LMFDB - Embedded newform 2034.3.d.a.2033.32

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2034,3,Mod(2033,2034)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2034.2033"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2034, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2034 = 2 \cdot 3^{2} \cdot 113 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2034.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$55.4224857709$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2033.32
Character	$\chi$	$=$	2034.2033
Dual form			2034.3.d.a.2033.31

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} -2.00000 q^{4} -3.08580 q^{5} +9.93072 q^{7} -2.82843i q^{8} -4.36398i q^{10} +6.64882i q^{11} -16.2542 q^{13} +14.0442i q^{14} +4.00000 q^{16} -11.4308 q^{17} -17.1180i q^{19} +6.17159 q^{20} -9.40285 q^{22} +25.9910 q^{23} -15.4779 q^{25} -22.9869i q^{26} -19.8614 q^{28} +39.6228 q^{29} -23.5666 q^{31} +5.65685i q^{32} -16.1657i q^{34} -30.6442 q^{35} +11.4764i q^{37} +24.2086 q^{38} +8.72795i q^{40} +13.2048i q^{41} -59.9593i q^{43} -13.2976i q^{44} +36.7568i q^{46} -18.9990 q^{47} +49.6192 q^{49} -21.8890i q^{50} +32.5084 q^{52} +18.3758i q^{53} -20.5169i q^{55} -28.0883i q^{56} +56.0350i q^{58} -4.79160 q^{59} -72.0736 q^{61} -33.3282i q^{62} -8.00000 q^{64} +50.1571 q^{65} +38.1779i q^{67} +22.8617 q^{68} -43.3374i q^{70} -78.0632 q^{71} -93.5234i q^{73} -16.2300 q^{74} +34.2361i q^{76} +66.0276i q^{77} -115.264i q^{79} -12.3432 q^{80} -18.6743 q^{82} +10.5559i q^{83} +35.2733 q^{85} +84.7953 q^{86} +18.8057 q^{88} +11.0370 q^{89} -161.416 q^{91} -51.9820 q^{92} -26.8687i q^{94} +52.8228i q^{95} -154.475 q^{97} +70.1721i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) $ 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2034\mathbb{Z}\right)^\times$.

$n$	$227$	$1585$
$\chi(n)$	$-1$	$-1$

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 $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$		0			0
$3$		0			0
$4$	−2.00000			−0.500000
$5$	−3.08580			−0.617159			−0.308580	−	0.951198i	$-0.599854\pi$
$5$	−3.08580			−0.617159			−0.308580	+	0.951198i	$0.599854\pi$
$6$		0			0
$7$	9.93072			1.41867			0.709337	−	0.704869i	$-0.248994\pi$
$7$	9.93072			1.41867			0.709337	+	0.704869i	$0.248994\pi$
$8$		−	2.82843i		−	0.353553i
$9$		0			0
$10$		−	4.36398i		−	0.436398i
$11$			6.64882i			0.604438i	0.953239	+	0.302219i	$0.0977274\pi$
$11$			6.64882i			0.604438i	−0.953239	+	0.302219i	$0.902273\pi$
$12$		0			0
$13$	−16.2542			−1.25032			−0.625161	−	0.780496i	$-0.714967\pi$
$13$	−16.2542			−1.25032			−0.625161	+	0.780496i	$0.714967\pi$
$14$			14.0442i			1.00315i
$15$		0			0
$16$	4.00000			0.250000
$17$	−11.4308			−0.672403			−0.336201	−	0.941790i	$-0.609142\pi$
$17$	−11.4308			−0.672403			−0.336201	+	0.941790i	$0.609142\pi$
$18$		0			0
$19$		−	17.1180i		−	0.900950i	−0.892789	−	0.450475i	$-0.851255\pi$
$19$		−	17.1180i		−	0.900950i	0.892789	−	0.450475i	$-0.148745\pi$
$20$	6.17159			0.308580
$21$		0			0
$22$	−9.40285			−0.427402
$23$	25.9910			1.13004			0.565021	−	0.825076i	$-0.308868\pi$
$23$	25.9910			1.13004			0.565021	+	0.825076i	$0.308868\pi$
$24$		0			0
$25$	−15.4779			−0.619114
$26$		−	22.9869i		−	0.884111i
$27$		0			0
$28$	−19.8614			−0.709337
$29$	39.6228			1.36630			0.683151	−	0.730277i	$-0.260609\pi$
$29$	39.6228			1.36630			0.683151	+	0.730277i	$0.260609\pi$
$30$		0			0
$31$	−23.5666			−0.760212			−0.380106	−	0.924943i	$-0.624112\pi$
$31$	−23.5666			−0.760212			−0.380106	+	0.924943i	$0.624112\pi$
$32$			5.65685i			0.176777i
$33$		0			0
$34$		−	16.1657i		−	0.475460i
$35$	−30.6442			−0.875548
$36$		0			0
$37$			11.4764i			0.310172i	0.987901	+	0.155086i	$0.0495655\pi$
$37$			11.4764i			0.310172i	−0.987901	+	0.155086i	$0.950435\pi$
$38$	24.2086			0.637068
$39$		0			0
$40$			8.72795i			0.218199i
$41$			13.2048i			0.322067i	0.986949	+	0.161034i	$0.0514827\pi$
$41$			13.2048i			0.322067i	−0.986949	+	0.161034i	$0.948517\pi$
$42$		0			0
$43$		−	59.9593i		−	1.39440i	−0.716875	−	0.697202i	$-0.754428\pi$
$43$		−	59.9593i		−	1.39440i	0.716875	−	0.697202i	$-0.245572\pi$
$44$		−	13.2976i		−	0.302219i
$45$		0			0
$46$			36.7568i			0.799061i
$47$	−18.9990			−0.404235			−0.202117	−	0.979361i	$-0.564782\pi$
$47$	−18.9990			−0.404235			−0.202117	+	0.979361i	$0.564782\pi$
$48$		0			0
$49$	49.6192			1.01264
$50$		−	21.8890i		−	0.437780i
$51$		0			0
$52$	32.5084			0.625161
$53$			18.3758i			0.346713i	0.984859	+	0.173357i	$0.0554613\pi$
$53$			18.3758i			0.346713i	−0.984859	+	0.173357i	$0.944539\pi$
$54$		0			0
$55$		−	20.5169i		−	0.373035i
$56$		−	28.0883i		−	0.501577i
$57$		0			0
$58$			56.0350i			0.966122i
$59$	−4.79160			−0.0812135			−0.0406068	−	0.999175i	$-0.512929\pi$
$59$	−4.79160			−0.0812135			−0.0406068	+	0.999175i	$0.512929\pi$
$60$		0			0
$61$	−72.0736			−1.18153			−0.590767	−	0.806842i	$-0.701175\pi$
$61$	−72.0736			−1.18153			−0.590767	+	0.806842i	$0.701175\pi$
$62$		−	33.3282i		−	0.537551i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	50.1571			0.771648
$66$		0			0
$67$			38.1779i			0.569820i	0.958554	+	0.284910i	$0.0919637\pi$
$67$			38.1779i			0.569820i	−0.958554	+	0.284910i	$0.908036\pi$
$68$	22.8617			0.336201
$69$		0			0
$70$		−	43.3374i		−	0.619106i
$71$	−78.0632			−1.09948			−0.549741	−	0.835335i	$-0.685273\pi$
$71$	−78.0632			−1.09948			−0.549741	+	0.835335i	$0.685273\pi$
$72$		0			0
$73$		−	93.5234i		−	1.28114i	−0.767899	−	0.640571i	$-0.778698\pi$
$73$		−	93.5234i		−	1.28114i	0.767899	−	0.640571i	$-0.221302\pi$
$74$	−16.2300			−0.219325
$75$		0			0
$76$			34.2361i			0.450475i
$77$			66.0276i			0.857501i
$78$		0			0
$79$		−	115.264i		−	1.45904i	−0.683959	−	0.729520i	$-0.739743\pi$
$79$		−	115.264i		−	1.45904i	0.683959	−	0.729520i	$-0.260257\pi$
$80$	−12.3432			−0.154290
$81$		0			0
$82$	−18.6743			−0.227736
$83$			10.5559i			0.127180i	0.997976	+	0.0635898i	$0.0202549\pi$
$83$			10.5559i			0.127180i	−0.997976	+	0.0635898i	$0.979745\pi$
$84$		0			0
$85$	35.2733			0.414980
$86$	84.7953			0.985992
$87$		0			0
$88$	18.8057			0.213701
$89$	11.0370			0.124011			0.0620057	−	0.998076i	$-0.480250\pi$
$89$	11.0370			0.124011			0.0620057	+	0.998076i	$0.480250\pi$
$90$		0			0
$91$	−161.416			−1.77380
$92$	−51.9820			−0.565021
$93$		0			0
$94$		−	26.8687i		−	0.285837i
$95$			52.8228i			0.556030i
$96$		0			0
$97$	−154.475			−1.59253			−0.796263	−	0.604950i	$-0.793193\pi$
$97$	−154.475			−1.59253			−0.796263	+	0.604950i	$0.793193\pi$
$98$			70.1721i			0.716042i
$99$		0			0
$100$	30.9557			0.309557
$101$	14.4811			0.143377			0.0716884	−	0.997427i	$-0.477161\pi$
$101$	14.4811			0.143377			0.0716884	+	0.997427i	$0.477161\pi$
$102$		0			0
$103$		−	130.454i		−	1.26654i	−0.773930	−	0.633271i	$-0.781712\pi$
$103$		−	130.454i		−	1.26654i	0.773930	−	0.633271i	$-0.218288\pi$
$104$			45.9738i			0.442056i
$105$		0			0
$106$	−25.9873			−0.245163
$107$	−135.187			−1.26343			−0.631716	−	0.775200i	$-0.717649\pi$
$107$	−135.187			−1.26343			−0.631716	+	0.775200i	$0.717649\pi$
$108$		0			0
$109$	136.445			1.25179			0.625896	−	0.779907i	$-0.284733\pi$
$109$	136.445			1.25179			0.625896	+	0.779907i	$0.284733\pi$
$110$	29.0153			0.263775
$111$		0			0
$112$	39.7229			0.354669
$113$	−98.6090	−	55.1839i	−0.872646	−	0.488353i
$113$	−98.6090	−	55.1839i	−0.872646	−	0.488353i
$114$		0			0
$115$	−80.2029			−0.697416
$116$	−79.2455			−0.683151
$117$		0			0
$118$		−	6.77634i		−	0.0574266i
$119$	−113.517			−0.953920
$120$		0			0
$121$	76.7932			0.634655
$122$		−	101.927i		−	0.835471i
$123$		0			0
$124$	47.1331			0.380106
$125$	124.906			0.999252
$126$		0			0
$127$	−90.2928			−0.710967			−0.355483	−	0.934683i	$-0.615684\pi$
$127$	−90.2928			−0.710967			−0.355483	+	0.934683i	$0.615684\pi$
$128$		−	11.3137i		−	0.0883883i
$129$		0			0
$130$			70.9329i			0.545637i
$131$		−	110.841i		−	0.846117i	−0.906102	−	0.423059i	$-0.860956\pi$
$131$		−	110.841i		−	0.846117i	0.906102	−	0.423059i	$-0.139044\pi$
$132$		0			0
$133$		−	169.995i		−	1.27815i
$134$	−53.9918			−0.402924
$135$		0			0
$136$			32.3313i			0.237730i
$137$	182.536			1.33238			0.666189	−	0.745783i	$-0.267924\pi$
$137$	182.536			1.33238			0.666189	+	0.745783i	$0.267924\pi$
$138$		0			0
$139$	112.153			0.806858			0.403429	−	0.915011i	$-0.367818\pi$
$139$	112.153			0.806858			0.403429	+	0.915011i	$0.367818\pi$
$140$	61.2884			0.437774
$141$		0			0
$142$		−	110.398i		−	0.777451i
$143$		−	108.071i		−	0.755742i
$144$		0			0
$145$	−122.268			−0.843226
$146$	132.262			0.905905
$147$		0			0
$148$		−	22.9527i		−	0.155086i
$149$		−	48.7209i		−	0.326986i	−0.986545	−	0.163493i	$-0.947724\pi$
$149$		−	48.7209i		−	0.326986i	0.986545	−	0.163493i	$-0.0522761\pi$
$150$		0			0
$151$		−	287.813i		−	1.90604i	−0.302902	−	0.953022i	$-0.597955\pi$
$151$		−	287.813i		−	1.90604i	0.302902	−	0.953022i	$-0.402045\pi$
$152$	−48.4171			−0.318534
$153$		0			0
$154$	−93.3771			−0.606345
$155$	72.7216			0.469172
$156$		0			0
$157$	272.135			1.73334			0.866672	−	0.498879i	$-0.166255\pi$
$157$	272.135			1.73334			0.866672	+	0.498879i	$0.166255\pi$
$158$	163.008			1.03170
$159$		0			0
$160$		−	17.4559i		−	0.109099i
$161$	258.109			1.60316
$162$		0			0
$163$	−44.3800			−0.272270			−0.136135	−	0.990690i	$-0.543468\pi$
$163$	−44.3800			−0.272270			−0.136135	+	0.990690i	$0.543468\pi$
$164$		−	26.4095i		−	0.161034i
$165$		0			0
$166$	−14.9283			−0.0899296
$167$	9.87958			0.0591592			0.0295796	−	0.999562i	$-0.490583\pi$
$167$	9.87958			0.0591592			0.0295796	+	0.999562i	$0.490583\pi$
$168$		0			0
$169$	95.1986			0.563305
$170$			49.8839i			0.293435i
$171$		0			0
$172$			119.919i			0.697202i
$173$		−	206.845i		−	1.19563i	−0.801633	−	0.597817i	$-0.796035\pi$
$173$		−	206.845i		−	1.19563i	0.801633	−	0.597817i	$-0.203965\pi$
$174$		0			0
$175$	−153.706			−0.878322
$176$			26.5953i			0.151110i
$177$		0			0
$178$			15.6087i			0.0876893i
$179$	20.2691			0.113235			0.0566176	−	0.998396i	$-0.481968\pi$
$179$	20.2691			0.113235			0.0566176	+	0.998396i	$0.481968\pi$
$180$		0			0
$181$		−	234.124i		−	1.29350i	−0.762701	−	0.646752i	$-0.776127\pi$
$181$		−	234.124i		−	1.29350i	0.762701	−	0.646752i	$-0.223873\pi$
$182$		−	228.276i		−	1.25427i
$183$		0			0
$184$		−	73.5136i		−	0.399530i
$185$		−	35.4137i		−	0.191426i
$186$		0			0
$187$		−	76.0016i		−	0.406426i
$188$	37.9981			0.202117
$189$		0			0
$190$	−74.7027			−0.393172
$191$	185.869			0.973139			0.486569	−	0.873642i	$-0.338248\pi$
$191$	185.869			0.973139			0.486569	+	0.873642i	$0.338248\pi$
$192$		0			0
$193$			172.224i			0.892355i	0.894945	+	0.446177i	$0.147215\pi$
$193$			172.224i			0.892355i	−0.894945	+	0.446177i	$0.852785\pi$
$194$		−	218.461i		−	1.12609i
$195$		0			0
$196$	−99.2384			−0.506318
$197$	280.846			1.42561			0.712807	−	0.701360i	$-0.247424\pi$
$197$	280.846			1.42561			0.712807	+	0.701360i	$0.247424\pi$
$198$		0			0
$199$		−	181.951i		−	0.914329i	−0.889382	−	0.457164i	$-0.848865\pi$
$199$		−	181.951i		−	0.914329i	0.889382	−	0.457164i	$-0.151135\pi$
$200$			43.7780i			0.218890i
$201$		0			0
$202$			20.4793i			0.101383i
$203$	393.483			1.93834
$204$		0			0
$205$		−	40.7472i		−	0.198767i
$206$	184.490			0.895580
$207$		0			0
$208$	−65.0167			−0.312580
$209$	113.815			0.544568
$210$		0			0
$211$	−297.802			−1.41138			−0.705692	−	0.708519i	$-0.749364\pi$
$211$	−297.802			−1.41138			−0.705692	+	0.708519i	$0.749364\pi$
$212$		−	36.7516i		−	0.173357i
$213$		0			0
$214$		−	191.184i		−	0.893382i
$215$			185.022i			0.860569i
$216$		0			0
$217$	−234.033			−1.07849
$218$			192.963i			0.885151i
$219$		0			0
$220$			41.0338i			0.186517i
$221$	185.799			0.840720
$222$		0			0
$223$		−	0.537524i		−	0.00241042i	−0.999999	−	0.00120521i	$-0.999616\pi$
$223$		−	0.537524i		−	0.00241042i	0.999999	−	0.00120521i	$-0.000383631\pi$
$224$			56.1766i			0.250789i
$225$		0			0
$226$	78.0419	−	139.454i	0.345318	−	0.617054i
$227$		−	329.581i		−	1.45190i	−0.687748	−	0.725950i	$-0.741400\pi$
$227$		−	329.581i		−	1.45190i	0.687748	−	0.725950i	$-0.258600\pi$
$228$		0			0
$229$			177.884i			0.776786i	0.921494	+	0.388393i	$0.126970\pi$
$229$			177.884i			0.776786i	−0.921494	+	0.388393i	$0.873030\pi$
$230$		−	113.424i		−	0.493148i
$231$		0			0
$232$		−	112.070i		−	0.483061i
$233$		−	161.269i		−	0.692144i	−0.938208	−	0.346072i	$-0.887515\pi$
$233$		−	161.269i		−	0.692144i	0.938208	−	0.346072i	$-0.112485\pi$
$234$		0			0
$235$	58.6272			0.249477
$236$	9.58320			0.0406068
$237$		0			0
$238$		−	160.537i		−	0.674523i
$239$			122.560i			0.512805i	0.966570	+	0.256402i	$0.0825372\pi$
$239$			122.560i			0.512805i	−0.966570	+	0.256402i	$0.917463\pi$
$240$		0			0
$241$	−215.979			−0.896180			−0.448090	−	0.893988i	$-0.647896\pi$
$241$	−215.979			−0.896180			−0.448090	+	0.893988i	$0.647896\pi$
$242$			108.602i			0.448769i
$243$		0			0
$244$	144.147			0.590767
$245$	−153.115			−0.624958
$246$		0			0
$247$			278.240i			1.12648i
$248$			66.6563i			0.268775i
$249$		0			0
$250$			176.644i			0.706578i
$251$		−	199.135i		−	0.793368i	−0.917955	−	0.396684i	$-0.870161\pi$
$251$		−	199.135i		−	0.793368i	0.917955	−	0.396684i	$-0.129839\pi$
$252$		0			0
$253$			172.809i			0.683041i
$254$		−	127.693i		−	0.502730i
$255$		0			0
$256$	16.0000			0.0625000
$257$			13.1179i			0.0510422i	0.999674	+	0.0255211i	$0.00812451\pi$
$257$			13.1179i			0.0510422i	−0.999674	+	0.0255211i	$0.991875\pi$
$258$		0			0
$259$			113.969i			0.440033i
$260$	−100.314			−0.385824
$261$		0			0
$262$	156.753			0.598295
$263$	−190.551			−0.724529			−0.362265	−	0.932075i	$-0.617996\pi$
$263$	−190.551			−0.724529			−0.362265	+	0.932075i	$0.617996\pi$
$264$		0			0
$265$		−	56.7040i		−	0.213977i
$266$	240.409			0.903791
$267$		0			0
$268$		−	76.3559i		−	0.284910i
$269$	−147.829			−0.549549			−0.274775	−	0.961509i	$-0.588603\pi$
$269$	−147.829			−0.549549			−0.274775	+	0.961509i	$0.588603\pi$
$270$		0			0
$271$		−	101.501i		−	0.374542i	−0.982308	−	0.187271i	$-0.940036\pi$
$271$		−	101.501i		−	0.374542i	0.982308	−	0.187271i	$-0.0599642\pi$
$272$	−45.7234			−0.168101
$273$		0			0
$274$			258.145i			0.942134i
$275$		−	102.909i		−	0.374216i
$276$		0			0
$277$	−63.8668			−0.230566			−0.115283	−	0.993333i	$-0.536778\pi$
$277$	−63.8668			−0.230566			−0.115283	+	0.993333i	$0.536778\pi$
$278$			158.609i			0.570535i
$279$		0			0
$280$			86.6748i			0.309553i
$281$	−188.990			−0.672563			−0.336281	−	0.941762i	$-0.609169\pi$
$281$	−188.990			−0.672563			−0.336281	+	0.941762i	$0.609169\pi$
$282$		0			0
$283$	−2.22240			−0.00785301			−0.00392651	−	0.999992i	$-0.501250\pi$
$283$	−2.22240			−0.00785301			−0.00392651	+	0.999992i	$0.501250\pi$
$284$	156.126			0.549741
$285$		0			0
$286$	152.836			0.534390
$287$			131.133i			0.456908i
$288$		0			0
$289$	−158.336			−0.547875
$290$		−	172.913i		−	0.596251i
$291$		0			0
$292$			187.047i			0.640571i
$293$	−490.066			−1.67258			−0.836291	−	0.548286i	$-0.815280\pi$
$293$	−490.066			−1.67258			−0.836291	+	0.548286i	$0.815280\pi$
$294$		0			0
$295$	14.7859			0.0501217
$296$	32.4601			0.109662
$297$		0			0
$298$	68.9017			0.231214
$299$	−422.462			−1.41292
$300$		0			0
$301$		−	595.439i		−	1.97820i
$302$	407.028			1.34778
$303$		0			0
$304$		−	68.4722i		−	0.225237i
$305$	222.405			0.729195
$306$		0			0
$307$	382.392			1.24558			0.622789	−	0.782390i	$-0.285999\pi$
$307$	382.392			1.24558			0.622789	+	0.782390i	$0.285999\pi$
$308$		−	132.055i		−	0.428750i
$309$		0			0
$310$			102.844i			0.331755i
$311$			42.9026i			0.137951i	0.997618	+	0.0689753i	$0.0219730\pi$
$311$			42.9026i			0.137951i	−0.997618	+	0.0689753i	$0.978027\pi$
$312$		0			0
$313$	−421.051			−1.34521			−0.672606	−	0.740001i	$-0.734825\pi$
$313$	−421.051			−1.34521			−0.672606	+	0.740001i	$0.734825\pi$
$314$			384.857i			1.22566i
$315$		0			0
$316$			230.528i			0.729520i
$317$			476.330i			1.50262i	0.659951	+	0.751309i	$0.270577\pi$
$317$			476.330i			1.50262i	−0.659951	+	0.751309i	$0.729423\pi$
$318$		0			0
$319$			263.445i			0.825845i
$320$	24.6864			0.0771449
$321$		0			0
$322$			365.021i			1.13361i
$323$			195.674i			0.605801i
$324$		0			0
$325$	251.580			0.774092
$326$		−	62.7628i		−	0.192524i
$327$		0			0
$328$	37.3487			0.113868
$329$	−188.674			−0.573478
$330$		0			0
$331$	−382.628			−1.15598			−0.577988	−	0.816045i	$-0.696162\pi$
$331$	−382.628			−1.15598			−0.577988	+	0.816045i	$0.696162\pi$
$332$		−	21.1118i		−	0.0635898i
$333$		0			0
$334$			13.9718i			0.0418318i
$335$		−	117.809i		−	0.351670i
$336$		0			0
$337$	3.07317			0.00911919			0.00455960	−	0.999990i	$-0.498549\pi$
$337$	3.07317			0.00911919			0.00455960	+	0.999990i	$0.498549\pi$
$338$			134.631i			0.398317i
$339$		0			0
$340$	−70.5465			−0.207490
$341$		−	156.690i		−	0.459501i
$342$		0			0
$343$	6.14906			0.0179273
$344$	−169.591			−0.492996
$345$		0			0
$346$	292.522			0.845441
$347$		−	173.667i		−	0.500480i	−0.968184	−	0.250240i	$-0.919490\pi$
$347$		−	173.667i		−	0.500480i	0.968184	−	0.250240i	$-0.0805095\pi$
$348$		0			0
$349$		−	110.693i		−	0.317172i	−0.987345	−	0.158586i	$-0.949306\pi$
$349$		−	110.693i		−	0.317172i	0.987345	−	0.158586i	$-0.0506935\pi$
$350$		−	217.373i		−	0.621067i
$351$		0			0
$352$	−37.6114			−0.106851
$353$		−	95.1212i		−	0.269465i	−0.990882	−	0.134733i	$-0.956982\pi$
$353$		−	95.1212i		−	0.269465i	0.990882	−	0.134733i	$-0.0430175\pi$
$354$		0			0
$355$	240.887			0.678555
$356$	−22.0740			−0.0620057
$357$		0			0
$358$			28.6648i			0.0800694i
$359$	332.555			0.926338			0.463169	−	0.886270i	$-0.346712\pi$
$359$	332.555			0.926338			0.463169	+	0.886270i	$0.346712\pi$
$360$		0			0
$361$	67.9726			0.188290
$362$	331.101			0.914645
$363$		0			0
$364$	322.832			0.886900
$365$			288.594i			0.790669i
$366$		0			0
$367$	166.688			0.454190			0.227095	−	0.973873i	$-0.427077\pi$
$367$	166.688			0.454190			0.227095	+	0.973873i	$0.427077\pi$
$368$	103.964			0.282511
$369$		0			0
$370$	50.0826			0.135358
$371$			182.485i			0.491873i
$372$		0			0
$373$			590.608i			1.58340i	0.610911	+	0.791699i	$0.290803\pi$
$373$			590.608i			1.58340i	−0.610911	+	0.791699i	$0.709197\pi$
$374$	107.483			0.287386
$375$		0			0
$376$			53.7374i			0.142919i
$377$	−644.036			−1.70832
$378$		0			0
$379$		−	384.629i		−	1.01485i	−0.861696	−	0.507426i	$-0.830597\pi$
$379$		−	384.629i		−	1.01485i	0.861696	−	0.507426i	$-0.169403\pi$
$380$		−	105.646i		−	0.278015i
$381$		0			0
$382$			262.859i			0.688113i
$383$			366.115i			0.955914i	0.878383	+	0.477957i	$0.158623\pi$
$383$			366.115i			0.955914i	−0.878383	+	0.477957i	$0.841377\pi$
$384$		0			0
$385$		−	203.748i		−	0.529215i
$386$	−243.562			−0.630990
$387$		0			0
$388$	308.950			0.796263
$389$			97.4432i			0.250497i	0.992125	+	0.125248i	$0.0399728\pi$
$389$			97.4432i			0.250497i	−0.992125	+	0.125248i	$0.960027\pi$
$390$		0			0
$391$	−297.099			−0.759843
$392$		−	140.344i		−	0.358021i
$393$		0			0
$394$			397.176i			1.00806i
$395$			355.682i			0.900461i
$396$		0			0
$397$		−	222.848i		−	0.561330i	−0.959806	−	0.280665i	$-0.909445\pi$
$397$		−	222.848i		−	0.561330i	0.959806	−	0.280665i	$-0.0905550\pi$
$398$	257.318			0.646528
$399$		0			0
$400$	−61.9114			−0.154779
$401$			93.1428i			0.232276i	0.993233	+	0.116138i	$0.0370515\pi$
$401$			93.1428i			0.232276i	−0.993233	+	0.116138i	$0.962948\pi$
$402$		0			0
$403$	383.055			0.950510
$404$	−28.9621			−0.0716884
$405$		0			0
$406$			556.468i			1.37061i
$407$	−76.3043			−0.187480
$408$		0			0
$409$		−	196.636i		−	0.480773i	−0.970677	−	0.240386i	$-0.922726\pi$
$409$		−	196.636i		−	0.480773i	0.970677	−	0.240386i	$-0.0772741\pi$
$410$	57.6252			0.140549
$411$		0			0
$412$			260.908i			0.633271i
$413$	−47.5840			−0.115216
$414$		0			0
$415$		−	32.5734i		−	0.0784901i
$416$		−	91.9476i		−	0.221028i
$417$		0			0
$418$			160.958i			0.385068i
$419$	256.014			0.611012			0.305506	−	0.952190i	$-0.401174\pi$
$419$	256.014			0.611012			0.305506	+	0.952190i	$0.401174\pi$
$420$		0			0
$421$	567.774			1.34863			0.674316	−	0.738443i	$-0.264439\pi$
$421$	567.774			1.34863			0.674316	+	0.738443i	$0.264439\pi$
$422$		−	421.155i		−	0.997999i
$423$		0			0
$424$	51.9746			0.122582
$425$	176.925			0.416294
$426$		0			0
$427$	−715.743			−1.67621
$428$	270.375			0.631716
$429$		0			0
$430$	−261.661			−0.608514
$431$	267.837			0.621430			0.310715	−	0.950503i	$-0.399431\pi$
$431$	267.837			0.621430			0.310715	+	0.950503i	$0.399431\pi$
$432$		0			0
$433$		−	335.852i		−	0.775639i	−0.921735	−	0.387820i	$-0.873228\pi$
$433$		−	335.852i		−	0.775639i	0.921735	−	0.387820i	$-0.126772\pi$
$434$		−	330.973i		−	0.762610i
$435$		0			0
$436$	−272.891			−0.625896
$437$		−	444.915i		−	1.01811i
$438$		0			0
$439$	149.156			0.339764			0.169882	−	0.985464i	$-0.445661\pi$
$439$	149.156			0.339764			0.169882	+	0.985464i	$0.445661\pi$
$440$	−58.0306			−0.131888
$441$		0			0
$442$			262.760i			0.594479i
$443$		−	192.893i		−	0.435423i	−0.976013	−	0.217712i	$-0.930141\pi$
$443$		−	192.893i		−	0.435423i	0.976013	−	0.217712i	$-0.0698592\pi$
$444$		0			0
$445$	−34.0580			−0.0765348
$446$	0.760174			0.00170443
$447$		0			0
$448$	−79.4458			−0.177334
$449$	−782.374			−1.74248			−0.871240	−	0.490857i	$-0.836684\pi$
$449$	−782.374			−1.74248			−0.871240	+	0.490857i	$0.836684\pi$
$450$		0			0
$451$	−87.7960			−0.194670
$452$	197.218	+	110.368i	0.436323	+	0.244177i
$453$		0			0
$454$	466.098			1.02665
$455$	498.096			1.09472
$456$		0			0
$457$		−	591.567i		−	1.29446i	−0.762296	−	0.647229i	$-0.775928\pi$
$457$		−	591.567i		−	1.29446i	0.762296	−	0.647229i	$-0.224072\pi$
$458$	−251.566			−0.549271
$459$		0			0
$460$	160.406			0.348708
$461$			39.4066i			0.0854807i	0.999086	+	0.0427403i	$0.0136088\pi$
$461$			39.4066i			0.0854807i	−0.999086	+	0.0427403i	$0.986391\pi$
$462$		0			0
$463$	35.5663			0.0768171			0.0384086	−	0.999262i	$-0.487771\pi$
$463$	35.5663			0.0768171			0.0384086	+	0.999262i	$0.487771\pi$
$464$	158.491			0.341576
$465$		0			0
$466$	228.069			0.489420
$467$		−	336.601i		−	0.720772i	−0.932803	−	0.360386i	$-0.882645\pi$
$467$		−	336.601i		−	0.720772i	0.932803	−	0.360386i	$-0.117355\pi$
$468$		0			0
$469$			379.134i			0.808389i
$470$			82.9113i			0.176407i
$471$		0			0
$472$			13.5527i			0.0287133i
$473$	398.659			0.842830
$474$		0			0
$475$			264.951i			0.557791i
$476$	227.033			0.476960
$477$		0			0
$478$	−173.327			−0.362608
$479$	−818.386			−1.70853			−0.854265	−	0.519838i	$-0.825992\pi$
$479$	−818.386			−1.70853			−0.854265	+	0.519838i	$0.825992\pi$
$480$		0			0
$481$		−	186.539i		−	0.387815i
$482$		−	305.441i		−	0.633695i
$483$		0			0
$484$	−153.586			−0.317327
$485$	476.679			0.982843
$486$		0			0
$487$			252.480i			0.518439i	0.965818	+	0.259219i	$0.0834653\pi$
$487$			252.480i			0.518439i	−0.965818	+	0.259219i	$0.916535\pi$
$488$			203.855i			0.417736i
$489$		0			0
$490$		−	216.537i		−	0.441912i
$491$	−400.644			−0.815976			−0.407988	−	0.912987i	$-0.633769\pi$
$491$	−400.644			−0.815976			−0.407988	+	0.912987i	$0.633769\pi$
$492$		0			0
$493$	−452.922			−0.918705
$494$	−393.491			−0.796540
$495$		0			0
$496$	−94.2663			−0.190053
$497$	−775.223			−1.55981
$498$		0			0
$499$			559.404i			1.12105i	0.828137	+	0.560525i	$0.189401\pi$
$499$			559.404i			1.12105i	−0.828137	+	0.560525i	$0.810599\pi$
$500$	−249.813			−0.499626
$501$		0			0
$502$	281.620			0.560996
$503$		−	718.782i		−	1.42899i	−0.699640	−	0.714495i	$-0.746656\pi$
$503$		−	718.782i		−	1.42899i	0.699640	−	0.714495i	$-0.253344\pi$
$504$		0			0
$505$	−44.6856			−0.0884863
$506$	−244.389			−0.482983
$507$		0			0
$508$	180.586			0.355483
$509$			551.808i			1.08410i	0.840345	+	0.542051i	$0.182352\pi$
$509$			551.808i			1.08410i	−0.840345	+	0.542051i	$0.817648\pi$
$510$		0			0
$511$		−	928.755i		−	1.81752i
$512$			22.6274i			0.0441942i
$513$		0			0
$514$	−18.5514			−0.0360923
$515$			402.554i			0.781658i
$516$		0			0
$517$		−	126.321i		−	0.244335i
$518$	−161.176			−0.311150
$519$		0			0
$520$		−	141.866i		−	0.272819i
$521$			771.969i			1.48171i	0.671667	+	0.740854i	$0.265579\pi$
$521$			771.969i			1.48171i	−0.671667	+	0.740854i	$0.734421\pi$
$522$		0			0
$523$			704.974i			1.34794i	0.738757	+	0.673972i	$0.235413\pi$
$523$			704.974i			1.34794i	−0.738757	+	0.673972i	$0.764587\pi$
$524$			221.683i			0.423059i
$525$		0			0
$526$		−	269.480i		−	0.512319i
$527$	269.386			0.511168
$528$		0			0
$529$	146.531			0.276996
$530$	80.1916			0.151305
$531$		0			0
$532$			339.989i			0.639077i
$533$		−	214.632i		−	0.402688i
$534$		0			0
$535$	417.161			0.779739
$536$	107.984			0.201462
$537$		0			0
$538$		−	209.061i		−	0.388590i
$539$			329.909i			0.612076i
$540$		0			0
$541$		−	234.260i		−	0.433013i	−0.976281	−	0.216506i	$-0.930534\pi$
$541$		−	234.260i		−	0.433013i	0.976281	−	0.216506i	$-0.0694662\pi$
$542$	143.544			0.264841
$543$		0			0
$544$		−	64.6626i		−	0.118865i
$545$	−421.043			−0.772555
$546$		0			0
$547$	427.068			0.780746			0.390373	−	0.920657i	$-0.372346\pi$
$547$	427.068			0.780746			0.390373	+	0.920657i	$0.372346\pi$
$548$	−365.072			−0.666189
$549$		0			0
$550$	145.536			0.264611
$551$		−	678.264i		−	1.23097i
$552$		0			0
$553$		−	1144.66i		−	2.06990i
$554$		−	90.3214i		−	0.163035i
$555$		0			0
$556$	−224.307			−0.403429
$557$		−	185.693i		−	0.333380i	−0.986009	−	0.166690i	$-0.946692\pi$
$557$		−	185.693i		−	0.333380i	0.986009	−	0.166690i	$-0.0533080\pi$
$558$		0			0
$559$			974.590i			1.74345i
$560$	−122.577			−0.218887
$561$		0			0
$562$		−	267.272i		−	0.475574i
$563$		−	681.947i		−	1.21127i	−0.795742	−	0.605636i	$-0.792919\pi$
$563$		−	681.947i		−	1.21127i	0.795742	−	0.605636i	$-0.207081\pi$
$564$		0			0
$565$	304.287	+	170.286i	0.538562	+	0.301392i
$566$		−	3.14295i		−	0.00555292i
$567$		0			0
$568$			220.796i			0.388725i
$569$			741.053i			1.30238i	0.758916	+	0.651189i	$0.225729\pi$
$569$			741.053i			1.30238i	−0.758916	+	0.651189i	$0.774271\pi$
$570$		0			0
$571$		−	629.420i		−	1.10231i	−0.834402	−	0.551156i	$-0.814187\pi$
$571$		−	629.420i		−	1.10231i	0.834402	−	0.551156i	$-0.185813\pi$
$572$			216.142i			0.377871i
$573$		0			0
$574$	−185.450			−0.323083
$575$	−402.285			−0.699625
$576$		0			0
$577$		−	331.909i		−	0.575232i	−0.957746	−	0.287616i	$-0.907137\pi$
$577$		−	331.909i		−	0.575232i	0.957746	−	0.287616i	$-0.0928626\pi$
$578$		−	223.921i		−	0.387406i
$579$		0			0
$580$	244.536			0.421613
$581$			104.828i			0.180426i
$582$		0			0
$583$	−122.177			−0.209567
$584$	−264.524			−0.452952
$585$		0			0
$586$		−	693.059i		−	1.18269i
$587$		−	226.511i		−	0.385879i	−0.981211	−	0.192940i	$-0.938198\pi$
$587$		−	226.511i		−	0.385879i	0.981211	−	0.192940i	$-0.0618022\pi$
$588$		0			0
$589$			403.414i			0.684913i
$590$			20.9104i			0.0354414i
$591$		0			0
$592$			45.9054i			0.0775430i
$593$			233.106i			0.393096i	0.980494	+	0.196548i	$0.0629732\pi$
$593$			233.106i			0.393096i	−0.980494	+	0.196548i	$0.937027\pi$
$594$		0			0
$595$	350.289			0.588721
$596$			97.4418i			0.163493i
$597$		0			0
$598$		−	597.452i		−	0.999083i
$599$	−824.557			−1.37656			−0.688278	−	0.725447i	$-0.741633\pi$
$599$	−824.557			−1.37656			−0.688278	+	0.725447i	$0.741633\pi$
$600$		0			0
$601$	−415.557			−0.691443			−0.345721	−	0.938337i	$-0.612366\pi$
$601$	−415.557			−0.691443			−0.345721	+	0.938337i	$0.612366\pi$
$602$	842.079			1.39880
$603$		0			0
$604$			575.625i			0.953022i
$605$	−236.968			−0.391683
$606$		0			0
$607$		−	920.990i		−	1.51728i	−0.651509	−	0.758641i	$-0.725864\pi$
$607$		−	920.990i		−	1.51728i	0.651509	−	0.758641i	$-0.274136\pi$
$608$	96.8343			0.159267
$609$		0			0
$610$			314.528i			0.515619i
$611$	308.814			0.505424
$612$		0			0
$613$			929.866i			1.51691i	0.651726	+	0.758455i	$0.274045\pi$
$613$			929.866i			1.51691i	−0.651726	+	0.758455i	$0.725955\pi$
$614$			540.785i			0.880757i
$615$		0			0
$616$	186.754			0.303172
$617$			700.176i			1.13481i	0.823440	+	0.567404i	$0.192052\pi$
$617$			700.176i			1.13481i	−0.823440	+	0.567404i	$0.807948\pi$
$618$		0			0
$619$			649.235i			1.04884i	0.851458	+	0.524422i	$0.175719\pi$
$619$			649.235i			1.04884i	−0.851458	+	0.524422i	$0.824281\pi$
$620$	−145.443			−0.234586
$621$		0			0
$622$	−60.6735			−0.0975458
$623$	109.606			0.175932
$624$		0			0
$625$	1.51054			0.00241686
$626$		−	595.456i		−	0.951208i
$627$		0			0
$628$	−544.270			−0.866672
$629$		−	131.185i		−	0.208560i
$630$		0			0
$631$			324.064i			0.513573i	0.966468	+	0.256786i	$0.0826637\pi$
$631$			324.064i			0.513573i	−0.966468	+	0.256786i	$0.917336\pi$
$632$	−326.016			−0.515849
$633$		0			0
$634$	−673.632			−1.06251
$635$	278.625			0.438780
$636$		0			0
$637$	−806.520			−1.26612
$638$	−372.567			−0.583961
$639$		0			0
$640$			34.9118i			0.0545497i
$641$	301.933			0.471034			0.235517	−	0.971870i	$-0.424322\pi$
$641$	301.933			0.471034			0.235517	+	0.971870i	$0.424322\pi$
$642$		0			0
$643$			746.370i			1.16076i	0.814345	+	0.580381i	$0.197096\pi$
$643$			746.370i			1.16076i	−0.814345	+	0.580381i	$0.802904\pi$
$644$	−516.218			−0.801581
$645$		0			0
$646$	−276.724			−0.428366
$647$			434.769i			0.671976i	0.941866	+	0.335988i	$0.109070\pi$
$647$			434.769i			0.671976i	−0.941866	+	0.335988i	$0.890930\pi$
$648$		0			0
$649$		−	31.8585i		−	0.0490885i
$650$			355.788i			0.547366i
$651$		0			0
$652$	88.7601			0.136135
$653$			1200.32i			1.83816i	0.394067	+	0.919082i	$0.371068\pi$
$653$			1200.32i			1.83816i	−0.394067	+	0.919082i	$0.628932\pi$
$654$		0			0
$655$			342.034i			0.522189i
$656$			52.8190i			0.0805168i
$657$		0			0
$658$		−	266.825i		−	0.405510i
$659$	124.323			0.188654			0.0943271	−	0.995541i	$-0.469930\pi$
$659$	124.323			0.188654			0.0943271	+	0.995541i	$0.469930\pi$
$660$		0			0
$661$			316.557i			0.478906i	0.970908	+	0.239453i	$0.0769681\pi$
$661$			316.557i			0.478906i	−0.970908	+	0.239453i	$0.923032\pi$
$662$		−	541.118i		−	0.817399i
$663$		0			0
$664$	29.8566			0.0449648
$665$			524.568i			0.788825i
$666$		0			0
$667$	1029.83			1.54398
$668$	−19.7592			−0.0295796
$669$		0			0
$670$	166.608			0.248668
$671$		−	479.204i		−	0.714165i
$672$		0			0
$673$		−	452.720i		−	0.672689i	−0.941739	−	0.336345i	$-0.890809\pi$
$673$		−	452.720i		−	0.672689i	0.941739	−	0.336345i	$-0.109191\pi$
$674$			4.34612i			0.00644824i
$675$		0			0
$676$	−190.397			−0.281653
$677$			1036.15i			1.53050i	0.643734	+	0.765249i	$0.277384\pi$
$677$			1036.15i			1.53050i	−0.643734	+	0.765249i	$0.722616\pi$
$678$		0			0
$679$	−1534.05			−2.25928
$680$		−	99.7679i		−	0.146717i
$681$		0			0
$682$	221.593			0.324916
$683$	683.266			1.00039			0.500194	−	0.865913i	$-0.333262\pi$
$683$	683.266			1.00039			0.500194	+	0.865913i	$0.333262\pi$
$684$		0			0
$685$	−563.269			−0.822290
$686$			8.69609i			0.0126765i
$687$		0			0
$688$		−	239.837i		−	0.348601i
$689$		−	298.684i		−	0.433503i
$690$		0			0
$691$	191.165			0.276650			0.138325	−	0.990387i	$-0.455828\pi$
$691$	191.165			0.276650			0.138325	+	0.990387i	$0.455828\pi$
$692$			413.689i			0.597817i
$693$		0			0
$694$	245.602			0.353893
$695$	−346.082			−0.497960
$696$		0			0
$697$		−	150.941i		−	0.216559i
$698$	156.544			0.224275
$699$		0			0
$700$	307.413			0.439161
$701$	−1011.63			−1.44313			−0.721563	−	0.692348i	$-0.756576\pi$
$701$	−1011.63			−1.44313			−0.721563	+	0.692348i	$0.756576\pi$
$702$		0			0
$703$	196.453			0.279449
$704$		−	53.1906i		−	0.0755548i
$705$		0			0
$706$	134.522			0.190541
$707$	143.807			0.203405
$708$		0			0
$709$	602.388			0.849630			0.424815	−	0.905280i	$-0.360339\pi$
$709$	602.388			0.849630			0.424815	+	0.905280i	$0.360339\pi$
$710$			340.666i			0.479811i
$711$		0			0
$712$		−	31.2174i		−	0.0438447i
$713$	−612.518			−0.859072
$714$		0			0
$715$			333.486i			0.466413i
$716$	−40.5382			−0.0566176
$717$		0			0
$718$			470.304i			0.655020i
$719$		−	196.665i		−	0.273525i	−0.990604	−	0.136763i	$-0.956330\pi$
$719$		−	196.665i		−	0.273525i	0.990604	−	0.136763i	$-0.0436698\pi$
$720$		0			0
$721$		−	1295.50i		−	1.79681i
$722$			96.1277i			0.133141i
$723$		0			0
$724$			468.248i			0.646752i
$725$	−613.275			−0.845897
$726$		0			0
$727$	−448.058			−0.616311			−0.308155	−	0.951336i	$-0.599712\pi$
$727$	−448.058			−0.616311			−0.308155	+	0.951336i	$0.599712\pi$
$728$			456.553i			0.627133i
$729$		0			0
$730$	−408.134			−0.559088
$731$			685.386i			0.937600i
$732$		0			0
$733$		−	732.969i		−	0.999957i	−0.866038	−	0.499979i	$-0.833341\pi$
$733$		−	732.969i		−	0.999957i	0.866038	−	0.499979i	$-0.166659\pi$
$734$			235.732i			0.321161i
$735$		0			0
$736$			147.027i			0.199765i
$737$	−253.838			−0.344421
$738$		0			0
$739$	570.332			0.771762			0.385881	−	0.922549i	$-0.373897\pi$
$739$	570.332			0.771762			0.385881	+	0.922549i	$0.373897\pi$
$740$			70.8274i			0.0957128i
$741$		0			0
$742$	−258.073			−0.347807
$743$	−964.878			−1.29862			−0.649312	−	0.760522i	$-0.724943\pi$
$743$	−964.878			−1.29862			−0.649312	+	0.760522i	$0.724943\pi$
$744$		0			0
$745$			150.343i			0.201802i
$746$	−835.245			−1.11963
$747$		0			0
$748$			152.003i			0.203213i
$749$	−1342.51			−1.79240
$750$		0			0
$751$			841.431i			1.12041i	0.828353	+	0.560207i	$0.189278\pi$
$751$			841.431i			1.12041i	−0.828353	+	0.560207i	$0.810722\pi$
$752$	−75.9961			−0.101059
$753$		0			0
$754$		−	910.804i		−	1.20796i
$755$			888.131i			1.17633i
$756$		0			0
$757$			905.223i			1.19580i	0.801570	+	0.597901i	$0.203999\pi$
$757$			905.223i			1.19580i	−0.801570	+	0.597901i	$0.796001\pi$
$758$	543.947			0.717608
$759$		0			0
$760$	149.405			0.196586
$761$		−	759.093i		−	0.997495i	−0.866747	−	0.498747i	$-0.833794\pi$
$761$		−	759.093i		−	0.997495i	0.866747	−	0.498747i	$-0.166206\pi$
$762$		0			0
$763$	1355.00			1.77588
$764$	−371.739			−0.486569
$765$		0			0
$766$	−517.765			−0.675933
$767$	77.8835			0.101543
$768$		0			0
$769$	−433.755			−0.564050			−0.282025	−	0.959407i	$-0.591006\pi$
$769$	−433.755			−0.564050			−0.282025	+	0.959407i	$0.591006\pi$
$770$	288.143			0.374211
$771$		0			0
$772$		−	344.449i		−	0.446177i
$773$		−	1393.98i		−	1.80334i	−0.432430	−	0.901668i	$-0.642344\pi$
$773$		−	1393.98i		−	1.80334i	0.432430	−	0.901668i	$-0.357656\pi$
$774$		0			0
$775$	364.760			0.470658
$776$			436.922i			0.563043i
$777$		0			0
$778$	−137.806			−0.177128
$779$	226.040			0.290166
$780$		0			0
$781$		−	519.028i		−	0.664568i
$782$		−	420.161i		−	0.537290i
$783$		0			0
$784$	198.477			0.253159
$785$	−839.753			−1.06975
$786$		0			0
$787$	1070.58			1.36033			0.680166	−	0.733059i	$-0.261908\pi$
$787$	1070.58			1.36033			0.680166	+	0.733059i	$0.261908\pi$
$788$	−561.692			−0.712807
$789$		0			0
$790$	−503.010			−0.636722
$791$	−979.258	−	548.016i	−1.23800	−	0.692814i
$792$		0			0
$793$	1171.50			1.47730
$794$	315.155			0.396920
$795$		0			0
$796$			363.903i			0.457164i
$797$	161.414			0.202527			0.101263	−	0.994860i	$-0.467712\pi$
$797$	161.414			0.202527			0.101263	+	0.994860i	$0.467712\pi$
$798$		0			0
$799$	217.175			0.271809
$800$		−	87.5560i		−	0.109445i
$801$		0			0
$802$	−131.724			−0.164244
$803$	621.820			0.774371
$804$		0			0
$805$	−796.472			−0.989407
$806$			541.722i			0.672112i
$807$		0			0
$808$		−	40.9586i		−	0.0506913i
$809$			1060.54i			1.31092i	0.755229	+	0.655461i	$0.227526\pi$
$809$			1060.54i			1.31092i	−0.755229	+	0.655461i	$0.772474\pi$
$810$		0			0
$811$			1011.36i			1.24705i	0.781804	+	0.623524i	$0.214300\pi$
$811$			1011.36i			1.24705i	−0.781804	+	0.623524i	$0.785700\pi$
$812$	−786.965			−0.969169
$813$		0			0
$814$		−	107.911i		−	0.132568i
$815$	136.948			0.168034
$816$		0			0
$817$	−1026.39			−1.25629
$818$	278.085			0.339958
$819$		0			0
$820$			81.4944i			0.0993834i
$821$		−	494.520i		−	0.602338i	−0.953571	−	0.301169i	$-0.902623\pi$
$821$		−	494.520i		−	0.602338i	0.953571	−	0.301169i	$-0.0973769\pi$
$822$		0			0
$823$	−1311.69			−1.59379			−0.796897	−	0.604115i	$-0.793527\pi$
$823$	−1311.69			−1.59379			−0.796897	+	0.604115i	$0.793527\pi$
$824$	−368.979			−0.447790
$825$		0			0
$826$		−	67.2940i		−	0.0814697i
$827$			1053.56i			1.27395i	0.770884	+	0.636975i	$0.219815\pi$
$827$			1053.56i			1.27395i	−0.770884	+	0.636975i	$0.780185\pi$
$828$		0			0
$829$			1378.27i			1.66257i	0.555845	+	0.831286i	$0.312395\pi$
$829$			1378.27i			1.66257i	−0.555845	+	0.831286i	$0.687605\pi$
$830$	46.0657			0.0555009
$831$		0			0
$832$	130.033			0.156290
$833$	−567.189			−0.680900
$834$		0			0
$835$	−30.4864			−0.0365106
$836$	−227.630			−0.272284
$837$		0			0
$838$			362.059i			0.432051i
$839$	−318.616			−0.379757			−0.189879	−	0.981808i	$-0.560809\pi$
$839$	−318.616			−0.379757			−0.189879	+	0.981808i	$0.560809\pi$
$840$		0			0
$841$	728.963			0.866782
$842$			802.954i			0.953627i
$843$		0			0
$844$	595.604			0.705692
$845$	−293.763			−0.347649
$846$		0			0
$847$	762.612			0.900368
$848$			73.5032i			0.0866783i
$849$		0			0
$850$			250.210i			0.294364i
$851$			298.282i			0.350507i
$852$		0			0
$853$	−662.642			−0.776837			−0.388419	−	0.921483i	$-0.626979\pi$
$853$	−662.642			−0.776837			−0.388419	+	0.921483i	$0.626979\pi$
$854$		−	1012.21i		−	1.18526i
$855$		0			0
$856$			382.367i			0.446691i
$857$	−346.429			−0.404235			−0.202117	−	0.979361i	$-0.564782\pi$
$857$	−346.429			−0.404235			−0.202117	+	0.979361i	$0.564782\pi$
$858$		0			0
$859$			228.923i			0.266499i	0.991082	+	0.133250i	$0.0425412\pi$
$859$			228.923i			0.266499i	−0.991082	+	0.133250i	$0.957459\pi$
$860$		−	370.045i		−	0.430285i
$861$		0			0
$862$			378.778i			0.439418i
$863$			1299.15i			1.50538i	0.658373	+	0.752692i	$0.271245\pi$
$863$			1299.15i			1.50538i	−0.658373	+	0.752692i	$0.728755\pi$
$864$		0			0
$865$			638.280i			0.737896i
$866$	474.966			0.548460
$867$		0			0
$868$	468.066			0.539246
$869$	766.371			0.881900
$870$		0			0
$871$		−	620.551i		−	0.712458i
$872$		−	385.926i		−	0.442575i
$873$		0			0
$874$	629.204			0.719913
$875$	1240.41			1.41761
$876$		0			0
$877$		−	1265.56i		−	1.44305i	−0.692387	−	0.721526i	$-0.743441\pi$
$877$		−	1265.56i		−	1.44305i	0.692387	−	0.721526i	$-0.256559\pi$
$878$			210.939i			0.240250i
$879$		0			0
$880$		−	82.0676i		−	0.0932587i
$881$	956.965			1.08623			0.543113	−	0.839660i	$-0.317246\pi$
$881$	956.965			1.08623			0.543113	+	0.839660i	$0.317246\pi$
$882$		0			0
$883$			1551.15i			1.75668i	0.478037	+	0.878340i	$0.341348\pi$
$883$			1551.15i			1.75668i	−0.478037	+	0.878340i	$0.658652\pi$
$884$	−371.598			−0.420360
$885$		0			0
$886$	272.791			0.307891
$887$	−1053.78			−1.18803			−0.594014	−	0.804455i	$-0.702458\pi$
$887$	−1053.78			−1.18803			−0.594014	+	0.804455i	$0.702458\pi$
$888$		0			0
$889$	−896.673			−1.00863
$890$		−	48.1653i		−	0.0541183i
$891$		0			0
$892$			1.07505i			0.00120521i
$893$			325.226i			0.364195i
$894$		0			0
$895$	−62.5463			−0.0698842
$896$		−	112.353i		−	0.125394i
$897$		0			0
$898$		−	1106.44i		−	1.23212i
$899$	−933.772			−1.03868
$900$		0			0
$901$		−	210.051i		−	0.233131i
$902$		−	124.162i		−	0.137652i
$903$		0			0
$904$	−156.084	+	278.908i	−0.172659	+	0.308527i
$905$			722.459i			0.798298i
$906$		0			0
$907$		−	328.354i		−	0.362022i	−0.983481	−	0.181011i	$-0.942063\pi$
$907$		−	328.354i		−	0.362022i	0.983481	−	0.181011i	$-0.0579370\pi$
$908$			659.163i			0.725950i
$909$		0			0
$910$			704.414i			0.774082i
$911$		−	232.472i		−	0.255184i	−0.991827	−	0.127592i	$-0.959275\pi$
$911$		−	232.472i		−	0.255184i	0.991827	−	0.127592i	$-0.0407248\pi$
$912$		0			0
$913$	−70.1843			−0.0768722
$914$	836.602			0.915320
$915$		0			0
$916$		−	355.768i		−	0.388393i
$917$		−	1100.73i		−	1.20036i
$918$		0			0
$919$	687.097			0.747658			0.373829	−	0.927498i	$-0.378045\pi$
$919$	687.097			0.747658			0.373829	+	0.927498i	$0.378045\pi$
$920$			226.848i			0.246574i
$921$		0			0
$922$	−55.7293			−0.0604440
$923$	1268.85			1.37471
$924$		0			0
$925$		−	177.630i		−	0.192032i
$926$			50.2984i			0.0543179i
$927$		0			0
$928$			224.140i			0.241530i
$929$			1075.68i			1.15789i	0.815366	+	0.578945i	$0.196536\pi$
$929$			1075.68i			1.15789i	−0.815366	+	0.578945i	$0.803464\pi$
$930$		0			0
$931$		−	849.384i		−	0.912335i
$932$			322.539i			0.346072i
$933$		0			0
$934$	476.025			0.509663
$935$			234.526i			0.250829i
$936$		0			0
$937$		−	291.373i		−	0.310964i	−0.987839	−	0.155482i	$-0.950307\pi$
$937$		−	291.373i		−	0.310964i	0.987839	−	0.155482i	$-0.0496930\pi$
$938$	−536.177			−0.571617
$939$		0			0
$940$	−117.254			−0.124739
$941$	−663.623			−0.705232			−0.352616	−	0.935768i	$-0.614708\pi$
$941$	−663.623			−0.705232			−0.352616	+	0.935768i	$0.614708\pi$
$942$		0			0
$943$			343.204i			0.363949i
$944$	−19.1664			−0.0203034
$945$		0			0
$946$			563.789i			0.595971i
$947$	1654.71			1.74732			0.873660	−	0.486537i	$-0.161740\pi$
$947$	1654.71			1.74732			0.873660	+	0.486537i	$0.161740\pi$
$948$		0			0
$949$			1520.15i			1.60184i
$950$	−374.697			−0.394418
$951$		0			0
$952$			321.073i			0.337262i
$953$		−	1728.14i		−	1.81337i	−0.421807	−	0.906686i	$-0.638604\pi$
$953$		−	1728.14i		−	1.81337i	0.421807	−	0.906686i	$-0.361396\pi$
$954$		0			0
$955$	−573.556			−0.600582
$956$		−	245.121i		−	0.256402i
$957$		0			0
$958$		−	1157.37i		−	1.20811i
$959$	1812.71			1.89021
$960$		0			0
$961$	−405.617			−0.422078
$962$	263.806			0.274226
$963$		0			0
$964$	431.959			0.448090
$965$		−	531.450i		−	0.550725i
$966$		0			0
$967$	−420.476			−0.434825			−0.217412	−	0.976080i	$-0.569762\pi$
$967$	−420.476			−0.434825			−0.217412	+	0.976080i	$0.569762\pi$
$968$		−	217.204i		−	0.224384i
$969$		0			0
$970$			674.126i			0.694975i
$971$	−1661.65			−1.71128			−0.855641	−	0.517570i	$-0.826837\pi$
$971$	−1661.65			−1.71128			−0.855641	+	0.517570i	$0.826837\pi$
$972$		0			0
$973$	1113.76			1.14467
$974$	−357.060			−0.366592
$975$		0			0
$976$	−288.294			−0.295384
$977$	−62.6869			−0.0641626			−0.0320813	−	0.999485i	$-0.510214\pi$
$977$	−62.6869			−0.0641626			−0.0320813	+	0.999485i	$0.510214\pi$
$978$		0			0
$979$			73.3831i			0.0749572i
$980$	306.230			0.312479
$981$		0			0
$982$		−	566.597i		−	0.576982i
$983$	−99.5296			−0.101251			−0.0506255	−	0.998718i	$-0.516121\pi$
$983$	−99.5296			−0.101251			−0.0506255	+	0.998718i	$0.516121\pi$
$984$		0			0
$985$	−866.633			−0.879831
$986$		−	640.528i		−	0.649623i
$987$		0			0
$988$		−	556.480i		−	0.563239i
$989$		−	1558.40i		−	1.57573i
$990$		0			0
$991$	196.425			0.198209			0.0991043	−	0.995077i	$-0.468402\pi$
$991$	196.425			0.198209			0.0991043	+	0.995077i	$0.468402\pi$
$992$		−	133.313i		−	0.134388i
$993$		0			0
$994$		−	1096.33i		−	1.10295i
$995$			561.465i			0.564286i
$996$		0			0
$997$		−	1180.54i		−	1.18410i	−0.805903	−	0.592048i	$-0.798320\pi$
$997$		−	1180.54i		−	1.18410i	0.805903	−	0.592048i	$-0.201680\pi$
$998$	−791.117			−0.792702
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2034.3.d.a.2033.32	yes	36
3.2	odd	2	inner	2034.3.d.a.2033.13	✓	36
113.112	even	2	inner	2034.3.d.a.2033.14	yes	36
339.338	odd	2	inner	2034.3.d.a.2033.31	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2034.3.d.a.2033.13	✓	36	3.2	odd	2	inner
2034.3.d.a.2033.14	yes	36	113.112	even	2	inner
2034.3.d.a.2033.31	yes	36	339.338	odd	2	inner
2034.3.d.a.2033.32	yes	36	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2034 = 2 \cdot 3^{2} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2034.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -3.08580 q^{5} +9.93072 q^{7} -2.82843i q^{8} -4.36398i q^{10} +6.64882i q^{11} -16.2542 q^{13} +14.0442i q^{14} +4.00000 q^{16} -11.4308 q^{17} -17.1180i q^{19} +6.17159 q^{20} -9.40285 q^{22} +25.9910 q^{23} -15.4779 q^{25} -22.9869i q^{26} -19.8614 q^{28} +39.6228 q^{29} -23.5666 q^{31} +5.65685i q^{32} -16.1657i q^{34} -30.6442 q^{35} +11.4764i q^{37} +24.2086 q^{38} +8.72795i q^{40} +13.2048i q^{41} -59.9593i q^{43} -13.2976i q^{44} +36.7568i q^{46} -18.9990 q^{47} +49.6192 q^{49} -21.8890i q^{50} +32.5084 q^{52} +18.3758i q^{53} -20.5169i q^{55} -28.0883i q^{56} +56.0350i q^{58} -4.79160 q^{59} -72.0736 q^{61} -33.3282i q^{62} -8.00000 q^{64} +50.1571 q^{65} +38.1779i q^{67} +22.8617 q^{68} -43.3374i q^{70} -78.0632 q^{71} -93.5234i q^{73} -16.2300 q^{74} +34.2361i q^{76} +66.0276i q^{77} -115.264i q^{79} -12.3432 q^{80} -18.6743 q^{82} +10.5559i q^{83} +35.2733 q^{85} +84.7953 q^{86} +18.8057 q^{88} +11.0370 q^{89} -161.416 q^{91} -51.9820 q^{92} -26.8687i q^{94} +52.8228i q^{95} -154.475 q^{97} +70.1721i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} - 8 q^{7} + 144 q^{16} + 36 q^{25} + 16 q^{28} + 64 q^{31} + 108 q^{49} - 80 q^{61} - 288 q^{64} + 248 q^{82} - 164 q^{85} + 516 q^{91} - 420 q^{97}+O(q^{100}) \) 36 * q - 72 * q^4 - 8 * q^7 + 144 * q^16 + 36 * q^25 + 16 * q^28 + 64 * q^31 + 108 * q^49 - 80 * q^61 - 288 * q^64 + 248 * q^82 - 164 * q^85 + 516 * q^91 - 420 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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