LMFDB - Embedded newform 1584.4.b.b.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,4,Mod(593,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1584.593");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1584.b (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:	$93.4590254491$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			593.1
Root			$-1.41421i$ of defining polynomial
Character	$\chi$	$=$	1584.593
Dual form			1584.4.b.b.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-19.7990i q^{5} -16.9706i q^{7} +O(q^{10})$	$q-19.7990i q^{5} -16.9706i q^{7} +(33.0000 - 15.5563i) q^{11} -29.6985i q^{13} -126.000 q^{17} -89.0955i q^{19} -120.208i q^{23} -267.000 q^{25} +24.0000 q^{29} +70.0000 q^{31} -336.000 q^{35} +182.000 q^{37} -294.000 q^{41} -4.24264i q^{43} +108.894i q^{47} +55.0000 q^{49} -147.078i q^{53} +(-308.000 - 653.367i) q^{55} -514.774i q^{59} -326.683i q^{61} -588.000 q^{65} +880.000 q^{67} +337.997i q^{71} -178.191i q^{73} +(-264.000 - 560.029i) q^{77} +772.161i q^{79} +1218.00 q^{83} +2494.67i q^{85} -1534.42i q^{89} -504.000 q^{91} -1764.00 q^{95} -196.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q + 66 q^{11} - 252 q^{17} - 534 q^{25} + 48 q^{29} + 140 q^{31} - 672 q^{35} + 364 q^{37} - 588 q^{41} + 110 q^{49} - 616 q^{55} - 1176 q^{65} + 1760 q^{67} - 528 q^{77} + 2436 q^{83} - 1008 q^{91} - 3528 q^{95} - 392 q^{97}+O(q^{100}) $ 2 * q + 66 * q^11 - 252 * q^17 - 534 * q^25 + 48 * q^29 + 140 * q^31 - 672 * q^35 + 364 * q^37 - 588 * q^41 + 110 * q^49 - 616 * q^55 - 1176 * q^65 + 1760 * q^67 - 528 * q^77 + 2436 * q^83 - 1008 * q^91 - 3528 * q^95 - 392 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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
$5$		−	19.7990i		−	1.77088i	−0.464758	−	0.885438i	$-0.653859\pi$
$5$		−	19.7990i		−	1.77088i	0.464758	−	0.885438i	$-0.346141\pi$
$6$		0			0
$7$		−	16.9706i		−	0.916324i	−0.888869	−	0.458162i	$-0.848508\pi$
$7$		−	16.9706i		−	0.916324i	0.888869	−	0.458162i	$-0.151492\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	33.0000	−	15.5563i	0.904534	−	0.426401i
$11$	33.0000	−	15.5563i	0.904534	−	0.426401i
$12$		0			0
$13$		−	29.6985i		−	0.633606i	−0.948491	−	0.316803i	$-0.897391\pi$
$13$		−	29.6985i		−	0.633606i	0.948491	−	0.316803i	$-0.102609\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−126.000			−1.79762			−0.898808	−	0.438342i	$-0.855566\pi$
$17$	−126.000			−1.79762			−0.898808	+	0.438342i	$0.855566\pi$
$18$		0			0
$19$		−	89.0955i		−	1.07578i	−0.843014	−	0.537892i	$-0.819221\pi$
$19$		−	89.0955i		−	1.07578i	0.843014	−	0.537892i	$-0.180779\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	120.208i		−	1.08979i	−0.838505	−	0.544894i	$-0.816570\pi$
$23$		−	120.208i		−	1.08979i	0.838505	−	0.544894i	$-0.183430\pi$
$24$		0			0
$25$	−267.000			−2.13600
$26$		0			0
$27$		0			0
$28$		0			0
$29$	24.0000			0.153679			0.0768395	−	0.997043i	$-0.475517\pi$
$29$	24.0000			0.153679			0.0768395	+	0.997043i	$0.475517\pi$
$30$		0			0
$31$	70.0000			0.405560			0.202780	−	0.979224i	$-0.435002\pi$
$31$	70.0000			0.405560			0.202780	+	0.979224i	$0.435002\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−336.000			−1.62270
$36$		0			0
$37$	182.000			0.808665			0.404333	−	0.914612i	$-0.367504\pi$
$37$	182.000			0.808665			0.404333	+	0.914612i	$0.367504\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−294.000			−1.11988			−0.559940	−	0.828533i	$-0.689176\pi$
$41$	−294.000			−1.11988			−0.559940	+	0.828533i	$0.689176\pi$
$42$		0			0
$43$		−	4.24264i		−	0.0150464i	−0.999972	−	0.00752322i	$-0.997605\pi$
$43$		−	4.24264i		−	0.0150464i	0.999972	−	0.00752322i	$-0.00239474\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			108.894i			0.337955i	0.985620	+	0.168978i	$0.0540465\pi$
$47$			108.894i			0.337955i	−0.985620	+	0.168978i	$0.945953\pi$
$48$		0			0
$49$	55.0000			0.160350
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	147.078i		−	0.381184i	−0.981669	−	0.190592i	$-0.938959\pi$
$53$		−	147.078i		−	0.381184i	0.981669	−	0.190592i	$-0.0610407\pi$
$54$		0			0
$55$	−308.000	−	653.367i	−0.755104	−	1.60182i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	514.774i		−	1.13590i	−0.823065	−	0.567948i	$-0.807738\pi$
$59$		−	514.774i		−	1.13590i	0.823065	−	0.567948i	$-0.192262\pi$
$60$		0			0
$61$		−	326.683i		−	0.685697i	−0.939391	−	0.342848i	$-0.888608\pi$
$61$		−	326.683i		−	0.685697i	0.939391	−	0.342848i	$-0.111392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−588.000			−1.12204
$66$		0			0
$67$	880.000			1.60461			0.802307	−	0.596912i	$-0.203606\pi$
$67$	880.000			1.60461			0.802307	+	0.596912i	$0.203606\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			337.997i			0.564970i	0.959272	+	0.282485i	$0.0911587\pi$
$71$			337.997i			0.564970i	−0.959272	+	0.282485i	$0.908841\pi$
$72$		0			0
$73$		−	178.191i		−	0.285694i	−0.989745	−	0.142847i	$-0.954374\pi$
$73$		−	178.191i		−	0.285694i	0.989745	−	0.142847i	$-0.0456257\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−264.000	−	560.029i	−0.390722	−	0.828846i
$78$		0			0
$79$			772.161i			1.09968i	0.835269	+	0.549841i	$0.185312\pi$
$79$			772.161i			1.09968i	−0.835269	+	0.549841i	$0.814688\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1218.00			1.61076			0.805379	−	0.592761i	$-0.201962\pi$
$83$	1218.00			1.61076			0.805379	+	0.592761i	$0.201962\pi$
$84$		0			0
$85$			2494.67i			3.18336i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1534.42i		−	1.82751i	−0.406266	−	0.913755i	$-0.633169\pi$
$89$		−	1534.42i		−	1.82751i	0.406266	−	0.913755i	$-0.366831\pi$
$90$		0			0
$91$	−504.000			−0.580589
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1764.00			−1.90508
$96$		0			0
$97$	−196.000			−0.205163			−0.102581	−	0.994725i	$-0.532710\pi$
$97$	−196.000			−0.205163			−0.102581	+	0.994725i	$0.532710\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	546.000			0.537911			0.268956	−	0.963153i	$-0.413322\pi$
$101$	546.000			0.537911			0.268956	+	0.963153i	$0.413322\pi$
$102$		0			0
$103$	826.000			0.790177			0.395088	−	0.918643i	$-0.370714\pi$
$103$	826.000			0.790177			0.395088	+	0.918643i	$0.370714\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	108.000			0.0975771			0.0487886	−	0.998809i	$-0.484464\pi$
$107$	108.000			0.0975771			0.0487886	+	0.998809i	$0.484464\pi$
$108$		0			0
$109$			1183.70i			1.04016i	0.854117	+	0.520081i	$0.174098\pi$
$109$			1183.70i			1.04016i	−0.854117	+	0.520081i	$0.825902\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1622.10i			1.35039i	0.737637	+	0.675197i	$0.235942\pi$
$113$			1622.10i			1.35039i	−0.737637	+	0.675197i	$0.764058\pi$
$114$		0			0
$115$	−2380.00			−1.92988
$116$		0			0
$117$		0			0
$118$		0			0
$119$			2138.29i			1.64720i
$120$		0			0
$121$	847.000	−	1026.72i	0.636364	−	0.771389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2811.46i			2.01171i
$126$		0			0
$127$			178.191i			0.124503i	0.998060	+	0.0622515i	$0.0198281\pi$
$127$			178.191i			0.124503i	−0.998060	+	0.0622515i	$0.980172\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−588.000			−0.392166			−0.196083	−	0.980587i	$-0.562822\pi$
$131$	−588.000			−0.392166			−0.196083	+	0.980587i	$0.562822\pi$
$132$		0			0
$133$	−1512.00			−0.985767
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	337.997i		−	0.210781i	−0.994431	−	0.105391i	$-0.966391\pi$
$137$		−	337.997i		−	0.210781i	0.994431	−	0.105391i	$-0.0336093\pi$
$138$		0			0
$139$			2880.75i			1.75786i	0.476952	+	0.878929i	$0.341741\pi$
$139$			2880.75i			1.75786i	−0.476952	+	0.878929i	$0.658259\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−462.000	−	980.050i	−0.270170	−	0.573118i
$144$		0			0
$145$		−	475.176i		−	0.272146i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	2082.00			1.14473			0.572363	−	0.820001i	$-0.306027\pi$
$149$	2082.00			1.14473			0.572363	+	0.820001i	$0.306027\pi$
$150$		0			0
$151$			1637.66i			0.882588i	0.897363	+	0.441294i	$0.145480\pi$
$151$			1637.66i			0.882588i	−0.897363	+	0.441294i	$0.854520\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	1385.93i		−	0.718197i
$156$		0			0
$157$	−2338.00			−1.18849			−0.594244	−	0.804285i	$-0.702549\pi$
$157$	−2338.00			−1.18849			−0.594244	+	0.804285i	$0.702549\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2040.00			−0.998600
$162$		0			0
$163$	−128.000			−0.0615076			−0.0307538	−	0.999527i	$-0.509791\pi$
$163$	−128.000			−0.0615076			−0.0307538	+	0.999527i	$0.509791\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1176.00			0.544920			0.272460	−	0.962167i	$-0.412163\pi$
$167$	1176.00			0.544920			0.272460	+	0.962167i	$0.412163\pi$
$168$		0			0
$169$	1315.00			0.598543
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1092.00			−0.479903			−0.239952	−	0.970785i	$-0.577132\pi$
$173$	−1092.00			−0.479903			−0.239952	+	0.970785i	$0.577132\pi$
$174$		0			0
$175$			4531.14i			1.95727i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	387.495i		−	0.161803i	−0.996722	−	0.0809014i	$-0.974220\pi$
$179$		−	387.495i		−	0.161803i	0.996722	−	0.0809014i	$-0.0257799\pi$
$180$		0			0
$181$	−1330.00			−0.546177			−0.273089	−	0.961989i	$-0.588045\pi$
$181$	−1330.00			−0.546177			−0.273089	+	0.961989i	$0.588045\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	3603.42i		−	1.43205i
$186$		0			0
$187$	−4158.00	+	1960.10i	−1.62601	+	0.766506i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3404.01i		−	1.28956i	−0.764369	−	0.644779i	$-0.776949\pi$
$191$		−	3404.01i		−	1.28956i	0.764369	−	0.644779i	$-0.223051\pi$
$192$		0			0
$193$			3419.57i			1.27537i	0.770298	+	0.637684i	$0.220107\pi$
$193$			3419.57i			1.27537i	−0.770298	+	0.637684i	$0.779893\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3798.00			1.37359			0.686793	−	0.726853i	$-0.259018\pi$
$197$	3798.00			1.37359			0.686793	+	0.726853i	$0.259018\pi$
$198$		0			0
$199$	−560.000			−0.199484			−0.0997421	−	0.995013i	$-0.531802\pi$
$199$	−560.000			−0.199484			−0.0997421	+	0.995013i	$0.531802\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	407.294i		−	0.140820i
$204$		0			0
$205$			5820.90i			1.98317i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1386.00	−	2940.15i	−0.458716	−	0.973083i
$210$		0			0
$211$		−	1904.95i		−	0.621525i	−0.950488	−	0.310763i	$-0.899416\pi$
$211$		−	1904.95i		−	0.621525i	0.950488	−	0.310763i	$-0.100584\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−84.0000			−0.0266454
$216$		0			0
$217$		−	1187.94i		−	0.371625i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			3742.01i			1.13898i
$222$		0			0
$223$	2968.00			0.891264			0.445632	−	0.895216i	$-0.352979\pi$
$223$	2968.00			0.891264			0.445632	+	0.895216i	$0.352979\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3444.00			−1.00699			−0.503494	−	0.863999i	$-0.667952\pi$
$227$	−3444.00			−1.00699			−0.503494	+	0.863999i	$0.667952\pi$
$228$		0			0
$229$	−574.000			−0.165638			−0.0828188	−	0.996565i	$-0.526392\pi$
$229$	−574.000			−0.165638			−0.0828188	+	0.996565i	$0.526392\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5310.00			−1.49300			−0.746501	−	0.665384i	$-0.768268\pi$
$233$	−5310.00			−1.49300			−0.746501	+	0.665384i	$0.768268\pi$
$234$		0			0
$235$	2156.00			0.598476
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1236.00			−0.334520			−0.167260	−	0.985913i	$-0.553492\pi$
$239$	−1236.00			−0.334520			−0.167260	+	0.985913i	$0.553492\pi$
$240$		0			0
$241$		−	3445.02i		−	0.920803i	−0.887711	−	0.460401i	$-0.847705\pi$
$241$		−	3445.02i		−	0.920803i	0.887711	−	0.460401i	$-0.152295\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	1088.94i		−	0.283960i
$246$		0			0
$247$	−2646.00			−0.681623
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	5682.31i		−	1.42894i	−0.699665	−	0.714471i	$-0.746668\pi$
$251$		−	5682.31i		−	1.42894i	0.699665	−	0.714471i	$-0.253332\pi$
$252$		0			0
$253$	−1870.00	−	3966.87i	−0.464687	−	0.985751i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1494.82i			0.362819i	0.983408	+	0.181410i	$0.0580660\pi$
$257$			1494.82i			0.362819i	−0.983408	+	0.181410i	$0.941934\pi$
$258$		0			0
$259$		−	3088.64i		−	0.741000i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4056.00			−0.950965			−0.475482	−	0.879725i	$-0.657726\pi$
$263$	−4056.00			−0.950965			−0.475482	+	0.879725i	$0.657726\pi$
$264$		0			0
$265$	−2912.00			−0.675029
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	6256.48i		−	1.41808i	−0.705167	−	0.709042i	$-0.749128\pi$
$269$		−	6256.48i		−	1.41808i	0.705167	−	0.709042i	$-0.250872\pi$
$270$		0			0
$271$			1603.72i			0.359479i	0.983714	+	0.179740i	$0.0575256\pi$
$271$			1603.72i			0.359479i	−0.983714	+	0.179740i	$0.942474\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8811.00	+	4153.55i	−1.93208	+	0.910793i
$276$		0			0
$277$			5019.04i			1.08868i	0.838864	+	0.544341i	$0.183220\pi$
$277$			5019.04i			1.08868i	−0.838864	+	0.544341i	$0.816780\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2874.00			−0.610137			−0.305068	−	0.952330i	$-0.598679\pi$
$281$	−2874.00			−0.610137			−0.305068	+	0.952330i	$0.598679\pi$
$282$		0			0
$283$			2167.99i			0.455384i	0.973733	+	0.227692i	$0.0731179\pi$
$283$			2167.99i			0.455384i	−0.973733	+	0.227692i	$0.926882\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			4989.35i			1.02617i
$288$		0			0
$289$	10963.0			2.23143
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9240.00			−1.84234			−0.921172	−	0.389157i	$-0.872766\pi$
$293$	−9240.00			−1.84234			−0.921172	+	0.389157i	$0.872766\pi$
$294$		0			0
$295$	−10192.0			−2.01153
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3570.00			−0.690496
$300$		0			0
$301$	−72.0000			−0.0137874
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6468.00			−1.21428
$306$		0			0
$307$			3831.10i			0.712224i	0.934443	+	0.356112i	$0.115898\pi$
$307$			3831.10i			0.712224i	−0.934443	+	0.356112i	$0.884102\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			287.085i			0.0523444i	0.999657	+	0.0261722i	$0.00833183\pi$
$311$			287.085i			0.0523444i	−0.999657	+	0.0261722i	$0.991668\pi$
$312$		0			0
$313$	−5992.00			−1.08207			−0.541035	−	0.841000i	$-0.681967\pi$
$313$	−5992.00			−1.08207			−0.541035	+	0.841000i	$0.681967\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	6307.39i		−	1.11753i	−0.829324	−	0.558767i	$-0.811275\pi$
$317$		−	6307.39i		−	1.11753i	0.829324	−	0.558767i	$-0.188725\pi$
$318$		0			0
$319$	792.000	−	373.352i	0.139008	−	0.0655289i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			11226.0i			1.93385i
$324$		0			0
$325$			7929.50i			1.35338i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1848.00			0.309676
$330$		0			0
$331$	1708.00			0.283626			0.141813	−	0.989893i	$-0.454707\pi$
$331$	1708.00			0.283626			0.141813	+	0.989893i	$0.454707\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	17423.1i		−	2.84157i
$336$		0			0
$337$		−	6864.59i		−	1.10961i	−0.831981	−	0.554804i	$-0.812793\pi$
$337$		−	6864.59i		−	1.10961i	0.831981	−	0.554804i	$-0.187207\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2310.00	−	1088.94i	0.366843	−	0.172932i
$342$		0			0
$343$		−	6754.28i		−	1.06326i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2922.00			0.452050			0.226025	−	0.974122i	$-0.427427\pi$
$347$	2922.00			0.452050			0.226025	+	0.974122i	$0.427427\pi$
$348$		0			0
$349$			8048.29i			1.23443i	0.786796	+	0.617214i	$0.211739\pi$
$349$			8048.29i			1.23443i	−0.786796	+	0.617214i	$0.788261\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			8087.89i			1.21948i	0.792603	+	0.609738i	$0.208725\pi$
$353$			8087.89i			1.21948i	−0.792603	+	0.609738i	$0.791275\pi$
$354$		0			0
$355$	6692.00			1.00049
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5436.00			0.799167			0.399584	−	0.916697i	$-0.369155\pi$
$359$	5436.00			0.799167			0.399584	+	0.916697i	$0.369155\pi$
$360$		0			0
$361$	−1079.00			−0.157312
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3528.00			−0.505929
$366$		0			0
$367$	11536.0			1.64080			0.820401	−	0.571789i	$-0.193750\pi$
$367$	11536.0			1.64080			0.820401	+	0.571789i	$0.193750\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2496.00			−0.349288
$372$		0			0
$373$			10305.4i			1.43054i	0.698847	+	0.715271i	$0.253697\pi$
$373$			10305.4i			1.43054i	−0.698847	+	0.715271i	$0.746303\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	712.764i		−	0.0973719i
$378$		0			0
$379$	3724.00			0.504720			0.252360	−	0.967633i	$-0.418793\pi$
$379$	3724.00			0.504720			0.252360	+	0.967633i	$0.418793\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			5098.24i			0.680177i	0.940393	+	0.340089i	$0.110457\pi$
$383$			5098.24i			0.680177i	−0.940393	+	0.340089i	$0.889543\pi$
$384$		0			0
$385$	−11088.0	+	5226.93i	−1.46778	+	0.691920i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	70.7107i		−	0.00921638i	−0.999989	−	0.00460819i	$-0.998533\pi$
$389$		−	70.7107i		−	0.00921638i	0.999989	−	0.00460819i	$-0.00146684\pi$
$390$		0			0
$391$			15146.2i			1.95902i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15288.0			1.94740
$396$		0			0
$397$	8498.00			1.07431			0.537157	−	0.843482i	$-0.319498\pi$
$397$	8498.00			1.07431			0.537157	+	0.843482i	$0.319498\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2614.88i			0.325638i	0.986656	+	0.162819i	$0.0520587\pi$
$401$			2614.88i			0.325638i	−0.986656	+	0.162819i	$0.947941\pi$
$402$		0			0
$403$		−	2078.89i		−	0.256965i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6006.00	−	2831.26i	0.731465	−	0.344816i
$408$		0			0
$409$		−	475.176i		−	0.0574473i	−0.999587	−	0.0287236i	$-0.990856\pi$
$409$		−	475.176i		−	0.0574473i	0.999587	−	0.0287236i	$-0.00914427\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−8736.00			−1.04085
$414$		0			0
$415$		−	24115.2i		−	2.85245i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	12809.9i		−	1.49357i	−0.665064	−	0.746786i	$-0.731596\pi$
$419$		−	12809.9i		−	1.49357i	0.665064	−	0.746786i	$-0.268404\pi$
$420$		0			0
$421$	6554.00			0.758723			0.379362	−	0.925249i	$-0.376144\pi$
$421$	6554.00			0.758723			0.379362	+	0.925249i	$0.376144\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	33642.0			3.83971
$426$		0			0
$427$	−5544.00			−0.628321
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10692.0			−1.19493			−0.597466	−	0.801894i	$-0.703826\pi$
$431$	−10692.0			−1.19493			−0.597466	+	0.801894i	$0.703826\pi$
$432$		0			0
$433$	7616.00			0.845269			0.422635	−	0.906300i	$-0.361105\pi$
$433$	7616.00			0.845269			0.422635	+	0.906300i	$0.361105\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−10710.0			−1.17238
$438$		0			0
$439$			13423.7i			1.45941i	0.683765	+	0.729703i	$0.260342\pi$
$439$			13423.7i			1.45941i	−0.683765	+	0.729703i	$0.739658\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			18322.6i			1.96508i	0.186050	+	0.982540i	$0.440431\pi$
$443$			18322.6i			1.96508i	−0.186050	+	0.982540i	$0.559569\pi$
$444$		0			0
$445$	−30380.0			−3.23629
$446$		0			0
$447$		0			0
$448$		0			0
$449$			654.781i			0.0688219i	0.999408	+	0.0344109i	$0.0109555\pi$
$449$			654.781i			0.0688219i	−0.999408	+	0.0344109i	$0.989044\pi$
$450$		0			0
$451$	−9702.00	+	4573.57i	−1.01297	+	0.477518i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			9978.69i			1.02815i
$456$		0			0
$457$			6024.55i			0.616666i	0.951278	+	0.308333i	$0.0997712\pi$
$457$			6024.55i			0.616666i	−0.951278	+	0.308333i	$0.900229\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4452.00			0.449784			0.224892	−	0.974384i	$-0.427797\pi$
$461$	4452.00			0.449784			0.224892	+	0.974384i	$0.427797\pi$
$462$		0			0
$463$	−14978.0			−1.50343			−0.751713	−	0.659490i	$-0.770772\pi$
$463$	−14978.0			−1.50343			−0.751713	+	0.659490i	$0.770772\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	9246.13i		−	0.916188i	−0.888904	−	0.458094i	$-0.848532\pi$
$467$		−	9246.13i		−	0.916188i	0.888904	−	0.458094i	$-0.151468\pi$
$468$		0			0
$469$		−	14934.1i		−	1.47035i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−66.0000	−	140.007i	−0.00641582	−	0.0136100i
$474$		0			0
$475$			23788.5i			2.29787i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5124.00			0.488771			0.244386	−	0.969678i	$-0.421414\pi$
$479$	5124.00			0.488771			0.244386	+	0.969678i	$0.421414\pi$
$480$		0			0
$481$		−	5405.12i		−	0.512375i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			3880.60i			0.363318i
$486$		0			0
$487$	−5096.00			−0.474172			−0.237086	−	0.971489i	$-0.576192\pi$
$487$	−5096.00			−0.474172			−0.237086	+	0.971489i	$0.576192\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	13164.0			1.20995			0.604973	−	0.796246i	$-0.293184\pi$
$491$	13164.0			1.20995			0.604973	+	0.796246i	$0.293184\pi$
$492$		0			0
$493$	−3024.00			−0.276256
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5736.00			0.517696
$498$		0			0
$499$	4228.00			0.379301			0.189651	−	0.981852i	$-0.439264\pi$
$499$	4228.00			0.379301			0.189651	+	0.981852i	$0.439264\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10332.0			0.915867			0.457934	−	0.888986i	$-0.348590\pi$
$503$	10332.0			0.915867			0.457934	+	0.888986i	$0.348590\pi$
$504$		0			0
$505$		−	10810.2i		−	0.952574i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			5504.12i			0.479304i	0.970859	+	0.239652i	$0.0770334\pi$
$509$			5504.12i			0.479304i	−0.970859	+	0.239652i	$0.922967\pi$
$510$		0			0
$511$	−3024.00			−0.261788
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	16354.0i		−	1.39930i
$516$		0			0
$517$	1694.00	+	3593.52i	0.144105	+	0.305692i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	643.467i		−	0.0541090i	−0.999634	−	0.0270545i	$-0.991387\pi$
$521$		−	643.467i		−	0.0541090i	0.999634	−	0.0270545i	$-0.00861277\pi$
$522$		0			0
$523$		−	11136.9i		−	0.931136i	−0.885012	−	0.465568i	$-0.845850\pi$
$523$		−	11136.9i		−	0.931136i	0.885012	−	0.465568i	$-0.154150\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8820.00			−0.729042
$528$		0			0
$529$	−2283.00			−0.187639
$530$		0			0
$531$		0			0
$532$		0			0
$533$			8731.35i			0.709563i
$534$		0			0
$535$		−	2138.29i		−	0.172797i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1815.00	−	855.599i	0.145042	−	0.0683734i
$540$		0			0
$541$		−	16304.5i		−	1.29572i	−0.761760	−	0.647859i	$-0.775664\pi$
$541$		−	16304.5i		−	1.29572i	0.761760	−	0.647859i	$-0.224336\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	23436.0			1.84200
$546$		0			0
$547$		−	5019.04i		−	0.392320i	−0.980572	−	0.196160i	$-0.937153\pi$
$547$		−	5019.04i		−	0.392320i	0.980572	−	0.196160i	$-0.0628471\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	2138.29i		−	0.165325i
$552$		0			0
$553$	13104.0			1.00767
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−4932.00			−0.375181			−0.187590	−	0.982247i	$-0.560068\pi$
$557$	−4932.00			−0.375181			−0.187590	+	0.982247i	$0.560068\pi$
$558$		0			0
$559$	−126.000			−0.00953351
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−17598.0			−1.31735			−0.658674	−	0.752428i	$-0.728882\pi$
$563$	−17598.0			−1.31735			−0.658674	+	0.752428i	$0.728882\pi$
$564$		0			0
$565$	32116.0			2.39138
$566$		0			0
$567$		0			0
$568$		0			0
$569$	18546.0			1.36641			0.683206	−	0.730225i	$-0.260585\pi$
$569$	18546.0			1.36641			0.683206	+	0.730225i	$0.260585\pi$
$570$		0			0
$571$		−	20105.9i		−	1.47356i	−0.676131	−	0.736782i	$-0.736345\pi$
$571$		−	20105.9i		−	1.47356i	0.676131	−	0.736782i	$-0.263655\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			32095.6i			2.32779i
$576$		0			0
$577$	−3220.00			−0.232323			−0.116161	−	0.993230i	$-0.537059\pi$
$577$	−3220.00			−0.232323			−0.116161	+	0.993230i	$0.537059\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	20670.1i		−	1.47598i
$582$		0			0
$583$	−2288.00	−	4853.58i	−0.162537	−	0.344794i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18017.1i			1.26686i	0.773802	+	0.633428i	$0.218353\pi$
$587$			18017.1i			1.26686i	−0.773802	+	0.633428i	$0.781647\pi$
$588$		0			0
$589$		−	6236.68i		−	0.436295i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6426.00			0.444999			0.222499	−	0.974933i	$-0.428578\pi$
$593$	6426.00			0.444999			0.222499	+	0.974933i	$0.428578\pi$
$594$		0			0
$595$	42336.0			2.91699
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	20968.5i		−	1.43030i	−0.698969	−	0.715152i	$-0.746358\pi$
$599$		−	20968.5i		−	1.43030i	0.698969	−	0.715152i	$-0.253642\pi$
$600$		0			0
$601$			9028.34i			0.612768i	0.951908	+	0.306384i	$0.0991192\pi$
$601$			9028.34i			0.612768i	−0.951908	+	0.306384i	$0.900881\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−20328.0	−	16769.7i	−1.36603	−	1.12692i
$606$		0			0
$607$		−	16809.3i		−	1.12400i	−0.827136	−	0.562002i	$-0.810031\pi$
$607$		−	16809.3i		−	1.12400i	0.827136	−	0.562002i	$-0.189969\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	3234.00			0.214130
$612$		0			0
$613$		−	5791.20i		−	0.381573i	−0.981632	−	0.190787i	$-0.938896\pi$
$613$		−	5791.20i		−	0.381573i	0.981632	−	0.190787i	$-0.0611039\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	57.9828i		−	0.00378330i	−0.999998	−	0.00189165i	$-0.999398\pi$
$617$		−	57.9828i		−	0.00378330i	0.999998	−	0.00189165i	$-0.000602132\pi$
$618$		0			0
$619$	−15680.0			−1.01815			−0.509073	−	0.860723i	$-0.670012\pi$
$619$	−15680.0			−1.01815			−0.509073	+	0.860723i	$0.670012\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−26040.0			−1.67459
$624$		0			0
$625$	22289.0			1.42650
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−22932.0			−1.45367
$630$		0			0
$631$	24334.0			1.53522			0.767608	−	0.640920i	$-0.221447\pi$
$631$	24334.0			1.53522			0.767608	+	0.640920i	$0.221447\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3528.00			0.220479
$636$		0			0
$637$		−	1633.42i		−	0.101599i
$638$		0			0
$639$		0			0
$640$		0			0
$641$			22215.9i			1.36892i	0.729053	+	0.684458i	$0.239961\pi$
$641$			22215.9i			1.36892i	−0.729053	+	0.684458i	$0.760039\pi$
$642$		0			0
$643$	−5096.00			−0.312545			−0.156273	−	0.987714i	$-0.549948\pi$
$643$	−5096.00			−0.312545			−0.156273	+	0.987714i	$0.549948\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			13829.6i			0.840336i	0.907446	+	0.420168i	$0.138029\pi$
$647$			13829.6i			0.840336i	−0.907446	+	0.420168i	$0.861971\pi$
$648$		0			0
$649$	−8008.00	−	16987.5i	−0.484347	−	1.02746i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			1813.02i			0.108651i	0.998523	+	0.0543254i	$0.0173008\pi$
$653$			1813.02i			0.108651i	−0.998523	+	0.0543254i	$0.982699\pi$
$654$		0			0
$655$			11641.8i			0.694478i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16566.0			−0.979241			−0.489620	−	0.871936i	$-0.662865\pi$
$659$	−16566.0			−0.979241			−0.489620	+	0.871936i	$0.662865\pi$
$660$		0			0
$661$	−6118.00			−0.360004			−0.180002	−	0.983666i	$-0.557610\pi$
$661$	−6118.00			−0.360004			−0.180002	+	0.983666i	$0.557610\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			29936.1i			1.74567i
$666$		0			0
$667$		−	2885.00i		−	0.167477i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−5082.00	−	10780.5i	−0.292382	−	0.620236i
$672$		0			0
$673$		−	20933.2i		−	1.19898i	−0.800381	−	0.599491i	$-0.795370\pi$
$673$		−	20933.2i		−	1.19898i	0.800381	−	0.599491i	$-0.204630\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−12558.0			−0.712915			−0.356457	−	0.934312i	$-0.616015\pi$
$677$	−12558.0			−0.712915			−0.356457	+	0.934312i	$0.616015\pi$
$678$		0			0
$679$			3326.23i			0.187996i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			2209.00i			0.123756i	0.998084	+	0.0618778i	$0.0197089\pi$
$683$			2209.00i			0.123756i	−0.998084	+	0.0618778i	$0.980291\pi$
$684$		0			0
$685$	−6692.00			−0.373267
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4368.00			−0.241520
$690$		0			0
$691$	−2828.00			−0.155691			−0.0778453	−	0.996965i	$-0.524804\pi$
$691$	−2828.00			−0.155691			−0.0778453	+	0.996965i	$0.524804\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	57036.0			3.11295
$696$		0			0
$697$	37044.0			2.01312
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8424.00			0.453880			0.226940	−	0.973909i	$-0.427128\pi$
$701$	8424.00			0.453880			0.226940	+	0.973909i	$0.427128\pi$
$702$		0			0
$703$		−	16215.4i		−	0.869949i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	9265.93i		−	0.492901i
$708$		0			0
$709$	−8890.00			−0.470904			−0.235452	−	0.971886i	$-0.575657\pi$
$709$	−8890.00			−0.470904			−0.235452	+	0.971886i	$0.575657\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	8414.57i		−	0.441975i
$714$		0			0
$715$	−19404.0	+	9147.13i	−1.01492	+	0.478438i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	23946.9i		−	1.24210i	−0.783772	−	0.621049i	$-0.786707\pi$
$719$		−	23946.9i		−	1.24210i	0.783772	−	0.621049i	$-0.213293\pi$
$720$		0			0
$721$		−	14017.7i		−	0.724058i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6408.00			−0.328258
$726$		0			0
$727$	10024.0			0.511375			0.255687	−	0.966759i	$-0.417698\pi$
$727$	10024.0			0.511375			0.255687	+	0.966759i	$0.417698\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			534.573i			0.0270477i
$732$		0			0
$733$			6206.98i			0.312770i	0.987696	+	0.156385i	$0.0499840\pi$
$733$			6206.98i			0.312770i	−0.987696	+	0.156385i	$0.950016\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29040.0	−	13689.6i	1.45143	−	0.684210i
$738$		0			0
$739$		−	37331.0i		−	1.85824i	−0.369772	−	0.929122i	$-0.620564\pi$
$739$		−	37331.0i		−	1.85824i	0.369772	−	0.929122i	$-0.379436\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−14340.0			−0.708053			−0.354027	−	0.935235i	$-0.615188\pi$
$743$	−14340.0			−0.708053			−0.354027	+	0.935235i	$0.615188\pi$
$744$		0			0
$745$		−	41221.5i		−	2.02717i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	1832.82i		−	0.0894123i
$750$		0			0
$751$	33334.0			1.61967			0.809837	−	0.586655i	$-0.199556\pi$
$751$	33334.0			1.61967			0.809837	+	0.586655i	$0.199556\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	32424.0			1.56295
$756$		0			0
$757$	−4354.00			−0.209047			−0.104524	−	0.994522i	$-0.533332\pi$
$757$	−4354.00			−0.209047			−0.104524	+	0.994522i	$0.533332\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	22638.0			1.07835			0.539177	−	0.842193i	$-0.318736\pi$
$761$	22638.0			1.07835			0.539177	+	0.842193i	$0.318736\pi$
$762$		0			0
$763$	20088.0			0.953125
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−15288.0			−0.719710
$768$		0			0
$769$		−	27025.6i		−	1.26732i	−0.773612	−	0.633660i	$-0.781552\pi$
$769$		−	27025.6i		−	1.26732i	0.773612	−	0.633660i	$-0.218448\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	20511.8i		−	0.954407i	−0.878793	−	0.477203i	$-0.841650\pi$
$773$		−	20511.8i		−	0.954407i	0.878793	−	0.477203i	$-0.158350\pi$
$774$		0			0
$775$	−18690.0			−0.866277
$776$		0			0
$777$		0			0
$778$		0			0
$779$			26194.1i			1.20475i
$780$		0			0
$781$	5258.00	+	11153.9i	0.240904	+	0.511035i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			46290.0i			2.10467i
$786$		0			0
$787$		−	37746.8i		−	1.70969i	−0.518882	−	0.854846i	$-0.673652\pi$
$787$		−	37746.8i		−	1.70969i	0.518882	−	0.854846i	$-0.326348\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	27528.0			1.23740
$792$		0			0
$793$	−9702.00			−0.434462
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	19264.4i		−	0.856187i	−0.903735	−	0.428093i	$-0.859185\pi$
$797$		−	19264.4i		−	0.856187i	0.903735	−	0.428093i	$-0.140815\pi$
$798$		0			0
$799$		−	13720.7i		−	0.607514i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2772.00	−	5880.30i	−0.121820	−	0.258420i
$804$		0			0
$805$			40389.9i			1.76840i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	27486.0			1.19451			0.597254	−	0.802052i	$-0.296259\pi$
$809$	27486.0			1.19451			0.597254	+	0.802052i	$0.296259\pi$
$810$		0			0
$811$			40894.8i			1.77067i	0.464956	+	0.885334i	$0.346070\pi$
$811$			40894.8i			1.77067i	−0.464956	+	0.885334i	$0.653930\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			2534.27i			0.108922i
$816$		0			0
$817$	−378.000			−0.0161867
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−25770.0			−1.09547			−0.547734	−	0.836653i	$-0.684509\pi$
$821$	−25770.0			−1.09547			−0.547734	+	0.836653i	$0.684509\pi$
$822$		0			0
$823$	42208.0			1.78770			0.893851	−	0.448365i	$-0.147993\pi$
$823$	42208.0			1.78770			0.893851	+	0.448365i	$0.147993\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−8082.00			−0.339829			−0.169915	−	0.985459i	$-0.554349\pi$
$827$	−8082.00			−0.339829			−0.169915	+	0.985459i	$0.554349\pi$
$828$		0			0
$829$	−11914.0			−0.499144			−0.249572	−	0.968356i	$-0.580290\pi$
$829$	−11914.0			−0.499144			−0.249572	+	0.968356i	$0.580290\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−6930.00			−0.288248
$834$		0			0
$835$		−	23283.6i		−	0.964985i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			27193.9i			1.11900i	0.828832	+	0.559498i	$0.189006\pi$
$839$			27193.9i			1.11900i	−0.828832	+	0.559498i	$0.810994\pi$
$840$		0			0
$841$	−23813.0			−0.976383
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	26035.7i		−	1.05995i
$846$		0			0
$847$	−17424.0	−	14374.1i	−0.706843	−	0.583115i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	21877.9i		−	0.881274i
$852$		0			0
$853$			5197.23i			0.208617i	0.994545	+	0.104308i	$0.0332629\pi$
$853$			5197.23i			0.208617i	−0.994545	+	0.104308i	$0.966737\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−24402.0			−0.972645			−0.486322	−	0.873779i	$-0.661662\pi$
$857$	−24402.0			−0.972645			−0.486322	+	0.873779i	$0.661662\pi$
$858$		0			0
$859$	−22232.0			−0.883057			−0.441529	−	0.897247i	$-0.645564\pi$
$859$	−22232.0			−0.883057			−0.441529	+	0.897247i	$0.645564\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	21223.1i		−	0.837130i	−0.908187	−	0.418565i	$-0.862533\pi$
$863$		−	21223.1i		−	0.837130i	0.908187	−	0.418565i	$-0.137467\pi$
$864$		0			0
$865$			21620.5i			0.849848i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12012.0	+	25481.3i	0.468906	+	0.994700i
$870$		0			0
$871$		−	26134.7i		−	1.01669i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	47712.0			1.84338
$876$		0			0
$877$			34183.0i			1.31616i	0.752946	+	0.658082i	$0.228632\pi$
$877$			34183.0i			1.31616i	−0.752946	+	0.658082i	$0.771368\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			28758.0i			1.09975i	0.835246	+	0.549877i	$0.185325\pi$
$881$			28758.0i			1.09975i	−0.835246	+	0.549877i	$0.814675\pi$
$882$		0			0
$883$	−8120.00			−0.309467			−0.154734	−	0.987956i	$-0.549452\pi$
$883$	−8120.00			−0.309467			−0.154734	+	0.987956i	$0.549452\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9996.00			0.378391			0.189196	−	0.981939i	$-0.439412\pi$
$887$	9996.00			0.378391			0.189196	+	0.981939i	$0.439412\pi$
$888$		0			0
$889$	3024.00			0.114085
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9702.00			0.363567
$894$		0			0
$895$	−7672.00			−0.286533
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1680.00			0.0623261
$900$		0			0
$901$			18531.9i			0.685223i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			26332.7i			0.967212i
$906$		0			0
$907$	37744.0			1.38177			0.690887	−	0.722963i	$-0.257220\pi$
$907$	37744.0			1.38177			0.690887	+	0.722963i	$0.257220\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	1800.29i		−	0.0654735i	−0.999464	−	0.0327368i	$-0.989578\pi$
$911$		−	1800.29i		−	0.0654735i	0.999464	−	0.0327368i	$-0.0104223\pi$
$912$		0			0
$913$	40194.0	−	18947.6i	1.45698	−	0.686829i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			9978.69i			0.359352i
$918$		0			0
$919$			25277.7i			0.907326i	0.891173	+	0.453663i	$0.149883\pi$
$919$			25277.7i			0.907326i	−0.891173	+	0.453663i	$0.850117\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	10038.0			0.357968
$924$		0			0
$925$	−48594.0			−1.72731
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	39667.3i		−	1.40091i	−0.713699	−	0.700453i	$-0.752981\pi$
$929$		−	39667.3i		−	1.40091i	0.713699	−	0.700453i	$-0.247019\pi$
$930$		0			0
$931$		−	4900.25i		−	0.172502i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	38808.0	+	82324.2i	1.35739	+	2.87945i
$936$		0			0
$937$		−	2851.05i		−	0.0994022i	−0.998764	−	0.0497011i	$-0.984173\pi$
$937$		−	2851.05i		−	0.0994022i	0.998764	−	0.0497011i	$-0.0158269\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	3990.00			0.138226			0.0691128	−	0.997609i	$-0.477983\pi$
$941$	3990.00			0.138226			0.0691128	+	0.997609i	$0.477983\pi$
$942$		0			0
$943$			35341.2i			1.22043i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			40774.6i			1.39915i	0.714558	+	0.699576i	$0.246628\pi$
$947$			40774.6i			1.39915i	−0.714558	+	0.699576i	$0.753372\pi$
$948$		0			0
$949$	−5292.00			−0.181017
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−11322.0			−0.384843			−0.192422	−	0.981312i	$-0.561634\pi$
$953$	−11322.0			−0.384843			−0.192422	+	0.981312i	$0.561634\pi$
$954$		0			0
$955$	−67396.0			−2.28365
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−5736.00			−0.193144
$960$		0			0
$961$	−24891.0			−0.835521
$962$		0			0
$963$		0			0
$964$		0			0
$965$	67704.0			2.25852
$966$		0			0
$967$			2791.66i			0.0928373i	0.998922	+	0.0464186i	$0.0147808\pi$
$967$			2791.66i			0.0928373i	−0.998922	+	0.0464186i	$0.985219\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	6216.88i		−	0.205468i	−0.994709	−	0.102734i	$-0.967241\pi$
$971$		−	6216.88i		−	0.205468i	0.994709	−	0.102734i	$-0.0327590\pi$
$972$		0			0
$973$	48888.0			1.61077
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	56468.1i		−	1.84911i	−0.381055	−	0.924553i	$-0.624439\pi$
$977$		−	56468.1i		−	1.84911i	0.381055	−	0.924553i	$-0.375561\pi$
$978$		0			0
$979$	−23870.0	−	50635.9i	−0.779253	−	1.65304i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			2781.76i			0.0902587i	0.998981	+	0.0451294i	$0.0143700\pi$
$983$			2781.76i			0.0902587i	−0.998981	+	0.0451294i	$0.985630\pi$
$984$		0			0
$985$		−	75196.6i		−	2.43245i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−510.000			−0.0163974
$990$		0			0
$991$	7882.00			0.252654			0.126327	−	0.991989i	$-0.459681\pi$
$991$	7882.00			0.252654			0.126327	+	0.991989i	$0.459681\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			11087.4i			0.353262i
$996$		0			0
$997$			5910.00i			0.187735i	0.995585	+	0.0938674i	$0.0299230\pi$
$997$			5910.00i			0.187735i	−0.995585	+	0.0938674i	$0.970077\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.4.b.b.593.1		2
3.2	odd	2		1584.4.b.a.593.2		2
4.3	odd	2		99.4.d.a.98.1	✓	2
11.10	odd	2		1584.4.b.a.593.1		2
12.11	even	2		99.4.d.b.98.2	yes	2
33.32	even	2	inner	1584.4.b.b.593.2		2
44.43	even	2		99.4.d.b.98.1	yes	2
132.131	odd	2		99.4.d.a.98.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.4.d.a.98.1	✓	2	4.3	odd	2
99.4.d.a.98.2	yes	2	132.131	odd	2
99.4.d.b.98.1	yes	2	44.43	even	2
99.4.d.b.98.2	yes	2	12.11	even	2
1584.4.b.a.593.1		2	11.10	odd	2
1584.4.b.a.593.2		2	3.2	odd	2
1584.4.b.b.593.1		2	1.1	even	1	trivial
1584.4.b.b.593.2		2	33.32	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1584.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-19.7990i q^{5} -16.9706i q^{7} +O(q^{10})\)	\(q-19.7990i q^{5} -16.9706i q^{7} +(33.0000 - 15.5563i) q^{11} -29.6985i q^{13} -126.000 q^{17} -89.0955i q^{19} -120.208i q^{23} -267.000 q^{25} +24.0000 q^{29} +70.0000 q^{31} -336.000 q^{35} +182.000 q^{37} -294.000 q^{41} -4.24264i q^{43} +108.894i q^{47} +55.0000 q^{49} -147.078i q^{53} +(-308.000 - 653.367i) q^{55} -514.774i q^{59} -326.683i q^{61} -588.000 q^{65} +880.000 q^{67} +337.997i q^{71} -178.191i q^{73} +(-264.000 - 560.029i) q^{77} +772.161i q^{79} +1218.00 q^{83} +2494.67i q^{85} -1534.42i q^{89} -504.000 q^{91} -1764.00 q^{95} -196.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q + 66 q^{11} - 252 q^{17} - 534 q^{25} + 48 q^{29} + 140 q^{31} - 672 q^{35} + 364 q^{37} - 588 q^{41} + 110 q^{49} - 616 q^{55} - 1176 q^{65} + 1760 q^{67} - 528 q^{77} + 2436 q^{83} - 1008 q^{91} - 3528 q^{95} - 392 q^{97}+O(q^{100}) \) 2 * q + 66 * q^11 - 252 * q^17 - 534 * q^25 + 48 * q^29 + 140 * q^31 - 672 * q^35 + 364 * q^37 - 588 * q^41 + 110 * q^49 - 616 * q^55 - 1176 * q^65 + 1760 * q^67 - 528 * q^77 + 2436 * q^83 - 1008 * q^91 - 3528 * q^95 - 392 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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