LMFDB - Embedded newform 855.3.e.a.721.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,3,Mod(721,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("855.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	855.e (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:	$23.2970626028$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 30x^{10} + 329x^{8} + 1620x^{6} + 3479x^{4} + 2470x^{2} + 55 $ x^12 + 30x^10 + 329x^8 + 1620x^6 + 3479x^4 + 2470*x^2 + 55
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.2
Root			$-2.84623i$ of defining polynomial
Character	$\chi$	$=$	855.721
Dual form			855.3.e.a.721.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.84623i q^{2} -4.10101 q^{4} +2.23607 q^{5} -9.15076 q^{7} +0.287487i q^{8} +O(q^{10})$	\(q-2.84623i q^{2} -4.10101 q^{4} +2.23607 q^{5} -9.15076 q^{7} +0.287487i q^{8} -6.36436i q^{10} -9.89207 q^{11} +12.3251i q^{13} +26.0451i q^{14} -15.5858 q^{16} +11.1895 q^{17} +(18.7659 - 2.97345i) q^{19} -9.17013 q^{20} +28.1551i q^{22} -20.8666 q^{23} +5.00000 q^{25} +35.0800 q^{26} +37.5273 q^{28} -10.5011i q^{29} +52.3075i q^{31} +45.5106i q^{32} -31.8479i q^{34} -20.4617 q^{35} -10.5041i q^{37} +(-8.46312 - 53.4120i) q^{38} +0.642841i q^{40} +42.6288i q^{41} +45.9768 q^{43} +40.5674 q^{44} +59.3912i q^{46} -4.22163 q^{47} +34.7363 q^{49} -14.2311i q^{50} -50.5453i q^{52} +95.8254i q^{53} -22.1193 q^{55} -2.63073i q^{56} -29.8886 q^{58} -6.38544i q^{59} +14.8591 q^{61} +148.879 q^{62} +67.1904 q^{64} +27.5598i q^{65} -38.0693i q^{67} -45.8882 q^{68} +58.2387i q^{70} +125.169i q^{71} +18.5252 q^{73} -29.8971 q^{74} +(-76.9590 + 12.1941i) q^{76} +90.5199 q^{77} +137.419i q^{79} -34.8508 q^{80} +121.331 q^{82} -127.881 q^{83} +25.0205 q^{85} -130.860i q^{86} -2.84385i q^{88} -19.8059i q^{89} -112.784i q^{91} +85.5743 q^{92} +12.0157i q^{94} +(41.9618 - 6.64884i) q^{95} -42.4473i q^{97} -98.8675i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{4} + 20 q^{7}+O(q^{10}) $ 12 * q - 12 * q^4 + 20 * q^7	$ 12 q - 12 q^{4} + 20 q^{7} - 32 q^{11} - 44 q^{16} + 44 q^{17} + 8 q^{19} - 36 q^{23} + 60 q^{25} - 108 q^{26} - 36 q^{28} + 40 q^{35} + 44 q^{38} + 320 q^{43} + 256 q^{44} + 56 q^{47} + 72 q^{49} + 68 q^{58} - 296 q^{61} + 376 q^{62} + 188 q^{64} + 340 q^{68} - 244 q^{73} - 136 q^{74} + 248 q^{76} + 200 q^{77} - 200 q^{80} + 424 q^{82} + 160 q^{83} + 160 q^{85} - 716 q^{92} + 80 q^{95}+O(q^{100}) $ 12 * q - 12 * q^4 + 20 * q^7 - 32 * q^11 - 44 * q^16 + 44 * q^17 + 8 * q^19 - 36 * q^23 + 60 * q^25 - 108 * q^26 - 36 * q^28 + 40 * q^35 + 44 * q^38 + 320 * q^43 + 256 * q^44 + 56 * q^47 + 72 * q^49 + 68 * q^58 - 296 * q^61 + 376 * q^62 + 188 * q^64 + 340 * q^68 - 244 * q^73 - 136 * q^74 + 248 * q^76 + 200 * q^77 - 200 * q^80 + 424 * q^82 + 160 * q^83 + 160 * q^85 - 716 * q^92 + 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	2.84623i		−	1.42311i	−0.702629	−	0.711557i	$-0.747990\pi$
$2$		−	2.84623i		−	1.42311i	0.702629	−	0.711557i	$-0.252010\pi$
$3$		0			0
$3$		0			0
$4$	−4.10101			−1.02525
$5$	2.23607			0.447214
$5$	2.23607			0.447214
$6$		0			0
$7$	−9.15076			−1.30725			−0.653625	−	0.756818i	$-0.726753\pi$
$7$	−9.15076			−1.30725			−0.653625	+	0.756818i	$0.726753\pi$
$8$			0.287487i			0.0359359i
$9$		0			0
$10$		−	6.36436i		−	0.636436i
$11$	−9.89207			−0.899279			−0.449640	−	0.893210i	$-0.648448\pi$
$11$	−9.89207			−0.899279			−0.449640	+	0.893210i	$0.648448\pi$
$12$		0			0
$13$			12.3251i			0.948085i	0.880502	+	0.474042i	$0.157206\pi$
$13$			12.3251i			0.948085i	−0.880502	+	0.474042i	$0.842794\pi$
$14$			26.0451i			1.86037i
$15$		0			0
$16$	−15.5858			−0.974111
$17$	11.1895			0.658206			0.329103	−	0.944294i	$-0.393254\pi$
$17$	11.1895			0.658206			0.329103	+	0.944294i	$0.393254\pi$
$18$		0			0
$19$	18.7659	−	2.97345i	0.987678	−	0.156497i
$19$	18.7659	−	2.97345i	0.987678	−	0.156497i
$20$	−9.17013			−0.458506
$21$		0			0
$22$			28.1551i			1.27978i
$23$	−20.8666			−0.907246			−0.453623	−	0.891194i	$-0.649869\pi$
$23$	−20.8666			−0.907246			−0.453623	+	0.891194i	$0.649869\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$	35.0800			1.34923
$27$		0			0
$28$	37.5273			1.34026
$29$		−	10.5011i		−	0.362107i	−0.983473	−	0.181054i	$-0.942049\pi$
$29$		−	10.5011i		−	0.362107i	0.983473	−	0.181054i	$-0.0579508\pi$
$30$		0			0
$31$			52.3075i			1.68734i	0.536864	+	0.843669i	$0.319609\pi$
$31$			52.3075i			1.68734i	−0.536864	+	0.843669i	$0.680391\pi$
$32$			45.5106i			1.42221i
$33$		0			0
$34$		−	31.8479i		−	0.936702i
$35$	−20.4617			−0.584620
$36$		0			0
$37$		−	10.5041i		−	0.283895i	−0.989874	−	0.141948i	$-0.954664\pi$
$37$		−	10.5041i		−	0.283895i	0.989874	−	0.141948i	$-0.0453364\pi$
$38$	−8.46312	−	53.4120i	−0.222714	−	1.40558i
$39$		0			0
$40$			0.642841i			0.0160710i
$41$			42.6288i			1.03973i	0.854250	+	0.519863i	$0.174017\pi$
$41$			42.6288i			1.03973i	−0.854250	+	0.519863i	$0.825983\pi$
$42$		0			0
$43$	45.9768			1.06923			0.534614	−	0.845097i	$-0.320457\pi$
$43$	45.9768			1.06923			0.534614	+	0.845097i	$0.320457\pi$
$44$	40.5674			0.921987
$45$		0			0
$46$			59.3912i			1.29111i
$47$	−4.22163			−0.0898220			−0.0449110	−	0.998991i	$-0.514300\pi$
$47$	−4.22163			−0.0898220			−0.0449110	+	0.998991i	$0.514300\pi$
$48$		0			0
$49$	34.7363			0.708905
$50$		−	14.2311i		−	0.284623i
$51$		0			0
$52$		−	50.5453i		−	0.972025i
$53$			95.8254i			1.80803i	0.427505	+	0.904013i	$0.359393\pi$
$53$			95.8254i			1.80803i	−0.427505	+	0.904013i	$0.640607\pi$
$54$		0			0
$55$	−22.1193			−0.402170
$56$		−	2.63073i		−	0.0469773i
$57$		0			0
$58$	−29.8886			−0.515320
$59$		−	6.38544i		−	0.108228i	−0.998535	−	0.0541139i	$-0.982767\pi$
$59$		−	6.38544i		−	0.108228i	0.998535	−	0.0541139i	$-0.0172334\pi$
$60$		0			0
$61$	14.8591			0.243593			0.121796	−	0.992555i	$-0.461135\pi$
$61$	14.8591			0.243593			0.121796	+	0.992555i	$0.461135\pi$
$62$	148.879			2.40127
$63$		0			0
$64$	67.1904			1.04985
$65$			27.5598i			0.423996i
$66$		0			0
$67$		−	38.0693i		−	0.568199i	−0.958795	−	0.284099i	$-0.908305\pi$
$67$		−	38.0693i		−	0.568199i	0.958795	−	0.284099i	$-0.0916946\pi$
$68$	−45.8882			−0.674827
$69$		0			0
$70$			58.2387i			0.831981i
$71$			125.169i			1.76295i	0.472230	+	0.881475i	$0.343449\pi$
$71$			125.169i			1.76295i	−0.472230	+	0.881475i	$0.656551\pi$
$72$		0			0
$73$	18.5252			0.253769			0.126885	−	0.991917i	$-0.459502\pi$
$73$	18.5252			0.253769			0.126885	+	0.991917i	$0.459502\pi$
$74$	−29.8971			−0.404015
$75$		0			0
$76$	−76.9590	+	12.1941i	−1.01262	+	0.160449i
$77$	90.5199			1.17558
$78$		0			0
$79$			137.419i			1.73948i	0.493510	+	0.869740i	$0.335714\pi$
$79$			137.419i			1.73948i	−0.493510	+	0.869740i	$0.664286\pi$
$80$	−34.8508			−0.435636
$81$		0			0
$82$	121.331			1.47965
$83$	−127.881			−1.54073			−0.770365	−	0.637603i	$-0.779926\pi$
$83$	−127.881			−1.54073			−0.770365	+	0.637603i	$0.779926\pi$
$84$		0			0
$85$	25.0205			0.294359
$86$		−	130.860i		−	1.52163i
$87$		0			0
$88$		−	2.84385i		−	0.0323164i
$89$		−	19.8059i		−	0.222538i	−0.993790	−	0.111269i	$-0.964508\pi$
$89$		−	19.8059i		−	0.222538i	0.993790	−	0.111269i	$-0.0354916\pi$
$90$		0			0
$91$		−	112.784i		−	1.23938i
$92$	85.5743			0.930155
$93$		0			0
$94$			12.0157i			0.127827i
$95$	41.9618	−	6.64884i	0.441703	−	0.0699878i
$96$		0			0
$97$		−	42.4473i		−	0.437601i	−0.975770	−	0.218800i	$-0.929786\pi$
$97$		−	42.4473i		−	0.437601i	0.975770	−	0.218800i	$-0.0702143\pi$
$98$		−	98.8675i		−	1.00885i
$99$		0			0
$100$	−20.5050			−0.205050
$101$	111.435			1.10332			0.551660	−	0.834069i	$-0.313995\pi$
$101$	111.435			1.10332			0.551660	+	0.834069i	$0.313995\pi$
$102$		0			0
$103$			5.71536i			0.0554889i	0.999615	+	0.0277445i	$0.00883247\pi$
$103$			5.71536i			0.0554889i	−0.999615	+	0.0277445i	$0.991168\pi$
$104$	−3.54331			−0.0340703
$105$		0			0
$106$	272.741			2.57303
$107$		−	164.563i		−	1.53797i	−0.639264	−	0.768987i	$-0.720761\pi$
$107$		−	164.563i		−	1.53797i	0.639264	−	0.768987i	$-0.279239\pi$
$108$		0			0
$109$		−	102.267i		−	0.938227i	−0.883138	−	0.469113i	$-0.844574\pi$
$109$		−	102.267i		−	0.938227i	0.883138	−	0.469113i	$-0.155426\pi$
$110$			62.9567i			0.572333i
$111$		0			0
$112$	142.622			1.27341
$113$			24.5050i			0.216858i	0.994104	+	0.108429i	$0.0345821\pi$
$113$			24.5050i			0.216858i	−0.994104	+	0.108429i	$0.965418\pi$
$114$		0			0
$115$	−46.6592			−0.405733
$116$			43.0651i			0.371251i
$117$		0			0
$118$	−18.1744			−0.154021
$119$	−102.392			−0.860440
$120$		0			0
$121$	−23.1469			−0.191297
$122$		−	42.2925i		−	0.346660i
$123$		0			0
$124$		−	214.513i		−	1.72995i
$125$	11.1803			0.0894427
$126$		0			0
$127$		−	58.5427i		−	0.460966i	−0.973076	−	0.230483i	$-0.925969\pi$
$127$		−	58.5427i		−	0.460966i	0.973076	−	0.230483i	$-0.0740307\pi$
$128$		−	9.19667i		−	0.0718489i
$129$		0			0
$130$	78.4413			0.603395
$131$	−223.883			−1.70903			−0.854516	−	0.519425i	$-0.826146\pi$
$131$	−223.883			−1.70903			−0.854516	+	0.519425i	$0.826146\pi$
$132$		0			0
$133$	−171.722	+	27.2093i	−1.29114	+	0.204581i
$134$	−108.354			−0.808611
$135$		0			0
$136$			3.21684i			0.0236532i
$137$	−6.24728			−0.0456006			−0.0228003	−	0.999740i	$-0.507258\pi$
$137$	−6.24728			−0.0456006			−0.0228003	+	0.999740i	$0.507258\pi$
$138$		0			0
$139$	−204.969			−1.47460			−0.737299	−	0.675567i	$-0.763899\pi$
$139$	−204.969			−1.47460			−0.737299	+	0.675567i	$0.763899\pi$
$140$	83.9136			0.599383
$141$		0			0
$142$	356.261			2.50888
$143$		−	121.921i		−	0.852593i
$144$		0			0
$145$		−	23.4812i		−	0.161939i
$146$		−	52.7268i		−	0.361143i
$147$		0			0
$148$			43.0775i			0.291064i
$149$	83.5502			0.560740			0.280370	−	0.959892i	$-0.409543\pi$
$149$	83.5502			0.560740			0.280370	+	0.959892i	$0.409543\pi$
$150$		0			0
$151$			189.386i			1.25421i	0.778934	+	0.627106i	$0.215761\pi$
$151$			189.386i			1.25421i	−0.778934	+	0.627106i	$0.784239\pi$
$152$	0.854830	+	5.39496i	0.00562388	+	0.0354931i
$153$		0			0
$154$		−	257.640i		−	1.67299i
$155$			116.963i			0.754600i
$156$		0			0
$157$	−195.060			−1.24242			−0.621212	−	0.783643i	$-0.713359\pi$
$157$	−195.060			−1.24242			−0.621212	+	0.783643i	$0.713359\pi$
$158$	391.126			2.47548
$159$		0			0
$160$			101.765i			0.636030i
$161$	190.946			1.18600
$162$		0			0
$163$	−171.677			−1.05323			−0.526616	−	0.850103i	$-0.676539\pi$
$163$	−171.677			−1.05323			−0.526616	+	0.850103i	$0.676539\pi$
$164$		−	174.821i		−	1.06598i
$165$		0			0
$166$			363.977i			2.19263i
$167$		−	40.7628i		−	0.244088i	−0.992525	−	0.122044i	$-0.961055\pi$
$167$		−	40.7628i		−	0.244088i	0.992525	−	0.122044i	$-0.0389450\pi$
$168$		0			0
$169$	17.0919			0.101136
$170$		−	71.2140i		−	0.418906i
$171$		0			0
$172$	−188.551			−1.09623
$173$		−	279.193i		−	1.61384i	−0.590664	−	0.806918i	$-0.701134\pi$
$173$		−	279.193i		−	1.61384i	0.590664	−	0.806918i	$-0.298866\pi$
$174$		0			0
$175$	−45.7538			−0.261450
$176$	154.176			0.875997
$177$		0			0
$178$	−56.3721			−0.316697
$179$			308.777i			1.72501i	0.506048	+	0.862505i	$0.331106\pi$
$179$			308.777i			1.72501i	−0.506048	+	0.862505i	$0.668894\pi$
$180$		0			0
$181$			252.084i			1.39273i	0.717688	+	0.696365i	$0.245201\pi$
$181$			252.084i			1.39273i	−0.717688	+	0.696365i	$0.754799\pi$
$182$	−321.009			−1.76378
$183$		0			0
$184$		−	5.99890i		−	0.0326027i
$185$		−	23.4879i		−	0.126962i
$186$		0			0
$187$	−110.687			−0.591911
$188$	17.3129			0.0920901
$189$		0			0
$190$	−18.9241	−	119.433i	−0.0996005	−	0.628594i
$191$	−210.693			−1.10311			−0.551553	−	0.834140i	$-0.685965\pi$
$191$	−210.693			−1.10311			−0.551553	+	0.834140i	$0.685965\pi$
$192$		0			0
$193$		−	113.346i		−	0.587285i	−0.955915	−	0.293643i	$-0.905132\pi$
$193$		−	113.346i		−	0.587285i	0.955915	−	0.293643i	$-0.0948676\pi$
$194$	−120.815			−0.622755
$195$		0			0
$196$	−142.454			−0.726806
$197$	206.358			1.04750			0.523752	−	0.851871i	$-0.324532\pi$
$197$	206.358			1.04750			0.523752	+	0.851871i	$0.324532\pi$
$198$		0			0
$199$	181.976			0.914450			0.457225	−	0.889351i	$-0.348843\pi$
$199$	181.976			0.914450			0.457225	+	0.889351i	$0.348843\pi$
$200$			1.43744i			0.00718719i
$201$		0			0
$202$		−	317.170i		−	1.57015i
$203$			96.0931i			0.473365i
$204$		0			0
$205$			95.3208i			0.464980i
$206$	16.2672			0.0789670
$207$		0			0
$208$		−	192.096i		−	0.923539i
$209$	−185.633	+	29.4136i	−0.888199	+	0.140735i
$210$		0			0
$211$		−	106.729i		−	0.505824i	−0.967489	−	0.252912i	$-0.918612\pi$
$211$		−	106.729i		−	0.505824i	0.967489	−	0.252912i	$-0.0813883\pi$
$212$		−	392.981i		−	1.85368i
$213$		0			0
$214$	−468.384			−2.18871
$215$	102.807			0.478173
$216$		0			0
$217$		−	478.653i		−	2.20577i
$218$	−291.074			−1.33520
$219$		0			0
$220$	90.7116			0.412325
$221$			137.912i			0.624035i
$222$		0			0
$223$		−	209.351i		−	0.938793i	−0.882987	−	0.469397i	$-0.844471\pi$
$223$		−	209.351i		−	0.938793i	0.882987	−	0.469397i	$-0.155529\pi$
$224$		−	416.456i		−	1.85918i
$225$		0			0
$226$	69.7467			0.308614
$227$			244.295i			1.07619i	0.842885	+	0.538094i	$0.180856\pi$
$227$			244.295i			1.07619i	−0.842885	+	0.538094i	$0.819144\pi$
$228$		0			0
$229$	254.372			1.11079			0.555397	−	0.831586i	$-0.312566\pi$
$229$	254.372			1.11079			0.555397	+	0.831586i	$0.312566\pi$
$230$			132.803i			0.577403i
$231$		0			0
$232$	3.01894			0.0130127
$233$	−347.037			−1.48943			−0.744714	−	0.667384i	$-0.767414\pi$
$233$	−347.037			−1.48943			−0.744714	+	0.667384i	$0.767414\pi$
$234$		0			0
$235$	−9.43986			−0.0401696
$236$			26.1867i			0.110961i
$237$		0			0
$238$			291.432i			1.22450i
$239$	−0.0478766			−0.000200320			−0.000100160	−	1.00000i	$-0.500032\pi$
$239$	−0.0478766			−0.000200320			−0.000100160		1.00000i	$0.500032\pi$
$240$		0			0
$241$			422.817i			1.75443i	0.480100	+	0.877214i	$0.340601\pi$
$241$			422.817i			1.75443i	−0.480100	+	0.877214i	$0.659399\pi$
$242$			65.8815i			0.272237i
$243$		0			0
$244$	−60.9375			−0.249744
$245$	77.6728			0.317032
$246$		0			0
$247$	36.6481	+	231.291i	0.148373	+	0.936403i
$248$	−15.0377			−0.0606361
$249$		0			0
$250$		−	31.8218i		−	0.127287i
$251$	−142.321			−0.567015			−0.283507	−	0.958970i	$-0.591498\pi$
$251$	−142.321			−0.567015			−0.283507	+	0.958970i	$0.591498\pi$
$252$		0			0
$253$	206.414			0.815867
$254$	−166.626			−0.656007
$255$		0			0
$256$	242.586			0.947600
$257$			198.179i			0.771123i	0.922682	+	0.385562i	$0.125992\pi$
$257$			198.179i			0.771123i	−0.922682	+	0.385562i	$0.874008\pi$
$258$		0			0
$259$			96.1206i			0.371122i
$260$		−	113.023i		−	0.434703i
$261$		0			0
$262$			637.223i			2.43215i
$263$	149.067			0.566794			0.283397	−	0.959003i	$-0.408539\pi$
$263$	149.067			0.566794			0.283397	+	0.959003i	$0.408539\pi$
$264$		0			0
$265$			214.272i			0.808574i
$266$	77.4439	+	488.760i	0.291143	+	1.83744i
$267$		0			0
$268$			156.123i			0.582547i
$269$			204.113i			0.758785i	0.925236	+	0.379392i	$0.123867\pi$
$269$			204.113i			0.758785i	−0.925236	+	0.379392i	$0.876133\pi$
$270$		0			0
$271$	−240.731			−0.888305			−0.444152	−	0.895951i	$-0.646495\pi$
$271$	−240.731			−0.888305			−0.444152	+	0.895951i	$0.646495\pi$
$272$	−174.397			−0.641166
$273$		0			0
$274$			17.7812i			0.0648948i
$275$	−49.4604			−0.179856
$276$		0			0
$277$	458.745			1.65612			0.828059	−	0.560641i	$-0.189445\pi$
$277$	458.745			1.65612			0.828059	+	0.560641i	$0.189445\pi$
$278$			583.388i			2.09852i
$279$		0			0
$280$		−	5.88249i		−	0.0210089i
$281$			249.476i			0.887816i	0.896072	+	0.443908i	$0.146408\pi$
$281$			249.476i			0.887816i	−0.896072	+	0.443908i	$0.853592\pi$
$282$		0			0
$283$	−365.612			−1.29192			−0.645958	−	0.763373i	$-0.723542\pi$
$283$	−365.612			−1.29192			−0.645958	+	0.763373i	$0.723542\pi$
$284$		−	513.321i		−	1.80747i
$285$		0			0
$286$	−347.014			−1.21334
$287$		−	390.086i		−	1.35918i
$288$		0			0
$289$	−163.795			−0.566765
$290$	−66.8328			−0.230458
$291$		0			0
$292$	−75.9718			−0.260177
$293$			173.088i			0.590745i	0.955382	+	0.295373i	$0.0954438\pi$
$293$			173.088i			0.590745i	−0.955382	+	0.295373i	$0.904556\pi$
$294$		0			0
$295$		−	14.2783i		−	0.0484010i
$296$	3.01980			0.0102020
$297$		0			0
$298$		−	237.803i		−	0.797996i
$299$		−	257.184i		−	0.860146i
$300$		0			0
$301$	−420.722			−1.39775
$302$	539.036			1.78489
$303$		0			0
$304$	−292.481	+	46.3435i	−0.962108	+	0.152446i
$305$	33.2261			0.108938
$306$		0			0
$307$		−	452.211i		−	1.47300i	−0.676438	−	0.736500i	$-0.736477\pi$
$307$		−	452.211i		−	1.47300i	0.676438	−	0.736500i	$-0.263523\pi$
$308$	−371.223			−1.20527
$309$		0			0
$310$	332.903			1.07388
$311$	−179.346			−0.576675			−0.288337	−	0.957529i	$-0.593102\pi$
$311$	−179.346			−0.576675			−0.288337	+	0.957529i	$0.593102\pi$
$312$		0			0
$313$	137.714			0.439981			0.219990	−	0.975502i	$-0.429397\pi$
$313$	137.714			0.439981			0.219990	+	0.975502i	$0.429397\pi$
$314$			555.186i			1.76811i
$315$		0			0
$316$		−	563.556i		−	1.78341i
$317$			602.099i			1.89936i	0.313215	+	0.949682i	$0.398594\pi$
$317$			602.099i			1.89936i	−0.313215	+	0.949682i	$0.601406\pi$
$318$		0			0
$319$			103.878i			0.325636i
$320$	150.242			0.469507
$321$		0			0
$322$		−	543.474i		−	1.68781i
$323$	209.981	−	33.2714i	0.650096	−	0.103008i
$324$		0			0
$325$			61.6255i			0.189617i
$326$			488.631i			1.49887i
$327$		0			0
$328$	−12.2552			−0.0373635
$329$	38.6311			0.117420
$330$		0			0
$331$			190.514i			0.575571i	0.957695	+	0.287785i	$0.0929189\pi$
$331$			190.514i			0.575571i	−0.957695	+	0.287785i	$0.907081\pi$
$332$	524.439			1.57964
$333$		0			0
$334$	−116.020			−0.347366
$335$		−	85.1256i		−	0.254106i
$336$		0			0
$337$			394.039i			1.16926i	0.811301	+	0.584628i	$0.198760\pi$
$337$			394.039i			1.16926i	−0.811301	+	0.584628i	$0.801240\pi$
$338$		−	48.6475i		−	0.143927i
$339$		0			0
$340$	−102.609			−0.301792
$341$		−	517.429i		−	1.51739i
$342$		0			0
$343$	130.523			0.380535
$344$			13.2177i			0.0384237i
$345$		0			0
$346$	−794.648			−2.29667
$347$	−205.061			−0.590953			−0.295476	−	0.955350i	$-0.595478\pi$
$347$	−205.061			−0.590953			−0.295476	+	0.955350i	$0.595478\pi$
$348$		0			0
$349$	−488.832			−1.40066			−0.700332	−	0.713817i	$-0.746965\pi$
$349$	−488.832			−1.40066			−0.700332	+	0.713817i	$0.746965\pi$
$350$			130.226i			0.372073i
$351$		0			0
$352$		−	450.194i		−	1.27896i
$353$	−466.761			−1.32227			−0.661134	−	0.750268i	$-0.729925\pi$
$353$	−466.761			−1.32227			−0.661134	+	0.750268i	$0.729925\pi$
$354$		0			0
$355$			279.888i			0.788416i
$356$			81.2242i			0.228158i
$357$		0			0
$358$	878.849			2.45489
$359$	−487.730			−1.35858			−0.679289	−	0.733871i	$-0.737712\pi$
$359$	−487.730			−1.35858			−0.679289	+	0.733871i	$0.737712\pi$
$360$		0			0
$361$	343.317	−	111.599i	0.951017	−	0.309138i
$362$	717.489			1.98201
$363$		0			0
$364$			462.528i			1.27068i
$365$	41.4235			0.113489
$366$		0			0
$367$	−63.0494			−0.171797			−0.0858983	−	0.996304i	$-0.527376\pi$
$367$	−63.0494			−0.171797			−0.0858983	+	0.996304i	$0.527376\pi$
$368$	325.223			0.883758
$369$		0			0
$370$	−66.8520			−0.180681
$371$		−	876.875i		−	2.36354i
$372$		0			0
$373$			137.833i			0.369526i	0.982783	+	0.184763i	$0.0591517\pi$
$373$			137.833i			0.369526i	−0.982783	+	0.184763i	$0.940848\pi$
$374$			315.041i			0.842356i
$375$		0			0
$376$		−	1.21367i		−	0.00322784i
$377$	129.427			0.343308
$378$		0			0
$379$		−	316.605i		−	0.835370i	−0.908592	−	0.417685i	$-0.862842\pi$
$379$		−	316.605i		−	0.835370i	0.908592	−	0.417685i	$-0.137158\pi$
$380$	−172.086	+	27.2669i	−0.452857	+	0.0717551i
$381$		0			0
$382$			599.681i			1.56985i
$383$		−	197.083i		−	0.514577i	−0.966335	−	0.257288i	$-0.917171\pi$
$383$		−	197.083i		−	0.514577i	0.966335	−	0.257288i	$-0.0828291\pi$
$384$		0			0
$385$	202.409			0.525737
$386$	−322.609			−0.835774
$387$		0			0
$388$			174.076i			0.448651i
$389$	−27.8203			−0.0715175			−0.0357587	−	0.999360i	$-0.511385\pi$
$389$	−27.8203			−0.0715175			−0.0357587	+	0.999360i	$0.511385\pi$
$390$		0			0
$391$	−233.487			−0.597155
$392$			9.98626i			0.0254751i
$393$		0			0
$394$		−	587.343i		−	1.49072i
$395$			307.278i			0.777919i
$396$		0			0
$397$	−702.745			−1.77014			−0.885069	−	0.465459i	$-0.845889\pi$
$397$	−702.745			−1.77014			−0.885069	+	0.465459i	$0.845889\pi$
$398$		−	517.944i		−	1.30137i
$399$		0			0
$400$	−77.9289			−0.194822
$401$		−	295.297i		−	0.736401i	−0.929746	−	0.368200i	$-0.879974\pi$
$401$		−	295.297i		−	0.736401i	0.929746	−	0.368200i	$-0.120026\pi$
$402$		0			0
$403$	−644.695			−1.59974
$404$	−456.997			−1.13118
$405$		0			0
$406$	273.503			0.673652
$407$			103.907i			0.255301i
$408$		0			0
$409$		−	371.735i		−	0.908887i	−0.890775	−	0.454444i	$-0.849838\pi$
$409$		−	371.735i		−	0.908887i	0.890775	−	0.454444i	$-0.150162\pi$
$410$	271.305			0.661719
$411$		0			0
$412$		−	23.4387i		−	0.0568901i
$413$			58.4316i			0.141481i
$414$		0			0
$415$	−285.950			−0.689035
$416$	−560.923			−1.34837
$417$		0			0
$418$	83.7177	+	528.355i	0.200282	+	1.26401i
$419$	109.242			0.260721			0.130360	−	0.991467i	$-0.458387\pi$
$419$	109.242			0.260721			0.130360	+	0.991467i	$0.458387\pi$
$420$		0			0
$421$		−	762.528i		−	1.81123i	−0.424100	−	0.905615i	$-0.639409\pi$
$421$		−	762.528i		−	1.81123i	0.424100	−	0.905615i	$-0.360591\pi$
$422$	−303.774			−0.719845
$423$		0			0
$424$	−27.5486			−0.0649731
$425$	55.9475			0.131641
$426$		0			0
$427$	−135.972			−0.318437
$428$			674.875i			1.57681i
$429$		0			0
$430$		−	292.613i		−	0.680494i
$431$			131.893i			0.306016i	0.988225	+	0.153008i	$0.0488960\pi$
$431$			131.893i			0.306016i	−0.988225	+	0.153008i	$0.951104\pi$
$432$		0			0
$433$		−	401.234i		−	0.926638i	−0.886191	−	0.463319i	$-0.846658\pi$
$433$		−	401.234i		−	0.926638i	0.886191	−	0.463319i	$-0.153342\pi$
$434$	−1362.35			−3.13907
$435$		0			0
$436$			419.397i			0.961919i
$437$	−391.581	+	62.0460i	−0.896067	+	0.141982i
$438$		0			0
$439$			264.338i			0.602136i	0.953603	+	0.301068i	$0.0973432\pi$
$439$			264.338i			0.602136i	−0.953603	+	0.301068i	$0.902657\pi$
$440$		−	6.35903i		−	0.0144523i
$441$		0			0
$442$	392.528			0.888073
$443$	451.274			1.01868			0.509338	−	0.860566i	$-0.329890\pi$
$443$	451.274			1.01868			0.509338	+	0.860566i	$0.329890\pi$
$444$		0			0
$445$		−	44.2874i		−	0.0995222i
$446$	−595.860			−1.33601
$447$		0			0
$448$	−614.843			−1.37242
$449$		−	156.631i		−	0.348844i	−0.984671	−	0.174422i	$-0.944194\pi$
$449$		−	156.631i		−	0.348844i	0.984671	−	0.174422i	$-0.0558057\pi$
$450$		0			0
$451$		−	421.687i		−	0.935004i
$452$		−	100.495i		−	0.222334i
$453$		0			0
$454$	695.318			1.53154
$455$		−	252.193i		−	0.554270i
$456$		0			0
$457$	906.098			1.98271			0.991355	−	0.131209i	$-0.0418859\pi$
$457$	906.098			1.98271			0.991355	+	0.131209i	$0.0418859\pi$
$458$		−	724.000i		−	1.58078i
$459$		0			0
$460$	191.350			0.415978
$461$	169.733			0.368184			0.184092	−	0.982909i	$-0.441066\pi$
$461$	169.733			0.368184			0.184092	+	0.982909i	$0.441066\pi$
$462$		0			0
$463$	207.991			0.449225			0.224613	−	0.974448i	$-0.427888\pi$
$463$	207.991			0.449225			0.224613	+	0.974448i	$0.427888\pi$
$464$			163.668i			0.352733i
$465$		0			0
$466$			987.745i			2.11962i
$467$	479.390			1.02653			0.513266	−	0.858230i	$-0.328435\pi$
$467$	479.390			1.02653			0.513266	+	0.858230i	$0.328435\pi$
$468$		0			0
$469$			348.363i			0.742778i
$470$			26.8680i			0.0571659i
$471$		0			0
$472$	1.83573			0.00388927
$473$	−454.805			−0.961534
$474$		0			0
$475$	93.8294	−	14.8673i	0.197536	−	0.0312995i
$476$	419.912			0.882168
$477$		0			0
$478$			0.136268i			0.000285079i
$479$	712.482			1.48744			0.743718	−	0.668493i	$-0.233060\pi$
$479$	712.482			1.48744			0.743718	+	0.668493i	$0.233060\pi$
$480$		0			0
$481$	129.464			0.269157
$482$	1203.43			2.49675
$483$		0			0
$484$	94.9258			0.196128
$485$		−	94.9150i		−	0.195701i
$486$		0			0
$487$			34.4191i			0.0706757i	0.999375	+	0.0353379i	$0.0112507\pi$
$487$			34.4191i			0.0706757i	−0.999375	+	0.0353379i	$0.988749\pi$
$488$			4.27182i			0.00875373i
$489$		0			0
$490$		−	221.074i		−	0.451172i
$491$	−361.608			−0.736472			−0.368236	−	0.929732i	$-0.620038\pi$
$491$	−361.608			−0.736472			−0.368236	+	0.929732i	$0.620038\pi$
$492$		0			0
$493$		−	117.502i		−	0.238341i
$494$	658.308	−	104.309i	1.33261	−	0.211151i
$495$		0			0
$496$		−	815.252i		−	1.64365i
$497$		−	1145.40i		−	2.30462i
$498$		0			0
$499$	−189.977			−0.380715			−0.190358	−	0.981715i	$-0.560965\pi$
$499$	−189.977			−0.380715			−0.190358	+	0.981715i	$0.560965\pi$
$500$	−45.8506			−0.0917013
$501$		0			0
$502$			405.077i			0.806926i
$503$	687.468			1.36674			0.683368	−	0.730074i	$-0.260514\pi$
$503$	687.468			1.36674			0.683368	+	0.730074i	$0.260514\pi$
$504$		0			0
$505$	249.177			0.493420
$506$		−	587.502i		−	1.16107i
$507$		0			0
$508$			240.084i			0.472607i
$509$		−	348.233i		−	0.684151i	−0.939672	−	0.342076i	$-0.888870\pi$
$509$		−	348.233i		−	0.684151i	0.939672	−	0.342076i	$-0.111130\pi$
$510$		0			0
$511$	−169.519			−0.331740
$512$		−	727.241i		−	1.42039i
$513$		0			0
$514$	564.061			1.09740
$515$			12.7799i			0.0248154i
$516$		0			0
$517$	41.7607			0.0807750
$518$	273.581			0.528149
$519$		0			0
$520$	−7.92309			−0.0152367
$521$			222.778i			0.427597i	0.976878	+	0.213798i	$0.0685835\pi$
$521$			222.778i			0.427597i	−0.976878	+	0.213798i	$0.931416\pi$
$522$		0			0
$523$			45.6247i			0.0872365i	0.999048	+	0.0436182i	$0.0138885\pi$
$523$			45.6247i			0.0872365i	−0.999048	+	0.0436182i	$0.986111\pi$
$524$	918.147			1.75219
$525$		0			0
$526$		−	424.278i		−	0.806612i
$527$			585.295i			1.11062i
$528$		0			0
$529$	−93.5830			−0.176905
$530$	609.867			1.15069
$531$		0			0
$532$	704.233	−	111.586i	1.32375	−	0.209747i
$533$	−525.404			−0.985749
$534$		0			0
$535$		−	367.975i		−	0.687803i
$536$	10.9444			0.0204187
$537$		0			0
$538$	580.952			1.07984
$539$	−343.614			−0.637503
$540$		0			0
$541$	671.919			1.24199			0.620997	−	0.783813i	$-0.286728\pi$
$541$	671.919			1.24199			0.620997	+	0.783813i	$0.286728\pi$
$542$			685.174i			1.26416i
$543$		0			0
$544$			509.241i			0.936105i
$545$		−	228.675i		−	0.419588i
$546$		0			0
$547$			910.710i			1.66492i	0.554087	+	0.832459i	$0.313068\pi$
$547$			910.710i			1.66492i	−0.554087	+	0.832459i	$0.686932\pi$
$548$	25.6201			0.0467520
$549$		0			0
$550$			140.775i			0.255955i
$551$	−31.2246	−	197.063i	−0.0566689	−	0.357646i
$552$		0			0
$553$		−	1257.49i		−	2.27394i
$554$		−	1305.69i		−	2.35684i
$555$		0			0
$556$	840.579			1.51183
$557$	246.347			0.442275			0.221138	−	0.975243i	$-0.429023\pi$
$557$	246.347			0.442275			0.221138	+	0.975243i	$0.429023\pi$
$558$		0			0
$559$			566.668i			1.01372i
$560$	318.912			0.569485
$561$		0			0
$562$	710.066			1.26346
$563$		−	772.769i		−	1.37259i	−0.727322	−	0.686296i	$-0.759235\pi$
$563$		−	772.769i		−	1.37259i	0.727322	−	0.686296i	$-0.240765\pi$
$564$		0			0
$565$			54.7948i			0.0969820i
$566$			1040.61i			1.83854i
$567$		0			0
$568$	−35.9847			−0.0633533
$569$		−	123.962i		−	0.217860i	−0.994049	−	0.108930i	$-0.965258\pi$
$569$		−	123.962i		−	0.217860i	0.994049	−	0.108930i	$-0.0347424\pi$
$570$		0			0
$571$	535.555			0.937925			0.468963	−	0.883218i	$-0.344628\pi$
$571$	535.555			0.937925			0.468963	+	0.883218i	$0.344628\pi$
$572$			499.998i			0.874122i
$573$		0			0
$574$	−1110.27			−1.93427
$575$	−104.333			−0.181449
$576$		0			0
$577$	−210.295			−0.364463			−0.182231	−	0.983256i	$-0.558332\pi$
$577$	−210.295			−0.364463			−0.182231	+	0.983256i	$0.558332\pi$
$578$			466.198i			0.806570i
$579$		0			0
$580$			96.2966i			0.166029i
$581$	1170.20			2.01412
$582$		0			0
$583$		−	947.912i		−	1.62592i
$584$			5.32575i			0.00911944i
$585$		0			0
$586$	492.649			0.840697
$587$	508.069			0.865534			0.432767	−	0.901506i	$-0.357537\pi$
$587$	508.069			0.865534			0.432767	+	0.901506i	$0.357537\pi$
$588$		0			0
$589$	155.534	+	981.596i	0.264064	+	1.66655i
$590$	−40.6392			−0.0688801
$591$		0			0
$592$			163.715i			0.276545i
$593$	32.8735			0.0554358			0.0277179	−	0.999616i	$-0.491176\pi$
$593$	32.8735			0.0554358			0.0277179	+	0.999616i	$0.491176\pi$
$594$		0			0
$595$	−228.956			−0.384801
$596$	−342.640			−0.574899
$597$		0			0
$598$	−732.003			−1.22408
$599$		−	225.201i		−	0.375961i	−0.982173	−	0.187981i	$-0.939806\pi$
$599$		−	225.201i		−	0.375961i	0.982173	−	0.187981i	$-0.0601942\pi$
$600$		0			0
$601$			169.930i			0.282746i	0.989956	+	0.141373i	$0.0451517\pi$
$601$			169.930i			0.282746i	−0.989956	+	0.141373i	$0.954848\pi$
$602$			1197.47i			1.98915i
$603$		0			0
$604$		−	776.674i		−	1.28588i
$605$	−51.7581			−0.0855506
$606$		0			0
$607$		−	522.217i		−	0.860325i	−0.902752	−	0.430162i	$-0.858456\pi$
$607$		−	522.217i		−	0.860325i	0.902752	−	0.430162i	$-0.141544\pi$
$608$	135.324	+	854.047i	0.222572	+	1.40468i
$609$		0			0
$610$		−	94.5689i		−	0.155031i
$611$		−	52.0320i		−	0.0851588i
$612$		0			0
$613$	−639.666			−1.04350			−0.521750	−	0.853098i	$-0.674721\pi$
$613$	−639.666			−1.04350			−0.521750	+	0.853098i	$0.674721\pi$
$614$	−1287.09			−2.09625
$615$		0			0
$616$			26.0233i			0.0422457i
$617$	−200.342			−0.324704			−0.162352	−	0.986733i	$-0.551908\pi$
$617$	−200.342			−0.324704			−0.162352	+	0.986733i	$0.551908\pi$
$618$		0			0
$619$	−347.156			−0.560833			−0.280417	−	0.959878i	$-0.590473\pi$
$619$	−347.156			−0.560833			−0.280417	+	0.959878i	$0.590473\pi$
$620$		−	479.666i		−	0.773655i
$621$		0			0
$622$			510.459i			0.820674i
$623$			181.239i			0.290914i
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	391.965i		−	0.626143i
$627$		0			0
$628$	799.944			1.27380
$629$		−	117.536i		−	0.186861i
$630$		0			0
$631$	204.105			0.323463			0.161732	−	0.986835i	$-0.448292\pi$
$631$	204.105			0.323463			0.161732	+	0.986835i	$0.448292\pi$
$632$	−39.5062			−0.0625099
$633$		0			0
$634$	1713.71			2.70301
$635$		−	130.906i		−	0.206150i
$636$		0			0
$637$			428.129i			0.672102i
$638$	295.660			0.463416
$639$		0			0
$640$		−	20.5644i		−	0.0321318i
$641$			334.435i			0.521740i	0.965374	+	0.260870i	$0.0840094\pi$
$641$			334.435i			0.521740i	−0.965374	+	0.260870i	$0.915991\pi$
$642$		0			0
$643$	−22.8316			−0.0355079			−0.0177540	−	0.999842i	$-0.505652\pi$
$643$	−22.8316			−0.0355079			−0.0177540	+	0.999842i	$0.505652\pi$
$644$	−783.069			−1.21595
$645$		0			0
$646$	−94.6981	−	597.653i	−0.146591	−	0.925160i
$647$	−280.968			−0.434263			−0.217131	−	0.976142i	$-0.569670\pi$
$647$	−280.968			−0.434263			−0.217131	+	0.976142i	$0.569670\pi$
$648$		0			0
$649$			63.1653i			0.0973271i
$650$	175.400			0.269846
$651$		0			0
$652$	704.048			1.07983
$653$	−891.423			−1.36512			−0.682559	−	0.730830i	$-0.739133\pi$
$653$	−891.423			−1.36512			−0.682559	+	0.730830i	$0.739133\pi$
$654$		0			0
$655$	−500.618			−0.764303
$656$		−	664.402i		−	1.01281i
$657$		0			0
$658$		−	109.953i		−	0.167102i
$659$			734.584i			1.11470i	0.830279	+	0.557348i	$0.188181\pi$
$659$			734.584i			1.11470i	−0.830279	+	0.557348i	$0.811819\pi$
$660$		0			0
$661$		−	968.622i		−	1.46539i	−0.680558	−	0.732694i	$-0.738263\pi$
$661$		−	968.622i		−	1.46539i	0.680558	−	0.732694i	$-0.261737\pi$
$662$	542.246			0.819102
$663$		0			0
$664$		−	36.7641i		−	0.0553676i
$665$	−383.982	+	60.8419i	−0.577417	+	0.0914916i
$666$		0			0
$667$			219.123i			0.328520i
$668$			167.168i			0.250252i
$669$		0			0
$670$	−242.287			−0.361622
$671$	−146.988			−0.219058
$672$		0			0
$673$			948.399i			1.40921i	0.709599	+	0.704606i	$0.248876\pi$
$673$			948.399i			1.40921i	−0.709599	+	0.704606i	$0.751124\pi$
$674$	1121.53			1.66398
$675$		0			0
$676$	−70.0940			−0.103689
$677$		−	82.3399i		−	0.121625i	−0.998149	−	0.0608123i	$-0.980631\pi$
$677$		−	82.3399i		−	0.121625i	0.998149	−	0.0608123i	$-0.0193691\pi$
$678$		0			0
$679$			388.425i			0.572054i
$680$			7.19308i			0.0105781i
$681$		0			0
$682$	−1472.72			−2.15941
$683$			10.8842i			0.0159359i	0.999968	+	0.00796796i	$0.00253631\pi$
$683$			10.8842i			0.0159359i	−0.999968	+	0.00796796i	$0.997464\pi$
$684$		0			0
$685$	−13.9693			−0.0203932
$686$		−	371.499i		−	0.541544i
$687$		0			0
$688$	−716.583			−1.04155
$689$	−1181.06			−1.71416
$690$		0			0
$691$	188.047			0.272138			0.136069	−	0.990699i	$-0.456553\pi$
$691$	188.047			0.272138			0.136069	+	0.990699i	$0.456553\pi$
$692$			1144.97i			1.65459i
$693$		0			0
$694$			583.649i			0.840993i
$695$	−458.325			−0.659460
$696$		0			0
$697$			476.995i			0.684354i
$698$			1391.33i			1.99330i
$699$		0			0
$700$	187.637			0.268052
$701$	1184.52			1.68976			0.844879	−	0.534957i	$-0.179672\pi$
$701$	1184.52			1.68976			0.844879	+	0.534957i	$0.179672\pi$
$702$		0			0
$703$	−31.2335	−	197.119i	−0.0444289	−	0.280397i
$704$	−664.652			−0.944108
$705$		0			0
$706$			1328.51i			1.88174i
$707$	−1019.72			−1.44232
$708$		0			0
$709$	−541.289			−0.763455			−0.381727	−	0.924275i	$-0.624671\pi$
$709$	−541.289			−0.763455			−0.381727	+	0.924275i	$0.624671\pi$
$710$	796.623			1.12200
$711$		0			0
$712$	5.69395			0.00799713
$713$		−	1091.48i		−	1.53083i
$714$		0			0
$715$		−	272.623i		−	0.381291i
$716$		−	1266.30i		−	1.76857i
$717$		0			0
$718$			1388.19i			1.93341i
$719$	388.854			0.540826			0.270413	−	0.962744i	$-0.412840\pi$
$719$	388.854			0.540826			0.270413	+	0.962744i	$0.412840\pi$
$720$		0			0
$721$		−	52.2998i		−	0.0725379i
$722$	−317.636	−	977.159i	−0.439939	−	1.35341i
$723$		0			0
$724$		−	1033.80i		−	1.42790i
$725$		−	52.5056i		−	0.0724215i
$726$		0			0
$727$	1450.46			1.99513			0.997565	−	0.0697458i	$-0.0222188\pi$
$727$	1450.46			1.99513			0.997565	+	0.0697458i	$0.0222188\pi$
$728$	32.4240			0.0445384
$729$		0			0
$730$		−	117.901i		−	0.161508i
$731$	514.457			0.703772
$732$		0			0
$733$	873.057			1.19107			0.595537	−	0.803328i	$-0.296939\pi$
$733$	873.057			1.19107			0.595537	+	0.803328i	$0.296939\pi$
$734$			179.453i			0.244486i
$735$		0			0
$736$		−	949.653i		−	1.29029i
$737$			376.584i			0.510969i
$738$		0			0
$739$	−456.992			−0.618393			−0.309196	−	0.950998i	$-0.600060\pi$
$739$	−456.992			−0.618393			−0.309196	+	0.950998i	$0.600060\pi$
$740$			96.3241i			0.130168i
$741$		0			0
$742$	−2495.78			−3.36359
$743$			60.6398i			0.0816148i	0.999167	+	0.0408074i	$0.0129930\pi$
$743$			60.6398i			0.0816148i	−0.999167	+	0.0408074i	$0.987007\pi$
$744$		0			0
$745$	186.824			0.250770
$746$	392.304			0.525877
$747$		0			0
$748$	453.930			0.606858
$749$			1505.88i			2.01052i
$750$		0			0
$751$		−	1177.72i		−	1.56821i	−0.620629	−	0.784104i	$-0.713123\pi$
$751$		−	1177.72i		−	1.56821i	0.620629	−	0.784104i	$-0.286877\pi$
$752$	65.7974			0.0874965
$753$		0			0
$754$		−	368.379i		−	0.488567i
$755$			423.480i			0.560901i
$756$		0			0
$757$	−112.664			−0.148830			−0.0744151	−	0.997227i	$-0.523709\pi$
$757$	−112.664			−0.148830			−0.0744151	+	0.997227i	$0.523709\pi$
$758$	−901.130			−1.18883
$759$		0			0
$760$	1.91146	+	12.0635i	0.00251508	+	0.0158730i
$761$	857.698			1.12707			0.563534	−	0.826093i	$-0.309442\pi$
$761$	857.698			1.12707			0.563534	+	0.826093i	$0.309442\pi$
$762$		0			0
$763$			935.818i			1.22650i
$764$	864.055			1.13096
$765$		0			0
$766$	−560.943			−0.732301
$767$	78.7012			0.102609
$768$		0			0
$769$	−146.990			−0.191144			−0.0955719	−	0.995423i	$-0.530468\pi$
$769$	−146.990			−0.191144			−0.0955719	+	0.995423i	$0.530468\pi$
$770$		−	576.101i		−	0.748183i
$771$		0			0
$772$			464.833i			0.602115i
$773$			624.217i			0.807525i	0.914864	+	0.403762i	$0.132298\pi$
$773$			624.217i			0.807525i	−0.914864	+	0.403762i	$0.867702\pi$
$774$		0			0
$775$			261.537i			0.337468i
$776$	12.2031			0.0157256
$777$		0			0
$778$			79.1829i			0.101778i
$779$	126.755	+	799.967i	0.162715	+	1.02692i
$780$		0			0
$781$		−	1238.19i		−	1.58538i
$782$			664.558i			0.849819i
$783$		0			0
$784$	−541.392			−0.690552
$785$	−436.168			−0.555629
$786$		0			0
$787$		−	551.370i		−	0.700597i	−0.936638	−	0.350298i	$-0.886080\pi$
$787$		−	551.370i		−	0.700597i	0.936638	−	0.350298i	$-0.113920\pi$
$788$	−846.277			−1.07396
$789$		0			0
$790$	874.583			1.10707
$791$		−	224.239i		−	0.283488i
$792$		0			0
$793$			183.140i			0.230946i
$794$			2000.17i			2.51911i
$795$		0			0
$796$	−746.283			−0.937541
$797$			202.295i			0.253821i	0.991914	+	0.126911i	$0.0405061\pi$
$797$			202.295i			0.253821i	−0.991914	+	0.126911i	$0.959494\pi$
$798$		0			0
$799$	−47.2380			−0.0591214
$800$			227.553i			0.284441i
$801$		0			0
$802$	−840.481			−1.04798
$803$	−183.252			−0.228209
$804$		0			0
$805$	426.967			0.530394
$806$			1834.95i			2.27661i
$807$		0			0
$808$			32.0363i			0.0396489i
$809$	1164.20			1.43906			0.719531	−	0.694461i	$-0.244357\pi$
$809$	1164.20			1.43906			0.719531	+	0.694461i	$0.244357\pi$
$810$		0			0
$811$			617.970i			0.761985i	0.924578	+	0.380993i	$0.124418\pi$
$811$			617.970i			0.761985i	−0.924578	+	0.380993i	$0.875582\pi$
$812$		−	394.079i		−	0.485318i
$813$		0			0
$814$	295.744			0.363322
$815$	−383.881			−0.471020
$816$		0			0
$817$	862.795	−	136.710i	1.05605	−	0.167331i
$818$	−1058.04			−1.29345
$819$		0			0
$820$		−	390.911i		−	0.476721i
$821$	−255.141			−0.310769			−0.155384	−	0.987854i	$-0.549662\pi$
$821$	−255.141			−0.310769			−0.155384	+	0.987854i	$0.549662\pi$
$822$		0			0
$823$	−658.923			−0.800636			−0.400318	−	0.916376i	$-0.631100\pi$
$823$	−658.923			−0.800636			−0.400318	+	0.916376i	$0.631100\pi$
$824$	−1.64309			−0.00199405
$825$		0			0
$826$	166.310			0.201343
$827$		−	633.264i		−	0.765736i	−0.923803	−	0.382868i	$-0.874936\pi$
$827$		−	633.264i		−	0.765736i	0.923803	−	0.382868i	$-0.125064\pi$
$828$		0			0
$829$		−	459.961i		−	0.554838i	−0.960749	−	0.277419i	$-0.910521\pi$
$829$		−	459.961i		−	0.554838i	0.960749	−	0.277419i	$-0.0894790\pi$
$830$			813.877i			0.980575i
$831$		0			0
$832$			828.128i			0.995346i
$833$	388.682			0.466605
$834$		0			0
$835$		−	91.1483i		−	0.109160i
$836$	761.284	−	120.625i	0.910627	−	0.144289i
$837$		0			0
$838$		−	310.927i		−	0.371035i
$839$			1033.89i			1.23228i	0.787635	+	0.616142i	$0.211305\pi$
$839$			1033.89i			1.23228i	−0.787635	+	0.616142i	$0.788695\pi$
$840$		0			0
$841$	730.727			0.868878
$842$	−2170.33			−2.57759
$843$		0			0
$844$			437.696i			0.518597i
$845$	38.2187			0.0452292
$846$		0			0
$847$	211.812			0.250073
$848$		−	1493.51i		−	1.76122i
$849$		0			0
$850$		−	159.239i		−	0.187340i
$851$			219.186i			0.257563i
$852$		0			0
$853$	−333.243			−0.390671			−0.195336	−	0.980736i	$-0.562580\pi$
$853$	−333.243			−0.390671			−0.195336	+	0.980736i	$0.562580\pi$
$854$			387.008i			0.453171i
$855$		0			0
$856$	47.3099			0.0552685
$857$		−	55.2946i		−	0.0645211i	−0.999479	−	0.0322605i	$-0.989729\pi$
$857$		−	55.2946i		−	0.0645211i	0.999479	−	0.0322605i	$-0.0102706\pi$
$858$		0			0
$859$	417.893			0.486488			0.243244	−	0.969965i	$-0.421788\pi$
$859$	417.893			0.486488			0.243244	+	0.969965i	$0.421788\pi$
$860$	−421.613			−0.490248
$861$		0			0
$862$	375.396			0.435495
$863$		−	1042.76i		−	1.20830i	−0.796871	−	0.604150i	$-0.793513\pi$
$863$		−	1042.76i		−	1.20830i	0.796871	−	0.604150i	$-0.206487\pi$
$864$		0			0
$865$		−	624.296i		−	0.721729i
$866$	−1142.00			−1.31871
$867$		0			0
$868$			1962.96i			2.26147i
$869$		−	1359.36i		−	1.56428i
$870$		0			0
$871$	469.208			0.538700
$872$	29.4004			0.0337161
$873$		0			0
$874$	176.597	+	1114.53i	0.202056	+	1.27520i
$875$	−102.309			−0.116924
$876$		0			0
$877$			143.302i			0.163400i	0.996657	+	0.0817002i	$0.0260350\pi$
$877$			143.302i			0.163400i	−0.996657	+	0.0817002i	$0.973965\pi$
$878$	752.366			0.856908
$879$		0			0
$880$	344.747			0.391758
$881$	944.916			1.07255			0.536275	−	0.844043i	$-0.319831\pi$
$881$	944.916			1.07255			0.536275	+	0.844043i	$0.319831\pi$
$882$		0			0
$883$	−1406.88			−1.59329			−0.796646	−	0.604446i	$-0.793395\pi$
$883$	−1406.88			−1.59329			−0.796646	+	0.604446i	$0.793395\pi$
$884$		−	565.577i		−	0.639793i
$885$		0			0
$886$		−	1284.43i		−	1.44969i
$887$		−	437.734i		−	0.493500i	−0.969079	−	0.246750i	$-0.920637\pi$
$887$		−	437.734i		−	0.493500i	0.969079	−	0.246750i	$-0.0793626\pi$
$888$		0			0
$889$			535.710i			0.602599i
$890$	−126.052			−0.141631
$891$		0			0
$892$			858.549i			0.962499i
$893$	−79.2227	+	12.5528i	−0.0887152	+	0.0140569i
$894$		0			0
$895$			690.446i			0.771448i
$896$			84.1564i			0.0939246i
$897$		0			0
$898$	−445.807			−0.496445
$899$	549.287			0.610998
$900$		0			0
$901$			1072.24i			1.19005i
$902$	−1200.22			−1.33062
$903$		0			0
$904$	−7.04488			−0.00779300
$905$			563.677i			0.622848i
$906$		0			0
$907$		−	174.898i		−	0.192832i	−0.995341	−	0.0964158i	$-0.969262\pi$
$907$		−	174.898i		−	0.192832i	0.995341	−	0.0964158i	$-0.0307378\pi$
$908$		−	1001.85i		−	1.10336i
$909$		0			0
$910$	−717.797			−0.788788
$911$		−	475.064i		−	0.521476i	−0.965410	−	0.260738i	$-0.916034\pi$
$911$		−	475.064i		−	0.521476i	0.965410	−	0.260738i	$-0.0839658\pi$
$912$		0			0
$913$	1265.00			1.38555
$914$		−	2578.96i		−	2.82162i
$915$		0			0
$916$	−1043.18			−1.13884
$917$	2048.70			2.23413
$918$		0			0
$919$	456.862			0.497130			0.248565	−	0.968615i	$-0.420041\pi$
$919$	456.862			0.497130			0.248565	+	0.968615i	$0.420041\pi$
$920$		−	13.4139i		−	0.0145804i
$921$		0			0
$922$		−	483.098i		−	0.523967i
$923$	−1542.73			−1.67143
$924$		0			0
$925$		−	52.5206i		−	0.0567790i
$926$		−	591.991i		−	0.639299i
$927$		0			0
$928$	477.912			0.514991
$929$	−714.272			−0.768861			−0.384430	−	0.923154i	$-0.625602\pi$
$929$	−714.272			−0.768861			−0.384430	+	0.923154i	$0.625602\pi$
$930$		0			0
$931$	651.858	−	103.287i	0.700170	−	0.110942i
$932$	1423.20			1.52704
$933$		0			0
$934$		−	1364.45i		−	1.46087i
$935$	−247.504			−0.264711
$936$		0			0
$937$	711.794			0.759652			0.379826	−	0.925058i	$-0.375984\pi$
$937$	711.794			0.759652			0.379826	+	0.925058i	$0.375984\pi$
$938$	991.520			1.05706
$939$		0			0
$940$	38.7129			0.0411840
$941$			1055.43i			1.12161i	0.827950	+	0.560803i	$0.189507\pi$
$941$			1055.43i			1.12161i	−0.827950	+	0.560803i	$0.810493\pi$
$942$		0			0
$943$		−	889.520i		−	0.943287i
$944$			99.5221i			0.105426i
$945$		0			0
$946$			1294.48i			1.36837i
$947$	611.188			0.645394			0.322697	−	0.946502i	$-0.395411\pi$
$947$	611.188			0.645394			0.322697	+	0.946502i	$0.395411\pi$
$948$		0			0
$949$			228.324i			0.240595i
$950$	−42.3156	−	267.060i	−0.0445427	−	0.281116i
$951$		0			0
$952$		−	29.4365i		−	0.0309207i
$953$			847.554i			0.889354i	0.895691	+	0.444677i	$0.146682\pi$
$953$			847.554i			0.889354i	−0.895691	+	0.444677i	$0.853318\pi$
$954$		0			0
$955$	−471.125			−0.493324
$956$	0.196342			0.000205379
$957$		0			0
$958$		−	2027.89i		−	2.11679i
$959$	57.1673			0.0596114
$960$		0			0
$961$	−1775.07			−1.84711
$962$		−	368.485i		−	0.383040i
$963$		0			0
$964$		−	1733.98i		−	1.79873i
$965$		−	253.449i		−	0.262642i
$966$		0			0
$967$	−293.882			−0.303911			−0.151955	−	0.988387i	$-0.548557\pi$
$967$	−293.882			−0.303911			−0.151955	+	0.988387i	$0.548557\pi$
$968$		−	6.65446i		−	0.00687444i
$969$		0			0
$970$	−270.150			−0.278505
$971$		−	1289.96i		−	1.32849i	−0.747517	−	0.664243i	$-0.768754\pi$
$971$		−	1289.96i		−	1.32849i	0.747517	−	0.664243i	$-0.231246\pi$
$972$		0			0
$973$	1875.62			1.92767
$974$	97.9645			0.100580
$975$		0			0
$976$	−231.591			−0.237286
$977$		−	362.429i		−	0.370961i	−0.982648	−	0.185481i	$-0.940616\pi$
$977$		−	362.429i		−	0.370961i	0.982648	−	0.185481i	$-0.0593842\pi$
$978$		0			0
$979$			195.922i			0.200124i
$980$	−318.537			−0.325037
$981$		0			0
$982$			1029.22i			1.04808i
$983$			236.553i			0.240644i	0.992735	+	0.120322i	$0.0383927\pi$
$983$			236.553i			0.240644i	−0.992735	+	0.120322i	$0.961607\pi$
$984$		0			0
$985$	461.431			0.468458
$986$	−334.438			−0.339187
$987$		0			0
$988$	−150.294	−	948.528i	−0.152119	−	0.960048i
$989$	−959.381			−0.970052
$990$		0			0
$991$			328.650i			0.331635i	0.986156	+	0.165817i	$0.0530262\pi$
$991$			328.650i			0.331635i	−0.986156	+	0.165817i	$0.946974\pi$
$992$	−2380.54			−2.39974
$993$		0			0
$994$	−3260.06			−3.27973
$995$	406.910			0.408955
$996$		0			0
$997$	−537.051			−0.538667			−0.269333	−	0.963047i	$-0.586803\pi$
$997$	−537.051			−0.538667			−0.269333	+	0.963047i	$0.586803\pi$
$998$			540.718i			0.541801i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.3.e.a.721.2		12
3.2	odd	2		95.3.c.a.56.11	yes	12
12.11	even	2		1520.3.h.a.721.5		12
15.2	even	4		475.3.d.c.474.4		24
15.8	even	4		475.3.d.c.474.21		24
15.14	odd	2		475.3.c.g.151.2		12
19.18	odd	2	inner	855.3.e.a.721.11		12
57.56	even	2		95.3.c.a.56.2	✓	12
228.227	odd	2		1520.3.h.a.721.8		12
285.113	odd	4		475.3.d.c.474.3		24
285.227	odd	4		475.3.d.c.474.22		24
285.284	even	2		475.3.c.g.151.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.c.a.56.2	✓	12	57.56	even	2
95.3.c.a.56.11	yes	12	3.2	odd	2
475.3.c.g.151.2		12	15.14	odd	2
475.3.c.g.151.11		12	285.284	even	2
475.3.d.c.474.3		24	285.113	odd	4
475.3.d.c.474.4		24	15.2	even	4
475.3.d.c.474.21		24	15.8	even	4
475.3.d.c.474.22		24	285.227	odd	4
855.3.e.a.721.2		12	1.1	even	1	trivial
855.3.e.a.721.11		12	19.18	odd	2	inner
1520.3.h.a.721.5		12	12.11	even	2
1520.3.h.a.721.8		12	228.227	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	855.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.84623i q^{2} -4.10101 q^{4} +2.23607 q^{5} -9.15076 q^{7} +0.287487i q^{8} +O(q^{10})\)	\(q-2.84623i q^{2} -4.10101 q^{4} +2.23607 q^{5} -9.15076 q^{7} +0.287487i q^{8} -6.36436i q^{10} -9.89207 q^{11} +12.3251i q^{13} +26.0451i q^{14} -15.5858 q^{16} +11.1895 q^{17} +(18.7659 - 2.97345i) q^{19} -9.17013 q^{20} +28.1551i q^{22} -20.8666 q^{23} +5.00000 q^{25} +35.0800 q^{26} +37.5273 q^{28} -10.5011i q^{29} +52.3075i q^{31} +45.5106i q^{32} -31.8479i q^{34} -20.4617 q^{35} -10.5041i q^{37} +(-8.46312 - 53.4120i) q^{38} +0.642841i q^{40} +42.6288i q^{41} +45.9768 q^{43} +40.5674 q^{44} +59.3912i q^{46} -4.22163 q^{47} +34.7363 q^{49} -14.2311i q^{50} -50.5453i q^{52} +95.8254i q^{53} -22.1193 q^{55} -2.63073i q^{56} -29.8886 q^{58} -6.38544i q^{59} +14.8591 q^{61} +148.879 q^{62} +67.1904 q^{64} +27.5598i q^{65} -38.0693i q^{67} -45.8882 q^{68} +58.2387i q^{70} +125.169i q^{71} +18.5252 q^{73} -29.8971 q^{74} +(-76.9590 + 12.1941i) q^{76} +90.5199 q^{77} +137.419i q^{79} -34.8508 q^{80} +121.331 q^{82} -127.881 q^{83} +25.0205 q^{85} -130.860i q^{86} -2.84385i q^{88} -19.8059i q^{89} -112.784i q^{91} +85.5743 q^{92} +12.0157i q^{94} +(41.9618 - 6.64884i) q^{95} -42.4473i q^{97} -98.8675i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{4} + 20 q^{7}+O(q^{10}) \) 12 * q - 12 * q^4 + 20 * q^7	\( 12 q - 12 q^{4} + 20 q^{7} - 32 q^{11} - 44 q^{16} + 44 q^{17} + 8 q^{19} - 36 q^{23} + 60 q^{25} - 108 q^{26} - 36 q^{28} + 40 q^{35} + 44 q^{38} + 320 q^{43} + 256 q^{44} + 56 q^{47} + 72 q^{49} + 68 q^{58} - 296 q^{61} + 376 q^{62} + 188 q^{64} + 340 q^{68} - 244 q^{73} - 136 q^{74} + 248 q^{76} + 200 q^{77} - 200 q^{80} + 424 q^{82} + 160 q^{83} + 160 q^{85} - 716 q^{92} + 80 q^{95}+O(q^{100}) \) 12 * q - 12 * q^4 + 20 * q^7 - 32 * q^11 - 44 * q^16 + 44 * q^17 + 8 * q^19 - 36 * q^23 + 60 * q^25 - 108 * q^26 - 36 * q^28 + 40 * q^35 + 44 * q^38 + 320 * q^43 + 256 * q^44 + 56 * q^47 + 72 * q^49 + 68 * q^58 - 296 * q^61 + 376 * q^62 + 188 * q^64 + 340 * q^68 - 244 * q^73 - 136 * q^74 + 248 * q^76 + 200 * q^77 - 200 * q^80 + 424 * q^82 + 160 * q^83 + 160 * q^85 - 716 * q^92 + 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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