LMFDB - Embedded newform 225.4.b.c.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.b (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:	$13.2754297513$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.4.b.c.199.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -8.00000 q^{4} -6.00000i q^{7} +O(q^{10})$	$q-4.00000i q^{2} -8.00000 q^{4} -6.00000i q^{7} -32.0000 q^{11} -38.0000i q^{13} -24.0000 q^{14} -64.0000 q^{16} +26.0000i q^{17} -100.000 q^{19} +128.000i q^{22} +78.0000i q^{23} -152.000 q^{26} +48.0000i q^{28} -50.0000 q^{29} -108.000 q^{31} +256.000i q^{32} +104.000 q^{34} -266.000i q^{37} +400.000i q^{38} -22.0000 q^{41} +442.000i q^{43} +256.000 q^{44} +312.000 q^{46} -514.000i q^{47} +307.000 q^{49} +304.000i q^{52} -2.00000i q^{53} +200.000i q^{58} +500.000 q^{59} -518.000 q^{61} +432.000i q^{62} +512.000 q^{64} -126.000i q^{67} -208.000i q^{68} -412.000 q^{71} -878.000i q^{73} -1064.00 q^{74} +800.000 q^{76} +192.000i q^{77} -600.000 q^{79} +88.0000i q^{82} -282.000i q^{83} +1768.00 q^{86} -150.000 q^{89} -228.000 q^{91} -624.000i q^{92} -2056.00 q^{94} -386.000i q^{97} -1228.00i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 16 q^{4}+O(q^{10}) $ 2 * q - 16 * q^4	$ 2 q - 16 q^{4} - 64 q^{11} - 48 q^{14} - 128 q^{16} - 200 q^{19} - 304 q^{26} - 100 q^{29} - 216 q^{31} + 208 q^{34} - 44 q^{41} + 512 q^{44} + 624 q^{46} + 614 q^{49} + 1000 q^{59} - 1036 q^{61} + 1024 q^{64} - 824 q^{71} - 2128 q^{74} + 1600 q^{76} - 1200 q^{79} + 3536 q^{86} - 300 q^{89} - 456 q^{91} - 4112 q^{94}+O(q^{100}) $ 2 * q - 16 * q^4 - 64 * q^11 - 48 * q^14 - 128 * q^16 - 200 * q^19 - 304 * q^26 - 100 * q^29 - 216 * q^31 + 208 * q^34 - 44 * q^41 + 512 * q^44 + 624 * q^46 + 614 * q^49 + 1000 * q^59 - 1036 * q^61 + 1024 * q^64 - 824 * q^71 - 2128 * q^74 + 1600 * q^76 - 1200 * q^79 + 3536 * q^86 - 300 * q^89 - 456 * q^91 - 4112 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.00000i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	− 4.00000i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$	0	0
$3$	0	0
$4$	−8.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 6.00000i	− 0.323970i	−0.986793	−	0.161985i	$-0.948210\pi$
$7$	− 6.00000i	− 0.323970i	0.986793	−	0.161985i	$-0.0517895\pi$
$8$	0	0
$9$	0	0
$10$	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	$-0.132853\pi$
$14$	−24.0000	−0.458162
$15$	0	0
$16$	−64.0000	−1.00000
$17$	26.0000i	0.370937i	0.982650	+	0.185468i	$0.0593802\pi$
$17$	26.0000i	0.370937i	−0.982650	+	0.185468i	$0.940620\pi$
$18$	0	0
$19$	−100.000	−1.20745	−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000	−1.20745	−0.603726	+	0.797192i	$0.706318\pi$
$20$	0	0
$21$	0	0
$22$	128.000i	1.24044i
$23$	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
$26$	−152.000	−1.14653
$27$	0	0
$28$	48.0000i	0.323970i
$29$	−50.0000	−0.320164	−0.160082	−	0.987104i	$-0.551176\pi$
$29$	−50.0000	−0.320164	−0.160082	+	0.987104i	$0.551176\pi$
$30$	0	0
$31$	−108.000	−0.625722	−0.312861	−	0.949799i	$-0.601287\pi$
$31$	−108.000	−0.625722	−0.312861	+	0.949799i	$0.601287\pi$
$32$	256.000i	1.41421i
$33$	0	0
$34$	104.000	0.524584
$35$	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	$-0.201246\pi$
$38$	400.000i	1.70759i
$39$	0	0
$40$	0	0
$41$	−22.0000	−0.0838006	−0.0419003	−	0.999122i	$-0.513341\pi$
$41$	−22.0000	−0.0838006	−0.0419003	+	0.999122i	$0.513341\pi$
$42$	0	0
$43$	442.000i	1.56754i	0.621049	+	0.783772i	$0.286707\pi$
$43$	442.000i	1.56754i	−0.621049	+	0.783772i	$0.713293\pi$
$44$	256.000	0.877124
$45$	0	0
$46$	312.000	1.00004
$47$	− 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
$51$	0	0
$52$	304.000i	0.810716i
$53$	− 2.00000i	− 0.00518342i	−0.999997	−	0.00259171i	$-0.999175\pi$
$53$	− 2.00000i	− 0.00518342i	0.999997	−	0.00259171i	$-0.000824967\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	200.000i	0.452781i
$59$	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	$0.682951\pi$
$62$	432.000i	0.884904i
$63$	0	0
$64$	512.000	1.00000
$65$	0	0
$66$	0	0
$67$	− 126.000i	− 0.229751i	−0.993380	−	0.114876i	$-0.963353\pi$
$67$	− 126.000i	− 0.229751i	0.993380	−	0.114876i	$-0.0366470\pi$
$68$	− 208.000i	− 0.370937i
$69$	0	0
$70$	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	$-0.248537\pi$
$74$	−1064.00	−1.67145
$75$	0	0
$76$	800.000	1.20745
$77$	192.000i	0.284161i
$78$	0	0
$79$	−600.000	−0.854497	−0.427249	−	0.904134i	$-0.640517\pi$
$79$	−600.000	−0.854497	−0.427249	+	0.904134i	$0.640517\pi$
$80$	0	0
$81$	0	0
$82$	88.0000i	0.118512i
$83$	− 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
$86$	1768.00	2.21684
$87$	0	0
$88$	0	0
$89$	−150.000	−0.178651	−0.0893257	−	0.996002i	$-0.528471\pi$
$89$	−150.000	−0.178651	−0.0893257	+	0.996002i	$0.528471\pi$
$90$	0	0
$91$	−228.000	−0.262647
$92$	− 624.000i	− 0.707136i
$93$	0	0
$94$	−2056.00	−2.25596
$95$	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	$-0.0647514\pi$
$98$	− 1228.00i	− 1.26578i
$99$	0	0
$100$	0	0
$101$	−702.000	−0.691600	−0.345800	−	0.938308i	$-0.612392\pi$
$101$	−702.000	−0.691600	−0.345800	+	0.938308i	$0.612392\pi$
$102$	0	0
$103$	− 598.000i	− 0.572065i	−0.958220	−	0.286032i	$-0.907663\pi$
$103$	− 598.000i	− 0.572065i	0.958220	−	0.286032i	$-0.0923365\pi$
$104$	0	0
$105$	0	0
$106$	−8.00000	−0.00733046
$107$	− 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
$111$	0	0
$112$	384.000i	0.323970i
$113$	− 1562.00i	− 1.30036i	−0.759781	−	0.650180i	$-0.774694\pi$
$113$	− 1562.00i	− 1.30036i	0.759781	−	0.650180i	$-0.225306\pi$
$114$	0	0
$115$	0	0
$116$	400.000	0.320164
$117$	0	0
$118$	− 2000.00i	− 1.56030i
$119$	156.000	0.120172
$120$	0	0
$121$	−307.000	−0.230654
$122$	2072.00i	1.53762i
$123$	0	0
$124$	864.000	0.625722
$125$	0	0
$126$	0	0
$127$	− 1846.00i	− 1.28981i	−0.764262	−	0.644906i	$-0.776897\pi$
$127$	− 1846.00i	− 1.28981i	0.764262	−	0.644906i	$-0.223103\pi$
$128$	0	0
$129$	0	0
$130$	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
$134$	−504.000	−0.324918
$135$	0	0
$136$	0	0
$137$	− 2334.00i	− 1.45553i	−0.685829	−	0.727763i	$-0.740560\pi$
$137$	− 2334.00i	− 1.45553i	0.685829	−	0.727763i	$-0.259440\pi$
$138$	0	0
$139$	700.000	0.427146	0.213573	−	0.976927i	$-0.431490\pi$
$139$	700.000	0.427146	0.213573	+	0.976927i	$0.431490\pi$
$140$	0	0
$141$	0	0
$142$	1648.00i	0.973923i
$143$	1216.00i	0.711098i
$144$	0	0
$145$	0	0
$146$	−3512.00	−1.99079
$147$	0	0
$148$	2128.00i	1.18190i
$149$	2050.00	1.12713	0.563566	−	0.826071i	$-0.309429\pi$
$149$	2050.00	1.12713	0.563566	+	0.826071i	$0.309429\pi$
$150$	0	0
$151$	1852.00	0.998103	0.499052	−	0.866572i	$-0.333682\pi$
$151$	1852.00	0.998103	0.499052	+	0.866572i	$0.333682\pi$
$152$	0	0
$153$	0	0
$154$	768.000	0.401865
$155$	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	$0.781455\pi$
$158$	2400.00i	1.20844i
$159$	0	0
$160$	0	0
$161$	468.000	0.229090
$162$	0	0
$163$	2762.00i	1.32722i	0.748080	+	0.663609i	$0.230976\pi$
$163$	2762.00i	1.32722i	−0.748080	+	0.663609i	$0.769024\pi$
$164$	176.000	0.0838006
$165$	0	0
$166$	−1128.00	−0.527408
$167$	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
$171$	0	0
$172$	− 3536.00i	− 1.56754i
$173$	78.0000i	0.0342788i	0.999853	+	0.0171394i	$0.00545591\pi$
$173$	78.0000i	0.0342788i	−0.999853	+	0.0171394i	$0.994544\pi$
$174$	0	0
$175$	0	0
$176$	2048.00	0.877124
$177$	0	0
$178$	600.000i	0.252651i
$179$	−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	$0.383566\pi$
$182$	912.000i	0.371439i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 832.000i	− 0.325358i
$188$	4112.00i	1.59520i
$189$	0	0
$190$	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	$-0.0212662\pi$
$194$	−1544.00	−0.571406
$195$	0	0
$196$	−2456.00	−0.895044
$197$	− 2214.00i	− 0.800716i	−0.916359	−	0.400358i	$-0.868886\pi$
$197$	− 2214.00i	− 0.800716i	0.916359	−	0.400358i	$-0.131114\pi$
$198$	0	0
$199$	2600.00	0.926176	0.463088	−	0.886312i	$-0.346741\pi$
$199$	2600.00	0.926176	0.463088	+	0.886312i	$0.346741\pi$
$200$	0	0
$201$	0	0
$202$	2808.00i	0.978070i
$203$	300.000i	0.103724i
$204$	0	0
$205$	0	0
$206$	−2392.00	−0.809022
$207$	0	0
$208$	2432.00i	0.810716i
$209$	3200.00	1.05908
$210$	0	0
$211$	−1168.00	−0.381083	−0.190541	−	0.981679i	$-0.561024\pi$
$211$	−1168.00	−0.381083	−0.190541	+	0.981679i	$0.561024\pi$
$212$	16.0000i	0.00518342i
$213$	0	0
$214$	−4776.00	−1.52561
$215$	0	0
$216$	0	0
$217$	648.000i	0.202715i
$218$	− 2200.00i	− 0.683499i
$219$	0	0
$220$	0	0
$221$	988.000	0.300724
$222$	0	0
$223$	− 6478.00i	− 1.94529i	−0.232303	−	0.972643i	$-0.574626\pi$
$223$	− 6478.00i	− 1.94529i	0.232303	−	0.972643i	$-0.425374\pi$
$224$	1536.00	0.458162
$225$	0	0
$226$	−6248.00	−1.83899
$227$	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
$231$	0	0
$232$	0	0
$233$	− 1482.00i	− 0.416691i	−0.978055	−	0.208346i	$-0.933192\pi$
$233$	− 1482.00i	− 0.416691i	0.978055	−	0.208346i	$-0.0668079\pi$
$234$	0	0
$235$	0	0
$236$	−4000.00	−1.10330
$237$	0	0
$238$	− 624.000i	− 0.169949i
$239$	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	$0.367666\pi$
$242$	1228.00i	0.326194i
$243$	0	0
$244$	4144.00	1.08726
$245$	0	0
$246$	0	0
$247$	3800.00i	0.978900i
$248$	0	0
$249$	0	0
$250$	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
$254$	−7384.00	−1.82407
$255$	0	0
$256$	4096.00	1.00000
$257$	2106.00i	0.511162i	0.966788	+	0.255581i	$0.0822668\pi$
$257$	2106.00i	0.511162i	−0.966788	+	0.255581i	$0.917733\pi$
$258$	0	0
$259$	−1596.00	−0.382898
$260$	0	0
$261$	0	0
$262$	− 8832.00i	− 2.08261i
$263$	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
$266$	2400.00	0.553208
$267$	0	0
$268$	1008.00i	0.229751i
$269$	−6550.00	−1.48461	−0.742306	−	0.670061i	$-0.766268\pi$
$269$	−6550.00	−1.48461	−0.742306	+	0.670061i	$0.766268\pi$
$270$	0	0
$271$	−4388.00	−0.983587	−0.491793	−	0.870712i	$-0.663658\pi$
$271$	−4388.00	−0.983587	−0.491793	+	0.870712i	$0.663658\pi$
$272$	− 1664.00i	− 0.370937i
$273$	0	0
$274$	−9336.00	−2.05842
$275$	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	$-0.0188602\pi$
$278$	− 2800.00i	− 0.604075i
$279$	0	0
$280$	0	0
$281$	6858.00	1.45592	0.727961	−	0.685619i	$-0.240468\pi$
$281$	6858.00	1.45592	0.727961	+	0.685619i	$0.240468\pi$
$282$	0	0
$283$	9282.00i	1.94967i	0.222920	+	0.974837i	$0.428441\pi$
$283$	9282.00i	1.94967i	−0.222920	+	0.974837i	$0.571559\pi$
$284$	3296.00	0.688668
$285$	0	0
$286$	4864.00	1.00564
$287$	132.000i	0.0271488i
$288$	0	0
$289$	4237.00	0.862406
$290$	0	0
$291$	0	0
$292$	7024.00i	1.40770i
$293$	− 4842.00i	− 0.965436i	−0.875776	−	0.482718i	$-0.839650\pi$
$293$	− 4842.00i	− 0.965436i	0.875776	−	0.482718i	$-0.160350\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	− 8200.00i	− 1.59400i
$299$	2964.00	0.573286
$300$	0	0
$301$	2652.00	0.507836
$302$	− 7408.00i	− 1.41153i
$303$	0	0
$304$	6400.00	1.20745
$305$	0	0
$306$	0	0
$307$	2594.00i	0.482239i	0.970495	+	0.241120i	$0.0775146\pi$
$307$	2594.00i	0.482239i	−0.970495	+	0.241120i	$0.922485\pi$
$308$	− 1536.00i	− 0.284161i
$309$	0	0
$310$	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	$0.954956\pi$
$314$	9976.00	1.79292
$315$	0	0
$316$	4800.00	0.854497
$317$	1426.00i	0.252657i	0.991988	+	0.126328i	$0.0403193\pi$
$317$	1426.00i	0.252657i	−0.991988	+	0.126328i	$0.959681\pi$
$318$	0	0
$319$	1600.00	0.280824
$320$	0	0
$321$	0	0
$322$	− 1872.00i	− 0.323983i
$323$	− 2600.00i	− 0.447888i
$324$	0	0
$325$	0	0
$326$	11048.0	1.87697
$327$	0	0
$328$	0	0
$329$	−3084.00	−0.516798
$330$	0	0
$331$	−4008.00	−0.665558	−0.332779	−	0.943005i	$-0.607986\pi$
$331$	−4008.00	−0.665558	−0.332779	+	0.943005i	$0.607986\pi$
$332$	2256.00i	0.372934i
$333$	0	0
$334$	12504.0	2.04847
$335$	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	$-0.254285\pi$
$338$	− 3012.00i	− 0.484708i
$339$	0	0
$340$	0	0
$341$	3456.00	0.548835
$342$	0	0
$343$	− 3900.00i	− 0.613936i
$344$	0	0
$345$	0	0
$346$	312.000	0.0484775
$347$	− 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
$351$	0	0
$352$	− 8192.00i	− 1.24044i
$353$	4398.00i	0.663122i	0.943434	+	0.331561i	$0.107575\pi$
$353$	4398.00i	0.663122i	−0.943434	+	0.331561i	$0.892425\pi$
$354$	0	0
$355$	0	0
$356$	1200.00	0.178651
$357$	0	0
$358$	5200.00i	0.767677i
$359$	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
$362$	− 6968.00i	− 1.01168i
$363$	0	0
$364$	1824.00	0.262647
$365$	0	0
$366$	0	0
$367$	5874.00i	0.835478i	0.908567	+	0.417739i	$0.137177\pi$
$367$	5874.00i	0.835478i	−0.908567	+	0.417739i	$0.862823\pi$
$368$	− 4992.00i	− 0.707136i
$369$	0	0
$370$	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	$-0.0460702\pi$
$374$	−3328.00	−0.460125
$375$	0	0
$376$	0	0
$377$	1900.00i	0.259562i
$378$	0	0
$379$	−7900.00	−1.07070	−0.535351	−	0.844630i	$-0.679821\pi$
$379$	−7900.00	−1.07070	−0.535351	+	0.844630i	$0.679821\pi$
$380$	0	0
$381$	0	0
$382$	15088.0i	2.02086i
$383$	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
$386$	−1432.00	−0.188826
$387$	0	0
$388$	3088.00i	0.404045i
$389$	−1950.00	−0.254162	−0.127081	−	0.991892i	$-0.540561\pi$
$389$	−1950.00	−0.254162	−0.127081	+	0.991892i	$0.540561\pi$
$390$	0	0
$391$	−2028.00	−0.262303
$392$	0	0
$393$	0	0
$394$	−8856.00	−1.13238
$395$	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	$-0.336794\pi$
$398$	− 10400.0i	− 1.30981i
$399$	0	0
$400$	0	0
$401$	−6402.00	−0.797258	−0.398629	−	0.917112i	$-0.630514\pi$
$401$	−6402.00	−0.797258	−0.398629	+	0.917112i	$0.630514\pi$
$402$	0	0
$403$	4104.00i	0.507282i
$404$	5616.00	0.691600
$405$	0	0
$406$	1200.00	0.146687
$407$	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
$411$	0	0
$412$	4784.00i	0.572065i
$413$	− 3000.00i	− 0.357434i
$414$	0	0
$415$	0	0
$416$	9728.00	1.14653
$417$	0	0
$418$	− 12800.0i	− 1.49777i
$419$	−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	$0.601926\pi$
$422$	4672.00i	0.538932i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3108.00i	0.352240i
$428$	9552.00i	1.07877i
$429$	0	0
$430$	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	$-0.0197610\pi$
$434$	2592.00	0.286682
$435$	0	0
$436$	−4400.00	−0.483307
$437$	− 7800.00i	− 0.853832i
$438$	0	0
$439$	2600.00	0.282668	0.141334	−	0.989962i	$-0.454861\pi$
$439$	2600.00	0.282668	0.141334	+	0.989962i	$0.454861\pi$
$440$	0	0
$441$	0	0
$442$	− 3952.00i	− 0.425288i
$443$	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
$446$	−25912.0	−2.75105
$447$	0	0
$448$	− 3072.00i	− 0.323970i
$449$	−17050.0	−1.79207	−0.896035	−	0.443984i	$-0.853565\pi$
$449$	−17050.0	−1.79207	−0.896035	+	0.443984i	$0.853565\pi$
$450$	0	0
$451$	704.000	0.0735035
$452$	12496.0i	1.30036i
$453$	0	0
$454$	2584.00	0.267121
$455$	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	$0.838493\pi$
$458$	15000.0i	1.53036i
$459$	0	0
$460$	0	0
$461$	11418.0	1.15356	0.576778	−	0.816901i	$-0.304310\pi$
$461$	11418.0	1.15356	0.576778	+	0.816901i	$0.304310\pi$
$462$	0	0
$463$	7962.00i	0.799191i	0.916692	+	0.399596i	$0.130849\pi$
$463$	7962.00i	0.799191i	−0.916692	+	0.399596i	$0.869151\pi$
$464$	3200.00	0.320164
$465$	0	0
$466$	−5928.00	−0.589290
$467$	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
$471$	0	0
$472$	0	0
$473$	− 14144.0i	− 1.37493i
$474$	0	0
$475$	0	0
$476$	−1248.00	−0.120172
$477$	0	0
$478$	− 5600.00i	− 0.535854i
$479$	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
$482$	− 12088.0i	− 1.14231i
$483$	0	0
$484$	2456.00	0.230654
$485$	0	0
$486$	0	0
$487$	− 1166.00i	− 0.108494i	−0.998528	−	0.0542469i	$-0.982724\pi$
$487$	− 1166.00i	− 0.108494i	0.998528	−	0.0542469i	$-0.0172758\pi$
$488$	0	0
$489$	0	0
$490$	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
$494$	15200.0	1.38437
$495$	0	0
$496$	6912.00	0.625722
$497$	2472.00i	0.223107i
$498$	0	0
$499$	−100.000	−0.00897117	−0.00448559	−	0.999990i	$-0.501428\pi$
$499$	−100.000	−0.00897117	−0.00448559	+	0.999990i	$0.501428\pi$
$500$	0	0
$501$	0	0
$502$	− 4992.00i	− 0.443832i
$503$	− 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
$506$	−9984.00	−0.877160
$507$	0	0
$508$	14768.0i	1.28981i
$509$	11150.0	0.970953	0.485476	−	0.874250i	$-0.338646\pi$
$509$	11150.0	0.970953	0.485476	+	0.874250i	$0.338646\pi$
$510$	0	0
$511$	−5268.00	−0.456052
$512$	− 16384.0i	− 1.41421i
$513$	0	0
$514$	8424.00	0.722892
$515$	0	0
$516$	0	0
$517$	16448.0i	1.39919i
$518$	6384.00i	0.541500i
$519$	0	0
$520$	0	0
$521$	3638.00	0.305919	0.152959	−	0.988232i	$-0.451120\pi$
$521$	3638.00	0.305919	0.152959	+	0.988232i	$0.451120\pi$
$522$	0	0
$523$	− 2078.00i	− 0.173737i	−0.996220	−	0.0868686i	$-0.972314\pi$
$523$	− 2078.00i	− 0.173737i	0.996220	−	0.0868686i	$-0.0276860\pi$
$524$	−17664.0	−1.47262
$525$	0	0
$526$	14552.0	1.20627
$527$	− 2808.00i	− 0.232103i
$528$	0	0
$529$	6083.00	0.499959
$530$	0	0
$531$	0	0
$532$	− 4800.00i	− 0.391177i
$533$	836.000i	0.0679384i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	26200.0i	2.09956i
$539$	−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	$0.428287\pi$
$542$	17552.0i	1.39100i
$543$	0	0
$544$	−6656.00	−0.524584
$545$	0	0
$546$	0	0
$547$	− 16486.0i	− 1.28865i	−0.764753	−	0.644324i	$-0.777139\pi$
$547$	− 16486.0i	− 1.28865i	0.764753	−	0.644324i	$-0.222861\pi$
$548$	18672.0i	1.45553i
$549$	0	0
$550$	0	0
$551$	5000.00	0.386583
$552$	0	0
$553$	3600.00i	0.276831i
$554$	−2184.00	−0.167490
$555$	0	0
$556$	−5600.00	−0.427146
$557$	11706.0i	0.890483i	0.895410	+	0.445242i	$0.146882\pi$
$557$	11706.0i	0.890483i	−0.895410	+	0.445242i	$0.853118\pi$
$558$	0	0
$559$	16796.0	1.27083
$560$	0	0
$561$	0	0
$562$	− 27432.0i	− 2.05898i
$563$	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
$566$	37128.0	2.75725
$567$	0	0
$568$	0	0
$569$	17550.0	1.29303	0.646515	−	0.762901i	$-0.276226\pi$
$569$	17550.0	1.29303	0.646515	+	0.762901i	$0.276226\pi$
$570$	0	0
$571$	10712.0	0.785084	0.392542	−	0.919734i	$-0.371596\pi$
$571$	10712.0	0.785084	0.392542	+	0.919734i	$0.371596\pi$
$572$	− 9728.00i	− 0.711098i
$573$	0	0
$574$	528.000	0.0383942
$575$	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	$0.836058\pi$
$578$	− 16948.0i	− 1.21963i
$579$	0	0
$580$	0	0
$581$	−1692.00	−0.120819
$582$	0	0
$583$	64.0000i	0.00454650i
$584$	0	0
$585$	0	0
$586$	−19368.0	−1.36533
$587$	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
$591$	0	0
$592$	17024.0i	1.18190i
$593$	− 17842.0i	− 1.23555i	−0.786354	−	0.617777i	$-0.788034\pi$
$593$	− 17842.0i	− 1.23555i	0.786354	−	0.617777i	$-0.211966\pi$
$594$	0	0
$595$	0	0
$596$	−16400.0	−1.12713
$597$	0	0
$598$	− 11856.0i	− 0.810749i
$599$	−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	$0.122789\pi$
$602$	− 10608.0i	− 0.718189i
$603$	0	0
$604$	−14816.0	−0.998103
$605$	0	0
$606$	0	0
$607$	3794.00i	0.253696i	0.991922	+	0.126848i	$0.0404861\pi$
$607$	3794.00i	0.253696i	−0.991922	+	0.126848i	$0.959514\pi$
$608$	− 25600.0i	− 1.70759i
$609$	0	0
$610$	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	$-0.143646\pi$
$614$	10376.0	0.681989
$615$	0	0
$616$	0	0
$617$	− 11574.0i	− 0.755189i	−0.925971	−	0.377595i	$-0.876751\pi$
$617$	− 11574.0i	− 0.755189i	0.925971	−	0.377595i	$-0.123249\pi$
$618$	0	0
$619$	−8300.00	−0.538942	−0.269471	−	0.963008i	$-0.586849\pi$
$619$	−8300.00	−0.538942	−0.269471	+	0.963008i	$0.586849\pi$
$620$	0	0
$621$	0	0
$622$	29328.0i	1.89059i
$623$	900.000i	0.0578776i
$624$	0	0
$625$	0	0
$626$	6248.00	0.398914
$627$	0	0
$628$	− 19952.0i	− 1.26779i
$629$	6916.00	0.438409
$630$	0	0
$631$	−7508.00	−0.473675	−0.236837	−	0.971549i	$-0.576111\pi$
$631$	−7508.00	−0.473675	−0.236837	+	0.971549i	$0.576111\pi$
$632$	0	0
$633$	0	0
$634$	5704.00	0.357310
$635$	0	0
$636$	0	0
$637$	− 11666.0i	− 0.725626i
$638$	− 6400.00i	− 0.397145i
$639$	0	0
$640$	0	0
$641$	27378.0	1.68700	0.843499	−	0.537130i	$-0.180492\pi$
$641$	27378.0	1.68700	0.843499	+	0.537130i	$0.180492\pi$
$642$	0	0
$643$	1842.00i	0.112973i	0.998403	+	0.0564863i	$0.0179897\pi$
$643$	1842.00i	0.112973i	−0.998403	+	0.0564863i	$0.982010\pi$
$644$	−3744.00	−0.229090
$645$	0	0
$646$	−10400.0	−0.633409
$647$	− 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
$651$	0	0
$652$	− 22096.0i	− 1.32722i
$653$	− 10402.0i	− 0.623372i	−0.950185	−	0.311686i	$-0.899106\pi$
$653$	− 10402.0i	− 0.623372i	0.950185	−	0.311686i	$-0.100894\pi$
$654$	0	0
$655$	0	0
$656$	1408.00	0.0838006
$657$	0	0
$658$	12336.0i	0.730862i
$659$	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	$0.567159\pi$
$662$	16032.0i	0.941241i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 3900.00i	− 0.226400i
$668$	− 25008.0i	− 1.44849i
$669$	0	0
$670$	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	$-0.353360\pi$
$674$	−35464.0	−2.02674
$675$	0	0
$676$	−6024.00	−0.342740
$677$	− 30054.0i	− 1.70616i	−0.521782	−	0.853079i	$-0.674732\pi$
$677$	− 30054.0i	− 1.70616i	0.521782	−	0.853079i	$-0.325268\pi$
$678$	0	0
$679$	−2316.00	−0.130898
$680$	0	0
$681$	0	0
$682$	− 13824.0i	− 0.776171i
$683$	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
$686$	−15600.0	−0.868237
$687$	0	0
$688$	− 28288.0i	− 1.56754i
$689$	−76.0000	−0.00420228
$690$	0	0
$691$	29272.0	1.61152	0.805759	−	0.592243i	$-0.201758\pi$
$691$	29272.0	1.61152	0.805759	+	0.592243i	$0.201758\pi$
$692$	− 624.000i	− 0.0342788i
$693$	0	0
$694$	−6856.00	−0.375000
$695$	0	0
$696$	0	0
$697$	− 572.000i	− 0.0310847i
$698$	4600.00i	0.249445i
$699$	0	0
$700$	0	0
$701$	5798.00	0.312393	0.156196	−	0.987726i	$-0.450077\pi$
$701$	5798.00	0.312393	0.156196	+	0.987726i	$0.450077\pi$
$702$	0	0
$703$	26600.0i	1.42708i
$704$	−16384.0	−0.877124
$705$	0	0
$706$	17592.0	0.937796
$707$	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
$711$	0	0
$712$	0	0
$713$	− 8424.00i	− 0.442470i
$714$	0	0
$715$	0	0
$716$	10400.0	0.542830
$717$	0	0
$718$	− 7200.00i	− 0.374236i
$719$	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
$722$	− 12564.0i	− 0.647623i
$723$	0	0
$724$	−13936.0	−0.715369
$725$	0	0
$726$	0	0
$727$	8554.00i	0.436383i	0.975906	+	0.218191i	$0.0700157\pi$
$727$	8554.00i	0.436383i	−0.975906	+	0.218191i	$0.929984\pi$
$728$	0	0
$729$	0	0
$730$	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	$0.976867\pi$
$734$	23496.0	1.18154
$735$	0	0
$736$	−19968.0	−1.00004
$737$	4032.00i	0.201521i
$738$	0	0
$739$	−18700.0	−0.930840	−0.465420	−	0.885090i	$-0.654097\pi$
$739$	−18700.0	−0.930840	−0.465420	+	0.885090i	$0.654097\pi$
$740$	0	0
$741$	0	0
$742$	48.0000i	0.00237485i
$743$	− 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
$746$	−8312.00	−0.407941
$747$	0	0
$748$	6656.00i	0.325358i
$749$	−7164.00	−0.349488
$750$	0	0
$751$	−31148.0	−1.51346	−0.756729	−	0.653729i	$-0.773204\pi$
$751$	−31148.0	−1.51346	−0.756729	+	0.653729i	$0.773204\pi$
$752$	32896.0i	1.59520i
$753$	0	0
$754$	7600.00	0.367076
$755$	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	$0.940867\pi$
$758$	31600.0i	1.51420i
$759$	0	0
$760$	0	0
$761$	4518.00	0.215213	0.107607	−	0.994194i	$-0.465681\pi$
$761$	4518.00	0.215213	0.107607	+	0.994194i	$0.465681\pi$
$762$	0	0
$763$	− 3300.00i	− 0.156577i
$764$	30176.0	1.42897
$765$	0	0
$766$	30072.0	1.41847
$767$	− 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
$771$	0	0
$772$	2864.00i	0.133520i
$773$	− 22122.0i	− 1.02933i	−0.857391	−	0.514666i	$-0.827916\pi$
$773$	− 22122.0i	− 1.02933i	0.857391	−	0.514666i	$-0.172084\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	7800.00i	0.359439i
$779$	2200.00	0.101185
$780$	0	0
$781$	13184.0	0.604047
$782$	8112.00i	0.370952i
$783$	0	0
$784$	−19648.0	−0.895044
$785$	0	0
$786$	0	0
$787$	16634.0i	0.753416i	0.926332	+	0.376708i	$0.122944\pi$
$787$	16634.0i	0.753416i	−0.926332	+	0.376708i	$0.877056\pi$
$788$	17712.0i	0.800716i
$789$	0	0
$790$	0	0
$791$	−9372.00	−0.421277
$792$	0	0
$793$	19684.0i	0.881462i
$794$	−55144.0	−2.46472
$795$	0	0
$796$	−20800.0	−0.926176
$797$	27586.0i	1.22603i	0.790071	+	0.613015i	$0.210044\pi$
$797$	27586.0i	1.22603i	−0.790071	+	0.613015i	$0.789956\pi$
$798$	0	0
$799$	13364.0	0.591720
$800$	0	0
$801$	0	0
$802$	25608.0i	1.12749i
$803$	28096.0i	1.23473i
$804$	0	0
$805$	0	0
$806$	16416.0	0.717406
$807$	0	0
$808$	0	0
$809$	3850.00	0.167316	0.0836581	−	0.996495i	$-0.473340\pi$
$809$	3850.00	0.167316	0.0836581	+	0.996495i	$0.473340\pi$
$810$	0	0
$811$	10032.0	0.434366	0.217183	−	0.976131i	$-0.430313\pi$
$811$	10032.0	0.434366	0.217183	+	0.976131i	$0.430313\pi$
$812$	− 2400.00i	− 0.103724i
$813$	0	0
$814$	34048.0	1.46607
$815$	0	0
$816$	0	0
$817$	− 44200.0i	− 1.89273i
$818$	44600.0i	1.90636i
$819$	0	0
$820$	0	0
$821$	−20562.0	−0.874079	−0.437039	−	0.899442i	$-0.643973\pi$
$821$	−20562.0	−0.874079	−0.437039	+	0.899442i	$0.643973\pi$
$822$	0	0
$823$	10322.0i	0.437184i	0.975816	+	0.218592i	$0.0701464\pi$
$823$	10322.0i	0.437184i	−0.975816	+	0.218592i	$0.929854\pi$
$824$	0	0
$825$	0	0
$826$	−12000.0	−0.505488
$827$	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
$831$	0	0
$832$	− 19456.0i	− 0.810716i
$833$	7982.00i	0.332005i
$834$	0	0
$835$	0	0
$836$	−25600.0	−1.05908
$837$	0	0
$838$	54800.0i	2.25899i
$839$	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
$842$	21752.0i	0.890289i
$843$	0	0
$844$	9344.00	0.381083
$845$	0	0
$846$	0	0
$847$	1842.00i	0.0747248i
$848$	128.000i	0.00518342i
$849$	0	0
$850$	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	$-0.110816\pi$
$854$	12432.0	0.498143
$855$	0	0
$856$	0	0
$857$	− 26494.0i	− 1.05603i	−0.849235	−	0.528015i	$-0.822936\pi$
$857$	− 26494.0i	− 1.05603i	0.849235	−	0.528015i	$-0.177064\pi$
$858$	0	0
$859$	21500.0	0.853982	0.426991	−	0.904256i	$-0.359574\pi$
$859$	21500.0	0.853982	0.426991	+	0.904256i	$0.359574\pi$
$860$	0	0
$861$	0	0
$862$	30768.0i	1.21573i
$863$	− 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
$866$	−4472.00	−0.175479
$867$	0	0
$868$	− 5184.00i	− 0.202715i
$869$	19200.0	0.749500
$870$	0	0
$871$	−4788.00	−0.186263
$872$	0	0
$873$	0	0
$874$	−31200.0	−1.20750
$875$	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	$-0.200110\pi$
$878$	− 10400.0i	− 0.399753i
$879$	0	0
$880$	0	0
$881$	−32942.0	−1.25976	−0.629878	−	0.776694i	$-0.716895\pi$
$881$	−32942.0	−1.25976	−0.629878	+	0.776694i	$0.716895\pi$
$882$	0	0
$883$	− 27118.0i	− 1.03351i	−0.856132	−	0.516757i	$-0.827139\pi$
$883$	− 27118.0i	− 1.03351i	0.856132	−	0.516757i	$-0.172861\pi$
$884$	−7904.00	−0.300724
$885$	0	0
$886$	47832.0	1.81371
$887$	− 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
$891$	0	0
$892$	51824.0i	1.94529i
$893$	51400.0i	1.92613i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	68200.0i	2.53437i
$899$	5400.00	0.200334
$900$	0	0
$901$	52.0000	0.00192272
$902$	− 2816.00i	− 0.103950i
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1794.00i	0.0656767i	0.999461	+	0.0328384i	$0.0104547\pi$
$907$	1794.00i	0.0656767i	−0.999461	+	0.0328384i	$0.989545\pi$
$908$	− 5168.00i	− 0.188883i
$909$	0	0
$910$	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
$914$	37976.0	1.37433
$915$	0	0
$916$	30000.0	1.08213
$917$	− 13248.0i	− 0.477086i
$918$	0	0
$919$	−29200.0	−1.04812	−0.524058	−	0.851682i	$-0.675583\pi$
$919$	−29200.0	−1.04812	−0.524058	+	0.851682i	$0.675583\pi$
$920$	0	0
$921$	0	0
$922$	− 45672.0i	− 1.63137i
$923$	15656.0i	0.558314i
$924$	0	0
$925$	0	0
$926$	31848.0	1.13023
$927$	0	0
$928$	− 12800.0i	− 0.452781i
$929$	−48650.0	−1.71814	−0.859071	−	0.511856i	$-0.828958\pi$
$929$	−48650.0	−1.71814	−0.859071	+	0.511856i	$0.828958\pi$
$930$	0	0
$931$	−30700.0	−1.08072
$932$	11856.0i	0.416691i
$933$	0	0
$934$	26104.0	0.914506
$935$	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	$0.936692\pi$
$938$	3024.00i	0.105263i
$939$	0	0
$940$	0	0
$941$	31178.0	1.08010	0.540050	−	0.841633i	$-0.318405\pi$
$941$	31178.0	1.08010	0.540050	+	0.841633i	$0.318405\pi$
$942$	0	0
$943$	− 1716.00i	− 0.0592584i
$944$	−32000.0	−1.10330
$945$	0	0
$946$	−56576.0	−1.94444
$947$	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
$951$	0	0
$952$	0	0
$953$	598.000i	0.0203265i	0.999948	+	0.0101632i	$0.00323511\pi$
$953$	598.000i	0.0203265i	−0.999948	+	0.0101632i	$0.996765\pi$
$954$	0	0
$955$	0	0
$956$	−11200.0	−0.378906
$957$	0	0
$958$	− 69600.0i	− 2.34726i
$959$	−14004.0	−0.471546
$960$	0	0
$961$	−18127.0	−0.608472
$962$	40432.0i	1.35507i
$963$	0	0
$964$	−24176.0	−0.807735
$965$	0	0
$966$	0	0
$967$	− 41726.0i	− 1.38761i	−0.720163	−	0.693804i	$-0.755933\pi$
$967$	− 41726.0i	− 1.38761i	0.720163	−	0.693804i	$-0.244067\pi$
$968$	0	0
$969$	0	0
$970$	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
$974$	−4664.00	−0.153433
$975$	0	0
$976$	33152.0	1.08726
$977$	40946.0i	1.34082i	0.741992	+	0.670409i	$0.233881\pi$
$977$	40946.0i	1.34082i	−0.741992	+	0.670409i	$0.766119\pi$
$978$	0	0
$979$	4800.00	0.156699
$980$	0	0
$981$	0	0
$982$	28288.0i	0.919253i
$983$	− 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
$986$	−5200.00	−0.167953
$987$	0	0
$988$	− 30400.0i	− 0.978900i
$989$	−34476.0	−1.10847
$990$	0	0
$991$	1172.00	0.0375679	0.0187840	−	0.999824i	$-0.494021\pi$
$991$	1172.00	0.0375679	0.0187840	+	0.999824i	$0.494021\pi$
$992$	− 27648.0i	− 0.884904i
$993$	0	0
$994$	9888.00	0.315521
$995$	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	$0.832554\pi$
$998$	400.000i	0.0126872i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.b.c.199.1		2
3.2	odd	2		25.4.b.a.24.2		2
5.2	odd	4		45.4.a.d.1.1		1
5.3	odd	4		225.4.a.b.1.1		1
5.4	even	2	inner	225.4.b.c.199.2		2
12.11	even	2		400.4.c.k.49.1		2
15.2	even	4		5.4.a.a.1.1	✓	1
15.8	even	4		25.4.a.c.1.1		1
15.14	odd	2		25.4.b.a.24.1		2
20.7	even	4		720.4.a.u.1.1		1
35.27	even	4		2205.4.a.q.1.1		1
45.2	even	12		405.4.e.l.271.1		2
45.7	odd	12		405.4.e.c.271.1		2
45.22	odd	12		405.4.e.c.136.1		2
45.32	even	12		405.4.e.l.136.1		2
60.23	odd	4		400.4.a.m.1.1		1
60.47	odd	4		80.4.a.d.1.1		1
60.59	even	2		400.4.c.k.49.2		2
105.2	even	12		245.4.e.f.116.1		2
105.17	odd	12		245.4.e.g.226.1		2
105.32	even	12		245.4.e.f.226.1		2
105.47	odd	12		245.4.e.g.116.1		2
105.62	odd	4		245.4.a.a.1.1		1
105.83	odd	4		1225.4.a.k.1.1		1
120.53	even	4		1600.4.a.bi.1.1		1
120.77	even	4		320.4.a.g.1.1		1
120.83	odd	4		1600.4.a.s.1.1		1
120.107	odd	4		320.4.a.h.1.1		1
165.32	odd	4		605.4.a.d.1.1		1
195.77	even	4		845.4.a.b.1.1		1
240.77	even	4		1280.4.d.e.641.2		2
240.107	odd	4		1280.4.d.l.641.2		2
240.197	even	4		1280.4.d.e.641.1		2
240.227	odd	4		1280.4.d.l.641.1		2
255.152	even	4		1445.4.a.a.1.1		1
285.227	odd	4		1805.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	15.2	even	4
25.4.a.c.1.1		1	15.8	even	4
25.4.b.a.24.1		2	15.14	odd	2
25.4.b.a.24.2		2	3.2	odd	2
45.4.a.d.1.1		1	5.2	odd	4
80.4.a.d.1.1		1	60.47	odd	4
225.4.a.b.1.1		1	5.3	odd	4
225.4.b.c.199.1		2	1.1	even	1	trivial
225.4.b.c.199.2		2	5.4	even	2	inner
245.4.a.a.1.1		1	105.62	odd	4
245.4.e.f.116.1		2	105.2	even	12
245.4.e.f.226.1		2	105.32	even	12
245.4.e.g.116.1		2	105.47	odd	12
245.4.e.g.226.1		2	105.17	odd	12
320.4.a.g.1.1		1	120.77	even	4
320.4.a.h.1.1		1	120.107	odd	4
400.4.a.m.1.1		1	60.23	odd	4
400.4.c.k.49.1		2	12.11	even	2
400.4.c.k.49.2		2	60.59	even	2
405.4.e.c.136.1		2	45.22	odd	12
405.4.e.c.271.1		2	45.7	odd	12
405.4.e.l.136.1		2	45.32	even	12
405.4.e.l.271.1		2	45.2	even	12
605.4.a.d.1.1		1	165.32	odd	4
720.4.a.u.1.1		1	20.7	even	4
845.4.a.b.1.1		1	195.77	even	4
1225.4.a.k.1.1		1	105.83	odd	4
1280.4.d.e.641.1		2	240.197	even	4
1280.4.d.e.641.2		2	240.77	even	4
1280.4.d.l.641.1		2	240.227	odd	4
1280.4.d.l.641.2		2	240.107	odd	4
1445.4.a.a.1.1		1	255.152	even	4
1600.4.a.s.1.1		1	120.83	odd	4
1600.4.a.bi.1.1		1	120.53	even	4
1805.4.a.h.1.1		1	285.227	odd	4
2205.4.a.q.1.1		1	35.27	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -8.00000 q^{4} -6.00000i q^{7} +O(q^{10})\)	\(q-4.00000i q^{2} -8.00000 q^{4} -6.00000i q^{7} -32.0000 q^{11} -38.0000i q^{13} -24.0000 q^{14} -64.0000 q^{16} +26.0000i q^{17} -100.000 q^{19} +128.000i q^{22} +78.0000i q^{23} -152.000 q^{26} +48.0000i q^{28} -50.0000 q^{29} -108.000 q^{31} +256.000i q^{32} +104.000 q^{34} -266.000i q^{37} +400.000i q^{38} -22.0000 q^{41} +442.000i q^{43} +256.000 q^{44} +312.000 q^{46} -514.000i q^{47} +307.000 q^{49} +304.000i q^{52} -2.00000i q^{53} +200.000i q^{58} +500.000 q^{59} -518.000 q^{61} +432.000i q^{62} +512.000 q^{64} -126.000i q^{67} -208.000i q^{68} -412.000 q^{71} -878.000i q^{73} -1064.00 q^{74} +800.000 q^{76} +192.000i q^{77} -600.000 q^{79} +88.0000i q^{82} -282.000i q^{83} +1768.00 q^{86} -150.000 q^{89} -228.000 q^{91} -624.000i q^{92} -2056.00 q^{94} -386.000i q^{97} -1228.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{4}+O(q^{10}) \) 2 * q - 16 * q^4	\( 2 q - 16 q^{4} - 64 q^{11} - 48 q^{14} - 128 q^{16} - 200 q^{19} - 304 q^{26} - 100 q^{29} - 216 q^{31} + 208 q^{34} - 44 q^{41} + 512 q^{44} + 624 q^{46} + 614 q^{49} + 1000 q^{59} - 1036 q^{61} + 1024 q^{64} - 824 q^{71} - 2128 q^{74} + 1600 q^{76} - 1200 q^{79} + 3536 q^{86} - 300 q^{89} - 456 q^{91} - 4112 q^{94}+O(q^{100}) \) 2 * q - 16 * q^4 - 64 * q^11 - 48 * q^14 - 128 * q^16 - 200 * q^19 - 304 * q^26 - 100 * q^29 - 216 * q^31 + 208 * q^34 - 44 * q^41 + 512 * q^44 + 624 * q^46 + 614 * q^49 + 1000 * q^59 - 1036 * q^61 + 1024 * q^64 - 824 * q^71 - 2128 * q^74 + 1600 * q^76 - 1200 * q^79 + 3536 * q^86 - 300 * q^89 - 456 * q^91 - 4112 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more