LMFDB - Embedded newform 1344.4.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(1,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1344.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.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:	$79.2985670477$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{137}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 34 $ x^2 - x - 34
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.35235$ of defining polynomial
Character	$\chi$	$=$	1344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -16.7047 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -16.7047 q^{5} +7.00000 q^{7} +9.00000 q^{9} -22.7047 q^{11} +11.4094 q^{13} +50.1141 q^{15} +3.29530 q^{17} -62.8188 q^{19} -21.0000 q^{21} +76.1141 q^{23} +154.047 q^{25} -27.0000 q^{27} +216.094 q^{29} +149.638 q^{31} +68.1141 q^{33} -116.933 q^{35} -187.866 q^{37} -34.2282 q^{39} -23.2953 q^{41} +214.094 q^{43} -150.342 q^{45} -187.960 q^{47} +49.0000 q^{49} -9.88590 q^{51} +122.685 q^{53} +379.275 q^{55} +188.456 q^{57} -225.409 q^{59} +546.188 q^{61} +63.0000 q^{63} -190.591 q^{65} +353.141 q^{67} -228.342 q^{69} +465.523 q^{71} +430.416 q^{73} -462.141 q^{75} -158.933 q^{77} +532.497 q^{79} +81.0000 q^{81} -185.181 q^{83} -55.0470 q^{85} -648.282 q^{87} -1597.31 q^{89} +79.8658 q^{91} -448.913 q^{93} +1049.37 q^{95} -1605.69 q^{97} -204.342 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9	$ 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} - 22 q^{11} - 24 q^{13} + 30 q^{15} + 30 q^{17} - 32 q^{19} - 42 q^{21} + 82 q^{23} + 74 q^{25} - 54 q^{27} - 36 q^{29} + 112 q^{31} + 66 q^{33} - 70 q^{35} - 48 q^{37} + 72 q^{39} - 70 q^{41} - 40 q^{43} - 90 q^{45} + 420 q^{47} + 98 q^{49} - 90 q^{51} - 176 q^{53} + 384 q^{55} + 96 q^{57} - 404 q^{59} + 156 q^{61} + 126 q^{63} - 428 q^{65} + 4 q^{67} - 246 q^{69} + 814 q^{71} - 216 q^{73} - 222 q^{75} - 154 q^{77} + 1580 q^{79} + 162 q^{81} - 464 q^{83} + 124 q^{85} + 108 q^{87} - 1158 q^{89} - 168 q^{91} - 336 q^{93} + 1256 q^{95} - 1760 q^{97} - 198 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9 - 22 * q^11 - 24 * q^13 + 30 * q^15 + 30 * q^17 - 32 * q^19 - 42 * q^21 + 82 * q^23 + 74 * q^25 - 54 * q^27 - 36 * q^29 + 112 * q^31 + 66 * q^33 - 70 * q^35 - 48 * q^37 + 72 * q^39 - 70 * q^41 - 40 * q^43 - 90 * q^45 + 420 * q^47 + 98 * q^49 - 90 * q^51 - 176 * q^53 + 384 * q^55 + 96 * q^57 - 404 * q^59 + 156 * q^61 + 126 * q^63 - 428 * q^65 + 4 * q^67 - 246 * q^69 + 814 * q^71 - 216 * q^73 - 222 * q^75 - 154 * q^77 + 1580 * q^79 + 162 * q^81 - 464 * q^83 + 124 * q^85 + 108 * q^87 - 1158 * q^89 - 168 * q^91 - 336 * q^93 + 1256 * q^95 - 1760 * q^97 - 198 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−16.7047	−1.49411	−0.747057	−	0.664760i	$-0.768534\pi$
$5$	−16.7047	−1.49411	−0.747057	+	0.664760i	$0.768534\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−22.7047	−0.622339	−0.311169	−	0.950354i	$-0.600721\pi$
$11$	−22.7047	−0.622339	−0.311169	+	0.950354i	$0.600721\pi$
$12$	0	0
$13$	11.4094	0.243415	0.121708	−	0.992566i	$-0.461163\pi$
$13$	11.4094	0.243415	0.121708	+	0.992566i	$0.461163\pi$
$14$	0	0
$15$	50.1141	0.862627
$16$	0	0
$17$	3.29530	0.0470134	0.0235067	−	0.999724i	$-0.492517\pi$
$17$	3.29530	0.0470134	0.0235067	+	0.999724i	$0.492517\pi$
$18$	0	0
$19$	−62.8188	−0.758506	−0.379253	−	0.925293i	$-0.623819\pi$
$19$	−62.8188	−0.758506	−0.379253	+	0.925293i	$0.623819\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	76.1141	0.690039	0.345019	−	0.938596i	$-0.387872\pi$
$23$	76.1141	0.690039	0.345019	+	0.938596i	$0.387872\pi$
$24$	0	0
$25$	154.047	1.23238
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	216.094	1.38371	0.691856	−	0.722036i	$-0.256793\pi$
$29$	216.094	1.38371	0.691856	+	0.722036i	$0.256793\pi$
$30$	0	0
$31$	149.638	0.866958	0.433479	−	0.901164i	$-0.357286\pi$
$31$	149.638	0.866958	0.433479	+	0.901164i	$0.357286\pi$
$32$	0	0
$33$	68.1141	0.359307
$34$	0	0
$35$	−116.933	−0.564722
$36$	0	0
$37$	−187.866	−0.834728	−0.417364	−	0.908739i	$-0.637046\pi$
$37$	−187.866	−0.834728	−0.417364	+	0.908739i	$0.637046\pi$
$38$	0	0
$39$	−34.2282	−0.140536
$40$	0	0
$41$	−23.2953	−0.0887345	−0.0443673	−	0.999015i	$-0.514127\pi$
$41$	−23.2953	−0.0887345	−0.0443673	+	0.999015i	$0.514127\pi$
$42$	0	0
$43$	214.094	0.759280	0.379640	−	0.925134i	$-0.376048\pi$
$43$	214.094	0.759280	0.379640	+	0.925134i	$0.376048\pi$
$44$	0	0
$45$	−150.342	−0.498038
$46$	0	0
$47$	−187.960	−0.583335	−0.291668	−	0.956520i	$-0.594210\pi$
$47$	−187.960	−0.583335	−0.291668	+	0.956520i	$0.594210\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−9.88590	−0.0271432
$52$	0	0
$53$	122.685	0.317963	0.158981	−	0.987282i	$-0.449179\pi$
$53$	122.685	0.317963	0.158981	+	0.987282i	$0.449179\pi$
$54$	0	0
$55$	379.275	0.929845
$56$	0	0
$57$	188.456	0.437924
$58$	0	0
$59$	−225.409	−0.497387	−0.248693	−	0.968582i	$-0.580001\pi$
$59$	−225.409	−0.497387	−0.248693	+	0.968582i	$0.580001\pi$
$60$	0	0
$61$	546.188	1.14643	0.573215	−	0.819405i	$-0.305696\pi$
$61$	546.188	1.14643	0.573215	+	0.819405i	$0.305696\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−190.591	−0.363690
$66$	0	0
$67$	353.141	0.643926	0.321963	−	0.946752i	$-0.395657\pi$
$67$	353.141	0.643926	0.321963	+	0.946752i	$0.395657\pi$
$68$	0	0
$69$	−228.342	−0.398394
$70$	0	0
$71$	465.523	0.778133	0.389067	−	0.921210i	$-0.372798\pi$
$71$	465.523	0.778133	0.389067	+	0.921210i	$0.372798\pi$
$72$	0	0
$73$	430.416	0.690088	0.345044	−	0.938587i	$-0.387864\pi$
$73$	430.416	0.690088	0.345044	+	0.938587i	$0.387864\pi$
$74$	0	0
$75$	−462.141	−0.711513
$76$	0	0
$77$	−158.933	−0.235222
$78$	0	0
$79$	532.497	0.758361	0.379181	−	0.925323i	$-0.376206\pi$
$79$	532.497	0.758361	0.379181	+	0.925323i	$0.376206\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−185.181	−0.244895	−0.122447	−	0.992475i	$-0.539074\pi$
$83$	−185.181	−0.244895	−0.122447	+	0.992475i	$0.539074\pi$
$84$	0	0
$85$	−55.0470	−0.0702434
$86$	0	0
$87$	−648.282	−0.798886
$88$	0	0
$89$	−1597.31	−1.90241	−0.951205	−	0.308561i	$-0.900153\pi$
$89$	−1597.31	−1.90241	−0.951205	+	0.308561i	$0.900153\pi$
$90$	0	0
$91$	79.8658	0.0920023
$92$	0	0
$93$	−448.913	−0.500539
$94$	0	0
$95$	1049.37	1.13329
$96$	0	0
$97$	−1605.69	−1.68076	−0.840378	−	0.542001i	$-0.817667\pi$
$97$	−1605.69	−1.68076	−0.840378	+	0.542001i	$0.817667\pi$
$98$	0	0
$99$	−204.342	−0.207446
$100$	0	0
$101$	−1359.94	−1.33979	−0.669896	−	0.742455i	$-0.733661\pi$
$101$	−1359.94	−1.33979	−0.669896	+	0.742455i	$0.733661\pi$
$102$	0	0
$103$	1821.56	1.74256	0.871278	−	0.490789i	$-0.163291\pi$
$103$	1821.56	1.74256	0.871278	+	0.490789i	$0.163291\pi$
$104$	0	0
$105$	350.799	0.326042
$106$	0	0
$107$	−1440.53	−1.30151	−0.650754	−	0.759289i	$-0.725547\pi$
$107$	−1440.53	−1.30151	−0.650754	+	0.759289i	$0.725547\pi$
$108$	0	0
$109$	−884.014	−0.776818	−0.388409	−	0.921487i	$-0.626975\pi$
$109$	−884.014	−0.776818	−0.388409	+	0.921487i	$0.626975\pi$
$110$	0	0
$111$	563.597	0.481931
$112$	0	0
$113$	−1493.46	−1.24330	−0.621651	−	0.783294i	$-0.713538\pi$
$113$	−1493.46	−1.24330	−0.621651	+	0.783294i	$0.713538\pi$
$114$	0	0
$115$	−1271.46	−1.03100
$116$	0	0
$117$	102.685	0.0811384
$118$	0	0
$119$	23.0671	0.0177694
$120$	0	0
$121$	−815.497	−0.612695
$122$	0	0
$123$	69.8859	0.0512309
$124$	0	0
$125$	−485.221	−0.347196
$126$	0	0
$127$	1421.41	0.993147	0.496574	−	0.867995i	$-0.334591\pi$
$127$	1421.41	0.993147	0.496574	+	0.867995i	$0.334591\pi$
$128$	0	0
$129$	−642.282	−0.438370
$130$	0	0
$131$	−755.087	−0.503605	−0.251803	−	0.967779i	$-0.581023\pi$
$131$	−755.087	−0.503605	−0.251803	+	0.967779i	$0.581023\pi$
$132$	0	0
$133$	−439.732	−0.286688
$134$	0	0
$135$	451.027	0.287542
$136$	0	0
$137$	1618.98	1.00963	0.504813	−	0.863229i	$-0.331561\pi$
$137$	1618.98	1.00963	0.504813	+	0.863229i	$0.331561\pi$
$138$	0	0
$139$	1458.93	0.890248	0.445124	−	0.895469i	$-0.353160\pi$
$139$	1458.93	0.890248	0.445124	+	0.895469i	$0.353160\pi$
$140$	0	0
$141$	563.879	0.336789
$142$	0	0
$143$	−259.047	−0.151487
$144$	0	0
$145$	−3609.79	−2.06742
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−3270.07	−1.79795	−0.898974	−	0.438001i	$-0.855687\pi$
$149$	−3270.07	−1.79795	−0.898974	+	0.438001i	$0.855687\pi$
$150$	0	0
$151$	1198.01	0.645649	0.322824	−	0.946459i	$-0.395368\pi$
$151$	1198.01	0.645649	0.322824	+	0.946459i	$0.395368\pi$
$152$	0	0
$153$	29.6577	0.0156711
$154$	0	0
$155$	−2499.65	−1.29533
$156$	0	0
$157$	1255.75	0.638340	0.319170	−	0.947697i	$-0.396596\pi$
$157$	1255.75	0.638340	0.319170	+	0.947697i	$0.396596\pi$
$158$	0	0
$159$	−368.054	−0.183576
$160$	0	0
$161$	532.799	0.260810
$162$	0	0
$163$	−1208.23	−0.580587	−0.290294	−	0.956938i	$-0.593753\pi$
$163$	−1208.23	−0.580587	−0.290294	+	0.956938i	$0.593753\pi$
$164$	0	0
$165$	−1137.83	−0.536846
$166$	0	0
$167$	−2785.14	−1.29054	−0.645271	−	0.763953i	$-0.723256\pi$
$167$	−2785.14	−1.29054	−0.645271	+	0.763953i	$0.723256\pi$
$168$	0	0
$169$	−2066.83	−0.940749
$170$	0	0
$171$	−565.369	−0.252835
$172$	0	0
$173$	−875.832	−0.384903	−0.192452	−	0.981306i	$-0.561644\pi$
$173$	−875.832	−0.384903	−0.192452	+	0.981306i	$0.561644\pi$
$174$	0	0
$175$	1078.33	0.465794
$176$	0	0
$177$	676.228	0.287166
$178$	0	0
$179$	1148.72	0.479661	0.239830	−	0.970815i	$-0.422908\pi$
$179$	1148.72	0.479661	0.239830	+	0.970815i	$0.422908\pi$
$180$	0	0
$181$	−1738.15	−0.713787	−0.356894	−	0.934145i	$-0.616164\pi$
$181$	−1738.15	−0.713787	−0.356894	+	0.934145i	$0.616164\pi$
$182$	0	0
$183$	−1638.56	−0.661891
$184$	0	0
$185$	3138.24	1.24718
$186$	0	0
$187$	−74.8188	−0.0292582
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−3779.34	−1.43174	−0.715872	−	0.698231i	$-0.753971\pi$
$191$	−3779.34	−1.43174	−0.715872	+	0.698231i	$0.753971\pi$
$192$	0	0
$193$	−3966.15	−1.47922	−0.739610	−	0.673035i	$-0.764990\pi$
$193$	−3966.15	−1.47922	−0.739610	+	0.673035i	$0.764990\pi$
$194$	0	0
$195$	571.772	0.209977
$196$	0	0
$197$	−2593.05	−0.937802	−0.468901	−	0.883251i	$-0.655350\pi$
$197$	−2593.05	−0.937802	−0.468901	+	0.883251i	$0.655350\pi$
$198$	0	0
$199$	−4407.76	−1.57014	−0.785070	−	0.619407i	$-0.787373\pi$
$199$	−4407.76	−1.57014	−0.785070	+	0.619407i	$0.787373\pi$
$200$	0	0
$201$	−1059.42	−0.371771
$202$	0	0
$203$	1512.66	0.522994
$204$	0	0
$205$	389.141	0.132579
$206$	0	0
$207$	685.027	0.230013
$208$	0	0
$209$	1426.28	0.472048
$210$	0	0
$211$	3745.13	1.22192	0.610960	−	0.791661i	$-0.290783\pi$
$211$	3745.13	1.22192	0.610960	+	0.791661i	$0.290783\pi$
$212$	0	0
$213$	−1396.57	−0.449256
$214$	0	0
$215$	−3576.38	−1.13445
$216$	0	0
$217$	1047.46	0.327679
$218$	0	0
$219$	−1291.25	−0.398422
$220$	0	0
$221$	37.5974	0.0114438
$222$	0	0
$223$	−1199.17	−0.360100	−0.180050	−	0.983657i	$-0.557626\pi$
$223$	−1199.17	−0.360100	−0.180050	+	0.983657i	$0.557626\pi$
$224$	0	0
$225$	1386.42	0.410792
$226$	0	0
$227$	−4378.08	−1.28010	−0.640052	−	0.768332i	$-0.721087\pi$
$227$	−4378.08	−1.28010	−0.640052	+	0.768332i	$0.721087\pi$
$228$	0	0
$229$	5518.82	1.59255	0.796275	−	0.604935i	$-0.206801\pi$
$229$	5518.82	1.59255	0.796275	+	0.604935i	$0.206801\pi$
$230$	0	0
$231$	476.799	0.135805
$232$	0	0
$233$	2511.79	0.706234	0.353117	−	0.935579i	$-0.385122\pi$
$233$	2511.79	0.706234	0.353117	+	0.935579i	$0.385122\pi$
$234$	0	0
$235$	3139.81	0.871569
$236$	0	0
$237$	−1597.49	−0.437840
$238$	0	0
$239$	7146.56	1.93419	0.967097	−	0.254408i	$-0.0818805\pi$
$239$	7146.56	1.93419	0.967097	+	0.254408i	$0.0818805\pi$
$240$	0	0
$241$	−332.563	−0.0888892	−0.0444446	−	0.999012i	$-0.514152\pi$
$241$	−332.563	−0.0888892	−0.0444446	+	0.999012i	$0.514152\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−818.530	−0.213445
$246$	0	0
$247$	−716.725	−0.184632
$248$	0	0
$249$	555.544	0.141390
$250$	0	0
$251$	4833.25	1.21543	0.607713	−	0.794157i	$-0.292087\pi$
$251$	4833.25	1.21543	0.607713	+	0.794157i	$0.292087\pi$
$252$	0	0
$253$	−1728.15	−0.429438
$254$	0	0
$255$	165.141	0.0405550
$256$	0	0
$257$	−1372.18	−0.333052	−0.166526	−	0.986037i	$-0.553255\pi$
$257$	−1372.18	−0.333052	−0.166526	+	0.986037i	$0.553255\pi$
$258$	0	0
$259$	−1315.06	−0.315498
$260$	0	0
$261$	1944.85	0.461237
$262$	0	0
$263$	−1170.60	−0.274457	−0.137228	−	0.990539i	$-0.543819\pi$
$263$	−1170.60	−0.274457	−0.137228	+	0.990539i	$0.543819\pi$
$264$	0	0
$265$	−2049.41	−0.475073
$266$	0	0
$267$	4791.93	1.09836
$268$	0	0
$269$	−4260.36	−0.965645	−0.482822	−	0.875718i	$-0.660388\pi$
$269$	−4260.36	−0.965645	−0.482822	+	0.875718i	$0.660388\pi$
$270$	0	0
$271$	−2574.85	−0.577161	−0.288581	−	0.957456i	$-0.593183\pi$
$271$	−2574.85	−0.577161	−0.288581	+	0.957456i	$0.593183\pi$
$272$	0	0
$273$	−239.597	−0.0531176
$274$	0	0
$275$	−3497.59	−0.766955
$276$	0	0
$277$	6453.13	1.39975	0.699875	−	0.714265i	$-0.253239\pi$
$277$	6453.13	1.39975	0.699875	+	0.714265i	$0.253239\pi$
$278$	0	0
$279$	1346.74	0.288986
$280$	0	0
$281$	349.664	0.0742321	0.0371160	−	0.999311i	$-0.488183\pi$
$281$	349.664	0.0742321	0.0371160	+	0.999311i	$0.488183\pi$
$282$	0	0
$283$	699.865	0.147006	0.0735029	−	0.997295i	$-0.476582\pi$
$283$	699.865	0.147006	0.0735029	+	0.997295i	$0.476582\pi$
$284$	0	0
$285$	−3148.11	−0.654308
$286$	0	0
$287$	−163.067	−0.0335385
$288$	0	0
$289$	−4902.14	−0.997790
$290$	0	0
$291$	4817.07	0.970385
$292$	0	0
$293$	−162.087	−0.0323182	−0.0161591	−	0.999869i	$-0.505144\pi$
$293$	−162.087	−0.0323182	−0.0161591	+	0.999869i	$0.505144\pi$
$294$	0	0
$295$	3765.40	0.743152
$296$	0	0
$297$	613.027	0.119769
$298$	0	0
$299$	868.416	0.167966
$300$	0	0
$301$	1498.66	0.286981
$302$	0	0
$303$	4079.82	0.773530
$304$	0	0
$305$	−9123.91	−1.71290
$306$	0	0
$307$	−2185.16	−0.406233	−0.203116	−	0.979155i	$-0.565107\pi$
$307$	−2185.16	−0.406233	−0.203116	+	0.979155i	$0.565107\pi$
$308$	0	0
$309$	−5464.67	−1.00607
$310$	0	0
$311$	−4763.42	−0.868517	−0.434259	−	0.900788i	$-0.642990\pi$
$311$	−4763.42	−0.868517	−0.434259	+	0.900788i	$0.642990\pi$
$312$	0	0
$313$	−1080.84	−0.195185	−0.0975926	−	0.995226i	$-0.531114\pi$
$313$	−1080.84	−0.195185	−0.0975926	+	0.995226i	$0.531114\pi$
$314$	0	0
$315$	−1052.40	−0.188241
$316$	0	0
$317$	5832.91	1.03347	0.516734	−	0.856146i	$-0.327148\pi$
$317$	5832.91	1.03347	0.516734	+	0.856146i	$0.327148\pi$
$318$	0	0
$319$	−4906.35	−0.861137
$320$	0	0
$321$	4321.59	0.751426
$322$	0	0
$323$	−207.007	−0.0356600
$324$	0	0
$325$	1757.58	0.299979
$326$	0	0
$327$	2652.04	0.448496
$328$	0	0
$329$	−1315.72	−0.220480
$330$	0	0
$331$	3483.11	0.578397	0.289198	−	0.957269i	$-0.406611\pi$
$331$	3483.11	0.578397	0.289198	+	0.957269i	$0.406611\pi$
$332$	0	0
$333$	−1690.79	−0.278243
$334$	0	0
$335$	−5899.11	−0.962099
$336$	0	0
$337$	−6625.03	−1.07089	−0.535443	−	0.844571i	$-0.679855\pi$
$337$	−6625.03	−1.07089	−0.535443	+	0.844571i	$0.679855\pi$
$338$	0	0
$339$	4480.39	0.717821
$340$	0	0
$341$	−3397.48	−0.539542
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	3814.39	0.595246
$346$	0	0
$347$	5567.83	0.861374	0.430687	−	0.902501i	$-0.358271\pi$
$347$	5567.83	0.861374	0.430687	+	0.902501i	$0.358271\pi$
$348$	0	0
$349$	−4552.55	−0.698259	−0.349130	−	0.937074i	$-0.613523\pi$
$349$	−4552.55	−0.698259	−0.349130	+	0.937074i	$0.613523\pi$
$350$	0	0
$351$	−308.054	−0.0468453
$352$	0	0
$353$	−4627.15	−0.697672	−0.348836	−	0.937184i	$-0.613423\pi$
$353$	−4627.15	−0.697672	−0.348836	+	0.937184i	$0.613423\pi$
$354$	0	0
$355$	−7776.43	−1.16262
$356$	0	0
$357$	−69.2013	−0.0102592
$358$	0	0
$359$	−8347.95	−1.22726	−0.613632	−	0.789592i	$-0.710292\pi$
$359$	−8347.95	−1.22726	−0.613632	+	0.789592i	$0.710292\pi$
$360$	0	0
$361$	−2912.80	−0.424668
$362$	0	0
$363$	2446.49	0.353739
$364$	0	0
$365$	−7189.97	−1.03107
$366$	0	0
$367$	−11015.9	−1.56683	−0.783417	−	0.621497i	$-0.786525\pi$
$367$	−11015.9	−1.56683	−0.783417	+	0.621497i	$0.786525\pi$
$368$	0	0
$369$	−209.658	−0.0295782
$370$	0	0
$371$	858.792	0.120179
$372$	0	0
$373$	3459.93	0.480291	0.240145	−	0.970737i	$-0.422805\pi$
$373$	3459.93	0.480291	0.240145	+	0.970737i	$0.422805\pi$
$374$	0	0
$375$	1455.66	0.200454
$376$	0	0
$377$	2465.50	0.336817
$378$	0	0
$379$	13137.3	1.78053	0.890264	−	0.455445i	$-0.150520\pi$
$379$	13137.3	1.78053	0.890264	+	0.455445i	$0.150520\pi$
$380$	0	0
$381$	−4264.23	−0.573394
$382$	0	0
$383$	−11542.6	−1.53995	−0.769973	−	0.638076i	$-0.779731\pi$
$383$	−11542.6	−1.53995	−0.769973	+	0.638076i	$0.779731\pi$
$384$	0	0
$385$	2654.93	0.351448
$386$	0	0
$387$	1926.85	0.253093
$388$	0	0
$389$	13580.9	1.77012	0.885061	−	0.465475i	$-0.154116\pi$
$389$	13580.9	1.77012	0.885061	+	0.465475i	$0.154116\pi$
$390$	0	0
$391$	250.819	0.0324411
$392$	0	0
$393$	2265.26	0.290757
$394$	0	0
$395$	−8895.20	−1.13308
$396$	0	0
$397$	−15560.6	−1.96717	−0.983584	−	0.180450i	$-0.942245\pi$
$397$	−15560.6	−1.96717	−0.983584	+	0.180450i	$0.942245\pi$
$398$	0	0
$399$	1319.19	0.165520
$400$	0	0
$401$	320.322	0.0398906	0.0199453	−	0.999801i	$-0.493651\pi$
$401$	320.322	0.0398906	0.0199453	+	0.999801i	$0.493651\pi$
$402$	0	0
$403$	1707.28	0.211031
$404$	0	0
$405$	−1353.08	−0.166013
$406$	0	0
$407$	4265.44	0.519484
$408$	0	0
$409$	−2303.79	−0.278521	−0.139260	−	0.990256i	$-0.544472\pi$
$409$	−2303.79	−0.278521	−0.139260	+	0.990256i	$0.544472\pi$
$410$	0	0
$411$	−4856.94	−0.582908
$412$	0	0
$413$	−1577.87	−0.187994
$414$	0	0
$415$	3093.40	0.365901
$416$	0	0
$417$	−4376.78	−0.513985
$418$	0	0
$419$	9770.19	1.13915	0.569576	−	0.821938i	$-0.307107\pi$
$419$	9770.19	1.13915	0.569576	+	0.821938i	$0.307107\pi$
$420$	0	0
$421$	918.792	0.106364	0.0531819	−	0.998585i	$-0.483064\pi$
$421$	918.792	0.106364	0.0531819	+	0.998585i	$0.483064\pi$
$422$	0	0
$423$	−1691.64	−0.194445
$424$	0	0
$425$	507.631	0.0579382
$426$	0	0
$427$	3823.32	0.433310
$428$	0	0
$429$	777.141	0.0874609
$430$	0	0
$431$	12570.7	1.40489	0.702446	−	0.711737i	$-0.252091\pi$
$431$	12570.7	1.40489	0.702446	+	0.711737i	$0.252091\pi$
$432$	0	0
$433$	8107.80	0.899852	0.449926	−	0.893066i	$-0.351450\pi$
$433$	8107.80	0.899852	0.449926	+	0.893066i	$0.351450\pi$
$434$	0	0
$435$	10829.4	1.19363
$436$	0	0
$437$	−4781.40	−0.523399
$438$	0	0
$439$	12395.2	1.34758	0.673792	−	0.738921i	$-0.264664\pi$
$439$	12395.2	1.34758	0.673792	+	0.738921i	$0.264664\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−934.503	−0.100225	−0.0501124	−	0.998744i	$-0.515958\pi$
$443$	−934.503	−0.100225	−0.0501124	+	0.998744i	$0.515958\pi$
$444$	0	0
$445$	26682.6	2.84242
$446$	0	0
$447$	9810.20	1.03805
$448$	0	0
$449$	13725.7	1.44266	0.721331	−	0.692591i	$-0.243531\pi$
$449$	13725.7	1.44266	0.721331	+	0.692591i	$0.243531\pi$
$450$	0	0
$451$	528.913	0.0552229
$452$	0	0
$453$	−3594.04	−0.372765
$454$	0	0
$455$	−1334.13	−0.137462
$456$	0	0
$457$	−15676.9	−1.60467	−0.802337	−	0.596872i	$-0.796410\pi$
$457$	−15676.9	−1.60467	−0.802337	+	0.596872i	$0.796410\pi$
$458$	0	0
$459$	−88.9731	−0.00904773
$460$	0	0
$461$	−19210.4	−1.94082	−0.970410	−	0.241463i	$-0.922373\pi$
$461$	−19210.4	−1.94082	−0.970410	+	0.241463i	$0.922373\pi$
$462$	0	0
$463$	−2935.18	−0.294621	−0.147310	−	0.989090i	$-0.547062\pi$
$463$	−2935.18	−0.294621	−0.147310	+	0.989090i	$0.547062\pi$
$464$	0	0
$465$	7498.95	0.747862
$466$	0	0
$467$	10685.1	1.05878	0.529389	−	0.848379i	$-0.322421\pi$
$467$	10685.1	1.05878	0.529389	+	0.848379i	$0.322421\pi$
$468$	0	0
$469$	2471.99	0.243381
$470$	0	0
$471$	−3767.24	−0.368546
$472$	0	0
$473$	−4860.94	−0.472529
$474$	0	0
$475$	−9677.05	−0.934765
$476$	0	0
$477$	1104.16	0.105988
$478$	0	0
$479$	6984.90	0.666280	0.333140	−	0.942877i	$-0.391892\pi$
$479$	6984.90	0.666280	0.333140	+	0.942877i	$0.391892\pi$
$480$	0	0
$481$	−2143.44	−0.203186
$482$	0	0
$483$	−1598.40	−0.150579
$484$	0	0
$485$	26822.6	2.51124
$486$	0	0
$487$	1157.50	0.107703	0.0538516	−	0.998549i	$-0.482850\pi$
$487$	1157.50	0.107703	0.0538516	+	0.998549i	$0.482850\pi$
$488$	0	0
$489$	3624.68	0.335202
$490$	0	0
$491$	−19036.2	−1.74968	−0.874839	−	0.484414i	$-0.839033\pi$
$491$	−19036.2	−1.74968	−0.874839	+	0.484414i	$0.839033\pi$
$492$	0	0
$493$	712.095	0.0650530
$494$	0	0
$495$	3413.48	0.309948
$496$	0	0
$497$	3258.66	0.294107
$498$	0	0
$499$	−13623.3	−1.22217	−0.611085	−	0.791565i	$-0.709267\pi$
$499$	−13623.3	−1.22217	−0.611085	+	0.791565i	$0.709267\pi$
$500$	0	0
$501$	8355.42	0.745095
$502$	0	0
$503$	−3211.73	−0.284700	−0.142350	−	0.989816i	$-0.545466\pi$
$503$	−3211.73	−0.284700	−0.142350	+	0.989816i	$0.545466\pi$
$504$	0	0
$505$	22717.4	2.00180
$506$	0	0
$507$	6200.48	0.543142
$508$	0	0
$509$	−5483.82	−0.477536	−0.238768	−	0.971077i	$-0.576744\pi$
$509$	−5483.82	−0.477536	−0.238768	+	0.971077i	$0.576744\pi$
$510$	0	0
$511$	3012.91	0.260829
$512$	0	0
$513$	1696.11	0.145975
$514$	0	0
$515$	−30428.6	−2.60358
$516$	0	0
$517$	4267.57	0.363032
$518$	0	0
$519$	2627.50	0.222224
$520$	0	0
$521$	16008.9	1.34618	0.673091	−	0.739560i	$-0.264966\pi$
$521$	16008.9	1.34618	0.673091	+	0.739560i	$0.264966\pi$
$522$	0	0
$523$	−12791.7	−1.06948	−0.534742	−	0.845016i	$-0.679591\pi$
$523$	−12791.7	−1.06948	−0.534742	+	0.845016i	$0.679591\pi$
$524$	0	0
$525$	−3234.99	−0.268926
$526$	0	0
$527$	493.101	0.0407586
$528$	0	0
$529$	−6373.64	−0.523847
$530$	0	0
$531$	−2028.68	−0.165796
$532$	0	0
$533$	−265.785	−0.0215993
$534$	0	0
$535$	24063.6	1.94460
$536$	0	0
$537$	−3446.15	−0.276932
$538$	0	0
$539$	−1112.53	−0.0889055
$540$	0	0
$541$	−7091.14	−0.563534	−0.281767	−	0.959483i	$-0.590921\pi$
$541$	−7091.14	−0.563534	−0.281767	+	0.959483i	$0.590921\pi$
$542$	0	0
$543$	5214.44	0.412105
$544$	0	0
$545$	14767.2	1.16065
$546$	0	0
$547$	4704.93	0.367766	0.183883	−	0.982948i	$-0.441133\pi$
$547$	4704.93	0.367766	0.183883	+	0.982948i	$0.441133\pi$
$548$	0	0
$549$	4915.69	0.382143
$550$	0	0
$551$	−13574.8	−1.04955
$552$	0	0
$553$	3727.48	0.286634
$554$	0	0
$555$	−9414.73	−0.720059
$556$	0	0
$557$	−3295.22	−0.250670	−0.125335	−	0.992114i	$-0.540001\pi$
$557$	−3295.22	−0.250670	−0.125335	+	0.992114i	$0.540001\pi$
$558$	0	0
$559$	2442.68	0.184820
$560$	0	0
$561$	224.456	0.0168923
$562$	0	0
$563$	−6620.17	−0.495572	−0.247786	−	0.968815i	$-0.579703\pi$
$563$	−6620.17	−0.495572	−0.247786	+	0.968815i	$0.579703\pi$
$564$	0	0
$565$	24947.9	1.85764
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	16960.6	1.24961	0.624804	−	0.780781i	$-0.285179\pi$
$569$	16960.6	1.24961	0.624804	+	0.780781i	$0.285179\pi$
$570$	0	0
$571$	15713.6	1.15165	0.575826	−	0.817572i	$-0.304680\pi$
$571$	15713.6	1.15165	0.575826	+	0.817572i	$0.304680\pi$
$572$	0	0
$573$	11338.0	0.826618
$574$	0	0
$575$	11725.1	0.850387
$576$	0	0
$577$	26793.7	1.93317	0.966583	−	0.256354i	$-0.0825214\pi$
$577$	26793.7	1.93317	0.966583	+	0.256354i	$0.0825214\pi$
$578$	0	0
$579$	11898.4	0.854028
$580$	0	0
$581$	−1296.27	−0.0925616
$582$	0	0
$583$	−2785.52	−0.197880
$584$	0	0
$585$	−1715.32	−0.121230
$586$	0	0
$587$	−25425.5	−1.78777	−0.893885	−	0.448296i	$-0.852031\pi$
$587$	−25425.5	−1.78777	−0.893885	+	0.448296i	$0.852031\pi$
$588$	0	0
$589$	−9400.05	−0.657593
$590$	0	0
$591$	7779.14	0.541440
$592$	0	0
$593$	−10089.0	−0.698660	−0.349330	−	0.937000i	$-0.613591\pi$
$593$	−10089.0	−0.698660	−0.349330	+	0.937000i	$0.613591\pi$
$594$	0	0
$595$	−385.329	−0.0265495
$596$	0	0
$597$	13223.3	0.906520
$598$	0	0
$599$	−17402.3	−1.18704	−0.593520	−	0.804819i	$-0.702262\pi$
$599$	−17402.3	−1.18704	−0.593520	+	0.804819i	$0.702262\pi$
$600$	0	0
$601$	2731.58	0.185397	0.0926985	−	0.995694i	$-0.470451\pi$
$601$	2731.58	0.185397	0.0926985	+	0.995694i	$0.470451\pi$
$602$	0	0
$603$	3178.27	0.214642
$604$	0	0
$605$	13622.6	0.915436
$606$	0	0
$607$	−5374.07	−0.359352	−0.179676	−	0.983726i	$-0.557505\pi$
$607$	−5374.07	−0.359352	−0.179676	+	0.983726i	$0.557505\pi$
$608$	0	0
$609$	−4537.97	−0.301951
$610$	0	0
$611$	−2144.51	−0.141993
$612$	0	0
$613$	24411.6	1.60844	0.804222	−	0.594330i	$-0.202583\pi$
$613$	24411.6	1.60844	0.804222	+	0.594330i	$0.202583\pi$
$614$	0	0
$615$	−1167.42	−0.0765448
$616$	0	0
$617$	8070.23	0.526572	0.263286	−	0.964718i	$-0.415194\pi$
$617$	8070.23	0.526572	0.263286	+	0.964718i	$0.415194\pi$
$618$	0	0
$619$	22451.4	1.45783	0.728915	−	0.684604i	$-0.240025\pi$
$619$	22451.4	1.45783	0.728915	+	0.684604i	$0.240025\pi$
$620$	0	0
$621$	−2055.08	−0.132798
$622$	0	0
$623$	−11181.2	−0.719043
$624$	0	0
$625$	−11150.4	−0.713625
$626$	0	0
$627$	−4278.85	−0.272537
$628$	0	0
$629$	−619.074	−0.0392434
$630$	0	0
$631$	23936.0	1.51011	0.755054	−	0.655662i	$-0.227610\pi$
$631$	23936.0	1.51011	0.755054	+	0.655662i	$0.227610\pi$
$632$	0	0
$633$	−11235.4	−0.705476
$634$	0	0
$635$	−23744.2	−1.48387
$636$	0	0
$637$	559.061	0.0347736
$638$	0	0
$639$	4189.71	0.259378
$640$	0	0
$641$	−10470.3	−0.645164	−0.322582	−	0.946542i	$-0.604551\pi$
$641$	−10470.3	−0.645164	−0.322582	+	0.946542i	$0.604551\pi$
$642$	0	0
$643$	−7322.20	−0.449081	−0.224541	−	0.974465i	$-0.572088\pi$
$643$	−7322.20	−0.449081	−0.224541	+	0.974465i	$0.572088\pi$
$644$	0	0
$645$	10729.1	0.654975
$646$	0	0
$647$	10714.9	0.651074	0.325537	−	0.945529i	$-0.394455\pi$
$647$	10714.9	0.651074	0.325537	+	0.945529i	$0.394455\pi$
$648$	0	0
$649$	5117.85	0.309543
$650$	0	0
$651$	−3142.39	−0.189186
$652$	0	0
$653$	−265.840	−0.0159313	−0.00796565	−	0.999968i	$-0.502536\pi$
$653$	−265.840	−0.0159313	−0.00796565	+	0.999968i	$0.502536\pi$
$654$	0	0
$655$	12613.5	0.752443
$656$	0	0
$657$	3873.75	0.230029
$658$	0	0
$659$	1989.35	0.117593	0.0587967	−	0.998270i	$-0.481274\pi$
$659$	1989.35	0.117593	0.0587967	+	0.998270i	$0.481274\pi$
$660$	0	0
$661$	5879.26	0.345956	0.172978	−	0.984926i	$-0.444661\pi$
$661$	5879.26	0.345956	0.172978	+	0.984926i	$0.444661\pi$
$662$	0	0
$663$	−112.792	−0.00660707
$664$	0	0
$665$	7345.58	0.428345
$666$	0	0
$667$	16447.8	0.954815
$668$	0	0
$669$	3597.50	0.207904
$670$	0	0
$671$	−12401.0	−0.713467
$672$	0	0
$673$	−16670.1	−0.954805	−0.477403	−	0.878685i	$-0.658422\pi$
$673$	−16670.1	−0.954805	−0.477403	+	0.878685i	$0.658422\pi$
$674$	0	0
$675$	−4159.27	−0.237171
$676$	0	0
$677$	13648.0	0.774795	0.387398	−	0.921913i	$-0.373374\pi$
$677$	13648.0	0.774795	0.387398	+	0.921913i	$0.373374\pi$
$678$	0	0
$679$	−11239.8	−0.635266
$680$	0	0
$681$	13134.2	0.739068
$682$	0	0
$683$	−811.575	−0.0454671	−0.0227336	−	0.999742i	$-0.507237\pi$
$683$	−811.575	−0.0454671	−0.0227336	+	0.999742i	$0.507237\pi$
$684$	0	0
$685$	−27044.6	−1.50850
$686$	0	0
$687$	−16556.5	−0.919459
$688$	0	0
$689$	1399.76	0.0773970
$690$	0	0
$691$	4615.20	0.254082	0.127041	−	0.991897i	$-0.459452\pi$
$691$	4615.20	0.254082	0.127041	+	0.991897i	$0.459452\pi$
$692$	0	0
$693$	−1430.40	−0.0784073
$694$	0	0
$695$	−24370.9	−1.33013
$696$	0	0
$697$	−76.7650	−0.00417171
$698$	0	0
$699$	−7535.36	−0.407744
$700$	0	0
$701$	6054.51	0.326214	0.163107	−	0.986608i	$-0.447848\pi$
$701$	6054.51	0.326214	0.163107	+	0.986608i	$0.447848\pi$
$702$	0	0
$703$	11801.5	0.633147
$704$	0	0
$705$	−9419.44	−0.503201
$706$	0	0
$707$	−9519.58	−0.506394
$708$	0	0
$709$	−19774.2	−1.04744	−0.523721	−	0.851890i	$-0.675457\pi$
$709$	−19774.2	−1.04744	−0.523721	+	0.851890i	$0.675457\pi$
$710$	0	0
$711$	4792.47	0.252787
$712$	0	0
$713$	11389.5	0.598235
$714$	0	0
$715$	4327.30	0.226338
$716$	0	0
$717$	−21439.7	−1.11671
$718$	0	0
$719$	−26139.8	−1.35584	−0.677920	−	0.735135i	$-0.737119\pi$
$719$	−26139.8	−1.35584	−0.677920	+	0.735135i	$0.737119\pi$
$720$	0	0
$721$	12750.9	0.658625
$722$	0	0
$723$	997.690	0.0513202
$724$	0	0
$725$	33288.6	1.70525
$726$	0	0
$727$	19081.7	0.973454	0.486727	−	0.873554i	$-0.338191\pi$
$727$	19081.7	0.973454	0.486727	+	0.873554i	$0.338191\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	705.504	0.0356963
$732$	0	0
$733$	34359.5	1.73138	0.865688	−	0.500584i	$-0.166881\pi$
$733$	34359.5	1.73138	0.865688	+	0.500584i	$0.166881\pi$
$734$	0	0
$735$	2455.59	0.123232
$736$	0	0
$737$	−8017.96	−0.400740
$738$	0	0
$739$	−4097.73	−0.203975	−0.101988	−	0.994786i	$-0.532520\pi$
$739$	−4097.73	−0.203975	−0.101988	+	0.994786i	$0.532520\pi$
$740$	0	0
$741$	2150.17	0.106597
$742$	0	0
$743$	5253.98	0.259421	0.129711	−	0.991552i	$-0.458595\pi$
$743$	5253.98	0.259421	0.129711	+	0.991552i	$0.458595\pi$
$744$	0	0
$745$	54625.5	2.68634
$746$	0	0
$747$	−1666.63	−0.0816316
$748$	0	0
$749$	−10083.7	−0.491924
$750$	0	0
$751$	−128.052	−0.00622195	−0.00311098	−	0.999995i	$-0.500990\pi$
$751$	−128.052	−0.00622195	−0.00311098	+	0.999995i	$0.500990\pi$
$752$	0	0
$753$	−14499.7	−0.701727
$754$	0	0
$755$	−20012.5	−0.964673
$756$	0	0
$757$	−16625.5	−0.798235	−0.399117	−	0.916900i	$-0.630683\pi$
$757$	−16625.5	−0.798235	−0.399117	+	0.916900i	$0.630683\pi$
$758$	0	0
$759$	5184.44	0.247936
$760$	0	0
$761$	−29091.6	−1.38577	−0.692885	−	0.721048i	$-0.743661\pi$
$761$	−29091.6	−1.38577	−0.692885	+	0.721048i	$0.743661\pi$
$762$	0	0
$763$	−6188.10	−0.293610
$764$	0	0
$765$	−495.423	−0.0234145
$766$	0	0
$767$	−2571.79	−0.121071
$768$	0	0
$769$	−2004.69	−0.0940063	−0.0470032	−	0.998895i	$-0.514967\pi$
$769$	−2004.69	−0.0940063	−0.0470032	+	0.998895i	$0.514967\pi$
$770$	0	0
$771$	4116.54	0.192288
$772$	0	0
$773$	−10983.8	−0.511071	−0.255536	−	0.966800i	$-0.582252\pi$
$773$	−10983.8	−0.511071	−0.255536	+	0.966800i	$0.582252\pi$
$774$	0	0
$775$	23051.2	1.06842
$776$	0	0
$777$	3945.18	0.182153
$778$	0	0
$779$	1463.38	0.0673057
$780$	0	0
$781$	−10569.6	−0.484262
$782$	0	0
$783$	−5834.54	−0.266295
$784$	0	0
$785$	−20976.8	−0.953753
$786$	0	0
$787$	21513.7	0.974436	0.487218	−	0.873280i	$-0.338012\pi$
$787$	21513.7	0.974436	0.487218	+	0.873280i	$0.338012\pi$
$788$	0	0
$789$	3511.79	0.158458
$790$	0	0
$791$	−10454.2	−0.469924
$792$	0	0
$793$	6231.68	0.279058
$794$	0	0
$795$	6148.23	0.274283
$796$	0	0
$797$	−5231.43	−0.232505	−0.116253	−	0.993220i	$-0.537088\pi$
$797$	−5231.43	−0.232505	−0.116253	+	0.993220i	$0.537088\pi$
$798$	0	0
$799$	−619.384	−0.0274246
$800$	0	0
$801$	−14375.8	−0.634136
$802$	0	0
$803$	−9772.47	−0.429468
$804$	0	0
$805$	−8900.24	−0.389680
$806$	0	0
$807$	12781.1	0.557515
$808$	0	0
$809$	−18118.0	−0.787384	−0.393692	−	0.919242i	$-0.628802\pi$
$809$	−18118.0	−0.787384	−0.393692	+	0.919242i	$0.628802\pi$
$810$	0	0
$811$	−34536.2	−1.49535	−0.747676	−	0.664064i	$-0.768830\pi$
$811$	−34536.2	−1.49535	−0.747676	+	0.664064i	$0.768830\pi$
$812$	0	0
$813$	7724.54	0.333224
$814$	0	0
$815$	20183.1	0.867464
$816$	0	0
$817$	−13449.1	−0.575919
$818$	0	0
$819$	718.792	0.0306674
$820$	0	0
$821$	10122.4	0.430297	0.215148	−	0.976581i	$-0.430976\pi$
$821$	10122.4	0.430297	0.215148	+	0.976581i	$0.430976\pi$
$822$	0	0
$823$	−1623.67	−0.0687696	−0.0343848	−	0.999409i	$-0.510947\pi$
$823$	−1623.67	−0.0687696	−0.0343848	+	0.999409i	$0.510947\pi$
$824$	0	0
$825$	10492.8	0.442802
$826$	0	0
$827$	23112.3	0.971819	0.485910	−	0.874009i	$-0.338488\pi$
$827$	23112.3	0.971819	0.485910	+	0.874009i	$0.338488\pi$
$828$	0	0
$829$	−23292.5	−0.975851	−0.487925	−	0.872885i	$-0.662246\pi$
$829$	−23292.5	−0.975851	−0.487925	+	0.872885i	$0.662246\pi$
$830$	0	0
$831$	−19359.4	−0.808146
$832$	0	0
$833$	161.470	0.00671620
$834$	0	0
$835$	46524.9	1.92822
$836$	0	0
$837$	−4040.22	−0.166846
$838$	0	0
$839$	−21603.7	−0.888968	−0.444484	−	0.895787i	$-0.646613\pi$
$839$	−21603.7	−0.888968	−0.444484	+	0.895787i	$0.646613\pi$
$840$	0	0
$841$	22307.6	0.914659
$842$	0	0
$843$	−1048.99	−0.0428579
$844$	0	0
$845$	34525.7	1.40559
$846$	0	0
$847$	−5708.48	−0.231577
$848$	0	0
$849$	−2099.60	−0.0848739
$850$	0	0
$851$	−14299.2	−0.575995
$852$	0	0
$853$	−38664.5	−1.55199	−0.775995	−	0.630739i	$-0.782752\pi$
$853$	−38664.5	−1.55199	−0.775995	+	0.630739i	$0.782752\pi$
$854$	0	0
$855$	9444.32	0.377765
$856$	0	0
$857$	9703.63	0.386779	0.193390	−	0.981122i	$-0.438052\pi$
$857$	9703.63	0.386779	0.193390	+	0.981122i	$0.438052\pi$
$858$	0	0
$859$	−21634.2	−0.859312	−0.429656	−	0.902993i	$-0.641365\pi$
$859$	−21634.2	−0.859312	−0.429656	+	0.902993i	$0.641365\pi$
$860$	0	0
$861$	489.201	0.0193635
$862$	0	0
$863$	27307.4	1.07712	0.538561	−	0.842587i	$-0.318968\pi$
$863$	27307.4	1.07712	0.538561	+	0.842587i	$0.318968\pi$
$864$	0	0
$865$	14630.5	0.575089
$866$	0	0
$867$	14706.4	0.576074
$868$	0	0
$869$	−12090.2	−0.471958
$870$	0	0
$871$	4029.13	0.156741
$872$	0	0
$873$	−14451.2	−0.560252
$874$	0	0
$875$	−3396.55	−0.131228
$876$	0	0
$877$	−32605.7	−1.25544	−0.627718	−	0.778441i	$-0.716011\pi$
$877$	−32605.7	−1.25544	−0.627718	+	0.778441i	$0.716011\pi$
$878$	0	0
$879$	486.261	0.0186589
$880$	0	0
$881$	−19306.3	−0.738303	−0.369151	−	0.929369i	$-0.620352\pi$
$881$	−19306.3	−0.738303	−0.369151	+	0.929369i	$0.620352\pi$
$882$	0	0
$883$	14203.0	0.541299	0.270650	−	0.962678i	$-0.412761\pi$
$883$	14203.0	0.541299	0.270650	+	0.962678i	$0.412761\pi$
$884$	0	0
$885$	−11296.2	−0.429059
$886$	0	0
$887$	1468.77	0.0555991	0.0277996	−	0.999614i	$-0.491150\pi$
$887$	1468.77	0.0555991	0.0277996	+	0.999614i	$0.491150\pi$
$888$	0	0
$889$	9949.87	0.375374
$890$	0	0
$891$	−1839.08	−0.0691487
$892$	0	0
$893$	11807.4	0.442463
$894$	0	0
$895$	−19189.0	−0.716667
$896$	0	0
$897$	−2605.25	−0.0969752
$898$	0	0
$899$	32335.8	1.19962
$900$	0	0
$901$	404.283	0.0149485
$902$	0	0
$903$	−4495.97	−0.165688
$904$	0	0
$905$	29035.2	1.06648
$906$	0	0
$907$	4131.84	0.151263	0.0756314	−	0.997136i	$-0.475903\pi$
$907$	4131.84	0.151263	0.0756314	+	0.997136i	$0.475903\pi$
$908$	0	0
$909$	−12239.5	−0.446598
$910$	0	0
$911$	−45434.5	−1.65237	−0.826187	−	0.563395i	$-0.809495\pi$
$911$	−45434.5	−1.65237	−0.826187	+	0.563395i	$0.809495\pi$
$912$	0	0
$913$	4204.48	0.152408
$914$	0	0
$915$	27371.7	0.988941
$916$	0	0
$917$	−5285.61	−0.190345
$918$	0	0
$919$	−14032.4	−0.503685	−0.251842	−	0.967768i	$-0.581036\pi$
$919$	−14032.4	−0.503685	−0.251842	+	0.967768i	$0.581036\pi$
$920$	0	0
$921$	6555.47	0.234538
$922$	0	0
$923$	5311.34	0.189410
$924$	0	0
$925$	−28940.2	−1.02870
$926$	0	0
$927$	16394.0	0.580852
$928$	0	0
$929$	−48732.4	−1.72105	−0.860527	−	0.509405i	$-0.829866\pi$
$929$	−48732.4	−1.72105	−0.860527	+	0.509405i	$0.829866\pi$
$930$	0	0
$931$	−3078.12	−0.108358
$932$	0	0
$933$	14290.3	0.501439
$934$	0	0
$935$	1249.83	0.0437152
$936$	0	0
$937$	12813.8	0.446755	0.223378	−	0.974732i	$-0.428292\pi$
$937$	12813.8	0.446755	0.223378	+	0.974732i	$0.428292\pi$
$938$	0	0
$939$	3242.53	0.112690
$940$	0	0
$941$	−28586.5	−0.990323	−0.495162	−	0.868801i	$-0.664891\pi$
$941$	−28586.5	−0.990323	−0.495162	+	0.868801i	$0.664891\pi$
$942$	0	0
$943$	−1773.10	−0.0612302
$944$	0	0
$945$	3157.19	0.108681
$946$	0	0
$947$	−32669.6	−1.12103	−0.560517	−	0.828143i	$-0.689397\pi$
$947$	−32669.6	−1.12103	−0.560517	+	0.828143i	$0.689397\pi$
$948$	0	0
$949$	4910.79	0.167978
$950$	0	0
$951$	−17498.7	−0.596672
$952$	0	0
$953$	−3583.93	−0.121821	−0.0609103	−	0.998143i	$-0.519400\pi$
$953$	−3583.93	−0.121821	−0.0609103	+	0.998143i	$0.519400\pi$
$954$	0	0
$955$	63132.7	2.13919
$956$	0	0
$957$	14719.0	0.497178
$958$	0	0
$959$	11332.9	0.381603
$960$	0	0
$961$	−7399.59	−0.248383
$962$	0	0
$963$	−12964.8	−0.433836
$964$	0	0
$965$	66253.3	2.21012
$966$	0	0
$967$	6332.28	0.210582	0.105291	−	0.994441i	$-0.466423\pi$
$967$	6332.28	0.210582	0.105291	+	0.994441i	$0.466423\pi$
$968$	0	0
$969$	621.020	0.0205883
$970$	0	0
$971$	21547.7	0.712152	0.356076	−	0.934457i	$-0.384114\pi$
$971$	21547.7	0.712152	0.356076	+	0.934457i	$0.384114\pi$
$972$	0	0
$973$	10212.5	0.336482
$974$	0	0
$975$	−5272.75	−0.173193
$976$	0	0
$977$	−15024.3	−0.491987	−0.245994	−	0.969271i	$-0.579114\pi$
$977$	−15024.3	−0.491987	−0.245994	+	0.969271i	$0.579114\pi$
$978$	0	0
$979$	36266.4	1.18394
$980$	0	0
$981$	−7956.12	−0.258939
$982$	0	0
$983$	−22229.5	−0.721271	−0.360636	−	0.932707i	$-0.617440\pi$
$983$	−22229.5	−0.721271	−0.360636	+	0.932707i	$0.617440\pi$
$984$	0	0
$985$	43316.1	1.40118
$986$	0	0
$987$	3947.16	0.127294
$988$	0	0
$989$	16295.6	0.523932
$990$	0	0
$991$	−37009.0	−1.18631	−0.593154	−	0.805089i	$-0.702117\pi$
$991$	−37009.0	−1.18631	−0.593154	+	0.805089i	$0.702117\pi$
$992$	0	0
$993$	−10449.3	−0.333937
$994$	0	0
$995$	73630.3	2.34597
$996$	0	0
$997$	15364.0	0.488048	0.244024	−	0.969769i	$-0.421532\pi$
$997$	15364.0	0.488048	0.244024	+	0.969769i	$0.421532\pi$
$998$	0	0
$999$	5072.38	0.160644

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.be.1.1		2
4.3	odd	2		1344.4.a.bm.1.1		2
8.3	odd	2		672.4.a.h.1.2	✓	2
8.5	even	2		672.4.a.m.1.2	yes	2
24.5	odd	2		2016.4.a.j.1.1		2
24.11	even	2		2016.4.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.h.1.2	✓	2	8.3	odd	2
672.4.a.m.1.2	yes	2	8.5	even	2
1344.4.a.be.1.1		2	1.1	even	1	trivial
1344.4.a.bm.1.1		2	4.3	odd	2
2016.4.a.i.1.1		2	24.11	even	2
2016.4.a.j.1.1		2	24.5	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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -16.7047 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -16.7047 q^{5} +7.00000 q^{7} +9.00000 q^{9} -22.7047 q^{11} +11.4094 q^{13} +50.1141 q^{15} +3.29530 q^{17} -62.8188 q^{19} -21.0000 q^{21} +76.1141 q^{23} +154.047 q^{25} -27.0000 q^{27} +216.094 q^{29} +149.638 q^{31} +68.1141 q^{33} -116.933 q^{35} -187.866 q^{37} -34.2282 q^{39} -23.2953 q^{41} +214.094 q^{43} -150.342 q^{45} -187.960 q^{47} +49.0000 q^{49} -9.88590 q^{51} +122.685 q^{53} +379.275 q^{55} +188.456 q^{57} -225.409 q^{59} +546.188 q^{61} +63.0000 q^{63} -190.591 q^{65} +353.141 q^{67} -228.342 q^{69} +465.523 q^{71} +430.416 q^{73} -462.141 q^{75} -158.933 q^{77} +532.497 q^{79} +81.0000 q^{81} -185.181 q^{83} -55.0470 q^{85} -648.282 q^{87} -1597.31 q^{89} +79.8658 q^{91} -448.913 q^{93} +1049.37 q^{95} -1605.69 q^{97} -204.342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9	\( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} - 22 q^{11} - 24 q^{13} + 30 q^{15} + 30 q^{17} - 32 q^{19} - 42 q^{21} + 82 q^{23} + 74 q^{25} - 54 q^{27} - 36 q^{29} + 112 q^{31} + 66 q^{33} - 70 q^{35} - 48 q^{37} + 72 q^{39} - 70 q^{41} - 40 q^{43} - 90 q^{45} + 420 q^{47} + 98 q^{49} - 90 q^{51} - 176 q^{53} + 384 q^{55} + 96 q^{57} - 404 q^{59} + 156 q^{61} + 126 q^{63} - 428 q^{65} + 4 q^{67} - 246 q^{69} + 814 q^{71} - 216 q^{73} - 222 q^{75} - 154 q^{77} + 1580 q^{79} + 162 q^{81} - 464 q^{83} + 124 q^{85} + 108 q^{87} - 1158 q^{89} - 168 q^{91} - 336 q^{93} + 1256 q^{95} - 1760 q^{97} - 198 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 - 10 * q^5 + 14 * q^7 + 18 * q^9 - 22 * q^11 - 24 * q^13 + 30 * q^15 + 30 * q^17 - 32 * q^19 - 42 * q^21 + 82 * q^23 + 74 * q^25 - 54 * q^27 - 36 * q^29 + 112 * q^31 + 66 * q^33 - 70 * q^35 - 48 * q^37 + 72 * q^39 - 70 * q^41 - 40 * q^43 - 90 * q^45 + 420 * q^47 + 98 * q^49 - 90 * q^51 - 176 * q^53 + 384 * q^55 + 96 * q^57 - 404 * q^59 + 156 * q^61 + 126 * q^63 - 428 * q^65 + 4 * q^67 - 246 * q^69 + 814 * q^71 - 216 * q^73 - 222 * q^75 - 154 * q^77 + 1580 * q^79 + 162 * q^81 - 464 * q^83 + 124 * q^85 + 108 * q^87 - 1158 * q^89 - 168 * q^91 - 336 * q^93 + 1256 * q^95 - 1760 * q^97 - 198 * q^99

Introduction

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