LMFDB - Embedded newform 6004.2.a.f.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.681472 q^{3} -3.07187 q^{5} -2.21919 q^{7} -2.53560 q^{9} +O(q^{10})$	$q+0.681472 q^{3} -3.07187 q^{5} -2.21919 q^{7} -2.53560 q^{9} +1.07728 q^{11} -1.31110 q^{13} -2.09339 q^{15} +7.22095 q^{17} -1.00000 q^{19} -1.51232 q^{21} +7.23144 q^{23} +4.43640 q^{25} -3.77235 q^{27} +10.0811 q^{29} -2.68021 q^{31} +0.734139 q^{33} +6.81708 q^{35} -7.13733 q^{37} -0.893478 q^{39} -1.76461 q^{41} -1.78442 q^{43} +7.78903 q^{45} +11.8703 q^{47} -2.07518 q^{49} +4.92087 q^{51} -6.43319 q^{53} -3.30928 q^{55} -0.681472 q^{57} -0.490279 q^{59} +2.00768 q^{61} +5.62698 q^{63} +4.02753 q^{65} +8.83542 q^{67} +4.92802 q^{69} -14.5953 q^{71} -6.54212 q^{73} +3.02328 q^{75} -2.39070 q^{77} +1.00000 q^{79} +5.03604 q^{81} -6.63031 q^{83} -22.1818 q^{85} +6.86999 q^{87} +0.754843 q^{89} +2.90959 q^{91} -1.82649 q^{93} +3.07187 q^{95} +7.63014 q^{97} -2.73156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	$ 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) $ 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.681472	0.393448	0.196724	−	0.980459i	$-0.436970\pi$
$3$	0.681472	0.393448	0.196724	+	0.980459i	$0.436970\pi$
$4$	0	0
$5$	−3.07187	−1.37378	−0.686891	−	0.726760i	$-0.741025\pi$
$5$	−3.07187	−1.37378	−0.686891	+	0.726760i	$0.741025\pi$
$6$	0	0
$7$	−2.21919	−0.838776	−0.419388	−	0.907807i	$-0.637755\pi$
$7$	−2.21919	−0.838776	−0.419388	+	0.907807i	$0.637755\pi$
$8$	0	0
$9$	−2.53560	−0.845199
$10$	0	0
$11$	1.07728	0.324814	0.162407	−	0.986724i	$-0.448074\pi$
$11$	1.07728	0.324814	0.162407	+	0.986724i	$0.448074\pi$
$12$	0	0
$13$	−1.31110	−0.363634	−0.181817	−	0.983332i	$-0.558198\pi$
$13$	−1.31110	−0.363634	−0.181817	+	0.983332i	$0.558198\pi$
$14$	0	0
$15$	−2.09339	−0.540512
$16$	0	0
$17$	7.22095	1.75134	0.875669	−	0.482912i	$-0.160421\pi$
$17$	7.22095	1.75134	0.875669	+	0.482912i	$0.160421\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.51232	−0.330015
$22$	0	0
$23$	7.23144	1.50786	0.753930	−	0.656955i	$-0.228156\pi$
$23$	7.23144	1.50786	0.753930	+	0.656955i	$0.228156\pi$
$24$	0	0
$25$	4.43640	0.887279
$26$	0	0
$27$	−3.77235	−0.725990
$28$	0	0
$29$	10.0811	1.87201	0.936007	−	0.351981i	$-0.114492\pi$
$29$	10.0811	1.87201	0.936007	+	0.351981i	$0.114492\pi$
$30$	0	0
$31$	−2.68021	−0.481380	−0.240690	−	0.970602i	$-0.577374\pi$
$31$	−2.68021	−0.481380	−0.240690	+	0.970602i	$0.577374\pi$
$32$	0	0
$33$	0.734139	0.127797
$34$	0	0
$35$	6.81708	1.15230
$36$	0	0
$37$	−7.13733	−1.17337	−0.586685	−	0.809815i	$-0.699567\pi$
$37$	−7.13733	−1.17337	−0.586685	+	0.809815i	$0.699567\pi$
$38$	0	0
$39$	−0.893478	−0.143071
$40$	0	0
$41$	−1.76461	−0.275585	−0.137793	−	0.990461i	$-0.544001\pi$
$41$	−1.76461	−0.275585	−0.137793	+	0.990461i	$0.544001\pi$
$42$	0	0
$43$	−1.78442	−0.272122	−0.136061	−	0.990700i	$-0.543444\pi$
$43$	−1.78442	−0.272122	−0.136061	+	0.990700i	$0.543444\pi$
$44$	0	0
$45$	7.78903	1.16112
$46$	0	0
$47$	11.8703	1.73146	0.865728	−	0.500514i	$-0.166856\pi$
$47$	11.8703	1.73146	0.865728	+	0.500514i	$0.166856\pi$
$48$	0	0
$49$	−2.07518	−0.296455
$50$	0	0
$51$	4.92087	0.689060
$52$	0	0
$53$	−6.43319	−0.883667	−0.441833	−	0.897097i	$-0.645672\pi$
$53$	−6.43319	−0.883667	−0.441833	+	0.897097i	$0.645672\pi$
$54$	0	0
$55$	−3.30928	−0.446223
$56$	0	0
$57$	−0.681472	−0.0902631
$58$	0	0
$59$	−0.490279	−0.0638289	−0.0319144	−	0.999491i	$-0.510160\pi$
$59$	−0.490279	−0.0638289	−0.0319144	+	0.999491i	$0.510160\pi$
$60$	0	0
$61$	2.00768	0.257057	0.128529	−	0.991706i	$-0.458975\pi$
$61$	2.00768	0.257057	0.128529	+	0.991706i	$0.458975\pi$
$62$	0	0
$63$	5.62698	0.708933
$64$	0	0
$65$	4.02753	0.499554
$66$	0	0
$67$	8.83542	1.07942	0.539709	−	0.841851i	$-0.318534\pi$
$67$	8.83542	1.07942	0.539709	+	0.841851i	$0.318534\pi$
$68$	0	0
$69$	4.92802	0.593264
$70$	0	0
$71$	−14.5953	−1.73214	−0.866069	−	0.499924i	$-0.833361\pi$
$71$	−14.5953	−1.73214	−0.866069	+	0.499924i	$0.833361\pi$
$72$	0	0
$73$	−6.54212	−0.765698	−0.382849	−	0.923811i	$-0.625057\pi$
$73$	−6.54212	−0.765698	−0.382849	+	0.923811i	$0.625057\pi$
$74$	0	0
$75$	3.02328	0.349098
$76$	0	0
$77$	−2.39070	−0.272446
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	5.03604	0.559560
$82$	0	0
$83$	−6.63031	−0.727771	−0.363886	−	0.931444i	$-0.618550\pi$
$83$	−6.63031	−0.727771	−0.363886	+	0.931444i	$0.618550\pi$
$84$	0	0
$85$	−22.1818	−2.40596
$86$	0	0
$87$	6.86999	0.736540
$88$	0	0
$89$	0.754843	0.0800132	0.0400066	−	0.999199i	$-0.487262\pi$
$89$	0.754843	0.0800132	0.0400066	+	0.999199i	$0.487262\pi$
$90$	0	0
$91$	2.90959	0.305007
$92$	0	0
$93$	−1.82649	−0.189398
$94$	0	0
$95$	3.07187	0.315167
$96$	0	0
$97$	7.63014	0.774723	0.387362	−	0.921928i	$-0.373386\pi$
$97$	7.63014	0.774723	0.387362	+	0.921928i	$0.373386\pi$
$98$	0	0
$99$	−2.73156	−0.274532
$100$	0	0
$101$	−12.4556	−1.23938	−0.619689	−	0.784847i	$-0.712741\pi$
$101$	−12.4556	−1.23938	−0.619689	+	0.784847i	$0.712741\pi$
$102$	0	0
$103$	−18.3392	−1.80702	−0.903508	−	0.428570i	$-0.859017\pi$
$103$	−18.3392	−1.80702	−0.903508	+	0.428570i	$0.859017\pi$
$104$	0	0
$105$	4.64565	0.453369
$106$	0	0
$107$	7.22157	0.698135	0.349068	−	0.937098i	$-0.386498\pi$
$107$	7.22157	0.698135	0.349068	+	0.937098i	$0.386498\pi$
$108$	0	0
$109$	15.7312	1.50677	0.753386	−	0.657578i	$-0.228419\pi$
$109$	15.7312	1.50677	0.753386	+	0.657578i	$0.228419\pi$
$110$	0	0
$111$	−4.86389	−0.461660
$112$	0	0
$113$	4.50819	0.424095	0.212047	−	0.977259i	$-0.431987\pi$
$113$	4.50819	0.424095	0.212047	+	0.977259i	$0.431987\pi$
$114$	0	0
$115$	−22.2141	−2.07147
$116$	0	0
$117$	3.32442	0.307343
$118$	0	0
$119$	−16.0247	−1.46898
$120$	0	0
$121$	−9.83946	−0.894496
$122$	0	0
$123$	−1.20253	−0.108429
$124$	0	0
$125$	1.73132	0.154854
$126$	0	0
$127$	−19.0779	−1.69289	−0.846443	−	0.532479i	$-0.821261\pi$
$127$	−19.0779	−1.69289	−0.846443	+	0.532479i	$0.821261\pi$
$128$	0	0
$129$	−1.21603	−0.107066
$130$	0	0
$131$	15.9419	1.39285	0.696426	−	0.717629i	$-0.254772\pi$
$131$	15.9419	1.39285	0.696426	+	0.717629i	$0.254772\pi$
$132$	0	0
$133$	2.21919	0.192428
$134$	0	0
$135$	11.5882	0.997352
$136$	0	0
$137$	−17.9259	−1.53152	−0.765758	−	0.643129i	$-0.777636\pi$
$137$	−17.9259	−1.53152	−0.765758	+	0.643129i	$0.777636\pi$
$138$	0	0
$139$	7.33358	0.622026	0.311013	−	0.950406i	$-0.399332\pi$
$139$	7.33358	0.622026	0.311013	+	0.950406i	$0.399332\pi$
$140$	0	0
$141$	8.08925	0.681238
$142$	0	0
$143$	−1.41243	−0.118113
$144$	0	0
$145$	−30.9679	−2.57174
$146$	0	0
$147$	−1.41418	−0.116639
$148$	0	0
$149$	−7.63258	−0.625285	−0.312643	−	0.949871i	$-0.601214\pi$
$149$	−7.63258	−0.625285	−0.312643	+	0.949871i	$0.601214\pi$
$150$	0	0
$151$	0.895624	0.0728849	0.0364424	−	0.999336i	$-0.488397\pi$
$151$	0.895624	0.0728849	0.0364424	+	0.999336i	$0.488397\pi$
$152$	0	0
$153$	−18.3094	−1.48023
$154$	0	0
$155$	8.23327	0.661312
$156$	0	0
$157$	−16.6868	−1.33175	−0.665876	−	0.746063i	$-0.731942\pi$
$157$	−16.6868	−1.33175	−0.665876	+	0.746063i	$0.731942\pi$
$158$	0	0
$159$	−4.38404	−0.347677
$160$	0	0
$161$	−16.0480	−1.26476
$162$	0	0
$163$	−20.5770	−1.61172	−0.805859	−	0.592107i	$-0.798296\pi$
$163$	−20.5770	−1.61172	−0.805859	+	0.592107i	$0.798296\pi$
$164$	0	0
$165$	−2.25518	−0.175566
$166$	0	0
$167$	−13.6589	−1.05695	−0.528477	−	0.848948i	$-0.677237\pi$
$167$	−13.6589	−1.05695	−0.528477	+	0.848948i	$0.677237\pi$
$168$	0	0
$169$	−11.2810	−0.867770
$170$	0	0
$171$	2.53560	0.193902
$172$	0	0
$173$	11.4417	0.869893	0.434947	−	0.900456i	$-0.356767\pi$
$173$	11.4417	0.869893	0.434947	+	0.900456i	$0.356767\pi$
$174$	0	0
$175$	−9.84522	−0.744229
$176$	0	0
$177$	−0.334111	−0.0251133
$178$	0	0
$179$	−12.3369	−0.922101	−0.461050	−	0.887374i	$-0.652527\pi$
$179$	−12.3369	−0.922101	−0.461050	+	0.887374i	$0.652527\pi$
$180$	0	0
$181$	6.78159	0.504072	0.252036	−	0.967718i	$-0.418900\pi$
$181$	6.78159	0.504072	0.252036	+	0.967718i	$0.418900\pi$
$182$	0	0
$183$	1.36818	0.101139
$184$	0	0
$185$	21.9250	1.61196
$186$	0	0
$187$	7.77902	0.568858
$188$	0	0
$189$	8.37158	0.608943
$190$	0	0
$191$	−9.52603	−0.689279	−0.344640	−	0.938735i	$-0.611999\pi$
$191$	−9.52603	−0.689279	−0.344640	+	0.938735i	$0.611999\pi$
$192$	0	0
$193$	17.2751	1.24349	0.621745	−	0.783220i	$-0.286424\pi$
$193$	17.2751	1.24349	0.621745	+	0.783220i	$0.286424\pi$
$194$	0	0
$195$	2.74465	0.196549
$196$	0	0
$197$	5.81388	0.414222	0.207111	−	0.978317i	$-0.433594\pi$
$197$	5.81388	0.414222	0.207111	+	0.978317i	$0.433594\pi$
$198$	0	0
$199$	−6.15157	−0.436073	−0.218036	−	0.975941i	$-0.569965\pi$
$199$	−6.15157	−0.436073	−0.218036	+	0.975941i	$0.569965\pi$
$200$	0	0
$201$	6.02109	0.424695
$202$	0	0
$203$	−22.3719	−1.57020
$204$	0	0
$205$	5.42065	0.378595
$206$	0	0
$207$	−18.3360	−1.27444
$208$	0	0
$209$	−1.07728	−0.0745174
$210$	0	0
$211$	3.32159	0.228668	0.114334	−	0.993442i	$-0.463527\pi$
$211$	3.32159	0.228668	0.114334	+	0.993442i	$0.463527\pi$
$212$	0	0
$213$	−9.94626	−0.681506
$214$	0	0
$215$	5.48152	0.373836
$216$	0	0
$217$	5.94791	0.403770
$218$	0	0
$219$	−4.45827	−0.301262
$220$	0	0
$221$	−9.46739	−0.636846
$222$	0	0
$223$	−17.0686	−1.14300	−0.571500	−	0.820602i	$-0.693638\pi$
$223$	−17.0686	−1.14300	−0.571500	+	0.820602i	$0.693638\pi$
$224$	0	0
$225$	−11.2489	−0.749927
$226$	0	0
$227$	6.01627	0.399314	0.199657	−	0.979866i	$-0.436017\pi$
$227$	6.01627	0.399314	0.199657	+	0.979866i	$0.436017\pi$
$228$	0	0
$229$	−14.6899	−0.970738	−0.485369	−	0.874309i	$-0.661315\pi$
$229$	−14.6899	−0.970738	−0.485369	+	0.874309i	$0.661315\pi$
$230$	0	0
$231$	−1.62920	−0.107193
$232$	0	0
$233$	18.0729	1.18400	0.591998	−	0.805939i	$-0.298339\pi$
$233$	18.0729	1.18400	0.591998	+	0.805939i	$0.298339\pi$
$234$	0	0
$235$	−36.4639	−2.37865
$236$	0	0
$237$	0.681472	0.0442663
$238$	0	0
$239$	−17.8962	−1.15761	−0.578805	−	0.815466i	$-0.696481\pi$
$239$	−17.8962	−1.15761	−0.578805	+	0.815466i	$0.696481\pi$
$240$	0	0
$241$	25.4944	1.64224	0.821119	−	0.570757i	$-0.193350\pi$
$241$	25.4944	1.64224	0.821119	+	0.570757i	$0.193350\pi$
$242$	0	0
$243$	14.7490	0.946147
$244$	0	0
$245$	6.37469	0.407264
$246$	0	0
$247$	1.31110	0.0834234
$248$	0	0
$249$	−4.51837	−0.286340
$250$	0	0
$251$	−5.49109	−0.346594	−0.173297	−	0.984870i	$-0.555442\pi$
$251$	−5.49109	−0.346594	−0.173297	+	0.984870i	$0.555442\pi$
$252$	0	0
$253$	7.79032	0.489773
$254$	0	0
$255$	−15.1163	−0.946619
$256$	0	0
$257$	3.16702	0.197553	0.0987767	−	0.995110i	$-0.468507\pi$
$257$	3.16702	0.197553	0.0987767	+	0.995110i	$0.468507\pi$
$258$	0	0
$259$	15.8391	0.984195
$260$	0	0
$261$	−25.5616	−1.58222
$262$	0	0
$263$	−0.576466	−0.0355464	−0.0177732	−	0.999842i	$-0.505658\pi$
$263$	−0.576466	−0.0355464	−0.0177732	+	0.999842i	$0.505658\pi$
$264$	0	0
$265$	19.7619	1.21397
$266$	0	0
$267$	0.514404	0.0314810
$268$	0	0
$269$	−6.16811	−0.376076	−0.188038	−	0.982162i	$-0.560213\pi$
$269$	−6.16811	−0.376076	−0.188038	+	0.982162i	$0.560213\pi$
$270$	0	0
$271$	−0.783772	−0.0476107	−0.0238054	−	0.999717i	$-0.507578\pi$
$271$	−0.783772	−0.0476107	−0.0238054	+	0.999717i	$0.507578\pi$
$272$	0	0
$273$	1.98280	0.120005
$274$	0	0
$275$	4.77926	0.288200
$276$	0	0
$277$	12.6380	0.759341	0.379671	−	0.925122i	$-0.376037\pi$
$277$	12.6380	0.759341	0.379671	+	0.925122i	$0.376037\pi$
$278$	0	0
$279$	6.79593	0.406862
$280$	0	0
$281$	−21.9153	−1.30736	−0.653678	−	0.756773i	$-0.726775\pi$
$281$	−21.9153	−1.30736	−0.653678	+	0.756773i	$0.726775\pi$
$282$	0	0
$283$	4.01520	0.238679	0.119339	−	0.992854i	$-0.461922\pi$
$283$	4.01520	0.238679	0.119339	+	0.992854i	$0.461922\pi$
$284$	0	0
$285$	2.09339	0.124002
$286$	0	0
$287$	3.91601	0.231154
$288$	0	0
$289$	35.1421	2.06718
$290$	0	0
$291$	5.19973	0.304813
$292$	0	0
$293$	−14.5013	−0.847175	−0.423587	−	0.905855i	$-0.639229\pi$
$293$	−14.5013	−0.847175	−0.423587	+	0.905855i	$0.639229\pi$
$294$	0	0
$295$	1.50607	0.0876870
$296$	0	0
$297$	−4.06390	−0.235811
$298$	0	0
$299$	−9.48115	−0.548309
$300$	0	0
$301$	3.95998	0.228249
$302$	0	0
$303$	−8.48814	−0.487631
$304$	0	0
$305$	−6.16734	−0.353141
$306$	0	0
$307$	−21.5105	−1.22767	−0.613833	−	0.789436i	$-0.710373\pi$
$307$	−21.5105	−1.22767	−0.613833	+	0.789436i	$0.710373\pi$
$308$	0	0
$309$	−12.4977	−0.710967
$310$	0	0
$311$	6.33725	0.359352	0.179676	−	0.983726i	$-0.442495\pi$
$311$	6.33725	0.359352	0.179676	+	0.983726i	$0.442495\pi$
$312$	0	0
$313$	−27.2341	−1.53936	−0.769680	−	0.638429i	$-0.779584\pi$
$313$	−27.2341	−1.53936	−0.769680	+	0.638429i	$0.779584\pi$
$314$	0	0
$315$	−17.2854	−0.973919
$316$	0	0
$317$	0.334081	0.0187639	0.00938193	−	0.999956i	$-0.497014\pi$
$317$	0.334081	0.0187639	0.00938193	+	0.999956i	$0.497014\pi$
$318$	0	0
$319$	10.8602	0.608056
$320$	0	0
$321$	4.92129	0.274680
$322$	0	0
$323$	−7.22095	−0.401784
$324$	0	0
$325$	−5.81656	−0.322645
$326$	0	0
$327$	10.7203	0.592837
$328$	0	0
$329$	−26.3424	−1.45230
$330$	0	0
$331$	14.9416	0.821263	0.410632	−	0.911801i	$-0.365308\pi$
$331$	14.9416	0.821263	0.410632	+	0.911801i	$0.365308\pi$
$332$	0	0
$333$	18.0974	0.991731
$334$	0	0
$335$	−27.1413	−1.48289
$336$	0	0
$337$	−32.2639	−1.75753	−0.878763	−	0.477259i	$-0.841630\pi$
$337$	−32.2639	−1.75753	−0.878763	+	0.477259i	$0.841630\pi$
$338$	0	0
$339$	3.07220	0.166859
$340$	0	0
$341$	−2.88735	−0.156359
$342$	0	0
$343$	20.1396	1.08744
$344$	0	0
$345$	−15.1382	−0.815016
$346$	0	0
$347$	−24.4521	−1.31266	−0.656328	−	0.754476i	$-0.727891\pi$
$347$	−24.4521	−1.31266	−0.656328	+	0.754476i	$0.727891\pi$
$348$	0	0
$349$	9.10801	0.487540	0.243770	−	0.969833i	$-0.421616\pi$
$349$	9.10801	0.487540	0.243770	+	0.969833i	$0.421616\pi$
$350$	0	0
$351$	4.94593	0.263994
$352$	0	0
$353$	8.90865	0.474159	0.237080	−	0.971490i	$-0.423810\pi$
$353$	8.90865	0.474159	0.237080	+	0.971490i	$0.423810\pi$
$354$	0	0
$355$	44.8348	2.37958
$356$	0	0
$357$	−10.9204	−0.577967
$358$	0	0
$359$	9.87903	0.521395	0.260698	−	0.965420i	$-0.416047\pi$
$359$	9.87903	0.521395	0.260698	+	0.965420i	$0.416047\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−6.70531	−0.351938
$364$	0	0
$365$	20.0966	1.05190
$366$	0	0
$367$	−10.7045	−0.558768	−0.279384	−	0.960179i	$-0.590130\pi$
$367$	−10.7045	−0.558768	−0.279384	+	0.960179i	$0.590130\pi$
$368$	0	0
$369$	4.47433	0.232924
$370$	0	0
$371$	14.2765	0.741199
$372$	0	0
$373$	−3.40278	−0.176190	−0.0880948	−	0.996112i	$-0.528078\pi$
$373$	−3.40278	−0.176190	−0.0880948	+	0.996112i	$0.528078\pi$
$374$	0	0
$375$	1.17984	0.0609268
$376$	0	0
$377$	−13.2173	−0.680728
$378$	0	0
$379$	17.8099	0.914831	0.457416	−	0.889253i	$-0.348775\pi$
$379$	17.8099	0.914831	0.457416	+	0.889253i	$0.348775\pi$
$380$	0	0
$381$	−13.0010	−0.666063
$382$	0	0
$383$	−6.52809	−0.333570	−0.166785	−	0.985993i	$-0.553339\pi$
$383$	−6.52809	−0.333570	−0.166785	+	0.985993i	$0.553339\pi$
$384$	0	0
$385$	7.34393	0.374282
$386$	0	0
$387$	4.52457	0.229997
$388$	0	0
$389$	−27.5062	−1.39462	−0.697310	−	0.716770i	$-0.745620\pi$
$389$	−27.5062	−1.39462	−0.697310	+	0.716770i	$0.745620\pi$
$390$	0	0
$391$	52.2179	2.64077
$392$	0	0
$393$	10.8640	0.548015
$394$	0	0
$395$	−3.07187	−0.154563
$396$	0	0
$397$	−8.26969	−0.415044	−0.207522	−	0.978230i	$-0.566540\pi$
$397$	−8.26969	−0.415044	−0.207522	+	0.978230i	$0.566540\pi$
$398$	0	0
$399$	1.51232	0.0757106
$400$	0	0
$401$	−3.63557	−0.181552	−0.0907760	−	0.995871i	$-0.528935\pi$
$401$	−3.63557	−0.181552	−0.0907760	+	0.995871i	$0.528935\pi$
$402$	0	0
$403$	3.51403	0.175046
$404$	0	0
$405$	−15.4701	−0.768714
$406$	0	0
$407$	−7.68894	−0.381127
$408$	0	0
$409$	8.90181	0.440166	0.220083	−	0.975481i	$-0.429367\pi$
$409$	8.90181	0.440166	0.220083	+	0.975481i	$0.429367\pi$
$410$	0	0
$411$	−12.2160	−0.602572
$412$	0	0
$413$	1.08802	0.0535381
$414$	0	0
$415$	20.3675	0.999799
$416$	0	0
$417$	4.99762	0.244735
$418$	0	0
$419$	3.19029	0.155856	0.0779280	−	0.996959i	$-0.475170\pi$
$419$	3.19029	0.155856	0.0779280	+	0.996959i	$0.475170\pi$
$420$	0	0
$421$	15.3432	0.747782	0.373891	−	0.927473i	$-0.378023\pi$
$421$	15.3432	0.747782	0.373891	+	0.927473i	$0.378023\pi$
$422$	0	0
$423$	−30.0982	−1.46343
$424$	0	0
$425$	32.0350	1.55393
$426$	0	0
$427$	−4.45543	−0.215614
$428$	0	0
$429$	−0.962531	−0.0464714
$430$	0	0
$431$	−10.3578	−0.498917	−0.249458	−	0.968386i	$-0.580253\pi$
$431$	−10.3578	−0.498917	−0.249458	+	0.968386i	$0.580253\pi$
$432$	0	0
$433$	15.4073	0.740426	0.370213	−	0.928947i	$-0.379285\pi$
$433$	15.4073	0.740426	0.370213	+	0.928947i	$0.379285\pi$
$434$	0	0
$435$	−21.1037	−1.01185
$436$	0	0
$437$	−7.23144	−0.345927
$438$	0	0
$439$	−23.3904	−1.11636	−0.558181	−	0.829719i	$-0.688501\pi$
$439$	−23.3904	−1.11636	−0.558181	+	0.829719i	$0.688501\pi$
$440$	0	0
$441$	5.26182	0.250563
$442$	0	0
$443$	19.4735	0.925213	0.462606	−	0.886564i	$-0.346914\pi$
$443$	19.4735	0.925213	0.462606	+	0.886564i	$0.346914\pi$
$444$	0	0
$445$	−2.31878	−0.109921
$446$	0	0
$447$	−5.20139	−0.246017
$448$	0	0
$449$	−30.2335	−1.42681	−0.713403	−	0.700754i	$-0.752847\pi$
$449$	−30.2335	−1.42681	−0.713403	+	0.700754i	$0.752847\pi$
$450$	0	0
$451$	−1.90099	−0.0895139
$452$	0	0
$453$	0.610343	0.0286764
$454$	0	0
$455$	−8.93788	−0.419014
$456$	0	0
$457$	12.2732	0.574116	0.287058	−	0.957913i	$-0.407323\pi$
$457$	12.2732	0.574116	0.287058	+	0.957913i	$0.407323\pi$
$458$	0	0
$459$	−27.2400	−1.27145
$460$	0	0
$461$	−26.3984	−1.22950	−0.614749	−	0.788723i	$-0.710743\pi$
$461$	−26.3984	−1.22950	−0.614749	+	0.788723i	$0.710743\pi$
$462$	0	0
$463$	16.3997	0.762157	0.381078	−	0.924543i	$-0.375553\pi$
$463$	16.3997	0.762157	0.381078	+	0.924543i	$0.375553\pi$
$464$	0	0
$465$	5.61074	0.260192
$466$	0	0
$467$	6.55871	0.303501	0.151750	−	0.988419i	$-0.451509\pi$
$467$	6.55871	0.303501	0.151750	+	0.988419i	$0.451509\pi$
$468$	0	0
$469$	−19.6075	−0.905391
$470$	0	0
$471$	−11.3716	−0.523975
$472$	0	0
$473$	−1.92233	−0.0883889
$474$	0	0
$475$	−4.43640	−0.203556
$476$	0	0
$477$	16.3120	0.746874
$478$	0	0
$479$	37.4669	1.71191	0.855953	−	0.517054i	$-0.172971\pi$
$479$	37.4669	1.71191	0.855953	+	0.517054i	$0.172971\pi$
$480$	0	0
$481$	9.35776	0.426677
$482$	0	0
$483$	−10.9362	−0.497616
$484$	0	0
$485$	−23.4388	−1.06430
$486$	0	0
$487$	−30.0923	−1.36361	−0.681805	−	0.731534i	$-0.738805\pi$
$487$	−30.0923	−1.36361	−0.681805	+	0.731534i	$0.738805\pi$
$488$	0	0
$489$	−14.0227	−0.634127
$490$	0	0
$491$	17.0378	0.768905	0.384452	−	0.923145i	$-0.374390\pi$
$491$	17.0378	0.768905	0.384452	+	0.923145i	$0.374390\pi$
$492$	0	0
$493$	72.7951	3.27853
$494$	0	0
$495$	8.39100	0.377147
$496$	0	0
$497$	32.3897	1.45288
$498$	0	0
$499$	18.6505	0.834909	0.417455	−	0.908698i	$-0.362922\pi$
$499$	18.6505	0.834909	0.417455	+	0.908698i	$0.362922\pi$
$500$	0	0
$501$	−9.30812	−0.415856
$502$	0	0
$503$	−11.8753	−0.529493	−0.264746	−	0.964318i	$-0.585288\pi$
$503$	−11.8753	−0.529493	−0.264746	+	0.964318i	$0.585288\pi$
$504$	0	0
$505$	38.2620	1.70264
$506$	0	0
$507$	−7.68769	−0.341422
$508$	0	0
$509$	−10.2326	−0.453553	−0.226776	−	0.973947i	$-0.572819\pi$
$509$	−10.2326	−0.453553	−0.226776	+	0.973947i	$0.572819\pi$
$510$	0	0
$511$	14.5182	0.642249
$512$	0	0
$513$	3.77235	0.166553
$514$	0	0
$515$	56.3357	2.48245
$516$	0	0
$517$	12.7877	0.562401
$518$	0	0
$519$	7.79717	0.342258
$520$	0	0
$521$	10.6007	0.464424	0.232212	−	0.972665i	$-0.425404\pi$
$521$	10.6007	0.464424	0.232212	+	0.972665i	$0.425404\pi$
$522$	0	0
$523$	43.0857	1.88400	0.942002	−	0.335607i	$-0.108941\pi$
$523$	43.0857	1.88400	0.942002	+	0.335607i	$0.108941\pi$
$524$	0	0
$525$	−6.70924	−0.292815
$526$	0	0
$527$	−19.3537	−0.843059
$528$	0	0
$529$	29.2937	1.27364
$530$	0	0
$531$	1.24315	0.0539481
$532$	0	0
$533$	2.31358	0.100212
$534$	0	0
$535$	−22.1837	−0.959086
$536$	0	0
$537$	−8.40722	−0.362799
$538$	0	0
$539$	−2.23556	−0.0962925
$540$	0	0
$541$	−4.08268	−0.175528	−0.0877641	−	0.996141i	$-0.527972\pi$
$541$	−4.08268	−0.175528	−0.0877641	+	0.996141i	$0.527972\pi$
$542$	0	0
$543$	4.62146	0.198326
$544$	0	0
$545$	−48.3241	−2.06998
$546$	0	0
$547$	13.3558	0.571054	0.285527	−	0.958371i	$-0.407831\pi$
$547$	13.3558	0.571054	0.285527	+	0.958371i	$0.407831\pi$
$548$	0	0
$549$	−5.09067	−0.217264
$550$	0	0
$551$	−10.0811	−0.429470
$552$	0	0
$553$	−2.21919	−0.0943697
$554$	0	0
$555$	14.9412	0.634220
$556$	0	0
$557$	−32.2410	−1.36609	−0.683047	−	0.730374i	$-0.739346\pi$
$557$	−32.2410	−1.36609	−0.683047	+	0.730374i	$0.739346\pi$
$558$	0	0
$559$	2.33956	0.0989528
$560$	0	0
$561$	5.30118	0.223816
$562$	0	0
$563$	35.3405	1.48942	0.744711	−	0.667387i	$-0.232587\pi$
$563$	35.3405	1.48942	0.744711	+	0.667387i	$0.232587\pi$
$564$	0	0
$565$	−13.8486	−0.582614
$566$	0	0
$567$	−11.1759	−0.469345
$568$	0	0
$569$	−11.5308	−0.483397	−0.241699	−	0.970351i	$-0.577705\pi$
$569$	−11.5308	−0.483397	−0.241699	+	0.970351i	$0.577705\pi$
$570$	0	0
$571$	22.3938	0.937149	0.468575	−	0.883424i	$-0.344768\pi$
$571$	22.3938	0.937149	0.468575	+	0.883424i	$0.344768\pi$
$572$	0	0
$573$	−6.49172	−0.271195
$574$	0	0
$575$	32.0815	1.33789
$576$	0	0
$577$	−34.0155	−1.41608	−0.708042	−	0.706170i	$-0.750421\pi$
$577$	−34.0155	−1.41608	−0.708042	+	0.706170i	$0.750421\pi$
$578$	0	0
$579$	11.7725	0.489248
$580$	0	0
$581$	14.7139	0.610437
$582$	0	0
$583$	−6.93038	−0.287027
$584$	0	0
$585$	−10.2122	−0.422223
$586$	0	0
$587$	2.65166	0.109446	0.0547228	−	0.998502i	$-0.482572\pi$
$587$	2.65166	0.109446	0.0547228	+	0.998502i	$0.482572\pi$
$588$	0	0
$589$	2.68021	0.110436
$590$	0	0
$591$	3.96199	0.162975
$592$	0	0
$593$	17.5658	0.721342	0.360671	−	0.932693i	$-0.382548\pi$
$593$	17.5658	0.721342	0.360671	+	0.932693i	$0.382548\pi$
$594$	0	0
$595$	49.2258	2.01806
$596$	0	0
$597$	−4.19212	−0.171572
$598$	0	0
$599$	−22.4683	−0.918031	−0.459015	−	0.888428i	$-0.651798\pi$
$599$	−22.4683	−0.918031	−0.459015	+	0.888428i	$0.651798\pi$
$600$	0	0
$601$	−21.9593	−0.895737	−0.447869	−	0.894099i	$-0.647817\pi$
$601$	−21.9593	−0.895737	−0.447869	+	0.894099i	$0.647817\pi$
$602$	0	0
$603$	−22.4031	−0.912323
$604$	0	0
$605$	30.2256	1.22884
$606$	0	0
$607$	−5.09538	−0.206815	−0.103408	−	0.994639i	$-0.532975\pi$
$607$	−5.09538	−0.206815	−0.103408	+	0.994639i	$0.532975\pi$
$608$	0	0
$609$	−15.2458	−0.617792
$610$	0	0
$611$	−15.5631	−0.629617
$612$	0	0
$613$	−38.9459	−1.57301	−0.786506	−	0.617583i	$-0.788112\pi$
$613$	−38.9459	−1.57301	−0.786506	+	0.617583i	$0.788112\pi$
$614$	0	0
$615$	3.69402	0.148957
$616$	0	0
$617$	−1.54397	−0.0621579	−0.0310790	−	0.999517i	$-0.509894\pi$
$617$	−1.54397	−0.0621579	−0.0310790	+	0.999517i	$0.509894\pi$
$618$	0	0
$619$	5.71727	0.229797	0.114898	−	0.993377i	$-0.463346\pi$
$619$	5.71727	0.229797	0.114898	+	0.993377i	$0.463346\pi$
$620$	0	0
$621$	−27.2795	−1.09469
$622$	0	0
$623$	−1.67514	−0.0671132
$624$	0	0
$625$	−27.5004	−1.10001
$626$	0	0
$627$	−0.734139	−0.0293187
$628$	0	0
$629$	−51.5383	−2.05497
$630$	0	0
$631$	−32.7324	−1.30306	−0.651528	−	0.758624i	$-0.725872\pi$
$631$	−32.7324	−1.30306	−0.651528	+	0.758624i	$0.725872\pi$
$632$	0	0
$633$	2.26357	0.0899689
$634$	0	0
$635$	58.6048	2.32566
$636$	0	0
$637$	2.72077	0.107801
$638$	0	0
$639$	37.0077	1.46400
$640$	0	0
$641$	12.7314	0.502859	0.251429	−	0.967876i	$-0.419099\pi$
$641$	12.7314	0.502859	0.251429	+	0.967876i	$0.419099\pi$
$642$	0	0
$643$	47.3988	1.86923	0.934614	−	0.355665i	$-0.115746\pi$
$643$	47.3988	1.86923	0.934614	+	0.355665i	$0.115746\pi$
$644$	0	0
$645$	3.73550	0.147085
$646$	0	0
$647$	0.941892	0.0370296	0.0185148	−	0.999829i	$-0.494106\pi$
$647$	0.941892	0.0370296	0.0185148	+	0.999829i	$0.494106\pi$
$648$	0	0
$649$	−0.528170	−0.0207325
$650$	0	0
$651$	4.05333	0.158863
$652$	0	0
$653$	−14.1464	−0.553591	−0.276796	−	0.960929i	$-0.589273\pi$
$653$	−14.1464	−0.553591	−0.276796	+	0.960929i	$0.589273\pi$
$654$	0	0
$655$	−48.9715	−1.91348
$656$	0	0
$657$	16.5882	0.647167
$658$	0	0
$659$	20.5398	0.800117	0.400059	−	0.916490i	$-0.368990\pi$
$659$	20.5398	0.800117	0.400059	+	0.916490i	$0.368990\pi$
$660$	0	0
$661$	−20.3412	−0.791182	−0.395591	−	0.918427i	$-0.629460\pi$
$661$	−20.3412	−0.791182	−0.395591	+	0.918427i	$0.629460\pi$
$662$	0	0
$663$	−6.45176	−0.250566
$664$	0	0
$665$	−6.81708	−0.264355
$666$	0	0
$667$	72.9009	2.82273
$668$	0	0
$669$	−11.6318	−0.449711
$670$	0	0
$671$	2.16285	0.0834957
$672$	0	0
$673$	19.0514	0.734378	0.367189	−	0.930146i	$-0.380320\pi$
$673$	19.0514	0.734378	0.367189	+	0.930146i	$0.380320\pi$
$674$	0	0
$675$	−16.7357	−0.644156
$676$	0	0
$677$	−33.2453	−1.27772	−0.638860	−	0.769323i	$-0.720594\pi$
$677$	−33.2453	−1.27772	−0.638860	+	0.769323i	$0.720594\pi$
$678$	0	0
$679$	−16.9328	−0.649819
$680$	0	0
$681$	4.09992	0.157109
$682$	0	0
$683$	−18.2221	−0.697249	−0.348624	−	0.937263i	$-0.613351\pi$
$683$	−18.2221	−0.697249	−0.348624	+	0.937263i	$0.613351\pi$
$684$	0	0
$685$	55.0662	2.10397
$686$	0	0
$687$	−10.0108	−0.381935
$688$	0	0
$689$	8.43456	0.321331
$690$	0	0
$691$	−31.3448	−1.19241	−0.596205	−	0.802832i	$-0.703326\pi$
$691$	−31.3448	−1.19241	−0.596205	+	0.802832i	$0.703326\pi$
$692$	0	0
$693$	6.06186	0.230271
$694$	0	0
$695$	−22.5278	−0.854528
$696$	0	0
$697$	−12.7421	−0.482643
$698$	0	0
$699$	12.3162	0.465841
$700$	0	0
$701$	−36.3515	−1.37298	−0.686488	−	0.727141i	$-0.740849\pi$
$701$	−36.3515	−1.37298	−0.686488	+	0.727141i	$0.740849\pi$
$702$	0	0
$703$	7.13733	0.269190
$704$	0	0
$705$	−24.8492	−0.935873
$706$	0	0
$707$	27.6414	1.03956
$708$	0	0
$709$	−13.7138	−0.515033	−0.257517	−	0.966274i	$-0.582904\pi$
$709$	−13.7138	−0.515033	−0.257517	+	0.966274i	$0.582904\pi$
$710$	0	0
$711$	−2.53560	−0.0950923
$712$	0	0
$713$	−19.3818	−0.725853
$714$	0	0
$715$	4.33880	0.162262
$716$	0	0
$717$	−12.1958	−0.455459
$718$	0	0
$719$	−2.29697	−0.0856627	−0.0428313	−	0.999082i	$-0.513638\pi$
$719$	−2.29697	−0.0856627	−0.0428313	+	0.999082i	$0.513638\pi$
$720$	0	0
$721$	40.6983	1.51568
$722$	0	0
$723$	17.3737	0.646135
$724$	0	0
$725$	44.7238	1.66100
$726$	0	0
$727$	−1.14700	−0.0425398	−0.0212699	−	0.999774i	$-0.506771\pi$
$727$	−1.14700	−0.0425398	−0.0212699	+	0.999774i	$0.506771\pi$
$728$	0	0
$729$	−5.05710	−0.187300
$730$	0	0
$731$	−12.8852	−0.476577
$732$	0	0
$733$	15.4567	0.570905	0.285452	−	0.958393i	$-0.407856\pi$
$733$	15.4567	0.570905	0.285452	+	0.958393i	$0.407856\pi$
$734$	0	0
$735$	4.34417	0.160237
$736$	0	0
$737$	9.51827	0.350610
$738$	0	0
$739$	−29.7889	−1.09580	−0.547901	−	0.836543i	$-0.684573\pi$
$739$	−29.7889	−1.09580	−0.547901	+	0.836543i	$0.684573\pi$
$740$	0	0
$741$	0.893478	0.0328227
$742$	0	0
$743$	33.5964	1.23253	0.616265	−	0.787539i	$-0.288645\pi$
$743$	33.5964	1.23253	0.616265	+	0.787539i	$0.288645\pi$
$744$	0	0
$745$	23.4463	0.859006
$746$	0	0
$747$	16.8118	0.615111
$748$	0	0
$749$	−16.0260	−0.585579
$750$	0	0
$751$	3.17778	0.115959	0.0579794	−	0.998318i	$-0.481534\pi$
$751$	3.17778	0.115959	0.0579794	+	0.998318i	$0.481534\pi$
$752$	0	0
$753$	−3.74202	−0.136367
$754$	0	0
$755$	−2.75124	−0.100128
$756$	0	0
$757$	−11.5467	−0.419672	−0.209836	−	0.977737i	$-0.567293\pi$
$757$	−11.5467	−0.419672	−0.209836	+	0.977737i	$0.567293\pi$
$758$	0	0
$759$	5.30888	0.192700
$760$	0	0
$761$	43.9575	1.59346	0.796730	−	0.604336i	$-0.206561\pi$
$761$	43.9575	1.59346	0.796730	+	0.604336i	$0.206561\pi$
$762$	0	0
$763$	−34.9105	−1.26384
$764$	0	0
$765$	56.2442	2.03351
$766$	0	0
$767$	0.642805	0.0232103
$768$	0	0
$769$	−24.0111	−0.865864	−0.432932	−	0.901427i	$-0.642521\pi$
$769$	−24.0111	−0.865864	−0.432932	+	0.901427i	$0.642521\pi$
$770$	0	0
$771$	2.15824	0.0777270
$772$	0	0
$773$	−21.4330	−0.770891	−0.385446	−	0.922731i	$-0.625952\pi$
$773$	−21.4330	−0.770891	−0.385446	+	0.922731i	$0.625952\pi$
$774$	0	0
$775$	−11.8905	−0.427119
$776$	0	0
$777$	10.7939	0.387229
$778$	0	0
$779$	1.76461	0.0632236
$780$	0	0
$781$	−15.7233	−0.562622
$782$	0	0
$783$	−38.0295	−1.35906
$784$	0	0
$785$	51.2597	1.82954
$786$	0	0
$787$	−33.4442	−1.19216	−0.596079	−	0.802926i	$-0.703275\pi$
$787$	−33.4442	−1.19216	−0.596079	+	0.802926i	$0.703275\pi$
$788$	0	0
$789$	−0.392845	−0.0139857
$790$	0	0
$791$	−10.0045	−0.355721
$792$	0	0
$793$	−2.63227	−0.0934748
$794$	0	0
$795$	13.4672	0.477633
$796$	0	0
$797$	−19.7890	−0.700963	−0.350482	−	0.936570i	$-0.613982\pi$
$797$	−19.7890	−0.700963	−0.350482	+	0.936570i	$0.613982\pi$
$798$	0	0
$799$	85.7146	3.03237
$800$	0	0
$801$	−1.91398	−0.0676271
$802$	0	0
$803$	−7.04773	−0.248709
$804$	0	0
$805$	49.2973	1.73750
$806$	0	0
$807$	−4.20339	−0.147966
$808$	0	0
$809$	−19.2881	−0.678133	−0.339066	−	0.940762i	$-0.610111\pi$
$809$	−19.2881	−0.678133	−0.339066	+	0.940762i	$0.610111\pi$
$810$	0	0
$811$	−16.4602	−0.577996	−0.288998	−	0.957330i	$-0.593322\pi$
$811$	−16.4602	−0.577996	−0.288998	+	0.957330i	$0.593322\pi$
$812$	0	0
$813$	−0.534118	−0.0187323
$814$	0	0
$815$	63.2101	2.21415
$816$	0	0
$817$	1.78442	0.0624290
$818$	0	0
$819$	−7.37754	−0.257792
$820$	0	0
$821$	−5.22914	−0.182498	−0.0912491	−	0.995828i	$-0.529086\pi$
$821$	−5.22914	−0.182498	−0.0912491	+	0.995828i	$0.529086\pi$
$822$	0	0
$823$	−43.4998	−1.51631	−0.758154	−	0.652075i	$-0.773899\pi$
$823$	−43.4998	−1.51631	−0.758154	+	0.652075i	$0.773899\pi$
$824$	0	0
$825$	3.25693	0.113392
$826$	0	0
$827$	−25.6930	−0.893434	−0.446717	−	0.894675i	$-0.647407\pi$
$827$	−25.6930	−0.893434	−0.446717	+	0.894675i	$0.647407\pi$
$828$	0	0
$829$	19.5603	0.679358	0.339679	−	0.940541i	$-0.389682\pi$
$829$	19.5603	0.679358	0.339679	+	0.940541i	$0.389682\pi$
$830$	0	0
$831$	8.61241	0.298761
$832$	0	0
$833$	−14.9848	−0.519192
$834$	0	0
$835$	41.9582	1.45202
$836$	0	0
$837$	10.1107	0.349477
$838$	0	0
$839$	−21.9619	−0.758208	−0.379104	−	0.925354i	$-0.623768\pi$
$839$	−21.9619	−0.758208	−0.379104	+	0.925354i	$0.623768\pi$
$840$	0	0
$841$	72.6287	2.50444
$842$	0	0
$843$	−14.9346	−0.514377
$844$	0	0
$845$	34.6538	1.19213
$846$	0	0
$847$	21.8357	0.750282
$848$	0	0
$849$	2.73625	0.0939077
$850$	0	0
$851$	−51.6132	−1.76928
$852$	0	0
$853$	18.7874	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$853$	18.7874	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$854$	0	0
$855$	−7.78903	−0.266379
$856$	0	0
$857$	−2.51160	−0.0857945	−0.0428972	−	0.999079i	$-0.513659\pi$
$857$	−2.51160	−0.0857945	−0.0428972	+	0.999079i	$0.513659\pi$
$858$	0	0
$859$	−18.4037	−0.627927	−0.313964	−	0.949435i	$-0.601657\pi$
$859$	−18.4037	−0.627927	−0.313964	+	0.949435i	$0.601657\pi$
$860$	0	0
$861$	2.66865	0.0909472
$862$	0	0
$863$	3.94185	0.134182	0.0670911	−	0.997747i	$-0.478628\pi$
$863$	3.94185	0.134182	0.0670911	+	0.997747i	$0.478628\pi$
$864$	0	0
$865$	−35.1473	−1.19504
$866$	0	0
$867$	23.9484	0.813329
$868$	0	0
$869$	1.07728	0.0365444
$870$	0	0
$871$	−11.5841	−0.392513
$872$	0	0
$873$	−19.3470	−0.654795
$874$	0	0
$875$	−3.84212	−0.129888
$876$	0	0
$877$	−3.24999	−0.109744	−0.0548722	−	0.998493i	$-0.517475\pi$
$877$	−3.24999	−0.109744	−0.0548722	+	0.998493i	$0.517475\pi$
$878$	0	0
$879$	−9.88223	−0.333319
$880$	0	0
$881$	19.9949	0.673644	0.336822	−	0.941568i	$-0.390648\pi$
$881$	19.9949	0.673644	0.336822	+	0.941568i	$0.390648\pi$
$882$	0	0
$883$	−15.3947	−0.518074	−0.259037	−	0.965867i	$-0.583405\pi$
$883$	−15.3947	−0.518074	−0.259037	+	0.965867i	$0.583405\pi$
$884$	0	0
$885$	1.02635	0.0345003
$886$	0	0
$887$	8.69825	0.292059	0.146029	−	0.989280i	$-0.453351\pi$
$887$	8.69825	0.292059	0.146029	+	0.989280i	$0.453351\pi$
$888$	0	0
$889$	42.3375	1.41995
$890$	0	0
$891$	5.42525	0.181753
$892$	0	0
$893$	−11.8703	−0.397223
$894$	0	0
$895$	37.8973	1.26677
$896$	0	0
$897$	−6.46113	−0.215731
$898$	0	0
$899$	−27.0195	−0.901150
$900$	0	0
$901$	−46.4538	−1.54760
$902$	0	0
$903$	2.69861	0.0898042
$904$	0	0
$905$	−20.8322	−0.692485
$906$	0	0
$907$	−32.9232	−1.09320	−0.546598	−	0.837395i	$-0.684078\pi$
$907$	−32.9232	−1.09320	−0.546598	+	0.837395i	$0.684078\pi$
$908$	0	0
$909$	31.5824	1.04752
$910$	0	0
$911$	25.6144	0.848642	0.424321	−	0.905512i	$-0.360513\pi$
$911$	25.6144	0.848642	0.424321	+	0.905512i	$0.360513\pi$
$912$	0	0
$913$	−7.14273	−0.236390
$914$	0	0
$915$	−4.20287	−0.138943
$916$	0	0
$917$	−35.3782	−1.16829
$918$	0	0
$919$	48.9317	1.61411	0.807054	−	0.590478i	$-0.201061\pi$
$919$	48.9317	1.61411	0.807054	+	0.590478i	$0.201061\pi$
$920$	0	0
$921$	−14.6588	−0.483023
$922$	0	0
$923$	19.1359	0.629864
$924$	0	0
$925$	−31.6640	−1.04111
$926$	0	0
$927$	46.5009	1.52729
$928$	0	0
$929$	18.9289	0.621037	0.310518	−	0.950567i	$-0.399497\pi$
$929$	18.9289	0.621037	0.310518	+	0.950567i	$0.399497\pi$
$930$	0	0
$931$	2.07518	0.0680114
$932$	0	0
$933$	4.31866	0.141386
$934$	0	0
$935$	−23.8962	−0.781488
$936$	0	0
$937$	−37.5936	−1.22813	−0.614065	−	0.789255i	$-0.710467\pi$
$937$	−37.5936	−1.22813	−0.614065	+	0.789255i	$0.710467\pi$
$938$	0	0
$939$	−18.5592	−0.605658
$940$	0	0
$941$	−39.4229	−1.28515	−0.642574	−	0.766223i	$-0.722134\pi$
$941$	−39.4229	−1.28515	−0.642574	+	0.766223i	$0.722134\pi$
$942$	0	0
$943$	−12.7607	−0.415544
$944$	0	0
$945$	−25.7164	−0.836555
$946$	0	0
$947$	−42.1932	−1.37110	−0.685548	−	0.728028i	$-0.740437\pi$
$947$	−42.1932	−1.37110	−0.685548	+	0.728028i	$0.740437\pi$
$948$	0	0
$949$	8.57738	0.278434
$950$	0	0
$951$	0.227667	0.00738260
$952$	0	0
$953$	38.9664	1.26225	0.631123	−	0.775683i	$-0.282595\pi$
$953$	38.9664	1.26225	0.631123	+	0.775683i	$0.282595\pi$
$954$	0	0
$955$	29.2627	0.946920
$956$	0	0
$957$	7.40093	0.239238
$958$	0	0
$959$	39.7811	1.28460
$960$	0	0
$961$	−23.8165	−0.768273
$962$	0	0
$963$	−18.3110	−0.590063
$964$	0	0
$965$	−53.0669	−1.70828
$966$	0	0
$967$	−24.7044	−0.794442	−0.397221	−	0.917723i	$-0.630025\pi$
$967$	−24.7044	−0.794442	−0.397221	+	0.917723i	$0.630025\pi$
$968$	0	0
$969$	−4.92087	−0.158081
$970$	0	0
$971$	−2.46029	−0.0789546	−0.0394773	−	0.999220i	$-0.512569\pi$
$971$	−2.46029	−0.0789546	−0.0394773	+	0.999220i	$0.512569\pi$
$972$	0	0
$973$	−16.2746	−0.521740
$974$	0	0
$975$	−3.96382	−0.126944
$976$	0	0
$977$	−41.1363	−1.31607	−0.658034	−	0.752989i	$-0.728611\pi$
$977$	−41.1363	−1.31607	−0.658034	+	0.752989i	$0.728611\pi$
$978$	0	0
$979$	0.813181	0.0259894
$980$	0	0
$981$	−39.8879	−1.27352
$982$	0	0
$983$	13.3783	0.426701	0.213350	−	0.976976i	$-0.431562\pi$
$983$	13.3783	0.426701	0.213350	+	0.976976i	$0.431562\pi$
$984$	0	0
$985$	−17.8595	−0.569051
$986$	0	0
$987$	−17.9516	−0.571406
$988$	0	0
$989$	−12.9039	−0.410321
$990$	0	0
$991$	−7.33256	−0.232926	−0.116463	−	0.993195i	$-0.537156\pi$
$991$	−7.33256	−0.232926	−0.116463	+	0.993195i	$0.537156\pi$
$992$	0	0
$993$	10.1823	0.323124
$994$	0	0
$995$	18.8968	0.599070
$996$	0	0
$997$	−14.7110	−0.465901	−0.232951	−	0.972489i	$-0.574838\pi$
$997$	−14.7110	−0.465901	−0.232951	+	0.972489i	$0.574838\pi$
$998$	0	0
$999$	26.9245	0.851854

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.f.1.14	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.f.1.14	✓	25	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.681472 q^{3} -3.07187 q^{5} -2.21919 q^{7} -2.53560 q^{9} +O(q^{10})\)	\(q+0.681472 q^{3} -3.07187 q^{5} -2.21919 q^{7} -2.53560 q^{9} +1.07728 q^{11} -1.31110 q^{13} -2.09339 q^{15} +7.22095 q^{17} -1.00000 q^{19} -1.51232 q^{21} +7.23144 q^{23} +4.43640 q^{25} -3.77235 q^{27} +10.0811 q^{29} -2.68021 q^{31} +0.734139 q^{33} +6.81708 q^{35} -7.13733 q^{37} -0.893478 q^{39} -1.76461 q^{41} -1.78442 q^{43} +7.78903 q^{45} +11.8703 q^{47} -2.07518 q^{49} +4.92087 q^{51} -6.43319 q^{53} -3.30928 q^{55} -0.681472 q^{57} -0.490279 q^{59} +2.00768 q^{61} +5.62698 q^{63} +4.02753 q^{65} +8.83542 q^{67} +4.92802 q^{69} -14.5953 q^{71} -6.54212 q^{73} +3.02328 q^{75} -2.39070 q^{77} +1.00000 q^{79} +5.03604 q^{81} -6.63031 q^{83} -22.1818 q^{85} +6.86999 q^{87} +0.754843 q^{89} +2.90959 q^{91} -1.82649 q^{93} +3.07187 q^{95} +7.63014 q^{97} -2.73156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9}+O(q^{10}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9	\( 25 q + 4 q^{3} - 8 q^{5} + 2 q^{7} + 13 q^{9} - 3 q^{11} + q^{13} - 5 q^{15} - 13 q^{17} - 25 q^{19} - 24 q^{21} - 31 q^{23} + 21 q^{25} + 7 q^{27} - 19 q^{29} - 7 q^{31} - 30 q^{33} - q^{35} - 29 q^{37} - 26 q^{39} - 40 q^{41} - 40 q^{45} - 8 q^{47} - 9 q^{49} + 12 q^{51} - 38 q^{53} - 29 q^{55} - 4 q^{57} + 18 q^{59} - 26 q^{61} - 40 q^{63} - 70 q^{65} - 13 q^{67} + q^{69} - 47 q^{71} - 8 q^{73} + 7 q^{75} - 19 q^{77} + 25 q^{79} - 19 q^{81} - 8 q^{83} - 33 q^{85} - 50 q^{87} - 54 q^{89} - 12 q^{91} - 24 q^{93} + 8 q^{95} - 4 q^{97} - 39 q^{99}+O(q^{100}) \) 25 * q + 4 * q^3 - 8 * q^5 + 2 * q^7 + 13 * q^9 - 3 * q^11 + q^13 - 5 * q^15 - 13 * q^17 - 25 * q^19 - 24 * q^21 - 31 * q^23 + 21 * q^25 + 7 * q^27 - 19 * q^29 - 7 * q^31 - 30 * q^33 - q^35 - 29 * q^37 - 26 * q^39 - 40 * q^41 - 40 * q^45 - 8 * q^47 - 9 * q^49 + 12 * q^51 - 38 * q^53 - 29 * q^55 - 4 * q^57 + 18 * q^59 - 26 * q^61 - 40 * q^63 - 70 * q^65 - 13 * q^67 + q^69 - 47 * q^71 - 8 * q^73 + 7 * q^75 - 19 * q^77 + 25 * q^79 - 19 * q^81 - 8 * q^83 - 33 * q^85 - 50 * q^87 - 54 * q^89 - 12 * q^91 - 24 * q^93 + 8 * q^95 - 4 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more