LMFDB - Embedded newform 825.3.b.d.76.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,3,Mod(76,825)] 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("825.76"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	825.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-8,0,0,0,0,48,0,-28,0,0,0,0,-16,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(22)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4796218097$
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} + 36x^{14} + 500x^{12} + 3364x^{10} + 11310x^{8} + 17932x^{6} + 12708x^{4} + 3244x^{2} + 121 $ x^16 + 36x^14 + 500x^12 + 3364x^10 + 11310x^8 + 17932x^6 + 12708x^4 + 3244*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			76.13
Root			$2.33103i$ of defining polynomial
Character	$\chi$	$=$	825.76
Dual form			825.3.b.d.76.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.33103i q^{2} +1.73205 q^{3} -1.43371 q^{4} +4.03747i q^{6} +6.32430i q^{7} +5.98210i q^{8} +3.00000 q^{9} +(-10.3943 + 3.59991i) q^{11} -2.48326 q^{12} +17.3965i q^{13} -14.7421 q^{14} -19.6793 q^{16} -12.2603i q^{17} +6.99310i q^{18} -10.6069i q^{19} +10.9540i q^{21} +(-8.39151 - 24.2294i) q^{22} +9.16729 q^{23} +10.3613i q^{24} -40.5519 q^{26} +5.19615 q^{27} -9.06722i q^{28} -0.592304i q^{29} +17.1631 q^{31} -21.9447i q^{32} +(-18.0034 + 6.23523i) q^{33} +28.5791 q^{34} -4.30114 q^{36} -55.3012 q^{37} +24.7249 q^{38} +30.1317i q^{39} -13.5046i q^{41} -25.5341 q^{42} +36.0017i q^{43} +(14.9024 - 5.16124i) q^{44} +21.3693i q^{46} +19.2531 q^{47} -34.0856 q^{48} +9.00325 q^{49} -21.2354i q^{51} -24.9416i q^{52} -70.4442 q^{53} +12.1124i q^{54} -37.8326 q^{56} -18.3716i q^{57} +1.38068 q^{58} +14.9479 q^{59} +36.1401i q^{61} +40.0077i q^{62} +18.9729i q^{63} -27.5634 q^{64} +(-14.5345 - 41.9665i) q^{66} -127.842 q^{67} +17.5777i q^{68} +15.8782 q^{69} -27.8440 q^{71} +17.9463i q^{72} +61.3643i q^{73} -128.909i q^{74} +15.2072i q^{76} +(-22.7669 - 65.7364i) q^{77} -70.2379 q^{78} +141.550i q^{79} +9.00000 q^{81} +31.4798 q^{82} -137.804i q^{83} -15.7049i q^{84} -83.9212 q^{86} -1.02590i q^{87} +(-21.5350 - 62.1795i) q^{88} +160.353 q^{89} -110.021 q^{91} -13.1433 q^{92} +29.7273 q^{93} +44.8797i q^{94} -38.0094i q^{96} +57.7240 q^{97} +20.9869i q^{98} +(-31.1828 + 10.7997i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} + 48 q^{9} - 28 q^{11} - 16 q^{16} + 20 q^{22} - 56 q^{23} - 88 q^{26} - 96 q^{31} + 12 q^{33} - 200 q^{34} - 24 q^{36} - 184 q^{37} - 296 q^{38} + 264 q^{42} + 300 q^{44} + 200 q^{47} - 96 q^{48}+ \cdots - 84 q^{99}+O(q^{100}) $ 16 * q - 8 * q^4 + 48 * q^9 - 28 * q^11 - 16 * q^16 + 20 * q^22 - 56 * q^23 - 88 * q^26 - 96 * q^31 + 12 * q^33 - 200 * q^34 - 24 * q^36 - 184 * q^37 - 296 * q^38 + 264 * q^42 + 300 * q^44 + 200 * q^47 - 96 * q^48 - 496 * q^49 + 80 * q^53 - 32 * q^56 + 16 * q^58 - 136 * q^59 + 456 * q^64 - 180 * q^66 - 256 * q^67 + 168 * q^69 + 216 * q^71 + 248 * q^77 - 240 * q^78 + 144 * q^81 - 232 * q^82 + 192 * q^86 + 116 * q^88 - 336 * q^89 + 256 * q^91 + 440 * q^92 + 144 * q^93 + 144 * q^97 - 84 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$			2.33103i			1.16552i	0.812646	+	0.582758i	$0.198026\pi$
$2$			2.33103i			1.16552i	−0.812646	+	0.582758i	$0.801974\pi$
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$	−1.43371			−0.358428
$5$		0			0
$5$		0			0
$6$			4.03747i			0.672911i
$7$			6.32430i			0.903471i	0.892152	+	0.451736i	$0.149195\pi$
$7$			6.32430i			0.903471i	−0.892152	+	0.451736i	$0.850805\pi$
$8$			5.98210i			0.747763i
$9$	3.00000			0.333333
$10$		0			0
$11$	−10.3943	+	3.59991i	−0.944933	+	0.327265i
$11$	−10.3943	+	3.59991i	−0.944933	+	0.327265i
$12$	−2.48326			−0.206938
$13$			17.3965i			1.33820i	0.743175	+	0.669098i	$0.233319\pi$
$13$			17.3965i			1.33820i	−0.743175	+	0.669098i	$0.766681\pi$
$14$	−14.7421			−1.05301
$15$		0			0
$16$	−19.6793			−1.22996
$17$		−	12.2603i		−	0.721192i	−0.932722	−	0.360596i	$-0.882573\pi$
$17$		−	12.2603i		−	0.721192i	0.932722	−	0.360596i	$-0.117427\pi$
$18$			6.99310i			0.388505i
$19$		−	10.6069i		−	0.558256i	−0.960254	−	0.279128i	$-0.909955\pi$
$19$		−	10.6069i		−	0.558256i	0.960254	−	0.279128i	$-0.0900454\pi$
$20$		0			0
$21$			10.9540i			0.521619i
$22$	−8.39151	−	24.2294i	−0.381432	−	1.10133i
$23$	9.16729			0.398578			0.199289	−	0.979941i	$-0.436137\pi$
$23$	9.16729			0.398578			0.199289	+	0.979941i	$0.436137\pi$
$24$			10.3613i			0.431721i
$25$		0			0
$26$	−40.5519			−1.55969
$27$	5.19615			0.192450
$28$		−	9.06722i		−	0.323829i
$29$		−	0.592304i		−	0.0204243i	−0.999948	−	0.0102121i	$-0.996749\pi$
$29$		−	0.592304i		−	0.0204243i	0.999948	−	0.0102121i	$-0.00325068\pi$
$30$		0			0
$31$	17.1631			0.553648			0.276824	−	0.960921i	$-0.410718\pi$
$31$	17.1631			0.553648			0.276824	+	0.960921i	$0.410718\pi$
$32$		−	21.9447i		−	0.685773i
$33$	−18.0034	+	6.23523i	−0.545557	+	0.188946i
$34$	28.5791			0.840561
$35$		0			0
$36$	−4.30114			−0.119476
$37$	−55.3012			−1.49463			−0.747314	−	0.664472i	$-0.768657\pi$
$37$	−55.3012			−1.49463			−0.747314	+	0.664472i	$0.768657\pi$
$38$	24.7249			0.650656
$39$			30.1317i			0.772607i
$40$		0			0
$41$		−	13.5046i		−	0.329382i	−0.986345	−	0.164691i	$-0.947337\pi$
$41$		−	13.5046i		−	0.329382i	0.986345	−	0.164691i	$-0.0526626\pi$
$42$	−25.5341			−0.607956
$43$			36.0017i			0.837249i	0.908159	+	0.418625i	$0.137488\pi$
$43$			36.0017i			0.837249i	−0.908159	+	0.418625i	$0.862512\pi$
$44$	14.9024	−	5.16124i	0.338690	−	0.117301i
$45$		0			0
$46$			21.3693i			0.464549i
$47$	19.2531			0.409641			0.204821	−	0.978800i	$-0.434339\pi$
$47$	19.2531			0.409641			0.204821	+	0.978800i	$0.434339\pi$
$48$	−34.0856			−0.710116
$49$	9.00325			0.183740
$50$		0			0
$51$		−	21.2354i		−	0.416380i
$52$		−	24.9416i		−	0.479646i
$53$	−70.4442			−1.32914			−0.664568	−	0.747228i	$-0.731384\pi$
$53$	−70.4442			−1.32914			−0.664568	+	0.747228i	$0.731384\pi$
$54$			12.1124i			0.224304i
$55$		0			0
$56$	−37.8326			−0.675582
$57$		−	18.3716i		−	0.322309i
$58$	1.38068			0.0238048
$59$	14.9479			0.253354			0.126677	−	0.991944i	$-0.459569\pi$
$59$	14.9479			0.253354			0.126677	+	0.991944i	$0.459569\pi$
$60$		0			0
$61$			36.1401i			0.592461i	0.955116	+	0.296231i	$0.0957297\pi$
$61$			36.1401i			0.592461i	−0.955116	+	0.296231i	$0.904270\pi$
$62$			40.0077i			0.645285i
$63$			18.9729i			0.301157i
$64$	−27.5634			−0.430678
$65$		0			0
$66$	−14.5345	−	41.9665i	−0.220220	−	0.635856i
$67$	−127.842			−1.90809			−0.954046	−	0.299660i	$-0.903127\pi$
$67$	−127.842			−1.90809			−0.954046	+	0.299660i	$0.903127\pi$
$68$			17.5777i			0.258495i
$69$	15.8782			0.230119
$70$		0			0
$71$	−27.8440			−0.392169			−0.196084	−	0.980587i	$-0.562823\pi$
$71$	−27.8440			−0.392169			−0.196084	+	0.980587i	$0.562823\pi$
$72$			17.9463i			0.249254i
$73$			61.3643i			0.840607i	0.907383	+	0.420304i	$0.138076\pi$
$73$			61.3643i			0.840607i	−0.907383	+	0.420304i	$0.861924\pi$
$74$		−	128.909i		−	1.74201i
$75$		0			0
$76$			15.2072i			0.200095i
$77$	−22.7669	−	65.7364i	−0.295674	−	0.853719i
$78$	−70.2379			−0.900486
$79$			141.550i			1.79177i	0.444289	+	0.895884i	$0.353456\pi$
$79$			141.550i			1.79177i	−0.444289	+	0.895884i	$0.646544\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	31.4798			0.383900
$83$		−	137.804i		−	1.66028i	−0.557552	−	0.830142i	$-0.688259\pi$
$83$		−	137.804i		−	1.66028i	0.557552	−	0.830142i	$-0.311741\pi$
$84$		−	15.7049i		−	0.186963i
$85$		0			0
$86$	−83.9212			−0.975828
$87$		−	1.02590i		−	0.0117920i
$88$	−21.5350	−	62.1795i	−0.244716	−	0.706585i
$89$	160.353			1.80172			0.900858	−	0.434113i	$-0.142938\pi$
$89$	160.353			1.80172			0.900858	+	0.434113i	$0.142938\pi$
$90$		0			0
$91$	−110.021			−1.20902
$92$	−13.1433			−0.142861
$93$	29.7273			0.319649
$94$			44.8797i			0.477443i
$95$		0			0
$96$		−	38.0094i		−	0.395931i
$97$	57.7240			0.595092			0.297546	−	0.954707i	$-0.403832\pi$
$97$	57.7240			0.595092			0.297546	+	0.954707i	$0.403832\pi$
$98$			20.9869i			0.214152i
$99$	−31.1828	+	10.7997i	−0.314978	+	0.109088i
$100$		0			0
$101$			147.055i			1.45599i	0.685585	+	0.727993i	$0.259547\pi$
$101$			147.055i			1.45599i	−0.685585	+	0.727993i	$0.740453\pi$
$102$	49.5004			0.485298
$103$	98.7792			0.959022			0.479511	−	0.877536i	$-0.340814\pi$
$103$	98.7792			0.959022			0.479511	+	0.877536i	$0.340814\pi$
$104$	−104.068			−1.00065
$105$		0			0
$106$		−	164.208i		−	1.54913i
$107$			107.953i			1.00891i	0.863439	+	0.504453i	$0.168306\pi$
$107$			107.953i			1.00891i	−0.863439	+	0.504453i	$0.831694\pi$
$108$	−7.44978			−0.0689795
$109$			33.0995i			0.303666i	0.988406	+	0.151833i	$0.0485175\pi$
$109$			33.0995i			0.303666i	−0.988406	+	0.151833i	$0.951482\pi$
$110$		0			0
$111$	−95.7845			−0.862923
$112$		−	124.458i		−	1.11123i
$113$	163.297			1.44510			0.722552	−	0.691316i	$-0.242969\pi$
$113$	163.297			1.44510			0.722552	+	0.691316i	$0.242969\pi$
$114$	42.8249			0.375657
$115$		0			0
$116$			0.849193i			0.00732063i
$117$			52.1896i			0.446065i
$118$			34.8441i			0.295289i
$119$	77.5376			0.651576
$120$		0			0
$121$	95.0813	−	74.8369i	0.785795	−	0.618486i
$122$	−84.2438			−0.690523
$123$		−	23.3907i		−	0.190169i
$124$	−24.6069			−0.198443
$125$		0			0
$126$	−44.2264			−0.351003
$127$			204.213i			1.60798i	0.594645	+	0.803988i	$0.297293\pi$
$127$			204.213i			1.60798i	−0.594645	+	0.803988i	$0.702707\pi$
$128$		−	152.030i		−	1.18774i
$129$			62.3568i			0.483386i
$130$		0			0
$131$		−	243.373i		−	1.85781i	−0.370316	−	0.928906i	$-0.620751\pi$
$131$		−	243.373i		−	1.85781i	0.370316	−	0.928906i	$-0.379249\pi$
$132$	25.8117	−	8.93953i	0.195543	−	0.0677237i
$133$	67.0810			0.504368
$134$		−	298.004i		−	2.22391i
$135$		0			0
$136$	73.3421			0.539280
$137$	−68.3154			−0.498652			−0.249326	−	0.968420i	$-0.580209\pi$
$137$	−68.3154			−0.498652			−0.249326	+	0.968420i	$0.580209\pi$
$138$			37.0126i			0.268208i
$139$		−	21.5172i		−	0.154800i	−0.997000	−	0.0774001i	$-0.975338\pi$
$139$		−	21.5172i		−	0.154800i	0.997000	−	0.0774001i	$-0.0246619\pi$
$140$		0			0
$141$	33.3474			0.236506
$142$		−	64.9052i		−	0.457079i
$143$	−62.6260	−	180.824i	−0.437944	−	1.26450i
$144$	−59.0380			−0.409986
$145$		0			0
$146$	−143.042			−0.979742
$147$	15.5941			0.106082
$148$	79.2860			0.535716
$149$			45.9449i			0.308355i	0.988043	+	0.154178i	$0.0492728\pi$
$149$			45.9449i			0.308355i	−0.988043	+	0.154178i	$0.950727\pi$
$150$		0			0
$151$		−	123.582i		−	0.818425i	−0.912439	−	0.409212i	$-0.865803\pi$
$151$		−	123.582i		−	0.818425i	0.912439	−	0.409212i	$-0.134197\pi$
$152$	63.4513			0.417443
$153$		−	36.7808i		−	0.240397i
$154$	153.234	−	53.0704i	0.995024	−	0.344613i
$155$		0			0
$156$		−	43.2001i		−	0.276924i
$157$	−65.4047			−0.416590			−0.208295	−	0.978066i	$-0.566791\pi$
$157$	−65.4047			−0.416590			−0.208295	+	0.978066i	$0.566791\pi$
$158$	−329.957			−2.08833
$159$	−122.013			−0.767377
$160$		0			0
$161$			57.9767i			0.360104i
$162$			20.9793i			0.129502i
$163$	318.348			1.95305			0.976527	−	0.215395i	$-0.0691037\pi$
$163$	318.348			1.95305			0.976527	+	0.215395i	$0.0691037\pi$
$164$			19.3618i			0.118060i
$165$		0			0
$166$	321.225			1.93509
$167$			17.2478i			0.103280i	0.998666	+	0.0516400i	$0.0164448\pi$
$167$			17.2478i			0.103280i	−0.998666	+	0.0516400i	$0.983555\pi$
$168$	−65.5280			−0.390047
$169$	−133.639			−0.790766
$170$		0			0
$171$		−	31.8206i		−	0.186085i
$172$		−	51.6161i		−	0.300094i
$173$		−	322.103i		−	1.86187i	−0.365190	−	0.930933i	$-0.618996\pi$
$173$		−	322.103i		−	1.86187i	0.365190	−	0.930933i	$-0.381004\pi$
$174$	2.39141			0.0137437
$175$		0			0
$176$	204.552	−	70.8438i	1.16223	−	0.402522i
$177$	25.8905			0.146274
$178$			373.788i			2.09993i
$179$	45.5484			0.254460			0.127230	−	0.991873i	$-0.459391\pi$
$179$	45.5484			0.254460			0.127230	+	0.991873i	$0.459391\pi$
$180$		0			0
$181$	134.271			0.741830			0.370915	−	0.928667i	$-0.379044\pi$
$181$	134.271			0.741830			0.370915	+	0.928667i	$0.379044\pi$
$182$		−	256.462i		−	1.40913i
$183$			62.5965i			0.342058i
$184$			54.8397i			0.298042i
$185$		0			0
$186$			69.2954i			0.372556i
$187$	44.1359	+	127.436i	0.236021	+	0.681478i
$188$	−27.6034			−0.146827
$189$			32.8620i			0.173873i
$190$		0			0
$191$	−68.0366			−0.356212			−0.178106	−	0.984011i	$-0.556997\pi$
$191$	−68.0366			−0.356212			−0.178106	+	0.984011i	$0.556997\pi$
$192$	−47.7412			−0.248652
$193$			127.465i			0.660443i	0.943904	+	0.330221i	$0.107123\pi$
$193$			127.465i			0.660443i	−0.943904	+	0.330221i	$0.892877\pi$
$194$			134.556i			0.693590i
$195$		0			0
$196$	−12.9081			−0.0658575
$197$			148.516i			0.753888i	0.926236	+	0.376944i	$0.123025\pi$
$197$			148.516i			0.753888i	−0.926236	+	0.376944i	$0.876975\pi$
$198$	−25.1745	−	72.6881i	−0.127144	−	0.367111i
$199$	63.0543			0.316856			0.158428	−	0.987371i	$-0.449357\pi$
$199$	63.0543			0.316856			0.158428	+	0.987371i	$0.449357\pi$
$200$		0			0
$201$	−221.429			−1.10164
$202$	−342.789			−1.69697
$203$	3.74591			0.0184527
$204$			30.4454i			0.149242i
$205$		0			0
$206$			230.258i			1.11776i
$207$	27.5019			0.132859
$208$		−	342.352i		−	1.64592i
$209$	38.1838	+	110.250i	0.182698	+	0.527514i
$210$		0			0
$211$			228.972i			1.08517i	0.840000	+	0.542587i	$0.182555\pi$
$211$			228.972i			1.08517i	−0.840000	+	0.542587i	$0.817445\pi$
$212$	100.997			0.476400
$213$	−48.2272			−0.226419
$214$	−251.642			−1.17590
$215$		0			0
$216$			31.0839i			0.143907i
$217$			108.544i			0.500205i
$218$	−77.1561			−0.353927
$219$			106.286i			0.485325i
$220$		0			0
$221$	213.286			0.965096
$222$		−	223.277i		−	1.00575i
$223$	101.764			0.456339			0.228169	−	0.973621i	$-0.426726\pi$
$223$	101.764			0.456339			0.228169	+	0.973621i	$0.426726\pi$
$224$	138.785			0.619576
$225$		0			0
$226$			380.650i			1.68429i
$227$		−	351.517i		−	1.54853i	−0.632861	−	0.774266i	$-0.718119\pi$
$227$		−	351.517i		−	1.54853i	0.632861	−	0.774266i	$-0.281881\pi$
$228$			26.3396i			0.115525i
$229$	141.322			0.617125			0.308563	−	0.951204i	$-0.400152\pi$
$229$	141.322			0.617125			0.308563	+	0.951204i	$0.400152\pi$
$230$		0			0
$231$	−39.4335	−	113.859i	−0.170708	−	0.492895i
$232$	3.54322			0.0152725
$233$			370.243i			1.58903i	0.607246	+	0.794514i	$0.292274\pi$
$233$			370.243i			1.58903i	−0.607246	+	0.794514i	$0.707726\pi$
$234$	−121.656			−0.519896
$235$		0			0
$236$	−21.4310			−0.0908093
$237$			245.171i			1.03448i
$238$			180.743i			0.759423i
$239$		−	68.3043i		−	0.285792i	−0.989738	−	0.142896i	$-0.954359\pi$
$239$		−	68.3043i		−	0.285792i	0.989738	−	0.142896i	$-0.0456414\pi$
$240$		0			0
$241$			200.283i			0.831052i	0.909581	+	0.415526i	$0.136402\pi$
$241$			200.283i			0.831052i	−0.909581	+	0.415526i	$0.863598\pi$
$242$	174.447	+	221.637i	0.720856	+	0.915857i
$243$	15.5885			0.0641500
$244$		−	51.8145i		−	0.212355i
$245$		0			0
$246$	54.5246			0.221645
$247$	184.523			0.747055
$248$			102.671i			0.413997i
$249$		−	238.683i		−	0.958565i
$250$		0			0
$251$	213.104			0.849019			0.424510	−	0.905423i	$-0.360447\pi$
$251$	213.104			0.849019			0.424510	+	0.905423i	$0.360447\pi$
$252$		−	27.2017i		−	0.107943i
$253$	−95.2872	+	33.0015i	−0.376629	+	0.130441i
$254$	−476.027			−1.87412
$255$		0			0
$256$	244.133			0.953646
$257$	30.3594			0.118130			0.0590650	−	0.998254i	$-0.481188\pi$
$257$	30.3594			0.118130			0.0590650	+	0.998254i	$0.481188\pi$
$258$	−145.356			−0.563394
$259$		−	349.741i		−	1.35035i
$260$		0			0
$261$		−	1.77691i		−	0.00680809i
$262$	567.311			2.16531
$263$			359.742i			1.36784i	0.729557	+	0.683920i	$0.239726\pi$
$263$			359.742i			1.36784i	−0.729557	+	0.683920i	$0.760274\pi$
$264$	−37.2998	−	107.698i	−0.141287	−	0.407947i
$265$		0			0
$266$			156.368i			0.587849i
$267$	277.739			1.04022
$268$	183.289			0.683914
$269$	−320.990			−1.19327			−0.596637	−	0.802512i	$-0.703497\pi$
$269$	−320.990			−1.19327			−0.596637	+	0.802512i	$0.703497\pi$
$270$		0			0
$271$			419.497i			1.54796i	0.633211	+	0.773979i	$0.281736\pi$
$271$			419.497i			1.54796i	−0.633211	+	0.773979i	$0.718264\pi$
$272$			241.274i			0.887036i
$273$	−190.562			−0.698028
$274$		−	159.245i		−	0.581187i
$275$		0			0
$276$	−22.7648			−0.0824811
$277$		−	86.8213i		−	0.313434i	−0.987644	−	0.156717i	$-0.949909\pi$
$277$		−	86.8213i		−	0.313434i	0.987644	−	0.156717i	$-0.0500911\pi$
$278$	50.1574			0.180422
$279$	51.4892			0.184549
$280$		0			0
$281$		−	300.755i		−	1.07030i	−0.844756	−	0.535152i	$-0.820255\pi$
$281$		−	300.755i		−	1.07030i	0.844756	−	0.535152i	$-0.179745\pi$
$282$			77.7339i			0.275652i
$283$		−	319.781i		−	1.12997i	−0.825102	−	0.564984i	$-0.808882\pi$
$283$		−	319.781i		−	1.12997i	0.825102	−	0.564984i	$-0.191118\pi$
$284$	39.9202			0.140564
$285$		0			0
$286$	421.507	−	145.983i	1.47380	−	0.510431i
$287$	85.4074			0.297587
$288$		−	65.8342i		−	0.228591i
$289$	138.686			0.479882
$290$		0			0
$291$	99.9808			0.343577
$292$		−	87.9788i		−	0.301297i
$293$			421.749i			1.43942i	0.694276	+	0.719709i	$0.255725\pi$
$293$			421.749i			1.43942i	−0.694276	+	0.719709i	$0.744275\pi$
$294$			36.3503i			0.123641i
$295$		0			0
$296$		−	330.817i		−	1.11763i
$297$	−54.0102	+	18.7057i	−0.181852	+	0.0629821i
$298$	−107.099			−0.359393
$299$			159.479i			0.533375i
$300$		0			0
$301$	−227.686			−0.756431
$302$	288.074			0.953887
$303$			254.706i			0.840614i
$304$			208.736i			0.686631i
$305$		0			0
$306$	85.7372			0.280187
$307$		−	65.1772i		−	0.212303i	−0.994350	−	0.106152i	$-0.966147\pi$
$307$		−	65.1772i		−	0.212303i	0.994350	−	0.106152i	$-0.0338529\pi$
$308$	32.6412	+	94.2470i	0.105978	+	0.305997i
$309$	171.091			0.553691
$310$		0			0
$311$	−296.311			−0.952768			−0.476384	−	0.879237i	$-0.658053\pi$
$311$	−296.311			−0.952768			−0.476384	+	0.879237i	$0.658053\pi$
$312$	−180.251			−0.577727
$313$	430.760			1.37623			0.688115	−	0.725601i	$-0.258438\pi$
$313$	430.760			1.37623			0.688115	+	0.725601i	$0.258438\pi$
$314$		−	152.460i		−	0.485543i
$315$		0			0
$316$		−	202.941i		−	0.642219i
$317$	248.606			0.784246			0.392123	−	0.919913i	$-0.371741\pi$
$317$	248.606			0.784246			0.392123	+	0.919913i	$0.371741\pi$
$318$		−	284.416i		−	0.894391i
$319$	2.13224	+	6.15656i	0.00668414	+	0.0192996i
$320$		0			0
$321$			186.980i			0.582492i
$322$	−135.146			−0.419707
$323$	−130.043			−0.402610
$324$	−12.9034			−0.0398253
$325$		0			0
$326$			742.079i			2.27632i
$327$			57.3301i			0.175321i
$328$	80.7862			0.246299
$329$			121.763i			0.370099i
$330$		0			0
$331$	−379.595			−1.14681			−0.573406	−	0.819271i	$-0.694378\pi$
$331$	−379.595			−1.14681			−0.573406	+	0.819271i	$0.694378\pi$
$332$			197.571i			0.595092i
$333$	−165.904			−0.498209
$334$	−40.2051			−0.120375
$335$		0			0
$336$		−	215.567i		−	0.641570i
$337$		−	528.010i		−	1.56680i	−0.621521	−	0.783398i	$-0.713485\pi$
$337$		−	528.010i		−	1.56680i	0.621521	−	0.783398i	$-0.286515\pi$
$338$		−	311.518i		−	0.921651i
$339$	282.838			0.834332
$340$		0			0
$341$	−178.398	+	61.7856i	−0.523160	+	0.181189i
$342$	74.1748			0.216885
$343$			366.830i			1.06947i
$344$	−215.366			−0.626064
$345$		0			0
$346$	750.832			2.17004
$347$			35.7638i			0.103066i	0.998671	+	0.0515328i	$0.0164107\pi$
$347$			35.7638i			0.103066i	−0.998671	+	0.0515328i	$0.983589\pi$
$348$			1.47084i			0.00422657i
$349$			453.183i			1.29852i	0.760568	+	0.649259i	$0.224921\pi$
$349$			453.183i			1.29852i	−0.760568	+	0.649259i	$0.775079\pi$
$350$		0			0
$351$			90.3951i			0.257536i
$352$	78.9991	+	228.099i	0.224429	+	0.648009i
$353$	76.8798			0.217790			0.108895	−	0.994053i	$-0.465269\pi$
$353$	76.8798			0.217790			0.108895	+	0.994053i	$0.465269\pi$
$354$			60.3517i			0.170485i
$355$		0			0
$356$	−229.900			−0.645786
$357$	134.299			0.376188
$358$			106.175i			0.296577i
$359$			406.327i			1.13183i	0.824463	+	0.565915i	$0.191477\pi$
$359$			406.327i			1.13183i	−0.824463	+	0.565915i	$0.808523\pi$
$360$		0			0
$361$	248.494			0.688350
$362$			312.991i			0.864615i
$363$	164.686	−	129.621i	0.453679	−	0.357083i
$364$	157.738			0.433347
$365$		0			0
$366$	−145.915			−0.398674
$367$	287.030			0.782098			0.391049	−	0.920370i	$-0.372112\pi$
$367$	287.030			0.782098			0.391049	+	0.920370i	$0.372112\pi$
$368$	−180.406			−0.490234
$369$		−	40.5139i		−	0.109794i
$370$		0			0
$371$		−	445.510i		−	1.20084i
$372$	−42.6204			−0.114571
$373$		−	359.705i		−	0.964356i	−0.876073	−	0.482178i	$-0.839846\pi$
$373$		−	359.705i		−	0.964356i	0.876073	−	0.482178i	$-0.160154\pi$
$374$	−297.058	+	102.882i	−0.794274	+	0.275086i
$375$		0			0
$376$			115.174i			0.306314i
$377$	10.3040			0.0273316
$378$	−76.6024			−0.202652
$379$	94.3980			0.249071			0.124536	−	0.992215i	$-0.460256\pi$
$379$	94.3980			0.249071			0.124536	+	0.992215i	$0.460256\pi$
$380$		0			0
$381$			353.707i			0.928366i
$382$		−	158.595i		−	0.415171i
$383$	92.3643			0.241160			0.120580	−	0.992704i	$-0.461525\pi$
$383$	92.3643			0.241160			0.120580	+	0.992704i	$0.461525\pi$
$384$		−	263.324i		−	0.685739i
$385$		0			0
$386$	−297.126			−0.769757
$387$			108.005i			0.279083i
$388$	−82.7595			−0.213298
$389$	126.237			0.324517			0.162259	−	0.986748i	$-0.448122\pi$
$389$	126.237			0.324517			0.162259	+	0.986748i	$0.448122\pi$
$390$		0			0
$391$		−	112.393i		−	0.287451i
$392$			53.8583i			0.137394i
$393$		−	421.535i		−	1.07261i
$394$	−346.195			−0.878668
$395$		0			0
$396$	44.7071	−	15.4837i	0.112897	−	0.0391003i
$397$	−732.111			−1.84411			−0.922054	−	0.387060i	$-0.873491\pi$
$397$	−732.111			−1.84411			−0.922054	+	0.387060i	$0.873491\pi$
$398$			146.982i			0.369300i
$399$	116.188			0.291197
$400$		0			0
$401$	288.306			0.718966			0.359483	−	0.933152i	$-0.382953\pi$
$401$	288.306			0.718966			0.359483	+	0.933152i	$0.382953\pi$
$402$		−	516.158i		−	1.28398i
$403$			298.578i			0.740889i
$404$		−	210.834i		−	0.521866i
$405$		0			0
$406$			8.73183i			0.0215070i
$407$	574.815	−	199.080i	1.41232	−	0.489139i
$408$	127.032			0.311354
$409$			536.385i			1.31146i	0.754997	+	0.655728i	$0.227638\pi$
$409$			536.385i			1.31146i	−0.754997	+	0.655728i	$0.772362\pi$
$410$		0			0
$411$	−118.326			−0.287897
$412$	−141.621			−0.343740
$413$			94.5350i			0.228898i
$414$			64.1078i			0.154850i
$415$		0			0
$416$	381.762			0.917697
$417$		−	37.2689i		−	0.0893739i
$418$	−256.997	+	89.0076i	−0.614826	+	0.212937i
$419$	772.235			1.84304			0.921521	−	0.388328i	$-0.126947\pi$
$419$	772.235			1.84304			0.921521	+	0.388328i	$0.126947\pi$
$420$		0			0
$421$	530.329			1.25969			0.629844	−	0.776721i	$-0.283119\pi$
$421$	530.329			1.25969			0.629844	+	0.776721i	$0.283119\pi$
$422$	−533.740			−1.26479
$423$	57.7594			0.136547
$424$		−	421.405i		−	0.993879i
$425$		0			0
$426$		−	112.419i		−	0.263895i
$427$	−228.561			−0.535272
$428$		−	154.773i		−	0.361620i
$429$	−108.471	−	313.197i	−0.252847	−	0.730062i
$430$		0			0
$431$		−	98.0885i		−	0.227584i	−0.993505	−	0.113792i	$-0.963700\pi$
$431$		−	98.0885i		−	0.227584i	0.993505	−	0.113792i	$-0.0362997\pi$
$432$	−102.257			−0.236705
$433$	467.819			1.08041			0.540206	−	0.841533i	$-0.318346\pi$
$433$	467.819			1.08041			0.540206	+	0.841533i	$0.318346\pi$
$434$	−253.021			−0.582997
$435$		0			0
$436$		−	47.4552i		−	0.108842i
$437$		−	97.2362i		−	0.222509i
$438$	−247.756			−0.565654
$439$		−	732.331i		−	1.66818i	−0.551628	−	0.834090i	$-0.685993\pi$
$439$		−	732.331i		−	1.66818i	0.551628	−	0.834090i	$-0.314007\pi$
$440$		0			0
$441$	27.0097			0.0612466
$442$			497.177i			1.12483i
$443$	−522.885			−1.18033			−0.590164	−	0.807284i	$-0.700937\pi$
$443$	−522.885			−1.18033			−0.590164	+	0.807284i	$0.700937\pi$
$444$	137.327			0.309296
$445$		0			0
$446$			237.214i			0.531870i
$447$			79.5790i			0.178029i
$448$		−	174.319i		−	0.389105i
$449$	119.788			0.266788			0.133394	−	0.991063i	$-0.457412\pi$
$449$	119.788			0.266788			0.133394	+	0.991063i	$0.457412\pi$
$450$		0			0
$451$	48.6156	+	140.371i	0.107795	+	0.311243i
$452$	−234.121			−0.517966
$453$		−	214.051i		−	0.472518i
$454$	819.397			1.80484
$455$		0			0
$456$	109.901			0.241011
$457$		−	4.81549i		−	0.0105372i	−0.999986	−	0.00526859i	$-0.998323\pi$
$457$		−	4.81549i		−	0.0105372i	0.999986	−	0.00526859i	$-0.00167705\pi$
$458$			329.425i			0.719269i
$459$		−	63.7062i		−	0.138793i
$460$		0			0
$461$		−	570.680i		−	1.23792i	−0.785424	−	0.618958i	$-0.787555\pi$
$461$		−	570.680i		−	1.23792i	0.785424	−	0.618958i	$-0.212445\pi$
$462$	265.409	−	91.9207i	0.574477	−	0.198963i
$463$	−131.935			−0.284956			−0.142478	−	0.989798i	$-0.545507\pi$
$463$	−131.935			−0.284956			−0.142478	+	0.989798i	$0.545507\pi$
$464$			11.6561i			0.0251210i
$465$		0			0
$466$	−863.050			−1.85204
$467$	89.8728			0.192447			0.0962235	−	0.995360i	$-0.469324\pi$
$467$	89.8728			0.192447			0.0962235	+	0.995360i	$0.469324\pi$
$468$		−	74.8249i		−	0.159882i
$469$		−	808.512i		−	1.72391i
$470$		0			0
$471$	−113.284			−0.240518
$472$			89.4199i			0.189449i
$473$	−129.603	−	374.211i	−0.274002	−	0.791144i
$474$	−571.502			−1.20570
$475$		0			0
$476$	−111.167			−0.233543
$477$	−211.333			−0.443045
$478$	159.219			0.333095
$479$			717.047i			1.49697i	0.663153	+	0.748484i	$0.269218\pi$
$479$			717.047i			1.49697i	−0.663153	+	0.748484i	$0.730782\pi$
$480$		0			0
$481$		−	962.049i		−	2.00010i
$482$	−466.867			−0.968604
$483$			100.419i			0.207906i
$484$	−136.319	+	107.294i	−0.281651	+	0.221683i
$485$		0			0
$486$			36.3372i			0.0747679i
$487$	−916.798			−1.88254			−0.941271	−	0.337653i	$-0.890367\pi$
$487$	−916.798			−1.88254			−0.941271	+	0.337653i	$0.890367\pi$
$488$	−216.194			−0.443020
$489$	551.395			1.12760
$490$		0			0
$491$		−	302.142i		−	0.615361i	−0.951490	−	0.307680i	$-0.900447\pi$
$491$		−	302.142i		−	0.615361i	0.951490	−	0.307680i	$-0.0995527\pi$
$492$			33.5356i			0.0681617i
$493$	−7.26180			−0.0147298
$494$			430.128i			0.870705i
$495$		0			0
$496$	−337.758			−0.680963
$497$		−	176.094i		−	0.354313i
$498$	556.377			1.11722
$499$	−102.798			−0.206008			−0.103004	−	0.994681i	$-0.532846\pi$
$499$	−102.798			−0.206008			−0.103004	+	0.994681i	$0.532846\pi$
$500$		0			0
$501$			29.8740i			0.0596287i
$502$			496.752i			0.989545i
$503$		−	567.728i		−	1.12868i	−0.825541	−	0.564342i	$-0.809130\pi$
$503$		−	567.728i		−	1.12868i	0.825541	−	0.564342i	$-0.190870\pi$
$504$	−113.498			−0.225194
$505$		0			0
$506$	−76.9275	−	222.118i	−0.152031	−	0.438968i
$507$	−231.470			−0.456549
$508$		−	292.783i		−	0.576344i
$509$	−423.197			−0.831428			−0.415714	−	0.909495i	$-0.636468\pi$
$509$	−423.197			−0.831428			−0.415714	+	0.909495i	$0.636468\pi$
$510$		0			0
$511$	−388.086			−0.759465
$512$		−	39.0375i		−	0.0762451i
$513$		−	55.1149i		−	0.107436i
$514$			70.7687i			0.137682i
$515$		0			0
$516$		−	89.4017i		−	0.173259i
$517$	−200.122	+	69.3096i	−0.387083	+	0.134061i
$518$	815.258			1.57386
$519$		−	557.898i		−	1.07495i
$520$		0			0
$521$	−658.340			−1.26361			−0.631805	−	0.775128i	$-0.717685\pi$
$521$	−658.340			−1.26361			−0.631805	+	0.775128i	$0.717685\pi$
$522$	4.14204			0.00793494
$523$		−	568.723i		−	1.08743i	−0.839272	−	0.543713i	$-0.817018\pi$
$523$		−	568.723i		−	1.08743i	0.839272	−	0.543713i	$-0.182982\pi$
$524$			348.927i			0.665892i
$525$		0			0
$526$	−838.570			−1.59424
$527$		−	210.424i		−	0.399286i
$528$	354.294	−	122.705i	0.671012	−	0.232396i
$529$	−444.961			−0.841136
$530$		0			0
$531$	44.8437			0.0844515
$532$	−96.1748			−0.180780
$533$	234.934			0.440777
$534$			647.419i			1.21240i
$535$		0			0
$536$		−	764.765i		−	1.42680i
$537$	78.8921			0.146913
$538$		−	748.239i		−	1.39078i
$539$	−93.5821	+	32.4109i	−0.173622	+	0.0601316i
$540$		0			0
$541$			451.918i			0.835339i	0.908599	+	0.417669i	$0.137153\pi$
$541$			451.918i			0.835339i	−0.908599	+	0.417669i	$0.862847\pi$
$542$	−977.860			−1.80417
$543$	232.565			0.428296
$544$	−269.048			−0.494574
$545$		0			0
$546$		−	444.206i		−	0.813563i
$547$		−	149.931i		−	0.274097i	−0.990564	−	0.137048i	$-0.956238\pi$
$547$		−	149.931i		−	0.274097i	0.990564	−	0.137048i	$-0.0437615\pi$
$548$	97.9446			0.178731
$549$			108.420i			0.197487i
$550$		0			0
$551$	−6.28248			−0.0114020
$552$			94.9851i			0.172074i
$553$	−895.202			−1.61881
$554$	202.383			0.365313
$555$		0			0
$556$			30.8495i			0.0554847i
$557$			14.5869i			0.0261883i	0.999914	+	0.0130941i	$0.00416812\pi$
$557$			14.5869i			0.0261883i	−0.999914	+	0.0130941i	$0.995832\pi$
$558$			120.023i			0.215095i
$559$	−626.305			−1.12040
$560$		0			0
$561$	76.4456	+	220.726i	0.136267	+	0.393451i
$562$	701.070			1.24746
$563$			123.461i			0.219291i	0.993971	+	0.109645i	$0.0349715\pi$
$563$			123.461i			0.219291i	−0.993971	+	0.109645i	$0.965028\pi$
$564$	−47.8106			−0.0847705
$565$		0			0
$566$	745.420			1.31700
$567$			56.9187i			0.100386i
$568$		−	166.565i		−	0.293249i
$569$		−	849.647i		−	1.49323i	−0.665257	−	0.746614i	$-0.731678\pi$
$569$		−	849.647i		−	1.49323i	0.665257	−	0.746614i	$-0.268322\pi$
$570$		0			0
$571$		−	607.460i		−	1.06385i	−0.846791	−	0.531926i	$-0.821468\pi$
$571$		−	607.460i		−	1.06385i	0.846791	−	0.531926i	$-0.178532\pi$
$572$	89.7876	+	259.250i	0.156971	+	0.453234i
$573$	−117.843			−0.205659
$574$			199.087i			0.346842i
$575$		0			0
$576$	−82.6902			−0.143559
$577$	345.393			0.598602			0.299301	−	0.954159i	$-0.403246\pi$
$577$	345.393			0.598602			0.299301	+	0.954159i	$0.403246\pi$
$578$			323.281i			0.559310i
$579$			220.777i			0.381307i
$580$		0			0
$581$	871.511			1.50002
$582$			233.059i			0.400444i
$583$	732.216	−	253.593i	1.25594	−	0.434980i
$584$	−367.088			−0.628575
$585$		0			0
$586$	−983.111			−1.67766
$587$	−800.135			−1.36309			−0.681546	−	0.731775i	$-0.738692\pi$
$587$	−800.135			−1.36309			−0.681546	+	0.731775i	$0.738692\pi$
$588$	−22.3574			−0.0380228
$589$		−	182.046i		−	0.309077i
$590$		0			0
$591$			257.237i			0.435257i
$592$	1088.29			1.83833
$593$		−	157.850i		−	0.266189i	−0.991103	−	0.133094i	$-0.957509\pi$
$593$		−	157.850i		−	0.266189i	0.991103	−	0.133094i	$-0.0424913\pi$
$594$	−43.6036	−	125.899i	−0.0734067	−	0.211952i
$595$		0			0
$596$		−	65.8718i		−	0.110523i
$597$	109.213			0.182937
$598$	−371.751			−0.621657
$599$	623.613			1.04109			0.520545	−	0.853834i	$-0.325729\pi$
$599$	623.613			1.04109			0.520545	+	0.853834i	$0.325729\pi$
$600$		0			0
$601$			140.001i			0.232947i	0.993194	+	0.116474i	$0.0371591\pi$
$601$			140.001i			0.232947i	−0.993194	+	0.116474i	$0.962841\pi$
$602$		−	530.743i		−	0.881632i
$603$	−383.527			−0.636031
$604$			177.181i			0.293346i
$605$		0			0
$606$	−593.728			−0.979749
$607$		−	758.571i		−	1.24971i	−0.780743	−	0.624853i	$-0.785159\pi$
$607$		−	758.571i		−	1.24971i	0.780743	−	0.624853i	$-0.214841\pi$
$608$	−232.765			−0.382837
$609$	6.48810			0.0106537
$610$		0			0
$611$			334.938i			0.548180i
$612$			52.7331i			0.0861651i
$613$			487.490i			0.795253i	0.917547	+	0.397627i	$0.130166\pi$
$613$			487.490i			0.795253i	−0.917547	+	0.397627i	$0.869834\pi$
$614$	151.930			0.247443
$615$		0			0
$616$	393.242	−	136.194i	0.638380	−	0.221094i
$617$	773.785			1.25411			0.627054	−	0.778975i	$-0.284260\pi$
$617$	773.785			1.25411			0.627054	+	0.778975i	$0.284260\pi$
$618$			398.818i			0.645336i
$619$	1182.12			1.90972			0.954859	−	0.297059i	$-0.0960058\pi$
$619$	1182.12			1.90972			0.954859	+	0.297059i	$0.0960058\pi$
$620$		0			0
$621$	47.6347			0.0767064
$622$		−	690.710i		−	1.11047i
$623$			1014.12i			1.62780i
$624$		−	592.971i		−	0.950274i
$625$		0			0
$626$			1004.12i			1.60402i
$627$	66.1363	+	190.959i	0.105480	+	0.304561i
$628$	93.7714			0.149318
$629$			678.007i			1.07791i
$630$		0			0
$631$	−1089.91			−1.72727			−0.863634	−	0.504120i	$-0.831817\pi$
$631$	−1089.91			−1.72727			−0.863634	+	0.504120i	$0.831817\pi$
$632$	−846.764			−1.33982
$633$			396.590i			0.626525i
$634$			579.509i			0.914051i
$635$		0			0
$636$	174.931			0.275049
$637$			156.625i			0.245880i
$638$	−14.3511	+	4.97032i	−0.0224939	+	0.00779048i
$639$	−83.5319			−0.130723
$640$		0			0
$641$	−661.586			−1.03212			−0.516058	−	0.856554i	$-0.672601\pi$
$641$	−661.586			−1.03212			−0.516058	+	0.856554i	$0.672601\pi$
$642$	−435.856			−0.678904
$643$	−700.467			−1.08937			−0.544687	−	0.838639i	$-0.683352\pi$
$643$	−700.467			−1.08937			−0.544687	+	0.838639i	$0.683352\pi$
$644$		−	83.1219i		−	0.129071i
$645$		0			0
$646$		−	303.134i		−	0.469248i
$647$	−128.968			−0.199332			−0.0996658	−	0.995021i	$-0.531777\pi$
$647$	−128.968			−0.199332			−0.0996658	+	0.995021i	$0.531777\pi$
$648$			53.8389i			0.0830847i
$649$	−155.372	+	53.8112i	−0.239403	+	0.0829140i
$650$		0			0
$651$			188.005i			0.288793i
$652$	−456.419			−0.700029
$653$	828.659			1.26900			0.634502	−	0.772921i	$-0.281205\pi$
$653$	828.659			1.26900			0.634502	+	0.772921i	$0.281205\pi$
$654$	−133.638			−0.204340
$655$		0			0
$656$			265.762i			0.405125i
$657$			184.093i			0.280202i
$658$	−283.832			−0.431356
$659$		−	216.327i		−	0.328265i	−0.986438	−	0.164133i	$-0.947517\pi$
$659$		−	216.327i		−	0.328265i	0.986438	−	0.164133i	$-0.0524825\pi$
$660$		0			0
$661$	114.149			0.172691			0.0863457	−	0.996265i	$-0.472481\pi$
$661$	114.149			0.172691			0.0863457	+	0.996265i	$0.472481\pi$
$662$		−	884.848i		−	1.33663i
$663$	369.422			0.557198
$664$	824.355			1.24150
$665$		0			0
$666$		−	386.727i		−	0.580671i
$667$		−	5.42982i		−	0.00814066i
$668$		−	24.7283i		−	0.0370184i
$669$	176.260			0.263467
$670$		0			0
$671$	−130.101	−	375.650i	−0.193892	−	0.559836i
$672$	240.383			0.357712
$673$			1173.35i			1.74346i	0.489989	+	0.871729i	$0.337001\pi$
$673$			1173.35i			1.74346i	−0.489989	+	0.871729i	$0.662999\pi$
$674$	1230.81			1.82613
$675$		0			0
$676$	191.600			0.283433
$677$			145.079i			0.214297i	0.994243	+	0.107148i	$0.0341720\pi$
$677$			145.079i			0.214297i	−0.994243	+	0.107148i	$0.965828\pi$
$678$			659.305i			0.972427i
$679$			365.064i			0.537649i
$680$		0			0
$681$		−	608.845i		−	0.894045i
$682$	−144.024	−	415.850i	−0.211179	−	0.609751i
$683$	−542.580			−0.794407			−0.397204	−	0.917731i	$-0.630019\pi$
$683$	−542.580			−0.794407			−0.397204	+	0.917731i	$0.630019\pi$
$684$			45.6216i			0.0666982i
$685$		0			0
$686$	−855.092			−1.24649
$687$	244.776			0.356297
$688$		−	708.489i		−	1.02978i
$689$		−	1225.49i		−	1.77864i
$690$		0			0
$691$	−190.564			−0.275780			−0.137890	−	0.990448i	$-0.544032\pi$
$691$	−190.564			−0.275780			−0.137890	+	0.990448i	$0.544032\pi$
$692$			461.803i			0.667345i
$693$	−68.3008	−	197.209i	−0.0985581	−	0.284573i
$694$	−83.3665			−0.120125
$695$		0			0
$696$	6.13704			0.00881758
$697$	−165.571			−0.237547
$698$	−1056.38			−1.51344
$699$			641.281i			0.917426i
$700$		0			0
$701$			149.736i			0.213603i	0.994280	+	0.106801i	$0.0340609\pi$
$701$			149.736i			0.213603i	−0.994280	+	0.106801i	$0.965939\pi$
$702$	−210.714			−0.300162
$703$			586.572i			0.834385i
$704$	286.501	−	99.2259i	0.406962	−	0.140946i
$705$		0			0
$706$			179.209i			0.253838i
$707$	−930.017			−1.31544
$708$	−37.1196			−0.0524288
$709$	−225.902			−0.318621			−0.159310	−	0.987229i	$-0.550927\pi$
$709$	−225.902			−0.318621			−0.159310	+	0.987229i	$0.550927\pi$
$710$		0			0
$711$			424.649i			0.597256i
$712$			959.247i			1.34726i
$713$	157.339			0.220672
$714$			313.055i			0.438453i
$715$		0			0
$716$	−65.3032			−0.0912056
$717$		−	118.306i		−	0.165002i
$718$	−947.162			−1.31917
$719$	508.963			0.707877			0.353938	−	0.935269i	$-0.384842\pi$
$719$	508.963			0.707877			0.353938	+	0.935269i	$0.384842\pi$
$720$		0			0
$721$			624.709i			0.866448i
$722$			579.249i			0.802283i
$723$			346.901i			0.479808i
$724$	−192.506			−0.265893
$725$		0			0
$726$	302.151	+	383.887i	0.416186	+	0.528770i
$727$	155.888			0.214426			0.107213	−	0.994236i	$-0.465807\pi$
$727$	155.888			0.214426			0.107213	+	0.994236i	$0.465807\pi$
$728$		−	658.156i		−	0.904061i
$729$	27.0000			0.0370370
$730$		0			0
$731$	441.391			0.603818
$732$		−	89.7454i		−	0.122603i
$733$		−	393.332i		−	0.536605i	−0.963335	−	0.268303i	$-0.913537\pi$
$733$		−	393.332i		−	0.536605i	0.963335	−	0.268303i	$-0.0864627\pi$
$734$			669.076i			0.911548i
$735$		0			0
$736$		−	201.174i		−	0.273334i
$737$	1328.82	−	460.221i	1.80302	−	0.624451i
$738$	94.4393			0.127967
$739$			224.197i			0.303379i	0.988428	+	0.151689i	$0.0484713\pi$
$739$			224.197i			0.303379i	−0.988428	+	0.151689i	$0.951529\pi$
$740$		0			0
$741$	319.603			0.431313
$742$	1038.50			1.39959
$743$			115.127i			0.154948i	0.996994	+	0.0774741i	$0.0246855\pi$
$743$			115.127i			0.154948i	−0.996994	+	0.0774741i	$0.975314\pi$
$744$			177.832i			0.239021i
$745$		0			0
$746$	838.484			1.12397
$747$		−	413.411i		−	0.553428i
$748$	−63.2781	−	182.707i	−0.0845964	−	0.244261i
$749$	−682.727			−0.911518
$750$		0			0
$751$	−494.453			−0.658392			−0.329196	−	0.944262i	$-0.606778\pi$
$751$	−494.453			−0.658392			−0.329196	+	0.944262i	$0.606778\pi$
$752$	−378.888			−0.503841
$753$	369.107			0.490181
$754$			24.0190i			0.0318555i
$755$		0			0
$756$		−	47.1147i		−	0.0623210i
$757$	−531.890			−0.702629			−0.351314	−	0.936258i	$-0.614265\pi$
$757$	−531.890			−0.702629			−0.351314	+	0.936258i	$0.614265\pi$
$758$			220.045i			0.290297i
$759$	−165.042	+	57.1602i	−0.217447	+	0.0753099i
$760$		0			0
$761$			1343.41i			1.76532i	0.470013	+	0.882660i	$0.344249\pi$
$761$			1343.41i			1.76532i	−0.470013	+	0.882660i	$0.655751\pi$
$762$	−824.503			−1.08203
$763$	−209.331			−0.274353
$764$	97.5448			0.127676
$765$		0			0
$766$			215.304i			0.281076i
$767$			260.042i			0.339038i
$768$	422.851			0.550588
$769$		−	735.802i		−	0.956830i	−0.878134	−	0.478415i	$-0.841212\pi$
$769$		−	735.802i		−	0.956830i	0.878134	−	0.478415i	$-0.158788\pi$
$770$		0			0
$771$	52.5840			0.0682023
$772$		−	182.749i		−	0.236721i
$773$	1135.40			1.46882			0.734410	−	0.678706i	$-0.237459\pi$
$773$	1135.40			1.46882			0.734410	+	0.678706i	$0.237459\pi$
$774$	−251.764			−0.325276
$775$		0			0
$776$			345.311i			0.444988i
$777$		−	605.770i		−	0.779626i
$778$			294.263i			0.378230i
$779$	−143.242			−0.183879
$780$		0			0
$781$	289.417	−	100.236i	0.370573	−	0.128343i
$782$	261.993			0.335029
$783$		−	3.07770i		−	0.00393065i
$784$	−177.178			−0.225992
$785$		0			0
$786$	982.612			1.25014
$787$		−	115.155i		−	0.146321i	−0.997320	−	0.0731605i	$-0.976691\pi$
$787$		−	115.155i		−	0.146321i	0.997320	−	0.0731605i	$-0.0233085\pi$
$788$		−	212.929i		−	0.270214i
$789$			623.091i			0.789723i
$790$		0			0
$791$			1032.74i			1.30561i
$792$	−64.6051	−	186.539i	−0.0815721	−	0.235528i
$793$	−628.713			−0.792828
$794$		−	1706.57i		−	2.14934i
$795$		0			0
$796$	−90.4017			−0.113570
$797$	1486.27			1.86483			0.932416	−	0.361387i	$-0.117697\pi$
$797$	1486.27			1.86483			0.932416	+	0.361387i	$0.117697\pi$
$798$			270.837i			0.339395i
$799$		−	236.049i		−	0.295430i
$800$		0			0
$801$	481.058			0.600572
$802$			672.049i			0.837967i
$803$	−220.906	−	637.837i	−0.275101	−	0.794317i
$804$	317.466			0.394858
$805$		0			0
$806$	−695.995			−0.863518
$807$	−555.972			−0.688937
$808$	−879.695			−1.08873
$809$			546.514i			0.675543i	0.941228	+	0.337771i	$0.109673\pi$
$809$			546.514i			0.675543i	−0.941228	+	0.337771i	$0.890327\pi$
$810$		0			0
$811$			1066.45i			1.31499i	0.753460	+	0.657494i	$0.228383\pi$
$811$			1066.45i			1.31499i	−0.753460	+	0.657494i	$0.771617\pi$
$812$	−5.37055			−0.00661398
$813$			726.589i			0.893714i
$814$	464.061	+	1339.91i	0.570099	+	1.64608i
$815$		0			0
$816$			417.898i			0.512130i
$817$	381.865			0.467399
$818$	−1250.33			−1.52852
$819$	−330.063			−0.403007
$820$		0			0
$821$		−	1206.66i		−	1.46974i	−0.678207	−	0.734871i	$-0.737243\pi$
$821$		−	1206.66i		−	1.46974i	0.678207	−	0.734871i	$-0.262757\pi$
$822$		−	275.821i		−	0.335549i
$823$	−312.518			−0.379730			−0.189865	−	0.981810i	$-0.560805\pi$
$823$	−312.518			−0.379730			−0.189865	+	0.981810i	$0.560805\pi$
$824$			590.907i			0.717120i
$825$		0			0
$826$	−220.364			−0.266785
$827$		−	1166.37i		−	1.41036i	−0.709029	−	0.705180i	$-0.750866\pi$
$827$		−	1166.37i		−	1.41036i	0.709029	−	0.705180i	$-0.249134\pi$
$828$	−39.4298			−0.0476205
$829$	−205.935			−0.248413			−0.124207	−	0.992256i	$-0.539639\pi$
$829$	−205.935			−0.248413			−0.124207	+	0.992256i	$0.539639\pi$
$830$		0			0
$831$		−	150.379i		−	0.180961i
$832$		−	479.508i		−	0.576332i
$833$		−	110.382i		−	0.132512i
$834$	86.8751			0.104167
$835$		0			0
$836$	−54.7445	−	158.067i	−0.0654839	−	0.189076i
$837$	89.1820			0.106550
$838$			1800.10i			2.14810i
$839$	−619.889			−0.738843			−0.369422	−	0.929262i	$-0.620444\pi$
$839$	−619.889			−0.738843			−0.369422	+	0.929262i	$0.620444\pi$
$840$		0			0
$841$	840.649			0.999583
$842$			1236.21i			1.46819i
$843$		−	520.923i		−	0.617940i
$844$		−	328.279i		−	0.388956i
$845$		0			0
$846$			134.639i			0.159148i
$847$	473.291	+	601.322i	0.558785	+	0.709944i
$848$	1386.29			1.63478
$849$		−	553.877i		−	0.652387i
$850$		0			0
$851$	−506.962			−0.595725
$852$	69.1439			0.0811548
$853$		−	1067.19i		−	1.25110i	−0.780185	−	0.625549i	$-0.784875\pi$
$853$		−	1067.19i		−	1.25110i	0.780185	−	0.625549i	$-0.215125\pi$
$854$		−	532.783i		−	0.623868i
$855$		0			0
$856$	−645.786			−0.754422
$857$		−	135.288i		−	0.157862i	−0.996880	−	0.0789310i	$-0.974849\pi$
$857$		−	135.288i		−	0.157862i	0.996880	−	0.0789310i	$-0.0251507\pi$
$858$	730.071	−	252.850i	0.850899	−	0.294697i
$859$	−88.1149			−0.102578			−0.0512892	−	0.998684i	$-0.516333\pi$
$859$	−88.1149			−0.102578			−0.0512892	+	0.998684i	$0.516333\pi$
$860$		0			0
$861$	147.930			0.171812
$862$	228.648			0.265252
$863$	1506.50			1.74566			0.872830	−	0.488024i	$-0.162282\pi$
$863$	1506.50			1.74566			0.872830	+	0.488024i	$0.162282\pi$
$864$		−	114.028i		−	0.131977i
$865$		0			0
$866$			1090.50i			1.25924i
$867$	240.211			0.277060
$868$		−	155.621i		−	0.179287i
$869$	−509.566	−	1471.30i	−0.586382	−	1.69310i
$870$		0			0
$871$		−	2224.01i		−	2.55340i
$872$	−198.005			−0.227070
$873$	173.172			0.198364
$874$	226.661			0.259337
$875$		0			0
$876$		−	152.384i		−	0.173954i
$877$			874.405i			0.997041i	0.866878	+	0.498521i	$0.166123\pi$
$877$			874.405i			0.997041i	−0.866878	+	0.498521i	$0.833877\pi$
$878$	1707.09			1.94429
$879$			730.491i			0.831048i
$880$		0			0
$881$	628.003			0.712830			0.356415	−	0.934328i	$-0.383999\pi$
$881$	628.003			0.712830			0.356415	+	0.934328i	$0.383999\pi$
$882$			62.9606i			0.0713839i
$883$	−1234.85			−1.39848			−0.699238	−	0.714889i	$-0.746477\pi$
$883$	−1234.85			−1.39848			−0.699238	+	0.714889i	$0.746477\pi$
$884$	−305.791			−0.345917
$885$		0			0
$886$		−	1218.86i		−	1.37569i
$887$			282.840i			0.318872i	0.987208	+	0.159436i	$0.0509676\pi$
$887$			282.840i			0.318872i	−0.987208	+	0.159436i	$0.949032\pi$
$888$		−	572.992i		−	0.645262i
$889$	−1291.50			−1.45276
$890$		0			0
$891$	−93.5483	+	32.3992i	−0.104993	+	0.0363628i
$892$	−145.900			−0.163565
$893$		−	204.215i		−	0.228685i
$894$	−185.501			−0.207496
$895$		0			0
$896$	961.484			1.07308
$897$			276.226i			0.307944i
$898$			279.229i			0.310946i
$899$		−	10.1658i		−	0.0113078i
$900$		0			0
$901$			863.665i			0.958563i
$902$	−327.209	+	113.324i	−0.362759	+	0.125637i
$903$	−394.363			−0.436725
$904$			976.858i			1.08060i
$905$		0			0
$906$	498.959			0.550727
$907$	−604.942			−0.666970			−0.333485	−	0.942755i	$-0.608225\pi$
$907$	−604.942			−0.666970			−0.333485	+	0.942755i	$0.608225\pi$
$908$			503.974i			0.555037i
$909$			441.164i			0.485329i
$910$		0			0
$911$	339.992			0.373208			0.186604	−	0.982435i	$-0.440252\pi$
$911$	339.992			0.373208			0.186604	+	0.982435i	$0.440252\pi$
$912$			361.541i			0.396427i
$913$	496.081	+	1432.37i	0.543352	+	1.56886i
$914$	11.2251			0.0122813
$915$		0			0
$916$	−202.615			−0.221195
$917$	1539.17			1.67848
$918$	148.501			0.161766
$919$		−	885.348i		−	0.963382i	−0.876341	−	0.481691i	$-0.840023\pi$
$919$		−	885.348i		−	0.963382i	0.876341	−	0.481691i	$-0.159977\pi$
$920$		0			0
$921$		−	112.890i		−	0.122573i
$922$	1330.27			1.44281
$923$		−	484.389i		−	0.524798i
$924$	56.5362	+	163.241i	0.0611864	+	0.176667i
$925$		0			0
$926$		−	307.544i		−	0.332121i
$927$	296.338			0.319674
$928$	−12.9979			−0.0140064
$929$	−821.063			−0.883814			−0.441907	−	0.897061i	$-0.645698\pi$
$929$	−821.063			−0.883814			−0.441907	+	0.897061i	$0.645698\pi$
$930$		0			0
$931$		−	95.4962i		−	0.102574i
$932$		−	530.822i		−	0.569552i
$933$	−513.225			−0.550081
$934$			209.496i			0.224300i
$935$		0			0
$936$	−312.203			−0.333551
$937$			1507.89i			1.60927i	0.593768	+	0.804636i	$0.297639\pi$
$937$			1507.89i			1.60927i	−0.593768	+	0.804636i	$0.702361\pi$
$938$	1884.67			2.00924
$939$	746.099			0.794567
$940$		0			0
$941$			138.310i			0.146982i	0.997296	+	0.0734909i	$0.0234140\pi$
$941$			138.310i			0.146982i	−0.997296	+	0.0734909i	$0.976586\pi$
$942$		−	264.069i		−	0.280328i
$943$		−	123.801i		−	0.131284i
$944$	−294.165			−0.311615
$945$		0			0
$946$	872.298	−	302.109i	0.922091	−	0.319354i
$947$	−1559.87			−1.64717			−0.823586	−	0.567192i	$-0.808030\pi$
$947$	−1559.87			−1.64717			−0.823586	+	0.567192i	$0.808030\pi$
$948$		−	351.505i		−	0.370786i
$949$	−1067.53			−1.12490
$950$		0			0
$951$	430.598			0.452785
$952$			463.838i			0.487224i
$953$		−	1073.64i		−	1.12659i	−0.826255	−	0.563296i	$-0.809533\pi$
$953$		−	1073.64i		−	1.12659i	0.826255	−	0.563296i	$-0.190467\pi$
$954$		−	492.623i		−	0.516377i
$955$		0			0
$956$			97.9286i			0.102436i
$957$	3.69315	+	10.6635i	0.00385909	+	0.0111426i
$958$	−1671.46			−1.74474
$959$		−	432.047i		−	0.450518i
$960$		0			0
$961$	−666.429			−0.693474
$962$	2242.57			2.33115
$963$			323.859i			0.336302i
$964$		−	287.149i		−	0.297872i
$965$		0			0
$966$	−234.079			−0.242318
$967$		−	1403.40i		−	1.45130i	−0.688066	−	0.725648i	$-0.741540\pi$
$967$		−	1403.40i		−	1.45130i	0.688066	−	0.725648i	$-0.258460\pi$
$968$	447.682	+	568.786i	0.462481	+	0.587588i
$969$	−225.241			−0.232447
$970$		0			0
$971$	724.260			0.745891			0.372945	−	0.927853i	$-0.378348\pi$
$971$	724.260			0.745891			0.372945	+	0.927853i	$0.378348\pi$
$972$	−22.3494			−0.0229932
$973$	136.081			0.139858
$974$		−	2137.09i		−	2.19413i
$975$		0			0
$976$		−	711.213i		−	0.728702i
$977$	−943.252			−0.965458			−0.482729	−	0.875770i	$-0.660354\pi$
$977$	−943.252			−0.965458			−0.482729	+	0.875770i	$0.660354\pi$
$978$			1285.32i			1.31423i
$979$	−1666.75	+	577.256i	−1.70250	+	0.589639i
$980$		0			0
$981$			99.2986i			0.101222i
$982$	704.303			0.717213
$983$	−1101.28			−1.12033			−0.560163	−	0.828383i	$-0.689261\pi$
$983$	−1101.28			−1.12033			−0.560163	+	0.828383i	$0.689261\pi$
$984$	139.926			0.142201
$985$		0			0
$986$		−	16.9275i		−	0.0171678i
$987$			210.899i			0.213677i
$988$	−264.552			−0.267766
$989$			330.038i			0.333709i
$990$		0			0
$991$	−639.463			−0.645271			−0.322635	−	0.946523i	$-0.604569\pi$
$991$	−639.463			−0.645271			−0.322635	+	0.946523i	$0.604569\pi$
$992$		−	376.639i		−	0.379676i
$993$	−657.478			−0.662112
$994$	410.480			0.412958
$995$		0			0
$996$			342.202i			0.343577i
$997$		−	351.666i		−	0.352724i	−0.984325	−	0.176362i	$-0.943567\pi$
$997$		−	351.666i		−	0.352724i	0.984325	−	0.176362i	$-0.0564329\pi$
$998$		−	239.626i		−	0.240106i
$999$	−287.353			−0.287641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.3.b.d.76.13		16
5.2	odd	4		825.3.h.b.274.8		32
5.3	odd	4		825.3.h.b.274.26		32
5.4	even	2		165.3.b.a.76.4	✓	16
11.10	odd	2	inner	825.3.b.d.76.4		16
15.14	odd	2		495.3.b.c.406.13		16
20.19	odd	2		2640.3.c.c.241.15		16
55.32	even	4		825.3.h.b.274.25		32
55.43	even	4		825.3.h.b.274.7		32
55.54	odd	2		165.3.b.a.76.13	yes	16
165.164	even	2		495.3.b.c.406.4		16
220.219	even	2		2640.3.c.c.241.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.3.b.a.76.4	✓	16	5.4	even	2
165.3.b.a.76.13	yes	16	55.54	odd	2
495.3.b.c.406.4		16	165.164	even	2
495.3.b.c.406.13		16	15.14	odd	2
825.3.b.d.76.4		16	11.10	odd	2	inner
825.3.b.d.76.13		16	1.1	even	1	trivial
825.3.h.b.274.7		32	55.43	even	4
825.3.h.b.274.8		32	5.2	odd	4
825.3.h.b.274.25		32	55.32	even	4
825.3.h.b.274.26		32	5.3	odd	4
2640.3.c.c.241.14		16	220.219	even	2
2640.3.c.c.241.15		16	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	825.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.33103i q^{2} +1.73205 q^{3} -1.43371 q^{4} +4.03747i q^{6} +6.32430i q^{7} +5.98210i q^{8} +3.00000 q^{9} +(-10.3943 + 3.59991i) q^{11} -2.48326 q^{12} +17.3965i q^{13} -14.7421 q^{14} -19.6793 q^{16} -12.2603i q^{17} +6.99310i q^{18} -10.6069i q^{19} +10.9540i q^{21} +(-8.39151 - 24.2294i) q^{22} +9.16729 q^{23} +10.3613i q^{24} -40.5519 q^{26} +5.19615 q^{27} -9.06722i q^{28} -0.592304i q^{29} +17.1631 q^{31} -21.9447i q^{32} +(-18.0034 + 6.23523i) q^{33} +28.5791 q^{34} -4.30114 q^{36} -55.3012 q^{37} +24.7249 q^{38} +30.1317i q^{39} -13.5046i q^{41} -25.5341 q^{42} +36.0017i q^{43} +(14.9024 - 5.16124i) q^{44} +21.3693i q^{46} +19.2531 q^{47} -34.0856 q^{48} +9.00325 q^{49} -21.2354i q^{51} -24.9416i q^{52} -70.4442 q^{53} +12.1124i q^{54} -37.8326 q^{56} -18.3716i q^{57} +1.38068 q^{58} +14.9479 q^{59} +36.1401i q^{61} +40.0077i q^{62} +18.9729i q^{63} -27.5634 q^{64} +(-14.5345 - 41.9665i) q^{66} -127.842 q^{67} +17.5777i q^{68} +15.8782 q^{69} -27.8440 q^{71} +17.9463i q^{72} +61.3643i q^{73} -128.909i q^{74} +15.2072i q^{76} +(-22.7669 - 65.7364i) q^{77} -70.2379 q^{78} +141.550i q^{79} +9.00000 q^{81} +31.4798 q^{82} -137.804i q^{83} -15.7049i q^{84} -83.9212 q^{86} -1.02590i q^{87} +(-21.5350 - 62.1795i) q^{88} +160.353 q^{89} -110.021 q^{91} -13.1433 q^{92} +29.7273 q^{93} +44.8797i q^{94} -38.0094i q^{96} +57.7240 q^{97} +20.9869i q^{98} +(-31.1828 + 10.7997i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} + 48 q^{9} - 28 q^{11} - 16 q^{16} + 20 q^{22} - 56 q^{23} - 88 q^{26} - 96 q^{31} + 12 q^{33} - 200 q^{34} - 24 q^{36} - 184 q^{37} - 296 q^{38} + 264 q^{42} + 300 q^{44} + 200 q^{47} - 96 q^{48}+ \cdots - 84 q^{99}+O(q^{100}) \) 16 * q - 8 * q^4 + 48 * q^9 - 28 * q^11 - 16 * q^16 + 20 * q^22 - 56 * q^23 - 88 * q^26 - 96 * q^31 + 12 * q^33 - 200 * q^34 - 24 * q^36 - 184 * q^37 - 296 * q^38 + 264 * q^42 + 300 * q^44 + 200 * q^47 - 96 * q^48 - 496 * q^49 + 80 * q^53 - 32 * q^56 + 16 * q^58 - 136 * q^59 + 456 * q^64 - 180 * q^66 - 256 * q^67 + 168 * q^69 + 216 * q^71 + 248 * q^77 - 240 * q^78 + 144 * q^81 - 232 * q^82 + 192 * q^86 + 116 * q^88 - 336 * q^89 + 256 * q^91 + 440 * q^92 + 144 * q^93 + 144 * q^97 - 84 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more