LMFDB - Embedded newform 7440.2.a.bm.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.28726 q^{7} +1.00000 q^{9} +2.96239 q^{11} -3.67513 q^{13} +1.00000 q^{15} -2.15633 q^{17} +2.38787 q^{19} -1.28726 q^{21} -4.80606 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.168544 q^{29} +1.00000 q^{31} -2.96239 q^{33} -1.28726 q^{35} +2.63752 q^{37} +3.67513 q^{39} +11.6629 q^{41} +3.73813 q^{43} -1.00000 q^{45} -12.3430 q^{47} -5.34297 q^{49} +2.15633 q^{51} -3.89446 q^{53} -2.96239 q^{55} -2.38787 q^{57} +13.8315 q^{59} -12.7005 q^{61} +1.28726 q^{63} +3.67513 q^{65} -12.5623 q^{67} +4.80606 q^{69} +0.481194 q^{71} +5.21203 q^{73} -1.00000 q^{75} +3.81336 q^{77} -15.4314 q^{79} +1.00000 q^{81} +10.7308 q^{83} +2.15633 q^{85} -0.168544 q^{87} +3.44358 q^{89} -4.73084 q^{91} -1.00000 q^{93} -2.38787 q^{95} -15.1998 q^{97} +2.96239 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^13 + 3 * q^15 + 4 * q^17 + 8 * q^19 + 2 * q^21 - 14 * q^23 + 3 * q^25 - 3 * q^27 + 16 * q^29 + 3 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 6 * q^39 + 4 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 + 7 * q^49 - 4 * q^51 + 8 * q^53 + 2 * q^55 - 8 * q^57 + 26 * q^59 - 18 * q^61 - 2 * q^63 + 6 * q^65 + 14 * q^69 - 4 * q^71 - 12 * q^73 - 3 * q^75 + 24 * q^77 - 4 * q^79 + 3 * q^81 + 10 * q^83 - 4 * q^85 - 16 * q^87 - 6 * q^89 + 8 * q^91 - 3 * q^93 - 8 * q^95 + 8 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.28726	0.486538	0.243269	−	0.969959i	$-0.421780\pi$
$7$	1.28726	0.486538	0.243269	+	0.969959i	$0.421780\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.96239	0.893194	0.446597	−	0.894735i	$-0.352636\pi$
$11$	2.96239	0.893194	0.446597	+	0.894735i	$0.352636\pi$
$12$	0	0
$13$	−3.67513	−1.01930	−0.509649	−	0.860382i	$-0.670225\pi$
$13$	−3.67513	−1.01930	−0.509649	+	0.860382i	$0.670225\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.15633	−0.522986	−0.261493	−	0.965205i	$-0.584215\pi$
$17$	−2.15633	−0.522986	−0.261493	+	0.965205i	$0.584215\pi$
$18$	0	0
$19$	2.38787	0.547816	0.273908	−	0.961756i	$-0.411684\pi$
$19$	2.38787	0.547816	0.273908	+	0.961756i	$0.411684\pi$
$20$	0	0
$21$	−1.28726	−0.280903
$22$	0	0
$23$	−4.80606	−1.00213	−0.501067	−	0.865409i	$-0.667059\pi$
$23$	−4.80606	−1.00213	−0.501067	+	0.865409i	$0.667059\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.168544	0.0312978	0.0156489	−	0.999878i	$-0.495019\pi$
$29$	0.168544	0.0312978	0.0156489	+	0.999878i	$0.495019\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−2.96239	−0.515686
$34$	0	0
$35$	−1.28726	−0.217586
$36$	0	0
$37$	2.63752	0.433606	0.216803	−	0.976215i	$-0.430437\pi$
$37$	2.63752	0.433606	0.216803	+	0.976215i	$0.430437\pi$
$38$	0	0
$39$	3.67513	0.588492
$40$	0	0
$41$	11.6629	1.82144	0.910720	−	0.413023i	$-0.135527\pi$
$41$	11.6629	1.82144	0.910720	+	0.413023i	$0.135527\pi$
$42$	0	0
$43$	3.73813	0.570060	0.285030	−	0.958519i	$-0.407996\pi$
$43$	3.73813	0.570060	0.285030	+	0.958519i	$0.407996\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−12.3430	−1.80041	−0.900203	−	0.435470i	$-0.856582\pi$
$47$	−12.3430	−1.80041	−0.900203	+	0.435470i	$0.856582\pi$
$48$	0	0
$49$	−5.34297	−0.763281
$50$	0	0
$51$	2.15633	0.301946
$52$	0	0
$53$	−3.89446	−0.534945	−0.267473	−	0.963565i	$-0.586188\pi$
$53$	−3.89446	−0.534945	−0.267473	+	0.963565i	$0.586188\pi$
$54$	0	0
$55$	−2.96239	−0.399448
$56$	0	0
$57$	−2.38787	−0.316282
$58$	0	0
$59$	13.8315	1.80070	0.900351	−	0.435164i	$-0.143310\pi$
$59$	13.8315	1.80070	0.900351	+	0.435164i	$0.143310\pi$
$60$	0	0
$61$	−12.7005	−1.62614	−0.813068	−	0.582169i	$-0.802204\pi$
$61$	−12.7005	−1.62614	−0.813068	+	0.582169i	$0.802204\pi$
$62$	0	0
$63$	1.28726	0.162179
$64$	0	0
$65$	3.67513	0.455844
$66$	0	0
$67$	−12.5623	−1.53473	−0.767364	−	0.641211i	$-0.778432\pi$
$67$	−12.5623	−1.53473	−0.767364	+	0.641211i	$0.778432\pi$
$68$	0	0
$69$	4.80606	0.578582
$70$	0	0
$71$	0.481194	0.0571073	0.0285536	−	0.999592i	$-0.490910\pi$
$71$	0.481194	0.0571073	0.0285536	+	0.999592i	$0.490910\pi$
$72$	0	0
$73$	5.21203	0.610023	0.305011	−	0.952349i	$-0.401340\pi$
$73$	5.21203	0.610023	0.305011	+	0.952349i	$0.401340\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	3.81336	0.434572
$78$	0	0
$79$	−15.4314	−1.73616	−0.868082	−	0.496421i	$-0.834647\pi$
$79$	−15.4314	−1.73616	−0.868082	+	0.496421i	$0.834647\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.7308	1.17786	0.588931	−	0.808183i	$-0.299549\pi$
$83$	10.7308	1.17786	0.588931	+	0.808183i	$0.299549\pi$
$84$	0	0
$85$	2.15633	0.233886
$86$	0	0
$87$	−0.168544	−0.0180698
$88$	0	0
$89$	3.44358	0.365019	0.182510	−	0.983204i	$-0.441578\pi$
$89$	3.44358	0.365019	0.182510	+	0.983204i	$0.441578\pi$
$90$	0	0
$91$	−4.73084	−0.495927
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	−2.38787	−0.244991
$96$	0	0
$97$	−15.1998	−1.54331	−0.771654	−	0.636043i	$-0.780570\pi$
$97$	−15.1998	−1.54331	−0.771654	+	0.636043i	$0.780570\pi$
$98$	0	0
$99$	2.96239	0.297731
$100$	0	0
$101$	14.0508	1.39811	0.699053	−	0.715070i	$-0.253605\pi$
$101$	14.0508	1.39811	0.699053	+	0.715070i	$0.253605\pi$
$102$	0	0
$103$	10.1744	1.00252	0.501258	−	0.865298i	$-0.332871\pi$
$103$	10.1744	1.00252	0.501258	+	0.865298i	$0.332871\pi$
$104$	0	0
$105$	1.28726	0.125623
$106$	0	0
$107$	−17.2447	−1.66711	−0.833555	−	0.552436i	$-0.813698\pi$
$107$	−17.2447	−1.66711	−0.833555	+	0.552436i	$0.813698\pi$
$108$	0	0
$109$	4.93207	0.472407	0.236203	−	0.971704i	$-0.424097\pi$
$109$	4.93207	0.472407	0.236203	+	0.971704i	$0.424097\pi$
$110$	0	0
$111$	−2.63752	−0.250342
$112$	0	0
$113$	−7.27504	−0.684378	−0.342189	−	0.939631i	$-0.611168\pi$
$113$	−7.27504	−0.684378	−0.342189	+	0.939631i	$0.611168\pi$
$114$	0	0
$115$	4.80606	0.448168
$116$	0	0
$117$	−3.67513	−0.339766
$118$	0	0
$119$	−2.77575	−0.254452
$120$	0	0
$121$	−2.22425	−0.202205
$122$	0	0
$123$	−11.6629	−1.05161
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.27504	0.113141	0.0565707	−	0.998399i	$-0.481983\pi$
$127$	1.27504	0.113141	0.0565707	+	0.998399i	$0.481983\pi$
$128$	0	0
$129$	−3.73813	−0.329124
$130$	0	0
$131$	−17.8070	−1.55581	−0.777903	−	0.628384i	$-0.783717\pi$
$131$	−17.8070	−1.55581	−0.777903	+	0.628384i	$0.783717\pi$
$132$	0	0
$133$	3.07381	0.266533
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	7.24472	0.618958	0.309479	−	0.950906i	$-0.399845\pi$
$137$	7.24472	0.618958	0.309479	+	0.950906i	$0.399845\pi$
$138$	0	0
$139$	14.8872	1.26271	0.631356	−	0.775493i	$-0.282498\pi$
$139$	14.8872	1.26271	0.631356	+	0.775493i	$0.282498\pi$
$140$	0	0
$141$	12.3430	1.03947
$142$	0	0
$143$	−10.8872	−0.910431
$144$	0	0
$145$	−0.168544	−0.0139968
$146$	0	0
$147$	5.34297	0.440681
$148$	0	0
$149$	−5.73813	−0.470086	−0.235043	−	0.971985i	$-0.575523\pi$
$149$	−5.73813	−0.470086	−0.235043	+	0.971985i	$0.575523\pi$
$150$	0	0
$151$	12.8568	1.04628	0.523138	−	0.852248i	$-0.324761\pi$
$151$	12.8568	1.04628	0.523138	+	0.852248i	$0.324761\pi$
$152$	0	0
$153$	−2.15633	−0.174329
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	1.73813	0.138718	0.0693591	−	0.997592i	$-0.477905\pi$
$157$	1.73813	0.138718	0.0693591	+	0.997592i	$0.477905\pi$
$158$	0	0
$159$	3.89446	0.308851
$160$	0	0
$161$	−6.18664	−0.487576
$162$	0	0
$163$	6.45088	0.505272	0.252636	−	0.967561i	$-0.418703\pi$
$163$	6.45088	0.505272	0.252636	+	0.967561i	$0.418703\pi$
$164$	0	0
$165$	2.96239	0.230622
$166$	0	0
$167$	−22.5745	−1.74687	−0.873434	−	0.486942i	$-0.838112\pi$
$167$	−22.5745	−1.74687	−0.873434	+	0.486942i	$0.838112\pi$
$168$	0	0
$169$	0.506587	0.0389682
$170$	0	0
$171$	2.38787	0.182605
$172$	0	0
$173$	9.35026	0.710887	0.355444	−	0.934698i	$-0.384330\pi$
$173$	9.35026	0.710887	0.355444	+	0.934698i	$0.384330\pi$
$174$	0	0
$175$	1.28726	0.0973075
$176$	0	0
$177$	−13.8315	−1.03964
$178$	0	0
$179$	24.1622	1.80597	0.902984	−	0.429674i	$-0.141372\pi$
$179$	24.1622	1.80597	0.902984	+	0.429674i	$0.141372\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	12.7005	0.938850
$184$	0	0
$185$	−2.63752	−0.193914
$186$	0	0
$187$	−6.38787	−0.467128
$188$	0	0
$189$	−1.28726	−0.0936342
$190$	0	0
$191$	−3.13093	−0.226546	−0.113273	−	0.993564i	$-0.536134\pi$
$191$	−3.13093	−0.226546	−0.113273	+	0.993564i	$0.536134\pi$
$192$	0	0
$193$	21.5877	1.55392	0.776958	−	0.629553i	$-0.216762\pi$
$193$	21.5877	1.55392	0.776958	+	0.629553i	$0.216762\pi$
$194$	0	0
$195$	−3.67513	−0.263182
$196$	0	0
$197$	4.60483	0.328081	0.164040	−	0.986454i	$-0.447547\pi$
$197$	4.60483	0.328081	0.164040	+	0.986454i	$0.447547\pi$
$198$	0	0
$199$	−7.76845	−0.550691	−0.275345	−	0.961345i	$-0.588792\pi$
$199$	−7.76845	−0.550691	−0.275345	+	0.961345i	$0.588792\pi$
$200$	0	0
$201$	12.5623	0.886076
$202$	0	0
$203$	0.216960	0.0152276
$204$	0	0
$205$	−11.6629	−0.814573
$206$	0	0
$207$	−4.80606	−0.334045
$208$	0	0
$209$	7.07381	0.489306
$210$	0	0
$211$	−0.337088	−0.0232061	−0.0116030	−	0.999933i	$-0.503693\pi$
$211$	−0.337088	−0.0232061	−0.0116030	+	0.999933i	$0.503693\pi$
$212$	0	0
$213$	−0.481194	−0.0329709
$214$	0	0
$215$	−3.73813	−0.254939
$216$	0	0
$217$	1.28726	0.0873847
$218$	0	0
$219$	−5.21203	−0.352197
$220$	0	0
$221$	7.92478	0.533078
$222$	0	0
$223$	2.83638	0.189938	0.0949690	−	0.995480i	$-0.469725\pi$
$223$	2.83638	0.189938	0.0949690	+	0.995480i	$0.469725\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	17.3054	1.14860	0.574298	−	0.818646i	$-0.305275\pi$
$227$	17.3054	1.14860	0.574298	+	0.818646i	$0.305275\pi$
$228$	0	0
$229$	−25.1998	−1.66525	−0.832625	−	0.553837i	$-0.813163\pi$
$229$	−25.1998	−1.66525	−0.832625	+	0.553837i	$0.813163\pi$
$230$	0	0
$231$	−3.81336	−0.250901
$232$	0	0
$233$	−4.08110	−0.267362	−0.133681	−	0.991024i	$-0.542680\pi$
$233$	−4.08110	−0.267362	−0.133681	+	0.991024i	$0.542680\pi$
$234$	0	0
$235$	12.3430	0.805166
$236$	0	0
$237$	15.4314	1.00237
$238$	0	0
$239$	−9.07381	−0.586936	−0.293468	−	0.955969i	$-0.594809\pi$
$239$	−9.07381	−0.586936	−0.293468	+	0.955969i	$0.594809\pi$
$240$	0	0
$241$	−21.9149	−1.41166	−0.705832	−	0.708379i	$-0.749427\pi$
$241$	−21.9149	−1.41166	−0.705832	+	0.708379i	$0.749427\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.34297	0.341350
$246$	0	0
$247$	−8.77575	−0.558387
$248$	0	0
$249$	−10.7308	−0.680039
$250$	0	0
$251$	−8.83638	−0.557747	−0.278874	−	0.960328i	$-0.589961\pi$
$251$	−8.83638	−0.557747	−0.278874	+	0.960328i	$0.589961\pi$
$252$	0	0
$253$	−14.2374	−0.895099
$254$	0	0
$255$	−2.15633	−0.135034
$256$	0	0
$257$	10.7816	0.672539	0.336270	−	0.941766i	$-0.390835\pi$
$257$	10.7816	0.672539	0.336270	+	0.941766i	$0.390835\pi$
$258$	0	0
$259$	3.39517	0.210965
$260$	0	0
$261$	0.168544	0.0104326
$262$	0	0
$263$	−21.2144	−1.30814	−0.654068	−	0.756436i	$-0.726939\pi$
$263$	−21.2144	−1.30814	−0.654068	+	0.756436i	$0.726939\pi$
$264$	0	0
$265$	3.89446	0.239235
$266$	0	0
$267$	−3.44358	−0.210744
$268$	0	0
$269$	−16.6072	−1.01256	−0.506279	−	0.862369i	$-0.668980\pi$
$269$	−16.6072	−1.01256	−0.506279	+	0.862369i	$0.668980\pi$
$270$	0	0
$271$	−22.8872	−1.39030	−0.695148	−	0.718866i	$-0.744661\pi$
$271$	−22.8872	−1.39030	−0.695148	+	0.718866i	$0.744661\pi$
$272$	0	0
$273$	4.73084	0.286324
$274$	0	0
$275$	2.96239	0.178639
$276$	0	0
$277$	−11.2628	−0.676717	−0.338359	−	0.941017i	$-0.609872\pi$
$277$	−11.2628	−0.676717	−0.338359	+	0.941017i	$0.609872\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−9.19982	−0.548815	−0.274408	−	0.961613i	$-0.588482\pi$
$281$	−9.19982	−0.548815	−0.274408	+	0.961613i	$0.588482\pi$
$282$	0	0
$283$	−21.4739	−1.27649	−0.638245	−	0.769833i	$-0.720339\pi$
$283$	−21.4739	−1.27649	−0.638245	+	0.769833i	$0.720339\pi$
$284$	0	0
$285$	2.38787	0.141445
$286$	0	0
$287$	15.0132	0.886200
$288$	0	0
$289$	−12.3503	−0.726486
$290$	0	0
$291$	15.1998	0.891029
$292$	0	0
$293$	21.6326	1.26379	0.631895	−	0.775054i	$-0.282277\pi$
$293$	21.6326	1.26379	0.631895	+	0.775054i	$0.282277\pi$
$294$	0	0
$295$	−13.8315	−0.805299
$296$	0	0
$297$	−2.96239	−0.171895
$298$	0	0
$299$	17.6629	1.02147
$300$	0	0
$301$	4.81194	0.277356
$302$	0	0
$303$	−14.0508	−0.807197
$304$	0	0
$305$	12.7005	0.727230
$306$	0	0
$307$	−20.6131	−1.17645	−0.588225	−	0.808697i	$-0.700173\pi$
$307$	−20.6131	−1.17645	−0.588225	+	0.808697i	$0.700173\pi$
$308$	0	0
$309$	−10.1744	−0.578803
$310$	0	0
$311$	−2.34534	−0.132992	−0.0664959	−	0.997787i	$-0.521182\pi$
$311$	−2.34534	−0.132992	−0.0664959	+	0.997787i	$0.521182\pi$
$312$	0	0
$313$	−24.1744	−1.36642	−0.683210	−	0.730222i	$-0.739416\pi$
$313$	−24.1744	−1.36642	−0.683210	+	0.730222i	$0.739416\pi$
$314$	0	0
$315$	−1.28726	−0.0725288
$316$	0	0
$317$	−6.68006	−0.375189	−0.187595	−	0.982247i	$-0.560069\pi$
$317$	−6.68006	−0.375189	−0.187595	+	0.982247i	$0.560069\pi$
$318$	0	0
$319$	0.499293	0.0279550
$320$	0	0
$321$	17.2447	0.962507
$322$	0	0
$323$	−5.14903	−0.286500
$324$	0	0
$325$	−3.67513	−0.203860
$326$	0	0
$327$	−4.93207	−0.272744
$328$	0	0
$329$	−15.8886	−0.875966
$330$	0	0
$331$	11.0581	0.607807	0.303904	−	0.952703i	$-0.401710\pi$
$331$	11.0581	0.607807	0.303904	+	0.952703i	$0.401710\pi$
$332$	0	0
$333$	2.63752	0.144535
$334$	0	0
$335$	12.5623	0.686352
$336$	0	0
$337$	6.11379	0.333039	0.166520	−	0.986038i	$-0.446747\pi$
$337$	6.11379	0.333039	0.166520	+	0.986038i	$0.446747\pi$
$338$	0	0
$339$	7.27504	0.395126
$340$	0	0
$341$	2.96239	0.160422
$342$	0	0
$343$	−15.8886	−0.857903
$344$	0	0
$345$	−4.80606	−0.258750
$346$	0	0
$347$	−31.5731	−1.69493	−0.847466	−	0.530849i	$-0.821873\pi$
$347$	−31.5731	−1.69493	−0.847466	+	0.530849i	$0.821873\pi$
$348$	0	0
$349$	−14.0811	−0.753744	−0.376872	−	0.926265i	$-0.623000\pi$
$349$	−14.0811	−0.753744	−0.376872	+	0.926265i	$0.623000\pi$
$350$	0	0
$351$	3.67513	0.196164
$352$	0	0
$353$	−29.8554	−1.58904	−0.794522	−	0.607235i	$-0.792279\pi$
$353$	−29.8554	−1.58904	−0.794522	+	0.607235i	$0.792279\pi$
$354$	0	0
$355$	−0.481194	−0.0255391
$356$	0	0
$357$	2.77575	0.146908
$358$	0	0
$359$	−8.74306	−0.461441	−0.230721	−	0.973020i	$-0.574108\pi$
$359$	−8.74306	−0.461441	−0.230721	+	0.973020i	$0.574108\pi$
$360$	0	0
$361$	−13.2981	−0.699898
$362$	0	0
$363$	2.22425	0.116743
$364$	0	0
$365$	−5.21203	−0.272810
$366$	0	0
$367$	−21.4010	−1.11713	−0.558563	−	0.829462i	$-0.688647\pi$
$367$	−21.4010	−1.11713	−0.558563	+	0.829462i	$0.688647\pi$
$368$	0	0
$369$	11.6629	0.607147
$370$	0	0
$371$	−5.01317	−0.260271
$372$	0	0
$373$	15.0640	0.779982	0.389991	−	0.920819i	$-0.372478\pi$
$373$	15.0640	0.779982	0.389991	+	0.920819i	$0.372478\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−0.619421	−0.0319018
$378$	0	0
$379$	2.91160	0.149559	0.0747795	−	0.997200i	$-0.476175\pi$
$379$	2.91160	0.149559	0.0747795	+	0.997200i	$0.476175\pi$
$380$	0	0
$381$	−1.27504	−0.0653222
$382$	0	0
$383$	9.83383	0.502485	0.251243	−	0.967924i	$-0.419161\pi$
$383$	9.83383	0.502485	0.251243	+	0.967924i	$0.419161\pi$
$384$	0	0
$385$	−3.81336	−0.194347
$386$	0	0
$387$	3.73813	0.190020
$388$	0	0
$389$	24.0082	1.21727	0.608633	−	0.793452i	$-0.291718\pi$
$389$	24.0082	1.21727	0.608633	+	0.793452i	$0.291718\pi$
$390$	0	0
$391$	10.3634	0.524101
$392$	0	0
$393$	17.8070	0.898245
$394$	0	0
$395$	15.4314	0.776436
$396$	0	0
$397$	−17.6873	−0.887703	−0.443851	−	0.896100i	$-0.646388\pi$
$397$	−17.6873	−0.887703	−0.443851	+	0.896100i	$0.646388\pi$
$398$	0	0
$399$	−3.07381	−0.153883
$400$	0	0
$401$	−33.6204	−1.67892	−0.839461	−	0.543420i	$-0.817129\pi$
$401$	−33.6204	−1.67892	−0.839461	+	0.543420i	$0.817129\pi$
$402$	0	0
$403$	−3.67513	−0.183071
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	7.81336	0.387294
$408$	0	0
$409$	−17.2243	−0.851685	−0.425842	−	0.904797i	$-0.640022\pi$
$409$	−17.2243	−0.851685	−0.425842	+	0.904797i	$0.640022\pi$
$410$	0	0
$411$	−7.24472	−0.357356
$412$	0	0
$413$	17.8046	0.876109
$414$	0	0
$415$	−10.7308	−0.526756
$416$	0	0
$417$	−14.8872	−0.729028
$418$	0	0
$419$	−8.79384	−0.429607	−0.214804	−	0.976657i	$-0.568911\pi$
$419$	−8.79384	−0.429607	−0.214804	+	0.976657i	$0.568911\pi$
$420$	0	0
$421$	−2.26774	−0.110523	−0.0552616	−	0.998472i	$-0.517599\pi$
$421$	−2.26774	−0.110523	−0.0552616	+	0.998472i	$0.517599\pi$
$422$	0	0
$423$	−12.3430	−0.600136
$424$	0	0
$425$	−2.15633	−0.104597
$426$	0	0
$427$	−16.3488	−0.791176
$428$	0	0
$429$	10.8872	0.525637
$430$	0	0
$431$	12.9805	0.625248	0.312624	−	0.949877i	$-0.398792\pi$
$431$	12.9805	0.625248	0.312624	+	0.949877i	$0.398792\pi$
$432$	0	0
$433$	−10.2642	−0.493268	−0.246634	−	0.969109i	$-0.579324\pi$
$433$	−10.2642	−0.493268	−0.246634	+	0.969109i	$0.579324\pi$
$434$	0	0
$435$	0.168544	0.00808106
$436$	0	0
$437$	−11.4763	−0.548984
$438$	0	0
$439$	21.4010	1.02142	0.510708	−	0.859754i	$-0.329383\pi$
$439$	21.4010	1.02142	0.510708	+	0.859754i	$0.329383\pi$
$440$	0	0
$441$	−5.34297	−0.254427
$442$	0	0
$443$	32.5705	1.54747	0.773737	−	0.633507i	$-0.218385\pi$
$443$	32.5705	1.54747	0.773737	+	0.633507i	$0.218385\pi$
$444$	0	0
$445$	−3.44358	−0.163241
$446$	0	0
$447$	5.73813	0.271404
$448$	0	0
$449$	−32.3209	−1.52532	−0.762659	−	0.646801i	$-0.776106\pi$
$449$	−32.3209	−1.52532	−0.762659	+	0.646801i	$0.776106\pi$
$450$	0	0
$451$	34.5501	1.62690
$452$	0	0
$453$	−12.8568	−0.604067
$454$	0	0
$455$	4.73084	0.221785
$456$	0	0
$457$	−20.7734	−0.971738	−0.485869	−	0.874032i	$-0.661497\pi$
$457$	−20.7734	−0.971738	−0.485869	+	0.874032i	$0.661497\pi$
$458$	0	0
$459$	2.15633	0.100649
$460$	0	0
$461$	−12.4206	−0.578483	−0.289242	−	0.957256i	$-0.593403\pi$
$461$	−12.4206	−0.578483	−0.289242	+	0.957256i	$0.593403\pi$
$462$	0	0
$463$	0.424070	0.0197082	0.00985410	−	0.999951i	$-0.496863\pi$
$463$	0.424070	0.0197082	0.00985410	+	0.999951i	$0.496863\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	14.2677	0.660232	0.330116	−	0.943940i	$-0.392912\pi$
$467$	14.2677	0.660232	0.330116	+	0.943940i	$0.392912\pi$
$468$	0	0
$469$	−16.1709	−0.746703
$470$	0	0
$471$	−1.73813	−0.0800890
$472$	0	0
$473$	11.0738	0.509174
$474$	0	0
$475$	2.38787	0.109563
$476$	0	0
$477$	−3.89446	−0.178315
$478$	0	0
$479$	30.8691	1.41044	0.705222	−	0.708987i	$-0.250847\pi$
$479$	30.8691	1.41044	0.705222	+	0.708987i	$0.250847\pi$
$480$	0	0
$481$	−9.69323	−0.441973
$482$	0	0
$483$	6.18664	0.281502
$484$	0	0
$485$	15.1998	0.690188
$486$	0	0
$487$	−10.5139	−0.476429	−0.238215	−	0.971213i	$-0.576562\pi$
$487$	−10.5139	−0.476429	−0.238215	+	0.971213i	$0.576562\pi$
$488$	0	0
$489$	−6.45088	−0.291719
$490$	0	0
$491$	−10.5139	−0.474485	−0.237242	−	0.971450i	$-0.576244\pi$
$491$	−10.5139	−0.474485	−0.237242	+	0.971450i	$0.576244\pi$
$492$	0	0
$493$	−0.363436	−0.0163683
$494$	0	0
$495$	−2.96239	−0.133149
$496$	0	0
$497$	0.619421	0.0277848
$498$	0	0
$499$	−30.5501	−1.36761	−0.683805	−	0.729665i	$-0.739676\pi$
$499$	−30.5501	−1.36761	−0.683805	+	0.729665i	$0.739676\pi$
$500$	0	0
$501$	22.5745	1.00855
$502$	0	0
$503$	−0.493413	−0.0220002	−0.0110001	−	0.999939i	$-0.503502\pi$
$503$	−0.493413	−0.0220002	−0.0110001	+	0.999939i	$0.503502\pi$
$504$	0	0
$505$	−14.0508	−0.625252
$506$	0	0
$507$	−0.506587	−0.0224983
$508$	0	0
$509$	−18.7285	−0.830125	−0.415062	−	0.909793i	$-0.636240\pi$
$509$	−18.7285	−0.830125	−0.415062	+	0.909793i	$0.636240\pi$
$510$	0	0
$511$	6.70923	0.296799
$512$	0	0
$513$	−2.38787	−0.105427
$514$	0	0
$515$	−10.1744	−0.448339
$516$	0	0
$517$	−36.5647	−1.60811
$518$	0	0
$519$	−9.35026	−0.410431
$520$	0	0
$521$	0.589104	0.0258091	0.0129046	−	0.999917i	$-0.495892\pi$
$521$	0.589104	0.0258091	0.0129046	+	0.999917i	$0.495892\pi$
$522$	0	0
$523$	9.48612	0.414799	0.207400	−	0.978256i	$-0.433500\pi$
$523$	9.48612	0.414799	0.207400	+	0.978256i	$0.433500\pi$
$524$	0	0
$525$	−1.28726	−0.0561805
$526$	0	0
$527$	−2.15633	−0.0939310
$528$	0	0
$529$	0.0982457	0.00427155
$530$	0	0
$531$	13.8315	0.600234
$532$	0	0
$533$	−42.8627	−1.85659
$534$	0	0
$535$	17.2447	0.745554
$536$	0	0
$537$	−24.1622	−1.04268
$538$	0	0
$539$	−15.8279	−0.681758
$540$	0	0
$541$	−0.493413	−0.0212135	−0.0106067	−	0.999944i	$-0.503376\pi$
$541$	−0.493413	−0.0212135	−0.0106067	+	0.999944i	$0.503376\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	−4.93207	−0.211267
$546$	0	0
$547$	−25.8011	−1.10318	−0.551589	−	0.834116i	$-0.685978\pi$
$547$	−25.8011	−1.10318	−0.551589	+	0.834116i	$0.685978\pi$
$548$	0	0
$549$	−12.7005	−0.542045
$550$	0	0
$551$	0.402462	0.0171454
$552$	0	0
$553$	−19.8641	−0.844709
$554$	0	0
$555$	2.63752	0.111956
$556$	0	0
$557$	7.70782	0.326591	0.163295	−	0.986577i	$-0.447788\pi$
$557$	7.70782	0.326591	0.163295	+	0.986577i	$0.447788\pi$
$558$	0	0
$559$	−13.7381	−0.581061
$560$	0	0
$561$	6.38787	0.269696
$562$	0	0
$563$	−5.05334	−0.212973	−0.106486	−	0.994314i	$-0.533960\pi$
$563$	−5.05334	−0.212973	−0.106486	+	0.994314i	$0.533960\pi$
$564$	0	0
$565$	7.27504	0.306063
$566$	0	0
$567$	1.28726	0.0540597
$568$	0	0
$569$	−13.2569	−0.555760	−0.277880	−	0.960616i	$-0.589632\pi$
$569$	−13.2569	−0.555760	−0.277880	+	0.960616i	$0.589632\pi$
$570$	0	0
$571$	−23.6629	−0.990262	−0.495131	−	0.868818i	$-0.664880\pi$
$571$	−23.6629	−0.990262	−0.495131	+	0.868818i	$0.664880\pi$
$572$	0	0
$573$	3.13093	0.130797
$574$	0	0
$575$	−4.80606	−0.200427
$576$	0	0
$577$	29.1002	1.21146	0.605728	−	0.795672i	$-0.292882\pi$
$577$	29.1002	1.21146	0.605728	+	0.795672i	$0.292882\pi$
$578$	0	0
$579$	−21.5877	−0.897154
$580$	0	0
$581$	13.8134	0.573075
$582$	0	0
$583$	−11.5369	−0.477810
$584$	0	0
$585$	3.67513	0.151948
$586$	0	0
$587$	34.2130	1.41212	0.706061	−	0.708151i	$-0.250471\pi$
$587$	34.2130	1.41212	0.706061	+	0.708151i	$0.250471\pi$
$588$	0	0
$589$	2.38787	0.0983906
$590$	0	0
$591$	−4.60483	−0.189418
$592$	0	0
$593$	−9.51247	−0.390630	−0.195315	−	0.980741i	$-0.562573\pi$
$593$	−9.51247	−0.390630	−0.195315	+	0.980741i	$0.562573\pi$
$594$	0	0
$595$	2.77575	0.113795
$596$	0	0
$597$	7.76845	0.317942
$598$	0	0
$599$	−26.1343	−1.06782	−0.533908	−	0.845542i	$-0.679277\pi$
$599$	−26.1343	−1.06782	−0.533908	+	0.845542i	$0.679277\pi$
$600$	0	0
$601$	27.6483	1.12780	0.563899	−	0.825844i	$-0.309301\pi$
$601$	27.6483	1.12780	0.563899	+	0.825844i	$0.309301\pi$
$602$	0	0
$603$	−12.5623	−0.511576
$604$	0	0
$605$	2.22425	0.0904288
$606$	0	0
$607$	−45.0010	−1.82653	−0.913266	−	0.407363i	$-0.866448\pi$
$607$	−45.0010	−1.82653	−0.913266	+	0.407363i	$0.866448\pi$
$608$	0	0
$609$	−0.216960	−0.00879164
$610$	0	0
$611$	45.3620	1.83515
$612$	0	0
$613$	−8.84860	−0.357392	−0.178696	−	0.983904i	$-0.557188\pi$
$613$	−8.84860	−0.357392	−0.178696	+	0.983904i	$0.557188\pi$
$614$	0	0
$615$	11.6629	0.470294
$616$	0	0
$617$	−41.5487	−1.67269	−0.836343	−	0.548206i	$-0.815311\pi$
$617$	−41.5487	−1.67269	−0.836343	+	0.548206i	$0.815311\pi$
$618$	0	0
$619$	2.49341	0.100219	0.0501094	−	0.998744i	$-0.484043\pi$
$619$	2.49341	0.100219	0.0501094	+	0.998744i	$0.484043\pi$
$620$	0	0
$621$	4.80606	0.192861
$622$	0	0
$623$	4.43278	0.177596
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.07381	−0.282501
$628$	0	0
$629$	−5.68735	−0.226769
$630$	0	0
$631$	−0.252016	−0.0100326	−0.00501630	−	0.999987i	$-0.501597\pi$
$631$	−0.252016	−0.0100326	−0.00501630	+	0.999987i	$0.501597\pi$
$632$	0	0
$633$	0.337088	0.0133980
$634$	0	0
$635$	−1.27504	−0.0505984
$636$	0	0
$637$	19.6361	0.778011
$638$	0	0
$639$	0.481194	0.0190358
$640$	0	0
$641$	28.7694	1.13632	0.568162	−	0.822917i	$-0.307655\pi$
$641$	28.7694	1.13632	0.568162	+	0.822917i	$0.307655\pi$
$642$	0	0
$643$	−31.6629	−1.24866	−0.624332	−	0.781159i	$-0.714629\pi$
$643$	−31.6629	−1.24866	−0.624332	+	0.781159i	$0.714629\pi$
$644$	0	0
$645$	3.73813	0.147189
$646$	0	0
$647$	12.4142	0.488053	0.244027	−	0.969769i	$-0.421532\pi$
$647$	12.4142	0.488053	0.244027	+	0.969769i	$0.421532\pi$
$648$	0	0
$649$	40.9741	1.60838
$650$	0	0
$651$	−1.28726	−0.0504516
$652$	0	0
$653$	43.7196	1.71088	0.855440	−	0.517903i	$-0.173287\pi$
$653$	43.7196	1.71088	0.855440	+	0.517903i	$0.173287\pi$
$654$	0	0
$655$	17.8070	0.695778
$656$	0	0
$657$	5.21203	0.203341
$658$	0	0
$659$	18.7431	0.730126	0.365063	−	0.930983i	$-0.381047\pi$
$659$	18.7431	0.730126	0.365063	+	0.930983i	$0.381047\pi$
$660$	0	0
$661$	−39.2750	−1.52762	−0.763811	−	0.645440i	$-0.776674\pi$
$661$	−39.2750	−1.52762	−0.763811	+	0.645440i	$0.776674\pi$
$662$	0	0
$663$	−7.92478	−0.307773
$664$	0	0
$665$	−3.07381	−0.119197
$666$	0	0
$667$	−0.810033	−0.0313646
$668$	0	0
$669$	−2.83638	−0.109661
$670$	0	0
$671$	−37.6239	−1.45245
$672$	0	0
$673$	2.03857	0.0785810	0.0392905	−	0.999228i	$-0.487490\pi$
$673$	2.03857	0.0785810	0.0392905	+	0.999228i	$0.487490\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	26.7572	1.02836	0.514181	−	0.857682i	$-0.328096\pi$
$677$	26.7572	1.02836	0.514181	+	0.857682i	$0.328096\pi$
$678$	0	0
$679$	−19.5661	−0.750877
$680$	0	0
$681$	−17.3054	−0.663143
$682$	0	0
$683$	30.1417	1.15334	0.576671	−	0.816977i	$-0.304352\pi$
$683$	30.1417	1.15334	0.576671	+	0.816977i	$0.304352\pi$
$684$	0	0
$685$	−7.24472	−0.276807
$686$	0	0
$687$	25.1998	0.961433
$688$	0	0
$689$	14.3127	0.545269
$690$	0	0
$691$	17.6483	0.671374	0.335687	−	0.941974i	$-0.391032\pi$
$691$	17.6483	0.671374	0.335687	+	0.941974i	$0.391032\pi$
$692$	0	0
$693$	3.81336	0.144857
$694$	0	0
$695$	−14.8872	−0.564702
$696$	0	0
$697$	−25.1490	−0.952587
$698$	0	0
$699$	4.08110	0.154361
$700$	0	0
$701$	−3.78892	−0.143106	−0.0715528	−	0.997437i	$-0.522795\pi$
$701$	−3.78892	−0.143106	−0.0715528	+	0.997437i	$0.522795\pi$
$702$	0	0
$703$	6.29806	0.237536
$704$	0	0
$705$	−12.3430	−0.464863
$706$	0	0
$707$	18.0870	0.680231
$708$	0	0
$709$	−21.8496	−0.820577	−0.410289	−	0.911956i	$-0.634572\pi$
$709$	−21.8496	−0.820577	−0.410289	+	0.911956i	$0.634572\pi$
$710$	0	0
$711$	−15.4314	−0.578721
$712$	0	0
$713$	−4.80606	−0.179988
$714$	0	0
$715$	10.8872	0.407157
$716$	0	0
$717$	9.07381	0.338868
$718$	0	0
$719$	11.9902	0.447157	0.223579	−	0.974686i	$-0.428226\pi$
$719$	11.9902	0.447157	0.223579	+	0.974686i	$0.428226\pi$
$720$	0	0
$721$	13.0971	0.487762
$722$	0	0
$723$	21.9149	0.815025
$724$	0	0
$725$	0.168544	0.00625957
$726$	0	0
$727$	24.6883	0.915639	0.457819	−	0.889045i	$-0.348631\pi$
$727$	24.6883	0.915639	0.457819	+	0.889045i	$0.348631\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.06063	−0.298133
$732$	0	0
$733$	16.9525	0.626156	0.313078	−	0.949727i	$-0.398640\pi$
$733$	16.9525	0.626156	0.313078	+	0.949727i	$0.398640\pi$
$734$	0	0
$735$	−5.34297	−0.197078
$736$	0	0
$737$	−37.2144	−1.37081
$738$	0	0
$739$	−27.0943	−0.996679	−0.498340	−	0.866982i	$-0.666057\pi$
$739$	−27.0943	−0.996679	−0.498340	+	0.866982i	$0.666057\pi$
$740$	0	0
$741$	8.77575	0.322385
$742$	0	0
$743$	18.9887	0.696629	0.348315	−	0.937378i	$-0.386754\pi$
$743$	18.9887	0.696629	0.348315	+	0.937378i	$0.386754\pi$
$744$	0	0
$745$	5.73813	0.210229
$746$	0	0
$747$	10.7308	0.392621
$748$	0	0
$749$	−22.1984	−0.811112
$750$	0	0
$751$	−15.2896	−0.557926	−0.278963	−	0.960302i	$-0.589991\pi$
$751$	−15.2896	−0.557926	−0.278963	+	0.960302i	$0.589991\pi$
$752$	0	0
$753$	8.83638	0.322016
$754$	0	0
$755$	−12.8568	−0.467909
$756$	0	0
$757$	18.0386	0.655623	0.327811	−	0.944743i	$-0.393689\pi$
$757$	18.0386	0.655623	0.327811	+	0.944743i	$0.393689\pi$
$758$	0	0
$759$	14.2374	0.516786
$760$	0	0
$761$	19.5794	0.709754	0.354877	−	0.934913i	$-0.384523\pi$
$761$	19.5794	0.709754	0.354877	+	0.934913i	$0.384523\pi$
$762$	0	0
$763$	6.34885	0.229844
$764$	0	0
$765$	2.15633	0.0779621
$766$	0	0
$767$	−50.8324	−1.83545
$768$	0	0
$769$	4.27759	0.154254	0.0771270	−	0.997021i	$-0.475425\pi$
$769$	4.27759	0.154254	0.0771270	+	0.997021i	$0.475425\pi$
$770$	0	0
$771$	−10.7816	−0.388291
$772$	0	0
$773$	27.7685	0.998762	0.499381	−	0.866383i	$-0.333561\pi$
$773$	27.7685	0.998762	0.499381	+	0.866383i	$0.333561\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−3.39517	−0.121801
$778$	0	0
$779$	27.8496	0.997814
$780$	0	0
$781$	1.42548	0.0510078
$782$	0	0
$783$	−0.168544	−0.00602327
$784$	0	0
$785$	−1.73813	−0.0620367
$786$	0	0
$787$	7.92478	0.282488	0.141244	−	0.989975i	$-0.454890\pi$
$787$	7.92478	0.282488	0.141244	+	0.989975i	$0.454890\pi$
$788$	0	0
$789$	21.2144	0.755253
$790$	0	0
$791$	−9.36485	−0.332976
$792$	0	0
$793$	46.6761	1.65752
$794$	0	0
$795$	−3.89446	−0.138122
$796$	0	0
$797$	−5.00729	−0.177367	−0.0886837	−	0.996060i	$-0.528266\pi$
$797$	−5.00729	−0.177367	−0.0886837	+	0.996060i	$0.528266\pi$
$798$	0	0
$799$	26.6155	0.941587
$800$	0	0
$801$	3.44358	0.121673
$802$	0	0
$803$	15.4401	0.544868
$804$	0	0
$805$	6.18664	0.218050
$806$	0	0
$807$	16.6072	0.584601
$808$	0	0
$809$	52.6697	1.85177	0.925885	−	0.377806i	$-0.123321\pi$
$809$	52.6697	1.85177	0.925885	+	0.377806i	$0.123321\pi$
$810$	0	0
$811$	12.2473	0.430060	0.215030	−	0.976607i	$-0.431015\pi$
$811$	12.2473	0.430060	0.215030	+	0.976607i	$0.431015\pi$
$812$	0	0
$813$	22.8872	0.802688
$814$	0	0
$815$	−6.45088	−0.225964
$816$	0	0
$817$	8.92619	0.312288
$818$	0	0
$819$	−4.73084	−0.165309
$820$	0	0
$821$	−22.3914	−0.781465	−0.390732	−	0.920504i	$-0.627778\pi$
$821$	−22.3914	−0.781465	−0.390732	+	0.920504i	$0.627778\pi$
$822$	0	0
$823$	−41.4227	−1.44390	−0.721951	−	0.691944i	$-0.756755\pi$
$823$	−41.4227	−1.44390	−0.721951	+	0.691944i	$0.756755\pi$
$824$	0	0
$825$	−2.96239	−0.103137
$826$	0	0
$827$	−27.5672	−0.958606	−0.479303	−	0.877649i	$-0.659111\pi$
$827$	−27.5672	−0.958606	−0.479303	+	0.877649i	$0.659111\pi$
$828$	0	0
$829$	46.4387	1.61288	0.806441	−	0.591315i	$-0.201391\pi$
$829$	46.4387	1.61288	0.806441	+	0.591315i	$0.201391\pi$
$830$	0	0
$831$	11.2628	0.390703
$832$	0	0
$833$	11.5212	0.399185
$834$	0	0
$835$	22.5745	0.781223
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	16.1540	0.557696	0.278848	−	0.960335i	$-0.410047\pi$
$839$	16.1540	0.557696	0.278848	+	0.960335i	$0.410047\pi$
$840$	0	0
$841$	−28.9716	−0.999020
$842$	0	0
$843$	9.19982	0.316859
$844$	0	0
$845$	−0.506587	−0.0174271
$846$	0	0
$847$	−2.86319	−0.0983803
$848$	0	0
$849$	21.4739	0.736982
$850$	0	0
$851$	−12.6761	−0.434531
$852$	0	0
$853$	5.36011	0.183527	0.0917634	−	0.995781i	$-0.470750\pi$
$853$	5.36011	0.183527	0.0917634	+	0.995781i	$0.470750\pi$
$854$	0	0
$855$	−2.38787	−0.0816635
$856$	0	0
$857$	19.9492	0.681452	0.340726	−	0.940163i	$-0.389327\pi$
$857$	19.9492	0.681452	0.340726	+	0.940163i	$0.389327\pi$
$858$	0	0
$859$	−48.4260	−1.65227	−0.826137	−	0.563470i	$-0.809466\pi$
$859$	−48.4260	−1.65227	−0.826137	+	0.563470i	$0.809466\pi$
$860$	0	0
$861$	−15.0132	−0.511648
$862$	0	0
$863$	−9.02776	−0.307309	−0.153654	−	0.988125i	$-0.549104\pi$
$863$	−9.02776	−0.307309	−0.153654	+	0.988125i	$0.549104\pi$
$864$	0	0
$865$	−9.35026	−0.317918
$866$	0	0
$867$	12.3503	0.419437
$868$	0	0
$869$	−45.7137	−1.55073
$870$	0	0
$871$	46.1681	1.56435
$872$	0	0
$873$	−15.1998	−0.514436
$874$	0	0
$875$	−1.28726	−0.0435173
$876$	0	0
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	0	0
$879$	−21.6326	−0.729649
$880$	0	0
$881$	40.4831	1.36391	0.681955	−	0.731394i	$-0.261130\pi$
$881$	40.4831	1.36391	0.681955	+	0.731394i	$0.261130\pi$
$882$	0	0
$883$	−8.85097	−0.297859	−0.148929	−	0.988848i	$-0.547583\pi$
$883$	−8.85097	−0.297859	−0.148929	+	0.988848i	$0.547583\pi$
$884$	0	0
$885$	13.8315	0.464939
$886$	0	0
$887$	−0.493413	−0.0165672	−0.00828359	−	0.999966i	$-0.502637\pi$
$887$	−0.493413	−0.0165672	−0.00828359	+	0.999966i	$0.502637\pi$
$888$	0	0
$889$	1.64130	0.0550476
$890$	0	0
$891$	2.96239	0.0992438
$892$	0	0
$893$	−29.4734	−0.986291
$894$	0	0
$895$	−24.1622	−0.807653
$896$	0	0
$897$	−17.6629	−0.589747
$898$	0	0
$899$	0.168544	0.00562126
$900$	0	0
$901$	8.39772	0.279769
$902$	0	0
$903$	−4.81194	−0.160131
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−29.3479	−0.974481	−0.487240	−	0.873268i	$-0.661996\pi$
$907$	−29.3479	−0.974481	−0.487240	+	0.873268i	$0.661996\pi$
$908$	0	0
$909$	14.0508	0.466035
$910$	0	0
$911$	−17.7842	−0.589216	−0.294608	−	0.955618i	$-0.595189\pi$
$911$	−17.7842	−0.589216	−0.294608	+	0.955618i	$0.595189\pi$
$912$	0	0
$913$	31.7889	1.05206
$914$	0	0
$915$	−12.7005	−0.419866
$916$	0	0
$917$	−22.9222	−0.756958
$918$	0	0
$919$	−13.8232	−0.455986	−0.227993	−	0.973663i	$-0.573216\pi$
$919$	−13.8232	−0.455986	−0.227993	+	0.973663i	$0.573216\pi$
$920$	0	0
$921$	20.6131	0.679224
$922$	0	0
$923$	−1.76845	−0.0582093
$924$	0	0
$925$	2.63752	0.0867211
$926$	0	0
$927$	10.1744	0.334172
$928$	0	0
$929$	56.1197	1.84123	0.920613	−	0.390476i	$-0.127689\pi$
$929$	56.1197	1.84123	0.920613	+	0.390476i	$0.127689\pi$
$930$	0	0
$931$	−12.7583	−0.418137
$932$	0	0
$933$	2.34534	0.0767829
$934$	0	0
$935$	6.38787	0.208906
$936$	0	0
$937$	30.8989	1.00942	0.504712	−	0.863288i	$-0.331599\pi$
$937$	30.8989	1.00942	0.504712	+	0.863288i	$0.331599\pi$
$938$	0	0
$939$	24.1744	0.788902
$940$	0	0
$941$	33.9937	1.10816	0.554081	−	0.832463i	$-0.313070\pi$
$941$	33.9937	1.10816	0.554081	+	0.832463i	$0.313070\pi$
$942$	0	0
$943$	−56.0527	−1.82533
$944$	0	0
$945$	1.28726	0.0418745
$946$	0	0
$947$	−8.90763	−0.289459	−0.144730	−	0.989471i	$-0.546231\pi$
$947$	−8.90763	−0.289459	−0.144730	+	0.989471i	$0.546231\pi$
$948$	0	0
$949$	−19.1549	−0.621795
$950$	0	0
$951$	6.68006	0.216616
$952$	0	0
$953$	−3.64718	−0.118144	−0.0590719	−	0.998254i	$-0.518814\pi$
$953$	−3.64718	−0.118144	−0.0590719	+	0.998254i	$0.518814\pi$
$954$	0	0
$955$	3.13093	0.101315
$956$	0	0
$957$	−0.499293	−0.0161398
$958$	0	0
$959$	9.32582	0.301147
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−17.2447	−0.555703
$964$	0	0
$965$	−21.5877	−0.694932
$966$	0	0
$967$	54.6859	1.75858	0.879291	−	0.476286i	$-0.158017\pi$
$967$	54.6859	1.75858	0.879291	+	0.476286i	$0.158017\pi$
$968$	0	0
$969$	5.14903	0.165411
$970$	0	0
$971$	−44.9072	−1.44114	−0.720570	−	0.693382i	$-0.756120\pi$
$971$	−44.9072	−1.44114	−0.720570	+	0.693382i	$0.756120\pi$
$972$	0	0
$973$	19.1636	0.614357
$974$	0	0
$975$	3.67513	0.117698
$976$	0	0
$977$	−25.6991	−0.822187	−0.411094	−	0.911593i	$-0.634853\pi$
$977$	−25.6991	−0.822187	−0.411094	+	0.911593i	$0.634853\pi$
$978$	0	0
$979$	10.2012	0.326033
$980$	0	0
$981$	4.93207	0.157469
$982$	0	0
$983$	−51.3376	−1.63741	−0.818707	−	0.574211i	$-0.805309\pi$
$983$	−51.3376	−1.63741	−0.818707	+	0.574211i	$0.805309\pi$
$984$	0	0
$985$	−4.60483	−0.146722
$986$	0	0
$987$	15.8886	0.505739
$988$	0	0
$989$	−17.9657	−0.571276
$990$	0	0
$991$	−19.6629	−0.624613	−0.312306	−	0.949981i	$-0.601102\pi$
$991$	−19.6629	−0.624613	−0.312306	+	0.949981i	$0.601102\pi$
$992$	0	0
$993$	−11.0581	−0.350918
$994$	0	0
$995$	7.76845	0.246276
$996$	0	0
$997$	15.4372	0.488902	0.244451	−	0.969662i	$-0.421392\pi$
$997$	15.4372	0.488902	0.244451	+	0.969662i	$0.421392\pi$
$998$	0	0
$999$	−2.63752	−0.0834474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bm.1.2		3
4.3	odd	2		465.2.a.g.1.3	✓	3
12.11	even	2		1395.2.a.h.1.1		3
20.3	even	4		2325.2.c.l.1024.1		6
20.7	even	4		2325.2.c.l.1024.6		6
20.19	odd	2		2325.2.a.p.1.1		3
60.59	even	2		6975.2.a.bi.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.g.1.3	✓	3	4.3	odd	2
1395.2.a.h.1.1		3	12.11	even	2
2325.2.a.p.1.1		3	20.19	odd	2
2325.2.c.l.1024.1		6	20.3	even	4
2325.2.c.l.1024.6		6	20.7	even	4
6975.2.a.bi.1.3		3	60.59	even	2
7440.2.a.bm.1.2		3	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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.28726 q^{7} +1.00000 q^{9} +2.96239 q^{11} -3.67513 q^{13} +1.00000 q^{15} -2.15633 q^{17} +2.38787 q^{19} -1.28726 q^{21} -4.80606 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.168544 q^{29} +1.00000 q^{31} -2.96239 q^{33} -1.28726 q^{35} +2.63752 q^{37} +3.67513 q^{39} +11.6629 q^{41} +3.73813 q^{43} -1.00000 q^{45} -12.3430 q^{47} -5.34297 q^{49} +2.15633 q^{51} -3.89446 q^{53} -2.96239 q^{55} -2.38787 q^{57} +13.8315 q^{59} -12.7005 q^{61} +1.28726 q^{63} +3.67513 q^{65} -12.5623 q^{67} +4.80606 q^{69} +0.481194 q^{71} +5.21203 q^{73} -1.00000 q^{75} +3.81336 q^{77} -15.4314 q^{79} +1.00000 q^{81} +10.7308 q^{83} +2.15633 q^{85} -0.168544 q^{87} +3.44358 q^{89} -4.73084 q^{91} -1.00000 q^{93} -2.38787 q^{95} -15.1998 q^{97} +2.96239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^13 + 3 * q^15 + 4 * q^17 + 8 * q^19 + 2 * q^21 - 14 * q^23 + 3 * q^25 - 3 * q^27 + 16 * q^29 + 3 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 6 * q^39 + 4 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 + 7 * q^49 - 4 * q^51 + 8 * q^53 + 2 * q^55 - 8 * q^57 + 26 * q^59 - 18 * q^61 - 2 * q^63 + 6 * q^65 + 14 * q^69 - 4 * q^71 - 12 * q^73 - 3 * q^75 + 24 * q^77 - 4 * q^79 + 3 * q^81 + 10 * q^83 - 4 * q^85 - 16 * q^87 - 6 * q^89 + 8 * q^91 - 3 * q^93 - 8 * q^95 + 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

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Database

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