LMFDB - Embedded newform 6004.2.a.d.1.2

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 15x^{6} + 56x^{5} + 87x^{4} - 248x^{3} - 241x^{2} + 340x + 248 $ x^8 - 4x^7 - 15x^6 + 56x^5 + 87x^4 - 248x^3 - 241x^2 + 340*x + 248
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.94482$ of defining polynomial
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)

$f(q)$	$=$	$q-1.94482 q^{5} -0.969069 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.94482 q^{5} -0.969069 q^{7} -3.00000 q^{9} +0.202183 q^{11} -5.56979 q^{13} +3.01217 q^{17} +1.00000 q^{19} +6.09105 q^{23} -1.21766 q^{25} -6.46811 q^{29} -5.83369 q^{31} +1.88467 q^{35} +8.65467 q^{37} -7.53148 q^{41} +2.75737 q^{43} +5.83447 q^{45} -5.67933 q^{47} -6.06091 q^{49} -12.1865 q^{53} -0.393210 q^{55} -10.5931 q^{59} +1.88233 q^{61} +2.90721 q^{63} +10.8323 q^{65} +8.95699 q^{67} +5.27497 q^{71} -9.05092 q^{73} -0.195929 q^{77} +1.00000 q^{79} +9.00000 q^{81} +4.23359 q^{83} -5.85814 q^{85} -7.91293 q^{89} +5.39751 q^{91} -1.94482 q^{95} -12.3246 q^{97} -0.606548 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + q^{7} - 24 q^{9}+O(q^{10}) $ 8 * q + 4 * q^5 + q^7 - 24 * q^9	$ 8 q + 4 q^{5} + q^{7} - 24 q^{9} - 7 q^{11} - 12 q^{13} - 7 q^{17} + 8 q^{19} + 10 q^{23} + 6 q^{25} + 5 q^{29} + 3 q^{31} - 11 q^{35} - 15 q^{37} - 5 q^{41} + 10 q^{43} - 12 q^{45} + 18 q^{47} + 31 q^{49} + 9 q^{53} - 17 q^{55} + 8 q^{59} + 13 q^{61} - 3 q^{63} + 4 q^{65} + 21 q^{67} + 44 q^{71} - 20 q^{73} + 15 q^{77} + 8 q^{79} + 72 q^{81} - 4 q^{83} - 9 q^{85} - 10 q^{89} + 48 q^{91} + 4 q^{95} - 22 q^{97} + 21 q^{99}+O(q^{100}) $ 8 * q + 4 * q^5 + q^7 - 24 * q^9 - 7 * q^11 - 12 * q^13 - 7 * q^17 + 8 * q^19 + 10 * q^23 + 6 * q^25 + 5 * q^29 + 3 * q^31 - 11 * q^35 - 15 * q^37 - 5 * q^41 + 10 * q^43 - 12 * q^45 + 18 * q^47 + 31 * q^49 + 9 * q^53 - 17 * q^55 + 8 * q^59 + 13 * q^61 - 3 * q^63 + 4 * q^65 + 21 * q^67 + 44 * q^71 - 20 * q^73 + 15 * q^77 + 8 * q^79 + 72 * q^81 - 4 * q^83 - 9 * q^85 - 10 * q^89 + 48 * q^91 + 4 * q^95 - 22 * q^97 + 21 * 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	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−1.94482	−0.869752	−0.434876	−	0.900490i	$-0.643208\pi$
$5$	−1.94482	−0.869752	−0.434876	+	0.900490i	$0.643208\pi$
$6$	0	0
$7$	−0.969069	−0.366274	−0.183137	−	0.983087i	$-0.558625\pi$
$7$	−0.969069	−0.366274	−0.183137	+	0.983087i	$0.558625\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	0.202183	0.0609604	0.0304802	−	0.999535i	$-0.490296\pi$
$11$	0.202183	0.0609604	0.0304802	+	0.999535i	$0.490296\pi$
$12$	0	0
$13$	−5.56979	−1.54478	−0.772391	−	0.635147i	$-0.780939\pi$
$13$	−5.56979	−1.54478	−0.772391	+	0.635147i	$0.780939\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.01217	0.730558	0.365279	−	0.930898i	$-0.380974\pi$
$17$	3.01217	0.730558	0.365279	+	0.930898i	$0.380974\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.09105	1.27007	0.635036	−	0.772483i	$-0.280985\pi$
$23$	6.09105	1.27007	0.635036	+	0.772483i	$0.280985\pi$
$24$	0	0
$25$	−1.21766	−0.243531
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.46811	−1.20110	−0.600549	−	0.799588i	$-0.705051\pi$
$29$	−6.46811	−1.20110	−0.600549	+	0.799588i	$0.705051\pi$
$30$	0	0
$31$	−5.83369	−1.04776	−0.523881	−	0.851791i	$-0.675516\pi$
$31$	−5.83369	−1.04776	−0.523881	+	0.851791i	$0.675516\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.88467	0.318567
$36$	0	0
$37$	8.65467	1.42282	0.711410	−	0.702777i	$-0.248057\pi$
$37$	8.65467	1.42282	0.711410	+	0.702777i	$0.248057\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.53148	−1.17622	−0.588110	−	0.808781i	$-0.700128\pi$
$41$	−7.53148	−1.17622	−0.588110	+	0.808781i	$0.700128\pi$
$42$	0	0
$43$	2.75737	0.420495	0.210247	−	0.977648i	$-0.432573\pi$
$43$	2.75737	0.420495	0.210247	+	0.977648i	$0.432573\pi$
$44$	0	0
$45$	5.83447	0.869752
$46$	0	0
$47$	−5.67933	−0.828416	−0.414208	−	0.910182i	$-0.635941\pi$
$47$	−5.67933	−0.828416	−0.414208	+	0.910182i	$0.635941\pi$
$48$	0	0
$49$	−6.06091	−0.865844
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.1865	−1.67395	−0.836975	−	0.547241i	$-0.815678\pi$
$53$	−12.1865	−1.67395	−0.836975	+	0.547241i	$0.815678\pi$
$54$	0	0
$55$	−0.393210	−0.0530204
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.5931	−1.37910	−0.689551	−	0.724237i	$-0.742192\pi$
$59$	−10.5931	−1.37910	−0.689551	+	0.724237i	$0.742192\pi$
$60$	0	0
$61$	1.88233	0.241007	0.120504	−	0.992713i	$-0.461549\pi$
$61$	1.88233	0.241007	0.120504	+	0.992713i	$0.461549\pi$
$62$	0	0
$63$	2.90721	0.366274
$64$	0	0
$65$	10.8323	1.34358
$66$	0	0
$67$	8.95699	1.09427	0.547135	−	0.837044i	$-0.315718\pi$
$67$	8.95699	1.09427	0.547135	+	0.837044i	$0.315718\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.27497	0.626024	0.313012	−	0.949749i	$-0.398662\pi$
$71$	5.27497	0.626024	0.313012	+	0.949749i	$0.398662\pi$
$72$	0	0
$73$	−9.05092	−1.05933	−0.529665	−	0.848207i	$-0.677682\pi$
$73$	−9.05092	−1.05933	−0.529665	+	0.848207i	$0.677682\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.195929	−0.0223282
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	4.23359	0.464697	0.232348	−	0.972633i	$-0.425359\pi$
$83$	4.23359	0.464697	0.232348	+	0.972633i	$0.425359\pi$
$84$	0	0
$85$	−5.85814	−0.635404
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.91293	−0.838768	−0.419384	−	0.907809i	$-0.637754\pi$
$89$	−7.91293	−0.838768	−0.419384	+	0.907809i	$0.637754\pi$
$90$	0	0
$91$	5.39751	0.565813
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.94482	−0.199535
$96$	0	0
$97$	−12.3246	−1.25138	−0.625689	−	0.780073i	$-0.715182\pi$
$97$	−12.3246	−1.25138	−0.625689	+	0.780073i	$0.715182\pi$
$98$	0	0
$99$	−0.606548	−0.0609604
$100$	0	0
$101$	6.68432	0.665115	0.332557	−	0.943083i	$-0.392089\pi$
$101$	6.68432	0.665115	0.332557	+	0.943083i	$0.392089\pi$
$102$	0	0
$103$	15.1474	1.49251	0.746257	−	0.665658i	$-0.231849\pi$
$103$	15.1474	1.49251	0.746257	+	0.665658i	$0.231849\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.2790	1.28373	0.641866	−	0.766817i	$-0.278161\pi$
$107$	13.2790	1.28373	0.641866	+	0.766817i	$0.278161\pi$
$108$	0	0
$109$	3.19533	0.306057	0.153029	−	0.988222i	$-0.451097\pi$
$109$	3.19533	0.306057	0.153029	+	0.988222i	$0.451097\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.6953	1.85277	0.926387	−	0.376573i	$-0.122898\pi$
$113$	19.6953	1.85277	0.926387	+	0.376573i	$0.122898\pi$
$114$	0	0
$115$	−11.8460	−1.10465
$116$	0	0
$117$	16.7094	1.54478
$118$	0	0
$119$	−2.91900	−0.267584
$120$	0	0
$121$	−10.9591	−0.996284
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0923	1.08156
$126$	0	0
$127$	20.7327	1.83973	0.919863	−	0.392240i	$-0.128300\pi$
$127$	20.7327	1.83973	0.919863	+	0.392240i	$0.128300\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.19154	0.366217	0.183108	−	0.983093i	$-0.441384\pi$
$131$	4.19154	0.366217	0.183108	+	0.983093i	$0.441384\pi$
$132$	0	0
$133$	−0.969069	−0.0840289
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.02851	0.515051	0.257525	−	0.966272i	$-0.417093\pi$
$137$	6.02851	0.515051	0.257525	+	0.966272i	$0.417093\pi$
$138$	0	0
$139$	−15.9788	−1.35531	−0.677654	−	0.735381i	$-0.737003\pi$
$139$	−15.9788	−1.35531	−0.677654	+	0.735381i	$0.737003\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.12612	−0.0941705
$144$	0	0
$145$	12.5793	1.04466
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.01337	0.574558	0.287279	−	0.957847i	$-0.407249\pi$
$149$	7.01337	0.574558	0.287279	+	0.957847i	$0.407249\pi$
$150$	0	0
$151$	14.7433	1.19979	0.599895	−	0.800079i	$-0.295209\pi$
$151$	14.7433	1.19979	0.599895	+	0.800079i	$0.295209\pi$
$152$	0	0
$153$	−9.03650	−0.730558
$154$	0	0
$155$	11.3455	0.911293
$156$	0	0
$157$	−7.06136	−0.563557	−0.281779	−	0.959479i	$-0.590924\pi$
$157$	−7.06136	−0.563557	−0.281779	+	0.959479i	$0.590924\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.90265	−0.465194
$162$	0	0
$163$	12.3905	0.970497	0.485248	−	0.874376i	$-0.338729\pi$
$163$	12.3905	0.970497	0.485248	+	0.874376i	$0.338729\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.8319	1.84417	0.922084	−	0.386990i	$-0.126485\pi$
$167$	23.8319	1.84417	0.922084	+	0.386990i	$0.126485\pi$
$168$	0	0
$169$	18.0226	1.38635
$170$	0	0
$171$	−3.00000	−0.229416
$172$	0	0
$173$	−10.0368	−0.763084	−0.381542	−	0.924352i	$-0.624607\pi$
$173$	−10.0368	−0.763084	−0.381542	+	0.924352i	$0.624607\pi$
$174$	0	0
$175$	1.17999	0.0891991
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.48749	−0.410155	−0.205077	−	0.978746i	$-0.565745\pi$
$179$	−5.48749	−0.410155	−0.205077	+	0.978746i	$0.565745\pi$
$180$	0	0
$181$	12.2243	0.908625	0.454312	−	0.890842i	$-0.349885\pi$
$181$	12.2243	0.908625	0.454312	+	0.890842i	$0.349885\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−16.8318	−1.23750
$186$	0	0
$187$	0.609008	0.0445351
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.55204	0.546447	0.273223	−	0.961951i	$-0.411910\pi$
$191$	7.55204	0.546447	0.273223	+	0.961951i	$0.411910\pi$
$192$	0	0
$193$	18.6413	1.34183	0.670914	−	0.741535i	$-0.265902\pi$
$193$	18.6413	1.34183	0.670914	+	0.741535i	$0.265902\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−14.1324	−1.00689	−0.503445	−	0.864027i	$-0.667934\pi$
$197$	−14.1324	−1.00689	−0.503445	+	0.864027i	$0.667934\pi$
$198$	0	0
$199$	−12.9218	−0.916001	−0.458000	−	0.888952i	$-0.651434\pi$
$199$	−12.9218	−0.916001	−0.458000	+	0.888952i	$0.651434\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.26804	0.439931
$204$	0	0
$205$	14.6474	1.02302
$206$	0	0
$207$	−18.2732	−1.27007
$208$	0	0
$209$	0.202183	0.0139853
$210$	0	0
$211$	12.4115	0.854444	0.427222	−	0.904147i	$-0.359492\pi$
$211$	12.4115	0.854444	0.427222	+	0.904147i	$0.359492\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.36260	−0.365726
$216$	0	0
$217$	5.65325	0.383768
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−16.7771	−1.12855
$222$	0	0
$223$	23.3449	1.56329	0.781645	−	0.623723i	$-0.214381\pi$
$223$	23.3449	1.56329	0.781645	+	0.623723i	$0.214381\pi$
$224$	0	0
$225$	3.65297	0.243531
$226$	0	0
$227$	−9.73366	−0.646046	−0.323023	−	0.946391i	$-0.604699\pi$
$227$	−9.73366	−0.646046	−0.323023	+	0.946391i	$0.604699\pi$
$228$	0	0
$229$	16.9689	1.12133	0.560667	−	0.828041i	$-0.310545\pi$
$229$	16.9689	1.12133	0.560667	+	0.828041i	$0.310545\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.09564	−0.530363	−0.265181	−	0.964199i	$-0.585432\pi$
$233$	−8.09564	−0.530363	−0.265181	+	0.964199i	$0.585432\pi$
$234$	0	0
$235$	11.0453	0.720517
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.56166	0.618492	0.309246	−	0.950982i	$-0.399923\pi$
$239$	9.56166	0.618492	0.309246	+	0.950982i	$0.399923\pi$
$240$	0	0
$241$	13.8100	0.889582	0.444791	−	0.895634i	$-0.353278\pi$
$241$	13.8100	0.889582	0.444791	+	0.895634i	$0.353278\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	11.7874	0.753069
$246$	0	0
$247$	−5.56979	−0.354397
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4.14231	0.261461	0.130730	−	0.991418i	$-0.458268\pi$
$251$	4.14231	0.261461	0.130730	+	0.991418i	$0.458268\pi$
$252$	0	0
$253$	1.23150	0.0774240
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.39737	−0.274301	−0.137150	−	0.990550i	$-0.543794\pi$
$257$	−4.39737	−0.274301	−0.137150	+	0.990550i	$0.543794\pi$
$258$	0	0
$259$	−8.38698	−0.521141
$260$	0	0
$261$	19.4043	1.20110
$262$	0	0
$263$	−12.0955	−0.745842	−0.372921	−	0.927863i	$-0.621644\pi$
$263$	−12.0955	−0.745842	−0.372921	+	0.927863i	$0.621644\pi$
$264$	0	0
$265$	23.7007	1.45592
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−0.368422	−0.0224631	−0.0112315	−	0.999937i	$-0.503575\pi$
$269$	−0.368422	−0.0224631	−0.0112315	+	0.999937i	$0.503575\pi$
$270$	0	0
$271$	−18.3494	−1.11465	−0.557324	−	0.830295i	$-0.688172\pi$
$271$	−18.3494	−1.11465	−0.557324	+	0.830295i	$0.688172\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.246189	−0.0148458
$276$	0	0
$277$	32.7165	1.96574	0.982871	−	0.184293i	$-0.0589995\pi$
$277$	32.7165	1.96574	0.982871	+	0.184293i	$0.0589995\pi$
$278$	0	0
$279$	17.5011	1.04776
$280$	0	0
$281$	−13.9586	−0.832703	−0.416351	−	0.909204i	$-0.636691\pi$
$281$	−13.9586	−0.832703	−0.416351	+	0.909204i	$0.636691\pi$
$282$	0	0
$283$	−22.7577	−1.35281	−0.676404	−	0.736531i	$-0.736463\pi$
$283$	−22.7577	−1.35281	−0.676404	+	0.736531i	$0.736463\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.29852	0.430818
$288$	0	0
$289$	−7.92685	−0.466285
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.6839	1.14995	0.574973	−	0.818172i	$-0.305012\pi$
$293$	19.6839	1.14995	0.574973	+	0.818172i	$0.305012\pi$
$294$	0	0
$295$	20.6017	1.19948
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−33.9259	−1.96198
$300$	0	0
$301$	−2.67208	−0.154016
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3.66079	−0.209616
$306$	0	0
$307$	−20.1331	−1.14906	−0.574528	−	0.818485i	$-0.694815\pi$
$307$	−20.1331	−1.14906	−0.574528	+	0.818485i	$0.694815\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.7448	−0.722689	−0.361344	−	0.932432i	$-0.617682\pi$
$311$	−12.7448	−0.722689	−0.361344	+	0.932432i	$0.617682\pi$
$312$	0	0
$313$	−11.0674	−0.625568	−0.312784	−	0.949824i	$-0.601262\pi$
$313$	−11.0674	−0.625568	−0.312784	+	0.949824i	$0.601262\pi$
$314$	0	0
$315$	−5.65401	−0.318567
$316$	0	0
$317$	−6.62769	−0.372248	−0.186124	−	0.982526i	$-0.559593\pi$
$317$	−6.62769	−0.372248	−0.186124	+	0.982526i	$0.559593\pi$
$318$	0	0
$319$	−1.30774	−0.0732194
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.01217	0.167601
$324$	0	0
$325$	6.78209	0.376203
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.50367	0.303427
$330$	0	0
$331$	−18.2445	−1.00281	−0.501405	−	0.865213i	$-0.667183\pi$
$331$	−18.2445	−1.00281	−0.501405	+	0.865213i	$0.667183\pi$
$332$	0	0
$333$	−25.9640	−1.42282
$334$	0	0
$335$	−17.4198	−0.951744
$336$	0	0
$337$	−30.8573	−1.68091	−0.840453	−	0.541884i	$-0.817711\pi$
$337$	−30.8573	−1.68091	−0.840453	+	0.541884i	$0.817711\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.17947	−0.0638720
$342$	0	0
$343$	12.6569	0.683409
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.7593	−0.631273	−0.315637	−	0.948880i	$-0.602218\pi$
$347$	−11.7593	−0.631273	−0.315637	+	0.948880i	$0.602218\pi$
$348$	0	0
$349$	−20.3591	−1.08980	−0.544898	−	0.838503i	$-0.683431\pi$
$349$	−20.3591	−1.08980	−0.544898	+	0.838503i	$0.683431\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	24.8354	1.32186	0.660929	−	0.750449i	$-0.270163\pi$
$353$	24.8354	1.32186	0.660929	+	0.750449i	$0.270163\pi$
$354$	0	0
$355$	−10.2589	−0.544486
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.1124	1.21983	0.609913	−	0.792469i	$-0.291205\pi$
$359$	23.1124	1.21983	0.609913	+	0.792469i	$0.291205\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	17.6024	0.921354
$366$	0	0
$367$	25.8920	1.35155	0.675775	−	0.737108i	$-0.263809\pi$
$367$	25.8920	1.35155	0.675775	+	0.737108i	$0.263809\pi$
$368$	0	0
$369$	22.5944	1.17622
$370$	0	0
$371$	11.8096	0.613124
$372$	0	0
$373$	−33.1282	−1.71531	−0.857656	−	0.514224i	$-0.828080\pi$
$373$	−33.1282	−1.71531	−0.857656	+	0.514224i	$0.828080\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	36.0260	1.85544
$378$	0	0
$379$	−4.51614	−0.231978	−0.115989	−	0.993250i	$-0.537004\pi$
$379$	−4.51614	−0.231978	−0.115989	+	0.993250i	$0.537004\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.7382	−1.11077	−0.555385	−	0.831593i	$-0.687429\pi$
$383$	−21.7382	−1.11077	−0.555385	+	0.831593i	$0.687429\pi$
$384$	0	0
$385$	0.381047	0.0194200
$386$	0	0
$387$	−8.27211	−0.420495
$388$	0	0
$389$	19.5420	0.990820	0.495410	−	0.868659i	$-0.335018\pi$
$389$	19.5420	0.990820	0.495410	+	0.868659i	$0.335018\pi$
$390$	0	0
$391$	18.3473	0.927861
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.94482	−0.0978548
$396$	0	0
$397$	−13.5986	−0.682496	−0.341248	−	0.939973i	$-0.610850\pi$
$397$	−13.5986	−0.682496	−0.341248	+	0.939973i	$0.610850\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.7974	1.78763	0.893817	−	0.448431i	$-0.148017\pi$
$401$	35.7974	1.78763	0.893817	+	0.448431i	$0.148017\pi$
$402$	0	0
$403$	32.4925	1.61856
$404$	0	0
$405$	−17.5034	−0.869752
$406$	0	0
$407$	1.74983	0.0867356
$408$	0	0
$409$	0.401479	0.0198519	0.00992594	−	0.999951i	$-0.496840\pi$
$409$	0.401479	0.0198519	0.00992594	+	0.999951i	$0.496840\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.2654	0.505129
$414$	0	0
$415$	−8.23359	−0.404171
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.924684	0.0451738	0.0225869	−	0.999745i	$-0.492810\pi$
$419$	0.924684	0.0451738	0.0225869	+	0.999745i	$0.492810\pi$
$420$	0	0
$421$	−19.3875	−0.944888	−0.472444	−	0.881361i	$-0.656628\pi$
$421$	−19.3875	−0.944888	−0.472444	+	0.881361i	$0.656628\pi$
$422$	0	0
$423$	17.0380	0.828416
$424$	0	0
$425$	−3.66778	−0.177914
$426$	0	0
$427$	−1.82410	−0.0882745
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.7203	0.516380	0.258190	−	0.966094i	$-0.416874\pi$
$431$	10.7203	0.516380	0.258190	+	0.966094i	$0.416874\pi$
$432$	0	0
$433$	33.5768	1.61360	0.806800	−	0.590825i	$-0.201198\pi$
$433$	33.5768	1.61360	0.806800	+	0.590825i	$0.201198\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.09105	0.291374
$438$	0	0
$439$	22.9305	1.09441	0.547206	−	0.836998i	$-0.315691\pi$
$439$	22.9305	1.09441	0.547206	+	0.836998i	$0.315691\pi$
$440$	0	0
$441$	18.1827	0.865844
$442$	0	0
$443$	22.7061	1.07880	0.539399	−	0.842050i	$-0.318651\pi$
$443$	22.7061	1.07880	0.539399	+	0.842050i	$0.318651\pi$
$444$	0	0
$445$	15.3893	0.729521
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13.4201	0.633335	0.316668	−	0.948537i	$-0.397436\pi$
$449$	13.4201	0.633335	0.316668	+	0.948537i	$0.397436\pi$
$450$	0	0
$451$	−1.52274	−0.0717028
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.4972	−0.492117
$456$	0	0
$457$	−5.34889	−0.250210	−0.125105	−	0.992143i	$-0.539927\pi$
$457$	−5.34889	−0.250210	−0.125105	+	0.992143i	$0.539927\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.0556	0.794360	0.397180	−	0.917741i	$-0.369989\pi$
$461$	17.0556	0.794360	0.397180	+	0.917741i	$0.369989\pi$
$462$	0	0
$463$	−17.2936	−0.803703	−0.401852	−	0.915705i	$-0.631633\pi$
$463$	−17.2936	−0.803703	−0.401852	+	0.915705i	$0.631633\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.0380	0.834701	0.417350	−	0.908746i	$-0.362959\pi$
$467$	18.0380	0.834701	0.417350	+	0.908746i	$0.362959\pi$
$468$	0	0
$469$	−8.67994	−0.400802
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0.557492	0.0256335
$474$	0	0
$475$	−1.21766	−0.0558699
$476$	0	0
$477$	36.5596	1.67395
$478$	0	0
$479$	−13.2513	−0.605468	−0.302734	−	0.953075i	$-0.597899\pi$
$479$	−13.2513	−0.605468	−0.302734	+	0.953075i	$0.597899\pi$
$480$	0	0
$481$	−48.2047	−2.19795
$482$	0	0
$483$	0	0
$484$	0	0
$485$	23.9693	1.08839
$486$	0	0
$487$	34.8466	1.57905	0.789525	−	0.613719i	$-0.210327\pi$
$487$	34.8466	1.57905	0.789525	+	0.613719i	$0.210327\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.28039	0.148042	0.0740210	−	0.997257i	$-0.476417\pi$
$491$	3.28039	0.148042	0.0740210	+	0.997257i	$0.476417\pi$
$492$	0	0
$493$	−19.4830	−0.877471
$494$	0	0
$495$	1.17963	0.0530204
$496$	0	0
$497$	−5.11181	−0.229296
$498$	0	0
$499$	−24.4043	−1.09248	−0.546242	−	0.837627i	$-0.683942\pi$
$499$	−24.4043	−1.09248	−0.546242	+	0.837627i	$0.683942\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.92788	−0.219723	−0.109862	−	0.993947i	$-0.535041\pi$
$503$	−4.92788	−0.219723	−0.109862	+	0.993947i	$0.535041\pi$
$504$	0	0
$505$	−12.9998	−0.578485
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.19457	−0.0972728	−0.0486364	−	0.998817i	$-0.515488\pi$
$509$	−2.19457	−0.0972728	−0.0486364	+	0.998817i	$0.515488\pi$
$510$	0	0
$511$	8.77096	0.388004
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−29.4590	−1.29812
$516$	0	0
$517$	−1.14826	−0.0505006
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.2667	0.975524	0.487762	−	0.872977i	$-0.337813\pi$
$521$	22.2667	0.975524	0.487762	+	0.872977i	$0.337813\pi$
$522$	0	0
$523$	14.2664	0.623826	0.311913	−	0.950111i	$-0.399030\pi$
$523$	14.2664	0.623826	0.311913	+	0.950111i	$0.399030\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.5721	−0.765451
$528$	0	0
$529$	14.1009	0.613082
$530$	0	0
$531$	31.7793	1.37910
$532$	0	0
$533$	41.9488	1.81700
$534$	0	0
$535$	−25.8254	−1.11653
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.22541	−0.0527822
$540$	0	0
$541$	−29.0710	−1.24986	−0.624931	−	0.780680i	$-0.714873\pi$
$541$	−29.0710	−1.24986	−0.624931	+	0.780680i	$0.714873\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.21436	−0.266194
$546$	0	0
$547$	−39.2386	−1.67772	−0.838862	−	0.544345i	$-0.816778\pi$
$547$	−39.2386	−1.67772	−0.838862	+	0.544345i	$0.816778\pi$
$548$	0	0
$549$	−5.64698	−0.241007
$550$	0	0
$551$	−6.46811	−0.275551
$552$	0	0
$553$	−0.969069	−0.0412090
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.35817	0.227033	0.113517	−	0.993536i	$-0.463789\pi$
$557$	5.35817	0.227033	0.113517	+	0.993536i	$0.463789\pi$
$558$	0	0
$559$	−15.3580	−0.649573
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−26.4894	−1.11640	−0.558198	−	0.829708i	$-0.688507\pi$
$563$	−26.4894	−1.11640	−0.558198	+	0.829708i	$0.688507\pi$
$564$	0	0
$565$	−38.3038	−1.61145
$566$	0	0
$567$	−8.72162	−0.366274
$568$	0	0
$569$	38.4626	1.61243	0.806217	−	0.591620i	$-0.201511\pi$
$569$	38.4626	1.61243	0.806217	+	0.591620i	$0.201511\pi$
$570$	0	0
$571$	−13.4314	−0.562088	−0.281044	−	0.959695i	$-0.590681\pi$
$571$	−13.4314	−0.562088	−0.281044	+	0.959695i	$0.590681\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−7.41681	−0.309302
$576$	0	0
$577$	26.0959	1.08639	0.543194	−	0.839607i	$-0.317215\pi$
$577$	26.0959	1.08639	0.543194	+	0.839607i	$0.317215\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.10264	−0.170206
$582$	0	0
$583$	−2.46391	−0.102045
$584$	0	0
$585$	−32.4968	−1.34358
$586$	0	0
$587$	29.5272	1.21872	0.609360	−	0.792894i	$-0.291427\pi$
$587$	29.5272	1.21872	0.609360	+	0.792894i	$0.291427\pi$
$588$	0	0
$589$	−5.83369	−0.240373
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−21.9870	−0.902898	−0.451449	−	0.892297i	$-0.649093\pi$
$593$	−21.9870	−0.902898	−0.451449	+	0.892297i	$0.649093\pi$
$594$	0	0
$595$	5.67694	0.232732
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.5233	−1.61488	−0.807440	−	0.589950i	$-0.799148\pi$
$599$	−39.5233	−1.61488	−0.807440	+	0.589950i	$0.799148\pi$
$600$	0	0
$601$	−3.24732	−0.132461	−0.0662305	−	0.997804i	$-0.521097\pi$
$601$	−3.24732	−0.132461	−0.0662305	+	0.997804i	$0.521097\pi$
$602$	0	0
$603$	−26.8710	−1.09427
$604$	0	0
$605$	21.3136	0.866520
$606$	0	0
$607$	−13.4957	−0.547773	−0.273887	−	0.961762i	$-0.588309\pi$
$607$	−13.4957	−0.547773	−0.273887	+	0.961762i	$0.588309\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.6327	1.27972
$612$	0	0
$613$	12.3846	0.500209	0.250104	−	0.968219i	$-0.419535\pi$
$613$	12.3846	0.500209	0.250104	+	0.968219i	$0.419535\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.3042	−1.66285	−0.831423	−	0.555640i	$-0.812473\pi$
$617$	−41.3042	−1.66285	−0.831423	+	0.555640i	$0.812473\pi$
$618$	0	0
$619$	−25.6149	−1.02955	−0.514775	−	0.857326i	$-0.672124\pi$
$619$	−25.6149	−1.02955	−0.514775	+	0.857326i	$0.672124\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.66817	0.307219
$624$	0	0
$625$	−17.4290	−0.697161
$626$	0	0
$627$	0	0
$628$	0	0
$629$	26.0693	1.03945
$630$	0	0
$631$	33.7156	1.34220	0.671099	−	0.741368i	$-0.265823\pi$
$631$	33.7156	1.34220	0.671099	+	0.741368i	$0.265823\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−40.3214	−1.60011
$636$	0	0
$637$	33.7580	1.33754
$638$	0	0
$639$	−15.8249	−0.626024
$640$	0	0
$641$	−1.97060	−0.0778342	−0.0389171	−	0.999242i	$-0.512391\pi$
$641$	−1.97060	−0.0778342	−0.0389171	+	0.999242i	$0.512391\pi$
$642$	0	0
$643$	−36.2746	−1.43053	−0.715266	−	0.698853i	$-0.753694\pi$
$643$	−36.2746	−1.43053	−0.715266	+	0.698853i	$0.753694\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.81188	−0.267803	−0.133901	−	0.990995i	$-0.542750\pi$
$647$	−6.81188	−0.267803	−0.133901	+	0.990995i	$0.542750\pi$
$648$	0	0
$649$	−2.14174	−0.0840706
$650$	0	0
$651$	0	0
$652$	0	0
$653$	35.3988	1.38526	0.692631	−	0.721292i	$-0.256452\pi$
$653$	35.3988	1.38526	0.692631	+	0.721292i	$0.256452\pi$
$654$	0	0
$655$	−8.15182	−0.318518
$656$	0	0
$657$	27.1527	1.05933
$658$	0	0
$659$	−11.4679	−0.446725	−0.223362	−	0.974735i	$-0.571703\pi$
$659$	−11.4679	−0.446725	−0.223362	+	0.974735i	$0.571703\pi$
$660$	0	0
$661$	−30.6931	−1.19382	−0.596911	−	0.802308i	$-0.703605\pi$
$661$	−30.6931	−1.19382	−0.596911	+	0.802308i	$0.703605\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.88467	0.0730843
$666$	0	0
$667$	−39.3976	−1.52548
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0.380574	0.0146919
$672$	0	0
$673$	−26.2566	−1.01212	−0.506059	−	0.862499i	$-0.668898\pi$
$673$	−26.2566	−1.01212	−0.506059	+	0.862499i	$0.668898\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	24.0683	0.925021	0.462510	−	0.886614i	$-0.346949\pi$
$677$	24.0683	0.925021	0.462510	+	0.886614i	$0.346949\pi$
$678$	0	0
$679$	11.9434	0.458347
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19.2822	−0.737813	−0.368907	−	0.929466i	$-0.620268\pi$
$683$	−19.2822	−0.737813	−0.368907	+	0.929466i	$0.620268\pi$
$684$	0	0
$685$	−11.7244	−0.447966
$686$	0	0
$687$	0	0
$688$	0	0
$689$	67.8765	2.58589
$690$	0	0
$691$	20.9343	0.796378	0.398189	−	0.917303i	$-0.369639\pi$
$691$	20.9343	0.796378	0.398189	+	0.917303i	$0.369639\pi$
$692$	0	0
$693$	0.587787	0.0223282
$694$	0	0
$695$	31.0761	1.17878
$696$	0	0
$697$	−22.6861	−0.859297
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−19.8817	−0.750922	−0.375461	−	0.926838i	$-0.622516\pi$
$701$	−19.8817	−0.750922	−0.375461	+	0.926838i	$0.622516\pi$
$702$	0	0
$703$	8.65467	0.326417
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.47756	−0.243614
$708$	0	0
$709$	−23.8869	−0.897090	−0.448545	−	0.893760i	$-0.648058\pi$
$709$	−23.8869	−0.897090	−0.448545	+	0.893760i	$0.648058\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	0	0
$713$	−35.5333	−1.33073
$714$	0	0
$715$	2.19010	0.0819050
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.9661	0.707316	0.353658	−	0.935375i	$-0.384938\pi$
$719$	18.9661	0.707316	0.353658	+	0.935375i	$0.384938\pi$
$720$	0	0
$721$	−14.6788	−0.546669
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.87594	0.292505
$726$	0	0
$727$	22.3342	0.828328	0.414164	−	0.910202i	$-0.364074\pi$
$727$	22.3342	0.828328	0.414164	+	0.910202i	$0.364074\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	8.30565	0.307196
$732$	0	0
$733$	44.6152	1.64790	0.823949	−	0.566663i	$-0.191766\pi$
$733$	44.6152	1.64790	0.823949	+	0.566663i	$0.191766\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.81095	0.0667071
$738$	0	0
$739$	−16.1349	−0.593531	−0.296766	−	0.954950i	$-0.595908\pi$
$739$	−16.1349	−0.593531	−0.296766	+	0.954950i	$0.595908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	45.1469	1.65628	0.828139	−	0.560523i	$-0.189400\pi$
$743$	45.1469	1.65628	0.828139	+	0.560523i	$0.189400\pi$
$744$	0	0
$745$	−13.6398	−0.499723
$746$	0	0
$747$	−12.7008	−0.464697
$748$	0	0
$749$	−12.8683	−0.470197
$750$	0	0
$751$	18.9280	0.690694	0.345347	−	0.938475i	$-0.387761\pi$
$751$	18.9280	0.690694	0.345347	+	0.938475i	$0.387761\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−28.6731	−1.04352
$756$	0	0
$757$	−44.8498	−1.63010	−0.815048	−	0.579394i	$-0.803289\pi$
$757$	−44.8498	−1.63010	−0.815048	+	0.579394i	$0.803289\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.18519	0.0792129	0.0396065	−	0.999215i	$-0.487390\pi$
$761$	2.18519	0.0792129	0.0396065	+	0.999215i	$0.487390\pi$
$762$	0	0
$763$	−3.09650	−0.112101
$764$	0	0
$765$	17.5744	0.635404
$766$	0	0
$767$	59.0013	2.13041
$768$	0	0
$769$	20.1606	0.727009	0.363504	−	0.931593i	$-0.381580\pi$
$769$	20.1606	0.727009	0.363504	+	0.931593i	$0.381580\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−16.4852	−0.592933	−0.296466	−	0.955043i	$-0.595808\pi$
$773$	−16.4852	−0.592933	−0.296466	+	0.955043i	$0.595808\pi$
$774$	0	0
$775$	7.10343	0.255163
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.53148	−0.269843
$780$	0	0
$781$	1.06651	0.0381626
$782$	0	0
$783$	0	0
$784$	0	0
$785$	13.7331	0.490155
$786$	0	0
$787$	−24.5662	−0.875691	−0.437846	−	0.899050i	$-0.644258\pi$
$787$	−24.5662	−0.875691	−0.437846	+	0.899050i	$0.644258\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19.0861	−0.678622
$792$	0	0
$793$	−10.4842	−0.372303
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.6913	−0.768347	−0.384173	−	0.923261i	$-0.625513\pi$
$797$	−21.6913	−0.768347	−0.384173	+	0.923261i	$0.625513\pi$
$798$	0	0
$799$	−17.1071	−0.605206
$800$	0	0
$801$	23.7388	0.838768
$802$	0	0
$803$	−1.82994	−0.0645771
$804$	0	0
$805$	11.4796	0.404603
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.94747	0.138786	0.0693928	−	0.997589i	$-0.477894\pi$
$809$	3.94747	0.138786	0.0693928	+	0.997589i	$0.477894\pi$
$810$	0	0
$811$	45.2087	1.58749	0.793747	−	0.608248i	$-0.208128\pi$
$811$	45.2087	1.58749	0.793747	+	0.608248i	$0.208128\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−24.0973	−0.844092
$816$	0	0
$817$	2.75737	0.0964681
$818$	0	0
$819$	−16.1925	−0.565813
$820$	0	0
$821$	50.2213	1.75273	0.876367	−	0.481643i	$-0.159960\pi$
$821$	50.2213	1.75273	0.876367	+	0.481643i	$0.159960\pi$
$822$	0	0
$823$	−29.1103	−1.01472	−0.507361	−	0.861733i	$-0.669379\pi$
$823$	−29.1103	−1.01472	−0.507361	+	0.861733i	$0.669379\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14.2082	0.494068	0.247034	−	0.969007i	$-0.420544\pi$
$827$	14.2082	0.494068	0.247034	+	0.969007i	$0.420544\pi$
$828$	0	0
$829$	38.9336	1.35222	0.676109	−	0.736801i	$-0.263665\pi$
$829$	38.9336	1.35222	0.676109	+	0.736801i	$0.263665\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.2565	−0.632549
$834$	0	0
$835$	−46.3489	−1.60397
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.01237	−0.311141	−0.155571	−	0.987825i	$-0.549722\pi$
$839$	−9.01237	−0.311141	−0.155571	+	0.987825i	$0.549722\pi$
$840$	0	0
$841$	12.8365	0.442637
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−35.0508	−1.20578
$846$	0	0
$847$	10.6201	0.364912
$848$	0	0
$849$	0	0
$850$	0	0
$851$	52.7161	1.80708
$852$	0	0
$853$	−19.0696	−0.652931	−0.326465	−	0.945209i	$-0.605858\pi$
$853$	−19.0696	−0.652931	−0.326465	+	0.945209i	$0.605858\pi$
$854$	0	0
$855$	5.83447	0.199535
$856$	0	0
$857$	−7.43388	−0.253937	−0.126968	−	0.991907i	$-0.540525\pi$
$857$	−7.43388	−0.253937	−0.126968	+	0.991907i	$0.540525\pi$
$858$	0	0
$859$	−27.5873	−0.941267	−0.470634	−	0.882329i	$-0.655975\pi$
$859$	−27.5873	−0.941267	−0.470634	+	0.882329i	$0.655975\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−4.59800	−0.156518	−0.0782588	−	0.996933i	$-0.524936\pi$
$863$	−4.59800	−0.156518	−0.0782588	+	0.996933i	$0.524936\pi$
$864$	0	0
$865$	19.5198	0.663694
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.202183	0.00685858
$870$	0	0
$871$	−49.8886	−1.69041
$872$	0	0
$873$	36.9739	1.25138
$874$	0	0
$875$	−11.7182	−0.396148
$876$	0	0
$877$	42.4654	1.43396	0.716978	−	0.697096i	$-0.245525\pi$
$877$	42.4654	1.43396	0.716978	+	0.697096i	$0.245525\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−44.8572	−1.51128	−0.755640	−	0.654988i	$-0.772674\pi$
$881$	−44.8572	−1.51128	−0.755640	+	0.654988i	$0.772674\pi$
$882$	0	0
$883$	−55.7323	−1.87554	−0.937770	−	0.347256i	$-0.887113\pi$
$883$	−55.7323	−1.87554	−0.937770	+	0.347256i	$0.887113\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.9354	1.77740	0.888699	−	0.458492i	$-0.151610\pi$
$887$	52.9354	1.77740	0.888699	+	0.458492i	$0.151610\pi$
$888$	0	0
$889$	−20.0914	−0.673843
$890$	0	0
$891$	1.81964	0.0609604
$892$	0	0
$893$	−5.67933	−0.190052
$894$	0	0
$895$	10.6722	0.356733
$896$	0	0
$897$	0	0
$898$	0	0
$899$	37.7330	1.25847
$900$	0	0
$901$	−36.7079	−1.22292
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.7741	−0.790278
$906$	0	0
$907$	−6.64552	−0.220661	−0.110330	−	0.993895i	$-0.535191\pi$
$907$	−6.64552	−0.220661	−0.110330	+	0.993895i	$0.535191\pi$
$908$	0	0
$909$	−20.0530	−0.665115
$910$	0	0
$911$	22.9453	0.760213	0.380107	−	0.924943i	$-0.375887\pi$
$911$	22.9453	0.760213	0.380107	+	0.924943i	$0.375887\pi$
$912$	0	0
$913$	0.855958	0.0283281
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.06189	−0.134136
$918$	0	0
$919$	−38.1037	−1.25693	−0.628463	−	0.777839i	$-0.716316\pi$
$919$	−38.1037	−1.25693	−0.628463	+	0.777839i	$0.716316\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−29.3805	−0.967071
$924$	0	0
$925$	−10.5384	−0.346501
$926$	0	0
$927$	−45.4421	−1.49251
$928$	0	0
$929$	35.0523	1.15003	0.575014	−	0.818143i	$-0.304996\pi$
$929$	35.0523	1.15003	0.575014	+	0.818143i	$0.304996\pi$
$930$	0	0
$931$	−6.06091	−0.198638
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.18441	−0.0387345
$936$	0	0
$937$	25.5571	0.834914	0.417457	−	0.908697i	$-0.362921\pi$
$937$	25.5571	0.834914	0.417457	+	0.908697i	$0.362921\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	32.6856	1.06552	0.532760	−	0.846266i	$-0.321155\pi$
$941$	32.6856	1.06552	0.532760	+	0.846266i	$0.321155\pi$
$942$	0	0
$943$	−45.8746	−1.49388
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.6671	0.346635	0.173318	−	0.984866i	$-0.444551\pi$
$947$	10.6671	0.346635	0.173318	+	0.984866i	$0.444551\pi$
$948$	0	0
$949$	50.4117	1.63643
$950$	0	0
$951$	0	0
$952$	0	0
$953$	47.5537	1.54041	0.770207	−	0.637794i	$-0.220153\pi$
$953$	47.5537	1.54041	0.770207	+	0.637794i	$0.220153\pi$
$954$	0	0
$955$	−14.6874	−0.475273
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.84204	−0.188649
$960$	0	0
$961$	3.03196	0.0978053
$962$	0	0
$963$	−39.8371	−1.28373
$964$	0	0
$965$	−36.2540	−1.16706
$966$	0	0
$967$	13.9357	0.448141	0.224071	−	0.974573i	$-0.428065\pi$
$967$	13.9357	0.448141	0.224071	+	0.974573i	$0.428065\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.2793	0.458245	0.229122	−	0.973398i	$-0.426414\pi$
$971$	14.2793	0.458245	0.229122	+	0.973398i	$0.426414\pi$
$972$	0	0
$973$	15.4846	0.496413
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.0764	1.05821	0.529104	−	0.848557i	$-0.322528\pi$
$977$	33.0764	1.05821	0.529104	+	0.848557i	$0.322528\pi$
$978$	0	0
$979$	−1.59986	−0.0511316
$980$	0	0
$981$	−9.58600	−0.306057
$982$	0	0
$983$	−34.6310	−1.10456	−0.552279	−	0.833660i	$-0.686242\pi$
$983$	−34.6310	−1.10456	−0.552279	+	0.833660i	$0.686242\pi$
$984$	0	0
$985$	27.4850	0.875745
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.7953	0.534059
$990$	0	0
$991$	21.1870	0.673026	0.336513	−	0.941679i	$-0.390752\pi$
$991$	21.1870	0.673026	0.336513	+	0.941679i	$0.390752\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.1306	0.796694
$996$	0	0
$997$	17.2962	0.547777	0.273889	−	0.961761i	$-0.411690\pi$
$997$	17.2962	0.547777	0.273889	+	0.961761i	$0.411690\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.d.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.d.1.2	✓	8	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)

\(f(q)\)	\(=\)	\(q-1.94482 q^{5} -0.969069 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.94482 q^{5} -0.969069 q^{7} -3.00000 q^{9} +0.202183 q^{11} -5.56979 q^{13} +3.01217 q^{17} +1.00000 q^{19} +6.09105 q^{23} -1.21766 q^{25} -6.46811 q^{29} -5.83369 q^{31} +1.88467 q^{35} +8.65467 q^{37} -7.53148 q^{41} +2.75737 q^{43} +5.83447 q^{45} -5.67933 q^{47} -6.06091 q^{49} -12.1865 q^{53} -0.393210 q^{55} -10.5931 q^{59} +1.88233 q^{61} +2.90721 q^{63} +10.8323 q^{65} +8.95699 q^{67} +5.27497 q^{71} -9.05092 q^{73} -0.195929 q^{77} +1.00000 q^{79} +9.00000 q^{81} +4.23359 q^{83} -5.85814 q^{85} -7.91293 q^{89} +5.39751 q^{91} -1.94482 q^{95} -12.3246 q^{97} -0.606548 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + q^{7} - 24 q^{9}+O(q^{10}) \) 8 * q + 4 * q^5 + q^7 - 24 * q^9	\( 8 q + 4 q^{5} + q^{7} - 24 q^{9} - 7 q^{11} - 12 q^{13} - 7 q^{17} + 8 q^{19} + 10 q^{23} + 6 q^{25} + 5 q^{29} + 3 q^{31} - 11 q^{35} - 15 q^{37} - 5 q^{41} + 10 q^{43} - 12 q^{45} + 18 q^{47} + 31 q^{49} + 9 q^{53} - 17 q^{55} + 8 q^{59} + 13 q^{61} - 3 q^{63} + 4 q^{65} + 21 q^{67} + 44 q^{71} - 20 q^{73} + 15 q^{77} + 8 q^{79} + 72 q^{81} - 4 q^{83} - 9 q^{85} - 10 q^{89} + 48 q^{91} + 4 q^{95} - 22 q^{97} + 21 q^{99}+O(q^{100}) \) 8 * q + 4 * q^5 + q^7 - 24 * q^9 - 7 * q^11 - 12 * q^13 - 7 * q^17 + 8 * q^19 + 10 * q^23 + 6 * q^25 + 5 * q^29 + 3 * q^31 - 11 * q^35 - 15 * q^37 - 5 * q^41 + 10 * q^43 - 12 * q^45 + 18 * q^47 + 31 * q^49 + 9 * q^53 - 17 * q^55 + 8 * q^59 + 13 * q^61 - 3 * q^63 + 4 * q^65 + 21 * q^67 + 44 * q^71 - 20 * q^73 + 15 * q^77 + 8 * q^79 + 72 * q^81 - 4 * q^83 - 9 * q^85 - 10 * q^89 + 48 * q^91 + 4 * q^95 - 22 * q^97 + 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more