LMFDB - Embedded newform 297.3.c.b.109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [297,3,Mod(109,297)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(297, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("297.109");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	297.c (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:	$8.09266385150$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 24x^{14} + 420x^{12} + 1908x^{10} + 20196x^{8} - 91800x^{6} + 597348x^{4} - 4428432x^{2} + 8714304 $ x^16 + 24x^14 + 420x^12 + 1908x^10 + 20196x^8 - 91800x^6 + 597348x^4 - 4428432*x^2 + 8714304
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.2
Root			$1.73205 - 3.81469i$ of defining polynomial
Character	$\chi$	$=$	297.109
Dual form			297.3.c.b.109.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.81469i q^{2} -10.5518 q^{4} +2.90548 q^{5} +12.6815i q^{7} +24.9932i q^{8} +O(q^{10})$	\(q-3.81469i q^{2} -10.5518 q^{4} +2.90548 q^{5} +12.6815i q^{7} +24.9932i q^{8} -11.0835i q^{10} +(8.23036 + 7.29802i) q^{11} -11.3193i q^{13} +48.3758 q^{14} +53.1339 q^{16} +9.32598i q^{17} +21.6555i q^{19} -30.6581 q^{20} +(27.8397 - 31.3962i) q^{22} -8.74379 q^{23} -16.5582 q^{25} -43.1797 q^{26} -133.813i q^{28} +26.0401i q^{29} +39.0366 q^{31} -102.716i q^{32} +35.5757 q^{34} +36.8457i q^{35} -33.2010 q^{37} +82.6090 q^{38} +72.6172i q^{40} -13.9464i q^{41} +9.48450i q^{43} +(-86.8454 - 77.0075i) q^{44} +33.3548i q^{46} +37.0771 q^{47} -111.820 q^{49} +63.1643i q^{50} +119.440i q^{52} +81.4480 q^{53} +(23.9131 + 21.2043i) q^{55} -316.950 q^{56} +99.3349 q^{58} +19.9192 q^{59} -55.6650i q^{61} -148.912i q^{62} -179.295 q^{64} -32.8880i q^{65} -85.6122 q^{67} -98.4062i q^{68} +140.555 q^{70} +85.9169 q^{71} +24.3691i q^{73} +126.651i q^{74} -228.505i q^{76} +(-92.5496 + 104.373i) q^{77} +28.1829i q^{79} +154.379 q^{80} -53.2010 q^{82} +46.2887i q^{83} +27.0964i q^{85} +36.1804 q^{86} +(-182.401 + 205.703i) q^{88} -23.0340 q^{89} +143.546 q^{91} +92.2631 q^{92} -141.437i q^{94} +62.9196i q^{95} +94.3033 q^{97} +426.557i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4}+O(q^{10}) $ 16 * q - 32 * q^4	$ 16 q - 32 q^{4} + 160 q^{16} + 102 q^{22} + 12 q^{25} + 68 q^{31} + 156 q^{34} - 124 q^{37} - 272 q^{49} + 182 q^{55} + 492 q^{58} - 680 q^{64} - 400 q^{67} + 324 q^{70} - 444 q^{82} - 510 q^{88} + 120 q^{91} + 152 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 160 * q^16 + 102 * q^22 + 12 * q^25 + 68 * q^31 + 156 * q^34 - 124 * q^37 - 272 * q^49 + 182 * q^55 + 492 * q^58 - 680 * q^64 - 400 * q^67 + 324 * q^70 - 444 * q^82 - 510 * q^88 + 120 * q^91 + 152 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	3.81469i		−	1.90734i	−0.300849	−	0.953672i	$-0.597270\pi$
$2$		−	3.81469i		−	1.90734i	0.300849	−	0.953672i	$-0.402730\pi$
$3$		0			0
$3$		0			0
$4$	−10.5518			−2.63796
$5$	2.90548			0.581096			0.290548	−	0.956860i	$-0.406162\pi$
$5$	2.90548			0.581096			0.290548	+	0.956860i	$0.406162\pi$
$6$		0			0
$7$			12.6815i			1.81164i	0.423665	+	0.905819i	$0.360743\pi$
$7$			12.6815i			1.81164i	−0.423665	+	0.905819i	$0.639257\pi$
$8$			24.9932i			3.12415i
$9$		0			0
$10$		−	11.0835i		−	1.10835i
$11$	8.23036	+	7.29802i	0.748215	+	0.663457i
$11$	8.23036	+	7.29802i	0.748215	+	0.663457i
$12$		0			0
$13$		−	11.3193i		−	0.870717i	−0.900257	−	0.435359i	$-0.856622\pi$
$13$		−	11.3193i		−	0.870717i	0.900257	−	0.435359i	$-0.143378\pi$
$14$	48.3758			3.45542
$15$		0			0
$16$	53.1339			3.32087
$17$			9.32598i			0.548587i	0.961646	+	0.274293i	$0.0884440\pi$
$17$			9.32598i			0.548587i	−0.961646	+	0.274293i	$0.911556\pi$
$18$		0			0
$19$			21.6555i			1.13976i	0.821727	+	0.569882i	$0.193011\pi$
$19$			21.6555i			1.13976i	−0.821727	+	0.569882i	$0.806989\pi$
$20$	−30.6581			−1.53291
$21$		0			0
$22$	27.8397	−	31.3962i	1.26544	−	1.42710i
$23$	−8.74379			−0.380165			−0.190082	−	0.981768i	$-0.560876\pi$
$23$	−8.74379			−0.380165			−0.190082	+	0.981768i	$0.560876\pi$
$24$		0			0
$25$	−16.5582			−0.662328
$26$	−43.1797			−1.66076
$27$		0			0
$28$		−	133.813i		−	4.77903i
$29$			26.0401i			0.897935i	0.893548	+	0.448967i	$0.148208\pi$
$29$			26.0401i			0.897935i	−0.893548	+	0.448967i	$0.851792\pi$
$30$		0			0
$31$	39.0366			1.25924			0.629622	−	0.776902i	$-0.283210\pi$
$31$	39.0366			1.25924			0.629622	+	0.776902i	$0.283210\pi$
$32$		−	102.716i		−	3.20988i
$33$		0			0
$34$	35.5757			1.04634
$35$			36.8457i			1.05274i
$36$		0			0
$37$	−33.2010			−0.897324			−0.448662	−	0.893702i	$-0.648099\pi$
$37$	−33.2010			−0.897324			−0.448662	+	0.893702i	$0.648099\pi$
$38$	82.6090			2.17392
$39$		0			0
$40$			72.6172i			1.81543i
$41$		−	13.9464i		−	0.340155i	−0.985431	−	0.170078i	$-0.945598\pi$
$41$		−	13.9464i		−	0.340155i	0.985431	−	0.170078i	$-0.0544018\pi$
$42$		0			0
$43$			9.48450i			0.220570i	0.993900	+	0.110285i	$0.0351763\pi$
$43$			9.48450i			0.220570i	−0.993900	+	0.110285i	$0.964824\pi$
$44$	−86.8454	−	77.0075i	−1.97376	−	1.75017i
$45$		0			0
$46$			33.3548i			0.725105i
$47$	37.0771			0.788873			0.394437	−	0.918923i	$-0.370940\pi$
$47$	37.0771			0.788873			0.394437	+	0.918923i	$0.370940\pi$
$48$		0			0
$49$	−111.820			−2.28203
$50$			63.1643i			1.26329i
$51$		0			0
$52$			119.440i			2.29692i
$53$	81.4480			1.53675			0.768377	−	0.639997i	$-0.221064\pi$
$53$	81.4480			1.53675			0.768377	+	0.639997i	$0.221064\pi$
$54$		0			0
$55$	23.9131	+	21.2043i	0.434784	+	0.385532i
$56$	−316.950			−5.65983
$57$		0			0
$58$	99.3349			1.71267
$59$	19.9192			0.337613			0.168807	−	0.985649i	$-0.446009\pi$
$59$	19.9192			0.337613			0.168807	+	0.985649i	$0.446009\pi$
$60$		0			0
$61$		−	55.6650i		−	0.912541i	−0.889841	−	0.456270i	$-0.849185\pi$
$61$		−	55.6650i		−	0.912541i	0.889841	−	0.456270i	$-0.150815\pi$
$62$		−	148.912i		−	2.40181i
$63$		0			0
$64$	−179.295			−2.80148
$65$		−	32.8880i		−	0.505970i
$66$		0			0
$67$	−85.6122			−1.27779			−0.638897	−	0.769292i	$-0.720609\pi$
$67$	−85.6122			−1.27779			−0.638897	+	0.769292i	$0.720609\pi$
$68$		−	98.4062i		−	1.44715i
$69$		0			0
$70$	140.555			2.00793
$71$	85.9169			1.21010			0.605049	−	0.796188i	$-0.293154\pi$
$71$	85.9169			1.21010			0.605049	+	0.796188i	$0.293154\pi$
$72$		0			0
$73$			24.3691i			0.333823i	0.985972	+	0.166912i	$0.0533795\pi$
$73$			24.3691i			0.333823i	−0.985972	+	0.166912i	$0.946621\pi$
$74$			126.651i			1.71150i
$75$		0			0
$76$		−	228.505i		−	3.00665i
$77$	−92.5496	+	104.373i	−1.20194	+	1.35549i
$78$		0			0
$79$			28.1829i			0.356746i	0.983963	+	0.178373i	$0.0570833\pi$
$79$			28.1829i			0.356746i	−0.983963	+	0.178373i	$0.942917\pi$
$80$	154.379			1.92974
$81$		0			0
$82$	−53.2010			−0.648792
$83$			46.2887i			0.557695i	0.960335	+	0.278848i	$0.0899524\pi$
$83$			46.2887i			0.557695i	−0.960335	+	0.278848i	$0.910048\pi$
$84$		0			0
$85$			27.0964i			0.318781i
$86$	36.1804			0.420702
$87$		0			0
$88$	−182.401	+	205.703i	−2.07274	+	2.33753i
$89$	−23.0340			−0.258809			−0.129404	−	0.991592i	$-0.541307\pi$
$89$	−23.0340			−0.258809			−0.129404	+	0.991592i	$0.541307\pi$
$90$		0			0
$91$	143.546			1.57742
$92$	92.2631			1.00286
$93$		0			0
$94$		−	141.437i		−	1.50465i
$95$			62.9196i			0.662312i
$96$		0			0
$97$	94.3033			0.972199			0.486099	−	0.873904i	$-0.338419\pi$
$97$	94.3033			0.972199			0.486099	+	0.873904i	$0.338419\pi$
$98$			426.557i			4.35262i
$99$		0			0
$100$	174.719			1.74719
$101$			9.43170i			0.0933832i	0.998909	+	0.0466916i	$0.0148678\pi$
$101$			9.43170i			0.0933832i	−0.998909	+	0.0466916i	$0.985132\pi$
$102$		0			0
$103$	−69.6730			−0.676437			−0.338218	−	0.941068i	$-0.609824\pi$
$103$	−69.6730			−0.676437			−0.338218	+	0.941068i	$0.609824\pi$
$104$	282.906			2.72025
$105$		0			0
$106$		−	310.699i		−	2.93112i
$107$		−	40.2389i		−	0.376065i	−0.982163	−	0.188032i	$-0.939789\pi$
$107$		−	40.2389i		−	0.376065i	0.982163	−	0.188032i	$-0.0602110\pi$
$108$		0			0
$109$			99.7387i			0.915034i	0.889201	+	0.457517i	$0.151261\pi$
$109$			99.7387i			0.915034i	−0.889201	+	0.457517i	$0.848739\pi$
$110$	80.8876	−	91.2211i	0.735342	−	0.829283i
$111$		0			0
$112$			673.815i			6.01621i
$113$	191.570			1.69531			0.847655	−	0.530548i	$-0.178014\pi$
$113$	191.570			1.69531			0.847655	+	0.530548i	$0.178014\pi$
$114$		0			0
$115$	−25.4049			−0.220912
$116$		−	274.771i		−	2.36871i
$117$		0			0
$118$		−	75.9854i		−	0.643944i
$119$	−118.267			−0.993841
$120$		0			0
$121$	14.4777	+	120.131i	0.119650	+	0.992816i
$122$	−212.344			−1.74053
$123$		0			0
$124$	−411.907			−3.32183
$125$	−120.746			−0.965972
$126$		0			0
$127$			128.963i			1.01546i	0.861517	+	0.507729i	$0.169515\pi$
$127$			128.963i			1.01546i	−0.861517	+	0.507729i	$0.830485\pi$
$128$			273.089i			2.13351i
$129$		0			0
$130$	−125.458			−0.965058
$131$		−	241.412i		−	1.84284i	−0.388571	−	0.921419i	$-0.627031\pi$
$131$		−	241.412i		−	1.84284i	0.388571	−	0.921419i	$-0.372969\pi$
$132$		0			0
$133$	−274.623			−2.06484
$134$			326.584i			2.43719i
$135$		0			0
$136$	−233.086			−1.71387
$137$	−152.097			−1.11020			−0.555098	−	0.831785i	$-0.687319\pi$
$137$	−152.097			−1.11020			−0.555098	+	0.831785i	$0.687319\pi$
$138$		0			0
$139$		−	191.365i		−	1.37673i	−0.725365	−	0.688365i	$-0.758329\pi$
$139$		−	191.365i		−	1.37673i	0.725365	−	0.688365i	$-0.241671\pi$
$140$		−	388.790i		−	2.77707i
$141$		0			0
$142$		−	327.746i		−	2.30807i
$143$	82.6087	−	93.1621i	0.577683	−	0.651483i
$144$		0			0
$145$			75.6590i			0.521786i
$146$	92.9605			0.636716
$147$		0			0
$148$	350.331			2.36710
$149$			169.402i			1.13692i	0.822710	+	0.568462i	$0.192461\pi$
$149$			169.402i			1.13692i	−0.822710	+	0.568462i	$0.807539\pi$
$150$		0			0
$151$			14.2932i			0.0946567i	0.998879	+	0.0473284i	$0.0150707\pi$
$151$			14.2932i			0.0946567i	−0.998879	+	0.0473284i	$0.984929\pi$
$152$	−541.240			−3.56079
$153$		0			0
$154$	398.150	+	353.048i	2.58539	+	2.29252i
$155$	113.420			0.731741
$156$		0			0
$157$	−65.8434			−0.419385			−0.209692	−	0.977767i	$-0.567246\pi$
$157$	−65.8434			−0.419385			−0.209692	+	0.977767i	$0.567246\pi$
$158$	107.509			0.680437
$159$		0			0
$160$		−	298.440i		−	1.86525i
$161$		−	110.884i		−	0.688721i
$162$		0			0
$163$	−67.7165			−0.415439			−0.207719	−	0.978188i	$-0.566604\pi$
$163$	−67.7165			−0.415439			−0.207719	+	0.978188i	$0.566604\pi$
$164$			147.160i			0.897315i
$165$		0			0
$166$	176.577			1.06372
$167$		−	144.470i		−	0.865089i	−0.901613	−	0.432544i	$-0.857616\pi$
$167$		−	144.470i		−	0.865089i	0.901613	−	0.432544i	$-0.142384\pi$
$168$		0			0
$169$	40.8729			0.241852
$170$	103.364			0.608026
$171$		0			0
$172$		−	100.079i		−	0.581854i
$173$		−	130.436i		−	0.753968i	−0.926220	−	0.376984i	$-0.876961\pi$
$173$		−	130.436i		−	0.753968i	0.926220	−	0.376984i	$-0.123039\pi$
$174$		0			0
$175$		−	209.982i		−	1.19990i
$176$	437.311	+	387.772i	2.48472	+	2.20325i
$177$		0			0
$178$			87.8674i			0.493637i
$179$	−265.776			−1.48478			−0.742391	−	0.669967i	$-0.766308\pi$
$179$	−265.776			−1.48478			−0.742391	+	0.669967i	$0.766308\pi$
$180$		0			0
$181$	42.9951			0.237542			0.118771	−	0.992922i	$-0.462105\pi$
$181$	42.9951			0.237542			0.118771	+	0.992922i	$0.462105\pi$
$182$		−	547.581i		−	3.00869i
$183$		0			0
$184$		−	218.535i		−	1.18769i
$185$	−96.4647			−0.521431
$186$		0			0
$187$	−68.0612	+	76.7562i	−0.363964	+	0.410461i
$188$	−391.231			−2.08102
$189$		0			0
$190$	240.019			1.26326
$191$	−28.4905			−0.149165			−0.0745824	−	0.997215i	$-0.523762\pi$
$191$	−28.4905			−0.149165			−0.0745824	+	0.997215i	$0.523762\pi$
$192$		0			0
$193$		−	56.6512i		−	0.293529i	−0.989171	−	0.146765i	$-0.953114\pi$
$193$		−	56.6512i		−	0.293529i	0.989171	−	0.146765i	$-0.0468860\pi$
$194$		−	359.738i		−	1.85432i
$195$		0			0
$196$	1179.90			6.01991
$197$		−	115.758i		−	0.587605i	−0.955866	−	0.293802i	$-0.905079\pi$
$197$		−	115.758i		−	0.587605i	0.955866	−	0.293802i	$-0.0949208\pi$
$198$		0			0
$199$	169.049			0.849494			0.424747	−	0.905312i	$-0.360363\pi$
$199$	169.049			0.849494			0.424747	+	0.905312i	$0.360363\pi$
$200$		−	413.842i		−	2.06921i
$201$		0			0
$202$	35.9790			0.178114
$203$	−330.227			−1.62673
$204$		0			0
$205$		−	40.5208i		−	0.197663i
$206$			265.781i			1.29020i
$207$		0			0
$208$		−	601.439i		−	2.89154i
$209$	−158.042	+	178.233i	−0.756184	+	0.852788i
$210$		0			0
$211$		−	254.304i		−	1.20523i	−0.798030	−	0.602617i	$-0.794125\pi$
$211$		−	254.304i		−	1.20523i	0.798030	−	0.602617i	$-0.205875\pi$
$212$	−859.426			−4.05390
$213$		0			0
$214$	−153.499			−0.717285
$215$			27.5570i			0.128172i
$216$		0			0
$217$			495.041i			2.28129i
$218$	380.472			1.74528
$219$		0			0
$220$	−252.327	−	223.744i	−1.14694	−	1.01702i
$221$	105.564			0.477664
$222$		0			0
$223$	−159.396			−0.714782			−0.357391	−	0.933955i	$-0.616334\pi$
$223$	−159.396			−0.714782			−0.357391	+	0.933955i	$0.616334\pi$
$224$	1302.59			5.81515
$225$		0			0
$226$		−	730.779i		−	3.23354i
$227$			329.438i			1.45127i	0.688080	+	0.725635i	$0.258454\pi$
$227$			329.438i			1.45127i	−0.688080	+	0.725635i	$0.741546\pi$
$228$		0			0
$229$	−388.164			−1.69504			−0.847519	−	0.530766i	$-0.821904\pi$
$229$	−388.164			−1.69504			−0.847519	+	0.530766i	$0.821904\pi$
$230$			96.9118i			0.421355i
$231$		0			0
$232$	−650.826			−2.80528
$233$		−	394.538i		−	1.69329i	−0.532155	−	0.846647i	$-0.678617\pi$
$233$		−	394.538i		−	1.69329i	0.532155	−	0.846647i	$-0.321383\pi$
$234$		0			0
$235$	107.727			0.458411
$236$	−210.184			−0.890610
$237$		0			0
$238$			451.152i			1.89560i
$239$		−	42.0226i		−	0.175827i	−0.996128	−	0.0879134i	$-0.971980\pi$
$239$		−	42.0226i		−	0.175827i	0.996128	−	0.0879134i	$-0.0280199\pi$
$240$		0			0
$241$			452.581i			1.87793i	0.344014	+	0.938964i	$0.388213\pi$
$241$			452.581i			1.87793i	−0.344014	+	0.938964i	$0.611787\pi$
$242$	458.261	−	55.2279i	1.89364	−	0.228214i
$243$		0			0
$244$			587.368i			2.40724i
$245$	−324.889			−1.32608
$246$		0			0
$247$	245.126			0.992411
$248$			975.649i			3.93407i
$249$		0			0
$250$			460.610i			1.84244i
$251$	209.156			0.833290			0.416645	−	0.909069i	$-0.363206\pi$
$251$	209.156			0.833290			0.416645	+	0.909069i	$0.363206\pi$
$252$		0			0
$253$	−71.9646	−	63.8124i	−0.284445	−	0.252223i
$254$	491.954			1.93683
$255$		0			0
$256$	324.569			1.26785
$257$	356.088			1.38556			0.692778	−	0.721151i	$-0.256386\pi$
$257$	356.088			1.38556			0.692778	+	0.721151i	$0.256386\pi$
$258$		0			0
$259$		−	421.037i		−	1.62563i
$260$			347.029i			1.33473i
$261$		0			0
$262$	−920.910			−3.51492
$263$		−	82.2857i		−	0.312873i	−0.987688	−	0.156437i	$-0.949999\pi$
$263$		−	82.2857i		−	0.312873i	0.987688	−	0.156437i	$-0.0500007\pi$
$264$		0			0
$265$	236.645			0.893002
$266$			1047.60i			3.93836i
$267$		0			0
$268$	903.366			3.37077
$269$	269.849			1.00316			0.501579	−	0.865112i	$-0.332753\pi$
$269$	269.849			1.00316			0.501579	+	0.865112i	$0.332753\pi$
$270$		0			0
$271$		−	150.338i		−	0.554753i	−0.960761	−	0.277377i	$-0.910535\pi$
$271$		−	150.338i		−	0.554753i	0.960761	−	0.277377i	$-0.0894650\pi$
$272$			495.525i			1.82178i
$273$		0			0
$274$			580.202i			2.11753i
$275$	−136.280	−	120.842i	−0.495563	−	0.439426i
$276$		0			0
$277$		−	331.268i		−	1.19591i	−0.801528	−	0.597957i	$-0.795979\pi$
$277$		−	331.268i		−	1.19591i	0.801528	−	0.597957i	$-0.204021\pi$
$278$	−729.999			−2.62590
$279$		0			0
$280$	−920.893			−3.28890
$281$		−	105.910i		−	0.376905i	−0.982082	−	0.188453i	$-0.939653\pi$
$281$		−	105.910i		−	0.376905i	0.982082	−	0.188453i	$-0.0603472\pi$
$282$		0			0
$283$			185.608i			0.655857i	0.944702	+	0.327929i	$0.106351\pi$
$283$			185.608i			0.655857i	−0.944702	+	0.327929i	$0.893649\pi$
$284$	−906.581			−3.19219
$285$		0			0
$286$	−355.384	−	315.126i	−1.24260	−	1.10184i
$287$	176.860			0.616238
$288$		0			0
$289$	202.026			0.699052
$290$	288.615			0.995225
$291$		0			0
$292$		−	257.139i		−	0.880612i
$293$		−	331.855i		−	1.13261i	−0.824195	−	0.566305i	$-0.808372\pi$
$293$		−	331.855i		−	1.13261i	0.824195	−	0.566305i	$-0.191628\pi$
$294$		0			0
$295$	57.8748			0.196186
$296$		−	829.799i		−	2.80337i
$297$		0			0
$298$	646.214			2.16850
$299$			98.9738i			0.331016i
$300$		0			0
$301$	−120.277			−0.399593
$302$	54.5240			0.180543
$303$		0			0
$304$			1150.64i			3.78500i
$305$		−	161.733i		−	0.530274i
$306$		0			0
$307$		−	140.978i		−	0.459212i	−0.973284	−	0.229606i	$-0.926256\pi$
$307$		−	140.978i		−	0.459212i	0.973284	−	0.229606i	$-0.0737438\pi$
$308$	976.569	−	1101.33i	3.17068	−	3.57574i
$309$		0			0
$310$		−	432.661i		−	1.39568i
$311$	344.913			1.10904			0.554522	−	0.832169i	$-0.312901\pi$
$311$	344.913			1.10904			0.554522	+	0.832169i	$0.312901\pi$
$312$		0			0
$313$	366.764			1.17177			0.585885	−	0.810395i	$-0.300747\pi$
$313$	366.764			1.17177			0.585885	+	0.810395i	$0.300747\pi$
$314$			251.172i			0.799911i
$315$		0			0
$316$		−	297.382i		−	0.941081i
$317$	47.3267			0.149295			0.0746477	−	0.997210i	$-0.476217\pi$
$317$	47.3267			0.149295			0.0746477	+	0.997210i	$0.476217\pi$
$318$		0			0
$319$	−190.041	+	214.319i	−0.595741	+	0.671848i
$320$	−520.938			−1.62793
$321$		0			0
$322$	−422.988			−1.31363
$323$	−201.959			−0.625259
$324$		0			0
$325$			187.428i			0.576700i
$326$			258.317i			0.792384i
$327$		0			0
$328$	348.564			1.06270
$329$			470.191i			1.42915i
$330$		0			0
$331$	600.137			1.81310			0.906552	−	0.422094i	$-0.138705\pi$
$331$	600.137			1.81310			0.906552	+	0.422094i	$0.138705\pi$
$332$		−	488.431i		−	1.47118i
$333$		0			0
$334$	−551.107			−1.65002
$335$	−248.745			−0.742521
$336$		0			0
$337$			31.0173i			0.0920395i	0.998941	+	0.0460197i	$0.0146537\pi$
$337$			31.0173i			0.0920395i	−0.998941	+	0.0460197i	$0.985346\pi$
$338$		−	155.917i		−	0.461294i
$339$		0			0
$340$		−	285.917i		−	0.840932i
$341$	321.285	+	284.890i	0.942185	+	0.835454i
$342$		0			0
$343$		−	796.644i		−	2.32258i
$344$	−237.048			−0.689093
$345$		0			0
$346$	−497.574			−1.43808
$347$		−	194.176i		−	0.559586i	−0.960060	−	0.279793i	$-0.909734\pi$
$347$		−	194.176i		−	0.559586i	0.960060	−	0.279793i	$-0.0902659\pi$
$348$		0			0
$349$			37.8073i			0.108330i	0.998532	+	0.0541652i	$0.0172498\pi$
$349$			37.8073i			0.108330i	−0.998532	+	0.0541652i	$0.982750\pi$
$350$	−801.016			−2.28862
$351$		0			0
$352$	749.626	−	845.392i	2.12962	−	2.40168i
$353$	−123.478			−0.349795			−0.174897	−	0.984587i	$-0.555959\pi$
$353$	−123.478			−0.349795			−0.174897	+	0.984587i	$0.555959\pi$
$354$		0			0
$355$	249.630			0.703183
$356$	243.051			0.682727
$357$		0			0
$358$			1013.85i			2.83199i
$359$			524.170i			1.46008i	0.683403	+	0.730042i	$0.260499\pi$
$359$			524.170i			1.46008i	−0.683403	+	0.730042i	$0.739501\pi$
$360$		0			0
$361$	−107.961			−0.299060
$362$		−	164.013i		−	0.453074i
$363$		0			0
$364$	−1514.67			−4.16118
$365$			70.8039i			0.193983i
$366$		0			0
$367$	143.669			0.391470			0.195735	−	0.980657i	$-0.437291\pi$
$367$	143.669			0.391470			0.195735	+	0.980657i	$0.437291\pi$
$368$	−464.592			−1.26248
$369$		0			0
$370$			367.983i			0.994548i
$371$			1032.88i			2.78404i
$372$		0			0
$373$		−	216.950i		−	0.581635i	−0.956779	−	0.290817i	$-0.906073\pi$
$373$		−	216.950i		−	0.581635i	0.956779	−	0.290817i	$-0.0939272\pi$
$374$	292.801	+	259.632i	0.782890	+	0.694204i
$375$		0			0
$376$			926.674i			2.46456i
$377$	294.756			0.781847
$378$		0			0
$379$	544.795			1.43746			0.718728	−	0.695292i	$-0.244725\pi$
$379$	544.795			1.43746			0.718728	+	0.695292i	$0.244725\pi$
$380$		−	663.917i		−	1.74715i
$381$		0			0
$382$			108.682i			0.284508i
$383$	−129.651			−0.338515			−0.169257	−	0.985572i	$-0.554137\pi$
$383$	−129.651			−0.338515			−0.169257	+	0.985572i	$0.554137\pi$
$384$		0			0
$385$	−268.901	+	303.254i	−0.698444	+	0.787672i
$386$	−216.106			−0.559861
$387$		0			0
$388$	−995.073			−2.56462
$389$	83.8532			0.215561			0.107780	−	0.994175i	$-0.465626\pi$
$389$	83.8532			0.215561			0.107780	+	0.994175i	$0.465626\pi$
$390$		0			0
$391$		−	81.5444i		−	0.208554i
$392$		−	2794.73i		−	7.12941i
$393$		0			0
$394$	−441.581			−1.12076
$395$			81.8849i			0.207303i
$396$		0			0
$397$	−506.041			−1.27466			−0.637331	−	0.770590i	$-0.719962\pi$
$397$	−506.041			−1.27466			−0.637331	+	0.770590i	$0.719962\pi$
$398$		−	644.870i		−	1.62028i
$399$		0			0
$400$	−879.801			−2.19950
$401$	−566.505			−1.41273			−0.706365	−	0.707848i	$-0.749666\pi$
$401$	−566.505			−1.41273			−0.706365	+	0.707848i	$0.749666\pi$
$402$		0			0
$403$		−	441.867i		−	1.09645i
$404$		−	99.5218i		−	0.246341i
$405$		0			0
$406$			1259.71i			3.10274i
$407$	−273.256	−	242.302i	−0.671391	−	0.595335i
$408$		0			0
$409$		−	159.785i		−	0.390673i	−0.980736	−	0.195336i	$-0.937420\pi$
$409$		−	159.785i		−	0.390673i	0.980736	−	0.195336i	$-0.0625799\pi$
$410$	−154.574			−0.377011
$411$		0			0
$412$	735.178			1.78441
$413$			252.604i			0.611633i
$414$		0			0
$415$			134.491i			0.324074i
$416$	−1162.68			−2.79490
$417$		0			0
$418$	679.902	+	602.882i	1.62656	+	1.44230i
$419$	583.330			1.39220			0.696098	−	0.717947i	$-0.254918\pi$
$419$	583.330			1.39220			0.696098	+	0.717947i	$0.254918\pi$
$420$		0			0
$421$	−200.149			−0.475414			−0.237707	−	0.971337i	$-0.576396\pi$
$421$	−200.149			−0.475414			−0.237707	+	0.971337i	$0.576396\pi$
$422$	−970.092			−2.29880
$423$		0			0
$424$			2035.65i			4.80105i
$425$		−	154.421i		−	0.363344i
$426$		0			0
$427$	705.914			1.65319
$428$			424.595i			0.992043i
$429$		0			0
$430$	105.121			0.244468
$431$		−	114.182i		−	0.264923i	−0.991188	−	0.132462i	$-0.957712\pi$
$431$		−	114.182i		−	0.264923i	0.991188	−	0.132462i	$-0.0422881\pi$
$432$		0			0
$433$	−409.553			−0.945851			−0.472925	−	0.881102i	$-0.656802\pi$
$433$	−409.553			−0.945851			−0.472925	+	0.881102i	$0.656802\pi$
$434$	1888.43			4.35121
$435$		0			0
$436$		−	1052.43i		−	2.41382i
$437$		−	189.351i		−	0.433298i
$438$		0			0
$439$			475.285i			1.08265i	0.840812	+	0.541327i	$0.182078\pi$
$439$			475.285i			1.08265i	−0.840812	+	0.541327i	$0.817922\pi$
$440$	−529.962	+	597.666i	−1.20446	+	1.35833i
$441$		0			0
$442$		−	402.693i		−	0.911069i
$443$	591.088			1.33428			0.667142	−	0.744930i	$-0.267517\pi$
$443$	591.088			1.33428			0.667142	+	0.744930i	$0.267517\pi$
$444$		0			0
$445$	−66.9247			−0.150393
$446$			608.047i			1.36333i
$447$		0			0
$448$		−	2273.72i		−	5.07528i
$449$	425.699			0.948104			0.474052	−	0.880497i	$-0.342791\pi$
$449$	425.699			0.948104			0.474052	+	0.880497i	$0.342791\pi$
$450$		0			0
$451$	101.781	−	114.784i	0.225678	−	0.254509i
$452$	−2021.41			−4.47216
$453$		0			0
$454$	1256.70			2.76807
$455$	417.069			0.916634
$456$		0			0
$457$		−	62.3827i		−	0.136505i	−0.997668	−	0.0682524i	$-0.978258\pi$
$457$		−	62.3827i		−	0.136505i	0.997668	−	0.0682524i	$-0.0217423\pi$
$458$			1480.72i			3.23302i
$459$		0			0
$460$	268.068			0.582757
$461$			171.033i			0.371004i	0.982644	+	0.185502i	$0.0593912\pi$
$461$			171.033i			0.371004i	−0.982644	+	0.185502i	$0.940609\pi$
$462$		0			0
$463$	211.699			0.457233			0.228616	−	0.973517i	$-0.426580\pi$
$463$	211.699			0.457233			0.228616	+	0.973517i	$0.426580\pi$
$464$			1383.61i			2.98192i
$465$		0			0
$466$	−1505.04			−3.22969
$467$	83.4527			0.178700			0.0893498	−	0.996000i	$-0.471521\pi$
$467$	83.4527			0.178700			0.0893498	+	0.996000i	$0.471521\pi$
$468$		0			0
$469$		−	1085.69i		−	2.31490i
$470$		−	410.943i		−	0.874347i
$471$		0			0
$472$			497.844i			1.05475i
$473$	−69.2181	+	78.0609i	−0.146339	+	0.165034i
$474$		0			0
$475$		−	358.576i		−	0.754897i
$476$	1247.93			2.62171
$477$		0			0
$478$	−160.303			−0.335362
$479$		−	819.053i		−	1.70992i	−0.518691	−	0.854962i	$-0.673581\pi$
$479$		−	819.053i		−	1.70992i	0.518691	−	0.854962i	$-0.326419\pi$
$480$		0			0
$481$			375.813i			0.781315i
$482$	1726.45			3.58186
$483$		0			0
$484$	−152.766	−	1267.60i	−0.315633	−	2.61901i
$485$	273.996			0.564941
$486$		0			0
$487$	35.7025			0.0733112			0.0366556	−	0.999328i	$-0.488330\pi$
$487$	35.7025			0.0733112			0.0366556	+	0.999328i	$0.488330\pi$
$488$	1391.25			2.85091
$489$		0			0
$490$			1239.35i			2.52929i
$491$			84.9109i			0.172935i	0.996255	+	0.0864673i	$0.0275578\pi$
$491$			84.9109i			0.172935i	−0.996255	+	0.0864673i	$0.972442\pi$
$492$		0			0
$493$	−242.849			−0.492595
$494$		−	935.077i		−	1.89287i
$495$		0			0
$496$	2074.16			4.18178
$497$			1089.55i			2.19226i
$498$		0			0
$499$	412.548			0.826750			0.413375	−	0.910561i	$-0.364350\pi$
$499$	412.548			0.826750			0.413375	+	0.910561i	$0.364350\pi$
$500$	1274.10			2.54819
$501$		0			0
$502$		−	797.864i		−	1.58937i
$503$			162.833i			0.323723i	0.986813	+	0.161862i	$0.0517498\pi$
$503$			162.833i			0.323723i	−0.986813	+	0.161862i	$0.948250\pi$
$504$		0			0
$505$			27.4036i			0.0542646i
$506$	−243.424	+	274.522i	−0.481076	+	0.542534i
$507$		0			0
$508$		−	1360.80i		−	2.67874i
$509$	−17.8846			−0.0351367			−0.0175684	−	0.999846i	$-0.505592\pi$
$509$	−17.8846			−0.0351367			−0.0175684	+	0.999846i	$0.505592\pi$
$510$		0			0
$511$	−309.036			−0.604767
$512$		−	145.774i		−	0.284714i
$513$		0			0
$514$		−	1358.36i		−	2.64273i
$515$	−202.433			−0.393075
$516$		0			0
$517$	305.158	+	270.589i	0.590247	+	0.523383i
$518$	−1606.12			−3.10063
$519$		0			0
$520$	821.977			1.58073
$521$	337.501			0.647795			0.323897	−	0.946092i	$-0.395007\pi$
$521$	337.501			0.647795			0.323897	+	0.946092i	$0.395007\pi$
$522$		0			0
$523$			142.623i			0.272702i	0.990661	+	0.136351i	$0.0435376\pi$
$523$			142.623i			0.272702i	−0.990661	+	0.136351i	$0.956462\pi$
$524$			2547.34i			4.86133i
$525$		0			0
$526$	−313.894			−0.596757
$527$			364.054i			0.690805i
$528$		0			0
$529$	−452.546			−0.855475
$530$		−	902.728i		−	1.70326i
$531$		0			0
$532$	2897.78			5.44696
$533$	−157.863			−0.296179
$534$		0			0
$535$		−	116.913i		−	0.218530i
$536$		−	2139.72i		−	3.99202i
$537$		0			0
$538$		−	1029.39i		−	1.91337i
$539$	−920.316	−	816.062i	−1.70745	−	1.51403i
$540$		0			0
$541$		−	159.909i		−	0.295580i	−0.989019	−	0.147790i	$-0.952784\pi$
$541$		−	159.909i		−	0.295580i	0.989019	−	0.147790i	$-0.0472160\pi$
$542$	−573.493			−1.05811
$543$		0			0
$544$	957.930			1.76090
$545$			289.789i			0.531722i
$546$		0			0
$547$		−	318.021i		−	0.581392i	−0.956815	−	0.290696i	$-0.906113\pi$
$547$		−	318.021i		−	0.581392i	0.956815	−	0.290696i	$-0.0938868\pi$
$548$	1604.90			2.92865
$549$		0			0
$550$	−460.975	+	519.865i	−0.838136	+	0.945209i
$551$	−563.912			−1.02343
$552$		0			0
$553$	−357.401			−0.646294
$554$	−1263.68			−2.28102
$555$		0			0
$556$			2019.26i			3.63176i
$557$		−	498.633i		−	0.895212i	−0.894231	−	0.447606i	$-0.852277\pi$
$557$		−	498.633i		−	0.895212i	0.894231	−	0.447606i	$-0.147723\pi$
$558$		0			0
$559$	107.358			0.192054
$560$			1957.76i			3.49599i
$561$		0			0
$562$	−404.015			−0.718887
$563$			194.087i			0.344737i	0.985033	+	0.172369i	$0.0551420\pi$
$563$			194.087i			0.344737i	−0.985033	+	0.172369i	$0.944858\pi$
$564$		0			0
$565$	556.602			0.985137
$566$	708.035			1.25095
$567$		0			0
$568$			2147.34i			3.78053i
$569$		−	141.139i		−	0.248048i	−0.992279	−	0.124024i	$-0.960420\pi$
$569$		−	141.139i		−	0.248048i	0.992279	−	0.124024i	$-0.0395800\pi$
$570$		0			0
$571$			897.712i			1.57217i	0.618116	+	0.786087i	$0.287896\pi$
$571$			897.712i			1.57217i	−0.618116	+	0.786087i	$0.712104\pi$
$572$	−871.673	+	983.031i	−1.52390	+	1.71859i
$573$		0			0
$574$		−	674.666i		−	1.17538i
$575$	144.781			0.251794
$576$		0			0
$577$	−876.761			−1.51952			−0.759758	−	0.650206i	$-0.774683\pi$
$577$	−876.761			−1.51952			−0.759758	+	0.650206i	$0.774683\pi$
$578$		−	770.666i		−	1.33333i
$579$		0			0
$580$		−	798.341i		−	1.37645i
$581$	−587.008			−1.01034
$582$		0			0
$583$	670.346	+	594.409i	1.14982	+	1.01957i
$584$	−609.062			−1.04291
$585$		0			0
$586$	−1265.92			−2.16028
$587$	200.889			0.342230			0.171115	−	0.985251i	$-0.445263\pi$
$587$	200.889			0.342230			0.171115	+	0.985251i	$0.445263\pi$
$588$		0			0
$589$			845.356i			1.43524i
$590$		−	220.774i		−	0.374193i
$591$		0			0
$592$	−1764.10			−2.97989
$593$			36.4646i			0.0614918i	0.999527	+	0.0307459i	$0.00978827\pi$
$593$			36.4646i			0.0614918i	−0.999527	+	0.0307459i	$0.990212\pi$
$594$		0			0
$595$	−343.622			−0.577517
$596$		−	1787.50i		−	2.99916i
$597$		0			0
$598$	377.554			0.631361
$599$	806.601			1.34658			0.673290	−	0.739379i	$-0.264881\pi$
$599$	806.601			1.34658			0.673290	+	0.739379i	$0.264881\pi$
$600$		0			0
$601$		−	290.419i		−	0.483226i	−0.970373	−	0.241613i	$-0.922324\pi$
$601$		−	290.419i		−	0.483226i	0.970373	−	0.241613i	$-0.0776764\pi$
$602$			458.821i			0.762160i
$603$		0			0
$604$		−	150.819i		−	0.249701i
$605$	42.0646	+	349.037i	0.0695283	+	0.576921i
$606$		0			0
$607$		−	856.374i		−	1.41083i	−0.708795	−	0.705415i	$-0.750761\pi$
$607$		−	856.374i		−	1.41083i	0.708795	−	0.705415i	$-0.249239\pi$
$608$	2224.37			3.65851
$609$		0			0
$610$	−616.962			−1.01141
$611$		−	419.687i		−	0.686886i
$612$		0			0
$613$			828.550i			1.35163i	0.737071	+	0.675815i	$0.236208\pi$
$613$			828.550i			1.35163i	−0.737071	+	0.675815i	$0.763792\pi$
$614$	−537.787			−0.875875
$615$		0			0
$616$	−2608.62	−	2313.11i	−4.23477	−	3.75505i
$617$	−430.144			−0.697155			−0.348577	−	0.937280i	$-0.613335\pi$
$617$	−430.144			−0.697155			−0.348577	+	0.937280i	$0.613335\pi$
$618$		0			0
$619$	1074.78			1.73631			0.868156	−	0.496292i	$-0.165305\pi$
$619$	1074.78			1.73631			0.868156	+	0.496292i	$0.165305\pi$
$620$	−1196.79			−1.93030
$621$		0			0
$622$		−	1315.73i		−	2.11533i
$623$		−	292.105i		−	0.468868i
$624$		0			0
$625$	63.1286			0.101006
$626$		−	1399.09i		−	2.23497i
$627$		0			0
$628$	694.769			1.10632
$629$		−	309.632i		−	0.492260i
$630$		0			0
$631$	−228.486			−0.362102			−0.181051	−	0.983474i	$-0.557950\pi$
$631$	−228.486			−0.362102			−0.181051	+	0.983474i	$0.557950\pi$
$632$	−704.381			−1.11453
$633$		0			0
$634$		−	180.536i		−	0.284758i
$635$			374.700i			0.590079i
$636$		0			0
$637$			1265.72i			1.98700i
$638$	817.562	+	724.948i	1.28144	+	1.13628i
$639$		0			0
$640$			793.454i			1.23977i
$641$	907.163			1.41523			0.707615	−	0.706598i	$-0.249771\pi$
$641$	907.163			1.41523			0.707615	+	0.706598i	$0.249771\pi$
$642$		0			0
$643$	−643.870			−1.00135			−0.500677	−	0.865634i	$-0.666915\pi$
$643$	−643.870			−1.00135			−0.500677	+	0.865634i	$0.666915\pi$
$644$			1170.03i			1.81682i
$645$		0			0
$646$			770.409i			1.19258i
$647$	−562.332			−0.869137			−0.434568	−	0.900639i	$-0.643099\pi$
$647$	−562.332			−0.869137			−0.434568	+	0.900639i	$0.643099\pi$
$648$		0			0
$649$	163.942	+	145.371i	0.252607	+	0.223992i
$650$	714.977			1.09997
$651$		0			0
$652$	714.533			1.09591
$653$	−896.515			−1.37292			−0.686459	−	0.727169i	$-0.740836\pi$
$653$	−896.515			−1.37292			−0.686459	+	0.727169i	$0.740836\pi$
$654$		0			0
$655$		−	701.416i		−	1.07086i
$656$		−	741.024i		−	1.12961i
$657$		0			0
$658$	1793.63			2.72589
$659$		−	562.023i		−	0.852842i	−0.904525	−	0.426421i	$-0.859774\pi$
$659$		−	562.023i		−	0.852842i	0.904525	−	0.426421i	$-0.140226\pi$
$660$		0			0
$661$	169.308			0.256139			0.128070	−	0.991765i	$-0.459122\pi$
$661$	169.308			0.256139			0.128070	+	0.991765i	$0.459122\pi$
$662$		−	2289.34i		−	3.45821i
$663$		0			0
$664$	−1156.90			−1.74232
$665$	−797.913			−1.19987
$666$		0			0
$667$		−	227.689i		−	0.341363i
$668$			1524.42i			2.28207i
$669$		0			0
$670$			948.883i			1.41624i
$671$	406.244	−	458.143i	0.605431	−	0.682776i
$672$		0			0
$673$		−	977.523i		−	1.45249i	−0.687438	−	0.726243i	$-0.741265\pi$
$673$		−	977.523i		−	1.45249i	0.687438	−	0.726243i	$-0.258735\pi$
$674$	118.321			0.175551
$675$		0			0
$676$	−431.285			−0.637995
$677$		−	159.768i		−	0.235994i	−0.993014	−	0.117997i	$-0.962353\pi$
$677$		−	159.768i		−	0.235994i	0.993014	−	0.117997i	$-0.0376473\pi$
$678$		0			0
$679$			1195.90i			1.76127i
$680$	−677.226			−0.995921
$681$		0			0
$682$	1086.77	−	1225.60i	1.59350	−	1.79707i
$683$	−978.547			−1.43272			−0.716359	−	0.697731i	$-0.754193\pi$
$683$	−978.547			−1.43272			−0.716359	+	0.697731i	$0.754193\pi$
$684$		0			0
$685$	−441.914			−0.645130
$686$	−3038.95			−4.42995
$687$		0			0
$688$			503.948i			0.732483i
$689$		−	921.936i		−	1.33808i
$690$		0			0
$691$	264.505			0.382786			0.191393	−	0.981513i	$-0.438699\pi$
$691$	264.505			0.382786			0.191393	+	0.981513i	$0.438699\pi$
$692$			1376.34i			1.98894i
$693$		0			0
$694$	−740.722			−1.06732
$695$		−	556.008i		−	0.800012i
$696$		0			0
$697$	130.063			0.186605
$698$	144.223			0.206623
$699$		0			0
$700$			2215.70i			3.16528i
$701$		−	1168.11i		−	1.66635i	−0.553012	−	0.833173i	$-0.686522\pi$
$701$		−	1168.11i		−	1.66635i	0.553012	−	0.833173i	$-0.313478\pi$
$702$		0			0
$703$		−	718.984i		−	1.02274i
$704$	−1475.66	−	1308.50i	−2.09611	−	1.85866i
$705$		0			0
$706$			471.028i			0.667179i
$707$	−119.608			−0.169177
$708$		0			0
$709$	−786.312			−1.10904			−0.554522	−	0.832169i	$-0.687099\pi$
$709$	−786.312			−1.10904			−0.554522	+	0.832169i	$0.687099\pi$
$710$		−	952.260i		−	1.34121i
$711$		0			0
$712$		−	575.693i		−	0.808557i
$713$	−341.328			−0.478720
$714$		0			0
$715$	240.018	−	270.681i	0.335689	−	0.378574i
$716$	2804.43			3.91679
$717$		0			0
$718$	1999.54			2.78488
$719$	1281.15			1.78185			0.890927	−	0.454146i	$-0.150056\pi$
$719$	1281.15			1.78185			0.890927	+	0.454146i	$0.150056\pi$
$720$		0			0
$721$		−	883.556i		−	1.22546i
$722$			411.836i			0.570410i
$723$		0			0
$724$	−453.677			−0.626626
$725$		−	431.177i		−	0.594727i
$726$		0			0
$727$	−761.461			−1.04740			−0.523701	−	0.851902i	$-0.675449\pi$
$727$	−761.461			−1.04740			−0.523701	+	0.851902i	$0.675449\pi$
$728$			3587.66i			4.92811i
$729$		0			0
$730$	270.095			0.369993
$731$	−88.4522			−0.121002
$732$		0			0
$733$			1026.88i			1.40093i	0.713687	+	0.700464i	$0.247024\pi$
$733$			1026.88i			1.40093i	−0.713687	+	0.700464i	$0.752976\pi$
$734$		−	548.054i		−	0.746668i
$735$		0			0
$736$			898.130i			1.22029i
$737$	−704.620	−	624.800i	−0.956065	−	0.847761i
$738$		0			0
$739$		−	1000.49i		−	1.35385i	−0.736052	−	0.676925i	$-0.763312\pi$
$739$		−	1000.49i		−	1.35385i	0.736052	−	0.676925i	$-0.236688\pi$
$740$	1017.88			1.37551
$741$		0			0
$742$	3940.11			5.31013
$743$		−	185.449i		−	0.249594i	−0.992182	−	0.124797i	$-0.960172\pi$
$743$		−	185.449i		−	0.249594i	0.992182	−	0.124797i	$-0.0398280\pi$
$744$		0			0
$745$			492.193i			0.660661i
$746$	−827.595			−1.10938
$747$		0			0
$748$	718.171	−	809.918i	0.960121	−	1.08278i
$749$	510.289			0.681293
$750$		0			0
$751$	−516.504			−0.687755			−0.343877	−	0.939015i	$-0.611740\pi$
$751$	−516.504			−0.687755			−0.343877	+	0.939015i	$0.611740\pi$
$752$	1970.05			2.61974
$753$		0			0
$754$		−	1124.40i		−	1.49125i
$755$			41.5285i			0.0550046i
$756$		0			0
$757$	599.543			0.791999			0.395999	−	0.918251i	$-0.370398\pi$
$757$	599.543			0.791999			0.395999	+	0.918251i	$0.370398\pi$
$758$		−	2078.22i		−	2.74172i
$759$		0			0
$760$	−1572.56			−2.06916
$761$			947.287i			1.24479i	0.782703	+	0.622396i	$0.213841\pi$
$761$			947.287i			1.24479i	−0.782703	+	0.622396i	$0.786159\pi$
$762$		0			0
$763$	−1264.83			−1.65771
$764$	300.627			0.393491
$765$		0			0
$766$			494.578i			0.645664i
$767$		−	225.472i		−	0.293966i
$768$		0			0
$769$		−	393.833i		−	0.512136i	−0.966659	−	0.256068i	$-0.917573\pi$
$769$		−	393.833i		−	0.512136i	0.966659	−	0.256068i	$-0.0824271\pi$
$770$	1156.82	+	1025.77i	1.50236	+	1.33217i
$771$		0			0
$772$			597.774i			0.774318i
$773$	−122.104			−0.157961			−0.0789807	−	0.996876i	$-0.525167\pi$
$773$	−122.104			−0.157961			−0.0789807	+	0.996876i	$0.525167\pi$
$774$		0			0
$775$	−646.375			−0.834032
$776$			2356.94i			3.03729i
$777$		0			0
$778$		−	319.874i		−	0.411149i
$779$	302.015			0.387696
$780$		0			0
$781$	707.127	+	627.024i	0.905413	+	0.802847i
$782$	−311.066			−0.397783
$783$		0			0
$784$	−5941.41			−7.57833
$785$	−191.307			−0.243703
$786$		0			0
$787$		−	751.793i		−	0.955264i	−0.878560	−	0.477632i	$-0.841495\pi$
$787$		−	751.793i		−	0.955264i	0.878560	−	0.477632i	$-0.158505\pi$
$788$			1221.46i			1.55008i
$789$		0			0
$790$	312.365			0.395399
$791$			2429.39i			3.07129i
$792$		0			0
$793$	−630.090			−0.794565
$794$			1930.39i			2.43122i
$795$		0			0
$796$	−1783.78			−2.24093
$797$	−754.763			−0.947005			−0.473503	−	0.880792i	$-0.657011\pi$
$797$	−754.763			−0.947005			−0.473503	+	0.880792i	$0.657011\pi$
$798$		0			0
$799$			345.780i			0.432766i
$800$			1700.80i			2.12600i
$801$		0			0
$802$			2161.04i			2.69456i
$803$	−177.846	+	200.566i	−0.221477	+	0.249771i
$804$		0			0
$805$		−	322.171i		−	0.400213i
$806$	−1685.59			−2.09130
$807$		0			0
$808$	−235.728			−0.291743
$809$			1330.23i			1.64429i	0.569277	+	0.822146i	$0.307223\pi$
$809$			1330.23i			1.64429i	−0.569277	+	0.822146i	$0.692777\pi$
$810$		0			0
$811$			625.877i			0.771735i	0.922554	+	0.385867i	$0.126098\pi$
$811$			625.877i			0.771735i	−0.922554	+	0.385867i	$0.873902\pi$
$812$	3484.50			4.29125
$813$		0			0
$814$	−924.304	+	1042.39i	−1.13551	+	1.28057i
$815$	−196.749			−0.241410
$816$		0			0
$817$	−205.392			−0.251397
$818$	−609.530			−0.745147
$819$		0			0
$820$			427.569i			0.521426i
$821$		−	238.063i		−	0.289967i	−0.989434	−	0.144983i	$-0.953687\pi$
$821$		−	238.063i		−	0.289967i	0.989434	−	0.144983i	$-0.0463129\pi$
$822$		0			0
$823$	360.039			0.437471			0.218736	−	0.975784i	$-0.429807\pi$
$823$	360.039			0.437471			0.218736	+	0.975784i	$0.429807\pi$
$824$		−	1741.35i		−	2.11329i
$825$		0			0
$826$	963.607			1.16659
$827$		−	86.7583i		−	0.104907i	−0.998623	−	0.0524536i	$-0.983296\pi$
$827$		−	86.7583i		−	0.104907i	0.998623	−	0.0524536i	$-0.0167042\pi$
$828$		0			0
$829$	482.406			0.581914			0.290957	−	0.956736i	$-0.406026\pi$
$829$	482.406			0.581914			0.290957	+	0.956736i	$0.406026\pi$
$830$	513.040			0.618121
$831$		0			0
$832$			2029.50i			2.43930i
$833$		−	1042.83i		−	1.25189i
$834$		0			0
$835$		−	419.754i		−	0.502699i
$836$	1667.64	−	1880.68i	1.99478	−	2.24962i
$837$		0			0
$838$		−	2225.22i		−	2.65539i
$839$	−1015.36			−1.21020			−0.605099	−	0.796151i	$-0.706866\pi$
$839$	−1015.36			−1.21020			−0.605099	+	0.796151i	$0.706866\pi$
$840$		0			0
$841$	162.913			0.193713
$842$			763.507i			0.906778i
$843$		0			0
$844$			2683.38i			3.17936i
$845$	118.755			0.140539
$846$		0			0
$847$	−1523.43	+	183.598i	−1.79862	+	0.216763i
$848$	4327.65			5.10336
$849$		0			0
$850$	−589.069			−0.693022
$851$	290.303			0.341131
$852$		0			0
$853$			1270.15i			1.48904i	0.667599	+	0.744521i	$0.267322\pi$
$853$			1270.15i			1.48904i	−0.667599	+	0.744521i	$0.732678\pi$
$854$		−	2692.84i		−	3.15321i
$855$		0			0
$856$	1005.70			1.17488
$857$		−	124.481i		−	0.145252i	−0.997359	−	0.0726258i	$-0.976862\pi$
$857$		−	124.481i		−	0.145252i	0.997359	−	0.0726258i	$-0.0231379\pi$
$858$		0			0
$859$	1351.85			1.57374			0.786872	−	0.617117i	$-0.211699\pi$
$859$	1351.85			1.57374			0.786872	+	0.617117i	$0.211699\pi$
$860$		−	290.777i		−	0.338113i
$861$		0			0
$862$	−435.568			−0.505299
$863$	−1611.73			−1.86759			−0.933796	−	0.357806i	$-0.883525\pi$
$863$	−1611.73			−1.86759			−0.933796	+	0.357806i	$0.883525\pi$
$864$		0			0
$865$		−	378.980i		−	0.438128i
$866$			1562.32i			1.80406i
$867$		0			0
$868$		−	5223.59i		−	6.01796i
$869$	−205.680	+	231.956i	−0.236685	+	0.266922i
$870$		0			0
$871$			969.073i			1.11260i
$872$	−2492.79			−2.85870
$873$		0			0
$874$	−722.316			−0.826448
$875$		−	1531.24i		−	1.74999i
$876$		0			0
$877$			370.623i			0.422603i	0.977421	+	0.211301i	$0.0677701\pi$
$877$			370.623i			0.422603i	−0.977421	+	0.211301i	$0.932230\pi$
$878$	1813.07			2.06499
$879$		0			0
$880$	1270.60	+	1126.66i	1.44386	+	1.28030i
$881$	719.594			0.816792			0.408396	−	0.912805i	$-0.366088\pi$
$881$	719.594			0.816792			0.408396	+	0.912805i	$0.366088\pi$
$882$		0			0
$883$	−100.661			−0.113999			−0.0569993	−	0.998374i	$-0.518153\pi$
$883$	−100.661			−0.113999			−0.0569993	+	0.998374i	$0.518153\pi$
$884$	−1113.89			−1.26006
$885$		0			0
$886$		−	2254.82i		−	2.54494i
$887$			1096.78i			1.23651i	0.785979	+	0.618253i	$0.212159\pi$
$887$			1096.78i			1.23651i	−0.785979	+	0.618253i	$0.787841\pi$
$888$		0			0
$889$	−1635.44			−1.83964
$890$			255.297i			0.286850i
$891$		0			0
$892$	1681.92			1.88556
$893$			802.922i			0.899129i
$894$		0			0
$895$	−772.207			−0.862801
$896$	−3463.17			−3.86514
$897$		0			0
$898$		−	1623.91i		−	1.80836i
$899$			1016.52i			1.13072i
$900$		0			0
$901$			759.582i			0.843044i
$902$	−437.863	−	388.262i	−0.485436	−	0.430446i
$903$		0			0
$904$			4787.95i			5.29640i
$905$	124.921			0.138035
$906$		0			0
$907$	787.806			0.868584			0.434292	−	0.900772i	$-0.356999\pi$
$907$	787.806			0.868584			0.434292	+	0.900772i	$0.356999\pi$
$908$		−	3476.18i		−	3.82839i
$909$		0			0
$910$		−	1590.99i		−	1.74834i
$911$	−205.640			−0.225730			−0.112865	−	0.993610i	$-0.536003\pi$
$911$	−205.640			−0.225730			−0.112865	+	0.993610i	$0.536003\pi$
$912$		0			0
$913$	−337.816	+	380.973i	−0.370007	+	0.417276i
$914$	−237.971			−0.260362
$915$		0			0
$916$	4095.84			4.47144
$917$	3061.45			3.33855
$918$		0			0
$919$			1487.91i			1.61906i	0.587079	+	0.809529i	$0.300278\pi$
$919$			1487.91i			1.61906i	−0.587079	+	0.809529i	$0.699722\pi$
$920$		−	634.950i		−	0.690163i
$921$		0			0
$922$	652.438			0.707633
$923$		−	972.522i		−	1.05365i
$924$		0			0
$925$	549.748			0.594322
$926$		−	807.564i		−	0.872099i
$927$		0			0
$928$	2674.74			2.88227
$929$	−628.753			−0.676807			−0.338403	−	0.941001i	$-0.609887\pi$
$929$	−628.753			−0.676807			−0.338403	+	0.941001i	$0.609887\pi$
$930$		0			0
$931$		−	2421.51i		−	2.60098i
$932$			4163.10i			4.46684i
$933$		0			0
$934$		−	318.346i		−	0.340841i
$935$	−197.750	+	223.013i	−0.211498	+	0.238517i
$936$		0			0
$937$			966.085i			1.03104i	0.856877	+	0.515520i	$0.172401\pi$
$937$			966.085i			1.03104i	−0.856877	+	0.515520i	$0.827599\pi$
$938$	−4141.56			−4.41531
$939$		0			0
$940$	−1136.71			−1.20927
$941$			1646.36i			1.74958i	0.484499	+	0.874792i	$0.339002\pi$
$941$			1646.36i			1.74958i	−0.484499	+	0.874792i	$0.660998\pi$
$942$		0			0
$943$			121.944i			0.129315i
$944$	1058.38			1.12117
$945$		0			0
$946$	297.778	+	264.045i	0.314776	+	0.279118i
$947$	1518.95			1.60396			0.801979	−	0.597352i	$-0.203780\pi$
$947$	1518.95			1.60396			0.801979	+	0.597352i	$0.203780\pi$
$948$		0			0
$949$	275.842			0.290666
$950$	−1367.86			−1.43985
$951$		0			0
$952$		−	2955.87i		−	3.10491i
$953$		−	1419.94i		−	1.48996i	−0.667084	−	0.744982i	$-0.732458\pi$
$953$		−	1419.94i		−	1.48996i	0.667084	−	0.744982i	$-0.267542\pi$
$954$		0			0
$955$	−82.7785			−0.0866790
$956$			443.415i			0.463824i
$957$		0			0
$958$	−3124.43			−3.26141
$959$		−	1928.81i		−	2.01127i
$960$		0			0
$961$	562.853			0.585696
$962$	1433.61			1.49024
$963$		0			0
$964$		−	4775.56i		−	4.95390i
$965$		−	164.599i		−	0.170569i
$966$		0			0
$967$			1598.67i			1.65323i	0.562768	+	0.826615i	$0.309736\pi$
$967$			1598.67i			1.65323i	−0.562768	+	0.826615i	$0.690264\pi$
$968$	−3002.45	+	361.844i	−3.10171	+	0.373806i
$969$		0			0
$970$		−	1045.21i		−	1.07754i
$971$	865.635			0.891488			0.445744	−	0.895160i	$-0.352939\pi$
$971$	865.635			0.891488			0.445744	+	0.895160i	$0.352939\pi$
$972$		0			0
$973$	2426.79			2.49414
$974$		−	136.194i		−	0.139830i
$975$		0			0
$976$		−	2957.70i		−	3.03043i
$977$	151.046			0.154602			0.0773011	−	0.997008i	$-0.475370\pi$
$977$	151.046			0.154602			0.0773011	+	0.997008i	$0.475370\pi$
$978$		0			0
$979$	−189.578	−	168.103i	−0.193645	−	0.171708i
$980$	3428.18			3.49814
$981$		0			0
$982$	323.908			0.329846
$983$	1195.67			1.21635			0.608173	−	0.793804i	$-0.291903\pi$
$983$	1195.67			1.21635			0.608173	+	0.793804i	$0.291903\pi$
$984$		0			0
$985$		−	336.333i		−	0.341455i
$986$			926.395i			0.939548i
$987$		0			0
$988$	−2586.52			−2.61794
$989$		−	82.9305i		−	0.0838529i
$990$		0			0
$991$	640.164			0.645977			0.322989	−	0.946403i	$-0.395312\pi$
$991$	640.164			0.645977			0.322989	+	0.946403i	$0.395312\pi$
$992$		−	4009.69i		−	4.04203i
$993$		0			0
$994$	4156.30			4.18139
$995$	491.169			0.493637
$996$		0			0
$997$		−	261.029i		−	0.261815i	−0.991395	−	0.130907i	$-0.958211\pi$
$997$		−	261.029i		−	0.261815i	0.991395	−	0.130907i	$-0.0417890\pi$
$998$		−	1573.74i		−	1.57690i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.3.c.b.109.2	yes	16
3.2	odd	2	inner	297.3.c.b.109.15	yes	16
11.10	odd	2	inner	297.3.c.b.109.16	yes	16
33.32	even	2	inner	297.3.c.b.109.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.3.c.b.109.1	✓	16	33.32	even	2	inner
297.3.c.b.109.2	yes	16	1.1	even	1	trivial
297.3.c.b.109.15	yes	16	3.2	odd	2	inner
297.3.c.b.109.16	yes	16	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	297.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.81469i q^{2} -10.5518 q^{4} +2.90548 q^{5} +12.6815i q^{7} +24.9932i q^{8} +O(q^{10})\)	\(q-3.81469i q^{2} -10.5518 q^{4} +2.90548 q^{5} +12.6815i q^{7} +24.9932i q^{8} -11.0835i q^{10} +(8.23036 + 7.29802i) q^{11} -11.3193i q^{13} +48.3758 q^{14} +53.1339 q^{16} +9.32598i q^{17} +21.6555i q^{19} -30.6581 q^{20} +(27.8397 - 31.3962i) q^{22} -8.74379 q^{23} -16.5582 q^{25} -43.1797 q^{26} -133.813i q^{28} +26.0401i q^{29} +39.0366 q^{31} -102.716i q^{32} +35.5757 q^{34} +36.8457i q^{35} -33.2010 q^{37} +82.6090 q^{38} +72.6172i q^{40} -13.9464i q^{41} +9.48450i q^{43} +(-86.8454 - 77.0075i) q^{44} +33.3548i q^{46} +37.0771 q^{47} -111.820 q^{49} +63.1643i q^{50} +119.440i q^{52} +81.4480 q^{53} +(23.9131 + 21.2043i) q^{55} -316.950 q^{56} +99.3349 q^{58} +19.9192 q^{59} -55.6650i q^{61} -148.912i q^{62} -179.295 q^{64} -32.8880i q^{65} -85.6122 q^{67} -98.4062i q^{68} +140.555 q^{70} +85.9169 q^{71} +24.3691i q^{73} +126.651i q^{74} -228.505i q^{76} +(-92.5496 + 104.373i) q^{77} +28.1829i q^{79} +154.379 q^{80} -53.2010 q^{82} +46.2887i q^{83} +27.0964i q^{85} +36.1804 q^{86} +(-182.401 + 205.703i) q^{88} -23.0340 q^{89} +143.546 q^{91} +92.2631 q^{92} -141.437i q^{94} +62.9196i q^{95} +94.3033 q^{97} +426.557i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4}+O(q^{10}) \) 16 * q - 32 * q^4	\( 16 q - 32 q^{4} + 160 q^{16} + 102 q^{22} + 12 q^{25} + 68 q^{31} + 156 q^{34} - 124 q^{37} - 272 q^{49} + 182 q^{55} + 492 q^{58} - 680 q^{64} - 400 q^{67} + 324 q^{70} - 444 q^{82} - 510 q^{88} + 120 q^{91} + 152 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 160 * q^16 + 102 * q^22 + 12 * q^25 + 68 * q^31 + 156 * q^34 - 124 * q^37 - 272 * q^49 + 182 * q^55 + 492 * q^58 - 680 * q^64 - 400 * q^67 + 324 * q^70 - 444 * q^82 - 510 * q^88 + 120 * q^91 + 152 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more