LMFDB - Embedded newform 6036.2.a.g.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 4 x^{14} - 27 x^{13} + 106 x^{12} + 266 x^{11} - 1004 x^{10} - 1105 x^{9} + 4076 x^{8} + \cdots - 44 $ x^15 - 4x^14 - 27x^13 + 106x^12 + 266x^11 - 1004x^10 - 1105x^9 + 4076x^8 + 1501x^7 - 7100x^6 - 134x^5 + 5356x^4 - 1041x^3 - 1381x^2 + 543x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$3.52096$ of defining polynomial
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.52096 q^{5} -1.02518 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.52096 q^{5} -1.02518 q^{7} +1.00000 q^{9} -3.26772 q^{11} -2.87517 q^{13} +2.52096 q^{15} +0.377785 q^{17} -1.62078 q^{19} -1.02518 q^{21} -6.17692 q^{23} +1.35524 q^{25} +1.00000 q^{27} -5.77725 q^{29} +6.59306 q^{31} -3.26772 q^{33} -2.58443 q^{35} +2.04591 q^{37} -2.87517 q^{39} +1.64121 q^{41} -1.25813 q^{43} +2.52096 q^{45} +12.2278 q^{47} -5.94901 q^{49} +0.377785 q^{51} -8.59305 q^{53} -8.23780 q^{55} -1.62078 q^{57} -9.35721 q^{59} -1.13492 q^{61} -1.02518 q^{63} -7.24820 q^{65} +0.942111 q^{67} -6.17692 q^{69} +0.988908 q^{71} -12.7507 q^{73} +1.35524 q^{75} +3.34999 q^{77} -3.81707 q^{79} +1.00000 q^{81} +4.87138 q^{83} +0.952380 q^{85} -5.77725 q^{87} -16.9858 q^{89} +2.94756 q^{91} +6.59306 q^{93} -4.08593 q^{95} +5.17800 q^{97} -3.26772 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	$ 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) $ 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.52096	1.12741	0.563704	−	0.825977i	$-0.309376\pi$
$5$	2.52096	1.12741	0.563704	+	0.825977i	$0.309376\pi$
$6$	0	0
$7$	−1.02518	−0.387480	−0.193740	−	0.981053i	$-0.562062\pi$
$7$	−1.02518	−0.387480	−0.193740	+	0.981053i	$0.562062\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.26772	−0.985255	−0.492628	−	0.870240i	$-0.663964\pi$
$11$	−3.26772	−0.985255	−0.492628	+	0.870240i	$0.663964\pi$
$12$	0	0
$13$	−2.87517	−0.797430	−0.398715	−	0.917075i	$-0.630544\pi$
$13$	−2.87517	−0.797430	−0.398715	+	0.917075i	$0.630544\pi$
$14$	0	0
$15$	2.52096	0.650909
$16$	0	0
$17$	0.377785	0.0916263	0.0458131	−	0.998950i	$-0.485412\pi$
$17$	0.377785	0.0916263	0.0458131	+	0.998950i	$0.485412\pi$
$18$	0	0
$19$	−1.62078	−0.371833	−0.185917	−	0.982566i	$-0.559525\pi$
$19$	−1.62078	−0.371833	−0.185917	+	0.982566i	$0.559525\pi$
$20$	0	0
$21$	−1.02518	−0.223712
$22$	0	0
$23$	−6.17692	−1.28798	−0.643989	−	0.765035i	$-0.722722\pi$
$23$	−6.17692	−1.28798	−0.643989	+	0.765035i	$0.722722\pi$
$24$	0	0
$25$	1.35524	0.271048
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.77725	−1.07281	−0.536404	−	0.843961i	$-0.680218\pi$
$29$	−5.77725	−1.07281	−0.536404	+	0.843961i	$0.680218\pi$
$30$	0	0
$31$	6.59306	1.18415	0.592074	−	0.805883i	$-0.298309\pi$
$31$	6.59306	1.18415	0.592074	+	0.805883i	$0.298309\pi$
$32$	0	0
$33$	−3.26772	−0.568837
$34$	0	0
$35$	−2.58443	−0.436848
$36$	0	0
$37$	2.04591	0.336345	0.168172	−	0.985758i	$-0.446213\pi$
$37$	2.04591	0.336345	0.168172	+	0.985758i	$0.446213\pi$
$38$	0	0
$39$	−2.87517	−0.460396
$40$	0	0
$41$	1.64121	0.256313	0.128157	−	0.991754i	$-0.459094\pi$
$41$	1.64121	0.256313	0.128157	+	0.991754i	$0.459094\pi$
$42$	0	0
$43$	−1.25813	−0.191863	−0.0959316	−	0.995388i	$-0.530583\pi$
$43$	−1.25813	−0.191863	−0.0959316	+	0.995388i	$0.530583\pi$
$44$	0	0
$45$	2.52096	0.375803
$46$	0	0
$47$	12.2278	1.78361	0.891803	−	0.452424i	$-0.149440\pi$
$47$	12.2278	1.78361	0.891803	+	0.452424i	$0.149440\pi$
$48$	0	0
$49$	−5.94901	−0.849859
$50$	0	0
$51$	0.377785	0.0529004
$52$	0	0
$53$	−8.59305	−1.18035	−0.590173	−	0.807277i	$-0.700941\pi$
$53$	−8.59305	−1.18035	−0.590173	+	0.807277i	$0.700941\pi$
$54$	0	0
$55$	−8.23780	−1.11078
$56$	0	0
$57$	−1.62078	−0.214678
$58$	0	0
$59$	−9.35721	−1.21820	−0.609102	−	0.793092i	$-0.708470\pi$
$59$	−9.35721	−1.21820	−0.609102	+	0.793092i	$0.708470\pi$
$60$	0	0
$61$	−1.13492	−0.145311	−0.0726555	−	0.997357i	$-0.523147\pi$
$61$	−1.13492	−0.145311	−0.0726555	+	0.997357i	$0.523147\pi$
$62$	0	0
$63$	−1.02518	−0.129160
$64$	0	0
$65$	−7.24820	−0.899029
$66$	0	0
$67$	0.942111	0.115097	0.0575486	−	0.998343i	$-0.481672\pi$
$67$	0.942111	0.115097	0.0575486	+	0.998343i	$0.481672\pi$
$68$	0	0
$69$	−6.17692	−0.743614
$70$	0	0
$71$	0.988908	0.117362	0.0586809	−	0.998277i	$-0.481311\pi$
$71$	0.988908	0.117362	0.0586809	+	0.998277i	$0.481311\pi$
$72$	0	0
$73$	−12.7507	−1.49236	−0.746178	−	0.665747i	$-0.768113\pi$
$73$	−12.7507	−1.49236	−0.746178	+	0.665747i	$0.768113\pi$
$74$	0	0
$75$	1.35524	0.156490
$76$	0	0
$77$	3.34999	0.381767
$78$	0	0
$79$	−3.81707	−0.429454	−0.214727	−	0.976674i	$-0.568886\pi$
$79$	−3.81707	−0.429454	−0.214727	+	0.976674i	$0.568886\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.87138	0.534703	0.267352	−	0.963599i	$-0.413851\pi$
$83$	4.87138	0.534703	0.267352	+	0.963599i	$0.413851\pi$
$84$	0	0
$85$	0.952380	0.103300
$86$	0	0
$87$	−5.77725	−0.619386
$88$	0	0
$89$	−16.9858	−1.80049	−0.900246	−	0.435382i	$-0.856613\pi$
$89$	−16.9858	−1.80049	−0.900246	+	0.435382i	$0.856613\pi$
$90$	0	0
$91$	2.94756	0.308988
$92$	0	0
$93$	6.59306	0.683668
$94$	0	0
$95$	−4.08593	−0.419208
$96$	0	0
$97$	5.17800	0.525746	0.262873	−	0.964830i	$-0.415330\pi$
$97$	5.17800	0.525746	0.262873	+	0.964830i	$0.415330\pi$
$98$	0	0
$99$	−3.26772	−0.328418
$100$	0	0
$101$	8.78904	0.874542	0.437271	−	0.899330i	$-0.355945\pi$
$101$	8.78904	0.874542	0.437271	+	0.899330i	$0.355945\pi$
$102$	0	0
$103$	−16.6235	−1.63796	−0.818980	−	0.573822i	$-0.805460\pi$
$103$	−16.6235	−1.63796	−0.818980	+	0.573822i	$0.805460\pi$
$104$	0	0
$105$	−2.58443	−0.252214
$106$	0	0
$107$	1.66433	0.160897	0.0804486	−	0.996759i	$-0.474365\pi$
$107$	1.66433	0.160897	0.0804486	+	0.996759i	$0.474365\pi$
$108$	0	0
$109$	−17.7496	−1.70010	−0.850050	−	0.526702i	$-0.823428\pi$
$109$	−17.7496	−1.70010	−0.850050	+	0.526702i	$0.823428\pi$
$110$	0	0
$111$	2.04591	0.194189
$112$	0	0
$113$	−4.24845	−0.399661	−0.199830	−	0.979831i	$-0.564039\pi$
$113$	−4.24845	−0.399661	−0.199830	+	0.979831i	$0.564039\pi$
$114$	0	0
$115$	−15.5718	−1.45208
$116$	0	0
$117$	−2.87517	−0.265810
$118$	0	0
$119$	−0.387296	−0.0355034
$120$	0	0
$121$	−0.321992	−0.0292720
$122$	0	0
$123$	1.64121	0.147983
$124$	0	0
$125$	−9.18830	−0.821826
$126$	0	0
$127$	10.3085	0.914731	0.457365	−	0.889279i	$-0.348793\pi$
$127$	10.3085	0.914731	0.457365	+	0.889279i	$0.348793\pi$
$128$	0	0
$129$	−1.25813	−0.110772
$130$	0	0
$131$	8.88317	0.776126	0.388063	−	0.921633i	$-0.373144\pi$
$131$	8.88317	0.776126	0.388063	+	0.921633i	$0.373144\pi$
$132$	0	0
$133$	1.66159	0.144078
$134$	0	0
$135$	2.52096	0.216970
$136$	0	0
$137$	−11.0387	−0.943097	−0.471549	−	0.881840i	$-0.656305\pi$
$137$	−11.0387	−0.943097	−0.471549	+	0.881840i	$0.656305\pi$
$138$	0	0
$139$	10.0393	0.851520	0.425760	−	0.904836i	$-0.360007\pi$
$139$	10.0393	0.851520	0.425760	+	0.904836i	$0.360007\pi$
$140$	0	0
$141$	12.2278	1.02977
$142$	0	0
$143$	9.39527	0.785672
$144$	0	0
$145$	−14.5642	−1.20949
$146$	0	0
$147$	−5.94901	−0.490666
$148$	0	0
$149$	4.47326	0.366463	0.183232	−	0.983070i	$-0.441344\pi$
$149$	4.47326	0.366463	0.183232	+	0.983070i	$0.441344\pi$
$150$	0	0
$151$	18.6111	1.51455	0.757275	−	0.653096i	$-0.226530\pi$
$151$	18.6111	1.51455	0.757275	+	0.653096i	$0.226530\pi$
$152$	0	0
$153$	0.377785	0.0305421
$154$	0	0
$155$	16.6208	1.33502
$156$	0	0
$157$	−14.3876	−1.14826	−0.574128	−	0.818766i	$-0.694659\pi$
$157$	−14.3876	−1.14826	−0.574128	+	0.818766i	$0.694659\pi$
$158$	0	0
$159$	−8.59305	−0.681473
$160$	0	0
$161$	6.33243	0.499066
$162$	0	0
$163$	−4.81245	−0.376940	−0.188470	−	0.982079i	$-0.560353\pi$
$163$	−4.81245	−0.376940	−0.188470	+	0.982079i	$0.560353\pi$
$164$	0	0
$165$	−8.23780	−0.641312
$166$	0	0
$167$	0.760554	0.0588534	0.0294267	−	0.999567i	$-0.490632\pi$
$167$	0.760554	0.0588534	0.0294267	+	0.999567i	$0.490632\pi$
$168$	0	0
$169$	−4.73337	−0.364106
$170$	0	0
$171$	−1.62078	−0.123944
$172$	0	0
$173$	−10.1595	−0.772410	−0.386205	−	0.922413i	$-0.626214\pi$
$173$	−10.1595	−0.772410	−0.386205	+	0.922413i	$0.626214\pi$
$174$	0	0
$175$	−1.38936	−0.105026
$176$	0	0
$177$	−9.35721	−0.703331
$178$	0	0
$179$	17.1094	1.27882	0.639409	−	0.768867i	$-0.279179\pi$
$179$	17.1094	1.27882	0.639409	+	0.768867i	$0.279179\pi$
$180$	0	0
$181$	3.61105	0.268407	0.134203	−	0.990954i	$-0.457152\pi$
$181$	3.61105	0.268407	0.134203	+	0.990954i	$0.457152\pi$
$182$	0	0
$183$	−1.13492	−0.0838954
$184$	0	0
$185$	5.15765	0.379198
$186$	0	0
$187$	−1.23450	−0.0902753
$188$	0	0
$189$	−1.02518	−0.0745706
$190$	0	0
$191$	0.296153	0.0214289	0.0107144	−	0.999943i	$-0.496589\pi$
$191$	0.296153	0.0214289	0.0107144	+	0.999943i	$0.496589\pi$
$192$	0	0
$193$	14.8084	1.06593	0.532966	−	0.846137i	$-0.321077\pi$
$193$	14.8084	1.06593	0.532966	+	0.846137i	$0.321077\pi$
$194$	0	0
$195$	−7.24820	−0.519054
$196$	0	0
$197$	−19.2880	−1.37421	−0.687105	−	0.726558i	$-0.741119\pi$
$197$	−19.2880	−1.37421	−0.687105	+	0.726558i	$0.741119\pi$
$198$	0	0
$199$	−15.1845	−1.07640	−0.538201	−	0.842817i	$-0.680896\pi$
$199$	−15.1845	−1.07640	−0.538201	+	0.842817i	$0.680896\pi$
$200$	0	0
$201$	0.942111	0.0664514
$202$	0	0
$203$	5.92270	0.415692
$204$	0	0
$205$	4.13742	0.288970
$206$	0	0
$207$	−6.17692	−0.429326
$208$	0	0
$209$	5.29627	0.366351
$210$	0	0
$211$	−22.1254	−1.52317	−0.761587	−	0.648063i	$-0.775579\pi$
$211$	−22.1254	−1.52317	−0.761587	+	0.648063i	$0.775579\pi$
$212$	0	0
$213$	0.988908	0.0677589
$214$	0	0
$215$	−3.17170	−0.216308
$216$	0	0
$217$	−6.75905	−0.458834
$218$	0	0
$219$	−12.7507	−0.861612
$220$	0	0
$221$	−1.08620	−0.0730655
$222$	0	0
$223$	−18.7967	−1.25872	−0.629359	−	0.777115i	$-0.716683\pi$
$223$	−18.7967	−1.25872	−0.629359	+	0.777115i	$0.716683\pi$
$224$	0	0
$225$	1.35524	0.0903493
$226$	0	0
$227$	−10.1902	−0.676350	−0.338175	−	0.941083i	$-0.609810\pi$
$227$	−10.1902	−0.676350	−0.338175	+	0.941083i	$0.609810\pi$
$228$	0	0
$229$	8.80690	0.581976	0.290988	−	0.956727i	$-0.406016\pi$
$229$	8.80690	0.581976	0.290988	+	0.956727i	$0.406016\pi$
$230$	0	0
$231$	3.34999	0.220413
$232$	0	0
$233$	−11.6965	−0.766263	−0.383131	−	0.923694i	$-0.625154\pi$
$233$	−11.6965	−0.766263	−0.383131	+	0.923694i	$0.625154\pi$
$234$	0	0
$235$	30.8258	2.01085
$236$	0	0
$237$	−3.81707	−0.247945
$238$	0	0
$239$	−23.7646	−1.53721	−0.768603	−	0.639725i	$-0.779048\pi$
$239$	−23.7646	−1.53721	−0.768603	+	0.639725i	$0.779048\pi$
$240$	0	0
$241$	4.06074	0.261575	0.130788	−	0.991410i	$-0.458249\pi$
$241$	4.06074	0.261575	0.130788	+	0.991410i	$0.458249\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−14.9972	−0.958138
$246$	0	0
$247$	4.66004	0.296511
$248$	0	0
$249$	4.87138	0.308711
$250$	0	0
$251$	29.8288	1.88278	0.941388	−	0.337326i	$-0.109522\pi$
$251$	29.8288	1.88278	0.941388	+	0.337326i	$0.109522\pi$
$252$	0	0
$253$	20.1845	1.26899
$254$	0	0
$255$	0.952380	0.0596404
$256$	0	0
$257$	−0.936638	−0.0584258	−0.0292129	−	0.999573i	$-0.509300\pi$
$257$	−0.936638	−0.0584258	−0.0292129	+	0.999573i	$0.509300\pi$
$258$	0	0
$259$	−2.09741	−0.130327
$260$	0	0
$261$	−5.77725	−0.357603
$262$	0	0
$263$	−3.88139	−0.239337	−0.119668	−	0.992814i	$-0.538183\pi$
$263$	−3.88139	−0.239337	−0.119668	+	0.992814i	$0.538183\pi$
$264$	0	0
$265$	−21.6627	−1.33073
$266$	0	0
$267$	−16.9858	−1.03951
$268$	0	0
$269$	32.5695	1.98580	0.992898	−	0.118971i	$-0.0379596\pi$
$269$	32.5695	1.98580	0.992898	+	0.118971i	$0.0379596\pi$
$270$	0	0
$271$	23.0539	1.40042	0.700211	−	0.713936i	$-0.253089\pi$
$271$	23.0539	1.40042	0.700211	+	0.713936i	$0.253089\pi$
$272$	0	0
$273$	2.94756	0.178395
$274$	0	0
$275$	−4.42854	−0.267051
$276$	0	0
$277$	24.4849	1.47115	0.735577	−	0.677442i	$-0.236911\pi$
$277$	24.4849	1.47115	0.735577	+	0.677442i	$0.236911\pi$
$278$	0	0
$279$	6.59306	0.394716
$280$	0	0
$281$	11.0539	0.659418	0.329709	−	0.944083i	$-0.393049\pi$
$281$	11.0539	0.659418	0.329709	+	0.944083i	$0.393049\pi$
$282$	0	0
$283$	17.3703	1.03256	0.516278	−	0.856421i	$-0.327317\pi$
$283$	17.3703	1.03256	0.516278	+	0.856421i	$0.327317\pi$
$284$	0	0
$285$	−4.08593	−0.242030
$286$	0	0
$287$	−1.68253	−0.0993164
$288$	0	0
$289$	−16.8573	−0.991605
$290$	0	0
$291$	5.17800	0.303540
$292$	0	0
$293$	2.92081	0.170636	0.0853178	−	0.996354i	$-0.472809\pi$
$293$	2.92081	0.170636	0.0853178	+	0.996354i	$0.472809\pi$
$294$	0	0
$295$	−23.5891	−1.37341
$296$	0	0
$297$	−3.26772	−0.189612
$298$	0	0
$299$	17.7597	1.02707
$300$	0	0
$301$	1.28981	0.0743432
$302$	0	0
$303$	8.78904	0.504917
$304$	0	0
$305$	−2.86108	−0.163825
$306$	0	0
$307$	23.5812	1.34585	0.672925	−	0.739711i	$-0.265038\pi$
$307$	23.5812	1.34585	0.672925	+	0.739711i	$0.265038\pi$
$308$	0	0
$309$	−16.6235	−0.945677
$310$	0	0
$311$	−15.0568	−0.853793	−0.426896	−	0.904301i	$-0.640393\pi$
$311$	−15.0568	−0.853793	−0.426896	+	0.904301i	$0.640393\pi$
$312$	0	0
$313$	17.4291	0.985153	0.492577	−	0.870269i	$-0.336055\pi$
$313$	17.4291	0.985153	0.492577	+	0.870269i	$0.336055\pi$
$314$	0	0
$315$	−2.58443	−0.145616
$316$	0	0
$317$	−6.21009	−0.348794	−0.174397	−	0.984675i	$-0.555798\pi$
$317$	−6.21009	−0.348794	−0.174397	+	0.984675i	$0.555798\pi$
$318$	0	0
$319$	18.8785	1.05699
$320$	0	0
$321$	1.66433	0.0928941
$322$	0	0
$323$	−0.612307	−0.0340697
$324$	0	0
$325$	−3.89655	−0.216142
$326$	0	0
$327$	−17.7496	−0.981553
$328$	0	0
$329$	−12.5356	−0.691112
$330$	0	0
$331$	−16.0705	−0.883314	−0.441657	−	0.897184i	$-0.645609\pi$
$331$	−16.0705	−0.883314	−0.441657	+	0.897184i	$0.645609\pi$
$332$	0	0
$333$	2.04591	0.112115
$334$	0	0
$335$	2.37502	0.129761
$336$	0	0
$337$	16.0344	0.873447	0.436724	−	0.899596i	$-0.356139\pi$
$337$	16.0344	0.873447	0.436724	+	0.899596i	$0.356139\pi$
$338$	0	0
$339$	−4.24845	−0.230744
$340$	0	0
$341$	−21.5443	−1.16669
$342$	0	0
$343$	13.2750	0.716784
$344$	0	0
$345$	−15.5718	−0.838356
$346$	0	0
$347$	32.1269	1.72466	0.862331	−	0.506345i	$-0.169004\pi$
$347$	32.1269	1.72466	0.862331	+	0.506345i	$0.169004\pi$
$348$	0	0
$349$	16.0352	0.858345	0.429173	−	0.903223i	$-0.358805\pi$
$349$	16.0352	0.858345	0.429173	+	0.903223i	$0.358805\pi$
$350$	0	0
$351$	−2.87517	−0.153465
$352$	0	0
$353$	−23.0322	−1.22588	−0.612940	−	0.790129i	$-0.710013\pi$
$353$	−23.0322	−1.22588	−0.612940	+	0.790129i	$0.710013\pi$
$354$	0	0
$355$	2.49300	0.132315
$356$	0	0
$357$	−0.387296	−0.0204979
$358$	0	0
$359$	−10.1002	−0.533067	−0.266533	−	0.963826i	$-0.585878\pi$
$359$	−10.1002	−0.533067	−0.266533	+	0.963826i	$0.585878\pi$
$360$	0	0
$361$	−16.3731	−0.861740
$362$	0	0
$363$	−0.321992	−0.0169002
$364$	0	0
$365$	−32.1440	−1.68249
$366$	0	0
$367$	9.20738	0.480622	0.240311	−	0.970696i	$-0.422751\pi$
$367$	9.20738	0.480622	0.240311	+	0.970696i	$0.422751\pi$
$368$	0	0
$369$	1.64121	0.0854378
$370$	0	0
$371$	8.80939	0.457361
$372$	0	0
$373$	21.8720	1.13249	0.566244	−	0.824237i	$-0.308396\pi$
$373$	21.8720	1.13249	0.566244	+	0.824237i	$0.308396\pi$
$374$	0	0
$375$	−9.18830	−0.474482
$376$	0	0
$377$	16.6106	0.855490
$378$	0	0
$379$	33.3983	1.71556	0.857779	−	0.514019i	$-0.171844\pi$
$379$	33.3983	1.71556	0.857779	+	0.514019i	$0.171844\pi$
$380$	0	0
$381$	10.3085	0.528120
$382$	0	0
$383$	20.2461	1.03453	0.517264	−	0.855826i	$-0.326950\pi$
$383$	20.2461	1.03453	0.517264	+	0.855826i	$0.326950\pi$
$384$	0	0
$385$	8.44519	0.430407
$386$	0	0
$387$	−1.25813	−0.0639544
$388$	0	0
$389$	−32.2980	−1.63758	−0.818788	−	0.574096i	$-0.805354\pi$
$389$	−32.2980	−1.63758	−0.818788	+	0.574096i	$0.805354\pi$
$390$	0	0
$391$	−2.33355	−0.118013
$392$	0	0
$393$	8.88317	0.448097
$394$	0	0
$395$	−9.62268	−0.484170
$396$	0	0
$397$	16.3523	0.820697	0.410348	−	0.911929i	$-0.365407\pi$
$397$	16.3523	0.820697	0.410348	+	0.911929i	$0.365407\pi$
$398$	0	0
$399$	1.66159	0.0831835
$400$	0	0
$401$	−27.6189	−1.37922	−0.689612	−	0.724179i	$-0.742219\pi$
$401$	−27.6189	−1.37922	−0.689612	+	0.724179i	$0.742219\pi$
$402$	0	0
$403$	−18.9562	−0.944275
$404$	0	0
$405$	2.52096	0.125268
$406$	0	0
$407$	−6.68545	−0.331385
$408$	0	0
$409$	−3.45875	−0.171024	−0.0855121	−	0.996337i	$-0.527253\pi$
$409$	−3.45875	−0.171024	−0.0855121	+	0.996337i	$0.527253\pi$
$410$	0	0
$411$	−11.0387	−0.544498
$412$	0	0
$413$	9.59279	0.472030
$414$	0	0
$415$	12.2806	0.602828
$416$	0	0
$417$	10.0393	0.491625
$418$	0	0
$419$	6.45922	0.315553	0.157777	−	0.987475i	$-0.449567\pi$
$419$	6.45922	0.315553	0.157777	+	0.987475i	$0.449567\pi$
$420$	0	0
$421$	−32.0739	−1.56319	−0.781593	−	0.623789i	$-0.785593\pi$
$421$	−32.0739	−1.56319	−0.781593	+	0.623789i	$0.785593\pi$
$422$	0	0
$423$	12.2278	0.594535
$424$	0	0
$425$	0.511989	0.0248351
$426$	0	0
$427$	1.16349	0.0563052
$428$	0	0
$429$	9.39527	0.453608
$430$	0	0
$431$	−21.9616	−1.05785	−0.528926	−	0.848668i	$-0.677405\pi$
$431$	−21.9616	−1.05785	−0.528926	+	0.848668i	$0.677405\pi$
$432$	0	0
$433$	20.1059	0.966226	0.483113	−	0.875558i	$-0.339506\pi$
$433$	20.1059	0.966226	0.483113	+	0.875558i	$0.339506\pi$
$434$	0	0
$435$	−14.5642	−0.698301
$436$	0	0
$437$	10.0115	0.478913
$438$	0	0
$439$	−0.533793	−0.0254766	−0.0127383	−	0.999919i	$-0.504055\pi$
$439$	−0.533793	−0.0254766	−0.0127383	+	0.999919i	$0.504055\pi$
$440$	0	0
$441$	−5.94901	−0.283286
$442$	0	0
$443$	−33.0425	−1.56989	−0.784947	−	0.619563i	$-0.787310\pi$
$443$	−33.0425	−1.56989	−0.784947	+	0.619563i	$0.787310\pi$
$444$	0	0
$445$	−42.8205	−2.02989
$446$	0	0
$447$	4.47326	0.211578
$448$	0	0
$449$	−18.1673	−0.857366	−0.428683	−	0.903455i	$-0.641022\pi$
$449$	−18.1673	−0.857366	−0.428683	+	0.903455i	$0.641022\pi$
$450$	0	0
$451$	−5.36301	−0.252534
$452$	0	0
$453$	18.6111	0.874426
$454$	0	0
$455$	7.43068	0.348356
$456$	0	0
$457$	23.1718	1.08393	0.541965	−	0.840401i	$-0.317680\pi$
$457$	23.1718	1.08393	0.541965	+	0.840401i	$0.317680\pi$
$458$	0	0
$459$	0.377785	0.0176335
$460$	0	0
$461$	−10.8369	−0.504725	−0.252363	−	0.967633i	$-0.581208\pi$
$461$	−10.8369	−0.504725	−0.252363	+	0.967633i	$0.581208\pi$
$462$	0	0
$463$	−5.90221	−0.274299	−0.137149	−	0.990550i	$-0.543794\pi$
$463$	−5.90221	−0.274299	−0.137149	+	0.990550i	$0.543794\pi$
$464$	0	0
$465$	16.6208	0.770773
$466$	0	0
$467$	−16.2479	−0.751864	−0.375932	−	0.926647i	$-0.622677\pi$
$467$	−16.2479	−0.751864	−0.375932	+	0.926647i	$0.622677\pi$
$468$	0	0
$469$	−0.965830	−0.0445979
$470$	0	0
$471$	−14.3876	−0.662946
$472$	0	0
$473$	4.11122	0.189034
$474$	0	0
$475$	−2.19655	−0.100785
$476$	0	0
$477$	−8.59305	−0.393449
$478$	0	0
$479$	20.2545	0.925452	0.462726	−	0.886501i	$-0.346871\pi$
$479$	20.2545	0.925452	0.462726	+	0.886501i	$0.346871\pi$
$480$	0	0
$481$	−5.88233	−0.268211
$482$	0	0
$483$	6.33243	0.288136
$484$	0	0
$485$	13.0535	0.592730
$486$	0	0
$487$	−6.32264	−0.286506	−0.143253	−	0.989686i	$-0.545756\pi$
$487$	−6.32264	−0.286506	−0.143253	+	0.989686i	$0.545756\pi$
$488$	0	0
$489$	−4.81245	−0.217627
$490$	0	0
$491$	22.0921	0.997002	0.498501	−	0.866889i	$-0.333884\pi$
$491$	22.0921	0.997002	0.498501	+	0.866889i	$0.333884\pi$
$492$	0	0
$493$	−2.18256	−0.0982975
$494$	0	0
$495$	−8.23780	−0.370261
$496$	0	0
$497$	−1.01381	−0.0454754
$498$	0	0
$499$	−26.8788	−1.20326	−0.601631	−	0.798774i	$-0.705482\pi$
$499$	−26.8788	−1.20326	−0.601631	+	0.798774i	$0.705482\pi$
$500$	0	0
$501$	0.760554	0.0339790
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	22.1568	0.985965
$506$	0	0
$507$	−4.73337	−0.210216
$508$	0	0
$509$	40.8050	1.80865	0.904325	−	0.426845i	$-0.140375\pi$
$509$	40.8050	1.80865	0.904325	+	0.426845i	$0.140375\pi$
$510$	0	0
$511$	13.0717	0.578258
$512$	0	0
$513$	−1.62078	−0.0715594
$514$	0	0
$515$	−41.9071	−1.84665
$516$	0	0
$517$	−39.9570	−1.75731
$518$	0	0
$519$	−10.1595	−0.445951
$520$	0	0
$521$	26.1863	1.14724	0.573621	−	0.819121i	$-0.305538\pi$
$521$	26.1863	1.14724	0.573621	+	0.819121i	$0.305538\pi$
$522$	0	0
$523$	32.3042	1.41256	0.706282	−	0.707930i	$-0.250371\pi$
$523$	32.3042	1.41256	0.706282	+	0.707930i	$0.250371\pi$
$524$	0	0
$525$	−1.38936	−0.0606366
$526$	0	0
$527$	2.49076	0.108499
$528$	0	0
$529$	15.1543	0.658885
$530$	0	0
$531$	−9.35721	−0.406068
$532$	0	0
$533$	−4.71876	−0.204392
$534$	0	0
$535$	4.19572	0.181397
$536$	0	0
$537$	17.1094	0.738326
$538$	0	0
$539$	19.4397	0.837328
$540$	0	0
$541$	−15.2223	−0.654458	−0.327229	−	0.944945i	$-0.606115\pi$
$541$	−15.2223	−0.654458	−0.327229	+	0.944945i	$0.606115\pi$
$542$	0	0
$543$	3.61105	0.154965
$544$	0	0
$545$	−44.7459	−1.91671
$546$	0	0
$547$	−22.6856	−0.969966	−0.484983	−	0.874524i	$-0.661174\pi$
$547$	−22.6856	−0.969966	−0.484983	+	0.874524i	$0.661174\pi$
$548$	0	0
$549$	−1.13492	−0.0484370
$550$	0	0
$551$	9.36368	0.398906
$552$	0	0
$553$	3.91317	0.166405
$554$	0	0
$555$	5.15765	0.218930
$556$	0	0
$557$	1.53786	0.0651614	0.0325807	−	0.999469i	$-0.489627\pi$
$557$	1.53786	0.0651614	0.0325807	+	0.999469i	$0.489627\pi$
$558$	0	0
$559$	3.61735	0.152998
$560$	0	0
$561$	−1.23450	−0.0521204
$562$	0	0
$563$	−36.8937	−1.55488	−0.777442	−	0.628954i	$-0.783483\pi$
$563$	−36.8937	−1.55488	−0.777442	+	0.628954i	$0.783483\pi$
$564$	0	0
$565$	−10.7102	−0.450581
$566$	0	0
$567$	−1.02518	−0.0430534
$568$	0	0
$569$	2.39067	0.100222	0.0501110	−	0.998744i	$-0.484042\pi$
$569$	2.39067	0.100222	0.0501110	+	0.998744i	$0.484042\pi$
$570$	0	0
$571$	13.1095	0.548616	0.274308	−	0.961642i	$-0.411551\pi$
$571$	13.1095	0.548616	0.274308	+	0.961642i	$0.411551\pi$
$572$	0	0
$573$	0.296153	0.0123720
$574$	0	0
$575$	−8.37120	−0.349103
$576$	0	0
$577$	−5.33586	−0.222135	−0.111067	−	0.993813i	$-0.535427\pi$
$577$	−5.33586	−0.222135	−0.111067	+	0.993813i	$0.535427\pi$
$578$	0	0
$579$	14.8084	0.615416
$580$	0	0
$581$	−4.99402	−0.207187
$582$	0	0
$583$	28.0797	1.16294
$584$	0	0
$585$	−7.24820	−0.299676
$586$	0	0
$587$	9.35300	0.386040	0.193020	−	0.981195i	$-0.438172\pi$
$587$	9.35300	0.386040	0.193020	+	0.981195i	$0.438172\pi$
$588$	0	0
$589$	−10.6859	−0.440306
$590$	0	0
$591$	−19.2880	−0.793401
$592$	0	0
$593$	−1.61989	−0.0665208	−0.0332604	−	0.999447i	$-0.510589\pi$
$593$	−1.61989	−0.0665208	−0.0332604	+	0.999447i	$0.510589\pi$
$594$	0	0
$595$	−0.976358	−0.0400268
$596$	0	0
$597$	−15.1845	−0.621461
$598$	0	0
$599$	−6.44660	−0.263401	−0.131700	−	0.991290i	$-0.542044\pi$
$599$	−6.44660	−0.263401	−0.131700	+	0.991290i	$0.542044\pi$
$600$	0	0
$601$	−12.9643	−0.528824	−0.264412	−	0.964410i	$-0.585178\pi$
$601$	−12.9643	−0.528824	−0.264412	+	0.964410i	$0.585178\pi$
$602$	0	0
$603$	0.942111	0.0383657
$604$	0	0
$605$	−0.811729	−0.0330015
$606$	0	0
$607$	−12.4087	−0.503654	−0.251827	−	0.967772i	$-0.581031\pi$
$607$	−12.4087	−0.503654	−0.251827	+	0.967772i	$0.581031\pi$
$608$	0	0
$609$	5.92270	0.240000
$610$	0	0
$611$	−35.1570	−1.42230
$612$	0	0
$613$	1.41520	0.0571593	0.0285796	−	0.999592i	$-0.490902\pi$
$613$	1.41520	0.0571593	0.0285796	+	0.999592i	$0.490902\pi$
$614$	0	0
$615$	4.13742	0.166837
$616$	0	0
$617$	−15.0686	−0.606638	−0.303319	−	0.952889i	$-0.598095\pi$
$617$	−15.0686	−0.606638	−0.303319	+	0.952889i	$0.598095\pi$
$618$	0	0
$619$	−7.87864	−0.316669	−0.158335	−	0.987386i	$-0.550612\pi$
$619$	−7.87864	−0.316669	−0.158335	+	0.987386i	$0.550612\pi$
$620$	0	0
$621$	−6.17692	−0.247871
$622$	0	0
$623$	17.4134	0.697655
$624$	0	0
$625$	−29.9395	−1.19758
$626$	0	0
$627$	5.29627	0.211513
$628$	0	0
$629$	0.772912	0.0308180
$630$	0	0
$631$	−8.63840	−0.343889	−0.171945	−	0.985107i	$-0.555005\pi$
$631$	−8.63840	−0.343889	−0.171945	+	0.985107i	$0.555005\pi$
$632$	0	0
$633$	−22.1254	−0.879405
$634$	0	0
$635$	25.9873	1.03127
$636$	0	0
$637$	17.1045	0.677703
$638$	0	0
$639$	0.988908	0.0391206
$640$	0	0
$641$	33.1040	1.30753	0.653764	−	0.756699i	$-0.273189\pi$
$641$	33.1040	1.30753	0.653764	+	0.756699i	$0.273189\pi$
$642$	0	0
$643$	−22.1720	−0.874377	−0.437188	−	0.899370i	$-0.644026\pi$
$643$	−22.1720	−0.874377	−0.437188	+	0.899370i	$0.644026\pi$
$644$	0	0
$645$	−3.17170	−0.124886
$646$	0	0
$647$	−5.66692	−0.222790	−0.111395	−	0.993776i	$-0.535532\pi$
$647$	−5.66692	−0.222790	−0.111395	+	0.993776i	$0.535532\pi$
$648$	0	0
$649$	30.5767	1.20024
$650$	0	0
$651$	−6.75905	−0.264908
$652$	0	0
$653$	−13.2652	−0.519108	−0.259554	−	0.965729i	$-0.583576\pi$
$653$	−13.2652	−0.519108	−0.259554	+	0.965729i	$0.583576\pi$
$654$	0	0
$655$	22.3941	0.875010
$656$	0	0
$657$	−12.7507	−0.497452
$658$	0	0
$659$	−41.7765	−1.62738	−0.813692	−	0.581297i	$-0.802545\pi$
$659$	−41.7765	−1.62738	−0.813692	+	0.581297i	$0.802545\pi$
$660$	0	0
$661$	−21.9461	−0.853605	−0.426802	−	0.904345i	$-0.640360\pi$
$661$	−21.9461	−0.853605	−0.426802	+	0.904345i	$0.640360\pi$
$662$	0	0
$663$	−1.08620	−0.0421844
$664$	0	0
$665$	4.18880	0.162435
$666$	0	0
$667$	35.6856	1.38175
$668$	0	0
$669$	−18.7967	−0.726721
$670$	0	0
$671$	3.70859	0.143168
$672$	0	0
$673$	−0.398443	−0.0153589	−0.00767943	−	0.999971i	$-0.502444\pi$
$673$	−0.398443	−0.0153589	−0.00767943	+	0.999971i	$0.502444\pi$
$674$	0	0
$675$	1.35524	0.0521632
$676$	0	0
$677$	−27.1248	−1.04249	−0.521246	−	0.853407i	$-0.674533\pi$
$677$	−27.1248	−1.04249	−0.521246	+	0.853407i	$0.674533\pi$
$678$	0	0
$679$	−5.30836	−0.203716
$680$	0	0
$681$	−10.1902	−0.390491
$682$	0	0
$683$	−0.255415	−0.00977319	−0.00488660	−	0.999988i	$-0.501555\pi$
$683$	−0.255415	−0.00977319	−0.00488660	+	0.999988i	$0.501555\pi$
$684$	0	0
$685$	−27.8281	−1.06326
$686$	0	0
$687$	8.80690	0.336004
$688$	0	0
$689$	24.7065	0.941244
$690$	0	0
$691$	11.6261	0.442279	0.221140	−	0.975242i	$-0.429022\pi$
$691$	11.6261	0.442279	0.221140	+	0.975242i	$0.429022\pi$
$692$	0	0
$693$	3.34999	0.127256
$694$	0	0
$695$	25.3086	0.960010
$696$	0	0
$697$	0.620023	0.0234850
$698$	0	0
$699$	−11.6965	−0.442402
$700$	0	0
$701$	12.8197	0.484193	0.242097	−	0.970252i	$-0.422165\pi$
$701$	12.8197	0.484193	0.242097	+	0.970252i	$0.422165\pi$
$702$	0	0
$703$	−3.31597	−0.125064
$704$	0	0
$705$	30.8258	1.16097
$706$	0	0
$707$	−9.01032	−0.338868
$708$	0	0
$709$	32.8398	1.23333	0.616663	−	0.787227i	$-0.288484\pi$
$709$	32.8398	1.23333	0.616663	+	0.787227i	$0.288484\pi$
$710$	0	0
$711$	−3.81707	−0.143151
$712$	0	0
$713$	−40.7248	−1.52516
$714$	0	0
$715$	23.6851	0.885773
$716$	0	0
$717$	−23.7646	−0.887507
$718$	0	0
$719$	−20.3717	−0.759735	−0.379867	−	0.925041i	$-0.624030\pi$
$719$	−20.3717	−0.759735	−0.379867	+	0.925041i	$0.624030\pi$
$720$	0	0
$721$	17.0420	0.634677
$722$	0	0
$723$	4.06074	0.151021
$724$	0	0
$725$	−7.82956	−0.290782
$726$	0	0
$727$	−24.8350	−0.921078	−0.460539	−	0.887640i	$-0.652344\pi$
$727$	−24.8350	−0.921078	−0.460539	+	0.887640i	$0.652344\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.475303	−0.0175797
$732$	0	0
$733$	23.4421	0.865853	0.432927	−	0.901429i	$-0.357481\pi$
$733$	23.4421	0.865853	0.432927	+	0.901429i	$0.357481\pi$
$734$	0	0
$735$	−14.9972	−0.553181
$736$	0	0
$737$	−3.07856	−0.113400
$738$	0	0
$739$	−13.3235	−0.490114	−0.245057	−	0.969509i	$-0.578807\pi$
$739$	−13.3235	−0.490114	−0.245057	+	0.969509i	$0.578807\pi$
$740$	0	0
$741$	4.66004	0.171191
$742$	0	0
$743$	−34.6296	−1.27044	−0.635218	−	0.772333i	$-0.719090\pi$
$743$	−34.6296	−1.27044	−0.635218	+	0.772333i	$0.719090\pi$
$744$	0	0
$745$	11.2769	0.413154
$746$	0	0
$747$	4.87138	0.178234
$748$	0	0
$749$	−1.70624	−0.0623445
$750$	0	0
$751$	20.2014	0.737158	0.368579	−	0.929596i	$-0.379844\pi$
$751$	20.2014	0.737158	0.368579	+	0.929596i	$0.379844\pi$
$752$	0	0
$753$	29.8288	1.08702
$754$	0	0
$755$	46.9179	1.70752
$756$	0	0
$757$	35.8622	1.30343	0.651716	−	0.758463i	$-0.274049\pi$
$757$	35.8622	1.30343	0.651716	+	0.758463i	$0.274049\pi$
$758$	0	0
$759$	20.1845	0.732650
$760$	0	0
$761$	14.8328	0.537690	0.268845	−	0.963184i	$-0.413358\pi$
$761$	14.8328	0.537690	0.268845	+	0.963184i	$0.413358\pi$
$762$	0	0
$763$	18.1964	0.658755
$764$	0	0
$765$	0.952380	0.0344334
$766$	0	0
$767$	26.9036	0.971433
$768$	0	0
$769$	1.01654	0.0366574	0.0183287	−	0.999832i	$-0.494165\pi$
$769$	1.01654	0.0366574	0.0183287	+	0.999832i	$0.494165\pi$
$770$	0	0
$771$	−0.936638	−0.0337322
$772$	0	0
$773$	−17.5917	−0.632731	−0.316365	−	0.948637i	$-0.602463\pi$
$773$	−17.5917	−0.632731	−0.316365	+	0.948637i	$0.602463\pi$
$774$	0	0
$775$	8.93517	0.320961
$776$	0	0
$777$	−2.09741	−0.0752443
$778$	0	0
$779$	−2.66004	−0.0953059
$780$	0	0
$781$	−3.23148	−0.115631
$782$	0	0
$783$	−5.77725	−0.206462
$784$	0	0
$785$	−36.2706	−1.29455
$786$	0	0
$787$	49.1247	1.75111	0.875553	−	0.483121i	$-0.160497\pi$
$787$	49.1247	1.75111	0.875553	+	0.483121i	$0.160497\pi$
$788$	0	0
$789$	−3.88139	−0.138181
$790$	0	0
$791$	4.35541	0.154861
$792$	0	0
$793$	3.26308	0.115875
$794$	0	0
$795$	−21.6627	−0.768298
$796$	0	0
$797$	30.3445	1.07486	0.537429	−	0.843309i	$-0.319396\pi$
$797$	30.3445	1.07486	0.537429	+	0.843309i	$0.319396\pi$
$798$	0	0
$799$	4.61947	0.163425
$800$	0	0
$801$	−16.9858	−0.600164
$802$	0	0
$803$	41.6657	1.47035
$804$	0	0
$805$	15.9638	0.562650
$806$	0	0
$807$	32.5695	1.14650
$808$	0	0
$809$	15.0475	0.529040	0.264520	−	0.964380i	$-0.414786\pi$
$809$	15.0475	0.529040	0.264520	+	0.964380i	$0.414786\pi$
$810$	0	0
$811$	−31.4681	−1.10499	−0.552497	−	0.833515i	$-0.686325\pi$
$811$	−31.4681	−1.10499	−0.552497	+	0.833515i	$0.686325\pi$
$812$	0	0
$813$	23.0539	0.808534
$814$	0	0
$815$	−12.1320	−0.424965
$816$	0	0
$817$	2.03916	0.0713412
$818$	0	0
$819$	2.94756	0.102996
$820$	0	0
$821$	−42.1869	−1.47233	−0.736167	−	0.676800i	$-0.763366\pi$
$821$	−42.1869	−1.47233	−0.736167	+	0.676800i	$0.763366\pi$
$822$	0	0
$823$	−0.0254204	−0.000886100 0	−0.000443050	−	1.00000i	$-0.500141\pi$
$823$	−0.0254204	−0.000886100 0	−0.000443050		1.00000i	$0.500141\pi$
$824$	0	0
$825$	−4.42854	−0.154182
$826$	0	0
$827$	−23.6664	−0.822961	−0.411480	−	0.911419i	$-0.634988\pi$
$827$	−23.6664	−0.822961	−0.411480	+	0.911419i	$0.634988\pi$
$828$	0	0
$829$	45.0883	1.56598	0.782990	−	0.622034i	$-0.213694\pi$
$829$	45.0883	1.56598	0.782990	+	0.622034i	$0.213694\pi$
$830$	0	0
$831$	24.4849	0.849371
$832$	0	0
$833$	−2.24745	−0.0778694
$834$	0	0
$835$	1.91733	0.0663518
$836$	0	0
$837$	6.59306	0.227889
$838$	0	0
$839$	−11.7537	−0.405782	−0.202891	−	0.979201i	$-0.565034\pi$
$839$	−11.7537	−0.405782	−0.202891	+	0.979201i	$0.565034\pi$
$840$	0	0
$841$	4.37664	0.150919
$842$	0	0
$843$	11.0539	0.380715
$844$	0	0
$845$	−11.9326	−0.410495
$846$	0	0
$847$	0.330099	0.0113423
$848$	0	0
$849$	17.3703	0.596147
$850$	0	0
$851$	−12.6374	−0.433204
$852$	0	0
$853$	17.4368	0.597026	0.298513	−	0.954406i	$-0.403509\pi$
$853$	17.4368	0.597026	0.298513	+	0.954406i	$0.403509\pi$
$854$	0	0
$855$	−4.08593	−0.139736
$856$	0	0
$857$	−9.57857	−0.327198	−0.163599	−	0.986527i	$-0.552310\pi$
$857$	−9.57857	−0.327198	−0.163599	+	0.986527i	$0.552310\pi$
$858$	0	0
$859$	−12.8356	−0.437944	−0.218972	−	0.975731i	$-0.570270\pi$
$859$	−12.8356	−0.437944	−0.218972	+	0.975731i	$0.570270\pi$
$860$	0	0
$861$	−1.68253	−0.0573403
$862$	0	0
$863$	−11.5875	−0.394443	−0.197221	−	0.980359i	$-0.563192\pi$
$863$	−11.5875	−0.394443	−0.197221	+	0.980359i	$0.563192\pi$
$864$	0	0
$865$	−25.6116	−0.870821
$866$	0	0
$867$	−16.8573	−0.572503
$868$	0	0
$869$	12.4731	0.423122
$870$	0	0
$871$	−2.70873	−0.0917819
$872$	0	0
$873$	5.17800	0.175249
$874$	0	0
$875$	9.41963	0.318441
$876$	0	0
$877$	−3.17868	−0.107336	−0.0536682	−	0.998559i	$-0.517091\pi$
$877$	−3.17868	−0.107336	−0.0536682	+	0.998559i	$0.517091\pi$
$878$	0	0
$879$	2.92081	0.0985165
$880$	0	0
$881$	33.9054	1.14230	0.571150	−	0.820845i	$-0.306497\pi$
$881$	33.9054	1.14230	0.571150	+	0.820845i	$0.306497\pi$
$882$	0	0
$883$	30.7707	1.03552	0.517758	−	0.855527i	$-0.326767\pi$
$883$	30.7707	1.03552	0.517758	+	0.855527i	$0.326767\pi$
$884$	0	0
$885$	−23.5891	−0.792940
$886$	0	0
$887$	23.4377	0.786962	0.393481	−	0.919333i	$-0.371271\pi$
$887$	23.4377	0.786962	0.393481	+	0.919333i	$0.371271\pi$
$888$	0	0
$889$	−10.5680	−0.354440
$890$	0	0
$891$	−3.26772	−0.109473
$892$	0	0
$893$	−19.8186	−0.663204
$894$	0	0
$895$	43.1322	1.44175
$896$	0	0
$897$	17.7597	0.592980
$898$	0	0
$899$	−38.0898	−1.27036
$900$	0	0
$901$	−3.24632	−0.108151
$902$	0	0
$903$	1.28981	0.0429221
$904$	0	0
$905$	9.10330	0.302604
$906$	0	0
$907$	23.7964	0.790148	0.395074	−	0.918649i	$-0.370719\pi$
$907$	23.7964	0.790148	0.395074	+	0.918649i	$0.370719\pi$
$908$	0	0
$909$	8.78904	0.291514
$910$	0	0
$911$	−39.6175	−1.31259	−0.656293	−	0.754506i	$-0.727877\pi$
$911$	−39.6175	−1.31259	−0.656293	+	0.754506i	$0.727877\pi$
$912$	0	0
$913$	−15.9183	−0.526819
$914$	0	0
$915$	−2.86108	−0.0945843
$916$	0	0
$917$	−9.10681	−0.300734
$918$	0	0
$919$	35.2164	1.16168	0.580840	−	0.814018i	$-0.302724\pi$
$919$	35.2164	1.16168	0.580840	+	0.814018i	$0.302724\pi$
$920$	0	0
$921$	23.5812	0.777027
$922$	0	0
$923$	−2.84328	−0.0935878
$924$	0	0
$925$	2.77269	0.0911655
$926$	0	0
$927$	−16.6235	−0.545987
$928$	0	0
$929$	−24.0219	−0.788134	−0.394067	−	0.919082i	$-0.628932\pi$
$929$	−24.0219	−0.788134	−0.394067	+	0.919082i	$0.628932\pi$
$930$	0	0
$931$	9.64206	0.316006
$932$	0	0
$933$	−15.0568	−0.492938
$934$	0	0
$935$	−3.11211	−0.101777
$936$	0	0
$937$	28.3384	0.925775	0.462888	−	0.886417i	$-0.346813\pi$
$937$	28.3384	0.925775	0.462888	+	0.886417i	$0.346813\pi$
$938$	0	0
$939$	17.4291	0.568778
$940$	0	0
$941$	−31.7155	−1.03390	−0.516948	−	0.856017i	$-0.672932\pi$
$941$	−31.7155	−1.03390	−0.516948	+	0.856017i	$0.672932\pi$
$942$	0	0
$943$	−10.1376	−0.330126
$944$	0	0
$945$	−2.58443	−0.0840715
$946$	0	0
$947$	−41.0291	−1.33327	−0.666634	−	0.745385i	$-0.732266\pi$
$947$	−41.0291	−1.33327	−0.666634	+	0.745385i	$0.732266\pi$
$948$	0	0
$949$	36.6605	1.19005
$950$	0	0
$951$	−6.21009	−0.201376
$952$	0	0
$953$	42.8897	1.38933	0.694667	−	0.719332i	$-0.255552\pi$
$953$	42.8897	1.38933	0.694667	+	0.719332i	$0.255552\pi$
$954$	0	0
$955$	0.746590	0.0241591
$956$	0	0
$957$	18.8785	0.610254
$958$	0	0
$959$	11.3166	0.365432
$960$	0	0
$961$	12.4684	0.402207
$962$	0	0
$963$	1.66433	0.0536324
$964$	0	0
$965$	37.3314	1.20174
$966$	0	0
$967$	−45.9167	−1.47658	−0.738291	−	0.674482i	$-0.764367\pi$
$967$	−45.9167	−1.47658	−0.738291	+	0.674482i	$0.764367\pi$
$968$	0	0
$969$	−0.612307	−0.0196701
$970$	0	0
$971$	−37.6867	−1.20942	−0.604712	−	0.796444i	$-0.706712\pi$
$971$	−37.6867	−1.20942	−0.604712	+	0.796444i	$0.706712\pi$
$972$	0	0
$973$	−10.2920	−0.329947
$974$	0	0
$975$	−3.89655	−0.124789
$976$	0	0
$977$	−5.87111	−0.187834	−0.0939168	−	0.995580i	$-0.529939\pi$
$977$	−5.87111	−0.187834	−0.0939168	+	0.995580i	$0.529939\pi$
$978$	0	0
$979$	55.5049	1.77394
$980$	0	0
$981$	−17.7496	−0.566700
$982$	0	0
$983$	−33.7567	−1.07667	−0.538336	−	0.842731i	$-0.680947\pi$
$983$	−33.7567	−1.07667	−0.538336	+	0.842731i	$0.680947\pi$
$984$	0	0
$985$	−48.6242	−1.54930
$986$	0	0
$987$	−12.5356	−0.399014
$988$	0	0
$989$	7.77138	0.247115
$990$	0	0
$991$	43.4635	1.38066	0.690332	−	0.723493i	$-0.257464\pi$
$991$	43.4635	1.38066	0.690332	+	0.723493i	$0.257464\pi$
$992$	0	0
$993$	−16.0705	−0.509982
$994$	0	0
$995$	−38.2796	−1.21354
$996$	0	0
$997$	11.9207	0.377532	0.188766	−	0.982022i	$-0.439551\pi$
$997$	11.9207	0.377532	0.188766	+	0.982022i	$0.439551\pi$
$998$	0	0
$999$	2.04591	0.0647296

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.g.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.g.1.14	✓	15	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.52096 q^{5} -1.02518 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.52096 q^{5} -1.02518 q^{7} +1.00000 q^{9} -3.26772 q^{11} -2.87517 q^{13} +2.52096 q^{15} +0.377785 q^{17} -1.62078 q^{19} -1.02518 q^{21} -6.17692 q^{23} +1.35524 q^{25} +1.00000 q^{27} -5.77725 q^{29} +6.59306 q^{31} -3.26772 q^{33} -2.58443 q^{35} +2.04591 q^{37} -2.87517 q^{39} +1.64121 q^{41} -1.25813 q^{43} +2.52096 q^{45} +12.2278 q^{47} -5.94901 q^{49} +0.377785 q^{51} -8.59305 q^{53} -8.23780 q^{55} -1.62078 q^{57} -9.35721 q^{59} -1.13492 q^{61} -1.02518 q^{63} -7.24820 q^{65} +0.942111 q^{67} -6.17692 q^{69} +0.988908 q^{71} -12.7507 q^{73} +1.35524 q^{75} +3.34999 q^{77} -3.81707 q^{79} +1.00000 q^{81} +4.87138 q^{83} +0.952380 q^{85} -5.77725 q^{87} -16.9858 q^{89} +2.94756 q^{91} +6.59306 q^{93} -4.08593 q^{95} +5.17800 q^{97} -3.26772 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9	\( 15 q + 15 q^{3} - 11 q^{5} - 4 q^{7} + 15 q^{9} - 5 q^{11} - 14 q^{13} - 11 q^{15} - 5 q^{17} + 3 q^{19} - 4 q^{21} - 32 q^{23} + 2 q^{25} + 15 q^{27} - 23 q^{29} - 13 q^{31} - 5 q^{33} - 16 q^{35} - 10 q^{37} - 14 q^{39} - 14 q^{41} + 4 q^{43} - 11 q^{45} - 20 q^{47} - 9 q^{49} - 5 q^{51} - 30 q^{53} - 10 q^{55} + 3 q^{57} - 14 q^{59} - 38 q^{61} - 4 q^{63} - 24 q^{65} - 8 q^{67} - 32 q^{69} - 41 q^{71} - 19 q^{73} + 2 q^{75} - 39 q^{77} - 27 q^{79} + 15 q^{81} - 17 q^{83} - 6 q^{85} - 23 q^{87} - 23 q^{89} + 4 q^{91} - 13 q^{93} - 30 q^{95} - 18 q^{97} - 5 q^{99}+O(q^{100}) \) 15 * q + 15 * q^3 - 11 * q^5 - 4 * q^7 + 15 * q^9 - 5 * q^11 - 14 * q^13 - 11 * q^15 - 5 * q^17 + 3 * q^19 - 4 * q^21 - 32 * q^23 + 2 * q^25 + 15 * q^27 - 23 * q^29 - 13 * q^31 - 5 * q^33 - 16 * q^35 - 10 * q^37 - 14 * q^39 - 14 * q^41 + 4 * q^43 - 11 * q^45 - 20 * q^47 - 9 * q^49 - 5 * q^51 - 30 * q^53 - 10 * q^55 + 3 * q^57 - 14 * q^59 - 38 * q^61 - 4 * q^63 - 24 * q^65 - 8 * q^67 - 32 * q^69 - 41 * q^71 - 19 * q^73 + 2 * q^75 - 39 * q^77 - 27 * q^79 + 15 * q^81 - 17 * q^83 - 6 * q^85 - 23 * q^87 - 23 * q^89 + 4 * q^91 - 13 * q^93 - 30 * q^95 - 18 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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