LMFDB - Embedded newform 441.8.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,8,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)

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:	yes
Analytic conductor:	$137.761796238$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 102x^{2} + 240x + 900 $ x^4 - x^3 - 102x^2 + 240x + 900
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$8.23552$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.18552 q^{2} -117.852 q^{4} -0.588975 q^{5} +783.169 q^{8} +O(q^{10})$	$q-3.18552 q^{2} -117.852 q^{4} -0.588975 q^{5} +783.169 q^{8} +1.87619 q^{10} +1199.54 q^{11} -7651.01 q^{13} +12590.3 q^{16} +35185.9 q^{17} +8573.44 q^{19} +69.4121 q^{20} -3821.17 q^{22} -78733.7 q^{23} -78124.7 q^{25} +24372.5 q^{26} +199760. q^{29} -137114. q^{31} -140352. q^{32} -112085. q^{34} +90084.3 q^{37} -27310.9 q^{38} -461.267 q^{40} -269601. q^{41} +602901. q^{43} -141369. q^{44} +250808. q^{46} -384915. q^{47} +248868. q^{50} +901690. q^{52} -664238. q^{53} -706.501 q^{55} -636341. q^{58} -2.30788e6 q^{59} -899876. q^{61} +436781. q^{62} -1.16446e6 q^{64} +4506.25 q^{65} +23758.8 q^{67} -4.14674e6 q^{68} -2.52558e6 q^{71} -3.52851e6 q^{73} -286966. q^{74} -1.01040e6 q^{76} +4.63782e6 q^{79} -7415.38 q^{80} +858819. q^{82} +7.71353e6 q^{83} -20723.6 q^{85} -1.92055e6 q^{86} +939444. q^{88} +8.78100e6 q^{89} +9.27896e6 q^{92} +1.22616e6 q^{94} -5049.54 q^{95} -4.42347e6 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8}+O(q^{10}) $ 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8	$ 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8} - 4774 q^{10} + 3972 q^{11} + 1176 q^{13} + 57264 q^{16} + 56364 q^{17} - 41748 q^{19} + 162372 q^{20} - 152954 q^{22} - 131748 q^{23} - 68016 q^{25} + 113652 q^{26} - 34056 q^{29} - 401212 q^{31} + 453408 q^{32} - 82110 q^{34} - 5396 q^{37} + 31794 q^{38} + 443688 q^{40} + 410424 q^{41} + 46544 q^{43} + 1465836 q^{44} + 1379202 q^{46} + 1470084 q^{47} - 1395528 q^{50} + 5269768 q^{52} + 642372 q^{53} + 3063340 q^{55} - 1972220 q^{58} + 752220 q^{59} - 1325772 q^{61} - 3016314 q^{62} - 865856 q^{64} + 4868808 q^{65} - 290916 q^{67} + 2453556 q^{68} - 3377760 q^{71} - 6706588 q^{73} - 3616878 q^{74} - 2923004 q^{76} + 3946244 q^{79} + 10695888 q^{80} + 17563252 q^{82} + 9542064 q^{83} + 7006068 q^{85} - 23237832 q^{86} - 8043336 q^{88} + 16165212 q^{89} + 19442628 q^{92} - 5720442 q^{94} + 3268452 q^{95} - 1533112 q^{97}+O(q^{100}) $ 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8 - 4774 * q^10 + 3972 * q^11 + 1176 * q^13 + 57264 * q^16 + 56364 * q^17 - 41748 * q^19 + 162372 * q^20 - 152954 * q^22 - 131748 * q^23 - 68016 * q^25 + 113652 * q^26 - 34056 * q^29 - 401212 * q^31 + 453408 * q^32 - 82110 * q^34 - 5396 * q^37 + 31794 * q^38 + 443688 * q^40 + 410424 * q^41 + 46544 * q^43 + 1465836 * q^44 + 1379202 * q^46 + 1470084 * q^47 - 1395528 * q^50 + 5269768 * q^52 + 642372 * q^53 + 3063340 * q^55 - 1972220 * q^58 + 752220 * q^59 - 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 + 4868808 * q^65 - 290916 * q^67 + 2453556 * q^68 - 3377760 * q^71 - 6706588 * q^73 - 3616878 * q^74 - 2923004 * q^76 + 3946244 * q^79 + 10695888 * q^80 + 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 - 23237832 * q^86 - 8043336 * q^88 + 16165212 * q^89 + 19442628 * q^92 - 5720442 * q^94 + 3268452 * q^95 - 1533112 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−3.18552	−0.281563	−0.140782	−	0.990041i	$-0.544962\pi$
$2$	−3.18552	−0.281563	−0.140782	+	0.990041i	$0.544962\pi$
$3$	0	0
$3$	0	0
$4$	−117.852	−0.920722
$5$	−0.588975	−0.00210718	−0.00105359	−	0.999999i	$-0.500335\pi$
$5$	−0.588975	−0.00210718	−0.00105359	+	0.999999i	$0.500335\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	783.169	0.540805
$9$	0	0
$10$	1.87619	0.000593304 0
$11$	1199.54	0.271732	0.135866	−	0.990727i	$-0.456618\pi$
$11$	1199.54	0.271732	0.135866	+	0.990727i	$0.456618\pi$
$12$	0	0
$13$	−7651.01	−0.965867	−0.482933	−	0.875657i	$-0.660429\pi$
$13$	−7651.01	−0.965867	−0.482933	+	0.875657i	$0.660429\pi$
$14$	0	0
$15$	0	0
$16$	12590.3	0.768452
$17$	35185.9	1.73699	0.868495	−	0.495698i	$-0.165088\pi$
$17$	35185.9	1.73699	0.868495	+	0.495698i	$0.165088\pi$
$18$	0	0
$19$	8573.44	0.286759	0.143380	−	0.989668i	$-0.454203\pi$
$19$	8573.44	0.286759	0.143380	+	0.989668i	$0.454203\pi$
$20$	69.4121	0.00194013
$21$	0	0
$22$	−3821.17	−0.0765098
$23$	−78733.7	−1.34931	−0.674657	−	0.738131i	$-0.735709\pi$
$23$	−78733.7	−1.34931	−0.674657	+	0.738131i	$0.735709\pi$
$24$	0	0
$25$	−78124.7	−0.999996
$26$	24372.5	0.271952
$27$	0	0
$28$	0	0
$29$	199760.	1.52095	0.760477	−	0.649365i	$-0.224965\pi$
$29$	199760.	1.52095	0.760477	+	0.649365i	$0.224965\pi$
$30$	0	0
$31$	−137114.	−0.826642	−0.413321	−	0.910585i	$-0.635631\pi$
$31$	−137114.	−0.826642	−0.413321	+	0.910585i	$0.635631\pi$
$32$	−140352.	−0.757172
$33$	0	0
$34$	−112085.	−0.489072
$35$	0	0
$36$	0	0
$37$	90084.3	0.292377	0.146188	−	0.989257i	$-0.453299\pi$
$37$	90084.3	0.292377	0.146188	+	0.989257i	$0.453299\pi$
$38$	−27310.9	−0.0807409
$39$	0	0
$40$	−461.267	−0.00113957
$41$	−269601.	−0.610911	−0.305455	−	0.952206i	$-0.598809\pi$
$41$	−269601.	−0.610911	−0.305455	+	0.952206i	$0.598809\pi$
$42$	0	0
$43$	602901.	1.15640	0.578198	−	0.815897i	$-0.303756\pi$
$43$	602901.	1.15640	0.578198	+	0.815897i	$0.303756\pi$
$44$	−141369.	−0.250190
$45$	0	0
$46$	250808.	0.379917
$47$	−384915.	−0.540783	−0.270391	−	0.962750i	$-0.587153\pi$
$47$	−384915.	−0.540783	−0.270391	+	0.962750i	$0.587153\pi$
$48$	0	0
$49$	0	0
$50$	248868.	0.281562
$51$	0	0
$52$	901690.	0.889295
$53$	−664238.	−0.612855	−0.306428	−	0.951894i	$-0.599134\pi$
$53$	−664238.	−0.612855	−0.306428	+	0.951894i	$0.599134\pi$
$54$	0	0
$55$	−706.501	−0.000572589 0
$56$	0	0
$57$	0	0
$58$	−636341.	−0.428244
$59$	−2.30788e6	−1.46296	−0.731478	−	0.681865i	$-0.761169\pi$
$59$	−2.30788e6	−1.46296	−0.731478	+	0.681865i	$0.761169\pi$
$60$	0	0
$61$	−899876.	−0.507608	−0.253804	−	0.967256i	$-0.581682\pi$
$61$	−899876.	−0.507608	−0.253804	+	0.967256i	$0.581682\pi$
$62$	436781.	0.232752
$63$	0	0
$64$	−1.16446e6	−0.555260
$65$	4506.25	0.00203526
$66$	0	0
$67$	23758.8	0.00965081	0.00482540	−	0.999988i	$-0.498464\pi$
$67$	23758.8	0.00965081	0.00482540	+	0.999988i	$0.498464\pi$
$68$	−4.14674e6	−1.59929
$69$	0	0
$70$	0	0
$71$	−2.52558e6	−0.837446	−0.418723	−	0.908114i	$-0.637522\pi$
$71$	−2.52558e6	−0.837446	−0.418723	+	0.908114i	$0.637522\pi$
$72$	0	0
$73$	−3.52851e6	−1.06160	−0.530800	−	0.847497i	$-0.678109\pi$
$73$	−3.52851e6	−1.06160	−0.530800	+	0.847497i	$0.678109\pi$
$74$	−286966.	−0.0823226
$75$	0	0
$76$	−1.01040e6	−0.264026
$77$	0	0
$78$	0	0
$79$	4.63782e6	1.05832	0.529162	−	0.848521i	$-0.322506\pi$
$79$	4.63782e6	1.05832	0.529162	+	0.848521i	$0.322506\pi$
$80$	−7415.38	−0.00161927
$81$	0	0
$82$	858819.	0.172010
$83$	7.71353e6	1.48074	0.740372	−	0.672197i	$-0.234649\pi$
$83$	7.71353e6	1.48074	0.740372	+	0.672197i	$0.234649\pi$
$84$	0	0
$85$	−20723.6	−0.00366015
$86$	−1.92055e6	−0.325598
$87$	0	0
$88$	939444.	0.146954
$89$	8.78100e6	1.32032	0.660160	−	0.751125i	$-0.270489\pi$
$89$	8.78100e6	1.32032	0.660160	+	0.751125i	$0.270489\pi$
$90$	0	0
$91$	0	0
$92$	9.27896e6	1.24234
$93$	0	0
$94$	1.22616e6	0.152264
$95$	−5049.54	−0.000604254 0
$96$	0	0
$97$	−4.42347e6	−0.492110	−0.246055	−	0.969256i	$-0.579134\pi$
$97$	−4.42347e6	−0.492110	−0.246055	+	0.969256i	$0.579134\pi$
$98$	0	0
$99$	0	0
$100$	9.20718e6	0.920718
$101$	5.87103e6	0.567008	0.283504	−	0.958971i	$-0.408503\pi$
$101$	5.87103e6	0.567008	0.283504	+	0.958971i	$0.408503\pi$
$102$	0	0
$103$	1.65362e6	0.149110	0.0745549	−	0.997217i	$-0.476246\pi$
$103$	1.65362e6	0.149110	0.0745549	+	0.997217i	$0.476246\pi$
$104$	−5.99203e6	−0.522345
$105$	0	0
$106$	2.11594e6	0.172557
$107$	−929383.	−0.0733418	−0.0366709	−	0.999327i	$-0.511675\pi$
$107$	−929383.	−0.0733418	−0.0366709	+	0.999327i	$0.511675\pi$
$108$	0	0
$109$	−9.25089e6	−0.684212	−0.342106	−	0.939661i	$-0.611140\pi$
$109$	−9.25089e6	−0.684212	−0.342106	+	0.939661i	$0.611140\pi$
$110$	2250.57	0.000161220 0
$111$	0	0
$112$	0	0
$113$	−1.16423e7	−0.759039	−0.379519	−	0.925184i	$-0.623911\pi$
$113$	−1.16423e7	−0.759039	−0.379519	+	0.925184i	$0.623911\pi$
$114$	0	0
$115$	46372.2	0.00284325
$116$	−2.35422e7	−1.40038
$117$	0	0
$118$	7.35180e6	0.411914
$119$	0	0
$120$	0	0
$121$	−1.80483e7	−0.926162
$122$	2.86658e6	0.142924
$123$	0	0
$124$	1.61593e7	0.761107
$125$	92027.1	0.00421435
$126$	0	0
$127$	437184.	0.0189387	0.00946937	−	0.999955i	$-0.496986\pi$
$127$	437184.	0.0189387	0.00946937	+	0.999955i	$0.496986\pi$
$128$	2.16745e7	0.913513
$129$	0	0
$130$	−14354.8	−0.000573053 0
$131$	2.91401e7	1.13251	0.566255	−	0.824230i	$-0.308392\pi$
$131$	2.91401e7	1.13251	0.566255	+	0.824230i	$0.308392\pi$
$132$	0	0
$133$	0	0
$134$	−75684.4	−0.00271731
$135$	0	0
$136$	2.75565e7	0.939372
$137$	2.12048e7	0.704550	0.352275	−	0.935896i	$-0.385408\pi$
$137$	2.12048e7	0.704550	0.352275	+	0.935896i	$0.385408\pi$
$138$	0	0
$139$	1.42359e7	0.449608	0.224804	−	0.974404i	$-0.427826\pi$
$139$	1.42359e7	0.449608	0.224804	+	0.974404i	$0.427826\pi$
$140$	0	0
$141$	0	0
$142$	8.04529e6	0.235794
$143$	−9.17771e6	−0.262457
$144$	0	0
$145$	−117654.	−0.00320492
$146$	1.12401e7	0.298907
$147$	0	0
$148$	−1.06167e7	−0.269198
$149$	3.56387e7	0.882611	0.441306	−	0.897357i	$-0.354516\pi$
$149$	3.56387e7	0.882611	0.441306	+	0.897357i	$0.354516\pi$
$150$	0	0
$151$	1.30093e7	0.307493	0.153747	−	0.988110i	$-0.450866\pi$
$151$	1.30093e7	0.307493	0.153747	+	0.988110i	$0.450866\pi$
$152$	6.71445e6	0.155081
$153$	0	0
$154$	0	0
$155$	80757.0	0.00174188
$156$	0	0
$157$	−6.61725e7	−1.36467	−0.682337	−	0.731038i	$-0.739036\pi$
$157$	−6.61725e7	−1.36467	−0.682337	+	0.731038i	$0.739036\pi$
$158$	−1.47739e7	−0.297985
$159$	0	0
$160$	82664.0	0.00159550
$161$	0	0
$162$	0	0
$163$	5.54341e7	1.00258	0.501292	−	0.865278i	$-0.332858\pi$
$163$	5.54341e7	1.00258	0.501292	+	0.865278i	$0.332858\pi$
$164$	3.17731e7	0.562479
$165$	0	0
$166$	−2.45716e7	−0.416923
$167$	1.10649e8	1.83840	0.919201	−	0.393788i	$-0.128836\pi$
$167$	1.10649e8	1.83840	0.919201	+	0.393788i	$0.128836\pi$
$168$	0	0
$169$	−4.21054e6	−0.0671019
$170$	66015.5	0.00103056
$171$	0	0
$172$	−7.10533e7	−1.06472
$173$	1.81428e7	0.266406	0.133203	−	0.991089i	$-0.457474\pi$
$173$	1.81428e7	0.266406	0.133203	+	0.991089i	$0.457474\pi$
$174$	0	0
$175$	0	0
$176$	1.51026e7	0.208813
$177$	0	0
$178$	−2.79721e7	−0.371753
$179$	−5.71114e7	−0.744282	−0.372141	−	0.928176i	$-0.621376\pi$
$179$	−5.71114e7	−0.744282	−0.372141	+	0.928176i	$0.621376\pi$
$180$	0	0
$181$	9.95371e7	1.24770	0.623850	−	0.781544i	$-0.285568\pi$
$181$	9.95371e7	1.24770	0.623850	+	0.781544i	$0.285568\pi$
$182$	0	0
$183$	0	0
$184$	−6.16618e7	−0.729716
$185$	−53057.4	−0.000616091 0
$186$	0	0
$187$	4.22070e7	0.471996
$188$	4.53632e7	0.497911
$189$	0	0
$190$	16085.4	0.000170136 0
$191$	4.61287e6	0.0479021	0.0239510	−	0.999713i	$-0.492375\pi$
$191$	4.61287e6	0.0479021	0.0239510	+	0.999713i	$0.492375\pi$
$192$	0	0
$193$	−7.86525e7	−0.787521	−0.393760	−	0.919213i	$-0.628826\pi$
$193$	−7.86525e7	−0.787521	−0.393760	+	0.919213i	$0.628826\pi$
$194$	1.40911e7	0.138560
$195$	0	0
$196$	0	0
$197$	1.37253e8	1.27906	0.639531	−	0.768765i	$-0.279129\pi$
$197$	1.37253e8	1.27906	0.639531	+	0.768765i	$0.279129\pi$
$198$	0	0
$199$	8.92592e7	0.802910	0.401455	−	0.915879i	$-0.368505\pi$
$199$	8.92592e7	0.802910	0.401455	+	0.915879i	$0.368505\pi$
$200$	−6.11848e7	−0.540802
$201$	0	0
$202$	−1.87023e7	−0.159649
$203$	0	0
$204$	0	0
$205$	158788.	0.00128730
$206$	−5.26765e6	−0.0419838
$207$	0	0
$208$	−9.63286e7	−0.742222
$209$	1.02842e7	0.0779218
$210$	0	0
$211$	1.16809e8	0.856031	0.428016	−	0.903771i	$-0.359213\pi$
$211$	1.16809e8	0.856031	0.428016	+	0.903771i	$0.359213\pi$
$212$	7.82820e7	0.564270
$213$	0	0
$214$	2.96057e6	0.0206503
$215$	−355094.	−0.00243673
$216$	0	0
$217$	0	0
$218$	2.94689e7	0.192649
$219$	0	0
$220$	83262.8	0.000527196 0
$221$	−2.69208e8	−1.67770
$222$	0	0
$223$	8.22050e7	0.496399	0.248200	−	0.968709i	$-0.420161\pi$
$223$	8.22050e7	0.496399	0.248200	+	0.968709i	$0.420161\pi$
$224$	0	0
$225$	0	0
$226$	3.70868e7	0.213717
$227$	2.10863e7	0.119649	0.0598247	−	0.998209i	$-0.480946\pi$
$227$	2.10863e7	0.119649	0.0598247	+	0.998209i	$0.480946\pi$
$228$	0	0
$229$	3.46629e8	1.90740	0.953698	−	0.300765i	$-0.0972420\pi$
$229$	3.46629e8	1.90740	0.953698	+	0.300765i	$0.0972420\pi$
$230$	−147720.	−0.000800554 0
$231$	0	0
$232$	1.56446e8	0.822538
$233$	3.32736e7	0.172328	0.0861638	−	0.996281i	$-0.472539\pi$
$233$	3.32736e7	0.172328	0.0861638	+	0.996281i	$0.472539\pi$
$234$	0	0
$235$	226706.	0.00113953
$236$	2.71989e8	1.34698
$237$	0	0
$238$	0	0
$239$	1.74413e8	0.826394	0.413197	−	0.910642i	$-0.364412\pi$
$239$	1.74413e8	0.826394	0.413197	+	0.910642i	$0.364412\pi$
$240$	0	0
$241$	2.85219e8	1.31256	0.656280	−	0.754518i	$-0.272129\pi$
$241$	2.85219e8	1.31256	0.656280	+	0.754518i	$0.272129\pi$
$242$	5.74932e7	0.260773
$243$	0	0
$244$	1.06053e8	0.467366
$245$	0	0
$246$	0	0
$247$	−6.55955e7	−0.276971
$248$	−1.07384e8	−0.447052
$249$	0	0
$250$	−293155.	−0.00118661
$251$	3.46951e8	1.38487	0.692437	−	0.721478i	$-0.256537\pi$
$251$	3.46951e8	1.38487	0.692437	+	0.721478i	$0.256537\pi$
$252$	0	0
$253$	−9.44445e7	−0.366652
$254$	−1.39266e6	−0.00533245
$255$	0	0
$256$	8.00067e7	0.298048
$257$	3.97183e7	0.145957	0.0729785	−	0.997334i	$-0.476750\pi$
$257$	3.97183e7	0.145957	0.0729785	+	0.997334i	$0.476750\pi$
$258$	0	0
$259$	0	0
$260$	−531073.	−0.00187390
$261$	0	0
$262$	−9.28266e7	−0.318873
$263$	−3.97976e8	−1.34900	−0.674499	−	0.738275i	$-0.735640\pi$
$263$	−3.97976e8	−1.34900	−0.674499	+	0.738275i	$0.735640\pi$
$264$	0	0
$265$	391219.	0.00129140
$266$	0	0
$267$	0	0
$268$	−2.80004e6	−0.00888571
$269$	3.94602e8	1.23602	0.618011	−	0.786169i	$-0.287939\pi$
$269$	3.94602e8	1.23602	0.618011	+	0.786169i	$0.287939\pi$
$270$	0	0
$271$	−3.23078e8	−0.986085	−0.493043	−	0.870005i	$-0.664115\pi$
$271$	−3.23078e8	−0.986085	−0.493043	+	0.870005i	$0.664115\pi$
$272$	4.43001e8	1.33479
$273$	0	0
$274$	−6.75484e7	−0.198375
$275$	−9.37139e7	−0.271731
$276$	0	0
$277$	−1.56919e8	−0.443604	−0.221802	−	0.975092i	$-0.571194\pi$
$277$	−1.56919e8	−0.443604	−0.221802	+	0.975092i	$0.571194\pi$
$278$	−4.53489e7	−0.126593
$279$	0	0
$280$	0	0
$281$	−4.96958e8	−1.33613	−0.668063	−	0.744105i	$-0.732876\pi$
$281$	−4.96958e8	−1.33613	−0.668063	+	0.744105i	$0.732876\pi$
$282$	0	0
$283$	−2.83547e8	−0.743655	−0.371828	−	0.928302i	$-0.621269\pi$
$283$	−2.83547e8	−0.743655	−0.371828	+	0.928302i	$0.621269\pi$
$284$	2.97646e8	0.771055
$285$	0	0
$286$	2.92358e7	0.0738983
$287$	0	0
$288$	0	0
$289$	8.27708e8	2.01713
$290$	374789.	0.000902388 0
$291$	0	0
$292$	4.15843e8	0.977439
$293$	3.75616e8	0.872384	0.436192	−	0.899854i	$-0.356327\pi$
$293$	3.75616e8	0.872384	0.436192	+	0.899854i	$0.356327\pi$
$294$	0	0
$295$	1.35928e6	0.00308271
$296$	7.05512e7	0.158119
$297$	0	0
$298$	−1.13528e8	−0.248511
$299$	6.02393e8	1.30326
$300$	0	0
$301$	0	0
$302$	−4.14415e7	−0.0865788
$303$	0	0
$304$	1.07942e8	0.220361
$305$	530004.	0.00106962
$306$	0	0
$307$	2.34442e8	0.462435	0.231217	−	0.972902i	$-0.425729\pi$
$307$	2.34442e8	0.462435	0.231217	+	0.972902i	$0.425729\pi$
$308$	0	0
$309$	0	0
$310$	−257253.	−0.000490450 0
$311$	4.62252e8	0.871401	0.435700	−	0.900092i	$-0.356501\pi$
$311$	4.62252e8	0.871401	0.435700	+	0.900092i	$0.356501\pi$
$312$	0	0
$313$	4.77219e8	0.879655	0.439827	−	0.898082i	$-0.355040\pi$
$313$	4.77219e8	0.879655	0.439827	+	0.898082i	$0.355040\pi$
$314$	2.10794e8	0.384242
$315$	0	0
$316$	−5.46578e8	−0.974423
$317$	5.04308e8	0.889177	0.444588	−	0.895735i	$-0.353350\pi$
$317$	5.04308e8	0.889177	0.444588	+	0.895735i	$0.353350\pi$
$318$	0	0
$319$	2.39621e8	0.413292
$320$	685840.	0.00117003
$321$	0	0
$322$	0	0
$323$	3.01664e8	0.498098
$324$	0	0
$325$	5.97733e8	0.965862
$326$	−1.76587e8	−0.282290
$327$	0	0
$328$	−2.11143e8	−0.330383
$329$	0	0
$330$	0	0
$331$	−9.78488e8	−1.48306	−0.741528	−	0.670922i	$-0.765899\pi$
$331$	−9.78488e8	−1.48306	−0.741528	+	0.670922i	$0.765899\pi$
$332$	−9.09059e8	−1.36335
$333$	0	0
$334$	−3.52476e8	−0.517626
$335$	−13993.4	−2.03360e−5 0
$336$	0	0
$337$	8.99273e8	1.27993	0.639966	−	0.768403i	$-0.278948\pi$
$337$	8.99273e8	1.27993	0.639966	+	0.768403i	$0.278948\pi$
$338$	1.34128e7	0.0188934
$339$	0	0
$340$	2.44233e6	0.00336998
$341$	−1.64475e8	−0.224625
$342$	0	0
$343$	0	0
$344$	4.72173e8	0.625384
$345$	0	0
$346$	−5.77944e7	−0.0750100
$347$	−2.39179e8	−0.307305	−0.153652	−	0.988125i	$-0.549104\pi$
$347$	−2.39179e8	−0.307305	−0.153652	+	0.988125i	$0.549104\pi$
$348$	0	0
$349$	9.79547e8	1.23349	0.616746	−	0.787162i	$-0.288451\pi$
$349$	9.79547e8	1.23349	0.616746	+	0.787162i	$0.288451\pi$
$350$	0	0
$351$	0	0
$352$	−1.68359e8	−0.205748
$353$	1.44109e7	0.0174373	0.00871864	−	0.999962i	$-0.497225\pi$
$353$	1.44109e7	0.0174373	0.00871864	+	0.999962i	$0.497225\pi$
$354$	0	0
$355$	1.48750e6	0.00176465
$356$	−1.03486e9	−1.21565
$357$	0	0
$358$	1.81930e8	0.209562
$359$	1.11085e7	0.0126714	0.00633569	−	0.999980i	$-0.497983\pi$
$359$	1.11085e7	0.0126714	0.00633569	+	0.999980i	$0.497983\pi$
$360$	0	0
$361$	−8.20368e8	−0.917769
$362$	−3.17078e8	−0.351306
$363$	0	0
$364$	0	0
$365$	2.07820e6	0.00223698
$366$	0	0
$367$	−6.14875e8	−0.649315	−0.324658	−	0.945832i	$-0.605249\pi$
$367$	−6.14875e8	−0.649315	−0.324658	+	0.945832i	$0.605249\pi$
$368$	−9.91282e8	−1.03688
$369$	0	0
$370$	169016.	0.000173469 0
$371$	0	0
$372$	0	0
$373$	4.02822e8	0.401913	0.200957	−	0.979600i	$-0.435595\pi$
$373$	4.02822e8	0.401913	0.200957	+	0.979600i	$0.435595\pi$
$374$	−1.34451e8	−0.132897
$375$	0	0
$376$	−3.01454e8	−0.292458
$377$	−1.52837e9	−1.46904
$378$	0	0
$379$	5.93933e7	0.0560403	0.0280201	−	0.999607i	$-0.491080\pi$
$379$	5.93933e7	0.0560403	0.0280201	+	0.999607i	$0.491080\pi$
$380$	595101.	0.000556350 0
$381$	0	0
$382$	−1.46944e7	−0.0134875
$383$	3.60401e8	0.327786	0.163893	−	0.986478i	$-0.447595\pi$
$383$	3.60401e8	0.327786	0.163893	+	0.986478i	$0.447595\pi$
$384$	0	0
$385$	0	0
$386$	2.50549e8	0.221737
$387$	0	0
$388$	5.21317e8	0.453097
$389$	1.70192e9	1.46594	0.732969	−	0.680263i	$-0.238134\pi$
$389$	1.70192e9	1.46594	0.732969	+	0.680263i	$0.238134\pi$
$390$	0	0
$391$	−2.77032e9	−2.34375
$392$	0	0
$393$	0	0
$394$	−4.37224e8	−0.360137
$395$	−2.73156e6	−0.00223008
$396$	0	0
$397$	−1.30027e9	−1.04296	−0.521478	−	0.853265i	$-0.674619\pi$
$397$	−1.30027e9	−1.04296	−0.521478	+	0.853265i	$0.674619\pi$
$398$	−2.84337e8	−0.226070
$399$	0	0
$400$	−9.83614e8	−0.768448
$401$	−6.72554e8	−0.520861	−0.260431	−	0.965493i	$-0.583865\pi$
$401$	−6.72554e8	−0.520861	−0.260431	+	0.965493i	$0.583865\pi$
$402$	0	0
$403$	1.04906e9	0.798425
$404$	−6.91915e8	−0.522057
$405$	0	0
$406$	0	0
$407$	1.08060e8	0.0794483
$408$	0	0
$409$	−2.29369e9	−1.65769	−0.828846	−	0.559477i	$-0.811002\pi$
$409$	−2.29369e9	−1.65769	−0.828846	+	0.559477i	$0.811002\pi$
$410$	−505823.	−0.000362456 0
$411$	0	0
$412$	−1.94883e8	−0.137289
$413$	0	0
$414$	0	0
$415$	−4.54308e6	−0.00312020
$416$	1.07384e9	0.731327
$417$	0	0
$418$	−3.27606e7	−0.0219399
$419$	8.65006e8	0.574473	0.287237	−	0.957860i	$-0.407263\pi$
$419$	8.65006e8	0.574473	0.287237	+	0.957860i	$0.407263\pi$
$420$	0	0
$421$	−3.17458e8	−0.207347	−0.103674	−	0.994611i	$-0.533060\pi$
$421$	−3.17458e8	−0.207347	−0.103674	+	0.994611i	$0.533060\pi$
$422$	−3.72099e8	−0.241027
$423$	0	0
$424$	−5.20210e8	−0.331435
$425$	−2.74889e9	−1.73698
$426$	0	0
$427$	0	0
$428$	1.09530e8	0.0675274
$429$	0	0
$430$	1.13116e6	0.000686094 0
$431$	1.33835e9	0.805193	0.402596	−	0.915378i	$-0.368108\pi$
$431$	1.33835e9	0.805193	0.402596	+	0.915378i	$0.368108\pi$
$432$	0	0
$433$	1.76343e9	1.04388	0.521939	−	0.852983i	$-0.325209\pi$
$433$	1.76343e9	1.04388	0.521939	+	0.852983i	$0.325209\pi$
$434$	0	0
$435$	0	0
$436$	1.09024e9	0.629970
$437$	−6.75019e8	−0.386929
$438$	0	0
$439$	2.91652e8	0.164528	0.0822640	−	0.996611i	$-0.473785\pi$
$439$	2.91652e8	0.164528	0.0822640	+	0.996611i	$0.473785\pi$
$440$	−553309.	−0.000309659 0
$441$	0	0
$442$	8.57567e8	0.472379
$443$	−1.68942e9	−0.923263	−0.461631	−	0.887072i	$-0.652736\pi$
$443$	−1.68942e9	−0.923263	−0.461631	+	0.887072i	$0.652736\pi$
$444$	0	0
$445$	−5.17179e6	−0.00278215
$446$	−2.61866e8	−0.139768
$447$	0	0
$448$	0	0
$449$	−2.63133e9	−1.37187	−0.685936	−	0.727662i	$-0.740607\pi$
$449$	−2.63133e9	−1.37187	−0.685936	+	0.727662i	$0.740607\pi$
$450$	0	0
$451$	−3.23398e8	−0.166004
$452$	1.37207e9	0.698864
$453$	0	0
$454$	−6.71710e7	−0.0336889
$455$	0	0
$456$	0	0
$457$	−6.09579e8	−0.298761	−0.149380	−	0.988780i	$-0.547728\pi$
$457$	−6.09579e8	−0.298761	−0.149380	+	0.988780i	$0.547728\pi$
$458$	−1.10420e9	−0.537053
$459$	0	0
$460$	−5.46508e6	−0.00261784
$461$	−1.07374e9	−0.510440	−0.255220	−	0.966883i	$-0.582148\pi$
$461$	−1.07374e9	−0.510440	−0.255220	+	0.966883i	$0.582148\pi$
$462$	0	0
$463$	−3.10301e9	−1.45295	−0.726474	−	0.687194i	$-0.758842\pi$
$463$	−3.10301e9	−1.45295	−0.726474	+	0.687194i	$0.758842\pi$
$464$	2.51504e9	1.16878
$465$	0	0
$466$	−1.05994e8	−0.0485211
$467$	2.12992e9	0.967728	0.483864	−	0.875143i	$-0.339233\pi$
$467$	2.12992e9	0.967728	0.483864	+	0.875143i	$0.339233\pi$
$468$	0	0
$469$	0	0
$470$	−722176.	−0.000320849 0
$471$	0	0
$472$	−1.80746e9	−0.791173
$473$	7.23205e8	0.314230
$474$	0	0
$475$	−6.69797e8	−0.286758
$476$	0	0
$477$	0	0
$478$	−5.55598e8	−0.232682
$479$	−2.57458e9	−1.07037	−0.535183	−	0.844736i	$-0.679757\pi$
$479$	−2.57458e9	−1.07037	−0.535183	+	0.844736i	$0.679757\pi$
$480$	0	0
$481$	−6.89236e8	−0.282397
$482$	−9.08572e8	−0.369568
$483$	0	0
$484$	2.12703e9	0.852738
$485$	2.60531e6	0.00103696
$486$	0	0
$487$	3.41020e9	1.33792	0.668958	−	0.743301i	$-0.266741\pi$
$487$	3.41020e9	1.33792	0.668958	+	0.743301i	$0.266741\pi$
$488$	−7.04755e8	−0.274517
$489$	0	0
$490$	0	0
$491$	−1.34263e9	−0.511884	−0.255942	−	0.966692i	$-0.582386\pi$
$491$	−1.34263e9	−0.511884	−0.255942	+	0.966692i	$0.582386\pi$
$492$	0	0
$493$	7.02874e9	2.64188
$494$	2.08956e8	0.0779849
$495$	0	0
$496$	−1.72631e9	−0.635234
$497$	0	0
$498$	0	0
$499$	−4.31704e7	−0.0155537	−0.00777686	−	0.999970i	$-0.502475\pi$
$499$	−4.31704e7	−0.0155537	−0.00777686	+	0.999970i	$0.502475\pi$
$500$	−1.08456e7	−0.00388025
$501$	0	0
$502$	−1.10522e9	−0.389929
$503$	−3.15009e9	−1.10366	−0.551829	−	0.833957i	$-0.686070\pi$
$503$	−3.15009e9	−1.10366	−0.551829	+	0.833957i	$0.686070\pi$
$504$	0	0
$505$	−3.45789e6	−0.00119479
$506$	3.00855e8	0.103236
$507$	0	0
$508$	−5.15232e7	−0.0174373
$509$	−2.29694e9	−0.772037	−0.386019	−	0.922491i	$-0.626150\pi$
$509$	−2.29694e9	−0.772037	−0.386019	+	0.922491i	$0.626150\pi$
$510$	0	0
$511$	0	0
$512$	−3.02920e9	−0.997432
$513$	0	0
$514$	−1.26524e8	−0.0410961
$515$	−973942.	−0.000314201 0
$516$	0	0
$517$	−4.61723e8	−0.146948
$518$	0	0
$519$	0	0
$520$	3.52916e6	0.00110068
$521$	3.93303e9	1.21841	0.609207	−	0.793011i	$-0.291488\pi$
$521$	3.93303e9	1.21841	0.609207	+	0.793011i	$0.291488\pi$
$522$	0	0
$523$	5.91976e9	1.80946	0.904729	−	0.425988i	$-0.140073\pi$
$523$	5.91976e9	1.80946	0.904729	+	0.425988i	$0.140073\pi$
$524$	−3.43424e9	−1.04273
$525$	0	0
$526$	1.26776e9	0.379828
$527$	−4.82449e9	−1.43587
$528$	0	0
$529$	2.79417e9	0.820651
$530$	−1.24624e6	−0.000363610 0
$531$	0	0
$532$	0	0
$533$	2.06272e9	0.590058
$534$	0	0
$535$	547383.	0.000154544 0
$536$	1.86072e7	0.00521920
$537$	0	0
$538$	−1.25701e9	−0.348018
$539$	0	0
$540$	0	0
$541$	−2.06626e9	−0.561041	−0.280520	−	0.959848i	$-0.590507\pi$
$541$	−2.06626e9	−0.561041	−0.280520	+	0.959848i	$0.590507\pi$
$542$	1.02917e9	0.277645
$543$	0	0
$544$	−4.93842e9	−1.31520
$545$	5.44854e6	0.00144176
$546$	0	0
$547$	−1.44090e9	−0.376425	−0.188213	−	0.982128i	$-0.560269\pi$
$547$	−1.44090e9	−0.376425	−0.188213	+	0.982128i	$0.560269\pi$
$548$	−2.49904e9	−0.648695
$549$	0	0
$550$	2.98528e8	0.0765095
$551$	1.71263e9	0.436148
$552$	0	0
$553$	0	0
$554$	4.99868e8	0.124903
$555$	0	0
$556$	−1.67774e9	−0.413964
$557$	5.51340e9	1.35184	0.675922	−	0.736973i	$-0.263746\pi$
$557$	5.51340e9	1.35184	0.675922	+	0.736973i	$0.263746\pi$
$558$	0	0
$559$	−4.61280e9	−1.11692
$560$	0	0
$561$	0	0
$562$	1.58307e9	0.376204
$563$	6.98195e8	0.164891	0.0824456	−	0.996596i	$-0.473727\pi$
$563$	6.98195e8	0.164891	0.0824456	+	0.996596i	$0.473727\pi$
$564$	0	0
$565$	6.85702e6	0.00159943
$566$	9.03244e8	0.209386
$567$	0	0
$568$	−1.97795e9	−0.452895
$569$	−2.85190e9	−0.648994	−0.324497	−	0.945887i	$-0.605195\pi$
$569$	−2.85190e9	−0.648994	−0.324497	+	0.945887i	$0.605195\pi$
$570$	0	0
$571$	−3.99872e9	−0.898865	−0.449433	−	0.893314i	$-0.648374\pi$
$571$	−3.99872e9	−0.898865	−0.449433	+	0.893314i	$0.648374\pi$
$572$	1.08162e9	0.241650
$573$	0	0
$574$	0	0
$575$	6.15104e9	1.34931
$576$	0	0
$577$	−3.69278e9	−0.800272	−0.400136	−	0.916456i	$-0.631037\pi$
$577$	−3.69278e9	−0.800272	−0.400136	+	0.916456i	$0.631037\pi$
$578$	−2.63668e9	−0.567951
$579$	0	0
$580$	1.38658e7	0.00295084
$581$	0	0
$582$	0	0
$583$	−7.96782e8	−0.166533
$584$	−2.76342e9	−0.574118
$585$	0	0
$586$	−1.19653e9	−0.245631
$587$	1.47103e9	0.300184	0.150092	−	0.988672i	$-0.452043\pi$
$587$	1.47103e9	0.300184	0.150092	+	0.988672i	$0.452043\pi$
$588$	0	0
$589$	−1.17554e9	−0.237047
$590$	−4.33003e6	−0.000867978 0
$591$	0	0
$592$	1.13419e9	0.224677
$593$	−6.79462e9	−1.33805	−0.669027	−	0.743238i	$-0.733289\pi$
$593$	−6.79462e9	−1.33805	−0.669027	+	0.743238i	$0.733289\pi$
$594$	0	0
$595$	0	0
$596$	−4.20010e9	−0.812640
$597$	0	0
$598$	−1.91894e9	−0.366949
$599$	4.16822e9	0.792423	0.396211	−	0.918159i	$-0.370325\pi$
$599$	4.16822e9	0.792423	0.396211	+	0.918159i	$0.370325\pi$
$600$	0	0
$601$	3.48946e9	0.655689	0.327845	−	0.944732i	$-0.393678\pi$
$601$	3.48946e9	0.655689	0.327845	+	0.944732i	$0.393678\pi$
$602$	0	0
$603$	0	0
$604$	−1.53318e9	−0.283116
$605$	1.06300e7	0.00195159
$606$	0	0
$607$	−4.58536e9	−0.832172	−0.416086	−	0.909325i	$-0.636599\pi$
$607$	−4.58536e9	−0.832172	−0.416086	+	0.909325i	$0.636599\pi$
$608$	−1.20330e9	−0.217126
$609$	0	0
$610$	−1.68834e6	−0.000301166 0
$611$	2.94499e9	0.522324
$612$	0	0
$613$	3.07599e9	0.539353	0.269676	−	0.962951i	$-0.413083\pi$
$613$	3.07599e9	0.539353	0.269676	+	0.962951i	$0.413083\pi$
$614$	−7.46819e8	−0.130205
$615$	0	0
$616$	0	0
$617$	−2.75456e9	−0.472123	−0.236061	−	0.971738i	$-0.575857\pi$
$617$	−2.75456e9	−0.472123	−0.236061	+	0.971738i	$0.575857\pi$
$618$	0	0
$619$	4.41678e9	0.748494	0.374247	−	0.927329i	$-0.377901\pi$
$619$	4.41678e9	0.748494	0.374247	+	0.927329i	$0.377901\pi$
$620$	−9.51741e6	−0.00160379
$621$	0	0
$622$	−1.47252e9	−0.245354
$623$	0	0
$624$	0	0
$625$	6.10343e9	0.999987
$626$	−1.52019e9	−0.247678
$627$	0	0
$628$	7.79859e9	1.25648
$629$	3.16970e9	0.507856
$630$	0	0
$631$	1.17209e10	1.85720	0.928598	−	0.371087i	$-0.121015\pi$
$631$	1.17209e10	1.85720	0.928598	+	0.371087i	$0.121015\pi$
$632$	3.63219e9	0.572347
$633$	0	0
$634$	−1.60648e9	−0.250359
$635$	−257490.	−3.99074e−5 0
$636$	0	0
$637$	0	0
$638$	−7.63318e8	−0.116368
$639$	0	0
$640$	−1.27658e7	−0.00192494
$641$	−8.89508e9	−1.33397	−0.666986	−	0.745070i	$-0.732416\pi$
$641$	−8.89508e9	−1.33397	−0.666986	+	0.745070i	$0.732416\pi$
$642$	0	0
$643$	7.02222e9	1.04168	0.520842	−	0.853653i	$-0.325618\pi$
$643$	7.02222e9	1.04168	0.520842	+	0.853653i	$0.325618\pi$
$644$	0	0
$645$	0	0
$646$	−9.60958e8	−0.140246
$647$	2.24007e9	0.325159	0.162579	−	0.986695i	$-0.448019\pi$
$647$	2.24007e9	0.325159	0.162579	+	0.986695i	$0.448019\pi$
$648$	0	0
$649$	−2.76840e9	−0.397532
$650$	−1.90409e9	−0.271951
$651$	0	0
$652$	−6.53304e9	−0.923101
$653$	4.68732e9	0.658761	0.329381	−	0.944197i	$-0.393160\pi$
$653$	4.68732e9	0.658761	0.329381	+	0.944197i	$0.393160\pi$
$654$	0	0
$655$	−1.71628e7	−0.00238640
$656$	−3.39436e9	−0.469455
$657$	0	0
$658$	0	0
$659$	1.65220e9	0.224886	0.112443	−	0.993658i	$-0.464132\pi$
$659$	1.65220e9	0.224886	0.112443	+	0.993658i	$0.464132\pi$
$660$	0	0
$661$	3.29848e9	0.444231	0.222116	−	0.975020i	$-0.428704\pi$
$661$	3.29848e9	0.444231	0.222116	+	0.975020i	$0.428704\pi$
$662$	3.11700e9	0.417574
$663$	0	0
$664$	6.04100e9	0.800793
$665$	0	0
$666$	0	0
$667$	−1.57279e10	−2.05224
$668$	−1.30403e10	−1.69266
$669$	0	0
$670$	44576.2	5.72587e−6 0
$671$	−1.07944e9	−0.137933
$672$	0	0
$673$	−3.03240e9	−0.383472	−0.191736	−	0.981447i	$-0.561412\pi$
$673$	−3.03240e9	−0.383472	−0.191736	+	0.981447i	$0.561412\pi$
$674$	−2.86466e9	−0.360382
$675$	0	0
$676$	4.96223e8	0.0617822
$677$	5.14349e9	0.637085	0.318542	−	0.947909i	$-0.396807\pi$
$677$	5.14349e9	0.637085	0.318542	+	0.947909i	$0.396807\pi$
$678$	0	0
$679$	0	0
$680$	−1.62301e7	−0.00197943
$681$	0	0
$682$	5.23938e8	0.0632462
$683$	1.35684e9	0.162950	0.0814752	−	0.996675i	$-0.474037\pi$
$683$	1.35684e9	0.162950	0.0814752	+	0.996675i	$0.474037\pi$
$684$	0	0
$685$	−1.24891e7	−0.00148462
$686$	0	0
$687$	0	0
$688$	7.59071e9	0.888634
$689$	5.08209e9	0.591937
$690$	0	0
$691$	−7.62426e9	−0.879072	−0.439536	−	0.898225i	$-0.644857\pi$
$691$	−7.62426e9	−0.879072	−0.439536	+	0.898225i	$0.644857\pi$
$692$	−2.13817e9	−0.245286
$693$	0	0
$694$	7.61909e8	0.0865257
$695$	−8.38460e6	−0.000947405 0
$696$	0	0
$697$	−9.48614e9	−1.06115
$698$	−3.12037e9	−0.347306
$699$	0	0
$700$	0	0
$701$	2.55816e9	0.280489	0.140244	−	0.990117i	$-0.455211\pi$
$701$	2.55816e9	0.280489	0.140244	+	0.990117i	$0.455211\pi$
$702$	0	0
$703$	7.72332e8	0.0838418
$704$	−1.39682e9	−0.150882
$705$	0	0
$706$	−4.59062e7	−0.00490969
$707$	0	0
$708$	0	0
$709$	−4.62364e9	−0.487216	−0.243608	−	0.969874i	$-0.578331\pi$
$709$	−4.62364e9	−0.487216	−0.243608	+	0.969874i	$0.578331\pi$
$710$	−4.73848e6	−0.000496861 0
$711$	0	0
$712$	6.87701e9	0.714035
$713$	1.07955e10	1.11540
$714$	0	0
$715$	5.40544e6	0.000553045 0
$716$	6.73072e9	0.685277
$717$	0	0
$718$	−3.53863e7	−0.00356779
$719$	1.07594e10	1.07954	0.539770	−	0.841813i	$-0.318511\pi$
$719$	1.07594e10	1.07954	0.539770	+	0.841813i	$0.318511\pi$
$720$	0	0
$721$	0	0
$722$	2.61330e9	0.258410
$723$	0	0
$724$	−1.17307e10	−1.14878
$725$	−1.56062e10	−1.52095
$726$	0	0
$727$	2.05793e9	0.198637	0.0993183	−	0.995056i	$-0.468334\pi$
$727$	2.05793e9	0.198637	0.0993183	+	0.995056i	$0.468334\pi$
$728$	0	0
$729$	0	0
$730$	−6.62016e6	−0.000629852 0
$731$	2.12136e10	2.00865
$732$	0	0
$733$	2.81644e9	0.264141	0.132071	−	0.991240i	$-0.457837\pi$
$733$	2.81644e9	0.264141	0.132071	+	0.991240i	$0.457837\pi$
$734$	1.95870e9	0.182823
$735$	0	0
$736$	1.10505e10	1.02166
$737$	2.84998e7	0.00262244
$738$	0	0
$739$	−4.65737e9	−0.424507	−0.212254	−	0.977215i	$-0.568080\pi$
$739$	−4.65737e9	−0.424507	−0.212254	+	0.977215i	$0.568080\pi$
$740$	6.25294e6	0.000567249 0
$741$	0	0
$742$	0	0
$743$	1.44668e10	1.29393	0.646964	−	0.762520i	$-0.276038\pi$
$743$	1.44668e10	1.29393	0.646964	+	0.762520i	$0.276038\pi$
$744$	0	0
$745$	−2.09903e7	−0.00185982
$746$	−1.28320e9	−0.113164
$747$	0	0
$748$	−4.97420e9	−0.434578
$749$	0	0
$750$	0	0
$751$	1.37575e10	1.18522	0.592611	−	0.805489i	$-0.298097\pi$
$751$	1.37575e10	1.18522	0.592611	+	0.805489i	$0.298097\pi$
$752$	−4.84621e9	−0.415565
$753$	0	0
$754$	4.86865e9	0.413627
$755$	−7.66217e6	−0.000647944 0
$756$	0	0
$757$	−4.61358e9	−0.386547	−0.193273	−	0.981145i	$-0.561910\pi$
$757$	−4.61358e9	−0.386547	−0.193273	+	0.981145i	$0.561910\pi$
$758$	−1.89199e8	−0.0157789
$759$	0	0
$760$	−3.95464e6	−0.000326783 0
$761$	2.19554e10	1.80590	0.902951	−	0.429742i	$-0.141396\pi$
$761$	2.19554e10	1.80590	0.902951	+	0.429742i	$0.141396\pi$
$762$	0	0
$763$	0	0
$764$	−5.43638e8	−0.0441045
$765$	0	0
$766$	−1.14807e9	−0.0922924
$767$	1.76576e10	1.41302
$768$	0	0
$769$	−1.61256e10	−1.27871	−0.639357	−	0.768910i	$-0.720799\pi$
$769$	−1.61256e10	−1.27871	−0.639357	+	0.768910i	$0.720799\pi$
$770$	0	0
$771$	0	0
$772$	9.26939e9	0.725088
$773$	3.18124e9	0.247724	0.123862	−	0.992299i	$-0.460472\pi$
$773$	3.18124e9	0.247724	0.123862	+	0.992299i	$0.460472\pi$
$774$	0	0
$775$	1.07120e10	0.826638
$776$	−3.46432e9	−0.266135
$777$	0	0
$778$	−5.42150e9	−0.412754
$779$	−2.31141e9	−0.175184
$780$	0	0
$781$	−3.02954e9	−0.227561
$782$	8.82490e9	0.659913
$783$	0	0
$784$	0	0
$785$	3.89739e7	0.00287561
$786$	0	0
$787$	7.54980e9	0.552108	0.276054	−	0.961142i	$-0.410973\pi$
$787$	7.54980e9	0.552108	0.276054	+	0.961142i	$0.410973\pi$
$788$	−1.61757e10	−1.17766
$789$	0	0
$790$	8.70145e6	0.000627909 0
$791$	0	0
$792$	0	0
$793$	6.88496e9	0.490281
$794$	4.14203e9	0.293658
$795$	0	0
$796$	−1.05194e10	−0.739258
$797$	2.52925e10	1.76965	0.884824	−	0.465926i	$-0.154279\pi$
$797$	2.52925e10	1.76965	0.884824	+	0.465926i	$0.154279\pi$
$798$	0	0
$799$	−1.35436e10	−0.939334
$800$	1.09650e10	0.757169
$801$	0	0
$802$	2.14244e9	0.146655
$803$	−4.23260e9	−0.288471
$804$	0	0
$805$	0	0
$806$	−3.34182e9	−0.224807
$807$	0	0
$808$	4.59800e9	0.306641
$809$	−1.48660e10	−0.987128	−0.493564	−	0.869709i	$-0.664306\pi$
$809$	−1.48660e10	−0.987128	−0.493564	+	0.869709i	$0.664306\pi$
$810$	0	0
$811$	2.03817e10	1.34174	0.670868	−	0.741577i	$-0.265922\pi$
$811$	2.03817e10	1.34174	0.670868	+	0.741577i	$0.265922\pi$
$812$	0	0
$813$	0	0
$814$	−3.44227e8	−0.0223697
$815$	−3.26493e7	−0.00211262
$816$	0	0
$817$	5.16894e9	0.331607
$818$	7.30661e9	0.466745
$819$	0	0
$820$	−1.87136e7	−0.00118524
$821$	3.80638e9	0.240055	0.120028	−	0.992771i	$-0.461702\pi$
$821$	3.80638e9	0.240055	0.120028	+	0.992771i	$0.461702\pi$
$822$	0	0
$823$	8.49039e9	0.530919	0.265459	−	0.964122i	$-0.414476\pi$
$823$	8.49039e9	0.530919	0.265459	+	0.964122i	$0.414476\pi$
$824$	1.29506e9	0.0806392
$825$	0	0
$826$	0	0
$827$	−2.17143e10	−1.33499	−0.667494	−	0.744615i	$-0.732633\pi$
$827$	−2.17143e10	−1.33499	−0.667494	+	0.744615i	$0.732633\pi$
$828$	0	0
$829$	−2.98362e9	−0.181888	−0.0909438	−	0.995856i	$-0.528988\pi$
$829$	−2.98362e9	−0.181888	−0.0909438	+	0.995856i	$0.528988\pi$
$830$	1.44721e7	0.000878532 0
$831$	0	0
$832$	8.90933e9	0.536307
$833$	0	0
$834$	0	0
$835$	−6.51696e7	−0.00387385
$836$	−1.21202e9	−0.0717443
$837$	0	0
$838$	−2.75549e9	−0.161750
$839$	2.09807e10	1.22646	0.613230	−	0.789905i	$-0.289870\pi$
$839$	2.09807e10	1.22646	0.613230	+	0.789905i	$0.289870\pi$
$840$	0	0
$841$	2.26542e10	1.31330
$842$	1.01127e9	0.0583814
$843$	0	0
$844$	−1.37663e10	−0.788167
$845$	2.47991e6	0.000141396 0
$846$	0	0
$847$	0	0
$848$	−8.36296e9	−0.470950
$849$	0	0
$850$	8.75664e9	0.489070
$851$	−7.09267e9	−0.394508
$852$	0	0
$853$	2.23064e10	1.23058	0.615288	−	0.788303i	$-0.289040\pi$
$853$	2.23064e10	1.23058	0.615288	+	0.788303i	$0.289040\pi$
$854$	0	0
$855$	0	0
$856$	−7.27864e8	−0.0396636
$857$	−1.71020e10	−0.928141	−0.464070	−	0.885798i	$-0.653612\pi$
$857$	−1.71020e10	−0.928141	−0.464070	+	0.885798i	$0.653612\pi$
$858$	0	0
$859$	2.26098e10	1.21708	0.608541	−	0.793522i	$-0.291755\pi$
$859$	2.26098e10	1.21708	0.608541	+	0.793522i	$0.291755\pi$
$860$	4.18486e7	0.00224356
$861$	0	0
$862$	−4.26335e9	−0.226713
$863$	−1.37995e10	−0.730847	−0.365423	−	0.930841i	$-0.619076\pi$
$863$	−1.37995e10	−0.730847	−0.365423	+	0.930841i	$0.619076\pi$
$864$	0	0
$865$	−1.06857e7	−0.000561365 0
$866$	−5.61743e9	−0.293917
$867$	0	0
$868$	0	0
$869$	5.56326e9	0.287581
$870$	0	0
$871$	−1.81779e8	−0.00932139
$872$	−7.24501e9	−0.370025
$873$	0	0
$874$	2.15029e9	0.108945
$875$	0	0
$876$	0	0
$877$	−2.15386e10	−1.07825	−0.539125	−	0.842226i	$-0.681245\pi$
$877$	−2.15386e10	−1.07825	−0.539125	+	0.842226i	$0.681245\pi$
$878$	−9.29066e8	−0.0463250
$879$	0	0
$880$	−8.89506e6	−0.000440007 0
$881$	−9.50136e8	−0.0468134	−0.0234067	−	0.999726i	$-0.507451\pi$
$881$	−9.50136e8	−0.0468134	−0.0234067	+	0.999726i	$0.507451\pi$
$882$	0	0
$883$	1.42541e10	0.696751	0.348376	−	0.937355i	$-0.386733\pi$
$883$	1.42541e10	0.696751	0.348376	+	0.937355i	$0.386733\pi$
$884$	3.17268e10	1.54470
$885$	0	0
$886$	5.38170e9	0.259957
$887$	−1.03291e10	−0.496971	−0.248486	−	0.968636i	$-0.579933\pi$
$887$	−1.03291e10	−0.496971	−0.248486	+	0.968636i	$0.579933\pi$
$888$	0	0
$889$	0	0
$890$	1.64749e7	0.000783352 0
$891$	0	0
$892$	−9.68805e9	−0.457046
$893$	−3.30005e9	−0.155075
$894$	0	0
$895$	3.36372e7	0.00156834
$896$	0	0
$897$	0	0
$898$	8.38217e9	0.386268
$899$	−2.73900e10	−1.25728
$900$	0	0
$901$	−2.33718e10	−1.06452
$902$	1.03019e9	0.0467406
$903$	0	0
$904$	−9.11788e9	−0.410492
$905$	−5.86248e7	−0.00262913
$906$	0	0
$907$	4.79520e9	0.213394	0.106697	−	0.994292i	$-0.465973\pi$
$907$	4.79520e9	0.213394	0.106697	+	0.994292i	$0.465973\pi$
$908$	−2.48508e9	−0.110164
$909$	0	0
$910$	0	0
$911$	2.81747e10	1.23465	0.617326	−	0.786707i	$-0.288216\pi$
$911$	2.81747e10	1.23465	0.617326	+	0.786707i	$0.288216\pi$
$912$	0	0
$913$	9.25271e9	0.402366
$914$	1.94183e9	0.0841200
$915$	0	0
$916$	−4.08511e10	−1.75618
$917$	0	0
$918$	0	0
$919$	3.60009e10	1.53006	0.765032	−	0.643992i	$-0.222723\pi$
$919$	3.60009e10	1.53006	0.765032	+	0.643992i	$0.222723\pi$
$920$	3.63172e7	0.00153764
$921$	0	0
$922$	3.42041e9	0.143721
$923$	1.93232e10	0.808861
$924$	0	0
$925$	−7.03780e9	−0.292376
$926$	9.88472e9	0.409097
$927$	0	0
$928$	−2.80368e10	−1.15162
$929$	2.25328e10	0.922063	0.461032	−	0.887384i	$-0.347479\pi$
$929$	2.25328e10	0.922063	0.461032	+	0.887384i	$0.347479\pi$
$930$	0	0
$931$	0	0
$932$	−3.92138e9	−0.158666
$933$	0	0
$934$	−6.78490e9	−0.272477
$935$	−2.48589e7	−0.000994582 0
$936$	0	0
$937$	−6.17708e9	−0.245298	−0.122649	−	0.992450i	$-0.539139\pi$
$937$	−6.17708e9	−0.245298	−0.122649	+	0.992450i	$0.539139\pi$
$938$	0	0
$939$	0	0
$940$	−2.67178e7	−0.00104919
$941$	3.76918e10	1.47463	0.737316	−	0.675548i	$-0.236093\pi$
$941$	3.76918e10	1.47463	0.737316	+	0.675548i	$0.236093\pi$
$942$	0	0
$943$	2.12267e10	0.824311
$944$	−2.90569e10	−1.12421
$945$	0	0
$946$	−2.30379e9	−0.0884756
$947$	−3.76562e10	−1.44082	−0.720412	−	0.693546i	$-0.756047\pi$
$947$	−3.76562e10	−1.44082	−0.720412	+	0.693546i	$0.756047\pi$
$948$	0	0
$949$	2.69966e10	1.02536
$950$	2.13365e9	0.0807405
$951$	0	0
$952$	0	0
$953$	5.18903e10	1.94205	0.971027	−	0.238971i	$-0.0768100\pi$
$953$	5.18903e10	1.94205	0.971027	+	0.238971i	$0.0768100\pi$
$954$	0	0
$955$	−2.71686e6	−0.000100938 0
$956$	−2.05550e10	−0.760879
$957$	0	0
$958$	8.20138e9	0.301375
$959$	0	0
$960$	0	0
$961$	−8.71224e9	−0.316664
$962$	2.19558e9	0.0795126
$963$	0	0
$964$	−3.36138e10	−1.20850
$965$	4.63243e7	0.00165945
$966$	0	0
$967$	2.54614e10	0.905501	0.452751	−	0.891637i	$-0.350443\pi$
$967$	2.54614e10	0.905501	0.452751	+	0.891637i	$0.350443\pi$
$968$	−1.41348e10	−0.500872
$969$	0	0
$970$	−8.29929e6	−0.000291971 0
$971$	−5.20386e10	−1.82414	−0.912070	−	0.410035i	$-0.865516\pi$
$971$	−5.20386e10	−1.82414	−0.912070	+	0.410035i	$0.865516\pi$
$972$	0	0
$973$	0	0
$974$	−1.08633e10	−0.376708
$975$	0	0
$976$	−1.13297e10	−0.390072
$977$	2.88776e10	0.990673	0.495336	−	0.868701i	$-0.335045\pi$
$977$	2.88776e10	0.990673	0.495336	+	0.868701i	$0.335045\pi$
$978$	0	0
$979$	1.05332e10	0.358774
$980$	0	0
$981$	0	0
$982$	4.27698e9	0.144128
$983$	4.42466e10	1.48574	0.742870	−	0.669435i	$-0.233464\pi$
$983$	4.42466e10	1.48574	0.742870	+	0.669435i	$0.233464\pi$
$984$	0	0
$985$	−8.08388e7	−0.00269521
$986$	−2.23902e10	−0.743856
$987$	0	0
$988$	7.73059e9	0.255014
$989$	−4.74686e10	−1.56034
$990$	0	0
$991$	−4.38355e10	−1.43077	−0.715383	−	0.698732i	$-0.753748\pi$
$991$	−4.38355e10	−1.43077	−0.715383	+	0.698732i	$0.753748\pi$
$992$	1.92443e10	0.625910
$993$	0	0
$994$	0	0
$995$	−5.25714e7	−0.00169188
$996$	0	0
$997$	−6.03353e9	−0.192814	−0.0964070	−	0.995342i	$-0.530735\pi$
$997$	−6.03353e9	−0.192814	−0.0964070	+	0.995342i	$0.530735\pi$
$998$	1.37520e8	0.00437935
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.8.a.t.1.2		4
3.2	odd	2		49.8.a.e.1.3		4
7.3	odd	6		63.8.e.b.37.3		8
7.5	odd	6		63.8.e.b.46.3		8
7.6	odd	2		441.8.a.s.1.2		4
21.2	odd	6		49.8.c.g.18.2		8
21.5	even	6		7.8.c.a.4.2	yes	8
21.11	odd	6		49.8.c.g.30.2		8
21.17	even	6		7.8.c.a.2.2	✓	8
21.20	even	2		49.8.a.f.1.3		4
84.47	odd	6		112.8.i.c.81.2		8
84.59	odd	6		112.8.i.c.65.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.8.c.a.2.2	✓	8	21.17	even	6
7.8.c.a.4.2	yes	8	21.5	even	6
49.8.a.e.1.3		4	3.2	odd	2
49.8.a.f.1.3		4	21.20	even	2
49.8.c.g.18.2		8	21.2	odd	6
49.8.c.g.30.2		8	21.11	odd	6
63.8.e.b.37.3		8	7.3	odd	6
63.8.e.b.46.3		8	7.5	odd	6
112.8.i.c.65.2		8	84.59	odd	6
112.8.i.c.81.2		8	84.47	odd	6
441.8.a.s.1.2		4	7.6	odd	2
441.8.a.t.1.2		4	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q-3.18552 q^{2} -117.852 q^{4} -0.588975 q^{5} +783.169 q^{8} +O(q^{10})\)	\(q-3.18552 q^{2} -117.852 q^{4} -0.588975 q^{5} +783.169 q^{8} +1.87619 q^{10} +1199.54 q^{11} -7651.01 q^{13} +12590.3 q^{16} +35185.9 q^{17} +8573.44 q^{19} +69.4121 q^{20} -3821.17 q^{22} -78733.7 q^{23} -78124.7 q^{25} +24372.5 q^{26} +199760. q^{29} -137114. q^{31} -140352. q^{32} -112085. q^{34} +90084.3 q^{37} -27310.9 q^{38} -461.267 q^{40} -269601. q^{41} +602901. q^{43} -141369. q^{44} +250808. q^{46} -384915. q^{47} +248868. q^{50} +901690. q^{52} -664238. q^{53} -706.501 q^{55} -636341. q^{58} -2.30788e6 q^{59} -899876. q^{61} +436781. q^{62} -1.16446e6 q^{64} +4506.25 q^{65} +23758.8 q^{67} -4.14674e6 q^{68} -2.52558e6 q^{71} -3.52851e6 q^{73} -286966. q^{74} -1.01040e6 q^{76} +4.63782e6 q^{79} -7415.38 q^{80} +858819. q^{82} +7.71353e6 q^{83} -20723.6 q^{85} -1.92055e6 q^{86} +939444. q^{88} +8.78100e6 q^{89} +9.27896e6 q^{92} +1.22616e6 q^{94} -5049.54 q^{95} -4.42347e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8}+O(q^{10}) \) 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8	\( 4 q + 6 q^{2} + 348 q^{4} + 252 q^{5} + 984 q^{8} - 4774 q^{10} + 3972 q^{11} + 1176 q^{13} + 57264 q^{16} + 56364 q^{17} - 41748 q^{19} + 162372 q^{20} - 152954 q^{22} - 131748 q^{23} - 68016 q^{25} + 113652 q^{26} - 34056 q^{29} - 401212 q^{31} + 453408 q^{32} - 82110 q^{34} - 5396 q^{37} + 31794 q^{38} + 443688 q^{40} + 410424 q^{41} + 46544 q^{43} + 1465836 q^{44} + 1379202 q^{46} + 1470084 q^{47} - 1395528 q^{50} + 5269768 q^{52} + 642372 q^{53} + 3063340 q^{55} - 1972220 q^{58} + 752220 q^{59} - 1325772 q^{61} - 3016314 q^{62} - 865856 q^{64} + 4868808 q^{65} - 290916 q^{67} + 2453556 q^{68} - 3377760 q^{71} - 6706588 q^{73} - 3616878 q^{74} - 2923004 q^{76} + 3946244 q^{79} + 10695888 q^{80} + 17563252 q^{82} + 9542064 q^{83} + 7006068 q^{85} - 23237832 q^{86} - 8043336 q^{88} + 16165212 q^{89} + 19442628 q^{92} - 5720442 q^{94} + 3268452 q^{95} - 1533112 q^{97}+O(q^{100}) \) 4 * q + 6 * q^2 + 348 * q^4 + 252 * q^5 + 984 * q^8 - 4774 * q^10 + 3972 * q^11 + 1176 * q^13 + 57264 * q^16 + 56364 * q^17 - 41748 * q^19 + 162372 * q^20 - 152954 * q^22 - 131748 * q^23 - 68016 * q^25 + 113652 * q^26 - 34056 * q^29 - 401212 * q^31 + 453408 * q^32 - 82110 * q^34 - 5396 * q^37 + 31794 * q^38 + 443688 * q^40 + 410424 * q^41 + 46544 * q^43 + 1465836 * q^44 + 1379202 * q^46 + 1470084 * q^47 - 1395528 * q^50 + 5269768 * q^52 + 642372 * q^53 + 3063340 * q^55 - 1972220 * q^58 + 752220 * q^59 - 1325772 * q^61 - 3016314 * q^62 - 865856 * q^64 + 4868808 * q^65 - 290916 * q^67 + 2453556 * q^68 - 3377760 * q^71 - 6706588 * q^73 - 3616878 * q^74 - 2923004 * q^76 + 3946244 * q^79 + 10695888 * q^80 + 17563252 * q^82 + 9542064 * q^83 + 7006068 * q^85 - 23237832 * q^86 - 8043336 * q^88 + 16165212 * q^89 + 19442628 * q^92 - 5720442 * q^94 + 3268452 * q^95 - 1533112 * q^97

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