LMFDB - Embedded newform 8024.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 29 x^{18} + 51 x^{17} + 341 x^{16} - 514 x^{15} - 2114 x^{14} + 2629 x^{13} + \cdots + 4 $ x^20 - 2x^19 - 29x^18 + 51x^17 + 341x^16 - 514x^15 - 2114x^14 + 2629x^13 + 7532x^12 - 7342x^11 - 15668x^10 + 11260x^9 + 18269x^8 - 9059x^7 - 10680x^6 + 3386x^5 + 2473x^4 - 427x^3 - 193x^2 + 18*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.69870$ of defining polynomial
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69870 q^{3} -1.98742 q^{5} -0.0153109 q^{7} +4.28301 q^{9} +O(q^{10})$	$q-2.69870 q^{3} -1.98742 q^{5} -0.0153109 q^{7} +4.28301 q^{9} -0.355869 q^{11} -0.443406 q^{13} +5.36346 q^{15} +1.00000 q^{17} +3.70136 q^{19} +0.0413197 q^{21} -8.46034 q^{23} -1.05017 q^{25} -3.46245 q^{27} -0.330360 q^{29} +0.619079 q^{31} +0.960384 q^{33} +0.0304292 q^{35} -7.74614 q^{37} +1.19662 q^{39} +3.94805 q^{41} +9.66425 q^{43} -8.51213 q^{45} +4.81834 q^{47} -6.99977 q^{49} -2.69870 q^{51} +3.98417 q^{53} +0.707260 q^{55} -9.98887 q^{57} -1.00000 q^{59} -3.54402 q^{61} -0.0655768 q^{63} +0.881233 q^{65} +1.94699 q^{67} +22.8320 q^{69} +4.41855 q^{71} +1.25271 q^{73} +2.83409 q^{75} +0.00544868 q^{77} -4.57478 q^{79} -3.50488 q^{81} +11.1941 q^{83} -1.98742 q^{85} +0.891544 q^{87} -12.2761 q^{89} +0.00678896 q^{91} -1.67071 q^{93} -7.35615 q^{95} -8.70382 q^{97} -1.52419 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	$ 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) $ 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * 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
$3$	−2.69870	−1.55810	−0.779049	−	0.626963i	$-0.784298\pi$
$3$	−2.69870	−1.55810	−0.779049	+	0.626963i	$0.784298\pi$
$4$	0	0
$5$	−1.98742	−0.888801	−0.444400	−	0.895828i	$-0.646583\pi$
$5$	−1.98742	−0.888801	−0.444400	+	0.895828i	$0.646583\pi$
$6$	0	0
$7$	−0.0153109	−0.00578699	−0.00289349	−	0.999996i	$-0.500921\pi$
$7$	−0.0153109	−0.00578699	−0.00289349	+	0.999996i	$0.500921\pi$
$8$	0	0
$9$	4.28301	1.42767
$10$	0	0
$11$	−0.355869	−0.107298	−0.0536492	−	0.998560i	$-0.517085\pi$
$11$	−0.355869	−0.107298	−0.0536492	+	0.998560i	$0.517085\pi$
$12$	0	0
$13$	−0.443406	−0.122979	−0.0614893	−	0.998108i	$-0.519585\pi$
$13$	−0.443406	−0.122979	−0.0614893	+	0.998108i	$0.519585\pi$
$14$	0	0
$15$	5.36346	1.38484
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.70136	0.849149	0.424575	−	0.905393i	$-0.360424\pi$
$19$	3.70136	0.849149	0.424575	+	0.905393i	$0.360424\pi$
$20$	0	0
$21$	0.0413197	0.00901669
$22$	0	0
$23$	−8.46034	−1.76410	−0.882051	−	0.471154i	$-0.843838\pi$
$23$	−8.46034	−1.76410	−0.882051	+	0.471154i	$0.843838\pi$
$24$	0	0
$25$	−1.05017	−0.210033
$26$	0	0
$27$	−3.46245	−0.666350
$28$	0	0
$29$	−0.330360	−0.0613463	−0.0306732	−	0.999529i	$-0.509765\pi$
$29$	−0.330360	−0.0613463	−0.0306732	+	0.999529i	$0.509765\pi$
$30$	0	0
$31$	0.619079	0.111190	0.0555949	−	0.998453i	$-0.482294\pi$
$31$	0.619079	0.111190	0.0555949	+	0.998453i	$0.482294\pi$
$32$	0	0
$33$	0.960384	0.167181
$34$	0	0
$35$	0.0304292	0.00514348
$36$	0	0
$37$	−7.74614	−1.27346	−0.636729	−	0.771088i	$-0.719713\pi$
$37$	−7.74614	−1.27346	−0.636729	+	0.771088i	$0.719713\pi$
$38$	0	0
$39$	1.19662	0.191613
$40$	0	0
$41$	3.94805	0.616582	0.308291	−	0.951292i	$-0.400243\pi$
$41$	3.94805	0.616582	0.308291	+	0.951292i	$0.400243\pi$
$42$	0	0
$43$	9.66425	1.47378	0.736892	−	0.676010i	$-0.236293\pi$
$43$	9.66425	1.47378	0.736892	+	0.676010i	$0.236293\pi$
$44$	0	0
$45$	−8.51213	−1.26891
$46$	0	0
$47$	4.81834	0.702827	0.351414	−	0.936220i	$-0.385701\pi$
$47$	4.81834	0.702827	0.351414	+	0.936220i	$0.385701\pi$
$48$	0	0
$49$	−6.99977	−0.999967
$50$	0	0
$51$	−2.69870	−0.377894
$52$	0	0
$53$	3.98417	0.547267	0.273634	−	0.961834i	$-0.411774\pi$
$53$	3.98417	0.547267	0.273634	+	0.961834i	$0.411774\pi$
$54$	0	0
$55$	0.707260	0.0953669
$56$	0	0
$57$	−9.98887	−1.32306
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−3.54402	−0.453765	−0.226883	−	0.973922i	$-0.572853\pi$
$61$	−3.54402	−0.453765	−0.226883	+	0.973922i	$0.572853\pi$
$62$	0	0
$63$	−0.0655768	−0.00826190
$64$	0	0
$65$	0.881233	0.109304
$66$	0	0
$67$	1.94699	0.237863	0.118932	−	0.992902i	$-0.462053\pi$
$67$	1.94699	0.237863	0.118932	+	0.992902i	$0.462053\pi$
$68$	0	0
$69$	22.8320	2.74864
$70$	0	0
$71$	4.41855	0.524385	0.262193	−	0.965016i	$-0.415554\pi$
$71$	4.41855	0.524385	0.262193	+	0.965016i	$0.415554\pi$
$72$	0	0
$73$	1.25271	0.146619	0.0733095	−	0.997309i	$-0.476644\pi$
$73$	1.25271	0.146619	0.0733095	+	0.997309i	$0.476644\pi$
$74$	0	0
$75$	2.83409	0.327252
$76$	0	0
$77$	0.00544868	0.000620935 0
$78$	0	0
$79$	−4.57478	−0.514703	−0.257352	−	0.966318i	$-0.582850\pi$
$79$	−4.57478	−0.514703	−0.257352	+	0.966318i	$0.582850\pi$
$80$	0	0
$81$	−3.50488	−0.389431
$82$	0	0
$83$	11.1941	1.22871	0.614354	−	0.789031i	$-0.289417\pi$
$83$	11.1941	1.22871	0.614354	+	0.789031i	$0.289417\pi$
$84$	0	0
$85$	−1.98742	−0.215566
$86$	0	0
$87$	0.891544	0.0955835
$88$	0	0
$89$	−12.2761	−1.30127	−0.650634	−	0.759392i	$-0.725497\pi$
$89$	−12.2761	−1.30127	−0.650634	+	0.759392i	$0.725497\pi$
$90$	0	0
$91$	0.00678896	0.000711676 0
$92$	0	0
$93$	−1.67071	−0.173245
$94$	0	0
$95$	−7.35615	−0.754725
$96$	0	0
$97$	−8.70382	−0.883739	−0.441869	−	0.897079i	$-0.645685\pi$
$97$	−8.70382	−0.883739	−0.441869	+	0.897079i	$0.645685\pi$
$98$	0	0
$99$	−1.52419	−0.153187
$100$	0	0
$101$	17.3925	1.73062	0.865309	−	0.501238i	$-0.167122\pi$
$101$	17.3925	1.73062	0.865309	+	0.501238i	$0.167122\pi$
$102$	0	0
$103$	14.3851	1.41741	0.708705	−	0.705505i	$-0.249280\pi$
$103$	14.3851	1.41741	0.708705	+	0.705505i	$0.249280\pi$
$104$	0	0
$105$	−0.0821195	−0.00801405
$106$	0	0
$107$	−1.72240	−0.166511	−0.0832553	−	0.996528i	$-0.526532\pi$
$107$	−1.72240	−0.166511	−0.0832553	+	0.996528i	$0.526532\pi$
$108$	0	0
$109$	7.95041	0.761511	0.380755	−	0.924676i	$-0.375664\pi$
$109$	7.95041	0.761511	0.380755	+	0.924676i	$0.375664\pi$
$110$	0	0
$111$	20.9045	1.98417
$112$	0	0
$113$	10.8026	1.01623	0.508113	−	0.861291i	$-0.330343\pi$
$113$	10.8026	1.01623	0.508113	+	0.861291i	$0.330343\pi$
$114$	0	0
$115$	16.8142	1.56794
$116$	0	0
$117$	−1.89911	−0.175573
$118$	0	0
$119$	−0.0153109	−0.00140355
$120$	0	0
$121$	−10.8734	−0.988487
$122$	0	0
$123$	−10.6546	−0.960694
$124$	0	0
$125$	12.0242	1.07548
$126$	0	0
$127$	11.8660	1.05294	0.526470	−	0.850194i	$-0.323515\pi$
$127$	11.8660	1.05294	0.526470	+	0.850194i	$0.323515\pi$
$128$	0	0
$129$	−26.0810	−2.29630
$130$	0	0
$131$	3.56143	0.311163	0.155582	−	0.987823i	$-0.450275\pi$
$131$	3.56143	0.311163	0.155582	+	0.987823i	$0.450275\pi$
$132$	0	0
$133$	−0.0566712	−0.00491402
$134$	0	0
$135$	6.88135	0.592252
$136$	0	0
$137$	3.48920	0.298102	0.149051	−	0.988829i	$-0.452378\pi$
$137$	3.48920	0.298102	0.149051	+	0.988829i	$0.452378\pi$
$138$	0	0
$139$	−2.72401	−0.231047	−0.115524	−	0.993305i	$-0.536855\pi$
$139$	−2.72401	−0.231047	−0.115524	+	0.993305i	$0.536855\pi$
$140$	0	0
$141$	−13.0033	−1.09507
$142$	0	0
$143$	0.157794	0.0131954
$144$	0	0
$145$	0.656564	0.0545246
$146$	0	0
$147$	18.8903	1.55805
$148$	0	0
$149$	7.86150	0.644039	0.322019	−	0.946733i	$-0.395638\pi$
$149$	7.86150	0.644039	0.322019	+	0.946733i	$0.395638\pi$
$150$	0	0
$151$	−18.2998	−1.48922	−0.744608	−	0.667502i	$-0.767364\pi$
$151$	−18.2998	−1.48922	−0.744608	+	0.667502i	$0.767364\pi$
$152$	0	0
$153$	4.28301	0.346261
$154$	0	0
$155$	−1.23037	−0.0988256
$156$	0	0
$157$	18.2196	1.45408	0.727040	−	0.686595i	$-0.240895\pi$
$157$	18.2196	1.45408	0.727040	+	0.686595i	$0.240895\pi$
$158$	0	0
$159$	−10.7521	−0.852696
$160$	0	0
$161$	0.129536	0.0102088
$162$	0	0
$163$	11.9265	0.934158	0.467079	−	0.884216i	$-0.345306\pi$
$163$	11.9265	0.934158	0.467079	+	0.884216i	$0.345306\pi$
$164$	0	0
$165$	−1.90869	−0.148591
$166$	0	0
$167$	−0.871415	−0.0674321	−0.0337161	−	0.999431i	$-0.510734\pi$
$167$	−0.871415	−0.0674321	−0.0337161	+	0.999431i	$0.510734\pi$
$168$	0	0
$169$	−12.8034	−0.984876
$170$	0	0
$171$	15.8529	1.21230
$172$	0	0
$173$	−6.54515	−0.497619	−0.248809	−	0.968552i	$-0.580039\pi$
$173$	−6.54515	−0.497619	−0.248809	+	0.968552i	$0.580039\pi$
$174$	0	0
$175$	0.0160790	0.00121546
$176$	0	0
$177$	2.69870	0.202847
$178$	0	0
$179$	−6.00946	−0.449168	−0.224584	−	0.974455i	$-0.572102\pi$
$179$	−6.00946	−0.449168	−0.224584	+	0.974455i	$0.572102\pi$
$180$	0	0
$181$	26.4423	1.96544	0.982721	−	0.185093i	$-0.0592588\pi$
$181$	26.4423	1.96544	0.982721	+	0.185093i	$0.0592588\pi$
$182$	0	0
$183$	9.56426	0.707011
$184$	0	0
$185$	15.3948	1.13185
$186$	0	0
$187$	−0.355869	−0.0260237
$188$	0	0
$189$	0.0530134	0.00385616
$190$	0	0
$191$	12.2401	0.885666	0.442833	−	0.896604i	$-0.353973\pi$
$191$	12.2401	0.885666	0.442833	+	0.896604i	$0.353973\pi$
$192$	0	0
$193$	−6.02333	−0.433569	−0.216785	−	0.976219i	$-0.569557\pi$
$193$	−6.02333	−0.433569	−0.216785	+	0.976219i	$0.569557\pi$
$194$	0	0
$195$	−2.37819	−0.170306
$196$	0	0
$197$	13.3960	0.954423	0.477212	−	0.878788i	$-0.341648\pi$
$197$	13.3960	0.954423	0.477212	+	0.878788i	$0.341648\pi$
$198$	0	0
$199$	−4.36598	−0.309496	−0.154748	−	0.987954i	$-0.549457\pi$
$199$	−4.36598	−0.309496	−0.154748	+	0.987954i	$0.549457\pi$
$200$	0	0
$201$	−5.25436	−0.370614
$202$	0	0
$203$	0.00505812	0.000355010 0
$204$	0	0
$205$	−7.84643	−0.548018
$206$	0	0
$207$	−36.2357	−2.51855
$208$	0	0
$209$	−1.31720	−0.0911124
$210$	0	0
$211$	−22.0386	−1.51720	−0.758600	−	0.651557i	$-0.774116\pi$
$211$	−22.0386	−1.51720	−0.758600	+	0.651557i	$0.774116\pi$
$212$	0	0
$213$	−11.9244	−0.817044
$214$	0	0
$215$	−19.2069	−1.30990
$216$	0	0
$217$	−0.00947868	−0.000643454 0
$218$	0	0
$219$	−3.38070	−0.228447
$220$	0	0
$221$	−0.443406	−0.0298267
$222$	0	0
$223$	9.34514	0.625797	0.312898	−	0.949787i	$-0.398700\pi$
$223$	9.34514	0.625797	0.312898	+	0.949787i	$0.398700\pi$
$224$	0	0
$225$	−4.49786	−0.299858
$226$	0	0
$227$	14.2903	0.948479	0.474240	−	0.880396i	$-0.342723\pi$
$227$	14.2903	0.948479	0.474240	+	0.880396i	$0.342723\pi$
$228$	0	0
$229$	−24.6805	−1.63093	−0.815467	−	0.578804i	$-0.803520\pi$
$229$	−24.6805	−1.63093	−0.815467	+	0.578804i	$0.803520\pi$
$230$	0	0
$231$	−0.0147044	−0.000967477 0
$232$	0	0
$233$	−9.85060	−0.645334	−0.322667	−	0.946513i	$-0.604579\pi$
$233$	−9.85060	−0.645334	−0.322667	+	0.946513i	$0.604579\pi$
$234$	0	0
$235$	−9.57606	−0.624673
$236$	0	0
$237$	12.3460	0.801958
$238$	0	0
$239$	29.1738	1.88710	0.943548	−	0.331234i	$-0.107465\pi$
$239$	29.1738	1.88710	0.943548	+	0.331234i	$0.107465\pi$
$240$	0	0
$241$	−26.6836	−1.71884	−0.859421	−	0.511269i	$-0.829176\pi$
$241$	−26.6836	−1.71884	−0.859421	+	0.511269i	$0.829176\pi$
$242$	0	0
$243$	19.8460	1.27312
$244$	0	0
$245$	13.9115	0.888771
$246$	0	0
$247$	−1.64120	−0.104427
$248$	0	0
$249$	−30.2094	−1.91445
$250$	0	0
$251$	−24.4633	−1.54411	−0.772053	−	0.635558i	$-0.780770\pi$
$251$	−24.4633	−1.54411	−0.772053	+	0.635558i	$0.780770\pi$
$252$	0	0
$253$	3.01077	0.189285
$254$	0	0
$255$	5.36346	0.335873
$256$	0	0
$257$	3.21912	0.200803	0.100402	−	0.994947i	$-0.467987\pi$
$257$	3.21912	0.200803	0.100402	+	0.994947i	$0.467987\pi$
$258$	0	0
$259$	0.118601	0.00736948
$260$	0	0
$261$	−1.41493	−0.0875822
$262$	0	0
$263$	25.2934	1.55966	0.779830	−	0.625991i	$-0.215305\pi$
$263$	25.2934	1.55966	0.779830	+	0.625991i	$0.215305\pi$
$264$	0	0
$265$	−7.91821	−0.486412
$266$	0	0
$267$	33.1297	2.02750
$268$	0	0
$269$	−15.4805	−0.943862	−0.471931	−	0.881636i	$-0.656443\pi$
$269$	−15.4805	−0.943862	−0.471931	+	0.881636i	$0.656443\pi$
$270$	0	0
$271$	−7.07425	−0.429730	−0.214865	−	0.976644i	$-0.568931\pi$
$271$	−7.07425	−0.429730	−0.214865	+	0.976644i	$0.568931\pi$
$272$	0	0
$273$	−0.0183214	−0.00110886
$274$	0	0
$275$	0.373721	0.0225362
$276$	0	0
$277$	−2.59431	−0.155877	−0.0779384	−	0.996958i	$-0.524834\pi$
$277$	−2.59431	−0.155877	−0.0779384	+	0.996958i	$0.524834\pi$
$278$	0	0
$279$	2.65152	0.158742
$280$	0	0
$281$	−26.8085	−1.59926	−0.799631	−	0.600491i	$-0.794972\pi$
$281$	−26.8085	−1.59926	−0.799631	+	0.600491i	$0.794972\pi$
$282$	0	0
$283$	−15.1950	−0.903247	−0.451623	−	0.892209i	$-0.649155\pi$
$283$	−15.1950	−0.903247	−0.451623	+	0.892209i	$0.649155\pi$
$284$	0	0
$285$	19.8521	1.17593
$286$	0	0
$287$	−0.0604483	−0.00356815
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	23.4890	1.37695
$292$	0	0
$293$	−24.5053	−1.43161	−0.715807	−	0.698298i	$-0.753941\pi$
$293$	−24.5053	−1.43161	−0.715807	+	0.698298i	$0.753941\pi$
$294$	0	0
$295$	1.98742	0.115712
$296$	0	0
$297$	1.23218	0.0714983
$298$	0	0
$299$	3.75136	0.216947
$300$	0	0
$301$	−0.147969	−0.00852877
$302$	0	0
$303$	−46.9372	−2.69647
$304$	0	0
$305$	7.04345	0.403307
$306$	0	0
$307$	−8.05561	−0.459758	−0.229879	−	0.973219i	$-0.573833\pi$
$307$	−8.05561	−0.459758	−0.229879	+	0.973219i	$0.573833\pi$
$308$	0	0
$309$	−38.8213	−2.20846
$310$	0	0
$311$	−16.0126	−0.907989	−0.453994	−	0.891005i	$-0.650001\pi$
$311$	−16.0126	−0.907989	−0.453994	+	0.891005i	$0.650001\pi$
$312$	0	0
$313$	−22.5581	−1.27506	−0.637529	−	0.770426i	$-0.720043\pi$
$313$	−22.5581	−1.27506	−0.637529	+	0.770426i	$0.720043\pi$
$314$	0	0
$315$	0.130329	0.00734319
$316$	0	0
$317$	8.15384	0.457965	0.228983	−	0.973430i	$-0.426460\pi$
$317$	8.15384	0.457965	0.228983	+	0.973430i	$0.426460\pi$
$318$	0	0
$319$	0.117565	0.00658236
$320$	0	0
$321$	4.64824	0.259440
$322$	0	0
$323$	3.70136	0.205949
$324$	0	0
$325$	0.465649	0.0258296
$326$	0	0
$327$	−21.4558	−1.18651
$328$	0	0
$329$	−0.0737733	−0.00406725
$330$	0	0
$331$	−1.96296	−0.107894	−0.0539469	−	0.998544i	$-0.517180\pi$
$331$	−1.96296	−0.107894	−0.0539469	+	0.998544i	$0.517180\pi$
$332$	0	0
$333$	−33.1768	−1.81808
$334$	0	0
$335$	−3.86949	−0.211413
$336$	0	0
$337$	−5.24113	−0.285503	−0.142751	−	0.989759i	$-0.545595\pi$
$337$	−5.24113	−0.285503	−0.142751	+	0.989759i	$0.545595\pi$
$338$	0	0
$339$	−29.1531	−1.58338
$340$	0	0
$341$	−0.220311	−0.0119305
$342$	0	0
$343$	0.214349	0.0115738
$344$	0	0
$345$	−45.3767	−2.44300
$346$	0	0
$347$	9.63627	0.517302	0.258651	−	0.965971i	$-0.416722\pi$
$347$	9.63627	0.517302	0.258651	+	0.965971i	$0.416722\pi$
$348$	0	0
$349$	−24.8754	−1.33155	−0.665776	−	0.746152i	$-0.731899\pi$
$349$	−24.8754	−1.33155	−0.665776	+	0.746152i	$0.731899\pi$
$350$	0	0
$351$	1.53527	0.0819468
$352$	0	0
$353$	−20.1231	−1.07104	−0.535522	−	0.844521i	$-0.679885\pi$
$353$	−20.1231	−1.07104	−0.535522	+	0.844521i	$0.679885\pi$
$354$	0	0
$355$	−8.78151	−0.466074
$356$	0	0
$357$	0.0413197	0.00218687
$358$	0	0
$359$	−23.2002	−1.22446	−0.612230	−	0.790679i	$-0.709728\pi$
$359$	−23.2002	−1.22446	−0.612230	+	0.790679i	$0.709728\pi$
$360$	0	0
$361$	−5.29996	−0.278945
$362$	0	0
$363$	29.3440	1.54016
$364$	0	0
$365$	−2.48967	−0.130315
$366$	0	0
$367$	−0.228698	−0.0119379	−0.00596897	−	0.999982i	$-0.501900\pi$
$367$	−0.228698	−0.0119379	−0.00596897	+	0.999982i	$0.501900\pi$
$368$	0	0
$369$	16.9095	0.880274
$370$	0	0
$371$	−0.0610013	−0.00316703
$372$	0	0
$373$	−36.7089	−1.90072	−0.950358	−	0.311160i	$-0.899282\pi$
$373$	−36.7089	−1.90072	−0.950358	+	0.311160i	$0.899282\pi$
$374$	0	0
$375$	−32.4498	−1.67570
$376$	0	0
$377$	0.146484	0.00754429
$378$	0	0
$379$	17.1899	0.882986	0.441493	−	0.897265i	$-0.354449\pi$
$379$	17.1899	0.882986	0.441493	+	0.897265i	$0.354449\pi$
$380$	0	0
$381$	−32.0229	−1.64058
$382$	0	0
$383$	31.7049	1.62004	0.810022	−	0.586400i	$-0.199455\pi$
$383$	31.7049	1.62004	0.810022	+	0.586400i	$0.199455\pi$
$384$	0	0
$385$	−0.0108288	−0.000551887 0
$386$	0	0
$387$	41.3920	2.10408
$388$	0	0
$389$	23.7682	1.20510	0.602548	−	0.798083i	$-0.294152\pi$
$389$	23.7682	1.20510	0.602548	+	0.798083i	$0.294152\pi$
$390$	0	0
$391$	−8.46034	−0.427858
$392$	0	0
$393$	−9.61124	−0.484823
$394$	0	0
$395$	9.09201	0.457469
$396$	0	0
$397$	−9.34221	−0.468872	−0.234436	−	0.972132i	$-0.575324\pi$
$397$	−9.34221	−0.468872	−0.234436	+	0.972132i	$0.575324\pi$
$398$	0	0
$399$	0.152939	0.00765652
$400$	0	0
$401$	0.781001	0.0390013	0.0195007	−	0.999810i	$-0.493792\pi$
$401$	0.781001	0.0390013	0.0195007	+	0.999810i	$0.493792\pi$
$402$	0	0
$403$	−0.274503	−0.0136740
$404$	0	0
$405$	6.96566	0.346126
$406$	0	0
$407$	2.75661	0.136640
$408$	0	0
$409$	12.4811	0.617149	0.308574	−	0.951200i	$-0.400148\pi$
$409$	12.4811	0.617149	0.308574	+	0.951200i	$0.400148\pi$
$410$	0	0
$411$	−9.41631	−0.464472
$412$	0	0
$413$	0.0153109	0.000753402 0
$414$	0	0
$415$	−22.2473	−1.09208
$416$	0	0
$417$	7.35130	0.359995
$418$	0	0
$419$	−29.1011	−1.42168	−0.710840	−	0.703353i	$-0.751685\pi$
$419$	−29.1011	−1.42168	−0.710840	+	0.703353i	$0.751685\pi$
$420$	0	0
$421$	15.7303	0.766648	0.383324	−	0.923614i	$-0.374779\pi$
$421$	15.7303	0.766648	0.383324	+	0.923614i	$0.374779\pi$
$422$	0	0
$423$	20.6370	1.00340
$424$	0	0
$425$	−1.05017	−0.0509405
$426$	0	0
$427$	0.0542623	0.00262593
$428$	0	0
$429$	−0.425840	−0.0205598
$430$	0	0
$431$	11.7993	0.568351	0.284175	−	0.958772i	$-0.408280\pi$
$431$	11.7993	0.568351	0.284175	+	0.958772i	$0.408280\pi$
$432$	0	0
$433$	−6.81385	−0.327453	−0.163726	−	0.986506i	$-0.552351\pi$
$433$	−6.81385	−0.327453	−0.163726	+	0.986506i	$0.552351\pi$
$434$	0	0
$435$	−1.77187	−0.0849547
$436$	0	0
$437$	−31.3147	−1.49799
$438$	0	0
$439$	33.8754	1.61679	0.808393	−	0.588644i	$-0.200338\pi$
$439$	33.8754	1.61679	0.808393	+	0.588644i	$0.200338\pi$
$440$	0	0
$441$	−29.9800	−1.42762
$442$	0	0
$443$	28.9385	1.37491	0.687456	−	0.726226i	$-0.258727\pi$
$443$	28.9385	1.37491	0.687456	+	0.726226i	$0.258727\pi$
$444$	0	0
$445$	24.3978	1.15657
$446$	0	0
$447$	−21.2159	−1.00348
$448$	0	0
$449$	32.0409	1.51210	0.756052	−	0.654512i	$-0.227126\pi$
$449$	32.0409	1.51210	0.756052	+	0.654512i	$0.227126\pi$
$450$	0	0
$451$	−1.40499	−0.0661582
$452$	0	0
$453$	49.3858	2.32035
$454$	0	0
$455$	−0.0134925	−0.000632538 0
$456$	0	0
$457$	17.4796	0.817662	0.408831	−	0.912610i	$-0.365937\pi$
$457$	17.4796	0.817662	0.408831	+	0.912610i	$0.365937\pi$
$458$	0	0
$459$	−3.46245	−0.161614
$460$	0	0
$461$	−24.3631	−1.13470	−0.567351	−	0.823476i	$-0.692032\pi$
$461$	−24.3631	−1.13470	−0.567351	+	0.823476i	$0.692032\pi$
$462$	0	0
$463$	−6.27799	−0.291763	−0.145881	−	0.989302i	$-0.546602\pi$
$463$	−6.27799	−0.291763	−0.145881	+	0.989302i	$0.546602\pi$
$464$	0	0
$465$	3.32040	0.153980
$466$	0	0
$467$	−0.847748	−0.0392291	−0.0196145	−	0.999808i	$-0.506244\pi$
$467$	−0.847748	−0.0392291	−0.0196145	+	0.999808i	$0.506244\pi$
$468$	0	0
$469$	−0.0298103	−0.00137651
$470$	0	0
$471$	−49.1692	−2.26560
$472$	0	0
$473$	−3.43920	−0.158135
$474$	0	0
$475$	−3.88704	−0.178349
$476$	0	0
$477$	17.0642	0.781316
$478$	0	0
$479$	−21.9742	−1.00403	−0.502013	−	0.864860i	$-0.667407\pi$
$479$	−21.9742	−1.00403	−0.502013	+	0.864860i	$0.667407\pi$
$480$	0	0
$481$	3.43468	0.156608
$482$	0	0
$483$	−0.349578	−0.0159064
$484$	0	0
$485$	17.2981	0.785468
$486$	0	0
$487$	1.25557	0.0568952	0.0284476	−	0.999595i	$-0.490944\pi$
$487$	1.25557	0.0568952	0.0284476	+	0.999595i	$0.490944\pi$
$488$	0	0
$489$	−32.1862	−1.45551
$490$	0	0
$491$	−0.132216	−0.00596681	−0.00298341	−	0.999996i	$-0.500950\pi$
$491$	−0.132216	−0.00596681	−0.00298341	+	0.999996i	$0.500950\pi$
$492$	0	0
$493$	−0.330360	−0.0148787
$494$	0	0
$495$	3.02920	0.136152
$496$	0	0
$497$	−0.0676521	−0.00303461
$498$	0	0
$499$	−26.1943	−1.17262	−0.586308	−	0.810088i	$-0.699419\pi$
$499$	−26.1943	−1.17262	−0.586308	+	0.810088i	$0.699419\pi$
$500$	0	0
$501$	2.35169	0.105066
$502$	0	0
$503$	5.12298	0.228422	0.114211	−	0.993457i	$-0.463566\pi$
$503$	5.12298	0.228422	0.114211	+	0.993457i	$0.463566\pi$
$504$	0	0
$505$	−34.5662	−1.53818
$506$	0	0
$507$	34.5526	1.53453
$508$	0	0
$509$	−0.0951659	−0.00421815	−0.00210908	−	0.999998i	$-0.500671\pi$
$509$	−0.0951659	−0.00421815	−0.00210908	+	0.999998i	$0.500671\pi$
$510$	0	0
$511$	−0.0191802	−0.000848483 0
$512$	0	0
$513$	−12.8158	−0.565830
$514$	0	0
$515$	−28.5893	−1.25980
$516$	0	0
$517$	−1.71470	−0.0754123
$518$	0	0
$519$	17.6634	0.775339
$520$	0	0
$521$	−28.4200	−1.24510	−0.622551	−	0.782579i	$-0.713904\pi$
$521$	−28.4200	−1.24510	−0.622551	+	0.782579i	$0.713904\pi$
$522$	0	0
$523$	−4.78994	−0.209449	−0.104725	−	0.994501i	$-0.533396\pi$
$523$	−4.78994	−0.209449	−0.104725	+	0.994501i	$0.533396\pi$
$524$	0	0
$525$	−0.0433925	−0.00189380
$526$	0	0
$527$	0.619079	0.0269675
$528$	0	0
$529$	48.5773	2.11206
$530$	0	0
$531$	−4.28301	−0.185867
$532$	0	0
$533$	−1.75059	−0.0758264
$534$	0	0
$535$	3.42313	0.147995
$536$	0	0
$537$	16.2178	0.699848
$538$	0	0
$539$	2.49100	0.107295
$540$	0	0
$541$	−17.4676	−0.750990	−0.375495	−	0.926824i	$-0.622527\pi$
$541$	−17.4676	−0.750990	−0.375495	+	0.926824i	$0.622527\pi$
$542$	0	0
$543$	−71.3600	−3.06235
$544$	0	0
$545$	−15.8008	−0.676831
$546$	0	0
$547$	11.7681	0.503166	0.251583	−	0.967836i	$-0.419049\pi$
$547$	11.7681	0.503166	0.251583	+	0.967836i	$0.419049\pi$
$548$	0	0
$549$	−15.1791	−0.647827
$550$	0	0
$551$	−1.22278	−0.0520922
$552$	0	0
$553$	0.0700442	0.00297858
$554$	0	0
$555$	−41.5461	−1.76353
$556$	0	0
$557$	37.1450	1.57389	0.786943	−	0.617026i	$-0.211663\pi$
$557$	37.1450	1.57389	0.786943	+	0.617026i	$0.211663\pi$
$558$	0	0
$559$	−4.28518	−0.181244
$560$	0	0
$561$	0.960384	0.0405475
$562$	0	0
$563$	−6.39357	−0.269457	−0.134728	−	0.990883i	$-0.543016\pi$
$563$	−6.39357	−0.269457	−0.134728	+	0.990883i	$0.543016\pi$
$564$	0	0
$565$	−21.4693	−0.903222
$566$	0	0
$567$	0.0536629	0.00225363
$568$	0	0
$569$	−37.5188	−1.57287	−0.786434	−	0.617674i	$-0.788075\pi$
$569$	−37.5188	−1.57287	−0.786434	+	0.617674i	$0.788075\pi$
$570$	0	0
$571$	−10.4969	−0.439283	−0.219641	−	0.975581i	$-0.570489\pi$
$571$	−10.4969	−0.439283	−0.219641	+	0.975581i	$0.570489\pi$
$572$	0	0
$573$	−33.0325	−1.37995
$574$	0	0
$575$	8.88475	0.370520
$576$	0	0
$577$	−38.4813	−1.60200	−0.800999	−	0.598666i	$-0.795698\pi$
$577$	−38.4813	−1.60200	−0.800999	+	0.598666i	$0.795698\pi$
$578$	0	0
$579$	16.2552	0.675543
$580$	0	0
$581$	−0.171391	−0.00711052
$582$	0	0
$583$	−1.41784	−0.0587209
$584$	0	0
$585$	3.77433	0.156049
$586$	0	0
$587$	−45.7102	−1.88666	−0.943330	−	0.331856i	$-0.892325\pi$
$587$	−45.7102	−1.88666	−0.943330	+	0.331856i	$0.892325\pi$
$588$	0	0
$589$	2.29143	0.0944168
$590$	0	0
$591$	−36.1518	−1.48708
$592$	0	0
$593$	−45.8593	−1.88322	−0.941608	−	0.336711i	$-0.890686\pi$
$593$	−45.8593	−1.88322	−0.941608	+	0.336711i	$0.890686\pi$
$594$	0	0
$595$	0.0304292	0.00124748
$596$	0	0
$597$	11.7825	0.482225
$598$	0	0
$599$	−0.132703	−0.00542211	−0.00271106	−	0.999996i	$-0.500863\pi$
$599$	−0.132703	−0.00542211	−0.00271106	+	0.999996i	$0.500863\pi$
$600$	0	0
$601$	−6.52061	−0.265981	−0.132991	−	0.991117i	$-0.542458\pi$
$601$	−6.52061	−0.265981	−0.132991	+	0.991117i	$0.542458\pi$
$602$	0	0
$603$	8.33898	0.339590
$604$	0	0
$605$	21.6099	0.878568
$606$	0	0
$607$	−31.9672	−1.29751	−0.648755	−	0.760997i	$-0.724710\pi$
$607$	−31.9672	−1.29751	−0.648755	+	0.760997i	$0.724710\pi$
$608$	0	0
$609$	−0.0136504	−0.000553141 0
$610$	0	0
$611$	−2.13648	−0.0864327
$612$	0	0
$613$	−18.1753	−0.734095	−0.367048	−	0.930202i	$-0.619631\pi$
$613$	−18.1753	−0.734095	−0.367048	+	0.930202i	$0.619631\pi$
$614$	0	0
$615$	21.1752	0.853866
$616$	0	0
$617$	27.6061	1.11138	0.555690	−	0.831389i	$-0.312454\pi$
$617$	27.6061	1.11138	0.555690	+	0.831389i	$0.312454\pi$
$618$	0	0
$619$	23.6183	0.949301	0.474651	−	0.880174i	$-0.342574\pi$
$619$	23.6183	0.949301	0.474651	+	0.880174i	$0.342574\pi$
$620$	0	0
$621$	29.2935	1.17551
$622$	0	0
$623$	0.187959	0.00753042
$624$	0	0
$625$	−18.6463	−0.745853
$626$	0	0
$627$	3.55472	0.141962
$628$	0	0
$629$	−7.74614	−0.308859
$630$	0	0
$631$	11.7751	0.468759	0.234380	−	0.972145i	$-0.424694\pi$
$631$	11.7751	0.468759	0.234380	+	0.972145i	$0.424694\pi$
$632$	0	0
$633$	59.4757	2.36395
$634$	0	0
$635$	−23.5828	−0.935854
$636$	0	0
$637$	3.10374	0.122975
$638$	0	0
$639$	18.9247	0.748649
$640$	0	0
$641$	−27.9297	−1.10316	−0.551579	−	0.834122i	$-0.685975\pi$
$641$	−27.9297	−1.10316	−0.551579	+	0.834122i	$0.685975\pi$
$642$	0	0
$643$	−31.1726	−1.22933	−0.614664	−	0.788789i	$-0.710708\pi$
$643$	−31.1726	−1.22933	−0.614664	+	0.788789i	$0.710708\pi$
$644$	0	0
$645$	51.8338	2.04095
$646$	0	0
$647$	28.9069	1.13645	0.568225	−	0.822873i	$-0.307630\pi$
$647$	28.9069	1.13645	0.568225	+	0.822873i	$0.307630\pi$
$648$	0	0
$649$	0.355869	0.0139691
$650$	0	0
$651$	0.0255801	0.00100256
$652$	0	0
$653$	−24.7831	−0.969836	−0.484918	−	0.874560i	$-0.661151\pi$
$653$	−24.7831	−0.969836	−0.484918	+	0.874560i	$0.661151\pi$
$654$	0	0
$655$	−7.07805	−0.276562
$656$	0	0
$657$	5.36538	0.209323
$658$	0	0
$659$	3.45472	0.134577	0.0672885	−	0.997734i	$-0.478565\pi$
$659$	3.45472	0.134577	0.0672885	+	0.997734i	$0.478565\pi$
$660$	0	0
$661$	31.7345	1.23433	0.617165	−	0.786834i	$-0.288281\pi$
$661$	31.7345	1.23433	0.617165	+	0.786834i	$0.288281\pi$
$662$	0	0
$663$	1.19662	0.0464729
$664$	0	0
$665$	0.112629	0.00436758
$666$	0	0
$667$	2.79496	0.108221
$668$	0	0
$669$	−25.2198	−0.975052
$670$	0	0
$671$	1.26121	0.0486883
$672$	0	0
$673$	−17.5326	−0.675831	−0.337916	−	0.941176i	$-0.609722\pi$
$673$	−17.5326	−0.675831	−0.337916	+	0.941176i	$0.609722\pi$
$674$	0	0
$675$	3.63615	0.139955
$676$	0	0
$677$	−0.210578	−0.00809317	−0.00404659	−	0.999992i	$-0.501288\pi$
$677$	−0.210578	−0.00809317	−0.00404659	+	0.999992i	$0.501288\pi$
$678$	0	0
$679$	0.133264	0.00511419
$680$	0	0
$681$	−38.5652	−1.47782
$682$	0	0
$683$	−2.53604	−0.0970389	−0.0485195	−	0.998822i	$-0.515450\pi$
$683$	−2.53604	−0.0970389	−0.0485195	+	0.998822i	$0.515450\pi$
$684$	0	0
$685$	−6.93450	−0.264953
$686$	0	0
$687$	66.6054	2.54115
$688$	0	0
$689$	−1.76660	−0.0673022
$690$	0	0
$691$	21.8644	0.831759	0.415880	−	0.909420i	$-0.363474\pi$
$691$	21.8644	0.831759	0.415880	+	0.909420i	$0.363474\pi$
$692$	0	0
$693$	0.0233367	0.000886489 0
$694$	0	0
$695$	5.41375	0.205355
$696$	0	0
$697$	3.94805	0.149543
$698$	0	0
$699$	26.5839	1.00549
$700$	0	0
$701$	42.2024	1.59396	0.796981	−	0.604005i	$-0.206429\pi$
$701$	42.2024	1.59396	0.796981	+	0.604005i	$0.206429\pi$
$702$	0	0
$703$	−28.6712	−1.08136
$704$	0	0
$705$	25.8430	0.973302
$706$	0	0
$707$	−0.266295	−0.0100151
$708$	0	0
$709$	33.3405	1.25213	0.626065	−	0.779771i	$-0.284665\pi$
$709$	33.3405	1.25213	0.626065	+	0.779771i	$0.284665\pi$
$710$	0	0
$711$	−19.5938	−0.734826
$712$	0	0
$713$	−5.23762	−0.196150
$714$	0	0
$715$	−0.313603	−0.0117281
$716$	0	0
$717$	−78.7315	−2.94028
$718$	0	0
$719$	−36.6592	−1.36716	−0.683580	−	0.729876i	$-0.739578\pi$
$719$	−36.6592	−1.36716	−0.683580	+	0.729876i	$0.739578\pi$
$720$	0	0
$721$	−0.220250	−0.00820254
$722$	0	0
$723$	72.0111	2.67812
$724$	0	0
$725$	0.346933	0.0128848
$726$	0	0
$727$	−20.5990	−0.763974	−0.381987	−	0.924168i	$-0.624760\pi$
$727$	−20.5990	−0.763974	−0.381987	+	0.924168i	$0.624760\pi$
$728$	0	0
$729$	−43.0438	−1.59422
$730$	0	0
$731$	9.66425	0.357445
$732$	0	0
$733$	−33.6047	−1.24122	−0.620609	−	0.784120i	$-0.713115\pi$
$733$	−33.6047	−1.24122	−0.620609	+	0.784120i	$0.713115\pi$
$734$	0	0
$735$	−37.5429	−1.38479
$736$	0	0
$737$	−0.692874	−0.0255223
$738$	0	0
$739$	33.4679	1.23114	0.615568	−	0.788084i	$-0.288927\pi$
$739$	33.4679	1.23114	0.615568	+	0.788084i	$0.288927\pi$
$740$	0	0
$741$	4.42912	0.162708
$742$	0	0
$743$	−0.520645	−0.0191006	−0.00955031	−	0.999954i	$-0.503040\pi$
$743$	−0.520645	−0.0191006	−0.00955031	+	0.999954i	$0.503040\pi$
$744$	0	0
$745$	−15.6241	−0.572422
$746$	0	0
$747$	47.9442	1.75419
$748$	0	0
$749$	0.0263715	0.000963594 0
$750$	0	0
$751$	−45.0623	−1.64435	−0.822173	−	0.569238i	$-0.807238\pi$
$751$	−45.0623	−1.64435	−0.822173	+	0.569238i	$0.807238\pi$
$752$	0	0
$753$	66.0191	2.40587
$754$	0	0
$755$	36.3694	1.32362
$756$	0	0
$757$	−12.0330	−0.437347	−0.218674	−	0.975798i	$-0.570173\pi$
$757$	−12.0330	−0.437347	−0.218674	+	0.975798i	$0.570173\pi$
$758$	0	0
$759$	−8.12518	−0.294925
$760$	0	0
$761$	−53.0378	−1.92262	−0.961309	−	0.275471i	$-0.911166\pi$
$761$	−53.0378	−1.92262	−0.961309	+	0.275471i	$0.911166\pi$
$762$	0	0
$763$	−0.121728	−0.00440685
$764$	0	0
$765$	−8.51213	−0.307757
$766$	0	0
$767$	0.443406	0.0160105
$768$	0	0
$769$	−40.1653	−1.44840	−0.724199	−	0.689591i	$-0.757790\pi$
$769$	−40.1653	−1.44840	−0.724199	+	0.689591i	$0.757790\pi$
$770$	0	0
$771$	−8.68746	−0.312871
$772$	0	0
$773$	13.7708	0.495300	0.247650	−	0.968850i	$-0.420342\pi$
$773$	13.7708	0.495300	0.247650	+	0.968850i	$0.420342\pi$
$774$	0	0
$775$	−0.650135	−0.0233535
$776$	0	0
$777$	−0.320068	−0.0114824
$778$	0	0
$779$	14.6131	0.523570
$780$	0	0
$781$	−1.57242	−0.0562657
$782$	0	0
$783$	1.14386	0.0408781
$784$	0	0
$785$	−36.2099	−1.29239
$786$	0	0
$787$	−50.3199	−1.79371	−0.896856	−	0.442323i	$-0.854154\pi$
$787$	−50.3199	−1.79371	−0.896856	+	0.442323i	$0.854154\pi$
$788$	0	0
$789$	−68.2595	−2.43010
$790$	0	0
$791$	−0.165398	−0.00588088
$792$	0	0
$793$	1.57144	0.0558034
$794$	0	0
$795$	21.3689	0.757877
$796$	0	0
$797$	−32.6291	−1.15578	−0.577891	−	0.816114i	$-0.696124\pi$
$797$	−32.6291	−1.15578	−0.577891	+	0.816114i	$0.696124\pi$
$798$	0	0
$799$	4.81834	0.170461
$800$	0	0
$801$	−52.5788	−1.85778
$802$	0	0
$803$	−0.445802	−0.0157320
$804$	0	0
$805$	−0.257442	−0.00907363
$806$	0	0
$807$	41.7773	1.47063
$808$	0	0
$809$	−0.827544	−0.0290949	−0.0145474	−	0.999894i	$-0.504631\pi$
$809$	−0.827544	−0.0290949	−0.0145474	+	0.999894i	$0.504631\pi$
$810$	0	0
$811$	−13.7416	−0.482534	−0.241267	−	0.970459i	$-0.577563\pi$
$811$	−13.7416	−0.482534	−0.241267	+	0.970459i	$0.577563\pi$
$812$	0	0
$813$	19.0913	0.669561
$814$	0	0
$815$	−23.7030	−0.830280
$816$	0	0
$817$	35.7708	1.25146
$818$	0	0
$819$	0.0290771	0.00101604
$820$	0	0
$821$	−6.26297	−0.218579	−0.109290	−	0.994010i	$-0.534858\pi$
$821$	−6.26297	−0.218579	−0.109290	+	0.994010i	$0.534858\pi$
$822$	0	0
$823$	15.3281	0.534305	0.267152	−	0.963654i	$-0.413917\pi$
$823$	15.3281	0.534305	0.267152	+	0.963654i	$0.413917\pi$
$824$	0	0
$825$	−1.00856	−0.0351136
$826$	0	0
$827$	−2.69845	−0.0938343	−0.0469171	−	0.998899i	$-0.514940\pi$
$827$	−2.69845	−0.0938343	−0.0469171	+	0.998899i	$0.514940\pi$
$828$	0	0
$829$	24.6399	0.855778	0.427889	−	0.903831i	$-0.359257\pi$
$829$	24.6399	0.855778	0.427889	+	0.903831i	$0.359257\pi$
$830$	0	0
$831$	7.00127	0.242871
$832$	0	0
$833$	−6.99977	−0.242528
$834$	0	0
$835$	1.73187	0.0599337
$836$	0	0
$837$	−2.14353	−0.0740913
$838$	0	0
$839$	−10.9375	−0.377605	−0.188803	−	0.982015i	$-0.560461\pi$
$839$	−10.9375	−0.377605	−0.188803	+	0.982015i	$0.560461\pi$
$840$	0	0
$841$	−28.8909	−0.996237
$842$	0	0
$843$	72.3483	2.49181
$844$	0	0
$845$	25.4457	0.875359
$846$	0	0
$847$	0.166481	0.00572036
$848$	0	0
$849$	41.0067	1.40735
$850$	0	0
$851$	65.5350	2.24651
$852$	0	0
$853$	30.4488	1.04255	0.521274	−	0.853389i	$-0.325457\pi$
$853$	30.4488	1.04255	0.521274	+	0.853389i	$0.325457\pi$
$854$	0	0
$855$	−31.5064	−1.07750
$856$	0	0
$857$	−22.9800	−0.784983	−0.392492	−	0.919756i	$-0.628387\pi$
$857$	−22.9800	−0.784983	−0.392492	+	0.919756i	$0.628387\pi$
$858$	0	0
$859$	19.8010	0.675601	0.337801	−	0.941218i	$-0.390317\pi$
$859$	19.8010	0.675601	0.337801	+	0.941218i	$0.390317\pi$
$860$	0	0
$861$	0.163132	0.00555953
$862$	0	0
$863$	−3.31300	−0.112776	−0.0563879	−	0.998409i	$-0.517958\pi$
$863$	−3.31300	−0.112776	−0.0563879	+	0.998409i	$0.517958\pi$
$864$	0	0
$865$	13.0080	0.442284
$866$	0	0
$867$	−2.69870	−0.0916528
$868$	0	0
$869$	1.62802	0.0552268
$870$	0	0
$871$	−0.863308	−0.0292521
$872$	0	0
$873$	−37.2785	−1.26169
$874$	0	0
$875$	−0.184102	−0.00622378
$876$	0	0
$877$	45.2611	1.52836	0.764180	−	0.645003i	$-0.223144\pi$
$877$	45.2611	1.52836	0.764180	+	0.645003i	$0.223144\pi$
$878$	0	0
$879$	66.1325	2.23059
$880$	0	0
$881$	37.5603	1.26544	0.632720	−	0.774381i	$-0.281939\pi$
$881$	37.5603	1.26544	0.632720	+	0.774381i	$0.281939\pi$
$882$	0	0
$883$	−8.58203	−0.288808	−0.144404	−	0.989519i	$-0.546127\pi$
$883$	−8.58203	−0.288808	−0.144404	+	0.989519i	$0.546127\pi$
$884$	0	0
$885$	−5.36346	−0.180291
$886$	0	0
$887$	27.6396	0.928046	0.464023	−	0.885823i	$-0.346405\pi$
$887$	27.6396	0.928046	0.464023	+	0.885823i	$0.346405\pi$
$888$	0	0
$889$	−0.181680	−0.00609335
$890$	0	0
$891$	1.24728	0.0417853
$892$	0	0
$893$	17.8344	0.596805
$894$	0	0
$895$	11.9433	0.399221
$896$	0	0
$897$	−10.1238	−0.338025
$898$	0	0
$899$	−0.204519	−0.00682109
$900$	0	0
$901$	3.98417	0.132732
$902$	0	0
$903$	0.399324	0.0132887
$904$	0	0
$905$	−52.5520	−1.74689
$906$	0	0
$907$	−46.5511	−1.54570	−0.772852	−	0.634586i	$-0.781171\pi$
$907$	−46.5511	−1.54570	−0.772852	+	0.634586i	$0.781171\pi$
$908$	0	0
$909$	74.4922	2.47075
$910$	0	0
$911$	−45.5758	−1.50999	−0.754997	−	0.655728i	$-0.772362\pi$
$911$	−45.5758	−1.50999	−0.754997	+	0.655728i	$0.772362\pi$
$912$	0	0
$913$	−3.98361	−0.131838
$914$	0	0
$915$	−19.0082	−0.628392
$916$	0	0
$917$	−0.0545288	−0.00180070
$918$	0	0
$919$	12.0961	0.399013	0.199507	−	0.979896i	$-0.436066\pi$
$919$	12.0961	0.399013	0.199507	+	0.979896i	$0.436066\pi$
$920$	0	0
$921$	21.7397	0.716348
$922$	0	0
$923$	−1.95921	−0.0644882
$924$	0	0
$925$	8.13473	0.267468
$926$	0	0
$927$	61.6117	2.02359
$928$	0	0
$929$	−21.9451	−0.719995	−0.359997	−	0.932953i	$-0.617222\pi$
$929$	−21.9451	−0.719995	−0.359997	+	0.932953i	$0.617222\pi$
$930$	0	0
$931$	−25.9086	−0.849121
$932$	0	0
$933$	43.2132	1.41474
$934$	0	0
$935$	0.707260	0.0231299
$936$	0	0
$937$	18.0276	0.588935	0.294467	−	0.955662i	$-0.404858\pi$
$937$	18.0276	0.588935	0.294467	+	0.955662i	$0.404858\pi$
$938$	0	0
$939$	60.8776	1.98667
$940$	0	0
$941$	−38.3786	−1.25111	−0.625554	−	0.780181i	$-0.715127\pi$
$941$	−38.3786	−1.25111	−0.625554	+	0.780181i	$0.715127\pi$
$942$	0	0
$943$	−33.4018	−1.08771
$944$	0	0
$945$	−0.105360	−0.00342736
$946$	0	0
$947$	22.3370	0.725856	0.362928	−	0.931817i	$-0.381777\pi$
$947$	22.3370	0.725856	0.362928	+	0.931817i	$0.381777\pi$
$948$	0	0
$949$	−0.555461	−0.0180310
$950$	0	0
$951$	−22.0048	−0.713555
$952$	0	0
$953$	6.80973	0.220589	0.110294	−	0.993899i	$-0.464821\pi$
$953$	6.80973	0.220589	0.110294	+	0.993899i	$0.464821\pi$
$954$	0	0
$955$	−24.3263	−0.787181
$956$	0	0
$957$	−0.317273	−0.0102560
$958$	0	0
$959$	−0.0534229	−0.00172511
$960$	0	0
$961$	−30.6167	−0.987637
$962$	0	0
$963$	−7.37704	−0.237722
$964$	0	0
$965$	11.9709	0.385357
$966$	0	0
$967$	49.6723	1.59735	0.798676	−	0.601761i	$-0.205534\pi$
$967$	49.6723	1.59735	0.798676	+	0.601761i	$0.205534\pi$
$968$	0	0
$969$	−9.98887	−0.320889
$970$	0	0
$971$	−1.62521	−0.0521556	−0.0260778	−	0.999660i	$-0.508302\pi$
$971$	−1.62521	−0.0521556	−0.0260778	+	0.999660i	$0.508302\pi$
$972$	0	0
$973$	0.0417071	0.00133707
$974$	0	0
$975$	−1.25665	−0.0402450
$976$	0	0
$977$	46.3321	1.48230	0.741148	−	0.671342i	$-0.234282\pi$
$977$	46.3321	1.48230	0.741148	+	0.671342i	$0.234282\pi$
$978$	0	0
$979$	4.36869	0.139624
$980$	0	0
$981$	34.0516	1.08719
$982$	0	0
$983$	−42.0767	−1.34204	−0.671019	−	0.741440i	$-0.734143\pi$
$983$	−42.0767	−1.34204	−0.671019	+	0.741440i	$0.734143\pi$
$984$	0	0
$985$	−26.6234	−0.848292
$986$	0	0
$987$	0.199092	0.00633718
$988$	0	0
$989$	−81.7628	−2.59991
$990$	0	0
$991$	−10.2231	−0.324748	−0.162374	−	0.986729i	$-0.551915\pi$
$991$	−10.2231	−0.324748	−0.162374	+	0.986729i	$0.551915\pi$
$992$	0	0
$993$	5.29744	0.168109
$994$	0	0
$995$	8.67702	0.275080
$996$	0	0
$997$	42.5132	1.34641	0.673204	−	0.739457i	$-0.264918\pi$
$997$	42.5132	1.34641	0.673204	+	0.739457i	$0.264918\pi$
$998$	0	0
$999$	26.8207	0.848568

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.w.1.2	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.w.1.2	✓	20	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69870 q^{3} -1.98742 q^{5} -0.0153109 q^{7} +4.28301 q^{9} +O(q^{10})\)	\(q-2.69870 q^{3} -1.98742 q^{5} -0.0153109 q^{7} +4.28301 q^{9} -0.355869 q^{11} -0.443406 q^{13} +5.36346 q^{15} +1.00000 q^{17} +3.70136 q^{19} +0.0413197 q^{21} -8.46034 q^{23} -1.05017 q^{25} -3.46245 q^{27} -0.330360 q^{29} +0.619079 q^{31} +0.960384 q^{33} +0.0304292 q^{35} -7.74614 q^{37} +1.19662 q^{39} +3.94805 q^{41} +9.66425 q^{43} -8.51213 q^{45} +4.81834 q^{47} -6.99977 q^{49} -2.69870 q^{51} +3.98417 q^{53} +0.707260 q^{55} -9.98887 q^{57} -1.00000 q^{59} -3.54402 q^{61} -0.0655768 q^{63} +0.881233 q^{65} +1.94699 q^{67} +22.8320 q^{69} +4.41855 q^{71} +1.25271 q^{73} +2.83409 q^{75} +0.00544868 q^{77} -4.57478 q^{79} -3.50488 q^{81} +11.1941 q^{83} -1.98742 q^{85} +0.891544 q^{87} -12.2761 q^{89} +0.00678896 q^{91} -1.67071 q^{93} -7.35615 q^{95} -8.70382 q^{97} -1.52419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9	\( 20 q - 2 q^{3} - 2 q^{5} + 5 q^{7} + 2 q^{9} + 8 q^{11} - 3 q^{13} - 2 q^{15} + 20 q^{17} - 9 q^{19} - 31 q^{21} - 8 q^{23} - 4 q^{25} - 17 q^{27} - 37 q^{29} - 9 q^{31} - 8 q^{33} - 8 q^{35} - 11 q^{37} - 12 q^{39} - 27 q^{41} + 17 q^{43} - 4 q^{45} + 7 q^{47} - 9 q^{49} - 2 q^{51} - q^{53} - 21 q^{55} - 57 q^{57} - 20 q^{59} - 12 q^{61} - 14 q^{63} - 59 q^{65} - 26 q^{67} + 14 q^{69} - 3 q^{71} - 45 q^{73} + 15 q^{75} + 6 q^{77} + 5 q^{79} + 12 q^{81} - 17 q^{83} - 2 q^{85} - 32 q^{87} - 40 q^{89} - 36 q^{91} - 4 q^{93} + 5 q^{95} - 77 q^{97} + 54 q^{99}+O(q^{100}) \) 20 * q - 2 * q^3 - 2 * q^5 + 5 * q^7 + 2 * q^9 + 8 * q^11 - 3 * q^13 - 2 * q^15 + 20 * q^17 - 9 * q^19 - 31 * q^21 - 8 * q^23 - 4 * q^25 - 17 * q^27 - 37 * q^29 - 9 * q^31 - 8 * q^33 - 8 * q^35 - 11 * q^37 - 12 * q^39 - 27 * q^41 + 17 * q^43 - 4 * q^45 + 7 * q^47 - 9 * q^49 - 2 * q^51 - q^53 - 21 * q^55 - 57 * q^57 - 20 * q^59 - 12 * q^61 - 14 * q^63 - 59 * q^65 - 26 * q^67 + 14 * q^69 - 3 * q^71 - 45 * q^73 + 15 * q^75 + 6 * q^77 + 5 * q^79 + 12 * q^81 - 17 * q^83 - 2 * q^85 - 32 * q^87 - 40 * q^89 - 36 * q^91 - 4 * q^93 + 5 * q^95 - 77 * q^97 + 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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