LMFDB - Embedded newform 16.22.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,22,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	16.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:	$44.7163750859$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2161}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 540 $ x^2 - x - 540
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8}\cdot 3\cdot 7 $
Twist minimal:	no (minimal twist has level 4)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$23.7433$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-157776. q^{3} +3.23357e7 q^{5} +8.65744e8 q^{7} +1.44329e10 q^{9} +O(q^{10})$	$q-157776. q^{3} +3.23357e7 q^{5} +8.65744e8 q^{7} +1.44329e10 q^{9} -3.34262e10 q^{11} +6.73870e11 q^{13} -5.10179e12 q^{15} -9.68320e12 q^{17} -2.47791e13 q^{19} -1.36594e14 q^{21} +2.38279e14 q^{23} +5.68758e14 q^{25} -6.26767e14 q^{27} +1.28080e15 q^{29} +5.18258e14 q^{31} +5.27385e15 q^{33} +2.79944e16 q^{35} -4.73720e16 q^{37} -1.06320e17 q^{39} +5.43523e16 q^{41} +1.52026e17 q^{43} +4.66696e17 q^{45} +4.11617e17 q^{47} +1.90967e17 q^{49} +1.52778e18 q^{51} -5.65716e17 q^{53} -1.08086e18 q^{55} +3.90954e18 q^{57} -1.43860e18 q^{59} +5.28870e17 q^{61} +1.24952e19 q^{63} +2.17900e19 q^{65} +1.27672e19 q^{67} -3.75946e19 q^{69} +1.85275e19 q^{71} +4.09540e19 q^{73} -8.97362e19 q^{75} -2.89386e19 q^{77} -4.81552e19 q^{79} -5.20842e19 q^{81} -2.70155e19 q^{83} -3.13113e20 q^{85} -2.02079e20 q^{87} +3.94268e19 q^{89} +5.83399e20 q^{91} -8.17687e19 q^{93} -8.01248e20 q^{95} +1.01835e21 q^{97} -4.82437e20 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9}+O(q^{10}) $ 2 * q - 65640 * q^3 + 13689324 * q^5 + 260508080 * q^7 + 12461535162 * q^9	$ 2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9} - 145435963320 q^{11} + 1428900417340 q^{13} - 6819782714352 q^{15} + 1840620576420 q^{17} + 16780743928568 q^{19} - 192357511002048 q^{21} + 319691925426960 q^{23} + 439606295919326 q^{25} - 17\!\cdots\!40 q^{27}+ \cdots - 26\!\cdots\!20 q^{99}+O(q^{100}) $ 2 * q - 65640 * q^3 + 13689324 * q^5 + 260508080 * q^7 + 12461535162 * q^9 - 145435963320 * q^11 + 1428900417340 * q^13 - 6819782714352 * q^15 + 1840620576420 * q^17 + 16780743928568 * q^19 - 192357511002048 * q^21 + 319691925426960 * q^23 + 439606295919326 * q^25 - 1772171769223440 * q^27 + 3742111775766492 * q^29 - 112042353462592 * q^31 - 5046260260851360 * q^33 + 39279847462424352 * q^35 - 33362705637547220 * q^37 - 36755051864070192 * q^39 + 175129744323133332 * q^41 - 15346613416528120 * q^43 + 503454557153115324 * q^45 + 684848819288455200 * q^47 - 1267753878311886 * q^49 + 2589533339352286896 * q^51 + 675305394244421580 * q^53 + 1007711578141009200 * q^55 + 7738689471355209120 * q^57 - 1042445250435434904 * q^59 + 9065997829736468764 * q^61 + 13688298514444201200 * q^63 + 7711482582413403048 * q^65 + 30464301046802775320 * q^67 - 30093532090729291584 * q^69 + 8199093502830518064 * q^71 + 25415086659374793940 * q^73 - 101635704862879905048 * q^75 + 38853770260581178560 * q^77 - 121204353225060164896 * q^79 - 136996120065469586382 * q^81 - 108742936757033809800 * q^83 - 527989845644919754344 * q^85 + 24695689277789874000 * q^87 - 184207999274965368972 * q^89 + 126427419539014666912 * q^91 - 139841964206001442560 * q^93 - 1576185911066925315504 * q^95 + 739497785188476467140 * q^97 - 261627767970577922520 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−157776.	−1.54265	−0.771325	−	0.636441i	$-0.780406\pi$
$3$	−157776.	−1.54265	−0.771325	+	0.636441i	$0.780406\pi$
$4$	0	0
$5$	3.23357e7	1.48080	0.740400	−	0.672166i	$-0.234636\pi$
$5$	3.23357e7	1.48080	0.740400	+	0.672166i	$0.234636\pi$
$6$	0	0
$7$	8.65744e8	1.15840	0.579202	−	0.815184i	$-0.303364\pi$
$7$	8.65744e8	1.15840	0.579202	+	0.815184i	$0.303364\pi$
$8$	0	0
$9$	1.44329e10	1.37977
$10$	0	0
$11$	−3.34262e10	−0.388566	−0.194283	−	0.980946i	$-0.562238\pi$
$11$	−3.34262e10	−0.388566	−0.194283	+	0.980946i	$0.562238\pi$
$12$	0	0
$13$	6.73870e11	1.35572	0.677861	−	0.735190i	$-0.262907\pi$
$13$	6.73870e11	1.35572	0.677861	+	0.735190i	$0.262907\pi$
$14$	0	0
$15$	−5.10179e12	−2.28436
$16$	0	0
$17$	−9.68320e12	−1.16494	−0.582472	−	0.812850i	$-0.697915\pi$
$17$	−9.68320e12	−1.16494	−0.582472	+	0.812850i	$0.697915\pi$
$18$	0	0
$19$	−2.47791e13	−0.927197	−0.463599	−	0.886045i	$-0.653442\pi$
$19$	−2.47791e13	−0.927197	−0.463599	+	0.886045i	$0.653442\pi$
$20$	0	0
$21$	−1.36594e14	−1.78701
$22$	0	0
$23$	2.38279e14	1.19934	0.599671	−	0.800247i	$-0.295298\pi$
$23$	2.38279e14	1.19934	0.599671	+	0.800247i	$0.295298\pi$
$24$	0	0
$25$	5.68758e14	1.19277
$26$	0	0
$27$	−6.26767e14	−0.585851
$28$	0	0
$29$	1.28080e15	0.565331	0.282665	−	0.959219i	$-0.408781\pi$
$29$	1.28080e15	0.565331	0.282665	+	0.959219i	$0.408781\pi$
$30$	0	0
$31$	5.18258e14	0.113566	0.0567830	−	0.998387i	$-0.481916\pi$
$31$	5.18258e14	0.113566	0.0567830	+	0.998387i	$0.481916\pi$
$32$	0	0
$33$	5.27385e15	0.599421
$34$	0	0
$35$	2.79944e16	1.71537
$36$	0	0
$37$	−4.73720e16	−1.61959	−0.809793	−	0.586716i	$-0.800420\pi$
$37$	−4.73720e16	−1.61959	−0.809793	+	0.586716i	$0.800420\pi$
$38$	0	0
$39$	−1.06320e17	−2.09141
$40$	0	0
$41$	5.43523e16	0.632393	0.316197	−	0.948694i	$-0.397594\pi$
$41$	5.43523e16	0.632393	0.316197	+	0.948694i	$0.397594\pi$
$42$	0	0
$43$	1.52026e17	1.07275	0.536376	−	0.843979i	$-0.319793\pi$
$43$	1.52026e17	1.07275	0.536376	+	0.843979i	$0.319793\pi$
$44$	0	0
$45$	4.66696e17	2.04316
$46$	0	0
$47$	4.11617e17	1.14147	0.570736	−	0.821133i	$-0.306658\pi$
$47$	4.11617e17	1.14147	0.570736	+	0.821133i	$0.306658\pi$
$48$	0	0
$49$	1.90967e17	0.341901
$50$	0	0
$51$	1.52778e18	1.79710
$52$	0	0
$53$	−5.65716e17	−0.444326	−0.222163	−	0.975010i	$-0.571312\pi$
$53$	−5.65716e17	−0.444326	−0.222163	+	0.975010i	$0.571312\pi$
$54$	0	0
$55$	−1.08086e18	−0.575388
$56$	0	0
$57$	3.90954e18	1.43034
$58$	0	0
$59$	−1.43860e18	−0.366432	−0.183216	−	0.983073i	$-0.558651\pi$
$59$	−1.43860e18	−0.366432	−0.183216	+	0.983073i	$0.558651\pi$
$60$	0	0
$61$	5.28870e17	0.0949262	0.0474631	−	0.998873i	$-0.484886\pi$
$61$	5.28870e17	0.0949262	0.0474631	+	0.998873i	$0.484886\pi$
$62$	0	0
$63$	1.24952e19	1.59833
$64$	0	0
$65$	2.17900e19	2.00756
$66$	0	0
$67$	1.27672e19	0.855679	0.427840	−	0.903855i	$-0.359275\pi$
$67$	1.27672e19	0.855679	0.427840	+	0.903855i	$0.359275\pi$
$68$	0	0
$69$	−3.75946e19	−1.85016
$70$	0	0
$71$	1.85275e19	0.675467	0.337733	−	0.941242i	$-0.390340\pi$
$71$	1.85275e19	0.675467	0.337733	+	0.941242i	$0.390340\pi$
$72$	0	0
$73$	4.09540e19	1.11534	0.557669	−	0.830064i	$-0.311696\pi$
$73$	4.09540e19	1.11534	0.557669	+	0.830064i	$0.311696\pi$
$74$	0	0
$75$	−8.97362e19	−1.84003
$76$	0	0
$77$	−2.89386e19	−0.450116
$78$	0	0
$79$	−4.81552e19	−0.572215	−0.286107	−	0.958198i	$-0.592361\pi$
$79$	−4.81552e19	−0.572215	−0.286107	+	0.958198i	$0.592361\pi$
$80$	0	0
$81$	−5.20842e19	−0.476007
$82$	0	0
$83$	−2.70155e19	−0.191115	−0.0955573	−	0.995424i	$-0.530463\pi$
$83$	−2.70155e19	−0.191115	−0.0955573	+	0.995424i	$0.530463\pi$
$84$	0	0
$85$	−3.13113e20	−1.72505
$86$	0	0
$87$	−2.02079e20	−0.872107
$88$	0	0
$89$	3.94268e19	0.134028	0.0670142	−	0.997752i	$-0.478653\pi$
$89$	3.94268e19	0.134028	0.0670142	+	0.997752i	$0.478653\pi$
$90$	0	0
$91$	5.83399e20	1.57048
$92$	0	0
$93$	−8.17687e19	−0.175193
$94$	0	0
$95$	−8.01248e20	−1.37299
$96$	0	0
$97$	1.01835e21	1.40214	0.701072	−	0.713090i	$-0.252705\pi$
$97$	1.01835e21	1.40214	0.701072	+	0.713090i	$0.252705\pi$
$98$	0	0
$99$	−4.82437e20	−0.536131
$100$	0	0
$101$	−4.35147e19	−0.0391978	−0.0195989	−	0.999808i	$-0.506239\pi$
$101$	−4.35147e19	−0.0391978	−0.0195989	+	0.999808i	$0.506239\pi$
$102$	0	0
$103$	1.05797e21	0.775683	0.387842	−	0.921726i	$-0.373221\pi$
$103$	1.05797e21	0.775683	0.387842	+	0.921726i	$0.373221\pi$
$104$	0	0
$105$	−4.41684e21	−2.64621
$106$	0	0
$107$	−2.63468e21	−1.29479	−0.647394	−	0.762155i	$-0.724141\pi$
$107$	−2.63468e21	−1.29479	−0.647394	+	0.762155i	$0.724141\pi$
$108$	0	0
$109$	3.69159e21	1.49360	0.746802	−	0.665046i	$-0.231588\pi$
$109$	3.69159e21	1.49360	0.746802	+	0.665046i	$0.231588\pi$
$110$	0	0
$111$	7.47416e21	2.49845
$112$	0	0
$113$	4.81011e21	1.33300	0.666502	−	0.745503i	$-0.267791\pi$
$113$	4.81011e21	1.33300	0.666502	+	0.745503i	$0.267791\pi$
$114$	0	0
$115$	7.70490e21	1.77599
$116$	0	0
$117$	9.72588e21	1.87058
$118$	0	0
$119$	−8.38318e21	−1.34948
$120$	0	0
$121$	−6.28294e21	−0.849017
$122$	0	0
$123$	−8.57547e21	−0.975561
$124$	0	0
$125$	2.97231e21	0.285456
$126$	0	0
$127$	9.74422e21	0.792151	0.396076	−	0.918218i	$-0.370372\pi$
$127$	9.74422e21	0.792151	0.396076	+	0.918218i	$0.370372\pi$
$128$	0	0
$129$	−2.39861e22	−1.65488
$130$	0	0
$131$	4.18119e21	0.245443	0.122722	−	0.992441i	$-0.460838\pi$
$131$	4.18119e21	0.245443	0.122722	+	0.992441i	$0.460838\pi$
$132$	0	0
$133$	−2.14523e22	−1.07407
$134$	0	0
$135$	−2.02669e22	−0.867528
$136$	0	0
$137$	4.98166e22	1.82729	0.913647	−	0.406508i	$-0.133254\pi$
$137$	4.98166e22	1.82729	0.913647	+	0.406508i	$0.133254\pi$
$138$	0	0
$139$	−8.54979e21	−0.269339	−0.134670	−	0.990891i	$-0.542997\pi$
$139$	−8.54979e21	−0.269339	−0.134670	+	0.990891i	$0.542997\pi$
$140$	0	0
$141$	−6.49432e22	−1.76089
$142$	0	0
$143$	−2.25249e22	−0.526787
$144$	0	0
$145$	4.14155e22	0.837142
$146$	0	0
$147$	−3.01300e22	−0.527433
$148$	0	0
$149$	5.89102e22	0.894817	0.447409	−	0.894330i	$-0.352347\pi$
$149$	5.89102e22	0.894817	0.447409	+	0.894330i	$0.352347\pi$
$150$	0	0
$151$	2.02460e22	0.267351	0.133676	−	0.991025i	$-0.457322\pi$
$151$	2.02460e22	0.267351	0.133676	+	0.991025i	$0.457322\pi$
$152$	0	0
$153$	−1.39756e23	−1.60735
$154$	0	0
$155$	1.67582e22	0.168169
$156$	0	0
$157$	−1.86777e23	−1.63823	−0.819117	−	0.573626i	$-0.805536\pi$
$157$	−1.86777e23	−1.63823	−0.819117	+	0.573626i	$0.805536\pi$
$158$	0	0
$159$	8.92563e22	0.685440
$160$	0	0
$161$	2.06288e23	1.38932
$162$	0	0
$163$	−1.77710e23	−1.05134	−0.525668	−	0.850690i	$-0.676185\pi$
$163$	−1.77710e23	−1.05134	−0.525668	+	0.850690i	$0.676185\pi$
$164$	0	0
$165$	1.70534e23	0.887623
$166$	0	0
$167$	3.06440e23	1.40547	0.702736	−	0.711451i	$-0.251962\pi$
$167$	3.06440e23	1.40547	0.702736	+	0.711451i	$0.251962\pi$
$168$	0	0
$169$	2.07036e23	0.837985
$170$	0	0
$171$	−3.57633e23	−1.27932
$172$	0	0
$173$	−3.16871e22	−0.100323	−0.0501613	−	0.998741i	$-0.515974\pi$
$173$	−3.16871e22	−0.100323	−0.0501613	+	0.998741i	$0.515974\pi$
$174$	0	0
$175$	4.92399e23	1.38171
$176$	0	0
$177$	2.26976e23	0.565277
$178$	0	0
$179$	6.42145e23	1.42127	0.710635	−	0.703561i	$-0.248408\pi$
$179$	6.42145e23	1.42127	0.710635	+	0.703561i	$0.248408\pi$
$180$	0	0
$181$	9.02174e23	1.77691	0.888455	−	0.458964i	$-0.151779\pi$
$181$	9.02174e23	1.77691	0.888455	+	0.458964i	$0.151779\pi$
$182$	0	0
$183$	−8.34430e22	−0.146438
$184$	0	0
$185$	−1.53181e24	−2.39828
$186$	0	0
$187$	3.23673e23	0.452657
$188$	0	0
$189$	−5.42620e23	−0.678652
$190$	0	0
$191$	−7.63677e23	−0.855184	−0.427592	−	0.903972i	$-0.640638\pi$
$191$	−7.63677e23	−0.855184	−0.427592	+	0.903972i	$0.640638\pi$
$192$	0	0
$193$	1.49796e23	0.150365	0.0751826	−	0.997170i	$-0.476046\pi$
$193$	1.49796e23	0.150365	0.0751826	+	0.997170i	$0.476046\pi$
$194$	0	0
$195$	−3.43794e24	−3.09696
$196$	0	0
$197$	4.46832e23	0.361617	0.180808	−	0.983518i	$-0.442129\pi$
$197$	4.46832e23	0.361617	0.180808	+	0.983518i	$0.442129\pi$
$198$	0	0
$199$	1.27167e23	0.0925587	0.0462794	−	0.998929i	$-0.485264\pi$
$199$	1.27167e23	0.0925587	0.0462794	+	0.998929i	$0.485264\pi$
$200$	0	0
$201$	−2.01436e24	−1.32001
$202$	0	0
$203$	1.10885e24	0.654881
$204$	0	0
$205$	1.75752e24	0.936448
$206$	0	0
$207$	3.43904e24	1.65481
$208$	0	0
$209$	8.28271e23	0.360277
$210$	0	0
$211$	−3.27555e24	−1.28920	−0.644598	−	0.764522i	$-0.722975\pi$
$211$	−3.27555e24	−1.28920	−0.644598	+	0.764522i	$0.722975\pi$
$212$	0	0
$213$	−2.92319e24	−1.04201
$214$	0	0
$215$	4.91587e24	1.58853
$216$	0	0
$217$	4.48679e23	0.131555
$218$	0	0
$219$	−6.46155e24	−1.72058
$220$	0	0
$221$	−6.52522e24	−1.57934
$222$	0	0
$223$	2.06204e24	0.454041	0.227021	−	0.973890i	$-0.427102\pi$
$223$	2.06204e24	0.454041	0.227021	+	0.973890i	$0.427102\pi$
$224$	0	0
$225$	8.20881e24	1.64575
$226$	0	0
$227$	−1.07994e25	−1.97300	−0.986501	−	0.163753i	$-0.947640\pi$
$227$	−1.07994e25	−1.97300	−0.986501	+	0.163753i	$0.947640\pi$
$228$	0	0
$229$	−1.58076e24	−0.263387	−0.131694	−	0.991290i	$-0.542041\pi$
$229$	−1.58076e24	−0.263387	−0.131694	+	0.991290i	$0.542041\pi$
$230$	0	0
$231$	4.56581e24	0.694371
$232$	0	0
$233$	1.13263e25	1.57344	0.786722	−	0.617307i	$-0.211776\pi$
$233$	1.13263e25	1.57344	0.786722	+	0.617307i	$0.211776\pi$
$234$	0	0
$235$	1.33099e25	1.69029
$236$	0	0
$237$	7.59773e24	0.882727
$238$	0	0
$239$	1.31199e25	1.39558	0.697789	−	0.716304i	$-0.254168\pi$
$239$	1.31199e25	1.39558	0.697789	+	0.716304i	$0.254168\pi$
$240$	0	0
$241$	−1.35031e25	−1.31600	−0.657999	−	0.753019i	$-0.728597\pi$
$241$	−1.35031e25	−1.31600	−0.657999	+	0.753019i	$0.728597\pi$
$242$	0	0
$243$	1.47738e25	1.32016
$244$	0	0
$245$	6.17505e24	0.506287
$246$	0	0
$247$	−1.66979e25	−1.25702
$248$	0	0
$249$	4.26240e24	0.294823
$250$	0	0
$251$	−2.69522e25	−1.71404	−0.857018	−	0.515286i	$-0.827686\pi$
$251$	−2.69522e25	−1.71404	−0.857018	+	0.515286i	$0.827686\pi$
$252$	0	0
$253$	−7.96476e24	−0.466023
$254$	0	0
$255$	4.94016e25	2.66115
$256$	0	0
$257$	−1.51030e24	−0.0749490	−0.0374745	−	0.999298i	$-0.511931\pi$
$257$	−1.51030e24	−0.0749490	−0.0374745	+	0.999298i	$0.511931\pi$
$258$	0	0
$259$	−4.10121e25	−1.87613
$260$	0	0
$261$	1.84856e25	0.780026
$262$	0	0
$263$	−3.87225e25	−1.50809	−0.754047	−	0.656820i	$-0.771901\pi$
$263$	−3.87225e25	−1.50809	−0.754047	+	0.656820i	$0.771901\pi$
$264$	0	0
$265$	−1.82928e25	−0.657959
$266$	0	0
$267$	−6.22060e24	−0.206759
$268$	0	0
$269$	−6.38040e24	−0.196087	−0.0980436	−	0.995182i	$-0.531258\pi$
$269$	−6.38040e24	−0.196087	−0.0980436	+	0.995182i	$0.531258\pi$
$270$	0	0
$271$	1.71712e25	0.488230	0.244115	−	0.969746i	$-0.421503\pi$
$271$	1.71712e25	0.488230	0.244115	+	0.969746i	$0.421503\pi$
$272$	0	0
$273$	−9.20463e25	−2.42269
$274$	0	0
$275$	−1.90114e25	−0.463470
$276$	0	0
$277$	−3.99150e25	−0.901776	−0.450888	−	0.892580i	$-0.648893\pi$
$277$	−3.99150e25	−0.901776	−0.450888	+	0.892580i	$0.648893\pi$
$278$	0	0
$279$	7.47996e24	0.156695
$280$	0	0
$281$	5.34680e25	1.03915	0.519574	−	0.854425i	$-0.326091\pi$
$281$	5.34680e25	1.03915	0.519574	+	0.854425i	$0.326091\pi$
$282$	0	0
$283$	6.31230e25	1.13875	0.569377	−	0.822077i	$-0.307184\pi$
$283$	6.31230e25	1.13875	0.569377	+	0.822077i	$0.307184\pi$
$284$	0	0
$285$	1.26418e26	2.11805
$286$	0	0
$287$	4.70552e25	0.732567
$288$	0	0
$289$	2.46725e25	0.357097
$290$	0	0
$291$	−1.60671e26	−2.16302
$292$	0	0
$293$	−1.16416e26	−1.45848	−0.729241	−	0.684257i	$-0.760126\pi$
$293$	−1.16416e26	−1.45848	−0.729241	+	0.684257i	$0.760126\pi$
$294$	0	0
$295$	−4.65181e25	−0.542613
$296$	0	0
$297$	2.09505e25	0.227641
$298$	0	0
$299$	1.60569e26	1.62597
$300$	0	0
$301$	1.31616e26	1.24268
$302$	0	0
$303$	6.86557e24	0.0604685
$304$	0	0
$305$	1.71014e25	0.140567
$306$	0	0
$307$	−8.12396e23	−0.00623469	−0.00311734	−	0.999995i	$-0.500992\pi$
$307$	−8.12396e23	−0.00623469	−0.00311734	+	0.999995i	$0.500992\pi$
$308$	0	0
$309$	−1.66923e26	−1.19661
$310$	0	0
$311$	−9.79448e25	−0.656141	−0.328071	−	0.944653i	$-0.606398\pi$
$311$	−9.79448e25	−0.656141	−0.328071	+	0.944653i	$0.606398\pi$
$312$	0	0
$313$	−3.18802e25	−0.199666	−0.0998332	−	0.995004i	$-0.531831\pi$
$313$	−3.18802e25	−0.199666	−0.0998332	+	0.995004i	$0.531831\pi$
$314$	0	0
$315$	4.04040e26	2.36681
$316$	0	0
$317$	−1.42256e26	−0.779736	−0.389868	−	0.920871i	$-0.627479\pi$
$317$	−1.42256e26	−0.779736	−0.389868	+	0.920871i	$0.627479\pi$
$318$	0	0
$319$	−4.28123e25	−0.219668
$320$	0	0
$321$	4.15689e26	1.99741
$322$	0	0
$323$	2.39941e26	1.08013
$324$	0	0
$325$	3.83269e26	1.61707
$326$	0	0
$327$	−5.82444e26	−2.30411
$328$	0	0
$329$	3.56355e26	1.32229
$330$	0	0
$331$	1.74752e26	0.608455	0.304228	−	0.952599i	$-0.401602\pi$
$331$	1.74752e26	0.608455	0.304228	+	0.952599i	$0.401602\pi$
$332$	0	0
$333$	−6.83714e26	−2.23465
$334$	0	0
$335$	4.12836e26	1.26709
$336$	0	0
$337$	−1.15176e26	−0.332084	−0.166042	−	0.986119i	$-0.553099\pi$
$337$	−1.15176e26	−0.332084	−0.166042	+	0.986119i	$0.553099\pi$
$338$	0	0
$339$	−7.58919e26	−2.05636
$340$	0	0
$341$	−1.73234e25	−0.0441279
$342$	0	0
$343$	−3.18229e26	−0.762345
$344$	0	0
$345$	−1.21565e27	−2.73972
$346$	0	0
$347$	−1.81281e26	−0.384496	−0.192248	−	0.981346i	$-0.561578\pi$
$347$	−1.81281e26	−0.384496	−0.192248	+	0.981346i	$0.561578\pi$
$348$	0	0
$349$	−1.30131e26	−0.259844	−0.129922	−	0.991524i	$-0.541473\pi$
$349$	−1.30131e26	−0.259844	−0.129922	+	0.991524i	$0.541473\pi$
$350$	0	0
$351$	−4.22360e26	−0.794251
$352$	0	0
$353$	−7.22702e26	−1.28034	−0.640170	−	0.768234i	$-0.721136\pi$
$353$	−7.22702e26	−1.28034	−0.640170	+	0.768234i	$0.721136\pi$
$354$	0	0
$355$	5.99099e26	1.00023
$356$	0	0
$357$	1.32266e27	2.08177
$358$	0	0
$359$	2.55601e26	0.379377	0.189689	−	0.981844i	$-0.439252\pi$
$359$	2.55601e26	0.379377	0.189689	+	0.981844i	$0.439252\pi$
$360$	0	0
$361$	−1.00207e26	−0.140305
$362$	0	0
$363$	9.91296e26	1.30974
$364$	0	0
$365$	1.32427e27	1.65159
$366$	0	0
$367$	−1.33233e27	−1.56898	−0.784492	−	0.620139i	$-0.787076\pi$
$367$	−1.33233e27	−1.56898	−0.784492	+	0.620139i	$0.787076\pi$
$368$	0	0
$369$	7.84459e26	0.872556
$370$	0	0
$371$	−4.89765e26	−0.514709
$372$	0	0
$373$	−1.19609e27	−1.18801	−0.594004	−	0.804462i	$-0.702454\pi$
$373$	−1.19609e27	−1.18801	−0.594004	+	0.804462i	$0.702454\pi$
$374$	0	0
$375$	−4.68959e26	−0.440358
$376$	0	0
$377$	8.63093e26	0.766432
$378$	0	0
$379$	1.92082e27	1.61352	0.806762	−	0.590877i	$-0.201218\pi$
$379$	1.92082e27	1.61352	0.806762	+	0.590877i	$0.201218\pi$
$380$	0	0
$381$	−1.53740e27	−1.22201
$382$	0	0
$383$	8.04197e26	0.605028	0.302514	−	0.953145i	$-0.402174\pi$
$383$	8.04197e26	0.605028	0.302514	+	0.953145i	$0.402174\pi$
$384$	0	0
$385$	−9.35748e26	−0.666532
$386$	0	0
$387$	2.19417e27	1.48015
$388$	0	0
$389$	2.45394e27	1.56817	0.784085	−	0.620653i	$-0.213132\pi$
$389$	2.45394e27	1.56817	0.784085	+	0.620653i	$0.213132\pi$
$390$	0	0
$391$	−2.30730e27	−1.39717
$392$	0	0
$393$	−6.59691e26	−0.378633
$394$	0	0
$395$	−1.55713e27	−0.847336
$396$	0	0
$397$	−5.79186e26	−0.298894	−0.149447	−	0.988770i	$-0.547749\pi$
$397$	−5.79186e26	−0.298894	−0.149447	+	0.988770i	$0.547749\pi$
$398$	0	0
$399$	3.38466e27	1.65691
$400$	0	0
$401$	3.53477e27	1.64190	0.820948	−	0.571003i	$-0.193445\pi$
$401$	3.53477e27	1.64190	0.820948	+	0.571003i	$0.193445\pi$
$402$	0	0
$403$	3.49239e26	0.153964
$404$	0	0
$405$	−1.68418e27	−0.704871
$406$	0	0
$407$	1.58347e27	0.629315
$408$	0	0
$409$	2.80199e27	1.05772	0.528861	−	0.848708i	$-0.322619\pi$
$409$	2.80199e27	1.05772	0.528861	+	0.848708i	$0.322619\pi$
$410$	0	0
$411$	−7.85986e27	−2.81888
$412$	0	0
$413$	−1.24546e27	−0.424477
$414$	0	0
$415$	−8.73565e26	−0.283003
$416$	0	0
$417$	1.34895e27	0.415497
$418$	0	0
$419$	2.59970e27	0.761509	0.380755	−	0.924676i	$-0.375664\pi$
$419$	2.59970e27	0.761509	0.380755	+	0.924676i	$0.375664\pi$
$420$	0	0
$421$	−3.07604e27	−0.857097	−0.428549	−	0.903519i	$-0.640975\pi$
$421$	−3.07604e27	−0.857097	−0.428549	+	0.903519i	$0.640975\pi$
$422$	0	0
$423$	5.94081e27	1.57497
$424$	0	0
$425$	−5.50740e27	−1.38951
$426$	0	0
$427$	4.57867e26	0.109963
$428$	0	0
$429$	3.55389e27	0.812648
$430$	0	0
$431$	−1.38695e27	−0.302029	−0.151014	−	0.988532i	$-0.548254\pi$
$431$	−1.38695e27	−0.302029	−0.151014	+	0.988532i	$0.548254\pi$
$432$	0	0
$433$	−5.00059e27	−1.03729	−0.518643	−	0.854991i	$-0.673563\pi$
$433$	−5.00059e27	−1.03729	−0.518643	+	0.854991i	$0.673563\pi$
$434$	0	0
$435$	−6.53437e27	−1.29142
$436$	0	0
$437$	−5.90432e27	−1.11203
$438$	0	0
$439$	−2.31775e26	−0.0416092	−0.0208046	−	0.999784i	$-0.506623\pi$
$439$	−2.31775e26	−0.0416092	−0.0208046	+	0.999784i	$0.506623\pi$
$440$	0	0
$441$	2.75621e27	0.471744
$442$	0	0
$443$	−9.82366e27	−1.60337	−0.801686	−	0.597746i	$-0.796063\pi$
$443$	−9.82366e27	−1.60337	−0.801686	+	0.597746i	$0.796063\pi$
$444$	0	0
$445$	1.27489e27	0.198469
$446$	0	0
$447$	−9.29460e27	−1.38039
$448$	0	0
$449$	−3.46353e27	−0.490832	−0.245416	−	0.969418i	$-0.578925\pi$
$449$	−3.46353e27	−0.490832	−0.245416	+	0.969418i	$0.578925\pi$
$450$	0	0
$451$	−1.81679e27	−0.245726
$452$	0	0
$453$	−3.19434e27	−0.412430
$454$	0	0
$455$	1.88646e28	2.32556
$456$	0	0
$457$	8.14165e27	0.958500	0.479250	−	0.877678i	$-0.340909\pi$
$457$	8.14165e27	0.958500	0.479250	+	0.877678i	$0.340909\pi$
$458$	0	0
$459$	6.06912e27	0.682484
$460$	0	0
$461$	1.74097e28	1.87039	0.935195	−	0.354133i	$-0.115224\pi$
$461$	1.74097e28	1.87039	0.935195	+	0.354133i	$0.115224\pi$
$462$	0	0
$463$	−1.17292e28	−1.20412	−0.602059	−	0.798452i	$-0.705653\pi$
$463$	−1.17292e28	−1.20412	−0.602059	+	0.798452i	$0.705653\pi$
$464$	0	0
$465$	−2.64404e27	−0.259425
$466$	0	0
$467$	2.05500e28	1.92745	0.963727	−	0.266891i	$-0.0859965\pi$
$467$	2.05500e28	1.92745	0.963727	+	0.266891i	$0.0859965\pi$
$468$	0	0
$469$	1.10531e28	0.991222
$470$	0	0
$471$	2.94688e28	2.52722
$472$	0	0
$473$	−5.08166e27	−0.416835
$474$	0	0
$475$	−1.40933e28	−1.10593
$476$	0	0
$477$	−8.16491e27	−0.613067
$478$	0	0
$479$	1.58389e28	1.13816	0.569078	−	0.822283i	$-0.307300\pi$
$479$	1.58389e28	1.13816	0.569078	+	0.822283i	$0.307300\pi$
$480$	0	0
$481$	−3.19226e28	−2.19571
$482$	0	0
$483$	−3.25473e28	−2.14324
$484$	0	0
$485$	3.29289e28	2.07630
$486$	0	0
$487$	−1.43251e28	−0.865053	−0.432527	−	0.901621i	$-0.642378\pi$
$487$	−1.43251e28	−0.865053	−0.432527	+	0.901621i	$0.642378\pi$
$488$	0	0
$489$	2.80383e28	1.62184
$490$	0	0
$491$	5.26401e27	0.291716	0.145858	−	0.989306i	$-0.453406\pi$
$491$	5.26401e27	0.291716	0.145858	+	0.989306i	$0.453406\pi$
$492$	0	0
$493$	−1.24022e28	−0.658579
$494$	0	0
$495$	−1.55999e28	−0.793903
$496$	0	0
$497$	1.60401e28	0.782464
$498$	0	0
$499$	2.03820e28	0.953217	0.476608	−	0.879116i	$-0.341866\pi$
$499$	2.03820e28	0.953217	0.476608	+	0.879116i	$0.341866\pi$
$500$	0	0
$501$	−4.83488e28	−2.16815
$502$	0	0
$503$	3.10287e26	0.0133444	0.00667222	−	0.999978i	$-0.497876\pi$
$503$	3.10287e26	0.0133444	0.00667222	+	0.999978i	$0.497876\pi$
$504$	0	0
$505$	−1.40708e27	−0.0580441
$506$	0	0
$507$	−3.26653e28	−1.29272
$508$	0	0
$509$	1.23140e28	0.467587	0.233794	−	0.972286i	$-0.424886\pi$
$509$	1.23140e28	0.467587	0.233794	+	0.972286i	$0.424886\pi$
$510$	0	0
$511$	3.54557e28	1.29201
$512$	0	0
$513$	1.55307e28	0.543199
$514$	0	0
$515$	3.42103e28	1.14863
$516$	0	0
$517$	−1.37588e28	−0.443537
$518$	0	0
$519$	4.99946e27	0.154763
$520$	0	0
$521$	1.59305e28	0.473623	0.236811	−	0.971556i	$-0.423898\pi$
$521$	1.59305e28	0.473623	0.236811	+	0.971556i	$0.423898\pi$
$522$	0	0
$523$	−2.35530e28	−0.672633	−0.336317	−	0.941749i	$-0.609181\pi$
$523$	−2.35530e28	−0.672633	−0.336317	+	0.941749i	$0.609181\pi$
$524$	0	0
$525$	−7.76886e28	−2.13150
$526$	0	0
$527$	−5.01840e27	−0.132298
$528$	0	0
$529$	1.73051e28	0.438420
$530$	0	0
$531$	−2.07631e28	−0.505592
$532$	0	0
$533$	3.66264e28	0.857350
$534$	0	0
$535$	−8.51942e28	−1.91732
$536$	0	0
$537$	−1.01315e29	−2.19252
$538$	0	0
$539$	−6.38332e27	−0.132851
$540$	0	0
$541$	−2.22176e28	−0.444760	−0.222380	−	0.974960i	$-0.571382\pi$
$541$	−2.22176e28	−0.444760	−0.222380	+	0.974960i	$0.571382\pi$
$542$	0	0
$543$	−1.42341e29	−2.74115
$544$	0	0
$545$	1.19370e29	2.21173
$546$	0	0
$547$	9.43340e28	1.68191	0.840953	−	0.541109i	$-0.181995\pi$
$547$	9.43340e28	1.68191	0.840953	+	0.541109i	$0.181995\pi$
$548$	0	0
$549$	7.63312e27	0.130976
$550$	0	0
$551$	−3.17370e28	−0.524173
$552$	0	0
$553$	−4.16901e28	−0.662856
$554$	0	0
$555$	2.41682e29	3.69971
$556$	0	0
$557$	2.41911e28	0.356595	0.178298	−	0.983977i	$-0.442941\pi$
$557$	2.41911e28	0.356595	0.178298	+	0.983977i	$0.442941\pi$
$558$	0	0
$559$	1.02446e29	1.45436
$560$	0	0
$561$	−5.10678e28	−0.698292
$562$	0	0
$563$	−4.36753e28	−0.575304	−0.287652	−	0.957735i	$-0.592875\pi$
$563$	−4.36753e28	−0.575304	−0.287652	+	0.957735i	$0.592875\pi$
$564$	0	0
$565$	1.55538e29	1.97391
$566$	0	0
$567$	−4.50916e28	−0.551408
$568$	0	0
$569$	−9.30684e28	−1.09679	−0.548395	−	0.836219i	$-0.684761\pi$
$569$	−9.30684e28	−1.09679	−0.548395	+	0.836219i	$0.684761\pi$
$570$	0	0
$571$	−7.64006e28	−0.867796	−0.433898	−	0.900962i	$-0.642862\pi$
$571$	−7.64006e28	−0.867796	−0.433898	+	0.900962i	$0.642862\pi$
$572$	0	0
$573$	1.20490e29	1.31925
$574$	0	0
$575$	1.35523e29	1.43054
$576$	0	0
$577$	−9.26568e28	−0.943043	−0.471521	−	0.881855i	$-0.656295\pi$
$577$	−9.26568e28	−0.943043	−0.471521	+	0.881855i	$0.656295\pi$
$578$	0	0
$579$	−2.36341e28	−0.231961
$580$	0	0
$581$	−2.33886e28	−0.221388
$582$	0	0
$583$	1.89098e28	0.172650
$584$	0	0
$585$	3.14493e29	2.76996
$586$	0	0
$587$	1.23846e29	1.05240	0.526202	−	0.850360i	$-0.323616\pi$
$587$	1.23846e29	1.05240	0.526202	+	0.850360i	$0.323616\pi$
$588$	0	0
$589$	−1.28420e28	−0.105298
$590$	0	0
$591$	−7.04993e28	−0.557848
$592$	0	0
$593$	−2.45974e28	−0.187852	−0.0939260	−	0.995579i	$-0.529942\pi$
$593$	−2.45974e28	−0.187852	−0.0939260	+	0.995579i	$0.529942\pi$
$594$	0	0
$595$	−2.71076e29	−1.99831
$596$	0	0
$597$	−2.00639e28	−0.142786
$598$	0	0
$599$	−8.88989e28	−0.610822	−0.305411	−	0.952221i	$-0.598794\pi$
$599$	−8.88989e28	−0.610822	−0.305411	+	0.952221i	$0.598794\pi$
$600$	0	0
$601$	−8.85037e28	−0.587191	−0.293596	−	0.955930i	$-0.594852\pi$
$601$	−8.85037e28	−0.587191	−0.293596	+	0.955930i	$0.594852\pi$
$602$	0	0
$603$	1.84268e29	1.18064
$604$	0	0
$605$	−2.03163e29	−1.25722
$606$	0	0
$607$	−3.72951e28	−0.222931	−0.111465	−	0.993768i	$-0.535554\pi$
$607$	−3.72951e28	−0.222931	−0.111465	+	0.993768i	$0.535554\pi$
$608$	0	0
$609$	−1.74949e29	−1.01025
$610$	0	0
$611$	2.77376e29	1.54752
$612$	0	0
$613$	−1.34336e29	−0.724197	−0.362099	−	0.932140i	$-0.617940\pi$
$613$	−1.34336e29	−0.724197	−0.362099	+	0.932140i	$0.617940\pi$
$614$	0	0
$615$	−2.77294e29	−1.44461
$616$	0	0
$617$	5.26542e28	0.265118	0.132559	−	0.991175i	$-0.457681\pi$
$617$	5.26542e28	0.265118	0.132559	+	0.991175i	$0.457681\pi$
$618$	0	0
$619$	−1.51228e29	−0.736006	−0.368003	−	0.929825i	$-0.619958\pi$
$619$	−1.51228e29	−0.736006	−0.368003	+	0.929825i	$0.619958\pi$
$620$	0	0
$621$	−1.49345e29	−0.702635
$622$	0	0
$623$	3.41336e28	0.155259
$624$	0	0
$625$	−1.75093e29	−0.770068
$626$	0	0
$627$	−1.30681e29	−0.555781
$628$	0	0
$629$	4.58713e29	1.88673
$630$	0	0
$631$	−7.93318e28	−0.315602	−0.157801	−	0.987471i	$-0.550440\pi$
$631$	−7.93318e28	−0.315602	−0.157801	+	0.987471i	$0.550440\pi$
$632$	0	0
$633$	5.16803e29	1.98878
$634$	0	0
$635$	3.15086e29	1.17302
$636$	0	0
$637$	1.28687e29	0.463523
$638$	0	0
$639$	2.67405e29	0.931988
$640$	0	0
$641$	−8.73743e28	−0.294696	−0.147348	−	0.989085i	$-0.547074\pi$
$641$	−8.73743e28	−0.294696	−0.147348	+	0.989085i	$0.547074\pi$
$642$	0	0
$643$	−4.43655e29	−1.44821	−0.724103	−	0.689692i	$-0.757746\pi$
$643$	−4.43655e29	−1.44821	−0.724103	+	0.689692i	$0.757746\pi$
$644$	0	0
$645$	−7.75605e29	−2.45055
$646$	0	0
$647$	−2.79563e29	−0.855035	−0.427518	−	0.904007i	$-0.640612\pi$
$647$	−2.79563e29	−0.855035	−0.427518	+	0.904007i	$0.640612\pi$
$648$	0	0
$649$	4.80870e28	0.142383
$650$	0	0
$651$	−7.07908e28	−0.202944
$652$	0	0
$653$	−2.84312e29	−0.789236	−0.394618	−	0.918845i	$-0.629123\pi$
$653$	−2.84312e29	−0.789236	−0.394618	+	0.918845i	$0.629123\pi$
$654$	0	0
$655$	1.35201e29	0.363453
$656$	0	0
$657$	5.91084e29	1.53891
$658$	0	0
$659$	−5.37621e29	−1.35575	−0.677875	−	0.735177i	$-0.737099\pi$
$659$	−5.37621e29	−1.35575	−0.677875	+	0.735177i	$0.737099\pi$
$660$	0	0
$661$	4.97933e28	0.121634	0.0608171	−	0.998149i	$-0.480629\pi$
$661$	4.97933e28	0.121634	0.0608171	+	0.998149i	$0.480629\pi$
$662$	0	0
$663$	1.02952e30	2.43637
$664$	0	0
$665$	−6.93676e29	−1.59048
$666$	0	0
$667$	3.05187e29	0.678024
$668$	0	0
$669$	−3.25340e29	−0.700427
$670$	0	0
$671$	−1.76781e28	−0.0368850
$672$	0	0
$673$	−5.14859e29	−1.04119	−0.520595	−	0.853804i	$-0.674290\pi$
$673$	−5.14859e29	−1.04119	−0.520595	+	0.853804i	$0.674290\pi$
$674$	0	0
$675$	−3.56479e29	−0.698786
$676$	0	0
$677$	9.61471e29	1.82707	0.913535	−	0.406760i	$-0.133341\pi$
$677$	9.61471e29	1.82707	0.913535	+	0.406760i	$0.133341\pi$
$678$	0	0
$679$	8.81629e29	1.62425
$680$	0	0
$681$	1.70388e30	3.04365
$682$	0	0
$683$	−3.34669e29	−0.579693	−0.289847	−	0.957073i	$-0.593604\pi$
$683$	−3.34669e29	−0.579693	−0.289847	+	0.957073i	$0.593604\pi$
$684$	0	0
$685$	1.61085e30	2.70586
$686$	0	0
$687$	2.49406e29	0.406314
$688$	0	0
$689$	−3.81219e29	−0.602383
$690$	0	0
$691$	1.01201e30	1.55119	0.775597	−	0.631229i	$-0.217449\pi$
$691$	1.01201e30	1.55119	0.775597	+	0.631229i	$0.217449\pi$
$692$	0	0
$693$	−4.17667e29	−0.621056
$694$	0	0
$695$	−2.76463e29	−0.398838
$696$	0	0
$697$	−5.26304e29	−0.736703
$698$	0	0
$699$	−1.78702e30	−2.42727
$700$	0	0
$701$	5.45526e29	0.719078	0.359539	−	0.933130i	$-0.382934\pi$
$701$	5.45526e29	0.719078	0.359539	+	0.933130i	$0.382934\pi$
$702$	0	0
$703$	1.17383e30	1.50168
$704$	0	0
$705$	−2.09998e30	−2.60753
$706$	0	0
$707$	−3.76726e28	−0.0454069
$708$	0	0
$709$	−3.39700e26	−0.000397475 0	−0.000198737	−	1.00000i	$-0.500063\pi$
$709$	−3.39700e26	−0.000397475 0	−0.000198737		1.00000i	$0.500063\pi$
$710$	0	0
$711$	−6.95018e29	−0.789524
$712$	0	0
$713$	1.23490e29	0.136204
$714$	0	0
$715$	−7.28359e29	−0.780067
$716$	0	0
$717$	−2.07001e30	−2.15289
$718$	0	0
$719$	−6.84006e29	−0.690886	−0.345443	−	0.938440i	$-0.612271\pi$
$719$	−6.84006e29	−0.690886	−0.345443	+	0.938440i	$0.612271\pi$
$720$	0	0
$721$	9.15936e29	0.898555
$722$	0	0
$723$	2.13047e30	2.03012
$724$	0	0
$725$	7.28465e29	0.674310
$726$	0	0
$727$	−5.14875e29	−0.463010	−0.231505	−	0.972834i	$-0.574365\pi$
$727$	−5.14875e29	−0.463010	−0.231505	+	0.972834i	$0.574365\pi$
$728$	0	0
$729$	−1.78614e30	−1.56054
$730$	0	0
$731$	−1.47210e30	−1.24970
$732$	0	0
$733$	−7.06925e29	−0.583151	−0.291576	−	0.956548i	$-0.594180\pi$
$733$	−7.06925e29	−0.583151	−0.291576	+	0.956548i	$0.594180\pi$
$734$	0	0
$735$	−9.74274e29	−0.781024
$736$	0	0
$737$	−4.26760e29	−0.332487
$738$	0	0
$739$	−7.87359e29	−0.596219	−0.298110	−	0.954532i	$-0.596356\pi$
$739$	−7.87359e29	−0.596219	−0.298110	+	0.954532i	$0.596356\pi$
$740$	0	0
$741$	2.63452e30	1.93915
$742$	0	0
$743$	1.46660e30	1.04937	0.524687	−	0.851295i	$-0.324182\pi$
$743$	1.46660e30	1.04937	0.524687	+	0.851295i	$0.324182\pi$
$744$	0	0
$745$	1.90490e30	1.32505
$746$	0	0
$747$	−3.89912e29	−0.263694
$748$	0	0
$749$	−2.28096e30	−1.49989
$750$	0	0
$751$	−1.47134e30	−0.940791	−0.470396	−	0.882456i	$-0.655889\pi$
$751$	−1.47134e30	−0.940791	−0.470396	+	0.882456i	$0.655889\pi$
$752$	0	0
$753$	4.25240e30	2.64416
$754$	0	0
$755$	6.54669e29	0.395894
$756$	0	0
$757$	−9.83325e29	−0.578350	−0.289175	−	0.957276i	$-0.593381\pi$
$757$	−9.83325e29	−0.578350	−0.289175	+	0.957276i	$0.593381\pi$
$758$	0	0
$759$	1.25665e30	0.718910
$760$	0	0
$761$	−1.59743e30	−0.888959	−0.444479	−	0.895789i	$-0.646611\pi$
$761$	−1.59743e30	−0.888959	−0.444479	+	0.895789i	$0.646611\pi$
$762$	0	0
$763$	3.19597e30	1.73020
$764$	0	0
$765$	−4.51912e30	−2.38017
$766$	0	0
$767$	−9.69429e29	−0.496781
$768$	0	0
$769$	−9.60134e29	−0.478746	−0.239373	−	0.970928i	$-0.576942\pi$
$769$	−9.60134e29	−0.478746	−0.239373	+	0.970928i	$0.576942\pi$
$770$	0	0
$771$	2.38289e29	0.115620
$772$	0	0
$773$	6.07964e29	0.287074	0.143537	−	0.989645i	$-0.454152\pi$
$773$	6.07964e29	0.287074	0.143537	+	0.989645i	$0.454152\pi$
$774$	0	0
$775$	2.94763e29	0.135458
$776$	0	0
$777$	6.47071e30	2.89422
$778$	0	0
$779$	−1.34680e30	−0.586353
$780$	0	0
$781$	−6.19304e29	−0.262463
$782$	0	0
$783$	−8.02764e29	−0.331199
$784$	0	0
$785$	−6.03954e30	−2.42590
$786$	0	0
$787$	−3.29277e29	−0.128774	−0.0643869	−	0.997925i	$-0.520509\pi$
$787$	−3.29277e29	−0.128774	−0.0643869	+	0.997925i	$0.520509\pi$
$788$	0	0
$789$	6.10948e30	2.32646
$790$	0	0
$791$	4.16433e30	1.54416
$792$	0	0
$793$	3.56390e29	0.128694
$794$	0	0
$795$	2.88616e30	1.01500
$796$	0	0
$797$	6.81986e29	0.233595	0.116797	−	0.993156i	$-0.462737\pi$
$797$	6.81986e29	0.233595	0.116797	+	0.993156i	$0.462737\pi$
$798$	0	0
$799$	−3.98577e30	−1.32975
$800$	0	0
$801$	5.69042e29	0.184928
$802$	0	0
$803$	−1.36894e30	−0.433382
$804$	0	0
$805$	6.67047e30	2.05731
$806$	0	0
$807$	1.00667e30	0.302494
$808$	0	0
$809$	3.74678e29	0.109698	0.0548490	−	0.998495i	$-0.482532\pi$
$809$	3.74678e29	0.109698	0.0548490	+	0.998495i	$0.482532\pi$
$810$	0	0
$811$	8.58232e29	0.244842	0.122421	−	0.992478i	$-0.460934\pi$
$811$	8.58232e29	0.244842	0.122421	+	0.992478i	$0.460934\pi$
$812$	0	0
$813$	−2.70921e30	−0.753168
$814$	0	0
$815$	−5.74636e30	−1.55682
$816$	0	0
$817$	−3.76707e30	−0.994654
$818$	0	0
$819$	8.42012e30	2.16689
$820$	0	0
$821$	−7.31777e30	−1.83559	−0.917795	−	0.397055i	$-0.870032\pi$
$821$	−7.31777e30	−1.83559	−0.917795	+	0.397055i	$0.870032\pi$
$822$	0	0
$823$	−4.58435e30	−1.12093	−0.560467	−	0.828177i	$-0.689378\pi$
$823$	−4.58435e30	−1.12093	−0.560467	+	0.828177i	$0.689378\pi$
$824$	0	0
$825$	2.99954e30	0.714972
$826$	0	0
$827$	2.33564e30	0.542748	0.271374	−	0.962474i	$-0.412522\pi$
$827$	2.33564e30	0.542748	0.271374	+	0.962474i	$0.412522\pi$
$828$	0	0
$829$	3.05277e30	0.691627	0.345813	−	0.938303i	$-0.387603\pi$
$829$	3.05277e30	0.691627	0.345813	+	0.938303i	$0.387603\pi$
$830$	0	0
$831$	6.29763e30	1.39113
$832$	0	0
$833$	−1.84917e30	−0.398296
$834$	0	0
$835$	9.90893e30	2.08122
$836$	0	0
$837$	−3.24827e29	−0.0665327
$838$	0	0
$839$	−9.26615e30	−1.85097	−0.925484	−	0.378787i	$-0.876341\pi$
$839$	−9.26615e30	−1.85097	−0.925484	+	0.378787i	$0.876341\pi$
$840$	0	0
$841$	−3.49239e30	−0.680401
$842$	0	0
$843$	−8.43596e30	−1.60304
$844$	0	0
$845$	6.69466e30	1.24089
$846$	0	0
$847$	−5.43942e30	−0.983505
$848$	0	0
$849$	−9.95928e30	−1.75670
$850$	0	0
$851$	−1.12877e31	−1.94244
$852$	0	0
$853$	−2.53001e30	−0.424774	−0.212387	−	0.977186i	$-0.568124\pi$
$853$	−2.53001e30	−0.424774	−0.212387	+	0.977186i	$0.568124\pi$
$854$	0	0
$855$	−1.15643e31	−1.89442
$856$	0	0
$857$	−1.53568e30	−0.245471	−0.122736	−	0.992439i	$-0.539167\pi$
$857$	−1.53568e30	−0.245471	−0.122736	+	0.992439i	$0.539167\pi$
$858$	0	0
$859$	6.68298e30	1.04242	0.521209	−	0.853429i	$-0.325481\pi$
$859$	6.68298e30	1.04242	0.521209	+	0.853429i	$0.325481\pi$
$860$	0	0
$861$	−7.42417e30	−1.13009
$862$	0	0
$863$	5.81975e29	0.0864552	0.0432276	−	0.999065i	$-0.486236\pi$
$863$	5.81975e29	0.0864552	0.0432276	+	0.999065i	$0.486236\pi$
$864$	0	0
$865$	−1.02462e30	−0.148558
$866$	0	0
$867$	−3.89272e30	−0.550875
$868$	0	0
$869$	1.60965e30	0.222343
$870$	0	0
$871$	8.60345e30	1.16006
$872$	0	0
$873$	1.46977e31	1.93464
$874$	0	0
$875$	2.57326e30	0.330673
$876$	0	0
$877$	−1.19007e30	−0.149306	−0.0746529	−	0.997210i	$-0.523785\pi$
$877$	−1.19007e30	−0.149306	−0.0746529	+	0.997210i	$0.523785\pi$
$878$	0	0
$879$	1.83676e31	2.24993
$880$	0	0
$881$	4.82538e30	0.577144	0.288572	−	0.957458i	$-0.406819\pi$
$881$	4.82538e30	0.577144	0.288572	+	0.957458i	$0.406819\pi$
$882$	0	0
$883$	−1.16938e31	−1.36574	−0.682871	−	0.730539i	$-0.739269\pi$
$883$	−1.16938e31	−1.36574	−0.682871	+	0.730539i	$0.739269\pi$
$884$	0	0
$885$	7.33943e30	0.837062
$886$	0	0
$887$	1.70621e31	1.90036	0.950179	−	0.311706i	$-0.100900\pi$
$887$	1.70621e31	1.90036	0.950179	+	0.311706i	$0.100900\pi$
$888$	0	0
$889$	8.43601e30	0.917632
$890$	0	0
$891$	1.74098e30	0.184960
$892$	0	0
$893$	−1.01995e31	−1.05837
$894$	0	0
$895$	2.07642e31	2.10462
$896$	0	0
$897$	−2.53339e31	−2.50831
$898$	0	0
$899$	6.63785e29	0.0642024
$900$	0	0
$901$	5.47794e30	0.517616
$902$	0	0
$903$	−2.07658e31	−1.91702
$904$	0	0
$905$	2.91724e31	2.63125
$906$	0	0
$907$	−9.46784e29	−0.0834400	−0.0417200	−	0.999129i	$-0.513284\pi$
$907$	−9.46784e29	−0.0834400	−0.0417200	+	0.999129i	$0.513284\pi$
$908$	0	0
$909$	−6.28042e29	−0.0540839
$910$	0	0
$911$	−1.97710e31	−1.66374	−0.831871	−	0.554969i	$-0.812730\pi$
$911$	−1.97710e31	−1.66374	−0.831871	+	0.554969i	$0.812730\pi$
$912$	0	0
$913$	9.03028e29	0.0742605
$914$	0	0
$915$	−2.69818e30	−0.216845
$916$	0	0
$917$	3.61984e30	0.284323
$918$	0	0
$919$	−1.34419e31	−1.03192	−0.515962	−	0.856611i	$-0.672565\pi$
$919$	−1.34419e31	−1.03192	−0.515962	+	0.856611i	$0.672565\pi$
$920$	0	0
$921$	1.28176e29	0.00961794
$922$	0	0
$923$	1.24851e31	0.915746
$924$	0	0
$925$	−2.69432e31	−1.93179
$926$	0	0
$927$	1.52696e31	1.07026
$928$	0	0
$929$	1.37918e31	0.945055	0.472528	−	0.881316i	$-0.343342\pi$
$929$	1.37918e31	0.945055	0.472528	+	0.881316i	$0.343342\pi$
$930$	0	0
$931$	−4.73199e30	−0.317010
$932$	0	0
$933$	1.54533e31	1.01220
$934$	0	0
$935$	1.04662e31	0.670296
$936$	0	0
$937$	9.41539e30	0.589621	0.294810	−	0.955556i	$-0.404744\pi$
$937$	9.41539e30	0.589621	0.294810	+	0.955556i	$0.404744\pi$
$938$	0	0
$939$	5.02992e30	0.308015
$940$	0	0
$941$	−2.69637e31	−1.61469	−0.807345	−	0.590080i	$-0.799096\pi$
$941$	−2.69637e31	−1.61469	−0.807345	+	0.590080i	$0.799096\pi$
$942$	0	0
$943$	1.29510e31	0.758455
$944$	0	0
$945$	−1.75460e31	−1.00495
$946$	0	0
$947$	−1.04077e31	−0.583018	−0.291509	−	0.956568i	$-0.594157\pi$
$947$	−1.04077e31	−0.583018	−0.291509	+	0.956568i	$0.594157\pi$
$948$	0	0
$949$	2.75977e31	1.51209
$950$	0	0
$951$	2.24445e31	1.20286
$952$	0	0
$953$	−1.24384e29	−0.00652062	−0.00326031	−	0.999995i	$-0.501038\pi$
$953$	−1.24384e29	−0.00652062	−0.00326031	+	0.999995i	$0.501038\pi$
$954$	0	0
$955$	−2.46940e31	−1.26636
$956$	0	0
$957$	6.75475e30	0.338871
$958$	0	0
$959$	4.31285e31	2.11675
$960$	0	0
$961$	−2.05569e31	−0.987103
$962$	0	0
$963$	−3.80260e31	−1.78651
$964$	0	0
$965$	4.84374e30	0.222661
$966$	0	0
$967$	−3.66804e31	−1.64989	−0.824947	−	0.565210i	$-0.808795\pi$
$967$	−3.66804e31	−1.64989	−0.824947	+	0.565210i	$0.808795\pi$
$968$	0	0
$969$	−3.78569e31	−1.66627
$970$	0	0
$971$	2.17312e31	0.936011	0.468005	−	0.883726i	$-0.344973\pi$
$971$	2.17312e31	0.936011	0.468005	+	0.883726i	$0.344973\pi$
$972$	0	0
$973$	−7.40193e30	−0.312004
$974$	0	0
$975$	−6.04706e31	−2.49457
$976$	0	0
$977$	−2.89893e31	−1.17043	−0.585215	−	0.810879i	$-0.698990\pi$
$977$	−2.89893e31	−1.17043	−0.585215	+	0.810879i	$0.698990\pi$
$978$	0	0
$979$	−1.31789e30	−0.0520788
$980$	0	0
$981$	5.32803e31	2.06083
$982$	0	0
$983$	1.61866e31	0.612835	0.306418	−	0.951897i	$-0.400870\pi$
$983$	1.61866e31	0.612835	0.306418	+	0.951897i	$0.400870\pi$
$984$	0	0
$985$	1.44486e31	0.535483
$986$	0	0
$987$	−5.62242e31	−2.03983
$988$	0	0
$989$	3.62246e31	1.28660
$990$	0	0
$991$	1.29555e31	0.450487	0.225243	−	0.974303i	$-0.427682\pi$
$991$	1.29555e31	0.450487	0.225243	+	0.974303i	$0.427682\pi$
$992$	0	0
$993$	−2.75717e31	−0.938634
$994$	0	0
$995$	4.11203e30	0.137061
$996$	0	0
$997$	−5.20161e31	−1.69761	−0.848806	−	0.528704i	$-0.822678\pi$
$997$	−5.20161e31	−1.69761	−0.848806	+	0.528704i	$0.822678\pi$
$998$	0	0
$999$	2.96912e31	0.948835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.22.a.d.1.1		2
4.3	odd	2		4.22.a.a.1.2	✓	2
8.3	odd	2		64.22.a.i.1.1		2
8.5	even	2		64.22.a.j.1.2		2
12.11	even	2		36.22.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.22.a.a.1.2	✓	2	4.3	odd	2
16.22.a.d.1.1		2	1.1	even	1	trivial
36.22.a.c.1.1		2	12.11	even	2
64.22.a.i.1.1		2	8.3	odd	2
64.22.a.j.1.2		2	8.5	even	2

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-157776. q^{3} +3.23357e7 q^{5} +8.65744e8 q^{7} +1.44329e10 q^{9} +O(q^{10})\)	\(q-157776. q^{3} +3.23357e7 q^{5} +8.65744e8 q^{7} +1.44329e10 q^{9} -3.34262e10 q^{11} +6.73870e11 q^{13} -5.10179e12 q^{15} -9.68320e12 q^{17} -2.47791e13 q^{19} -1.36594e14 q^{21} +2.38279e14 q^{23} +5.68758e14 q^{25} -6.26767e14 q^{27} +1.28080e15 q^{29} +5.18258e14 q^{31} +5.27385e15 q^{33} +2.79944e16 q^{35} -4.73720e16 q^{37} -1.06320e17 q^{39} +5.43523e16 q^{41} +1.52026e17 q^{43} +4.66696e17 q^{45} +4.11617e17 q^{47} +1.90967e17 q^{49} +1.52778e18 q^{51} -5.65716e17 q^{53} -1.08086e18 q^{55} +3.90954e18 q^{57} -1.43860e18 q^{59} +5.28870e17 q^{61} +1.24952e19 q^{63} +2.17900e19 q^{65} +1.27672e19 q^{67} -3.75946e19 q^{69} +1.85275e19 q^{71} +4.09540e19 q^{73} -8.97362e19 q^{75} -2.89386e19 q^{77} -4.81552e19 q^{79} -5.20842e19 q^{81} -2.70155e19 q^{83} -3.13113e20 q^{85} -2.02079e20 q^{87} +3.94268e19 q^{89} +5.83399e20 q^{91} -8.17687e19 q^{93} -8.01248e20 q^{95} +1.01835e21 q^{97} -4.82437e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9}+O(q^{10}) \) 2 * q - 65640 * q^3 + 13689324 * q^5 + 260508080 * q^7 + 12461535162 * q^9	\( 2 q - 65640 q^{3} + 13689324 q^{5} + 260508080 q^{7} + 12461535162 q^{9} - 145435963320 q^{11} + 1428900417340 q^{13} - 6819782714352 q^{15} + 1840620576420 q^{17} + 16780743928568 q^{19} - 192357511002048 q^{21} + 319691925426960 q^{23} + 439606295919326 q^{25} - 17\!\cdots\!40 q^{27}+ \cdots - 26\!\cdots\!20 q^{99}+O(q^{100}) \) 2 * q - 65640 * q^3 + 13689324 * q^5 + 260508080 * q^7 + 12461535162 * q^9 - 145435963320 * q^11 + 1428900417340 * q^13 - 6819782714352 * q^15 + 1840620576420 * q^17 + 16780743928568 * q^19 - 192357511002048 * q^21 + 319691925426960 * q^23 + 439606295919326 * q^25 - 1772171769223440 * q^27 + 3742111775766492 * q^29 - 112042353462592 * q^31 - 5046260260851360 * q^33 + 39279847462424352 * q^35 - 33362705637547220 * q^37 - 36755051864070192 * q^39 + 175129744323133332 * q^41 - 15346613416528120 * q^43 + 503454557153115324 * q^45 + 684848819288455200 * q^47 - 1267753878311886 * q^49 + 2589533339352286896 * q^51 + 675305394244421580 * q^53 + 1007711578141009200 * q^55 + 7738689471355209120 * q^57 - 1042445250435434904 * q^59 + 9065997829736468764 * q^61 + 13688298514444201200 * q^63 + 7711482582413403048 * q^65 + 30464301046802775320 * q^67 - 30093532090729291584 * q^69 + 8199093502830518064 * q^71 + 25415086659374793940 * q^73 - 101635704862879905048 * q^75 + 38853770260581178560 * q^77 - 121204353225060164896 * q^79 - 136996120065469586382 * q^81 - 108742936757033809800 * q^83 - 527989845644919754344 * q^85 + 24695689277789874000 * q^87 - 184207999274965368972 * q^89 + 126427419539014666912 * q^91 - 139841964206001442560 * q^93 - 1576185911066925315504 * q^95 + 739497785188476467140 * q^97 - 261627767970577922520 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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