LMFDB - Embedded newform 1856.4.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 $ x^9 - 4x^8 - 138x^7 + 394x^6 + 5872x^5 - 10822x^4 - 85158x^3 + 30654x^2 + 439999x + 396802
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 928)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.38920$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38920 q^{3} +3.17647 q^{5} +28.0947 q^{7} -21.2917 q^{9} +O(q^{10})$	$q-2.38920 q^{3} +3.17647 q^{5} +28.0947 q^{7} -21.2917 q^{9} -8.12273 q^{11} +77.0237 q^{13} -7.58923 q^{15} -24.9268 q^{17} -98.5864 q^{19} -67.1238 q^{21} -1.00724 q^{23} -114.910 q^{25} +115.379 q^{27} +29.0000 q^{29} -139.131 q^{31} +19.4068 q^{33} +89.2419 q^{35} +1.27594 q^{37} -184.025 q^{39} -400.215 q^{41} -104.788 q^{43} -67.6326 q^{45} -273.025 q^{47} +446.311 q^{49} +59.5552 q^{51} +461.075 q^{53} -25.8016 q^{55} +235.543 q^{57} -553.184 q^{59} -755.592 q^{61} -598.184 q^{63} +244.663 q^{65} +768.582 q^{67} +2.40649 q^{69} +977.428 q^{71} -428.764 q^{73} +274.543 q^{75} -228.205 q^{77} -267.455 q^{79} +299.214 q^{81} -243.674 q^{83} -79.1794 q^{85} -69.2868 q^{87} -259.043 q^{89} +2163.95 q^{91} +332.412 q^{93} -313.157 q^{95} -1217.71 q^{97} +172.947 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	$ 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) $ 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * 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
$3$	−2.38920	−0.459802	−0.229901	−	0.973214i	$-0.573840\pi$
$3$	−2.38920	−0.459802	−0.229901	+	0.973214i	$0.573840\pi$
$4$	0	0
$5$	3.17647	0.284112	0.142056	−	0.989859i	$-0.454629\pi$
$5$	3.17647	0.284112	0.142056	+	0.989859i	$0.454629\pi$
$6$	0	0
$7$	28.0947	1.51697	0.758485	−	0.651691i	$-0.225940\pi$
$7$	28.0947	1.51697	0.758485	+	0.651691i	$0.225940\pi$
$8$	0	0
$9$	−21.2917	−0.788582
$10$	0	0
$11$	−8.12273	−0.222645	−0.111322	−	0.993784i	$-0.535509\pi$
$11$	−8.12273	−0.222645	−0.111322	+	0.993784i	$0.535509\pi$
$12$	0	0
$13$	77.0237	1.64327	0.821635	−	0.570014i	$-0.193062\pi$
$13$	77.0237	1.64327	0.821635	+	0.570014i	$0.193062\pi$
$14$	0	0
$15$	−7.58923	−0.130635
$16$	0	0
$17$	−24.9268	−0.355626	−0.177813	−	0.984064i	$-0.556902\pi$
$17$	−24.9268	−0.355626	−0.177813	+	0.984064i	$0.556902\pi$
$18$	0	0
$19$	−98.5864	−1.19038	−0.595191	−	0.803584i	$-0.702924\pi$
$19$	−98.5864	−1.19038	−0.595191	+	0.803584i	$0.702924\pi$
$20$	0	0
$21$	−67.1238	−0.697505
$22$	0	0
$23$	−1.00724	−0.00913144	−0.00456572	−	0.999990i	$-0.501453\pi$
$23$	−1.00724	−0.00913144	−0.00456572	+	0.999990i	$0.501453\pi$
$24$	0	0
$25$	−114.910	−0.919280
$26$	0	0
$27$	115.379	0.822393
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−139.131	−0.806087	−0.403043	−	0.915181i	$-0.632048\pi$
$31$	−139.131	−0.806087	−0.403043	+	0.915181i	$0.632048\pi$
$32$	0	0
$33$	19.4068	0.102372
$34$	0	0
$35$	89.2419	0.430990
$36$	0	0
$37$	1.27594	0.00566928	0.00283464	−	0.999996i	$-0.499098\pi$
$37$	1.27594	0.00566928	0.00283464	+	0.999996i	$0.499098\pi$
$38$	0	0
$39$	−184.025	−0.755579
$40$	0	0
$41$	−400.215	−1.52447	−0.762233	−	0.647303i	$-0.775897\pi$
$41$	−400.215	−1.52447	−0.762233	+	0.647303i	$0.775897\pi$
$42$	0	0
$43$	−104.788	−0.371628	−0.185814	−	0.982585i	$-0.559492\pi$
$43$	−104.788	−0.371628	−0.185814	+	0.982585i	$0.559492\pi$
$44$	0	0
$45$	−67.6326	−0.224046
$46$	0	0
$47$	−273.025	−0.847336	−0.423668	−	0.905817i	$-0.639258\pi$
$47$	−273.025	−0.847336	−0.423668	+	0.905817i	$0.639258\pi$
$48$	0	0
$49$	446.311	1.30120
$50$	0	0
$51$	59.5552	0.163517
$52$	0	0
$53$	461.075	1.19497	0.597486	−	0.801879i	$-0.296166\pi$
$53$	461.075	1.19497	0.597486	+	0.801879i	$0.296166\pi$
$54$	0	0
$55$	−25.8016	−0.0632562
$56$	0	0
$57$	235.543	0.547340
$58$	0	0
$59$	−553.184	−1.22065	−0.610326	−	0.792150i	$-0.708961\pi$
$59$	−553.184	−1.22065	−0.610326	+	0.792150i	$0.708961\pi$
$60$	0	0
$61$	−755.592	−1.58596	−0.792980	−	0.609247i	$-0.791472\pi$
$61$	−755.592	−1.58596	−0.792980	+	0.609247i	$0.791472\pi$
$62$	0	0
$63$	−598.184	−1.19626
$64$	0	0
$65$	244.663	0.466873
$66$	0	0
$67$	768.582	1.40145	0.700726	−	0.713431i	$-0.252860\pi$
$67$	768.582	1.40145	0.700726	+	0.713431i	$0.252860\pi$
$68$	0	0
$69$	2.40649	0.00419865
$70$	0	0
$71$	977.428	1.63379	0.816897	−	0.576784i	$-0.195692\pi$
$71$	977.428	1.63379	0.816897	+	0.576784i	$0.195692\pi$
$72$	0	0
$73$	−428.764	−0.687438	−0.343719	−	0.939072i	$-0.611687\pi$
$73$	−428.764	−0.687438	−0.343719	+	0.939072i	$0.611687\pi$
$74$	0	0
$75$	274.543	0.422687
$76$	0	0
$77$	−228.205	−0.337746
$78$	0	0
$79$	−267.455	−0.380899	−0.190450	−	0.981697i	$-0.560995\pi$
$79$	−267.455	−0.380899	−0.190450	+	0.981697i	$0.560995\pi$
$80$	0	0
$81$	299.214	0.410445
$82$	0	0
$83$	−243.674	−0.322250	−0.161125	−	0.986934i	$-0.551512\pi$
$83$	−243.674	−0.322250	−0.161125	+	0.986934i	$0.551512\pi$
$84$	0	0
$85$	−79.1794	−0.101038
$86$	0	0
$87$	−69.2868	−0.0853830
$88$	0	0
$89$	−259.043	−0.308523	−0.154261	−	0.988030i	$-0.549300\pi$
$89$	−259.043	−0.308523	−0.154261	+	0.988030i	$0.549300\pi$
$90$	0	0
$91$	2163.95	2.49279
$92$	0	0
$93$	332.412	0.370640
$94$	0	0
$95$	−313.157	−0.338202
$96$	0	0
$97$	−1217.71	−1.27463	−0.637317	−	0.770601i	$-0.719956\pi$
$97$	−1217.71	−1.27463	−0.637317	+	0.770601i	$0.719956\pi$
$98$	0	0
$99$	172.947	0.175574
$100$	0	0
$101$	525.853	0.518063	0.259031	−	0.965869i	$-0.416597\pi$
$101$	525.853	0.518063	0.259031	+	0.965869i	$0.416597\pi$
$102$	0	0
$103$	−247.734	−0.236990	−0.118495	−	0.992955i	$-0.537807\pi$
$103$	−247.734	−0.236990	−0.118495	+	0.992955i	$0.537807\pi$
$104$	0	0
$105$	−213.217	−0.198170
$106$	0	0
$107$	1936.40	1.74952	0.874761	−	0.484554i	$-0.161018\pi$
$107$	1936.40	1.74952	0.874761	+	0.484554i	$0.161018\pi$
$108$	0	0
$109$	−23.9022	−0.0210038	−0.0105019	−	0.999945i	$-0.503343\pi$
$109$	−23.9022	−0.0210038	−0.0105019	+	0.999945i	$0.503343\pi$
$110$	0	0
$111$	−3.04848	−0.00260675
$112$	0	0
$113$	−752.572	−0.626513	−0.313257	−	0.949669i	$-0.601420\pi$
$113$	−752.572	−0.626513	−0.313257	+	0.949669i	$0.601420\pi$
$114$	0	0
$115$	−3.19946	−0.00259435
$116$	0	0
$117$	−1639.97	−1.29585
$118$	0	0
$119$	−700.311	−0.539474
$120$	0	0
$121$	−1265.02	−0.950429
$122$	0	0
$123$	956.193	0.700952
$124$	0	0
$125$	−762.068	−0.545291
$126$	0	0
$127$	636.389	0.444649	0.222324	−	0.974973i	$-0.428636\pi$
$127$	636.389	0.444649	0.222324	+	0.974973i	$0.428636\pi$
$128$	0	0
$129$	250.359	0.170875
$130$	0	0
$131$	941.744	0.628096	0.314048	−	0.949407i	$-0.398315\pi$
$131$	941.744	0.628096	0.314048	+	0.949407i	$0.398315\pi$
$132$	0	0
$133$	−2769.75	−1.80577
$134$	0	0
$135$	366.497	0.233652
$136$	0	0
$137$	−1995.47	−1.24441	−0.622206	−	0.782854i	$-0.713763\pi$
$137$	−1995.47	−1.24441	−0.622206	+	0.782854i	$0.713763\pi$
$138$	0	0
$139$	147.456	0.0899786	0.0449893	−	0.998987i	$-0.485675\pi$
$139$	147.456	0.0899786	0.0449893	+	0.998987i	$0.485675\pi$
$140$	0	0
$141$	652.312	0.389607
$142$	0	0
$143$	−625.642	−0.365866
$144$	0	0
$145$	92.1177	0.0527583
$146$	0	0
$147$	−1066.32	−0.598293
$148$	0	0
$149$	493.930	0.271572	0.135786	−	0.990738i	$-0.456644\pi$
$149$	493.930	0.271572	0.135786	+	0.990738i	$0.456644\pi$
$150$	0	0
$151$	791.634	0.426638	0.213319	−	0.976983i	$-0.431573\pi$
$151$	791.634	0.426638	0.213319	+	0.976983i	$0.431573\pi$
$152$	0	0
$153$	530.735	0.280441
$154$	0	0
$155$	−441.946	−0.229019
$156$	0	0
$157$	−2099.45	−1.06722	−0.533612	−	0.845729i	$-0.679166\pi$
$157$	−2099.45	−1.06722	−0.533612	+	0.845729i	$0.679166\pi$
$158$	0	0
$159$	−1101.60	−0.549450
$160$	0	0
$161$	−28.2980	−0.0138521
$162$	0	0
$163$	281.346	0.135194	0.0675972	−	0.997713i	$-0.478467\pi$
$163$	281.346	0.135194	0.0675972	+	0.997713i	$0.478467\pi$
$164$	0	0
$165$	61.6452	0.0290853
$166$	0	0
$167$	−2919.08	−1.35261	−0.676303	−	0.736623i	$-0.736419\pi$
$167$	−2919.08	−1.35261	−0.676303	+	0.736623i	$0.736419\pi$
$168$	0	0
$169$	3735.64	1.70034
$170$	0	0
$171$	2099.07	0.938715
$172$	0	0
$173$	1967.06	0.864466	0.432233	−	0.901762i	$-0.357726\pi$
$173$	1967.06	0.864466	0.432233	+	0.901762i	$0.357726\pi$
$174$	0	0
$175$	−3228.36	−1.39452
$176$	0	0
$177$	1321.67	0.561258
$178$	0	0
$179$	3086.17	1.28867	0.644333	−	0.764745i	$-0.277135\pi$
$179$	3086.17	1.28867	0.644333	+	0.764745i	$0.277135\pi$
$180$	0	0
$181$	493.048	0.202475	0.101237	−	0.994862i	$-0.467720\pi$
$181$	493.048	0.202475	0.101237	+	0.994862i	$0.467720\pi$
$182$	0	0
$183$	1805.26	0.729227
$184$	0	0
$185$	4.05299	0.00161071
$186$	0	0
$187$	202.474	0.0791784
$188$	0	0
$189$	3241.52	1.24755
$190$	0	0
$191$	643.911	0.243936	0.121968	−	0.992534i	$-0.461079\pi$
$191$	643.911	0.243936	0.121968	+	0.992534i	$0.461079\pi$
$192$	0	0
$193$	241.962	0.0902427	0.0451213	−	0.998982i	$-0.485633\pi$
$193$	241.962	0.0902427	0.0451213	+	0.998982i	$0.485633\pi$
$194$	0	0
$195$	−584.550	−0.214669
$196$	0	0
$197$	−3386.28	−1.22468	−0.612340	−	0.790594i	$-0.709772\pi$
$197$	−3386.28	−1.22468	−0.612340	+	0.790594i	$0.709772\pi$
$198$	0	0
$199$	1312.07	0.467389	0.233694	−	0.972310i	$-0.424919\pi$
$199$	1312.07	0.467389	0.233694	+	0.972310i	$0.424919\pi$
$200$	0	0
$201$	−1836.30	−0.644390
$202$	0	0
$203$	814.745	0.281694
$204$	0	0
$205$	−1271.27	−0.433119
$206$	0	0
$207$	21.4458	0.00720089
$208$	0	0
$209$	800.790	0.265033
$210$	0	0
$211$	−5497.48	−1.79366	−0.896830	−	0.442375i	$-0.854136\pi$
$211$	−5497.48	−1.79366	−0.896830	+	0.442375i	$0.854136\pi$
$212$	0	0
$213$	−2335.27	−0.751221
$214$	0	0
$215$	−332.856	−0.105584
$216$	0	0
$217$	−3908.84	−1.22281
$218$	0	0
$219$	1024.40	0.316085
$220$	0	0
$221$	−1919.96	−0.584390
$222$	0	0
$223$	−5856.29	−1.75859	−0.879296	−	0.476276i	$-0.841986\pi$
$223$	−5856.29	−1.75859	−0.879296	+	0.476276i	$0.841986\pi$
$224$	0	0
$225$	2446.63	0.724928
$226$	0	0
$227$	1103.18	0.322559	0.161279	−	0.986909i	$-0.448438\pi$
$227$	1103.18	0.322559	0.161279	+	0.986909i	$0.448438\pi$
$228$	0	0
$229$	−3571.63	−1.03065	−0.515327	−	0.856994i	$-0.672330\pi$
$229$	−3571.63	−1.03065	−0.515327	+	0.856994i	$0.672330\pi$
$230$	0	0
$231$	545.228	0.155296
$232$	0	0
$233$	773.527	0.217491	0.108746	−	0.994070i	$-0.465317\pi$
$233$	773.527	0.217491	0.108746	+	0.994070i	$0.465317\pi$
$234$	0	0
$235$	−867.257	−0.240739
$236$	0	0
$237$	639.003	0.175138
$238$	0	0
$239$	−1185.92	−0.320965	−0.160483	−	0.987039i	$-0.551305\pi$
$239$	−1185.92	−0.320965	−0.160483	+	0.987039i	$0.551305\pi$
$240$	0	0
$241$	−6155.63	−1.64531	−0.822653	−	0.568543i	$-0.807507\pi$
$241$	−6155.63	−1.64531	−0.822653	+	0.568543i	$0.807507\pi$
$242$	0	0
$243$	−3830.10	−1.01112
$244$	0	0
$245$	1417.69	0.369686
$246$	0	0
$247$	−7593.48	−1.95612
$248$	0	0
$249$	582.187	0.148171
$250$	0	0
$251$	−1349.84	−0.339447	−0.169724	−	0.985492i	$-0.554288\pi$
$251$	−1349.84	−0.339447	−0.169724	+	0.985492i	$0.554288\pi$
$252$	0	0
$253$	8.18150	0.00203307
$254$	0	0
$255$	189.175	0.0464573
$256$	0	0
$257$	−7281.01	−1.76723	−0.883613	−	0.468218i	$-0.844896\pi$
$257$	−7281.01	−1.76723	−0.883613	+	0.468218i	$0.844896\pi$
$258$	0	0
$259$	35.8472	0.00860013
$260$	0	0
$261$	−617.460	−0.146436
$262$	0	0
$263$	5891.64	1.38135	0.690673	−	0.723167i	$-0.257314\pi$
$263$	5891.64	1.38135	0.690673	+	0.723167i	$0.257314\pi$
$264$	0	0
$265$	1464.59	0.339506
$266$	0	0
$267$	618.906	0.141859
$268$	0	0
$269$	1987.63	0.450512	0.225256	−	0.974300i	$-0.427678\pi$
$269$	1987.63	0.450512	0.225256	+	0.974300i	$0.427678\pi$
$270$	0	0
$271$	4737.00	1.06182	0.530908	−	0.847430i	$-0.321851\pi$
$271$	4737.00	1.06182	0.530908	+	0.847430i	$0.321851\pi$
$272$	0	0
$273$	−5170.12	−1.14619
$274$	0	0
$275$	933.383	0.204673
$276$	0	0
$277$	2115.97	0.458975	0.229487	−	0.973312i	$-0.426295\pi$
$277$	2115.97	0.458975	0.229487	+	0.973312i	$0.426295\pi$
$278$	0	0
$279$	2962.34	0.635666
$280$	0	0
$281$	−1498.10	−0.318040	−0.159020	−	0.987275i	$-0.550833\pi$
$281$	−1498.10	−0.318040	−0.159020	+	0.987275i	$0.550833\pi$
$282$	0	0
$283$	−8830.44	−1.85482	−0.927412	−	0.374040i	$-0.877972\pi$
$283$	−8830.44	−1.85482	−0.927412	+	0.374040i	$0.877972\pi$
$284$	0	0
$285$	748.194	0.155506
$286$	0	0
$287$	−11243.9	−2.31257
$288$	0	0
$289$	−4291.65	−0.873530
$290$	0	0
$291$	2909.35	0.586079
$292$	0	0
$293$	−7802.72	−1.55577	−0.777884	−	0.628408i	$-0.783707\pi$
$293$	−7802.72	−1.55577	−0.777884	+	0.628408i	$0.783707\pi$
$294$	0	0
$295$	−1757.17	−0.346802
$296$	0	0
$297$	−937.188	−0.183102
$298$	0	0
$299$	−77.5810	−0.0150054
$300$	0	0
$301$	−2943.99	−0.563749
$302$	0	0
$303$	−1256.37	−0.238206
$304$	0	0
$305$	−2400.12	−0.450591
$306$	0	0
$307$	−630.036	−0.117127	−0.0585636	−	0.998284i	$-0.518652\pi$
$307$	−630.036	−0.117127	−0.0585636	+	0.998284i	$0.518652\pi$
$308$	0	0
$309$	591.885	0.108968
$310$	0	0
$311$	2670.57	0.486927	0.243464	−	0.969910i	$-0.421716\pi$
$311$	2670.57	0.486927	0.243464	+	0.969910i	$0.421716\pi$
$312$	0	0
$313$	−9102.91	−1.64386	−0.821928	−	0.569591i	$-0.807102\pi$
$313$	−9102.91	−1.64386	−0.821928	+	0.569591i	$0.807102\pi$
$314$	0	0
$315$	−1900.11	−0.339871
$316$	0	0
$317$	2053.14	0.363772	0.181886	−	0.983320i	$-0.441780\pi$
$317$	2053.14	0.363772	0.181886	+	0.983320i	$0.441780\pi$
$318$	0	0
$319$	−235.559	−0.0413441
$320$	0	0
$321$	−4626.45	−0.804434
$322$	0	0
$323$	2457.45	0.423331
$324$	0	0
$325$	−8850.79	−1.51063
$326$	0	0
$327$	57.1071	0.00965759
$328$	0	0
$329$	−7670.55	−1.28538
$330$	0	0
$331$	−10798.9	−1.79324	−0.896622	−	0.442796i	$-0.853986\pi$
$331$	−10798.9	−1.79324	−0.896622	+	0.442796i	$0.853986\pi$
$332$	0	0
$333$	−27.1670	−0.00447070
$334$	0	0
$335$	2441.38	0.398170
$336$	0	0
$337$	−493.745	−0.0798101	−0.0399050	−	0.999203i	$-0.512706\pi$
$337$	−493.745	−0.0798101	−0.0399050	+	0.999203i	$0.512706\pi$
$338$	0	0
$339$	1798.04	0.288072
$340$	0	0
$341$	1130.12	0.179471
$342$	0	0
$343$	2902.48	0.456907
$344$	0	0
$345$	7.64414	0.00119289
$346$	0	0
$347$	2550.22	0.394533	0.197266	−	0.980350i	$-0.436794\pi$
$347$	2550.22	0.394533	0.197266	+	0.980350i	$0.436794\pi$
$348$	0	0
$349$	−12819.7	−1.96626	−0.983131	−	0.182904i	$-0.941450\pi$
$349$	−12819.7	−1.96626	−0.983131	+	0.182904i	$0.941450\pi$
$350$	0	0
$351$	8886.88	1.35141
$352$	0	0
$353$	9781.99	1.47491	0.737455	−	0.675397i	$-0.236028\pi$
$353$	9781.99	1.47491	0.737455	+	0.675397i	$0.236028\pi$
$354$	0	0
$355$	3104.77	0.464181
$356$	0	0
$357$	1673.18	0.248051
$358$	0	0
$359$	8439.35	1.24070	0.620351	−	0.784325i	$-0.286990\pi$
$359$	8439.35	1.24070	0.620351	+	0.784325i	$0.286990\pi$
$360$	0	0
$361$	2860.28	0.417011
$362$	0	0
$363$	3022.39	0.437009
$364$	0	0
$365$	−1361.96	−0.195310
$366$	0	0
$367$	10532.9	1.49812	0.749062	−	0.662500i	$-0.230505\pi$
$367$	10532.9	1.49812	0.749062	+	0.662500i	$0.230505\pi$
$368$	0	0
$369$	8521.27	1.20217
$370$	0	0
$371$	12953.7	1.81274
$372$	0	0
$373$	−10474.5	−1.45401	−0.727006	−	0.686631i	$-0.759089\pi$
$373$	−10474.5	−1.45401	−0.727006	+	0.686631i	$0.759089\pi$
$374$	0	0
$375$	1820.73	0.250726
$376$	0	0
$377$	2233.69	0.305148
$378$	0	0
$379$	−2508.32	−0.339957	−0.169979	−	0.985448i	$-0.554370\pi$
$379$	−2508.32	−0.339957	−0.169979	+	0.985448i	$0.554370\pi$
$380$	0	0
$381$	−1520.46	−0.204450
$382$	0	0
$383$	−1437.22	−0.191745	−0.0958724	−	0.995394i	$-0.530564\pi$
$383$	−1437.22	−0.191745	−0.0958724	+	0.995394i	$0.530564\pi$
$384$	0	0
$385$	−724.888	−0.0959577
$386$	0	0
$387$	2231.12	0.293060
$388$	0	0
$389$	9214.69	1.20104	0.600518	−	0.799611i	$-0.294961\pi$
$389$	9214.69	1.20104	0.600518	+	0.799611i	$0.294961\pi$
$390$	0	0
$391$	25.1072	0.00324738
$392$	0	0
$393$	−2250.01	−0.288800
$394$	0	0
$395$	−849.563	−0.108218
$396$	0	0
$397$	14164.9	1.79072	0.895362	−	0.445339i	$-0.146917\pi$
$397$	14164.9	1.79072	0.895362	+	0.445339i	$0.146917\pi$
$398$	0	0
$399$	6617.49	0.830298
$400$	0	0
$401$	4293.98	0.534740	0.267370	−	0.963594i	$-0.413845\pi$
$401$	4293.98	0.534740	0.267370	+	0.963594i	$0.413845\pi$
$402$	0	0
$403$	−10716.4	−1.32462
$404$	0	0
$405$	950.446	0.116612
$406$	0	0
$407$	−10.3641	−0.00126224
$408$	0	0
$409$	11914.6	1.44044	0.720218	−	0.693748i	$-0.244042\pi$
$409$	11914.6	1.44044	0.720218	+	0.693748i	$0.244042\pi$
$410$	0	0
$411$	4767.57	0.572182
$412$	0	0
$413$	−15541.5	−1.85169
$414$	0	0
$415$	−774.025	−0.0915552
$416$	0	0
$417$	−352.301	−0.0413723
$418$	0	0
$419$	−15388.7	−1.79424	−0.897119	−	0.441789i	$-0.854344\pi$
$419$	−15388.7	−1.79424	−0.897119	+	0.441789i	$0.854344\pi$
$420$	0	0
$421$	−3324.80	−0.384895	−0.192448	−	0.981307i	$-0.561643\pi$
$421$	−3324.80	−0.384895	−0.192448	+	0.981307i	$0.561643\pi$
$422$	0	0
$423$	5813.18	0.668195
$424$	0	0
$425$	2864.34	0.326920
$426$	0	0
$427$	−21228.1	−2.40585
$428$	0	0
$429$	1494.78	0.168226
$430$	0	0
$431$	9257.03	1.03456	0.517280	−	0.855816i	$-0.326945\pi$
$431$	9257.03	1.03456	0.517280	+	0.855816i	$0.326945\pi$
$432$	0	0
$433$	4596.59	0.510158	0.255079	−	0.966920i	$-0.417899\pi$
$433$	4596.59	0.510158	0.255079	+	0.966920i	$0.417899\pi$
$434$	0	0
$435$	−220.088	−0.0242584
$436$	0	0
$437$	99.2997	0.0108699
$438$	0	0
$439$	−5002.69	−0.543884	−0.271942	−	0.962314i	$-0.587666\pi$
$439$	−5002.69	−0.543884	−0.271942	+	0.962314i	$0.587666\pi$
$440$	0	0
$441$	−9502.72	−1.02610
$442$	0	0
$443$	−7882.71	−0.845415	−0.422708	−	0.906266i	$-0.638920\pi$
$443$	−7882.71	−0.845415	−0.422708	+	0.906266i	$0.638920\pi$
$444$	0	0
$445$	−822.844	−0.0876551
$446$	0	0
$447$	−1180.10	−0.124869
$448$	0	0
$449$	−14050.5	−1.47680	−0.738399	−	0.674364i	$-0.764418\pi$
$449$	−14050.5	−1.47680	−0.738399	+	0.674364i	$0.764418\pi$
$450$	0	0
$451$	3250.84	0.339414
$452$	0	0
$453$	−1891.37	−0.196169
$454$	0	0
$455$	6873.74	0.708233
$456$	0	0
$457$	2322.29	0.237707	0.118853	−	0.992912i	$-0.462078\pi$
$457$	2322.29	0.237707	0.118853	+	0.992912i	$0.462078\pi$
$458$	0	0
$459$	−2876.02	−0.292465
$460$	0	0
$461$	8371.09	0.845727	0.422864	−	0.906193i	$-0.361025\pi$
$461$	8371.09	0.845727	0.422864	+	0.906193i	$0.361025\pi$
$462$	0	0
$463$	18624.1	1.86940	0.934701	−	0.355434i	$-0.115667\pi$
$463$	18624.1	1.86940	0.934701	+	0.355434i	$0.115667\pi$
$464$	0	0
$465$	1055.90	0.105303
$466$	0	0
$467$	3060.76	0.303287	0.151643	−	0.988435i	$-0.451543\pi$
$467$	3060.76	0.303287	0.151643	+	0.988435i	$0.451543\pi$
$468$	0	0
$469$	21593.1	2.12596
$470$	0	0
$471$	5016.00	0.490711
$472$	0	0
$473$	851.164	0.0827412
$474$	0	0
$475$	11328.6	1.09430
$476$	0	0
$477$	−9817.08	−0.942334
$478$	0	0
$479$	−5013.01	−0.478184	−0.239092	−	0.970997i	$-0.576850\pi$
$479$	−5013.01	−0.478184	−0.239092	+	0.970997i	$0.576850\pi$
$480$	0	0
$481$	98.2777	0.00931617
$482$	0	0
$483$	67.6095	0.00636923
$484$	0	0
$485$	−3868.02	−0.362139
$486$	0	0
$487$	−9700.73	−0.902633	−0.451316	−	0.892364i	$-0.649045\pi$
$487$	−9700.73	−0.902633	−0.451316	+	0.892364i	$0.649045\pi$
$488$	0	0
$489$	−672.191	−0.0621626
$490$	0	0
$491$	4847.67	0.445565	0.222782	−	0.974868i	$-0.428486\pi$
$491$	4847.67	0.445565	0.222782	+	0.974868i	$0.428486\pi$
$492$	0	0
$493$	−722.878	−0.0660381
$494$	0	0
$495$	549.361	0.0498827
$496$	0	0
$497$	27460.5	2.47842
$498$	0	0
$499$	−3055.35	−0.274101	−0.137051	−	0.990564i	$-0.543762\pi$
$499$	−3055.35	−0.274101	−0.137051	+	0.990564i	$0.543762\pi$
$500$	0	0
$501$	6974.27	0.621931
$502$	0	0
$503$	−18006.3	−1.59615	−0.798074	−	0.602559i	$-0.794148\pi$
$503$	−18006.3	−1.59615	−0.798074	+	0.602559i	$0.794148\pi$
$504$	0	0
$505$	1670.36	0.147188
$506$	0	0
$507$	−8925.20	−0.781818
$508$	0	0
$509$	−5232.20	−0.455625	−0.227812	−	0.973705i	$-0.573157\pi$
$509$	−5232.20	−0.455625	−0.227812	+	0.973705i	$0.573157\pi$
$510$	0	0
$511$	−12046.0	−1.04282
$512$	0	0
$513$	−11374.8	−0.978962
$514$	0	0
$515$	−786.919	−0.0673316
$516$	0	0
$517$	2217.71	0.188655
$518$	0	0
$519$	−4699.69	−0.397483
$520$	0	0
$521$	−16382.3	−1.37758	−0.688792	−	0.724959i	$-0.741859\pi$
$521$	−16382.3	−1.37758	−0.688792	+	0.724959i	$0.741859\pi$
$522$	0	0
$523$	−1115.37	−0.0932534	−0.0466267	−	0.998912i	$-0.514847\pi$
$523$	−1115.37	−0.0932534	−0.0466267	+	0.998912i	$0.514847\pi$
$524$	0	0
$525$	7713.19	0.641203
$526$	0	0
$527$	3468.10	0.286666
$528$	0	0
$529$	−12166.0	−0.999917
$530$	0	0
$531$	11778.2	0.962585
$532$	0	0
$533$	−30826.0	−2.50511
$534$	0	0
$535$	6150.92	0.497061
$536$	0	0
$537$	−7373.48	−0.592531
$538$	0	0
$539$	−3625.26	−0.289705
$540$	0	0
$541$	20229.9	1.60768	0.803838	−	0.594849i	$-0.202788\pi$
$541$	20229.9	1.60768	0.803838	+	0.594849i	$0.202788\pi$
$542$	0	0
$543$	−1177.99	−0.0930982
$544$	0	0
$545$	−75.9247	−0.00596744
$546$	0	0
$547$	910.852	0.0711979	0.0355989	−	0.999366i	$-0.488666\pi$
$547$	910.852	0.0711979	0.0355989	+	0.999366i	$0.488666\pi$
$548$	0	0
$549$	16087.9	1.25066
$550$	0	0
$551$	−2859.01	−0.221048
$552$	0	0
$553$	−7514.06	−0.577812
$554$	0	0
$555$	−9.68341	−0.000740609 0
$556$	0	0
$557$	−14758.8	−1.12271	−0.561356	−	0.827575i	$-0.689720\pi$
$557$	−14758.8	−1.12271	−0.561356	+	0.827575i	$0.689720\pi$
$558$	0	0
$559$	−8071.16	−0.610686
$560$	0	0
$561$	−483.750	−0.0364063
$562$	0	0
$563$	14342.3	1.07363	0.536817	−	0.843699i	$-0.319627\pi$
$563$	14342.3	1.07363	0.536817	+	0.843699i	$0.319627\pi$
$564$	0	0
$565$	−2390.52	−0.178000
$566$	0	0
$567$	8406.32	0.622632
$568$	0	0
$569$	−4793.57	−0.353175	−0.176588	−	0.984285i	$-0.556506\pi$
$569$	−4793.57	−0.353175	−0.176588	+	0.984285i	$0.556506\pi$
$570$	0	0
$571$	−473.379	−0.0346941	−0.0173470	−	0.999850i	$-0.505522\pi$
$571$	−473.379	−0.0346941	−0.0173470	+	0.999850i	$0.505522\pi$
$572$	0	0
$573$	−1538.43	−0.112162
$574$	0	0
$575$	115.741	0.00839435
$576$	0	0
$577$	14551.8	1.04991	0.524956	−	0.851129i	$-0.324082\pi$
$577$	14551.8	1.04991	0.524956	+	0.851129i	$0.324082\pi$
$578$	0	0
$579$	−578.097	−0.0414937
$580$	0	0
$581$	−6845.95	−0.488843
$582$	0	0
$583$	−3745.18	−0.266054
$584$	0	0
$585$	−5209.31	−0.368168
$586$	0	0
$587$	−5617.81	−0.395011	−0.197506	−	0.980302i	$-0.563284\pi$
$587$	−5617.81	−0.395011	−0.197506	+	0.980302i	$0.563284\pi$
$588$	0	0
$589$	13716.4	0.959552
$590$	0	0
$591$	8090.49	0.563110
$592$	0	0
$593$	18403.8	1.27446	0.637229	−	0.770675i	$-0.280081\pi$
$593$	18403.8	1.27446	0.637229	+	0.770675i	$0.280081\pi$
$594$	0	0
$595$	−2224.52	−0.153271
$596$	0	0
$597$	−3134.80	−0.214906
$598$	0	0
$599$	2372.91	0.161860	0.0809301	−	0.996720i	$-0.474211\pi$
$599$	2372.91	0.161860	0.0809301	+	0.996720i	$0.474211\pi$
$600$	0	0
$601$	−198.041	−0.0134414	−0.00672069	−	0.999977i	$-0.502139\pi$
$601$	−198.041	−0.0134414	−0.00672069	+	0.999977i	$0.502139\pi$
$602$	0	0
$603$	−16364.4	−1.10516
$604$	0	0
$605$	−4018.30	−0.270029
$606$	0	0
$607$	20749.1	1.38745	0.693723	−	0.720242i	$-0.255969\pi$
$607$	20749.1	1.38745	0.693723	+	0.720242i	$0.255969\pi$
$608$	0	0
$609$	−1946.59	−0.129523
$610$	0	0
$611$	−21029.4	−1.39240
$612$	0	0
$613$	−15150.7	−0.998257	−0.499128	−	0.866528i	$-0.666346\pi$
$613$	−15150.7	−0.998257	−0.499128	+	0.866528i	$0.666346\pi$
$614$	0	0
$615$	3037.32	0.199149
$616$	0	0
$617$	−30486.4	−1.98920	−0.994601	−	0.103771i	$-0.966909\pi$
$617$	−30486.4	−1.98920	−0.994601	+	0.103771i	$0.966909\pi$
$618$	0	0
$619$	9756.70	0.633530	0.316765	−	0.948504i	$-0.397403\pi$
$619$	9756.70	0.633530	0.316765	+	0.948504i	$0.397403\pi$
$620$	0	0
$621$	−116.213	−0.00750963
$622$	0	0
$623$	−7277.74	−0.468020
$624$	0	0
$625$	11943.1	0.764356
$626$	0	0
$627$	−1913.25	−0.121862
$628$	0	0
$629$	−31.8052	−0.00201615
$630$	0	0
$631$	−2808.74	−0.177202	−0.0886009	−	0.996067i	$-0.528240\pi$
$631$	−2808.74	−0.177202	−0.0886009	+	0.996067i	$0.528240\pi$
$632$	0	0
$633$	13134.6	0.824728
$634$	0	0
$635$	2021.47	0.126330
$636$	0	0
$637$	34376.5	2.13822
$638$	0	0
$639$	−20811.1	−1.28838
$640$	0	0
$641$	20273.9	1.24925	0.624626	−	0.780924i	$-0.285251\pi$
$641$	20273.9	1.24925	0.624626	+	0.780924i	$0.285251\pi$
$642$	0	0
$643$	−14975.1	−0.918444	−0.459222	−	0.888322i	$-0.651872\pi$
$643$	−14975.1	−0.918444	−0.459222	+	0.888322i	$0.651872\pi$
$644$	0	0
$645$	795.260	0.0485478
$646$	0	0
$647$	14975.9	0.909990	0.454995	−	0.890494i	$-0.349641\pi$
$647$	14975.9	0.909990	0.454995	+	0.890494i	$0.349641\pi$
$648$	0	0
$649$	4493.36	0.271772
$650$	0	0
$651$	9339.01	0.562250
$652$	0	0
$653$	−3696.33	−0.221514	−0.110757	−	0.993848i	$-0.535328\pi$
$653$	−3696.33	−0.221514	−0.110757	+	0.993848i	$0.535328\pi$
$654$	0	0
$655$	2991.42	0.178450
$656$	0	0
$657$	9129.12	0.542102
$658$	0	0
$659$	13594.1	0.803568	0.401784	−	0.915734i	$-0.368390\pi$
$659$	13594.1	0.803568	0.401784	+	0.915734i	$0.368390\pi$
$660$	0	0
$661$	3982.38	0.234337	0.117168	−	0.993112i	$-0.462618\pi$
$661$	3982.38	0.234337	0.117168	+	0.993112i	$0.462618\pi$
$662$	0	0
$663$	4587.16	0.268703
$664$	0	0
$665$	−8798.04	−0.513043
$666$	0	0
$667$	−29.2098	−0.00169567
$668$	0	0
$669$	13991.8	0.808604
$670$	0	0
$671$	6137.46	0.353106
$672$	0	0
$673$	16130.3	0.923887	0.461944	−	0.886909i	$-0.347152\pi$
$673$	16130.3	0.923887	0.461944	+	0.886909i	$0.347152\pi$
$674$	0	0
$675$	−13258.2	−0.756010
$676$	0	0
$677$	−7743.60	−0.439602	−0.219801	−	0.975545i	$-0.570541\pi$
$677$	−7743.60	−0.439602	−0.219801	+	0.975545i	$0.570541\pi$
$678$	0	0
$679$	−34211.1	−1.93358
$680$	0	0
$681$	−2635.73	−0.148313
$682$	0	0
$683$	22344.0	1.25178	0.625892	−	0.779910i	$-0.284735\pi$
$683$	22344.0	1.25178	0.625892	+	0.779910i	$0.284735\pi$
$684$	0	0
$685$	−6338.55	−0.353553
$686$	0	0
$687$	8533.33	0.473897
$688$	0	0
$689$	35513.7	1.96366
$690$	0	0
$691$	17619.0	0.969985	0.484993	−	0.874518i	$-0.338822\pi$
$691$	17619.0	0.969985	0.484993	+	0.874518i	$0.338822\pi$
$692$	0	0
$693$	4858.89	0.266340
$694$	0	0
$695$	468.389	0.0255640
$696$	0	0
$697$	9976.09	0.542140
$698$	0	0
$699$	−1848.11	−0.100003
$700$	0	0
$701$	8754.26	0.471675	0.235837	−	0.971793i	$-0.424217\pi$
$701$	8754.26	0.471675	0.235837	+	0.971793i	$0.424217\pi$
$702$	0	0
$703$	−125.790	−0.00674862
$704$	0	0
$705$	2072.05	0.110692
$706$	0	0
$707$	14773.7	0.785886
$708$	0	0
$709$	−16008.3	−0.847960	−0.423980	−	0.905672i	$-0.639367\pi$
$709$	−16008.3	−0.847960	−0.423980	+	0.905672i	$0.639367\pi$
$710$	0	0
$711$	5694.58	0.300370
$712$	0	0
$713$	140.138	0.00736074
$714$	0	0
$715$	−1987.33	−0.103947
$716$	0	0
$717$	2833.40	0.147580
$718$	0	0
$719$	−17746.6	−0.920497	−0.460248	−	0.887790i	$-0.652240\pi$
$719$	−17746.6	−0.920497	−0.460248	+	0.887790i	$0.652240\pi$
$720$	0	0
$721$	−6960.00	−0.359506
$722$	0	0
$723$	14707.0	0.756515
$724$	0	0
$725$	−3332.39	−0.170706
$726$	0	0
$727$	−29765.4	−1.51848	−0.759241	−	0.650810i	$-0.774430\pi$
$727$	−29765.4	−1.51848	−0.759241	+	0.650810i	$0.774430\pi$
$728$	0	0
$729$	1072.10	0.0544682
$730$	0	0
$731$	2612.03	0.132161
$732$	0	0
$733$	22724.3	1.14508	0.572538	−	0.819878i	$-0.305959\pi$
$733$	22724.3	1.14508	0.572538	+	0.819878i	$0.305959\pi$
$734$	0	0
$735$	−3387.15	−0.169982
$736$	0	0
$737$	−6242.98	−0.312026
$738$	0	0
$739$	29125.1	1.44978	0.724889	−	0.688866i	$-0.241891\pi$
$739$	29125.1	1.44978	0.724889	+	0.688866i	$0.241891\pi$
$740$	0	0
$741$	18142.3	0.899427
$742$	0	0
$743$	3626.82	0.179078	0.0895391	−	0.995983i	$-0.471461\pi$
$743$	3626.82	0.179078	0.0895391	+	0.995983i	$0.471461\pi$
$744$	0	0
$745$	1568.95	0.0771571
$746$	0	0
$747$	5188.25	0.254121
$748$	0	0
$749$	54402.6	2.65397
$750$	0	0
$751$	31279.4	1.51984	0.759922	−	0.650014i	$-0.225237\pi$
$751$	31279.4	1.51984	0.759922	+	0.650014i	$0.225237\pi$
$752$	0	0
$753$	3225.04	0.156078
$754$	0	0
$755$	2514.60	0.121213
$756$	0	0
$757$	3829.60	0.183869	0.0919347	−	0.995765i	$-0.470695\pi$
$757$	3829.60	0.183869	0.0919347	+	0.995765i	$0.470695\pi$
$758$	0	0
$759$	−19.5472	−0.000934808 0
$760$	0	0
$761$	26478.2	1.26128	0.630640	−	0.776075i	$-0.282792\pi$
$761$	26478.2	1.26128	0.630640	+	0.776075i	$0.282792\pi$
$762$	0	0
$763$	−671.525	−0.0318622
$764$	0	0
$765$	1685.87	0.0796766
$766$	0	0
$767$	−42608.3	−2.00586
$768$	0	0
$769$	−11354.6	−0.532453	−0.266226	−	0.963911i	$-0.585777\pi$
$769$	−11354.6	−0.532453	−0.266226	+	0.963911i	$0.585777\pi$
$770$	0	0
$771$	17395.8	0.812573
$772$	0	0
$773$	23525.3	1.09462	0.547312	−	0.836928i	$-0.315651\pi$
$773$	23525.3	1.09462	0.547312	+	0.836928i	$0.315651\pi$
$774$	0	0
$775$	15987.6	0.741020
$776$	0	0
$777$	−85.6460	−0.00395435
$778$	0	0
$779$	39455.7	1.81470
$780$	0	0
$781$	−7939.38	−0.363756
$782$	0	0
$783$	3345.98	0.152715
$784$	0	0
$785$	−6668.84	−0.303211
$786$	0	0
$787$	−1211.31	−0.0548648	−0.0274324	−	0.999624i	$-0.508733\pi$
$787$	−1211.31	−0.0548648	−0.0274324	+	0.999624i	$0.508733\pi$
$788$	0	0
$789$	−14076.3	−0.635145
$790$	0	0
$791$	−21143.3	−0.950401
$792$	0	0
$793$	−58198.4	−2.60616
$794$	0	0
$795$	−3499.20	−0.156105
$796$	0	0
$797$	18935.2	0.841557	0.420778	−	0.907163i	$-0.361757\pi$
$797$	18935.2	0.841557	0.420778	+	0.907163i	$0.361757\pi$
$798$	0	0
$799$	6805.65	0.301335
$800$	0	0
$801$	5515.48	0.243296
$802$	0	0
$803$	3482.73	0.153055
$804$	0	0
$805$	−89.8877	−0.00393556
$806$	0	0
$807$	−4748.83	−0.207146
$808$	0	0
$809$	21361.8	0.928357	0.464179	−	0.885742i	$-0.346350\pi$
$809$	21361.8	0.928357	0.464179	+	0.885742i	$0.346350\pi$
$810$	0	0
$811$	20447.2	0.885324	0.442662	−	0.896688i	$-0.354034\pi$
$811$	20447.2	0.885324	0.442662	+	0.896688i	$0.354034\pi$
$812$	0	0
$813$	−11317.6	−0.488224
$814$	0	0
$815$	893.686	0.0384104
$816$	0	0
$817$	10330.7	0.442380
$818$	0	0
$819$	−46074.3	−1.96577
$820$	0	0
$821$	−9122.39	−0.387788	−0.193894	−	0.981023i	$-0.562112\pi$
$821$	−9122.39	−0.387788	−0.193894	+	0.981023i	$0.562112\pi$
$822$	0	0
$823$	5633.88	0.238621	0.119310	−	0.992857i	$-0.461932\pi$
$823$	5633.88	0.238621	0.119310	+	0.992857i	$0.461932\pi$
$824$	0	0
$825$	−2230.04	−0.0941090
$826$	0	0
$827$	−15933.2	−0.669955	−0.334977	−	0.942226i	$-0.608729\pi$
$827$	−15933.2	−0.669955	−0.334977	+	0.942226i	$0.608729\pi$
$828$	0	0
$829$	1523.83	0.0638419	0.0319210	−	0.999490i	$-0.489838\pi$
$829$	1523.83	0.0638419	0.0319210	+	0.999490i	$0.489838\pi$
$830$	0	0
$831$	−5055.46	−0.211037
$832$	0	0
$833$	−11125.1	−0.462740
$834$	0	0
$835$	−9272.38	−0.384292
$836$	0	0
$837$	−16052.8	−0.662920
$838$	0	0
$839$	−10697.2	−0.440178	−0.220089	−	0.975480i	$-0.570635\pi$
$839$	−10697.2	−0.440178	−0.220089	+	0.975480i	$0.570635\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	3579.27	0.146235
$844$	0	0
$845$	11866.2	0.483087
$846$	0	0
$847$	−35540.4	−1.44177
$848$	0	0
$849$	21097.7	0.852852
$850$	0	0
$851$	−1.28517	−5.17687e−5 0
$852$	0	0
$853$	−20289.2	−0.814405	−0.407202	−	0.913338i	$-0.633496\pi$
$853$	−20289.2	−0.814405	−0.407202	+	0.913338i	$0.633496\pi$
$854$	0	0
$855$	6667.65	0.266700
$856$	0	0
$857$	5811.62	0.231647	0.115823	−	0.993270i	$-0.463049\pi$
$857$	5811.62	0.231647	0.115823	+	0.993270i	$0.463049\pi$
$858$	0	0
$859$	15247.1	0.605615	0.302807	−	0.953052i	$-0.402076\pi$
$859$	15247.1	0.605615	0.302807	+	0.953052i	$0.402076\pi$
$860$	0	0
$861$	26863.9	1.06332
$862$	0	0
$863$	−45402.4	−1.79086	−0.895432	−	0.445198i	$-0.853133\pi$
$863$	−45402.4	−1.79086	−0.895432	+	0.445198i	$0.853133\pi$
$864$	0	0
$865$	6248.31	0.245606
$866$	0	0
$867$	10253.6	0.401651
$868$	0	0
$869$	2172.46	0.0848052
$870$	0	0
$871$	59199.0	2.30296
$872$	0	0
$873$	25927.1	1.00515
$874$	0	0
$875$	−21410.0	−0.827190
$876$	0	0
$877$	−31125.1	−1.19842	−0.599212	−	0.800590i	$-0.704519\pi$
$877$	−31125.1	−1.19842	−0.599212	+	0.800590i	$0.704519\pi$
$878$	0	0
$879$	18642.3	0.715345
$880$	0	0
$881$	−6954.05	−0.265934	−0.132967	−	0.991120i	$-0.542450\pi$
$881$	−6954.05	−0.265934	−0.132967	+	0.991120i	$0.542450\pi$
$882$	0	0
$883$	21416.7	0.816228	0.408114	−	0.912931i	$-0.366187\pi$
$883$	21416.7	0.816228	0.408114	+	0.912931i	$0.366187\pi$
$884$	0	0
$885$	4198.24	0.159460
$886$	0	0
$887$	−38576.5	−1.46028	−0.730142	−	0.683296i	$-0.760546\pi$
$887$	−38576.5	−1.46028	−0.730142	+	0.683296i	$0.760546\pi$
$888$	0	0
$889$	17879.1	0.674518
$890$	0	0
$891$	−2430.44	−0.0913834
$892$	0	0
$893$	26916.6	1.00865
$894$	0	0
$895$	9803.14	0.366126
$896$	0	0
$897$	185.356	0.00689952
$898$	0	0
$899$	−4034.80	−0.149687
$900$	0	0
$901$	−11493.1	−0.424963
$902$	0	0
$903$	7033.77	0.259213
$904$	0	0
$905$	1566.15	0.0575256
$906$	0	0
$907$	−14563.6	−0.533160	−0.266580	−	0.963813i	$-0.585894\pi$
$907$	−14563.6	−0.533160	−0.266580	+	0.963813i	$0.585894\pi$
$908$	0	0
$909$	−11196.3	−0.408535
$910$	0	0
$911$	−17964.1	−0.653324	−0.326662	−	0.945141i	$-0.605924\pi$
$911$	−17964.1	−0.653324	−0.326662	+	0.945141i	$0.605924\pi$
$912$	0	0
$913$	1979.30	0.0717473
$914$	0	0
$915$	5734.36	0.207182
$916$	0	0
$917$	26458.0	0.952802
$918$	0	0
$919$	11751.4	0.421809	0.210905	−	0.977507i	$-0.432359\pi$
$919$	11751.4	0.421809	0.210905	+	0.977507i	$0.432359\pi$
$920$	0	0
$921$	1505.28	0.0538553
$922$	0	0
$923$	75285.1	2.68477
$924$	0	0
$925$	−146.618	−0.00521166
$926$	0	0
$927$	5274.68	0.186886
$928$	0	0
$929$	41247.4	1.45671	0.728354	−	0.685201i	$-0.240286\pi$
$929$	41247.4	1.45671	0.728354	+	0.685201i	$0.240286\pi$
$930$	0	0
$931$	−44000.1	−1.54892
$932$	0	0
$933$	−6380.54	−0.223890
$934$	0	0
$935$	643.152	0.0224955
$936$	0	0
$937$	31197.5	1.08770	0.543851	−	0.839182i	$-0.316966\pi$
$937$	31197.5	1.08770	0.543851	+	0.839182i	$0.316966\pi$
$938$	0	0
$939$	21748.7	0.755848
$940$	0	0
$941$	15450.2	0.535241	0.267621	−	0.963524i	$-0.413763\pi$
$941$	15450.2	0.535241	0.267621	+	0.963524i	$0.413763\pi$
$942$	0	0
$943$	403.111	0.0139206
$944$	0	0
$945$	10296.6	0.354443
$946$	0	0
$947$	−6274.17	−0.215294	−0.107647	−	0.994189i	$-0.534332\pi$
$947$	−6274.17	−0.215294	−0.107647	+	0.994189i	$0.534332\pi$
$948$	0	0
$949$	−33025.0	−1.12965
$950$	0	0
$951$	−4905.36	−0.167263
$952$	0	0
$953$	−39562.3	−1.34475	−0.672376	−	0.740210i	$-0.734726\pi$
$953$	−39562.3	−1.34475	−0.672376	+	0.740210i	$0.734726\pi$
$954$	0	0
$955$	2045.37	0.0693052
$956$	0	0
$957$	562.798	0.0190101
$958$	0	0
$959$	−56062.0	−1.88773
$960$	0	0
$961$	−10433.5	−0.350224
$962$	0	0
$963$	−41229.3	−1.37964
$964$	0	0
$965$	768.587	0.0256391
$966$	0	0
$967$	26067.3	0.866874	0.433437	−	0.901184i	$-0.357301\pi$
$967$	26067.3	0.866874	0.433437	+	0.901184i	$0.357301\pi$
$968$	0	0
$969$	−5871.33	−0.194648
$970$	0	0
$971$	−20737.8	−0.685384	−0.342692	−	0.939448i	$-0.611339\pi$
$971$	−20737.8	−0.685384	−0.342692	+	0.939448i	$0.611339\pi$
$972$	0	0
$973$	4142.72	0.136495
$974$	0	0
$975$	21146.3	0.694588
$976$	0	0
$977$	15813.3	0.517821	0.258910	−	0.965901i	$-0.416637\pi$
$977$	15813.3	0.517821	0.258910	+	0.965901i	$0.416637\pi$
$978$	0	0
$979$	2104.14	0.0686910
$980$	0	0
$981$	508.919	0.0165632
$982$	0	0
$983$	−39712.3	−1.28853	−0.644265	−	0.764803i	$-0.722836\pi$
$983$	−39712.3	−1.28853	−0.644265	+	0.764803i	$0.722836\pi$
$984$	0	0
$985$	−10756.4	−0.347947
$986$	0	0
$987$	18326.5	0.591022
$988$	0	0
$989$	105.546	0.00339350
$990$	0	0
$991$	−45275.1	−1.45127	−0.725636	−	0.688079i	$-0.758454\pi$
$991$	−45275.1	−1.45127	−0.725636	+	0.688079i	$0.758454\pi$
$992$	0	0
$993$	25800.8	0.824537
$994$	0	0
$995$	4167.76	0.132791
$996$	0	0
$997$	−42466.5	−1.34897	−0.674487	−	0.738287i	$-0.735635\pi$
$997$	−42466.5	−1.34897	−0.674487	+	0.738287i	$0.735635\pi$
$998$	0	0
$999$	147.216	0.00466238

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.bg.1.3		9
4.3	odd	2		1856.4.a.bf.1.7		9
8.3	odd	2		928.4.a.e.1.3	yes	9
8.5	even	2		928.4.a.d.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.4.a.d.1.7	✓	9	8.5	even	2
928.4.a.e.1.3	yes	9	8.3	odd	2
1856.4.a.bf.1.7		9	4.3	odd	2
1856.4.a.bg.1.3		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38920 q^{3} +3.17647 q^{5} +28.0947 q^{7} -21.2917 q^{9} +O(q^{10})\)	\(q-2.38920 q^{3} +3.17647 q^{5} +28.0947 q^{7} -21.2917 q^{9} -8.12273 q^{11} +77.0237 q^{13} -7.58923 q^{15} -24.9268 q^{17} -98.5864 q^{19} -67.1238 q^{21} -1.00724 q^{23} -114.910 q^{25} +115.379 q^{27} +29.0000 q^{29} -139.131 q^{31} +19.4068 q^{33} +89.2419 q^{35} +1.27594 q^{37} -184.025 q^{39} -400.215 q^{41} -104.788 q^{43} -67.6326 q^{45} -273.025 q^{47} +446.311 q^{49} +59.5552 q^{51} +461.075 q^{53} -25.8016 q^{55} +235.543 q^{57} -553.184 q^{59} -755.592 q^{61} -598.184 q^{63} +244.663 q^{65} +768.582 q^{67} +2.40649 q^{69} +977.428 q^{71} -428.764 q^{73} +274.543 q^{75} -228.205 q^{77} -267.455 q^{79} +299.214 q^{81} -243.674 q^{83} -79.1794 q^{85} -69.2868 q^{87} -259.043 q^{89} +2163.95 q^{91} +332.412 q^{93} -313.157 q^{95} -1217.71 q^{97} +172.947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9	\( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 9 * q + 4 * q^3 - 10 * q^5 - 12 * q^7 + 49 * q^9 + 64 * q^11 - 70 * q^13 - 170 * q^15 - 66 * q^17 + 42 * q^19 - 76 * q^21 - 40 * q^23 + 111 * q^25 + 322 * q^27 + 261 * q^29 + 64 * q^31 - 52 * q^33 + 496 * q^35 + 54 * q^37 - 590 * q^39 - 378 * q^41 - 32 * q^43 - 1046 * q^45 - 1164 * q^47 - 351 * q^49 + 376 * q^51 - 278 * q^53 - 614 * q^55 + 28 * q^57 + 640 * q^59 - 1054 * q^61 - 1660 * q^63 - 708 * q^65 + 1184 * q^67 - 188 * q^69 - 1988 * q^71 - 750 * q^73 + 3126 * q^75 - 1260 * q^77 - 2916 * q^79 + 293 * q^81 + 2832 * q^83 - 56 * q^85 + 116 * q^87 - 370 * q^89 + 3016 * q^91 + 1696 * q^93 - 4412 * q^95 - 2234 * q^97 + 4118 * q^99

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