LMFDB - Embedded newform 3528.2.s.bm.3313.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3528,2,Mod(361,3528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3528, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3528.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3528.s (of order $3$, degree $2$, 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:	$28.1712218331$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1176)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3313.1
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	3528.3313
Dual form			3528.2.s.bm.361.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.292893 - 0.507306i) q^{5} +O(q^{10})$	$q+(0.292893 - 0.507306i) q^{5} +(-0.414214 - 0.717439i) q^{11} -1.41421 q^{13} +(1.12132 + 1.94218i) q^{17} +(-3.41421 + 5.91359i) q^{19} +(2.41421 - 4.18154i) q^{23} +(2.32843 + 4.03295i) q^{25} -8.48528 q^{29} +(-2.58579 - 4.47871i) q^{31} +(-0.828427 + 1.43488i) q^{37} +0.585786 q^{41} -8.00000 q^{43} +(3.41421 - 5.91359i) q^{47} +(-6.65685 - 11.5300i) q^{53} -0.485281 q^{55} +(2.58579 + 4.47871i) q^{59} +(-6.94975 + 12.0373i) q^{61} +(-0.414214 + 0.717439i) q^{65} +(4.00000 + 6.92820i) q^{67} +0.828427 q^{71} +(5.53553 + 9.58783i) q^{73} +(1.17157 - 2.02922i) q^{79} -15.3137 q^{83} +1.31371 q^{85} +(-5.36396 + 9.29065i) q^{89} +(2.00000 + 3.46410i) q^{95} -7.75736 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5}+O(q^{10}) $ 4 * q + 4 * q^5	$ 4 q + 4 q^{5} + 4 q^{11} - 4 q^{17} - 8 q^{19} + 4 q^{23} - 2 q^{25} - 16 q^{31} + 8 q^{37} + 8 q^{41} - 32 q^{43} + 8 q^{47} - 4 q^{53} + 32 q^{55} + 16 q^{59} - 8 q^{61} + 4 q^{65} + 16 q^{67} - 8 q^{71} + 8 q^{73} + 16 q^{79} - 16 q^{83} - 40 q^{85} + 4 q^{89} + 8 q^{95} - 48 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^11 - 4 * q^17 - 8 * q^19 + 4 * q^23 - 2 * q^25 - 16 * q^31 + 8 * q^37 + 8 * q^41 - 32 * q^43 + 8 * q^47 - 4 * q^53 + 32 * q^55 + 16 * q^59 - 8 * q^61 + 4 * q^65 + 16 * q^67 - 8 * q^71 + 8 * q^73 + 16 * q^79 - 16 * q^83 - 40 * q^85 + 4 * q^89 + 8 * q^95 - 48 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$1765$	$2647$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.292893	−	0.507306i	0.130986	−	0.226874i	−0.793071	−	0.609129i	$-0.791519\pi$
$5$	0.292893	−	0.507306i	0.130986	−	0.226874i	0.924057	+	0.382255i	$0.124852\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.414214	−	0.717439i	−0.124890	−	0.216316i	0.796800	−	0.604243i	$-0.206524\pi$
$11$	−0.414214	−	0.717439i	−0.124890	−	0.216316i	−0.921690	+	0.387927i	$0.873191\pi$
$12$		0			0
$13$	−1.41421			−0.392232			−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421			−0.392232			−0.196116	+	0.980581i	$0.562833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.12132	+	1.94218i	0.271960	+	0.471049i	0.969364	−	0.245630i	$-0.0789948\pi$
$17$	1.12132	+	1.94218i	0.271960	+	0.471049i	−0.697404	+	0.716679i	$0.745661\pi$
$18$		0			0
$19$	−3.41421	+	5.91359i	−0.783274	+	1.35667i	0.146750	+	0.989174i	$0.453119\pi$
$19$	−3.41421	+	5.91359i	−0.783274	+	1.35667i	−0.930025	+	0.367497i	$0.880215\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.41421	−	4.18154i	0.503398	−	0.871911i	−0.496594	−	0.867983i	$-0.665416\pi$
$23$	2.41421	−	4.18154i	0.503398	−	0.871911i	0.999992	−	0.00392850i	$-0.00125049\pi$
$24$		0			0
$25$	2.32843	+	4.03295i	0.465685	+	0.806591i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−8.48528			−1.57568			−0.787839	−	0.615882i	$-0.788800\pi$
$29$	−8.48528			−1.57568			−0.787839	+	0.615882i	$0.788800\pi$
$30$		0			0
$31$	−2.58579	−	4.47871i	−0.464421	−	0.804401i	0.534754	−	0.845008i	$-0.320404\pi$
$31$	−2.58579	−	4.47871i	−0.464421	−	0.804401i	−0.999175	+	0.0406069i	$0.987071\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.828427	+	1.43488i	−0.136193	+	0.235892i	−0.926052	−	0.377395i	$-0.876820\pi$
$37$	−0.828427	+	1.43488i	−0.136193	+	0.235892i	0.789860	+	0.613287i	$0.210153\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.585786			0.0914845			0.0457422	−	0.998953i	$-0.485435\pi$
$41$	0.585786			0.0914845			0.0457422	+	0.998953i	$0.485435\pi$
$42$		0			0
$43$	−8.00000			−1.21999			−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000			−1.21999			−0.609994	+	0.792406i	$0.708828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.41421	−	5.91359i	0.498014	−	0.862586i	−0.501983	−	0.864877i	$-0.667396\pi$
$47$	3.41421	−	5.91359i	0.498014	−	0.862586i	0.999997	+	0.00229145i	$0.000729392\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.65685	−	11.5300i	−0.914389	−	1.58377i	−0.807793	−	0.589466i	$-0.799338\pi$
$53$	−6.65685	−	11.5300i	−0.914389	−	1.58377i	−0.106596	−	0.994302i	$-0.533995\pi$
$54$		0			0
$55$	−0.485281			−0.0654353
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.58579	+	4.47871i	0.336641	+	0.583079i	0.983799	−	0.179277i	$-0.0573759\pi$
$59$	2.58579	+	4.47871i	0.336641	+	0.583079i	−0.647158	+	0.762356i	$0.724043\pi$
$60$		0			0
$61$	−6.94975	+	12.0373i	−0.889824	+	1.54122i	−0.0497412	+	0.998762i	$0.515840\pi$
$61$	−6.94975	+	12.0373i	−0.889824	+	1.54122i	−0.840083	+	0.542458i	$0.817494\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.414214	+	0.717439i	−0.0513769	+	0.0889873i
$66$		0			0
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	0.999915	−	0.0130248i	$-0.00414604\pi$
$67$	4.00000	+	6.92820i	0.488678	+	0.846415i	−0.511237	+	0.859440i	$0.670813\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.828427			0.0983162			0.0491581	−	0.998791i	$-0.484346\pi$
$71$	0.828427			0.0983162			0.0491581	+	0.998791i	$0.484346\pi$
$72$		0			0
$73$	5.53553	+	9.58783i	0.647885	+	1.12217i	0.983627	+	0.180216i	$0.0576797\pi$
$73$	5.53553	+	9.58783i	0.647885	+	1.12217i	−0.335742	+	0.941954i	$0.608987\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1.17157	−	2.02922i	0.131812	−	0.228306i	−0.792563	−	0.609790i	$-0.791254\pi$
$79$	1.17157	−	2.02922i	0.131812	−	0.228306i	0.924375	+	0.381485i	$0.124587\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−15.3137			−1.68090			−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137			−1.68090			−0.840449	+	0.541891i	$0.817709\pi$
$84$		0			0
$85$	1.31371			0.142492
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.36396	+	9.29065i	−0.568579	+	0.984807i	0.428128	+	0.903718i	$0.359173\pi$
$89$	−5.36396	+	9.29065i	−0.568579	+	0.984807i	−0.996707	+	0.0810892i	$0.974160\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	+	3.46410i	0.205196	+	0.355409i
$96$		0			0
$97$	−7.75736			−0.787641			−0.393820	−	0.919187i	$-0.628847\pi$
$97$	−7.75736			−0.787641			−0.393820	+	0.919187i	$0.628847\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.70711	+	2.95680i	0.169863	+	0.294212i	0.938372	−	0.345628i	$-0.112334\pi$
$101$	1.70711	+	2.95680i	0.169863	+	0.294212i	−0.768508	+	0.639840i	$0.779001\pi$
$102$		0			0
$103$	−5.41421	+	9.37769i	−0.533478	+	0.924012i	0.465757	+	0.884913i	$0.345782\pi$
$103$	−5.41421	+	9.37769i	−0.533478	+	0.924012i	−0.999235	+	0.0390989i	$0.987551\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	7.24264	−	12.5446i	0.700173	−	1.21273i	−0.268233	−	0.963354i	$-0.586440\pi$
$107$	7.24264	−	12.5446i	0.700173	−	1.21273i	0.968406	−	0.249380i	$-0.0802269\pi$
$108$		0			0
$109$	5.65685	+	9.79796i	0.541828	+	0.938474i	0.998799	+	0.0489926i	$0.0156011\pi$
$109$	5.65685	+	9.79796i	0.541828	+	0.938474i	−0.456971	+	0.889482i	$0.651066\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−1.41421	−	2.44949i	−0.131876	−	0.228416i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.15685	−	8.93193i	0.468805	−	0.811994i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	5.65685			0.505964
$126$		0			0
$127$	−15.3137			−1.35887			−0.679436	−	0.733735i	$-0.737775\pi$
$127$	−15.3137			−1.35887			−0.679436	+	0.733735i	$0.737775\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.65685	+	6.33386i	−0.319501	+	0.553392i	−0.980384	−	0.197097i	$-0.936849\pi$
$131$	−3.65685	+	6.33386i	−0.319501	+	0.553392i	0.660883	+	0.750489i	$0.270182\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.24264	+	3.88437i	0.191602	+	0.331864i	0.945781	−	0.324804i	$-0.105298\pi$
$137$	2.24264	+	3.88437i	0.191602	+	0.331864i	−0.754179	+	0.656668i	$0.771965\pi$
$138$		0			0
$139$	−1.65685			−0.140533			−0.0702663	−	0.997528i	$-0.522385\pi$
$139$	−1.65685			−0.140533			−0.0702663	+	0.997528i	$0.522385\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.585786	+	1.01461i	0.0489859	+	0.0848461i
$144$		0			0
$145$	−2.48528	+	4.30463i	−0.206391	+	0.357480i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	−0.994847	−	0.101391i	$-0.967671\pi$
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	0.585231	+	0.810867i	$0.301004\pi$
$150$		0			0
$151$	−4.82843	−	8.36308i	−0.392932	−	0.680578i	0.599903	−	0.800073i	$-0.295206\pi$
$151$	−4.82843	−	8.36308i	−0.392932	−	0.680578i	−0.992835	+	0.119495i	$0.961872\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.02944			−0.243330
$156$		0			0
$157$	6.94975	+	12.0373i	0.554650	+	0.960682i	0.997931	+	0.0642994i	$0.0204813\pi$
$157$	6.94975	+	12.0373i	0.554650	+	0.960682i	−0.443280	+	0.896383i	$0.646185\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.82843	−	11.8272i	0.534844	−	0.926376i	−0.464327	−	0.885664i	$-0.653704\pi$
$163$	6.82843	−	11.8272i	0.534844	−	0.926376i	0.999171	−	0.0407127i	$-0.0129628\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.17157			−0.0906590			−0.0453295	−	0.998972i	$-0.514434\pi$
$167$	−1.17157			−0.0906590			−0.0453295	+	0.998972i	$0.514434\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.70711	+	2.95680i	−0.129789	+	0.224801i	−0.923595	−	0.383370i	$-0.874763\pi$
$173$	−1.70711	+	2.95680i	−0.129789	+	0.224801i	0.793806	+	0.608171i	$0.208097\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.89949	+	15.4144i	0.665179	+	1.15212i	0.979237	+	0.202721i	$0.0649783\pi$
$179$	8.89949	+	15.4144i	0.665179	+	1.15212i	−0.314057	+	0.949404i	$0.601688\pi$
$180$		0			0
$181$	−9.89949			−0.735824			−0.367912	−	0.929861i	$-0.619927\pi$
$181$	−9.89949			−0.735824			−0.367912	+	0.929861i	$0.619927\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.485281	+	0.840532i	0.0356786	+	0.0617971i
$186$		0			0
$187$	0.928932	−	1.60896i	0.0679302	−	0.117659i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	7.58579	−	13.1390i	0.548888	−	0.950702i	−0.449463	−	0.893299i	$-0.648385\pi$
$191$	7.58579	−	13.1390i	0.548888	−	0.950702i	0.998351	−	0.0574033i	$-0.0182821\pi$
$192$		0			0
$193$	−12.3137	−	21.3280i	−0.886360	−	1.53522i	−0.844147	−	0.536112i	$-0.819892\pi$
$193$	−12.3137	−	21.3280i	−0.886360	−	1.53522i	−0.0422134	−	0.999109i	$-0.513441\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	−2.82843	−	4.89898i	−0.200502	−	0.347279i	0.748188	−	0.663486i	$-0.230924\pi$
$199$	−2.82843	−	4.89898i	−0.200502	−	0.347279i	−0.948690	+	0.316207i	$0.897591\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	0.171573	−	0.297173i	0.0119832	−	0.0207555i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.65685			0.391293
$210$		0			0
$211$	−14.3431			−0.987423			−0.493711	−	0.869626i	$-0.664360\pi$
$211$	−14.3431			−0.987423			−0.493711	+	0.869626i	$0.664360\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.34315	+	4.05845i	−0.159801	+	0.276784i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.58579	−	2.74666i	−0.106672	−	0.184761i
$222$		0			0
$223$	−13.6569			−0.914531			−0.457265	−	0.889330i	$-0.651171\pi$
$223$	−13.6569			−0.914531			−0.457265	+	0.889330i	$0.651171\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.89949	−	17.1464i	−0.657053	−	1.13805i	−0.981375	−	0.192102i	$-0.938470\pi$
$227$	−9.89949	−	17.1464i	−0.657053	−	1.13805i	0.324322	−	0.945947i	$-0.394864\pi$
$228$		0			0
$229$	−7.53553	+	13.0519i	−0.497962	+	0.862496i	−0.999997	−	0.00235157i	$-0.999251\pi$
$229$	−7.53553	+	13.0519i	−0.497962	+	0.862496i	0.502035	+	0.864847i	$0.332585\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.2426	+	24.6690i	−0.933066	+	1.61612i	−0.155020	+	0.987911i	$0.549544\pi$
$233$	−14.2426	+	24.6690i	−0.933066	+	1.61612i	−0.778046	+	0.628207i	$0.783789\pi$
$234$		0			0
$235$	−2.00000	−	3.46410i	−0.130466	−	0.225973i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.17157			0.205152			0.102576	−	0.994725i	$-0.467292\pi$
$239$	3.17157			0.205152			0.102576	+	0.994725i	$0.467292\pi$
$240$		0			0
$241$	10.9497	+	18.9655i	0.705335	+	1.22168i	0.966570	+	0.256401i	$0.0825369\pi$
$241$	10.9497	+	18.9655i	0.705335	+	1.22168i	−0.261235	+	0.965275i	$0.584130\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	4.82843	−	8.36308i	0.307225	−	0.532130i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−8.48528			−0.535586			−0.267793	−	0.963476i	$-0.586294\pi$
$251$	−8.48528			−0.535586			−0.267793	+	0.963476i	$0.586294\pi$
$252$		0			0
$253$	−4.00000			−0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.1213	+	26.1909i	−0.943242	+	1.63374i	−0.184009	+	0.982925i	$0.558907\pi$
$257$	−15.1213	+	26.1909i	−0.943242	+	1.63374i	−0.759233	+	0.650819i	$0.774426\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.4142	+	21.5020i	0.765493	+	1.32587i	0.939985	+	0.341215i	$0.110838\pi$
$263$	12.4142	+	21.5020i	0.765493	+	1.32587i	−0.174492	+	0.984659i	$0.555828\pi$
$264$		0			0
$265$	−7.79899			−0.479088
$266$		0			0
$267$		0			0
$268$		0			0
$269$	15.0208	+	26.0168i	0.915835	+	1.58627i	0.805674	+	0.592360i	$0.201803\pi$
$269$	15.0208	+	26.0168i	0.915835	+	1.58627i	0.110162	+	0.993914i	$0.464863\pi$
$270$		0			0
$271$	6.58579	−	11.4069i	0.400058	−	0.692921i	−0.593674	−	0.804705i	$-0.702323\pi$
$271$	6.58579	−	11.4069i	0.400058	−	0.692921i	0.993732	+	0.111784i	$0.0356566\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.92893	−	3.34101i	0.116319	−	0.201470i
$276$		0			0
$277$	3.00000	+	5.19615i	0.180253	+	0.312207i	0.941966	−	0.335707i	$-0.108975\pi$
$277$	3.00000	+	5.19615i	0.180253	+	0.312207i	−0.761714	+	0.647913i	$0.775642\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	12.4853			0.744809			0.372405	−	0.928070i	$-0.378533\pi$
$281$	12.4853			0.744809			0.372405	+	0.928070i	$0.378533\pi$
$282$		0			0
$283$	7.41421	+	12.8418i	0.440729	+	0.763365i	0.997744	−	0.0671373i	$-0.0213866\pi$
$283$	7.41421	+	12.8418i	0.440729	+	0.763365i	−0.557014	+	0.830503i	$0.688053\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.98528	−	10.3668i	0.352075	−	0.609812i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.07107			−0.0625724			−0.0312862	−	0.999510i	$-0.509960\pi$
$293$	−1.07107			−0.0625724			−0.0312862	+	0.999510i	$0.509960\pi$
$294$		0			0
$295$	3.02944			0.176381
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.41421	+	5.91359i	−0.197449	+	0.341992i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.07107	+	7.05130i	0.233109	+	0.403756i
$306$		0			0
$307$	−11.5147			−0.657180			−0.328590	−	0.944473i	$-0.606573\pi$
$307$	−11.5147			−0.657180			−0.328590	+	0.944473i	$0.606573\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−13.0711	−	22.6398i	−0.741192	−	1.28378i	−0.951953	−	0.306245i	$-0.900927\pi$
$311$	−13.0711	−	22.6398i	−0.741192	−	1.28378i	0.210761	−	0.977538i	$-0.432406\pi$
$312$		0			0
$313$	8.70711	−	15.0812i	0.492155	−	0.852437i	−0.507804	−	0.861472i	$-0.669543\pi$
$313$	8.70711	−	15.0812i	0.492155	−	0.852437i	0.999959	+	0.00903532i	$0.00287607\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	10.6569	−	18.4582i	0.598549	−	1.03672i	−0.394487	−	0.918902i	$-0.629078\pi$
$317$	10.6569	−	18.4582i	0.598549	−	1.03672i	0.993036	−	0.117815i	$-0.0375890\pi$
$318$		0			0
$319$	3.51472	+	6.08767i	0.196786	+	0.340844i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−15.3137			−0.852078
$324$		0			0
$325$	−3.29289	−	5.70346i	−0.182657	−	0.316371i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.34315	−	7.52255i	0.238721	−	0.413477i	−0.721627	−	0.692282i	$-0.756605\pi$
$331$	4.34315	−	7.52255i	0.238721	−	0.413477i	0.960348	+	0.278806i	$0.0899386\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.68629			0.256039
$336$		0			0
$337$	−16.9706			−0.924445			−0.462223	−	0.886764i	$-0.652948\pi$
$337$	−16.9706			−0.924445			−0.462223	+	0.886764i	$0.652948\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.14214	+	3.71029i	−0.116003	+	0.200923i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.07107	−	3.58719i	−0.111181	−	0.192571i	0.805066	−	0.593185i	$-0.202130\pi$
$347$	−2.07107	−	3.58719i	−0.111181	−	0.192571i	−0.916247	+	0.400615i	$0.868797\pi$
$348$		0			0
$349$	30.3848			1.62646			0.813230	−	0.581943i	$-0.197707\pi$
$349$	30.3848			1.62646			0.813230	+	0.581943i	$0.197707\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−7.94975	−	13.7694i	−0.423122	−	0.732869i	0.573121	−	0.819471i	$-0.305733\pi$
$353$	−7.94975	−	13.7694i	−0.423122	−	0.732869i	−0.996243	+	0.0866016i	$0.972399\pi$
$354$		0			0
$355$	0.242641	−	0.420266i	0.0128780	−	0.0223054i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−17.7279	+	30.7057i	−0.935644	+	1.62058i	−0.162162	+	0.986764i	$0.551847\pi$
$359$	−17.7279	+	30.7057i	−0.935644	+	1.62058i	−0.773482	+	0.633819i	$0.781487\pi$
$360$		0			0
$361$	−13.8137	−	23.9260i	−0.727037	−	1.25927i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	6.48528			0.339455
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−7.34315	+	12.7187i	−0.380214	+	0.658549i	−0.991093	−	0.133175i	$-0.957483\pi$
$373$	−7.34315	+	12.7187i	−0.380214	+	0.658549i	0.610879	+	0.791724i	$0.290816\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.0000			0.618031
$378$		0			0
$379$	−0.686292			−0.0352524			−0.0176262	−	0.999845i	$-0.505611\pi$
$379$	−0.686292			−0.0352524			−0.0176262	+	0.999845i	$0.505611\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.4853	−	21.6251i	0.637968	−	1.10499i	−0.347910	−	0.937528i	$-0.613108\pi$
$383$	12.4853	−	21.6251i	0.637968	−	1.10499i	0.985878	−	0.167465i	$-0.0535582\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.07107	−	12.2474i	−0.358517	−	0.620970i	0.629196	−	0.777247i	$-0.283384\pi$
$389$	−7.07107	−	12.2474i	−0.358517	−	0.620970i	−0.987713	+	0.156276i	$0.950051\pi$
$390$		0			0
$391$	10.8284			0.547617
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−0.686292	−	1.18869i	−0.0345311	−	0.0598096i
$396$		0			0
$397$	−3.63604	+	6.29780i	−0.182488	+	0.316078i	−0.942727	−	0.333565i	$-0.891748\pi$
$397$	−3.63604	+	6.29780i	−0.182488	+	0.316078i	0.760239	+	0.649643i	$0.225082\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6.24264	+	10.8126i	−0.311743	+	0.539954i	−0.978740	−	0.205106i	$-0.934246\pi$
$401$	−6.24264	+	10.8126i	−0.311743	+	0.539954i	0.666997	+	0.745060i	$0.267579\pi$
$402$		0			0
$403$	3.65685	+	6.33386i	0.182161	+	0.315512i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.37258			0.0680364
$408$		0			0
$409$	−8.12132	−	14.0665i	−0.401573	−	0.695546i	0.592343	−	0.805686i	$-0.298203\pi$
$409$	−8.12132	−	14.0665i	−0.401573	−	0.695546i	−0.993916	+	0.110141i	$0.964870\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−4.48528	+	7.76874i	−0.220174	+	0.381352i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10.8284			−0.529003			−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284			−0.529003			−0.264502	+	0.964385i	$0.585207\pi$
$420$		0			0
$421$	6.68629			0.325870			0.162935	−	0.986637i	$-0.447904\pi$
$421$	6.68629			0.325870			0.162935	+	0.986637i	$0.447904\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.22183	+	9.04447i	−0.253296	+	0.438721i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−15.2426	−	26.4010i	−0.734212	−	1.27169i	−0.955068	−	0.296386i	$-0.904218\pi$
$431$	−15.2426	−	26.4010i	−0.734212	−	1.27169i	0.220856	−	0.975306i	$-0.429115\pi$
$432$		0			0
$433$	26.3848			1.26797			0.633986	−	0.773345i	$-0.281418\pi$
$433$	26.3848			1.26797			0.633986	+	0.773345i	$0.281418\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	16.4853	+	28.5533i	0.788598	+	1.36589i
$438$		0			0
$439$	1.65685	−	2.86976i	0.0790773	−	0.136966i	−0.823775	−	0.566917i	$-0.808136\pi$
$439$	1.65685	−	2.86976i	0.0790773	−	0.136966i	0.902852	+	0.429951i	$0.141469\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	19.2426	−	33.3292i	0.914245	−	1.58352i	0.106243	−	0.994340i	$-0.466118\pi$
$443$	19.2426	−	33.3292i	0.914245	−	1.58352i	0.808002	−	0.589179i	$-0.200549\pi$
$444$		0			0
$445$	3.14214	+	5.44234i	0.148952	+	0.257992i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	10.0000			0.471929			0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000			0.471929			0.235965	+	0.971762i	$0.424175\pi$
$450$		0			0
$451$	−0.242641	−	0.420266i	−0.0114255	−	0.0197896i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−10.3137	+	17.8639i	−0.482455	+	0.835636i	−0.999797	−	0.0201422i	$-0.993588\pi$
$457$	−10.3137	+	17.8639i	−0.482455	+	0.835636i	0.517342	+	0.855779i	$0.326921\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−18.2426			−0.849644			−0.424822	−	0.905277i	$-0.639663\pi$
$461$	−18.2426			−0.849644			−0.424822	+	0.905277i	$0.639663\pi$
$462$		0			0
$463$	20.9706			0.974585			0.487292	−	0.873239i	$-0.337985\pi$
$463$	20.9706			0.974585			0.487292	+	0.873239i	$0.337985\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5.41421	+	9.37769i	−0.250540	+	0.433948i	−0.963675	−	0.267079i	$-0.913941\pi$
$467$	−5.41421	+	9.37769i	−0.250540	+	0.433948i	0.713135	+	0.701027i	$0.247275\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.31371	+	5.73951i	0.152364	+	0.263903i
$474$		0			0
$475$	−31.7990			−1.45904
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−21.5563	−	37.3367i	−0.984935	−	1.70596i	−0.642226	−	0.766515i	$-0.721989\pi$
$479$	−21.5563	−	37.3367i	−0.984935	−	1.70596i	−0.342709	−	0.939442i	$-0.611344\pi$
$480$		0			0
$481$	1.17157	−	2.02922i	0.0534191	−	0.0925246i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.27208	+	3.93535i	−0.103170	+	0.178695i
$486$		0			0
$487$	10.4853	+	18.1610i	0.475133	+	0.822955i	0.999594	−	0.0284792i	$-0.00906645\pi$
$487$	10.4853	+	18.1610i	0.475133	+	0.822955i	−0.524461	+	0.851435i	$0.675733\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−24.1421			−1.08952			−0.544760	−	0.838592i	$-0.683379\pi$
$491$	−24.1421			−1.08952			−0.544760	+	0.838592i	$0.683379\pi$
$492$		0			0
$493$	−9.51472	−	16.4800i	−0.428521	−	0.742221i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.1421	−	24.4949i	0.633089	−	1.09654i	−0.353828	−	0.935311i	$-0.615120\pi$
$499$	14.1421	−	24.4949i	0.633089	−	1.09654i	0.986917	−	0.161232i	$-0.0515466\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14.6274			0.652204			0.326102	−	0.945335i	$-0.394265\pi$
$503$	14.6274			0.652204			0.326102	+	0.945335i	$0.394265\pi$
$504$		0			0
$505$	2.00000			0.0889988
$506$		0			0
$507$		0			0
$508$		0			0
$509$	18.5355	−	32.1045i	0.821573	−	1.42301i	−0.0829373	−	0.996555i	$-0.526430\pi$
$509$	18.5355	−	32.1045i	0.821573	−	1.42301i	0.904510	−	0.426452i	$-0.140237\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	3.17157	+	5.49333i	0.139756	+	0.242065i
$516$		0			0
$517$	−5.65685			−0.248788
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.3640	+	26.6112i	0.673107	+	1.16586i	0.977018	+	0.213156i	$0.0683742\pi$
$521$	15.3640	+	26.6112i	0.673107	+	1.16586i	−0.303911	+	0.952700i	$0.598293\pi$
$522$		0			0
$523$	−4.82843	+	8.36308i	−0.211132	+	0.365692i	−0.952069	−	0.305883i	$-0.901048\pi$
$523$	−4.82843	+	8.36308i	−0.211132	+	0.365692i	0.740937	+	0.671575i	$0.234382\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.79899	−	10.0441i	0.252608	−	0.437530i
$528$		0			0
$529$	−0.156854	−	0.271680i	−0.00681975	−	0.0118122i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.828427			−0.0358832
$534$		0			0
$535$	−4.24264	−	7.34847i	−0.183425	−	0.317702i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−18.3137	+	31.7203i	−0.787368	+	1.36376i	0.140206	+	0.990122i	$0.455223\pi$
$541$	−18.3137	+	31.7203i	−0.787368	+	1.36376i	−0.927574	+	0.373639i	$0.878110\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.62742			0.283887
$546$		0			0
$547$	28.9706			1.23869			0.619346	−	0.785118i	$-0.287398\pi$
$547$	28.9706			1.23869			0.619346	+	0.785118i	$0.287398\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	28.9706	−	50.1785i	1.23419	−	2.13768i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11.9706	−	20.7336i	−0.507209	−	0.878512i	−0.999965	−	0.00834436i	$-0.997344\pi$
$557$	−11.9706	−	20.7336i	−0.507209	−	0.878512i	0.492756	−	0.870167i	$-0.335989\pi$
$558$		0			0
$559$	11.3137			0.478519
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.5858	+	18.3351i	0.446138	+	0.772733i	0.998131	−	0.0611156i	$-0.0194658\pi$
$563$	10.5858	+	18.3351i	0.446138	+	0.772733i	−0.551993	+	0.833849i	$0.686132\pi$
$564$		0			0
$565$	−1.75736	+	3.04384i	−0.0739327	+	0.128055i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.585786	−	1.01461i	0.0245574	−	0.0425347i	−0.853486	−	0.521117i	$-0.825516\pi$
$569$	0.585786	−	1.01461i	0.0245574	−	0.0425347i	0.878043	+	0.478582i	$0.158849\pi$
$570$		0			0
$571$	8.14214	+	14.1026i	0.340738	+	0.590175i	0.984570	−	0.174992i	$-0.0559898\pi$
$571$	8.14214	+	14.1026i	0.340738	+	0.590175i	−0.643832	+	0.765167i	$0.722657\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	22.4853			0.937701
$576$		0			0
$577$	3.29289	+	5.70346i	0.137085	+	0.237438i	0.926392	−	0.376561i	$-0.122893\pi$
$577$	3.29289	+	5.70346i	0.137085	+	0.237438i	−0.789307	+	0.613999i	$0.789560\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−5.51472	+	9.55177i	−0.228396	+	0.395594i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	21.1716			0.873844			0.436922	−	0.899499i	$-0.356069\pi$
$587$	21.1716			0.873844			0.436922	+	0.899499i	$0.356069\pi$
$588$		0			0
$589$	35.3137			1.45508
$590$		0			0
$591$		0			0
$592$		0			0
$593$	7.46447	−	12.9288i	0.306529	−	0.530924i	−0.671072	−	0.741392i	$-0.734166\pi$
$593$	7.46447	−	12.9288i	0.306529	−	0.530924i	0.977601	+	0.210469i	$0.0674991\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.07107	−	10.5154i	−0.248057	−	0.429648i	0.714930	−	0.699196i	$-0.246459\pi$
$599$	−6.07107	−	10.5154i	−0.248057	−	0.429648i	−0.962987	+	0.269549i	$0.913125\pi$
$600$		0			0
$601$	−3.75736			−0.153266			−0.0766329	−	0.997059i	$-0.524417\pi$
$601$	−3.75736			−0.153266			−0.0766329	+	0.997059i	$0.524417\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.02082	−	5.23221i	−0.122814	−	0.212719i
$606$		0			0
$607$	23.7990	−	41.2211i	0.965971	−	1.67311i	0.258988	−	0.965880i	$-0.416611\pi$
$607$	23.7990	−	41.2211i	0.965971	−	1.67311i	0.706983	−	0.707231i	$-0.250056\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.82843	+	8.36308i	−0.195337	+	0.338334i
$612$		0			0
$613$	6.82843	+	11.8272i	0.275798	+	0.477695i	0.970336	−	0.241760i	$-0.0777247\pi$
$613$	6.82843	+	11.8272i	0.275798	+	0.477695i	−0.694538	+	0.719456i	$0.744391\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	29.4558			1.18585			0.592924	−	0.805259i	$-0.297974\pi$
$617$	29.4558			1.18585			0.592924	+	0.805259i	$0.297974\pi$
$618$		0			0
$619$	−8.14214	−	14.1026i	−0.327260	−	0.566831i	0.654707	−	0.755883i	$-0.272792\pi$
$619$	−8.14214	−	14.1026i	−0.327260	−	0.566831i	−0.981967	+	0.189052i	$0.939459\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−9.98528	+	17.2950i	−0.399411	+	0.691801i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.71573			−0.148156
$630$		0			0
$631$	48.2843			1.92217			0.961083	−	0.276259i	$-0.0890947\pi$
$631$	48.2843			1.92217			0.961083	+	0.276259i	$0.0890947\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.48528	+	7.76874i	−0.177993	+	0.308293i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.4142	+	26.6982i	0.608825	+	1.05452i	0.991434	+	0.130605i	$0.0416921\pi$
$641$	15.4142	+	26.6982i	0.608825	+	1.05452i	−0.382610	+	0.923910i	$0.624975\pi$
$642$		0			0
$643$	36.4853			1.43884			0.719420	−	0.694576i	$-0.244408\pi$
$643$	36.4853			1.43884			0.719420	+	0.694576i	$0.244408\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	4.58579	+	7.94282i	0.180286	+	0.312264i	0.941978	−	0.335675i	$-0.108964\pi$
$647$	4.58579	+	7.94282i	0.180286	+	0.312264i	−0.761692	+	0.647939i	$0.775631\pi$
$648$		0			0
$649$	2.14214	−	3.71029i	0.0840862	−	0.145642i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.41421	+	2.44949i	−0.0553425	+	0.0958559i	−0.892369	−	0.451306i	$-0.850958\pi$
$653$	−1.41421	+	2.44949i	−0.0553425	+	0.0958559i	0.837027	+	0.547162i	$0.184292\pi$
$654$		0			0
$655$	2.14214	+	3.71029i	0.0837002	+	0.144973i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−42.4853			−1.65499			−0.827496	−	0.561472i	$-0.810235\pi$
$659$	−42.4853			−1.65499			−0.827496	+	0.561472i	$0.810235\pi$
$660$		0			0
$661$	−23.6777	−	41.0109i	−0.920955	−	1.59514i	−0.797941	−	0.602735i	$-0.794078\pi$
$661$	−23.6777	−	41.0109i	−0.920955	−	1.59514i	−0.123013	−	0.992405i	$-0.539256\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−20.4853	+	35.4815i	−0.793193	+	1.37385i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	11.5147			0.444521
$672$		0			0
$673$	−7.31371			−0.281923			−0.140961	−	0.990015i	$-0.545019\pi$
$673$	−7.31371			−0.281923			−0.140961	+	0.990015i	$0.545019\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.7782	−	25.5965i	0.567971	−	0.983755i	−0.428795	−	0.903402i	$-0.641062\pi$
$677$	14.7782	−	25.5965i	0.567971	−	0.983755i	0.996766	−	0.0803535i	$-0.0256049\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8.07107	−	13.9795i	−0.308831	−	0.534911i	0.669276	−	0.743014i	$-0.266604\pi$
$683$	−8.07107	−	13.9795i	−0.308831	−	0.534911i	−0.978107	+	0.208103i	$0.933271\pi$
$684$		0			0
$685$	2.62742			0.100388
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.41421	+	16.3059i	0.358653	+	0.621205i
$690$		0			0
$691$	−14.0000	+	24.2487i	−0.532585	+	0.922464i	0.466691	+	0.884420i	$0.345446\pi$
$691$	−14.0000	+	24.2487i	−0.532585	+	0.922464i	−0.999276	+	0.0380440i	$0.987887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.485281	+	0.840532i	−0.0184078	+	0.0318832i
$696$		0			0
$697$	0.656854	+	1.13770i	0.0248801	+	0.0430936i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	5.17157			0.195328			0.0976638	−	0.995219i	$-0.468863\pi$
$701$	5.17157			0.195328			0.0976638	+	0.995219i	$0.468863\pi$
$702$		0			0
$703$	−5.65685	−	9.79796i	−0.213352	−	0.369537i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.00000	−	6.92820i	0.150223	−	0.260194i	−0.781086	−	0.624423i	$-0.785334\pi$
$709$	4.00000	−	6.92820i	0.150223	−	0.260194i	0.931309	+	0.364229i	$0.118667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−24.9706			−0.935155
$714$		0			0
$715$	0.686292			0.0256658
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2.34315	+	4.05845i	−0.0873846	+	0.151355i	−0.906405	−	0.422410i	$-0.861184\pi$
$719$	−2.34315	+	4.05845i	−0.0873846	+	0.151355i	0.819020	+	0.573765i	$0.194518\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−19.7574	−	34.2208i	−0.733770	−	1.27093i
$726$		0			0
$727$	25.4558			0.944105			0.472052	−	0.881570i	$-0.343513\pi$
$727$	25.4558			0.944105			0.472052	+	0.881570i	$0.343513\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.97056	−	15.5375i	−0.331788	−	0.574674i
$732$		0			0
$733$	−5.87868	+	10.1822i	−0.217134	+	0.376087i	−0.953931	−	0.300027i	$-0.903004\pi$
$733$	−5.87868	+	10.1822i	−0.217134	+	0.376087i	0.736797	+	0.676114i	$0.236338\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.31371	−	5.73951i	0.122062	−	0.211418i
$738$		0			0
$739$	−10.1421	−	17.5667i	−0.373084	−	0.646201i	0.616954	−	0.786999i	$-0.288367\pi$
$739$	−10.1421	−	17.5667i	−0.373084	−	0.646201i	−0.990038	+	0.140798i	$0.955033\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11.1716			0.409845			0.204923	−	0.978778i	$-0.434306\pi$
$743$	11.1716			0.409845			0.204923	+	0.978778i	$0.434306\pi$
$744$		0			0
$745$	2.92893	+	5.07306i	0.107308	+	0.185863i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−14.8284	+	25.6836i	−0.541097	+	0.937207i	0.457744	+	0.889084i	$0.348657\pi$
$751$	−14.8284	+	25.6836i	−0.541097	+	0.937207i	−0.998841	+	0.0481236i	$0.984676\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.65685			−0.205874
$756$		0			0
$757$	−11.3137			−0.411204			−0.205602	−	0.978636i	$-0.565915\pi$
$757$	−11.3137			−0.411204			−0.205602	+	0.978636i	$0.565915\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.87868	+	8.45012i	−0.176852	+	0.306317i	−0.940801	−	0.338960i	$-0.889925\pi$
$761$	−4.87868	+	8.45012i	−0.176852	+	0.306317i	0.763949	+	0.645277i	$0.223258\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.65685	−	6.33386i	−0.132041	−	0.228702i
$768$		0			0
$769$	15.0711			0.543477			0.271738	−	0.962371i	$-0.412401\pi$
$769$	15.0711			0.543477			0.271738	+	0.962371i	$0.412401\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1.70711	−	2.95680i	−0.0614004	−	0.106349i	0.833691	−	0.552231i	$-0.186223\pi$
$773$	−1.70711	−	2.95680i	−0.0614004	−	0.106349i	−0.895092	+	0.445882i	$0.852890\pi$
$774$		0			0
$775$	12.0416	−	20.8567i	0.432548	−	0.749195i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.00000	+	3.46410i	−0.0716574	+	0.124114i
$780$		0			0
$781$	−0.343146	−	0.594346i	−0.0122787	−	0.0212674i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.14214			0.290605
$786$		0			0
$787$	5.31371	+	9.20361i	0.189413	+	0.328073i	0.945055	−	0.326912i	$-0.106008\pi$
$787$	5.31371	+	9.20361i	0.189413	+	0.328073i	−0.755642	+	0.654985i	$0.772675\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	9.82843	−	17.0233i	0.349018	−	0.604516i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	43.2132			1.53069			0.765345	−	0.643620i	$-0.222568\pi$
$797$	43.2132			1.53069			0.765345	+	0.643620i	$0.222568\pi$
$798$		0			0
$799$	15.3137			0.541760
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.58579	−	7.94282i	0.161829	−	0.280296i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−15.0000	−	25.9808i	−0.527372	−	0.913435i	−0.999491	−	0.0319002i	$-0.989844\pi$
$809$	−15.0000	−	25.9808i	−0.527372	−	0.913435i	0.472119	−	0.881535i	$-0.343489\pi$
$810$		0			0
$811$	−20.9706			−0.736376			−0.368188	−	0.929751i	$-0.620022\pi$
$811$	−20.9706			−0.736376			−0.368188	+	0.929751i	$0.620022\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.00000	−	6.92820i	−0.140114	−	0.242684i
$816$		0			0
$817$	27.3137	−	47.3087i	0.955586	−	1.65512i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	5.00000	−	8.66025i	0.174501	−	0.302245i	−0.765487	−	0.643451i	$-0.777502\pi$
$821$	5.00000	−	8.66025i	0.174501	−	0.302245i	0.939989	+	0.341206i	$0.110835\pi$
$822$		0			0
$823$	1.17157	+	2.02922i	0.0408385	+	0.0707343i	0.885722	−	0.464216i	$-0.153664\pi$
$823$	1.17157	+	2.02922i	0.0408385	+	0.0707343i	−0.844884	+	0.534950i	$0.820330\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	19.4558			0.676546			0.338273	−	0.941048i	$-0.390157\pi$
$827$	19.4558			0.676546			0.338273	+	0.941048i	$0.390157\pi$
$828$		0			0
$829$	−1.63604	−	2.83370i	−0.0568220	−	0.0984186i	0.836215	−	0.548401i	$-0.184763\pi$
$829$	−1.63604	−	2.83370i	−0.0568220	−	0.0984186i	−0.893037	+	0.449983i	$0.851430\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.343146	+	0.594346i	−0.0118750	+	0.0205682i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−49.1716			−1.69759			−0.848796	−	0.528721i	$-0.822672\pi$
$839$	−49.1716			−1.69759			−0.848796	+	0.528721i	$0.822672\pi$
$840$		0			0
$841$	43.0000			1.48276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.22183	+	5.58037i	−0.110834	+	0.191970i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.00000	+	6.92820i	0.137118	+	0.237496i
$852$		0			0
$853$	36.0416			1.23404			0.617021	−	0.786947i	$-0.288339\pi$
$853$	36.0416			1.23404			0.617021	+	0.786947i	$0.288339\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21.8492	−	37.8440i	−0.746356	−	1.29273i	−0.949559	−	0.313590i	$-0.898468\pi$
$857$	−21.8492	−	37.8440i	−0.746356	−	1.29273i	0.203203	−	0.979137i	$-0.434865\pi$
$858$		0			0
$859$	19.4142	−	33.6264i	0.662404	−	1.14732i	−0.317578	−	0.948232i	$-0.602869\pi$
$859$	19.4142	−	33.6264i	0.662404	−	1.14732i	0.979982	−	0.199086i	$-0.0637973\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.7574	+	25.5605i	−0.502346	+	0.870089i	0.497650	+	0.867378i	$0.334196\pi$
$863$	−14.7574	+	25.5605i	−0.502346	+	0.870089i	−0.999996	+	0.00271146i	$0.999137\pi$
$864$		0			0
$865$	1.00000	+	1.73205i	0.0340010	+	0.0588915i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.94113			−0.0658482
$870$		0			0
$871$	−5.65685	−	9.79796i	−0.191675	−	0.331991i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−10.1421	+	17.5667i	−0.342476	+	0.593185i	−0.984892	−	0.173171i	$-0.944599\pi$
$877$	−10.1421	+	17.5667i	−0.342476	+	0.593185i	0.642416	+	0.766356i	$0.277932\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25.0711			0.844666			0.422333	−	0.906441i	$-0.361211\pi$
$881$	25.0711			0.844666			0.422333	+	0.906441i	$0.361211\pi$
$882$		0			0
$883$	18.3431			0.617296			0.308648	−	0.951176i	$-0.400123\pi$
$883$	18.3431			0.617296			0.308648	+	0.951176i	$0.400123\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.7279	−	18.5813i	0.360208	−	0.623899i	−0.627787	−	0.778386i	$-0.716039\pi$
$887$	10.7279	−	18.5813i	0.360208	−	0.623899i	0.987995	+	0.154486i	$0.0493723\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.3137	+	40.3805i	0.780164	+	1.35128i
$894$		0			0
$895$	10.4264			0.348516
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21.9411	+	38.0031i	0.731778	+	1.26748i
$900$		0			0
$901$	14.9289	−	25.8577i	0.497355	−	0.861444i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.89949	+	5.02207i	−0.0963825	+	0.166939i
$906$		0			0
$907$	3.51472	+	6.08767i	0.116704	+	0.202138i	0.918460	−	0.395514i	$-0.129434\pi$
$907$	3.51472	+	6.08767i	0.116704	+	0.202138i	−0.801755	+	0.597652i	$0.796100\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	54.4853			1.80518			0.902589	−	0.430503i	$-0.141664\pi$
$911$	54.4853			1.80518			0.902589	+	0.430503i	$0.141664\pi$
$912$		0			0
$913$	6.34315	+	10.9867i	0.209927	+	0.363605i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	24.6274	−	42.6559i	0.812384	−	1.40709i	−0.0988080	−	0.995107i	$-0.531503\pi$
$919$	24.6274	−	42.6559i	0.812384	−	1.40709i	0.911192	−	0.411983i	$-0.135164\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.17157			−0.0385628
$924$		0			0
$925$	−7.71573			−0.253692
$926$		0			0
$927$		0			0
$928$		0			0
$929$	30.4350	−	52.7150i	0.998541	−	1.72952i	0.452505	−	0.891762i	$-0.350531\pi$
$929$	30.4350	−	52.7150i	0.998541	−	1.72952i	0.546036	−	0.837762i	$-0.316136\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.544156	−	0.942506i	−0.0177958	−	0.0308232i
$936$		0			0
$937$	−6.10051			−0.199295			−0.0996474	−	0.995023i	$-0.531771\pi$
$937$	−6.10051			−0.199295			−0.0996474	+	0.995023i	$0.531771\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−21.9497	−	38.0181i	−0.715541	−	1.23935i	−0.962750	−	0.270392i	$-0.912847\pi$
$941$	−21.9497	−	38.0181i	−0.715541	−	1.23935i	0.247209	−	0.968962i	$-0.420487\pi$
$942$		0			0
$943$	1.41421	−	2.44949i	0.0460531	−	0.0797664i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	10.0711	−	17.4436i	0.327266	−	0.566841i	−0.654703	−	0.755887i	$-0.727206\pi$
$947$	10.0711	−	17.4436i	0.327266	−	0.566841i	0.981968	+	0.189046i	$0.0605394\pi$
$948$		0			0
$949$	−7.82843	−	13.5592i	−0.254121	−	0.440151i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.37258			0.238821			0.119411	−	0.992845i	$-0.461899\pi$
$953$	7.37258			0.238821			0.119411	+	0.992845i	$0.461899\pi$
$954$		0			0
$955$	−4.44365	−	7.69663i	−0.143793	−	0.249057i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.12742	−	3.68479i	0.0686264	−	0.118864i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−14.4264			−0.464402
$966$		0			0
$967$	−34.6274			−1.11354			−0.556771	−	0.830666i	$-0.687960\pi$
$967$	−34.6274			−1.11354			−0.556771	+	0.830666i	$0.687960\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.6274	+	49.5841i	−0.918698	+	1.59123i	−0.117302	+	0.993096i	$0.537424\pi$
$971$	−28.6274	+	49.5841i	−0.918698	+	1.59123i	−0.801396	+	0.598134i	$0.795909\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10.9289	+	18.9295i	0.349648	+	0.605607i	0.986187	−	0.165637i	$-0.0529680\pi$
$977$	10.9289	+	18.9295i	0.349648	+	0.605607i	−0.636539	+	0.771244i	$0.719635\pi$
$978$		0			0
$979$	8.88730			0.284039
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.31371	−	12.6677i	−0.233271	−	0.404037i	0.725498	−	0.688225i	$-0.241610\pi$
$983$	−7.31371	−	12.6677i	−0.233271	−	0.404037i	−0.958769	+	0.284187i	$0.908276\pi$
$984$		0			0
$985$	−0.585786	+	1.01461i	−0.0186647	+	0.0323282i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.3137	+	33.4523i	−0.614140	+	1.06372i
$990$		0			0
$991$	−11.6569	−	20.1903i	−0.370292	−	0.641365i	0.619318	−	0.785140i	$-0.287409\pi$
$991$	−11.6569	−	20.1903i	−0.370292	−	0.641365i	−0.989610	+	0.143775i	$0.954076\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.31371			−0.105052
$996$		0			0
$997$	17.2929	+	29.9522i	0.547671	+	0.948595i	0.998434	+	0.0559506i	$0.0178189\pi$
$997$	17.2929	+	29.9522i	0.547671	+	0.948595i	−0.450762	+	0.892644i	$0.648848\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3528.2.s.bm.3313.1		4
3.2	odd	2		1176.2.q.k.961.2		4
7.2	even	3		3528.2.a.bb.1.2		2
7.3	odd	6		3528.2.s.bd.361.2		4
7.4	even	3	inner	3528.2.s.bm.361.1		4
7.5	odd	6		3528.2.a.bl.1.1		2
7.6	odd	2		3528.2.s.bd.3313.2		4
12.11	even	2		2352.2.q.be.961.2		4
21.2	odd	6		1176.2.a.o.1.1	yes	2
21.5	even	6		1176.2.a.j.1.2	✓	2
21.11	odd	6		1176.2.q.k.361.2		4
21.17	even	6		1176.2.q.o.361.1		4
21.20	even	2		1176.2.q.o.961.1		4
28.19	even	6		7056.2.a.cx.1.1		2
28.23	odd	6		7056.2.a.cg.1.2		2
84.11	even	6		2352.2.q.be.1537.2		4
84.23	even	6		2352.2.a.bb.1.1		2
84.47	odd	6		2352.2.a.bd.1.2		2
84.59	odd	6		2352.2.q.bc.1537.1		4
84.83	odd	2		2352.2.q.bc.961.1		4
168.5	even	6		9408.2.a.ee.1.1		2
168.107	even	6		9408.2.a.du.1.2		2
168.131	odd	6		9408.2.a.ds.1.1		2
168.149	odd	6		9408.2.a.dg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.a.j.1.2	✓	2	21.5	even	6
1176.2.a.o.1.1	yes	2	21.2	odd	6
1176.2.q.k.361.2		4	21.11	odd	6
1176.2.q.k.961.2		4	3.2	odd	2
1176.2.q.o.361.1		4	21.17	even	6
1176.2.q.o.961.1		4	21.20	even	2
2352.2.a.bb.1.1		2	84.23	even	6
2352.2.a.bd.1.2		2	84.47	odd	6
2352.2.q.bc.961.1		4	84.83	odd	2
2352.2.q.bc.1537.1		4	84.59	odd	6
2352.2.q.be.961.2		4	12.11	even	2
2352.2.q.be.1537.2		4	84.11	even	6
3528.2.a.bb.1.2		2	7.2	even	3
3528.2.a.bl.1.1		2	7.5	odd	6
3528.2.s.bd.361.2		4	7.3	odd	6
3528.2.s.bd.3313.2		4	7.6	odd	2
3528.2.s.bm.361.1		4	7.4	even	3	inner
3528.2.s.bm.3313.1		4	1.1	even	1	trivial
7056.2.a.cg.1.2		2	28.23	odd	6
7056.2.a.cx.1.1		2	28.19	even	6
9408.2.a.dg.1.2		2	168.149	odd	6
9408.2.a.ds.1.1		2	168.131	odd	6
9408.2.a.du.1.2		2	168.107	even	6
9408.2.a.ee.1.1		2	168.5	even	6

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:	\( N \)	\(=\)	\( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3528.s (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.292893 - 0.507306i) q^{5} +O(q^{10})\)	\(q+(0.292893 - 0.507306i) q^{5} +(-0.414214 - 0.717439i) q^{11} -1.41421 q^{13} +(1.12132 + 1.94218i) q^{17} +(-3.41421 + 5.91359i) q^{19} +(2.41421 - 4.18154i) q^{23} +(2.32843 + 4.03295i) q^{25} -8.48528 q^{29} +(-2.58579 - 4.47871i) q^{31} +(-0.828427 + 1.43488i) q^{37} +0.585786 q^{41} -8.00000 q^{43} +(3.41421 - 5.91359i) q^{47} +(-6.65685 - 11.5300i) q^{53} -0.485281 q^{55} +(2.58579 + 4.47871i) q^{59} +(-6.94975 + 12.0373i) q^{61} +(-0.414214 + 0.717439i) q^{65} +(4.00000 + 6.92820i) q^{67} +0.828427 q^{71} +(5.53553 + 9.58783i) q^{73} +(1.17157 - 2.02922i) q^{79} -15.3137 q^{83} +1.31371 q^{85} +(-5.36396 + 9.29065i) q^{89} +(2.00000 + 3.46410i) q^{95} -7.75736 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5}+O(q^{10}) \) 4 * q + 4 * q^5	\( 4 q + 4 q^{5} + 4 q^{11} - 4 q^{17} - 8 q^{19} + 4 q^{23} - 2 q^{25} - 16 q^{31} + 8 q^{37} + 8 q^{41} - 32 q^{43} + 8 q^{47} - 4 q^{53} + 32 q^{55} + 16 q^{59} - 8 q^{61} + 4 q^{65} + 16 q^{67} - 8 q^{71} + 8 q^{73} + 16 q^{79} - 16 q^{83} - 40 q^{85} + 4 q^{89} + 8 q^{95} - 48 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^11 - 4 * q^17 - 8 * q^19 + 4 * q^23 - 2 * q^25 - 16 * q^31 + 8 * q^37 + 8 * q^41 - 32 * q^43 + 8 * q^47 - 4 * q^53 + 32 * q^55 + 16 * q^59 - 8 * q^61 + 4 * q^65 + 16 * q^67 - 8 * q^71 + 8 * q^73 + 16 * q^79 - 16 * q^83 - 40 * q^85 + 4 * q^89 + 8 * q^95 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more