LMFDB - Embedded newform 3040.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3040,2,Mod(1,3040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3040.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3040 = 2^{5} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3040.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-4,0,-5,0,4,0,7,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$24.2745222145$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.387268.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 $ x^5 - x^4 - 7x^3 + 4x^2 + 12*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.160536$ of defining polynomial
Character	$\chi$	$=$	3040.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.13476 q^{3} -1.00000 q^{5} +0.878290 q^{7} +1.55722 q^{9} -4.50981 q^{11} -2.05398 q^{13} -2.13476 q^{15} -4.31693 q^{17} +1.00000 q^{19} +1.87494 q^{21} -2.15430 q^{23} +1.00000 q^{25} -3.08000 q^{27} +9.01549 q^{29} +4.35137 q^{31} -9.62738 q^{33} -0.878290 q^{35} -8.67901 q^{37} -4.38475 q^{39} +4.26953 q^{41} -12.3806 q^{43} -1.55722 q^{45} +2.76620 q^{47} -6.22861 q^{49} -9.21564 q^{51} -3.98510 q^{53} +4.50981 q^{55} +2.13476 q^{57} -11.5800 q^{59} -1.51630 q^{61} +1.36769 q^{63} +2.05398 q^{65} -1.10217 q^{67} -4.59893 q^{69} -14.4854 q^{71} -6.03444 q^{73} +2.13476 q^{75} -3.96092 q^{77} +7.34489 q^{79} -11.2467 q^{81} +1.34175 q^{83} +4.31693 q^{85} +19.2459 q^{87} -15.5455 q^{89} -1.80399 q^{91} +9.28915 q^{93} -1.00000 q^{95} +7.55630 q^{97} -7.02276 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9} - 2 q^{11} - 4 q^{13} + 4 q^{15} - 12 q^{17} + 5 q^{19} - 10 q^{21} + 8 q^{23} + 5 q^{25} - 16 q^{27} - 6 q^{29} + 10 q^{31} - 18 q^{33} - 4 q^{35} - 6 q^{37}+ \cdots + 14 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 - 5 * q^5 + 4 * q^7 + 7 * q^9 - 2 * q^11 - 4 * q^13 + 4 * q^15 - 12 * q^17 + 5 * q^19 - 10 * q^21 + 8 * q^23 + 5 * q^25 - 16 * q^27 - 6 * q^29 + 10 * q^31 - 18 * q^33 - 4 * q^35 - 6 * q^37 + 18 * q^39 - 8 * q^41 - 12 * q^43 - 7 * q^45 + 16 * q^47 + 7 * q^49 + 14 * q^51 - 18 * q^53 + 2 * q^55 - 4 * q^57 - 8 * q^59 + 2 * q^61 + 36 * q^63 + 4 * q^65 - 10 * q^67 - 22 * q^69 - 18 * q^71 - 28 * q^73 - 4 * q^75 - 28 * q^77 + 14 * q^79 + 25 * q^81 - 8 * q^83 + 12 * q^85 + 24 * q^87 - 30 * q^89 - 28 * q^91 - 24 * q^93 - 5 * q^95 - 18 * q^97 + 14 * q^99

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
$3$	2.13476	1.23251	0.616253	−	0.787548i	$-0.288650\pi$
$3$	2.13476	1.23251	0.616253	+	0.787548i	$0.288650\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.878290	0.331962	0.165981	−	0.986129i	$-0.446921\pi$
$7$	0.878290	0.331962	0.165981	+	0.986129i	$0.446921\pi$
$8$	0	0
$9$	1.55722	0.519073
$10$	0	0
$11$	−4.50981	−1.35976	−0.679880	−	0.733324i	$-0.737968\pi$
$11$	−4.50981	−1.35976	−0.679880	+	0.733324i	$0.737968\pi$
$12$	0	0
$13$	−2.05398	−0.569670	−0.284835	−	0.958577i	$-0.591939\pi$
$13$	−2.05398	−0.569670	−0.284835	+	0.958577i	$0.591939\pi$
$14$	0	0
$15$	−2.13476	−0.551194
$16$	0	0
$17$	−4.31693	−1.04701	−0.523505	−	0.852022i	$-0.675376\pi$
$17$	−4.31693	−1.04701	−0.523505	+	0.852022i	$0.675376\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.87494	0.409146
$22$	0	0
$23$	−2.15430	−0.449203	−0.224602	−	0.974451i	$-0.572108\pi$
$23$	−2.15430	−0.449203	−0.224602	+	0.974451i	$0.572108\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.08000	−0.592746
$28$	0	0
$29$	9.01549	1.67413	0.837067	−	0.547101i	$-0.184268\pi$
$29$	9.01549	1.67413	0.837067	+	0.547101i	$0.184268\pi$
$30$	0	0
$31$	4.35137	0.781529	0.390765	−	0.920491i	$-0.372211\pi$
$31$	4.35137	0.781529	0.390765	+	0.920491i	$0.372211\pi$
$32$	0	0
$33$	−9.62738	−1.67591
$34$	0	0
$35$	−0.878290	−0.148458
$36$	0	0
$37$	−8.67901	−1.42682	−0.713410	−	0.700746i	$-0.752850\pi$
$37$	−8.67901	−1.42682	−0.713410	+	0.700746i	$0.752850\pi$
$38$	0	0
$39$	−4.38475	−0.702123
$40$	0	0
$41$	4.26953	0.666788	0.333394	−	0.942788i	$-0.391806\pi$
$41$	4.26953	0.666788	0.333394	+	0.942788i	$0.391806\pi$
$42$	0	0
$43$	−12.3806	−1.88803	−0.944013	−	0.329908i	$-0.892982\pi$
$43$	−12.3806	−1.88803	−0.944013	+	0.329908i	$0.892982\pi$
$44$	0	0
$45$	−1.55722	−0.232136
$46$	0	0
$47$	2.76620	0.403492	0.201746	−	0.979438i	$-0.435338\pi$
$47$	2.76620	0.403492	0.201746	+	0.979438i	$0.435338\pi$
$48$	0	0
$49$	−6.22861	−0.889801
$50$	0	0
$51$	−9.21564	−1.29045
$52$	0	0
$53$	−3.98510	−0.547396	−0.273698	−	0.961816i	$-0.588247\pi$
$53$	−3.98510	−0.547396	−0.273698	+	0.961816i	$0.588247\pi$
$54$	0	0
$55$	4.50981	0.608103
$56$	0	0
$57$	2.13476	0.282756
$58$	0	0
$59$	−11.5800	−1.50758	−0.753792	−	0.657113i	$-0.771778\pi$
$59$	−11.5800	−1.50758	−0.753792	+	0.657113i	$0.771778\pi$
$60$	0	0
$61$	−1.51630	−0.194142	−0.0970709	−	0.995277i	$-0.530947\pi$
$61$	−1.51630	−0.194142	−0.0970709	+	0.995277i	$0.530947\pi$
$62$	0	0
$63$	1.36769	0.172313
$64$	0	0
$65$	2.05398	0.254764
$66$	0	0
$67$	−1.10217	−0.134652	−0.0673258	−	0.997731i	$-0.521447\pi$
$67$	−1.10217	−0.134652	−0.0673258	+	0.997731i	$0.521447\pi$
$68$	0	0
$69$	−4.59893	−0.553646
$70$	0	0
$71$	−14.4854	−1.71910	−0.859552	−	0.511048i	$-0.829257\pi$
$71$	−14.4854	−1.71910	−0.859552	+	0.511048i	$0.829257\pi$
$72$	0	0
$73$	−6.03444	−0.706277	−0.353139	−	0.935571i	$-0.614886\pi$
$73$	−6.03444	−0.706277	−0.353139	+	0.935571i	$0.614886\pi$
$74$	0	0
$75$	2.13476	0.246501
$76$	0	0
$77$	−3.96092	−0.451389
$78$	0	0
$79$	7.34489	0.826364	0.413182	−	0.910648i	$-0.364417\pi$
$79$	7.34489	0.826364	0.413182	+	0.910648i	$0.364417\pi$
$80$	0	0
$81$	−11.2467	−1.24964
$82$	0	0
$83$	1.34175	0.147276	0.0736381	−	0.997285i	$-0.476539\pi$
$83$	1.34175	0.147276	0.0736381	+	0.997285i	$0.476539\pi$
$84$	0	0
$85$	4.31693	0.468237
$86$	0	0
$87$	19.2459	2.06338
$88$	0	0
$89$	−15.5455	−1.64782	−0.823912	−	0.566718i	$-0.808213\pi$
$89$	−15.5455	−1.64782	−0.823912	+	0.566718i	$0.808213\pi$
$90$	0	0
$91$	−1.80399	−0.189109
$92$	0	0
$93$	9.28915	0.963240
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	7.55630	0.767226	0.383613	−	0.923494i	$-0.374680\pi$
$97$	7.55630	0.767226	0.383613	+	0.923494i	$0.374680\pi$
$98$	0	0
$99$	−7.02276	−0.705814
$100$	0	0
$101$	−8.50981	−0.846758	−0.423379	−	0.905953i	$-0.639156\pi$
$101$	−8.50981	−0.846758	−0.423379	+	0.905953i	$0.639156\pi$
$102$	0	0
$103$	−0.763993	−0.0752784	−0.0376392	−	0.999291i	$-0.511984\pi$
$103$	−0.763993	−0.0752784	−0.0376392	+	0.999291i	$0.511984\pi$
$104$	0	0
$105$	−1.87494	−0.182976
$106$	0	0
$107$	16.4092	1.58633	0.793167	−	0.609005i	$-0.208431\pi$
$107$	16.4092	1.58633	0.793167	+	0.609005i	$0.208431\pi$
$108$	0	0
$109$	19.0286	1.82261	0.911306	−	0.411730i	$-0.135075\pi$
$109$	19.0286	1.82261	0.911306	+	0.411730i	$0.135075\pi$
$110$	0	0
$111$	−18.5276	−1.75857
$112$	0	0
$113$	6.40121	0.602175	0.301088	−	0.953596i	$-0.402650\pi$
$113$	6.40121	0.602175	0.301088	+	0.953596i	$0.402650\pi$
$114$	0	0
$115$	2.15430	0.200890
$116$	0	0
$117$	−3.19849	−0.295700
$118$	0	0
$119$	−3.79152	−0.347568
$120$	0	0
$121$	9.33840	0.848946
$122$	0	0
$123$	9.11444	0.821821
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.1135	1.25237	0.626184	−	0.779675i	$-0.284616\pi$
$127$	14.1135	1.25237	0.626184	+	0.779675i	$0.284616\pi$
$128$	0	0
$129$	−26.4297	−2.32700
$130$	0	0
$131$	10.7289	0.937384	0.468692	−	0.883362i	$-0.344725\pi$
$131$	10.7289	0.937384	0.468692	+	0.883362i	$0.344725\pi$
$132$	0	0
$133$	0.878290	0.0761574
$134$	0	0
$135$	3.08000	0.265084
$136$	0	0
$137$	−21.3366	−1.82291	−0.911453	−	0.411405i	$-0.865038\pi$
$137$	−21.3366	−1.82291	−0.911453	+	0.411405i	$0.865038\pi$
$138$	0	0
$139$	6.49684	0.551055	0.275527	−	0.961293i	$-0.411147\pi$
$139$	6.49684	0.551055	0.275527	+	0.961293i	$0.411147\pi$
$140$	0	0
$141$	5.90519	0.497306
$142$	0	0
$143$	9.26304	0.774615
$144$	0	0
$145$	−9.01549	−0.748695
$146$	0	0
$147$	−13.2966	−1.09669
$148$	0	0
$149$	−13.4904	−1.10517	−0.552587	−	0.833456i	$-0.686359\pi$
$149$	−13.4904	−1.10517	−0.552587	+	0.833456i	$0.686359\pi$
$150$	0	0
$151$	19.7890	1.61040	0.805201	−	0.593001i	$-0.202057\pi$
$151$	19.7890	1.61040	0.805201	+	0.593001i	$0.202057\pi$
$152$	0	0
$153$	−6.72241	−0.543475
$154$	0	0
$155$	−4.35137	−0.349511
$156$	0	0
$157$	2.37748	0.189744	0.0948718	−	0.995490i	$-0.469756\pi$
$157$	2.37748	0.189744	0.0948718	+	0.995490i	$0.469756\pi$
$158$	0	0
$159$	−8.50725	−0.674669
$160$	0	0
$161$	−1.89210	−0.149119
$162$	0	0
$163$	−8.86118	−0.694061	−0.347031	−	0.937854i	$-0.612810\pi$
$163$	−8.86118	−0.694061	−0.347031	+	0.937854i	$0.612810\pi$
$164$	0	0
$165$	9.62738	0.749491
$166$	0	0
$167$	6.26190	0.484561	0.242280	−	0.970206i	$-0.422105\pi$
$167$	6.26190	0.484561	0.242280	+	0.970206i	$0.422105\pi$
$168$	0	0
$169$	−8.78118	−0.675476
$170$	0	0
$171$	1.55722	0.119083
$172$	0	0
$173$	−18.2673	−1.38884	−0.694419	−	0.719571i	$-0.744339\pi$
$173$	−18.2673	−1.38884	−0.694419	+	0.719571i	$0.744339\pi$
$174$	0	0
$175$	0.878290	0.0663925
$176$	0	0
$177$	−24.7205	−1.85811
$178$	0	0
$179$	−14.1211	−1.05546	−0.527730	−	0.849412i	$-0.676957\pi$
$179$	−14.1211	−1.05546	−0.527730	+	0.849412i	$0.676957\pi$
$180$	0	0
$181$	−3.34489	−0.248623	−0.124312	−	0.992243i	$-0.539672\pi$
$181$	−3.34489	−0.248623	−0.124312	+	0.992243i	$0.539672\pi$
$182$	0	0
$183$	−3.23694	−0.239281
$184$	0	0
$185$	8.67901	0.638094
$186$	0	0
$187$	19.4686	1.42368
$188$	0	0
$189$	−2.70513	−0.196769
$190$	0	0
$191$	3.76558	0.272468	0.136234	−	0.990677i	$-0.456500\pi$
$191$	3.76558	0.272468	0.136234	+	0.990677i	$0.456500\pi$
$192$	0	0
$193$	14.7505	1.06176	0.530881	−	0.847446i	$-0.321861\pi$
$193$	14.7505	1.06176	0.530881	+	0.847446i	$0.321861\pi$
$194$	0	0
$195$	4.38475	0.313999
$196$	0	0
$197$	−22.0976	−1.57439	−0.787193	−	0.616706i	$-0.788467\pi$
$197$	−22.0976	−1.57439	−0.787193	+	0.616706i	$0.788467\pi$
$198$	0	0
$199$	13.2198	0.937129	0.468564	−	0.883429i	$-0.344771\pi$
$199$	13.2198	0.937129	0.468564	+	0.883429i	$0.344771\pi$
$200$	0	0
$201$	−2.35288	−0.165959
$202$	0	0
$203$	7.91821	0.555749
$204$	0	0
$205$	−4.26953	−0.298197
$206$	0	0
$207$	−3.35472	−0.233169
$208$	0	0
$209$	−4.50981	−0.311950
$210$	0	0
$211$	4.41192	0.303729	0.151864	−	0.988401i	$-0.451472\pi$
$211$	4.41192	0.303729	0.151864	+	0.988401i	$0.451472\pi$
$212$	0	0
$213$	−30.9230	−2.11881
$214$	0	0
$215$	12.3806	0.844351
$216$	0	0
$217$	3.82177	0.259438
$218$	0	0
$219$	−12.8821	−0.870492
$220$	0	0
$221$	8.86688	0.596451
$222$	0	0
$223$	26.0319	1.74322	0.871612	−	0.490196i	$-0.163075\pi$
$223$	26.0319	1.74322	0.871612	+	0.490196i	$0.163075\pi$
$224$	0	0
$225$	1.55722	0.103815
$226$	0	0
$227$	−3.44727	−0.228803	−0.114402	−	0.993435i	$-0.536495\pi$
$227$	−3.44727	−0.228803	−0.114402	+	0.993435i	$0.536495\pi$
$228$	0	0
$229$	17.1177	1.13117	0.565586	−	0.824689i	$-0.308650\pi$
$229$	17.1177	1.13117	0.565586	+	0.824689i	$0.308650\pi$
$230$	0	0
$231$	−8.45564	−0.556340
$232$	0	0
$233$	0.838212	0.0549131	0.0274565	−	0.999623i	$-0.491259\pi$
$233$	0.838212	0.0549131	0.0274565	+	0.999623i	$0.491259\pi$
$234$	0	0
$235$	−2.76620	−0.180447
$236$	0	0
$237$	15.6796	1.01850
$238$	0	0
$239$	−9.10560	−0.588993	−0.294496	−	0.955653i	$-0.595152\pi$
$239$	−9.10560	−0.588993	−0.294496	+	0.955653i	$0.595152\pi$
$240$	0	0
$241$	−10.4099	−0.670558	−0.335279	−	0.942119i	$-0.608831\pi$
$241$	−10.4099	−0.670558	−0.335279	+	0.942119i	$0.608831\pi$
$242$	0	0
$243$	−14.7691	−0.947439
$244$	0	0
$245$	6.22861	0.397931
$246$	0	0
$247$	−2.05398	−0.130691
$248$	0	0
$249$	2.86432	0.181519
$250$	0	0
$251$	18.1237	1.14396	0.571978	−	0.820269i	$-0.306176\pi$
$251$	18.1237	1.14396	0.571978	+	0.820269i	$0.306176\pi$
$252$	0	0
$253$	9.71550	0.610808
$254$	0	0
$255$	9.21564	0.577106
$256$	0	0
$257$	3.34726	0.208797	0.104398	−	0.994536i	$-0.466708\pi$
$257$	3.34726	0.208797	0.104398	+	0.994536i	$0.466708\pi$
$258$	0	0
$259$	−7.62269	−0.473651
$260$	0	0
$261$	14.0391	0.868997
$262$	0	0
$263$	−10.3379	−0.637459	−0.318730	−	0.947846i	$-0.603256\pi$
$263$	−10.3379	−0.637459	−0.318730	+	0.947846i	$0.603256\pi$
$264$	0	0
$265$	3.98510	0.244803
$266$	0	0
$267$	−33.1861	−2.03095
$268$	0	0
$269$	3.82804	0.233400	0.116700	−	0.993167i	$-0.462768\pi$
$269$	3.82804	0.233400	0.116700	+	0.993167i	$0.462768\pi$
$270$	0	0
$271$	0.727573	0.0441969	0.0220985	−	0.999756i	$-0.492965\pi$
$271$	0.727573	0.0441969	0.0220985	+	0.999756i	$0.492965\pi$
$272$	0	0
$273$	−3.85109	−0.233078
$274$	0	0
$275$	−4.50981	−0.271952
$276$	0	0
$277$	−20.8347	−1.25184	−0.625918	−	0.779889i	$-0.715275\pi$
$277$	−20.8347	−1.25184	−0.625918	+	0.779889i	$0.715275\pi$
$278$	0	0
$279$	6.77603	0.405671
$280$	0	0
$281$	27.8475	1.66124	0.830620	−	0.556840i	$-0.187986\pi$
$281$	27.8475	1.66124	0.830620	+	0.556840i	$0.187986\pi$
$282$	0	0
$283$	−29.1356	−1.73193	−0.865965	−	0.500104i	$-0.833295\pi$
$283$	−29.1356	−1.73193	−0.865965	+	0.500104i	$0.833295\pi$
$284$	0	0
$285$	−2.13476	−0.126453
$286$	0	0
$287$	3.74988	0.221349
$288$	0	0
$289$	1.63592	0.0962309
$290$	0	0
$291$	16.1309	0.945611
$292$	0	0
$293$	10.3105	0.602348	0.301174	−	0.953569i	$-0.402622\pi$
$293$	10.3105	0.602348	0.301174	+	0.953569i	$0.402622\pi$
$294$	0	0
$295$	11.5800	0.674212
$296$	0	0
$297$	13.8902	0.805992
$298$	0	0
$299$	4.42489	0.255898
$300$	0	0
$301$	−10.8738	−0.626754
$302$	0	0
$303$	−18.1664	−1.04363
$304$	0	0
$305$	1.51630	0.0868229
$306$	0	0
$307$	15.6872	0.895317	0.447659	−	0.894205i	$-0.352258\pi$
$307$	15.6872	0.895317	0.447659	+	0.894205i	$0.352258\pi$
$308$	0	0
$309$	−1.63094	−0.0927812
$310$	0	0
$311$	34.6721	1.96608	0.983038	−	0.183404i	$-0.0587116\pi$
$311$	34.6721	1.96608	0.983038	+	0.183404i	$0.0587116\pi$
$312$	0	0
$313$	21.9918	1.24305	0.621526	−	0.783393i	$-0.286513\pi$
$313$	21.9918	1.24305	0.621526	+	0.783393i	$0.286513\pi$
$314$	0	0
$315$	−1.36769	−0.0770605
$316$	0	0
$317$	−20.3496	−1.14295	−0.571474	−	0.820620i	$-0.693628\pi$
$317$	−20.3496	−1.14295	−0.571474	+	0.820620i	$0.693628\pi$
$318$	0	0
$319$	−40.6581	−2.27642
$320$	0	0
$321$	35.0297	1.95517
$322$	0	0
$323$	−4.31693	−0.240201
$324$	0	0
$325$	−2.05398	−0.113934
$326$	0	0
$327$	40.6216	2.24638
$328$	0	0
$329$	2.42953	0.133944
$330$	0	0
$331$	−9.45748	−0.519830	−0.259915	−	0.965631i	$-0.583695\pi$
$331$	−9.45748	−0.519830	−0.259915	+	0.965631i	$0.583695\pi$
$332$	0	0
$333$	−13.5151	−0.740624
$334$	0	0
$335$	1.10217	0.0602180
$336$	0	0
$337$	2.96821	0.161689	0.0808443	−	0.996727i	$-0.474238\pi$
$337$	2.96821	0.161689	0.0808443	+	0.996727i	$0.474238\pi$
$338$	0	0
$339$	13.6651	0.742185
$340$	0	0
$341$	−19.6239	−1.06269
$342$	0	0
$343$	−11.6186	−0.627343
$344$	0	0
$345$	4.59893	0.247598
$346$	0	0
$347$	12.4197	0.666724	0.333362	−	0.942799i	$-0.391817\pi$
$347$	12.4197	0.666724	0.333362	+	0.942799i	$0.391817\pi$
$348$	0	0
$349$	−26.9381	−1.44196	−0.720981	−	0.692955i	$-0.756308\pi$
$349$	−26.9381	−1.44196	−0.720981	+	0.692955i	$0.756308\pi$
$350$	0	0
$351$	6.32624	0.337670
$352$	0	0
$353$	−12.3858	−0.659230	−0.329615	−	0.944115i	$-0.606919\pi$
$353$	−12.3858	−0.659230	−0.329615	+	0.944115i	$0.606919\pi$
$354$	0	0
$355$	14.4854	0.768807
$356$	0	0
$357$	−8.09400	−0.428380
$358$	0	0
$359$	−1.35216	−0.0713643	−0.0356822	−	0.999363i	$-0.511360\pi$
$359$	−1.35216	−0.0713643	−0.0356822	+	0.999363i	$0.511360\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	19.9353	1.04633
$364$	0	0
$365$	6.03444	0.315857
$366$	0	0
$367$	0.362613	0.0189283	0.00946413	−	0.999955i	$-0.496987\pi$
$367$	0.362613	0.0189283	0.00946413	+	0.999955i	$0.496987\pi$
$368$	0	0
$369$	6.64859	0.346112
$370$	0	0
$371$	−3.50008	−0.181715
$372$	0	0
$373$	−6.79352	−0.351755	−0.175878	−	0.984412i	$-0.556276\pi$
$373$	−6.79352	−0.351755	−0.175878	+	0.984412i	$0.556276\pi$
$374$	0	0
$375$	−2.13476	−0.110239
$376$	0	0
$377$	−18.5176	−0.953704
$378$	0	0
$379$	3.29889	0.169453	0.0847263	−	0.996404i	$-0.472998\pi$
$379$	3.29889	0.169453	0.0847263	+	0.996404i	$0.472998\pi$
$380$	0	0
$381$	30.1289	1.54355
$382$	0	0
$383$	4.62625	0.236390	0.118195	−	0.992990i	$-0.462289\pi$
$383$	4.62625	0.236390	0.118195	+	0.992990i	$0.462289\pi$
$384$	0	0
$385$	3.96092	0.201867
$386$	0	0
$387$	−19.2793	−0.980023
$388$	0	0
$389$	−18.6792	−0.947074	−0.473537	−	0.880774i	$-0.657023\pi$
$389$	−18.6792	−0.947074	−0.473537	+	0.880774i	$0.657023\pi$
$390$	0	0
$391$	9.29999	0.470320
$392$	0	0
$393$	22.9036	1.15533
$394$	0	0
$395$	−7.34489	−0.369561
$396$	0	0
$397$	14.1602	0.710678	0.355339	−	0.934737i	$-0.384365\pi$
$397$	14.1602	0.710678	0.355339	+	0.934737i	$0.384365\pi$
$398$	0	0
$399$	1.87494	0.0938645
$400$	0	0
$401$	30.5975	1.52797	0.763984	−	0.645235i	$-0.223240\pi$
$401$	30.5975	1.52797	0.763984	+	0.645235i	$0.223240\pi$
$402$	0	0
$403$	−8.93761	−0.445214
$404$	0	0
$405$	11.2467	0.558854
$406$	0	0
$407$	39.1407	1.94013
$408$	0	0
$409$	−4.69447	−0.232127	−0.116063	−	0.993242i	$-0.537028\pi$
$409$	−4.69447	−0.232127	−0.116063	+	0.993242i	$0.537028\pi$
$410$	0	0
$411$	−45.5485	−2.24674
$412$	0	0
$413$	−10.1706	−0.500461
$414$	0	0
$415$	−1.34175	−0.0658639
$416$	0	0
$417$	13.8692	0.679179
$418$	0	0
$419$	−4.93761	−0.241218	−0.120609	−	0.992700i	$-0.538485\pi$
$419$	−4.93761	−0.241218	−0.120609	+	0.992700i	$0.538485\pi$
$420$	0	0
$421$	24.4531	1.19177	0.595885	−	0.803070i	$-0.296801\pi$
$421$	24.4531	1.19177	0.595885	+	0.803070i	$0.296801\pi$
$422$	0	0
$423$	4.30758	0.209442
$424$	0	0
$425$	−4.31693	−0.209402
$426$	0	0
$427$	−1.33175	−0.0644478
$428$	0	0
$429$	19.7744	0.954718
$430$	0	0
$431$	25.8908	1.24712	0.623558	−	0.781777i	$-0.285686\pi$
$431$	25.8908	1.24712	0.623558	+	0.781777i	$0.285686\pi$
$432$	0	0
$433$	−23.5184	−1.13022	−0.565111	−	0.825015i	$-0.691167\pi$
$433$	−23.5184	−1.13022	−0.565111	+	0.825015i	$0.691167\pi$
$434$	0	0
$435$	−19.2459	−0.922772
$436$	0	0
$437$	−2.15430	−0.103054
$438$	0	0
$439$	−6.16158	−0.294076	−0.147038	−	0.989131i	$-0.546974\pi$
$439$	−6.16158	−0.294076	−0.147038	+	0.989131i	$0.546974\pi$
$440$	0	0
$441$	−9.69930	−0.461871
$442$	0	0
$443$	28.8889	1.37255	0.686276	−	0.727341i	$-0.259244\pi$
$443$	28.8889	1.37255	0.686276	+	0.727341i	$0.259244\pi$
$444$	0	0
$445$	15.5455	0.736929
$446$	0	0
$447$	−28.7987	−1.36213
$448$	0	0
$449$	−23.7011	−1.11853	−0.559263	−	0.828991i	$-0.688916\pi$
$449$	−23.7011	−1.11853	−0.559263	+	0.828991i	$0.688916\pi$
$450$	0	0
$451$	−19.2548	−0.906672
$452$	0	0
$453$	42.2448	1.98483
$454$	0	0
$455$	1.80399	0.0845722
$456$	0	0
$457$	12.0585	0.564075	0.282037	−	0.959403i	$-0.408990\pi$
$457$	12.0585	0.564075	0.282037	+	0.959403i	$0.408990\pi$
$458$	0	0
$459$	13.2962	0.620611
$460$	0	0
$461$	−18.6778	−0.869912	−0.434956	−	0.900452i	$-0.643236\pi$
$461$	−18.6778	−0.869912	−0.434956	+	0.900452i	$0.643236\pi$
$462$	0	0
$463$	−20.3186	−0.944286	−0.472143	−	0.881522i	$-0.656519\pi$
$463$	−20.3186	−0.944286	−0.472143	+	0.881522i	$0.656519\pi$
$464$	0	0
$465$	−9.28915	−0.430774
$466$	0	0
$467$	−27.7921	−1.28606	−0.643032	−	0.765839i	$-0.722324\pi$
$467$	−27.7921	−1.28606	−0.643032	+	0.765839i	$0.722324\pi$
$468$	0	0
$469$	−0.968026	−0.0446993
$470$	0	0
$471$	5.07536	0.233860
$472$	0	0
$473$	55.8342	2.56726
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	−6.20567	−0.284138
$478$	0	0
$479$	−24.4130	−1.11546	−0.557729	−	0.830023i	$-0.688327\pi$
$479$	−24.4130	−1.11546	−0.557729	+	0.830023i	$0.688327\pi$
$480$	0	0
$481$	17.8265	0.812818
$482$	0	0
$483$	−4.03919	−0.183790
$484$	0	0
$485$	−7.55630	−0.343114
$486$	0	0
$487$	11.1955	0.507317	0.253659	−	0.967294i	$-0.418366\pi$
$487$	11.1955	0.507317	0.253659	+	0.967294i	$0.418366\pi$
$488$	0	0
$489$	−18.9165	−0.855435
$490$	0	0
$491$	−26.8953	−1.21377	−0.606884	−	0.794791i	$-0.707581\pi$
$491$	−26.8953	−1.21377	−0.606884	+	0.794791i	$0.707581\pi$
$492$	0	0
$493$	−38.9193	−1.75284
$494$	0	0
$495$	7.02276	0.315650
$496$	0	0
$497$	−12.7224	−0.570678
$498$	0	0
$499$	11.1629	0.499721	0.249861	−	0.968282i	$-0.419615\pi$
$499$	11.1629	0.499721	0.249861	+	0.968282i	$0.419615\pi$
$500$	0	0
$501$	13.3677	0.597224
$502$	0	0
$503$	0.979268	0.0436634	0.0218317	−	0.999762i	$-0.493050\pi$
$503$	0.979268	0.0436634	0.0218317	+	0.999762i	$0.493050\pi$
$504$	0	0
$505$	8.50981	0.378682
$506$	0	0
$507$	−18.7458	−0.832528
$508$	0	0
$509$	−21.5263	−0.954136	−0.477068	−	0.878866i	$-0.658301\pi$
$509$	−21.5263	−0.954136	−0.477068	+	0.878866i	$0.658301\pi$
$510$	0	0
$511$	−5.29999	−0.234458
$512$	0	0
$513$	−3.08000	−0.135985
$514$	0	0
$515$	0.763993	0.0336655
$516$	0	0
$517$	−12.4750	−0.548652
$518$	0	0
$519$	−38.9964	−1.71175
$520$	0	0
$521$	29.2184	1.28008	0.640042	−	0.768340i	$-0.278917\pi$
$521$	29.2184	1.28008	0.640042	+	0.768340i	$0.278917\pi$
$522$	0	0
$523$	−2.90627	−0.127082	−0.0635412	−	0.997979i	$-0.520239\pi$
$523$	−2.90627	−0.127082	−0.0635412	+	0.997979i	$0.520239\pi$
$524$	0	0
$525$	1.87494	0.0818292
$526$	0	0
$527$	−18.7846	−0.818269
$528$	0	0
$529$	−18.3590	−0.798216
$530$	0	0
$531$	−18.0326	−0.782546
$532$	0	0
$533$	−8.76951	−0.379850
$534$	0	0
$535$	−16.4092	−0.709430
$536$	0	0
$537$	−30.1452	−1.30086
$538$	0	0
$539$	28.0898	1.20992
$540$	0	0
$541$	31.0618	1.33545	0.667726	−	0.744407i	$-0.267268\pi$
$541$	31.0618	1.33545	0.667726	+	0.744407i	$0.267268\pi$
$542$	0	0
$543$	−7.14054	−0.306430
$544$	0	0
$545$	−19.0286	−0.815097
$546$	0	0
$547$	8.26218	0.353265	0.176633	−	0.984277i	$-0.443480\pi$
$547$	8.26218	0.353265	0.176633	+	0.984277i	$0.443480\pi$
$548$	0	0
$549$	−2.36120	−0.100774
$550$	0	0
$551$	9.01549	0.384073
$552$	0	0
$553$	6.45094	0.274322
$554$	0	0
$555$	18.5276	0.786455
$556$	0	0
$557$	8.83469	0.374338	0.187169	−	0.982328i	$-0.440069\pi$
$557$	8.83469	0.374338	0.187169	+	0.982328i	$0.440069\pi$
$558$	0	0
$559$	25.4295	1.07555
$560$	0	0
$561$	41.5608	1.75470
$562$	0	0
$563$	−46.0855	−1.94227	−0.971137	−	0.238524i	$-0.923337\pi$
$563$	−46.0855	−1.94227	−0.971137	+	0.238524i	$0.923337\pi$
$564$	0	0
$565$	−6.40121	−0.269301
$566$	0	0
$567$	−9.87789	−0.414832
$568$	0	0
$569$	−13.1041	−0.549350	−0.274675	−	0.961537i	$-0.588570\pi$
$569$	−13.1041	−0.549350	−0.274675	+	0.961537i	$0.588570\pi$
$570$	0	0
$571$	27.3020	1.14255	0.571277	−	0.820757i	$-0.306448\pi$
$571$	27.3020	1.14255	0.571277	+	0.820757i	$0.306448\pi$
$572$	0	0
$573$	8.03863	0.335819
$574$	0	0
$575$	−2.15430	−0.0898406
$576$	0	0
$577$	5.56014	0.231472	0.115736	−	0.993280i	$-0.463077\pi$
$577$	5.56014	0.231472	0.115736	+	0.993280i	$0.463077\pi$
$578$	0	0
$579$	31.4888	1.30863
$580$	0	0
$581$	1.17845	0.0488902
$582$	0	0
$583$	17.9721	0.744327
$584$	0	0
$585$	3.19849	0.132241
$586$	0	0
$587$	9.47739	0.391174	0.195587	−	0.980686i	$-0.437339\pi$
$587$	9.47739	0.391174	0.195587	+	0.980686i	$0.437339\pi$
$588$	0	0
$589$	4.35137	0.179295
$590$	0	0
$591$	−47.1731	−1.94044
$592$	0	0
$593$	−15.5002	−0.636517	−0.318258	−	0.948004i	$-0.603098\pi$
$593$	−15.5002	−0.636517	−0.318258	+	0.948004i	$0.603098\pi$
$594$	0	0
$595$	3.79152	0.155437
$596$	0	0
$597$	28.2212	1.15502
$598$	0	0
$599$	2.67436	0.109271	0.0546356	−	0.998506i	$-0.482600\pi$
$599$	2.67436	0.109271	0.0546356	+	0.998506i	$0.482600\pi$
$600$	0	0
$601$	39.0353	1.59229	0.796143	−	0.605109i	$-0.206871\pi$
$601$	39.0353	1.59229	0.796143	+	0.605109i	$0.206871\pi$
$602$	0	0
$603$	−1.71632	−0.0698940
$604$	0	0
$605$	−9.33840	−0.379660
$606$	0	0
$607$	5.07601	0.206029	0.103014	−	0.994680i	$-0.467151\pi$
$607$	5.07601	0.206029	0.103014	+	0.994680i	$0.467151\pi$
$608$	0	0
$609$	16.9035	0.684965
$610$	0	0
$611$	−5.68171	−0.229857
$612$	0	0
$613$	−29.4734	−1.19042	−0.595210	−	0.803570i	$-0.702931\pi$
$613$	−29.4734	−1.19042	−0.595210	+	0.803570i	$0.702931\pi$
$614$	0	0
$615$	−9.11444	−0.367530
$616$	0	0
$617$	−34.5806	−1.39216	−0.696081	−	0.717963i	$-0.745075\pi$
$617$	−34.5806	−1.39216	−0.696081	+	0.717963i	$0.745075\pi$
$618$	0	0
$619$	−10.1975	−0.409873	−0.204936	−	0.978775i	$-0.565699\pi$
$619$	−10.1975	−0.409873	−0.204936	+	0.978775i	$0.565699\pi$
$620$	0	0
$621$	6.63525	0.266263
$622$	0	0
$623$	−13.6535	−0.547016
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−9.62738	−0.384481
$628$	0	0
$629$	37.4667	1.49390
$630$	0	0
$631$	−35.5500	−1.41522	−0.707612	−	0.706601i	$-0.750228\pi$
$631$	−35.5500	−1.41522	−0.707612	+	0.706601i	$0.750228\pi$
$632$	0	0
$633$	9.41840	0.374348
$634$	0	0
$635$	−14.1135	−0.560076
$636$	0	0
$637$	12.7934	0.506893
$638$	0	0
$639$	−22.5570	−0.892340
$640$	0	0
$641$	22.9448	0.906263	0.453132	−	0.891444i	$-0.350307\pi$
$641$	22.9448	0.906263	0.453132	+	0.891444i	$0.350307\pi$
$642$	0	0
$643$	21.5175	0.848566	0.424283	−	0.905530i	$-0.360526\pi$
$643$	21.5175	0.848566	0.424283	+	0.905530i	$0.360526\pi$
$644$	0	0
$645$	26.4297	1.04067
$646$	0	0
$647$	−2.75737	−0.108403	−0.0542017	−	0.998530i	$-0.517261\pi$
$647$	−2.75737	−0.108403	−0.0542017	+	0.998530i	$0.517261\pi$
$648$	0	0
$649$	52.2235	2.04995
$650$	0	0
$651$	8.15857	0.319760
$652$	0	0
$653$	3.04942	0.119333	0.0596665	−	0.998218i	$-0.480996\pi$
$653$	3.04942	0.119333	0.0596665	+	0.998218i	$0.480996\pi$
$654$	0	0
$655$	−10.7289	−0.419211
$656$	0	0
$657$	−9.39693	−0.366609
$658$	0	0
$659$	−35.9817	−1.40165	−0.700825	−	0.713334i	$-0.747184\pi$
$659$	−35.9817	−1.40165	−0.700825	+	0.713334i	$0.747184\pi$
$660$	0	0
$661$	14.3411	0.557805	0.278903	−	0.960319i	$-0.410029\pi$
$661$	14.3411	0.557805	0.278903	+	0.960319i	$0.410029\pi$
$662$	0	0
$663$	18.9287	0.735130
$664$	0	0
$665$	−0.878290	−0.0340586
$666$	0	0
$667$	−19.4221	−0.752026
$668$	0	0
$669$	55.5720	2.14854
$670$	0	0
$671$	6.83821	0.263986
$672$	0	0
$673$	41.5033	1.59984	0.799918	−	0.600110i	$-0.204876\pi$
$673$	41.5033	1.59984	0.799918	+	0.600110i	$0.204876\pi$
$674$	0	0
$675$	−3.08000	−0.118549
$676$	0	0
$677$	−32.4315	−1.24644	−0.623221	−	0.782046i	$-0.714176\pi$
$677$	−32.4315	−1.24644	−0.623221	+	0.782046i	$0.714176\pi$
$678$	0	0
$679$	6.63662	0.254690
$680$	0	0
$681$	−7.35911	−0.282002
$682$	0	0
$683$	21.8723	0.836921	0.418461	−	0.908235i	$-0.362570\pi$
$683$	21.8723	0.836921	0.418461	+	0.908235i	$0.362570\pi$
$684$	0	0
$685$	21.3366	0.815228
$686$	0	0
$687$	36.5423	1.39418
$688$	0	0
$689$	8.18530	0.311835
$690$	0	0
$691$	−40.6915	−1.54798	−0.773989	−	0.633199i	$-0.781741\pi$
$691$	−40.6915	−1.54798	−0.773989	+	0.633199i	$0.781741\pi$
$692$	0	0
$693$	−6.16802	−0.234304
$694$	0	0
$695$	−6.49684	−0.246439
$696$	0	0
$697$	−18.4313	−0.698134
$698$	0	0
$699$	1.78938	0.0676807
$700$	0	0
$701$	−33.7482	−1.27465	−0.637326	−	0.770594i	$-0.719960\pi$
$701$	−33.7482	−1.27465	−0.637326	+	0.770594i	$0.719960\pi$
$702$	0	0
$703$	−8.67901	−0.327335
$704$	0	0
$705$	−5.90519	−0.222402
$706$	0	0
$707$	−7.47408	−0.281092
$708$	0	0
$709$	−4.17013	−0.156612	−0.0783062	−	0.996929i	$-0.524951\pi$
$709$	−4.17013	−0.156612	−0.0783062	+	0.996929i	$0.524951\pi$
$710$	0	0
$711$	11.4376	0.428943
$712$	0	0
$713$	−9.37417	−0.351066
$714$	0	0
$715$	−9.26304	−0.346418
$716$	0	0
$717$	−19.4383	−0.725937
$718$	0	0
$719$	6.29675	0.234829	0.117414	−	0.993083i	$-0.462539\pi$
$719$	6.29675	0.234829	0.117414	+	0.993083i	$0.462539\pi$
$720$	0	0
$721$	−0.671007	−0.0249896
$722$	0	0
$723$	−22.2226	−0.826467
$724$	0	0
$725$	9.01549	0.334827
$726$	0	0
$727$	32.6776	1.21195	0.605973	−	0.795485i	$-0.292784\pi$
$727$	32.6776	1.21195	0.605973	+	0.795485i	$0.292784\pi$
$728$	0	0
$729$	2.21161	0.0819115
$730$	0	0
$731$	53.4463	1.97678
$732$	0	0
$733$	−48.1735	−1.77933	−0.889664	−	0.456616i	$-0.849061\pi$
$733$	−48.1735	−1.77933	−0.889664	+	0.456616i	$0.849061\pi$
$734$	0	0
$735$	13.2966	0.490453
$736$	0	0
$737$	4.97058	0.183094
$738$	0	0
$739$	−2.86274	−0.105308	−0.0526538	−	0.998613i	$-0.516768\pi$
$739$	−2.86274	−0.105308	−0.0526538	+	0.998613i	$0.516768\pi$
$740$	0	0
$741$	−4.38475	−0.161078
$742$	0	0
$743$	−28.1718	−1.03352	−0.516761	−	0.856130i	$-0.672863\pi$
$743$	−28.1718	−1.03352	−0.516761	+	0.856130i	$0.672863\pi$
$744$	0	0
$745$	13.4904	0.494248
$746$	0	0
$747$	2.08940	0.0764471
$748$	0	0
$749$	14.4120	0.526603
$750$	0	0
$751$	33.1632	1.21014	0.605072	−	0.796171i	$-0.293144\pi$
$751$	33.1632	1.21014	0.605072	+	0.796171i	$0.293144\pi$
$752$	0	0
$753$	38.6898	1.40993
$754$	0	0
$755$	−19.7890	−0.720194
$756$	0	0
$757$	−10.0716	−0.366060	−0.183030	−	0.983107i	$-0.558590\pi$
$757$	−10.0716	−0.366060	−0.183030	+	0.983107i	$0.558590\pi$
$758$	0	0
$759$	20.7403	0.752825
$760$	0	0
$761$	−51.0524	−1.85065	−0.925324	−	0.379178i	$-0.876207\pi$
$761$	−51.0524	−1.85065	−0.925324	+	0.379178i	$0.876207\pi$
$762$	0	0
$763$	16.7127	0.605039
$764$	0	0
$765$	6.72241	0.243049
$766$	0	0
$767$	23.7850	0.858826
$768$	0	0
$769$	31.1271	1.12247	0.561236	−	0.827656i	$-0.310326\pi$
$769$	31.1271	1.12247	0.561236	+	0.827656i	$0.310326\pi$
$770$	0	0
$771$	7.14562	0.257343
$772$	0	0
$773$	16.2291	0.583721	0.291860	−	0.956461i	$-0.405726\pi$
$773$	16.2291	0.583721	0.291860	+	0.956461i	$0.405726\pi$
$774$	0	0
$775$	4.35137	0.156306
$776$	0	0
$777$	−16.2726	−0.583778
$778$	0	0
$779$	4.26953	0.152972
$780$	0	0
$781$	65.3266	2.33757
$782$	0	0
$783$	−27.7677	−0.992336
$784$	0	0
$785$	−2.37748	−0.0848559
$786$	0	0
$787$	−36.1329	−1.28800	−0.644000	−	0.765026i	$-0.722726\pi$
$787$	−36.1329	−1.28800	−0.644000	+	0.765026i	$0.722726\pi$
$788$	0	0
$789$	−22.0689	−0.785673
$790$	0	0
$791$	5.62212	0.199900
$792$	0	0
$793$	3.11444	0.110597
$794$	0	0
$795$	8.50725	0.301721
$796$	0	0
$797$	13.3349	0.472345	0.236172	−	0.971711i	$-0.424107\pi$
$797$	13.3349	0.472345	0.236172	+	0.971711i	$0.424107\pi$
$798$	0	0
$799$	−11.9415	−0.422460
$800$	0	0
$801$	−24.2078	−0.855340
$802$	0	0
$803$	27.2142	0.960367
$804$	0	0
$805$	1.89210	0.0666879
$806$	0	0
$807$	8.17196	0.287667
$808$	0	0
$809$	−18.5912	−0.653630	−0.326815	−	0.945088i	$-0.605975\pi$
$809$	−18.5912	−0.653630	−0.326815	+	0.945088i	$0.605975\pi$
$810$	0	0
$811$	−4.56880	−0.160432	−0.0802161	−	0.996777i	$-0.525561\pi$
$811$	−4.56880	−0.160432	−0.0802161	+	0.996777i	$0.525561\pi$
$812$	0	0
$813$	1.55320	0.0544730
$814$	0	0
$815$	8.86118	0.310394
$816$	0	0
$817$	−12.3806	−0.433143
$818$	0	0
$819$	−2.80920	−0.0981614
$820$	0	0
$821$	2.15671	0.0752698	0.0376349	−	0.999292i	$-0.488018\pi$
$821$	2.15671	0.0752698	0.0376349	+	0.999292i	$0.488018\pi$
$822$	0	0
$823$	8.69218	0.302990	0.151495	−	0.988458i	$-0.451591\pi$
$823$	8.69218	0.302990	0.151495	+	0.988458i	$0.451591\pi$
$824$	0	0
$825$	−9.62738	−0.335183
$826$	0	0
$827$	−14.6572	−0.509680	−0.254840	−	0.966983i	$-0.582023\pi$
$827$	−14.6572	−0.509680	−0.254840	+	0.966983i	$0.582023\pi$
$828$	0	0
$829$	37.7966	1.31273	0.656364	−	0.754444i	$-0.272093\pi$
$829$	37.7966	1.31273	0.656364	+	0.754444i	$0.272093\pi$
$830$	0	0
$831$	−44.4772	−1.54290
$832$	0	0
$833$	26.8885	0.931631
$834$	0	0
$835$	−6.26190	−0.216702
$836$	0	0
$837$	−13.4022	−0.463249
$838$	0	0
$839$	46.1806	1.59433	0.797165	−	0.603761i	$-0.206332\pi$
$839$	46.1806	1.59433	0.797165	+	0.603761i	$0.206332\pi$
$840$	0	0
$841$	52.2790	1.80272
$842$	0	0
$843$	59.4477	2.04749
$844$	0	0
$845$	8.78118	0.302082
$846$	0	0
$847$	8.20182	0.281818
$848$	0	0
$849$	−62.1976	−2.13462
$850$	0	0
$851$	18.6972	0.640933
$852$	0	0
$853$	8.42055	0.288314	0.144157	−	0.989555i	$-0.453953\pi$
$853$	8.42055	0.288314	0.144157	+	0.989555i	$0.453953\pi$
$854$	0	0
$855$	−1.55722	−0.0532557
$856$	0	0
$857$	36.8489	1.25873	0.629367	−	0.777108i	$-0.283314\pi$
$857$	36.8489	1.25873	0.629367	+	0.777108i	$0.283314\pi$
$858$	0	0
$859$	7.62728	0.260239	0.130120	−	0.991498i	$-0.458464\pi$
$859$	7.62728	0.260239	0.130120	+	0.991498i	$0.458464\pi$
$860$	0	0
$861$	8.00512	0.272814
$862$	0	0
$863$	56.8931	1.93666	0.968332	−	0.249665i	$-0.0803205\pi$
$863$	56.8931	1.93666	0.968332	+	0.249665i	$0.0803205\pi$
$864$	0	0
$865$	18.2673	0.621107
$866$	0	0
$867$	3.49231	0.118605
$868$	0	0
$869$	−33.1241	−1.12366
$870$	0	0
$871$	2.26383	0.0767070
$872$	0	0
$873$	11.7668	0.398246
$874$	0	0
$875$	−0.878290	−0.0296916
$876$	0	0
$877$	47.2084	1.59411	0.797057	−	0.603904i	$-0.206389\pi$
$877$	47.2084	1.59411	0.797057	+	0.603904i	$0.206389\pi$
$878$	0	0
$879$	22.0106	0.742398
$880$	0	0
$881$	−45.2188	−1.52346	−0.761729	−	0.647895i	$-0.775649\pi$
$881$	−45.2188	−1.52346	−0.761729	+	0.647895i	$0.775649\pi$
$882$	0	0
$883$	−12.9525	−0.435886	−0.217943	−	0.975962i	$-0.569935\pi$
$883$	−12.9525	−0.435886	−0.217943	+	0.975962i	$0.569935\pi$
$884$	0	0
$885$	24.7205	0.830971
$886$	0	0
$887$	−5.58397	−0.187491	−0.0937457	−	0.995596i	$-0.529884\pi$
$887$	−5.58397	−0.187491	−0.0937457	+	0.995596i	$0.529884\pi$
$888$	0	0
$889$	12.3957	0.415739
$890$	0	0
$891$	50.7206	1.69920
$892$	0	0
$893$	2.76620	0.0925674
$894$	0	0
$895$	14.1211	0.472016
$896$	0	0
$897$	9.44609	0.315396
$898$	0	0
$899$	39.2297	1.30838
$900$	0	0
$901$	17.2034	0.573129
$902$	0	0
$903$	−23.2129	−0.772478
$904$	0	0
$905$	3.34489	0.111188
$906$	0	0
$907$	−53.7305	−1.78409	−0.892046	−	0.451944i	$-0.850731\pi$
$907$	−53.7305	−1.78409	−0.892046	+	0.451944i	$0.850731\pi$
$908$	0	0
$909$	−13.2516	−0.439529
$910$	0	0
$911$	−23.6398	−0.783222	−0.391611	−	0.920131i	$-0.628082\pi$
$911$	−23.6398	−0.783222	−0.391611	+	0.920131i	$0.628082\pi$
$912$	0	0
$913$	−6.05104	−0.200260
$914$	0	0
$915$	3.23694	0.107010
$916$	0	0
$917$	9.42304	0.311176
$918$	0	0
$919$	−5.86850	−0.193584	−0.0967920	−	0.995305i	$-0.530858\pi$
$919$	−5.86850	−0.193584	−0.0967920	+	0.995305i	$0.530858\pi$
$920$	0	0
$921$	33.4885	1.10348
$922$	0	0
$923$	29.7527	0.979323
$924$	0	0
$925$	−8.67901	−0.285364
$926$	0	0
$927$	−1.18970	−0.0390750
$928$	0	0
$929$	42.1417	1.38263	0.691313	−	0.722556i	$-0.257033\pi$
$929$	42.1417	1.38263	0.691313	+	0.722556i	$0.257033\pi$
$930$	0	0
$931$	−6.22861	−0.204134
$932$	0	0
$933$	74.0168	2.42320
$934$	0	0
$935$	−19.4686	−0.636690
$936$	0	0
$937$	13.0515	0.426373	0.213187	−	0.977012i	$-0.431616\pi$
$937$	13.0515	0.426373	0.213187	+	0.977012i	$0.431616\pi$
$938$	0	0
$939$	46.9474	1.53207
$940$	0	0
$941$	3.91219	0.127534	0.0637668	−	0.997965i	$-0.479689\pi$
$941$	3.91219	0.127534	0.0637668	+	0.997965i	$0.479689\pi$
$942$	0	0
$943$	−9.19786	−0.299523
$944$	0	0
$945$	2.70513	0.0879980
$946$	0	0
$947$	46.8887	1.52368	0.761840	−	0.647766i	$-0.224296\pi$
$947$	46.8887	1.52368	0.761840	+	0.647766i	$0.224296\pi$
$948$	0	0
$949$	12.3946	0.402345
$950$	0	0
$951$	−43.4416	−1.40869
$952$	0	0
$953$	−36.0745	−1.16857	−0.584284	−	0.811549i	$-0.698625\pi$
$953$	−36.0745	−1.16857	−0.584284	+	0.811549i	$0.698625\pi$
$954$	0	0
$955$	−3.76558	−0.121851
$956$	0	0
$957$	−86.7956	−2.80570
$958$	0	0
$959$	−18.7397	−0.605136
$960$	0	0
$961$	−12.0656	−0.389212
$962$	0	0
$963$	25.5526	0.823422
$964$	0	0
$965$	−14.7505	−0.474834
$966$	0	0
$967$	−5.14385	−0.165415	−0.0827075	−	0.996574i	$-0.526357\pi$
$967$	−5.14385	−0.165415	−0.0827075	+	0.996574i	$0.526357\pi$
$968$	0	0
$969$	−9.21564	−0.296049
$970$	0	0
$971$	−1.55134	−0.0497848	−0.0248924	−	0.999690i	$-0.507924\pi$
$971$	−1.55134	−0.0497848	−0.0248924	+	0.999690i	$0.507924\pi$
$972$	0	0
$973$	5.70611	0.182930
$974$	0	0
$975$	−4.38475	−0.140425
$976$	0	0
$977$	−33.4684	−1.07075	−0.535374	−	0.844615i	$-0.679829\pi$
$977$	−33.4684	−1.07075	−0.535374	+	0.844615i	$0.679829\pi$
$978$	0	0
$979$	70.1075	2.24064
$980$	0	0
$981$	29.6317	0.946068
$982$	0	0
$983$	−11.5436	−0.368184	−0.184092	−	0.982909i	$-0.558934\pi$
$983$	−11.5436	−0.368184	−0.184092	+	0.982909i	$0.558934\pi$
$984$	0	0
$985$	22.0976	0.704087
$986$	0	0
$987$	5.18647	0.165087
$988$	0	0
$989$	26.6716	0.848108
$990$	0	0
$991$	0.624895	0.0198504	0.00992522	−	0.999951i	$-0.496841\pi$
$991$	0.624895	0.0198504	0.00992522	+	0.999951i	$0.496841\pi$
$992$	0	0
$993$	−20.1895	−0.640694
$994$	0	0
$995$	−13.2198	−0.419097
$996$	0	0
$997$	−19.6074	−0.620974	−0.310487	−	0.950578i	$-0.600492\pi$
$997$	−19.6074	−0.620974	−0.310487	+	0.950578i	$0.600492\pi$
$998$	0	0
$999$	26.7314	0.845743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3040.2.a.u.1.5	✓	5
4.3	odd	2		3040.2.a.x.1.1	yes	5
8.3	odd	2		6080.2.a.ci.1.5		5
8.5	even	2		6080.2.a.cl.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.u.1.5	✓	5	1.1	even	1	trivial
3040.2.a.x.1.1	yes	5	4.3	odd	2
6080.2.a.ci.1.5		5	8.3	odd	2
6080.2.a.cl.1.1		5	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3040.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13476 q^{3} -1.00000 q^{5} +0.878290 q^{7} +1.55722 q^{9} -4.50981 q^{11} -2.05398 q^{13} -2.13476 q^{15} -4.31693 q^{17} +1.00000 q^{19} +1.87494 q^{21} -2.15430 q^{23} +1.00000 q^{25} -3.08000 q^{27} +9.01549 q^{29} +4.35137 q^{31} -9.62738 q^{33} -0.878290 q^{35} -8.67901 q^{37} -4.38475 q^{39} +4.26953 q^{41} -12.3806 q^{43} -1.55722 q^{45} +2.76620 q^{47} -6.22861 q^{49} -9.21564 q^{51} -3.98510 q^{53} +4.50981 q^{55} +2.13476 q^{57} -11.5800 q^{59} -1.51630 q^{61} +1.36769 q^{63} +2.05398 q^{65} -1.10217 q^{67} -4.59893 q^{69} -14.4854 q^{71} -6.03444 q^{73} +2.13476 q^{75} -3.96092 q^{77} +7.34489 q^{79} -11.2467 q^{81} +1.34175 q^{83} +4.31693 q^{85} +19.2459 q^{87} -15.5455 q^{89} -1.80399 q^{91} +9.28915 q^{93} -1.00000 q^{95} +7.55630 q^{97} -7.02276 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} - 5 q^{5} + 4 q^{7} + 7 q^{9} - 2 q^{11} - 4 q^{13} + 4 q^{15} - 12 q^{17} + 5 q^{19} - 10 q^{21} + 8 q^{23} + 5 q^{25} - 16 q^{27} - 6 q^{29} + 10 q^{31} - 18 q^{33} - 4 q^{35} - 6 q^{37}+ \cdots + 14 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 - 5 * q^5 + 4 * q^7 + 7 * q^9 - 2 * q^11 - 4 * q^13 + 4 * q^15 - 12 * q^17 + 5 * q^19 - 10 * q^21 + 8 * q^23 + 5 * q^25 - 16 * q^27 - 6 * q^29 + 10 * q^31 - 18 * q^33 - 4 * q^35 - 6 * q^37 + 18 * q^39 - 8 * q^41 - 12 * q^43 - 7 * q^45 + 16 * q^47 + 7 * q^49 + 14 * q^51 - 18 * q^53 + 2 * q^55 - 4 * q^57 - 8 * q^59 + 2 * q^61 + 36 * q^63 + 4 * q^65 - 10 * q^67 - 22 * q^69 - 18 * q^71 - 28 * q^73 - 4 * q^75 - 28 * q^77 + 14 * q^79 + 25 * q^81 - 8 * q^83 + 12 * q^85 + 24 * q^87 - 30 * q^89 - 28 * q^91 - 24 * q^93 - 5 * q^95 - 18 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more