LMFDB - Embedded newform 475.3.c.i.151.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(151,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.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:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 42x^{12} + 677x^{10} + 5313x^{8} + 21125x^{6} + 40138x^{4} + 30565x^{2} + 3675 $ x^14 + 42x^12 + 677x^10 + 5313x^8 + 21125x^6 + 40138x^4 + 30565x^2 + 3675
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			151.13
Root			$3.01757i$ of defining polynomial
Character	$\chi$	$=$	475.151
Dual form			475.3.c.i.151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.01757i q^{2} -4.83172i q^{3} -5.10575 q^{4} +14.5801 q^{6} -6.15539 q^{7} -3.33669i q^{8} -14.3455 q^{9} +O(q^{10})$	\(q+3.01757i q^{2} -4.83172i q^{3} -5.10575 q^{4} +14.5801 q^{6} -6.15539 q^{7} -3.33669i q^{8} -14.3455 q^{9} +5.09534 q^{11} +24.6696i q^{12} +4.20326i q^{13} -18.5743i q^{14} -10.3543 q^{16} +1.29941 q^{17} -43.2887i q^{18} +(18.3714 + 4.84663i) q^{19} +29.7411i q^{21} +15.3756i q^{22} -37.1355 q^{23} -16.1219 q^{24} -12.6837 q^{26} +25.8282i q^{27} +31.4279 q^{28} +42.9209i q^{29} +52.1594i q^{31} -44.5916i q^{32} -24.6193i q^{33} +3.92107i q^{34} +73.2448 q^{36} +41.8600i q^{37} +(-14.6251 + 55.4372i) q^{38} +20.3090 q^{39} +7.13498i q^{41} -89.7461 q^{42} -59.2867 q^{43} -26.0155 q^{44} -112.059i q^{46} -57.1573 q^{47} +50.0291i q^{48} -11.1112 q^{49} -6.27840i q^{51} -21.4608i q^{52} +61.6167i q^{53} -77.9384 q^{54} +20.5386i q^{56} +(23.4176 - 88.7657i) q^{57} -129.517 q^{58} +54.5223i q^{59} +38.4303 q^{61} -157.395 q^{62} +88.3024 q^{63} +93.1413 q^{64} +74.2904 q^{66} -44.7851i q^{67} -6.63448 q^{68} +179.428i q^{69} -13.0537i q^{71} +47.8666i q^{72} +134.087 q^{73} -126.316 q^{74} +(-93.8001 - 24.7457i) q^{76} -31.3638 q^{77} +61.2839i q^{78} -128.146i q^{79} -4.31527 q^{81} -21.5303 q^{82} -23.2187 q^{83} -151.851i q^{84} -178.902i q^{86} +207.382 q^{87} -17.0015i q^{88} -21.0659i q^{89} -25.8727i q^{91} +189.605 q^{92} +252.020 q^{93} -172.476i q^{94} -215.454 q^{96} -75.9288i q^{97} -33.5288i q^{98} -73.0954 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9}+O(q^{10}) $ 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9	$ 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} - 22 q^{17} + 39 q^{19} - 12 q^{23} - 44 q^{24} + 30 q^{26} - 98 q^{28} + 4 q^{36} - 37 q^{38} - 32 q^{39} - 250 q^{42} - 90 q^{43} - 52 q^{44} - 148 q^{47} + 234 q^{49} + 98 q^{54} + 195 q^{57} + 274 q^{58} + 222 q^{61} - 518 q^{62} - 198 q^{63} - 218 q^{64} + 92 q^{66} - 80 q^{68} + 228 q^{73} - 92 q^{74} - 351 q^{76} + 260 q^{77} + 402 q^{81} - 58 q^{82} + 280 q^{83} + 282 q^{87} - 302 q^{92} + 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) $ 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9 - 4 * q^11 + 36 * q^16 - 22 * q^17 + 39 * q^19 - 12 * q^23 - 44 * q^24 + 30 * q^26 - 98 * q^28 + 4 * q^36 - 37 * q^38 - 32 * q^39 - 250 * q^42 - 90 * q^43 - 52 * q^44 - 148 * q^47 + 234 * q^49 + 98 * q^54 + 195 * q^57 + 274 * q^58 + 222 * q^61 - 518 * q^62 - 198 * q^63 - 218 * q^64 + 92 * q^66 - 80 * q^68 + 228 * q^73 - 92 * q^74 - 351 * q^76 + 260 * q^77 + 402 * q^81 - 58 * q^82 + 280 * q^83 + 282 * q^87 - 302 * q^92 + 358 * q^93 + 190 * q^96 - 180 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$1$	$-1$

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 $

$2$			3.01757i			1.50879i	0.656423	+	0.754393i	$0.272069\pi$
$2$			3.01757i			1.50879i	−0.656423	+	0.754393i	$0.727931\pi$
$3$		−	4.83172i		−	1.61057i	−0.592885	−	0.805287i	$-0.702011\pi$
$3$		−	4.83172i		−	1.61057i	0.592885	−	0.805287i	$-0.297989\pi$
$4$	−5.10575			−1.27644
$5$		0			0
$5$		0			0
$6$	14.5801			2.43001
$7$	−6.15539			−0.879341			−0.439671	−	0.898159i	$-0.644905\pi$
$7$	−6.15539			−0.879341			−0.439671	+	0.898159i	$0.644905\pi$
$8$		−	3.33669i		−	0.417086i
$9$	−14.3455			−1.59395
$10$		0			0
$11$	5.09534			0.463213			0.231606	−	0.972810i	$-0.425602\pi$
$11$	5.09534			0.463213			0.231606	+	0.972810i	$0.425602\pi$
$12$			24.6696i			2.05580i
$13$			4.20326i			0.323328i	0.986846	+	0.161664i	$0.0516861\pi$
$13$			4.20326i			0.323328i	−0.986846	+	0.161664i	$0.948314\pi$
$14$		−	18.5743i		−	1.32674i
$15$		0			0
$16$	−10.3543			−0.647144
$17$	1.29941			0.0764361			0.0382180	−	0.999269i	$-0.487832\pi$
$17$	1.29941			0.0764361			0.0382180	+	0.999269i	$0.487832\pi$
$18$		−	43.2887i		−	2.40493i
$19$	18.3714	+	4.84663i	0.966918	+	0.255086i
$19$	18.3714	+	4.84663i	0.966918	+	0.255086i
$20$		0			0
$21$			29.7411i			1.41624i
$22$			15.3756i			0.698889i
$23$	−37.1355			−1.61459			−0.807293	−	0.590151i	$-0.799068\pi$
$23$	−37.1355			−1.61459			−0.807293	+	0.590151i	$0.799068\pi$
$24$	−16.1219			−0.671748
$25$		0			0
$26$	−12.6837			−0.487833
$27$			25.8282i			0.956599i
$28$	31.4279			1.12242
$29$			42.9209i			1.48003i	0.672590	+	0.740015i	$0.265182\pi$
$29$			42.9209i			1.48003i	−0.672590	+	0.740015i	$0.734818\pi$
$30$		0			0
$31$			52.1594i			1.68256i	0.540599	+	0.841280i	$0.318198\pi$
$31$			52.1594i			1.68256i	−0.540599	+	0.841280i	$0.681802\pi$
$32$		−	44.5916i		−	1.39349i
$33$		−	24.6193i		−	0.746038i
$34$			3.92107i			0.115326i
$35$		0			0
$36$	73.2448			2.03458
$37$			41.8600i			1.13135i	0.824628	+	0.565676i	$0.191385\pi$
$37$			41.8600i			1.13135i	−0.824628	+	0.565676i	$0.808615\pi$
$38$	−14.6251	+	55.4372i	−0.384870	+	1.45887i
$39$	20.3090			0.520744
$40$		0			0
$41$			7.13498i			0.174024i	0.996207	+	0.0870119i	$0.0277318\pi$
$41$			7.13498i			0.174024i	−0.996207	+	0.0870119i	$0.972268\pi$
$42$	−89.7461			−2.13681
$43$	−59.2867			−1.37876			−0.689380	−	0.724400i	$-0.742117\pi$
$43$	−59.2867			−1.37876			−0.689380	+	0.724400i	$0.742117\pi$
$44$	−26.0155			−0.591262
$45$		0			0
$46$		−	112.059i		−	2.43607i
$47$	−57.1573			−1.21611			−0.608057	−	0.793894i	$-0.708051\pi$
$47$	−57.1573			−1.21611			−0.608057	+	0.793894i	$0.708051\pi$
$48$			50.0291i			1.04227i
$49$	−11.1112			−0.226759
$50$		0			0
$51$		−	6.27840i		−	0.123106i
$52$		−	21.4608i		−	0.412708i
$53$			61.6167i			1.16258i	0.813697	+	0.581290i	$0.197452\pi$
$53$			61.6167i			1.16258i	−0.813697	+	0.581290i	$0.802548\pi$
$54$	−77.9384			−1.44330
$55$		0			0
$56$			20.5386i			0.366761i
$57$	23.4176	−	88.7657i	0.410835	−	1.55729i
$58$	−129.517			−2.23305
$59$			54.5223i			0.924106i	0.886852	+	0.462053i	$0.152887\pi$
$59$			54.5223i			0.924106i	−0.886852	+	0.462053i	$0.847113\pi$
$60$		0			0
$61$	38.4303			0.630006			0.315003	−	0.949091i	$-0.397995\pi$
$61$	38.4303			0.630006			0.315003	+	0.949091i	$0.397995\pi$
$62$	−157.395			−2.53863
$63$	88.3024			1.40163
$64$	93.1413			1.45533
$65$		0			0
$66$	74.2904			1.12561
$67$		−	44.7851i		−	0.668435i	−0.942496	−	0.334217i	$-0.891528\pi$
$67$		−	44.7851i		−	0.668435i	0.942496	−	0.334217i	$-0.108472\pi$
$68$	−6.63448			−0.0975659
$69$			179.428i			2.60041i
$70$		0			0
$71$		−	13.0537i		−	0.183855i	−0.995766	−	0.0919277i	$-0.970697\pi$
$71$		−	13.0537i		−	0.183855i	0.995766	−	0.0919277i	$-0.0293029\pi$
$72$			47.8666i			0.664814i
$73$	134.087			1.83681			0.918406	−	0.395638i	$-0.129477\pi$
$73$	134.087			1.83681			0.918406	+	0.395638i	$0.129477\pi$
$74$	−126.316			−1.70697
$75$		0			0
$76$	−93.8001	−	24.7457i	−1.23421	−	0.325601i
$77$	−31.3638			−0.407322
$78$			61.2839i			0.785691i
$79$		−	128.146i		−	1.62210i	−0.584977	−	0.811050i	$-0.698897\pi$
$79$		−	128.146i		−	1.62210i	0.584977	−	0.811050i	$-0.301103\pi$
$80$		0			0
$81$	−4.31527			−0.0532750
$82$	−21.5303			−0.262565
$83$	−23.2187			−0.279743			−0.139872	−	0.990170i	$-0.544669\pi$
$83$	−23.2187			−0.279743			−0.139872	+	0.990170i	$0.544669\pi$
$84$		−	151.851i		−	1.80775i
$85$		0			0
$86$		−	178.902i		−	2.08026i
$87$	207.382			2.38370
$88$		−	17.0015i		−	0.193199i
$89$		−	21.0659i		−	0.236695i	−0.992972	−	0.118348i	$-0.962240\pi$
$89$		−	21.0659i		−	0.236695i	0.992972	−	0.118348i	$-0.0377597\pi$
$90$		0			0
$91$		−	25.8727i		−	0.284316i
$92$	189.605			2.06092
$93$	252.020			2.70989
$94$		−	172.476i		−	1.83486i
$95$		0			0
$96$	−215.454			−2.24432
$97$		−	75.9288i		−	0.782771i	−0.920227	−	0.391385i	$-0.871996\pi$
$97$		−	75.9288i		−	0.782771i	0.920227	−	0.391385i	$-0.128004\pi$
$98$		−	33.5288i		−	0.342130i
$99$	−73.0954			−0.738337
$100$		0			0
$101$	137.524			1.36162			0.680810	−	0.732460i	$-0.261628\pi$
$101$	137.524			1.36162			0.680810	+	0.732460i	$0.261628\pi$
$102$	18.9455			0.185741
$103$		−	13.7280i		−	0.133281i	−0.997777	−	0.0666407i	$-0.978772\pi$
$103$		−	13.7280i		−	0.133281i	0.997777	−	0.0666407i	$-0.0212281\pi$
$104$	14.0250			0.134856
$105$		0			0
$106$	−185.933			−1.75408
$107$		−	64.4608i		−	0.602437i	−0.953555	−	0.301219i	$-0.902607\pi$
$107$		−	64.4608i		−	0.602437i	0.953555	−	0.301219i	$-0.0973934\pi$
$108$		−	131.872i		−	1.22104i
$109$		−	46.6388i		−	0.427879i	−0.976847	−	0.213939i	$-0.931371\pi$
$109$		−	46.6388i		−	0.427879i	0.976847	−	0.213939i	$-0.0686295\pi$
$110$		0			0
$111$	202.256			1.82213
$112$	63.7348			0.569061
$113$		−	56.7151i		−	0.501904i	−0.968000	−	0.250952i	$-0.919256\pi$
$113$		−	56.7151i		−	0.501904i	0.968000	−	0.250952i	$-0.0807436\pi$
$114$	267.857	+	70.6643i	2.34962	+	0.619862i
$115$		0			0
$116$		−	219.143i		−	1.88917i
$117$		−	60.2981i		−	0.515368i
$118$	−164.525			−1.39428
$119$	−7.99839			−0.0672134
$120$		0			0
$121$	−95.0375			−0.785434
$122$			115.966i			0.950544i
$123$	34.4742			0.280278
$124$		−	266.313i		−	2.14768i
$125$		0			0
$126$			266.459i			2.11475i
$127$			238.179i			1.87542i	0.347417	+	0.937711i	$0.387059\pi$
$127$			238.179i			1.87542i	−0.347417	+	0.937711i	$0.612941\pi$
$128$			102.694i			0.802299i
$129$			286.457i			2.22060i
$130$		0			0
$131$	−41.0781			−0.313573			−0.156787	−	0.987633i	$-0.550113\pi$
$131$	−41.0781			−0.313573			−0.156787	+	0.987633i	$0.550113\pi$
$132$			125.700i			0.952271i
$133$	−113.083	−	29.8329i	−0.850251	−	0.224308i
$134$	135.142			1.00853
$135$		0			0
$136$		−	4.33573i		−	0.0318804i
$137$	−141.611			−1.03366			−0.516829	−	0.856088i	$-0.672888\pi$
$137$	−141.611			−1.03366			−0.516829	+	0.856088i	$0.672888\pi$
$138$	−541.438			−3.92347
$139$	−133.167			−0.958038			−0.479019	−	0.877805i	$-0.659007\pi$
$139$	−133.167			−0.958038			−0.479019	+	0.877805i	$0.659007\pi$
$140$		0			0
$141$			276.168i			1.95864i
$142$	39.3906			0.277398
$143$			21.4171i			0.149770i
$144$	148.538			1.03152
$145$		0			0
$146$			404.618i			2.77136i
$147$			53.6861i			0.365211i
$148$		−	213.727i		−	1.44410i
$149$	−275.472			−1.84881			−0.924403	−	0.381417i	$-0.875436\pi$
$149$	−275.472			−1.84881			−0.924403	+	0.381417i	$0.875436\pi$
$150$		0			0
$151$		−	141.099i		−	0.934430i	−0.884144	−	0.467215i	$-0.845257\pi$
$151$		−	141.099i		−	0.934430i	0.884144	−	0.467215i	$-0.154743\pi$
$152$	16.1717	−	61.2998i	0.106393	−	0.403288i
$153$	−18.6408			−0.121835
$154$		−	94.6426i		−	0.614562i
$155$		0			0
$156$	−103.693			−0.664697
$157$	−134.179			−0.854644			−0.427322	−	0.904099i	$-0.640543\pi$
$157$	−134.179			−0.854644			−0.427322	+	0.904099i	$0.640543\pi$
$158$	386.690			2.44740
$159$	297.715			1.87242
$160$		0			0
$161$	228.583			1.41977
$162$		−	13.0217i		−	0.0803806i
$163$	176.426			1.08237			0.541183	−	0.840905i	$-0.317976\pi$
$163$	176.426			1.08237			0.541183	+	0.840905i	$0.317976\pi$
$164$		−	36.4294i		−	0.222131i
$165$		0			0
$166$		−	70.0641i		−	0.422073i
$167$		−	174.834i		−	1.04691i	−0.852053	−	0.523455i	$-0.824643\pi$
$167$		−	174.834i		−	1.04691i	0.852053	−	0.523455i	$-0.175357\pi$
$168$	99.2369			0.590696
$169$	151.333			0.895459
$170$		0			0
$171$	−263.548	−	69.5276i	−1.54122	−	0.406594i
$172$	302.703			1.75990
$173$			254.288i			1.46987i	0.678135	+	0.734937i	$0.262788\pi$
$173$			254.288i			1.46987i	−0.678135	+	0.734937i	$0.737212\pi$
$174$			625.790i			3.59649i
$175$		0			0
$176$	−52.7587			−0.299765
$177$	263.437			1.48834
$178$	63.5678			0.357122
$179$			207.084i			1.15689i	0.815720	+	0.578447i	$0.196341\pi$
$179$			207.084i			1.15689i	−0.815720	+	0.578447i	$0.803659\pi$
$180$		0			0
$181$		−	137.706i		−	0.760805i	−0.924821	−	0.380402i	$-0.875786\pi$
$181$		−	137.706i		−	0.760805i	0.924821	−	0.380402i	$-0.124214\pi$
$182$	78.0729			0.428972
$183$		−	185.685i		−	1.01467i
$184$			123.909i			0.673421i
$185$		0			0
$186$			760.488i			4.08864i
$187$	6.62095			0.0354061
$188$	291.831			1.55229
$189$		−	158.983i		−	0.841178i
$190$		0			0
$191$	28.0282			0.146745			0.0733723	−	0.997305i	$-0.476624\pi$
$191$	28.0282			0.146745			0.0733723	+	0.997305i	$0.476624\pi$
$192$		−	450.033i		−	2.34392i
$193$			250.152i			1.29612i	0.761588	+	0.648062i	$0.224420\pi$
$193$			250.152i			1.29612i	−0.761588	+	0.648062i	$0.775580\pi$
$194$	229.121			1.18103
$195$		0			0
$196$	56.7309			0.289443
$197$	48.2838			0.245096			0.122548	−	0.992463i	$-0.460894\pi$
$197$	48.2838			0.245096			0.122548	+	0.992463i	$0.460894\pi$
$198$		−	220.571i		−	1.11399i
$199$	−265.858			−1.33597			−0.667984	−	0.744175i	$-0.732843\pi$
$199$	−265.858			−1.33597			−0.667984	+	0.744175i	$0.732843\pi$
$200$		0			0
$201$	−216.389			−1.07656
$202$			414.988i			2.05440i
$203$		−	264.195i		−	1.30145i
$204$			32.0560i			0.157137i
$205$		0			0
$206$	41.4252			0.201093
$207$	532.729			2.57357
$208$		−	43.5219i		−	0.209240i
$209$	93.6087	+	24.6952i	0.447889	+	0.118159i
$210$		0			0
$211$		−	182.771i		−	0.866213i	−0.901343	−	0.433107i	$-0.857417\pi$
$211$		−	182.771i		−	0.866213i	0.901343	−	0.433107i	$-0.142583\pi$
$212$		−	314.600i		−	1.48396i
$213$	−63.0720			−0.296113
$214$	194.515			0.908949
$215$		0			0
$216$	86.1806			0.398984
$217$		−	321.061i		−	1.47955i
$218$	140.736			0.645577
$219$		−	647.873i		−	2.95832i
$220$		0			0
$221$			5.46178i			0.0247139i
$222$			610.323i			2.74920i
$223$		−	106.620i		−	0.478118i	−0.971005	−	0.239059i	$-0.923161\pi$
$223$		−	106.620i		−	0.478118i	0.971005	−	0.239059i	$-0.0768390\pi$
$224$			274.479i			1.22535i
$225$		0			0
$226$	171.142			0.757266
$227$			35.0010i			0.154189i	0.997024	+	0.0770947i	$0.0245644\pi$
$227$			35.0010i			0.154189i	−0.997024	+	0.0770947i	$0.975436\pi$
$228$	−119.564	+	453.216i	−0.524405	+	1.98779i
$229$	−77.3701			−0.337861			−0.168930	−	0.985628i	$-0.554031\pi$
$229$	−77.3701			−0.337861			−0.168930	+	0.985628i	$0.554031\pi$
$230$		0			0
$231$			151.541i			0.656022i
$232$	143.214			0.617300
$233$	89.0034			0.381989			0.190994	−	0.981591i	$-0.438829\pi$
$233$	89.0034			0.381989			0.190994	+	0.981591i	$0.438829\pi$
$234$	181.954			0.777581
$235$		0			0
$236$		−	278.377i		−	1.17956i
$237$	−619.165			−2.61251
$238$		−	24.1357i		−	0.101411i
$239$	47.2189			0.197569			0.0987843	−	0.995109i	$-0.468505\pi$
$239$	47.2189			0.197569			0.0987843	+	0.995109i	$0.468505\pi$
$240$		0			0
$241$		−	73.1493i		−	0.303524i	−0.988417	−	0.151762i	$-0.951505\pi$
$241$		−	73.1493i		−	0.303524i	0.988417	−	0.151762i	$-0.0484947\pi$
$242$		−	286.783i		−	1.18505i
$243$			253.304i			1.04240i
$244$	−196.216			−0.804163
$245$		0			0
$246$			104.029i			0.422880i
$247$	−20.3717	+	77.2200i	−0.0824765	+	0.312632i
$248$	174.039			0.701772
$249$			112.186i			0.450547i
$250$		0			0
$251$	266.248			1.06075			0.530374	−	0.847764i	$-0.322051\pi$
$251$	266.248			1.06075			0.530374	+	0.847764i	$0.322051\pi$
$252$	−450.850			−1.78909
$253$	−189.218			−0.747896
$254$	−718.721			−2.82961
$255$		0			0
$256$	62.6778			0.244835
$257$			76.0071i			0.295747i	0.989006	+	0.147874i	$0.0472429\pi$
$257$			76.0071i			0.295747i	−0.989006	+	0.147874i	$0.952757\pi$
$258$	−864.405			−3.35041
$259$		−	257.665i		−	0.994845i
$260$		0			0
$261$		−	615.723i		−	2.35909i
$262$		−	123.956i		−	0.473115i
$263$	291.378			1.10790			0.553951	−	0.832549i	$-0.313119\pi$
$263$	291.378			1.10790			0.553951	+	0.832549i	$0.313119\pi$
$264$	−82.1468			−0.311162
$265$		0			0
$266$	90.0231	−	341.238i	0.338433	−	1.28285i
$267$	−101.784			−0.381215
$268$			228.662i			0.853215i
$269$			406.215i			1.51009i	0.655672	+	0.755046i	$0.272385\pi$
$269$			406.215i			1.51009i	−0.655672	+	0.755046i	$0.727615\pi$
$270$		0			0
$271$	−289.751			−1.06919			−0.534596	−	0.845108i	$-0.679536\pi$
$271$	−289.751			−1.06919			−0.534596	+	0.845108i	$0.679536\pi$
$272$	−13.4545			−0.0494652
$273$	−125.010			−0.457912
$274$		−	427.322i		−	1.55957i
$275$		0			0
$276$		−	916.116i		−	3.31926i
$277$	102.717			0.370820			0.185410	−	0.982661i	$-0.440639\pi$
$277$	102.717			0.370820			0.185410	+	0.982661i	$0.440639\pi$
$278$		−	401.842i		−	1.44547i
$279$		−	748.255i		−	2.68192i
$280$		0			0
$281$			0.126667i			0.000450774i	1.00000		0.000225387i	$7.17429e-5\pi$
$281$			0.126667i			0.000450774i	−1.00000		0.000225387i	$0.999928\pi$
$282$	−833.358			−2.95517
$283$	84.2579			0.297731			0.148866	−	0.988857i	$-0.452438\pi$
$283$	84.2579			0.297731			0.148866	+	0.988857i	$0.452438\pi$
$284$			66.6491i			0.234680i
$285$		0			0
$286$	−64.6275			−0.225970
$287$		−	43.9186i		−	0.153026i
$288$			639.691i			2.22115i
$289$	−287.312			−0.994158
$290$		0			0
$291$	−366.867			−1.26071
$292$	−684.617			−2.34458
$293$			404.118i			1.37924i	0.724171	+	0.689621i	$0.242223\pi$
$293$			404.118i			1.37924i	−0.724171	+	0.689621i	$0.757777\pi$
$294$	−162.002			−0.551026
$295$		0			0
$296$	139.674			0.471871
$297$			131.603i			0.443109i
$298$		−	831.258i		−	2.78946i
$299$		−	156.090i		−	0.522041i
$300$		0			0
$301$	364.933			1.21240
$302$	425.777			1.40986
$303$		−	664.476i		−	2.19299i
$304$	−190.224	−	50.1835i	−0.625736	−	0.165077i
$305$		0			0
$306$		−	56.2499i		−	0.183823i
$307$			158.822i			0.517335i	0.965966	+	0.258667i	$0.0832833\pi$
$307$			158.822i			0.517335i	−0.965966	+	0.258667i	$0.916717\pi$
$308$	160.136			0.519921
$309$	−66.3298			−0.214659
$310$		0			0
$311$	256.822			0.825793			0.412897	−	0.910778i	$-0.364517\pi$
$311$	256.822			0.825793			0.412897	+	0.910778i	$0.364517\pi$
$312$		−	67.7648i		−	0.217195i
$313$	−429.290			−1.37153			−0.685766	−	0.727822i	$-0.740533\pi$
$313$	−429.290			−1.37153			−0.685766	+	0.727822i	$0.740533\pi$
$314$		−	404.895i		−	1.28948i
$315$		0			0
$316$			654.281i			2.07051i
$317$			230.765i			0.727966i	0.931406	+	0.363983i	$0.118583\pi$
$317$			230.765i			0.727966i	−0.931406	+	0.363983i	$0.881417\pi$
$318$			898.376i			2.82508i
$319$			218.696i			0.685569i
$320$		0			0
$321$	−311.457			−0.970270
$322$			689.767i			2.14213i
$323$	23.8721	+	6.29778i	0.0739074	+	0.0194978i
$324$	22.0327			0.0680022
$325$		0			0
$326$			532.378i			1.63306i
$327$	−225.346			−0.689130
$328$	23.8072			0.0725829
$329$	351.826			1.06938
$330$		0			0
$331$		−	528.588i		−	1.59694i	−0.602032	−	0.798472i	$-0.705642\pi$
$331$		−	528.588i		−	1.59694i	0.602032	−	0.798472i	$-0.294358\pi$
$332$	118.549			0.357075
$333$		−	600.505i		−	1.80332i
$334$	527.575			1.57957
$335$		0			0
$336$		−	307.949i		−	0.916515i
$337$		−	61.4407i		−	0.182317i	−0.995836	−	0.0911583i	$-0.970943\pi$
$337$		−	61.4407i		−	0.182317i	0.995836	−	0.0911583i	$-0.0290569\pi$
$338$			456.657i			1.35106i
$339$	−274.032			−0.808353
$340$		0			0
$341$			265.770i			0.779383i
$342$	209.805	−	795.277i	0.613464	−	2.32537i
$343$	370.008			1.07874
$344$			197.821i			0.575061i
$345$		0			0
$346$	−767.333			−2.21773
$347$	555.017			1.59947			0.799737	−	0.600351i	$-0.204972\pi$
$347$	555.017			1.59947			0.799737	+	0.600351i	$0.204972\pi$
$348$	−1058.84			−3.04264
$349$	34.6618			0.0993176			0.0496588	−	0.998766i	$-0.484187\pi$
$349$	34.6618			0.0993176			0.0496588	+	0.998766i	$0.484187\pi$
$350$		0			0
$351$	−108.563			−0.309295
$352$		−	227.209i		−	0.645481i
$353$	493.002			1.39661			0.698303	−	0.715802i	$-0.253939\pi$
$353$	493.002			1.39661			0.698303	+	0.715802i	$0.253939\pi$
$354$			794.939i			2.24559i
$355$		0			0
$356$			107.557i			0.302127i
$357$			38.6460i			0.108252i
$358$	−624.892			−1.74551
$359$	145.520			0.405349			0.202675	−	0.979246i	$-0.435037\pi$
$359$	145.520			0.405349			0.202675	+	0.979246i	$0.435037\pi$
$360$		0			0
$361$	314.020	+	178.079i	0.869862	+	0.493295i
$362$	415.537			1.14789
$363$			459.195i			1.26500i
$364$			132.100i			0.362911i
$365$		0			0
$366$	560.317			1.53092
$367$	−342.929			−0.934413			−0.467206	−	0.884148i	$-0.654739\pi$
$367$	−342.929			−0.934413			−0.467206	+	0.884148i	$0.654739\pi$
$368$	384.512			1.04487
$369$		−	102.355i		−	0.277385i
$370$		0			0
$371$		−	379.275i		−	1.02230i
$372$	−1286.75			−3.45900
$373$		−	16.6464i		−	0.0446283i	−0.999751	−	0.0223141i	$-0.992897\pi$
$373$		−	16.6464i		−	0.0446283i	0.999751	−	0.0223141i	$-0.00710340\pi$
$374$			19.9792i			0.0534203i
$375$		0			0
$376$			190.716i			0.507224i
$377$	−180.408			−0.478535
$378$	479.742			1.26916
$379$		−	312.804i		−	0.825341i	−0.910880	−	0.412671i	$-0.864596\pi$
$379$		−	312.804i		−	0.825341i	0.910880	−	0.412671i	$-0.135404\pi$
$380$		0			0
$381$	1150.81			3.02051
$382$			84.5772i			0.221406i
$383$		−	493.251i		−	1.28786i	−0.765083	−	0.643931i	$-0.777302\pi$
$383$		−	493.251i		−	1.28786i	0.765083	−	0.643931i	$-0.222698\pi$
$384$	496.190			1.29216
$385$		0			0
$386$	−754.852			−1.95557
$387$	850.500			2.19767
$388$			387.673i			0.999158i
$389$	−407.055			−1.04641			−0.523207	−	0.852205i	$-0.675265\pi$
$389$	−407.055			−1.04641			−0.523207	+	0.852205i	$0.675265\pi$
$390$		0			0
$391$	−48.2543			−0.123413
$392$			37.0745i			0.0945778i
$393$			198.478i			0.505033i
$394$			145.700i			0.369797i
$395$		0			0
$396$	373.207			0.942442
$397$	−208.967			−0.526366			−0.263183	−	0.964746i	$-0.584772\pi$
$397$	−208.967			−0.526366			−0.263183	+	0.964746i	$0.584772\pi$
$398$		−	802.245i		−	2.01569i
$399$	−144.144	+	546.388i	−0.361264	+	1.36939i
$400$		0			0
$401$			240.966i			0.600913i	0.953796	+	0.300456i	$0.0971390\pi$
$401$			240.966i			0.600913i	−0.953796	+	0.300456i	$0.902861\pi$
$402$		−	652.971i		−	1.62430i
$403$	−219.240			−0.544019
$404$	−702.162			−1.73802
$405$		0			0
$406$	797.227			1.96361
$407$			213.291i			0.524056i
$408$	−20.9491			−0.0513457
$409$		−	259.859i		−	0.635352i	−0.948199	−	0.317676i	$-0.897098\pi$
$409$		−	259.859i		−	0.635352i	0.948199	−	0.317676i	$-0.102902\pi$
$410$		0			0
$411$			684.226i			1.66478i
$412$			70.0916i			0.170125i
$413$		−	335.606i		−	0.812605i
$414$			1607.55i			3.88297i
$415$		0			0
$416$	187.430			0.450554
$417$			643.427i			1.54299i
$418$	−74.5197	+	282.471i	−0.178277	+	0.675769i
$419$	−729.115			−1.74013			−0.870066	−	0.492935i	$-0.835924\pi$
$419$	−729.115			−1.74013			−0.870066	+	0.492935i	$0.835924\pi$
$420$		0			0
$421$			608.641i			1.44570i	0.691003	+	0.722851i	$0.257169\pi$
$421$			608.641i			1.44570i	−0.691003	+	0.722851i	$0.742831\pi$
$422$	551.525			1.30693
$423$	819.953			1.93842
$424$	205.596			0.484895
$425$		0			0
$426$		−	190.324i		−	0.446771i
$427$	−236.554			−0.553990
$428$			329.121i			0.768973i
$429$	103.481			0.241215
$430$		0			0
$431$			537.936i			1.24811i	0.781380	+	0.624056i	$0.214516\pi$
$431$			537.936i			1.24811i	−0.781380	+	0.624056i	$0.785484\pi$
$432$		−	267.433i		−	0.619058i
$433$		−	255.532i		−	0.590144i	−0.955475	−	0.295072i	$-0.904656\pi$
$433$		−	255.532i		−	0.590144i	0.955475	−	0.295072i	$-0.0953437\pi$
$434$	968.826			2.23232
$435$		0			0
$436$			238.126i			0.546160i
$437$	−682.233	−	179.982i	−1.56117	−	0.411858i
$438$	1955.00			4.46348
$439$			699.710i			1.59387i	0.604063	+	0.796937i	$0.293548\pi$
$439$			699.710i			1.59387i	−0.604063	+	0.796937i	$0.706452\pi$
$440$		0			0
$441$	159.396			0.361442
$442$	−16.4813			−0.0372880
$443$	−682.143			−1.53983			−0.769913	−	0.638149i	$-0.779701\pi$
$443$	−682.143			−1.53983			−0.769913	+	0.638149i	$0.779701\pi$
$444$	−1032.67			−2.32583
$445$		0			0
$446$	321.735			0.721378
$447$			1331.01i			2.97764i
$448$	−573.321			−1.27973
$449$			675.642i			1.50477i	0.658723	+	0.752386i	$0.271097\pi$
$449$			675.642i			1.50477i	−0.658723	+	0.752386i	$0.728903\pi$
$450$		0			0
$451$			36.3551i			0.0806100i
$452$			289.573i			0.640649i
$453$	−681.751			−1.50497
$454$	−105.618			−0.232639
$455$		0			0
$456$	−296.183	−	78.1372i	−0.649525	−	0.171353i
$457$	−124.772			−0.273024			−0.136512	−	0.990638i	$-0.543589\pi$
$457$	−124.772			−0.273024			−0.136512	+	0.990638i	$0.543589\pi$
$458$		−	233.470i		−	0.509759i
$459$			33.5615i			0.0731187i
$460$		0			0
$461$	328.840			0.713320			0.356660	−	0.934234i	$-0.383916\pi$
$461$	328.840			0.713320			0.356660	+	0.934234i	$0.383916\pi$
$462$	−457.287			−0.989798
$463$	−122.860			−0.265356			−0.132678	−	0.991159i	$-0.542358\pi$
$463$	−122.860			−0.265356			−0.132678	+	0.991159i	$0.542358\pi$
$464$		−	444.416i		−	0.957793i
$465$		0			0
$466$			268.574i			0.576340i
$467$	653.452			1.39925			0.699627	−	0.714508i	$-0.253349\pi$
$467$	653.452			1.39925			0.699627	+	0.714508i	$0.253349\pi$
$468$			307.867i			0.657836i
$469$			275.670i			0.587782i
$470$		0			0
$471$			648.316i			1.37647i
$472$	181.924			0.385432
$473$	−302.086			−0.638659
$474$		−	1868.38i		−	3.94172i
$475$		0			0
$476$	40.8378			0.0857937
$477$		−	883.925i		−	1.85309i
$478$			142.487i			0.298089i
$479$	360.148			0.751874			0.375937	−	0.926645i	$-0.377321\pi$
$479$	360.148			0.751874			0.375937	+	0.926645i	$0.377321\pi$
$480$		0			0
$481$	−175.949			−0.365798
$482$	220.733			0.457953
$483$		−	1104.45i		−	2.28665i
$484$	485.238			1.00256
$485$		0			0
$486$	−764.363			−1.57276
$487$		−	47.1304i		−	0.0967770i	−0.998829	−	0.0483885i	$-0.984591\pi$
$487$		−	47.1304i		−	0.0967770i	0.998829	−	0.0483885i	$-0.0154085\pi$
$488$		−	128.230i		−	0.262766i
$489$		−	852.440i		−	1.74323i
$490$		0			0
$491$	−744.260			−1.51580			−0.757902	−	0.652369i	$-0.773775\pi$
$491$	−744.260			−1.51580			−0.757902	+	0.652369i	$0.773775\pi$
$492$	−176.017			−0.357758
$493$			55.7720i			0.113128i
$494$	−233.017	−	61.4731i	−0.471695	−	0.124439i
$495$		0			0
$496$		−	540.074i		−	1.08886i
$497$			80.3508i			0.161672i
$498$	−338.530			−0.679779
$499$	814.737			1.63274			0.816370	−	0.577529i	$-0.195983\pi$
$499$	814.737			1.63274			0.816370	+	0.577529i	$0.195983\pi$
$500$		0			0
$501$	−844.750			−1.68613
$502$			803.423i			1.60044i
$503$	−618.711			−1.23004			−0.615020	−	0.788511i	$-0.710852\pi$
$503$	−618.711			−1.23004			−0.615020	+	0.788511i	$0.710852\pi$
$504$		−	294.637i		−	0.584598i
$505$		0			0
$506$		−	570.979i		−	1.12842i
$507$		−	731.197i		−	1.44220i
$508$		−	1216.08i		−	2.39386i
$509$		−	309.300i		−	0.607661i	−0.952726	−	0.303831i	$-0.901734\pi$
$509$		−	309.300i		−	0.607661i	0.952726	−	0.303831i	$-0.0982657\pi$
$510$		0			0
$511$	−825.360			−1.61519
$512$			599.912i			1.17170i
$513$	−125.180	+	474.501i	−0.244015	+	0.924954i
$514$	−229.357			−0.446220
$515$		0			0
$516$		−	1462.58i		−	2.83445i
$517$	−291.236			−0.563319
$518$	777.523			1.50101
$519$	1228.65			2.36734
$520$		0			0
$521$		−	163.664i		−	0.314134i	−0.987588	−	0.157067i	$-0.949796\pi$
$521$		−	163.664i		−	0.314134i	0.987588	−	0.157067i	$-0.0502039\pi$
$522$	1857.99			3.55937
$523$			579.994i			1.10897i	0.832192	+	0.554487i	$0.187086\pi$
$523$			579.994i			1.10897i	−0.832192	+	0.554487i	$0.812914\pi$
$524$	209.734			0.400257
$525$		0			0
$526$			879.256i			1.67159i
$527$			67.7766i			0.128608i
$528$			254.915i			0.482794i
$529$	850.044			1.60689
$530$		0			0
$531$		−	782.152i		−	1.47298i
$532$	577.376	+	152.320i	1.08529	+	0.286315i
$533$	−29.9902			−0.0562668
$534$		−	307.142i		−	0.575172i
$535$		0			0
$536$	−149.434			−0.278795
$537$	1000.57			1.86327
$538$	−1225.78			−2.27841
$539$	−56.6152			−0.105037
$540$		0			0
$541$	−706.641			−1.30618			−0.653088	−	0.757282i	$-0.726527\pi$
$541$	−706.641			−1.30618			−0.653088	+	0.757282i	$0.726527\pi$
$542$		−	874.345i		−	1.61318i
$543$	−665.356			−1.22533
$544$		−	57.9429i		−	0.106513i
$545$		0			0
$546$		−	377.226i		−	0.690891i
$547$		−	726.986i		−	1.32904i	−0.747269	−	0.664521i	$-0.768636\pi$
$547$		−	726.986i		−	1.32904i	0.747269	−	0.664521i	$-0.231364\pi$
$548$	723.032			1.31940
$549$	−551.304			−1.00420
$550$		0			0
$551$	−208.022	+	788.519i	−0.377535	+	1.43107i
$552$	598.696			1.08459
$553$			788.788i			1.42638i
$554$			309.957i			0.559489i
$555$		0			0
$556$	679.919			1.22288
$557$	−12.5711			−0.0225692			−0.0112846	−	0.999936i	$-0.503592\pi$
$557$	−12.5711			−0.0225692			−0.0112846	+	0.999936i	$0.503592\pi$
$558$	2257.91			4.04644
$559$		−	249.198i		−	0.445792i
$560$		0			0
$561$		−	31.9906i		−	0.0570242i
$562$	−0.382228			−0.000680122
$563$			635.525i			1.12882i	0.825495	+	0.564410i	$0.190896\pi$
$563$			635.525i			1.12882i	−0.825495	+	0.564410i	$0.809104\pi$
$564$		−	1410.05i		−	2.50008i
$565$		0			0
$566$			254.255i			0.449213i
$567$	26.5622			0.0468469
$568$	−43.5562			−0.0766834
$569$		−	306.296i		−	0.538306i	−0.963097	−	0.269153i	$-0.913256\pi$
$569$		−	306.296i		−	0.538306i	0.963097	−	0.269153i	$-0.0867438\pi$
$570$		0			0
$571$	127.895			0.223984			0.111992	−	0.993709i	$-0.464277\pi$
$571$	127.895			0.223984			0.111992	+	0.993709i	$0.464277\pi$
$572$		−	109.350i		−	0.191172i
$573$		−	135.425i		−	0.236343i
$574$	132.528			0.230884
$575$		0			0
$576$	−1336.16			−2.31973
$577$	558.970			0.968752			0.484376	−	0.874860i	$-0.339047\pi$
$577$	558.970			0.968752			0.484376	+	0.874860i	$0.339047\pi$
$578$		−	866.984i		−	1.49997i
$579$	1208.66			2.08750
$580$		0			0
$581$	142.920			0.245990
$582$		−	1107.05i		−	1.90214i
$583$			313.958i			0.538521i
$584$		−	447.407i		−	0.766109i
$585$		0			0
$586$	−1219.46			−2.08098
$587$	−140.817			−0.239893			−0.119946	−	0.992780i	$-0.538272\pi$
$587$	−140.817			−0.239893			−0.119946	+	0.992780i	$0.538272\pi$
$588$		−	274.108i		−	0.466170i
$589$	−252.797	+	958.243i	−0.429198	+	1.62690i
$590$		0			0
$591$		−	233.294i		−	0.394745i
$592$		−	433.432i		−	0.732148i
$593$	−340.511			−0.574218			−0.287109	−	0.957898i	$-0.592694\pi$
$593$	−340.511			−0.574218			−0.287109	+	0.957898i	$0.592694\pi$
$594$	−397.123			−0.668557
$595$		0			0
$596$	1406.49			2.35989
$597$			1284.55i			2.15168i
$598$	471.014			0.787648
$599$		−	694.681i		−	1.15973i	−0.814711	−	0.579867i	$-0.803104\pi$
$599$		−	694.681i		−	1.15973i	0.814711	−	0.579867i	$-0.196896\pi$
$600$		0			0
$601$		−	243.254i		−	0.404749i	−0.979308	−	0.202375i	$-0.935134\pi$
$601$		−	243.254i		−	0.404749i	0.979308	−	0.202375i	$-0.0648658\pi$
$602$			1101.21i			1.82925i
$603$			642.467i			1.06545i
$604$			720.416i			1.19274i
$605$		0			0
$606$	2005.11			3.30876
$607$		−	942.849i		−	1.55329i	−0.629936	−	0.776647i	$-0.716919\pi$
$607$		−	942.849i		−	1.55329i	0.629936	−	0.776647i	$-0.283081\pi$
$608$	216.119	−	819.213i	0.355459	−	1.34739i
$609$	−1276.52			−2.09609
$610$		0			0
$611$		−	240.247i		−	0.393203i
$612$	95.1752			0.155515
$613$	402.930			0.657308			0.328654	−	0.944450i	$-0.393405\pi$
$613$	402.930			0.657308			0.328654	+	0.944450i	$0.393405\pi$
$614$	−479.256			−0.780548
$615$		0			0
$616$			104.651i			0.169888i
$617$	−964.671			−1.56349			−0.781743	−	0.623600i	$-0.785669\pi$
$617$	−964.671			−1.56349			−0.781743	+	0.623600i	$0.785669\pi$
$618$		−	200.155i		−	0.323875i
$619$	830.004			1.34088			0.670439	−	0.741964i	$-0.266106\pi$
$619$	830.004			1.34088			0.670439	+	0.741964i	$0.266106\pi$
$620$		0			0
$621$		−	959.142i		−	1.54451i
$622$			774.978i			1.24595i
$623$			129.669i			0.208136i
$624$	−210.286			−0.336996
$625$		0			0
$626$		−	1295.41i		−	2.06935i
$627$	119.321	−	452.291i	0.190304	−	0.721358i
$628$	685.085			1.09090
$629$			54.3935i			0.0864761i
$630$		0			0
$631$	775.784			1.22945			0.614726	−	0.788741i	$-0.289267\pi$
$631$	775.784			1.22945			0.614726	+	0.788741i	$0.289267\pi$
$632$	−427.583			−0.676555
$633$	−883.099			−1.39510
$634$	−696.351			−1.09834
$635$		0			0
$636$	−1520.06			−2.39003
$637$		−	46.7032i		−	0.0733174i
$638$	−659.933			−1.03438
$639$			187.263i			0.293056i
$640$		0			0
$641$			277.516i			0.432943i	0.976289	+	0.216471i	$0.0694548\pi$
$641$			277.516i			0.432943i	−0.976289	+	0.216471i	$0.930545\pi$
$642$		−	939.843i		−	1.46393i
$643$	617.315			0.960055			0.480027	−	0.877254i	$-0.340627\pi$
$643$	617.315			0.960055			0.480027	+	0.877254i	$0.340627\pi$
$644$	−1167.09			−1.81225
$645$		0			0
$646$	−19.0040	+	72.0358i	−0.0294180	+	0.111511i
$647$	−642.107			−0.992437			−0.496218	−	0.868198i	$-0.665278\pi$
$647$	−642.107			−0.992437			−0.496218	+	0.868198i	$0.665278\pi$
$648$			14.3987i			0.0222202i
$649$			277.809i			0.428058i
$650$		0			0
$651$	−1551.28			−2.38292
$652$	−900.786			−1.38157
$653$	803.876			1.23105			0.615525	−	0.788117i	$-0.288944\pi$
$653$	803.876			1.23105			0.615525	+	0.788117i	$0.288944\pi$
$654$		−	679.997i		−	1.03975i
$655$		0			0
$656$		−	73.8777i		−	0.112619i
$657$	−1923.56			−2.92779
$658$			1061.66i			1.61346i
$659$			1024.56i			1.55472i	0.629054	+	0.777362i	$0.283442\pi$
$659$			1024.56i			1.55472i	−0.629054	+	0.777362i	$0.716558\pi$
$660$		0			0
$661$		−	1059.45i		−	1.60281i	−0.598125	−	0.801403i	$-0.704087\pi$
$661$		−	1059.45i		−	1.60281i	0.598125	−	0.801403i	$-0.295913\pi$
$662$	1595.05			2.40945
$663$	26.3898			0.0398036
$664$			77.4734i			0.116677i
$665$		0			0
$666$	1812.07			2.72082
$667$		−	1593.89i		−	2.38964i
$668$			892.659i			1.33632i
$669$	−515.160			−0.770045
$670$		0			0
$671$	195.816			0.291826
$672$	1326.21			1.97352
$673$			766.057i			1.13827i	0.822243	+	0.569136i	$0.192722\pi$
$673$			766.057i			1.13827i	−0.822243	+	0.569136i	$0.807278\pi$
$674$	185.402			0.275077
$675$		0			0
$676$	−772.666			−1.14300
$677$			386.540i			0.570960i	0.958385	+	0.285480i	$0.0921531\pi$
$677$			386.540i			0.570960i	−0.958385	+	0.285480i	$0.907847\pi$
$678$		−	826.911i		−	1.21963i
$679$			467.371i			0.688323i
$680$		0			0
$681$	169.115			0.248334
$682$	−801.979			−1.17592
$683$			1160.28i			1.69880i	0.527746	+	0.849402i	$0.323037\pi$
$683$			1160.28i			1.69880i	−0.527746	+	0.849402i	$0.676963\pi$
$684$	1345.61	+	354.991i	1.96727	+	0.518992i
$685$		0			0
$686$			1116.53i			1.62759i
$687$			373.831i			0.544149i
$688$	613.873			0.892257
$689$	−258.991			−0.375894
$690$		0			0
$691$	−180.805			−0.261657			−0.130828	−	0.991405i	$-0.541764\pi$
$691$	−180.805			−0.261657			−0.130828	+	0.991405i	$0.541764\pi$
$692$		−	1298.33i		−	1.87620i
$693$	449.931			0.649251
$694$			1674.81i			2.41327i
$695$		0			0
$696$		−	691.968i		−	0.994207i
$697$			9.27128i			0.0133017i
$698$			104.595i			0.149849i
$699$		−	430.040i		−	0.615221i
$700$		0			0
$701$	1195.12			1.70488			0.852442	−	0.522821i	$-0.175121\pi$
$701$	1195.12			1.70488			0.852442	+	0.522821i	$0.175121\pi$
$702$		−	327.596i		−	0.466661i
$703$	−202.880	+	769.029i	−0.288592	+	1.09393i
$704$	474.586			0.674128
$705$		0			0
$706$			1487.67i			2.10718i
$707$	−846.512			−1.19733
$708$	−1345.04			−1.89978
$709$	352.953			0.497818			0.248909	−	0.968527i	$-0.419928\pi$
$709$	352.953			0.497818			0.248909	+	0.968527i	$0.419928\pi$
$710$		0			0
$711$			1838.32i			2.58554i
$712$	−70.2902			−0.0987221
$713$		−	1936.96i		−	2.71664i
$714$	−116.617			−0.163329
$715$		0			0
$716$		−	1057.32i		−	1.47670i
$717$		−	228.149i		−	0.318199i
$718$			439.118i			0.611585i
$719$	388.649			0.540541			0.270271	−	0.962784i	$-0.412887\pi$
$719$	388.649			0.540541			0.270271	+	0.962784i	$0.412887\pi$
$720$		0			0
$721$			84.5010i			0.117200i
$722$	−537.368	+	947.579i	−0.744277	+	1.31244i
$723$	−353.437			−0.488848
$724$			703.091i			0.971120i
$725$		0			0
$726$	−1385.65			−1.90862
$727$	1159.46			1.59486			0.797431	−	0.603410i	$-0.206192\pi$
$727$	1159.46			1.59486			0.797431	+	0.603410i	$0.206192\pi$
$728$	−86.3292			−0.118584
$729$	1185.06			1.62559
$730$		0			0
$731$	−77.0379			−0.105387
$732$			948.060i			1.29516i
$733$	−420.415			−0.573554			−0.286777	−	0.957997i	$-0.592584\pi$
$733$	−420.415			−0.573554			−0.286777	+	0.957997i	$0.592584\pi$
$734$		−	1034.81i		−	1.40983i
$735$		0			0
$736$			1655.93i			2.24991i
$737$		−	228.195i		−	0.309627i
$738$	308.864			0.418515
$739$	117.628			0.159173			0.0795863	−	0.996828i	$-0.474640\pi$
$739$	117.628			0.159173			0.0795863	+	0.996828i	$0.474640\pi$
$740$		0			0
$741$	373.106	+	98.4303i	0.503517	+	0.132834i
$742$	1144.49			1.54244
$743$			131.785i			0.177368i	0.996060	+	0.0886842i	$0.0282662\pi$
$743$			131.785i			0.177368i	−0.996060	+	0.0886842i	$0.971734\pi$
$744$		−	840.911i		−	1.13026i
$745$		0			0
$746$	50.2316			0.0673346
$747$	333.084			0.445896
$748$	−33.8049			−0.0451937
$749$			396.781i			0.529748i
$750$		0			0
$751$			162.721i			0.216672i	0.994114	+	0.108336i	$0.0345522\pi$
$751$			162.721i			0.216672i	−0.994114	+	0.108336i	$0.965448\pi$
$752$	591.824			0.787001
$753$		−	1286.44i		−	1.70841i
$754$		−	544.394i		−	0.722008i
$755$		0			0
$756$			811.725i			1.07371i
$757$	1220.77			1.61264			0.806322	−	0.591477i	$-0.201455\pi$
$757$	1220.77			1.61264			0.806322	+	0.591477i	$0.201455\pi$
$758$	943.910			1.24526
$759$			914.248i			1.20454i
$760$		0			0
$761$	−164.940			−0.216741			−0.108370	−	0.994111i	$-0.534563\pi$
$761$	−164.940			−0.216741			−0.108370	+	0.994111i	$0.534563\pi$
$762$			3472.66i			4.55730i
$763$			287.080i			0.376251i
$764$	−143.105			−0.187310
$765$		0			0
$766$	1488.42			1.94311
$767$	−229.172			−0.298789
$768$		−	302.842i		−	0.394325i
$769$	−1070.57			−1.39216			−0.696079	−	0.717965i	$-0.745074\pi$
$769$	−1070.57			−1.39216			−0.696079	+	0.717965i	$0.745074\pi$
$770$		0			0
$771$	367.245			0.476323
$772$		−	1277.21i		−	1.65442i
$773$		−	181.214i		−	0.234430i	−0.993107	−	0.117215i	$-0.962603\pi$
$773$		−	181.214i		−	0.234430i	0.993107	−	0.117215i	$-0.0373967\pi$
$774$			2566.45i			3.31582i
$775$		0			0
$776$	−253.350			−0.326483
$777$	−1244.96			−1.60227
$778$		−	1228.32i		−	1.57882i
$779$	−34.5806	+	131.080i	−0.0443910	+	0.168267i
$780$		0			0
$781$		−	66.5132i		−	0.0851641i
$782$		−	145.611i		−	0.186203i
$783$	−1108.57			−1.41580
$784$	115.048			0.146745
$785$		0			0
$786$	−598.922			−0.761987
$787$		−	1339.67i		−	1.70225i	−0.524966	−	0.851123i	$-0.675922\pi$
$787$		−	1339.67i		−	1.70225i	0.524966	−	0.851123i	$-0.324078\pi$
$788$	−246.525			−0.312849
$789$		−	1407.86i		−	1.78436i
$790$		0			0
$791$			349.104i			0.441345i
$792$			243.896i			0.307950i
$793$			161.533i			0.203698i
$794$		−	630.574i		−	0.794174i
$795$		0			0
$796$	1357.40			1.70528
$797$			456.100i			0.572272i	0.958189	+	0.286136i	$0.0923708\pi$
$797$			456.100i			0.572272i	−0.958189	+	0.286136i	$0.907629\pi$
$798$	−1648.77	−	434.966i	−2.06612	−	0.545071i
$799$	−74.2710			−0.0929549
$800$		0			0
$801$			302.201i			0.377280i
$802$	−727.133			−0.906649
$803$	683.220			0.850835
$804$	1104.83			1.37417
$805$		0			0
$806$		−	661.572i		−	0.820809i
$807$	1962.72			2.43212
$808$		−	458.873i		−	0.567913i
$809$	−415.566			−0.513678			−0.256839	−	0.966454i	$-0.582681\pi$
$809$	−415.566			−0.513678			−0.256839	+	0.966454i	$0.582681\pi$
$810$		0			0
$811$		−	891.629i		−	1.09942i	−0.835356	−	0.549710i	$-0.814738\pi$
$811$		−	891.629i		−	1.09942i	0.835356	−	0.549710i	$-0.185262\pi$
$812$			1348.91i			1.66122i
$813$			1400.00i			1.72201i
$814$	−643.621			−0.790690
$815$		0			0
$816$			65.0085i			0.0796673i
$817$	−1089.18	−	287.341i	−1.33315	−	0.351702i
$818$	784.143			0.958611
$819$			371.158i			0.453185i
$820$		0			0
$821$	−241.578			−0.294249			−0.147124	−	0.989118i	$-0.547002\pi$
$821$	−241.578			−0.294249			−0.147124	+	0.989118i	$0.547002\pi$
$822$	−2064.70			−2.51180
$823$	585.888			0.711894			0.355947	−	0.934506i	$-0.384158\pi$
$823$	585.888			0.711894			0.355947	+	0.934506i	$0.384158\pi$
$824$	−45.8060			−0.0555897
$825$		0			0
$826$	1012.72			1.22605
$827$			61.7508i			0.0746684i	0.999303	+	0.0373342i	$0.0118866\pi$
$827$			61.7508i			0.0746684i	−0.999303	+	0.0373342i	$0.988113\pi$
$828$	−2719.98			−3.28500
$829$		−	776.948i		−	0.937211i	−0.883408	−	0.468606i	$-0.844757\pi$
$829$		−	776.948i		−	0.937211i	0.883408	−	0.468606i	$-0.155243\pi$
$830$		0			0
$831$		−	496.301i		−	0.597233i
$832$			391.498i			0.470550i
$833$	−14.4380			−0.0173325
$834$	−1941.59			−2.32804
$835$		0			0
$836$	−477.943	−	126.088i	−0.571702	−	0.150823i
$837$	−1347.18			−1.60954
$838$		−	2200.16i		−	2.62549i
$839$			997.057i			1.18839i	0.804322	+	0.594194i	$0.202529\pi$
$839$			997.057i			1.18839i	−0.804322	+	0.594194i	$0.797471\pi$
$840$		0			0
$841$	−1001.20			−1.19049
$842$	−1836.62			−2.18126
$843$	0.612022			0.000726005
$844$			933.183i			1.10567i
$845$		0			0
$846$			2474.27i			2.92467i
$847$	584.993			0.690665
$848$		−	637.998i		−	0.752357i
$849$		−	407.111i		−	0.479518i
$850$		0			0
$851$		−	1554.49i		−	1.82667i
$852$	322.030			0.377969
$853$	56.0425			0.0657004			0.0328502	−	0.999460i	$-0.489542\pi$
$853$	56.0425			0.0657004			0.0328502	+	0.999460i	$0.489542\pi$
$854$		−	713.818i		−	0.835853i
$855$		0			0
$856$	−215.085			−0.251268
$857$			926.670i			1.08129i	0.841249	+	0.540647i	$0.181821\pi$
$857$			926.670i			1.08129i	−0.841249	+	0.540647i	$0.818179\pi$
$858$			312.262i			0.363942i
$859$	83.5317			0.0972429			0.0486215	−	0.998817i	$-0.484517\pi$
$859$	83.5317			0.0972429			0.0486215	+	0.998817i	$0.484517\pi$
$860$		0			0
$861$	−212.202			−0.246460
$862$	−1623.26			−1.88313
$863$			187.981i			0.217823i	0.994051	+	0.108911i	$0.0347364\pi$
$863$			187.981i			0.217823i	−0.994051	+	0.108911i	$0.965264\pi$
$864$	1151.72			1.33301
$865$		0			0
$866$	771.088			0.890402
$867$			1388.21i			1.60116i
$868$			1639.26i			1.88855i
$869$		−	652.946i		−	0.751377i
$870$		0			0
$871$	188.244			0.216124
$872$	−155.619			−0.178462
$873$			1089.24i			1.24770i
$874$	543.109	−	2058.69i	0.621406	−	2.35548i
$875$		0			0
$876$			3307.88i			3.77612i
$877$			732.648i			0.835402i	0.908585	+	0.417701i	$0.137164\pi$
$877$			732.648i			0.835402i	−0.908585	+	0.417701i	$0.862836\pi$
$878$	−2111.43			−2.40482
$879$	1952.59			2.22137
$880$		0			0
$881$	1157.50			1.31385			0.656925	−	0.753956i	$-0.271857\pi$
$881$	1157.50			1.31385			0.656925	+	0.753956i	$0.271857\pi$
$882$			480.988i			0.545338i
$883$	1565.09			1.77247			0.886234	−	0.463237i	$-0.153312\pi$
$883$	1565.09			1.77247			0.886234	+	0.463237i	$0.153312\pi$
$884$		−	27.8865i		−	0.0315458i
$885$		0			0
$886$		−	2058.42i		−	2.32327i
$887$		−	618.630i		−	0.697440i	−0.937227	−	0.348720i	$-0.886616\pi$
$887$		−	618.630i		−	0.697440i	0.937227	−	0.348720i	$-0.113384\pi$
$888$		−	674.865i		−	0.759983i
$889$		−	1466.08i		−	1.64914i
$890$		0			0
$891$	−21.9878			−0.0246776
$892$			544.377i			0.610288i
$893$	−1050.06	−	277.021i	−1.17588	−	0.310213i
$894$	−4016.41			−4.49262
$895$		0			0
$896$		−	632.123i		−	0.705495i
$897$	−754.185			−0.840786
$898$	−2038.80			−2.27038
$899$	−2238.73			−2.49024
$900$		0			0
$901$			80.0655i			0.0888630i
$902$	−109.704			−0.121623
$903$		−	1763.25i		−	1.95266i
$904$	−189.241			−0.209337
$905$		0			0
$906$		−	2057.23i		−	2.27068i
$907$			289.512i			0.319197i	0.987182	+	0.159598i	$0.0510199\pi$
$907$			289.512i			0.319197i	−0.987182	+	0.159598i	$0.948980\pi$
$908$		−	178.706i		−	0.196813i
$909$	−1972.85			−2.17035
$910$		0			0
$911$			1095.19i			1.20219i	0.799179	+	0.601093i	$0.205268\pi$
$911$			1095.19i			1.20219i	−0.799179	+	0.601093i	$0.794732\pi$
$912$	−242.473	+	919.108i	−0.265870	+	1.00779i
$913$	−118.307			−0.129580
$914$		−	376.509i		−	0.411935i
$915$		0			0
$916$	395.032			0.431258
$917$	252.852			0.275738
$918$	−101.274			−0.110321
$919$	−393.630			−0.428324			−0.214162	−	0.976798i	$-0.568702\pi$
$919$	−393.630			−0.428324			−0.214162	+	0.976798i	$0.568702\pi$
$920$		0			0
$921$	767.382			0.833206
$922$			992.300i			1.07625i
$923$	54.8683			0.0594456
$924$		−	773.731i		−	0.837372i
$925$		0			0
$926$		−	370.739i		−	0.400366i
$927$			196.935i			0.212444i
$928$	1913.91			2.06241
$929$	−826.818			−0.890008			−0.445004	−	0.895529i	$-0.646798\pi$
$929$	−826.818			−0.890008			−0.445004	+	0.895529i	$0.646798\pi$
$930$		0			0
$931$	−204.128	−	53.8518i	−0.219257	−	0.0578429i
$932$	−454.429			−0.487585
$933$		−	1240.89i		−	1.33000i
$934$			1971.84i			2.11118i
$935$		0			0
$936$	−201.196			−0.214953
$937$	360.005			0.384210			0.192105	−	0.981374i	$-0.438469\pi$
$937$	360.005			0.384210			0.192105	+	0.981374i	$0.438469\pi$
$938$	−831.854			−0.886838
$939$			2074.21i			2.20896i
$940$		0			0
$941$		−	499.144i		−	0.530440i	−0.964188	−	0.265220i	$-0.914555\pi$
$941$		−	499.144i		−	0.530440i	0.964188	−	0.265220i	$-0.0854445\pi$
$942$	−1956.34			−2.07680
$943$		−	264.961i		−	0.280976i
$944$		−	564.540i		−	0.598030i
$945$		0			0
$946$		−	911.566i		−	0.963600i
$947$	404.548			0.427189			0.213594	−	0.976922i	$-0.431483\pi$
$947$	404.548			0.427189			0.213594	+	0.976922i	$0.431483\pi$
$948$	3161.30			3.33471
$949$			563.604i			0.593893i
$950$		0			0
$951$	1114.99			1.17244
$952$			26.6881i			0.0280338i
$953$		−	441.098i		−	0.462852i	−0.972852	−	0.231426i	$-0.925661\pi$
$953$		−	441.098i		−	0.462852i	0.972852	−	0.231426i	$-0.0743392\pi$
$954$	2667.31			2.79592
$955$		0			0
$956$	−241.088			−0.252184
$957$	1056.68			1.10416
$958$			1086.77i			1.13442i
$959$	871.673			0.908939
$960$		0			0
$961$	−1759.60			−1.83101
$962$		−	530.938i		−	0.551911i
$963$			924.725i			0.960254i
$964$			373.482i			0.387429i
$965$		0			0
$966$	3332.76			3.45007
$967$	1146.66			1.18579			0.592897	−	0.805279i	$-0.297984\pi$
$967$	1146.66			1.18579			0.592897	+	0.805279i	$0.297984\pi$
$968$			317.110i			0.327593i
$969$	30.4291	−	115.343i	0.0314026	−	0.119033i
$970$		0			0
$971$		−	37.3147i		−	0.0384292i	−0.999815	−	0.0192146i	$-0.993883\pi$
$971$		−	37.3147i		−	0.0384292i	0.999815	−	0.0192146i	$-0.00611657\pi$
$972$		−	1293.31i		−	1.33056i
$973$	819.696			0.842442
$974$	142.219			0.146016
$975$		0			0
$976$	−397.920			−0.407705
$977$		−	892.499i		−	0.913509i	−0.889593	−	0.456755i	$-0.849012\pi$
$977$		−	892.499i		−	0.913509i	0.889593	−	0.456755i	$-0.150988\pi$
$978$	2572.30			2.63017
$979$		−	107.338i		−	0.109640i
$980$		0			0
$981$			669.058i			0.682017i
$982$		−	2245.86i		−	2.28702i
$983$		−	336.089i		−	0.341902i	−0.985280	−	0.170951i	$-0.945316\pi$
$983$		−	336.089i		−	0.341902i	0.985280	−	0.170951i	$-0.0546839\pi$
$984$		−	115.030i		−	0.116900i
$985$		0			0
$986$	−168.296			−0.170686
$987$		−	1699.92i		−	1.72231i
$988$	104.013	−	394.266i	0.105276	−	0.399055i
$989$	2201.64			2.22613
$990$		0			0
$991$			552.478i			0.557495i	0.960364	+	0.278747i	$0.0899192\pi$
$991$			552.478i			0.557495i	−0.960364	+	0.278747i	$0.910081\pi$
$992$	2325.87			2.34463
$993$	−2553.99			−2.57200
$994$	−242.464			−0.243928
$995$		0			0
$996$		−	572.795i		−	0.575095i
$997$	−459.052			−0.460433			−0.230217	−	0.973139i	$-0.573943\pi$
$997$	−459.052			−0.460433			−0.230217	+	0.973139i	$0.573943\pi$
$998$			2458.53i			2.46346i
$999$	−1081.17			−1.08225

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.c.i.151.13	yes	14
5.2	odd	4		475.3.d.d.474.3		28
5.3	odd	4		475.3.d.d.474.26		28
5.4	even	2		475.3.c.h.151.2	✓	14
19.18	odd	2	inner	475.3.c.i.151.2	yes	14
95.18	even	4		475.3.d.d.474.4		28
95.37	even	4		475.3.d.d.474.25		28
95.94	odd	2		475.3.c.h.151.13	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.3.c.h.151.2	✓	14	5.4	even	2
475.3.c.h.151.13	yes	14	95.94	odd	2
475.3.c.i.151.2	yes	14	19.18	odd	2	inner
475.3.c.i.151.13	yes	14	1.1	even	1	trivial
475.3.d.d.474.3		28	5.2	odd	4
475.3.d.d.474.4		28	95.18	even	4
475.3.d.d.474.25		28	95.37	even	4
475.3.d.d.474.26		28	5.3	odd	4

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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.01757i q^{2} -4.83172i q^{3} -5.10575 q^{4} +14.5801 q^{6} -6.15539 q^{7} -3.33669i q^{8} -14.3455 q^{9} +O(q^{10})\)	\(q+3.01757i q^{2} -4.83172i q^{3} -5.10575 q^{4} +14.5801 q^{6} -6.15539 q^{7} -3.33669i q^{8} -14.3455 q^{9} +5.09534 q^{11} +24.6696i q^{12} +4.20326i q^{13} -18.5743i q^{14} -10.3543 q^{16} +1.29941 q^{17} -43.2887i q^{18} +(18.3714 + 4.84663i) q^{19} +29.7411i q^{21} +15.3756i q^{22} -37.1355 q^{23} -16.1219 q^{24} -12.6837 q^{26} +25.8282i q^{27} +31.4279 q^{28} +42.9209i q^{29} +52.1594i q^{31} -44.5916i q^{32} -24.6193i q^{33} +3.92107i q^{34} +73.2448 q^{36} +41.8600i q^{37} +(-14.6251 + 55.4372i) q^{38} +20.3090 q^{39} +7.13498i q^{41} -89.7461 q^{42} -59.2867 q^{43} -26.0155 q^{44} -112.059i q^{46} -57.1573 q^{47} +50.0291i q^{48} -11.1112 q^{49} -6.27840i q^{51} -21.4608i q^{52} +61.6167i q^{53} -77.9384 q^{54} +20.5386i q^{56} +(23.4176 - 88.7657i) q^{57} -129.517 q^{58} +54.5223i q^{59} +38.4303 q^{61} -157.395 q^{62} +88.3024 q^{63} +93.1413 q^{64} +74.2904 q^{66} -44.7851i q^{67} -6.63448 q^{68} +179.428i q^{69} -13.0537i q^{71} +47.8666i q^{72} +134.087 q^{73} -126.316 q^{74} +(-93.8001 - 24.7457i) q^{76} -31.3638 q^{77} +61.2839i q^{78} -128.146i q^{79} -4.31527 q^{81} -21.5303 q^{82} -23.2187 q^{83} -151.851i q^{84} -178.902i q^{86} +207.382 q^{87} -17.0015i q^{88} -21.0659i q^{89} -25.8727i q^{91} +189.605 q^{92} +252.020 q^{93} -172.476i q^{94} -215.454 q^{96} -75.9288i q^{97} -33.5288i q^{98} -73.0954 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9}+O(q^{10}) \) 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9	\( 14 q - 28 q^{4} - 4 q^{6} + 20 q^{7} - 36 q^{9} - 4 q^{11} + 36 q^{16} - 22 q^{17} + 39 q^{19} - 12 q^{23} - 44 q^{24} + 30 q^{26} - 98 q^{28} + 4 q^{36} - 37 q^{38} - 32 q^{39} - 250 q^{42} - 90 q^{43} - 52 q^{44} - 148 q^{47} + 234 q^{49} + 98 q^{54} + 195 q^{57} + 274 q^{58} + 222 q^{61} - 518 q^{62} - 198 q^{63} - 218 q^{64} + 92 q^{66} - 80 q^{68} + 228 q^{73} - 92 q^{74} - 351 q^{76} + 260 q^{77} + 402 q^{81} - 58 q^{82} + 280 q^{83} + 282 q^{87} - 302 q^{92} + 358 q^{93} + 190 q^{96} - 180 q^{99}+O(q^{100}) \) 14 * q - 28 * q^4 - 4 * q^6 + 20 * q^7 - 36 * q^9 - 4 * q^11 + 36 * q^16 - 22 * q^17 + 39 * q^19 - 12 * q^23 - 44 * q^24 + 30 * q^26 - 98 * q^28 + 4 * q^36 - 37 * q^38 - 32 * q^39 - 250 * q^42 - 90 * q^43 - 52 * q^44 - 148 * q^47 + 234 * q^49 + 98 * q^54 + 195 * q^57 + 274 * q^58 + 222 * q^61 - 518 * q^62 - 198 * q^63 - 218 * q^64 + 92 * q^66 - 80 * q^68 + 228 * q^73 - 92 * q^74 - 351 * q^76 + 260 * q^77 + 402 * q^81 - 58 * q^82 + 280 * q^83 + 282 * q^87 - 302 * q^92 + 358 * q^93 + 190 * q^96 - 180 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more