LMFDB - Embedded newform 6240.2.a.bp.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6240,2,Mod(1,6240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6240, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6240.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:	$49.8266508613$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	6240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} -6.37228 q^{11} -1.00000 q^{13} -1.00000 q^{15} +0.372281 q^{17} +4.74456 q^{19} +2.37228 q^{21} +1.62772 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.74456 q^{29} -6.37228 q^{33} -2.37228 q^{35} -3.62772 q^{37} -1.00000 q^{39} +9.11684 q^{41} -0.744563 q^{43} -1.00000 q^{45} +4.00000 q^{47} -1.37228 q^{49} +0.372281 q^{51} +4.37228 q^{53} +6.37228 q^{55} +4.74456 q^{57} -4.00000 q^{59} +8.37228 q^{61} +2.37228 q^{63} +1.00000 q^{65} +13.4891 q^{67} +1.62772 q^{69} +5.62772 q^{71} +10.0000 q^{73} +1.00000 q^{75} -15.1168 q^{77} +2.37228 q^{79} +1.00000 q^{81} -13.4891 q^{83} -0.372281 q^{85} -2.74456 q^{87} -3.62772 q^{89} -2.37228 q^{91} -4.74456 q^{95} -0.372281 q^{97} -6.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 7 q^{33} + q^{35} - 13 q^{37} - 2 q^{39} + q^{41} + 10 q^{43} - 2 q^{45} + 8 q^{47} + 3 q^{49} - 5 q^{51} + 3 q^{53} + 7 q^{55} - 2 q^{57} - 8 q^{59} + 11 q^{61} - q^{63} + 2 q^{65} + 4 q^{67} + 9 q^{69} + 17 q^{71} + 20 q^{73} + 2 q^{75} - 13 q^{77} - q^{79} + 2 q^{81} - 4 q^{83} + 5 q^{85} + 6 q^{87} - 13 q^{89} + q^{91} + 2 q^{95} + 5 q^{97} - 7 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 2 * q^19 - q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 7 * q^33 + q^35 - 13 * q^37 - 2 * q^39 + q^41 + 10 * q^43 - 2 * q^45 + 8 * q^47 + 3 * q^49 - 5 * q^51 + 3 * q^53 + 7 * q^55 - 2 * q^57 - 8 * q^59 + 11 * q^61 - q^63 + 2 * q^65 + 4 * q^67 + 9 * q^69 + 17 * q^71 + 20 * q^73 + 2 * q^75 - 13 * q^77 - q^79 + 2 * q^81 - 4 * q^83 + 5 * q^85 + 6 * q^87 - 13 * q^89 + q^91 + 2 * q^95 + 5 * q^97 - 7 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.37228	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$7$	2.37228	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.37228	−1.92132	−0.960658	−	0.277736i	$-0.910416\pi$
$11$	−6.37228	−1.92132	−0.960658	+	0.277736i	$0.910416\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0.372281	0.0902915	0.0451457	−	0.998980i	$-0.485625\pi$
$17$	0.372281	0.0902915	0.0451457	+	0.998980i	$0.485625\pi$
$18$	0	0
$19$	4.74456	1.08848	0.544239	−	0.838930i	$-0.316819\pi$
$19$	4.74456	1.08848	0.544239	+	0.838930i	$0.316819\pi$
$20$	0	0
$21$	2.37228	0.517674
$22$	0	0
$23$	1.62772	0.339403	0.169701	−	0.985496i	$-0.445720\pi$
$23$	1.62772	0.339403	0.169701	+	0.985496i	$0.445720\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.74456	−0.509652	−0.254826	−	0.966987i	$-0.582018\pi$
$29$	−2.74456	−0.509652	−0.254826	+	0.966987i	$0.582018\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−6.37228	−1.10927
$34$	0	0
$35$	−2.37228	−0.400989
$36$	0	0
$37$	−3.62772	−0.596393	−0.298197	−	0.954504i	$-0.596385\pi$
$37$	−3.62772	−0.596393	−0.298197	+	0.954504i	$0.596385\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	9.11684	1.42381	0.711906	−	0.702275i	$-0.247832\pi$
$41$	9.11684	1.42381	0.711906	+	0.702275i	$0.247832\pi$
$42$	0	0
$43$	−0.744563	−0.113545	−0.0567724	−	0.998387i	$-0.518081\pi$
$43$	−0.744563	−0.113545	−0.0567724	+	0.998387i	$0.518081\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−1.37228	−0.196040
$50$	0	0
$51$	0.372281	0.0521298
$52$	0	0
$53$	4.37228	0.600579	0.300290	−	0.953848i	$-0.402917\pi$
$53$	4.37228	0.600579	0.300290	+	0.953848i	$0.402917\pi$
$54$	0	0
$55$	6.37228	0.859238
$56$	0	0
$57$	4.74456	0.628433
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	8.37228	1.07196	0.535980	−	0.844230i	$-0.319942\pi$
$61$	8.37228	1.07196	0.535980	+	0.844230i	$0.319942\pi$
$62$	0	0
$63$	2.37228	0.298879
$64$	0	0
$65$	1.00000	0.124035
$66$	0	0
$67$	13.4891	1.64796	0.823979	−	0.566620i	$-0.191749\pi$
$67$	13.4891	1.64796	0.823979	+	0.566620i	$0.191749\pi$
$68$	0	0
$69$	1.62772	0.195954
$70$	0	0
$71$	5.62772	0.667887	0.333944	−	0.942593i	$-0.391620\pi$
$71$	5.62772	0.667887	0.333944	+	0.942593i	$0.391620\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−15.1168	−1.72272
$78$	0	0
$79$	2.37228	0.266903	0.133451	−	0.991055i	$-0.457394\pi$
$79$	2.37228	0.266903	0.133451	+	0.991055i	$0.457394\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.4891	−1.48062	−0.740312	−	0.672264i	$-0.765322\pi$
$83$	−13.4891	−1.48062	−0.740312	+	0.672264i	$0.765322\pi$
$84$	0	0
$85$	−0.372281	−0.0403796
$86$	0	0
$87$	−2.74456	−0.294248
$88$	0	0
$89$	−3.62772	−0.384537	−0.192269	−	0.981342i	$-0.561585\pi$
$89$	−3.62772	−0.384537	−0.192269	+	0.981342i	$0.561585\pi$
$90$	0	0
$91$	−2.37228	−0.248683
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.74456	−0.486782
$96$	0	0
$97$	−0.372281	−0.0377994	−0.0188997	−	0.999821i	$-0.506016\pi$
$97$	−0.372281	−0.0377994	−0.0188997	+	0.999821i	$0.506016\pi$
$98$	0	0
$99$	−6.37228	−0.640438
$100$	0	0
$101$	5.25544	0.522936	0.261468	−	0.965212i	$-0.415793\pi$
$101$	5.25544	0.522936	0.261468	+	0.965212i	$0.415793\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−2.37228	−0.231511
$106$	0	0
$107$	−6.37228	−0.616032	−0.308016	−	0.951381i	$-0.599665\pi$
$107$	−6.37228	−0.616032	−0.308016	+	0.951381i	$0.599665\pi$
$108$	0	0
$109$	1.25544	0.120249	0.0601245	−	0.998191i	$-0.480850\pi$
$109$	1.25544	0.120249	0.0601245	+	0.998191i	$0.480850\pi$
$110$	0	0
$111$	−3.62772	−0.344328
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	−1.62772	−0.151786
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0.883156	0.0809588
$120$	0	0
$121$	29.6060	2.69145
$122$	0	0
$123$	9.11684	0.822038
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	−0.744563	−0.0655551
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	11.2554	0.975970
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	5.62772	0.477337	0.238668	−	0.971101i	$-0.423289\pi$
$139$	5.62772	0.477337	0.238668	+	0.971101i	$0.423289\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	6.37228	0.532877
$144$	0	0
$145$	2.74456	0.227924
$146$	0	0
$147$	−1.37228	−0.113184
$148$	0	0
$149$	13.8614	1.13557	0.567785	−	0.823177i	$-0.307800\pi$
$149$	13.8614	1.13557	0.567785	+	0.823177i	$0.307800\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	0	0
$153$	0.372281	0.0300972
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	4.37228	0.346744
$160$	0	0
$161$	3.86141	0.304321
$162$	0	0
$163$	6.37228	0.499116	0.249558	−	0.968360i	$-0.419715\pi$
$163$	6.37228	0.499116	0.249558	+	0.968360i	$0.419715\pi$
$164$	0	0
$165$	6.37228	0.496081
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.74456	0.362826
$172$	0	0
$173$	2.00000	0.152057	0.0760286	−	0.997106i	$-0.475776\pi$
$173$	2.00000	0.152057	0.0760286	+	0.997106i	$0.475776\pi$
$174$	0	0
$175$	2.37228	0.179328
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	8.74456	0.653599	0.326800	−	0.945094i	$-0.394030\pi$
$179$	8.74456	0.653599	0.326800	+	0.945094i	$0.394030\pi$
$180$	0	0
$181$	−9.11684	−0.677650	−0.338825	−	0.940849i	$-0.610029\pi$
$181$	−9.11684	−0.677650	−0.338825	+	0.940849i	$0.610029\pi$
$182$	0	0
$183$	8.37228	0.618897
$184$	0	0
$185$	3.62772	0.266715
$186$	0	0
$187$	−2.37228	−0.173478
$188$	0	0
$189$	2.37228	0.172558
$190$	0	0
$191$	14.2337	1.02991	0.514957	−	0.857216i	$-0.327808\pi$
$191$	14.2337	1.02991	0.514957	+	0.857216i	$0.327808\pi$
$192$	0	0
$193$	−22.6060	−1.62721	−0.813607	−	0.581416i	$-0.802499\pi$
$193$	−22.6060	−1.62721	−0.813607	+	0.581416i	$0.802499\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	−15.4891	−1.10355	−0.551777	−	0.833992i	$-0.686050\pi$
$197$	−15.4891	−1.10355	−0.551777	+	0.833992i	$0.686050\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	13.4891	0.951450
$202$	0	0
$203$	−6.51087	−0.456974
$204$	0	0
$205$	−9.11684	−0.636748
$206$	0	0
$207$	1.62772	0.113134
$208$	0	0
$209$	−30.2337	−2.09131
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	5.62772	0.385605
$214$	0	0
$215$	0.744563	0.0507788
$216$	0	0
$217$	0	0
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−0.372281	−0.0250424
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−8.74456	−0.580397	−0.290199	−	0.956966i	$-0.593721\pi$
$227$	−8.74456	−0.580397	−0.290199	+	0.956966i	$0.593721\pi$
$228$	0	0
$229$	10.7446	0.710021	0.355010	−	0.934862i	$-0.384477\pi$
$229$	10.7446	0.710021	0.355010	+	0.934862i	$0.384477\pi$
$230$	0	0
$231$	−15.1168	−0.994615
$232$	0	0
$233$	−21.8614	−1.43219	−0.716094	−	0.698004i	$-0.754072\pi$
$233$	−21.8614	−1.43219	−0.716094	+	0.698004i	$0.754072\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	2.37228	0.154096
$238$	0	0
$239$	13.6277	0.881504	0.440752	−	0.897629i	$-0.354712\pi$
$239$	13.6277	0.881504	0.440752	+	0.897629i	$0.354712\pi$
$240$	0	0
$241$	−9.25544	−0.596195	−0.298098	−	0.954535i	$-0.596352\pi$
$241$	−9.25544	−0.596195	−0.298098	+	0.954535i	$0.596352\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.37228	0.0876718
$246$	0	0
$247$	−4.74456	−0.301889
$248$	0	0
$249$	−13.4891	−0.854839
$250$	0	0
$251$	29.4891	1.86134	0.930669	−	0.365863i	$-0.119226\pi$
$251$	29.4891	1.86134	0.930669	+	0.365863i	$0.119226\pi$
$252$	0	0
$253$	−10.3723	−0.652100
$254$	0	0
$255$	−0.372281	−0.0233132
$256$	0	0
$257$	−11.4891	−0.716672	−0.358336	−	0.933593i	$-0.616656\pi$
$257$	−11.4891	−0.716672	−0.358336	+	0.933593i	$0.616656\pi$
$258$	0	0
$259$	−8.60597	−0.534749
$260$	0	0
$261$	−2.74456	−0.169884
$262$	0	0
$263$	5.48913	0.338474	0.169237	−	0.985575i	$-0.445870\pi$
$263$	5.48913	0.338474	0.169237	+	0.985575i	$0.445870\pi$
$264$	0	0
$265$	−4.37228	−0.268587
$266$	0	0
$267$	−3.62772	−0.222013
$268$	0	0
$269$	25.7228	1.56835	0.784174	−	0.620541i	$-0.213087\pi$
$269$	25.7228	1.56835	0.784174	+	0.620541i	$0.213087\pi$
$270$	0	0
$271$	−12.7446	−0.774177	−0.387089	−	0.922043i	$-0.626519\pi$
$271$	−12.7446	−0.774177	−0.387089	+	0.922043i	$0.626519\pi$
$272$	0	0
$273$	−2.37228	−0.143577
$274$	0	0
$275$	−6.37228	−0.384263
$276$	0	0
$277$	16.2337	0.975388	0.487694	−	0.873015i	$-0.337838\pi$
$277$	16.2337	0.975388	0.487694	+	0.873015i	$0.337838\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.48913	−0.446764	−0.223382	−	0.974731i	$-0.571710\pi$
$281$	−7.48913	−0.446764	−0.223382	+	0.974731i	$0.571710\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	−4.74456	−0.281044
$286$	0	0
$287$	21.6277	1.27664
$288$	0	0
$289$	−16.8614	−0.991847
$290$	0	0
$291$	−0.372281	−0.0218235
$292$	0	0
$293$	−26.7446	−1.56243	−0.781217	−	0.624260i	$-0.785401\pi$
$293$	−26.7446	−1.56243	−0.781217	+	0.624260i	$0.785401\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−6.37228	−0.369757
$298$	0	0
$299$	−1.62772	−0.0941334
$300$	0	0
$301$	−1.76631	−0.101809
$302$	0	0
$303$	5.25544	0.301917
$304$	0	0
$305$	−8.37228	−0.479395
$306$	0	0
$307$	6.37228	0.363685	0.181843	−	0.983328i	$-0.441794\pi$
$307$	6.37228	0.363685	0.181843	+	0.983328i	$0.441794\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	11.2554	0.638237	0.319119	−	0.947715i	$-0.396613\pi$
$311$	11.2554	0.638237	0.319119	+	0.947715i	$0.396613\pi$
$312$	0	0
$313$	−7.48913	−0.423310	−0.211655	−	0.977344i	$-0.567885\pi$
$313$	−7.48913	−0.423310	−0.211655	+	0.977344i	$0.567885\pi$
$314$	0	0
$315$	−2.37228	−0.133663
$316$	0	0
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	17.4891	0.979203
$320$	0	0
$321$	−6.37228	−0.355666
$322$	0	0
$323$	1.76631	0.0982802
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	0	0
$327$	1.25544	0.0694258
$328$	0	0
$329$	9.48913	0.523152
$330$	0	0
$331$	−22.2337	−1.22207	−0.611037	−	0.791602i	$-0.709247\pi$
$331$	−22.2337	−1.22207	−0.611037	+	0.791602i	$0.709247\pi$
$332$	0	0
$333$	−3.62772	−0.198798
$334$	0	0
$335$	−13.4891	−0.736990
$336$	0	0
$337$	−15.4891	−0.843746	−0.421873	−	0.906655i	$-0.638627\pi$
$337$	−15.4891	−0.843746	−0.421873	+	0.906655i	$0.638627\pi$
$338$	0	0
$339$	−2.00000	−0.108625
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.8614	−1.07242
$344$	0	0
$345$	−1.62772	−0.0876334
$346$	0	0
$347$	27.1168	1.45571	0.727854	−	0.685732i	$-0.240518\pi$
$347$	27.1168	1.45571	0.727854	+	0.685732i	$0.240518\pi$
$348$	0	0
$349$	15.4891	0.829114	0.414557	−	0.910023i	$-0.363937\pi$
$349$	15.4891	0.829114	0.414557	+	0.910023i	$0.363937\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−25.7228	−1.36909	−0.684544	−	0.728972i	$-0.739998\pi$
$353$	−25.7228	−1.36909	−0.684544	+	0.728972i	$0.739998\pi$
$354$	0	0
$355$	−5.62772	−0.298688
$356$	0	0
$357$	0.883156	0.0467416
$358$	0	0
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	29.6060	1.55391
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	11.2554	0.587529	0.293765	−	0.955878i	$-0.405092\pi$
$367$	11.2554	0.587529	0.293765	+	0.955878i	$0.405092\pi$
$368$	0	0
$369$	9.11684	0.474604
$370$	0	0
$371$	10.3723	0.538502
$372$	0	0
$373$	35.4891	1.83756	0.918779	−	0.394773i	$-0.129177\pi$
$373$	35.4891	1.83756	0.918779	+	0.394773i	$0.129177\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	2.74456	0.141352
$378$	0	0
$379$	−1.48913	−0.0764912	−0.0382456	−	0.999268i	$-0.512177\pi$
$379$	−1.48913	−0.0764912	−0.0382456	+	0.999268i	$0.512177\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	−8.74456	−0.446826	−0.223413	−	0.974724i	$-0.571720\pi$
$383$	−8.74456	−0.446826	−0.223413	+	0.974724i	$0.571720\pi$
$384$	0	0
$385$	15.1168	0.770426
$386$	0	0
$387$	−0.744563	−0.0378482
$388$	0	0
$389$	−9.25544	−0.469269	−0.234635	−	0.972084i	$-0.575389\pi$
$389$	−9.25544	−0.469269	−0.234635	+	0.972084i	$0.575389\pi$
$390$	0	0
$391$	0.605969	0.0306452
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−2.37228	−0.119362
$396$	0	0
$397$	−11.6277	−0.583578	−0.291789	−	0.956483i	$-0.594251\pi$
$397$	−11.6277	−0.583578	−0.291789	+	0.956483i	$0.594251\pi$
$398$	0	0
$399$	11.2554	0.563477
$400$	0	0
$401$	−31.4891	−1.57249	−0.786246	−	0.617914i	$-0.787978\pi$
$401$	−31.4891	−1.57249	−0.786246	+	0.617914i	$0.787978\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	23.1168	1.14586
$408$	0	0
$409$	33.7228	1.66749	0.833743	−	0.552153i	$-0.186194\pi$
$409$	33.7228	1.66749	0.833743	+	0.552153i	$0.186194\pi$
$410$	0	0
$411$	−2.00000	−0.0986527
$412$	0	0
$413$	−9.48913	−0.466929
$414$	0	0
$415$	13.4891	0.662155
$416$	0	0
$417$	5.62772	0.275591
$418$	0	0
$419$	10.2337	0.499948	0.249974	−	0.968253i	$-0.419578\pi$
$419$	10.2337	0.499948	0.249974	+	0.968253i	$0.419578\pi$
$420$	0	0
$421$	−8.23369	−0.401285	−0.200643	−	0.979664i	$-0.564303\pi$
$421$	−8.23369	−0.401285	−0.200643	+	0.979664i	$0.564303\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	0.372281	0.0180583
$426$	0	0
$427$	19.8614	0.961161
$428$	0	0
$429$	6.37228	0.307657
$430$	0	0
$431$	−10.9783	−0.528804	−0.264402	−	0.964413i	$-0.585175\pi$
$431$	−10.9783	−0.528804	−0.264402	+	0.964413i	$0.585175\pi$
$432$	0	0
$433$	38.4674	1.84862	0.924312	−	0.381638i	$-0.124640\pi$
$433$	38.4674	1.84862	0.924312	+	0.381638i	$0.124640\pi$
$434$	0	0
$435$	2.74456	0.131592
$436$	0	0
$437$	7.72281	0.369432
$438$	0	0
$439$	24.6060	1.17438	0.587189	−	0.809450i	$-0.300234\pi$
$439$	24.6060	1.17438	0.587189	+	0.809450i	$0.300234\pi$
$440$	0	0
$441$	−1.37228	−0.0653467
$442$	0	0
$443$	−15.8614	−0.753598	−0.376799	−	0.926295i	$-0.622975\pi$
$443$	−15.8614	−0.753598	−0.376799	+	0.926295i	$0.622975\pi$
$444$	0	0
$445$	3.62772	0.171970
$446$	0	0
$447$	13.8614	0.655622
$448$	0	0
$449$	−16.3723	−0.772656	−0.386328	−	0.922362i	$-0.626257\pi$
$449$	−16.3723	−0.772656	−0.386328	+	0.922362i	$0.626257\pi$
$450$	0	0
$451$	−58.0951	−2.73559
$452$	0	0
$453$	8.00000	0.375873
$454$	0	0
$455$	2.37228	0.111214
$456$	0	0
$457$	17.1168	0.800692	0.400346	−	0.916364i	$-0.368890\pi$
$457$	17.1168	0.800692	0.400346	+	0.916364i	$0.368890\pi$
$458$	0	0
$459$	0.372281	0.0173766
$460$	0	0
$461$	1.11684	0.0520166	0.0260083	−	0.999662i	$-0.491720\pi$
$461$	1.11684	0.0520166	0.0260083	+	0.999662i	$0.491720\pi$
$462$	0	0
$463$	18.3723	0.853832	0.426916	−	0.904291i	$-0.359600\pi$
$463$	18.3723	0.853832	0.426916	+	0.904291i	$0.359600\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.1168	0.884622	0.442311	−	0.896862i	$-0.354159\pi$
$467$	19.1168	0.884622	0.442311	+	0.896862i	$0.354159\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	4.74456	0.218155
$474$	0	0
$475$	4.74456	0.217695
$476$	0	0
$477$	4.37228	0.200193
$478$	0	0
$479$	−7.11684	−0.325177	−0.162588	−	0.986694i	$-0.551984\pi$
$479$	−7.11684	−0.325177	−0.162588	+	0.986694i	$0.551984\pi$
$480$	0	0
$481$	3.62772	0.165410
$482$	0	0
$483$	3.86141	0.175700
$484$	0	0
$485$	0.372281	0.0169044
$486$	0	0
$487$	12.1386	0.550052	0.275026	−	0.961437i	$-0.411314\pi$
$487$	12.1386	0.550052	0.275026	+	0.961437i	$0.411314\pi$
$488$	0	0
$489$	6.37228	0.288165
$490$	0	0
$491$	2.23369	0.100805	0.0504025	−	0.998729i	$-0.483950\pi$
$491$	2.23369	0.100805	0.0504025	+	0.998729i	$0.483950\pi$
$492$	0	0
$493$	−1.02175	−0.0460173
$494$	0	0
$495$	6.37228	0.286413
$496$	0	0
$497$	13.3505	0.598853
$498$	0	0
$499$	−11.2554	−0.503863	−0.251931	−	0.967745i	$-0.581066\pi$
$499$	−11.2554	−0.503863	−0.251931	+	0.967745i	$0.581066\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	32.4674	1.44765	0.723824	−	0.689984i	$-0.242383\pi$
$503$	32.4674	1.44765	0.723824	+	0.689984i	$0.242383\pi$
$504$	0	0
$505$	−5.25544	−0.233864
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	10.8832	0.482387	0.241194	−	0.970477i	$-0.422461\pi$
$509$	10.8832	0.482387	0.241194	+	0.970477i	$0.422461\pi$
$510$	0	0
$511$	23.7228	1.04944
$512$	0	0
$513$	4.74456	0.209478
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−25.4891	−1.12101
$518$	0	0
$519$	2.00000	0.0877903
$520$	0	0
$521$	4.97825	0.218101	0.109051	−	0.994036i	$-0.465219\pi$
$521$	4.97825	0.218101	0.109051	+	0.994036i	$0.465219\pi$
$522$	0	0
$523$	−34.2337	−1.49693	−0.748467	−	0.663172i	$-0.769210\pi$
$523$	−34.2337	−1.49693	−0.748467	+	0.663172i	$0.769210\pi$
$524$	0	0
$525$	2.37228	0.103535
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−20.3505	−0.884806
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−9.11684	−0.394894
$534$	0	0
$535$	6.37228	0.275498
$536$	0	0
$537$	8.74456	0.377356
$538$	0	0
$539$	8.74456	0.376655
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	0	0
$543$	−9.11684	−0.391241
$544$	0	0
$545$	−1.25544	−0.0537770
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	0	0
$549$	8.37228	0.357320
$550$	0	0
$551$	−13.0217	−0.554745
$552$	0	0
$553$	5.62772	0.239315
$554$	0	0
$555$	3.62772	0.153988
$556$	0	0
$557$	25.7228	1.08991	0.544955	−	0.838465i	$-0.316547\pi$
$557$	25.7228	1.08991	0.544955	+	0.838465i	$0.316547\pi$
$558$	0	0
$559$	0.744563	0.0314916
$560$	0	0
$561$	−2.37228	−0.100158
$562$	0	0
$563$	12.6060	0.531278	0.265639	−	0.964073i	$-0.414417\pi$
$563$	12.6060	0.531278	0.265639	+	0.964073i	$0.414417\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	2.37228	0.0996265
$568$	0	0
$569$	4.97825	0.208699	0.104350	−	0.994541i	$-0.466724\pi$
$569$	4.97825	0.208699	0.104350	+	0.994541i	$0.466724\pi$
$570$	0	0
$571$	−8.60597	−0.360149	−0.180074	−	0.983653i	$-0.557634\pi$
$571$	−8.60597	−0.360149	−0.180074	+	0.983653i	$0.557634\pi$
$572$	0	0
$573$	14.2337	0.594621
$574$	0	0
$575$	1.62772	0.0678806
$576$	0	0
$577$	31.6277	1.31668	0.658340	−	0.752721i	$-0.271259\pi$
$577$	31.6277	1.31668	0.658340	+	0.752721i	$0.271259\pi$
$578$	0	0
$579$	−22.6060	−0.939472
$580$	0	0
$581$	−32.0000	−1.32758
$582$	0	0
$583$	−27.8614	−1.15390
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−10.2337	−0.422390	−0.211195	−	0.977444i	$-0.567735\pi$
$587$	−10.2337	−0.422390	−0.211195	+	0.977444i	$0.567735\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−15.4891	−0.637137
$592$	0	0
$593$	45.7228	1.87761	0.938805	−	0.344448i	$-0.111934\pi$
$593$	45.7228	1.87761	0.938805	+	0.344448i	$0.111934\pi$
$594$	0	0
$595$	−0.883156	−0.0362059
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−44.4674	−1.81689	−0.908444	−	0.418007i	$-0.862729\pi$
$599$	−44.4674	−1.81689	−0.908444	+	0.418007i	$0.862729\pi$
$600$	0	0
$601$	−35.3505	−1.44198	−0.720989	−	0.692946i	$-0.756312\pi$
$601$	−35.3505	−1.44198	−0.720989	+	0.692946i	$0.756312\pi$
$602$	0	0
$603$	13.4891	0.549320
$604$	0	0
$605$	−29.6060	−1.20365
$606$	0	0
$607$	−22.2337	−0.902438	−0.451219	−	0.892413i	$-0.649011\pi$
$607$	−22.2337	−0.902438	−0.451219	+	0.892413i	$0.649011\pi$
$608$	0	0
$609$	−6.51087	−0.263834
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	−27.3505	−1.10468	−0.552339	−	0.833620i	$-0.686265\pi$
$613$	−27.3505	−1.10468	−0.552339	+	0.833620i	$0.686265\pi$
$614$	0	0
$615$	−9.11684	−0.367627
$616$	0	0
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	0	0
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	1.62772	0.0653181
$622$	0	0
$623$	−8.60597	−0.344791
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−30.2337	−1.20742
$628$	0	0
$629$	−1.35053	−0.0538492
$630$	0	0
$631$	−2.97825	−0.118562	−0.0592811	−	0.998241i	$-0.518881\pi$
$631$	−2.97825	−0.118562	−0.0592811	+	0.998241i	$0.518881\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	0	0
$635$	−16.0000	−0.634941
$636$	0	0
$637$	1.37228	0.0543718
$638$	0	0
$639$	5.62772	0.222629
$640$	0	0
$641$	43.4891	1.71772	0.858859	−	0.512213i	$-0.171174\pi$
$641$	43.4891	1.71772	0.858859	+	0.512213i	$0.171174\pi$
$642$	0	0
$643$	31.8614	1.25649	0.628246	−	0.778015i	$-0.283773\pi$
$643$	31.8614	1.25649	0.628246	+	0.778015i	$0.283773\pi$
$644$	0	0
$645$	0.744563	0.0293171
$646$	0	0
$647$	41.3505	1.62566	0.812829	−	0.582503i	$-0.197927\pi$
$647$	41.3505	1.62566	0.812829	+	0.582503i	$0.197927\pi$
$648$	0	0
$649$	25.4891	1.00054
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.51087	−0.176524	−0.0882621	−	0.996097i	$-0.528131\pi$
$653$	−4.51087	−0.176524	−0.0882621	+	0.996097i	$0.528131\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	0	0
$663$	−0.372281	−0.0144582
$664$	0	0
$665$	−11.2554	−0.436467
$666$	0	0
$667$	−4.46738	−0.172977
$668$	0	0
$669$	16.0000	0.618596
$670$	0	0
$671$	−53.3505	−2.05957
$672$	0	0
$673$	46.7446	1.80187	0.900935	−	0.433954i	$-0.142882\pi$
$673$	46.7446	1.80187	0.900935	+	0.433954i	$0.142882\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	10.8832	0.418274	0.209137	−	0.977886i	$-0.432935\pi$
$677$	10.8832	0.418274	0.209137	+	0.977886i	$0.432935\pi$
$678$	0	0
$679$	−0.883156	−0.0338924
$680$	0	0
$681$	−8.74456	−0.335092
$682$	0	0
$683$	−18.2337	−0.697693	−0.348846	−	0.937180i	$-0.613426\pi$
$683$	−18.2337	−0.697693	−0.348846	+	0.937180i	$0.613426\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	10.7446	0.409931
$688$	0	0
$689$	−4.37228	−0.166571
$690$	0	0
$691$	−19.2554	−0.732511	−0.366256	−	0.930514i	$-0.619360\pi$
$691$	−19.2554	−0.732511	−0.366256	+	0.930514i	$0.619360\pi$
$692$	0	0
$693$	−15.1168	−0.574241
$694$	0	0
$695$	−5.62772	−0.213472
$696$	0	0
$697$	3.39403	0.128558
$698$	0	0
$699$	−21.8614	−0.826874
$700$	0	0
$701$	−44.2337	−1.67068	−0.835342	−	0.549731i	$-0.814730\pi$
$701$	−44.2337	−1.67068	−0.835342	+	0.549731i	$0.814730\pi$
$702$	0	0
$703$	−17.2119	−0.649161
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	12.4674	0.468884
$708$	0	0
$709$	−43.2119	−1.62286	−0.811429	−	0.584451i	$-0.801310\pi$
$709$	−43.2119	−1.62286	−0.811429	+	0.584451i	$0.801310\pi$
$710$	0	0
$711$	2.37228	0.0889675
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−6.37228	−0.238310
$716$	0	0
$717$	13.6277	0.508936
$718$	0	0
$719$	44.4674	1.65835	0.829177	−	0.558987i	$-0.188810\pi$
$719$	44.4674	1.65835	0.829177	+	0.558987i	$0.188810\pi$
$720$	0	0
$721$	18.9783	0.706787
$722$	0	0
$723$	−9.25544	−0.344213
$724$	0	0
$725$	−2.74456	−0.101930
$726$	0	0
$727$	31.7228	1.17653	0.588267	−	0.808667i	$-0.299810\pi$
$727$	31.7228	1.17653	0.588267	+	0.808667i	$0.299810\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.277187	−0.0102521
$732$	0	0
$733$	−14.6060	−0.539484	−0.269742	−	0.962933i	$-0.586938\pi$
$733$	−14.6060	−0.539484	−0.269742	+	0.962933i	$0.586938\pi$
$734$	0	0
$735$	1.37228	0.0506174
$736$	0	0
$737$	−85.9565	−3.16625
$738$	0	0
$739$	33.2119	1.22172	0.610860	−	0.791738i	$-0.290824\pi$
$739$	33.2119	1.22172	0.610860	+	0.791738i	$0.290824\pi$
$740$	0	0
$741$	−4.74456	−0.174296
$742$	0	0
$743$	35.7228	1.31054	0.655271	−	0.755393i	$-0.272554\pi$
$743$	35.7228	1.31054	0.655271	+	0.755393i	$0.272554\pi$
$744$	0	0
$745$	−13.8614	−0.507843
$746$	0	0
$747$	−13.4891	−0.493541
$748$	0	0
$749$	−15.1168	−0.552357
$750$	0	0
$751$	5.62772	0.205358	0.102679	−	0.994715i	$-0.467258\pi$
$751$	5.62772	0.205358	0.102679	+	0.994715i	$0.467258\pi$
$752$	0	0
$753$	29.4891	1.07464
$754$	0	0
$755$	−8.00000	−0.291150
$756$	0	0
$757$	−15.7663	−0.573036	−0.286518	−	0.958075i	$-0.592498\pi$
$757$	−15.7663	−0.573036	−0.286518	+	0.958075i	$0.592498\pi$
$758$	0	0
$759$	−10.3723	−0.376490
$760$	0	0
$761$	24.5109	0.888519	0.444259	−	0.895898i	$-0.353467\pi$
$761$	24.5109	0.888519	0.444259	+	0.895898i	$0.353467\pi$
$762$	0	0
$763$	2.97825	0.107820
$764$	0	0
$765$	−0.372281	−0.0134599
$766$	0	0
$767$	4.00000	0.144432
$768$	0	0
$769$	−0.978251	−0.0352766	−0.0176383	−	0.999844i	$-0.505615\pi$
$769$	−0.978251	−0.0352766	−0.0176383	+	0.999844i	$0.505615\pi$
$770$	0	0
$771$	−11.4891	−0.413771
$772$	0	0
$773$	−33.2554	−1.19611	−0.598057	−	0.801453i	$-0.704061\pi$
$773$	−33.2554	−1.19611	−0.598057	+	0.801453i	$0.704061\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−8.60597	−0.308737
$778$	0	0
$779$	43.2554	1.54979
$780$	0	0
$781$	−35.8614	−1.28322
$782$	0	0
$783$	−2.74456	−0.0980827
$784$	0	0
$785$	−2.00000	−0.0713831
$786$	0	0
$787$	−21.4891	−0.766005	−0.383002	−	0.923747i	$-0.625110\pi$
$787$	−21.4891	−0.766005	−0.383002	+	0.923747i	$0.625110\pi$
$788$	0	0
$789$	5.48913	0.195418
$790$	0	0
$791$	−4.74456	−0.168697
$792$	0	0
$793$	−8.37228	−0.297308
$794$	0	0
$795$	−4.37228	−0.155069
$796$	0	0
$797$	23.6277	0.836937	0.418468	−	0.908231i	$-0.362567\pi$
$797$	23.6277	0.836937	0.418468	+	0.908231i	$0.362567\pi$
$798$	0	0
$799$	1.48913	0.0526815
$800$	0	0
$801$	−3.62772	−0.128179
$802$	0	0
$803$	−63.7228	−2.24873
$804$	0	0
$805$	−3.86141	−0.136097
$806$	0	0
$807$	25.7228	0.905486
$808$	0	0
$809$	−18.4674	−0.649278	−0.324639	−	0.945838i	$-0.605243\pi$
$809$	−18.4674	−0.649278	−0.324639	+	0.945838i	$0.605243\pi$
$810$	0	0
$811$	−43.2554	−1.51890	−0.759452	−	0.650563i	$-0.774533\pi$
$811$	−43.2554	−1.51890	−0.759452	+	0.650563i	$0.774533\pi$
$812$	0	0
$813$	−12.7446	−0.446971
$814$	0	0
$815$	−6.37228	−0.223211
$816$	0	0
$817$	−3.53262	−0.123591
$818$	0	0
$819$	−2.37228	−0.0828942
$820$	0	0
$821$	−46.3288	−1.61689	−0.808443	−	0.588575i	$-0.799689\pi$
$821$	−46.3288	−1.61689	−0.808443	+	0.588575i	$0.799689\pi$
$822$	0	0
$823$	−2.97825	−0.103815	−0.0519076	−	0.998652i	$-0.516530\pi$
$823$	−2.97825	−0.103815	−0.0519076	+	0.998652i	$0.516530\pi$
$824$	0	0
$825$	−6.37228	−0.221854
$826$	0	0
$827$	−22.9783	−0.799032	−0.399516	−	0.916726i	$-0.630822\pi$
$827$	−22.9783	−0.799032	−0.399516	+	0.916726i	$0.630822\pi$
$828$	0	0
$829$	26.4674	0.919250	0.459625	−	0.888113i	$-0.347984\pi$
$829$	26.4674	0.919250	0.459625	+	0.888113i	$0.347984\pi$
$830$	0	0
$831$	16.2337	0.563140
$832$	0	0
$833$	−0.510875	−0.0177008
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−54.8397	−1.89328	−0.946638	−	0.322300i	$-0.895544\pi$
$839$	−54.8397	−1.89328	−0.946638	+	0.322300i	$0.895544\pi$
$840$	0	0
$841$	−21.4674	−0.740254
$842$	0	0
$843$	−7.48913	−0.257939
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	70.2337	2.41326
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	−5.90491	−0.202418
$852$	0	0
$853$	17.1168	0.586070	0.293035	−	0.956102i	$-0.405335\pi$
$853$	17.1168	0.586070	0.293035	+	0.956102i	$0.405335\pi$
$854$	0	0
$855$	−4.74456	−0.162261
$856$	0	0
$857$	−12.3723	−0.422629	−0.211315	−	0.977418i	$-0.567774\pi$
$857$	−12.3723	−0.422629	−0.211315	+	0.977418i	$0.567774\pi$
$858$	0	0
$859$	53.0733	1.81084	0.905420	−	0.424518i	$-0.139556\pi$
$859$	53.0733	1.81084	0.905420	+	0.424518i	$0.139556\pi$
$860$	0	0
$861$	21.6277	0.737071
$862$	0	0
$863$	2.23369	0.0760356	0.0380178	−	0.999277i	$-0.487896\pi$
$863$	2.23369	0.0760356	0.0380178	+	0.999277i	$0.487896\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	−16.8614	−0.572643
$868$	0	0
$869$	−15.1168	−0.512804
$870$	0	0
$871$	−13.4891	−0.457062
$872$	0	0
$873$	−0.372281	−0.0125998
$874$	0	0
$875$	−2.37228	−0.0801977
$876$	0	0
$877$	−26.4674	−0.893740	−0.446870	−	0.894599i	$-0.647461\pi$
$877$	−26.4674	−0.893740	−0.446870	+	0.894599i	$0.647461\pi$
$878$	0	0
$879$	−26.7446	−0.902072
$880$	0	0
$881$	−29.7228	−1.00139	−0.500694	−	0.865625i	$-0.666922\pi$
$881$	−29.7228	−1.00139	−0.500694	+	0.865625i	$0.666922\pi$
$882$	0	0
$883$	2.51087	0.0844977	0.0422488	−	0.999107i	$-0.486548\pi$
$883$	2.51087	0.0844977	0.0422488	+	0.999107i	$0.486548\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	20.8832	0.701188	0.350594	−	0.936528i	$-0.385980\pi$
$887$	20.8832	0.701188	0.350594	+	0.936528i	$0.385980\pi$
$888$	0	0
$889$	37.9565	1.27302
$890$	0	0
$891$	−6.37228	−0.213479
$892$	0	0
$893$	18.9783	0.635083
$894$	0	0
$895$	−8.74456	−0.292298
$896$	0	0
$897$	−1.62772	−0.0543479
$898$	0	0
$899$	0	0
$900$	0	0
$901$	1.62772	0.0542272
$902$	0	0
$903$	−1.76631	−0.0587792
$904$	0	0
$905$	9.11684	0.303054
$906$	0	0
$907$	32.7446	1.08727	0.543633	−	0.839323i	$-0.317048\pi$
$907$	32.7446	1.08727	0.543633	+	0.839323i	$0.317048\pi$
$908$	0	0
$909$	5.25544	0.174312
$910$	0	0
$911$	−3.25544	−0.107857	−0.0539287	−	0.998545i	$-0.517174\pi$
$911$	−3.25544	−0.107857	−0.0539287	+	0.998545i	$0.517174\pi$
$912$	0	0
$913$	85.9565	2.84474
$914$	0	0
$915$	−8.37228	−0.276779
$916$	0	0
$917$	9.48913	0.313359
$918$	0	0
$919$	−10.0951	−0.333006	−0.166503	−	0.986041i	$-0.553248\pi$
$919$	−10.0951	−0.333006	−0.166503	+	0.986041i	$0.553248\pi$
$920$	0	0
$921$	6.37228	0.209974
$922$	0	0
$923$	−5.62772	−0.185239
$924$	0	0
$925$	−3.62772	−0.119279
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−24.0951	−0.790534	−0.395267	−	0.918566i	$-0.629348\pi$
$929$	−24.0951	−0.790534	−0.395267	+	0.918566i	$0.629348\pi$
$930$	0	0
$931$	−6.51087	−0.213385
$932$	0	0
$933$	11.2554	0.368486
$934$	0	0
$935$	2.37228	0.0775819
$936$	0	0
$937$	−39.2119	−1.28100	−0.640499	−	0.767959i	$-0.721272\pi$
$937$	−39.2119	−1.28100	−0.640499	+	0.767959i	$0.721272\pi$
$938$	0	0
$939$	−7.48913	−0.244398
$940$	0	0
$941$	31.3505	1.02200	0.510999	−	0.859581i	$-0.329276\pi$
$941$	31.3505	1.02200	0.510999	+	0.859581i	$0.329276\pi$
$942$	0	0
$943$	14.8397	0.483246
$944$	0	0
$945$	−2.37228	−0.0771703
$946$	0	0
$947$	−36.0000	−1.16984	−0.584921	−	0.811090i	$-0.698875\pi$
$947$	−36.0000	−1.16984	−0.584921	+	0.811090i	$0.698875\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	−14.0000	−0.453981
$952$	0	0
$953$	−41.1168	−1.33191	−0.665953	−	0.745994i	$-0.731975\pi$
$953$	−41.1168	−1.33191	−0.665953	+	0.745994i	$0.731975\pi$
$954$	0	0
$955$	−14.2337	−0.460591
$956$	0	0
$957$	17.4891	0.565343
$958$	0	0
$959$	−4.74456	−0.153210
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−6.37228	−0.205344
$964$	0	0
$965$	22.6060	0.727712
$966$	0	0
$967$	−61.9565	−1.99239	−0.996193	−	0.0871708i	$-0.972217\pi$
$967$	−61.9565	−1.99239	−0.996193	+	0.0871708i	$0.972217\pi$
$968$	0	0
$969$	1.76631	0.0567421
$970$	0	0
$971$	11.7228	0.376203	0.188101	−	0.982150i	$-0.439767\pi$
$971$	11.7228	0.376203	0.188101	+	0.982150i	$0.439767\pi$
$972$	0	0
$973$	13.3505	0.427998
$974$	0	0
$975$	−1.00000	−0.0320256
$976$	0	0
$977$	26.4674	0.846766	0.423383	−	0.905951i	$-0.360842\pi$
$977$	26.4674	0.846766	0.423383	+	0.905951i	$0.360842\pi$
$978$	0	0
$979$	23.1168	0.738818
$980$	0	0
$981$	1.25544	0.0400830
$982$	0	0
$983$	−45.4891	−1.45088	−0.725439	−	0.688287i	$-0.758363\pi$
$983$	−45.4891	−1.45088	−0.725439	+	0.688287i	$0.758363\pi$
$984$	0	0
$985$	15.4891	0.493525
$986$	0	0
$987$	9.48913	0.302042
$988$	0	0
$989$	−1.21194	−0.0385374
$990$	0	0
$991$	20.1386	0.639724	0.319862	−	0.947464i	$-0.396364\pi$
$991$	20.1386	0.639724	0.319862	+	0.947464i	$0.396364\pi$
$992$	0	0
$993$	−22.2337	−0.705565
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−23.7663	−0.752687	−0.376343	−	0.926480i	$-0.622819\pi$
$997$	−23.7663	−0.752687	−0.376343	+	0.926480i	$0.622819\pi$
$998$	0	0
$999$	−3.62772	−0.114776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6240.2.a.bp.1.2	yes	2
4.3	odd	2		6240.2.a.bj.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6240.2.a.bj.1.1	✓	2	4.3	odd	2
6240.2.a.bp.1.2	yes	2	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:	\( N \)	\(=\)	\( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +2.37228 q^{7} +1.00000 q^{9} -6.37228 q^{11} -1.00000 q^{13} -1.00000 q^{15} +0.372281 q^{17} +4.74456 q^{19} +2.37228 q^{21} +1.62772 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.74456 q^{29} -6.37228 q^{33} -2.37228 q^{35} -3.62772 q^{37} -1.00000 q^{39} +9.11684 q^{41} -0.744563 q^{43} -1.00000 q^{45} +4.00000 q^{47} -1.37228 q^{49} +0.372281 q^{51} +4.37228 q^{53} +6.37228 q^{55} +4.74456 q^{57} -4.00000 q^{59} +8.37228 q^{61} +2.37228 q^{63} +1.00000 q^{65} +13.4891 q^{67} +1.62772 q^{69} +5.62772 q^{71} +10.0000 q^{73} +1.00000 q^{75} -15.1168 q^{77} +2.37228 q^{79} +1.00000 q^{81} -13.4891 q^{83} -0.372281 q^{85} -2.74456 q^{87} -3.62772 q^{89} -2.37228 q^{91} -4.74456 q^{95} -0.372281 q^{97} -6.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - q^{7} + 2 q^{9} - 7 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} - q^{21} + 9 q^{23} + 2 q^{25} + 2 q^{27} + 6 q^{29} - 7 q^{33} + q^{35} - 13 q^{37} - 2 q^{39} + q^{41} + 10 q^{43} - 2 q^{45} + 8 q^{47} + 3 q^{49} - 5 q^{51} + 3 q^{53} + 7 q^{55} - 2 q^{57} - 8 q^{59} + 11 q^{61} - q^{63} + 2 q^{65} + 4 q^{67} + 9 q^{69} + 17 q^{71} + 20 q^{73} + 2 q^{75} - 13 q^{77} - q^{79} + 2 q^{81} - 4 q^{83} + 5 q^{85} + 6 q^{87} - 13 q^{89} + q^{91} + 2 q^{95} + 5 q^{97} - 7 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - q^7 + 2 * q^9 - 7 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 - 2 * q^19 - q^21 + 9 * q^23 + 2 * q^25 + 2 * q^27 + 6 * q^29 - 7 * q^33 + q^35 - 13 * q^37 - 2 * q^39 + q^41 + 10 * q^43 - 2 * q^45 + 8 * q^47 + 3 * q^49 - 5 * q^51 + 3 * q^53 + 7 * q^55 - 2 * q^57 - 8 * q^59 + 11 * q^61 - q^63 + 2 * q^65 + 4 * q^67 + 9 * q^69 + 17 * q^71 + 20 * q^73 + 2 * q^75 - 13 * q^77 - q^79 + 2 * q^81 - 4 * q^83 + 5 * q^85 + 6 * q^87 - 13 * q^89 + q^91 + 2 * q^95 + 5 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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