LMFDB - Embedded newform 336.3.m.c.127.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,3,Mod(127,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	336.m (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:	$9.15533688251$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - x^{2} - 2x + 4 $ x^4 - x^3 - x^2 - 2*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.2
Root			$-0.895644 + 1.09445i$ of defining polynomial
Character	$\chi$	$=$	336.127
Dual form			336.3.m.c.127.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +5.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.73205i q^{3} +5.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} -4.37780i q^{11} +17.1652 q^{13} -9.66930i q^{15} +0.747727 q^{17} -3.65480i q^{19} -4.58258 q^{21} -9.86001i q^{23} +6.16515 q^{25} +5.19615i q^{27} -2.00000 q^{29} -17.8926i q^{31} -7.58258 q^{33} -14.7701i q^{35} -7.49545 q^{37} -29.7309i q^{39} +76.5735 q^{41} -70.0448i q^{43} -16.7477 q^{45} -40.1232i q^{47} -7.00000 q^{49} -1.29510i q^{51} +49.8258 q^{53} -24.4394i q^{55} -6.33030 q^{57} +89.3834i q^{59} -120.991 q^{61} +7.93725i q^{63} +95.8258 q^{65} +23.3748i q^{67} -17.0780 q^{69} +66.0484i q^{71} -40.5045 q^{73} -10.6784i q^{75} -11.5826 q^{77} +95.5488i q^{79} +9.00000 q^{81} +115.650i q^{83} +4.17424 q^{85} +3.46410i q^{87} +34.9038 q^{89} -45.4147i q^{91} -30.9909 q^{93} -20.4032i q^{95} +52.5045 q^{97} +13.1334i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 12 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 - 12 * q^9	$ 4 q + 4 q^{5} - 12 q^{9} + 32 q^{13} - 52 q^{17} - 12 q^{25} - 8 q^{29} - 12 q^{33} + 80 q^{37} + 68 q^{41} - 12 q^{45} - 28 q^{49} + 16 q^{53} + 48 q^{57} - 264 q^{61} + 200 q^{65} + 60 q^{69} - 272 q^{73} - 28 q^{77} + 36 q^{81} + 200 q^{85} - 172 q^{89} + 96 q^{93} + 320 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 12 * q^9 + 32 * q^13 - 52 * q^17 - 12 * q^25 - 8 * q^29 - 12 * q^33 + 80 * q^37 + 68 * q^41 - 12 * q^45 - 28 * q^49 + 16 * q^53 + 48 * q^57 - 264 * q^61 + 200 * q^65 + 60 * q^69 - 272 * q^73 - 28 * q^77 + 36 * q^81 + 200 * q^85 - 172 * q^89 + 96 * q^93 + 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	0	0
$5$	5.58258	1.11652	0.558258	−	0.829668i	$-0.311470\pi$
$5$	5.58258	1.11652	0.558258	+	0.829668i	$0.311470\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 4.37780i	− 0.397982i	−0.980001	−	0.198991i	$-0.936234\pi$
$11$	− 4.37780i	− 0.397982i	0.980001	−	0.198991i	$-0.0637665\pi$
$12$	0	0
$13$	17.1652	1.32040	0.660198	−	0.751091i	$-0.270472\pi$
$13$	17.1652	1.32040	0.660198	+	0.751091i	$0.270472\pi$
$14$	0	0
$15$	− 9.66930i	− 0.644620i
$16$	0	0
$17$	0.747727	0.0439839	0.0219920	−	0.999758i	$-0.492999\pi$
$17$	0.747727	0.0439839	0.0219920	+	0.999758i	$0.492999\pi$
$18$	0	0
$19$	− 3.65480i	− 0.192358i	−0.995364	−	0.0961790i	$-0.969338\pi$
$19$	− 3.65480i	− 0.192358i	0.995364	−	0.0961790i	$-0.0306621\pi$
$20$	0	0
$21$	−4.58258	−0.218218
$22$	0	0
$23$	− 9.86001i	− 0.428696i	−0.976757	−	0.214348i	$-0.931237\pi$
$23$	− 9.86001i	− 0.428696i	0.976757	−	0.214348i	$-0.0687626\pi$
$24$	0	0
$25$	6.16515	0.246606
$26$	0	0
$27$	5.19615i	0.192450i
$28$	0	0
$29$	−2.00000	−0.0689655	−0.0344828	−	0.999405i	$-0.510978\pi$
$29$	−2.00000	−0.0689655	−0.0344828	+	0.999405i	$0.510978\pi$
$30$	0	0
$31$	− 17.8926i	− 0.577181i	−0.957453	−	0.288590i	$-0.906813\pi$
$31$	− 17.8926i	− 0.577181i	0.957453	−	0.288590i	$-0.0931866\pi$
$32$	0	0
$33$	−7.58258	−0.229775
$34$	0	0
$35$	− 14.7701i	− 0.422003i
$36$	0	0
$37$	−7.49545	−0.202580	−0.101290	−	0.994857i	$-0.532297\pi$
$37$	−7.49545	−0.202580	−0.101290	+	0.994857i	$0.532297\pi$
$38$	0	0
$39$	− 29.7309i	− 0.762331i
$40$	0	0
$41$	76.5735	1.86765	0.933823	−	0.357735i	$-0.116451\pi$
$41$	76.5735	1.86765	0.933823	+	0.357735i	$0.116451\pi$
$42$	0	0
$43$	− 70.0448i	− 1.62895i	−0.580199	−	0.814475i	$-0.697025\pi$
$43$	− 70.0448i	− 1.62895i	0.580199	−	0.814475i	$-0.302975\pi$
$44$	0	0
$45$	−16.7477	−0.372172
$46$	0	0
$47$	− 40.1232i	− 0.853686i	−0.904326	−	0.426843i	$-0.859626\pi$
$47$	− 40.1232i	− 0.853686i	0.904326	−	0.426843i	$-0.140374\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	− 1.29510i	− 0.0253941i
$52$	0	0
$53$	49.8258	0.940109	0.470054	−	0.882637i	$-0.344234\pi$
$53$	49.8258	0.940109	0.470054	+	0.882637i	$0.344234\pi$
$54$	0	0
$55$	− 24.4394i	− 0.444353i
$56$	0	0
$57$	−6.33030	−0.111058
$58$	0	0
$59$	89.3834i	1.51497i	0.652850	+	0.757487i	$0.273573\pi$
$59$	89.3834i	1.51497i	−0.652850	+	0.757487i	$0.726427\pi$
$60$	0	0
$61$	−120.991	−1.98346	−0.991729	−	0.128351i	$-0.959031\pi$
$61$	−120.991	−1.98346	−0.991729	+	0.128351i	$0.959031\pi$
$62$	0	0
$63$	7.93725i	0.125988i
$64$	0	0
$65$	95.8258	1.47424
$66$	0	0
$67$	23.3748i	0.348878i	0.984668	+	0.174439i	$0.0558112\pi$
$67$	23.3748i	0.348878i	−0.984668	+	0.174439i	$0.944189\pi$
$68$	0	0
$69$	−17.0780	−0.247508
$70$	0	0
$71$	66.0484i	0.930260i	0.885242	+	0.465130i	$0.153992\pi$
$71$	66.0484i	0.930260i	−0.885242	+	0.465130i	$0.846008\pi$
$72$	0	0
$73$	−40.5045	−0.554857	−0.277428	−	0.960746i	$-0.589482\pi$
$73$	−40.5045	−0.554857	−0.277428	+	0.960746i	$0.589482\pi$
$74$	0	0
$75$	− 10.6784i	− 0.142378i
$76$	0	0
$77$	−11.5826	−0.150423
$78$	0	0
$79$	95.5488i	1.20948i	0.796423	+	0.604740i	$0.206723\pi$
$79$	95.5488i	1.20948i	−0.796423	+	0.604740i	$0.793277\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	115.650i	1.39338i	0.717374	+	0.696688i	$0.245344\pi$
$83$	115.650i	1.39338i	−0.717374	+	0.696688i	$0.754656\pi$
$84$	0	0
$85$	4.17424	0.0491087
$86$	0	0
$87$	3.46410i	0.0398173i
$88$	0	0
$89$	34.9038	0.392177	0.196089	−	0.980586i	$-0.437176\pi$
$89$	34.9038	0.392177	0.196089	+	0.980586i	$0.437176\pi$
$90$	0	0
$91$	− 45.4147i	− 0.499063i
$92$	0	0
$93$	−30.9909	−0.333236
$94$	0	0
$95$	− 20.4032i	− 0.214771i
$96$	0	0
$97$	52.5045	0.541284	0.270642	−	0.962680i	$-0.412764\pi$
$97$	52.5045	0.541284	0.270642	+	0.962680i	$0.412764\pi$
$98$	0	0
$99$	13.1334i	0.132661i
$100$	0	0
$101$	−113.583	−1.12458	−0.562290	−	0.826940i	$-0.690080\pi$
$101$	−113.583	−1.12458	−0.562290	+	0.826940i	$0.690080\pi$
$102$	0	0
$103$	50.0230i	0.485660i	0.970069	+	0.242830i	$0.0780758\pi$
$103$	50.0230i	0.485660i	−0.970069	+	0.242830i	$0.921924\pi$
$104$	0	0
$105$	−25.5826	−0.243644
$106$	0	0
$107$	160.533i	1.50031i	0.661265	+	0.750153i	$0.270020\pi$
$107$	160.533i	1.50031i	−0.661265	+	0.750153i	$0.729980\pi$
$108$	0	0
$109$	−205.303	−1.88351	−0.941757	−	0.336294i	$-0.890826\pi$
$109$	−205.303	−1.88351	−0.941757	+	0.336294i	$0.890826\pi$
$110$	0	0
$111$	12.9825i	0.116960i
$112$	0	0
$113$	179.321	1.58691	0.793457	−	0.608627i	$-0.208279\pi$
$113$	179.321	1.58691	0.793457	+	0.608627i	$0.208279\pi$
$114$	0	0
$115$	− 55.0442i	− 0.478645i
$116$	0	0
$117$	−51.4955	−0.440132
$118$	0	0
$119$	− 1.97830i	− 0.0166244i
$120$	0	0
$121$	101.835	0.841610
$122$	0	0
$123$	− 132.629i	− 1.07829i
$124$	0	0
$125$	−105.147	−0.841176
$126$	0	0
$127$	130.571i	1.02812i	0.857754	+	0.514060i	$0.171859\pi$
$127$	130.571i	1.02812i	−0.857754	+	0.514060i	$0.828141\pi$
$128$	0	0
$129$	−121.321	−0.940475
$130$	0	0
$131$	71.1890i	0.543428i	0.962378	+	0.271714i	$0.0875904\pi$
$131$	71.1890i	0.543428i	−0.962378	+	0.271714i	$0.912410\pi$
$132$	0	0
$133$	−9.66970	−0.0727045
$134$	0	0
$135$	29.0079i	0.214873i
$136$	0	0
$137$	−99.8439	−0.728788	−0.364394	−	0.931245i	$-0.618724\pi$
$137$	−99.8439	−0.728788	−0.364394	+	0.931245i	$0.618724\pi$
$138$	0	0
$139$	− 162.400i	− 1.16834i	−0.811630	−	0.584172i	$-0.801419\pi$
$139$	− 162.400i	− 1.16834i	0.811630	−	0.584172i	$-0.198581\pi$
$140$	0	0
$141$	−69.4955	−0.492876
$142$	0	0
$143$	− 75.1456i	− 0.525494i
$144$	0	0
$145$	−11.1652	−0.0770010
$146$	0	0
$147$	12.1244i	0.0824786i
$148$	0	0
$149$	293.165	1.96755	0.983776	−	0.179403i	$-0.0574166\pi$
$149$	293.165	1.96755	0.983776	+	0.179403i	$0.0574166\pi$
$150$	0	0
$151$	120.449i	0.797677i	0.917021	+	0.398839i	$0.130587\pi$
$151$	120.449i	0.797677i	−0.917021	+	0.398839i	$0.869413\pi$
$152$	0	0
$153$	−2.24318	−0.0146613
$154$	0	0
$155$	− 99.8868i	− 0.644431i
$156$	0	0
$157$	−163.982	−1.04447	−0.522235	−	0.852802i	$-0.674902\pi$
$157$	−163.982	−1.04447	−0.522235	+	0.852802i	$0.674902\pi$
$158$	0	0
$159$	− 86.3007i	− 0.542772i
$160$	0	0
$161$	−26.0871	−0.162032
$162$	0	0
$163$	226.803i	1.39143i	0.718317	+	0.695716i	$0.244913\pi$
$163$	226.803i	1.39143i	−0.718317	+	0.695716i	$0.755087\pi$
$164$	0	0
$165$	−42.3303	−0.256547
$166$	0	0
$167$	226.803i	1.35810i	0.734090	+	0.679052i	$0.237609\pi$
$167$	226.803i	1.35810i	−0.734090	+	0.679052i	$0.762391\pi$
$168$	0	0
$169$	125.642	0.743446
$170$	0	0
$171$	10.9644i	0.0641193i
$172$	0	0
$173$	−108.922	−0.629607	−0.314803	−	0.949157i	$-0.601939\pi$
$173$	−108.922	−0.629607	−0.314803	+	0.949157i	$0.601939\pi$
$174$	0	0
$175$	− 16.3115i	− 0.0932083i
$176$	0	0
$177$	154.817	0.874670
$178$	0	0
$179$	− 91.2506i	− 0.509780i	−0.966970	−	0.254890i	$-0.917961\pi$
$179$	− 91.2506i	− 0.509780i	0.966970	−	0.254890i	$-0.0820393\pi$
$180$	0	0
$181$	135.495	0.748594	0.374297	−	0.927309i	$-0.377884\pi$
$181$	135.495	0.748594	0.374297	+	0.927309i	$0.377884\pi$
$182$	0	0
$183$	209.562i	1.14515i
$184$	0	0
$185$	−41.8439	−0.226183
$186$	0	0
$187$	− 3.27340i	− 0.0175048i
$188$	0	0
$189$	13.7477	0.0727393
$190$	0	0
$191$	− 106.934i	− 0.559866i	−0.960020	−	0.279933i	$-0.909688\pi$
$191$	− 106.934i	− 0.559866i	0.960020	−	0.279933i	$-0.0903123\pi$
$192$	0	0
$193$	−116.505	−0.603650	−0.301825	−	0.953363i	$-0.597596\pi$
$193$	−116.505	−0.603650	−0.301825	+	0.953363i	$0.597596\pi$
$194$	0	0
$195$	− 165.975i	− 0.851154i
$196$	0	0
$197$	38.8348	0.197131	0.0985656	−	0.995131i	$-0.468575\pi$
$197$	38.8348	0.197131	0.0985656	+	0.995131i	$0.468575\pi$
$198$	0	0
$199$	− 353.116i	− 1.77445i	−0.461334	−	0.887227i	$-0.652629\pi$
$199$	− 353.116i	− 1.77445i	0.461334	−	0.887227i	$-0.347371\pi$
$200$	0	0
$201$	40.4864	0.201425
$202$	0	0
$203$	5.29150i	0.0260665i
$204$	0	0
$205$	427.477	2.08525
$206$	0	0
$207$	29.5800i	0.142899i
$208$	0	0
$209$	−16.0000	−0.0765550
$210$	0	0
$211$	− 54.0592i	− 0.256205i	−0.991761	−	0.128102i	$-0.959111\pi$
$211$	− 54.0592i	− 0.256205i	0.991761	−	0.128102i	$-0.0408886\pi$
$212$	0	0
$213$	114.399	0.537086
$214$	0	0
$215$	− 391.031i	− 1.81875i
$216$	0	0
$217$	−47.3394	−0.218154
$218$	0	0
$219$	70.1559i	0.320347i
$220$	0	0
$221$	12.8348	0.0580762
$222$	0	0
$223$	− 93.3400i	− 0.418565i	−0.977855	−	0.209283i	$-0.932887\pi$
$223$	− 93.3400i	− 0.418565i	0.977855	−	0.209283i	$-0.0671129\pi$
$224$	0	0
$225$	−18.4955	−0.0822020
$226$	0	0
$227$	− 170.233i	− 0.749927i	−0.927040	−	0.374964i	$-0.877655\pi$
$227$	− 170.233i	− 0.749927i	0.927040	−	0.374964i	$-0.122345\pi$
$228$	0	0
$229$	−227.844	−0.994952	−0.497476	−	0.867478i	$-0.665740\pi$
$229$	−227.844	−0.994952	−0.497476	+	0.867478i	$0.665740\pi$
$230$	0	0
$231$	20.0616i	0.0868468i
$232$	0	0
$233$	378.486	1.62440	0.812202	−	0.583376i	$-0.198268\pi$
$233$	378.486	1.62440	0.812202	+	0.583376i	$0.198268\pi$
$234$	0	0
$235$	− 223.991i	− 0.953153i
$236$	0	0
$237$	165.495	0.698293
$238$	0	0
$239$	− 205.535i	− 0.859977i	−0.902834	−	0.429989i	$-0.858518\pi$
$239$	− 205.535i	− 0.859977i	0.902834	−	0.429989i	$-0.141482\pi$
$240$	0	0
$241$	40.4682	0.167918	0.0839589	−	0.996469i	$-0.473244\pi$
$241$	40.4682	0.167918	0.0839589	+	0.996469i	$0.473244\pi$
$242$	0	0
$243$	− 15.5885i	− 0.0641500i
$244$	0	0
$245$	−39.0780	−0.159502
$246$	0	0
$247$	− 62.7352i	− 0.253989i
$248$	0	0
$249$	200.312	0.804466
$250$	0	0
$251$	332.856i	1.32612i	0.748567	+	0.663059i	$0.230742\pi$
$251$	332.856i	1.32612i	−0.748567	+	0.663059i	$0.769258\pi$
$252$	0	0
$253$	−43.1652	−0.170613
$254$	0	0
$255$	− 7.23000i	− 0.0283529i
$256$	0	0
$257$	−205.372	−0.799113	−0.399556	−	0.916709i	$-0.630836\pi$
$257$	−205.372	−0.799113	−0.399556	+	0.916709i	$0.630836\pi$
$258$	0	0
$259$	19.8311i	0.0765680i
$260$	0	0
$261$	6.00000	0.0229885
$262$	0	0
$263$	45.4064i	0.172648i	0.996267	+	0.0863240i	$0.0275120\pi$
$263$	45.4064i	0.172648i	−0.996267	+	0.0863240i	$0.972488\pi$
$264$	0	0
$265$	278.156	1.04965
$266$	0	0
$267$	− 60.4551i	− 0.226424i
$268$	0	0
$269$	−202.069	−0.751186	−0.375593	−	0.926785i	$-0.622561\pi$
$269$	−202.069	−0.751186	−0.375593	+	0.926785i	$0.622561\pi$
$270$	0	0
$271$	− 279.798i	− 1.03246i	−0.856449	−	0.516232i	$-0.827334\pi$
$271$	− 279.798i	− 1.03246i	0.856449	−	0.516232i	$-0.172666\pi$
$272$	0	0
$273$	−78.6606	−0.288134
$274$	0	0
$275$	− 26.9898i	− 0.0981448i
$276$	0	0
$277$	135.459	0.489022	0.244511	−	0.969647i	$-0.421373\pi$
$277$	135.459	0.489022	0.244511	+	0.969647i	$0.421373\pi$
$278$	0	0
$279$	53.6778i	0.192394i
$280$	0	0
$281$	20.8530	0.0742101	0.0371050	−	0.999311i	$-0.488186\pi$
$281$	20.8530	0.0742101	0.0371050	+	0.999311i	$0.488186\pi$
$282$	0	0
$283$	− 113.743i	− 0.401920i	−0.979599	−	0.200960i	$-0.935594\pi$
$283$	− 113.743i	− 0.401920i	0.979599	−	0.200960i	$-0.0644061\pi$
$284$	0	0
$285$	−35.3394	−0.123998
$286$	0	0
$287$	− 202.594i	− 0.705904i
$288$	0	0
$289$	−288.441	−0.998065
$290$	0	0
$291$	− 90.9405i	− 0.312510i
$292$	0	0
$293$	−121.931	−0.416147	−0.208073	−	0.978113i	$-0.566719\pi$
$293$	−121.931	−0.416147	−0.208073	+	0.978113i	$0.566719\pi$
$294$	0	0
$295$	498.990i	1.69149i
$296$	0	0
$297$	22.7477	0.0765917
$298$	0	0
$299$	− 169.248i	− 0.566048i
$300$	0	0
$301$	−185.321	−0.615685
$302$	0	0
$303$	196.731i	0.649277i
$304$	0	0
$305$	−675.441	−2.21456
$306$	0	0
$307$	337.131i	1.09815i	0.835775	+	0.549073i	$0.185019\pi$
$307$	337.131i	1.09815i	−0.835775	+	0.549073i	$0.814981\pi$
$308$	0	0
$309$	86.6424	0.280396
$310$	0	0
$311$	− 246.762i	− 0.793447i	−0.917938	−	0.396724i	$-0.870147\pi$
$311$	− 246.762i	− 0.793447i	0.917938	−	0.396724i	$-0.129853\pi$
$312$	0	0
$313$	274.624	0.877394	0.438697	−	0.898635i	$-0.355440\pi$
$313$	274.624	0.877394	0.438697	+	0.898635i	$0.355440\pi$
$314$	0	0
$315$	44.3103i	0.140668i
$316$	0	0
$317$	−129.441	−0.408331	−0.204165	−	0.978936i	$-0.565448\pi$
$317$	−129.441	−0.408331	−0.204165	+	0.978936i	$0.565448\pi$
$318$	0	0
$319$	8.75560i	0.0274470i
$320$	0	0
$321$	278.051	0.866202
$322$	0	0
$323$	− 2.73279i	− 0.00846066i
$324$	0	0
$325$	105.826	0.325618
$326$	0	0
$327$	355.595i	1.08745i
$328$	0	0
$329$	−106.156	−0.322663
$330$	0	0
$331$	219.573i	0.663363i	0.943391	+	0.331682i	$0.107616\pi$
$331$	219.573i	0.663363i	−0.943391	+	0.331682i	$0.892384\pi$
$332$	0	0
$333$	22.4864	0.0675266
$334$	0	0
$335$	130.492i	0.389527i
$336$	0	0
$337$	−132.955	−0.394524	−0.197262	−	0.980351i	$-0.563205\pi$
$337$	−132.955	−0.394524	−0.197262	+	0.980351i	$0.563205\pi$
$338$	0	0
$339$	− 310.593i	− 0.916205i
$340$	0	0
$341$	−78.3303	−0.229708
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	−95.3394	−0.276346
$346$	0	0
$347$	453.282i	1.30629i	0.757234	+	0.653144i	$0.226550\pi$
$347$	453.282i	1.30629i	−0.757234	+	0.653144i	$0.773450\pi$
$348$	0	0
$349$	525.615	1.50606	0.753030	−	0.657986i	$-0.228591\pi$
$349$	525.615	1.50606	0.753030	+	0.657986i	$0.228591\pi$
$350$	0	0
$351$	89.1927i	0.254110i
$352$	0	0
$353$	−389.858	−1.10441	−0.552207	−	0.833707i	$-0.686214\pi$
$353$	−389.858	−1.10441	−0.552207	+	0.833707i	$0.686214\pi$
$354$	0	0
$355$	368.720i	1.03865i
$356$	0	0
$357$	−3.42652	−0.00959808
$358$	0	0
$359$	− 692.718i	− 1.92958i	−0.263030	−	0.964788i	$-0.584722\pi$
$359$	− 692.718i	− 1.92958i	0.263030	−	0.964788i	$-0.415278\pi$
$360$	0	0
$361$	347.642	0.962998
$362$	0	0
$363$	− 176.383i	− 0.485904i
$364$	0	0
$365$	−226.120	−0.619506
$366$	0	0
$367$	556.863i	1.51734i	0.651476	+	0.758669i	$0.274150\pi$
$367$	556.863i	1.51734i	−0.651476	+	0.758669i	$0.725850\pi$
$368$	0	0
$369$	−229.720	−0.622549
$370$	0	0
$371$	− 131.827i	− 0.355328i
$372$	0	0
$373$	549.267	1.47256	0.736282	−	0.676674i	$-0.236580\pi$
$373$	549.267	1.47256	0.736282	+	0.676674i	$0.236580\pi$
$374$	0	0
$375$	182.120i	0.485653i
$376$	0	0
$377$	−34.3303	−0.0910618
$378$	0	0
$379$	− 232.667i	− 0.613897i	−0.951726	−	0.306948i	$-0.900692\pi$
$379$	− 232.667i	− 0.613897i	0.951726	−	0.306948i	$-0.0993079\pi$
$380$	0	0
$381$	226.156	0.593585
$382$	0	0
$383$	− 394.526i	− 1.03009i	−0.857162	−	0.515047i	$-0.827774\pi$
$383$	− 394.526i	− 1.03009i	0.857162	−	0.515047i	$-0.172226\pi$
$384$	0	0
$385$	−64.6606	−0.167950
$386$	0	0
$387$	210.135i	0.542983i
$388$	0	0
$389$	3.98182	0.0102360	0.00511802	−	0.999987i	$-0.498371\pi$
$389$	3.98182	0.0102360	0.00511802	+	0.999987i	$0.498371\pi$
$390$	0	0
$391$	− 7.37259i	− 0.0188557i
$392$	0	0
$393$	123.303	0.313748
$394$	0	0
$395$	533.409i	1.35040i
$396$	0	0
$397$	−476.606	−1.20052	−0.600260	−	0.799805i	$-0.704936\pi$
$397$	−476.606	−1.20052	−0.600260	+	0.799805i	$0.704936\pi$
$398$	0	0
$399$	16.7484i	0.0419760i
$400$	0	0
$401$	−157.441	−0.392621	−0.196310	−	0.980542i	$-0.562896\pi$
$401$	−157.441	−0.392621	−0.196310	+	0.980542i	$0.562896\pi$
$402$	0	0
$403$	− 307.129i	− 0.762108i
$404$	0	0
$405$	50.2432	0.124057
$406$	0	0
$407$	32.8136i	0.0806231i
$408$	0	0
$409$	−112.156	−0.274220	−0.137110	−	0.990556i	$-0.543781\pi$
$409$	−112.156	−0.274220	−0.137110	+	0.990556i	$0.543781\pi$
$410$	0	0
$411$	172.935i	0.420766i
$412$	0	0
$413$	236.486	0.572606
$414$	0	0
$415$	645.626i	1.55573i
$416$	0	0
$417$	−281.285	−0.674544
$418$	0	0
$419$	699.065i	1.66841i	0.551452	+	0.834207i	$0.314074\pi$
$419$	699.065i	1.66841i	−0.551452	+	0.834207i	$0.685926\pi$
$420$	0	0
$421$	198.762	0.472119	0.236060	−	0.971739i	$-0.424144\pi$
$421$	198.762	0.472119	0.236060	+	0.971739i	$0.424144\pi$
$422$	0	0
$423$	120.370i	0.284562i
$424$	0	0
$425$	4.60985	0.0108467
$426$	0	0
$427$	320.112i	0.749676i
$428$	0	0
$429$	−130.156	−0.303394
$430$	0	0
$431$	513.871i	1.19228i	0.802882	+	0.596138i	$0.203299\pi$
$431$	513.871i	1.19228i	−0.802882	+	0.596138i	$0.796701\pi$
$432$	0	0
$433$	−654.900	−1.51247	−0.756236	−	0.654299i	$-0.772964\pi$
$433$	−654.900	−1.51247	−0.756236	+	0.654299i	$0.772964\pi$
$434$	0	0
$435$	19.3386i	0.0444566i
$436$	0	0
$437$	−36.0364	−0.0824631
$438$	0	0
$439$	143.141i	0.326061i	0.986621	+	0.163031i	$0.0521269\pi$
$439$	143.141i	0.326061i	−0.986621	+	0.163031i	$0.947873\pi$
$440$	0	0
$441$	21.0000	0.0476190
$442$	0	0
$443$	46.8690i	0.105799i	0.998600	+	0.0528996i	$0.0168463\pi$
$443$	46.8690i	0.105799i	−0.998600	+	0.0528996i	$0.983154\pi$
$444$	0	0
$445$	194.853	0.437872
$446$	0	0
$447$	− 507.777i	− 1.13597i
$448$	0	0
$449$	−292.955	−0.652460	−0.326230	−	0.945290i	$-0.605778\pi$
$449$	−292.955	−0.652460	−0.326230	+	0.945290i	$0.605778\pi$
$450$	0	0
$451$	− 335.224i	− 0.743289i
$452$	0	0
$453$	208.624	0.460539
$454$	0	0
$455$	− 253.531i	− 0.557211i
$456$	0	0
$457$	−424.018	−0.927830	−0.463915	−	0.885880i	$-0.653556\pi$
$457$	−424.018	−0.927830	−0.463915	+	0.885880i	$0.653556\pi$
$458$	0	0
$459$	3.88530i	0.00846471i
$460$	0	0
$461$	−344.886	−0.748125	−0.374062	−	0.927404i	$-0.622035\pi$
$461$	−344.886	−0.748125	−0.374062	+	0.927404i	$0.622035\pi$
$462$	0	0
$463$	319.381i	0.689807i	0.938638	+	0.344903i	$0.112088\pi$
$463$	319.381i	0.689807i	−0.938638	+	0.344903i	$0.887912\pi$
$464$	0	0
$465$	−173.009	−0.372063
$466$	0	0
$467$	306.065i	0.655385i	0.944784	+	0.327692i	$0.106271\pi$
$467$	306.065i	0.655385i	−0.944784	+	0.327692i	$0.893729\pi$
$468$	0	0
$469$	61.8439	0.131863
$470$	0	0
$471$	284.025i	0.603025i
$472$	0	0
$473$	−306.642	−0.648293
$474$	0	0
$475$	− 22.5324i	− 0.0474366i
$476$	0	0
$477$	−149.477	−0.313370
$478$	0	0
$479$	355.769i	0.742734i	0.928486	+	0.371367i	$0.121111\pi$
$479$	355.769i	0.742734i	−0.928486	+	0.371367i	$0.878889\pi$
$480$	0	0
$481$	−128.661	−0.267486
$482$	0	0
$483$	45.1842i	0.0935491i
$484$	0	0
$485$	293.111	0.604352
$486$	0	0
$487$	− 765.313i	− 1.57148i	−0.618554	−	0.785742i	$-0.712281\pi$
$487$	− 765.313i	− 1.57148i	0.618554	−	0.785742i	$-0.287719\pi$
$488$	0	0
$489$	392.835	0.803343
$490$	0	0
$491$	− 975.169i	− 1.98609i	−0.117749	−	0.993043i	$-0.537568\pi$
$491$	− 975.169i	− 1.98609i	0.117749	−	0.993043i	$-0.462432\pi$
$492$	0	0
$493$	−1.49545	−0.00303338
$494$	0	0
$495$	73.3182i	0.148118i
$496$	0	0
$497$	174.748	0.351605
$498$	0	0
$499$	− 472.358i	− 0.946610i	−0.880899	−	0.473305i	$-0.843061\pi$
$499$	− 472.358i	− 0.946610i	0.880899	−	0.473305i	$-0.156939\pi$
$500$	0	0
$501$	392.835	0.784101
$502$	0	0
$503$	− 513.165i	− 1.02021i	−0.860113	−	0.510104i	$-0.829607\pi$
$503$	− 513.165i	− 1.02021i	0.860113	−	0.510104i	$-0.170393\pi$
$504$	0	0
$505$	−634.083	−1.25561
$506$	0	0
$507$	− 217.619i	− 0.429229i
$508$	0	0
$509$	361.546	0.710307	0.355153	−	0.934808i	$-0.384429\pi$
$509$	361.546	0.710307	0.355153	+	0.934808i	$0.384429\pi$
$510$	0	0
$511$	107.165i	0.209716i
$512$	0	0
$513$	18.9909	0.0370193
$514$	0	0
$515$	279.257i	0.542247i
$516$	0	0
$517$	−175.652	−0.339751
$518$	0	0
$519$	188.658i	0.363504i
$520$	0	0
$521$	71.4265	0.137095	0.0685475	−	0.997648i	$-0.478164\pi$
$521$	71.4265	0.137095	0.0685475	+	0.997648i	$0.478164\pi$
$522$	0	0
$523$	− 587.246i	− 1.12284i	−0.827531	−	0.561420i	$-0.810255\pi$
$523$	− 587.246i	− 1.12284i	0.827531	−	0.561420i	$-0.189745\pi$
$524$	0	0
$525$	−28.2523	−0.0538139
$526$	0	0
$527$	− 13.3788i	− 0.0253867i
$528$	0	0
$529$	431.780	0.816220
$530$	0	0
$531$	− 268.150i	− 0.504991i
$532$	0	0
$533$	1314.40	2.46603
$534$	0	0
$535$	896.186i	1.67511i
$536$	0	0
$537$	−158.051	−0.294322
$538$	0	0
$539$	30.6446i	0.0568546i
$540$	0	0
$541$	−436.505	−0.806848	−0.403424	−	0.915013i	$-0.632180\pi$
$541$	−436.505	−0.806848	−0.403424	+	0.915013i	$0.632180\pi$
$542$	0	0
$543$	− 234.685i	− 0.432201i
$544$	0	0
$545$	−1146.12	−2.10297
$546$	0	0
$547$	25.2188i	0.0461039i	0.999734	+	0.0230519i	$0.00733831\pi$
$547$	25.2188i	0.0461039i	−0.999734	+	0.0230519i	$0.992662\pi$
$548$	0	0
$549$	362.973	0.661153
$550$	0	0
$551$	7.30960i	0.0132661i
$552$	0	0
$553$	252.798	0.457140
$554$	0	0
$555$	72.4758i	0.130587i
$556$	0	0
$557$	58.1742	0.104442	0.0522210	−	0.998636i	$-0.483370\pi$
$557$	58.1742	0.104442	0.0522210	+	0.998636i	$0.483370\pi$
$558$	0	0
$559$	− 1202.33i	− 2.15086i
$560$	0	0
$561$	−5.66970	−0.0101064
$562$	0	0
$563$	− 565.396i	− 1.00426i	−0.864793	−	0.502128i	$-0.832551\pi$
$563$	− 565.396i	− 1.00426i	0.864793	−	0.502128i	$-0.167449\pi$
$564$	0	0
$565$	1001.07	1.77181
$566$	0	0
$567$	− 23.8118i	− 0.0419961i
$568$	0	0
$569$	573.056	1.00713	0.503564	−	0.863958i	$-0.332022\pi$
$569$	573.056	1.00713	0.503564	+	0.863958i	$0.332022\pi$
$570$	0	0
$571$	− 685.750i	− 1.20096i	−0.799639	−	0.600481i	$-0.794976\pi$
$571$	− 685.750i	− 1.20096i	0.799639	−	0.600481i	$-0.205024\pi$
$572$	0	0
$573$	−185.216	−0.323239
$574$	0	0
$575$	− 60.7884i	− 0.105719i
$576$	0	0
$577$	−278.900	−0.483362	−0.241681	−	0.970356i	$-0.577699\pi$
$577$	−278.900	−0.483362	−0.241681	+	0.970356i	$0.577699\pi$
$578$	0	0
$579$	201.792i	0.348518i
$580$	0	0
$581$	305.982	0.526647
$582$	0	0
$583$	− 218.127i	− 0.374146i
$584$	0	0
$585$	−287.477	−0.491414
$586$	0	0
$587$	1169.74i	1.99274i	0.0851245	+	0.996370i	$0.472871\pi$
$587$	1169.74i	1.99274i	−0.0851245	+	0.996370i	$0.527129\pi$
$588$	0	0
$589$	−65.3939	−0.111025
$590$	0	0
$591$	− 67.2639i	− 0.113814i
$592$	0	0
$593$	969.023	1.63410	0.817052	−	0.576564i	$-0.195607\pi$
$593$	969.023	1.63410	0.817052	+	0.576564i	$0.195607\pi$
$594$	0	0
$595$	− 11.0440i	− 0.0185614i
$596$	0	0
$597$	−611.615	−1.02448
$598$	0	0
$599$	− 656.488i	− 1.09597i	−0.836487	−	0.547987i	$-0.815394\pi$
$599$	− 656.488i	− 1.09597i	0.836487	−	0.547987i	$-0.184606\pi$
$600$	0	0
$601$	312.955	0.520723	0.260362	−	0.965511i	$-0.416158\pi$
$601$	312.955	0.520723	0.260362	+	0.965511i	$0.416158\pi$
$602$	0	0
$603$	− 70.1244i	− 0.116293i
$604$	0	0
$605$	568.501	0.939671
$606$	0	0
$607$	113.584i	0.187124i	0.995613	+	0.0935618i	$0.0298253\pi$
$607$	113.584i	0.187124i	−0.995613	+	0.0935618i	$0.970175\pi$
$608$	0	0
$609$	9.16515	0.0150495
$610$	0	0
$611$	− 688.721i	− 1.12720i
$612$	0	0
$613$	10.0000	0.0163132	0.00815661	−	0.999967i	$-0.497404\pi$
$613$	10.0000	0.0163132	0.00815661	+	0.999967i	$0.497404\pi$
$614$	0	0
$615$	− 740.412i	− 1.20392i
$616$	0	0
$617$	984.323	1.59534	0.797668	−	0.603096i	$-0.206067\pi$
$617$	984.323	1.59534	0.797668	+	0.603096i	$0.206067\pi$
$618$	0	0
$619$	− 266.864i	− 0.431120i	−0.976491	−	0.215560i	$-0.930842\pi$
$619$	− 266.864i	− 0.431120i	0.976491	−	0.215560i	$-0.0691578\pi$
$620$	0	0
$621$	51.2341	0.0825026
$622$	0	0
$623$	− 92.3467i	− 0.148229i
$624$	0	0
$625$	−741.120	−1.18579
$626$	0	0
$627$	27.7128i	0.0441991i
$628$	0	0
$629$	−5.60455	−0.00891026
$630$	0	0
$631$	609.443i	0.965837i	0.875665	+	0.482918i	$0.160423\pi$
$631$	609.443i	0.965837i	−0.875665	+	0.482918i	$0.839577\pi$
$632$	0	0
$633$	−93.6333	−0.147920
$634$	0	0
$635$	728.924i	1.14791i
$636$	0	0
$637$	−120.156	−0.188628
$638$	0	0
$639$	− 198.145i	− 0.310087i
$640$	0	0
$641$	528.120	0.823900	0.411950	−	0.911207i	$-0.364848\pi$
$641$	528.120	0.823900	0.411950	+	0.911207i	$0.364848\pi$
$642$	0	0
$643$	− 812.666i	− 1.26387i	−0.775023	−	0.631933i	$-0.782262\pi$
$643$	− 812.666i	− 1.26387i	0.775023	−	0.631933i	$-0.217738\pi$
$644$	0	0
$645$	−677.285	−1.05005
$646$	0	0
$647$	1153.37i	1.78265i	0.453369	+	0.891323i	$0.350222\pi$
$647$	1153.37i	1.78265i	−0.453369	+	0.891323i	$0.649778\pi$
$648$	0	0
$649$	391.303	0.602932
$650$	0	0
$651$	81.9942i	0.125951i
$652$	0	0
$653$	−994.000	−1.52221	−0.761103	−	0.648632i	$-0.775342\pi$
$653$	−994.000	−1.52221	−0.761103	+	0.648632i	$0.775342\pi$
$654$	0	0
$655$	397.418i	0.606745i
$656$	0	0
$657$	121.514	0.184952
$658$	0	0
$659$	− 1223.14i	− 1.85605i	−0.372516	−	0.928026i	$-0.621505\pi$
$659$	− 1223.14i	− 1.85605i	0.372516	−	0.928026i	$-0.378495\pi$
$660$	0	0
$661$	24.0545	0.0363911	0.0181956	−	0.999834i	$-0.494208\pi$
$661$	24.0545	0.0363911	0.0181956	+	0.999834i	$0.494208\pi$
$662$	0	0
$663$	− 22.2306i	− 0.0335303i
$664$	0	0
$665$	−53.9818	−0.0811757
$666$	0	0
$667$	19.7200i	0.0295652i
$668$	0	0
$669$	−161.670	−0.241659
$670$	0	0
$671$	529.674i	0.789380i
$672$	0	0
$673$	653.789	0.971455	0.485728	−	0.874110i	$-0.338555\pi$
$673$	653.789	0.971455	0.485728	+	0.874110i	$0.338555\pi$
$674$	0	0
$675$	32.0351i	0.0474594i
$676$	0	0
$677$	366.069	0.540722	0.270361	−	0.962759i	$-0.412857\pi$
$677$	366.069	0.540722	0.270361	+	0.962759i	$0.412857\pi$
$678$	0	0
$679$	− 138.914i	− 0.204586i
$680$	0	0
$681$	−294.853	−0.432971
$682$	0	0
$683$	971.912i	1.42300i	0.702684	+	0.711502i	$0.251985\pi$
$683$	971.912i	1.42300i	−0.702684	+	0.711502i	$0.748015\pi$
$684$	0	0
$685$	−557.386	−0.813703
$686$	0	0
$687$	394.637i	0.574436i
$688$	0	0
$689$	855.267	1.24132
$690$	0	0
$691$	1028.20i	1.48799i	0.668184	+	0.743996i	$0.267072\pi$
$691$	1028.20i	1.48799i	−0.668184	+	0.743996i	$0.732928\pi$
$692$	0	0
$693$	34.7477	0.0501410
$694$	0	0
$695$	− 906.610i	− 1.30447i
$696$	0	0
$697$	57.2561	0.0821464
$698$	0	0
$699$	− 655.558i	− 0.937851i
$700$	0	0
$701$	−276.918	−0.395033	−0.197517	−	0.980300i	$-0.563288\pi$
$701$	−276.918	−0.395033	−0.197517	+	0.980300i	$0.563288\pi$
$702$	0	0
$703$	27.3944i	0.0389679i
$704$	0	0
$705$	−387.964	−0.550303
$706$	0	0
$707$	300.511i	0.425051i
$708$	0	0
$709$	51.2485	0.0722828	0.0361414	−	0.999347i	$-0.488493\pi$
$709$	51.2485	0.0722828	0.0361414	+	0.999347i	$0.488493\pi$
$710$	0	0
$711$	− 286.647i	− 0.403160i
$712$	0	0
$713$	−176.421	−0.247435
$714$	0	0
$715$	− 419.506i	− 0.586722i
$716$	0	0
$717$	−355.996	−0.496508
$718$	0	0
$719$	− 898.093i	− 1.24909i	−0.780990	−	0.624543i	$-0.785285\pi$
$719$	− 898.093i	− 1.24909i	0.780990	−	0.624543i	$-0.214715\pi$
$720$	0	0
$721$	132.348	0.183562
$722$	0	0
$723$	− 70.0929i	− 0.0969474i
$724$	0	0
$725$	−12.3303	−0.0170073
$726$	0	0
$727$	− 107.100i	− 0.147318i	−0.997283	−	0.0736590i	$-0.976532\pi$
$727$	− 107.100i	− 0.147318i	0.997283	−	0.0736590i	$-0.0234677\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	− 52.3744i	− 0.0716476i
$732$	0	0
$733$	−541.368	−0.738565	−0.369283	−	0.929317i	$-0.620397\pi$
$733$	−541.368	−0.738565	−0.369283	+	0.929317i	$0.620397\pi$
$734$	0	0
$735$	67.6851i	0.0920886i
$736$	0	0
$737$	102.330	0.138847
$738$	0	0
$739$	241.423i	0.326688i	0.986569	+	0.163344i	$0.0522281\pi$
$739$	241.423i	0.326688i	−0.986569	+	0.163344i	$0.947772\pi$
$740$	0	0
$741$	−108.661	−0.146640
$742$	0	0
$743$	− 593.753i	− 0.799129i	−0.916705	−	0.399564i	$-0.869161\pi$
$743$	− 593.753i	− 0.799129i	0.916705	−	0.399564i	$-0.130839\pi$
$744$	0	0
$745$	1636.62	2.19680
$746$	0	0
$747$	− 346.951i	− 0.464459i
$748$	0	0
$749$	424.730	0.567062
$750$	0	0
$751$	− 918.781i	− 1.22341i	−0.791086	−	0.611705i	$-0.790484\pi$
$751$	− 918.781i	− 1.22341i	0.791086	−	0.611705i	$-0.209516\pi$
$752$	0	0
$753$	576.523	0.765634
$754$	0	0
$755$	672.417i	0.890619i
$756$	0	0
$757$	−1407.80	−1.85971	−0.929855	−	0.367927i	$-0.880068\pi$
$757$	−1407.80	−1.85971	−0.929855	+	0.367927i	$0.880068\pi$
$758$	0	0
$759$	74.7642i	0.0985036i
$760$	0	0
$761$	−295.042	−0.387703	−0.193851	−	0.981031i	$-0.562098\pi$
$761$	−295.042	−0.387703	−0.193851	+	0.981031i	$0.562098\pi$
$762$	0	0
$763$	543.181i	0.711901i
$764$	0	0
$765$	−12.5227	−0.0163696
$766$	0	0
$767$	1534.28i	2.00037i
$768$	0	0
$769$	116.991	0.152134	0.0760669	−	0.997103i	$-0.475764\pi$
$769$	116.991	0.152134	0.0760669	+	0.997103i	$0.475764\pi$
$770$	0	0
$771$	355.715i	0.461368i
$772$	0	0
$773$	105.408	0.136363	0.0681813	−	0.997673i	$-0.478280\pi$
$773$	105.408	0.136363	0.0681813	+	0.997673i	$0.478280\pi$
$774$	0	0
$775$	− 110.311i	− 0.142336i
$776$	0	0
$777$	34.3485	0.0442065
$778$	0	0
$779$	− 279.861i	− 0.359257i
$780$	0	0
$781$	289.147	0.370227
$782$	0	0
$783$	− 10.3923i	− 0.0132724i
$784$	0	0
$785$	−915.441	−1.16617
$786$	0	0
$787$	− 542.148i	− 0.688879i	−0.938809	−	0.344439i	$-0.888069\pi$
$787$	− 542.148i	− 0.688879i	0.938809	−	0.344439i	$-0.111931\pi$
$788$	0	0
$789$	78.6462	0.0996783
$790$	0	0
$791$	− 474.439i	− 0.599797i
$792$	0	0
$793$	−2076.83	−2.61895
$794$	0	0
$795$	− 481.780i	− 0.606013i
$796$	0	0
$797$	−296.886	−0.372504	−0.186252	−	0.982502i	$-0.559634\pi$
$797$	−296.886	−0.372504	−0.186252	+	0.982502i	$0.559634\pi$
$798$	0	0
$799$	− 30.0012i	− 0.0375485i
$800$	0	0
$801$	−104.711	−0.130726
$802$	0	0
$803$	177.321i	0.220823i
$804$	0	0
$805$	−145.633	−0.180911
$806$	0	0
$807$	349.994i	0.433697i
$808$	0	0
$809$	1066.83	1.31870	0.659349	−	0.751837i	$-0.270832\pi$
$809$	1066.83	1.31870	0.659349	+	0.751837i	$0.270832\pi$
$810$	0	0
$811$	9.82020i	0.0121088i	0.999982	+	0.00605438i	$0.00192718\pi$
$811$	9.82020i	0.0121088i	−0.999982	+	0.00605438i	$0.998073\pi$
$812$	0	0
$813$	−484.624	−0.596094
$814$	0	0
$815$	1266.15i	1.55355i
$816$	0	0
$817$	−256.000	−0.313341
$818$	0	0
$819$	136.244i	0.166354i
$820$	0	0
$821$	−928.395	−1.13081	−0.565405	−	0.824813i	$-0.691280\pi$
$821$	−928.395	−1.13081	−0.565405	+	0.824813i	$0.691280\pi$
$822$	0	0
$823$	396.291i	0.481520i	0.970585	+	0.240760i	$0.0773966\pi$
$823$	396.291i	0.481520i	−0.970585	+	0.240760i	$0.922603\pi$
$824$	0	0
$825$	−46.7477	−0.0566639
$826$	0	0
$827$	1094.28i	1.32320i	0.749858	+	0.661599i	$0.230122\pi$
$827$	1094.28i	1.32320i	−0.749858	+	0.661599i	$0.769878\pi$
$828$	0	0
$829$	−107.074	−0.129161	−0.0645804	−	0.997913i	$-0.520571\pi$
$829$	−107.074	−0.129161	−0.0645804	+	0.997913i	$0.520571\pi$
$830$	0	0
$831$	− 234.622i	− 0.282337i
$832$	0	0
$833$	−5.23409	−0.00628342
$834$	0	0
$835$	1266.15i	1.51634i
$836$	0	0
$837$	92.9727	0.111079
$838$	0	0
$839$	1110.28i	1.32333i	0.749798	+	0.661667i	$0.230151\pi$
$839$	1110.28i	1.32333i	−0.749798	+	0.661667i	$0.769849\pi$
$840$	0	0
$841$	−837.000	−0.995244
$842$	0	0
$843$	− 36.1185i	− 0.0428452i
$844$	0	0
$845$	701.408	0.830069
$846$	0	0
$847$	− 269.430i	− 0.318099i
$848$	0	0
$849$	−197.009	−0.232048
$850$	0	0
$851$	73.9052i	0.0868451i
$852$	0	0
$853$	657.506	0.770816	0.385408	−	0.922746i	$-0.374061\pi$
$853$	657.506	0.770816	0.385408	+	0.922746i	$0.374061\pi$
$854$	0	0
$855$	61.2096i	0.0715902i
$856$	0	0
$857$	664.958	0.775914	0.387957	−	0.921677i	$-0.373181\pi$
$857$	664.958	0.775914	0.387957	+	0.921677i	$0.373181\pi$
$858$	0	0
$859$	1071.96i	1.24792i	0.781456	+	0.623961i	$0.214477\pi$
$859$	1071.96i	1.24792i	−0.781456	+	0.623961i	$0.785523\pi$
$860$	0	0
$861$	−350.904	−0.407554
$862$	0	0
$863$	− 612.073i	− 0.709239i	−0.935011	−	0.354619i	$-0.884610\pi$
$863$	− 612.073i	− 0.709239i	0.935011	−	0.354619i	$-0.115390\pi$
$864$	0	0
$865$	−608.065	−0.702965
$866$	0	0
$867$	499.594i	0.576233i
$868$	0	0
$869$	418.294	0.481351
$870$	0	0
$871$	401.232i	0.460657i
$872$	0	0
$873$	−157.514	−0.180428
$874$	0	0
$875$	278.193i	0.317935i
$876$	0	0
$877$	−1385.19	−1.57947	−0.789734	−	0.613449i	$-0.789782\pi$
$877$	−1385.19	−1.57947	−0.789734	+	0.613449i	$0.789782\pi$
$878$	0	0
$879$	211.191i	0.240263i
$880$	0	0
$881$	−623.325	−0.707520	−0.353760	−	0.935336i	$-0.615097\pi$
$881$	−623.325	−0.707520	−0.353760	+	0.935336i	$0.615097\pi$
$882$	0	0
$883$	− 636.791i	− 0.721168i	−0.932727	−	0.360584i	$-0.882577\pi$
$883$	− 636.791i	− 0.721168i	0.932727	−	0.360584i	$-0.117423\pi$
$884$	0	0
$885$	864.276	0.976583
$886$	0	0
$887$	282.866i	0.318902i	0.987206	+	0.159451i	$0.0509723\pi$
$887$	282.866i	0.318902i	−0.987206	+	0.159451i	$0.949028\pi$
$888$	0	0
$889$	345.459	0.388593
$890$	0	0
$891$	− 39.4002i	− 0.0442202i
$892$	0	0
$893$	−146.642	−0.164213
$894$	0	0
$895$	− 509.414i	− 0.569177i
$896$	0	0
$897$	−293.147	−0.326808
$898$	0	0
$899$	35.7852i	0.0398056i
$900$	0	0
$901$	37.2561	0.0413497
$902$	0	0
$903$	320.986i	0.355466i
$904$	0	0
$905$	756.414	0.835816
$906$	0	0
$907$	1419.65i	1.56521i	0.622517	+	0.782607i	$0.286110\pi$
$907$	1419.65i	1.56521i	−0.622517	+	0.782607i	$0.713890\pi$
$908$	0	0
$909$	340.748	0.374860
$910$	0	0
$911$	1170.46i	1.28481i	0.766365	+	0.642405i	$0.222063\pi$
$911$	1170.46i	1.28481i	−0.766365	+	0.642405i	$0.777937\pi$
$912$	0	0
$913$	506.294	0.554539
$914$	0	0
$915$	1169.90i	1.27858i
$916$	0	0
$917$	188.348	0.205396
$918$	0	0
$919$	− 944.239i	− 1.02746i	−0.857951	−	0.513732i	$-0.828263\pi$
$919$	− 944.239i	− 1.02746i	0.857951	−	0.513732i	$-0.171737\pi$
$920$	0	0
$921$	583.927	0.634014
$922$	0	0
$923$	1133.73i	1.22831i
$924$	0	0
$925$	−46.2106	−0.0499574
$926$	0	0
$927$	− 150.069i	− 0.161887i
$928$	0	0
$929$	1253.20	1.34897	0.674487	−	0.738286i	$-0.264365\pi$
$929$	1253.20	1.34897	0.674487	+	0.738286i	$0.264365\pi$
$930$	0	0
$931$	25.5836i	0.0274797i
$932$	0	0
$933$	−427.405	−0.458097
$934$	0	0
$935$	− 18.2740i	− 0.0195444i
$936$	0	0
$937$	470.936	0.502600	0.251300	−	0.967909i	$-0.419142\pi$
$937$	470.936	0.502600	0.251300	+	0.967909i	$0.419142\pi$
$938$	0	0
$939$	− 475.663i	− 0.506564i
$940$	0	0
$941$	−816.951	−0.868173	−0.434086	−	0.900871i	$-0.642929\pi$
$941$	−816.951	−0.868173	−0.434086	+	0.900871i	$0.642929\pi$
$942$	0	0
$943$	− 755.015i	− 0.800652i
$944$	0	0
$945$	76.7477	0.0812145
$946$	0	0
$947$	− 994.889i	− 1.05057i	−0.850927	−	0.525284i	$-0.823959\pi$
$947$	− 994.889i	− 1.05057i	0.850927	−	0.525284i	$-0.176041\pi$
$948$	0	0
$949$	−695.267	−0.732631
$950$	0	0
$951$	224.198i	0.235750i
$952$	0	0
$953$	−1293.85	−1.35766	−0.678832	−	0.734293i	$-0.737514\pi$
$953$	−1293.85	−1.35766	−0.678832	+	0.734293i	$0.737514\pi$
$954$	0	0
$955$	− 596.970i	− 0.625099i
$956$	0	0
$957$	15.1652	0.0158466
$958$	0	0
$959$	264.162i	0.275456i
$960$	0	0
$961$	640.855	0.666862
$962$	0	0
$963$	− 481.598i	− 0.500102i
$964$	0	0
$965$	−650.395	−0.673985
$966$	0	0
$967$	1071.60i	1.10817i	0.832460	+	0.554085i	$0.186932\pi$
$967$	1071.60i	1.10817i	−0.832460	+	0.554085i	$0.813068\pi$
$968$	0	0
$969$	−4.73334	−0.00488477
$970$	0	0
$971$	− 1433.12i	− 1.47593i	−0.674842	−	0.737963i	$-0.735788\pi$
$971$	− 1433.12i	− 1.47593i	0.674842	−	0.737963i	$-0.264212\pi$
$972$	0	0
$973$	−429.670	−0.441593
$974$	0	0
$975$	− 183.296i	− 0.187995i
$976$	0	0
$977$	−126.559	−0.129538	−0.0647692	−	0.997900i	$-0.520631\pi$
$977$	−126.559	−0.129538	−0.0647692	+	0.997900i	$0.520631\pi$
$978$	0	0
$979$	− 152.802i	− 0.156080i
$980$	0	0
$981$	615.909	0.627838
$982$	0	0
$983$	− 18.1480i	− 0.0184619i	−0.999957	−	0.00923094i	$-0.997062\pi$
$983$	− 18.1480i	− 0.0184619i	0.999957	−	0.00923094i	$-0.00293834\pi$
$984$	0	0
$985$	216.798	0.220100
$986$	0	0
$987$	183.868i	0.186289i
$988$	0	0
$989$	−690.642	−0.698324
$990$	0	0
$991$	643.338i	0.649181i	0.945855	+	0.324590i	$0.105226\pi$
$991$	643.338i	0.649181i	−0.945855	+	0.324590i	$0.894774\pi$
$992$	0	0
$993$	380.312	0.382993
$994$	0	0
$995$	− 1971.30i	− 1.98120i
$996$	0	0
$997$	631.358	0.633257	0.316629	−	0.948550i	$-0.397449\pi$
$997$	631.358	0.633257	0.316629	+	0.948550i	$0.397449\pi$
$998$	0	0
$999$	− 38.9475i	− 0.0389865i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.3.m.c.127.2	✓	4
3.2	odd	2		1008.3.m.b.127.1		4
4.3	odd	2	inner	336.3.m.c.127.4	yes	4
7.6	odd	2		2352.3.m.f.1471.3		4
8.3	odd	2		1344.3.m.a.127.1		4
8.5	even	2		1344.3.m.a.127.3		4
12.11	even	2		1008.3.m.b.127.2		4
28.27	even	2		2352.3.m.f.1471.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.3.m.c.127.2	✓	4	1.1	even	1	trivial
336.3.m.c.127.4	yes	4	4.3	odd	2	inner
1008.3.m.b.127.1		4	3.2	odd	2
1008.3.m.b.127.2		4	12.11	even	2
1344.3.m.a.127.1		4	8.3	odd	2
1344.3.m.a.127.3		4	8.5	even	2
2352.3.m.f.1471.1		4	28.27	even	2
2352.3.m.f.1471.3		4	7.6	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	336.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +5.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.73205i q^{3} +5.58258 q^{5} -2.64575i q^{7} -3.00000 q^{9} -4.37780i q^{11} +17.1652 q^{13} -9.66930i q^{15} +0.747727 q^{17} -3.65480i q^{19} -4.58258 q^{21} -9.86001i q^{23} +6.16515 q^{25} +5.19615i q^{27} -2.00000 q^{29} -17.8926i q^{31} -7.58258 q^{33} -14.7701i q^{35} -7.49545 q^{37} -29.7309i q^{39} +76.5735 q^{41} -70.0448i q^{43} -16.7477 q^{45} -40.1232i q^{47} -7.00000 q^{49} -1.29510i q^{51} +49.8258 q^{53} -24.4394i q^{55} -6.33030 q^{57} +89.3834i q^{59} -120.991 q^{61} +7.93725i q^{63} +95.8258 q^{65} +23.3748i q^{67} -17.0780 q^{69} +66.0484i q^{71} -40.5045 q^{73} -10.6784i q^{75} -11.5826 q^{77} +95.5488i q^{79} +9.00000 q^{81} +115.650i q^{83} +4.17424 q^{85} +3.46410i q^{87} +34.9038 q^{89} -45.4147i q^{91} -30.9909 q^{93} -20.4032i q^{95} +52.5045 q^{97} +13.1334i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 12 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 - 12 * q^9	\( 4 q + 4 q^{5} - 12 q^{9} + 32 q^{13} - 52 q^{17} - 12 q^{25} - 8 q^{29} - 12 q^{33} + 80 q^{37} + 68 q^{41} - 12 q^{45} - 28 q^{49} + 16 q^{53} + 48 q^{57} - 264 q^{61} + 200 q^{65} + 60 q^{69} - 272 q^{73} - 28 q^{77} + 36 q^{81} + 200 q^{85} - 172 q^{89} + 96 q^{93} + 320 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 12 * q^9 + 32 * q^13 - 52 * q^17 - 12 * q^25 - 8 * q^29 - 12 * q^33 + 80 * q^37 + 68 * q^41 - 12 * q^45 - 28 * q^49 + 16 * q^53 + 48 * q^57 - 264 * q^61 + 200 * q^65 + 60 * q^69 - 272 * q^73 - 28 * q^77 + 36 * q^81 + 200 * q^85 - 172 * q^89 + 96 * q^93 + 320 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more