LMFDB - Embedded newform 6040.2.a.p.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 5 x^{18} - 29 x^{17} + 165 x^{16} + 325 x^{15} - 2208 x^{14} - 1891 x^{13} + 15895 x^{12} + \cdots - 5628 $ x^19 - 5x^18 - 29x^17 + 165x^16 + 325x^15 - 2208x^14 - 1891x^13 + 15895x^12 + 6652x^11 - 67665x^10 - 17345x^9 + 174105x^8 + 41499x^7 - 262172x^6 - 80919x^5 + 206783x^4 + 91643x^3 - 59750x^2 - 40224x - 5628
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-0.435585$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.435585 q^{3} +1.00000 q^{5} -2.87898 q^{7} -2.81027 q^{9} +O(q^{10})$	$q+0.435585 q^{3} +1.00000 q^{5} -2.87898 q^{7} -2.81027 q^{9} -2.29647 q^{11} -0.0953190 q^{13} +0.435585 q^{15} +3.92440 q^{17} +6.59133 q^{19} -1.25404 q^{21} -0.150319 q^{23} +1.00000 q^{25} -2.53086 q^{27} +2.49904 q^{29} -0.535578 q^{31} -1.00031 q^{33} -2.87898 q^{35} -11.7988 q^{37} -0.0415195 q^{39} +9.05597 q^{41} +8.32623 q^{43} -2.81027 q^{45} -1.88440 q^{47} +1.28851 q^{49} +1.70941 q^{51} +10.4032 q^{53} -2.29647 q^{55} +2.87108 q^{57} -2.46632 q^{59} -13.0317 q^{61} +8.09069 q^{63} -0.0953190 q^{65} -4.05767 q^{67} -0.0654768 q^{69} -14.0463 q^{71} +5.10385 q^{73} +0.435585 q^{75} +6.61150 q^{77} -17.1799 q^{79} +7.32839 q^{81} -8.49380 q^{83} +3.92440 q^{85} +1.08855 q^{87} -7.92067 q^{89} +0.274421 q^{91} -0.233289 q^{93} +6.59133 q^{95} +10.3248 q^{97} +6.45371 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	$ 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) $ 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * 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
$3$	0.435585	0.251485	0.125742	−	0.992063i	$-0.459869\pi$
$3$	0.435585	0.251485	0.125742	+	0.992063i	$0.459869\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.87898	−1.08815	−0.544076	−	0.839036i	$-0.683120\pi$
$7$	−2.87898	−1.08815	−0.544076	+	0.839036i	$0.683120\pi$
$8$	0	0
$9$	−2.81027	−0.936755
$10$	0	0
$11$	−2.29647	−0.692413	−0.346207	−	0.938158i	$-0.612530\pi$
$11$	−2.29647	−0.692413	−0.346207	+	0.938158i	$0.612530\pi$
$12$	0	0
$13$	−0.0953190	−0.0264367	−0.0132184	−	0.999913i	$-0.504208\pi$
$13$	−0.0953190	−0.0264367	−0.0132184	+	0.999913i	$0.504208\pi$
$14$	0	0
$15$	0.435585	0.112467
$16$	0	0
$17$	3.92440	0.951808	0.475904	−	0.879497i	$-0.342121\pi$
$17$	3.92440	0.951808	0.475904	+	0.879497i	$0.342121\pi$
$18$	0	0
$19$	6.59133	1.51216	0.756078	−	0.654482i	$-0.227113\pi$
$19$	6.59133	1.51216	0.756078	+	0.654482i	$0.227113\pi$
$20$	0	0
$21$	−1.25404	−0.273654
$22$	0	0
$23$	−0.150319	−0.0313438	−0.0156719	−	0.999877i	$-0.504989\pi$
$23$	−0.150319	−0.0313438	−0.0156719	+	0.999877i	$0.504989\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.53086	−0.487065
$28$	0	0
$29$	2.49904	0.464061	0.232030	−	0.972709i	$-0.425463\pi$
$29$	2.49904	0.464061	0.232030	+	0.972709i	$0.425463\pi$
$30$	0	0
$31$	−0.535578	−0.0961926	−0.0480963	−	0.998843i	$-0.515315\pi$
$31$	−0.535578	−0.0961926	−0.0480963	+	0.998843i	$0.515315\pi$
$32$	0	0
$33$	−1.00031	−0.174132
$34$	0	0
$35$	−2.87898	−0.486636
$36$	0	0
$37$	−11.7988	−1.93971	−0.969853	−	0.243691i	$-0.921642\pi$
$37$	−11.7988	−1.93971	−0.969853	+	0.243691i	$0.921642\pi$
$38$	0	0
$39$	−0.0415195	−0.00664844
$40$	0	0
$41$	9.05597	1.41430	0.707152	−	0.707061i	$-0.249980\pi$
$41$	9.05597	1.41430	0.707152	+	0.707061i	$0.249980\pi$
$42$	0	0
$43$	8.32623	1.26974	0.634869	−	0.772619i	$-0.281054\pi$
$43$	8.32623	1.26974	0.634869	+	0.772619i	$0.281054\pi$
$44$	0	0
$45$	−2.81027	−0.418930
$46$	0	0
$47$	−1.88440	−0.274869	−0.137434	−	0.990511i	$-0.543886\pi$
$47$	−1.88440	−0.274869	−0.137434	+	0.990511i	$0.543886\pi$
$48$	0	0
$49$	1.28851	0.184073
$50$	0	0
$51$	1.70941	0.239365
$52$	0	0
$53$	10.4032	1.42899	0.714495	−	0.699641i	$-0.246656\pi$
$53$	10.4032	1.42899	0.714495	+	0.699641i	$0.246656\pi$
$54$	0	0
$55$	−2.29647	−0.309657
$56$	0	0
$57$	2.87108	0.380284
$58$	0	0
$59$	−2.46632	−0.321088	−0.160544	−	0.987029i	$-0.551325\pi$
$59$	−2.46632	−0.321088	−0.160544	+	0.987029i	$0.551325\pi$
$60$	0	0
$61$	−13.0317	−1.66854	−0.834272	−	0.551353i	$-0.814112\pi$
$61$	−13.0317	−1.66854	−0.834272	+	0.551353i	$0.814112\pi$
$62$	0	0
$63$	8.09069	1.01933
$64$	0	0
$65$	−0.0953190	−0.0118229
$66$	0	0
$67$	−4.05767	−0.495723	−0.247862	−	0.968795i	$-0.579728\pi$
$67$	−4.05767	−0.495723	−0.247862	+	0.968795i	$0.579728\pi$
$68$	0	0
$69$	−0.0654768	−0.00788248
$70$	0	0
$71$	−14.0463	−1.66699	−0.833496	−	0.552526i	$-0.813664\pi$
$71$	−14.0463	−1.66699	−0.833496	+	0.552526i	$0.813664\pi$
$72$	0	0
$73$	5.10385	0.597360	0.298680	−	0.954353i	$-0.403454\pi$
$73$	5.10385	0.597360	0.298680	+	0.954353i	$0.403454\pi$
$74$	0	0
$75$	0.435585	0.0502970
$76$	0	0
$77$	6.61150	0.753450
$78$	0	0
$79$	−17.1799	−1.93290	−0.966448	−	0.256864i	$-0.917311\pi$
$79$	−17.1799	−1.93290	−0.966448	+	0.256864i	$0.917311\pi$
$80$	0	0
$81$	7.32839	0.814266
$82$	0	0
$83$	−8.49380	−0.932316	−0.466158	−	0.884702i	$-0.654362\pi$
$83$	−8.49380	−0.932316	−0.466158	+	0.884702i	$0.654362\pi$
$84$	0	0
$85$	3.92440	0.425661
$86$	0	0
$87$	1.08855	0.116704
$88$	0	0
$89$	−7.92067	−0.839590	−0.419795	−	0.907619i	$-0.637898\pi$
$89$	−7.92067	−0.839590	−0.419795	+	0.907619i	$0.637898\pi$
$90$	0	0
$91$	0.274421	0.0287672
$92$	0	0
$93$	−0.233289	−0.0241910
$94$	0	0
$95$	6.59133	0.676256
$96$	0	0
$97$	10.3248	1.04833	0.524164	−	0.851617i	$-0.324378\pi$
$97$	10.3248	1.04833	0.524164	+	0.851617i	$0.324378\pi$
$98$	0	0
$99$	6.45371	0.648622
$100$	0	0
$101$	12.8058	1.27422	0.637111	−	0.770772i	$-0.280129\pi$
$101$	12.8058	1.27422	0.637111	+	0.770772i	$0.280129\pi$
$102$	0	0
$103$	−9.53362	−0.939375	−0.469688	−	0.882833i	$-0.655633\pi$
$103$	−9.53362	−0.939375	−0.469688	+	0.882833i	$0.655633\pi$
$104$	0	0
$105$	−1.25404	−0.122382
$106$	0	0
$107$	−8.03299	−0.776578	−0.388289	−	0.921538i	$-0.626934\pi$
$107$	−8.03299	−0.776578	−0.388289	+	0.921538i	$0.626934\pi$
$108$	0	0
$109$	−16.2799	−1.55934	−0.779668	−	0.626194i	$-0.784612\pi$
$109$	−16.2799	−1.55934	−0.779668	+	0.626194i	$0.784612\pi$
$110$	0	0
$111$	−5.13936	−0.487807
$112$	0	0
$113$	−11.2119	−1.05473	−0.527365	−	0.849639i	$-0.676820\pi$
$113$	−11.2119	−1.05473	−0.527365	+	0.849639i	$0.676820\pi$
$114$	0	0
$115$	−0.150319	−0.0140174
$116$	0	0
$117$	0.267872	0.0247647
$118$	0	0
$119$	−11.2983	−1.03571
$120$	0	0
$121$	−5.72620	−0.520564
$122$	0	0
$123$	3.94464	0.355676
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.5298	1.20058	0.600288	−	0.799784i	$-0.295053\pi$
$127$	13.5298	1.20058	0.600288	+	0.799784i	$0.295053\pi$
$128$	0	0
$129$	3.62678	0.319320
$130$	0	0
$131$	−13.9965	−1.22288	−0.611440	−	0.791291i	$-0.709410\pi$
$131$	−13.9965	−1.22288	−0.611440	+	0.791291i	$0.709410\pi$
$132$	0	0
$133$	−18.9763	−1.64545
$134$	0	0
$135$	−2.53086	−0.217822
$136$	0	0
$137$	−13.6311	−1.16458	−0.582290	−	0.812981i	$-0.697843\pi$
$137$	−13.6311	−1.16458	−0.582290	+	0.812981i	$0.697843\pi$
$138$	0	0
$139$	5.05697	0.428927	0.214463	−	0.976732i	$-0.431200\pi$
$139$	5.05697	0.428927	0.214463	+	0.976732i	$0.431200\pi$
$140$	0	0
$141$	−0.820818	−0.0691253
$142$	0	0
$143$	0.218898	0.0183051
$144$	0	0
$145$	2.49904	0.207534
$146$	0	0
$147$	0.561256	0.0462916
$148$	0	0
$149$	10.7804	0.883163	0.441582	−	0.897221i	$-0.354418\pi$
$149$	10.7804	0.883163	0.441582	+	0.897221i	$0.354418\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−11.0286	−0.891611
$154$	0	0
$155$	−0.535578	−0.0430186
$156$	0	0
$157$	−6.79705	−0.542464	−0.271232	−	0.962514i	$-0.587431\pi$
$157$	−6.79705	−0.542464	−0.271232	+	0.962514i	$0.587431\pi$
$158$	0	0
$159$	4.53148	0.359370
$160$	0	0
$161$	0.432766	0.0341068
$162$	0	0
$163$	−10.9477	−0.857490	−0.428745	−	0.903426i	$-0.641044\pi$
$163$	−10.9477	−0.857490	−0.428745	+	0.903426i	$0.641044\pi$
$164$	0	0
$165$	−1.00031	−0.0778740
$166$	0	0
$167$	−18.1080	−1.40124	−0.700619	−	0.713536i	$-0.747093\pi$
$167$	−18.1080	−1.40124	−0.700619	+	0.713536i	$0.747093\pi$
$168$	0	0
$169$	−12.9909	−0.999301
$170$	0	0
$171$	−18.5234	−1.41652
$172$	0	0
$173$	−10.0264	−0.762293	−0.381147	−	0.924515i	$-0.624471\pi$
$173$	−10.0264	−0.762293	−0.381147	+	0.924515i	$0.624471\pi$
$174$	0	0
$175$	−2.87898	−0.217630
$176$	0	0
$177$	−1.07429	−0.0807488
$178$	0	0
$179$	−6.55432	−0.489893	−0.244946	−	0.969537i	$-0.578770\pi$
$179$	−6.55432	−0.489893	−0.244946	+	0.969537i	$0.578770\pi$
$180$	0	0
$181$	−1.98392	−0.147464	−0.0737319	−	0.997278i	$-0.523491\pi$
$181$	−1.98392	−0.147464	−0.0737319	+	0.997278i	$0.523491\pi$
$182$	0	0
$183$	−5.67643	−0.419614
$184$	0	0
$185$	−11.7988	−0.867463
$186$	0	0
$187$	−9.01229	−0.659044
$188$	0	0
$189$	7.28630	0.530000
$190$	0	0
$191$	−23.2637	−1.68331	−0.841653	−	0.540019i	$-0.818417\pi$
$191$	−23.2637	−1.68331	−0.841653	+	0.540019i	$0.818417\pi$
$192$	0	0
$193$	−6.70918	−0.482937	−0.241469	−	0.970409i	$-0.577629\pi$
$193$	−6.70918	−0.482937	−0.241469	+	0.970409i	$0.577629\pi$
$194$	0	0
$195$	−0.0415195	−0.00297327
$196$	0	0
$197$	11.4808	0.817973	0.408987	−	0.912540i	$-0.365882\pi$
$197$	11.4808	0.817973	0.408987	+	0.912540i	$0.365882\pi$
$198$	0	0
$199$	−0.798470	−0.0566021	−0.0283010	−	0.999599i	$-0.509010\pi$
$199$	−0.798470	−0.0566021	−0.0283010	+	0.999599i	$0.509010\pi$
$200$	0	0
$201$	−1.76746	−0.124667
$202$	0	0
$203$	−7.19469	−0.504968
$204$	0	0
$205$	9.05597	0.632496
$206$	0	0
$207$	0.422438	0.0293614
$208$	0	0
$209$	−15.1368	−1.04704
$210$	0	0
$211$	−1.44453	−0.0994458	−0.0497229	−	0.998763i	$-0.515834\pi$
$211$	−1.44453	−0.0994458	−0.0497229	+	0.998763i	$0.515834\pi$
$212$	0	0
$213$	−6.11836	−0.419223
$214$	0	0
$215$	8.32623	0.567844
$216$	0	0
$217$	1.54192	0.104672
$218$	0	0
$219$	2.22316	0.150227
$220$	0	0
$221$	−0.374070	−0.0251627
$222$	0	0
$223$	−26.8447	−1.79766	−0.898829	−	0.438300i	$-0.855581\pi$
$223$	−26.8447	−1.79766	−0.898829	+	0.438300i	$0.855581\pi$
$224$	0	0
$225$	−2.81027	−0.187351
$226$	0	0
$227$	−20.8675	−1.38503	−0.692514	−	0.721404i	$-0.743497\pi$
$227$	−20.8675	−1.38503	−0.692514	+	0.721404i	$0.743497\pi$
$228$	0	0
$229$	17.3475	1.14635	0.573177	−	0.819431i	$-0.305711\pi$
$229$	17.3475	1.14635	0.573177	+	0.819431i	$0.305711\pi$
$230$	0	0
$231$	2.87987	0.189481
$232$	0	0
$233$	13.6111	0.891695	0.445848	−	0.895109i	$-0.352902\pi$
$233$	13.6111	0.891695	0.445848	+	0.895109i	$0.352902\pi$
$234$	0	0
$235$	−1.88440	−0.122925
$236$	0	0
$237$	−7.48332	−0.486094
$238$	0	0
$239$	14.8966	0.963583	0.481792	−	0.876286i	$-0.339986\pi$
$239$	14.8966	0.963583	0.481792	+	0.876286i	$0.339986\pi$
$240$	0	0
$241$	22.0346	1.41937	0.709687	−	0.704517i	$-0.248836\pi$
$241$	22.0346	1.41937	0.709687	+	0.704517i	$0.248836\pi$
$242$	0	0
$243$	10.7847	0.691840
$244$	0	0
$245$	1.28851	0.0823200
$246$	0	0
$247$	−0.628279	−0.0399764
$248$	0	0
$249$	−3.69977	−0.234463
$250$	0	0
$251$	10.9392	0.690474	0.345237	−	0.938515i	$-0.387798\pi$
$251$	10.9392	0.690474	0.345237	+	0.938515i	$0.387798\pi$
$252$	0	0
$253$	0.345205	0.0217028
$254$	0	0
$255$	1.70941	0.107047
$256$	0	0
$257$	−12.7485	−0.795229	−0.397615	−	0.917553i	$-0.630162\pi$
$257$	−12.7485	−0.795229	−0.397615	+	0.917553i	$0.630162\pi$
$258$	0	0
$259$	33.9684	2.11069
$260$	0	0
$261$	−7.02298	−0.434711
$262$	0	0
$263$	23.7818	1.46645	0.733226	−	0.679986i	$-0.238014\pi$
$263$	23.7818	1.46645	0.733226	+	0.679986i	$0.238014\pi$
$264$	0	0
$265$	10.4032	0.639064
$266$	0	0
$267$	−3.45012	−0.211144
$268$	0	0
$269$	21.5279	1.31258	0.656289	−	0.754509i	$-0.272125\pi$
$269$	21.5279	1.31258	0.656289	+	0.754509i	$0.272125\pi$
$270$	0	0
$271$	−15.8224	−0.961142	−0.480571	−	0.876956i	$-0.659571\pi$
$271$	−15.8224	−0.961142	−0.480571	+	0.876956i	$0.659571\pi$
$272$	0	0
$273$	0.119534	0.00723451
$274$	0	0
$275$	−2.29647	−0.138483
$276$	0	0
$277$	15.7192	0.944473	0.472236	−	0.881472i	$-0.343447\pi$
$277$	15.7192	0.944473	0.472236	+	0.881472i	$0.343447\pi$
$278$	0	0
$279$	1.50512	0.0901089
$280$	0	0
$281$	25.2396	1.50567	0.752836	−	0.658209i	$-0.228686\pi$
$281$	25.2396	1.50567	0.752836	+	0.658209i	$0.228686\pi$
$282$	0	0
$283$	20.2312	1.20262	0.601309	−	0.799017i	$-0.294646\pi$
$283$	20.2312	1.20262	0.601309	+	0.799017i	$0.294646\pi$
$284$	0	0
$285$	2.87108	0.170068
$286$	0	0
$287$	−26.0719	−1.53898
$288$	0	0
$289$	−1.59906	−0.0940621
$290$	0	0
$291$	4.49734	0.263639
$292$	0	0
$293$	1.18611	0.0692935	0.0346467	−	0.999400i	$-0.488969\pi$
$293$	1.18611	0.0692935	0.0346467	+	0.999400i	$0.488969\pi$
$294$	0	0
$295$	−2.46632	−0.143595
$296$	0	0
$297$	5.81206	0.337250
$298$	0	0
$299$	0.0143283	0.000828627 0
$300$	0	0
$301$	−23.9710	−1.38167
$302$	0	0
$303$	5.57800	0.320448
$304$	0	0
$305$	−13.0317	−0.746195
$306$	0	0
$307$	16.8285	0.960452	0.480226	−	0.877145i	$-0.340555\pi$
$307$	16.8285	0.960452	0.480226	+	0.877145i	$0.340555\pi$
$308$	0	0
$309$	−4.15270	−0.236239
$310$	0	0
$311$	−24.3198	−1.37905	−0.689524	−	0.724263i	$-0.742180\pi$
$311$	−24.3198	−1.37905	−0.689524	+	0.724263i	$0.742180\pi$
$312$	0	0
$313$	−11.3054	−0.639018	−0.319509	−	0.947583i	$-0.603518\pi$
$313$	−11.3054	−0.639018	−0.319509	+	0.947583i	$0.603518\pi$
$314$	0	0
$315$	8.09069	0.455859
$316$	0	0
$317$	−11.1520	−0.626358	−0.313179	−	0.949694i	$-0.601394\pi$
$317$	−11.1520	−0.626358	−0.313179	+	0.949694i	$0.601394\pi$
$318$	0	0
$319$	−5.73899	−0.321322
$320$	0	0
$321$	−3.49905	−0.195298
$322$	0	0
$323$	25.8670	1.43928
$324$	0	0
$325$	−0.0953190	−0.00528735
$326$	0	0
$327$	−7.09129	−0.392149
$328$	0	0
$329$	5.42516	0.299099
$330$	0	0
$331$	−30.4711	−1.67484	−0.837421	−	0.546558i	$-0.815938\pi$
$331$	−30.4711	−1.67484	−0.837421	+	0.546558i	$0.815938\pi$
$332$	0	0
$333$	33.1577	1.81703
$334$	0	0
$335$	−4.05767	−0.221694
$336$	0	0
$337$	−7.12539	−0.388145	−0.194072	−	0.980987i	$-0.562170\pi$
$337$	−7.12539	−0.388145	−0.194072	+	0.980987i	$0.562170\pi$
$338$	0	0
$339$	−4.88375	−0.265249
$340$	0	0
$341$	1.22994	0.0666050
$342$	0	0
$343$	16.4432	0.887852
$344$	0	0
$345$	−0.0654768	−0.00352515
$346$	0	0
$347$	16.6155	0.891968	0.445984	−	0.895041i	$-0.352854\pi$
$347$	16.6155	0.891968	0.445984	+	0.895041i	$0.352854\pi$
$348$	0	0
$349$	−2.03464	−0.108912	−0.0544560	−	0.998516i	$-0.517342\pi$
$349$	−2.03464	−0.108912	−0.0544560	+	0.998516i	$0.517342\pi$
$350$	0	0
$351$	0.241239	0.0128764
$352$	0	0
$353$	−5.89772	−0.313904	−0.156952	−	0.987606i	$-0.550167\pi$
$353$	−5.89772	−0.313904	−0.156952	+	0.987606i	$0.550167\pi$
$354$	0	0
$355$	−14.0463	−0.745501
$356$	0	0
$357$	−4.92135	−0.260466
$358$	0	0
$359$	34.5127	1.82151	0.910755	−	0.412947i	$-0.135501\pi$
$359$	34.5127	1.82151	0.910755	+	0.412947i	$0.135501\pi$
$360$	0	0
$361$	24.4456	1.28661
$362$	0	0
$363$	−2.49425	−0.130914
$364$	0	0
$365$	5.10385	0.267148
$366$	0	0
$367$	28.4303	1.48405	0.742026	−	0.670372i	$-0.233865\pi$
$367$	28.4303	1.48405	0.742026	+	0.670372i	$0.233865\pi$
$368$	0	0
$369$	−25.4497	−1.32486
$370$	0	0
$371$	−29.9506	−1.55496
$372$	0	0
$373$	1.71457	0.0887769	0.0443884	−	0.999014i	$-0.485866\pi$
$373$	1.71457	0.0887769	0.0443884	+	0.999014i	$0.485866\pi$
$374$	0	0
$375$	0.435585	0.0224935
$376$	0	0
$377$	−0.238206	−0.0122682
$378$	0	0
$379$	12.4490	0.639462	0.319731	−	0.947508i	$-0.396407\pi$
$379$	12.4490	0.639462	0.319731	+	0.947508i	$0.396407\pi$
$380$	0	0
$381$	5.89338	0.301927
$382$	0	0
$383$	5.31980	0.271829	0.135915	−	0.990721i	$-0.456603\pi$
$383$	5.31980	0.271829	0.135915	+	0.990721i	$0.456603\pi$
$384$	0	0
$385$	6.61150	0.336953
$386$	0	0
$387$	−23.3989	−1.18943
$388$	0	0
$389$	31.9844	1.62167	0.810837	−	0.585272i	$-0.199012\pi$
$389$	31.9844	1.62167	0.810837	+	0.585272i	$0.199012\pi$
$390$	0	0
$391$	−0.589914	−0.0298332
$392$	0	0
$393$	−6.09666	−0.307536
$394$	0	0
$395$	−17.1799	−0.864417
$396$	0	0
$397$	9.29137	0.466320	0.233160	−	0.972438i	$-0.425093\pi$
$397$	9.29137	0.466320	0.233160	+	0.972438i	$0.425093\pi$
$398$	0	0
$399$	−8.26578	−0.413807
$400$	0	0
$401$	−37.7900	−1.88714	−0.943571	−	0.331170i	$-0.892557\pi$
$401$	−37.7900	−1.88714	−0.943571	+	0.331170i	$0.892557\pi$
$402$	0	0
$403$	0.0510507	0.00254302
$404$	0	0
$405$	7.32839	0.364151
$406$	0	0
$407$	27.0956	1.34308
$408$	0	0
$409$	−2.84897	−0.140873	−0.0704363	−	0.997516i	$-0.522439\pi$
$409$	−2.84897	−0.140873	−0.0704363	+	0.997516i	$0.522439\pi$
$410$	0	0
$411$	−5.93748	−0.292874
$412$	0	0
$413$	7.10049	0.349392
$414$	0	0
$415$	−8.49380	−0.416944
$416$	0	0
$417$	2.20274	0.107869
$418$	0	0
$419$	−3.33413	−0.162883	−0.0814413	−	0.996678i	$-0.525952\pi$
$419$	−3.33413	−0.162883	−0.0814413	+	0.996678i	$0.525952\pi$
$420$	0	0
$421$	−30.6477	−1.49368	−0.746838	−	0.665006i	$-0.768429\pi$
$421$	−30.6477	−1.49368	−0.746838	+	0.665006i	$0.768429\pi$
$422$	0	0
$423$	5.29568	0.257485
$424$	0	0
$425$	3.92440	0.190362
$426$	0	0
$427$	37.5181	1.81563
$428$	0	0
$429$	0.0953485	0.00460347
$430$	0	0
$431$	−33.7009	−1.62332	−0.811658	−	0.584133i	$-0.801435\pi$
$431$	−33.7009	−1.62332	−0.811658	+	0.584133i	$0.801435\pi$
$432$	0	0
$433$	−5.02391	−0.241434	−0.120717	−	0.992687i	$-0.538519\pi$
$433$	−5.02391	−0.241434	−0.120717	+	0.992687i	$0.538519\pi$
$434$	0	0
$435$	1.08855	0.0521918
$436$	0	0
$437$	−0.990805	−0.0473966
$438$	0	0
$439$	−33.1409	−1.58173	−0.790864	−	0.611992i	$-0.790368\pi$
$439$	−33.1409	−1.58173	−0.790864	+	0.611992i	$0.790368\pi$
$440$	0	0
$441$	−3.62106	−0.172431
$442$	0	0
$443$	18.3387	0.871298	0.435649	−	0.900117i	$-0.356519\pi$
$443$	18.3387	0.871298	0.435649	+	0.900117i	$0.356519\pi$
$444$	0	0
$445$	−7.92067	−0.375476
$446$	0	0
$447$	4.69577	0.222102
$448$	0	0
$449$	32.2443	1.52170	0.760852	−	0.648925i	$-0.224781\pi$
$449$	32.2443	1.52170	0.760852	+	0.648925i	$0.224781\pi$
$450$	0	0
$451$	−20.7968	−0.979283
$452$	0	0
$453$	0.435585	0.0204656
$454$	0	0
$455$	0.274421	0.0128651
$456$	0	0
$457$	33.3685	1.56091	0.780457	−	0.625210i	$-0.214987\pi$
$457$	33.3685	1.56091	0.780457	+	0.625210i	$0.214987\pi$
$458$	0	0
$459$	−9.93213	−0.463592
$460$	0	0
$461$	−18.8368	−0.877317	−0.438659	−	0.898654i	$-0.644546\pi$
$461$	−18.8368	−0.877317	−0.438659	+	0.898654i	$0.644546\pi$
$462$	0	0
$463$	−30.5919	−1.42173	−0.710863	−	0.703330i	$-0.751695\pi$
$463$	−30.5919	−1.42173	−0.710863	+	0.703330i	$0.751695\pi$
$464$	0	0
$465$	−0.233289	−0.0108185
$466$	0	0
$467$	−4.66251	−0.215755	−0.107878	−	0.994164i	$-0.534405\pi$
$467$	−4.66251	−0.215755	−0.107878	+	0.994164i	$0.534405\pi$
$468$	0	0
$469$	11.6819	0.539422
$470$	0	0
$471$	−2.96069	−0.136421
$472$	0	0
$473$	−19.1210	−0.879184
$474$	0	0
$475$	6.59133	0.302431
$476$	0	0
$477$	−29.2358	−1.33861
$478$	0	0
$479$	33.7934	1.54406	0.772030	−	0.635586i	$-0.219241\pi$
$479$	33.7934	1.54406	0.772030	+	0.635586i	$0.219241\pi$
$480$	0	0
$481$	1.12465	0.0512795
$482$	0	0
$483$	0.188506	0.00857734
$484$	0	0
$485$	10.3248	0.468826
$486$	0	0
$487$	−24.3168	−1.10190	−0.550950	−	0.834538i	$-0.685734\pi$
$487$	−24.3168	−1.10190	−0.550950	+	0.834538i	$0.685734\pi$
$488$	0	0
$489$	−4.76865	−0.215646
$490$	0	0
$491$	17.9100	0.808268	0.404134	−	0.914700i	$-0.367573\pi$
$491$	17.9100	0.808268	0.404134	+	0.914700i	$0.367573\pi$
$492$	0	0
$493$	9.80726	0.441697
$494$	0	0
$495$	6.45371	0.290072
$496$	0	0
$497$	40.4390	1.81394
$498$	0	0
$499$	4.95485	0.221809	0.110905	−	0.993831i	$-0.464625\pi$
$499$	4.95485	0.221809	0.110905	+	0.993831i	$0.464625\pi$
$500$	0	0
$501$	−7.88756	−0.352390
$502$	0	0
$503$	−15.5787	−0.694621	−0.347311	−	0.937750i	$-0.612905\pi$
$503$	−15.5787	−0.694621	−0.347311	+	0.937750i	$0.612905\pi$
$504$	0	0
$505$	12.8058	0.569850
$506$	0	0
$507$	−5.65864	−0.251309
$508$	0	0
$509$	−40.7094	−1.80441	−0.902207	−	0.431304i	$-0.858054\pi$
$509$	−40.7094	−1.80441	−0.902207	+	0.431304i	$0.858054\pi$
$510$	0	0
$511$	−14.6939	−0.650018
$512$	0	0
$513$	−16.6818	−0.736518
$514$	0	0
$515$	−9.53362	−0.420101
$516$	0	0
$517$	4.32749	0.190323
$518$	0	0
$519$	−4.36735	−0.191705
$520$	0	0
$521$	26.4160	1.15731	0.578654	−	0.815573i	$-0.303578\pi$
$521$	26.4160	1.15731	0.578654	+	0.815573i	$0.303578\pi$
$522$	0	0
$523$	19.6940	0.861157	0.430578	−	0.902553i	$-0.358310\pi$
$523$	19.6940	0.861157	0.430578	+	0.902553i	$0.358310\pi$
$524$	0	0
$525$	−1.25404	−0.0547307
$526$	0	0
$527$	−2.10182	−0.0915568
$528$	0	0
$529$	−22.9774	−0.999018
$530$	0	0
$531$	6.93102	0.300781
$532$	0	0
$533$	−0.863206	−0.0373896
$534$	0	0
$535$	−8.03299	−0.347296
$536$	0	0
$537$	−2.85496	−0.123201
$538$	0	0
$539$	−2.95903	−0.127455
$540$	0	0
$541$	−25.9581	−1.11603	−0.558013	−	0.829832i	$-0.688436\pi$
$541$	−25.9581	−1.11603	−0.558013	+	0.829832i	$0.688436\pi$
$542$	0	0
$543$	−0.864166	−0.0370849
$544$	0	0
$545$	−16.2799	−0.697356
$546$	0	0
$547$	16.0398	0.685811	0.342905	−	0.939370i	$-0.388589\pi$
$547$	16.0398	0.685811	0.342905	+	0.939370i	$0.388589\pi$
$548$	0	0
$549$	36.6227	1.56302
$550$	0	0
$551$	16.4720	0.701732
$552$	0	0
$553$	49.4607	2.10328
$554$	0	0
$555$	−5.13936	−0.218154
$556$	0	0
$557$	−3.91035	−0.165687	−0.0828435	−	0.996563i	$-0.526400\pi$
$557$	−3.91035	−0.165687	−0.0828435	+	0.996563i	$0.526400\pi$
$558$	0	0
$559$	−0.793648	−0.0335677
$560$	0	0
$561$	−3.92562	−0.165740
$562$	0	0
$563$	−20.3282	−0.856729	−0.428365	−	0.903606i	$-0.640910\pi$
$563$	−20.3282	−0.856729	−0.428365	+	0.903606i	$0.640910\pi$
$564$	0	0
$565$	−11.2119	−0.471690
$566$	0	0
$567$	−21.0983	−0.886044
$568$	0	0
$569$	−35.2925	−1.47954	−0.739768	−	0.672862i	$-0.765065\pi$
$569$	−35.2925	−1.47954	−0.739768	+	0.672862i	$0.765065\pi$
$570$	0	0
$571$	−38.6754	−1.61852	−0.809259	−	0.587453i	$-0.800131\pi$
$571$	−38.6754	−1.61852	−0.809259	+	0.587453i	$0.800131\pi$
$572$	0	0
$573$	−10.1333	−0.423326
$574$	0	0
$575$	−0.150319	−0.00626875
$576$	0	0
$577$	19.4030	0.807756	0.403878	−	0.914813i	$-0.367662\pi$
$577$	19.4030	0.807756	0.403878	+	0.914813i	$0.367662\pi$
$578$	0	0
$579$	−2.92242	−0.121451
$580$	0	0
$581$	24.4535	1.01450
$582$	0	0
$583$	−23.8907	−0.989452
$584$	0	0
$585$	0.267872	0.0110751
$586$	0	0
$587$	5.30229	0.218849	0.109424	−	0.993995i	$-0.465099\pi$
$587$	5.30229	0.218849	0.109424	+	0.993995i	$0.465099\pi$
$588$	0	0
$589$	−3.53017	−0.145458
$590$	0	0
$591$	5.00086	0.205708
$592$	0	0
$593$	33.1485	1.36125	0.680623	−	0.732634i	$-0.261709\pi$
$593$	33.1485	1.36125	0.680623	+	0.732634i	$0.261709\pi$
$594$	0	0
$595$	−11.2983	−0.463184
$596$	0	0
$597$	−0.347801	−0.0142346
$598$	0	0
$599$	−1.07126	−0.0437704	−0.0218852	−	0.999760i	$-0.506967\pi$
$599$	−1.07126	−0.0437704	−0.0218852	+	0.999760i	$0.506967\pi$
$600$	0	0
$601$	−45.9381	−1.87386	−0.936928	−	0.349524i	$-0.886343\pi$
$601$	−45.9381	−1.87386	−0.936928	+	0.349524i	$0.886343\pi$
$602$	0	0
$603$	11.4031	0.464371
$604$	0	0
$605$	−5.72620	−0.232803
$606$	0	0
$607$	35.3792	1.43600	0.717999	−	0.696044i	$-0.245058\pi$
$607$	35.3792	1.43600	0.717999	+	0.696044i	$0.245058\pi$
$608$	0	0
$609$	−3.13390	−0.126992
$610$	0	0
$611$	0.179619	0.00726663
$612$	0	0
$613$	9.71728	0.392477	0.196239	−	0.980556i	$-0.437127\pi$
$613$	9.71728	0.392477	0.196239	+	0.980556i	$0.437127\pi$
$614$	0	0
$615$	3.94464	0.159063
$616$	0	0
$617$	26.5619	1.06934	0.534670	−	0.845061i	$-0.320436\pi$
$617$	26.5619	1.06934	0.534670	+	0.845061i	$0.320436\pi$
$618$	0	0
$619$	−30.6930	−1.23366	−0.616828	−	0.787098i	$-0.711583\pi$
$619$	−30.6930	−1.23366	−0.616828	+	0.787098i	$0.711583\pi$
$620$	0	0
$621$	0.380438	0.0152664
$622$	0	0
$623$	22.8034	0.913601
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.59337	−0.263314
$628$	0	0
$629$	−46.3031	−1.84623
$630$	0	0
$631$	−10.9175	−0.434618	−0.217309	−	0.976103i	$-0.569728\pi$
$631$	−10.9175	−0.434618	−0.217309	+	0.976103i	$0.569728\pi$
$632$	0	0
$633$	−0.629216	−0.0250091
$634$	0	0
$635$	13.5298	0.536914
$636$	0	0
$637$	−0.122820	−0.00486629
$638$	0	0
$639$	39.4739	1.56156
$640$	0	0
$641$	−13.0137	−0.514011	−0.257005	−	0.966410i	$-0.582736\pi$
$641$	−13.0137	−0.514011	−0.257005	+	0.966410i	$0.582736\pi$
$642$	0	0
$643$	−37.1789	−1.46619	−0.733096	−	0.680125i	$-0.761925\pi$
$643$	−37.1789	−1.46619	−0.733096	+	0.680125i	$0.761925\pi$
$644$	0	0
$645$	3.62678	0.142804
$646$	0	0
$647$	−40.7533	−1.60218	−0.801088	−	0.598547i	$-0.795745\pi$
$647$	−40.7533	−1.60218	−0.801088	+	0.598547i	$0.795745\pi$
$648$	0	0
$649$	5.66385	0.222326
$650$	0	0
$651$	0.671635	0.0263234
$652$	0	0
$653$	9.45694	0.370079	0.185039	−	0.982731i	$-0.440759\pi$
$653$	9.45694	0.370079	0.185039	+	0.982731i	$0.440759\pi$
$654$	0	0
$655$	−13.9965	−0.546889
$656$	0	0
$657$	−14.3432	−0.559580
$658$	0	0
$659$	−19.9956	−0.778917	−0.389458	−	0.921044i	$-0.627338\pi$
$659$	−19.9956	−0.778917	−0.389458	+	0.921044i	$0.627338\pi$
$660$	0	0
$661$	30.6198	1.19097	0.595486	−	0.803366i	$-0.296959\pi$
$661$	30.6198	1.19097	0.595486	+	0.803366i	$0.296959\pi$
$662$	0	0
$663$	−0.162939	−0.00632804
$664$	0	0
$665$	−18.9763	−0.735869
$666$	0	0
$667$	−0.375655	−0.0145454
$668$	0	0
$669$	−11.6932	−0.452084
$670$	0	0
$671$	29.9271	1.15532
$672$	0	0
$673$	18.4372	0.710700	0.355350	−	0.934733i	$-0.384362\pi$
$673$	18.4372	0.710700	0.355350	+	0.934733i	$0.384362\pi$
$674$	0	0
$675$	−2.53086	−0.0974130
$676$	0	0
$677$	−31.2683	−1.20174	−0.600870	−	0.799347i	$-0.705179\pi$
$677$	−31.2683	−1.20174	−0.600870	+	0.799347i	$0.705179\pi$
$678$	0	0
$679$	−29.7250	−1.14074
$680$	0	0
$681$	−9.08959	−0.348314
$682$	0	0
$683$	−20.9300	−0.800866	−0.400433	−	0.916326i	$-0.631140\pi$
$683$	−20.9300	−0.800866	−0.400433	+	0.916326i	$0.631140\pi$
$684$	0	0
$685$	−13.6311	−0.520816
$686$	0	0
$687$	7.55630	0.288291
$688$	0	0
$689$	−0.991623	−0.0377778
$690$	0	0
$691$	25.0385	0.952510	0.476255	−	0.879307i	$-0.341994\pi$
$691$	25.0385	0.952510	0.476255	+	0.879307i	$0.341994\pi$
$692$	0	0
$693$	−18.5801	−0.705799
$694$	0	0
$695$	5.05697	0.191822
$696$	0	0
$697$	35.5393	1.34615
$698$	0	0
$699$	5.92880	0.224248
$700$	0	0
$701$	12.6976	0.479583	0.239792	−	0.970824i	$-0.422921\pi$
$701$	12.6976	0.479583	0.239792	+	0.970824i	$0.422921\pi$
$702$	0	0
$703$	−77.7696	−2.93314
$704$	0	0
$705$	−0.820818	−0.0309138
$706$	0	0
$707$	−36.8676	−1.38655
$708$	0	0
$709$	35.3293	1.32682	0.663410	−	0.748256i	$-0.269109\pi$
$709$	35.3293	1.32682	0.663410	+	0.748256i	$0.269109\pi$
$710$	0	0
$711$	48.2802	1.81065
$712$	0	0
$713$	0.0805077	0.00301504
$714$	0	0
$715$	0.218898	0.00818631
$716$	0	0
$717$	6.48875	0.242327
$718$	0	0
$719$	6.63766	0.247543	0.123771	−	0.992311i	$-0.460501\pi$
$719$	6.63766	0.247543	0.123771	+	0.992311i	$0.460501\pi$
$720$	0	0
$721$	27.4471	1.02218
$722$	0	0
$723$	9.59794	0.356951
$724$	0	0
$725$	2.49904	0.0928122
$726$	0	0
$727$	−35.6453	−1.32201	−0.661005	−	0.750381i	$-0.729870\pi$
$727$	−35.6453	−1.32201	−0.661005	+	0.750381i	$0.729870\pi$
$728$	0	0
$729$	−17.2875	−0.640278
$730$	0	0
$731$	32.6755	1.20855
$732$	0	0
$733$	27.7386	1.02455	0.512275	−	0.858821i	$-0.328803\pi$
$733$	27.7386	1.02455	0.512275	+	0.858821i	$0.328803\pi$
$734$	0	0
$735$	0.561256	0.0207022
$736$	0	0
$737$	9.31834	0.343245
$738$	0	0
$739$	−27.7731	−1.02165	−0.510825	−	0.859685i	$-0.670660\pi$
$739$	−27.7731	−1.02165	−0.510825	+	0.859685i	$0.670660\pi$
$740$	0	0
$741$	−0.273669	−0.0100535
$742$	0	0
$743$	−33.2912	−1.22133	−0.610667	−	0.791887i	$-0.709099\pi$
$743$	−33.2912	−1.22133	−0.610667	+	0.791887i	$0.709099\pi$
$744$	0	0
$745$	10.7804	0.394963
$746$	0	0
$747$	23.8698	0.873352
$748$	0	0
$749$	23.1268	0.845035
$750$	0	0
$751$	−19.1238	−0.697839	−0.348919	−	0.937153i	$-0.613451\pi$
$751$	−19.1238	−0.697839	−0.348919	+	0.937153i	$0.613451\pi$
$752$	0	0
$753$	4.76494	0.173644
$754$	0	0
$755$	1.00000	0.0363937
$756$	0	0
$757$	38.8844	1.41328	0.706639	−	0.707574i	$-0.250210\pi$
$757$	38.8844	1.41328	0.706639	+	0.707574i	$0.250210\pi$
$758$	0	0
$759$	0.150366	0.00545794
$760$	0	0
$761$	28.6053	1.03694	0.518471	−	0.855095i	$-0.326501\pi$
$761$	28.6053	1.03694	0.518471	+	0.855095i	$0.326501\pi$
$762$	0	0
$763$	46.8696	1.69679
$764$	0	0
$765$	−11.0286	−0.398741
$766$	0	0
$767$	0.235087	0.00848851
$768$	0	0
$769$	−1.10018	−0.0396735	−0.0198368	−	0.999803i	$-0.506315\pi$
$769$	−1.10018	−0.0396735	−0.0198368	+	0.999803i	$0.506315\pi$
$770$	0	0
$771$	−5.55305	−0.199988
$772$	0	0
$773$	−20.1611	−0.725146	−0.362573	−	0.931955i	$-0.618102\pi$
$773$	−20.1611	−0.725146	−0.362573	+	0.931955i	$0.618102\pi$
$774$	0	0
$775$	−0.535578	−0.0192385
$776$	0	0
$777$	14.7961	0.530808
$778$	0	0
$779$	59.6909	2.13865
$780$	0	0
$781$	32.2570	1.15425
$782$	0	0
$783$	−6.32474	−0.226028
$784$	0	0
$785$	−6.79705	−0.242597
$786$	0	0
$787$	−36.1630	−1.28907	−0.644537	−	0.764573i	$-0.722950\pi$
$787$	−36.1630	−1.28907	−0.644537	+	0.764573i	$0.722950\pi$
$788$	0	0
$789$	10.3590	0.368790
$790$	0	0
$791$	32.2789	1.14771
$792$	0	0
$793$	1.24217	0.0441108
$794$	0	0
$795$	4.53148	0.160715
$796$	0	0
$797$	−0.711463	−0.0252013	−0.0126007	−	0.999921i	$-0.504011\pi$
$797$	−0.711463	−0.0252013	−0.0126007	+	0.999921i	$0.504011\pi$
$798$	0	0
$799$	−7.39516	−0.261622
$800$	0	0
$801$	22.2592	0.786490
$802$	0	0
$803$	−11.7209	−0.413620
$804$	0	0
$805$	0.432766	0.0152530
$806$	0	0
$807$	9.37722	0.330094
$808$	0	0
$809$	33.0952	1.16357	0.581783	−	0.813344i	$-0.302355\pi$
$809$	33.0952	1.16357	0.581783	+	0.813344i	$0.302355\pi$
$810$	0	0
$811$	−16.9967	−0.596835	−0.298418	−	0.954435i	$-0.596459\pi$
$811$	−16.9967	−0.596835	−0.298418	+	0.954435i	$0.596459\pi$
$812$	0	0
$813$	−6.89199	−0.241713
$814$	0	0
$815$	−10.9477	−0.383481
$816$	0	0
$817$	54.8810	1.92004
$818$	0	0
$819$	−0.771197	−0.0269478
$820$	0	0
$821$	−40.8714	−1.42642	−0.713211	−	0.700949i	$-0.752760\pi$
$821$	−40.8714	−1.42642	−0.713211	+	0.700949i	$0.752760\pi$
$822$	0	0
$823$	13.8495	0.482763	0.241382	−	0.970430i	$-0.422399\pi$
$823$	13.8495	0.482763	0.241382	+	0.970430i	$0.422399\pi$
$824$	0	0
$825$	−1.00031	−0.0348263
$826$	0	0
$827$	−4.02260	−0.139879	−0.0699397	−	0.997551i	$-0.522281\pi$
$827$	−4.02260	−0.139879	−0.0699397	+	0.997551i	$0.522281\pi$
$828$	0	0
$829$	−57.0302	−1.98074	−0.990371	−	0.138439i	$-0.955791\pi$
$829$	−57.0302	−1.98074	−0.990371	+	0.138439i	$0.955791\pi$
$830$	0	0
$831$	6.84703	0.237521
$832$	0	0
$833$	5.05664	0.175202
$834$	0	0
$835$	−18.1080	−0.626652
$836$	0	0
$837$	1.35547	0.0468520
$838$	0	0
$839$	−37.4269	−1.29212	−0.646060	−	0.763286i	$-0.723585\pi$
$839$	−37.4269	−1.29212	−0.646060	+	0.763286i	$0.723585\pi$
$840$	0	0
$841$	−22.7548	−0.784648
$842$	0	0
$843$	10.9940	0.378654
$844$	0	0
$845$	−12.9909	−0.446901
$846$	0	0
$847$	16.4856	0.566452
$848$	0	0
$849$	8.81238	0.302440
$850$	0	0
$851$	1.77358	0.0607977
$852$	0	0
$853$	−8.50988	−0.291373	−0.145686	−	0.989331i	$-0.546539\pi$
$853$	−8.50988	−0.291373	−0.145686	+	0.989331i	$0.546539\pi$
$854$	0	0
$855$	−18.5234	−0.633487
$856$	0	0
$857$	−7.12565	−0.243407	−0.121704	−	0.992566i	$-0.538836\pi$
$857$	−7.12565	−0.243407	−0.121704	+	0.992566i	$0.538836\pi$
$858$	0	0
$859$	22.1830	0.756873	0.378437	−	0.925627i	$-0.376462\pi$
$859$	22.1830	0.756873	0.378437	+	0.925627i	$0.376462\pi$
$860$	0	0
$861$	−11.3565	−0.387030
$862$	0	0
$863$	27.3622	0.931420	0.465710	−	0.884937i	$-0.345799\pi$
$863$	27.3622	0.931420	0.465710	+	0.884937i	$0.345799\pi$
$864$	0	0
$865$	−10.0264	−0.340908
$866$	0	0
$867$	−0.696524	−0.0236552
$868$	0	0
$869$	39.4533	1.33836
$870$	0	0
$871$	0.386773	0.0131053
$872$	0	0
$873$	−29.0155	−0.982027
$874$	0	0
$875$	−2.87898	−0.0973272
$876$	0	0
$877$	27.6885	0.934973	0.467487	−	0.884000i	$-0.345160\pi$
$877$	27.6885	0.934973	0.467487	+	0.884000i	$0.345160\pi$
$878$	0	0
$879$	0.516653	0.0174263
$880$	0	0
$881$	−34.6743	−1.16821	−0.584103	−	0.811679i	$-0.698554\pi$
$881$	−34.6743	−1.16821	−0.584103	+	0.811679i	$0.698554\pi$
$882$	0	0
$883$	51.3340	1.72753	0.863763	−	0.503898i	$-0.168101\pi$
$883$	51.3340	1.72753	0.863763	+	0.503898i	$0.168101\pi$
$884$	0	0
$885$	−1.07429	−0.0361120
$886$	0	0
$887$	−1.81524	−0.0609498	−0.0304749	−	0.999536i	$-0.509702\pi$
$887$	−1.81524	−0.0609498	−0.0304749	+	0.999536i	$0.509702\pi$
$888$	0	0
$889$	−38.9520	−1.30641
$890$	0	0
$891$	−16.8295	−0.563808
$892$	0	0
$893$	−12.4207	−0.415644
$894$	0	0
$895$	−6.55432	−0.219087
$896$	0	0
$897$	0.00624119	0.000208387 0
$898$	0	0
$899$	−1.33843	−0.0446392
$900$	0	0
$901$	40.8264	1.36012
$902$	0	0
$903$	−10.4414	−0.347469
$904$	0	0
$905$	−1.98392	−0.0659478
$906$	0	0
$907$	4.41840	0.146711	0.0733553	−	0.997306i	$-0.476629\pi$
$907$	4.41840	0.146711	0.0733553	+	0.997306i	$0.476629\pi$
$908$	0	0
$909$	−35.9876	−1.19363
$910$	0	0
$911$	4.72012	0.156385	0.0781923	−	0.996938i	$-0.475085\pi$
$911$	4.72012	0.156385	0.0781923	+	0.996938i	$0.475085\pi$
$912$	0	0
$913$	19.5058	0.645548
$914$	0	0
$915$	−5.67643	−0.187657
$916$	0	0
$917$	40.2956	1.33068
$918$	0	0
$919$	34.5048	1.13821	0.569105	−	0.822265i	$-0.307290\pi$
$919$	34.5048	1.13821	0.569105	+	0.822265i	$0.307290\pi$
$920$	0	0
$921$	7.33023	0.241539
$922$	0	0
$923$	1.33888	0.0440698
$924$	0	0
$925$	−11.7988	−0.387941
$926$	0	0
$927$	26.7920	0.879965
$928$	0	0
$929$	20.8723	0.684799	0.342399	−	0.939555i	$-0.388760\pi$
$929$	20.8723	0.684799	0.342399	+	0.939555i	$0.388760\pi$
$930$	0	0
$931$	8.49301	0.278347
$932$	0	0
$933$	−10.5933	−0.346810
$934$	0	0
$935$	−9.01229	−0.294734
$936$	0	0
$937$	−22.2913	−0.728226	−0.364113	−	0.931355i	$-0.618628\pi$
$937$	−22.2913	−0.728226	−0.364113	+	0.931355i	$0.618628\pi$
$938$	0	0
$939$	−4.92445	−0.160703
$940$	0	0
$941$	−52.8416	−1.72259	−0.861293	−	0.508109i	$-0.830345\pi$
$941$	−52.8416	−1.72259	−0.861293	+	0.508109i	$0.830345\pi$
$942$	0	0
$943$	−1.36129	−0.0443296
$944$	0	0
$945$	7.28630	0.237023
$946$	0	0
$947$	16.6008	0.539454	0.269727	−	0.962937i	$-0.413067\pi$
$947$	16.6008	0.539454	0.269727	+	0.962937i	$0.413067\pi$
$948$	0	0
$949$	−0.486494	−0.0157923
$950$	0	0
$951$	−4.85763	−0.157520
$952$	0	0
$953$	54.7708	1.77420	0.887100	−	0.461577i	$-0.152716\pi$
$953$	54.7708	1.77420	0.887100	+	0.461577i	$0.152716\pi$
$954$	0	0
$955$	−23.2637	−0.752797
$956$	0	0
$957$	−2.49982	−0.0808076
$958$	0	0
$959$	39.2435	1.26724
$960$	0	0
$961$	−30.7132	−0.990747
$962$	0	0
$963$	22.5748	0.727464
$964$	0	0
$965$	−6.70918	−0.215976
$966$	0	0
$967$	−2.82755	−0.0909277	−0.0454639	−	0.998966i	$-0.514477\pi$
$967$	−2.82755	−0.0909277	−0.0454639	+	0.998966i	$0.514477\pi$
$968$	0	0
$969$	11.2673	0.361957
$970$	0	0
$971$	−32.4830	−1.04243	−0.521215	−	0.853425i	$-0.674521\pi$
$971$	−32.4830	−1.04243	−0.521215	+	0.853425i	$0.674521\pi$
$972$	0	0
$973$	−14.5589	−0.466737
$974$	0	0
$975$	−0.0415195	−0.00132969
$976$	0	0
$977$	16.9936	0.543673	0.271837	−	0.962343i	$-0.412369\pi$
$977$	16.9936	0.543673	0.271837	+	0.962343i	$0.412369\pi$
$978$	0	0
$979$	18.1896	0.581343
$980$	0	0
$981$	45.7510	1.46072
$982$	0	0
$983$	−36.9257	−1.17775	−0.588874	−	0.808225i	$-0.700429\pi$
$983$	−36.9257	−1.17775	−0.588874	+	0.808225i	$0.700429\pi$
$984$	0	0
$985$	11.4808	0.365809
$986$	0	0
$987$	2.36312	0.0752188
$988$	0	0
$989$	−1.25159	−0.0397984
$990$	0	0
$991$	29.2820	0.930172	0.465086	−	0.885265i	$-0.346023\pi$
$991$	29.2820	0.930172	0.465086	+	0.885265i	$0.346023\pi$
$992$	0	0
$993$	−13.2727	−0.421198
$994$	0	0
$995$	−0.798470	−0.0253132
$996$	0	0
$997$	52.3315	1.65736	0.828678	−	0.559726i	$-0.189094\pi$
$997$	52.3315	1.65736	0.828678	+	0.559726i	$0.189094\pi$
$998$	0	0
$999$	29.8611	0.944763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.p.1.11	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.p.1.11	✓	19	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q+0.435585 q^{3} +1.00000 q^{5} -2.87898 q^{7} -2.81027 q^{9} +O(q^{10})\)	\(q+0.435585 q^{3} +1.00000 q^{5} -2.87898 q^{7} -2.81027 q^{9} -2.29647 q^{11} -0.0953190 q^{13} +0.435585 q^{15} +3.92440 q^{17} +6.59133 q^{19} -1.25404 q^{21} -0.150319 q^{23} +1.00000 q^{25} -2.53086 q^{27} +2.49904 q^{29} -0.535578 q^{31} -1.00031 q^{33} -2.87898 q^{35} -11.7988 q^{37} -0.0415195 q^{39} +9.05597 q^{41} +8.32623 q^{43} -2.81027 q^{45} -1.88440 q^{47} +1.28851 q^{49} +1.70941 q^{51} +10.4032 q^{53} -2.29647 q^{55} +2.87108 q^{57} -2.46632 q^{59} -13.0317 q^{61} +8.09069 q^{63} -0.0953190 q^{65} -4.05767 q^{67} -0.0654768 q^{69} -14.0463 q^{71} +5.10385 q^{73} +0.435585 q^{75} +6.61150 q^{77} -17.1799 q^{79} +7.32839 q^{81} -8.49380 q^{83} +3.92440 q^{85} +1.08855 q^{87} -7.92067 q^{89} +0.274421 q^{91} -0.233289 q^{93} +6.59133 q^{95} +10.3248 q^{97} +6.45371 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9}+O(q^{10}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9	\( 19 q - 5 q^{3} + 19 q^{5} - 8 q^{7} + 26 q^{9} - 18 q^{11} + 5 q^{13} - 5 q^{15} - 4 q^{17} - 27 q^{19} - 18 q^{21} - 25 q^{23} + 19 q^{25} - 35 q^{27} - 35 q^{29} - 26 q^{31} - 8 q^{35} - 10 q^{37} - 48 q^{39} - 14 q^{41} - 21 q^{43} + 26 q^{45} - 40 q^{47} + 23 q^{49} - 32 q^{51} - 3 q^{53} - 18 q^{55} - 13 q^{57} - 28 q^{59} - 46 q^{61} - 53 q^{63} + 5 q^{65} - 42 q^{67} - 31 q^{69} - 46 q^{71} + 31 q^{73} - 5 q^{75} + 15 q^{77} - 56 q^{79} + 31 q^{81} - 25 q^{83} - 4 q^{85} - 20 q^{87} - 7 q^{89} - 61 q^{91} + 29 q^{93} - 27 q^{95} + 39 q^{97} - 52 q^{99}+O(q^{100}) \) 19 * q - 5 * q^3 + 19 * q^5 - 8 * q^7 + 26 * q^9 - 18 * q^11 + 5 * q^13 - 5 * q^15 - 4 * q^17 - 27 * q^19 - 18 * q^21 - 25 * q^23 + 19 * q^25 - 35 * q^27 - 35 * q^29 - 26 * q^31 - 8 * q^35 - 10 * q^37 - 48 * q^39 - 14 * q^41 - 21 * q^43 + 26 * q^45 - 40 * q^47 + 23 * q^49 - 32 * q^51 - 3 * q^53 - 18 * q^55 - 13 * q^57 - 28 * q^59 - 46 * q^61 - 53 * q^63 + 5 * q^65 - 42 * q^67 - 31 * q^69 - 46 * q^71 + 31 * q^73 - 5 * q^75 + 15 * q^77 - 56 * q^79 + 31 * q^81 - 25 * q^83 - 4 * q^85 - 20 * q^87 - 7 * q^89 - 61 * q^91 + 29 * q^93 - 27 * q^95 + 39 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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