LMFDB - Embedded newform 160.4.c.d.129.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(129,160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 160 = 2^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	160.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:	$9.44030560092$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.359712057600.22
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 17x^{6} + 82x^{4} + 96x^{2} + 9 $ x^8 + 17x^6 + 82x^4 + 96*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.7
Root			$-3.03888i$ of defining polynomial
Character	$\chi$	$=$	160.129
Dual form			160.4.c.d.129.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.57295i q^{3} +(-0.582576 - 11.1652i) q^{5} +22.1403i q^{7} -46.4955 q^{9} +O(q^{10})$	$q+8.57295i q^{3} +(-0.582576 - 11.1652i) q^{5} +22.1403i q^{7} -46.4955 q^{9} -27.1347 q^{11} +70.3303i q^{13} +(95.7183 - 4.99439i) q^{15} -73.3212i q^{17} -110.033 q^{19} -189.808 q^{21} -107.870i q^{23} +(-124.321 + 13.0091i) q^{25} -167.134i q^{27} -68.6424 q^{29} +137.167 q^{31} -232.624i q^{33} +(247.200 - 12.8984i) q^{35} +60.3121i q^{37} -602.938 q^{39} +95.1470 q^{41} +501.479i q^{43} +(27.0871 + 519.129i) q^{45} +439.305i q^{47} -147.192 q^{49} +628.579 q^{51} +286.955i q^{53} +(15.8080 + 302.963i) q^{55} -943.303i q^{57} -547.175 q^{59} +511.459 q^{61} -1029.42i q^{63} +(785.248 - 40.9727i) q^{65} -301.547i q^{67} +924.762 q^{69} -82.8978 q^{71} +763.267i q^{73} +(-111.526 - 1065.80i) q^{75} -600.770i q^{77} +1011.45 q^{79} +177.450 q^{81} +704.554i q^{83} +(-818.642 + 42.7152i) q^{85} -588.468i q^{87} +743.212 q^{89} -1557.13 q^{91} +1175.93i q^{93} +(64.1023 + 1228.53i) q^{95} -1136.00i q^{97} +1261.64 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 32 q^{5} - 152 q^{9}+O(q^{10}) $ 8 * q + 32 * q^5 - 152 * q^9	$ 8 q + 32 q^{5} - 152 q^{9} - 272 q^{21} - 408 q^{25} + 624 q^{29} - 192 q^{41} + 400 q^{45} - 2424 q^{49} + 2112 q^{61} + 2176 q^{65} + 3952 q^{69} - 1000 q^{81} - 5376 q^{85} + 80 q^{89}+O(q^{100}) $ 8 * q + 32 * q^5 - 152 * q^9 - 272 * q^21 - 408 * q^25 + 624 * q^29 - 192 * q^41 + 400 * q^45 - 2424 * q^49 + 2112 * q^61 + 2176 * q^65 + 3952 * q^69 - 1000 * q^81 - 5376 * q^85 + 80 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$101$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$			8.57295i			1.64986i	0.565231	+	0.824932i	$0.308787\pi$
$3$			8.57295i			1.64986i	−0.565231	+	0.824932i	$0.691213\pi$
$4$		0			0
$5$	−0.582576	−	11.1652i	−0.0521072	−	0.998641i
$5$	−0.582576	−	11.1652i	−0.0521072	−	0.998641i
$6$		0			0
$7$			22.1403i			1.19546i	0.801696	+	0.597732i	$0.203931\pi$
$7$			22.1403i			1.19546i	−0.801696	+	0.597732i	$0.796069\pi$
$8$		0			0
$9$	−46.4955			−1.72205
$10$		0			0
$11$	−27.1347			−0.743765			−0.371882	−	0.928280i	$-0.621288\pi$
$11$	−27.1347			−0.743765			−0.371882	+	0.928280i	$0.621288\pi$
$12$		0			0
$13$			70.3303i			1.50047i	0.661171	+	0.750235i	$0.270060\pi$
$13$			70.3303i			1.50047i	−0.661171	+	0.750235i	$0.729940\pi$
$14$		0			0
$15$	95.7183	−	4.99439i	1.64762	−	0.0859698i
$16$		0			0
$17$		−	73.3212i		−	1.04606i	−0.852315	−	0.523030i	$-0.824802\pi$
$17$		−	73.3212i		−	1.04606i	0.852315	−	0.523030i	$-0.175198\pi$
$18$		0			0
$19$	−110.033			−1.32859			−0.664294	−	0.747471i	$-0.731268\pi$
$19$	−110.033			−1.32859			−0.664294	+	0.747471i	$0.731268\pi$
$20$		0			0
$21$	−189.808			−1.97235
$22$		0			0
$23$		−	107.870i		−	0.977931i	−0.872303	−	0.488965i	$-0.837374\pi$
$23$		−	107.870i		−	0.977931i	0.872303	−	0.488965i	$-0.162626\pi$
$24$		0			0
$25$	−124.321	+	13.0091i	−0.994570	+	0.104073i
$26$		0			0
$27$		−	167.134i		−	1.19129i
$28$		0			0
$29$	−68.6424			−0.439537			−0.219769	−	0.975552i	$-0.570530\pi$
$29$	−68.6424			−0.439537			−0.219769	+	0.975552i	$0.570530\pi$
$30$		0			0
$31$	137.167			0.794708			0.397354	−	0.917665i	$-0.369928\pi$
$31$	137.167			0.794708			0.397354	+	0.917665i	$0.369928\pi$
$32$		0			0
$33$		−	232.624i		−	1.22711i
$34$		0			0
$35$	247.200	−	12.8984i	1.19384	−	0.0622922i
$36$		0			0
$37$			60.3121i			0.267980i	0.990983	+	0.133990i	$0.0427790\pi$
$37$			60.3121i			0.267980i	−0.990983	+	0.133990i	$0.957221\pi$
$38$		0			0
$39$	−602.938			−2.47557
$40$		0			0
$41$	95.1470			0.362426			0.181213	−	0.983444i	$-0.441998\pi$
$41$	95.1470			0.362426			0.181213	+	0.983444i	$0.441998\pi$
$42$		0			0
$43$			501.479i			1.77848i	0.457438	+	0.889241i	$0.348767\pi$
$43$			501.479i			1.77848i	−0.457438	+	0.889241i	$0.651233\pi$
$44$		0			0
$45$	27.0871	+	519.129i	0.0897313	+	1.71971i
$46$		0			0
$47$			439.305i			1.36339i	0.731637	+	0.681694i	$0.238756\pi$
$47$			439.305i			1.36339i	−0.731637	+	0.681694i	$0.761244\pi$
$48$		0			0
$49$	−147.192			−0.429132
$50$		0			0
$51$	628.579			1.72586
$52$		0			0
$53$			286.955i			0.743703i	0.928292	+	0.371851i	$0.121277\pi$
$53$			286.955i			0.743703i	−0.928292	+	0.371851i	$0.878723\pi$
$54$		0			0
$55$	15.8080	+	302.963i	0.0387555	+	0.742755i
$56$		0			0
$57$		−	943.303i		−	2.19199i
$58$		0			0
$59$	−547.175			−1.20739			−0.603696	−	0.797215i	$-0.706306\pi$
$59$	−547.175			−1.20739			−0.603696	+	0.797215i	$0.706306\pi$
$60$		0			0
$61$	511.459			1.07353			0.536767	−	0.843730i	$-0.319645\pi$
$61$	511.459			1.07353			0.536767	+	0.843730i	$0.319645\pi$
$62$		0			0
$63$		−	1029.42i		−	2.05865i
$64$		0			0
$65$	785.248	−	40.9727i	1.49843	−	0.0781852i
$66$		0			0
$67$		−	301.547i		−	0.549848i	−0.961466	−	0.274924i	$-0.911347\pi$
$67$		−	301.547i		−	0.549848i	0.961466	−	0.274924i	$-0.0886527\pi$
$68$		0			0
$69$	924.762			1.61345
$70$		0			0
$71$	−82.8978			−0.138566			−0.0692828	−	0.997597i	$-0.522071\pi$
$71$	−82.8978			−0.138566			−0.0692828	+	0.997597i	$0.522071\pi$
$72$		0			0
$73$			763.267i			1.22375i	0.790955	+	0.611874i	$0.209584\pi$
$73$			763.267i			1.22375i	−0.790955	+	0.611874i	$0.790416\pi$
$74$		0			0
$75$	−111.526	−	1065.80i	−0.171706	−	1.64091i
$76$		0			0
$77$		−	600.770i		−	0.889144i
$78$		0			0
$79$	1011.45			1.44047			0.720236	−	0.693729i	$-0.244034\pi$
$79$	1011.45			1.44047			0.720236	+	0.693729i	$0.244034\pi$
$80$		0			0
$81$	177.450			0.243416
$82$		0			0
$83$			704.554i			0.931745i	0.884852	+	0.465872i	$0.154259\pi$
$83$			704.554i			0.931745i	−0.884852	+	0.465872i	$0.845741\pi$
$84$		0			0
$85$	−818.642	+	42.7152i	−1.04464	+	0.0545072i
$86$		0			0
$87$		−	588.468i		−	0.725177i
$88$		0			0
$89$	743.212			0.885172			0.442586	−	0.896726i	$-0.354061\pi$
$89$	743.212			0.885172			0.442586	+	0.896726i	$0.354061\pi$
$90$		0			0
$91$	−1557.13			−1.79376
$92$		0			0
$93$			1175.93i			1.31116i
$94$		0			0
$95$	64.1023	+	1228.53i	0.0692290	+	1.32678i
$96$		0			0
$97$		−	1136.00i		−	1.18911i	−0.804056	−	0.594553i	$-0.797329\pi$
$97$		−	1136.00i		−	1.18911i	0.804056	−	0.594553i	$-0.202671\pi$
$98$		0			0
$99$	1261.64			1.28080
$100$		0			0
$101$	291.212			0.286898			0.143449	−	0.989658i	$-0.454181\pi$
$101$	291.212			0.286898			0.143449	+	0.989658i	$0.454181\pi$
$102$		0			0
$103$			200.445i			0.191751i	0.995393	+	0.0958757i	$0.0305651\pi$
$103$			200.445i			0.191751i	−0.995393	+	0.0958757i	$0.969435\pi$
$104$		0			0
$105$	110.577	+	2119.23i	0.102774	+	1.96967i
$106$		0			0
$107$			754.342i			0.681542i	0.940146	+	0.340771i	$0.110688\pi$
$107$			754.342i			0.681542i	−0.940146	+	0.340771i	$0.889312\pi$
$108$		0			0
$109$	−501.680			−0.440846			−0.220423	−	0.975404i	$-0.570744\pi$
$109$	−501.680			−0.440846			−0.220423	+	0.975404i	$0.570744\pi$
$110$		0			0
$111$	−517.053			−0.442130
$112$		0			0
$113$		−	497.248i		−	0.413958i	−0.978345	−	0.206979i	$-0.933637\pi$
$113$		−	497.248i		−	0.413958i	0.978345	−	0.206979i	$-0.0663631\pi$
$114$		0			0
$115$	−1204.38	+	62.8423i	−0.976602	+	0.0509572i
$116$		0			0
$117$		−	3270.04i		−	2.58389i
$118$		0			0
$119$	1623.35			1.25053
$120$		0			0
$121$	−594.709			−0.446814
$122$		0			0
$123$			815.690i			0.597954i
$124$		0			0
$125$	217.675	+	1380.49i	0.155756	+	0.987796i
$126$		0			0
$127$			915.221i			0.639470i	0.947507	+	0.319735i	$0.103594\pi$
$127$			915.221i			0.639470i	−0.947507	+	0.319735i	$0.896406\pi$
$128$		0			0
$129$	−4299.15			−2.93426
$130$		0			0
$131$	−607.419			−0.405118			−0.202559	−	0.979270i	$-0.564926\pi$
$131$	−607.419			−0.405118			−0.202559	+	0.979270i	$0.564926\pi$
$132$		0			0
$133$		−	2436.15i		−	1.58828i
$134$		0			0
$135$	−1866.07	+	97.3679i	−1.18967	+	0.0620748i
$136$		0			0
$137$			1418.98i			0.884900i	0.896793	+	0.442450i	$0.145891\pi$
$137$			1418.98i			0.884900i	−0.896793	+	0.442450i	$0.854109\pi$
$138$		0			0
$139$	3035.85			1.85250			0.926250	−	0.376910i	$-0.123013\pi$
$139$	3035.85			1.85250			0.926250	+	0.376910i	$0.123013\pi$
$140$		0			0
$141$	−3766.14			−2.24941
$142$		0			0
$143$		−	1908.39i		−	1.11600i
$144$		0			0
$145$	39.9894	+	766.403i	0.0229030	+	0.438940i
$146$		0			0
$147$		−	1261.87i		−	0.708011i
$148$		0			0
$149$	2649.26			1.45662			0.728308	−	0.685250i	$-0.240307\pi$
$149$	2649.26			1.45662			0.728308	+	0.685250i	$0.240307\pi$
$150$		0			0
$151$	−283.297			−0.152678			−0.0763390	−	0.997082i	$-0.524323\pi$
$151$	−283.297			−0.152678			−0.0763390	+	0.997082i	$0.524323\pi$
$152$		0			0
$153$			3409.10i			1.80137i
$154$		0			0
$155$	−79.9103	−	1531.49i	−0.0414100	−	0.793629i
$156$		0			0
$157$		−	1414.88i		−	0.719235i	−0.933100	−	0.359617i	$-0.882907\pi$
$157$		−	1414.88i		−	0.719235i	0.933100	−	0.359617i	$-0.117093\pi$
$158$		0			0
$159$	−2460.05			−1.22701
$160$		0			0
$161$	2388.27			1.16908
$162$		0			0
$163$		−	1725.68i		−	0.829239i	−0.909995	−	0.414619i	$-0.863915\pi$
$163$		−	1725.68i		−	0.829239i	0.909995	−	0.414619i	$-0.136085\pi$
$164$		0			0
$165$	−2597.28	+	135.521i	−1.22544	+	0.0639413i
$166$		0			0
$167$		−	1979.92i		−	0.917428i	−0.888584	−	0.458714i	$-0.848310\pi$
$167$		−	1979.92i		−	0.917428i	0.888584	−	0.458714i	$-0.151690\pi$
$168$		0			0
$169$	−2749.35			−1.25141
$170$		0			0
$171$	5116.01			2.28790
$172$		0			0
$173$			1468.72i			0.645460i	0.946491	+	0.322730i	$0.104601\pi$
$173$			1468.72i			0.645460i	−0.946491	+	0.322730i	$0.895399\pi$
$174$		0			0
$175$	−288.025	−	2752.51i	−0.124415	−	1.18897i
$176$		0			0
$177$		−	4690.90i		−	1.99203i
$178$		0			0
$179$	−2978.59			−1.24375			−0.621873	−	0.783118i	$-0.713628\pi$
$179$	−2978.59			−1.24375			−0.621873	+	0.783118i	$0.713628\pi$
$180$		0			0
$181$	−1682.07			−0.690760			−0.345380	−	0.938463i	$-0.612250\pi$
$181$	−1682.07			−0.690760			−0.345380	+	0.938463i	$0.612250\pi$
$182$		0			0
$183$			4384.71i			1.77119i
$184$		0			0
$185$	673.394	−	35.1364i	0.267616	−	0.0139637i
$186$		0			0
$187$			1989.55i			0.778022i
$188$		0			0
$189$	3700.38			1.42414
$190$		0			0
$191$	−274.334			−0.103927			−0.0519637	−	0.998649i	$-0.516548\pi$
$191$	−274.334			−0.103927			−0.0519637	+	0.998649i	$0.516548\pi$
$192$		0			0
$193$			2898.50i			1.08103i	0.841335	+	0.540514i	$0.181770\pi$
$193$			2898.50i			1.08103i	−0.841335	+	0.540514i	$0.818230\pi$
$194$		0			0
$195$	351.257	+	6731.90i	0.128995	+	2.47221i
$196$		0			0
$197$		−	2330.37i		−	0.842801i	−0.906875	−	0.421400i	$-0.861539\pi$
$197$		−	2330.37i		−	0.842801i	0.906875	−	0.421400i	$-0.138461\pi$
$198$		0			0
$199$	1105.05			0.393644			0.196822	−	0.980439i	$-0.436938\pi$
$199$	1105.05			0.393644			0.196822	+	0.980439i	$0.436938\pi$
$200$		0			0
$201$	2585.15			0.907175
$202$		0			0
$203$		−	1519.76i		−	0.525451i
$204$		0			0
$205$	−55.4303	−	1062.33i	−0.0188850	−	0.361933i
$206$		0			0
$207$			5015.45i			1.68405i
$208$		0			0
$209$	2985.70			0.988158
$210$		0			0
$211$	−4749.82			−1.54972			−0.774860	−	0.632133i	$-0.782180\pi$
$211$	−4749.82			−1.54972			−0.774860	+	0.632133i	$0.782180\pi$
$212$		0			0
$213$		−	710.679i		−	0.228615i
$214$		0			0
$215$	5599.08	−	292.149i	1.77607	−	0.0926717i
$216$		0			0
$217$			3036.92i			0.950044i
$218$		0			0
$219$	−6543.45			−2.01902
$220$		0			0
$221$	5156.70			1.56958
$222$		0			0
$223$			1889.71i			0.567462i	0.958904	+	0.283731i	$0.0915723\pi$
$223$			1889.71i			0.567462i	−0.958904	+	0.283731i	$0.908428\pi$
$224$		0			0
$225$	5780.37	−	604.864i	1.71270	−	0.179219i
$226$		0			0
$227$		−	4154.11i		−	1.21462i	−0.794466	−	0.607309i	$-0.792249\pi$
$227$		−	4154.11i		−	1.21462i	0.794466	−	0.607309i	$-0.207751\pi$
$228$		0			0
$229$	888.642			0.256433			0.128216	−	0.991746i	$-0.459075\pi$
$229$	888.642			0.256433			0.128216	+	0.991746i	$0.459075\pi$
$230$		0			0
$231$	5150.37			1.46697
$232$		0			0
$233$		−	4919.38i		−	1.38317i	−0.722294	−	0.691586i	$-0.756912\pi$
$233$		−	4919.38i		−	1.38317i	0.722294	−	0.691586i	$-0.243088\pi$
$234$		0			0
$235$	4904.91	−	255.929i	1.36154	−	0.0710423i
$236$		0			0
$237$			8671.13i			2.37658i
$238$		0			0
$239$	−2178.00			−0.589468			−0.294734	−	0.955579i	$-0.595231\pi$
$239$	−2178.00			−0.589468			−0.294734	+	0.955579i	$0.595231\pi$
$240$		0			0
$241$	−3156.12			−0.843583			−0.421792	−	0.906693i	$-0.638599\pi$
$241$	−3156.12			−0.843583			−0.421792	+	0.906693i	$0.638599\pi$
$242$		0			0
$243$		−	2991.34i		−	0.789688i
$244$		0			0
$245$	85.7507	+	1643.43i	0.0223609	+	0.428549i
$246$		0			0
$247$		−	7738.62i		−	1.99351i
$248$		0			0
$249$	−6040.10			−1.53725
$250$		0			0
$251$	−2719.20			−0.683801			−0.341901	−	0.939736i	$-0.611071\pi$
$251$	−2719.20			−0.683801			−0.341901	+	0.939736i	$0.611071\pi$
$252$		0			0
$253$			2927.01i			0.727350i
$254$		0			0
$255$	−366.195	−	7018.18i	−0.0899295	−	1.72351i
$256$		0			0
$257$		−	749.067i		−	0.181811i	−0.995860	−	0.0909056i	$-0.971024\pi$
$257$		−	749.067i		−	0.181811i	0.995860	−	0.0909056i	$-0.0289762\pi$
$258$		0			0
$259$	−1335.33			−0.320360
$260$		0			0
$261$	3191.56			0.756907
$262$		0			0
$263$			2546.04i			0.596942i	0.954419	+	0.298471i	$0.0964766\pi$
$263$			2546.04i			0.596942i	−0.954419	+	0.298471i	$0.903523\pi$
$264$		0			0
$265$	3203.89	−	167.173i	0.742692	−	0.0387522i
$266$		0			0
$267$			6371.52i			1.46041i
$268$		0			0
$269$	7982.07			1.80920			0.904601	−	0.426260i	$-0.140169\pi$
$269$	7982.07			1.80920			0.904601	+	0.426260i	$0.140169\pi$
$270$		0			0
$271$	−1686.58			−0.378054			−0.189027	−	0.981972i	$-0.560533\pi$
$271$	−1686.58			−0.378054			−0.189027	+	0.981972i	$0.560533\pi$
$272$		0			0
$273$		−	13349.2i		−	2.95946i
$274$		0			0
$275$	3373.42	−	352.998i	0.739726	−	0.0774056i
$276$		0			0
$277$		−	1423.07i		−	0.308679i	−0.988018	−	0.154339i	$-0.950675\pi$
$277$		−	1423.07i		−	0.308679i	0.988018	−	0.154339i	$-0.0493249\pi$
$278$		0			0
$279$	−6377.65			−1.36853
$280$		0			0
$281$	5418.67			1.15036			0.575180	−	0.818027i	$-0.304932\pi$
$281$	5418.67			1.15036			0.575180	+	0.818027i	$0.304932\pi$
$282$		0			0
$283$		−	8343.38i		−	1.75252i	−0.481842	−	0.876258i	$-0.660032\pi$
$283$		−	8343.38i		−	1.75252i	0.481842	−	0.876258i	$-0.339968\pi$
$284$		0			0
$285$	−10532.1	+	549.545i	−2.18901	+	0.114218i
$286$		0			0
$287$			2106.58i			0.433267i
$288$		0			0
$289$	−463.000			−0.0942398
$290$		0			0
$291$	9738.87			1.96186
$292$		0			0
$293$			2331.73i			0.464919i	0.972606	+	0.232459i	$0.0746772\pi$
$293$			2331.73i			0.464919i	−0.972606	+	0.232459i	$0.925323\pi$
$294$		0			0
$295$	318.771	+	6109.29i	0.0629137	+	1.20575i
$296$		0			0
$297$			4535.12i			0.886041i
$298$		0			0
$299$	7586.51			1.46736
$300$		0			0
$301$	−11102.9			−2.12611
$302$		0			0
$303$			2496.55i			0.473343i
$304$		0			0
$305$	−297.964	−	5710.52i	−0.0559388	−	1.07208i
$306$		0			0
$307$			2100.00i			0.390401i	0.980763	+	0.195200i	$0.0625357\pi$
$307$			2100.00i			0.390401i	−0.980763	+	0.195200i	$0.937464\pi$
$308$		0			0
$309$	−1718.40			−0.316364
$310$		0			0
$311$	−8501.38			−1.55006			−0.775030	−	0.631924i	$-0.782265\pi$
$311$	−8501.38			−1.55006			−0.775030	+	0.631924i	$0.782265\pi$
$312$		0			0
$313$			7257.73i			1.31064i	0.755350	+	0.655321i	$0.227467\pi$
$313$			7257.73i			1.31064i	−0.755350	+	0.655321i	$0.772533\pi$
$314$		0			0
$315$	−11493.7	+	599.717i	−2.05586	+	0.107271i
$316$		0			0
$317$			9639.14i			1.70785i	0.520397	+	0.853925i	$0.325784\pi$
$317$			9639.14i			1.70785i	−0.520397	+	0.853925i	$0.674216\pi$
$318$		0			0
$319$	1862.59			0.326912
$320$		0			0
$321$	−6466.93			−1.12445
$322$		0			0
$323$			8067.72i			1.38978i
$324$		0			0
$325$	−914.933	−	8743.55i	−0.156158	−	1.49232i
$326$		0			0
$327$		−	4300.88i		−	0.727337i
$328$		0			0
$329$	−9726.34			−1.62988
$330$		0			0
$331$	360.466			0.0598581			0.0299290	−	0.999552i	$-0.490472\pi$
$331$	360.466			0.0598581			0.0299290	+	0.999552i	$0.490472\pi$
$332$		0			0
$333$		−	2804.24i		−	0.461476i
$334$		0			0
$335$	−3366.82	+	175.674i	−0.549101	+	0.0286510i
$336$		0			0
$337$			2820.47i			0.455908i	0.973672	+	0.227954i	$0.0732036\pi$
$337$			2820.47i			0.455908i	−0.973672	+	0.227954i	$0.926796\pi$
$338$		0			0
$339$	4262.89			0.682974
$340$		0			0
$341$	−3721.99			−0.591076
$342$		0			0
$343$			4335.24i			0.682451i
$344$		0			0
$345$	−538.744	−	10325.1i	−0.0840725	−	1.61126i
$346$		0			0
$347$			8617.62i			1.33319i	0.745419	+	0.666597i	$0.232250\pi$
$347$			8617.62i			1.33319i	−0.745419	+	0.666597i	$0.767750\pi$
$348$		0			0
$349$	−735.067			−0.112743			−0.0563714	−	0.998410i	$-0.517953\pi$
$349$	−735.067			−0.112743			−0.0563714	+	0.998410i	$0.517953\pi$
$350$		0			0
$351$	11754.6			1.78750
$352$		0			0
$353$			4535.35i			0.683830i	0.939731	+	0.341915i	$0.111076\pi$
$353$			4535.35i			0.683830i	−0.939731	+	0.341915i	$0.888924\pi$
$354$		0			0
$355$	48.2943	+	925.567i	0.00722026	+	0.138377i
$356$		0			0
$357$			13916.9i			2.06320i
$358$		0			0
$359$	5426.44			0.797762			0.398881	−	0.917003i	$-0.369399\pi$
$359$	5426.44			0.797762			0.398881	+	0.917003i	$0.369399\pi$
$360$		0			0
$361$	5248.15			0.765148
$362$		0			0
$363$		−	5098.41i		−	0.737182i
$364$		0			0
$365$	8521.99	−	444.661i	1.22209	−	0.0637660i
$366$		0			0
$367$		−	609.673i		−	0.0867157i	−0.999060	−	0.0433579i	$-0.986194\pi$
$367$		−	609.673i		−	0.0867157i	0.999060	−	0.0433579i	$-0.0138056\pi$
$368$		0			0
$369$	−4423.90			−0.624117
$370$		0			0
$371$	−6353.26			−0.889069
$372$		0			0
$373$		−	5089.42i		−	0.706489i	−0.935531	−	0.353244i	$-0.885078\pi$
$373$		−	5089.42i		−	0.706489i	0.935531	−	0.353244i	$-0.114922\pi$
$374$		0			0
$375$	−11834.8	+	1866.12i	−1.62973	+	0.256976i
$376$		0			0
$377$		−	4827.64i		−	0.659513i
$378$		0			0
$379$	−910.876			−0.123453			−0.0617263	−	0.998093i	$-0.519661\pi$
$379$	−910.876			−0.123453			−0.0617263	+	0.998093i	$0.519661\pi$
$380$		0			0
$381$	−7846.14			−1.05504
$382$		0			0
$383$		−	7686.98i		−	1.02555i	−0.858522	−	0.512776i	$-0.828617\pi$
$383$		−	7686.98i		−	1.02555i	0.858522	−	0.512776i	$-0.171383\pi$
$384$		0			0
$385$	−6707.68	+	349.994i	−0.887936	+	0.0463307i
$386$		0			0
$387$		−	23316.5i		−	3.06264i
$388$		0			0
$389$	−4372.77			−0.569943			−0.284972	−	0.958536i	$-0.591984\pi$
$389$	−4372.77			−0.569943			−0.284972	+	0.958536i	$0.591984\pi$
$390$		0			0
$391$	−7909.14			−1.02297
$392$		0			0
$393$		−	5207.38i		−	0.668390i
$394$		0			0
$395$	−589.247	−	11293.0i	−0.0750589	−	1.43851i
$396$		0			0
$397$		−	12591.9i		−	1.59186i	−0.605391	−	0.795928i	$-0.706983\pi$
$397$		−	12591.9i		−	1.59186i	0.605391	−	0.795928i	$-0.293017\pi$
$398$		0			0
$399$	20885.0			2.62045
$400$		0			0
$401$	3614.48			0.450122			0.225061	−	0.974345i	$-0.427742\pi$
$401$	3614.48			0.450122			0.225061	+	0.974345i	$0.427742\pi$
$402$		0			0
$403$			9647.01i			1.19244i
$404$		0			0
$405$	−103.378	−	1981.26i	−0.0126837	−	0.243085i
$406$		0			0
$407$		−	1636.55i		−	0.199314i
$408$		0			0
$409$	−639.314			−0.0772910			−0.0386455	−	0.999253i	$-0.512304\pi$
$409$	−639.314			−0.0772910			−0.0386455	+	0.999253i	$0.512304\pi$
$410$		0			0
$411$	−12164.8			−1.45997
$412$		0			0
$413$		−	12114.6i		−	1.44339i
$414$		0			0
$415$	7866.45	−	410.456i	0.930479	−	0.0485506i
$416$		0			0
$417$			26026.2i			3.05637i
$418$		0			0
$419$	11018.4			1.28469			0.642346	−	0.766415i	$-0.277961\pi$
$419$	11018.4			1.28469			0.642346	+	0.766415i	$0.277961\pi$
$420$		0			0
$421$	16513.1			1.91164			0.955820	−	0.293952i	$-0.0949707\pi$
$421$	16513.1			1.91164			0.955820	+	0.293952i	$0.0949707\pi$
$422$		0			0
$423$		−	20425.7i		−	2.34783i
$424$		0			0
$425$	953.842	+	9115.38i	0.108866	+	1.04038i
$426$		0			0
$427$			11323.9i			1.28337i
$428$		0			0
$429$	16360.5			1.84124
$430$		0			0
$431$	−6106.80			−0.682492			−0.341246	−	0.939974i	$-0.610849\pi$
$431$	−6106.80			−0.682492			−0.341246	+	0.939974i	$0.610849\pi$
$432$		0			0
$433$			1757.88i			0.195100i	0.995231	+	0.0975500i	$0.0311006\pi$
$433$			1757.88i			0.195100i	−0.995231	+	0.0975500i	$0.968899\pi$
$434$		0			0
$435$	−6570.33	+	342.827i	−0.724192	+	0.0377869i
$436$		0			0
$437$			11869.2i			1.29927i
$438$		0			0
$439$	1055.02			0.114700			0.0573500	−	0.998354i	$-0.481735\pi$
$439$	1055.02			0.114700			0.0573500	+	0.998354i	$0.481735\pi$
$440$		0			0
$441$	6843.78			0.738989
$442$		0			0
$443$			7775.51i			0.833918i	0.908925	+	0.416959i	$0.136904\pi$
$443$			7775.51i			0.833918i	−0.908925	+	0.416959i	$0.863096\pi$
$444$		0			0
$445$	−432.977	−	8298.08i	−0.0461238	−	0.883970i
$446$		0			0
$447$			22712.0i			2.40322i
$448$		0			0
$449$	11245.1			1.18194			0.590969	−	0.806694i	$-0.298745\pi$
$449$	11245.1			1.18194			0.590969	+	0.806694i	$0.298745\pi$
$450$		0			0
$451$	−2581.78			−0.269560
$452$		0			0
$453$		−	2428.69i		−	0.251898i
$454$		0			0
$455$	907.148	+	17385.6i	0.0934676	+	1.79132i
$456$		0			0
$457$			13576.9i			1.38972i	0.719145	+	0.694860i	$0.244534\pi$
$457$			13576.9i			1.38972i	−0.719145	+	0.694860i	$0.755466\pi$
$458$		0			0
$459$	−12254.4			−1.24616
$460$		0			0
$461$	−9605.08			−0.970397			−0.485199	−	0.874404i	$-0.661253\pi$
$461$	−9605.08			−0.970397			−0.485199	+	0.874404i	$0.661253\pi$
$462$		0			0
$463$			3226.71i			0.323883i	0.986800	+	0.161942i	$0.0517756\pi$
$463$			3226.71i			0.323883i	−0.986800	+	0.161942i	$0.948224\pi$
$464$		0			0
$465$	13129.4	−	685.067i	1.30938	−	0.0683209i
$466$		0			0
$467$		−	20.6790i		−	0.00204906i	−0.999999	−	0.00102453i	$-0.999674\pi$
$467$		−	20.6790i		−	0.00204906i	0.999999	−	0.00102453i	$-0.000326118\pi$
$468$		0			0
$469$	6676.34			0.657323
$470$		0			0
$471$	12129.7			1.18664
$472$		0			0
$473$		−	13607.5i		−	1.32277i
$474$		0			0
$475$	13679.4	−	1431.42i	1.32137	−	0.138270i
$476$		0			0
$477$		−	13342.1i		−	1.28070i
$478$		0			0
$479$	−12492.9			−1.19168			−0.595841	−	0.803102i	$-0.703181\pi$
$479$	−12492.9			−1.19168			−0.595841	+	0.803102i	$0.703181\pi$
$480$		0			0
$481$	−4241.77			−0.402096
$482$		0			0
$483$			20474.5i			1.92882i
$484$		0			0
$485$	−12683.6	+	661.806i	−1.18749	+	0.0619610i
$486$		0			0
$487$			4944.60i			0.460084i	0.973181	+	0.230042i	$0.0738864\pi$
$487$			4944.60i			0.460084i	−0.973181	+	0.230042i	$0.926114\pi$
$488$		0			0
$489$	14794.2			1.36813
$490$		0			0
$491$	7712.73			0.708902			0.354451	−	0.935075i	$-0.384668\pi$
$491$	7712.73			0.708902			0.354451	+	0.935075i	$0.384668\pi$
$492$		0			0
$493$			5032.95i			0.459782i
$494$		0			0
$495$	−735.000	−	14086.4i	−0.0667390	−	1.27906i
$496$		0			0
$497$		−	1835.38i		−	0.165650i
$498$		0			0
$499$	−1492.41			−0.133886			−0.0669432	−	0.997757i	$-0.521325\pi$
$499$	−1492.41			−0.133886			−0.0669432	+	0.997757i	$0.521325\pi$
$500$		0			0
$501$	16973.7			1.51363
$502$		0			0
$503$		−	3105.13i		−	0.275250i	−0.990484	−	0.137625i	$-0.956053\pi$
$503$		−	3105.13i		−	0.275250i	0.990484	−	0.137625i	$-0.0439469\pi$
$504$		0			0
$505$	−169.653	−	3251.43i	−0.0149494	−	0.286508i
$506$		0			0
$507$		−	23570.1i		−	2.06466i
$508$		0			0
$509$	−15363.5			−1.33787			−0.668933	−	0.743322i	$-0.733249\pi$
$509$	−15363.5			−1.33787			−0.668933	+	0.743322i	$0.733249\pi$
$510$		0			0
$511$	−16898.9			−1.46295
$512$		0			0
$513$			18390.1i			1.58274i
$514$		0			0
$515$	2237.99	−	116.774i	0.191491	−	0.00999162i
$516$		0			0
$517$		−	11920.4i		−	1.01404i
$518$		0			0
$519$	−12591.2			−1.06492
$520$		0			0
$521$	−9924.25			−0.834529			−0.417264	−	0.908785i	$-0.637011\pi$
$521$	−9924.25			−0.834529			−0.417264	+	0.908785i	$0.637011\pi$
$522$		0			0
$523$		−	455.146i		−	0.0380538i	−0.999819	−	0.0190269i	$-0.993943\pi$
$523$		−	455.146i		−	0.0380538i	0.999819	−	0.0190269i	$-0.00605681\pi$
$524$		0			0
$525$	23597.1	−	2469.22i	1.96164	−	0.205268i
$526$		0			0
$527$		−	10057.3i		−	0.831312i
$528$		0			0
$529$	531.111			0.0436517
$530$		0			0
$531$	25441.1			2.07919
$532$		0			0
$533$			6691.72i			0.543809i
$534$		0			0
$535$	8422.34	−	439.461i	0.680616	−	0.0355132i
$536$		0			0
$537$		−	25535.3i		−	2.05201i
$538$		0			0
$539$	3994.02			0.319174
$540$		0			0
$541$	23383.9			1.85832			0.929160	−	0.369678i	$-0.120532\pi$
$541$	23383.9			1.85832			0.929160	+	0.369678i	$0.120532\pi$
$542$		0			0
$543$		−	14420.3i		−	1.13966i
$544$		0			0
$545$	292.267	+	5601.34i	0.0229713	+	0.440248i
$546$		0			0
$547$		−	6908.03i		−	0.539974i	−0.962864	−	0.269987i	$-0.912981\pi$
$547$		−	6908.03i		−	0.539974i	0.962864	−	0.269987i	$-0.0870195\pi$
$548$		0			0
$549$	−23780.5			−1.84868
$550$		0			0
$551$	7552.90			0.583964
$552$		0			0
$553$			22393.8i			1.72203i
$554$		0			0
$555$	301.222	+	5772.97i	0.0230382	+	0.441530i
$556$		0			0
$557$			3221.81i			0.245085i	0.992463	+	0.122543i	$0.0391048\pi$
$557$			3221.81i			0.245085i	−0.992463	+	0.122543i	$0.960895\pi$
$558$		0			0
$559$	−35269.1			−2.66856
$560$		0			0
$561$	−17056.3			−1.28363
$562$		0			0
$563$		−	14822.7i		−	1.10959i	−0.831986	−	0.554796i	$-0.812796\pi$
$563$		−	14822.7i		−	1.10959i	0.831986	−	0.554796i	$-0.187204\pi$
$564$		0			0
$565$	−5551.85	+	289.685i	−0.413395	+	0.0215701i
$566$		0			0
$567$			3928.79i			0.290994i
$568$		0			0
$569$	−6434.10			−0.474045			−0.237022	−	0.971504i	$-0.576172\pi$
$569$	−6434.10			−0.474045			−0.237022	+	0.971504i	$0.576172\pi$
$570$		0			0
$571$	17760.3			1.30165			0.650827	−	0.759226i	$-0.274422\pi$
$571$	17760.3			1.30165			0.650827	+	0.759226i	$0.274422\pi$
$572$		0			0
$573$		−	2351.85i		−	0.171466i
$574$		0			0
$575$	1403.29	+	13410.5i	0.101776	+	0.972620i
$576$		0			0
$577$			1341.96i			0.0968227i	0.998827	+	0.0484113i	$0.0154158\pi$
$577$			1341.96i			0.0968227i	−0.998827	+	0.0484113i	$0.984584\pi$
$578$		0			0
$579$	−24848.7			−1.78355
$580$		0			0
$581$	−15599.0			−1.11387
$582$		0			0
$583$		−	7786.42i		−	0.553140i
$584$		0			0
$585$	−36510.5	+	1905.05i	−2.58038	+	0.134639i
$586$		0			0
$587$			12957.3i			0.911082i	0.890215	+	0.455541i	$0.150554\pi$
$587$			12957.3i			0.911082i	−0.890215	+	0.455541i	$0.849446\pi$
$588$		0			0
$589$	−15092.8			−1.05584
$590$		0			0
$591$	19978.1			1.39051
$592$		0			0
$593$		−	5966.63i		−	0.413187i	−0.978427	−	0.206594i	$-0.933762\pi$
$593$		−	5966.63i		−	0.413187i	0.978427	−	0.206594i	$-0.0662378\pi$
$594$		0			0
$595$	−945.726	−	18125.0i	−0.0651613	−	1.24883i
$596$		0			0
$597$			9473.56i			0.649459i
$598$		0			0
$599$	10311.9			0.703396			0.351698	−	0.936114i	$-0.385604\pi$
$599$	10311.9			0.703396			0.351698	+	0.936114i	$0.385604\pi$
$600$		0			0
$601$	2594.60			0.176100			0.0880499	−	0.996116i	$-0.471936\pi$
$601$	2594.60			0.176100			0.0880499	+	0.996116i	$0.471936\pi$
$602$		0			0
$603$			14020.6i			0.946868i
$604$		0			0
$605$	346.463	+	6640.02i	0.0232822	+	0.446207i
$606$		0			0
$607$		−	7796.08i		−	0.521306i	−0.965432	−	0.260653i	$-0.916062\pi$
$607$		−	7796.08i		−	0.521306i	0.965432	−	0.260653i	$-0.0839379\pi$
$608$		0			0
$609$	13028.9			0.866922
$610$		0			0
$611$	−30896.5			−2.04572
$612$		0			0
$613$			10443.5i			0.688106i	0.938950	+	0.344053i	$0.111800\pi$
$613$			10443.5i			0.688106i	−0.938950	+	0.344053i	$0.888200\pi$
$614$		0			0
$615$	9107.30	−	475.201i	0.597141	−	0.0311577i
$616$		0			0
$617$		−	18306.1i		−	1.19445i	−0.802073	−	0.597226i	$-0.796270\pi$
$617$		−	18306.1i		−	1.19445i	0.802073	−	0.597226i	$-0.203730\pi$
$618$		0			0
$619$	−149.857			−0.00973066			−0.00486533	−	0.999988i	$-0.501549\pi$
$619$	−149.857			−0.00973066			−0.00486533	+	0.999988i	$0.501549\pi$
$620$		0			0
$621$	−18028.7			−1.16500
$622$		0			0
$623$			16454.9i			1.05819i
$624$		0			0
$625$	15286.5	−	3234.61i	0.978338	−	0.207015i
$626$		0			0
$627$			25596.2i			1.63033i
$628$		0			0
$629$	4422.16			0.280323
$630$		0			0
$631$	24466.5			1.54358			0.771789	−	0.635879i	$-0.219362\pi$
$631$	24466.5			1.54358			0.771789	+	0.635879i	$0.219362\pi$
$632$		0			0
$633$		−	40719.9i		−	2.55683i
$634$		0			0
$635$	10218.6	−	533.185i	0.638601	−	0.0333210i
$636$		0			0
$637$		−	10352.1i		−	0.643901i
$638$		0			0
$639$	3854.37			0.238618
$640$		0			0
$641$	−20274.7			−1.24930			−0.624652	−	0.780903i	$-0.714759\pi$
$641$	−20274.7			−1.24930			−0.624652	+	0.780903i	$0.714759\pi$
$642$		0			0
$643$		−	9167.25i		−	0.562241i	−0.959672	−	0.281121i	$-0.909294\pi$
$643$		−	9167.25i		−	0.562241i	0.959672	−	0.281121i	$-0.0907061\pi$
$644$		0			0
$645$	2504.58	+	48000.7i	0.152896	+	2.93027i
$646$		0			0
$647$		−	5459.82i		−	0.331758i	−0.986146	−	0.165879i	$-0.946954\pi$
$647$		−	5459.82i		−	0.331758i	0.986146	−	0.165879i	$-0.0530462\pi$
$648$		0			0
$649$	14847.4			0.898016
$650$		0			0
$651$	−26035.4			−1.56744
$652$		0			0
$653$		−	16280.5i		−	0.975659i	−0.872939	−	0.487830i	$-0.837789\pi$
$653$		−	16280.5i		−	0.975659i	0.872939	−	0.487830i	$-0.162211\pi$
$654$		0			0
$655$	353.868	+	6781.93i	0.0211096	+	0.404568i
$656$		0			0
$657$		−	35488.4i		−	2.10736i
$658$		0			0
$659$	−23975.9			−1.41725			−0.708625	−	0.705585i	$-0.750684\pi$
$659$	−23975.9			−1.41725			−0.708625	+	0.705585i	$0.750684\pi$
$660$		0			0
$661$	−13876.4			−0.816532			−0.408266	−	0.912863i	$-0.633866\pi$
$661$	−13876.4			−0.816532			−0.408266	+	0.912863i	$0.633866\pi$
$662$		0			0
$663$			44208.2i			2.58960i
$664$		0			0
$665$	−27200.0	+	1419.24i	−1.58612	+	0.0827607i
$666$		0			0
$667$			7404.44i			0.429837i
$668$		0			0
$669$	−16200.3			−0.936236
$670$		0			0
$671$	−13878.3			−0.798458
$672$		0			0
$673$			30526.1i			1.74843i	0.485537	+	0.874216i	$0.338624\pi$
$673$			30526.1i			1.74843i	−0.485537	+	0.874216i	$0.661376\pi$
$674$		0			0
$675$	2174.26	+	20778.2i	0.123981	+	1.18482i
$676$		0			0
$677$		−	6992.09i		−	0.396939i	−0.980107	−	0.198470i	$-0.936403\pi$
$677$		−	6992.09i		−	0.396939i	0.980107	−	0.198470i	$-0.0635971\pi$
$678$		0			0
$679$	25151.4			1.42153
$680$		0			0
$681$	35613.0			2.00395
$682$		0			0
$683$		−	22479.2i		−	1.25936i	−0.776854	−	0.629681i	$-0.783186\pi$
$683$		−	22479.2i		−	1.25936i	0.776854	−	0.629681i	$-0.216814\pi$
$684$		0			0
$685$	15843.1	−	826.661i	0.883698	−	0.0461096i
$686$		0			0
$687$			7618.29i			0.423080i
$688$		0			0
$689$	−20181.6			−1.11590
$690$		0			0
$691$	−5536.97			−0.304828			−0.152414	−	0.988317i	$-0.548705\pi$
$691$	−5536.97			−0.304828			−0.152414	+	0.988317i	$0.548705\pi$
$692$		0			0
$693$			27933.1i			1.53115i
$694$		0			0
$695$	−1768.61	−	33895.7i	−0.0965285	−	1.84998i
$696$		0			0
$697$		−	6976.29i		−	0.379119i
$698$		0			0
$699$	42173.6			2.28205
$700$		0			0
$701$	−16934.8			−0.912436			−0.456218	−	0.889868i	$-0.650796\pi$
$701$	−16934.8			−0.912436			−0.456218	+	0.889868i	$0.650796\pi$
$702$		0			0
$703$		−	6636.29i		−	0.356035i
$704$		0			0
$705$	2194.06	+	42049.5i	0.117210	+	2.24635i
$706$		0			0
$707$			6447.52i			0.342976i
$708$		0			0
$709$	4112.16			0.217821			0.108911	−	0.994052i	$-0.465264\pi$
$709$	4112.16			0.217821			0.108911	+	0.994052i	$0.465264\pi$
$710$		0			0
$711$	−47027.9			−2.48057
$712$		0			0
$713$		−	14796.2i		−	0.777169i
$714$		0			0
$715$	−21307.5	+	1111.78i	−1.11448	+	0.0581514i
$716$		0			0
$717$		−	18671.9i		−	0.972543i
$718$		0			0
$719$	34938.3			1.81221			0.906105	−	0.423053i	$-0.139042\pi$
$719$	34938.3			1.81221			0.906105	+	0.423053i	$0.139042\pi$
$720$		0			0
$721$	−4437.90			−0.229232
$722$		0			0
$723$		−	27057.3i		−	1.39180i
$724$		0			0
$725$	8533.71	−	892.976i	0.437150	−	0.0457438i
$726$		0			0
$727$		−	34679.2i		−	1.76916i	−0.466389	−	0.884580i	$-0.654445\pi$
$727$		−	34679.2i		−	1.76916i	0.466389	−	0.884580i	$-0.345555\pi$
$728$		0			0
$729$	30435.7			1.54629
$730$		0			0
$731$	36769.0			1.86040
$732$		0			0
$733$			30802.4i			1.55213i	0.630652	+	0.776065i	$0.282787\pi$
$733$			30802.4i			1.55213i	−0.630652	+	0.776065i	$0.717213\pi$
$734$		0			0
$735$	−14089.0	+	735.137i	−0.707049	+	0.0368924i
$736$		0			0
$737$			8182.38i			0.408958i
$738$		0			0
$739$	16364.2			0.814570			0.407285	−	0.913301i	$-0.366476\pi$
$739$	16364.2			0.814570			0.407285	+	0.913301i	$0.366476\pi$
$740$		0			0
$741$	66342.8			3.28902
$742$		0			0
$743$		−	4464.06i		−	0.220418i	−0.993908	−	0.110209i	$-0.964848\pi$
$743$		−	4464.06i		−	0.220418i	0.993908	−	0.110209i	$-0.0351520\pi$
$744$		0			0
$745$	−1543.39	−	29579.4i	−0.0759001	−	1.45464i
$746$		0			0
$747$		−	32758.5i		−	1.60451i
$748$		0			0
$749$	−16701.3			−0.814758
$750$		0			0
$751$	−17873.6			−0.868463			−0.434231	−	0.900801i	$-0.642980\pi$
$751$	−17873.6			−0.868463			−0.434231	+	0.900801i	$0.642980\pi$
$752$		0			0
$753$		−	23311.5i		−	1.12818i
$754$		0			0
$755$	165.042	+	3163.05i	0.00795562	+	0.152471i
$756$		0			0
$757$		−	14305.3i		−	0.686834i	−0.939183	−	0.343417i	$-0.888416\pi$
$757$		−	14305.3i		−	0.686834i	0.939183	−	0.343417i	$-0.111584\pi$
$758$		0			0
$759$	−25093.1			−1.20003
$760$		0			0
$761$	21819.9			1.03938			0.519692	−	0.854354i	$-0.326047\pi$
$761$	21819.9			1.03938			0.519692	+	0.854354i	$0.326047\pi$
$762$		0			0
$763$		−	11107.3i		−	0.527016i
$764$		0			0
$765$	38063.2	−	1986.06i	1.79892	−	0.0938643i
$766$		0			0
$767$		−	38483.0i		−	1.81166i
$768$		0			0
$769$	29446.6			1.38085			0.690423	−	0.723406i	$-0.257425\pi$
$769$	29446.6			1.38085			0.690423	+	0.723406i	$0.257425\pi$
$770$		0			0
$771$	6421.71			0.299964
$772$		0			0
$773$			30232.4i			1.40671i	0.710840	+	0.703354i	$0.248315\pi$
$773$			30232.4i			1.40671i	−0.710840	+	0.703354i	$0.751685\pi$
$774$		0			0
$775$	−17052.8	+	1784.42i	−0.790393	+	0.0827075i
$776$		0			0
$777$		−	11447.7i		−	0.528551i
$778$		0			0
$779$	−10469.3			−0.481515
$780$		0			0
$781$	2249.41			0.103060
$782$		0			0
$783$			11472.5i			0.523617i
$784$		0			0
$785$	−15797.4	+	824.276i	−0.718258	+	0.0374773i
$786$		0			0
$787$			30871.6i			1.39829i	0.714980	+	0.699145i	$0.246436\pi$
$787$			30871.6i			1.39829i	−0.714980	+	0.699145i	$0.753564\pi$
$788$		0			0
$789$	−21827.1			−0.984873
$790$		0			0
$791$	11009.2			0.494871
$792$		0			0
$793$			35971.1i			1.61081i
$794$		0			0
$795$	1433.16	+	27466.8i	0.0639359	+	1.22534i
$796$		0			0
$797$			4612.35i			0.204991i	0.994733	+	0.102496i	$0.0326828\pi$
$797$			4612.35i			0.204991i	−0.994733	+	0.102496i	$0.967317\pi$
$798$		0			0
$799$	32210.4			1.42618
$800$		0			0
$801$	−34556.0			−1.52431
$802$		0			0
$803$		−	20711.0i		−	0.910181i
$804$		0			0
$805$	−1391.35	−	26665.4i	−0.0609174	−	1.16749i
$806$		0			0
$807$			68429.8i			2.98494i
$808$		0			0
$809$	5893.44			0.256122			0.128061	−	0.991766i	$-0.459125\pi$
$809$	5893.44			0.256122			0.128061	+	0.991766i	$0.459125\pi$
$810$		0			0
$811$	−28114.3			−1.21730			−0.608648	−	0.793441i	$-0.708288\pi$
$811$	−28114.3			−1.21730			−0.608648	+	0.793441i	$0.708288\pi$
$812$		0			0
$813$		−	14459.0i		−	0.623739i
$814$		0			0
$815$	−19267.5	+	1005.34i	−0.828112	+	0.0432093i
$816$		0			0
$817$		−	55178.9i		−	2.36287i
$818$		0			0
$819$	72399.6			3.08895
$820$		0			0
$821$	28087.6			1.19399			0.596994	−	0.802246i	$-0.296362\pi$
$821$	28087.6			1.19399			0.596994	+	0.802246i	$0.296362\pi$
$822$		0			0
$823$			43426.8i			1.83932i	0.392712	+	0.919661i	$0.371537\pi$
$823$			43426.8i			1.83932i	−0.392712	+	0.919661i	$0.628463\pi$
$824$		0			0
$825$	3026.23	+	28920.1i	0.127709	+	1.22045i
$826$		0			0
$827$			9481.07i			0.398657i	0.979933	+	0.199328i	$0.0638760\pi$
$827$			9481.07i			0.398657i	−0.979933	+	0.199328i	$0.936124\pi$
$828$		0			0
$829$	31634.3			1.32534			0.662668	−	0.748913i	$-0.269424\pi$
$829$	31634.3			1.32534			0.662668	+	0.748913i	$0.269424\pi$
$830$		0			0
$831$	12199.9			0.509278
$832$		0			0
$833$			10792.3i			0.448898i
$834$		0			0
$835$	−22106.1	+	1153.45i	−0.916182	+	0.0478046i
$836$		0			0
$837$		−	22925.2i		−	0.946729i
$838$		0			0
$839$	−24661.7			−1.01480			−0.507399	−	0.861711i	$-0.669393\pi$
$839$	−24661.7			−1.01480			−0.507399	+	0.861711i	$0.669393\pi$
$840$		0			0
$841$	−19677.2			−0.806807
$842$		0			0
$843$			46454.0i			1.89794i
$844$		0			0
$845$	1601.71	+	30696.9i	0.0652075	+	1.24971i
$846$		0			0
$847$		−	13167.0i		−	0.534149i
$848$		0			0
$849$	71527.3			2.89142
$850$		0			0
$851$	6505.86			0.262066
$852$		0			0
$853$		−	24125.3i		−	0.968386i	−0.874961	−	0.484193i	$-0.839113\pi$
$853$		−	24125.3i		−	0.968386i	0.874961	−	0.484193i	$-0.160887\pi$
$854$		0			0
$855$	−2980.46	−	57121.0i	−0.119216	−	2.28479i
$856$		0			0
$857$			23911.4i			0.953089i	0.879150	+	0.476544i	$0.158111\pi$
$857$			23911.4i			0.953089i	−0.879150	+	0.476544i	$0.841889\pi$
$858$		0			0
$859$	−22987.8			−0.913079			−0.456539	−	0.889703i	$-0.650911\pi$
$859$	−22987.8			−0.913079			−0.456539	+	0.889703i	$0.650911\pi$
$860$		0			0
$861$	−18059.6			−0.714832
$862$		0			0
$863$			29565.5i			1.16619i	0.812404	+	0.583095i	$0.198159\pi$
$863$			29565.5i			1.16619i	−0.812404	+	0.583095i	$0.801841\pi$
$864$		0			0
$865$	16398.5	−	855.640i	0.644583	−	0.0336331i
$866$		0			0
$867$		−	3969.28i		−	0.155483i
$868$		0			0
$869$	−27445.4			−1.07137
$870$		0			0
$871$	21207.9			0.825031
$872$		0			0
$873$			52818.8i			2.04771i
$874$		0			0
$875$	−30564.4	+	4819.39i	−1.18087	+	0.186200i
$876$		0			0
$877$			10218.0i			0.393427i	0.980461	+	0.196714i	$0.0630269\pi$
$877$			10218.0i			0.393427i	−0.980461	+	0.196714i	$0.936973\pi$
$878$		0			0
$879$	−19989.8			−0.767053
$880$		0			0
$881$	−3269.43			−0.125028			−0.0625141	−	0.998044i	$-0.519912\pi$
$881$	−3269.43			−0.125028			−0.0625141	+	0.998044i	$0.519912\pi$
$882$		0			0
$883$		−	13275.6i		−	0.505956i	−0.967472	−	0.252978i	$-0.918590\pi$
$883$		−	13275.6i		−	0.505956i	0.967472	−	0.252978i	$-0.0814101\pi$
$884$		0			0
$885$	−52374.6	+	2732.81i	−1.98933	+	0.103799i
$886$		0			0
$887$			23809.7i			0.901298i	0.892701	+	0.450649i	$0.148807\pi$
$887$			23809.7i			0.901298i	−0.892701	+	0.450649i	$0.851193\pi$
$888$		0			0
$889$	−20263.3			−0.764463
$890$		0			0
$891$	−4815.05			−0.181044
$892$		0			0
$893$		−	48337.8i		−	1.81138i
$894$		0			0
$895$	1735.26	+	33256.4i	0.0648081	+	1.24206i
$896$		0			0
$897$			65038.8i			2.42094i
$898$		0			0
$899$	−9415.49			−0.349304
$900$		0			0
$901$	21039.9			0.777957
$902$		0			0
$903$		−	95184.4i		−	3.50780i
$904$		0			0
$905$	979.935	+	18780.6i	0.0359935	+	0.689821i
$906$		0			0
$907$		−	39404.3i		−	1.44255i	−0.692646	−	0.721277i	$-0.743555\pi$
$907$		−	39404.3i		−	1.44255i	0.692646	−	0.721277i	$-0.256445\pi$
$908$		0			0
$909$	−13540.0			−0.494054
$910$		0			0
$911$	48325.4			1.75751			0.878755	−	0.477273i	$-0.158375\pi$
$911$	48325.4			1.75751			0.878755	+	0.477273i	$0.158375\pi$
$912$		0			0
$913$		−	19117.8i		−	0.692999i
$914$		0			0
$915$	48956.0	−	2554.43i	1.76878	−	0.0922915i
$916$		0			0
$917$		−	13448.4i		−	0.484304i
$918$		0			0
$919$	35431.2			1.27178			0.635891	−	0.771779i	$-0.280633\pi$
$919$	35431.2			1.27178			0.635891	+	0.771779i	$0.280633\pi$
$920$		0			0
$921$	−18003.2			−0.644109
$922$		0			0
$923$		−	5830.23i		−	0.207914i
$924$		0			0
$925$	−784.606	−	7498.08i	−0.0278894	−	0.266525i
$926$		0			0
$927$		−	9319.76i		−	0.330206i
$928$		0			0
$929$	−40430.1			−1.42785			−0.713923	−	0.700225i	$-0.753083\pi$
$929$	−40430.1			−1.42785			−0.713923	+	0.700225i	$0.753083\pi$
$930$		0			0
$931$	16196.0			0.570141
$932$		0			0
$933$		−	72881.9i		−	2.55739i
$934$		0			0
$935$	22213.6	−	1159.06i	0.776965	−	0.0405405i
$936$		0			0
$937$			3421.64i			0.119296i	0.998219	+	0.0596479i	$0.0189978\pi$
$937$			3421.64i			0.119296i	−0.998219	+	0.0596479i	$0.981002\pi$
$938$		0			0
$939$	−62220.1			−2.16238
$940$		0			0
$941$	−33075.2			−1.14582			−0.572912	−	0.819617i	$-0.694186\pi$
$941$	−33075.2			−1.14582			−0.572912	+	0.819617i	$0.694186\pi$
$942$		0			0
$943$		−	10263.5i		−	0.354427i
$944$		0			0
$945$	−2155.75	−	41315.4i	−0.0742081	−	1.42221i
$946$		0			0
$947$			21896.5i			0.751364i	0.926749	+	0.375682i	$0.122591\pi$
$947$			21896.5i			0.751364i	−0.926749	+	0.375682i	$0.877409\pi$
$948$		0			0
$949$	−53680.8			−1.83620
$950$		0			0
$951$	−82635.9			−2.81772
$952$		0			0
$953$		−	48287.2i		−	1.64132i	−0.571417	−	0.820660i	$-0.693606\pi$
$953$		−	48287.2i		−	1.64132i	0.571417	−	0.820660i	$-0.306394\pi$
$954$		0			0
$955$	159.821	+	3062.98i	0.00541536	+	0.103786i
$956$		0			0
$957$			15967.9i			0.539361i
$958$		0			0
$959$	−31416.5			−1.05787
$960$		0			0
$961$	−10976.2			−0.368439
$962$		0			0
$963$		−	35073.5i		−	1.17365i
$964$		0			0
$965$	32362.2	−	1688.59i	1.07956	−	0.0563293i
$966$		0			0
$967$			13708.7i			0.455886i	0.973674	+	0.227943i	$0.0732000\pi$
$967$			13708.7i			0.455886i	−0.973674	+	0.227943i	$0.926800\pi$
$968$		0			0
$969$	−69164.1			−2.29295
$970$		0			0
$971$	−24503.4			−0.809836			−0.404918	−	0.914353i	$-0.632700\pi$
$971$	−24503.4			−0.809836			−0.404918	+	0.914353i	$0.632700\pi$
$972$		0			0
$973$			67214.6i			2.21460i
$974$		0			0
$975$	74958.0	−	7843.68i	2.46213	−	0.257640i
$976$		0			0
$977$			27660.1i			0.905758i	0.891572	+	0.452879i	$0.149603\pi$
$977$			27660.1i			0.905758i	−0.891572	+	0.452879i	$0.850397\pi$
$978$		0			0
$979$	−20166.8			−0.658360
$980$		0			0
$981$	23325.9			0.759161
$982$		0			0
$983$			56556.7i			1.83507i	0.397651	+	0.917537i	$0.369825\pi$
$983$			56556.7i			1.83507i	−0.397651	+	0.917537i	$0.630175\pi$
$984$		0			0
$985$	−26018.9	+	1357.61i	−0.841656	+	0.0439159i
$986$		0			0
$987$		−	83383.4i		−	2.68908i
$988$		0			0
$989$	54094.4			1.73923
$990$		0			0
$991$	47154.6			1.51152			0.755760	−	0.654848i	$-0.227268\pi$
$991$	47154.6			1.51152			0.755760	+	0.654848i	$0.227268\pi$
$992$		0			0
$993$			3090.26i			0.0987578i
$994$		0			0
$995$	−643.777	−	12338.1i	−0.0205117	−	0.393109i
$996$		0			0
$997$			13606.6i			0.432223i	0.976369	+	0.216111i	$0.0693375\pi$
$997$			13606.6i			0.432223i	−0.976369	+	0.216111i	$0.930663\pi$
$998$		0			0
$999$	10080.2			0.319242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	160.4.c.d.129.7	yes	8
3.2	odd	2		1440.4.f.k.289.8		8
4.3	odd	2	inner	160.4.c.d.129.1	✓	8
5.2	odd	4		800.4.a.z.1.4		4
5.3	odd	4		800.4.a.y.1.1		4
5.4	even	2	inner	160.4.c.d.129.2	yes	8
8.3	odd	2		320.4.c.j.129.8		8
8.5	even	2		320.4.c.j.129.2		8
12.11	even	2		1440.4.f.k.289.7		8
15.14	odd	2		1440.4.f.k.289.5		8
20.3	even	4		800.4.a.y.1.4		4
20.7	even	4		800.4.a.z.1.1		4
20.19	odd	2	inner	160.4.c.d.129.8	yes	8
40.3	even	4		1600.4.a.cv.1.1		4
40.13	odd	4		1600.4.a.cv.1.4		4
40.19	odd	2		320.4.c.j.129.1		8
40.27	even	4		1600.4.a.cu.1.4		4
40.29	even	2		320.4.c.j.129.7		8
40.37	odd	4		1600.4.a.cu.1.1		4
60.59	even	2		1440.4.f.k.289.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.d.129.1	✓	8	4.3	odd	2	inner
160.4.c.d.129.2	yes	8	5.4	even	2	inner
160.4.c.d.129.7	yes	8	1.1	even	1	trivial
160.4.c.d.129.8	yes	8	20.19	odd	2	inner
320.4.c.j.129.1		8	40.19	odd	2
320.4.c.j.129.2		8	8.5	even	2
320.4.c.j.129.7		8	40.29	even	2
320.4.c.j.129.8		8	8.3	odd	2
800.4.a.y.1.1		4	5.3	odd	4
800.4.a.y.1.4		4	20.3	even	4
800.4.a.z.1.1		4	20.7	even	4
800.4.a.z.1.4		4	5.2	odd	4
1440.4.f.k.289.5		8	15.14	odd	2
1440.4.f.k.289.6		8	60.59	even	2
1440.4.f.k.289.7		8	12.11	even	2
1440.4.f.k.289.8		8	3.2	odd	2
1600.4.a.cu.1.1		4	40.37	odd	4
1600.4.a.cu.1.4		4	40.27	even	4
1600.4.a.cv.1.1		4	40.3	even	4
1600.4.a.cv.1.4		4	40.13	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:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	160.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+8.57295i q^{3} +(-0.582576 - 11.1652i) q^{5} +22.1403i q^{7} -46.4955 q^{9} +O(q^{10})\)	\(q+8.57295i q^{3} +(-0.582576 - 11.1652i) q^{5} +22.1403i q^{7} -46.4955 q^{9} -27.1347 q^{11} +70.3303i q^{13} +(95.7183 - 4.99439i) q^{15} -73.3212i q^{17} -110.033 q^{19} -189.808 q^{21} -107.870i q^{23} +(-124.321 + 13.0091i) q^{25} -167.134i q^{27} -68.6424 q^{29} +137.167 q^{31} -232.624i q^{33} +(247.200 - 12.8984i) q^{35} +60.3121i q^{37} -602.938 q^{39} +95.1470 q^{41} +501.479i q^{43} +(27.0871 + 519.129i) q^{45} +439.305i q^{47} -147.192 q^{49} +628.579 q^{51} +286.955i q^{53} +(15.8080 + 302.963i) q^{55} -943.303i q^{57} -547.175 q^{59} +511.459 q^{61} -1029.42i q^{63} +(785.248 - 40.9727i) q^{65} -301.547i q^{67} +924.762 q^{69} -82.8978 q^{71} +763.267i q^{73} +(-111.526 - 1065.80i) q^{75} -600.770i q^{77} +1011.45 q^{79} +177.450 q^{81} +704.554i q^{83} +(-818.642 + 42.7152i) q^{85} -588.468i q^{87} +743.212 q^{89} -1557.13 q^{91} +1175.93i q^{93} +(64.1023 + 1228.53i) q^{95} -1136.00i q^{97} +1261.64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 32 q^{5} - 152 q^{9}+O(q^{10}) \) 8 * q + 32 * q^5 - 152 * q^9	\( 8 q + 32 q^{5} - 152 q^{9} - 272 q^{21} - 408 q^{25} + 624 q^{29} - 192 q^{41} + 400 q^{45} - 2424 q^{49} + 2112 q^{61} + 2176 q^{65} + 3952 q^{69} - 1000 q^{81} - 5376 q^{85} + 80 q^{89}+O(q^{100}) \) 8 * q + 32 * q^5 - 152 * q^9 - 272 * q^21 - 408 * q^25 + 624 * q^29 - 192 * q^41 + 400 * q^45 - 2424 * q^49 + 2112 * q^61 + 2176 * q^65 + 3952 * q^69 - 1000 * q^81 - 5376 * q^85 + 80 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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