LMFDB - Embedded newform 768.6.a.v.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,6,Mod(1,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,18,0,40,0,16,0,162,0,344] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$123.174773616$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{61}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 15 $ x^2 - x - 15
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 384)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.40512$ of defining polynomial
Character	$\chi$	$=$	768.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -11.2410 q^{5} -148.205 q^{7} +81.0000 q^{9} +47.0360 q^{11} +545.374 q^{13} -101.169 q^{15} -1162.53 q^{17} +606.748 q^{19} -1333.84 q^{21} +1104.70 q^{23} -2998.64 q^{25} +729.000 q^{27} +2355.82 q^{29} +2949.39 q^{31} +423.324 q^{33} +1665.97 q^{35} +3567.52 q^{37} +4908.37 q^{39} -14129.3 q^{41} -17714.4 q^{43} -910.521 q^{45} -11597.8 q^{47} +5157.72 q^{49} -10462.8 q^{51} +1514.95 q^{53} -528.732 q^{55} +5460.73 q^{57} +10085.1 q^{59} +35214.0 q^{61} -12004.6 q^{63} -6130.55 q^{65} +28755.0 q^{67} +9942.28 q^{69} +53443.0 q^{71} +43934.9 q^{73} -26987.8 q^{75} -6970.97 q^{77} -26786.0 q^{79} +6561.00 q^{81} +105107. q^{83} +13068.0 q^{85} +21202.4 q^{87} +47927.2 q^{89} -80827.1 q^{91} +26544.5 q^{93} -6820.45 q^{95} +38270.6 q^{97} +3809.92 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 40 q^{5} + 16 q^{7} + 162 q^{9} + 344 q^{11} + 216 q^{13} + 360 q^{15} + 924 q^{17} - 536 q^{19} + 144 q^{21} + 3584 q^{23} - 3498 q^{25} + 1458 q^{27} + 8648 q^{29} - 3536 q^{31} + 3096 q^{33}+ \cdots + 27864 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 40 * q^5 + 16 * q^7 + 162 * q^9 + 344 * q^11 + 216 * q^13 + 360 * q^15 + 924 * q^17 - 536 * q^19 + 144 * q^21 + 3584 * q^23 - 3498 * q^25 + 1458 * q^27 + 8648 * q^29 - 3536 * q^31 + 3096 * q^33 + 10080 * q^35 - 3112 * q^37 + 1944 * q^39 - 2516 * q^41 - 3688 * q^43 + 3240 * q^45 - 4576 * q^47 + 15314 * q^49 + 8316 * q^51 + 968 * q^53 + 14688 * q^55 - 4824 * q^57 - 27816 * q^59 + 28440 * q^61 + 1296 * q^63 - 23008 * q^65 + 78504 * q^67 + 32256 * q^69 + 75520 * q^71 - 17100 * q^73 - 31482 * q^75 + 41792 * q^77 - 100496 * q^79 + 13122 * q^81 + 70504 * q^83 + 119984 * q^85 + 77832 * q^87 + 90356 * q^89 - 134912 * q^91 - 31824 * q^93 - 65376 * q^95 - 99908 * q^97 + 27864 * q^99

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−11.2410	−0.201085	−0.100543	−	0.994933i	$-0.532058\pi$
$5$	−11.2410	−0.201085	−0.100543	+	0.994933i	$0.532058\pi$
$6$	0	0
$7$	−148.205	−1.14319	−0.571594	−	0.820537i	$-0.693675\pi$
$7$	−148.205	−1.14319	−0.571594	+	0.820537i	$0.693675\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	47.0360	0.117206	0.0586028	−	0.998281i	$-0.481335\pi$
$11$	47.0360	0.117206	0.0586028	+	0.998281i	$0.481335\pi$
$12$	0	0
$13$	545.374	0.895027	0.447513	−	0.894277i	$-0.352310\pi$
$13$	545.374	0.895027	0.447513	+	0.894277i	$0.352310\pi$
$14$	0	0
$15$	−101.169	−0.116097
$16$	0	0
$17$	−1162.53	−0.975624	−0.487812	−	0.872949i	$-0.662205\pi$
$17$	−1162.53	−0.975624	−0.487812	+	0.872949i	$0.662205\pi$
$18$	0	0
$19$	606.748	0.385589	0.192794	−	0.981239i	$-0.438245\pi$
$19$	606.748	0.385589	0.192794	+	0.981239i	$0.438245\pi$
$20$	0	0
$21$	−1333.84	−0.660020
$22$	0	0
$23$	1104.70	0.435436	0.217718	−	0.976012i	$-0.430139\pi$
$23$	1104.70	0.435436	0.217718	+	0.976012i	$0.430139\pi$
$24$	0	0
$25$	−2998.64	−0.959565
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	2355.82	0.520172	0.260086	−	0.965586i	$-0.416249\pi$
$29$	2355.82	0.520172	0.260086	+	0.965586i	$0.416249\pi$
$30$	0	0
$31$	2949.39	0.551224	0.275612	−	0.961269i	$-0.411120\pi$
$31$	2949.39	0.551224	0.275612	+	0.961269i	$0.411120\pi$
$32$	0	0
$33$	423.324	0.0676687
$34$	0	0
$35$	1665.97	0.229878
$36$	0	0
$37$	3567.52	0.428413	0.214206	−	0.976788i	$-0.431283\pi$
$37$	3567.52	0.428413	0.214206	+	0.976788i	$0.431283\pi$
$38$	0	0
$39$	4908.37	0.516744
$40$	0	0
$41$	−14129.3	−1.31269	−0.656343	−	0.754463i	$-0.727897\pi$
$41$	−14129.3	−1.31269	−0.656343	+	0.754463i	$0.727897\pi$
$42$	0	0
$43$	−17714.4	−1.46102	−0.730510	−	0.682902i	$-0.760717\pi$
$43$	−17714.4	−1.46102	−0.730510	+	0.682902i	$0.760717\pi$
$44$	0	0
$45$	−910.521	−0.0670284
$46$	0	0
$47$	−11597.8	−0.765829	−0.382914	−	0.923784i	$-0.625080\pi$
$47$	−11597.8	−0.765829	−0.382914	+	0.923784i	$0.625080\pi$
$48$	0	0
$49$	5157.72	0.306879
$50$	0	0
$51$	−10462.8	−0.563277
$52$	0	0
$53$	1514.95	0.0740814	0.0370407	−	0.999314i	$-0.488207\pi$
$53$	1514.95	0.0740814	0.0370407	+	0.999314i	$0.488207\pi$
$54$	0	0
$55$	−528.732	−0.0235683
$56$	0	0
$57$	5460.73	0.222620
$58$	0	0
$59$	10085.1	0.377181	0.188590	−	0.982056i	$-0.439608\pi$
$59$	10085.1	0.377181	0.188590	+	0.982056i	$0.439608\pi$
$60$	0	0
$61$	35214.0	1.21169	0.605843	−	0.795584i	$-0.292836\pi$
$61$	35214.0	1.21169	0.605843	+	0.795584i	$0.292836\pi$
$62$	0	0
$63$	−12004.6	−0.381063
$64$	0	0
$65$	−6130.55	−0.179977
$66$	0	0
$67$	28755.0	0.782576	0.391288	−	0.920268i	$-0.372030\pi$
$67$	28755.0	0.782576	0.391288	+	0.920268i	$0.372030\pi$
$68$	0	0
$69$	9942.28	0.251399
$70$	0	0
$71$	53443.0	1.25819	0.629093	−	0.777330i	$-0.283426\pi$
$71$	53443.0	1.25819	0.629093	+	0.777330i	$0.283426\pi$
$72$	0	0
$73$	43934.9	0.964944	0.482472	−	0.875911i	$-0.339739\pi$
$73$	43934.9	0.964944	0.482472	+	0.875911i	$0.339739\pi$
$74$	0	0
$75$	−26987.8	−0.554005
$76$	0	0
$77$	−6970.97	−0.133988
$78$	0	0
$79$	−26786.0	−0.482881	−0.241441	−	0.970416i	$-0.577620\pi$
$79$	−26786.0	−0.482881	−0.241441	+	0.970416i	$0.577620\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	105107.	1.67470	0.837348	−	0.546671i	$-0.184105\pi$
$83$	105107.	1.67470	0.837348	+	0.546671i	$0.184105\pi$
$84$	0	0
$85$	13068.0	0.196183
$86$	0	0
$87$	21202.4	0.300321
$88$	0	0
$89$	47927.2	0.641368	0.320684	−	0.947186i	$-0.396087\pi$
$89$	47927.2	0.641368	0.320684	+	0.947186i	$0.396087\pi$
$90$	0	0
$91$	−80827.1	−1.02318
$92$	0	0
$93$	26544.5	0.318249
$94$	0	0
$95$	−6820.45	−0.0775361
$96$	0	0
$97$	38270.6	0.412986	0.206493	−	0.978448i	$-0.433795\pi$
$97$	38270.6	0.412986	0.206493	+	0.978448i	$0.433795\pi$
$98$	0	0
$99$	3809.92	0.0390686
$100$	0	0
$101$	128974.	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$101$	128974.	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$102$	0	0
$103$	202213.	1.87809	0.939044	−	0.343798i	$-0.111713\pi$
$103$	202213.	1.87809	0.939044	+	0.343798i	$0.111713\pi$
$104$	0	0
$105$	14993.7	0.132720
$106$	0	0
$107$	−50910.8	−0.429883	−0.214942	−	0.976627i	$-0.568956\pi$
$107$	−50910.8	−0.429883	−0.214942	+	0.976627i	$0.568956\pi$
$108$	0	0
$109$	−86786.3	−0.699656	−0.349828	−	0.936814i	$-0.613760\pi$
$109$	−86786.3	−0.699656	−0.349828	+	0.936814i	$0.613760\pi$
$110$	0	0
$111$	32107.7	0.247344
$112$	0	0
$113$	10281.4	0.0757457	0.0378729	−	0.999283i	$-0.487942\pi$
$113$	10281.4	0.0757457	0.0378729	+	0.999283i	$0.487942\pi$
$114$	0	0
$115$	−12417.9	−0.0875596
$116$	0	0
$117$	44175.3	0.298342
$118$	0	0
$119$	172293.	1.11532
$120$	0	0
$121$	−158839.	−0.986263
$122$	0	0
$123$	−127164.	−0.757879
$124$	0	0
$125$	68835.8	0.394039
$126$	0	0
$127$	−118323.	−0.650966	−0.325483	−	0.945548i	$-0.605527\pi$
$127$	−118323.	−0.650966	−0.325483	+	0.945548i	$0.605527\pi$
$128$	0	0
$129$	−159430.	−0.843520
$130$	0	0
$131$	328293.	1.67141	0.835705	−	0.549179i	$-0.185060\pi$
$131$	328293.	1.67141	0.835705	+	0.549179i	$0.185060\pi$
$132$	0	0
$133$	−89923.1	−0.440800
$134$	0	0
$135$	−8194.69	−0.0386988
$136$	0	0
$137$	11172.7	0.0508579	0.0254289	−	0.999677i	$-0.491905\pi$
$137$	11172.7	0.0508579	0.0254289	+	0.999677i	$0.491905\pi$
$138$	0	0
$139$	−109510.	−0.480748	−0.240374	−	0.970680i	$-0.577270\pi$
$139$	−109510.	−0.480748	−0.240374	+	0.970680i	$0.577270\pi$
$140$	0	0
$141$	−104380.	−0.442151
$142$	0	0
$143$	25652.2	0.104902
$144$	0	0
$145$	−26481.7	−0.104599
$146$	0	0
$147$	46419.5	0.177177
$148$	0	0
$149$	−309729.	−1.14292	−0.571461	−	0.820629i	$-0.693623\pi$
$149$	−309729.	−1.14292	−0.571461	+	0.820629i	$0.693623\pi$
$150$	0	0
$151$	461953.	1.64875	0.824377	−	0.566041i	$-0.191526\pi$
$151$	461953.	1.64875	0.824377	+	0.566041i	$0.191526\pi$
$152$	0	0
$153$	−94165.1	−0.325208
$154$	0	0
$155$	−33154.1	−0.110843
$156$	0	0
$157$	434846.	1.40795	0.703973	−	0.710227i	$-0.251408\pi$
$157$	434846.	1.40795	0.703973	+	0.710227i	$0.251408\pi$
$158$	0	0
$159$	13634.6	0.0427709
$160$	0	0
$161$	−163722.	−0.497785
$162$	0	0
$163$	119362.	0.351883	0.175941	−	0.984401i	$-0.443703\pi$
$163$	119362.	0.351883	0.175941	+	0.984401i	$0.443703\pi$
$164$	0	0
$165$	−4758.59	−0.0136072
$166$	0	0
$167$	460259.	1.27706	0.638530	−	0.769597i	$-0.279543\pi$
$167$	460259.	1.27706	0.638530	+	0.769597i	$0.279543\pi$
$168$	0	0
$169$	−73860.2	−0.198927
$170$	0	0
$171$	49146.6	0.128530
$172$	0	0
$173$	−465904.	−1.18353	−0.591767	−	0.806109i	$-0.701569\pi$
$173$	−465904.	−1.18353	−0.591767	+	0.806109i	$0.701569\pi$
$174$	0	0
$175$	444413.	1.09696
$176$	0	0
$177$	90765.8	0.217765
$178$	0	0
$179$	−143810.	−0.335471	−0.167736	−	0.985832i	$-0.553645\pi$
$179$	−143810.	−0.335471	−0.167736	+	0.985832i	$0.553645\pi$
$180$	0	0
$181$	−281826.	−0.639418	−0.319709	−	0.947516i	$-0.603585\pi$
$181$	−281826.	−0.639418	−0.319709	+	0.947516i	$0.603585\pi$
$182$	0	0
$183$	316926.	0.699567
$184$	0	0
$185$	−40102.5	−0.0861475
$186$	0	0
$187$	−54680.9	−0.114349
$188$	0	0
$189$	−108041.	−0.220007
$190$	0	0
$191$	319313.	0.633335	0.316667	−	0.948537i	$-0.397436\pi$
$191$	319313.	0.633335	0.316667	+	0.948537i	$0.397436\pi$
$192$	0	0
$193$	755250.	1.45948	0.729739	−	0.683726i	$-0.239642\pi$
$193$	755250.	1.45948	0.729739	+	0.683726i	$0.239642\pi$
$194$	0	0
$195$	−55174.9	−0.103910
$196$	0	0
$197$	69303.0	0.127229	0.0636146	−	0.997975i	$-0.479737\pi$
$197$	69303.0	0.127229	0.0636146	+	0.997975i	$0.479737\pi$
$198$	0	0
$199$	16621.2	0.0297530	0.0148765	−	0.999889i	$-0.495264\pi$
$199$	16621.2	0.0297530	0.0148765	+	0.999889i	$0.495264\pi$
$200$	0	0
$201$	258795.	0.451821
$202$	0	0
$203$	−349144.	−0.594654
$204$	0	0
$205$	158827.	0.263962
$206$	0	0
$207$	89480.5	0.145145
$208$	0	0
$209$	28539.0	0.0451932
$210$	0	0
$211$	412143.	0.637297	0.318648	−	0.947873i	$-0.396771\pi$
$211$	412143.	0.637297	0.318648	+	0.947873i	$0.396771\pi$
$212$	0	0
$213$	480987.	0.726414
$214$	0	0
$215$	199128.	0.293789
$216$	0	0
$217$	−437114.	−0.630153
$218$	0	0
$219$	395414.	0.557111
$220$	0	0
$221$	−634015.	−0.873210
$222$	0	0
$223$	1.19123e6	1.60411	0.802053	−	0.597252i	$-0.203741\pi$
$223$	1.19123e6	1.60411	0.802053	+	0.597252i	$0.203741\pi$
$224$	0	0
$225$	−242890.	−0.319855
$226$	0	0
$227$	−1.41130e6	−1.81783	−0.908916	−	0.416980i	$-0.863089\pi$
$227$	−1.41130e6	−1.81783	−0.908916	+	0.416980i	$0.863089\pi$
$228$	0	0
$229$	−546374.	−0.688496	−0.344248	−	0.938879i	$-0.611866\pi$
$229$	−546374.	−0.688496	−0.344248	+	0.938879i	$0.611866\pi$
$230$	0	0
$231$	−62738.7	−0.0773581
$232$	0	0
$233$	919704.	1.10984	0.554918	−	0.831905i	$-0.312750\pi$
$233$	919704.	1.10984	0.554918	+	0.831905i	$0.312750\pi$
$234$	0	0
$235$	130371.	0.153997
$236$	0	0
$237$	−241074.	−0.278792
$238$	0	0
$239$	1.38063e6	1.56345	0.781724	−	0.623624i	$-0.214340\pi$
$239$	1.38063e6	1.56345	0.781724	+	0.623624i	$0.214340\pi$
$240$	0	0
$241$	650375.	0.721308	0.360654	−	0.932700i	$-0.382553\pi$
$241$	650375.	0.721308	0.360654	+	0.932700i	$0.382553\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−57977.9	−0.0617089
$246$	0	0
$247$	330905.	0.345112
$248$	0	0
$249$	945962.	0.966886
$250$	0	0
$251$	393270.	0.394009	0.197005	−	0.980403i	$-0.436879\pi$
$251$	393270.	0.394009	0.197005	+	0.980403i	$0.436879\pi$
$252$	0	0
$253$	51960.6	0.0510355
$254$	0	0
$255$	117612.	0.113267
$256$	0	0
$257$	−1.23955e6	−1.17066	−0.585331	−	0.810794i	$-0.699036\pi$
$257$	−1.23955e6	−1.17066	−0.585331	+	0.810794i	$0.699036\pi$
$258$	0	0
$259$	−528725.	−0.489757
$260$	0	0
$261$	190821.	0.173391
$262$	0	0
$263$	1.47951e6	1.31895	0.659476	−	0.751726i	$-0.270778\pi$
$263$	1.47951e6	1.31895	0.659476	+	0.751726i	$0.270778\pi$
$264$	0	0
$265$	−17029.6	−0.0148967
$266$	0	0
$267$	431345.	0.370294
$268$	0	0
$269$	345680.	0.291269	0.145634	−	0.989338i	$-0.453478\pi$
$269$	345680.	0.291269	0.145634	+	0.989338i	$0.453478\pi$
$270$	0	0
$271$	1.53008e6	1.26558	0.632791	−	0.774322i	$-0.281909\pi$
$271$	1.53008e6	1.26558	0.632791	+	0.774322i	$0.281909\pi$
$272$	0	0
$273$	−727444.	−0.590736
$274$	0	0
$275$	−141044.	−0.112466
$276$	0	0
$277$	−464109.	−0.363429	−0.181715	−	0.983351i	$-0.558165\pi$
$277$	−464109.	−0.363429	−0.181715	+	0.983351i	$0.558165\pi$
$278$	0	0
$279$	238901.	0.183741
$280$	0	0
$281$	−847312.	−0.640144	−0.320072	−	0.947393i	$-0.603707\pi$
$281$	−847312.	−0.640144	−0.320072	+	0.947393i	$0.603707\pi$
$282$	0	0
$283$	2.08212e6	1.54540	0.772698	−	0.634773i	$-0.218907\pi$
$283$	2.08212e6	1.54540	0.772698	+	0.634773i	$0.218907\pi$
$284$	0	0
$285$	−61384.1	−0.0447655
$286$	0	0
$287$	2.09403e6	1.50065
$288$	0	0
$289$	−68376.5	−0.0481573
$290$	0	0
$291$	344435.	0.238438
$292$	0	0
$293$	1.23146e6	0.838017	0.419008	−	0.907982i	$-0.362378\pi$
$293$	1.23146e6	0.838017	0.419008	+	0.907982i	$0.362378\pi$
$294$	0	0
$295$	−113366.	−0.0758454
$296$	0	0
$297$	34289.2	0.0225562
$298$	0	0
$299$	602474.	0.389727
$300$	0	0
$301$	2.62537e6	1.67022
$302$	0	0
$303$	1.16076e6	0.726335
$304$	0	0
$305$	−395840.	−0.243652
$306$	0	0
$307$	−1.70679e6	−1.03356	−0.516778	−	0.856120i	$-0.672869\pi$
$307$	−1.70679e6	−1.03356	−0.516778	+	0.856120i	$0.672869\pi$
$308$	0	0
$309$	1.81992e6	1.08431
$310$	0	0
$311$	−2.97615e6	−1.74484	−0.872418	−	0.488761i	$-0.837449\pi$
$311$	−2.97615e6	−1.74484	−0.872418	+	0.488761i	$0.837449\pi$
$312$	0	0
$313$	−3.41990e6	−1.97312	−0.986558	−	0.163411i	$-0.947750\pi$
$313$	−3.41990e6	−1.97312	−0.986558	+	0.163411i	$0.947750\pi$
$314$	0	0
$315$	134944.	0.0766260
$316$	0	0
$317$	−3.30728e6	−1.84851	−0.924257	−	0.381770i	$-0.875315\pi$
$317$	−3.30728e6	−1.84851	−0.924257	+	0.381770i	$0.875315\pi$
$318$	0	0
$319$	110808.	0.0609671
$320$	0	0
$321$	−458197.	−0.248193
$322$	0	0
$323$	−705364.	−0.376190
$324$	0	0
$325$	−1.63538e6	−0.858836
$326$	0	0
$327$	−781077.	−0.403947
$328$	0	0
$329$	1.71885e6	0.875486
$330$	0	0
$331$	858062.	0.430476	0.215238	−	0.976562i	$-0.430947\pi$
$331$	858062.	0.430476	0.215238	+	0.976562i	$0.430947\pi$
$332$	0	0
$333$	288969.	0.142804
$334$	0	0
$335$	−323235.	−0.157364
$336$	0	0
$337$	2.31479e6	1.11029	0.555146	−	0.831753i	$-0.312662\pi$
$337$	2.31479e6	1.11029	0.555146	+	0.831753i	$0.312662\pi$
$338$	0	0
$339$	92533.0	0.0437318
$340$	0	0
$341$	138728.	0.0646066
$342$	0	0
$343$	1.72648e6	0.792367
$344$	0	0
$345$	−111761.	−0.0505526
$346$	0	0
$347$	−42663.1	−0.0190208	−0.00951039	−	0.999955i	$-0.503027\pi$
$347$	−42663.1	−0.0190208	−0.00951039	+	0.999955i	$0.503027\pi$
$348$	0	0
$349$	4.06310e6	1.78564	0.892820	−	0.450413i	$-0.148723\pi$
$349$	4.06310e6	1.78564	0.892820	+	0.450413i	$0.148723\pi$
$350$	0	0
$351$	397578.	0.172248
$352$	0	0
$353$	1.25470e6	0.535922	0.267961	−	0.963430i	$-0.413650\pi$
$353$	1.25470e6	0.535922	0.267961	+	0.963430i	$0.413650\pi$
$354$	0	0
$355$	−600752.	−0.253002
$356$	0	0
$357$	1.55064e6	0.643932
$358$	0	0
$359$	1.93310e6	0.791621	0.395811	−	0.918332i	$-0.370464\pi$
$359$	1.93310e6	0.791621	0.395811	+	0.918332i	$0.370464\pi$
$360$	0	0
$361$	−2.10796e6	−0.851321
$362$	0	0
$363$	−1.42955e6	−0.569419
$364$	0	0
$365$	−493872.	−0.194036
$366$	0	0
$367$	932855.	0.361534	0.180767	−	0.983526i	$-0.442142\pi$
$367$	932855.	0.361534	0.180767	+	0.983526i	$0.442142\pi$
$368$	0	0
$369$	−1.14447e6	−0.437562
$370$	0	0
$371$	−224524.	−0.0846890
$372$	0	0
$373$	636513.	0.236884	0.118442	−	0.992961i	$-0.462210\pi$
$373$	636513.	0.236884	0.118442	+	0.992961i	$0.462210\pi$
$374$	0	0
$375$	619522.	0.227499
$376$	0	0
$377$	1.28480e6	0.465567
$378$	0	0
$379$	−436637.	−0.156143	−0.0780715	−	0.996948i	$-0.524876\pi$
$379$	−436637.	−0.156143	−0.0780715	+	0.996948i	$0.524876\pi$
$380$	0	0
$381$	−1.06490e6	−0.375835
$382$	0	0
$383$	2.77960e6	0.968244	0.484122	−	0.875001i	$-0.339139\pi$
$383$	2.77960e6	0.968244	0.484122	+	0.875001i	$0.339139\pi$
$384$	0	0
$385$	78360.7	0.0269430
$386$	0	0
$387$	−1.43487e6	−0.487006
$388$	0	0
$389$	−4.22186e6	−1.41459	−0.707294	−	0.706919i	$-0.750084\pi$
$389$	−4.22186e6	−1.41459	−0.707294	+	0.706919i	$0.750084\pi$
$390$	0	0
$391$	−1.28425e6	−0.424822
$392$	0	0
$393$	2.95463e6	0.964989
$394$	0	0
$395$	301102.	0.0971002
$396$	0	0
$397$	3.89554e6	1.24048	0.620242	−	0.784411i	$-0.287034\pi$
$397$	3.89554e6	1.24048	0.620242	+	0.784411i	$0.287034\pi$
$398$	0	0
$399$	−809308.	−0.254496
$400$	0	0
$401$	−82294.6	−0.0255570	−0.0127785	−	0.999918i	$-0.504068\pi$
$401$	−82294.6	−0.0255570	−0.0127785	+	0.999918i	$0.504068\pi$
$402$	0	0
$403$	1.60852e6	0.493360
$404$	0	0
$405$	−73752.2	−0.0223428
$406$	0	0
$407$	167802.	0.0502124
$408$	0	0
$409$	133394.	0.0394300	0.0197150	−	0.999806i	$-0.493724\pi$
$409$	133394.	0.0394300	0.0197150	+	0.999806i	$0.493724\pi$
$410$	0	0
$411$	100555.	0.0293628
$412$	0	0
$413$	−1.49466e6	−0.431189
$414$	0	0
$415$	−1.18151e6	−0.336756
$416$	0	0
$417$	−985591.	−0.277560
$418$	0	0
$419$	4.85459e6	1.35088	0.675441	−	0.737414i	$-0.263953\pi$
$419$	4.85459e6	1.35088	0.675441	+	0.737414i	$0.263953\pi$
$420$	0	0
$421$	674802.	0.185554	0.0927772	−	0.995687i	$-0.470426\pi$
$421$	674802.	0.185554	0.0927772	+	0.995687i	$0.470426\pi$
$422$	0	0
$423$	−939423.	−0.255276
$424$	0	0
$425$	3.48601e6	0.936175
$426$	0	0
$427$	−5.21888e6	−1.38519
$428$	0	0
$429$	230870.	0.0605653
$430$	0	0
$431$	−5.62432e6	−1.45840	−0.729200	−	0.684301i	$-0.760108\pi$
$431$	−5.62432e6	−1.45840	−0.729200	+	0.684301i	$0.760108\pi$
$432$	0	0
$433$	5.89978e6	1.51222	0.756112	−	0.654442i	$-0.227097\pi$
$433$	5.89978e6	1.51222	0.756112	+	0.654442i	$0.227097\pi$
$434$	0	0
$435$	−238336.	−0.0603901
$436$	0	0
$437$	670273.	0.167899
$438$	0	0
$439$	417376.	0.103363	0.0516817	−	0.998664i	$-0.483542\pi$
$439$	417376.	0.103363	0.0516817	+	0.998664i	$0.483542\pi$
$440$	0	0
$441$	417775.	0.102293
$442$	0	0
$443$	5.30672e6	1.28474	0.642372	−	0.766393i	$-0.277950\pi$
$443$	5.30672e6	1.28474	0.642372	+	0.766393i	$0.277950\pi$
$444$	0	0
$445$	−538750.	−0.128969
$446$	0	0
$447$	−2.78756e6	−0.659866
$448$	0	0
$449$	−2.85104e6	−0.667402	−0.333701	−	0.942679i	$-0.608298\pi$
$449$	−2.85104e6	−0.667402	−0.333701	+	0.942679i	$0.608298\pi$
$450$	0	0
$451$	−664585.	−0.153854
$452$	0	0
$453$	4.15758e6	0.951909
$454$	0	0
$455$	908578.	0.205747
$456$	0	0
$457$	−2.94965e6	−0.660662	−0.330331	−	0.943865i	$-0.607160\pi$
$457$	−2.94965e6	−0.660662	−0.330331	+	0.943865i	$0.607160\pi$
$458$	0	0
$459$	−847486.	−0.187759
$460$	0	0
$461$	−7.87075e6	−1.72490	−0.862449	−	0.506143i	$-0.831071\pi$
$461$	−7.87075e6	−1.72490	−0.862449	+	0.506143i	$0.831071\pi$
$462$	0	0
$463$	−3.22083e6	−0.698257	−0.349128	−	0.937075i	$-0.613522\pi$
$463$	−3.22083e6	−0.698257	−0.349128	+	0.937075i	$0.613522\pi$
$464$	0	0
$465$	−298387.	−0.0639952
$466$	0	0
$467$	3.23362e6	0.686115	0.343057	−	0.939314i	$-0.388537\pi$
$467$	3.23362e6	0.686115	0.343057	+	0.939314i	$0.388537\pi$
$468$	0	0
$469$	−4.26164e6	−0.894632
$470$	0	0
$471$	3.91361e6	0.812878
$472$	0	0
$473$	−833216.	−0.171240
$474$	0	0
$475$	−1.81942e6	−0.369997
$476$	0	0
$477$	122711.	0.0246938
$478$	0	0
$479$	−7.98448e6	−1.59004	−0.795020	−	0.606583i	$-0.792540\pi$
$479$	−7.98448e6	−1.59004	−0.795020	+	0.606583i	$0.792540\pi$
$480$	0	0
$481$	1.94563e6	0.383441
$482$	0	0
$483$	−1.47350e6	−0.287396
$484$	0	0
$485$	−430200.	−0.0830454
$486$	0	0
$487$	−4.14364e6	−0.791698	−0.395849	−	0.918316i	$-0.629550\pi$
$487$	−4.14364e6	−0.791698	−0.395849	+	0.918316i	$0.629550\pi$
$488$	0	0
$489$	1.07426e6	0.203160
$490$	0	0
$491$	−8.09427e6	−1.51521	−0.757607	−	0.652711i	$-0.773632\pi$
$491$	−8.09427e6	−1.51521	−0.757607	+	0.652711i	$0.773632\pi$
$492$	0	0
$493$	−2.73871e6	−0.507492
$494$	0	0
$495$	−42827.3	−0.00785611
$496$	0	0
$497$	−7.92052e6	−1.43834
$498$	0	0
$499$	6.18633e6	1.11220	0.556098	−	0.831117i	$-0.312298\pi$
$499$	6.18633e6	1.11220	0.556098	+	0.831117i	$0.312298\pi$
$500$	0	0
$501$	4.14233e6	0.737311
$502$	0	0
$503$	−1.12950e6	−0.199052	−0.0995260	−	0.995035i	$-0.531733\pi$
$503$	−1.12950e6	−0.199052	−0.0995260	+	0.995035i	$0.531733\pi$
$504$	0	0
$505$	−1.44979e6	−0.252975
$506$	0	0
$507$	−664742.	−0.114851
$508$	0	0
$509$	−5.44872e6	−0.932181	−0.466091	−	0.884737i	$-0.654338\pi$
$509$	−5.44872e6	−0.932181	−0.466091	+	0.884737i	$0.654338\pi$
$510$	0	0
$511$	−6.51137e6	−1.10311
$512$	0	0
$513$	442319.	0.0742066
$514$	0	0
$515$	−2.27308e6	−0.377655
$516$	0	0
$517$	−545515.	−0.0897595
$518$	0	0
$519$	−4.19313e6	−0.683314
$520$	0	0
$521$	−5.87834e6	−0.948769	−0.474385	−	0.880318i	$-0.657329\pi$
$521$	−5.87834e6	−0.948769	−0.474385	+	0.880318i	$0.657329\pi$
$522$	0	0
$523$	−1.00787e7	−1.61120	−0.805601	−	0.592458i	$-0.798158\pi$
$523$	−1.00787e7	−1.61120	−0.805601	+	0.592458i	$0.798158\pi$
$524$	0	0
$525$	3.99972e6	0.633332
$526$	0	0
$527$	−3.42876e6	−0.537787
$528$	0	0
$529$	−5.21599e6	−0.810396
$530$	0	0
$531$	816892.	0.125727
$532$	0	0
$533$	−7.70575e6	−1.17489
$534$	0	0
$535$	572288.	0.0864431
$536$	0	0
$537$	−1.29429e6	−0.193684
$538$	0	0
$539$	242599.	0.0359680
$540$	0	0
$541$	1.07470e7	1.57869	0.789343	−	0.613953i	$-0.210421\pi$
$541$	1.07470e7	1.57869	0.789343	+	0.613953i	$0.210421\pi$
$542$	0	0
$543$	−2.53643e6	−0.369168
$544$	0	0
$545$	975565.	0.140690
$546$	0	0
$547$	1.90847e6	0.272719	0.136360	−	0.990659i	$-0.456460\pi$
$547$	1.90847e6	0.272719	0.136360	+	0.990659i	$0.456460\pi$
$548$	0	0
$549$	2.85233e6	0.403895
$550$	0	0
$551$	1.42939e6	0.200572
$552$	0	0
$553$	3.96982e6	0.552024
$554$	0	0
$555$	−360923.	−0.0497373
$556$	0	0
$557$	−9.16492e6	−1.25167	−0.625837	−	0.779954i	$-0.715242\pi$
$557$	−9.16492e6	−1.25167	−0.625837	+	0.779954i	$0.715242\pi$
$558$	0	0
$559$	−9.66099e6	−1.30765
$560$	0	0
$561$	−492128.	−0.0660193
$562$	0	0
$563$	7.86037e6	1.04513	0.522567	−	0.852598i	$-0.324974\pi$
$563$	7.86037e6	1.04513	0.522567	+	0.852598i	$0.324974\pi$
$564$	0	0
$565$	−115574.	−0.0152313
$566$	0	0
$567$	−972373.	−0.127021
$568$	0	0
$569$	3.16208e6	0.409442	0.204721	−	0.978820i	$-0.434371\pi$
$569$	3.16208e6	0.409442	0.204721	+	0.978820i	$0.434371\pi$
$570$	0	0
$571$	1.36016e7	1.74582	0.872910	−	0.487881i	$-0.162230\pi$
$571$	1.36016e7	1.74582	0.872910	+	0.487881i	$0.162230\pi$
$572$	0	0
$573$	2.87382e6	0.365656
$574$	0	0
$575$	−3.31259e6	−0.417829
$576$	0	0
$577$	−1.53482e6	−0.191919	−0.0959594	−	0.995385i	$-0.530592\pi$
$577$	−1.53482e6	−0.191919	−0.0959594	+	0.995385i	$0.530592\pi$
$578$	0	0
$579$	6.79725e6	0.842630
$580$	0	0
$581$	−1.55774e7	−1.91449
$582$	0	0
$583$	71257.3	0.00868277
$584$	0	0
$585$	−496574.	−0.0599922
$586$	0	0
$587$	−7.35171e6	−0.880629	−0.440315	−	0.897844i	$-0.645133\pi$
$587$	−7.35171e6	−0.880629	−0.440315	+	0.897844i	$0.645133\pi$
$588$	0	0
$589$	1.78954e6	0.212546
$590$	0	0
$591$	623727.	0.0734558
$592$	0	0
$593$	7.72521e6	0.902139	0.451069	−	0.892489i	$-0.351043\pi$
$593$	7.72521e6	0.902139	0.451069	+	0.892489i	$0.351043\pi$
$594$	0	0
$595$	−1.93675e6	−0.224275
$596$	0	0
$597$	149591.	0.0171779
$598$	0	0
$599$	−1.53129e7	−1.74378	−0.871890	−	0.489702i	$-0.837106\pi$
$599$	−1.53129e7	−1.74378	−0.871890	+	0.489702i	$0.837106\pi$
$600$	0	0
$601$	−3.69885e6	−0.417715	−0.208857	−	0.977946i	$-0.566974\pi$
$601$	−3.69885e6	−0.417715	−0.208857	+	0.977946i	$0.566974\pi$
$602$	0	0
$603$	2.32916e6	0.260859
$604$	0	0
$605$	1.78550e6	0.198323
$606$	0	0
$607$	−4.37471e6	−0.481923	−0.240962	−	0.970535i	$-0.577463\pi$
$607$	−4.37471e6	−0.481923	−0.240962	+	0.970535i	$0.577463\pi$
$608$	0	0
$609$	−3.14229e6	−0.343324
$610$	0	0
$611$	−6.32515e6	−0.685437
$612$	0	0
$613$	1.46082e7	1.57016	0.785082	−	0.619392i	$-0.212621\pi$
$613$	1.46082e7	1.57016	0.785082	+	0.619392i	$0.212621\pi$
$614$	0	0
$615$	1.42945e6	0.152398
$616$	0	0
$617$	−576151.	−0.0609289	−0.0304644	−	0.999536i	$-0.509699\pi$
$617$	−576151.	−0.0609289	−0.0304644	+	0.999536i	$0.509699\pi$
$618$	0	0
$619$	4.66461e6	0.489315	0.244657	−	0.969610i	$-0.421325\pi$
$619$	4.66461e6	0.489315	0.244657	+	0.969610i	$0.421325\pi$
$620$	0	0
$621$	805325.	0.0837996
$622$	0	0
$623$	−7.10305e6	−0.733204
$624$	0	0
$625$	8.59697e6	0.880329
$626$	0	0
$627$	256851.	0.0260923
$628$	0	0
$629$	−4.14736e6	−0.417970
$630$	0	0
$631$	−4.93025e6	−0.492942	−0.246471	−	0.969150i	$-0.579271\pi$
$631$	−4.93025e6	−0.492942	−0.246471	+	0.969150i	$0.579271\pi$
$632$	0	0
$633$	3.70929e6	0.367944
$634$	0	0
$635$	1.33006e6	0.130900
$636$	0	0
$637$	2.81289e6	0.274665
$638$	0	0
$639$	4.32888e6	0.419395
$640$	0	0
$641$	7.42308e6	0.713574	0.356787	−	0.934186i	$-0.383872\pi$
$641$	7.42308e6	0.713574	0.356787	+	0.934186i	$0.383872\pi$
$642$	0	0
$643$	−1.28737e6	−0.122794	−0.0613969	−	0.998113i	$-0.519556\pi$
$643$	−1.28737e6	−0.122794	−0.0613969	+	0.998113i	$0.519556\pi$
$644$	0	0
$645$	1.79215e6	0.169619
$646$	0	0
$647$	−5.25779e6	−0.493790	−0.246895	−	0.969042i	$-0.579410\pi$
$647$	−5.25779e6	−0.493790	−0.246895	+	0.969042i	$0.579410\pi$
$648$	0	0
$649$	474362.	0.0442077
$650$	0	0
$651$	−3.93403e6	−0.363819
$652$	0	0
$653$	−1.03822e7	−0.952811	−0.476405	−	0.879226i	$-0.658061\pi$
$653$	−1.03822e7	−0.952811	−0.476405	+	0.879226i	$0.658061\pi$
$654$	0	0
$655$	−3.69034e6	−0.336096
$656$	0	0
$657$	3.55873e6	0.321648
$658$	0	0
$659$	−5.31093e6	−0.476384	−0.238192	−	0.971218i	$-0.576555\pi$
$659$	−5.31093e6	−0.476384	−0.238192	+	0.971218i	$0.576555\pi$
$660$	0	0
$661$	5.12917e6	0.456608	0.228304	−	0.973590i	$-0.426682\pi$
$661$	5.12917e6	0.456608	0.228304	+	0.973590i	$0.426682\pi$
$662$	0	0
$663$	−5.70613e6	−0.504148
$664$	0	0
$665$	1.01083e6	0.0886384
$666$	0	0
$667$	2.60247e6	0.226501
$668$	0	0
$669$	1.07211e7	0.926132
$670$	0	0
$671$	1.65632e6	0.142017
$672$	0	0
$673$	7.93890e6	0.675652	0.337826	−	0.941209i	$-0.390309\pi$
$673$	7.93890e6	0.675652	0.337826	+	0.941209i	$0.390309\pi$
$674$	0	0
$675$	−2.18601e6	−0.184668
$676$	0	0
$677$	−381860.	−0.0320208	−0.0160104	−	0.999872i	$-0.505096\pi$
$677$	−381860.	−0.0320208	−0.0160104	+	0.999872i	$0.505096\pi$
$678$	0	0
$679$	−5.67189e6	−0.472121
$680$	0	0
$681$	−1.27017e7	−1.04953
$682$	0	0
$683$	−8.87069e6	−0.727621	−0.363811	−	0.931473i	$-0.618524\pi$
$683$	−8.87069e6	−0.727621	−0.363811	+	0.931473i	$0.618524\pi$
$684$	0	0
$685$	−125593.	−0.0102268
$686$	0	0
$687$	−4.91736e6	−0.397503
$688$	0	0
$689$	826216.	0.0663049
$690$	0	0
$691$	1.54329e6	0.122957	0.0614783	−	0.998108i	$-0.480419\pi$
$691$	1.54329e6	0.122957	0.0614783	+	0.998108i	$0.480419\pi$
$692$	0	0
$693$	−564649.	−0.0446627
$694$	0	0
$695$	1.23100e6	0.0966712
$696$	0	0
$697$	1.64258e7	1.28069
$698$	0	0
$699$	8.27734e6	0.640764
$700$	0	0
$701$	−7.51477e6	−0.577592	−0.288796	−	0.957391i	$-0.593255\pi$
$701$	−7.51477e6	−0.577592	−0.288796	+	0.957391i	$0.593255\pi$
$702$	0	0
$703$	2.16459e6	0.165191
$704$	0	0
$705$	1.17334e6	0.0889101
$706$	0	0
$707$	−1.91145e7	−1.43819
$708$	0	0
$709$	1.54954e7	1.15768	0.578839	−	0.815442i	$-0.303506\pi$
$709$	1.54954e7	1.15768	0.578839	+	0.815442i	$0.303506\pi$
$710$	0	0
$711$	−2.16967e6	−0.160960
$712$	0	0
$713$	3.25819e6	0.240023
$714$	0	0
$715$	−288356.	−0.0210943
$716$	0	0
$717$	1.24257e7	0.902657
$718$	0	0
$719$	1.10624e7	0.798045	0.399022	−	0.916941i	$-0.369350\pi$
$719$	1.10624e7	0.798045	0.399022	+	0.916941i	$0.369350\pi$
$720$	0	0
$721$	−2.99690e7	−2.14701
$722$	0	0
$723$	5.85337e6	0.416448
$724$	0	0
$725$	−7.06425e6	−0.499138
$726$	0	0
$727$	−1.63858e7	−1.14982	−0.574912	−	0.818216i	$-0.694964\pi$
$727$	−1.63858e7	−1.14982	−0.574912	+	0.818216i	$0.694964\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.05936e7	1.42541
$732$	0	0
$733$	4.09602e6	0.281580	0.140790	−	0.990039i	$-0.455036\pi$
$733$	4.09602e6	0.281580	0.140790	+	0.990039i	$0.455036\pi$
$734$	0	0
$735$	−521801.	−0.0356276
$736$	0	0
$737$	1.35252e6	0.0917224
$738$	0	0
$739$	−2.35232e7	−1.58447	−0.792237	−	0.610213i	$-0.791084\pi$
$739$	−2.35232e7	−1.58447	−0.792237	+	0.610213i	$0.791084\pi$
$740$	0	0
$741$	2.97814e6	0.199251
$742$	0	0
$743$	−8.62364e6	−0.573084	−0.286542	−	0.958068i	$-0.592506\pi$
$743$	−8.62364e6	−0.573084	−0.286542	+	0.958068i	$0.592506\pi$
$744$	0	0
$745$	3.48167e6	0.229825
$746$	0	0
$747$	8.51366e6	0.558232
$748$	0	0
$749$	7.54524e6	0.491438
$750$	0	0
$751$	1.62269e7	1.04987	0.524937	−	0.851141i	$-0.324089\pi$
$751$	1.62269e7	1.04987	0.524937	+	0.851141i	$0.324089\pi$
$752$	0	0
$753$	3.53943e6	0.227481
$754$	0	0
$755$	−5.19282e6	−0.331540
$756$	0	0
$757$	2.05486e7	1.30330	0.651648	−	0.758522i	$-0.274078\pi$
$757$	2.05486e7	1.30330	0.651648	+	0.758522i	$0.274078\pi$
$758$	0	0
$759$	467645.	0.0294654
$760$	0	0
$761$	−4.33531e6	−0.271368	−0.135684	−	0.990752i	$-0.543323\pi$
$761$	−4.33531e6	−0.271368	−0.135684	+	0.990752i	$0.543323\pi$
$762$	0	0
$763$	1.28622e7	0.799839
$764$	0	0
$765$	1.05851e6	0.0653945
$766$	0	0
$767$	5.50014e6	0.337587
$768$	0	0
$769$	−1.72697e7	−1.05310	−0.526548	−	0.850145i	$-0.676514\pi$
$769$	−1.72697e7	−1.05310	−0.526548	+	0.850145i	$0.676514\pi$
$770$	0	0
$771$	−1.11560e7	−0.675882
$772$	0	0
$773$	−2.64326e7	−1.59108	−0.795539	−	0.605903i	$-0.792812\pi$
$773$	−2.64326e7	−1.59108	−0.795539	+	0.605903i	$0.792812\pi$
$774$	0	0
$775$	−8.84416e6	−0.528935
$776$	0	0
$777$	−4.75852e6	−0.282761
$778$	0	0
$779$	−8.57292e6	−0.506157
$780$	0	0
$781$	2.51374e6	0.147467
$782$	0	0
$783$	1.71739e6	0.100107
$784$	0	0
$785$	−4.88810e6	−0.283117
$786$	0	0
$787$	2.17967e7	1.25445	0.627226	−	0.778837i	$-0.284190\pi$
$787$	2.17967e7	1.25445	0.627226	+	0.778837i	$0.284190\pi$
$788$	0	0
$789$	1.33156e7	0.761497
$790$	0	0
$791$	−1.52376e6	−0.0865916
$792$	0	0
$793$	1.92048e7	1.08449
$794$	0	0
$795$	−153266.	−0.00860060
$796$	0	0
$797$	−2.45761e7	−1.37046	−0.685231	−	0.728326i	$-0.740299\pi$
$797$	−2.45761e7	−1.37046	−0.685231	+	0.728326i	$0.740299\pi$
$798$	0	0
$799$	1.34828e7	0.747161
$800$	0	0
$801$	3.88210e6	0.213789
$802$	0	0
$803$	2.06652e6	0.113097
$804$	0	0
$805$	1.84040e6	0.100097
$806$	0	0
$807$	3.11112e6	0.168164
$808$	0	0
$809$	−2.10063e7	−1.12844	−0.564221	−	0.825624i	$-0.690823\pi$
$809$	−2.10063e7	−1.12844	−0.564221	+	0.825624i	$0.690823\pi$
$810$	0	0
$811$	−2.68688e7	−1.43448	−0.717242	−	0.696824i	$-0.754596\pi$
$811$	−2.68688e7	−1.43448	−0.717242	+	0.696824i	$0.754596\pi$
$812$	0	0
$813$	1.37707e7	0.730685
$814$	0	0
$815$	−1.34175e6	−0.0707583
$816$	0	0
$817$	−1.07482e7	−0.563353
$818$	0	0
$819$	−6.54700e6	−0.341061
$820$	0	0
$821$	−1.90654e7	−0.987161	−0.493581	−	0.869700i	$-0.664312\pi$
$821$	−1.90654e7	−0.987161	−0.493581	+	0.869700i	$0.664312\pi$
$822$	0	0
$823$	−3.64112e6	−0.187385	−0.0936926	−	0.995601i	$-0.529867\pi$
$823$	−3.64112e6	−0.187385	−0.0936926	+	0.995601i	$0.529867\pi$
$824$	0	0
$825$	−1.26940e6	−0.0649325
$826$	0	0
$827$	3.44261e7	1.75034	0.875172	−	0.483812i	$-0.160748\pi$
$827$	3.44261e7	1.75034	0.875172	+	0.483812i	$0.160748\pi$
$828$	0	0
$829$	−2.51490e7	−1.27097	−0.635485	−	0.772114i	$-0.719200\pi$
$829$	−2.51490e7	−1.27097	−0.635485	+	0.772114i	$0.719200\pi$
$830$	0	0
$831$	−4.17698e6	−0.209826
$832$	0	0
$833$	−5.99601e6	−0.299399
$834$	0	0
$835$	−5.17377e6	−0.256798
$836$	0	0
$837$	2.15011e6	0.106083
$838$	0	0
$839$	−1.37585e7	−0.674785	−0.337392	−	0.941364i	$-0.609545\pi$
$839$	−1.37585e7	−0.674785	−0.337392	+	0.941364i	$0.609545\pi$
$840$	0	0
$841$	−1.49613e7	−0.729422
$842$	0	0
$843$	−7.62581e6	−0.369587
$844$	0	0
$845$	830263.	0.0400013
$846$	0	0
$847$	2.35407e7	1.12748
$848$	0	0
$849$	1.87391e7	0.892235
$850$	0	0
$851$	3.94104e6	0.186546
$852$	0	0
$853$	2.71986e7	1.27989	0.639946	−	0.768419i	$-0.278957\pi$
$853$	2.71986e7	1.27989	0.639946	+	0.768419i	$0.278957\pi$
$854$	0	0
$855$	−552457.	−0.0258454
$856$	0	0
$857$	−1.90034e7	−0.883851	−0.441925	−	0.897052i	$-0.645704\pi$
$857$	−1.90034e7	−0.883851	−0.441925	+	0.897052i	$0.645704\pi$
$858$	0	0
$859$	4.23596e7	1.95871	0.979353	−	0.202157i	$-0.0647950\pi$
$859$	4.23596e7	1.95871	0.979353	+	0.202157i	$0.0647950\pi$
$860$	0	0
$861$	1.88463e7	0.866399
$862$	0	0
$863$	1.43661e6	0.0656616	0.0328308	−	0.999461i	$-0.489548\pi$
$863$	1.43661e6	0.0656616	0.0328308	+	0.999461i	$0.489548\pi$
$864$	0	0
$865$	5.23722e6	0.237991
$866$	0	0
$867$	−615389.	−0.0278036
$868$	0	0
$869$	−1.25991e6	−0.0565964
$870$	0	0
$871$	1.56822e7	0.700427
$872$	0	0
$873$	3.09992e6	0.137662
$874$	0	0
$875$	−1.02018e7	−0.450461
$876$	0	0
$877$	−1.32043e7	−0.579718	−0.289859	−	0.957069i	$-0.593608\pi$
$877$	−1.32043e7	−0.579718	−0.289859	+	0.957069i	$0.593608\pi$
$878$	0	0
$879$	1.10832e7	0.483829
$880$	0	0
$881$	9.00664e6	0.390952	0.195476	−	0.980709i	$-0.437375\pi$
$881$	9.00664e6	0.390952	0.195476	+	0.980709i	$0.437375\pi$
$882$	0	0
$883$	−2.18359e7	−0.942473	−0.471237	−	0.882007i	$-0.656192\pi$
$883$	−2.18359e7	−0.942473	−0.471237	+	0.882007i	$0.656192\pi$
$884$	0	0
$885$	−1.02030e6	−0.0437894
$886$	0	0
$887$	4.58096e7	1.95500	0.977502	−	0.210926i	$-0.0676479\pi$
$887$	4.58096e7	1.95500	0.977502	+	0.210926i	$0.0676479\pi$
$888$	0	0
$889$	1.75360e7	0.744176
$890$	0	0
$891$	308603.	0.0130229
$892$	0	0
$893$	−7.03695e6	−0.295295
$894$	0	0
$895$	1.61656e6	0.0674582
$896$	0	0
$897$	5.42226e6	0.225009
$898$	0	0
$899$	6.94823e6	0.286731
$900$	0	0
$901$	−1.76118e6	−0.0722757
$902$	0	0
$903$	2.36283e7	0.964302
$904$	0	0
$905$	3.16801e6	0.128577
$906$	0	0
$907$	−3.25677e7	−1.31452	−0.657262	−	0.753662i	$-0.728286\pi$
$907$	−3.25677e7	−1.31452	−0.657262	+	0.753662i	$0.728286\pi$
$908$	0	0
$909$	1.04469e7	0.419350
$910$	0	0
$911$	4.24433e7	1.69439	0.847194	−	0.531283i	$-0.178290\pi$
$911$	4.24433e7	1.69439	0.847194	+	0.531283i	$0.178290\pi$
$912$	0	0
$913$	4.94381e6	0.196284
$914$	0	0
$915$	−3.56256e6	−0.140673
$916$	0	0
$917$	−4.86546e7	−1.91074
$918$	0	0
$919$	3.77474e7	1.47434	0.737172	−	0.675705i	$-0.236161\pi$
$919$	3.77474e7	1.47434	0.737172	+	0.675705i	$0.236161\pi$
$920$	0	0
$921$	−1.53611e7	−0.596724
$922$	0	0
$923$	2.91464e7	1.12611
$924$	0	0
$925$	−1.06977e7	−0.411090
$926$	0	0
$927$	1.63792e7	0.626029
$928$	0	0
$929$	3.41047e7	1.29651	0.648254	−	0.761424i	$-0.275499\pi$
$929$	3.41047e7	1.29651	0.648254	+	0.761424i	$0.275499\pi$
$930$	0	0
$931$	3.12944e6	0.118329
$932$	0	0
$933$	−2.67854e7	−1.00738
$934$	0	0
$935$	614667.	0.0229938
$936$	0	0
$937$	2.26592e7	0.843130	0.421565	−	0.906798i	$-0.361481\pi$
$937$	2.26592e7	0.843130	0.421565	+	0.906798i	$0.361481\pi$
$938$	0	0
$939$	−3.07791e7	−1.13918
$940$	0	0
$941$	1.44909e7	0.533484	0.266742	−	0.963768i	$-0.414053\pi$
$941$	1.44909e7	0.533484	0.266742	+	0.963768i	$0.414053\pi$
$942$	0	0
$943$	−1.56086e7	−0.571590
$944$	0	0
$945$	1.21449e6	0.0442401
$946$	0	0
$947$	3.43018e7	1.24292	0.621459	−	0.783447i	$-0.286540\pi$
$947$	3.43018e7	1.24292	0.621459	+	0.783447i	$0.286540\pi$
$948$	0	0
$949$	2.39609e7	0.863651
$950$	0	0
$951$	−2.97655e7	−1.06724
$952$	0	0
$953$	3.23134e7	1.15252	0.576262	−	0.817265i	$-0.304511\pi$
$953$	3.23134e7	1.15252	0.576262	+	0.817265i	$0.304511\pi$
$954$	0	0
$955$	−3.58940e6	−0.127354
$956$	0	0
$957$	997274.	0.0351994
$958$	0	0
$959$	−1.65586e6	−0.0581401
$960$	0	0
$961$	−1.99302e7	−0.696152
$962$	0	0
$963$	−4.12378e6	−0.143294
$964$	0	0
$965$	−8.48976e6	−0.293479
$966$	0	0
$967$	2.73182e7	0.939477	0.469738	−	0.882806i	$-0.344348\pi$
$967$	2.73182e7	0.939477	0.469738	+	0.882806i	$0.344348\pi$
$968$	0	0
$969$	−6.34827e6	−0.217193
$970$	0	0
$971$	−2.99922e7	−1.02085	−0.510423	−	0.859923i	$-0.670511\pi$
$971$	−2.99922e7	−1.02085	−0.510423	+	0.859923i	$0.670511\pi$
$972$	0	0
$973$	1.62300e7	0.549585
$974$	0	0
$975$	−1.47184e7	−0.495849
$976$	0	0
$977$	−2.33987e7	−0.784251	−0.392126	−	0.919912i	$-0.628260\pi$
$977$	−2.33987e7	−0.784251	−0.392126	+	0.919912i	$0.628260\pi$
$978$	0	0
$979$	2.25430e6	0.0751720
$980$	0	0
$981$	−7.02969e6	−0.233219
$982$	0	0
$983$	1.38086e7	0.455792	0.227896	−	0.973686i	$-0.426815\pi$
$983$	1.38086e7	0.455792	0.227896	+	0.973686i	$0.426815\pi$
$984$	0	0
$985$	−779035.	−0.0255839
$986$	0	0
$987$	1.54697e7	0.505462
$988$	0	0
$989$	−1.95691e7	−0.636180
$990$	0	0
$991$	3.17280e6	0.102626	0.0513131	−	0.998683i	$-0.483659\pi$
$991$	3.17280e6	0.102626	0.0513131	+	0.998683i	$0.483659\pi$
$992$	0	0
$993$	7.72256e6	0.248535
$994$	0	0
$995$	−186839.	−0.00598288
$996$	0	0
$997$	4.02543e6	0.128255	0.0641276	−	0.997942i	$-0.479574\pi$
$997$	4.02543e6	0.128255	0.0641276	+	0.997942i	$0.479574\pi$
$998$	0	0
$999$	2.60072e6	0.0824481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.6.a.v.1.1		2
4.3	odd	2		768.6.a.q.1.1		2
8.3	odd	2		768.6.a.r.1.2		2
8.5	even	2		768.6.a.m.1.2		2
16.3	odd	4		384.6.d.h.193.2	yes	4
16.5	even	4		384.6.d.g.193.1	✓	4
16.11	odd	4		384.6.d.h.193.3	yes	4
16.13	even	4		384.6.d.g.193.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.6.d.g.193.1	✓	4	16.5	even	4
384.6.d.g.193.4	yes	4	16.13	even	4
384.6.d.h.193.2	yes	4	16.3	odd	4
384.6.d.h.193.3	yes	4	16.11	odd	4
768.6.a.m.1.2		2	8.5	even	2
768.6.a.q.1.1		2	4.3	odd	2
768.6.a.r.1.2		2	8.3	odd	2
768.6.a.v.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -11.2410 q^{5} -148.205 q^{7} +81.0000 q^{9} +47.0360 q^{11} +545.374 q^{13} -101.169 q^{15} -1162.53 q^{17} +606.748 q^{19} -1333.84 q^{21} +1104.70 q^{23} -2998.64 q^{25} +729.000 q^{27} +2355.82 q^{29} +2949.39 q^{31} +423.324 q^{33} +1665.97 q^{35} +3567.52 q^{37} +4908.37 q^{39} -14129.3 q^{41} -17714.4 q^{43} -910.521 q^{45} -11597.8 q^{47} +5157.72 q^{49} -10462.8 q^{51} +1514.95 q^{53} -528.732 q^{55} +5460.73 q^{57} +10085.1 q^{59} +35214.0 q^{61} -12004.6 q^{63} -6130.55 q^{65} +28755.0 q^{67} +9942.28 q^{69} +53443.0 q^{71} +43934.9 q^{73} -26987.8 q^{75} -6970.97 q^{77} -26786.0 q^{79} +6561.00 q^{81} +105107. q^{83} +13068.0 q^{85} +21202.4 q^{87} +47927.2 q^{89} -80827.1 q^{91} +26544.5 q^{93} -6820.45 q^{95} +38270.6 q^{97} +3809.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 40 q^{5} + 16 q^{7} + 162 q^{9} + 344 q^{11} + 216 q^{13} + 360 q^{15} + 924 q^{17} - 536 q^{19} + 144 q^{21} + 3584 q^{23} - 3498 q^{25} + 1458 q^{27} + 8648 q^{29} - 3536 q^{31} + 3096 q^{33}+ \cdots + 27864 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 40 * q^5 + 16 * q^7 + 162 * q^9 + 344 * q^11 + 216 * q^13 + 360 * q^15 + 924 * q^17 - 536 * q^19 + 144 * q^21 + 3584 * q^23 - 3498 * q^25 + 1458 * q^27 + 8648 * q^29 - 3536 * q^31 + 3096 * q^33 + 10080 * q^35 - 3112 * q^37 + 1944 * q^39 - 2516 * q^41 - 3688 * q^43 + 3240 * q^45 - 4576 * q^47 + 15314 * q^49 + 8316 * q^51 + 968 * q^53 + 14688 * q^55 - 4824 * q^57 - 27816 * q^59 + 28440 * q^61 + 1296 * q^63 - 23008 * q^65 + 78504 * q^67 + 32256 * q^69 + 75520 * q^71 - 17100 * q^73 - 31482 * q^75 + 41792 * q^77 - 100496 * q^79 + 13122 * q^81 + 70504 * q^83 + 119984 * q^85 + 77832 * q^87 + 90356 * q^89 - 134912 * q^91 - 31824 * q^93 - 65376 * q^95 - 99908 * q^97 + 27864 * q^99

Introduction

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