LMFDB - Embedded newform 36.4.e.a.25.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,4,Mod(13,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("36.13");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 36 = 2^{2} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	36.e (of order $3$, degree $2$, 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:	$2.12406876021$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.6831243.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 13x^{4} + 49x^{2} + 48 $ x^6 + 13x^4 + 49x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			25.1
Root			$-1.23396i$ of defining polynomial
Character	$\chi$	$=$	36.25
Dual form			36.4.e.a.13.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.14739 + 3.13036i) q^{3} +(6.92194 + 11.9892i) q^{5} +(-15.3540 + 26.5939i) q^{7} +(7.40171 - 25.9656i) q^{9} +O(q^{10})$	\(q+(-4.14739 + 3.13036i) q^{3} +(6.92194 + 11.9892i) q^{5} +(-15.3540 + 26.5939i) q^{7} +(7.40171 - 25.9656i) q^{9} +(21.9523 - 38.0225i) q^{11} +(6.11853 + 10.5976i) q^{13} +(-66.2383 - 28.0555i) q^{15} +76.0286 q^{17} -44.1789 q^{19} +(-19.5694 - 158.359i) q^{21} +(39.3135 + 68.0930i) q^{23} +(-33.3265 + 57.7232i) q^{25} +(50.5840 + 130.860i) q^{27} +(46.3903 - 80.3504i) q^{29} +(71.5329 + 123.899i) q^{31} +(27.9793 + 226.413i) q^{33} -425.118 q^{35} -32.4741 q^{37} +(-58.5502 - 24.7992i) q^{39} +(167.778 + 290.599i) q^{41} +(249.166 - 431.567i) q^{43} +(362.540 - 90.9925i) q^{45} +(140.882 - 244.016i) q^{47} +(-299.990 - 519.599i) q^{49} +(-315.320 + 237.997i) q^{51} -628.565 q^{53} +607.810 q^{55} +(183.227 - 138.296i) q^{57} +(-252.327 - 437.043i) q^{59} +(-185.951 + 322.076i) q^{61} +(576.882 + 595.517i) q^{63} +(-84.7041 + 146.712i) q^{65} +(81.3304 + 140.868i) q^{67} +(-376.204 - 159.343i) q^{69} +433.512 q^{71} -629.645 q^{73} +(-42.4763 - 343.725i) q^{75} +(674.111 + 1167.59i) q^{77} +(-86.3649 + 149.588i) q^{79} +(-619.429 - 384.380i) q^{81} +(87.4288 - 151.431i) q^{83} +(526.265 + 911.518i) q^{85} +(59.1267 + 478.463i) q^{87} +336.716 q^{89} -375.775 q^{91} +(-684.522 - 289.933i) q^{93} +(-305.804 - 529.668i) q^{95} +(-42.1594 + 73.0222i) q^{97} +(-824.794 - 851.437i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 + 6 * q^5 - 6 * q^7 + 39 * q^9	$ 6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9} + 51 q^{11} + 12 q^{13} - 180 q^{15} - 222 q^{17} + 30 q^{19} - 120 q^{21} + 210 q^{23} - 3 q^{25} + 648 q^{27} + 456 q^{29} + 48 q^{31} - 603 q^{33} - 1104 q^{35} - 96 q^{37} - 36 q^{39} + 897 q^{41} + 129 q^{43} + 1494 q^{45} + 522 q^{47} - 225 q^{49} - 1647 q^{51} - 2208 q^{53} - 216 q^{55} - 645 q^{57} + 453 q^{59} - 402 q^{61} + 1896 q^{63} + 1110 q^{65} - 213 q^{67} - 198 q^{69} + 120 q^{71} + 750 q^{73} + 921 q^{75} + 1128 q^{77} + 552 q^{79} - 549 q^{81} - 612 q^{83} + 1188 q^{85} - 1386 q^{87} - 924 q^{89} - 264 q^{91} - 1998 q^{93} - 2184 q^{95} + 93 q^{97} - 1854 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 + 6 * q^5 - 6 * q^7 + 39 * q^9 + 51 * q^11 + 12 * q^13 - 180 * q^15 - 222 * q^17 + 30 * q^19 - 120 * q^21 + 210 * q^23 - 3 * q^25 + 648 * q^27 + 456 * q^29 + 48 * q^31 - 603 * q^33 - 1104 * q^35 - 96 * q^37 - 36 * q^39 + 897 * q^41 + 129 * q^43 + 1494 * q^45 + 522 * q^47 - 225 * q^49 - 1647 * q^51 - 2208 * q^53 - 216 * q^55 - 645 * q^57 + 453 * q^59 - 402 * q^61 + 1896 * q^63 + 1110 * q^65 - 213 * q^67 - 198 * q^69 + 120 * q^71 + 750 * q^73 + 921 * q^75 + 1128 * q^77 + 552 * q^79 - 549 * q^81 - 612 * q^83 + 1188 * q^85 - 1386 * q^87 - 924 * q^89 - 264 * q^91 - 1998 * q^93 - 2184 * q^95 + 93 * q^97 - 1854 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$.

$n$	$19$	$29$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	−4.14739	+	3.13036i	−0.798166	+	0.602438i
$3$	−4.14739	+	3.13036i	−0.798166	+	0.602438i
$4$		0			0
$5$	6.92194	+	11.9892i	0.619117	+	1.07234i	0.989647	+	0.143522i	$0.0458427\pi$
$5$	6.92194	+	11.9892i	0.619117	+	1.07234i	−0.370530	+	0.928821i	$0.620824\pi$
$6$		0			0
$7$	−15.3540	+	26.5939i	−0.829038	+	1.43594i	0.0697558	+	0.997564i	$0.477778\pi$
$7$	−15.3540	+	26.5939i	−0.829038	+	1.43594i	−0.898794	+	0.438372i	$0.855555\pi$
$8$		0			0
$9$	7.40171	−	25.9656i	0.274137	−	0.961691i
$10$		0			0
$11$	21.9523	−	38.0225i	0.601715	−	1.04220i	−0.390846	−	0.920456i	$-0.627818\pi$
$11$	21.9523	−	38.0225i	0.601715	−	1.04220i	0.992561	−	0.121745i	$-0.0388490\pi$
$12$		0			0
$13$	6.11853	+	10.5976i	0.130536	+	0.226096i	0.923883	−	0.382674i	$-0.124997\pi$
$13$	6.11853	+	10.5976i	0.130536	+	0.226096i	−0.793347	+	0.608770i	$0.791663\pi$
$14$		0			0
$15$	−66.2383	−	28.0555i	−1.14018	−	0.482927i
$16$		0			0
$17$	76.0286			1.08468			0.542342	−	0.840158i	$-0.317538\pi$
$17$	76.0286			1.08468			0.542342	+	0.840158i	$0.317538\pi$
$18$		0			0
$19$	−44.1789			−0.533439			−0.266720	−	0.963774i	$-0.585940\pi$
$19$	−44.1789			−0.533439			−0.266720	+	0.963774i	$0.585940\pi$
$20$		0			0
$21$	−19.5694	−	158.359i	−0.203352	−	1.64556i
$22$		0			0
$23$	39.3135	+	68.0930i	0.356410	+	0.617321i	0.987358	−	0.158504i	$-0.0506672\pi$
$23$	39.3135	+	68.0930i	0.356410	+	0.617321i	−0.630948	+	0.775825i	$0.717334\pi$
$24$		0			0
$25$	−33.3265	+	57.7232i	−0.266612	+	0.461786i
$26$		0			0
$27$	50.5840	+	130.860i	0.360552	+	0.932739i
$28$		0			0
$29$	46.3903	−	80.3504i	0.297050	−	0.514506i	−0.678409	−	0.734684i	$-0.737330\pi$
$29$	46.3903	−	80.3504i	0.297050	−	0.514506i	0.975460	+	0.220178i	$0.0706637\pi$
$30$		0			0
$31$	71.5329	+	123.899i	0.414442	+	0.717834i	0.995370	−	0.0961208i	$-0.0306435\pi$
$31$	71.5329	+	123.899i	0.414442	+	0.717834i	−0.580928	+	0.813955i	$0.697310\pi$
$32$		0			0
$33$	27.9793	+	226.413i	0.147593	+	1.19434i
$34$		0			0
$35$	−425.118			−2.05309
$36$		0			0
$37$	−32.4741			−0.144289			−0.0721447	−	0.997394i	$-0.522984\pi$
$37$	−32.4741			−0.144289			−0.0721447	+	0.997394i	$0.522984\pi$
$38$		0			0
$39$	−58.5502	−	24.7992i	−0.240398	−	0.101822i
$40$		0			0
$41$	167.778	+	290.599i	0.639084	+	1.10693i	0.985634	+	0.168895i	$0.0540197\pi$
$41$	167.778	+	290.599i	0.639084	+	1.10693i	−0.346550	+	0.938031i	$0.612647\pi$
$42$		0			0
$43$	249.166	−	431.567i	0.883660	−	1.53054i	0.0364186	−	0.999337i	$-0.488405\pi$
$43$	249.166	−	431.567i	0.883660	−	1.53054i	0.847242	−	0.531208i	$-0.178262\pi$
$44$		0			0
$45$	362.540	−	90.9925i	1.20098	−	0.301430i
$46$		0			0
$47$	140.882	−	244.016i	0.437230	−	0.757305i	−0.560244	−	0.828327i	$-0.689293\pi$
$47$	140.882	−	244.016i	0.437230	−	0.757305i	0.997475	+	0.0710223i	$0.0226261\pi$
$48$		0			0
$49$	−299.990	−	519.599i	−0.874608	−	1.51487i
$50$		0			0
$51$	−315.320	+	237.997i	−0.865758	+	0.653455i
$52$		0			0
$53$	−628.565			−1.62906			−0.814529	−	0.580123i	$-0.803005\pi$
$53$	−628.565			−1.62906			−0.814529	+	0.580123i	$0.803005\pi$
$54$		0			0
$55$	607.810			1.49013
$56$		0			0
$57$	183.227	−	138.296i	0.425773	−	0.321364i
$58$		0			0
$59$	−252.327	−	437.043i	−0.556782	−	0.964375i	−0.997762	−	0.0668584i	$-0.978702\pi$
$59$	−252.327	−	437.043i	−0.556782	−	0.964375i	0.440980	−	0.897517i	$-0.354631\pi$
$60$		0			0
$61$	−185.951	+	322.076i	−0.390304	+	0.676026i	−0.992489	−	0.122330i	$-0.960963\pi$
$61$	−185.951	+	322.076i	−0.390304	+	0.676026i	0.602186	+	0.798356i	$0.294297\pi$
$62$		0			0
$63$	576.882	+	595.517i	1.15366	+	1.19092i
$64$		0			0
$65$	−84.7041	+	146.712i	−0.161635	+	0.279960i
$66$		0			0
$67$	81.3304	+	140.868i	0.148300	+	0.256863i	0.930599	−	0.366040i	$-0.119287\pi$
$67$	81.3304	+	140.868i	0.148300	+	0.256863i	−0.782299	+	0.622903i	$0.785953\pi$
$68$		0			0
$69$	−376.204	−	159.343i	−0.656372	−	0.278009i
$70$		0			0
$71$	433.512			0.724626			0.362313	−	0.932056i	$-0.381987\pi$
$71$	433.512			0.724626			0.362313	+	0.932056i	$0.381987\pi$
$72$		0			0
$73$	−629.645			−1.00951			−0.504756	−	0.863262i	$-0.668418\pi$
$73$	−629.645			−1.00951			−0.504756	+	0.863262i	$0.668418\pi$
$74$		0			0
$75$	−42.4763	−	343.725i	−0.0653965	−	0.529199i
$76$		0			0
$77$	674.111	+	1167.59i	0.997689	+	1.72805i
$78$		0			0
$79$	−86.3649	+	149.588i	−0.122998	+	0.213038i	−0.920948	−	0.389684i	$-0.872584\pi$
$79$	−86.3649	+	149.588i	−0.122998	+	0.213038i	0.797951	+	0.602723i	$0.205917\pi$
$80$		0			0
$81$	−619.429	−	384.380i	−0.849697	−	0.527271i
$82$		0			0
$83$	87.4288	−	151.431i	0.115621	−	0.200262i	−0.802407	−	0.596778i	$-0.796447\pi$
$83$	87.4288	−	151.431i	0.115621	−	0.200262i	0.918028	+	0.396516i	$0.129781\pi$
$84$		0			0
$85$	526.265	+	911.518i	0.671547	+	1.16315i
$86$		0			0
$87$	59.1267	+	478.463i	0.0728626	+	0.589616i
$88$		0			0
$89$	336.716			0.401032			0.200516	−	0.979690i	$-0.435738\pi$
$89$	336.716			0.401032			0.200516	+	0.979690i	$0.435738\pi$
$90$		0			0
$91$	−375.775			−0.432879
$92$		0			0
$93$	−684.522	−	289.933i	−0.763244	−	0.323275i
$94$		0			0
$95$	−305.804	−	529.668i	−0.330261	−	0.572029i
$96$		0			0
$97$	−42.1594	+	73.0222i	−0.0441303	+	0.0764358i	−0.887247	−	0.461295i	$-0.847385\pi$
$97$	−42.1594	+	73.0222i	−0.0441303	+	0.0764358i	0.843117	+	0.537731i	$0.180718\pi$
$98$		0			0
$99$	−824.794	−	851.437i	−0.837322	−	0.864370i
$100$		0			0
$101$	874.902	−	1515.38i	0.861941	−	1.49293i	−0.00811161	−	0.999967i	$-0.502582\pi$
$101$	874.902	−	1515.38i	0.861941	−	1.49293i	0.870053	−	0.492959i	$-0.164085\pi$
$102$		0			0
$103$	−55.6978	−	96.4714i	−0.0532822	−	0.0922875i	0.838154	−	0.545433i	$-0.183635\pi$
$103$	−55.6978	−	96.4714i	−0.0532822	−	0.0922875i	−0.891436	+	0.453146i	$0.850302\pi$
$104$		0			0
$105$	1763.13	−	1330.77i	1.63870	−	1.23686i
$106$		0			0
$107$	895.520			0.809095			0.404548	−	0.914517i	$-0.367429\pi$
$107$	895.520			0.809095			0.404548	+	0.914517i	$0.367429\pi$
$108$		0			0
$109$	−716.957			−0.630019			−0.315009	−	0.949089i	$-0.602008\pi$
$109$	−716.957			−0.630019			−0.315009	+	0.949089i	$0.602008\pi$
$110$		0			0
$111$	134.683	−	101.656i	0.115167	−	0.0869254i
$112$		0			0
$113$	−115.526	−	200.096i	−0.0961746	−	0.166579i	0.813924	−	0.580972i	$-0.197327\pi$
$113$	−115.526	−	200.096i	−0.0961746	−	0.166579i	−0.910098	+	0.414393i	$0.863994\pi$
$114$		0			0
$115$	−544.252	+	942.672i	−0.441320	+	0.764388i
$116$		0			0
$117$	320.461	−	80.4312i	0.253219	−	0.0635544i
$118$		0			0
$119$	−1167.34	+	2021.90i	−0.899245	+	1.55754i
$120$		0			0
$121$	−298.306	−	516.681i	−0.224122	−	0.388190i
$122$		0			0
$123$	−1605.52	−	680.025i	−1.17695	−	0.498502i
$124$		0			0
$125$	807.748			0.577978
$126$		0			0
$127$	1715.22			1.19843			0.599217	−	0.800586i	$-0.295478\pi$
$127$	1715.22			1.19843			0.599217	+	0.800586i	$0.295478\pi$
$128$		0			0
$129$	317.574	+	2569.86i	0.216750	+	1.75398i
$130$		0			0
$131$	−666.435	−	1154.30i	−0.444479	−	0.769859i	0.553537	−	0.832824i	$-0.313278\pi$
$131$	−666.435	−	1154.30i	−0.444479	−	0.769859i	−0.998016	+	0.0629650i	$0.979944\pi$
$132$		0			0
$133$	678.323	−	1174.89i	0.442241	−	0.765984i
$134$		0			0
$135$	−1218.76	+	1512.26i	−0.776992	+	0.964110i
$136$		0			0
$137$	−1259.49	+	2181.50i	−0.785443	+	1.36043i	0.143292	+	0.989680i	$0.454231\pi$
$137$	−1259.49	+	2181.50i	−0.785443	+	1.36043i	−0.928734	+	0.370746i	$0.879102\pi$
$138$		0			0
$139$	−311.442	−	539.433i	−0.190044	−	0.329166i	0.755220	−	0.655471i	$-0.227530\pi$
$139$	−311.442	−	539.433i	−0.190044	−	0.329166i	−0.945265	+	0.326305i	$0.894196\pi$
$140$		0			0
$141$	179.562	+	1453.04i	0.107247	+	0.867859i
$142$		0			0
$143$	537.263			0.314183
$144$		0			0
$145$	1284.44			0.735636
$146$		0			0
$147$	2870.71	+	1215.90i	1.61069	+	0.682217i
$148$		0			0
$149$	−1204.47	−	2086.21i	−0.662243	−	1.14704i	−0.980025	−	0.198875i	$-0.936271\pi$
$149$	−1204.47	−	2086.21i	−0.662243	−	1.14704i	0.317781	−	0.948164i	$-0.397062\pi$
$150$		0			0
$151$	−33.8218	+	58.5810i	−0.0182277	+	0.0315712i	−0.874995	−	0.484131i	$-0.839136\pi$
$151$	−33.8218	+	58.5810i	−0.0182277	+	0.0315712i	0.856768	+	0.515703i	$0.172469\pi$
$152$		0			0
$153$	562.741	−	1974.13i	0.297352	−	1.04313i
$154$		0			0
$155$	−990.293	+	1715.24i	−0.513176	+	0.888847i
$156$		0			0
$157$	2.19676	+	3.80490i	0.00111669	+	0.00193417i	0.866583	−	0.499033i	$-0.166311\pi$
$157$	2.19676	+	3.80490i	0.00111669	+	0.00193417i	−0.865467	+	0.500967i	$0.832978\pi$
$158$		0			0
$159$	2606.91	−	1967.64i	1.30026	−	0.981406i
$160$		0			0
$161$	−2414.48			−1.18191
$162$		0			0
$163$	−2863.88			−1.37617			−0.688087	−	0.725628i	$-0.741549\pi$
$163$	−2863.88			−1.37617			−0.688087	+	0.725628i	$0.741549\pi$
$164$		0			0
$165$	−2520.82	+	1902.66i	−1.18937	+	0.897710i
$166$		0			0
$167$	−214.757	−	371.970i	−0.0995114	−	0.172359i	0.811971	−	0.583698i	$-0.198395\pi$
$167$	−214.757	−	371.970i	−0.0995114	−	0.172359i	−0.911483	+	0.411339i	$0.865061\pi$
$168$		0			0
$169$	1023.63	−	1772.97i	0.465920	−	0.806998i
$170$		0			0
$171$	−327.000	+	1147.13i	−0.146236	+	0.513003i
$172$		0			0
$173$	1287.14	−	2229.40i	0.565663	−	0.979756i	−0.431325	−	0.902197i	$-0.641954\pi$
$173$	1287.14	−	2229.40i	0.565663	−	0.979756i	0.996988	−	0.0775599i	$-0.0247129\pi$
$174$		0			0
$175$	−1023.39	−	1772.56i	−0.442063	−	0.765676i
$176$		0			0
$177$	2414.60	+	1022.71i	1.02538	+	0.434305i
$178$		0			0
$179$	807.448			0.337159			0.168580	−	0.985688i	$-0.446082\pi$
$179$	807.448			0.337159			0.168580	+	0.985688i	$0.446082\pi$
$180$		0			0
$181$	4296.57			1.76443			0.882215	−	0.470847i	$-0.156052\pi$
$181$	4296.57			1.76443			0.882215	+	0.470847i	$0.156052\pi$
$182$		0			0
$183$	−237.003	−	1917.87i	−0.0957365	−	0.774714i
$184$		0			0
$185$	−224.784	−	389.337i	−0.0893321	−	0.154728i
$186$		0			0
$187$	1669.00	−	2890.79i	0.652671	−	1.13046i
$188$		0			0
$189$	−4256.74	−	663.993i	−1.63826	−	0.255547i
$190$		0			0
$191$	153.147	−	265.259i	0.0580176	−	0.100489i	−0.835558	−	0.549402i	$-0.814855\pi$
$191$	153.147	−	265.259i	0.0580176	−	0.100489i	0.893575	+	0.448913i	$0.148189\pi$
$192$		0			0
$193$	856.177	+	1482.94i	0.319321	+	0.553080i	0.980347	−	0.197283i	$-0.0632119\pi$
$193$	856.177	+	1482.94i	0.319321	+	0.553080i	−0.661026	+	0.750363i	$0.729879\pi$
$194$		0			0
$195$	−107.960	−	873.626i	−0.0396469	−	0.320829i
$196$		0			0
$197$	−263.636			−0.0953466			−0.0476733	−	0.998863i	$-0.515181\pi$
$197$	−263.636			−0.0953466			−0.0476733	+	0.998863i	$0.515181\pi$
$198$		0			0
$199$	−3835.87			−1.36642			−0.683211	−	0.730221i	$-0.739417\pi$
$199$	−3835.87			−1.36642			−0.683211	+	0.730221i	$0.739417\pi$
$200$		0			0
$201$	−778.278	−	329.643i	−0.273112	−	0.115678i
$202$		0			0
$203$	1424.55	+	2467.40i	0.492532	+	0.853091i
$204$		0			0
$205$	−2322.69	+	4023.02i	−0.791336	+	1.37063i
$206$		0			0
$207$	2059.07	−	516.796i	0.691377	−	0.173526i
$208$		0			0
$209$	−969.829	+	1679.79i	−0.320978	+	0.555951i
$210$		0			0
$211$	2034.66	+	3524.14i	0.663848	+	1.14982i	0.979596	+	0.200976i	$0.0644112\pi$
$211$	2034.66	+	3524.14i	0.663848	+	1.14982i	−0.315748	+	0.948843i	$0.602255\pi$
$212$		0			0
$213$	−1797.95	+	1357.05i	−0.578372	+	0.436542i
$214$		0			0
$215$	6898.84			2.18836
$216$		0			0
$217$	−4393.27			−1.37435
$218$		0			0
$219$	2611.38	−	1971.01i	0.805758	−	0.608168i
$220$		0			0
$221$	465.183	+	805.720i	0.141591	+	0.245243i
$222$		0			0
$223$	−3078.07	+	5331.38i	−0.924318	+	1.60097i	−0.131664	+	0.991294i	$0.542032\pi$
$223$	−3078.07	+	5331.38i	−0.924318	+	1.60097i	−0.792654	+	0.609672i	$0.791301\pi$
$224$		0			0
$225$	1252.15	+	1292.60i	0.371007	+	0.382991i
$226$		0			0
$227$	−12.1307	+	21.0110i	−0.00354689	+	0.00614339i	−0.867793	−	0.496925i	$-0.834462\pi$
$227$	−12.1307	+	21.0110i	−0.00354689	+	0.00614339i	0.864247	+	0.503069i	$0.167796\pi$
$228$		0			0
$229$	316.209	+	547.691i	0.0912476	+	0.158045i	0.908036	−	0.418891i	$-0.137581\pi$
$229$	316.209	+	547.691i	0.0912476	+	0.158045i	−0.816789	+	0.576937i	$0.804248\pi$
$230$		0			0
$231$	−6450.79	−	2732.26i	−1.83736	−	0.778223i
$232$		0			0
$233$	−3163.59			−0.889502			−0.444751	−	0.895654i	$-0.646708\pi$
$233$	−3163.59			−0.889502			−0.444751	+	0.895654i	$0.646708\pi$
$234$		0			0
$235$	3900.72			1.08279
$236$		0			0
$237$	−110.076	−	890.755i	−0.0301697	−	0.244138i
$238$		0			0
$239$	−1110.57	−	1923.57i	−0.300573	−	0.520608i	0.675693	−	0.737183i	$-0.263845\pi$
$239$	−1110.57	−	1923.57i	−0.300573	−	0.520608i	−0.976266	+	0.216575i	$0.930511\pi$
$240$		0			0
$241$	300.730	−	520.879i	0.0803805	−	0.139223i	−0.823033	−	0.567994i	$-0.807720\pi$
$241$	300.730	−	520.879i	0.0803805	−	0.139223i	0.903413	+	0.428771i	$0.141053\pi$
$242$		0			0
$243$	3772.26	−	344.861i	0.995847	−	0.0910406i
$244$		0			0
$245$	4153.03	−	7193.26i	1.08297	−	1.87576i
$246$		0			0
$247$	−270.310	−	468.191i	−0.0696332	−	0.120608i
$248$		0			0
$249$	111.432	+	901.727i	0.0283604	+	0.229497i
$250$		0			0
$251$	1350.71			0.339665			0.169833	−	0.985473i	$-0.445677\pi$
$251$	1350.71			0.339665			0.169833	+	0.985473i	$0.445677\pi$
$252$		0			0
$253$	3452.09			0.857830
$254$		0			0
$255$	−5036.01	−	2133.02i	−1.23673	−	0.523824i
$256$		0			0
$257$	3521.95	+	6100.20i	0.854838	+	1.48062i	0.876795	+	0.480865i	$0.159677\pi$
$257$	3521.95	+	6100.20i	0.854838	+	1.48062i	−0.0219564	+	0.999759i	$0.506990\pi$
$258$		0			0
$259$	498.607	−	863.613i	0.119621	−	0.207190i
$260$		0			0
$261$	−1742.98	−	1799.28i	−0.413363	−	0.426716i
$262$		0			0
$263$	−1162.13	+	2012.87i	−0.272472	+	0.471935i	−0.969494	−	0.245114i	$-0.921175\pi$
$263$	−1162.13	+	2012.87i	−0.272472	+	0.471935i	0.697022	+	0.717050i	$0.254508\pi$
$264$		0			0
$265$	−4350.89	−	7535.97i	−1.00858	−	1.74691i
$266$		0			0
$267$	−1396.49	+	1054.04i	−0.320090	+	0.241597i
$268$		0			0
$269$	−3675.87			−0.833167			−0.416584	−	0.909097i	$-0.636773\pi$
$269$	−3675.87			−0.833167			−0.416584	+	0.909097i	$0.636773\pi$
$270$		0			0
$271$	−1881.48			−0.421741			−0.210871	−	0.977514i	$-0.567630\pi$
$271$	−1881.48			−0.421741			−0.210871	+	0.977514i	$0.567630\pi$
$272$		0			0
$273$	1558.49	−	1176.31i	0.345509	−	0.260782i
$274$		0			0
$275$	1463.19	+	2534.31i	0.320849	+	0.555727i
$276$		0			0
$277$	−732.008	+	1267.87i	−0.158780	+	0.275015i	−0.934429	−	0.356149i	$-0.884089\pi$
$277$	−732.008	+	1267.87i	−0.158780	+	0.275015i	0.775649	+	0.631165i	$0.217423\pi$
$278$		0			0
$279$	3746.58	−	940.337i	0.803948	−	0.201780i
$280$		0			0
$281$	4116.28	−	7129.61i	0.873867	−	1.51358i	0.0159023	−	0.999874i	$-0.494938\pi$
$281$	4116.28	−	7129.61i	0.873867	−	1.51358i	0.857965	−	0.513709i	$-0.171729\pi$
$282$		0			0
$283$	−844.364	−	1462.48i	−0.177358	−	0.307192i	0.763617	−	0.645669i	$-0.223422\pi$
$283$	−844.364	−	1462.48i	−0.177358	−	0.307192i	−0.940975	+	0.338477i	$0.890088\pi$
$284$		0			0
$285$	2926.34	+	1239.46i	0.608215	+	0.257612i
$286$		0			0
$287$	−10304.2			−2.11930
$288$		0			0
$289$	867.343			0.176540
$290$		0			0
$291$	−53.7342	−	434.825i	−0.0108246	−	0.0875942i
$292$		0			0
$293$	−255.866	−	443.173i	−0.0510166	−	0.0883633i	0.839389	−	0.543531i	$-0.182913\pi$
$293$	−255.866	−	443.173i	−0.0510166	−	0.0883633i	−0.890406	+	0.455167i	$0.849579\pi$
$294$		0			0
$295$	3493.18	−	6050.37i	0.689427	−	1.19412i
$296$		0			0
$297$	6086.04	+	949.340i	1.18905	+	0.185476i
$298$		0			0
$299$	−481.082	+	833.258i	−0.0930491	+	0.161166i
$300$		0			0
$301$	7651.37	+	13252.6i	1.46518	+	2.53776i
$302$		0			0
$303$	1115.11	+	9023.61i	0.211423	+	1.71087i
$304$		0			0
$305$	−5148.55			−0.966575
$306$		0			0
$307$	−3171.98			−0.589690			−0.294845	−	0.955545i	$-0.595268\pi$
$307$	−3171.98			−0.589690			−0.294845	+	0.955545i	$0.595268\pi$
$308$		0			0
$309$	532.991	+	225.751i	0.0981255	+	0.0415615i
$310$		0			0
$311$	−2530.39	−	4382.76i	−0.461367	−	0.799112i	0.537662	−	0.843160i	$-0.319308\pi$
$311$	−2530.39	−	4382.76i	−0.461367	−	0.799112i	−0.999029	+	0.0440487i	$0.985974\pi$
$312$		0			0
$313$	4732.92	−	8197.66i	0.854698	−	1.48038i	−0.0222263	−	0.999753i	$-0.507075\pi$
$313$	4732.92	−	8197.66i	0.854698	−	1.48038i	0.876925	−	0.480628i	$-0.159591\pi$
$314$		0			0
$315$	−3146.60	+	11038.5i	−0.562828	+	1.97443i
$316$		0			0
$317$	−3208.57	+	5557.40i	−0.568489	+	0.984652i	0.428226	+	0.903671i	$0.359139\pi$
$317$	−3208.57	+	5557.40i	−0.568489	+	0.984652i	−0.996716	+	0.0809808i	$0.974195\pi$
$318$		0			0
$319$	−2036.75	−	3527.75i	−0.357479	−	0.619172i
$320$		0			0
$321$	−3714.07	+	2803.30i	−0.645792	+	0.487430i
$322$		0			0
$323$	−3358.86			−0.578613
$324$		0			0
$325$	−815.637			−0.139210
$326$		0			0
$327$	2973.50	−	2244.33i	0.502860	−	0.379547i
$328$		0			0
$329$	4326.22	+	7493.23i	0.724961	+	1.25567i
$330$		0			0
$331$	−3332.37	+	5771.83i	−0.553364	+	0.958455i	0.444665	+	0.895697i	$0.353323\pi$
$331$	−3332.37	+	5771.83i	−0.553364	+	0.958455i	−0.998029	+	0.0627575i	$0.980011\pi$
$332$		0			0
$333$	−240.364	+	843.211i	−0.0395551	+	0.138762i
$334$		0			0
$335$	−1125.93	+	1950.17i	−0.183630	+	0.318056i
$336$		0			0
$337$	−5244.05	−	9082.95i	−0.847660	−	1.46819i	−0.883291	−	0.468825i	$-0.844678\pi$
$337$	−5244.05	−	9082.95i	−0.847660	−	1.46819i	0.0356314	−	0.999365i	$-0.488656\pi$
$338$		0			0
$339$	1105.50	+	468.241i	0.177117	+	0.0750187i
$340$		0			0
$341$	6281.25			0.997503
$342$		0			0
$343$	7891.37			1.24226
$344$		0			0
$345$	−693.676	−	5613.33i	−0.108250	−	0.875976i
$346$		0			0
$347$	−1074.48	−	1861.05i	−0.166228	−	0.287915i	0.770863	−	0.637001i	$-0.219825\pi$
$347$	−1074.48	−	1861.05i	−0.166228	−	0.287915i	−0.937091	+	0.349086i	$0.886492\pi$
$348$		0			0
$349$	1388.53	−	2405.00i	0.212969	−	0.368874i	−0.739673	−	0.672966i	$-0.765020\pi$
$349$	1388.53	−	2405.00i	0.212969	−	0.368874i	0.952642	+	0.304093i	$0.0983532\pi$
$350$		0			0
$351$	−1077.30	+	1336.74i	−0.163823	+	0.203276i
$352$		0			0
$353$	−3898.72	+	6752.78i	−0.587841	+	1.01817i	0.406673	+	0.913574i	$0.366689\pi$
$353$	−3898.72	+	6752.78i	−0.587841	+	1.01817i	−0.994515	+	0.104597i	$0.966645\pi$
$354$		0			0
$355$	3000.75	+	5197.45i	0.448629	+	0.777047i
$356$		0			0
$357$	−1487.84	−	12039.8i	−0.220573	−	1.78491i
$358$		0			0
$359$	2309.31			0.339500			0.169750	−	0.985487i	$-0.445704\pi$
$359$	2309.31			0.339500			0.169750	+	0.985487i	$0.445704\pi$
$360$		0			0
$361$	−4907.22			−0.715443
$362$		0			0
$363$	2854.59	+	1209.07i	0.412747	+	0.174821i
$364$		0			0
$365$	−4358.36	−	7548.91i	−0.625006	−	1.08254i
$366$		0			0
$367$	3105.66	−	5379.16i	0.441728	−	0.765095i	−0.556090	−	0.831122i	$-0.687699\pi$
$367$	3105.66	−	5379.16i	0.441728	−	0.765095i	0.997818	+	0.0660271i	$0.0210324\pi$
$368$		0			0
$369$	8787.44	−	2205.52i	1.23972	−	0.311151i
$370$		0			0
$371$	9650.99	−	16716.0i	1.35055	−	2.33922i
$372$		0			0
$373$	−4349.61	−	7533.75i	−0.603792	−	1.04580i	−0.992241	−	0.124328i	$-0.960323\pi$
$373$	−4349.61	−	7533.75i	−0.603792	−	1.04580i	0.388450	−	0.921470i	$-0.373011\pi$
$374$		0			0
$375$	−3350.05	+	2528.54i	−0.461322	+	0.348196i
$376$		0			0
$377$	1135.36			0.155104
$378$		0			0
$379$	5954.77			0.807061			0.403531	−	0.914966i	$-0.367783\pi$
$379$	5954.77			0.807061			0.403531	+	0.914966i	$0.367783\pi$
$380$		0			0
$381$	−7113.69	+	5369.26i	−0.956550	+	0.721983i
$382$		0			0
$383$	−2128.35	−	3686.41i	−0.283952	−	0.491819i	0.688403	−	0.725329i	$-0.258312\pi$
$383$	−2128.35	−	3686.41i	−0.283952	−	0.491819i	−0.972354	+	0.233510i	$0.924979\pi$
$384$		0			0
$385$	−9332.31	+	16164.0i	−1.23537	+	2.13973i
$386$		0			0
$387$	−9361.67	−	9664.08i	−1.22967	−	1.26939i
$388$		0			0
$389$	−801.019	+	1387.41i	−0.104404	+	0.180834i	−0.913495	−	0.406851i	$-0.866627\pi$
$389$	−801.019	+	1387.41i	−0.104404	+	0.180834i	0.809090	+	0.587684i	$0.199960\pi$
$390$		0			0
$391$	2988.95	+	5177.02i	0.386593	+	0.669598i
$392$		0			0
$393$	6377.34	+	2701.15i	0.818560	+	0.346705i
$394$		0			0
$395$	−2391.25			−0.304600
$396$		0			0
$397$	−7987.87			−1.00982			−0.504912	−	0.863171i	$-0.668475\pi$
$397$	−7987.87			−1.00982			−0.504912	+	0.863171i	$0.668475\pi$
$398$		0			0
$399$	864.557	+	6996.13i	0.108476	+	0.877805i
$400$		0			0
$401$	3534.75	+	6122.36i	0.440191	+	0.762434i	0.997703	−	0.0677350i	$-0.0215772\pi$
$401$	3534.75	+	6122.36i	0.440191	+	0.762434i	−0.557512	+	0.830169i	$0.688244\pi$
$402$		0			0
$403$	−875.352	+	1516.15i	−0.108200	+	0.187407i
$404$		0			0
$405$	320.739	−	10087.1i	0.0393523	−	1.23761i
$406$		0			0
$407$	−712.881	+	1234.75i	−0.0868211	+	0.150379i
$408$		0			0
$409$	5932.38	+	10275.2i	0.717207	+	1.24224i	0.962102	+	0.272689i	$0.0879129\pi$
$409$	5932.38	+	10275.2i	0.717207	+	1.24224i	−0.244896	+	0.969549i	$0.578754\pi$
$410$		0			0
$411$	−1605.28	−	12990.2i	−0.192659	−	1.55903i
$412$		0			0
$413$	15496.9			1.84637
$414$		0			0
$415$	2420.71			0.286332
$416$		0			0
$417$	2980.29	+	1262.31i	0.349989	+	0.148239i
$418$		0			0
$419$	6171.30	+	10689.0i	0.719541	+	1.24628i	0.961182	+	0.275916i	$0.0889810\pi$
$419$	6171.30	+	10689.0i	0.719541	+	1.24628i	−0.241641	+	0.970366i	$0.577686\pi$
$420$		0			0
$421$	−2732.93	+	4733.58i	−0.316377	+	0.547982i	−0.979729	−	0.200326i	$-0.935800\pi$
$421$	−2732.93	+	4733.58i	−0.316377	+	0.547982i	0.663352	+	0.748308i	$0.269133\pi$
$422$		0			0
$423$	−5293.25	−	5464.24i	−0.608432	−	0.628086i
$424$		0			0
$425$	−2533.77	+	4388.61i	−0.289190	+	0.500892i
$426$		0			0
$427$	−5710.17	−	9890.30i	−0.647153	−	1.12090i
$428$		0			0
$429$	−2228.24	+	1681.82i	−0.250770	+	0.189276i
$430$		0			0
$431$	8997.67			1.00557			0.502787	−	0.864410i	$-0.332308\pi$
$431$	8997.67			1.00557			0.502787	+	0.864410i	$0.332308\pi$
$432$		0			0
$433$	−10967.4			−1.21723			−0.608615	−	0.793466i	$-0.708275\pi$
$433$	−10967.4			−1.21723			−0.608615	+	0.793466i	$0.708275\pi$
$434$		0			0
$435$	−5327.09	+	4020.77i	−0.587160	+	0.443175i
$436$		0			0
$437$	−1736.83	−	3008.28i	−0.190123	−	0.329303i
$438$		0			0
$439$	−3563.51	+	6172.18i	−0.387419	+	0.671029i	−0.992102	−	0.125437i	$-0.959967\pi$
$439$	−3563.51	+	6172.18i	−0.387419	+	0.671029i	0.604683	+	0.796467i	$0.293300\pi$
$440$		0			0
$441$	−15712.2	+	3943.53i	−1.69659	+	0.425821i
$442$		0			0
$443$	−4434.28	+	7680.39i	−0.475573	+	0.823716i	−0.999608	−	0.0279799i	$-0.991093\pi$
$443$	−4434.28	+	7680.39i	−0.475573	+	0.823716i	0.524036	+	0.851696i	$0.324426\pi$
$444$		0			0
$445$	2330.73	+	4036.94i	0.248286	+	0.430043i
$446$		0			0
$447$	11526.0	+	4881.89i	1.21960	+	0.516567i
$448$		0			0
$449$	9835.05			1.03373			0.516865	−	0.856067i	$-0.327099\pi$
$449$	9835.05			1.03373			0.516865	+	0.856067i	$0.327099\pi$
$450$		0			0
$451$	14732.4			1.53819
$452$		0			0
$453$	−43.1075	−	348.833i	−0.00447101	−	0.0361801i
$454$		0			0
$455$	−2601.09	−	4505.23i	−0.268003	−	0.464194i
$456$		0			0
$457$	4443.25	−	7695.93i	0.454806	−	0.787747i	−0.543871	−	0.839169i	$-0.683042\pi$
$457$	4443.25	−	7695.93i	0.454806	−	0.787747i	0.998677	+	0.0514218i	$0.0163753\pi$
$458$		0			0
$459$	3845.83	+	9949.07i	0.391085	+	1.01173i
$460$		0			0
$461$	−932.894	+	1615.82i	−0.0942499	+	0.163246i	−0.909295	−	0.416151i	$-0.863379\pi$
$461$	−932.894	+	1615.82i	−0.0942499	+	0.163246i	0.815045	+	0.579397i	$0.196712\pi$
$462$		0			0
$463$	6811.70	+	11798.2i	0.683729	+	1.18425i	0.973834	+	0.227259i	$0.0729763\pi$
$463$	6811.70	+	11798.2i	0.683729	+	1.18425i	−0.290105	+	0.956995i	$0.593690\pi$
$464$		0			0
$465$	−1262.18	−	10213.7i	−0.125875	−	1.01860i
$466$		0			0
$467$	−9328.97			−0.924397			−0.462199	−	0.886776i	$-0.652939\pi$
$467$	−9328.97			−0.924397			−0.462199	+	0.886776i	$0.652939\pi$
$468$		0			0
$469$	−4994.99			−0.491785
$470$		0			0
$471$	−21.0215	−	8.90377i	−0.00205652	−	0.000871048i
$472$		0			0
$473$	−10939.5	−	18947.8i	−1.06342	−	1.84190i
$474$		0			0
$475$	1472.33	−	2550.15i	0.142221	−	0.246335i
$476$		0			0
$477$	−4652.46	+	16321.1i	−0.446586	+	1.56665i
$478$		0			0
$479$	−1330.15	+	2303.89i	−0.126881	+	0.219765i	−0.922467	−	0.386077i	$-0.873830\pi$
$479$	−1330.15	+	2303.89i	−0.126881	+	0.219765i	0.795586	+	0.605841i	$0.207163\pi$
$480$		0			0
$481$	−198.694	−	344.147i	−0.0188350	−	0.0326232i
$482$		0			0
$483$	10013.8	−	7558.19i	0.943361	−	0.712028i
$484$		0			0
$485$	−1167.30			−0.109287
$486$		0			0
$487$	−20450.7			−1.90289			−0.951447	−	0.307813i	$-0.900403\pi$
$487$	−20450.7			−1.90289			−0.951447	+	0.307813i	$0.900403\pi$
$488$		0			0
$489$	11877.6	−	8964.97i	1.09842	−	0.829059i
$490$		0			0
$491$	−10327.4	−	17887.7i	−0.949228	−	1.64411i	−0.747055	−	0.664762i	$-0.768533\pi$
$491$	−10327.4	−	17887.7i	−0.949228	−	1.64411i	−0.202173	−	0.979350i	$-0.564800\pi$
$492$		0			0
$493$	3526.99	−	6108.92i	0.322206	−	0.558077i
$494$		0			0
$495$	4498.83	−	15782.2i	0.408500	−	1.43304i
$496$		0			0
$497$	−6656.15	+	11528.8i	−0.600743	+	1.04052i
$498$		0			0
$499$	−8733.31	−	15126.5i	−0.783480	−	1.35703i	−0.929903	−	0.367805i	$-0.880109\pi$
$499$	−8733.31	−	15126.5i	−0.783480	−	1.35703i	0.146423	−	0.989222i	$-0.453224\pi$
$500$		0			0
$501$	2055.08	+	870.439i	0.183262	+	0.0776215i
$502$		0			0
$503$	19951.2			1.76855			0.884275	−	0.466966i	$-0.154653\pi$
$503$	19951.2			1.76855			0.884275	+	0.466966i	$0.154653\pi$
$504$		0			0
$505$	24224.1			2.13457
$506$		0			0
$507$	1304.66	+	10557.5i	0.114284	+	0.924806i
$508$		0			0
$509$	3638.88	+	6302.73i	0.316878	+	0.548848i	0.979835	−	0.199809i	$-0.0640322\pi$
$509$	3638.88	+	6302.73i	0.316878	+	0.548848i	−0.662957	+	0.748657i	$0.730699\pi$
$510$		0			0
$511$	9667.57	−	16744.7i	0.836924	−	1.44959i
$512$		0			0
$513$	−2234.75	−	5781.24i	−0.192332	−	0.497560i
$514$		0			0
$515$	771.074	−	1335.54i	0.0659759	−	0.114274i
$516$		0			0
$517$	−6185.39	−	10713.4i	−0.526176	−	0.911364i
$518$		0			0
$519$	1640.53	+	13275.4i	0.138750	+	1.12278i
$520$		0			0
$521$	−11664.5			−0.980866			−0.490433	−	0.871479i	$-0.663161\pi$
$521$	−11664.5			−0.980866			−0.490433	+	0.871479i	$0.663161\pi$
$522$		0			0
$523$	−8925.51			−0.746243			−0.373122	−	0.927782i	$-0.621713\pi$
$523$	−8925.51			−0.746243			−0.373122	+	0.927782i	$0.621713\pi$
$524$		0			0
$525$	9793.16	+	4147.94i	0.814112	+	0.344821i
$526$		0			0
$527$	5438.55	+	9419.84i	0.449539	+	0.778624i
$528$		0			0
$529$	2992.39	−	5182.98i	0.245943	−	0.425986i
$530$		0			0
$531$	−13215.8	+	3316.97i	−1.08007	+	0.271081i
$532$		0			0
$533$	−2053.10	+	3556.08i	−0.166848	+	0.288988i
$534$		0			0
$535$	6198.74	+	10736.5i	0.500925	+	0.867627i
$536$		0			0
$537$	−3348.80	+	2527.60i	−0.269109	+	0.203118i
$538$		0			0
$539$	−26341.9			−2.10506
$540$		0			0
$541$	12334.1			0.980193			0.490097	−	0.871668i	$-0.336962\pi$
$541$	12334.1			0.980193			0.490097	+	0.871668i	$0.336962\pi$
$542$		0			0
$543$	−17819.6	+	13449.8i	−1.40831	+	1.06296i
$544$		0			0
$545$	−4962.74	−	8595.71i	−0.390056	−	0.675596i
$546$		0			0
$547$	−2284.81	+	3957.41i	−0.178595	+	0.309336i	−0.941400	−	0.337293i	$-0.890489\pi$
$547$	−2284.81	+	3957.41i	−0.178595	+	0.309336i	0.762804	+	0.646629i	$0.223822\pi$
$548$		0			0
$549$	6986.55	+	7212.24i	0.543131	+	0.560675i
$550$		0			0
$551$	−2049.47	+	3549.79i	−0.158458	+	0.274458i
$552$		0			0
$553$	−2652.09	−	4593.56i	−0.203939	−	0.353233i
$554$		0			0
$555$	2151.03	+	911.079i	0.164516	+	0.0696813i
$556$		0			0
$557$	−8154.40			−0.620310			−0.310155	−	0.950686i	$-0.600381\pi$
$557$	−8154.40			−0.620310			−0.310155	+	0.950686i	$0.600381\pi$
$558$		0			0
$559$	6098.10			0.461399
$560$		0			0
$561$	2127.22	+	17213.8i	0.160092	+	1.29549i
$562$		0			0
$563$	8086.82	+	14006.8i	0.605362	+	1.04852i	0.991994	+	0.126284i	$0.0403050\pi$
$563$	8086.82	+	14006.8i	0.605362	+	1.04852i	−0.386632	+	0.922234i	$0.626362\pi$
$564$		0			0
$565$	1599.32	−	2770.11i	0.119087	−	0.206264i
$566$		0			0
$567$	19732.9	−	10571.3i	1.46156	−	0.782984i
$568$		0			0
$569$	4548.76	−	7878.69i	0.335139	−	0.580478i	−0.648372	−	0.761323i	$-0.724550\pi$
$569$	4548.76	−	7878.69i	0.335139	−	0.580478i	0.983512	+	0.180845i	$0.0578834\pi$
$570$		0			0
$571$	1647.54	+	2853.63i	0.120749	+	0.209143i	0.920063	−	0.391770i	$-0.128137\pi$
$571$	1647.54	+	2853.63i	0.120749	+	0.209143i	−0.799314	+	0.600913i	$0.794804\pi$
$572$		0			0
$573$	195.194	+	1579.54i	0.0142310	+	0.115159i
$574$		0			0
$575$	−5240.73			−0.380093
$576$		0			0
$577$	15544.7			1.12155			0.560775	−	0.827968i	$-0.310503\pi$
$577$	15544.7			1.12155			0.560775	+	0.827968i	$0.310503\pi$
$578$		0			0
$579$	−8193.04	−	3470.20i	−0.588067	−	0.249079i
$580$		0			0
$581$	2684.76	+	4650.14i	0.191709	+	0.332049i
$582$		0			0
$583$	−13798.4	+	23899.6i	−0.980229	+	1.69781i
$584$		0			0
$585$	3182.51	+	3285.32i	0.224924	+	0.232190i
$586$		0			0
$587$	2344.33	−	4060.50i	0.164840	−	0.285511i	−0.771759	−	0.635916i	$-0.780623\pi$
$587$	2344.33	−	4060.50i	0.164840	−	0.285511i	0.936598	+	0.350405i	$0.113956\pi$
$588$		0			0
$589$	−3160.25	−	5473.71i	−0.221079	−	0.382921i
$590$		0			0
$591$	1093.40	−	825.275i	0.0761024	−	0.0574404i
$592$		0			0
$593$	16637.6			1.15215			0.576075	−	0.817397i	$-0.304584\pi$
$593$	16637.6			1.15215			0.576075	+	0.817397i	$0.304584\pi$
$594$		0			0
$595$	−32321.1			−2.22695
$596$		0			0
$597$	15908.9	−	12007.7i	1.09063	−	0.823184i
$598$		0			0
$599$	2933.73	+	5081.37i	0.200115	+	0.346610i	0.948565	−	0.316581i	$-0.102535\pi$
$599$	2933.73	+	5081.37i	0.200115	+	0.346610i	−0.748450	+	0.663191i	$0.769202\pi$
$600$		0			0
$601$	−10337.9	+	17905.8i	−0.701649	+	1.21529i	0.266238	+	0.963907i	$0.414219\pi$
$601$	−10337.9	+	17905.8i	−0.701649	+	1.21529i	−0.967887	+	0.251385i	$0.919114\pi$
$602$		0			0
$603$	4259.72	−	1069.13i	0.287677	−	0.0722029i
$604$		0			0
$605$	4129.71	−	7152.87i	0.277515	−	0.480671i
$606$		0			0
$607$	−6303.86	−	10918.6i	−0.421525	−	0.730103i	0.574564	−	0.818460i	$-0.305172\pi$
$607$	−6303.86	−	10918.6i	−0.421525	−	0.730103i	−0.996089	+	0.0883567i	$0.971838\pi$
$608$		0			0
$609$	−13632.0	−	5773.90i	−0.907056	−	0.384188i
$610$		0			0
$611$	3447.97			0.228298
$612$		0			0
$613$	12469.7			0.821608			0.410804	−	0.911724i	$-0.365248\pi$
$613$	12469.7			0.821608			0.410804	+	0.911724i	$0.365248\pi$
$614$		0			0
$615$	−2960.39	−	23955.9i	−0.194104	−	1.57072i
$616$		0			0
$617$	−9293.48	−	16096.8i	−0.606388	−	1.05030i	−0.991830	−	0.127563i	$-0.959284\pi$
$617$	−9293.48	−	16096.8i	−0.606388	−	1.05030i	0.385442	−	0.922732i	$-0.374049\pi$
$618$		0			0
$619$	5972.96	−	10345.5i	0.387841	−	0.671761i	−0.604318	−	0.796743i	$-0.706554\pi$
$619$	5972.96	−	10345.5i	0.387841	−	0.671761i	0.992159	+	0.124983i	$0.0398876\pi$
$620$		0			0
$621$	−6922.00	+	8588.98i	−0.447295	+	0.555014i
$622$		0			0
$623$	−5169.94	+	8954.60i	−0.332471	+	0.575856i
$624$		0			0
$625$	9757.00	+	16899.6i	0.624448	+	1.08158i
$626$		0			0
$627$	−1236.09	−	10002.7i	−0.0787318	−	0.637110i
$628$		0			0
$629$	−2468.96			−0.156509
$630$		0			0
$631$	4285.35			0.270360			0.135180	−	0.990821i	$-0.456839\pi$
$631$	4285.35			0.270360			0.135180	+	0.990821i	$0.456839\pi$
$632$		0			0
$633$	−19470.4	−	8246.76i	−1.22256	−	0.517819i
$634$		0			0
$635$	11872.7	+	20564.0i	0.741972	+	1.28513i
$636$		0			0
$637$	3671.00	−	6358.36i	0.228336	−	0.395490i
$638$		0			0
$639$	3208.73	−	11256.4i	0.198647	−	0.696866i
$640$		0			0
$641$	10879.1	−	18843.2i	0.670357	−	1.16109i	−0.307446	−	0.951566i	$-0.599474\pi$
$641$	10879.1	−	18843.2i	0.670357	−	1.16109i	0.977803	−	0.209527i	$-0.0671925\pi$
$642$		0			0
$643$	3671.54	+	6359.30i	0.225181	+	0.390025i	0.956374	−	0.292146i	$-0.0943692\pi$
$643$	3671.54	+	6359.30i	0.225181	+	0.390025i	−0.731193	+	0.682171i	$0.761036\pi$
$644$		0			0
$645$	−28612.2	+	21595.8i	−1.74667	+	1.31835i
$646$		0			0
$647$	9483.24			0.576236			0.288118	−	0.957595i	$-0.406970\pi$
$647$	9483.24			0.576236			0.288118	+	0.957595i	$0.406970\pi$
$648$		0			0
$649$	−22156.6			−1.34010
$650$		0			0
$651$	18220.6	−	13752.5i	1.09696	−	0.827962i
$652$		0			0
$653$	−2392.09	−	4143.23i	−0.143353	−	0.248295i	0.785404	−	0.618984i	$-0.212455\pi$
$653$	−2392.09	−	4143.23i	−0.143353	−	0.248295i	−0.928757	+	0.370688i	$0.879122\pi$
$654$		0			0
$655$	9226.05	−	15980.0i	0.550369	−	0.953266i
$656$		0			0
$657$	−4660.45	+	16349.1i	−0.276745	+	0.970838i
$658$		0			0
$659$	16034.0	−	27771.6i	0.947791	−	1.64162i	0.197727	−	0.980257i	$-0.436644\pi$
$659$	16034.0	−	27771.6i	0.947791	−	1.64162i	0.750064	−	0.661365i	$-0.230023\pi$
$660$		0			0
$661$	3762.64	+	6517.09i	0.221407	+	0.383488i	0.955235	−	0.295847i	$-0.0956019\pi$
$661$	3762.64	+	6517.09i	0.221407	+	0.383488i	−0.733829	+	0.679335i	$0.762269\pi$
$662$		0			0
$663$	−4451.49	−	1885.45i	−0.260756	−	0.110445i
$664$		0			0
$665$	18781.3			1.09520
$666$		0			0
$667$	7295.07			0.423487
$668$		0			0
$669$	−3923.16	−	31746.8i	−0.226723	−	1.83468i
$670$		0			0
$671$	8164.08	+	14140.6i	0.469703	+	0.813550i
$672$		0			0
$673$	3670.98	−	6358.32i	0.210261	−	0.364183i	−0.741535	−	0.670914i	$-0.765902\pi$
$673$	3670.98	−	6358.32i	0.210261	−	0.364183i	0.951796	+	0.306731i	$0.0992352\pi$
$674$		0			0
$675$	−9239.43	−	1441.23i	−0.526853	−	0.0821819i
$676$		0			0
$677$	−4760.52	+	8245.45i	−0.270253	+	0.468092i	−0.968927	−	0.247348i	$-0.920441\pi$
$677$	−4760.52	+	8245.45i	−0.270253	+	0.468092i	0.698673	+	0.715441i	$0.253774\pi$
$678$		0			0
$679$	−1294.63	−	2242.36i	−0.0731713	−	0.126736i
$680$		0			0
$681$	−15.4612	−	125.114i	−0.000870006	−	0.00704022i
$682$		0			0
$683$	−1475.06			−0.0826377			−0.0413188	−	0.999146i	$-0.513156\pi$
$683$	−1475.06			−0.0826377			−0.0413188	+	0.999146i	$0.513156\pi$
$684$		0			0
$685$	−34872.5			−1.94512
$686$		0			0
$687$	−3025.91	−	1281.64i	−0.168043	−	0.0711755i
$688$		0			0
$689$	−3845.89	−	6661.28i	−0.212651	−	0.368323i
$690$		0			0
$691$	−17080.4	+	29584.0i	−0.940329	+	1.62870i	−0.175484	+	0.984482i	$0.556149\pi$
$691$	−17080.4	+	29584.0i	−0.940329	+	1.62870i	−0.764844	+	0.644215i	$0.777184\pi$
$692$		0			0
$693$	35306.9	−	8861.53i	1.93535	−	0.485746i
$694$		0			0
$695$	4311.56	−	7467.84i	0.235319	−	0.407585i
$696$		0			0
$697$	12755.9	+	22093.8i	0.693205	+	1.20067i
$698$		0			0
$699$	13120.7	−	9903.19i	0.709970	−	0.535870i
$700$		0			0
$701$	−29427.2			−1.58552			−0.792759	−	0.609535i	$-0.791356\pi$
$701$	−29427.2			−1.58552			−0.792759	+	0.609535i	$0.791356\pi$
$702$		0			0
$703$	1434.67			0.0769696
$704$		0			0
$705$	−16177.8	+	12210.7i	−0.864244	+	0.652312i
$706$		0			0
$707$	26866.5	+	46534.1i	1.42916	+	2.47538i
$708$		0			0
$709$	−9846.43	+	17054.5i	−0.521566	+	0.903379i	0.478119	+	0.878295i	$0.341319\pi$
$709$	−9846.43	+	17054.5i	−0.521566	+	0.903379i	−0.999685	+	0.0250840i	$0.992015\pi$
$710$		0			0
$711$	3244.91	+	3349.73i	0.171159	+	0.176687i
$712$		0			0
$713$	−5624.42	+	9741.79i	−0.295423	+	0.511687i
$714$		0			0
$715$	3718.90	+	6441.32i	0.194516	+	0.336912i
$716$		0			0
$717$	10627.4	+	4501.30i	0.553541	+	0.234455i
$718$		0			0
$719$	−33746.9			−1.75041			−0.875206	−	0.483751i	$-0.839274\pi$
$719$	−33746.9			−1.75041			−0.875206	+	0.483751i	$0.839274\pi$
$720$		0			0
$721$	3420.74			0.176692
$722$		0			0
$723$	383.295	+	3101.68i	0.0197163	+	0.159547i
$724$		0			0
$725$	3092.05	+	5355.60i	0.158395	+	0.274347i
$726$		0			0
$727$	−10380.4	+	17979.4i	−0.529558	+	0.917221i	0.469848	+	0.882748i	$0.344309\pi$
$727$	−10380.4	+	17979.4i	−0.529558	+	0.917221i	−0.999406	+	0.0344737i	$0.989025\pi$
$728$		0			0
$729$	−14565.5	+	13238.8i	−0.740005	+	0.672602i
$730$		0			0
$731$	18943.7	−	32811.4i	0.958492	−	1.66016i
$732$		0			0
$733$	−7122.02	−	12335.7i	−0.358878	−	0.621596i	0.628895	−	0.777490i	$-0.283507\pi$
$733$	−7122.02	−	12335.7i	−0.358878	−	0.621596i	−0.987774	+	0.155894i	$0.950174\pi$
$734$		0			0
$735$	5293.24	+	42833.8i	0.265638	+	2.14959i
$736$		0			0
$737$	7141.55			0.356937
$738$		0			0
$739$	27490.2			1.36839			0.684196	−	0.729298i	$-0.260153\pi$
$739$	27490.2			1.36839			0.684196	+	0.729298i	$0.260153\pi$
$740$		0			0
$741$	2586.69	+	1095.60i	0.128238	+	0.0543157i
$742$		0			0
$743$	−18234.0	−	31582.3i	−0.900326	−	1.55941i	−0.827072	−	0.562096i	$-0.809995\pi$
$743$	−18234.0	−	31582.3i	−0.900326	−	1.55941i	−0.0732537	−	0.997313i	$-0.523338\pi$
$744$		0			0
$745$	16674.6	−	28881.2i	0.820013	−	1.42030i
$746$		0			0
$747$	−3284.88	−	3390.99i	−0.160894	−	0.166091i
$748$		0			0
$749$	−13749.8	+	23815.4i	−0.670771	+	1.16181i
$750$		0			0
$751$	−19341.5	−	33500.5i	−0.939791	−	1.62777i	−0.765860	−	0.643007i	$-0.777686\pi$
$751$	−19341.5	−	33500.5i	−0.939791	−	1.62777i	−0.173931	−	0.984758i	$-0.555647\pi$
$752$		0			0
$753$	−5601.92	+	4228.20i	−0.271109	+	0.204627i
$754$		0			0
$755$	−936.449			−0.0451402
$756$		0			0
$757$	37768.4			1.81337			0.906683	−	0.421813i	$-0.138606\pi$
$757$	37768.4			1.81337			0.906683	+	0.421813i	$0.138606\pi$
$758$		0			0
$759$	−14317.2	+	10806.3i	−0.684690	+	0.516789i
$760$		0			0
$761$	10406.6	+	18024.7i	0.495714	+	0.858603i	0.999988	−	0.00494154i	$-0.00157295\pi$
$761$	10406.6	+	18024.7i	0.495714	+	0.858603i	−0.504273	+	0.863544i	$0.668240\pi$
$762$		0			0
$763$	11008.2	−	19066.7i	0.522310	−	0.904667i
$764$		0			0
$765$	27563.4	−	6918.03i	1.30269	−	0.326957i
$766$		0			0
$767$	3087.74	−	5348.12i	0.145361	−	0.251772i
$768$		0			0
$769$	−3060.85	−	5301.55i	−0.143533	−	0.248607i	0.785291	−	0.619126i	$-0.212513\pi$
$769$	−3060.85	−	5301.55i	−0.143533	−	0.248607i	−0.928825	+	0.370519i	$0.879180\pi$
$770$		0			0
$771$	−33702.7	−	14275.0i	−1.57429	−	0.666796i
$772$		0			0
$773$	−34251.2			−1.59370			−0.796850	−	0.604177i	$-0.793502\pi$
$773$	−34251.2			−1.59370			−0.796850	+	0.604177i	$0.793502\pi$
$774$		0			0
$775$	−9535.78			−0.441981
$776$		0			0
$777$	635.500	+	5142.56i	0.0293416	+	0.237437i
$778$		0			0
$779$	−7412.23	−	12838.4i	−0.340912	−	0.590478i
$780$		0			0
$781$	9516.59	−	16483.2i	0.436018	−	0.755206i
$782$		0			0
$783$	12861.2	+	2006.18i	0.587002	+	0.0915644i
$784$		0			0
$785$	−30.4117	+	52.6746i	−0.00138273	+	0.00239495i
$786$		0

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:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	36.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.14739 + 3.13036i) q^{3} +(6.92194 + 11.9892i) q^{5} +(-15.3540 + 26.5939i) q^{7} +(7.40171 - 25.9656i) q^{9} +O(q^{10})\)	\(q+(-4.14739 + 3.13036i) q^{3} +(6.92194 + 11.9892i) q^{5} +(-15.3540 + 26.5939i) q^{7} +(7.40171 - 25.9656i) q^{9} +(21.9523 - 38.0225i) q^{11} +(6.11853 + 10.5976i) q^{13} +(-66.2383 - 28.0555i) q^{15} +76.0286 q^{17} -44.1789 q^{19} +(-19.5694 - 158.359i) q^{21} +(39.3135 + 68.0930i) q^{23} +(-33.3265 + 57.7232i) q^{25} +(50.5840 + 130.860i) q^{27} +(46.3903 - 80.3504i) q^{29} +(71.5329 + 123.899i) q^{31} +(27.9793 + 226.413i) q^{33} -425.118 q^{35} -32.4741 q^{37} +(-58.5502 - 24.7992i) q^{39} +(167.778 + 290.599i) q^{41} +(249.166 - 431.567i) q^{43} +(362.540 - 90.9925i) q^{45} +(140.882 - 244.016i) q^{47} +(-299.990 - 519.599i) q^{49} +(-315.320 + 237.997i) q^{51} -628.565 q^{53} +607.810 q^{55} +(183.227 - 138.296i) q^{57} +(-252.327 - 437.043i) q^{59} +(-185.951 + 322.076i) q^{61} +(576.882 + 595.517i) q^{63} +(-84.7041 + 146.712i) q^{65} +(81.3304 + 140.868i) q^{67} +(-376.204 - 159.343i) q^{69} +433.512 q^{71} -629.645 q^{73} +(-42.4763 - 343.725i) q^{75} +(674.111 + 1167.59i) q^{77} +(-86.3649 + 149.588i) q^{79} +(-619.429 - 384.380i) q^{81} +(87.4288 - 151.431i) q^{83} +(526.265 + 911.518i) q^{85} +(59.1267 + 478.463i) q^{87} +336.716 q^{89} -375.775 q^{91} +(-684.522 - 289.933i) q^{93} +(-305.804 - 529.668i) q^{95} +(-42.1594 + 73.0222i) q^{97} +(-824.794 - 851.437i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 + 6 * q^5 - 6 * q^7 + 39 * q^9	\( 6 q - 3 q^{3} + 6 q^{5} - 6 q^{7} + 39 q^{9} + 51 q^{11} + 12 q^{13} - 180 q^{15} - 222 q^{17} + 30 q^{19} - 120 q^{21} + 210 q^{23} - 3 q^{25} + 648 q^{27} + 456 q^{29} + 48 q^{31} - 603 q^{33} - 1104 q^{35} - 96 q^{37} - 36 q^{39} + 897 q^{41} + 129 q^{43} + 1494 q^{45} + 522 q^{47} - 225 q^{49} - 1647 q^{51} - 2208 q^{53} - 216 q^{55} - 645 q^{57} + 453 q^{59} - 402 q^{61} + 1896 q^{63} + 1110 q^{65} - 213 q^{67} - 198 q^{69} + 120 q^{71} + 750 q^{73} + 921 q^{75} + 1128 q^{77} + 552 q^{79} - 549 q^{81} - 612 q^{83} + 1188 q^{85} - 1386 q^{87} - 924 q^{89} - 264 q^{91} - 1998 q^{93} - 2184 q^{95} + 93 q^{97} - 1854 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 + 6 * q^5 - 6 * q^7 + 39 * q^9 + 51 * q^11 + 12 * q^13 - 180 * q^15 - 222 * q^17 + 30 * q^19 - 120 * q^21 + 210 * q^23 - 3 * q^25 + 648 * q^27 + 456 * q^29 + 48 * q^31 - 603 * q^33 - 1104 * q^35 - 96 * q^37 - 36 * q^39 + 897 * q^41 + 129 * q^43 + 1494 * q^45 + 522 * q^47 - 225 * q^49 - 1647 * q^51 - 2208 * q^53 - 216 * q^55 - 645 * q^57 + 453 * q^59 - 402 * q^61 + 1896 * q^63 + 1110 * q^65 - 213 * q^67 - 198 * q^69 + 120 * q^71 + 750 * q^73 + 921 * q^75 + 1128 * q^77 + 552 * q^79 - 549 * q^81 - 612 * q^83 + 1188 * q^85 - 1386 * q^87 - 924 * q^89 - 264 * q^91 - 1998 * q^93 - 2184 * q^95 + 93 * q^97 - 1854 * q^99

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