LMFDB - Embedded newform 528.5.j.a.241.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,5,Mod(241,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("528.241");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	528.j (of order $2$, degree $1$, not 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:	$54.5793405083$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 102x^{6} + 2913x^{4} + 23292x^{2} + 41364 $ x^8 + 102x^6 + 2913x^4 + 23292*x^2 + 41364
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.7
Root			$1.57474i$ of defining polynomial
Character	$\chi$	$=$	528.241
Dual form			528.5.j.a.241.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} +15.5526 q^{5} -93.8006i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} +15.5526 q^{5} -93.8006i q^{7} +27.0000 q^{9} +(-60.9369 + 104.536i) q^{11} -29.4527i q^{13} +80.8135 q^{15} -251.915i q^{17} -80.2497i q^{19} -487.402i q^{21} +702.557 q^{23} -383.118 q^{25} +140.296 q^{27} -1449.00i q^{29} -1279.46 q^{31} +(-316.638 + 543.183i) q^{33} -1458.84i q^{35} +115.552 q^{37} -153.041i q^{39} +1076.75i q^{41} +1887.47i q^{43} +419.919 q^{45} -1591.75 q^{47} -6397.55 q^{49} -1308.99i q^{51} +1190.69 q^{53} +(-947.726 + 1625.80i) q^{55} -416.990i q^{57} -1895.56 q^{59} -3776.15i q^{61} -2532.62i q^{63} -458.065i q^{65} +1253.00 q^{67} +3650.59 q^{69} +3591.39 q^{71} -1750.75i q^{73} -1990.74 q^{75} +(9805.50 + 5715.92i) q^{77} -6749.08i q^{79} +729.000 q^{81} -8616.78i q^{83} -3917.92i q^{85} -7529.24i q^{87} -9334.05 q^{89} -2762.68 q^{91} -6648.27 q^{93} -1248.09i q^{95} +11068.8 q^{97} +(-1645.30 + 2822.46i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 36 q^{5} + 216 q^{9}+O(q^{10}) $ 8 * q - 36 * q^5 + 216 * q^9	$ 8 q - 36 q^{5} + 216 q^{9} - 36 q^{11} - 108 q^{15} - 516 q^{23} - 2280 q^{25} - 2752 q^{31} + 1008 q^{33} + 5296 q^{37} - 972 q^{45} - 420 q^{47} - 6832 q^{49} + 3540 q^{53} - 3784 q^{55} + 16632 q^{59} + 3656 q^{67} + 9036 q^{69} + 13212 q^{71} + 9288 q^{75} + 23268 q^{77} + 5832 q^{81} + 15528 q^{89} + 19752 q^{91} - 21384 q^{93} + 7624 q^{97} - 972 q^{99}+O(q^{100}) $ 8 * q - 36 * q^5 + 216 * q^9 - 36 * q^11 - 108 * q^15 - 516 * q^23 - 2280 * q^25 - 2752 * q^31 + 1008 * q^33 + 5296 * q^37 - 972 * q^45 - 420 * q^47 - 6832 * q^49 + 3540 * q^53 - 3784 * q^55 + 16632 * q^59 + 3656 * q^67 + 9036 * q^69 + 13212 * q^71 + 9288 * q^75 + 23268 * q^77 + 5832 * q^81 + 15528 * q^89 + 19752 * q^91 - 21384 * q^93 + 7624 * q^97 - 972 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$.

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$		0			0
$5$	15.5526			0.622103			0.311051	−	0.950393i	$-0.399319\pi$
$5$	15.5526			0.622103			0.311051	+	0.950393i	$0.399319\pi$
$6$		0			0
$7$		−	93.8006i		−	1.91430i	−0.289596	−	0.957149i	$-0.593521\pi$
$7$		−	93.8006i		−	1.91430i	0.289596	−	0.957149i	$-0.406479\pi$
$8$		0			0
$9$	27.0000			0.333333
$10$		0			0
$11$	−60.9369	+	104.536i	−0.503611	+	0.863931i
$11$	−60.9369	+	104.536i	−0.503611	+	0.863931i
$12$		0			0
$13$		−	29.4527i		−	0.174276i	−0.996196	−	0.0871382i	$-0.972228\pi$
$13$		−	29.4527i		−	0.174276i	0.996196	−	0.0871382i	$-0.0277722\pi$
$14$		0			0
$15$	80.8135			0.359171
$16$		0			0
$17$		−	251.915i		−	0.871677i	−0.900025	−	0.435839i	$-0.856452\pi$
$17$		−	251.915i		−	0.871677i	0.900025	−	0.435839i	$-0.143548\pi$
$18$		0			0
$19$		−	80.2497i		−	0.222298i	−0.993804	−	0.111149i	$-0.964547\pi$
$19$		−	80.2497i		−	0.222298i	0.993804	−	0.111149i	$-0.0354531\pi$
$20$		0			0
$21$		−	487.402i		−	1.10522i
$22$		0			0
$23$	702.557			1.32809			0.664043	−	0.747695i	$-0.268839\pi$
$23$	702.557			1.32809			0.664043	+	0.747695i	$0.268839\pi$
$24$		0			0
$25$	−383.118			−0.612988
$26$		0			0
$27$	140.296			0.192450
$28$		0			0
$29$		−	1449.00i		−	1.72295i	−0.507798	−	0.861476i	$-0.669540\pi$
$29$		−	1449.00i		−	1.72295i	0.507798	−	0.861476i	$-0.330460\pi$
$30$		0			0
$31$	−1279.46			−1.33138			−0.665692	−	0.746227i	$-0.731863\pi$
$31$	−1279.46			−1.33138			−0.665692	+	0.746227i	$0.731863\pi$
$32$		0			0
$33$	−316.638	+	543.183i	−0.290760	+	0.498791i
$34$		0			0
$35$		−	1458.84i		−	1.19089i
$36$		0			0
$37$	115.552			0.0844063			0.0422031	−	0.999109i	$-0.486562\pi$
$37$	115.552			0.0844063			0.0422031	+	0.999109i	$0.486562\pi$
$38$		0			0
$39$		−	153.041i		−	0.100618i
$40$		0			0
$41$			1076.75i			0.640541i	0.947326	+	0.320270i	$0.103774\pi$
$41$			1076.75i			0.640541i	−0.947326	+	0.320270i	$0.896226\pi$
$42$		0			0
$43$			1887.47i			1.02080i	0.859936	+	0.510402i	$0.170503\pi$
$43$			1887.47i			1.02080i	−0.859936	+	0.510402i	$0.829497\pi$
$44$		0			0
$45$	419.919			0.207368
$46$		0			0
$47$	−1591.75			−0.720575			−0.360287	−	0.932841i	$-0.617321\pi$
$47$	−1591.75			−0.720575			−0.360287	+	0.932841i	$0.617321\pi$
$48$		0			0
$49$	−6397.55			−2.66454
$50$		0			0
$51$		−	1308.99i		−	0.503263i
$52$		0			0
$53$	1190.69			0.423886			0.211943	−	0.977282i	$-0.432021\pi$
$53$	1190.69			0.423886			0.211943	+	0.977282i	$0.432021\pi$
$54$		0			0
$55$	−947.726	+	1625.80i	−0.313298	+	0.537454i
$56$		0			0
$57$		−	416.990i		−	0.128344i
$58$		0			0
$59$	−1895.56			−0.544544			−0.272272	−	0.962220i	$-0.587775\pi$
$59$	−1895.56			−0.544544			−0.272272	+	0.962220i	$0.587775\pi$
$60$		0			0
$61$		−	3776.15i		−	1.01482i	−0.861704	−	0.507411i	$-0.830603\pi$
$61$		−	3776.15i		−	1.01482i	0.861704	−	0.507411i	$-0.169397\pi$
$62$		0			0
$63$		−	2532.62i		−	0.638099i
$64$		0			0
$65$		−	458.065i		−	0.108418i
$66$		0			0
$67$	1253.00			0.279127			0.139564	−	0.990213i	$-0.455430\pi$
$67$	1253.00			0.279127			0.139564	+	0.990213i	$0.455430\pi$
$68$		0			0
$69$	3650.59			0.766770
$70$		0			0
$71$	3591.39			0.712437			0.356218	−	0.934403i	$-0.384066\pi$
$71$	3591.39			0.712437			0.356218	+	0.934403i	$0.384066\pi$
$72$		0			0
$73$		−	1750.75i		−	0.328533i	−0.986416	−	0.164267i	$-0.947474\pi$
$73$		−	1750.75i		−	0.328533i	0.986416	−	0.164267i	$-0.0525257\pi$
$74$		0			0
$75$	−1990.74			−0.353909
$76$		0			0
$77$	9805.50	+	5715.92i	1.65382	+	0.964061i
$78$		0			0
$79$		−	6749.08i		−	1.08141i	−0.841212	−	0.540705i	$-0.818157\pi$
$79$		−	6749.08i		−	1.08141i	0.841212	−	0.540705i	$-0.181843\pi$
$80$		0			0
$81$	729.000			0.111111
$82$		0			0
$83$		−	8616.78i		−	1.25080i	−0.780303	−	0.625401i	$-0.784935\pi$
$83$		−	8616.78i		−	1.25080i	0.780303	−	0.625401i	$-0.215065\pi$
$84$		0			0
$85$		−	3917.92i		−	0.542273i
$86$		0			0
$87$		−	7529.24i		−	0.994747i
$88$		0			0
$89$	−9334.05			−1.17839			−0.589196	−	0.807990i	$-0.700556\pi$
$89$	−9334.05			−1.17839			−0.589196	+	0.807990i	$0.700556\pi$
$90$		0			0
$91$	−2762.68			−0.333617
$92$		0			0
$93$	−6648.27			−0.768675
$94$		0			0
$95$		−	1248.09i		−	0.138292i
$96$		0			0
$97$	11068.8			1.17640			0.588202	−	0.808714i	$-0.299836\pi$
$97$	11068.8			1.17640			0.588202	+	0.808714i	$0.299836\pi$
$98$		0			0
$99$	−1645.30	+	2822.46i	−0.167870	+	0.287977i
$100$		0			0
$101$		−	9732.30i		−	0.954053i	−0.878889	−	0.477027i	$-0.841715\pi$
$101$		−	9732.30i		−	0.954053i	0.878889	−	0.477027i	$-0.158285\pi$
$102$		0			0
$103$	−2229.35			−0.210138			−0.105069	−	0.994465i	$-0.533506\pi$
$103$	−2229.35			−0.210138			−0.105069	+	0.994465i	$0.533506\pi$
$104$		0			0
$105$		−	7580.36i		−	0.687561i
$106$		0			0
$107$			8911.94i			0.778404i	0.921152	+	0.389202i	$0.127249\pi$
$107$			8911.94i			0.778404i	−0.921152	+	0.389202i	$0.872751\pi$
$108$		0			0
$109$			8075.42i			0.679692i	0.940481	+	0.339846i	$0.110375\pi$
$109$			8075.42i			0.679692i	−0.940481	+	0.339846i	$0.889625\pi$
$110$		0			0
$111$	600.427			0.0487320
$112$		0			0
$113$	−16548.3			−1.29597			−0.647987	−	0.761652i	$-0.724389\pi$
$113$	−16548.3			−1.29597			−0.647987	+	0.761652i	$0.724389\pi$
$114$		0			0
$115$	10926.6			0.826206
$116$		0			0
$117$		−	795.223i		−	0.0580921i
$118$		0			0
$119$	−23629.8			−1.66865
$120$		0			0
$121$	−7214.38	−	12740.2i	−0.492752	−	0.870170i
$122$		0			0
$123$			5594.95i			0.369816i
$124$		0			0
$125$	−15678.8			−1.00344
$126$		0			0
$127$			2165.25i			0.134246i	0.997745	+	0.0671228i	$0.0213819\pi$
$127$			2165.25i			0.134246i	−0.997745	+	0.0671228i	$0.978618\pi$
$128$		0			0
$129$			9807.56i			0.589361i
$130$		0			0
$131$		−	15267.4i		−	0.889658i	−0.895616	−	0.444829i	$-0.853264\pi$
$131$		−	15267.4i		−	0.889658i	0.895616	−	0.444829i	$-0.146736\pi$
$132$		0			0
$133$	−7527.47			−0.425545
$134$		0			0
$135$	2181.97			0.119724
$136$		0			0
$137$	10391.2			0.553636			0.276818	−	0.960922i	$-0.410720\pi$
$137$	10391.2			0.553636			0.276818	+	0.960922i	$0.410720\pi$
$138$		0			0
$139$		−	21983.9i		−	1.13782i	−0.822399	−	0.568911i	$-0.807365\pi$
$139$		−	21983.9i		−	1.13782i	0.822399	−	0.568911i	$-0.192635\pi$
$140$		0			0
$141$	−8270.98			−0.416024
$142$		0			0
$143$	3078.86	+	1794.76i	0.150563	+	0.0877675i
$144$		0			0
$145$		−	22535.7i		−	1.07185i
$146$		0			0
$147$	−33242.7			−1.53837
$148$		0			0
$149$			10316.9i			0.464705i	0.972632	+	0.232353i	$0.0746423\pi$
$149$			10316.9i			0.464705i	−0.972632	+	0.232353i	$0.925358\pi$
$150$		0			0
$151$			20057.9i			0.879694i	0.898073	+	0.439847i	$0.144967\pi$
$151$			20057.9i			0.879694i	−0.898073	+	0.439847i	$0.855033\pi$
$152$		0			0
$153$		−	6801.70i		−	0.290559i
$154$		0			0
$155$	−19898.9			−0.828258
$156$		0			0
$157$	−5742.18			−0.232958			−0.116479	−	0.993193i	$-0.537161\pi$
$157$	−5742.18			−0.232958			−0.116479	+	0.993193i	$0.537161\pi$
$158$		0			0
$159$	6187.03			0.244730
$160$		0			0
$161$		−	65900.3i		−	2.54235i
$162$		0			0
$163$	24660.6			0.928174			0.464087	−	0.885790i	$-0.346383\pi$
$163$	24660.6			0.928174			0.464087	+	0.885790i	$0.346383\pi$
$164$		0			0
$165$	−4924.53	+	8447.89i	−0.180883	+	0.310299i
$166$		0			0
$167$		−	47391.6i		−	1.69930i	−0.527351	−	0.849648i	$-0.676815\pi$
$167$		−	47391.6i		−	1.69930i	0.527351	−	0.849648i	$-0.323185\pi$
$168$		0			0
$169$	27693.5			0.969628
$170$		0			0
$171$		−	2166.74i		−	0.0740995i
$172$		0			0
$173$			1326.29i			0.0443147i	0.999754	+	0.0221573i	$0.00705347\pi$
$173$			1326.29i			0.0443147i	−0.999754	+	0.0221573i	$0.992947\pi$
$174$		0			0
$175$			35936.7i			1.17344i
$176$		0			0
$177$	−9849.60			−0.314392
$178$		0			0
$179$	53822.7			1.67981			0.839903	−	0.542736i	$-0.182612\pi$
$179$	53822.7			1.67981			0.839903	+	0.542736i	$0.182612\pi$
$180$		0			0
$181$	9961.35			0.304061			0.152031	−	0.988376i	$-0.451419\pi$
$181$	9961.35			0.304061			0.152031	+	0.988376i	$0.451419\pi$
$182$		0			0
$183$		−	19621.5i		−	0.585907i
$184$		0			0
$185$	1797.13			0.0525094
$186$		0			0
$187$	26334.1	+	15350.9i	0.753069	+	0.438986i
$188$		0			0
$189$		−	13159.9i		−	0.368407i
$190$		0			0
$191$	54964.8			1.50667			0.753335	−	0.657637i	$-0.228444\pi$
$191$	54964.8			1.50667			0.753335	+	0.657637i	$0.228444\pi$
$192$		0			0
$193$			1722.76i			0.0462498i	0.999733	+	0.0231249i	$0.00736154\pi$
$193$			1722.76i			0.0462498i	−0.999733	+	0.0231249i	$0.992638\pi$
$194$		0			0
$195$		−	2380.18i		−	0.0625950i
$196$		0			0
$197$			10387.5i			0.267656i	0.991005	+	0.133828i	$0.0427270\pi$
$197$			10387.5i			0.267656i	−0.991005	+	0.133828i	$0.957273\pi$
$198$		0			0
$199$	−50964.3			−1.28695			−0.643473	−	0.765469i	$-0.722507\pi$
$199$	−50964.3			−1.28695			−0.643473	+	0.765469i	$0.722507\pi$
$200$		0			0
$201$	6510.79			0.161154
$202$		0			0
$203$	−135917.			−3.29824
$204$		0			0
$205$			16746.2i			0.398482i
$206$		0			0
$207$	18969.0			0.442695
$208$		0			0
$209$	8388.95	+	4890.17i	0.192050	+	0.111952i
$210$		0			0
$211$		−	68681.5i		−	1.54268i	−0.636426	−	0.771338i	$-0.719588\pi$
$211$		−	68681.5i		−	1.54268i	0.636426	−	0.771338i	$-0.280412\pi$
$212$		0			0
$213$	18661.4			0.411326
$214$		0			0
$215$			29355.0i			0.635045i
$216$		0			0
$217$			120014.i			2.54866i
$218$		0			0
$219$		−	9097.18i		−	0.189679i
$220$		0			0
$221$	−7419.57			−0.151913
$222$		0			0
$223$	1628.89			0.0327553			0.0163777	−	0.999866i	$-0.494787\pi$
$223$	1628.89			0.0327553			0.0163777	+	0.999866i	$0.494787\pi$
$224$		0			0
$225$	−10344.2			−0.204329
$226$		0			0
$227$		−	42599.1i		−	0.826701i	−0.910572	−	0.413351i	$-0.864358\pi$
$227$		−	42599.1i		−	0.826701i	0.910572	−	0.413351i	$-0.135642\pi$
$228$		0			0
$229$	32387.8			0.617605			0.308802	−	0.951126i	$-0.400072\pi$
$229$	32387.8			0.617605			0.308802	+	0.951126i	$0.400072\pi$
$230$		0			0
$231$	50950.9	+	29700.8i	0.954834	+	0.556601i
$232$		0			0
$233$		−	57867.1i		−	1.06591i	−0.846144	−	0.532954i	$-0.821082\pi$
$233$		−	57867.1i		−	1.06591i	0.846144	−	0.532954i	$-0.178918\pi$
$234$		0			0
$235$	−24755.8			−0.448272
$236$		0			0
$237$		−	35069.3i		−	0.624352i
$238$		0			0
$239$			3758.46i			0.0657982i	0.999459	+	0.0328991i	$0.0104740\pi$
$239$			3758.46i			0.0657982i	−0.999459	+	0.0328991i	$0.989526\pi$
$240$		0			0
$241$			53393.2i			0.919288i	0.888103	+	0.459644i	$0.152023\pi$
$241$			53393.2i			0.919288i	−0.888103	+	0.459644i	$0.847977\pi$
$242$		0			0
$243$	3788.00			0.0641500
$244$		0			0
$245$	−99498.4			−1.65762
$246$		0			0
$247$	−2363.57			−0.0387414
$248$		0			0
$249$		−	44774.1i		−	0.722151i
$250$		0			0
$251$	−1857.31			−0.0294807			−0.0147403	−	0.999891i	$-0.504692\pi$
$251$	−1857.31			−0.0294807			−0.0147403	+	0.999891i	$0.504692\pi$
$252$		0			0
$253$	−42811.7	+	73442.2i	−0.668838	+	1.14737i
$254$		0			0
$255$		−	20358.1i		−	0.313081i
$256$		0			0
$257$	−1262.77			−0.0191187			−0.00955933	−	0.999954i	$-0.503043\pi$
$257$	−1262.77			−0.0191187			−0.00955933	+	0.999954i	$0.503043\pi$
$258$		0			0
$259$		−	10838.9i		−	0.161579i
$260$		0			0
$261$		−	39123.1i		−	0.574317i
$262$		0			0
$263$			118343.i			1.71092i	0.517870	+	0.855459i	$0.326725\pi$
$263$			118343.i			1.71092i	−0.517870	+	0.855459i	$0.673275\pi$
$264$		0			0
$265$	18518.4			0.263700
$266$		0			0
$267$	−48501.1			−0.680345
$268$		0			0
$269$	100514.			1.38906			0.694528	−	0.719465i	$-0.255613\pi$
$269$	100514.			1.38906			0.694528	+	0.719465i	$0.255613\pi$
$270$		0			0
$271$			138354.i			1.88389i	0.335773	+	0.941943i	$0.391003\pi$
$271$			138354.i			1.88389i	−0.335773	+	0.941943i	$0.608997\pi$
$272$		0			0
$273$	−14355.3			−0.192614
$274$		0			0
$275$	23346.0	−	40049.4i	0.308707	−	0.529579i
$276$		0			0
$277$			40696.2i			0.530388i	0.964195	+	0.265194i	$0.0854360\pi$
$277$			40696.2i			0.530388i	−0.964195	+	0.265194i	$0.914564\pi$
$278$		0			0
$279$	−34545.4			−0.443795
$280$		0			0
$281$			17381.6i			0.220129i	0.993924	+	0.110065i	$0.0351058\pi$
$281$			17381.6i			0.220129i	−0.993924	+	0.110065i	$0.964894\pi$
$282$		0			0
$283$			102799.i			1.28355i	0.766891	+	0.641777i	$0.221802\pi$
$283$			102799.i			1.28355i	−0.766891	+	0.641777i	$0.778198\pi$
$284$		0			0
$285$		−	6485.26i		−	0.0798432i
$286$		0			0
$287$	101000.			1.22619
$288$		0			0
$289$	20059.9			0.240178
$290$		0			0
$291$	57515.1			0.679197
$292$		0			0
$293$			106779.i			1.24380i	0.783098	+	0.621898i	$0.213638\pi$
$293$			106779.i			1.24380i	−0.783098	+	0.621898i	$0.786362\pi$
$294$		0			0
$295$	−29480.8			−0.338762
$296$		0			0
$297$	−8549.21	+	14665.9i	−0.0969200	+	0.166264i
$298$		0			0
$299$		−	20692.2i		−	0.231454i
$300$		0			0
$301$	177045.			1.95412
$302$		0			0
$303$		−	50570.5i		−	0.550823i
$304$		0			0
$305$		−	58728.8i		−	0.631323i
$306$		0			0
$307$		−	78416.1i		−	0.832010i	−0.909362	−	0.416005i	$-0.863430\pi$
$307$		−	78416.1i		−	0.832010i	0.909362	−	0.416005i	$-0.136570\pi$
$308$		0			0
$309$	−11584.1			−0.121323
$310$		0			0
$311$	60684.2			0.627415			0.313707	−	0.949520i	$-0.398429\pi$
$311$	60684.2			0.627415			0.313707	+	0.949520i	$0.398429\pi$
$312$		0			0
$313$	−85739.0			−0.875165			−0.437582	−	0.899178i	$-0.644165\pi$
$313$	−85739.0			−0.875165			−0.437582	+	0.899178i	$0.644165\pi$
$314$		0			0
$315$		−	39388.7i		−	0.396963i
$316$		0			0
$317$	117644.			1.17071			0.585357	−	0.810776i	$-0.300955\pi$
$317$	117644.			1.17071			0.585357	+	0.810776i	$0.300955\pi$
$318$		0			0
$319$	151472.	+	88297.8i	1.48851	+	0.867698i
$320$		0			0
$321$			46307.8i			0.449412i
$322$		0			0
$323$	−20216.1			−0.193773
$324$		0			0
$325$			11283.8i			0.106829i
$326$		0			0
$327$			41961.1i			0.392420i
$328$		0			0
$329$			149307.i			1.37940i
$330$		0			0
$331$	109718.			1.00143			0.500717	−	0.865611i	$-0.333070\pi$
$331$	109718.			1.00143			0.500717	+	0.865611i	$0.333070\pi$
$332$		0			0
$333$	3119.91			0.0281354
$334$		0			0
$335$	19487.4			0.173646
$336$		0			0
$337$			73221.5i			0.644731i	0.946615	+	0.322365i	$0.104478\pi$
$337$			73221.5i			0.644731i	−0.946615	+	0.322365i	$0.895522\pi$
$338$		0			0
$339$	−85987.4			−0.748230
$340$		0			0
$341$	77966.3	−	133749.i	0.670499	−	1.15022i
$342$		0			0
$343$			374879.i			3.18642i
$344$		0			0
$345$	56776.1			0.477010
$346$		0			0
$347$			160256.i			1.33093i	0.746428	+	0.665466i	$0.231767\pi$
$347$			160256.i			1.33093i	−0.746428	+	0.665466i	$0.768233\pi$
$348$		0			0
$349$			24791.6i			0.203542i	0.994808	+	0.101771i	$0.0324509\pi$
$349$			24791.6i			0.203542i	−0.994808	+	0.101771i	$0.967549\pi$
$350$		0			0
$351$		−	4132.10i		−	0.0335395i
$352$		0			0
$353$	−140695.			−1.12909			−0.564547	−	0.825401i	$-0.690949\pi$
$353$	−140695.			−1.12909			−0.564547	+	0.825401i	$0.690949\pi$
$354$		0			0
$355$	55855.4			0.443209
$356$		0			0
$357$	−122784.			−0.963396
$358$		0			0
$359$			14603.5i			0.113310i	0.998394	+	0.0566552i	$0.0180436\pi$
$359$			14603.5i			0.113310i	−0.998394	+	0.0566552i	$0.981956\pi$
$360$		0			0
$361$	123881.			0.950583
$362$		0			0
$363$	−37487.0	−	66199.8i	−0.284491	−	0.502393i
$364$		0			0
$365$		−	27228.7i		−	0.204381i
$366$		0			0
$367$	168034.			1.24757			0.623785	−	0.781596i	$-0.285594\pi$
$367$	168034.			1.24757			0.623785	+	0.781596i	$0.285594\pi$
$368$		0			0
$369$			29072.2i			0.213514i
$370$		0			0
$371$		−	111688.i		−	0.811443i
$372$		0			0
$373$		−	179739.i		−	1.29189i	−0.763385	−	0.645944i	$-0.776464\pi$
$373$		−	179739.i		−	1.29189i	0.763385	−	0.645944i	$-0.223536\pi$
$374$		0			0
$375$	−81469.5			−0.579339
$376$		0			0
$377$	−42677.0			−0.300270
$378$		0			0
$379$	88424.0			0.615590			0.307795	−	0.951453i	$-0.400409\pi$
$379$	88424.0			0.615590			0.307795	+	0.951453i	$0.400409\pi$
$380$		0			0
$381$			11251.0i			0.0775067i
$382$		0			0
$383$	−95956.7			−0.654151			−0.327075	−	0.944998i	$-0.606063\pi$
$383$	−95956.7			−0.654151			−0.327075	+	0.944998i	$0.606063\pi$
$384$		0			0
$385$	152501.	+	88897.3i	1.02885	+	0.599745i
$386$		0			0
$387$			50961.6i			0.340268i
$388$		0			0
$389$	−118749.			−0.784747			−0.392373	−	0.919806i	$-0.628346\pi$
$389$	−118749.			−0.784747			−0.392373	+	0.919806i	$0.628346\pi$
$390$		0			0
$391$		−	176985.i		−	1.15766i
$392$		0			0
$393$		−	79331.8i		−	0.513644i
$394$		0			0
$395$		−	104966.i		−	0.672748i
$396$		0			0
$397$	−159427.			−1.01154			−0.505768	−	0.862670i	$-0.668791\pi$
$397$	−159427.			−1.01154			−0.505768	+	0.862670i	$0.668791\pi$
$398$		0			0
$399$	−39113.9			−0.245689
$400$		0			0
$401$	−41646.5			−0.258994			−0.129497	−	0.991580i	$-0.541336\pi$
$401$	−41646.5			−0.258994			−0.129497	+	0.991580i	$0.541336\pi$
$402$		0			0
$403$			37683.5i			0.232029i
$404$		0			0
$405$	11337.8			0.0691225
$406$		0			0
$407$	−7041.40	+	12079.3i	−0.0425079	+	0.0729212i
$408$		0			0
$409$			265054.i			1.58449i	0.610206	+	0.792243i	$0.291087\pi$
$409$			265054.i			1.58449i	−0.610206	+	0.792243i	$0.708913\pi$
$410$		0			0
$411$	53994.3			0.319642
$412$		0			0
$413$			177804.i			1.04242i
$414$		0			0
$415$		−	134013.i		−	0.778128i
$416$		0			0
$417$		−	114232.i		−	0.656922i
$418$		0			0
$419$	−255711.			−1.45654			−0.728270	−	0.685291i	$-0.759675\pi$
$419$	−255711.			−1.45654			−0.728270	+	0.685291i	$0.759675\pi$
$420$		0			0
$421$	189209.			1.06753			0.533763	−	0.845634i	$-0.320778\pi$
$421$	189209.			1.06753			0.533763	+	0.845634i	$0.320778\pi$
$422$		0			0
$423$	−42977.3			−0.240192
$424$		0			0
$425$			96513.0i			0.534328i
$426$		0			0
$427$	−354205.			−1.94267
$428$		0			0
$429$	15998.2	+	9325.83i	0.0869274	+	0.0506726i
$430$		0			0
$431$		−	110749.i		−	0.596190i	−0.954536	−	0.298095i	$-0.903649\pi$
$431$		−	110749.i		−	0.596190i	0.954536	−	0.298095i	$-0.0963512\pi$
$432$		0			0
$433$	172877.			0.922062			0.461031	−	0.887384i	$-0.347480\pi$
$433$	172877.			0.922062			0.461031	+	0.887384i	$0.347480\pi$
$434$		0			0
$435$		−	117099.i		−	0.618835i
$436$		0			0
$437$		−	56380.0i		−	0.295231i
$438$		0			0
$439$			155214.i			0.805383i	0.915336	+	0.402691i	$0.131925\pi$
$439$			155214.i			0.805383i	−0.915336	+	0.402691i	$0.868075\pi$
$440$		0			0
$441$	−172734.			−0.888179
$442$		0			0
$443$	−104175.			−0.530828			−0.265414	−	0.964134i	$-0.585509\pi$
$443$	−104175.			−0.530828			−0.265414	+	0.964134i	$0.585509\pi$
$444$		0			0
$445$	−145168.			−0.733082
$446$		0			0
$447$			53608.3i			0.268298i
$448$		0			0
$449$	88086.4			0.436934			0.218467	−	0.975844i	$-0.429894\pi$
$449$	88086.4			0.436934			0.218467	+	0.975844i	$0.429894\pi$
$450$		0			0
$451$	−112559.	−	65613.8i	−0.553383	−	0.322583i
$452$		0			0
$453$			104224.i			0.507891i
$454$		0			0
$455$	−42966.8			−0.207544
$456$		0			0
$457$		−	77946.3i		−	0.373219i	−0.982434	−	0.186609i	$-0.940250\pi$
$457$		−	77946.3i		−	0.373219i	0.982434	−	0.186609i	$-0.0597499\pi$
$458$		0			0
$459$		−	35342.7i		−	0.167754i
$460$		0			0
$461$		−	115963.i		−	0.545653i	−0.962063	−	0.272827i	$-0.912041\pi$
$461$		−	115963.i		−	0.545653i	0.962063	−	0.272827i	$-0.0879586\pi$
$462$		0			0
$463$	313839.			1.46401			0.732007	−	0.681297i	$-0.238584\pi$
$463$	313839.			1.46401			0.732007	+	0.681297i	$0.238584\pi$
$464$		0			0
$465$	−103398.			−0.478195
$466$		0			0
$467$	96033.3			0.440340			0.220170	−	0.975462i	$-0.429339\pi$
$467$	96033.3			0.440340			0.220170	+	0.975462i	$0.429339\pi$
$468$		0			0
$469$		−	117532.i		−	0.534333i
$470$		0			0
$471$	−29837.2			−0.134498
$472$		0			0
$473$	−197307.	−	115016.i	−0.881904	−	0.514088i
$474$		0			0
$475$			30745.1i			0.136266i
$476$		0			0
$477$	32148.7			0.141295
$478$		0			0
$479$			215693.i			0.940082i	0.882645	+	0.470041i	$0.155761\pi$
$479$			215693.i			0.940082i	−0.882645	+	0.470041i	$0.844239\pi$
$480$		0			0
$481$		−	3403.32i		−	0.0147100i
$482$		0			0
$483$		−	342428.i		−	1.46783i
$484$		0			0
$485$	172148.			0.731844
$486$		0			0
$487$	281250.			1.18587			0.592933	−	0.805252i	$-0.297970\pi$
$487$	281250.			1.18587			0.592933	+	0.805252i	$0.297970\pi$
$488$		0			0
$489$	128140.			0.535881
$490$		0			0
$491$		−	229094.i		−	0.950279i	−0.879910	−	0.475139i	$-0.842398\pi$
$491$		−	229094.i		−	0.950279i	0.879910	−	0.475139i	$-0.157602\pi$
$492$		0			0
$493$	−365025.			−1.50186
$494$		0			0
$495$	−25588.6	+	43896.5i	−0.104433	+	0.179151i
$496$		0			0
$497$		−	336875.i		−	1.36382i
$498$		0			0
$499$	−37652.6			−0.151215			−0.0756073	−	0.997138i	$-0.524090\pi$
$499$	−37652.6			−0.151215			−0.0756073	+	0.997138i	$0.524090\pi$
$500$		0			0
$501$		−	246254.i		−	0.981088i
$502$		0			0
$503$		−	353544.i		−	1.39736i	−0.715435	−	0.698679i	$-0.753772\pi$
$503$		−	353544.i		−	1.39736i	0.715435	−	0.698679i	$-0.246228\pi$
$504$		0			0
$505$		−	151362.i		−	0.593519i
$506$		0			0
$507$	143900.			0.559815
$508$		0			0
$509$	−240356.			−0.927727			−0.463864	−	0.885907i	$-0.653537\pi$
$509$	−240356.			−0.927727			−0.463864	+	0.885907i	$0.653537\pi$
$510$		0			0
$511$	−164222.			−0.628910
$512$		0			0
$513$		−	11258.7i		−	0.0427814i
$514$		0			0
$515$	−34672.2			−0.130727
$516$		0			0
$517$	96996.4	−	166395.i	0.362889	−	0.622527i
$518$		0			0
$519$			6891.62i			0.0255851i
$520$		0			0
$521$	484627.			1.78539			0.892693	−	0.450666i	$-0.148813\pi$
$521$	484627.			1.78539			0.892693	+	0.450666i	$0.148813\pi$
$522$		0			0
$523$			188211.i			0.688084i	0.938954	+	0.344042i	$0.111796\pi$
$523$			188211.i			0.688084i	−0.938954	+	0.344042i	$0.888204\pi$
$524$		0			0
$525$			186732.i			0.677487i
$526$		0			0
$527$			322315.i			1.16054i
$528$		0			0
$529$	213746.			0.763811
$530$		0			0
$531$	−51180.0			−0.181515
$532$		0			0
$533$	31713.2			0.111631
$534$		0			0
$535$			138604.i			0.484247i
$536$		0			0
$537$	279671.			0.969837
$538$		0			0
$539$	389847.	−	668772.i	1.34189	−	2.30197i
$540$		0			0
$541$			329143.i			1.12458i	0.826940	+	0.562290i	$0.190080\pi$
$541$			329143.i			1.12458i	−0.826940	+	0.562290i	$0.809920\pi$
$542$		0			0
$543$	51760.7			0.175550
$544$		0			0
$545$			125594.i			0.422838i
$546$		0			0
$547$			201214.i			0.672486i	0.941775	+	0.336243i	$0.109156\pi$
$547$			201214.i			0.672486i	−0.941775	+	0.336243i	$0.890844\pi$
$548$		0			0
$549$		−	101956.i		−	0.338274i
$550$		0			0
$551$	−116282.			−0.383010
$552$		0			0
$553$	−633068.			−2.07014
$554$		0			0
$555$	9338.18			0.0303163
$556$		0			0
$557$		−	52859.7i		−	0.170378i	−0.996365	−	0.0851892i	$-0.972851\pi$
$557$		−	52859.7i		−	0.170378i	0.996365	−	0.0851892i	$-0.0271495\pi$
$558$		0			0
$559$	55591.0			0.177902
$560$		0			0
$561$	136836.	+	79765.7i	0.434784	+	0.253449i
$562$		0			0
$563$		−	143681.i		−	0.453295i	−0.973977	−	0.226648i	$-0.927223\pi$
$563$		−	143681.i		−	0.453295i	0.973977	−	0.226648i	$-0.0727766\pi$
$564$		0			0
$565$	−257368.			−0.806229
$566$		0			0
$567$		−	68380.6i		−	0.212700i
$568$		0			0
$569$		−	568025.i		−	1.75446i	−0.480071	−	0.877230i	$-0.659389\pi$
$569$		−	568025.i		−	1.75446i	0.480071	−	0.877230i	$-0.340611\pi$
$570$		0			0
$571$		−	165896.i		−	0.508818i	−0.967097	−	0.254409i	$-0.918119\pi$
$571$		−	165896.i		−	0.508818i	0.967097	−	0.254409i	$-0.0818810\pi$
$572$		0			0
$573$	285606.			0.869876
$574$		0			0
$575$	−269162.			−0.814100
$576$		0			0
$577$	157238.			0.472286			0.236143	−	0.971718i	$-0.424117\pi$
$577$	157238.			0.472286			0.236143	+	0.971718i	$0.424117\pi$
$578$		0			0
$579$			8951.72i			0.0267023i
$580$		0			0
$581$	−808259.			−2.39441
$582$		0			0
$583$	−72557.3	+	124470.i	−0.213473	+	0.366208i
$584$		0			0
$585$		−	12367.8i		−	0.0361393i
$586$		0			0
$587$	−9552.74			−0.0277237			−0.0138619	−	0.999904i	$-0.504413\pi$
$587$	−9552.74			−0.0277237			−0.0138619	+	0.999904i	$0.504413\pi$
$588$		0			0
$589$			102676.i			0.295964i
$590$		0			0
$591$			53974.9i			0.154531i
$592$		0			0
$593$			164511.i			0.467827i	0.972257	+	0.233913i	$0.0751532\pi$
$593$			164511.i			0.467827i	−0.972257	+	0.233913i	$0.924847\pi$
$594$		0			0
$595$	−367503.			−1.03807
$596$		0			0
$597$	−264818.			−0.743018
$598$		0			0
$599$	−318614.			−0.887996			−0.443998	−	0.896028i	$-0.646440\pi$
$599$	−318614.			−0.887996			−0.443998	+	0.896028i	$0.646440\pi$
$600$		0			0
$601$			385872.i			1.06830i	0.845389	+	0.534151i	$0.179369\pi$
$601$			385872.i			1.06830i	−0.845389	+	0.534151i	$0.820631\pi$
$602$		0			0
$603$	33831.1			0.0930425
$604$		0			0
$605$	−112202.	−	198142.i	−0.306542	−	0.541335i
$606$		0			0
$607$			218753.i			0.593712i	0.954922	+	0.296856i	$0.0959381\pi$
$607$			218753.i			0.593712i	−0.954922	+	0.296856i	$0.904062\pi$
$608$		0			0
$609$	−706247.			−1.90424
$610$		0			0
$611$			46881.3i			0.125579i
$612$		0			0
$613$		−	348766.i		−	0.928140i	−0.885799	−	0.464070i	$-0.846389\pi$
$613$		−	348766.i		−	0.928140i	0.885799	−	0.464070i	$-0.153611\pi$
$614$		0			0
$615$			87015.9i			0.230064i
$616$		0			0
$617$	222234.			0.583767			0.291884	−	0.956454i	$-0.405718\pi$
$617$	222234.			0.583767			0.291884	+	0.956454i	$0.405718\pi$
$618$		0			0
$619$	−478523.			−1.24888			−0.624441	−	0.781072i	$-0.714673\pi$
$619$	−478523.			−1.24888			−0.624441	+	0.781072i	$0.714673\pi$
$620$		0			0
$621$	98566.0			0.255590
$622$		0			0
$623$			875539.i			2.25579i
$624$		0			0
$625$	−4397.54			−0.0112577
$626$		0			0
$627$	43590.3	+	25410.1i	0.110880	+	0.0646355i
$628$		0			0
$629$		−	29109.3i		−	0.0735750i
$630$		0			0
$631$	332157.			0.834229			0.417115	−	0.908854i	$-0.363041\pi$
$631$	332157.			0.834229			0.417115	+	0.908854i	$0.363041\pi$
$632$		0			0
$633$		−	356879.i		−	0.890664i
$634$		0			0
$635$			33675.2i			0.0835146i
$636$		0			0
$637$			188425.i			0.464366i
$638$		0			0
$639$	96967.6			0.237479
$640$		0			0
$641$	−194677.			−0.473803			−0.236902	−	0.971534i	$-0.576132\pi$
$641$	−194677.			−0.473803			−0.236902	+	0.971534i	$0.576132\pi$
$642$		0			0
$643$	−598433.			−1.44742			−0.723709	−	0.690106i	$-0.757564\pi$
$643$	−598433.			−1.44742			−0.723709	+	0.690106i	$0.757564\pi$
$644$		0			0
$645$			152533.i			0.366643i
$646$		0			0
$647$	789504.			1.88602			0.943008	−	0.332769i	$-0.107983\pi$
$647$	789504.			1.88602			0.943008	+	0.332769i	$0.107983\pi$
$648$		0			0
$649$	115509.	−	198153.i	0.274238	−	0.470448i
$650$		0			0
$651$			623611.i			1.47147i
$652$		0			0
$653$	704147.			1.65134			0.825671	−	0.564151i	$-0.190797\pi$
$653$	704147.			1.65134			0.825671	+	0.564151i	$0.190797\pi$
$654$		0			0
$655$		−	237448.i		−	0.553459i
$656$		0			0
$657$		−	47270.3i		−	0.109511i
$658$		0			0
$659$		−	603395.i		−	1.38941i	−0.719294	−	0.694706i	$-0.755535\pi$
$659$		−	603395.i		−	1.38941i	0.719294	−	0.694706i	$-0.244465\pi$
$660$		0			0
$661$	257502.			0.589356			0.294678	−	0.955597i	$-0.404788\pi$
$661$	257502.			0.589356			0.294678	+	0.955597i	$0.404788\pi$
$662$		0			0
$663$	−38553.2			−0.0877069
$664$		0			0
$665$	−117072.			−0.264733
$666$		0			0
$667$		−	1.01801e6i		−	2.28823i
$668$		0			0
$669$	8463.96			0.0189113
$670$		0			0
$671$	394742.	+	230107.i	0.876735	+	0.511075i
$672$		0			0
$673$			10491.1i			0.0231629i	0.999933	+	0.0115814i	$0.00368656\pi$
$673$			10491.1i			0.0231629i	−0.999933	+	0.0115814i	$0.996313\pi$
$674$		0			0
$675$	−53749.9			−0.117970
$676$		0			0
$677$			243377.i			0.531008i	0.964110	+	0.265504i	$0.0855384\pi$
$677$			243377.i			0.531008i	−0.964110	+	0.265504i	$0.914462\pi$
$678$		0			0
$679$		−	1.03826e6i		−	2.25199i
$680$		0			0
$681$		−	221351.i		−	0.477296i
$682$		0			0
$683$	−94210.5			−0.201957			−0.100978	−	0.994889i	$-0.532197\pi$
$683$	−94210.5			−0.201957			−0.100978	+	0.994889i	$0.532197\pi$
$684$		0			0
$685$	161610.			0.344419
$686$		0			0
$687$	168292.			0.356574
$688$		0			0
$689$		−	35069.2i		−	0.0738732i
$690$		0			0
$691$	49743.2			0.104178			0.0520892	−	0.998642i	$-0.483412\pi$
$691$	49743.2			0.104178			0.0520892	+	0.998642i	$0.483412\pi$
$692$		0			0
$693$	264749.	+	154330.i	0.551273	+	0.321354i
$694$		0			0
$695$		−	341906.i		−	0.707843i
$696$		0			0
$697$	271249.			0.558345
$698$		0			0
$699$		−	300686.i		−	0.615402i
$700$		0			0
$701$		−	337972.i		−	0.687772i	−0.939011	−	0.343886i	$-0.888257\pi$
$701$		−	337972.i		−	0.687772i	0.939011	−	0.343886i	$-0.111743\pi$
$702$		0			0
$703$		−	9273.03i		−	0.0187634i
$704$		0			0
$705$	−128635.			−0.258810
$706$		0			0
$707$	−912895.			−1.82634
$708$		0			0
$709$	209306.			0.416379			0.208189	−	0.978089i	$-0.433243\pi$
$709$	209306.			0.416379			0.208189	+	0.978089i	$0.433243\pi$
$710$		0			0
$711$		−	182225.i		−	0.360470i
$712$		0			0
$713$	−898894.			−1.76819
$714$		0			0
$715$	47884.1	+	27913.1i	0.0936655	+	0.0546004i
$716$		0			0
$717$			19529.5i			0.0379886i
$718$		0			0
$719$	−382242.			−0.739403			−0.369701	−	0.929151i	$-0.620540\pi$
$719$	−382242.			−0.739403			−0.369701	+	0.929151i	$0.620540\pi$
$720$		0			0
$721$			209115.i			0.402267i
$722$		0			0
$723$			277439.i			0.530751i
$724$		0			0
$725$			555138.i			1.05615i
$726$		0			0
$727$	573664.			1.08540			0.542698	−	0.839928i	$-0.317403\pi$
$727$	573664.			1.08540			0.542698	+	0.839928i	$0.317403\pi$
$728$		0			0
$729$	19683.0			0.0370370
$730$		0			0
$731$	475481.			0.889812
$732$		0			0
$733$		−	898874.i		−	1.67298i	−0.547982	−	0.836490i	$-0.684604\pi$
$733$		−	898874.i		−	1.67298i	0.547982	−	0.836490i	$-0.315396\pi$
$734$		0			0
$735$	−517009.			−0.957025
$736$		0			0
$737$	−76354.1	+	130983.i	−0.140572	+	0.241147i
$738$		0			0
$739$		−	716712.i		−	1.31237i	−0.754601	−	0.656184i	$-0.772170\pi$
$739$		−	716712.i		−	1.31237i	0.754601	−	0.656184i	$-0.227830\pi$
$740$		0			0
$741$	−12281.5			−0.0223673
$742$		0			0
$743$		−	327330.i		−	0.592937i	−0.955043	−	0.296468i	$-0.904191\pi$
$743$		−	327330.i		−	0.592937i	0.955043	−	0.296468i	$-0.0958089\pi$
$744$		0			0
$745$			160455.i			0.289094i
$746$		0			0
$747$		−	232653.i		−	0.416934i
$748$		0			0
$749$	835946.			1.49010
$750$		0			0
$751$	−473695.			−0.839883			−0.419941	−	0.907551i	$-0.637949\pi$
$751$	−473695.			−0.839883			−0.419941	+	0.907551i	$0.637949\pi$
$752$		0			0
$753$	−9650.88			−0.0170207
$754$		0			0
$755$			311952.i			0.547260i
$756$		0			0
$757$	168385.			0.293841			0.146920	−	0.989148i	$-0.453064\pi$
$757$	168385.			0.293841			0.146920	+	0.989148i	$0.453064\pi$
$758$		0			0
$759$	−222456.	+	381617.i	−0.386154	+	0.662436i
$760$		0			0
$761$		−	216244.i		−	0.373400i	−0.982417	−	0.186700i	$-0.940221\pi$
$761$		−	216244.i		−	0.373400i	0.982417	−	0.186700i	$-0.0597792\pi$
$762$		0			0
$763$	757479.			1.30113
$764$		0			0
$765$		−	105784.i		−	0.180758i
$766$		0			0
$767$			55829.3i			0.0949011i
$768$		0			0
$769$		−	356784.i		−	0.603328i	−0.953414	−	0.301664i	$-0.902458\pi$
$769$		−	356784.i		−	0.603328i	0.953414	−	0.301664i	$-0.0975420\pi$
$770$		0			0
$771$	−6561.53			−0.0110382
$772$		0			0
$773$	−91587.2			−0.153277			−0.0766383	−	0.997059i	$-0.524419\pi$
$773$	−91587.2			−0.153277			−0.0766383	+	0.997059i	$0.524419\pi$
$774$		0			0
$775$	490183.			0.816122
$776$		0			0
$777$		−	56320.4i		−	0.0932875i
$778$		0			0
$779$	86408.8			0.142391
$780$		0			0
$781$	−218849.	+	375429.i	−0.358791	+	0.615496i
$782$		0			0
$783$		−	203289.i		−	0.331582i
$784$		0			0
$785$	−89305.7			−0.144924
$786$		0			0
$787$			77739.6i			0.125514i	0.998029	+	0.0627571i	$0.0199894\pi$
$787$			77739.6i			0.125514i	−0.998029	+	0.0627571i	$0.980011\pi$
$788$		0			0
$789$			614926.i			0.987799i
$790$		0			0
$791$			1.55224e6i			2.48088i
$792$		0			0
$793$	−111218.			−0.176859
$794$		0			0
$795$	96224.2			0.152247
$796$		0			0
$797$	125376.			0.197377			0.0986887	−	0.995118i	$-0.468535\pi$
$797$	125376.			0.197377			0.0986887	+	0.995118i	$0.468535\pi$
$798$		0			0
$799$			400985.i			0.628109i
$800$		0			0
$801$	−252019.			−0.392798
$802$		0			0
$803$	183016.	+	106685.i	0.283830	+	0.165453i
$804$		0			0
$805$		−	1.02492e6i		−	1.58160i
$806$		0			0
$807$	522284.			0.801972
$808$		0			0
$809$			466047.i			0.712087i	0.934469	+	0.356044i	$0.115875\pi$
$809$			466047.i			0.712087i	−0.934469	+	0.356044i	$0.884125\pi$
$810$		0			0
$811$			1.07207e6i			1.62998i	0.579473	+	0.814992i	$0.303259\pi$
$811$			1.07207e6i			1.62998i	−0.579473	+	0.814992i	$0.696741\pi$
$812$		0			0
$813$			718911.i			1.08766i
$814$		0			0
$815$	383536.			0.577420
$816$		0			0
$817$	151469.			0.226923
$818$		0			0
$819$	−74592.4			−0.111206
$820$		0			0
$821$			423155.i			0.627789i	0.949458	+	0.313894i	$0.101634\pi$
$821$			423155.i			0.627789i	−0.949458	+	0.313894i	$0.898366\pi$
$822$		0			0
$823$	−1.01630e6			−1.50045			−0.750227	−	0.661180i	$-0.770056\pi$
$823$	−1.01630e6			−1.50045			−0.750227	+	0.661180i	$0.770056\pi$
$824$		0			0
$825$	121309.	−	208103.i	0.178232	−	0.305753i
$826$		0			0
$827$		−	83099.6i		−	0.121503i	−0.998153	−	0.0607517i	$-0.980650\pi$
$827$		−	83099.6i		−	0.121503i	0.998153	−	0.0607517i	$-0.0193498\pi$
$828$		0			0
$829$	−293001.			−0.426343			−0.213172	−	0.977015i	$-0.568379\pi$
$829$	−293001.			−0.426343			−0.213172	+	0.977015i	$0.568379\pi$
$830$		0			0
$831$			211463.i			0.306220i
$832$		0			0
$833$			1.61164e6i			2.32262i
$834$		0			0
$835$		−	737062.i		−	1.05714i
$836$		0			0
$837$	−179503.			−0.256225
$838$		0			0
$839$	−426202.			−0.605468			−0.302734	−	0.953075i	$-0.597899\pi$
$839$	−426202.			−0.605468			−0.302734	+	0.953075i	$0.597899\pi$
$840$		0			0
$841$	−1.39233e6			−1.96856
$842$		0			0
$843$			90317.5i			0.127092i
$844$		0			0
$845$	430706.			0.603208
$846$		0			0
$847$	−1.19503e6	+	676713.i	−1.66576	+	0.943274i
$848$		0			0
$849$			534157.i			0.741060i
$850$		0			0
$851$	81182.0			0.112099
$852$		0			0
$853$		−	1.33250e6i		−	1.83134i	−0.401926	−	0.915672i	$-0.631659\pi$
$853$		−	1.33250e6i		−	1.83134i	0.401926	−	0.915672i	$-0.368341\pi$
$854$		0			0
$855$		−	33698.4i		−	0.0460975i
$856$		0			0
$857$		−	295654.i		−	0.402552i	−0.979535	−	0.201276i	$-0.935491\pi$
$857$		−	295654.i		−	0.402552i	0.979535	−	0.201276i	$-0.0645088\pi$
$858$		0			0
$859$	931988.			1.26306			0.631530	−	0.775352i	$-0.282427\pi$
$859$	931988.			1.26306			0.631530	+	0.775352i	$0.282427\pi$
$860$		0			0
$861$	524810.			0.707939
$862$		0			0
$863$	−159744.			−0.214488			−0.107244	−	0.994233i	$-0.534203\pi$
$863$	−159744.			−0.214488			−0.107244	+	0.994233i	$0.534203\pi$
$864$		0			0
$865$			20627.3i			0.0275683i
$866$		0			0
$867$	104235.			0.138667
$868$		0			0
$869$	705519.	+	411268.i	0.934263	+	0.544610i
$870$		0			0
$871$		−	36904.3i		−	0.0486453i
$872$		0			0
$873$	298857.			0.392135
$874$		0			0
$875$			1.47068e6i			1.92089i
$876$		0			0
$877$			1.43711e6i			1.86849i	0.356636	+	0.934244i	$0.383924\pi$
$877$			1.43711e6i			1.86849i	−0.356636	+	0.934244i	$0.616076\pi$
$878$		0			0
$879$			554838.i			0.718106i
$880$		0			0
$881$	478465.			0.616451			0.308225	−	0.951313i	$-0.400265\pi$
$881$	478465.			0.616451			0.308225	+	0.951313i	$0.400265\pi$
$882$		0			0
$883$	−280446.			−0.359689			−0.179845	−	0.983695i	$-0.557560\pi$
$883$	−280446.			−0.359689			−0.179845	+	0.983695i	$0.557560\pi$
$884$		0			0
$885$	−153187.			−0.195584
$886$		0			0
$887$		−	1.20382e6i		−	1.53008i	−0.643982	−	0.765041i	$-0.722719\pi$
$887$		−	1.20382e6i		−	1.53008i	0.643982	−	0.765041i	$-0.277281\pi$
$888$		0			0
$889$	203101.			0.256986
$890$		0			0
$891$	−44423.0	+	76206.5i	−0.0559568	+	0.0959923i
$892$		0			0
$893$			127738.i			0.160183i
$894$		0			0
$895$	837081.			1.04501
$896$		0			0
$897$		−	107520.i		−	0.133630i
$898$		0			0
$899$			1.85394e6i			2.29391i
$900$		0			0
$901$		−	299954.i		−	0.369491i
$902$		0			0
$903$	919955.			1.12821
$904$		0			0
$905$	154925.			0.189157
$906$		0			0
$907$	−224182.			−0.272512			−0.136256	−	0.990674i	$-0.543507\pi$
$907$	−224182.			−0.272512			−0.136256	+	0.990674i	$0.543507\pi$
$908$		0			0
$909$		−	262772.i		−	0.318018i
$910$		0			0
$911$	815416.			0.982522			0.491261	−	0.871012i	$-0.336536\pi$
$911$	815416.			0.982522			0.491261	+	0.871012i	$0.336536\pi$
$912$		0			0
$913$	900760.	+	525080.i	1.08061	+	0.629918i
$914$		0			0
$915$		−	305164.i		−	0.364495i
$916$		0			0
$917$	−1.43209e6			−1.70307
$918$		0			0
$919$		−	858763.i		−	1.01682i	−0.861116	−	0.508408i	$-0.830234\pi$
$919$		−	858763.i		−	1.01682i	0.861116	−	0.508408i	$-0.169766\pi$
$920$		0			0
$921$		−	407462.i		−	0.480361i
$922$		0			0
$923$		−	105776.i		−	0.124161i
$924$		0			0
$925$	−44270.1			−0.0517400
$926$		0			0
$927$	−60192.6			−0.0700460
$928$		0			0
$929$	489087.			0.566702			0.283351	−	0.959016i	$-0.408554\pi$
$929$	489087.			0.566702			0.283351	+	0.959016i	$0.408554\pi$
$930$		0			0
$931$			513402.i			0.592322i
$932$		0			0
$933$	315324.			0.362238
$934$		0			0
$935$	409562.	+	238746.i	0.468486	+	0.273095i
$936$		0			0
$937$			276854.i			0.315335i	0.987492	+	0.157667i	$0.0503974\pi$
$937$			276854.i			0.315335i	−0.987492	+	0.157667i	$0.949603\pi$
$938$		0			0
$939$	−445513.			−0.505277
$940$		0			0
$941$		−	61582.8i		−	0.0695472i	−0.999395	−	0.0347736i	$-0.988929\pi$
$941$		−	61582.8i		−	0.0695472i	0.999395	−	0.0347736i	$-0.0110710\pi$
$942$		0			0
$943$			756478.i			0.850693i
$944$		0			0
$945$		−	204670.i		−	0.229187i
$946$		0			0
$947$	1.23913e6			1.38171			0.690856	−	0.722992i	$-0.257234\pi$
$947$	1.23913e6			1.38171			0.690856	+	0.722992i	$0.257234\pi$
$948$		0			0
$949$	−51564.4			−0.0572555
$950$		0			0
$951$	611295.			0.675912
$952$		0			0
$953$		−	1.44690e6i		−	1.59314i	−0.604548	−	0.796569i	$-0.706646\pi$
$953$		−	1.44690e6i		−	1.59314i	0.604548	−	0.796569i	$-0.293354\pi$
$954$		0			0
$955$	854844.			0.937303
$956$		0			0
$957$	787074.	+	458809.i	0.859392	+	0.500966i
$958$		0			0
$959$		−	974701.i		−	1.05982i
$960$		0			0
$961$	713496.			0.772582
$962$		0			0
$963$			240622.i			0.259468i
$964$		0			0
$965$			26793.3i			0.0287721i
$966$		0			0
$967$			382759.i			0.409328i	0.978832	+	0.204664i	$0.0656102\pi$
$967$			382759.i			0.409328i	−0.978832	+	0.204664i	$0.934390\pi$
$968$		0			0
$969$	−105046.			−0.111875
$970$		0			0
$971$	−571623.			−0.606277			−0.303138	−	0.952947i	$-0.598034\pi$
$971$	−571623.			−0.606277			−0.303138	+	0.952947i	$0.598034\pi$
$972$		0			0
$973$	−2.06210e6			−2.17813
$974$		0			0
$975$			58632.6i			0.0616779i
$976$		0			0
$977$	1.22248e6			1.28072			0.640360	−	0.768075i	$-0.278785\pi$
$977$	1.22248e6			1.28072			0.640360	+	0.768075i	$0.278785\pi$
$978$		0			0
$979$	568788.	−	975740.i	0.593451	−	1.01805i
$980$		0			0
$981$			218036.i			0.226564i
$982$		0			0
$983$	1.01427e6			1.04966			0.524829	−	0.851208i	$-0.324129\pi$
$983$	1.01427e6			1.04966			0.524829	+	0.851208i	$0.324129\pi$
$984$		0			0
$985$			161552.i			0.166510i
$986$		0			0
$987$			775822.i			0.796394i
$988$		0			0
$989$			1.32605e6i			1.35571i
$990$		0			0
$991$	−609717.			−0.620842			−0.310421	−	0.950599i	$-0.600470\pi$
$991$	−609717.			−0.620842			−0.310421	+	0.950599i	$0.600470\pi$
$992$		0			0
$993$	570112.			0.578178
$994$		0			0
$995$	−792626.			−0.800612
$996$		0			0
$997$		−	134059.i		−	0.134867i	−0.997724	−	0.0674335i	$-0.978519\pi$
$997$		−	134059.i		−	0.134867i	0.997724	−	0.0674335i	$-0.0214811\pi$
$998$		0			0
$999$	16211.5			0.0162440

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.5.j.a.241.7		8
4.3	odd	2		33.5.c.a.10.5	yes	8
11.10	odd	2	inner	528.5.j.a.241.8		8
12.11	even	2		99.5.c.c.10.4		8
44.43	even	2		33.5.c.a.10.4	✓	8
132.131	odd	2		99.5.c.c.10.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.c.a.10.4	✓	8	44.43	even	2
33.5.c.a.10.5	yes	8	4.3	odd	2
99.5.c.c.10.4		8	12.11	even	2
99.5.c.c.10.5		8	132.131	odd	2
528.5.j.a.241.7		8	1.1	even	1	trivial
528.5.j.a.241.8		8	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	528.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} +15.5526 q^{5} -93.8006i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} +15.5526 q^{5} -93.8006i q^{7} +27.0000 q^{9} +(-60.9369 + 104.536i) q^{11} -29.4527i q^{13} +80.8135 q^{15} -251.915i q^{17} -80.2497i q^{19} -487.402i q^{21} +702.557 q^{23} -383.118 q^{25} +140.296 q^{27} -1449.00i q^{29} -1279.46 q^{31} +(-316.638 + 543.183i) q^{33} -1458.84i q^{35} +115.552 q^{37} -153.041i q^{39} +1076.75i q^{41} +1887.47i q^{43} +419.919 q^{45} -1591.75 q^{47} -6397.55 q^{49} -1308.99i q^{51} +1190.69 q^{53} +(-947.726 + 1625.80i) q^{55} -416.990i q^{57} -1895.56 q^{59} -3776.15i q^{61} -2532.62i q^{63} -458.065i q^{65} +1253.00 q^{67} +3650.59 q^{69} +3591.39 q^{71} -1750.75i q^{73} -1990.74 q^{75} +(9805.50 + 5715.92i) q^{77} -6749.08i q^{79} +729.000 q^{81} -8616.78i q^{83} -3917.92i q^{85} -7529.24i q^{87} -9334.05 q^{89} -2762.68 q^{91} -6648.27 q^{93} -1248.09i q^{95} +11068.8 q^{97} +(-1645.30 + 2822.46i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 36 q^{5} + 216 q^{9}+O(q^{10}) \) 8 * q - 36 * q^5 + 216 * q^9	\( 8 q - 36 q^{5} + 216 q^{9} - 36 q^{11} - 108 q^{15} - 516 q^{23} - 2280 q^{25} - 2752 q^{31} + 1008 q^{33} + 5296 q^{37} - 972 q^{45} - 420 q^{47} - 6832 q^{49} + 3540 q^{53} - 3784 q^{55} + 16632 q^{59} + 3656 q^{67} + 9036 q^{69} + 13212 q^{71} + 9288 q^{75} + 23268 q^{77} + 5832 q^{81} + 15528 q^{89} + 19752 q^{91} - 21384 q^{93} + 7624 q^{97} - 972 q^{99}+O(q^{100}) \) 8 * q - 36 * q^5 + 216 * q^9 - 36 * q^11 - 108 * q^15 - 516 * q^23 - 2280 * q^25 - 2752 * q^31 + 1008 * q^33 + 5296 * q^37 - 972 * q^45 - 420 * q^47 - 6832 * q^49 + 3540 * q^53 - 3784 * q^55 + 16632 * q^59 + 3656 * q^67 + 9036 * q^69 + 13212 * q^71 + 9288 * q^75 + 23268 * q^77 + 5832 * q^81 + 15528 * q^89 + 19752 * q^91 - 21384 * q^93 + 7624 * q^97 - 972 * q^99

Introduction

L-functions

Modular forms

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