LMFDB - Embedded newform 630.3.f.c.181.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [630,3,Mod(181,630)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(630, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("630.181");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	630.f (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:	$17.1662566547$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.3317760000.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 $ x^8 - 4x^6 + 7x^4 - 36*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			181.1
Root			$-1.01575 + 1.40294i$ of defining polynomial
Character	$\chi$	$=$	630.181
Dual form			630.3.f.c.181.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +(-1.04456 + 6.92163i) q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +(-1.04456 + 6.92163i) q^{7} -2.82843 q^{8} +3.16228i q^{10} -1.00806 q^{11} +7.05322i q^{13} +(1.47723 - 9.78866i) q^{14} +4.00000 q^{16} -7.09185i q^{17} -4.04257i q^{19} -4.47214i q^{20} +1.42561 q^{22} -25.4766 q^{23} -5.00000 q^{25} -9.97476i q^{26} +(-2.08911 + 13.8433i) q^{28} -17.2909 q^{29} -23.6516i q^{31} -5.65685 q^{32} +10.0294i q^{34} +(15.4772 + 2.33570i) q^{35} -29.2171 q^{37} +5.71705i q^{38} +6.32456i q^{40} +5.16217i q^{41} +45.6529 q^{43} -2.01611 q^{44} +36.0294 q^{46} +61.0391i q^{47} +(-46.8178 - 14.4601i) q^{49} +7.07107 q^{50} +14.1064i q^{52} -74.8237 q^{53} +2.25408i q^{55} +(2.95445 - 19.5773i) q^{56} +24.4531 q^{58} +50.5639i q^{59} -58.0943i q^{61} +33.4485i q^{62} +8.00000 q^{64} +15.7715 q^{65} -73.5827 q^{67} -14.1837i q^{68} +(-21.8881 - 3.30318i) q^{70} -99.8513 q^{71} +1.15350i q^{73} +41.3192 q^{74} -8.08514i q^{76} +(1.05297 - 6.97738i) q^{77} -102.445 q^{79} -8.94427i q^{80} -7.30042i q^{82} +83.4227i q^{83} -15.8579 q^{85} -64.5629 q^{86} +2.85121 q^{88} -69.0342i q^{89} +(-48.8198 - 7.36749i) q^{91} -50.9533 q^{92} -86.3223i q^{94} -9.03946 q^{95} +152.172i q^{97} +(66.2104 + 20.4496i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 16 q^{11} - 32 q^{14} + 32 q^{16} + 96 q^{22} - 40 q^{25} + 144 q^{29} + 80 q^{35} - 48 q^{37} - 64 q^{43} + 32 q^{44} + 128 q^{46} - 24 q^{49} - 128 q^{53} - 64 q^{56} + 224 q^{58} + 64 q^{64} - 80 q^{65} - 192 q^{67} - 176 q^{71} + 160 q^{74} - 192 q^{77} - 288 q^{79} - 240 q^{85} + 64 q^{86} + 192 q^{88} + 64 q^{91} - 80 q^{95}+O(q^{100}) $ 8 * q + 16 * q^4 + 16 * q^11 - 32 * q^14 + 32 * q^16 + 96 * q^22 - 40 * q^25 + 144 * q^29 + 80 * q^35 - 48 * q^37 - 64 * q^43 + 32 * q^44 + 128 * q^46 - 24 * q^49 - 128 * q^53 - 64 * q^56 + 224 * q^58 + 64 * q^64 - 80 * q^65 - 192 * q^67 - 176 * q^71 + 160 * q^74 - 192 * q^77 - 288 * q^79 - 240 * q^85 + 64 * q^86 + 192 * q^88 + 64 * q^91 - 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/630\mathbb{Z}\right)^\times$.

$n$	$127$	$281$	$451$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$		−	2.23607i		−	0.447214i
$5$		−	2.23607i		−	0.447214i
$6$		0			0
$7$	−1.04456	+	6.92163i	−0.149222	+	0.988804i
$7$	−1.04456	+	6.92163i	−0.149222	+	0.988804i
$8$	−2.82843			−0.353553
$9$		0			0
$10$			3.16228i			0.316228i
$11$	−1.00806			−0.0916414			−0.0458207	−	0.998950i	$-0.514590\pi$
$11$	−1.00806			−0.0916414			−0.0458207	+	0.998950i	$0.514590\pi$
$12$		0			0
$13$			7.05322i			0.542556i	0.962501	+	0.271278i	$0.0874462\pi$
$13$			7.05322i			0.542556i	−0.962501	+	0.271278i	$0.912554\pi$
$14$	1.47723	−	9.78866i	0.105516	−	0.699190i
$15$		0			0
$16$	4.00000			0.250000
$17$		−	7.09185i		−	0.417168i	−0.978004	−	0.208584i	$-0.933115\pi$
$17$		−	7.09185i		−	0.417168i	0.978004	−	0.208584i	$-0.0668854\pi$
$18$		0			0
$19$		−	4.04257i		−	0.212767i	−0.994325	−	0.106383i	$-0.966073\pi$
$19$		−	4.04257i		−	0.212767i	0.994325	−	0.106383i	$-0.0339271\pi$
$20$		−	4.47214i		−	0.223607i
$21$		0			0
$22$	1.42561			0.0648003
$23$	−25.4766			−1.10768			−0.553840	−	0.832623i	$-0.686838\pi$
$23$	−25.4766			−1.10768			−0.553840	+	0.832623i	$0.686838\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		−	9.97476i		−	0.383645i
$27$		0			0
$28$	−2.08911	+	13.8433i	−0.0746112	+	0.494402i
$29$	−17.2909			−0.596239			−0.298119	−	0.954529i	$-0.596359\pi$
$29$	−17.2909			−0.596239			−0.298119	+	0.954529i	$0.596359\pi$
$30$		0			0
$31$		−	23.6516i		−	0.762956i	−0.924378	−	0.381478i	$-0.875415\pi$
$31$		−	23.6516i		−	0.762956i	0.924378	−	0.381478i	$-0.124585\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$			10.0294i			0.294982i
$35$	15.4772	+	2.33570i	0.442206	+	0.0667342i
$36$		0			0
$37$	−29.2171			−0.789651			−0.394826	−	0.918756i	$-0.629195\pi$
$37$	−29.2171			−0.789651			−0.394826	+	0.918756i	$0.629195\pi$
$38$			5.71705i			0.150449i
$39$		0			0
$40$			6.32456i			0.158114i
$41$			5.16217i			0.125907i	0.998016	+	0.0629533i	$0.0200519\pi$
$41$			5.16217i			0.125907i	−0.998016	+	0.0629533i	$0.979948\pi$
$42$		0			0
$43$	45.6529			1.06169			0.530847	−	0.847467i	$-0.321874\pi$
$43$	45.6529			1.06169			0.530847	+	0.847467i	$0.321874\pi$
$44$	−2.01611			−0.0458207
$45$		0			0
$46$	36.0294			0.783248
$47$			61.0391i			1.29870i	0.760488	+	0.649352i	$0.224960\pi$
$47$			61.0391i			1.29870i	−0.760488	+	0.649352i	$0.775040\pi$
$48$		0			0
$49$	−46.8178	−	14.4601i	−0.955465	−	0.295103i
$50$	7.07107			0.141421
$51$		0			0
$52$			14.1064i			0.271278i
$53$	−74.8237			−1.41177			−0.705884	−	0.708328i	$-0.749450\pi$
$53$	−74.8237			−1.41177			−0.705884	+	0.708328i	$0.749450\pi$
$54$		0			0
$55$			2.25408i			0.0409833i
$56$	2.95445	−	19.5773i	0.0527581	−	0.349595i
$57$		0			0
$58$	24.4531			0.421604
$59$			50.5639i			0.857014i	0.903538	+	0.428507i	$0.140960\pi$
$59$			50.5639i			0.857014i	−0.903538	+	0.428507i	$0.859040\pi$
$60$		0			0
$61$		−	58.0943i		−	0.952365i	−0.879346	−	0.476183i	$-0.842020\pi$
$61$		−	58.0943i		−	0.952365i	0.879346	−	0.476183i	$-0.157980\pi$
$62$			33.4485i			0.539491i
$63$		0			0
$64$	8.00000			0.125000
$65$	15.7715			0.242638
$66$		0			0
$67$	−73.5827			−1.09825			−0.549125	−	0.835741i	$-0.685039\pi$
$67$	−73.5827			−1.09825			−0.549125	+	0.835741i	$0.685039\pi$
$68$		−	14.1837i		−	0.208584i
$69$		0			0
$70$	−21.8881	−	3.30318i	−0.312687	−	0.0471882i
$71$	−99.8513			−1.40636			−0.703178	−	0.711013i	$-0.748236\pi$
$71$	−99.8513			−1.40636			−0.703178	+	0.711013i	$0.748236\pi$
$72$		0			0
$73$			1.15350i			0.0158014i	0.999969	+	0.00790069i	$0.00251489\pi$
$73$			1.15350i			0.0158014i	−0.999969	+	0.00790069i	$0.997485\pi$
$74$	41.3192			0.558368
$75$		0			0
$76$		−	8.08514i		−	0.106383i
$77$	1.05297	−	6.97738i	0.0136749	−	0.0906154i
$78$		0			0
$79$	−102.445			−1.29677			−0.648387	−	0.761311i	$-0.724556\pi$
$79$	−102.445			−1.29677			−0.648387	+	0.761311i	$0.724556\pi$
$80$		−	8.94427i		−	0.111803i
$81$		0			0
$82$		−	7.30042i		−	0.0890295i
$83$			83.4227i			1.00509i	0.864550	+	0.502546i	$0.167603\pi$
$83$			83.4227i			1.00509i	−0.864550	+	0.502546i	$0.832397\pi$
$84$		0			0
$85$	−15.8579			−0.186563
$86$	−64.5629			−0.750732
$87$		0			0
$88$	2.85121			0.0324001
$89$		−	69.0342i		−	0.775666i	−0.921730	−	0.387833i	$-0.873224\pi$
$89$		−	69.0342i		−	0.775666i	0.921730	−	0.387833i	$-0.126776\pi$
$90$		0			0
$91$	−48.8198	−	7.36749i	−0.536481	−	0.0809614i
$92$	−50.9533			−0.553840
$93$		0			0
$94$		−	86.3223i		−	0.918323i
$95$	−9.03946			−0.0951522
$96$		0			0
$97$			152.172i			1.56878i	0.620268	+	0.784390i	$0.287024\pi$
$97$			152.172i			1.56878i	−0.620268	+	0.784390i	$0.712976\pi$
$98$	66.2104	+	20.4496i	0.675616	+	0.208669i
$99$		0			0
$100$	−10.0000			−0.100000
$101$		−	66.8354i		−	0.661737i	−0.943677	−	0.330869i	$-0.892658\pi$
$101$		−	66.8354i		−	0.661737i	0.943677	−	0.330869i	$-0.107342\pi$
$102$		0			0
$103$		−	65.0648i		−	0.631697i	−0.948810	−	0.315848i	$-0.897711\pi$
$103$		−	65.0648i		−	0.631697i	0.948810	−	0.315848i	$-0.102289\pi$
$104$		−	19.9495i		−	0.191822i
$105$		0			0
$106$	105.817			0.998270
$107$	−10.8093			−0.101021			−0.0505106	−	0.998724i	$-0.516085\pi$
$107$	−10.8093			−0.101021			−0.0505106	+	0.998724i	$0.516085\pi$
$108$		0			0
$109$	−152.035			−1.39482			−0.697410	−	0.716673i	$-0.745664\pi$
$109$	−152.035			−1.39482			−0.697410	+	0.716673i	$0.745664\pi$
$110$		−	3.18775i		−	0.0289796i
$111$		0			0
$112$	−4.17822	+	27.6865i	−0.0373056	+	0.247201i
$113$	−223.980			−1.98212			−0.991060	−	0.133419i	$-0.957404\pi$
$113$	−223.980			−1.98212			−0.991060	+	0.133419i	$0.957404\pi$
$114$		0			0
$115$			56.9675i			0.495370i
$116$	−34.5818			−0.298119
$117$		0			0
$118$		−	71.5081i		−	0.606001i
$119$	49.0871	+	7.40784i	0.412497	+	0.0622507i
$120$		0			0
$121$	−119.984			−0.991602
$122$			82.1577i			0.673424i
$123$		0			0
$124$		−	47.3033i		−	0.381478i
$125$			11.1803i			0.0894427i
$126$		0			0
$127$	178.242			1.40348			0.701739	−	0.712434i	$-0.252407\pi$
$127$	178.242			1.40348			0.701739	+	0.712434i	$0.252407\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	−22.3042			−0.171571
$131$		−	124.116i		−	0.947453i	−0.880672	−	0.473726i	$-0.842909\pi$
$131$		−	124.116i		−	0.947453i	0.880672	−	0.473726i	$-0.157091\pi$
$132$		0			0
$133$	27.9811	+	4.22269i	0.210385	+	0.0317495i
$134$	104.062			0.776579
$135$		0			0
$136$			20.0588i			0.147491i
$137$	82.7435			0.603967			0.301984	−	0.953313i	$-0.402351\pi$
$137$	82.7435			0.603967			0.301984	+	0.953313i	$0.402351\pi$
$138$		0			0
$139$			112.631i			0.810298i	0.914251	+	0.405149i	$0.132780\pi$
$139$			112.631i			0.810298i	−0.914251	+	0.405149i	$0.867220\pi$
$140$	30.9545	+	4.67140i	0.221103	+	0.0333671i
$141$		0			0
$142$	141.211			0.994444
$143$		−	7.11004i		−	0.0497206i
$144$		0			0
$145$			38.6637i			0.266646i
$146$		−	1.63130i		−	0.0111733i
$147$		0			0
$148$	−58.4342			−0.394826
$149$	32.3830			0.217336			0.108668	−	0.994078i	$-0.465342\pi$
$149$	32.3830			0.217336			0.108668	+	0.994078i	$0.465342\pi$
$150$		0			0
$151$	−45.7944			−0.303274			−0.151637	−	0.988436i	$-0.548455\pi$
$151$	−45.7944			−0.303274			−0.151637	+	0.988436i	$0.548455\pi$
$152$			11.4341i			0.0752244i
$153$		0			0
$154$	−1.48913	+	9.86751i	−0.00966965	+	0.0640748i
$155$	−52.8867			−0.341204
$156$		0			0
$157$			251.015i			1.59882i	0.600784	+	0.799411i	$0.294855\pi$
$157$			251.015i			1.59882i	−0.600784	+	0.799411i	$0.705145\pi$
$158$	144.879			0.916958
$159$		0			0
$160$			12.6491i			0.0790569i
$161$	26.6118	−	176.340i	0.165291	−	1.09528i
$162$		0			0
$163$	138.555			0.850032			0.425016	−	0.905186i	$-0.360268\pi$
$163$	138.555			0.850032			0.425016	+	0.905186i	$0.360268\pi$
$164$			10.3243i			0.0629533i
$165$		0			0
$166$		−	117.977i		−	0.710708i
$167$			180.489i			1.08077i	0.841417	+	0.540386i	$0.181722\pi$
$167$			180.489i			1.08077i	−0.841417	+	0.540386i	$0.818278\pi$
$168$		0			0
$169$	119.252			0.705633
$170$	22.4264			0.131920
$171$		0			0
$172$	91.3058			0.530847
$173$		−	292.353i		−	1.68990i	−0.534846	−	0.844950i	$-0.679630\pi$
$173$		−	292.353i		−	1.68990i	0.534846	−	0.844950i	$-0.320370\pi$
$174$		0			0
$175$	5.22278	−	34.6081i	0.0298445	−	0.197761i
$176$	−4.03222			−0.0229104
$177$		0			0
$178$			97.6292i			0.548478i
$179$	273.253			1.52655			0.763277	−	0.646071i	$-0.223589\pi$
$179$	273.253			1.52655			0.763277	+	0.646071i	$0.223589\pi$
$180$		0			0
$181$			156.148i			0.862698i	0.902185	+	0.431349i	$0.141962\pi$
$181$			156.148i			0.862698i	−0.902185	+	0.431349i	$0.858038\pi$
$182$	69.0416	+	10.4192i	0.379349	+	0.0572484i
$183$		0			0
$184$	72.0588			0.391624
$185$			65.3314i			0.353143i
$186$		0			0
$187$			7.14898i			0.0382299i
$188$			122.078i			0.649352i
$189$		0			0
$190$	12.7837			0.0672827
$191$	275.462			1.44221			0.721104	−	0.692827i	$-0.243635\pi$
$191$	275.462			1.44221			0.721104	+	0.692827i	$0.243635\pi$
$192$		0			0
$193$	−9.69062			−0.0502105			−0.0251052	−	0.999685i	$-0.507992\pi$
$193$	−9.69062			−0.0502105			−0.0251052	+	0.999685i	$0.507992\pi$
$194$		−	215.203i		−	1.10929i
$195$		0			0
$196$	−93.6356	−	28.9201i	−0.477733	−	0.147552i
$197$	65.1041			0.330478			0.165239	−	0.986254i	$-0.447161\pi$
$197$	65.1041			0.330478			0.165239	+	0.986254i	$0.447161\pi$
$198$		0			0
$199$		−	220.166i		−	1.10636i	−0.833062	−	0.553180i	$-0.813414\pi$
$199$		−	220.166i		−	1.10636i	0.833062	−	0.553180i	$-0.186586\pi$
$200$	14.1421			0.0707107
$201$		0			0
$202$			94.5196i			0.467919i
$203$	18.0613	−	119.681i	0.0889721	−	0.589563i
$204$		0			0
$205$	11.5430			0.0563072
$206$			92.0155i			0.446677i
$207$		0			0
$208$			28.2129i			0.135639i
$209$			4.07513i			0.0194982i
$210$		0			0
$211$	9.08550			0.0430593			0.0215296	−	0.999768i	$-0.493146\pi$
$211$	9.08550			0.0430593			0.0215296	+	0.999768i	$0.493146\pi$
$212$	−149.647			−0.705884
$213$		0			0
$214$	15.2866			0.0714328
$215$		−	102.083i		−	0.474804i
$216$		0			0
$217$	163.708	+	24.7055i	0.754414	+	0.113850i
$218$	215.010			0.986286
$219$		0			0
$220$			4.50816i			0.0204916i
$221$	50.0204			0.226337
$222$		0			0
$223$		−	438.428i		−	1.96604i	−0.183489	−	0.983022i	$-0.558739\pi$
$223$		−	438.428i		−	1.96604i	0.183489	−	0.983022i	$-0.441261\pi$
$224$	5.90890	−	39.1546i	0.0263790	−	0.174797i
$225$		0			0
$226$	316.755			1.40157
$227$			235.641i			1.03807i	0.854754	+	0.519033i	$0.173708\pi$
$227$			235.641i			1.03807i	−0.854754	+	0.519033i	$0.826292\pi$
$228$		0			0
$229$		−	301.350i		−	1.31594i	−0.753044	−	0.657970i	$-0.771416\pi$
$229$		−	301.350i		−	1.31594i	0.753044	−	0.657970i	$-0.228584\pi$
$230$		−	80.5642i		−	0.350279i
$231$		0			0
$232$	48.9061			0.210802
$233$	−48.9562			−0.210113			−0.105056	−	0.994466i	$-0.533502\pi$
$233$	−48.9562			−0.210113			−0.105056	+	0.994466i	$0.533502\pi$
$234$		0			0
$235$	136.488			0.580798
$236$			101.128i			0.428507i
$237$		0			0
$238$	−69.4197	−	10.4763i	−0.291679	−	0.0440179i
$239$	−39.0966			−0.163584			−0.0817921	−	0.996649i	$-0.526064\pi$
$239$	−39.0966			−0.163584			−0.0817921	+	0.996649i	$0.526064\pi$
$240$		0			0
$241$		−	163.337i		−	0.677745i	−0.940832	−	0.338873i	$-0.889954\pi$
$241$		−	163.337i		−	0.677745i	0.940832	−	0.338873i	$-0.110046\pi$
$242$	169.683			0.701168
$243$		0			0
$244$		−	116.189i		−	0.476183i
$245$	−32.3337	+	104.688i	−0.131974	+	0.427297i
$246$		0			0
$247$	28.5131			0.115438
$248$			66.8969i			0.269746i
$249$		0			0
$250$		−	15.8114i		−	0.0632456i
$251$		−	2.50618i		−	0.00998480i	−0.999988	−	0.00499240i	$-0.998411\pi$
$251$		−	2.50618i		−	0.00998480i	0.999988	−	0.00499240i	$-0.00158914\pi$
$252$		0			0
$253$	25.6819			0.101509
$254$	−252.072			−0.992409
$255$		0			0
$256$	16.0000			0.0625000
$257$			205.951i			0.801365i	0.916217	+	0.400682i	$0.131227\pi$
$257$			205.951i			0.801365i	−0.916217	+	0.400682i	$0.868773\pi$
$258$		0			0
$259$	30.5189	−	202.230i	0.117834	−	0.780810i
$260$	31.5430			0.121319
$261$		0			0
$262$			175.527i			0.669950i
$263$	431.680			1.64137			0.820685	−	0.571381i	$-0.193592\pi$
$263$	431.680			1.64137			0.820685	+	0.571381i	$0.193592\pi$
$264$		0			0
$265$			167.311i			0.631362i
$266$	−39.5713	−	5.97178i	−0.148764	−	0.0224503i
$267$		0			0
$268$	−147.165			−0.549125
$269$			401.016i			1.49077i	0.666636	+	0.745384i	$0.267734\pi$
$269$			401.016i			1.49077i	−0.666636	+	0.745384i	$0.732266\pi$
$270$		0			0
$271$		−	409.154i		−	1.50979i	−0.655844	−	0.754897i	$-0.727687\pi$
$271$		−	409.154i		−	1.50979i	0.655844	−	0.754897i	$-0.272313\pi$
$272$		−	28.3674i		−	0.104292i
$273$		0			0
$274$	−117.017			−0.427070
$275$	5.04028			0.0183283
$276$		0			0
$277$	−311.824			−1.12572			−0.562859	−	0.826553i	$-0.690299\pi$
$277$	−311.824			−1.12572			−0.562859	+	0.826553i	$0.690299\pi$
$278$		−	159.285i		−	0.572968i
$279$		0			0
$280$	−43.7762	−	6.60635i	−0.156344	−	0.0235941i
$281$	131.505			0.467990			0.233995	−	0.972238i	$-0.424820\pi$
$281$	131.505			0.467990			0.233995	+	0.972238i	$0.424820\pi$
$282$		0			0
$283$			50.6362i			0.178926i	0.995990	+	0.0894632i	$0.0285151\pi$
$283$			50.6362i			0.178926i	−0.995990	+	0.0894632i	$0.971485\pi$
$284$	−199.703			−0.703178
$285$		0			0
$286$			10.0551i			0.0351578i
$287$	−35.7306	−	5.39218i	−0.124497	−	0.0187881i
$288$		0			0
$289$	238.706			0.825971
$290$		−	54.6787i		−	0.188547i
$291$		0			0
$292$			2.30700i			0.00790069i
$293$			176.891i			0.603725i	0.953351	+	0.301863i	$0.0976084\pi$
$293$			176.891i			0.603725i	−0.953351	+	0.301863i	$0.902392\pi$
$294$		0			0
$295$	113.064			0.383269
$296$	82.6384			0.279184
$297$		0			0
$298$	−45.7965			−0.153680
$299$		−	179.692i		−	0.600978i
$300$		0			0
$301$	−47.6870	+	315.992i	−0.158429	+	1.04981i
$302$	64.7631			0.214447
$303$		0			0
$304$		−	16.1703i		−	0.0531917i
$305$	−129.903			−0.425911
$306$		0			0
$307$			569.023i			1.85349i	0.375686	+	0.926747i	$0.377407\pi$
$307$			569.023i			1.85349i	−0.375686	+	0.926747i	$0.622593\pi$
$308$	2.10594	−	13.9548i	0.00683747	−	0.0453077i
$309$		0			0
$310$	74.7931			0.241268
$311$			322.176i			1.03593i	0.855401	+	0.517967i	$0.173311\pi$
$311$			322.176i			1.03593i	−0.855401	+	0.517967i	$0.826689\pi$
$312$		0			0
$313$		−	21.5138i		−	0.0687340i	−0.999409	−	0.0343670i	$-0.989058\pi$
$313$		−	21.5138i		−	0.0687340i	0.999409	−	0.0343670i	$-0.0109415\pi$
$314$		−	354.989i		−	1.13054i
$315$		0			0
$316$	−204.890			−0.648387
$317$	384.271			1.21221			0.606106	−	0.795384i	$-0.292731\pi$
$317$	384.271			1.21221			0.606106	+	0.795384i	$0.292731\pi$
$318$		0			0
$319$	17.4302			0.0546402
$320$		−	17.8885i		−	0.0559017i
$321$		0			0
$322$	−37.6348	+	249.382i	−0.116878	+	0.774479i
$323$	−28.6693			−0.0887594
$324$		0			0
$325$		−	35.2661i		−	0.108511i
$326$	−195.947			−0.601063
$327$		0			0
$328$		−	14.6008i		−	0.0445147i
$329$	−422.490	−	63.7588i	−1.28416	−	0.193796i
$330$		0			0
$331$	28.0523			0.0847503			0.0423751	−	0.999102i	$-0.486508\pi$
$331$	28.0523			0.0847503			0.0423751	+	0.999102i	$0.486508\pi$
$332$			166.845i			0.502546i
$333$		0			0
$334$		−	255.250i		−	0.764221i
$335$			164.536i			0.491152i
$336$		0			0
$337$	−481.854			−1.42983			−0.714917	−	0.699210i	$-0.753535\pi$
$337$	−481.854			−1.42983			−0.714917	+	0.699210i	$0.753535\pi$
$338$	−168.648			−0.498958
$339$		0			0
$340$	−31.7157			−0.0932816
$341$			23.8422i			0.0699184i
$342$		0			0
$343$	148.991	−	308.951i	0.434376	−	0.900732i
$344$	−129.126			−0.375366
$345$		0			0
$346$			413.449i			1.19494i
$347$	−376.261			−1.08432			−0.542162	−	0.840274i	$-0.682394\pi$
$347$	−376.261			−1.08432			−0.542162	+	0.840274i	$0.682394\pi$
$348$		0			0
$349$			260.936i			0.747668i	0.927496	+	0.373834i	$0.121957\pi$
$349$			260.936i			0.747668i	−0.927496	+	0.373834i	$0.878043\pi$
$350$	−7.38613	+	48.9433i	−0.0211032	+	0.139838i
$351$		0			0
$352$	5.70242			0.0162001
$353$		−	142.203i		−	0.402841i	−0.979505	−	0.201421i	$-0.935444\pi$
$353$		−	142.203i		−	0.402841i	0.979505	−	0.201421i	$-0.0645558\pi$
$354$		0			0
$355$			223.274i			0.628942i
$356$		−	138.068i		−	0.387833i
$357$		0			0
$358$	−386.438			−1.07944
$359$	−214.757			−0.598209			−0.299104	−	0.954220i	$-0.596688\pi$
$359$	−214.757			−0.598209			−0.299104	+	0.954220i	$0.596688\pi$
$360$		0			0
$361$	344.658			0.954730
$362$		−	220.827i		−	0.610019i
$363$		0			0
$364$	−97.6395	−	14.7350i	−0.268240	−	0.0404807i
$365$	2.57931			0.00706659
$366$		0			0
$367$			291.094i			0.793172i	0.917997	+	0.396586i	$0.129805\pi$
$367$			291.094i			0.793172i	−0.917997	+	0.396586i	$0.870195\pi$
$368$	−101.907			−0.276920
$369$		0			0
$370$		−	92.3926i		−	0.249710i
$371$	78.1575	−	517.901i	0.210667	−	1.39596i
$372$		0			0
$373$	−677.342			−1.81593			−0.907966	−	0.419044i	$-0.862365\pi$
$373$	−677.342			−1.81593			−0.907966	+	0.419044i	$0.862365\pi$
$374$		−	10.1102i		−	0.0270326i
$375$		0			0
$376$		−	172.645i		−	0.459161i
$377$		−	121.957i		−	0.323493i
$378$		0			0
$379$	679.912			1.79396			0.896981	−	0.442069i	$-0.145755\pi$
$379$	679.912			1.79396			0.896981	+	0.442069i	$0.145755\pi$
$380$	−18.0789			−0.0475761
$381$		0			0
$382$	−389.562			−1.01979
$383$		−	742.938i		−	1.93979i	−0.243530	−	0.969893i	$-0.578305\pi$
$383$		−	742.938i		−	1.93979i	0.243530	−	0.969893i	$-0.421695\pi$
$384$		0			0
$385$	−15.6019	−	2.35451i	−0.0405244	−	0.00611562i
$386$	13.7046			0.0355042
$387$		0			0
$388$			304.343i			0.784390i
$389$	−497.964			−1.28011			−0.640056	−	0.768328i	$-0.721089\pi$
$389$	−497.964			−1.28011			−0.640056	+	0.768328i	$0.721089\pi$
$390$		0			0
$391$			180.677i			0.462088i
$392$	132.421	+	40.8992i	0.337808	+	0.104335i
$393$		0			0
$394$	−92.0711			−0.233683
$395$			229.074i			0.579935i
$396$		0			0
$397$		−	79.8232i		−	0.201066i	−0.994934	−	0.100533i	$-0.967945\pi$
$397$		−	79.8232i		−	0.201066i	0.994934	−	0.100533i	$-0.0320548\pi$
$398$			311.361i			0.782315i
$399$		0			0
$400$	−20.0000			−0.0500000
$401$	−30.9575			−0.0772008			−0.0386004	−	0.999255i	$-0.512290\pi$
$401$	−30.9575			−0.0772008			−0.0386004	+	0.999255i	$0.512290\pi$
$402$		0			0
$403$	166.820			0.413946
$404$		−	133.671i		−	0.330869i
$405$		0			0
$406$	−25.5426	+	169.255i	−0.0629128	+	0.416884i
$407$	29.4525			0.0723648
$408$		0			0
$409$		−	179.566i		−	0.439036i	−0.975608	−	0.219518i	$-0.929552\pi$
$409$		−	179.566i		−	0.439036i	0.975608	−	0.219518i	$-0.0704484\pi$
$410$	−16.3242			−0.0398152
$411$		0			0
$412$		−	130.130i		−	0.315848i
$413$	−349.984	−	52.8168i	−0.847419	−	0.127886i
$414$		0			0
$415$	186.539			0.449491
$416$		−	39.8991i		−	0.0959112i
$417$		0			0
$418$		−	5.76311i		−	0.0137873i
$419$		−	283.259i		−	0.676035i	−0.941140	−	0.338017i	$-0.890244\pi$
$419$		−	283.259i		−	0.676035i	0.941140	−	0.338017i	$-0.109756\pi$
$420$		0			0
$421$	−600.315			−1.42593			−0.712963	−	0.701202i	$-0.752647\pi$
$421$	−600.315			−1.42593			−0.712963	+	0.701202i	$0.752647\pi$
$422$	−12.8488			−0.0304475
$423$		0			0
$424$	211.633			0.499135
$425$			35.4593i			0.0834336i
$426$		0			0
$427$	402.107	+	60.6827i	0.941702	+	0.142114i
$428$	−21.6185			−0.0505106
$429$		0			0
$430$			144.367i			0.335737i
$431$	−267.667			−0.621036			−0.310518	−	0.950567i	$-0.600503\pi$
$431$	−267.667			−0.621036			−0.310518	+	0.950567i	$0.600503\pi$
$432$		0			0
$433$		−	312.397i		−	0.721471i	−0.932668	−	0.360736i	$-0.882526\pi$
$433$		−	312.397i		−	0.721471i	0.932668	−	0.360736i	$-0.117474\pi$
$434$	−231.518	−	34.9388i	−0.533451	−	0.0805042i
$435$		0			0
$436$	−304.071			−0.697410
$437$			102.991i			0.235677i
$438$		0			0
$439$			72.4481i			0.165030i	0.996590	+	0.0825150i	$0.0262952\pi$
$439$			72.4481i			0.165030i	−0.996590	+	0.0825150i	$0.973705\pi$
$440$		−	6.37550i		−	0.0144898i
$441$		0			0
$442$	−70.7396			−0.160044
$443$	279.795			0.631591			0.315795	−	0.948827i	$-0.397729\pi$
$443$	279.795			0.631591			0.315795	+	0.948827i	$0.397729\pi$
$444$		0			0
$445$	−154.365			−0.346888
$446$			620.030i			1.39020i
$447$		0			0
$448$	−8.35645	+	55.3730i	−0.0186528	+	0.123600i
$449$	165.129			0.367770			0.183885	−	0.982948i	$-0.441133\pi$
$449$	165.129			0.367770			0.183885	+	0.982948i	$0.441133\pi$
$450$		0			0
$451$		−	5.20376i		−	0.0115383i
$452$	−447.959			−0.991060
$453$		0			0
$454$		−	333.246i		−	0.734023i
$455$	−16.4742	+	109.164i	−0.0362070	+	0.239922i
$456$		0			0
$457$	−387.165			−0.847189			−0.423594	−	0.905852i	$-0.639232\pi$
$457$	−387.165			−0.847189			−0.423594	+	0.905852i	$0.639232\pi$
$458$			426.173i			0.930510i
$459$		0			0
$460$			113.935i			0.247685i
$461$		−	74.1988i		−	0.160952i	−0.996757	−	0.0804759i	$-0.974356\pi$
$461$		−	74.1988i		−	0.160952i	0.996757	−	0.0804759i	$-0.0256440\pi$
$462$		0			0
$463$	−423.801			−0.915336			−0.457668	−	0.889123i	$-0.651315\pi$
$463$	−423.801			−0.915336			−0.457668	+	0.889123i	$0.651315\pi$
$464$	−69.1637			−0.149060
$465$		0			0
$466$	69.2345			0.148572
$467$		−	327.240i		−	0.700727i	−0.936614	−	0.350364i	$-0.886058\pi$
$467$		−	327.240i		−	0.700727i	0.936614	−	0.350364i	$-0.113942\pi$
$468$		0			0
$469$	76.8613	−	509.312i	0.163883	−	1.08595i
$470$	−193.023			−0.410686
$471$		0			0
$472$		−	143.016i		−	0.303000i
$473$	−46.0207			−0.0972953
$474$		0			0
$475$			20.2128i			0.0425533i
$476$	98.1743	+	14.8157i	0.206249	+	0.0311254i
$477$		0			0
$478$	55.2910			0.115671
$479$			427.880i			0.893277i	0.894715	+	0.446638i	$0.147379\pi$
$479$			427.880i			0.893277i	−0.894715	+	0.446638i	$0.852621\pi$
$480$		0			0
$481$		−	206.075i		−	0.428430i
$482$			230.993i			0.479238i
$483$		0			0
$484$	−239.968			−0.495801
$485$	340.266			0.701579
$486$		0			0
$487$	815.490			1.67452			0.837259	−	0.546807i	$-0.184157\pi$
$487$	815.490			1.67452			0.837259	+	0.546807i	$0.184157\pi$
$488$			164.315i			0.336712i
$489$		0			0
$490$	45.7267	−	148.051i	0.0933198	−	0.302145i
$491$	−6.57128			−0.0133835			−0.00669173	−	0.999978i	$-0.502130\pi$
$491$	−6.57128			−0.0133835			−0.00669173	+	0.999978i	$0.502130\pi$
$492$		0			0
$493$			122.625i			0.248732i
$494$	−40.3237			−0.0816268
$495$		0			0
$496$		−	94.6066i		−	0.190739i
$497$	104.300	−	691.134i	0.209860	−	1.39061i
$498$		0			0
$499$	−468.850			−0.939579			−0.469790	−	0.882778i	$-0.655670\pi$
$499$	−468.850			−0.939579			−0.469790	+	0.882778i	$0.655670\pi$
$500$			22.3607i			0.0447214i
$501$		0			0
$502$			3.54428i			0.00706032i
$503$			567.093i			1.12742i	0.825972	+	0.563711i	$0.190627\pi$
$503$			567.093i			1.12742i	−0.825972	+	0.563711i	$0.809373\pi$
$504$		0			0
$505$	−149.449			−0.295938
$506$	−36.3197			−0.0717780
$507$		0			0
$508$	356.483			0.701739
$509$			464.735i			0.913036i	0.889714	+	0.456518i	$0.150904\pi$
$509$			464.735i			0.913036i	−0.889714	+	0.456518i	$0.849096\pi$
$510$		0			0
$511$	−7.98410	−	1.20490i	−0.0156245	−	0.00235792i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$		−	291.258i		−	0.566651i
$515$	−145.489			−0.282503
$516$		0			0
$517$		−	61.5308i		−	0.119015i
$518$	−43.1602	+	285.996i	−0.0833209	+	0.552116i
$519$		0			0
$520$	−44.6085			−0.0857856
$521$			925.284i			1.77598i	0.459866	+	0.887988i	$0.347897\pi$
$521$			925.284i			1.77598i	−0.459866	+	0.887988i	$0.652103\pi$
$522$		0			0
$523$			743.118i			1.42088i	0.703760	+	0.710438i	$0.251503\pi$
$523$			743.118i			1.42088i	−0.703760	+	0.710438i	$0.748497\pi$
$524$		−	248.233i		−	0.473726i
$525$		0			0
$526$	−610.488			−1.16062
$527$	−167.734			−0.318281
$528$		0			0
$529$	120.059			0.226955
$530$		−	236.613i		−	0.446440i
$531$		0			0
$532$	55.9623	+	8.44538i	0.105192	+	0.0158748i
$533$	−36.4100			−0.0683114
$534$		0			0
$535$			24.1703i			0.0451781i
$536$	208.123			0.388290
$537$		0			0
$538$		−	567.123i		−	1.05413i
$539$	47.1950	+	14.5765i	0.0875602	+	0.0270437i
$540$		0			0
$541$	224.274			0.414555			0.207277	−	0.978282i	$-0.433540\pi$
$541$	224.274			0.414555			0.207277	+	0.978282i	$0.433540\pi$
$542$			578.631i			1.06759i
$543$		0			0
$544$			40.1176i			0.0737455i
$545$			339.961i			0.623782i
$546$		0			0
$547$	636.899			1.16435			0.582175	−	0.813064i	$-0.302202\pi$
$547$	636.899			1.16435			0.582175	+	0.813064i	$0.302202\pi$
$548$	165.487			0.301984
$549$		0			0
$550$	−7.12803			−0.0129601
$551$			69.8997i			0.126860i
$552$		0			0
$553$	107.010	−	709.087i	0.193508	−	1.28226i
$554$	440.986			0.796003
$555$		0			0
$556$			225.263i			0.405149i
$557$	−715.978			−1.28542			−0.642709	−	0.766111i	$-0.722189\pi$
$557$	−715.978			−1.28542			−0.642709	+	0.766111i	$0.722189\pi$
$558$		0			0
$559$			322.000i			0.576029i
$560$	61.9089	+	9.34279i	0.110552	+	0.0166836i
$561$		0			0
$562$	−185.976			−0.330919
$563$		−	911.747i		−	1.61944i	−0.586813	−	0.809722i	$-0.699618\pi$
$563$		−	911.747i		−	1.61944i	0.586813	−	0.809722i	$-0.300382\pi$
$564$		0			0
$565$			500.833i			0.886431i
$566$		−	71.6103i		−	0.126520i
$567$		0			0
$568$	282.422			0.497222
$569$	−83.3213			−0.146435			−0.0732173	−	0.997316i	$-0.523327\pi$
$569$	−83.3213			−0.146435			−0.0732173	+	0.997316i	$0.523327\pi$
$570$		0			0
$571$	112.867			0.197666			0.0988328	−	0.995104i	$-0.468489\pi$
$571$	112.867			0.197666			0.0988328	+	0.995104i	$0.468489\pi$
$572$		−	14.2201i		−	0.0248603i
$573$		0			0
$574$	50.5307	+	7.62570i	0.0880327	+	0.0132852i
$575$	127.383			0.221536
$576$		0			0
$577$		−	187.954i		−	0.325743i	−0.986647	−	0.162872i	$-0.947924\pi$
$577$		−	187.954i		−	0.325743i	0.986647	−	0.162872i	$-0.0520757\pi$
$578$	−337.581			−0.584050
$579$		0			0
$580$			77.3274i			0.133323i
$581$	−577.420	−	87.1397i	−0.993839	−	0.149982i
$582$		0			0
$583$	75.4264			0.129376
$584$		−	3.26259i		−	0.00558663i
$585$		0			0
$586$		−	250.162i		−	0.426898i
$587$		−	137.244i		−	0.233806i	−0.993143	−	0.116903i	$-0.962703\pi$
$587$		−	137.244i		−	0.233806i	0.993143	−	0.116903i	$-0.0372967\pi$
$588$		0			0
$589$	−95.6134			−0.162332
$590$	−159.897			−0.271012
$591$		0			0
$592$	−116.868			−0.197413
$593$		−	691.207i		−	1.16561i	−0.812612	−	0.582805i	$-0.801955\pi$
$593$		−	691.207i		−	1.16561i	0.812612	−	0.582805i	$-0.198045\pi$
$594$		0			0
$595$	16.5644	−	109.762i	0.0278394	−	0.184474i
$596$	64.7660			0.108668
$597$		0			0
$598$			254.123i			0.424956i
$599$	915.774			1.52884			0.764419	−	0.644720i	$-0.223026\pi$
$599$	915.774			1.52884			0.764419	+	0.644720i	$0.223026\pi$
$600$		0			0
$601$			178.510i			0.297022i	0.988911	+	0.148511i	$0.0474480\pi$
$601$			178.510i			0.297022i	−0.988911	+	0.148511i	$0.952552\pi$
$602$	67.4396	−	446.880i	0.112026	−	0.742326i
$603$		0			0
$604$	−91.5889			−0.151637
$605$			268.292i			0.443458i
$606$		0			0
$607$		−	673.634i		−	1.10978i	−0.831925	−	0.554888i	$-0.812761\pi$
$607$		−	673.634i		−	1.10978i	0.831925	−	0.554888i	$-0.187239\pi$
$608$			22.8682i			0.0376122i
$609$		0			0
$610$	183.710			0.301164
$611$	−430.522			−0.704619
$612$		0			0
$613$	859.658			1.40238			0.701189	−	0.712976i	$-0.252653\pi$
$613$	859.658			1.40238			0.701189	+	0.712976i	$0.252653\pi$
$614$		−	804.720i		−	1.31062i
$615$		0			0
$616$	−2.97825	+	19.7350i	−0.00483482	+	0.0320374i
$617$	695.263			1.12684			0.563422	−	0.826169i	$-0.309484\pi$
$617$	695.263			1.12684			0.563422	+	0.826169i	$0.309484\pi$
$618$		0			0
$619$			1036.53i			1.67452i	0.546808	+	0.837258i	$0.315843\pi$
$619$			1036.53i			1.67452i	−0.546808	+	0.837258i	$0.684157\pi$
$620$	−105.773			−0.170602
$621$		0			0
$622$		−	455.625i		−	0.732516i
$623$	477.829	+	72.1102i	0.766981	+	0.115747i
$624$		0			0
$625$	25.0000			0.0400000
$626$			30.4250i			0.0486023i
$627$		0			0
$628$			502.030i			0.799411i
$629$			207.203i			0.329417i
$630$		0			0
$631$	702.851			1.11387			0.556934	−	0.830557i	$-0.311978\pi$
$631$	702.851			1.11387			0.556934	+	0.830557i	$0.311978\pi$
$632$	289.759			0.458479
$633$		0			0
$634$	−543.442			−0.857164
$635$		−	398.561i		−	0.627655i
$636$		0			0
$637$	101.990	−	330.216i	0.160110	−	0.518393i
$638$	−24.6500			−0.0386364
$639$		0			0
$640$			25.2982i			0.0395285i
$641$	421.315			0.657277			0.328639	−	0.944456i	$-0.393410\pi$
$641$	421.315			0.657277			0.328639	+	0.944456i	$0.393410\pi$
$642$		0			0
$643$			414.352i			0.644404i	0.946671	+	0.322202i	$0.104423\pi$
$643$			414.352i			0.644404i	−0.946671	+	0.322202i	$0.895577\pi$
$644$	53.2236	−	352.680i	0.0826453	−	0.547639i
$645$		0			0
$646$	40.5445			0.0627624
$647$		−	116.692i		−	0.180358i	−0.995926	−	0.0901790i	$-0.971256\pi$
$647$		−	116.692i		−	0.180358i	0.995926	−	0.0901790i	$-0.0287439\pi$
$648$		0			0
$649$		−	50.9712i		−	0.0785380i
$650$			49.8738i			0.0767289i
$651$		0			0
$652$	277.110			0.425016
$653$	−168.591			−0.258178			−0.129089	−	0.991633i	$-0.541205\pi$
$653$	−168.591			−0.258178			−0.129089	+	0.991633i	$0.541205\pi$
$654$		0			0
$655$	−277.533			−0.423714
$656$			20.6487i			0.0314767i
$657$		0			0
$658$	597.491	+	90.1685i	0.908041	+	0.137034i
$659$	1009.52			1.53189			0.765946	−	0.642904i	$-0.222271\pi$
$659$	1009.52			1.53189			0.765946	+	0.642904i	$0.222271\pi$
$660$		0			0
$661$		−	646.529i		−	0.978107i	−0.872254	−	0.489054i	$-0.837342\pi$
$661$		−	646.529i		−	0.978107i	0.872254	−	0.489054i	$-0.162658\pi$
$662$	−39.6720			−0.0599275
$663$		0			0
$664$		−	235.955i		−	0.355354i
$665$	9.44222	−	62.5677i	0.0141988	−	0.0940868i
$666$		0			0
$667$	440.515			0.660442
$668$			360.978i			0.540386i
$669$		0			0
$670$		−	232.689i		−	0.347297i
$671$			58.5623i			0.0872761i
$672$		0			0
$673$	−840.576			−1.24900			−0.624500	−	0.781025i	$-0.714697\pi$
$673$	−840.576			−1.24900			−0.624500	+	0.781025i	$0.714697\pi$
$674$	681.444			1.01104
$675$		0			0
$676$	238.504			0.352817
$677$			379.146i			0.560038i	0.959994	+	0.280019i	$0.0903408\pi$
$677$			379.146i			0.560038i	−0.959994	+	0.280019i	$0.909659\pi$
$678$		0			0
$679$	−1053.27	−	158.952i	−1.55121	−	0.234097i
$680$	44.8528			0.0659600
$681$		0			0
$682$		−	33.7179i		−	0.0494398i
$683$	115.942			0.169754			0.0848770	−	0.996391i	$-0.472950\pi$
$683$	115.942			0.169754			0.0848770	+	0.996391i	$0.472950\pi$
$684$		0			0
$685$		−	185.020i		−	0.270102i
$686$	−210.705	+	436.923i	−0.307150	+	0.636914i
$687$		0			0
$688$	182.612			0.265424
$689$		−	527.748i		−	0.765962i
$690$		0			0
$691$		−	112.794i		−	0.163233i	−0.996664	−	0.0816167i	$-0.973992\pi$
$691$		−	112.794i		−	0.163233i	0.996664	−	0.0816167i	$-0.0260083\pi$
$692$		−	584.705i		−	0.844950i
$693$		0			0
$694$	532.113			0.766733
$695$	251.852			0.362377
$696$		0			0
$697$	36.6094			0.0525242
$698$		−	369.020i		−	0.528681i
$699$		0			0
$700$	10.4456	−	69.2163i	0.0149222	−	0.0988804i
$701$	−192.764			−0.274985			−0.137492	−	0.990503i	$-0.543904\pi$
$701$	−192.764			−0.274985			−0.137492	+	0.990503i	$0.543904\pi$
$702$		0			0
$703$			118.112i			0.168012i
$704$	−8.06445			−0.0114552
$705$		0			0
$706$			201.105i			0.284852i
$707$	462.610	+	69.8134i	0.654328	+	0.0987459i
$708$		0			0
$709$	−317.352			−0.447604			−0.223802	−	0.974635i	$-0.571847\pi$
$709$	−317.352			−0.447604			−0.223802	+	0.974635i	$0.571847\pi$
$710$		−	315.758i		−	0.444729i
$711$		0			0
$712$			195.258i			0.274239i
$713$			602.564i			0.845111i
$714$		0			0
$715$	−15.8985			−0.0222357
$716$	546.506			0.763277
$717$		0			0
$718$	303.712			0.422998
$719$			668.666i			0.929995i	0.885312	+	0.464997i	$0.153945\pi$
$719$			668.666i			0.929995i	−0.885312	+	0.464997i	$0.846055\pi$
$720$		0			0
$721$	450.354	+	67.9638i	0.624624	+	0.0942633i
$722$	−487.420			−0.675096
$723$		0			0
$724$			312.297i			0.431349i
$725$	86.4546			0.119248
$726$		0			0
$727$		−	164.229i		−	0.225900i	−0.993601	−	0.112950i	$-0.963970\pi$
$727$		−	164.229i		−	0.225900i	0.993601	−	0.112950i	$-0.0360300\pi$
$728$	138.083	+	20.8384i	0.189675	+	0.0286242i
$729$		0			0
$730$	−3.64769			−0.00499683
$731$		−	323.764i		−	0.442905i
$732$		0			0
$733$			986.499i			1.34584i	0.739717	+	0.672919i	$0.234960\pi$
$733$			986.499i			1.34584i	−0.739717	+	0.672919i	$0.765040\pi$
$734$		−	411.669i		−	0.560858i
$735$		0			0
$736$	144.118			0.195812
$737$	74.1755			0.100645
$738$		0			0
$739$	−645.708			−0.873759			−0.436880	−	0.899520i	$-0.643916\pi$
$739$	−645.708			−0.873759			−0.436880	+	0.899520i	$0.643916\pi$
$740$			130.663i			0.176571i
$741$		0			0
$742$	−110.531	+	732.423i	−0.148964	+	0.987093i
$743$	−45.9927			−0.0619013			−0.0309507	−	0.999521i	$-0.509853\pi$
$743$	−45.9927			−0.0619013			−0.0309507	+	0.999521i	$0.509853\pi$
$744$		0			0
$745$		−	72.4106i		−	0.0971955i
$746$	957.907			1.28406
$747$		0			0
$748$			14.2980i			0.0191149i
$749$	11.2909	−	74.8177i	0.0150746	−	0.0998902i
$750$		0			0
$751$	638.813			0.850616			0.425308	−	0.905049i	$-0.360166\pi$
$751$	638.813			0.850616			0.425308	+	0.905049i	$0.360166\pi$
$752$			244.156i			0.324676i
$753$		0			0
$754$			172.473i			0.228744i
$755$			102.399i			0.135628i
$756$		0			0
$757$	−1133.23			−1.49700			−0.748502	−	0.663133i	$-0.769226\pi$
$757$	−1133.23			−1.49700			−0.748502	+	0.663133i	$0.769226\pi$
$758$	−961.540			−1.26852
$759$		0			0
$760$	25.5674			0.0336414
$761$			846.462i			1.11230i	0.831081	+	0.556151i	$0.187722\pi$
$761$			846.462i			1.11230i	−0.831081	+	0.556151i	$0.812278\pi$
$762$		0			0
$763$	158.809	−	1052.33i	0.208138	−	1.37920i
$764$	550.923			0.721104
$765$		0			0
$766$			1050.67i			1.37164i
$767$	−356.638			−0.464978
$768$		0			0
$769$			894.921i			1.16375i	0.813280	+	0.581873i	$0.197680\pi$
$769$			894.921i			1.16375i	−0.813280	+	0.581873i	$0.802320\pi$
$770$	22.0644	+	3.32979i	0.0286551	+	0.00432440i
$771$		0			0
$772$	−19.3812			−0.0251052
$773$			104.513i			0.135204i	0.997712	+	0.0676019i	$0.0215348\pi$
$773$			104.513i			0.135204i	−0.997712	+	0.0676019i	$0.978465\pi$
$774$		0			0
$775$			118.258i			0.152591i
$776$		−	430.406i		−	0.554647i
$777$		0			0
$778$	704.227			0.905176
$779$	20.8684			0.0267888
$780$		0			0
$781$	100.656			0.128881
$782$		−	255.515i		−	0.326746i
$783$		0			0
$784$	−187.271	−	57.8402i	−0.238866	−	0.0737758i
$785$	561.287			0.715015
$786$		0			0
$787$		−	349.479i		−	0.444065i	−0.975039	−	0.222032i	$-0.928731\pi$
$787$		−	349.479i		−	0.444065i	0.975039	−	0.222032i	$-0.0712691\pi$
$788$	130.208			0.165239
$789$		0			0
$790$		−	323.960i		−	0.410076i
$791$	233.959	−	1550.30i	0.295776	−	1.95993i
$792$		0			0
$793$	409.752			0.516711
$794$			112.887i			0.142175i
$795$		0			0
$796$		−	440.332i		−	0.553180i
$797$		−	606.585i		−	0.761085i	−0.924763	−	0.380543i	$-0.875737\pi$
$797$		−	606.585i		−	0.761085i	0.924763	−	0.380543i	$-0.124263\pi$
$798$		0			0
$799$	432.880			0.541778
$800$	28.2843			0.0353553
$801$		0			0
$802$	43.7805			0.0545892
$803$		−	1.16279i		−	0.00144806i
$804$		0			0
$805$	−394.308	−	59.5058i	−0.489823	−	0.0739202i
$806$	−235.920			−0.292704
$807$		0			0
$808$			189.039i			0.233959i
$809$	157.390			0.194548			0.0972742	−	0.995258i	$-0.468988\pi$
$809$	157.390			0.194548			0.0972742	+	0.995258i	$0.468988\pi$
$810$		0			0
$811$			443.035i			0.546283i	0.961974	+	0.273141i	$0.0880626\pi$
$811$			443.035i			0.546283i	−0.961974	+	0.273141i	$0.911937\pi$
$812$	36.1227	−	239.363i	0.0444861	−	0.294782i
$813$		0			0
$814$	−41.6521			−0.0511696
$815$		−	309.819i		−	0.380146i
$816$		0			0
$817$		−	184.555i		−	0.225893i
$818$			253.944i			0.310445i
$819$		0			0
$820$	23.0859			0.0281536
$821$	−768.304			−0.935815			−0.467907	−	0.883777i	$-0.654992\pi$
$821$	−768.304			−0.935815			−0.467907	+	0.883777i	$0.654992\pi$
$822$		0			0
$823$	−1094.49			−1.32988			−0.664938	−	0.746899i	$-0.731542\pi$
$823$	−1094.49			−1.32988			−0.664938	+	0.746899i	$0.731542\pi$
$824$			184.031i			0.223339i
$825$		0			0
$826$	494.952	+	74.6942i	0.599216	+	0.0904288i
$827$	−1258.41			−1.52165			−0.760826	−	0.648956i	$-0.775206\pi$
$827$	−1258.41			−1.52165			−0.760826	+	0.648956i	$0.775206\pi$
$828$		0			0
$829$		−	182.499i		−	0.220143i	−0.993924	−	0.110072i	$-0.964892\pi$
$829$		−	182.499i		−	0.220143i	0.993924	−	0.110072i	$-0.0351081\pi$
$830$	−263.806			−0.317838
$831$		0			0
$832$			56.4258i			0.0678195i
$833$	−102.549	+	332.025i	−0.123108	+	0.398589i
$834$		0			0
$835$	403.585			0.483336
$836$			8.15027i			0.00974912i
$837$		0			0
$838$			400.588i			0.478029i
$839$			1157.55i			1.37968i	0.723963	+	0.689839i	$0.242319\pi$
$839$			1157.55i			1.37968i	−0.723963	+	0.689839i	$0.757681\pi$
$840$		0			0
$841$	−542.024			−0.644499
$842$	848.973			1.00828
$843$		0			0
$844$	18.1710			0.0215296
$845$		−	266.656i		−	0.315569i
$846$		0			0
$847$	125.330	−	830.483i	0.147969	−	0.980500i
$848$	−299.295			−0.352942
$849$		0			0
$850$		−	50.1470i		−	0.0589964i
$851$	744.354			0.874681
$852$		0			0
$853$		−	1038.69i		−	1.21769i	−0.793290	−	0.608844i	$-0.791634\pi$
$853$		−	1038.69i		−	1.21769i	0.793290	−	0.608844i	$-0.208366\pi$
$854$	−568.665	−	85.8184i	−0.665884	−	0.100490i
$855$		0			0
$856$	30.5732			0.0357164
$857$		−	1206.41i		−	1.40771i	−0.710344	−	0.703854i	$-0.751461\pi$
$857$		−	1206.41i		−	1.40771i	0.710344	−	0.703854i	$-0.248539\pi$
$858$		0			0
$859$			333.020i			0.387683i	0.981033	+	0.193841i	$0.0620947\pi$
$859$			333.020i			0.387683i	−0.981033	+	0.193841i	$0.937905\pi$
$860$		−	204.166i		−	0.237402i
$861$		0			0
$862$	378.538			0.439139
$863$	−683.072			−0.791508			−0.395754	−	0.918357i	$-0.629517\pi$
$863$	−683.072			−0.791508			−0.395754	+	0.918357i	$0.629517\pi$
$864$		0			0
$865$	−653.720			−0.755746
$866$			441.796i			0.510157i
$867$		0			0
$868$	327.416	+	49.4109i	0.377207	+	0.0569250i
$869$	103.270			0.118838
$870$		0			0
$871$		−	518.995i		−	0.595861i
$872$	430.021			0.493143
$873$		0			0
$874$		−	145.651i		−	0.166649i
$875$	−77.3861	−	11.6785i	−0.0884413	−	0.0133468i
$876$		0			0
$877$	−792.582			−0.903743			−0.451871	−	0.892083i	$-0.649243\pi$
$877$	−792.582			−0.903743			−0.451871	+	0.892083i	$0.649243\pi$
$878$		−	102.457i		−	0.116694i
$879$		0			0
$880$			9.01633i			0.0102458i
$881$		−	260.584i		−	0.295782i	−0.989004	−	0.147891i	$-0.952752\pi$
$881$		−	260.584i		−	0.295782i	0.989004	−	0.147891i	$-0.0472485\pi$
$882$		0			0
$883$	25.2235			0.0285657			0.0142828	−	0.999898i	$-0.495453\pi$
$883$	25.2235			0.0285657			0.0142828	+	0.999898i	$0.495453\pi$
$884$	100.041			0.113168
$885$		0			0
$886$	−395.690			−0.446602
$887$			1430.63i			1.61289i	0.591308	+	0.806446i	$0.298612\pi$
$887$			1430.63i			1.61289i	−0.591308	+	0.806446i	$0.701388\pi$
$888$		0			0
$889$	−186.184	+	1233.72i	−0.209430	+	1.38776i
$890$	218.305			0.245287
$891$		0			0
$892$		−	876.855i		−	0.983022i
$893$	246.755			0.276321
$894$		0			0
$895$		−	611.013i		−	0.682696i
$896$	11.8178	−	78.3093i	0.0131895	−	0.0873987i
$897$		0			0
$898$	−233.528			−0.260053
$899$			408.959i			0.454904i
$900$		0			0
$901$			530.638i			0.588944i
$902$			7.35923i			0.00815879i
$903$		0			0
$904$	633.510			0.700785
$905$	349.158			0.385810
$906$		0			0
$907$	−1208.91			−1.33287			−0.666436	−	0.745563i	$-0.732181\pi$
$907$	−1208.91			−1.33287			−0.666436	+	0.745563i	$0.732181\pi$
$908$			471.282i			0.519033i
$909$		0			0
$910$	23.2980	−	154.382i	0.0256022	−	0.169650i
$911$	−604.287			−0.663323			−0.331661	−	0.943398i	$-0.607609\pi$
$911$	−604.287			−0.663323			−0.331661	+	0.943398i	$0.607609\pi$
$912$		0			0
$913$		−	84.0947i		−	0.0921081i
$914$	547.534			0.599053
$915$		0			0
$916$		−	602.700i		−	0.657970i
$917$	859.087	+	129.646i	0.936845	+	0.141381i
$918$		0			0
$919$	264.211			0.287498			0.143749	−	0.989614i	$-0.454084\pi$
$919$	264.211			0.287498			0.143749	+	0.989614i	$0.454084\pi$
$920$		−	161.128i		−	0.175140i
$921$		0			0
$922$			104.933i			0.113810i
$923$		−	704.274i		−	0.763027i
$924$		0			0
$925$	146.086			0.157930
$926$	599.345			0.647240
$927$		0			0
$928$	97.8122			0.105401
$929$			828.146i			0.891439i	0.895173	+	0.445719i	$0.147052\pi$
$929$			828.146i			0.891439i	−0.895173	+	0.445719i	$0.852948\pi$
$930$		0			0
$931$	−58.4558	+	189.264i	−0.0627881	+	0.203291i
$932$	−97.9124			−0.105056
$933$		0			0
$934$			462.787i			0.495489i
$935$	15.9856			0.0170969
$936$		0			0
$937$			396.006i			0.422632i	0.977418	+	0.211316i	$0.0677749\pi$
$937$			396.006i			0.422632i	−0.977418	+	0.211316i	$0.932225\pi$
$938$	−108.698	+	720.276i	−0.115883	+	0.767885i
$939$		0			0
$940$	272.975			0.290399
$941$			457.424i			0.486105i	0.970013	+	0.243052i	$0.0781487\pi$
$941$			457.424i			0.486105i	−0.970013	+	0.243052i	$0.921851\pi$
$942$		0			0
$943$		−	131.515i		−	0.139464i
$944$			202.255i			0.214254i
$945$		0			0
$946$	65.0830			0.0687981
$947$	−203.596			−0.214991			−0.107495	−	0.994206i	$-0.534283\pi$
$947$	−203.596			−0.214991			−0.107495	+	0.994206i	$0.534283\pi$
$948$		0			0
$949$	−8.13590			−0.00857313
$950$		−	28.5853i		−	0.0300898i
$951$		0			0
$952$	−138.839	−	20.9525i	−0.145840	−	0.0220090i
$953$	−119.616			−0.125516			−0.0627578	−	0.998029i	$-0.519990\pi$
$953$	−119.616			−0.125516			−0.0627578	+	0.998029i	$0.519990\pi$
$954$		0			0
$955$		−	615.951i		−	0.644975i
$956$	−78.1932			−0.0817921
$957$		0			0
$958$		−	605.113i		−	0.631642i
$959$	−86.4303	+	572.720i	−0.0901254	+	0.597205i
$960$		0			0
$961$	401.600			0.417898
$962$			291.434i			0.302946i
$963$		0			0
$964$		−	326.673i		−	0.338873i
$965$			21.6689i			0.0224548i
$966$		0			0
$967$	427.535			0.442126			0.221063	−	0.975260i	$-0.429047\pi$
$967$	427.535			0.442126			0.221063	+	0.975260i	$0.429047\pi$
$968$	339.366			0.350584
$969$		0			0
$970$	−481.209			−0.496092
$971$		−	6.82181i		−	0.00702555i	−0.999994	−	0.00351278i	$-0.998882\pi$
$971$		−	6.82181i		−	0.00702555i	0.999994	−	0.00351278i	$-0.00111815\pi$
$972$		0			0
$973$	−779.593	−	117.650i	−0.801226	−	0.120915i
$974$	−1153.28			−1.18406
$975$		0			0
$976$		−	232.377i		−	0.238091i
$977$	−251.640			−0.257564			−0.128782	−	0.991673i	$-0.541107\pi$
$977$	−251.640			−0.257564			−0.128782	+	0.991673i	$0.541107\pi$
$978$		0			0
$979$			69.5904i			0.0710831i
$980$	−64.6673	+	209.376i	−0.0659871	+	0.213649i
$981$		0			0
$982$	9.29320			0.00946354
$983$			743.441i			0.756298i	0.925745	+	0.378149i	$0.123439\pi$
$983$			743.441i			0.756298i	−0.925745	+	0.378149i	$0.876561\pi$
$984$		0			0
$985$		−	145.577i		−	0.147794i
$986$		−	173.417i		−	0.175880i
$987$		0			0
$988$	57.0263			0.0577189
$989$	−1163.08			−1.17602
$990$		0			0
$991$	−579.815			−0.585080			−0.292540	−	0.956253i	$-0.594501\pi$
$991$	−579.815			−0.585080			−0.292540	+	0.956253i	$0.594501\pi$
$992$			133.794i			0.134873i
$993$		0			0
$994$	−147.503	+	977.410i	−0.148393	+	0.983310i
$995$	−492.306			−0.494780
$996$		0			0
$997$		−	723.701i		−	0.725878i	−0.931813	−	0.362939i	$-0.881773\pi$
$997$		−	723.701i		−	0.725878i	0.931813	−	0.362939i	$-0.118227\pi$
$998$	663.054			0.664383
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	630.3.f.c.181.1		8
3.2	odd	2		210.3.f.a.181.8	yes	8
7.6	odd	2	inner	630.3.f.c.181.3		8
12.11	even	2		1680.3.s.a.1441.3		8
15.2	even	4		1050.3.h.b.349.13		16
15.8	even	4		1050.3.h.b.349.4		16
15.14	odd	2		1050.3.f.b.601.2		8
21.20	even	2		210.3.f.a.181.5	✓	8
84.83	odd	2		1680.3.s.a.1441.5		8
105.62	odd	4		1050.3.h.b.349.12		16
105.83	odd	4		1050.3.h.b.349.5		16
105.104	even	2		1050.3.f.b.601.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.f.a.181.5	✓	8	21.20	even	2
210.3.f.a.181.8	yes	8	3.2	odd	2
630.3.f.c.181.1		8	1.1	even	1	trivial
630.3.f.c.181.3		8	7.6	odd	2	inner
1050.3.f.b.601.2		8	15.14	odd	2
1050.3.f.b.601.4		8	105.104	even	2
1050.3.h.b.349.4		16	15.8	even	4
1050.3.h.b.349.5		16	105.83	odd	4
1050.3.h.b.349.12		16	105.62	odd	4
1050.3.h.b.349.13		16	15.2	even	4
1680.3.s.a.1441.3		8	12.11	even	2
1680.3.s.a.1441.5		8	84.83	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:	\( N \)	\(=\)	\( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	630.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +(-1.04456 + 6.92163i) q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} -2.23607i q^{5} +(-1.04456 + 6.92163i) q^{7} -2.82843 q^{8} +3.16228i q^{10} -1.00806 q^{11} +7.05322i q^{13} +(1.47723 - 9.78866i) q^{14} +4.00000 q^{16} -7.09185i q^{17} -4.04257i q^{19} -4.47214i q^{20} +1.42561 q^{22} -25.4766 q^{23} -5.00000 q^{25} -9.97476i q^{26} +(-2.08911 + 13.8433i) q^{28} -17.2909 q^{29} -23.6516i q^{31} -5.65685 q^{32} +10.0294i q^{34} +(15.4772 + 2.33570i) q^{35} -29.2171 q^{37} +5.71705i q^{38} +6.32456i q^{40} +5.16217i q^{41} +45.6529 q^{43} -2.01611 q^{44} +36.0294 q^{46} +61.0391i q^{47} +(-46.8178 - 14.4601i) q^{49} +7.07107 q^{50} +14.1064i q^{52} -74.8237 q^{53} +2.25408i q^{55} +(2.95445 - 19.5773i) q^{56} +24.4531 q^{58} +50.5639i q^{59} -58.0943i q^{61} +33.4485i q^{62} +8.00000 q^{64} +15.7715 q^{65} -73.5827 q^{67} -14.1837i q^{68} +(-21.8881 - 3.30318i) q^{70} -99.8513 q^{71} +1.15350i q^{73} +41.3192 q^{74} -8.08514i q^{76} +(1.05297 - 6.97738i) q^{77} -102.445 q^{79} -8.94427i q^{80} -7.30042i q^{82} +83.4227i q^{83} -15.8579 q^{85} -64.5629 q^{86} +2.85121 q^{88} -69.0342i q^{89} +(-48.8198 - 7.36749i) q^{91} -50.9533 q^{92} -86.3223i q^{94} -9.03946 q^{95} +152.172i q^{97} +(66.2104 + 20.4496i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 16 q^{11} - 32 q^{14} + 32 q^{16} + 96 q^{22} - 40 q^{25} + 144 q^{29} + 80 q^{35} - 48 q^{37} - 64 q^{43} + 32 q^{44} + 128 q^{46} - 24 q^{49} - 128 q^{53} - 64 q^{56} + 224 q^{58} + 64 q^{64} - 80 q^{65} - 192 q^{67} - 176 q^{71} + 160 q^{74} - 192 q^{77} - 288 q^{79} - 240 q^{85} + 64 q^{86} + 192 q^{88} + 64 q^{91} - 80 q^{95}+O(q^{100}) \) 8 * q + 16 * q^4 + 16 * q^11 - 32 * q^14 + 32 * q^16 + 96 * q^22 - 40 * q^25 + 144 * q^29 + 80 * q^35 - 48 * q^37 - 64 * q^43 + 32 * q^44 + 128 * q^46 - 24 * q^49 - 128 * q^53 - 64 * q^56 + 224 * q^58 + 64 * q^64 - 80 * q^65 - 192 * q^67 - 176 * q^71 + 160 * q^74 - 192 * q^77 - 288 * q^79 - 240 * q^85 + 64 * q^86 + 192 * q^88 + 64 * q^91 - 80 * q^95

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