LMFDB - Embedded newform 768.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, 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("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.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:	$45.3134668844$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1436.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 12 $ x^3 - 11*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.76644$ 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)

$f(q)$	$=$	$q-3.00000 q^{3} -0.612661 q^{5} -22.7441 q^{7} +9.00000 q^{9} +60.2630 q^{11} -52.9062 q^{13} +1.83798 q^{15} +47.1643 q^{17} +29.1643 q^{19} +68.2324 q^{21} +109.488 q^{23} -124.625 q^{25} -27.0000 q^{27} +10.4250 q^{29} +220.881 q^{31} -180.789 q^{33} +13.9345 q^{35} -408.348 q^{37} +158.718 q^{39} +360.742 q^{41} -236.414 q^{43} -5.51395 q^{45} -129.113 q^{47} +174.296 q^{49} -141.493 q^{51} -117.819 q^{53} -36.9208 q^{55} -87.4928 q^{57} -262.854 q^{59} -273.465 q^{61} -204.697 q^{63} +32.4135 q^{65} +89.4077 q^{67} -328.465 q^{69} -350.521 q^{71} -532.610 q^{73} +373.874 q^{75} -1370.63 q^{77} +166.561 q^{79} +81.0000 q^{81} -361.934 q^{83} -28.8957 q^{85} -31.2750 q^{87} -40.3285 q^{89} +1203.31 q^{91} -662.642 q^{93} -17.8678 q^{95} -614.921 q^{97} +542.367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9} - 52 q^{13} + 30 q^{15} + 26 q^{17} - 28 q^{19} - 42 q^{21} + 164 q^{23} + 53 q^{25} - 81 q^{27} - 174 q^{29} + 318 q^{31} + 92 q^{35} - 296 q^{37} + 156 q^{39}+ \cdots - 1222 q^{97}+O(q^{100}) $ 3 * q - 9 * q^3 - 10 * q^5 + 14 * q^7 + 27 * q^9 - 52 * q^13 + 30 * q^15 + 26 * q^17 - 28 * q^19 - 42 * q^21 + 164 * q^23 + 53 * q^25 - 81 * q^27 - 174 * q^29 + 318 * q^31 + 92 * q^35 - 296 * q^37 + 156 * q^39 - 118 * q^41 + 260 * q^43 - 90 * q^45 + 204 * q^47 + 327 * q^49 - 78 * q^51 - 1086 * q^53 + 512 * q^55 + 84 * q^57 - 196 * q^59 - 1536 * q^61 + 126 * q^63 - 872 * q^65 + 660 * q^67 - 492 * q^69 - 852 * q^71 - 478 * q^73 - 159 * q^75 - 2304 * q^77 + 22 * q^79 + 243 * q^81 - 1136 * q^83 - 2732 * q^85 + 522 * q^87 + 110 * q^89 + 632 * q^91 - 954 * q^93 - 2552 * q^95 - 1222 * q^97

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−0.612661	−0.0547981	−0.0273990	−	0.999625i	$-0.508722\pi$
$5$	−0.612661	−0.0547981	−0.0273990	+	0.999625i	$0.508722\pi$
$6$	0	0
$7$	−22.7441	−1.22807	−0.614034	−	0.789279i	$-0.710454\pi$
$7$	−22.7441	−1.22807	−0.614034	+	0.789279i	$0.710454\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	60.2630	1.65182	0.825908	−	0.563805i	$-0.190663\pi$
$11$	60.2630	1.65182	0.825908	+	0.563805i	$0.190663\pi$
$12$	0	0
$13$	−52.9062	−1.12873	−0.564367	−	0.825524i	$-0.690880\pi$
$13$	−52.9062	−1.12873	−0.564367	+	0.825524i	$0.690880\pi$
$14$	0	0
$15$	1.83798	0.0316377
$16$	0	0
$17$	47.1643	0.672883	0.336442	−	0.941704i	$-0.390777\pi$
$17$	47.1643	0.672883	0.336442	+	0.941704i	$0.390777\pi$
$18$	0	0
$19$	29.1643	0.352144	0.176072	−	0.984377i	$-0.443661\pi$
$19$	29.1643	0.352144	0.176072	+	0.984377i	$0.443661\pi$
$20$	0	0
$21$	68.2324	0.709026
$22$	0	0
$23$	109.488	0.992604	0.496302	−	0.868150i	$-0.334691\pi$
$23$	109.488	0.992604	0.496302	+	0.868150i	$0.334691\pi$
$24$	0	0
$25$	−124.625	−0.996997
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	10.4250	0.0667542	0.0333771	−	0.999443i	$-0.489374\pi$
$29$	10.4250	0.0667542	0.0333771	+	0.999443i	$0.489374\pi$
$30$	0	0
$31$	220.881	1.27972	0.639860	−	0.768492i	$-0.278992\pi$
$31$	220.881	1.27972	0.639860	+	0.768492i	$0.278992\pi$
$32$	0	0
$33$	−180.789	−0.953676
$34$	0	0
$35$	13.9345	0.0672958
$36$	0	0
$37$	−408.348	−1.81438	−0.907188	−	0.420725i	$-0.861776\pi$
$37$	−408.348	−1.81438	−0.907188	+	0.420725i	$0.861776\pi$
$38$	0	0
$39$	158.718	0.651674
$40$	0	0
$41$	360.742	1.37411	0.687054	−	0.726606i	$-0.258903\pi$
$41$	360.742	1.37411	0.687054	+	0.726606i	$0.258903\pi$
$42$	0	0
$43$	−236.414	−0.838436	−0.419218	−	0.907886i	$-0.637696\pi$
$43$	−236.414	−0.838436	−0.419218	+	0.907886i	$0.637696\pi$
$44$	0	0
$45$	−5.51395	−0.0182660
$46$	0	0
$47$	−129.113	−0.400703	−0.200352	−	0.979724i	$-0.564208\pi$
$47$	−129.113	−0.400703	−0.200352	+	0.979724i	$0.564208\pi$
$48$	0	0
$49$	174.296	0.508152
$50$	0	0
$51$	−141.493	−0.388489
$52$	0	0
$53$	−117.819	−0.305353	−0.152677	−	0.988276i	$-0.548789\pi$
$53$	−117.819	−0.305353	−0.152677	+	0.988276i	$0.548789\pi$
$54$	0	0
$55$	−36.9208	−0.0905163
$56$	0	0
$57$	−87.4928	−0.203311
$58$	0	0
$59$	−262.854	−0.580012	−0.290006	−	0.957025i	$-0.593657\pi$
$59$	−262.854	−0.580012	−0.290006	+	0.957025i	$0.593657\pi$
$60$	0	0
$61$	−273.465	−0.573993	−0.286996	−	0.957932i	$-0.592657\pi$
$61$	−273.465	−0.573993	−0.286996	+	0.957932i	$0.592657\pi$
$62$	0	0
$63$	−204.697	−0.409356
$64$	0	0
$65$	32.4135	0.0618524
$66$	0	0
$67$	89.4077	0.163028	0.0815141	−	0.996672i	$-0.474024\pi$
$67$	89.4077	0.163028	0.0815141	+	0.996672i	$0.474024\pi$
$68$	0	0
$69$	−328.465	−0.573080
$70$	0	0
$71$	−350.521	−0.585904	−0.292952	−	0.956127i	$-0.594638\pi$
$71$	−350.521	−0.585904	−0.292952	+	0.956127i	$0.594638\pi$
$72$	0	0
$73$	−532.610	−0.853936	−0.426968	−	0.904267i	$-0.640418\pi$
$73$	−532.610	−0.853936	−0.426968	+	0.904267i	$0.640418\pi$
$74$	0	0
$75$	373.874	0.575617
$76$	0	0
$77$	−1370.63	−2.02854
$78$	0	0
$79$	166.561	0.237210	0.118605	−	0.992942i	$-0.462158\pi$
$79$	166.561	0.237210	0.118605	+	0.992942i	$0.462158\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−361.934	−0.478644	−0.239322	−	0.970940i	$-0.576925\pi$
$83$	−361.934	−0.478644	−0.239322	+	0.970940i	$0.576925\pi$
$84$	0	0
$85$	−28.8957	−0.0368727
$86$	0	0
$87$	−31.2750	−0.0385405
$88$	0	0
$89$	−40.3285	−0.0480316	−0.0240158	−	0.999712i	$-0.507645\pi$
$89$	−40.3285	−0.0480316	−0.0240158	+	0.999712i	$0.507645\pi$
$90$	0	0
$91$	1203.31	1.38616
$92$	0	0
$93$	−662.642	−0.738846
$94$	0	0
$95$	−17.8678	−0.0192968
$96$	0	0
$97$	−614.921	−0.643667	−0.321834	−	0.946796i	$-0.604299\pi$
$97$	−614.921	−0.643667	−0.321834	+	0.946796i	$0.604299\pi$
$98$	0	0
$99$	542.367	0.550605
$100$	0	0
$101$	−1664.99	−1.64033	−0.820163	−	0.572130i	$-0.806117\pi$
$101$	−1664.99	−1.64033	−0.820163	+	0.572130i	$0.806117\pi$
$102$	0	0
$103$	396.858	0.379647	0.189823	−	0.981818i	$-0.439208\pi$
$103$	396.858	0.379647	0.189823	+	0.981818i	$0.439208\pi$
$104$	0	0
$105$	−41.8034	−0.0388532
$106$	0	0
$107$	350.630	0.316791	0.158396	−	0.987376i	$-0.449368\pi$
$107$	350.630	0.316791	0.158396	+	0.987376i	$0.449368\pi$
$108$	0	0
$109$	−597.009	−0.524615	−0.262308	−	0.964984i	$-0.584484\pi$
$109$	−597.009	−0.524615	−0.262308	+	0.964984i	$0.584484\pi$
$110$	0	0
$111$	1225.04	1.04753
$112$	0	0
$113$	496.422	0.413270	0.206635	−	0.978418i	$-0.433749\pi$
$113$	496.422	0.413270	0.206635	+	0.978418i	$0.433749\pi$
$114$	0	0
$115$	−67.0792	−0.0543928
$116$	0	0
$117$	−476.155	−0.376244
$118$	0	0
$119$	−1072.71	−0.826346
$120$	0	0
$121$	2300.63	1.72849
$122$	0	0
$123$	−1082.23	−0.793342
$124$	0	0
$125$	152.935	0.109432
$126$	0	0
$127$	−1799.85	−1.25756	−0.628782	−	0.777581i	$-0.716446\pi$
$127$	−1799.85	−1.25756	−0.628782	+	0.777581i	$0.716446\pi$
$128$	0	0
$129$	709.241	0.484071
$130$	0	0
$131$	1121.45	0.747949	0.373974	−	0.927439i	$-0.377995\pi$
$131$	1121.45	0.747949	0.373974	+	0.927439i	$0.377995\pi$
$132$	0	0
$133$	−663.316	−0.432457
$134$	0	0
$135$	16.5419	0.0105459
$136$	0	0
$137$	−2449.55	−1.52759	−0.763793	−	0.645461i	$-0.776665\pi$
$137$	−2449.55	−1.52759	−0.763793	+	0.645461i	$0.776665\pi$
$138$	0	0
$139$	−2457.56	−1.49962	−0.749811	−	0.661652i	$-0.769856\pi$
$139$	−2457.56	−1.49962	−0.749811	+	0.661652i	$0.769856\pi$
$140$	0	0
$141$	387.339	0.231346
$142$	0	0
$143$	−3188.28	−1.86446
$144$	0	0
$145$	−6.38698	−0.00365800
$146$	0	0
$147$	−522.888	−0.293382
$148$	0	0
$149$	−2084.96	−1.14635	−0.573177	−	0.819432i	$-0.694289\pi$
$149$	−2084.96	−1.14635	−0.573177	+	0.819432i	$0.694289\pi$
$150$	0	0
$151$	1057.80	0.570084	0.285042	−	0.958515i	$-0.407992\pi$
$151$	1057.80	0.570084	0.285042	+	0.958515i	$0.407992\pi$
$152$	0	0
$153$	424.478	0.224294
$154$	0	0
$155$	−135.325	−0.0701262
$156$	0	0
$157$	−3193.01	−1.62312	−0.811559	−	0.584270i	$-0.801381\pi$
$157$	−3193.01	−1.62312	−0.811559	+	0.584270i	$0.801381\pi$
$158$	0	0
$159$	353.458	0.176296
$160$	0	0
$161$	−2490.22	−1.21899
$162$	0	0
$163$	−846.854	−0.406937	−0.203469	−	0.979081i	$-0.565221\pi$
$163$	−846.854	−0.406937	−0.203469	+	0.979081i	$0.565221\pi$
$164$	0	0
$165$	110.762	0.0522596
$166$	0	0
$167$	−2630.15	−1.21873	−0.609363	−	0.792892i	$-0.708575\pi$
$167$	−2630.15	−1.21873	−0.609363	+	0.792892i	$0.708575\pi$
$168$	0	0
$169$	602.062	0.274038
$170$	0	0
$171$	262.478	0.117381
$172$	0	0
$173$	−429.843	−0.188904	−0.0944519	−	0.995529i	$-0.530110\pi$
$173$	−429.843	−0.188904	−0.0944519	+	0.995529i	$0.530110\pi$
$174$	0	0
$175$	2834.48	1.22438
$176$	0	0
$177$	788.563	0.334870
$178$	0	0
$179$	−1516.30	−0.633149	−0.316574	−	0.948568i	$-0.602533\pi$
$179$	−1516.30	−0.633149	−0.316574	+	0.948568i	$0.602533\pi$
$180$	0	0
$181$	−3380.20	−1.38811	−0.694056	−	0.719921i	$-0.744178\pi$
$181$	−3380.20	−1.38811	−0.694056	+	0.719921i	$0.744178\pi$
$182$	0	0
$183$	820.394	0.331395
$184$	0	0
$185$	250.179	0.0994244
$186$	0	0
$187$	2842.26	1.11148
$188$	0	0
$189$	614.092	0.236342
$190$	0	0
$191$	2799.71	1.06063	0.530314	−	0.847801i	$-0.322074\pi$
$191$	2799.71	1.06063	0.530314	+	0.847801i	$0.322074\pi$
$192$	0	0
$193$	624.106	0.232768	0.116384	−	0.993204i	$-0.462870\pi$
$193$	624.106	0.232768	0.116384	+	0.993204i	$0.462870\pi$
$194$	0	0
$195$	−97.2406	−0.0357105
$196$	0	0
$197$	4779.25	1.72846	0.864232	−	0.503094i	$-0.167805\pi$
$197$	4779.25	1.72846	0.864232	+	0.503094i	$0.167805\pi$
$198$	0	0
$199$	2615.92	0.931846	0.465923	−	0.884825i	$-0.345722\pi$
$199$	2615.92	0.931846	0.465923	+	0.884825i	$0.345722\pi$
$200$	0	0
$201$	−268.223	−0.0941244
$202$	0	0
$203$	−237.107	−0.0819787
$204$	0	0
$205$	−221.013	−0.0752985
$206$	0	0
$207$	985.395	0.330868
$208$	0	0
$209$	1757.52	0.581677
$210$	0	0
$211$	−1745.78	−0.569595	−0.284798	−	0.958588i	$-0.591926\pi$
$211$	−1745.78	−0.569595	−0.284798	+	0.958588i	$0.591926\pi$
$212$	0	0
$213$	1051.56	0.338272
$214$	0	0
$215$	144.841	0.0459447
$216$	0	0
$217$	−5023.74	−1.57158
$218$	0	0
$219$	1597.83	0.493020
$220$	0	0
$221$	−2495.28	−0.759505
$222$	0	0
$223$	3385.60	1.01667	0.508333	−	0.861161i	$-0.330262\pi$
$223$	3385.60	1.01667	0.508333	+	0.861161i	$0.330262\pi$
$224$	0	0
$225$	−1121.62	−0.332332
$226$	0	0
$227$	3847.72	1.12503	0.562515	−	0.826787i	$-0.309834\pi$
$227$	3847.72	1.12503	0.562515	+	0.826787i	$0.309834\pi$
$228$	0	0
$229$	1335.15	0.385279	0.192640	−	0.981270i	$-0.438295\pi$
$229$	1335.15	0.385279	0.192640	+	0.981270i	$0.438295\pi$
$230$	0	0
$231$	4111.89	1.17118
$232$	0	0
$233$	5146.38	1.44700	0.723499	−	0.690325i	$-0.242532\pi$
$233$	5146.38	1.44700	0.723499	+	0.690325i	$0.242532\pi$
$234$	0	0
$235$	79.1025	0.0219578
$236$	0	0
$237$	−499.683	−0.136953
$238$	0	0
$239$	−7085.07	−1.91755	−0.958777	−	0.284160i	$-0.908285\pi$
$239$	−7085.07	−1.91755	−0.958777	+	0.284160i	$0.908285\pi$
$240$	0	0
$241$	2538.40	0.678476	0.339238	−	0.940701i	$-0.389831\pi$
$241$	2538.40	0.678476	0.339238	+	0.940701i	$0.389831\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−106.784	−0.0278458
$246$	0	0
$247$	−1542.97	−0.397477
$248$	0	0
$249$	1085.80	0.276345
$250$	0	0
$251$	1696.99	0.426746	0.213373	−	0.976971i	$-0.431555\pi$
$251$	1696.99	0.426746	0.213373	+	0.976971i	$0.431555\pi$
$252$	0	0
$253$	6598.09	1.63960
$254$	0	0
$255$	86.6871	0.0212885
$256$	0	0
$257$	−382.902	−0.0929369	−0.0464685	−	0.998920i	$-0.514797\pi$
$257$	−382.902	−0.0929369	−0.0464685	+	0.998920i	$0.514797\pi$
$258$	0	0
$259$	9287.52	2.22818
$260$	0	0
$261$	93.8249	0.0222514
$262$	0	0
$263$	−5002.02	−1.17277	−0.586383	−	0.810034i	$-0.699449\pi$
$263$	−5002.02	−1.17277	−0.586383	+	0.810034i	$0.699449\pi$
$264$	0	0
$265$	72.1833	0.0167328
$266$	0	0
$267$	120.986	0.0277311
$268$	0	0
$269$	6117.47	1.38658	0.693288	−	0.720661i	$-0.256161\pi$
$269$	6117.47	1.38658	0.693288	+	0.720661i	$0.256161\pi$
$270$	0	0
$271$	3956.12	0.886780	0.443390	−	0.896329i	$-0.353776\pi$
$271$	3956.12	0.886780	0.443390	+	0.896329i	$0.353776\pi$
$272$	0	0
$273$	−3609.92	−0.800301
$274$	0	0
$275$	−7510.25	−1.64686
$276$	0	0
$277$	4842.17	1.05032	0.525158	−	0.851004i	$-0.324006\pi$
$277$	4842.17	1.05032	0.525158	+	0.851004i	$0.324006\pi$
$278$	0	0
$279$	1987.92	0.426573
$280$	0	0
$281$	−1878.68	−0.398835	−0.199417	−	0.979915i	$-0.563905\pi$
$281$	−1878.68	−0.398835	−0.199417	+	0.979915i	$0.563905\pi$
$282$	0	0
$283$	−5724.87	−1.20250	−0.601251	−	0.799060i	$-0.705331\pi$
$283$	−5724.87	−1.20250	−0.601251	+	0.799060i	$0.705331\pi$
$284$	0	0
$285$	53.6034	0.0111410
$286$	0	0
$287$	−8204.77	−1.68750
$288$	0	0
$289$	−2688.53	−0.547228
$290$	0	0
$291$	1844.76	0.371622
$292$	0	0
$293$	−5088.75	−1.01464	−0.507318	−	0.861759i	$-0.669363\pi$
$293$	−5088.75	−1.01464	−0.507318	+	0.861759i	$0.669363\pi$
$294$	0	0
$295$	161.041	0.0317836
$296$	0	0
$297$	−1627.10	−0.317892
$298$	0	0
$299$	−5792.61	−1.12038
$300$	0	0
$301$	5377.02	1.02966
$302$	0	0
$303$	4994.98	0.947043
$304$	0	0
$305$	167.541	0.0314537
$306$	0	0
$307$	−7219.21	−1.34209	−0.671046	−	0.741416i	$-0.734155\pi$
$307$	−7219.21	−1.34209	−0.671046	+	0.741416i	$0.734155\pi$
$308$	0	0
$309$	−1190.58	−0.219189
$310$	0	0
$311$	−1537.06	−0.280252	−0.140126	−	0.990134i	$-0.544751\pi$
$311$	−1537.06	−0.280252	−0.140126	+	0.990134i	$0.544751\pi$
$312$	0	0
$313$	2200.93	0.397456	0.198728	−	0.980055i	$-0.436319\pi$
$313$	2200.93	0.397456	0.198728	+	0.980055i	$0.436319\pi$
$314$	0	0
$315$	125.410	0.0224319
$316$	0	0
$317$	2840.41	0.503260	0.251630	−	0.967824i	$-0.419033\pi$
$317$	2840.41	0.503260	0.251630	+	0.967824i	$0.419033\pi$
$318$	0	0
$319$	628.240	0.110266
$320$	0	0
$321$	−1051.89	−0.182899
$322$	0	0
$323$	1375.51	0.236952
$324$	0	0
$325$	6593.41	1.12534
$326$	0	0
$327$	1791.03	0.302887
$328$	0	0
$329$	2936.56	0.492091
$330$	0	0
$331$	2118.52	0.351795	0.175898	−	0.984408i	$-0.443717\pi$
$331$	2118.52	0.351795	0.175898	+	0.984408i	$0.443717\pi$
$332$	0	0
$333$	−3675.13	−0.604792
$334$	0	0
$335$	−54.7766	−0.00893364
$336$	0	0
$337$	−659.599	−0.106619	−0.0533096	−	0.998578i	$-0.516977\pi$
$337$	−659.599	−0.106619	−0.0533096	+	0.998578i	$0.516977\pi$
$338$	0	0
$339$	−1489.27	−0.238601
$340$	0	0
$341$	13310.9	2.11386
$342$	0	0
$343$	3837.03	0.604023
$344$	0	0
$345$	201.238	0.0314037
$346$	0	0
$347$	8377.42	1.29603	0.648017	−	0.761626i	$-0.275599\pi$
$347$	8377.42	1.29603	0.648017	+	0.761626i	$0.275599\pi$
$348$	0	0
$349$	3254.18	0.499119	0.249559	−	0.968360i	$-0.419714\pi$
$349$	3254.18	0.499119	0.249559	+	0.968360i	$0.419714\pi$
$350$	0	0
$351$	1428.47	0.217225
$352$	0	0
$353$	11117.5	1.67627	0.838137	−	0.545459i	$-0.183645\pi$
$353$	11117.5	1.67627	0.838137	+	0.545459i	$0.183645\pi$
$354$	0	0
$355$	214.750	0.0321064
$356$	0	0
$357$	3218.13	0.477091
$358$	0	0
$359$	−4756.56	−0.699281	−0.349640	−	0.936884i	$-0.613696\pi$
$359$	−4756.56	−0.699281	−0.349640	+	0.936884i	$0.613696\pi$
$360$	0	0
$361$	−6008.45	−0.875994
$362$	0	0
$363$	−6901.88	−0.997946
$364$	0	0
$365$	326.310	0.0467940
$366$	0	0
$367$	−1837.40	−0.261339	−0.130670	−	0.991426i	$-0.541713\pi$
$367$	−1837.40	−0.261339	−0.130670	+	0.991426i	$0.541713\pi$
$368$	0	0
$369$	3246.68	0.458036
$370$	0	0
$371$	2679.70	0.374995
$372$	0	0
$373$	5598.07	0.777097	0.388549	−	0.921428i	$-0.372977\pi$
$373$	5598.07	0.777097	0.388549	+	0.921428i	$0.372977\pi$
$374$	0	0
$375$	−458.806	−0.0631804
$376$	0	0
$377$	−551.546	−0.0753476
$378$	0	0
$379$	3460.18	0.468965	0.234482	−	0.972120i	$-0.424661\pi$
$379$	3460.18	0.468965	0.234482	+	0.972120i	$0.424661\pi$
$380$	0	0
$381$	5399.55	0.726055
$382$	0	0
$383$	−5059.63	−0.675027	−0.337513	−	0.941321i	$-0.609586\pi$
$383$	−5059.63	−0.675027	−0.337513	+	0.941321i	$0.609586\pi$
$384$	0	0
$385$	839.732	0.111160
$386$	0	0
$387$	−2127.72	−0.279479
$388$	0	0
$389$	2192.22	0.285732	0.142866	−	0.989742i	$-0.454368\pi$
$389$	2192.22	0.285732	0.142866	+	0.989742i	$0.454368\pi$
$390$	0	0
$391$	5163.93	0.667906
$392$	0	0
$393$	−3364.34	−0.431828
$394$	0	0
$395$	−102.045	−0.0129986
$396$	0	0
$397$	−5519.94	−0.697828	−0.348914	−	0.937155i	$-0.613449\pi$
$397$	−5519.94	−0.697828	−0.348914	+	0.937155i	$0.613449\pi$
$398$	0	0
$399$	1989.95	0.249679
$400$	0	0
$401$	−7352.64	−0.915645	−0.457822	−	0.889044i	$-0.651370\pi$
$401$	−7352.64	−0.915645	−0.457822	+	0.889044i	$0.651370\pi$
$402$	0	0
$403$	−11685.9	−1.44446
$404$	0	0
$405$	−49.6256	−0.00608868
$406$	0	0
$407$	−24608.2	−2.99702
$408$	0	0
$409$	11311.2	1.36749	0.683745	−	0.729721i	$-0.260350\pi$
$409$	11311.2	1.36749	0.683745	+	0.729721i	$0.260350\pi$
$410$	0	0
$411$	7348.66	0.881952
$412$	0	0
$413$	5978.40	0.712295
$414$	0	0
$415$	221.743	0.0262288
$416$	0	0
$417$	7372.68	0.865808
$418$	0	0
$419$	13042.2	1.52065	0.760325	−	0.649543i	$-0.225040\pi$
$419$	13042.2	1.52065	0.760325	+	0.649543i	$0.225040\pi$
$420$	0	0
$421$	−4544.38	−0.526080	−0.263040	−	0.964785i	$-0.584725\pi$
$421$	−4544.38	−0.526080	−0.263040	+	0.964785i	$0.584725\pi$
$422$	0	0
$423$	−1162.02	−0.133568
$424$	0	0
$425$	−5877.83	−0.670863
$426$	0	0
$427$	6219.72	0.704903
$428$	0	0
$429$	9564.85	1.07645
$430$	0	0
$431$	−7713.83	−0.862093	−0.431047	−	0.902330i	$-0.641856\pi$
$431$	−7713.83	−0.862093	−0.431047	+	0.902330i	$0.641856\pi$
$432$	0	0
$433$	−15068.3	−1.67237	−0.836183	−	0.548451i	$-0.815218\pi$
$433$	−15068.3	−1.67237	−0.836183	+	0.548451i	$0.815218\pi$
$434$	0	0
$435$	19.1609	0.00211195
$436$	0	0
$437$	3193.14	0.349540
$438$	0	0
$439$	11004.7	1.19642	0.598208	−	0.801341i	$-0.295880\pi$
$439$	11004.7	1.19642	0.598208	+	0.801341i	$0.295880\pi$
$440$	0	0
$441$	1568.67	0.169384
$442$	0	0
$443$	−2513.04	−0.269522	−0.134761	−	0.990878i	$-0.543027\pi$
$443$	−2513.04	−0.269522	−0.134761	+	0.990878i	$0.543027\pi$
$444$	0	0
$445$	24.7077	0.00263204
$446$	0	0
$447$	6254.89	0.661848
$448$	0	0
$449$	−15752.7	−1.65571	−0.827855	−	0.560942i	$-0.810439\pi$
$449$	−15752.7	−1.65571	−0.827855	+	0.560942i	$0.810439\pi$
$450$	0	0
$451$	21739.4	2.26977
$452$	0	0
$453$	−3173.40	−0.329138
$454$	0	0
$455$	−737.218	−0.0759590
$456$	0	0
$457$	5257.06	0.538107	0.269053	−	0.963125i	$-0.413289\pi$
$457$	5257.06	0.538107	0.269053	+	0.963125i	$0.413289\pi$
$458$	0	0
$459$	−1273.43	−0.129496
$460$	0	0
$461$	8066.31	0.814936	0.407468	−	0.913220i	$-0.366412\pi$
$461$	8066.31	0.814936	0.407468	+	0.913220i	$0.366412\pi$
$462$	0	0
$463$	−5683.43	−0.570478	−0.285239	−	0.958456i	$-0.592073\pi$
$463$	−5683.43	−0.570478	−0.285239	+	0.958456i	$0.592073\pi$
$464$	0	0
$465$	405.975	0.0404874
$466$	0	0
$467$	−11139.3	−1.10378	−0.551891	−	0.833916i	$-0.686094\pi$
$467$	−11139.3	−1.10378	−0.551891	+	0.833916i	$0.686094\pi$
$468$	0	0
$469$	−2033.50	−0.200210
$470$	0	0
$471$	9579.02	0.937108
$472$	0	0
$473$	−14247.0	−1.38494
$474$	0	0
$475$	−3634.59	−0.351087
$476$	0	0
$477$	−1060.37	−0.101784
$478$	0	0
$479$	−3477.35	−0.331699	−0.165850	−	0.986151i	$-0.553037\pi$
$479$	−3477.35	−0.331699	−0.165850	+	0.986151i	$0.553037\pi$
$480$	0	0
$481$	21604.1	2.04795
$482$	0	0
$483$	7470.65	0.703782
$484$	0	0
$485$	376.738	0.0352717
$486$	0	0
$487$	−478.797	−0.0445510	−0.0222755	−	0.999752i	$-0.507091\pi$
$487$	−478.797	−0.0445510	−0.0222755	+	0.999752i	$0.507091\pi$
$488$	0	0
$489$	2540.56	0.234945
$490$	0	0
$491$	−16601.8	−1.52592	−0.762961	−	0.646444i	$-0.776255\pi$
$491$	−16601.8	−1.52592	−0.762961	+	0.646444i	$0.776255\pi$
$492$	0	0
$493$	491.687	0.0449178
$494$	0	0
$495$	−332.287	−0.0301721
$496$	0	0
$497$	7972.29	0.719530
$498$	0	0
$499$	−9482.20	−0.850664	−0.425332	−	0.905037i	$-0.639843\pi$
$499$	−9482.20	−0.850664	−0.425332	+	0.905037i	$0.639843\pi$
$500$	0	0
$501$	7890.45	0.703631
$502$	0	0
$503$	16561.2	1.46805	0.734023	−	0.679124i	$-0.237640\pi$
$503$	16561.2	1.46805	0.734023	+	0.679124i	$0.237640\pi$
$504$	0	0
$505$	1020.08	0.0898867
$506$	0	0
$507$	−1806.19	−0.158216
$508$	0	0
$509$	−4197.35	−0.365509	−0.182755	−	0.983159i	$-0.558501\pi$
$509$	−4197.35	−0.365509	−0.182755	+	0.983159i	$0.558501\pi$
$510$	0	0
$511$	12113.8	1.04869
$512$	0	0
$513$	−787.435	−0.0677702
$514$	0	0
$515$	−243.140	−0.0208039
$516$	0	0
$517$	−7780.73	−0.661888
$518$	0	0
$519$	1289.53	0.109064
$520$	0	0
$521$	15755.5	1.32488	0.662440	−	0.749115i	$-0.269521\pi$
$521$	15755.5	1.32488	0.662440	+	0.749115i	$0.269521\pi$
$522$	0	0
$523$	−11555.1	−0.966098	−0.483049	−	0.875593i	$-0.660471\pi$
$523$	−11555.1	−0.966098	−0.483049	+	0.875593i	$0.660471\pi$
$524$	0	0
$525$	−8503.44	−0.706897
$526$	0	0
$527$	10417.7	0.861102
$528$	0	0
$529$	−179.314	−0.0147378
$530$	0	0
$531$	−2365.69	−0.193337
$532$	0	0
$533$	−19085.5	−1.55100
$534$	0	0
$535$	−214.817	−0.0173595
$536$	0	0
$537$	4548.90	0.365549
$538$	0	0
$539$	10503.6	0.839373
$540$	0	0
$541$	−7475.65	−0.594091	−0.297045	−	0.954863i	$-0.596001\pi$
$541$	−7475.65	−0.594091	−0.297045	+	0.954863i	$0.596001\pi$
$542$	0	0
$543$	10140.6	0.801427
$544$	0	0
$545$	365.764	0.0287479
$546$	0	0
$547$	−6028.08	−0.471192	−0.235596	−	0.971851i	$-0.575704\pi$
$547$	−6028.08	−0.471192	−0.235596	+	0.971851i	$0.575704\pi$
$548$	0	0
$549$	−2461.18	−0.191331
$550$	0	0
$551$	304.037	0.0235071
$552$	0	0
$553$	−3788.29	−0.291310
$554$	0	0
$555$	−750.536	−0.0574027
$556$	0	0
$557$	−19381.5	−1.47436	−0.737181	−	0.675696i	$-0.763843\pi$
$557$	−19381.5	−1.47436	−0.737181	+	0.675696i	$0.763843\pi$
$558$	0	0
$559$	12507.7	0.946370
$560$	0	0
$561$	−8526.77	−0.641712
$562$	0	0
$563$	−20565.0	−1.53946	−0.769728	−	0.638372i	$-0.779608\pi$
$563$	−20565.0	−1.53946	−0.769728	+	0.638372i	$0.779608\pi$
$564$	0	0
$565$	−304.139	−0.0226464
$566$	0	0
$567$	−1842.28	−0.136452
$568$	0	0
$569$	−15252.9	−1.12379	−0.561895	−	0.827209i	$-0.689927\pi$
$569$	−15252.9	−1.12379	−0.561895	+	0.827209i	$0.689927\pi$
$570$	0	0
$571$	16492.8	1.20876	0.604379	−	0.796697i	$-0.293421\pi$
$571$	16492.8	1.20876	0.604379	+	0.796697i	$0.293421\pi$
$572$	0	0
$573$	−8399.13	−0.612354
$574$	0	0
$575$	−13644.9	−0.989623
$576$	0	0
$577$	−10298.2	−0.743016	−0.371508	−	0.928430i	$-0.621159\pi$
$577$	−10298.2	−0.743016	−0.371508	+	0.928430i	$0.621159\pi$
$578$	0	0
$579$	−1872.32	−0.134388
$580$	0	0
$581$	8231.89	0.587808
$582$	0	0
$583$	−7100.14	−0.504387
$584$	0	0
$585$	291.722	0.0206175
$586$	0	0
$587$	13104.8	0.921453	0.460727	−	0.887542i	$-0.347589\pi$
$587$	13104.8	0.921453	0.460727	+	0.887542i	$0.347589\pi$
$588$	0	0
$589$	6441.82	0.450646
$590$	0	0
$591$	−14337.7	−0.997929
$592$	0	0
$593$	4163.34	0.288310	0.144155	−	0.989555i	$-0.453954\pi$
$593$	4163.34	0.288310	0.144155	+	0.989555i	$0.453954\pi$
$594$	0	0
$595$	657.208	0.0452822
$596$	0	0
$597$	−7847.75	−0.538001
$598$	0	0
$599$	−5718.60	−0.390076	−0.195038	−	0.980796i	$-0.562483\pi$
$599$	−5718.60	−0.390076	−0.195038	+	0.980796i	$0.562483\pi$
$600$	0	0
$601$	−17473.0	−1.18592	−0.592959	−	0.805233i	$-0.702040\pi$
$601$	−17473.0	−1.18592	−0.592959	+	0.805233i	$0.702040\pi$
$602$	0	0
$603$	804.670	0.0543428
$604$	0	0
$605$	−1409.50	−0.0947181
$606$	0	0
$607$	5647.60	0.377643	0.188821	−	0.982011i	$-0.439533\pi$
$607$	5647.60	0.377643	0.188821	+	0.982011i	$0.439533\pi$
$608$	0	0
$609$	711.322	0.0473304
$610$	0	0
$611$	6830.87	0.452287
$612$	0	0
$613$	16023.6	1.05577	0.527884	−	0.849317i	$-0.322986\pi$
$613$	16023.6	1.05577	0.527884	+	0.849317i	$0.322986\pi$
$614$	0	0
$615$	663.038	0.0434736
$616$	0	0
$617$	21022.1	1.37167	0.685834	−	0.727758i	$-0.259438\pi$
$617$	21022.1	1.37167	0.685834	+	0.727758i	$0.259438\pi$
$618$	0	0
$619$	−17824.2	−1.15737	−0.578686	−	0.815550i	$-0.696434\pi$
$619$	−17824.2	−1.15737	−0.578686	+	0.815550i	$0.696434\pi$
$620$	0	0
$621$	−2956.18	−0.191027
$622$	0	0
$623$	917.238	0.0589861
$624$	0	0
$625$	15484.4	0.991001
$626$	0	0
$627$	−5272.57	−0.335831
$628$	0	0
$629$	−19259.4	−1.22086
$630$	0	0
$631$	−22339.0	−1.40935	−0.704677	−	0.709528i	$-0.748908\pi$
$631$	−22339.0	−1.40935	−0.704677	+	0.709528i	$0.748908\pi$
$632$	0	0
$633$	5237.35	0.328856
$634$	0	0
$635$	1102.70	0.0689121
$636$	0	0
$637$	−9221.34	−0.573568
$638$	0	0
$639$	−3154.69	−0.195301
$640$	0	0
$641$	−5268.43	−0.324634	−0.162317	−	0.986739i	$-0.551897\pi$
$641$	−5268.43	−0.324634	−0.162317	+	0.986739i	$0.551897\pi$
$642$	0	0
$643$	21965.8	1.34719	0.673597	−	0.739099i	$-0.264748\pi$
$643$	21965.8	1.34719	0.673597	+	0.739099i	$0.264748\pi$
$644$	0	0
$645$	−434.524	−0.0265262
$646$	0	0
$647$	3165.40	0.192341	0.0961706	−	0.995365i	$-0.469341\pi$
$647$	3165.40	0.192341	0.0961706	+	0.995365i	$0.469341\pi$
$648$	0	0
$649$	−15840.4	−0.958073
$650$	0	0
$651$	15071.2	0.907354
$652$	0	0
$653$	−12094.7	−0.724814	−0.362407	−	0.932020i	$-0.618045\pi$
$653$	−12094.7	−0.724814	−0.362407	+	0.932020i	$0.618045\pi$
$654$	0	0
$655$	−687.067	−0.0409861
$656$	0	0
$657$	−4793.49	−0.284645
$658$	0	0
$659$	7523.17	0.444706	0.222353	−	0.974966i	$-0.428626\pi$
$659$	7523.17	0.444706	0.222353	+	0.974966i	$0.428626\pi$
$660$	0	0
$661$	−24141.1	−1.42054	−0.710271	−	0.703928i	$-0.751428\pi$
$661$	−24141.1	−1.42054	−0.710271	+	0.703928i	$0.751428\pi$
$662$	0	0
$663$	7485.84	0.438501
$664$	0	0
$665$	406.388	0.0236978
$666$	0	0
$667$	1141.41	0.0662604
$668$	0	0
$669$	−10156.8	−0.586972
$670$	0	0
$671$	−16479.8	−0.948130
$672$	0	0
$673$	944.143	0.0540773	0.0270387	−	0.999634i	$-0.491392\pi$
$673$	944.143	0.0540773	0.0270387	+	0.999634i	$0.491392\pi$
$674$	0	0
$675$	3364.87	0.191872
$676$	0	0
$677$	−4450.51	−0.252654	−0.126327	−	0.991989i	$-0.540319\pi$
$677$	−4450.51	−0.252654	−0.126327	+	0.991989i	$0.540319\pi$
$678$	0	0
$679$	13985.8	0.790468
$680$	0	0
$681$	−11543.1	−0.649536
$682$	0	0
$683$	−1726.40	−0.0967189	−0.0483594	−	0.998830i	$-0.515399\pi$
$683$	−1726.40	−0.0967189	−0.0483594	+	0.998830i	$0.515399\pi$
$684$	0	0
$685$	1500.75	0.0837088
$686$	0	0
$687$	−4005.44	−0.222441
$688$	0	0
$689$	6233.37	0.344662
$690$	0	0
$691$	683.143	0.0376092	0.0188046	−	0.999823i	$-0.494014\pi$
$691$	683.143	0.0376092	0.0188046	+	0.999823i	$0.494014\pi$
$692$	0	0
$693$	−12335.7	−0.676181
$694$	0	0
$695$	1505.65	0.0821764
$696$	0	0
$697$	17014.1	0.924614
$698$	0	0
$699$	−15439.1	−0.835425
$700$	0	0
$701$	−19533.2	−1.05244	−0.526220	−	0.850349i	$-0.676391\pi$
$701$	−19533.2	−1.05244	−0.526220	+	0.850349i	$0.676391\pi$
$702$	0	0
$703$	−11909.2	−0.638922
$704$	0	0
$705$	−237.307	−0.0126773
$706$	0	0
$707$	37868.8	2.01443
$708$	0	0
$709$	−26081.9	−1.38156	−0.690779	−	0.723066i	$-0.742732\pi$
$709$	−26081.9	−1.38156	−0.690779	+	0.723066i	$0.742732\pi$
$710$	0	0
$711$	1499.05	0.0790699
$712$	0	0
$713$	24183.8	1.27025
$714$	0	0
$715$	1953.34	0.102169
$716$	0	0
$717$	21255.2	1.10710
$718$	0	0
$719$	30077.3	1.56007	0.780036	−	0.625734i	$-0.215200\pi$
$719$	30077.3	1.56007	0.780036	+	0.625734i	$0.215200\pi$
$720$	0	0
$721$	−9026.21	−0.466232
$722$	0	0
$723$	−7615.20	−0.391718
$724$	0	0
$725$	−1299.21	−0.0665537
$726$	0	0
$727$	23049.9	1.17589	0.587946	−	0.808900i	$-0.299937\pi$
$727$	23049.9	1.17589	0.587946	+	0.808900i	$0.299937\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−11150.3	−0.564169
$732$	0	0
$733$	−4444.57	−0.223962	−0.111981	−	0.993710i	$-0.535720\pi$
$733$	−4444.57	−0.223962	−0.111981	+	0.993710i	$0.535720\pi$
$734$	0	0
$735$	320.353	0.0160768
$736$	0	0
$737$	5387.98	0.269293
$738$	0	0
$739$	−28465.1	−1.41692	−0.708462	−	0.705749i	$-0.750610\pi$
$739$	−28465.1	−1.41692	−0.708462	+	0.705749i	$0.750610\pi$
$740$	0	0
$741$	4628.91	0.229483
$742$	0	0
$743$	4389.76	0.216749	0.108374	−	0.994110i	$-0.465435\pi$
$743$	4389.76	0.216749	0.108374	+	0.994110i	$0.465435\pi$
$744$	0	0
$745$	1277.38	0.0628180
$746$	0	0
$747$	−3257.41	−0.159548
$748$	0	0
$749$	−7974.77	−0.389041
$750$	0	0
$751$	−14121.9	−0.686171	−0.343085	−	0.939304i	$-0.611472\pi$
$751$	−14121.9	−0.686171	−0.343085	+	0.939304i	$0.611472\pi$
$752$	0	0
$753$	−5090.98	−0.246382
$754$	0	0
$755$	−648.074	−0.0312395
$756$	0	0
$757$	−17006.3	−0.816516	−0.408258	−	0.912866i	$-0.633864\pi$
$757$	−17006.3	−0.816516	−0.408258	+	0.912866i	$0.633864\pi$
$758$	0	0
$759$	−19794.3	−0.946622
$760$	0	0
$761$	4603.38	0.219280	0.109640	−	0.993971i	$-0.465030\pi$
$761$	4603.38	0.219280	0.109640	+	0.993971i	$0.465030\pi$
$762$	0	0
$763$	13578.5	0.644264
$764$	0	0
$765$	−260.061	−0.0122909
$766$	0	0
$767$	13906.6	0.654679
$768$	0	0
$769$	−12459.1	−0.584248	−0.292124	−	0.956380i	$-0.594362\pi$
$769$	−12459.1	−0.584248	−0.292124	+	0.956380i	$0.594362\pi$
$770$	0	0
$771$	1148.71	0.0536571
$772$	0	0
$773$	−27068.1	−1.25947	−0.629737	−	0.776808i	$-0.716837\pi$
$773$	−27068.1	−1.25947	−0.629737	+	0.776808i	$0.716837\pi$
$774$	0	0
$775$	−27527.2	−1.27588
$776$	0	0
$777$	−27862.6	−1.28644
$778$	0	0
$779$	10520.8	0.483884
$780$	0	0
$781$	−21123.4	−0.967804
$782$	0	0
$783$	−281.475	−0.0128468
$784$	0	0
$785$	1956.23	0.0889438
$786$	0	0
$787$	−6986.86	−0.316461	−0.158230	−	0.987402i	$-0.550579\pi$
$787$	−6986.86	−0.316461	−0.158230	+	0.987402i	$0.550579\pi$
$788$	0	0
$789$	15006.1	0.677097
$790$	0	0
$791$	−11290.7	−0.507523
$792$	0	0
$793$	14468.0	0.647885
$794$	0	0
$795$	−216.550	−0.00966067
$796$	0	0
$797$	27271.5	1.21205	0.606027	−	0.795444i	$-0.292762\pi$
$797$	27271.5	1.21205	0.606027	+	0.795444i	$0.292762\pi$
$798$	0	0
$799$	−6089.52	−0.269627
$800$	0	0
$801$	−362.957	−0.0160105
$802$	0	0
$803$	−32096.7	−1.41054
$804$	0	0
$805$	1525.66	0.0667981
$806$	0	0
$807$	−18352.4	−0.800540
$808$	0	0
$809$	−23785.9	−1.03371	−0.516853	−	0.856074i	$-0.672897\pi$
$809$	−23785.9	−1.03371	−0.516853	+	0.856074i	$0.672897\pi$
$810$	0	0
$811$	−21703.5	−0.939718	−0.469859	−	0.882741i	$-0.655695\pi$
$811$	−21703.5	−0.939718	−0.469859	+	0.882741i	$0.655695\pi$
$812$	0	0
$813$	−11868.4	−0.511982
$814$	0	0
$815$	518.835	0.0222994
$816$	0	0
$817$	−6894.83	−0.295250
$818$	0	0
$819$	10829.7	0.462054
$820$	0	0
$821$	33240.4	1.41303	0.706515	−	0.707698i	$-0.250266\pi$
$821$	33240.4	1.41303	0.706515	+	0.707698i	$0.250266\pi$
$822$	0	0
$823$	−17227.5	−0.729665	−0.364832	−	0.931073i	$-0.618874\pi$
$823$	−17227.5	−0.729665	−0.364832	+	0.931073i	$0.618874\pi$
$824$	0	0
$825$	22530.8	0.950812
$826$	0	0
$827$	−21678.4	−0.911528	−0.455764	−	0.890101i	$-0.650634\pi$
$827$	−21678.4	−0.911528	−0.455764	+	0.890101i	$0.650634\pi$
$828$	0	0
$829$	−34269.8	−1.43575	−0.717877	−	0.696170i	$-0.754886\pi$
$829$	−34269.8	−1.43575	−0.717877	+	0.696170i	$0.754886\pi$
$830$	0	0
$831$	−14526.5	−0.606401
$832$	0	0
$833$	8220.55	0.341927
$834$	0	0
$835$	1611.39	0.0667838
$836$	0	0
$837$	−5963.77	−0.246282
$838$	0	0
$839$	33444.1	1.37618	0.688092	−	0.725623i	$-0.258448\pi$
$839$	33444.1	1.37618	0.688092	+	0.725623i	$0.258448\pi$
$840$	0	0
$841$	−24280.3	−0.995544
$842$	0	0
$843$	5636.04	0.230267
$844$	0	0
$845$	−368.860	−0.0150168
$846$	0	0
$847$	−52325.8	−2.12271
$848$	0	0
$849$	17174.6	0.694265
$850$	0	0
$851$	−44709.3	−1.80096
$852$	0	0
$853$	37037.3	1.48667	0.743337	−	0.668917i	$-0.233242\pi$
$853$	37037.3	1.48667	0.743337	+	0.668917i	$0.233242\pi$
$854$	0	0
$855$	−160.810	−0.00643227
$856$	0	0
$857$	−35794.2	−1.42673	−0.713365	−	0.700793i	$-0.752830\pi$
$857$	−35794.2	−1.42673	−0.713365	+	0.700793i	$0.752830\pi$
$858$	0	0
$859$	20582.1	0.817522	0.408761	−	0.912641i	$-0.365961\pi$
$859$	20582.1	0.817522	0.408761	+	0.912641i	$0.365961\pi$
$860$	0	0
$861$	24614.3	0.974278
$862$	0	0
$863$	25677.7	1.01284	0.506420	−	0.862287i	$-0.330969\pi$
$863$	25677.7	1.01284	0.506420	+	0.862287i	$0.330969\pi$
$864$	0	0
$865$	263.348	0.0103516
$866$	0	0
$867$	8065.60	0.315942
$868$	0	0
$869$	10037.5	0.391827
$870$	0	0
$871$	−4730.22	−0.184015
$872$	0	0
$873$	−5534.29	−0.214556
$874$	0	0
$875$	−3478.38	−0.134389
$876$	0	0
$877$	−32783.0	−1.26226	−0.631131	−	0.775676i	$-0.717409\pi$
$877$	−32783.0	−1.26226	−0.631131	+	0.775676i	$0.717409\pi$
$878$	0	0
$879$	15266.3	0.585800
$880$	0	0
$881$	26615.7	1.01783	0.508914	−	0.860818i	$-0.330047\pi$
$881$	26615.7	1.01783	0.508914	+	0.860818i	$0.330047\pi$
$882$	0	0
$883$	19203.4	0.731875	0.365937	−	0.930639i	$-0.380748\pi$
$883$	19203.4	0.731875	0.365937	+	0.930639i	$0.380748\pi$
$884$	0	0
$885$	−483.122	−0.0183503
$886$	0	0
$887$	23198.0	0.878144	0.439072	−	0.898452i	$-0.355307\pi$
$887$	23198.0	0.878144	0.439072	+	0.898452i	$0.355307\pi$
$888$	0	0
$889$	40936.0	1.54438
$890$	0	0
$891$	4881.30	0.183535
$892$	0	0
$893$	−3765.48	−0.141105
$894$	0	0
$895$	928.979	0.0346953
$896$	0	0
$897$	17377.8	0.646854
$898$	0	0
$899$	2302.68	0.0854266
$900$	0	0
$901$	−5556.86	−0.205467
$902$	0	0
$903$	−16131.1	−0.594472
$904$	0	0
$905$	2070.92	0.0760659
$906$	0	0
$907$	16151.3	0.591285	0.295643	−	0.955299i	$-0.404466\pi$
$907$	16151.3	0.591285	0.295643	+	0.955299i	$0.404466\pi$
$908$	0	0
$909$	−14984.9	−0.546776
$910$	0	0
$911$	3487.34	0.126829	0.0634143	−	0.997987i	$-0.479801\pi$
$911$	3487.34	0.126829	0.0634143	+	0.997987i	$0.479801\pi$
$912$	0	0
$913$	−21811.2	−0.790632
$914$	0	0
$915$	−502.624	−0.0181598
$916$	0	0
$917$	−25506.3	−0.918532
$918$	0	0
$919$	17055.5	0.612198	0.306099	−	0.952000i	$-0.400976\pi$
$919$	17055.5	0.612198	0.306099	+	0.952000i	$0.400976\pi$
$920$	0	0
$921$	21657.6	0.774857
$922$	0	0
$923$	18544.7	0.661329
$924$	0	0
$925$	50890.2	1.80893
$926$	0	0
$927$	3571.73	0.126549
$928$	0	0
$929$	−55339.3	−1.95438	−0.977192	−	0.212359i	$-0.931885\pi$
$929$	−55339.3	−1.95438	−0.977192	+	0.212359i	$0.931885\pi$
$930$	0	0
$931$	5083.22	0.178943
$932$	0	0
$933$	4611.17	0.161804
$934$	0	0
$935$	−1741.34	−0.0609069
$936$	0	0
$937$	20457.6	0.713255	0.356627	−	0.934247i	$-0.383927\pi$
$937$	20457.6	0.713255	0.356627	+	0.934247i	$0.383927\pi$
$938$	0	0
$939$	−6602.78	−0.229471
$940$	0	0
$941$	55891.0	1.93623	0.968116	−	0.250502i	$-0.0805958\pi$
$941$	55891.0	1.93623	0.968116	+	0.250502i	$0.0805958\pi$
$942$	0	0
$943$	39497.0	1.36395
$944$	0	0
$945$	−376.230	−0.0129511
$946$	0	0
$947$	54727.0	1.87792	0.938960	−	0.344027i	$-0.111791\pi$
$947$	54727.0	1.87792	0.938960	+	0.344027i	$0.111791\pi$
$948$	0	0
$949$	28178.4	0.963865
$950$	0	0
$951$	−8521.23	−0.290557
$952$	0	0
$953$	−29958.8	−1.01832	−0.509160	−	0.860672i	$-0.670044\pi$
$953$	−29958.8	−1.01832	−0.509160	+	0.860672i	$0.670044\pi$
$954$	0	0
$955$	−1715.27	−0.0581204
$956$	0	0
$957$	−1884.72	−0.0636619
$958$	0	0
$959$	55713.0	1.87598
$960$	0	0
$961$	18997.2	0.637682
$962$	0	0
$963$	3155.67	0.105597
$964$	0	0
$965$	−382.365	−0.0127552
$966$	0	0
$967$	13498.5	0.448896	0.224448	−	0.974486i	$-0.427942\pi$
$967$	13498.5	0.448896	0.224448	+	0.974486i	$0.427942\pi$
$968$	0	0
$969$	−4126.53	−0.136804
$970$	0	0
$971$	9652.36	0.319010	0.159505	−	0.987197i	$-0.449010\pi$
$971$	9652.36	0.319010	0.159505	+	0.987197i	$0.449010\pi$
$972$	0	0
$973$	55895.1	1.84164
$974$	0	0
$975$	−19780.2	−0.649717
$976$	0	0
$977$	12169.8	0.398511	0.199256	−	0.979948i	$-0.436148\pi$
$977$	12169.8	0.398511	0.199256	+	0.979948i	$0.436148\pi$
$978$	0	0
$979$	−2430.32	−0.0793394
$980$	0	0
$981$	−5373.08	−0.174872
$982$	0	0
$983$	47545.2	1.54268	0.771341	−	0.636423i	$-0.219587\pi$
$983$	47545.2	1.54268	0.771341	+	0.636423i	$0.219587\pi$
$984$	0	0
$985$	−2928.06	−0.0947165
$986$	0	0
$987$	−8809.69	−0.284109
$988$	0	0
$989$	−25884.5	−0.832234
$990$	0	0
$991$	892.350	0.0286039	0.0143019	−	0.999898i	$-0.495447\pi$
$991$	892.350	0.0286039	0.0143019	+	0.999898i	$0.495447\pi$
$992$	0	0
$993$	−6355.55	−0.203109
$994$	0	0
$995$	−1602.67	−0.0510634
$996$	0	0
$997$	−4458.57	−0.141629	−0.0708146	−	0.997489i	$-0.522560\pi$
$997$	−4458.57	−0.141629	−0.0708146	+	0.997489i	$0.522560\pi$
$998$	0	0
$999$	11025.4	0.349177

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.a.q.1.2		3
3.2	odd	2		2304.4.a.bw.1.2		3
4.3	odd	2		768.4.a.s.1.2		3
8.3	odd	2		768.4.a.r.1.2		3
8.5	even	2		768.4.a.t.1.2		3
12.11	even	2		2304.4.a.bv.1.2		3
16.3	odd	4		24.4.d.a.13.5	✓	6
16.5	even	4		96.4.d.a.49.5		6
16.11	odd	4		24.4.d.a.13.6	yes	6
16.13	even	4		96.4.d.a.49.2		6
24.5	odd	2		2304.4.a.bu.1.2		3
24.11	even	2		2304.4.a.bt.1.2		3
48.5	odd	4		288.4.d.d.145.4		6
48.11	even	4		72.4.d.d.37.1		6
48.29	odd	4		288.4.d.d.145.3		6
48.35	even	4		72.4.d.d.37.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.4.d.a.13.5	✓	6	16.3	odd	4
24.4.d.a.13.6	yes	6	16.11	odd	4
72.4.d.d.37.1		6	48.11	even	4
72.4.d.d.37.2		6	48.35	even	4
96.4.d.a.49.2		6	16.13	even	4
96.4.d.a.49.5		6	16.5	even	4
288.4.d.d.145.3		6	48.29	odd	4
288.4.d.d.145.4		6	48.5	odd	4
768.4.a.q.1.2		3	1.1	even	1	trivial
768.4.a.r.1.2		3	8.3	odd	2
768.4.a.s.1.2		3	4.3	odd	2
768.4.a.t.1.2		3	8.5	even	2
2304.4.a.bt.1.2		3	24.11	even	2
2304.4.a.bu.1.2		3	24.5	odd	2
2304.4.a.bv.1.2		3	12.11	even	2
2304.4.a.bw.1.2		3	3.2	odd	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -0.612661 q^{5} -22.7441 q^{7} +9.00000 q^{9} +60.2630 q^{11} -52.9062 q^{13} +1.83798 q^{15} +47.1643 q^{17} +29.1643 q^{19} +68.2324 q^{21} +109.488 q^{23} -124.625 q^{25} -27.0000 q^{27} +10.4250 q^{29} +220.881 q^{31} -180.789 q^{33} +13.9345 q^{35} -408.348 q^{37} +158.718 q^{39} +360.742 q^{41} -236.414 q^{43} -5.51395 q^{45} -129.113 q^{47} +174.296 q^{49} -141.493 q^{51} -117.819 q^{53} -36.9208 q^{55} -87.4928 q^{57} -262.854 q^{59} -273.465 q^{61} -204.697 q^{63} +32.4135 q^{65} +89.4077 q^{67} -328.465 q^{69} -350.521 q^{71} -532.610 q^{73} +373.874 q^{75} -1370.63 q^{77} +166.561 q^{79} +81.0000 q^{81} -361.934 q^{83} -28.8957 q^{85} -31.2750 q^{87} -40.3285 q^{89} +1203.31 q^{91} -662.642 q^{93} -17.8678 q^{95} -614.921 q^{97} +542.367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{3} - 10 q^{5} + 14 q^{7} + 27 q^{9} - 52 q^{13} + 30 q^{15} + 26 q^{17} - 28 q^{19} - 42 q^{21} + 164 q^{23} + 53 q^{25} - 81 q^{27} - 174 q^{29} + 318 q^{31} + 92 q^{35} - 296 q^{37} + 156 q^{39}+ \cdots - 1222 q^{97}+O(q^{100}) \) 3 * q - 9 * q^3 - 10 * q^5 + 14 * q^7 + 27 * q^9 - 52 * q^13 + 30 * q^15 + 26 * q^17 - 28 * q^19 - 42 * q^21 + 164 * q^23 + 53 * q^25 - 81 * q^27 - 174 * q^29 + 318 * q^31 + 92 * q^35 - 296 * q^37 + 156 * q^39 - 118 * q^41 + 260 * q^43 - 90 * q^45 + 204 * q^47 + 327 * q^49 - 78 * q^51 - 1086 * q^53 + 512 * q^55 + 84 * q^57 - 196 * q^59 - 1536 * q^61 + 126 * q^63 - 872 * q^65 + 660 * q^67 - 492 * q^69 - 852 * q^71 - 478 * q^73 - 159 * q^75 - 2304 * q^77 + 22 * q^79 + 243 * q^81 - 1136 * q^83 - 2732 * q^85 + 522 * q^87 + 110 * q^89 + 632 * q^91 - 954 * q^93 - 2552 * q^95 - 1222 * q^97

Introduction

L-functions

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