LMFDB - Embedded newform 1344.4.a.bp.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.54138$ 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)

$f(q)$	$=$	$q+3.00000 q^{3} +14.1655 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +14.1655 q^{5} -7.00000 q^{7} +9.00000 q^{9} -11.8345 q^{11} -22.6621 q^{13} +42.4966 q^{15} +79.1587 q^{17} -55.6689 q^{19} -21.0000 q^{21} +84.8276 q^{23} +75.6621 q^{25} +27.0000 q^{27} +117.007 q^{29} -126.979 q^{31} -35.5034 q^{33} -99.1587 q^{35} +201.311 q^{37} -67.9863 q^{39} -10.4966 q^{41} -2.64840 q^{43} +127.490 q^{45} +494.290 q^{47} +49.0000 q^{49} +237.476 q^{51} +763.959 q^{53} -167.642 q^{55} -167.007 q^{57} +55.6416 q^{59} +457.007 q^{61} -63.0000 q^{63} -321.021 q^{65} +291.034 q^{67} +254.483 q^{69} +956.442 q^{71} -517.669 q^{73} +226.986 q^{75} +82.8413 q^{77} -830.290 q^{79} +81.0000 q^{81} -346.621 q^{83} +1121.32 q^{85} +351.021 q^{87} -584.401 q^{89} +158.635 q^{91} -380.938 q^{93} -788.580 q^{95} +85.6963 q^{97} -106.510 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 4 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 + 4 * q^5 - 14 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} + 4 q^{5} - 14 q^{7} + 18 q^{9} - 48 q^{11} + 52 q^{13} + 12 q^{15} - 12 q^{17} - 160 q^{19} - 42 q^{21} + 48 q^{23} + 54 q^{25} + 54 q^{27} + 380 q^{29} + 184 q^{31} - 144 q^{33} - 28 q^{35} - 84 q^{37} + 156 q^{39} + 52 q^{41} + 384 q^{43} + 36 q^{45} + 64 q^{47} + 98 q^{49} - 36 q^{51} + 652 q^{53} + 200 q^{55} - 480 q^{57} - 424 q^{59} + 1060 q^{61} - 126 q^{63} - 1080 q^{65} + 1312 q^{67} + 144 q^{69} + 672 q^{71} - 1084 q^{73} + 162 q^{75} + 336 q^{77} - 736 q^{79} + 162 q^{81} + 280 q^{83} + 2048 q^{85} + 1140 q^{87} + 948 q^{89} - 364 q^{91} + 552 q^{93} + 272 q^{95} + 804 q^{97} - 432 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 4 * q^5 - 14 * q^7 + 18 * q^9 - 48 * q^11 + 52 * q^13 + 12 * q^15 - 12 * q^17 - 160 * q^19 - 42 * q^21 + 48 * q^23 + 54 * q^25 + 54 * q^27 + 380 * q^29 + 184 * q^31 - 144 * q^33 - 28 * q^35 - 84 * q^37 + 156 * q^39 + 52 * q^41 + 384 * q^43 + 36 * q^45 + 64 * q^47 + 98 * q^49 - 36 * q^51 + 652 * q^53 + 200 * q^55 - 480 * q^57 - 424 * q^59 + 1060 * q^61 - 126 * q^63 - 1080 * q^65 + 1312 * q^67 + 144 * q^69 + 672 * q^71 - 1084 * q^73 + 162 * q^75 + 336 * q^77 - 736 * q^79 + 162 * q^81 + 280 * q^83 + 2048 * q^85 + 1140 * q^87 + 948 * q^89 - 364 * q^91 + 552 * q^93 + 272 * q^95 + 804 * q^97 - 432 * 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
$5$	14.1655	1.26700	0.633502	−	0.773741i	$-0.281617\pi$
$5$	14.1655	1.26700	0.633502	+	0.773741i	$0.281617\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.8345	−0.324384	−0.162192	−	0.986759i	$-0.551856\pi$
$11$	−11.8345	−0.324384	−0.162192	+	0.986759i	$0.551856\pi$
$12$	0	0
$13$	−22.6621	−0.483487	−0.241744	−	0.970340i	$-0.577719\pi$
$13$	−22.6621	−0.483487	−0.241744	+	0.970340i	$0.577719\pi$
$14$	0	0
$15$	42.4966	0.731505
$16$	0	0
$17$	79.1587	1.12934	0.564671	−	0.825316i	$-0.309003\pi$
$17$	79.1587	1.12934	0.564671	+	0.825316i	$0.309003\pi$
$18$	0	0
$19$	−55.6689	−0.672175	−0.336088	−	0.941831i	$-0.609104\pi$
$19$	−55.6689	−0.672175	−0.336088	+	0.941831i	$0.609104\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	84.8276	0.769034	0.384517	−	0.923118i	$-0.374368\pi$
$23$	84.8276	0.769034	0.384517	+	0.923118i	$0.374368\pi$
$24$	0	0
$25$	75.6621	0.605297
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	117.007	0.749229	0.374614	−	0.927181i	$-0.377775\pi$
$29$	117.007	0.749229	0.374614	+	0.927181i	$0.377775\pi$
$30$	0	0
$31$	−126.979	−0.735683	−0.367842	−	0.929888i	$-0.619903\pi$
$31$	−126.979	−0.735683	−0.367842	+	0.929888i	$0.619903\pi$
$32$	0	0
$33$	−35.5034	−0.187283
$34$	0	0
$35$	−99.1587	−0.478882
$36$	0	0
$37$	201.311	0.894466	0.447233	−	0.894417i	$-0.352409\pi$
$37$	201.311	0.894466	0.447233	+	0.894417i	$0.352409\pi$
$38$	0	0
$39$	−67.9863	−0.279142
$40$	0	0
$41$	−10.4966	−0.0399827	−0.0199913	−	0.999800i	$-0.506364\pi$
$41$	−10.4966	−0.0399827	−0.0199913	+	0.999800i	$0.506364\pi$
$42$	0	0
$43$	−2.64840	−0.00939250	−0.00469625	−	0.999989i	$-0.501495\pi$
$43$	−2.64840	−0.00939250	−0.00469625	+	0.999989i	$0.501495\pi$
$44$	0	0
$45$	127.490	0.422334
$46$	0	0
$47$	494.290	1.53403	0.767017	−	0.641627i	$-0.221740\pi$
$47$	494.290	1.53403	0.767017	+	0.641627i	$0.221740\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	237.476	0.652025
$52$	0	0
$53$	763.959	1.97996	0.989979	−	0.141211i	$-0.0450997\pi$
$53$	763.959	1.97996	0.989979	+	0.141211i	$0.0450997\pi$
$54$	0	0
$55$	−167.642	−0.410996
$56$	0	0
$57$	−167.007	−0.388081
$58$	0	0
$59$	55.6416	0.122778	0.0613891	−	0.998114i	$-0.480447\pi$
$59$	55.6416	0.122778	0.0613891	+	0.998114i	$0.480447\pi$
$60$	0	0
$61$	457.007	0.959241	0.479621	−	0.877476i	$-0.340774\pi$
$61$	457.007	0.959241	0.479621	+	0.877476i	$0.340774\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−321.021	−0.612580
$66$	0	0
$67$	291.034	0.530679	0.265339	−	0.964155i	$-0.414516\pi$
$67$	291.034	0.530679	0.265339	+	0.964155i	$0.414516\pi$
$68$	0	0
$69$	254.483	0.444002
$70$	0	0
$71$	956.442	1.59871	0.799357	−	0.600856i	$-0.205173\pi$
$71$	956.442	1.59871	0.799357	+	0.600856i	$0.205173\pi$
$72$	0	0
$73$	−517.669	−0.829980	−0.414990	−	0.909826i	$-0.636215\pi$
$73$	−517.669	−0.829980	−0.414990	+	0.909826i	$0.636215\pi$
$74$	0	0
$75$	226.986	0.349468
$76$	0	0
$77$	82.8413	0.122606
$78$	0	0
$79$	−830.290	−1.18247	−0.591234	−	0.806500i	$-0.701359\pi$
$79$	−830.290	−1.18247	−0.591234	+	0.806500i	$0.701359\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−346.621	−0.458393	−0.229196	−	0.973380i	$-0.573610\pi$
$83$	−346.621	−0.458393	−0.229196	+	0.973380i	$0.573610\pi$
$84$	0	0
$85$	1121.32	1.43088
$86$	0	0
$87$	351.021	0.432567
$88$	0	0
$89$	−584.401	−0.696026	−0.348013	−	0.937490i	$-0.613144\pi$
$89$	−584.401	−0.696026	−0.348013	+	0.937490i	$0.613144\pi$
$90$	0	0
$91$	158.635	0.182741
$92$	0	0
$93$	−380.938	−0.424747
$94$	0	0
$95$	−788.580	−0.851648
$96$	0	0
$97$	85.6963	0.0897025	0.0448513	−	0.998994i	$-0.485719\pi$
$97$	85.6963	0.0897025	0.0448513	+	0.998994i	$0.485719\pi$
$98$	0	0
$99$	−106.510	−0.108128
$100$	0	0
$101$	519.876	0.512174	0.256087	−	0.966654i	$-0.417567\pi$
$101$	519.876	0.512174	0.256087	+	0.966654i	$0.417567\pi$
$102$	0	0
$103$	986.235	0.943462	0.471731	−	0.881742i	$-0.343629\pi$
$103$	986.235	0.943462	0.471731	+	0.881742i	$0.343629\pi$
$104$	0	0
$105$	−297.476	−0.276483
$106$	0	0
$107$	1154.79	1.04334	0.521670	−	0.853147i	$-0.325309\pi$
$107$	1154.79	1.04334	0.521670	+	0.853147i	$0.325309\pi$
$108$	0	0
$109$	−181.890	−0.159834	−0.0799172	−	0.996802i	$-0.525466\pi$
$109$	−181.890	−0.159834	−0.0799172	+	0.996802i	$0.525466\pi$
$110$	0	0
$111$	603.932	0.516420
$112$	0	0
$113$	985.228	0.820199	0.410099	−	0.912041i	$-0.365494\pi$
$113$	985.228	0.820199	0.410099	+	0.912041i	$0.365494\pi$
$114$	0	0
$115$	1201.63	0.974368
$116$	0	0
$117$	−203.959	−0.161162
$118$	0	0
$119$	−554.111	−0.426851
$120$	0	0
$121$	−1190.95	−0.894775
$122$	0	0
$123$	−31.4897	−0.0230840
$124$	0	0
$125$	−698.897	−0.500090
$126$	0	0
$127$	−1818.98	−1.27093	−0.635466	−	0.772129i	$-0.719192\pi$
$127$	−1818.98	−1.27093	−0.635466	+	0.772129i	$0.719192\pi$
$128$	0	0
$129$	−7.94520	−0.00542276
$130$	0	0
$131$	−949.269	−0.633115	−0.316557	−	0.948573i	$-0.602527\pi$
$131$	−949.269	−0.633115	−0.316557	+	0.948573i	$0.602527\pi$
$132$	0	0
$133$	389.683	0.254058
$134$	0	0
$135$	382.469	0.243835
$136$	0	0
$137$	−95.6553	−0.0596524	−0.0298262	−	0.999555i	$-0.509495\pi$
$137$	−95.6553	−0.0596524	−0.0298262	+	0.999555i	$0.509495\pi$
$138$	0	0
$139$	−1845.82	−1.12634	−0.563168	−	0.826343i	$-0.690417\pi$
$139$	−1845.82	−1.12634	−0.563168	+	0.826343i	$0.690417\pi$
$140$	0	0
$141$	1482.87	0.885675
$142$	0	0
$143$	268.194	0.156836
$144$	0	0
$145$	1657.46	0.949275
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	2463.93	1.35472	0.677360	−	0.735652i	$-0.263124\pi$
$149$	2463.93	1.35472	0.677360	+	0.735652i	$0.263124\pi$
$150$	0	0
$151$	−3093.79	−1.66735	−0.833674	−	0.552257i	$-0.813767\pi$
$151$	−3093.79	−1.66735	−0.833674	+	0.552257i	$0.813767\pi$
$152$	0	0
$153$	712.428	0.376447
$154$	0	0
$155$	−1798.73	−0.932113
$156$	0	0
$157$	651.655	0.331260	0.165630	−	0.986188i	$-0.447034\pi$
$157$	651.655	0.331260	0.165630	+	0.986188i	$0.447034\pi$
$158$	0	0
$159$	2291.88	1.14313
$160$	0	0
$161$	−593.793	−0.290668
$162$	0	0
$163$	1832.83	0.880725	0.440363	−	0.897820i	$-0.354850\pi$
$163$	1832.83	0.880725	0.440363	+	0.897820i	$0.354850\pi$
$164$	0	0
$165$	−502.925	−0.237289
$166$	0	0
$167$	−322.153	−0.149275	−0.0746376	−	0.997211i	$-0.523780\pi$
$167$	−322.153	−0.149275	−0.0746376	+	0.997211i	$0.523780\pi$
$168$	0	0
$169$	−1683.43	−0.766240
$170$	0	0
$171$	−501.021	−0.224058
$172$	0	0
$173$	3928.43	1.72643	0.863216	−	0.504834i	$-0.168446\pi$
$173$	3928.43	1.72643	0.863216	+	0.504834i	$0.168446\pi$
$174$	0	0
$175$	−529.635	−0.228781
$176$	0	0
$177$	166.925	0.0708860
$178$	0	0
$179$	417.076	0.174155	0.0870775	−	0.996202i	$-0.472247\pi$
$179$	417.076	0.174155	0.0870775	+	0.996202i	$0.472247\pi$
$180$	0	0
$181$	2318.42	0.952080	0.476040	−	0.879424i	$-0.342072\pi$
$181$	2318.42	0.952080	0.476040	+	0.879424i	$0.342072\pi$
$182$	0	0
$183$	1371.02	0.553818
$184$	0	0
$185$	2851.67	1.13329
$186$	0	0
$187$	−936.801	−0.366341
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−161.462	−0.0611676	−0.0305838	−	0.999532i	$-0.509737\pi$
$191$	−161.462	−0.0611676	−0.0305838	+	0.999532i	$0.509737\pi$
$192$	0	0
$193$	1919.38	0.715855	0.357927	−	0.933749i	$-0.383484\pi$
$193$	1919.38	0.715855	0.357927	+	0.933749i	$0.383484\pi$
$194$	0	0
$195$	−963.062	−0.353673
$196$	0	0
$197$	418.137	0.151223	0.0756117	−	0.997137i	$-0.475909\pi$
$197$	418.137	0.151223	0.0756117	+	0.997137i	$0.475909\pi$
$198$	0	0
$199$	2393.82	0.852731	0.426366	−	0.904551i	$-0.359794\pi$
$199$	2393.82	0.852731	0.426366	+	0.904551i	$0.359794\pi$
$200$	0	0
$201$	873.103	0.306388
$202$	0	0
$203$	−819.048	−0.283182
$204$	0	0
$205$	−148.689	−0.0506582
$206$	0	0
$207$	763.449	0.256345
$208$	0	0
$209$	658.813	0.218043
$210$	0	0
$211$	−3207.62	−1.04655	−0.523274	−	0.852165i	$-0.675289\pi$
$211$	−3207.62	−1.04655	−0.523274	+	0.852165i	$0.675289\pi$
$212$	0	0
$213$	2869.33	0.923018
$214$	0	0
$215$	−37.5160	−0.0119003
$216$	0	0
$217$	888.856	0.278062
$218$	0	0
$219$	−1553.01	−0.479189
$220$	0	0
$221$	−1793.90	−0.546022
$222$	0	0
$223$	−2189.24	−0.657410	−0.328705	−	0.944433i	$-0.606612\pi$
$223$	−2189.24	−0.657410	−0.328705	+	0.944433i	$0.606612\pi$
$224$	0	0
$225$	680.959	0.201766
$226$	0	0
$227$	1961.30	0.573463	0.286731	−	0.958011i	$-0.407431\pi$
$227$	1961.30	0.573463	0.286731	+	0.958011i	$0.407431\pi$
$228$	0	0
$229$	2616.73	0.755102	0.377551	−	0.925989i	$-0.376766\pi$
$229$	2616.73	0.755102	0.377551	+	0.925989i	$0.376766\pi$
$230$	0	0
$231$	248.524	0.0707865
$232$	0	0
$233$	1818.28	0.511241	0.255621	−	0.966777i	$-0.417720\pi$
$233$	1818.28	0.511241	0.255621	+	0.966777i	$0.417720\pi$
$234$	0	0
$235$	7001.88	1.94363
$236$	0	0
$237$	−2490.87	−0.682698
$238$	0	0
$239$	−772.223	−0.209000	−0.104500	−	0.994525i	$-0.533324\pi$
$239$	−772.223	−0.209000	−0.104500	+	0.994525i	$0.533324\pi$
$240$	0	0
$241$	4031.57	1.07758	0.538789	−	0.842441i	$-0.318882\pi$
$241$	4031.57	1.07758	0.538789	+	0.842441i	$0.318882\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	694.111	0.181000
$246$	0	0
$247$	1261.58	0.324988
$248$	0	0
$249$	−1039.86	−0.264653
$250$	0	0
$251$	−7216.30	−1.81470	−0.907349	−	0.420379i	$-0.861897\pi$
$251$	−7216.30	−1.81470	−0.907349	+	0.420379i	$0.861897\pi$
$252$	0	0
$253$	−1003.89	−0.249463
$254$	0	0
$255$	3363.97	0.826118
$256$	0	0
$257$	134.220	0.0325776	0.0162888	−	0.999867i	$-0.494815\pi$
$257$	134.220	0.0325776	0.0162888	+	0.999867i	$0.494815\pi$
$258$	0	0
$259$	−1409.17	−0.338076
$260$	0	0
$261$	1053.06	0.249743
$262$	0	0
$263$	−4533.82	−1.06299	−0.531497	−	0.847060i	$-0.678370\pi$
$263$	−4533.82	−1.06299	−0.531497	+	0.847060i	$0.678370\pi$
$264$	0	0
$265$	10821.9	2.50861
$266$	0	0
$267$	−1753.20	−0.401851
$268$	0	0
$269$	−5497.56	−1.24607	−0.623034	−	0.782195i	$-0.714100\pi$
$269$	−5497.56	−1.24607	−0.623034	+	0.782195i	$0.714100\pi$
$270$	0	0
$271$	−8439.31	−1.89170	−0.945852	−	0.324598i	$-0.894771\pi$
$271$	−8439.31	−1.89170	−0.945852	+	0.324598i	$0.894771\pi$
$272$	0	0
$273$	475.904	0.105506
$274$	0	0
$275$	−895.421	−0.196349
$276$	0	0
$277$	27.2420	0.00590907	0.00295454	−	0.999996i	$-0.499060\pi$
$277$	27.2420	0.00590907	0.00295454	+	0.999996i	$0.499060\pi$
$278$	0	0
$279$	−1142.82	−0.245228
$280$	0	0
$281$	9036.54	1.91842	0.959208	−	0.282702i	$-0.0912306\pi$
$281$	9036.54	1.91842	0.959208	+	0.282702i	$0.0912306\pi$
$282$	0	0
$283$	6847.23	1.43825	0.719127	−	0.694879i	$-0.244542\pi$
$283$	6847.23	1.43825	0.719127	+	0.694879i	$0.244542\pi$
$284$	0	0
$285$	−2365.74	−0.491699
$286$	0	0
$287$	73.4760	0.0151120
$288$	0	0
$289$	1353.10	0.275411
$290$	0	0
$291$	257.089	0.0517898
$292$	0	0
$293$	485.725	0.0968476	0.0484238	−	0.998827i	$-0.484580\pi$
$293$	485.725	0.0968476	0.0484238	+	0.998827i	$0.484580\pi$
$294$	0	0
$295$	788.192	0.155560
$296$	0	0
$297$	−319.531	−0.0624278
$298$	0	0
$299$	−1922.37	−0.371818
$300$	0	0
$301$	18.5388	0.00355003
$302$	0	0
$303$	1559.63	0.295704
$304$	0	0
$305$	6473.74	1.21536
$306$	0	0
$307$	6598.73	1.22674	0.613371	−	0.789795i	$-0.289813\pi$
$307$	6598.73	1.22674	0.613371	+	0.789795i	$0.289813\pi$
$308$	0	0
$309$	2958.71	0.544708
$310$	0	0
$311$	6183.29	1.12740	0.563701	−	0.825979i	$-0.309377\pi$
$311$	6183.29	1.12740	0.563701	+	0.825979i	$0.309377\pi$
$312$	0	0
$313$	7614.03	1.37498	0.687492	−	0.726192i	$-0.258711\pi$
$313$	7614.03	1.37498	0.687492	+	0.726192i	$0.258711\pi$
$314$	0	0
$315$	−892.428	−0.159627
$316$	0	0
$317$	1013.78	0.179620	0.0898101	−	0.995959i	$-0.471374\pi$
$317$	1013.78	0.179620	0.0898101	+	0.995959i	$0.471374\pi$
$318$	0	0
$319$	−1384.71	−0.243038
$320$	0	0
$321$	3464.36	0.602373
$322$	0	0
$323$	−4406.68	−0.759115
$324$	0	0
$325$	−1714.66	−0.292653
$326$	0	0
$327$	−545.671	−0.0922804
$328$	0	0
$329$	−3460.03	−0.579810
$330$	0	0
$331$	9076.91	1.50729	0.753644	−	0.657283i	$-0.228294\pi$
$331$	9076.91	1.50729	0.753644	+	0.657283i	$0.228294\pi$
$332$	0	0
$333$	1811.79	0.298155
$334$	0	0
$335$	4122.65	0.672372
$336$	0	0
$337$	−8980.63	−1.45165	−0.725825	−	0.687880i	$-0.758542\pi$
$337$	−8980.63	−1.45165	−0.725825	+	0.687880i	$0.758542\pi$
$338$	0	0
$339$	2955.68	0.473542
$340$	0	0
$341$	1502.74	0.238644
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	3604.88	0.562552
$346$	0	0
$347$	−7827.62	−1.21098	−0.605488	−	0.795855i	$-0.707022\pi$
$347$	−7827.62	−1.21098	−0.605488	+	0.795855i	$0.707022\pi$
$348$	0	0
$349$	−8425.15	−1.29223	−0.646114	−	0.763241i	$-0.723607\pi$
$349$	−8425.15	−1.29223	−0.646114	+	0.763241i	$0.723607\pi$
$350$	0	0
$351$	−611.877	−0.0930472
$352$	0	0
$353$	−6227.22	−0.938927	−0.469464	−	0.882952i	$-0.655553\pi$
$353$	−6227.22	−0.938927	−0.469464	+	0.882952i	$0.655553\pi$
$354$	0	0
$355$	13548.5	2.02558
$356$	0	0
$357$	−1662.33	−0.246442
$358$	0	0
$359$	2523.17	0.370941	0.185471	−	0.982650i	$-0.440619\pi$
$359$	2523.17	0.370941	0.185471	+	0.982650i	$0.440619\pi$
$360$	0	0
$361$	−3759.97	−0.548180
$362$	0	0
$363$	−3572.84	−0.516598
$364$	0	0
$365$	−7333.05	−1.05159
$366$	0	0
$367$	8802.48	1.25201	0.626003	−	0.779821i	$-0.284690\pi$
$367$	8802.48	1.25201	0.626003	+	0.779821i	$0.284690\pi$
$368$	0	0
$369$	−94.4692	−0.0133276
$370$	0	0
$371$	−5347.71	−0.748354
$372$	0	0
$373$	−5714.80	−0.793301	−0.396651	−	0.917970i	$-0.629828\pi$
$373$	−5714.80	−0.793301	−0.396651	+	0.917970i	$0.629828\pi$
$374$	0	0
$375$	−2096.69	−0.288727
$376$	0	0
$377$	−2651.62	−0.362243
$378$	0	0
$379$	12339.0	1.67233	0.836166	−	0.548476i	$-0.184792\pi$
$379$	12339.0	1.67233	0.836166	+	0.548476i	$0.184792\pi$
$380$	0	0
$381$	−5456.94	−0.733773
$382$	0	0
$383$	−12155.2	−1.62168	−0.810838	−	0.585271i	$-0.800988\pi$
$383$	−12155.2	−1.62168	−0.810838	+	0.585271i	$0.800988\pi$
$384$	0	0
$385$	1173.49	0.155342
$386$	0	0
$387$	−23.8356	−0.00313083
$388$	0	0
$389$	−6364.82	−0.829587	−0.414793	−	0.909916i	$-0.636146\pi$
$389$	−6364.82	−0.829587	−0.414793	+	0.909916i	$0.636146\pi$
$390$	0	0
$391$	6714.84	0.868502
$392$	0	0
$393$	−2847.81	−0.365529
$394$	0	0
$395$	−11761.5	−1.49819
$396$	0	0
$397$	6439.24	0.814046	0.407023	−	0.913418i	$-0.366567\pi$
$397$	6439.24	0.814046	0.407023	+	0.913418i	$0.366567\pi$
$398$	0	0
$399$	1169.05	0.146681
$400$	0	0
$401$	−10320.0	−1.28518	−0.642591	−	0.766209i	$-0.722141\pi$
$401$	−10320.0	−1.28518	−0.642591	+	0.766209i	$0.722141\pi$
$402$	0	0
$403$	2877.62	0.355694
$404$	0	0
$405$	1147.41	0.140778
$406$	0	0
$407$	−2382.40	−0.290151
$408$	0	0
$409$	−2940.29	−0.355472	−0.177736	−	0.984078i	$-0.556877\pi$
$409$	−2940.29	−0.355472	−0.177736	+	0.984078i	$0.556877\pi$
$410$	0	0
$411$	−286.966	−0.0344403
$412$	0	0
$413$	−389.491	−0.0464058
$414$	0	0
$415$	−4910.07	−0.580785
$416$	0	0
$417$	−5537.47	−0.650290
$418$	0	0
$419$	−9721.24	−1.13345	−0.566723	−	0.823909i	$-0.691789\pi$
$419$	−9721.24	−1.13345	−0.566723	+	0.823909i	$0.691789\pi$
$420$	0	0
$421$	6572.79	0.760898	0.380449	−	0.924802i	$-0.375769\pi$
$421$	6572.79	0.760898	0.380449	+	0.924802i	$0.375769\pi$
$422$	0	0
$423$	4448.61	0.511345
$424$	0	0
$425$	5989.31	0.683587
$426$	0	0
$427$	−3199.05	−0.362559
$428$	0	0
$429$	804.582	0.0905492
$430$	0	0
$431$	−8087.14	−0.903814	−0.451907	−	0.892065i	$-0.649256\pi$
$431$	−8087.14	−0.903814	−0.451907	+	0.892065i	$0.649256\pi$
$432$	0	0
$433$	−12450.4	−1.38182	−0.690908	−	0.722943i	$-0.742789\pi$
$433$	−12450.4	−1.38182	−0.690908	+	0.722943i	$0.742789\pi$
$434$	0	0
$435$	4972.39	0.548064
$436$	0	0
$437$	−4722.26	−0.516926
$438$	0	0
$439$	−985.374	−0.107128	−0.0535642	−	0.998564i	$-0.517058\pi$
$439$	−985.374	−0.107128	−0.0535642	+	0.998564i	$0.517058\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−5058.57	−0.542528	−0.271264	−	0.962505i	$-0.587442\pi$
$443$	−5058.57	−0.542528	−0.271264	+	0.962505i	$0.587442\pi$
$444$	0	0
$445$	−8278.34	−0.881868
$446$	0	0
$447$	7391.79	0.782147
$448$	0	0
$449$	5471.79	0.575122	0.287561	−	0.957762i	$-0.407156\pi$
$449$	5471.79	0.575122	0.287561	+	0.957762i	$0.407156\pi$
$450$	0	0
$451$	124.221	0.0129698
$452$	0	0
$453$	−9281.38	−0.962643
$454$	0	0
$455$	2247.14	0.231533
$456$	0	0
$457$	−2636.51	−0.269870	−0.134935	−	0.990854i	$-0.543083\pi$
$457$	−2636.51	−0.269870	−0.134935	+	0.990854i	$0.543083\pi$
$458$	0	0
$459$	2137.28	0.217342
$460$	0	0
$461$	13996.7	1.41408	0.707040	−	0.707174i	$-0.250030\pi$
$461$	13996.7	1.41408	0.707040	+	0.707174i	$0.250030\pi$
$462$	0	0
$463$	5693.68	0.571507	0.285754	−	0.958303i	$-0.407756\pi$
$463$	5693.68	0.571507	0.285754	+	0.958303i	$0.407756\pi$
$464$	0	0
$465$	−5396.19	−0.538156
$466$	0	0
$467$	11337.7	1.12344	0.561719	−	0.827328i	$-0.310140\pi$
$467$	11337.7	1.12344	0.561719	+	0.827328i	$0.310140\pi$
$468$	0	0
$469$	−2037.24	−0.200578
$470$	0	0
$471$	1954.97	0.191253
$472$	0	0
$473$	31.3424	0.00304678
$474$	0	0
$475$	−4212.03	−0.406866
$476$	0	0
$477$	6875.63	0.659986
$478$	0	0
$479$	10309.1	0.983369	0.491684	−	0.870773i	$-0.336381\pi$
$479$	10309.1	0.983369	0.491684	+	0.870773i	$0.336381\pi$
$480$	0	0
$481$	−4562.12	−0.432463
$482$	0	0
$483$	−1781.38	−0.167817
$484$	0	0
$485$	1213.93	0.113653
$486$	0	0
$487$	−9312.94	−0.866550	−0.433275	−	0.901262i	$-0.642642\pi$
$487$	−9312.94	−0.866550	−0.433275	+	0.901262i	$0.642642\pi$
$488$	0	0
$489$	5498.49	0.508487
$490$	0	0
$491$	−7516.17	−0.690835	−0.345417	−	0.938449i	$-0.612263\pi$
$491$	−7516.17	−0.690835	−0.345417	+	0.938449i	$0.612263\pi$
$492$	0	0
$493$	9262.11	0.846135
$494$	0	0
$495$	−1508.77	−0.136999
$496$	0	0
$497$	−6695.09	−0.604257
$498$	0	0
$499$	−14415.1	−1.29320	−0.646600	−	0.762829i	$-0.723810\pi$
$499$	−14415.1	−1.29320	−0.646600	+	0.762829i	$0.723810\pi$
$500$	0	0
$501$	−966.459	−0.0861840
$502$	0	0
$503$	−12056.0	−1.06869	−0.534345	−	0.845267i	$-0.679442\pi$
$503$	−12056.0	−1.06869	−0.534345	+	0.845267i	$0.679442\pi$
$504$	0	0
$505$	7364.31	0.648926
$506$	0	0
$507$	−5050.29	−0.442389
$508$	0	0
$509$	−148.871	−0.0129638	−0.00648192	−	0.999979i	$-0.502063\pi$
$509$	−148.871	−0.0129638	−0.00648192	+	0.999979i	$0.502063\pi$
$510$	0	0
$511$	3623.68	0.313703
$512$	0	0
$513$	−1503.06	−0.129360
$514$	0	0
$515$	13970.5	1.19537
$516$	0	0
$517$	−5849.66	−0.497617
$518$	0	0
$519$	11785.3	0.996756
$520$	0	0
$521$	1986.75	0.167066	0.0835328	−	0.996505i	$-0.473380\pi$
$521$	1986.75	0.167066	0.0835328	+	0.996505i	$0.473380\pi$
$522$	0	0
$523$	13315.0	1.11324	0.556620	−	0.830767i	$-0.312098\pi$
$523$	13315.0	1.11324	0.556620	+	0.830767i	$0.312098\pi$
$524$	0	0
$525$	−1588.90	−0.132087
$526$	0	0
$527$	−10051.5	−0.830837
$528$	0	0
$529$	−4971.27	−0.408587
$530$	0	0
$531$	500.774	0.0409261
$532$	0	0
$533$	237.874	0.0193311
$534$	0	0
$535$	16358.2	1.32192
$536$	0	0
$537$	1251.23	0.100548
$538$	0	0
$539$	−579.889	−0.0463406
$540$	0	0
$541$	−23218.6	−1.84519	−0.922594	−	0.385773i	$-0.873935\pi$
$541$	−23218.6	−1.84519	−0.922594	+	0.385773i	$0.873935\pi$
$542$	0	0
$543$	6955.25	0.549684
$544$	0	0
$545$	−2576.57	−0.202511
$546$	0	0
$547$	−394.870	−0.0308655	−0.0154327	−	0.999881i	$-0.504913\pi$
$547$	−394.870	−0.0308655	−0.0154327	+	0.999881i	$0.504913\pi$
$548$	0	0
$549$	4113.06	0.319747
$550$	0	0
$551$	−6513.65	−0.503613
$552$	0	0
$553$	5812.03	0.446931
$554$	0	0
$555$	8555.01	0.654306
$556$	0	0
$557$	4689.32	0.356719	0.178360	−	0.983965i	$-0.442921\pi$
$557$	4689.32	0.356719	0.178360	+	0.983965i	$0.442921\pi$
$558$	0	0
$559$	60.0183	0.00454115
$560$	0	0
$561$	−2810.40	−0.211507
$562$	0	0
$563$	−8411.83	−0.629692	−0.314846	−	0.949143i	$-0.601953\pi$
$563$	−8411.83	−0.629692	−0.314846	+	0.949143i	$0.601953\pi$
$564$	0	0
$565$	13956.3	1.03919
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−12.9521	−0.000954268 0	−0.000477134	−	1.00000i	$-0.500152\pi$
$569$	−12.9521	−0.000954268 0	−0.000477134		1.00000i	$0.500152\pi$
$570$	0	0
$571$	13632.8	0.999153	0.499576	−	0.866270i	$-0.333489\pi$
$571$	13632.8	0.999153	0.499576	+	0.866270i	$0.333489\pi$
$572$	0	0
$573$	−484.387	−0.0353151
$574$	0	0
$575$	6418.24	0.465494
$576$	0	0
$577$	−4633.04	−0.334274	−0.167137	−	0.985934i	$-0.553452\pi$
$577$	−4633.04	−0.334274	−0.167137	+	0.985934i	$0.553452\pi$
$578$	0	0
$579$	5758.14	0.413299
$580$	0	0
$581$	2426.35	0.173256
$582$	0	0
$583$	−9041.05	−0.642268
$584$	0	0
$585$	−2889.18	−0.204193
$586$	0	0
$587$	−1511.42	−0.106274	−0.0531370	−	0.998587i	$-0.516922\pi$
$587$	−1511.42	−0.106274	−0.0531370	+	0.998587i	$0.516922\pi$
$588$	0	0
$589$	7068.81	0.494508
$590$	0	0
$591$	1254.41	0.0873089
$592$	0	0
$593$	−26971.7	−1.86778	−0.933890	−	0.357560i	$-0.883609\pi$
$593$	−26971.7	−1.86778	−0.933890	+	0.357560i	$0.883609\pi$
$594$	0	0
$595$	−7849.27	−0.540821
$596$	0	0
$597$	7181.47	0.492325
$598$	0	0
$599$	−25436.6	−1.73508	−0.867539	−	0.497369i	$-0.834300\pi$
$599$	−25436.6	−1.73508	−0.867539	+	0.497369i	$0.834300\pi$
$600$	0	0
$601$	−17018.5	−1.15508	−0.577538	−	0.816364i	$-0.695986\pi$
$601$	−17018.5	−1.15508	−0.577538	+	0.816364i	$0.695986\pi$
$602$	0	0
$603$	2619.31	0.176893
$604$	0	0
$605$	−16870.4	−1.13368
$606$	0	0
$607$	−7000.28	−0.468093	−0.234047	−	0.972225i	$-0.575197\pi$
$607$	−7000.28	−0.468093	−0.234047	+	0.972225i	$0.575197\pi$
$608$	0	0
$609$	−2457.14	−0.163495
$610$	0	0
$611$	−11201.6	−0.741686
$612$	0	0
$613$	9819.22	0.646973	0.323487	−	0.946233i	$-0.395145\pi$
$613$	9819.22	0.646973	0.323487	+	0.946233i	$0.395145\pi$
$614$	0	0
$615$	−446.068	−0.0292475
$616$	0	0
$617$	6479.85	0.422802	0.211401	−	0.977399i	$-0.432197\pi$
$617$	6479.85	0.422802	0.211401	+	0.977399i	$0.432197\pi$
$618$	0	0
$619$	−6671.60	−0.433205	−0.216603	−	0.976260i	$-0.569498\pi$
$619$	−6671.60	−0.433205	−0.216603	+	0.976260i	$0.569498\pi$
$620$	0	0
$621$	2290.35	0.148001
$622$	0	0
$623$	4090.80	0.263073
$624$	0	0
$625$	−19358.0	−1.23891
$626$	0	0
$627$	1976.44	0.125887
$628$	0	0
$629$	15935.5	1.01016
$630$	0	0
$631$	22410.8	1.41388	0.706942	−	0.707272i	$-0.250074\pi$
$631$	22410.8	1.41388	0.706942	+	0.707272i	$0.250074\pi$
$632$	0	0
$633$	−9622.85	−0.604224
$634$	0	0
$635$	−25766.8	−1.61027
$636$	0	0
$637$	−1110.44	−0.0690696
$638$	0	0
$639$	8607.98	0.532905
$640$	0	0
$641$	−30568.6	−1.88360	−0.941800	−	0.336175i	$-0.890867\pi$
$641$	−30568.6	−1.88360	−0.941800	+	0.336175i	$0.890867\pi$
$642$	0	0
$643$	−27501.7	−1.68672	−0.843360	−	0.537349i	$-0.819426\pi$
$643$	−27501.7	−1.68672	−0.843360	+	0.537349i	$0.819426\pi$
$644$	0	0
$645$	−112.548	−0.00687065
$646$	0	0
$647$	−14954.8	−0.908706	−0.454353	−	0.890822i	$-0.650129\pi$
$647$	−14954.8	−0.908706	−0.454353	+	0.890822i	$0.650129\pi$
$648$	0	0
$649$	−658.489	−0.0398273
$650$	0	0
$651$	2666.57	0.160539
$652$	0	0
$653$	13435.0	0.805132	0.402566	−	0.915391i	$-0.368118\pi$
$653$	13435.0	0.805132	0.402566	+	0.915391i	$0.368118\pi$
$654$	0	0
$655$	−13446.9	−0.802159
$656$	0	0
$657$	−4659.02	−0.276660
$658$	0	0
$659$	25599.1	1.51320	0.756600	−	0.653877i	$-0.226859\pi$
$659$	25599.1	1.51320	0.756600	+	0.653877i	$0.226859\pi$
$660$	0	0
$661$	21512.6	1.26587	0.632936	−	0.774204i	$-0.281849\pi$
$661$	21512.6	1.26587	0.632936	+	0.774204i	$0.281849\pi$
$662$	0	0
$663$	−5381.71	−0.315246
$664$	0	0
$665$	5520.06	0.321893
$666$	0	0
$667$	9925.41	0.576182
$668$	0	0
$669$	−6567.73	−0.379556
$670$	0	0
$671$	−5408.44	−0.311163
$672$	0	0
$673$	−18508.5	−1.06010	−0.530052	−	0.847965i	$-0.677828\pi$
$673$	−18508.5	−1.06010	−0.530052	+	0.847965i	$0.677828\pi$
$674$	0	0
$675$	2042.88	0.116489
$676$	0	0
$677$	−30.9144	−0.00175500	−0.000877500	−	1.00000i	$-0.500279\pi$
$677$	−30.9144	−0.00175500	−0.000877500		1.00000i	$0.500279\pi$
$678$	0	0
$679$	−599.874	−0.0339044
$680$	0	0
$681$	5883.90	0.331089
$682$	0	0
$683$	30454.8	1.70618	0.853090	−	0.521764i	$-0.174726\pi$
$683$	30454.8	1.70618	0.853090	+	0.521764i	$0.174726\pi$
$684$	0	0
$685$	−1355.01	−0.0755798
$686$	0	0
$687$	7850.19	0.435959
$688$	0	0
$689$	−17312.9	−0.957285
$690$	0	0
$691$	−23855.9	−1.31334	−0.656671	−	0.754177i	$-0.728036\pi$
$691$	−23855.9	−1.31334	−0.656671	+	0.754177i	$0.728036\pi$
$692$	0	0
$693$	745.572	0.0408686
$694$	0	0
$695$	−26147.0	−1.42707
$696$	0	0
$697$	−830.895	−0.0451541
$698$	0	0
$699$	5454.83	0.295165
$700$	0	0
$701$	−28607.6	−1.54136	−0.770682	−	0.637220i	$-0.780084\pi$
$701$	−28607.6	−1.54136	−0.770682	+	0.637220i	$0.780084\pi$
$702$	0	0
$703$	−11206.7	−0.601238
$704$	0	0
$705$	21005.6	1.12215
$706$	0	0
$707$	−3639.13	−0.193583
$708$	0	0
$709$	−25897.9	−1.37181	−0.685907	−	0.727689i	$-0.740595\pi$
$709$	−25897.9	−1.37181	−0.685907	+	0.727689i	$0.740595\pi$
$710$	0	0
$711$	−7472.61	−0.394156
$712$	0	0
$713$	−10771.4	−0.565765
$714$	0	0
$715$	3799.11	0.198711
$716$	0	0
$717$	−2316.67	−0.120666
$718$	0	0
$719$	−38378.4	−1.99065	−0.995323	−	0.0966059i	$-0.969201\pi$
$719$	−38378.4	−1.99065	−0.995323	+	0.0966059i	$0.969201\pi$
$720$	0	0
$721$	−6903.65	−0.356595
$722$	0	0
$723$	12094.7	0.622140
$724$	0	0
$725$	8852.98	0.453506
$726$	0	0
$727$	30693.9	1.56585	0.782927	−	0.622114i	$-0.213726\pi$
$727$	30693.9	1.56585	0.782927	+	0.622114i	$0.213726\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−209.644	−0.0106073
$732$	0	0
$733$	−12664.6	−0.638169	−0.319084	−	0.947726i	$-0.603375\pi$
$733$	−12664.6	−0.638169	−0.319084	+	0.947726i	$0.603375\pi$
$734$	0	0
$735$	2082.33	0.104501
$736$	0	0
$737$	−3444.24	−0.172144
$738$	0	0
$739$	26458.6	1.31704	0.658522	−	0.752562i	$-0.271182\pi$
$739$	26458.6	1.31704	0.658522	+	0.752562i	$0.271182\pi$
$740$	0	0
$741$	3784.73	0.187632
$742$	0	0
$743$	34022.1	1.67988	0.839940	−	0.542680i	$-0.182590\pi$
$743$	34022.1	1.67988	0.839940	+	0.542680i	$0.182590\pi$
$744$	0	0
$745$	34902.9	1.71643
$746$	0	0
$747$	−3119.59	−0.152798
$748$	0	0
$749$	−8083.51	−0.394346
$750$	0	0
$751$	1731.20	0.0841175	0.0420588	−	0.999115i	$-0.486608\pi$
$751$	1731.20	0.0841175	0.0420588	+	0.999115i	$0.486608\pi$
$752$	0	0
$753$	−21648.9	−1.04772
$754$	0	0
$755$	−43825.2	−2.11253
$756$	0	0
$757$	−6692.85	−0.321342	−0.160671	−	0.987008i	$-0.551366\pi$
$757$	−6692.85	−0.321342	−0.160671	+	0.987008i	$0.551366\pi$
$758$	0	0
$759$	−3011.67	−0.144027
$760$	0	0
$761$	9615.60	0.458036	0.229018	−	0.973422i	$-0.426449\pi$
$761$	9615.60	0.458036	0.229018	+	0.973422i	$0.426449\pi$
$762$	0	0
$763$	1273.23	0.0604117
$764$	0	0
$765$	10091.9	0.476960
$766$	0	0
$767$	−1260.95	−0.0593617
$768$	0	0
$769$	−20565.1	−0.964366	−0.482183	−	0.876071i	$-0.660156\pi$
$769$	−20565.1	−0.964366	−0.482183	+	0.876071i	$0.660156\pi$
$770$	0	0
$771$	402.661	0.0188087
$772$	0	0
$773$	3168.70	0.147439	0.0737195	−	0.997279i	$-0.476513\pi$
$773$	3168.70	0.147439	0.0737195	+	0.997279i	$0.476513\pi$
$774$	0	0
$775$	−9607.53	−0.445307
$776$	0	0
$777$	−4227.52	−0.195188
$778$	0	0
$779$	584.333	0.0268754
$780$	0	0
$781$	−11319.0	−0.518598
$782$	0	0
$783$	3159.18	0.144189
$784$	0	0
$785$	9231.04	0.419707
$786$	0	0
$787$	−34703.8	−1.57186	−0.785932	−	0.618313i	$-0.787817\pi$
$787$	−34703.8	−1.57186	−0.785932	+	0.618313i	$0.787817\pi$
$788$	0	0
$789$	−13601.5	−0.613720
$790$	0	0
$791$	−6896.60	−0.310006
$792$	0	0
$793$	−10356.7	−0.463781
$794$	0	0
$795$	32465.6	1.44835
$796$	0	0
$797$	−31850.2	−1.41555	−0.707775	−	0.706438i	$-0.750301\pi$
$797$	−31850.2	−1.41555	−0.707775	+	0.706438i	$0.750301\pi$
$798$	0	0
$799$	39127.3	1.73245
$800$	0	0
$801$	−5259.61	−0.232009
$802$	0	0
$803$	6126.34	0.269233
$804$	0	0
$805$	−8411.39	−0.368277
$806$	0	0
$807$	−16492.7	−0.719417
$808$	0	0
$809$	−2097.32	−0.0911471	−0.0455735	−	0.998961i	$-0.514512\pi$
$809$	−2097.32	−0.0911471	−0.0455735	+	0.998961i	$0.514512\pi$
$810$	0	0
$811$	−21316.7	−0.922973	−0.461487	−	0.887147i	$-0.652684\pi$
$811$	−21316.7	−0.922973	−0.461487	+	0.887147i	$0.652684\pi$
$812$	0	0
$813$	−25317.9	−1.09218
$814$	0	0
$815$	25963.0	1.11588
$816$	0	0
$817$	147.434	0.00631341
$818$	0	0
$819$	1427.71	0.0609137
$820$	0	0
$821$	−4249.23	−0.180632	−0.0903162	−	0.995913i	$-0.528788\pi$
$821$	−4249.23	−0.180632	−0.0903162	+	0.995913i	$0.528788\pi$
$822$	0	0
$823$	−6491.53	−0.274946	−0.137473	−	0.990506i	$-0.543898\pi$
$823$	−6491.53	−0.274946	−0.137473	+	0.990506i	$0.543898\pi$
$824$	0	0
$825$	−2686.26	−0.113362
$826$	0	0
$827$	−19598.8	−0.824082	−0.412041	−	0.911165i	$-0.635184\pi$
$827$	−19598.8	−0.824082	−0.412041	+	0.911165i	$0.635184\pi$
$828$	0	0
$829$	−21954.5	−0.919795	−0.459898	−	0.887972i	$-0.652114\pi$
$829$	−21954.5	−0.919795	−0.459898	+	0.887972i	$0.652114\pi$
$830$	0	0
$831$	81.7260	0.00341161
$832$	0	0
$833$	3878.78	0.161334
$834$	0	0
$835$	−4563.47	−0.189132
$836$	0	0
$837$	−3428.45	−0.141582
$838$	0	0
$839$	29714.0	1.22270	0.611348	−	0.791362i	$-0.290628\pi$
$839$	29714.0	1.22270	0.611348	+	0.791362i	$0.290628\pi$
$840$	0	0
$841$	−10698.4	−0.438657
$842$	0	0
$843$	27109.6	1.10760
$844$	0	0
$845$	−23846.7	−0.970828
$846$	0	0
$847$	8336.62	0.338193
$848$	0	0
$849$	20541.7	0.830376
$850$	0	0
$851$	17076.7	0.687875
$852$	0	0
$853$	16510.0	0.662709	0.331354	−	0.943506i	$-0.392494\pi$
$853$	16510.0	0.662709	0.331354	+	0.943506i	$0.392494\pi$
$854$	0	0
$855$	−7097.22	−0.283883
$856$	0	0
$857$	−16114.4	−0.642307	−0.321154	−	0.947027i	$-0.604071\pi$
$857$	−16114.4	−0.642307	−0.321154	+	0.947027i	$0.604071\pi$
$858$	0	0
$859$	−45784.6	−1.81857	−0.909285	−	0.416174i	$-0.863371\pi$
$859$	−45784.6	−1.81857	−0.909285	+	0.416174i	$0.863371\pi$
$860$	0	0
$861$	220.428	0.00872493
$862$	0	0
$863$	−32002.7	−1.26232	−0.631162	−	0.775651i	$-0.717422\pi$
$863$	−32002.7	−1.26232	−0.631162	+	0.775651i	$0.717422\pi$
$864$	0	0
$865$	55648.2	2.18740
$866$	0	0
$867$	4059.29	0.159009
$868$	0	0
$869$	9826.05	0.383574
$870$	0	0
$871$	−6595.45	−0.256577
$872$	0	0
$873$	771.267	0.0299008
$874$	0	0
$875$	4892.28	0.189016
$876$	0	0
$877$	−43761.0	−1.68495	−0.842476	−	0.538733i	$-0.818903\pi$
$877$	−43761.0	−1.68495	−0.842476	+	0.538733i	$0.818903\pi$
$878$	0	0
$879$	1457.17	0.0559150
$880$	0	0
$881$	28283.2	1.08160	0.540798	−	0.841153i	$-0.318122\pi$
$881$	28283.2	1.08160	0.540798	+	0.841153i	$0.318122\pi$
$882$	0	0
$883$	11859.2	0.451973	0.225987	−	0.974130i	$-0.427439\pi$
$883$	11859.2	0.451973	0.225987	+	0.974130i	$0.427439\pi$
$884$	0	0
$885$	2364.58	0.0898128
$886$	0	0
$887$	15526.8	0.587755	0.293877	−	0.955843i	$-0.405054\pi$
$887$	15526.8	0.587755	0.293877	+	0.955843i	$0.405054\pi$
$888$	0	0
$889$	12732.9	0.480367
$890$	0	0
$891$	−958.592	−0.0360427
$892$	0	0
$893$	−27516.6	−1.03114
$894$	0	0
$895$	5908.11	0.220655
$896$	0	0
$897$	−5767.12	−0.214669
$898$	0	0
$899$	−14857.5	−0.551195
$900$	0	0
$901$	60474.0	2.23605
$902$	0	0
$903$	55.6164	0.00204961
$904$	0	0
$905$	32841.6	1.20629
$906$	0	0
$907$	39330.9	1.43987	0.719935	−	0.694041i	$-0.244171\pi$
$907$	39330.9	1.43987	0.719935	+	0.694041i	$0.244171\pi$
$908$	0	0
$909$	4678.88	0.170725
$910$	0	0
$911$	23942.5	0.870749	0.435374	−	0.900250i	$-0.356616\pi$
$911$	23942.5	0.870749	0.435374	+	0.900250i	$0.356616\pi$
$912$	0	0
$913$	4102.08	0.148695
$914$	0	0
$915$	19421.2	0.701690
$916$	0	0
$917$	6644.89	0.239295
$918$	0	0
$919$	−5531.04	−0.198533	−0.0992667	−	0.995061i	$-0.531650\pi$
$919$	−5531.04	−0.198533	−0.0992667	+	0.995061i	$0.531650\pi$
$920$	0	0
$921$	19796.2	0.708259
$922$	0	0
$923$	−21675.0	−0.772958
$924$	0	0
$925$	15231.6	0.541417
$926$	0	0
$927$	8876.12	0.314487
$928$	0	0
$929$	39364.5	1.39021	0.695105	−	0.718908i	$-0.255358\pi$
$929$	39364.5	1.39021	0.695105	+	0.718908i	$0.255358\pi$
$930$	0	0
$931$	−2727.78	−0.0960251
$932$	0	0
$933$	18549.9	0.650906
$934$	0	0
$935$	−13270.3	−0.464155
$936$	0	0
$937$	4028.47	0.140453	0.0702265	−	0.997531i	$-0.477628\pi$
$937$	4028.47	0.140453	0.0702265	+	0.997531i	$0.477628\pi$
$938$	0	0
$939$	22842.1	0.793848
$940$	0	0
$941$	−6784.62	−0.235040	−0.117520	−	0.993071i	$-0.537494\pi$
$941$	−6784.62	−0.235040	−0.117520	+	0.993071i	$0.537494\pi$
$942$	0	0
$943$	−890.400	−0.0307480
$944$	0	0
$945$	−2677.28	−0.0921609
$946$	0	0
$947$	6550.25	0.224767	0.112384	−	0.993665i	$-0.464151\pi$
$947$	6550.25	0.224767	0.112384	+	0.993665i	$0.464151\pi$
$948$	0	0
$949$	11731.5	0.401285
$950$	0	0
$951$	3041.34	0.103704
$952$	0	0
$953$	−5566.28	−0.189202	−0.0946010	−	0.995515i	$-0.530158\pi$
$953$	−5566.28	−0.189202	−0.0946010	+	0.995515i	$0.530158\pi$
$954$	0	0
$955$	−2287.20	−0.0774995
$956$	0	0
$957$	−4154.14	−0.140318
$958$	0	0
$959$	669.587	0.0225465
$960$	0	0
$961$	−13667.2	−0.458770
$962$	0	0
$963$	10393.1	0.347780
$964$	0	0
$965$	27189.0	0.906990
$966$	0	0
$967$	49783.4	1.65556	0.827781	−	0.561052i	$-0.189603\pi$
$967$	49783.4	1.65556	0.827781	+	0.561052i	$0.189603\pi$
$968$	0	0
$969$	−13220.0	−0.438275
$970$	0	0
$971$	−32492.4	−1.07387	−0.536937	−	0.843623i	$-0.680419\pi$
$971$	−32492.4	−1.07387	−0.536937	+	0.843623i	$0.680419\pi$
$972$	0	0
$973$	12920.8	0.425715
$974$	0	0
$975$	−5143.99	−0.168963
$976$	0	0
$977$	−17797.1	−0.582785	−0.291392	−	0.956604i	$-0.594119\pi$
$977$	−17797.1	−0.582785	−0.291392	+	0.956604i	$0.594119\pi$
$978$	0	0
$979$	6916.08	0.225780
$980$	0	0
$981$	−1637.01	−0.0532781
$982$	0	0
$983$	−20155.7	−0.653985	−0.326992	−	0.945027i	$-0.606035\pi$
$983$	−20155.7	−0.653985	−0.326992	+	0.945027i	$0.606035\pi$
$984$	0	0
$985$	5923.13	0.191601
$986$	0	0
$987$	−10380.1	−0.334754
$988$	0	0
$989$	−224.658	−0.00722315
$990$	0	0
$991$	44715.4	1.43333	0.716665	−	0.697417i	$-0.245668\pi$
$991$	44715.4	1.43333	0.716665	+	0.697417i	$0.245668\pi$
$992$	0	0
$993$	27230.7	0.870233
$994$	0	0
$995$	33909.7	1.08041
$996$	0	0
$997$	−6924.95	−0.219975	−0.109988	−	0.993933i	$-0.535081\pi$
$997$	−6924.95	−0.219975	−0.109988	+	0.993933i	$0.535081\pi$
$998$	0	0
$999$	5435.38	0.172140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.a.bp.1.2		2
4.3	odd	2		1344.4.a.bh.1.2		2
8.3	odd	2		672.4.a.k.1.1	yes	2
8.5	even	2		672.4.a.f.1.1	✓	2
24.5	odd	2		2016.4.a.m.1.2		2
24.11	even	2		2016.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.4.a.f.1.1	✓	2	8.5	even	2
672.4.a.k.1.1	yes	2	8.3	odd	2
1344.4.a.bh.1.2		2	4.3	odd	2
1344.4.a.bp.1.2		2	1.1	even	1	trivial
2016.4.a.m.1.2		2	24.5	odd	2
2016.4.a.n.1.2		2	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1344.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +14.1655 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +14.1655 q^{5} -7.00000 q^{7} +9.00000 q^{9} -11.8345 q^{11} -22.6621 q^{13} +42.4966 q^{15} +79.1587 q^{17} -55.6689 q^{19} -21.0000 q^{21} +84.8276 q^{23} +75.6621 q^{25} +27.0000 q^{27} +117.007 q^{29} -126.979 q^{31} -35.5034 q^{33} -99.1587 q^{35} +201.311 q^{37} -67.9863 q^{39} -10.4966 q^{41} -2.64840 q^{43} +127.490 q^{45} +494.290 q^{47} +49.0000 q^{49} +237.476 q^{51} +763.959 q^{53} -167.642 q^{55} -167.007 q^{57} +55.6416 q^{59} +457.007 q^{61} -63.0000 q^{63} -321.021 q^{65} +291.034 q^{67} +254.483 q^{69} +956.442 q^{71} -517.669 q^{73} +226.986 q^{75} +82.8413 q^{77} -830.290 q^{79} +81.0000 q^{81} -346.621 q^{83} +1121.32 q^{85} +351.021 q^{87} -584.401 q^{89} +158.635 q^{91} -380.938 q^{93} -788.580 q^{95} +85.6963 q^{97} -106.510 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 4 q^{5} - 14 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 + 4 * q^5 - 14 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} + 4 q^{5} - 14 q^{7} + 18 q^{9} - 48 q^{11} + 52 q^{13} + 12 q^{15} - 12 q^{17} - 160 q^{19} - 42 q^{21} + 48 q^{23} + 54 q^{25} + 54 q^{27} + 380 q^{29} + 184 q^{31} - 144 q^{33} - 28 q^{35} - 84 q^{37} + 156 q^{39} + 52 q^{41} + 384 q^{43} + 36 q^{45} + 64 q^{47} + 98 q^{49} - 36 q^{51} + 652 q^{53} + 200 q^{55} - 480 q^{57} - 424 q^{59} + 1060 q^{61} - 126 q^{63} - 1080 q^{65} + 1312 q^{67} + 144 q^{69} + 672 q^{71} - 1084 q^{73} + 162 q^{75} + 336 q^{77} - 736 q^{79} + 162 q^{81} + 280 q^{83} + 2048 q^{85} + 1140 q^{87} + 948 q^{89} - 364 q^{91} + 552 q^{93} + 272 q^{95} + 804 q^{97} - 432 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 4 * q^5 - 14 * q^7 + 18 * q^9 - 48 * q^11 + 52 * q^13 + 12 * q^15 - 12 * q^17 - 160 * q^19 - 42 * q^21 + 48 * q^23 + 54 * q^25 + 54 * q^27 + 380 * q^29 + 184 * q^31 - 144 * q^33 - 28 * q^35 - 84 * q^37 + 156 * q^39 + 52 * q^41 + 384 * q^43 + 36 * q^45 + 64 * q^47 + 98 * q^49 - 36 * q^51 + 652 * q^53 + 200 * q^55 - 480 * q^57 - 424 * q^59 + 1060 * q^61 - 126 * q^63 - 1080 * q^65 + 1312 * q^67 + 144 * q^69 + 672 * q^71 - 1084 * q^73 + 162 * q^75 + 336 * q^77 - 736 * q^79 + 162 * q^81 + 280 * q^83 + 2048 * q^85 + 1140 * q^87 + 948 * q^89 - 364 * q^91 + 552 * q^93 + 272 * q^95 + 804 * q^97 - 432 * q^99

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