LMFDB - Embedded newform 6288.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6288,2,Mod(1,6288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6288, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6288 = 2^{4} \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6288.a (trivial)

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:	yes
Analytic conductor:	$50.2099327910$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.81589.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} + 8x - 1 $ x^5 - 6x^3 + 8x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3144)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.55053$ of defining polynomial
Character	$\chi$	$=$	6288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.92391 q^{5} +3.07030 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.92391 q^{5} +3.07030 q^{7} +1.00000 q^{9} +0.362242 q^{11} +0.550527 q^{13} +1.92391 q^{15} +6.35066 q^{17} -1.63542 q^{19} -3.07030 q^{21} -2.45087 q^{23} -1.29858 q^{25} -1.00000 q^{27} -5.32354 q^{29} -4.38141 q^{31} -0.362242 q^{33} -5.90698 q^{35} -8.27156 q^{37} -0.550527 q^{39} -7.54958 q^{41} -9.69486 q^{43} -1.92391 q^{45} +12.3202 q^{47} +2.42675 q^{49} -6.35066 q^{51} -0.0848914 q^{53} -0.696920 q^{55} +1.63542 q^{57} +3.23736 q^{59} +3.16076 q^{61} +3.07030 q^{63} -1.05916 q^{65} +8.85930 q^{67} +2.45087 q^{69} +2.03636 q^{71} -8.89510 q^{73} +1.29858 q^{75} +1.11219 q^{77} +16.8199 q^{79} +1.00000 q^{81} +7.26739 q^{83} -12.2181 q^{85} +5.32354 q^{87} -0.206324 q^{89} +1.69028 q^{91} +4.38141 q^{93} +3.14639 q^{95} -7.16472 q^{97} +0.362242 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 3 q^{11} - 5 q^{13} + 5 q^{15} - q^{17} + q^{19} - 3 q^{21} + 7 q^{23} - 8 q^{25} - 5 q^{27} - 8 q^{29} + 14 q^{31} - 3 q^{33} - 6 q^{35} - 27 q^{37} + 5 q^{39} + 5 q^{41} + 3 q^{43} - 5 q^{45} + 4 q^{47} - 16 q^{49} + q^{51} + q^{53} + 12 q^{55} - q^{57} - 4 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} + 5 q^{67} - 7 q^{69} + 22 q^{71} - 18 q^{73} + 8 q^{75} - 8 q^{77} + 24 q^{79} + 5 q^{81} + 9 q^{83} - 14 q^{85} + 8 q^{87} - 12 q^{89} + 5 q^{91} - 14 q^{93} + 8 q^{95} - 20 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 3 * q^11 - 5 * q^13 + 5 * q^15 - q^17 + q^19 - 3 * q^21 + 7 * q^23 - 8 * q^25 - 5 * q^27 - 8 * q^29 + 14 * q^31 - 3 * q^33 - 6 * q^35 - 27 * q^37 + 5 * q^39 + 5 * q^41 + 3 * q^43 - 5 * q^45 + 4 * q^47 - 16 * q^49 + q^51 + q^53 + 12 * q^55 - q^57 - 4 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 + 5 * q^67 - 7 * q^69 + 22 * q^71 - 18 * q^73 + 8 * q^75 - 8 * q^77 + 24 * q^79 + 5 * q^81 + 9 * q^83 - 14 * q^85 + 8 * q^87 - 12 * q^89 + 5 * q^91 - 14 * q^93 + 8 * q^95 - 20 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.92391	−0.860398	−0.430199	−	0.902734i	$-0.641557\pi$
$5$	−1.92391	−0.860398	−0.430199	+	0.902734i	$0.641557\pi$
$6$	0	0
$7$	3.07030	1.16047	0.580233	−	0.814451i	$-0.302962\pi$
$7$	3.07030	1.16047	0.580233	+	0.814451i	$0.302962\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.362242	0.109220	0.0546100	−	0.998508i	$-0.482608\pi$
$11$	0.362242	0.109220	0.0546100	+	0.998508i	$0.482608\pi$
$12$	0	0
$13$	0.550527	0.152689	0.0763443	−	0.997082i	$-0.475675\pi$
$13$	0.550527	0.152689	0.0763443	+	0.997082i	$0.475675\pi$
$14$	0	0
$15$	1.92391	0.496751
$16$	0	0
$17$	6.35066	1.54026	0.770131	−	0.637886i	$-0.220191\pi$
$17$	6.35066	1.54026	0.770131	+	0.637886i	$0.220191\pi$
$18$	0	0
$19$	−1.63542	−0.375191	−0.187595	−	0.982246i	$-0.560069\pi$
$19$	−1.63542	−0.375191	−0.187595	+	0.982246i	$0.560069\pi$
$20$	0	0
$21$	−3.07030	−0.669995
$22$	0	0
$23$	−2.45087	−0.511041	−0.255521	−	0.966804i	$-0.582247\pi$
$23$	−2.45087	−0.511041	−0.255521	+	0.966804i	$0.582247\pi$
$24$	0	0
$25$	−1.29858	−0.259716
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.32354	−0.988556	−0.494278	−	0.869304i	$-0.664568\pi$
$29$	−5.32354	−0.988556	−0.494278	+	0.869304i	$0.664568\pi$
$30$	0	0
$31$	−4.38141	−0.786925	−0.393463	−	0.919341i	$-0.628723\pi$
$31$	−4.38141	−0.786925	−0.393463	+	0.919341i	$0.628723\pi$
$32$	0	0
$33$	−0.362242	−0.0630583
$34$	0	0
$35$	−5.90698	−0.998462
$36$	0	0
$37$	−8.27156	−1.35984	−0.679918	−	0.733288i	$-0.737985\pi$
$37$	−8.27156	−1.35984	−0.679918	+	0.733288i	$0.737985\pi$
$38$	0	0
$39$	−0.550527	−0.0881548
$40$	0	0
$41$	−7.54958	−1.17905	−0.589523	−	0.807751i	$-0.700684\pi$
$41$	−7.54958	−1.17905	−0.589523	+	0.807751i	$0.700684\pi$
$42$	0	0
$43$	−9.69486	−1.47845	−0.739227	−	0.673457i	$-0.764809\pi$
$43$	−9.69486	−1.47845	−0.739227	+	0.673457i	$0.764809\pi$
$44$	0	0
$45$	−1.92391	−0.286799
$46$	0	0
$47$	12.3202	1.79709	0.898543	−	0.438886i	$-0.144627\pi$
$47$	12.3202	1.79709	0.898543	+	0.438886i	$0.144627\pi$
$48$	0	0
$49$	2.42675	0.346679
$50$	0	0
$51$	−6.35066	−0.889271
$52$	0	0
$53$	−0.0848914	−0.0116607	−0.00583037	−	0.999983i	$-0.501856\pi$
$53$	−0.0848914	−0.0116607	−0.00583037	+	0.999983i	$0.501856\pi$
$54$	0	0
$55$	−0.696920	−0.0939727
$56$	0	0
$57$	1.63542	0.216616
$58$	0	0
$59$	3.23736	0.421468	0.210734	−	0.977543i	$-0.432415\pi$
$59$	3.23736	0.421468	0.210734	+	0.977543i	$0.432415\pi$
$60$	0	0
$61$	3.16076	0.404694	0.202347	−	0.979314i	$-0.435143\pi$
$61$	3.16076	0.404694	0.202347	+	0.979314i	$0.435143\pi$
$62$	0	0
$63$	3.07030	0.386822
$64$	0	0
$65$	−1.05916	−0.131373
$66$	0	0
$67$	8.85930	1.08234	0.541168	−	0.840915i	$-0.317982\pi$
$67$	8.85930	1.08234	0.541168	+	0.840915i	$0.317982\pi$
$68$	0	0
$69$	2.45087	0.295050
$70$	0	0
$71$	2.03636	0.241672	0.120836	−	0.992672i	$-0.461443\pi$
$71$	2.03636	0.241672	0.120836	+	0.992672i	$0.461443\pi$
$72$	0	0
$73$	−8.89510	−1.04109	−0.520546	−	0.853833i	$-0.674272\pi$
$73$	−8.89510	−1.04109	−0.520546	+	0.853833i	$0.674272\pi$
$74$	0	0
$75$	1.29858	0.149947
$76$	0	0
$77$	1.11219	0.126746
$78$	0	0
$79$	16.8199	1.89238	0.946191	−	0.323610i	$-0.104896\pi$
$79$	16.8199	1.89238	0.946191	+	0.323610i	$0.104896\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.26739	0.797699	0.398850	−	0.917016i	$-0.369410\pi$
$83$	7.26739	0.797699	0.398850	+	0.917016i	$0.369410\pi$
$84$	0	0
$85$	−12.2181	−1.32524
$86$	0	0
$87$	5.32354	0.570743
$88$	0	0
$89$	−0.206324	−0.0218703	−0.0109352	−	0.999940i	$-0.503481\pi$
$89$	−0.206324	−0.0218703	−0.0109352	+	0.999940i	$0.503481\pi$
$90$	0	0
$91$	1.69028	0.177190
$92$	0	0
$93$	4.38141	0.454331
$94$	0	0
$95$	3.14639	0.322813
$96$	0	0
$97$	−7.16472	−0.727467	−0.363733	−	0.931503i	$-0.618498\pi$
$97$	−7.16472	−0.727467	−0.363733	+	0.931503i	$0.618498\pi$
$98$	0	0
$99$	0.362242	0.0364067
$100$	0	0
$101$	9.17909	0.913353	0.456677	−	0.889633i	$-0.349040\pi$
$101$	9.17909	0.913353	0.456677	+	0.889633i	$0.349040\pi$
$102$	0	0
$103$	17.0774	1.68269	0.841343	−	0.540502i	$-0.181766\pi$
$103$	17.0774	1.68269	0.841343	+	0.540502i	$0.181766\pi$
$104$	0	0
$105$	5.90698	0.576462
$106$	0	0
$107$	−13.1731	−1.27349	−0.636747	−	0.771073i	$-0.719720\pi$
$107$	−13.1731	−1.27349	−0.636747	+	0.771073i	$0.719720\pi$
$108$	0	0
$109$	−16.2778	−1.55913	−0.779563	−	0.626324i	$-0.784559\pi$
$109$	−16.2778	−1.55913	−0.779563	+	0.626324i	$0.784559\pi$
$110$	0	0
$111$	8.27156	0.785102
$112$	0	0
$113$	10.3147	0.970322	0.485161	−	0.874425i	$-0.338761\pi$
$113$	10.3147	0.970322	0.485161	+	0.874425i	$0.338761\pi$
$114$	0	0
$115$	4.71524	0.439699
$116$	0	0
$117$	0.550527	0.0508962
$118$	0	0
$119$	19.4985	1.78742
$120$	0	0
$121$	−10.8688	−0.988071
$122$	0	0
$123$	7.54958	0.680723
$124$	0	0
$125$	12.1179	1.08386
$126$	0	0
$127$	−8.80498	−0.781316	−0.390658	−	0.920536i	$-0.627752\pi$
$127$	−8.80498	−0.781316	−0.390658	+	0.920536i	$0.627752\pi$
$128$	0	0
$129$	9.69486	0.853585
$130$	0	0
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	0	0
$133$	−5.02123	−0.435396
$134$	0	0
$135$	1.92391	0.165584
$136$	0	0
$137$	−9.94363	−0.849541	−0.424771	−	0.905301i	$-0.639645\pi$
$137$	−9.94363	−0.849541	−0.424771	+	0.905301i	$0.639645\pi$
$138$	0	0
$139$	−9.31061	−0.789715	−0.394858	−	0.918742i	$-0.629206\pi$
$139$	−9.31061	−0.789715	−0.394858	+	0.918742i	$0.629206\pi$
$140$	0	0
$141$	−12.3202	−1.03755
$142$	0	0
$143$	0.199424	0.0166767
$144$	0	0
$145$	10.2420	0.850552
$146$	0	0
$147$	−2.42675	−0.200155
$148$	0	0
$149$	−17.5889	−1.44094	−0.720470	−	0.693486i	$-0.756074\pi$
$149$	−17.5889	−1.44094	−0.720470	+	0.693486i	$0.756074\pi$
$150$	0	0
$151$	−17.5468	−1.42793	−0.713967	−	0.700179i	$-0.753104\pi$
$151$	−17.5468	−1.42793	−0.713967	+	0.700179i	$0.753104\pi$
$152$	0	0
$153$	6.35066	0.513421
$154$	0	0
$155$	8.42944	0.677069
$156$	0	0
$157$	−22.1882	−1.77081	−0.885405	−	0.464820i	$-0.846119\pi$
$157$	−22.1882	−1.77081	−0.885405	+	0.464820i	$0.846119\pi$
$158$	0	0
$159$	0.0848914	0.00673233
$160$	0	0
$161$	−7.52490	−0.593046
$162$	0	0
$163$	4.17535	0.327039	0.163519	−	0.986540i	$-0.447715\pi$
$163$	4.17535	0.327039	0.163519	+	0.986540i	$0.447715\pi$
$164$	0	0
$165$	0.696920	0.0542552
$166$	0	0
$167$	−7.04274	−0.544983	−0.272492	−	0.962158i	$-0.587848\pi$
$167$	−7.04274	−0.544983	−0.272492	+	0.962158i	$0.587848\pi$
$168$	0	0
$169$	−12.6969	−0.976686
$170$	0	0
$171$	−1.63542	−0.125064
$172$	0	0
$173$	−0.792828	−0.0602776	−0.0301388	−	0.999546i	$-0.509595\pi$
$173$	−0.792828	−0.0602776	−0.0301388	+	0.999546i	$0.509595\pi$
$174$	0	0
$175$	−3.98703	−0.301391
$176$	0	0
$177$	−3.23736	−0.243335
$178$	0	0
$179$	8.06807	0.603036	0.301518	−	0.953461i	$-0.402507\pi$
$179$	8.06807	0.603036	0.301518	+	0.953461i	$0.402507\pi$
$180$	0	0
$181$	−13.5679	−1.00849	−0.504247	−	0.863559i	$-0.668230\pi$
$181$	−13.5679	−1.00849	−0.504247	+	0.863559i	$0.668230\pi$
$182$	0	0
$183$	−3.16076	−0.233650
$184$	0	0
$185$	15.9137	1.17000
$186$	0	0
$187$	2.30048	0.168228
$188$	0	0
$189$	−3.07030	−0.223332
$190$	0	0
$191$	−24.6242	−1.78175	−0.890874	−	0.454251i	$-0.849907\pi$
$191$	−24.6242	−1.78175	−0.890874	+	0.454251i	$0.849907\pi$
$192$	0	0
$193$	5.12128	0.368638	0.184319	−	0.982867i	$-0.440992\pi$
$193$	5.12128	0.368638	0.184319	+	0.982867i	$0.440992\pi$
$194$	0	0
$195$	1.05916	0.0758482
$196$	0	0
$197$	21.3906	1.52401	0.762007	−	0.647569i	$-0.224214\pi$
$197$	21.3906	1.52401	0.762007	+	0.647569i	$0.224214\pi$
$198$	0	0
$199$	−11.2900	−0.800330	−0.400165	−	0.916443i	$-0.631047\pi$
$199$	−11.2900	−0.800330	−0.400165	+	0.916443i	$0.631047\pi$
$200$	0	0
$201$	−8.85930	−0.624887
$202$	0	0
$203$	−16.3449	−1.14719
$204$	0	0
$205$	14.5247	1.01445
$206$	0	0
$207$	−2.45087	−0.170347
$208$	0	0
$209$	−0.592417	−0.0409784
$210$	0	0
$211$	0.148229	0.0102045	0.00510224	−	0.999987i	$-0.498376\pi$
$211$	0.148229	0.0102045	0.00510224	+	0.999987i	$0.498376\pi$
$212$	0	0
$213$	−2.03636	−0.139529
$214$	0	0
$215$	18.6520	1.27206
$216$	0	0
$217$	−13.4523	−0.913199
$218$	0	0
$219$	8.89510	0.601075
$220$	0	0
$221$	3.49621	0.235180
$222$	0	0
$223$	11.8988	0.796805	0.398402	−	0.917211i	$-0.369565\pi$
$223$	11.8988	0.796805	0.398402	+	0.917211i	$0.369565\pi$
$224$	0	0
$225$	−1.29858	−0.0865718
$226$	0	0
$227$	−11.7117	−0.777332	−0.388666	−	0.921379i	$-0.627064\pi$
$227$	−11.7117	−0.777332	−0.388666	+	0.921379i	$0.627064\pi$
$228$	0	0
$229$	3.33249	0.220217	0.110108	−	0.993920i	$-0.464880\pi$
$229$	3.33249	0.220217	0.110108	+	0.993920i	$0.464880\pi$
$230$	0	0
$231$	−1.11219	−0.0731769
$232$	0	0
$233$	−5.58447	−0.365850	−0.182925	−	0.983127i	$-0.558557\pi$
$233$	−5.58447	−0.365850	−0.182925	+	0.983127i	$0.558557\pi$
$234$	0	0
$235$	−23.7029	−1.54621
$236$	0	0
$237$	−16.8199	−1.09257
$238$	0	0
$239$	0.0395923	0.00256101	0.00128051	−	0.999999i	$-0.499592\pi$
$239$	0.0395923	0.00256101	0.00128051	+	0.999999i	$0.499592\pi$
$240$	0	0
$241$	5.36261	0.345436	0.172718	−	0.984971i	$-0.444745\pi$
$241$	5.36261	0.345436	0.172718	+	0.984971i	$0.444745\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−4.66885	−0.298282
$246$	0	0
$247$	−0.900341	−0.0572873
$248$	0	0
$249$	−7.26739	−0.460552
$250$	0	0
$251$	−23.7190	−1.49713	−0.748564	−	0.663063i	$-0.769256\pi$
$251$	−23.7190	−1.49713	−0.748564	+	0.663063i	$0.769256\pi$
$252$	0	0
$253$	−0.887807	−0.0558160
$254$	0	0
$255$	12.2181	0.765126
$256$	0	0
$257$	−8.26537	−0.515580	−0.257790	−	0.966201i	$-0.582994\pi$
$257$	−8.26537	−0.515580	−0.257790	+	0.966201i	$0.582994\pi$
$258$	0	0
$259$	−25.3962	−1.57804
$260$	0	0
$261$	−5.32354	−0.329519
$262$	0	0
$263$	23.7023	1.46155	0.730774	−	0.682619i	$-0.239159\pi$
$263$	23.7023	1.46155	0.730774	+	0.682619i	$0.239159\pi$
$264$	0	0
$265$	0.163323	0.0100329
$266$	0	0
$267$	0.206324	0.0126268
$268$	0	0
$269$	21.3939	1.30441	0.652205	−	0.758042i	$-0.273844\pi$
$269$	21.3939	1.30441	0.652205	+	0.758042i	$0.273844\pi$
$270$	0	0
$271$	−9.33906	−0.567308	−0.283654	−	0.958927i	$-0.591547\pi$
$271$	−9.33906	−0.567308	−0.283654	+	0.958927i	$0.591547\pi$
$272$	0	0
$273$	−1.69028	−0.102301
$274$	0	0
$275$	−0.470400	−0.0283662
$276$	0	0
$277$	−27.6115	−1.65902	−0.829509	−	0.558494i	$-0.811379\pi$
$277$	−27.6115	−1.65902	−0.829509	+	0.558494i	$0.811379\pi$
$278$	0	0
$279$	−4.38141	−0.262308
$280$	0	0
$281$	25.2325	1.50524	0.752622	−	0.658453i	$-0.228789\pi$
$281$	25.2325	1.50524	0.752622	+	0.658453i	$0.228789\pi$
$282$	0	0
$283$	29.9072	1.77780	0.888898	−	0.458104i	$-0.151471\pi$
$283$	29.9072	1.77780	0.888898	+	0.458104i	$0.151471\pi$
$284$	0	0
$285$	−3.14639	−0.186376
$286$	0	0
$287$	−23.1795	−1.36824
$288$	0	0
$289$	23.3309	1.37241
$290$	0	0
$291$	7.16472	0.420003
$292$	0	0
$293$	1.14617	0.0669602	0.0334801	−	0.999439i	$-0.489341\pi$
$293$	1.14617	0.0669602	0.0334801	+	0.999439i	$0.489341\pi$
$294$	0	0
$295$	−6.22838	−0.362630
$296$	0	0
$297$	−0.362242	−0.0210194
$298$	0	0
$299$	−1.34927	−0.0780302
$300$	0	0
$301$	−29.7662	−1.71569
$302$	0	0
$303$	−9.17909	−0.527325
$304$	0	0
$305$	−6.08102	−0.348198
$306$	0	0
$307$	−7.24654	−0.413582	−0.206791	−	0.978385i	$-0.566302\pi$
$307$	−7.24654	−0.413582	−0.206791	+	0.978385i	$0.566302\pi$
$308$	0	0
$309$	−17.0774	−0.971499
$310$	0	0
$311$	−6.82586	−0.387059	−0.193530	−	0.981094i	$-0.561994\pi$
$311$	−6.82586	−0.387059	−0.193530	+	0.981094i	$0.561994\pi$
$312$	0	0
$313$	−15.0353	−0.849847	−0.424924	−	0.905229i	$-0.639699\pi$
$313$	−15.0353	−0.849847	−0.424924	+	0.905229i	$0.639699\pi$
$314$	0	0
$315$	−5.90698	−0.332821
$316$	0	0
$317$	−10.3135	−0.579261	−0.289631	−	0.957139i	$-0.593532\pi$
$317$	−10.3135	−0.579261	−0.289631	+	0.957139i	$0.593532\pi$
$318$	0	0
$319$	−1.92841	−0.107970
$320$	0	0
$321$	13.1731	0.735252
$322$	0	0
$323$	−10.3860	−0.577892
$324$	0	0
$325$	−0.714902	−0.0396556
$326$	0	0
$327$	16.2778	0.900162
$328$	0	0
$329$	37.8267	2.08545
$330$	0	0
$331$	1.74795	0.0960762	0.0480381	−	0.998846i	$-0.484703\pi$
$331$	1.74795	0.0960762	0.0480381	+	0.998846i	$0.484703\pi$
$332$	0	0
$333$	−8.27156	−0.453279
$334$	0	0
$335$	−17.0445	−0.931239
$336$	0	0
$337$	22.6824	1.23559	0.617796	−	0.786339i	$-0.288026\pi$
$337$	22.6824	1.23559	0.617796	+	0.786339i	$0.288026\pi$
$338$	0	0
$339$	−10.3147	−0.560216
$340$	0	0
$341$	−1.58713	−0.0859480
$342$	0	0
$343$	−14.0412	−0.758156
$344$	0	0
$345$	−4.71524	−0.253860
$346$	0	0
$347$	−4.41228	−0.236864	−0.118432	−	0.992962i	$-0.537787\pi$
$347$	−4.41228	−0.236864	−0.118432	+	0.992962i	$0.537787\pi$
$348$	0	0
$349$	−25.0155	−1.33905	−0.669525	−	0.742790i	$-0.733502\pi$
$349$	−25.0155	−1.33905	−0.669525	+	0.742790i	$0.733502\pi$
$350$	0	0
$351$	−0.550527	−0.0293849
$352$	0	0
$353$	11.3134	0.602152	0.301076	−	0.953600i	$-0.402654\pi$
$353$	11.3134	0.602152	0.301076	+	0.953600i	$0.402654\pi$
$354$	0	0
$355$	−3.91778	−0.207934
$356$	0	0
$357$	−19.4985	−1.03197
$358$	0	0
$359$	28.7406	1.51687	0.758435	−	0.651749i	$-0.225964\pi$
$359$	28.7406	1.51687	0.758435	+	0.651749i	$0.225964\pi$
$360$	0	0
$361$	−16.3254	−0.859232
$362$	0	0
$363$	10.8688	0.570463
$364$	0	0
$365$	17.1134	0.895754
$366$	0	0
$367$	−4.20578	−0.219540	−0.109770	−	0.993957i	$-0.535011\pi$
$367$	−4.20578	−0.219540	−0.109770	+	0.993957i	$0.535011\pi$
$368$	0	0
$369$	−7.54958	−0.393015
$370$	0	0
$371$	−0.260642	−0.0135319
$372$	0	0
$373$	21.6156	1.11921	0.559607	−	0.828758i	$-0.310952\pi$
$373$	21.6156	1.11921	0.559607	+	0.828758i	$0.310952\pi$
$374$	0	0
$375$	−12.1179	−0.625765
$376$	0	0
$377$	−2.93075	−0.150941
$378$	0	0
$379$	−33.0525	−1.69779	−0.848896	−	0.528560i	$-0.822732\pi$
$379$	−33.0525	−1.69779	−0.848896	+	0.528560i	$0.822732\pi$
$380$	0	0
$381$	8.80498	0.451093
$382$	0	0
$383$	−5.03321	−0.257185	−0.128593	−	0.991698i	$-0.541046\pi$
$383$	−5.03321	−0.257185	−0.128593	+	0.991698i	$0.541046\pi$
$384$	0	0
$385$	−2.13976	−0.109052
$386$	0	0
$387$	−9.69486	−0.492818
$388$	0	0
$389$	−7.99594	−0.405410	−0.202705	−	0.979240i	$-0.564973\pi$
$389$	−7.99594	−0.405410	−0.202705	+	0.979240i	$0.564973\pi$
$390$	0	0
$391$	−15.5646	−0.787137
$392$	0	0
$393$	1.00000	0.0504433
$394$	0	0
$395$	−32.3599	−1.62820
$396$	0	0
$397$	29.0880	1.45989	0.729944	−	0.683507i	$-0.239546\pi$
$397$	29.0880	1.45989	0.729944	+	0.683507i	$0.239546\pi$
$398$	0	0
$399$	5.02123	0.251376
$400$	0	0
$401$	−34.9414	−1.74489	−0.872445	−	0.488712i	$-0.837467\pi$
$401$	−34.9414	−1.74489	−0.872445	+	0.488712i	$0.837467\pi$
$402$	0	0
$403$	−2.41208	−0.120154
$404$	0	0
$405$	−1.92391	−0.0955998
$406$	0	0
$407$	−2.99631	−0.148521
$408$	0	0
$409$	−26.1420	−1.29264	−0.646319	−	0.763068i	$-0.723692\pi$
$409$	−26.1420	−1.29264	−0.646319	+	0.763068i	$0.723692\pi$
$410$	0	0
$411$	9.94363	0.490483
$412$	0	0
$413$	9.93967	0.489099
$414$	0	0
$415$	−13.9818	−0.686339
$416$	0	0
$417$	9.31061	0.455942
$418$	0	0
$419$	−6.45048	−0.315127	−0.157563	−	0.987509i	$-0.550364\pi$
$419$	−6.45048	−0.315127	−0.157563	+	0.987509i	$0.550364\pi$
$420$	0	0
$421$	−30.2258	−1.47312	−0.736558	−	0.676375i	$-0.763550\pi$
$421$	−30.2258	−1.47312	−0.736558	+	0.676375i	$0.763550\pi$
$422$	0	0
$423$	12.3202	0.599028
$424$	0	0
$425$	−8.24683	−0.400030
$426$	0	0
$427$	9.70449	0.469633
$428$	0	0
$429$	−0.199424	−0.00962828
$430$	0	0
$431$	30.6972	1.47863	0.739315	−	0.673360i	$-0.235149\pi$
$431$	30.6972	1.47863	0.739315	+	0.673360i	$0.235149\pi$
$432$	0	0
$433$	1.13646	0.0546146	0.0273073	−	0.999627i	$-0.491307\pi$
$433$	1.13646	0.0546146	0.0273073	+	0.999627i	$0.491307\pi$
$434$	0	0
$435$	−10.2420	−0.491066
$436$	0	0
$437$	4.00819	0.191738
$438$	0	0
$439$	11.8082	0.563575	0.281788	−	0.959477i	$-0.409073\pi$
$439$	11.8082	0.563575	0.281788	+	0.959477i	$0.409073\pi$
$440$	0	0
$441$	2.42675	0.115560
$442$	0	0
$443$	27.0975	1.28744	0.643721	−	0.765260i	$-0.277390\pi$
$443$	27.0975	1.28744	0.643721	+	0.765260i	$0.277390\pi$
$444$	0	0
$445$	0.396948	0.0188172
$446$	0	0
$447$	17.5889	0.831927
$448$	0	0
$449$	−8.94544	−0.422161	−0.211081	−	0.977469i	$-0.567698\pi$
$449$	−8.94544	−0.422161	−0.211081	+	0.977469i	$0.567698\pi$
$450$	0	0
$451$	−2.73478	−0.128776
$452$	0	0
$453$	17.5468	0.824419
$454$	0	0
$455$	−3.25195	−0.152454
$456$	0	0
$457$	−10.8406	−0.507101	−0.253551	−	0.967322i	$-0.581598\pi$
$457$	−10.8406	−0.507101	−0.253551	+	0.967322i	$0.581598\pi$
$458$	0	0
$459$	−6.35066	−0.296424
$460$	0	0
$461$	−3.29039	−0.153249	−0.0766244	−	0.997060i	$-0.524414\pi$
$461$	−3.29039	−0.153249	−0.0766244	+	0.997060i	$0.524414\pi$
$462$	0	0
$463$	−9.22453	−0.428700	−0.214350	−	0.976757i	$-0.568763\pi$
$463$	−9.22453	−0.428700	−0.214350	+	0.976757i	$0.568763\pi$
$464$	0	0
$465$	−8.42944	−0.390906
$466$	0	0
$467$	−39.3963	−1.82304	−0.911521	−	0.411253i	$-0.865091\pi$
$467$	−39.3963	−1.82304	−0.911521	+	0.411253i	$0.865091\pi$
$468$	0	0
$469$	27.2007	1.25601
$470$	0	0
$471$	22.1882	1.02238
$472$	0	0
$473$	−3.51189	−0.161477
$474$	0	0
$475$	2.12372	0.0974428
$476$	0	0
$477$	−0.0848914	−0.00388691
$478$	0	0
$479$	−13.4494	−0.614517	−0.307258	−	0.951626i	$-0.599412\pi$
$479$	−13.4494	−0.614517	−0.307258	+	0.951626i	$0.599412\pi$
$480$	0	0
$481$	−4.55371	−0.207631
$482$	0	0
$483$	7.52490	0.342395
$484$	0	0
$485$	13.7843	0.625911
$486$	0	0
$487$	20.9928	0.951274	0.475637	−	0.879642i	$-0.342218\pi$
$487$	20.9928	0.951274	0.475637	+	0.879642i	$0.342218\pi$
$488$	0	0
$489$	−4.17535	−0.188816
$490$	0	0
$491$	−4.07678	−0.183982	−0.0919911	−	0.995760i	$-0.529323\pi$
$491$	−4.07678	−0.183982	−0.0919911	+	0.995760i	$0.529323\pi$
$492$	0	0
$493$	−33.8080	−1.52264
$494$	0	0
$495$	−0.696920	−0.0313242
$496$	0	0
$497$	6.25225	0.280452
$498$	0	0
$499$	26.6644	1.19366	0.596832	−	0.802366i	$-0.296426\pi$
$499$	26.6644	1.19366	0.596832	+	0.802366i	$0.296426\pi$
$500$	0	0
$501$	7.04274	0.314646
$502$	0	0
$503$	−1.69390	−0.0755274	−0.0377637	−	0.999287i	$-0.512023\pi$
$503$	−1.69390	−0.0755274	−0.0377637	+	0.999287i	$0.512023\pi$
$504$	0	0
$505$	−17.6597	−0.785847
$506$	0	0
$507$	12.6969	0.563890
$508$	0	0
$509$	−9.05577	−0.401390	−0.200695	−	0.979654i	$-0.564320\pi$
$509$	−9.05577	−0.401390	−0.200695	+	0.979654i	$0.564320\pi$
$510$	0	0
$511$	−27.3106	−1.20815
$512$	0	0
$513$	1.63542	0.0722055
$514$	0	0
$515$	−32.8553	−1.44778
$516$	0	0
$517$	4.46289	0.196278
$518$	0	0
$519$	0.792828	0.0348013
$520$	0	0
$521$	−36.3672	−1.59328	−0.796639	−	0.604456i	$-0.793391\pi$
$521$	−36.3672	−1.59328	−0.796639	+	0.604456i	$0.793391\pi$
$522$	0	0
$523$	−24.7750	−1.08333	−0.541667	−	0.840593i	$-0.682207\pi$
$523$	−24.7750	−1.08333	−0.541667	+	0.840593i	$0.682207\pi$
$524$	0	0
$525$	3.98703	0.174008
$526$	0	0
$527$	−27.8249	−1.21207
$528$	0	0
$529$	−16.9932	−0.738837
$530$	0	0
$531$	3.23736	0.140489
$532$	0	0
$533$	−4.15625	−0.180027
$534$	0	0
$535$	25.3439	1.09571
$536$	0	0
$537$	−8.06807	−0.348163
$538$	0	0
$539$	0.879072	0.0378643
$540$	0	0
$541$	−8.77898	−0.377438	−0.188719	−	0.982031i	$-0.560434\pi$
$541$	−8.77898	−0.377438	−0.188719	+	0.982031i	$0.560434\pi$
$542$	0	0
$543$	13.5679	0.582255
$544$	0	0
$545$	31.3169	1.34147
$546$	0	0
$547$	33.7493	1.44302	0.721508	−	0.692406i	$-0.243449\pi$
$547$	33.7493	1.44302	0.721508	+	0.692406i	$0.243449\pi$
$548$	0	0
$549$	3.16076	0.134898
$550$	0	0
$551$	8.70621	0.370897
$552$	0	0
$553$	51.6420	2.19604
$554$	0	0
$555$	−15.9137	−0.675500
$556$	0	0
$557$	−10.1211	−0.428843	−0.214422	−	0.976741i	$-0.568787\pi$
$557$	−10.1211	−0.428843	−0.214422	+	0.976741i	$0.568787\pi$
$558$	0	0
$559$	−5.33728	−0.225743
$560$	0	0
$561$	−2.30048	−0.0971262
$562$	0	0
$563$	−36.1558	−1.52379	−0.761893	−	0.647703i	$-0.775730\pi$
$563$	−36.1558	−1.52379	−0.761893	+	0.647703i	$0.775730\pi$
$564$	0	0
$565$	−19.8445	−0.834863
$566$	0	0
$567$	3.07030	0.128941
$568$	0	0
$569$	−33.0779	−1.38670	−0.693348	−	0.720603i	$-0.743865\pi$
$569$	−33.0779	−1.38670	−0.693348	+	0.720603i	$0.743865\pi$
$570$	0	0
$571$	28.5143	1.19329	0.596643	−	0.802507i	$-0.296501\pi$
$571$	28.5143	1.19329	0.596643	+	0.802507i	$0.296501\pi$
$572$	0	0
$573$	24.6242	1.02869
$574$	0	0
$575$	3.18264	0.132725
$576$	0	0
$577$	−28.1611	−1.17236	−0.586181	−	0.810180i	$-0.699369\pi$
$577$	−28.1611	−1.17236	−0.586181	+	0.810180i	$0.699369\pi$
$578$	0	0
$579$	−5.12128	−0.212833
$580$	0	0
$581$	22.3131	0.925702
$582$	0	0
$583$	−0.0307512	−0.00127359
$584$	0	0
$585$	−1.05916	−0.0437910
$586$	0	0
$587$	−16.3109	−0.673224	−0.336612	−	0.941643i	$-0.609281\pi$
$587$	−16.3109	−0.673224	−0.336612	+	0.941643i	$0.609281\pi$
$588$	0	0
$589$	7.16544	0.295247
$590$	0	0
$591$	−21.3906	−0.879890
$592$	0	0
$593$	−31.1979	−1.28114	−0.640571	−	0.767899i	$-0.721302\pi$
$593$	−31.1979	−1.28114	−0.640571	+	0.767899i	$0.721302\pi$
$594$	0	0
$595$	−37.5132	−1.53789
$596$	0	0
$597$	11.2900	0.462071
$598$	0	0
$599$	30.3110	1.23847	0.619237	−	0.785204i	$-0.287442\pi$
$599$	30.3110	1.23847	0.619237	+	0.785204i	$0.287442\pi$
$600$	0	0
$601$	6.50836	0.265482	0.132741	−	0.991151i	$-0.457622\pi$
$601$	6.50836	0.265482	0.132741	+	0.991151i	$0.457622\pi$
$602$	0	0
$603$	8.85930	0.360779
$604$	0	0
$605$	20.9105	0.850134
$606$	0	0
$607$	19.1501	0.777280	0.388640	−	0.921390i	$-0.372945\pi$
$607$	19.1501	0.777280	0.388640	+	0.921390i	$0.372945\pi$
$608$	0	0
$609$	16.3449	0.662328
$610$	0	0
$611$	6.78259	0.274394
$612$	0	0
$613$	12.5682	0.507626	0.253813	−	0.967253i	$-0.418315\pi$
$613$	12.5682	0.507626	0.253813	+	0.967253i	$0.418315\pi$
$614$	0	0
$615$	−14.5247	−0.585692
$616$	0	0
$617$	12.6306	0.508489	0.254245	−	0.967140i	$-0.418173\pi$
$617$	12.6306	0.508489	0.254245	+	0.967140i	$0.418173\pi$
$618$	0	0
$619$	−17.0823	−0.686594	−0.343297	−	0.939227i	$-0.611544\pi$
$619$	−17.0823	−0.686594	−0.343297	+	0.939227i	$0.611544\pi$
$620$	0	0
$621$	2.45087	0.0983499
$622$	0	0
$623$	−0.633477	−0.0253797
$624$	0	0
$625$	−16.8208	−0.672832
$626$	0	0
$627$	0.592417	0.0236589
$628$	0	0
$629$	−52.5299	−2.09450
$630$	0	0
$631$	−20.4320	−0.813384	−0.406692	−	0.913565i	$-0.633318\pi$
$631$	−20.4320	−0.813384	−0.406692	+	0.913565i	$0.633318\pi$
$632$	0	0
$633$	−0.148229	−0.00589156
$634$	0	0
$635$	16.9400	0.672242
$636$	0	0
$637$	1.33599	0.0529340
$638$	0	0
$639$	2.03636	0.0805573
$640$	0	0
$641$	9.75949	0.385477	0.192738	−	0.981250i	$-0.438263\pi$
$641$	9.75949	0.385477	0.192738	+	0.981250i	$0.438263\pi$
$642$	0	0
$643$	−2.87885	−0.113531	−0.0567653	−	0.998388i	$-0.518079\pi$
$643$	−2.87885	−0.113531	−0.0567653	+	0.998388i	$0.518079\pi$
$644$	0	0
$645$	−18.6520	−0.734423
$646$	0	0
$647$	−36.1475	−1.42110	−0.710552	−	0.703645i	$-0.751555\pi$
$647$	−36.1475	−1.42110	−0.710552	+	0.703645i	$0.751555\pi$
$648$	0	0
$649$	1.17271	0.0460328
$650$	0	0
$651$	13.4523	0.527236
$652$	0	0
$653$	30.9341	1.21054	0.605272	−	0.796018i	$-0.293064\pi$
$653$	30.9341	1.21054	0.605272	+	0.796018i	$0.293064\pi$
$654$	0	0
$655$	1.92391	0.0751733
$656$	0	0
$657$	−8.89510	−0.347031
$658$	0	0
$659$	−24.4094	−0.950857	−0.475428	−	0.879754i	$-0.657707\pi$
$659$	−24.4094	−0.950857	−0.475428	+	0.879754i	$0.657707\pi$
$660$	0	0
$661$	48.4736	1.88541	0.942703	−	0.333633i	$-0.108275\pi$
$661$	48.4736	1.88541	0.942703	+	0.333633i	$0.108275\pi$
$662$	0	0
$663$	−3.49621	−0.135781
$664$	0	0
$665$	9.66038	0.374613
$666$	0	0
$667$	13.0473	0.505193
$668$	0	0
$669$	−11.8988	−0.460035
$670$	0	0
$671$	1.14496	0.0442007
$672$	0	0
$673$	14.3668	0.553801	0.276901	−	0.960899i	$-0.410693\pi$
$673$	14.3668	0.553801	0.276901	+	0.960899i	$0.410693\pi$
$674$	0	0
$675$	1.29858	0.0499823
$676$	0	0
$677$	34.2398	1.31594	0.657972	−	0.753042i	$-0.271415\pi$
$677$	34.2398	1.31594	0.657972	+	0.753042i	$0.271415\pi$
$678$	0	0
$679$	−21.9978	−0.844200
$680$	0	0
$681$	11.7117	0.448793
$682$	0	0
$683$	−29.6395	−1.13412	−0.567062	−	0.823675i	$-0.691920\pi$
$683$	−29.6395	−1.13412	−0.567062	+	0.823675i	$0.691920\pi$
$684$	0	0
$685$	19.1306	0.730944
$686$	0	0
$687$	−3.33249	−0.127142
$688$	0	0
$689$	−0.0467350	−0.00178046
$690$	0	0
$691$	−12.9351	−0.492073	−0.246037	−	0.969261i	$-0.579128\pi$
$691$	−12.9351	−0.492073	−0.246037	+	0.969261i	$0.579128\pi$
$692$	0	0
$693$	1.11219	0.0422487
$694$	0	0
$695$	17.9128	0.679469
$696$	0	0
$697$	−47.9448	−1.81604
$698$	0	0
$699$	5.58447	0.211224
$700$	0	0
$701$	−28.7226	−1.08484	−0.542419	−	0.840108i	$-0.682492\pi$
$701$	−28.7226	−1.08484	−0.542419	+	0.840108i	$0.682492\pi$
$702$	0	0
$703$	13.5275	0.510198
$704$	0	0
$705$	23.7029	0.892704
$706$	0	0
$707$	28.1826	1.05991
$708$	0	0
$709$	16.8837	0.634081	0.317040	−	0.948412i	$-0.397311\pi$
$709$	16.8837	0.634081	0.317040	+	0.948412i	$0.397311\pi$
$710$	0	0
$711$	16.8199	0.630794
$712$	0	0
$713$	10.7383	0.402151
$714$	0	0
$715$	−0.383673	−0.0143486
$716$	0	0
$717$	−0.0395923	−0.00147860
$718$	0	0
$719$	24.0712	0.897705	0.448853	−	0.893606i	$-0.351833\pi$
$719$	24.0712	0.897705	0.448853	+	0.893606i	$0.351833\pi$
$720$	0	0
$721$	52.4327	1.95270
$722$	0	0
$723$	−5.36261	−0.199437
$724$	0	0
$725$	6.91303	0.256743
$726$	0	0
$727$	8.60834	0.319266	0.159633	−	0.987176i	$-0.448969\pi$
$727$	8.60834	0.319266	0.159633	+	0.987176i	$0.448969\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−61.5688	−2.27720
$732$	0	0
$733$	7.13816	0.263654	0.131827	−	0.991273i	$-0.457916\pi$
$733$	7.13816	0.263654	0.131827	+	0.991273i	$0.457916\pi$
$734$	0	0
$735$	4.66885	0.172213
$736$	0	0
$737$	3.20921	0.118213
$738$	0	0
$739$	−3.83378	−0.141028	−0.0705140	−	0.997511i	$-0.522464\pi$
$739$	−3.83378	−0.141028	−0.0705140	+	0.997511i	$0.522464\pi$
$740$	0	0
$741$	0.900341	0.0330749
$742$	0	0
$743$	−27.7225	−1.01704	−0.508520	−	0.861050i	$-0.669807\pi$
$743$	−27.7225	−1.01704	−0.508520	+	0.861050i	$0.669807\pi$
$744$	0	0
$745$	33.8394	1.23978
$746$	0	0
$747$	7.26739	0.265900
$748$	0	0
$749$	−40.4454	−1.47784
$750$	0	0
$751$	−6.00161	−0.219002	−0.109501	−	0.993987i	$-0.534925\pi$
$751$	−6.00161	−0.219002	−0.109501	+	0.993987i	$0.534925\pi$
$752$	0	0
$753$	23.7190	0.864367
$754$	0	0
$755$	33.7583	1.22859
$756$	0	0
$757$	30.3450	1.10291	0.551453	−	0.834206i	$-0.314073\pi$
$757$	30.3450	1.10291	0.551453	+	0.834206i	$0.314073\pi$
$758$	0	0
$759$	0.887807	0.0322254
$760$	0	0
$761$	15.3997	0.558238	0.279119	−	0.960257i	$-0.409958\pi$
$761$	15.3997	0.558238	0.279119	+	0.960257i	$0.409958\pi$
$762$	0	0
$763$	−49.9776	−1.80931
$764$	0	0
$765$	−12.2181	−0.441746
$766$	0	0
$767$	1.78225	0.0643534
$768$	0	0
$769$	−4.37913	−0.157916	−0.0789578	−	0.996878i	$-0.525159\pi$
$769$	−4.37913	−0.157916	−0.0789578	+	0.996878i	$0.525159\pi$
$770$	0	0
$771$	8.26537	0.297670
$772$	0	0
$773$	−13.1596	−0.473319	−0.236659	−	0.971593i	$-0.576053\pi$
$773$	−13.1596	−0.473319	−0.236659	+	0.971593i	$0.576053\pi$
$774$	0	0
$775$	5.68961	0.204377
$776$	0	0
$777$	25.3962	0.911083
$778$	0	0
$779$	12.3467	0.442367
$780$	0	0
$781$	0.737656	0.0263954
$782$	0	0
$783$	5.32354	0.190248
$784$	0	0
$785$	42.6880	1.52360
$786$	0	0
$787$	−22.0106	−0.784592	−0.392296	−	0.919839i	$-0.628319\pi$
$787$	−22.0106	−0.784592	−0.392296	+	0.919839i	$0.628319\pi$
$788$	0	0
$789$	−23.7023	−0.843826
$790$	0	0
$791$	31.6691	1.12603
$792$	0	0
$793$	1.74008	0.0617922
$794$	0	0
$795$	−0.163323	−0.00579248
$796$	0	0
$797$	−9.35048	−0.331211	−0.165606	−	0.986192i	$-0.552958\pi$
$797$	−9.35048	−0.331211	−0.165606	+	0.986192i	$0.552958\pi$
$798$	0	0
$799$	78.2414	2.76798
$800$	0	0
$801$	−0.206324	−0.00729010
$802$	0	0
$803$	−3.22218	−0.113708
$804$	0	0
$805$	14.4772	0.510255
$806$	0	0
$807$	−21.3939	−0.753102
$808$	0	0
$809$	49.0327	1.72390	0.861949	−	0.506995i	$-0.169244\pi$
$809$	49.0327	1.72390	0.861949	+	0.506995i	$0.169244\pi$
$810$	0	0
$811$	36.2852	1.27415	0.637073	−	0.770804i	$-0.280145\pi$
$811$	36.2852	1.27415	0.637073	+	0.770804i	$0.280145\pi$
$812$	0	0
$813$	9.33906	0.327535
$814$	0	0
$815$	−8.03299	−0.281383
$816$	0	0
$817$	15.8552	0.554702
$818$	0	0
$819$	1.69028	0.0590633
$820$	0	0
$821$	34.5796	1.20683	0.603417	−	0.797425i	$-0.293805\pi$
$821$	34.5796	1.20683	0.603417	+	0.797425i	$0.293805\pi$
$822$	0	0
$823$	−0.783939	−0.0273264	−0.0136632	−	0.999907i	$-0.504349\pi$
$823$	−0.783939	−0.0273264	−0.0136632	+	0.999907i	$0.504349\pi$
$824$	0	0
$825$	0.470400	0.0163772
$826$	0	0
$827$	10.2122	0.355115	0.177557	−	0.984110i	$-0.443180\pi$
$827$	10.2122	0.355115	0.177557	+	0.984110i	$0.443180\pi$
$828$	0	0
$829$	−11.7643	−0.408592	−0.204296	−	0.978909i	$-0.565491\pi$
$829$	−11.7643	−0.408592	−0.204296	+	0.978909i	$0.565491\pi$
$830$	0	0
$831$	27.6115	0.957834
$832$	0	0
$833$	15.4115	0.533977
$834$	0	0
$835$	13.5496	0.468903
$836$	0	0
$837$	4.38141	0.151444
$838$	0	0
$839$	57.6331	1.98972	0.994858	−	0.101280i	$-0.0322939\pi$
$839$	57.6331	1.98972	0.994858	+	0.101280i	$0.0322939\pi$
$840$	0	0
$841$	−0.659933	−0.0227563
$842$	0	0
$843$	−25.2325	−0.869053
$844$	0	0
$845$	24.4277	0.840339
$846$	0	0
$847$	−33.3704	−1.14662
$848$	0	0
$849$	−29.9072	−1.02641
$850$	0	0
$851$	20.2725	0.694932
$852$	0	0
$853$	22.7692	0.779601	0.389801	−	0.920899i	$-0.372544\pi$
$853$	22.7692	0.779601	0.389801	+	0.920899i	$0.372544\pi$
$854$	0	0
$855$	3.14639	0.107604
$856$	0	0
$857$	−0.0257536	−0.000879727 0	−0.000439864	−	1.00000i	$-0.500140\pi$
$857$	−0.0257536	−0.000879727 0	−0.000439864		1.00000i	$0.500140\pi$
$858$	0	0
$859$	−45.1069	−1.53903	−0.769513	−	0.638631i	$-0.779501\pi$
$859$	−45.1069	−1.53903	−0.769513	+	0.638631i	$0.779501\pi$
$860$	0	0
$861$	23.1795	0.789955
$862$	0	0
$863$	−28.7484	−0.978606	−0.489303	−	0.872114i	$-0.662749\pi$
$863$	−28.7484	−0.978606	−0.489303	+	0.872114i	$0.662749\pi$
$864$	0	0
$865$	1.52533	0.0518627
$866$	0	0
$867$	−23.3309	−0.792359
$868$	0	0
$869$	6.09286	0.206686
$870$	0	0
$871$	4.87728	0.165260
$872$	0	0
$873$	−7.16472	−0.242489
$874$	0	0
$875$	37.2056	1.25778
$876$	0	0
$877$	22.5161	0.760313	0.380157	−	0.924922i	$-0.375870\pi$
$877$	22.5161	0.760313	0.380157	+	0.924922i	$0.375870\pi$
$878$	0	0
$879$	−1.14617	−0.0386595
$880$	0	0
$881$	31.1378	1.04906	0.524529	−	0.851393i	$-0.324241\pi$
$881$	31.1378	1.04906	0.524529	+	0.851393i	$0.324241\pi$
$882$	0	0
$883$	−32.8652	−1.10600	−0.553000	−	0.833181i	$-0.686517\pi$
$883$	−32.8652	−1.10600	−0.553000	+	0.833181i	$0.686517\pi$
$884$	0	0
$885$	6.22838	0.209365
$886$	0	0
$887$	−39.9612	−1.34176	−0.670882	−	0.741564i	$-0.734084\pi$
$887$	−39.9612	−1.34176	−0.670882	+	0.741564i	$0.734084\pi$
$888$	0	0
$889$	−27.0339	−0.906689
$890$	0	0
$891$	0.362242	0.0121356
$892$	0	0
$893$	−20.1487	−0.674249
$894$	0	0
$895$	−15.5222	−0.518851
$896$	0	0
$897$	1.34927	0.0450507
$898$	0	0
$899$	23.3246	0.777920
$900$	0	0
$901$	−0.539117	−0.0179606
$902$	0	0
$903$	29.7662	0.990556
$904$	0	0
$905$	26.1034	0.867706
$906$	0	0
$907$	2.56940	0.0853155	0.0426577	−	0.999090i	$-0.486418\pi$
$907$	2.56940	0.0853155	0.0426577	+	0.999090i	$0.486418\pi$
$908$	0	0
$909$	9.17909	0.304451
$910$	0	0
$911$	27.8906	0.924057	0.462028	−	0.886865i	$-0.347122\pi$
$911$	27.8906	0.924057	0.462028	+	0.886865i	$0.347122\pi$
$912$	0	0
$913$	2.63255	0.0871248
$914$	0	0
$915$	6.08102	0.201032
$916$	0	0
$917$	−3.07030	−0.101390
$918$	0	0
$919$	2.10494	0.0694354	0.0347177	−	0.999397i	$-0.488947\pi$
$919$	2.10494	0.0694354	0.0347177	+	0.999397i	$0.488947\pi$
$920$	0	0
$921$	7.24654	0.238782
$922$	0	0
$923$	1.12107	0.0369005
$924$	0	0
$925$	10.7413	0.353171
$926$	0	0
$927$	17.0774	0.560895
$928$	0	0
$929$	−3.94739	−0.129510	−0.0647549	−	0.997901i	$-0.520627\pi$
$929$	−3.94739	−0.129510	−0.0647549	+	0.997901i	$0.520627\pi$
$930$	0	0
$931$	−3.96876	−0.130071
$932$	0	0
$933$	6.82586	0.223469
$934$	0	0
$935$	−4.42591	−0.144743
$936$	0	0
$937$	−51.1612	−1.67136	−0.835682	−	0.549214i	$-0.814927\pi$
$937$	−51.1612	−1.67136	−0.835682	+	0.549214i	$0.814927\pi$
$938$	0	0
$939$	15.0353	0.490660
$940$	0	0
$941$	0.412264	0.0134394	0.00671971	−	0.999977i	$-0.497861\pi$
$941$	0.412264	0.0134394	0.00671971	+	0.999977i	$0.497861\pi$
$942$	0	0
$943$	18.5030	0.602541
$944$	0	0
$945$	5.90698	0.192154
$946$	0	0
$947$	58.1470	1.88952	0.944761	−	0.327759i	$-0.106294\pi$
$947$	58.1470	1.88952	0.944761	+	0.327759i	$0.106294\pi$
$948$	0	0
$949$	−4.89699	−0.158963
$950$	0	0
$951$	10.3135	0.334437
$952$	0	0
$953$	36.8130	1.19249	0.596245	−	0.802803i	$-0.296659\pi$
$953$	36.8130	1.19249	0.596245	+	0.802803i	$0.296659\pi$
$954$	0	0
$955$	47.3748	1.53301
$956$	0	0
$957$	1.92841	0.0623366
$958$	0	0
$959$	−30.5299	−0.985863
$960$	0	0
$961$	−11.8032	−0.380749
$962$	0	0
$963$	−13.1731	−0.424498
$964$	0	0
$965$	−9.85286	−0.317175
$966$	0	0
$967$	−8.99483	−0.289254	−0.144627	−	0.989486i	$-0.546198\pi$
$967$	−8.99483	−0.289254	−0.144627	+	0.989486i	$0.546198\pi$
$968$	0	0
$969$	10.3860	0.333646
$970$	0	0
$971$	−24.8731	−0.798215	−0.399108	−	0.916904i	$-0.630680\pi$
$971$	−24.8731	−0.798215	−0.399108	+	0.916904i	$0.630680\pi$
$972$	0	0
$973$	−28.5864	−0.916437
$974$	0	0
$975$	0.714902	0.0228952
$976$	0	0
$977$	−23.3518	−0.747091	−0.373545	−	0.927612i	$-0.621858\pi$
$977$	−23.3518	−0.747091	−0.373545	+	0.927612i	$0.621858\pi$
$978$	0	0
$979$	−0.0747392	−0.00238868
$980$	0	0
$981$	−16.2778	−0.519709
$982$	0	0
$983$	−5.91035	−0.188511	−0.0942554	−	0.995548i	$-0.530047\pi$
$983$	−5.91035	−0.188511	−0.0942554	+	0.995548i	$0.530047\pi$
$984$	0	0
$985$	−41.1535	−1.31126
$986$	0	0
$987$	−37.8267	−1.20404
$988$	0	0
$989$	23.7608	0.755550
$990$	0	0
$991$	−53.4971	−1.69939	−0.849696	−	0.527273i	$-0.823215\pi$
$991$	−53.4971	−1.69939	−0.849696	+	0.527273i	$0.823215\pi$
$992$	0	0
$993$	−1.74795	−0.0554696
$994$	0	0
$995$	21.7210	0.688603
$996$	0	0
$997$	−43.9788	−1.39282	−0.696411	−	0.717643i	$-0.745221\pi$
$997$	−43.9788	−1.39282	−0.696411	+	0.717643i	$0.745221\pi$
$998$	0	0
$999$	8.27156	0.261701

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6288.2.a.bd.1.2		5
4.3	odd	2		3144.2.a.h.1.2	✓	5
12.11	even	2		9432.2.a.p.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3144.2.a.h.1.2	✓	5	4.3	odd	2
6288.2.a.bd.1.2		5	1.1	even	1	trivial
9432.2.a.p.1.4		5	12.11	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:	\( N \)	\(=\)	\( 6288 = 2^{4} \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6288.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.92391 q^{5} +3.07030 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.92391 q^{5} +3.07030 q^{7} +1.00000 q^{9} +0.362242 q^{11} +0.550527 q^{13} +1.92391 q^{15} +6.35066 q^{17} -1.63542 q^{19} -3.07030 q^{21} -2.45087 q^{23} -1.29858 q^{25} -1.00000 q^{27} -5.32354 q^{29} -4.38141 q^{31} -0.362242 q^{33} -5.90698 q^{35} -8.27156 q^{37} -0.550527 q^{39} -7.54958 q^{41} -9.69486 q^{43} -1.92391 q^{45} +12.3202 q^{47} +2.42675 q^{49} -6.35066 q^{51} -0.0848914 q^{53} -0.696920 q^{55} +1.63542 q^{57} +3.23736 q^{59} +3.16076 q^{61} +3.07030 q^{63} -1.05916 q^{65} +8.85930 q^{67} +2.45087 q^{69} +2.03636 q^{71} -8.89510 q^{73} +1.29858 q^{75} +1.11219 q^{77} +16.8199 q^{79} +1.00000 q^{81} +7.26739 q^{83} -12.2181 q^{85} +5.32354 q^{87} -0.206324 q^{89} +1.69028 q^{91} +4.38141 q^{93} +3.14639 q^{95} -7.16472 q^{97} +0.362242 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 3 q^{11} - 5 q^{13} + 5 q^{15} - q^{17} + q^{19} - 3 q^{21} + 7 q^{23} - 8 q^{25} - 5 q^{27} - 8 q^{29} + 14 q^{31} - 3 q^{33} - 6 q^{35} - 27 q^{37} + 5 q^{39} + 5 q^{41} + 3 q^{43} - 5 q^{45} + 4 q^{47} - 16 q^{49} + q^{51} + q^{53} + 12 q^{55} - q^{57} - 4 q^{59} - 12 q^{61} + 3 q^{63} + 9 q^{65} + 5 q^{67} - 7 q^{69} + 22 q^{71} - 18 q^{73} + 8 q^{75} - 8 q^{77} + 24 q^{79} + 5 q^{81} + 9 q^{83} - 14 q^{85} + 8 q^{87} - 12 q^{89} + 5 q^{91} - 14 q^{93} + 8 q^{95} - 20 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 3 * q^11 - 5 * q^13 + 5 * q^15 - q^17 + q^19 - 3 * q^21 + 7 * q^23 - 8 * q^25 - 5 * q^27 - 8 * q^29 + 14 * q^31 - 3 * q^33 - 6 * q^35 - 27 * q^37 + 5 * q^39 + 5 * q^41 + 3 * q^43 - 5 * q^45 + 4 * q^47 - 16 * q^49 + q^51 + q^53 + 12 * q^55 - q^57 - 4 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 + 5 * q^67 - 7 * q^69 + 22 * q^71 - 18 * q^73 + 8 * q^75 - 8 * q^77 + 24 * q^79 + 5 * q^81 + 9 * q^83 - 14 * q^85 + 8 * q^87 - 12 * q^89 + 5 * q^91 - 14 * q^93 + 8 * q^95 - 20 * q^97 + 3 * q^99

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