LMFDB - Embedded newform 400.4.c.k.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not 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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.k.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000i q^{3} +6.00000i q^{7} +23.0000 q^{9} +O(q^{10})$	$q-2.00000i q^{3} +6.00000i q^{7} +23.0000 q^{9} -32.0000 q^{11} -38.0000i q^{13} -26.0000i q^{17} +100.000 q^{19} +12.0000 q^{21} +78.0000i q^{23} -100.000i q^{27} +50.0000 q^{29} +108.000 q^{31} +64.0000i q^{33} -266.000i q^{37} -76.0000 q^{39} +22.0000 q^{41} -442.000i q^{43} -514.000i q^{47} +307.000 q^{49} -52.0000 q^{51} +2.00000i q^{53} -200.000i q^{57} +500.000 q^{59} -518.000 q^{61} +138.000i q^{63} +126.000i q^{67} +156.000 q^{69} -412.000 q^{71} -878.000i q^{73} -192.000i q^{77} +600.000 q^{79} +421.000 q^{81} -282.000i q^{83} -100.000i q^{87} +150.000 q^{89} +228.000 q^{91} -216.000i q^{93} -386.000i q^{97} -736.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 46 q^{9}+O(q^{10}) $ 2 * q + 46 * q^9	$ 2 q + 46 q^{9} - 64 q^{11} + 200 q^{19} + 24 q^{21} + 100 q^{29} + 216 q^{31} - 152 q^{39} + 44 q^{41} + 614 q^{49} - 104 q^{51} + 1000 q^{59} - 1036 q^{61} + 312 q^{69} - 824 q^{71} + 1200 q^{79} + 842 q^{81} + 300 q^{89} + 456 q^{91} - 1472 q^{99}+O(q^{100}) $ 2 * q + 46 * q^9 - 64 * q^11 + 200 * q^19 + 24 * q^21 + 100 * q^29 + 216 * q^31 - 152 * q^39 + 44 * q^41 + 614 * q^49 - 104 * q^51 + 1000 * q^59 - 1036 * q^61 + 312 * q^69 - 824 * q^71 + 1200 * q^79 + 842 * q^81 + 300 * q^89 + 456 * q^91 - 1472 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 2.00000i	− 0.384900i	−0.981307	−	0.192450i	$-0.938357\pi$
$3$	− 2.00000i	− 0.384900i	0.981307	−	0.192450i	$-0.0616434\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.00000i	0.323970i	0.986793	+	0.161985i	$0.0517895\pi$
$7$	6.00000i	0.323970i	−0.986793	+	0.161985i	$0.948210\pi$
$8$	0	0
$9$	23.0000	0.851852
$10$	0	0
$11$	−32.0000	−0.877124	−0.438562	−	0.898701i	$-0.644512\pi$
$11$	−32.0000	−0.877124	−0.438562	+	0.898701i	$0.644512\pi$
$12$	0	0
$13$	− 38.0000i	− 0.810716i	−0.914158	−	0.405358i	$-0.867147\pi$
$13$	− 38.0000i	− 0.810716i	0.914158	−	0.405358i	$-0.132853\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 26.0000i	− 0.370937i	−0.982650	−	0.185468i	$-0.940620\pi$
$17$	− 26.0000i	− 0.370937i	0.982650	−	0.185468i	$-0.0593802\pi$
$18$	0	0
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	0	0
$21$	12.0000	0.124696
$22$	0	0
$23$	78.0000i	0.707136i	0.935409	+	0.353568i	$0.115032\pi$
$23$	78.0000i	0.707136i	−0.935409	+	0.353568i	$0.884968\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 100.000i	− 0.712778i
$28$	0	0
$29$	50.0000	0.320164	0.160082	−	0.987104i	$-0.448824\pi$
$29$	50.0000	0.320164	0.160082	+	0.987104i	$0.448824\pi$
$30$	0	0
$31$	108.000	0.625722	0.312861	−	0.949799i	$-0.398713\pi$
$31$	108.000	0.625722	0.312861	+	0.949799i	$0.398713\pi$
$32$	0	0
$33$	64.0000i	0.337605i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 266.000i	− 1.18190i	−0.806710	−	0.590948i	$-0.798754\pi$
$37$	− 266.000i	− 1.18190i	0.806710	−	0.590948i	$-0.201246\pi$
$38$	0	0
$39$	−76.0000	−0.312045
$40$	0	0
$41$	22.0000	0.0838006	0.0419003	−	0.999122i	$-0.486659\pi$
$41$	22.0000	0.0838006	0.0419003	+	0.999122i	$0.486659\pi$
$42$	0	0
$43$	− 442.000i	− 1.56754i	−0.621049	−	0.783772i	$-0.713293\pi$
$43$	− 442.000i	− 1.56754i	0.621049	−	0.783772i	$-0.286707\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 514.000i	− 1.59520i	−0.603184	−	0.797602i	$-0.706101\pi$
$47$	− 514.000i	− 1.59520i	0.603184	−	0.797602i	$-0.293899\pi$
$48$	0	0
$49$	307.000	0.895044
$50$	0	0
$51$	−52.0000	−0.142774
$52$	0	0
$53$	2.00000i	0.00518342i	0.999997	+	0.00259171i	$0.000824967\pi$
$53$	2.00000i	0.00518342i	−0.999997	+	0.00259171i	$0.999175\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 200.000i	− 0.464748i
$58$	0	0
$59$	500.000	1.10330	0.551648	−	0.834077i	$-0.313999\pi$
$59$	500.000	1.10330	0.551648	+	0.834077i	$0.313999\pi$
$60$	0	0
$61$	−518.000	−1.08726	−0.543632	−	0.839324i	$-0.682951\pi$
$61$	−518.000	−1.08726	−0.543632	+	0.839324i	$0.682951\pi$
$62$	0	0
$63$	138.000i	0.275974i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	126.000i	0.229751i	0.993380	+	0.114876i	$0.0366470\pi$
$67$	126.000i	0.229751i	−0.993380	+	0.114876i	$0.963353\pi$
$68$	0	0
$69$	156.000	0.272177
$70$	0	0
$71$	−412.000	−0.688668	−0.344334	−	0.938847i	$-0.611895\pi$
$71$	−412.000	−0.688668	−0.344334	+	0.938847i	$0.611895\pi$
$72$	0	0
$73$	− 878.000i	− 1.40770i	−0.710348	−	0.703850i	$-0.751463\pi$
$73$	− 878.000i	− 1.40770i	0.710348	−	0.703850i	$-0.248537\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 192.000i	− 0.284161i
$78$	0	0
$79$	600.000	0.854497	0.427249	−	0.904134i	$-0.359483\pi$
$79$	600.000	0.854497	0.427249	+	0.904134i	$0.359483\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	− 282.000i	− 0.372934i	−0.982461	−	0.186467i	$-0.940296\pi$
$83$	− 282.000i	− 0.372934i	0.982461	−	0.186467i	$-0.0597037\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 100.000i	− 0.123231i
$88$	0	0
$89$	150.000	0.178651	0.0893257	−	0.996002i	$-0.471529\pi$
$89$	150.000	0.178651	0.0893257	+	0.996002i	$0.471529\pi$
$90$	0	0
$91$	228.000	0.262647
$92$	0	0
$93$	− 216.000i	− 0.240840i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 386.000i	− 0.404045i	−0.979381	−	0.202022i	$-0.935249\pi$
$97$	− 386.000i	− 0.404045i	0.979381	−	0.202022i	$-0.0647514\pi$
$98$	0	0
$99$	−736.000	−0.747180
$100$	0	0
$101$	702.000	0.691600	0.345800	−	0.938308i	$-0.387608\pi$
$101$	702.000	0.691600	0.345800	+	0.938308i	$0.387608\pi$
$102$	0	0
$103$	598.000i	0.572065i	0.958220	+	0.286032i	$0.0923365\pi$
$103$	598.000i	0.572065i	−0.958220	+	0.286032i	$0.907663\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1194.00i	− 1.07877i	−0.842059	−	0.539385i	$-0.818657\pi$
$107$	− 1194.00i	− 1.07877i	0.842059	−	0.539385i	$-0.181343\pi$
$108$	0	0
$109$	550.000	0.483307	0.241653	−	0.970363i	$-0.422310\pi$
$109$	550.000	0.483307	0.241653	+	0.970363i	$0.422310\pi$
$110$	0	0
$111$	−532.000	−0.454912
$112$	0	0
$113$	1562.00i	1.30036i	0.759781	+	0.650180i	$0.225306\pi$
$113$	1562.00i	1.30036i	−0.759781	+	0.650180i	$0.774694\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 874.000i	− 0.690610i
$118$	0	0
$119$	156.000	0.120172
$120$	0	0
$121$	−307.000	−0.230654
$122$	0	0
$123$	− 44.0000i	− 0.0322548i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1846.00i	1.28981i	0.764262	+	0.644906i	$0.223103\pi$
$127$	1846.00i	1.28981i	−0.764262	+	0.644906i	$0.776897\pi$
$128$	0	0
$129$	−884.000	−0.603348
$130$	0	0
$131$	2208.00	1.47262	0.736312	−	0.676642i	$-0.236565\pi$
$131$	2208.00	1.47262	0.736312	+	0.676642i	$0.236565\pi$
$132$	0	0
$133$	600.000i	0.391177i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2334.00i	1.45553i	0.685829	+	0.727763i	$0.259440\pi$
$137$	2334.00i	1.45553i	−0.685829	+	0.727763i	$0.740560\pi$
$138$	0	0
$139$	−700.000	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$139$	−700.000	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$140$	0	0
$141$	−1028.00	−0.613994
$142$	0	0
$143$	1216.00i	0.711098i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 614.000i	− 0.344502i
$148$	0	0
$149$	−2050.00	−1.12713	−0.563566	−	0.826071i	$-0.690571\pi$
$149$	−2050.00	−1.12713	−0.563566	+	0.826071i	$0.690571\pi$
$150$	0	0
$151$	−1852.00	−0.998103	−0.499052	−	0.866572i	$-0.666318\pi$
$151$	−1852.00	−0.998103	−0.499052	+	0.866572i	$0.666318\pi$
$152$	0	0
$153$	− 598.000i	− 0.315983i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2494.00i	1.26779i	0.773420	+	0.633894i	$0.218545\pi$
$157$	2494.00i	1.26779i	−0.773420	+	0.633894i	$0.781455\pi$
$158$	0	0
$159$	4.00000	0.00199510
$160$	0	0
$161$	−468.000	−0.229090
$162$	0	0
$163$	− 2762.00i	− 1.32722i	−0.748080	−	0.663609i	$-0.769024\pi$
$163$	− 2762.00i	− 1.32722i	0.748080	−	0.663609i	$-0.230976\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3126.00i	1.44849i	0.689545	+	0.724243i	$0.257811\pi$
$167$	3126.00i	1.44849i	−0.689545	+	0.724243i	$0.742189\pi$
$168$	0	0
$169$	753.000	0.342740
$170$	0	0
$171$	2300.00	1.02857
$172$	0	0
$173$	− 78.0000i	− 0.0342788i	−0.999853	−	0.0171394i	$-0.994544\pi$
$173$	− 78.0000i	− 0.0342788i	0.999853	−	0.0171394i	$-0.00545591\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1000.00i	− 0.424659i
$178$	0	0
$179$	−1300.00	−0.542830	−0.271415	−	0.962462i	$-0.587492\pi$
$179$	−1300.00	−0.542830	−0.271415	+	0.962462i	$0.587492\pi$
$180$	0	0
$181$	1742.00	0.715369	0.357685	−	0.933842i	$-0.383566\pi$
$181$	1742.00	0.715369	0.357685	+	0.933842i	$0.383566\pi$
$182$	0	0
$183$	1036.00i	0.418488i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	832.000i	0.325358i
$188$	0	0
$189$	600.000	0.230918
$190$	0	0
$191$	−3772.00	−1.42897	−0.714483	−	0.699653i	$-0.753338\pi$
$191$	−3772.00	−1.42897	−0.714483	+	0.699653i	$0.753338\pi$
$192$	0	0
$193$	− 358.000i	− 0.133520i	−0.997769	−	0.0667601i	$-0.978734\pi$
$193$	− 358.000i	− 0.133520i	0.997769	−	0.0667601i	$-0.0212662\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2214.00i	0.800716i	0.916359	+	0.400358i	$0.131114\pi$
$197$	2214.00i	0.800716i	−0.916359	+	0.400358i	$0.868886\pi$
$198$	0	0
$199$	−2600.00	−0.926176	−0.463088	−	0.886312i	$-0.653259\pi$
$199$	−2600.00	−0.926176	−0.463088	+	0.886312i	$0.653259\pi$
$200$	0	0
$201$	252.000	0.0884314
$202$	0	0
$203$	300.000i	0.103724i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1794.00i	0.602375i
$208$	0	0
$209$	−3200.00	−1.05908
$210$	0	0
$211$	1168.00	0.381083	0.190541	−	0.981679i	$-0.438976\pi$
$211$	1168.00	0.381083	0.190541	+	0.981679i	$0.438976\pi$
$212$	0	0
$213$	824.000i	0.265068i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	648.000i	0.202715i
$218$	0	0
$219$	−1756.00	−0.541824
$220$	0	0
$221$	−988.000	−0.300724
$222$	0	0
$223$	6478.00i	1.94529i	0.232303	+	0.972643i	$0.425374\pi$
$223$	6478.00i	1.94529i	−0.232303	+	0.972643i	$0.574626\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	646.000i	0.188883i	0.995530	+	0.0944417i	$0.0301066\pi$
$227$	646.000i	0.188883i	−0.995530	+	0.0944417i	$0.969893\pi$
$228$	0	0
$229$	−3750.00	−1.08213	−0.541063	−	0.840982i	$-0.681978\pi$
$229$	−3750.00	−1.08213	−0.541063	+	0.840982i	$0.681978\pi$
$230$	0	0
$231$	−384.000	−0.109374
$232$	0	0
$233$	1482.00i	0.416691i	0.978055	+	0.208346i	$0.0668079\pi$
$233$	1482.00i	0.416691i	−0.978055	+	0.208346i	$0.933192\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1200.00i	− 0.328896i
$238$	0	0
$239$	1400.00	0.378906	0.189453	−	0.981890i	$-0.439329\pi$
$239$	1400.00	0.378906	0.189453	+	0.981890i	$0.439329\pi$
$240$	0	0
$241$	3022.00	0.807735	0.403867	−	0.914817i	$-0.367666\pi$
$241$	3022.00	0.807735	0.403867	+	0.914817i	$0.367666\pi$
$242$	0	0
$243$	− 3542.00i	− 0.935059i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3800.00i	− 0.978900i
$248$	0	0
$249$	−564.000	−0.143542
$250$	0	0
$251$	1248.00	0.313837	0.156918	−	0.987612i	$-0.449844\pi$
$251$	1248.00	0.313837	0.156918	+	0.987612i	$0.449844\pi$
$252$	0	0
$253$	− 2496.00i	− 0.620246i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2106.00i	− 0.511162i	−0.966788	−	0.255581i	$-0.917733\pi$
$257$	− 2106.00i	− 0.511162i	0.966788	−	0.255581i	$-0.0822668\pi$
$258$	0	0
$259$	1596.00	0.382898
$260$	0	0
$261$	1150.00	0.272733
$262$	0	0
$263$	3638.00i	0.852961i	0.904497	+	0.426480i	$0.140247\pi$
$263$	3638.00i	0.852961i	−0.904497	+	0.426480i	$0.859753\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 300.000i	− 0.0687629i
$268$	0	0
$269$	6550.00	1.48461	0.742306	−	0.670061i	$-0.233732\pi$
$269$	6550.00	1.48461	0.742306	+	0.670061i	$0.233732\pi$
$270$	0	0
$271$	4388.00	0.983587	0.491793	−	0.870712i	$-0.336342\pi$
$271$	4388.00	0.983587	0.491793	+	0.870712i	$0.336342\pi$
$272$	0	0
$273$	− 456.000i	− 0.101093i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 546.000i	− 0.118433i	−0.998245	−	0.0592165i	$-0.981140\pi$
$277$	− 546.000i	− 0.118433i	0.998245	−	0.0592165i	$-0.0188602\pi$
$278$	0	0
$279$	2484.00	0.533022
$280$	0	0
$281$	−6858.00	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$281$	−6858.00	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$282$	0	0
$283$	− 9282.00i	− 1.94967i	−0.222920	−	0.974837i	$-0.571559\pi$
$283$	− 9282.00i	− 1.94967i	0.222920	−	0.974837i	$-0.428441\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	132.000i	0.0271488i
$288$	0	0
$289$	4237.00	0.862406
$290$	0	0
$291$	−772.000	−0.155517
$292$	0	0
$293$	4842.00i	0.965436i	0.875776	+	0.482718i	$0.160350\pi$
$293$	4842.00i	0.965436i	−0.875776	+	0.482718i	$0.839650\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3200.00i	0.625195i
$298$	0	0
$299$	2964.00	0.573286
$300$	0	0
$301$	2652.00	0.507836
$302$	0	0
$303$	− 1404.00i	− 0.266197i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2594.00i	− 0.482239i	−0.970495	−	0.241120i	$-0.922485\pi$
$307$	− 2594.00i	− 0.482239i	0.970495	−	0.241120i	$-0.0775146\pi$
$308$	0	0
$309$	1196.00	0.220188
$310$	0	0
$311$	−7332.00	−1.33685	−0.668424	−	0.743781i	$-0.733031\pi$
$311$	−7332.00	−1.33685	−0.668424	+	0.743781i	$0.733031\pi$
$312$	0	0
$313$	1562.00i	0.282075i	0.990004	+	0.141037i	$0.0450438\pi$
$313$	1562.00i	0.282075i	−0.990004	+	0.141037i	$0.954956\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1426.00i	− 0.252657i	−0.991988	−	0.126328i	$-0.959681\pi$
$317$	− 1426.00i	− 0.252657i	0.991988	−	0.126328i	$-0.0403193\pi$
$318$	0	0
$319$	−1600.00	−0.280824
$320$	0	0
$321$	−2388.00	−0.415219
$322$	0	0
$323$	− 2600.00i	− 0.447888i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1100.00i	− 0.186025i
$328$	0	0
$329$	3084.00	0.516798
$330$	0	0
$331$	4008.00	0.665558	0.332779	−	0.943005i	$-0.392014\pi$
$331$	4008.00	0.665558	0.332779	+	0.943005i	$0.392014\pi$
$332$	0	0
$333$	− 6118.00i	− 1.00680i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 8866.00i	− 1.43312i	−0.697525	−	0.716561i	$-0.745715\pi$
$337$	− 8866.00i	− 1.43312i	0.697525	−	0.716561i	$-0.254285\pi$
$338$	0	0
$339$	3124.00	0.500509
$340$	0	0
$341$	−3456.00	−0.548835
$342$	0	0
$343$	3900.00i	0.613936i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1714.00i	− 0.265165i	−0.991172	−	0.132583i	$-0.957673\pi$
$347$	− 1714.00i	− 0.265165i	0.991172	−	0.132583i	$-0.0423270\pi$
$348$	0	0
$349$	−1150.00	−0.176384	−0.0881921	−	0.996103i	$-0.528109\pi$
$349$	−1150.00	−0.176384	−0.0881921	+	0.996103i	$0.528109\pi$
$350$	0	0
$351$	−3800.00	−0.577860
$352$	0	0
$353$	− 4398.00i	− 0.663122i	−0.943434	−	0.331561i	$-0.892425\pi$
$353$	− 4398.00i	− 0.663122i	0.943434	−	0.331561i	$-0.107575\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 312.000i	− 0.0462543i
$358$	0	0
$359$	1800.00	0.264625	0.132312	−	0.991208i	$-0.457760\pi$
$359$	1800.00	0.264625	0.132312	+	0.991208i	$0.457760\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	614.000i	0.0887786i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 5874.00i	− 0.835478i	−0.908567	−	0.417739i	$-0.862823\pi$
$367$	− 5874.00i	− 0.835478i	0.908567	−	0.417739i	$-0.137177\pi$
$368$	0	0
$369$	506.000	0.0713857
$370$	0	0
$371$	−12.0000	−0.00167927
$372$	0	0
$373$	− 2078.00i	− 0.288458i	−0.989544	−	0.144229i	$-0.953930\pi$
$373$	− 2078.00i	− 0.288458i	0.989544	−	0.144229i	$-0.0460702\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1900.00i	− 0.259562i
$378$	0	0
$379$	7900.00	1.07070	0.535351	−	0.844630i	$-0.320179\pi$
$379$	7900.00	1.07070	0.535351	+	0.844630i	$0.320179\pi$
$380$	0	0
$381$	3692.00	0.496449
$382$	0	0
$383$	7518.00i	1.00301i	0.865155	+	0.501504i	$0.167220\pi$
$383$	7518.00i	1.00301i	−0.865155	+	0.501504i	$0.832780\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 10166.0i	− 1.33531i
$388$	0	0
$389$	1950.00	0.254162	0.127081	−	0.991892i	$-0.459439\pi$
$389$	1950.00	0.254162	0.127081	+	0.991892i	$0.459439\pi$
$390$	0	0
$391$	2028.00	0.262303
$392$	0	0
$393$	− 4416.00i	− 0.566814i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 13786.0i	− 1.74282i	−0.490555	−	0.871410i	$-0.663206\pi$
$397$	− 13786.0i	− 1.74282i	0.490555	−	0.871410i	$-0.336794\pi$
$398$	0	0
$399$	1200.00	0.150564
$400$	0	0
$401$	6402.00	0.797258	0.398629	−	0.917112i	$-0.369486\pi$
$401$	6402.00	0.797258	0.398629	+	0.917112i	$0.369486\pi$
$402$	0	0
$403$	− 4104.00i	− 0.507282i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8512.00i	1.03667i
$408$	0	0
$409$	−11150.0	−1.34800	−0.674000	−	0.738731i	$-0.735425\pi$
$409$	−11150.0	−1.34800	−0.674000	+	0.738731i	$0.735425\pi$
$410$	0	0
$411$	4668.00	0.560232
$412$	0	0
$413$	3000.00i	0.357434i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1400.00i	0.164408i
$418$	0	0
$419$	−13700.0	−1.59735	−0.798674	−	0.601764i	$-0.794465\pi$
$419$	−13700.0	−1.59735	−0.798674	+	0.601764i	$0.794465\pi$
$420$	0	0
$421$	−5438.00	−0.629529	−0.314765	−	0.949170i	$-0.601926\pi$
$421$	−5438.00	−0.629529	−0.314765	+	0.949170i	$0.601926\pi$
$422$	0	0
$423$	− 11822.0i	− 1.35888i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3108.00i	− 0.352240i
$428$	0	0
$429$	2432.00	0.273702
$430$	0	0
$431$	−7692.00	−0.859653	−0.429827	−	0.902911i	$-0.641425\pi$
$431$	−7692.00	−0.859653	−0.429827	+	0.902911i	$0.641425\pi$
$432$	0	0
$433$	− 1118.00i	− 0.124082i	−0.998074	−	0.0620412i	$-0.980239\pi$
$433$	− 1118.00i	− 0.124082i	0.998074	−	0.0620412i	$-0.0197610\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7800.00i	0.853832i
$438$	0	0
$439$	−2600.00	−0.282668	−0.141334	−	0.989962i	$-0.545139\pi$
$439$	−2600.00	−0.282668	−0.141334	+	0.989962i	$0.545139\pi$
$440$	0	0
$441$	7061.00	0.762445
$442$	0	0
$443$	11958.0i	1.28249i	0.767337	+	0.641243i	$0.221581\pi$
$443$	11958.0i	1.28249i	−0.767337	+	0.641243i	$0.778419\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4100.00i	0.433833i
$448$	0	0
$449$	17050.0	1.79207	0.896035	−	0.443984i	$-0.146435\pi$
$449$	17050.0	1.79207	0.896035	+	0.443984i	$0.146435\pi$
$450$	0	0
$451$	−704.000	−0.0735035
$452$	0	0
$453$	3704.00i	0.384170i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9494.00i	0.971796i	0.874016	+	0.485898i	$0.161507\pi$
$457$	9494.00i	0.971796i	−0.874016	+	0.485898i	$0.838493\pi$
$458$	0	0
$459$	−2600.00	−0.264396
$460$	0	0
$461$	−11418.0	−1.15356	−0.576778	−	0.816901i	$-0.695690\pi$
$461$	−11418.0	−1.15356	−0.576778	+	0.816901i	$0.695690\pi$
$462$	0	0
$463$	− 7962.00i	− 0.799191i	−0.916692	−	0.399596i	$-0.869151\pi$
$463$	− 7962.00i	− 0.799191i	0.916692	−	0.399596i	$-0.130849\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6526.00i	0.646654i	0.946287	+	0.323327i	$0.104801\pi$
$467$	6526.00i	0.646654i	−0.946287	+	0.323327i	$0.895199\pi$
$468$	0	0
$469$	−756.000	−0.0744325
$470$	0	0
$471$	4988.00	0.487972
$472$	0	0
$473$	14144.0i	1.37493i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	46.0000i	0.00441550i
$478$	0	0
$479$	17400.0	1.65976	0.829881	−	0.557940i	$-0.188408\pi$
$479$	17400.0	1.65976	0.829881	+	0.557940i	$0.188408\pi$
$480$	0	0
$481$	−10108.0	−0.958181
$482$	0	0
$483$	936.000i	0.0881770i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1166.00i	0.108494i	0.998528	+	0.0542469i	$0.0172758\pi$
$487$	1166.00i	0.108494i	−0.998528	+	0.0542469i	$0.982724\pi$
$488$	0	0
$489$	−5524.00	−0.510846
$490$	0	0
$491$	−7072.00	−0.650010	−0.325005	−	0.945712i	$-0.605366\pi$
$491$	−7072.00	−0.650010	−0.325005	+	0.945712i	$0.605366\pi$
$492$	0	0
$493$	− 1300.00i	− 0.118761i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2472.00i	− 0.223107i
$498$	0	0
$499$	100.000	0.00897117	0.00448559	−	0.999990i	$-0.498572\pi$
$499$	100.000	0.00897117	0.00448559	+	0.999990i	$0.498572\pi$
$500$	0	0
$501$	6252.00	0.557522
$502$	0	0
$503$	− 2602.00i	− 0.230651i	−0.993328	−	0.115325i	$-0.963209\pi$
$503$	− 2602.00i	− 0.230651i	0.993328	−	0.115325i	$-0.0367911\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1506.00i	− 0.131921i
$508$	0	0
$509$	−11150.0	−0.970953	−0.485476	−	0.874250i	$-0.661354\pi$
$509$	−11150.0	−0.970953	−0.485476	+	0.874250i	$0.661354\pi$
$510$	0	0
$511$	5268.00	0.456052
$512$	0	0
$513$	− 10000.0i	− 0.860645i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16448.0i	1.39919i
$518$	0	0
$519$	−156.000	−0.0131939
$520$	0	0
$521$	−3638.00	−0.305919	−0.152959	−	0.988232i	$-0.548880\pi$
$521$	−3638.00	−0.305919	−0.152959	+	0.988232i	$0.548880\pi$
$522$	0	0
$523$	2078.00i	0.173737i	0.996220	+	0.0868686i	$0.0276860\pi$
$523$	2078.00i	0.173737i	−0.996220	+	0.0868686i	$0.972314\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2808.00i	− 0.232103i
$528$	0	0
$529$	6083.00	0.499959
$530$	0	0
$531$	11500.0	0.939845
$532$	0	0
$533$	− 836.000i	− 0.0679384i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2600.00i	0.208935i
$538$	0	0
$539$	−9824.00	−0.785064
$540$	0	0
$541$	5622.00	0.446781	0.223391	−	0.974729i	$-0.428287\pi$
$541$	5622.00	0.446781	0.223391	+	0.974729i	$0.428287\pi$
$542$	0	0
$543$	− 3484.00i	− 0.275346i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16486.0i	1.28865i	0.764753	+	0.644324i	$0.222861\pi$
$547$	16486.0i	1.28865i	−0.764753	+	0.644324i	$0.777139\pi$
$548$	0	0
$549$	−11914.0	−0.926188
$550$	0	0
$551$	5000.00	0.386583
$552$	0	0
$553$	3600.00i	0.276831i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 11706.0i	− 0.890483i	−0.895410	−	0.445242i	$-0.853118\pi$
$557$	− 11706.0i	− 0.890483i	0.895410	−	0.445242i	$-0.146882\pi$
$558$	0	0
$559$	−16796.0	−1.27083
$560$	0	0
$561$	1664.00	0.125230
$562$	0	0
$563$	25038.0i	1.87429i	0.348939	+	0.937146i	$0.386542\pi$
$563$	25038.0i	1.87429i	−0.348939	+	0.937146i	$0.613458\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2526.00i	0.187094i
$568$	0	0
$569$	−17550.0	−1.29303	−0.646515	−	0.762901i	$-0.723774\pi$
$569$	−17550.0	−1.29303	−0.646515	+	0.762901i	$0.723774\pi$
$570$	0	0
$571$	−10712.0	−0.785084	−0.392542	−	0.919734i	$-0.628404\pi$
$571$	−10712.0	−0.785084	−0.392542	+	0.919734i	$0.628404\pi$
$572$	0	0
$573$	7544.00i	0.550009i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13654.0i	0.985136i	0.870274	+	0.492568i	$0.163942\pi$
$577$	13654.0i	0.985136i	−0.870274	+	0.492568i	$0.836058\pi$
$578$	0	0
$579$	−716.000	−0.0513920
$580$	0	0
$581$	1692.00	0.120819
$582$	0	0
$583$	− 64.0000i	− 0.00454650i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14166.0i	0.996071i	0.867157	+	0.498035i	$0.165945\pi$
$587$	14166.0i	0.996071i	−0.867157	+	0.498035i	$0.834055\pi$
$588$	0	0
$589$	10800.0	0.755528
$590$	0	0
$591$	4428.00	0.308196
$592$	0	0
$593$	17842.0i	1.23555i	0.786354	+	0.617777i	$0.211966\pi$
$593$	17842.0i	1.23555i	−0.786354	+	0.617777i	$0.788034\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5200.00i	0.356485i
$598$	0	0
$599$	−17600.0	−1.20053	−0.600264	−	0.799802i	$-0.704938\pi$
$599$	−17600.0	−1.20053	−0.600264	+	0.799802i	$0.704938\pi$
$600$	0	0
$601$	27302.0	1.85303	0.926516	−	0.376256i	$-0.122789\pi$
$601$	27302.0	1.85303	0.926516	+	0.376256i	$0.122789\pi$
$602$	0	0
$603$	2898.00i	0.195714i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 3794.00i	− 0.253696i	−0.991922	−	0.126848i	$-0.959514\pi$
$607$	− 3794.00i	− 0.253696i	0.991922	−	0.126848i	$-0.0404861\pi$
$608$	0	0
$609$	600.000	0.0399232
$610$	0	0
$611$	−19532.0	−1.29326
$612$	0	0
$613$	− 13238.0i	− 0.872231i	−0.899891	−	0.436116i	$-0.856354\pi$
$613$	− 13238.0i	− 0.872231i	0.899891	−	0.436116i	$-0.143646\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11574.0i	0.755189i	0.925971	+	0.377595i	$0.123249\pi$
$617$	11574.0i	0.755189i	−0.925971	+	0.377595i	$0.876751\pi$
$618$	0	0
$619$	8300.00	0.538942	0.269471	−	0.963008i	$-0.413151\pi$
$619$	8300.00	0.538942	0.269471	+	0.963008i	$0.413151\pi$
$620$	0	0
$621$	7800.00	0.504031
$622$	0	0
$623$	900.000i	0.0578776i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6400.00i	0.407642i
$628$	0	0
$629$	−6916.00	−0.438409
$630$	0	0
$631$	7508.00	0.473675	0.236837	−	0.971549i	$-0.423889\pi$
$631$	7508.00	0.473675	0.236837	+	0.971549i	$0.423889\pi$
$632$	0	0
$633$	− 2336.00i	− 0.146679i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 11666.0i	− 0.725626i
$638$	0	0
$639$	−9476.00	−0.586643
$640$	0	0
$641$	−27378.0	−1.68700	−0.843499	−	0.537130i	$-0.819508\pi$
$641$	−27378.0	−1.68700	−0.843499	+	0.537130i	$0.819508\pi$
$642$	0	0
$643$	− 1842.00i	− 0.112973i	−0.998403	−	0.0564863i	$-0.982010\pi$
$643$	− 1842.00i	− 0.112973i	0.998403	−	0.0564863i	$-0.0179897\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 10114.0i	− 0.614563i	−0.951619	−	0.307282i	$-0.900581\pi$
$647$	− 10114.0i	− 0.614563i	0.951619	−	0.307282i	$-0.0994193\pi$
$648$	0	0
$649$	−16000.0	−0.967727
$650$	0	0
$651$	1296.00	0.0780250
$652$	0	0
$653$	10402.0i	0.623372i	0.950185	+	0.311686i	$0.100894\pi$
$653$	10402.0i	0.623372i	−0.950185	+	0.311686i	$0.899106\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 20194.0i	− 1.19915i
$658$	0	0
$659$	7100.00	0.419692	0.209846	−	0.977734i	$-0.432704\pi$
$659$	7100.00	0.419692	0.209846	+	0.977734i	$0.432704\pi$
$660$	0	0
$661$	−7118.00	−0.418847	−0.209424	−	0.977825i	$-0.567159\pi$
$661$	−7118.00	−0.418847	−0.209424	+	0.977825i	$0.567159\pi$
$662$	0	0
$663$	1976.00i	0.115749i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3900.00i	0.226400i
$668$	0	0
$669$	12956.0	0.748741
$670$	0	0
$671$	16576.0	0.953665
$672$	0	0
$673$	− 31278.0i	− 1.79150i	−0.444560	−	0.895749i	$-0.646640\pi$
$673$	− 31278.0i	− 1.79150i	0.444560	−	0.895749i	$-0.353360\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	30054.0i	1.70616i	0.521782	+	0.853079i	$0.325268\pi$
$677$	30054.0i	1.70616i	−0.521782	+	0.853079i	$0.674732\pi$
$678$	0	0
$679$	2316.00	0.130898
$680$	0	0
$681$	1292.00	0.0727012
$682$	0	0
$683$	4518.00i	0.253113i	0.991959	+	0.126557i	$0.0403926\pi$
$683$	4518.00i	0.253113i	−0.991959	+	0.126557i	$0.959607\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7500.00i	0.416511i
$688$	0	0
$689$	76.0000	0.00420228
$690$	0	0
$691$	−29272.0	−1.61152	−0.805759	−	0.592243i	$-0.798242\pi$
$691$	−29272.0	−1.61152	−0.805759	+	0.592243i	$0.798242\pi$
$692$	0	0
$693$	− 4416.00i	− 0.242063i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 572.000i	− 0.0310847i
$698$	0	0
$699$	2964.00	0.160385
$700$	0	0
$701$	−5798.00	−0.312393	−0.156196	−	0.987726i	$-0.549923\pi$
$701$	−5798.00	−0.312393	−0.156196	+	0.987726i	$0.549923\pi$
$702$	0	0
$703$	− 26600.0i	− 1.42708i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4212.00i	0.224057i
$708$	0	0
$709$	−8950.00	−0.474082	−0.237041	−	0.971500i	$-0.576178\pi$
$709$	−8950.00	−0.474082	−0.237041	+	0.971500i	$0.576178\pi$
$710$	0	0
$711$	13800.0	0.727905
$712$	0	0
$713$	8424.00i	0.442470i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2800.00i	− 0.145841i
$718$	0	0
$719$	7800.00	0.404577	0.202289	−	0.979326i	$-0.435162\pi$
$719$	7800.00	0.404577	0.202289	+	0.979326i	$0.435162\pi$
$720$	0	0
$721$	−3588.00	−0.185332
$722$	0	0
$723$	− 6044.00i	− 0.310897i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8554.00i	− 0.436383i	−0.975906	−	0.218191i	$-0.929984\pi$
$727$	− 8554.00i	− 0.436383i	0.975906	−	0.218191i	$-0.0700157\pi$
$728$	0	0
$729$	4283.00	0.217599
$730$	0	0
$731$	−11492.0	−0.581460
$732$	0	0
$733$	2882.00i	0.145224i	0.997360	+	0.0726119i	$0.0231335\pi$
$733$	2882.00i	0.145224i	−0.997360	+	0.0726119i	$0.976867\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4032.00i	− 0.201521i
$738$	0	0
$739$	18700.0	0.930840	0.465420	−	0.885090i	$-0.345903\pi$
$739$	18700.0	0.930840	0.465420	+	0.885090i	$0.345903\pi$
$740$	0	0
$741$	−7600.00	−0.376779
$742$	0	0
$743$	− 12242.0i	− 0.604462i	−0.953235	−	0.302231i	$-0.902269\pi$
$743$	− 12242.0i	− 0.604462i	0.953235	−	0.302231i	$-0.0977314\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 6486.00i	− 0.317685i
$748$	0	0
$749$	7164.00	0.349488
$750$	0	0
$751$	31148.0	1.51346	0.756729	−	0.653729i	$-0.226796\pi$
$751$	31148.0	1.51346	0.756729	+	0.653729i	$0.226796\pi$
$752$	0	0
$753$	− 2496.00i	− 0.120796i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7694.00i	0.369410i	0.982794	+	0.184705i	$0.0591329\pi$
$757$	7694.00i	0.369410i	−0.982794	+	0.184705i	$0.940867\pi$
$758$	0	0
$759$	−4992.00	−0.238733
$760$	0	0
$761$	−4518.00	−0.215213	−0.107607	−	0.994194i	$-0.534319\pi$
$761$	−4518.00	−0.215213	−0.107607	+	0.994194i	$0.534319\pi$
$762$	0	0
$763$	3300.00i	0.156577i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 19000.0i	− 0.894459i
$768$	0	0
$769$	39550.0	1.85463	0.927314	−	0.374283i	$-0.122111\pi$
$769$	39550.0	1.85463	0.927314	+	0.374283i	$0.122111\pi$
$770$	0	0
$771$	−4212.00	−0.196746
$772$	0	0
$773$	22122.0i	1.02933i	0.857391	+	0.514666i	$0.172084\pi$
$773$	22122.0i	1.02933i	−0.857391	+	0.514666i	$0.827916\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3192.00i	− 0.147378i
$778$	0	0
$779$	2200.00	0.101185
$780$	0	0
$781$	13184.0	0.604047
$782$	0	0
$783$	− 5000.00i	− 0.228206i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16634.0i	− 0.753416i	−0.926332	−	0.376708i	$-0.877056\pi$
$787$	− 16634.0i	− 0.753416i	0.926332	−	0.376708i	$-0.122944\pi$
$788$	0	0
$789$	7276.00	0.328305
$790$	0	0
$791$	−9372.00	−0.421277
$792$	0	0
$793$	19684.0i	0.881462i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 27586.0i	− 1.22603i	−0.790071	−	0.613015i	$-0.789956\pi$
$797$	− 27586.0i	− 1.22603i	0.790071	−	0.613015i	$-0.210044\pi$
$798$	0	0
$799$	−13364.0	−0.591720
$800$	0	0
$801$	3450.00	0.152184
$802$	0	0
$803$	28096.0i	1.23473i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13100.0i	− 0.571427i
$808$	0	0
$809$	−3850.00	−0.167316	−0.0836581	−	0.996495i	$-0.526660\pi$
$809$	−3850.00	−0.167316	−0.0836581	+	0.996495i	$0.526660\pi$
$810$	0	0
$811$	−10032.0	−0.434366	−0.217183	−	0.976131i	$-0.569687\pi$
$811$	−10032.0	−0.434366	−0.217183	+	0.976131i	$0.569687\pi$
$812$	0	0
$813$	− 8776.00i	− 0.378583i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 44200.0i	− 1.89273i
$818$	0	0
$819$	5244.00	0.223736
$820$	0	0
$821$	20562.0	0.874079	0.437039	−	0.899442i	$-0.356027\pi$
$821$	20562.0	0.874079	0.437039	+	0.899442i	$0.356027\pi$
$822$	0	0
$823$	− 10322.0i	− 0.437184i	−0.975816	−	0.218592i	$-0.929854\pi$
$823$	− 10322.0i	− 0.437184i	0.975816	−	0.218592i	$-0.0701464\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8846.00i	0.371954i	0.982554	+	0.185977i	$0.0595449\pi$
$827$	8846.00i	0.371954i	−0.982554	+	0.185977i	$0.940455\pi$
$828$	0	0
$829$	25350.0	1.06205	0.531026	−	0.847355i	$-0.321806\pi$
$829$	25350.0	1.06205	0.531026	+	0.847355i	$0.321806\pi$
$830$	0	0
$831$	−1092.00	−0.0455849
$832$	0	0
$833$	− 7982.00i	− 0.332005i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 10800.0i	− 0.446001i
$838$	0	0
$839$	46000.0	1.89284	0.946422	−	0.322932i	$-0.104669\pi$
$839$	46000.0	1.89284	0.946422	+	0.322932i	$0.104669\pi$
$840$	0	0
$841$	−21889.0	−0.897495
$842$	0	0
$843$	13716.0i	0.560385i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1842.00i	− 0.0747248i
$848$	0	0
$849$	−18564.0	−0.750430
$850$	0	0
$851$	20748.0	0.835761
$852$	0	0
$853$	− 16998.0i	− 0.682298i	−0.940009	−	0.341149i	$-0.889184\pi$
$853$	− 16998.0i	− 0.682298i	0.940009	−	0.341149i	$-0.110816\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26494.0i	1.05603i	0.849235	+	0.528015i	$0.177064\pi$
$857$	26494.0i	1.05603i	−0.849235	+	0.528015i	$0.822936\pi$
$858$	0	0
$859$	−21500.0	−0.853982	−0.426991	−	0.904256i	$-0.640426\pi$
$859$	−21500.0	−0.853982	−0.426991	+	0.904256i	$0.640426\pi$
$860$	0	0
$861$	264.000	0.0104496
$862$	0	0
$863$	− 25762.0i	− 1.01616i	−0.861309	−	0.508082i	$-0.830355\pi$
$863$	− 25762.0i	− 1.01616i	0.861309	−	0.508082i	$-0.169645\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 8474.00i	− 0.331940i
$868$	0	0
$869$	−19200.0	−0.749500
$870$	0	0
$871$	4788.00	0.186263
$872$	0	0
$873$	− 8878.00i	− 0.344186i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 30546.0i	− 1.17613i	−0.808814	−	0.588064i	$-0.799890\pi$
$877$	− 30546.0i	− 1.17613i	0.808814	−	0.588064i	$-0.200110\pi$
$878$	0	0
$879$	9684.00	0.371596
$880$	0	0
$881$	32942.0	1.25976	0.629878	−	0.776694i	$-0.283105\pi$
$881$	32942.0	1.25976	0.629878	+	0.776694i	$0.283105\pi$
$882$	0	0
$883$	27118.0i	1.03351i	0.856132	+	0.516757i	$0.172861\pi$
$883$	27118.0i	1.03351i	−0.856132	+	0.516757i	$0.827139\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 38634.0i	− 1.46246i	−0.682131	−	0.731230i	$-0.738946\pi$
$887$	− 38634.0i	− 1.46246i	0.682131	−	0.731230i	$-0.261054\pi$
$888$	0	0
$889$	−11076.0	−0.417860
$890$	0	0
$891$	−13472.0	−0.506542
$892$	0	0
$893$	− 51400.0i	− 1.92613i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5928.00i	− 0.220658i
$898$	0	0
$899$	5400.00	0.200334
$900$	0	0
$901$	52.0000	0.00192272
$902$	0	0
$903$	− 5304.00i	− 0.195466i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1794.00i	− 0.0656767i	−0.999461	−	0.0328384i	$-0.989545\pi$
$907$	− 1794.00i	− 0.0656767i	0.999461	−	0.0328384i	$-0.0104547\pi$
$908$	0	0
$909$	16146.0	0.589141
$910$	0	0
$911$	−41732.0	−1.51772	−0.758860	−	0.651254i	$-0.774243\pi$
$911$	−41732.0	−1.51772	−0.758860	+	0.651254i	$0.774243\pi$
$912$	0	0
$913$	9024.00i	0.327109i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	13248.0i	0.477086i
$918$	0	0
$919$	29200.0	1.04812	0.524058	−	0.851682i	$-0.324417\pi$
$919$	29200.0	1.04812	0.524058	+	0.851682i	$0.324417\pi$
$920$	0	0
$921$	−5188.00	−0.185614
$922$	0	0
$923$	15656.0i	0.558314i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13754.0i	0.487315i
$928$	0	0
$929$	48650.0	1.71814	0.859071	−	0.511856i	$-0.171042\pi$
$929$	48650.0	1.71814	0.859071	+	0.511856i	$0.171042\pi$
$930$	0	0
$931$	30700.0	1.08072
$932$	0	0
$933$	14664.0i	0.514553i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11334.0i	0.395161i	0.980287	+	0.197580i	$0.0633083\pi$
$937$	11334.0i	0.395161i	−0.980287	+	0.197580i	$0.936692\pi$
$938$	0	0
$939$	3124.00	0.108571
$940$	0	0
$941$	−31178.0	−1.08010	−0.540050	−	0.841633i	$-0.681595\pi$
$941$	−31178.0	−1.08010	−0.540050	+	0.841633i	$0.681595\pi$
$942$	0	0
$943$	1716.00i	0.0592584i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4686.00i	0.160797i	0.996763	+	0.0803984i	$0.0256193\pi$
$947$	4686.00i	0.160797i	−0.996763	+	0.0803984i	$0.974381\pi$
$948$	0	0
$949$	−33364.0	−1.14124
$950$	0	0
$951$	−2852.00	−0.0972476
$952$	0	0
$953$	− 598.000i	− 0.0203265i	−0.999948	−	0.0101632i	$-0.996765\pi$
$953$	− 598.000i	− 0.0203265i	0.999948	−	0.0101632i	$-0.00323511\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3200.00i	0.108089i
$958$	0	0
$959$	−14004.0	−0.471546
$960$	0	0
$961$	−18127.0	−0.608472
$962$	0	0
$963$	− 27462.0i	− 0.918952i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41726.0i	1.38761i	0.720163	+	0.693804i	$0.244067\pi$
$967$	41726.0i	1.38761i	−0.720163	+	0.693804i	$0.755933\pi$
$968$	0	0
$969$	−5200.00	−0.172392
$970$	0	0
$971$	−24312.0	−0.803511	−0.401756	−	0.915747i	$-0.631600\pi$
$971$	−24312.0	−0.803511	−0.401756	+	0.915747i	$0.631600\pi$
$972$	0	0
$973$	− 4200.00i	− 0.138382i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 40946.0i	− 1.34082i	−0.741992	−	0.670409i	$-0.766119\pi$
$977$	− 40946.0i	− 1.34082i	0.741992	−	0.670409i	$-0.233881\pi$
$978$	0	0
$979$	−4800.00	−0.156699
$980$	0	0
$981$	12650.0	0.411706
$982$	0	0
$983$	− 42282.0i	− 1.37191i	−0.727645	−	0.685954i	$-0.759385\pi$
$983$	− 42282.0i	− 1.37191i	0.727645	−	0.685954i	$-0.240615\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6168.00i	− 0.198916i
$988$	0	0
$989$	34476.0	1.10847
$990$	0	0
$991$	−1172.00	−0.0375679	−0.0187840	−	0.999824i	$-0.505979\pi$
$991$	−1172.00	−0.0375679	−0.0187840	+	0.999824i	$0.505979\pi$
$992$	0	0
$993$	− 8016.00i	− 0.256173i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31614.0i	1.00424i	0.864798	+	0.502119i	$0.167446\pi$
$997$	31614.0i	1.00424i	−0.864798	+	0.502119i	$0.832554\pi$
$998$	0	0
$999$	−26600.0	−0.842429

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.k.49.1		2
4.3	odd	2		25.4.b.a.24.2		2
5.2	odd	4		80.4.a.d.1.1		1
5.3	odd	4		400.4.a.m.1.1		1
5.4	even	2	inner	400.4.c.k.49.2		2
12.11	even	2		225.4.b.c.199.1		2
15.2	even	4		720.4.a.u.1.1		1
20.3	even	4		25.4.a.c.1.1		1
20.7	even	4		5.4.a.a.1.1	✓	1
20.19	odd	2		25.4.b.a.24.1		2
40.3	even	4		1600.4.a.bi.1.1		1
40.13	odd	4		1600.4.a.s.1.1		1
40.27	even	4		320.4.a.g.1.1		1
40.37	odd	4		320.4.a.h.1.1		1
60.23	odd	4		225.4.a.b.1.1		1
60.47	odd	4		45.4.a.d.1.1		1
60.59	even	2		225.4.b.c.199.2		2
80.27	even	4		1280.4.d.e.641.1		2
80.37	odd	4		1280.4.d.l.641.2		2
80.67	even	4		1280.4.d.e.641.2		2
80.77	odd	4		1280.4.d.l.641.1		2
140.27	odd	4		245.4.a.a.1.1		1
140.47	odd	12		245.4.e.g.116.1		2
140.67	even	12		245.4.e.f.226.1		2
140.83	odd	4		1225.4.a.k.1.1		1
140.87	odd	12		245.4.e.g.226.1		2
140.107	even	12		245.4.e.f.116.1		2
180.7	even	12		405.4.e.l.271.1		2
180.47	odd	12		405.4.e.c.271.1		2
180.67	even	12		405.4.e.l.136.1		2
180.167	odd	12		405.4.e.c.136.1		2
220.87	odd	4		605.4.a.d.1.1		1
260.207	even	4		845.4.a.b.1.1		1
340.67	even	4		1445.4.a.a.1.1		1
380.227	odd	4		1805.4.a.h.1.1		1
420.167	even	4		2205.4.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	20.7	even	4
25.4.a.c.1.1		1	20.3	even	4
25.4.b.a.24.1		2	20.19	odd	2
25.4.b.a.24.2		2	4.3	odd	2
45.4.a.d.1.1		1	60.47	odd	4
80.4.a.d.1.1		1	5.2	odd	4
225.4.a.b.1.1		1	60.23	odd	4
225.4.b.c.199.1		2	12.11	even	2
225.4.b.c.199.2		2	60.59	even	2
245.4.a.a.1.1		1	140.27	odd	4
245.4.e.f.116.1		2	140.107	even	12
245.4.e.f.226.1		2	140.67	even	12
245.4.e.g.116.1		2	140.47	odd	12
245.4.e.g.226.1		2	140.87	odd	12
320.4.a.g.1.1		1	40.27	even	4
320.4.a.h.1.1		1	40.37	odd	4
400.4.a.m.1.1		1	5.3	odd	4
400.4.c.k.49.1		2	1.1	even	1	trivial
400.4.c.k.49.2		2	5.4	even	2	inner
405.4.e.c.136.1		2	180.167	odd	12
405.4.e.c.271.1		2	180.47	odd	12
405.4.e.l.136.1		2	180.67	even	12
405.4.e.l.271.1		2	180.7	even	12
605.4.a.d.1.1		1	220.87	odd	4
720.4.a.u.1.1		1	15.2	even	4
845.4.a.b.1.1		1	260.207	even	4
1225.4.a.k.1.1		1	140.83	odd	4
1280.4.d.e.641.1		2	80.27	even	4
1280.4.d.e.641.2		2	80.67	even	4
1280.4.d.l.641.1		2	80.77	odd	4
1280.4.d.l.641.2		2	80.37	odd	4
1445.4.a.a.1.1		1	340.67	even	4
1600.4.a.s.1.1		1	40.13	odd	4
1600.4.a.bi.1.1		1	40.3	even	4
1805.4.a.h.1.1		1	380.227	odd	4
2205.4.a.q.1.1		1	420.167	even	4

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000i q^{3} +6.00000i q^{7} +23.0000 q^{9} +O(q^{10})\)	\(q-2.00000i q^{3} +6.00000i q^{7} +23.0000 q^{9} -32.0000 q^{11} -38.0000i q^{13} -26.0000i q^{17} +100.000 q^{19} +12.0000 q^{21} +78.0000i q^{23} -100.000i q^{27} +50.0000 q^{29} +108.000 q^{31} +64.0000i q^{33} -266.000i q^{37} -76.0000 q^{39} +22.0000 q^{41} -442.000i q^{43} -514.000i q^{47} +307.000 q^{49} -52.0000 q^{51} +2.00000i q^{53} -200.000i q^{57} +500.000 q^{59} -518.000 q^{61} +138.000i q^{63} +126.000i q^{67} +156.000 q^{69} -412.000 q^{71} -878.000i q^{73} -192.000i q^{77} +600.000 q^{79} +421.000 q^{81} -282.000i q^{83} -100.000i q^{87} +150.000 q^{89} +228.000 q^{91} -216.000i q^{93} -386.000i q^{97} -736.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 46 q^{9}+O(q^{10}) \) 2 * q + 46 * q^9	\( 2 q + 46 q^{9} - 64 q^{11} + 200 q^{19} + 24 q^{21} + 100 q^{29} + 216 q^{31} - 152 q^{39} + 44 q^{41} + 614 q^{49} - 104 q^{51} + 1000 q^{59} - 1036 q^{61} + 312 q^{69} - 824 q^{71} + 1200 q^{79} + 842 q^{81} + 300 q^{89} + 456 q^{91} - 1472 q^{99}+O(q^{100}) \) 2 * q + 46 * q^9 - 64 * q^11 + 200 * q^19 + 24 * q^21 + 100 * q^29 + 216 * q^31 - 152 * q^39 + 44 * q^41 + 614 * q^49 - 104 * q^51 + 1000 * q^59 - 1036 * q^61 + 312 * q^69 - 824 * q^71 + 1200 * q^79 + 842 * q^81 + 300 * q^89 + 456 * q^91 - 1472 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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