LMFDB - Embedded newform 1344.4.c.e.673.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,4,Mod(673,1344)]

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, 1, 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("1344.673");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1344.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$79.2985670477$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 386x^{10} + 54793x^{8} + 3447408x^{6} + 90154296x^{4} + 707138208x^{2} + 525876624 $ x^12 + 386x^10 + 54793x^8 + 3447408x^6 + 90154296x^4 + 707138208*x^2 + 525876624
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			673.3
Root			$10.8906i$ of defining polynomial
Character	$\chi$	$=$	1344.673
Dual form			1344.4.c.e.673.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -1.28149i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -1.28149i q^{5} -7.00000 q^{7} -9.00000 q^{9} -58.8368i q^{11} -11.2921i q^{13} -3.84446 q^{15} -104.187 q^{17} -51.5021i q^{19} +21.0000i q^{21} -131.383 q^{23} +123.358 q^{25} +27.0000i q^{27} +32.6564i q^{29} +108.814 q^{31} -176.510 q^{33} +8.97041i q^{35} +5.75772i q^{37} -33.8763 q^{39} -198.587 q^{41} +97.7382i q^{43} +11.5334i q^{45} +422.128 q^{47} +49.0000 q^{49} +312.561i q^{51} +209.333i q^{53} -75.3986 q^{55} -154.506 q^{57} +74.8335i q^{59} -447.218i q^{61} +63.0000 q^{63} -14.4707 q^{65} -328.017i q^{67} +394.149i q^{69} +148.641 q^{71} -35.9094 q^{73} -370.073i q^{75} +411.858i q^{77} -1302.70 q^{79} +81.0000 q^{81} +393.931i q^{83} +133.514i q^{85} +97.9692 q^{87} -480.693 q^{89} +79.0448i q^{91} -326.443i q^{93} -65.9994 q^{95} -1278.12 q^{97} +529.531i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) $ 12 * q - 84 * q^7 - 108 * q^9	$ 12 q - 84 q^{7} - 108 q^{9} - 24 q^{15} + 24 q^{17} + 80 q^{23} - 564 q^{25} - 640 q^{31} - 408 q^{33} + 120 q^{39} + 1416 q^{41} - 1536 q^{47} + 588 q^{49} + 1392 q^{55} - 336 q^{57} + 756 q^{63} - 2880 q^{65} + 1392 q^{71} + 2472 q^{73} - 544 q^{79} + 972 q^{81} + 720 q^{87} + 888 q^{89} + 2368 q^{95} - 2712 q^{97}+O(q^{100}) $ 12 * q - 84 * q^7 - 108 * q^9 - 24 * q^15 + 24 * q^17 + 80 * q^23 - 564 * q^25 - 640 * q^31 - 408 * q^33 + 120 * q^39 + 1416 * q^41 - 1536 * q^47 + 588 * q^49 + 1392 * q^55 - 336 * q^57 + 756 * q^63 - 2880 * q^65 + 1392 * q^71 + 2472 * q^73 - 544 * q^79 + 972 * q^81 + 720 * q^87 + 888 * q^89 + 2368 * q^95 - 2712 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	− 1.28149i	− 0.114620i	−0.998356	−	0.0573099i	$-0.981748\pi$
$5$	− 1.28149i	− 0.114620i	0.998356	−	0.0573099i	$-0.0182523\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	− 58.8368i	− 1.61272i	−0.591422	−	0.806362i	$-0.701433\pi$
$11$	− 58.8368i	− 1.61272i	0.591422	−	0.806362i	$-0.298567\pi$
$12$	0	0
$13$	− 11.2921i	− 0.240913i	−0.992719	−	0.120456i	$-0.961564\pi$
$13$	− 11.2921i	− 0.240913i	0.992719	−	0.120456i	$-0.0384358\pi$
$14$	0	0
$15$	−3.84446	−0.0661757
$16$	0	0
$17$	−104.187	−1.48642	−0.743208	−	0.669060i	$-0.766697\pi$
$17$	−104.187	−1.48642	−0.743208	+	0.669060i	$0.766697\pi$
$18$	0	0
$19$	− 51.5021i	− 0.621863i	−0.950432	−	0.310932i	$-0.899359\pi$
$19$	− 51.5021i	− 0.621863i	0.950432	−	0.310932i	$-0.100641\pi$
$20$	0	0
$21$	21.0000i	0.218218i
$22$	0	0
$23$	−131.383	−1.19110	−0.595549	−	0.803319i	$-0.703066\pi$
$23$	−131.383	−1.19110	−0.595549	+	0.803319i	$0.703066\pi$
$24$	0	0
$25$	123.358	0.986862
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	32.6564i	0.209108i	0.994519	+	0.104554i	$0.0333415\pi$
$29$	32.6564i	0.209108i	−0.994519	+	0.104554i	$0.966658\pi$
$30$	0	0
$31$	108.814	0.630440	0.315220	−	0.949019i	$-0.397922\pi$
$31$	108.814	0.630440	0.315220	+	0.949019i	$0.397922\pi$
$32$	0	0
$33$	−176.510	−0.931107
$34$	0	0
$35$	8.97041i	0.0433222i
$36$	0	0
$37$	5.75772i	0.0255828i	0.999918	+	0.0127914i	$0.00407174\pi$
$37$	5.75772i	0.0255828i	−0.999918	+	0.0127914i	$0.995928\pi$
$38$	0	0
$39$	−33.8763	−0.139091
$40$	0	0
$41$	−198.587	−0.756442	−0.378221	−	0.925715i	$-0.623464\pi$
$41$	−198.587	−0.756442	−0.378221	+	0.925715i	$0.623464\pi$
$42$	0	0
$43$	97.7382i	0.346626i	0.984867	+	0.173313i	$0.0554473\pi$
$43$	97.7382i	0.346626i	−0.984867	+	0.173313i	$0.944553\pi$
$44$	0	0
$45$	11.5334i	0.0382066i
$46$	0	0
$47$	422.128	1.31008	0.655039	−	0.755595i	$-0.272652\pi$
$47$	422.128	1.31008	0.655039	+	0.755595i	$0.272652\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	312.561i	0.858183i
$52$	0	0
$53$	209.333i	0.542530i	0.962505	+	0.271265i	$0.0874420\pi$
$53$	209.333i	0.542530i	−0.962505	+	0.271265i	$0.912558\pi$
$54$	0	0
$55$	−75.3986	−0.184850
$56$	0	0
$57$	−154.506	−0.359033
$58$	0	0
$59$	74.8335i	0.165127i	0.996586	+	0.0825635i	$0.0263107\pi$
$59$	74.8335i	0.165127i	−0.996586	+	0.0825635i	$0.973689\pi$
$60$	0	0
$61$	− 447.218i	− 0.938694i	−0.883014	−	0.469347i	$-0.844489\pi$
$61$	− 447.218i	− 0.938694i	0.883014	−	0.469347i	$-0.155511\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−14.4707	−0.0276134
$66$	0	0
$67$	− 328.017i	− 0.598115i	−0.954235	−	0.299057i	$-0.903328\pi$
$67$	− 328.017i	− 0.598115i	0.954235	−	0.299057i	$-0.0966722\pi$
$68$	0	0
$69$	394.149i	0.687681i
$70$	0	0
$71$	148.641	0.248457	0.124228	−	0.992254i	$-0.460354\pi$
$71$	148.641	0.248457	0.124228	+	0.992254i	$0.460354\pi$
$72$	0	0
$73$	−35.9094	−0.0575737	−0.0287868	−	0.999586i	$-0.509164\pi$
$73$	−35.9094	−0.0575737	−0.0287868	+	0.999586i	$0.509164\pi$
$74$	0	0
$75$	− 370.073i	− 0.569765i
$76$	0	0
$77$	411.858i	0.609552i
$78$	0	0
$79$	−1302.70	−1.85526	−0.927628	−	0.373505i	$-0.878156\pi$
$79$	−1302.70	−1.85526	−0.927628	+	0.373505i	$0.878156\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	393.931i	0.520958i	0.965479	+	0.260479i	$0.0838805\pi$
$83$	393.931i	0.520958i	−0.965479	+	0.260479i	$0.916119\pi$
$84$	0	0
$85$	133.514i	0.170373i
$86$	0	0
$87$	97.9692	0.120729
$88$	0	0
$89$	−480.693	−0.572509	−0.286255	−	0.958154i	$-0.592410\pi$
$89$	−480.693	−0.572509	−0.286255	+	0.958154i	$0.592410\pi$
$90$	0	0
$91$	79.0448i	0.0910565i
$92$	0	0
$93$	− 326.443i	− 0.363985i
$94$	0	0
$95$	−65.9994	−0.0712778
$96$	0	0
$97$	−1278.12	−1.33787	−0.668936	−	0.743320i	$-0.733250\pi$
$97$	−1278.12	−1.33787	−0.668936	+	0.743320i	$0.733250\pi$
$98$	0	0
$99$	529.531i	0.537575i
$100$	0	0
$101$	1506.28i	1.48397i	0.670417	+	0.741985i	$0.266115\pi$
$101$	1506.28i	1.48397i	−0.670417	+	0.741985i	$0.733885\pi$
$102$	0	0
$103$	1101.07	1.05331	0.526656	−	0.850078i	$-0.323446\pi$
$103$	1101.07	1.05331	0.526656	+	0.850078i	$0.323446\pi$
$104$	0	0
$105$	26.9112	0.0250121
$106$	0	0
$107$	837.935i	0.757068i	0.925587	+	0.378534i	$0.123572\pi$
$107$	837.935i	0.757068i	−0.925587	+	0.378534i	$0.876428\pi$
$108$	0	0
$109$	1278.36i	1.12334i	0.827361	+	0.561671i	$0.189841\pi$
$109$	1278.36i	1.12334i	−0.827361	+	0.561671i	$0.810159\pi$
$110$	0	0
$111$	17.2732	0.0147702
$112$	0	0
$113$	−1523.07	−1.26795	−0.633976	−	0.773353i	$-0.718578\pi$
$113$	−1523.07	−1.26795	−0.633976	+	0.773353i	$0.718578\pi$
$114$	0	0
$115$	168.366i	0.136523i
$116$	0	0
$117$	101.629i	0.0803043i
$118$	0	0
$119$	729.309	0.561812
$120$	0	0
$121$	−2130.77	−1.60088
$122$	0	0
$123$	595.762i	0.436732i
$124$	0	0
$125$	− 318.267i	− 0.227734i
$126$	0	0
$127$	451.549	0.315500	0.157750	−	0.987479i	$-0.449576\pi$
$127$	451.549	0.315500	0.157750	+	0.987479i	$0.449576\pi$
$128$	0	0
$129$	293.215	0.200125
$130$	0	0
$131$	− 946.680i	− 0.631388i	−0.948861	−	0.315694i	$-0.897763\pi$
$131$	− 946.680i	− 0.631388i	0.948861	−	0.315694i	$-0.102237\pi$
$132$	0	0
$133$	360.515i	0.235042i
$134$	0	0
$135$	34.6002	0.0220586
$136$	0	0
$137$	−467.626	−0.291621	−0.145810	−	0.989313i	$-0.546579\pi$
$137$	−467.626	−0.291621	−0.145810	+	0.989313i	$0.546579\pi$
$138$	0	0
$139$	962.541i	0.587350i	0.955905	+	0.293675i	$0.0948784\pi$
$139$	962.541i	0.587350i	−0.955905	+	0.293675i	$0.905122\pi$
$140$	0	0
$141$	− 1266.38i	− 0.756374i
$142$	0	0
$143$	−664.392	−0.388526
$144$	0	0
$145$	41.8488	0.0239679
$146$	0	0
$147$	− 147.000i	− 0.0824786i
$148$	0	0
$149$	− 1466.66i	− 0.806399i	−0.915112	−	0.403200i	$-0.867898\pi$
$149$	− 1466.66i	− 0.806399i	0.915112	−	0.403200i	$-0.132102\pi$
$150$	0	0
$151$	3483.73	1.87750	0.938749	−	0.344603i	$-0.111986\pi$
$151$	3483.73	1.87750	0.938749	+	0.344603i	$0.111986\pi$
$152$	0	0
$153$	937.683	0.495472
$154$	0	0
$155$	− 139.444i	− 0.0722609i
$156$	0	0
$157$	2687.47i	1.36614i	0.730353	+	0.683070i	$0.239355\pi$
$157$	2687.47i	1.36614i	−0.730353	+	0.683070i	$0.760645\pi$
$158$	0	0
$159$	627.999	0.313230
$160$	0	0
$161$	919.681	0.450193
$162$	0	0
$163$	2442.97i	1.17391i	0.809618	+	0.586957i	$0.199674\pi$
$163$	2442.97i	1.17391i	−0.809618	+	0.586957i	$0.800326\pi$
$164$	0	0
$165$	226.196i	0.106723i
$166$	0	0
$167$	−1671.70	−0.774610	−0.387305	−	0.921952i	$-0.626594\pi$
$167$	−1671.70	−0.774610	−0.387305	+	0.921952i	$0.626594\pi$
$168$	0	0
$169$	2069.49	0.941961
$170$	0	0
$171$	463.519i	0.207288i
$172$	0	0
$173$	2590.55i	1.13847i	0.822173	+	0.569237i	$0.192761\pi$
$173$	2590.55i	1.13847i	−0.822173	+	0.569237i	$0.807239\pi$
$174$	0	0
$175$	−863.505	−0.372999
$176$	0	0
$177$	224.501	0.0953361
$178$	0	0
$179$	1898.74i	0.792840i	0.918069	+	0.396420i	$0.129748\pi$
$179$	1898.74i	0.792840i	−0.918069	+	0.396420i	$0.870252\pi$
$180$	0	0
$181$	− 4009.08i	− 1.64637i	−0.567776	−	0.823183i	$-0.692196\pi$
$181$	− 4009.08i	− 1.64637i	0.567776	−	0.823183i	$-0.307804\pi$
$182$	0	0
$183$	−1341.65	−0.541955
$184$	0	0
$185$	7.37845	0.00293229
$186$	0	0
$187$	6130.03i	2.39718i
$188$	0	0
$189$	− 189.000i	− 0.0727393i
$190$	0	0
$191$	4203.58	1.59246	0.796232	−	0.604991i	$-0.206823\pi$
$191$	4203.58	1.59246	0.796232	+	0.604991i	$0.206823\pi$
$192$	0	0
$193$	−3161.70	−1.17919	−0.589597	−	0.807698i	$-0.700713\pi$
$193$	−3161.70	−1.17919	−0.589597	+	0.807698i	$0.700713\pi$
$194$	0	0
$195$	43.4121i	0.0159426i
$196$	0	0
$197$	− 22.8747i	− 0.00827287i	−0.999991	−	0.00413643i	$-0.998683\pi$
$197$	− 22.8747i	− 0.00827287i	0.999991	−	0.00413643i	$-0.00131667\pi$
$198$	0	0
$199$	4942.15	1.76050	0.880251	−	0.474509i	$-0.157374\pi$
$199$	4942.15	1.76050	0.880251	+	0.474509i	$0.157374\pi$
$200$	0	0
$201$	−984.051	−0.345322
$202$	0	0
$203$	− 228.595i	− 0.0790355i
$204$	0	0
$205$	254.487i	0.0867032i
$206$	0	0
$207$	1182.45	0.397033
$208$	0	0
$209$	−3030.22	−1.00289
$210$	0	0
$211$	5076.99i	1.65647i	0.560384	+	0.828233i	$0.310654\pi$
$211$	5076.99i	1.65647i	−0.560384	+	0.828233i	$0.689346\pi$
$212$	0	0
$213$	− 445.922i	− 0.143446i
$214$	0	0
$215$	125.250	0.0397302
$216$	0	0
$217$	−761.701	−0.238284
$218$	0	0
$219$	107.728i	0.0332402i
$220$	0	0
$221$	1176.49i	0.358097i
$222$	0	0
$223$	2689.06	0.807500	0.403750	−	0.914869i	$-0.367707\pi$
$223$	2689.06	0.807500	0.403750	+	0.914869i	$0.367707\pi$
$224$	0	0
$225$	−1110.22	−0.328954
$226$	0	0
$227$	1351.24i	0.395086i	0.980294	+	0.197543i	$0.0632963\pi$
$227$	1351.24i	0.395086i	−0.980294	+	0.197543i	$0.936704\pi$
$228$	0	0
$229$	− 1017.83i	− 0.293714i	−0.989158	−	0.146857i	$-0.953084\pi$
$229$	− 1017.83i	− 0.293714i	0.989158	−	0.146857i	$-0.0469156\pi$
$230$	0	0
$231$	1235.57	0.351925
$232$	0	0
$233$	−5123.55	−1.44058	−0.720290	−	0.693673i	$-0.755991\pi$
$233$	−5123.55	−1.44058	−0.720290	+	0.693673i	$0.755991\pi$
$234$	0	0
$235$	− 540.951i	− 0.150161i
$236$	0	0
$237$	3908.10i	1.07113i
$238$	0	0
$239$	−3568.88	−0.965906	−0.482953	−	0.875646i	$-0.660436\pi$
$239$	−3568.88	−0.965906	−0.482953	+	0.875646i	$0.660436\pi$
$240$	0	0
$241$	−1056.55	−0.282399	−0.141200	−	0.989981i	$-0.545096\pi$
$241$	−1056.55	−0.282399	−0.141200	+	0.989981i	$0.545096\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	− 62.7929i	− 0.0163742i
$246$	0	0
$247$	−581.568	−0.149815
$248$	0	0
$249$	1181.79	0.300775
$250$	0	0
$251$	4004.08i	1.00691i	0.864020	+	0.503457i	$0.167939\pi$
$251$	4004.08i	1.00691i	−0.864020	+	0.503457i	$0.832061\pi$
$252$	0	0
$253$	7730.16i	1.92091i
$254$	0	0
$255$	400.543	0.0983647
$256$	0	0
$257$	−2498.49	−0.606425	−0.303213	−	0.952923i	$-0.598059\pi$
$257$	−2498.49	−0.606425	−0.303213	+	0.952923i	$0.598059\pi$
$258$	0	0
$259$	− 40.3040i	− 0.00966939i
$260$	0	0
$261$	− 293.907i	− 0.0697027i
$262$	0	0
$263$	51.9986	0.0121915	0.00609577	−	0.999981i	$-0.498060\pi$
$263$	51.9986	0.0121915	0.00609577	+	0.999981i	$0.498060\pi$
$264$	0	0
$265$	268.258	0.0621846
$266$	0	0
$267$	1442.08i	0.330538i
$268$	0	0
$269$	− 99.8880i	− 0.0226404i	−0.999936	−	0.0113202i	$-0.996397\pi$
$269$	− 99.8880i	− 0.0226404i	0.999936	−	0.0113202i	$-0.00360341\pi$
$270$	0	0
$271$	−7653.72	−1.71561	−0.857806	−	0.513974i	$-0.828173\pi$
$271$	−7653.72	−1.71561	−0.857806	+	0.513974i	$0.828173\pi$
$272$	0	0
$273$	237.134	0.0525715
$274$	0	0
$275$	− 7257.98i	− 1.59154i
$276$	0	0
$277$	− 3373.81i	− 0.731814i	−0.930651	−	0.365907i	$-0.880759\pi$
$277$	− 3373.81i	− 0.731814i	0.930651	−	0.365907i	$-0.119241\pi$
$278$	0	0
$279$	−979.330	−0.210147
$280$	0	0
$281$	−4769.05	−1.01245	−0.506224	−	0.862402i	$-0.668959\pi$
$281$	−4769.05	−1.01245	−0.506224	+	0.862402i	$0.668959\pi$
$282$	0	0
$283$	− 4315.56i	− 0.906480i	−0.891389	−	0.453240i	$-0.850268\pi$
$283$	− 4315.56i	− 0.906480i	0.891389	−	0.453240i	$-0.149732\pi$
$284$	0	0
$285$	197.998i	0.0411523i
$286$	0	0
$287$	1390.11	0.285908
$288$	0	0
$289$	5941.94	1.20943
$290$	0	0
$291$	3834.36i	0.772421i
$292$	0	0
$293$	6184.45i	1.23310i	0.787314	+	0.616552i	$0.211471\pi$
$293$	6184.45i	1.23310i	−0.787314	+	0.616552i	$0.788529\pi$
$294$	0	0
$295$	95.8982	0.0189268
$296$	0	0
$297$	1588.59	0.310369
$298$	0	0
$299$	1483.59i	0.286951i
$300$	0	0
$301$	− 684.167i	− 0.131012i
$302$	0	0
$303$	4518.85	0.856770
$304$	0	0
$305$	−573.104	−0.107593
$306$	0	0
$307$	− 997.239i	− 0.185392i	−0.995694	−	0.0926962i	$-0.970451\pi$
$307$	− 997.239i	− 0.185392i	0.995694	−	0.0926962i	$-0.0295485\pi$
$308$	0	0
$309$	− 3303.20i	− 0.608130i
$310$	0	0
$311$	−1979.15	−0.360860	−0.180430	−	0.983588i	$-0.557749\pi$
$311$	−1979.15	−0.360860	−0.180430	+	0.983588i	$0.557749\pi$
$312$	0	0
$313$	−3819.96	−0.689830	−0.344915	−	0.938634i	$-0.612092\pi$
$313$	−3819.96	−0.689830	−0.344915	+	0.938634i	$0.612092\pi$
$314$	0	0
$315$	− 80.7337i	− 0.0144407i
$316$	0	0
$317$	5469.80i	0.969132i	0.874755	+	0.484566i	$0.161022\pi$
$317$	5469.80i	0.969132i	−0.874755	+	0.484566i	$0.838978\pi$
$318$	0	0
$319$	1921.40	0.337234
$320$	0	0
$321$	2513.81	0.437093
$322$	0	0
$323$	5365.86i	0.924347i
$324$	0	0
$325$	− 1392.97i	− 0.237748i
$326$	0	0
$327$	3835.07	0.648562
$328$	0	0
$329$	−2954.89	−0.495163
$330$	0	0
$331$	1451.21i	0.240983i	0.992714	+	0.120492i	$0.0384471\pi$
$331$	1451.21i	0.240983i	−0.992714	+	0.120492i	$0.961553\pi$
$332$	0	0
$333$	− 51.8195i	− 0.00852760i
$334$	0	0
$335$	−420.350	−0.0685557
$336$	0	0
$337$	6599.01	1.06668	0.533340	−	0.845901i	$-0.320937\pi$
$337$	6599.01	1.06668	0.533340	+	0.845901i	$0.320937\pi$
$338$	0	0
$339$	4569.21i	0.732052i
$340$	0	0
$341$	− 6402.29i	− 1.01673i
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	505.097	0.0788218
$346$	0	0
$347$	− 6659.56i	− 1.03027i	−0.857109	−	0.515135i	$-0.827742\pi$
$347$	− 6659.56i	− 1.03027i	0.857109	−	0.515135i	$-0.172258\pi$
$348$	0	0
$349$	− 7006.27i	− 1.07460i	−0.843390	−	0.537302i	$-0.819443\pi$
$349$	− 7006.27i	− 1.07460i	0.843390	−	0.537302i	$-0.180557\pi$
$350$	0	0
$351$	304.887	0.0463637
$352$	0	0
$353$	−9955.17	−1.50102	−0.750510	−	0.660859i	$-0.770192\pi$
$353$	−9955.17	−1.50102	−0.750510	+	0.660859i	$0.770192\pi$
$354$	0	0
$355$	− 190.481i	− 0.0284780i
$356$	0	0
$357$	− 2187.93i	− 0.324363i
$358$	0	0
$359$	5535.13	0.813741	0.406870	−	0.913486i	$-0.366620\pi$
$359$	5535.13	0.813741	0.406870	+	0.913486i	$0.366620\pi$
$360$	0	0
$361$	4206.53	0.613286
$362$	0	0
$363$	6392.31i	0.924268i
$364$	0	0
$365$	46.0175i	0.00659908i
$366$	0	0
$367$	−3303.44	−0.469859	−0.234930	−	0.972012i	$-0.575486\pi$
$367$	−3303.44	−0.469859	−0.234930	+	0.972012i	$0.575486\pi$
$368$	0	0
$369$	1787.29	0.252147
$370$	0	0
$371$	− 1465.33i	− 0.205057i
$372$	0	0
$373$	12293.9i	1.70658i	0.521435	+	0.853291i	$0.325397\pi$
$373$	12293.9i	1.70658i	−0.521435	+	0.853291i	$0.674603\pi$
$374$	0	0
$375$	−954.802	−0.131482
$376$	0	0
$377$	368.760	0.0503769
$378$	0	0
$379$	− 5187.37i	− 0.703054i	−0.936178	−	0.351527i	$-0.885663\pi$
$379$	− 5187.37i	− 0.703054i	0.936178	−	0.351527i	$-0.114337\pi$
$380$	0	0
$381$	− 1354.65i	− 0.182154i
$382$	0	0
$383$	−8112.41	−1.08231	−0.541155	−	0.840923i	$-0.682013\pi$
$383$	−8112.41	−1.08231	−0.541155	+	0.840923i	$0.682013\pi$
$384$	0	0
$385$	527.790	0.0698667
$386$	0	0
$387$	− 879.644i	− 0.115542i
$388$	0	0
$389$	6411.90i	0.835723i	0.908511	+	0.417861i	$0.137220\pi$
$389$	6411.90i	0.835723i	−0.908511	+	0.417861i	$0.862780\pi$
$390$	0	0
$391$	13688.4	1.77047
$392$	0	0
$393$	−2840.04	−0.364532
$394$	0	0
$395$	1669.39i	0.212649i
$396$	0	0
$397$	− 11946.9i	− 1.51032i	−0.655540	−	0.755160i	$-0.727559\pi$
$397$	− 11946.9i	− 1.51032i	0.655540	−	0.755160i	$-0.272441\pi$
$398$	0	0
$399$	1081.55	0.135702
$400$	0	0
$401$	2194.25	0.273256	0.136628	−	0.990622i	$-0.456374\pi$
$401$	2194.25	0.273256	0.136628	+	0.990622i	$0.456374\pi$
$402$	0	0
$403$	− 1228.74i	− 0.151881i
$404$	0	0
$405$	− 103.800i	− 0.0127355i
$406$	0	0
$407$	338.766	0.0412580
$408$	0	0
$409$	1599.68	0.193396	0.0966982	−	0.995314i	$-0.469172\pi$
$409$	1599.68	0.193396	0.0966982	+	0.995314i	$0.469172\pi$
$410$	0	0
$411$	1402.88i	0.168367i
$412$	0	0
$413$	− 523.835i	− 0.0624121i
$414$	0	0
$415$	504.818	0.0597121
$416$	0	0
$417$	2887.62	0.339107
$418$	0	0
$419$	− 16378.8i	− 1.90969i	−0.297109	−	0.954843i	$-0.596023\pi$
$419$	− 16378.8i	− 1.90969i	0.297109	−	0.954843i	$-0.403977\pi$
$420$	0	0
$421$	6202.51i	0.718033i	0.933331	+	0.359016i	$0.116888\pi$
$421$	6202.51i	0.718033i	−0.933331	+	0.359016i	$0.883112\pi$
$422$	0	0
$423$	−3799.15	−0.436693
$424$	0	0
$425$	−12852.3	−1.46689
$426$	0	0
$427$	3130.52i	0.354793i
$428$	0	0
$429$	1993.18i	0.224316i
$430$	0	0
$431$	−10846.2	−1.21217	−0.606084	−	0.795400i	$-0.707261\pi$
$431$	−10846.2	−1.21217	−0.606084	+	0.795400i	$0.707261\pi$
$432$	0	0
$433$	−11096.6	−1.23156	−0.615782	−	0.787917i	$-0.711160\pi$
$433$	−11096.6	−1.23156	−0.615782	+	0.787917i	$0.711160\pi$
$434$	0	0
$435$	− 125.546i	− 0.0138379i
$436$	0	0
$437$	6766.51i	0.740700i
$438$	0	0
$439$	−3308.76	−0.359723	−0.179862	−	0.983692i	$-0.557565\pi$
$439$	−3308.76	−0.359723	−0.179862	+	0.983692i	$0.557565\pi$
$440$	0	0
$441$	−441.000	−0.0476190
$442$	0	0
$443$	− 9065.37i	− 0.972254i	−0.873888	−	0.486127i	$-0.838409\pi$
$443$	− 9065.37i	− 0.972254i	0.873888	−	0.486127i	$-0.161591\pi$
$444$	0	0
$445$	616.002i	0.0656209i
$446$	0	0
$447$	−4399.98	−0.465575
$448$	0	0
$449$	1298.44	0.136475	0.0682374	−	0.997669i	$-0.478262\pi$
$449$	1298.44	0.136475	0.0682374	+	0.997669i	$0.478262\pi$
$450$	0	0
$451$	11684.2i	1.21993i
$452$	0	0
$453$	− 10451.2i	− 1.08397i
$454$	0	0
$455$	101.295	0.0104369
$456$	0	0
$457$	8174.80	0.836764	0.418382	−	0.908271i	$-0.362597\pi$
$457$	8174.80	0.836764	0.418382	+	0.908271i	$0.362597\pi$
$458$	0	0
$459$	− 2813.05i	− 0.286061i
$460$	0	0
$461$	4936.55i	0.498738i	0.968409	+	0.249369i	$0.0802231\pi$
$461$	4936.55i	0.498738i	−0.968409	+	0.249369i	$0.919777\pi$
$462$	0	0
$463$	−12083.1	−1.21285	−0.606425	−	0.795141i	$-0.707397\pi$
$463$	−12083.1	−1.21285	−0.606425	+	0.795141i	$0.707397\pi$
$464$	0	0
$465$	−418.333	−0.0417198
$466$	0	0
$467$	− 14461.5i	− 1.43298i	−0.697599	−	0.716488i	$-0.745748\pi$
$467$	− 14461.5i	− 1.43298i	0.697599	−	0.716488i	$-0.254252\pi$
$468$	0	0
$469$	2296.12i	0.226066i
$470$	0	0
$471$	8062.42	0.788741
$472$	0	0
$473$	5750.60	0.559013
$474$	0	0
$475$	− 6353.19i	− 0.613693i
$476$	0	0
$477$	− 1884.00i	− 0.180843i
$478$	0	0
$479$	−15170.7	−1.44711	−0.723555	−	0.690266i	$-0.757493\pi$
$479$	−15170.7	−1.44711	−0.723555	+	0.690266i	$0.757493\pi$
$480$	0	0
$481$	65.0168	0.00616323
$482$	0	0
$483$	− 2759.04i	− 0.259919i
$484$	0	0
$485$	1637.90i	0.153347i
$486$	0	0
$487$	4020.50	0.374099	0.187050	−	0.982350i	$-0.440108\pi$
$487$	4020.50	0.374099	0.187050	+	0.982350i	$0.440108\pi$
$488$	0	0
$489$	7328.90	0.677759
$490$	0	0
$491$	− 18396.4i	− 1.69087i	−0.534078	−	0.845435i	$-0.679341\pi$
$491$	− 18396.4i	− 1.69087i	0.534078	−	0.845435i	$-0.320659\pi$
$492$	0	0
$493$	− 3402.37i	− 0.310822i
$494$	0	0
$495$	678.588	0.0616167
$496$	0	0
$497$	−1040.49	−0.0939078
$498$	0	0
$499$	− 4588.93i	− 0.411681i	−0.978586	−	0.205840i	$-0.934007\pi$
$499$	− 4588.93i	− 0.411681i	0.978586	−	0.205840i	$-0.0659927\pi$
$500$	0	0
$501$	5015.09i	0.447221i
$502$	0	0
$503$	6342.89	0.562257	0.281129	−	0.959670i	$-0.409291\pi$
$503$	6342.89	0.562257	0.281129	+	0.959670i	$0.409291\pi$
$504$	0	0
$505$	1930.28	0.170092
$506$	0	0
$507$	− 6208.46i	− 0.543841i
$508$	0	0
$509$	− 16127.4i	− 1.40439i	−0.711984	−	0.702196i	$-0.752203\pi$
$509$	− 16127.4i	− 1.40439i	0.711984	−	0.702196i	$-0.247797\pi$
$510$	0	0
$511$	251.366	0.0217608
$512$	0	0
$513$	1390.56	0.119678
$514$	0	0
$515$	− 1411.00i	− 0.120730i
$516$	0	0
$517$	− 24836.6i	− 2.11279i
$518$	0	0
$519$	7771.66	0.657298
$520$	0	0
$521$	10521.2	0.884723	0.442362	−	0.896837i	$-0.354141\pi$
$521$	10521.2	0.884723	0.442362	+	0.896837i	$0.354141\pi$
$522$	0	0
$523$	4897.33i	0.409455i	0.978819	+	0.204728i	$0.0656309\pi$
$523$	4897.33i	0.409455i	−0.978819	+	0.204728i	$0.934369\pi$
$524$	0	0
$525$	2590.51i	0.215351i
$526$	0	0
$527$	−11337.1	−0.937096
$528$	0	0
$529$	5094.51	0.418715
$530$	0	0
$531$	− 673.502i	− 0.0550423i
$532$	0	0
$533$	2242.47i	0.182237i
$534$	0	0
$535$	1073.80	0.0867749
$536$	0	0
$537$	5696.22	0.457746
$538$	0	0
$539$	− 2883.00i	− 0.230389i
$540$	0	0
$541$	− 3638.72i	− 0.289170i	−0.989492	−	0.144585i	$-0.953815\pi$
$541$	− 3638.72i	− 0.289170i	0.989492	−	0.144585i	$-0.0461847\pi$
$542$	0	0
$543$	−12027.2	−0.950530
$544$	0	0
$545$	1638.20	0.128757
$546$	0	0
$547$	18255.3i	1.42695i	0.700681	+	0.713475i	$0.252880\pi$
$547$	18255.3i	1.42695i	−0.700681	+	0.713475i	$0.747120\pi$
$548$	0	0
$549$	4024.96i	0.312898i
$550$	0	0
$551$	1681.87	0.130037
$552$	0	0
$553$	9118.90	0.701221
$554$	0	0
$555$	− 22.1353i	− 0.00169296i
$556$	0	0
$557$	18481.5i	1.40590i	0.711240	+	0.702950i	$0.248134\pi$
$557$	18481.5i	1.40590i	−0.711240	+	0.702950i	$0.751866\pi$
$558$	0	0
$559$	1103.67	0.0835068
$560$	0	0
$561$	18390.1	1.38401
$562$	0	0
$563$	− 20507.1i	− 1.53512i	−0.640980	−	0.767558i	$-0.721472\pi$
$563$	− 20507.1i	− 1.53512i	0.640980	−	0.767558i	$-0.278528\pi$
$564$	0	0
$565$	1951.80i	0.145332i
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	−7707.00	−0.567828	−0.283914	−	0.958850i	$-0.591633\pi$
$569$	−7707.00	−0.567828	−0.283914	+	0.958850i	$0.591633\pi$
$570$	0	0
$571$	21912.4i	1.60596i	0.596005	+	0.802981i	$0.296754\pi$
$571$	21912.4i	1.60596i	−0.596005	+	0.802981i	$0.703246\pi$
$572$	0	0
$573$	− 12610.8i	− 0.919410i
$574$	0	0
$575$	−16207.1	−1.17545
$576$	0	0
$577$	21547.3	1.55464	0.777319	−	0.629107i	$-0.216579\pi$
$577$	21547.3	1.55464	0.777319	+	0.629107i	$0.216579\pi$
$578$	0	0
$579$	9485.11i	0.680808i
$580$	0	0
$581$	− 2757.52i	− 0.196904i
$582$	0	0
$583$	12316.5	0.874951
$584$	0	0
$585$	130.236	0.00920446
$586$	0	0
$587$	− 2902.83i	− 0.204110i	−0.994779	−	0.102055i	$-0.967458\pi$
$587$	− 2902.83i	− 0.204110i	0.994779	−	0.102055i	$-0.0325418\pi$
$588$	0	0
$589$	− 5604.18i	− 0.392048i
$590$	0	0
$591$	−68.6241	−0.00477634
$592$	0	0
$593$	−7451.82	−0.516036	−0.258018	−	0.966140i	$-0.583069\pi$
$593$	−7451.82	−0.516036	−0.258018	+	0.966140i	$0.583069\pi$
$594$	0	0
$595$	− 934.601i	− 0.0643948i
$596$	0	0
$597$	− 14826.5i	− 1.01643i
$598$	0	0
$599$	3734.85	0.254761	0.127381	−	0.991854i	$-0.459343\pi$
$599$	3734.85	0.254761	0.127381	+	0.991854i	$0.459343\pi$
$600$	0	0
$601$	5577.40	0.378547	0.189274	−	0.981924i	$-0.439387\pi$
$601$	5577.40	0.378547	0.189274	+	0.981924i	$0.439387\pi$
$602$	0	0
$603$	2952.15i	0.199372i
$604$	0	0
$605$	2730.56i	0.183492i
$606$	0	0
$607$	10660.7	0.712858	0.356429	−	0.934322i	$-0.383994\pi$
$607$	10660.7	0.712858	0.356429	+	0.934322i	$0.383994\pi$
$608$	0	0
$609$	−685.784	−0.0456312
$610$	0	0
$611$	− 4766.71i	− 0.315615i
$612$	0	0
$613$	634.143i	0.0417827i	0.999782	+	0.0208914i	$0.00665041\pi$
$613$	634.143i	0.0417827i	−0.999782	+	0.0208914i	$0.993350\pi$
$614$	0	0
$615$	763.461	0.0500581
$616$	0	0
$617$	7716.38	0.503484	0.251742	−	0.967794i	$-0.418997\pi$
$617$	7716.38	0.503484	0.251742	+	0.967794i	$0.418997\pi$
$618$	0	0
$619$	− 6849.48i	− 0.444756i	−0.974961	−	0.222378i	$-0.928618\pi$
$619$	− 6849.48i	− 0.444756i	0.974961	−	0.222378i	$-0.0713819\pi$
$620$	0	0
$621$	− 3547.34i	− 0.229227i
$622$	0	0
$623$	3364.85	0.216388
$624$	0	0
$625$	15011.9	0.960760
$626$	0	0
$627$	9090.67i	0.579021i
$628$	0	0
$629$	− 599.880i	− 0.0380267i
$630$	0	0
$631$	−27119.1	−1.71092	−0.855462	−	0.517866i	$-0.826727\pi$
$631$	−27119.1	−1.71092	−0.855462	+	0.517866i	$0.826727\pi$
$632$	0	0
$633$	15231.0	0.956361
$634$	0	0
$635$	− 578.655i	− 0.0361625i
$636$	0	0
$637$	− 553.313i	− 0.0344161i
$638$	0	0
$639$	−1337.77	−0.0828189
$640$	0	0
$641$	1587.63	0.0978275	0.0489137	−	0.998803i	$-0.484424\pi$
$641$	1587.63	0.0978275	0.0489137	+	0.998803i	$0.484424\pi$
$642$	0	0
$643$	− 16849.4i	− 1.03340i	−0.856167	−	0.516699i	$-0.827161\pi$
$643$	− 16849.4i	− 1.03340i	0.856167	−	0.516699i	$-0.172839\pi$
$644$	0	0
$645$	− 375.751i	− 0.0229383i
$646$	0	0
$647$	3808.51	0.231419	0.115709	−	0.993283i	$-0.463086\pi$
$647$	3808.51	0.231419	0.115709	+	0.993283i	$0.463086\pi$
$648$	0	0
$649$	4402.96	0.266304
$650$	0	0
$651$	2285.10i	0.137573i
$652$	0	0
$653$	− 17569.4i	− 1.05290i	−0.850206	−	0.526451i	$-0.823523\pi$
$653$	− 17569.4i	− 1.05290i	0.850206	−	0.526451i	$-0.176477\pi$
$654$	0	0
$655$	−1213.16	−0.0723695
$656$	0	0
$657$	323.185	0.0191912
$658$	0	0
$659$	16528.3i	0.977011i	0.872561	+	0.488506i	$0.162458\pi$
$659$	16528.3i	0.977011i	−0.872561	+	0.488506i	$0.837542\pi$
$660$	0	0
$661$	− 5472.10i	− 0.321997i	−0.986955	−	0.160999i	$-0.948529\pi$
$661$	− 5472.10i	− 0.321997i	0.986955	−	0.160999i	$-0.0514714\pi$
$662$	0	0
$663$	3529.48	0.206747
$664$	0	0
$665$	461.995	0.0269405
$666$	0	0
$667$	− 4290.50i	− 0.249068i
$668$	0	0
$669$	− 8067.17i	− 0.466210i
$670$	0	0
$671$	−26312.9	−1.51385
$672$	0	0
$673$	4565.96	0.261523	0.130761	−	0.991414i	$-0.458258\pi$
$673$	4565.96	0.261523	0.130761	+	0.991414i	$0.458258\pi$
$674$	0	0
$675$	3330.66i	0.189922i
$676$	0	0
$677$	− 11267.7i	− 0.639667i	−0.947474	−	0.319834i	$-0.896373\pi$
$677$	− 11267.7i	− 0.639667i	0.947474	−	0.319834i	$-0.103627\pi$
$678$	0	0
$679$	8946.85	0.505668
$680$	0	0
$681$	4053.71	0.228103
$682$	0	0
$683$	− 33282.6i	− 1.86460i	−0.361680	−	0.932302i	$-0.617797\pi$
$683$	− 33282.6i	− 1.86460i	0.361680	−	0.932302i	$-0.382203\pi$
$684$	0	0
$685$	599.257i	0.0334255i
$686$	0	0
$687$	−3053.50	−0.169576
$688$	0	0
$689$	2363.81	0.130702
$690$	0	0
$691$	− 26803.4i	− 1.47561i	−0.675012	−	0.737807i	$-0.735861\pi$
$691$	− 26803.4i	− 1.47561i	0.675012	−	0.737807i	$-0.264139\pi$
$692$	0	0
$693$	− 3706.72i	− 0.203184i
$694$	0	0
$695$	1233.48	0.0673219
$696$	0	0
$697$	20690.2	1.12439
$698$	0	0
$699$	15370.7i	0.831719i
$700$	0	0
$701$	15482.4i	0.834185i	0.908864	+	0.417093i	$0.136951\pi$
$701$	15482.4i	0.834185i	−0.908864	+	0.417093i	$0.863049\pi$
$702$	0	0
$703$	296.535	0.0159090
$704$	0	0
$705$	−1622.85	−0.0866953
$706$	0	0
$707$	− 10544.0i	− 0.560888i
$708$	0	0
$709$	− 21219.1i	− 1.12398i	−0.827144	−	0.561989i	$-0.810036\pi$
$709$	− 21219.1i	− 1.12398i	0.827144	−	0.561989i	$-0.189964\pi$
$710$	0	0
$711$	11724.3	0.618419
$712$	0	0
$713$	−14296.4	−0.750916
$714$	0	0
$715$	851.410i	0.0445328i
$716$	0	0
$717$	10706.6i	0.557666i
$718$	0	0
$719$	−298.549	−0.0154854	−0.00774270	−	0.999970i	$-0.502465\pi$
$719$	−298.549	−0.0154854	−0.00774270	+	0.999970i	$0.502465\pi$
$720$	0	0
$721$	−7707.46	−0.398115
$722$	0	0
$723$	3169.65i	0.163043i
$724$	0	0
$725$	4028.42i	0.206361i
$726$	0	0
$727$	1760.30	0.0898021	0.0449010	−	0.998991i	$-0.485703\pi$
$727$	1760.30	0.0898021	0.0449010	+	0.998991i	$0.485703\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	− 10183.1i	− 0.515231i
$732$	0	0
$733$	13537.8i	0.682170i	0.940032	+	0.341085i	$0.110794\pi$
$733$	13537.8i	0.682170i	−0.940032	+	0.341085i	$0.889206\pi$
$734$	0	0
$735$	−188.379	−0.00945368
$736$	0	0
$737$	−19299.5	−0.964594
$738$	0	0
$739$	− 32498.8i	− 1.61771i	−0.588008	−	0.808855i	$-0.700088\pi$
$739$	− 32498.8i	− 1.61771i	0.588008	−	0.808855i	$-0.299912\pi$
$740$	0	0
$741$	1744.70i	0.0864957i
$742$	0	0
$743$	−18031.9	−0.890344	−0.445172	−	0.895445i	$-0.646858\pi$
$743$	−18031.9	−0.890344	−0.445172	+	0.895445i	$0.646858\pi$
$744$	0	0
$745$	−1879.51	−0.0924293
$746$	0	0
$747$	− 3545.38i	− 0.173653i
$748$	0	0
$749$	− 5865.55i	− 0.286145i
$750$	0	0
$751$	−4040.23	−0.196312	−0.0981559	−	0.995171i	$-0.531294\pi$
$751$	−4040.23	−0.196312	−0.0981559	+	0.995171i	$0.531294\pi$
$752$	0	0
$753$	12012.2	0.581342
$754$	0	0
$755$	− 4464.36i	− 0.215198i
$756$	0	0
$757$	− 13359.2i	− 0.641413i	−0.947179	−	0.320706i	$-0.896080\pi$
$757$	− 13359.2i	− 0.641413i	0.947179	−	0.320706i	$-0.103920\pi$
$758$	0	0
$759$	23190.5	1.10904
$760$	0	0
$761$	12534.3	0.597066	0.298533	−	0.954399i	$-0.403503\pi$
$761$	12534.3	0.597066	0.298533	+	0.954399i	$0.403503\pi$
$762$	0	0
$763$	− 8948.49i	− 0.424583i
$764$	0	0
$765$	− 1201.63i	− 0.0567909i
$766$	0	0
$767$	845.028	0.0397812
$768$	0	0
$769$	6633.90	0.311085	0.155543	−	0.987829i	$-0.450287\pi$
$769$	6633.90	0.311085	0.155543	+	0.987829i	$0.450287\pi$
$770$	0	0
$771$	7495.46i	0.350120i
$772$	0	0
$773$	26826.6i	1.24824i	0.781330	+	0.624118i	$0.214542\pi$
$773$	26826.6i	1.24824i	−0.781330	+	0.624118i	$0.785458\pi$
$774$	0	0
$775$	13423.1	0.622158
$776$	0	0
$777$	−120.912	−0.00558262
$778$	0	0
$779$	10227.7i	0.470403i
$780$	0	0
$781$	− 8745.55i	− 0.400692i
$782$	0	0
$783$	−881.722	−0.0402429
$784$	0	0
$785$	3443.97	0.156586
$786$	0	0
$787$	17011.8i	0.770527i	0.922807	+	0.385264i	$0.125889\pi$
$787$	17011.8i	0.770527i	−0.922807	+	0.385264i	$0.874111\pi$
$788$	0	0
$789$	− 155.996i	− 0.00703878i
$790$	0	0
$791$	10661.5	0.479240
$792$	0	0
$793$	−5050.03	−0.226144
$794$	0	0
$795$	− 804.773i	− 0.0359023i
$796$	0	0
$797$	− 32459.1i	− 1.44261i	−0.692617	−	0.721306i	$-0.743542\pi$
$797$	− 32459.1i	− 1.44261i	0.692617	−	0.721306i	$-0.256458\pi$
$798$	0	0
$799$	−43980.2	−1.94732
$800$	0	0
$801$	4326.23	0.190836
$802$	0	0
$803$	2112.80i	0.0928505i
$804$	0	0
$805$	− 1178.56i	− 0.0516010i
$806$	0	0
$807$	−299.664	−0.0130715
$808$	0	0
$809$	−26684.3	−1.15966	−0.579832	−	0.814736i	$-0.696882\pi$
$809$	−26684.3	−1.15966	−0.579832	+	0.814736i	$0.696882\pi$
$810$	0	0
$811$	6621.72i	0.286708i	0.989672	+	0.143354i	$0.0457887\pi$
$811$	6621.72i	0.286708i	−0.989672	+	0.143354i	$0.954211\pi$
$812$	0	0
$813$	22961.2i	0.990509i
$814$	0	0
$815$	3130.63	0.134554
$816$	0	0
$817$	5033.73	0.215554
$818$	0	0
$819$	− 711.403i	− 0.0303522i
$820$	0	0
$821$	3728.98i	0.158517i	0.996854	+	0.0792583i	$0.0252552\pi$
$821$	3728.98i	0.158517i	−0.996854	+	0.0792583i	$0.974745\pi$
$822$	0	0
$823$	4227.28	0.179045	0.0895224	−	0.995985i	$-0.471466\pi$
$823$	4227.28	0.179045	0.0895224	+	0.995985i	$0.471466\pi$
$824$	0	0
$825$	−21773.9	−0.918874
$826$	0	0
$827$	− 13173.2i	− 0.553903i	−0.960884	−	0.276951i	$-0.910676\pi$
$827$	− 13173.2i	− 0.553903i	0.960884	−	0.276951i	$-0.0893241\pi$
$828$	0	0
$829$	30108.2i	1.26140i	0.776026	+	0.630701i	$0.217233\pi$
$829$	30108.2i	1.26140i	−0.776026	+	0.630701i	$0.782767\pi$
$830$	0	0
$831$	−10121.4	−0.422513
$832$	0	0
$833$	−5105.17	−0.212345
$834$	0	0
$835$	2142.26i	0.0887855i
$836$	0	0
$837$	2937.99i	0.121328i
$838$	0	0
$839$	19630.4	0.807769	0.403885	−	0.914810i	$-0.367660\pi$
$839$	19630.4	0.807769	0.403885	+	0.914810i	$0.367660\pi$
$840$	0	0
$841$	23322.6	0.956274
$842$	0	0
$843$	14307.2i	0.584537i
$844$	0	0
$845$	− 2652.02i	− 0.107967i
$846$	0	0
$847$	14915.4	0.605075
$848$	0	0
$849$	−12946.7	−0.523356
$850$	0	0
$851$	− 756.467i	− 0.0304716i
$852$	0	0
$853$	− 5126.66i	− 0.205784i	−0.994693	−	0.102892i	$-0.967190\pi$
$853$	− 5126.66i	− 0.205784i	0.994693	−	0.102892i	$-0.0328096\pi$
$854$	0	0
$855$	593.994	0.0237593
$856$	0	0
$857$	−9742.87	−0.388343	−0.194172	−	0.980968i	$-0.562202\pi$
$857$	−9742.87	−0.388343	−0.194172	+	0.980968i	$0.562202\pi$
$858$	0	0
$859$	− 4584.46i	− 0.182095i	−0.995847	−	0.0910475i	$-0.970978\pi$
$859$	− 4584.46i	− 0.182095i	0.995847	−	0.0910475i	$-0.0290215\pi$
$860$	0	0
$861$	− 4170.33i	− 0.165069i
$862$	0	0
$863$	−47428.1	−1.87076	−0.935382	−	0.353638i	$-0.884944\pi$
$863$	−47428.1	−1.87076	−0.935382	+	0.353638i	$0.884944\pi$
$864$	0	0
$865$	3319.76	0.130492
$866$	0	0
$867$	− 17825.8i	− 0.698266i
$868$	0	0
$869$	76646.7i	2.99202i
$870$	0	0
$871$	−3704.01	−0.144094
$872$	0	0
$873$	11503.1	0.445957
$874$	0	0
$875$	2227.87i	0.0860752i
$876$	0	0
$877$	− 32425.1i	− 1.24848i	−0.781232	−	0.624241i	$-0.785408\pi$
$877$	− 32425.1i	− 1.24848i	0.781232	−	0.624241i	$-0.214592\pi$
$878$	0	0
$879$	18553.3	0.711932
$880$	0	0
$881$	−13989.5	−0.534982	−0.267491	−	0.963560i	$-0.586195\pi$
$881$	−13989.5	−0.534982	−0.267491	+	0.963560i	$0.586195\pi$
$882$	0	0
$883$	44760.1i	1.70589i	0.522003	+	0.852944i	$0.325185\pi$
$883$	44760.1i	1.70589i	−0.522003	+	0.852944i	$0.674815\pi$
$884$	0	0
$885$	− 287.695i	− 0.0109274i
$886$	0	0
$887$	8460.71	0.320274	0.160137	−	0.987095i	$-0.448806\pi$
$887$	8460.71	0.320274	0.160137	+	0.987095i	$0.448806\pi$
$888$	0	0
$889$	−3160.84	−0.119248
$890$	0	0
$891$	− 4765.78i	− 0.179192i
$892$	0	0
$893$	− 21740.5i	− 0.814689i
$894$	0	0
$895$	2433.21	0.0908751
$896$	0	0
$897$	4450.78	0.165671
$898$	0	0
$899$	3553.49i	0.131830i
$900$	0	0
$901$	− 21809.8i	− 0.806425i
$902$	0	0
$903$	−2052.50	−0.0756401
$904$	0	0
$905$	−5137.58	−0.188706
$906$	0	0
$907$	19962.9i	0.730825i	0.930846	+	0.365412i	$0.119072\pi$
$907$	19962.9i	0.730825i	−0.930846	+	0.365412i	$0.880928\pi$
$908$	0	0
$909$	− 13556.6i	− 0.494656i
$910$	0	0
$911$	−33633.5	−1.22319	−0.611596	−	0.791170i	$-0.709472\pi$
$911$	−33633.5	−1.22319	−0.611596	+	0.791170i	$0.709472\pi$
$912$	0	0
$913$	23177.6	0.840162
$914$	0	0
$915$	1719.31i	0.0621188i
$916$	0	0
$917$	6626.76i	0.238642i
$918$	0	0
$919$	36166.7	1.29818	0.649092	−	0.760710i	$-0.275149\pi$
$919$	36166.7	1.29818	0.649092	+	0.760710i	$0.275149\pi$
$920$	0	0
$921$	−2991.72	−0.107036
$922$	0	0
$923$	− 1678.47i	− 0.0598564i
$924$	0	0
$925$	710.260i	0.0252467i
$926$	0	0
$927$	−9909.59	−0.351104
$928$	0	0
$929$	−24869.0	−0.878282	−0.439141	−	0.898418i	$-0.644717\pi$
$929$	−24869.0	−0.878282	−0.439141	+	0.898418i	$0.644717\pi$
$930$	0	0
$931$	− 2523.61i	− 0.0888376i
$932$	0	0
$933$	5937.45i	0.208342i
$934$	0	0
$935$	7855.56	0.274764
$936$	0	0
$937$	13606.6	0.474396	0.237198	−	0.971461i	$-0.423771\pi$
$937$	13606.6	0.474396	0.237198	+	0.971461i	$0.423771\pi$
$938$	0	0
$939$	11459.9i	0.398274i
$940$	0	0
$941$	− 24937.2i	− 0.863899i	−0.901898	−	0.431949i	$-0.857826\pi$
$941$	− 24937.2i	− 0.863899i	0.901898	−	0.431949i	$-0.142174\pi$
$942$	0	0
$943$	26091.0	0.900997
$944$	0	0
$945$	−242.201	−0.00833736
$946$	0	0
$947$	− 21167.6i	− 0.726351i	−0.931721	−	0.363176i	$-0.881692\pi$
$947$	− 21167.6i	− 0.726351i	0.931721	−	0.363176i	$-0.118308\pi$
$948$	0	0
$949$	405.493i	0.0138702i
$950$	0	0
$951$	16409.4	0.559528
$952$	0	0
$953$	16473.2	0.559937	0.279968	−	0.960009i	$-0.409676\pi$
$953$	16473.2	0.559937	0.279968	+	0.960009i	$0.409676\pi$
$954$	0	0
$955$	− 5386.84i	− 0.182528i
$956$	0	0
$957$	− 5764.19i	− 0.194702i
$958$	0	0
$959$	3273.38	0.110222
$960$	0	0
$961$	−17950.4	−0.602545
$962$	0	0
$963$	− 7541.42i	− 0.252356i
$964$	0	0
$965$	4051.68i	0.135159i
$966$	0	0
$967$	14680.9	0.488219	0.244109	−	0.969748i	$-0.421504\pi$
$967$	14680.9	0.488219	0.244109	+	0.969748i	$0.421504\pi$
$968$	0	0
$969$	16097.6	0.533672
$970$	0	0
$971$	− 49712.5i	− 1.64300i	−0.570211	−	0.821498i	$-0.693138\pi$
$971$	− 49712.5i	− 1.64300i	0.570211	−	0.821498i	$-0.306862\pi$
$972$	0	0
$973$	− 6737.79i	− 0.221998i
$974$	0	0
$975$	−4178.91	−0.137264
$976$	0	0
$977$	15124.5	0.495266	0.247633	−	0.968854i	$-0.420347\pi$
$977$	15124.5	0.495266	0.247633	+	0.968854i	$0.420347\pi$
$978$	0	0
$979$	28282.4i	0.923300i
$980$	0	0
$981$	− 11505.2i	− 0.374447i
$982$	0	0
$983$	−43510.2	−1.41176	−0.705881	−	0.708331i	$-0.749448\pi$
$983$	−43510.2	−1.41176	−0.705881	+	0.708331i	$0.749448\pi$
$984$	0	0
$985$	−29.3136	−0.000948234 0
$986$	0	0
$987$	8864.68i	0.285882i
$988$	0	0
$989$	− 12841.1i	− 0.412866i
$990$	0	0
$991$	−50363.6	−1.61438	−0.807191	−	0.590291i	$-0.799013\pi$
$991$	−50363.6	−1.61438	−0.807191	+	0.590291i	$0.799013\pi$
$992$	0	0
$993$	4353.62	0.139132
$994$	0	0
$995$	− 6333.30i	− 0.201788i
$996$	0	0
$997$	39782.3i	1.26371i	0.775087	+	0.631855i	$0.217706\pi$
$997$	39782.3i	1.26371i	−0.775087	+	0.631855i	$0.782294\pi$
$998$	0	0
$999$	−155.458	−0.00492341

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.4.c.e.673.3	✓	12
4.3	odd	2		1344.4.c.h.673.9	yes	12
8.3	odd	2		1344.4.c.h.673.4	yes	12
8.5	even	2	inner	1344.4.c.e.673.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1344.4.c.e.673.3	✓	12	1.1	even	1	trivial
1344.4.c.e.673.10	yes	12	8.5	even	2	inner
1344.4.c.h.673.4	yes	12	8.3	odd	2
1344.4.c.h.673.9	yes	12	4.3	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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -1.28149i q^{5} -7.00000 q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -1.28149i q^{5} -7.00000 q^{7} -9.00000 q^{9} -58.8368i q^{11} -11.2921i q^{13} -3.84446 q^{15} -104.187 q^{17} -51.5021i q^{19} +21.0000i q^{21} -131.383 q^{23} +123.358 q^{25} +27.0000i q^{27} +32.6564i q^{29} +108.814 q^{31} -176.510 q^{33} +8.97041i q^{35} +5.75772i q^{37} -33.8763 q^{39} -198.587 q^{41} +97.7382i q^{43} +11.5334i q^{45} +422.128 q^{47} +49.0000 q^{49} +312.561i q^{51} +209.333i q^{53} -75.3986 q^{55} -154.506 q^{57} +74.8335i q^{59} -447.218i q^{61} +63.0000 q^{63} -14.4707 q^{65} -328.017i q^{67} +394.149i q^{69} +148.641 q^{71} -35.9094 q^{73} -370.073i q^{75} +411.858i q^{77} -1302.70 q^{79} +81.0000 q^{81} +393.931i q^{83} +133.514i q^{85} +97.9692 q^{87} -480.693 q^{89} +79.0448i q^{91} -326.443i q^{93} -65.9994 q^{95} -1278.12 q^{97} +529.531i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 84 q^{7} - 108 q^{9}+O(q^{10}) \) 12 * q - 84 * q^7 - 108 * q^9	\( 12 q - 84 q^{7} - 108 q^{9} - 24 q^{15} + 24 q^{17} + 80 q^{23} - 564 q^{25} - 640 q^{31} - 408 q^{33} + 120 q^{39} + 1416 q^{41} - 1536 q^{47} + 588 q^{49} + 1392 q^{55} - 336 q^{57} + 756 q^{63} - 2880 q^{65} + 1392 q^{71} + 2472 q^{73} - 544 q^{79} + 972 q^{81} + 720 q^{87} + 888 q^{89} + 2368 q^{95} - 2712 q^{97}+O(q^{100}) \) 12 * q - 84 * q^7 - 108 * q^9 - 24 * q^15 + 24 * q^17 + 80 * q^23 - 564 * q^25 - 640 * q^31 - 408 * q^33 + 120 * q^39 + 1416 * q^41 - 1536 * q^47 + 588 * q^49 + 1392 * q^55 - 336 * q^57 + 756 * q^63 - 2880 * q^65 + 1392 * q^71 + 2472 * q^73 - 544 * q^79 + 972 * q^81 + 720 * q^87 + 888 * q^89 + 2368 * q^95 - 2712 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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