$553$			382.394i			0.691490i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−445.491			−0.799805			−0.399903	−	0.916558i	$-0.630956\pi$
$557$	−445.491			−0.799805			−0.399903	+	0.916558i	$0.630956\pi$
$558$		0			0
$559$		−	574.343i		−	1.02745i
$560$		0			0
$561$	−165.113			−0.294319
$562$		0			0
$563$	315.518			0.560422			0.280211	−	0.959938i	$-0.409595\pi$
$563$	315.518			0.560422			0.280211	+	0.959938i	$0.409595\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	518.351			0.914199
$568$		0			0
$569$		−	954.594i		−	1.67767i	−0.544386	−	0.838835i	$-0.683237\pi$
$569$		−	954.594i		−	1.67767i	0.544386	−	0.838835i	$-0.316763\pi$
$570$		0			0
$571$		−	266.616i		−	0.466929i	−0.972365	−	0.233464i	$-0.924994\pi$
$571$		−	266.616i		−	0.466929i	0.972365	−	0.233464i	$-0.0750062\pi$
$572$		0			0
$573$	381.998			0.666663
$574$		0			0
$575$		0			0
$576$		0			0
$577$			1069.07i			1.85280i	0.376539	+	0.926401i	$0.377114\pi$
$577$			1069.07i			1.85280i	−0.376539	+	0.926401i	$0.622886\pi$
$578$		0			0
$579$	−1438.62			−2.48466
$580$		0			0
$581$	−575.557			−0.990632
$582$		0			0
$583$			229.138i			0.393033i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1065.52i			1.81520i	0.419835	+	0.907600i	$0.362088\pi$
$587$			1065.52i			1.81520i	−0.419835	+	0.907600i	$0.637912\pi$
$588$		0			0
$589$		−	108.666i		−	0.184492i
$590$		0			0
$591$	1230.93			2.08280
$592$		0			0
$593$		−	337.936i		−	0.569875i	−0.958546	−	0.284938i	$-0.908027\pi$
$593$		−	337.936i		−	0.569875i	0.958546	−	0.284938i	$-0.0919729\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1554.59			−2.60400
$598$		0			0
$599$	−549.708			−0.917709			−0.458854	−	0.888511i	$-0.651740\pi$
$599$	−549.708			−0.917709			−0.458854	+	0.888511i	$0.651740\pi$
$600$		0			0
$601$	−375.387			−0.624605			−0.312302	−	0.949983i	$-0.601100\pi$
$601$	−375.387			−0.624605			−0.312302	+	0.949983i	$0.601100\pi$
$602$		0			0
$603$	1888.85			3.13242
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	440.263i		−	0.725309i	−0.931924	−	0.362654i	$-0.881871\pi$
$607$		−	440.263i		−	0.725309i	0.931924	−	0.362654i	$-0.118129\pi$
$608$		0			0
$609$			27.8978i			0.0458093i
$610$		0			0
$611$	−802.452			−1.31334
$612$		0			0
$613$	−49.5676			−0.0808607			−0.0404303	−	0.999182i	$-0.512873\pi$
$613$	−49.5676			−0.0808607			−0.0404303	+	0.999182i	$0.512873\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−785.550			−1.27318			−0.636588	−	0.771204i	$-0.719655\pi$
$617$	−785.550			−1.27318			−0.636588	+	0.771204i	$0.719655\pi$
$618$		0			0
$619$		−	693.014i		−	1.11957i	−0.828638	−	0.559785i	$-0.810884\pi$
$619$		−	693.014i		−	1.11957i	0.828638	−	0.559785i	$-0.189116\pi$
$620$		0			0
$621$	207.886	−	974.028i	0.334761	−	1.56848i
$622$		0			0
$623$			732.432i			1.17565i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	60.7201i		−	0.0968422i
$628$		0			0
$629$	−29.2938			−0.0465721
$630$		0			0
$631$			9.68025i			0.0153411i	0.999971	+	0.00767056i	$0.00244164\pi$
$631$			9.68025i			0.0153411i	−0.999971	+	0.00767056i	$0.997558\pi$
$632$		0			0
$633$		−	1609.63i		−	2.54285i
$634$		0			0
$635$		0			0
$636$		0			0
$637$			176.965i			0.277810i
$638$		0			0
$639$	1178.19			1.84381
$640$		0			0
$641$		−	174.812i		−	0.272718i	−0.990659	−	0.136359i	$-0.956460\pi$
$641$		−	174.812i		−	0.272718i	0.990659	−	0.136359i	$-0.0435401\pi$
$642$		0			0
$643$	−285.997			−0.444785			−0.222393	−	0.974957i	$-0.571387\pi$
$643$	−285.997			−0.444785			−0.222393	+	0.974957i	$0.571387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	698.951i		−	1.08030i	−0.841570	−	0.540148i	$-0.818368\pi$
$647$		−	698.951i		−	1.08030i	0.841570	−	0.540148i	$-0.181632\pi$
$648$		0			0
$649$		−	101.064i		−	0.155722i
$650$		0			0
$651$			1484.53i			2.28039i
$652$		0			0
$653$		−	639.144i		−	0.978781i	−0.872065	−	0.489390i	$-0.837219\pi$
$653$		−	639.144i		−	0.978781i	0.872065	−	0.489390i	$-0.162781\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			1779.87i			2.70909i
$658$		0			0
$659$			1204.48i			1.82774i	0.406004	+	0.913871i	$0.366922\pi$
$659$			1204.48i			1.82774i	−0.406004	+	0.913871i	$0.633078\pi$
$660$		0			0
$661$			442.012i			0.668702i	0.942449	+	0.334351i	$0.108517\pi$
$661$			442.012i			0.668702i	−0.942449	+	0.334351i	$0.891483\pi$
$662$		0			0
$663$	560.029			0.844690
$664$		0			0
$665$		0			0
$666$		0			0
$667$	15.4915	+	3.30633i	0.0232256	+	0.00495702i
$668$		0			0
$669$	−218.948			−0.327276
$670$		0			0
$671$	−126.772			−0.188930
$672$		0			0
$673$			401.763i			0.596974i	0.954414	+	0.298487i	$0.0964819\pi$
$673$			401.763i			0.596974i	−0.954414	+	0.298487i	$0.903518\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−575.340			−0.849838			−0.424919	−	0.905231i	$-0.639697\pi$
$677$	−575.340			−0.849838			−0.424919	+	0.905231i	$0.639697\pi$
$678$		0			0
$679$	−1091.71			−1.60782
$680$		0			0
$681$			1268.23i			1.86230i
$682$		0			0
$683$		−	148.111i		−	0.216854i	−0.994104	−	0.108427i	$-0.965419\pi$
$683$		−	148.111i		−	0.216854i	0.994104	−	0.108427i	$-0.0345814\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	989.715			1.44063
$688$		0			0
$689$		−	777.191i		−	1.12800i
$690$		0			0
$691$	−176.640			−0.255629			−0.127815	−	0.991798i	$-0.540796\pi$
$691$	−176.640			−0.255629			−0.127815	+	0.991798i	$0.540796\pi$
$692$		0			0
$693$			546.989i			0.789306i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−261.747			−0.375533
$698$		0			0
$699$	1267.15			1.81281
$700$		0			0
$701$			179.122i			0.255524i	0.991805	+	0.127762i	$0.0407793\pi$
$701$			179.122i			0.255524i	−0.991805	+	0.127762i	$0.959221\pi$
$702$		0			0
$703$		−	10.7728i		−	0.0153240i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	561.312			0.793935
$708$		0			0
$709$		−	612.124i		−	0.863363i	−0.902026	−	0.431681i	$-0.857920\pi$
$709$		−	612.124i		−	0.863363i	0.902026	−	0.431681i	$-0.142080\pi$
$710$		0			0
$711$		−	845.521i		−	1.18920i
$712$		0			0
$713$	824.349	+	175.940i	1.15617	+	0.246761i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	185.595i		−	0.258850i
$718$		0			0
$719$	332.970			0.463102			0.231551	−	0.972823i	$-0.425620\pi$
$719$	332.970			0.463102			0.231551	+	0.972823i	$0.425620\pi$
$720$		0			0
$721$	−1367.85			−1.89716
$722$		0			0
$723$	707.693			0.978829
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−363.921			−0.500579			−0.250290	−	0.968171i	$-0.580526\pi$
$727$	−363.921			−0.500579			−0.250290	+	0.968171i	$0.580526\pi$
$728$		0			0
$729$	857.219			1.17588
$730$		0			0
$731$	−342.711			−0.468825
$732$		0			0
$733$	414.694			0.565749			0.282875	−	0.959157i	$-0.408712\pi$
$733$	414.694			0.565749			0.282875	+	0.959157i	$0.408712\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			431.864i			0.585975i
$738$		0			0
$739$	750.501			1.01556			0.507782	−	0.861486i	$-0.330466\pi$
$739$	750.501			1.01556			0.507782	+	0.861486i	$0.330466\pi$
$740$		0			0
$741$			205.950i			0.277936i
$742$		0			0
$743$	−1301.85			−1.75215			−0.876077	−	0.482171i	$-0.839848\pi$
$743$	−1301.85			−1.75215			−0.876077	+	0.482171i	$0.839848\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1272.63			1.70365
$748$		0			0
$749$	−696.317			−0.929662
$750$		0			0
$751$		−	860.771i		−	1.14617i	−0.819497	−	0.573083i	$-0.805747\pi$
$751$		−	860.771i		−	1.14617i	0.819497	−	0.573083i	$-0.194253\pi$
$752$		0			0
$753$	−308.484			−0.409674
$754$		0			0
$755$		0			0
$756$		0			0
$757$	281.079			0.371306			0.185653	−	0.982615i	$-0.440560\pi$
$757$	281.079			0.371306			0.185653	+	0.982615i	$0.440560\pi$
$758$		0			0
$759$	460.629	+	98.3118i	0.606889	+	0.129528i
$760$		0			0
$761$	155.150			0.203877			0.101938	−	0.994791i	$-0.467496\pi$
$761$	155.150			0.203877			0.101938	+	0.994791i	$0.467496\pi$
$762$		0			0
$763$		−	1124.07i		−	1.47322i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			342.787i			0.446920i
$768$		0			0
$769$		−	1242.19i		−	1.61533i	−0.589642	−	0.807665i	$-0.700731\pi$
$769$		−	1242.19i		−	1.61533i	0.589642	−	0.807665i	$-0.299269\pi$
$770$		0			0
$771$	835.556			1.08373
$772$		0			0
$773$	1105.56			1.43022			0.715108	−	0.699014i	$-0.246377\pi$
$773$	1105.56			1.43022			0.715108	+	0.699014i	$0.246377\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$			147.172i			0.189411i
$778$		0			0
$779$		−	96.2572i		−	0.123565i
$780$		0			0
$781$			269.381i			0.344918i
$782$		0			0
$783$		−	29.8232i		−	0.0380884i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	166.440			0.211486			0.105743	−	0.994393i	$-0.466278\pi$
$787$	166.440			0.211486			0.105743	+	0.994393i	$0.466278\pi$
$788$		0			0
$789$		−	2143.33i		−	2.71652i
$790$		0			0
$791$	721.098			0.911629
$792$		0			0
$793$	429.986			0.542227
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−702.946			−0.881990			−0.440995	−	0.897510i	$-0.645374\pi$
$797$	−702.946			−0.881990			−0.440995	+	0.897510i	$0.645374\pi$
$798$		0			0
$799$			478.824i			0.599279i
$800$		0			0
$801$		−	1619.50i		−	2.02185i
$802$		0			0
$803$	−406.948			−0.506785
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	1821.68i		−	2.25735i
$808$		0			0
$809$	−612.933			−0.757643			−0.378822	−	0.925470i	$-0.623671\pi$
$809$	−612.933			−0.757643			−0.378822	+	0.925470i	$0.623671\pi$
$810$		0			0
$811$	−664.035			−0.818785			−0.409393	−	0.912358i	$-0.634259\pi$
$811$	−664.035			−0.818785			−0.409393	+	0.912358i	$0.634259\pi$
$812$		0			0
$813$		−	2215.12i		−	2.72462i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	126.032i		−	0.154262i
$818$		0			0
$819$		−	1855.28i		−	2.26530i
$820$		0			0
$821$	598.546			0.729046			0.364523	−	0.931194i	$-0.381232\pi$
$821$	598.546			0.729046			0.364523	+	0.931194i	$0.381232\pi$
$822$		0			0
$823$			1216.57i			1.47821i	0.673591	+	0.739105i	$0.264751\pi$
$823$			1216.57i			1.47821i	−0.673591	+	0.739105i	$0.735249\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−719.387			−0.869875			−0.434937	−	0.900461i	$-0.643230\pi$
$827$	−719.387			−0.869875			−0.434937	+	0.900461i	$0.643230\pi$
$828$		0			0
$829$	340.645			0.410911			0.205455	−	0.978666i	$-0.434132\pi$
$829$	340.645			0.410911			0.205455	+	0.978666i	$0.434132\pi$
$830$		0			0
$831$	91.5608			0.110181
$832$		0			0
$833$	105.595			0.126765
$834$		0			0
$835$		0			0
$836$		0			0
$837$		−	1586.98i		−	1.89604i
$838$		0			0
$839$		−	9.11326i		−	0.0108621i	−0.999985	−	0.00543103i	$-0.998271\pi$
$839$		−	9.11326i		−	0.0108621i	0.999985	−	0.00543103i	$-0.00172876\pi$
$840$		0			0
$841$	−840.526			−0.999436
$842$		0			0
$843$	2762.21			3.27664
$844$		0			0
$845$		0			0
$846$		0			0
$847$	828.434			0.978081
$848$		0			0
$849$		−	1168.17i		−	1.37593i
$850$		0			0
$851$	81.7235	+	17.4422i	0.0960323	+	0.0204961i
$852$		0			0
$853$		−	1029.00i		−	1.20633i	−0.797618	−	0.603163i	$-0.793907\pi$
$853$		−	1029.00i		−	1.20633i	0.797618	−	0.603163i	$-0.206093\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	606.460i		−	0.707654i	−0.935311	−	0.353827i	$-0.884880\pi$
$857$		−	606.460i		−	0.707654i	0.935311	−	0.353827i	$-0.115120\pi$
$858$		0			0
$859$	−1147.62			−1.33600			−0.668001	−	0.744161i	$-0.732850\pi$
$859$	−1147.62			−1.33600			−0.668001	+	0.744161i	$0.732850\pi$
$860$		0			0
$861$			1315.01i			1.52731i
$862$		0			0
$863$		−	1299.38i		−	1.50566i	−0.658216	−	0.752829i	$-0.728689\pi$
$863$		−	1299.38i		−	1.50566i	0.658216	−	0.752829i	$-0.271311\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1151.41i			1.32804i
$868$		0			0
$869$	193.319			0.222461
$870$		0			0
$871$		−	1464.80i		−	1.68174i
$872$		0			0
$873$	2413.90			2.76507
$874$		0			0
$875$		0			0
$876$		0			0
$877$			869.529i			0.991482i	0.868470	+	0.495741i	$0.165103\pi$
$877$			869.529i			0.991482i	−0.868470	+	0.495741i	$0.834897\pi$
$878$		0			0
$879$			2933.02i			3.33677i
$880$		0			0
$881$			1210.10i			1.37355i	0.726869	+	0.686776i	$0.240975\pi$
$881$			1210.10i			1.37355i	−0.726869	+	0.686776i	$0.759025\pi$
$882$		0			0
$883$			275.674i			0.312202i	0.987741	+	0.156101i	$0.0498925\pi$
$883$			275.674i			0.312202i	−0.987741	+	0.156101i	$0.950107\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	742.524i		−	0.837119i	−0.908189	−	0.418559i	$-0.862535\pi$
$887$		−	742.524i		−	0.837119i	0.908189	−	0.418559i	$-0.137465\pi$
$888$		0			0
$889$		−	799.769i		−	0.899628i
$890$		0			0
$891$		−	262.052i		−	0.294109i
$892$		0			0
$893$	−176.087			−0.197186
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1562.36	−	333.454i	−1.74176	−	0.371744i
$898$		0			0
$899$	25.2402			0.0280759
$900$		0			0
$901$	−463.751			−0.514707
$902$		0			0
$903$			1721.78i			1.90673i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1335.53			1.47247			0.736234	−	0.676727i	$-0.236602\pi$
$907$	1335.53			1.47247			0.736234	+	0.676727i	$0.236602\pi$
$908$		0			0
$909$	−1241.13			−1.36538
$910$		0			0
$911$		−	511.312i		−	0.561265i	−0.959815	−	0.280632i	$-0.909456\pi$
$911$		−	511.312i		−	0.561265i	0.959815	−	0.280632i	$-0.0905442\pi$
$912$		0			0
$913$			290.972i			0.318699i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−346.013			−0.377332
$918$		0			0
$919$		−	712.793i		−	0.775618i	−0.921740	−	0.387809i	$-0.873232\pi$
$919$		−	712.793i		−	0.775618i	0.921740	−	0.387809i	$-0.126768\pi$
$920$		0			0
$921$	89.7903			0.0974921
$922$		0			0
$923$		−	913.686i		−	0.989909i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	3024.49			3.26266
$928$		0			0
$929$	−86.3665			−0.0929672			−0.0464836	−	0.998919i	$-0.514802\pi$
$929$	−86.3665			−0.0929672			−0.0464836	+	0.998919i	$0.514802\pi$
$930$		0			0
$931$			38.8326i			0.0417106i
$932$		0			0
$933$			213.618i			0.228958i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	994.559			1.06143			0.530714	−	0.847551i	$-0.321924\pi$
$937$	994.559			1.06143			0.530714	+	0.847551i	$0.321924\pi$
$938$		0			0
$939$		−	995.960i		−	1.06066i
$940$		0			0
$941$			1456.29i			1.54760i	0.633429	+	0.773800i	$0.281647\pi$
$941$			1456.29i			1.54760i	−0.633429	+	0.773800i	$0.718353\pi$
$942$		0			0
$943$	730.217	+	155.850i	0.774356	+	0.165270i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			206.079i			0.217613i	0.994063	+	0.108806i	$0.0347029\pi$
$947$			206.079i			0.217613i	−0.994063	+	0.108806i	$0.965297\pi$
$948$		0			0
$949$	1380.29			1.45446
$950$		0			0
$951$	−1501.46			−1.57882
$952$		0			0
$953$	−457.750			−0.480325			−0.240163	−	0.970733i	$-0.577201\pi$
$953$	−457.750			−0.480325			−0.240163	+	0.970733i	$0.577201\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	14.1037			0.0147374
$958$		0			0
$959$	1695.77			1.76827
$960$		0			0
$961$	382.112			0.397619
$962$		0			0
$963$	1539.64			1.59880
$964$		0			0
$965$		0			0
$966$		0			0
$967$			942.336i			0.974495i	0.873264	+	0.487247i	$0.161999\pi$
$967$			942.336i			0.974495i	−0.873264	+	0.487247i	$0.838001\pi$
$968$		0			0
$969$	122.891			0.126822
$970$		0			0
$971$			218.442i			0.224966i	0.993654	+	0.112483i	$0.0358804\pi$
$971$			218.442i			0.224966i	−0.993654	+	0.112483i	$0.964120\pi$
$972$		0			0
$973$	−501.607			−0.515526
$974$		0			0
$975$		0			0
$976$		0			0
$977$	378.433			0.387342			0.193671	−	0.981067i	$-0.437961\pi$
$977$	378.433			0.387342			0.193671	+	0.981067i	$0.437961\pi$
$978$		0			0
$979$	370.280			0.378223
$980$		0			0
$981$			2485.45i			2.53359i
$982$		0			0
$983$	1359.48			1.38299			0.691495	−	0.722381i	$-0.256952\pi$
$983$	1359.48			1.38299			0.691495	+	0.722381i	$0.256952\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	2405.61			2.43729
$988$		0			0
$989$	956.090	+	204.058i	0.966724	+	0.206327i
$990$		0			0
$991$	−648.537			−0.654426			−0.327213	−	0.944951i	$-0.606110\pi$
$991$	−648.537			−0.654426			−0.327213	+	0.944951i	$0.606110\pi$
$992$		0			0
$993$		−	2893.47i		−	2.91387i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			155.843i			0.156312i	0.996941	+	0.0781562i	$0.0249033\pi$
$997$			155.843i			0.156312i	−0.996941	+	0.0781562i	$0.975097\pi$
$998$		0			0
$999$		−	157.329i		−	0.157486i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.3.d.b.1149.3		32
5.2	odd	4		460.3.f.a.321.2	yes	16
5.3	odd	4		2300.3.f.e.1701.15		16
5.4	even	2	inner	2300.3.d.b.1149.30		32
15.2	even	4		4140.3.d.a.2161.7		16
20.7	even	4		1840.3.k.c.321.16		16
23.22	odd	2	inner	2300.3.d.b.1149.29		32
115.22	even	4		460.3.f.a.321.1	✓	16
115.68	even	4		2300.3.f.e.1701.16		16
115.114	odd	2	inner	2300.3.d.b.1149.4		32
345.137	odd	4		4140.3.d.a.2161.10		16
460.367	odd	4		1840.3.k.c.321.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.3.f.a.321.1	✓	16	115.22	even	4
460.3.f.a.321.2	yes	16	5.2	odd	4
1840.3.k.c.321.15		16	460.367	odd	4
1840.3.k.c.321.16		16	20.7	even	4
2300.3.d.b.1149.3		32	1.1	even	1	trivial
2300.3.d.b.1149.4		32	115.114	odd	2	inner
2300.3.d.b.1149.29		32	23.22	odd	2	inner
2300.3.d.b.1149.30		32	5.4	even	2	inner
2300.3.f.e.1701.15		16	5.3	odd	4
2300.3.f.e.1701.16		16	115.68	even	4
4140.3.d.a.2161.7		16	15.2	even	4
4140.3.d.a.2161.10		16	345.137	odd	4

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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2300.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.14043i q^{3} +7.88014 q^{7} -17.4240 q^{9} +O(q^{10})\)	\(q-5.14043i q^{3} +7.88014 q^{7} -17.4240 q^{9} -3.98380i q^{11} +13.5122i q^{13} +8.06277 q^{17} +2.96508i q^{19} -40.5073i q^{21} +(-22.4934 - 4.80075i) q^{23} +43.3029i q^{27} -0.688711 q^{29} -36.6485 q^{31} -20.4784 q^{33} -3.63322 q^{37} +69.4587 q^{39} -32.4636 q^{41} -42.5054 q^{43} +59.3870i q^{47} +13.0966 q^{49} -41.4461i q^{51} -57.5176 q^{53} +15.2418 q^{57} +25.3686 q^{59} -31.8220i q^{61} -137.303 q^{63} -108.405 q^{67} +(-24.6779 + 115.626i) q^{69} -67.6191 q^{71} -102.151i q^{73} -31.3929i q^{77} +48.5263i q^{79} +65.7793 q^{81} -73.0389 q^{83} +3.54027i q^{87} +92.9465i q^{89} +106.478i q^{91} +188.389i q^{93} -138.539 q^{97} +69.4136i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 128 q^{9}+O(q^{10}) \) 32 * q - 128 * q^9	\( 32 q - 128 q^{9} - 180 q^{29} + 20 q^{31} - 40 q^{39} + 372 q^{41} - 4 q^{49} + 180 q^{59} + 464 q^{69} - 476 q^{71} + 1408 q^{81}+O(q^{100}) \) 32 * q - 128 * q^9 - 180 * q^29 + 20 * q^31 - 40 * q^39 + 372 * q^41 - 4 * q^49 + 180 * q^59 + 464 * q^69 - 476 * q^71 + 1408 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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