LMFDB - Embedded newform 35.4.b.a.29.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [35,4,Mod(29,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("35.29");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 35 = 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	35.b (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:	$2.06506685020$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 $ x^10 + 55x^8 + 983x^6 + 6409x^4 + 13560x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			29.10
Root			$4.31366i$ of defining polynomial
Character	$\chi$	$=$	35.29
Dual form			35.4.b.a.29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.31366i q^{2} -1.93939i q^{3} -20.2350 q^{4} +(-2.32771 + 10.9353i) q^{5} +10.3052 q^{6} +7.00000i q^{7} -65.0123i q^{8} +23.2388 q^{9} +(-58.1067 - 12.3686i) q^{10} +25.5420 q^{11} +39.2434i q^{12} +64.1014i q^{13} -37.1956 q^{14} +(21.2079 + 4.51433i) q^{15} +183.574 q^{16} -27.6952i q^{17} +123.483i q^{18} -0.792436 q^{19} +(47.1011 - 221.276i) q^{20} +13.5757 q^{21} +135.721i q^{22} -108.606i q^{23} -126.084 q^{24} +(-114.164 - 50.9086i) q^{25} -340.613 q^{26} -97.4324i q^{27} -141.645i q^{28} +234.000 q^{29} +(-23.9876 + 112.691i) q^{30} +129.204 q^{31} +455.349i q^{32} -49.5357i q^{33} +147.163 q^{34} +(-76.5474 - 16.2940i) q^{35} -470.236 q^{36} +38.3108i q^{37} -4.21073i q^{38} +124.317 q^{39} +(710.932 + 151.330i) q^{40} -403.216 q^{41} +72.1366i q^{42} -172.895i q^{43} -516.840 q^{44} +(-54.0931 + 254.124i) q^{45} +577.097 q^{46} +206.943i q^{47} -356.020i q^{48} -49.0000 q^{49} +(270.511 - 606.626i) q^{50} -53.7117 q^{51} -1297.09i q^{52} -144.031i q^{53} +517.722 q^{54} +(-59.4542 + 279.310i) q^{55} +455.086 q^{56} +1.53684i q^{57} +1243.40i q^{58} +679.086 q^{59} +(-429.140 - 91.3471i) q^{60} -574.717 q^{61} +686.544i q^{62} +162.671i q^{63} -950.977 q^{64} +(-700.971 - 149.209i) q^{65} +263.216 q^{66} +515.640i q^{67} +560.411i q^{68} -210.630 q^{69} +(86.5805 - 406.747i) q^{70} +556.612 q^{71} -1510.81i q^{72} +173.243i q^{73} -203.571 q^{74} +(-98.7314 + 221.407i) q^{75} +16.0349 q^{76} +178.794i q^{77} +660.580i q^{78} -79.3290 q^{79} +(-427.306 + 2007.44i) q^{80} +438.488 q^{81} -2142.55i q^{82} -1043.56i q^{83} -274.704 q^{84} +(302.856 + 64.4663i) q^{85} +918.703 q^{86} -453.817i q^{87} -1660.54i q^{88} -652.060 q^{89} +(-1350.33 - 287.432i) q^{90} -448.710 q^{91} +2197.65i q^{92} -250.576i q^{93} -1099.62 q^{94} +(1.84456 - 8.66556i) q^{95} +883.097 q^{96} +515.714i q^{97} -260.369i q^{98} +593.564 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9} - 16 q^{10} + 84 q^{11} - 56 q^{14} + 8 q^{15} + 148 q^{16} + 72 q^{19} - 68 q^{20} + 140 q^{21} + 72 q^{24} - 362 q^{25} - 620 q^{26} + 88 q^{29} + 52 q^{30}+ \cdots - 5304 q^{99}+O(q^{100}) $ 10 * q - 36 * q^4 + 6 * q^5 + 12 * q^6 - 46 * q^9 - 16 * q^10 + 84 * q^11 - 56 * q^14 + 8 * q^15 + 148 * q^16 + 72 * q^19 - 68 * q^20 + 140 * q^21 + 72 * q^24 - 362 * q^25 - 620 * q^26 + 88 * q^29 + 52 * q^30 + 120 * q^31 + 964 * q^34 - 28 * q^35 - 420 * q^36 + 212 * q^39 + 1396 * q^40 - 852 * q^41 - 1424 * q^44 - 510 * q^45 + 176 * q^46 - 490 * q^49 + 1644 * q^50 + 1276 * q^51 - 996 * q^54 - 1136 * q^55 + 588 * q^56 + 864 * q^59 - 1692 * q^60 - 884 * q^61 - 2724 * q^64 + 520 * q^65 + 1148 * q^66 + 4808 * q^69 + 756 * q^70 + 880 * q^71 + 3024 * q^74 + 720 * q^75 - 2672 * q^76 - 3580 * q^79 - 2316 * q^80 - 302 * q^81 - 1904 * q^84 + 1572 * q^85 + 2432 * q^86 - 1492 * q^89 - 7748 * q^90 + 476 * q^91 + 1652 * q^94 - 1628 * q^95 + 4080 * q^96 - 5304 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$.

$n$	$22$	$31$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			5.31366i			1.87866i	0.343013	+	0.939331i	$0.388553\pi$
$2$			5.31366i			1.87866i	−0.343013	+	0.939331i	$0.611447\pi$
$3$		−	1.93939i		−	0.373235i	−0.982433	−	0.186617i	$-0.940247\pi$
$3$		−	1.93939i		−	0.373235i	0.982433	−	0.186617i	$-0.0597525\pi$
$4$	−20.2350			−2.52937
$5$	−2.32771	+	10.9353i	−0.208197	+	0.978087i
$5$	−2.32771	+	10.9353i	−0.208197	+	0.978087i
$6$	10.3052			0.701182
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$		−	65.0123i		−	2.87317i
$9$	23.2388			0.860696
$10$	−58.1067	−	12.3686i	−1.83749	−	0.391131i
$11$	25.5420			0.700108			0.350054	−	0.936730i	$-0.386163\pi$
$11$	25.5420			0.700108			0.350054	+	0.936730i	$0.386163\pi$
$12$			39.2434i			0.944049i
$13$			64.1014i			1.36758i	0.729679	+	0.683790i	$0.239670\pi$
$13$			64.1014i			1.36758i	−0.729679	+	0.683790i	$0.760330\pi$
$14$	−37.1956			−0.710067
$15$	21.2079	+	4.51433i	0.365056	+	0.0777063i
$16$	183.574			2.86834
$17$		−	27.6952i		−	0.395122i	−0.980291	−	0.197561i	$-0.936698\pi$
$17$		−	27.6952i		−	0.395122i	0.980291	−	0.197561i	$-0.0633020\pi$
$18$			123.483i			1.61696i
$19$	−0.792436			−0.00956828			−0.00478414	−	0.999989i	$-0.501523\pi$
$19$	−0.792436			−0.00956828			−0.00478414	+	0.999989i	$0.501523\pi$
$20$	47.1011	−	221.276i	0.526606	−	2.47394i
$21$	13.5757			0.141070
$22$			135.721i			1.31527i
$23$		−	108.606i		−	0.984609i	−0.870423	−	0.492305i	$-0.836155\pi$
$23$		−	108.606i		−	0.984609i	0.870423	−	0.492305i	$-0.163845\pi$
$24$	−126.084			−1.07237
$25$	−114.164	−	50.9086i	−0.913308	−	0.407269i
$26$	−340.613			−2.56922
$27$		−	97.4324i		−	0.694477i
$28$		−	141.645i		−	0.956012i
$29$	234.000			1.49837			0.749186	−	0.662360i	$-0.230445\pi$
$29$	234.000			1.49837			0.749186	+	0.662360i	$0.230445\pi$
$30$	−23.9876	+	112.691i	−0.145984	+	0.685817i
$31$	129.204			0.748570			0.374285	−	0.927314i	$-0.377888\pi$
$31$	129.204			0.748570			0.374285	+	0.927314i	$0.377888\pi$
$32$			455.349i			2.51547i
$33$		−	49.5357i		−	0.261305i
$34$	147.163			0.742300
$35$	−76.5474	−	16.2940i	−0.369682	−	0.0786909i
$36$	−470.236			−2.17702
$37$			38.3108i			0.170223i	0.996371	+	0.0851116i	$0.0271247\pi$
$37$			38.3108i			0.170223i	−0.996371	+	0.0851116i	$0.972875\pi$
$38$		−	4.21073i		−	0.0179756i
$39$	124.317			0.510429
$40$	710.932	+	151.330i	2.81021	+	0.598183i
$41$	−403.216			−1.53590			−0.767949	−	0.640511i	$-0.778722\pi$
$41$	−403.216			−1.53590			−0.767949	+	0.640511i	$0.778722\pi$
$42$			72.1366i			0.265022i
$43$		−	172.895i		−	0.613167i	−0.951844	−	0.306584i	$-0.900814\pi$
$43$		−	172.895i		−	0.613167i	0.951844	−	0.306584i	$-0.0991859\pi$
$44$	−516.840			−1.77083
$45$	−54.0931	+	254.124i	−0.179194	+	0.841835i
$46$	577.097			1.84975
$47$			206.943i			0.642250i	0.947037	+	0.321125i	$0.104061\pi$
$47$			206.943i			0.642250i	−0.947037	+	0.321125i	$0.895939\pi$
$48$		−	356.020i		−	1.07056i
$49$	−49.0000			−0.142857
$50$	270.511	−	606.626i	0.765120	−	1.71580i
$51$	−53.7117			−0.147473
$52$		−	1297.09i		−	3.45911i
$53$		−	144.031i		−	0.373287i	−0.982428	−	0.186643i	$-0.940239\pi$
$53$		−	144.031i		−	0.373287i	0.982428	−	0.186643i	$-0.0597609\pi$
$54$	517.722			1.30469
$55$	−59.4542	+	279.310i	−0.145760	+	0.684767i
$56$	455.086			1.08595
$57$			1.53684i			0.00357122i
$58$			1243.40i			2.81493i
$59$	679.086			1.49846			0.749232	−	0.662307i	$-0.230423\pi$
$59$	679.086			1.49846			0.749232	+	0.662307i	$0.230423\pi$
$60$	−429.140	−	91.3471i	−0.923362	−	0.196548i
$61$	−574.717			−1.20631			−0.603155	−	0.797624i	$-0.706090\pi$
$61$	−574.717			−1.20631			−0.603155	+	0.797624i	$0.706090\pi$
$62$			686.544i			1.40631i
$63$			162.671i			0.325312i
$64$	−950.977			−1.85738
$65$	−700.971	−	149.209i	−1.33761	−	0.284725i
$66$	263.216			0.490903
$67$			515.640i			0.940230i	0.882605	+	0.470115i	$0.155788\pi$
$67$			515.640i			0.940230i	−0.882605	+	0.470115i	$0.844212\pi$
$68$			560.411i			0.999409i
$69$	−210.630			−0.367491
$70$	86.5805	−	406.747i	0.147834	−	0.694508i
$71$	556.612			0.930391			0.465195	−	0.885208i	$-0.345984\pi$
$71$	556.612			0.930391			0.465195	+	0.885208i	$0.345984\pi$
$72$		−	1510.81i		−	2.47292i
$73$			173.243i			0.277762i	0.990309	+	0.138881i	$0.0443505\pi$
$73$			173.243i			0.277762i	−0.990309	+	0.138881i	$0.955650\pi$
$74$	−203.571			−0.319792
$75$	−98.7314	+	221.407i	−0.152007	+	0.340879i
$76$	16.0349			0.0242017
$77$			178.794i			0.264616i
$78$			660.580i			0.958923i
$79$	−79.3290			−0.112977			−0.0564887	−	0.998403i	$-0.517990\pi$
$79$	−79.3290			−0.112977			−0.0564887	+	0.998403i	$0.517990\pi$
$80$	−427.306	+	2007.44i	−0.597178	+	2.80548i
$81$	438.488			0.601493
$82$		−	2142.55i		−	2.88543i
$83$		−	1043.56i		−	1.38007i	−0.723777	−	0.690034i	$-0.757596\pi$
$83$		−	1043.56i		−	1.38007i	0.723777	−	0.690034i	$-0.242404\pi$
$84$	−274.704			−0.356817
$85$	302.856	+	64.4663i	0.386463	+	0.0822630i
$86$	918.703			1.15193
$87$		−	453.817i		−	0.559245i
$88$		−	1660.54i		−	2.01153i
$89$	−652.060			−0.776609			−0.388304	−	0.921531i	$-0.626939\pi$
$89$	−652.060			−0.776609			−0.388304	+	0.921531i	$0.626939\pi$
$90$	−1350.33	−	287.432i	−1.58152	−	0.336645i
$91$	−448.710			−0.516897
$92$			2197.65i			2.49044i
$93$		−	250.576i		−	0.279393i
$94$	−1099.62			−1.20657
$95$	1.84456	−	8.66556i	0.00199208	−	0.00935861i
$96$	883.097			0.938861
$97$			515.714i			0.539823i	0.962885	+	0.269912i	$0.0869945\pi$
$97$			515.714i			0.539823i	−0.962885	+	0.269912i	$0.913005\pi$
$98$		−	260.369i		−	0.268380i
$99$	593.564			0.602580
$100$	2310.09	+	1030.13i	2.31009	+	1.03013i
$101$	−536.339			−0.528394			−0.264197	−	0.964469i	$-0.585107\pi$
$101$	−536.339			−0.528394			−0.264197	+	0.964469i	$0.585107\pi$
$102$		−	285.405i		−	0.277052i
$103$		−	381.693i		−	0.365139i	−0.983193	−	0.182570i	$-0.941559\pi$
$103$		−	381.693i		−	0.365139i	0.983193	−	0.182570i	$-0.0584415\pi$
$104$	4167.38			3.92928
$105$	−31.6003	+	148.455i	−0.0293702	+	0.137978i
$106$	765.332			0.701279
$107$		−	1381.12i		−	1.24783i	−0.781492	−	0.623915i	$-0.785541\pi$
$107$		−	1381.12i		−	1.24783i	0.781492	−	0.623915i	$-0.214459\pi$
$108$			1971.54i			1.75659i
$109$	390.582			0.343220			0.171610	−	0.985165i	$-0.445103\pi$
$109$	390.582			0.343220			0.171610	+	0.985165i	$0.445103\pi$
$110$	−1484.16	−	315.919i	−1.28644	−	0.273834i
$111$	74.2995			0.0635333
$112$			1285.02i			1.08413i
$113$		−	1643.15i		−	1.36792i	−0.729521	−	0.683958i	$-0.760257\pi$
$113$		−	1643.15i		−	1.36792i	0.729521	−	0.683958i	$-0.239743\pi$
$114$	−8.16624			−0.00670911
$115$	1187.65	+	252.804i	0.963033	+	0.204992i
$116$	−4734.99			−3.78994
$117$			1489.64i			1.17707i
$118$			3608.43i			2.81511i
$119$	193.866			0.149342
$120$	293.487	−	1378.77i	0.223263	−	1.04887i
$121$	−678.609			−0.509849
$122$		−	3053.85i		−	2.26625i
$123$			781.992i			0.573251i
$124$	−2614.43			−1.89341
$125$	822.443	−	1129.92i	0.588492	−	0.808503i
$126$	−864.380			−0.611152
$127$		−	192.032i		−	0.134174i	−0.997747	−	0.0670869i	$-0.978630\pi$
$127$		−	192.032i		−	0.134174i	0.997747	−	0.0670869i	$-0.0213705\pi$
$128$		−	1410.38i		−	0.973914i
$129$	−335.310			−0.228856
$130$	792.848	−	3724.72i	0.534903	−	2.51292i
$131$	2082.90			1.38919			0.694594	−	0.719402i	$-0.255584\pi$
$131$	2082.90			1.38919			0.694594	+	0.719402i	$0.255584\pi$
$132$			1002.35i			0.660936i
$133$		−	5.54705i		−	0.00361647i
$134$	−2739.93			−1.76637
$135$	1065.46	+	226.794i	0.679259	+	0.144588i
$136$	−1800.53			−1.13525
$137$		−	78.1709i		−	0.0487488i	−0.999703	−	0.0243744i	$-0.992241\pi$
$137$		−	78.1709i		−	0.0487488i	0.999703	−	0.0243744i	$-0.00775939\pi$
$138$		−	1119.21i		−	0.690390i
$139$	−1393.67			−0.850426			−0.425213	−	0.905093i	$-0.639801\pi$
$139$	−1393.67			−0.850426			−0.425213	+	0.905093i	$0.639801\pi$
$140$	1548.93	+	329.707i	0.935062	+	0.199038i
$141$	401.342			0.239710
$142$			2957.65i			1.74789i
$143$			1637.28i			0.957454i
$144$	4266.03			2.46877
$145$	−544.685	+	2558.88i	−0.311956	+	1.46554i
$146$	−920.556			−0.521820
$147$			95.0299i			0.0533193i
$148$		−	775.217i		−	0.430557i
$149$	−32.5002			−0.0178693			−0.00893463	−	0.999960i	$-0.502844\pi$
$149$	−32.5002			−0.0178693			−0.00893463	+	0.999960i	$0.502844\pi$
$150$	−1176.48	−	524.625i	−0.640396	−	0.285570i
$151$	466.762			0.251553			0.125777	−	0.992059i	$-0.459858\pi$
$151$	466.762			0.251553			0.125777	+	0.992059i	$0.459858\pi$
$152$			51.5181i			0.0274913i
$153$		−	643.602i		−	0.340080i
$154$	−950.048			−0.497124
$155$	−300.749	+	1412.89i	−0.155850	+	0.732167i
$156$	−2515.56			−1.29106
$157$			1673.50i			0.850701i	0.905029	+	0.425351i	$0.139849\pi$
$157$			1673.50i			0.850701i	−0.905029	+	0.425351i	$0.860151\pi$
$158$		−	421.527i		−	0.212246i
$159$	−279.332			−0.139324
$160$	−4979.39	−	1059.92i	−2.46035	−	0.523712i
$161$	760.245			0.372147
$162$			2329.98i			1.13000i
$163$		−	1869.36i		−	0.898279i	−0.893462	−	0.449139i	$-0.851731\pi$
$163$		−	1869.36i		−	0.898279i	0.893462	−	0.449139i	$-0.148269\pi$
$164$	8159.06			3.88485
$165$	541.690	+	115.305i	0.255579	+	0.0544028i
$166$	5545.12			2.59268
$167$			46.5250i			0.0215581i	0.999942	+	0.0107791i	$0.00343115\pi$
$167$			46.5250i			0.0215581i	−0.999942	+	0.0107791i	$0.996569\pi$
$168$		−	882.588i		−	0.405316i
$169$	−1911.99			−0.870275
$170$	−342.552	+	1609.28i	−0.154544	+	0.726034i
$171$	−18.4152			−0.00823538
$172$			3498.52i			1.55093i
$173$		−	2496.28i		−	1.09704i	−0.836137	−	0.548521i	$-0.815191\pi$
$173$		−	2496.28i		−	1.09704i	0.836137	−	0.548521i	$-0.184809\pi$
$174$	2411.43			1.05063
$175$	356.360	−	799.145i	0.153933	−	0.345198i
$176$	4688.83			2.00815
$177$		−	1317.01i		−	0.559279i
$178$		−	3464.82i		−	1.45898i
$179$	−2975.70			−1.24254			−0.621269	−	0.783598i	$-0.713382\pi$
$179$	−2975.70			−1.24254			−0.621269	+	0.783598i	$0.713382\pi$
$180$	1094.57	−	5142.19i	0.453247	−	2.12931i
$181$	966.273			0.396809			0.198405	−	0.980120i	$-0.436424\pi$
$181$	966.273			0.396809			0.198405	+	0.980120i	$0.436424\pi$
$182$		−	2384.29i		−	0.971074i
$183$			1114.60i			0.450237i
$184$	−7060.76			−2.82895
$185$	−418.942	−	89.1764i	−0.166493	−	0.0354399i
$186$	1331.47			0.524884
$187$		−	707.389i		−	0.276628i
$188$		−	4187.48i		−	1.62449i
$189$	682.027			0.262488
$190$	46.0458	+	9.80136i	0.0175817	+	0.00374245i
$191$	−1545.50			−0.585488			−0.292744	−	0.956191i	$-0.594568\pi$
$191$	−1545.50			−0.585488			−0.292744	+	0.956191i	$0.594568\pi$
$192$			1844.31i			0.693238i
$193$			2304.05i			0.859322i	0.902990	+	0.429661i	$0.141367\pi$
$193$			2304.05i			0.859322i	−0.902990	+	0.429661i	$0.858633\pi$
$194$	−2740.33			−1.01415
$195$	−289.375	+	1359.45i	−0.106270	+	0.499244i
$196$	991.513			0.361338
$197$		−	222.021i		−	0.0802960i	−0.999194	−	0.0401480i	$-0.987217\pi$
$197$		−	222.021i		−	0.0802960i	0.999194	−	0.0401480i	$-0.0127829\pi$
$198$			3153.99i			1.13204i
$199$	−3580.56			−1.27547			−0.637736	−	0.770255i	$-0.720129\pi$
$199$	−3580.56			−1.27547			−0.637736	+	0.770255i	$0.720129\pi$
$200$	−3309.69	+	7422.04i	−1.17015	+	2.62409i
$201$	1000.02			0.350927
$202$		−	2849.92i		−	0.992673i
$203$			1638.00i			0.566331i
$204$	1086.85			0.373014
$205$	938.570	−	4409.31i	0.319769	−	1.50224i
$206$	2028.19			0.685973
$207$		−	2523.88i		−	0.847449i
$208$			11767.3i			3.92268i
$209$	−20.2404			−0.00669883
$210$	−788.839	−	167.913i	−0.259215	−	0.0551767i
$211$	−4181.04			−1.36415			−0.682073	−	0.731284i	$-0.738921\pi$
$211$	−4181.04			−1.36415			−0.682073	+	0.731284i	$0.738921\pi$
$212$			2914.46i			0.944180i
$213$		−	1079.49i		−	0.347254i
$214$	7338.79			2.34425
$215$	1890.66	+	402.449i	0.599731	+	0.127659i
$216$	−6334.31			−1.99535
$217$			904.426i			0.282933i
$218$			2075.42i			0.644794i
$219$	335.986			0.103670
$220$	1203.05	−	5651.83i	0.368681	−	1.73203i
$221$	1775.30			0.540361
$222$			394.802i			0.119357i
$223$		−	2361.52i		−	0.709145i	−0.935028	−	0.354573i	$-0.884626\pi$
$223$		−	2361.52i		−	0.709145i	0.935028	−	0.354573i	$-0.115374\pi$
$224$	−3187.44			−0.950758
$225$	−2653.02	−	1183.05i	−0.786081	−	0.350534i
$226$	8731.14			2.56985
$227$			586.877i			0.171596i	0.996313	+	0.0857982i	$0.0273440\pi$
$227$			586.877i			0.171596i	−0.996313	+	0.0857982i	$0.972656\pi$
$228$		−	31.0979i		−	0.00903292i
$229$	4619.55			1.33305			0.666526	−	0.745482i	$-0.267781\pi$
$229$	4619.55			1.33305			0.666526	+	0.745482i	$0.267781\pi$
$230$	−1343.31	+	6310.76i	−0.385111	+	1.80921i
$231$	346.750			0.0987639
$232$		−	15212.9i		−	4.30507i
$233$			5120.44i			1.43971i	0.694127	+	0.719853i	$0.255791\pi$
$233$			5120.44i			1.43971i	−0.694127	+	0.719853i	$0.744209\pi$
$234$	−7915.43			−2.21132
$235$	−2262.99	−	481.703i	−0.628176	−	0.133714i
$236$	−13741.3			−3.79017
$237$			153.850i			0.0421671i
$238$			1030.14i			0.280563i
$239$	−1127.51			−0.305158			−0.152579	−	0.988291i	$-0.548758\pi$
$239$	−1127.51			−0.305158			−0.152579	+	0.988291i	$0.548758\pi$
$240$	3893.20	+	828.711i	1.04710	+	0.222888i
$241$	−3549.53			−0.948736			−0.474368	−	0.880327i	$-0.657323\pi$
$241$	−3549.53			−0.948736			−0.474368	+	0.880327i	$0.657323\pi$
$242$		−	3605.89i		−	0.957833i
$243$		−	3481.07i		−	0.918975i
$244$	11629.4			3.05120
$245$	114.058	−	535.832i	0.0297424	−	0.139727i
$246$	−4155.24			−1.07694
$247$		−	50.7963i		−	0.0130854i
$248$		−	8399.84i		−	2.15077i
$249$	−2023.87			−0.515090
$250$	6003.99	+	4370.18i	1.51890	+	1.10558i
$251$	4717.19			1.18624			0.593120	−	0.805114i	$-0.297896\pi$
$251$	4717.19			1.18624			0.593120	+	0.805114i	$0.297896\pi$
$252$		−	3291.65i		−	0.822835i
$253$		−	2774.02i		−	0.689333i
$254$	1020.39			0.252067
$255$	125.025	−	587.355i	0.0307034	−	0.144242i
$256$	−113.552			−0.0277227
$257$			6260.31i			1.51948i	0.650224	+	0.759742i	$0.274675\pi$
$257$			6260.31i			1.51948i	−0.650224	+	0.759742i	$0.725325\pi$
$258$		−	1781.72i		−	0.429942i
$259$	−268.176			−0.0643383
$260$	14184.1	+	3019.25i	3.38331	+	0.720176i
$261$	5437.88			1.28964
$262$			11067.8i			2.60981i
$263$			5753.04i			1.34885i	0.738344	+	0.674425i	$0.235608\pi$
$263$			5753.04i			1.34885i	−0.738344	+	0.674425i	$0.764392\pi$
$264$	−3220.43			−0.750772
$265$	1575.03	+	335.262i	0.365107	+	0.0777170i
$266$	29.4751			0.00679412
$267$			1264.60i			0.289858i
$268$		−	10433.9i		−	2.37819i
$269$	7059.21			1.60003			0.800014	−	0.599982i	$-0.204826\pi$
$269$	7059.21			1.60003			0.800014	+	0.599982i	$0.204826\pi$
$270$	−1205.11	+	5661.47i	−0.271631	+	1.27610i
$271$	−8534.52			−1.91305			−0.956523	−	0.291658i	$-0.905793\pi$
$271$	−8534.52			−1.91305			−0.956523	+	0.291658i	$0.905793\pi$
$272$		−	5084.11i		−	1.13334i
$273$			870.222i			0.192924i
$274$	415.373			0.0915826
$275$	−2915.96	−	1300.30i	−0.639415	−	0.285132i
$276$	4262.08			0.929519
$277$		−	1313.94i		−	0.285008i	−0.989794	−	0.142504i	$-0.954485\pi$
$277$		−	1313.94i		−	0.285008i	0.989794	−	0.142504i	$-0.0455154\pi$
$278$		−	7405.47i		−	1.59766i
$279$	3002.54			0.644291
$280$	−1059.31	+	4976.53i	−0.226092	+	1.06216i
$281$	−247.229			−0.0524856			−0.0262428	−	0.999656i	$-0.508354\pi$
$281$	−247.229			−0.0524856			−0.0262428	+	0.999656i	$0.508354\pi$
$282$			2132.60i			0.450334i
$283$			9074.90i			1.90617i	0.302697	+	0.953087i	$0.402113\pi$
$283$			9074.90i			1.90617i	−0.302697	+	0.953087i	$0.597887\pi$
$284$	−11263.0			−2.35330
$285$	−16.8059	−	3.57731i	−0.00349296	−	0.000743515i
$286$	−8699.92			−1.79873
$287$		−	2822.51i		−	0.580515i
$288$			10581.7i			2.16505i
$289$	4145.98			0.843879
$290$	−13597.0	−	2894.27i	−2.75325	−	0.586060i
$291$	1000.17			0.201481
$292$		−	3505.57i		−	0.702562i
$293$			2740.72i			0.546466i	0.961948	+	0.273233i	$0.0880930\pi$
$293$			2740.72i			0.546466i	−0.961948	+	0.273233i	$0.911907\pi$
$294$	−504.956			−0.100169
$295$	−1580.71	+	7426.04i	−0.311975	+	1.46563i
$296$	2490.68			0.489080
$297$		−	2488.61i		−	0.486209i
$298$		−	172.695i		−	0.0335703i
$299$	6961.83			1.34653
$300$	1997.83	−	4480.16i	0.384482	−	0.862208i
$301$	1210.26			0.231755
$302$			2480.21i			0.472584i
$303$			1040.17i			0.197215i
$304$	−145.470			−0.0274451
$305$	1337.77	−	6284.73i	0.251150	−	1.17988i
$306$	3419.88			0.638894
$307$		−	6985.46i		−	1.29864i	−0.760517	−	0.649318i	$-0.775054\pi$
$307$		−	6985.46i		−	1.29864i	0.760517	−	0.649318i	$-0.224946\pi$
$308$		−	3617.88i		−	0.669311i
$309$	−740.250			−0.136283
$310$	−7507.60	−	1598.08i	−1.37549	−	0.292789i
$311$	−356.841			−0.0650630			−0.0325315	−	0.999471i	$-0.510357\pi$
$311$	−356.841			−0.0650630			−0.0325315	+	0.999471i	$0.510357\pi$
$312$		−	8082.16i		−	1.46655i
$313$		−	6630.12i		−	1.19731i	−0.801009	−	0.598653i	$-0.795703\pi$
$313$		−	6630.12i		−	1.19731i	0.801009	−	0.598653i	$-0.204297\pi$
$314$	−8892.42			−1.59818
$315$	−1778.87	−	378.652i	−0.318184	−	0.0677289i
$316$	1605.22			0.285762
$317$			2494.05i			0.441891i	0.975286	+	0.220946i	$0.0709144\pi$
$317$			2494.05i			0.441891i	−0.975286	+	0.220946i	$0.929086\pi$
$318$		−	1484.27i		−	0.261742i
$319$	5976.83			1.04902
$320$	2213.60	−	10399.3i	0.386700	−	1.81668i
$321$	−2678.52			−0.465734
$322$			4039.68i			0.699139i
$323$			21.9467i			0.00378063i
$324$	−8872.79			−1.52140
$325$	3263.31	−	7318.05i	0.556973	−	1.24902i
$326$	9933.13			1.68756
$327$		−	757.489i		−	0.128102i
$328$			26214.0i			4.41289i
$329$	−1448.60			−0.242748
$330$	−612.690	+	2878.35i	−0.102204	+	0.480146i
$331$	−4682.47			−0.777558			−0.388779	−	0.921331i	$-0.627103\pi$
$331$	−4682.47			−0.777558			−0.388779	+	0.921331i	$0.627103\pi$
$332$			21116.4i			3.49070i
$333$			890.297i			0.146510i
$334$	−247.218			−0.0405005
$335$	−5638.70	−	1200.26i	−0.919627	−	0.195753i
$336$	2492.14			0.404635
$337$		−	3596.60i		−	0.581363i	−0.956820	−	0.290681i	$-0.906118\pi$
$337$		−	3596.60i		−	0.581363i	0.956820	−	0.290681i	$-0.0938820\pi$
$338$		−	10159.7i		−	1.63495i
$339$	−3186.70			−0.510554
$340$	−6128.28	−	1304.47i	−0.977509	−	0.208073i
$341$	3300.12			0.524080
$342$		−	97.8523i		−	0.0154715i
$343$		−	343.000i		−	0.0539949i
$344$	−11240.3			−1.76173
$345$	490.285	−	2303.31i	0.0765103	−	0.359438i
$346$	13264.4			2.06097
$347$		−	1899.51i		−	0.293865i	−0.989147	−	0.146932i	$-0.953060\pi$
$347$		−	1899.51i		−	0.293865i	0.989147	−	0.146932i	$-0.0469400\pi$
$348$			9182.97i			1.41454i
$349$	1037.55			0.159137			0.0795683	−	0.996829i	$-0.474646\pi$
$349$	1037.55			0.159137			0.0795683	+	0.996829i	$0.474646\pi$
$350$	4246.38	+	1893.58i	0.648510	+	0.289188i
$351$	6245.56			0.949752
$352$			11630.5i			1.76110i
$353$		−	4087.08i		−	0.616242i	−0.951347	−	0.308121i	$-0.900300\pi$
$353$		−	4087.08i		−	0.616242i	0.951347	−	0.308121i	$-0.0997001\pi$
$354$	6998.13			1.05070
$355$	−1295.63	+	6086.75i	−0.193704	+	0.910003i
$356$	13194.4			1.96433
$357$		−	375.982i		−	0.0557397i
$358$		−	15811.8i		−	2.33431i
$359$	−3472.67			−0.510530			−0.255265	−	0.966871i	$-0.582163\pi$
$359$	−3472.67			−0.510530			−0.255265	+	0.966871i	$0.582163\pi$
$360$	16521.2	+	3516.72i	2.41873	+	0.514854i
$361$	−6858.37			−0.999908
$362$			5134.44i			0.745470i
$363$			1316.08i			0.190293i
$364$	9079.63			1.30742
$365$	−1894.48	−	403.260i	−0.271675	−	0.0578290i
$366$	−5922.59			−0.845844
$367$		−	8769.14i		−	1.24726i	−0.781719	−	0.623631i	$-0.785657\pi$
$367$		−	8769.14i		−	1.24726i	0.781719	−	0.623631i	$-0.214343\pi$
$368$		−	19937.3i		−	2.82419i
$369$	−9370.25			−1.32194
$370$	473.853	−	2226.11i	0.0665796	−	0.312784i
$371$	1008.22			0.141089
$372$			5070.39i			0.706687i
$373$			11368.9i			1.57817i	0.614284	+	0.789085i	$0.289445\pi$
$373$			11368.9i			1.57817i	−0.614284	+	0.789085i	$0.710555\pi$
$374$	3758.82			0.519690
$375$	−2191.35	−	1595.03i	−0.301762	−	0.219646i
$376$	13453.8			1.84529
$377$			14999.8i			2.04914i
$378$			3624.06i			0.493125i
$379$	12137.4			1.64500			0.822501	−	0.568764i	$-0.192578\pi$
$379$	12137.4			1.64500			0.822501	+	0.568764i	$0.192578\pi$
$380$	−37.3246	+	175.347i	−0.00503871	+	0.0236714i
$381$	−372.424			−0.0500784
$382$		−	8212.24i		−	1.09993i
$383$		−	9869.61i		−	1.31675i	−0.752692	−	0.658373i	$-0.771245\pi$
$383$		−	9869.61i		−	1.31675i	0.752692	−	0.658373i	$-0.228755\pi$
$384$	−2735.27			−0.363499
$385$	−1955.17	−	416.180i	−0.258817	−	0.0550921i
$386$	−12242.9			−1.61438
$387$		−	4017.86i		−	0.527751i
$388$		−	10435.5i		−	1.36541i
$389$	−57.1166			−0.00744454			−0.00372227	−	0.999993i	$-0.501185\pi$
$389$	−57.1166			−0.00744454			−0.00372227	+	0.999993i	$0.501185\pi$
$390$	−7223.67	−	1537.64i	−0.937910	−	0.199644i
$391$	−3007.88			−0.389041
$392$			3185.60i			0.410452i
$393$		−	4039.54i		−	0.518494i
$394$	1179.74			0.150849
$395$	184.655	−	867.490i	0.0235215	−	0.110502i
$396$	−12010.7			−1.52415
$397$			7436.17i			0.940078i	0.882646	+	0.470039i	$0.155760\pi$
$397$			7436.17i			0.940078i	−0.882646	+	0.470039i	$0.844240\pi$
$398$		−	19025.8i		−	2.39618i
$399$	−10.7579			−0.00134979
$400$	−20957.4	−	9345.47i	−2.61968	−	1.16818i
$401$	−12465.0			−1.55230			−0.776149	−	0.630550i	$-0.782829\pi$
$401$	−12465.0			−1.55230			−0.776149	+	0.630550i	$0.782829\pi$
$402$			5313.79i			0.659273i
$403$			8282.15i			1.02373i
$404$	10852.8			1.33650
$405$	−1020.67	+	4795.02i	−0.125229	+	0.588312i
$406$	−8703.79			−1.06394
$407$			978.533i			0.119175i
$408$			3491.92i			0.423715i
$409$	1708.72			0.206578			0.103289	−	0.994651i	$-0.467063\pi$
$409$	1708.72			0.206578			0.103289	+	0.994651i	$0.467063\pi$
$410$	23429.6	+	4987.24i	2.82220	+	0.600737i
$411$	−151.604			−0.0181948
$412$			7723.54i			0.923571i
$413$			4753.60i			0.566366i
$414$	13411.0			1.59207
$415$	11411.7	+	2429.10i	1.34983	+	0.287325i
$416$	−29188.5			−3.44011
$417$			2702.86i			0.317409i
$418$		−	107.550i		−	0.0125848i
$419$	−10618.8			−1.23810			−0.619050	−	0.785352i	$-0.712482\pi$
$419$	−10618.8			−1.23810			−0.619050	+	0.785352i	$0.712482\pi$
$420$	639.430	−	3003.98i	0.0742881	−	0.348998i
$421$	13273.5			1.53661			0.768304	−	0.640085i	$-0.221101\pi$
$421$	13273.5			1.53661			0.768304	+	0.640085i	$0.221101\pi$
$422$		−	22216.6i		−	2.56277i
$423$			4809.10i			0.552781i
$424$	−9363.80			−1.07251
$425$	−1409.92	+	3161.78i	−0.160921	+	0.360868i
$426$	5736.02			0.652373
$427$		−	4023.02i		−	0.455943i
$428$			27946.9i			3.15622i
$429$	3175.31			0.357355
$430$	−2138.47	+	10046.3i	−0.239829	+	1.12669i
$431$	7918.20			0.884934			0.442467	−	0.896785i	$-0.354103\pi$
$431$	7918.20			0.884934			0.442467	+	0.896785i	$0.354103\pi$
$432$		−	17886.0i		−	1.99199i
$433$		−	4433.34i		−	0.492038i	−0.969265	−	0.246019i	$-0.920877\pi$
$433$		−	4433.34i		−	0.492038i	0.969265	−	0.246019i	$-0.0791226\pi$
$434$	−4805.81			−0.531535
$435$	4962.65	+	1056.35i	0.546990	+	0.116433i
$436$	−7903.40			−0.868129
$437$			86.0637i			0.00942102i
$438$			1785.31i			0.194761i
$439$	−12958.4			−1.40882			−0.704408	−	0.709796i	$-0.748787\pi$
$439$	−12958.4			−1.40882			−0.704408	+	0.709796i	$0.748787\pi$
$440$	18158.6	+	3865.26i	1.96745	+	0.418793i
$441$	−1138.70			−0.122957
$442$			9433.34i			1.01515i
$443$			12040.4i			1.29133i	0.763621	+	0.645664i	$0.223420\pi$
$443$			12040.4i			1.29133i	−0.763621	+	0.645664i	$0.776580\pi$
$444$	−1503.45			−0.160699
$445$	1517.80	−	7130.50i	0.161687	−	0.759591i
$446$	12548.3			1.33224
$447$			63.0304i			0.00666943i
$448$		−	6656.84i		−	0.702023i
$449$	−11586.3			−1.21780			−0.608899	−	0.793247i	$-0.708389\pi$
$449$	−11586.3			−1.21780			−0.608899	+	0.793247i	$0.708389\pi$
$450$	6286.34	−	14097.2i	0.658536	−	1.47678i
$451$	−10298.9			−1.07529
$452$			33249.1i			3.45997i
$453$		−	905.232i		−	0.0938886i
$454$	−3118.46			−0.322371
$455$	1044.47	−	4906.80i	0.107616	−	0.505570i
$456$	99.9135			0.0102607
$457$			9734.34i			0.996396i	0.867063	+	0.498198i	$0.166005\pi$
$457$			9734.34i			0.996396i	−0.867063	+	0.498198i	$0.833995\pi$
$458$			24546.7i			2.50435i
$459$	−2698.41			−0.274403
$460$	−24032.0	−	5115.48i	−2.43587	−	0.518501i
$461$	−1343.41			−0.135724			−0.0678621	−	0.997695i	$-0.521618\pi$
$461$	−1343.41			−0.135724			−0.0678621	+	0.997695i	$0.521618\pi$
$462$			1842.51i			0.185544i
$463$			6613.72i			0.663857i	0.943305	+	0.331929i	$0.107699\pi$
$463$			6613.72i			0.663857i	−0.943305	+	0.331929i	$0.892301\pi$
$464$	42956.3			4.29784
$465$	2740.13	+	583.268i	0.273270	+	0.0581686i
$466$	−27208.3			−2.70472
$467$		−	14688.9i		−	1.45551i	−0.685838	−	0.727755i	$-0.740564\pi$
$467$		−	14688.9i		−	1.45551i	0.685838	−	0.727755i	$-0.259436\pi$
$468$		−	30142.8i		−	2.97724i
$469$	−3609.48			−0.355374
$470$	2559.60	−	12024.8i	0.251204	−	1.18013i
$471$	3245.57			0.317511
$472$		−	44148.9i		−	4.30534i
$473$		−	4416.07i		−	0.429283i
$474$	−817.504			−0.0792177
$475$	90.4673	+	40.3418i	0.00873879	+	0.00389686i
$476$	−3922.88			−0.377741
$477$		−	3347.11i		−	0.321286i
$478$		−	5991.22i		−	0.573288i
$479$	15298.6			1.45931			0.729657	−	0.683813i	$-0.239680\pi$
$479$	15298.6			1.45931			0.729657	+	0.683813i	$0.239680\pi$
$480$	−2055.59	+	9656.97i	−0.195468	+	0.918288i
$481$	−2455.78			−0.232794
$482$		−	18861.0i		−	1.78235i
$483$		−	1474.41i		−	0.138898i
$484$	13731.6			1.28960
$485$	−5639.52	−	1200.43i	−0.527994	−	0.112389i
$486$	18497.2			1.72644
$487$		−	9653.80i		−	0.898266i	−0.893465	−	0.449133i	$-0.851733\pi$
$487$		−	9653.80i		−	0.898266i	0.893465	−	0.449133i	$-0.148267\pi$
$488$			37363.7i			3.46593i
$489$	−3625.41			−0.335269
$490$	2847.23	+	606.064i	0.262499	+	0.0558758i
$491$	20142.6			1.85137			0.925684	−	0.378297i	$-0.123490\pi$
$491$	20142.6			1.85137			0.925684	+	0.378297i	$0.123490\pi$
$492$		−	15823.6i		−	1.44996i
$493$		−	6480.69i		−	0.592039i
$494$	269.914			0.0245830
$495$	−1381.64	+	6490.83i	−0.125455	+	0.589376i
$496$	23718.4			2.14715
$497$			3896.29i			0.351655i
$498$		−	10754.1i		−	0.967679i
$499$	1309.29			0.117459			0.0587293	−	0.998274i	$-0.481295\pi$
$499$	1309.29			0.117459			0.0587293	+	0.998274i	$0.481295\pi$
$500$	−16642.1	+	22863.8i	−1.48851	+	2.04500i
$501$	90.2299			0.00804625
$502$			25065.5i			2.22855i
$503$		−	2186.17i		−	0.193791i	−0.995295	−	0.0968953i	$-0.969109\pi$
$503$		−	2186.17i		−	0.193791i	0.995295	−	0.0968953i	$-0.0308912\pi$
$504$	10575.7			0.934677
$505$	1248.44	−	5865.06i	0.110010	−	0.516815i
$506$	14740.2			1.29502
$507$			3708.09i			0.324817i
$508$			3885.76i			0.339375i
$509$	−3591.11			−0.312718			−0.156359	−	0.987700i	$-0.549976\pi$
$509$	−3591.11			−0.312718			−0.156359	+	0.987700i	$0.549976\pi$
$510$	3121.01	+	664.340i	0.270981	+	0.0576813i
$511$	−1212.70			−0.104984
$512$		−	11886.4i		−	1.02600i
$513$			77.2089i			0.00664495i
$514$	−33265.2			−2.85460
$515$	4173.94	+	888.470i	0.357138	+	0.0760207i
$516$	6784.97			0.578860
$517$			5285.73i			0.449644i
$518$		−	1424.99i		−	0.120870i
$519$	−4841.24			−0.409455
$520$	−9700.46	+	45571.8i	−0.818064	+	3.84318i
$521$	−8605.34			−0.723621			−0.361811	−	0.932252i	$-0.617841\pi$
$521$	−8605.34			−0.723621			−0.361811	+	0.932252i	$0.617841\pi$
$522$			28895.1i			2.42280i
$523$		−	22536.4i		−	1.88423i	−0.335297	−	0.942113i	$-0.608837\pi$
$523$		−	22536.4i		−	1.88423i	0.335297	−	0.942113i	$-0.391163\pi$
$524$	−42147.3			−3.51377
$525$	−1549.85	−	691.120i	−0.128840	−	0.0574532i
$526$	−30569.7			−2.53403
$527$		−	3578.32i		−	0.295776i
$528$		−	9093.45i		−	0.749510i
$529$	371.637			0.0305447
$530$	−1781.47	+	8369.17i	−0.146004	+	0.685912i
$531$	15781.1			1.28972
$532$			112.244i			0.00914738i
$533$		−	25846.7i		−	2.10046i
$534$	−6719.63			−0.544544
$535$	15103.0	+	3214.84i	1.22049	+	0.259794i
$536$	33522.9			2.70144
$537$			5771.03i			0.463758i
$538$			37510.2i			3.00591i
$539$	−1251.56			−0.100015
$540$	−21559.5	−	4589.17i	−1.71810	−	0.365716i
$541$	8782.44			0.697942			0.348971	−	0.937134i	$-0.386531\pi$
$541$	8782.44			0.697942			0.348971	+	0.937134i	$0.386531\pi$
$542$		−	45349.5i		−	3.59396i
$543$		−	1873.98i		−	0.148103i
$544$	12611.0			0.993917
$545$	−909.161	+	4271.15i	−0.0714572	+	0.335699i
$546$	−4624.06			−0.362439
$547$			22593.0i			1.76601i	0.469366	+	0.883004i	$0.344482\pi$
$547$			22593.0i			1.76601i	−0.469366	+	0.883004i	$0.655518\pi$
$548$			1581.78i			0.123304i
$549$	−13355.7			−1.03827
$550$	6909.37	−	15494.4i	0.535667	−	1.20124i
$551$	−185.430			−0.0143368
$552$			13693.5i			1.05586i
$553$		−	555.303i		−	0.0427014i
$554$	6981.84			0.535433
$555$	−172.947	+	812.490i	−0.0132274	+	0.0621411i
$556$	28200.8			2.15104
$557$		−	7264.13i		−	0.552587i	−0.961073	−	0.276294i	$-0.910894\pi$
$557$		−	7264.13i		−	0.552587i	0.961073	−	0.276294i	$-0.0891062\pi$
$558$			15954.5i			1.21040i
$559$	11082.8			0.838555
$560$	−14052.1	−	2991.14i	−1.06037	−	0.225712i
$561$	−1371.90			−0.103247
$562$		−	1313.69i		−	0.0986027i
$563$		−	3851.42i		−	0.288309i	−0.989555	−	0.144154i	$-0.953954\pi$
$563$		−	3851.42i		−	0.288309i	0.989555	−	0.144154i	$-0.0460462\pi$
$564$	−8121.14			−0.606315
$565$	17968.4	+	3824.78i	1.33794	+	0.284796i
$566$	−48220.9			−3.58105
$567$			3069.42i			0.227343i
$568$		−	36186.7i		−	2.67317i
$569$	−16580.3			−1.22158			−0.610792	−	0.791791i	$-0.709149\pi$
$569$	−16580.3			−1.22158			−0.610792	+	0.791791i	$0.709149\pi$
$570$	19.0086	−	89.3006i	0.00139681	−	0.00656209i
$571$	6385.86			0.468021			0.234010	−	0.972234i	$-0.424815\pi$
$571$	6385.86			0.468021			0.234010	+	0.972234i	$0.424815\pi$
$572$		−	33130.2i		−	2.42175i
$573$			2997.32i			0.218525i
$574$	14997.9			1.09059
$575$	−5529.00	+	12398.9i	−0.401001	+	0.899252i
$576$	−22099.6			−1.59864
$577$		−	11059.1i		−	0.797912i	−0.916970	−	0.398956i	$-0.869373\pi$
$577$		−	11059.1i		−	0.797912i	0.916970	−	0.398956i	$-0.130627\pi$
$578$			22030.3i			1.58536i
$579$	4468.44			0.320729
$580$	11021.7	−	51778.7i	0.789052	−	3.70689i
$581$	7304.92			0.521617
$582$			5314.56i			0.378515i
$583$		−	3678.84i		−	0.261341i
$584$	11263.0			0.798055
$585$	−16289.7	−	3467.45i	−1.15128	−	0.245062i
$586$	−14563.2			−1.02662
$587$		−	7871.25i		−	0.553461i	−0.960948	−	0.276730i	$-0.910749\pi$
$587$		−	7871.25i		−	0.553461i	0.960948	−	0.276730i	$-0.0892509\pi$
$588$		−	1922.93i		−	0.134864i
$589$	−102.386			−0.00716253
$590$	−39459.4	−	8399.37i	−2.75342	−	0.586096i
$591$	−430.584			−0.0299693
$592$			7032.85i			0.488258i
$593$			2018.06i			0.139750i	0.997556	+	0.0698750i	$0.0222600\pi$
$593$			2018.06i			0.139750i	−0.997556	+	0.0698750i	$0.977740\pi$
$594$	13223.6			0.913422
$595$	−451.264	+	2120.00i	−0.0310925	+	0.146069i
$596$	657.640			0.0451980
$597$			6944.08i			0.476051i
$598$			36992.8i			2.52968i
$599$	1356.67			0.0925409			0.0462705	−	0.998929i	$-0.485266\pi$
$599$	1356.67			0.0925409			0.0462705	+	0.998929i	$0.485266\pi$
$600$	14394.2	+	6418.76i	0.979401	+	0.436741i
$601$	11178.7			0.758715			0.379358	−	0.925250i	$-0.376145\pi$
$601$	11178.7			0.758715			0.379358	+	0.925250i	$0.376145\pi$
$602$			6430.92i			0.435390i
$603$			11982.8i			0.809252i
$604$	−9444.91			−0.636272
$605$	1579.60	−	7420.82i	0.106149	−	0.498676i
$606$	−5527.10			−0.370500
$607$			9404.67i			0.628870i	0.949279	+	0.314435i	$0.101815\pi$
$607$			9404.67i			0.628870i	−0.949279	+	0.314435i	$0.898185\pi$
$608$		−	360.835i		−	0.0240687i
$609$	3176.72			0.211375
$610$	33394.9	+	7108.47i	2.21659	+	0.471825i
$611$	−13265.3			−0.878328
$612$			13023.3i			0.860187i
$613$			18938.0i			1.24779i	0.781507	+	0.623897i	$0.214452\pi$
$613$			18938.0i			1.24779i	−0.781507	+	0.623897i	$0.785548\pi$
$614$	37118.4			2.43970
$615$	−8551.35	−	1820.25i	−0.560689	−	0.119349i
$616$	11623.8			0.760286
$617$			17716.9i			1.15600i	0.816036	+	0.578001i	$0.196167\pi$
$617$			17716.9i			1.15600i	−0.816036	+	0.578001i	$0.803833\pi$
$618$		−	3933.43i		−	0.256029i
$619$	6240.33			0.405202			0.202601	−	0.979261i	$-0.435061\pi$
$619$	6240.33			0.405202			0.202601	+	0.979261i	$0.435061\pi$
$620$	6085.63	−	28589.7i	0.394202	−	1.85192i
$621$	−10581.8			−0.683788
$622$		−	1896.13i		−	0.122231i
$623$		−	4564.42i		−	0.293531i
$624$	22821.4			1.46408
$625$	10441.6	+	11623.8i	0.668264	+	0.743924i
$626$	35230.2			2.24933
$627$			39.2539i			0.00250024i
$628$		−	33863.2i		−	2.15174i
$629$	1061.03			0.0672589
$630$	2012.03	−	9452.30i	0.127240	−	0.597760i
$631$	−25887.7			−1.63323			−0.816617	−	0.577179i	$-0.804153\pi$
$631$	−25887.7			−1.63323			−0.816617	+	0.577179i	$0.804153\pi$
$632$			5157.37i			0.324603i
$633$			8108.65i			0.509147i
$634$	−13252.5			−0.830165
$635$	2099.94	+	446.994i	0.131234	+	0.0279345i
$636$	5652.27			0.352401
$637$		−	3140.97i		−	0.195369i
$638$			31758.8i			1.97076i
$639$	12935.0			0.800783
$640$	15423.0	+	3282.95i	0.952573	+	0.202766i
$641$	18798.4			1.15834			0.579168	−	0.815208i	$-0.303378\pi$
$641$	18798.4			1.15834			0.579168	+	0.815208i	$0.303378\pi$
$642$		−	14232.8i		−	0.874956i
$643$			2287.70i			0.140308i	0.997536	+	0.0701541i	$0.0223491\pi$
$643$			2287.70i			0.140308i	−0.997536	+	0.0701541i	$0.977651\pi$
$644$	−15383.5			−0.941298
$645$	780.503	−	3666.73i	0.0476469	−	0.223841i
$646$	−116.617			−0.00710253
$647$			1769.31i			0.107510i	0.998554	+	0.0537548i	$0.0171189\pi$
$647$			1769.31i			0.107510i	−0.998554	+	0.0537548i	$0.982881\pi$
$648$		−	28507.1i		−	1.72819i
$649$	17345.2			1.04909
$650$	38885.6	+	17340.1i	2.34649	+	1.04636i
$651$	1754.03			0.105600
$652$			37826.4i			2.27208i
$653$		−	3891.48i		−	0.233209i	−0.993178	−	0.116604i	$-0.962799\pi$
$653$		−	3891.48i		−	0.233209i	0.993178	−	0.116604i	$-0.0372010\pi$
$654$	4025.04			0.240660
$655$	−4848.38	+	22777.2i	−0.289224	+	1.35875i
$656$	−74019.8			−4.40547
$657$			4025.96i			0.239068i
$658$		−	7697.37i		−	0.456040i
$659$	−20097.6			−1.18800			−0.594001	−	0.804465i	$-0.702452\pi$
$659$	−20097.6			−1.18800			−0.594001	+	0.804465i	$0.702452\pi$
$660$	−10961.1	−	2333.18i	−0.646453	−	0.137605i
$661$	−27167.9			−1.59865			−0.799326	−	0.600898i	$-0.794810\pi$
$661$	−27167.9			−1.59865			−0.799326	+	0.600898i	$0.794810\pi$
$662$		−	24881.0i		−	1.46077i
$663$		−	3442.99i		−	0.201681i
$664$	−67844.3			−3.96516
$665$	60.6589	+	12.9119i	0.00353722	+	0.000752937i
$666$	−4730.73			−0.275243
$667$		−	25414.0i		−	1.47531i
$668$		−	941.430i		−	0.0545285i
$669$	−4579.91			−0.264678
$670$	6377.77	−	29962.1i	0.367753	−	1.72767i
$671$	−14679.4			−0.844548
$672$			6181.68i			0.354856i
$673$		−	25909.7i		−	1.48402i	−0.670389	−	0.742010i	$-0.733873\pi$
$673$		−	25909.7i		−	1.48402i	0.670389	−	0.742010i	$-0.266127\pi$
$674$	19111.1			1.09218
$675$	−4960.15	+	11123.2i	−0.282839	+	0.634271i
$676$	38689.1			2.20125
$677$			4359.22i			0.247472i	0.992315	+	0.123736i	$0.0394875\pi$
$677$			4359.22i			0.247472i	−0.992315	+	0.123736i	$0.960512\pi$
$678$		−	16933.0i		−	0.959159i
$679$	−3610.00			−0.204034
$680$	4191.11	−	19689.4i	0.236355	−	1.11037i
$681$	1138.18			0.0640458
$682$			17535.7i			0.984569i
$683$		−	29721.9i		−	1.66512i	−0.553937	−	0.832559i	$-0.686875\pi$
$683$		−	29721.9i		−	1.66512i	0.553937	−	0.832559i	$-0.313125\pi$
$684$	372.632			0.0208303
$685$	854.826	+	181.959i	0.0476806	+	0.0101493i
$686$	1822.58			0.101438
$687$		−	8959.10i		−	0.497541i
$688$		−	31738.9i		−	1.75877i
$689$	9232.60			0.510499
$690$	12239.0	+	2605.21i	0.675262	+	0.143737i
$691$	−4929.13			−0.271365			−0.135682	−	0.990752i	$-0.543323\pi$
$691$	−4929.13			−0.271365			−0.135682	+	0.990752i	$0.543323\pi$
$692$			50512.0i			2.77482i
$693$			4154.95i			0.227754i
$694$	10093.4			0.552073
$695$	3244.05	−	15240.2i	0.177056	−	0.831791i
$696$	−29503.7			−1.60680
$697$			11167.1i			0.606866i
$698$			5513.18i			0.298964i
$699$	9930.51			0.537348
$700$	−7210.93	+	16170.7i	−0.389354	+	0.873133i
$701$	−19358.8			−1.04304			−0.521520	−	0.853239i	$-0.674635\pi$
$701$	−19358.8			−1.04304			−0.521520	+	0.853239i	$0.674635\pi$
$702$			33186.7i			1.78426i
$703$		−	30.3589i		−	0.00162874i
$704$	−24289.8			−1.30036
$705$	−934.208	+	4388.82i	−0.0499068	+	0.234457i
$706$	21717.3			1.15771
$707$		−	3754.38i		−	0.199714i
$708$			26649.6i			1.41462i
$709$	17186.3			0.910359			0.455180	−	0.890400i	$-0.349575\pi$
$709$	17186.3			0.910359			0.455180	+	0.890400i	$0.349575\pi$
$710$	−32342.9	−	6884.54i	−1.70959	−	0.363905i
$711$	−1843.51			−0.0972392
$712$			42391.9i			2.23133i
$713$		−	14032.4i		−	0.737049i
$714$	1997.84			0.104716
$715$	−17904.2	−	3811.10i	−0.936473	−	0.199339i
$716$	60213.1			3.14284
$717$			2186.68i			0.113896i
$718$		−	18452.6i		−	0.959113i
$719$	15107.3			0.783598			0.391799	−	0.920051i	$-0.371853\pi$
$719$	15107.3			0.783598			0.391799	+	0.920051i	$0.371853\pi$
$720$	−9930.07	+	46650.5i	−0.513989	+	2.41467i
$721$	2671.85			0.138010
$722$		−	36443.0i		−	1.87849i
$723$			6883.91i			0.354102i
$724$	−19552.5			−1.00368
$725$	−26714.3	−	11912.6i	−1.36848	−	0.610240i
$726$	−6993.22			−0.357497
$727$		−	15840.9i		−	0.808124i	−0.914732	−	0.404062i	$-0.867598\pi$
$727$		−	15840.9i		−	0.808124i	0.914732	−	0.404062i	$-0.132402\pi$
$728$			29171.7i			1.48513i
$729$	5088.04			0.258499
$730$	2142.79	−	10066.6i	0.108641	−	0.510385i
$731$	−4788.35			−0.242276
$732$		−	22553.8i		−	1.13882i
$733$			27639.7i			1.39276i	0.717671	+	0.696382i	$0.245208\pi$
$733$			27639.7i			1.39276i	−0.717671	+	0.696382i	$0.754792\pi$
$734$	46596.2			2.34318
$735$	−1039.18	−	221.202i	−0.0521509	−	0.0111009i
$736$	49453.8			2.47675
$737$			13170.4i			0.658263i
$738$		−	49790.3i		−	2.48348i
$739$	−34874.4			−1.73596			−0.867982	−	0.496597i	$-0.834583\pi$
$739$	−34874.4			−1.73596			−0.867982	+	0.496597i	$0.834583\pi$
$740$	8477.27	+	1804.48i	0.421123	+	0.0896406i
$741$	−98.5136			−0.00488392
$742$			5357.32i			0.265059i
$743$			27686.7i			1.36706i	0.729922	+	0.683530i	$0.239556\pi$
$743$			27686.7i			1.36706i	−0.729922	+	0.683530i	$0.760444\pi$
$744$	−16290.5			−0.802741
$745$	75.6510	−	355.401i	0.00372032	−	0.0174777i
$746$	−60410.2			−2.96485
$747$		−	24251.1i		−	1.18782i
$748$			14314.0i			0.699694i
$749$	9667.83			0.471635
$750$	8475.46	−	11644.1i	0.412640	−	0.566908i
$751$	4806.85			0.233561			0.116781	−	0.993158i	$-0.462743\pi$
$751$	4806.85			0.233561			0.116781	+	0.993158i	$0.462743\pi$
$752$			37989.3i			1.84219i
$753$		−	9148.45i		−	0.442747i
$754$	−79703.6			−3.84965
$755$	−1086.49	+	5104.21i	−0.0523726	+	0.246041i
$756$	−13800.8			−0.663928
$757$		−	40166.6i		−	1.92851i	−0.264983	−	0.964253i	$-0.585366\pi$
$757$		−	40166.6i		−	1.92851i	0.264983	−	0.964253i	$-0.414634\pi$
$758$			64493.9i			3.09040i
$759$	−5379.90			−0.257283
$760$	−563.368	−	119.919i	−0.0268888	−	0.00572359i
$761$	25912.3			1.23432			0.617162	−	0.786836i	$-0.288282\pi$
$761$	25912.3			1.23432			0.617162	+	0.786836i	$0.288282\pi$
$762$		−	1978.93i		−	0.0940803i
$763$			2734.07i			0.129725i
$764$	31273.1			1.48092
$765$	7038.01	+	1498.12i	0.332627	+	0.0708034i
$766$	52443.7			2.47372
$767$			43530.4i			2.04927i
$768$			220.221i			0.0103471i
$769$	23231.3			1.08939			0.544697	−	0.838633i	$-0.316645\pi$
$769$	23231.3			1.08939			0.544697	+	0.838633i	$0.316645\pi$
$770$	2211.44	−	10389.1i	0.103499	−	0.486230i
$771$	12141.2			0.567125
$772$		−	46622.4i		−	2.17354i
$773$		−	836.306i		−	0.0389131i	−0.999811	−	0.0194566i	$-0.993806\pi$
$773$		−	836.306i		−	0.0389131i	0.999811	−	0.0194566i	$-0.00619360\pi$
$774$	21349.5			0.991465
$775$	−14750.4	−	6577.58i	−0.683675	−	0.304869i
$776$	33527.8			1.55100
$777$			520.096i			0.0240133i
$778$		−	303.498i		−	0.0139858i
$779$	319.523

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:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	35.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.31366i q^{2} -1.93939i q^{3} -20.2350 q^{4} +(-2.32771 + 10.9353i) q^{5} +10.3052 q^{6} +7.00000i q^{7} -65.0123i q^{8} +23.2388 q^{9} +(-58.1067 - 12.3686i) q^{10} +25.5420 q^{11} +39.2434i q^{12} +64.1014i q^{13} -37.1956 q^{14} +(21.2079 + 4.51433i) q^{15} +183.574 q^{16} -27.6952i q^{17} +123.483i q^{18} -0.792436 q^{19} +(47.1011 - 221.276i) q^{20} +13.5757 q^{21} +135.721i q^{22} -108.606i q^{23} -126.084 q^{24} +(-114.164 - 50.9086i) q^{25} -340.613 q^{26} -97.4324i q^{27} -141.645i q^{28} +234.000 q^{29} +(-23.9876 + 112.691i) q^{30} +129.204 q^{31} +455.349i q^{32} -49.5357i q^{33} +147.163 q^{34} +(-76.5474 - 16.2940i) q^{35} -470.236 q^{36} +38.3108i q^{37} -4.21073i q^{38} +124.317 q^{39} +(710.932 + 151.330i) q^{40} -403.216 q^{41} +72.1366i q^{42} -172.895i q^{43} -516.840 q^{44} +(-54.0931 + 254.124i) q^{45} +577.097 q^{46} +206.943i q^{47} -356.020i q^{48} -49.0000 q^{49} +(270.511 - 606.626i) q^{50} -53.7117 q^{51} -1297.09i q^{52} -144.031i q^{53} +517.722 q^{54} +(-59.4542 + 279.310i) q^{55} +455.086 q^{56} +1.53684i q^{57} +1243.40i q^{58} +679.086 q^{59} +(-429.140 - 91.3471i) q^{60} -574.717 q^{61} +686.544i q^{62} +162.671i q^{63} -950.977 q^{64} +(-700.971 - 149.209i) q^{65} +263.216 q^{66} +515.640i q^{67} +560.411i q^{68} -210.630 q^{69} +(86.5805 - 406.747i) q^{70} +556.612 q^{71} -1510.81i q^{72} +173.243i q^{73} -203.571 q^{74} +(-98.7314 + 221.407i) q^{75} +16.0349 q^{76} +178.794i q^{77} +660.580i q^{78} -79.3290 q^{79} +(-427.306 + 2007.44i) q^{80} +438.488 q^{81} -2142.55i q^{82} -1043.56i q^{83} -274.704 q^{84} +(302.856 + 64.4663i) q^{85} +918.703 q^{86} -453.817i q^{87} -1660.54i q^{88} -652.060 q^{89} +(-1350.33 - 287.432i) q^{90} -448.710 q^{91} +2197.65i q^{92} -250.576i q^{93} -1099.62 q^{94} +(1.84456 - 8.66556i) q^{95} +883.097 q^{96} +515.714i q^{97} -260.369i q^{98} +593.564 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9} - 16 q^{10} + 84 q^{11} - 56 q^{14} + 8 q^{15} + 148 q^{16} + 72 q^{19} - 68 q^{20} + 140 q^{21} + 72 q^{24} - 362 q^{25} - 620 q^{26} + 88 q^{29} + 52 q^{30}+ \cdots - 5304 q^{99}+O(q^{100}) \) 10 * q - 36 * q^4 + 6 * q^5 + 12 * q^6 - 46 * q^9 - 16 * q^10 + 84 * q^11 - 56 * q^14 + 8 * q^15 + 148 * q^16 + 72 * q^19 - 68 * q^20 + 140 * q^21 + 72 * q^24 - 362 * q^25 - 620 * q^26 + 88 * q^29 + 52 * q^30 + 120 * q^31 + 964 * q^34 - 28 * q^35 - 420 * q^36 + 212 * q^39 + 1396 * q^40 - 852 * q^41 - 1424 * q^44 - 510 * q^45 + 176 * q^46 - 490 * q^49 + 1644 * q^50 + 1276 * q^51 - 996 * q^54 - 1136 * q^55 + 588 * q^56 + 864 * q^59 - 1692 * q^60 - 884 * q^61 - 2724 * q^64 + 520 * q^65 + 1148 * q^66 + 4808 * q^69 + 756 * q^70 + 880 * q^71 + 3024 * q^74 + 720 * q^75 - 2672 * q^76 - 3580 * q^79 - 2316 * q^80 - 302 * q^81 - 1904 * q^84 + 1572 * q^85 + 2432 * q^86 - 1492 * q^89 - 7748 * q^90 + 476 * q^91 + 1652 * q^94 - 1628 * q^95 + 4080 * q^96 - 5304 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more