LMFDB - Embedded newform 6040.2.a.m.1.4

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} + \cdots - 16 $ x^12 - 3x^11 - 18x^10 + 54x^9 + 110x^8 - 335x^7 - 258x^6 + 825x^5 + 168x^4 - 669x^3 + 39x^2 + 106*x - 16
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			1.4
Root			$-1.08777$ 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)

$f(q)$	$=$	$q-1.08777 q^{3} -1.00000 q^{5} +2.80815 q^{7} -1.81675 q^{9} +O(q^{10})$	$q-1.08777 q^{3} -1.00000 q^{5} +2.80815 q^{7} -1.81675 q^{9} +2.75610 q^{11} +4.91070 q^{13} +1.08777 q^{15} +2.79690 q^{17} +1.91242 q^{19} -3.05463 q^{21} -2.47816 q^{23} +1.00000 q^{25} +5.23953 q^{27} +3.90049 q^{29} +4.75999 q^{31} -2.99801 q^{33} -2.80815 q^{35} +5.10673 q^{37} -5.34172 q^{39} +6.19747 q^{41} -10.6311 q^{43} +1.81675 q^{45} +5.90047 q^{47} +0.885692 q^{49} -3.04239 q^{51} -8.75987 q^{53} -2.75610 q^{55} -2.08028 q^{57} -0.0340130 q^{59} -4.60402 q^{61} -5.10170 q^{63} -4.91070 q^{65} -3.15091 q^{67} +2.69567 q^{69} -5.26329 q^{71} -5.59014 q^{73} -1.08777 q^{75} +7.73954 q^{77} -3.41716 q^{79} -0.249170 q^{81} +13.0891 q^{83} -2.79690 q^{85} -4.24285 q^{87} +17.9649 q^{89} +13.7900 q^{91} -5.17779 q^{93} -1.91242 q^{95} -0.569338 q^{97} -5.00715 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * 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
$3$	−1.08777	−0.628026	−0.314013	−	0.949419i	$-0.601674\pi$
$3$	−1.08777	−0.628026	−0.314013	+	0.949419i	$0.601674\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.80815	1.06138	0.530690	−	0.847566i	$-0.321933\pi$
$7$	2.80815	1.06138	0.530690	+	0.847566i	$0.321933\pi$
$8$	0	0
$9$	−1.81675	−0.605583
$10$	0	0
$11$	2.75610	0.830996	0.415498	−	0.909594i	$-0.363607\pi$
$11$	2.75610	0.830996	0.415498	+	0.909594i	$0.363607\pi$
$12$	0	0
$13$	4.91070	1.36198	0.680991	−	0.732292i	$-0.261549\pi$
$13$	4.91070	1.36198	0.680991	+	0.732292i	$0.261549\pi$
$14$	0	0
$15$	1.08777	0.280862
$16$	0	0
$17$	2.79690	0.678348	0.339174	−	0.940724i	$-0.389852\pi$
$17$	2.79690	0.678348	0.339174	+	0.940724i	$0.389852\pi$
$18$	0	0
$19$	1.91242	0.438739	0.219370	−	0.975642i	$-0.429600\pi$
$19$	1.91242	0.438739	0.219370	+	0.975642i	$0.429600\pi$
$20$	0	0
$21$	−3.05463	−0.666574
$22$	0	0
$23$	−2.47816	−0.516731	−0.258366	−	0.966047i	$-0.583184\pi$
$23$	−2.47816	−0.516731	−0.258366	+	0.966047i	$0.583184\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.23953	1.00835
$28$	0	0
$29$	3.90049	0.724303	0.362151	−	0.932119i	$-0.382042\pi$
$29$	3.90049	0.724303	0.362151	+	0.932119i	$0.382042\pi$
$30$	0	0
$31$	4.75999	0.854919	0.427460	−	0.904034i	$-0.359409\pi$
$31$	4.75999	0.854919	0.427460	+	0.904034i	$0.359409\pi$
$32$	0	0
$33$	−2.99801	−0.521887
$34$	0	0
$35$	−2.80815	−0.474664
$36$	0	0
$37$	5.10673	0.839541	0.419770	−	0.907630i	$-0.362111\pi$
$37$	5.10673	0.839541	0.419770	+	0.907630i	$0.362111\pi$
$38$	0	0
$39$	−5.34172	−0.855360
$40$	0	0
$41$	6.19747	0.967882	0.483941	−	0.875101i	$-0.339205\pi$
$41$	6.19747	0.967882	0.483941	+	0.875101i	$0.339205\pi$
$42$	0	0
$43$	−10.6311	−1.62123	−0.810614	−	0.585581i	$-0.800866\pi$
$43$	−10.6311	−1.62123	−0.810614	+	0.585581i	$0.800866\pi$
$44$	0	0
$45$	1.81675	0.270825
$46$	0	0
$47$	5.90047	0.860672	0.430336	−	0.902669i	$-0.358395\pi$
$47$	5.90047	0.860672	0.430336	+	0.902669i	$0.358395\pi$
$48$	0	0
$49$	0.885692	0.126527
$50$	0	0
$51$	−3.04239	−0.426020
$52$	0	0
$53$	−8.75987	−1.20326	−0.601630	−	0.798775i	$-0.705482\pi$
$53$	−8.75987	−1.20326	−0.601630	+	0.798775i	$0.705482\pi$
$54$	0	0
$55$	−2.75610	−0.371633
$56$	0	0
$57$	−2.08028	−0.275540
$58$	0	0
$59$	−0.0340130	−0.00442812	−0.00221406	−	0.999998i	$-0.500705\pi$
$59$	−0.0340130	−0.00442812	−0.00221406	+	0.999998i	$0.500705\pi$
$60$	0	0
$61$	−4.60402	−0.589484	−0.294742	−	0.955577i	$-0.595234\pi$
$61$	−4.60402	−0.589484	−0.294742	+	0.955577i	$0.595234\pi$
$62$	0	0
$63$	−5.10170	−0.642754
$64$	0	0
$65$	−4.91070	−0.609097
$66$	0	0
$67$	−3.15091	−0.384945	−0.192472	−	0.981302i	$-0.561651\pi$
$67$	−3.15091	−0.384945	−0.192472	+	0.981302i	$0.561651\pi$
$68$	0	0
$69$	2.69567	0.324521
$70$	0	0
$71$	−5.26329	−0.624637	−0.312319	−	0.949977i	$-0.601106\pi$
$71$	−5.26329	−0.624637	−0.312319	+	0.949977i	$0.601106\pi$
$72$	0	0
$73$	−5.59014	−0.654276	−0.327138	−	0.944977i	$-0.606084\pi$
$73$	−5.59014	−0.654276	−0.327138	+	0.944977i	$0.606084\pi$
$74$	0	0
$75$	−1.08777	−0.125605
$76$	0	0
$77$	7.73954	0.882003
$78$	0	0
$79$	−3.41716	−0.384461	−0.192231	−	0.981350i	$-0.561572\pi$
$79$	−3.41716	−0.384461	−0.192231	+	0.981350i	$0.561572\pi$
$80$	0	0
$81$	−0.249170	−0.0276856
$82$	0	0
$83$	13.0891	1.43672	0.718360	−	0.695671i	$-0.244893\pi$
$83$	13.0891	1.43672	0.718360	+	0.695671i	$0.244893\pi$
$84$	0	0
$85$	−2.79690	−0.303367
$86$	0	0
$87$	−4.24285	−0.454881
$88$	0	0
$89$	17.9649	1.90427	0.952136	−	0.305674i	$-0.0988817\pi$
$89$	17.9649	1.90427	0.952136	+	0.305674i	$0.0988817\pi$
$90$	0	0
$91$	13.7900	1.44558
$92$	0	0
$93$	−5.17779	−0.536911
$94$	0	0
$95$	−1.91242	−0.196210
$96$	0	0
$97$	−0.569338	−0.0578075	−0.0289038	−	0.999582i	$-0.509202\pi$
$97$	−0.569338	−0.0578075	−0.0289038	+	0.999582i	$0.509202\pi$
$98$	0	0
$99$	−5.00715	−0.503237
$100$	0	0
$101$	0.788165	0.0784253	0.0392127	−	0.999231i	$-0.487515\pi$
$101$	0.788165	0.0784253	0.0392127	+	0.999231i	$0.487515\pi$
$102$	0	0
$103$	6.87981	0.677888	0.338944	−	0.940807i	$-0.389930\pi$
$103$	6.87981	0.677888	0.338944	+	0.940807i	$0.389930\pi$
$104$	0	0
$105$	3.05463	0.298101
$106$	0	0
$107$	0.400821	0.0387488	0.0193744	−	0.999812i	$-0.493833\pi$
$107$	0.400821	0.0387488	0.0193744	+	0.999812i	$0.493833\pi$
$108$	0	0
$109$	−2.65258	−0.254071	−0.127035	−	0.991898i	$-0.540546\pi$
$109$	−2.65258	−0.254071	−0.127035	+	0.991898i	$0.540546\pi$
$110$	0	0
$111$	−5.55496	−0.527253
$112$	0	0
$113$	0.0880902	0.00828683	0.00414341	−	0.999991i	$-0.498681\pi$
$113$	0.0880902	0.00828683	0.00414341	+	0.999991i	$0.498681\pi$
$114$	0	0
$115$	2.47816	0.231089
$116$	0	0
$117$	−8.92151	−0.824794
$118$	0	0
$119$	7.85411	0.719985
$120$	0	0
$121$	−3.40390	−0.309446
$122$	0	0
$123$	−6.74144	−0.607855
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.35474	−0.297685	−0.148843	−	0.988861i	$-0.547555\pi$
$127$	−3.35474	−0.297685	−0.148843	+	0.988861i	$0.547555\pi$
$128$	0	0
$129$	11.5642	1.01817
$130$	0	0
$131$	3.10618	0.271388	0.135694	−	0.990751i	$-0.456674\pi$
$131$	3.10618	0.271388	0.135694	+	0.990751i	$0.456674\pi$
$132$	0	0
$133$	5.37036	0.465669
$134$	0	0
$135$	−5.23953	−0.450947
$136$	0	0
$137$	−8.11149	−0.693012	−0.346506	−	0.938048i	$-0.612632\pi$
$137$	−8.11149	−0.693012	−0.346506	+	0.938048i	$0.612632\pi$
$138$	0	0
$139$	−7.36089	−0.624343	−0.312171	−	0.950026i	$-0.601056\pi$
$139$	−7.36089	−0.624343	−0.312171	+	0.950026i	$0.601056\pi$
$140$	0	0
$141$	−6.41837	−0.540524
$142$	0	0
$143$	13.5344	1.13180
$144$	0	0
$145$	−3.90049	−0.323918
$146$	0	0
$147$	−0.963432	−0.0794625
$148$	0	0
$149$	−5.67920	−0.465258	−0.232629	−	0.972566i	$-0.574733\pi$
$149$	−5.67920	−0.465258	−0.232629	+	0.972566i	$0.574733\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−5.08127	−0.410796
$154$	0	0
$155$	−4.75999	−0.382331
$156$	0	0
$157$	−15.1632	−1.21016	−0.605078	−	0.796166i	$-0.706858\pi$
$157$	−15.1632	−1.21016	−0.605078	+	0.796166i	$0.706858\pi$
$158$	0	0
$159$	9.52875	0.755679
$160$	0	0
$161$	−6.95903	−0.548448
$162$	0	0
$163$	10.0404	0.786425	0.393212	−	0.919448i	$-0.371364\pi$
$163$	10.0404	0.786425	0.393212	+	0.919448i	$0.371364\pi$
$164$	0	0
$165$	2.99801	0.233395
$166$	0	0
$167$	4.31644	0.334016	0.167008	−	0.985956i	$-0.446589\pi$
$167$	4.31644	0.334016	0.167008	+	0.985956i	$0.446589\pi$
$168$	0	0
$169$	11.1150	0.854996
$170$	0	0
$171$	−3.47439	−0.265693
$172$	0	0
$173$	0.825246	0.0627423	0.0313711	−	0.999508i	$-0.490013\pi$
$173$	0.825246	0.0627423	0.0313711	+	0.999508i	$0.490013\pi$
$174$	0	0
$175$	2.80815	0.212276
$176$	0	0
$177$	0.0369984	0.00278097
$178$	0	0
$179$	14.5003	1.08381	0.541904	−	0.840441i	$-0.317704\pi$
$179$	14.5003	1.08381	0.541904	+	0.840441i	$0.317704\pi$
$180$	0	0
$181$	−10.4484	−0.776621	−0.388310	−	0.921529i	$-0.626941\pi$
$181$	−10.4484	−0.776621	−0.388310	+	0.921529i	$0.626941\pi$
$182$	0	0
$183$	5.00813	0.370211
$184$	0	0
$185$	−5.10673	−0.375454
$186$	0	0
$187$	7.70855	0.563705
$188$	0	0
$189$	14.7134	1.07024
$190$	0	0
$191$	2.21165	0.160029	0.0800146	−	0.996794i	$-0.474503\pi$
$191$	2.21165	0.160029	0.0800146	+	0.996794i	$0.474503\pi$
$192$	0	0
$193$	−23.5349	−1.69408	−0.847038	−	0.531532i	$-0.821617\pi$
$193$	−23.5349	−1.69408	−0.847038	+	0.531532i	$0.821617\pi$
$194$	0	0
$195$	5.34172	0.382529
$196$	0	0
$197$	25.4669	1.81444	0.907220	−	0.420656i	$-0.138200\pi$
$197$	25.4669	1.81444	0.907220	+	0.420656i	$0.138200\pi$
$198$	0	0
$199$	−25.9054	−1.83639	−0.918194	−	0.396130i	$-0.870353\pi$
$199$	−25.9054	−1.83639	−0.918194	+	0.396130i	$0.870353\pi$
$200$	0	0
$201$	3.42747	0.241755
$202$	0	0
$203$	10.9531	0.768760
$204$	0	0
$205$	−6.19747	−0.432850
$206$	0	0
$207$	4.50219	0.312924
$208$	0	0
$209$	5.27083	0.364591
$210$	0	0
$211$	18.2601	1.25708	0.628539	−	0.777778i	$-0.283653\pi$
$211$	18.2601	1.25708	0.628539	+	0.777778i	$0.283653\pi$
$212$	0	0
$213$	5.72526	0.392288
$214$	0	0
$215$	10.6311	0.725035
$216$	0	0
$217$	13.3667	0.907394
$218$	0	0
$219$	6.08080	0.410902
$220$	0	0
$221$	13.7347	0.923898
$222$	0	0
$223$	3.03066	0.202948	0.101474	−	0.994838i	$-0.467644\pi$
$223$	3.03066	0.202948	0.101474	+	0.994838i	$0.467644\pi$
$224$	0	0
$225$	−1.81675	−0.121117
$226$	0	0
$227$	13.9590	0.926492	0.463246	−	0.886230i	$-0.346685\pi$
$227$	13.9590	0.926492	0.463246	+	0.886230i	$0.346685\pi$
$228$	0	0
$229$	12.0666	0.797381	0.398691	−	0.917085i	$-0.369465\pi$
$229$	12.0666	0.797381	0.398691	+	0.917085i	$0.369465\pi$
$230$	0	0
$231$	−8.41886	−0.553921
$232$	0	0
$233$	−6.13025	−0.401606	−0.200803	−	0.979632i	$-0.564355\pi$
$233$	−6.13025	−0.401606	−0.200803	+	0.979632i	$0.564355\pi$
$234$	0	0
$235$	−5.90047	−0.384904
$236$	0	0
$237$	3.71710	0.241452
$238$	0	0
$239$	3.25612	0.210621	0.105310	−	0.994439i	$-0.466416\pi$
$239$	3.25612	0.210621	0.105310	+	0.994439i	$0.466416\pi$
$240$	0	0
$241$	6.04926	0.389667	0.194833	−	0.980836i	$-0.437583\pi$
$241$	6.04926	0.389667	0.194833	+	0.980836i	$0.437583\pi$
$242$	0	0
$243$	−15.4476	−0.990961
$244$	0	0
$245$	−0.885692	−0.0565848
$246$	0	0
$247$	9.39132	0.597555
$248$	0	0
$249$	−14.2380	−0.902298
$250$	0	0
$251$	5.75298	0.363125	0.181563	−	0.983379i	$-0.441885\pi$
$251$	5.75298	0.363125	0.181563	+	0.983379i	$0.441885\pi$
$252$	0	0
$253$	−6.83005	−0.429402
$254$	0	0
$255$	3.04239	0.190522
$256$	0	0
$257$	1.64057	0.102336	0.0511681	−	0.998690i	$-0.483706\pi$
$257$	1.64057	0.102336	0.0511681	+	0.998690i	$0.483706\pi$
$258$	0	0
$259$	14.3404	0.891072
$260$	0	0
$261$	−7.08621	−0.438626
$262$	0	0
$263$	19.1443	1.18049	0.590244	−	0.807225i	$-0.299032\pi$
$263$	19.1443	1.18049	0.590244	+	0.807225i	$0.299032\pi$
$264$	0	0
$265$	8.75987	0.538115
$266$	0	0
$267$	−19.5417	−1.19593
$268$	0	0
$269$	14.1320	0.861644	0.430822	−	0.902437i	$-0.358224\pi$
$269$	14.1320	0.861644	0.430822	+	0.902437i	$0.358224\pi$
$270$	0	0
$271$	20.5018	1.24540	0.622699	−	0.782462i	$-0.286036\pi$
$271$	20.5018	1.24540	0.622699	+	0.782462i	$0.286036\pi$
$272$	0	0
$273$	−15.0003	−0.907862
$274$	0	0
$275$	2.75610	0.166199
$276$	0	0
$277$	−1.99520	−0.119880	−0.0599400	−	0.998202i	$-0.519091\pi$
$277$	−1.99520	−0.119880	−0.0599400	+	0.998202i	$0.519091\pi$
$278$	0	0
$279$	−8.64771	−0.517725
$280$	0	0
$281$	0.853722	0.0509288	0.0254644	−	0.999676i	$-0.491894\pi$
$281$	0.853722	0.0509288	0.0254644	+	0.999676i	$0.491894\pi$
$282$	0	0
$283$	19.8922	1.18247	0.591234	−	0.806500i	$-0.298641\pi$
$283$	19.8922	1.18247	0.591234	+	0.806500i	$0.298641\pi$
$284$	0	0
$285$	2.08028	0.123225
$286$	0	0
$287$	17.4034	1.02729
$288$	0	0
$289$	−9.17734	−0.539844
$290$	0	0
$291$	0.619311	0.0363046
$292$	0	0
$293$	14.5383	0.849338	0.424669	−	0.905349i	$-0.360390\pi$
$293$	14.5383	0.849338	0.424669	+	0.905349i	$0.360390\pi$
$294$	0	0
$295$	0.0340130	0.00198031
$296$	0	0
$297$	14.4407	0.837933
$298$	0	0
$299$	−12.1695	−0.703779
$300$	0	0
$301$	−29.8537	−1.72074
$302$	0	0
$303$	−0.857344	−0.0492532
$304$	0	0
$305$	4.60402	0.263625
$306$	0	0
$307$	19.4459	1.10984	0.554919	−	0.831904i	$-0.312749\pi$
$307$	19.4459	1.10984	0.554919	+	0.831904i	$0.312749\pi$
$308$	0	0
$309$	−7.48367	−0.425731
$310$	0	0
$311$	5.73137	0.324996	0.162498	−	0.986709i	$-0.448045\pi$
$311$	5.73137	0.324996	0.162498	+	0.986709i	$0.448045\pi$
$312$	0	0
$313$	17.3563	0.981036	0.490518	−	0.871431i	$-0.336808\pi$
$313$	17.3563	0.981036	0.490518	+	0.871431i	$0.336808\pi$
$314$	0	0
$315$	5.10170	0.287448
$316$	0	0
$317$	−4.42268	−0.248402	−0.124201	−	0.992257i	$-0.539637\pi$
$317$	−4.42268	−0.248402	−0.124201	+	0.992257i	$0.539637\pi$
$318$	0	0
$319$	10.7501	0.601893
$320$	0	0
$321$	−0.436002	−0.0243353
$322$	0	0
$323$	5.34885	0.297618
$324$	0	0
$325$	4.91070	0.272396
$326$	0	0
$327$	2.88540	0.159563
$328$	0	0
$329$	16.5694	0.913500
$330$	0	0
$331$	−25.4053	−1.39640	−0.698202	−	0.715901i	$-0.746016\pi$
$331$	−25.4053	−1.39640	−0.698202	+	0.715901i	$0.746016\pi$
$332$	0	0
$333$	−9.27764	−0.508412
$334$	0	0
$335$	3.15091	0.172153
$336$	0	0
$337$	7.32651	0.399100	0.199550	−	0.979888i	$-0.436052\pi$
$337$	7.32651	0.399100	0.199550	+	0.979888i	$0.436052\pi$
$338$	0	0
$339$	−0.0958221	−0.00520434
$340$	0	0
$341$	13.1190	0.710434
$342$	0	0
$343$	−17.1699	−0.927086
$344$	0	0
$345$	−2.69567	−0.145130
$346$	0	0
$347$	19.9048	1.06855	0.534274	−	0.845311i	$-0.320585\pi$
$347$	19.9048	1.06855	0.534274	+	0.845311i	$0.320585\pi$
$348$	0	0
$349$	26.4691	1.41686	0.708430	−	0.705781i	$-0.249404\pi$
$349$	26.4691	1.41686	0.708430	+	0.705781i	$0.249404\pi$
$350$	0	0
$351$	25.7297	1.37335
$352$	0	0
$353$	−17.1890	−0.914878	−0.457439	−	0.889241i	$-0.651233\pi$
$353$	−17.1890	−0.914878	−0.457439	+	0.889241i	$0.651233\pi$
$354$	0	0
$355$	5.26329	0.279346
$356$	0	0
$357$	−8.54349	−0.452169
$358$	0	0
$359$	13.2642	0.700057	0.350028	−	0.936739i	$-0.386172\pi$
$359$	13.2642	0.700057	0.350028	+	0.936739i	$0.386172\pi$
$360$	0	0
$361$	−15.3426	−0.807508
$362$	0	0
$363$	3.70267	0.194340
$364$	0	0
$365$	5.59014	0.292601
$366$	0	0
$367$	31.3401	1.63594	0.817970	−	0.575260i	$-0.195099\pi$
$367$	31.3401	1.63594	0.817970	+	0.575260i	$0.195099\pi$
$368$	0	0
$369$	−11.2592	−0.586133
$370$	0	0
$371$	−24.5990	−1.27712
$372$	0	0
$373$	−13.1693	−0.681883	−0.340941	−	0.940085i	$-0.610746\pi$
$373$	−13.1693	−0.681883	−0.340941	+	0.940085i	$0.610746\pi$
$374$	0	0
$375$	1.08777	0.0561724
$376$	0	0
$377$	19.1541	0.986487
$378$	0	0
$379$	15.2678	0.784253	0.392126	−	0.919911i	$-0.371740\pi$
$379$	15.2678	0.784253	0.392126	+	0.919911i	$0.371740\pi$
$380$	0	0
$381$	3.64920	0.186954
$382$	0	0
$383$	3.60084	0.183994	0.0919971	−	0.995759i	$-0.470675\pi$
$383$	3.60084	0.183994	0.0919971	+	0.995759i	$0.470675\pi$
$384$	0	0
$385$	−7.73954	−0.394444
$386$	0	0
$387$	19.3140	0.981788
$388$	0	0
$389$	−29.7835	−1.51008	−0.755041	−	0.655678i	$-0.772383\pi$
$389$	−29.7835	−1.51008	−0.755041	+	0.655678i	$0.772383\pi$
$390$	0	0
$391$	−6.93116	−0.350524
$392$	0	0
$393$	−3.37882	−0.170439
$394$	0	0
$395$	3.41716	0.171936
$396$	0	0
$397$	14.3007	0.717729	0.358865	−	0.933390i	$-0.383164\pi$
$397$	14.3007	0.717729	0.358865	+	0.933390i	$0.383164\pi$
$398$	0	0
$399$	−5.84173	−0.292452
$400$	0	0
$401$	−5.06897	−0.253132	−0.126566	−	0.991958i	$-0.540396\pi$
$401$	−5.06897	−0.253132	−0.126566	+	0.991958i	$0.540396\pi$
$402$	0	0
$403$	23.3749	1.16438
$404$	0	0
$405$	0.249170	0.0123814
$406$	0	0
$407$	14.0747	0.697655
$408$	0	0
$409$	−23.0984	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$409$	−23.0984	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$410$	0	0
$411$	8.82346	0.435229
$412$	0	0
$413$	−0.0955135	−0.00469991
$414$	0	0
$415$	−13.0891	−0.642521
$416$	0	0
$417$	8.00698	0.392104
$418$	0	0
$419$	8.88670	0.434144	0.217072	−	0.976156i	$-0.430349\pi$
$419$	8.88670	0.434144	0.217072	+	0.976156i	$0.430349\pi$
$420$	0	0
$421$	−20.8111	−1.01427	−0.507136	−	0.861866i	$-0.669296\pi$
$421$	−20.8111	−1.01427	−0.507136	+	0.861866i	$0.669296\pi$
$422$	0	0
$423$	−10.7197	−0.521208
$424$	0	0
$425$	2.79690	0.135670
$426$	0	0
$427$	−12.9288	−0.625667
$428$	0	0
$429$	−14.7223	−0.710801
$430$	0	0
$431$	−32.4734	−1.56419	−0.782095	−	0.623159i	$-0.785849\pi$
$431$	−32.4734	−1.56419	−0.782095	+	0.623159i	$0.785849\pi$
$432$	0	0
$433$	−35.5130	−1.70665	−0.853324	−	0.521382i	$-0.825417\pi$
$433$	−35.5130	−1.70665	−0.853324	+	0.521382i	$0.825417\pi$
$434$	0	0
$435$	4.24285	0.203429
$436$	0	0
$437$	−4.73928	−0.226710
$438$	0	0
$439$	22.4618	1.07204	0.536022	−	0.844204i	$-0.319926\pi$
$439$	22.4618	1.07204	0.536022	+	0.844204i	$0.319926\pi$
$440$	0	0
$441$	−1.60908	−0.0766229
$442$	0	0
$443$	18.7890	0.892694	0.446347	−	0.894860i	$-0.352725\pi$
$443$	18.7890	0.892694	0.446347	+	0.894860i	$0.352725\pi$
$444$	0	0
$445$	−17.9649	−0.851617
$446$	0	0
$447$	6.17768	0.292194
$448$	0	0
$449$	−36.8673	−1.73988	−0.869939	−	0.493160i	$-0.835842\pi$
$449$	−36.8673	−1.73988	−0.869939	+	0.493160i	$0.835842\pi$
$450$	0	0
$451$	17.0809	0.804306
$452$	0	0
$453$	−1.08777	−0.0511080
$454$	0	0
$455$	−13.7900	−0.646483
$456$	0	0
$457$	29.9130	1.39927	0.699636	−	0.714500i	$-0.253346\pi$
$457$	29.9130	1.39927	0.699636	+	0.714500i	$0.253346\pi$
$458$	0	0
$459$	14.6545	0.684011
$460$	0	0
$461$	−22.1999	−1.03395	−0.516977	−	0.856000i	$-0.672943\pi$
$461$	−22.1999	−1.03395	−0.516977	+	0.856000i	$0.672943\pi$
$462$	0	0
$463$	−19.3795	−0.900641	−0.450320	−	0.892867i	$-0.648690\pi$
$463$	−19.3795	−0.900641	−0.450320	+	0.892867i	$0.648690\pi$
$464$	0	0
$465$	5.17779	0.240114
$466$	0	0
$467$	35.2986	1.63342	0.816712	−	0.577046i	$-0.195795\pi$
$467$	35.2986	1.63342	0.816712	+	0.577046i	$0.195795\pi$
$468$	0	0
$469$	−8.84822	−0.408573
$470$	0	0
$471$	16.4941	0.760009
$472$	0	0
$473$	−29.3004	−1.34723
$474$	0	0
$475$	1.91242	0.0877479
$476$	0	0
$477$	15.9145	0.728675
$478$	0	0
$479$	−22.8501	−1.04405	−0.522024	−	0.852931i	$-0.674823\pi$
$479$	−22.8501	−1.04405	−0.522024	+	0.852931i	$0.674823\pi$
$480$	0	0
$481$	25.0776	1.14344
$482$	0	0
$483$	7.56984	0.344440
$484$	0	0
$485$	0.569338	0.0258523
$486$	0	0
$487$	−15.2095	−0.689207	−0.344604	−	0.938748i	$-0.611987\pi$
$487$	−15.2095	−0.689207	−0.344604	+	0.938748i	$0.611987\pi$
$488$	0	0
$489$	−10.9217	−0.493895
$490$	0	0
$491$	29.0196	1.30964	0.654819	−	0.755786i	$-0.272745\pi$
$491$	29.0196	1.30964	0.654819	+	0.755786i	$0.272745\pi$
$492$	0	0
$493$	10.9093	0.491329
$494$	0	0
$495$	5.00715	0.225055
$496$	0	0
$497$	−14.7801	−0.662978
$498$	0	0
$499$	39.1439	1.75232	0.876160	−	0.482021i	$-0.160097\pi$
$499$	39.1439	1.75232	0.876160	+	0.482021i	$0.160097\pi$
$500$	0	0
$501$	−4.69530	−0.209771
$502$	0	0
$503$	14.6137	0.651591	0.325796	−	0.945440i	$-0.394368\pi$
$503$	14.6137	0.651591	0.325796	+	0.945440i	$0.394368\pi$
$504$	0	0
$505$	−0.788165	−0.0350729
$506$	0	0
$507$	−12.0905	−0.536960
$508$	0	0
$509$	42.9539	1.90390	0.951948	−	0.306259i	$-0.0990773\pi$
$509$	42.9539	1.90390	0.951948	+	0.306259i	$0.0990773\pi$
$510$	0	0
$511$	−15.6979	−0.694435
$512$	0	0
$513$	10.0202	0.442402
$514$	0	0
$515$	−6.87981	−0.303161
$516$	0	0
$517$	16.2623	0.715215
$518$	0	0
$519$	−0.897680	−0.0394038
$520$	0	0
$521$	−31.5428	−1.38192	−0.690959	−	0.722894i	$-0.742811\pi$
$521$	−31.5428	−1.38192	−0.690959	+	0.722894i	$0.742811\pi$
$522$	0	0
$523$	3.01988	0.132050	0.0660250	−	0.997818i	$-0.478968\pi$
$523$	3.01988	0.132050	0.0660250	+	0.997818i	$0.478968\pi$
$524$	0	0
$525$	−3.05463	−0.133315
$526$	0	0
$527$	13.3132	0.579933
$528$	0	0
$529$	−16.8587	−0.732989
$530$	0	0
$531$	0.0617931	0.00268159
$532$	0	0
$533$	30.4339	1.31824
$534$	0	0
$535$	−0.400821	−0.0173290
$536$	0	0
$537$	−15.7731	−0.680659
$538$	0	0
$539$	2.44106	0.105144
$540$	0	0
$541$	−36.3231	−1.56165	−0.780825	−	0.624749i	$-0.785201\pi$
$541$	−36.3231	−1.56165	−0.780825	+	0.624749i	$0.785201\pi$
$542$	0	0
$543$	11.3654	0.487738
$544$	0	0
$545$	2.65258	0.113624
$546$	0	0
$547$	0.954624	0.0408168	0.0204084	−	0.999792i	$-0.493503\pi$
$547$	0.954624	0.0408168	0.0204084	+	0.999792i	$0.493503\pi$
$548$	0	0
$549$	8.36435	0.356982
$550$	0	0
$551$	7.45938	0.317780
$552$	0	0
$553$	−9.59590	−0.408059
$554$	0	0
$555$	5.55496	0.235795
$556$	0	0
$557$	−5.07842	−0.215179	−0.107590	−	0.994195i	$-0.534313\pi$
$557$	−5.07842	−0.215179	−0.107590	+	0.994195i	$0.534313\pi$
$558$	0	0
$559$	−52.2061	−2.20808
$560$	0	0
$561$	−8.38515	−0.354021
$562$	0	0
$563$	14.9990	0.632132	0.316066	−	0.948737i	$-0.397638\pi$
$563$	14.9990	0.632132	0.316066	+	0.948737i	$0.397638\pi$
$564$	0	0
$565$	−0.0880902	−0.00370598
$566$	0	0
$567$	−0.699707	−0.0293849
$568$	0	0
$569$	2.13374	0.0894510	0.0447255	−	0.998999i	$-0.485759\pi$
$569$	2.13374	0.0894510	0.0447255	+	0.998999i	$0.485759\pi$
$570$	0	0
$571$	37.7513	1.57984	0.789922	−	0.613207i	$-0.210121\pi$
$571$	37.7513	1.57984	0.789922	+	0.613207i	$0.210121\pi$
$572$	0	0
$573$	−2.40577	−0.100503
$574$	0	0
$575$	−2.47816	−0.103346
$576$	0	0
$577$	28.0702	1.16858	0.584288	−	0.811546i	$-0.301374\pi$
$577$	28.0702	1.16858	0.584288	+	0.811546i	$0.301374\pi$
$578$	0	0
$579$	25.6006	1.06392
$580$	0	0
$581$	36.7563	1.52491
$582$	0	0
$583$	−24.1431	−0.999905
$584$	0	0
$585$	8.92151	0.368859
$586$	0	0
$587$	−31.0670	−1.28227	−0.641135	−	0.767428i	$-0.721536\pi$
$587$	−31.0670	−1.28227	−0.641135	+	0.767428i	$0.721536\pi$
$588$	0	0
$589$	9.10310	0.375087
$590$	0	0
$591$	−27.7022	−1.13952
$592$	0	0
$593$	22.8441	0.938095	0.469047	−	0.883173i	$-0.344597\pi$
$593$	22.8441	0.938095	0.469047	+	0.883173i	$0.344597\pi$
$594$	0	0
$595$	−7.85411	−0.321987
$596$	0	0
$597$	28.1792	1.15330
$598$	0	0
$599$	−23.5538	−0.962381	−0.481190	−	0.876616i	$-0.659795\pi$
$599$	−23.5538	−0.962381	−0.481190	+	0.876616i	$0.659795\pi$
$600$	0	0
$601$	12.2443	0.499457	0.249728	−	0.968316i	$-0.419659\pi$
$601$	12.2443	0.499457	0.249728	+	0.968316i	$0.419659\pi$
$602$	0	0
$603$	5.72442	0.233116
$604$	0	0
$605$	3.40390	0.138388
$606$	0	0
$607$	−17.0485	−0.691976	−0.345988	−	0.938239i	$-0.612456\pi$
$607$	−17.0485	−0.691976	−0.345988	+	0.938239i	$0.612456\pi$
$608$	0	0
$609$	−11.9145	−0.482801
$610$	0	0
$611$	28.9754	1.17222
$612$	0	0
$613$	14.0322	0.566757	0.283378	−	0.959008i	$-0.408545\pi$
$613$	14.0322	0.566757	0.283378	+	0.959008i	$0.408545\pi$
$614$	0	0
$615$	6.74144	0.271841
$616$	0	0
$617$	24.7211	0.995234	0.497617	−	0.867397i	$-0.334209\pi$
$617$	24.7211	0.995234	0.497617	+	0.867397i	$0.334209\pi$
$618$	0	0
$619$	−16.2217	−0.652004	−0.326002	−	0.945369i	$-0.605702\pi$
$619$	−16.2217	−0.652004	−0.326002	+	0.945369i	$0.605702\pi$
$620$	0	0
$621$	−12.9844	−0.521045
$622$	0	0
$623$	50.4480	2.02116
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.73346	−0.228972
$628$	0	0
$629$	14.2830	0.569501
$630$	0	0
$631$	−21.1916	−0.843624	−0.421812	−	0.906683i	$-0.638606\pi$
$631$	−21.1916	−0.843624	−0.421812	+	0.906683i	$0.638606\pi$
$632$	0	0
$633$	−19.8629	−0.789477
$634$	0	0
$635$	3.35474	0.133129
$636$	0	0
$637$	4.34937	0.172328
$638$	0	0
$639$	9.56208	0.378270
$640$	0	0
$641$	34.3614	1.35719	0.678597	−	0.734511i	$-0.262588\pi$
$641$	34.3614	1.35719	0.678597	+	0.734511i	$0.262588\pi$
$642$	0	0
$643$	44.9333	1.77200	0.885998	−	0.463690i	$-0.153475\pi$
$643$	44.9333	1.77200	0.885998	+	0.463690i	$0.153475\pi$
$644$	0	0
$645$	−11.5642	−0.455341
$646$	0	0
$647$	20.3367	0.799519	0.399759	−	0.916620i	$-0.369094\pi$
$647$	20.3367	0.799519	0.399759	+	0.916620i	$0.369094\pi$
$648$	0	0
$649$	−0.0937433	−0.00367975
$650$	0	0
$651$	−14.5400	−0.569867
$652$	0	0
$653$	11.4083	0.446443	0.223221	−	0.974768i	$-0.428343\pi$
$653$	11.4083	0.446443	0.223221	+	0.974768i	$0.428343\pi$
$654$	0	0
$655$	−3.10618	−0.121369
$656$	0	0
$657$	10.1559	0.396219
$658$	0	0
$659$	−24.1563	−0.940996	−0.470498	−	0.882401i	$-0.655926\pi$
$659$	−24.1563	−0.940996	−0.470498	+	0.882401i	$0.655926\pi$
$660$	0	0
$661$	−16.2486	−0.631998	−0.315999	−	0.948759i	$-0.602340\pi$
$661$	−16.2486	−0.631998	−0.315999	+	0.948759i	$0.602340\pi$
$662$	0	0
$663$	−14.9403	−0.580232
$664$	0	0
$665$	−5.37036	−0.208254
$666$	0	0
$667$	−9.66602	−0.374270
$668$	0	0
$669$	−3.29667	−0.127457
$670$	0	0
$671$	−12.6891	−0.489859
$672$	0	0
$673$	5.75656	0.221899	0.110950	−	0.993826i	$-0.464611\pi$
$673$	5.75656	0.221899	0.110950	+	0.993826i	$0.464611\pi$
$674$	0	0
$675$	5.23953	0.201670
$676$	0	0
$677$	28.5466	1.09713	0.548567	−	0.836107i	$-0.315174\pi$
$677$	28.5466	1.09713	0.548567	+	0.836107i	$0.315174\pi$
$678$	0	0
$679$	−1.59879	−0.0613557
$680$	0	0
$681$	−15.1842	−0.581861
$682$	0	0
$683$	−31.7234	−1.21386	−0.606932	−	0.794754i	$-0.707600\pi$
$683$	−31.7234	−1.21386	−0.606932	+	0.794754i	$0.707600\pi$
$684$	0	0
$685$	8.11149	0.309924
$686$	0	0
$687$	−13.1257	−0.500776
$688$	0	0
$689$	−43.0171	−1.63882
$690$	0	0
$691$	22.6537	0.861786	0.430893	−	0.902403i	$-0.358199\pi$
$691$	22.6537	0.861786	0.430893	+	0.902403i	$0.358199\pi$
$692$	0	0
$693$	−14.0608	−0.534126
$694$	0	0
$695$	7.36089	0.279215
$696$	0	0
$697$	17.3337	0.656561
$698$	0	0
$699$	6.66832	0.252219
$700$	0	0
$701$	3.17524	0.119927	0.0599635	−	0.998201i	$-0.480902\pi$
$701$	3.17524	0.119927	0.0599635	+	0.998201i	$0.480902\pi$
$702$	0	0
$703$	9.76621	0.368340
$704$	0	0
$705$	6.41837	0.241730
$706$	0	0
$707$	2.21328	0.0832391
$708$	0	0
$709$	23.4477	0.880597	0.440298	−	0.897852i	$-0.354873\pi$
$709$	23.4477	0.880597	0.440298	+	0.897852i	$0.354873\pi$
$710$	0	0
$711$	6.20813	0.232823
$712$	0	0
$713$	−11.7960	−0.441763
$714$	0	0
$715$	−13.5344	−0.506157
$716$	0	0
$717$	−3.54192	−0.132275
$718$	0	0
$719$	−24.7167	−0.921776	−0.460888	−	0.887458i	$-0.652469\pi$
$719$	−24.7167	−0.921776	−0.460888	+	0.887458i	$0.652469\pi$
$720$	0	0
$721$	19.3195	0.719496
$722$	0	0
$723$	−6.58022	−0.244721
$724$	0	0
$725$	3.90049	0.144861
$726$	0	0
$727$	−38.5106	−1.42828	−0.714140	−	0.700003i	$-0.753182\pi$
$727$	−38.5106	−1.42828	−0.714140	+	0.700003i	$0.753182\pi$
$728$	0	0
$729$	17.5509	0.650035
$730$	0	0
$731$	−29.7341	−1.09976
$732$	0	0
$733$	−8.15755	−0.301306	−0.150653	−	0.988587i	$-0.548138\pi$
$733$	−8.15755	−0.301306	−0.150653	+	0.988587i	$0.548138\pi$
$734$	0	0
$735$	0.963432	0.0355367
$736$	0	0
$737$	−8.68423	−0.319888
$738$	0	0
$739$	−3.17075	−0.116638	−0.0583190	−	0.998298i	$-0.518574\pi$
$739$	−3.17075	−0.116638	−0.0583190	+	0.998298i	$0.518574\pi$
$740$	0	0
$741$	−10.2156	−0.375280
$742$	0	0
$743$	2.15494	0.0790570	0.0395285	−	0.999218i	$-0.487414\pi$
$743$	2.15494	0.0790570	0.0395285	+	0.999218i	$0.487414\pi$
$744$	0	0
$745$	5.67920	0.208070
$746$	0	0
$747$	−23.7797	−0.870054
$748$	0	0
$749$	1.12556	0.0411272
$750$	0	0
$751$	17.1640	0.626324	0.313162	−	0.949700i	$-0.398612\pi$
$751$	17.1640	0.626324	0.313162	+	0.949700i	$0.398612\pi$
$752$	0	0
$753$	−6.25794	−0.228052
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	18.0615	0.656455	0.328227	−	0.944599i	$-0.393549\pi$
$757$	18.0615	0.656455	0.328227	+	0.944599i	$0.393549\pi$
$758$	0	0
$759$	7.42954	0.269675
$760$	0	0
$761$	3.76642	0.136533	0.0682663	−	0.997667i	$-0.478253\pi$
$761$	3.76642	0.136533	0.0682663	+	0.997667i	$0.478253\pi$
$762$	0	0
$763$	−7.44883	−0.269666
$764$	0	0
$765$	5.08127	0.183714
$766$	0	0
$767$	−0.167028	−0.00603102
$768$	0	0
$769$	−48.5648	−1.75129	−0.875646	−	0.482954i	$-0.839564\pi$
$769$	−48.5648	−1.75129	−0.875646	+	0.482954i	$0.839564\pi$
$770$	0	0
$771$	−1.78457	−0.0642697
$772$	0	0
$773$	33.0145	1.18745	0.593724	−	0.804669i	$-0.297657\pi$
$773$	33.0145	1.18745	0.593724	+	0.804669i	$0.297657\pi$
$774$	0	0
$775$	4.75999	0.170984
$776$	0	0
$777$	−15.5991	−0.559616
$778$	0	0
$779$	11.8522	0.424648
$780$	0	0
$781$	−14.5062	−0.519071
$782$	0	0
$783$	20.4367	0.730349
$784$	0	0
$785$	15.1632	0.541198
$786$	0	0
$787$	−30.4514	−1.08548	−0.542738	−	0.839902i	$-0.682612\pi$
$787$	−30.4514	−1.08548	−0.542738	+	0.839902i	$0.682612\pi$
$788$	0	0
$789$	−20.8246	−0.741377
$790$	0	0
$791$	0.247370	0.00879547
$792$	0	0
$793$	−22.6089	−0.802867
$794$	0	0
$795$	−9.52875	−0.337950
$796$	0	0
$797$	9.97570	0.353357	0.176679	−	0.984269i	$-0.443465\pi$
$797$	9.97570	0.353357	0.176679	+	0.984269i	$0.443465\pi$
$798$	0	0
$799$	16.5030	0.583835
$800$	0	0
$801$	−32.6377	−1.15320
$802$	0	0
$803$	−15.4070	−0.543701
$804$	0	0
$805$	6.95903	0.245273
$806$	0	0
$807$	−15.3724	−0.541135
$808$	0	0
$809$	1.35937	0.0477929	0.0238964	−	0.999714i	$-0.492393\pi$
$809$	1.35937	0.0477929	0.0238964	+	0.999714i	$0.492393\pi$
$810$	0	0
$811$	14.9984	0.526665	0.263332	−	0.964705i	$-0.415178\pi$
$811$	14.9984	0.526665	0.263332	+	0.964705i	$0.415178\pi$
$812$	0	0
$813$	−22.3013	−0.782142
$814$	0	0
$815$	−10.0404	−0.351700
$816$	0	0
$817$	−20.3311	−0.711296
$818$	0	0
$819$	−25.0529	−0.875420
$820$	0	0
$821$	−9.05088	−0.315878	−0.157939	−	0.987449i	$-0.550485\pi$
$821$	−9.05088	−0.315878	−0.157939	+	0.987449i	$0.550485\pi$
$822$	0	0
$823$	−16.6427	−0.580129	−0.290064	−	0.957007i	$-0.593677\pi$
$823$	−16.6427	−0.580129	−0.290064	+	0.957007i	$0.593677\pi$
$824$	0	0
$825$	−2.99801	−0.104377
$826$	0	0
$827$	−10.4588	−0.363688	−0.181844	−	0.983327i	$-0.558207\pi$
$827$	−10.4588	−0.363688	−0.181844	+	0.983327i	$0.558207\pi$
$828$	0	0
$829$	−14.1748	−0.492311	−0.246155	−	0.969230i	$-0.579167\pi$
$829$	−14.1748	−0.492311	−0.246155	+	0.969230i	$0.579167\pi$
$830$	0	0
$831$	2.17032	0.0752877
$832$	0	0
$833$	2.47719	0.0858297
$834$	0	0
$835$	−4.31644	−0.149377
$836$	0	0
$837$	24.9401	0.862056
$838$	0	0
$839$	−7.36382	−0.254227	−0.127114	−	0.991888i	$-0.540571\pi$
$839$	−7.36382	−0.254227	−0.127114	+	0.991888i	$0.540571\pi$
$840$	0	0
$841$	−13.7862	−0.475386
$842$	0	0
$843$	−0.928656	−0.0319846
$844$	0	0
$845$	−11.1150	−0.382366
$846$	0	0
$847$	−9.55866	−0.328439
$848$	0	0
$849$	−21.6382	−0.742621
$850$	0	0
$851$	−12.6553	−0.433817
$852$	0	0
$853$	17.3231	0.593133	0.296566	−	0.955012i	$-0.404158\pi$
$853$	17.3231	0.593133	0.296566	+	0.955012i	$0.404158\pi$
$854$	0	0
$855$	3.47439	0.118822
$856$	0	0
$857$	−42.0079	−1.43496	−0.717481	−	0.696578i	$-0.754705\pi$
$857$	−42.0079	−1.43496	−0.717481	+	0.696578i	$0.754705\pi$
$858$	0	0
$859$	−58.3067	−1.98940	−0.994700	−	0.102824i	$-0.967212\pi$
$859$	−58.3067	−1.98940	−0.994700	+	0.102824i	$0.967212\pi$
$860$	0	0
$861$	−18.9310	−0.645165
$862$	0	0
$863$	−1.58892	−0.0540874	−0.0270437	−	0.999634i	$-0.508609\pi$
$863$	−1.58892	−0.0540874	−0.0270437	+	0.999634i	$0.508609\pi$
$864$	0	0
$865$	−0.825246	−0.0280592
$866$	0	0
$867$	9.98287	0.339036
$868$	0	0
$869$	−9.41805	−0.319486
$870$	0	0
$871$	−15.4732	−0.524288
$872$	0	0
$873$	1.03434	0.0350073
$874$	0	0
$875$	−2.80815	−0.0949327
$876$	0	0
$877$	20.3174	0.686071	0.343036	−	0.939322i	$-0.388545\pi$
$877$	20.3174	0.686071	0.343036	+	0.939322i	$0.388545\pi$
$878$	0	0
$879$	−15.8144	−0.533406
$880$	0	0
$881$	−18.6018	−0.626710	−0.313355	−	0.949636i	$-0.601453\pi$
$881$	−18.6018	−0.626710	−0.313355	+	0.949636i	$0.601453\pi$
$882$	0	0
$883$	−9.59233	−0.322808	−0.161404	−	0.986888i	$-0.551602\pi$
$883$	−9.59233	−0.322808	−0.161404	+	0.986888i	$0.551602\pi$
$884$	0	0
$885$	−0.0369984	−0.00124369
$886$	0	0
$887$	1.40904	0.0473111	0.0236556	−	0.999720i	$-0.492470\pi$
$887$	1.40904	0.0473111	0.0236556	+	0.999720i	$0.492470\pi$
$888$	0	0
$889$	−9.42061	−0.315957
$890$	0	0
$891$	−0.686739	−0.0230066
$892$	0	0
$893$	11.2842	0.377611
$894$	0	0
$895$	−14.5003	−0.484693
$896$	0	0
$897$	13.2376	0.441991
$898$	0	0
$899$	18.5663	0.619220
$900$	0	0
$901$	−24.5005	−0.816230
$902$	0	0
$903$	32.4740	1.08067
$904$	0	0
$905$	10.4484	0.347315
$906$	0	0
$907$	47.0101	1.56094	0.780472	−	0.625191i	$-0.214979\pi$
$907$	47.0101	1.56094	0.780472	+	0.625191i	$0.214979\pi$
$908$	0	0
$909$	−1.43190	−0.0474931
$910$	0	0
$911$	−15.5441	−0.514998	−0.257499	−	0.966279i	$-0.582898\pi$
$911$	−15.5441	−0.514998	−0.257499	+	0.966279i	$0.582898\pi$
$912$	0	0
$913$	36.0750	1.19391
$914$	0	0
$915$	−5.00813	−0.165564
$916$	0	0
$917$	8.72262	0.288046
$918$	0	0
$919$	45.2844	1.49379	0.746897	−	0.664940i	$-0.231543\pi$
$919$	45.2844	1.49379	0.746897	+	0.664940i	$0.231543\pi$
$920$	0	0
$921$	−21.1528	−0.697007
$922$	0	0
$923$	−25.8464	−0.850745
$924$	0	0
$925$	5.10673	0.167908
$926$	0	0
$927$	−12.4989	−0.410517
$928$	0	0
$929$	−13.2107	−0.433429	−0.216715	−	0.976235i	$-0.569534\pi$
$929$	−13.2107	−0.433429	−0.216715	+	0.976235i	$0.569534\pi$
$930$	0	0
$931$	1.69382	0.0555126
$932$	0	0
$933$	−6.23443	−0.204106
$934$	0	0
$935$	−7.70855	−0.252096
$936$	0	0
$937$	−44.6619	−1.45904	−0.729520	−	0.683959i	$-0.760257\pi$
$937$	−44.6619	−1.45904	−0.729520	+	0.683959i	$0.760257\pi$
$938$	0	0
$939$	−18.8797	−0.616116
$940$	0	0
$941$	−54.7971	−1.78633	−0.893166	−	0.449726i	$-0.851522\pi$
$941$	−54.7971	−1.78633	−0.893166	+	0.449726i	$0.851522\pi$
$942$	0	0
$943$	−15.3583	−0.500135
$944$	0	0
$945$	−14.7134	−0.478626
$946$	0	0
$947$	16.0145	0.520401	0.260200	−	0.965555i	$-0.416211\pi$
$947$	16.0145	0.520401	0.260200	+	0.965555i	$0.416211\pi$
$948$	0	0
$949$	−27.4515	−0.891112
$950$	0	0
$951$	4.81087	0.156003
$952$	0	0
$953$	27.0445	0.876056	0.438028	−	0.898961i	$-0.355677\pi$
$953$	27.0445	0.876056	0.438028	+	0.898961i	$0.355677\pi$
$954$	0	0
$955$	−2.21165	−0.0715672
$956$	0	0
$957$	−11.6937	−0.378004
$958$	0	0
$959$	−22.7783	−0.735549
$960$	0	0
$961$	−8.34251	−0.269113
$962$	0	0
$963$	−0.728191	−0.0234656
$964$	0	0
$965$	23.5349	0.757614
$966$	0	0
$967$	15.1699	0.487833	0.243916	−	0.969796i	$-0.421568\pi$
$967$	15.1699	0.487833	0.243916	+	0.969796i	$0.421568\pi$
$968$	0	0
$969$	−5.81834	−0.186912
$970$	0	0
$971$	−16.9263	−0.543190	−0.271595	−	0.962412i	$-0.587551\pi$
$971$	−16.9263	−0.543190	−0.271595	+	0.962412i	$0.587551\pi$
$972$	0	0
$973$	−20.6705	−0.662665
$974$	0	0
$975$	−5.34172	−0.171072
$976$	0	0
$977$	20.1236	0.643812	0.321906	−	0.946772i	$-0.395676\pi$
$977$	20.1236	0.643812	0.321906	+	0.946772i	$0.395676\pi$
$978$	0	0
$979$	49.5130	1.58244
$980$	0	0
$981$	4.81907	0.153861
$982$	0	0
$983$	−25.2017	−0.803811	−0.401905	−	0.915681i	$-0.631652\pi$
$983$	−25.2017	−0.803811	−0.401905	+	0.915681i	$0.631652\pi$
$984$	0	0
$985$	−25.4669	−0.811443
$986$	0	0
$987$	−18.0237	−0.573701
$988$	0	0
$989$	26.3455	0.837739
$990$	0	0
$991$	5.69235	0.180823	0.0904117	−	0.995904i	$-0.471182\pi$
$991$	5.69235	0.180823	0.0904117	+	0.995904i	$0.471182\pi$
$992$	0	0
$993$	27.6352	0.876978
$994$	0	0
$995$	25.9054	0.821258
$996$	0	0
$997$	12.3544	0.391266	0.195633	−	0.980677i	$-0.437324\pi$
$997$	12.3544	0.391266	0.195633	+	0.980677i	$0.437324\pi$
$998$	0	0
$999$	26.7569	0.846549

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.m.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.m.1.4	✓	12	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)

\(f(q)\)	\(=\)	\(q-1.08777 q^{3} -1.00000 q^{5} +2.80815 q^{7} -1.81675 q^{9} +O(q^{10})\)	\(q-1.08777 q^{3} -1.00000 q^{5} +2.80815 q^{7} -1.81675 q^{9} +2.75610 q^{11} +4.91070 q^{13} +1.08777 q^{15} +2.79690 q^{17} +1.91242 q^{19} -3.05463 q^{21} -2.47816 q^{23} +1.00000 q^{25} +5.23953 q^{27} +3.90049 q^{29} +4.75999 q^{31} -2.99801 q^{33} -2.80815 q^{35} +5.10673 q^{37} -5.34172 q^{39} +6.19747 q^{41} -10.6311 q^{43} +1.81675 q^{45} +5.90047 q^{47} +0.885692 q^{49} -3.04239 q^{51} -8.75987 q^{53} -2.75610 q^{55} -2.08028 q^{57} -0.0340130 q^{59} -4.60402 q^{61} -5.10170 q^{63} -4.91070 q^{65} -3.15091 q^{67} +2.69567 q^{69} -5.26329 q^{71} -5.59014 q^{73} -1.08777 q^{75} +7.73954 q^{77} -3.41716 q^{79} -0.249170 q^{81} +13.0891 q^{83} -2.79690 q^{85} -4.24285 q^{87} +17.9649 q^{89} +13.7900 q^{91} -5.17779 q^{93} -1.91242 q^{95} -0.569338 q^{97} -5.00715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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