LMFDB - Embedded newform 600.8.f.a.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,8,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$187.431015290$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.8.f.a.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000i q^{3} -120.000i q^{7} -729.000 q^{9} +O(q^{10})$	$q+27.0000i q^{3} -120.000i q^{7} -729.000 q^{9} -7196.00 q^{11} -9626.00i q^{13} -18674.0i q^{17} -7004.00 q^{19} +3240.00 q^{21} -63704.0i q^{23} -19683.0i q^{27} -29334.0 q^{29} +87968.0 q^{31} -194292. i q^{33} -227982. i q^{37} +259902. q^{39} -160806. q^{41} +136132. i q^{43} +1.20696e6i q^{47} +809143. q^{49} +504198. q^{51} -398786. i q^{53} -189108. i q^{57} -1.15244e6 q^{59} -2.07060e6 q^{61} +87480.0i q^{63} +4.07343e6i q^{67} +1.72001e6 q^{69} -383752. q^{71} +3.00601e6i q^{73} +863520. i q^{77} +4.94811e6 q^{79} +531441. q^{81} -9.16349e6i q^{83} -792018. i q^{87} -7.30411e6 q^{89} -1.15512e6 q^{91} +2.37514e6i q^{93} +690526. i q^{97} +5.24588e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1458 q^{9}+O(q^{10}) $ 2 * q - 1458 * q^9	$ 2 q - 1458 q^{9} - 14392 q^{11} - 14008 q^{19} + 6480 q^{21} - 58668 q^{29} + 175936 q^{31} + 519804 q^{39} - 321612 q^{41} + 1618286 q^{49} + 1008396 q^{51} - 2304872 q^{59} - 4141204 q^{61} + 3440016 q^{69} - 767504 q^{71} + 9896224 q^{79} + 1062882 q^{81} - 14608212 q^{89} - 2310240 q^{91} + 10491768 q^{99}+O(q^{100}) $ 2 * q - 1458 * q^9 - 14392 * q^11 - 14008 * q^19 + 6480 * q^21 - 58668 * q^29 + 175936 * q^31 + 519804 * q^39 - 321612 * q^41 + 1618286 * q^49 + 1008396 * q^51 - 2304872 * q^59 - 4141204 * q^61 + 3440016 * q^69 - 767504 * q^71 + 9896224 * q^79 + 1062882 * q^81 - 14608212 * q^89 - 2310240 * q^91 + 10491768 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	27.0000i	0.577350i
$3$	27.0000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 120.000i	− 0.132232i	−0.997812	−	0.0661162i	$-0.978939\pi$
$7$	− 120.000i	− 0.132232i	0.997812	−	0.0661162i	$-0.0210608\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	−7196.00	−1.63011	−0.815055	−	0.579384i	$-0.803293\pi$
$11$	−7196.00	−1.63011	−0.815055	+	0.579384i	$0.803293\pi$
$12$	0	0
$13$	− 9626.00i	− 1.21519i	−0.794247	−	0.607595i	$-0.792134\pi$
$13$	− 9626.00i	− 1.21519i	0.794247	−	0.607595i	$-0.207866\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 18674.0i	− 0.921862i	−0.887436	−	0.460931i	$-0.847515\pi$
$17$	− 18674.0i	− 0.921862i	0.887436	−	0.460931i	$-0.152485\pi$
$18$	0	0
$19$	−7004.00	−0.234266	−0.117133	−	0.993116i	$-0.537370\pi$
$19$	−7004.00	−0.234266	−0.117133	+	0.993116i	$0.537370\pi$
$20$	0	0
$21$	3240.00	0.0763445
$22$	0	0
$23$	− 63704.0i	− 1.09174i	−0.837870	−	0.545870i	$-0.816199\pi$
$23$	− 63704.0i	− 1.09174i	0.837870	−	0.545870i	$-0.183801\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 19683.0i	− 0.192450i
$28$	0	0
$29$	−29334.0	−0.223346	−0.111673	−	0.993745i	$-0.535621\pi$
$29$	−29334.0	−0.223346	−0.111673	+	0.993745i	$0.535621\pi$
$30$	0	0
$31$	87968.0	0.530345	0.265173	−	0.964201i	$-0.414571\pi$
$31$	87968.0	0.530345	0.265173	+	0.964201i	$0.414571\pi$
$32$	0	0
$33$	− 194292.i	− 0.941144i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 227982.i	− 0.739937i	−0.929044	−	0.369968i	$-0.879369\pi$
$37$	− 227982.i	− 0.739937i	0.929044	−	0.369968i	$-0.120631\pi$
$38$	0	0
$39$	259902.	0.701590
$40$	0	0
$41$	−160806.	−0.364384	−0.182192	−	0.983263i	$-0.558319\pi$
$41$	−160806.	−0.364384	−0.182192	+	0.983263i	$0.558319\pi$
$42$	0	0
$43$	136132.i	0.261108i	0.991441	+	0.130554i	$0.0416756\pi$
$43$	136132.i	0.261108i	−0.991441	+	0.130554i	$0.958324\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.20696e6i	1.69571i	0.530232	+	0.847853i	$0.322105\pi$
$47$	1.20696e6i	1.69571i	−0.530232	+	0.847853i	$0.677895\pi$
$48$	0	0
$49$	809143.	0.982515
$50$	0	0
$51$	504198.	0.532238
$52$	0	0
$53$	− 398786.i	− 0.367938i	−0.982932	−	0.183969i	$-0.941105\pi$
$53$	− 398786.i	− 0.367938i	0.982932	−	0.183969i	$-0.0588946\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 189108.i	− 0.135253i
$58$	0	0
$59$	−1.15244e6	−0.730524	−0.365262	−	0.930905i	$-0.619021\pi$
$59$	−1.15244e6	−0.730524	−0.365262	+	0.930905i	$0.619021\pi$
$60$	0	0
$61$	−2.07060e6	−1.16800	−0.583999	−	0.811754i	$-0.698513\pi$
$61$	−2.07060e6	−1.16800	−0.583999	+	0.811754i	$0.698513\pi$
$62$	0	0
$63$	87480.0i	0.0440775i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.07343e6i	1.65462i	0.561746	+	0.827310i	$0.310130\pi$
$67$	4.07343e6i	1.65462i	−0.561746	+	0.827310i	$0.689870\pi$
$68$	0	0
$69$	1.72001e6	0.630316
$70$	0	0
$71$	−383752.	−0.127247	−0.0636233	−	0.997974i	$-0.520266\pi$
$71$	−383752.	−0.127247	−0.0636233	+	0.997974i	$0.520266\pi$
$72$	0	0
$73$	3.00601e6i	0.904400i	0.891917	+	0.452200i	$0.149361\pi$
$73$	3.00601e6i	0.904400i	−0.891917	+	0.452200i	$0.850639\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	863520.i	0.215553i
$78$	0	0
$79$	4.94811e6	1.12913	0.564566	−	0.825388i	$-0.309044\pi$
$79$	4.94811e6	1.12913	0.564566	+	0.825388i	$0.309044\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	− 9.16349e6i	− 1.75909i	−0.475817	−	0.879544i	$-0.657848\pi$
$83$	− 9.16349e6i	− 1.75909i	0.475817	−	0.879544i	$-0.342152\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 792018.i	− 0.128949i
$88$	0	0
$89$	−7.30411e6	−1.09825	−0.549126	−	0.835740i	$-0.685039\pi$
$89$	−7.30411e6	−1.09825	−0.549126	+	0.835740i	$0.685039\pi$
$90$	0	0
$91$	−1.15512e6	−0.160688
$92$	0	0
$93$	2.37514e6i	0.306195i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	690526.i	0.0768208i	0.999262	+	0.0384104i	$0.0122294\pi$
$97$	690526.i	0.0768208i	−0.999262	+	0.0384104i	$0.987771\pi$
$98$	0	0
$99$	5.24588e6	0.543370
$100$	0	0
$101$	1.32667e7	1.28126	0.640631	−	0.767849i	$-0.278673\pi$
$101$	1.32667e7	1.28126	0.640631	+	0.767849i	$0.278673\pi$
$102$	0	0
$103$	4.86825e6i	0.438978i	0.975615	+	0.219489i	$0.0704389\pi$
$103$	4.86825e6i	0.438978i	−0.975615	+	0.219489i	$0.929561\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 7.91860e6i	− 0.624892i	−0.949936	−	0.312446i	$-0.898852\pi$
$107$	− 7.91860e6i	− 0.624892i	0.949936	−	0.312446i	$-0.101148\pi$
$108$	0	0
$109$	4.93030e6	0.364654	0.182327	−	0.983238i	$-0.441637\pi$
$109$	4.93030e6	0.364654	0.182327	+	0.983238i	$0.441637\pi$
$110$	0	0
$111$	6.15551e6	0.427203
$112$	0	0
$113$	2.31707e6i	0.151066i	0.997143	+	0.0755328i	$0.0240657\pi$
$113$	2.31707e6i	0.151066i	−0.997143	+	0.0755328i	$0.975934\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	7.01735e6i	0.405063i
$118$	0	0
$119$	−2.24088e6	−0.121900
$120$	0	0
$121$	3.22952e7	1.65726
$122$	0	0
$123$	− 4.34176e6i	− 0.210377i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.25119e7i	0.975214i	0.873063	+	0.487607i	$0.162130\pi$
$127$	2.25119e7i	0.975214i	−0.873063	+	0.487607i	$0.837870\pi$
$128$	0	0
$129$	−3.67556e6	−0.150751
$130$	0	0
$131$	−1.35895e7	−0.528147	−0.264073	−	0.964503i	$-0.585066\pi$
$131$	−1.35895e7	−0.528147	−0.264073	+	0.964503i	$0.585066\pi$
$132$	0	0
$133$	840480.i	0.0309775i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 714618.i	− 0.0237439i	−0.999930	−	0.0118719i	$-0.996221\pi$
$137$	− 714618.i	− 0.0237439i	0.999930	−	0.0118719i	$-0.00377905\pi$
$138$	0	0
$139$	−1.78816e7	−0.564747	−0.282373	−	0.959305i	$-0.591122\pi$
$139$	−1.78816e7	−0.564747	−0.282373	+	0.959305i	$0.591122\pi$
$140$	0	0
$141$	−3.25879e7	−0.979016
$142$	0	0
$143$	6.92687e7i	1.98089i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.18469e7i	0.567255i
$148$	0	0
$149$	2.00391e7	0.496279	0.248139	−	0.968724i	$-0.420181\pi$
$149$	2.00391e7	0.496279	0.248139	+	0.968724i	$0.420181\pi$
$150$	0	0
$151$	−4.07634e7	−0.963499	−0.481749	−	0.876309i	$-0.659998\pi$
$151$	−4.07634e7	−0.963499	−0.481749	+	0.876309i	$0.659998\pi$
$152$	0	0
$153$	1.36133e7i	0.307287i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.54702e7i	− 1.14396i	−0.820268	−	0.571980i	$-0.806176\pi$
$157$	− 5.54702e7i	− 1.14396i	0.820268	−	0.571980i	$-0.193824\pi$
$158$	0	0
$159$	1.07672e7	0.212429
$160$	0	0
$161$	−7.64448e6	−0.144363
$162$	0	0
$163$	− 1.80344e6i	− 0.0326172i	−0.999867	−	0.0163086i	$-0.994809\pi$
$163$	− 1.80344e6i	− 0.0326172i	0.999867	−	0.0163086i	$-0.00519141\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.99273e6i	− 0.0995673i	−0.998760	−	0.0497837i	$-0.984147\pi$
$167$	− 5.99273e6i	− 0.0995673i	0.998760	−	0.0497837i	$-0.0158532\pi$
$168$	0	0
$169$	−2.99114e7	−0.476686
$170$	0	0
$171$	5.10592e6	0.0780885
$172$	0	0
$173$	1.25631e8i	1.84474i	0.386313	+	0.922368i	$0.373748\pi$
$173$	1.25631e8i	1.84474i	−0.386313	+	0.922368i	$0.626252\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3.11158e7i	− 0.421769i
$178$	0	0
$179$	−4.86498e7	−0.634009	−0.317004	−	0.948424i	$-0.602677\pi$
$179$	−4.86498e7	−0.634009	−0.317004	+	0.948424i	$0.602677\pi$
$180$	0	0
$181$	−4.97548e7	−0.623677	−0.311838	−	0.950135i	$-0.600945\pi$
$181$	−4.97548e7	−0.623677	−0.311838	+	0.950135i	$0.600945\pi$
$182$	0	0
$183$	− 5.59063e7i	− 0.674344i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.34378e8i	1.50274i
$188$	0	0
$189$	−2.36196e6	−0.0254482
$190$	0	0
$191$	1.11324e8	1.15604	0.578021	−	0.816022i	$-0.303825\pi$
$191$	1.11324e8	1.15604	0.578021	+	0.816022i	$0.303825\pi$
$192$	0	0
$193$	− 1.34786e8i	− 1.34957i	−0.738014	−	0.674786i	$-0.764236\pi$
$193$	− 1.34786e8i	− 1.34957i	0.738014	−	0.674786i	$-0.235764\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.97557e7i	− 0.370482i	−0.982693	−	0.185241i	$-0.940693\pi$
$197$	− 3.97557e7i	− 0.370482i	0.982693	−	0.185241i	$-0.0593066\pi$
$198$	0	0
$199$	1.03321e8	0.929398	0.464699	−	0.885469i	$-0.346162\pi$
$199$	1.03321e8	0.929398	0.464699	+	0.885469i	$0.346162\pi$
$200$	0	0
$201$	−1.09983e8	−0.955295
$202$	0	0
$203$	3.52008e6i	0.0295336i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.64402e7i	0.363913i
$208$	0	0
$209$	5.04008e7	0.381879
$210$	0	0
$211$	−1.79475e8	−1.31527	−0.657634	−	0.753337i	$-0.728443\pi$
$211$	−1.79475e8	−1.31527	−0.657634	+	0.753337i	$0.728443\pi$
$212$	0	0
$213$	− 1.03613e7i	− 0.0734659i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.05562e7i	− 0.0701289i
$218$	0	0
$219$	−8.11623e7	−0.522155
$220$	0	0
$221$	−1.79756e8	−1.12024
$222$	0	0
$223$	2.85311e8i	1.72286i	0.507872	+	0.861432i	$0.330432\pi$
$223$	2.85311e8i	1.72286i	−0.507872	+	0.861432i	$0.669568\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.74798e7i	0.155928i	0.996956	+	0.0779638i	$0.0248419\pi$
$227$	2.74798e7i	0.155928i	−0.996956	+	0.0779638i	$0.975158\pi$
$228$	0	0
$229$	8.56562e7	0.471341	0.235670	−	0.971833i	$-0.424271\pi$
$229$	8.56562e7	0.471341	0.235670	+	0.971833i	$0.424271\pi$
$230$	0	0
$231$	−2.33150e7	−0.124450
$232$	0	0
$233$	1.04907e8i	0.543322i	0.962393	+	0.271661i	$0.0875729\pi$
$233$	1.04907e8i	0.543322i	−0.962393	+	0.271661i	$0.912427\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.33599e8i	0.651905i
$238$	0	0
$239$	2.77184e7	0.131333	0.0656667	−	0.997842i	$-0.479083\pi$
$239$	2.77184e7	0.131333	0.0656667	+	0.997842i	$0.479083\pi$
$240$	0	0
$241$	−2.95271e8	−1.35882	−0.679409	−	0.733760i	$-0.737764\pi$
$241$	−2.95271e8	−1.35882	−0.679409	+	0.733760i	$0.737764\pi$
$242$	0	0
$243$	1.43489e7i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.74205e7i	0.284677i
$248$	0	0
$249$	2.47414e8	1.01561
$250$	0	0
$251$	−3.02885e8	−1.20898	−0.604491	−	0.796612i	$-0.706624\pi$
$251$	−3.02885e8	−1.20898	−0.604491	+	0.796612i	$0.706624\pi$
$252$	0	0
$253$	4.58414e8i	1.77966i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.14636e7i	− 0.152370i	−0.997094	−	0.0761852i	$-0.975726\pi$
$257$	− 4.14636e7i	− 0.152370i	0.997094	−	0.0761852i	$-0.0242740\pi$
$258$	0	0
$259$	−2.73578e7	−0.0978436
$260$	0	0
$261$	2.13845e7	0.0744487
$262$	0	0
$263$	4.20007e8i	1.42368i	0.702343	+	0.711839i	$0.252137\pi$
$263$	4.20007e8i	1.42368i	−0.702343	+	0.711839i	$0.747863\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.97211e8i	− 0.634076i
$268$	0	0
$269$	−1.56477e8	−0.490138	−0.245069	−	0.969506i	$-0.578811\pi$
$269$	−1.56477e8	−0.490138	−0.245069	+	0.969506i	$0.578811\pi$
$270$	0	0
$271$	−2.71491e8	−0.828633	−0.414316	−	0.910133i	$-0.635979\pi$
$271$	−2.71491e8	−0.828633	−0.414316	+	0.910133i	$0.635979\pi$
$272$	0	0
$273$	− 3.11882e7i	− 0.0927730i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.80318e7i	0.277133i	0.990353	+	0.138566i	$0.0442494\pi$
$277$	9.80318e7i	0.277133i	−0.990353	+	0.138566i	$0.955751\pi$
$278$	0	0
$279$	−6.41287e7	−0.176782
$280$	0	0
$281$	−4.04839e8	−1.08845	−0.544227	−	0.838938i	$-0.683177\pi$
$281$	−4.04839e8	−1.08845	−0.544227	+	0.838938i	$0.683177\pi$
$282$	0	0
$283$	2.41439e8i	0.633220i	0.948556	+	0.316610i	$0.102545\pi$
$283$	2.41439e8i	0.633220i	−0.948556	+	0.316610i	$0.897455\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.92967e7i	0.0481833i
$288$	0	0
$289$	6.16204e7	0.150170
$290$	0	0
$291$	−1.86442e7	−0.0443525
$292$	0	0
$293$	− 1.95052e8i	− 0.453016i	−0.974009	−	0.226508i	$-0.927269\pi$
$293$	− 1.95052e8i	− 0.453016i	0.974009	−	0.226508i	$-0.0727310\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.41639e8i	0.313715i
$298$	0	0
$299$	−6.13215e8	−1.32667
$300$	0	0
$301$	1.63358e7	0.0345270
$302$	0	0
$303$	3.58201e8i	0.739737i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.02082e8i	0.201356i	0.994919	+	0.100678i	$0.0321012\pi$
$307$	1.02082e8i	0.201356i	−0.994919	+	0.100678i	$0.967899\pi$
$308$	0	0
$309$	−1.31443e8	−0.253444
$310$	0	0
$311$	3.04913e8	0.574797	0.287398	−	0.957811i	$-0.407210\pi$
$311$	3.04913e8	0.574797	0.287398	+	0.957811i	$0.407210\pi$
$312$	0	0
$313$	6.47441e8i	1.19342i	0.802455	+	0.596712i	$0.203527\pi$
$313$	6.47441e8i	1.19342i	−0.802455	+	0.596712i	$0.796473\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.48935e8i	0.262596i	0.991343	+	0.131298i	$0.0419145\pi$
$317$	1.48935e8i	0.262596i	−0.991343	+	0.131298i	$0.958086\pi$
$318$	0	0
$319$	2.11087e8	0.364079
$320$	0	0
$321$	2.13802e8	0.360782
$322$	0	0
$323$	1.30793e8i	0.215961i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.33118e8i	0.210533i
$328$	0	0
$329$	1.44835e8	0.224227
$330$	0	0
$331$	9.68290e8	1.46760	0.733800	−	0.679366i	$-0.237745\pi$
$331$	9.68290e8	1.46760	0.733800	+	0.679366i	$0.237745\pi$
$332$	0	0
$333$	1.66199e8i	0.246646i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.65424e8i	− 0.235447i	−0.993046	−	0.117723i	$-0.962440\pi$
$337$	− 1.65424e8i	− 0.235447i	0.993046	−	0.117723i	$-0.0375596\pi$
$338$	0	0
$339$	−6.25610e7	−0.0872177
$340$	0	0
$341$	−6.33018e8	−0.864521
$342$	0	0
$343$	− 1.95922e8i	− 0.262153i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.12455e9i	1.44486i	0.691443	+	0.722431i	$0.256975\pi$
$347$	1.12455e9i	1.44486i	−0.691443	+	0.722431i	$0.743025\pi$
$348$	0	0
$349$	2.55383e8	0.321590	0.160795	−	0.986988i	$-0.448594\pi$
$349$	2.55383e8	0.321590	0.160795	+	0.986988i	$0.448594\pi$
$350$	0	0
$351$	−1.89469e8	−0.233863
$352$	0	0
$353$	3.45151e8i	0.417635i	0.977955	+	0.208818i	$0.0669615\pi$
$353$	3.45151e8i	0.417635i	−0.977955	+	0.208818i	$0.933038\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.05038e7i	− 0.0703791i
$358$	0	0
$359$	8.11307e8	0.925453	0.462727	−	0.886501i	$-0.346871\pi$
$359$	8.11307e8	0.925453	0.462727	+	0.886501i	$0.346871\pi$
$360$	0	0
$361$	−8.44816e8	−0.945120
$362$	0	0
$363$	8.71972e8i	0.956818i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.76625e8i	0.820125i	0.912057	+	0.410062i	$0.134493\pi$
$367$	7.76625e8i	0.820125i	−0.912057	+	0.410062i	$0.865507\pi$
$368$	0	0
$369$	1.17228e8	0.121461
$370$	0	0
$371$	−4.78543e7	−0.0486533
$372$	0	0
$373$	1.66790e9i	1.66414i	0.554669	+	0.832071i	$0.312845\pi$
$373$	1.66790e9i	1.66414i	−0.554669	+	0.832071i	$0.687155\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.82369e8i	0.271408i
$378$	0	0
$379$	1.25007e9	1.17949	0.589747	−	0.807588i	$-0.299228\pi$
$379$	1.25007e9	1.17949	0.589747	+	0.807588i	$0.299228\pi$
$380$	0	0
$381$	−6.07822e8	−0.563040
$382$	0	0
$383$	1.43374e9i	1.30399i	0.758223	+	0.651995i	$0.226068\pi$
$383$	1.43374e9i	1.30399i	−0.758223	+	0.651995i	$0.773932\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 9.92402e7i	− 0.0870361i
$388$	0	0
$389$	−7.31613e8	−0.630170	−0.315085	−	0.949063i	$-0.602033\pi$
$389$	−7.31613e8	−0.630170	−0.315085	+	0.949063i	$0.602033\pi$
$390$	0	0
$391$	−1.18961e9	−1.00643
$392$	0	0
$393$	− 3.66917e8i	− 0.304926i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.19059e9i	− 0.954983i	−0.878636	−	0.477492i	$-0.841546\pi$
$397$	− 1.19059e9i	− 0.954983i	0.878636	−	0.477492i	$-0.158454\pi$
$398$	0	0
$399$	−2.26930e7	−0.0178849
$400$	0	0
$401$	−2.44638e9	−1.89460	−0.947302	−	0.320343i	$-0.896202\pi$
$401$	−2.44638e9	−1.89460	−0.947302	+	0.320343i	$0.896202\pi$
$402$	0	0
$403$	− 8.46780e8i	− 0.644470i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.64056e9i	1.20618i
$408$	0	0
$409$	−1.83663e9	−1.32737	−0.663684	−	0.748013i	$-0.731008\pi$
$409$	−1.83663e9	−1.32737	−0.663684	+	0.748013i	$0.731008\pi$
$410$	0	0
$411$	1.92947e7	0.0137085
$412$	0	0
$413$	1.38292e8i	0.0965991i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.82802e8i	− 0.326057i
$418$	0	0
$419$	2.77854e9	1.84530	0.922651	−	0.385636i	$-0.126018\pi$
$419$	2.77854e9	1.84530	0.922651	+	0.385636i	$0.126018\pi$
$420$	0	0
$421$	1.16625e9	0.761736	0.380868	−	0.924629i	$-0.375625\pi$
$421$	1.16625e9	0.761736	0.380868	+	0.924629i	$0.375625\pi$
$422$	0	0
$423$	− 8.79874e8i	− 0.565235i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.48472e8i	0.154447i
$428$	0	0
$429$	−1.87025e9	−1.14367
$430$	0	0
$431$	−2.90479e8	−0.174761	−0.0873806	−	0.996175i	$-0.527850\pi$
$431$	−2.90479e8	−0.174761	−0.0873806	+	0.996175i	$0.527850\pi$
$432$	0	0
$433$	2.20651e9i	1.30616i	0.757288	+	0.653082i	$0.226524\pi$
$433$	2.20651e9i	1.30616i	−0.757288	+	0.653082i	$0.773476\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.46183e8i	0.255757i
$438$	0	0
$439$	−9.97564e7	−0.0562749	−0.0281375	−	0.999604i	$-0.508958\pi$
$439$	−9.97564e7	−0.0562749	−0.0281375	+	0.999604i	$0.508958\pi$
$440$	0	0
$441$	−5.89865e8	−0.327505
$442$	0	0
$443$	− 8.18948e8i	− 0.447552i	−0.974641	−	0.223776i	$-0.928162\pi$
$443$	− 8.18948e8i	− 0.447552i	0.974641	−	0.223776i	$-0.0718383\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.41055e8i	0.286527i
$448$	0	0
$449$	2.46241e9	1.28380	0.641900	−	0.766788i	$-0.278146\pi$
$449$	2.46241e9	1.28380	0.641900	+	0.766788i	$0.278146\pi$
$450$	0	0
$451$	1.15716e9	0.593985
$452$	0	0
$453$	− 1.10061e9i	− 0.556276i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.76135e9i	1.35337i	0.736274	+	0.676683i	$0.236583\pi$
$457$	2.76135e9i	1.35337i	−0.736274	+	0.676683i	$0.763417\pi$
$458$	0	0
$459$	−3.67560e8	−0.177413
$460$	0	0
$461$	1.48308e9	0.705036	0.352518	−	0.935805i	$-0.385326\pi$
$461$	1.48308e9	0.705036	0.352518	+	0.935805i	$0.385326\pi$
$462$	0	0
$463$	− 1.42591e9i	− 0.667666i	−0.942632	−	0.333833i	$-0.891658\pi$
$463$	− 1.42591e9i	− 0.667666i	0.942632	−	0.333833i	$-0.108342\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.99579e9i	− 0.906788i	−0.891310	−	0.453394i	$-0.850213\pi$
$467$	− 1.99579e9i	− 0.906788i	0.891310	−	0.453394i	$-0.149787\pi$
$468$	0	0
$469$	4.88811e8	0.218794
$470$	0	0
$471$	1.49769e9	0.660465
$472$	0	0
$473$	− 9.79606e8i	− 0.425635i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.90715e8i	0.122646i
$478$	0	0
$479$	−5.66385e8	−0.235471	−0.117736	−	0.993045i	$-0.537564\pi$
$479$	−5.66385e8	−0.235471	−0.117736	+	0.993045i	$0.537564\pi$
$480$	0	0
$481$	−2.19455e9	−0.899163
$482$	0	0
$483$	− 2.06401e8i	− 0.0833483i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.23609e9i	0.484953i	0.970157	+	0.242476i	$0.0779597\pi$
$487$	1.23609e9i	0.484953i	−0.970157	+	0.242476i	$0.922040\pi$
$488$	0	0
$489$	4.86930e7	0.0188315
$490$	0	0
$491$	−7.49612e8	−0.285793	−0.142896	−	0.989738i	$-0.545642\pi$
$491$	−7.49612e8	−0.285793	−0.142896	+	0.989738i	$0.545642\pi$
$492$	0	0
$493$	5.47783e8i	0.205894i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.60502e7i	0.0168261i
$498$	0	0
$499$	3.65642e9	1.31736	0.658680	−	0.752423i	$-0.271115\pi$
$499$	3.65642e9	1.31736	0.658680	+	0.752423i	$0.271115\pi$
$500$	0	0
$501$	1.61804e8	0.0574852
$502$	0	0
$503$	− 2.19390e9i	− 0.768649i	−0.923198	−	0.384324i	$-0.874434\pi$
$503$	− 2.19390e9i	− 0.768649i	0.923198	−	0.384324i	$-0.125566\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 8.07607e8i	− 0.275215i
$508$	0	0
$509$	−4.24865e9	−1.42804	−0.714018	−	0.700127i	$-0.753127\pi$
$509$	−4.24865e9	−1.42804	−0.714018	+	0.700127i	$0.753127\pi$
$510$	0	0
$511$	3.60721e8	0.119591
$512$	0	0
$513$	1.37860e8i	0.0450844i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 8.68528e9i	− 2.76419i
$518$	0	0
$519$	−3.39203e9	−1.06506
$520$	0	0
$521$	−3.70617e8	−0.114814	−0.0574068	−	0.998351i	$-0.518283\pi$
$521$	−3.70617e8	−0.114814	−0.0574068	+	0.998351i	$0.518283\pi$
$522$	0	0
$523$	6.33645e9i	1.93682i	0.249356	+	0.968412i	$0.419781\pi$
$523$	6.33645e9i	1.93682i	−0.249356	+	0.968412i	$0.580219\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.64271e9i	− 0.488905i
$528$	0	0
$529$	−6.53374e8	−0.191897
$530$	0	0
$531$	8.40126e8	0.243508
$532$	0	0
$533$	1.54792e9i	0.442795i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.31354e9i	− 0.366045i
$538$	0	0
$539$	−5.82259e9	−1.60161
$540$	0	0
$541$	−4.00792e9	−1.08825	−0.544125	−	0.839004i	$-0.683138\pi$
$541$	−4.00792e9	−1.08825	−0.544125	+	0.839004i	$0.683138\pi$
$542$	0	0
$543$	− 1.34338e9i	− 0.360080i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3.66046e9i	0.956269i	0.878287	+	0.478135i	$0.158687\pi$
$547$	3.66046e9i	0.956269i	−0.878287	+	0.478135i	$0.841313\pi$
$548$	0	0
$549$	1.50947e9	0.389333
$550$	0	0
$551$	2.05455e8	0.0523223
$552$	0	0
$553$	− 5.93773e8i	− 0.149308i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.25630e9i	− 1.28880i	−0.764687	−	0.644402i	$-0.777106\pi$
$557$	− 5.25630e9i	− 1.28880i	0.764687	−	0.644402i	$-0.222894\pi$
$558$	0	0
$559$	1.31041e9	0.317296
$560$	0	0
$561$	−3.62821e9	−0.867605
$562$	0	0
$563$	− 5.01985e9i	− 1.18553i	−0.805376	−	0.592764i	$-0.798037\pi$
$563$	− 5.01985e9i	− 1.18553i	0.805376	−	0.592764i	$-0.201963\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 6.37729e7i	− 0.0146925i
$568$	0	0
$569$	−2.80495e9	−0.638311	−0.319155	−	0.947702i	$-0.603399\pi$
$569$	−2.80495e9	−0.638311	−0.319155	+	0.947702i	$0.603399\pi$
$570$	0	0
$571$	6.10454e9	1.37223	0.686115	−	0.727493i	$-0.259315\pi$
$571$	6.10454e9	1.37223	0.686115	+	0.727493i	$0.259315\pi$
$572$	0	0
$573$	3.00576e9i	0.667442i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	3.02864e8i	0.0656346i	0.999461	+	0.0328173i	$0.0104479\pi$
$577$	3.02864e8i	0.0656346i	−0.999461	+	0.0328173i	$0.989552\pi$
$578$	0	0
$579$	3.63924e9	0.779175
$580$	0	0
$581$	−1.09962e9	−0.232609
$582$	0	0
$583$	2.86966e9i	0.599779i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.42370e9i	0.494590i	0.968940	+	0.247295i	$0.0795417\pi$
$587$	2.42370e9i	0.494590i	−0.968940	+	0.247295i	$0.920458\pi$
$588$	0	0
$589$	−6.16128e8	−0.124242
$590$	0	0
$591$	1.07340e9	0.213898
$592$	0	0
$593$	− 6.85538e9i	− 1.35002i	−0.737809	−	0.675010i	$-0.764139\pi$
$593$	− 6.85538e9i	− 1.35002i	0.737809	−	0.675010i	$-0.235861\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.78966e9i	0.536588i
$598$	0	0
$599$	−1.47203e8	−0.0279847	−0.0139924	−	0.999902i	$-0.504454\pi$
$599$	−1.47203e8	−0.0279847	−0.0139924	+	0.999902i	$0.504454\pi$
$600$	0	0
$601$	7.15884e9	1.34519	0.672593	−	0.740013i	$-0.265181\pi$
$601$	7.15884e9	1.34519	0.672593	+	0.740013i	$0.265181\pi$
$602$	0	0
$603$	− 2.96953e9i	− 0.551540i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.08333e9i	− 0.741062i	−0.928820	−	0.370531i	$-0.879176\pi$
$607$	− 4.08333e9i	− 0.741062i	0.928820	−	0.370531i	$-0.120824\pi$
$608$	0	0
$609$	−9.50422e7	−0.0170512
$610$	0	0
$611$	1.16182e10	2.06060
$612$	0	0
$613$	− 6.81081e9i	− 1.19423i	−0.802157	−	0.597113i	$-0.796314\pi$
$613$	− 6.81081e9i	− 1.19423i	0.802157	−	0.597113i	$-0.203686\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.95161e9i	− 0.677292i	−0.940914	−	0.338646i	$-0.890031\pi$
$617$	− 3.95161e9i	− 0.677292i	0.940914	−	0.338646i	$-0.109969\pi$
$618$	0	0
$619$	1.07835e10	1.82744	0.913722	−	0.406339i	$-0.133195\pi$
$619$	1.07835e10	1.82744	0.913722	+	0.406339i	$0.133195\pi$
$620$	0	0
$621$	−1.25389e9	−0.210105
$622$	0	0
$623$	8.76493e8i	0.145225i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.36082e9i	0.220478i
$628$	0	0
$629$	−4.25734e9	−0.682120
$630$	0	0
$631$	1.10443e10	1.75000	0.874999	−	0.484125i	$-0.160862\pi$
$631$	1.10443e10	1.75000	0.874999	+	0.484125i	$0.160862\pi$
$632$	0	0
$633$	− 4.84582e9i	− 0.759371i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 7.78881e9i	− 1.19394i
$638$	0	0
$639$	2.79755e8	0.0424156
$640$	0	0
$641$	−5.93797e9	−0.890502	−0.445251	−	0.895406i	$-0.646886\pi$
$641$	−5.93797e9	−0.890502	−0.445251	+	0.895406i	$0.646886\pi$
$642$	0	0
$643$	− 8.24945e9i	− 1.22373i	−0.790961	−	0.611866i	$-0.790419\pi$
$643$	− 8.24945e9i	− 1.22373i	0.790961	−	0.611866i	$-0.209581\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.49145e9i	0.942272i	0.882061	+	0.471136i	$0.156156\pi$
$647$	6.49145e9i	0.942272i	−0.882061	+	0.471136i	$0.843844\pi$
$648$	0	0
$649$	8.29293e9	1.19083
$650$	0	0
$651$	2.85016e8	0.0404889
$652$	0	0
$653$	− 1.04406e9i	− 0.146734i	−0.997305	−	0.0733668i	$-0.976626\pi$
$653$	− 1.04406e9i	− 0.146734i	0.997305	−	0.0733668i	$-0.0233744\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.19138e9i	− 0.301467i
$658$	0	0
$659$	−1.11126e10	−1.51257	−0.756285	−	0.654243i	$-0.772988\pi$
$659$	−1.11126e10	−1.51257	−0.756285	+	0.654243i	$0.772988\pi$
$660$	0	0
$661$	−1.70105e9	−0.229093	−0.114547	−	0.993418i	$-0.536541\pi$
$661$	−1.70105e9	−0.229093	−0.114547	+	0.993418i	$0.536541\pi$
$662$	0	0
$663$	− 4.85341e9i	− 0.646770i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.86869e9i	0.243836i
$668$	0	0
$669$	−7.70339e9	−0.994696
$670$	0	0
$671$	1.49001e10	1.90397
$672$	0	0
$673$	− 7.45938e9i	− 0.943301i	−0.881786	−	0.471650i	$-0.843658\pi$
$673$	− 7.45938e9i	− 0.943301i	0.881786	−	0.471650i	$-0.156342\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.80473e9i	0.718988i	0.933147	+	0.359494i	$0.117051\pi$
$677$	5.80473e9i	0.718988i	−0.933147	+	0.359494i	$0.882949\pi$
$678$	0	0
$679$	8.28631e7	0.0101582
$680$	0	0
$681$	−7.41954e8	−0.0900248
$682$	0	0
$683$	− 7.26903e9i	− 0.872979i	−0.899709	−	0.436490i	$-0.856222\pi$
$683$	− 7.26903e9i	− 0.872979i	0.899709	−	0.436490i	$-0.143778\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.31272e9i	0.272129i
$688$	0	0
$689$	−3.83871e9	−0.447114
$690$	0	0
$691$	1.17298e10	1.35244	0.676219	−	0.736701i	$-0.263617\pi$
$691$	1.17298e10	1.35244	0.676219	+	0.736701i	$0.263617\pi$
$692$	0	0
$693$	− 6.29506e8i	− 0.0718511i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.00289e9i	0.335912i
$698$	0	0
$699$	−2.83248e9	−0.313687
$700$	0	0
$701$	−9.88743e9	−1.08410	−0.542051	−	0.840345i	$-0.682352\pi$
$701$	−9.88743e9	−1.08410	−0.542051	+	0.840345i	$0.682352\pi$
$702$	0	0
$703$	1.59679e9i	0.173342i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.59200e9i	− 0.169425i
$708$	0	0
$709$	1.43284e10	1.50986	0.754931	−	0.655804i	$-0.227670\pi$
$709$	1.43284e10	1.50986	0.754931	+	0.655804i	$0.227670\pi$
$710$	0	0
$711$	−3.60717e9	−0.376377
$712$	0	0
$713$	− 5.60391e9i	− 0.578999i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.48396e8i	0.0758254i
$718$	0	0
$719$	1.73607e10	1.74187	0.870936	−	0.491396i	$-0.163513\pi$
$719$	1.73607e10	1.74187	0.870936	+	0.491396i	$0.163513\pi$
$720$	0	0
$721$	5.84190e8	0.0580471
$722$	0	0
$723$	− 7.97232e9i	− 0.784514i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.27740e10i	1.23298i	0.787364	+	0.616489i	$0.211446\pi$
$727$	1.27740e10i	1.23298i	−0.787364	+	0.616489i	$0.788554\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	2.54213e9	0.240706
$732$	0	0
$733$	6.12419e9i	0.574361i	0.957877	+	0.287180i	$0.0927179\pi$
$733$	6.12419e9i	0.574361i	−0.957877	+	0.287180i	$0.907282\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.93124e10i	− 2.69721i
$738$	0	0
$739$	−1.05335e10	−0.960104	−0.480052	−	0.877240i	$-0.659382\pi$
$739$	−1.05335e10	−0.960104	−0.480052	+	0.877240i	$0.659382\pi$
$740$	0	0
$741$	−1.82035e9	−0.164358
$742$	0	0
$743$	7.55392e9i	0.675634i	0.941212	+	0.337817i	$0.109688\pi$
$743$	7.55392e9i	0.675634i	−0.941212	+	0.337817i	$0.890312\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.68019e9i	0.586363i
$748$	0	0
$749$	−9.50232e8	−0.0826310
$750$	0	0
$751$	−1.31273e10	−1.13093	−0.565466	−	0.824772i	$-0.691304\pi$
$751$	−1.31273e10	−1.13093	−0.565466	+	0.824772i	$0.691304\pi$
$752$	0	0
$753$	− 8.17790e9i	− 0.698007i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.72126e9i	0.479353i	0.970853	+	0.239677i	$0.0770414\pi$
$757$	5.72126e9i	0.479353i	−0.970853	+	0.239677i	$0.922959\pi$
$758$	0	0
$759$	−1.23772e10	−1.02748
$760$	0	0
$761$	1.63771e10	1.34708	0.673538	−	0.739153i	$-0.264774\pi$
$761$	1.63771e10	1.34708	0.673538	+	0.739153i	$0.264774\pi$
$762$	0	0
$763$	− 5.91636e8i	− 0.0482190i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.10933e10i	0.887726i
$768$	0	0
$769$	6.59634e9	0.523071	0.261536	−	0.965194i	$-0.415771\pi$
$769$	6.59634e9	0.523071	0.261536	+	0.965194i	$0.415771\pi$
$770$	0	0
$771$	1.11952e9	0.0879711
$772$	0	0
$773$	− 6.24126e9i	− 0.486009i	−0.970025	−	0.243004i	$-0.921867\pi$
$773$	− 6.24126e9i	− 0.486009i	0.970025	−	0.243004i	$-0.0781329\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.38662e8i	− 0.0564901i
$778$	0	0
$779$	1.12629e9	0.0853625
$780$	0	0
$781$	2.76148e9	0.207426
$782$	0	0
$783$	5.77381e8i	0.0429830i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.69328e10i	− 1.23827i	−0.785284	−	0.619136i	$-0.787483\pi$
$787$	− 1.69328e10i	− 1.23827i	0.785284	−	0.619136i	$-0.212517\pi$
$788$	0	0
$789$	−1.13402e10	−0.821960
$790$	0	0
$791$	2.78049e8	0.0199758
$792$	0	0
$793$	1.99316e10i	1.41934i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.25579e10i	− 0.878641i	−0.898330	−	0.439321i	$-0.855219\pi$
$797$	− 1.25579e10i	− 0.878641i	0.898330	−	0.439321i	$-0.144781\pi$
$798$	0	0
$799$	2.25388e10	1.56321
$800$	0	0
$801$	5.32469e9	0.366084
$802$	0	0
$803$	− 2.16312e10i	− 1.47427i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.22489e9i	− 0.282981i
$808$	0	0
$809$	2.51860e9	0.167239	0.0836197	−	0.996498i	$-0.473352\pi$
$809$	2.51860e9	0.167239	0.0836197	+	0.996498i	$0.473352\pi$
$810$	0	0
$811$	2.70760e10	1.78243	0.891213	−	0.453585i	$-0.149855\pi$
$811$	2.70760e10	1.78243	0.891213	+	0.453585i	$0.149855\pi$
$812$	0	0
$813$	− 7.33024e9i	− 0.478411i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.53469e8i	− 0.0611687i
$818$	0	0
$819$	8.42082e8	0.0535625
$820$	0	0
$821$	2.70089e10	1.70336	0.851678	−	0.524066i	$-0.175585\pi$
$821$	2.70089e10	1.70336	0.851678	+	0.524066i	$0.175585\pi$
$822$	0	0
$823$	1.23883e10i	0.774662i	0.921941	+	0.387331i	$0.126603\pi$
$823$	1.23883e10i	0.774662i	−0.921941	+	0.387331i	$0.873397\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.28562e10i	1.40519i	0.711589	+	0.702596i	$0.247976\pi$
$827$	2.28562e10i	1.40519i	−0.711589	+	0.702596i	$0.752024\pi$
$828$	0	0
$829$	8.68742e9	0.529602	0.264801	−	0.964303i	$-0.414694\pi$
$829$	8.68742e9	0.529602	0.264801	+	0.964303i	$0.414694\pi$
$830$	0	0
$831$	−2.64686e9	−0.160003
$832$	0	0
$833$	− 1.51099e10i	− 0.905743i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.73147e9i	− 0.102065i
$838$	0	0
$839$	−1.16792e10	−0.682724	−0.341362	−	0.939932i	$-0.610888\pi$
$839$	−1.16792e10	−0.682724	−0.341362	+	0.939932i	$0.610888\pi$
$840$	0	0
$841$	−1.63894e10	−0.950117
$842$	0	0
$843$	− 1.09307e10i	− 0.628419i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 3.87543e9i	− 0.219143i
$848$	0	0
$849$	−6.51884e9	−0.365589
$850$	0	0
$851$	−1.45234e10	−0.807819
$852$	0	0
$853$	2.25452e9i	0.124375i	0.998064	+	0.0621875i	$0.0198077\pi$
$853$	2.25452e9i	0.124375i	−0.998064	+	0.0621875i	$0.980192\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.96605e9i	0.269512i	0.990879	+	0.134756i	$0.0430250\pi$
$857$	4.96605e9i	0.269512i	−0.990879	+	0.134756i	$0.956975\pi$
$858$	0	0
$859$	−1.54039e10	−0.829192	−0.414596	−	0.910006i	$-0.636077\pi$
$859$	−1.54039e10	−0.829192	−0.414596	+	0.910006i	$0.636077\pi$
$860$	0	0
$861$	−5.21011e8	−0.0278187
$862$	0	0
$863$	4.61225e9i	0.244273i	0.992513	+	0.122136i	$0.0389746\pi$
$863$	4.61225e9i	0.244273i	−0.992513	+	0.122136i	$0.961025\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.66375e9i	0.0867005i
$868$	0	0
$869$	−3.56066e10	−1.84061
$870$	0	0
$871$	3.92108e10	2.01068
$872$	0	0
$873$	− 5.03393e8i	− 0.0256069i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 3.57505e10i	− 1.78971i	−0.446355	−	0.894856i	$-0.647278\pi$
$877$	− 3.57505e10i	− 1.78971i	0.446355	−	0.894856i	$-0.352722\pi$
$878$	0	0
$879$	5.26640e9	0.261549
$880$	0	0
$881$	−2.48309e10	−1.22342	−0.611711	−	0.791082i	$-0.709518\pi$
$881$	−2.48309e10	−1.22342	−0.611711	+	0.791082i	$0.709518\pi$
$882$	0	0
$883$	9.37996e9i	0.458499i	0.973368	+	0.229249i	$0.0736272\pi$
$883$	9.37996e9i	0.458499i	−0.973368	+	0.229249i	$0.926373\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 5.98940e9i	− 0.288171i	−0.989565	−	0.144086i	$-0.953976\pi$
$887$	− 5.98940e9i	− 0.288171i	0.989565	−	0.144086i	$-0.0460241\pi$
$888$	0	0
$889$	2.70143e9	0.128955
$890$	0	0
$891$	−3.82425e9	−0.181123
$892$	0	0
$893$	− 8.45355e9i	− 0.397245i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.65568e10i	− 0.765954i
$898$	0	0
$899$	−2.58045e9	−0.118451
$900$	0	0
$901$	−7.44693e9	−0.339188
$902$	0	0
$903$	4.41068e8i	0.0199342i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.76568e9i	− 0.0785755i	−0.999228	−	0.0392878i	$-0.987491\pi$
$907$	− 1.76568e9i	− 0.0785755i	0.999228	−	0.0392878i	$-0.0125089\pi$
$908$	0	0
$909$	−9.67143e9	−0.427088
$910$	0	0
$911$	−4.18232e10	−1.83275	−0.916373	−	0.400325i	$-0.868897\pi$
$911$	−4.18232e10	−1.83275	−0.916373	+	0.400325i	$0.868897\pi$
$912$	0	0
$913$	6.59405e10i	2.86751i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.63074e9i	0.0698382i
$918$	0	0
$919$	3.71658e10	1.57957	0.789785	−	0.613384i	$-0.210192\pi$
$919$	3.71658e10	1.57957	0.789785	+	0.613384i	$0.210192\pi$
$920$	0	0
$921$	−2.75622e9	−0.116253
$922$	0	0
$923$	3.69400e9i	0.154629i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.54895e9i	− 0.146326i
$928$	0	0
$929$	−2.56580e10	−1.04995	−0.524974	−	0.851118i	$-0.675925\pi$
$929$	−2.56580e10	−1.04995	−0.524974	+	0.851118i	$0.675925\pi$
$930$	0	0
$931$	−5.66724e9	−0.230169
$932$	0	0
$933$	8.23264e9i	0.331859i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.75353e9i	0.228479i	0.993453	+	0.114239i	$0.0364431\pi$
$937$	5.75353e9i	0.228479i	−0.993453	+	0.114239i	$0.963557\pi$
$938$	0	0
$939$	−1.74809e10	−0.689024
$940$	0	0
$941$	−4.60599e10	−1.80202	−0.901010	−	0.433799i	$-0.857173\pi$
$941$	−4.60599e10	−1.80202	−0.901010	+	0.433799i	$0.857173\pi$
$942$	0	0
$943$	1.02440e10i	0.397812i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.34709e7i	0.00357645i	0.999998	+	0.00178822i	$0.000569209\pi$
$947$	9.34709e7i	0.00357645i	−0.999998	+	0.00178822i	$0.999431\pi$
$948$	0	0
$949$	2.89359e10	1.09902
$950$	0	0
$951$	−4.02123e9	−0.151610
$952$	0	0
$953$	1.17289e10i	0.438968i	0.975616	+	0.219484i	$0.0704373\pi$
$953$	1.17289e10i	0.438968i	−0.975616	+	0.219484i	$0.929563\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.69936e9i	0.210201i
$958$	0	0
$959$	−8.57542e7	−0.00313971
$960$	0	0
$961$	−1.97742e10	−0.718734
$962$	0	0
$963$	5.77266e9i	0.208297i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.64786e10i	0.586040i	0.956106	+	0.293020i	$0.0946602\pi$
$967$	1.64786e10i	0.586040i	−0.956106	+	0.293020i	$0.905340\pi$
$968$	0	0
$969$	−3.53140e9	−0.124685
$970$	0	0
$971$	−1.88436e10	−0.660536	−0.330268	−	0.943887i	$-0.607139\pi$
$971$	−1.88436e10	−0.660536	−0.330268	+	0.943887i	$0.607139\pi$
$972$	0	0
$973$	2.14579e9i	0.0746779i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.26661e10i	1.46370i	0.681467	+	0.731849i	$0.261343\pi$
$977$	4.26661e10i	1.46370i	−0.681467	+	0.731849i	$0.738657\pi$
$978$	0	0
$979$	5.25603e10	1.79027
$980$	0	0
$981$	−3.59419e9	−0.121551
$982$	0	0
$983$	4.34527e9i	0.145908i	0.997335	+	0.0729541i	$0.0232426\pi$
$983$	4.34527e9i	0.145908i	−0.997335	+	0.0729541i	$0.976757\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.91055e9i	0.129458i
$988$	0	0
$989$	8.67215e9	0.285062
$990$	0	0
$991$	1.36239e10	0.444675	0.222338	−	0.974970i	$-0.428631\pi$
$991$	1.36239e10	0.444675	0.222338	+	0.974970i	$0.428631\pi$
$992$	0	0
$993$	2.61438e10i	0.847319i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	4.32788e10i	1.38306i	0.722346	+	0.691531i	$0.243064\pi$
$997$	4.32788e10i	1.38306i	−0.722346	+	0.691531i	$0.756936\pi$
$998$	0	0
$999$	−4.48737e9	−0.142401

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.8.f.a.49.2		2
5.2	odd	4		24.8.a.b.1.1	✓	1
5.3	odd	4		600.8.a.b.1.1		1
5.4	even	2	inner	600.8.f.a.49.1		2
15.2	even	4		72.8.a.e.1.1		1
20.7	even	4		48.8.a.a.1.1		1
40.27	even	4		192.8.a.p.1.1		1
40.37	odd	4		192.8.a.h.1.1		1
60.47	odd	4		144.8.a.k.1.1		1
120.77	even	4		576.8.a.c.1.1		1
120.107	odd	4		576.8.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.a.b.1.1	✓	1	5.2	odd	4
48.8.a.a.1.1		1	20.7	even	4
72.8.a.e.1.1		1	15.2	even	4
144.8.a.k.1.1		1	60.47	odd	4
192.8.a.h.1.1		1	40.37	odd	4
192.8.a.p.1.1		1	40.27	even	4
576.8.a.b.1.1		1	120.107	odd	4
576.8.a.c.1.1		1	120.77	even	4
600.8.a.b.1.1		1	5.3	odd	4
600.8.f.a.49.1		2	5.4	even	2	inner
600.8.f.a.49.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+27.0000i q^{3} -120.000i q^{7} -729.000 q^{9} +O(q^{10})\)	\(q+27.0000i q^{3} -120.000i q^{7} -729.000 q^{9} -7196.00 q^{11} -9626.00i q^{13} -18674.0i q^{17} -7004.00 q^{19} +3240.00 q^{21} -63704.0i q^{23} -19683.0i q^{27} -29334.0 q^{29} +87968.0 q^{31} -194292. i q^{33} -227982. i q^{37} +259902. q^{39} -160806. q^{41} +136132. i q^{43} +1.20696e6i q^{47} +809143. q^{49} +504198. q^{51} -398786. i q^{53} -189108. i q^{57} -1.15244e6 q^{59} -2.07060e6 q^{61} +87480.0i q^{63} +4.07343e6i q^{67} +1.72001e6 q^{69} -383752. q^{71} +3.00601e6i q^{73} +863520. i q^{77} +4.94811e6 q^{79} +531441. q^{81} -9.16349e6i q^{83} -792018. i q^{87} -7.30411e6 q^{89} -1.15512e6 q^{91} +2.37514e6i q^{93} +690526. i q^{97} +5.24588e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1458 q^{9}+O(q^{10}) \) 2 * q - 1458 * q^9	\( 2 q - 1458 q^{9} - 14392 q^{11} - 14008 q^{19} + 6480 q^{21} - 58668 q^{29} + 175936 q^{31} + 519804 q^{39} - 321612 q^{41} + 1618286 q^{49} + 1008396 q^{51} - 2304872 q^{59} - 4141204 q^{61} + 3440016 q^{69} - 767504 q^{71} + 9896224 q^{79} + 1062882 q^{81} - 14608212 q^{89} - 2310240 q^{91} + 10491768 q^{99}+O(q^{100}) \) 2 * q - 1458 * q^9 - 14392 * q^11 - 14008 * q^19 + 6480 * q^21 - 58668 * q^29 + 175936 * q^31 + 519804 * q^39 - 321612 * q^41 + 1618286 * q^49 + 1008396 * q^51 - 2304872 * q^59 - 4141204 * q^61 + 3440016 * q^69 - 767504 * q^71 + 9896224 * q^79 + 1062882 * q^81 - 14608212 * q^89 - 2310240 * q^91 + 10491768 * q^99

Introduction

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