LMFDB - Embedded newform 528.6.o.b.175.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,6,Mod(175,528)]

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([1, 0, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("528.175");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	528.o (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:	$84.6826568613$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.5
Character	$\chi$	$=$	528.175
Dual form			528.6.o.b.175.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000i q^{3} +89.8188 q^{5} -112.074 q^{7} -81.0000 q^{9} +O(q^{10})$	$q-9.00000i q^{3} +89.8188 q^{5} -112.074 q^{7} -81.0000 q^{9} +(-346.715 - 202.088i) q^{11} -898.935i q^{13} -808.369i q^{15} -13.3656i q^{17} -2209.77 q^{19} +1008.66i q^{21} +2287.32i q^{23} +4942.41 q^{25} +729.000i q^{27} +2528.16i q^{29} +6300.41i q^{31} +(-1818.79 + 3120.44i) q^{33} -10066.3 q^{35} +6835.68 q^{37} -8090.42 q^{39} -7881.85i q^{41} +77.7558 q^{43} -7275.32 q^{45} +17705.3i q^{47} -4246.48 q^{49} -120.290 q^{51} -33455.4 q^{53} +(-31141.6 - 18151.3i) q^{55} +19888.0i q^{57} +6095.26i q^{59} +40208.4i q^{61} +9077.97 q^{63} -80741.2i q^{65} +30700.1i q^{67} +20585.9 q^{69} -13371.9i q^{71} -54521.6i q^{73} -44481.7i q^{75} +(38857.7 + 22648.7i) q^{77} +104657. q^{79} +6561.00 q^{81} +12243.3 q^{83} -1200.48i q^{85} +22753.4 q^{87} -105422. q^{89} +100747. i q^{91} +56703.7 q^{93} -198479. q^{95} +23279.4 q^{97} +(28083.9 + 16369.1i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) $ 40 * q + 88 * q^5 - 3240 * q^9	$ 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) $ 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

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
$3$		−	9.00000i		−	0.577350i
$3$		−	9.00000i		−	0.577350i
$4$		0			0
$5$	89.8188			1.60673			0.803364	−	0.595489i	$-0.203042\pi$
$5$	89.8188			1.60673			0.803364	+	0.595489i	$0.203042\pi$
$6$		0			0
$7$	−112.074			−0.864487			−0.432244	−	0.901757i	$-0.642278\pi$
$7$	−112.074			−0.864487			−0.432244	+	0.901757i	$0.642278\pi$
$8$		0			0
$9$	−81.0000			−0.333333
$10$		0			0
$11$	−346.715	−	202.088i	−0.863956	−	0.503568i
$11$	−346.715	−	202.088i	−0.863956	−	0.503568i
$12$		0			0
$13$		−	898.935i		−	1.47526i	−0.675203	−	0.737632i	$-0.735944\pi$
$13$		−	898.935i		−	1.47526i	0.675203	−	0.737632i	$-0.264056\pi$
$14$		0			0
$15$		−	808.369i		−	0.927644i
$16$		0			0
$17$		−	13.3656i		−	0.0112167i	−0.999984	−	0.00560836i	$-0.998215\pi$
$17$		−	13.3656i		−	0.0112167i	0.999984	−	0.00560836i	$-0.00178521\pi$
$18$		0			0
$19$	−2209.77			−1.40431			−0.702156	−	0.712023i	$-0.747779\pi$
$19$	−2209.77			−1.40431			−0.702156	+	0.712023i	$0.747779\pi$
$20$		0			0
$21$			1008.66i			0.499112i
$22$		0			0
$23$			2287.32i			0.901588i	0.892628	+	0.450794i	$0.148859\pi$
$23$			2287.32i			0.901588i	−0.892628	+	0.450794i	$0.851141\pi$
$24$		0			0
$25$	4942.41			1.58157
$26$		0			0
$27$			729.000i			0.192450i
$28$		0			0
$29$			2528.16i			0.558224i	0.960258	+	0.279112i	$0.0900401\pi$
$29$			2528.16i			0.558224i	−0.960258	+	0.279112i	$0.909960\pi$
$30$		0			0
$31$			6300.41i			1.17751i	0.808311	+	0.588755i	$0.200382\pi$
$31$			6300.41i			1.17751i	−0.808311	+	0.588755i	$0.799618\pi$
$32$		0			0
$33$	−1818.79	+	3120.44i	−0.290735	+	0.498805i
$34$		0			0
$35$	−10066.3			−1.38900
$36$		0			0
$37$	6835.68			0.820875			0.410438	−	0.911889i	$-0.365376\pi$
$37$	6835.68			0.820875			0.410438	+	0.911889i	$0.365376\pi$
$38$		0			0
$39$	−8090.42			−0.851744
$40$		0			0
$41$		−	7881.85i		−	0.732265i	−0.930563	−	0.366133i	$-0.880682\pi$
$41$		−	7881.85i		−	0.732265i	0.930563	−	0.366133i	$-0.119318\pi$
$42$		0			0
$43$	77.7558			0.00641301			0.00320650	−	0.999995i	$-0.498979\pi$
$43$	77.7558			0.00641301			0.00320650	+	0.999995i	$0.498979\pi$
$44$		0			0
$45$	−7275.32			−0.535576
$46$		0			0
$47$			17705.3i			1.16912i	0.811351	+	0.584559i	$0.198732\pi$
$47$			17705.3i			1.16912i	−0.811351	+	0.584559i	$0.801268\pi$
$48$		0			0
$49$	−4246.48			−0.252661
$50$		0			0
$51$	−120.290			−0.00647598
$52$		0			0
$53$	−33455.4			−1.63597			−0.817986	−	0.575238i	$-0.804910\pi$
$53$	−33455.4			−1.63597			−0.817986	+	0.575238i	$0.804910\pi$
$54$		0			0
$55$	−31141.6	−	18151.3i	−1.38814	−	0.809096i
$56$		0			0
$57$			19888.0i			0.810780i
$58$		0			0
$59$			6095.26i			0.227962i	0.993483	+	0.113981i	$0.0363603\pi$
$59$			6095.26i			0.227962i	−0.993483	+	0.113981i	$0.963640\pi$
$60$		0			0
$61$			40208.4i			1.38354i	0.722117	+	0.691771i	$0.243169\pi$
$61$			40208.4i			1.38354i	−0.722117	+	0.691771i	$0.756831\pi$
$62$		0			0
$63$	9077.97			0.288162
$64$		0			0
$65$		−	80741.2i		−	2.37035i
$66$		0			0
$67$			30700.1i			0.835513i	0.908559	+	0.417757i	$0.137184\pi$
$67$			30700.1i			0.835513i	−0.908559	+	0.417757i	$0.862816\pi$
$68$		0			0
$69$	20585.9			0.520532
$70$		0			0
$71$		−	13371.9i		−	0.314809i	−0.987534	−	0.157404i	$-0.949687\pi$
$71$		−	13371.9i		−	0.314809i	0.987534	−	0.157404i	$-0.0503126\pi$
$72$		0			0
$73$		−	54521.6i		−	1.19746i	−0.800951	−	0.598731i	$-0.795672\pi$
$73$		−	54521.6i		−	1.19746i	0.800951	−	0.598731i	$-0.204328\pi$
$74$		0			0
$75$		−	44481.7i		−	0.913121i
$76$		0			0
$77$	38857.7	+	22648.7i	0.746879	+	0.435328i
$78$		0			0
$79$	104657.			1.88669			0.943343	−	0.331818i	$-0.107662\pi$
$79$	104657.			1.88669			0.943343	+	0.331818i	$0.107662\pi$
$80$		0			0
$81$	6561.00			0.111111
$82$		0			0
$83$	12243.3			0.195076			0.0975382	−	0.995232i	$-0.468903\pi$
$83$	12243.3			0.195076			0.0975382	+	0.995232i	$0.468903\pi$
$84$		0			0
$85$		−	1200.48i		−	0.0180222i
$86$		0			0
$87$	22753.4			0.322291
$88$		0			0
$89$	−105422.			−1.41078			−0.705388	−	0.708821i	$-0.749227\pi$
$89$	−105422.			−1.41078			−0.705388	+	0.708821i	$0.749227\pi$
$90$		0			0
$91$			100747.i			1.27535i
$92$		0			0
$93$	56703.7			0.679836
$94$		0			0
$95$	−198479.			−2.25635
$96$		0			0
$97$	23279.4			0.251213			0.125606	−	0.992080i	$-0.459912\pi$
$97$	23279.4			0.251213			0.125606	+	0.992080i	$0.459912\pi$
$98$		0			0
$99$	28083.9	+	16369.1i	0.287985	+	0.167856i
$100$		0			0
$101$		−	38128.6i		−	0.371918i	−0.982557	−	0.185959i	$-0.940461\pi$
$101$		−	38128.6i		−	0.371918i	0.982557	−	0.185959i	$-0.0595392\pi$
$102$		0			0
$103$		−	103196.i		−	0.958453i	−0.877691	−	0.479227i	$-0.840917\pi$
$103$		−	103196.i		−	0.958453i	0.877691	−	0.479227i	$-0.159083\pi$
$104$		0			0
$105$			90596.9i			0.801937i
$106$		0			0
$107$	−183197.			−1.54689			−0.773443	−	0.633865i	$-0.781467\pi$
$107$	−183197.			−1.54689			−0.773443	+	0.633865i	$0.781467\pi$
$108$		0			0
$109$			191091.i			1.54054i	0.637718	+	0.770270i	$0.279879\pi$
$109$			191091.i			1.54054i	−0.637718	+	0.770270i	$0.720121\pi$
$110$		0			0
$111$		−	61521.1i		−	0.473932i
$112$		0			0
$113$	56260.5			0.414484			0.207242	−	0.978290i	$-0.433551\pi$
$113$	56260.5			0.414484			0.207242	+	0.978290i	$0.433551\pi$
$114$		0			0
$115$			205445.i			1.44861i
$116$		0			0
$117$			72813.7i			0.491755i
$118$		0			0
$119$			1497.93i			0.00969672i
$120$		0			0
$121$	79372.1	+	140134.i	0.492838	+	0.870121i
$122$		0			0
$123$	−70936.6			−0.422774
$124$		0			0
$125$	163238.			0.934427
$126$		0			0
$127$	−113747.			−0.625793			−0.312896	−	0.949787i	$-0.601299\pi$
$127$	−113747.			−0.625793			−0.312896	+	0.949787i	$0.601299\pi$
$128$		0			0
$129$		−	699.802i		−	0.00370255i
$130$		0			0
$131$	−113226.			−0.576460			−0.288230	−	0.957561i	$-0.593067\pi$
$131$	−113226.			−0.576460			−0.288230	+	0.957561i	$0.593067\pi$
$132$		0			0
$133$	247658.			1.21401
$134$		0			0
$135$			65477.9i			0.309215i
$136$		0			0
$137$	−186747.			−0.850065			−0.425033	−	0.905178i	$-0.639737\pi$
$137$	−186747.			−0.850065			−0.425033	+	0.905178i	$0.639737\pi$
$138$		0			0
$139$	−338688.			−1.48684			−0.743419	−	0.668826i	$-0.766797\pi$
$139$	−338688.			−1.48684			−0.743419	+	0.668826i	$0.766797\pi$
$140$		0			0
$141$	159347.			0.674990
$142$		0			0
$143$	−181664.	+	311675.i	−0.742896	+	1.27456i
$144$		0			0
$145$			227076.i			0.896914i
$146$		0			0
$147$			38218.3i			0.145874i
$148$		0			0
$149$			533173.i			1.96744i	0.179698	+	0.983722i	$0.442488\pi$
$149$			533173.i			1.96744i	−0.179698	+	0.983722i	$0.557512\pi$
$150$		0			0
$151$	−324307.			−1.15748			−0.578741	−	0.815512i	$-0.696456\pi$
$151$	−324307.			−1.15748			−0.578741	+	0.815512i	$0.696456\pi$
$152$		0			0
$153$			1082.61i			0.00373891i
$154$		0			0
$155$			565895.i			1.89194i
$156$		0			0
$157$	355587.			1.15132			0.575661	−	0.817688i	$-0.304745\pi$
$157$	355587.			1.15132			0.575661	+	0.817688i	$0.304745\pi$
$158$		0			0
$159$			301098.i			0.944529i
$160$		0			0
$161$		−	256349.i		−	0.779412i
$162$		0			0
$163$		−	117160.i		−	0.345389i	−0.984975	−	0.172695i	$-0.944753\pi$
$163$		−	117160.i		−	0.345389i	0.984975	−	0.172695i	$-0.0552474\pi$
$164$		0			0
$165$	−163361.	+	280274.i	−0.467132	+	0.801443i
$166$		0			0
$167$	213101.			0.591282			0.295641	−	0.955299i	$-0.404467\pi$
$167$	213101.			0.591282			0.295641	+	0.955299i	$0.404467\pi$
$168$		0			0
$169$	−436791.			−1.17641
$170$		0			0
$171$	178992.			0.468104
$172$		0			0
$173$			436755.i			1.10949i	0.832021	+	0.554744i	$0.187184\pi$
$173$			436755.i			1.10949i	−0.832021	+	0.554744i	$0.812816\pi$
$174$		0			0
$175$	−553915.			−1.36725
$176$		0			0
$177$	54857.3			0.131614
$178$		0			0
$179$		−	327566.i		−	0.764128i	−0.924136	−	0.382064i	$-0.875213\pi$
$179$		−	327566.i		−	0.764128i	0.924136	−	0.382064i	$-0.124787\pi$
$180$		0			0
$181$	−628199.			−1.42528			−0.712641	−	0.701529i	$-0.752501\pi$
$181$	−628199.			−1.42528			−0.712641	+	0.701529i	$0.752501\pi$
$182$		0			0
$183$	361876.			0.798788
$184$		0			0
$185$	613972.			1.31892
$186$		0			0
$187$	−2701.02	+	4634.06i	−0.00564838	+	0.00969075i
$188$		0			0
$189$		−	81701.7i		−	0.166371i
$190$		0			0
$191$			221656.i			0.439638i	0.975541	+	0.219819i	$0.0705467\pi$
$191$			221656.i			0.439638i	−0.975541	+	0.219819i	$0.929453\pi$
$192$		0			0
$193$		−	206584.i		−	0.399213i	−0.979876	−	0.199606i	$-0.936034\pi$
$193$		−	206584.i		−	0.399213i	0.979876	−	0.199606i	$-0.0639663\pi$
$194$		0			0
$195$	−726671.			−1.36852
$196$		0			0
$197$		−	44725.3i		−	0.0821084i	−0.999157	−	0.0410542i	$-0.986928\pi$
$197$		−	44725.3i		−	0.0821084i	0.999157	−	0.0410542i	$-0.0130716\pi$
$198$		0			0
$199$		−	275353.i		−	0.492898i	−0.969156	−	0.246449i	$-0.920736\pi$
$199$		−	275353.i		−	0.492898i	0.969156	−	0.246449i	$-0.0792638\pi$
$200$		0			0
$201$	276301.			0.482384
$202$		0			0
$203$		−	283340.i		−	0.482578i
$204$		0			0
$205$		−	707938.i		−	1.17655i
$206$		0			0
$207$		−	185273.i		−	0.300529i
$208$		0			0
$209$	766162.	+	446568.i	1.21326	+	0.707167i
$210$		0			0
$211$	−596512.			−0.922386			−0.461193	−	0.887300i	$-0.652578\pi$
$211$	−596512.			−0.922386			−0.461193	+	0.887300i	$0.652578\pi$
$212$		0			0
$213$	−120347.			−0.181755
$214$		0			0
$215$	6983.93			0.0103040
$216$		0			0
$217$		−	706111.i		−	1.01794i
$218$		0			0
$219$	−490694.			−0.691354
$220$		0			0
$221$	−12014.8			−0.0165476
$222$		0			0
$223$		−	457754.i		−	0.616411i	−0.951320	−	0.308206i	$-0.900272\pi$
$223$		−	457754.i		−	0.616411i	0.951320	−	0.308206i	$-0.0997284\pi$
$224$		0			0
$225$	−400335.			−0.527191
$226$		0			0
$227$	637308.			0.820890			0.410445	−	0.911885i	$-0.365373\pi$
$227$	637308.			0.820890			0.410445	+	0.911885i	$0.365373\pi$
$228$		0			0
$229$	11576.4			0.0145877			0.00729383	−	0.999973i	$-0.497678\pi$
$229$	11576.4			0.0145877			0.00729383	+	0.999973i	$0.497678\pi$
$230$		0			0
$231$	203838.	−	349719.i	0.251337	−	0.431211i
$232$		0			0
$233$			320414.i			0.386653i	0.981134	+	0.193327i	$0.0619277\pi$
$233$			320414.i			0.386653i	−0.981134	+	0.193327i	$0.938072\pi$
$234$		0			0
$235$			1.59027e6i			1.87845i
$236$		0			0
$237$		−	941911.i		−	1.08928i
$238$		0			0
$239$	143049.			0.161991			0.0809956	−	0.996714i	$-0.474190\pi$
$239$	143049.			0.161991			0.0809956	+	0.996714i	$0.474190\pi$
$240$		0			0
$241$			883632.i			0.980007i	0.871721	+	0.490003i	$0.163004\pi$
$241$			883632.i			0.980007i	−0.871721	+	0.490003i	$0.836996\pi$
$242$		0			0
$243$		−	59049.0i		−	0.0641500i
$244$		0			0
$245$	−381414.			−0.405958
$246$		0			0
$247$			1.98644e6i			2.07173i
$248$		0			0
$249$		−	110190.i		−	0.112627i
$250$		0			0
$251$		−	835376.i		−	0.836946i	−0.908229	−	0.418473i	$-0.862565\pi$
$251$		−	835376.i		−	0.836946i	0.908229	−	0.418473i	$-0.137435\pi$
$252$		0			0
$253$	462240.	−	793051.i	0.454011	−	0.778932i
$254$		0			0
$255$	−10804.3			−0.0104051
$256$		0			0
$257$	1.98007e6			1.87002			0.935011	−	0.354618i	$-0.115389\pi$
$257$	1.98007e6			1.87002			0.935011	+	0.354618i	$0.115389\pi$
$258$		0			0
$259$	−766100.			−0.709636
$260$		0			0
$261$		−	204781.i		−	0.186075i
$262$		0			0
$263$	−188475.			−0.168022			−0.0840108	−	0.996465i	$-0.526773\pi$
$263$	−188475.			−0.168022			−0.0840108	+	0.996465i	$0.526773\pi$
$264$		0			0
$265$	−3.00492e6			−2.62856
$266$		0			0
$267$			948802.i			0.814512i
$268$		0			0
$269$	−1.97605e6			−1.66501			−0.832506	−	0.554015i	$-0.813095\pi$
$269$	−1.97605e6			−1.66501			−0.832506	+	0.554015i	$0.813095\pi$
$270$		0			0
$271$	1.63323e6			1.35090			0.675452	−	0.737404i	$-0.263948\pi$
$271$	1.63323e6			1.35090			0.675452	+	0.737404i	$0.263948\pi$
$272$		0			0
$273$	906723.			0.736322
$274$		0			0
$275$	−1.71361e6	−	998801.i	−1.36641	−	0.796429i
$276$		0			0
$277$		−	1.78983e6i		−	1.40156i	−0.713376	−	0.700782i	$-0.752835\pi$
$277$		−	1.78983e6i		−	1.40156i	0.713376	−	0.700782i	$-0.247165\pi$
$278$		0			0
$279$		−	510333.i		−	0.392504i
$280$		0			0
$281$			1.54996e6i			1.17099i	0.810675	+	0.585496i	$0.199100\pi$
$281$			1.54996e6i			1.17099i	−0.810675	+	0.585496i	$0.800900\pi$
$282$		0			0
$283$	−1.97417e6			−1.46527			−0.732636	−	0.680620i	$-0.761710\pi$
$283$	−1.97417e6			−1.46527			−0.732636	+	0.680620i	$0.761710\pi$
$284$		0			0
$285$			1.78631e6i			1.30270i
$286$		0			0
$287$			883348.i			0.633034i
$288$		0			0
$289$	1.41968e6			0.999874
$290$		0			0
$291$		−	209514.i		−	0.145038i
$292$		0			0
$293$			2.02687e6i			1.37930i	0.724145	+	0.689648i	$0.242235\pi$
$293$			2.02687e6i			1.37930i	−0.724145	+	0.689648i	$0.757765\pi$
$294$		0			0
$295$			547469.i			0.366272i
$296$		0			0
$297$	147322.	−	252756.i	0.0969117	−	0.166268i
$298$		0			0
$299$	2.05616e6			1.33008
$300$		0			0
$301$	−8714.38			−0.00554396
$302$		0			0
$303$	−343158.			−0.214727
$304$		0			0
$305$			3.61147e6i			2.22297i
$306$		0			0
$307$	−1.29240e6			−0.782621			−0.391311	−	0.920259i	$-0.627978\pi$
$307$	−1.29240e6			−0.782621			−0.391311	+	0.920259i	$0.627978\pi$
$308$		0			0
$309$	−928766.			−0.553363
$310$		0			0
$311$		−	2.35695e6i		−	1.38181i	−0.722945	−	0.690906i	$-0.757212\pi$
$311$		−	2.35695e6i		−	1.38181i	0.722945	−	0.690906i	$-0.242788\pi$
$312$		0			0
$313$	−1.42898e6			−0.824454			−0.412227	−	0.911081i	$-0.635249\pi$
$313$	−1.42898e6			−0.824454			−0.412227	+	0.911081i	$0.635249\pi$
$314$		0			0
$315$	815372.			0.462998
$316$		0			0
$317$	2.11227e6			1.18060			0.590298	−	0.807185i	$-0.299010\pi$
$317$	2.11227e6			1.18060			0.590298	+	0.807185i	$0.299010\pi$
$318$		0			0
$319$	510909.	−	876551.i	0.281104	−	0.482281i
$320$		0			0
$321$			1.64877e6i			0.893096i
$322$		0			0
$323$			29534.9i			0.0157518i
$324$		0			0
$325$		−	4.44291e6i		−	2.33324i
$326$		0			0
$327$	1.71982e6			0.889431
$328$		0			0
$329$		−	1.98430e6i		−	1.01069i
$330$		0			0
$331$		−	125700.i		−	0.0630614i	−0.999503	−	0.0315307i	$-0.989962\pi$
$331$		−	125700.i		−	0.0630614i	0.999503	−	0.0315307i	$-0.0100382\pi$
$332$		0			0
$333$	−553690.			−0.273625
$334$		0			0
$335$			2.75745e6i			1.34244i
$336$		0			0
$337$			30271.8i			0.0145199i	0.999974	+	0.00725994i	$0.00231093\pi$
$337$			30271.8i			0.0145199i	−0.999974	+	0.00725994i	$0.997689\pi$
$338$		0			0
$339$		−	506344.i		−	0.239302i
$340$		0			0
$341$	1.27324e6	−	2.18445e6i	0.592957	−	1.01732i
$342$		0			0
$343$	2.35954e6			1.08291
$344$		0			0
$345$	1.84900e6			0.836353
$346$		0			0
$347$	2.63473e6			1.17466			0.587330	−	0.809347i	$-0.300179\pi$
$347$	2.63473e6			1.17466			0.587330	+	0.809347i	$0.300179\pi$
$348$		0			0
$349$			836966.i			0.367827i	0.982942	+	0.183914i	$0.0588767\pi$
$349$			836966.i			0.367827i	−0.982942	+	0.183914i	$0.941123\pi$
$350$		0			0
$351$	655324.			0.283915
$352$		0			0
$353$	−3.12349e6			−1.33414			−0.667072	−	0.744993i	$-0.732453\pi$
$353$	−3.12349e6			−1.33414			−0.667072	+	0.744993i	$0.732453\pi$
$354$		0			0
$355$		−	1.20105e6i		−	0.505811i
$356$		0			0
$357$	13481.4			0.00559840
$358$		0			0
$359$	−3.53925e6			−1.44936			−0.724679	−	0.689087i	$-0.758012\pi$
$359$	−3.53925e6			−1.44936			−0.724679	+	0.689087i	$0.758012\pi$
$360$		0			0
$361$	2.40700e6			0.972093
$362$		0			0
$363$	1.26120e6	−	714349.i	0.502364	−	0.284540i
$364$		0			0
$365$		−	4.89706e6i		−	1.92399i
$366$		0			0
$367$		−	3.42498e6i		−	1.32737i	−0.748011	−	0.663686i	$-0.768991\pi$
$367$		−	3.42498e6i		−	1.32737i	0.748011	−	0.663686i	$-0.231009\pi$
$368$		0			0
$369$			638430.i			0.244088i
$370$		0			0
$371$	3.74947e6			1.41428
$372$		0			0
$373$		−	4.86288e6i		−	1.80976i	−0.425666	−	0.904880i	$-0.639960\pi$
$373$		−	4.86288e6i		−	1.80976i	0.425666	−	0.904880i	$-0.360040\pi$
$374$		0			0
$375$		−	1.46914e6i		−	0.539492i
$376$		0			0
$377$	2.27265e6			0.823529
$378$		0			0
$379$			1.95198e6i			0.698036i	0.937116	+	0.349018i	$0.113485\pi$
$379$			1.95198e6i			0.698036i	−0.937116	+	0.349018i	$0.886515\pi$
$380$		0			0
$381$			1.02372e6i			0.361302i
$382$		0			0
$383$		−	4.90493e6i		−	1.70858i	−0.519794	−	0.854291i	$-0.673991\pi$
$383$		−	4.90493e6i		−	1.70858i	0.519794	−	0.854291i	$-0.326009\pi$
$384$		0			0
$385$	3.49015e6	+	2.03428e6i	1.20003	+	0.699454i
$386$		0			0
$387$	−6298.22			−0.00213767
$388$		0			0
$389$	−4.84168e6			−1.62227			−0.811133	−	0.584862i	$-0.801149\pi$
$389$	−4.84168e6			−1.62227			−0.811133	+	0.584862i	$0.801149\pi$
$390$		0			0
$391$	30571.5			0.0101129
$392$		0			0
$393$			1.01904e6i			0.332819i
$394$		0			0
$395$	9.40015e6			3.03139
$396$		0			0
$397$	−3.59071e6			−1.14342			−0.571708	−	0.820457i	$-0.693719\pi$
$397$	−3.59071e6			−1.14342			−0.571708	+	0.820457i	$0.693719\pi$
$398$		0			0
$399$		−	2.22892e6i		−	0.700909i
$400$		0			0
$401$	−2.05871e6			−0.639344			−0.319672	−	0.947528i	$-0.603573\pi$
$401$	−2.05871e6			−0.639344			−0.319672	+	0.947528i	$0.603573\pi$
$402$		0			0
$403$	5.66366e6			1.73714
$404$		0			0
$405$	589301.			0.178525
$406$		0			0
$407$	−2.37003e6	−	1.38141e6i	−0.709200	−	0.413366i
$408$		0			0
$409$		−	3.43925e6i		−	1.01661i	−0.861176	−	0.508307i	$-0.830272\pi$
$409$		−	3.43925e6i		−	1.01661i	0.861176	−	0.508307i	$-0.169728\pi$
$410$		0			0
$411$			1.68072e6i			0.490785i
$412$		0			0
$413$		−	683118.i		−	0.197070i
$414$		0			0
$415$	1.09968e6			0.313435
$416$		0			0
$417$			3.04820e6i			0.858426i
$418$		0			0
$419$		−	5.64367e6i		−	1.57046i	−0.619204	−	0.785230i	$-0.712545\pi$
$419$		−	5.64367e6i		−	1.57046i	0.619204	−	0.785230i	$-0.287455\pi$
$420$		0			0
$421$	−5.14508e6			−1.41477			−0.707386	−	0.706827i	$-0.750126\pi$
$421$	−5.14508e6			−1.41477			−0.707386	+	0.706827i	$0.750126\pi$
$422$		0			0
$423$		−	1.43413e6i		−	0.389706i
$424$		0			0
$425$		−	66058.3i		−	0.0177401i
$426$		0			0
$427$		−	4.50631e6i		−	1.19605i
$428$		0			0
$429$	2.80507e6	+	1.63497e6i	0.735869	+	0.428911i
$430$		0			0
$431$	5.52903e6			1.43369			0.716846	−	0.697232i	$-0.245585\pi$
$431$	5.52903e6			1.43369			0.716846	+	0.697232i	$0.245585\pi$
$432$		0			0
$433$	−5.77341e6			−1.47983			−0.739917	−	0.672698i	$-0.765135\pi$
$433$	−5.77341e6			−1.47983			−0.739917	+	0.672698i	$0.765135\pi$
$434$		0			0
$435$	2.04368e6			0.517834
$436$		0			0
$437$		−	5.05447e6i		−	1.26611i
$438$		0			0
$439$	−383253.			−0.0949126			−0.0474563	−	0.998873i	$-0.515111\pi$
$439$	−383253.			−0.0949126			−0.0474563	+	0.998873i	$0.515111\pi$
$440$		0			0
$441$	343965.			0.0842205
$442$		0			0
$443$			3.20492e6i			0.775904i	0.921680	+	0.387952i	$0.126817\pi$
$443$			3.20492e6i			0.775904i	−0.921680	+	0.387952i	$0.873183\pi$
$444$		0			0
$445$	−9.46892e6			−2.26673
$446$		0			0
$447$	4.79856e6			1.13590
$448$		0			0
$449$	2.55939e6			0.599130			0.299565	−	0.954076i	$-0.403158\pi$
$449$	2.55939e6			0.599130			0.299565	+	0.954076i	$0.403158\pi$
$450$		0			0
$451$	−1.59282e6	+	2.73276e6i	−0.368745	+	0.632645i
$452$		0			0
$453$			2.91876e6i			0.668272i
$454$		0			0
$455$			9.04897e6i			2.04914i
$456$		0			0
$457$		−	34813.7i		−	0.00779759i	−0.999992	−	0.00389879i	$-0.998759\pi$
$457$		−	34813.7i		−	0.00779759i	0.999992	−	0.00389879i	$-0.00124103\pi$
$458$		0			0
$459$	9743.52			0.00215866
$460$		0			0
$461$			17245.1i			0.00377933i	0.999998	+	0.00188966i	$0.000601499\pi$
$461$			17245.1i			0.00377933i	−0.999998	+	0.00188966i	$0.999399\pi$
$462$		0			0
$463$			1.91586e6i			0.415348i	0.978198	+	0.207674i	$0.0665893\pi$
$463$			1.91586e6i			0.415348i	−0.978198	+	0.207674i	$0.933411\pi$
$464$		0			0
$465$	5.09306e6			1.09231
$466$		0			0
$467$			2.79411e6i			0.592859i	0.955055	+	0.296429i	$0.0957960\pi$
$467$			2.79411e6i			0.592859i	−0.955055	+	0.296429i	$0.904204\pi$
$468$		0			0
$469$		−	3.44068e6i		−	0.722291i
$470$		0			0
$471$		−	3.20028e6i		−	0.664716i
$472$		0			0
$473$	−26959.1	−	15713.5i	−0.00554055	−	0.00322939i
$474$		0			0
$475$	−1.09216e7			−2.22102
$476$		0			0
$477$	2.70988e6			0.545324
$478$		0			0
$479$	−7.46636e6			−1.48686			−0.743430	−	0.668814i	$-0.766802\pi$
$479$	−7.46636e6			−1.48686			−0.743430	+	0.668814i	$0.766802\pi$
$480$		0			0
$481$		−	6.14483e6i		−	1.21101i
$482$		0			0
$483$	−2.30714e6			−0.449994
$484$		0			0
$485$	2.09092e6			0.403631
$486$		0			0
$487$		−	3.16711e6i		−	0.605118i	−0.953131	−	0.302559i	$-0.902159\pi$
$487$		−	3.16711e6i		−	0.605118i	0.953131	−	0.302559i	$-0.0978409\pi$
$488$		0			0
$489$	−1.05444e6			−0.199411
$490$		0			0
$491$	−3.44388e6			−0.644681			−0.322340	−	0.946624i	$-0.604470\pi$
$491$	−3.44388e6			−0.644681			−0.322340	+	0.946624i	$0.604470\pi$
$492$		0			0
$493$	33790.3			0.00626145
$494$		0			0
$495$	2.52247e6	+	1.47025e6i	0.462714	+	0.269699i
$496$		0			0
$497$			1.49864e6i			0.272148i
$498$		0			0
$499$			963935.i			0.173299i	0.996239	+	0.0866495i	$0.0276160\pi$
$499$			963935.i			0.173299i	−0.996239	+	0.0866495i	$0.972384\pi$
$500$		0			0
$501$		−	1.91791e6i		−	0.341377i
$502$		0			0
$503$	−238270.			−0.0419904			−0.0209952	−	0.999780i	$-0.506683\pi$
$503$	−238270.			−0.0419904			−0.0209952	+	0.999780i	$0.506683\pi$
$504$		0			0
$505$		−	3.42467e6i		−	0.597571i
$506$		0			0
$507$			3.93112e6i			0.679198i
$508$		0			0
$509$	4.49077e6			0.768292			0.384146	−	0.923272i	$-0.374496\pi$
$509$	4.49077e6			0.768292			0.384146	+	0.923272i	$0.374496\pi$
$510$		0			0
$511$			6.11044e6i			1.03519i
$512$		0			0
$513$		−	1.61092e6i		−	0.270260i
$514$		0			0
$515$		−	9.26896e6i		−	1.53997i
$516$		0			0
$517$	3.57802e6	−	6.13869e6i	0.588730	−	1.01007i
$518$		0			0
$519$	3.93079e6			0.640563
$520$		0			0
$521$	5.56150e6			0.897631			0.448816	−	0.893624i	$-0.351846\pi$
$521$	5.56150e6			0.897631			0.448816	+	0.893624i	$0.351846\pi$
$522$		0			0
$523$	3.58354e6			0.572872			0.286436	−	0.958099i	$-0.407529\pi$
$523$	3.58354e6			0.572872			0.286436	+	0.958099i	$0.407529\pi$
$524$		0			0
$525$			4.98523e6i			0.789382i
$526$		0			0
$527$	84208.8			0.0132078
$528$		0			0
$529$	1.20449e6			0.187139
$530$		0			0
$531$		−	493716.i		−	0.0759873i
$532$		0			0
$533$	−7.08527e6			−1.08029
$534$		0			0
$535$	−1.64545e7			−2.48542
$536$		0			0
$537$	−2.94809e6			−0.441169
$538$		0			0
$539$	1.47232e6	+	858161.i	0.218288	+	0.127232i
$540$		0			0
$541$			7.48601e6i			1.09966i	0.835277	+	0.549829i	$0.185307\pi$
$541$			7.48601e6i			1.09966i	−0.835277	+	0.549829i	$0.814693\pi$
$542$		0			0
$543$			5.65379e6i			0.822887i
$544$		0			0
$545$			1.71635e7i			2.47523i
$546$		0			0
$547$	854126.			0.122055			0.0610273	−	0.998136i	$-0.480562\pi$
$547$	854126.			0.122055			0.0610273	+	0.998136i	$0.480562\pi$
$548$		0			0
$549$		−	3.25688e6i		−	0.461181i
$550$		0			0
$551$		−	5.58665e6i		−	0.783921i
$552$		0			0
$553$	−1.17293e7			−1.63102
$554$		0			0
$555$		−	5.52575e6i		−	0.761480i
$556$		0			0
$557$			3.34510e6i			0.456847i	0.973562	+	0.228424i	$0.0733571\pi$
$557$			3.34510e6i			0.456847i	−0.973562	+	0.228424i	$0.926643\pi$
$558$		0			0
$559$		−	69897.4i		−	0.00946088i
$560$		0			0
$561$	41706.5	+	24309.2i	0.00559496	+	0.00326110i
$562$		0			0
$563$	−3.12950e6			−0.416106			−0.208053	−	0.978118i	$-0.566713\pi$
$563$	−3.12950e6			−0.416106			−0.208053	+	0.978118i	$0.566713\pi$
$564$		0			0
$565$	5.05325e6			0.665962
$566$		0			0
$567$	−735316.			−0.0960542
$568$		0			0
$569$		−	2.83867e6i		−	0.367565i	−0.982967	−	0.183783i	$-0.941166\pi$
$569$		−	2.83867e6i		−	0.367565i	0.982967	−	0.183783i	$-0.0588342\pi$
$570$		0			0
$571$	−8.17864e6			−1.04976			−0.524881	−	0.851175i	$-0.675890\pi$
$571$	−8.17864e6			−1.04976			−0.524881	+	0.851175i	$0.675890\pi$
$572$		0			0
$573$	1.99490e6			0.253825
$574$		0			0
$575$			1.13049e7i			1.42593i
$576$		0			0
$577$	9.84164e6			1.23063			0.615316	−	0.788281i	$-0.289028\pi$
$577$	9.84164e6			1.23063			0.615316	+	0.788281i	$0.289028\pi$
$578$		0			0
$579$	−1.85926e6			−0.230485
$580$		0			0
$581$	−1.37216e6			−0.168641
$582$		0			0
$583$	1.15995e7	+	6.76092e6i	1.41341	+	0.823824i
$584$		0			0
$585$			6.54004e6i			0.790116i
$586$		0			0
$587$		−	1.10586e7i		−	1.32467i	−0.749209	−	0.662334i	$-0.769566\pi$
$587$		−	1.10586e7i		−	1.32467i	0.749209	−	0.662334i	$-0.230434\pi$
$588$		0			0
$589$		−	1.39225e7i		−	1.65359i
$590$		0			0
$591$	−402528.			−0.0474053
$592$		0			0
$593$			1.24782e7i			1.45719i	0.684945	+	0.728595i	$0.259826\pi$
$593$			1.24782e7i			1.45719i	−0.684945	+	0.728595i	$0.740174\pi$
$594$		0			0
$595$			134542.i			0.0155800i
$596$		0			0
$597$	−2.47818e6			−0.284575
$598$		0			0
$599$		−	6.36046e6i		−	0.724305i	−0.932119	−	0.362152i	$-0.882042\pi$
$599$		−	6.36046e6i		−	0.724305i	0.932119	−	0.362152i	$-0.117958\pi$
$600$		0			0
$601$		−	1.59521e7i		−	1.80149i	−0.434352	−	0.900743i	$-0.643023\pi$
$601$		−	1.59521e7i		−	1.80149i	0.434352	−	0.900743i	$-0.356977\pi$
$602$		0			0
$603$		−	2.48671e6i		−	0.278504i
$604$		0			0
$605$	7.12911e6	+	1.25866e7i	0.791857	+	1.39805i
$606$		0			0
$607$	−492273.			−0.0542293			−0.0271147	−	0.999632i	$-0.508632\pi$
$607$	−492273.			−0.0542293			−0.0271147	+	0.999632i	$0.508632\pi$
$608$		0			0
$609$	−2.55006e6			−0.278617
$610$		0			0
$611$	1.59159e7			1.72476
$612$		0			0
$613$		−	1.51759e7i		−	1.63118i	−0.578628	−	0.815591i	$-0.696412\pi$
$613$		−	1.51759e7i		−	1.63118i	0.578628	−	0.815591i	$-0.303588\pi$
$614$		0			0
$615$	−6.37144e6			−0.679282
$616$		0			0
$617$	−6.39757e6			−0.676553			−0.338277	−	0.941047i	$-0.609844\pi$
$617$	−6.39757e6			−0.676553			−0.338277	+	0.941047i	$0.609844\pi$
$618$		0			0
$619$			2.70573e6i			0.283830i	0.989879	+	0.141915i	$0.0453259\pi$
$619$			2.70573e6i			0.283830i	−0.989879	+	0.141915i	$0.954674\pi$
$620$		0			0
$621$	−1.66746e6			−0.173511
$622$		0			0
$623$	1.18151e7			1.21960
$624$		0			0
$625$	−783225.			−0.0802022
$626$		0			0
$627$	4.01911e6	−	6.89546e6i	0.408283	−	0.700478i
$628$		0			0
$629$		−	91362.9i		−	0.00920753i
$630$		0			0
$631$			1.05673e6i			0.105655i	0.998604	+	0.0528277i	$0.0168234\pi$
$631$			1.05673e6i			0.105655i	−0.998604	+	0.0528277i	$0.983177\pi$
$632$		0			0
$633$			5.36860e6i			0.532540i
$634$		0			0
$635$	−1.02166e7			−1.00548
$636$		0			0
$637$			3.81731e6i			0.372742i
$638$		0			0
$639$			1.08312e6i			0.104936i
$640$		0			0
$641$	6.49935e6			0.624777			0.312388	−	0.949955i	$-0.398871\pi$
$641$	6.49935e6			0.624777			0.312388	+	0.949955i	$0.398871\pi$
$642$		0			0
$643$			1.77783e7i			1.69575i	0.530194	+	0.847876i	$0.322119\pi$
$643$			1.77783e7i			1.69575i	−0.530194	+	0.847876i	$0.677881\pi$
$644$		0			0
$645$		−	62855.4i		−	0.00594899i
$646$		0			0
$647$			1.19193e7i			1.11942i	0.828690	+	0.559708i	$0.189087\pi$
$647$			1.19193e7i			1.11942i	−0.828690	+	0.559708i	$0.810913\pi$
$648$		0			0
$649$	1.23178e6	−	2.11332e6i	0.114794	−	0.196949i
$650$		0			0
$651$	−6.35500e6			−0.587710
$652$		0			0
$653$	1.02018e7			0.936257			0.468129	−	0.883660i	$-0.344928\pi$
$653$	1.02018e7			0.936257			0.468129	+	0.883660i	$0.344928\pi$
$654$		0			0
$655$	−1.01698e7			−0.926214
$656$		0			0
$657$			4.41625e6i			0.399154i
$658$		0			0
$659$	1.37890e7			1.23686			0.618428	−	0.785842i	$-0.287770\pi$
$659$	1.37890e7			1.23686			0.618428	+	0.785842i	$0.287770\pi$
$660$		0			0
$661$	−1.53761e7			−1.36880			−0.684402	−	0.729105i	$-0.739937\pi$
$661$	−1.53761e7			−1.36880			−0.684402	+	0.729105i	$0.739937\pi$
$662$		0			0
$663$			108133.i			0.00955378i
$664$		0			0
$665$	2.22443e7			1.95058
$666$		0			0
$667$	−5.78271e6			−0.503289
$668$		0			0
$669$	−4.11979e6			−0.355885
$670$		0			0
$671$	8.12563e6	−	1.39409e7i	0.696708	−	1.19532i
$672$		0			0
$673$			1.40262e7i			1.19372i	0.802346	+	0.596859i	$0.203585\pi$
$673$			1.40262e7i			1.19372i	−0.802346	+	0.596859i	$0.796415\pi$
$674$		0			0
$675$			3.60302e6i			0.304374i
$676$		0			0
$677$			1.07455e7i			0.901066i	0.892760	+	0.450533i	$0.148766\pi$
$677$			1.07455e7i			0.901066i	−0.892760	+	0.450533i	$0.851234\pi$
$678$		0			0
$679$	−2.60901e6			−0.217170
$680$		0			0
$681$		−	5.73577e6i		−	0.473941i
$682$		0			0
$683$		−	7.28300e6i		−	0.597390i	−0.954349	−	0.298695i	$-0.903449\pi$
$683$		−	7.28300e6i		−	0.597390i	0.954349	−	0.298695i	$-0.0965514\pi$
$684$		0			0
$685$	−1.67734e7			−1.36582
$686$		0			0
$687$		−	104188.i		−	0.00842219i
$688$		0			0
$689$			3.00742e7i			2.41349i
$690$		0			0
$691$			8.21467e6i			0.654478i	0.944942	+	0.327239i	$0.106118\pi$
$691$			8.21467e6i			0.654478i	−0.944942	+	0.327239i	$0.893882\pi$
$692$		0			0
$693$	−3.14747e6	−	1.83455e6i	−0.248960	−	0.145109i
$694$		0			0
$695$	−3.04206e7			−2.38894
$696$		0			0
$697$	−105346.			−0.00821362
$698$		0			0
$699$	2.88373e6			0.223234
$700$		0			0
$701$		−	1.37576e7i		−	1.05742i	−0.848802	−	0.528710i	$-0.822676\pi$
$701$		−	1.37576e7i		−	1.05742i	0.848802	−	0.528710i	$-0.177324\pi$
$702$		0			0
$703$	−1.51053e7			−1.15277
$704$		0			0
$705$	1.43124e7			1.08453
$706$		0			0
$707$			4.27322e6i			0.321519i
$708$		0			0
$709$	1.63987e7			1.22517			0.612583	−	0.790407i	$-0.290131\pi$
$709$	1.63987e7			1.22517			0.612583	+	0.790407i	$0.290131\pi$
$710$		0			0
$711$	−8.47720e6			−0.628896
$712$		0			0
$713$	−1.44111e7			−1.06163
$714$		0			0
$715$	−1.63168e7	+	2.79942e7i	−1.19363	+	2.04788i
$716$		0			0
$717$		−	1.28744e6i		−	0.0935257i
$718$		0			0
$719$			2.35444e7i			1.69850i	0.527993	+	0.849248i	$0.322945\pi$
$719$			2.35444e7i			1.69850i	−0.527993	+	0.849248i	$0.677055\pi$
$720$		0			0
$721$			1.15656e7i			0.828571i
$722$		0			0
$723$	7.95269e6			0.565807
$724$		0			0
$725$			1.24952e7i			0.882872i
$726$		0			0
$727$		−	2.79400e7i		−	1.96061i	−0.197493	−	0.980304i	$-0.563280\pi$
$727$		−	2.79400e7i		−	1.96061i	0.197493	−	0.980304i	$-0.436720\pi$
$728$		0			0
$729$	−531441.			−0.0370370
$730$		0			0
$731$		−	1039.25i		−	7.19329e-5i
$732$		0			0
$733$		−	5.85874e6i		−	0.402758i	−0.979513	−	0.201379i	$-0.935458\pi$
$733$		−	5.85874e6i		−	0.402758i	0.979513	−	0.201379i	$-0.0645423\pi$
$734$		0			0
$735$			3.43272e6i			0.234380i
$736$		0			0
$737$	6.20412e6	−	1.06442e7i	0.420738	−	0.721846i
$738$		0			0
$739$	−172556.			−0.0116231			−0.00581153	−	0.999983i	$-0.501850\pi$
$739$	−172556.			−0.0116231			−0.00581153	+	0.999983i	$0.501850\pi$
$740$		0			0
$741$	1.78780e7			1.19612
$742$		0			0
$743$	−3.98111e6			−0.264565			−0.132283	−	0.991212i	$-0.542231\pi$
$743$	−3.98111e6			−0.264565			−0.132283	+	0.991212i	$0.542231\pi$
$744$		0			0
$745$			4.78889e7i			3.16115i
$746$		0			0
$747$	−991711.			−0.0650255
$748$		0			0
$749$	2.05316e7			1.33726
$750$		0			0
$751$		−	1.14439e7i		−	0.740413i	−0.928949	−	0.370207i	$-0.879287\pi$
$751$		−	1.14439e7i		−	0.740413i	0.928949	−	0.370207i	$-0.120713\pi$
$752$		0			0
$753$	−7.51838e6			−0.483211
$754$		0			0
$755$	−2.91289e7			−1.85976
$756$		0			0
$757$	−1.48825e7			−0.943922			−0.471961	−	0.881620i	$-0.656454\pi$
$757$	−1.48825e7			−0.943922			−0.471961	+	0.881620i	$0.656454\pi$
$758$		0			0
$759$	−7.13746e6	−	4.16016e6i	−0.449717	−	0.262123i
$760$		0			0
$761$			3.37860e6i			0.211483i	0.994394	+	0.105741i	$0.0337216\pi$
$761$			3.37860e6i			0.211483i	−0.994394	+	0.105741i	$0.966278\pi$
$762$		0			0
$763$		−	2.14162e7i		−	1.33178i
$764$		0			0
$765$			97239.0i			0.00600741i
$766$		0			0
$767$	5.47924e6			0.336304
$768$		0			0
$769$		−	2.17586e7i		−	1.32683i	−0.748253	−	0.663413i	$-0.769107\pi$
$769$		−	2.17586e7i		−	1.32683i	0.748253	−	0.663413i	$-0.230893\pi$
$770$		0			0
$771$		−	1.78206e7i		−	1.07966i
$772$		0			0
$773$	−2.18333e7			−1.31423			−0.657115	−	0.753791i	$-0.728223\pi$
$773$	−2.18333e7			−1.31423			−0.657115	+	0.753791i	$0.728223\pi$
$774$		0			0
$775$			3.11392e7i			1.86232i
$776$		0			0
$777$			6.89490e6i			0.409709i
$778$		0			0
$779$			1.74171e7i			1.02833i
$780$		0			0
$781$	−2.70229e6	+	4.63624e6i	−0.158528	+	0.271981i
$782$		0			0
$783$	−1.84303e6			−0.107430
$784$		0			0
$785$	3.19384e7			1.84986
$786$		0			0
$787$	−2.59948e7			−1.49606			−0.748031	−	0.663664i	$-0.769001\pi$
$787$	−2.59948e7			−1.49606			−0.748031	+	0.663664i	$0.769001\pi$
$788$		0			0
$789$			1.69628e6i			0.0970074i
$790$		0			0
$791$	−6.30532e6			−0.358316
$792$		0			0
$793$	3.61448e7			2.04109
$794$		0			0
$795$			2.70443e7i			1.51760i
$796$		0			0
$797$	−1.08248e7			−0.603633			−0.301816	−	0.953366i	$-0.597593\pi$
$797$	−1.08248e7			−0.603633			−0.301816	+	0.953366i	$0.597593\pi$
$798$		0			0
$799$	236642.			0.0131137
$800$		0			0
$801$	8.53922e6			0.470259
$802$		0			0
$803$	−1.10181e7	+	1.89035e7i	−0.603003	+	1.03455i
$804$		0			0
$805$		−	2.30250e7i		−	1.25230i
$806$		0			0
$807$			1.77845e7i			0.961296i
$808$		0			0
$809$		−	2.11825e7i		−	1.13790i	−0.822371	−	0.568951i	$-0.807349\pi$
$809$		−	2.11825e7i		−	1.13790i	0.822371	−	0.568951i	$-0.192651\pi$
$810$		0			0
$811$	5.50202e6			0.293745			0.146872	−	0.989155i	$-0.453079\pi$
$811$	5.50202e6			0.293745			0.146872	+	0.989155i	$0.453079\pi$
$812$		0			0
$813$		−	1.46991e7i		−	0.779945i
$814$		0			0
$815$		−	1.05231e7i		−	0.554946i
$816$		0			0
$817$	−171823.			−0.00900587
$818$		0			0
$819$		−	8.16051e6i		−	0.425116i
$820$		0			0
$821$		−	3.02867e7i		−	1.56817i	−0.620651	−	0.784087i	$-0.713132\pi$
$821$		−	3.02867e7i		−	1.56817i	0.620651	−	0.784087i	$-0.286868\pi$
$822$		0			0
$823$			3.19043e7i			1.64191i	0.570991	+	0.820956i	$0.306559\pi$
$823$			3.19043e7i			1.64191i	−0.570991	+	0.820956i	$0.693441\pi$
$824$		0			0
$825$	−8.98921e6	+	1.54225e7i	−0.459819	+	0.788896i
$826$		0			0
$827$	−3.93519e6			−0.200079			−0.100040	−	0.994983i	$-0.531897\pi$
$827$	−3.93519e6			−0.200079			−0.100040	+	0.994983i	$0.531897\pi$
$828$		0			0
$829$	−1.98042e7			−1.00085			−0.500427	−	0.865779i	$-0.666824\pi$
$829$	−1.98042e7			−1.00085			−0.500427	+	0.865779i	$0.666824\pi$
$830$		0			0
$831$	−1.61085e7			−0.809193
$832$		0			0
$833$			56756.7i			0.00283403i
$834$		0			0
$835$	1.91405e7			0.950029
$836$		0			0
$837$	−4.59300e6			−0.226612
$838$		0			0
$839$			2.18090e6i			0.106962i	0.998569	+	0.0534811i	$0.0170317\pi$
$839$			2.18090e6i			0.106962i	−0.998569	+	0.0534811i	$0.982968\pi$
$840$		0			0
$841$	1.41196e7			0.688385
$842$		0			0
$843$	1.39496e7			0.676072
$844$		0			0
$845$	−3.92321e7			−1.89016
$846$		0			0
$847$	−8.89553e6	−	1.57053e7i	−0.426053	−	0.752209i
$848$		0			0
$849$			1.77675e7i			0.845975i
$850$		0			0
$851$			1.56354e7i			0.740091i
$852$		0			0
$853$			286288.i			0.0134720i	0.999977	+	0.00673598i	$0.00214415\pi$
$853$			286288.i			0.0134720i	−0.999977	+	0.00673598i	$0.997856\pi$
$854$		0			0
$855$	1.60768e7			0.752116
$856$		0			0
$857$			2.15658e7i			1.00303i	0.865150	+	0.501514i	$0.167223\pi$
$857$			2.15658e7i			1.00303i	−0.865150	+	0.501514i	$0.832777\pi$
$858$		0			0
$859$		−	1.75311e7i		−	0.810638i	−0.914175	−	0.405319i	$-0.867160\pi$
$859$		−	1.75311e7i		−	0.810638i	0.914175	−	0.405319i	$-0.132840\pi$
$860$		0			0
$861$	7.95013e6			0.365482
$862$		0			0
$863$		−	3.07586e7i		−	1.40585i	−0.711263	−	0.702926i	$-0.751877\pi$
$863$		−	3.07586e7i		−	1.40585i	0.711263	−	0.702926i	$-0.248123\pi$
$864$		0			0
$865$			3.92288e7i			1.78264i
$866$		0			0
$867$		−	1.27771e7i		−	0.577278i
$868$		0			0
$869$	−3.62861e7	−	2.11499e7i	−1.63001	−	0.950075i
$870$		0			0
$871$	2.75974e7			1.23260
$872$		0			0
$873$	−1.88563e6			−0.0837376
$874$		0			0
$875$	−1.82947e7			−0.807801
$876$		0			0
$877$		−	2.73236e7i		−	1.19961i	−0.800147	−	0.599804i	$-0.795245\pi$
$877$		−	2.73236e7i		−	1.19961i	0.800147	−	0.599804i	$-0.204755\pi$
$878$		0			0
$879$	1.82419e7			0.796337
$880$		0			0
$881$	3.03772e7			1.31858			0.659291	−	0.751887i	$-0.270856\pi$
$881$	3.03772e7			1.31858			0.659291	+	0.751887i	$0.270856\pi$
$882$		0			0
$883$		−	9.66964e6i		−	0.417358i	−0.977984	−	0.208679i	$-0.933084\pi$
$883$		−	9.66964e6i		−	0.417358i	0.977984	−	0.208679i	$-0.0669164\pi$
$884$		0			0
$885$	4.92722e6			0.211467
$886$		0			0
$887$	3.51155e7			1.49861			0.749307	−	0.662222i	$-0.230387\pi$
$887$	3.51155e7			1.49861			0.749307	+	0.662222i	$0.230387\pi$
$888$		0			0
$889$	1.27481e7			0.540990
$890$		0			0
$891$	−2.27480e6	−	1.32590e6i	−0.0959951	−	0.0559520i
$892$		0			0
$893$		−	3.91246e7i		−	1.64181i
$894$		0			0
$895$		−	2.94216e7i		−	1.22774i
$896$		0			0
$897$		−	1.85054e7i		−	0.767923i
$898$		0			0
$899$	−1.59284e7			−0.657315
$900$		0			0
$901$			447151.i			0.0183503i
$902$		0			0
$903$			78429.5i			0.00320081i
$904$		0			0
$905$	−5.64240e7			−2.29004
$906$		0			0
$907$		−	2.65431e7i		−	1.07136i	−0.844422	−	0.535678i	$-0.820056\pi$
$907$		−	2.65431e7i		−	1.07136i	0.844422	−	0.535678i	$-0.179944\pi$
$908$		0			0
$909$			3.08842e6i			0.123973i
$910$		0			0
$911$		−	3.44323e7i		−	1.37458i	−0.726383	−	0.687290i	$-0.758800\pi$
$911$		−	3.44323e7i		−	1.37458i	0.726383	−	0.687290i	$-0.241200\pi$
$912$		0			0
$913$	−4.24495e6	−	2.47423e6i	−0.168537	−	0.0982342i
$914$		0			0
$915$	3.25032e7			1.28343
$916$		0			0
$917$	1.26897e7			0.498342
$918$		0			0
$919$	1.18671e7			0.463507			0.231754	−	0.972775i	$-0.425554\pi$
$919$	1.18671e7			0.463507			0.231754	+	0.972775i	$0.425554\pi$
$920$		0			0
$921$			1.16316e7i			0.451846i
$922$		0			0
$923$	−1.20204e7			−0.464426
$924$		0			0
$925$	3.37847e7			1.29827
$926$		0			0
$927$			8.35890e6i			0.319484i
$928$		0			0
$929$	−2.83595e7			−1.07810			−0.539050	−	0.842274i	$-0.681217\pi$
$929$	−2.83595e7			−1.07810			−0.539050	+	0.842274i	$0.681217\pi$
$930$		0			0
$931$	9.38376e6			0.354816
$932$		0			0
$933$	−2.12125e7			−0.797790
$934$		0			0
$935$	−242603.	+	416225.i	−0.00907541	+	0.0155704i
$936$		0			0
$937$			4.11828e6i			0.153238i	0.997060	+	0.0766190i	$0.0244125\pi$
$937$			4.11828e6i			0.153238i	−0.997060	+	0.0766190i	$0.975587\pi$
$938$		0			0
$939$			1.28609e7i			0.475999i
$940$		0			0
$941$		−	1.42245e7i		−	0.523677i	−0.965112	−	0.261838i	$-0.915671\pi$
$941$		−	1.42245e7i		−	0.523677i	0.965112	−	0.261838i	$-0.0843287\pi$
$942$		0			0
$943$	1.80283e7			0.660202
$944$		0			0
$945$		−	7.33835e6i		−	0.267312i
$946$		0			0
$947$			2.22364e7i			0.805728i	0.915260	+	0.402864i	$0.131985\pi$
$947$			2.22364e7i			0.805728i	−0.915260	+	0.402864i	$0.868015\pi$
$948$		0			0
$949$	−4.90114e7			−1.76657
$950$		0			0
$951$		−	1.90104e7i		−	0.681618i
$952$		0			0
$953$			7.66528e6i			0.273398i	0.990613	+	0.136699i	$0.0436493\pi$
$953$			7.66528e6i			0.273398i	−0.990613	+	0.136699i	$0.956351\pi$
$954$		0			0
$955$			1.99088e7i			0.706379i
$956$		0			0
$957$	−7.88895e6	−	4.59818e6i	−0.278445	−	0.162295i
$958$		0			0
$959$	2.09294e7			0.734871
$960$		0			0
$961$	−1.10661e7			−0.386531
$962$		0			0
$963$	1.48389e7			0.515629
$964$		0			0
$965$		−	1.85552e7i		−	0.641426i
$966$		0			0
$967$	4.59274e7			1.57945			0.789725	−	0.613461i	$-0.210223\pi$
$967$	4.59274e7			1.57945			0.789725	+	0.613461i	$0.210223\pi$
$968$		0			0
$969$	265814.			0.00909430
$970$		0			0
$971$			4.40365e7i			1.49887i	0.662076	+	0.749437i	$0.269676\pi$
$971$			4.40365e7i			1.49887i	−0.662076	+	0.749437i	$0.730324\pi$
$972$		0			0
$973$	3.79581e7			1.28535
$974$		0			0
$975$	−3.99862e7			−1.34710
$976$		0			0
$977$	−2.41462e7			−0.809306			−0.404653	−	0.914470i	$-0.632608\pi$
$977$	−2.41462e7			−0.809306			−0.404653	+	0.914470i	$0.632608\pi$
$978$		0			0
$979$	3.65516e7	+	2.13046e7i	1.21885	+	0.710422i
$980$		0			0
$981$		−	1.54783e7i		−	0.513513i
$982$		0			0
$983$			2.56398e7i			0.846312i	0.906057	+	0.423156i	$0.139078\pi$
$983$			2.56398e7i			0.846312i	−0.906057	+	0.423156i	$0.860922\pi$
$984$		0			0
$985$		−	4.01717e6i		−	0.131926i
$986$		0			0
$987$	−1.78587e7			−0.583521
$988$		0			0
$989$			177853.i			0.00578189i
$990$		0			0
$991$		−	5.52281e7i		−	1.78639i	−0.449671	−	0.893194i	$-0.648459\pi$
$991$		−	5.52281e7i		−	1.78639i	0.449671	−	0.893194i	$-0.351541\pi$
$992$		0			0
$993$	−1.13130e6			−0.0364085
$994$		0			0
$995$		−	2.47319e7i		−	0.791953i
$996$		0			0
$997$			5.63606e7i			1.79572i	0.440284	+	0.897859i	$0.354878\pi$
$997$			5.63606e7i			1.79572i	−0.440284	+	0.897859i	$0.645122\pi$
$998$		0			0
$999$			4.98321e6i			0.157977i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.6.o.b.175.5	yes	40
4.3	odd	2	inner	528.6.o.b.175.4	yes	40
11.10	odd	2	inner	528.6.o.b.175.3	✓	40
44.43	even	2	inner	528.6.o.b.175.6	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.6.o.b.175.3	✓	40	11.10	odd	2	inner
528.6.o.b.175.4	yes	40	4.3	odd	2	inner
528.6.o.b.175.5	yes	40	1.1	even	1	trivial
528.6.o.b.175.6	yes	40	44.43	even	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	528.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.00000i q^{3} +89.8188 q^{5} -112.074 q^{7} -81.0000 q^{9} +O(q^{10})\)	\(q-9.00000i q^{3} +89.8188 q^{5} -112.074 q^{7} -81.0000 q^{9} +(-346.715 - 202.088i) q^{11} -898.935i q^{13} -808.369i q^{15} -13.3656i q^{17} -2209.77 q^{19} +1008.66i q^{21} +2287.32i q^{23} +4942.41 q^{25} +729.000i q^{27} +2528.16i q^{29} +6300.41i q^{31} +(-1818.79 + 3120.44i) q^{33} -10066.3 q^{35} +6835.68 q^{37} -8090.42 q^{39} -7881.85i q^{41} +77.7558 q^{43} -7275.32 q^{45} +17705.3i q^{47} -4246.48 q^{49} -120.290 q^{51} -33455.4 q^{53} +(-31141.6 - 18151.3i) q^{55} +19888.0i q^{57} +6095.26i q^{59} +40208.4i q^{61} +9077.97 q^{63} -80741.2i q^{65} +30700.1i q^{67} +20585.9 q^{69} -13371.9i q^{71} -54521.6i q^{73} -44481.7i q^{75} +(38857.7 + 22648.7i) q^{77} +104657. q^{79} +6561.00 q^{81} +12243.3 q^{83} -1200.48i q^{85} +22753.4 q^{87} -105422. q^{89} +100747. i q^{91} +56703.7 q^{93} -198479. q^{95} +23279.4 q^{97} +(28083.9 + 16369.1i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 88 q^{5} - 3240 q^{9}+O(q^{10}) \) 40 * q + 88 * q^5 - 3240 * q^9	\( 40 q + 88 q^{5} - 3240 q^{9} + 5464 q^{25} + 6840 q^{33} + 4576 q^{37} - 7128 q^{45} + 58760 q^{49} - 165336 q^{53} + 16632 q^{69} + 28136 q^{77} + 262440 q^{81} - 304608 q^{89} + 102672 q^{93} + 15664 q^{97}+O(q^{100}) \) 40 * q + 88 * q^5 - 3240 * q^9 + 5464 * q^25 + 6840 * q^33 + 4576 * q^37 - 7128 * q^45 + 58760 * q^49 - 165336 * q^53 + 16632 * q^69 + 28136 * q^77 + 262440 * q^81 - 304608 * q^89 + 102672 * q^93 + 15664 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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